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+{"text":"\\section{Introduction}\n\n\\footnotetext[1]\npheligenius@yahoo.com}\\footnotetext[2]\nfarida\\_tahir@comsats.edu.pk}One of the most flummoxing problems in modern\nphysics that kept scientists at alert and has been hotly debated since $1929\n, is the realization of the expansion of the universe, established when\nEdwin Hubble published his revolutionary paper. Astronomical observations\nand study of universe, in the past few decades, strongly invalidated\nastronomers' view point that the universe was entirely composed of \\emph\n\\textquotedblleft baryonic matter\\textquotedblright }. The latest\nconformation of the accelerating universe \\cite{DN,AG,SP,SW,SE} endorsed the\nfact that the universe is infused with an unknown form of energy density $(\ndubbed as dark energy $\\left( \\rho _{\\Lambda }\\right) )$ which makes up for\nabout $75\\%$ of the total energy density of the universe. It is this $75\\%$\nmysterious $\\rho _{\\Lambda }$, which conditions our three-dimensional\nspatial curvature to be zero, that is responsible for the acceleration of\nthe universe. This discovery provided the first direct evidence that $\\rho\n_{\\Lambda }$ is non-zero, with $\\rho _{\\Lambda }\\approx \\left( 2.3\\times\n10^{-3}eV\\right) ^{4}$\\cite{FR,JZ}.\n\nHowever, the theoretical expectations for the $\\rho _{\\Lambda }$ exceed\nobservational limits by some $120$ orders of magnitude \\cite{JC}. This huge\ndiscrepancy between theory and observation, hitherto, constitutes a serious\nproblem for theoretical physics community. In fact, Steven Weinberg puts it\nmore succinctly by saying that the small non-zero value of $\\rho _{\\Lambda }$\nis \\emph{\\textquotedblleft a bone in the throat of theoretical\nphysics\\textquotedblright .} \\ Considering this huge discrepancy may\nun-shroud something fundamental, yet to be unveiled, about the hidden nature\nof the universe. This paper is one of such attempts.\n\nThe most elegant and comprehensible endeavour in order to solve this\nproblem, in our view, was put forward by F. R. Urban and A. R. Zhitnitsky \n\\cite{FR}. These authors approached the problem from the angle of the\neffective theory of gravity interacting with standard model fields by using\nthe solution of the $U(1)$ problem as put forward by G. Veneziano and E.\nWitten \\cite{GV,EW}.\\ In this framework, the basic problem of why the dark\nenergy is $120$ orders of magnitude smaller than its Planck scale \nM_{planck}^{4}$, is replaced by fundamentally different questions:\n\\textquotedblleft $(i)$ What is the relevant scale which enters the\neffective theory of gravitation? $(ii)$ How does this scale appear in the\neffective quantum field theory for gravity?\\textquotedblright\\ In their\nview, this effective scale has nothing to do with the cutoff ultraviolet \n(UV)$ scale $M_{planck}$: the appropriate effective scale must emerge as a\nresult of a subtraction at which some infrared (IR) scale enters the\nphysics. They completely turned the problem on its head!\n\nThough their attempt being cognizant, yet it fails to reproduce, exactly,\nthe measured value of $\\rho _{\\Lambda }$ \\cite{MT}. We observe here that\ntheir assumption $g\\equiv c=C_{QCD}\\times C_{grav}=1$ is debatable, since it\nis valid only for $C_{QCD}$ but not for $C_{grav}$, as proved in this paper.\nHere $g$ is the Minkowski metric in vacuum, $C_{QCD}$ is the Quantum\nChromodynamic $(QCD)$ coupling constant, and $C_{grav}$ is the gravitational\ncoupling constant.\n\nFrom Ref.[7], the value of $C_{grav}$ was wrongly computed to be \nC_{grav}=0.0588$\\ (which is approximately one-third of the value we proved\nin our calculation) but for obvious reason the authors neglected this value\nand used a position dependent Minkowski metric distance $g(x^{2})$\\ instead.\nThey computed $g(x^{2})$ to be $g(x^{2})=1\/6.25.$ For no clear reason, they\napproximated the value of $g(x^{2})$ to $g(x^{2})\\approx 1\/6$ by truncating \n0.25$ from their original \\ value of $g(x^{2}).$ This approach is totally\nunacceptable in the field of computational cosmology where every minuscule\nvalue counts.\n\nIn this paper, we have proved the value of $C_{grav}$ to an order of\nmagnitude less than one i.e. $1.797\\times 10^{-1}$; this leads towards the\nexact measured \\cite{MT} value of $\\rho _{\\Lambda }$. In order to get this\nvalue, we have used finite temperature and density $(FTD)$ correction\ntechnique. Here, the $FTD$ background acts as highly energetic medium \n\\left( M_{planck}^{4}\\right) $ controlling the particle propagation. Our\nbasic guiding idea is that the finite temperature field theory $(FTFT)$,\nsimilar to the physics of superconductivity (quantum field theory at $T=0$),\nis linked to the infrared sector of the effective theory of gravity\ninteracting with standard model fields, specifically with $QCD$ fields \\cit\n{FR}. In this case, the statistical background effects are incorporated in\npropagators through the Bose-Einstein distribution function \\cite{KR,FE}: it\nis worth noting that the Bose-Einstein distribution function is the\nmathematical tool for understanding the essential feature of the theory of\nsuperconductivity \\cite{FE}. The general attribute of a successful theory of\nsuperconductivity is the existence of degenerate vacuum\/broken symmetry\nmechanism. A characteristic feature of such a theory is the possible\nexistence of \\textquotedblleft unphysical\\textquotedblright\\ zero-mass\nbosons which tend to preserve the underlying symmetry of the theory.\\ The\nmasslessness of these singularities is protected in the limit \nq\\longrightarrow 0$. This means that it should cost no energy to create a\nYang-Mills quantum at $q=0$ and thus the mass is zero \\cite{GS}. In the\npreceding Ref. the Goldstone-Salam-Weinberg theorem is valid for a zero-mass\npole, which is protected. That pole is not physical and is purely gauge,\nhence unphysical. This is precisely the highly celebrated Veneziano ghost \n\\cite{FR}, which is analogous to the Kogut-Susskind \\emph{(KS)} ghost in the\nSchwinger model \\emph{(distinctive unphysical degree of freedom which is\nmassless and can propagate to arbitrary large distances)}.\n\n\\qquad It is imperative to note that this set of unphysical massless bosons\ntends to transform as a basis for a representation of a compact Lie group \n\\cite{FE} thereby, forming a compact manifold. We do not make any specific\nassumptions on the topological nature of the manifold; we only assume that\nthere is at least one Minkowski metric distance that defines a general\ncovariance of comoving coordinates \\cite{SW} with size $L_{M}=2\\times\nEuclidean$ metric distance.\n\nIn the next section, we derive the finite temperature and density relation\nfor the Veneziano ghosts by using Bose-Einstein distribution function. It\nshould be noted here that Veneziano ghosts are treated as unphysical\nmassless bosons due to the fact that they both have the same propagator \n(+ig_{\\mu \\nu }\/q^{2})$\\cite{FR}: the propagator for unphysical massless\nboson is obtained from $(+\\left( 2\\pi \\right) ^{4}ie^{2}g_{\\mu \\nu\n}\/q^{2})\\langle \\phi \\rangle ^{2}$ \\cite{FE}.\n\n\\section{ Veneziano-Ghost Density}\n\nFrom Fermi-Dirac and Bose-Einstein distribution functions, we hav\n\\begin{equation}\nn_{r}=\\frac{g_{r}}{e^{\\alpha +\\beta \\varepsilon _{r}}\\pm 1}  \\tag{1}\n\\end{equation}\n\nThe positive sign applies to fermions and the negative to bosons. $g_{r}$ is\nthe degenerate parameter, $\\alpha $ is the coefficient of expansion of the\nboson gas inside the volume $(V)$, $\\beta $ is the Lagrange undetermined\nmultiplier, $n_{r}$ and $\\varepsilon _{r}$ are the numbers of particles and\nthe energy of the $r-th$ state respectively. The value of $\\alpha $ for\nboson gas at a given temperature is determined by the normalization\ncondition \\cite{PC\n\\begin{equation}\nN=\\underset{r}{\\sum }\\frac{g_{r}}{e^{\\alpha +\\beta \\varepsilon _{r}}-1} \n\\tag{2}\n\\end{equation}\n\nThis sum can be converted into an integral, because for a particle in a box,\nthe states of the system have been found to be very close together i.e. \n\\left( \\Delta \\varepsilon _{vac}\\equiv d\\varepsilon \\rightarrow 0\\right) $.\nUsing the density of single-particle states function, $Eq.(2)$ reduces to\n\n\\begin{equation}\nN=\\overset{\\infty }{\\underset{0}{\\int }}\\frac{D\\left( \\varepsilon \\right)\nd\\varepsilon }{e^{\\alpha +\\beta \\varepsilon }-1}  \\tag{3}\n\\end{equation}\n\nWhere $D\\left( \\varepsilon \\right) d\\varepsilon $ is the number of allowed\nstates in the energy range $\\varepsilon $ to $\\varepsilon +d\\varepsilon $\nand $\\varepsilon $ is the energy of the single-particle states. Using the\ndensity of states as a function of energy, we have \\cite{PC}\n\n\\begin{equation*}\nD\\left( \\varepsilon \\right) d\\varepsilon =\\frac{4\\pi V}{h^{3}}2m\\varepsilon\n\\left( \\frac{m}{p}\\right) d\\varepsilon\n\\end{equation*}\n\nwith $p=\\sqrt{2m\\varepsilon }$\n\n\\begin{equation}\nD\\left( \\varepsilon \\right) d\\varepsilon =2\\pi V\\left( \\frac{2m}{h^{2}\n\\right) ^{3\/2}\\varepsilon ^{1\/2}d\\varepsilon  \\tag{4}\n\\end{equation}\n\nPutting $Eq.(4)$ into $Eq.(3),$ we get\n\n\\begin{equation}\nN=2\\pi V\\left( \\frac{2m}{h^{2}}\\right) ^{3\/2}\\overset{\\infty }{\\underset{0}\n\\int }}\\frac{\\varepsilon ^{1\/2}d\\varepsilon }{e^{\\alpha +\\beta \\varepsilon\n}-1}  \\tag{5}\n\\end{equation}\n\nWhere $m$ is the mass of boson and $h$ is the Planck constant. $\\alpha\n=\\beta \\mu $ and $\\beta =1\/kT$. $\\mu $ is the chemical potential, $k$ is the\nBoltzmann constant and $T$ is temperature. Since there is no restriction on\nthe total number of bosons, the chemical potential is always equals to zero.\nThus $Eq.(5)$ reads as:\n\n\\begin{equation}\nN=2\\pi V\\left( \\frac{2m}{h^{2}}\\right) ^{3\/2}\\overset{\\infty }{\\underset{0}\n\\int }}\\frac{\\varepsilon ^{1\/2}d\\varepsilon }{e^{\\varepsilon \/kT}-1}  \\tag{6}\n\\end{equation}\n\nBy using standard integral\n\n\\begin{equation*}\n\\overset{\\infty }{\\underset{0}{\\int }}\\frac{x^{z-1}dx}{e^{x}-1}=\\varsigma\n\\left( z\\right) \\Gamma \\left( z\\right)\n\\end{equation*}\n\nwhere $\\varsigma \\left( z\\right) $ is the Riemann zeta function and $\\Gamma\n\\left( z\\right) $ is the gamma function. $Eq.(6)$ takes the form\n\n\\begin{equation*}\nN=2.61V\\left( \\frac{2\\pi mkT}{h^{2}}\\right) ^{3\/2}\n\\end{equation*}\n\nLet $n_{gv}=N\/V$\n\n\\begin{equation}\nn_{gv}=2.61\\left( \\frac{2\\pi mkT}{h^{2}}\\right) ^{3\/2}  \\tag{7}\n\\end{equation}\n\n\\bigskip Recall that \\ $m=\\Delta \\varepsilon _{vac}\/c^{2}$ and the average\nkinetic energy of gas in three-dimensional space is given by $\\Delta\n\\varepsilon _{vac}=\\frac{3kT}{2}$. Thus $Eq.(7)$ becomes\n\n\\begin{equation*}\nn_{gv}=\\left( \\frac{\\left( 2.61\\right) \\left( 3\\pi \\right) ^{3\/2}k^{3}}\n\\left( hc\\right) ^{3}}\\right) T^{3}\n\\end{equation*}\n\nDefine\n\n\\begin{eqnarray*}\n\\xi &\\equiv &\\left( \\frac{\\left( 2.61\\right) \\left( 3\\pi \\right) ^{3\/2}k^{3\n}{\\left( hc\\right) ^{3}}\\right) \\\\\n&=&2.522\\times 10^{7}\\left( mk\\right) ^{-3}\n\\end{eqnarray*}\n\nHence, the Veneziano-ghost density$\\left( n_{gv}\\right) $ can be\nre-expressed in more elegant form as:\n\n\\begin{equation}\nn_{gv}=\\xi T^{3}  \\tag{8}\n\\end{equation}\n\n\\bigskip $Eq.(8)$ is the required result for the finite temperature and\ndensity relation for the Veneziano ghost(s).\n\n\\section{Gravitational Coupling Constant From Veneziano-Ghost Density}\n\nThe principle of general covariance tells us that the energy-momentum tensor\nin the vacuum must take the form\n\n\\begin{equation}\n\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert 0\\right\\rangle\n=T_{\\mu \\nu }^{vac}=g\\left\\langle \\rho \\right\\rangle  \\tag{9}\n\\end{equation}\n\n\\bigskip Here $\\left\\langle \\rho \\right\\rangle $ has the dimension of energy\ndensity and $g$ describes a real gravitational field \\cite{SE}. Thus $Eq.(9)$\ncan be written as\n\n\\begin{equation}\n\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert 0\\right\\rangle\n=g\\left( \\Delta \\varepsilon _{vac}\\right) ^{4}  \\tag{10}\n\\end{equation}\n\nWhere \\textquotedblleft $g$\\textquotedblright\\ in Ref.\\cite{FR,JZ}, is\ndefined as $g\\equiv c=C_{QCD}\\times C_{grav}.$ Therefore, $Eq.(10)$ can be\nwritten as\n\n\\begin{equation*}\n\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert 0\\right\\rangle\n=C_{QCD}\\times C_{grav}\\times \\left( \\Delta \\varepsilon _{vac}\\right) ^{4}\n\\end{equation*}\n\nWhere, $C_{QCD}=1$ as quoted by \\cite{JZ}, and references within, thus\n\n\\begin{equation}\n\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert 0\\right\\rangle\n=C_{grav}\\times \\left( \\Delta \\varepsilon _{vac}\\right) ^{4}  \\tag{11}\n\\end{equation}\n\nNow, the energy density can be written as\n\n\\begin{equation}\n\\rho _{vac}=\\frac{\\Delta \\varepsilon _{vac}}{V}=V^{-1}\\times \\Delta\n\\varepsilon _{vac}  \\tag{12}\n\\end{equation}\n\n$Eq.(12)$ is justified by the standard box-quantization procedure \\cite{SE}.\nBy comparing $Eq.(12)$ with $Eq.(8)$, we get\n\n\\begin{equation}\n\\rho _{vac}=n_{gv}\\times \\Delta \\varepsilon _{vac}  \\tag{13}\n\\end{equation}\n\nWith $n_{gv}\\equiv V^{-1},$ From the average kinetic energy for gas in\nthree-dimensional space, we have $T=2\\Delta \\varepsilon _{vac}\/3k.$ Hence \nEq.(8)$ becomes\n\n\\begin{equation}\nn_{gv}=\\frac{8\\xi \\left( \\Delta \\varepsilon _{vac}\\right) ^{3}}{27k^{3}} \n\\tag{14}\n\\end{equation}\n\nPutting the value of $n_{gv}$ in $Eq.(13)$, we get\n\n\\begin{equation}\n\\rho _{vac}=\\frac{8\\xi \\left( \\Delta \\varepsilon _{vac}\\right) ^{4}}{27k^{3}}\n\\tag{15}\n\\end{equation}\n\n$Eq.(15)$ represents the energy density of a vacuum state.\n\nThe natural demand of the Lorentz invariance of the vacuum state is bedecked\nin the structure of (effective) quantum field theory in Minkowski space-time\ngeometry \\cite{SE,RM}. Hence, if $\\left\\vert 0\\right\\rangle $\\ is a vacuum\nstate in a reference frame $S$ and $\\left\\vert \n{\\acute{}\n\\right\\rangle $ refers to the same vacuum state observed from a reference\nframe $\n{\\acute{}\n,$\\ which moves with uniform velocity relative to $S$, then the quantum\nexpression for Lorentz invariance of the vacuum state read\n\\begin{equation}\n\\left\\vert \n{\\acute{}\n\\right\\rangle =u\\left( L\\right) \\left\\vert 0\\right\\rangle =\\left\\vert\n0\\right\\rangle  \\tag{16}\n\\end{equation}\n\nWhere $u\\left( L\\right) $ is the unitary transformation (acting on the\nquantum state $\\left\\vert 0\\right\\rangle $) corresponding to a Lorentz\ntransformation $L$. All the physical properties that can be extracted from\nthis vacuum state, such as the value of energy density, should also remain\ninvariant under Lorentz transformations \\cite{SE}. If the Lorentz\ntransformation is initiated by $\\rho _{vac}$, then $2\\times \\rho _{vac}$ is\nneeded for a unitary transformation to take place. The logic behind this\nassumption is simple: if $\\rho _{vac}$ defines the Lorentz invariant length \n(L)$ (Euclidean metric distance) of $\\left\\vert 0\\right\\rangle $, then the\nLorentz transformation from $\\left\\vert 0\\right\\rangle $ to $\\left\\vert \n{\\acute{}\n\\right\\rangle $ (with continuous excitation) requires \\ $2\\times \\rho _{vac}\n: \\ $\\left\\vert 0\\right\\rangle \\overset{2\\times \\rho _{vac}}{\\longrightarrow \n}$\\ $\\left\\vert \n{\\acute{}\n\\right\\rangle $. This leads to the principle of general covariance as\napriori stated in the introduction \\cite{SE}. Thus,\n\n\\begin{equation}\n\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert 0\\right\\rangle\n=2\\times \\rho _{vac}=\\frac{16\\xi \\left( \\Delta \\varepsilon _{vac}\\right) ^{4\n}{27k^{3}}  \\tag{17}\n\\end{equation}\n\n\\bigskip $Eq.(17)$ is also justified by the standard box-quantization\nprocedure \\cite{SE}. Now by combining $Eq.(11)$ and $Eq.(17)$, we have\n\n\\begin{equation*}\nC_{grav}=\\frac{16\\xi }{27k^{3}}=2.336\\times 10^{19}\\left( m.eV\\right) ^{-3}\n\\end{equation*}\n\n\\bigskip As $1m=5.07\\times 10^{15}GeV^{-1}$. This leads to\n\n\\begin{equation}\nC_{grav}=1.797\\times 10^{-1}  \\tag{18}\n\\end{equation}\n\nwhich is the required gravitational coupling constant.\n\n\\section{Dark Energy From The Veneziano-Ghost: A Review}\n\nThe major ingredient of standard Witten-Veneziano resolution of $U(1)$\nproblem is the existence of topological susceptibility $\\chi $. In Ref.\\cit\n{FR}, it has been proved that the deviation in $\\chi $, i.e. $\\Delta \\chi $,\nrepresents the vacuum energy density \\emph{(dark energy)}. We review this\nresult by making use of $Eq.(9)$ and resolve the inherent hitch in this\napproach with the help of $Eq.(18)$. Thus from $Eq.(9)$ we have,\n\n\\begin{equation}\ni\\int dx\\left\\langle 0\\left\\vert \\widehat{T}_{\\mu \\nu }\\right\\vert\n0\\right\\rangle =i\\int dxT_{\\mu \\nu }^{vac}  \\tag{19}\n\\end{equation}\n\nBy using the standard Witten-Veneziano relations\n\n\\begin{equation*}\n\\widehat{T}_{\\mu \\nu }\\equiv T\\left\\{ Q\\left( x\\right) ,Q\\left( 0\\right)\n\\right\\}\n\\end{equation*}\n\n\\bigskip Wher\n\\begin{equation*}\nQ\\equiv \\frac{\\alpha _{s}}{16\\pi }\\epsilon ^{\\mu \\nu \\rho \\sigma }G_{\\mu \\nu\n}^{a}G_{\\rho \\sigma }^{a}\\equiv \\frac{\\alpha _{s}}{8\\pi }G_{\\mu \\nu }^{a\n\\widetilde{G}^{\\mu \\nu a}\\equiv \\partial _{\\mu }K^{\\mu }\n\\end{equation*}\n\nAnd\n\n\\begin{equation}\nK^{\\mu }\\equiv \\frac{\\Gamma ^{2}}{16\\pi ^{2}}\\epsilon ^{\\mu \\nu \\lambda\n\\sigma }A_{\\nu }^{a}\\left( \\partial _{\\lambda }A_{\\sigma }^{a}+\\frac{\\Gamma \n}{3}f^{abc}A_{\\lambda }^{b}A_{\\sigma }^{c}\\right)  \\tag{20}\n\\end{equation}\n\n\\bigskip Where $A_{\\mu }^{a}$\\ are the conventional $QCD$ color gluon fields\nand $Q$ is the topological charge density, and $\\alpha _{s}=$ $\\frac{\\Gamma\n^{2}}{4\\pi }$. Thus we have\n\n\\begin{equation*}\ni\\int dx\\left\\langle 0\\left\\vert T\\left\\{ Q\\left( x\\right) ,Q\\left( 0\\right)\n\\right\\} \\right\\vert 0\\right\\rangle =i\\int dxT_{\\mu \\nu }^{vac}\n\\end{equation*}\n\n\\begin{equation}\n\\underset{q\\longrightarrow 0}{\\lim }i\\int dxe^{iqx}\\left\\langle 0\\left\\vert\nT\\left\\{ Q\\left( x\\right) ,Q\\left( 0\\right) \\right\\} \\right\\vert\n0\\right\\rangle =\\underset{q\\longrightarrow 0}{\\lim }i\\int dxe^{iqx}T_{\\mu\n\\nu }^{vac}  \\tag{21}\n\\end{equation}\n\n\\bigskip Le\n\\begin{equation*}\n\\underset{q\\longrightarrow 0}{\\lim }i\\int dxe^{iqx}T_{\\mu \\nu }^{vac}=\\chi\n\\end{equation*}\n\nHence $Eq.(21)$ becomes\n\n\\begin{equation*}\n\\chi =\\underset{q\\longrightarrow 0}{\\lim }i\\int dxe^{iqx}\\left\\langle\n0\\left\\vert T\\left\\{ Q\\left( x\\right) ,Q\\left( 0\\right) \\right\\} \\right\\vert\n0\\right\\rangle\n\\end{equation*}\n\nAnd\n\n\\begin{equation}\n\\Delta \\chi =\\Delta \\left[ \\underset{q\\longrightarrow 0}{\\lim }i\\int\ndxe^{iqx}\\left\\langle 0\\left\\vert T\\left\\{ Q\\left( x\\right) ,Q\\left(\n0\\right) \\right\\} \\right\\vert 0\\right\\rangle \\right]  \\tag{22}\n\\end{equation}\n\nUsing $\\Delta =c\\left( H\/m_{\\eta }\\right) $ and $\\left[ \\underset\nq\\longrightarrow 0}{\\lim }i\\int dxe^{iqx}\\left\\langle 0\\left\\vert T\\left\\{\nQ\\left( x\\right) ,Q\\left( 0\\right) \\right\\} \\right\\vert 0\\right\\rangle\n\\right] =-\\left[ \\lambda _{YM}^{2}\\left( q^{2}-m_{0}^{2}\\right) \/\\left(\nq^{2}-m_{0}^{2}-\\frac{\\lambda _{\\eta }^{2}}{N_{c}}\\right) \\right] $ from \\\nRef.\\cite{FR}, $Eq.(22)$ can be written as\n\n\\begin{equation}\n\\Delta \\chi =-c\\left( \\frac{2H}{m_{\\eta }}\\right) .\\frac{\\lambda\n_{YM}^{2}\\left( q^{2}-m_{0}^{2}\\right) }{\\left( q^{2}-m_{0}^{2}-\\frac\n\\lambda _{\\eta }^{2}}{N_{c}}\\right) }  \\tag{23}\n\\end{equation}\n\nThe standard Witten-Veneziano solution of $U(1)$ problem is based on the\nwell-established assumption (confirmed by various lattice computations) that\n\\ $\\chi $ does not vanish, despite the fact that $Q$ is a total derivative \nQ\\equiv \\partial _{\\mu }K^{\\mu }$. This suggests that there is an unphysical\npole at $q=0$ in the correlation function of $K^{\\mu }$, similar to \\emph{KS}\nghost in the Schwinger model \\cite{FR}. Thus $Eq.(23)$ becomes\n\n\\begin{equation}\n\\Delta \\chi =-c\\left( \\frac{2H}{m_{\\eta }}\\right) .\\frac{\\lambda\n_{YM}^{2}m_{0}^{2}}{m_{\\eta }^{2}}  \\tag{24}\n\\end{equation}\n\nwhere $m_{\\eta }^{2}=m_{0}^{2}+\\frac{\\lambda _{\\eta }^{2}}{N_{c}}$ is the\nmass of physical $\\eta $ field and the reason for a factor of $2$ in \nEq.(24) $ follows from the principle of general covariance as we have\nalready established. Using Witten-Veneziano relation \\ $4\\lambda _{YM}^{2}$\\ \n$=f_{\\pi }^{2}m_{\\eta }^{2}$ and chiral condensate $m_{0}^{2}f_{\\pi\n}^{2}=-4m_{q}\\left\\langle \\overline{q}q\\right\\rangle ,$ \\ $Eq.(24)$ \\ can be\nwritten as\n\n\\begin{equation}\n\\Delta \\chi =c\\left( \\frac{2H}{m_{\\eta }}\\right) \\left\\vert\nm_{q}\\left\\langle \\overline{q}q\\right\\rangle \\right\\vert  \\tag{25}\n\\end{equation}\n\nwhere $H$ is Hubble constant and $m_{q}$\\ is the mass of a single light\nquark. From Ref.\\cite{FR} $c\\left( \\frac{2H}{m_{\\eta }}\\right) \\left\\vert\nm_{q}\\left\\langle \\overline{q}q\\right\\rangle \\right\\vert \\approx c\\left(\n3.6\\times 10^{-3}eV\\right) ^{4}$ leads to\n\n\\begin{equation}\n\\Delta \\chi \\approx c\\left( 3.6\\times 10^{-3}eV\\right) ^{4}  \\tag{26}\n\\end{equation}\n\nBy using $c=C_{QCD}\\times C_{grav}\\approx C_{grav}$ from \\cite{JZ} and\nreference within, $Eq.(26)$ can be written as\n\n\\begin{equation}\n\\Delta \\chi \\approx C_{grav}\\left( 3.6\\times 10^{-3}eV\\right) ^{4}  \\tag{27}\n\\end{equation}\n\n\\bigskip Comparision of $Eq.(18)$ with $Eq.(27)$ gives \n\\begin{equation}\n\\rho _{\\Lambda }\\equiv \\Delta \\chi \\approx \\left( 2.3\\times 10^{-3}eV\\right)\n^{4}  \\tag{28}\n\\end{equation}\n\n$Eq.(28)$ is the measured value of $\\rho _{\\Lambda }$ that is responsible\nfor the acceleration of the universe.\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Using Planck scale $M_{Pl}^{4}$ as the cutoff\ncorrection, $Eq.(8)$ becomes\n\n\\begin{equation}\nn_{gv}^{planck}=7\\times 10^{103}m^{-3}  \\tag{29}\n\\end{equation}\n\nFrom the standard box-quantization procedure \\cite{SE}, we have\n\n\\begin{equation}\n2\\times \\rho _{\\Lambda }^{total}=\\frac{1}{V}\\underset{k}{\\sum \\text\nh{\\hskip-.2em}\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}{\\hskip.2em\n}\\omega _{k}}  \\tag{30}\n\\end{equation}\n\nBy imposing Lorentz invariance of vacuum state formalism on $Eq.(30)$, we\nhave\n\n\\begin{equation}\n2\\times \\rho _{\\Lambda }^{total}=\\frac{1}{V}n\\text\nh{\\hskip-.2em}\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}{\\hskip.2em\n}\\omega =n\\left[ n_{gv}\\times \\Delta \\varepsilon _{vac}\\right]  \\tag{31}\n\\end{equation}\n\nWhere $n_{gv}\\equiv \\frac{1}{V}$ and $\\Delta \\varepsilon _{vac}\\equiv \nh{\\hskip-.2em}\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}{\\hskip.2em\n$\\omega .$ Note that $Eq.(31)$ reduces to $Eq.(17)$ for $n=1,$ therefore \nEq.(31)$ can be rewritten for Planck scale cutoff correction (where Planck\nseries of energy $(n\nh{\\hskip-.2em}\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}{\\hskip.2em\n$\\omega )$ is taken to be the Planck energy $(E_{Pl})$):\n\n\\begin{equation}\nn\\left[ n_{gv}\\times \\Delta \\varepsilon _{vac}\\right] =n_{gv}^{Planck}\\times\nE_{Pl}  \\tag{32}\n\\end{equation}\n\nFrom $Eqs.(13),(14),(15),(17)$ and $(28),$ we have\n\n\\begin{equation*}\nn\\left[ \\frac{8\\xi \\left( \\Delta \\varepsilon _{vac}\\right) ^{4}}{27k^{3}\n\\right] =n_{gv}^{Planck}\\times E_{Pl}\n\\end{equation*}\n\n\\begin{equation}\nn\\left[ \\frac{\\rho _{\\Lambda }}{2}\\right] =n_{gv}^{Planck}\\times\nE_{Pl}=M_{Pl}^{4}  \\tag{33}\n\\end{equation}\n\nWhere $\\rho _{vac}=\\rho _{\\Lambda }\/2$ is the energy density of each\ninfrared sector. $Eq.(33)$ shows how cutoff $UV$ scale $M_{Pl}^{4}$\\\nmanifests itself as linearly independent infrared sectors of the effective\ntheory of gravity interacting with $QCD$ fields.\n\nBy combining $Eqs.(28),(29)$ and $(33)$ we have\n\n\\begin{equation}\nn=4\\times 10^{122}\\approx 10^{122}  \\tag{34}\n\\end{equation}\n\nWhere $n_{gv}^{Planck}\\times E_{Pl}=M_{Pl}^{4}=1.4\\times 10^{113}J\/m^{3}.$\nThus $Eq.(34)$ suggests that there are $\\approx 10^{122}$(degenerate) vacuum\nstates. These vacuum states ($n-$torus) are called \\emph{\\textquotedblleft\nsubuniverses or multiverse\"\\ }\\cite{EB,SWH,SC,AV,SW97}. An $n-$torus is an\nexample of $n-$dimensional compact manifold or a compact Abelian Lie group \nU(1).$ In this sense, it is a product of $n$ circles i.e $T^{n}=S^{1}\\times\nS^{1}\\times ..........\\times S^{1}=T^{1}\\times T^{1}\\times ........\\times\nT^{1}$ \\cite{BR,DL,RP}. In this paper, $n$ circles, which are the elements\nof $U(1)$ group, represent $n$ linearly independent infrared sectors or the\nunphysical massless gauge bosons dubbed as Veneziano ghosts.\n\nIt is important to notice that the existence of non-vanishing \\ and linearly\nindependent infrared sectors of the effective theory of gravity interacting\nwith $QCD$ fields is parametrically proportional to the Planck cutoff\nenergy. Therefore, our simple extension of Veneziano ghost theory of $QCD$\nto accommodate FTFT has striking consequences: it predicts, accurately, the\nvalue of \\ $C_{grav},$ which leads towards the $100\\%$ consistency between\ntheory and experimental value of $\\rho _{\\Lambda }.$ As an offshoot, it\nfotifies the idea of multiverse and paints a new picture of quantum\ncosmological paradigm.\n\n\\section{Summary and Conclusion}\n\nThe computational analysis of the dark energy problem from the combined\nframeworks of finite temperature-density correction technique and the\nVeneziano ghost theory of $QCD$ conditions $FTD$ background to behave like a\nreservoir for the infrared sectors of the effective theory of gravity\ninteracting with $QCD$ fields. These infrared sectors (unphysical massless\nbosons) transform as a basis for a representation of a compact manifold.\nThis is analogous to the process of quantizing on manifold $M$ (such as a\ntorus group $T^{n}=T^{1}\\times ..........\\times T^{1}$ $=T^{10^{122}}$), in\nwhich all the submanifolds (tori) are linearly independent of each other.\nThis means that an \\emph{\\textquotedblleft observer\\textquotedblright }\\\ntrapped in one of such tori would think his torus is the whole Universe. An\nimportant prediction of this is that the vacuum energy $\\Delta \\varepsilon\n_{vac}$ owes its existence to the degenerate nature of vacuum (or to the\nasymmetric nature of the universe). The effect of this is a direct\nconsequence of the embedding of our subuniverse on a non-trivial manifold $M$\nwith (minuscule) different linear sizes.\n\nThe main result of the present study is that the effective scales obviously\nhave something to do with the cutoff Ultraviolet $(UV)$ scale $M_{Pl}$.\nBased on the standard box-quantization procedure, the $UV$ scale $M_{Pl}$ is\na collection of infrared $(IR)$ scales. Undoubtedly, the relevant effective\nscales appear as a result of energy differences (subtractions) at which the \nIR$ scales enter the physics of $UV$ scale $M_{Pl}$. It is therefore\nimpossible to compute the value of $\\rho _{\\Lambda }$ without putting into\nconsideration the statistical effect of the $UV$ scale $M_{Pl}$ which\nmanifests itself through the existence of the linearly independent $IR$\nsectors of the effective theory of quantum field theory $(QFT)$: this is the \n\\emph{\\textquotedblleft stone\\textquotedblright } that confirms the\ninterrelationship between $FTFT$ and the theory of superconductivity $(QFT$\nat $T=0)$.\n\nThus, if you buy the idea of Lorentz invariance of vacuum state formalism or\nthe degenerate vacuum mechanism, then $\\sim 10^{122}$ subuniverses come as\nfree gifts!\n\n\\subsubsection{\\textbf{Acknowledgement}}\n\nMr. O. F. Akinto is indebted to the Department of Physics, CIIT, Islamabad \\\nand the National Mathematical Center Abuja, Nigeria \\ for their finacial\nsupport.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStereo matching\\cite{DBLP:journals\/pami\/SunZS03,Ke2009Cross} is a core technique in many 3D computer vision applications, such as autonomous driving, robot navigation, object detection and recognition\\cite{DBLP:conf\/nips\/ChenKZBMFU15,DBLP:conf\/iccv\/ZhangLCCCR15}. It aims to get the depth information for the reference image by calculating the disparity map of the two images (\\emph{i.e.} left image and right image, sized $H\\times W$, respectively) captured by the stereo camera. The reference image can be either left image or right image. In the rest of this manuscript, we assume the left image to be the reference image, and the right image is regarded as the source image accordingly. \n\nThe disparity of a target point in the reference image is the horizontal displacement between the target point and its most similar point in the source image\\cite{DBLP:journals\/ijcv\/ScharsteinS02,DBLP:conf\/cvpr\/LuoSU16,DBLP:conf\/cvpr\/SekiP17} . In order to find the most similar point for a target point, the similarity scores between this point and the candidate points in the source image are calculated(\\emph{i.e.} similarity distribution)\\cite{DBLP:conf\/cvpr\/Hirschmuller05,DBLP:journals\/pami\/BrownBH03}. When there are $D_{max}$ candidate points(\\emph{i.e.} matching range), a 3D cost volume of size $D_{max}\\times H \\times W$ containing all similarity scores for $H \\times W$ target points is calculated. \n\nTo obtain such 3D cost volume, recent cost aggregation network based learning methods\\cite{DBLP:conf\/iccv\/KendallMDH17,DBLP:conf\/cvpr\/ChangC18,DBLP:conf\/cvpr\/GuoYYWL19,DBLP:conf\/cvpr\/ZhangPYT19} first form a 4D volume of size $F\\times \\frac{1}{n}D_{max}\\times \\frac{1}{n}H \\times \\frac{1}{n}W$(where $F$ is the dimension of the correlation feature and $n$ is the ratio of downsampling) by associating each unary feature with their corresponding unary from the opposite source image across $ \\frac{1}{n}D_{max}$ disparity levels. They obtain a high quality low-resolution 3D cost volume through focusing on optimizing the low-resolution 4D volume by cost aggregation network, and then get the high precision performance on final high-resolution disparity map ($H\\times W$). The process of most cost aggregation networks contains multiple identical 4D volume aggregation stages to refine the correlation features multiple times. These methods only output low-resolution 3D cost volume containing similarity scores of partial candidates. To obtain high-resolution disparity,  the widely accepted method is to use the linear interpolation to get the  complete 3D cost volume firstly.  However, the similarity score is not a linear function of the disparity, which causes inaccurate estimation of the final disparity. Although some methods add additional refinement modules to refine the disparity, they still cannot get satisfactory results due to the lack of correlation features between matching pairs.\n \nActually, due to the nature of CNN, each point in the low resolution features contains the information of all pixels in the patch of the original resolution images where its located.  Therefore, all $D_{max}$ correlation features between target points and candidates are included in the low-resolution 4D volume. Leveraging convolutional layers for decoupling all $D_{max}$ similarity scores from the 4D volume is naturally a better solution for complete 3D cost volume. Early methods, such as GC-Net\\cite{DBLP:conf\/iccv\/KendallMDH17}, decouple all $D_{max}$ similarity scores by \\emph{Transposed convolution}. However, the implementation of \\emph{Transposed convolution} introduces additional computation. More notably, as the network deepens, details in 4D Volume will be lost. In addition, learning $D_{max}$ similarity scores from the optimized low scale 4D volume, containing $\\frac{1}{n}D_{max}$ correlation features for each target point, means that one correlation feature outputs $n$ similarity scores.  This is an internal competitive task, because each feature essentially represents the degree of the correlation between the target point and $n$ different candidates. It is too difficult for the network to compute a universal correlation features to predict $n$ optimal similarity scores of $n$ different candidates simultaneously.   \n\nBased on the above analysis, we design a new Multistage Full Matching scheme (MFM) in this work through simply decomposing the full matching task into multiple stages. Each stage estimate a different $\\frac{1}{n}D_{max}$ similarity scores. In this decomposing way, we can not only learning all similarity scores directly from the low-resolution 4D volume, but also keep one similarity score learning from one correlation feature. \n\nWhile it is noteworthy that we share the similar insight with the existing multistage matching methods\\cite{DBLP:conf\/cvpr\/TonioniTPMS19,DBLP:conf\/iccv\/DuggalWMHU19,DBLP:conf\/cvpr\/YangMAL20,DBLP:conf\/cvpr\/YinDY19}, as decomposing the matching task into multiple stages. Such methods first obtain a coarse disparity and then perform residual disparity search from the neighbor of the current disparity by constructing a partial cost volume. The later stage strongly depend on  the previous stage. In contrast, the previous stage only provide a reference for the later stage in MFM. \n\nMultiple tasks in the proposed MFM are equally important. However, serial multistage framework, which is designed for more sufficient cost aggregation, results in unbalanced prediction of multiple stages. Aiming at this problem, we propose the strategy of \\emph{Stages Mutual Aid}. Specifically, we take advantage of the close distribution of the similarities predicted at each stage, and merge the output of other $(n-1)$ stages to obtain a voting result of the current similarity distribution for reference in the current stage. In this way, not only can the shallower network exploit the output of the deeper network, but also the voting result provides a correction message for the current prediction. \n\nThe contributions of our work are summarized as follows:\n\\begin{itemize} \n\\item A Multistage Full Matching disparity estimation scheme (MFM) is proposed to decompose the full matching learning task into multiple stages and decouple all $D_{max}$ similarity scores from the low-resolution 4D volume step by step, which improves the stereo matching precision accordingly. \n\\item A \\emph{Stages Mutual Aid} strategy is proposed to solve the unbalance between the predictions of each stage in the serial multistage framework.\n\\end{itemize}\nWe evaluate the proposed method on three challenging datasets, \\emph{i.e.} SceneFlow\\cite{DBLP:conf\/cvpr\/MayerIHFCDB16}, KITTI 2012\\cite{DBLP:conf\/cvpr\/GeigerLU12} and KITTI 2015 datasets\\cite{DBLP:conf\/cvpr\/MenzeG15}. The results demonstrate that our MFM scheme achieves state-of-the-art.\n\n\\section{Related Work}\nThis section reviews recent end-to-end supervised deep learning stereo matching methods.\n\\subsection{2D CNN Regression Module Based Methods}\nDispNetC\\cite{DBLP:conf\/cvpr\/MayerIHFCDB16} is the first end-to-end trainable disparity estimation network. It forms a low-resolution 3D cost volume  by calculating  cosine distance of each unary feature with their corresponding unary from the opposite stereo image across each disparity level. Then the 3D cost volume is input to 2D CNN with left features for disparity regeression. Following DispNetC, CRL\\cite{DBLP:conf\/iccvw\/PangSRYY17} and iRes-Net\\cite{DBLP:conf\/cvpr\/LiangFGLCQZZ18} introduce stack refinement sub-networks to further improve the performance. SegStereo\\cite{DBLP:conf\/eccv\/YangZSDJ18} and EdgeStereo\\cite{DBLP:journals\/corr\/abs-1903-01700} both design multiple tasks frameworks for the disparity regression task. The former introduces semantic information in the refinement stage and the latter applies edge information in guiding disparity optimization. \n\nThe 2D CNN regression network fails to make good use of geometric principles in stereo matching to regress accurate disparity. More recent works focus on directly optimize and compute 3D cost volume by cost aggregation networks.\n\n\\subsection{Cost Aggregation Network Based Methods}\nCost aggregation network based methods study how to optimize the low-resolution 4D volume to obtain more accurate similarity scores in the low-resolution 3D cost volume and output a better disparity map accordingly. Yu \\emph{et al.}\\cite{DBLP:conf\/aaai\/YuWWJ18} propose an explicit cost aggregation sub-network to provide better contextual information. PSMNet\\cite{DBLP:conf\/cvpr\/ChangC18} introduces a pyramid pooling module for incorporating global context information into image features, and stacked 3D CNN hourglasses to extend the regional support of context information in cost volume. In order to make full use of the features, GwcNet\\cite{DBLP:conf\/cvpr\/GuoYYWL19} builds the cost volume by concatenating the cost volume constructed in different ways. GANet\\cite{DBLP:conf\/cvpr\/ZhangPYT19} proposes two new neural net layers to capture the local and the whole-image cost dependencies and to replace the 3D convolutional layer. AANet\\cite{DBLP:journals\/corr\/abs-2004-09548} proposes a sparse points based intra-scale cost aggregation method to achieve fast inference speed while maintaining comparable accuracy. \n\nThese methods  only output low-resolution 3D cost volume containing similarity scores of partial candidates from the iterative cost aggregation network. However, the low-resolution 3D cost volume is inadequate for calculating high-resolution disparity without correlation features.  Although the high-resolution 3D cost volume in GC-Net\\cite{DBLP:conf\/iccv\/KendallMDH17} is obtained by \\emph{Transposed convolution}, additional calculations are introduced because of the way the \\emph{Transposed convolution} is implemented. In addition, it is an internal competitive task to calculate $D_{max}$ similarity features for the high-resolution 3D cost volume from only one 4D volume with $\\frac{1}{n}D_{max}$ correlation features. The proposed MFM in this work decomposes the full matching task into these multiple cost aggregation stages and decouple the high-resolution 3D cost volume directly from the correlation features, which solved the aforementioned problem.\n\n\\subsection{Multistage Matching Methods}\nMultistage matching methods\\cite{DBLP:conf\/cvpr\/TonioniTPMS19,DBLP:conf\/cvpr\/YinDY19} first obtain a coarse disparity and then perform residual disparity search from the neighbor of the current disparity by constructing a partial cost volume. DeepPruner\\cite{DBLP:conf\/iccv\/DuggalWMHU19}  develops a differentiable PatchMatch module that allows to discard most disparities without requiring full cost volume evaluation in the second stage. CVP-MVSNet\\cite{DBLP:conf\/cvpr\/YangMAL20} proposes a cost volume pyramid based Multi-View Stereo Network for depth inference. \n\nThese methods improve the stereo matching speed and avoid the process of obtaining high-resolution disparity from low-resolution 3D cost volume. However, if the cues obtained in the first stage is wrong, the subsequent fine matching will also be wrong. In contrast, the previous stage in our multistage methods has only guidance for the latter stage and does not limit the estimation range of the latter stage, which guarantees the freedom of estimation in the latter stage.\n\n\\section{Multistage Full Matching Network}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{architecture}\n\t\\caption{The architecture of the Multistage Full Matching network. LR denotes low-resolution and HR represents high-resolution. The Multistage estimation module is composed of $n$ stages. Here takes $n=3$ as an example. Each stage estimates the similarity scores of different candidate points($\\frac{1}{n}D_{max}$ points) in the matching range($D_{max}$ points). The high-resolution similarity distribution($D_{max}\\times H\\times W$) is obtained by combining the low-resolution similarity distribution output from each stage.}\n\t\\label{fig1}\n\\end{figure*}\n\nAs shown in Figure 1, the proposed framework has the following main structure: \n\nFirst, the MFM network extracts low-resolution feature maps of left and right images with shared 2D convolution. For feature extraction network, we adopt the ResNet-like network. We control the scale of the output features by controlling the stride of each convolution layer. The scale of the features is $\\frac{1}{n}$ the size of the input image.  \n\nSecond, low-resolution 4D volume is calculated by \\emph{Group-wise correlation}\\cite{DBLP:conf\/cvpr\/GuoYYWL19}. The left features and the right features are divided into groups along the channel dimension, and correlation maps are computed among each group to obtain multiple cost volumes, which are then packed into one cost volume called \\emph{g-cost}. Another cost volume called \\emph{cat-cost} is obtained by concatenating the features of each target point and candidate points in the matching range. The final cost volume is obtained by concatenating the corresponding correlation features of \\emph{g-cost} and \\emph{cat-cost}.\n\nThen, the low-resolution 4D volume from the second step is fed to the cost aggregation process for the high-resolution 3D cost volume by multistage 3D cost volume estimation module. \n\nAt last, the final disparity is obtained by the modified parabolic fitting\\cite{Raffel1998Particle,DBLP:journals\/scjapan\/ShimizuO02,nishiguti} sub-pixel estimation method on the high-resolution 3D cost volume.\n\n\\subsection{Multistage 3D Cost Volume Estimation Module} \nThe proposed method divides the matching range into $k$ cells, and each cell contains $n$ points, where $k=D_{max}\/n$. $D_{max}$ is the length of the matching range. $n$ is the same as the ratio of downsampling. In this way, the candidate set can be represented as $\\{c_{0}^{0}, c_{0}^{1},c_{0}^{2}, \\cdots, c_{0}^{n-1}, c_{1}^{0}, c_{1}^{1}, \\cdots,  c_{m}^s, \\cdots, c_{k-1}^{n-1}\\}$. Each stage of the multistage full matching module learns similarity scores of specific $k$ candidates, \\emph{i.e.} $\\{c_{0}^{s}, c_{1}^{s}, \\cdots, c_{k-1}^{s}\\}$($s$ is the stage number of current stage), from $k$ correlation features of the low-resolution 4D volume. The candidates learned in the $s$-th stage is adjacent to $(s-1)$ stage and $(s+1)$ stage. $D_{max}$ high-resolution similarity scores are obtained after $n$ stages.\n\nIn order to obtain $D_{max}$ high-resolution similarity scores, we decouple $D_{max}$ similarity features from different low-resolution 4D volumes. Let $f(x)$-$C$ be the coordinate system set up with $f(x)$ as the origin in the C dimensional feature space, and $f(x)$  is the feature corresponding to the point $I_r(x)$ of reference image. The similarity features between $I_r(x)$ and its candidates, \\emph{i.e.} $\\{fc(x+0), fc(x+1), \\cdots, fc(x+d), \\cdots, fc(x+D_{max}-1)\\}$, is a series points in $f(x)$-$C$. \n\nIn camera coordinate system, the position difference between $c_m^s(x)$ can be represented as $\\Delta x$. Similarly, in $f(x)$-$C$, the position difference between $fc(x+d)$ can be represented as $\\Delta fc$, where $\\Delta fc \\in \\mathbb{R}^C$. Therefore, when the position of $fc(x+d)$ is known, the position of $fc(x+d+\\Delta x)$ can be obtained by:\n\\begin{equation}\nfc(x+d+\\Delta x) = fc(x+d) + \\Delta fc\n\\end{equation}\nwhere $\\Delta x$ represents the position difference between  $c_m^s(x)$ and  $c_m^s(x+\\Delta x)$.\n\n$fc(x+d)$ in high-dimensional feature space not only contains the information of $c_m^s(x)$, but also include the features of points surround it. Consequently, $\\Delta fc$ from $fc(x+d)$ to $fc(x+d+\\Delta x)$ can be decoupled from $fc(x+d)$ when $\\Delta x$ is small. The success of our experiment verifies the correctness of this inference.\n\nBased on the above analysis, we design the follow framework:\n\\paragraph{Step 1} we first initialize a similarity feature $F^{ori}=\\{fc^{ori}_m | m \\in N, m < k\\}$ for all stage in the space. After that, we will estimate $\\Delta F^{s}=\\{\\Delta fc^{s}_m | m \\in N, m < k\\}$ from $F^{ori}$ to $F^{s}=\\{fc^{s}_m | m \\in N, m < k\\}$ for each stage in the second step, where $s$ is the stage number.\n\nDue to the nature of CNN, each point in the low scale features contains the information of all pixels in the patch of the original resolution images where its located.  Such features will first be associated  across $\\frac{1}{n}D_{max}$ disparity levels as follows:\n\\begin{equation}\nF_{raw}(f_{r},f_{so}^{d}) = Corr(f_{r},f_{so}^{d})\n\\end{equation}\nwhere $Corr(\\cdot, \\cdot)$ represent \\emph{Group-wise correlation}. $f_{r}$ is the feature of a target point on reference feature map, and $f_{so}^{d}$ is the feature of the $d$-th candidate point on source feature map. $F_{raw}(f_{r},f_{so}^{d})$ is the raw correlation feature of $f_{r}$ and $f_{so}^{d}$.\n\nTherefore, all $D_{max}$  correlation features between target points and candidates are included in the raw 4D volume composed of $F_{raw}(f_{r},f_{so}^{d})$. Then, $F_{raw}(f_{r},f_{so}^{d})$ in such raw 4D volume is converted to  $F^{ori}$ by two 3D convolutions. In this way, the $F^{ori}$ can provide an excellent initial value for better $\\Delta F^{s} $.\n\n\\paragraph{Step 2} The multistage similarity features estimation process is divided into $n$ stages. Except the first stage which takes $F_{ori}$ as input, each stage takes  $F^{s-1}$ as the input to get the shift  $\\Delta F^{s} $ between $F^{s}$ and $F^{ori}$.\n\\begin{equation}\n \\Delta F^{s} = De(F^{s-1} )\n\\end{equation}\nwhere $De(\\cdot)$ is the decouple function, which is implemented by an hourglass network composed of 3D CNN. The structure of   $De(\\cdot)$ is the same as the basic block in most cost aggregation network\\cite{DBLP:conf\/cvpr\/ChangC18,DBLP:conf\/cvpr\/GuoYYWL19}. Then, the simlairty features of candidates $\\{c_{0}^{s}, c_{1}^{s}, \\cdots, c_{k-1}^{s}\\}$ in the $s$-th stage can be obtained by\n\\begin{equation}\nF^{s} =F^{ori} + \\Delta F^{s}\n\\end{equation}\n\n\\paragraph{Step 3} Another role of $De(\\cdot)$ is to update each similarity feature by referring to the information in a larger receptive field. Therefore, a serial network composed of the $n$ $De(\\cdot)$ is necessary to obtain more sufficient aggregation. However, a serial network will lead to unbalance between the predictions of each stage, for each $De(\\cdot)$ is responsible for a different task. In this step,  as shown in Fig.1, a \\emph{Stages Mutual Aid} operation is conducted on $F^{s}$.\n \n We design the formula(5) to construct similarity scores for supervising each stage:\n\\begin{equation}\nS\\left(m,i\\right)=e^{-(m \\times n+i-d_{gt})^2}\n\\end{equation}\nwhere $m$ stands for the $m$-th cell, and $m \\in \\{0,1,2,\\ldots, k-1\\}$.  $n$ is the length of each cell. $i$ is the order of the candidate point in each cell, and it also represents the stage order. $d_{gt}$ is the ground-truth.  $S(m,i)$ represents the similarity score that the $i$-th point in the $m$-th cell of the $i$-th stage should have. The similarity distribution ground-truth of different stages can be obtained by changing the value of $i$. If the similarity peak falls in the $i$-th position of the $m$-th bin, the similarity peak of each stage will fall in the $m$-th bin or the ajacent bin of $m$. Therefore, the similarity distributions output of each stage is close, the similarity peak difference of which is 1 or 0. Accordingly, a voting result can be obtained from the similarity features output of the other $(n-1)$ stages for the estimation of the $s$-th stage.\n\nFirst, we obtain the voting results $V$ for the $s$-th stage by \n\\begin{equation}\nV= \\sum_{i=0, i\\ne s}^{n-1}(F^i)\n\\end{equation}\nThen, the $V$ is optimized by one 3D convolution layer, obtaining $V^s$.\nBecause it's hard to decide artificially the percentage of $V^s$ and $F^s$ when fusion the two features, we directly let the network learn how to merge the two features. The two features are concatenated along $d$ dimension and fed to the distance network $D(\\cdot)$ to calculate the similarity scores. The $D(\\cdot)$ network is consist of two 3D convolution layers.\n\\begin{equation}\nP(I_r(x), c^s(x))) = D(Concat(V^s, F^s)), \n\\end{equation}\nwhere $P(I_r(x), c^s(x)))$ is the 3D cost volume output from the $s$-th stage, which represents the predicted similarity scores between $I_r(x)$ and $c^s(x)$ of the $s$-th stage. $Concat(\\cdot)$ is the concatenation operation. \n\n \\paragraph{Step 4}\n Latitude $H$ and latitude $W$ of the $n$ 3D cost volumes are restored to the original scale by linear interpolation. The full resolution cost volume is obtained by combining the $n$ low-resolution 3D cost volume along the similarity dimension. Finally,  high-resolution cost volume is normalized by \\emph{softmax} operation along the similarity dimension and rearranged in the similarity dimension to obtain the final high-resolution cost volume.\n\n\\subsection{Supervision Strategy and Loss Function}\n\nWe design the formula(5) to construct similarity scores for supervising each stage. By using formula(5), our supervision strategy can guarantee the relationship between the output results of multiple stages. \n\nThe full loss of the proposed MFM network can be represented as the following:\n\\begin{equation}\nLoss= L_{stage}+L_{1}\n\\end{equation}\n\n$L_{1}$ loss is utilized to optimize the final disparity calculated through the high-resolution similarity distribution. We denote the predicted disparity as $d$ and the disparity ground-truth as $d_{gt}$, $L_{1}$ loss can be represented as the following:\n\\begin{equation}\nL_{1} = \\sum|d-d_{gt}|\n\\end{equation}\n\n$L_{stage}$ is designed to guide each stage to learn specific $k$ similarity scores from $k$ correlation features of the low-resolution 4D volume. We need to align the predicted similarity distributions of $n$ stages with the aforementioned supervision similarity distributions of the $n$ stages, respectively, therefore the Cross Entropy Error Function is selected for supervision. $L_{stage}$ loss is designed as follows:\n\\begin{equation}\nL_{stage}=\\sum_{i=0}^n\\sum_{m=0}^{k-1}S(m,i)\\cdot logP(m,i)\n\\end{equation}\nwhere $P(m,i)$ represents the similarity score between the target point and the $i$-th candidate point in the $m$-th cell output from the $i$-th stage, and $k=D_{max}\/n$. \n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[scale=0.43]{kitti12.pdf}\n  \\caption{Error map visualization of AcfNet\\cite{DBLP:conf\/aaai\/0005C0YYLY20}, AANet+\\cite{DBLP:journals\/corr\/abs-2004-09548}  and our method on KITTI 2012. Darker represents lower error.}\n \\end{figure*}\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[scale=0.43]{kitti15.pdf}\n  \\caption{ Error map visualization of AcfNet\\cite{DBLP:conf\/aaai\/0005C0YYLY20}, AANet+\\cite{DBLP:journals\/corr\/abs-2004-09548}  and our method on KITTI 2015. Darker blue represents lower error.}\n\\end{figure*}\n\\section{Experiments}\n\\subsection{Implementation Details}\n\\paragraph{Datasets.} Scene Flow datasets\\cite{DBLP:conf\/cvpr\/MayerIHFCDB16} provide 35,454 training and 4,370 testing images of size $960\\times540$ with accurate ground-truth. We use the Finalpass of the Scene Flow datasets, since it contains more motion blur and defocus, and is more real than the Cleanpass. KITTI 2012\\cite{DBLP:conf\/cvpr\/GeigerLU12} and KITTI 2015\\cite{DBLP:conf\/cvpr\/MenzeG15} are driving scene datasets. KITTI 2012 contains 194 training image pairs with sparse ground truth and 195 testing image pairs with ground truth disparities held by evaluation server for submission evaluation only. KITTI 2015 contains 200 training stereo image pairs with sparse ground-truth and 200 testing image pairs with ground truth disparities held by evaluation server for submission evaluation only. \n\n\\paragraph{Evaluation Indicators.} For Scene Flow datasets, the evaluation metrics are the end-point error (EPE), which is the mean average disparity error in pixels, and the error rates $>1px$ and $>3px$ are the percentage of pixels whose error are greater than 1 pixel and 3 pixels, respectively. For KITTI 2015, we use the percentage of disparity outliers D1 as evaluation indicators. The outliers are defined as the pixels whose disparity errors are larger than $max(3px, 0.05\\cdot d_{gt})$, where $d_{gt}$ denotes the ground-truth disparity. For KITTI 2012, we use the error rates $ >2px, >3px, >4px$ and $>5px$ as evaluation indicators.\n\n\\paragraph{Training.} Our network is implemented with PyTorch. We use Adam optimizer, with $\\beta_{1} = 0.9, \\beta_{2} = 0.999$. The batch size is fixed to 8. For Scene Flow datasets, we train the network for 16 epochs in total. The initial learning rate is set as 0.001 and it is down-scaled by 2 after epoch 10, 12, 14. We set the $D_{max}$ as 192 and $n$ as 4, respectively. For KITTI 2015 and KITTI 2012, we fine-tune the network pre-trained on Scene Flow datasets for another 300 epochs. The learning rate is 0.001 and it is down-scaled by 10 after 210 epochs.\n\n\\subsection{Performance Comparison}\n\n\\paragraph{Quantitative Comparison.}\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Performance comparison on Scene Flow datasets.}\n\t\\begin{tabular}{lccc}\n\t\t\\hline\n\t\tModel & EPE & \\textgreater{}1px  & \\textgreater{}3px \\\\ \n\t\t\\hline\n\t\tiResNet-i3\\shortcite{DBLP:conf\/cvpr\/LiangFGLCQZZ18}&2.45&9.28\\%&4.57\\% \\\\\n\t\tCRL\\shortcite{DBLP:conf\/iccvw\/PangSRYY17}&1.32&-&6.20\\% \\\\\n\t\t\\hline\n\t\tStereoNet\\shortcite{DBLP:conf\/eccv\/KhamisFRKVI18}&1.10&21.33\\%&8.80\\%   \\\\\n\t\tPSMNet\\shortcite{DBLP:conf\/cvpr\/ChangC18}&1.03& 10.32\\%&4.12\\%        \\\\\n\t\tGANet\\shortcite{DBLP:conf\/cvpr\/ZhangPYT19}&0.81& 9.00\\%&3.49\\%         \\\\\n\t\tGwcNet\\shortcite{DBLP:conf\/cvpr\/GuoYYWL19}&0.77&8.03\\%  &3.30\\%       \\\\\n\t\tAANet\\shortcite{DBLP:journals\/corr\/abs-2004-09548}&0.87&9.30\\%&-         \\\\\n\t\tAcfNet\\shortcite{DBLP:conf\/aaai\/0005C0YYLY20}&0.87&-&4.31\\%          \\\\   \n\t\t\\hline\n\t\tDeepPruner-Best\\shortcite{DBLP:conf\/iccv\/DuggalWMHU19}&0.86&-&- \\\\\n\t\t\\hline\n\t\tOurs&\\textbf{0.66}&\\textbf{4.95\\%}&\\textbf{2.50\\%}                    \\\\ \n\t\t\\hline\n\t\\end{tabular}\n    \\label{table1}\n\\end{table}\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Performance comparison on KITTI 2015 datasets.}\n\t\\begin{tabular}{llll}\n\t\t\\hline \n\t\tModel&D1-bg&D1-fg&D1-all \\\\ \n\t\t\\hline\n\t\tCRL\\shortcite{DBLP:conf\/iccvw\/PangSRYY17}&2.48\\%&3.59\\%&2.67\\% \\\\\n\t\tEdgeStereo-v2\\shortcite{DBLP:journals\/corr\/abs-1903-01700}&1.84\\%&3.30\\%&2.08\\% \\\\\n\t\tSegStereo\\shortcite{DBLP:conf\/eccv\/YangZSDJ18}&1.88\\%&4.07\\%&2.25\\% \\\\\n\t\t\\hline\n\t\tGCNet\\shortcite{DBLP:conf\/iccv\/KendallMDH17}&2.21\\%&6.16\\%&2.87\\% \\\\\n\t\tGwcNet-g\\shortcite{DBLP:conf\/cvpr\/GuoYYWL19}&1.74\\%&3.93\\%&2.11\\%   \\\\\n\t\tAcfNet\\shortcite{DBLP:conf\/aaai\/0005C0YYLY20}&1.51\\%&3.80\\%&1.89\\%   \\\\\n\t\tBi3D\\shortcite{DBLP:conf\/cvpr\/BadkiTKKSG20}&1.95\\%&\\textbf{3.48}\\%&2.21\\%         \\\\\n\t\tAANet+\\shortcite{DBLP:journals\/corr\/abs-2004-09548}&1.65\\%&3.96\\%&2.03\\% \\\\\n\t\t\\hline\n\t\tHD\\^ \\ 3\\shortcite{DBLP:conf\/cvpr\/YinDY19}&1.70\\%&3.63\\%&2.02\\%         \\\\\n\t\tCSN\\shortcite{DBLP:conf\/cvpr\/GuFZDTT20}&1.59\\%&4.03\\%&2.00\\% \\\\\n\t\t\\hline\n\t\tOurs&\\textbf{1.51\\%}&3.67\\%&\\textbf{1.87}\\% \\\\\n\t\t\\hline\n\t\\end{tabular}\n    \\label{table2}\n\\end{table}\n\\begin{table*}[t]\n\t\\centering\n\t\\caption{Performance comparison on KITTI 2012 datasets.}\n\t\\begin{tabular}{l|cc|cc|cc|cc}\n\t\t\\hline\n\t\t\\multirow{2}{*}{Model} & \\multicolumn{2}{c|}{ \\textgreater{}2px}&\\multicolumn{2}{c|}{ \\textgreater{}3px}&\\multicolumn{2}{c|}{ \\textgreater{}4px}& \\multicolumn{2}{c}{ \\textgreater{}5px} \\\\\n\t\t\\cline{2-9} \n\t\t&Noc&All&Noc&All&Noc&All&Noc&All \\\\\n\t\t\\hlin\n\t\tEdgeStereo-v2\\shortcite{DBLP:journals\/corr\/abs-1903-01700}&2.32\\%&2.88\\%&1.46\\%&1.83\\%&1.07\\%&1.34\\%&0.83\\%&1.04\\%\\\\\n\t\tSegStereo\\shortcite{DBLP:conf\/eccv\/YangZSDJ18}&2.66\\%&3.19\\%&1.68\\%&2.03\\%&1.25\\%&1.52\\%&1.00\\%&1.21\\%\\\\%&0.5\\%\\\\\n\t\t\\hline\n\t\tGwcNet-gc\\shortcite{DBLP:conf\/cvpr\/GuoYYWL19}&2.16\\%&2.71\\%&1.32\\%&1.70\\%&0.99\\%&1.27\\%&0.80\\%&1.03\\%\\\\\n\t\tGANet-deep\\shortcite{DBLP:conf\/cvpr\/ZhangPYT19}&1.89\\%&2.50\\%&1.19\\%&1.60\\%&0.91\\%&1.23\\%&0.76\\%&1.02\\%\\\\\n\t\tAMNet\\shortcite{DBLP:journals\/corr\/abs-1904-09099}&2.12\\%&2.71\\%&1.32\\%&1.73\\%&0.99\\%&1.31\\%&0.80\\%&1.06\\%\\\\\n\t\tAcfNet\\shortcite{DBLP:conf\/aaai\/0005C0YYLY20}&1.83\\%&2.35\\%&1.17\\%&1.54\\%&0.92\\%&1.21\\%&0.77\\%&1.01\\%\\\\\n\t\tAANet+\\shortcite{DBLP:journals\/corr\/abs-2004-09548}&2.30\\%&2.96\\%&1.55\\%&2.04\\%&1.20\\%&1.58\\%&0.98\\%&1.30\\%\\\\\n\t\t\\hline\n\t        HD\\^ \\ 3\\shortcite{DBLP:conf\/cvpr\/YinDY19}&2.00\\%&2.56\\%&1.40\\%&1.80\\%&1.12\\%&1.43\\%&0.94\\%&1.19\\%\\\\\n\t\t\\hline\n\t\tOurs&\\textbf{1.68\\%}&\\textbf{2.16\\%}&\\textbf{1.15\\%}&\\textbf{1.47\\%}&\\textbf{0.91\\%}&\\textbf{1.16\\%}&\\textbf{0.76\\%}&\\textbf{0.97\\%}\\\\%&\\textbf{0.4}&\\textbf{0.5}   \n\t\t\\hline       \n\t\\end{tabular}\n    \\label{table3}\n\\end{table*}\n\nThe quantitative comparison focuses on deep learning stereo matching methods. Table 1, Table 2 and Table 3 show the performance of different methods on Scene Flow datasets,  KITTI 2012 dataset, and KITTI 2015 dataset, respectively. In each table from top to bottom, the methods are separated into four groups: (1)2D CNN regression based methods, (2)cost aggregation network based methods, (3)multistage matching methods, (4)our MFM method.\n\n(1) methods refine the low-resolution coarse disparity with much error and noise without referring to the correlation features. Thus, such methods have lower precision. (2) methods obtain the high quality 3D cost volume iteratively from the 4D volume, which provides satisfied geometry context. Therefore, compared with (1) methods, the EPE value of (2) methods is significantly reduced to below 1.0. However, the $>1px$ error rates is still high. Because the multistage cost aggregation module only outputs a low-resolution 3D cost volume, and the high-resolution disparity is obtained from the low-resolution 3D cost volume without the geometry correlation features. (3) methods match within the narrow matching range obtained from the previous stage other than obtaining all similarity scores directly. The miscalculated narrow matching range obtained in the previous stage results in misestimating in the latter stage, which causes its low accuracy. As shown in Table1,Table2 and Table3, the proposed MFM method performs better, which demonstrates the effectiveness of the multistage 3D cost volume estimation module.\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{scene}\n\t\\caption{Error map visualization of GwcNet(top row)\\cite{DBLP:conf\/cvpr\/GuoYYWL19} and our method(bottom row) on Scene Flow datasets. Darker blue represents lower error.}\n\t\\label{figure2}\n\\end{figure*}\n\n\n\n\\paragraph{Qualitative Comparison.}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{similarity}\n\t\\caption{Peak position estimation comparison with existing classic  cost aggregation network based methods. }\n\t\\label{figure3}\n\\end{figure}\nFigure 4 visualizes the error maps of our method and the GwcNet\\cite{DBLP:conf\/cvpr\/GuoYYWL19} on Scene Flow datasets. As shown in Figure 4, light blue area takes up less area in the error map of our MFM method, which illustrates that our method has better prediction accuracy by decompose the matching task into multiple stage for high-resolution 3D cost volume directly.\n\nFigure 2 and Figure 3 visualize the error maps of our method and other methods  on KITTI 2012 and KITTI 2015, respectively. The error area takes up less area in the error map of our MFM method, which also demonstrates the effectiveness of the proposed MFM mechanism.\n\nThe prediction of the peak position in the similarity distribution determines the accuracy of the disparity obtained through 3D cost volume refinement, which is the first and the most important phase of the disparity prediction task. We count the proportion of the points with peak position deviating from the ground truth by more than 1 pixel and 3 pixels, respectively. As shown in Figure 5, PSMNet, GANet, and GwcNet all have a larger number of error points than our MFM method, especially on the deviated more than 1 pixel figure. Well begun is half done. The remarkable advantage of our method shown in Figure 5 largely determines our final state-of-art performance.\n\n\\subsection{Detailed Analysis of Proposed Method}\n\\paragraph{Ablation Study.}\n\\begin{table}[t]\n\t\\centering\n\t\\caption{The ablative prediction results of different variants of MFM on Scene Flow datasets. B:Baseline, de:decouple, ms:\\emph{Multistage Matching}, sma: \\emph{Stages Mutual Aid}.}\n\n\t\\begin{tabular}{ccccccl}\n\t\t\\hline\n\t\tModel&EPE& \\textgreater{}1px  & \\textgreater{}3 px \\\\\n\t\t\\hline\n\t\tBaseline &0.77&8.03\\% &3.30\\%\\\\\n\t\t\\hline\n\t\tB+de&0.86&6.30\\%&3.10\\%\\\\\n\t\t\\hline\n\t\tB+de+ms&0.74&5.18\\%&2.66\\%\\\\\n\t\tB+de+ms+sma&\\textbf{0.66}&\\textbf{4.95\\%}& \\textbf{2.50\\%} \\\\\n\t\t\\hline\n\t\\end{tabular}\n    \\label{table3}\n\\end{table}\nWe conduct ablation studies to understand the influence of different designs in our proposed method. We design different runs on Scene Flow datasets and report the results in Table 4. First, GwcNet\\cite{DBLP:conf\/cvpr\/GuoYYWL19} is adopted as our baseline. The \"Baseline\" refine the correlation features multiple times for a low-resolution 3D cost volume and the high-resolution 3D cost volume is obtained by linear interpolation. When we introduce the learning mechanism(Baseline+decouple) to learn all similarity scores from the last stage of the cost volume aggregation module, the $> 1px$ and $> 3px$ prediction error on Scene Flow datasets improve $1.73\\%$ and $0.20\\%$, respectively. However, the EPE drops slightly for the competition of simultaneously learning multiple similarity scores from one correlation feature, which may influence the prediction results on a large pixel error range. Then, we take account into the multistage decomposition of the learning task(Baseline+decouple +\\emph{Multistage matching}), and achieve the state-of-the-art result on all accuracy metrics. This demonstrates that decoupling all similarity scores step by step from different 4D volume makes the task easier to learn. Finally, the \\emph{Stages Mutual Aid} operation is added to 'Baseline+decouple +\\emph{Multistage matching}'. The performance is significantly improved, which demonstrates  the simplicity and effectiveness of the \\emph{Stages Mutual Aid} module. Ablation experiments have verified that the proposed MFM scheme indeed learns the more accurate similarity scores by decomposing the task into multiple stages, thus effectively improves the prediction performance of the high-resolution disparity map.  \n\n\\section{Conclusion}\nIn this paper, we propose the Multistage Full Matching framework, which directly estimates high-resolution 3D cost volume from the low-resolution 4D volume by decomposing the matching task into multiple stages. First, a serial network is designed for sufficient cost aggregation and multistage high-resolution 3D cost volume estimation. Then, the  \\emph{Stages Mutual Aid} is proposed to solve the unbalanced prediction of the multiple stages resulted by serial network. The last but the most important, our MFM scheme achieves state-of-the-art on three popular datasets.\n\n\n\n\n\\bibliographystyle{aaai21}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCurrent advances in robotics and autonomous systems have expanded the use of robots in a wide range of robotic tasks including assembly, advanced manufacturing, human-robot or robot-robot collaboration. In order for robots to efficiently perform these tasks, they need to have the ability to adapt to the changing environment while interacting with their surroundings, and a key component of this interaction is the reliable grasping of arbitrary objects. Consequently, a recent trend in robotics research has focused on object detection and pose estimation for the purpose of dynamic robotic grasping.\n\nHowever, identifying objects and recovering their poses are particularly challenging tasks as objects in the real world are extremely varied in shape and appearance. Moreover, cluttered scenes, occlusion between objects, and variance in lighting conditions make it even more difficult. Additionally, the system needs to be sufficiently fast to facilitate real-time robotic tasks. As a result, a generic solution that can address all these problems remains an open challenge.\n\nWhile classification~\\cite{resnet, vgg, inception,flexnet, overfeat, spatial}, detection~\\cite{fast-rcnn, faster-rcnn, ssd, yolo, yolo-9000, fldnet}, and segmentation~\\cite{segnet, maskrcnn, unet}  of objects from images have taken a significant step forward - thanks to deep learning, the same has not yet happened to 3D localization and pose estimation. One primary reason was the lack of labeled data in the past as it is not practical to manually infer, thus   As a result, the recent research trend in the deep learning community for such applications has shifted towards synthetic datasets~\\cite{butler,mayer,qiu,zhang,mccormac}. Several pose estimation methods leveraging deep learning techniques~\\cite{posecnn,dope,brachmann,wang,hu} use these synthetic datasets for training and have shown satisfactory accuracy. \n\nAlthough synthetic data is a promising alternative, capable of generating large amounts of labeled data, it requires photorealistic 3D models of the objects to mirror the real-world scenario. Hence, generating synthetic data for each newly introduced object needs photo-realistic 3D models and thus significant effort from skilled 3D artists. Furthermore, training and running deep learning models are not feasible without high computing resources as well. As a result, object detection and pose estimation in real-time with computationally moderate machines remain a challenging problem. To address these issues, we have devised a simpler pipeline that does not rely on high computing resources and focuses on planar objects, requiring only an RGB image and the depth information in order to infer real-time object detection and pose estimation.\n\nIn this work, we present a feature-detector-descriptor based method for detection and a homography based pose estimation technique where, by utilizing the depth information, we estimate the pose of an object in terms of a 2D planar representation in 3D space. The robot is pre-trained to perform a set of canonical grasps; a canonical grasp describes how a robotic end-effector should be placed relative to an object in a fixed pose so that it can securely grasp it. Afterward, the robot is able to detect objects and estimates their pose in real-time, and then adapt the pre-trained canonical grasp to the new pose of the object of interest. We demonstrate that the proposed method can detect a well-textured planar object and estimate its accurate pose within a tolerable amount of out-of-plane rotation. We also conducted experiments with the humanoid PR2 robot to show the applicability of the framework where the robot grasped objects by adapting to a range of different poses.\n\n\n\\section{Related Work}\nOur work constitutes of three modules: object detection, planar pose estimation, and adaptive grasping. In the following sub-sections, several fields of research that are closely related to our work are reviewed.\n\\subsection{Object Detection}\nObject detection has been one of the fundamental challenges in the field of computer vision and in that aspect, the introduction of feature detectors and descriptors represents a great achievement. Over the past decades, many detectors, descriptors, and their numerous variants have been presented in the literature. The applications of these methods have widely extended to numerous other vision applications such as panorama stitching, tracking, visual navigation, etc.\n\nOne of the first feature detectors was proposed by Harris et al.~\\cite{harris} (widely known as the Harris corner detector). Later Tomasi et al.~\\cite{tomasi} developed the KLT (Kanade-Lucas-Tomasi) tracker based on the Harris corner detector. Shi and Tomasi introduced a new detection metric GFTT~\\cite{shi} (Good Features To Track) and argued that it offered superior performance. Hall et al. introduced the concept of saliency~\\cite{hall} in terms of the change in scale and evaluated the Harris method proposed in~\\cite{lindeberg} and the Harris Laplacian corner detector~\\cite{mikolajczyk} where a Harris detector and a Laplacian function are combined.\n\nMotivated by the need for a scale-invariant feature detector, in 2004 Lowe~\\cite{SIFT} published one of the most influential papers in computer vision, SIFT (Scale Invariant Feature Transform). SIFT is both a feature point detector and descriptor. H. Bay et al.~\\cite{SURF} proposed SURF (Speeded Up Robust Features) in 2008. But both of these methods are computationally expensive as SIFT detector leverages the difference of Gaussians (DoG) in different scales while SURF detector uses a Haar wavelet approximation of the determinant of the Hessian matrix to speed up the detection process. Many variants of SIFT~\\cite{sift_v_1, sift_v_2, sift_v_3, sift_v_4} and SURF~\\cite{surf_v_1, surf_v_2, surf_v_3} were proposed, either targeting a different problem or reporting improvements in matching, however, the execution time remained a persisting problem for several vision applications.\n\nTo improve execution time, several other detectors such as FAST~\\cite{fast} and AGAST~\\cite{agast} have been introduced. Calonder et al. developed the BRIEF~\\cite{brief} (Binary Robust Independent Elementary Features) descriptor of binary strings that has a fast execution time and is very useful for matching images. E. Rublee et al. presented ORB~\\cite{ORB} (Oriented FAST and Rotated Brief) which is a combination of modified FAST (Features from Accelerated Segment Test) for feature detection and BRIEF for description. S. Leutnegger et al. designed BRISK~\\cite{BRISK} (Binary Robust Invariant Scale Keypoint) that detects corners using AGAST and filters them using FAST. On the other hand, FREAK (Fast Retina Key-point), introduced by Alahi et al.~\\cite{FREAK} generates retinal sampling patterns using a circular sampling grid and uses a binary descriptor, formed by a one bit difference of Gaussians (DoG). Alcantarilla et al. introduced KAZE~\\cite{KAZE} features that exploit non-linear scale-space using non-linear diffusion filtering and later extended it to AKAZE~\\cite{AKAZE} where they replaced it with a more computationally efficient method called FED (Fast Explicit Diffusion)~\\cite{fed1, fed2}.\n\nIn our work, we have selected four methods to investigate: SIFT, SURF, FAST+BRISK, AKAZE. \n\n\\subsection{Planar Pose Estimation}\n\nAmong the many techniques in literature on pose estimation, we focus our review on those related to planar pose estimation. In recent years, planar pose estimation has been increasingly becoming popular in many fields, such as robotics and augmented reality. \n\nSimon et. al~\\cite{simon} proposed a pose estimation technique for planar structures using homography projection and by computing camera pose from consecutive images. Changhai et. al~\\cite{changhai} presented a method to robustly estimate 3D poses of planes by applying a weighted incremental normal estimation method that uses Bayesian inference. Donoser et al.~\\cite{donoser} utilized the properties of Maximally Stable Extremal Regions (MSERs~\\cite{LMSER}) to construct a perspectively invariant frame on the closed contour to estimate the planar pose. In our approach, we applied perspective transformation to approximate a set of corresponding points on the test image for estimating the basis vectors of the object surface and used the depth information to estimate the 3D pose by computing the normal to the planar object.\n\n\\subsection{Adaptive Grasping}\n\nDesigning an adaptive grasping system is challenging due to the complex nature of the shapes of objects. In early times, analytical methods were used where the system would analyze the geometric structure of the object and would try to predict suitable grasping points. Sahbani et al.~\\cite{sahbani} did an in depth review on the existing analytical approaches for 3D object grasping. However, with the analytical approach it is difficult to compute force and not suitable for autonomous manipulation. Later, as the number of 3D models increased, numerous data driven methods were introduced that would analyze grasps in the 3D model database and then transfer to the target object. Bohg et al.~\\cite{bohg} reviewed data driven grasping method methods where they divided the approach into three groups based on the familiarity of the object.\n\nKehoe et al. \\cite{Kehoe2013} used a candidate grasp from the candidate grasp set based on the feasibility score determined by the grasp planner. The grasps weren't very accurate in situations where the objects had stable horizontal poses and were close to the width of the robot's gripper. Huebner et al. \\cite{Huebner2008} also take a similar approach as they perform grasp candidate simulation. They created a sequence of grasps by approximating the shape of the objects and then computed a random grasp evaluation for each model of objects. In both works, a grasp has been chosen from a list of candidate grasps.\n\nThe recent advances in deep learning also made it possible to regress grasp configuration through deep convolutional networks. A number of deep learning-based methods were reviewed in~\\cite{caldera} where the authors also discussed how each element in deep learning-based methods enhances the robotic grasping detection. \\cite{Yu2013} presented a system where deep neural networks were used to learn hierarchical features to detect and estimate the pose of an object, and then use the centers of the defined pose classes to grasps the objects. Kroemer et al.~\\cite{Kroemer2009} introduced an active learning approach where the robot observes a few good grasps by demonstration and learns a value function for these grasps using Gaussian process regression. Aleotti et al.~\\cite{Aleotti2011} proposed a grasping model that is capable of grasping objects by their parts which learns new tasks from human demonstration with automatic 3D shape segmentation for object recognition and semantic modeling. \\cite{Saxena2008} and \\cite{Montesano2012} used supervised learning to predict grasp locations from RGB images. In~\\cite{Nogueira2016}, as an alternative to a trial and error exploration strategy, the authors proposed a Bayesian optimization technique to address the robot grasp optimization problem of unknown objects. These methods emphasized developing and using learning models for obtaining accurate grasps. \n\nIn our work, we focus on pre-defining a suitable grasp relative to an object that can adapt to a new grasp based on the change of position and orientation of the object.\n\n\\section{Method}\nThe proposed method is divided into two parts. The first part outlines the process of simultaneous object detection and pose estimation of multiple objects and the second part describes the process of generating an adaptive grasp using the pre-trained canonical grasp and the object pose.\nThe following sections describe the architecture of the proposed framework (figure~\\ref{fig:sysarch}) in detail.\n\n\n\n\\subsection{Object Detection and Pose Estimation}\n\nWe present a planar pose estimation algorithm (algorithm \\ref{algorithm}) for adaptive grasping that consists of four phases: (i) feature extraction and matching, (ii) homography estimation and perspective transformation, (iii) directional vectors estimation on the object surface, (iv) planar pose estimation using the depth data. In the following sections, we will focus on the detailed description of the aforementioned steps.\n\n\\subsubsection{Feature extraction and matching}\n\nOur object detection starts with extracting features from the images of the planar objects and then matching them with the features found in the images acquired from the camera. Image features are patterns in images based on which we can describe the image. A feature detecting algorithm takes an image and returns the locations of these patterns - they can be edges, corners or interest points, blobs or regions of interest points, ridges, etc. This feature information then needs to be transformed into a vector space using a feature descriptor, so that it gives us the possibility to execute numerical operations on them. A feature descriptor encodes these patterns into a series of numerical values that can be used to match, compare, and differentiate one feature to another; for example, we can use these feature vectors to find the similarities in different images which can lead us to detect objects in the image. In theory, this information would be invariant to image transformations. In our work, we have investigated SIFT~\\cite{SIFT}, SURF~\\cite{SURF}, AKAZE~\\cite{AKAZE}, and BRISK~\\cite{BRISK} descriptors. SIFT, SURF, AKAZE are both feature detectors and descriptors, but BRISK uses FAST~\\cite{fast} algorithm for feature detection. These descriptors were selected after carefully reviewing the comparisons done in the recent literature~\\cite{andersson2016comparison, karami2017image, tareen2018comparative}.\n\nOnce the features are extracted and transformed into vectors, we compare the features to determine the presence of an object in the scene. For non-binary feature descriptors (SIFT, SURF) we find matches using the Nearest Neighbor algorithm. However, finding the nearest neighbor matches within high dimensional data is computationally expensive, and with more objects introduced it can affect the process of updating the pose in real-time. To counter this issue to some extent, we used the FLANN~\\cite{muja_flann_2009} implementation of K-d Nearest Neighbor Search, which is an approximation of the K-Nearest Neighbor algorithm that is optimized for high dimensional features. For binary features (AKAZE, BRISK), we used the Hamming distance ratio method to find the matches. Finally, if we have more than ten matches, we presume the object is present in the scene. \n\n\\RestyleAlgo{boxruled}\n\\begin{algorithm}[ht]\n\\fontsize{8}{8}\\selectfont\n\\DontPrintSemicolon\n   \n    \\KwIn{Training images of planar objects, $\\mathcal{I}$}\n    $Detector \\gets \\text{Define feature detector}$\\;\n    $Descriptor \\gets \\text{Define feature descriptor}$\\;\n    \\tcc{\\fontsize{8}{8}\\selectfont retrieve feature descriptor}\n    \\tcc{\\fontsize{8}{8}\\selectfont for each image in $\\mathcal{I}$}\n    \\For{i in $\\mathcal{I}$}{\n        \\tcc{\\fontsize{7}{7}\\selectfont $\\mathcal{K}$ is set of detected keypoints for image i}\n       \\fontsize{8}{8}\\selectfont $\\mathcal{K} \\gets \\texttt{DetectKeypoints($i, Detector$)}$\\;\n        \\tcc{\\fontsize{7}{7}\\selectfont $\\mathcal{D}[i]$ is the corresponding descriptor set for image i }\n        $\\mathcal{D}[i] \\gets \\texttt{GetDescriptors( $\\mathcal{K}, Descriptor$)}$\\;\n    }\n\n    \\While{$\\text{camera is on}$}\n    { \\fontsize{8}{8}\\selectfont\n        $f \\gets \\text{RGB image frame}$\\;\n        $PC \\gets \\text{Point cloud data}$\\;\n        \\tcc{\\fontsize{7}{7}\\selectfont $K_F$ is set of detected keypoints for image frame $f$}\n        $K_F \\gets \\texttt{DetectKeypoints($f, Detector$)}$\\;\n        \\tcc{\\fontsize{7}{7}\\selectfont $D_F$ is the corresponding descriptor set for rgb image $f$}\n        $D_F \\gets \\texttt{GetDescriptors( $K_F, Descriptor$)}$\\;\n        \\For{i in $\\mathcal{I}$}\n        {\n            $matches \\gets \\texttt{FindMatches( $\\mathcal{D}[i]$, $D_F$)}$\\;\n            \\tcc{\\fontsize{7}{7}\\selectfont If there is at least 10 matches then we have the object (described in image $i$) in the scene}\n            \\uIf{\\text{Total number of }$matches \\geq 10$}\n            {\n               \n                \\tcc{\\fontsize{7}{7}\\selectfont extract matched keypoints pair $(kp_{i},kp_{f})$ from the corresponding descriptors matches.}\n                $kp_{i}, kp_{f} \\gets \\texttt{ExtractKeypoints($matches$)}$\\;\n                $\\mathbf{H} \\gets \\texttt{EstimateHomography($kp_{i}, kp_{f}$)}$\\;\n                $p_c, p_x, p_y \\gets \\text{points on the planar object }\\newline \\text{~~~~~~~~~~~~~ obtained using  equation (\\ref{eqn:axis})}$\\;\n                $p_c^{'}, p_x^{'}, p_y^{'} \\gets \\text{corresponding projected points}\\newline \\text{~~~~~~~~~~~~~ of $p_c, p_x, p_y$ on image frame $f$}\\newline \\text{~~~~~~~~~~~~~  estimated using equations}\\newline \\text{~~~~~~~~~~~~~ (\\ref{eqn:homography}) and (\\ref{eqn:projection})}$\\;\n                \\tcc{\\fontsize{7}{7}\\selectfont $\\vec{c}$ denotes the origin of the object frame with respect to the base\/world frame}\n                $\\Vec{c}, \\Vec{x}, \\Vec{y} \\gets \\text{corresponding 3d locations }\\newline \\text{~~~~~~~~~ of $p_c^{'}, p_x^{'}, p_y^{'}$ from point cloud $PC$}$\\;\n                \\tcc{\\fontsize{7}{7}\\selectfont shift $\\vec{x}, \\vec{y}$ to the  origin of the base or the world frame}\n                $\\vec{x} \\gets \\vec{x}-\\vec{c}$\\; \n                $\\vec{y} \\gets \\vec{y}-\\vec{c}$\\;\n                \\tcc{\\fontsize{7}{7}\\selectfont estimate the object frame in terms of three orthonormal vectors \n                $\\hat{i}, \\hat{j}$, and $\\hat{k}$.}\n                $\\hat{i}, \\hat{j}, \\hat{k} \\gets \\text{from equation (\\ref{eqn:unitv})}$\\;\n                \\tcc{\\fontsize{7}{7}\\selectfont compute the rotation  $\\phi_i,\\theta_i,\\psi_i$ of the object frame $\\hat{i}, \\hat{j}$, $\\hat{k}$ with respect to the base or the world frame $\\vec{X}, \\vec{Y}, \\vec{Z}$.}\n                $\\phi_i,\\theta_i,\\psi_i \\gets \\text{from equation (\\ref{eqn:eulerangles})}$\\;\n               \n               \n                \\tcc{\\fontsize{7}{7}\\selectfont finally, publish the position and orientation of the object.}\n                \\texttt{publish$(\\vec{c},\\phi_i,\\theta_i,\\psi_i)$}\\;\n            }\n        }\n    }\n  \\caption{Planar Pose Estimation}\n  \\label{algorithm}\n\\end{algorithm}\n\n\\subsubsection{Homography Estimation and Perspective Transformation}\nA homography is an invertible mapping of points and lines on the projective plane that describes a 2D planar projective transformation~(figure~\\ref{fig:homography}) that can be estimated from a given pair of images. In simple terms, a homography is a matrix that maps a set of points in one image to the corresponding set of points in another image. We can use a homography matrix $\\mathbf{H}$ to find the corresponding points using equation \\ref{eqn:homography} and~\\ref{eqn:projection}, which defines the relation of projected point $(x^{'}, y^{'})$ (figure \\ref{fig:homography}) on the rotated plane to the reference point $(x,y)$. \n\nA 2D point $(x,y)$ in an image can be represented as a 3D vector $(x, y, 1)$ which is called the homogeneous representation of a point that lies on the reference plane or image of the planar object. In equation (\\ref{eqn:homography}), $\\mathbf{H}$ represents the homography matrix and $[x~y~1]^{T}$ is the homogeneous representation of the reference point $(x,y)$ and we can use the values of $a,b,c$ to estimate the projected point $(x^{'},y^{'})$ in equation (\\ref{eqn:projection}). \n\n\\begin{align}\n \\left [ \\begin{matrix} a \\\\ b \\\\  c \\end{matrix} \\right ] = \\mathbf{H}\\begin{bmatrix} x\\\\ y\\\\ 1\\\\ \\end{bmatrix} = \\begin{bmatrix} h_{11}&h_{12}&h_{13}\\\\ h_{21}&h_{22}&h_{23}\\\\ h_{31}&h_{32}&h_{33}\\\\ \\end{bmatrix} \\begin{bmatrix} x\\\\ y\\\\ 1\\\\ \\end{bmatrix}\n\\label{eqn:homography}   \n\\end{align}\n\n\\begin{equation}\n    \\begin{aligned}\n\\left \\lbrace \\begin{aligned}\n    x^{'} = \\frac{a}{c} \\\\\n    y^{'} = \\frac{b}{c}  \n\\end{aligned} \\right . \n\\end{aligned}\n\\label{eqn:projection}\n\\end{equation}\n\nWe estimate the homography using the matches found from the nearest neighbor search as input; often these matches can have completely false correspondences, meaning they don't correspond to the same real-world feature at all which can be a problem in estimating the homography. So, we chose RANSAC~\\cite{ransac} to robustly estimate the homography by considering only inlier matches as it tries to estimate the underlying model parameters and detect outliers by generating candidate solutions through random sampling using a minimum number of observations.\n\nWhile the other techniques use as much data as possible to find the model parameters and then pruning the outliers, RANSAC uses the smallest set of data point possible to estimate the model, thus making it faster and more efficient than the conventional solutions. \n\n\\begin{figure}[h]\n\\begin{center}\n\\graphicspath{ {.\/images\/} }\n\\includegraphics[height=6cm]{images\/homography.png}\n\\end{center}\n  \\caption{Object in different orientation from the camera}\n\\label{fig:homography}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\linewidth, height=0.50\\linewidth]{images\/sys_arch_2.png}\n\\end{center}\n  \\caption{System architecture.}\n\\label{fig:sysarch}\n\n\\end{figure*}\n\n\\subsubsection{Finding directional vectors on the object}\n\nIn order to find the pose of a planar object, we need to find the three orthonormal vectors on the planar object that describe the object coordinate frame and consequently, the orientation of the object relative to the world coordinate system. We start by estimating the vectors on the planar object that form the basis of the plane, illustrated in figure~\\ref{eqn:axis}. Then, we take the cross product of these two vectors to find the third directional vector which is the normal to the object surface. Let's denote the world coordinate system as $XYZ$, and the object coordinate system as $xyz$. We define the axes of the  orientation in relation to a body as: \n\\\\\n\n\\qquad \\qquad \\qquad \\qquad $x \\to \\text{right}$\n\n\\qquad \\qquad \\qquad \\qquad $y \\to \\text{up}$ \n\n\\qquad \\qquad \\qquad \\qquad $z \\to \\text{towards the camera}$ \n\nFirst, we retrieve the locations of the three points $p_c, p_x, p_y$ on the planar object from the reference image using equation (\\ref{eqn:axis}) and then locate the corresponding points $p_{c}^{'}, p_{x}^{'}, p_{y}^{'}$ on the image acquired from the Microsoft Kinect sensor. We estimate the locations of these points using the homography matrix $\\mathbf{H}$ as shown in equation~\\ref{eqn:homography}, \\ref{eqn:projection}. Then we find the corresponding 3D locations of $p_{c}^{'}, p_{x}^{'}, p_{y}^{'}$ from the point cloud data also obtained from the Microsoft Kinect sensor. We denote them as vectors $\\vec{c}$,$\\vec{x}$, and $\\vec{y}$. Here, $\\vec{c}$ represents the translation vector from the object frame to the world frame and also the position of the object in the world frame. Next, we subtract $\\vec{c}$ from $\\vec{x}$, $\\vec{y}$ which essentially gives us two vectors $\\vec{x}$ and $\\vec{y}$ centered at the origin of the world frame. We take the cross product of these two vectors $\\vec{x}, \\vec{y}$ to find the third axis $\\vec{z}$. But, depending on the homography matrix the estimated axes $\\vec{x}$ and $\\vec{y}$ might not be exactly orthogonal, so we take the cross product of $\\vec{y}$ and $\\vec{z}$ to recalculate the vector $\\vec{x}$. Now that we have three orthogonal vectors, we compute the three unit vectors $\\hat{i}$, $\\hat{j}$, and $\\hat{k}$ along the $\\vec{x}$, $\\vec{y}$, and $\\vec{z}$ vectors respectively using equation~\\ref{eqn:unitv}. These three orthonormal vectors describe the object frame. These vectors were projected onto the image plane to give a visual confirmation of the methods applied; figure~\\ref{fig:posevizcam} shows the orthogonal axes projected onto the object plane.\n\n    \n\n\\begin{equation}\n\\vcenter{\\hbox{\\begin{minipage}{5cm}\n\\centering\n\\includegraphics[width=4cm,height=4cm]{images\/box_axis.png}\n\\captionof{figure}{Axis on the reference plane}\n\\end{minipage}}}\n\\begin{aligned}\n\\left \\lbrace \\begin{aligned}\np_c &= (w\/2, h\/2)\n\\\\\np_x &= (w, h\/2)\n\\\\\np_y &= (w\/2, 0)\n\\end{aligned} \\right . \n\\end{aligned}\n\\label{eqn:axis}\n\\end{equation}\n\n\\begin{figure}[h]       \n\\centering\n    {\\includegraphics[width=78pt, height=78pt]{images\/pose1.png}}   \n    \\hspace{1px}\n    {\\includegraphics[width=78pt, height=78pt]{images\/pose2.png}}\n    \\hspace{1px}\n    {\\includegraphics[width=78pt, height=78pt]{images\/pose3.png}}\n    \\caption{Computed third directional axis projected onto image plane}\n    \\label{fig:posevizcam}\n\\end{figure}\n\n\\begin{align}\n\\begin{split}\n\\hat{j} = \\frac{\\Vec{y}}{|\\Vec{y}|} = [j_X \\hspace{0.15cm} j_Y \\hspace{0.15cm} j_Z]\n\\\\\n\\hat{k} = \\frac{\\Vec{x} \\times \\Vec{y}}{|\\Vec{x} \\times \\Vec{y}|} = [k_X \\hspace{0.15cm} k_Y \\hspace{0.15cm} k_Z]\n\\\\\n\\hat{i} = \\frac{\\Vec{y} \\times \\Vec{z}}{|\\Vec{y} \\times \\Vec{z}|} = [i_X \\hspace{0.15cm} i_Y \\hspace{0.15cm} i_Z]\n\\end{split}\n\\label{eqn:unitv}\n\\end{align}\n\n\\subsubsection{Planar pose computation}\nWe compute the pose of the object in terms of the Euler angles. Euler angles are three angles that describe the orientation of a rigid body with respect to a fixed coordinate system. The rotation matrix $\\mathbf{R}$ in equation (\\ref{eqn:rotR}) rotates X axis to $\\hat{i}$, Y axis to $\\hat{j}$, and Z axis to $\\hat{k}$. \n\n\\begin{align}\n \\mathbf{R} = \\left [ \\begin{matrix} i_X & j_X & k_X \\\\ i_Y & j_Y & k_Y \\\\ i_Z & j_Z & k_Z \\end{matrix} \\right ]\n\\label{eqn:rotR}   \n\\end{align}\n\nEuler angles are combinations of the three axis rotations (equation~\\ref{eqn:euler-axis}), where $\\phi$, $\\theta$, and $\\psi$ specify the intrinsic rotations around the X, Y, and Z axis respectively.  The combined rotation matrix is a product of three matrices: $\\mathbf{R} = \\mathbf{R}_z \\mathbf{R}_y \\mathbf{R}_x$ (equation~\\ref{eqn:rotcomb}); the first intrinsic rotation rightmost, last leftmost.\n\n\\begin{align}\\medmath{\n    \\left\\lbrace \\begin{aligned}\n\\mathbf{R}_x &= \\colvec {1 & 0 & 0 \\\\ 0 & \\cos\\phi & -\\sin\\phi \\\\ 0 & \\sin\\phi & \\cos\\phi }  \\\\\n\\mathbf{R}_y &= \\colvec {\\cos\\theta & 0 & \\sin\\theta \\\\ 0 & 1 & 0 \\\\ -\\sin\\theta & 0 & \\cos\\theta }  \\\\\n\\mathbf{R}_z &= \\colvec {\\cos\\psi & -\\sin\\psi & 0 \\\\ \\sin\\psi & \\cos\\psi & 0 \\\\ 0 & 0 & 1 }\n    \\end{aligned} \\right .}\n    \\label{eqn:euler-axis}\n\\end{align}\n\n\\begin{align}\n\\mathbf{R} = \n    \\begin{bmatrix*}\n        c\\theta c\\psi\n        &  s\\phi s\\theta c\\psi - c\\phi s\\psi\n        &  c\\phi s\\theta c\\psi + s\\phi s\\psi\n        \\\\ c\\theta s\\psi\n        &  s\\phi s\\theta s\\psi + c\\phi c\\psi\n        &  c\\phi s\\theta s\\psi - s\\phi c\\psi\n        \\\\ -s\\theta\n        &  s\\phi c\\theta\n        &  c\\phi c\\theta\n    \\end{bmatrix*}\n\\label{eqn:rotcomb}\n\\end{align}\n\nIn equation~\\ref{eqn:rotcomb}, $c$ and $s$ represents $\\cos$ and $\\sin$ respectively.\n\nSolving for $\\phi, \\theta$, and $\\psi$ from (\\ref{eqn:rotR}) and (\\ref{eqn:rotcomb}), we get,\n\n\\begin{align}\\medmath{\n    \\left\\lbrace \\begin{aligned}\n    \\phi &= \\tan^{-1}\\left(\\frac{j_Z}{k_Z}\\right) \\\\\n    \\theta &= \\tan^{-1}\\left(\\frac{-i_Z}{\\sqrt{1-i_Z^2}}\\right) = \\sin^{-1}\\left(-i_Z\\right) \\\\\n    \\psi &= \\tan^{-1}\\left(\\frac{i_Y}{i_X}\\right)\n    \\end{aligned} \\right .}\n    \\label{eqn:eulerangles}\n\\end{align}\n\n\\begin{figure*}\n\\centering\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_robot-head_1.jpg}}}\\hspace{10px}\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_robot-head_2.jpg}} }\\hspace{10px}\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_robot-head_3.jpg}}}\n\n\\vspace{0pt}\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_rviz_1.jpg}}} \\hspace{10px}\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_rviz_2.jpg}}} \\hspace{10px}\n\\subfloat[]{\\includegraphics[width = 135pt, height=80pt]{{images\/box_rviz_3.jpg}}}\n\n\\vspace{-5pt}\n   \\caption{(a),(b),(c) are recovered poses from robot's camera and (d),(e),(f) are corresponding poses visualized in RViz}\n\\label{fig:poseviz}\n\\end{figure*}\n\n\n\n\n\\subsection{Training Grasps for Humanoid Robots}\nTo ensure that the robot can grasp objects in an adaptive manner, we pre-train the robot to perform a set of canonical grasps. We place the object and the robot's gripper close to each other and record the relative pose. This essentially gives us the pose of the gripper with respect to the object. Figure~\\ref{fig:can_grasp} illustrates the training process in which the robot's gripper and a cracker box have been placed in close proximity and the relative poses have been recorded for grasping the objects from the side. \n\n\\begin{equation}\n    \\textbf{T}_{s}^{d}\n    = \\begin{bmatrix} \\textbf{R}_{s}^{d} & P_{s}^{d} \\\\ 0 & 1 \\end{bmatrix}\n    =\\begin{bmatrix} r_{11} & r_{12} & r_{13} & X_t \\\\\n                r_{21} & r_{22} & r_{23} & Y_t \\\\\n                r_{31} & r_{32} & r_{33} & Z_t \\\\\n                0 & 0 & 0 & 1 \\end{bmatrix} \n\\label{eqn:transmat}\n\\end{equation}\n\nEquation~\\ref{eqn:transmat} outlines the structure of a transformation matrix $\\textbf{T}_{s}^{d}$ that describes the rotation and translation of frame  $d$ with respect to frame $s$; $\\textbf{R}_{s}^{d}$ represents the rotation matrix similar to equation~\\ref{eqn:rotcomb} and $P_{s}^{d}=[X_{t},Y_{t},Z_{t}]^{T}$ is the translation matrix which is the 3D location of the origin of frame $d$ in frame $s$.\n\nDuring the training phase, we first formulate the transformation matrix $\\textbf{T}_{b}^{o}$ using the rotation matrix and the object location. We take the inverse of $\\textbf{T}_{b}^{o}$ which gives us the transformation matrix $\\textbf{T}_{o}^{b}$. We then use the equation~\\ref{eqn:graspmat} to record the transformation $\\mathbf{T}_{o}^{g}$ of the robot's wrist relative to the object.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.65\\linewidth]{images\/canonical_grasp.png}\n    \\caption{Pre-training canonical grasp}\n    \\label{fig:can_grasp}\n\\end{figure}\n\n\\begin{equation} \\label{eqn:graspmat}\nT_{o}^{g} = T_{o}^{b} \\times T_{b}^{g} \\  \\text{where} \\ T_{o}^{b} = (T_{b}^{o})^{-1}\n\\end{equation}\n\\par\n\nIn the equation~\\ref{eqn:graspmat}, $b$ refers to the robot's base, $o$ refers to the object, and $g$ refers to the wrist of the robot to which the gripper is attached. Once we record the matrix, we get a new pose of the object from the vision in the testing phase and generate the final matrix using the equation~\\ref{eqn:fingrasp} that has the new position and orientation of the robot's wrist in matrix form .\n\\begin{equation} \\label{eqn:fingrasp}\nT_{b}^{g} = T_{b}^{o} \\times T_{o}^{g}\n\\end{equation}\n\\par\n\nWe then extract the rotational angles $\\gamma$, $\\beta$, $\\alpha$~(roll, pitch, yaw)  of the grasp pose from matrix $\\mathbf{T}_{b}^{g}$ using equation~\\ref{eqn:grippereulerangles}\n\n\\begin{align}\\medmath{\n    \\left\\lbrace \\begin{aligned}\n    \\gamma=tan^{-1}(r_{32}\/r_{33}) \\\\\n    \\beta=tan^{-1}\\frac{-r_{31}}{\\sqrt {{r_{32}}^2+{r_{33}}^2}}\\\\\n    \\alpha=tan^{-1}(r_{21}\/r_{11})\n    \\end{aligned} \\right .}\n    \\label{eqn:grippereulerangles}\n\\end{align}\n\n\\section{Evaluation}\nThe proposed object recognition and pose estimation algorithm was implemented on an Ubuntu 14.04 platform equipped with 3.0 GHz Intel R Core(TM) i5-7400 CPU and 8GB system memory. The RGB-D camera used in the experiments was a Microsoft Kinect sensor v1. We evaluated the proposed algorithm by comparing the accuracy of object recognition, pose estimation, and execution time of four different feature descriptors. We also validated the effectiveness of our approach for adaptive grasping by conducting experiments with the PR2 robot.\n\n\\subsection{Object detection and pose estimation}\n\nWithout enough observable features, the system would fail to find good matches that are required for accurate homography estimation. Consequently, our object detection and pose estimation approach has a constraint on the out-of-plane rotation $\\theta$, illustrated in figure~\\ref{fig:theta}. In other words, if the out-of-plane rotation of the object is more than $\\theta$, the system would not be able to recognize the object. Fast execution is also a crucial aspect to facilitate multiple object detection and pose estimation for real-time applications. We experimented with four different descriptors on several planar objects and the comparative result is shown in table~\\ref{tbl:comparisondescriptor}. The execution time was measured for the object detection and pose estimation step. AKAZE and BRISK had much lower processing time for detection and pose estimation, thus would have a better frame rate, but SIFT and SURF had larger out-of-plane rotational freedom.\n\n\\begin{figure}[h]\n\\begin{center}\n\\graphicspath{ {.\/images\/} }\n\\includegraphics[width=5cm]{images\/rot_all.png}\n\\end{center}\n  \\caption{Out of plane rotation}\n\\label{fig:theta}\n\\end{figure}\n\n\\begin{table}[h!]\n\\centering\n    \\caption{Comparison of feature descriptors}\n   \n    \\begin{tabular}{l|c|s}\n      \\toprule\n   \n      \\textbf{Descriptor} & \\begin{tabular}[x]{@{}c@{}}\\textbf{Maximum out of}\\\\ \\textbf{plane rotation} (degree)\\end{tabular} & \\begin{tabular}[x]{@{}c@{}}\\textbf{Execution time}\\\\ (second)\\end{tabular}\\\\\n   \n      \\midrule\n      \n      SIFT & $48^{\\circ}\\pm2^{\\circ}$ & \\text{~~~~~~~0.21s}\\\\ \\hline\n      SURF & $37^{\\circ}\\pm2^{\\circ}$ & \\text{~~~~~~~0.27s}\\\\ \\hline\n      AKAZE & $18^{\\circ}\\pm1^{\\circ}$ & \\text{~~~~~~~0.05s}\\\\ \\hline\n      BRISK & $22^{\\circ}\\pm2^{\\circ}$ & \\text{~~~~~~~0.06s}\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\label{tbl:comparisondescriptor}\n\\end{table}\n\nWe also compared the \\textit{RMS} difference $\\epsilon$~(equation~\\ref{eqn:epsilon}) of re-calculated $\\vec{x}$ to original $\\vec{x}$ ($\\vec{x}^{'}$ in the equation) for increasing out-of-plane rotation of the planar objects to assess the homography estimation. Ideally, the two estimated vectors $\\vec{x}$ and $\\vec{y}$, which describe the basis of the plane of the planar object, should be orthogonal to each other, but often they are not. So, the values of $\\epsilon$ in figure~\\ref{fig:epsilon} give us an indication of the average error in homography estimation for different out-of-plane rotations. In figure~\\ref{fig:epsilon}, we can see AKAZE has much higher $\\epsilon$ values while the rest remained within a close range. This tells us AKAZE results in a much larger error in estimating the homography than the other methods. \n\n\\begin{figure}[h]\n\\begin{center}\n\\graphicspath{ {.\/images\/} }\n\\includegraphics[width=\\linewidth]{images\/chart.png}\n\\end{center}\n  \\caption{Out of plane rotation vs $\\epsilon$}\n\\label{fig:epsilon}\n\\end{figure}\n\nWe chose SIFT and SURF to evaluate how the execution time for detection scales up while increasing the number of objects. From table~\\ref{tbl:multobjcompdesc}, which shows the mean processing time for object detection, we can see that SURF had a detection time around 50\\% more than SIFT in all the cases. This outcome coupled with the previous results prompted us to select SIFT for the subsequent experiments.\n\nThe system was capable of detecting multiple objects in real-time and at the same time could estimate their corresponding poses. Figure~\\ref{fig:multobjdet} shows detected objects with estimated directional planar vectors. We can also observe that the system was robust to in-plane rotation and partial occlusion.\n\n\\begin{table}[h]\n\\centering\n\\caption{\\centering Execution time of SIFT and SURF for multiple object detection}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Number of\\\\ Objects\\end{tabular}} & \\multicolumn{2}{c|}{\\begin{tabular}[c]{@{}c@{}}Detection time\\\\ (second)\\end{tabular}} \\\\ \\cline{2-3} \n                                                                             & SIFT                                       & SURF                                       \\\\ \\hline\n1                                                                            & 0.06s                                       & 0.09s                                       \\\\ \\hline\n2                                                                            & 0.11s                                       & 0.17s                                       \\\\ \\hline\n3                                                                            & 0.17s                                       & 0.26s                                       \\\\ \\hline\n4                                                                            & 0.22s                                       & 0.35s                                       \\\\ \\hline\n5                                                                            & 0.28s                                       & 0.4s5                                       \\\\ \\hline\n6                                                                            & 0.34s                                       & 0.54s                                       \\\\ \\hline\n\\end{tabular}\n\\label{tbl:multobjcompdesc}\n\\end{table}\n\n\\begin{figure}[h]       \n\\centering\n    {\\includegraphics[width=78pt, height=78pt]{images\/col1.png}}   \n    \\hspace{1px}\n    {\\includegraphics[width=78pt, height=78pt]{images\/col2.png}}\n    \\hspace{1px}\n    {\\includegraphics[width=78pt, height=78pt]{images\/col3.png}}\n    \\caption{\\centering Multiple object detection with estimated planar vectors}\n    \\label{fig:multobjdet}\n\\end{figure}\n\nWe used RViz~\\cite{RViz}, a 3D visualizer for the Robot Operating System (ROS)~\\cite{ros}, to validate the pose estimation. The calculated directional axes were projected onto the image and the estimated poses were visualized in RViz. As shown in figure~\\ref{fig:poseviz}, we qualitatively verified the accuracy of the detection and the estimated pose by comparing the two outputs. We can see that both the outputs render similar results. We conducted experiments with multiple objects and human held objects as well. Figure~\\ref{fig:pose_check} illustrates the simultaneous detection and pose estimation of two different boxes and an object held by a human, respectively.\n\n\\begin{figure}[h]       \n\\centering\n    {\\includegraphics[width=100pt, height=100pt]{images\/Pose_Detection.png}}   \n    \\hspace{1px}\n    {\\includegraphics[width=100pt, height=100pt]{images\/pose_detection_human.png}}\n   \n   \n    \\caption{(a) Pose estimation of multiple objects (b) Estimated pose of an object held by a human}\n   \n    \\label{fig:pose_check}\n\\end{figure}\n\n\n\\begin{equation}\n    \\epsilon = \\frac{1}{N}\\sum_{i=1}^{N}||\\vec{x_i}^{'}-\\vec{x_i}||, \\text{\\fontsize{8}{8}\\selectfont where N is the number of frames}\n    \\label{eqn:epsilon}\n\\end{equation}\n\n\\subsection{Adaptive grasping}\nWe assessed our approach for adaptive grasping keeping two different aspects of the robotic application in mind; robotic tasks that require 1) interacting with a static environment, and 2) interacting with humans.\n\nWe first tested our system for static objects where the object was attached to a tripod. Next, we set up experiments where the object was held by a human. We used a sticker book and a cartoon book and evaluated our system on a comprehensive set of poses. In almost all the experiments, the robot successfully grasped the object in a manner consistent with its training. There were some poses that were not reachable by the robot - for instance, when the object was pointing inward along the X-axis in the robot reference frame, it was not possible for the end-effector to make a top grasp. Figure~\\ref{fig:tripod_results} and \\ref{fig:human_results} show the successful grasping of the robot for both types of experiments.\n\n\\begin{figure}\n    \\centering\n    \\captionsetup[subfigure]{labelformat=empty}\n    \n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_3.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_3.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_3.3.jpg}}\\\\[-5ex]\n    \n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_4.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_4.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_4.3.jpg}}\\\\[-5ex]\n    \n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_5.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_5.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.3\\linewidth, height=0.3\\linewidth]{images\/exp_5.3.jpg}}\n    \n    \\caption{Robot grasping an object from a tripod. Left: initial position of the robot's gripper, middle: gripper adapting to the object's pose, right: grasping of the object.}\n    \\label{fig:tripod_results}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\captionsetup[subfigure]{labelformat=empty}\n    \n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_6.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_6.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_6.3.jpg}}\\\\[-5ex]\n    \n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_7.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_7.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_7.3.jpg}}\\\\[-5ex]\n    \n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_8.1.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_8.2.jpg}}\n    \\vspace{5px}\n    \\subfloat[]{\\includegraphics[width=0.30\\linewidth, height=0.35\\linewidth]{images\/exp_8.3.jpg}}\n    \n    \\caption{Robot grasping an object held by a human. Left: initial position of the robot's gripper, middle: gripper adapting to the object's pose, right: grasping of the object.}\n    \\label{fig:human_results}\n\\end{figure}\n\n\n\n\n\\section{Conclusion and Future Work}\n\nThis work presents an approach that enables humanoid robots to grasp objects using planar pose estimation based on RGB image and depth data. We examined the performance of four feature-detector-descriptors for object recognition and found SIFT to be the best solution. We used FLANN's K-d Tree Nearest Neighbor implementation, and Bruteforce Hamming to find the keypoint matches and employed RANSAC to estimate the homography. The homography matrix was used to approximate the three orthonormal directional vectors on the planar object using perspective transformation. The pose of the planar object was estimated from the three directional vectors. The system was able to detect multiple objects and estimate the pose of the objects in real-time. We also conducted experiments with the humanoid PR2 robot to show the practical applicability of the framework where the robot grasped objects by adapting to a range of different poses.\n\nIn the future, we plan to add GPU acceleration for the proposed algorithm that would further improve the overall computational efficiency of the system. We would like to extend the algorithm to automatically prioritize certain objects and limit the number of objects needed for detection based on different scheduled tasks. Finally, we would like to incorporate transferring grasp configuration for familiar objects and explore other feature matching technique e.g. multi probe LSH, hierarchical k-means tree, etc.\n\n \\bibliographystyle{unsrt2}\n \n \\footnotesize{\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSupport Vector Data Description (SVDD) is a machine learning technique used\nfor single class classification and outlier detection. SVDD technique is\nsimilar to Support Vector Machines and was first introduced by Tax and Duin\n\\cite{tax2004support}. It can be used to build a flexible boundary around\nsingle class data. Data boundary is characterized by observations designated\nas support vectors. SVDD is used in domains where majority of data belongs to\na single class. Several researchers have proposed use of SVDD for multivariate\nprocess control \\cite{sukchotrat2009one}. Other applications of SVDD involve\nmachine condition monitoring \\cite{widodo2007support, ypma1999robust} and image\nclassification \\cite{sanchez2007one}.\n\n\\subsection{\\bf Mathematical Formulation of SVDD}\n\\label{mfsvdd}\n\\paragraph*{\\bf Normal Data Description}\\mbox{}\\\\\nThe SVDD model for normal data description builds a minimum radius hypersphere around the data.\n\\paragraph*{\\bf Primal Form}\\mbox{}\\\\\nObjective Function:\n\\begin{equation}\n\\min R^{2} + C\\sum_{i=1}^{n}\\xi _{i}, \n\\end{equation}\nsubject to: \n\\begin{align}\n\\|x _{i}-a\\|^2 \\leq R^{2} + \\xi_{i}, \\forall i=1,\\dots,n,\\\\\n\\xi _{i}\\geq 0, \\forall i=1,...n.\n\\end{align}\nwhere:\\\\\n$x_{i} \\in {\\mathbb{R}}^{m}, i=1,\\dots,n  $ represents the training data,\\\\\n$ R:$ radius, represents the decision variable,\\\\\n$\\xi_{i}:$ is the slack for each variable,\\\\\n$a$: is the center, a decision variable, \\\\\n$C=\\frac{1}{nf}:$ is the penalty constant that controls the trade-off between the volume and the errors, and,\\\\\n$f:$ is the expected outlier fraction.\n\\paragraph*{\\bf Dual Form}\\mbox{}\\\\\nThe dual formulation is obtained using the Lagrange multipliers.\\\\ \nObjective Function:\n\\begin{equation} \n\\max\\ \\sum_{i=1}^{n}\\alpha _{i}(x_{i}.x_{i}) - \\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}(x_{i}.x_{j}) ,\n\\end{equation}\nsubject to:\n\\begin{align}\n& &  \\sum_{i=1}^{n}\\alpha _{i}  = 1,\\label{sv:s}\\\\\n& & 0 \\leq  \\alpha_{i}\\leq C,\\forall i=1,\\dots,n.\n\\end{align}\nwhere:\\\\\n$\\alpha_{i}\\in \\mathbb{R}$: are the Lagrange constants,\\\\\n$C=\\frac{1}{nf}:$ is the penalty constant.\n\\paragraph*{\\bf Duality Information}\\mbox{}\\\\\nDepending upon the position of the observation, the following results hold:\nCenter Position: \\begin{equation} \\sum_{i=1}^{n}\\alpha _{i}x_{i}=a. \\label{sv:0} \\end{equation}\nInside Position: \\begin{equation} \\left \\| x_{i}-a \\right \\| < R \\rightarrow \\alpha _{i}=0.\\end{equation}\nBoundary Position: \\begin{equation} \\left \\| x_{i}-a \\right \\| = R \\rightarrow 0< \\alpha _{i}< C.\\end{equation}\nOutside Position: \\begin{equation}\\left \\| x_{i}-a \\right \\| > R \\rightarrow \\alpha _{i}= C. \\label{sv:1} \\end{equation}\nThe radius of the hypersphere is calculated as follows:\\\\\n\\begin{equation}   \nR^{2}=(x_{k}.x_{k})-2\\sum_{i}^{ }\\alpha _{i}(x_{i}.x_{k})+\\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}(x_{i}.x_{j}).\n\\label{eq:a}\n\\end{equation}\n using any $ x_{k} \\in SV_{<C} $\n, where $SV_{<C}$  is the set of support vectors that have $ \\alpha _{k} < C $.\n\n\\paragraph*{\\bf Scoring}\\mbox{}\\\\\nFor each observation $ z $  in the scoring data set, the distance $ \\textrm{dist}^{2}(z) $ is calculated as: \n\\begin{equation}  dist^{2}(z)=(z.z) - 2\\sum_{i}^{ }\\alpha _{i}(x_{i}.z) +\\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}(x_{i}.x_{j}) \\end{equation}\nand observations with $ dist^{2}(z) > R^{2} $ are designated as outliers.\n\nThe spherical data boundary can include a significant amount of space with a very sparse distribution of training\nobservations which leads to a large number of falses positives. The use of kernel functions leads to better compact\nrepresentation of the training data.\n\\paragraph*{\\bf Flexible Data Description}\\mbox{}\\\\\nThe Support Vector Data Description is made flexible by replacing the inner product $ (x_{i}.x_{j}) $ in equation \\eqref{eq:a} with a \nsuitable kernel function $ K(x_{i},x_{j}) $. The Gaussian kernel function used in this paper is defined as:\n\\begin{equation}  \nK(x_{i}, x_{j})= \\exp  \\dfrac{ -\\|x_i - x_j\\|^2}{2s^2}\n\\label{eq:b}\n\\end{equation}\nwhere $s$: Gaussian bandwidth parameter.\n\nThe modified mathematical formulation of SVDD with kernel function is:\n\nObjective function:\n\\begin{equation}  \\label{eq:1}\n\\max\\ \\sum_{i=1}^{n}\\alpha _{i}K(x_{i},x_{i}) - \\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}K(x_{i},x_{j}),\n\\end{equation}\nSubject to:\n\\begin{align}\n& &\\sum_{i=1}^{n}\\alpha _{i} = 1, \\label{eq:2} \\\\\n& & 0 \\leq  \\alpha_{i}\\leq C = \\frac{1}{nf} , \\forall i=1,\\dots,n. \\label{eq:3}\n\\end{align}\nConditions similar to \\eqref{sv:0} to \\eqref{sv:1} continue to hold even when the kernel function is used.\\\\\nThe threshold $R^{2}$ is calculated as :\n\\begin{multline}\nR^{2} = K(x_{k},x_{k})-2\\sum_{i}^{ }\\alpha _{i}K(x_{i},x_{k})+\\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}K(x_{i},x_{j})\n\\end{multline}\nusing any $ x_{k} \\in SV_{<C} $\n, where $SV_{<C}$  is the set of support vectors that have $ \\alpha _{k} < C $.\n\n\\paragraph*{\\bf Scoring}\\mbox{}\\\\\nFor each observation $z$ in the scoring dataset, the distance $ dist^{2}(z) $ is calculated as: \n\\begin{equation} dist^{2}(z)= K(z,z) - 2\\sum_{i}^{ }\\alpha _{i}K(x_{i},z) +\\sum_{i,j}^{ }\\alpha _{i}\\alpha _{j}K(x_{i},x_{j}),\\end{equation}\nand the observations with $ dist^{2}(z) > R^{2} $ are designated as outliers.\n\n\\section{Need for a Sampling-based Approach}\nAs outlined in Section \\ref{mfsvdd}, SVDD of a data set is obtained\nby solving a quadratic programming problem. The time required to solve\nthe quadratic programming problem is directly related to the number of\nobservations in the training data set. The actual time complexity depends\nupon the implementation of the underlying Quadratic Programming solver.\nWe used LIBSVM  to evaluate SVDD training time as\na function of the training data set size.\nWe have used C++ code that uses LIBSVM ~\\cite{chang2011libsvm} implementation of SVDD\nthe examples in this paper, we have also provided a Python implmentation which uses Scikit-learn~\\cite{scikit-learn} \nat \\cite{smsvddg}.\nFigure~\\ref{fig:image_0} shows\nprocessing time as a function of training data set size for the two donut\ndata set (see Figure \\ref{fig:image_3} for a scatterplot of the two donut\ndata). In Figure~\\ref{fig:image_0} the x-axis indicates the training data set\nsize and the y-axis indicates processing time in minutes. As indicated in\nFigure~\\ref{fig:image_0}, the SVDD training time is low for small or moderately\nsized training data but gets prohibitively high for large datasets.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[scale=0.25]{image0}\n\t\\caption{SVDD Training Time: Two Donut data}\n\t\\label{fig:image_0}\n\\end{figure}\n\nThere are applications of SVDD in areas such as process control and equipment\nhealth monitoring where size of training data set can be very large, consisting\nof few million observations. The training data set consists of sensors readings\nmeasuring multiple key health or process parameters at a very high frequency.\nFor example, a typical airplane currently has $\\approx$7,000 sensors\nmeasuring critical health parameters and creates 2.5 terabytes of data per day.\nBy 2020, this number is expected to triple or quadruple to over 7.5 terabytes\n\\cite{ege2015IoT}. In such applications, multiple SVDD training models are\ndeveloped, each representing separate operating mode of the equipment or process\nsettings. The success of SVDD in these applications require algorithms which can\ntrain using huge amounts of training data in an efficient manner.\n\n\nTo improve performance of SVDD training on large data sets, we propose a new\nsampling based method. Instead of using all observations from the training data\nset, the algorithm computes the training data SVDD by iteratively computing SVDD\non independent random samples obtained from the training data set and combining\nthem. The method works well even when the random samples have few observations.\nWe also provide a criteria for detecting convergence. At convergence the our\nmethod provides a data description that compares favorably with result obtained\nby using all the training data set observations.\n\n\nThe rest of this document is organized as follows: Section~\\ref{sbm} provides\ndetails of the proposed sampling-based iterative method. Results of training\nwith the proposed method are provided in section~\\ref{res}; the analysis of high\ndimensional data is provided in section~\\ref{pd}; the results of a simulation\nstudy on random polygons is provided in section Section~\\ref{ss} and we provide\nour conclusions in section~\\ref{cn}.\n\n\\textbf{Note:\\underline{\\underline{}}} \\textit{In the remainder of this paper, we refer to the training method using all\nobservations in one iteration as the full SVDD method.\\textit{}}\n\n\\section{Sampling-based Method}\n\\label{sbm}\nThe Decomposition and Combination method of Luo et.al.\\cite{luo2010fast} and\nK-means Clustering Method of Kim et.al.\\cite{kim2007fast}, both use sampling for\nfast SVDD training, but are computationally expensive. The first method by Lou\net.al. uses an iterative approach and requires one scoring action on the entire\ntraining data set per iteration. The second method by Kim et.al. is a classic\ndivide and conquer algorithm. It uses each observation from the training data\nset to arrive at the final solution.\n\n\nIn this section we describe our sampling-based method for fast SVDD training.\nThe method iteratively samples from the training data set with the objective\nof updating a set of support vectors called as the master set of support\nvectors ($SV^{*}$). During each iteration, the method updates $SV^{*}$ and\ncorresponding threshold $R^{2}$ value and center $a$. As the threshold value\n$R^{2}$ increases, the volume enclosed by the $SV^{*}$ increases. The method\nstops iterating and provides a solution when the threshold value $R^{2}$ and the\ncenter $a$ converge. At convergence,\nthe members of the master set of support vectors $SV^{*}$, characterize\nthe description of the training data set. For all test cases, our method\nprovided a good approximation to the solution that can be obtained by using all\nobservations in the training data set.\n\nOur method addresses drawbacks of existing sampling based methods proposed\nby Luo et.al.\\cite{luo2010fast} and Kim et.al.\\cite{kim2007fast}. In each\niteration, our method learns using very a small sample from the training data\nset during each step and typically uses a very small subset of the training data set. The method\ndoes not require any scoring actions while it trains.\n\nThe sampling method works well for different sample sizes for the random draws in the iterations. \nIt also provides a better alternative to training SVDD on one large random sample from the training data\nset, since establishing a right size, especially with high dimensional data, is\na challenge.\n\nThe important steps in this algorithm are outlined below:\\mbox{}\\\\\n\\textbf{Step 1:}\nThe algorithm is initialized by selecting a random sample $S_{0}$ of size $n$\nfrom the training data set of $M$ observations ($n \\ll M$). SVDD of $S_{0}$ is\ncomputed to obtain the corresponding set of support vectors $SV_{0}$. The set\n$SV_{0}$ initializes the master set of support vectors $SV^{*}$. The iteration\nnumber $i$ is set to 1.\\\\\n\\textbf{Step 2:}\nDuring this step, the algorithm updates the master set of support vectors,\n$SV^{*}$ until the convergence criteria is satisfied. In each iteration $i$,\nfollowing steps are executed:\n\\begin{adjustwidth}{2mm}{0pt} \\textbf{Step 2.1:}  A random sample $S_{i}$ of size $n$ is selected and its SVDD is computed. The corresponding support vectors are designated as $SV_{i}$.\\\\\n\\textbf{Step 2.2}: A union of $SV_{i}$ with the current master set of support vectors,  $SV^{*}$ is taken to obtain a set $S_{i}^{'}$ ($S_{i}^{'}=SV_{i} \\bigcup SV^{*}$).\\\\\n\\textbf{Step 2.3: }SVDD of $S_{i}^{'}$ is computed to obtain corresponding support vectors $SV_{i}^{'}$, threshold value\n$R_{i}^{2}$ and ``center'' $a_{i}$ (which we define as $\\sum_{i}\\alpha_i x_i$ even when a Kernel is used).  The set $SV_{i}^{'}$, is designated as the new master set of support vectors $SV^{*}$.  \n\\end{adjustwidth}\n\\textbf{ Convergence Criteria:} \nAt the end of each iteration $i$, the following conditions are checked to determine convergence.\n\\begin{adjustwidth}{2mm}{0pt}\n\\begin{enumerate}\n\\item  $i$ = $maxiter$, where $maxiter$ is the maximum number of iteration; or\\\\\n\\item $ \\| a_{i} - a_{i-1} \\| \\le  \\epsilon_1 \\|a_{i-1}\\|$,  and\n   $\\left \\| R_{i}^{2}-R_{i-1}^{2} \\right \\| \\le \\epsilon_2 R_{i-1}^{2}$ where $\\epsilon_1,\\epsilon_2$ are appropriately\nchosen tolerance parameters.\n\\end{enumerate}\n\\end{adjustwidth}\nIf the maximum number of iterations is reached or the second condition satisfied for $t$ consecutive iterations,\nconvergence is declared. In many cases checking the convergence of just $R_i^2$ suffices.\n\nThe pseudo-code for this method is provided in algorithm~\\ref{alg:the_alg1}. The pseudo-code uses following notations:\n\\begin{enumerate}\n\\item $S_{i} \\leftarrow SAMPLE (T, n)$ denotes the data set $S_{i}$ obtained by selecting random sample of size $n$ from data set $T$.\n\\item $\\delta S_{i}$ denotes SVDD computation on data set $S_{i}$.\n\\item $\\langle SV_{i}, R_{i}^{2}, a_{i} \\rangle \\leftarrow \\delta S_{i} $ denotes the set of support vectors $SV_{i}$, threshold value $R_{i}^{2}$ and center $a_{i}$ obtained by performing SVDD computations on data set $S_{i}$.\n\\end{enumerate}\n\\begin{algorithm}\n\t\\caption{Sampling-based iterative method}\\label{euclid}\n\t\\label{alg:the_alg1}\n\t\\begin{algorithmic}[1]\n\t\t\\State  $T$ (training data set) , $n$ (sample size), convergence criteria, $s$ (Gaussian bandwidth parameter),\n$f$ (fraction of outliers) and $t$ (required number of consecutive observations satisfying convergence criteria ).\n\t\t\\State  $S_{0} \\leftarrow SAMPLE(T, n)$\n\t\t\\State  $ \\langle SV_{0}, R_{0}^{2}, a_{0} \\rangle \\leftarrow \\delta S_{0}$ \n\t\t\\State  $SV^{*} \\leftarrow SV_{0}$\n\t\t\\State  $i=1$\n\t\t\\While {(Convergence criteria not satisfied for $t$ consecutive obs)} \n\t\t\\State  $S_{i} \\leftarrow SAMPLE(T, n)$\n\t\t\\State  $\\langle SV_{i}, R_{i}^{2}, a_{i} \\rangle \\leftarrow \\delta S_{i}$\n\t\t\\State $S_{i}^{'} \\leftarrow SV_{i} \\bigcup SV^{*}$.\t\t\n\t\t\\State  $\\langle SV_{i}^{'}, R_{i}^{2'}, a_{i}^{'}\\rangle \\leftarrow \\delta S_{i}^{'}$\n\t\t\\State Test for convergence\n\t\t\\State $SV^{*} \\leftarrow SV_{i}^{'}$\n\t\t\\State $i=i+1$\n\t\t\\EndWhile\n\t\t\\RETURN { $SV^{*}$}\n\t\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\nAs outlined in steps 1 and 2, the algorithm obtains the final training data\ndescription by incrementally updating the master set of support vectors $SV^{*}$.\nDuring each iteration, the algorithm first selects a small random\nsample $S_{i}$, computes its SVDD and obtains corresponding set of support\nvectors, $SV_{i}$. The support vectors of set $SV_{i}$ are included in the\nmaster set of support vectors $SV^{*}$ to obtain $S_{i}^{'}$ ($S_{i}^{'}=SV_{i}\n\\bigcup SV^{*}$). The set $S_{i}^{'}$ thus represents an incremental expansion\nof the current master set of support vectors $SV^{*}$. Some members of $SV_{i}$\ncan be potentially ``inside'' the data boundary characterized by $SV^{*}$ the\nnext SVDD computation on $S_{i}^{'}$ eliminates such ``inside'' points.\nDuring initial iterations as $SV^{*}$ gets updated, its threshold value $R_{i}^{2'}$ typically increases and \nthe master set of support vectors expands to describe the entire data set.\n\nEach iteration of our algorithm involves two small SVDD computations and one union operation.  The first SVDD\ncomputation is fast since it is perfomed on a small sample of training data set. For the remaining two operations, our\nmethod exploits the fact that for most data sets support vectors obtained from SVDD are a tiny fraction\nof the input data set and both the union operation and the second SVDD computation are fast. So our method consists\nof three fast operations per iteration. For most large datasets we have experimented on the time to convergence is fast\nand we achieve a reasonable approximation to full SVDD in a fraction to time needed compute SVDD with the full dataset.\n\n\n\n\\subsubsection{Distributed Implementation}\nFor extremely large training datasets, efficiency gains using distributed\nimplementation are possible. Figure \\ref{fig:image_511} describes SVDD solution\nusing the sampling method outlined in section \\ref{sbm} utilizing a distributed\narchitecture. The training data set with $M$ observations is first distributed\nover $p$ worker nodes. Each worker node computes SVDD of its $\\dfrac{M}{p}$\nobservations using the sampling method to obtain its own master set of support\nvectors $SV_{i}^*$. Once SVDD computations are completed, each worker node\npromotes its own master set of support vectors $SV_{i}^*$, to the controller\nnode. The controller node takes a union of all worker node master sets of\nsupport vectors, $SV_{i}^*$ to create data set $S^{'}$. Finally, solution is\nobtained by performing SVDD computation on $S^{'}$. The corresponding set of\nsupport vectors $SV^{*}$ are used to approximate the original training data set\ndescription.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[scale=0.25]{image1}\n\t\\caption{Distributed Implementation}\n\t\\label{fig:image_511}\n\\end{figure}\n\n\\section{Results}\n\\label{res}\nTo test our method we experimented with three data sets of known geometry which we call the Banana-shaped, Star-shaped,\nand Two-Donut-shaped\ndata. The figures \\ref{fig:image_1}-\\ref{fig:image_3} illustrate these three data sets. \n\\begin{figure}\n\t\\centering    \n\t\\subfloat[Banana-shaped data]{\\label{fig:image_1}\\includegraphics[scale=0.11]{image2}}\n\t\\subfloat[Star-shaped data]{\\label{fig:image_2}\\includegraphics[scale=0.11]{image3}}\n\t\\subfloat[Two-donut-shaped data]{\\label{fig:image_3}\\includegraphics[scale=0.11]{image4}}\n\t\\caption{Scatter plots}\\label{fig:image_4}\n\\end{figure}\nFor each data set, we first obtained SVDD using all observations.\nTable~\\ref{table:t2} summarizes the results.\\\\\nFor each data set, we varied the value of the sample size $n$ from 3 to 20\nand obtained multiple SVDD using the sampling method. For each sample size\nvalue, the total processing time and number of iterations till convergence was\nnoted. Figures \\ref{fig:image_5} to \\ref{fig:image_7} illustrate the results.\nThe vertical reference line indicates the sample size corresponding to the\nminimum processing time. Table~\\ref{table:t3} provides the minimum processing\ntime, corresponding sample size and other details for all three data sets.\nFigure \\ref{fig:image_106} shows the convergence of threshold $R^2$ for the\nBanana-shaped data trained using sampling method.\n\n\\begin{figure}\n\\centering    \n\\subfloat[Run time vs. sample size]{\\label{fig:image_51}\\includegraphics[width=3.00in]{image5_1}}\\mbox{}\\\\\n\\subfloat[\\# iterations vs. sample size]{\\label{fig:image_52}\\includegraphics[width=3.00in]{image5_2}}\n\\caption{Banana-shaped data}\\label{fig:image_5}\n\\end{figure}\n\n\\begin{figure}\n\\centering    \n\\subfloat[Run time vs. sample size]{\\label{fig:image_61}\\includegraphics[width=3.00in]{image6_1}}\\mbox{}\\\\\n\\subfloat[\\# iterations vs. sample size]{\\label{fig:image_62}\\includegraphics[width=3.00in]{image6_2}}\n\\caption{Star-shaped data}\\label{fig:image_6}\n\\end{figure}\n\n\\begin{figure}\n\\centering    \n\\subfloat[Run time vs. sample size]{\\label{fig:image_71}\\includegraphics[width=3.00in]{image7_1}}\\\\\n\\subfloat[\\# iterations vs. sample size]{\\label{fig:image_72}\\includegraphics[width=3.00in]{image7_2}}\n\\caption{Two Donut data}\\label{fig:image_7}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.00in]{image106}\n\\caption{Plot of threshold $R^2$ - Banana shaped data (Sample size = 6) }\n\\label{fig:image_106}\n\\end{figure}\n\n\\begin{table}[h!]\n\\begin{minipage}{.5\\textwidth}\n    \\begin{tabular}{||c c c c c||} \n    \\hline\n        Data & \\#Obs & $R^{2}$ & \\#SV & Time \\\\ [0.5ex] \n        \\hline\\hline\n        Banana & 11,016  & 0.8789 & 21 & 1.98 sec   \\\\ \n        TwoDonut &1,333,334  & 0.8982 &178  & 32 min \\\\ \n        Star & 64,000 & 0.9362  &76  &11.55 sec   \\\\ [1ex] \n    \\hline\n    \\end{tabular}\n    \\caption{SVDD Training using full SVDD method}\\label{table:t2}\n\\end{minipage}\\\\\n\\mbox{}\\\\\n\\begin{minipage}{.4\\textwidth}\n\t\\begin{tabular}{||ccccc||} \n\t\t\\hline\n\t\tData&Iterations & $R^{2}$ & \\#SV & Time\\\\ [0.5ex] \n\t\t\\hline\\hline\n\t\tBanana(6)&119&0.872&19&0.32 sec\\\\ \n\t\tTwoDonut(11)&157&0.897&37&0.29 sec\\\\\n\t\tStar(11)&141&0.932&44&0.28 sec\\\\[1ex] \n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{SVDD Results using Sampling Method (sample size in parenthesis)}\\label{table:t3} \n\\end{minipage}\n\\end{table}\nResults provided in Table \\ref{table:t2} and Table \\ref{table:t3} indicate\nthat our method provides an order of magnitude performance improvement as\ncompared to training using all observations in a single iteration. The threshold\n$R^{2}$ values obtained using the sampling-based method are approximately equal\nto the values that can be obtained by training using all observations in a\nsingle iteration. Although the radius values are same, to confirm if the data\nboundary defined using support vectors is similar, we performed scoring on a\n$200\\times200$ data grid. Figure \\ref{fig:image_8} provides the scoring results\nfor all data sets. The scoring results for the Banana-shaped and the Two-Donut-shaped are very similar\nfor both the method, the scoring results for the Star-shaped shaped data for the two methods are also similar\nexcept for a region near the center.\n\n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\includegraphics[width=1.5in]{image100_1}  &   \\includegraphics[width=1.5in]{image100_2} \\\\\n(a)  & (b)  \\\\[6pt]\n\\includegraphics[width=1.5in]{image101_1} &   \\includegraphics[width=1.5in]{image101_2} \\\\\n(a)  & (b)  \\\\[6pt]\n\\includegraphics[width=1.5in]{image102_1} &   \\includegraphics[width=1.5in]{image102_2} \\\\\n(a)  & (b)  \\\\[6pt]\nFull SVDD Method  & Sampling method \\\\[6pt]\n\\end{tabular}\n\\caption{Scoring results. Above figures show results of scoring on a 200x200 data grid. Light gray color indicates outside points and black color indicates inside points. Figure (a) used full SVDD method for training. Figure (b) used sampling method for training. }\\label{fig:image_8}\n\\end{figure}\n\n\n\\section{Analysis of High Dimensional Data}\n\\label{pd}\nSection \\ref{res} provided comparison of our sampling method with full SVDD\nmethod. For two-dimensional data sets the performance of sampling method can\nbe visually judged using the scoring results. We tested the sampling method\nwith high dimensional datasets, where such visual feedback about classification\naccuracy of sampling method is not available. We compared classification\naccuracy of the sampling method with the accuracy of training with full SVDD\nmethod. We use the $F_{1}$-measure to quantify the classification accuracy\n\\cite{zhuang2006parameter}.\nThe $F_{1}$-measure is defined as follows:\n\\begin{equation}  \nF_{1}=\\dfrac{2\\times \\text{Precision}\\times \\text{Recall}}{\\text {Precision}+\\text {Recall}},\n\\end{equation}\nwhere:\n\\begin{align}\n\\text {Precision}=\\dfrac{\\text{true positives}}{\\text{true positives} + \\text{false positives}}\\\\\n\\text {Recall}=\\dfrac{\\text{true positives}}{\\text{true positives} + \\text{false negatives}}.\n\\end{align} \nThus high precision relates to a low false positive rate, and high recall\nrelates to a low false negative rate. We chose the $F_{1}$-measure because it is\na composite measure that takes into account both the Precision and the Recall.\nModels with higher values of $F_{1}$-measure provide a better fit. \\\\\n\n\\subsection{Analysis of Shuttle Data}\nIn this section we provide results of our experiments with Statlog (shuttle)\ndataset \\cite{Lichman:2013}. This is a high dimensional data consists of nine\nnumeric attributes and one class attribute. Out of 58,000 total observations,\n80\\% of the observations belong to class one.\nWe created a training data set of randomly selected 2,000 observations belonging\nto class one. The remaining 56,000 observations were used to create a scoring\ndata set. SVDD model was first trained using all observations in the training\ndata set. The training results were used to score the observations in the\nscoring data set to determine if the model could accurately classify an\nobservation as belonging to class one and the accuracy of scoring was measured\nusing the $F_{1}$-measure. We then trained using the sampling-based method,\nfollowed by scoring to compute the $F_{1}$-measure again. The sample size for\nthe sampling-based method was set to 10 (number of variables + 1). We measured\nthe performance of the sampling method using the $F_{1}$-measure ratio defined\nas $F_{\\text{Sampling}}\/F_{\\text{Allobs}}$ where\n$F_{\\text{Sampling}}$ is the $F_1$-measure obtained when the value obtained using the sampling method for training, and\n$F_{\\text{Allobs}}$ is the value of $F_1$-measure computed when all observations were used for training. A value close to 1 indicate that sampling method is competitive with full SVDD method.\nWe repeated the above steps varying the training data set of size from 3,000\nto 40,000 in the increments of 1,000. The corresponding scoring data set size\nchanged from 55,000 to 18,000. Figure \\ref{fig:image_9_1_1} provides the plot of\n$F_{1}$-measure ratio. The plot of $F_1$-measure ratio is constant, very close\nto 1 for all training data set sizes, provides the evidence that our sampling\nmethod provides near identical classification accuracy as compared to full SVDD\nmethod. Figure \\ref{fig:image_9_1_2} provides the plot of the processing time\nfor the sampling method and training using all obsrvations. As the training\ndata set size increased, the processing time for full SVDD method increased\nalmost linearly to a value of about 5 seconds for training data set of 40,000\nobservations. In comparison, the processing time of the sampling based method\nwas in the range of 0.24 to 0.35 sec. The results prove that the sampling-based\nmethod is efficient and it provides near identical results to full SVDD method.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=3.45in]{image9_1_b}\n\t\\caption{$F_1$-measure plot: Shuttle data. \\textit{Sample size for sampling method=10}}\n\t\\label{fig:image_9_1_1}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.45in]{image9_2_b}\n\\caption{Processing time plot: Shuttle data. \\textit{Sample size for sampling method=10}}\n\\label{fig:image_9_1_2}\n\\end{figure}\n\n\n\\subsection{Analysis of Tennessee Eastman Data}\nIn this section we provide results of our experiments with high dimensional\nTennessee Eastman data. The data was generated using the MATLAB simulation code\n\\cite{Ricker:2002} which provides a model of an industrial chemical process\n\\cite{downs1993plant}. The data was generated for normal operations of the\nprocess and twenty faulty processes. Each observation consists of 41 variables,\nout of which 22 are measured continuously, on an average, every 6 seconds\nand remaining 19 sampled at a specified interval either every 0.1 or 0.25\nhours. We interpolated the 22 observations which are measured continuously\nusing SAS\\textsuperscript{\\textregistered} \\textit{EXPAND} procedure. The\ninterpolation increased the observation frequency and generated 20 observations\nper second. The interpolation ensured that we have adequate data volume to\ncompare performance our sampling method with full SVDD method.\n\n\nWe created a training data set of 5,000 randomly selected observations belonging\nto the normal operations of the process. From the remaining observations, we\ncreated a scoring data of 228,000 observations by randomly selecting 108,000\nobservations belonging to the normal operations and 120,000 observations\nbelonging to the faulty processes. A SVDD model was first trained using all\nobservations in the training data set. The training results were used to score\nthe observations in the scoring data set to determine if the model could\naccurately classify an observation as belonging to the normal operations. The\naccuracy of scoring was measured using the $F_{1}$-measure. We then trained\nusing the sampling method, followed by scoring to compute the $F_{1}$-measure\nagain. The sample size for the sampling based method was set to 42 (number\nof variables + 1). Similar to the Shuttle data analysis, we measured the\nperformance of the sampling method using the $F_{1}$-measure ratio defined as\n$F_{\\text{Sampling}}\/F_{\\text{Allobs}}$ where $F_{\\text{Sampling}}$ is the\n$F_1$-measure obtained when the value obtained using the sampling method for\ntraining, and $F_{\\text{Allobs}}$ is the value of $F_1$-measure computed when\nall observations were used for training. A value close to 1 indicate that\nsampling method is competitive with full SVDD method.\n\nWe repeated the above steps varying the training data set of size from 10,000\nto 100,000 in the increments of 5,000. The scoring data set was kept unchanged\nduring each iteration. Figure \\ref{fig:image_9_1} provides the plot of\n$F_{1}$-measure ratio. The plot of $F_{1}$-measure ratio was constant, very\nclose to 1 for all training data set sizes, provides the evidence that the\nsampling method provides near identical classification accuracy as compared\nto full SVDD method. Figure \\ref{fig:image_9_2} provides the plot of the\nprocessing time for the sampling-based method and the all obsrvation method. As\nthe training data set size increased, the processing time for full SVDD method\nincreased almost linearly to a value of about one minute for training data set\nof 100,000 observations. In comparison, the processing time of the sampling\nbased method was in the range of 0.5 to 2.0 sec. The results prove that the\nsampling-based method is efficient and it provides and closely approximates\nthe results obtained from full SVDD method.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.45in]{image12_1}\n\t\\caption{$F_1$-measure ratio plot: Tennessee Eastman data. Sample size for sampling method=42}\n\t\\label{fig:image_9_1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.45in]{image12_2}\n\\caption{Processing time plot: Tennessee Eastman data. Sample size for sampling method=42}\n\\label{fig:image_9_2}\n\\end{figure}\n\n\\section{Simulation Study}\n\\label{ss}\nIn this section we measure the accuracy of Sampling method when it is\napplied to randomly generated polygons. Given the number of vertices, $k$,we\ngenerate the vertices of a randomly generated polygon in the anticlockwise\nsense as $r_1\\exp i \\theta_{(1)}, \\dots, r_k \\exp i \\theta_{(k)}.$ Here\n$\\theta_{(i)}$'s are the order statistics of an i.i.d sample uniformly\ndrawn from $(0,2\\pi)$ and $r_i$'s are uniformly drawn from an interval\n$[\\text{r}_{\\text{min}},\\text{r}_{\\text{max}}].$ For this simulation we chose\n$\\text{r}_{\\text{min}}=3$ and $\\text{r}_{\\text{max}}=5$ and varied the number\nof vertices from $5$ to $30$. We generated $20$ random polygons for each vertex\nsize. Figure \\ref{fig:image_10} shows two random polygons. Having determined a\npolygon we randomly selected $600$ points uniformly from the interior of the\npolygon to construct a training data set.\n\nTo create the scoring data set we the divided the bounding rectangle of each\npolygon into a $200 \\times 200$ grid. We labeled each point on this grid\nas an ``inside'' or an ``outside'' point. We then fit SVDD on the training\ndata set and scored the corresponding scoring data set and calculated the\n$F_1$-measure. The process of training and scoring was first performed using the\nfull SVDD method, followed by the sampling method. For sampling method we\nused sample size of 5. We trained and scored each instance of a polygon 10 times\nby changing the value of the Gaussian bandwidth parameter, $s$. We used $s$\nvalues from the following set:\\linebreak $s=[1, 1.44, 1.88, 2.33, 2.77, 3.22, 3.66,\n4.11, 4.55, 5].$\n\nAs in previous examples we used the $F_!$ measure ratio to judge the accuracy of the sampling method.\n\n\\begin{figure}\n\t\\centering    \n\t\\subfloat[Number of Vertices = 5]{\\label{fig:image_10}\\includegraphics[width=3.45in]{p5}}\\\\\n\t\\subfloat[Number of Vertices = 25]{\\label{fig:image_11}\\includegraphics[width=3.45in]{p25}}\n\t\\caption{Random Polygons}\\label{fig:image_10}\n\\end{figure}\n\nThe Box-whisker plots in figures \\ref{fig:image_11_a} to \\ref{fig:image_11_c}\nsummarize the simulation study results. The x- axis shows the number of\nvertices of the ploygon and y-axis shows the $F_1$-measure ratio. The bottom and\nthe top of the box shows the first and the third quartile values. The ends of\nthe whiskers represent the minimum and the maximum value of the $F_1$-measure\nratio. The diamond shape indicates the mean value and the horizontal line in the\nbox indicates the second quartile.\n\\subsection{Comparison of the best fit across $s$}\nFor each instance of a polygon we looked at $s$ value which provides the best\nfit in terms of the $F_1$-ratio for each of the methods. The plot in Figure\n\\ref{fig:image_11_a} shows the plot of $F_{1}$ measure ratio computed using the\nmaximum values of $F_{1}$ measures. The plot shows that $F_1$-measure ratio\nis greater than $\\approx 0.92$ across all values of number of vertices. The\n$F_1$ measure ratio in the top three quartiles is greater than $\\approx$ 0.97\nacross all values of the number of vertices. Using best possible value of s, the\nsampling method provides comparable results with full SVDD method.\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.45in]{image104}\n\\caption{Box-whisker plot: Number of vertices vs. Ratio of max $F_1$ measures \\label{fig:image_11_a}}\n\\end{figure}\n\n\\subsection{Results Using Same Value of $s$}\nWe evaluated sampling method against full SVDD method, for the same value\nof $s$. The plots in Figure \\ref{fig:image_11_b} illustrate the results for\ndifferent six different values of $s$. The plot shows that except for one outlier result in Figure \\ref{fig:image_11_b} (d), $F_1$-measure\nratio is greater than 0.9 across number of vertices and $s$. In Figures\n\\ref{fig:image_11_b} (c) to (f), the top three quartiles of $F_{1}$ measure\nratio was consistently greater than $\\approx 0.95$. Training using sampling method\nand full SVDD method, using same $s$ value, provide similar results.\\\\\n\n\\begin{figure}\n    \\centering\n\t\\subfloat[$s$=1]{\\includegraphics[width=3.00in]{image105_1}}\\\\\n\t\\subfloat[$s$=1.4]{\\includegraphics[width=3.00in]{image105_2}}\\\\\n\t\\subfloat[$s$=2.3]{\\includegraphics[width=3.00in]{image105_3}}\\\\\n\t\\subfloat[$s$=3.4]{\\includegraphics[width=3.00in]{image105_4}}\n    \\phantomcaption\n\\end{figure}\n\\begin{figure}\n    \\ContinuedFloat\n    \\centering\n\t\\subfloat[$s$=4.1]{\\includegraphics[width=3.00in]{image105_5}}\\\\\n\t\\subfloat[$s$=5.0]{\\includegraphics[width=3.00in]{image105_6}}\n    \\caption[]{Box-whisker plot: Number of vertices vs. $F_1$ measure ratio for different s values \\label{fig:image_11_b}}\n\\end{figure}\n\n\\subsection{Overall Results}\nFigure \\ref{fig:image_11_c} provides summary of all simulation performed for\ndifferent polygon instances and varying values of $s$. The plot shows that\nexcept for one outlier result, $F_1$-measure ratio is greater than 0.9 across number of vertice. The $F_1$\nmeasure ratio in the top three quartiles is greater than $\\approx 0.98$ across\nall values of the number of vertices. The accuracy of sampling method is\ncomaprable to full SVDD method.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.45in]{image103}\n\\caption{Box-whisker plot: Number of vertices vs. $F_1$ measure ratio  }\n\\label{fig:image_11_c}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{cn}\nWe propose a simple sampling-based iterative method for training SVDD. The\nmethod incrementally learns during each iteration by utilizing information\ncontained in the current master set of support vectors and new information\nprovided by the random sample. After a certain number of iterations, the\nthreshold $R^2$ value and the center $a$ start to converge. At this point, the\nSVDD of the master set of support vectors is close to the SVDD of training data\nset. We provide a mechanism to detect convergence and establish a stopping\ncriteria. The simplicity of proposed method ensures ease of implementation.\nThe implementation involves writing additional code for calling SVDD training\ncode iteratively, maintaining a master set of support vectors and implementing\nconvergence criteria based on threshold $R^2$ and center $a$. We do not\npropose any changes to the core SVDD training algorithm as outlined in section\n\\ref{mfsvdd}. The method is fast. The number of observations used for\nfinding the SVDD in each iteration can be a very small fraction of the number\nof observations in the training data set. The algorithm provides good results\nin many cases with sample size as small as $m+1$, where $m$ is the number of variables in\nthe training data set. The small sample size ensures that each iteration of the\nalgorithm is extremely fast. The proposed method provides a fast alternative\nto traditional SVDD training method which uses information from all observations\nin one iteration. Even though the sampling based method provides an approximation \nof the data description but in applications where training data set is large, fast\napproximation is often preferred to an exact description which takes more time to determine.\nWithin the broader realm of Internet of Things (IoT) we\nexpect to see multiple applications of SVDD especially to monitor industrial\nprocesses and equipment health and many of these applications will require fast periodic training\nusing large data sets. This can be done very efficiently with our\nmethod.\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe two gas giants of the \\replaced{solar}{Solar} system, Jupiter and Saturn, are host to a large number of satellites and rings. The satellites of both planets follow a similar progression pattern. The inner region of each system consists of small icy satellites, with an accompanying ring system \\citep{Thomas1998InnerSatJup, Throop2004JupiterRings, Porco2005CassiniRingSat,Thomas2013InnerSatSatu}. Further out, there are larger icy\/silicate satellites \\citep{Thomas2010SaturnSatsCassiniProps, Deienno2014OrbitGalileanSat}. In the outer system, both planets have a series of irregular satellites, small satellites with high eccentricities and inclinations \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature, Jewitt2007IrregularSats}. It is thought that these satellites were captured from other populations of small \\replaced{solar}{Solar} system bodies \\citep{Colombo1971JupSatsForm, Heppenheimer1977Capture, Pollack1979SatGasDrag, Sheppard2003IrrSatNature, Nesvorny2004IrrSatFamilyOrigin, Johnson2005PhoebeKuiper, Nesvorny2007IrrSatCap, Nesvorny2014IrrCapture}. This is in contrast to the inner satellites, which are thought to have accreted in a circumplanetary disk \\citep[e.g.][]{Canup2002GalSatAcc, Canup2010SaturnSatOrigin}. Such a formation mechanism is thought to resemble the accretion of planets in a protoplanetary disk around a young star \\citep{Lissauer1987PlanetAccretion}, a conclusion that is supported by the recent discovery of the TRAPPIST-1 planetary system \\citep{Gillon2016Trapist1}. That system features at least seven Earth-mass planets orbiting a very low mass star. The star itself, TRAPPIST-1, is within two orders of magnitude more massive than Jupiter, and similar in size. The seven planets span an area comparable to that of Jupiter's regular satellite system. Studying and understanding the gas giant systems in our own \\replaced{solar}{Solar} system, can therefore provide context for future exploration of low-mass exoplanetary systems. \n\n\\subsection{The Jovian System}\nHistorically, \\citet{Galileo1610SidereusNuncius} discovered the first satellites in the Jovian system, the large Galileans, Io, Europa, Ganymede and Callisto. Our knowledge of these satellites has increased greatly, as a result of both improved ground-based instrumentation \\citep[e.g.][]{Sparks2016EuropaHST,Vasundhara2017GalileansGround} and spacecraft visitations \\citep[e.g.][]{Smith1979Jupiter, Grundy2007NewHorizonsJupiterSats, Greenberg2010IcyJovian}\\deleted{have given us a more detailed understanding of these objects}. \n\nAmalthea, one of the inner set of Jovian satellites, was discovered by \\citet{Barnard1892Amalthea}. A few years later, the first two small irregular satellites, Himalia \\citep{Perrine1905Himalia} and Elara \\citep{Perrine1905Elara}, were discovered in inclined, prograde orbits. The discovery of Pasiphae three years later by \\citet{Melotte1908Pasiphae} is significant as this was only the second satellite in the Solar system to be found on a retrograde orbit, and the first such object found in the Jovian system. Several other irregular satellites were discovered in the first half of the 20th century, Sinope \\citep{Nicholson1914Sinope}, Lysithea \\citep{Nicholson1938LysitheaCarme}, Carme \\citep{Nicholson1938LysitheaCarme} and Ananke \\citep{Nicholson1951Ananke}. Leda, another small prograde irregular, was discovered 20 years later by \\citet{Kowal1975Leda}. Themisto, the first Jovian satellite smaller than 10km to be discovered, was found that same year \\citep{Kowal1975Themisto} and subsequently lost. Themisto was rediscovered by \\citet{Sheppard2000ThemistoReDis} nearly 20 years later. The Voyager visitations of Jupiter discovered the remaining three inner satellites, Metis \\citep{Synnott1981MetisDiscov}, Adrastea \\citep{Jewitt1979AdrasteaDiscov} and Thebe \\citep{Synnott1980ThebeDiscov}, along with a ring system \\citep{Smith1979Jupiter}. These three satellites, Amalthea and the ring system, would be imaged again by the Galileo \\citep{OckertBell1999JupiterRing} and Cassini \\citep{Porco2005CassiniRingSat} spacecraft during their missions. \n\nThe irregular Jovian satellites orbit the planet with semi-major axes an order of magnitude greater than the Galilean moons, and have large eccentricities and inclinations. In the early years of the 21st century, extensive surveys were carried out to search for the Jovian irregular satellites \\citep{Scotti2000Callirrhoe, Sheppard2001SatsJupDiscovery,\nSheppard2002JupSatDisc, Gladman2003JupIAU1, Gladman2003JupIAU2,\nSheppard2003JupIAU1, Sheppard2003JupIAU2, Sheppard2003JupIAU3,\nSheppard2003IrrSatNature,\nSheppard2004JupIAU1,Sheppard2004JupIAU2, Beauge2007IrregSatRsonance,\nJacobson2011NewJupSats, Sheppard2012JupIAU}. These surveys increased the number of known Jovian satellites from 14 after Voyager, to the 67 known today.  The inner five irregular satellites, Leda, Himalia, Lystea, Elara and Dia, have prograde orbits and have previously been classified into the Himalia group \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature}. Themisto and Carpo were proposed as single members of their own  groups by \\citet{Sheppard2003IrrSatNature}. The remainder of the irregular satellites have retrograde orbits. Based on similarities in semi-major axis, inclination and eccentricity, these satellites have been grouped into families by \\citet{Sheppard2003IrrSatNature} and \\citet{Nesvorny2003IrrSatEvol}. These dynamical families are typified by their largest member, Himalia representing the inner prograde satellites, with the retrograde ones being broken down into the Ananke, Pasiphae and Carme families. Recently, several additional small irregular satellites have been discovered \\citep{Jacobson2011NewJupSats, Sheppard2012JupIAU} which are yet to be named or classified. With the discovery of new satellites \\citep{Scotti2000Callirrhoe, Sheppard2001SatsJupDiscovery, Beauge2007IrregSatRsonance,\nJacobson2011NewJupSats, Sheppard2012JupIAU} and additional information from the Cassini spacecraft \\citep{Porco2005CassiniRingSat}, a revisitation of the classification of the Jovian irregular satellites \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature, Jewitt2007IrregularSats} is warranted. \n\n\\subsection{The Saturnian System}\nThe Saturnian system is broadly similar to that of Jupiter, but exhibits greater complexity. One of the most striking features, visible to even the most modest telescope, is Saturn's ring system. First observed by Galileo in 1610, it was \\citet{Huygens1659systema} that observed that the objects surrounding Saturn were in fact rings. The rings themselves are composed of individual particles, from micrometer to meter size \\citep{Zebker1985SaturnRingParticle}.  Embedded within several of the main rings are a series of small moonlets \\citep{Tiscareno2006SaturnAringMoonlets} and several shepherd satellites \\citep{Showalter1991Pan, Porco2007SaturnSmallSats, Cuzzi2014FringPromethius}. The co-orbitals Janus and Epimetheus \\citep{Yoder1983SaturnCoorobiting, Yoder1989JanusEpiMassOrbit, Nicholson1992CoorbitalSaturn, Treffenstdt2015JanusEpiFormation, ElMoutamid2016JansSwapAring}, and their associated faint ring system \\citep{Winter2016JanusEpiRing} are unique to the Saturn system. Just beyond the Janus\/Epimetheus orbit, there is a diffuse G-ring, the source of which is the satellite Aegaeon \\citep{Hedman2007Gring}.\n\n\\citet{Huygens1659systema} also discovered Saturn's largest satellite, Titan. Earth-based observations highlighted the methane based atmosphere of Titan \\citep{Kuiper1944TitanAtmos, Karkoschka1994TitanESO}, with further characterization by the Cassini spacecraft \\citep{Niemann2005TitanAtmos} and Huygens lander \\citep{Lebreton2005HuygensTiten}. The bulk composition of Titan is analogous to that of the other icy satellites, with an icy shell, subsurface water ocean and silicate core \\citep{Hemingway2013TitanIceshell}. There are seven other mid-sized icy satellites, with semi-major axes on a similar order of magnitude to \\added{that of} Titan. The five largest, Mimas, Enceladus, Tethys, Dione and Rhea are large enough to be in hydrostatic equilibrium. All of the mid-sized satellites are thought to be predominantly composed of water ice, with some contribution from silicate rock, and may contain subsurface liquid oceans \\citep{Matson2009SaturnSat, Filacchione2012VIMS3}. Those satellites closer to Saturn than Titan, Mimas, Enceladus, Tethys, Dione and Rhea, are embedded in the E-ring \\citep{Feibelman1967SatEring, Baum1981SatEring, Hillier2007EringComp, Hedman2012EringStruc}. The Cassini mission identified the source of this ring as the southern cryo-plumes of Enceladus \\citep{Sphan2006EnceladusEring}. \n\nIn addition to the larger icy satellites, there are four small Trojan satellites \\citep{Porco2005CassiniRingSat}, situated at the leading and trailing Lagrange points, 60\\degree \\ ahead or behind the parent satellites in their orbit. Tethys has Telesto and Calypso as Trojan satellites, while Helene and Polydeuces are Trojan satellites of Dione. So far, these Trojan satellites are unique to the Saturnian system. Between the orbits of Mimas and Enceladus, there are the Alkyonides, Methone, Anthe and Pallene, recently discovered by the Cassini spacecraft \\citep{Porco2005CassiniRingSat}. Each of the Alkyonides have their own faint ring arcs \\citep{Hedman2009SatRingArcs} comprised of similar material to the satellite. Dynamical modeling by \\citet{Sun2017MethoneDust} supports the theory of \\citet{Hedman2009SatRingArcs}, that the parent satellite is the source of the rings.\n\nIn the outer Saturnian system there are a large number of smaller irregular satellites, with 38 known to date. The first of these irregular satellites to be discovered was Phoebe, which was the first planetary satellite to be discovered photographically \\citep{Pickering1899Phoebe}. Phoebe was also the first satellite to be discovered moving on a retrograde orbit \\citep{Pickering1905Phoebe, Ross1905Phoebe}. Phoebe is the best studied irregular satellite and the only one for which in-situ observations have been obtained \\citep{Clark2005Phoebe}. Recently, a large outer ring associated with Phoebe and the other irregular satellites has been discovered \\citep{Verbiscer2009SatLarRing}. It has been suggested that Phoebe may have originated in the Edgeworth-Kuiper Belt and captured into orbit around Saturn \\citep{Johnson2005PhoebeKuiper}. The other Saturnian irregular satellites were discovered in extensive surveys during the early 21st century \\citep{Gladman200112Sat, Sheppard2003IAUJupSat, Jewitt2005IAUCSat,\nSheppard2006SatIAUC, Sheppard2007SatIAUC}. \\replaced{With}{Due to} the small size of the majority of these satellites, only their orbital information is available. There are nine prograde and 29 retrograde outer satellites, of which attempts have been made to place into families based on dynamical \\citep{Gladman200112Sat, Jewitt2007IrregularSats, Turrini2008IrregularSatsSaturn} and photometric \\citep{Grav2003IrregSatPhoto, Grav2007IrregSatCol} information. In the traditional naming convention \\citep{Grav2003IrregSatPhoto}, the Inuit family, Ijiraq, Kiviuq, Paaliaq, Siarnaq and Tarqeq, are small prograde satellites, whose inclination is between 45\\degree and 50\\degree . The Gallic family, Albiorix, Bebhionn, Erriapus and Tarvos, is a similar, prograde group, but with inclinations between 35\\degree and 40\\degree . The retrograde satellites are all grouped into the Norse family, including Phoebe. There is a possibility that the Norse family could be further split into subfamilies, based on photometric studies \\citep{Grav2003IrregSatPhoto, Grav2007IrregSatCol}. The convention of using names from respective mythologies for the satellite clusters \\citep{Jewitt2007IrregularSats}, has become the default standard for the irregular satellite families of Saturn. \n\n\\subsection{Formation Theories}\n\nThe purpose of taxonomy and classification, beyond simple grouping, is to investigate the origin of objects. The origin of the irregular satellites is a major topic of ongoing study \\citep{Nesvorny2012JumpingJupiter, Nesvorny2014IrrCapture}. Here we present an overview for context. There are three main theories in the formation of the Jovian satellites: formation via disk accretion \\citep{Canup2002GalSatAcc}; via nebula drag \\citep{Pollack1979SatGasDrag}; or via dynamic capture \\citep{Nesvorny2003IrrSatEvol, Nesvorny2007IrrSatCap}. The satellites that are captured either by nebula drag or through dynamical means, are thought to be from \\replaced{solar}{Solar} system debris, such as asteroids and comets. \n\nThe disk accretion theory has generally been accepted as the mechanism for the formation of the inner prograde satellites of Jupiter \\citep{Canup2002GalSatAcc}. The satellites form from dust surrounding proto-Jupiter in a process analogous to the formation of planets around a star \\citep{Lissauer1987PlanetAccretion}. This surrounding disk would have lain in the equatorial plane of Jupiter, with material being accreted to the planet itself through the disk. This would explain both the prograde, coplanar orbits of the regular satellites and their near circular orbits.\n\nThe second theory requires satellites to be captured in the original Jovian nebula \\citep{Pollack1979SatGasDrag, Cuk2004HimaliaGasDrag}. Before it coalesced into a planet, Jupiter is proposed to have had a greater radius, and lower density than now. There was a `nebula' surrounding this proto-Jupiter. As other pieces of \\replaced{solar}{Solar} system debris crossed into the Hill sphere of this nebula, they would be slowed down by friction and be captured as a satellite. Related to this is the concept of a pull down mechanism \\citep{Heppenheimer1977Capture}. As a gas giant increases in mass from accretion \\citep{Pollack1996GiantPlanetAccretion}, the hills sphere increases. As a subsequent effect, small \\replaced{solar}{Solar} system bodies can possibly be captured as irregular satellites.\n\nDynamical capture can explain the retrograde orbits of the Jovian satellites \\citep{Nesvorny2003IrrSatEvol}. The Hill sphere of a planet dictates the limit of its gravitational influence over other bodies. The theory \\citep{Nesvorny2003IrrSatEvol, Nesvorny2007IrrSatCap} states that is it impossible for a satellite to be captured in a three body system (Sun, planet and satellite). The Nice model of the \\replaced{solar}{Solar} system \\citep{Tsiganis2005NICEplanets, Nesvorny2007IrrSatCap, Nesvorny2014IrrCapture} has a fourth body interaction placing the satellite into a stable orbit inside the Hill sphere of the gas giant. Recently the Nice model was updated to include a fifth giant planet \\citep{Nesvorny2012JumpingJupiter}. This updated theory has the new planet interacting with Jupiter and allowing for the capture of the satellites, before the fifth giant planet is ejected from the \\replaced{solar}{Solar} system. Collisions between objects could also play a part in the dynamical capture of the irregular satellites \\citep{Colombo1971JupSatsForm}.\n\nThe formation of the Saturnian satellite system is thought to be similarly complex. The inner satellites are possibly formed from accretion within the ring system \\citep{Charnoz2010SaturnMooletsfromMainRings} or from the breakup of a large, lost satellite \\citep{Canup2010SaturnSatOrigin}. Modeling of the Saturnian system by \\citet{Salmon2017SaturnMidAccretion} has \\replaced{indicated}{shown} that the mid-sized satellites could have formed from a large ice-dominated ring, with contamination of \\replaced{asteroids}{cometary material} during the Late Heavy Bombardment, delivering the requisite silicate rock. Being the largest satellite in the Saturnian system, Titan is thought to have formed from accretion of proto-satellites \\citep{Asphaug2013SatMerger}. The Saturnian irregular satellites are predicted to be captured objects \\citep{Jewitt2007IrregularSats}, though their origins are still in dispute. Collisions are thought to have played a part in the capture of the irregular satellites of Saturn \\citep{Turrini2009IrregularSatsSaturn}. The cratering data provided by the Cassini spacecraft \\citep{Giese2006PhoebeTopo} supports this hypothesis.\n\n\\subsection{This Project}\nWith the discovery of several new irregular satellites \\citep{Scotti2000Callirrhoe, Gladman200112Sat, Sheppard2001SatsJupDiscovery,\nSheppard2002JupSatDisc, Gladman2003JupIAU1,Gladman2003JupIAU2,\nSheppard2003IAUJupSat, Sheppard2003JupIAU1,Sheppard2003JupIAU2,Sheppard2003JupIAU3,\nSheppard2003IrrSatNature,\nSheppard2004JupIAU1,Sheppard2004JupIAU2, Jewitt2005IAUCSat, Sheppard2006SatIAUC,Sheppard2007SatIAUC, \nJacobson2011NewJupSats, Sheppard2012JupIAU}, along with the detailed examination of the Jovian and Saturnian system by the Cassini spacecraft \\citep{Brown2003CassiniJupiter, \nPorco2005CassiniRingSat, Cooper2006CassiniAmaltheaThebe, Giese2006PhoebeTopo, Porco2006EnceladusPlume, Sphan2006EnceladusEring, Filacchione2007VIMS1, Nicholson2008VIMSRings, Matson2009SaturnSat, Buratti2010SatInnerSat, Filacchione2010VIMS2, Thomas2010SaturnSatsCassiniProps, \nClark2012VIMSIapetus, Filacchione2012VIMS3, Spitale2012s2009s1, Tosi2010IapetusDark, Hirtzig2013VIMSTitan, Brown2014Rayleigh, Filacchione2014VIMSrings, Filacchione2016VIMS4}, there is an opportunity to revisit the classification of the satellite systems of the gas giants. We apply a technique called \\textit{cladistics} to characteristics of the Jovian and Saturnian satellites, in order to examine the relationships between objects in the systems. The purpose of this is two fold. First, due to their \\replaced{well established}{well-established} classification systems, the Jovian and Saturnian satellite systems offer an opportunity to test the cladistical technique in a planetary science context. This project is an extension of \\citet{Holt2016JovSatCald} and together they form the first use of cladistics for planetary bodies. The second aim of the project is to classify recently discovered satellites, as well as providing context for future work.\n\nIn Section \\ref{Methods}, we introduce the cladistical technique, and how it is used in this paper. The resulting taxonomic trees for the Jovian and Saturnian systems, along with their implications for the taxonomy of the satellites, are presented in Sections \\ref{JupiterTax} and \\ref{SaturnTax} respectively. Section \\ref{Discussion} discusses the implications of cladistics in a planetary science context, along with some remarks on origins of the gas giant satellites and possible future work.\n\n\\section{Methods}\n\\label{Methods}\nIn this section, we present an overview of the cladistical method and how it is applied to the Jovian and Saturnian satellite systems. Following a general overview of cladistics, the section progresses into the specifics of this study, including characteristics used in the paper. The section concludes with an explanation on the specific matrices of the Jovian and Saturnian satellites and how they are applied to the cladistical method. \n\\subsection{Cladistics}\n\\label{cladistics}\n\nCladistics is an analytical technique, originally developed to examine the relationships between living organisms \\citep{Hennig1965PhylogeneticSystem}. A \\textit{clade} is the term used for a cluster of objects or \\textit{taxa}, that are related to each other at some level. In astronomy\/astrophysics, the technique has been used to look at the relationships between stars \\citep{FraixBurnet2015StarClads,Jofre2017StarsClads}, gamma-ray bursts \\citep{Cardone2013GRBClads}, globular clusters \\citep{FraixBurnet2009GlobularClusters} and galaxies \\citep{FraixBurnet2006DwarfGalaxies, FraixBurnet2010EarlyGalx, FraixBurnet2012SixPermGal, FraixBurnet2015GalClad}. These works, along with this study, form a  body of work in the new field of `Astrocladistics' \\citep{FraixBurnet2015GalClad}. There are good reasons to believe that cladistics can provide sensible groupings in a planetary science context. Objects that have similar formation mechanisms should have comparable characteristics. Daughter objects that are formed by breaking pieces off a larger object should also have similar characteristics. The advantage of this method over other multivariate analysis systems is the inclusion of a larger number of characteristics, enabling us to infer more detailed relationships. \n\nThe vast majority of work in cladistics and phylogenetics has been undertaken in the Biological and Paleontological sciences. Biologists and Paleontologists use cladistics as a method to investigate the common origins\\replaced{ or `tree', of life}{, or `tree' of life} \\citep{Darwin1859Origin, Hennig1965PhylogeneticSystem, Hug2016TreeLife}, and how different species are related to one another \\citep[e.g.][]{Van1993new, Salisbury2006originCrocs, Vrivcan2011two, Smith2017new, Aria2017burgess}. Historically, the investigation into relationships between different organisms reaches back to \\citet{Darwin1859Origin}. Early attempts at using tree analysis techniques occurred in the early 20th century \\citep{Mitchell1901BridsClads, Tillyard1926insects, Zimmermann1931arbeitsweise}. \\citet{Hennig1965PhylogeneticSystem} is regarded as one of the first to propose `phylogenetic systematics', the technique that would become modern cladistical\/phylogenetic analysis. The technique was quickly adopted by the biological community and used to analyze every form of life, from Bacteria \\citep[e.g.][]{Olsen1994winds} to Dinosauria \\citep[e.g.][]{Bakker1974dinosaur} and our own ancestors \\citep[e.g.][]{Chamberlain1987EarlyHominid}. Recently, the use of DNA led to the expansion of the technique to become molecular phylogenetics \\citep{Suarez2008HistoryPhylo}. As computing power improves, larger datasets can be examined, and our understanding of the `Tree of Life' improves \\citep{Hug2016TreeLife}. For a detailed examination of the history of cladistics and pyholgenetics, we refer the interested reader to \\citet{Hamilton2014EvolSystem}.\n\n\nThe cladisitcal methodology begins with the creation of a taxon-character matrix. Each matrix is a 2-d array, with the taxa, the objects of interest, in the rows, and each characteristic in the columns. The taxa used in this study are the rings and satellites of the Jovian and Saturnian Systems. The orbital, physical and compositional properties of the rings and satellites are used as characteristics, see Section \\ref{Characteristics}. For a given taxa, each corresponding characteristic is defined as a numerical state, usually a 0 or 1, though multiple, discrete states may be used. A 0 numerical state is used to indicate the original or `base' state. An \\textit{outgroup}, or a taxa outside the area of interest, is used to dictate the 0 base state of a characteristic. For this study, we use the Sun as an outgroup. An unknown character state can be accounted for, with a question mark (?). This taxon-character matrix is created using the Mesquite software package \\citep{Mesquite}.\n\nA set of phylogenetic trees are subsequently created  from the Mesquite taxon-character matrix, using Tree analysis using New Technology (TNT) 1.5 \\citep{Goloboff2008TNT, Golboff2016TNT15}, via the Zephyr Mesquite package \\citep{MesquiteZephyr}. The trees are created on the concept of maximum parsimony \\citep{Maddison1984outgroup}, that the tree with the shortest lengths, the smallest number of changes, is most likely to show the true relationships. TNT uses a method of indirect tree length estimation \\citep{Goloboff1994Treelengths, Goloboff1996FastPasrimony}, in its heuristic search for trees with the smallest length. TNT starts the drift algorithm \\citep{Goloboff1996FastPasrimony} search by generating 100 Wagner trees \\citep{Farris1970MethodsComp}, with 10 drifting trees per replicate. These starting trees are then checked using a Tree bisection and reconnection (TBR) algorithm \\citep{Goloboff1996FastPasrimony} to generate a block of \\replaced{equality}{equally} parsimonious trees. Closely related taxa are grouped together in the tree. Ideally, all equally parsimonious trees would be stored, but this is computationally prohibitive. For this analysis, 10000 equally parsimonious trees are requested from TNT, to create the tree block. Once a tree block has been generated and imported into Mesquite \\citep{Mesquite} for analysis, a 0.5 majority-rules consensus tree can be constructed using \\replaced{the a well established}{a well-established} algorithm \\citep{Margush1981MajorityRules}. This tree is generated as a consensus of the block, with a tree branch being preserved if it is present in the majority of the trees. The resulting branching taxonomic tree is then a hypothesis for the relations between taxa, the satellites and rings of the gas giants.  \n\nWe can assess how accurately a tree represents true relationships between taxa. The number of steps it takes to create a tree is call the \\textit{tree length}. A smaller tree length \\replaced{indicates}{implies} a more likely tree, as it is more parsimonious. Tree length estimation algorithms \\citep{Farris1970MethodsComp} continue to be improved, and are fully explored in a modern context by \\cite{Goloboff2015Parsimony}. Two other tree metrics, the consistency and retention indices, are a measure of \\textit{homoplasy}, or the independent loss or gain of a characteristic \\citep{Givnish1997consistency}. High amounts of homoplasy in a tree is \\replaced{indicative}{suggestive} of random events, rather than the desired relationships between taxa \\citep{Brandley2009Homoplasy}. Mathematically, homoplasy can be represented by the consistency index ($CI$) of a tree,  (equation (\\ref{ConIndexEq}) \\citep{Kluge1969Cladistics}) and is related to the minimum number of changes ($M$) and the number of changes on the tree actually observed ($S$). \n\n\\begin{equation}\nCI = M\/S\n\\label{ConIndexEq}\n\\end{equation}\n\nA tree with no \\textit{homoplasy} would have a consistency index of 1. \\added{One of the criticisms of the consistency index is that it shows a negative correlation with the number of taxa and characteristics \\citep{Archie1989homoplasy, Naylor1995RetentionIndex}. In order to combat the issues with the consistency index, a new measure of homoplasy, the retention index, was created \\citep{Farris1989retention}.} The retention index ($RI$) \\citep{Farris1989retention} \\deleted{, a second measure of homoplasy} introduces the maximum number of changes ($G$) required into equation (\\ref{RetentionIndexEq}). \n\n\\begin{equation}\nRI = \\frac{G - M}{G - S}\n\\label{RetentionIndexEq}\n\\end{equation}\n\nAs with the consistency index, a tree with a retention index of 1 indicates a perfectly reliable tree. Both of these metrics \\replaced{give an indication of}{show} how confidently the tree represents the \\replaced{true}{most plausible} relationships between taxa. Values closer to 1 of both the consistency and retention indices indicate that the tree represents the true relationships between taxa \\citep{Sanderson1989VaiationHomoplasy}. For a detailed examination of the mathematics behind the algorithms and statistics used in cladistical analysis, we direct the interested reader to \\cite{Gascuel2005MathEvolPhylogeny}.\n\nA traditional form of multivariate hierarchical clustering is used in the detection of asteroid collisional families \\citep{Zappala1990HierarchicalClustering1, Zappala1994HierarchicalClustering2}.\nThis method of clustering uses Gauss equations to find clusters in n parameter space, typically using semi-major axis, eccentricity and inclination \\citep{Zappala1990HierarchicalClustering1}. Work has also been undertaken incorporating the known colors \\citep{Parker2008AsteroidFamSDSS} and albedo \\citep{Carruba2013AsteroidFamilies} of the asteroids \\citep{Milani2014AsteroidFamilies} into the classical method, though this reduces the dataset significantly. The classical method of multivariate hierarchical clustering was used by \\citep{Nesvorny2003IrrSatEvol} to identify the Jovian irregular satellite families. \\citet{Turrini2008IrregularSatsSaturn} expanded the classical method into the Saturnian irregular satellites, and utilized the Gauss equations, solved for velocities, in a similar way to \\cite{Nesvorny2003IrrSatEvol} to verify the families found, using semi-major axis ($a$), eccentricity ($e$) and inclination ($i$) of the satellites. The rational behind these calculations is that the dispersal velocities of the clusters would be similar to the escape velocities of the parent body. In this work we use the inverse Gauss equations, equations \\ref{InvGauss1}, \\ref{InvGauss2} and \\ref{InvGauss3}, substituted into equation \\ref{VelocityEq}, to test the dispersal velocities of the clusters found through cladistics. $\\delta a$, $\\delta e$ and $\\delta i$ are the difference between the individual satellites and the reference object. $a_r$, $e_r$, $i_r$ and orbital frequency ($n_r$) are parameters of the reference object. In this case, the reference object is taken as the largest member of the cluster. The true anomaly ($f$) and perihelion argument ($w + f$) at the time of disruption are unknown. Only in special cases, for example, for young asteroid families \\citep[e.g.]{Nesvorny2002AsteroidBreakup}, the values of ($f$) and ($w + f$) can be inferred from observations. In this work we adopt $f = 90 \\degree $ and $(f+w) = 45 \\degree $ respectively as reasonable assumptions. Previous works by \\cite{Nesvorny2003IrrSatEvol} and \\cite{Turrini2008IrregularSatsSaturn} using this method, do not \\replaced{indicate}{specify} the true anomaly ($f$) and perihelion argument ($w + f$) used, nor the central reference point, making any comparisons between them and this work relative rather than absolute. The final $\\delta V_d$ for the cluster is composed of the velocities in the direction of orbital motion ($\\delta V_T$), the radial direction ($\\delta V_R$) and perpendicular to the orbital plane ($\\delta V_W$). \n\n\\begin{equation}\n\\delta V_T = \\frac{n_r a_r (1+e_r \\cos f)}{\\sqrt{1-e_r^2}} \\cdot \\left [ \\frac{\\delta a}{2 a_r} - \\frac{e_r \\delta e}{1-e_r^2} \\right ]\n\\label{InvGauss1}\n\\end{equation}\n\n\\begin{equation}\n\\delta V_R =  \\frac{n_r a_r}{(\\sqrt{1-e_r^2})\\sin f} \\cdot \\left [ \\frac{\\delta e_r (1 + e_r \\cos f)^2}{1-e_r^2} - \\frac{\\delta a (e_r + e_r \\cos^2 f + 2\\cos f)}{2a_r}   \\right ]\n\\label{InvGauss2}\n\\end{equation}\n\n\\begin{equation}\n\\delta V_W = \\frac{\\delta i \\cdot n_r a_r }{\\sqrt{1-e_r^2}} \\cdot \\frac{1+e_r \\cos f}{\\cos (w+f)}\n\\label{InvGauss3}\n\\end{equation}\n\n\\begin{equation}\n\\delta V_d = \\sqrt{\\delta V_T^2 + \\delta V_R^2 +  \\delta V_W^2}\n\\label{VelocityEq}\n\\end{equation}\n\nCladistics offers a fundamental advantage over this primarily dynamics based clustering, and that is the incorporation of unknown values. Classical multivariate hierarchical clustering \\citep{Zappala1990HierarchicalClustering1} requires the use of a complete dataset, and as such a choice is required. The parameters are either restricted to only known dynamical elements, or the dataset is reduced to well studied objects. Cladistical analysis can incorporate objects with large amounts of unknown information, originally fossil organisms \\citep{Cobbett2007FossilsCladistics}, without a reduction in the number of parameters. \n\n\n\\subsection{Characteristics}\n\\label{Characteristics}\nWe define 38 characteristics that can be broken into three broad categories: orbital, physical and compositional parameters. All numerical states are considered having equal weight. The discrete character sets are unordered. Any continuous characteristics are broken into bins, as cladistical analysis requires discrete characteristics. We developed a Python program to establish the binning of continuous characteristics. \\added{The pandas Cut module \\citep{Mckinney2010Pandas} is used to create the bins.} Each characteristic is binned independently of each other and for each of the Jovian and Saturnian systems. The aforementioned \\replaced{python}{Python} program iterates the number of bins until \\replaced{an $r^2$ score of $>0.99$ is reached for that characteristic set}{a linear regression model between binned and unbinned sets achieves a coefficient of determination ($r^2$) score of $>0.99$. This is calculated using the stats package in SciPy \\citep{Jones2010SciPy}}. Thus each character set will have a different number of bins, $r^2$ score and delimiters. All characteristics are binned in a linear fashion, with the majority increasing in progression. The exception to the linear increase is the density character set, with a reversed profile. All of the continuous, binned characteristic sets are ordered, as used by \\cite{FraixBurnet2006DwarfGalaxies}. A full list of the characteristics used, the $r^2$ score for each of the binned characteristics, along with the delimiters are listed in Appendix \\ref{ListCharacters}.\n\nThe first broad category includes the five orbital characteristics (Appendix \\ref{listOrbitChar}). This category is comprised of two discrete characteristics, presence in orbit around the gas giant, and prograde or retrograde orbit. The three remaining characteristics, semi-major axis (a), orbital inclination (i) and eccentricity (e), are continuous and require binning using the aforementioned \\replaced{python}{Python} program. \n\nThe second category used to construct the matrix consists of two continuous physical characteristics, density and visual geometric albedo (Appendix \\ref{listPhysChar}). We chose to not include mass, or any properties related to mass, as characters in the analysis. The inclusion of these characteristics could hide any relationships between a massive object and any daughter objects, as the result of collisions. \n\nThe third category describes the discrete compositional characteristics and details the presence or absence of 31 different chemical species (Appendix \\ref{listCompChar}). In order to account for any positional bias, the fundamental state, solid, liquid, gas or plasma was not considered. In this anyalysis, we make no distinction between surface, bulk and trace compositions. This is to account for the potential of daughter objects to have their bulk composition comprised of surface material from the parent. The majority of compounds have absence as a base state (0), and presence as the derived (1). The exception are the first three molecules, elemental hydrogen (eH), hydrogen (H$_2$) and helium (He), all of which are found in the Sun. As the Sun is the designated outgroup, the base state (0) indicates the presence of these species. With the exception of elemental hydrogen, the remaining single element species are those found in compounds. The spectroscopy of an object often only reports on the presence of an ion, as opposed to a full chemical analysis. \\deleted{As the full chemical composition of a body.} As more detailed analysis becomes available, characters may be added to the matrix. Several chemical species are used in this particular matrix which are either not present in any of the satellites or unknown. These are included for future comparisons with other orbital bodies.\n\n\\subsection{Matrices}\nThe Jovian taxon-character matrix holds 68 taxa consisting of: the Sun (outgroup), four inner satellites, the main ring, four Galilean satellites and 59 irregular satellites. Appendix \\ref{JupiterMatrix} contains the matrix, along with the references used in its construction.\n\nThe Saturnian matrix, presented in Appendix \\ref{SaturnMatrix}, is created with 76 taxa. These taxa are the Sun (outgroup), six main rings, nine inner small satellites, four minor rings, eight large icy satellites, four Trojan satellites, three Alkynoids and their associated rings, and the 38 irregular satellites. The references used in the construction of the Saturnian matrix are located in Appendix \\ref{SaturnMatrix}. Both matricies use the same characteristics, as discussed in Section \\ref{Characteristics}, and are available in machine readable format. \n\n\n\\section{Results}\n\nIn this section we present the resulting taxonomic trees from the analysis of the Jovian and Saturnian satellites. The taxonomic trees are used to form the systematic classification, of the Jovian (Table \\ref{JupiterClassTable}) and Saturnian (Table \\ref{SaturnClassTable}) satellite systems. Using inverse Gauss equations \\citep{Zappala1990HierarchicalClustering1}, in a similar method to \\cite{Nesvorny2003IrrSatEvol} and \\cite{Turrini2008IrregularSatsSaturn}, we show in Tables \\ref{JupiterClassTable} and \\ref{SaturnClassTable}, dispersal velocities ($\\delta V$) for each of the taxonomic groups where a single origin object is hypothesized, namely the irregular satellites. For these calculations we assume the largest representative of the cluster as the origin point. See section \\ref{cladistics} for further discussion.\n\n\\subsection{Jovian Taxonomy}\n\\label{JupiterTax}\nThe results of the cladistical analysis of the individual Jovian satellites is shown in Figure \\ref{JupiterTree}. This 0.5 majority-rules consensus tree has a tree length score of 128, with a consistency index of 0.46 and a retention index of 0.85. \\added{The low value of the consistency index is possibly due to the mixed use of ordered, multi-state, continuous characteristics and bi-modal compositional characteristics \\citep{Farris1990phenetics}.}  \\replaced{These values indicate}{The high retention index suggests} that the consensus tree is robust and \\replaced{indicative of the true}{demonstrates the most likely} relationships between the satellites. \n\n\\begin{figure*}\n\\includegraphics[height=0.65\\paperheight]{Fig1JupiterCladColourResub.pdf} \n\\caption{Majority consensus taxonomic tree of objects in the Jovian system. This tree has a tree length score of 128, with a consistency index of 0.46 and a retention index of 0.85. Numbers indicate frequency of the node in the 10000 most parsimonious tree block. \\replaced{Colors are indicative of}{Colors represent terminology used in} traditional classification: \\textcolor{Amalthea}{Amalthea inner regular family} ;      \n \\textcolor{Galilians}{Galilean family};\n  \\textcolor{Themisto}{Themisto prograde irregular}  ;\n   \\textcolor{Himalia}{Himalia prograde irregular family};\n  \\textcolor{Carpo}{Carpo prograde irregular} ;     \n  \\textcolor{Ananke}{Ananke irregular family};\n  \\textcolor{Carme}{Carme irregular family}  ;           \n  \\textcolor{Pasiphae}{Pasiphae irregular group};\n Unnamed and unclassified. Proposed groups and families are \\replaced{indicated}{shown} on the right.} \n\\label{JupiterTree}\n\\end{figure*}\n\n\\begin{deluxetable}{p{3cm}p{3.8cm}ccccccp{2cm}cc}\n\\label{JupiterClassTable}\n\n\\rotate\n\n\\tabletypesize{\\scriptsize}\n\n\n\\tablecaption{Jovian Satellite Systematic Classification}\n\n\n\\tablenum{1}\n\n\n\\tablehead{\\colhead{Taxonomy} & \\colhead{Members} & \\colhead{Orbit} & \\colhead{Semi-major Axis} & \\colhead{Inclination} & \\colhead{Eccentricity} & \\colhead{Density} & \\colhead{Albedo} & \\colhead{Composition} & \\colhead{Velocity ($\\delta V$)} & \\colhead{Ref.} \\\\ \n\\colhead{} & \\colhead{} & \\colhead{} & \\colhead{(km)} & \\colhead{} & \\colhead{} & \\colhead{($kg~m^{-3}$)} & \\colhead{} & \\colhead{} & \\colhead{($m~s^{-1}$)} & \\colhead{} } \n\n\\startdata\nAmalthea family  & Thebe, Amalthea, Metis and Adrastea & Prograde & $< 3.0 \\times 10^5$ & $< 0.02\\degree$ & $< 2\\degree$ & $< 900$ & $< 0.1$  & predominately water ice and silicates & $3570.4  \\pm 491.8 $ & 1 \\\\\nGalilean family  & Io, Ganymede, Europa, Callisto & Prograde & $4.0\\times 10^5$ \u2013 $2.0\\times 10^6$ & $< 0.5\\degree$ & $< 0.01$ & $> 1800$ & $> 0.18$ & water ice and silicates dominate; presence of SO$_2$; other chemical species present & \\nodata & 2 \\\\\nJovian Irregular Satellite group  &  &  &  &  &  &  &  &  &  &  \\\\\nHimalia family  & Leda, Elara, Lyithea, Himalia and Themisto. & Prograde & $7.5 \\times 10^6$ - $1.8 \\times 10^6$  & $25\\degree$ - $55\\degree$ & $0.1$ - $0.3$ & \\nodata & $< 0.1$ & silicate based & $623.8 \\pm 750.3$ & 3,4 \\\\\nAnanke\/Carme Family & S\/2003 J3, S\/2003 J9, Ananke subfamily, Carme subfamily and Sinope subfamily. & Retrograde & $1.88 \\times 10^7$ - $2.5\\times 10^7$  & $143\\degree$ - $166\\degree$ & $0.2$ - $0.4$ & \\nodata & $< 0.07$ & \\nodata & $457.2 \\pm 445.7$ & 3,4 \\\\\nAnanke Subfamily & Euanthe, Thyone, Mneme, Harpalyke, Praxidike, Thelxinoe and Ananke. & Retrograde & $2.0 \\times 10^7$ - $2.15 \\times 10^7$  & $145\\degree$ - $152\\degree$ & $0.2$ - $0.25$ & \\nodata & $< 0.07$ & \\nodata & $61.0 \\pm 45.6$ & 3,4 \\\\\nCarme Subfamily & Arche, Pasithee, Chaldene, Isonoe, Kale, Aitne, Erinome, Taygete, Carme, Kalyke, Eukelade and Kallichore. & Retrograde & $2.2 \\times 10^7$ - $2.4 \\times 10^7$  & $164\\degree$ - $166\\degree$ & $0.24$ - $0.27$ & \\nodata & $< 0.07$ & \\nodata & $36.1 \\pm 13.1$ & 3,4 \\\\\nSinope Subfamily & Eurydome, Autonoe, Sinope and Callirrhoe. & Retrograde & $2.2 \\times 10^7$ - $2.42 \\times 10^7$  & $147\\degree$ - $159\\degree$ & $0.27$ - $0.35$ & \\nodata & $< 0.06$ & \\nodata & $323.9 \\pm 97.3$ &  \\\\\nIocaste Family & Euporie, S\/2003 J18, Hermippe, Helike, Iocaste, S\/2003 J15, Herse, S\/2003 J4, Aoede, S\/2003 J5 and S\/2003 J10 & Retrograde & $1.9 \\times 10^7$ - $2.5 \\times 10^7$  & $140\\degree$ - $165\\degree$ & $0.1$ - $0.45$ & \\nodata & $< 0.05$ & \\nodata & $510.2 \\pm 303.3$ &  \\\\\nPasiphae Family & S\/2003 J12, S\/2011 J1, S\/2010 J2, S\/2003 J19, S\/2010 J1, S\/2011 J2, Sponde, Pasiphae, Megaclite, Hegemone, S\/2003 J23, Cyllene, Kore and S\/2003 J2. & Retrograde & $1.9 \\times 10^7$ - $2.9 \\times 10^7$  & $145\\degree$ - $164\\degree$ & $0.30$ - $0.421$ & \\nodata & $< 0.1$ & \\nodata & $412.3 \\pm 224.5$ & 3,4 \\\\\n\\enddata\n\n\n\\tablerefs{(1) \\citet{Barnard1892Amalthea};\n(2) \\citet{Galileo1610SidereusNuncius};\n(3) \\citet{Nesvorny2003IrrSatEvol};\n(4) \\citet{Sheppard2003IrrSatNature}.}\n\n\\end{deluxetable}\n\n\nAs can be seen in the Jovian taxonomic tree in Figure \\ref{JupiterTree}, the satellites cluster in to clades resembling the taxonomy proposed by \\citet{Nesvorny2003IrrSatEvol} and \\citet{Sheppard2003IrrSatNature}. The irregular satellites are a separate cluster to the prograde regular satellites. \n\nWe maintain the closest family to Jupiter, the Amalthea family, as a valid taxonomic cluster. The dispersal velocity is very large and may \\replaced{indicative}{suggest} that the Amalthea family did not form from a single object. This family, along with Jupiter's main ring, \\replaced{are}{is} associated with the well known Galilean family. \n\nIn the analysis, we maintain the 'irregular' satellite group. The Himalia family clusters with the retrograde satellites, separate to the other prograde satellites.  The Himalia family has relatively low inclinations, in comparison with the Jovian retrograde satellites and their high eccentricity could be explained by disruptions \\citep{Christou2005HimaliaScattering}. The small satellites Themisto and Carpo cluster together with the other prograde satellites in the Himalia family. We propose that Themisto and Carpo be included in the Himalia family, as they are the sole members of the groups proposed by \\citet{Sheppard2003IrrSatNature}, and show similar orbital characteristics. The large mean dispersal velocity calculated for the Himalia family (see Table \\ref{JupiterClassTable}) was also noticed by \\citet{Nesvorny2003IrrSatEvol} for the Prograde satellites. The large mean dispersal velocity due to the dispersal velocities of Themisto and Carpo. Without including these members, the mean dispersal velocity for the classical Himalia family is $154.6 \\pm 72.5 m\/s$, close to the escape velocity of Himalia, $121.14 m\/s$. This dispersal velocity of the classical Himalia family, was explained via gravitational scattering from Himalia by \\citet{Christou2005HimaliaScattering}. Disruption and scattering could also be used to explain the large dispersal velocities of Themisto and Carpo, though further modeling is required. \n\nThe term `irregular' is maintained through the retrograde family for consistency with the literature \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature, Nesvorny2004IrrSatFamilyOrigin,  Beauge2007IrregSatRsonance, Jewitt2007IrregularSats}. The retrograde irregular satellites are separate, but related cluster to the Himalia, prograde irregulars. The broad classifications introduced by \\citet{Sheppard2003IrrSatNature} and \\citet{Nesvorny2003IrrSatEvol} are preserved, though the Ananke\/Carme family is unresolved and may be split into subfamilies. Separating out the traditional families \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature}, see colors in figure \\ref{JupiterTree}, give smaller dispersal velocities. The traditional Ananke (escape velocity (eV) $23.10 m\/s$) family has a $\\delta V$ of $61.0 \\pm 45.6 m\/s$, traditional Carme (eV $29.83 m\/s$) has $36.2 \\pm 13.1 m\/s$, and a created Sinope (eV $27.62 m\/s$) family has $323.9 \\pm 97.3 m\/s$. These are smaller than the $\\delta V$ of our unresolved Ananke\/Carme Family ($457.2 \\pm 445.7 m\/s$, see table \\ref{JupiterClassTable}). \\citet{Nesvorny2003IrrSatEvol} used similar small $\\delta V$ values to establish the Ananke and Carme dynamical families. The dynamical situation could be explained through a more recent capture and breakup event for Ananke, Carme and Sinope, that disrupted the ancestral irregular satellites. The identified Iocaste and Pasiphae families also have large dispersal velocities, \\replaced{indicative}{suggestive} of disruptions. Following the nomenclature of \\citet{Sheppard2003IrrSatNature}, each of the families and subfamilies are represented by the name of the largest contained satellite. Satellites within families are related by their retrograde orbit, high inclinations and eccentricities. In addition to their linked orbital characteristics, the satellites of the retrograde irregular group all show a low albedo \\citep{Beauge2007IrregSatRsonance}. \n\nThe Ananke subfamily is tightly constrained in its orbital characteristics, with a small dispersal velocity. While the characteristics listed in Table \\ref{JupiterClassTable} would preclude them from being included in the Pasiphae family, their clustering around a common semi-major axis, inclination and eccentricity, \\replaced{indicated}{suggesting} that they are a distinct young dynamical family. The members we include in the Ananke family for this analysis are all historical members of the family \\citep{Jewitt2007IrregularSats}. Some of the satellites that have been historically included in the Ananke family \\citep{Jewitt2007IrregularSats} are moved to other families. We do not add any new satellites to this family. \n\nThe orbital characteristics of the Carme subfamily are tightly constrained. Satellites in this family orbit further from Jupiter, with higher orbital inclinations, but similar eccentricities to the Ananke family. As with the Ananke family, it is the highly constrained orbital characteristics and low mean dispersal velocity, that justify the classification of this traditional family \\citep{Jewitt2007IrregularSats}. According to the tree presented in Figure \\ref{JupiterTree}, there is a continuum between the Ananke and Carme families. However, differences in orbital characteristics, broken down in Table \\ref{JupiterClassTable}, distinguish both of these families from each other.\n\nA new cluster, the Iocaste family, is defined as shown in Figure \\ref{JupiterTree} and Table \\ref{JupiterClassTable}. The semi-major axis of this family spans most of the orbital space where irregular satellites have been discovered. The lower eccentricities and albedo are used to separate this family from the Pasiphae family. As with the Passiphae family, the Iocaste family has a high mean dispersal velocity ($510.2 \\pm 303.3$ compared with a escape velocity of $3.16 m\/s$), \\replaced{indicative}{suggestive} of disruptions taking place at some point since the break-up of the original object \\citep{Christou2005HimaliaScattering}. Iocaste, being the largest member of this family, is proposed as the representative object. Also included are several members that have been previously included in other families \\citep{Jewitt2007IrregularSats}, along with new unnamed satellites. For full details on included satellites and the descriptive properties of the family, see Table \\ref{JupiterClassTable}.\n\nThe Pasiphae family show a broad range of orbital characteristics that, along with the large dispersal velocity ($412.3 \\pm 224.5$ compared with an escape velocity of $47.16 m\/s$), are \\replaced{indicative}{suggestive} of disruptions during the family's life-time \\citep{Christou2005HimaliaScattering}. The Pasiphae family has a broad range semi-major axies and inclinations, with the Pasiphae family orbiting further from Jupiter and having larger eccentricities on average than the new Iocaste family, see table \\ref{JupiterClassTable}. A Pasiphae subfamily, see figure \\ref{JupiterTree}, with a $\\delta V$ of $230.1 \\pm 174.3 m\/s$, can be identified. This may \\replaced{indicate}{imply} a secondary, more recent break-up from Pasiphae. In addition, many of the unnamed satellites from recent observations  \\citep{Gladman2003JupIAU1, Gladman2003JupIAU2, Sheppard2003JupIAU1, Sheppard2003JupIAU2, Sheppard2003JupIAU3, Sheppard2003IAUJupSat,\nSheppard2004JupIAU1,Sheppard2004JupIAU2,\nJacobson2011NewJupSats, Sheppard2012JupIAU} are associated with this family, see Table \\ref{JupiterClassTable} and Figure \\ref{JupiterTree} for a complete list. \n\n\n\\subsection{Saturnian Taxonomy}\n\\label{SaturnTax}\nCladistical analysis of the Saturnian \\replaced{System}{system} yields the 0.5 majority-rules consensus tree, Figure \\ref{SaturnTree}, constructed from the 10000 parsimonious trees, with a tree length score of 186. The tree has a consistency index of 0.30 and a retention index of 0.81. The consistency index of the Saturnian tree is lower than that of the Jovian tree, though this could be due to the number of taxa used \\citep{Sanderson1989VaiationHomoplasy}. \\added{As with the Jovian tree, this low consistency index could be due to the mixed character states. This effect is to be explored further in a future paper.} The high retention index indicates that the tree \\replaced{is indicative}{is suggestive} of the true relationships \\citep{Farris1989retention}. \n\n\n\\begin{figure*}\n\n\\includegraphics[height=0.65\\paperheight]{Fig2SaturnCladColourReSub.pdf} \n\\caption{Majority Consensus taxonomic tree of objects in the Saturnian system. The tree has a consistency index of 0.30 and a retention index of 0.81. Numbers indicate frequency of the node in the 10000 most parsimonious tree block. \\replaced{Colors are indicative of}{Colors represent terminology used in} classical classification: \\textcolor{MainRing}{Main ring group, with associated shepherd satellites} ;      \n \\textcolor{IcySats}{Mid-sized Icy satellites and Titan};\n  \\textcolor{Trojans}{Trojan satellites}  ;\n   \\textcolor{Alkanoids}{Alkanoids and associated rings};\n  \\textcolor{Inuit}{`Inuit' prograde irregular family} ;     \n  \\textcolor{Gallic}{`Gallic' prograde irregular family};\n  \\textcolor{Norse}{`Norse' retrograde irregular family}  ; \n  Unnamed and unclassified.           \n  Proposed groups and families are \\replaced{indicated}{shown} to the right. }\n\\label{SaturnTree}\n\\end{figure*}\n\n\n\\begin{deluxetable}{p{3cm}p{3.8cm}ccccccp{2cm}cc}\n\n\\rotate\n\n\\tabletypesize{\\scriptsize}\n\n\\tablecaption{Saturnian Satellite Systematic Classification}\n\\label{SaturnClassTable}\n\n\\tablenum{2}\n\n\n\\tablehead{\\colhead{Taxonomy} & \\colhead{Members} & \\colhead{Orbit} & \\colhead{Semi-major Axis} & \\colhead{Inclination} & \\colhead{Eccentricity} & \\colhead{Density} & \\colhead{Albedo} & \\colhead{Composition} & \\colhead{Velocity ($\\delta V$)} & \\colhead{Ref.} \\\\ \n\\colhead{} & \\colhead{} & \\colhead{} & \\colhead{(km)} & \\colhead{} & \\colhead{} & \\colhead{($kg~m^{-3}$)} & \\colhead{} & \\colhead{} & \\colhead{($m~s^{-1}$)} & \\colhead{} } \n\n\\startdata\nSaturnian Inner system Group, Main ring and Icy satellites & Atlas, Janus, Epimetheus, Prometheus, Janus\/Epimetheus ring, G ring, D ring, Pan, Aegaeon, S\/2009 S1, F ring, B ring, Cassini Division, C ring, Daphnis and A ring. Possible members: Telesto, Calypso, Methone ring arc, Anthe ring arc, Pallene ring arc, Methone, Anthe, Pallene,  Polydeuces  Mimas, Tethys,  Enceladus family, Hyperion, Titan and Iapetus ; see Section \\ref{SaturnTax} for discussion. & Prograde & $< 4.0 \\times 10^6$ & $< 15\\degree$ & $< 0.03$ & $550$ \u2013 $1900$ & $0.1$ - $1$ & Composition of water ice with silicates and presence of CO$_2$. Other chemical species may be present  & \\nodata & 1, 2 \\\\\nEnceladus Family & E ring, Enceladus, Rhea, Dione and Helene. & Prograde & $1.8 \\times 10^5$ - $5.3 \\times 10^5$ & $< 0.5\\degree$ & $0$ & $1200$ \u2013 $1700$ & $> 0.7$ & Complex composition, predominately water ice and silicates, with Hydrocarbons and CO$_2$ present & \\nodata &  \\\\\nSaturnian Irregular Satellite group  &  &  &  &  &  &  &  &  &  &  \\\\\nAlbiorix family  & Bebhionn, Erriapus, Albiorix and Tarvos & Prograde & $1.6 \\times 10^7$ - $1.8 \\times 10^7$ & $30\\degree$ - $40\\degree$ & $0.4$ - $0.6$ & \\nodata & $< 0.1$ & \\nodata & $80.9 \\pm 1.6 $ & 3,4,5 \\\\\nSiarnaq family  & Tarqeq, Kiviuq, Ijiraq, Paaliaq and Siarnaq & Prograde & $1.1 \\times 10^7$ - $1.9 \\times 10^7$ & $40\\degree$ - $50\\degree$ & $0.1$ \u2013 $0.4$ & \\nodata & $< 0.1$ & \\nodata & $266.8 \\pm 60.0$ & 3,4,5 \\\\\nPhoebe family  & Phoebe Ring, Phoebe,  Fenrir, Loge, Aegir subfamily, and Ymir subfamily. & Retrograde & $1.1 \\times 10^7$ - $2.51 \\times 10^7$ & $> 145\\degree$ & $> 0.1$ & \\nodata & $< 0.1$ & \\nodata & $763.3 \\pm 259.0 $ & 3,4,5 \\\\\nAegir subfamily & S\/2007 S2, Mundilfari, Jarnsaxa, S\/2006 S1, Bergelmir, Suttungr, Farbauti, S\/2007 S3, Aegir and Fornjot. & Retrograde & $1.6 \\times 10^7$ - $2.51 \\times 10^7$ & $> 150\\degree$ & $0.1$ \u2013 $0.25$ & \\nodata & \\nodata & \\nodata & $295.1 \\pm 125.0$ & 5 \\\\\nYmir subfamily & Skathi, Skoll, Greip, Hyrrokkin, S\/2004 S13, S\/2004 S17, Narvi, S\/2004 S12, S\/2004 S07, Hati, Bestla, Thrymr, S\/2006 S3, Kari, Surtur and Ymir & Retrograde & $1.55 \\times 10^7$ - $2.30 \\times 10^7$  & $> 145\\degree$ & $0.25$ - $0.6$ & \\nodata & $< 0.1$ & \\nodata & $497.5 \\pm 247.7$ & 5 \\\\\n\\enddata\n\n\n\\tablerefs{(1) \\citet{Huygens1659systema};\n(2) \\citet{Cassini1673Sat2Sats,Cassini1686Sat2Sats}\n(3) \\citet{Nesvorny2003IrrSatEvol};\n(4) \\citet{Sheppard2003IrrSatNature};\n(5) \\citet{Turrini2008IrregularSatsSaturn}.}\n\n\\end{deluxetable}\n\n\n\n\nThe tree shown in Figure \\ref{SaturnTree} highlights the diversity of structures found in the orbit of Saturn. Satellites cluster into two main grouping around Saturn, the Inner group, comprised of rings and icy satellites, and the Irregular satellite group, see Table \\ref{SaturnClassTable} for members and diagnostic properties of each clade. While the traditional classification nomenclature \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature, Jewitt2007IrregularSats} is broadly conserved, several \\replaced{anomalies}{discrepancies} require discussion. Table \\ref{SaturnClassTable} shows our new taxonomy, along with included members of the families and their descriptive properties.\n\nThe Main ring and Icy satellite group form an unresolved, inner system group. This group includes the Saturnian ring system, the Alkynoids and their associated ring arcs, as well as the larger Icy satellites and their Trojans. We have confirmed the recently discovered S\/2009 S1 \\citep{Spitale2012s2009s1} is part of this group due to its orbital characteristics. Within this large group, there is the resolved Enceladus family.\n\nOur results suggest the traditionally classified Alkyonides, Methone, Anthe and Pallene, along with their associated rings, are not clustered with the the Enceladus family, as would be expected by their orbital location, between Mimas and Enceladus, within the E-ring. Due to their bulk water ice composition, the Alkynoides associate with the Main ring objects, see Figure \\ref{SaturnTree}. The low density and mid-range albedo of Pallene and Methone \\citep{Hedman2009SatRingArcs} \\replaced{indicates}{suggests} that the association with the Main ring group is genuine. The dynamic resonances of both Methone and Anthe \\citep{Callegari2010SmallSaturnSatsDynamics} \\replaced{are also indicative of these objects being}{implies that these objects where} captured, rather than forming in-situ. As there is very little known about the composition of these objects, beyond their bulk water ice composition \\citep{Hedman2009SatRingArcs}, further study and dynamical modeling of the capture process is required to resolve their true origins.\n \nLike the Alkynoids, the Trojan satellites of Tethys, Calypso and Telesto, also form an association with the main rings. The reason for this could be that Calypso and Telesto, like the Alkynoids, are also possible captured main ring objects. The capture dynamics could be similar to that of the Jovian Trojan asteroids \\citep{Morbidelli2005TrojanCapture, Lykawka2010TrojanCaputer, Nesvorny2013TrojanCaptureJJ}. Both of the Tethys Trojans \\citep{Buratti2010SatInnerSat} and main ring objects, are chiefly comprised of water ice, \\replaced{indicative of}{implying} a common origin. The bulk composition of Tethys is also prominently water ice \\citep{Buratti2010SatInnerSat}, with a very small fraction of silicates. Trojans may instead have formed from the same material as Tethys itself, either during accretion \\citep{Charnoz2011SaturnSatAccretion} or in the same orbit from a large debris disk \\citep{Canup2010SaturnSatOrigin}. As Tethys is also in the unresolved Main ring and Satellite group, we can not differentiate between the two scenarios. Further compositional information about the Tethys Trojans could shed light on this issue. Polydeuces, a Trojan of Dione, also forms an association with the Main ring group in our analysis. This could be due to overemphasis on orbital and physical characteristics, since the bulk composition of Polydeauces is unknown \\citep{Thomas2013InnerSatSatu}. Helene, the more well studied Trojan of Dione \\citep{Thomas2013InnerSatSatu}, is well within the Enceladus Family. Helene and Dione are closely associated in our analysis, \\replaced{indicating}{implying} that Helene is a daughter object of Dione.\n\nThe outer icy satellites, Titan, Hyperion and Iapetus, do not form a single cluster, and are therefore not considered a valid taxonomic group. They are associated with the Main ring and Icy Satellite group. The Enceladus family is formed by the known association between the E-ring, Enceladus and Icy Satellites \\citep{Verbiscer2007Enceladus}, which is mainly due to the detection of volatile chemicals, such as NH$_3$, CH$_4$ and other hydrocarbons. Plumes from Enceleadus containing these chemicals \\citep{Porco2006EnceladusPlume}, thought to be representative  of the subcrust ocean \\citep{Porco2006EnceladusPlume}, \\deleted{and} are the source of the E ring \\citep{Sphan2006EnceladusEring}. Titan itself also has an abundance of these volatiles \\citep{Hirtzig2013VIMSTitan}, \\replaced{indicating}{implying} a possible association between the Icy satellites of Saturn that remains unresolved in our analysis.\nMaterial from the outer satellites, particularly Pheobe and its associated ring \\citep{Tosi2010IapetusDark, Tamayo2011IapetusDust} is thought to play a role in the observed hemispherical dichotomy on Iapetus \\citep{Tosi2010IapetusDark}. In Figure \\ref{SaturnTree}, Iapetus is unresolved in the Main ring and Icy Satellite group.\n\nThe irregular satellites form a major cluster with each other separate from the inner Saturnian system, and are therefore collected under the Irregular satellite group. Along with their high inclinations, eccentricities and semi-major axes, the Irregular satellite group is characterized by a dark albedo, comparative to the other objects in the Saturnian system. We follow the naming convention introduced with the Jovian satellites, Section \\ref{JupiterTax},  where each irregular satellite family is represented by the largest member \\citep{Jewitt2007IrregularSats}. We therefore rename the classical Inuit group \\citep{Blunck2010SaturnSats} to the Siarnaq family and the Gallic group \\citep{Blunck2010SaturnSats} to the Albiorix family. Though this does change the formal name of the clusters, we encourage the discoverers of the unnamed satellites \\citep{Gladman200112Sat,Sheppard2003IAUJupSat,Jewitt2005IAUCSat,Sheppard2006SatIAUC,Sheppard2007SatIAUC} and any future discoveries that are placed in these groups, to follow IAU convention and use names from Inuit and Gallic mythology for satellites in the Siarnaq and Albiorix families respectively. \nAs in \\cite{Turrini2008IrregularSatsSaturn}, the Albiorix family is distinct and has a low mean dispersal velocity ($\\delta V$). The Siarnaq family has a higher $\\delta V$, again \\replaced{indicative}{suggestive} of disruptions \\citep{Christou2005HimaliaScattering}. The mean $\\delta V$ of all prograde satellites is $364.8 \\pm 114.9 m\/s$, only slightly higher than that of the Siarnaq family \\citep{Turrini2008IrregularSatsSaturn}. This could \\replaced{be indicative of}{imply} a disruption scenario, with a more recent capture of the Albiorix family parent body disrupting the older Siarnaq family. Our cladistical analysis supports this scenario, as the Siarnaq family shows a more branching structure than the Albiorix family. Further compositional information about these bodies, as well as dynamical modeling, could resolve this complex situation.\n\n\\replaced{Our study shows that the classically named retrograde Norse group \\citep{Blunck2010SaturnSats} is determined to be the Phoebe family, which can be split into at least two subfamilies.}{In our analysis, we separate out the retrograde irregular satellites, including Phoebe, from the prograde irregular satellites. In previous taxonomy, this group has been classified as the 'Norse' group \\citep{Blunck2010SaturnSats}. In our revised nomenclature, this group should be termed the Phoebe family. We further separate out two clades, distinct from Phoebe and its associated ring.} The first clade, the unresolved Aegir subfamily\\added{ (previously identified as the S\/2004 S10 group in \\cite{Turrini2008IrregularSatsSaturn})}, is characterized as having on average, orbits further from Saturn, with low eccentricities and higher inclinations. The second clade is the Ymir subfamily and is categorized, on average, by being closer to Saturn, but with high eccentricities. This subfamily shows a branching structure and may be further split \\citep{Grav2007IrregSatCol}. \\added{This family was also identified by \\cite{Turrini2008IrregularSatsSaturn}.} We identify an association between Fenrir and Loge, with a low dispersal velocity ($\\delta V = 114.4 m\/s$), \\replaced{indicative}{suggestive} of a recent breakup. The high dispersal velocity ($\\delta V$) of the Phoebe family is due to the selection of Phoebe as a reference point. If Phoebe and the \\replaced{associate}{associated} ring are removed from the \\replaced{subfamily}{family}, \\added{and Ymir (with an escape velocity of $8.56 m\/s$) selected as the reference object,} the $\\delta V$ is halved from $763.3 \\pm 259.0 m\/s$ to $439.9 \\pm 215.1 m\/s$. The satellite with the lowest $\\delta V$ to Phoebe is S\/2007 S2, with $\\delta V = 248.0 m\/s$, still \\added{significantly}larger than the escape velocity of Phoebe ($100.8 m\/s$). \\deleted{If Phoebe is removed from the family, and Ymir (escape velocity $8.56 m\/s$) selected as the reference object, the mean $\\delta V$ of the cluster is $439.9 \\pm 215.1 m\/s$, lower than the $\\delta V$ of the Ymir subfamily.} \\added{\\cite{Turrini2008IrregularSatsSaturn} also found a dynamical separation between Phoebe and the other retrograde satellites.} This is supportive of the narrative \\replaced{for a different origin of Phoebe and}{that Phoebe has a different origin to} the other retrograde irregular satellites of Saturn \\citep{Turrini2008IrregularSatsSaturn}. The high $\\delta V$ among all the subfamilies shows \\added{that} a complex dynamical situation is present in the Saturnian irregular satellites. Phoebe has been shown to clear its orbital parameter space \\citep{Turrini2008IrregularSatsSaturn}, which could have had a major disruptive effect on those remaining satellites \\citep{Turrini2008IrregularSatsSaturn}. \\added{The similarities between our analysis and that of \\cite{Turrini2008IrregularSatsSaturn} further validates cladistics as a method suitable for applications in Solar system astronomy.} The addition of \\replaced{detail}{detailed} compositional information from the other irregular satellites to an updated cladistical analysis could solve some of the \\added{minor} discrepancies found between this analysis and that of \\cite{Turrini2008IrregularSatsSaturn}.\n\nWe assign the currently unnamed irregular satellites to each of the subfamilies. S\/2006 S1, S\/2007 S2 and S\/2007 S3 are part of the Aegir subfamily. We include S\/2004 S13, S\/2004 S17, S\/2004 S12, S\/2006 S3 and S\/2007 S7 in the Ymir subfamily. See Table \\ref{SaturnClassTable} for a full list of members in each subfamily. As with the Albiorix and Siarnaq families, we encourage discoverers of new satellites that fall within the Phoebe family to follow the Norse mythological naming convention as set by the IAU. \n\n\n\n\\section{Discussion}\n\\label{Discussion}\n\nIn this study we have shown, using the Jovian and Saturnian satellite systems, that cladistics can be used in a planetary science context. We have ensured that the technique is objective, by statistically creating bins for characteristics that are continuous in nature, see Section \\ref{Characteristics}. By thus ensuring the objectivity of our analysis, we increase the confidence that cladistics is a valid technique that can be applied in the planetary sciences. Our results largely support the traditional classifications used in both the Jovian and Saturnian systems. However, the power of cladistics is shown in the ease of classifying new satellites as well as identifying substructures within larger clusters. Cladistics also offers a method of analysis where limited information is available. In our study we have examined well studied satellites, as well as those where only dynamical information available. In traditional methods of analysis, either only dynamical information is considered, or the dataset is truncated in favor of more well studied bodies. Cladistics offers a method that can incorporate as much information about an object as is available, while accounting for any unknown characteristics. As more detailed information becomes available, either of known or newly discovered satellites, cladistics offers a systematic method for inclusion or revision of the classification system. \n\nThe relationships that we noted between the satellites suggest common formation scenarios within the clusters. The prograde, inner families of Jupiter are the products of accretion from a circumplanetary disk \\citep{Canup2002GalSatAcc}. The association of the Amalthea and Galilean families, along with the Main ring of Jupiter, in our analysis supports this hypothesis. Clustering of the Himalia family with other 'irregular' satellites, \\replaced{indicates}{implying} a capture scenario. The prograde nature of the Himalia family is possibly explained via a nebula drag capture mechanism \\citep{Cuk2004HimaliaGasDrag}. Further modeling of the Himalia family is required to ascertain their true origins, particularly in light of the Jovian pebble formation hypothesis that may not include an extended nebula \\citep{Levison2015GasGiantsPebbles}. \n\nWith the proposal that Sinope forms its own subfamily, each Jovian irregular satellite subfamilies contain only a single large satellite. This strengthens the hypothesis that each of the families represents a capture event and subsequent breakup \\citep{Nesvorny2007IrrSatCap} of an object external to the Jovian system. Two of the subfamiles, the Pasiphae and Sinope subfamiles, show a broad range of orbital characteristics and larger dispersal velocities. The other two, the Ananke and Carme subfamiles, show much more constrained characteristics and smaller dispersal velocities. This dichotomy between the two types of subfamiles, broad versus constrained, could \\replaced{indicate}{imply} at least two capture events, with the earlier Pasiphae and Sinope families being disrupted by later Ananke and Carme captures. The Iocaste family does not contain a large progenitor satellite, but has high dispersal velocities. This is \\replaced{indicative}{suggestive} of a possible ejection scenario. An alternative hypothesis is that the capture events happen simultaneously, but there were multiple disruption events. Both scenarios are supported by the dichotomy in dispersal velocities. Future analysis and simulations into the origins of the irregular satellites could help determine which theory is more probably correct. \n\nAs with the Jovian satellites, there are multiple origins for the origin of the Saturnian rings and satellites. The results from our analysis support a growing body of work showing the complexity of formation scenarios in the Saturnian system. The rings themselves possibly formed after the breakup of an inner icy satellite \\citep{Canup2010SaturnSatOrigin}. \n\nThe unresolved nature of the inner Saturnian system shows a complexity of formation scenarios. The main ring satellites, along with the Alkyonides and Tethys Trojans possibly formed via accretion from the current ring system \\citep{Charnoz2010SaturnMooletsfromMainRings}. The Alkynoides and Tethys Trojans are then secondarily captured in their current orbits. The major icy satellites, those in the E-ring and outer satellites, probably formed in an accretion scenario, with delivery of the silicate from the outer system \\citep{Salmon2017SaturnMidAccretion}. Titan could be secondarily derived from multiple subsatellites that formed in the same disk \\citep{Asphaug2013SatMerger}. The volatiles are delivered from comets, with at least one, Phoebe, being captured in orbit. The size of Phoebe is not traditionally associated with comet nuclei, but atleast one comet, C\/2002 VQ94 with a similar ~100km diameter has been observed \\citep{Korsun2014c2002vq94100kmComet}. The irregular satellite families and subfamiles form from collisional breakup events \\citep{Nesvorny2004IrrSatFamilyOrigin} resulting from the captured comet nuclei. The large dispersal velocities of the subfamilies \\replaced{indicate}{imply} that this capture and disruption process is complex and requires detailed modeling.\n\nWe have shown that cladistics can be used in the classification of the Jovian and Saturnian satellite systems. Consequently, several related studies may be attempted in the future. Uranus and Neptune have similarly complex satellite systems to those of Jupiter and Saturn \\citep{Jewitt2007IrregularSats}. These satellite systems could also be classified using cladistics, particularly the irregular satellites. Such a study is hampered by a lack of completeness in the irregular satellite dataset \\citep{Sheppard2005UransIrr, Sheppard2006NeptuneIrr}, but may become practical as observational technology improves and the hypothesized small irregular satellites are discovered. Cladistics could be used to further investigate the origins of the irregular satellites of Saturn and Jupiter. As the irregular satellites are thought to be captured bodies \\replaced{\\citep{Nesvorny2007IrrSatCap}}{\\citep[e.g.][]{Nesvorny2007IrrSatCap}}, the question becomes from \\replaced{what}{which} small body population they originated. Comparisons between the well studied irregular satellites and other \\replaced{solar}{Solar} system bodies could help constrain the origins of these satellites. \n\n\n\\section{Conclusions}\n\\label{Conclusion}\n\nWe have shown that the new application of cladistics on the Jovian and Saturnian satellite systems is valid for investigating the relationships between orbital bodies. In the Jovian system, the traditional classification categories \\citep{Nesvorny2003IrrSatEvol,Sheppard2003IrrSatNature,Jewitt2007IrregularSats} are preserved. We support the hypothesis put forward by \\cite{Nesvorny2007IrrSatCap} that each Jovian irregular satellite family can be represented by the largest member, and that each family is the remnants of a dynamical capture event, and subsequent breakup. We can also assign recently discovered, as yet unnamed, satellites to each of their respective Jovian families. Cladistical analysis of the Saturnian system broadly preserves the traditional classifications \\citep{Nesvorny2003IrrSatEvol, Sheppard2003IrrSatNature, Jewitt2007IrregularSats,Turrini2008IrregularSatsSaturn}, strengthening the validity of the cladistical method. In the Phoebe family of retrograde, irregular satellites, we assign two subfamilies\\added{similar to those found by \\citep{Turrini2008IrregularSatsSaturn}}. We rename the classical mythological designations for the Saturnian irregular satellites, to represent the largest member of the subfamily, in order to be consistent with the Jovian naming convention. Newly discovered, unnamed Saturnian satellites are easily assigned to various subfamiles. Through the application of the technique to the Jovian and Saturnian systems, we show that cladistics can be used as a valuable tool in a planetary science context, providing a systematic method for future classification.\n\n\n\\acknowledgments\nThis research was in part supported by the University of Southern Queensland's Strategic Research Initiative program. We wish to thank an anonymous reviewer for \\replaced{their}{his\/her} comments, particularly on Multivariate Hierarchical Clustering. The AAS Statistics Reviewer provided valuable feedback on the methodology. Dr. Guido Grimm assisted with the cladistical methodology and terminology used in this paper. Dr. Pablo Goloboff provided assistance with TNT, which is subsidized by the Willi Hennig Society, as well as additional comments on the methodology. We would like to thank \\added{Dr.} Henry Throop for discussions regarding the Ring systems.\n\n\\software{\nMesquite 3.10 \\citep{Mesquite}, \nPython 3.5, \nSpyder 2.3.8 \\citep{Spyder238}, \nAnaconda Python distribution package 2.40 \\citep{Anaconda240},\n\\added{pandas Python package \\citep{Mckinney2010Pandas},\nScyPy Python package \\citep{Jones2010SciPy},}\nTexMaker 4.1.1, \nTree analysis using New Technology (TNT) 1.5 \\citep{Goloboff2008TNT, Golboff2016TNT15}.\nZephyr 1.1: Mesquite package \\citep{MesquiteZephyr}.\n}\n\n\n\\bibliographystyle{aasjournal} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Basic Results} \\label{sec:basic_results}\n\nIn this section, we prove some basic results concerning the complexity of \\textsc{ListIso}\\xspace and \\textsc{ListAut}\\xspace.\n\n\\begin{lemma} \\label{lem:lautlgi_equiv}\nBoth problems \\textsc{ListAut}\\xspace and \\textsc{ListIso}\\xspace are polynomially equivalent.\n\\end{lemma}\n\n\\begin{proof}\nTo see that \\textsc{ListAut}\\xspace is polynomially reducible to \\textsc{ListIso}\\xspace just set $H$ to be a copy of $G$ and keep the\nlists for all vertices of $G$. It is straightforward to check that these two instances are\nequivalent.  For the other direction, we build an instance $G'$ and $\\frakL'$ of \\textsc{ListAut}\\xspace as follows. Let\n$G'$ be a disjoint union of $G$ and $H$. And let $\\frakL'(v) = \\frakL(v)$ for all $v\\in V(G)$ and set\n$\\frakL'(w) = V(G)$ for all $w \\in V(H)$. It is easy to see that there exists  list-compatible\nisomorphism from $G$ to $H$, if and only if there exists a list-compatible automorphism of $G'$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:list_size_two}\nThe problem \\textsc{ListIso}\\xspace can be solved in time $\\calO(n+m)$ when all lists are of size at\nmost two.\n\\end{lemma}\n\n\\begin{proof}\nWe construct a list-compatible isomorphism $\\pi : G \\to H$ by solving a 2-SAT formula which can be\ndone in linear time~\\cite{even_2sat,aspvall_2sat}.  When $w \\in \\frakL(v)$, we assume that $\\deg(v) =\n\\deg(w)$, otherwise we remove $w$ from $\\frakL(v)$.  Notice that if $\\frakL(u) = \\{w\\}$, we can set $\\pi(u)\n= w$ and for every $v \\in N(u)$, we modify $\\frakL(v) := L(v) \\cap N(w)$. Now, for every vertex $u_i$\nwith $\\frakL(u_i) = \\{w^0_i, w^1_i\\}$, we introduce a variable $x_i$ such that $\\pi(u_i) = w^{x_i}_i$.\nClearly, the mapping $\\pi$ is compatible with the lists.\n\nWe construct a 2-SAT formula such that there exists a list-compatible isomorphism if and only if it\nis satisfiable. First, if $\\frakL(u_i) \\cap \\frakL(u_j) \\ne \\emptyset$, we add implications for $x_i$ and\n$x_j$ such that $\\pi(u_i) \\ne \\pi(u_j)$. Next, when $\\pi(u_i) = w^j_i$, we add implications that\nevery $u_j \\in N(u_i)$ is mapped to $N(w^j_i)$. If $\\frakL(u_j) \\cap N(w^j_i) \\ne \\emptyset$, otherwise\n$u_i$ cannot be mapped to $w^j_i$ and $x_i \\ne j$. Therefore, $\\pi$ obtained from a satisfiable\nassignment maps $N[u]$ bijectively to $N[\\pi(u)]$ and it is an isomorphism. The total number of\nvariables in $n$, and the total number of clauses is $\\calO(n+m)$, so the running time is\n$\\calO(n+m)$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:disconnected}\nLet $G_1,\\dots,G_k$ be the components of $G$ and $H_1,\\dots,H_k$ be the components of $H$.\nIf we can decide \\textsc{ListIso}\\xspace in polynomial time for all pairs $G_i$ and $H_j$, then we can\nsolve \\textsc{ListIso}\\xspace for $G$ and $H$ in polynomial time.\n\\end{lemma}\n\n\\begin{proof}\nLet $G_1,\\dots,G_k$ be the components of $G$ and $H_1,\\dots,H_k$ be the components of $H$.  For each\ncomponent $G_i$, we find all components $H_j$ such that there exists a list-compatible isomorphism\nfrom $G_i$ to $H_j$. Notice that a necessary condition is that every vertex in $G_i$ contains one\nvertex of $H_j$ in its list. So we can go through all lists of $G_i$ and find all candidates $H_j$,\nin total time $\\calO(\\ell)$ for all components $G_1,\\dots,G_k$.  Let $n' = |V(G_i)|$, $m' = |E(G_i)|$,\nand $\\ell'$ be the total size of lists of $G_i$ restricted to $H_j$. We test existence of a\nlist-compatible isomorphism in time $\\varphi(n',m',\\ell')$. Then we form the bipartite graph $B$\nbetween $G_1,\\dots,G_k$ and $H_1,\\dots,H_k$ such that $G_iH_j \\in E(B)$ if and only if there exists\na list-compatible isomorphism from $G_i$ to $H_j$.  There exists a list-compatible isomorphism from\n$G$ to $H$, if and only if there exists a perfect matching in $B$.  Using Lemma~\\ref{lem:bipmatch},\nthis can be tested in time $\\calO(\\sqrt k \\ell)$. The total running time depends on the running time of\ntesting \\textsc{ListIso}\\xspace of the components, and we note that the sum of the lengths of lists in these test is at\nmost $\\ell$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:cycles}\nThe problem \\textsc{ListIso}\\xspace can be solved for cycles in time $\\calO(\\ell)$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume that $|V(G)| = |V(H)|$.  Let $u \\in V(G)$ be a vertex with a smallest list and let $k\n= |\\frakL(u)|$.  Since $\\ell = \\calO(kn)$, it suffices to show that we can find a list-compatible\nisomorphism in time $\\calO(kn)$. We test all the $k$ possible mappings $\\pi \\colon G \\to H$ with\n$\\pi(u) \\in \\frakL(u)$. For $u \\in V(G)$ and $v \\in \\frakL(u)$, there are at most two possible isomorphisms\nthat map $u$ to $v$. For each of these isomorphism, we test whether they are list-compatible.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:max_degree_2}\nThe problem \\textsc{ListIso}\\xspace can be solved for graphs of maximum degree 2 in time $\\calO(\\sqrt n \\ell)$.\n\\end{lemma}\n\n\\begin{proof}\nBoth graphs $G$ and $H$ are disjoint unions of paths and cycles of various lengths.  For each two\nconnected components, we can decide in time $\\calO(\\ell')$  whether there exists a list-compatible\nisomorphism between them, where $\\ell'$ is the total size of lists when restricted to these\ncomponents: for paths trivially, and for cycles by Lemma~\\ref{lem:cycles}. The rest follows from\nLemma~\\ref{lem:disconnected}, where the running time is of each test in $\\calO(\\ell')$ where $\\ell'$ is\nthe total length of lists restricted to two components.\\qed\n\\end{proof}\n\n\n\\section{Bounded Genus Graphs} \\label{sec:bounded_genus}\n\nIn this section, we describe an FPT algorithm solving \\textsc{ListIso}\\xspace when parameterized by the Euler genus\n$g$. We modify the recent paper of Kawarabayashi~\\cite{kawarabayashi} solving graph isomorphism in\nlinear time for a fixed genus $g$. The harder part of this paper are structural results, described\nbelow, which transfer to list-compatible isomorphisms without any change. Using these structural\nresults, we can build our algorithm.\n\n\\begin{theorem} \\label{thm:bounded_genus}\nFor every integer $g$, the problem \\textsc{ListIso}\\xspace can be solved on graphs of Euler genus at most $g$ in time\n$\\calO(\\sqrt n \\ell)$.\n\\end{theorem}\n\n\\begin{proof}\nSee~\\cite[p.~14]{kawarabayashi} for overview of the main steps. We show that these steps can be\nmodified to deal with lists. We prove this result by induction on $g$, where the base case for $g=0$\nis Theorem~\\ref{thm:planar}. Next, we assume that both graphs $G$ and $H$ are 3-connected, otherwise\nwe apply Theorem~\\ref{thm:3conn_reduction}.  By~\\cite[Theorem 1.2]{kawarabayashi}, if $G$ and $H$\nhave no polyhedral embeddings, then the face-width is at most two.\n\t \n\\emph{Case 1: $G$ and $H$ have polyhedral embeddings.} Following~\\cite[Theorem 1.2]{kawarabayashi},\nwe have at most $f(g)$ possible embeddings of $G$ and $H$. We choose one embedding of $G$ and we\ntest all embeddings of $H$. It is known that the average degree is $\\calO(g)$.  Therefore, we can apply\nthe same idea as in the proof of Lemma~\\ref{lem:planar_3conn} and test isomorphism of all these\nembeddings in time $\\calO(\\ell)$. \n\n\\emph{Case 2: $G$ and $H$ have no polyhedral embedding, but have embeddings of face-width exactly two.}\nThen we split $G$ into a pair of graphs $(G',L)$. The graph $L$\nare called \\emph{cylinders} and the graph $G'$ correspond to the remainder of $G$. The following\nproperties hold~\\cite[p.~5]{kawarabayashi}:\n\\begin{packed_itemize}\n\\item We have $G = G' \\cup L$ and for $\\bo L = V(G' \\cap L)$, we have $|\\bo L| = 4$.\n\\item The graph $G'$ can be embedded to a surface of genus at most $g-1$, and $L$ is\nplanar~\\cite[p.~4]{kawarabayashi}.\n\\item This pair $(G',L)$ is canonical, i.e., every isomorphism from $G$ to $H$ maps $(G',L)$ to another\npair $(H',L')$ in $H$.\n\\end{packed_itemize}\nIt is proved~\\cite[Theorem 5.1]{kawarabayashi} that there exists some function $q'(g)$ bounding the\nnumber of these pairs both in $G$ and $H$, and can be found in time $\\calO(n)$. We fix a pair $(G',L)$\nin $G$ and iterate over all pairs $(H'_i,L'_i)$ in $H$.  Following~\\cite[p.~36]{kawarabayashi}, we\nget that $G \\cong H$, if and only if there exists a pair $(H'_i,L'_i)$ in $H$ such that $G' \\cong\nH'_i$, $L \\cong L'_i$, and $G' \\cap L$ is mapped to $H'_i \\cap L'_i$. To test this, we run at most\n$2q'(g)$ instances of \\textsc{ListIso}\\xspace on smaller graphs with modified lists.\n\nSuppose that we want to test whether $G' \\cong H'_i$ and $L \\cong L'_i$. First, we modify the lists:\nfor $u \\in V(G')$, put $\\frakL'(u) = \\frakL(u) \\cap H'_i$, and for $v \\in V(L)$, put $\\frakL'(v) = \\frakL(v) \\cap\nL'_i$, and similarly for lists of darts. Further, for all vertices $u \\in \\bo L$ in both $G'$ and\n$L$, we put $\\frakL'(u) = \\frakL(u) \\cap \\bo L$. We test existence of list-compatible isomorphisms from $G'$\nto $H'_i$ and from $L$ to $L'_i$. There exists a list-compatible isomorphism from $G$ to $H$, if and\nonly if these list-compatible isomorphisms exist at least for one pair $(H'_i,L'_i)$.\n\nWe note that when $g=2$, a special case is described in~\\cite[Theorem 5.3]{kawarabayashi}, which is\nslightly easier and can be modified similarly.\n\n\\emph{Case 3: $G$ and $H$ have no polyhedral embedding and have only embeddings of face-width one.}\nLet $V$ be the set of vertices in $G$ such that for each $u \\in V$, there exists a non-contractible\ncurve passing only through $u$. By~\\cite[Lemma 6.3]{kawarabayashi}, $|V| \\le q(g)$ for some function\n$q$. For $u$, the non-contractible curve divides its edges to two sides, so we can cut $G$ at $u$,\nand split the incident edges. We obtain a graph $G'$ which can be embedded to a surface of genus at\nmost $g-1$.\n\nBy~\\cite[Lemma 6.3]{kawarabayashi}, we can find all these vertices $V$ and $V'$ in $G$ and $H$ in\ntime $\\calO(n)$. We choose $u \\in V$ arbitrarily, and we test all possible vertices $v \\in V'$.\nLet $G'$ be constructed from $G$ by splitting $u$ into new vertices $u'$ and $u''$, and similarly\n$H'$ be constructed from $H$ by splitting $v$ into new vertices $v'$ and $v''$. \nIn~\\cite[p.~36]{kawarabayashi}, it is stated that $G \\cong H$, if and only if there exists a choice\nof $v \\in V'$ such that $G' \\cong H'$ and $\\{u',u''\\}$ is mapped to $\\{v',v''\\}$.\nTherefore, we run at most $q(g)$ instances of \\textsc{ListIso}\\xspace on smaller graphs with modified lists.\n\nIf $v \\notin L(u)$, clearly a list-compatible isomorphism is not possible for this choice of $v \\in\nV'$. If $v \\in L(u)$, we put $L'(u') = L'(u'') = \\{v',v''\\}$. Then there exists a list-compatible\nisomorphism from $G$ to $H$, if and only if there exists a list-compatible isomorphism from $G'$ to\n$H'$.\n\nThe correctness of our algorithm follows from~\\cite{kawarabayashi}.  It remains to argue the\ncomplexity. Throughout the algorithm, we produce at most $w(g)$ subgraphs of $G$ and $H$, for some\nfunction $w$, for which we test list-compatible isomorphisms. Assuming the induction hypothesis, the\nreduction of graphs to 3-connected graphs can be done in time $\\calO(\\sqrt n \\ell)$. Case 1 can be\nsolved in time $\\calO(\\ell)$.  Case 2 can be solved in time $\\calO(\\sqrt n \\ell)$. Case 3 can be solved in\ntime $\\calO(\\sqrt n \\ell)$.\\qed\n\\end{proof}\n\n\\section{Bounded Treewidth Graphs} \\label{sec:bounded_treewidth}\n\nIn this section, we prove that \\textsc{ListIso}\\xspace can be solved in \\hbox{\\rm \\sffamily FPT}\\xspace with respect to the parameter treewidth\n$\\tw(G)$. Unlike in Sections~\\ref{sec:planar_graphs} and~\\ref{sec:intersection}, the difficulty\nof graph isomorphism on bounded treewidth graphs raises from the fact that\ntree decomposition is not uniquely determined.  We follow the approach of\nBodlaender~\\cite{iso_xp_treewidth} which describes an \\hbox{\\rm \\sffamily XP}\\xspace algorithm for \\textsc{GraphIso}\\xspace of bounded treewidth\ngraphs, running in time $n^{\\calO(\\tw(G))}$. Then we show that the recent breakthrough by Lokshtanov\net al.~\\cite{iso_fpt_treewidth}, giving an \\hbox{\\rm \\sffamily FPT}\\xspace algorithm for \\textsc{GraphIso}\\xspace, translates as well.\n\n\\begin{definition}\nA \\emph{tree decomposition} of a graph $G$ is a pair ${\\cal T} =\n(\\{B_i\\colon i\\in I\\},T = (I,F)),$ where $T$ is a rooted tree and $\\{B_i\\colon\ni\\in I\\}$ is a family of subsets of $V,$ such that\n\\begin{enumerate}\n\\item for each $v\\in V(G)$ there exists an $i \\in I$ such that $v\\in B_i$,\n\\item for each $e\\in E(G)$ there exists an $i \\in I$ such that $e\\subseteq B_i$,\n\\item for each $v\\in V(G), I_v = \\{i \\in I\\colon v\\in B_i\\}$ induces a subtree of $T.$\n\\end{enumerate} \nWe call the elements $B_i$ the \\emph{nodes}, and the elements of the set $F$ the\ndecomposition edges.\n\\end{definition}\n\nWe define the width of a tree decomposition ${\\cal T} = (\\{B_i\\colon i\\in I\\},\nT)$ as $\\max_{i\\in I}|B_i|-1$ and the \\emph{treewidth} $\\tw(G)$ of a graph $G$\nas the minimum width of a tree decomposition of the graph $G$. \n\n\\heading{Nice Tree Decompositions.}\nIt is common to define a {\\it nice tree decomposition} of the graph~\\cite{kloks}.  We naturally\norient the decomposition edges towards the root and for an oriented decomposition edge $(B_j,B_i)$\nfrom $B_j$ to $B_i$ we call $B_i$ the {\\it parent} of $B_j$ and $B_j$ a {\\it child} of $B_i$.  If\nthere is an oriented path from $B_j$ to $B_i$ we say that $B_j$ is a {\\it descendant} of $B_i$.\n\nWe also adjust a tree decomposition such that for each decomposition edge $(B_i,B_j)$ it holds that\n$\\big| |B_i|-|B_j| \\big| \\le 1$ (i.e. it joins nodes that differ in at most one vertex). The\nin-degree of each node is at most $2$ and if the in-degree of the node $B_k$ is $2$ then for its\nchildren $B_i,B_j$ holds that $B_i = B_j = B_k$ (i.e. they represent the same vertex set).\n\nWe classify the nodes of a nice decomposition into four classes---namely {\\it introduce nodes}, {\\it\nforget nodes}, {\\it join nodes} and {\\it leaf nodes}. We call the node $B_i$ an introduce node of\nthe vertex $v$, if it has a single child $B_j$ and $B_i\\setminus B_j = \\{v\\}$. We call the node\n$B_i$ a forget node of the vertex $v$, if it has a single child $B_j$ and $B_j\\setminus B_i =\n\\{v\\}$. If the node $B_k$ has two children, we call it a join node (of nodes $B_i$ and $B_j$).\nFinally we call a node $B_i$ a leaf node, if it has no child.\n\n\\heading{Bodlaender's Algorithm.}\nA graph $G$ has treewidth at most $k$ if either $|V(G)| \\le k$, or there exists a cut set $U\n\\subseteq V(G)$ such that $|U| \\le k$ and each component of $G \\setminus U$ together with $U$ has\ntreewidth at most $k$.  The set $U$ corresponds to a bag in some tree decomposition of $G$.\nBodlaender's algorithm~\\cite{iso_xp_treewidth} enumerates all possible cut sets $U$ of size at most\n$k$ in $G$ (resp. $H$), we denote these $C_i$ (resp. $D_i$). Furthermore, it enumerates all\nconnected components of $G\\setminus C_i$ as $C_i^j$ (resp.~of $H \\setminus D_i$ as $D_i^j$). We\ndenote by $G[U,W]$ the graph induced by $U \\mathbin{\\dot\\cup} W$.  The set $W$ is either a connected component\nor a collection of connected components. We call $U$ the \\emph{border set}.\n\n\\begin{lemma}[\\cite{ACP87:partialktrees,iso_xp_treewidth}]\n\\label{lem:partialKTree}\nA graph $G[U, W]$ with at least $k$ vertices has a treewidth at most $k$ with the border set $U$ if\nand only if there exists a vertex $v\\in W$ such that for each connected component $A$ of $G[W\n\\setminus v]$, there is a $k$-vertex cut $C_s \\subseteq U\\cup\\{v\\}$ such that no vertex in $A$ is\nadjacent to the (unique) vertex in $(U \\cup\\{v\\})\\setminus C_s$, and $G[C_s, A]$ has treewidth at\nmost $k$.\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:xp_treewidth}\nThe problem \\textsc{ListIso}\\xspace can be solved in \\hbox{\\rm \\sffamily XP}\\xspace with respect to the parameter treewidth.\n\\end{lemma}\n\n\\begin{proof}\nWe modify the algorithm of Bodlaender~\\cite{iso_xp_treewidth}.  Let $k = \\tw(G) = \\tw(H)$. We\ncompute the sets $C_i, C_i^j$ for $G$ and the sets $D_{i'}, D_{i'}^{j'}$ for $H$; there are\n$n^{\\calO(k)}$ pairs $(C_i,C_i^j)$.  The pair $(C_i, C_i^j)$ is \\emph{compatible} if $C_i^j$ is a\nconnected component of $G'\\setminus C_i$ for some $G'\\subseteq G$ that arises during the recursive\ndefinition of treewidth. Let $f \\colon C_i\\to D_{i'}$ be an isomorphism. We say that $(C_i,\nC_i^j)\\equiv_f(D_{i'},D_{i'}^{j'})$ if and only if there exists an isomorphism $\\varphi\\colon\nC_i\\cup C_i^j\\to D_{i'}\\cup D_{i'}^{j'}$ such that $\\varphi|_{C_i} = f$.  In other words, $\\varphi$\nis a partial isomorphism from $G$ to $H$.  The change for \\textsc{ListIso}\\xspace is that we also require that both $f$\nand $\\varphi$ are list-compatible.\n\nThe algorithm resolves $(C_i, C_i^j)\\equiv_f(D_{i'},D_{i'}^{j'})$ by the dynamic programming,\naccording to the size of $D_{i'}^{j'}$. If $|C_i^j| = |D_{i'}^{j'}| \\le 1$, we can check it\ntrivially in time $k^{\\calO(k)}$. Otherwise, suppose that $|C_i^j| = |D_{i'}^{j'}| > 1$, and let\n$m$ be the number of components of $C_i^j$ (and thus $D_{i'}^{j'}$). We test whether $f : C_i \\to\nD_{i'}$ is a list-compatible isomorphism.  Let $v\\in C_i^j$ be a vertex given by\nLemma~\\ref{lem:partialKTree} (with $U = C_i$ and $W = C_i^j$) and let $C_s$ be the corresponding\nextension of $v$ to a cut set. We compute for all $w \\in D_{i'}^{j'} \\cap \\frakL(v)$ all connected\ncomponents $B_q$. From the dynamic programming, we know for all possible extensions $D'$ of $w$ to a\ncut set whether $(C_m, A_p)\\equiv_{f'}(D',B_q)$ with $f'(x) = f(x)$ for $x\\in C_i$ and $f'(v) = w$.\nFinally, we decide whether there exists a perfect matching in the bipartite graph between $(C_m,\nA_p)$'s and $(D',B_q)$'s where the edges are according to the equivalence.\\qed\n\\end{proof}\n\n\\heading{Reducing The Number of Possible Bags.}\nOtachi and Schweitzer~\\cite{GIReductionTechniques} proposed the idea of pruning the family of\npotential bags which finally led to an \\hbox{\\rm \\sffamily FPT}\\xspace algorithm~\\cite{iso_fpt_treewidth}.  A family\n$\\mathcal{B}(G)$, whose definition depends on the graph, is called \\emph{isomorphism-invariant} if\nfor an isomorphism $\\phi : G \\to G'$, we get $\\mathcal{B}(G') = \\phi(\\mathcal{B}(G))$, where\n$\\phi(\\mathcal{B}(G))$ denotes the family $\\mathcal{B}(G)$ with all the vertices of $G$ replaced by\ntheir images under $\\phi$. \n\nFor a graph $G$, a pair $(A,B)$ with $A \\cup B = V$ is called a {\\em\nseparation} if there are no edges between $A\\setminus B$ and $B\\setminus A$ in\n$G$. The order of $(A,B)$ is $|A\\cap B|$.  For two vertices $u,v \\in V(G)$, by\n$\\mu(u,v)$ we denote the minimum order of separation $(A,B)$ with $u\\in\nA\\setminus B$ and $v\\in B\\setminus A$.  We say a graph $G$ is {\\em\n$k$-complemented} if $\\mu_G(u,v) \\ge k) \\implies uv \\in E(G)$ holds for\nevery two vertices $u,v\\in V$. We may canonically modify the input graphs $G$ and $H$ \\textsc{ListIso}\\xspace, by\nadding these additional edges and making them $k$-complemented.\n\n\n\\begin{theorem}[\\cite{iso_fpt_treewidth}, Theorem~5.5] \\label{thm:pruned_bags}\nLet $k$ be a positive integer, and let $G$ be a graph on $n$ vertices that is connected and\n$k$-complemented. There exists an algorithm that computes in time $2^{\\calO(k^5\\log k)}\\cdot n^3$ an\nisomorphism-invariant family of bags $\\mathcal{B}$ with the following properties:\n\\begin{enumerate}\n  \\item $|B|\\le \\calO(k^4)$ for each $B\\in\\mathcal{B}$,\n  \\item $|\\mathcal{B}|\\le 2^{\\calO(k^5\\log k)}\\cdot n^2$,\n  \\item Assuming $\\tw(G)<k$, the family $\\mathcal{B}$ captures some tree\n        decomposition of $G$ that has width $\\calO(k^4)$.\n  \\item The family $\\mathcal{B}$ is closed under taking subsets.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:fpt_treewidth}\nThe problem \\textsc{ListIso}\\xspace can be solved in \\hbox{\\rm \\sffamily FPT}\\xspace time $2^{\\calO(k^5 \\log k)} n^5$ where $k = \\tw(G)$.\n\\end{theorem}\n\n\\begin{proof}\nWe use the algorithm of Lemma~\\ref{lem:xp_treewidth}, where $C_i$'s and $D_i$'s are from the\ncollection $\\calB$ of Theorem~\\ref{thm:pruned_bags}.  The total number of pairs $(C_i,C_i^j)$ and\n$(D_{i'},D_{i'}^{j'})$ is bounded by $2^{\\calO(k^5 \\log k)} n^3$~\\cite[p. 20]{iso_fpt_treewidth}.  The\ndynamic programming in~\\cite[Theorem 6.2]{iso_fpt_treewidth} is done according to the potential\nfunction $\\Phi(D_{i'},D_{i'}^{j'}) = 2|D_{i'}^{j'}| + |D_{i'}|$. We use nice tree decompositions,\nso in each step, the dynamic programming either introduces a new node into the bag $D_{i'}$, or\nmoves a node from the bag $D_{i'}$ to $D_{i'}^{j'}$, or joins several pairs with the same bag\n$D_{i'}$. In all these operations, we check existence of a list-compatible isomorphism, using\ndynamic programming, exactly as in Lemma~\\ref{lem:xp_treewidth}.\\qed\n\\end{proof}\n\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nWe conclude this paper with description of related results and open problems.\n\n\\heading{Forbidden Images.}\nWe note that Lubiw~\\cite{lubiw1981some} used a different definition of \\textsc{ListIso}\\xspace: for every vertex $u \\in\nV(G)$, we are given a \\emph{list of forbidden images} $\\calF(u) \\subseteq V(H)$ and we want to find\nan isomorphism $\\pi : G \\to H$ such that $\\pi(u) \\notin \\calF(u)$. The advantage of forbidden lists\nis that we can express \\textsc{GraphIso}\\xspace in space $\\calO(n+m)$, but the input for \\textsc{ListIso}\\xspace is of size $\\calO(n^2)$. On\nthe other hand, we consider lists of allowed images more natural (for instance, list coloring is\ndefined similarly) and also such a definition appears naturally in~\\cite{fkkn}. Both statements are\nclearly polynomially equivalent, and the main focus of our paper is to distinguish between tractable\nand intractable cases for \\textsc{ListIso}\\xspace.\n\n\\heading{Group Reformulation.}\nLuks~\\cite{luks1982isomorphism} described the following group problem which\ngeneralizes computing automorphism groups of graphs. Let $\\Omega$ be a ground\nset and let $\\Gamma$ be a group acting on $\\Omega$.  Further, let $\\Omega$ be\ncolored. We want to compute the subgroup of $\\Gamma$ which is color preserving.\nWhen $\\Gamma$ is the symmetric group acting on all pairs of vertices $V(G)$\nwhich are colored by two colors (corresponding to edges and non-edges in $G$),\nthen the computed subgroup is ${\\rm Aut}(G)$. To generalize graph isomorphism in\nthis language, we have two colorings and we want to find a color-preserving\npermutation $g \\in \\Gamma$. We note that Babai~\\cite{babai_quasipoly} calls\nthese generalization as the \\emph{string automorphism\/isomorphism problems}.\n\nA similar generalization of \\textsc{ListIso}\\xspace was suggested to us by Ponomarenko. We are given a group\n$\\Gamma$ acting on a ground set $\\Omega$ and for every $x \\in \\Omega$, we have a list $\\frakL(x)\n\\subseteq \\Omega$. We ask whether there exists a permutation $g \\in \\Gamma$ such that $g(x) \\in\n\\frakL(x)$ for every $x \\in \\Omega$. We obtain \\textsc{ListIso}\\xspace either similarly as above, or when $\\Omega = V(G)$\nand $\\Gamma = {\\rm Aut}(G)$.\n\nWe may interpret our results for \\textsc{ListIso}\\xspace using this group reformulation.  The robust combinatorial\nalgorithms work because the groups $\\Gamma$ are highly restricted. In particular, for trees,\nJordan~\\cite{jordan1869assemblages} proved that ${\\rm Aut}(G)$ is formed by a series of direct products\nand wreath products with symmetric groups, to it has a tree structure. Therefore, the algorithm of\nTheorem~\\ref{thm:trees} solves \\textsc{ListIso}\\xspace on ${\\rm Aut}(G)$ by a bottom-up dynamic algorithm.  Similar\ncharacterizations were recently proved for interval, permutation and circle graphs~\\cite{kz,kz15}\nand for planar graphs~\\cite{knz}, and are used in the algorithms of Theorems~\\ref{thm:planar},\n\\ref{thm:int}, \\ref{thm:perm}, and~\\ref{thm:circle}. For graphs of bounded genus or bounded\ntreewidth, no such detailed description of the automorphism groups is yet known, but they are likely\nrestricted as well. On the other hand, for cubic graphs, the automorphism groups ${\\rm Aut}(G)$ may be\narbitrary, so this approach fails, and actually \\textsc{ListIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete by Theorem~\\ref{thm:3reg_npc}.\n\nTo attack \\textsc{ListIso}\\xspace from the point of this group reformulation, instead of different graph classes, we\nmay study it for different combinations of $\\Gamma$ and the lists $\\frakL$. First, for which\ngroups $\\Gamma$, it can be solved efficiently for all possible lists $\\frakL$? Second, for which lists\n$\\frakL$, the problem can be solved efficiently? We did not try to attack the problem much\nin the second direction, aside Lemma~\\ref{lem:list_size_two} and Theorem~\\ref{thm:3reg_npc}. For\ninstance, when $\\frakL$ is a partitioning of $\\Omega$, the problem is easy since we get the usual\ncolor-preserving isomorphism problem.\n\n\\heading{$\\boldsymbol k$-dimensional Weisfieler-Leman refinement ($\\boldsymbol k$-WL).} The classical\n$2$-WL~\\cite{wl,weisfeiler} colors vertices of a graph and it initiates with different colors for\neach degree. In each step, it takes vertices of one color class and partitions them by different\nnumbers of neighbors of other color classes. It stops when no partitioning longer occurs. Its\ngeneralization $k$-WL~\\cite{babai77,immerman_lander} colors and partitions $(k-1)$-tuples according\nto their adjacencies.\n\nCertainly, when $G$ and $H$ are isomorphic graphs, they are partitioned and\ncolored the same.  So when $G \\not\\cong H$, $k$-WL distinguishes $G$ and $H$,\nfor a suitable value of $k$. If we prove for a graph class that $k$ is small\nenough, we obtain a combinatorial algorithm for \\textsc{GraphIso}\\xspace.  For instance,\nGrohe~\\cite{grohe_graph_structure} proves that for every graph $X$, there\nexists a value $k$ such that two $X$-minor free graphs are either isomorphic,\nor distinguished by $k$-WL.  This does not translate to \\textsc{ListIso}\\xspace since $k$-WL\napplied on $G$ only estimates the orbits of ${\\rm Aut}(G)$.  When $G \\cong H$, and\nwe may test this, assuming that \\textsc{GraphIso}\\xspace can be decided efficiently for $G$ and $H$,\nwe obtain two identical partitions. They may be used to reduce sizes of the\nlists, but we still end up with the question whether there exists a\nlist-compatible isomorphism. In Section~\\ref{sec:group_theory}, we show that it\nis \\hbox{\\rm \\sffamily NP}\\xspace-complete to decide \\textsc{ListIso}\\xspace even when sizes of all lists are bounded by 3.\n\n\\heading{Bounded Rankwidth.} Rankwidth generalizes treewidth in the way that bounded treewidth\nimplies bounded rankwidth, but not the opposite. Very recently, the first \\hbox{\\rm \\sffamily XP}\\xspace algorithm for graph\nisomorphism of graphs of bounded rankwidth was described~\\cite{bounded_rankwidth}.  The approach is\nby computing automorphism-invariant tree decomposition (which should translate to \\textsc{ListIso}\\xspace), but then\nfurther group theory is applied to test whether the decompositions are isomorphic.  It is a very\ninteresting question whether group theory can be avoided and the problem can be solved in a purely\ncombinatoric way. Therefore, determining the complexity of \\textsc{ListIso}\\xspace for graphs of bounded rankwidth is\none of major open problems and it might give an insight into this question as well.\n\nAlso, rankwidth is closely related to cliquewidth: when one parameter is bounded, the other is\nbounded as well. The graphs of cliquewidth at most 2 are called \\emph{cographs} and can be\nrepresented by \\emph{cotrees}. Their isomorphism can therefore be reduced to isomorphism of cotrees\nand solved in polynomial time~\\cite{cographs}, and this approach should translate to \\textsc{ListIso}\\xspace. Very\nrecently, a combinatorial polynomial-time algorithm for graph isomorphism of graphs of cliquewidth\nat most 3 was described~\\cite{cliquewidth_three} which might translate to \\textsc{ListIso}\\xspace as well.\n\n\\heading{Excluded Minors.} Another major open problem for \\textsc{ListIso}\\xspace is its complexity for graphs with\nexcluded minors. As described in Introduction, the original polynomial-time algorithm of\nPonomarenko~\\cite{ponomarenko_minor} for \\textsc{GraphIso}\\xspace is based heavily on group theory, and his technique\nunlikely translates to \\textsc{ListIso}\\xspace. But it seems doubtful that the problem will be \\hbox{\\rm \\sffamily NP}\\xspace-complete, since\nnew combinatorial structural and algorithmic results may be applied.\n\nRobertson and Seymour~\\cite{robertson_seymour} proved that every graph $G$ with an excluded minor\ncan be decomposed into pieces which are ``almost embeddable'' to a surface of genus $g$, where $g$\ndepends on the minor. The recent book of Grohe~\\cite{grohe_graph_structure} describes the seminal\nidea of automorphism-invariant treelike decompositions. A \\emph{treelike decomposition} generalizes\na classical tree decomposition by replacing a tree of bags by a directed acyclic graph of bags.\nUnlike tree decompositions, a treelike decomposition $D$ of $G$ can be constructed with two\nadditional properties. Firstly, it is \\emph{automorphism-invariant}, meaning that every automorphism of $G$\ninduces an automorphism of $D$. Secondly, it is \\emph{canonical}, meaning that for two isomorphic graphs\n$G$ and $G'$, isomorphic treelike decompositions $D$ and $D'$ are constructed. \n\nThe main structural result of Grohe~\\cite{grohe_graph_structure} is that every graph $G$ with an\nexcluded minor has a canonical automorphism-invariant treelike decomposition for which the graphs\ninduced by bags called \\emph{torsos} are ``almost embeddable'' to a surface of genus $g$, where $g$\ndepends on the minor. Therefore, to solve \\textsc{ListIso}\\xspace on graphs with excluded minors, we need to prove the\nfollowing:\n\\begin{packed_enum}\n\\item We need to show that \\textsc{ListIso}\\xspace can be solved on almost embeddable graphs in polynomial time. We may\nuse the results of Theorems~\\ref{thm:bounded_genus} and~\\ref{thm:fpt_treewidth} to do so.\n\\item We need to prove Lifting Lemma for \\textsc{ListIso}\\xspace, stating the following: If we can compute a canonical\nautomorphism-invariant treelike decomposition of a class $\\calC$ in polynomial time and we can solve\n\\textsc{ListIso}\\xspace for its torsos ${\\mathcal A}} \\def\\calB{{\\mathcal B}} \\def\\calC{{\\mathcal C}$ in polynomial time, then we can solve \\textsc{ListIso}\\xspace for $\\calC$ in polynomial\ntime as well. Here, we might modify the algorithm for lifting of canonization by\nGrohe~\\cite{grohe_graph_structure}.\n\\end{packed_enum}\n\nWe note that it is quite difficult to understand and describe everything in detail. The book of\nGrohe~\\cite{grohe_graph_structure} is very extensive (almost 500 pages) and described in the\nlanguage of graph logics. Unfortunately, no purely combinatorial description of the results is\navailable, and we believe that such a description of a combinatorial algorithm for solving graph\nisomorphism of graphs with excluded minors would be desirable.  A combinatorial description of\ntreelike decompositions is described by Grohe and Marx~\\cite{grohe_marx}, but an algorithm for the\ngraph isomorphism problem of graphs with excluded minors is only used as a black box.\n\n\\heading{Bounded Eigenvalue Multiplicity.} The polynomial-time algorithms for \\textsc{GraphIso}\\xspace of graphs of\nbounded eigenvalue multiplicity~\\cite{babai_eig_multiplicity,ponomarenko_eig_multiplicity} are\nheavily based on group theory. Actually, already the case of multiplicity one is non-trivial. It\nseems unlikely that these results will translate to \\textsc{ListIso}\\xspace, but constructing an \\hbox{\\rm \\sffamily NP}\\xspace-hardness\nreduction with bounded eigenvalue multiplicity seems non-trivial.\n\n\\heading{Forbidden Subgraphs, Induced Subgraphs and Induced Minors.} There are several papers\ndealing with \\textsc{GraphIso}\\xspace for classes of graphs with excluded subgraphs, induced subgraphs and induced\nminors, and again the question is which results translate to \\textsc{ListIso}\\xspace. Otachi and\nSchweitzer~\\cite{otachi_schweitzer_subgraph} prove a dichotomy for excluded subgraphs. The\n\\hbox{\\rm \\sffamily GI}\\xspace-complete cases translate by Theorem~\\ref{thm:vertex_gadget_reductions}, but the polynomial\ncases follow from~\\cite{grohe_marx} which does not seem to translate. More complicated\ncharacterizations are known for forbidden induced\nsubgraphs~\\cite{booth_colbourn_GI,kratsch_schweitzer,schweitzer_dichotomy}.  Belmonte et\nal.~\\cite{belmonte_otachi_schweitzer} describe dichotomy for forbidden induced minors.\n\n\\heading{Logspace Results.} For some graph classes, \\textsc{GraphIso}\\xspace is known to be solvable in \\hbox{\\rm \\sffamily LogSpace}\\xspace and\nother subclasses of \\hbox{\\rm \\sffamily P}\\xspace. It is a natural question to ask whether these results translate to \\textsc{ListIso}\\xspace.\nFor instance, graph isomorphism of trees~\\cite{tree_log_iso} can be solved in \\hbox{\\rm \\sffamily LogSpace}\\xspace, with a\nsimilar bottom-up procedure as in the proof of Theorem~\\ref{thm:trees}. The celebrated result of\nReingold~\\cite{reingold}, stating that undirected reachability can be solved in \\hbox{\\rm \\sffamily LogSpace}\\xspace, allowed\nmany other graph algorithms to be translated to \\hbox{\\rm \\sffamily LogSpace}\\xspace. In particular, \\textsc{GraphIso}\\xspace is known to be solvable\nin \\hbox{\\rm \\sffamily LogSpace}\\xspace for planar graphs~\\cite{planar_3conn_log_iso,planar_log_iso},\n$k$-trees~\\cite{ktree_log_iso}, interval graphs~\\cite{int_log_iso}, and bounded treewidth\ngraphs~\\cite{treewidth_log_iso}.\n\n\n\\section{Graphs with Forbidden Minors} \\label{sec:forbidden_minors}\n\nIn this section, we sketch a modification of graph isomorphism testing for graph classes with\nexcluded minors of Grohe~\\cite{grohe_graph_structure} to \\textsc{ListIso}\\xspace.\n\n\\section{GI-completeness Implies NP-completeness} \\label{sec:gi_completeness}\n\nSuppose that graph isomorphism is \\hbox{\\rm \\sffamily GI}\\xspace-complete for some class of graphs $\\calC'$. We want to show\nthat in most cases, this translates in \\hbox{\\rm \\sffamily NP}\\xspace-completeness of \\textsc{ListIso}\\xspace for $\\calC'$.\n\n\\heading{Vertex-gadget Reductions.}\nSuppose that \\textsc{GraphIso}\\xspace is \\hbox{\\rm \\sffamily GI}\\xspace-complete for a class $\\calC$. To show that \\textsc{GraphIso}\\xspace is \\hbox{\\rm \\sffamily GI}\\xspace-complete for another\nclass $\\calC'$, one builds a polynomial-time reduction $\\psi$ from \\textsc{GraphIso}\\xspace of $\\calC$:\ngiven graphs $G, H \\in \\calC$, we construct graphs $G',H' \\in \\calC'$ in\npolynomial time such that $G \\cong H$ if and only if $G' \\cong H'$. We say that $\\psi$ uses\n\\emph{vertex-gadgets}, if to every vertex $u \\in V(G)$ (resp. $u \\in V(H)$), it \nassigns a \\emph{vertex-gadget} ${\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u$, and these gadgets are \nsubgraphs of $G'$ (resp. of $H'$), and satisfies the following two conditions:\n\\begin{packed_enum}\n\\item Every isomorphism $\\pi : G \\to H$ induces an isomorphism $\\pi' : G' \\to H'$ such that $\\pi(u)\n= v$ implies $\\pi'({\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u) = {\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_v$.\n\\item Every isomorphism $\\pi' : G' \\to H'$ maps vertex-gadgets to vertex-gadgets and induces an\nisomorphism $\\pi : G \\to H$ such that $\\pi'({\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u) = {\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_v$ implies $\\pi(u) = v$.\n\\end{packed_enum}\n\n\\begin{theorem} \\label{thm:vertex_gadget_reductions}\nLet $\\calC$ and $\\calC'$ be classes of graphs. Suppose that there exists a polynomial-time reduction\n$\\psi$ using vertex-gadgets from \\textsc{GraphIso}\\xspace of $\\calC$ to \\textsc{GraphIso}\\xspace of $\\calC'$.  Then there exists a\npolynomial-time reduction from \\textsc{ListIso}\\xspace of $\\calC$ to \\textsc{ListIso}\\xspace of $\\calC'$.\n\\end{theorem}\n\n\\begin{proof}\nLet $G,H \\in \\calC$ be an instance of \\textsc{ListIso}\\xspace. Using the reduction $\\psi$, we construct the\ncorresponding graphs $G',H' \\in \\calC'$ with vertex-gadgets. We need to add lists for $V(G')$, we\ninitiate them empty. Let $u \\in V(G)$. To all vertices $w$ of ${\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u$, we add $\\bigcup_{v \\in\n\\frakL(u)} V({\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_v)$ to $\\frakL(w)$. For the vertices of $G'$ outside vertex-gadgets, we set the lists\nequal to the union of all remaining vertices of $H'$.\n\nWe want to argue that there exists a list-compatible isomorphism $\\pi' : G' \\to H'$, if and only if\nthere exists a list-compatible isomorphism $\\pi : G \\to H$. If $\\pi$ exists, by the first assumption\nof the reduction, it induces $\\pi'$ which is list-compatible by our construction of lists. On the\nother hand, suppose that there exists a list-compatible isomorphism $\\pi'$. By the second\nassumption, $\\pi'$ maps vertex-gadgets to vertex-gadgets and induces an isomorphism $\\pi : G \\to H$\nwhich is list-compatible by our construction.\\qed\n\\end{proof}\n\n\\begin{corollary} \\label{cor:gi_completeness_translates}\nLet $\\calC$ be a class of graphs with \\hbox{\\rm \\sffamily NP}\\xspace-complete \\textsc{ListIso}\\xspace. Suppose that there exists a reduction\n$\\psi$ using vertex-gadgets from \\textsc{GraphIso}\\xspace of $\\calC$ to \\textsc{GraphIso}\\xspace of $\\calC'$. Then \\textsc{ListIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete for\n$\\calC'$.\\qed\n\\end{corollary}\n\nAmong others, this implies \\hbox{\\rm \\sffamily NP}\\xspace-completeness of \\textsc{ListIso}\\xspace for the following graph classes:\n\n\\begin{corollary} \\label{cor:NPc_lgi}\nThe problem \\textsc{ListIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete for bipartite graphs, split and chordal graphs, chordal bipartite\nand strongly chordal graphs, trapezoid graphs, comparability graphs of dimension 4, grid intersection\ngraphs, and line graphs.\n\\end{corollary}\n\n\\begin{proof}\nWe use Corollary~\\ref{cor:gi_completeness_translates} together with Theorem~\\ref{thm:lubiw}.\nWe briefly describe \\hbox{\\rm \\sffamily GI}\\xspace-hardness reductions for every mentioned class. It is easy to check that,\nexcept for line graphs, all these reductions use vertex-gadgets, where ${\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u = \\{u\\}$ for every\n$u \\in V(G) \\cup V(H)$.\n\\begin{packed_itemize}\n\\item \\emph{Bipartite graphs.} Assuming the graphs are not cycles, we subdivide every edge in the\ninput graphs $G$ and $H$.\n\\item \\emph{Split and chordal graphs~\\cite{lueker1979linear}.} We subdivide every edge in $G$ and\n$H$ and add the complete graphs on the original vertices.\n\\item \\emph{Chordal bipartite and strongly chordal graphs~\\cite{strongly_chordal}.} For bipartite\ngraphs $G$ and $H$, we subdivide all edges $e_i$ twice, by adding vertices $a_i$ and $b_i$, we add paths\nof length three from $a_i$ to $b_i$, and we add the complete bipartite graph between $a_i$'s and\n$b_i$'s.\n\\item \\emph{Trapezoid graphs~\\cite{trapezoid}.} For bipartite graphs $G$ and $H$, we subdivide every\nedge and add the complete bipartite graph on the original vertices.\n\\item \\emph{Comparability graphs of dimension at most 4~\\cite{kz15}.} Assuming the graphs are not\ncycles, we replace every edge in $G$ and $H$ by a path of length 8.\n\\item \\emph{Grid intersection graphs~\\cite{grid_GI_complete}.} For bipartite graphs $G$ and $H$, we\nsubdivide every edge twice and add the complete bipartite graph on the original vertices.\n\\item \\emph{Line graphs~\\cite{line_graphs}.} Assuming the graphs are not $K_3$ and $K_{1,3}$, we\nconsider $G'$ and $H'$ being the line graphs of $G$ and $H$. For every $u \\in V(G)$, we put ${\\mathcal V}} \\def\\calW{{\\mathcal W}} \\def\\calX{{\\mathcal X}_u\n= \\{e : e \\in E(G), u \\in e\\}$, and similarly for $u \\in V(H)$. By Whitney Theorem~\\cite{line_graphs}, $G\n\\cong H$ if and only if $G' \\cong H'$, and it is easy to observe that it is a reduction using\nvertex-gadgets.\n\\qed\n\\end{packed_itemize}\n\\end{proof}\n\n\\section{Group Theory Techniques Do Not Translate} \\label{sec:group_theory}\n\nUsing group theory techniques, graph isomorphism can be solved in polynomial time for graphs of\nbounded degree~\\cite{luks1982isomorphism} and for colored graphs with color classes of bounded\nsize~\\cite{furst_hopcroft_luks}. In this section, we modify the reduction of\nLubiw~\\cite{lubiw1981some} to show that \\textsc{ListIso}\\xspace remains \\hbox{\\rm \\sffamily NP}\\xspace-complete even for 3-regular colored graphs\nwith color classes of size at most 16 and each list of size at most 3.\n\n\\subsection{Modifying the Reduction of Lubiw} \\label{sec:lubiw}\n\nWe modify the original \\hbox{\\rm \\sffamily NP}\\xspace-hardness reduction of \\textsc{ListIso}\\xspace by Lubiw~\\cite{lubiw1981some}. The original\nreduction is from 3-SAT, but instead we use \\emph{1-in-3 SAT} which is \\hbox{\\rm \\sffamily NP}\\xspace-complete by\nSchaefer~\\cite{1in3_sat}: all literals are positive, each clause is of size $3$ and a satisfying\nassignment has exactly one true literal in each clause.  We show that an instance of 1-in-3 SAT can\nbe solved using \\textsc{ListAut}\\xspace.\n\n\\heading{Variable Gadget.}\nFor each variable $u_i$, we construct the \\emph{variable gadget} $H_i$ which is a 4-cycle with the\nvertices labeled as in Fig.~\\ref{fig:truth_gadget}, and let $H$ be the disjoint union of these\ncycles. Consider two automorphisms of $H_i$: the $180^{\\circ}$ rotation $\\alpha_i$ and the\nreflection $\\beta_i$. The automorphism $\\alpha_i$ swaps $u_i(j)$ with $u'_i(1-j)$ while the\nautomorphism $\\beta_i$ swaps $u_i(j)$ with $u'_i(j)$, for $j = 0,1$.  To a vertex $v \\in V(H_i)$, we\nassign the list $\\frakL(v) = \\{\\alpha_i(v),\\beta_i(v)\\}$.\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics{group_theory_truth_gadget}\n\\caption{Variable and clause gadgets.}\n\\label{fig:truth_gadget}\n\\end{figure}\n\n\\heading{Clause Gadget.} Let $c_j$ be a clause with the literals $q_j$, $r_j$, and $s_j$.  For every\nsuch clause $c_j$, the \\emph{clause gadget} $G_j$ consists of the isolated vertices $c_j(0), \\dots,\nc_j(7)$. For every $k = 0,\\dots,7$, we consider its binary representation $k = abc_2$, for $a,b,c \\in\n\\{0,1\\}$. We add an edge between $c_j(k)$ and the vertices $q_j(a), q_j'(a), r_j(b), r_j'(b),\ns_j(c), s_j'(c)$ (these vertices belong to variable gadgets); see Fig.~\\ref{fig:truth_gadget}b. We\nassign the list $$\\frakL(c_j(k)) = \\{c_j(k \\oplus 100_2), c_j(k \\oplus 010_2), c_j(k \\oplus 001_2)\\},$$ where\n$\\oplus$ denotes the bitwise XOR; i.e., $\\frakL(c_j(k))$ contains all $c_j(k')$ in which $k'$\ndiffers from $k$ in exactly one bit.  Let $G$ be the resulting graph.\n\n\\begin{lemma} \\label{lem:unique_extension}\nSuppose that $\\pi'$ is a partial automorphism obtained by choosing $\\alpha_i$ or $\\beta_i$ on each\nvariable gadget $H_i$. There exists a unique automorphism $\\pi$ extending $\\pi'$ such that\n$\\pi(G_j) = G_j$.\n\\end{lemma}\n\n\\begin{proof}\nLet $c_j$ be a clause with the literals $q_j$, $r_j$, and $s_j$. We claim that $\\pi(c_j(k))$ is\ndetermined by the images of its neighbors.  Recall that $\\beta_i$ preserves the numbers in brackets\nof $H_i$, but $\\alpha_i$ swaps them. Therefore, two neighbors of $\\pi(c_j(k))$ are different from\nthe neighbors of $c_j(k)$ for every application of $\\alpha_i$ on $q_j$, $r_j$ and $s_j$. Let $p =\nabc_2$ such that $a = 1$, $b=1$ and $c=1$ if and only if $\\alpha_i$ is applied on the variable\ngadget of $q_j$, $r_j$, and $s_j$, respectively. Therefore, $\\pi(c_j(k)) = c_j(k \\oplus p)$;\notherwise $\\pi$ would not be an automorphism.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:reduction_correctness}\nThe 1-in-3 SAT formula is satisfiable if and only if there exists a list-compatible automorphism of $G$.\n\\end{lemma}\n\n\\begin{proof}\nLet $T$ be a truth value assignment satisfying the input formula. We construct a list-compatible\nautomorphism $\\pi$ of $G$. If $T(u_i) = 1$, we put $\\pi|_{H_i} =\n\\alpha_i$, and if $T(u_i) = 0$, we put $\\pi|_{H_i} = \\beta_i$. By Lemma~\\ref{lem:unique_extension},\nthis partial isomorphism has a unique extension to an automorphism $\\pi$ of $G$. It is\nlist-compatible since $\\pi(c_j(k)) = c_j(k \\oplus p)$ and $p \\in \\{100_2,010_2,001_2\\}$ (since $T$\nsatisfies the 1-in-3 condition). \n\nFor the other implication, let $\\pi$ be a list-compatible automorphism. Then $\\pi|_{H_i}$ is either\nequal $\\alpha_i$, or $\\beta_i$, which gives the values $T(u_i)$. By\nLemma~\\ref{lem:unique_extension}, $\\pi(c_j(k)) = c_j(k \\oplus p)$ and since $\\pi$ is a\nlist-compatible isomorphism, we have $p \\in \\{100_2,010_2,001_2\\}$. Therefore, exactly one literal in\neach clause is true, so all clauses are satisfied in $T$.\\qed\n\\end{proof}\n\nThe described reduction is clearly polynomial, so we have established a proof of Theorem~\\ref{thm:lubiw}.\n\n\n\\subsection{NP-hardness Proof}\n\nFor colored graphs, we require that automorphisms preserve colors.  By a simple modification of the\nabove reduction, we get the following: \n\n\\begin{theorem} \\label{thm:3reg_npc}\nThe problem \\textsc{ListIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete for 3-connected colored graphs for which each color class is of size\nat most 16 and each list is of size at most 3.\n\\end{theorem}\n\n\\begin{proof}\nWe modify the graph $G$ to a 3-regular graph. For a clause gadget $G_j$ representing $c_j$, we add\nthree new vertices $c_{j,q}(k)$, $c_{j,r}(k)$ and $c_{j,s}(k)$, each adjacent to $c_j(k)$ and two\nvertices of the variable gadget of the literal $q_j$, $r_j$, and $s_j$, respectively; see\nFig.~\\ref{fig:treeReplacement}b. For the newly added vertices, we set lists of size 3 compatibly\nwith $\\frakL(c_j(k))$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics{group_theory_reducing_degree}\n\\caption{The common vertices are depicted in white. (a) The variable gadget $H_i$. (b) The clause\ngadget $G_j$.}\n\\label{fig:treeReplacement}\n\\end{figure}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics{group_theory_connecting_gadgets}\n\\caption{Suppose that the variable $u_i$ is the literal $s_j$ of the clause $c_j$, so $k = yzx_2$.\nWe connect $H_i$ with $G_j$ as depicted. Suppose that an automorphism $\\pi$\nmaps $c_{j,s}(k)$ to $c_{j,s}(k \\oplus p)$. We depict the action of $\\pi$ on the vertices of $H_i$\nwhen $p = 001$ (in white), $p = 010$ (in gray), and $p = 100$ (in black).}\n\\label{fig:connecting_gadgets}\n\\end{figure}\n\nSuppose that a variable $u_i$ has $o$ literals in the formula. We replace $H_i$\nby a $4o$-cycle, together with a small gadget attached to each vertex as\ndepicted in Fig.~\\ref{fig:treeReplacement}a.  Suppose that the vertices\n$u_{i,1}(j),\\dots,u_{i,o}(j)$ correspond to $u_i(j)$ and similarly\n$u'_{i,1}(j),\\dots,u'_{i,o}(j)$ correspond to $u'_i(j)$. The vertices of\n$4o$-cycle are ordered:\n$$u_{i,1}(0),\\dots,u_{i,o}(0),u_{i,o}(1),\\dots,u_{i,1}(1),u'_{i,1}(1),\\dots,u'_{i,o}(1),\nu'_{i,o}(0),\\dots,u'_{i,1}(0).$$ We similarly define the automorphisms\n$\\alpha_i$ and $\\beta_i$ and lists for the vertices on the cycle.\n\nConsider the attached gadgets to four vertices corresponding to one literal of a clause $c_j$.  The\nvertices depicted in white are adjacent to the added vertices $c_{j,u_i}(k)$ of $G_j$, as depicted\nin Fig.~\\ref{fig:connecting_gadgets}. Each $k$ consists of three bits, denoted $x$, $y$ and $z$ (in\nsome order). The bit $x$ corresponds to the literal (i.e, the first bit for $u_i = q_j$, and so on).\nThe white vertices of gadgets attached to $u_{i,t}(j)$ and $u'_{i,t}(j)$ are attached to\n$c_{j,u_i}(k)$ where $x = j$.  Adjacent pairs of white vertices are connected to $c_j(k)$ where $k$\ndiffers in another bit $y$.  Non-adjacent pairs of white vertices in one gadget are connected to\n$c_j(k)$ where $k$ differs in the last bit $z$.\n\nIn Fig.~\\ref{fig:connecting_gadgets}, the action of ${\\mathbb Z}_2^3$ is depicted. When $\\alpha_i$\nor $\\beta_i$ is used on $H_i$, they are further composed with the vertical reflection. Therefore,\nLemma~\\ref{lem:unique_extension} translated to the modified definitions of variable and clause\ngadgets which implies correctness of the reduction. The lists for the vertices of the attached\ngadgets are created by composition of three depicted automorphisms together with $\\alpha_i$ and\n$\\beta_i$; observe that they are of size at most $3$.\n\nThe constructed graph $G$ is 3-regular and all lists of $G$ are of size at most $3$.  We color the\nvertices by the orbits of all list-compatible automorphisms and their compositions. Notice that each\ncolor class is of size at most $16$. More precisely, all color classes on each $H_i$ are of size $16$,\nand all color classes of clause gadgets $G_j$ are of size $8$.\n\\qed\n\\end{proof}\n\nWith Lemma~\\ref{lem:max_degree_2}, we get a dichotomy for the maximum degree: \\textsc{ListIso}\\xspace can be\nsolved in time $\\calO(\\sqrt n \\ell)$ for the maximum degree 2, and it is \\hbox{\\rm \\sffamily NP}\\xspace-complete for the maximum\ndegree 3. Similarly, Lemma~\\ref{lem:list_size_two} implies a dichotomy for the list sizes: \\textsc{ListIso}\\xspace can\nbe solved in time $\\calO(n+m)$ where all lists are of size 2, and it is \\hbox{\\rm \\sffamily NP}\\xspace-complete for lists of size\nat most 3. For the last parameter, the maximum size of color classes, there is a gap.\nLemma~\\ref{lem:list_size_two} implies that \\textsc{ListIso}\\xspace can be solved in time $\\calO(n+m)$ when all color\nclasses are of size 2 while it is \\hbox{\\rm \\sffamily NP}\\xspace-complete for size at most 16.\n\n\\section{Interval, Permutation and Circle Graphs} \\label{sec:intersection}\n\nIn this section, we prove that the standard algorithms solving \\textsc{GraphIso}\\xspace on interval, circle and\npermutation graphs can be modified to solve \\textsc{ListIso}\\xspace on them. The key idea is that the structure of\nthese graph classes can be captured by graph-labeled trees which are unique up to isomorphism and\nwhich capture the structure of all automorphisms; see~\\cite{kz,kz15} and the references therein.\n\nFor interval graphs, we use MPQ-tree. For circle graphs, we use split trees. For permutation graphs,\nwe use modular trees. On these trees, we apply bottom-up procedure similarly as in the proof of\nTheorem~\\ref{thm:trees}. The key difference is that nodes correspond to either prime, or degenerate\ngraphs. Degenerate graphs are simpler and lead to perfect matchings in bipartite graphs.  Prime\ngraphs have a small number of automorphisms~\\cite{kz,kz15}, so all of them can be tested.\n\n\\subsection{Interval Graphs}\n\nTo each interval graph $G$, a unique MPQ-tree $T_G$ is assigned. Two interval graphs $G$ and $H$ are\nisomorphic if and only if $T_G$ and $T_H$ are equivalent, and these trees capture all isomorphisms.\nTherefore, we apply a bottom-up proceduce to test \\textsc{ListIso}\\xspace for MPQ-trees, similarly as in\nTheorem~\\ref{thm:trees}.\n\n\\heading{MPQ-trees.} Booth and Lueker~\\cite{PQ_trees} invented a data structure called a\n\\emph{PQ-tree} which capture the structure of an interval graph.  We use \\emph{modified PQ-trees}\n(MPQ-trees) due to Korte and M\\\"ohring~\\cite{incremental_linear_int_recognition}.  Let $G$ be an\ninterval graph.  A rooted tree $T$ is an MPQ-tree if the following holds.  It has two types of inner\nnodes: \\emph{P-nodes} and \\emph{Q-nodes}. For every inner node, its children are ordered from left\nto right. Each P-node has at least two children and each Q-node at least three.  The leaves of $T$\ncorrespond one-to-one to the maximal cliques in $G$.\n\nTwo MPQ-trees are \\emph{equivalent} if one can be obtained from the other by a sequence of two\n\\emph{equivalence transformations}: (i) an arbitrary permutation of the order of the children of a\nP-node, and (ii) the reversal of the order of the children of a Q-node.  Booth and\nLueker~\\cite{PQ_trees} proved the existence and uniqueness of PQ-trees (up to equivalence\ntransformations); see Fig.~\\ref{fig:mpq_tree}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1]{intersection_mpq_tree}\n\\caption{An ordering of the maximal cliques, and the corresponding MPQ-tree. The P-nodes\nare denoted by circles, the Q-nodes by rectangles. There are four equivalent MPQ-trees.}\n\\label{fig:mpq_tree}\n\\end{figure}\n\nWe assign subsets of $V(G)$, called \\emph{sections}, to the\nnodes of $T$; see Fig.~\\ref{fig:mpq_tree}. The leaves and the P-nodes have\neach assigned exactly one section while the Q-nodes have one section per child.\nWe assign these sections as follows:\n\\begin{itemize}\n\\item For a leaf $L$, the section ${\\rm sec}(L)$ contains those vertices that are only\nin the maximal clique represented by $L$, and no other maximal clique.\n\\item For a P-node $P$, the section ${\\rm sec}(P)$ contains those vertices that are in\nall maximal cliques of the subtree of $P$, and no other maximal clique.\n\\item For a Q-node $Q$ and its children $T_1, \\dots, T_n$, the section ${\\rm sec}_i(Q)$\ncontains those vertices that are in the maximal cliques represented by the\nleaves of the subtree of $T_i$ and also some other $T_j$, but not in any other\nmaximal clique outside the subtree of $Q$. We put ${\\rm sec}(Q) = {\\rm sec}_1(Q) \\cup\n\\cdots \\cup {\\rm sec}_n(Q)$.\n\\end{itemize}\nEach vertex appears in sections of exactly one node and in the case of a Q-node in consecutive\nsections. Two vertices are in the same sections if and only if they belong to precisely the same\nmaximal cliques. Figure~\\ref{fig:mpq_tree} shows an example. MPQ-tree can be constructed in\ntime~\\cite{incremental_linear_int_recognition}. \n\n\\heading{Testing \\textsc{ListIso}\\xspace.}\nLet $G$ and $H$ be two isomorphic interval graphs. From~\\cite[Lemma 4.3]{kz15}, it follows that\n$T_G$ and $T_H$ are equivalent, and every isomorphism $\\pi : G \\to H$ is obtained by an equivalence\ntransformation of $T_G$ and some permutation of the vertices in identical sections.  Now, we are\nready to show \\textsc{ListIso}\\xspace can be solved on interval graphs in time $\\calO(\\sqrt{n}\\ell + m)$:\n\n\\begin{theorem} \\label{thm:int}\nThe problem \\textsc{ListIso}\\xspace can be solved for interval graphs in time $\\calO(\\sqrt{n}\\ell + m)$.\n\\end{theorem}\n\n\\begin{proof}\nWe proceed similarly as in Theorem~\\ref{thm:trees}.  We compute MPQ-trees representing $T_G$\nand $T_H$ representing the graphs $G$ and $H$ in linear\ntime~\\cite{incremental_linear_int_recognition}. Then we compute lists $\\frakL(N)$ for every node $N$ of\n$T_G$ from the bottom. We distinguish three types of nodes.\n\\begin{itemize}\n\\item \\emph{Leaf nodes.} Let $L_G$ be a leaf node in $T_G$ and let $L_H$ be a leaf node in $T_H$.\nThen $L_H \\in \\frakL(L_G)$ if there exists a list-compatible isomorphism between the induced complete\nsubgraphs $G[{\\rm sec}(L_G)]$ and $H[{\\rm sec}(L_H)]$.\n\\item \\emph{P-nodes.} Let $N$ and $M$ be P-nodes of $T_G$ and $T_H$, respectively. We want to decide\nwhether $M \\in \\frakL(N)$. Let $N_1,\\dots,N_k$ be the children of $N$ and let\n$M_1,\\dots,M_k$ be the children of $M_k$. We construct a bipartite graph\nsimilarly as in Theorem~\\ref{thm:trees}.  Then $M \\in \\frakL(N)$ if there exists a\nperfect matching in the bipartite graph and a perfect matching between the lists of \n$G[{\\rm sec}(N)]$ and $H[{\\rm sec}(M)]$ (which are complete graphs).\n\\item \\emph{Q-nodes.} Let $N$ and $M$ be Q-nodes of $T_G$ and $T_H$ and let $N_1,\\dots,N_k$ and\n$M_1,\\dots,M_k$ be their children. Here we have at most two possible\nisomorphisms. In particular, an isomorphism can either map the subtree of $N_i$\non the subtree of $M_i$, or in the reversed order, and we can test for both possibilities whether\nthe lists $\\frakL(N_i)$ are compatible. Moreover, we consider all sets of intervals belonging to\nexactly the same sections of the Q-node, and we test by perfect matchings between pairs of them\nwhether there exists a list-compatible isomorphism between them.\n\\end{itemize}\nThe MPQ-trees have $\\calO(n)$ nodes and $\\calO(n)$ intervals in their sections. For leaf nodes and\nP-nodes, the analysis is exactly the same as in the proof of Theorem~\\ref{thm:trees}. For Q-nodes,\nwe just test two possible mappings and bipartite matchings for sections. We get the total running\ntime $\\calO(\\sqrt n \\ell + m)$.\\qed\n\\end{proof}\n\n\\subsection{Permutation Graphs}\n\nA \\emph{module} $M$ of a graph $G$ is a set of vertices such that each $x \\in V(G) \\setminus M$ is\neither adjacent to all vertices in $M$, or to none of them.  See Fig.~\\ref{fig:modules}a for\nexamples. A module $M$ is called \\emph{trivial} if $M=V(G)$ or $|M|=1$, and \\emph{non-trivial}\notherwise. If $M$ and $M'$ are two disjoint modules, then either the edges between $M$ and $M'$\nform the complete bipartite graph, or there are no edges at all; see Fig.~\\ref{fig:modules}a.  In\nthe former case, $M$ and $M'$ are called \\emph{adjacent}, otherwise they are \\emph{non-adjacent}.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=1]{intersection_modules}\n\\caption{(a) A graph $G$ with a modular partition ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$. (b) The quotient graph $G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ is prime.}\n\\label{fig:modules}\n\\end{figure}\n\nLet ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R} = \\{M_1, \\dots, M_k\\}$ be a \\emph{modular partition} of $V(G)$, i.e., each $M_i$ is a\nmodule of $G$, $M_i \\cap M_j = \\emptyset$ for every $i\\neq j$, and $M_1 \\cup \\cdots \\cup M_k =\nV(G)$. We define the \\emph{quotient graph} $G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ with the vertices $m_1,\\dots,m_k$ corresponding\nto $M_1,\\dots,M_k$ where $m_im_j \\in E(G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R})$ if and only if $M_i$ and $M_j$ are adjacent.  In\nother words, the quotient graph is obtained by contracting each module $M_i$ into the single vertex\n$m_i$; see Fig.~\\ref{fig:modules}b.\n\n\\heading{Modular Decomposition.}\nTo decompose $G$, we find some modular partition ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R} = \\{M_1, \\dots, M_k\\}$, compute $G \/ {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$\nand recursively decompose $G \/ {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ and each $G[M_i]$. The recursive process terminates on\n\\emph{prime graphs} which are graphs containing only trivial modules.  There might be many such\ndecompositions for different choices of ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ in each step.  In 1960s,\nGallai~\\cite{gallai1967transitiv} described the \\emph{modular decomposition} in which special\nmodular partitions are chosen and which encodes all other decompositions.\n\nThe key is the following observation. Let $M$ be a module of $G$ and let $M' \\subseteq M$. Then $M'$\nis a module of $G$ if and only if it is a module of $G[M]$.  A graph $G$ is called \\emph{degenerate}\nif it is $K_n$ or $\\overline{K}_n$.\tWe construct the modular decomposition of a graph $G$ in the\nfollowing way, see Fig.~\\ref{fig:modular_tree}a for an example:\n\\begin{itemize}\n\\item If $G$ is a prime or a degenerate graph, then we terminate the modular decomposition on $G$.\nWe stop on degenerate graphs since every subset of vertices forms a module, so it is not useful to\nfurther decompose them.\n\\item Let $G$ and $\\overline{G}$ be connected graphs.  Gallai~\\cite{gallai1967transitiv} shows that\nthe inclusion maximal proper subsets of $V(G)$ which are modules form a modular partition ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ of\n$V(G)$, and the quotient graph $G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ is a prime graph; see Fig.~\\ref{fig:modules}. We\nrecursively decompose $G[M]$ for each $M \\in {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$.\n\\item If $G$ is disconnected and $\\overline{G}$ is connected, then every union of connected\ncomponents is a module. Therefore the connected components form a modular partition ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ of\n$V(G)$, and the quotient graph $G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ is an independent set. We recursively decompose $G[M]$ for\neach $M \\in {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$.\n\\item If $\\overline{G}$ is disconnected and $G$ is connected, then the modular decomposition is\ndefined in the same way on the connected components of $\\overline{G}$.  They form a modular\npartition ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ and the quotient graph $G\/{\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ is a complete graph. We recursively decompose\n$G[M]$ for each $M \\in {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$.\n\\end{itemize}\n\n\\heading{Modular Tree.}\nWe encode the modular decomposition by the \\emph{modular tree} $T$.  The modular tree $T$ is a graph\nwith two types of vertices (normal and \\emph{marker vertices}) and two types of edges (normal and\n\\emph{directed tree edges}). The directed tree edges connect the prime and degenerate graphs\nencountered in the modular decomposition (as quotients and terminal graphs) into a rooted tree. \n\nWe give a recursive definition.  Every modular tree has an induced subgraph called \\emph{root node}.\nIf $G$ is a prime or a degenerate graph, we define $T = G$ and its root node equals $T$.  Otherwise,\nlet ${\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R} = \\{M_1,\\dots,M_k\\}$ be the used modular partition of $G$ and let $T_1,\\dots,T_k$ be the\nmodular trees corresponding to $G[M_1],\\dots,G[M_k]$. The modular tree $T$ is the disjoint union of\n$T_1,\\dots,T_k$ and of $G \/ {\\mathcal P}} \\def\\calQ{{\\mathcal Q}} \\def\\calR{{\\mathcal R}$ with the marker vertices $m_1,\\dots,m_k$. To every graph $T_i$,\nwe add a new marker vertex $m'_i$ such that $m'_i$ is adjacent exactly to the vertices of the root\nnode of $T_i$. We further add a tree edge oriented from $m_i$ to $m'_i$.  For an example, see\nFig.~\\ref{fig:modular_tree}b.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1]{intersection_modular_tree}\n\\caption{(a) The graph $G$ from Fig.~\\ref{fig:modules} with the modular partitions used in the\nmodular decomposition. (b) The modular tree $T$ of $G$, the marker vertices are white, the tree\nedges are dashed.}\n\\label{fig:modular_tree}\n\\end{figure}\n\nThe modular tree of $G$ is unique.  The graphs encountered in the modular decomposition are called\n\\emph{nodes of $T$}, or alternatively root nodes of some modular trees in the construction of $T$.\nFor a node $N$, its subtree is the modular tree which has $N$ as the root node. \\emph{Leaf nodes}\ncorrespond to the terminal graphs in the modular decomposition, and \\emph{inner nodes} are the\nquotients in the modular decomposition. All vertices of $G$ are in leaf nodes and all marker\nvertices correspond to modules of $G$. \\emph{All inner nodes consist of marker vertices}.\n\n\\heading{Testing \\textsc{ListIso}\\xspace.}\nNow, we are ready to show that the problem \\textsc{ListIso}\\xspace can be decided in $\\calO(\\sqrt{n}\\ell + m)$ time for\npermutation graphs.\n\n\\begin{theorem} \\label{thm:perm}\nThe problem \\textsc{ListIso}\\xspace can be solved for permutation graphs in time $\\calO(\\sqrt{n}\\ell + m)$.\n\\end{theorem}\n\n\\begin{proof}\nFor input graph $G$ and $H$, we first compute the modular trees $T_G$ and\n$T_H$, respectively, in time $\\calO(n+m)$~\\cite{mcconnell_spinrad}. We again apply\nthe idea of Theorem~\\ref{thm:trees}.  We compute the list $\\frakL(N)$ for every\nnode $N$ of $T_G$. Note that all inner nodes consist only of marker vertices\nwhich have no lists. Therefore, we first compute $\\frakL(L)$, for every leaf node.\nA leaf node $K$ is in $\\frakL(L)$ if every non-marker vertex of $L$ has a\nnon-marker vertex of $K$ in its list. These candidate nodes for $\\frakL(L)$ can be\nfound in linear time in the total size of lists by iterating through the lists\nof vertices of $L$.\n\nSuppose that a node $N$ has the children $N_1,\\dots,N_k$ with computed lists $\\frakL(N_1),\\dots,\\frakL(N_k)$\nand $M$ has the children $M_1,\\dots,M_k$. There exist a list-compatible isomorphism mapping the\nsubtree of $N_i$ to the subtree of $M_j$ if and only if $M_j \\in \\frakL(N_i)$. Moreover these subtrees\nhave to be compatible with a list isomorphism from $N$ to $M$. We compute $\\frakL(N)$ according to the\ntype of $N$.\n\\begin{packed_itemize}\n\\item \\emph{Degenerate nodes.} For degenerate nodes, we proceed similarly as for trees in\nTheorem~\\ref{thm:trees}. We construct a bipartite graph between the nodes nodes $N_1,\\dots,N_k$ and\n$M_1,\\dots,M_k$ and test for a perfect matching using Lemma~\\ref{lem:bipmatch}.\n\\item \\emph{Prime nodes.} For prime nodes, there are at most four possible isomorphisms mapping $N$\nto $M$~\\cite[Lemma 6.6]{kz15}. We test for these four possible isomorphisms $\\pi$ whether $\\pi(M_i)\n\\in \\frakL(M_i)$ for every $M_i$.\n\\end{packed_itemize}\nA list compatible isomorphism exists if $M \\in \\frakL(N)$, for the root nodes $N$ and $M$ of $T_G$ and\n$T_H$.  The correctness of the algorithm follows from the fact that all\nautomorphisms of a permutation\ngraph are captured by the modular tree~\\cite{kz15}. A similar argument as in the proofs of\nTheorems~\\ref{thm:trees} and~\\ref{thm:int} gives the running time.\\qed\n\\end{proof}\n\n\\subsection{Circle Graphs}\n\nFor a given circle graph, we define the split tree which captures its automorphism group.  A\n\\emph{split} is a partition $(A,B,A',B')$ of $V(G)$ such that:\n\\begin{itemize}\n\\item For every $a \\in A$ and $b \\in B$, we have $ab \\in E(G)$.\n\\item There are no edges between $A'$ and $B \\cup B'$, and between $B'$ and $A \\cup A'$.\n\\item Both sides have at least two vertices: $|A \\cup A'| \\ge 2$ and $|B \\cup B'| \\ge 2$.\n\\end{itemize}\n\nThe split decomposition of $G$ is constructed by taking a split of $G$ and replacing $G$ by the\ngraphs $G_A$ and $G_B$ defined as follows. The graph $G_A$ is created from $G[A \\cup A']$ together\nwith a new \\emph{marker vertex} $m_A$ adjacent exactly to the vertices in $A$.  The graph $G_B$ is\ndefined analogously for $B$, $B'$ and $m_B$; see Fig.~\\ref{fig:split_graph_split-tree}a. The\ndecomposition is then applied recursively on $G_A$ and $G_B$. Graphs containing no splits are called\n\\emph{prime graphs}. We stop the split decomposition also on \\emph{degenerate graphs} which are\ncomplete graphs $K_n$ and stars $K_{1,n}$. A split decomposition is called \\emph{minimal} if it is\nconstructed by the least number of splits. Cunningham~\\cite{cunningham} proved that the minimal\nsplit decomposition of a connected graph is unique.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=1]{intersection_split_graph_tree}\n\\caption{(a) An example of a split of the graph $G$. The marker vertices are depicted in white.  The\ntree edge is depicted by a dashed line.  (b) The split tree $S$ of the graph $G$. We have that\n${\\rm Aut}(S) \\cong \\mathbb{Z}_2^5 \\rtimes \\mathbb{D}_5$.}\n\\label{fig:split_graph_split-tree}\n\\end{figure}\n\n\\heading{Split tree.}\nThe \\emph{split tree} $T$ representing a graph $G$ encodes the minimal split decomposition. A split\ntree is a graph with two types of vertices (normal and marker vertices) and two types of edges\n(normal and tree edges). We initially put $T = G$ and modify it according to the minimal split\ndecomposition. If the minimal decomposition contains a split $(A, B, A', B')$ in $G$, then we\nreplace $G$ in $T$ by the graphs $G_A$ and $G_B$, and connect the marker vertices $m_A$ and $m_B$ by\na \\emph{tree edge} (see Fig.~\\ref{fig:split_graph_split-tree}a). We repeat this recursively on $G_A$\nand $G_B$; see Fig.~\\ref{fig:split_graph_split-tree}b. Each prime and degenerate graph is a\n\\emph{node} of the split tree. A node that is incident with exactly one tree edge is called a\n\\emph{leaf node}.\n\nSince the minimal split decomposition is unique, we also have that the split tree is unique.\nFurther, each automorphism $\\pi$ of $G$ induces an automorphism $\\pi'$ of the split tree $T$\nrepresenting $G$. Similarly as for trees, there exists a \\emph{center} of $T$ which is either a tree\nedge, or a prime or degenerate node. The automorphism $\\pi'$ preserves the center, so we can regard\n$T$ as rooted by the center. Every automorphism of $G$ can be reconstructed from the root of $T$ to\nthe bottom.\n\n\\heading{Testing \\textsc{ListIso}\\xspace.}\nNext, we show that the problem \\textsc{ListIso}\\xspace can be solved on circle graphs in time $\\calO(\\sqrt{n}\\ell +\nm\\cdot \\alpha(m)$.\n\n\\begin{theorem} \\label{thm:circle}\nThe problem \\textsc{ListIso}\\xspace can be solved for circle graphs in time $\\calO(\\sqrt{n}\\ell + m\\cdot \\alpha(m))$,\nwhere $\\alpha$ is the inverse Ackermann function.\n\\end{theorem}\n\n\\begin{proof}\nFor input graph $G$ and $H$, we first compute the split trees $T_G$ and $T_H$, in time\n$\\calO((n+m)\\cdot \\alpha(n+m))$~\\cite{split_trees_linear}. We assume that the trees $T_G$ and $T_H$ are\nrooted and we can also assume that the roots are prime or degenerate nodes. We again apply the idea\nof Theorem~\\ref{thm:trees}.\n\nWe compute the list $\\frakL(N)$ for every node $N$ of $T_G$. Let $M$ be a leaf node of $T_H$ and let\n$m_N \\in V(N)$ and $m_M \\in V(M)$ be the marker vertices incident to a tree edge closer to the root.\nThen $M$ is in $\\frakL(N)$ if there is a list-compatible isomorphism from $N$ to $M$ which maps $m_N$\nto $m_M$.\n\nSuppose that a node $N$ has the children $N_1,\\dots,N_k$ with computed lists $\\frakL(N_1),\\dots,\\frakL(N_k)$\nand $M$ has the children $M_1,\\dots,M_k$. There exist a list-compatible isomorphism mapping the\nsubtree of $N_i$ to the subtree of $M_j$ if and only if $M_j \\in \\frakL(N_i)$. Moreover these subtrees\nhave to be compatible with an isomorphism from $N$ to $M$. We compute $\\frakL(N)$ according to the\ntype of $N$.\n\n\\begin{packed_itemize}\n\\item \\emph{Degenerate nodes.} For degenerate nodes, we proceed similarly as for trees in\nTheorem~\\ref{thm:trees}. We construct a bipartite graph between the nodes nodes $N_1,\\dots,N_k$ and\n$M_1,\\dots,M_k$ and test for a perfect matching using Lemma~\\ref{lem:bipmatch}.\n\\item \\emph{Prime nodes.} For prime nodes, there are at most four possible isomorphisms mapping\n$m_N$ to $m_M$~\\cite[Lemma 5.6]{kz}. We test those four possible isomorphisms, construct four bipartite graphs\nand test existence of perfect matchings.\n\\item \\emph{The root node.} If it is degenerate, we proceed as above. If it is prime, then its\nautomorphism groups is a subgroup of a dihedral group~\\cite[Lemma 5.5]{kz}; essentially it behaves\nas a cycle. Therefore, we approach it similarly as in Lemma~\\ref{lem:cycles}.\n\\end{packed_itemize}\nA list compatible isomorphism exists if $M \\in \\frakL(N)$, for the root nodes $N$\nand $M$ of $T_G$ and $T_H$.\n\nThe correctness of the algorithm follows from the fact that all automorphisms of a circle graph are\ncaptured by the split tree~\\cite{kz}. The running time can be argued as in Theorems~\\ref{thm:trees},\n\\ref{thm:int}, and~\\ref{thm:perm}.\\qed\n\\end{proof}\n\n\n\\section{Introduction} \\label{sec:introduction}\n\nFor graphs $G$ and $H$, a bijection $\\pi : G \\to H$ is called an \\emph{isomorphism} if $uv \\in E(G)\n\\iff \\pi(u)\\pi(v) \\in E(H)$. The \\emph{graph isomorphism problem} (\\textsc{GraphIso}\\xspace) asks whether there exists an\nisomorphism from $G$ to $H$. It obviously belongs to \\hbox{\\rm \\sffamily NP}\\xspace, and no polynomial-time algorithm is\nknown. It is a prime candidate for an intermediate problem with complexity between \\hbox{\\rm \\sffamily P}\\xspace\\ and\n\\hbox{\\rm \\sffamily NP}\\xspace-complete.  There are threefold evidences that \\textsc{GraphIso}\\xspace is unlikely to be \\hbox{\\rm \\sffamily NP}\\xspace-complete: equivalence of\nexistence and counting~\\cite{babai77,mathon1979note}, \\textsc{GraphIso}\\xspace belongs to \\hbox{\\rm \\sffamily coAM}\\xspace, so the\npolynomial-hierarchy collapses if \\textsc{GraphIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete~\\cite{goldreich1986,schoning1988graph}, and\n\\textsc{GraphIso}\\xspace can be solved in quasipolynomial time~\\cite{babai_quasipoly}.  For a survey,\nsee~\\cite{babai1996automorphism}.\n\n\\subsection{Graph Isomorphism Problem for Restricted Graph Classes and Parameters}\n\nThe graph isomorphism problem is solved efficiently for various restricted graph classes and\nparameters, see Fig.~\\ref{fig:diagram}.\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[scale=1]{introduction_diagram}\n\\caption{Important graph classes for which the graph isomorphism problem can be solved in\npolynomial time. Our complexity results for the list restricted graph isomorphism problem are\ndepicted.}\n\\label{fig:diagram}\n\\end{figure}\n\n\\heading{Combinatorial Algorithms.}\nA prime example is the linear-time algorithm for testing graph isomorphism of (rooted) trees. It is\na bottom-up procedure comparing subtrees. This algorithm is very robust and captures all possible\nisomorphisms.  For many other graph classes, graph isomorphism reduces to graph isomorphism of\nlabeled trees: for planar\ngraphs~\\cite{hopcroft_tarjan_dividing,hopcroft_tarjan_planar_iso,hopcroft1974linear}, interval\ngraphs~\\cite{lueker1979linear}, circle graphs~\\cite{hsu1995m}, and permutation\ngraphs~\\cite{permutation_isomorphism,spinrad}.  Involved combinatorial arguments are used to solve\ngraph isomorphism for bounded genus graphs~\\cite{pp_iso,filotti_mayer,miller_iso,kawarabayashi} and\nbounded treewidth graphs~\\cite{iso_xp_treewidth,iso_fpt_treewidth}.\n\n\\heading{Algorithms Based on Group Theory.}\nThe graph isomorphism problem is closely related to group theory, in particular to computing\ngenerators of automorphism groups of graphs.  Assuming that $G$ and $H$ are connected, we can test\n$G \\cong H$ by computing generators of ${\\rm Aut}(G \\mathbin{\\dot\\cup} H)$ and checking whether there exists a\ngenerator which swaps $G$ and $H$.  For the converse relation, Mathon~\\cite{mathon1979note} proved\nthat generators of the automorphism group can be computed using $\\calO(n^3)$ instances of graph\nisomorphism. \n\nTherefore, \\textsc{GraphIso}\\xspace can be attacked by techniques of group theory. A prime example is the seminal result\nof Luks~\\cite{luks1982isomorphism} which uses group theory to solve \\textsc{GraphIso}\\xspace for graphs of bounded degree\nin polynomial time. If $G$ has bounded degree, its automorphism group ${\\rm Aut}(G)$ may be arbitrary,\nbut the stabilizer ${\\rm Aut}_e(G)$ of an edge $e$ is restricted. Luks' algorithm tests \\textsc{GraphIso}\\xspace by an\niterative process which determines ${\\rm Aut}_e(G)$ in steps, by adding layers around $e$.\n\nGroup theory can be used to solve \\textsc{GraphIso}\\xspace of colored graphs with bounded sizes of color\nclasses~\\cite{furst_hopcroft_luks} and of graphs with bounded eigenvalue\nmultiplicity~\\cite{babai_eig_multiplicity,ponomarenko_eig_multiplicity}.\nMiller~\\cite{miller_k_contractible} solved \\textsc{GraphIso}\\xspace of $k$-contractible graphs (which generalize both\nbounded degree and bounded genus graphs), and his results are used by\nPonomarenko~\\cite{ponomarenko_minor} to show that \\textsc{GraphIso}\\xspace can be decided in polynomial time for\ngraphs with excluded minors. Luks' algorithm~\\cite{luks1982isomorphism} for bounded degree graphs is also\nused by Grohe and Marx~\\cite{grohe_marx} as a subroutine to solve \\textsc{GraphIso}\\xspace on graphs with excluded\ntopological subgraphs.  The recent breakthrough of Babai~\\cite{babai_quasipoly} heavily uses group\ntheory to solve the graph isomorphism problem in quasipolynomial time.\n\n\\heading{Is Group Theory Needed?} One of the fundamental problems for understanding the graph\nisomorphism problem is to understand in which cases group theory is really needed, and in which\ncases it can be avoided.\\footnote{Ilya Ponomarenko in personal communication.} For instance, for\nwhich graph classes can \\textsc{GraphIso}\\xspace be decided by the classical combinatorial algorithm called\n$k$-dimensional Weisfieler-Leman refinement ($k$-WL)? (Described in Conclusions.)\n\nPonomarenko~\\cite{ponomarenko_minor} used group theory to solve \\textsc{GraphIso}\\xspace in polynomial time on graphs\nwith excluded minors.  Robertson and Seymour~\\cite{robertson_seymour} proved that a graph $G$ with\nan excluded minor can be decomposed into pieces which are ``almost embeddable'' to a surface of genus\n$g$, where $g$ depends on this minor. Recently, Grohe~\\cite{grohe_graph_structure} generalized this\nto show that for $G$, there exists a treelike decomposition into almost embeddable pieces which is\n\\emph{automorphism-invariant} (every automorphism of $G$ induces an automorphism of the treelike\ndecomposition). Using this decomposition, it is possible to solve graph isomorphism in polynomial\ntime and to avoid group theory techniques. In particular, $k$-WL can decide graph isomorphism on\ngraphs with excluded minors where $k$ depends on the minor.\n\nIt is a long-standing open problem whether the graph isomorphism problem for bounded degree graphs,\nand in particular for cubic graphs, can be solved in polynomial time without group theory. For\ninstance, can cubic graph isomorphism be decided by $k$-WL for a suitable value $k$? Very recently,\nfixed parameter tractable algorithms for graphs of bounded treewidth~\\cite{iso_fpt_treewidth} and\nfor graphs of bounded genus~\\cite{kawarabayashi} were constructed. On the other hand, the best known\nparameterized algorithm for graphs of bounded degree is the \\hbox{\\rm \\sffamily XP}\\xspace algorithm of\nLuks~\\cite{luks1982isomorphism}, and it is a major open problem whether an \\hbox{\\rm \\sffamily FPT}\\xspace algorithm exists.\n\nIn this paper, we propose a different approach to show limitations of techniques used to attack the\ngraph isomorphism problem. We study its generalization called \\emph{list restricted graph\nisomorphism} (\\textsc{ListIso}\\xspace) which is \\hbox{\\rm \\sffamily NP}\\xspace-complete for general graphs. \n\n\\begin{quote}\n{\\bf Implications for {\\sc \\bfseries GraphIso}\\xspace.} The study for \\textsc{ListIso}\\xspace allows to classify the results for the graph isomorphism problem.\nAn algorithm for \\textsc{GraphIso}\\xspace is called \\emph{robust} if it\ncan be modified to solve \\textsc{ListIso}\\xspace while preserving the complexity. (Say, it remains a polynomial-time\nalgorithm, fixed parameter tractable algorithm, etc.)\n\\end{quote}\n\nWe show that combinatorial algorithms for graph isomorphism are robust. On the other hand, hardness\nresults for \\textsc{ListIso}\\xspace imply non-existence of robust algorithms for \\textsc{GraphIso}\\xspace. In particular, we show that \\textsc{ListIso}\\xspace is\n\\hbox{\\rm \\sffamily NP}\\xspace-complete for cubic graphs, so no robust algorithm for cubic graph isomorphism exists, unless\n$\\hbox{\\rm \\sffamily P}\\xspace = \\hbox{\\rm \\sffamily NP}\\xspace$. Similarly, no robust \\hbox{\\rm \\sffamily FPT}\\xspace algorithm for graph isomorphism of graphs of bounded degree\nexists.\n\n\\subsection{List Restricted Graph Isomorphism}\n\nIn 1981, Lubiw~\\cite{lubiw1981some} introduced the following computational problems.  Let $G$ and\n$H$ be graphs, and the vertices of $G$ be equipped with lists: each vertex $u \\in V(G)$ has a\nlist $\\frakL(u) \\subseteq V(H)$. We say that an isomorphism $\\pi : G \\to H$ is\n\\emph{list-compatible} if, for all vertices $u \\in V(G)$, we have $\\pi(u) \\in \\frakL(u)$; see\nFig.~\\ref{fig:example}a. A list-compatible isomorphism $\\pi : G \\to G$ is called a\n\\emph{list-compatible automorphism}. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=1]{introduction_example}\n\\caption{(a) Two isomorphic graphs $G$ and $H$ with no list-compatible isomorphism.  (b) It does not\nexist because there is no perfect matching between the lists of the leaves of $G$ and the leaves of\n$H$.}\n\\label{fig:example}\n\\end{figure}\n\n\\computationproblem\n{List restricted graph isomorphism -- \\textsc{ListIso}\\xspace}\n{Graphs $G$ and $H$, and the vertices of $G$ are equipped by lists $\\frakL(u) \\subseteq V(H)$.}\n{Is there a list-compatible isomorphism $\\pi : G \\to H$?}\n{13cm}\n\n\\computationproblem\n{List restricted graph automorphism -- \\textsc{ListAut}\\xspace}\n{A graph $G$ with vertices equipped with lists $\\frakL(u) \\subseteq V(G)$.}\n{Is there a list-compatible automorphism $\\pi : G \\to G$?}\n{10cm}\n\nThese two problems are polynomially equivalent (see Lemma~\\ref{lem:lautlgi_equiv}).\nLubiw~\\cite{lubiw1981some} proved the following surprising result:\n\n\\begin{theorem}[Lubiw \\cite{lubiw1981some}] \\label{thm:lubiw}\nThe problems \\textsc{ListIso}\\xspace and \\textsc{ListAut}\\xspace are \\hbox{\\rm \\sffamily NP}\\xspace-complete.\n\\end{theorem}\n\n\\noindent Moreover, she proved that finding a fixed-point free involutory automorphism of a graph is\n\\hbox{\\rm \\sffamily NP}\\xspace-complete.  Lalonde~\\cite{lalonde} showed that it is \\hbox{\\rm \\sffamily NP}\\xspace-complete to decide whether a bipartite\ngraph has an involutory automorphism exchanging the parts; see~\\cite{star_systems}.\n\nIndependently, \\textsc{ListIso}\\xspace was rediscovered in~\\cite{fkkn,fkkn16b}. Given two graphs $G$ and $H$,\nwe say that $G$ \\emph{regularly covers} $H$ if there exists a semiregular subgroup $\\Gamma \\le\n{\\rm Aut}(G)$ such that $G \/ \\Gamma \\cong H$. The list restricted graph isomorphism problem was used as a\nsubroutine in~\\cite{fkkn,fkkn16b} for 3-connected planar and projective graphs to test regular\ncovering when $G$ is a planar graph. The key idea is that a planar graph $G$ can be reduced to a\n3-connected planar graph $G_r$, for which ${\\rm Aut}(G_r)$ is a spherical group.  Therefore, we can\ncompute all regular quotients $G_r \/ \\Gamma_r$.  Next, we reduce $H$ towards $G_r \/ \\Gamma_r$. The\nproblem is that subgraphs of $H$ may correspond to several different parts in $G$, so we compute\nlists of all possibilities. One subroutine of the reduction leads to \\textsc{ListIso}\\xspace of 3-connected planar and\nprojective planar graphs, while the other leads to a generalization of bipartite perfect\nmatching~\\cite{iv_matching}.\n\nWe note that other computational problems restricted by lists are frequently studied. \\emph{List\ncoloring}, introduced by Vizing~\\cite{list_coloring}, is \\hbox{\\rm \\sffamily NP}\\xspace-complete even for planar\ngraphs~\\cite{list_coloring_planar} and interval graphs~\\cite{list_coloring_int}. \\emph{List\n$H$-homomorphisms}, having a similar setting as \\textsc{ListIso}\\xspace, were also considered;\nsee~\\cite{graphs_and_homomorphisms,list_hom_dichotomy,list_hom_removing}.\n\n\\subsection{Our Results} \\label{sec:our_results}\n\nWe revive the study of list restricted graph isomorphism. The goal is to determine which techniques\nfor \\textsc{GraphIso}\\xspace translate to \\textsc{ListIso}\\xspace. We believe that \\textsc{ListIso}\\xspace is a very natural computational problem, as\nevidenced by its application in~\\cite{fkkn,fkkn16b}.  Further, its hardness results prove\nnon-existence of robust algorithms for the graph isomorphism problem itself. For instance, it is\nbelieved that no \\hbox{\\rm \\sffamily NP}\\xspace-complete problem can be solved in quasipolynomial time. Therefore, some\ntechniques used by Babai~\\cite{babai_quasipoly} to solve \\textsc{GraphIso}\\xspace in quasipolynomial time do not\ntranslate to \\textsc{ListIso}\\xspace. To solve \\textsc{GraphIso}\\xspace efficiently, one necessarily has to apply such techniques.\n\nThe described algorithms for \\textsc{ListIso}\\xspace are a straightforward modification of previously known algorithms\nfor the graph isomorphism problem. The main point of this paper is \\emph{not to develop new\nalgorithmic techniques}, but to \\emph{classify known techniques for \\textsc{GraphIso}\\xspace from a different\nviewpoint}. This viewpoint is the robustness of algorithms with respect to \\textsc{ListIso}\\xspace, and our results\ngive an insight into its meaning.\n\nWe prove the following three \\emph{informal results} in this paper; see Fig.~\\ref{fig:diagram} for\nan overview:\n\n\\begin{result} \\label{res:gi_completeness}\n$ $\\hbox{\\rm \\sffamily GI}\\xspace-completeness results for \\textsc{GraphIso}\\xspace translate to \\hbox{\\rm \\sffamily NP}\\xspace-completeness for \\textsc{ListIso}\\xspace.\n\\end{result}\n\nFor many classes $\\calC$ of graphs, it is known that \\textsc{GraphIso}\\xspace is equally hard for them as for general\ngraphs, i.e., it is \\hbox{\\rm \\sffamily GI}\\xspace-complete. For instance, \\textsc{GraphIso}\\xspace is \\hbox{\\rm \\sffamily GI}\\xspace-complete for bipartite graphs, split\nand chordal graphs~\\cite{lueker1979linear}, chordal bipartite and strongly chordal\ngraphs~\\cite{strongly_chordal}, trapezoid graphs~\\cite{trapezoid}, comparability graphs of dimension\n4~\\cite{kz}, grid intersection graphs~\\cite{grid_GI_complete}, and line graphs~\\cite{line_graphs}.\nThe polynomial-time reductions are often done in a way that all graphs are encoded into $\\calC$, by\nreplacing each vertex with a small vertex-gadget.  (The constructions are quite simple, and the\nnon-trivial part is to prove that the constructed graph belongs to $\\calC$.) As we prove in\nTheorem~\\ref{thm:vertex_gadget_reductions}, such reductions using vertex-gadgets also translate to\n\\textsc{ListIso}\\xspace: they imply \\hbox{\\rm \\sffamily NP}\\xspace-completness of \\textsc{ListIso}\\xspace for $\\calC$. For instance, \\textsc{ListIso}\\xspace is \\hbox{\\rm \\sffamily NP}\\xspace-complete for all\ngraph classes mentioned above (Corollary~\\ref{cor:NPc_lgi}).\n\n\\begin{result} \\label{res:combinatorial}\nCombinatorial techniques for \\textsc{GraphIso}\\xspace translate to \\textsc{ListIso}\\xspace.\n\\end{result}\n\nAs a by-product, our paper gives a nice overview of the main combinatorial techniques involved in\nattacking the graph isomorphism problem. These combinatorial techniques for \\textsc{GraphIso}\\xspace are often robust and\ntranslate to \\textsc{ListIso}\\xspace straightforwardly.  Moreover, we can describe them more naturally with lists.\n\nFor example, the bottom-up linear-time algorithm for testing graph isomorphism of (rooted) trees\ntranslates to \\textsc{ListIso}\\xspace in Theorem~\\ref{thm:trees}, since it captures all possible isomorphisms. The key\ndifference is that the algorithm for \\textsc{ListIso}\\xspace finds perfect matchings in bipartite graphs, in order to\ndecide whether lists of several subtrees are simultaneously compatible; see Fig.~\\ref{fig:example}b.\nWe use the algorithm of Hopcroft and Karp~\\cite{hopcroft_karp}, running in time $\\calO(\\sqrt n m)$.\n\nThe algorithms for graph isomorphism of planar, interval, permutation and circle graphs based on\ntree decompositions and translate to \\textsc{ListIso}\\xspace, as we show in Theorems~\\ref{thm:planar}, \\ref{thm:int},\n\\ref{thm:perm}, and~\\ref{thm:circle}. Even more involved algorithms for graphs isomorphism of\nbounded genus and bounded treewidth graphs translate to \\textsc{ListIso}\\xspace in Theorems~\\ref{thm:bounded_genus}\nand~\\ref{thm:fpt_treewidth}. The complexity for graphs with bounded rankwidth and graphs with\nexcluded minors remains open, see Conclusions for details.\n\n\\begin{result} \\label{res:group_theory}\nGroup theory techniques for \\textsc{GraphIso}\\xspace do not translate to \\textsc{ListIso}\\xspace.\n\\end{result}\n\nGroup theory techniques do not translate to \\textsc{ListIso}\\xspace since list-compatible automorphisms of a graph $G$\ndo not form a subgroup of ${\\rm Aut}(G)$. In Section~\\ref{sec:group_theory}, when automorphism groups are\nsufficiently rich, we show that \\textsc{ListIso}\\xspace remains \\hbox{\\rm \\sffamily NP}\\xspace-complete. In particular, we describe a non-trivial\nmodification of the original \\hbox{\\rm \\sffamily NP}\\xspace-hardness reduction of Lubiw~\\cite{lubiw1981some} to show that \\textsc{ListIso}\\xspace\nis \\hbox{\\rm \\sffamily NP}\\xspace-complete even for cubic colored graphs with color classes of size bounded by 16\n(Theorem~\\ref{thm:3reg_npc}). Therefore, no robust polynomial-time algorithm for cubic graph\nisomorphism exists.\n\n\n\\section{Planar Graphs} \\label{sec:planar_graphs}\n\nIn this section, we describe how to solve \\textsc{ListIso}\\xspace on planar graphs.\n\nFor the purpose of this section, we need to consider a more general definition of a graph. We work\nwith multigraphs and we admit pendant edges with free ends (which are edges attached to single\nvertices). Also, each edge $uv$ gives rise to two incident darts,\\footnote{In the standard\ndefinition of graphs, the primary objects are vertices and the secondary objects are edges. The\ndefinition via darts, from algebraic and topological graph theory, makes edges (or more precisely\ntheir halves) the primary objects, while the vertices are secondary objects. It is important because\nwe need to distinguish between an isomorphism which maps an edge and which also reflects it.}\none attached to $u$, the other to $v$.  Every isomorphism maps vertices and darts while preserving\nincidencies.  We consider the problem \\textsc{ListIso}\\xspace with lists on both vertices and darts.\n\n\\heading{3-connected Planar Graphs.}\nWe have a unique embedding into the sphere (up to the reflection). This embeddings can be described\nin the language of flags, which are pairs $(d,f)$ where $d$ is a dart and $f$ is an incident face.\nEvery automorphism of $G$ corresponds either to a direct map automorphism, or to a indirect map\nautomorphism (composed with a reflection). In particular, ${\\rm Aut}(G) \\cong {\\rm Aut}({\\mathcal M}} \\def\\calN{{\\mathcal N}} \\def\\calO{{\\mathcal O})$ acts\nsemiregularly on the set of flags of ${\\mathcal M}} \\def\\calN{{\\mathcal N}} \\def\\calO{{\\mathcal O}$.  See~\\cite{knz} for more details and references.\nTherefore, if the images of two consecutive darts in the rotational scheme are set, the entire\nmapping is determined and we just need to check whether it is an isomorphism.\n\n\\begin{lemma} \\label{lem:planar_3conn}\nThe problem \\textsc{ListIso}\\xspace (with lists on both vertices and darts) can be solved for 3-connected planar graphs\nin time $\\calO(\\ell)$.\n\\end{lemma}\n\n\\begin{proof}\nWe start by computing embeddings of both $G$ and $H$, in time $\\calO(n)$. It remains to decide whether\nthere exists a list-compatible isomorphism which has to be a map isomorphism. By Euler Theorem, we\nknow that the average degree is less than six. Consider all vertices of degree at most 5, let $u$ be\nsuch a vertex with a smallest list, and let $k = |\\frakL(u)|$. We have $\\ell = \\Omega(kn)$ and we show\nthat we can decide existence of a list-compatible isomorphism in time $\\calO(kn)$.\n\nWe test all possible mappings $\\pi : G \\to H$ having $\\pi(u) \\in \\frakL(u)$. For each, we have at\nmost 10 possible ways how to extend this mapping on the neighbors of $u$, and the rest of the\nmapping is uniquely determined by the embeddings and can be computed in time $\\calO(n)$. In the end, we\ntest whether the constructed mapping $\\pi$ is an isomorphism and whether it is list-compatible.\\qed\n\\end{proof}\n\n\\heading{3-connected Reduction.}\nA seminal paper by Trakhtenbrot~\\cite{trakhtenbrot} introduced reduction which decomposes a graph\ninto its \\emph{3-connected components}. This idea was further extended\nin~\\cite{tutte_connectivity,quadratic_isomorphism_planar,hopcroft_tarjan_dividing,cunnigham_edmonds,walsh}.\nWe use an augmentation described in~\\cite{fkkn,fkkn16,knz} which behaves well with respect to\nautomorphism groups. \n\nThe reduction is constructed by replacing atoms by colored possibly directed edges. \\emph{Atoms} are\nsubgraphs of the following three types (for precise definitions, see~\\cite{fkkn16}):\n\\begin{packed_itemize}\n\\item \\emph{Block atom.} Either a pendant star, or a pendant block with attached single pendant\nedges.\n\\item \\emph{Proper atom.} Inclusion minimal subgraphs separated by a 2-cut.\n\\item \\emph{Dipoles.} They are two vertices together with all (at least two) parallel edges between\nthem.\n\\end{packed_itemize}\nFurther, each atom $A$ has the \\emph{boundary} $\\bo A$ (of size at most 2) and the \\emph{interior}\n$\\hbox{\\rm \\sffamily INT}\\xspace A$. A graph is called \\emph{essentially 3-connected} if it is a 3-connected graph with\nattached single pendant edges attached. Similarly, a graph is called \\emph{essentially a cycle} if\nit is a cycle with attached single pendant edges.  It follows from~\\cite{fkkn16} that each block\natom is either a star, or essentially a cycle, or essentially 3-connected, or $K_2$ with a single\npendant edge attached. For a proper atom $A$ with $\\bo A = \\{u,v\\}$, we denote by $A^+$ the graph\nwith the added edge $uv$. The graph $A^+$ is always either essentially a cycle, or essentially\n3-connected.\n\nA proper atom or a dipole $A$ is called \\emph{symmetric} if there exists an automorphism in\n${\\rm Aut}(A)$ exchanging $\\bo A$, and \\emph{asymmetric} otherwise. Every block atom is symmetric by the\ndefinition. The reduction is done by finding all atoms in $G$ (by~\\cite{fkkn16}, they have disjoint\ninteriors) and replacing their interiors by edges. Further, we color these edges to code isomorphism\ntypes of atoms, and we use directed edges for asymmetric atoms. Block atoms are replaced by pendant\nedges with free ends.\n\nWe repeat this reductions over and over, which gives a sequence of graphs $G = G_0, \\dots, G_r$\nwhere $G_r$ is called \\emph{primitive} and contains no atoms. By~\\cite{fkkn16}, it is either\nessentially 3-connected, essentially a cycle, $K_2$ possibly with attached single pendant edges, or\n$K_1$ with an attached single pendant edge with a free end. Further, this reduction process can be\nencoded by the \\emph{reduction tree} $T_G$; see Fig.~\\ref{fig:reduction_tree} for an example.\nIt is a rooted tree, where each node is labeled by a graph. The root of $T_G$ is the primitive graph\n$G_r$. The other nodes correspond to atoms obtained in the reductions. When the interior of an atom\n$A$ is replaced by an edge $e$, we attach the node representing to $A$ to the edge $e$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=1]{planar_graphs_reduction_tree}\n\\caption{A graph $G$ together with its reduction tree $T_G$.}\n\\label{fig:reduction_tree}\n\\end{figure}\n\nIt easily follows that the reduction tree is unique and canonical. Further each automorphism $\\pi$\nof $G$ induces automorphisms of $G_1,\\dots,G_r$ by permuting edges exactly as atoms. Therefore, \nit induces an automorphism $\\pi'$ of $T_G$ which permutes the nodes of isomorphic graphs, and when\nit maps a colored edge $e$ to a colored edge $e'$, it maps the subtree attached to $e$ to the\nisomorphic subtree attached to $e'$. And every automorphism of $G$ can be constructed in this way,\nfrom the root of $T_G$ to the bottom. We can use this to solve \\textsc{ListIso}\\xspace.\n\n\\begin{theorem} \\label{thm:3conn_reduction}\nLet $\\calC$ be a class of graphs closed under contractions and removing vertices. Suppose that \\textsc{ListIso}\\xspace\nwith lists on both vertices and darts can be solved for 3-connected graphs in $\\calC$ in time\n$\\varphi(n,m,\\ell)$. We can solve \\textsc{ListIso}\\xspace on $\\calC$ in time $\\calO(\\sqrt m \\ell + m +\n\\varphi(n,m,\\ell))$.\n\\end{theorem}\n\n\\begin{proof}\nWe compute reduction trees $T_G$ and $T_H$ for both $G$ and $H$ in time $\\calO(n+m)$.  We apply the\nidea of Theorem~\\ref{thm:trees} to test list-compatible isomorphism of $T_G$ and $T_H$. We compute\nthe lists $\\frakL(N)$ for the nodes $N$ of $T_G$, from the bottom to the root. A node $M \\in \\frakL(N)$ if\nthere exists a list-compatible isomorphism from $N$ to $M$ mapping $\\bo N$ to $\\bo M$ and there\nexists list-compatible isomorphism between attached subtrees. (Further, if $|\\bo N| = |\\bo M| = 2$,\nwe remember in $\\frakL(N)$ which of both possible mappings of $\\bo N$ to $\\bo M$ can be extended as\nlist-compatible isomorphisms.)\n\nSuppose that $N$ has the children $N_1,\\dots,N_k$ with computed lists and $M$ has the children\n$M_1,\\dots,M_k$.  There exists a list-compatible isomorphism mapping the subtree of $N_i$ to the\nsubtree of $M_j$, if and only if $M_j \\in \\frakL(N_i)$. The difference from Theorem~\\ref{thm:trees} is\nthese subtrees have to be compatible with a list-isomorphism from $N$ to $M$; so it depends on the\nstructure of the nodes $N$ and $M$.\n\nThere, we compute $\\frakL(N)$ differently according to the type of $N$:\n\\begin{packed_itemize}\n\\item \\emph{Star block atoms or dipoles.} For star block atoms, similarly as in\nTheorem~\\ref{thm:trees}, we construct a bipartite graph between $N_1,\\dots,N_k$ and $M_1,\\dots,M_k$\nand test existence of a perfect matching using Lemma~\\ref{lem:bipmatch}. For dipoles, we test two\npossible isomorphisms, construct two bipartite graph and test existence of perfect matchings.\n\\item \\emph{Non-star block or proper atoms.} We modify the lists of $\\bo N$ to the vertices of $\\bo\nM$ only. (When they are proper atoms, we run this in two different ways.) We encode the lists\n$\\frakL(N_1),\\dots,\\frakL(N_k)$ by lists on the corresponding darts of $N$ (depending on which of two\npossible list-isomorphisms of $\\bo N_i$ are possible), and we remove single pendant edges, and\nintersect their lists with the lists of the incident vertices. For a proper atom, we further\nconsider $N^+$ and $M^+$ with added edges $e$ and $f$ such that $\\frakL(e) = \\{f\\}$. If the nodes are\n$K_2$ or cycles, and we can test existence of a list-compatible isomorphism using\nLemma~\\ref{lem:cycles}. If both are 3-connected, we can test it by our assumption in time\n$\\varphi(n',m',\\ell')$. If this list-compatible isomorphism exists, we add $M$ to $\\frakL(N)$.\n\\item \\emph{The root primitive graphs.} We use the same approach as above, ignoring the part\nabout $\\bo N$ and $\\bo M$.\n\\end{packed_itemize}\nA list-compatible isomorphism from $G$ to $H$ exists, if and only if $M \\in \\frakL(N)$ for the root\nnodes $N$ and $M$ of $T_G$ and $T_H$.\n\nThe correctness of the algorithm can be argued from the fact that all automorphisms are captured by\nthe reduction trees~\\cite{fkkn16}, inductively from the top to the bottom as in\nTheorem~\\ref{thm:trees}.  It remains to discuss the running time. The reduction trees can be\ncomputed in linear time~\\cite{hopcroft_tarjan_dividing}. When computing $\\frakL(N)$, we first consider\nthe lists of all vertices and edges of $N$. A node $M$ is a candidate for $\\frakL(N)$, if every vertex\nand every edge of $N$ has a vertex\/edge of $M$ in its list. Therefore, we can find all these\ncandidate nodes by iterating these lists, in linear time with respect to their total size. Let $M$\nbe one of them, and let $n' = |V(N)|$, $m' = |E(N)|$ and $\\ell'$ be the total size of lists of the\nvertices and edges of $N$ when restricted only to the vertices and edges of $M$. Either we construct\na bipartite graph and test existence of a perfect matching in time $\\calO(\\sqrt m' \\ell')$, or we test\nexistence of a list-compatible isomorphism in time $\\varphi(n',m',\\ell')$. The total running time\nspent on the tree is $\\calO(\\ell)$, the total running time spent testing perfect matchings is $\\calO(\\sqrt\nm \\ell)$, and the total running time testing list-compatible isomorphisms of 3-connected graphs is\n$\\calO(\\varphi(n,m,\\ell))$.\\qed\n\\end{proof} \n\n\\heading{General Planar Graphs.}\nBy putting both results together, we get the following result:\n\n\\begin{theorem} \\label{thm:planar}\nThe problem \\textsc{ListIso}\\xspace can be solved for planar graphs in time $\\calO(\\sqrt n \\ell)$.\n\\end{theorem}\n\n\\begin{proof}\nIf $G$ and $H$ are connected, we use Theorem~\\ref{thm:3conn_reduction}.  By\nLemma~\\ref{lem:planar_3conn}, the function $\\varphi(n,m,\\ell)$ is $\\calO(\\ell)$. If $G$ and $H$ are\ndisconnected, we apply Lemma~\\ref{lem:disconnected} on all connected components of $G$ and $H$, and\nby analysing the proof, the total running time is $\\calO(\\sqrt n \\ell)$.\\qed\n\\end{proof}\n\n\\section{Preliminaries and Outline}\n\nLet $G$ be an input graph of \\textsc{ListIso}\\xspace or \\textsc{ListAut}\\xspace.  We denote the set of its vertices by $V(G)$ and the set\nof its edges by $E(G)$.  Let $n = |V(G)|$, $m = |E(G)|$ and $\\ell$ be the total size of all lists.\nTo make the problem non-trivial, we can assume that $\\ell \\ge n$.\n\n\\heading{Bipartite Perfect Matchings.}\nAs a subroutine, we frequently solve \\emph{bipartite perfect matching}:\n\n\\begin{lemma}[Hopcroft and Karp~\\cite{hopcroft_karp}] \\label{lem:bipmatch} The bipartite perfect\nmatching problem can be solved in time $\\calO(\\sqrt{n}m)$, where $n$ is the number of vertices and $m$\nis the number of edges.  \\end{lemma}\n\n\\noindent For instance, when both $G$ and $H$ are independent sets, existence of a list-compatible\nisomorphism is equivalent to existence of a perfect matching between the lists of $G$ and the\nvertices of $H$. Therefore, using the algorithm of Hopcroft and Karp, we get the running time\n$\\calO(\\sqrt n \\ell)$ while the input size is $\\Omega(n+\\ell)$. Finding bipartite perfect matchings is\nthe bottleneck in many of our algorithms and cannot be avoided: if it cannot be solved in linear\ntime, \\textsc{ListIso}\\xspace for many graph classes cannot be solved in linear time as well.\n\n\\heading{Outline: Main Points of This Paper.}\nIn Section~\\ref{sec:basic_results}, we prove some basic results for \\textsc{ListIso}\\xspace such as polynomial-time\nequivalence of \\textsc{ListIso}\\xspace and \\textsc{ListAut}\\xspace and polynomial-time algorithms when maximum degree is 2 or all lists\nare of size at most 2.\n\nIn Section~\\ref{sec:gi_completeness}, we give a formal description of\nResult~\\ref{res:gi_completeness}.  We study polynomial-time reductions $\\psi$ for \\textsc{GraphIso}\\xspace from a graph\nclass $\\calC$ to another graph class $\\calC'$: for each graph $G \\in \\calC$, the reduction $\\psi$\nproduces another graph $G' \\in \\calC'$ such that $G \\cong H$ if and only if $G' \\cong H'$.  When\n$\\psi$ uses vertex-gadgets, it can be modified to a polynomial-time reduction for \\textsc{ListIso}\\xspace from $\\calC$ to\n$\\calC'$. The vertex-gadget assumption means the following: in $G'$, each vertex $V(G)$ is replaced\nby a small vertex-gadget while all automorphisms of $G'$ preserve vertex-gadgets and automorphisms\nof $G$ induce automorphisms of $G'$ and vice versa.\n\nIn Section~\\ref{sec:group_theory}, we give a formal description of Result~\\ref{res:group_theory}. We\nshow that \\textsc{ListIso}\\xspace remains \\hbox{\\rm \\sffamily NP}\\xspace-complete for cubic colored graphs when each color class is of size at\nmost 16 and each list is of size at most 3. We modify the reduction of Lubiw~\\cite{lubiw1981some} in\ntwo steps. First, we reduce the problem from (positive) 1-in-3 SAT instead of 3-SAT. Therefore, only\npositive literals appear and we reduce the sizes of lists from 7 to 3. Second, we modify variable\nand clause gadgets to make the graph cubic.\n\nIn the remaining sections, we give a formal description of Result~\\ref{res:combinatorial}.\nIn Section~\\ref{sec:trees}, we modify the basic algorithm for graph isomorphism of (rooted) trees.\nTo deal with lists, we solve several bipartite perfect matching subroutines to test whether subtrees\nare simultaneously list-compatible. The idea of this algorithm for \\textsc{ListIso}\\xspace of trees is used in some\nother combinatorial algorithms.\n\nIn Section~\\ref{sec:planar_graphs}, we describe that every planar graph can be decomposed into a tree\nof its 3-connected components. Since \\textsc{ListIso}\\xspace can be easily solved on 3-connected planar graphs (using\ngeometry and uniqueness of embedding), we apply dynamic programming on this tree and solve \\textsc{ListIso}\\xspace for\ngeneral planar graphs as well.\n\nIn Section~\\ref{sec:intersection}, we describe how to modify the algorithms for graph isomorphism of\ninterval, permutation and circle graphs. Similarly, they can be represented by MPQ-trees, modular\ntrees and split trees. Since we can solve \\textsc{ListIso}\\xspace on the graphs induced by nodes, we apply dynamic\nprogramming on these trees and solve \\textsc{ListIso}\\xspace on interval, permutation and circle graphs as well.\n\nIn Section~\\ref{sec:bounded_genus}, we modify the algorithm of Kawarabayashi~\\cite{kawarabayashi}\nto solve \\textsc{ListIso}\\xspace on graphs of bounded genus. This algorithm either uses a small number of possible\nembeddings (which translates to \\textsc{ListIso}\\xspace), or finds a small cut of size at most 4 which is canonical,\nsplits the graph and test graph isomorphism of both pieces (which again translates to \\textsc{ListIso}\\xspace).\n\nIn Section~\\ref{sec:bounded_treewidth}, we modify Bodlaender's \\hbox{\\rm \\sffamily XP}\\xspace\nalgorithm~\\cite{iso_xp_treewidth} for graph isomorphism of graphs of bounded treewidth. The problem\nis non-trivial since tree decomposition is not canonical. Therefore, it is a dynamic algorithm\nrunning over all potential bags, and it can be easily modified with lists. Lokshtanov et\nal.~\\cite{iso_fpt_treewidth} obtain an \\hbox{\\rm \\sffamily FPT}\\xspace running time by computing a smaller set of potential\nbags which is canonical.\n\nIn Section~\\ref{sec:conclusions}, we conclude this paper with a group reformulation, related results\nand open problems.\n\n\\section{Trees} \\label{sec:trees}\n\nIn this section, we modify the standard algorithm for tree isomorphism to solve list restricted\nisomorphism of trees. We may assume that both trees $G$ and $H$ are rooted, otherwise we root them\nby their centers (and possibly subdivide the central edges). The algorithm for \\textsc{GraphIso}\\xspace process both\ntrees from bottom to the top. Using dynamic programming, it computes for every vertex possible\nimages using possible images of its children. This algorithm can be modified to \\textsc{ListIso}\\xspace.\n\n\\begin{theorem} \\label{thm:trees}\nThe problem \\textsc{ListIso}\\xspace can be solved for trees in time $\\calO(\\sqrt{n}\\ell)$.\n\\end{theorem}\n\n\\begin{proof}\nWe apply the same approach with lists and update these lists as we go from bottom to the top.\nAfter processing a vertex $u$, we compute an updated list $\\frakL'(u)$ which contains all elements\nof $\\frakL(u)$ to which $u$ can be mapped compatibly with its descendants. To initiate, each leaf\n$u$ of $G$ has $\\frakL'(u) = \\{w : \\text{$w$ is a leaf and $w \\in \\frakL(u)$}\\}$.\n\nNext, we want to compute $\\frakL'(u)$ and we know $\\frakL'(u_i)$ of all children $U =\n\\{u_1,\\dots,u_k\\}$ of $u$.  For each $w \\in \\frakL(u)$ with $k$ children $w_1,\\dots,w_k$, we want to\ndecide whether to put $w \\in \\frakL'(u)$.  Let $W = \\{w_1,\\dots,w_k\\}$. Each $u_i$ can be mapped to\nall vertices in $\\frakL'(u_i) \\cap W$. We need to decide whether all $u_i$'s can be mapped\nsimultaneously.  Therefore, we form a bipartite graph $B(U,W)$ between $U$ and $W$: we put an edge\n$u_iw_j$ if and only if $w_j \\in \\frakL'(u_i)$. Simultaneous mapping is possible if and only if there\nexists a perfect matching in this bipartite graph.\n\nLet $r$ be the root of $G$ and $r'$ be the root of $H$.  We claim that there is a list-compatible\nisomorphism $\\pi : G \\to H$, if and only if $\\frakL'(r) = \\{r'\\}$. Suppose that $\\pi$ exists. When\n$\\pi(u) = w$, its children $U$ are mapped to $W$. Since this mapping is compatible with the lists,\n$w \\in \\frakL(u)$, and the mapping of $u_1,\\dots,u_k$ gives a perfect matching in $B(U,W)$.\nTherefore, $w \\in \\frakL'(u)$, and by induction $r' \\in \\frakL'(r)$. On the other hand, we can construct\n$\\pi$ from the top to the bottom. We start by putting $\\pi(r) = r'$. When $\\pi(u) = w$, we map its\nchildren $U$ to $W$ according to some perfect matching in $B(U,W)$ which exists from the fact that\n$w \\in \\frakL'(u)$.\n\nIt remains to argue details of the complexity. We process the tree which takes time $\\calO(\\ell)$\n(assuming $n \\le \\ell$) and we process each list constantly many times which takes $\\calO(\\ell)$.\nSuppose that we want to compute $\\frakL'(u)$. We consider all vertices $w^1,\\dots,w^p \\in \\frakL(u)$, and\nlet $W^j$ be the children of $w^j$.  We go through all lists of $\\frakL'(u_1),\\dots,\\frakL'(u_k)$ in linear\ntime, and split them into sublists $\\frakL'(u_i^j)$ of vertices whose parent is $w^j$. Only these\nsublists are used in the construction of the bipartite graph $B(U,W^j)$.  Using\nLemma~\\ref{lem:bipmatch}, we decide existence of a perfect matching in time $\\calO(\\sqrt{k}\\ell_j)$\nwhich is at most $\\calO(\\sqrt{n} \\ell_j)$, where $\\ell_j$ is the total size of all sublists\n$\\frakL'(u_i^j)$.  When we sum this complexity for all vertices $u$, we get the total running time\n$\\calO(\\sqrt{n} \\ell)$.\\qed\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nDynamic stability analysis of power grids concerns the investigation of transient stability and power quality. Especially with more intermittent renewable generation sudden changes in power followed by the transient reaction of frequency and voltage dynamics may present a challenge to power system stability. At the same time, fluctuations also challenge power quality, which refers to whether voltage and frequency stay within safety bounds and whether the waveform has undesired harmonic distortions. \nFurthermore, the wide deployment of power electronics, witch RES are connected to the grid, introduces new features such as delayed reaction into future power grids.\n \nThese challenges have been the topic of recent research on the stability of renewable power grids \\cite{auer2017stability,auer2018stability,schafer2016taming}. However, the usage of state-of-the-art Open-Source software repeatedly posed limitations to the possible modeling scope. One reason is the lack of solvers for the different types of dynamics which need to be described a stochastic differential equations, delay differential equations and differential algebraic equations. At the same time to network size was constrained by computational performance. This means a realistic model implementation and hence reliable stability analysis has been constrained by the available OS software framework. \n\nIn this conference paper, we give a sneak preview to \\pd{}, an new Open-Source approach to dynamic power grid modeling\\footnote{This paper is meant as a sneak preview, as the planned publication date is Oct 15, 2018, i.e.\\ before the Wind Integration Workshop 2018 will take place.}.\nWhile there are multiple modeling suites available they are either proprietary (e.g.\\  PSS\\textregistered{} \\cite{PSS}) or need proprietary tools to be used (e.g.\\ PSAT \\cite{PSAT, milano2008open}). With \\pd{}, we aim to provide a modeling framework that tackles all the needs a modeler for power grids with high shares of renewables to execute dynamic analyses and the first publication will provide tools for time-domain modeling.\n\nAt the moment, \\pd{} is under high development, with the basic software architecture already being developed and implemented. We are currently adding new types of dynamics and controls to represent different buses, e.g. droop control, multiple inverter types, higher-order representations of synchronous machines, and in particular intermittency representations of renewable energy sources. A second focus of effort goes into usability, in order to improve the experience of modelers and decrease their time they have to concern with software-related problems so they can focus on their modeling instead. Our vision is to provide a modular, fully flexible library where modelers can decide whether they (a) want to simply use repdefined bus models to analyze a power grid quickly, (b) want to into all the detail of modeling each bus type, or (c) want to find their right position in-between (a) and (b). All of that is done in the context of an Open-Source context in order to make it available to and invitre contributions from research and industry.\n\n\\pd{} is written in Julia \\cite{bezanson2017julia}, a new, fast developing programming language targeting the scientific computing community. With its recent release to Julia 1.0, it has matured enough to become a powerful player competing with more traditional programming languages like Python and C in the modeling world.\nIn the wider context, there is a current movement within the Open Energy Modeling community \\cite{openmod} to switch to Julia and unite modeling efforts. With \\pd{} we are aiming to support these efforts by contributing to dynamic power system analysis.\n\nThe rest of the paper is structured as follows: in \\Cref{sec:julia} ``\\nameref{sec:julia}'' we give a detailed reasoning why we find Julia to be the correct choice as the programming language for our modeling effort; in \\Cref{sec:powerdynamics} ``\\nameref{sec:powerdynamics}'' we introduce the basic mathematical notions we use in \\pd{}; in \\Cref{sec:ieee14} ``\\nameref{sec:ieee14}'' we provide the details of the example system used in this paper; in \\Cref{sec:implementation} ``\\nameref{sec:implementation}'' we show how to implement the IEEE 14-bus feeder in \\pd{}; in \\Cref{sec:modeling} ``\\nameref{sec:modeling}'' we present the results for our example model; and in \\Cref{sec:conclusion} ``\\nameref{sec:conclusion}'' we conclude the paper.\n\n\\subsection*{Convention \\& Notation}\n\nWithin this paper, we take up the convention of \\pd{} where minor letters are used as variables for dynamic variables and capital letters denote (constant) parameters and all variables are in defined in the co-rotating frame.\n\nFurther, we take $u_a$ to to be the complex voltage, $v_a = |u_a|$ to be the voltage magnitude, $\\varphi_a = \\arg(u_a)$ to be the voltage angle, $i_a = \\sum_b \\LY{ab}u_b$ to be the complex current, $s_a = u_a \\cdot i_a^*$ to be the complex power, $p_a = \\Re(s_a)$ the active power and $q_a = \\Im(s_a)$ to be the reactive power of the \\nth{$a$} bus. \\LY{} is the admittance Laplacian. The imaginary element is denoted as $j = \\sqrt{-1}$.\n\n\\section{Julia for Scientific Modeling}\n\\label{sec:julia}\n\nThe advantages of Julia for scientific modeling can be summarized as (adapted from \\cite{julia}):\n\\begin{itemize}\n\t\\item \\textbf{Julia is fast!} It was designed from the beginning as a high-performance language. Using the just-in-time (JIT) compilation method, it produces efficient native code for multiple platforms.\n\t\\item \\textbf{Julia allows for rapid development!} It uses dynamic typing, so it is easy to use, feels like a scripting language and has good interactive use without losing any performance. That way, it saves the developer a lot of time.\n\t\\item \\textbf{Julia is technical!} It is developed for numerical computing, the syntax is focusing on precise mathematics and many datatypes and even parallelism is available out of the box.\n\t\\item \\textbf{Julia is general!} Using multiple dispatch as the fundamental paradigm, it is easy to express object-oriented and functional programming patterns at the same time.\n\t\\item \\textbf{Julia is composable!} Julia has been designed such that independent packages work well together. Hence, we can use matrices of unit quantities or differentiation with sparse matrices without having these types ever defined explicitly before.\n\t\\item \\textbf{Julia has a growing ecosystem!} The aforementioned advantages attract a growing community\\footnote{The community growth recently reached the point where the organizers had to cut off ticket sales for this year's JuliaCon at some point.} and with that a growing ecosystem of Julia packages. Particularly important for modeling is the \\diffeq{} library which provides high-performance solvers and interfaces to industrial grade solvers like \\sundials{} \\cite{hindmarsh2005sundials} for solving multiple kinds of differential equations (ordinary, algebraic, delayed, stochastic).\n\t\\item \\textbf{Julia allows for metaprogramming!} This means, at execution time, the source code is available as data and one can easily modify it. Furthermore, one can even add own syntax to Julia. Within \\pd{}, we make heavy use of metaprogramming to make it as simple as possible for a user to implement her\/his own kind of bus model (see \\Cref{sec:implementation-nodes}). Also, it allows to write very simple code that can automatically be modified to highly optimized code.\n\\end{itemize}\n\nOf course, this is just a simply overview and we would strongly recommend the reader to try out Julia and convince herself\/himself\\footnote{\\url{https:\/\/julialang.org\/learning\/}}.\n\n\\section{Mathematics of \\pd{}}\n\\label{sec:powerdynamics}\n\nWithin this paper, we take the approach to describe the dynamics of a power grid by a (set of) semi-explicit differential algebraic equation(s). We explicitly distinguish between the complex voltages $u_a$ (of the \\nth{$a$} bus), whose information is transmitted between buses directly through the complex current $i$, and internal variables $x_{ab}$ (the \\nth{$b$} internal variable of the \\nth{$a$} bus), that exist locally at a node. An example for an internal variable would be the frequency of a synchronous machine. While the phase angle $\\varphi$ is transmitted directly via the current, its derivative is the angle velocity whose dynamics is defined locally by the the rotating mass. We are aware that a multitude of approaches are possible to write down the dynamics and we have chosen this particular one as it can be easily translated into source code. Writing this idea in formulas yields\n\\begin{subequations}\n\\begin{align}\ni_a = \\sum_c \\LY{ac} \\cdot u_c , \\label{eq:i-definition} \\\\\nm^u_a \\dot u_a = f_a(u_a, x_a, i_a)  ,\\label{eq:u-general} \\\\\nm^x_{ab} \\dot x_{ab} = g_{ab}(u_a, x_a, i_a) , \\label{eq:x-general}\n\\end{align}\n\\end{subequations}\nwhere $m^u$ \/ $m^x$ are the masses for the voltages \/ internal variables respectively\\footnote{Within \\pd{} we allow masses to be $1$ or $0$.}. Also, $f_a$ encodes the voltage dynamics of the \\nth{$a$} bus and $g_{ab}$ the dynamics of the \\nth{$b$} internal variable of the \\nth{$a$} bus. Note the convention of using minor letters for dynamical variables.\n\n\\LY{} is the admittance Laplacian. So having the admittances encoded as $Y_{ab}$ where $Y_{ab} = 0$ if there is no line, or $Y_{ab}$ with a complex value (with $\\Re(Y_{ab})\\geq 0$) for existing lines, then the admittance Laplacian is defined as \n\\begin{align}\n\\LY{ab} = \\delta_{ab}\\sum_c Y_{ac} - Y_{ab}, \n\\end{align}\nwhere $\\delta_{ab}$ is Kronecker-$\\delta$.\nParticular examples would be an algebraic slack bus as the \\nth{$a$} bus with a fixed complex voltage $U_a$. There are no internal variables and the voltage mass is $m_a^u = 0$ (as its an algebraic equation), so it can simply be written as\n\\begin{align}\n0\\cdot \\dot u_a = f_a(u_a, x_a, i_a) = u_a - U_a , \\label{eq:slack-bus}\n\\end{align}\nwhere we have explicitly not removed $\\dot u$ in order to keep a strong analogy the the implementation shown later.\nSimilarly, we can treat an algebraic load at the \\nth{$a$} bus with a constant complex power $-S_a$ flowing out\\footnote{The $-$ is due to the definition of all variables such that the power flows always from the respective node into the power grid.}. Again, there are no internal variables and the voltage mass is $0$ so it reduces to\n\\begin{align}\n0\\cdot \\dot u_a = f_a(u_a, x_a, i_a) = s_a - S_a = u_a\\cdot i_a^* - S_a .\n\\end{align}\nA synchronous machine as the \\nth{$a$} bus can be represented by the Swing equation (or $2^{\\text{nd}}$-order synchronous machine model) \\cite{sauer1998power,kundur1994power} with a produced active power $P_a$, the damping constant $D_a$, the rated frequency $\\Omega$ and the inertia $H_a$. In this case, the voltage mass is $1$, there is one internal variable, the frequency, $x_{a1} = \\omega_a$ with a mass of $1$ because it is dynamic and not an algebraic constraint\\footnote{In vector notation this gives $x_{a} = (\\omega_a)$ for the Swing equation.}. Hence, the equations reduce to\n\\begin{subequations}\n\\begin{align}\n&\\dot u_a = f_a(u_a, (\\omega_a), i_a) = u_aj\\omega_a , \\label{eq:swing-u}  \\\\\n&\\begin{aligned}\n\\dot x_{a1} = \\dot\\omega_a &= g_{a1}(u_a, (\\omega_a), i_a)\\\\\n&= \\frac{2\\pi\\Omega}{H_a}\\left( P_a - D_a\\omega_a - \\Re(u_a\\cdot i_a^*)  \\right) \n\\end{aligned} \\label{eq:swing-omega}\n\\end{align}\n\\end{subequations}\nwhere we would like to remind the reader of $j$ being used as the imaginary element in this paper. Note that with $u_a = e^{j\\varphi_a}\\ \\implies\\ du_a = ju_ad\\varphi_a$, \\Cref{eq:swing-u} reduces to $\\dot\\varphi_a = \\omega_a$ and one recovers the usual version in terms of the voltage angle.\n\nIn the following section, we will introduce the example system in order to demonstrate afterwards, how these kind of dynamics are then described in \\pd{}.\n\n\\section{The IEEE 14-bus distribution grid feeder}\n\\label{sec:ieee14}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{ieee14grid}\n\t\\caption{Technical representation of the IEEE 14-bus test system, extracted from \\cite{kodsi2003modeling}. It contains five synchronous machines of which two are generators. The one at bus 2 is taken as the slack bus for the modeling in this paper. Furthermore, there are 8 loads (buses 4, 5, 8, 9, 10, 11, 12, 13, and 14). Bus 7 is for the representation of the transformer.}\n\t\\label{fig:ieee14grid}\n\\end{figure}\n\nThe IEEE 14-bus system is a representation of a medium-voltage distribution grid. A graphical representation is in \\Cref{fig:ieee14grid}, which is extracted from \\cite{kodsi2003modeling}. There are two generators, where the one at bus 2 is the slack bus, three synchronous compensators, and eight (complex) loads. Bus 7 is for the representation of the transformer only.\n\nThe generator at bus 1 and the synchronous compensators will be modeled with the swing equation, the slack bus 2 with the algebraic constraint introduced in \\Cref{sec:powerdynamics} and the loads with the algebraic constraints for complex loads (see \\Cref{sec:powerdynamics} also). The bus parameters haven been extracted from \\cite{kodsi2003modeling} and are presented in \\Cref{tab:ieee14-bus-parameters}. The resistance $R$ and reactance $X$ for the lines (taken from \\cite{kodsi2003modeling} as well) are summarized in \\Cref{tab:ieee14-line-parameters}.\n\nHaving a description of the power grid in question enables us to have a look at the implementation in \\pd{} now.\n\n\\setlength\\tabcolsep{1.5mm}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{table}[!t]\n\\caption{Bus parameters of the IEEE 14-bus test system. INERTIA UNIT IS?}\n\\label{tab:ieee14-bus-parameters}\n\\centering\n\\begin{tabular}{|c|c||c|c|c|c|c|}\n\\toprule\nBus & Type & $U\\,[p.u.]$ & $P\\,[p.u.]$ & $Q\\,[p.u.]$ & $D\\,[p.u.]$ & $H[p.u.\\cdot s]$ \\\\\\midrule\n1 & Generator &  & 2.32 & & 2 & 5.148\\\\\\hline\n2 & Slack & 1 & & & & \\\\\\hline\n3 & Syn. Comp. & & -0.942 & & 2 & 6.54 \\\\\\hline\n4 & Load & & -0.478 & 0 & & \\\\\\hline\n5 & Load & & -0.076 & -0.016 & & \\\\\\hline\n6 & Syn. Comp. & & -0.112 & & 2 & 5.06 \\\\\\hline\n7 & Load & & 0 & 0 & & \\\\\\hline\n8 & Syn. Comp. & & 0 & & 2 & 5.06 \\\\\\hline\n9 & Load & & -0.295 & -0.166 & & \\\\\\hline\n10 & Load & & -0.09 & -0.058 & & \\\\\\hline\n11 & Load & & -0.035 & -0.018 & & \\\\\\hline\n12 & Load & & -0.061 & -0.016 & & \\\\\\hline\n13 & Load & & -0.135 & -0.058 & & \\\\\\hline\n14 & Load & & -0.149 & -0.05 & & \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!t]\n\\caption{Line parameters of the IEEE 14-bus test system.}\n\\label{tab:ieee14-line-parameters}\n\\centering\n\\begin{tabular}{|c||c|c|c|c|}\n\\toprule\nline number & from bus & to bus & $ R\\, [p.u.]$ & $ X\\, [p.u.]$ \\\\\\midrule1 & 1 & 2 & 0.01938 & 0.05917\\\\\\hline2 & 1 & 5 & 0.05403 & 0.22304\\\\\\hline3 & 2 & 3 & 0.04699 &0.19797\\\\\\hline4 & 2 & 4 & 0.05811 & 0.17632\\\\\\hline5 & 2 & 5 & 0.05695 & 0.17388\\\\\\hline6 & 3 & 4 & 0.06701 & 0.17103\\\\\\hline7 & 4 & 5 & 0.01335 & 0.04211\\\\\\hline8 & 4 & 7 & 0.0 & 0.20912\\\\\\hline9 & 4 & 9 & 0.0 & 0.55618\\\\\\hline10 & 5 & 6 & 0.0 & 0.25202\\\\\\hline11 & 6 & 11 & 0.09498 & 0.1989\\\\\\hline12 & 6 & 12 & 0.12291 & 0.25581\\\\\\hline13 & 6 & 13 & 0.06615 & 0.13027\\\\\\hline14 & 7 & 8 & 0.0 & 0.17615\\\\\\hline15 & 7 & 9 & 0.0 & 0.11001\\\\\\hline16 & 9 & 10 & 0.03181 & 0.0845\\\\\\hline17 & 9 & 14 & 0.12711 & 0.27038\\\\\\hline18 & 10 & 11 & 0.08205 & 0.19207\\\\\\hline19 & 12 & 13 & 0.22092 & 0.19988\\\\\\hline20 & 13 & 14 & 0.17093 & 0.34802\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Implementation in PowerDynamics.jl}\n\\label{sec:implementation}\n\nIn this section, we will show first how to implement the dynamics for the different buses presented in \\Cref{sec:powerdynamics} in \\pd{} and then how create a power grid model from these bus dynamics definitions, in this case the IEEE 14-bus system.\n\n\\subsection{Defining the dynamics}\n\\label{sec:implementation-nodes}\n\nThe implementation of such a power grid in \\pd{} is strongly based on the \\dynamicnodemacro{} macro. A macro is a metaprogramming function that takes part of the source codes and modifies it at parsing time before giving it to the compiler, that creates the actual bitcode. In case of \\dynamicnodemacro{}, this is used to hide all the complicated internals of \\pd{} and make the definition of a type of dynamics as easy as possible. The following examples are already provided by default in \\pd{} but using them as an example, any kind of dynamics can be implemented.\n\n\nUsing \\dynamicnodemacro{}, the slack bus constraint from \\Cref{eq:slack-bus} is implemented as\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\n@DynamicNode SlackAlgebraic(U) <: OrdinaryNodeDynamicsWithMass(m_u=false) begin\nend [] begin\n    du = u - U\nend\n\\end{lstlisting}\nwhere line breaks and line numbers have been added for easier orientation.\nThe left part on the first line states the name of the new type \\texttt{SlackAlgebraic} and the parameter name \\texttt{U}. Separated by the subtyping operator (\\texttt{<:}) comes the statement that it can be represented by an ordinary differential equation with a mass term, where the mass for the voltage \\verb|m_u| is set to \\texttt{false}, meaning $m^u = 0$ from \\Cref{eq:u-general} as it is a constraint.  The \\verb|[]| in line 2 states that there are no internal variables. And line 3 simply provides the formula for the dynamics and assigns it to \\texttt{du}. As \\verb|m_u = false| is given, this simply translates it to a constraint.\n\nIn a similar manner, the load is defined as an algebraic PQ-constraint\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\n@DynamicNode PQAlgebraic(S) <: OrdinaryNodeDynamicsWithMass(m_u=false)  begin\nend [] begin\n    s = u*conj(i)\n    du = S - s\nend\n\\end{lstlisting}\nIn this case, an additional line was added to calculate the outflowing complex power \\texttt{s} from the voltage \\texttt{u} and the current \\texttt{i} and then the definition of the constraint is written in line 4.\n\nConcerning the generator and the synchronous compensators, we implemented the swing equation as\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\n@DynamicNode SwingEq(H, P, D, SOmega) <: OrdinaryNodeDynamics() begin\n\t@assert D > 0 \"damping (D) should be >0\"\n\t@assert H > 0 \"inertia (H) should be >0\" \n\tSOmega_H = SOmega * 2pi \/ H\nend [[Somega, dSomega]] begin\n\tp = real(u * conj(i))\n\tdu = u * im * Somega\n\tdSomega = (P - D*Somega - p)*SOmega_H\nend\n\\end{lstlisting}\nAgain, the left part of the first line provides the name of the new type and the name of the parameters. Lines 2 to 4 are code that should be run only once. In this case these are consistency checks, that the damping and inertia are positive, and the reduction of the rated frequency $\\mathtt{\\Omega}$ and the inertia $\\mathtt{H}$ to a single variable $\\mathtt{\\Omega\\_H}$. In line 5, the variable name of the internal variable $\\mathtt{\\omega}$ and the name for its derivative $\\mathtt{d\\omega}$ are given. Finally, lines 6 to 8 implement \\Cref{eq:swing-u,eq:swing-omega} by simply writing down the mathematical terms.\n\n\\subsection{Instantiating the power grid model}\n\\label{sec:implementation-grid}\n\nIn order to create the grid model, we first have to instantiate the bus models simply by calling them with the corresponding parameter values from \\Cref{tab:ieee14-bus-parameters}, e.g.:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nSwingEq(H=5.148, P=2.32, D=2, SOmega=50) # for bus 1\nPQAlgebraic(S=-0.295-0.166im) # for bus 9\n\\end{lstlisting}\nWithin the actual code we simply loaded the data from a \\texttt{.csv} file into and automatized this instantiation\\footnote{The full source code is not part of this paper as it would be too long, but it will be published along with \\pd{}.}. The instantiated bus models should then be saved in an array called e.g. \\texttt{nodes}. Similarly, the admittance Laplacian should be generated from the line data in \\Cref{tab:ieee14-line-parameters} and saved in a matrix called e.g. \\texttt{LY}. The actual grid model instatiation is then simply one line where the model is saved in the variable \\texttt{g}:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\ng = GridDynamics(node_list, LY)\n\\end{lstlisting}\nNow, \\texttt{g} contains all the information of the power grid and in the following section we will show how to solve it.\n\n\n\\section{Modeling Results}\n\\label{sec:modeling}\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\subfloat{\\includegraphics[width=\\columnwidth]{ieee14-frequency-perturbation}\n\t\\label{fig:ieee14-frequency-perturbation}}\n\t\\subfloat{\\includegraphics[width=\\columnwidth]{ieee14-line-tripping}\n\t\\label{fig:ieee14-line-tripping}}\n\t\\hfil\n\t\\caption{The two subfigures show the outflowing power (top) at each node and the angular frequency (bottom) for the buses modeled by the swing equation (1, 3, 6, 8) in for the two perturbation scenarios: (left) frequency perturbation at bus 1 and (right) line tripping of line 2 (between bus 1 and 5). The letter in bracets refers to the modeling type, i.e. $G$ for generator \/ synchronous compensator as swing equation, $S$ for the slack bus, and $L$ for a load as algebraic PQ-constraint.}\n\\end{figure*}\n\nWithin this section, we will analyze two simple cases: (a) a frequency perturbation at the largest generator at bus 1 and (b) a line tripping event at of line 2 (between bus 1 and 5) (cmp.\\ \\Cref{fig:ieee14grid}).\n\nBefore we find the normal point of operation for the power grid, i.e.\\ the fixed point of \\Cref{eq:i-definition,eq:u-general,eq:x-general} or synchronous state for the IEEE 14-bus system. For that, we use the grid model \\texttt{g} generated in the previous \\Cref{sec:implementation} and use the function provided by \\pd{}:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nfp = operationpoint(g, ones(SystemSize(g)))\n\\end{lstlisting}\nwhere $\\mathtt{ones(SystemSize(g))}$ is a vector of the correct length for the initial condition of the fixed point search. \\texttt{fp} is now a \\texttt{State} object that we can use as initial condition for the solving the differential equations corresponding to the power grid.\n\n\\subsection{Frequency perturbation}\n\nIn order to model a frequency perturbation, one can simply take a copy of the fixed point as found before and adjust the initial frequency value:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nx0 = copy(fp)\nx0[1, :int, 1] += 0.2 \n\\end{lstlisting}\nThe second line can be read as ``add 0.2 to the 1$^\\text{st}$ internal variable of the 1$^\\text{st}$ node'' which is the frequency $\\omega$ as there is only one internal variable. Note that the first \\texttt{1} refers to the node and then second \\texttt{1} to the internal variable counter. The power grid with the initial condition \\texttt{x0} can then be solved for a time span of 0.5 seconds by calling:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nsol = solve(g, x0, (0.0,.5))\n\\end{lstlisting}\nThe solution is shown in \\Cref{fig:ieee14-frequency-perturbation}. It shows that the system is very stable against frequency perturbations. The actual dynamics is not so exciting as the system is very stable. Please note that this system was taken as an example to present on how easily one can model a power grid using \\pd{}, not to find new exciting dynamics.\n\n\\subsection{Line tripping}\n\nTo show some more dynamic behavior we simulated a line tripping as well. We model this effect by taking the operation point of the full power grid (\\texttt{fp}) as initial condition but defining a new admittance Laplacian where line 2 (between bus 1 and 5, see \\Cref{tab:ieee14-line-parameters,fig:ieee14grid}) has been taken out (i.e. the admittance is set to $0$). Running the model with this new Laplacian yields \\Cref{fig:ieee14-line-tripping}.\n\nIn the frequency plot we identify how the frequency of bus 1, where the line tripping happened, compensates for the momentarily excess power at the bus. The lacking power in the rest of the grid is matched by the synchronous compensators whose frequency decreases in turn. Note that the angular frequency $\\omega$ is shown, so with a division by $2\\pi$ the maximal frequency deviation $f$ is $\\approx 0.05\\,$Hz. After about one second, the system recovers to the normal state of operation.\n\n\n\\section{Conclusion \\& Outlook}\n\\label{sec:conclusion}\n\nWithin this paper, we have seen how one one can use the Open-Source library \\pd{} in order to model the dynamics of a power grid with just a few lines of code. We have seen how the fundamental mathematical equations (given in \\Cref{sec:powerdynamics}) translate to source code that reads exactly the same. We employed \\pd{} for the IEEE 14-bus distribution grid feeder in order to demonstrate how one can easily simulate faults and analyze the transient reaction of the power grid dynamics. As example scenarios we used a frequency perturbation and a line tripping.\n\nFinally, this paper is really just a sneak preview with a simple example the publication of \\pd{} is planned for October 15, 2018. By then, we will have added more inverter control schemes and stochastic descriptions of intermittency due to renewable energy sources.\n\n\n\n\n\n\n\\section*{Acknowledgment}\n\nThis paper was presented at the 19th Wind Integration Workshop and published in the workshop's proceedings.\n\nWe would like to thank the German Academic Exchange Service for the opportunity to participate at the Wind Integration Workshop 2018 in Stockholm via the funding program ``Kongressreisen 2018''.\nThis paper is based on work developed within the Climate-KIC Pathfinder project ``elena -- electricity network analysis'' funded by the European Institute of Innovation \\& Technology.\nThis work was conducted in the framework of the Complex Energy Networks research group at the Potsdam Institute for Climate Impact Research.\n\nWe would like to thank Frank Hellmann and Paul Schultz for the discussions on structuring an Open-Source library for dynamic power grid modeling.\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro-sec}\n\\setcounter{equation}{0}\n\nA significant stage in the formation of living systems was the \ntransition from a symmetric chemistry involving mirror-symmetric and \napproximately equal numbers of left- and right-handed chiral species \ninto a system involving just one-handedness of chiral molecules.  \n\nIn this paper we focus on mathematical models of one example of a \nphysicochemical system which undergoes such a symmetry-breaking \ntransition, namely the crystal grinding processes investigated by \nViedma \\cite{viedma} and Noorduin {\\em et al.}\\  \\cite{wim}, \nwhich have been recently reviewed by McBride \\& Tully \n\\cite{mcbride-nature}. Our aim is to describe this process by way \nof a detailed microscopic model of the nucleation and growth processes \nand then to simplify the model, retaining only the bare essential \nmechanisms responsible for the symmetry-breaking bifurcation. \n\nWe start by reviewing the processes which are already known to \ncause a symmetry-breaking bifurcation. By this we mean that a system \nwhich starts off in a racemic state (one in which both left-handed and \nright-handed structures occur with approximately equal frequencies) \nand, as the system evolves, the two handednesses grow differently, \nso that at a later time, one handedness is predominant in the system.  \n\n\\subsection{Models for homochiralisation}\n\nMany models have been proposed for the emergence of homochirality \nfrom an initially racemic mixture of precursors.   Frank \\cite{frank} \nproposed an open system into which $R$ and $S$ particles are \ncontinually introduced, and combine to form one of two possible \nproducts: left-  or right-handed species, $X,Y$.   Each of these \nproducts acts as a catalyst for its own production (autocatalysis), and \neach combines with the opposing handed product (cross-inhibition) to \nform an inert product ($P$) which is removed from the system at some \nrate.   These processes are summarised by the following reaction scheme: \n\\begin{equation} \\begin{array}{rclcrclcl} \n&&&& \\hspace*{-9mm}{\\rm external \\;\\;\\; source} & \\rightarrow &R,S& \n\t\\;\\; & {\\rm input}, k_0, \\\\ \nR+S & \\rightleftharpoons & X && \nR+S & \\rightleftharpoons & Y &\\qquad &\\mbox{slow}, k_1 , \\\\ \nR+S+X & \\rightleftharpoons & 2 X && \nR+S+Y & \\rightleftharpoons & 2 Y &\\quad& \\mbox{fast, autocatalytic}, k_2 \\\\ \n&&&&X + Y & \\rightarrow & P  &\\qquad& \\mbox{cross-inhibition}, k_3 , \\\\\n&&&& P &\\rightarrow & & \\qquad & {\\rm removal}, k_4 . \n\\end{array}\\end{equation} \nIgnoring the reversible reactions (for simplicity), \nthis system can be modelled by the differential equations \n\\begin{eqnarray}\n\\frac{{\\rm d} r}{{\\rm d} t} & =& k_0 - 2 k_1 r s - k_2 r s (x\\!+\\!y) \n\t+ k_{-1} (x\\!+\\!y) + k_{-2} (x^2\\!+\\!y^2) , \\\\\n\\frac{{\\rm d} s}{{\\rm d} t} & = & k_0 - 2 k_1 r s - k_2 r s (x\\!+\\!y) \n\t+ k_{-1} (x\\!+\\!y) + k_{-2} (x^2\\!+\\!y^2) , \\\\ \n\\frac{{\\rm d} x}{{\\rm d} t} & = & \n\tk_1 r s + k_2 r s x - k_3 x y - k_{-1} x - k_{-2} x^2 , \\\\ \n\\frac{{\\rm d} y}{{\\rm d} t} & = & \n\tk_1 r s + k_2 r s y - k_3 x y - k_{-1} y - k_{-2} y^2 , \\\\ \n\\frac{{\\rm d} p}{{\\rm d} t} & = & k_{3} x y - k_4 p , \n\\end{eqnarray}\nfrom which we note that at steady-state we have \n\\begin{equation}\nrs=\\frac{k_0+k_{-1}(x+y) + k_{-1}(x^2+y^2)}{2k_1+k_2(x+y)}.\n\\end{equation}\nWe write the absolute enantiomeric excess as $ee=x-y$ and the \ntotal concentration as $\\sigma=x+y$;  adding and subtracting \nthe equations for ${\\rm d} x\/{\\rm d} t$ and ${\\rm d} y\/{\\rm d} t$, we find \n\\begin{equation} \n\\sigma^2 = \\frac{2k_0}{k_3} + ee^2 , \n\\end{equation} \n\\begin{equation} \nee \\left[ \n\\frac{k_2(k_{-2}ee^2+k_{-2}\\sigma^2+2k_{-1}\\sigma+2k_0)}\n{2(2k_1+k_2\\sigma)} - k_{-1} - k_{-2} \\sigma \\right] = 0 . \n\\end{equation} \nHence $ee=0$ is always a solution, and there are other solutions with \n$ee\\neq0$ if the rate constants $k_*$ satisfy certain conditions (these \ninclude $k_3>k_{-2}$ and $k_0$ being sufficiently large). \n\nThe important issues to note here are: \n\\begin{description}\n\\item[(i)] this system is {\\em open}, it requires the continual supply of \nfresh $R,S$ to maintain the asymmetric steady-state. Also, the \nremoval of products is required to avoid the input terms \ncausing the total amount of material to increase indefinitely;  \n\\item[(ii)] the forcing input term drives the system away from \nan equilibrium solution, into a distinct steady-state solution;  \n\\item[(iii)] the system has cross-inhibition which removes equal \nnumbers of $X$ and $Y$, amplifying any differences caused by \nrandom fluctuations in the initial data or in the input rates. \n\\end{description} \n\nSaito \\& Hyuga \\cite{saito} discuss a sequence of toy models \ndescribing homochirality caused by nonlinear autocatalysis and \nrecycling.  Their family of models can be summarised by \n\\begin{eqnarray}\n\\frac{{\\rm d} r}{{\\rm d} t} & =& k r^2 (1-r-s) - \\lambda r , \\\\ \n\\frac{{\\rm d} s}{{\\rm d} t} & =& k s^2 (1-r-s) - \\lambda s , \n\\end{eqnarray}\nwhere $r$ and $s$ are the concentrations of the two enantiomers. \nInitially they consider $k_r=k_s=k$ and $\\lambda=0$ and find that \nenantiomeric exess, $r-s$ is constant.   Next the case $k_r=kr$, \n$k_s=ks$, $\\lambda=0$ is analysed, wherein the relative enantiomeric \nexcess $\\frac{r-s}{r+s}$ is constant.  Then the more complex case of \n$k_r=k r^2$, $k_s=k s^2$, $\\lambda=0$ is analysed, and amplification \nof the enantiomeric excess is obtained.  This amplification persists \nwhen the case $\\lambda>0$ is finally analysed.  This shows us strong \nautocatalysis may cause homochiralisation, but in any given \nexperiment, it is not clear which form of rate coefficients \n($k_r,k_s,\\lambda$) should be used. \n\nSaito \\& Hyuga (2005) analyse a series of models of crystallisation \nwhich include some of features present in our more general model.  \nThey note that a model truncated at tetramers exhibits different \nbehaviour from one truncated at hexamers.  In particular, the \nsymmetry-breaking phenomena is {\\em not} present in the tetramer \nmodel, but {\\em is} exhibited by the hexamer model.  Hence, later, \nwe will consider models truncated at the tetramer and the hexamer \nlevels and investigate the differences in symmetry-breaking \nbehaviour (Sections \\ref{tetra-sec} and \\ref{hex-sec}).  \n\nDenoting monomers by $c$, small and large left-handed clusters by \n$x_1,x_2$ respectively and right-handed by $y_1,y_2$, Uwaha \n\\cite{uwaha} writes down the scheme \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& - 2 k_0 z^2 k_1 z (x_1+y_1) + \n\t\\lambda_1(x_2+y_2) + \\lambda_0(x_1+y_1) , \n\\\\ \n\\frac{{\\rm d} x_1}{{\\rm d} t} & = & k_0 z^2 - k_u x_1 x_2 \n\t- k_c x_1^2 + \\lambda_u x_2 + \\lambda_0 x_1 , \n\\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & =& k_1 x_2 c + k_u x_1 x_2 + \n\tk_c x_1^2 - \\lambda_1 x_2 - \\lambda_u x_2 , \n\\\\ \n\\frac{{\\rm d} y_1}{{\\rm d} t} & = & k_0 z^2 - k_u y_1 y_2 \n\t- k_c y_1^2 + \\lambda_u y_2 + \\lambda_0 y_1 , \n\\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & =& k_1 y_2 c + k_u y_1 y_2 + \n\tk_c y_1^2 - \\lambda_1 y_2 - \\lambda_u y_2 , \n\\end{eqnarray}\nwhich models \n\\begin{itemize}\n\\item the formation of small chiral clusters \n\t($x_1,y_1$) from an achiral monomer ($c$) at rate $k_0$, \n\\item small chiral clusters ($x_1,y_1$) of the same handedness \n\tcombining to form larger chiral clusters (rate $k_c$), \n\\item small and larger clusters combining to form larger clusters \n(rate $k_u$), \n\\item large clusters combining with achiral monomers to form more \n\tlarge clusters at the rate $k_1$, \n\\item the break up of larger clusters into smaller clusters (rate $\\lambda_u$), \n\\item the break up of small clusters into achiral monomers (rate $\\lambda_0$), \n\\item the break up of larger clusters into achiral monomers (rate $\\lambda_1$). \n\\end{itemize}\n\nSuch a model can exhibit symmetry-breaking to a solution in which \n$x_1\\neq x_2$ and $x_2\\neq y_2$.  Uwaha points out that the \nrecycling part of the model (the $\\lambda_*$ parameters) are crucial \nto the formation of a `completely' homochiral state. One problem with \nsuch a model is that since the variables are all total masses in the \nsystem, the size of clusters is not explicitly included.  In asymmetric \ndistributions, the typical size of left- and right- handed clusters may \ndiffer drastically, hence the rates of reactions will proceed differently \nin the cases of a few large crystals or many smaller crystals. \n\nSandars has proposed a model of symmetry-breaking in the \nformation of chiral polymers \\cite{sandars}. His model has an \nachiral substrate ($S$) which splits into chiral monomers $L_1,R_1$ \nboth spontaneously at a slow rate and at a faster rate, when catalysed \nby the presence of long homochiral chains.  This catalytic effect has \nboth autocatalytic and crosscatalytic components, that is, for example, \nthe presence of long right-handed chains $R_n$ autocatalyses the \nproduction of right-handed monomers $R_1$ from $S$, (autocatalysis) \nas well as the production of left-handed monomers, $L_1$ (crosscatalysis).   \nSandars assumes the growth rates of chains are linear and not \ncatalysed; the other mechanism required to produce a symmetry-breaking \nbifurcation to a chiral state is cross-inhibition, by which chains of \nopposite handednesses interact and prevent either from further \ngrowth. These mechanisms are summarised by \n\\begin{eqnarray}\nS \\rightarrow L_1 , && S \\rightarrow R_1 , \n\t\\qquad \\mbox{slow} , \\nonumber \\\\ \nS \\!+\\! L_n \\rightarrow L_1 \\!+\\! L_n , && \nS \\!+\\! R_n \\rightarrow R_1 \\!+\\! R_n , \\quad \n\\mbox{autocatalytic, rate $\\propto 1\\!+\\!f$} , \\nonumber \n\\\\\nS \\!+\\! R_n \\rightarrow L_1 \\!+\\! R_n , && \nS \\!+\\! L_n \\rightarrow R_1 \\!+\\! L_n , \\quad \n\\mbox{cross-catalytic, rate $\\propto 1\\!-\\!f$} , \\nonumber \n\\\\\nL_n + L_1 \\rightarrow L_{n+1} , && \nR_n + R_1 \\rightarrow R_{n+1} , \\qquad \n\\mbox{chain growth, rate $=a$} , \\nonumber\n\\\\\nL_n + R_1 \\rightarrow Q_{n+1} , && \nR_n + L_1 \\rightarrow P_{n+1} , \\qquad \n\\mbox{cross-inhibition, rate $=a\\chi$} . \\nonumber\n\\end{eqnarray}\nThis model and generalisations of it have been analysed \nby Sandars \\cite{sandars},  Brandenburg {\\em et al.}\\  \\cite{axel,axel3}, \nMultimaki \\& Brandenburg \\cite{axel2}, \nWattis \\& Coveney \\cite{jw-sandars,jw-ch-rna-rev}, \nGleiser \\& Walker \\cite{sara-bup}, Gleiser {\\em et al.}\\  \\cite{sara-punc}. \nTypically a classic pitchfork bifurcation is found when the fidelity \n($f$) of the autocatalysis over the cross-catalysis is increased.  \nOne counterintuitive effect is that increasing the cross-inhibition \neffect ($\\chi$) aids the bifurcation, allowing it to occur at lower \nvalues of the fidelity parameter $f$. \n\n\n\\subsection{Experimental results on homochiralisation}\n\nThe Soai reaction was one of the first experiments which \ndemonstrated that a chemical reaction could amplify initial small \nimbalances in chiral balance;  that is, a small enantiomeric exess in \ncatalyst at the start of the experiment led to a much larger imbalance \nin the chiralities of the products at the end of the reaction. Soai {\\em et al.}\\ \n\\cite{soai} was able to achieve an enantiomeric exess exceeding \n85\\% in the asymmetric autocatalysis of chiral pyrimidyl alkanol.  \n\nThe first work showing that crystallisation experiments could exhibit \nsymmetry breaking was that of Kondepudi \\& Nelson \n\\cite{kon-sci}.  Later Kondepudi {\\em et al.}\\ \\cite{kon-jacs} \nshowed that the stirring rate was a good bifurcation parameter to \nanalyse the final distribution of chiralities of crystals emerging from a \nsupersaturated solution of sodium chlorate.  With no stirring, there \nwere approximately equal numbers of left- and right-handed crystals.  \nAbove a critical (threshold) stirring rate, the imbalance in the numbers \nof each handedness increased, until, at large enough stirring rates, \ntotal chiral purity was achieved.  This is due to all crystals in the \nsystem being derived from the same `mother' crystal, which is the first \ncrystal to become  established in the system; all other crystals grow \nfrom fragments removed from it (either directly or indirectly). \nBefore this, Kondepudi \\& Nelson \\cite{kon-pla,kon-nat}  \nworked on the theory of chiral symmetry-breaking mechanisms \nwith the aim of predicting how parity-violating perturbations could \nbe amplified to give an enantiomeric exess in prebiotic chemistry, \nand the timescales involved.  Their results suggest \na timescale of approximately $10^4$ years.   More recently, \nKondepudi and Asakura \\cite{kon+a} have summarised \nboth the experimental and theoretical aspects of this work. \n\nViedma \\cite{viedma} was the first to observe that grinding a \nmixture of chiral crystals eventually led to a distribution of crystals \nwhich were all of the same handedness.  The crystalline material \nused was sodium chlorate, as used by Kondepudi {\\em et al.}\\ \n\\cite{kon-sci}.  \nSamples of L and D crystals are mixed with water in round-bottomed \nflasks and the system is stirred by a magnetic bar (of length 3-20mm) \nat 600rpm.  The system is maintained in a supersaturated state; \nsmall glass balls are added to continually crush the crystals.  \nThe grinding is thus continuous, and crystals are maintained below \na size of 200 $\\mu$m. The chirality of the resulting crystals was \ndetermined by removing them from the flask, allowing them to grow \nand measuring their optical activity. \nThe results show that, over time, the percentages of left- and \nright-handed crystals steadily change from about 50\/50 to 100\/0 or \n0\/100  -- a state which is described as complete chiral purity. \nWith stirring only and no glass balls, the systems conserve their \ninitial chiral excesses; with glass balls present and stirring, the chiral\nexcess increases, and this occurs more rapidly if more balls are \npresent or the speed of stirring is increased. \n\nMore recently, Noorduin {\\em et al.}\\ \\cite{wim} have observed a \nsimilar effect with amino acids -- a much more relevant molecule \nin the study of origins of life.  This work has been reviewed by \nMcBride \\& Tully \\cite{mcbride-nature}, who add to the \nspeculation on the mechanisms responsible for the phenomenon.  \nNoorduin {\\em et al.}\\ describe grinding as `dynamic \ndissolution\/crystallization processes that result in the conversion \nof one solid enantiomorph into the other'.  They also note that `once \na state of single chirality is achieved, the system is ``locked'' \nbecause primary nucleation to form and sustain new crystals from \nthe opposite enantiomer is kinetically prohibited'.  Both these quotes \ninclude the crucial fact that the process evolves {\\em not} towards \nan equilibrium solution (which would be racemic), but towards a \ndifferent, dynamic steady-state solution. As noted by Plasson \n(personal communication, 2008), this nonequilibrium state is \nmaintained due to the constant input of energy into the system \nthrough the grinding process. \n\nMcBride \\& Tully \\cite{mcbride-nature} discuss  the growth \nof one enantiomorph, and the dissolution of the other as a type of \nOstwald ripening process; with the large surface area to volume ratio \nof smaller crystals giving a rapid dissolution rate, whilst larger crystals, \nhave a lower surface area to volume ratio meaning that they dissolve \nmore slowly. However appealing such an argument maybe, since \nsurface area arguments can equally well be applied to the growth \nside of the process, it is not clear that this is either necessary or \nsufficient.   Infact, the model analysed later in this paper will show \nthat a critical cluster size is not necessary to explain homochiralisation \nthrough grinding.  \n\n\\subsection{Our aims}\n\nWe aim to describe the results of the crystal grinding phenomenon \nthrough a model which recycles mass through grinding, which \ncauses crystals to fragment, rather than having explicit mass input \nand removal.  Simultaneously we need crystal growth processes \nto maintain a distribution of sizeable crystals.  \n\nWe assume that the crystals are solids formed in an aqueous \nenvironment, however, we leave open questions as to whether \nthey are crystals of some mineral of {\\em direct} biological \nrelevance (such as amino acids), or whether they are some other \nmaterial, which after growing, will later provide a chirally \nselective surface for biomolecules to crystallise on, or be a catalyst \nfor chiral polymerisation to occur.  Following Darwin's \n\\cite{darwin} ``warm little pond'', an attractive scenario might \nbe a tidal rock pool, where waves agitating pebbles provide the \nenergetic input for grinding.  Taking more account of recent work, \na more likely place is a suboceanic hydrothermal vent where \nthe rapid convection of hot water impels growing nucleii into the \nvent's rough walls as well as breaking particles off the walls and \nentraining them into the fluid flow, simultaneously grinding any \ngrowing crystals. \n\nIn Section \\ref{model-sec} we propose a detailed microscopic \nmodel of the nucleation and crystal growth of several species \nsimultaneously.  This has the form of a generalised Becker-D\\\"{o}ring \nsystem of equations \\cite{bd}.  Due to the complexity of the \nmodel we immediately simplify it, making assumptions on the rate \ncoefficients.   Furthermore, to elucidate those processes which \nare responsible for homochiralisation, we remove some processes \ncompletely so as to obtain a simple system of ordinary differential \nequations which can be analysed theoretically.  \n\nThe simplest model which might be expected to show \nhomochiralisation is one which has small and large clusters of each \nhandedness.  Such a truncated model is considered in Section \n\\ref{tetra-sec} wherein it is shown that such a model might lead \nto amplification of enantiomeric exess in the short time, but that \nin the long-time limit, only the racemic state can be approached. \nThis model has the structure akin to that of Saito \\& Hyuga \n\\cite{saito2} truncated at the tetramer level. \n\nHence, in Section \\ref{hex-sec} we consider a more complex model \nwith a cut-off at larger sizes (one can think of small, medium, and large \nclusters of each handedness).  Such a model has a similar structure to \nthe hexamer truncation analysed by Saito \\& Hyuga \\cite{saito2}.  \nWe find that such a model does allow a final steady-state in which one \nchirality dominates the system and the other is present only in \nvanishingly small amounts. \n\nHowever, as discussed earlier, there may be subtle effects whereby it \nis not just the {\\em number} of crystals of each type that is important \nto the effect, but a combination of size and number of each handedness \nof crystal that is important to the evolution of the process.  Hence, in \nSection \\ref{new-sec} we introduce an alternative reduction of the \nsystem of governing equations.  In this, instead of truncating and \nkeeping only clusters of a small size, we postulate a form for the \ndistribution which includes information on both the number and size \nof crystals, and use these two quantities to construct a system of \nfive ordinary differential equations for the system's evolution. \n\nWe discuss the results in Sections \\ref{disc-sec} and \\ref{conc-sec} \nwhich conclude the paper.  The Appendix \\ref{app} shows how, \nby removing the symmetry in the growth rates of the two \nhandednesses, the model could be generalised to account for \nthe competitive nucleation of different polymorphs growing from \na common supply of monomer. \n\n\\section{The BD model with dimer interactions and \nan amorphous metastable phase} \n\\label{model-sec}\n\\setcounter{equation}{0}\n\n\\subsection{Preliminaries}\n\nSmoluchowski \\cite{smol} proposed a model in which clusters \nof any sizes could combine pairwise to form larger clusters. Chemically \nthis process is written $C_r + C_s \\rightarrow C_{r+s}$ where $C_r$ \nrepresents a cluster of size $r$.  Assuming this process is reversible \nand occurs with a forward rate given by $a_{r,s}$ and a reverse rate \ngiven by $b_{r,s}$, the law of mass action yields the kinetic equations \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} &\\!=\\!&\\! \\mbox{$\\frac{1}{2}$} \\sum_{s=1}^{r-1} \n \\left( a_{s,r-s} c_s c_{r-s} - b_{s,r-s} c_r \\right) \n - \\sum_{s=1}^\\infty \\left( a_{r,s} c_r c_s - \n b_{r,s} c_{r+s} \\right) . \\nonumber \\\\ && \\lbl{smol-eq}\n\\end{eqnarray}\nThese are known as the coagulation-fragmentation equations.  \nThere are simplifications in which only interactions between clusters \nof particular sizes are permitted to occur, for example when only \ncluster-monomer interactions can occur, the Becker-D\\\"{o}ring \nequations \\cite{bd} are obtained.  Da Costa has formulated a \nsystem in which only clusters upto a certain size ($N$) are permitted \nto coalesce with or fragment from other clusters.  In the case of \n$N=2$, which is pertinent to the current study, only cluster-monomer \nand cluster-dimer interactions are allowed, for example \n\\begin{equation}\nC_r + C_1 \\rightleftharpoons C_{r+1} , \n\\qquad \nC_r + C_2 \\rightleftharpoons C_{r+2} . \n\\end{equation}  \nThis leads to a system of kinetic equations of the form \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} & = & J_{r-1} - J_r + K_{r-2} - K_r , \n\t\\qquad (r\\geq3) , \\lbl{gbd-eq1} \\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & J_1 - J_2 - K_2 \n\t- \\displaystyle\\sum_{r=1}^\\infty K_r , \\\\ \n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & - J_1 - K_2 \n\t- \\displaystyle\\sum_{r=1}^\\infty J_r , \\\\ \nJ_r &=& a_r c_r c_1 - b_{r+1} c_{r+1} , \\qquad \nK_r = \\alpha_r c_r c_2 - \\beta_{r+2} c_{r+2} . \n\\lbl{gbd-eq4} \\end{eqnarray}\nA simple example of such a system has been analysed \npreviously by Bolton \\& Wattis \\cite{bw-dimers}. \n\nIn the next subsection we generalise the model (\\ref{smol-eq}) to \ninclude a variety of `species' or `morphologies' of cluster, \nrepresenting left-handed, right-handed and achiral clusters.  We \nsimplify the model in stages to one in which only monomer and dimer \ninteractions are described, and then one in which only dimer \ninteractions occur. \n\n\\subsection{A full microscopic model of chiral crystallisation}\n\nWe start by outlining all the possible cluster growth, fragmentation \nand transformation processes.  We denote the two handed clusters \nby $X_r$, $Y_r$, where the subscript $r$ specifies the size of cluster. \nAchiral clusters are denoted by $C_r$, and we allow clusters to \nchange their morphology spontaneously according to \n\\begin{equation} \\begin{array}{rclclccrclcl}\nC_r & \\rightarrow & X_r & \\quad& {\\rm rate} = \\mu_r , &&\nX_r & \\rightarrow & C_r & \\quad& {\\rm rate} = \\mu_r \\nu_r , \\\\ \nC_r & \\rightarrow & Y_r & \\quad& {\\rm rate} = \\mu_r , && \nY_r & \\rightarrow & C_r & \\quad& {\\rm rate} = \\mu_r \\nu_r .  \n\\end{array} \\end{equation}\nWe allow clusters to grow by coalescing with clusters \nof similar handedness or an achiral cluster.  In the case of \nthe latter process, we assume that the cluster produced \nis chiral with the same chirality as the parent. Thus \n\\begin{equation} \\begin{array}{rclcl}\nX_r + X_s & \\rightarrow & X_{r+s} , && {\\rm rate} = \\xi_{r,s}, \\\\ \nX_r + C_s & \\rightarrow & X_{r+s} , && {\\rm rate} = \\alpha_{r,s},\\\\ \nC_r + C_s & \\rightarrow & C_{r+s} , && {\\rm rate} = \\delta_{r,s},\\\\ \nY_r + C_s & \\rightarrow & Y_{r+s} , && {\\rm rate} = \\alpha_{r,s},\\\\ \nY_r + Y_s & \\rightarrow & Y_{r+s} , && {\\rm rate} = \\xi_{r,s} . \n\\end{array} \\end{equation}\nWe do not permit clusters of opposite to chirality to merge. \nFinally we describe fragmentation: all clusters may \nfragment, producing two smaller clusters each of \nthe same chirality as the parent cluster \n\\begin{equation} \\begin{array}{rclcl}\nX_{r+s} & \\rightarrow & X_r + X_s && {\\rm rate} = \\beta_{r,s}, \\\\ \nC_{r+s} & \\rightarrow & C_r + C_s && {\\rm rate} = \\epsilon_{r,s}, \\\\ \nY_{r+s} & \\rightarrow & Y_r + Y_s &\\quad& {\\rm rate} = \\beta_{r,s} . \n\\end{array} \\end{equation}\nSetting up concentration variables for each size and each type of \ncluster by defining $c_r(t) = [C_r]$, $x_r(t) = [X_r]$, $y_r(t) = [Y_r]$ \nand applying the law of mass action, we obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} &\\!=\\!& -2\\mu_r c_r + \\mu_r\\nu_r(x_r+y_r) \n\t- \\sum_{k=1}^\\infty \\alpha_{k,r} c_r (x_k+y_k) \\\\ && \\nonumber \n\t+ \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\delta_{k,r-k} c_k c_{r-k} \n\t- \\epsilon_{k,r-k} c_k c_{r-k} \\right) \n\t- \\sum_{k=1}^\\infty \\left( \\delta_{k,r} c_k c_r \n\t- \\epsilon_{k,r} c_{r+k} \\right) , \n\\lbl{gbd1} \\\\ \n\\frac{{\\rm d} x_r}{{\\rm d} t} &\\!=\\!&\\! \\mu_r c_r \\!-\\! \\mu_r \\nu_r x_r \n\t+ \\sum_{k=1}^{r-1} \\alpha_{k,r-k} c_k x_{r-k} \n\t\\!-\\! \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\xi_{k,r-k} x_k x_{r-k} \n\t\\!-\\! \\beta_{k,r\\!-\\!k} x_r \\right) \\nonumber \\\\ && \n\t- \\sum_{k=1}^\\infty \\left( \\xi_{k,r} x_k x_r \n\t- \\beta_{k,r} x_{r+k} \\right) , \n\\\\ \n\\frac{{\\rm d} y_r}{{\\rm d} t} &\\!=\\!&\\! \\mu_r c_r \\!-\\! \\mu_r \\nu_r y_r  \n\t+ \\sum_{k=1}^{r-1} \\alpha_{k,r-k} c_k y_{r-k} \n\t\\!-\\! \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\xi_{k,r-k} y_k y_{r-k} \n\t\\!-\\! \\beta_{k,r\\!-\\!k} y_r \\right) \\nonumber \\\\ && \n\t- \\sum_{k=1}^\\infty \\left( \\xi_{k,r} y_k y_r \n\t- \\beta_{k,r} y_{r+k} \\right) . \n\\lbl{gbd3} \\end{eqnarray}\nThe main problem with such a model is the vast number \nof parameters that have been introduced ($\\alpha_{r,k}$, \n$\\xi_{r,k}$, $\\beta_{r,k}$, $\\mu_r$, $\\nu_r$, $\\delta_{r,k}$, \n$\\epsilon_{r,k}$, for all $k,r$).  \n\nHence we make several simplifications: \n\\begin{description} \n\\item[(i)] \nwe assume that the dominant coagulation and fragmentation \nprocesses are between large and very small clusters (rather \nthan large clusters and other large clusters).   Specifically, we \nassume that only coalescences involving $C_1$ and $C_2$ \nneed to be retained in the model, and fragmentation always \nyields either a monomer or a dimer fragment.  This assumption \nmeans that the system can be reduced to a generalised \nBecker-D\\\"{o}ring equation closer to the form of \n(\\ref{gbd-eq1})--(\\ref{gbd-eq4}) rather than (\\ref{smol-eq}); \n\n\\item[(ii)] \nwe also assume that the achiral clusters are unstable at larger size, \nso that their presence is only relevant at small sizes.  \nTypically at small sizes, clusters are amorphous and do not take \non the properties of the bulk phase, hence at small sizes clusters \ncan be considered achiral.   We assume that there is a regime of \ncluster sizes where there is a transition to chiral structures, and \nwhere clusters can take on the bulk structure (which is chiral) as well \nas exist in amorphous form.   At even larger sizes, we assume that \nonly the chiral forms exist, and no achiral structure can be adopted;  \n\n\\item[(iv)] \nfurthermore, we assume that all rates are independent of cluster size, \nspecifically, \n\\begin{eqnarray}\n\\alpha_{_{k,1}} & = & a , \\qquad \\qquad \n\\alpha_{_{k,2}} = \\alpha , \\qquad \\quad \n\\alpha_{_{k,r}} =0 , \\quad (r\\geq2) \t\\\\ \n\\mu_2 &=& \\mu , \\qquad \\qquad \n\\mu_r=0 , \\quad (r\\geq3) , \t\\\\ \n\\nu_2 &=& \\nu , \\qquad \\qquad \n\\nu_r=0 , \\quad (r\\geq3) , \t\\\\ \n\\delta_{1,1} & = & \\delta , \\qquad \n\\delta_{k,r} = 0 , \\quad ({\\rm otherwise}) \t\\\\ \n\\epsilon_{1,1} & = & \\epsilon , \\qquad \n\\epsilon_{k,r} = 0 , \\quad ({\\rm  otherwise})\t\\\\ \n\\xi_{k,2} &=& \\xi_{2,k} = \\xi , \\qquad \n\\xi_{k,r} = 0 , \\quad ({\\rm otherwise})\t\t\\\\\n\\beta_{k,1} & = & \\beta_{1,k} = b , \\qquad \n\\beta_{k,2} = \\beta_{2,k} = \\beta , \\qquad \n\\beta_{k,r} = 0 , \\quad ({\\rm otherwise}), \n\\nonumber \\\\ && \\end{eqnarray}\nUltimately we will set $a=b=0=\\delta=\\epsilon$ so that we have only \nfive parameters to consider ($\\alpha$, $\\xi$, $\\beta$, $\\mu$, $\\nu$). \n\n\\end{description} \n\n\\begin{figure}[!ht]\n\\begin{picture}(500,160)(35,-35)\n\\put(48,50){\\circle{15}}\n\\multiput(80,50)(40,0){6}{\\circle{15}}\n\\multiput(80,10)(40,0){8}{\\circle{15}}\n\\multiput(80,90)(40,0){8}{\\circle{15}}\n\\put(044,48){$c_1$}\n\\put(076,48){$c_2$}\n\\put(116,48){$c_3$}\n\\put(156,48){$c_4$}\n\\put(196,48){$c_5$}\n\\put(236,48){$c_6$}\n\\put(276,48){$c_7$}\n\\put(076,88){$x_2$}\n\\put(116,88){$x_3$}\n\\put(156,88){$x_4$}\n\\put(196,88){$x_5$}\n\\put(236,88){$x_6$}\n\\put(276,88){$x_7$}\n\\put(316,88){$x_8$}\n\\put(356,88){$x_9$}\n\\put(076,08){$y_2$}\n\\put(116,08){$y_3$}\n\\put(156,08){$y_4$}\n\\put(196,08){$y_5$}\n\\put(236,08){$y_6$}\n\\put(276,08){$y_7$}\n\\put(316,08){$y_8$}\n\\put(356,08){$y_9$}\n\\multiput(83,20)(40,0){6}{\\vector(0,1){20}}\n\\multiput(77,40)(40,0){6}{\\vector(0,-1){20}}\n\\multiput(77,60)(40,0){6}{\\vector(0,1){20}}\n\\multiput(83,80)(40,0){6}{\\vector(0,-1){20}}\n\\multiput(69,30)(40,0){6}{$\\mu$}\n\\multiput(85,30)(40,0){6}{$\\nu$}\n\\multiput(69,70)(40,0){6}{$\\mu$}\n\\multiput(85,70)(40,0){6}{$\\nu$}\n\\put(058,53){\\vector(1,0){10}}\n\\put(068,50){\\vector(-1,0){10}}\n\\multiput(095,53)(40,0){5}{\\vector(1,0){10}}\n\\multiput(105,50)(40,0){5}{\\vector(-1,0){10}}\n\\multiput(095,95)(40,0){7}{\\vector(1,0){10}}\n\\multiput(105,90)(40,0){7}{\\vector(-1,0){10}}\n\\multiput(095,15)(40,0){7}{\\vector(1,0){10}}\n\\multiput(105,10)(40,0){7}{\\vector(-1,0){10}}\n\\put(060,59){$\\delta$}\n\\put(060,40){$\\epsilon$}\n\\multiput(097,59)(40,0){5}{$\\delta$}\n\\multiput(097,40)(40,0){5}{$\\epsilon$}\n\\multiput(097,97)(40,0){5}{$a$}\n\\multiput(097,80)(40,0){5}{$b$}\n\\multiput(097,17)(40,0){5}{$a$}\n\\multiput(097,00)(40,0){5}{$b$}\n\\put(118,119){$\\beta$}\n\\put(127,115){\\vector(-1,0){10}}\n\\put(120,100){\\oval(80,30)[t]}\n\\put(120,  0){\\oval(80,30)[b]}\n\\put(127,-15){\\vector(-1,0){10}}\n\\put(118,-27){$\\beta$}\n\\put(158,124){$\\beta$}\n\\put(167,120){\\vector(-1,0){10}}\n\\put(160,100){\\oval(80,40)[t]}\n\\put(160,  0){\\oval(80,40)[b]}\n\\put(167,-20){\\vector(-1,0){10}}\n\\put(158,-32){$\\beta$}\n\\put(198,119){$\\beta$}\n\\put(207,115){\\vector(-1,0){10}}\n\\put(200,100){\\oval(80,30)[t]}\n\\put(200,  0){\\oval(80,30)[b]}\n\\put(207,-15){\\vector(-1,0){10}}\n\\put(198,-27){$\\beta$}\n\\put(238,124){$\\beta$}\n\\put(247,120){\\vector(-1,0){10}}\n\\put(240,100){\\oval(80,40)[t]}\n\\put(240,  0){\\oval(80,40)[b]}\n\\put(247,-20){\\vector(-1,0){10}}\n\\put(238,-32){$\\beta$}\n\\put(278,119){$\\beta$}\n\\put(287,115){\\vector(-1,0){10}}\n\\put(280,100){\\oval(80,30)[t]}\n\\put(280,  0){\\oval(80,30)[b]}\n\\put(287,-15){\\vector(-1,0){10}}\n\\put(278,-27){$\\beta$}\n\\put(318,124){$\\beta$}\n\\put(327,120){\\vector(-1,0){10}}\n\\put(320,100){\\oval(80,40)[t]}\n\\put(320,  0){\\oval(80,40)[b]}\n\\put(327,-20){\\vector(-1,0){10}}\n\\put(318,-32){$\\beta$}\n\\end{picture} \n\\caption{Reaction scheme involving monomer and \ndimer aggregation and fragmentation of achiral clusters \nand those of both handednesses (right and left). \nThe aggregation of achiral and chiral clusters \nis not shown (rates $\\alpha$, $\\xi$). }\n\\label{rec-sch-fig}\n\\end{figure}\n\nThis scheme is illustrated in Figure \\ref{rec-sch-fig}.  However, \nbefore writing down a further system of equations,  we make one \nfurther simplification. We take the transition region described in (ii), \nabove, to be just the dimers.  Thus the only types of achiral cluster \nare the monomer and the dimer ($c_1$, $c_2$); dimers exist in achiral, \nright- and left-handed forms ($c_2$, $x_2$, $y_2$); at larger sizes \nonly left- and right-handed clusters exist ($x_r$, $y_r$, $r\\geq2$). \n\nThe kinetic equations can be reduced to \n\\begin{eqnarray} \n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & 2 \\varepsilon c_2 - 2 \\delta c_1^2 \n\t- \\sum_{r=2}^\\infty ( a c_1 x_r + a c_1 y_r - b x_{r+1} \n\t- b y_{r+1} ) , \\lbl{gbd-c1}\n\t\\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & \\delta c_1^2 - \\varepsilon c_2 - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\sum_{r=2}^\\infty \\alpha c_2 (x_r+y_r) , \n\t\\\\\n\\frac{{\\rm d} x_r}{{\\rm d} t} & = & a c_1 x_{r-1} - b x_r \n\t- a c_1 x_r + b x_{r+1} + \\alpha c_2 x_{r-2} \n\t- \\alpha c_2 x_r  \\nonumber\\\\ && \n\t- \\beta x_r + \\beta x_{r+2} + \\xi x_2 x_{r-2}  \n\t- \\xi x_2 x_r , \\qquad \\hfill (r\\geq4) , \n\t\\\\ \n\\frac{{\\rm d} x_3}{{\\rm d} t} & = & a c_1 x_2 - b x_3 - a c_1 x_3 \n\t+ b x_4 - \\alpha c_2 x_3 - \\xi x_2 x_3 + \\beta x_5 , \n\t\\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 + b x_3 \n\t- a c_1 x_2 - \\alpha x_2 c_2 \n\t+ \\beta x_4 \\nonumber \\\\ && + \\sum_{r=2}^\\infty \\beta x_{r+2} \n\t- \\sum_{r=2}^\\infty \\xi x_2 x_r - \\xi x_2^2 , \n\t\\\\ \n\\frac{{\\rm d} y_r}{{\\rm d} t} & = & a c_1 y_{r-1} - b y_r \n\t- a c_1 y_r + b y_{r+1} + \\alpha c_2 y_{r-2} \n\t- \\alpha c_2 y_r   \\nonumber\\\\ && \n\t- \\beta y_r + \\beta y_{r+2} + \\xi y_2 y_{r-2} \n\t- \\xi y_2 y_r , \\qquad \\hfill (r\\geq4), \n\t\\\\ \n\\frac{{\\rm d} y_3}{{\\rm d} t} & = & a c_1 y_2 - b y_3 - a c_1 y_3 \n\t+ b y_4 - \\alpha c_2 y_3 - \\xi y_2 y_3 + \\beta y_5  , \n\t\\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 + b y_3 \n\t- a c_1 y_2 - \\alpha y_2 c_2 \n\t+ \\beta y_4 \\nonumber \\\\ && + \\sum_{r=2}^\\infty \\beta y_{r+2} \n\t- \\sum_{r=2}^\\infty \\xi y_2 y_r - \\xi y_2^2 . \\lbl{gbd-y2}\n\\end{eqnarray} \n\n\\subsection{Summary and simulations of the macroscopic model}\n\\label{macro-sec} \n\nThe advantage of the above simplifications is that certain sums \nappear repeatedly; by defining new quantities as these sums, \nthe system can be written in a simpler fashion. We define $N_x = \n\\sum_{r=2}^\\infty x_r$, $N_y = \\sum_{r=2}^\\infty y_r$, then \n\\begin{eqnarray}\n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & 2 \\varepsilon c_2 - 2 \\delta c_1^2 \n\t- a c_1 (N_x+N_y) + b (N_x-x_2+N_y-y_2) , \\lbl{macro-c1}\\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & \\delta c_1^2 - \\varepsilon c_2 - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\alpha c_2(N_x+N_y) ,\\\\\n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 \n\t+ \\beta (N_x-x_3-x_2) - \\xi x_2 N_x , \\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 + b x_3 \n\t- a c_1 x_2 - \\alpha x_2 c_2 + \\beta (x_4+N_x-x_2-x_3) \n\t\\nonumber \\\\ && -\\xi x_2^2 - \\xi x_2 N_x , \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 \n\t+ \\beta (N_y-y_3-y_2) - \\xi y_2 N_y , \\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 + b y_3 \n\t- a c_1 y_2 - \\alpha y_2 c_2 + \\beta (y_4+N_y-y_2-y_3) \n\t\\nonumber\\\\ && - \\xi y_2^2 - \\xi y_2 N_y . \\lbl{macro-y2}\n\\end{eqnarray} \nHowever, such a system of equations is not `closed'.  The equations \ncontain $x_3,y_3,x_4,y_4$, and yet we have no expressions for \nthese; reintroducing equations for $x_3,y_3$ would introduce \n$x_5,y_5$ and so an infinite regression would be entered into.  \n\n\\begin{figure}[!ht]\n\\vspace*{65mm}\n\\special{psfile=fig-fullt.eps \n\thscale=85 vscale=60 hoffset=-80 voffset=-160}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, \n$N_x$, $N_y$, $N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, \n$\\varrho_x+\\varrho_y$ and $\\varrho_x+\\varrho_y+2c_2+c1$ \nagainst time, $t$ on a logarithmic timescale.  Since model equations \nare in nondimensional form, the time units are arbitrary.  \nParameter values $\\mu=1.0$, $\\nu=0.5$, $\\delta=1$, $\\varepsilon=5$, \n$a=4$, $b=0.02$, $\\alpha=10$, $\\xi=10$, $\\beta=0.03$, with \ninitial conditions $c_2=0.49$, $x_4(0)=0.004$, $y_4(0)=0.006$, \nand all other concentrations zero.  }\n\\label{fig-fullt}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{73mm}\n\\special{psfile=fig-fullxy.eps \n\thscale=80 vscale=60 hoffset=-80 voffset=-145}\n\\caption{Plot of the cluster size distribution at \n$t=0$ (dashed line), $t=112$ (dotted line) and \n$t=9.4\\times10^5$. Parameters and initial conditions \nas in Figure \\protect\\ref{fig-fullt}. }\n\\label{fig-fullxy}\n\\end{figure}\n\nHence we need to find some suitable alternative expressions for \n$x_3,y_3,x_4,y_4$; or an alternative way of reducing the system to \njust a few ordinary differential equations that can easily be analysed. \nSuch systems are considered in Sections \\ref{tetra-sec}, \n\\ref{hex-sec} and \\ref{new-sec}. Before that, however, we illustrate \nthe behaviour of the system by briefly presenting the results of some \nnumerical simulations.    In Figures \\ref{fig-fullt} and \\ref{fig-fullxy} \nwe show the results of a simulation of (\\ref{macro-c1})--(\\ref{macro-y2}). \nThe former shows the evolution of the concentrations $c_1$ which \nrises then decays, $c_2$ which decays since the parameters have \nbeen chosen to reflect a cluster-dominated system. Also plotted are \nthe numbers of clusters $N_x,N_y$ and the mass of material in \nclusters $\\varrho_x$, $\\varrho_y$ defined by \n\\begin{equation} \n\\varrho_x = \\sum_{j=2}^K j x_j , \\qquad \\varrho_y = \\sum_{j=2}^K j y_j . \n\\end{equation} \nNote that under this definition $\\varrho_x+\\varrho_y+c_1+2c_2$ \nis conserved, and this is plotted as {\\sl rho}.   Both the total number \nof clusters, $N_x+N_y$, and total mass of material in handed \nclusters $\\varrho_x+\\varrho_y$ appear to equilibrate by $t=10^2$, \nhowever, at a much later time ($t\\sim 10^4 - 10^5$) a \nsymmetry-breaking bifurcation occurs, and the system changes from \nalmost racemic (that is, symmetric) to asymmetric.   This is more \nclearly seen in Figure \\ref{fig-fullxy}, where we plot the cluster size \ndistribution at three time points.   At $t=0$ there are only dimers \npresent (dashed line), and we impose a small difference in the \nconcentrations of $x_2$ and $y_2$.  At a later time, $t=112$ \n(dotted line), there is almost no difference between the $X$- and \n$Y$-distributions, however by the end of the simulation ($t\\sim10^6$, \nsolid line) one distribution clearly completely dominates the other. \n\n\\subsection{Simplified macroscopic model}\n\nTo obtain the simplest model which involves three polymorphs \ncorresponding to right-handed and left-handed chiral clusters and \nachiral clusters, we now aim to simplify the processes of cluster \naggregation and fragmentation in (\\ref{macro-c1})--(\\ref{macro-y2}).  \nOur aim is to retain the symmetry-breaking phenomenon but \neliminate physical processes which are not necessary for it to occur. \n\nOur first simplification is to remove all clusters of odd size from the \nmodel, and just consider dimers, tetramers, hexamers, {\\em etc}. \nThis corresponds to putting $a=0$, $b=0$ which removes $x_3$ \nand $y_3$ from the system. \nFurthermore, we put $\\varepsilon=0$ and make $\\delta$ large, so that the \nachiral monomer is rapidly and irreversibly converted to achiral dimer.  \nSince the monomers do not then influence the evolution of any of \nthe other variables, we further simplify the system by ignoring $c_1$ \n(or, more simply, just impose initial data in which $c_1(0)=0$).  \nThus we are left with \n\\begin{eqnarray}\n\\!\\!\\!\\frac{{\\rm d} c_2}{{\\rm d} t} & \\!=\\! & - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\alpha c_2(N_x+N_y) , \\lbl{smm2} \\\\\n\\!\\!\\!\\frac{{\\rm d} N_x}{{\\rm d} t} & \\!=\\! & \\mu c_2 - \\mu\\nu x_2 \n\t+ \\beta (N_x-x_2) - \\xi x_2 N_x , \\\\ \n\\!\\!\\!\\frac{{\\rm d} x_2}{{\\rm d} t} & \\!=\\! & \\!\\mu c_2 - \\mu\\nu x_2 - \\alpha x_2 c_2 \n\t+ \\beta (N_x\\!-\\!x_2 \\!+\\! x_4 ) - \\xi x_2^2 - \\xi x_2 N_x , \\\\ \n\\!\\!\\!\\frac{{\\rm d} N_y}{{\\rm d} t} & \\!=\\! & \\mu c_2 - \\mu\\nu y_2 \n\t+ \\beta (N_y-y_2) - \\xi y_2 N_y , \\\\ \n\\!\\!\\!\\frac{{\\rm d} y_2}{{\\rm d} t} & \\!=\\! & \\!\\mu c_2 - \\mu\\nu y_2  - \\alpha y_2 c_2 \n\t+ \\beta (N_y\\!-\\!y_2 \\!+\\! y_4) - \\xi y_2^2 - \\xi y_2 N_y . \\lbl{smmy2}\n\\end{eqnarray} \nSince we have removed four parameters from the model, \nand halved the number of dependent variables, we show a couple \nof numerical simulations just to show that the system \nabove does still exhibit symmetry-breaking behaviour. \n\n\\begin{figure}[!ht]\n\\vspace*{69mm}\n\\special{psfile=fig-dimt.eps \n\thscale=85 vscale=58 hoffset=-80 voffset=-150}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, \n$N_x$, $N_y$, $N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, \n$\\varrho_x+\\varrho_y$ and $\\varrho_x+\\varrho_y+2c_2+c_1$ \nagainst time, $t$ on a logarithmic timescale.  Since model \nequations are in nondimensional form, the time units are \narbitrary.  Parameter values $\\mu=1$, $\\nu=0.5$,  \n$\\alpha=10$, $\\xi=10$, $\\beta=0.03$, with initial \nconditions $c_2=0.49$, $x_4(0)=0.004$, $y_4(0)=0.006$, \nall other concentrations zero.  }\n\\label{fig-dimt}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{71mm}\n\\special{psfile=fig-dimxy.eps \n\thscale=80 vscale=58 hoffset=-80 voffset=-140}\n\\caption{Plot of the cluster size distribution at \n$t=0$ (dashed line), $t=250$ (dotted line) and  \n$t=6\\times10^5$. Parameters and initial conditions \nas in Figure \\protect\\ref{fig-dimt}. }\n\\label{fig-dimxy}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{70mm}\n\\special{psfile=fig-dimtt.eps \n\thscale=80 vscale=60 hoffset=-80 voffset=-150}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, $N_x$, $N_y$, \n$N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, $\\varrho_x+\\varrho_y$ and \n$\\varrho_x+\\varrho_y+2c_2+c_1$ against time, $t$ on a logarithmic timescale.  \nParameters and initial conditions as in Figure \\protect\\ref{fig-dimt}. }\n\\label{fig-dimtt}\n\\end{figure}\n\nFigure \\ref{fig-dimt} appears similar to Figure \\ref{fig-fullt}, \nsuggesting that removing the monomer interactions has changed the \nunderlying dynamics little.  We still observe the characteristic \nequilibration of cluster numbers and cluster masses as $c_2$ decays, \nand then a period of quiesence ($t\\sim10$ to $10^4$) before a later \nsymmetry-breaking event, around $t\\sim10^5$. \nAt first sight, the distribution of $X$- and $Y$-clusters displayed in \nFigure \\ref{fig-dimxy} is quite different to Figure \\ref{fig-fullxy}; this \nis due to the absence of monomers from the system, meaning that \nonly even-sized clusters can now be formed.  If one only looks at the \neven-sized clusters in Figure \\ref{fig-dimxy}, we once again see only \na slight difference at $t=0$ (dashed line), almost no difference at \n$t\\approx250$ (dotted line) but a significant difference at \n$t=6\\times10^5$ (solid line). \nWe include one further graph here, Figure \\ref{fig-dimtt} similar to \nFigure \\ref{fig-dimt} but on a linear rather than a logarithmic timescale.  \nThis should be compared with Figures such as Figures 3 and 4 of \nViedma \\cite{viedma} and Figure 1 of Noorduin {\\em et al.}\\ \\cite{wim}. \n\n\\section{The truncation at tetramers} \n\\label{tetra-sec} \n\\setcounter{equation}{0}\n\n\\begin{figure}[!ht]\n\\begin{picture}(500,120)(0,-10)\n\\put(80,50){\\circle{17}}\n\\put(077,48){$c_2$}\n\\put(080,10){\\circle{17}}\n\\put(077,08){$y_2$}\n\\put(080,90){\\circle{17}}\n\\put(077,88){$x_2$}\n\\put(140,10){\\circle{20}}\n\\put(137,08){$y_4$}\n\\put(140,90){\\circle{20}}\n\\put(137,88){$x_4$}\n\\put(77,20){\\vector(0,1){20}}\n\\put(83,40){\\vector(0,-1){20}}\n\\put(83,60){\\vector(0,1){20}}\n\\put(77,80){\\vector(0,-1){20}}\n\\put(63,30){$\\nu\\mu$}\n\\put(85,30){$\\mu$}\n\\put(63,70){$\\nu\\mu$}\n\\put(85,70){$\\mu$}\n\\put(107,97){$\\beta$}\n\\put(125,95){\\vector(-1,0){30}}\n\\put(095,90){\\vector(1,0){30}}\n\\put(107,81){$\\xi$}\n\\put(110,85){\\oval(30,20)[b]}\n\\put(125,85){\\vector(0,1){5}}\n\\put(113,58){$\\alpha$}\n\\put(090,52){\\vector(3,+2){40}}\n\\put(090,48){\\vector(3,-2){40}}\n\\put(113,38){$\\alpha$}\n\\put(125,17){\\vector(0,-1){5}}\n\\put(110,17){\\oval(30,20)[t]}\n\\put(107,12){$\\xi$}\n\\put(095,10){\\vector(1,0){30}}\n\\put(125,05){\\vector(-1,0){30}}\n\\put(107,-5){$\\beta$}\n\\end{picture} \n\\caption{Simplest possible reaction scheme which might \nexhibit chiral symmetry-breaking. }\n\\label{simp-rec-sch-fig}\n\\end{figure}\n\nThe simplest possible reaction scheme of the form \n(\\ref{gbd-c1})--(\\ref{gbd-y2}) which we might expect to exhibit \nsymmetry-breaking to homochirality is the system truncated at \ntetramers, namely \n\\begin{eqnarray} \n\\displaystyle\\frac{{\\rm d} c_2}{{\\rm d} t} & = & - 2\\mu c_2 + \\mu\\nu (x_2+y_2) \n\t-\\alpha c_2(x_2+y_2) , \\lbl{dim-c2dot} \\\\ \n\\displaystyle\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 \n\t- \\alpha c_2 x_2 - 2 \\xi x_2^2 + 2 \\beta x_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 \n\t- \\alpha c_2 y_2 - 2 \\xi y_2^2 + 2 \\beta y_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} x_4}{{\\rm d} t} & = & \\alpha x_2 c_2 + \\xi x_2^2 - \\beta x_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} y_4}{{\\rm d} t} & = & \\alpha y_2 c_2 + \\xi y_2^2 - \\beta y_4 . \n\\lbl{dim-y4dot}  \\end{eqnarray} \n\nWe investigate the symmetry-breaking by transforming \nthe variables $x_2$, $x_4$, $y_2$, $y_4$ according to \n\\begin{eqnarray}\nx_2 = \\mbox{$\\frac{1}{2}$} z (1+\\theta) , &\\quad& y_2 = \\mbox{$\\frac{1}{2}$} z (1-\\theta) , \n\\lbl{tet-ztheta} \\\\  \nx_4 = \\mbox{$\\frac{1}{2}$} w (1+\\phi) , && y_4 = \\mbox{$\\frac{1}{2}$} w (1-\\phi) , \n\\lbl{tet-wphi} \n\\end{eqnarray}\nwhere $z=x_2+y_2$ is the total concentration of chiral dimers, \n$w=x_4+y_4$ is the total tetramer concentration, $\\theta=(x_2-y_2)\/z$ \nis the relative chirality of the dimers, $\\phi=(x_4-y_4)\/w$ is the \nrelative chirality of tetramers.  Hence \n\\begin{eqnarray}\n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & - 2\\mu c_2 + \\mu\\nu z - \\alpha c_2 z , \n\t\\lbl{c23} \\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & = & 2 \\mu c_2 - \\mu\\nu z - \\alpha c_2 z \n\t- \\xi z^2 (1+\\theta^2) + 2 \\beta w , \\\\ \n\\frac{{\\rm d} w}{{\\rm d} t} & = & \\alpha z c_2 + \\mbox{$\\frac{1}{2}$} \\xi z^2 (1+\\theta^2) \n\t- \\beta w , \\lbl{w3}\\\\ \n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & - \\theta \\left( \\frac{2\\mu c}{z} + \n\t\\frac{2\\beta w}{z}+ \\xi z (1-\\theta^2) \\right) + \n\t\\frac{2\\beta w\\phi}{z} , \\\\\n\\frac{{\\rm d} \\phi}{{\\rm d} t} & = & \\theta \\frac{z}{w} ( \\alpha c + \\xi z ) \n\t- \\left( \\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z (1+\\theta^2) \\right) \\frac{z}{w} \\phi . \n\\end{eqnarray}\nThe stability of the evolving symmetric-state ($\\theta=\\phi=0$) \nis given by the eigenvalues ($q$) of the matrix \n\\begin{equation}\n\\left( \\begin{array}{cc} \n- \\left( \\frac{2\\mu c}{z} + \\frac{2\\beta w}{z} + \\xi z \\right) & \n\\frac{2\\beta w}{z} \\\\ \n(\\alpha c + \\xi z) \\frac{z}{w} & \n- (\\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z) \\frac{z}{w} \n\\end{array} \\right) , \n\\end{equation}\nwhich are given by\n\\begin{eqnarray}\nq^2 + q \\left( \\frac{\\alpha c z}{w} + \\frac{\\xi z^2}{w} \n+ \\frac{2\\mu c}{z} + \\xi z + \\frac{2\\beta w}{z} \\right) + && \\nonumber \\\\ \n\\frac{1}{w} \\left( 2\\mu c \\alpha c + \\mu c \\xi z + \n\\alpha c \\xi z^2 + \\mbox{$\\frac{1}{2}$} \\xi^2 z^3 - \\beta \\xi z w \\right) &=&0 . \n\\end{eqnarray}\nHence there is an instability if \n\\begin{equation}\n\\beta \\xi z w > 2\\mu c \\alpha c + \\mu c \\xi z + \n\\alpha c \\xi z^2 + \\mbox{$\\frac{1}{2}$} \\xi^2 z^3 , \n\\lbl{crude-instab}\n\\end{equation}\nusing the steady-state result that $2\\beta w = z(2\\alpha c + \\xi z)$ \nand factorising ($2\\alpha c + \\xi z$) out of the result, reduces the \ninstability (\\ref{crude-instab}) to the contradictory $\\xi z^2 > \n\\xi z^2 + 2\\mu c$. Hence the racemic steady-state of the system \nis stable for all choices of parameter values and is approached \nfrom all initial conditions. However, initial perturbations, \nmay be amplified due to the presence of nonlinear terms. \n\n\\begin{figure}[!ht]\n\\vspace*{75mm}\n\\special{psfile=fig-tet3.eps \nhscale=80 vscale=67 hoffset=-60 voffset=-175}\n\\caption{The concentrations $c_2$, $z$ and $w$ \n(\\protect\\ref{tet-ztheta})--(\\protect\\ref{tet-wphi}) plotted against \ntime, for the tetramer-truncated system with the two sets of initial \ndata (\\protect\\ref{tet-ics}). Since model equations \nare in nondimensional form, the time units are arbitrary. The \nparameter values are $\\mu=1$, $\\nu=0.5$, $\\alpha=\\xi=10$, \n$\\beta=0.1$. }\n\\label{fig-tet3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{75mm}\n\\special{psfile=fig-tet4.eps \nhscale=80 vscale=67 hoffset=-60 voffset=-175}\n\\caption{The chiralities $\\theta$, $\\phi$ \n(\\protect\\ref{tet-ztheta})--(\\protect\\ref{tet-wphi}) plotted against \ntime, for the tetramer-truncated system with \nthe two sets of initial data (\\protect\\ref{tet-ics}). Since model equations \nare in nondimensional form, the time units are arbitrary. The \nparameter values are the same as in Figure \\ref{fig-tet3}.}\n\\label{fig-tet4}\n\\end{figure}\n\nEvolution from two sets of initial conditions of the system \n(\\ref{dim-c2dot})--(\\ref{dim-y4dot}) are shown in each of Figures \n\\ref{fig-tet3}, \\ref{fig-tet4}.   The continuous and dotted lines \ncorrespond to the initial data \n\\begin{equation} \\begin{array}{c} \nc_2(0) = 0.29 , \\quad x_2(0) = 0.0051, \\quad y_2(0) = 0.0049, \\\\ \nx_4(0) = 0.051 , \\quad y_4(0) = 0.049 ; \\quad {\\rm and} \\\\ \nc_2(0) = 0 , \\quad x_2(0) = 0.051 \\quad y_2(0) = 0.049, \\\\  \nx_4(0) = 0.1 , \\quad y_4(0) = 0.1 ;  \n\\end{array} \\lbl{tet-ics} \\end{equation}\nrespectively. \nIn the former case, the system starts with considerable amount \nof amorphous dimer, which is converted into clusters, and initially \nthere is a slight chiral imbalance in favour of $x_2$ and $x_4$ over \n$y_2$ and $y_4$.  Over time this imbalance reduces (see figure \n\\ref{fig-tet4}); although there is a region around $t=1$ where \n$\\theta$ increases, both $\\theta$ and $\\phi$ eventually \napproach the zero steady-state.  \n\nFor both sets of initial conditions we note that the chiralities evolve \nover a significantly longer timescale than the concentrations, the \nlatter having reached steady-state before $t=10$ and the former \nstill evolving when $t={\\cal O}(10^2)$.   In the second set of initial \ndata, there is no $c_2$ present initially and there are exactly equal \nnumbers of the two chiral forms of the larger cluster, but a slight \nexess of $x_2$ over $y_2$. In time an imbalance in larger clusters \nis produced, but over larger timescales, both $\\theta$ and $\\phi$ \nagain approach the zero steady-state.  \n\nHence, we observe that the truncated system \n(\\ref{dim-c2dot})--(\\ref{dim-y4dot}) does {\\em not} yield a chirally \nasymmetric steady-state.  Even though in the early stages of the \nreaction chiral perturbations may be amplified, at the end of the \nreaction there is a slower timescale over which the system returns \nto a racemic state. In the next section we consider a system \ntruncated at hexamers to investigate whether that system allows \nsymmetry-breaking of the steady-state. \n\n\\section{The truncation at hexamers}\n\\label{hex-sec}  \\setcounter{equation}{0}\n\nThe above analysis has shown that the truncation of the model \n(\\ref{gbd-c1})--(\\ref{gbd-y2}) to (\\ref{dim-c2dot})--(\\ref{dim-y4dot}) \nresults in a model which always ultimately approaches the \nsymmetric (racemic) steady-state. In this section, we show that a \nmore complex model, the truncation at hexamers retains enough \ncomplexity to demonstrate the symmetry-breaking bifurcation which \noccurs in the full system. In this case the governing equations are \n\\begin{eqnarray} \n\\displaystyle\\frac{{\\rm d} c_2}{{\\rm d} t} &=&  -2 \\mu c_2 + \\mu \\nu (x_2+y_2) \n- \\alpha c_2 (x_2+y_2) - \\alpha c_2 (x_4+y_4) , \n\\lbl{hex-c2} \\\\ \n\\displaystyle\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu \\nu x_2 \n- \\alpha c_2 x_2 - 2 \\xi x_2^2 - \\xi x_2 x_4 + 2\\beta x_4 \n+ \\beta x_6 , \n\\\\ \n\\displaystyle\\frac{{\\rm d} x_4}{{\\rm d} t} & = & \\alpha x_2 c_2 + \\xi x_2^2 \n- \\beta x_4 - \\alpha c_2 x_4 - \\xi x_2 x_4 + \\beta x_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} x_6}{{\\rm d} t} & = & \\alpha x_4 c_2 + \\xi x_2 x_4 \n- \\beta x_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu \\nu y_2 \n- \\alpha c_2 y_2 - 2 \\xi y_2^2 - \\xi y_2 y_4 + 2\\beta y_4 \n+ \\beta y_6 , \n\\\\ \n\\displaystyle\\frac{{\\rm d} y_4}{{\\rm d} t} & = & \\alpha y_2 c_2 + \\xi y_2^2 \n- \\beta y_4 - \\alpha c_2 y_4 - \\xi y_2 y_4 + \\beta y_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} y_6}{{\\rm d} t} & = & \\alpha y_4 c_2 + \\xi y_2 y_4 \n- \\beta y_6 . \n\\lbl{hex-y6} \\end{eqnarray}\n\nTo analyse the symmetry-breaking in the system we transform the \ndependent coordinates from $x_2,x_4,x_6,y_2,y_4,y_6$ to total \nconcentrations $z,w,u$ and relative chiralities $\\theta,\\phi,\\psi$ \naccording to \n\\begin{equation} \\begin{array}{rclcrclcrcl}\nx_2 &=& \\mbox{$\\frac{1}{2}$} z (1 + \\theta) , & \\quad & \nx_4 &=& \\mbox{$\\frac{1}{2}$} w (1 + \\phi) , & \\quad & \nx_6 &=& \\mbox{$\\frac{1}{2}$} u (1 + \\psi) , \\\\[2ex]   \ny_2 &=& \\mbox{$\\frac{1}{2}$} z (1 - \\theta) , & \\quad & \ny_4 &=& \\mbox{$\\frac{1}{2}$} w (1 - \\phi) , & \\quad & \ny_6 &=& \\mbox{$\\frac{1}{2}$} u (1 - \\psi) . \n\\end{array} \\end{equation}\n\nWe now separate the governing equations for the total concentrations \nof dimers ($c,z$), tetramers ($w$) and hexamers ($u$)\n\\begin{eqnarray}\n\\displaystyle\\frac{{\\rm d} c}{{\\rm d} t} & = & - 2 \\mu c \n+ \\mu \\nu z - \\alpha c z - \\alpha c w , \n\\lbl{hex-cdot}\\\\ \n\\displaystyle\\frac{{\\rm d} z}{{\\rm d} t} & =& 2\\mu c - \\mu \\nu z \n- \\alpha c z - \\xi z^2 (1+\\theta^2) - \\mbox{$\\frac{1}{2}$} z w (1+\\theta\\phi) \n+ \\beta u + 2 \\beta w  , \\nonumber \\\\ && \n\\\\ \n\\displaystyle\\frac{{\\rm d} w}{{\\rm d} t} & = & \\alpha c z  \n+ \\mbox{$\\frac{1}{2}$} \\xi z^2 (1+\\theta^2) - \\beta w + \\beta u \n- \\alpha c w - \\mbox{$\\frac{1}{2}$} \\xi z w (1+\\theta\\phi)  , \n\\\\ \n\\displaystyle\\frac{{\\rm d} u}{{\\rm d} t} & = & \\alpha c w \n+ \\mbox{$\\frac{1}{2}$} \\xi z w (1+\\theta\\phi) - \\beta u , \n\\lbl{hex-udot}\\end{eqnarray}\nfrom those for the chiralities \n\\begin{eqnarray}\n\\displaystyle \\frac{{\\rm d} \\psi}{{\\rm d} t} & =& \n\\frac{\\alpha c w}{u} (\\phi-\\psi) + \n\\frac{\\xi z w}{2u} ( \\theta+\\phi-\\psi-\\psi\\phi\\theta )\n\\lbl{hex-psi-dot} \\\\ \n\\displaystyle \\frac{{\\rm d} \\phi}{{\\rm d} t} & = & \n\\frac{\\alpha c z }{w} (\\theta-\\phi) + \n\\frac{\\xi z^2}{2w} ( 2\\theta -\\phi-\\phi\\theta^2) + \n\\frac{\\beta u}{w} (\\psi-\\phi) - \n\\mbox{$\\frac{1}{2}$} \\xi z \\theta (1-\\phi^2) , \\nonumber \\\\ && \n\\\\ \n\\displaystyle \\frac{{\\rm d} \\theta}{{\\rm d} t} & = & \n-\\frac{2\\mu c \\theta}{z} - \\xi z \\theta(1\\!-\\!\\theta^2) - \n\\mbox{$\\frac{1}{2}$} \\xi w \\phi (1\\!-\\!\\theta^2) + \\frac{\\beta u\\psi}{z} - \n\\frac{\\beta u \\theta}{z} \\nonumber \\\\ && + \\frac{2\\beta w\\phi}{z} \n- \\frac{2\\beta w \\theta}{z} .   \\lbl{hex-th-dot}\n\\end{eqnarray}\n\nIn applications, we expect $\\nu<1$, so that the small amorphous \nclusters (dimers) prefer to adopt one of their chiral states rather \nthan the achiral structure.   In addition, we note that the grinding \nprocess observed in experiments is much longer than the \ncrystallisation process, and that there are many larger, macroscopic \ncrystals hence we consider two limits in which $\\beta \\ll \\alpha \\xi$. \nWe will consider the case of small $\\beta$ with all other parameters \nbeing ${\\cal O}(1)$ and then the case where $\\alpha\\sim\\xi\\gg1$ and \nall other parameters are ${\\cal O}(1)$. \n\n\\subsection{Symmetric steady-state for the concentrations}\n\nFirstly, let us solve for the symmetric steady-state.  \nIn this case we assume $\\theta=0=\\phi=\\psi$, simplifying \nequations (\\ref{hex-cdot})--(\\ref{hex-udot}).   One of \nthese is a redundant equation, hence we have the solution \n\\begin{equation}\nw = \\frac{z}{\\beta}(\\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z) , \\qquad \nu = \\frac{z}{\\beta^2}(\\alpha c+\\mbox{$\\frac{1}{2}$}\\xi z)^2 , \n\\lbl{hex-wu-sol} \\end{equation}\n\\begin{equation}\nc = \\frac{1}{\\alpha} \\left(\\sqrt{ \\left( \\frac{\\beta}{2} + \n\\frac{\\beta\\mu}{\\alpha z} + \\frac{\\xi z}{4} \\right)^2 \n+ \\beta\\mu\\nu}  - \\frac{\\beta}{2} - \\frac{\\beta\\mu}\n{\\alpha z} - \\frac{\\xi z}{4} \\right) , \n\\lbl{hex-c-sol} \\end{equation}\nwith $z$ being determined by conservation of total mass in the system \n\\begin{equation} \n2c + 2 z + 4 w + 6 u = \\varrho . \\lbl{hex-roo} \n\\end{equation} \n\nIn the case of small grinding, ($\\beta\\ll1$), with $\\varrho$ \nand all other parameters being ${\\cal O}(1)$, we find \n\\begin{equation}  \\begin{array}{rclcrcl}\nz & = & \\left( \\displaystyle\\frac{2\\varrho \\beta^2}{3 (\\alpha\\nu+\\xi)^2} \n\t\\right)^{1\/3} , &\\qquad& \nc & = & \\nu \\left( \\displaystyle\\frac{\\varrho \\beta^2}{12 (\\alpha\\nu+\\xi)^2} \n\t\\right)^{1\/3} , \\\\ \nw & = & \\left( \\displaystyle\\frac{\\varrho^2 \\beta}{18 (\\alpha\\nu+\\xi)} \n\t\\right)^{1\/3} , &\\qquad& \nu & = & \\displaystyle\\frac{\\varrho}{6} . \n\\end{array}  \\lbl{hex-ssss-asymp}  \\end{equation} \nIn this case most of the mass is in hexamers with a little in \ntetramers and very little in dimers. \n\nIn the asymptotic limit of $\\alpha \\sim \\xi \\gg 1$ \nand all other parameters ${\\cal O}(1)$, we find \n\\begin{eqnarray} & \nc  = \\displaystyle\\frac{\\mu\\nu}{\\alpha} \\left( \n\\displaystyle\\frac{12\\beta}{\\varrho\\xi} \\right)^{1\/3} , \\quad  \nz = \\left( \\displaystyle\\frac{2\\beta^2\\varrho}{3\\xi^2} \\right)^{1\/3} , \\quad \nw = \\left( \\displaystyle\\frac{\\beta\\varrho^2}{18\\xi} \\right)^{1\/3} ,  \\quad\nu = \\displaystyle\\frac{\\varrho}{6} . \n& \\nonumber \\\\ && \\lbl{hex-sss2-asymp} \n\\end{eqnarray} \nThis differs significantly from the other asymptotic scaling as, not only \nare $c$ and $z$ both small, they are now different orders of magnitude, \nwith $c\\ll z$. We next analyse the stability of these symmetric states. \n\n\\subsection{Stability of symmetric state}\n\nIn deriving the above solutions (\\ref{hex-wu-sol})--(\\ref{hex-c-sol}), \nwe have assumed chiral symmetry, that is, $\\theta=0=\\psi=\\phi$.   \nWe now turn to analyse the validity of this assumption. Linearising \nthe system of equations (\\ref{hex-psi-dot})--(\\ref{hex-th-dot}) \nwhich govern the chiralities, we determine whether the symmetric \nsolution is stable from \n\\begin{equation} \\!\\! \n\\frac{{\\rm d} }{{\\rm d} t} \\!\\! \\left( \\!\\! \\begin{array}{c} \n\\psi \\\\ \\phi \\\\ \\theta \\end{array} \\!\\! \\right) \\!=\\! \n\\left( \\begin{array}{ccc}  \n\\!\\!\\!- \\displaystyle\\frac{\\alpha c w}{u} \\!-\\! \\displaystyle\\frac{\\xi z w}{2u} \\!\\!& \n\\displaystyle\\frac{\\alpha c w}{u} \\!+\\! \\displaystyle\\frac{\\xi z w}{2u} & \n\\displaystyle\\frac{\\xi z w}{2u} \\\\[2ex] \n\\displaystyle\\frac{\\beta u}{w} & -\\displaystyle\\frac{\\alpha c z}{w} \n\\!-\\! \\displaystyle\\frac{\\xi z^2}{2w} \\!-\\! \\displaystyle\\frac{\\beta u}{w} & \n\\displaystyle\\frac{\\alpha c z}{w} \\!+\\! \\displaystyle\\frac{\\xi z^2}{w} \n\\!-\\! \\mbox{$\\frac{1}{2}$} \\xi z \\\\[2ex] \n\\displaystyle\\frac{\\beta u}{z} & \n\\displaystyle\\frac{2\\beta w}{z} \\!-\\! \\displaystyle\\frac{\\xi w}{2}  & \\!\\!\n- \\displaystyle\\frac{2\\mu c}{z} \\!-\\! \\xi z \\!-\\! \\frac{\\beta u}{z} \n\\!-\\! \\displaystyle\\frac{2\\beta w}{z} \\!\\!\n\\end{array} \\! \\right) \\!\\!\n\\left( \\!\\begin{array}{c} \\psi \\\\ \\phi \\\\ \\theta \n\\end{array} \\!\\right) \\!.\\!\\!\n\\lbl{hex-stab-mat} \\end{equation}\nFor later calculations it is useful to know the determinant of this matrix.  \nUsing the steady-state solutions (\\ref{hex-wu-sol}), the determinant \nsimplifies to \n\\begin{equation}\nD = \\frac{3 c}{4 \\beta \\rho} ( 2 \\alpha c + \\xi z )^2 \n( \\alpha \\xi z^2 - 4 \\beta \\mu ) . \n\\end{equation}\n\nFor general parameter values, the signs of the real parts of the \neigenvalues of the matrix in (\\ref{hex-stab-mat}) are not clear. \nHowever, using the asymptotic result (\\ref{hex-ssss-asymp}), \nfor $\\beta\\ll1$, we obtain the simpler matrix \n\\begin{equation}\n\\left( \\!\\!\\begin{array}{ccc} \n-\\beta & \\beta & \\displaystyle \\frac{\\beta\\xi}{\\xi\\!+\\!\\alpha\\nu} \n\\\\[2ex] \n\\left( \\displaystyle\\frac{\\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu) }{12} \\right)^{1\/3} & \n- \\left( \\displaystyle\\frac{\\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu) }{12} \\right)^{1\/3} & \n-\\frac{\\xi}{2} \\left( \\displaystyle\\frac{2\\beta^2\\varrho}{3(\\xi\\!+\\!\\alpha\\nu)^2} \\right) ^{1\/3} \n\\\\[2ex]  \n\\beta^{1\/3} \\left( \\displaystyle\\frac{\\xi\\!+\\!\\alpha\\nu}{12\\varrho} \\right)^{2\/3} & \n- \\frac{\\xi}{2} \\left( \\displaystyle\\frac{\\beta\\varrho^2}{18(\\xi\\!+\\!\\alpha\\nu)} \\right)^{1\/3} & \n- \\mu \\nu - \\beta^{1\/3} \\left( \\displaystyle\\frac{\\xi\\!+\\!\\alpha\\nu}{12\\varrho} \\right)^{2\/3} \n\\end{array} \\!\\!\\right) \\! , \\lbl{hex-asy1-mat}\n\\end{equation}\nwhose characteristic polynomial is \n\\begin{equation} \n0 = q^3 + \\mu\\nu q^2 + \\mu\\nu \\left( \\rec{12} \\beta^2 \\varrho \n(\\xi\\!+\\!\\alpha\\nu) \\right)^{1\/3} q - D , \\lbl{hex-asy1-cp} \n\\end{equation} \nFormally $D$ is the determinant of the matrix in (\\ref{hex-asy1-mat}), \nwhich is zero, giving a zero eigenvalue, which indicates marginal \nstability.   Hence, we return to the more accurate matrix in \n(\\ref{hex-stab-mat}),  which gives $D \\sim -\\beta^2\\mu\\nu$.   \nThe polynomial (\\ref{hex-asy1-cp}) thus has roots \n\\begin{equation} \nq_1 \\sim -\\mu\\nu, \\quad \nq_2 \\sim - \\left( \\frac{ \\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu)}{12} \\right)^{1\/3} , \n\\quad \nq_3 \\sim - \\left( \\frac{12 \\beta^4}{\\varrho(\\alpha\\nu\\!+\\!\\xi)} \\right)^{1\/3} . \n\\lbl{hex-ev1} \\end{equation} \nThis means that the symmetric state is always linearly stable \nfor this asymptotic scaling.  We expect to observe evolution on \nthree distinct timescales, one of ${\\cal O}(1)$, one of \n${\\cal O}(\\beta^{-2\/3})$ and one of ${\\cal O}(\\beta^{-4\/3})$. \n\nWe now consider the other asymptotic limit, namely, \n$\\alpha\\sim\\xi\\gg1$ and all other parameters are ${\\cal O}(1)$.  \nIn this case, taking the leading order terms in each row, the stability \nmatrix in (\\ref{hex-stab-mat}) reduces to \n\\begin{equation}\n\\left( \\begin{array}{ccc} \n-6 \\mu \\nu \\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{2\/3} & \n6 \\mu \\nu \\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{2\/3} & 0\n\\\\\n\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} & \n-\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} & \n-\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} \n\\\\ \n\\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} &  \n- \\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} &  \n- \\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} \n\\end{array} \\right) , \n\\end{equation}\nwhich again formally has a zero determinant. \nThe characteristic polynomial is \n\\begin{equation}\n0 = q^3 + q^2 + 6 \\beta \\mu\\nu q - D , \n\\end{equation}\nwherein we again take the more accurate determinant \nobtained from a higher-order expansion of \n(\\ref{hex-stab-mat}), namely $D=\\beta^2\\mu\\nu$. \nThe eigenvalues are then given by \n\\begin{equation}\nq_1 \\sim - \\left( \\frac{\\beta\\varrho^2\\xi^2}{144} \\right)^{1\/3} , \n\\qquad \nq_{2,3} \\sim \\pm \\sqrt{\\beta\\mu\\nu} \n\\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{1\/3} . \n\\lbl{hex-ev2} \\end{equation}\nWe now observe that there is always one stable and \ntwo unstable eigenvalues, so we deduce that the system \nbreaks symmetry in the case $\\alpha \\sim \\xi \\gg 1$.  \nThe first eigenvalue corresponds to a faster timescale \nwhere $t\\sim {\\cal O}(\\xi^{-2\/3})$ whilst the latter two \ncorrespond to the slow timescale where $t={\\cal O}(\\xi^{1\/3})$. \n\n\\subsection{Simulation results} \n\n\\begin{figure}[!ht] \n\\vspace*{64mm} \n\\special{psfile=fig-hex1.eps \n\thscale=60 vscale=45 hoffset=00 voffset=-96} \n\\caption{Illustration of the evolution of the total concentrations \n$c_2,z,w,u$ for a numerical solution of the system truncated \nat hexamers (\\ref{hex-c2})--(\\ref{hex-y6}) in the limit \n$\\alpha\\sim\\xi \\gg1$.  Since model equations \nare in nondimensional form, the time units are arbitrary. \nThe parameters are $\\alpha=\\xi=30$, $\\nu=0.5$, $\\beta=\\mu=1$, \nand the initial data is $x_6(0)=y_6(0)=0.06$, \n$x_4(0)=y_4(0)=0.01$, $x_2(0) = 0.051$, $y_2(0) = 0.049$, \n$c_2(0) = 0$. Note the time axis has a logarithmic scale. } \n\\label{fig-hex-1} \n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{64mm}\n\\special{psfile=fig-hex2.eps \n\thscale=60 vscale=45 hoffset=00 voffset=-96}\n\\caption{Graph of the evolution of the chiralities against time \non a log-log scale; results of numerical simulation of the same \nhexamer-truncated system, with identical initial data and \nparameters as in Figure \\protect\\ref{fig-hex-1}. }\n\\label{fig-hex-2}\n\\end{figure}\n\nWe briefly review the results of a numerical simulation of \n(\\ref{hex-c2})--(\\ref{hex-y6}) in the case $\\alpha\\sim\\xi\\gg1$ \nto illustrate the symmetry-breaking observed therein. \nAlthough the numerical simulation used the variables $x_k$ and \n$y_k$ ($k=2,4,6$) and $c_2$, we plot the total concentrations \n$z,w,u$ in Figure \\ref{fig-hex-1}.   The initial conditions have a \nslight imbalance in the handedness of small crystals ($x_2,y_2$).  \nThe chiralities of small ($x_2,y_2,z$), medium ($x_4,y_4,w$), and \nlarger ($x_6,y_6,u$) are plotted in Figure \\ref{fig-hex-2} on a \nlog-log scale. Whilst Figure \\ref{fig-hex-1} shows the \nconcentrations in the system has equilibrated by $t=10$, at this \nstage the chiralities are in a metastable state, that is, a long plateau \nin the chiralities between $t=10$ and $t=10^3$ where little appears \nto change.  There then follows a period of equilibration of chirality \non the longer timescale when $t\\sim 10^4$.   We have observed \nthis significant delay between the equilibration of concentrations \nand that of chiralities in a large number of simulations.  The reason \nfor this difference in timescales is due to the differences in the sizes \nof the eigenvalues in (\\ref{hex-ev1}). \n\nWe have also investigated the case $\\beta \\ll1$ with all other \nparameters ${\\cal O}(1)$ to verify that this case does indeed approach \nthe racemic state at large times (that is, $\\theta,\\phi,\\zeta \\rightarrow0$ \nas $t\\rightarrow\\infty$).   However, once again the difference in \ntimescales can be observed, with the concentrations reaching \nequilibration on a faster timescale than the chiralities, due to the \ndifferent magnitudes of eigenvalues (\\ref{hex-ev2}).\n\n\\section{New simplifications of the system}\n\\label{new-sec}\n\\setcounter{equation}{0}\n\nWe return to the equations (\\ref{smm2})--(\\ref{smmy2}) \nin the case $\\delta=0$, now writing $x_2=x$ and $y=y_2$ \nto obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & = & - 2 \\mu c \n\t+ \\mu\\nu (x+y) - \\alpha c(N_x+N_y) , \\lbl{newcdot} \\\\\n\\frac{{\\rm d} x}{{\\rm d} t} & = & \\mu c - \\mu\\nu x - \\alpha x c \n\t+ \\beta (N_x-x + x_4) - \\xi x^2 - \\xi x N_x , \\lbl{newxdot} \\\\ \n\\frac{{\\rm d} y}{{\\rm d} t} & = & \\mu c - \\mu\\nu y  - \\alpha y c \n\t+ \\beta (N_y-y + y_4) - \\xi y^2 - \\xi y N_y , \\lbl{newydot}\\\\ \n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mu c - \\mu\\nu x \n\t+ \\beta (N_x-x) - \\xi x N_x , \\lbl{newnxdot} \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mu c - \\mu\\nu y \n\t+ \\beta (N_y-y) - \\xi y N_y , \\lbl{newnydot} \n\\end{eqnarray} \nwhich are not closed, since $x_4,y_4$ appear \non the {\\sc rhs}'s of (\\ref{newxdot}) and (\\ref{newydot}), \nhence we need to find formulae to determine $x_4$ and $y_4$ \nin terms of $x,y,N_x,N_y$.\n\nOne way of achieving this is to expand the system to include other \nproperties of the distribution of cluster sizes.  For example, equations \ngoverning the mass of crystals in each chirality can be derived as \n\\begin{equation}\n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t}=2\\mu c-2\\mu\\nu x+2\\alpha c N_x ,\n\\quad\n\\frac{{\\rm d} \\varrho_y}{{\\rm d} t}=2\\mu c-2\\mu\\nu y+2\\alpha c N_y . \n\\lbl{new-roxy-dot} \\end{equation}\nThese introduce no more new new quantities into the \nmacroscopic system of equations, and do not rely on knowing \n$x_4$ or $y_4$, (although they do require knowledge of $x$ and $y$). \n\nIn the remainder of this section we consider various potential formulae \nfor $x_4$, $y_4$ in terms of macroscopic quantities so that a \nmacroscopic system can be constructed.  We then analyse such  \nmacroscopic systems in two specific limits to show that predictions \nrelating to symmetry-breaking can be made. \n\n\\subsection{Reductions}\n\nThe equations governing the larger cluster sizes $x_k$, $y_k$, are \n\\begin{equation} \n\\frac{{\\rm d} x_{2k}}{{\\rm d} t} = \\beta( x_{2k+2} - x_{2k} ) \n- (x_{2k}-x_{2k-2})(\\alpha c + \\xi x) ;  \n\\lbl{5xkdot} \\end{equation} \nin general this has solutions of the form $x_{2k} = \\sum_j A_j(t) \n\\Lambda_j^{k-1}$, \nwhere $\\Lambda_j$ are parameters (typically taking values between \nunity (corresponding to a steady-state in which mass is being \nadded to the distribution) and \n$\\mfrac{\\alpha c+\\xi x}{\\beta}$ (the equilibrium value); and $A_j(t)$ \nare time-dependent; for some $\\Lambda_j$, $A_j$ will be constant. \n\nWe assume that the distribution of each chirality of cluster is given by \n\\begin{equation} \nx_{2k} = x \\left( 1 - \\frac{1}{\\lambda_x} \\right)^{k-1} ,\\qquad\\qquad\ny_{2k} = y \\left( 1 - \\frac{1}{\\lambda_y} \\right)^{k-1} , \n\\end{equation}\nsince solutions of this form may be steady-states of the governing \nequations (\\ref{5xkdot}).  However, in our approximations for \n$x_4$ and $y_4$ the parameters $\\lambda_x$, $\\lambda_y$ are \npermitted to vary with time in some way that depends on other \nquantities in the model equations.   The resulting expressions for \nthe macroscopic number and mass quantities are \n\\begin{eqnarray}\nN_x = \\sum_{k=1}^\\infty x_{2k} = x \\lambda_x , &\\qquad&  \nN_y = \\sum_{k=1}^\\infty y_{2k} = y \\lambda_y , \\\\   \n\\varrho_x = \\sum_{k=1}^\\infty 2 k x_{2k} = 2 x \\lambda_x^2 , &\\qquad& \n\\varrho_y = \\sum_{k=1}^\\infty 2 k y_{2k} = 2 y \\lambda_y^2 . \n\\end{eqnarray}\nOur aim is to find a simpler expression for the terms $x_4$ \nand $y_4$ which occur in (\\ref{newxdot})--(\\ref{newydot}), \nthese are given by $x_4=x(1-1\/\\lambda_x)$ where \n\\begin{equation}\n\\lambda_x = \\frac{N_x}{x} = \\frac{\\varrho_x}{2N_x} \n= \\sqrt{\\frac{\\varrho_x}{2x}} , \\lbl{lambda-eqs}\n\\end{equation}\nhence\n\\begin{equation}\nx_4 = x - \\frac{x^2}{N_x} , \\quad \nx_4 = x - \\frac{2 x N_x}{\\varrho_x} ,\\quad \n{\\rm or} \\;\\;\\; \nx_4 = x - x\\sqrt{\\frac{2x}{\\varrho_x}} . \n\\end{equation}\n\nThere are thus three possible reductions of the equations \n(\\ref{newcdot})--(\\ref{newnydot}), each eliminating one of \n$x,N_x,\\varrho_x$ (and the corresponding $y,N_y,\\varrho_y$). We consider \neach reduction in turn in the following subsections.  Since some \nof these reductions involve $\\varrho_x, \\varrho_y$, we also use the \nevolution equations (\\ref{new-roxy-dot}) for these quantities. \n\n\\subsection{Reduction 1: to $x,y,N_x,N_y$}\n\nHere we assume $\\lambda_x = N_x\/x$, $\\lambda_y = N_y\/y$, \nso, in addition to (\\ref{newcdot}), (\\ref{newnxdot})--(\\ref{newnydot}) \nthe equations of motion are \n\\begin{eqnarray}\n\\frac{{\\rm d} x}{{\\rm d} t} & = & \\mu c - \\mu \\nu x + \\beta N_x \n\t- \\frac{\\beta x^2}{N_x} - \\xi x^2 - \\xi x N_x ,  \\\\\n\\frac{{\\rm d} y}{ {\\rm d} t} & = & \\mu c - \\mu \\nu y + \\beta N_y \n\t- \\frac{\\beta y^2}{N_y} - \\xi y^2 - \\xi y N_y ; \n\\end{eqnarray}\nwe have no need of the densities $\\varrho_x,\\varrho_y$ in this formulation. \n\nThe disadvantage of this reduction is that, due to \n(\\ref{lambda-eqs}), the total mass is given by \n\\begin{equation}\n\\varrho = 2c + \\varrho_x+\\varrho_y = 2 c + \\frac{2 N_x^2}{x} \n+ \\frac{2 N_y^2}{y} , \n\\end{equation}\nand there is no guarantee that this will be conserved. \n\nWe once again consider the system in terms of total concentrations \nand relative chiralities by applying the transformation \n\\begin{eqnarray}&\nx = \\mbox{$\\frac{1}{2}$} z (1\\!+\\!\\theta) , \\quad \ny = \\mbox{$\\frac{1}{2}$} z (1\\!-\\!\\theta) , \\quad \nN_x = \\mbox{$\\frac{1}{2}$} N (1\\!+\\!\\phi) , \\quad \nN_y = \\mbox{$\\frac{1}{2}$} N (1\\!-\\!\\phi) , & \\nonumber \\\\ && \n\\end{eqnarray}\nto obtain the equations\n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& - 2 \\mu c + \\mu \\nu z - \\alpha c N , \n\\lbl{R1cdot} \\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & =& 2\\mu c - \\mu \\nu z - \\alpha c z + \\beta N \n\t-\\frac{\\beta z^2(1+\\theta^2-2\\theta\\phi)}{N(1-\\phi^2)} \\nonumber \\\\ && \n\t- \\mbox{$\\frac{1}{2}$} \\xi z^2(1+\\theta^2) - \\mbox{$\\frac{1}{2}$} \\xi z N (1+\\theta\\phi) , \n\t\\lbl{R1zdot} \\\\\n\\frac{{\\rm d} N}{{\\rm d} t} & = & 2\\mu c - \\mu \\nu z + \\beta N - \\beta z \n\t- \\mbox{$\\frac{1}{2}$} \\xi z N (1+\\theta\\phi) . \\lbl{R1Ndot} \\\\ \n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & \n\t- \\left( \\mu \\nu + \\alpha c + \\xi z + \\mbox{$\\frac{1}{2}$} \\xi N \n\t\t+ \\frac{2\\beta z}{N(1\\!-\\!\\phi^2)} \n\t\t+ \\frac{1}{z} \\frac{{\\rm d} z}{{\\rm d} t} \\right) \\theta \\nonumber \\\\ && \n\t+ \\left( \\frac{\\beta N}{z} - \\mbox{$\\frac{1}{2}$} \\xi N \n\t+ \\frac{\\beta z (1\\!+\\!\\theta^2)}{N(1\\!-\\!\\phi^2)} \n\t\\right) \\phi , \\\\ \n\\frac{{\\rm d} \\phi}{{\\rm d} t} & =& \n\t- \\left( \\mu\\nu + \\beta + \\mbox{$\\frac{1}{2}$} \\xi N \\right) \\frac{z}{N} \\theta \n\t+ \\left( \\beta - \\mbox{$\\frac{1}{2}$} \\xi z - \\frac{1}{N}\\frac{{\\rm d} N}{{\\rm d} t} \\right) \\phi . \n\t\\nonumber \\\\ && \n\\end{eqnarray}\nThese equations have the symmetric steady-state given by \n$\\theta=0=\\phi$ and $c,z,N$ satisfying \n\\begin{equation}\nc = \\frac{\\mu\\nu z}{2\\mu+\\alpha N}  , \\qquad\nz = \\frac{2\\beta N (2\\mu+\\alpha N) }\n{(2\\beta + \\xi N)(2\\mu+\\alpha N)+2\\alpha\\mu\\nu N} , \n\\lbl{R1sss} \\end{equation}\nfrom (\\ref{R1cdot}) and (\\ref{R1Ndot}).   Note that the steady \nstate value of $N$ will depend upon the initial conditions, it is \nnot determined by (\\ref{R1zdot}). This is because the \nsteady-state equations obtained by setting the time derivatives \nin (\\ref{R1cdot})--(\\ref{R1Ndot}) are not independent.  \nThe difference (\\ref{R1zdot})--(\\ref{R1Ndot}) is equal \nto $z\/N$ times the sum (\\ref{R1cdot})$+$(\\ref{R1Ndot}).  \n\nIn subsections \\ref{5R1A1-sec} and \\ref{5R1A2-sec} below, \nso as to discuss the stability of a solution in the two asymptotic \nregimes $\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$, we augment the \nsteady-state equations (\\ref{R1cdot})--(\\ref{R1Ndot}) with the \ncondition $\\varrho=2N^2\/z$, with $\\varrho$ assumed to be ${\\cal O}(1)$.\n\nThe linear stability of $\\theta=0=\\phi$ is given by assuming \n$\\theta$ and $\\phi$ are small, yielding the system \n\\begin{equation} \n\\frac{{\\rm d}}{{\\rm d} t} \\!\\!\\left(\\!\\! \\begin{array}{c} \n\\theta \\\\[2ex] \\phi \\end{array}\\!\\! \\right) \\!\n= \\! \\left( \\begin{array}{cc} \n- \\left( \\displaystyle\\frac{2\\mu c}{z} + \\displaystyle\\frac{\\xi z}{2} \n\t+ \\displaystyle\\frac{\\beta z}{N} \n\t+ \\displaystyle\\frac{\\beta N}{z} \\right) &\n\\left(\\displaystyle\\frac{\\beta N}{z} + \\displaystyle\\frac{\\beta z}{N} \n\t- \\displaystyle\\frac{\\xi N}{2} \\right) \\\\ \n- ( \\mu \\nu + \\beta + \\mbox{$\\frac{1}{2}$} \\xi N ) \\displaystyle\\frac{z}{N} & \n\\left( \\beta + \\mu\\nu - \\displaystyle\\frac{2\\mu c}{z} \\right) \n\\displaystyle\\frac{z}{N} \\end{array} \\right) \\!\\! \\left( \\!\\!\n\\begin{array}{c} \\theta \\\\[2ex] \\phi \\end{array} \\!\\!\\right)\\! . \n\\lbl{5stabmat} \\end{equation}\nAn instability of the symmetric solution is indicated by the \ndeterminant of this matrix being negative. Substituting (\\ref{R1sss}) \ninto the determinant, yields \n\\begin{equation}\n\\mbox{det} = \\frac{ \\beta \\mu \\nu ( 4 \\beta \\mu - \\alpha \\xi N^2 )}\n{4\\beta\\mu + 2 \\alpha \\beta N + 2 \\mu \\xi N + 2 \\alpha \\mu \\nu N \n+ \\alpha \\xi N^2} . \n\\lbl{5detsimp} \\end{equation}\nHence we find that the symmetric (racemic) state is unstable if \n$N > 2 \\sqrt{ \\mu\\beta \/ \\alpha \\xi }$, that is,  large aggregation rates \n($\\alpha,\\xi$) and slow grinding ($\\beta$) are preferable for \nsymmetry-breaking. \n\nWe consider two specific asymptotic limits of parameter values \nso as to derive specific results for steady-states and conditions on \nstability.  In both limits, we have that the aggregation rates dominate \nfragmentation ($\\alpha \\sim \\xi \\gg \\beta$), so that the system is \nstrongly biased towards the formation of crystals and the dimer \nconcentrations are small.   In the first case we assume that the \nfragmentation is small and the aggregation rates are of a similar \nscale to the interconversion of dimers ($\\beta \\ll \\mu \\sim \\alpha \\sim \n\\xi = {\\cal O}(1)$); whilst the second has a fragmentation rate of \nsimilar size to the dimer conversion rates and larger aggregation rates \n($\\alpha \\sim \\xi \\gg \\mu \\sim \\beta = {\\cal O}(1)$). \n\n\\subsubsection{Asymptotic limit 1: $\\beta\\ll1$} \n\\label{5R1A1-sec} \n\nIn the case of  asymptotic limit 1, $\\beta\\ll1$, we find the \nsteady-state solution \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi+\\alpha\\nu}} , \\quad \nz \\sim \\frac{2\\beta}{\\xi+\\alpha\\nu} , \\quad \nc \\sim \\frac{\\beta\\nu}{\\xi+\\alpha\\nu} . \n\\end{equation}\nFrom (\\ref{5detsimp}), we find an instability if $\\varrho > \\varrho_c := \n4 \\mu (\\xi+\\alpha\\nu) \/ \\alpha\\xi$. That is, larger masses ($\\varrho$) \nfavour symmetry-breaking, as do larger aggregation rates \n($\\alpha,\\xi$).  The eigenvalues of (\\ref{5stabmat}) in this limit are \n$q_1 = -\\mu\\nu$ -- a fast stable mode of the dynamics and \n\\begin{equation}\nq_2 = \\frac{\\alpha \\xi \\beta^{3\/2}}{2\\mu \\sqrt{\\varrho} (\\xi+\\alpha\\nu)^{3\/2}} \n\\left( \\varrho - \\frac{4\\mu(\\xi+\\alpha\\nu)}{\\alpha\\xi} \\right) , \n\\end{equation}\nwhich indicates a slowly growing instability when $\\varrho>\\varrho_c$.  Hence \nthe balace of achiral to chiral morphologies of smaller clusters ($\\nu$) \nalso influences the propensity for non-racemic solution.  However, \nsince the dynamics described by this model does not conserve total \nmass, the results from this should be treated with some caution, \nand we now analyse models which do conserve total mass. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha\\sim\\xi\\gg1$}\n\\label{5R1A2-sec}\n\nIn this case we find the steady-state solution is given by \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \\quad  \nz \\sim \\frac{2\\beta}{\\xi} , \\quad \nc \\sim \\frac{4\\mu\\nu}{\\alpha} \\sqrt{\\frac{\\beta}{\\xi\\varrho}} . \n\\end{equation}\nThe condition following from (\\ref{5detsimp}) then implies that we \nhave an instability if $\\varrho>\\varrho_c := 4\\mu\/\\alpha \\ll 1$. The eigenvalues \nof the stability matrix are $q_1 = - \\mbox{$\\frac{1}{2}$} \\sqrt{\\beta\\varrho\\xi}$, which is \nlarge and negative, indicating attraction to some lower dimensional \nsolution over a relatively fast timescale; the eigenvector being \n$(1,0)^T$ showing that $\\theta\\rightarrow0$. The other eigenvalue \nis $q_2 = 2\\mu\\nu \\sqrt{\\beta\/\\varrho\\xi} \\ll 1$, and  corresponds to a slow \ngrowth of the chirality of the solution, since it relates to the \neigenvector $(0,1)^T$. Assuming the system is initiated near its \nsymmetric solution ($\\theta=\\phi=0$), this shows that the distribution \nof clusters changes its chirality first, whilst the dimer concentrations \nremain, at least to leading order, racemic.  We expect that at a later \nstage the chirality of the dimers too will become nonzero. \n\n\\subsection{Reduction 2: to $x,y,\\varrho_x,\\varrho_y$}\n\nHere we eliminate $x_4=x(1-1\/\\lambda_x)$, \n$y_4=y(1-1\/\\lambda_y)$ together with $N_x$ and $N_y$ using \n\\begin{equation}\n\\lambda_x=\\sqrt{\\frac{\\varrho_x}{2x}}, \\quad \n\\lambda_y=\\sqrt{\\frac{\\varrho_y}{2y}}, \\quad \nN_x = \\sqrt{\\frac{x\\varrho_x}{2}}, \\quad \nN_y = \\sqrt{\\frac{y\\varrho_y}{2}}, \n\\end{equation}\nleaving a system of equations for $(c,x,y,\\varrho_x,\\varrho_y)$ \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & = & \\mu\\nu(x+y) - 2\\mu c \n- \\sqrt{2} \\alpha c \\left( \\sqrt{x\\varrho_x} + \\sqrt{y \\varrho_y} \\right) , \\\\ \n\\frac{{\\rm d} x}{{\\rm d} t} & =& \\mu c - \\mu \\nu x - \\alpha c x - \n\\xi x^2 - \\xi x \\sqrt{\\frac{x\\varrho_x}{2}} + \\beta \\sqrt{\\frac{x\\varrho_x}{2}} \n- \\beta x \\sqrt{\\frac{2x}{\\varrho_x}} , \\nonumber \\\\ && \\\\ \n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t} & = & - 2 \\mu \\nu x + 2 \\mu c \n+ 2 \\alpha c \\sqrt{\\frac{x\\varrho_x}{2}} , \n\\end{eqnarray} \nwith similar equations for $y,\\varrho_y$.  Transforming to total \nconcentrations and relative chiralities by way of \n\\begin{eqnarray}&\nx = \\mbox{$\\frac{1}{2}$} z (1+\\theta) , \\quad \ny = \\mbox{$\\frac{1}{2}$} z (1-\\theta) , \\quad \n\\varrho_x = \\mbox{$\\frac{1}{2}$} R (1+\\zeta) , \\quad \n\\varrho_y = \\mbox{$\\frac{1}{2}$} R (1-\\zeta) , \n&\\nonumber\\\\&&\n\\end{eqnarray}\nwe find \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& \\mu \\nu z - 2 \\mu c \n\t- \\frac{\\alpha c \\sqrt{z R}}{2\\sqrt{2}} \\left[ \n\t\\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \n\t\\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \\right] , \\lbl{new-r2-cdot} \n\\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & = & 2\\mu c - \\mu \\nu z - \\alpha c z \n\t- \\mbox{$\\frac{1}{2}$} \\xi z^2 (1\\!+\\!\\theta^2) \\nonumber \\\\ && \n\t+ \\frac{\\beta \\sqrt{zR}}{2\\sqrt{2}} \n\t\\left[ \\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \n\t\\right] \\nonumber \\\\ && \n\t- \\frac{\\xi z^{3\/2} R^{1\/2}}{4\\sqrt{2}} \\left[ \n\t(1\\!+\\!\\theta)^{3\/2} (1\\!+\\!\\zeta)^{1\/2} + (1\\!-\\!\\theta)^{3\/2} \n\t(1\\!-\\!\\zeta)^{1\/2} \\right] \\nonumber \\\\ && \n\t- \\frac{\\beta z^{3\/2} }{\\sqrt{2R}} \n\t\\left[ \\frac{(1\\!+\\!\\theta)^{3\/2}}{(1\\!+\\!\\zeta)^{1\/2}} + \n\t\\frac{(1\\!-\\!\\theta)^{3\/2}}{(1\\!-\\!\\zeta)^{1\/2}} \\right] , \n\\lbl{new-r2-zdot} \n\\\\ \n\\frac{{\\rm d} R}{{\\rm d} t} & = & - 2\\mu\\nu z + 4 \\mu c \n\t+ \\mbox{$\\frac{1}{2}$} \\alpha c \\sqrt{2zR} \\left[ \n\t\\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \n\t\\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \\right] , \n\\nonumber \\\\ && \\lbl{new-r2-Rdot} \n\\end{eqnarray}\ntogether with the equations \n(\\ref{new-r2-thetadot})--(\\ref{new-r2-zetadot}) for the relative \nchiralities $\\theta$ and $\\zeta$,  which will be analysed later. \n\nSince the equations for ${\\rm d} R\/dd t$ and ${\\rm d} c\/{\\rm d} t$ are \nessentially the same, we obtain a third piece of information from \nthe requirement that the total mass in the system is unchanged \nfrom the initial data, hence the new middle equation above. \nSolving these we find $c=\\mbox{$\\frac{1}{2}$} (\\varrho-R)$ and use this in place \nof the equation for $c$. \n\nIn the symmetric case ($\\theta=\\zeta=0$) we obtain the \nsteady-state conditions \n\\begin{eqnarray}\n0 & = & 2\\mu\\nu z - 4\\mu c - \\alpha c \\sqrt{2zR} , \\qquad\\qquad \n\\varrho \\; = \\; R + 2 c , \\lbl{R2-ssss1} \\\\ \n0 & = & 2\\mu c - \\mu \\nu z - \\alpha c z - \\mbox{$\\frac{1}{2}$} \\xi z^2 \n\t+ \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{2zR} - \\beta z \\sqrt{\\frac{2z}{R}} \n\t- \\frac{\\xi z}{2} \\sqrt{\\frac{zR}{2}} . \\nonumber \\\\ && \\lbl{R2-ssss2} \n\\end{eqnarray}\nFor small $\\theta,\\zeta$, the equations for the chiralities \ncan be approximated by \n\\begin{eqnarray}\n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & - \\left( \\frac{2\\mu c}{z} \n\t+ \\mbox{$\\frac{1}{2}$} \\xi z + \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{\\frac{R}{2z}} \n\t+ \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{\\frac{2z}{R}} \n\t+ \\rec{4} \\xi \\sqrt{\\frac{zR}{2}} \\right) \\theta \\nonumber \\\\ && \n\t+ \\left( \\frac{\\beta(R+2z)}{2\\sqrt{2zR}} \n\t- \\frac{\\xi}{4} \\sqrt{\\frac{Rz}{2}} \\right) \\zeta , \n\\lbl{new-r2-thetadot}  \\\\ \n\\frac{{\\rm d} \\zeta}{{\\rm d} t} & = & \\left( \\frac{2\\mu\\nu z}{R} \n- \\alpha c \\sqrt{\\frac{zR}{2}} \\right) \\theta \n- \\left( \\frac{2\\mu\\nu z}{R} - \\frac{4\\mu c}{R} \\right) \\zeta , \n\\lbl{new-r2-zetadot} \n\\end{eqnarray}\nWe analyse the stability of the symmetric (racemic) state in the two \nlimits $\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$ in the next subsections. \n\n\\subsubsection{Asymptotic limit 1: $\\beta\\ll1$}\n\nIn this case, solving the conditions \n(\\ref{R2-ssss1})--(\\ref{R2-ssss2}) asymptotically, \nwe find \n\\begin{equation}\nz \\sim \\frac{2\\beta}{\\xi+\\alpha\\nu} , \\qquad \nc \\sim \\frac{\\beta\\nu}{\\xi+\\alpha\\nu} , \\qquad \nR \\sim \\varrho - 2c . \n\\end{equation}\nSubstituting these values into the differential equations \nwhich determine the stability of the racemic state leads to \n\\begin{equation}\n\\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\theta \\\\[3ex] \\zeta \\end{array} \\right) \n\\left( \\begin{array}{cc} \n-\\mu\\nu & \n\\displaystyle\\frac{\\alpha\\nu}{4} \\sqrt{\\displaystyle\\frac{\\beta\\varrho}{\\xi+\\alpha\\nu}}\\\\ \n-\\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho(\\xi+\\alpha\\nu)} & \n\\displaystyle\\frac{\\alpha\\nu\\beta^{3\/2}}{(\\xi+\\alpha\\nu)^{3\/2} \\sqrt{\\varrho}}  \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\theta \\\\[3ex] \\zeta \\end{array} \\right) . \n\\end{equation}\nFormally this matrix has eigenvalues of zero and $-\\mu\\nu$. \nSince the zero eigenvalue indicates marginal stability of the \nracemic solution, we need to consider higher-order terms to \nobtain a more definite result. \n\nGoing to higher order, gives the determinant of the resulting matrix \nas $-\\alpha \\xi \\nu \/ (\\alpha\\nu+\\xi)^2$ hence the eigenvalues are \n\\begin{equation}\nq_1 = -\\mu\\nu , \\qquad {\\rm and} \\quad \nq_2 = \\frac{ \\alpha \\xi }{\\mu (\\alpha\\nu+\\xi)^2 } , \n\\end{equation}\nthe former indicating a rapid decay of $\\theta$ (corresponding to the \neigenvector $(1,0)^T$), and the latter showing a slow divergence from \nthe racemic state in the $\\zeta$-direction, at leading order, according to \n\\begin{equation}\n\\left( \\begin{array}{c} \\theta \\\\ \\zeta \\end{array} \\right) \n\\sim C_1 \\left( \\begin{array}{c} 0 \\\\ 1 \\end{array} \\right) \n\\exp \\left( \\frac{ \\alpha \\xi t }{\\mu (\\alpha\\nu+\\xi)^2 } \\right) . \n\\end{equation}\nHence in the case $\\beta\\ll1$, we find an instability of the \nsymmetric solution for all other parameter values. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha\\sim\\xi\\gg1$}\n\nIn this case, solving the conditions \n(\\ref{R2-ssss1})--(\\ref{R2-ssss2}) asymptotically, we find \n\\begin{equation} \nz \\sim \\frac{2\\beta}{\\xi} , \\qquad \nc \\sim \\frac{2\\mu\\nu}{\\alpha} \\sqrt{\\frac{\\beta}{\\varrho\\xi}} , \\qquad \nR \\sim \\varrho - 2c . \n\\end{equation} \nSubstituting these values into the differential equations \n(\\ref{new-r2-thetadot})--(\\ref{new-r2-zetadot}) \nwhich determine the stability of the racemic state leads to \n\\begin{equation} \n\\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\theta \\\\[1ex] \\zeta \\end{array} \\right) \n\\left( \\begin{array}{ccc} \n- \\mbox{$\\frac{1}{2}$} \\sqrt{\\beta\\xi\\varrho} && o(\\sqrt{\\xi}) \\\\[1ex] \n- \\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho\\xi} && \\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho\\xi} \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\theta \\\\[1ex] \\zeta \\end{array} \\right) , \n\\end{equation} \nhence the eigenvalues are $q_1=-\\mbox{$\\frac{1}{2}$}\\sqrt{\\beta\\varrho\\xi}$ and \n$q_2 = 4\\mu\\nu\\beta\/\\varrho\\xi$, (in the above $o(\\sqrt{\\xi})$ means \na quantity $q$ satisfying $q\\ll\\sqrt{\\xi}$ as $\\xi\\rightarrow\\infty$).  \nWhilst the former indicates the existence of a stable manifold (with \na fast rate of attraction), the latter shows that there is also an unstable \nmanifold. Although the timescale associated with this is much slower, \nit shows that the symmetric (racemic) state is unstable. \n\n\\subsection{Reduction 3: to $N_x,N_y,\\varrho_x,\\varrho_y$}\n\nIn this case our aim is to retain only information on the number \nand typical size of crystal distribution, so we eliminate the dimer \nconcentrations $x,y$, using \n\\begin{equation} \n\\lambda_x = \\frac{\\varrho_x}{2 N_x} , \\quad \n\\lambda_y = \\frac{\\varrho_y}{2 N_y} , \\quad \nx = \\frac{2 N_x^2}{\\varrho_x} , \\quad \ny = \\frac{2 N_y^2}{\\varrho_y} . \n\\end{equation} \nThese transformations reformulate the governing equations \n(\\ref{newcdot})--(\\ref{new-roxy-dot}) to \n\\begin{eqnarray}\n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mbox{$\\frac{1}{2}$} \\mu (\\varrho -R) + \\beta N_x  \n\t- 2 (\\mu\\nu+\\beta) \\frac{N_x^2}{\\varrho_x} \n\t- \\frac{2\\xi N_x^3}{\\varrho_x} , \\lbl{r3-nxdot} \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mbox{$\\frac{1}{2}$} \\mu (\\varrho - R) + \\beta N_y\n\t- 2 (\\mu\\nu+\\beta) \\frac{N_y^2}{\\varrho_y} \n\t- \\frac{2\\xi N_y^3}{\\varrho_y} , \\\\ \n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t} & = & (\\varrho-R)(\\mu+\\alpha N_x) \n\t- \\frac{4\\mu\\nu N_x^2}{\\varrho_x} , \\lbl{r3-roxdot} \\\\ \n\\frac{{\\rm d} \\varrho_y}{{\\rm d} t} & = & (\\varrho-R)(\\mu+\\alpha N_y) \n\t- \\frac{4\\mu\\nu N_y^2}{\\varrho_y} , \\lbl{r3-roydot}\n\\end{eqnarray}\nwhere $R := \\varrho_x + \\varrho_y$. \nWe now transform to total concentrations ($N$, $R$) \nand relative chiralities ($\\phi$ and $\\zeta$) {\\em via}  \n\\begin{equation} \nN_x = \\mbox{$\\frac{1}{2}$} N (1+\\phi) , \\quad \nN_y = \\mbox{$\\frac{1}{2}$} N (1-\\phi) , \\quad \n\\varrho_x = \\mbox{$\\frac{1}{2}$} R (1+\\zeta) , \\quad \n\\varrho_y = \\mbox{$\\frac{1}{2}$} R (1-\\zeta) , \n\\end{equation}\ntogether with $c = \\mbox{$\\frac{1}{2}$} (\\varrho - R)$, to obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} R}{{\\rm d} t} & = & (\\varrho-R)(2\\mu+ \\alpha N) \n- \\frac{4\\mu\\nu N^2(1+\\phi^2-2\\phi\\zeta)}{R (1-\\zeta^2)} , \n\\lbl{r3Rd} \\\\ \\lbl{r3Nd}  \n\\frac{{\\rm d} N}{{\\rm d} t} & = & \\!\\!\\mu (\\varrho \\! - \\! R) + \\beta N \n\t\\\\ && \\! - \\frac{N^2}{R(1\\!-\\!\\zeta^2)} \\left[ \n\t2(\\mu\\nu\\!+\\!\\beta) (1\\!+\\!\\phi^2\\!-\\!2\\phi\\zeta) + \n\t\\xi N (1\\!+\\!3\\phi^2\\!-\\!3\\phi\\zeta\\!-\\!\\phi^3\\zeta) \\right] ,\n\\nonumber \\\\ \n\\frac{{\\rm d}\\phi}{{\\rm d} t} &=& \\beta\\phi - \\frac{1}{N}\\frac{{\\rm d} N}{{\\rm d} t}\\phi\n\t\\\\&& \\!\\!- \\frac{N}{R(1\\!-\\!\\zeta^2)} \\left[ \n\t2(\\beta\\!+\\!\\mu\\nu)(2\\phi\\!-\\!\\zeta\\!-\\!\\phi^2\\zeta)\n\t+ \\xi N (3\\phi\\!-\\!\\zeta\\!+\\!\\phi^3\\!-\\!3\\phi^2\\zeta) \\right] , \\nonumber \n\\\\ \n\\frac{{\\rm d} \\zeta}{{\\rm d} t} & =& \\frac{\\alpha (\\varrho-R) N \\phi}{R} \n\t- \\frac{1}{R}\\frac{{\\rm d} R}{{\\rm d} t} \\zeta - \\frac{4\\mu\\nu N^2 \n\t(2\\phi-\\zeta-\\phi^2\\zeta)}{R^2 (1-\\zeta^2)} . \n\\end{eqnarray}\nWe now analyse this system in more detail, since this set of \nequations conserves mass, and is easier to analyse than \n(\\ref{new-r2-cdot})--(\\ref{new-r2-Rdot}) due to the absence of \nsquare roots.  We consider the two asymptotic limits ($\\beta\\ll1$ \nand $\\alpha\\sim\\xi\\gg1$) in which, at steady-state, the majority of \nmass is in the form of clusters. \n\n\\subsubsection{The symmetric steady-state}\n\nPutting $\\zeta=0=\\phi$, we find the symmetric steady-state is given by \n\\begin{eqnarray} \n0 &=& (\\varrho-R)(2\\mu+\\alpha N) - \\frac{4\\mu\\nu N^2}{R} , \n\\lbl{r3ssss1} \\\\ \n0 &=& \\mu (\\varrho-R) + \\beta N \n- 2(\\mu\\nu+\\beta)\\frac{N^2}{R} - \\frac{\\xi N^3}{R} . \n\\lbl{r3ssss2} \\end{eqnarray} \nthe former is solved by one of \n\\begin{equation}\nR = \\mbox{$\\frac{1}{2}$} \\varrho \\left( 1 \\pm \\sqrt{ 1 - \\frac{16\\mu\\nu N^2} \n{ (2\\mu+\\alpha N) \\varrho^2 } } \\right) , \\qquad \n\\end{equation}\n\\begin{equation}\nN = \\frac{\\alpha R(\\varrho-R)}{8\\mu\\nu} \\left( 1 + \n\\sqrt{1 + \\frac{32\\mu^2\\nu}{\\alpha^2 R(\\varrho-R)}} \\right) . \n\\end{equation}\nMore complete asymptotic solutions will be derived in Sections \n\\ref{r3-a1-sec} and \\ref{r3-a2-sec}.  \n\n\\subsubsection{Stability of the symmetric state} \n\nWe now consider the stability of the \nsymmetric steady-state. For small $\\phi,\\zeta$ we have \n\\begin{eqnarray} \\!\\!\\!\\!\\!& \n\\displaystyle\\frac{R}{N} \\displaystyle\\frac{{\\rm d}}{{\\rm d} t} \\!\\!\n\\left( \\!\\!\\begin{array}{c} \\phi \\\\ \\\\ \\zeta \\end{array} \\!\\!\\right) \n\\!=\\!\\! \\left( \\!\\!\\begin{array}{cc} \n\\!\\! - \\! 2\\beta \\!-\\! 2\\mu\\nu \\!-\\! 2 \\xi N \n\t\\!-\\! \\displaystyle\\frac{\\mu (\\varrho\\!-\\!R) R}{N^2} \\!\\!&\\!  \n2\\beta \\!+\\! 2\\mu\\nu \\!+\\! \\xi N   \n\\\\  \n\\left( \\alpha (\\varrho\\!-\\!R) \n\t\\!-\\! \\displaystyle\\frac{8\\mu\\nu N}{R} \\right) \\! \\!&\\! \n\\!8\\mu\\nu \\!-\\! \\displaystyle\\frac{(\\varrho\\!\\!-\\!\\!R)(2\\mu\\!\\!+\\!\\!\\alpha N)R}{N^2} \\!  \n\\end{array} \\!\\!\\right) \\!\\!\\!\n\\left(\\!\\! \\begin{array}{c} \\phi \\\\ \\\\ \\zeta \\end{array} \\!\\!\\right) \\!\\!,\\!\\! \n& \\nonumber \\\\ \\!\\!\\!\\!\\!\\! && \\!\\!\\!\\!\\lbl{r3-stab}\n\\end{eqnarray}\nand this is unstable if the determinant of this matrix is negative. \nNow we consider the two asymptotic limits in more detail. \n\n\\subsubsection{Asymptotic limit 1: $\\beta \\ll1$}\n\\label{r3-a1-sec}\n\nWhen fragmentation is slow, that is, $\\beta\\ll1$, at steady-state we \nhave $N={\\cal O}(\\sqrt{\\beta})$ and $R = \\varrho - {\\cal O}(\\beta)$. \nBalancing terms in (\\ref{r3ssss1})--(\\ref{r3ssss2}) we find the same \nleading order equation twice, namely $2\\nu N^2=\\beta\\varrho(\\varrho-R) $. \nTaking the difference of the two yields an independent equation \nfrom higher order terms, hence we obtain \n\\begin{equation} \nN \\sim \\sqrt{\\frac{\\beta \\varrho}{\\xi+\\alpha\\nu}} , \\qquad \nR \\sim \\varrho - \\frac{2\\nu\\beta}{\\xi+\\alpha\\nu} . \n\\end{equation} \nNote that this result implies that the dimer concentrations \nare small, with $c\\sim z$ and $c \\sim \\beta\\nu \/ (\\xi+\\alpha\\nu)$, \n$z\\sim 2\\beta\/(\\xi+\\alpha\\nu)$.  \n\nSubstituting these expressions into those for the stability of the \nsymmetric steady-state (\\ref{r3-stab}), we find \n\\begin{equation}\n\\frac{R}{4\\mu\\nu N} \\frac{{\\rm d}}{{\\rm d} t} \n\\left( \\begin{array}{c} \\phi \\\\[1ex] \\zeta \\end{array} \\right) = \n\\left( \\begin{array}{cc} -1 & \\quad \\frac{1}{2} \\\\ \n-2\\sqrt{\\displaystyle\\frac{\\beta}{\\varrho(\\xi\\!+\\!\\alpha\\nu)}} & \\quad 1 \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\phi \\\\[1ex] \\zeta \\end{array} \\right) . \n\\end{equation}\nThis matrix has one stable eigenvalue (corresponding to \n$(1,0)^T$ and hence the decay of $\\phi$ whilst $\\zeta$ remains \ninvariant), the unstable eigenvector is $(1,4)^T$, hence we find \n\\begin{equation}\n\\left( \\begin{array}{c} \\phi(t) \\\\ \\zeta(t) \\end{array} \\right) \\sim C \n\\left( \\begin{array}{c} 1 \\\\ 4 \\end{array} \\right) \\exp \\left( \n\\frac{4\\mu\\nu t \\sqrt{\\beta}}{\\sqrt{\\varrho(\\xi+\\alpha\\nu)}} \\right) . \n\\lbl{r3-chir-rate} \\end{equation} \nIf we compare the timescale of this solution to that over which the \nconcentrations $N,R$ vary, we find that symmetry-breaking \noccurs on a slower timescale than the evolution of cluster masses \nand numbers. This is illustrated in the numerical simulation of \nequations (\\ref{r3-nxdot})--(\\ref{r3-roydot}) shown in Figure \n\\ref{fig-r3alpha}.  More specifically, the time-scale increases with \nthe mass in the system, and with the ratio of aggregation to \nfragmentation rates, $(\\alpha\\nu+\\xi)\/\\beta$, and is inversely related \nto the chiral switching rate of small clusters ($\\mu\\nu$). \n\n\\begin{figure}[!ht]\n\\vspace*{68mm}\n\\special{psfile=fig-r3beta.eps \n\thscale=80 vscale=60 hoffset=-70 voffset=-150}\n\\caption{Graph of concentrations $N_x,N_y,\\varrho_x,\\varrho_y,c$ \nagainst time on a logarithmic time for the asymptotic limit 1, \nwith initial conditions $N_x=0.2=N_y$, $\\varrho_x=0.45$, $\\varrho_y=0.44$, \nother parameters given by $\\alpha=1=\\xi=\\mu$, $\\beta=0.01$ , \n$\\varrho=8$. Since model equations are in nondimensional form, \nthe time units are arbitrary.  }\n\\label{fig-r3alpha}\n\\end{figure}\n\n\\subsubsection{Asymptotic limit 2: $\\alpha \\sim \\xi \\gg 1$}\n\\label{r3-a2-sec}\n\nIn this case we retain the assumptions that $\\mu,\\nu={\\cal O}(1)$, \nhowever, we now impose $\\beta={\\cal O}(1)$ and \n$\\alpha \\sim \\xi \\gg1$.  For a steady-state, we require the scalings \n$N ={\\cal O}(1\/\\sqrt{\\xi})$ and $\\varrho-R={\\cal O}(1\/\\xi^{3\/2})$. \nSpecifically, solving (\\ref{r3ssss1})--(\\ref{r3ssss2}) we find \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \\qquad \nR \\sim \\varrho - \\frac{4\\mu\\nu}{\\alpha\\varrho} \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \n\\lbl{r3a2-sss} \\end{equation}\nhence the dimer concentrations $c = \\mbox{$\\frac{1}{2}$} (\\varrho-R) \\sim N^3 = \n{\\cal O}(1\/\\xi^{3\/2})$ and $z = 2 N^2\/\\varrho \\sim N^2 = {\\cal O}(1\/\\xi)$. \nMore precisely, $c\\sim (2\\mu\\nu\/\\alpha)\\sqrt{\\beta\/\\varrho\\xi}$ and \n$z\\sim 2\\beta\/\\xi$, in contrast with the previous asymptotic scaling \nwhich gave $z\\sim N^2$). \n\nTo determine the timescales for crystal growth and dissolution, \nwe use (\\ref{r3a2-sss}) to define\n\\begin{equation} \nN \\sim n(t) \\sqrt{\\beta \\varrho\/\\xi} , \\quad \nR \\sim \\varrho - \\frac{4\\mu\\nu r(t)}{\\alpha \\varrho} \n\\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \n\\end{equation} \nand so rewrite the governing equations (\\ref{r3Rd})--(\\ref{r3Nd}) as \n\\begin{eqnarray}\n\\frac{{\\rm d} n}{{\\rm d} t} & = & \\beta n \\left( 1 - n^2 - \n\\frac{2 n (\\beta+\\mu\\nu)}{\\sqrt{\\varrho\\xi\\beta}} \\right) , \\\\ \n\\frac{{\\rm d} r}{{\\rm d} t} & = & \\alpha \\sqrt{\\frac{\\beta\\varrho}{\\xi}} \n\\left( n^2 -r - \\frac{2\\mu r}{\\alpha} \n\\sqrt{\\frac{\\xi}{\\beta\\varrho}} \\right) . \n\\end{eqnarray}\nHere, the former equation for $n(t)$ corresponds to the \nslower timescale, with a rate $\\beta$, the rate of \nequilibration of $r(t)$ being $\\alpha \\sqrt{\\beta\\varrho\/\\xi}$. \n\nThe stability of the symmetric state is determined by \n\\begin{equation}\n\\frac{R}{N} \\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\phi(t) \\\\ \\zeta(t) \\end{array} \\right) = \n\\left( \\begin{array}{cc} -2 \\sqrt{\\beta\\varrho\\xi} & \\sqrt{\\beta\\varrho\\xi} \\\\ \n-4\\mu\\nu \\sqrt{\\beta \/ \\xi \\varrho} & 4\\mu\\nu \\end{array} \\right) \n\\left( \\begin{array}{c} \\phi \\\\ \\zeta \\end{array} \\right) . \n\\lbl{r3a2-phi-zeta-sys} \\end{equation} \nThis matrix has one large negative eigenvalue ($\\sim -2\\sqrt{\\beta\\varrho\\xi}$) \nand one (smaller) positive eigenvalue ($\\sim 4\\mu\\nu$); the former \ncorresponds to $(1,0)^T$ hence the decay of $\\phi$, whilst the latter \ncorresponds to the eigenvector $(1,2)^T$.  Hence the system \n(\\ref{r3a2-phi-zeta-sys}) has the solution \n\\begin{equation} \n\\left( \\begin{array}{c} \\phi \\\\ \\zeta \\end{array} \\right) \\sim \nC \\left( \\begin{array}{c} 1 \\\\ 2 \\end{array} \\right) \n\\exp \\left( 4 \\mu \\nu t \\sqrt{ \\frac{\\beta}{\\varrho\\xi}} \\right) . \n\\lbl{r3a2urate} \\end{equation} \nThe chiralities evolve on two timescales, the faster being \n$2\\beta$ corresponding to the stable eigenvalue of \n(\\ref{r3a2-phi-zeta-sys}) and the slower unstable rate \nbeing $4\\mu\\nu\\sqrt{\\beta\/\\xi\\varrho}$.  This timescale is \nsimilar to (\\ref{r3-chir-rate}), being dependent on mass and \nthe ratio of aggregation to fragmentation, and inversely \nproportional to the chiral switching rate of dimers ($\\mu\\nu$).  \n\n\\begin{figure}[!ht]\n\\vspace*{68mm}\n\\special{psfile=fig-r3alpha.eps \n\thscale=80 vscale=60 hoffset=-70 voffset=-150}\n\\caption{Graph of the concentrations $N_x,N_y,\\varrho_x,\\varrho_y,c$ \nagainst time on a logarithmic time for the asymptotic limit 2, \nwith initial conditions $N_x=0.2=N_y$, $\\varrho_x=0.45$, $\\varrho_y=0.44$, \nother parameters given by $\\alpha=10=\\xi$, $\\beta=1=\\mu$, \n$\\nu=0.5$, $\\varrho=2$. Since model equations \nare in nondimensional form, the time units are arbitrary. }\n\\label{fig-r3beta}\n\\end{figure}\n\n\\subsection{The asymmetric steady-state}\n\nSince the symmetric state can be unstable, there must be some \nother large-time asymmetric attractor(s) for the system, \nwhich we now aim to find.  From (\\ref{r3-nxdot}) and \n(\\ref{r3-roxdot}), at steady-state, we have \n\\begin{equation}\n2c_2 (2\\mu+\\alpha N_x) = \\frac{4\\mu\\nu N_x^2}{\\varrho_x} , \n\\qquad \\mu c_2 + \\beta N_x = \n2 (\\mu\\nu+\\beta+\\xi N_x) \\frac{N_x^2}{\\varrho_x} . \n\\lbl{r3a-eqs} \\end{equation} \nTaking the ratio of these we find a single quadratic \nequation for $N_x$\n\\begin{equation}\n0 = \\alpha \\xi N_x^2 - \\left( \\frac{\\beta\\mu\\nu}{c_2} \n- \\alpha\\beta - \\alpha\\mu\\nu - \\xi\\mu \\right) N_x + \\beta\\mu , \n\\lbl{r3a-Neq}\n\\end{equation}\nwith an identical one for $N_y$.   Hence there is the possibility \nof distinct solutions for $N_x$ and $N_y$ if both roots of \n(\\ref{r3a-Neq}) are positive; this occurs if \n\\begin{equation}\nc_2 < \\frac{\\beta\\mu\\nu}{\\alpha\\beta + \\xi\\mu \n+ \\alpha\\mu\\nu + 2\\sqrt{\\alpha\\beta\\xi\\mu} } . \n\\lbl{r3a-ineq} \\end{equation}\nGiven $N_x$ ($N_y$), we then have to solve one of \n(\\ref{r3a-eqs}) to find $\\varrho_x$ ($\\varrho_y$), {\\em via} \n\\begin{equation}\n\\varrho_x = \\frac{2 \\mu \\nu N_x^2}{c_2 (\\mu+\\alpha N_x)} ,  \n\\lbl{r3a-a1-rox} \\end{equation} \nand then satisfy the consistency condition that \n$\\varrho_x + \\varrho_y + 2 c_2 = \\varrho$.  After some algebra, \nthis condition reduces to \n\\begin{eqnarray}\n\\mbox{$\\frac{1}{2}$} \\alpha^2 \\xi c_2^2 (\\beta \\!-\\! \\alpha c_2 ) (\\varrho\\!-\\!2c_2) \n&\\!=\\!& \\beta^2\\mu^2\\nu^2 - \\beta\\mu\\nu c_2 [ \\alpha\\beta \n+ 2\\alpha\\mu\\nu + 2\\xi\\mu ] \\nonumber \\\\ && \n+ \\mu c_2^2 [ \\mu (\\alpha\\nu\\!+\\!\\xi)^2 + \\alpha\\beta (\\alpha\\nu\\!-\\!\\xi) ] . \n\\lbl{r3a-consistency} \\end{eqnarray}\nBeing a cubic, it is not straightforward to write down explicit \nsolutions of this equation, hence we once again consider the two \nasymptotic limits ($\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$).  \n\n\\subsubsection{Asymptotic limit 1: $\\beta \\ll 1$}\n\nIn this case, $c_2 = {\\cal O}(\\beta)$ hence we put $c_2=\\beta C$ \nand the consistency condition (\\ref{r3a-consistency}) yields  \n\\begin{equation} \n{\\cal O}(\\beta^3) = \\beta^2 \\left[ \\nu - (\\alpha\\nu+\\xi) C \\right]^2 , \n\\lbl{r3a-a1-C} \\end{equation} \nhence, to leading order, $C=\\nu\/(\\alpha\\nu+\\xi)$ . Unfortunately, the \nresulting value for $c_2$ leads to all the leading order terms in the \nlinear equation (\\ref{r3a-Neq}) for $N_x$ to cancel. We thus have to \nfind higher order terms in the expansion for $c_2$; due to the form of \n(\\ref{r3a-a1-C}), the next correction term is ${\\cal O}(\\beta^{3\/2})$. \nPutting $c_2=\\beta C(1+\\tilde C \\sqrt{\\beta})$, we find \n\\begin{equation} \n\\tilde C^2 = \\frac{\\alpha\\xi \\,\\left[ \\, \\alpha\\xi\\varrho + 4 \\mu (\\alpha\\nu+\\xi) \n\\, \\right] }{2\\mu^2 (\\alpha\\nu+\\xi)^3} . \n\\end{equation} \nIn order to satisfy the inequality (\\ref{r3a-ineq}), we require the \nnegative root, that is, $\\tilde C<0$. \n\nAlthough the formulae for $N_x,N_y$ are lengthy, \ntheir sum and products simplify to \n\\begin{equation}\n\\Sigma = N_x + N_y = \n\\frac{\\mu \\tilde C \\sqrt{\\beta} (\\alpha\\nu+\\xi)}{\\alpha\\xi} , \\qquad \n\\Pi = N_x N_y = \\frac{\\beta\\mu}{\\alpha\\xi} . \n\\end{equation}\nThe chirality $\\phi$ can be simplified using $\\phi^2=1-4\\Pi\/\\Sigma^2$ \nwhich implies  \n\\begin{equation}\n\\phi^2 = \\frac{\\alpha\\varrho \\xi - 4\\mu(\\alpha\\nu+\\xi)}\n{\\alpha\\varrho\\xi+4\\mu (\\alpha\\nu+\\xi)}  . \n\\end{equation}\nHence we require $\\varrho > \\varrho_c := 4\\mu(\\alpha\\nu+\\xi)\/\\alpha\\xi$ in \norder for the system to have nonsymmetric steady-states, that is, the \nsystem undergoes a symmetry-breaking bifurcation as $\\varrho$ increases \nthrough $\\varrho=\\varrho_c$.  As the mass in the system increases further, \nthe chirality $\\phi$ approaches ($\\pm$) unity, indicating a state in \nwhich one handedness of crystal completely dominates the other. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha \\sim \\xi \\gg 1$}\n\nIn this case,  the left-hand side of the consistency condition \n(\\ref{r3a-consistency}) is ${\\cal O}(\\alpha^2\\xi c_2^2)$ whilst the \nright-hand side is ${\\cal O}(1)+{\\cal O}(\\alpha c_2^2)$, which implies \nthe balance $c_2={\\cal O}(\\xi^{-3\/2})$.  Solving for $c_2$ leads to \n\\begin{equation} \nc_2 \\sim \\frac{\\mu\\nu}{\\alpha} \\sqrt{ \\frac{2\\beta}{\\varrho\\xi} } . \n\\end{equation} \nThe leading order equation for $N_x,N_y$ is then \n\\begin{equation} \n0 = \\alpha\\xi N^2 - \\alpha N \\sqrt{\\mbox{$\\frac{1}{2}$}\\beta\\varrho\\xi} + \\beta\\mu , \n\\end{equation} \nhence we find the roots \n\\begin{equation} \nN_x,N_y \\sim \\sqrt{\\frac{\\beta\\varrho}{2\\xi}} , \n\\frac{2\\mu}{\\alpha} \\sqrt{\\frac{\\beta}{2\\xi\\varrho}} , \\qquad \n\\varrho_x , \\varrho_y \\sim \\varrho , \\frac{2\\mu}{\\alpha} . \n\\end{equation} \nSince we have either $\\varrho_x \\gg N_x \\gg \\varrho_y \\gg N_y$ \nor $\\varrho_y \\gg N_y \\gg \\varrho_x \\gg N_x$, in this asymptotic limit, \nthe system is completely dominated by one species or the other. \nPutting $\\Sigma=N_x+N_y$ and $\\Pi=N_xN_y$ we have \n$\\phi^2=1-4\\Pi\/\\Sigma^2 \\sim 1 - 8 \\mu\/\\alpha\\varrho$.  \n\n\\section{Discussion}\n\\label{disc-sec}\n\\setcounter{equation}{0}\n\nWe now try to use the above theory and experimental \nresults of Viedma \\cite{viedma} to estimate the relevant \ntimescales for symmetry-breaking in a prebiotic world. \nExtrapolating the data of time against grinding rate in rpm \nfrom Figure 2 of Viedma \\cite{viedma} \nsuggests times of $2\\times10^5$ hours using a straight line \nfit to log(time) against log(rpm) or 1000--3000 hours if \nlog(time) against rpm or time against log(rpm) is fitted. \nA reduction in the speed of grinding in prebiotic circumstances \nis expected since natural processes such as water waves \nare much more likely to operate at the order of a few seconds$^{-1}$ \nor minutes$^{-1}$ rather than 600 rpm.   \n\nSimilar extrapolations on the number and mass of balls \nused to much lower amounts gives a further reduction \nof about 3, using a linear fit to log(time) against mass of balls \nfrom Figure 1 of Viedma \\cite{viedma}.   There is an \nequally good straight line fit to time against log(ball-mass) \nbut it is then difficult to know how small a mass of balls \nwould be appropriate in the prebiotic scenario. \nThere is an additional factor due to the experiments \nof Viedma being on a small volume of 10 ml, whereas a \nsensible volume for prebiotic chemistry is 1000 l, \ngiving an additional factor of $10^5$. \nCombining these three factors ($10^3$, 3, and $10^5$) with \nthe 10 days of the original experiment, we estimate that the \ntimescale for prebiotic symmetry breaking is ${\\cal O}(3\\times10^9)$ \ndays, which is equivalent to the order of about ten million years.  \n\nThis extrapolation ignores the time required to arrive \nat the initial enantiomeric excesses of 5\\% used by Viedma \n\\cite{viedma} from a small asymmetry caused by \neither a random fluctuation or by the parity-violation.   \nAlthough the observed chiral structures are the minimum energy \nconfigurations as predicted by parity violation, there is an evens \nprobability that the observed handedness could simply be the result \nof a random fluctuation which was amplified by the same mechanisms. \nIn order to perform an example calculation, we take a random \nfluctuation of the size predicted by parity violation, which is \nof the order of $10^{-17}$, as suggested by Kondepudi \\& Nelson \n\\cite{kon-pla}.  Our goal is now to find the time taken to \namplify this to an ${\\cal O}(1)$ (5\\%) enantiomeric excess. \n\nThe models derived in this paper, for example in Section \n\\ref{r3-a2-sec}, predict that the chiral excess grows \nexponentially in time.  Assuming, from (\\ref{r3a2urate}), \nthat $\\phi(t_0)=10^{-17}$ and $\\phi(t_1)= 0.1$, then \nthe timescale for the growth of this small perturbation is \n\\[ t_1 - t_0 = \\frac{1}{4\\mu\\nu} \\sqrt{\\frac{\\xi\\varrho}{\\beta}} \n\\log \\frac{10^{-1}}{10^{-17}} . \\] \nSince the growth of enantiomeric excess is exponential, \nit only takes 16 times as long for the perturbation to grow \nfrom $10^{-17}$ to $10^{-1}$ as from $10^{-1}$ to 1. \nHence we only need to increase our estimate of the timescale \nby one power of ten, to 100 million years. \n\nThis estimate should be taken as a very rough estimate, since it \nrelies on extrapolating results by many orders of magnitude. \nAlso, given the vast differences in temperature from the \nputative subzero prebiotic world to a tentative hot hydrothermal \nvent, there could easily be changes in timescale by a factor of \nseveral orders of magnitude. \n\n\\section{Conclusions}\n\\label{conc-sec}\n\\setcounter{equation}{0}\n\nAfter summarising the existing models of chiral symmetry-breaking \nprocesses we have systematically derived a model in which through \naggregation and fragmentation chiral clusters compete for achiral \nmaterial.  The model is closed, in that there is no input of mass \ninto the system, although the form of the aggregation and \nfragmentation rate coefficients mean that there is an input of energy, \nkeeping the system away from equilibrium. Furthermore, there is no \ndirect interaction of clusters of opposite handedness; rather \njust through a simple competition for achiral substrate, the system \ncan spontaneously undergo chiral symmetry-breaking.  This model \nhelps explain the experimental results of Viedma \\cite{viedma} \nand Noorduin {\\em et al.}\\ \\cite{wim}. \n\nThe microscopic model originally derived has been simplified \nsuccessively to a minimalistic model, which, numerical results show, \nexhibits symmetry-breaking. Even after this reduction, the model is \nextremely complex to analyse due to the large number of cluster \nsizes retained in the model. Hence we construct two truncated \nmodels, one truncated at tetramers, which shows no \nsymmetry-breaking and one at hexamers which shows \nsymmetry-breaking under certain conditions on the parameter values.  \nAlternative reductions are proposed: instead of retaining the \nconcentrations of just a few cluster sizes, we retain information \nabout the shape of the distribution, such as the number of clusters \nand the total mass of material in clusters of each handedness.  \nThese reduced models are as simple to analyse as truncated models \nyet, since they more accurately account for the shape of the \nsize-distribution than a truncated model, are expected to give models \nwhich more easily fit to experimental data.  Of course, other \nansatzes for the shape of the size distributions could be made, \nand will lead to modified conditions for symmetry-breaking; \nhowever, we believe that the qualitative results outlined here will \nnot be contradicted by analyses of other macroscopic reductions. \n\nOne noteworthy feature of the results shown herein is that the \nsymmetry-breaking is inherently a product of the two handednesses \ncompeting for achiral material.  The symmetry-breaking does not rely \non critical cluster sizes, which are a common feature of theories of \ncrystallisation, or on complicated arguments about surface area to \nvolume ratios to make the symmetric state unstable.  We do not \ndeny that these aspects of crystallisation are genuine, these features \nare present in the phenomena of crystal growth, but they are not \nthe fundamental cause of chiral symmetry-breaking. \n\nMore accurate fitting of the models to experimental data could be \nacheived if one were to fit the generalised Becker-D\\\"{o}ring model \n(\\ref{gbd1})--(\\ref{gbd3}) with realistic rate coefficients.  Questions \nto address include elucidating how the number and size distribution \nat the start of the grinding influences the end state.  For example, if \none were to start with a few large right-handed crystals and many \nsmall left-handed crystals, would the system convert to entirely \nleft- or entirely right-handed crystals ?  Answers to these more \ncomplex questions may rely on higher moments of the size distributions, \nsurface area to volume ratios and critical cluster nuclei sizes. \n\n\\subsection*{Acknowledgments}\n\nI would particularly like to thank Professors Axel Brandenburg and \nRaphael Plasson for inviting me to an extended programme of \nstudy on homochirality at Nordita (Stockholm, Sweden) in February 2008.  \nThere I met and benefited greatly from discussions with Professors \nMeir Lahav, Mike McBride, Wim Noorduin, as well as many others.  \nThe models described here are a product of the stimulating \ndiscussions held there.  I am also grateful for funding under \nEPSRC springboard fellowship EP\/E032362\/1. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\\emph{Seriation} is an important ordering problem\nwhose aim is to find the best enumeration order of a set of units, according\nto a given correlation function. The desired order can be characteristic of the\ndata, a chronological order, a gradient or any sequential structure of the\ndata. \n\nThe concept of seriation has been formulated in many different ways and appears\nin various fields, such as archaeology, anthropology, psychology, and \nbiology~\\cite{brusco06,genome,matharcheo,mirkin1984}.\nIn this paper we use the archaeological setting as a metaphor for the seriation\nproblem.\nAn important aim of archaeological investigation is to date excavation\nsites on the basis of found objects and determine their relative chronology,\ni.e., a dating which indicates if a given site is chronologically preceding or\nsubsequent to another.\nIn general, relative chronologies are devoid of a direction, in the sense that\nthe units are placed in a sequence which can be read in both directions.\nRelative dating methods can be used where absolute dating methods, such as\ncarbon dating, cannot be applied.\n\nThe available data are usually represented by a \\emph{data matrix}, in which\nthe rows are the archaeological units (e.g., the sites) and the columns\nrepresent the types (the archaeological finds).\nEach unit is characterized by the presence of certain artefacts, which are in\nturn classified in types. In~\\cite{ps05}, the authors refer to the data matrix as either \\emph{incidence matrix} or \\emph{abundance\nmatrix}, depending on the archaeological data representation. In the first case, the data are reported by using a binary\nrepresentation, i.e., an element in the position $(i,j)$ is equal to $1$ if\ntype $j$ is present in the unit $i$, and $0$ otherwise.\nIn the second second case, the data matrix reports the number of objects\nbelonging to a certain type in a given unit, or its percentage. In this paper,\nwe will follow the usual terminology used in \\emph{complex networks theory} and\nwe will refer to a binary representation as an \\emph{adjacency matrix}, an\nexample of which is given in Table~\\ref{tab:adjacency}. More details can be\nfound in Section~\\ref{sec:mathback}. If\nthe data matrix represents types of found objects as columns and the locations\n(graves, pits, etc.) in which they are found as rows, we can find a\nchronological order for the locations by assuming that the types were produced,\nor were ``fashionable'', only for a limited period of time. In the light of\nthis assumption, the purpose of determining a relative chronology results in\nobtaining an ordering of the rows and columns of the data matrix that places\nthe nonzero entries close to the diagonal of the data matrix.\n\n\\begin{table}\n\\caption{Adjacency matrix for archaeological data originated from female\nburials at the Bornholm site, Germany; see~\\cite{ps05} and the references\ntherein. The rows report the names of the tombs, the columns the identification\ncodes of the found \\emph{fibulae}.}\n\\label{tab:adjacency}\n\\scriptsize\n\\begin{center}\n\\begin{tabular}{rcccccccccccc}\n\\hline\\noalign{\\smallskip}\n&G3&F27&S1&F26&N2&F24&P6&F25&P5&P4&N1&F23\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textsf{Mollebakken 2} & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\\textsf{Kobbea 11}&0&1&1&0&1&1&0&0&0&0&0&0\\\\ \n\\textsf{Mollebakken 1}&1&1&0&1&1&0&1&1&0&0&0&0\\\\ \n\\textsf{Levka 2}&0&1&1&0&1&0&0&1&1&0&0&0\\\\ \n\\textsf{Grodbygard 324}&0&0&0&0&1&1&0&0&0&1&0&0\\\\ \n\\textsf{Melsted 8}&0&0&1&1&0&0&1&1&0&1&0&0\\\\ \n\\textsf{Bokul 7}&0&0&0&0&0&0&1&1&0&0&1&0\\\\ \n\\textsf{Heslergaard 11}&0&0&0&0&0&0&0&1&0&1&0&0\\\\ \n\\textsf{Bokul 12}&0&0&0&0&0&0&0&1&1&0&0&1\\\\ \n\\textsf{Slamrebjerg 142}&0&0&0&0&0&0&0&0&0&1&0&1\\\\ \n\\textsf{Nexo 6} &0&0&0&0&0&0&0&0&0&1&1&1\\\\ \n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nA closely related problem is the \\emph{consecutive ones problem}\n(C1P)~\\cite{fulkerson65,or09}, whose aim is to find all the permutations of the\nrows of a binary matrix that place the $1$'s consecutively in each column. If\nsuch permutations exist, then the matrix is said to have the \\emph{consecutive\nones property} for columns. The equivalent property for rows can be similarly\ndefined. The problem of verifying if a matrix possesses this property has\napplications in different fields, such as computational biology and recognition\nof interval graphs~\\cite{booth1976testing,cor98}.\nThe connection between C1P and seriation has been investigated by Kendall \nin~\\cite{kendall69}.\n\nThe first systematic formalization of the seriation problem was made by Petrie\nin 1899~\\cite{petrie}, even if the term seriation was used before in\narchaeology. The subject was later resumed by Breinerd and\nRobinson~\\cite{brainerd1951place,robinson1951method}, who also proposed a\npractical method for its solution, and by\nKendall~\\cite{kendall63,kendall69,kendall70}.\nNice reviews on seriation are given in~\\cite{liiv2010},~\\cite{ps05}, \nand~\\cite{bl02}, where its application in stratigraphy is discussed, \nwhile~\\cite{bb15,matharcheo} describe other applications of mathematics in\narchaeology.\n\nGiven the variety of applications, some software packages have been developed in\nthe past to manipulate seriation data. Some of these packages have not undergo a\nregular maintenance, and does not seem to be easily usable on modern computers,\nlike the Bonn Archaeological Software Package (BASP)\n(\\url{http:\/\/www.uni-koeln.de\/~al001\/}).\nA software specifically designed for the seriation problem in bioinformatics\nhas been developed by Caraux and Pinloche~\\cite{permutsoft}, and is available\nfor free download.\n\nOther implementations of the spectral algorithm from~\\cite{atkins1998spectral}\nhave been discussed in~\\cite{fogel2014}, \\cite{hahsler2008}, which describes an\nR package available at \\url{http:\/\/cran.r-project.org\/web\/packages\/seriation\/},\nand~\\cite{seminaroti2016}.\nThe paper~\\cite{fogel2015} proposes an interesting method, based on quadratic\nprogramming, aimed at treating ``noisy cases''.\n\nIn this paper we present a Matlab implementation of a spectral method for the\nsolution of the seriation problem which appeared in~\\cite{atkins1998spectral},\nbased on the use of the Fiedler vector of the Laplacian associated to the\nproblem, and which describes the results in terms of a particular data\nstructure called a PQ-tree.\nWe further develop some numerical aspects of the algorithm, concerning the\ndetection of equal components in the Fiedler vector and the computation of the\neigensystem of the Laplacian associated to a large scale problem.\nWe also provide a parallel version of the method.\nThe package, named the \\texttt{PQser}  toolbox, also defines a data structure to store\na PQ-tree and provides the Matlab functions to manipulate and visualize it.\nFinally, we discuss the implications of the presence of a multiple Fiedler\nvalue, an issue which has been disregarded up to now, and we illustrate some\nnumerical experiments.\n\nThe plan of the paper is the following. Section \\ref{sec:mathback} reviews the\nnecessary mathematical background and sets up the terminology to be used in the\nrest of the paper. Section~\\ref{sec:pqtrees} describes the data structures\nused to store the solutions of the seriation problem.\nThe spectral algorithm is discussed in Section~\\ref{sec:seriation} and the\nspecial case of a multiple Fiedler values is analyzed in\nSection~\\ref{sec:mult}.\nSection~\\ref{sec:numexp} reports some numerical results and\nSection~\\ref{sec:last} contains concluding remarks. \n\n\n\\section{Mathematical background}\\label{sec:mathback}\n\nWith the aim of making this paper self-contained, we review some mathematical\nconcepts that will be used in the following.\nWe denote matrices by upper case roman letters and their elements by\nlower case double indexed letters.\n\nLet $G$ be a simple graph formed by $n$ nodes.\nEach entry $f_{ij}$ of the adjacency matrix $F\\in{\\mathbb{R}}^{n\\times n}$ associated to\n$G$ is taken to be the weight of the edge connecting node $i$ to node $j$. \nIf the two nodes are not connected, then $f_{ij}=0$.\nA graph is unweighted if the weights are either 0 or 1.\nThe adjacency matrix is symmetric if and only if the graph is undirected.\n\nThe (unnormalized) \\emph{graph Laplacian} of a symmetric, matrix \n$F\\in{\\mathbb{R}}^{n\\times n}$ is the symmetric, positive semidefinite matrix \n$$\nL=D-F,\n$$\nwhere $D=\\diag(d_1,\\ldots,d_n)$ is the \\emph{degree matrix}, whose $i$th\ndiagonal element equals the sum of the weights of the edges starting from node\n$i$ in the undirected network defined by $F$, that is, \n$d_i=\\sum_{j=1}^n f_{ij}$.\nIn the case of an unweighted graph, $d_i$ is the number of nodes connected to\nnode $i$.\n\nSetting $\\bm{e}=[1,\\dots,1]^T\\in{\\mathbb{R}}^n$, it is immediate to observe that\n$$\nL\\bm{e} = (D-F)\\bm{e}= \\bm{0},\n$$\nwhere $\\bm{0}\\in{\\mathbb{R}}^n$ is the zero vector.\nHence, $0$ is an eigenvalue of the graph Laplacian with eigenvector $\\bm{e}$.\n\nThe Gershgorin circle theorem implies all the eigenvalues are non-negative, so\nwe order them as $\\lambda_1=0\\leq\\lambda_2\\leq\\dots\\leq\\lambda_n$, with\ncorresponding eigenvectors $\\bm{v}_1=\\bm{e},\\bm{v}_2,\\dots, \\bm{v}_n$.\nThe smallest eigenvalue of $L$ with associated eigenvector orthogonal to\n$\\bm{e}$ is called the \\emph{Fiedler value}, or the \\emph{algebraic\nconnectivity}, of the graph described by $F$.\nThe corresponding eigenvector is the \\emph{Fiedler\nvector}~\\cite{fiedler1973algebraic,fiedler1975property,fiedler1989laplacian}.\n\nAlternatively, the Fiedler value may be defined by\n$$\n\\min_{\\bm{x}^{T}\\bm{e}=0,\\ \\bm{x}^{T}\\bm{x}=1}\\bm{x}^{T}L\\bm{x}.\n$$\nThen, a Fiedler vector is any vector $\\bm{x}$ that achieves the minimum.\n\nFrom the Kirchhoff matrix-tree theorem it follows that the Fiedler value is\nzero if and only if the graph is not connected~\\cite{de2007old}; in particular,\nthe number of times 0 appears as an eigenvalue of the Laplacian is the number\nof connected components of the graph.\nSo, if the considered adjacency matrix is irreducible, that is, if the\ngraph is connected, the Fiedler vector corresponds to the first non-zero\neigenvalue of the Laplacian matrix.\n\nA \\emph{bipartite graph} $G$ is a graph whose vertices can be divided into two\ndisjoint sets $U$ and $V$ containing $n$ and $m$ nodes, respectively, such\nthat every edge connects a node in $U$ to one in $V$.\n\nIn our archaeological metaphor, the nodes sets $U$ (units) and $V$\n(types) represent the excavation sites and the found artifacts, respectively.\nThen, as already outlined in the Introduction, \nthe associated adjacency matrix $A$, of size $n\\times m$, is obtained by\nsetting $a_{ij}=1$ if the unit $i$ contains objects of type $j$, and 0\notherwise. \nIf the element $a_{ij}$ takes value different from 1, we consider it as a\nweight indicating the number of objects of type $j$ contained in unit $i$, or\ntheir percentage. In this case, we denote $A$ as the \\emph{abundance matrix}.\n\nThe first mathematical definition of seriation was based on the construction of\na symmetric matrix $S$ known as \\emph{similarity\nmatrix}~\\cite{brainerd1951place,robinson1951method}, where $s_{ij}$ describes,\nin some way, the likeness of the nodes $i,j\\in U$. One possible definition is\nthrough the product $S=AA^T$, being $A$ the adjacency matrix of the problem.\nIn this case, $s_{ij}$ equals the number of types shared between unit $i$ and\nunit $j$. \nThe largest value on each row is the diagonal element, which reports the\nnumber of types associated to each unit. By permuting the rows and columns of\n$S$ in order to cluster the largest values close to the main diagonal, one \nobtains a permutation of the corresponding rows of $A$ that places closer the\nunits similar in types.\nIt is worth noting that this operation of permuting rows and columns of $S$ is\nnot uniquely defined.\n\nThe \\emph{Robinson method}~\\cite{robinson1951method} is a statistical technique\nbased on a different similarity matrix. It is based on the concept that each\ntype of artifact used in a certain period eventually decreases in popularity\nuntil it becomes forgotten. This method is probably the first documented\nexample of a practical procedure based on the use of the similarity matrix, so\nits description is interesting in a historical perspective.\n\nThe method, starting from an abundance matrix $A \\in {\\mathbb{R}}^{n \\times m}$ whose\nentries are in percentage form (the sum of each row is 100), computes the\nsimilarity matrix $S$ by a particular rule, leading to a symmetric matrix of\norder $n$ with entries between $0$ (rows with no types in common) and $200$,\nwhich corresponds to units containing exactly the same types.\nThen, the method searches for a permutation matrix $P$ such that $PSP^T$ has\nits largest entries as close as possible to the main diagonal. The same\npermutation is applied to the rows of the data matrix $A$ to obtain a\nchronological order for the archaeological units. Since, as already remarked,\nthe sequence can be read in both directions, external information must be used\nto choose an orientation.\n\nThe procedure of finding a permutation matrix $P$ is not uniquely specified.\nOne way to deal with it is given by the so called \\emph{Robinson's form}, which\nplaces larger values close to the main diagonal, and lets off-diagonal entries\nbe nonincreasing moving away from the main diagonal.\nMore in detail, a symmetric matrix $S$ is in Robinson's form, or is an\nR-matrix, if and only if\n\\begin{eqnarray}\ns_{ij}\\leqslant s_{ik}, \\quad \\text{if } j\\leqslant k\\leqslant i, \n\\label{rmatrix1} \\\\\ns_{ij}\\geqslant s_{ik}, \\quad \\text{if } i\\leqslant j\\leqslant k.\n\\label{rmatrix2} \n\\end{eqnarray}\nA symmetric matrix is pre-$R$ if and only if there exists a simultaneous\npermutation of its rows and columns which transforms it in Robinson's form, so\nit corresponds to a well-posed ordering problem.\nFor other interesting references on R-matrices and their detection, \nsee~\\cite{chepoi1997,laurent2017lex,laurent2017,prea2014,seston2008}.\n\n\n\n\\section{PQ-trees}\\label{sec:pqtrees}\n\nA \\emph{PQ-tree} is a data structure introduced by Booth and\nLueker~\\cite{booth1976testing} to encode a family of permutations of a\nset of elements, and solve problems connected to finding admissible permutations\naccording to specific rules.\n\nA PQ-tree $T$ over a set $U = \\{u_1,u_2,\\dots,u_n\\}$ is a rooted tree whose\nleaves are elements of $U$ and whose internal (non-leaf) nodes are\ndistinguished as either P-nodes or Q-nodes. \nThe only difference between them is the way in which their children are\ntreated: for Q-nodes only one order and its reverse are allowed, whereas in the\ncase of P-node all possible permutations of the children leaves are permitted.\nThe root of the tree can be either a P or a Q-node.\n\nWe will represent graphically a P-node by a circle, and a Q-node by a\nrectangle. \nThe leaves of $T$ will be displayed as triangles, and labeled by the elements\nof $U$. The frontier of $T$ is one possible permutation of the elements of $U$,\nobtained by reading the labels of the leaves from left to right.\n\nWe recall two definitions from~\\cite{booth1976testing}.\n\n\\begin{definition}\\label{def:proper}\nA PQ-tree is \\textit{proper} when the following conditions hold:\n\\begin{itemize}\n\\item[i)] every $u_{i}\\in U$ appears precisely once as a leaf;\n\\item[ii)] every P-node has at least two children;\n\\item[iii)] every Q-node has at least three children. \n\\end{itemize}\n\\end{definition}\n\nAs we observed above, the only difference between a P-node and a Q-node is the\ntreatment of their children, and in the case of exactly two children there is\nno real distinction between a P-node and a Q-node.\nThis justifies the second and third conditions of Definition~\\ref{def:proper}.\n\n\\begin{definition}\\label{def:equiv}\nTwo PQ-trees are said to be \\textit{equivalent} if one can be transformed into\nthe other by applying a sequence of the following two transformations:\n\\begin{itemize}\n\\item[i)] arbitrarily permute the children of a P-node;\n\\item[ii)] reverse the children of a Q-node.\n\\end{itemize} \n\\end{definition}\n\nA PQ-tree represents permutations of the elements of a set through\nadmissible reorderings of its leaves.\nEach transformation in Definition~\\ref{def:equiv} specifies an admissible\nreordering of the nodes within a PQ-tree.\nFor example, a tree with a single P-node represents the equivalence\nclass of all permutations of the elements of $U$, while a tree with a single\nQ-node represents both the left-to-right and right-to-left orderings of the\nleaves.\nA tree with a mixed P-node and Q-node structure represents the equivalence\nclass of a constrained permutation, where the exact structure of the tree\ndetermines the constraints.\nFigure~\\ref{fig:pqtree} displays a PQ-tree and the admissible permutations\nit represents.\n\n\\begin{figure}\n\\begin{minipage}{.52\\textwidth}\n\\includegraphics[width=\\textwidth]{pqtree}\n\\end{minipage}\n\\begin{minipage}{.45\\textwidth}\n\\tiny\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\hline\\noalign{\\smallskip}\n1 & 2 & 3 & 4 & 5 & 6 \\\\\n1 & 2 & 3 & 6 & 5 & 4 \\\\\n1 & 3 & 2 & 4 & 5 & 6 \\\\\n1 & 3 & 2 & 6 & 5 & 4 \\\\\n2 & 1 & 3 & 4 & 5 & 6 \\\\\n2 & 1 & 3 & 6 & 5 & 4 \\\\\n2 & 3 & 1 & 4 & 5 & 6 \\\\\n2 & 3 & 1 & 6 & 5 & 4 \\\\\n3 & 1 & 2 & 4 & 5 & 6 \\\\\n3 & 1 & 2 & 6 & 5 & 4 \\\\\n3 & 2 & 1 & 4 & 5 & 6 \\\\\n3 & 2 & 1 & 6 & 5 & 4 \\\\\n4 & 5 & 6 & 1 & 2 & 3 \\\\\n4 & 5 & 6 & 1 & 3 & 2 \\\\\n4 & 5 & 6 & 2 & 1 & 3 \\\\\n4 & 5 & 6 & 2 & 3 & 1 \\\\\n4 & 5 & 6 & 3 & 1 & 2 \\\\\n4 & 5 & 6 & 3 & 2 & 1 \\\\\n6 & 5 & 4 & 1 & 2 & 3 \\\\\n6 & 5 & 4 & 1 & 3 & 2 \\\\\n6 & 5 & 4 & 2 & 1 & 3 \\\\\n6 & 5 & 4 & 2 & 3 & 1 \\\\\n6 & 5 & 4 & 3 & 1 & 2 \\\\\n6 & 5 & 4 & 3 & 2 & 1 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\caption{On the left, a PQ-tree over the set $U=\\{1,\\ldots,6\\}$; on the right,\nthe 24 admissible permutations encoded in the tree.}\n\\label{fig:pqtree}\n\\end{figure}\n\nThe PQ-tree data structure has been exploited in a variety of applications,\nfrom archaeology and chronology reconstruction~\\cite{atkins1998spectral} to\nmolecular biology with DNA mapping and sequence\nassembly~\\cite{greenberg1995physical}.\nThe first problem to which it was applied is the consecutive ones property\n(C1P) for matrices~\\cite{booth1976testing}, mentioned in\nSection~\\ref{sec:intro}.\n\nGiven a pre-R matrix, the spectral algorithm from~\\cite{atkins1998spectral},\nthat will be discussed in Section~\\ref{sec:seriation}, constructs a PQ-tree\ndescribing the set of all the permutations of rows and columns that lead to an\nR-matrix.\n\n\n\\subsection{Implementation of PQ-trees}\n\nThe \\texttt{PQser}  toolbox for Matlab is available as a compressed archive that can be downloaded from the authors webpages (see, e.g., \n\\url{http:\/\/bugs.unica.it\/~gppe\/soft\/}).\nBy uncompressing it, the directory \\texttt{PQser}  will be created. \nIt must be added to Matlab search path, either by the command \\ttt{addpath} or\nusing the graphical interface menus.\nThe sub-directory \\texttt{demo} contains a tutorial for the toolbox and the\nscripts which were used to construct the examples reported in the paper.\nThe installation procedure and the toolbox content are described in detail in\nthe README.txt file, which can be found in the main directory.\n\nIn the \\texttt{PQser}  toolbox, a PQ-tree $T$ is a \\ttt{struct} variable (i.e., a\nrecord) composed by two fields.\nThe first field, \\ttt{T.type}, specifies the type of the node, i.e., P, Q,\nor a leaf, in the case of a trivial tree.\nThe second field, \\ttt{T.value}, is a vector which provides a list of PQ-trees,\nrecursively defined.\nIn the case of a leaf, this field contains the index of the unit it represents.\n\nFor example, the graph in Figure~\\ref{fig:pqtree} was obtained by the following\npiece of code\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nv(1) = pnode([1 2 3]);\nv(2) = qnode([4 5 6]);\nT = pnode(v);\t\npqtreeplot(T)\t\n\\end{verbatim}\n\\end{quote}\nthe resulting data structure for the PQ-tree is\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nT = \n  struct with fields:\n     type: 'P'\n    value: [1x2 struct]\n\\end{verbatim}\n\\end{quote}\nand the permutations encoded in $T$ are computed by\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nperms_matrix = pqtreeperms(T)\n\\end{verbatim}\n\\end{quote}\nThese instructions are contained in the script \\texttt{graf1.m}, in the\n\\texttt{demo} sub-directory.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{pnode}         & create a P-node \\\\\n\\ttt{qnode}         & create a Q-node \\\\\n\\ttt{lnode}         & create a leaf \\\\\n\\ttt{mnode}         & create an M-node \\\\\n\\ttt{pqtreeplot}    & plot a PQ-tree \\\\\n\\ttt{pqtreeNperm}   & number of admissible permutations in a PQ-tree \\\\\n\\ttt{pqtreeperms}   & extract all admissible permutations from a PQ-tree \\\\\n\\ttt{pqtree1perm}   & extract one admissible permutation from a PQ-tree \\\\\n\\ttt{pqtreegetnode} & extract a subtree from a PQ-tree \\\\\n\\ttt{pqtreenodes}   & converts a PQ-tree to Matlab \\ttt{treeplot} format \\\\\n\\hline\n\\end{tabular}\n\\caption{Functions in the \\texttt{PQser}  toolbox devoted to the manipulation of\nPQ-trees.}\n\\label{tab:pqfuncs}\n\\end{table}\n\nThe functions intended for creating and manipulating a PQ-tree are listed in\nTable~\\ref{tab:pqfuncs}.\nThe function \\ttt{mnode} creates an additional type of node, an M-node, which\nis intended to deal with multiple Fiedler values; we will comment on it in\nSection~\\ref{sec:mult};\n\\ttt{pqtreegetnode} and \\ttt{pqtreenodes} are utility functions for\n\\ttt{pqtreeplot}, they are not intended to be called directly by the user.\nAll the functions are documented via the usual Matlab \\ttt{help} command, e.g.,\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nhelp pnode\nhelp pqtreeplot\n\\end{verbatim}\n\\end{quote}\n\nAs an example, we report in Algorithm~\\ref{alg:pqtreeNperm} the structure of\n\\ttt{pqtreeNperm}, a function which returns the number $N$ of all the\npermutations contained in the tree whose root $T$ is given in input.\nIn the particular case of a leaf, only one permutation is possible \n(line~\\ref{l1}--\\ref{l2}).\nOtherwise, we consider the vector $\\bm{c}$ of size $k$, containing the children\nnodes of the root of T (line~\\ref{l5}).\nThe algorithm calls itself recursively on each component of $\\bm{c}$\n(line~\\ref{l8}).\nIn the case of a Q-node the number of permutations is doubled, because only one\nordering and its reverse are admissible, whereas for a P-node the number is\nmultiplied by the factorial of $k$, since in this case all the possible\npermutations of the children are allowed.\nThe same procedure is applied to an M-node; see Section~\\ref{sec:mult} for\ndetails.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $N = \\pqtreeNperm(T)$\n\\IF $T$ is a leaf\n\\label{l1}\n\\STATE $N =1$\n\\label{l2}\n\\ELSE \n\t\\STATE $\\bm{c}=T\\ttt{.value}$, $k=\\length(\\bm{c})$\n\\label{l5}\n\t\\STATE $p=1$\n\t\\FOR $i = 1,\\dots,k$\n\t\t\\STATE $p = p*\\pqtreeNperm(c_i)$\n\\label{l8}\n\t\\ENDFOR\n\t\\IF $T$ is a Q-node\n\t\t\\STATE $N = 2*p$\n\t\\ELSE \n\t\t\\STATE $N = \\factorial(k)*p$\n\t\\ENDIF\n\\ENDIF\t \n\\end{algo}\n\\caption{Compute the number of admissible permutations in a PQ-tree.}\n\\label{alg:pqtreeNperm}\n\\end{algorithm}\n\nThe toolbox includes an interactive graphical tool for exploring a PQ-tree $T$.\nAfter displaying $T$ by \\ttt{pqtreeplot}, it is possible to extract a subtree\nby clicking on one node with the left mouse button. In this case, the\ncorresponding subtree is extracted, it is plotted in a new figure, and it is\nsaved to the variable \\ttt{PQsubtree} in the workspace.\nThis feature is particularly useful when analyzing a large PQ-tree.\nThe function \\ttt{pqtreeplot} allows to set some attributes of the plot; see\nthe help page.\n\n\n\n\\section{A spectral algorithm for the seriation problem}\\label{sec:seriation}\n\nIn this section we briefly review the spectral algorithm for the seriation\nproblem introduced in~\\cite{atkins1998spectral}, and describe our\nimplementation.\n\nGiven the set of units $U=\\{u_1,u_2,\\dots,u_n\\}$, we will write \n$i\\preccurlyeq j$ if $u_i$ precedes $u_j$ in a chosen ordering.\nIn~\\cite{atkins1998spectral}, the authors consider a symmetric bivariate\n\\emph{correlation function} $f$ reflecting the desire for units $i$ and $j$ to\nbe close to each other in the sought sequence.\nThe point is to find all index permutation vectors\n${\\boldsymbol{\\pi}}=(\\pi_1,\\ldots,\\pi_n)^T$ such that \n\\begin{equation}\\label{fperm}\n\\pi_i\\preccurlyeq\\pi_j\\preccurlyeq\\pi_k \\quad \\iff \\quad \nf(\\pi_i,\\pi_j)\\geq f(\\pi_i,\\pi_k) \\quad \\text{and} \\quad\nf(\\pi_j,\\pi_k)\\geq f(\\pi_i,\\pi_k).\n\\end{equation}\nIt is natural to associate to such correlation function a real symmetric matrix\n$F$, whose entries are defined by $f_{ij}=f(i,j)$.\nThis matrix plays exactly the role of the similarity matrix $S$ discussed in\nSection~\\ref{sec:mathback}, as the following theorem states.\n\n\\begin{theorem}\\label{theo:fs}\nA matrix $F$ is an R-matrix if and only if \\eqref{fperm} holds.\n\\end{theorem}\n\n\\begin{proof}\nLet us assume that the permutation ${\\boldsymbol{\\pi}}$ which realizes \\eqref{fperm} has\nalready been applied to the units. \nThen, since a permutation of the units corresponds to a simultaneous\npermutation of the rows and columns of the matrix $F$, we obtain\n$$\ni\\leq j\\leq k \\quad \\iff \\quad \nf_{ij}\\geq f_{ik} \\quad \\text{and} \\quad\nf_{jk}\\geq f_{ik}.\n$$\nThe first inequality $f_{ij}\\geq f_{ik}$ is exactly \\eqref{rmatrix2}.\nKeeping into account the symmetry of $F$ and cyclically permuting the indexes,\nfrom the second inequality we get\n$$\nj\\leq k\\leq i \\quad \\iff \\quad \nf_{ij}\\leq f_{ik},\n$$\nwhich corresponds to \\eqref{rmatrix1}.\n\\end{proof}\n\nIf a seriation data set is described by an adjacency (or abundance) matrix $A$,\nwe will set $F=AA^T$.\nIf $F$ is pre-$R$ (see Section~\\ref{sec:mathback}), there exists a rows\/columns\npermutation that takes it in $R$-form. Unfortunately, this property cannot be\nstated in advance, in general. This property can be ascertained, e.g., after\napplying the algorithm discussed in this section; see below.\n\nThe authors approach in~\\cite{atkins1998spectral}, see\nalso~\\cite{estrada2010network}, is to consider the minimization of the\nfollowing penalty function \n$$\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i-x_j)^2, \n\\quad \\bm{x}\\in {\\mathbb{R}}^n,\n$$ \nwhose value is small for a vector $\\bm{x}$ such that each pair $(i,j)$ of\nhighly correlated units is associated to components $x_i$ and $x_j$ with close\nvalues.\nOnce the minimizing vector $\\bm{x}_{\\min}$ is computed, it is sorted in\neither nonincreasing or nondecreasing value order, yielding\n$\\bm{x}_{\\boldsymbol{\\pi}}=(x_{\\pi_1},\\ldots,x_{\\pi_n})^T$.\nThe permutation of the units ${\\boldsymbol{\\pi}}$ realizes \\eqref{fperm}.\n\nNote that $h$ does not have a unique minimizer, since its value does not change\nif a constant is added to each of the components $x_i$ of the vector\n$\\bm{x}$.\nIn order to ensure uniqueness and to rule out the trivial solution, it is\nnecessary to impose two suitable constraints on the components of the vector\n$\\mathbf{x}$.\nThe resulting minimization problem is:\n$$\n\\begin{aligned}\n&\\text{minimize} & &\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i-x_j)^2 \\\\\n&\\text{subject to} & & \\sum_i x_i = 0 \\quad \\text{and} \\quad \\sum_i x_i^2 = 1.\n\\end{aligned}\n$$\n\nThe solution to this approximated problem may be obtained from the Fiedler\nvector of the Laplacian $L$ of the correlation matrix $F$. \nLetting $D = \\diag(d_i)$ be the degree matrix, $d_i=\\sum_{j=1}^n f_{ij}$, it is\nimmediate to observe that\n$$\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i^2+x_j^2-2x_ix_j)\n= \\bm{x}^T D \\bm{x} - \\bm{x}^T F \\bm{x}.\n$$ \nThis shows that the previous minimization problem can be rewritten as \n\\begin{eqnarray*}\n\\min_{\\|\\mathbf{x}\\|=1,\\ \\bm{x}^T\\bm{e} = 0}\n\\mathbf{x}^TL\\mathbf{x}\n\\end{eqnarray*}\nwhere $L=D-F$.\nThe constraints require $\\mathbf{x}$ to be a unit vector orthogonal to\n$\\mathbf{e}$.\nBeing $L$ symmetric, all the eigenvectors except $\\bm{e}$ satisfy the\nconstraints.\nConsequently, a Fiedler vector is a solution to the constrained minimization\nproblem.\n\nIn fact, Theorem~3.2 from~\\cite{atkins1998spectral} proves that an R-matrix has\na monotone Fiedler vector, while Theorem~3.3, under suitable assumptions,\nimplies that a reordering of the Fiedler vector takes a pre-R matrix to R-form.\nThis confirms that the problem is well posed only when $F$ is pre-R.\nNevertheless, real data sets may be inconsistent, in the sense that do not\nnecessarily lead to pre-R similarity matrices.\nIn such cases, it may be useful to construct an approximate solution to the\nseriation problem, and sorting the entries of the Fiedler vector generates an\nordering that tries to keep highly correlated elements close to each other.\nThis is relevant because techniques based on Fiedler vectors are being used for\nthe solution of different sequencing\nproblems~\\cite{barnard1995spectral,greenberg1995physical,higham2007,juvan1992optimal}. \nIn particular, they are employed in complex network analysis, e.g., for\ncommunity detection and partitioning of graphs~\\cite{estrada2012,estrada2015}.\n\nThe algorithm proposed in~\\cite{atkins1998spectral} is based upon the above\nidea, and uses a PQ-tree to store the permutations of the units that produce a\nsolution to the seriation problem; its implementation is described in\nAlgorithm~\\ref{alg:spectrsort}.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $T = \\spectrsort(F,U)$\n\\STATE $n=$ row size of $F$\n\\STATE $\\alpha = \\min_{i,j} f_{i,j}$, \\If $\\alpha\\neq 0$, \n\t$\\bm{e}=(1,\\ldots,1)^T$, $F=F-\\alpha \\bm{e}\\bm{e}^{T}$, \\End\n\\label{line2}\n\\STATE call $\\getconcomp$ to construct the connected components \n$\\{F_1,\\dots,F_k\\}$ of $F$\n\\label{line3}\n\\STATE \\phantom{XXX} and the corresponding index sets $U=\\{U_1,\\dots,U_k\\}$ \n\\label{line4}\n\\IF {$k>1$}\n\t\\FOR {$j=1,\\dots,k$}\n\\label{line6}\n\t\t\\STATE $v(j) = \\spectrsort(F_j,U_j)$\n\t\\ENDFOR\n\t\\STATE $T = \\pnode(v)$\n\\label{line9}\n\\ELSE \n\t\\IF {$n=1$}\n\\label{line11}\n\t\t\\STATE $T=\\lnode(U)$\n\t\\ELSEIF {$n=2$}\n\t\t\\STATE $T=\\pnode(U)$\n\t\\ELSE\n\\label{line15}\n\t\t\\STATE $L=$ Laplacian matrix of $F$\n\\label{line16}\n\t\t\\STATE compute (part of) the eigenvalues and eigenvectors of $L$\n\\label{line17}\n\t\t\\STATE determine multiplicity $n_F$ of the Fiedler value\n\t\t\taccording to a tolerance $\\tau$\n\t\t\\IF {$n_F = 1$}\n\t\t\t\\STATE $\\bm{x}=$ sorted Fiedler vector\n\t\t\t\\STATE $t$ number of distinct values in $\\bm{x}$\n\t\t\t\taccording to a tolerance $\\tau$\n\\label{line19}\n\t\t\t\\FOR {$j=1,\\dots,t$}\n\t\t\t\t\\STATE $u_j$ indices of elements in $\\bm{x}$ \n\t\t\t\t\twith value $x_j$\n\t\t\t\t\\IF $u_j$ has just one element\n\t\t\t\t\t\\STATE $v_j=\\lnode(u_j)$\n\\label{line26}\n\t\t\t\t\\ELSE \n\t\t\t\t\t\\STATE $v(j) = \\spectrsort(F(u_j,u_j),U(u_j,u_j))$\n\\label{line28}\n\t\t\t\t\\ENDIF\n\t\t\t\\ENDFOR\n\t\t\t\\STATE $T = \\qnode(v)$\n\t\t\\ELSE\n\t\t\t\\STATE $T=\\mnode(U)$\n\\label{line33}\n\t\t\\ENDIF\n\t\\ENDIF\n\\ENDIF\n\\end{algo}\n\\caption{Spectral sort algorithm.}\n\\label{alg:spectrsort}\n\\end{algorithm}\n\nThe algorithm starts translating all the entries of the correlation matrix so\nthat the smallest is 0 , i.e., \n\\begin{equation}\\label{trans}\n\\tilde{F}=F-\\alpha \\bm{e}\\bm{e}^{T}, \\qquad \\alpha = \\min_{i,j} f_{ij};\n\\end{equation}\nsee line~\\ref{line2} of Algorithm~\\ref{alg:spectrsort}.\nThis is justified by the fact that $F$ and $\\tilde{F}$ have the same Fiedler\nvectors and that if $F$ is an irreducible R-matrix such translation ensures\nthat the Fiedler value is a simple eigenvalue of\n$L$~\\cite[Lemma~4.1~and~Theorem~4.6]{atkins1998spectral}.\nOur software allows the user to disable this procedure (see\nTable~\\ref{tab:opts} below)\nas he may decide to\nsuitably preprocess the similarity matrix in order to reduce the\ncomputational load.\nIndeed, the translation procedure is repeated each time the algorithm calls\nitself recursively.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $U = \\getconcomp(F)$\n\\STATE preallocate the cell-array $U$, $chlist=$empty vector\n\\STATE $root=$\\{node 1\\}, $list=root$, $n=$row size of $F$\n\\STATE $i=0$, $flag=true$ (logical variable)\n\\WHILE $flag$\n\\STATE $i =i+1$\n\\STATE $list =\\graphvisit(root,list)$\n\\STATE $U\\{i\\}=list$\n\\STATE update $chlist$ adding the nodes in $list$ and sort the vector\n\\STATE $flag=true$ if the number of elements in $chlist$ is different from $n$\n\\STATE \\phantom{XXX} otherwise $flag=false$\n\\IF $flag$\n\t\\STATE choose the $root$ for a new connected component\n\t\\IF there are no connected components left\n\t\t\\STATE exit\n\t\\ENDIF\n\t\\STATE $list=root$\n\\ENDIF\n\\ENDWHILE\n\\end{algo}\n\\caption{Detect the connected components of a graph.}\n\\label{alg:getconcomp}\n\\end{algorithm}\n\nIf the matrix $F$ is reducible, then the seriation problem can be\ndecoupled~\\cite[Lemma~4.2]{atkins1998spectral}.\nLines~\\ref{line3}--\\ref{line4} of the algorithm detect the irreducible blocks\nof the correlation matrix by using the function \\textsf{getconcomp.m}, which\nalso identifies the corresponding index sets.\nThe function, described in Algorithm~\\ref{alg:getconcomp}, constructs a\n\\emph{cell array} containing the indices which identify each connected\ncomponent of a graph.\nIt calls the function \\textsf{graphvisit.m}, which visits a graph starting from\na chosen node; see Algorithm~\\ref{alg:graphvisit}.\nNote that these two functions, in order to reduce the stack consumption due to\nrecursion, use a global variable to store the correlation matrix.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $list = \\graphvisit(root,list)$\n\\STATE construct the list $l$ of the indices of the nodes connected to the root\n\\STATE initialize an empty list $nlist$\n\\STATE find the elements of $l$ which are not in $list$\n\\STATE add the new elements to $list$ and to $nlist$\n\\IF $nlist$ is not empty\n\t\\STATE sort $list$\n\t\\FOR each node $i$ in $nlist$\n\t\t\\STATE $list=\\graphvisit(nlist(i),list)$\n\t\\ENDFOR\n\\ENDIF\n\\end{algo}\n\\caption{Visit a graph starting from a node.}\n\\label{alg:graphvisit}\n\\end{algorithm}\n\nIf more than one connected component is found, then the function calls itself\non each component, and stores the returned lists of nodes as children of\na P-node (lines \\ref{line6}--\\ref{line9}).\nIf the matrix is irreducible, the dimension $n$ of the matrix is\nconsidered (lines~\\ref{line11}--\\ref{line15}).\nThe cases $n=1,2$ are trivial.\nIf $n>2$, the Laplacian matrix $L$ is computed, as well as the Fiedler value\nand vector (lines~\\ref{line16}--\\ref{line17}). \nDepending on the matrix being ``small'' or ``large'' different algorithms are\nused.\nFor a small scale problem the full spectral decomposition of the Laplacian is\ncomputed by the \\ttt{eig} function of Matlab.\nFor a large scale problem only a small subset of the eigenvalues and\neigenvectors are evaluated using the \\ttt{eigs} function, which is based on a\nKrylov space projection method.\nThe \\texttt{PQser}  toolbox computes by default the eigenpairs corresponding to the\nthree eigenvalues of smallest magnitude, since they are sufficient to\nunderstand if the Fiedler value is simple or multiple, but the default value\ncan be modified.\nThe choice between the two approaches is automatically performed and it may be\ninfluenced by the user; see Table~\\ref{tab:opts} in Section~\\ref{sec:implser}.\n\nThen, the algorithm determines the multiplicity of the Fiedler value according\nto a given tolerance.\nIf the Fiedler value is a simple eigenvalue of $L$, the algorithm sorts the\nelements of the current list according to the reordering of the Fiedler vector\nand stores them as the children of a Q-node.\nIf two or more values of the Fiedler vector are repeated the function invokes\nitself recursively (line~\\ref{line28}), in accordance\nwith~\\cite[Theorem~4.7]{atkins1998spectral}; on the contrary, the corresponding\nnode becomes a leaf (line~\\ref{line26}).\nIn our implementation we introduce a tolerance $\\tau$ to distinguish ``equal''\nand ``different'' numbers: $a$ and $b$ are considered ``equal'' if\n$|a-b|<\\tau$. The default value for $\\tau$ is $10^{-8}$.\n\nIn the case of a multiple Fiedler value, the algorithm conventionally\nconstructs an ``M-node'' (line~\\ref{line33}).\nThis new type of node has been introduced in order to flag this particular\nsituation, which will be further discussed in Section~\\ref{sec:mult}.\n\nAlgorithm~\\ref{alg:spectrsort} produces a PQ-tree whether $F$ is a pre-R matrix\nor not.\nIf all the Fiedler values computed are simple, the starting matrix is pre-R and\nany permutation encoded in the PQ-tree will take it to R-form.\nIn the presence of a multiple Fiedler vector the problem is not well posed and\nan approximate solution is computed.\n\nThe number $N$ of all the admissible permutations generated by the algorithm\ncan be obtained by counting all the admissible boundaries of the tree.\nIn the case of a PQ-tree consisting of a single Q-node $N$ is equal to 2,\nbecause only the left-to-right order of the children leaves and its reverse\nare possible. \nFor a single P-node, the number of all the permutations is the factorial of the\nnumber of the children.\nAn M-node is temporarily treated as a P-node, although we experimentally\nobserved that not all the permutations are admissible; this aspect is discussed\nin Section~\\ref{sec:mult}.\n\n\n\\subsection{Implementation of spectral seriation}\\label{sec:implser}\n\nThe functions included in the \\texttt{PQser}  toolbox are listed in\nTable~\\ref{tab:serfuncs}.\nBesides the function \\ttt{spectrsort}, which implements \nAlgorithm~\\ref{alg:spectrsort}, there is a parallel version of the same method,\n\\ttt{pspectrsort}, which distributes the \\ttt{for} loop at\nline~\\ref{line6} of the algorithm among the available processing units.\nIn order to execute the function \\ttt{pspectrsort}, the Parallel Computing\nToolbox must be present in the current Matlab installation.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{spectrsort}   & spectral sort for the seriation problem \\\\\n\\ttt{pspectrsort}  & parallel version of \\ttt{spectrsort} \\\\\n\\ttt{fiedvecs}     & compute the Fiedler vectors and values of a Laplacian \\\\\n\\ttt{getconcomp}   & determine the connected components of a graph \\\\\n\\ttt{graphvisit}   & visit a graph starting from a node \\\\\n\\ttt{distinct}     & sort and level the elements of a vector \\\\\n\\ttt{lapl}         & construct the graph Laplacian of a matrix \\\\\n\\ttt{testmatr}     & test matrices for PQser \\\\\n\\hline\n\\end{tabular}\n\\caption{Functions in the \\texttt{PQser}  toolbox devoted to the solution of the\nseriation problem.}\n\\label{tab:serfuncs}\n\\end{table}\n\nThe function \\ttt{testmatr} allows one to create some simple test problems.\nThe remaining functions of Table~\\ref{tab:serfuncs} are not likely to be used\nin the common use of the toolbox. \nThey are made available to the expert user, who may decide to call them\ndirectly or to modify their content.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{tau} & tolerance used to distinguish between ``equal'' and ``different'' \\\\\n\t  & values (\\ttt{spectrsort} and \\ttt{fiedvecs}, def. $10^{-8}$) \\\\\n\\ttt{translate} & applies translation \\eqref{trans} (\\ttt{spectrsort}, def. 1) \\\\\n\\ttt{lrg} & used to select small scale or large scale algorithm (\\ttt{fiedvecs}, \\\\\n\t& true if the input matrix is sparse) \\\\\n\\ttt{nlarge} & if matrix size is below this value, the small scale algorithm \\\\\n\t     & is used (\\ttt{fiedvecs}, def. 1000) \\\\\n\\ttt{neig} & number of eigenpairs to be computed when the large scale \\\\\n\t   & algorithm is used (\\ttt{fiedvecs}, def. 3) \\\\\n\\ttt{maxncomp} & maximum number of connected components (\\ttt{getconcomp}, def. 100) \\\\\n\\ttt{bw} & half bandwidth of test matrix (\\ttt{testmatr}, type 2 example, def. 2) \\\\\n\\ttt{spar} & construct a sparse test matrix (\\ttt{testmatr}, type 2 example, def. 1) \\\\\n\\hline\n\\end{tabular}\n\\caption{Tuning parameters for the \\texttt{PQser}  toolbox; the functions affected are\nreported in parentheses, together with the default value of each parameter.}\n\\label{tab:opts}\n\\end{table}\n\nThe toolbox has some tuning parameters, which are set to a default value, but\ncan be modified by the user. This can be done by passing to a function, as an\noptional argument, a variable of type \\ttt{struct} with fields chosen among the\nones listed in Table~\\ref{tab:opts}.\nFor example:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nopts.translate = 0;\nT = spectrsort(F,opts);\n\\end{verbatim}\n\\end{quote}\napplies Algorithm~\\ref{alg:spectrsort} to a similarity matrix $F$ omitting\nthe translation process described in \\eqref{trans}.\n\nTo illustrate the use of the toolbox, we consider a similarity matrix $R$\nsatisfying the Robinson criterion \n$$\nR = \\begin{bmatrix}\n200 & 150 & 120 & 80 & 40 & 0 & 0 & 0 & 0 & 0\\\\\n150 & 200 & 160 & 120 & 80 & 40 & 0 & 0 & 0 & 0\\\\\n120 & 160 & 200 & 160 & 120 & 80 & 40 & 0 & 0 & 0\\\\\n80 & 120 & 160 & 200 & 160 & 120 & 80 & 40 & 0 & 0\\\\\n40 & 80 & 120 & 160 & 200 & 160 & 120 & 80 & 40 & 0\\\\\n0 & 40 & 80 & 120 & 160 & 200 & 160 & 120 & 80 & 40\\\\\n0 & 0 & 40 & 80 & 120 & 160 & 200 & 160 & 120 & 80\\\\\n0 & 0 & 0 & 40 & 80 & 120 & 160 & 200 & 160 & 120\\\\\n0 & 0 & 0 & 0 & 40 & 80 & 120 & 160 & 200 & 150\\\\\n0 & 0 & 0 & 0 & 0 & 40 & 80 & 120 & 150 & 200\n\\end{bmatrix}\n$$\nand the pre-R matrix obtained by applying to the rows and columns of $R$ a\nrandom permutation \n$$\nF = \\begin{bmatrix} \n200 & 0 & 0 & 150 & 120 & 0 & 160 & 40 & 0 & 80\\\\\n0 & 200 & 150 & 0 & 0 & 120 & 0 & 80 & 160 & 40\\\\ \n0 & 150 & 200 & 0 & 0 & 80 & 0 & 40 & 120 & 0\\\\ \n150 & 0 & 0 & 200 & 80 & 0 & 120 & 0 & 0 & 40\\\\ \n120 & 0 & 0 & 80 & 200 & 80 & 160 & 120 & 40 & 160\\\\ \n0 & 120 & 80 & 0 & 80 & 200 & 40 & 160 & 160 & 120\\\\ \n160 & 0 & 0 & 120 & 160 & 40 & 200 & 80 & 0 & 120\\\\ \n40 & 80 & 40 & 0 & 120 & 160 & 80 & 200 & 120 & 160\\\\ \n0 & 160 & 120 & 0 & 40 & 160 & 0 & 120 & 200 & 80\\\\ \n80 & 40 & 0 & 40 & 160 & 120 & 120 & 160 & 80 & 200 \n\\end{bmatrix}.\n$$\n\nThe PQ-tree $T$ containing the solution of the reordering problem is\nconstructed by calling the function \\ttt{spectrsort}, which returns the\nresulting data structure:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nT = spectrsort(F,opts)\nT = \n  struct with fields:\n     type: 'Q'\n    value: [1x10 struct]\n\\end{verbatim}\n\\end{quote}\nUsing the function \\ttt{pqtreeplot} \n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\npqtreeplot(T)\n\\end{verbatim}\n\\end{quote}\nwe obtain the representation of the PQ-tree displayed in Figure~\\ref{pqtree2}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.52\\textwidth]{pqtree2}\n\\caption{A PQ-tree corresponding to a pre-R matrix of dimension 10.}\n\\label{pqtree2}\n\\end{center}\n\\end{figure}\n\nIn this particular case, the PQ-tree $T$ consists of just a Q-node as a root,\nso only two permutations of the leaves are allowed.\nThey can be extracted from the tree using the function \\ttt{pqtreeperms}, whose\noutput is\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nperms_matrix = pqtreeperms(T)\nperms_matrix =\n     4     1     7     5    10     8     6     9     2     3\n     3     2     9     6     8    10     5     7     1     4\n\\end{verbatim}\n\\end{quote}   \nIn some occasions, a PQ-tree may contain a very large number of permutations.\nIn such cases, the function \\ttt{pqtree1perm} extracts just one of the\npermutations, in order to apply it to the rows and columns of the matrix $F$:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nseq = pqtree1perm(T);\nAR = F(seq,seq);\n\\end{verbatim}\n\\end{quote}   \nSince $F$ is pre-R, we clearly reconstruct the starting similarity matrix $R$.\n\nThis experiment is contained in the script \\texttt{graf2.m}, in the\n\\texttt{demo} sub-directory.\nThe script \\texttt{tutorial.m}, in the same directory, illustrates the use of\nthe toolbox on other numerical examples.\n\n\n\\section{The case of a multiple Fiedler value}\\label{sec:mult}\n\nIn this section we discuss the case where the Fiedler value is a multiple root\nof the characteristic polynomial of the Laplacian $L$.\nWhen this happens, the eigenspace corresponding to the smallest nonzero\neigenvalue of $L$ has dimension larger than one, so there is no uniqueness in\nthe choice of the Fiedler vector.\n\nWe conjecture that sorting the entries of a Fiedler vector, that is, of any\nvector in the eigenspace of the Fiedler value, does not necessarily lead to all\npossible indexes permutations, i.e., the factorial of the number $n$ of units.\nWe observed that there may be some constraints that limit the number of\npermutations deriving from the Fiedler vector, and this number does not appear\nto be related to the multiplicity of the Fiedler value by a simple formula.\nWe will illustrate this issue by a numerical experiment.\n\nAs we did not find any reference to this problem in the literature, we plan\nto study it in a subsequent paper.\nThis is the reason why the \\texttt{PQser}  toolbox conventionally associates an\n\\emph{M-node}, to the presence of a multiple Fiedler value.\nIn the software, the new type of node is temporarily treated as a P-node.\nThis leaves the possibility to implement the correct treatment of the case,\nonce the problem will be understood.\n\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tikzpicture}[->,>=stealth',auto,node distance=1cm,\n\ton grid,semithick, every state\/.style={fill=cyan!20!white,draw=none,\n\tcircular drop shadow,text=black,inner sep=0pt}]\n\t\\tiny\n\\node[state](A){1};\n\\node[state](B)[below=of A]{2};\n\\node[state](C)[below=of B]{3};\n\\node[state](D)[below=of C]{4};\n\\node[state](E)[below=of D]{5};\n\\node[state,fill=red!20!white,node distance=4cm](L)[right=of A]{1};\n\\node[state,fill=red!20!white,node distance=4cm](M)[right=of B]{2};\n\\node[state,fill=red!20!white,node distance=4cm](N)[right=of C]{3};\n\\node[state,fill=red!20!white,node distance=4cm](O)[right=of D]{4};\n\\node[state,fill=red!20!white,node distance=4cm](P)[right=of E]{5};\n\\path (A) edge (L);\n\\path (A) edge (M);\n\\path (B) edge (M);\n\\path (B) edge (N);\n\\path (C) edge (N);\n\\path (C) edge (O);\n\\path (D) edge (O);\n\\path (D) edge (P);\n\\path (E) edge (P);\n\\path (E) edge (L);\n\\end{tikzpicture}\n\\caption{The \\emph{cycle} seriation problem; the units are on the left, the\ntypes on the right.}\n\\label{fig:cycle}\n\\end{center}\n\\end{figure}\n\nHere we present a simple example to justify our conjecture.\nLet us consider the seriation problem described by the bipartite graph depicted\nin Figure~\\ref{fig:cycle}.\nThe nodes on the left are the units, e.g., the excavation sites; on the right\nthere are the types, which may be seen as the archeological findings. The\nrelationships between units and types are represented by edges connecting the\nnodes.\n\nThe problem is clearly unsolvable, as the associated graph describes a\n\\emph{cycle}: each unit is related to surrounding units by a connection to a\ncommon type, and the two extremal units are related to each other in the same\nway.\n\nAt the same time, not all the units permutations are admissible.\nFor example, one may argue that the permutation ${\\boldsymbol{\\pi}}_1=(3,4,5,1,2)^T$ should\nbe considered partially feasible, as it breaks only one of the constraints\ncontained in the bipartite graph, while the ordering ${\\boldsymbol{\\pi}}_2=(1,4,2,5,3)^T$ has\nnothing to do with the problem considered.\n\nThe adjacency matrix associated to the graph in Figure~\\ref{fig:cycle} is the\nfollowing\n\\begin{equation}\\label{incidCycle}\nA = \\begin{bmatrix}\n1 & 1 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 1 & 0\\\\\n0 & 0 & 0 & 1 & 1\\\\\n1 & 0 & 0 & 0 & 1\n\\end{bmatrix}.\n\\end{equation}\nWe can associate to \\eqref{incidCycle} the similarity matrix\n\\begin{equation}\\label{cycleF}\nF = AA^T = \\begin{bmatrix}\n2 & 1 & 0 & 0 & 1\\\\\n1 & 2 & 1 & 0 & 0\\\\\n0 & 1 & 2 & 1 & 0\\\\\n0 & 0 & 1 & 2 & 1\\\\\n1 & 0 & 0 & 1 & 2\n\\end{bmatrix},\n\\end{equation}\nwhose Laplacian is\n$$\nL = D - F = \\begin{bmatrix}\n2 & -1 & 0 & 0 & -1\\\\\n-1 & 2 & -1 & 0 & 0\\\\\n0 & -1 & 2 & -1 & 0\\\\\n0 & 0 & -1 & 2 & -1\\\\\n-1 & 0 & 0 & -1 & 2\n\\end{bmatrix}.\n$$\n\nThe matrix $L$ is circulant, that is, it is fully specified by its first\ncolumn, while the other columns are cyclic permutations of the first one\nwith an offset equal to the column index. \nA complete treatment of circulant matrices can be found\nin~\\cite{davis1979circulant}, while~\\cite{redivo2012smt} implements a Matlab\nclass for optimized circulant matrix computations.\n\nOne of the basic properties of circulant matrices is that their spectrum is\nanalytically known. In particular, the eigenvalues of $L$ are given by\n$$\n\\{\\widehat{L}(1), \\widehat{L}(\\omega), \\widehat{L}(\\omega^2),\n\\widehat{L}(\\omega^3), \\widehat{L}(\\omega^4) \\},\n$$\nwhere $\\widehat{L}(\\zeta)$ is the discrete Fourier transform of the first\ncolumn of $L$\n$$\n\\hat{L}(\\zeta) = 2 -\\zeta ^{-1}-\\zeta ^{-4},\n$$\nand $\\omega = {\\mathrm{e}}^{\\frac{2\\pi{\\mathrm{i}}}{5}}$ is the minimal phase $5$th root of\nunity; see~\\cite{davis1979circulant}.\nA simple computation shows that\n$$\n\\widehat{L}(1)=0, \\quad\n\\widehat{L}(\\omega)=\\widehat{L}(\\omega^4)=2-2\\cos\\frac{2\\pi}{5}, \\quad\n\\widehat{L}(\\omega^2)=\\widehat{L}(\\omega^3)=2-2\\cos\\frac{4\\pi}{5}, \n$$\nso that the Fiedler value $\\widehat{L}(\\omega)$ has multiplicity 2.\n\nTo explore this situation we performed the following numerical experiment.\nWe considered 10000 random linear combinations of an orthonormal basis for the\neigenspace corresponding to the Fiedler value. This produces a set of random\nvectors, belonging to a plane immersed in ${\\mathbb{R}}^5$, which can all be considered\nas legitimate ``Fiedler vectors''.\n\nEach vector was sorted, and the corresponding permutations of indexes were\nstored in the columns of a matrix.\nIn the end, all the repeated permutations were removed.\nWe obtained 10 permutations, reported in the columns of the following matrix\n\\begin{equation}\\label{permat}\n\\begin{bmatrix}\n3 & 2 & 5 & 3 & 2 & 4 & 4 & 1 & 5 & 1 \\\\\n2 & 1 & 4 & 4 & 3 & 3 & 5 & 5 & 1 & 2 \\\\\n4 & 3 & 1 & 2 & 1 & 5 & 3 & 2 & 4 & 5 \\\\\n1 & 5 & 3 & 5 & 4 & 2 & 1 & 4 & 2 & 3 \\\\\n5 & 4 & 2 & 1 & 5 & 1 & 2 & 3 & 3 & 4\n\\end{bmatrix}.\n\\end{equation}\nThey are much less than the $5!=120$ possible permutations, and reduce to 5 if\nwe remove the columns which are the reverse of another column.\nThis confirms our conjecture: when a Fiedler value is multiple some constraints\nare imposed on the admissible permutations of the units.\n\nIt is relevant to notice that matrix \\eqref{permat} does not contain the cyclic\npermutations of the units depicted in Figure~\\ref{fig:cycle}.\nIndeed, the spectral algorithm aims at moving the nonzero components close to\nthe main diagonal, and this contrasts with the presence of nonzeros in the\nelements $f_{51}$ and $f_{15}$ of matrix \\eqref{cycleF}.\nAll permutations contained in \\eqref{permat}, when applied to the similarity\nmatrix $F$, produce the same matrix\n$$\n\\tilde{F} = \\begin{bmatrix}\n2 & 1 & 1 & 0 & 0 \\\\\n1 & 2 & 0 & 1 & 0 \\\\\n1 & 0 & 2 & 0 & 1 \\\\\n0 & 1 & 0 & 2 & 1 \\\\\n0 & 0 & 1 & 1 & 2\n\\end{bmatrix},\n$$\nwhich exhibits a smaller bandwidth than \\eqref{cycleF}.\n\nThis experiment can be reproduced by executing the script \\texttt{mfiedval.m},\nwhich is found in the \\texttt{demo} sub-directory.\n\n\n\\section{Numerical experiments}\\label{sec:numexp}\n\nIn this section we illustrate the application of the \\texttt{PQser}  toolbox to some\nnumerical examples.\nThe experiments can be repeated by running the related Matlab scripts located\nin the \\texttt{demo} sub-directory of the toolbox.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.30\\textwidth]{bornmat}\\hfil \n\\includegraphics[width=.30\\textwidth]{bornmix}\\hfil \n\\includegraphics[width=.30\\textwidth]{bornord}\n\\caption{Processing of the Bornholm data set: the spy plot on the left shows\nthe original matrix, a permuted version is reported in the central graph, on\nthe right we display the matrix reordered by the spectral algorithm.}\n\\label{bornholm}\n\\end{center}\n\\end{figure}\n\nThe first example is the numerical processing of the Bornholm data set,\npresented in Table~\\ref{tab:adjacency}.\nWe randomly permute the rows of the adjacency matrix and apply the\nspectral algorithm to the similarity matrix associated to the permuted matrix. \nThe resulting PQ-tree contains just a Q-node, so there is only one solution\n(actually, this is a proof that the matrix is pre-R) which we use to reorder\nthe permuted matrix.\nThe computational code is contained in the file \\ttt{exper1.m}.\n\nFigure~\\ref{bornholm} reports the spy plots which represent the nonzero\nentries of the initial matrix, its permuted version, and the final reordering.\nIt is immediate to observe that the lower band of the reordered matrix is\nslightly narrower than the initial matrix, showing that the spectral algorithm\nwas able to improve the results obtained empirically by archaeologists.\n\nThe second example concerns the comparison of the spectral algorithm with its\nparallel version in the solution of a large scale problem.\nThe experiments were performed on a dual Xeon CPU E5-2620 system (12 cores),\nrunning the Debian GNU\/Linux operating system and Matlab 9.2.\n\nThe function \\ttt{testmatr} of the toolbox allows the user to create a block\ndiagonal matrix, formed by $m$ banded blocks whose size is chosen using a\nsecond input parameter.\nThe matrix is randomly permuted in order to hide its reducible structure.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{exper2time}\\hfil \n\\includegraphics[width=.47\\textwidth]{exper2spup}\n\\caption{Comparison between the sequential and the parallel versions of\nAlgorithm~\\ref{alg:spectrsort}: on the left the execution time in seconds, on\nthe right the parallel speedup, defined as the ratio between the sequential and\nthe parallel timings. The test matrix is of dimension $2^{15}=32768$, the size\nof each reducible block is $2^j$, where $j$ is  the index reported in the\nhorizontal axis.}\n\\label{exper2}\n\\end{center}\n\\end{figure}\n\nWe let the size of the problem be $n=2^{15}=32768$ and, for $j=1,2\\ldots,15$,\nwe generate a sequence of test matrices containing $n\\cdot 2^{-j}$ blocks, each\nof size $2^j$.\n\nWe apply the function \\ttt{spectrsort} that implements\nAlgorithm~\\ref{alg:spectrsort} to the above problems, as well as its parallel\nversion \\ttt{pspectrsort}, and record the execution time; see the file\n\\ttt{exper2.m}.\nThe number of processors available on our computer was 12.\n\nThe graph on the left of Figure~\\ref{exper2} shows that there is a significant\nadvantage from running the toolbox on a parallel computing system when the\nnetwork associated to the problem is composed by a small number of large\nconnected components.\nThis is confirmed by the plot of the parallel speedup, that is, the ratio\nbetween the timings of the sequential and the parallel implementations,\ndisplayed in the graph on the right in the same figure.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.30\\textwidth]{exper3data}\\hfil \n\\includegraphics[width=.30\\textwidth]{exper3spec}\\hfil \n\\includegraphics[width=.30\\textwidth]{exper3rcm}\n\\caption{Bandwidth reduction of a sparse matrix of size 1024: the three spy\nplots display the initial matrix, the reordered matrix resulting from the\nspectral algorithm, and the one produced by the \\ttt{symrcm} function of\nMatlab.}\n\\label{exper3}\n\\end{center}\n\\end{figure}\n\nTo conclude, we consider another important application of the reordering\ndefined by the Fiedler vector of the Laplacian, namely, the reduction of the\nbandwidth for a sparse matrix; see~\\cite{barnard1995spectral}.\n\nWe generate a sparse symmetric matrix of size $n=1024$, having approximately\n0.2\\% nonzero elements, and reorder its rows and columns by\nAlgorithm~\\ref{alg:spectrsort}.\nNotice that in this case the spectral algorithm must be applied to a matrix\nwhose elements are taken in absolute value.\nThe computation is described in the script \\ttt{exper3.m}.\n\nThe resulting matrix is depicted by displaying its nonzero pattern in\nFigure~\\ref{exper3}, where it is compared to the reverse Cuthill-McKee\nordering, as implemented in the \\ttt{symrcm} function of Matlab.\nThe spectral algorithm appears to be less effective than \\ttt{symrcm},\nleading to a reordered matrix with a wider band.\nThis is due to the fact that \\ttt{spectrsort} aims at placing the largest\nentries close to the diagonal, and this does not necessarily produce the\nmaximal bandwidth reduction.\nExperimenting with sparser matrices we observed that often the two methods\nproduce similar results.\n\nWe remark that, also in this application, the presence of a multiple Fiedler\nvalue may constitute a problem.\nFor example, we were not able to correctly process the Matlab test matrix\n\\ttt{bucky} (the connectivity graph of the Buckminster Fuller geodesic dome),\nbecause the associated Laplacian possesses a triple Fiedler value.\n\n\n\\section{Conclusions}\\label{sec:last}\n\nIn this paper we present a new Matlab toolbox principally aimed to the\nsolution of the seriation problem, but which can be applied to other related\nproblems.\n\nIt is based on a spectral algorithm introduced in~\\cite{atkins1998spectral},\nand contains an implementation of PQ-trees as well as some tools for their\nmanipulation, including an interactive visualization tool.\nThe implemented algorithm includes the possibility to choose between a small\nscale and a large scale algorithm for the computation of the Fiedler vector,\nand to detect equal components in the same vector according to a chosen\ntolerance. Further, a parallel version of the method is provided.\n\nWe also point out the importance of the presence of multiple Fiedler values, a\nproblem which has not been considered before in the literature and which has a\nsignificant influence on the computation of an approximate solution to the\nseriation problem.\n\nThe use of the toolbox is illustrated by a few practical examples, and its\nperformance is investigated through a set of numerical experiments, both of\nsmall and large scale.\n\n\n\\section{Acknowledgements}\n\nWe would like to thank Matteo Sommacal for pointing our attention to the\nproblem of seriation and to its application in archaeology.\nThe paper~\\cite{ps05}, which he coauthored, was our principal source of\ninformation when the research which lead to this paper started.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n Herbig-Haro (HH) jets are the optical manifestations of\ncollimated outflows from young stellar objects (YSOs) in their early\nstages of evolution and, along with typical HH objects, are apparent\ntracers of active star formation. YSOs, unfortunately, are usually deeply\nembedded in dense, extended envelopes and\/or opaque molecular cloud\ncores that strongly impede efforts for investigations in the optical \nregime and make our view of the early stages of star formation still \nquite a puzzle. In some cases, however, YSOs are located within\nthe confines of HII regions (Reipurth et al. 1998) and their power sources\nsuffer very low extinction and are therefore optically visible.  Similar\njets immersed in a photoionized medium were identified in the Orion\nnebula and near the late B stars in NGC~1333 (Bally et al. 2000;\nBally \\& Reipurth 2001).\nHowever, some properties of photoionized jets in HII regions are\nfound to be highly dependent on their environment.  Those immersed in the \noutskirts of HII regions, or near soft-UV radiation sources such as late B stars,\ntend to emerge as bipolar jets which usually share a ``C\" shaped appearance\n(Bally et al. 2000; Bally \\& Reipurth 2001), whereas jets deeply embedded \nin HII regions \nendure a more disruptive environment and seem to favor highly asymmetric\nor even monopolar jet formation (Reipurth et al. 1998).\n\nThe Rosette Nebula is a spectacular HII region excavated by the strong\nstellar winds from O stars in the center of the young open\ncluster NGC~2244, which has an age of about 3 x $10^6$ yr (Ogura \\& Ishida 1981).\nThis on-going star forming region (Meaburn \\& Walsh 1986;\nClayton et al. 1998; Phelps \\& Lada 1997; Li et al. 2002; Li 2003) is located\nat a distance of $\\sim$1500 pc at the tip of a giant molecular cloud\ncomplex with an extent of around 100 pc (Dorland \\& Montmerle 1987).\nIn this paper, we present the discovery of an optical jet system embedded \nin the Rosette Nebula with a fascinating morphology that displays exceptional \nnew features not known before.\n\n\\section{Observations and Data Reduction}\n\n\\subsection{Narrowband Imaging}\n\nNarrow band images of the Rosette Nebula were obtained 3 March 1999 with the\nKitt Peak National Observatory 0.9-meter telescope and the 8k$\\times$8k MOSAIC I\ncamera (Muller et al. 1998). Images were taken with the H${\\alpha}$,\n[OIII] and [SII] filters, whose central wavelengths\/FWHMs are 6569\\AA\/80\\AA, 5021\\AA\/55\\AA,\nand 6730\\AA\/80\\AA, respectively.  For each filter, five exposures of 600~seconds\nwere taken, each image slightly offset to fill in physical gaps between the\nMOSAIC CCDs.  The pixel scale is 0.423\\arcsec~pixel$^{-1}$, resulting in a 59\\arcmin~x~59\\arcmin\\\nfield of view. An astrometric solution for these images was determined with\nthe use of stars in the Guide Star Catalog 1.2 (Roser et al. 1998). The astrometric\naccuracy in the optical images presented is $\\sim$1\\arcsec.\n\n\\subsection{Optical Spectroscopy}\n\nOptical spectroscopy of the jet system, with a\n200\\AA~mm$^{-1}$, 4.8\\AA~pixel$^{-1}$ dispersion, and a 2.5\\arcsec slit,\nwas carried out with the 2.16m telescope of the National Astronomical\nObservatories of China on 10 and 16 January 2003, with slit positions\nalong and orthogonal to the jet direction. A possible\nphysical companion of the apparent power source was also observed with the\northogonal slit position to avoid contamination from the jet system.\nAn OMR (Optomechanics Research Inc.) spectrograph and a Tektronix 1024${\\times}$1024 CCD\ndetector were used in this run of observations.\n\nThe spectral data were reduced following\nstandard procedures in the NOAO Image Reduction and Analysis Facility\n(IRAF, version 2.11) software package. The CCD reductions included bias\nand flat-field correction, successful nebular background subtraction, and\ncosmic rays removal. Wavelength calibration was performed based on helium-argon\nlamps exposed at both the beginning and the end of the observations every night.\nFlux density calibration of each spectrum was conducted based on observations of at\nleast two of the KPNO spectral standards (Massey et al. 1988) per night.\n\n\\section{Results and Discussion}\n\n\\subsection{The Jet System}\n\nAn optical jet system with a striking morphology was identified\nwhen scrutinizing the narrow band images of the Rosette Nebula (Fig. 1). A\nprominent jet with a position angle of 312\\arcdeg\\ is clearly seen traced back\nto a faint visible star, indicating rather low extinction along the\nline of sight. The circumstellar envelope of the YSO\nmight have been stripped away, leaving a photoablating disk exposed\nto the strong UV radiation field of the exciting sources in the HII region.\nThe optical jet remains highly collimated for a projected distance\nof $>$8000~AU  from the energy source if a mean distance\nof 1500 pc to Rosette is adopted.  At the end of the collimated jet is a \nside-impacted shock structure, probably due to severe radiation pressure from the\nstrong UV field present and strong stellar winds encountered.   \nA prominent knot or compact mass bullet resembling a point source is immediately\nnoticed in the highly collimated part of the jet with a projected separation of\n7\\arcsec\\ from the source.  Another one or \ntwo knots can be marginally noticed in the bow shaped extension of the jet, \nindicating episodic eruption events from the driving source.\n\n We infer the possible existence of \na counterjet from the presence of a promising bow shock\nstructure on the opposite side of the jet.  Successive episodes\nof counterjet ejection may have ceased or may be too faint. \nAlternatively, this structure may simply be shocked\ninterstellar material that was swept up and photoionized by the strong\nradiation field, which happen to be projected onto the vicinity of\nthe jet exciting source. Another possible explanation is that it is\ndisrupted jet material from a possible T Tauri companion (please refer to\nFig.~1) or from nearby young stars. If it is indeed a \ndegenerated counterjet, however, it may be the only existing observational \nevidence for how bipolar jets evolve into monopolar or \nhighly asymmetric jets.\nWe further argue that the long-puzzling high-velocity components from\nthe photoionized gas fronts aggregated in the Rosette (Meaburn \\& Walsh 1986; Clayton \\& Meaburn 1995; Clayton, Meaburn \\& Lopez et al. 1998) could\nactually be from rapidly disrupted young disk-jet systems in the inner\npart of the HII region.  Diffuse [OIII] emission is prominent from \nall parts of the jet system, which is likely due to its photo-dissipating nature.  \n[SII] emission from the highly-collimated part\nof the jet decreases rapidly from the base of the jet and is undetected\nin the shocked structures, indicating a steep decline of internal\nshock effects or rather overwhelming external ionization.\n\nSpectroscopic observation along the jet direction reveals prominent H$\\alpha$ \n($W_{\\lambda} \\sim$~35\\AA) as well as [SII] emission.  The presence of\nsignificant [OIII] $\\lambda\\lambda$4959, 5007 line emission \nconfirms our suspicion that the jet is in a high-excitation state.\nFig. 2 clearly shows the existence of a weak counterjet as discussed above.\nNote that enhanced [NII] emission is detected along the jet, which is not\nshown here.\n\n\\subsection{The exciting source}\n\nSpectroscopic observations of the exciting source reveal primarily the \nspectrum of a normal F8Ve star with weak H$\\alpha$ ($W_{\\lambda} \\sim$~6.7\\AA)\nand prominent [OIII] emission.  However, the spectrum also shows a \nsignificant red-displaced absorption component in H$\\alpha$, with \na receding velocity of 500$\\pm$50~km~s$^{-1}$ (Fig. 3). Two weak\nbut significant emission features with unclear nature are also present immediately\nto the red of the rest H$\\alpha$ emission and its red-displaced absorption. One is\ncentered at 6582.5 \\AA~ with an accuracy of $\\pm$ 1 \\AA~, and can be \nthe pronounced [NII] $\\lambda$ 6583.6 emission detected also along the jet. \nThis gives a high [NII] to H$\\alpha$ flux ratio of about 1:3.\nAlbeit the other with a central wavelength of 6613.3 \\AA~ cannot be identified \nwith any known emission lines with rest velocities. A possible interpretation\nis a red-displaced emission component of H$\\alpha$. This however would suggest\nan extraordinarily high receding velocity of 2300$\\pm$50~km~s$^{-1}$ and should\nbe treated with great caution. The possible ejection of high velocity, compact \nmass bullets from the energy source, if true, can be partially supported by the \nclear existence of a compact knot in the highly collimated part of the jet\nas mentioned in the previous subsection, and may offer a big challenge to \ncurrently available disk-jet models. This scenario, however, seems to match well\nwith Meaburn's (2003) recent speculations on the formation of ablated jets\nin HII regions.\n\n  Spectral energy distribution (Kurucz 1991) fitting of the energy source\nbased on spectroscopic observations and data extracted from 2MASS indicated a relic\ndisk mass of only 0.006 M$_{\\odot}$, consistent with a photoablating\norigin of the system. It is noteworthy that Inverse P Cygni (IPC) profiles\nseldom appear in low Balmer\nemission lines such as H$\\alpha$.  The most well known case is the YY Orionis\ntype, weak-lined T Tauri star T Cha (Alcala, Covino \\& Franchini et al. 1993), \nfrom which, at most, a relic circumstellar\ndisk can be expected. An IPC profile in H$\\alpha$ is usually attributed to its\noptically thin nature, which normally leads to the conclusion of a high\ninclination angle of the relic disk. Furthermore, this must happen under\ntight conditions such as low-density in the residual circumstellar\nmaterial and weak H$\\alpha$ emission (Alcala, Covino \\& Franchini et al. 1993).\nBoth are consistent with properties of this jet system.\n\n\\subsection{Implications}\n\nA similar red-displaced absorption profile is also extracted \naround the power source from the spectra taken along the jet direction, \nwhich helps to elucidate the reliability of the spectral reduction and time \nelapsed mass accretion of the central YSO.  The simultaneous existence of \nsignatures of mass inflow and outflow, along with the absence of apparent \nveiling or blue excess, the absence of signatures of chromospheric \nactivity, and its weak or absent Balmer emission in combination is strong evidence \nthat the forming star is highly affected by its location in the strong UV \nradiation fields.  Indeed, Reipurth et al. (1998) state that external ionization may \nhelp on the feeding and\/or launching of the jet, which makes jet formation in such\nsystems survive. We further speculate that the aforementioned effects eventually \nchange the configuration of the stellar-disk magnetosphere \nsuch that photodissipated material in the relic disk is \neasily loaded onto the magnetic channels, especially on the side facing the\nstrong radiation fields.  It is conceivable that most or all of this material could \nhave been ejected in the form of a jet on the opposite side of the \nmass loading, instead of eventually ramming into the\ncontracting infant star embarrassed with starvation. \nIt's therefore hard for the energy source to further grow in mass, its accretion\ndisk could be dissipated on a comparatively short time scale and its evolution\ntoward the main sequence is either highly accelerated or in some cases even led \nto the formation of a failed star that would have evolved into a higher mass\nstar under normal conditions.\nMany young, very low mass stars and brown dwarfs \nare found to show evidence of mass accretion from its circumstellar environment \n(Fernandez \\& Comeron 2001; Luhman, Briceno \\& Stauffer 2003; Jayawardhana, \nMohanty \\& Basri 2003; White \\& Basri, 2003).  This study presents\nfurther implications on how incipience of \nmassive stars in giant molecular clouds inhibits further generations of low-mass \nstar formation.  It could also well serve as an inspiring case of how isolated \nsubstellar\/planetary mass objects in regions of massive star formation, especially \nthose found in HII regions (Zapatero Osorio et al. 2000), were prohibited from \ngrowing in mass, ceased their accretion process and came into being.\n\n{\\flushleft \\bf Acknowledgments~}\n\nWe are grateful to an anonymous referee for the \nmany helpful comments on the paper. Thanks prof. You-Hua Chu very much\nfor her valuable comments and suggestions.  Appreciations \nto prof. W. P. Chen and W. H. Ip for their kind accommodations and help \nduring my two years stay at NCU. This work has made use of the 2MASS database.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn recent years, significant advances have been made in learning representations of graph-structured data and predicting quantities of interest for nodes, edges or graphs themselves~\\cite{gcn,gat}. This new subfield has attracted an increasing amount of interest, leading to the development of numerous methods~\\cite{2019survey}. However, several earlier works have noted issues with existing standard benchmarks, which make it difficult to rigorously compare results and accurately distinguish between the performance of competing architectures~\\cite{oleks2018pitfalls,appnp}.\n\nOur primary focus is semi-supervised node classification: given labels of a small subset of nodes (typically 1--5\\%) and features of all nodes, as well as their connectivity information, the task is to predict all other labels. This setup is often used to assess the performance of various Graph Neural Network (GNN) architectures~\\cite{gcn,gat}. These methods are usually evaluated on three citation network datasets (Cora, CiteSeer, PubMed) introduced by Yang et al~\\yrcite{planetoid}. Unfortunately, different training splits are used across studies, which makes comparisons challenging, especially since the performance on this task is very sensitive to the choice of training split~\\cite{oleks2018pitfalls}. Furthermore, all three benchmarks are derived from the same domain, with similar structural properties. In contrast, \\textsc{Wiki-CS} has a much higher connectivity rate, hence it provides a different kind of distribution for these methods to be tested on.\n\nSimilar datasets have also been used in single-relation link prediction~\\cite{vgae,graphstar}. We further use \\textsc{Wiki-CS} to benchmark relational methods for this task, along with a non-structural SVM baseline. \n\n\\section{Related Work}\nThe most commonly used semi-supervised node classification benchmarks are the previously-described citation network graphs, proposed by Yang et al.~\\yrcite{planetoid}. Larger datasets have also been used, such as Reddit and PPI~\\cite{graphsage}. However, due to the standard split sizes proposed for these benchmarks, state-of-the-art methods have already achieved F1 scores of 0.995 and 0.97 respectively~\\cite{graphsaint}, making it difficult for further improvements to be properly gauged.\n\nDue to the issues with existing datasets, there has been significant concurrent work on establishing robust GNN benchmarks:\n\\begin{itemize}\n    \\item The Open Graph Benchmark~\\cite{open-graph-benchmark} (OGB) has recently developed a range of datasets, focusing on diversity of domains, graph sizes and types of tasks and unified evaluation methods. A Wikidata Knowledge Graph is included for a link prediction task---note that this source material is entirely different from the article hyperlink graph used for \\textsc{Wiki-CS}. OGB also proposes challenging domain-specific splits based on some aspect of the data (for exaxmple, time or molecular structure), instead of selecting this randomly.\n    \\item Dwivedi et al.~\\yrcite{dwivedi2020benchmarking} similarly proposed several datasets to rigorously distinguish the aspects of GNN architectures that significantly contribute to good performance on challenging benchmarks. To achieve this, they used largely synthetic graphs.\n\\end{itemize}  \nOur contribution complements the existing ones by providing a dataset and experimental results based on a new domain. We thus further establish the generality of GNN methods and extend the range of available benchmarks.\n\n\\section{The Dataset}\n\n\\subsection{Article Selection and Label Generation}\nWe processed Wikipedia datadumps from August 2019 to extract a subgraph where accurate class labels could be provided, based on the categories of each article. Unfortunately, these category tags were not suitable for direct use as class labels, as most of them are highly specific and inconsistently applied to a small number of pages---there were around 1.5 million different categories defined on the 6 million pages, at the time of the snapshot that we used.\n\nThis problem was mitigated by using the category sanitizer tool made available by Boldi \\& Monti~\\yrcite{category-sanitizer}, with some modifications. Their method relies on the subcategory relation to aggregate articles belonging to subcategories to their parent categories. A small set of prominent categories is selected based on harmonic centrality measures in the subcategory graph; other nodes in the subcategory graph are aggregated to one of their nearest ancestors (see Figure \\ref{fig:cs-categories} for an example subgraph). See Boldi \\& Monti~\\yrcite{category-sanitizer} for the details of the process. This avoids aggregation through many indirect steps, which would often lead to articles being mapped to categories which they have little semantic overlap with.\n\nHowever, the output still required further clean-up: some aggregated categories still contained unrelated articles. Additionally, if the subcategory graph related to some topic is very dense, the selected prominent categories and the aggregation choices can be very arbitrary.\n\n\\textsc{Wiki-CS} was created by inspecting the list of $10,000$ prominent categories selected by the sanitizer and picking a subject area with few such issues. We identified three possible candidate subjects (branches of biology, US states, branches of CS), and sampled 20 pages from every class of these candidates. Although all of these had some issues, we were able to clean up the CS data by dropping some categories and manually disabling aggregation across specific subcategories to prune bad pages from others. This resulted in a dataset with 10 classes corresponding to branches of computer science, with very high connectivity. See Appendix \\ref{app:categories} for the set of prominent categories we used for each label. Finally, we dropped the articles that would have been mapped to multiple classes.\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\centering\n\n\\includegraphics[width=0.9\\columnwidth]{cs-cat.pdf} \n\n\\caption{A subgraph of the subcategory relation graph. Nodes with dark borders are the prominent categories chosen based on centrality. The others were aggregated to the nearest marked ancestor as denoted by their colors, with ties broken arbitrarily.}\n\n\\label{fig:cs-categories}\n\\vskip -0.2in\n\\end{figure}\n\n\\subsection{Node Features}\n\nSimilarly to previous work~\\cite{planetoid}, our node features were derived from the text of the corresponding articles. However, they were calculated as the average of pre-trained GloVe word embeddings~\\cite{glove} instead of using binary bag-of-words vectors. This allowed us to encode rich features corresponding to a large vocabulary in relatively small 300-dimensional input vectors, which can be an advantage for training large models on a GPU.\n\n\\subsection{Training Splits}\n\\label{subsec:training-splits}\n\nIt has been shown that the choice of the training split can seriously affect model performance for semi-supervised node classification~\\cite{oleks2018pitfalls}. Therefore, using multiple training splits can improve the robustness of a benchmark~\\cite{appnp}. For this reason, we randomly selected 20 different training splits from the data that was not used for testing.\n\nMore specifically, we split the nodes in each class into two sets, 50\\% for the test set and 50\\% potentially visible. From the visible set, we generated 20 different splits of training, validation and early-stopping sets: 5\\% of the nodes in each class were used for training in each split, 22.5\\% were used to evaluate the early-stopping criterion, and 22.5\\% were used as the validation set for hyperparameter tuning. We stored the resulting mask vectors with the rest of the dataset, so that they can be used consistently across all future work.\n\n\\subsection{Statistics and Structural Properties}\n\n\\begin{table}[t]\n\\caption{Comparison of key dataset statistics between \\textsc{Wiki-CS} and standard citation network benchmarks. SP stands for shortest path length.}\n\\label{tab:dataset-statistics}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{l|ccc|c}\n\\toprule\n                                & \\bf{Cora} & \\bf{CiteSeer} & \\bf{PubMed} & \\bf{Wiki-CS}\\\\\n\\midrule\n         \\bf{Classes}               &   7   &       6       &       3     &     10   \\\\\n         \\bf{Nodes}                 &  2708 &       3327    &   19717     &  11701   \\\\\n         \\bf{Edges}                 &  5429 &       4732    &   44338     & 216123   \\\\\n         \\bf{Features dim.}         &  1433 &       3703    &     500     &    300   \\\\\n         \\bf{Label rate}            & 3.6\\% &       5.2\\%   &   0.3\\%     &    5\\%   \\\\\n         \\bf{Mean degree}           &  4.00 &       2.84    &   4.50      &  36.94   \\\\\n         \\bf{\\shortstack{Average SP}}  & 6.31  &       9.32    &       6.34  &    3.01    \\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nTable \\ref{tab:dataset-statistics} summarises the key statistics of the citation network and the \\textsc{Wiki-CS} datasets. Note the significantly higher rate of connectivity compared to existing benchmarks and the short average distance between any two nodes. This suggests that progress could be made on the benchmark by designing more involved computations within the neighborhood of a node, rather than focusing on long-range connections. This makes \\textsc{Wiki-CS} a useful and complementary addition to existing node classification datasets.\n\nThis connectivity also leads to more varied node neighborhoods: for each node, we calculated the proportion of neighbors that belong to the same class as the node itself, and plotted this distribution for \\textsc{Wiki-CS} as well as the existing citation network benchmarks. The results shown in Figure \\ref{fig:same-class-neighbors} show that the existing datasets have a large share of nodes in homogeneous neighborhoods, while \\textsc{Wiki-CS} is significantly more varied.\n\n\\begin{figure}\n\\vskip 0.2in\n\\centering\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/cora.png} \n\\caption{Cora}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/citeseer.png} \n\\caption{CiteSeer}\n\\end{subfigure}\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/pubmed.png} \n\\caption{PubMed}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/wiki.png} \n\\caption{Wiki-CS}\n\\label{fig:same-class-neighbors}\n\\end{subfigure}\n\n\\caption{Distribution of the ratio of neighbors belonging to the same class. In all three the citation network datasets, almost two-thirds of all nodes have all neighbors belonging to the same class. The distribution of \\textsc{Wiki-CS} is considerably more balanced.}\n\\label{fig:same-class-neighbors}\n\\end{figure}\n\n\nWe also visualized the structure of all four datasets using Deep Graph Mapper~\\cite{dgm}, an unsupervised GNN-based visualisation technique. The results shown in Figure \\ref{fig:dgm-vis} suggest that \\textsc{Wiki-CS} might have a more centralized, hierarchical structure than the citation networks, which seems plausible considering the different source domains.\n\\begin{figure}\n\\vskip 0.2in\n\\centering\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/cora.png} \n\\caption{Cora}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/citeseer.png} \n\\caption{CiteSeer}\n\\end{subfigure}\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/pubmed.png} \n\\caption{PubMed}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/wiki.png} \n\\caption{Wiki-CS}\n\\label{fig:dgm-vis-wiki-cs}\n\\end{subfigure}\n\n\\caption{Deep Graph Mapper (DGM) visualisation of benchmarks. Each node in the figure corresponds to a cluster of similar nodes in the original graph, with edge thickness representing the amount of connections between clusters. Colors represent the most frequent class in each cluster. The DGM unsupervised embedding process did not take labels into account, only relying on the node features and edges. The hyperparameters are described in Appendix \\ref{app:hyperparameters}.}\n\\label{fig:dgm-vis}\n\\vskip -0.2in\n\\end{figure}\n\n\\section{Experiments}\n\n\\subsection{Semi-Supervised Node Classification}\n\nAs described in Section \\ref{subsec:training-splits}, 20 different training splits were created for the node classification task, each consisting of 5\\% of nodes from each class. The same test set (50\\% of the nodes) was evaluated for all splits. In each split, a different 22.5\\% of nodes is used for early-stopping: we finish training when the loss calculated on this set has not improved for 100 epochs, and evaluate the model snapshot that produced the lowest loss.\n\nThis evaluation was performed 5 times on each of the 20 splits; we report the mean accuracy with a 95\\% confidence interval based on bootstrap resampling from these results with 1,000 samples.\n\nThree GNN models were evaluated: GCN~\\cite{gcn}, GAT~\\cite{gat} and APPNP~\\cite{appnp}. Hyperparameter tuning was performed using the same training setup and measuring validation performance on 22.5\\% of the nodes disjoint from the training and early-stopping sets. For efficiency, only the first 10 (out of 20) splits were used for hyperparameter tuning. The model configurations are described in Appendix \\ref{app:hyperparameters}.\n\nTwo non-structural baselines were also included: a multi-layer perceptron (MLP) and a support vector machine (SVM). These predicted the class for each node individually, based on the node features. Since SVMs are deterministic, we only had a single data point from each training split and report the mean accuracy.\n\nThe results are shown in Table \\ref{tab:node-classification}. The relative model performances align well with the results on citation network benchmarks, providing evidence that these are indeed good general-purpose methods. It is perhaps surprising that the attention mechanism of GAT improved very little on the GCN result despite the large neighborhoods---one reason might be that it is difficult to learn what to attend to in the semi-supervised setting, as discussed in-depth by Knyazev et al.~\\yrcite{attention}.\n\nThe model predictions were also visualised with Deep Graph Mapper, and are included in Appendix \\ref{app:dgm-preds}. This was based on training each model once, on the first of the 20 training splits. As expected, the mistakes and disagreements are largely located near boundaries of classes. This reinforces the idea that more complex neighborhood aggregation methods might be able to improve prediction accuracy. There are also some less connected clusters that seem to produce consistent incorrect predictions under all models---this might be due to not having good training samples in their proximity.\n\n\\begin{table}[ht]\n\\caption{Performance of semi-supervised node classification methods on the \\textsc{Wiki-CS} dataset. Accuracies are represented as the average over 100 runs, with 95\\% confidence intervals calculated by bootstrapping.}\n\\label{tab:node-classification}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c}\n\\toprule\n          & \\textbf{Accuracy} \\\\\n\\midrule\n         \\textbf{SVM} & 72.63\\%\\\\\n         \\textbf{MLP} & 73.17 $\\pm$ 0.19\\%\\\\\n\\midrule\n         \\textbf{GCN} & 79.07 $\\pm$ 0.10\\%\\\\\n         \\textbf{GAT} & 79.63 $\\pm$ 0.10\\%\\\\\n         \\textbf{APPNP}&79.84 $\\pm$ 0.10\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\subsection{Link Prediction}\n\nFor the link prediction benchmark, we followed the experimental setup of studies performing single-relation link prediction on the Cora, CiteSeer and PubMed datasets~\\cite{vgae, graphstar}. We split the data as follows: 85\\% of the real edges for training, 5\\% for validation and 10\\% for testing. For each group, the same number of negative examples (non-edge node pairs) was sampled uniformly at random. \n\nTwo GNN methods were benchmarked for link prediction: GraphStar~\\cite{graphstar} and VGAE~\\cite{vgae}. They were trained using the configurations reported in the original works, except for the hidden layer size of GraphStar: a maximum size of 256 would fit on the GPU. Details are included in Appendix \\ref{app:hyperparameters}. An MLP baseline was also trained using concatenated pairs of node feature vectors. \n\nThe results are shown in Table \\ref{tab:link-prediction}. Note the extremely high performance of all models, even the MLP baseline. It appears that randomly selected false edges are very easy to distinguish from true edges in this dataset, and harder negative samples would be needed for more meaningful benchmarking. The large number of edges aggravates this, but it is not the main cause: we performed an experiment where we trained the models on just $10000$ examples of each class, and found the metrics to be still comfortably above $0.9$. See Table \\ref{tab:lp-10k} for the results.we\n\n\\begin{table}[t]\n\\caption{Performance of link prediction methods on the \\textsc{Wiki-CS} dataset. Metrics are represented as the average over 50 runs of VGAE, 20 runs of the MLP and 10 runs of GraphStar, with 95\\% confidence intervals calculated by bootstrapping.}\n\\label{tab:link-prediction}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c c}\n\\toprule\n          & \\textbf{ROC-AUC} & \\textbf{AP}  \\\\\n\\midrule\n         \\textbf{MLP} & $0.9785 \\pm 0.0001$ & $0.9761 \\pm 0.0002$ \\\\\n\\midrule\n         \\textbf{VGAE} & $0.9553 \\pm 0.0008$ & $0.9608 \\pm 0.0007$ \\\\\n         \\textbf{GraphStar} & $0.9793 \\pm 0.0002$ & $0.9896 \\pm 0.0001$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\begin{table}[t]\n\\caption{Performance of link prediction methods trained on only $10,000$ examples of each class.}\n\\label{tab:lp-10k}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c c}\n\\toprule\n          & \\textbf{ROC-AUC} & \\textbf{AP}  \\\\\n\\midrule\n         \\textbf{MLP} & $0.9192 \\pm 0.0004$ & $0.9119 \\pm 0.0006$ \\\\\n\\midrule\n         \\textbf{VGAE} & $0.8546 \\pm 0.0024$ & $0.8427 \\pm 0.0032$ \\\\\n         \\textbf{GraphStar} & $0.9577 \\pm 0.0006$ & $0.9795 \\pm 0.0003$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\section{Conclusion}\n\nWe have presented \\textsc{Wiki-CS}, a new benchmark for GNN methods. We have described how its structural properties are significantly different from commonly used datasets. Our experiments show existing GNN architectures for semi-supervised node classification and link prediction performing similarly to their results on other benchmarks, which is further evidence that they are good general-purpose methods for graph-learning tasks. Our dataset is available for further study, broadening the range of available benchmarks.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nSpacetime singularities are among the most spectacular predictions of classical general relativity.\nIn the context of cosmology, the presence of an initial singularity signals a fundamental incompleteness in our understanding of the Universe, whose origin and beginning can not be explained within a classical or semiclassical treatment.\nThe singularity theorems of general relativity require energy conditions which may be violated in the early Universe if inflation was present, but one can nevertheless show that inflationary spacetimes are past incomplete \\cite{BGV03}.\n\\footnote{See e.g.\\ \\cite{YQ18} for an analysis of conditions under which continuous or differentiable extensions of the spacetime metric beyond the past incomplete region may exist for such inflationary spacetimes.}\nIt is widely expected that a quantum theory of gravity is required to resolve the classical Big Bang singularity and give a fundamental basis to theoretical cosmology; a full quantum treatment of spacetime may indeed be needed to justify the assumption that the early Universe can be studied in terms of quantum fields on a classical background \\cite{FLT17,TFLT19}.\n\nOn the other hand, the spacetime geometry of the early Universe was presumably very simple, describable in terms of a homogeneous isotropic background with only small perturbations.\nThis simplicity must ultimately be explained more fundamentally, but it also results in significant practical simplifications: at least as a first step, one can study simple homogeneous, isotropic spacetimes in quantum gravity to learn about singularity resolution.\nOne could hope that, as in the classical theory, this does not require understanding the nonlinear and presumably complicated dynamics of full quantum gravity.\nThis philosophy, systematically applied to \\ac{lqg} as a candidate theory of quantum gravity, gave birth to the field of \\ac{lqc} where, for models of quantum gravity coupled to a massless scalar field, the classical Big Bang singularity is indeed resolved and replaced by a `big bounce' \\cite{Boj01,AS11}.\nMore recently \\ac{lqc} has made direct contact with inflation, providing a quantum-gravitational extension of the usual semiclassical framework \\cite{AAN12}.\nThe precise relation between \\ac{lqg} and \\ac{lqc} has been the focus of much work in recent years \\cite{AC15,ADLP18,DLP19}.\n\nThe origin of singularity resolution in \\ac{lqc} is the fundamental discreteness of the theory, manifest in discrete spectra for areas and volumes with a gap away from zero \\cite{Rov04,BH11}.\nThis key feature of the \\ac{lqg} kinematics is shared by its reformulation in terms of \\ac{gft} \\cite{Ori17}.\n\\ac{gft} interprets the {\\em quanta of spacetime} seen in \\ac{lqg} as excitations of a quantum field (also then a quantum field {\\em of}, not {\\em on} spacetime), while the dynamics of a \\ac{gft} are generally defined such that its perturbative expansion corresponds to a sum over {\\em spin foams}, or discrete spacetime histories of \\ac{lqg} states \\cite{RR01}.\nThe continuum limit of this sum, needed to obtain continuum  quantum geometry, is to be taken in a way similar to matrix and tensor models \\cite{Ori12}.\nGiven the very close relation of \\ac{gft} to \\ac{lqg}, it is natural to ask whether cosmological models of \\ac{gft} dynamics can resolve the classical Big Bang singularity in a way similar to what is seen in \\ac{lqc}.\n\nThe key idea that led to the derivation of cosmological models in the \\ac{gft} approach was that a spatially homogeneous quantum geometry could be understood as a type of Bose--Einstein condensate in \\ac{gft} \\cite{GOS14,GS16,Ori17a,PS19}.\nAs in other quantum field theories, such a condensate can be understood as a nonperturbative vacuum of the theory, characterised by a common quantum state for a very large number of quanta with respect to the original Fock vacuum.\nThe idea that spacetime could be a kind of Bose--Einstein condensate of geometric quanta had been formulated in other approaches before \\cite{Hu05,KS12}, but the quantum field theory framework of \\ac{gft} allows studying such a condensate with relatively conventional methods, adapted to a background-independent quantum gravity context.\nIn the simplest (mean-field) approximation, the equation of motion of the condensate mean field is the analogue of the usual Gross--Pitaevskii equation in condensed matter physics.\nFrom a solution to this equation of motion, one can compute geometric observables such as the total volume of the condensate.\nDynamics are introduced, just as in \\ac{lqc} and many other models of quantum cosmology \\cite{BI75}, by coupling a relational matter clock, given by a free massless scalar field.\nConcretely, one extracts equations for the relational volume observable $V(\\phi)$, the three-dimensional volume of a condensate given a particular value of the relational clock field, and its derivatives.\nThese are then interpreted as effective Friedmann equations derived from the \\ac{gft} condensate dynamics.\nThese steps were first fully implemented in \\cite{OSW16,OSW17} where it was shown how such effective Friedmann equations, for a wide class of \\ac{gft} models and under various simplifying assumptions, are consistent with the classical Friedmann equations at large volume while showing a bouncing behaviour at high densities very similar to the one in \\ac{lqc}.\nIn particular, these effective Friedmann equations can reproduce the preferred `improved dynamics' form of \\ac{lqc} \\cite{APS06} whose derivation from Hamiltonian formulations of \\ac{lqg} is a largely outstanding challenge \\cite{DLP19}.\n\nThese very promising results for effective cosmological dynamics from \\ac{gft} relied on assuming the emergence of a condensate regime in which the mean-field approximation is valid and \\ac{gft} interactions are subdominant with respect to the quadratic (kinetic) term.\nIncluding interactions leads to interesting modifications to effective cosmological equations, which can serve as a starting point for \\ac{gft} phenomenology \\cite{CPS16,PS17}.\nTo better understand the dependence of \\ac{gft} cosmology on a mean-field approximation, a simple toy model was then studied in \\cite{AGW18}.\nIn this model, only a single mode of a \\ac{gft} field is excited and a {\\em squeezing} Hamiltonian generates evolution in relational matter time $\\phi$, so that a squeezed state emerges dynamically even from the Fock vacuum without assuming a mean-field regime.\nThe general features of a bounce at high densities and agreement with classical cosmology at large volume could be reproduced in this simpler setting.\nThe choice of Hamiltonian was rather ad-hoc, motivated by agreement with classical cosmology and the properties of squeezed states.\nIt was then shown in \\cite{WEw19} that a Legendre transformation of the free \\ac{gft} action, taking again the matter clock $\\phi$ to define time, leads essentially to a squeezing Hamiltonian, the latter representing the dynamics of an `upside-down' harmonic oscillator with negative quadratic potential.\nThe results of \\cite{WEw19} hence explained the agreement between the effective cosmological dynamics of a squeezing Hamiltonian and those of previous results for \\ac{gft} condensates.\nA different argument explaining the close connection of \\ac{gft} cosmology to \\ac{lqc} was given in \\cite{BBC19} where it was argued that the canonical \\ac{lqc} framework and the field-theoretic (bosonic) \\ac{gft} cosmology can be seen as different realisations of the same $\\liealg{su}(1,1)$ algebra.\n\nThe aim of this paper is to extend many of these recent results to further clarify the connection between fundamental \\ac{gft} dynamics and effective cosmological equations.\nRather than using mean-field approximations, we derive general dynamical equations for operators and therefore expectation values of these operators; we work similarly to the `toy model' analysis of \\cite{AGW18} but consider a more general form of the \\ac{gft} dynamics.\nIn particular, we will go beyond quadratic Hamiltonians and add simple interactions, allowing us to connect to results such as those of \\cite{CPS16} for \\ac{gft} interactions which were obtained in a mean-field approximation.\nWe will also extend the algebraic viewpoint of \\cite{BBC19} to the case of an interacting Hamiltonian.\nAn issue in the derivation of effective dynamical equations that we will encounter is that, in the general case, these equations involve additional expectation values or higher moments that are not directly identifiable with the variables of a classical \\ac{flrw} cosmology (which is fully characterised by the scale factor or volume and the energy density in the massless scalar field).\nIn other words, dynamical equations for quantum expectation values require knowledge of additional initial conditions compared to a classical cosmology.\nThis is of course the usual situation for effective equations for quantum systems.\n(See e.g.~\\cite{BHT11} for a systematic discussion.)\nChoosing a specific class of initial conditions, for instance by focussing on a class of coherent states, then simplifies these equations.\nWe will illustrate this trade-off between obtaining effective equations that are as general as possible but include a dependence on additional variables, and more specific equations in which a particular choice of state (or family of states) allows deriving equations with more direct (semiclassical) physical interpretation.\nAs concrete examples, we will discuss Fock coherent states which have been studied in most of the existing literature on \\ac{gft} cosmology, but also coherent states based on the $\\liealg{su}(1,1)$ algebra satisfied by basic \\ac{gft} operators (in particular the well-known \\ac{pg} coherent states).\n\nWe show a key property of simple Fock coherent states, which is that relative uncertainties of quantities like volume and energy can be made arbitrarily small at late times, so that these states are as semiclassical as desired.\nThis gives further justification to their interpretation as semiclassical macroscopic geometries \\cite{GOS14}.\n\\acl{pg} states that can be thought of as elements of a \\ac{gft}-like Fock space do not admit such a semiclassical interpretation and are therefore disfavoured for \\ac{gft} cosmology.\n \n\\section{Group field theory cosmology}\n\\label{sec:gft_cosmo}\nIn this section we summarise previous work on the derivation of effective cosmological equations from group field theory, in particular the previous papers we are building on in this work.\nFor more details and background we point to \\cite{GOS14,GS16,Ori17a,PS19}.\n\n\\subsection{The group field theory approach to quantum gravity}\n\\Acfp{gft} are a nonperturbative approach to quantum gravity, aiming to extend the successes of matrix and tensor models to a theory of quantum geometry in higher (in particular four) dimensions by incorporating the kinematical and dynamical structure of loop quantum gravity and spin foams \\cite{Ori17,Ori12,Ori06,Fre05,Kra11}.\n\nConcretely, rather than being based on a matrix or tensor with a number of discrete indices of finite range, a \\ac{gft} is defined in terms of a field $\\varphi$ depending on a number of continuous variables taking values in a Lie group.\nIn this sense, the purely combinatorial structure of matrix models is enriched by additional group-valued degrees of freedom, interpreted as parallel transports akin to the fundamental variables in lattice gauge theory.\nNevertheless, the main ideas are similar; a \\ac{gft} perturbative expansion should generate a sum over quantum geometries and admit a consistent continuum limit.\n\nThe prototype for \\ac{gft} as an approach to quantum gravity is the {\\em Boulatov model} in three dimensions \\cite{Bou92}.\nOne defines a real field\n\\begin{equation}\n  \\varphi:\n  G^3 \\rightarrow \\field{R}\n  \\mathcomma\n  \\quad\n  \\varphi(g_1,g_2,g_3)\n  = \\varphi(g_2,g_3,g_1)\n  = \\varphi(g_3,g_1,g_2)\n\\end{equation}\nwith an action\n\\begin{equation}\n\\eqalign{\n  S[\\varphi]\n  =\n&\n  \\frac{1}{2}\\int\n  \\intmeasure[3]{g} \\,\n  \\varphi(g_1,g_2,g_3)\n  \\varphi(g_1,g_2,g_3)\n\\\\\n&\n  -\\frac{\\lambda}{4!}\\int\n  \\intmeasure[6]{g} \\,\n  \\varphi(g_1,g_2,g_3)\\varphi(g_1,g_4,g_5)\\varphi(g_2,g_5,g_6)\\varphi(g_3,g_6,g_4)\n  \\mathcomma\n}\n\\end{equation}\nwhere $G$ is a Lie group and $\\intmeasure{g}$ is the Haar measure on $G$.\nNotice the `nonlocal' combinatorial pairing of field arguments in the interaction term which is the generalisation of trace invariants such as ${\\rm tr}M^n$ in the case of a matrix model.\nOne can then show that, for $G=\\liegroup{SU}(2)$, the \\ac{gft} partition function admits a perturbative expansion of the form\n\\begin{equation}\n  \\label{eq:gft_boulatov_exp}\n  \\int \\mathcal{D}\\varphi\\;e^{-S[\\varphi]}=\\sum_{C}\n  \\lambda^{N_{T}(C)}\\sum_{\\{j_l\\}\\in{\\rm Irrep}}\\prod_{l\\in C} (2j_l+1)\\sum_{T\\in\n  C}\\left\\{ \\matrix{\n    j_{T_1} & j_{T_2} & j_{T_3}\n    \\cr\n    j_{T_4} & j_{T_5} & j_{T_6}\n  }\\right\\}\n\\end{equation}\nwhere the first sum is over all oriented three-dimensional simplicial complexes $C$, $N_T(C)$ is the number of tetrahedra in $C$, $j_l$ is an irreducible representation of ${\\liegroup{SU}(2)}$ assigned to each link $l\\in C$, and $\\{\\cdot\\}$ is the Wigner $6j$-symbol associated to a tetrahedron $T\\in C$ (involving its six links).\nUp to the factor $\\lambda^{N_{T}(C)}$, each complex $C$ appearing in \\eref{eq:gft_boulatov_exp} is weighted by its {\\em Ponzano--Regge state sum} \\cite{PR68}, a possible definition of discrete three-dimensional quantum gravity (see e.g.~\\cite{BN09}).\nIn this sense, the perturbative expansion of the Boulatov model generates all possible triangulations (including all topologies) each weighted by a partition function for quantum gravity on this triangulation.\nThis expansion is highly divergent without further regularisation \\cite{BN09}.\nThe \\ac{gft} programme aims to extend \\eref{eq:gft_boulatov_exp} to more complicated models, in particular candidates for quantum gravity in four dimensions, where the Ponzano--Regge state sum is replaced by a general {\\em spin foam amplitude} \\cite{Rov04}: the amplitudes of the Barrett--Crane model \\cite{BC98} can be obtained as Feynman amplitudes of a \\ac{gft} defined on the three-sphere $S^3=\\liegroup{SO}(4)\/\\liegroup{SO}(3)$ \\cite{PFKR00} and this extends to a one-to-one correspondence between general spin foam amplitudes and their realisations as the perturbative expansion of a \\ac{gft} \\cite{RR01}.\nThis correspondence extends from spin foam models for Euclidean quantum gravity to models with Lorentzian signature such as \\cite{BC00} which can be defined through a \\ac{gft} on a noncompact group such as $\\liegroup{SO}(3,1)$ (see e.g.~\\cite{LO07}).\nIn this sense one could say that a \\ac{gft} defines a completion of the spin foam programme in that it not only generates spin foam amplitudes for quantum gravity on a given discretisation, but also the weights in a sum over discretisations.\n\n\\subsection{Cosmology from group field theory}\nFor general \\ac{gft} models for quantum gravity, it is difficult to make sense of a perturbative expansion of the form \\eref{eq:gft_boulatov_exp}.\nThe number of terms quickly grows out of control as the number of building blocks is increased and there is no obvious physical meaning to truncating such an expansion to the first few terms, i.e. to discretisations with very few building blocks.\n\\Eref{eq:gft_boulatov_exp} is really an expansion around a `no-space' vacuum in which no geometry is present at all.\n\nHowever, there is more to quantum field theory than a perturbative expansion\naround vanishing field value: interacting field theories often exhibit phase transitions to a {\\em condensate} characterised by a nonvanishing field expectation value.\nWith respect to the original Fock vacuum in which the field vanishes, a condensate has a very large number of quanta all characterised by a single quantum state (the `condensate wavefunction').\nThis is a quantum state of high symmetry and quantum coherence.\nThe key idea of {\\em \\ac{gft} condensates} is that such a configuration in \\ac{gft} is a candidate for a macroscopic, nearly homogeneous Universe, and hence  a starting point for effective cosmology.\nWe refer the reader to \\cite{GOS14} for details and arguments for this geometric interpretation.\n\nWe generally define a \\ac{gft} field for a four-dimensional quantum gravity model coupled to scalar matter by\n\\begin{equation}\n  \\varphi:\n  G^4\\times \\field{R} \\rightarrow \\field{K}\n  \\mathcomma\n  \\quad\\varphi(g_1,\\ldots,g_4,\\phi)=\\varphi(g_1h,\\ldots,g_4h,\\phi)\\;\\forall h\\in G\n  \\mathperiod\n\\end{equation}\nwhere $G$ is the gauge group of gravity (often assumed to be $\\liegroup{SU}(2)$) and $\\field{K}$ is either the real or complex numbers.\nThe action takes the general form\n\\begin{equation}\n  S\n  =\n  \\int\n  \\intmeasure[4]{g}\\,\n  \\intmeasure{\\phi}\\,\n  \\bar\\varphi(g_I,\\phi)\\,\\mathcal{K}\\varphi(g_I,\\phi)\n  + \\mathcal{V}[\\varphi]\n\\end{equation}\nwhere for a real field $\\bar\\varphi=\\varphi$ (and, to obtain the usual normalisation of a kinetic term, one has to also rescale the field), $\\mathcal{K}$ is a kinetic operator which in general contains derivatives with respect to all arguments, and all terms that  are higher order than quadratic are part of $\\mathcal{V}[\\varphi]$.\nIn general, $\\mathcal{V}[\\varphi]$ is also nonlocal in a way similar to the Boulatov \\ac{gft} defined above.\nIn its Feynman expansion, such a field theory will generate graphs whose edges are labelled by $g_I$ (interpreted as parallel transports of a $G$-connection) and whose vertices are labelled by $\\phi$ interpreted as the values of a matter scalar field.\n\nIf we denote the expectation value of the field operator by\n\\begin{equation}\n  \\expval{\\op{\\varphi}(g_I, \\phi)}\n  =\n  \\sigma(g_I,\\phi)\n  \\mathcomma\n\\end{equation}\na condensate phase is then characterised by a nonvanishing $\\sigma(g_I,\\phi)$.\n\nThe {\\em mean-field approximation} which is used in most work on \\ac{gft} cosmology so far requires that the mean field $\\sigma(g_I,\\phi)$, for a \\ac{gft} with complex field, satisfies the classical \\ac{gft} equation of motion\n\\begin{equation}\n  \\label{eq:gft_cosmo_meanfield_eom}\n  \\mathcal{K}\\sigma(g_I,\\phi)\n  +\n  \\frac{\n    \\delta\\mathcal{V}[\\sigma]\n  }{\n    \\delta\\bar\\sigma(g_I,\\phi)\n  }\n  =\n  0\n  \\mathcomma\n\\end{equation}\nthe analogue of the Gross--Pitaevskii equation for the condensate wavefunction in condensed matter physics.\nFrom a solution $\\sigma(g_I,\\phi)$ to the equation of motion, one can then extract an observable corresponding to the total condensate volume as a function of the matter field `clock',\n\\begin{equation}\n  \\label{eq:gft_cosmo_gft_volumeop}\n  \\expval{\\op{V}(\\phi)}\n  \\equiv\n  \\int\n  \\intmeasure[4]{g}\\,\n  \\intmeasure[4]{g'}\\,\n  V(g_I,g'_I)\n  \\bar\\sigma(g_I,\\phi)\n  \\sigma(g'_I,\\phi)\n  \\mathcomma\n\\end{equation}\nwhere $V(g_I,g'_I)$ are matrix elements of the \\ac{gft} volume operator between `single-particle' states $\\ket{g_I}$ and $\\ket{g'_I}$; such an operator can be defined from the action of a volume operator in \\ac{lqg} on open spin networks with a single vertex and four links.\nDynamical equations satisfied by $\\expval{\\op{V}(\\phi)}$ and its derivatives with respect to $\\phi$ are then interpreted as effective cosmological (Friedmann) equations for the three-volume of (a patch of) the Universe, derived directly from a prescription for the microscopic dynamics of a \\ac{gft}.\n\nThe most concrete derivation of this type, for models of quantum gravity coupled to massless scalar matter, was given in \\cite{OSW16}.\nFirst the nonlinear, nonlocal equation of motion \\eref{eq:gft_cosmo_meanfield_eom} was simplified by making an `isotropic' ansatz\n\\begin{equation}\n  \\sigma(g_I,\\phi)\n  =\n  \\sum_{j\\in{\\rm Irrep}}\n  \\sigma_j(\\phi){\\bf D}^j(g_I)\n  \\mathcomma\n\\end{equation}\nwhere the \\ac{gft} gauge group is taken to be $\\liegroup{SU}(2)$ and ${\\bf D}^j(g_I)$ is a fixed convolution of four Wigner $D$-matrices for the irreducible representation $j$, encoding the `shape' of the \\ac{gft} building blocks.\n(${\\bf D}^j(g_I)$ requires a choice of intertwiner $j\\otimes j\\otimes j\\otimes j\\rightarrow {\\bf 0}$; in \\cite{OSW16} this is taken to be the intertwiner with maximum eigenvalue for the volume, see \\eref{eq:gft_volume_expv}.)\n\nAssuming a quintic potential as is done for many spin foam models related to \\ac{lqg}, this reduces \\eref{eq:gft_cosmo_meanfield_eom} to a decoupled form\n\\begin{equation}\n  \\label{eq:gft_cosmo_isotropic_eom}\n  A_j\n  \\partial_\\phi^2\\sigma_j(\\phi)\n  - B_j\\sigma_j(\\phi)\n  + w_j\\bar\\sigma_j(\\phi)^4\n  =\n  0\n  \\mathcomma\n\\end{equation}\nwhere $A_j$, $B_j$ and $w_j$ are determined by the couplings in the \\ac{gft} action.\nSince the volume operator is diagonal when written in terms of $\\liegroup{SU}(2)$ representations, the volume of a condensate in such a state is given by\n\\begin{equation}\n  \\expval{\n    \\op{V}(\\phi)\n  }\n  =\n  \\sum_{j\\in{\\rm Irrep}}\n  V_j\\,|\\sigma_j(\\phi)|^2\n\\label{eq:gft_volume_expv}\n\\end{equation}\nwhere $V_j$ is the volume eigenvalue assigned to the spin $j$ (which in general depends on the intertwiner used to define ${\\bf D}^j(g_I)$).\nIn a regime in which interactions can be neglected, and assuming that the ratios $B_j\/A_j$ take a positive maximum for some $j=j_0$, it is easy to see that for almost any solution to \\eref{eq:gft_cosmo_isotropic_eom}\\footnote{The only cases for which $V(\\phi)$ does not have the given asymptotics are solutions for which one only uses the exponentially growing or the exponentially decaying solution to \\eref{eq:gft_cosmo_isotropic_eom} for $j=j_0$.} the volume $V(\\phi) \\equiv \\expval{\\op{V}(\\phi)}$ satisfies\n\\begin{equation}\n  V(\\phi)\n  \\stackrel{\\phi\\rightarrow -\\infty}{\\sim}\n  c_1 \\exp\\left(-2\\sqrt{\\frac{B_{j_0}}{A_{j_0}}}\\phi\\right)\n  \\mathcomma\n  \\quad\n  V(\\phi)\n  \\stackrel{\\phi\\rightarrow +\\infty}{\\sim}\n  c_2 \\exp\\left(2\\sqrt{\\frac{B_{j_0}}{A_{j_0}}}\\phi\\right)\n\\end{equation}\nfor some constants $c_1$ and $c_2$ \\cite{Gie16}.\nMoreover, $V(\\phi)$ can only ever reach zero for very special initial conditions (although this case becomes generic if the \\ac{gft} field is taken to be real-valued \\cite{PS17}).\nWith the identification $\\frac{B_{j_0}}{A_{j_0}}=:3\\pi G$, this corresponds to a bounce solution interpolating between the expanding and contracting solutions to the {\\em classical} Friedmann equations for a flat \\ac{flrw} Universe filled with a massless scalar field, $V(\\phi)=V_0\\exp(\\pm\\sqrt{12 \\pi G}\\phi)$.\nSimilar conclusions apply if one considers condensates only formed by a single $j$ component, again denoted by $j_0$.\nIn the latter case, one can show that the volume satisfies an effective Friedmann equation \\cite{OSW16}\n\\begin{equation}\n  \\label{eq:gft_cosmo_friedmann}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  =\n  12\\pi G\n  \\left(\n    1-\\frac{\\rho(\\phi)}{\\rho_\\critical}\n  \\right)\n  + \\frac{4V_{j_0}E}{V(\\phi)}\n\\end{equation}\nwhere $\\rho=\\pi_\\phi^2\/(2V^2)$, with $\\pi_\\phi$ the conserved momentum conjugate to $\\phi$, is the energy density of the massless scalar field, and $\\rho_\\critical$ is a maximal (critical) energy density similar to the one found in \\ac{lqc} \\cite{Boj01,AS11} (and we have again set $\\frac{B_{j_0}}{A_{j_0}}=:3\\pi G$).\nThe last term, involving a conserved quantity $E$ (`\\ac{gft} energy'), represents a slight modification with respect to the usual \\ac{lqc} effective dynamics.\nAgain, clearly at large volumes or late times such effective dynamics reduce to the classical Friedmann equation $(V'\/V)^2=12\\pi G$.\n\nIn this article, we strengthen the foundations of these past results.\nWe aim to obtain effective cosmological dynamics from \\ac{gft} without several assumptions that were necessary to obtain \\eref{eq:gft_cosmo_friedmann}, namely: the validity of a mean-field regime in which one solves equations for the mean field; restriction of the effective equations to simple expectation values such as $\\expval{\\op{V}(\\phi)}$ without taking into account fluctuations around these expectation values; neglecting \\ac{gft} interactions by effectively setting $w_j=0$.\n\\footnote{The work of \\cite{CPS16} included \\ac{gft} interactions into the analysis, leading to additional terms on the right-hand side of \\eref{eq:gft_cosmo_friedmann}, while working in a mean-field regime.}\nIndirectly, the results outlined so far also assumed a given Fock space structure used to define a \\ac{gft} volume operator, which has not been derived from the canonical analysis of a \\ac{gft} action.\n\n\\subsection{Toy model for group field cosmology, and a Hamiltonian for \\ac{gft}}\nA first step towards deriving effective cosmological dynamics from \\ac{gft} outside of a mean-field regime was taken in \\cite{AGW18}.\nOne motivation for this work was to develop a toy model in which some of the successes of \\ac{gft} cosmology could be obtained in a simpler setting, but there was also a new technical assumption: the massless scalar field $\\phi$ was proposed as a (relational) time variable, with a Hamiltonian generating evolution with respect to this clock.\nThat is, the idea was to define a {\\em deparametrised} setting in which some degrees of freedom serve as coordinates parametrising the remaining ones, a strategy widely employed in canonical quantum gravity \\cite{BK95,DGKL10}.\\footnote{This strategy can be extended to a \\ac{gft} coupled to four massless scalar fields serving as relational coordinates for both time and space \\cite{Gie18}.}\nThis approach was different from previous work on \\ac{gft} cosmology in which the fundamental \\ac{gft} formalism treated all arguments of the field on the same footing.\nThe Hamiltonian itself was chosen so as to reproduce the correct cosmological dynamics at large volume.\n\nClassical \\ac{flrw} cosmology can be defined by a volume variable $V(\\phi)$ and conjugate momentum $p_V(\\phi)$ subject to a Hamiltonian\n\\begin{equation}\n  \\label{eq:gft_cosmo_dilat}\n  \\mathcal{H}\n  =\n  \\sqrt{12 \\pi G}\\,Vp_V\n  \\mathcomma\n\\end{equation}\ngenerating a dilatation as its time evolution, i.e.~the exponential solutions in $\\phi$ mentioned above.\nIn \\cite{AGW18} it was then observed that, for a Fock space generated by annihilation operators $\\op{A}^i$ and creation operators $\\hermconj{\\op{A}}_j$ (here $i,j$ run from 1 to 5) with algebra\n\\begin{equation}\n  \\commutator{\n    \\op{A}^i\n  }{\n    \\hermconj{\\op{A}}_j\n  }\n  =\n  \\delta^i_j\n  \\mathcomma\n\\end{equation}\na discrete analogue of the dilatation operator is given by a {\\em squeezing Hamiltonian}\n\\begin{equation}\n  \\label{eq:gft_cosmo_squeezing}\n  \\op{\\mathcal{H}}\n  =\n  \\frac{\\mathup{i}}{2}\n  \\lambda\n  \\left(\n    \\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j \\epsilon^{ij}\n    - \\op{A}^i \\op{A}^j \\epsilon_{ij}\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere $\\epsilon^{ij}$ is an appropriate symmetric tensor.\nIndeed, for a volume operator taken to be the multiple of the number operator\n\\begin{equation}\n  \\label{eq:gft_cosmo_volume_via_number_op}\n  \\op{V}\n  =\n  v_0 \\op{N}\n  :=\n  v_0 \\hermconj{\\op{A}}_i \\op{A}^i\n\\end{equation}\none can show that, for suitable states characterised by the eigenvalues of $\\op{V}$, $\\op{\\mathcal{H}}$ acts as\n\\begin{equation}\n  (\\op{\\mathcal{H} }\\Psi)(V)\n  \\stackrel{v_0\\rightarrow 0}{\\rightarrow}\n  - \\mathup{i}\\lambda\\left(V\\frac{\\partial}{\\partial V}\n  + \\frac{\\partial}{\\partial V}V\\right)\\Psi(V)\n  \\mathperiod\n\\end{equation}\nThus, with the identification $\\lambda:=\\sqrt{3\\pi G}$ the continuum limit of  squeezing \\eref{eq:gft_cosmo_squeezing} is compatible with the classical dilatation Hamiltonian \\eref{eq:gft_cosmo_dilat}.\nThe picture of a Fock space of `quanta of geometry' in which each quantum carries a given volume mimics the Fock space structure of \\ac{gft}, with the simplification that here each quantum comes with a fixed $v_0$ rather than a general state-dependent volume as in \\eref{eq:gft_cosmo_gft_volumeop}.\n\nGiven that the Hamiltonian is quadratic, expressions for the time evolution of observables of interest can be computed analytically.\nOne finds that\n\\begin{equation}\n  \\label{eq:gft_cosmo_toymodel_n}\n  \\expval{\\op{N}(\\phi)}\n  =\n  - \\frac{5}{2}\n  +\n  \\left(\n    N_0 + \\frac{5}{2}\n  \\right)\n  \\cosh(2\\lambda\\phi)\n  + Q \\sinh(2\\lambda\\phi)\n\\end{equation}\nwith\n\\begin{equation}\n  N_0\n  :=\n  \\left.\n    \\expval{\\op{N}}\n  \\right|_{\\phi=0}\n  \\mathcomma\n  \\quad\n  Q\n  :=\n  \\frac{1}{2}\n  \\left.\n    \\left(\n      \\epsilon^{ij} \\expval{\\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j}\n      + \\epsilon_{ij} \\expval{\\op{A}^i \\op{A}^j}\n    \\right)\n  \\right|_{\\phi=0}\n\\end{equation}\n(computed equivalently in the Schr\u00f6dinger or Heisenberg picture).\nAt late or early times $\\phi\\rightarrow\\pm\\infty$ (and with $\\lambda:=\\sqrt{3\\pi G}$), the expectation value $V(\\phi)\\equiv\\expval{\\op{V}(\\phi)}$ then again reproduces the classical solution $V(\\phi)=V_0\\exp(\\pm\\sqrt{12\\pi G}\\phi)$\\,.\nMoreover, one can show that only special initial conditions (such as choosing the Fock vacuum as initial state) lead to a solution that ever encounters a singularity where $V(\\phi_0)=0$ for some $\\phi_0$.\nGeneric solutions avoid the classical singularity and describe a bounce connecting the classical expanding and contracting branches.\n\nThe quantity $Q$ cannot be directly interpreted in terms of the volume or energy density of the corresponding classical cosmology; its presence leads to an asymmetry in the solution \\eref{eq:gft_cosmo_toymodel_n}. For simplicity, the further analysis of \\cite{AGW18} only considered the case $Q=0$, for which one obtains the effective Friedmann equation\n\\begin{equation}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  =\n  4\\lambda^2\n  \\left(\n    1\n    + \\frac{5v_0}{V(\\phi)}\n    - \\frac{N_0(N_0+5)v_0^2}{V(\\phi)^2}\n  \\right)\n  \\mathperiod\n\\end{equation}\nThe similarity to the effective Friedmann equation \\eref{eq:gft_cosmo_friedmann} of \\ac{gft} in the mean-field setting is apparent.\nIn this sense, the toy model based on a squeezing Hamiltonian \\eref{eq:gft_cosmo_squeezing} already reproduced several previous results in \\ac{gft} cosmology.\n\nOne reason for this close connection was uncovered in \\cite{WEw19,GPW19} where, taking again a deparametrised viewpoint, a Hamiltonian formalism was derived from a Legendre transformation of the full (free) \\ac{gft} action in which the `matter' argument $\\phi$ of the \\ac{gft} field is taken as a time coordinate.\nThe starting point is an action for real \\ac{gft} fields of the form\n\\begin{equation}\n  \\label{eq:gft_cosmo_gft_hamiltonian_action}\n  S\n  =\n  \\frac{1}{2}\n  \\int\n  \\intmeasure{\\phi}\n  \\sum_{\\vec{\\jmath},\\vec{m},\\iota}\n  \\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}(\\phi)\n  \\left[\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    + \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\\partial_\\phi^2\n  \\right]\n  \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\n  + \\mathcal{V}[\\varphi]\n  \\mathcomma\n\\end{equation}\nwhere the field $\\varphi(g_I,\\phi)$ has been decomposed into Peter--Weyl modes according to\n\\begin{equation}\n  \\varphi(g_I,\\phi)\n  =\n  \\sum_{j_I\\in{\\rm Irrep}}\n  \\sum_{m_I,n_I=-j_I}^{j_I}\n  \\sum_{\\iota}\\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\\,\n  \\intertwiner^{\\vec{\\jmath},\\iota}_{\\vec{n}}\n  \\prod_{I=1}^4\n  \\sqrt{2j_I+1}\\,\n  D^{j_I}_{m_I n_I}(g_I)\n\\end{equation}\nand $\\iota$ labels a basis of intertwiners $\\intertwiner$ for the representation labels $\\{j_I\\}$ (and the sum over $(\\vec{\\jmath},\\vec{m},\\iota)$ in \\eref{eq:gft_cosmo_gft_hamiltonian_action} is a shorthand for the sums appearing in this decomposition).\nFor a real field, these Peter--Weyl coefficients satisfy the reality condition\n\\begin{equation}\n  \\compconj*{\n    \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\n  }\n  =\n  (-1)^{\\sum_I (j_I-m_I)}\\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}(\\phi)\n  \\mathperiod\n\\end{equation}\nThe Hamiltonian can then be written as\n\\begin{equation}\n  \\mathcal{H}\n  =\n  - \\frac{1}{2}\n  \\sum_{\\vec{\\jmath},\\vec{m},\\iota}\n  \\left[\n    \\frac{\n      \\pi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\\pi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n    }{\n      \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\n    }\n    +\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n  \\right]\n  - \\mathcal{V}[\\varphi]\n\\end{equation}\nwhose free part corresponds to either a harmonic oscillator or an `upside down' harmonic oscillator for each mode, depending on the signs of the couplings $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}$ and $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}$.\nAnnihilation and creation operators can then be defined by\n\\begin{eqnarray}\n  \\label{eq:gft_cosmo_annihilation}\n  \\op{a}_{\\vec{\\jmath},\\vec{m},\\iota}\n  & = &\n  \\frac{1}{\n    \\sqrt{\n      2\n      \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n      \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    }\n  }\n  \\left(\n    \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n    \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\op\\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    +\\mathup{i} (-1)^{\\sum_I (j_I-m_I)}\\op\\pi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n  \\right)\n  \\\\\n  \\label{eq:gft_cosmo_creation}\n  \\hermconj{\\op{a}}_{\\vec{\\jmath},\\vec{m},\\iota}\n  & = &\n  \\frac{1}{\n    \\sqrt{\n      2\n      \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n      \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    }\n  }\n  \\left(\n    (-1)^{\\sum_I (j_I-m_I)}\n    \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n    \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\op\\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n    -\\mathup{i} \\op\\pi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n  \\right)\n  \\mathcomma\n\\end{eqnarray}\nwhere\n$\n\\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n=\n\\sqrt{\n  \\abs{\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    \/ \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\n  }\n}\n$.\nThe free Hamiltonian is then $\\op{\\mathcal{H}} = \\sum_{\\vec{\\jmath},\\vec{m},\\iota} \\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}$ written as a sum of single-mode Hamiltonians $\\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}$.\nFor a mode for which $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}$ and $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}$ have different signs the single-mode Hamiltonian is given by\n\\begin{equation}\n  \\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}\n  =\n  -\\frac{1}{2}\n  {\\rm sgn}(\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)})\n  \\omega_{\\vec{m}}^{\\vec{j}, \\iota}\n  \\left(\n    \\hermconj{\\op{a}}_{\\vec{\\jmath},\\vec{m},\\iota}\n    \\hermconj{\\op{a}}_{\\vec{\\jmath},-\\vec{m},\\iota}\n    + \\op{a}_{\\vec{\\jmath},\\vec{m},\\iota}\n    \\op{a}_{\\vec{\\jmath},-\\vec{m},\\iota}\n  \\right)\n\\end{equation}\nwhich is analogous to the squeezing operator \\eref{eq:gft_cosmo_squeezing} (after redefinition by a phase $\\op{A}^i \\rightarrow e^{\\mathup{i}\\pi\/4}\\op{A}^i$ and $\\hermconj{\\op{A}}_j \\rightarrow e^{-\\mathup{i}\\pi\/4}\\hermconj{\\op{A}}_j$, \\eref{eq:gft_cosmo_squeezing} becomes $\\op{\\mathcal{H}}=\\frac{1}{2}\\lambda(\\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j\\epsilon^{ij}+\\op{A}^i\\op{A}^j\\epsilon_{ij})$).\nIn this sense, at least for modes with magnetic indices $m_i=0$ for which there is no coupling between modes, the Hamiltonian dynamics coming from the quadratic part of the full \\ac{gft} action is exactly of squeezing type.\n \n\\section{A toy model revisited}\n\\label{sec:tm}\nIn this section, we  study the dynamics of \\ac{gft} for a single field mode, in the approximation where \\ac{gft} interactions are neglected.\nThe observation that a squeezing operator as used in \\cite{AGW18} is already the (free) \\ac{gft} Hamiltonian for a mode in which all the magnetic indices are zero motivates us to revisit the model studied in \\cite{AGW18}.\nThe restriction of this model to a single field mode can be partially motivated by results in \\cite{Gie16} that suggest \\ac{gft} dynamics are generically dominated by a single value for the spin $j$.\nIn the next section, we will add interactions and go beyond the assumption of free dynamics.\n\nWe make use of the observation of \\cite{BBC19} that the fundamental operators appearing in this model (representing the Hamiltonian and volume operators) generate an $\\liealg{su}(1, 1)$ algebra.\nWe will extend some results both of the toy model analysis \\cite{AGW18} and of the algebraic discussion in \\cite{BBC19}: we will discuss general algebraic expressions representing effective Friedmann equations, and then specify by choosing different classes of coherent states.\nImportantly, we will compute relative uncertainties for the main physical quantities and use them as a criterion for the selection of good semiclassical states.\n\nThe Hamiltonian we consider is the one-mode squeezing Hamiltonian\n\\begin{equation}\n  \\label{eq:tm_hamiltonian}\n  \\op{H}\n  =\n  - \\frac{\\omega}{2}\n  (\n    \\op{a}^2\n    + \\hermconj{\\op{a}}{}^2\n  )\n  \\mathperiod\n\\end{equation}\nAs in previous work the main observable of interest is the volume operator\n\\begin{equation}\n  \\label{eq:tm_n_v_relation}\n  \\op{V}\n  =\n  v_0\n  \\op{N}\n  :=\n  v_0\n  \\hermconj{\\op{a}}\n  \\op{a}\n  \\mathcomma\n\\end{equation}\nwhere $v_0$ would be the eigenvalue for the \\ac{gft} volume operator for the representation (and intertwiner) chosen for the model, i.e., the volume associated to a single quantum in this mode.\nWe are working in a deparametrised framework in which the Hamiltonian generates time evolution with respect to scalar field time $\\phi$.\nThe energy expectation value $\\expval{\\op{H}}$ thus physically represents the conjugate momentum $\\pi_\\phi$ of $\\phi$.\nWe can then define an effective energy density of the matter scalar field $\\phi$, at the level of expectation values, by\n\\begin{equation}\n  \\label{eq:tm_energy_density}\n  \\rho_\\phi(\\phi)\n  =\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    2 \\expval{\\op{V}(\\phi)}^2\n  }\n  \\mathcomma\n\\end{equation}\ngiven the classical expression $\\rho_\\phi=\\pi_\\phi^2\/(2V^2)$.\nThis definition extends the one used in the mean-field setting (see \\eref{eq:gft_cosmo_friedmann}) which also only included expectation values of elementary operators $\\op{H}$ and $\\op{V}$.\nOther definitions using composite operators would be possible and would differ from  \\eref{eq:tm_energy_density} by higher moments such as $\\expval{\\op{H^2}}-\\expval{\\op{H}}^2$.\nNotice that inverse operators such as $\\op{V}(\\phi)^{-2}$ are not obviously well-defined in the \\ac{gft} formalism.\n\n\\subsection{\\texorpdfstring{$\\liealg{su}(1, 1)$}{su(1, 1)} structure of the system}\nAs was first pointed out for \\ac{gft} cosmology in \\cite{BBC19}, the operators $\\op{V}$ and $\\op{H}$ generate the Lie algebra $\\liealg{su}(1, 1)$, which appears frequently in the context of quantum cosmology for a flat \\ac{flrw} Universe filled with a free scalar field, see e.g.\\ \\cite{Boj07,EM12,AL19}.\nThe three independent quadratic products of creation and annihilation operators form a realisation of the $\\liealg{su}(1,1)$ algebra.\nIn particular the identifications\n\\begin{equation}\n  \\label{eq:su11_bosonic_realisation}\n  \\op{K}_0\n  =\n  \\frac{1}{4}\n  \\left(\n    \\hermconj{\\op{a}}\n    \\op{a}\n    +\n    \\op{a}\n    \\hermconj{\\op{a}}\n  \\right)\n  = \\frac{1}{2}\\op{N}+\\frac{1}{4}\\op{I}\n  \\mathcomma\n  \\qquad\n  \\op{K}_+\n  =\n  \\frac{1}{2}\n  \\hermconj{\\op{a}}{}^2\n  \\mathcomma\n  \\qquad\n  \\op{K}_-\n  =\n  \\frac{1}{2}\n  \\op{a}^2\n\\end{equation}\n(where $\\op{I}$ denotes the identity)\ngive the $\\liealg{su}(1,1)$ relations with the usual normalisation\n\\begin{equation}\n  \\commutator{\n    \\op{K}_0\n  }{\n    \\op{K}_\\pm\n  }\n  =\n  \\pm \\op{K}_\\pm\n  \\mathcomma\n  \\qquad\n  \\commutator{\n    \\op{K}_-\n  }{\n    \\op{K}_+\n  }\n  =\n  2 \\op{K}_0\n  \\mathperiod\n\\end{equation}\nThe Casimir of $\\liealg{su}(1, 1)$ is given by\n\\begin{equation}\n  \\label{eq:su11_casimir}\n  \\op{C}\n  =\n  (\\op{K}_0)^2\n  -\n  \\frac{1}{2}\n  (\n    \\op{K}_+\n    \\op{K}_-\n    +\n    \\op{K}_-\n    \\op{K}_+\n  )\n  \\mathperiod\n\\end{equation}\nIn terms of the $\\liealg{su}(1, 1)$ generators the Hamiltonian \\eref{eq:tm_hamiltonian} reads\n\\begin{equation}\n  \\label{eq:tm_hamiltonian_su11}\n  \\op{H}\n  =\n  - \\omega\n  (\n    \\op{K}_+\n    +\n    \\op{K}_-\n  )\n  \\mathperiod\n\\end{equation}\nAs one can see from \\eref{eq:su11_bosonic_realisation} the dynamics of the operator $\\op{K}_0$ are intimately related to the dynamics of the volume operator $\\op{V} = v_0 \\op{N}$.\nWe consider here only the $\\liealg{su}(1, 1)$ representations of the discrete ascending series in which the operator $\\op{K}_0$, and hence the volume, is bounded from below.\\footnote{In general, for such representations one can only say that $\\expval{\\op{N}}>-\\frac{1}{2}$.\nBelow we mostly focus on Fock representations, for which $\\op{N}$ is always nonnegative.}\n(A more general discussion would also include other types of representations, for which there is no such lower bound. See also the comments in \\cite[Sec.~4]{BBC19}.)\n\nThese representations are labelled by a real positive number $k$, the so-called Bargmann index, and satisfy\n\\begin{eqnarray}\n  &\n  \\op{K}_-\n  \\ket{k, 0}\n  =\n  0\n  \\mathcomma\n  \\\\\n  &\n  \\op{K}_0\n  \\ket{k, m}\n  =\n  (k + m)\n  \\ket{k, m}\n  \\mathcomma\n  \\\\\n  &\n  \\op{C}\n  \\ket{k, m}\n  =\n  k (k-1)\n  \\ket{k, m}\n  \\mathcomma\n\\end{eqnarray}\nwhere the states $\\ket{k, m}$ are the normalised states proportional to\n$(\\op{K}_+)^m \\ket{k, 0}$.\nSee \\aref{app:su11} for more details.\n\nWhen one inserts the realisation of the $\\liealg{su}(1, 1)$ operators in terms of bosonic creation and annihilation operators \\eref{eq:su11_bosonic_realisation} into the Casimir \\eref{eq:su11_casimir}, one finds that the Casimir is $\\op{C} = - 3\/16 \\op{I}$ which implies a Bargmann index of either $k=1\/4$ or $k=3\/4$.\nThese two cases respectively correspond to representations spanned by the eigenstates of the number operator with even or odd eigenvalues.\nThe choice $k=1\/4$ appears more interesting physically since it contains the Fock vacuum (or cosmological `singularity') in which no quanta are present.\nSince we are interested in studying the Fock space representations of the \\ac{gft} field, we will mostly restrict the Bargmann index to these cases.\n\n\\subsection{Classes of coherent states and relative uncertainties}\n\\label{sec:tm_class_of_coh_stat_and_rel_uncert}\nThe time evolution of a system can be quite sensitive to its initial state.\nIn this section we discuss classes of coherent states and comment on their usefulness in the context of \\ac{gft} cosmology.\nThe coherent states we consider are the following:\n\\begin{itemize}\n  \\item\n    Fock coherent states,\n  \\item\n    \\acf{pg} coherent states of $\\liealg{su}(1, 1)$,\n  \\item\n    \\acf{bg} coherent states of $\\liealg{su}(1, 1)$.\n\\end{itemize}\n\nThe Fock coherent states correspond to the well-known coherent states of the harmonic oscillator, labelled by a complex number $\\sigma$.\nOne possible way to define the Fock coherent states is by acting with the displacement operator on the Fock vacuum,\n\\begin{equation}\n  \\label{eq:tm_coh_state_fock}\n  \\ket{\\sigma}\n  =\n  \\exp\\left(\n    \\sigma \\hermconj{\\op{a}}\n    - \\compconj{\\sigma} \\op{a}\n  \\right)\n  \\ket{0}\n  \\mathperiod\n\\end{equation}\n\nThe \\ac{pg} coherent states are obtained by acting on the $\\liealg{su}(1, 1)$ ground state $\\ket{k, 0}$ with a squeezing operator\n\\begin{equation}\n  \\op{S}(\\xi)\n  =\n  \\exp\\left(\n    \\xi\n    \\op{K}_+\n    -\n    \\compconj{\\xi}\n    \\op{K}_-\n  \\right)\n  \\mathperiod\n\\end{equation}\nWe will denote the \\ac{pg} coherent states by $\\ket{\\zeta, k}$ and they are obtained by the following choice of squeezing parameter\n\\begin{equation}\n  \\ket{\n    \\zeta, k\n  }\n  =\n  \\op{S}\\left(\n    \\frac{\\zeta}{\\abs{\\zeta}}\n    \\artanh{\\abs{\\zeta}}\n  \\right)\n  \\ket{k, 0}\n  \\mathcomma\n  \\qquad\n  \\abs{\\zeta} < 1\n  \\mathperiod\n\\end{equation}\n\nThe \\ac{bg} coherent states will be denoted by $\\ket{\\chi, k}$ and are defined to be the eigenstates of the lowering operator $\\op{K}_-$,\n\\begin{equation}\n  \\op{K}_-\n  \\ket{\\chi, k}\n  =\n  \\chi\n  \\ket{\\chi, k}\n  \\mathperiod\n\\end{equation}\n\nWe now turn to the question of which class of coherent states should be considered in the context of \\ac{gft} cosmology.\nIn the context of quantum cosmology a commonly studied quantity is the relative uncertainty of the volume operator.\nIt is argued that the magnitude of the relative uncertainty corresponds to a measure of `quantumness' of the system at some given time and it is therefore important that the theory allows for initial states which give a (comparatively) small value for the relative uncertainty at late times since then the system has become (semi-)classical.\nFor previous works commenting on this in the context of \\ac{lqc} see, e.g., \\cite{AG15} and in the context of \\ac{gft} see, e.g., \\cite{PS17}.\nIn addition one would also require the relative uncertainty of the energy (which is a constant of motion) to be very small.\n\nWe define the relative uncertainty of an operator $\\op{O}$ for a given state $\\ket{\\psi}$ as\n\\begin{equation}\n  r(\\op{O}, \\ket{\\psi})\n  =\n  \\frac{\n    \\bra{\\psi}\n    \\op{O}^2\n    \\ket{\\psi}\n    -\n    \\bra{\\psi}\n    \\op{O}\n    \\ket{\\psi}^2\n  }{\n    \\bra{\\psi}\n    \\op{O}\n    \\ket{\\psi}^2\n  }\n  \\mathperiod\n\\end{equation}\nIn the following we state the relative uncertainty of the Hamiltonian and the asymptotic relative uncertainty of the volume operator at large volumes (i.e., for $\\phi\\rightarrow\\pm\\infty$), for the three classes of coherent states that we are interested in.\n\nFor the Fock coherent states one obtains for the relative uncertainty of the Hamiltonian and the asymptotic relative uncertainty of the volume operator\n\\begin{eqnarray}\n  \\label{eq:tm_rel_uncertainty_h_fock}\n  r(\n    \\op{H},\n    \\ket{\\sigma}\n  )\n  =\n  \\frac{\n    2 (1 + 2 \\abs{\\sigma}^2)\n  }{\n    (\\sigma^2 + \\compconj{\\sigma}^2)^2\n  }\n  \\mathcomma\n  \\\\\n  \\label{eq:tm_rel_uncertainty_v_fock}\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\sigma}\n  )\n  =\n  \\frac{\n    2\n    \\left(\n      1 \\mp 2 \\mathup{i} ( \\sigma \\pm \\mathup{i} \\compconj{\\sigma} )^2\n    \\right)\n  }{\n    \\left(\n      1 \\mp \\mathup{i} ( \\sigma \\pm \\mathup{i} \\compconj{\\sigma} )^2\n    \\right)^2\n  }\n  \\mathperiod\n\\end{eqnarray}\nIn principle the value of the parameter $\\sigma$ is arbitrary and therefore for suitable choices of $\\sigma$ the asymptotic relative uncertainty in both energy and volume becomes arbitrarily small.\nThese states can hence be interpreted as becoming semiclassical, consistent with arguments from \\ac{gft} that suggest that such `condensates' are good candidates for effective semiclassical macroscopic geometries \\cite{GOS14,GS16,Ori17a,PS19}.\n\nFor the \\ac{pg} coherent states the relative uncertainties of interest are\n\\begin{eqnarray}\n  r(\n    \\op{H},\n    \\ket{\\zeta, k}\n  )\n  =\n  \\frac{1}{2k}\n  \\frac{\n    (1 + \\zeta^2)\n    (1 + \\compconj{\\zeta}^2)\n  }{\n    (\\zeta + \\compconj{\\zeta})^2\n  }\n  \\mathcomma\n  \\\\\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\zeta, k}\n  )\n  =\n  \\frac{1}{2k}\n  \\mathperiod\n\\end{eqnarray}\nThe asymptotic relative uncertainty of the volume operator is independent of the parameter labelling the different \\ac{pg} coherent states.\nThis suggests that for \\ac{pg} coherent states the classical limit is reached for $k \\rightarrow \\infty$.\nHowever, we saw before that if we want to consider coherent states living in a bosonic Fock representation (rather than a more general $\\liealg{su}(1,1)$ representation), this restricts the values of the Bargmann index to either $k = 1\/4$ or $k = 3\/4$.\nThus we conclude that in the context of \\ac{gft} cosmology the class of \\ac{pg} coherent states do not `classicalise' at late times and hence, even though these states are naturally suggested by the  $\\liealg{su}(1,1)$ structure, they do not appear to be good candidate states for effective macroscopic cosmologies in \\ac{gft}.\n\nFor completeness we also state the relative uncertainties for the \\ac{bg} coherent states\n\\begin{eqnarray}\n  &r(\n    \\op{H},\n    \\ket{\\chi, k}\n  )\n  =\n  \\frac{2}{\n    (\\chi + \\compconj{\\chi})^2\n  }\n  \\left[\n    k\n    +\n    \\abs{\\chi}\n    \\frac{\n      I_{2k}(2 \\abs{\\chi})\n    }{\n      I_{2k-1}(2 \\abs{\\chi})\n    }\n  \\right]\n  \\mathcomma\n  \\\\\n&\n\\eqalign{\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\chi, k}\n  )\n  =\n  &\n  2\n  \\Big[\n    - 2 \\abs{\\chi}^2\n    I_{2k}(2 \\abs{\\chi})^2\n    +\n    (3 - 4k)\n    \\abs{\\chi}\n    I_{2k}(2\\abs{\\chi})\n    I_{2k-1}(2\\abs{\\chi})\n    \\\\\n  &\n    \\qquad\n    +\n    (k \\mp \\mathup{i}(\\chi - \\compconj{\\chi}) + 2 \\abs{\\chi}^2)\n    I_{2k-1}(2\\abs{\\chi})^2\n  \\Big]\n  \\\\\n  &\n  \\times\n  \\Big[\n    2 \\abs{\\chi}\n    I_{2k}(2 \\abs{\\chi})\n    +\n    (2 k \\mp \\mathup{i}(\\chi - \\compconj{\\chi}))\n    I_{2k-1}(2 \\abs{\\chi})\n  \\Big]^{-2}\n  \\mathcomma\n}\n\\end{eqnarray}\nwhere $I_\\alpha(x)$ is the modified Bessel function of the first kind.\nWe can use an asymptotic expansion of the modified Bessel functions to get the relative uncertainties for large values of $\\abs{\\chi}$,\n\\begin{eqnarray}\n  &r(\n    \\op{H},\n    \\ket{\\chi, k}\n  )\n  \\stackrel{\\abs{\\chi} \\rightarrow \\infty}{\\sim}\n  \\frac{\n    2\n    \\abs{\\chi}\n  }{\n    (\\chi + \\compconj{\\chi})^2\n  }\n  \\mathcomma\n\\\\\n&\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\chi, k}\n  )\n  \\stackrel{\\abs{\\chi} \\rightarrow \\infty}{\\sim}\n  \\frac{\n    2\n  }{\n    2 \\abs{\\chi} \\mp \\mathup{i} (\\chi - \\compconj{\\chi})\n  }\n  \\mathcomma\n\\end{eqnarray}\nwhich shows that the asymptotic relative uncertainties for the \\ac{bg} coherent states are also arbitrarily small for large values of $\\abs{\\chi}$.\nHence these states would also be suitable states for \\ac{gft} cosmology.\nHowever, the ubiquitous appearance of the modified Bessel functions makes calculations with the \\ac{bg} states quite cumbersome.\nBelow we mostly focus on Fock coherent states which are easier to calculate with.\n\nIn \\fref{fig:tm_rel_uncertainty_comp} we provide an overview of the time dependence of the relative uncertainty of the various states discussed here.\nOne notable aspect is that the uncertainties are asymmetric with respect to time.\nWhile the uncertainty can be asymptotically small in the future, it might have been asymptotically large in the past.\nIn particular, one could in general not conclude from the emergence of a semiclassical regime at late times, in which the relative uncertainties remain small, that the same was true at early times in the collapsing pre-bounce phase.\nIn order to quantify this asymmetry of the asymptotic relative uncertainty, we define an `asymptotic asymmetry parameter'\n\\begin{equation}\n  \\label{eq:tm_asymmetry_parameter}\n  \\eta(\\ket{\\psi})\n  =\n  1\n  -\n  \\min\n  \\left\\{\n    \\frac{\n      r(\\op{V}(+\\infty), \\ket{\\psi})\n    }{\n      r(\\op{V}(-\\infty), \\ket{\\psi})\n    }\n    ,\n    \\frac{\n      r(\\op{V}(-\\infty), \\ket{\\psi})\n    }{\n      r(\\op{V}(+\\infty), \\ket{\\psi})\n    }\n  \\right\\}\n  \\mathperiod\n\\end{equation}\nThe values this parameter can take lie between zero and one, where small values signify that the relative uncertainty is the same in the asymptotic past and future whereas values close to one correspond to a large past-future asymmetry.\nIn \\fref{fig:tm_asymmetry_parameter} the asymptotic asymmetry parameter is shown for Fock and \\ac{bg} coherent states as a function of the argument $\\theta$ of the complex parameters characterising the state, i.e.\\ $\\chi = \\abs{\\chi} \\exp(\\mathup{i} \\theta)$ and $\\sigma = \\abs{\\sigma} \\exp(\\mathup{i} \\theta)$.\n    Note that even though the plot is done for some specific value of the absolute value of the coherent state parameters, the situation is generic.\n    Only for very small absolute values ($\\abs{\\sigma} \\ll 1, \\abs{\\chi} \\ll 1$) is the asymmetry parameter close to one for all values of the argument $\\theta$.\nFrom this it becomes apparent that the past-future asymmetry is rather generic.\nSimilar questions have been discussed previously in the context of \\ac{lqc} \\cite{Boj07a,CP08,Boj08}; our analysis here extends them from the mean-field calculations in \\cite{PS17} to broader classes of states of interest for \\ac{gft} cosmology.\n\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-tm_rel_uncertainty_comp.pdf}\n  \\caption{The relative uncertainty of the volume operator as a function of $\\omega \\phi$ for $k=1\/4$.\n    The complex parameter $x$ given in the figure is related to the parameters of the coherent states in the following manner.\n    \\ac{pg}:\n    $\\ket{\\zeta, k} \\equiv \\ket{(x\/\\abs{x}) \\tanh(\\abs{x}),1\/4}$\n    \\ac{bg}:\n    $\\ket{\\chi, k} \\equiv \\ket{x, 1\/4}$,\n    Fock:\n    $\\ket{\\sigma} \\equiv \\ket{x}$.\n    For the bottom-left plot, $|x|\\ll 1$ and these states have very small volume around $\\phi=0$, which leads to the large relative uncertainties.\n  }\\label{fig:tm_rel_uncertainty_comp}\n\\end{figure}\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-tm_asymmetry_parameter.pdf}\n  \\caption{The asymptotic asymmetry parameter \\eref{eq:tm_asymmetry_parameter} as a function of the argument of the complex coherent state parameters for $k=1\/4$.\n    The argument $\\theta$ is related to the parameters of the coherent states in the following way.\n    \\ac{bg}:\n    $\\ket{\\chi, k} \\equiv \\ket{100 \\exp(\\mathup{i} \\theta), 1\/4}$,\n    Fock:\n    $\\ket{\\sigma} \\equiv \\ket{100 \\exp(\\mathup{i} \\theta)}$.\n  }\\label{fig:tm_asymmetry_parameter}\n\\end{figure}\n\n\\subsection{Effective Friedmann equations}\nIn order to derive the cosmological implications of the model, we derive in this section an effective Friedmann equation\n\\begin{equation}\n  \\label{eq:tm_eff_fried_formal}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  f[V(\\phi)]\n  \\mathcomma\n\\end{equation}\nwhere we introduced the compact notation $V(\\phi) \\equiv \\expval{\\op{V}(\\phi)}$ and $f[V(\\phi)]$ is some functional to be specified later.\nThe method we will be employing to solve this problem is an algebraic approach introduced in \\cite{BBC19} which we extend to noncommuting variables and connect to the Fock representation underlying the kinematics of \\ac{gft}.\n\nWe work in the Heisenberg picture and assume that the Schr\u00f6dinger and Heisenberg picture coincide at $\\phi = 0$.\nOperators without argument denote the Schr\u00f6dinger picture operators.\nThe equations of motion for $\\op{K}_0$ and $\\op{K}_+ - \\op{K}_-$ are given by ($\\op{K}_+ + \\op{K}_-$ is proportional to the Hamiltonian and therefore constant under time evolution)\n\\begin{eqnarray}\n  &\n  \\label{eq:tm_k0_eom}\n  \\op{K}_0'\n  (\\phi)\n  =\n  \\mathup{i} \\omega\n  (\n    \\op{K}_+\n    -\n    \\op{K}_-\n  )\n  (\\phi)\n  \\mathcomma\n  \\\\\n  &\n  (\\op{K}_+ - \\op{K}_-)'(\\phi)\n  =\n  - 4 \\mathup{i} \\omega \\op{K}_0\n\\end{eqnarray}\nwhich are solved by\n\\begin{eqnarray}\n  \\label{eq:tm_k0_heisenberg}\n  &\n  \\op{K}_0(\\phi)\n  =\n  \\op{K}_0\n  \\cosh(2 \\omega \\phi)\n  +\n  \\frac{\\mathup{i}}{2}\n  (\\op{K}_+ - \\op{K}_-)\n  \\sinh(2 \\omega \\phi)\n  \\mathcomma\n  \\\\\n  &\n  (\\op{K}_+ - \\op{K}_-)(\\phi)\n  =\n  (\\op{K}_+ - \\op{K}_-)\n  \\cosh(2 \\omega \\phi)\n  -\n  2 \\mathup{i} \\op{K}_0\n  \\sinh(2 \\omega \\phi)\n  \\mathperiod\n\\end{eqnarray}\nFrom this one gets for the time dependence of the number operator\n\\begin{equation}\n  \\label{eq:tm_n_heisenberg}\n  \\op{N}(\\phi)\n  =\n  -\\frac{1}{2}\n  +\n  \\left(\n    \\op{N}\n    + \\frac{1}{2} \\op{I}\n  \\right)\n  \\cosh(2 \\omega \\phi)\n  +\n  \\mathup{i}\n  (\\op{K}_+ - \\op{K}_-)\n  \\sinh(2 \\omega \\phi)\n  \\mathperiod\n\\end{equation}\nThe expectation value of this is nonnegative for all Fock states and grows exponentially at early or late times ($|\\omega\\phi|\\gg 1$).\nA nonvanishing expectation value $\\expval{\\mathup{i}(\\op{K}_+ - \\op{K}_-)}$ implies a time asymmetry in the resulting effective cosmological history, i.e., different pre- and post-bounce phases.\nFor generic states $\\expval{\\op{N}(\\phi)}$ is positive for all $\\phi$; the only cases for which it becomes zero at some point during the evolution is for states which satisfy $ \\abs{\\expval{\\op{K}_+ - \\op{K}_-}}^2 \\geq \\expval{\\op{N}} (\\expval{\\op{N}} + 1)$.\nFor Fock coherent states this is only the case for the Fock vacuum for which both sides are zero.\nFor both \\ac{pg} and \\ac{bg} coherent states it depends on the value of the Bargmann index $k$.\nFor $k > 1\/4$ this inequality never holds, for $k < 1\/4$, however, there are states for which the inequality is satisfied and for $k=1\/4$ it is only the ground state (analogue to the Fock vacuum) for which the inequality holds.\n\\Eref{eq:tm_n_heisenberg} reproduces the result obtained in the `toy model' context of \\cite{AGW18} (cf.~\\eref{eq:gft_cosmo_toymodel_n} and below), the only difference being that a factor 5 is replaced by 1 since we consider a single field mode, not five modes as in the toy model.\n\nOne can derive an effective Friedmann equation directly by taking the expectation value of the explicit expression \\eref{eq:tm_n_heisenberg}.\nOne then finds\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_from_heisenberg}\n  \\fl\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4\\omega^2\n  \\left(\n    1\n    +\n    \\frac{v_0}{V(\\phi)}\n    -\n    \\frac{v_0^2}{V(\\phi)^2}\n    \\left[\n      \\expval{\\op{N}}^2 + \\expval{\\op{N}}\n      -\n      \\expval{\\mathup{i}(\\op{K}_+ - \\op{K}_-)}^2\n    \\right]\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere $V(\\phi) \\equiv v_0 \\expval{\\op{N}(\\phi)}$ (cf.~\\eref{eq:tm_n_v_relation}).\n\nOne can, however, also obtain this effective Friedmann equation without knowing the time dependence of the number operator explicitly by using the algebraic structure of the system.\nThis was shown in \\cite{BBC19} for the corresponding classical system, where the variables commute.\nHere we extend this algebraic approach to the noncommutative case.\n\nStarting from \\eref{eq:tm_k0_eom} and the definition of the Casimir \\eref{eq:su11_casimir} one arrives at\n\\begin{equation}\n    \\op{K}_0' (\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    \\op{K}_0(\\phi)^2\n    -\n    \\left(\n      \\frac{\\op{H}^2}{4 \\omega^2}\n      +\n      \\op{C}\n    \\right)\n  \\right]\n\\end{equation}\nor written in terms of the number operator\n\\begin{equation}\n  \\label{eq:gft_alg_friedmann}\n    \\op{N}' (\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    \\op{N}(\\phi)^2\n    +\n    \\op{N}(\\phi)\n    -\n    \\left(\n      \\frac{\\op{H}^2}{\\omega^2}\n      +\n      4 \\op{C}\n      -\n      \\frac{1}{4}\n      \\op{I}\n    \\right)\n  \\right]\n\\mathperiod\n\\end{equation}\n\nIn order to get the effective Friedmann equation one has to take the expectation value of \\eref{eq:gft_alg_friedmann}.\nHowever, it is crucial to note that in \\eref{eq:tm_eff_fried_formal} the expectation value of the volume operator enters, rather than the expectation value of the volume operator squared.\nThe difference between the two is related to the variance of the volume, which is in general state-dependent.\nIndeed, rearranging the expectation value of \\eref{eq:gft_alg_friedmann} gives\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_k0}\n  N'(\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    N(\\phi)^2\n    +\n    N(\\phi)\n    +\n    X\n  \\right]\n\\end{equation}\nwith $N(\\phi) \\equiv \\expval{\\op{N}(\\phi)}$ and $X$ being given by\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n  -\n  \\frac{1}{4\\omega^2}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}'\n    (\\phi)\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}^2}\n  }{\n    \\omega^2\n  }\n  -\n  4 \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathcomma\n\\end{equation}\nwhere the covariance $\\covariance{\\op{A}}{\\op{B}}$ is defined as\n\\begin{equation}\n  \\covariance{\n    \\op{A}\n  }{\n    \\op{B}\n  }\n  =\n  \\frac{1}{2}\n  \\expval{\n    \\op{A}\n    \\op{B}\n    +\n    \\op{B}\n    \\op{A}\n  }\n  -\n  \\expval{\\op{A}}\n  \\expval{\\op{B}}\n\\end{equation}\n(and for the case $\\op{A}=\\op{B}$ this would be called the variance of $\\op{A}$).\n\nWe will now show that the quantity $X$ is indeed time-independent as suggested by the notation.\nNoting that the variance of $\\op{N}'(\\phi)$ can be written as\n\\begin{equation}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}'\n    (\\phi)\n  }\n  =\n  - 4 \\covariance{\\op{H}}{\\op{H}}\n  + 16 \\omega^2\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  \\mathcomma\n\\end{equation}\none can write $X$ as\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n  -\n  4\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    \\omega^2\n  }\n  -\n  4 \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nA quick calculation shows that\n\\begin{equation}\n  \\dfrac{\\phi}\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  =\n  \\frac{1}{2}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n\\end{equation}\nwhich shows that $X$ is time-independent, since both $\\op{H}$ and $\\op{C}$ are constants\nof motion.\nIt follows that $X$ can be written as\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n  }{\n    \\op{N}\n  }\n  -\n  4\n  \\covariance{\n    \\op{K}_+\n  }{\n    \\op{K}_-\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    \\omega^2\n  }\n  -\n  4\n  \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\n\nUsing the definition of the Casimir \\eref{eq:su11_casimir} one can show that $X$ can be written in the alternative form\n\\begin{equation}\n  X\n  =\n  - \\expval{\n    \\op{N}\n  }^2\n  -\n  \\expval{\n    \\op{N}\n  }\n  +\n  4\n  \\expval{\n    \\op{K}_+\n  }\n  \\expval{\n    \\op{K}_-\n  }\n  -\n  \\frac{\n    \\expval{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  \\mathperiod\n\\end{equation}\nYet another form can be obtained by inserting the Hamiltonian \\eref{eq:tm_hamiltonian_su11} to get\n\\begin{equation}\n  X\n  =\n  -\n  \\expval{\n    \\op{N}\n    +\n    \\frac{1}{2}\n    \\op{I}\n  }^2\n  +\n  \\expval{\n    \\mathup{i}\n    (\n      \\op{K}_+\n      -\n      \\op{K}_-\n    )\n  }^2\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nThe operators appearing in this last expression for $X$ are exactly those appearing in the formula for the operator $\\op{K}_0(\\phi)$ in the Heisenberg picture \\eref{eq:tm_n_heisenberg} and one recovers the form of the effective Friedmann equation given in \\eref{eq:tm_eff_friedmann_from_heisenberg}.\nFrom the exact solution \\eref{eq:tm_n_heisenberg}, one can see that $X\\le 0$ in all Fock states, since $X>0$ would be equivalent to the number operator $N(\\phi)$ taking a negative expectation value somewhere.\nThere are Fock states with $X=0$; these states encounter a singularity in their geometric interpretation, in the sense that the expectation value of the volume reaches zero somewhere and hence the effective energy density defined according to \\eref{eq:tm_energy_density} diverges, even though the quantum evolution is completely regular even for these states.\n\nRecalling that the volume operator is the rescaled number operator, i.e.\\ $\\op{V} = v_0 \\op{N}$, one arrives at the effective Friedmann equation\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_full}\n  \\fl\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval{\n        \\op{N}\n      }\n      (\n        \\expval{\n          \\op{N}\n        }\n        +\n        1\n      )\n    }{\n      V(\\phi)^2\n    }\n    +\n    \\frac{\n      4\n      v_0^2\n      \\expval{\n        \\op{K}_+\n      }\n      \\expval{\n        \\op{K}_-\n      }\n    }{\n      V(\\phi)^2\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n  \\right)\n\\end{equation}\nTaking the late time limit (corresponding to large volumes) suggests that one should identify $12 \\pi G := 4 \\omega^2$ in order for the leading term to be compatible with the classical Friedmann dynamics.\nThis identification of fundamental couplings with an `emergent' Newton's constant is common in \\ac{gft} cosmology \\cite{OSW16,AGW18}.\nFurthermore, identifying an energy density as in \\eref{eq:tm_energy_density} and defining a critical energy density\n\\begin{equation}\n  \\rho_\\critical\n  =\n  \\frac{\n    \\omega^2\n  }{\n    2 v_0^2\n  }\n  =\n  \\frac{\n    3 \\pi G\n  }{\n    2 v_0^2\n  }\n  =\n  \\frac{3\\pi}{2}\n  \\rho_\\planck\n  \\left(\n    \\frac{v_\\planck}{v_0}\n  \\right)^2\n  \\mathcomma\n\\end{equation}\nwhere $\\rho_\\planck$ is the Planck mass density and $v_\\planck$ is the Planck volume, the last term inside the parentheses in \\eref{eq:tm_eff_friedmann_full} takes the form $-\\rho\/\\rho_\\critical$ familiar from \\ac{lqc}.\nThe value for $\\rho_\\critical$ appearing here agrees with the critical density found in \\cite{OSW16,OSW17}.\nIn close analogy to the results obtained in \\ac{lqc} we then write for the effective Friedmann equation (see also \\eref{eq:gft_cosmo_friedmann})\n\\begin{equation}\n\\label{eq:tm_friedman_with_rho_eff}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    -\n    \\frac{\n      \\rho_\\effective(\\phi)\n    }{\n      \\rho_\\critical\n    }\n  \\right)\n  +\n  4 \\omega^2\n  \\frac{\n    v_0\n  }{\n    V(\\phi)\n  }\n  \\mathcomma\n\\end{equation}\nwhere the effective energy density $\\rho_\\effective(\\phi)$ is defined as\n\\begin{equation}\n  \\rho_\\effective(\\phi)\n  =\n  \\rho_\\phi(\\phi)\n  +\n  \\frac{\n    \\omega^2\n    \\expval{\n      \\op{N}\n    }\n    (\n      \\expval{\n        \\op{N}\n      }\n      +\n      1\n    )\n  }{\n    2V(\\phi)^2\n  }\n  -\n  \\frac{\n    2\n    \\omega^2\n    \\expval{\n      \\op{K}_+\n    }\n    \\expval{\n      \\op{K}_-\n    }\n  }{\n    V(\\phi)^2\n  }\n  \\mathperiod\n\\end{equation}\nThe first contribution to this effective energy density is given by the energy density $\\rho_\\phi(\\phi)=\\expval{\\op{H}}^2\/(2V(\\phi)^2)$ associated to a massless scalar field (as defined in  \\eref{eq:tm_energy_density}), but there are two additional contributions depending on the expectation values $\\expval{\\op{N}},\\,\\expval{\\op{K}_+}$ and\n    $\\expval{\n      \\op{K}_-\n    }$\nin the initial state.\nAs these additional contributions to $\\rho_\\effective$ also scale as $V(\\phi)^{-2}$, their effect is equivalent to a shift in the scalar field momentum compared to its classical value $\\expval{\\op{H}}$.\nThe last term in \\eref{eq:tm_friedman_with_rho_eff}, scaling as $1\/V(\\phi)$, is similar to a correction found in mean-field calculations and takes the form of an effective matter contribution for matter with equation of state $p=2\\rho$ (cf.~\\cite{CPS16}).\n\nDepending on the initial state, the quantity $X$ (or, alternatively, the additional contributions to the effective energy density) can take different forms.\nFor the \\ac{pg} coherent states one finds the following form\n\\begin{equation}\n  \\label{eq:gft_su11_x_pg}\n  X_{\\mathsubscript{PG}}\n  =\n  - 4 k^2\n  \\frac{\n    (1 + \\zeta^2)\n    (1 + \\compconj{\\zeta}^2)\n  }{\n    (1 - \\abs{\\zeta}^2)^2\n  }\n  +\n  \\frac{1}{4}\n  =\n  -\n  4 k^2\n  -\n  \\frac{\n    \\expval[\\mathsubscript{PG}]{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nWe note that for $k = 1\/4$ this reduces to\n$\n  X_{\\mathsubscript{PG}}\n  =\n  -\n  \\expval[\\mathsubscript{PG}]{\n    \\op{H}\n  }^2\n  \/\n  \\omega^2\n$, and the effective energy density and classical energy density exactly coincide.\nThe effective Friedmann equation for \\ac{pg} coherent states therefore reads\n(for general $k$)\n\\begin{equation}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{PG}]{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n    - \\frac{\n      v_0^2\\left(16 k^2\n    - 1\\right)\n    }{\n     4V(\\phi)^2\n    }\n  \\right)\n  \\mathperiod\n\\end{equation}\n\nFor Fock coherent states one gets for $X$\n\\begin{equation}\n  X_{\\mathsubscript{F}}\n  =\n  - \\frac{1}{4} (\\sigma^2 + \\compconj{\\sigma}^2)^2\n  - \\abs{\\sigma}^2\n  =\n  -\n  \\expval[\\mathsubscript{F}]{\n    \\op{N}\n  }\n  -\n  \\frac{\n    \\expval[\\mathsubscript{F}]{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  \\mathperiod\n\\end{equation}\nTherefore the Friedmann equation for Fock coherent states is given by\n\\begin{equation}\n  \\label{eq:tm_eff_fried_fock}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{F}]{\n        \\op{N}\n      }\n    }{\n      V(\\phi)^2\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{F}]{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n  \\right)\n  \\mathperiod\n\\end{equation}\n\nFor completeness we also state the value of $X$ one gets for \\ac{bg} coherent states,\n\\begin{equation}\n  X_{\\mathsubscript{BG}}\n  =\n  -\n  (\\chi - \\compconj{\\chi})^2\n  -\n  4\n  \\left(\n    k\n    +\n    \\abs{\\chi}\n    \\frac{\n      I_{2k}(2 \\abs{\\chi})\n    }{\n      I_{2k - 1}(2 \\abs{\\chi})\n    }\n  \\right)^2\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\n\nThe Friedmann equations derived here are compatible with previous results in \\ac{gft} cosmology \\cite{OSW16,OSW17,AGW18,WEw19} where either a mean-field approach was used or a simplifying assumption was imposed on the initial conditions.\nWe emphasise that no approximations were used which resulted in the appearance of extra terms.\nIn particular, we were able to identify one of those extra terms with the energy density of the real scalar field acting as a relational clock variable.\nCorrections to the classical Friedmann dynamics of the \\ac{lqc}-like form $-\\rho\/\\rho_\\critical$, which lead to effective repulsive behaviour and a bounce at high energies, were found for all coherent states considered.\nWe also found that in general the `effective' energy density appearing in the Friedmann equation \\eref{eq:tm_friedman_with_rho_eff} is not equal to the classical energy density $\\rho_\\phi=\\pi_\\phi^2\/(2V^2)$ of a massless scalar field, but contains additional terms depending on the initial conditions chosen.\n \n\\section{Interacting toy model}\nIn this section we extend the toy model discussed in \\sref{sec:tm} by adding an interaction term to the Hamiltonian.\nThe resulting interacting model still represents a simplification of the dynamics of full \\ac{gft}, since we continue to assume that only one mode is relevant.\nWhile the general expectation is that the dynamics should depend on the coupling of different modes, studying this simpler model can provide insights on how \\ac{gft} interactions can change the interpretation of the dynamics in terms of effective cosmology.\nA similar model, which included polynomial interactions for a single \\ac{gft} field mode, was previously studied in \\cite{CPS16} in a mean-field approximation (see also \\cite{PS17}), leading to corrections to the effective Friedmann equations coming from these interactions.\nThese corrections become more significant at late times as the impact of \\ac{gft} interactions grows with the number of quanta.\nHere we will be able to contrast these mean-field results with effective modified Friedmann equations obtained in a more general setting.\n\nWe now consider a Hamiltonian given in terms of the $\\liealg{su}(1, 1)$ variables by\n\\begin{equation}\n  \\label{eq:int_hamiltonian}\n  \\op{H}\n  =\n  -\\omega\n  (\n    \\op{K}_+\n    + \\op{K}_-\n  )\n  +\n  \\lambda \\omega\n  (\n    \\op{K}_+\n    + \\op{K}_-\n    + 2 \\op{K}_0\n  )^2\n  \\mathperiod\n\\end{equation}\nIn terms of the bosonic realisation of the $\\liealg{su}(1, 1)$ algebra \\eref{eq:su11_bosonic_realisation} the Hamiltonian reads\n\\begin{equation}\n  \\label{eq:int_hamiltonian_fock_vars}\n  \\op{H}\n  =\n  - \\frac{\\omega}{2}\n  (\n    \\op{a}^2\n    + \\hermconj{\\op{a}}{}^2\n  )\n  +\n  \\frac{\n    \\lambda \\omega\n  }{\n    4\n  }\n  (\n    \\op{a}\n    + \\hermconj{\\op{a}}\n  )^4\n  \\mathperiod\n\\end{equation}\nRecalling the definitions of the creation and annihilation operators in terms of the \\ac{gft} field and its conjugate momentum \\eref{eq:gft_cosmo_annihilation}, \\eref{eq:gft_cosmo_creation} one can rewrite the Hamiltonian as\n\\begin{equation}\n  \\label{eq:int_hamiltonian_gft_vars}\n  \\op{H}\n  =\n  \\frac{1}{2 \\abs{\\mathcal{K}^{(2)}}}\n  \\op{\\pi}^2\n  -\n  \\frac{1}{2}\n  \\abs{\\mathcal{K}^{(0)}}\n  \\op{\\varphi}^2\n  +\n  \\lambda\n  \\abs{\\mathcal{K}^{(0)}}^{3\/2}\n  \\abs{\\mathcal{K}^{(2)}}^{1\/2}\n  \\op{\\varphi}^4\n  \\mathcomma\n\\end{equation}\nwhere we suppressed the Peter--Weyl representation labels and we also assume the mode to be of the type discussed at the end of \\sref{sec:gft_cosmo} with magnetic indices $m_i = 0$.\nFrom this expression one sees that the interaction term would correspond to a $\\varphi^4$ interaction term in an appropriately defined \\ac{gft} action.\n\nThe dynamics of this system crucially depend on the sign of $\\lambda$.\nIndeed, for positive $\\lambda$ this Hamiltonian will be bounded from below, whereas for the case that $\\lambda$ is negative the Hamiltonian is unbounded as it is in the free case.\nInterpreting \\eref{eq:int_hamiltonian_gft_vars} as a mechanical system with kinetic and potential terms, one sees that for positive $\\lambda$ one gets a `Mexican hat' type potential, whereas in the case of negative $\\lambda$ the potential is an `upside-down' anharmonic oscillator.\nIn the cosmological context one expects from this that for negative $\\lambda$ the Universe will undergo an enhanced exponential expansion and for positive $\\lambda$ the Universe will recollapse after some time leading to a cyclic cosmology.\nSuch a cyclic cosmology was indeed found in \\cite{CPS16}, where $\\lambda > 0$ was assumed.\nIn \\sref{sec:int_alg_poisson} we argue that this expectation is correct if \\eref{eq:int_hamiltonian} is seen as the Hamiltonian of a classical system.\n\nWhen one tries to use the algebraic approach detailed in \\sref{sec:tm} to derive an effective Friedmann equation for the interacting model \\eref{eq:int_hamiltonian} one faces several challenges.\nFirstly, the noncommutativity does not allow the reduction to a small set of `basis operators'.\nSecondly, the expressions involved feature products of three operators and it is technically challenging to relate them to the expectation values of `simple' operators such as the Hamiltonian and number operator.\n\nTo begin with, we restrict ourselves to the classical case, where the variables commute and one can employ the algebraic approach to derive an effective Friedmann equation.\nWe find an exact Friedmann equation whose limits at early and late times are given.\nAfter that we turn to the general (quantum) case, where the operators do not commute.\nIn that case we resort to a perturbative treatment which is valid at early times.\nFurthermore, we perform a numerical analysis for Fock coherent states.\nWe find that a linear (perturbative) correction to the effective Friedmann equation can capture the effect of the interaction term for a short time, after which the dynamics become nonperturbative.\n\n\\subsection{Algebraic approach for classical analogue system}\n\\label{sec:int_alg_poisson}\nIn this section we study a classical dynamical system with time evolution generated by the Hamiltonian \\eref{eq:int_hamiltonian}.\nIn this approach the $\\liealg{su}(1,1)$ variables $K_0$, $K_+$ and $K_-$ are not viewed as quantum operators but as coordinates on a Poisson manifold, subject to an $\\liealg{su}(1,1)$ Poisson algebra which then defines the Hamiltonian dynamics.\nIn contrast to the full quantum case, these variables commute and we interpret the variables themselves as the observables of interest, i.e.\\ one does not have to take expectation values.\nIn this sense, this approximation neglects all quantum corrections coming from operator orderings and uncertainties in a quantum state.\nWe will identify the variable $K_0$ with the particle number $N$ such that $N \\equiv 2 K_0$.\nAs above, we assume the total volume to be proportional to the particle number, $V = v_0 N$, and switch between $N$ and $V$ freely.\n\nThe equation of motion of the variable $N$ is given by\n\\begin{equation}\n  N'(\\phi)\n  =\n  2 \\mathup{i} \\omega\n  (K_+(\\phi) - K_-(\\phi))\n  (\n    1 - 2 \\lambda (K_+(\\phi) + K_-(\\phi) + N(\\phi))\n  )\n  \\mathperiod\n\\end{equation}\nAfter squaring this equation one can use the Casimir \\eref{eq:su11_casimir} to replace the combination $(K_+ - K_-)$.\nOne is then left with an expression where only the combination $(K_+ + K_-)$ appears,\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_intermediate}\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    &\n    4 \\omega^2\n    \\left(\n      N(\\phi)^2\n      - 4C\n      - (K_+(\\phi) + K_-(\\phi))^2\n    \\right)\n    \\\\\n    &\n    \\qquad\n    \\times\n    \\left(\n      1\n      - 2 \\lambda (K_+(\\phi) + K_-(\\phi) + N(\\phi))\n    \\right)^2\n    \\mathperiod\n  }\n\\end{equation}\nFrom the Hamiltonian one can derive an explicit formula for $(K_+ + K_-)$\n\\begin{equation}\n  \\label{eq:int_kP_plus_kM}\n  K_+(\\phi)\n  +\n  K_-(\\phi)\n  =\n  \\frac{1}{2 \\lambda}\n  \\left(\n    1\n    - 2 \\lambda N(\\phi)\n    -\n    \\sqrt{\n      1\n      + 4 \\lambda \\left( \\frac{H}{\\omega} - N(\\phi) \\right)\n    }\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere we chose the solution connected to the free theory.\nInserting this into \\eref{eq:int_classical_eff_friedmann_intermediate} one gets the nonperturbative effective Friedmann equation\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_nonpert}\n  \\fl\n\\eqalign{\n  N'(\\phi)^2\n  =\n  &\n  -\n  \\frac{\n    2\n    \\omega^2\n  }{\n    \\lambda^2\n  }\n  \\left(\n    1 + 4 \\lambda\n    \\left(\n      \\frac{H}{\\omega} - N(\\phi)\n    \\right)\n  \\right)\n    \\times\n    \\Bigg[\n      1 - 4 \\lambda N(\\phi)\n  \\\\\n  &\n  \\qquad\n  \\qquad\n      - (1 - 2 \\lambda N(\\phi))\n      \\sqrt{\n        1 + 4 \\lambda\n        \\left(\n          \\frac{H}{\\omega} - N(\\phi)\n        \\right)\n      }\n      + 2 \\lambda\n      \\left(\n        4 \\lambda C\n        + \\frac{H}{\\omega}\n      \\right)\n    \\Bigg]\n    \\mathperiod\n}\n\\end{equation}\nWe already see that $\\lambda > 0$ implies an upper bound on the value of $N$ and hence the volume, since the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} has to be real and positive.\nIndeed, at exactly that upper limit the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} becomes zero leading to a recollapse.\nIn the case $\\lambda < 0$, for the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} to be real and positive $N$ has to be greater than some minimal value (which can be zero).\nThe right-hand side remains real and positive for all values of $N$ greater than that minimal value, implying that the Universe expands indefinitely.\nFor clarity, we recall that $H$ here and throughout the paper denotes the Hamiltonian or energy (interpreted as the canonical momentum conjugate to the scalar field $\\phi$), not a Hubble rate in cosmology.\n\nWe would like to interpret \\eref{eq:int_classical_eff_friedmann_nonpert} in terms of a cosmological model given by one or several matter components which contribute to the matter energy density on the right-hand side.\nFor a general such model in usual classical cosmology, with $n$ matter components labelled by an index $i$ viewed as perfect fluids with each having an equation of state $p_i=w_i \\rho_i$, the Friedmann equation for the volume as a function of relational time $\\phi$ would be of the form\n\\begin{equation}\n  \\label{eq:int_friedmann_eos}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  = \\sum_{i=1}^n\n  A_i V(\\phi)^{1-w_i}\n\\end{equation}\nwhere $A_i$ are constants of motion.\nWhile \\eref{eq:int_classical_eff_friedmann_nonpert} is valid at all times, its interpretation in terms of cosmological models of the form \\eref{eq:int_friedmann_eos} is not clear due to the appearance of a square root on the right-hand side.\nMoreover, additional matter components as in \\eref{eq:int_friedmann_eos} would come with new conserved quantities $A_i$, whose values can be varied independently (or, in other words, are determined by the initial conditions).\nIn our model, the presence of \\ac{gft} interactions does not introduce new parameters set by initial conditions, only a new coupling constant $\\lambda$; these interactions therefore modify the dynamics of gravity rather than matter.\nHere we follow the convention in which quantum gravity corrections are written as modifying the right-hand side rather than the left-hand side of Friedmann equations (as is usually done in \\ac{lqc}) in order to give intuition for the effective dynamics.\\footnote{An alternative possible interpretation of effective Friedmann equations obtained from quantum gravity models is to view them as equivalent to Friedmann equations of a modified theory of gravity.\n  A general reconstruction method of this type for mimetic gravity theories was developed in \\cite{Ces19}.\n}\n\nOne might be interested in the Friedmann equation valid at relatively small volumes where interactions have become relevant but not dominant, so that one can employ perturbation theory.\nExpanding \\eref{eq:int_classical_eff_friedmann_nonpert} as a series around $\\lambda=0$ one gets\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_series}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4\\omega^2\n    \\Bigg\\{\n  &\n      N(\\phi)^2\n      - \\frac{H^2}{\\omega^2}\n      - 4C\n      \\left(\n        1\n        -\n        4 \\lambda\n        \\left(\n          N(\\phi) - \\frac{H}{\\omega}\n        \\right)\n      \\right)\n  \\\\\n  &\n    \\eqalign{\n      -\n      \\sum_{n=1}^\\infty\n      \\bigg[\n    &\n      12 \\lambda^n\n      \\frac{\n        (2n - 2)!\n      }{\n        (n - 1)!\n        (n + 2)!\n      }\n      \\left(\n        (2n - 1) \\frac{H}{\\omega}\n        + (3 - n) N(\\phi)\n      \\right)\n    \\\\\n    &\n      \\times\n      \\left(\n        N(\\phi)\n        - \\frac{H}{\\omega}\n      \\right)^{n + 1}\n      \\bigg]\n      \\Bigg\\}\n      \\mathperiod\n    }\n  }\n\\end{equation}\nFrom this form one can see that it is the product $\\lambda N(\\phi)$ that must be small for the perturbative expansion to make sense.\nComparing with \\eref{eq:int_friedmann_eos}, one could interpret the leading (linear) correction coming from the interaction term as an effective matter component with an equation of state parameter $w=0$, i.e., a dust component.\nThis result would then agree with the results of a mean-field calculation in \\cite{CPS16} where adding a $\\varphi^4$ interaction to the \\ac{gft} Lagrangian led to such a dust-like contribution in the effective cosmology.\nHowever, the full expansion given in \\eref{eq:int_classical_eff_friedmann_series} shows that such an interpretation would only be valid in an intermediate regime in which the product $\\lambda N(\\phi)$ is no longer negligible, but also not yet large enough for higher orders to contribute.\nIndeed, as the volume grows further, $\\lambda N(\\phi)$ would soon be $\\bigO{1}$ and the perturbative expansion receives contributions from \\emph{all} orders.\nIn particular, there is never a regime in which the effective Friedmann equation is dominated by the `dust-like' component, as it would be in the mean-field form obtained in \\cite{CPS16}.\nOne possible interpretation of this discrepancy is that mean-field methods are strictly only valid in the free theory, since they assume the absence of correlations between quanta.\nHence one would expect them to become inaccurate as the contribution to the effective dynamics coming from \\ac{gft} interactions becomes strong.\nCare has to be taken when interpreting \\eref{eq:int_classical_eff_friedmann_series} up to some order.\nFor instance, if one only considers terms up to order $\\lambda^2$ in \\eref{eq:int_classical_eff_friedmann_series}, one would conclude that there is always a recollapse (even for negative $\\lambda$), which is not true for the full solution.\n\nTo understand the late-time behaviour of the system, recall that $N(\\phi)$ grows without bound while the other dynamical quantities on the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert}, $C$ and $H$, are constants of motion.\nIn the limit of large $N(\\phi)$ (corresponding to late times) the leading order contribution of \\eref{eq:int_classical_eff_friedmann_nonpert} is given by\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_asymptotic}\n  \\left(\n    \\frac{N'(\\phi)}{N(\\phi)}\n  \\right)^2\n  =\n  32 \\omega^2\n  \\sqrt{-\\lambda N(\\phi)}\n  +\n  \\bigO{1}\n  \\mathperiod\n\\end{equation}\nInterpreting the right-hand side of \\eref{eq:int_classical_eff_friedmann_asymptotic} as an energy density of matter, one finds from \\eref{eq:int_friedmann_eos} that it corresponds to matter with an equation of state parameter $w = 1\/2$.\nThe solutions to this asymptotic form of the effective Friedmann equation behave as $N(\\phi)\\sim|\\phi-\\phi_0|^{-4}$, diverging at some $\\phi=\\phi_0$.\nWe would hence expect the evolution of our system to terminate at some finite value of $\\phi$, depending on initial conditions.\nNotice that in this interpretation the energy density of this new `matter' is fixed by the \\ac{gft} coupling $\\lambda$ and hence, as we mentioned before, the effect of adding \\ac{gft} interactions should rather be seen as modifying the dynamics of gravity on large scales.\nThe scale at which such a modification would become relevant depends on the chosen value of $\\lambda$.\n\nTo characterise the dynamics of the effective cosmology at arbitrary times, from \\eref{eq:int_friedmann_eos} we define an `effective equation of state parameter'\n\\begin{equation}\n  \\label{eq:int_eos_eff}\n  w_{\\effective}(\\phi)\n  =\n  1\n  -\n  \\frac{\n    \\mathrm{d} \\log \\left( N'(\\phi)\/N(\\phi) \\right)^2\n  }{\n    \\mathrm{d} \\log N(\\phi)\n  }\n\\end{equation}\nwhich is plotted in \\fref{fig:int_eos_poisson} as a function of $N(\\phi)$ for various truncations of \\eref{eq:int_classical_eff_friedmann_series}.\nIn the plot one sees that for $\\abs{\\lambda N(\\phi)} \\lesssim 10^{-2}$ the interactions become relevant and that for $\\abs{\\lambda N(\\phi)} \\lesssim 1$ the difference between the exact result and the first order truncation becomes large as expected.\nAnother observation is that while for small values of $\\abs{\\lambda N(\\phi)}$ the second order truncation is in better agreement with the nonperturbative results, this truncation quickly diverges for $\\abs{\\lambda N(\\phi)} \\gtrsim 1$.\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-int_poisson.pdf}\n  \\caption{The relative relational expansion rate squared \\eref{eq:int_classical_eff_friedmann_nonpert} and the `effective equation of state parameter' \\eref{eq:int_eos_eff} as functions of $N$.\n    The solid lines correspond to a truncation at zeroth order in $\\lambda$.\n    The dashed lines correspond to a truncation at first order in $\\lambda$.\n    The dotted lines correspond to a truncation at second order in $\\lambda$.\n    The dash-dotted lines correspond to the full nonperturbative case.\n    The parameters are:\n    $\\omega = 1$,\n    $\\lambda = - 10^{-7}$,\n    $H = -10040$,\n    $C = - 3\/16$.\n    (The choice of $H$ corresponds to $\\bra{\\sigma} \\op{H} \\ket{\\sigma}$ for $\\sigma = 100$.)\n  }\n  \\label{fig:int_eos_poisson}\n\\end{figure}\n\n\\subsection{Quantum calculation}\nWe are not able to derive an exact solution for $\\hat{N}(\\phi)$ in the interacting quantum mechanical case, where operators do not commute and higher moments and simple expectation values are independent.\nTo give approximate solutions, we first give a perturbative analytical method which we then contrast with numerical results.\nWe expand the number operator as a series with expansion parameter $\\lambda$, i.e.\\\n\\begin{equation}\n  \\label{eq:int_n_pert_series}\n  \\op{N}(\\phi)\n  =\n  \\sum_{n=0}^\\infty\n  \\lambda^n\n  \\op{N}_n(\\phi)\n  \\mathperiod\n\\end{equation}\nSplitting the Hamiltonian $\\op{H}$ into a $\\lambda$-independent part, $\\op{H}_0$, and a $\\lambda$-dependent part, $\\op{H}_1$, one can split the time evolution operator in a way similar to what is done in the interaction picture as $\\op{U}(\\phi) = \\op{U}_0(\\phi) \\op{U}_\\interaction(\\phi)$, where $\\op{U}_0(\\phi) = \\exp(-\\mathup{i} \\op{H}_0 \\phi)$ is the time evolution operator of the free system and the interaction time evolution operator is defined by the time-ordered exponential\n\\begin{equation}\n  \\op{U}_\\interaction(\\phi)\n  =\n  \\timeordering\n  \\exp\n  \\left(\n    -\\mathup{i}\n    \\int_0^\\phi\n    \\intmeasure{\\phi'}\\,\n    \\op{U}_0^{-1}(\\phi')\n    \\op{H}_1\n    \\op{U}_0(\\phi')\n  \\right)\n  \\mathperiod\n\\end{equation}\nNote that the inverse interaction time evolution operator requires anti-time-ordering.\nExpanding the interaction time evolution operator as a series in $\\lambda$,\n\\begin{equation}\n  \\op{U}_\\interaction(\\phi)\n  =\n  \\sum_{n=0}^\\infty\n  \\lambda^n\n  (\\op{U}_\\interaction)_n(\\phi)\n  \\mathcomma\n\\end{equation}\nand inserting into \\eref{eq:int_n_pert_series} gives for the $\\op{N}_n(\\phi)$\n\\begin{equation}\n  \\op{N}_n(\\phi)\n  =\n  \\sum_{m=0}^n\n  (\\op{U}_\\interaction^{-1})_m(\\phi)\n  \\op{U}_0^{-1}(\\phi)\n  \\op{N}\n  \\op{U}_0(\\phi)\n  (\\op{U}_\\interaction)_{n-m}(\\phi)\n  \\mathperiod\n\\end{equation}\nThe strategy would be to find the exact form of these $\\op{N}_n(\\phi)$ up to some order and then derive an effective Friedmann equation from these.\nWe will only do this to first order.\n\nThe leading term $\\op{N}_0(\\phi)$ is of course the same as in the free theory \\eref{eq:tm_n_heisenberg}.\nThe first order term $\\op{N}_1(\\phi)$ is given by\n\\begin{equation}\n\\eqalign{\n  \\fl\n  \\op{N}_1(\\phi)\n  =\n  \\frac{\\lambda}{2}\n  \\Big\\{\n    &\n    \\mkern-12mu (\\op{K}_+)^2\n    \\left[\n      3\n      - 2 (2 + 3\\mathup{i}\\omega\\phi) \\cosh(2\\omega\\phi)\n      + \\cosh(4\\omega\\phi)\n    \\right]\n    + \\hermconjtext\n    \\\\\n    &\n    +\n    \\op{K}_+\n    (2\\op{N} + 3)\n    \\left[\n      (\\mathup{i} - 3\\omega\\phi) \\sinh(2\\omega\\phi)\n      - \\mathup{i} \\sinh(4\\omega\\phi)\n    \\right]\n    + \\hermconjtext\n    \\\\\n    &\n    -\n    \\frac{1}{2}\n    \\sinh^2(2\\omega\\phi)\n    [\n      3 + 4 \\op{N}^2 + 8 \\op{N} + 8 \\op{K}_+ \\op{K}_-\n    ]\n  \\Big\\}\n  \\mathperiod\n}\n\\end{equation}\n\nAs outlined above, what one would like to do is take the expectation values of the perturbative expansion \\eref{eq:int_n_pert_series} and derive an effective Friedmann equation for arbitrary states, valid up to some order in $\\lambda$.\nHowever, already the first order expression for $\\op{N}(\\phi)$ is quite complicated and we were not able to derive a corresponding effective Friedmann equation for general states.\nTo ameliorate this we resort to taking the expectation values for some specific classes of coherent states.\n\nFirstly, we turn to the Fock coherent states.\nWriting for the parameter of the Fock coherent states \\eref{eq:tm_coh_state_fock} $\\sigma = \\sigma_1 + \\mathup{i} \\sigma_2$, one finds for the case $\\sigma_2 = 0$, i.e.\\ for real $\\sigma$,\n\\begin{equation}\n  \\label{eq:int_eff_fried_fock_real_order1}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4 \\omega^2\n    \\bigg(\n    &\n    N(\\phi)^2\n    + N(\\phi)\n    - \\sigma_1^2 (1 + \\sigma_1^2)\n    \\\\\n    &\n    \\eqalign{\n      +\n      \\frac{\n        \\lambda\n      }{\n        (1 + 2 \\sigma_1^2)^2\n      }\n      &\n      \\Big[\n        -4 N(\\phi)^3 (3 + 12 \\sigma_1^2 + 4 \\sigma_1^4)\n        \\\\\n        &\n        - 6 N(\\phi)^2 (3 + 15 \\sigma_1^2 + 12 \\sigma_1^4 + 4 \\sigma_1^6)\n        \\\\\n        &\n        + 6 N(\\phi) (-1 - 5 \\sigma_1^2 + 4 \\sigma_1^6)\n        \\\\\n        &\n        +  2 \\sigma_1^2 (3 + 24 \\sigma_1^2 + 51 \\sigma_1^4 + 48 \\sigma_1^6 +\n        20 \\sigma_1^8)\n      \\Big]\n      + \\bigO{\\lambda^2}\n      \\bigg)\n      \\mathperiod\n    }\n  }\n\\end{equation}\nFor the case $\\sigma_1 = 0$, i.e.\\ imaginary $\\sigma$, one finds the similar expression\n\\begin{equation}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4 \\omega^2\n    \\bigg(\n    &\n    N(\\phi)^2\n    + N(\\phi)\n    - \\sigma_2^2 (1 + \\sigma_2^2)\n    \\\\\n    &\n    \\eqalign{\n      +\n      \\frac{\n        \\lambda\n      }{\n        (1 + 2 \\sigma_2^2)^2\n      }\n      &\n      \\Big[\n        -4 N(\\phi)^3 (3 + 12 \\sigma_2^2 + 4 \\sigma_2^4)\n        \\\\\n        &\n        - 6 N(\\phi)^2 (3 + 9 \\sigma_2^2 - 4 \\sigma_2^4 - 4 \\sigma_2^6)\n        \\\\\n        &\n        + 6 N(\\phi) (-1 + \\sigma_2^2 + 16 \\sigma_2^4 + 12 \\sigma_2^6)\n        \\\\\n        &\n        +  2 \\sigma_2^2 (3 + 6 \\sigma_2^2 - 15 \\sigma_2^4 - 24 \\sigma_2^6\n          - 4 \\sigma_2^8)\n      \\Big]\n      + \\bigO{\\lambda^2}\n      \\bigg)\n      \\mathperiod\n    }\n  }\n\\end{equation}\nThe expression for the general case, where $\\sigma$ can be any complex number, is quite involved and we do not state it here.\n\nSecondly, we turn to the \\ac{pg} coherent states.\nFor the these states one finds the following general expression\n\\begin{equation}\n  \\fl\n  \\eqalign{\n  N'(\\phi)^2\n  =\n  4 \\omega^2\n  \\Bigg\\{\n  &\n    \\left(\n      N(\\phi)\n      + \\frac{1}{2}\n    \\right)^2\n    - 4 k^2\n    - \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}^2}{\\omega^2}\n  \\\\\n  &\n    \\eqalign{\n    +\n    \\lambda\n    \\frac{2k + 1}{4k}\n    \\Big[\n    &\n      - 8 N(\\phi)^3\n      + 12 N(\\phi)^2\n      \\left(\n        \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega} - 1\n      \\right)\n    \\\\\n    &\n      + 2 N(\\phi)\n      \\left(\n        6 \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        + 16 k^2\n        - 3\n      \\right)\n    \\\\\n    &\n      -\n      \\left(\n        \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        - 1\n      \\right)\n      \\left(\n        2 \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}^2}{\\omega^2}\n        + \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        + 16 k^2\n        - 1\n      \\right)\n    \\Big]\n    }\n  \\\\\n  &\n    +\n    \\bigO{\\lambda^2}\n    \\Bigg\\}\n    \\mathperiod\n  }\n\\end{equation}\nIt is remarkable that it is possible to write the right-hand side only in terms of $N(\\phi)$ and $\\expval[\\mathsubscript{PG}]{H}$.\nAs before, we would expect all higher order corrections to become relevant as soon as a regime is reached in which $\\abs{\\lambda N(\\phi)} \\lesssim 1$.\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-int_quantum.pdf}\n  \\caption{\n    The relative relational expansion rate squared and the `effective equation of state parameter' \\eref{eq:int_eos_eff} as functions of $N$ for Fock coherent states with real parameter $\\sigma$.\n    The solid lines correspond to a truncation at zeroth order in $\\lambda$.\n    The dashed lines correspond to a truncation at first order in $\\lambda$.\n    The dotted lines correspond to a truncation at second order in $\\lambda$.\n    The dash-dotted lines correspond to the full nonperturbative case.\n    The parameters are:\n    $\\omega = 1$,\n    $\\lambda = - 10^{-3}$,\n    $\\sigma = 10$\n  }\\label{fig:int_quantum}\n\\end{figure}\n\nIn \\fref{fig:int_quantum} the relative relational expansion rate squared and the effective equation of state parameter \\eref{eq:int_eos_eff} are plotted as a function of $N$ for different truncations of the perturbative expansion and for the result of a numerical calculation.\\footnote{The numerical results were obtained by solving the time-dependent Schr\u00f6dinger equation of the Hamiltonian \\eref{eq:int_hamiltonian_fock_vars} in the position representation.}\nThe zeroth order truncation corresponds to the free case and corresponds to \\eref{eq:tm_eff_fried_fock} and the first order truncation is given in \\eref{eq:int_eff_fried_fock_real_order1}.\nNote that we do not state the second order truncation explicitly, since the expression is rather convoluted.\nThe state considered in this plot is a Fock coherent state with real parameter $\\sigma$.\nThe numerical results are in good agreement with the second order truncation.\nHowever, the second order truncation diverges quickly for values $\\abs{\\lambda N(\\phi)} \\gtrsim 1$, whereas the first order truncation does not.\nNote that the parameters chosen do not accommodate a regime in which $\\abs{\\lambda N(\\phi)} \\ll 1$, explaining the mismatch with the linearised theory.\nDue to numerical limitations we were not able to enter the asymptotic regime and determine the corresponding effective equation of state parameter for late times.\n\nWe conclude the present discussion of the quantum behaviour of the interacting toy model by revisiting the relative uncertainties of the volume as a function of relational time presented in \\sref{sec:tm_class_of_coh_stat_and_rel_uncert} and  illustrated in \\fref{fig:tm_rel_uncertainty_comp}.\nHere we restrict ourselves to the class of Fock coherent states.\nThe results of numerical calculations comparing the free and interacting cases for a range of parameters are given in \\fref{fig:int_rel_uncertainty_comp}.\nAn immediate effect of the interactions is that the expectation values diverge at finite relational time, resulting in different ranges of the relational time coordinate.\nThese divergences can already be anticipated from the discussion below \\eref{eq:int_classical_eff_friedmann_asymptotic}.\nThe key observation is that when interactions are present the relative uncertainties are not asymptotically constant but start growing as soon as the interactions become dominant.\nThe general statement that Fock coherent states become semiclassical at late times can therefore not be extended to this interacting case in an obvious way.\nAgain, this is also consistent with the expectation that mean-field methods break down in the interacting case when interactions begin to dominate over the quadratic Hamiltonian.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics{fig-int_rel_uncertainty_comp.pdf}\n  \\caption{The relative uncertainty of the volume operator as a function of $\\omega \\phi$ for Fock coherent states.\n    The solid line corresponds to the free case, $\\lambda = 0$.\n    The dashed line corresponds to the interacting case with $\\lambda = -10^{-3}$.\n  }\n  \\label{fig:int_rel_uncertainty_comp}\n\\end{figure}\n \n\\section{Conclusions}\nOur aim was to present a general perspective on the derivation of reliable effective Friedmann equations from given quantum dynamics of a \\ac{gft} model, building on various recent developments in the derivation of effective cosmological dynamics from \\ac{gft}.\nWithin this general perspective, the most important assumption was to restrict the \\ac{gft} dynamics to those of a single field mode, i.e., fixed values of the representation labels in the Peter--Weyl decomposition.\nThis simplification of the full dynamics in which all modes would be present can be seen as the most important limitation of our work.\nWhile there are arguments suggesting the dynamical emergence of a regime dominated by a single field mode in \\ac{gft}, showing such an emergence in models of interest for four-dimensional quantum gravity remains an outstanding challenge.\nOn the other hand, we were able to derive effective cosmological dynamics without relying on a mean-field approximation, and in general no assumptions needed to be made on the initial state.\n\nWe first focussed on the case of dynamics defined by a free (quadratic) Hamiltonian.\nSuch a Hamiltonian can be of harmonic form or of `upside-down' harmonic form; the latter case in which the Hamiltonian is unbounded from below is most relevant to \\ac{gft} cosmology, since it admits solutions expanding to infinity \\cite{WEw19,GPW19}.\nFor this case, we recovered and extended results of \\cite{AGW18} and \\cite{BBC19} for the resulting effective Friedmann equations.\nGeneric solutions exhibit singularity resolution in the sense of a minimal non-zero value for the volume, and interpolate between a collapsing and an expanding branch which both at large volumes approach classical \\ac{flrw} solutions.\nThese solutions depend on a parameter determined by the initial conditions (with no obvious classical analogue) which generates an asymmetry between the solution before and after the minimum for the volume.\nThe main new result in this part was a discussion of relative uncertainties in the two main physical observables---volume and energy---in three different classes of coherent states: Fock coherent states which have been used previously in \\ac{gft} cosmology, and \\acf{pg} and \\acf{bg} coherent states of $\\liealg{su}(1, 1)$.\nWe found that Fock coherent states approach a semiclassical regime at large volume, where relative uncertainties can be arbitrarily small, while PG states of interest here never reach such a regime.\nThe difference in our treatment compared with works such as \\cite{EM12} was that we assumed the Fock space structure of \\ac{gft} and only considered PG states that live in bosonic Fock representations.\nIn \\ac{gft}, Fock coherent states are a good choice for initial conditions that become semiclassical at low curvature.\nThis part of our analysis could be broadened by going to more general types of coherent states for which only a Casimir condition and the saturation of uncertainty relations are assumed, as was done for $\\liealg{su}(1, 1)$ in \\cite{BT14}.\n\nWe then added a quartic interaction term to the Hamiltonian to extend the derivation of effective Friedmann equations for \\ac{gft} models with polynomial interactions given in \\cite{CPS16} to situations where no mean-field approximation is assumed.\nAs for the free case, one has a choice between a quartic term for which the Hamiltonian is bounded from below, which leads to a recollapse and cyclic cosmology \\cite{CPS16}, or an interaction that admits solutions escaping to infinity, corresponding to a Universe expanding forever.\nWe focussed on the second case, the more common one in usual cosmology.\nTo understand the dynamics for this system, we first used a classical approximation in which one considers the basic dynamical variables as commuting phase space functions, with commutators replaced by Poisson brackets.\nIn this case we could derive an exact nonperturbative Friedmann equation; its unusual feature is the appearance of a square root involving the volume and energy on the right-hand side, preventing its straightforward interpretation in terms of effective perfect fluids.\nLinearising this equation in the interaction constant leads to a Friedmann equation with an effective dust term, which would reproduce the mean-field result of \\cite{CPS16}.\nThis linear correction only describes an intermediate regime in the expansion history, after which all orders become relevant.\nWe interpret this as signifying a failure of mean-field methods as soon as interactions become strong.\nThe asymptotic form of the effective Friedmann equation at late times would correspond to a matter component with equation of state $p=\\frac{1}{2}\\rho$ (instead of dust with $p=0$), or a modification of gravity on large scales in this model.\nWe then turned to the full quantum case in which one has to resort to a perturbative or numerical treatment.\nFor Fock coherent states, we find qualitatively similar results to the classical case: we derived a linearised correction to the effective Friedmann equation, and the full numerical solution quickly deviates from this regime as interactions become stronger.\nWe found numerical evidence that relative uncertainties in the volume start growing for Fock coherent states when the interactions become relevant, spoiling the property of these states to become semiclassical at late times that we observed for a quadratic Hamiltonian.\n\nA main direction for future work will be lifting the assumption that only a single field mode contributes to the \\ac{gft} dynamics.\nSince the quadratic part of the full \\ac{gft} Hamiltonian only couples pairs of modes, in the free case it would not be difficult to include additional modes into the analysis.\nFor this case one question would be whether some modes would always dominate asymptotically, as in the mean-field analysis of \\cite{Gie16}.\nWhen adding interaction terms however, we would expect the interaction of different modes to lead to a substantial modification of the effective cosmological dynamics away from the effectively free regime described by an \\ac{lqc}-like bounce.\nIn particular this could apply to the recent proposal of \\cite{GO18} for the generation of cosmological perturbations through quantum fluctuations in \\ac{gft} cosmology.\nIn the long term, we would then also aim to bring the interacting \\ac{gft} `toy' models studied in cosmological applications closer to candidate theories for full quantum gravity.\n\nAn entirely different but conceptually important direction would be to contrast the deparametrised framework used here, in which the scalar field $\\phi$ is used as a clock from the beginning, with a covariant setting in which one is free to choose different clocks, following e.g. the ideas of \\cite{Van18,Hoe18}.\nWe plan to investigate this in models which include multiple candidate matter clocks.\n\n \n\\section*{Acknowledgments}\nWe thank Martin Bojowald, Marco de Cesare, Daniele Oriti, Andreas Pithis and Edward Wilson-Ewing for helpful comments on an earlier version of the manuscript.\nWe also thank the referees for helpful suggestions for improvement.\nThe work of SG was funded by the Royal Society through a University Research Fellowship (UF160622) and a Research Grant for Research Fellows (RGF\\textbackslash R1\\textbackslash 180030).\nAP was supported by the same Research Grant for Research Fellows awarded to SG.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn \\cite{Lo2}, J.-L. Loday introduced a non-antisymmetric version of\nLie algebras, whose bracket satisfies the Leibniz relation (see\n(2.5)), therefore  called {\\it Leibniz algebra}. The Leibniz\nrelation, combined with antisymmetry, is a variation of the Jacobi\nidentity, hence Lie algebras are anti-symmetric Leibniz algebras. In\n\\cite{Lo4}, Loday also introduced an 'associative' version of\nLeibniz algebras, called {\\it associative dialgebras}, equipped with\ntwo binary operations, $\\vdash$ and $\\dashv$, which satisfy the five\nrelations (see the axiom (Ass) in section 2). These identities are\nall variations of the associative law, so associative algebras are\ndialgebras for which the two products coincide. The peculiar point\nis that the bracket $[a, b]=:a\\dashv b-b\\vdash a$ defines a Leibniz\nalgebra which is not antisymmetric, unless the left and right\nproducts coincide. Hence dialgebras yield a commutative diagram of\ncategories and functors\n\\begin{eqnarray*}\n{\\bf Dias}&\\stackrel{-}{\\to}& {\\bf Leib}\\\\\n\\downarrow&&\\downarrow\\\\\n{\\bf Assoc}&\\stackrel{-}{\\to}& {\\bf Lie}\n\\end{eqnarray*}\n\n\nSteinberg Lie algebras come from Steinberg groups, which are closely\nconnected with K-theory, and play a key role in the study of Lie\nalgebras graded by finite root systems of type $A$. By definition,\nthe {\\it Steinberg Lie algebra} $\\frak{st}\\,(n, A)$ over a $K$-algebra $A$\nis a Lie algebra generated by symbols $v_{ij}(a)$, $1\\le i\\ne j\\le\nn$, $a\\in A$, subject to the relations\n\\begin{eqnarray}\n&&v_{ij}(k_1a+k_2b)=k_1v_{ij}(a)+k_2v_{ij}(b),\\  \\hbox{ for } \\ a, b\\in D, \\ k_1,\nk_2\\in K;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=v_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k.\n\\end{eqnarray}\nIt is clear that the relation (3) makes sense only if $n\\ge3$.\n\n From \\cite{Fa} we see that the map $\\eta: a\\to v_{ij}(a)$ is one-to-one if and\n only if $A$ is an associative algebra for $n\\ge 4$ and  $A$ is an alternative\n algebra for $n=3$.\n\n\nIn 1992, S. Berman and R.V. Moody (\\cite{BM}) studied Lie algebras\ngraded by finite root systems of $A_l\\,(l\\ge 2)$, $D_l\\, (l\\ge 4)$,\n$E_l\\, (l=6, 7, 8)$ and obtained the structure of a Lie algebra over\n$K$ graded by the root system $\\Delta$ of type $X_l\\, ( l\\ge2)\\, (\nX_l=A_l, D_l, E_l)$.\n\n\nThe universal central extensions of Lie algebras graded by finite\nroot systems were studied in several papers (\\cite{B}, \\cite{Gar},\n\\cite{KL}, \\cite{Gao1}, \\cite{ABG1}, etc.).\n\n In this paper we shall consider Leibniz algebras graded by finite root systems of\n types $A, D$ and $E$. We also prove that\n\\begin{theo}{\\bf(Recognition Theorem).}\n Let $L$ be a Leibniz algebra over $K$ graded by the root system $\\Delta$ of\n type $X_l(l\\ge2)\\, (X_l=A_l, D_l, E_l)$.\n\n(1) If $X_l=A_l, l\\ge 3$, then there exists a unital associative\n$K$-dialgebra $R$ such that $L$ is centrally isogenous with\n$\\frak{sl}\\,(l+1, R)$;\n\n(2)  If $X_l=A_l, l=2$, then there exists a unital alternative\n$K$-dialgebra $R$ such that $L$ is centrally isogenous with\n$\\frak{stl}\\,(l+1, R)$, where $\\frak{stl}\\,(n, R)$ is defined in Section 2.4;\n\n(3)  If $X_l=D_l\\, (l\\ge 4), E_l\\, (l=6, 7, 8)$, then there exists a\nunital associative commutative $K$-dialgebra $R$ such that $L$ is\ncentrally isogenous with $\\dot{\\mg}\\otimes R$.\n\n\\end{theo}\n{\\bf Remark.} Two perfect Lie algebras $L_1$ and $L_2$ are called\n{\\it centrally isogenous} if they have the same universal central\nextension (up to isomorphism).\n\n\nThe paper is organized as follows. In Section 2, we recall some\nnotions of Leibniz algebras and dialgebras. In Section 3, we give\nthe definition of Leibniz algebras graded by finite root systems. In\nSections 4 and 5, we mainly prove the Recognition Theorem (Theorem\n1.1). Throughout this paper, $K$ denotes a field of characteristic\n0, $R$ a unital dialgebra over $K$.\n\n\n\n\n\\section{Dialgebras and Leibniz algebras}\n\n\nWe recall the notions of associative dialgebras, alternative\ndialgebras, Leibniz algebras and their (co)homology as defined in\n\\cite{Lo1}---\\cite{Lo4} and \\cite{L}.\n\n\n\\subsection{Dialgebras.}\n\n\\begin{defi}\\cite{Lo4}  A {\\it dialgebra} $D$ over $K$ is a $K$-vector space\n$D$ with two operations $\\dashv, \\vdash:D\\otimes D\\to D$, called left\nand right products.\n\\end{defi}\n\nA dialgebra is called unital if it is given a specified bar-unit: an\nelement $1\\in D$ which is a unit for the left and right products\nonly on the bar-side, that is, $1\\vdash a=a=a\\dashv 1$, for any\n$a\\in D$. A morphism of dialgebras is a $K$-linear map $f:D\\to D'$\nwhich preserves the products, i.e. $f(a\\star b)=f(a)\\star f(b)$,\nwhere $\\star$ denotes either the product $\\dashv$ or the product\n$\\vdash$.\n\n\n\\begin{defi} \\cite{Lo4} A dialgebra $D$ over $K$ is called {\\it associative}\nif the two operators $\\dashv$ and $\\vdash$ satisfy the following\nfive axioms:\n$$\\cases{a\\dashv(b\\dashv c)=(a\\dashv b)\\dashv c=a\\dashv(b\\vdash c),\\cr\n        (a\\vdash b)\\dashv c=a\\vdash(b\\dashv c),\\cr\n         (a\\vdash b)\\vdash c=a\\vdash (b\\vdash c)=(a\\dashv b)\\vdash c.  }\\leqno(Ass)$$\n\\end{defi}\n\n\nDenote by {\\bf Dias, Assoc} the categories of associative dialgebras\nand associative algebras over $K$ respectively. Then the category\n{\\bf Assoc} is a full subcategory of {\\bf Dias}.\n\nObviously, an associative dialgebra is an associative algebra if and\nonly if $a\\dashv b=a\\vdash b=ab$.\n\nThe concept of alternative dialgebras was introduced in \\cite{L} for\nthe study of the Steinberg Leibniz algebras.\n\n\\begin{defi}\\cite {L} A dialgebra $D$ over $K$ is called {\\it alternative}\nif the two operators $\\dashv$ and $\\vdash$ satisfying the following five axioms:\n$$\\cases{J_{\\dashv}(a, b, c)=-J_{\\vdash}(c, b, a), \\quad J_{\\dashv}(a, b, c)=J_{\\vdash}(b, c, a),\\cr\n        J_{\\times}(a, b, c)=-J_{\\vdash}(a, c, b),\\cr\n         (a\\vdash b)\\vdash c=(a\\dashv b)\\vdash c,\\quad  a\\dashv (b\\vdash c)\n         =a\\dashv (b\\dashv c), }\\leqno(Alt)$$\nwhere $J_{\\dashv}(a, b, c)=(a\\dashv b)\\dashv c-a\\dashv(b\\dashv c),\\  J_{\\vdash}(a, b, c)\n=(a\\vdash b)\\vdash c-a\\vdash(b\\vdash c)$ and $J_{\\times}(a, b, c)\n=(a\\vdash b)\\dashv c-a\\vdash(b\\dashv c)$.\n\\end{defi}\n\nObviously, an alternative dialgebra is an alternative algebra if\n$a\\dashv b=a\\vdash b=ab$. Moreover, the following formulae are clear\nfor an alternative dialgebras according to the definition.\n$$J_{\\dashv}(a, b, c)=-J_{\\dashv}(a, c, b),\\eqno(2.1)$$\n$$J_{\\vdash}(a, b, c)=-J_{\\vdash}(b, a, c),\\eqno(2.2)$$\n$$J_{\\times}(a, b, c)=-J_{\\times}(c, b, a).\\eqno(2.3)$$\nSo we also have $$J_{\\dashv}(a, b, b)=0, \\  J_{\\vdash}(a, a, b)=0, \\\nJ_{\\times}(a, b, a)=0.\\eqno(2.4)$$ {\\bf Examples.}\n\n1. Obviously, an associative (alternative) dialgebra is an\nassociative (alternative) algebra if and only if $a\\dashv b=a\\vdash\nb=ab$.\n\n2. {\\it Differential associative (alternative) dialgebra.} Let $(A, d)$ be\na differential associative (alternative) algebra. So by hypothesis,\n$d(ab)=(da)b+adb$ and $d^2=0$. Define left and right products on\n$A$ by the formulas\n$$x\\dashv y=xdy, \\quad x\\vdash y=(dx)y.$$\nThen $A$ equipped with these two products is an associative (alternative) dialgebra.\n\n3. Tensor product. Let $D$ and $D'$ be two associative dialgebras,\nthen $D\\otimes D'$ with multiplication $(a\\otimes a')\\star (b\\otimes b')=(a\\star\nb)\\otimes (a'\\star b')$, $\\star=\\dashv, \\vdash$, is also an associative\ndialgebra. Especially, if $D$ is a unital associative dialgebra,\nthen $M_n(D)=M_n(K)\\otimes D$ is also a unital associative dialgebra.\n\n4. Let $D$ be an associative (alternative) algebra.  On the module of $n$-space $D=A^{\\otimes n}$ one puts\n$$(x\\dashv y)_i=x_i(\\sum_{j=1}^ny_j), \\  i=1, \\cdots, n\\quad \\hbox {and}$$\n$$(x\\vdash y )_i=(\\sum_{j=1}^nx_j)y_i, \\  i=1, \\cdots, n. $$\nThen $(D, \\dashv, \\vdash)$ is an associative (alternative) dialgebra. For $n=1$, this is example 1.\n\n\n\n\\subsection{Leibniz algebras.} A {\\it Leibniz algebra} \\cite {Lo2} $L$\nis a vector space over a field $K$ equipped with a $K$-bilinear\nmap\n$$[-,-]: L\\times L\\to L$$\nsatisfying the Leibniz identity\n$$[x, [y, z]]= [[x, y], z]-[[x, z], y], \\quad \\forall \\;x, \\,y, \\,z\\in L.\\eqno(2.5)$$\n\nObviously, a Lie algebra is a Leibniz algebra. A Leibniz algebra is a Lie algebra if\nand only if\n$[x, x]=0$ for all $x\\in L$.\n\n\nSuppose that $L$ is a Leibniz algebra over $K$. For any $z\\in L$,\nwe define $\\hbox{\\rm ad}\\, z\\in \\hbox{End}_kL$ by\n$$\\hbox{\\rm ad}\\, z(x)=-[x, z], \\quad\\forall x\\in L.$$\nIt follows (2.5) that\n$$\\hbox{\\rm ad}\\, z([x, y])=[\\hbox{\\rm ad}\\, z(x), y]+[x, \\hbox{\\rm ad}\\, z(y)]$$\nfor all $x, y\\in L$. This says that $\\hbox{\\rm ad}\\, z$ is a derivation of\n$L$. We also call it an inner derivation of $L$.\n\n\nSimilarly, we also have the definition of general derivation of a\nLeibniz algebra and we denote by $\\hbox{\\rm Inn}\\,(L)$, $\\hbox{Der}\\,(L)$ the set of all\ninner derivations, derivations of $L$ respectively. They are also\nLeibniz algebras.\n\n\nLet $L$ be a Leibniz algebra over $K$. Consider the boundary map: $\\delta_n:L^{\\otimes n}\\to L^{\\otimes (n-1)}$ defined by\n$$\\delta_n(x_1\\otimes\\cdots\\otimes x_n)=\\sum_{1\\le i<j\\le n}(-1)^{j+1}x_1\\otimes \\cdots \\otimes x_{i-1}\\otimes[x_i, x_j]\\otimes x_{i+1}\\otimes\\cdots\\otimes\\hat x_j\\otimes\\cdots\\otimes x_n,$$\nwhere $\\hat x_j$ indicates that the term $x_j$ is omitted. One can\nshow that $\\delta^2=0$ (see \\cite{LP1}) and the complex $(L^{\\otimes\nn},\\, \\delta)\\ (L^0=K,\\, \\delta_1=0)$ gives the Leibniz homology\n$HL_*(L)$ of the Leibniz algebra $L$.\n\nLet $L$ be a Leibniz algebra over $K$. It is called perfect if\n$[L, L]=L$. A central extension of $L$ is a pair $(\\hat L, \\pi)$\nwhere $\\hat L$ is a Leibniz algebra and $\\pi: \\hat L\\to L$ is a\nsurjective homomorphism such that $\\hbox{Ker}\\,\\pi$ lies in the center of\n$\\hat L$ and the exact sequence $0\\to \\hbox{Ker}\\,\\pi\\to \\hat L\\to L\\to 0$\nsplits as $K$-module. The pair $(\\hat L, \\pi)$ is a universal\ncentral extension of $L$ if for every central extension $(\\tilde\nL, \\tau)$ of $L$ there is a unique homomorphism $\\psi:\\hat L\\to\n\\tilde L$ for which $\\tau\\circ\\psi=\\pi$. So the universal central\nextension is unique, up to isomorphism. A Leibniz algebra $L$ has\na universal central extension if and only if $L$ is perfect. If\n$(\\hat L, \\pi)$ is the universal central extension of $L$, then\n$$HL_2(L)\\cong \\hbox{Ker}\\,\\pi.\\eqno(2.6)$$\n\n\n\\noindent{\\bf Remark.} In \\cite{LP1}, \\cite{Gao1}, \\cite{Gao2} and\n\\cite{LH1}, the universal central extensions of  many infinite\ndimensional Lie algebras in the category of Leibniz algebras are\ndetermined.\n\nWe also denote by {\\bf Leib} and {\\bf Lie} the categories of Leibniz\nalgebras and Lie algebras over $K$ respectively.\n\n\nFor any associative dialgebra $D$, define $$[x, y]=x\\dashv y-y\\vdash x,$$\nthen $D$ equipped with this bracket is a Leibniz algebra. We denote it by $D_L$.\n The canonical map $D\\mapsto D_{L}$ induces a functor $(-):$ {\\bf Dias}$\\to${\\bf Leib}.\n\n\nFor a Leibniz algebra $L$, let $L_{Lie}$ be the quotient of $L$ by\nthe ideal generated by the elements $[x, y]+[y, x]$, for all $x,\ny\\in L$. It is clear that $L_{Lie}$ is a Lie algebra. The canonical\nprojection $L\\to L_{Lie}$ is universal among the maps from $L$ to\nLie algebras. In other words, the functor $(-)_{Lie}:$ {\\bf\nLeib}$\\to${\\bf Lie} is left adjoint to $inc:$ {\\bf Lie}$\\to${\\bf\nLeib}.\n\nMoreover, we have the following commutative diagram of categories\nand functors\n\\begin{eqnarray*}\n{\\bf Dias}&\\stackrel{-}{\\to}& {\\bf Leib}\\\\\n\\downarrow&&\\downarrow\\\\\n{\\bf Assoc}&\\stackrel{-}{\\to}& {\\bf Lie}\n\\end{eqnarray*}\n\nAs in the Lie algebra case, the universal enveloping associative\ndialgebra of a Leibniz algebra $L$ is defined as\n$$Ud(L):=(T(L)\\otimes L\\otimes T(L))\/\\{[x, y]-x\\dashv y+y\\vdash x|x, y\\in L\\},$$\nwhere elements $x,y$ of $L$ are regarded as elements in $K\\otimes\nL\\otimes K$.\n\n\\begin{prop}$\\cite{Lo4}$\n\n The functor $Ud:{\\bf Leib}\\to {\\bf Dias}$ is left adjoint to the functor $-:{\\bf Dias}\\to {\\bf Leib}$.\n\\end{prop}\n\n\n\nLet $L$ be a Leibniz algebra, then $M$ is said to be a right $L$-module if $M$ is a $K$-vector space  equipped with the action of $L$:\n$$[-,-]: M\\times L\\to M$$\nsatisfying\n\n$$[m,[x,y]]=[[m,x],y]-[[m,y],x],\\ \\forall x,\\, y\\in L, m\\in M.$$\n\n\n\nSo for a Lie algebra ${\\bf \\frak g}$, any right ${\\bf \\frak g}$-module in the Leibniz algebra case is just the right ${\\bf \\frak g}$-module in the Lie algebra case.\n\n\n\n\\subsection{Lie algebras graded by finite root system.}\n\nFirst we introduce the Steinberg Lie algebra. Steinberg Lie algebras\ncome from Steinberg groups, which are closely connected with\nK-theory.\n\nIf $R$ is a (nonassociative) ring with 1, and $n\\ge3$, the {\\it\nSteinberg group} (see \\cite{Fa}) St$_n(R)$ is the group generated by\nthe symbols $x_{ij}(a), 1\\le i\\ne j\\le n, a\\in R$, subject to the\nrelations\n\\begin{eqnarray*}\n&&x_{ij}(a+b)=x_{ij}(a)x_{ij}(b),\\  \\hbox{ for } \\ a, b\\in R, \\ k_1, k_2\\in K;\\\\\n&&[x_{ij}(a), x_{kl}(b)]=1\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[x_{ij}(a), x_{kl}(b)]=x_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k,\n\\end{eqnarray*} where $[x, y]=xyx^{-1}y^{-1}$ is the group commutator.\n\nBy definition, the {\\it Steinberg Lie algebra} (see \\cite{Fa})\n$\\frak{st}\\,(n, A)$ $(n\\ge 3)$ over a $K$-algebra $A$ is a Lie algebra\ngenerated by symbols $u_{ij}(a)$, $1\\le i\\ne j\\le n$, $a\\in A$,\nsubject to the relations\n\\begin{eqnarray*}\n&&u_{ij}(k_1a+k_2b)=k_1u_{ij}(a)+k_2u_{ij}(b),\\  \\hbox{ for } \\ a, b\\in A, \\ k_1, k_2\\in K;\\\\\n&&[u_{ij}(a), u_{kl}(b)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[u_{ij}(a), u_{kl}(b)]=u_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k.\n\\end{eqnarray*}\n\nDefine $i_n(A)=\\{a\\in A\\mid u_{ij}(a)=0 \\}$. Clearly, it is an ideal\nof $A$ and does not depend on the choice of $i\\ne j$.\n\n From \\cite{Fa}, we see that $i_n(A)=0$ if and only if $A$ is an associative algebra\n for $n\\ge 4$ and  $A$ is an alternative algebra for $n=3$.\n\nBy definition, a $K$-algebra $A$ is called {\\it alternative} if $(a, b,\nc)=-(c, b, a)=(b, c, a)$, where $(a, b, c)=(ab)c-a(bc)$.\n\n\nA Lie algebra $L$ over a field $K$ of characteristic 0 is {\\it\ngraded by the (reduced) root system} (see \\cite{BM}) $\\Delta$ or\nis {\\it $\\Delta$-graded} if\n\n(1) L contains as a subalgebra a finite-dimensional simple Lie\nalgebra $\\dot{\\mg}=H\\oplus\\bigoplus_{\\alpha\\in\\Delta}\\dot{\\mg}_{\\alpha}$ whose root\nsystem is $\\Delta$ relative to a split Cartan subalgebra $H=\\dot{\\mg}_0$;\n\n(2) $L=\\bigoplus_{\\alpha\\in\\Delta\\cup\\{0\\}}L_{\\alpha}$, where\n$L_{\\alpha}=\\{x\\in L\\mid [h, x]=\\alpha(h)x, \\forall h\\in H\\}$ for $\\alpha\\in \\Delta\\cup\\{0\\}$; and\n\n(3) $L_0=\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}]$.\n\n\n\n\n\\begin{theo} \\cite{BM}\n Let $L$ be a Lie algebra over $K$ graded by the root system $\\Delta$ of\n type $X_l\\, ( l\\ge2)\\, ( X_l=A_l, D_l, E_l)$ .\n\n(1) If $X_l=A_l,\\, l\\ge 3$, then there exists a unital associative\n$K$-algebra $A$ such that $L$ is centrally isogenous with\n$\\frak{sl}\\,(l+1, A)$.\n\n(2)  If $X_l=A_l,\\, l=2$, then there exists a unital alternative\n$K$-algebra $A$ such that $L$ is centrally isogenous with\n$\\frak{st}(l+1, A)$.\n\n(3)  If $X_l=D_l\\,(l\\ge 4), \\, E_l\\,(l=6, 7, 8)$, then there\nexists a unital associative commutative $K$-algebra $A$ such that\n$L$ is centrally isogenous with $\\dot{\\mg}\\otimes A$.\n\\end{theo}\n\n\n\n\n\n\n\n\\subsection{Steinberg Leibniz algebras }\n\n\nThe matrix Leibniz algebra\n$\\frak{gl}(n, D)$ is generated by all $n\\times n$ matrices with coefficients from a unital\nassociative dialgebra $D$, and $n\\ge 3$ with the bracket\n$$[E_{ij}(a), E_{kl}(b)]=\\delta_{jk}E_{il}(a\\dashv b)-\\delta_{il}E_{kj}(b\\vdash a),$$ for all $a, b\\in D$, where $E_{ij}(a)$ is the $n\\times n$ matrix with coefficient $a$ on $(i, j)$-th position and 0 in all others.\n\nClearly, $\\frak{gl}(n, D)$ is a Leibniz algebra. If $D$ is an associative algebra, then  $\\frak{gl}(n, D)$ becomes a Lie algebra.\n\n\nNow we consider the subalgebra $\\frak{sl}\\,(n, D):=[\\,\\frak{gl}(n, D), \\frak{gl}(n, D)\\,]$, which is called the {\\it special linear Leibniz algebra} with coefficients in $D$, of $\\frak{gl}(n, D)$.\n\n\nBy definition, the {\\it special linear Leibniz algebra} $\\frak{sl}\\,(n, D)$\nhas generators $E_{ij}(a),\\, 1\\le i\\ne j\\le n, a\\in D$, which\nsatisfy the following relations:\n\\begin{eqnarray*}\n&&[E_{ij}(a), E_{kl}(b)]=0\\  \\hbox{if}\\ i\\ne l\\ \\hbox{and}\\  j\\ne k;\\\\\n&&[E_{ij}(a), E_{kl}(b)]=E_{il}(a\\dashv b)\\  \\hbox{if}\\ i\\ne l\\ \\hbox{and}\\  j= k;\\\\\n&&[E_{ij}(a), E_{kl}(b)]=-E_{kj}(b\\vdash a)\\ \\hbox{if}\\ i= l\\\n\\hbox{and}\\  j\\ne k.\n\\end{eqnarray*}\n\n\nThe Steinberg Leibniz algebra was first introduced in \\cite{LP1} for\nassociative algebras and in \\cite{L} for associative dialgebras. By\ndefinition, the {\\it Steinberg Leibniz algebra} $\\frak{stl}\\,(n, D)$ is a\nLeibniz algebra generated by symbols $v_{ij}(a)$, $1\\le i\\ne j\\le\nn$, $a\\in D$, subject to the relations\n\\begin{eqnarray}\n&&v_{ij}(k_1a+k_2b)=k_1v_{ij}(a)+k_2v_{ij}(b),\\  \\hbox{ for } \\ a, b\\in D, \\ k_1, k_2\\in K;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=0,\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=v_{il}(a\\dashv b)\\  \\hbox{ if } \\ i\\ne l\\ \\hbox{and } \\  j= k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=-v_{kj}(b\\vdash a)\\  \\hbox{ if } \\ i= l\\\n\\hbox{and} \\  j\\ne k.\n\\end{eqnarray}\nIt is clear that the relations (6)---(7) make sense only if\n$n\\ge3$.\n\n\n\nLet $H_{ij}(a, b):=[v_{ij}(a), v_{ji}(b)]$ for $1\\le i\\ne j\\le n, a,\nb\\in D$, and $H$ the submodule of $\\frak{stl}\\,(n, D)$ generated by\n$H_{ij}(a, b), i\\ne j, a, b\\in D$. Define $i_n(D)=\\{a\\in D\\mid\nv_{ij}(a)=0 \\}$. Clearly, it is an ideal of $D$ and does not depend\non the choice of $i\\ne j$. The same consideration as in Steinberg\nLie algebras, we have\n\\begin{prop}\\cite{L}\n For a unital dialgebra $D$, $i_n(D)=0$ in $\\frak{stl}\\,(n, D)$ if and only if $D$ is\n associative for $n\\ge 4$ and  $D$ is alternative for $n=3$.\n\\end{prop}\n\nThe Steinberg Leibniz algebra $\\frak{stl}\\,(n, D)$ with $n\\ge3$ is perfect.\n\nThe homomorphism $\\psi$ of Leibniz algebras\n$$\\psi: \\quad \\frak{stl}\\,(n, D)\\to \\frak{sl}\\,(n, D)$$ by the rule\n$\\psi(v_{ij}(a))=E_{ij}(a)$ is a surjective homomorphism.\n\n\\begin{theo}\\cite{L}\nIf $n\\ge3$, then $(\\frak{stl}\\,(n, D), \\psi)$ is the universal central\nextension of the Leibniz algebra $\\frak{sl}\\,(n, D)$ with kernel $HHS_1(D)$\nfor a unital associative dialgebra $D$, where $HHS_1(D)$ is the\nfirst homology group of chain complex $(CS_*(D), d)$ defined in\n\\cite{Fa} (or see \\cite{L}) .\n\\end{theo}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{ Leibniz algebras graded by finite root systems}\n\n\\begin{defi}\nA Leibniz algebra $L$ over a field $K$ of characteristic 0 is\ngraded by the $($reduced$)$ root system $\\Delta$ or is\n$\\Delta$-graded if\n\n$(1)$ L contains as a subalgebra a finite-dimensional simple Lie\nalgebra $\\dot{\\mg}=H\\oplus\\bigoplus_{\\alpha\\in\\Delta}\\dot{\\mg}_{\\alpha}$ whose root\nsystem is $\\Delta$ relative to a split Cartan subalgebra $H=\\dot{\\mg}_0$;\n\n$(2)$ $L=\\bigoplus_{\\alpha\\in\\Delta\\cup\\{0\\}}L_{\\alpha}$, where\n$L_{\\alpha}=\\{x\\in L\\mid \\hbox{\\rm ad}\\, h(x)=-[x, h]=\\alpha(h)x, \\forall h\\in H\\}$\nfor $\\alpha\\in \\Delta\\cup\\{0\\}$; and\n\n$(3)$ $L_0=\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}]$.\n\\end{defi}\n\n{\\bf Remarks.}\n\n1. The conditions for being a $\\Delta$-graded Leibniz algebra imply\nthat $L$ is a direct sum of finite-dimensional irreducible right\n$\\dot{\\mg}$-modules whose highest weights are roots, hence either the\nhighest long root or short root or $0$.\n\n2. If $L$ is $\\Delta$-graded, then $L$ is perfect. Indeed,  the\nresult follows from $L_\\alpha=[L_\\alpha, H]$ for all $\\alpha\\in\\Delta$ and (3)\nas above.\n\n3. The Steinberg Leibniz algebra $\\frak{stl}\\,(n, D)$ is graded by the\nroot system of type $A_{n-1}$. Let $D$ be a commutative dialgebra,\nthen the Leibniz algebra $\\dot{\\mg}\\otimes D$ and its central extensions\nare graded by the root system of type $\\dot{\\mg}$ (see \\cite{LL}).\n\nNow we shall prove the Recognition Theorem (Theorem 1.1). So from\nnow on, we always set $\\Delta$ to be the root system of type $A_l\\,\n(l\\ge2)$, $D_l\\, (l\\ge 4)$ or $E_l \\,(l=6, 7, 8)$.\n\nSuch as that in \\cite{BM}, we also have the following results.\n\\begin{defi}\nAn ordered pair $(\\beta, \\gamma)\\in \\Delta\\times\\Delta$ is an $A_2$-pair if\n$(\\beta, \\gamma)=-1$. Thus $(\\beta, \\gamma)$ is an $A_2$-pair if and only if it is\na base for an $A_2$ subroot system\n of $\\Delta$.\n Two $A_2$-pairs $(\\beta, \\gamma)$ and $(\\beta', \\gamma')$ are equivalent,\n written  $(\\beta, \\gamma)\\sim(\\beta', \\gamma')$,\n if there is an element $w$ of the Weyl group $W$ of $\\Delta$ such that $\\beta'=w\\beta$\n and $\\gamma'=w\\gamma$.\nThe equivalent class of $(\\beta, \\gamma)$ is denoted by $[(\\beta, \\gamma)]$.\n\\end{defi}\n\n\\begin{lemm}\\cite{BM}\n$(1)$ If $\\Delta$ is of type $D$ or $E$, then there is only one\nequivalent class of $A_2$-pairs.\n\n$(2)$ If $\\Delta$ is of type $A$, then there are exactly two\nequivalent classes of $A_2$-pairs and furthermore, if $(\\beta, \\gamma)$ is\nan $A_2$-pair, then\n$$(\\beta, \\gamma)\\sim(-\\gamma, -\\beta), \\  (\\beta, \\gamma)\\not\\sim(\\gamma, \\beta).$$\n\n$(3)$ In all cases if $(\\beta, \\gamma)$ and $(\\gamma, \\delta)$ are $A_2$-pairs\nwith $(\\beta\\mid\\delta)=0$, then\n$$(\\beta, \\gamma)\\sim(\\gamma, \\delta)\\sim(\\beta, \\gamma+\\delta)\\sim(\\beta+\\gamma, \\delta).\\quad  \\rule[-.23ex]{1.0ex}{2.0ex}$$\n\\end{lemm}\n\n\n{\\bf Remark.} In type $A_l$, for definiteness, we distinguish the\ntwo classes as follows. We choose a base $\\Pi=\\{\\alpha_1, \\cdots,\n\\alpha_l\\}$ for $\\Delta$ once and for all with Coxeter-Dynkin diagram\n$\\alpha_1\\circ$----$\\circ$----$\\cdots$----$\\circ\\alpha_l$. Then $[(\\alpha_1,\n\\alpha_2)]$ is the positive class and an $A_2$-pair $(\\alpha, \\beta)\\in [\\alpha_1,\n\\alpha_2]$ is called positive pair, $[(\\alpha_2, \\alpha_1)]$ is the negative\nclass and an $A_2$-pair $(\\alpha, \\beta)\\in [\\alpha_2, \\alpha_1]$ is called\nnegative pair. For convenience, any $A_2$-pair in types $D_l$ and\n$E_l$ is either positive and negative.\n\nLet $L$ be a Leibniz algebra graded by $\\Delta$ and $\\dot{\\mg}\\subset L$\nthe split simple Lie algebra of Definition 3.1. Let $\\{e_\\alpha, H_i\\mid\n\\alpha\\in\\Delta, i=1, \\cdots, l\\}$ be a Chevalley basis of $\\dot{\\mg}$ so\nthat $e_\\alpha\\in L_\\alpha$ for any $\\alpha\\in\\Delta$, and $\\{H_i, i=1, \\cdots,\nl\\}$ a basis of $H$. Let $G=\\langle \\exp te_\\alpha\\mid t\\in K\\rangle$ be\nthe corresponding simply connected Chevalley group. For each\n$\\alpha\\in\\Delta$, $\\{e_\\alpha,\\, \\alpha^{\\vee}=[e_\\alpha, e_{-\\alpha}],\\, e_{-\\alpha}\\}$ is\nan $\\frak{sl}\\,_2$-triplet. Let\n$$n_\\alpha(t)=\\exp te_\\alpha\\exp(-t^{-1}e_{-\\alpha})\\exp te_\\alpha$$ and\nset $$N=\\langle n_\\alpha(t)\\mid \\alpha\\in\\Delta,\\, t\\in K^{\\times}\\rangle,$$\nwhere $K^{\\times}$ is the set of non-zero elements of $K$.\n\nLet $h_\\alpha(t)=n_\\alpha(t)n_\\alpha(1)^{-1}$ and $T=\\langle h_\\alpha(t)\\mid\n\\alpha\\in\\Delta,\\, t\\in K^{\\times}\\rangle$. Then $$T\\lhd N,\\, N\/T\\cong\nW.$$\n\nClearly, from the $\\Delta$-grading $\\hbox{\\rm ad}\\, e_\\alpha$ and  $\\hbox{\\rm ad}\\, e_{-\\alpha}$ are\nnilpotent on $L$, so we can define a homomorphism\n$$\\hbox{\\rm Ad}\\,: G\\to \\hbox{Aut}\\,(L)\\quad \\hbox{by}\\quad \\exp te_\\alpha\\to \\exp\\hbox{\\rm ad}\\, te_\\alpha, \\  \\alpha\\in \\Delta,\\, t\\in K.$$\n\nNext, recall that if $M$ is any integrable (right) $\\dot{\\mg}$-module\n(one on which $e_\\alpha$ acts locally nilpotent for all $\\alpha\\in\\Delta$)\nwith weight space decomposition $M=\\oplus M_\\lambda$ relative to $\\dot{\\mg}$\nthen, letting $r_\\alpha$ denote the reflection $r_\\alpha\\lambda=\\lambda-\\langle \\lambda,\n\\alpha^{\\vee}\\rangle\\alpha$, we have\n$$n_\\alpha(t)M_\\lambda=M_{r_\\alpha\\lambda} \\quad \\hbox{for all weights}\\ \\lambda,\\ \\hbox{and} \\eqno(3.1)$$\n$$\\hbox{the action of}\\  h_\\alpha(t)\\ \\hbox{restricted to}\\  M_\\lambda\\  \\hbox{is a\nscalar-multiplication by}\\  t^{\\langle \\lambda, \\alpha^{\\vee}\\rangle}. \\eqno(3.2)$$\n\nIn particular, the adjoint representation restricted to $\\dot{\\mg}$ acts\non $L$ in this way.\n\nFix any $\\alpha\\in\\Delta$. Let $W^\\alpha$ denote the stabilizer of $\\alpha$ in\n$W$ and let $N^\\alpha:=W^{\\alpha}T$. Then\n$$W^\\alpha=\\langle r_\\beta\\mid \\beta\\in\\Delta, (\\beta|\\alpha)=0\\rangle, \\eqno(3.3)$$\n$$N^\\alpha=\\langle n_\\beta(1)\\mid \\beta\\in\\Delta, (\\beta|\\alpha)=0\\rangle\\cdot T.\\eqno(3.4)$$\n\nFix any $\\beta\\in \\Delta$ and choose $w\\in W$ with $w\\alpha=\\beta$. Choose any\n$n\\in N$ with $nT\\leftrightarrow w$ in the isomorphism $N\/T\\cong W$.\nThen $\\hbox{\\rm Ad}\\,(n)L_\\alpha=L_\\beta$ by (3.1). Let the restriction of $\\hbox{\\rm Ad}\\,(n)$ to\n$L_\\alpha$ be denoted by $\\tilde\\lambda_{\\beta, \\alpha}$. If also $w'\\alpha=\\beta$ and\n$n'T\\leftrightarrow w'$, then $w^{-1}w'\\in W^\\alpha$ and $n^{-1}n'\\in\nN^\\alpha$. Thus $n'\\in nN^\\alpha=n\\langle n_\\gamma(1)\\mid \\gamma\\in\\Delta,\\,\n(\\beta|\\alpha)=0\\rangle T$. Elements of $T$ acts as scalar multiplications\non $L_\\alpha$. Furthermore,\n$$(\\gamma|\\alpha)=0\\Rightarrow (\\alpha\\pm\\gamma, \\alpha\\pm\\gamma)=4\\Rightarrow \\alpha\\pm\\gamma\\not\\in\\Delta\\cup\\{0\\}$$\n$$\\Rightarrow \\hbox{\\rm Ad}\\,\\exp(te_{\\pm\\gamma})\\ \\hbox{acts as the identity on}\\ L_\\alpha, \\forall\\, t\\in K^{\\times}\\Rightarrow n_\\gamma(1)\\ \\hbox{acts as the identity on}\\ L_\\alpha.$$\nThis establishes that $\\tilde\\lambda_{\\beta, \\alpha}$ is determined, up to a\nnonzero scalar multiple, by $\\alpha$ and $\\beta$ and does not otherwise\ndepend on our choice of $w$ or $n$. Since also $n\\dot{\\mg}_\\alpha=\\dot{\\mg}_\\beta, \\\nne_\\alpha=\\varepsilon e_\\beta$ for some $\\varepsilon\\in K^{\\times}$ (actually $\\varepsilon=\\pm\n1$).\n\n\\begin{defi}  $\\lambda_{\\beta, \\alpha}: L_\\alpha\\to L_\\beta$ is the $K$-linear map\n$\\varepsilon^{-1}\\tilde\\lambda_{\\beta, \\alpha}.$\n\\end{defi}\n\nClearly, $\\lambda_{\\beta, \\alpha}$ depends only on $\\alpha, \\beta$ and the choice of\nChevally basis. We see that\n$$\\lambda_{\\alpha, \\beta}=\\lambda_{\\beta, \\alpha}^{-1}, \\eqno(3.5)$$\n$$\\lambda_{\\alpha, \\beta}\\lambda_{\\beta, \\gamma}=\\lambda_{\\alpha, \\gamma}, \\eqno(3.6)$$\n$$\\lambda_{\\alpha, \\alpha}=1. \\eqno(3.7)$$\n\nWith $\\alpha$ fixed as above, define $R=L_\\alpha$ (as a $K$-space).\nEventually, $R$ will have a life of its own, independent of $\\alpha$.\nFor this reason, given $r\\in R$, we shall write it as $e_\\alpha(r)$ when\nwe think of it as being in $L_\\alpha$. We identify $K$ as a subspace of\n$L_\\alpha$ by $ae_\\alpha=e_\\alpha(a)$ for some $a\\in K$.\n\nNow if $\\beta\\in \\Delta $ is arbitrary, then the $K$-linear map\n$\\lambda_{\\beta, \\alpha}:L_\\alpha\\to L_\\beta$ is defined and we define\n$$e_\\beta(r)=\\lambda_{\\beta, \\alpha}e_\\alpha(r).\\eqno(3.8)$$\n From (3.7), we see that this definition is consistent when $\\beta=\\alpha$.\nFrom (3.5) and (3.6), we see that (3.8) holds for all pairs of roots\n $\\alpha, \\beta\\in\\Delta.$\n\nWe note that for all $a, b\\in K,\\, r, s\\in R$, we have\n$e_\\beta(ar+bs)=ae_\\beta(r)+be_\\beta(s)$. This allows us to write $x_\\beta(r),\\,\nr\\in R$, for any $x_\\beta\\in \\dot{\\mg}_\\beta$ unambiguously. Clearly,\n$$x_\\beta(r)=x_\\beta(s)\\Longleftrightarrow r=s.$$\n\n\\begin{lemm}\nFor all $n\\in N,\\, \\beta\\in\\Delta,\\, r\\in R$, we have\n$$(\\hbox{\\rm Ad}\\, n)(e_\\beta(r))=((\\hbox{\\rm Ad}\\, n)e_\\beta)(r).$$\n\\end{lemm}\n{\\bf Proof.} Let $nT\\leftrightarrow w\\in W$ and $\\gamma=w\\beta$, so both\nsides of the asserted equality lie in $L_\\gamma$. Since $\\lambda_{\\gamma, \\beta}$\nequals, up to a scalar, $\\hbox{\\rm Ad}\\,(n)|_{L_\\beta}$, $b\\lambda_{\\gamma, \\alpha}=\\hbox{\\rm Ad}\\,\nn(\\lambda_{\\beta, \\alpha})$, for some $b\\in K^{\\times}$. Then\n$$be_\\gamma=b\\lambda_{\\gamma, \\alpha}e_\\alpha=(\\hbox{\\rm Ad}\\, n)\\lambda_{\\beta, \\alpha}e_\\alpha=(\\hbox{\\rm Ad}\\, n)e_\\beta$$ and hence\n$$(\\hbox{\\rm Ad}\\, n)e_\\beta(r)=(\\hbox{\\rm Ad}\\, n)\\lambda_{\\beta, \\alpha}e_\\alpha(r)=b\\lambda_{\\gamma, \\alpha}e_\\alpha(r)=be_\\gamma(r)\n=((\\hbox{\\rm Ad}\\, n)e_\\beta)(r).\\quad \\rule[-.23ex]{1.0ex}{2.0ex}$$\n\nNow we set $$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](m_{(\\beta, \\gamma)}(r,\ns)).\\eqno(3.9)$$\n\n\\begin{lemm}\nIf $(\\beta', \\gamma')\\in [(\\beta, \\gamma)]$, then $m_{(\\beta', \\gamma')}=m_{(\\beta, \\gamma)}$.\n\\end{lemm}\n{\\bf Proof.}  Let $m'=m_{(\\beta', \\gamma')}, m=m_{(\\beta, \\gamma)}$. Choose $w\\in\nW$ with $\\beta'=w\\beta$ and $\\gamma'=w\\gamma$. Let $a, b, c, \\varepsilon, \\varepsilon'\\in\nK^{\\times}$ be chosen such that\n$$a\\hbox{\\rm Ad}\\,(n)e_\\beta=e_{\\beta'}, \\  b\\hbox{\\rm Ad}\\,(n)e_\\gamma=e_{\\gamma'},\\  c\\hbox{\\rm Ad}\\,(n)e_{\\beta+\\gamma}=e_{\\beta'+\\gamma'},$$\n$$[e_\\beta, e_\\gamma]=\\varepsilon e_{\\beta+\\gamma}, \\  [e_{\\beta'}, e_{\\gamma'}]=\\varepsilon' e_{\\beta'+\\gamma'}.$$\nSince $\\hbox{\\rm Ad}\\,(n)$ is an automorphism, $\\varepsilon'=abc^{-1}\\varepsilon$. Now for\n$r, s\\in R$,\n\\begin{eqnarray*}\n&&[e_{\\beta'}, e_{\\gamma'}]m'(r, s)=[e_{\\beta'}(r), e_{\\gamma'}(s)]\\\\\n&=&[\\lambda_{\\beta', \\beta}e_{\\beta}(r), \\lambda_{\\gamma', \\gamma}e_{\\gamma}(s)]\\\\\n&=&[a\\hbox{\\rm Ad}\\,(n)e_\\beta(r), b\\hbox{\\rm Ad}\\,(n)e_\\gamma(s)]=ab\\hbox{\\rm Ad}\\,(n)[e_\\beta(r), e_\\gamma(s)]\\\\\n&=&ab\\hbox{\\rm Ad}\\,(n)[e_\\beta, e_\\gamma](m(r, s))\\\\\n&=&ab\\varepsilon\\hbox{\\rm Ad}\\,(n)e_{\\beta+\\gamma}(m(r, s))=ab\\varepsilon c^{-1} e_{\\beta'+\\gamma'}(m(r, s))\\\\\n&=&\\varepsilon'e_{\\beta'+\\gamma'}(m(r, s))=[e_{\\beta'}, e_{\\gamma'}](m(r, s)).\n\\end{eqnarray*}\nThus  $m_{(\\beta', \\gamma')}=m_{(\\beta, \\gamma)}$  and $m=m'$.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n From the above Lemma, we can define two multiplications on $R$:\n\n(1) For a positive $A_2$-pair $(\\beta, \\gamma)$ (see the Remark after Lemma\n3.3), we define $\\dashv: R\\times R\\to R$ given by\n$$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](r\\dashv s).$$\n\n(2) For a negative $A_2$-pair $(\\beta, \\gamma)$, we define $\\vdash: R\\times\nR\\to R$ given by\n$$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](s\\vdash r).$$\n\nClearly from Lemmas 3.5 and 3.6, we know that the above definition\nis well-defined on $D$. Moreover, if $\\Delta$ is  of type $D_l\\;\n(l\\ge4)$ or $E_l\\; (l=6, 7, 8)$, then $r\\dashv s=s\\dashv r$, i.e.,\n$R$ is a commutative dialgebra since $m_{\\beta, \\gamma}(r, s)=m_{\\gamma, \\beta}(r,\ns)$.\n\n\nLet $(\\alpha, \\beta)$ be a $A_2$-pair. Then $r_\\beta\\alpha=\\alpha+\\beta$. Suppose that\n$e_\\alpha, e_\\beta$ have already been chosen as our Chevalley basis. Let\n$n_\\beta=\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})\\exp\\hbox{\\rm ad}\\, e_\\beta$. Then by\ndefinition, we get that\n\\begin{eqnarray*}\nn_\\beta e_\\alpha(r)=n_\\beta(e_\\alpha(r))&=&\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})\\exp\\hbox{\\rm ad}\\, e_\\beta(e_\\alpha(r))\\\\\n&=&\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})(e_\\alpha(r)-[e_\\alpha(r), e_\\beta])\\\\\n&=&\\exp\\hbox{\\rm ad}\\, e_\\beta(e_\\alpha(r)-[e_\\alpha(r), e_\\beta]-[e_\\alpha(r)+[e_\\alpha(r), e_\\beta], e_{-\\beta}])\\\\\n&=&-[e_\\alpha(r), e_\\beta].\n\\end{eqnarray*}\nThus $$(n_\\beta e_\\alpha)(r)=-[e_\\alpha(r), e_\\beta].$$ In particular, $$n_\\beta\ne_\\alpha=-[e_\\alpha, e_\\beta].$$\n\nThe element $e_\\alpha\\in L_\\alpha$ defines an element of $R$ that has been\nidentified with $1\\in K: e_\\alpha(1)=1e_\\alpha$. For any $\\beta\\in \\Delta$ we\nhave $e_\\beta(1)=\\lambda_{\\beta, \\alpha}(e_\\alpha(1))=e_\\beta$, so every basis element\n$e_\\beta$ defines the same element of $R$.\n\nNote that $1$ is bar-unit of $R$. For a positive $A_2$-pair $(\\alpha,\n\\beta)$, we have $[e_\\alpha(r), e_\\beta(1)]=[e_\\alpha, e_\\beta](r\\dashv 1)$. But\n\\begin{eqnarray*}\n[e_\\alpha(r), e_\\beta(1)]&=&[e_\\alpha(r), e_\\beta]=-n_\\beta(e_\\alpha(r))\\\\\n&=&-(n_\\beta e_\\alpha)(r)=[e_\\alpha, e_\\beta](r).\n\\end{eqnarray*}\nThus $r\\dashv 1=r$. Similarly, for a negative $A_2$-pair $(\\alpha, \\beta)$,\nwe can prove that $1\\vdash r=r.$ So $R$ is a unital dialgebra.\n\nNext, we investigate the associativity of the multiplication on $R$.\n\nNow we assume that $l\\ge 3$. Let $(\\beta, \\gamma)$ and $(\\gamma, \\delta)$ be two\npositive $A_2$-pairs with $(\\beta, \\delta)=0$ (such pairs exist if\n$l\\ge3$). By Lemma 3.3 (3), $m_{(\\beta, \\gamma)}=m_{(\\gamma, \\delta)}$. By the\nJacobi identity and $[L_\\beta, L_\\delta]=0$, we obtain that\n$$[[e_\\beta(r), e_\\gamma(s)], e_\\delta(t)]=[e_\\beta(r), [e_\\gamma(s),  e_\\delta(t)]],$$\n$$[e_\\beta(r), [e_\\delta(t), e_\\gamma(s)]]=-[[e_\\beta(r), e_\\gamma(s)],  e_\\delta(t)],$$\n$$[[e_\\gamma(s), e_\\beta(r)], e_\\delta(t)]=-[[e_\\gamma(r), e_\\delta(t)],  e_\\gamma(s)].$$\nIt follows that $(r\\dashv s)\\dashv t=r\\dashv (s\\dashv\nt)=r\\dashv(s\\vdash t)$ and $(r\\vdash s)\\dashv t=r\\vdash (s\\dashv\nt)$. Similarly, we can prove the left identities in the axiom(Ass).\nThe $l=2$ case will be handled in \\S5.2.\n\n\nFrom the above, we have proved the following theorem.\n\\begin{theo}\nLet $L$ be a $\\Delta$-graded Leibniz algebra over a field $K$ of\ncharacteristic $0$, where $\\Delta$ is a simply-laced finite\nindecomposable root system of rank $l\\ge 2$. Let $\\dot{\\mg}$ be the\nassociated split simple Lie subalgebra with root system $\\Delta$.\nFor any root $\\alpha\\in\\Delta$ and let $R=L_\\alpha$ as a $K$-vector space.\nRelative to a Chevally basis $\\{e_\\beta\\mid \\beta\\in\\Delta\\}\\cup\\{H_i\\mid\ni=1, \\cdots, l\\}$, define the map $\\lambda_{\\beta, \\alpha}:L_\\alpha\\to L_\\beta$ of\nDefinition 3.4 and the element $e_\\beta(r), r\\in R$ of (3.8). Then $R$\nis a unital $K$-dialgebra. Moreover, $R$ is associative if the rank\n$l\\ge 3$. If $\\Delta$ is of type $D$ or $E$, then $R$ is\ncommutative.\n\\end{theo}\n\n\n\n\n\n\\section{Centrally isogenous of $\\Delta$-graded Leibniz algebra}\n\nLet $L$ and $L'$ be $\\Delta$-graded Leibniz algebras over $K$ for\nthe same finite root system $\\Delta$. Then their associated split\nsimple subalgebras $\\dot{\\mg}$ and $\\dot{\\mg}'$ are isomorphic. Simply denoted\nthem by $\\dot{\\mg}$, so $\\dot{\\mg}$ is a subalgebra both of $L$ and $L'$.\n\n\\begin{defi}\nA homomorphism $\\varphi: L\\to L'$ is $\\Delta$-homomorphic if\n$\\varphi|_{\\dot{\\mg}}=id_{\\dot{\\mg}}$.\n\\end{defi}\n\nLet $\\varphi: L\\to L'$ be a $\\Delta$-homomorphism. From the definition,\nit follows at once that $\\varphi(L_\\alpha)\\subset L_\\alpha'$ for all $\\alpha\\in\n\\Delta\\cup\\{0\\}$. Let $R$ and $R'$ be the $K$-dialgebras associated\nto $L$ and $L'$ respectively defined by a certain choice of\nChevalley basis for $\\dot{\\mg}$. For each $\\alpha\\in \\Delta$, we have\n$R=L_\\alpha\\stackrel{\\varphi}{\\to}L_\\alpha'=R'$, so $\\varphi$ determines a map\n$$\\bar\\varphi_\\alpha: R\\to R'.$$\n\\begin{prop}\n$(1)$ $\\bar\\varphi_\\alpha: R\\to R'$ is independent of the choice of\n$\\alpha\\in\\Delta$ (so we can denote it by $\\bar\\varphi$) and is a\nhomomorphism of dialgebras. Furthermore, $\\bar\\varphi$ is injective\n(resp. surjective, bijective) if $\\varphi$ is injective (resp.\nsurjective, bijective).\n\n$(2)$ If $\\bar\\varphi$ is an isomorphism, then the $\\Delta$-homomorphism\n$\\varphi$ is central and $L$ and $L'$ are centrally isogenous.\n\\end{prop}\n\n{\\bf Proof.} (1) Let $\\{e_\\beta\\}_{\\beta\\in\\Delta}\\cup \\{H_i\\}_{i=1}^l$ be\nthe Chevalley basis of $\\dot{\\mg}$. We have $L_\\alpha=\\{e_\\alpha(r)\\mid r\\in R\\}$\nand $L_\\alpha'=\\{e_\\alpha(r)\\mid r\\in R'\\}$. Furthermore, $ke_\\alpha=e_\\alpha(k)$\nfor all $k\\in K$ and $e_\\alpha(k_1r+k_2s)=k_1e_\\alpha(r)+k_2e_\\alpha(s)$ for all\n$r, s\\in R, R'$, $ k_1, k_2\\in K$. By definition,\n$\\varphi(e_\\alpha(r))=e_\\alpha(\\bar\\varphi_\\alpha(r))$ for all $r\\in R$, and it follows\nthat $\\bar\\varphi_\\alpha$ is $K$-linear and maps $1\\in R$ to $1\\in R'$.\n\nNow suppose that $(\\alpha, \\beta)$ is a positive $A_2$-pair. Then\n\\begin{eqnarray*}\n[e_\\alpha, e_\\beta](\\bar\\varphi_{\\alpha+\\beta}(r\\dashv s))&=&\\varphi([e_\\alpha, e_\\beta](r\\dashv s))\n=\\varphi([e_\\alpha(r), e_\\beta(s)])\\\\\n&=&[e_\\alpha(\\bar\\varphi_\\alpha(r)), e_\\beta(\\bar\\varphi_\\beta(s))]=[e_\\alpha,\ne_\\beta](\\bar\\varphi_\\alpha(r)\\dashv\\bar\\varphi_\\beta(s)).\n\\end{eqnarray*}\nThus  $$\\bar\\varphi_{\\alpha+\\beta}(r\\dashv\ns)=\\bar\\varphi_\\alpha(r)\\dashv\\bar\\varphi_\\beta(s).\\eqno(4.1)$$ Similarly, we have\n$$\\bar\\varphi_{\\alpha+\\beta}(r\\vdash\ns)=\\bar\\varphi_\\alpha(r)\\vdash\\bar\\varphi_\\beta(s).\\eqno(4.2)$$\n\n\nWith $s=1$, we get $\\bar\\varphi_{\\alpha+\\beta}(r)=\\bar\\varphi_\\alpha(r)$. From this, it\nis easy to see that $\\bar\\varphi$ is independent of $\\alpha$. Thus, $(4.1)$\nand (4.2) show the homomorphism property of $\\bar\\varphi$. The remaining\nparts of (1) are obvious.\n\n(2) If $\\bar\\varphi$ is an isomorphism, then  $\\bar\\varphi_\\alpha$ is an\nisomorphism for each $\\alpha\\in\\Delta$, so $\\hbox{Ker}\\,\\varphi\\subset L_0$. Since\n$[L_\\alpha, \\hbox{Ker}\\,\\varphi]\\subset L_\\alpha\\cap\\hbox{Ker}\\,\\varphi=\\{0\\}$, we see that\n$\\hbox{Ker}\\,\\varphi$ lies in the center of $L$ from Definition 3.1. Since $L$\nand $L'$ are perfect and $L'\\cong L\/Z$ for some central ideal $Z$,\nthey have the same universal central extension.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\n\\begin{prop}\nLet $L$ be a Leibniz algebra graded by $\\Delta$ and $(\\frak u, \\varphi)$\nthe universal central extension of $L$. Then $\\frak u$ is graded by\n$\\Delta$ and has the same associated dialgebra as that of $L$.\nFurthermore, $\\varphi$ is a $\\Delta$-homomorphism and $\\varphi: \\frak\nu_\\alpha\\to L_\\alpha$ is a homomorphism for all $\\alpha\\in\\Delta$. In\nparticular, $\\hbox{Ker}\\,\\varphi\\subset \\frak u_0$.\n\\end{prop}\n{\\bf Proof.} It is well known that $\\dot{\\mg}$ is centrally closed in the\ncategory of Leibniz algebras (\\cite{Gao2}). Thus the central\nextension $\\varphi: \\varphi^{-1}(\\dot{\\mg})\\to \\dot{\\mg}$ splits and we may view\n$\\dot{\\mg}$ as a subalgebra of $\\frak u$. In particular, $H$ is a\nsubalgebra of $\\frak u$. We define\n\\begin{eqnarray*}\n&&\\tilde\\frak u_\\alpha:=\\varphi^{-1}(L_\\alpha), \\quad \\alpha\\in \\Delta\\cup\\{0\\},\\\\\n&&\\frak u_\\alpha:=\\cases{[\\tilde\\frak u_\\alpha, H], \\quad \\alpha\\in\\Delta,\\cr\n\\tilde\\frak u_0, \\quad \\alpha=0.}\n\\end{eqnarray*}\nFor all $h_1, h_2\\in H, x\\in \\tilde\\frak u_\\alpha$, we have\n$$\\hbox{\\rm ad}\\, h_1([x, h_2])=-[[x, h_2], h_1]=-[[x, h_1], h_2]=[\\alpha(h_1)x+z, h_2]=\\alpha(h_1)[x, h_2],$$\nwhere $z\\in\\hbox{Ker}\\,\\varphi$. This proves that\n$$\\frak u_\\alpha \\ \\hbox{is an  $\\alpha$-weight space for}\\  \\hbox{\\rm ad}\\,_{\\frak u}H,\\, \\alpha\\in\\Delta.\\eqno(4.3)$$\n It follows that for $\\alpha\\in\\Delta$, $\\frak u_\\alpha\\cap \\hbox{Ker}\\,\\varphi=\\{0\\}$ and hence\n$$\\varphi|_{\\frak u_\\alpha}:\\frak u_\\alpha\\to L_\\alpha\\eqno(4.4)$$ is an isomorphism of vector spaces.\n\nLet $x\\in\\frak u$ and write $x=\\sum_{\\alpha\\in\\Delta\\cup\\{0\\}}\\tilde x_\\alpha$,\nwhere $\\tilde x_\\alpha\\in\\tilde\\frak u_\\alpha$. Fix $h\\in H$ so that for all\n$\\alpha\\in\\Delta, \\alpha(h)\\ne 0$. Then\n$$\\tilde x_\\alpha-\\alpha(h)^{-1}[\\tilde x_\\alpha, x]\\in\\hbox{Ker}\\,\\varphi\\subset \\tilde \\frak u_0=\\frak u_0.\\eqno(4.5)$$\nThus we may rewrite $x$ as $\\sum_{\\alpha\\in\\Delta\\cup\\{0\\}} x_\\alpha$, where\n$x_\\alpha\\in\\frak u_\\alpha$ and $x_0\\in \\frak u_0$. It follows that\n$$\\frak u=\\frak u_0+\\sum_{\\alpha\\in\\Delta}\\frak u_\\alpha.$$\n\nNow\n$$\\frak u_0=\\tilde\\frak u_0=\\varphi^{-1}(\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}])\n=\\sum_{\\alpha\\in\\Delta}[\\tilde \\frak u_\\alpha, \\tilde \\frak u_{-\\alpha}]+\\hbox{Ker}\\,\\varphi.$$\nSince $\\tilde \\frak u_\\alpha=\\frak u_\\alpha+\\hbox{Ker}\\,\\varphi$ from (4.3), we have\n$$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak u_{-\\alpha}]+\\hbox{Ker}\\,\\varphi.$$\nThis proves that $\\frak u_0$ is a $0$-eigenspace for $H$. Now we see\nthat $\\frak u_\\alpha$ is exactly the $\\alpha$-eigenspace for $H$ for all\n$\\alpha\\in\\Delta\\cup\\{0\\}$ and $[\\frak u_\\alpha, \\frak u_\\beta]\\subset \\frak\nu_{\\alpha+\\beta}$, whenever $\\alpha+\\beta\\in\\Delta\\cup\\{0\\}$. Also since $\\frak\nu=[\\frak u, \\frak u]$, we see that\n$$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak u_{-\\alpha}]+[\\frak u_0, \\frak u_0].$$\nBut\n$$[\\frak u_0, \\frak u_0] =\\sum_{\\alpha, \\beta\\in\\Delta}[[\\frak u_\\alpha, \\frak u_{-\\alpha}],\n[\\frak u_\\beta, \\frak u_{-\\beta}]]\\subset \\sum_{\\gamma\\in\\Delta} [\\frak u_\\gamma,\n\\frak u_{-\\gamma}],$$\n so $$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak\nu_{-\\alpha}].$$ This proves that $\\frak u$ is graded by $\\Delta$. From\n(4.4) and the construction of $\\varphi$, we see that $\\varphi$ is a\n$\\Delta$-homomorphism and $\\bar\\varphi$ is an isomorphism. Thus the\ndialgebra associated to $\\frak u$ is the same as that associated to\n$L$. \\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\\begin{prop}\nLet $L$ be a Leibniz algebra graded by $\\Delta$, $Z$ the center of\n$L$ and $Z'$ any subspapce of $Z$. Then\n\n$(1)$  $Z\\subset L_0$.\n\n$(2)$  $L\/Z'$ has a unique structure as a Leibniz algebra graded\nby $\\Delta$ that makes the natural map $\\pi: L\\to L\/Z'$ a\n$\\Delta$-homomorphism.\n\n$(3)$  For all $\\alpha\\in\\Delta$, $L_\\alpha\\stackrel{\\pi}{\\cong}(L\/Z')_\\alpha$ as\nvector spaces.\n\n$(4)$  The dialgebras associated to $L$ and $L\/Z'$ are isomorphic\nby the map $\\bar\\pi$ induced by $\\pi$ as in Proposition 4.2.\n\\end{prop}\n{\\bf Proof.} Since for all $h\\in H$, $\\hbox{\\rm ad}\\, h|_{L_\\alpha}$ is a scalar by\n$\\langle \\alpha, h \\rangle$, we see that $Z\\subset L_0$. The remaining\nresults are now obvious. \\rule[-.23ex]{1.0ex}{2.0ex}\n\n\\begin{prop}\nLet $L$ and $L'$ be centrally isogenous Leibniz algebras and suppose\nthat $L$ is graded by a finite root system $\\Delta$. Then $L'$ is\nalso graded by $\\Delta$ and in such a way that the associated\ndialgebras are isomorphic.\n\\end{prop}\n{\\bf Proof.} Let $\\frak u$ be a universal central extension of $L$.\nBy Proposition 4.3, $\\frak u$ is graded by $\\Delta$. By assumption,\n$L'\\cong \\frak u\/Z_1$ for some subspace of the center of $\\frak u$.\nBy Proposition 4.4, $L'$ is graded by $\\Delta$. The associated\ndialgebras are isomorphic by Propositions 4.3 and 4.4.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\nIn short, all Leibniz algebras in a given isogeny class are\n$\\Delta$-graded if one of them is, and all have isomorphic root spaces\nfor all $\\alpha\\in\\Delta$. They differ only by central elements in the\n$0$-root space.\n\n\\section{Proof of Recognition Theorem}\n\nIn this section we shall complete the proof of the Recognition\nTheorem. We use the results and notation of $\\S 3$ and $\\S4$ freely.\n\n\\subsection{The Recognition Theorem for types $D$ and $E$}\n\nNow we assume that $L$ is a Leibniz algerba graded by $\\Delta$, where\n$\\Delta$ is a finite root system of type $D_l$, $l\\ge 4$, or $E_6, E_7,\nE_8$.\n\n\\begin{lemm} Let $\\alpha, \\beta\\in\\Delta, r, s \\in R $ and set $\\rho=:\\hbox{\\rm ad}\\,([e_\\alpha(r), e_{-\\alpha}(s)])$. Then\n$$\\rho(e_\\beta(t))=\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv s).$$\n\\end{lemm}\n{\\bf Proof.} Case 1. $(\\alpha|\\beta)=-1$:  Then $\\beta-\\alpha\\not\\in\\Delta$ and we\nhave\n\\begin{eqnarray*}\n\\rho(e_\\beta(t))&=&-[e_\\beta(t), [e_\\alpha(r), e_{-\\alpha}(s)]]\\\\\n             &=&-[[e_\\beta(t), e_\\alpha(r)], e_{-\\alpha}(s)]\\\\\n             &=&-[[e_\\beta, e_\\alpha](t\\dashv r), e_{-\\alpha}(s)]\\\\\n             &=&-[[e_\\beta, e_\\alpha], e_{-\\alpha}]((t\\dashv r)\\dashv s)\\\\\n             &=&-[[e_\\beta, [e_\\alpha, e_{-\\alpha}]]((t\\dashv r)\\dashv s)\\\\\n             &=&\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv\n             s).\n\\end{eqnarray*}\n\nCase 2. $(\\alpha|\\beta)=1$: Then $(-\\alpha|\\beta)=-1$ and we may use Case 1.\n\nCase 3. $(\\alpha|\\beta)=0$: Then neither $\\beta+\\alpha$ nor $\\beta-\\alpha$ is a root\nand both sides of our equation are 0.\n\nCase 4. $\\alpha=\\beta$:  It is easy to find a pair of roots $\\gamma,\n\\varepsilon\\in\\Delta$ with $\\gamma+\\varepsilon=\\alpha$, $\\gamma, \\varepsilon\\not\\in\\{\\alpha, -\\alpha\\}$. Then\n$e_\\beta(t)=e_\\alpha(t)=[e_\\gamma, e_\\varepsilon](t')$ for some $t'\\in R$ which is\nsome $K$-multiple of $t$. Applying $\\rho$ and using the previous\ncases,\n\\begin{eqnarray*}\n\\rho(e_\\beta(t))&=&\\rho([e_\\gamma, e_\\varepsilon](t'))=\\rho([e_\\gamma(t'), e_\\varepsilon(1)])\\\\\n             &=&\\langle \\gamma, \\alpha^{\\vee}\\rangle [e_\\gamma((t'\\dashv r)\\dashv s), e_\\varepsilon(1)]+\\langle \\varepsilon, \\alpha^{\\vee}\\rangle [e_\\gamma(t'), e_\\varepsilon(r\\dashv s)]\\\\\n             &=&\\langle \\gamma+\\varepsilon, \\alpha^{\\vee}\\rangle [e_\\gamma, e_\\varepsilon](t'\\dashv(r\\dashv s))\\\\\n             &=&\\langle \\alpha, \\alpha^{\\vee}\\rangle e_\\beta(t\\dashv (r\\dashv s))\\\\\n             &=&\\langle \\alpha, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv\n             s).\n\\end{eqnarray*}\n\nCase 5. $-\\alpha=\\beta$: This is similar to Case 4.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\nWe wish to define a homomorphism\n$$\\phi:L\\to \\frak t(R, \\Delta)=\\dot{\\mg}\\otimes R\\eqno(5.1)$$\nby defining\n$$\\phi(e_\\alpha(r))=e_\\alpha\\otimes r, \\quad \\hbox{for all}\\ \\alpha\\in\\Delta, r\\in R.\\eqno(5.2)$$\n\nSince $L$ is generated by the subspace $L_\\alpha, \\alpha\\in\\Delta$, it is clear\nthat (5.2) uniquely defines $\\phi$.\n\nSuppose that $\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}[e_\\alpha(r(\\alpha, i)),\ne_{-\\alpha}(s(\\alpha, i))]=0$ for some $r(\\alpha, i), s(\\alpha, i)\\in R$. Then using\nLemma 5.1, we have for all $\\beta\\in\\Delta$\n$$\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta(r(\\alpha, i)\\dashv\ns(\\alpha, i))=0$$\nand hence\n$$\\sum_{\\alpha\\in\\Delta}\\langle \\beta, \\alpha^{\\vee}\\rangle\\sum_{i=1}^{n_\\alpha} e_\\beta(r(\\alpha, i)\n\\dashv s(\\alpha, i))=0.$$ Since $e_\\beta(r)=e_\\beta(s)\\Leftrightarrow r=s$. It\nfollows that in $H\\otimes R$,\n$\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\alpha^{\\vee}\\otimes(r(\\alpha, i)\\dashv s(\\alpha,\ni))=0$ (it is killed by all the functionals $1\\otimes \\beta$). Thus we may\nextend $\\phi$ to $L_0$ by\n$$\\phi: \\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}[e_\\alpha(r(\\alpha, i)), e_{-\\alpha}(s(\\alpha, i))]\\mapsto \\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\alpha^{\\vee}\\otimes (r(\\alpha, i)\\dashv s(\\alpha, i)) .$$\n\nWe check that $\\phi$ is a homomorphism: If $(\\alpha, \\beta)$ is an\n$A_2$-pair, then\n\\begin{eqnarray*}\n\\phi([e_\\alpha(r), e_\\beta(s)])&=&\\phi([e_\\alpha, e_\\beta](r\\dashv s))\\\\\n                        &=&[e_\\alpha, e_\\beta]\\otimes(r\\dashv s)\\\\\n                        &=&[\\phi(e_\\alpha(r)), \\phi(e_\\beta(s))].\n\\end{eqnarray*}\n\nIf $(\\alpha, \\beta)\\ge 0$, then $\\phi([e_\\alpha(r),\ne_\\beta(s)])=0=[\\phi(e_\\alpha(r)), \\phi(e_\\beta(s))]$.\n\nIf $\\beta=-\\alpha$ then  $\\phi([e_\\alpha(r),\ne_{-\\alpha}(s)])=\\alpha^{\\vee}\\otimes(r\\dashv s)=[e_\\alpha\\otimes r, e_\\beta\\otimes s]$. Now\nlet $h=[e_\\alpha(r), e_{-\\alpha}(s)]$. We have\n\\begin{eqnarray*}\n\\phi([e_\\beta(t), h])&=&-\\phi(\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv s))\\\\\n                  &=&-\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta\\otimes ((t\\dashv r)\\dashv s)\\\\\n                  &=&[e_\\beta\\otimes t, \\alpha^{\\vee}\\otimes (r\\dashv s)]\\\\\n                  &=&[\\phi(e_\\beta(t)), \\phi(h)],\n\\end{eqnarray*} and\n\\begin{eqnarray*}\n&&\\phi([[e_\\beta(t), e_{-\\beta}(u)], h])\\\\\n&=&-\\phi(\\langle \\beta, \\alpha^{\\vee}\\rangle [e_{\\beta}(t), e_{-\\beta}((u\\dashv r)\\dashv s)]+\\langle \\beta, \\alpha^{\\vee}\\rangle [e_\\beta((t\\dashv r)\\dashv s), e_{-\\beta}(u)])\\\\\n                  &=&-\\langle \\beta, \\alpha^{\\vee}\\rangle \\Big(\\beta^{\\vee}\\otimes (t\\dashv ((u\\dashv r)\\dashv s))-\\beta^{\\vee}\\otimes (((t\\dashv r)\\dashv s)\\dashv u)\\Big)\\\\\n                  &=&0=[\\phi([e_\\beta(t), e_{-\\beta}(u)]), \\phi(h)]\n\\end{eqnarray*}\nsince $ ((t\\dashv r)\\dashv s)\\dashv u=u\\vdash((t\\dashv r)\\dashv\ns)=(u\\vdash(t\\dashv r))\\dashv s=((u\\vdash t)\\dashv r)\\dashv\ns=((t\\dashv u)\\dashv r)\\dashv s=t\\dashv ((u\\dashv r)\\dashv s)$,\nwhere we have used the commutativity and associativity of $R$.\n\n\\begin{prop}\nThe homomorphism $\\phi$ define by (5.1) and (5.2) is a surjective\n$\\Delta$-homomorphism and $\\hbox{Ker}\\,\\varphi$ is contained in the center of\n$L$.\n\\end{prop}\n{\\bf Proof.} Since $L_\\alpha=e_\\alpha(R)\\stackrel{\\phi}{\\to}e_\\alpha\\otimes R$, it\nis clear that $\\phi$ is bijective on the root spaces of $L_\\alpha,\n\\alpha\\in\\Delta$ and hence $\\phi$ is surjective and $\\hbox{Ker}\\,\\varphi\\subset L_0$.\nThus $[\\hbox{Ker}\\,\\phi, L_\\alpha]\\subset L_\\alpha\\cap (\\hbox{Ker}\\,\\phi)=(0)$ and hence\n$\\hbox{Ker}\\,\\phi$ is central by (2) of Definition 3.1.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\nWe have proved that $L$ is a central extension of $\\dot{\\mg}\\otimes R$, thus\nproving the third part of the Recognition Theorem 1.1.  The\nuniversal central extension of $\\dot{\\mg}\\otimes R$ is given in \\cite{LL} for\na unital commutative associative dialgebra $R$.\n\n\n\\subsection{The Recognition Theorem for type $A$}\n\nNow we assume that $L$ is a Leibniz algerba graded by $\\Delta$, where\n$\\Delta$ is a finite root system of type $A_l$, $l\\ge 2$.\n\nLet $n=l+1$ and $\\varepsilon_1, \\cdots, \\varepsilon_n$ be an orthonormal basis\nfor $\\mathbb R^n$. Identify $\\Delta$ with $\\{\\varepsilon_i-\\varepsilon_j\\mid\ni\\ne j\\}$ and define a base $\\{\\alpha_1, \\cdots, \\alpha_l\\}$ of $\\Delta$ by\n$\\alpha_i=\\varepsilon_i-\\varepsilon_{i+1}$. The positive class of $A_2$-pair in\n$\\Delta$ will be taken as $[(\\alpha_1, \\alpha_2)]$.\n\nFor $\\alpha=\\varepsilon_i-\\varepsilon_j\\in\\Delta$ we let $L_{ij}=L_\\alpha$. The simple Lie\nalgebra $\\dot{\\mg}$ over $K$ of type $A_l$ may be identified with\n$\\frak{sl}\\,(n, K)$. We choose as our Chevalley basis the matrix units\n$E_{ij}, i\\ne j$ and the elements $h_i=[E_{i, i+1}, E_{i+1, i}],\ni=1, \\cdots, l$. To bring the notation in line with $\\S2.4$, we\nwrite $e_{ij}$ for $E_{ij}$. Let $R$ be the $K$-dialgebra derived\nfrom $L$ with this choice of positive $A_2$-pairs and the given\nChevalley basis. Then we have\n$$[(\\alpha_1, \\alpha_2)]=\\{\\varepsilon_i-\\varepsilon_j, \\varepsilon_j-\\varepsilon_k\\mid i, j, k\\ \\hbox{distinct}\\}$$ and\n$$[(\\alpha_2, \\alpha_1)]=\\{\\varepsilon_i-\\varepsilon_j, \\varepsilon_k-\\varepsilon_i\\mid i, j, k\\ \\hbox{distinct}\\}.$$\n\nThus by results in $\\S4$, we have\n\\begin{eqnarray}\n&(i)&  L_{ij}=K\\{e_{ij}(r)\\mid r\\in R\\}, \\,i\\ne j;\\\\\n&(ii)& e_{ij}(k_1r+k_2s)=k_1e_{ij}(r)+k_2e_{ij}(s) ;\\\\\n&(iii)& [e_{ij}(r), e_{kl}(s)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&(iv)& [e_{ij}(r), e_{kl}(s)]=e_{il}(r\\dashv s)\\  \\hbox{ if } \\ i\\ne l\\ \\hbox{and } \\  j= k;\\\\\n&(v)& [e_{ij}(r), e_{kl}(s)]=-e_{kj}(s\\vdash r)\\  \\hbox{ if } \\ i=l\\\n\\hbox{and} \\  j\\ne k,\n\\end{eqnarray}\nfor all  $r, s\\in R, \\ k_1, k_2\\in K$.\n\nNow whether $R$ is associative or not, $L$ is homomorphic to the\nimage of $\\frak{stl}\\,(n, R)$, under the map\n$$\\phi: v_{ij}(r)\\to e_{ij}(r), \\quad r\\in R,\\, i\\ne j.\\eqno(5.6)$$\n\nIn Section 3, we show that $R$ is associative if $l\\ge3$. Similar to\nthe proof of Proposition 3.1 in \\cite{L}, we can show that $R$ is\nalternative if $l=2$.\n\n\n\\begin{prop}\nThe homomorphism of $(5.6)$ is central.\n\\end{prop}\n{\\bf Proof.} It is clear that $\\frak{stl}\\,(n, R)$ is graded by $\\Delta$ and\nits associated dialgebra is $R$. The homomorphism (5.6) sends\n$v_{ij}(R)\\to e_{ij}(R)=L_{ij}$ isomorphically and hence the\n$\\hbox{Ker}\\,(\\phi) \\subset \\frak{stl}\\,(n, R)_0$.  The same argument as used in\nProposition 5.2 gives that $\\phi$ is central.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\nThis completes (1) and (2) of the Recognition Theorem 1.1.\n\n\n\\noindent{\\bf Remark.} 1. The structures of Leibniz algebras graded\nby finite root systems of other types are determined in \\cite{LH2}\nby using the methods in \\cite{BZ} and \\cite{ABG2}.\n\n2. By Theorem 1.3 in \\cite{MZ} and Theorem 1.1, we can easily obtain\nthe following theorem.\n\\begin{theo}\n A $Q(n)$-graded Leibniz superalgebra over $K$ is centrally isogenous to $\\frak{stl}\\,(n+1, D)$,\n where $D=D_{\\bar0}+D_{\\bar1}$ is an associative or an alternative (if $n=2$)\n unital dialgebra such that there exists $\\nu\\in D_{\\bar1},\n \\nu^2=1$. (The definition of $Q(n)$-graded Leibniz superalgebra immediately follows\n from Definition 1.3 in \\cite{MZ} and\n Definition 3.1 in section 3, also see \\cite{LH3}).\n\\end{theo}\n\n\n\n\n\n\n\\vskip45pt \\centerline{\\bf ACKNOWLEDGMENTS}\n\n\\vskip15pt    Part of the work was done during the first author's\nvisit in the Department of Mathematics at University of Bielefeld\nand study in the Department of Mathematics at Shanghai Jiaotong\nUniversity for his Post-doctor study. He expresses his gratitude to\nthe `AsiaLink Project' for financial support. He is also indebted to\nC.M. Ringel and C.P. Jiang for the hospitality and encouragement.\nAuthors give their thanks to Y. Gao for his useful comments. Liu is\nsupported by the NNSF (Grant 10671027, 10701019 and 10571119), the\nZJNSF(Grant Y607136), and Qianjiang Excellence Project of Zhejiang\nProvince (No. 2007R10031).  Hu is supported in part by the NNSF\n(Grants 10431040, 10728102), the PCSIRT, the Priority Academic\nDiscipline from the MOE of China,  the Shanghai Priority Academic\nDiscipline from the SMEC. Authors are grateful to the referee for\ncorrection in some errors and invaluable suggestions.\n\n\n\n\n\\vskip30pt\n\\def\\cen{\\normalsize\\bf REFERENCES}{\\centerline{\\normalsize\\bf REFERENCES}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Cherenkov Telescope Array (CTA) project design study \n\\cite{cta-url} is under way to\ndesign and optimise the next generation of stereoscopic Imaging\nAtmospheric Cherenkov Telescope (IACT) arrays. CTA aims to achieve\nan order of magnitude improvement in point-source sensitivity \nover current instruments and a\nwider energy coverage in order to study astrophysical processes of\nmore sources in more detail.\n\nWhile extrapolation from current generation instruments may be a first\nguide to what is needed to achieve the CTA goals, only detailed\nsimulations can give definite answers. Physical processes in air\nshowers may ultimately limit the gamma{}-hadron separation power of\nthe instruments, as well as their angular and energy resolution.\nDepending on area coverage and instrumental capabilities, any\naffordable solution will typically not reach such ultimate limits.\nThe approach in this paper is the evaluation of realistic -- or conservative --\nperformance parameters by detailed simulations of air showers and\nthe potential instruments, followed by analysis methods well tested\non current instruments.\n\nThe air{}-shower simulation is based on the CORSIKA \\cite{CORSIKA} code with its\nchoice of different interaction models. With the IACT option of CORSIKA\nwe can simulate the Cherenkov light hitting arbitrary arrays of\ntelescopes, each defined separately by its position and a fiducial radius. \nSince the CORSIKA stage requires the largest amount of\nCPU time, each shower is re-used at different random displacements.\nDue to the excellent gamma-hadron separation of all the\nconfigurations envisaged, a huge number of\nhadron showers has to be simulated in order to estimate the\nremaining backgrounds -- in most cases billions of events.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{pixel_size_gray}\n\\caption{Different pixel sizes can have an\nimpact on the image obtained from the same shower (here\n0.07{\\textdegree} to 0.28{\\textdegree} pixels on a 23 m telescope, only\npart of f.o.v. shown). CORSIKA output can be piped directly into\nmultiple telescope simulations running in parallel, allowing for\nefficient simulation of different configurations. \n(See Figure \\ref{fig:pixel-size-results} for results.)\n\\label{fig:pixel-size-image}}\n\\end{figure}\n\nIn a second processing stage, the atmospheric extinction\nand the details of the detector response are simulated by the\n{\\tt sim\\_telarray} program \\cite{simtelarray}. \nThis program is very flexible and\ndifferent detectors with different read{}-out, different triggering\nschemes, or just different reflectors are treated by the same code,\nby just specifying different configuration files. Simulations include\noptical ray{}-tracing, night{}-sky background, all relevant electronic\npulse shapes, switching behaviour of comparators or discriminators, as\nwell as many other details. Telescope configurations are tested\nfor random night-sky trigger rates before being used for shower\nsimulations.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{arrays}\n\\caption{Selection of different array\nconfigurations simulated for altitudes of 1800 to 2000 m.\nSee text for more details.\n\\label{fig:arrays}}\n\\end{figure}\n\nFor efficiency reasons, the CORSIKA output data is usually piped into the\ntelescope simulation, without intermediate on-disk storage.\nBy placing a utility program termed {\\tt multipipe\\_corsika} into\nthis pipe, the CORSIKA data can be piped into multiple telescope\nsimulations at the same time, e.g. for different pixel sizes (see\nFigure \\ref{fig:pixel-size-image}),\ndifferent offset angles etc.\n\n\n\\section{A variety of candidate configurations}\n\nSince the performance is a complex function of array layout (which may\nconsist of multiple types of telescopes), and of a variety of\ntelescope parameters, detailed simulations can be performed only\nfor a limited {--} and therefore rather complementary {--} subset of layout and\ntelescope parameters. These candidate configurations \n(see Figure \\ref{fig:arrays}) include:\n\\begin{itemize}\n\\item A 9{}-telescope array aiming at low energies (420 m{\\texttwosuperior}\nmirror area each, with 0.1{\\textdegree} pixels covering a\n5{\\textdegree} field{}-of{}-view (f.o.v.). This configuration was\nsimulated at different altitudes (2000 m, 3500 m, and 5000 m\na.s.l.). The 2000 m simulations were carried out with a variety of\npixel sizes (from 0.07{\\textdegree} up to 0.28{\\textdegree}, see \nFigure~\\ref{fig:pixel-size-image}). It was also cross-checked with\nan different telescope simulation code \\cite{ICRC2007paper}.\n\\item \nAn array of 41 H.E.S.S.{}-I type telescopes (106 m{\\texttwosuperior}),\nto see how gamma{}-hadron rejection can be improved by better coverage\nwith current{}-generation instruments. It also allows to test\narrays of 4, 5, 9, or 16 telescopes with different inter-telescope\nseparations by selecting subset of the 41 telescopes in the\nanalysis.\n\\item \nAn array consisting of telescopes of two different sizes (600\nm{\\texttwosuperior} and 106 m{\\texttwosuperior} mirror area,\n5{\\textdegree} and 7{\\textdegree} f.o.v.) aiming at a rather wide\nenergy coverage. For the large telescopes, three sets of four\ntelescopes each, with different separations, were included in\nthe simulation -- usually selecting only one set of four large\nplus the 85 small telescopes in the analysis.\nAll telescopes were assumed to have a 50\\% higher quantum efficiency\nthan H.E.S.S.\n\\item \nDifferent arrays of rather small (37 m{\\texttwosuperior}) telescopes\nwith large separations and wide{}-f.o.v. cameras (up to 8{\\textdegree}\nwith 0.3{\\textdegree} pixels), aiming at high energies. \n\\end{itemize}\n\nSimulations include gammas, protons, and electrons as\nprimaries {--} nuclei, being generally easy to distinguish from\ngammas, were not considered. \nAll simulations were carried out at the Max Planck Institute in\nHeidelberg, except for the 3500 m 9{}-telescope configuration which was\nprocessed at SLAC.\n\n\n\\section{Analysis}\n\nThe analysis of simulation data is based on methods similar to the\nH.E.S.S. Hillas{}-parameter based standard analysis, with some\nextensions like using the height of shower maximum and how well the\nlateral distribution conforms with expectations for gamma showers.\nImage shape cuts include the mean reduced scaled width and length\n\\cite{HESS-Crab-paper}. Since, after geometrical shower reconstruction,\nthe image in each telescope can be used for an energy estimate,\nthe consistency of the individual estimates for the same event \nis used as another cut. Showers where the expected accuracy of\nthe final (weighted mean) energy estimate is poor for the\ngiven energy, e.g. showers outside the array, are also discarded.\nAt the lowest energies, a few tens of GeV, the image shape cuts\ntypically have little gamma-hadron discrimination power. \nThe most important parameter in this energy range turned out to\nbe the height of shower maximum.\nSelection cuts were manually optimised as simple functions of\nreconstructed energy or telescope multiplicity (e.g. of the\nform $a+b\\;\\log E$, with additional lower and upper bounds).\nFurther improvements\nby more sophisticated shower reconstruction and advanced cut\noptimisations may be feasible. In that sense, the following\nperformance results are very conservative.\n\nThe set of cuts from the H.E.S.S. standard analysis was also\nused on H.E.S.S.-I simulations as well as corresponding\nfour-telescope subsets of the 41-telescope array, with \ncorresponding sensitivities matching the actual H.E.S.S.\nsensitivity \\cite{HESS-Crab-paper}.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{sens_var3}\n\\caption{Differential sensitivity limit\n(with 5 sigma, 10 events in each energy bin) in HEGRA Crab units \n\\cite{HEGRA-Crab} for\nthe 9{}-telescope array at 2000 m altitude, at a zenith\nangle of 20{\\textdegree}. Each curve is for one of\nthe pixel sizes in Figure \\ref{fig:pixel-size-image}. Since the\nlarger pixels will see more night-sky background noise, it is\nimportant to adapt image cleaning thresholds to this noise.\nOnly at very low energies any improvement\ncould be achieved by small pixels.\n\\label{fig:pixel-size-results}}\n\\end{figure}\n\n\n\\section {Performance results}\n\nThe performance of the tested CTA candidate configurations was\nevaluated for point sources, usually taking advantage of the angular\nresolution improving with increasing number of telescopes with usable\nimages. Since the rejection of hadronic background improves with\nincreasing energy, most selection cuts have to be rather strict at\nlow energies and get looser towards higher energies. Both proton and\nelectron background was included, with the electron flux\nextrapolated ${\\propto}{E}^{-3.3}$ beyond available measurements. \nIn contrast to na\\\"ive expectations, electrons never dominated\nthe background at the lowest energies, due to the energy-dependent\nhadron rejection efficiency.\nIn the evaluation of flux limits, the\nLi\\&Ma formula was used to find the minimum number of gammas to achieve\na 5{}-sigma significance. In addition, a minimum of 10 gammas was required. At\nvery low energies, another limitation had to be taken into account: the\nsystematics in the subtraction of remaining background, here\nassumed at a level of 1\\% being equivalent to a 1{}-sigma fluctuation.\n\nIt turns out that with the arrays of nine 420~m{\\texttwosuperior} telescopes\nthe energy threshold can in fact be reduced to some 20~GeV. Rather strict\nselection cuts are needed at the lowest energies to achieve\nsufficient gamma-hadron separation to avoid running into the\nlimitation by background systematics. \n\nThe 41-telescope array of \n106~m{\\texttwosuperior} telescopes would not substantially improve the \nenergy threshold compared to the four-telescope H.E.S.S. phase I array.\nThanks to the much better sampling of the showers it would however\nimprove gamma-hadron separation, angular resolution, and effective\narea compared to H.E.S.S.-I, resulting in a sensitivity improvement\nmuch faster than the na\\\"ive $1\/\\sqrt{N}$ expectation ($N$ being\nthe number of telescopes).\n\nThe 97-telescope array (of which usually 85 small plus 4 large\ntelescopes were used in the analysis) combines low energy threshold\nwith much improved gamma-hadron rejection. In fact, the presence\nof the smaller telescopes helps to improve the gamma-hadron\nrejection even in the energy domain normally only accessible\nto the large telescopes -- by rejecting hadron showers where\na sub-shower may look like a low-energy gamma-ray shower.\nAt intermediate energies, this configuration fully meets\ninitial goals for CTA (`` milli-Crab sensitivity'').\n\nAt the highest energies envisaged for CTA, even the square kilometre\narea of the 97-telescope configuration may not be sufficient to\ndetect enough events. It could be supplemented with additional\nsmall telescopes instrumented with much coarser pixels (up to \n0.3{\\textdegree}) and operated at larger inter-telescope separations.\nThis is demonstrated by the assumed 39-telescope array.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{cta-he-sens-prelim}\n\\caption{Point source sensitivity limits\nF({\\textgreater}E) of the four arrays from Figure \\ref{fig:arrays} \nat 20{\\textdegree} zenith angle, \ncompared with current instruments \n\\cite{HESS-Crab-paper,MAGIC-II-sens,VERITAS-proposal,GLAST-sens}. \nOne C.U. (Crab unit) in this case is\n$F_{\\rm{cu}}(>E)=1.78\\cdot10^{-7}(E\/\\mathrm{TeV})^{-1.57}\\mathrm{photons}\/(\\mathrm{m}^{2}\\;\\mathrm{s})$\n\\cite{HEGRA-Crab}.\n\\label{fig:sensitivity-comparison}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{differential2rn}\n\\caption{Differential sensitivity limit\n(with 5 sigma, 10 events in each energy bin of 0.2 dex) for the 97{}-tel.\nconfiguration, with different exposure times, from 3 minutes to 50 hours. \nA 1 hour exposure would\nbe enough for a high{}-quality spectrum of a 0.1 Crab source over two\norders of magnitude in energy. 50 hours would be needed for the\nspectrum of a milli{}-Crab source.\nOne C.U. (Crab unit) is assumed here as \n$dF_{\\rm{cu}}\/dE=2.79\\cdot10^{-7}(E\/\\mathrm{TeV})^{-2.57}\\mathrm{photons}\/(\\mathrm{m}^{2}\\;\\mathrm{s}\\;\\mathrm{TeV})$\n\\cite{HEGRA-Crab}.\n\\label{fig:sensitivity-time}}\n\\end{figure}\n\n\n\\section {Conclusions and outlook}\n\nThe combination of the CORSIKA and sim\\_telarray programs is very well\nsuited for simulations of large telescope arrays, even with hundreds\nof telescopes. It is also very\nflexible and can be adapted to arbitrary array configurations and\ntelescope setups just by means of run{}-time configuration. The\nsubsequent analysis with its Hillas{}-parameter based shower\nreconstruction results in rather conservative performance\nestimates. Arrays consisting of multiple telescope types are easily\nhandled in all stages. First simulations included a range of\ndifferent test configurations, with emphasis on different energy\nranges, and demonstrated that the CTA sensitivity goal can be\nachieved \\cite{ICRC2007paper} {--} at least for energies above 50 to 100 GeV. Future\nsimulations will include more realistic CTA configurations (within\nbudget constraints) as well as some corner cases needed to improve the\nCTA design optimisation scheme.\n\n\n\n\\begin{theacknowledgments}\nI would like to thank Emiliano Carmona, Jim Hinton, and many others\nfor useful discussions on the CTA configurations. Stefan Funk has\ncomplemented my 9-telescope simulations for 2000 and 5000~m altitude\nwith corresonding ones for 3500~m altitude, carried out at the\nSLAC computing center.\n\\end{theacknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nThe large hadron collider (LHC) has discovered the  Higgs boson and measured its properties consistent with the precision measurements of the Standard Model (SM). The discovery of the last particle of the standard model strengthened the foundation of the SM. However, overwhelming evidence and hints require physics beyond the Standard Model (BSM). While the intensive searches for new particles at the weak scale or heavier have been performed experimentally, we have not found any convincing evidence of such new particles yet. In these circumstances, there is growing attention to feebly interacting light particles from both theoretical and experimental points of view. \n\n\nNew light particles with mass less than tens of GeV are compelling addition to the SM as they can directly resolve the central issues of the SM, such as the strong CP problem and the finite neutrino masses,  explain  many flavor physics anomalies, and be a portal to the dark sector accommodating the dark matter and the baryon asymmetry  of the universe. On the experimental grounds, the light particles, if they exist, need to be feebly interacting with the SM sector due to various experimental constraints. However, one should be aware that the bounds are strongly model-dependent because the signature depends on the nature of the light degrees of freedom. This situation is contrasted with heavy new particle exploration with mass beyond the reach of current collider energies, which can be probed through  Effective Field Theories~(EFT). Therefore, there is growing interest in various scenarios based on the current experimental data and the potential for future high-intensity experiments. The status of recent progress is well-summarized in the report of {\\it Physics Beyond Collider}~\\cite{Beacham:2019nyx}. \n\n\nIn this paper, we study the physics potential of the dump facilities of the International Linear Collider (ILC) to hunt new light particles. The ILC is a proposed future $e^+e^-$ linear collider with beam-energy of 125~GeV, which can be upgraded to 500~GeV. The main goal of the ILC is to study events of $e^+e^-$ collision to perform high precision measurements of the Higgs bosons and search for new particles produced by the electroweak interaction.   In the linear collider the beam has to go into the beam dump after the collision, and it provides an excellent opportunity for a high-intensity experiment to produce feebly interacting light particles. The number of electrons on the beam dump is enormous, $N_{\\rm EOT}=4\\times 10^{21}\/\\rm$~per~year, and the electromagnetic (EM) shower also leads to a high-intensity photon carrying $\\cal O$(10\\%) of the beam energy. The setup is equivalent to the other fixed target experiments if a detector is placed downstream of the beam dump. Hereafter we refer to such setup as {\\it the ILC beam dump experiment}. One can search for light particles produced in the dump, fly beyond the muon shield, then decay to the SM particles or scatter the detector material. The potential of the ILC beam dump project has been investigated in Refs~\\cite{Izaguirre:2013uxa,Kanemura:2015cxa,Sakaki:2020mqb,Asai:2021ehn,Asai:2021xtg,Moroi:2022qwz}. \n\n\nIn the previous works of the electron beam dump experiments, the searches for light particles  dominantly interacting with electron or photon were mainly studied since they are produced by  the primary electron and positron or the secondary shower particles. In the ILC beam dump experiment, the initial beam energy is much higher than those of the past  electron beam dump experiments. In addition to the EM showers, heavy mesons and $\\tau$ lepton can be produced by the shower photon  hitting nuclei. The decay of the produced SM particles is another promising source of light particles. This paper examines the production yield and spectrum of the mesons and $\\tau$ lepton at the ILC beam dump setup for the first time. \n\n \nAs an application of this study, we investigate a projected sensitivity of heavy neutral leptons  (HNLs, called sterile neutrinos in some literature). The HNLs mix with the SM neutrinos with an angle of $U_\\ell \\ (\\ell=e,\\mu, \\tau)$, resulting in a suppressed weak interaction to the SM. Therefore it is very natural to consider the HNL production from the meson and $\\tau$ lepton decays where  the weak interaction dominates. The current experimental constraints are mostly for mixing with $\\nu_e$ and $\\nu_\\mu$, and the high intensity of mesons would further reach the under-explored parameter space. Also the sensitivity to the $\\tau$ neutrino mixing angle $U_\\tau$ can be significantly improved because $\\tau$ lepton is accessible. \n\n\nThe phenomenology of the HNLs is well-reviewed in Refs~\\cite{Alekhin:2015byh, Dasgupta:2021ies}, and see references therein. We stress that feebly interacting HNLs in a GeV mass range are a motivated and well-defined physics target. The seesaw mechanism can explain the neutrino masses observed by the  neutrino oscillations with at least two HNLs.  If the two HNLs are almost degenerate,  the sum of the mixing angles have the lower bound, approximately $U^2\\equiv \\sum_l |U_l|^2\\gtrsim m_{\\rm atm}\/m_N \\sim 10^{-11} (m_N\/{\\rm 1~GeV})^{-1}$~\\cite{Shaposhnikov:2006nn,Alekhin:2015byh}. Furthermore the degenerate HNLs in the early universe can produce the baryon asymmetry via the HNL oscillations~\\cite{Akhmedov:1998qx, Asaka:2005pn}. The feeble interactions are necessary for the departure of the thermal equilibrium which is one of Sakharov's conditions for generating the baryon asymmetry. Together with the neutrino mass constraints, the interesting parameter space is in a range of $10^{-11}\\lesssim U^2\\lesssim10^{-6}$.~\\footnote{The benchmark values depend on the number of HNLs involved in the seesaw mechanism. Three or more HNLs will allow a smaller value of $|U_l|^2$. However, it is important to examine the parameter space of the minimal scenarios.}\n\n\nThis paper is organized as follows. In Sec.~\\ref{sec:dumpsetup}, we briefly review the ILC beam dump experiment, and in the following section, we evaluate the spectra of mesons and $\\tau$ lepton. In Sec.~\\ref{sec:HNL}, we study the HNLs at the ILC beam dump experiment. In addition to  the HNL production from the SM particle decays, we  consider an HNL  production via deep inelastic scattering and another production from $Z$ decays at the interaction point.  We finish with the discussion in Sec.~\\ref{sec:discussion}. \n\n\n\\section{ILC Beam Dump Experiment}\\label{sec:dumpsetup}\nThe ILC  main beam dump has to absorb 2.6~MW ($125~{\\rm GeV}\\times 21~{\\rm \\mu A}$) of the $e^\\pm$ beam energy for 125~GeV in the initial stage and 13.6~MW ($500~{\\rm GeV}\\times 27.3~{\\rm \\mu A}$) for 500~GeV in an upgrade stage. Following a water dump designed with the length of $l_{\\rm dump}=11~{\\rm m}$ and the diameter of 1.8~m, a muon shield of length $l_{\\rm sh}=70~{\\rm m}$ would be placed as proposed in \\cite{Sakaki:2020mqb,Asai:2021ehn}, to remove the secondary muon background. The cylindrical decay volume of length $l_{\\rm dec}=50~{\\rm m}$ and radius $r_{\\rm det}=3$~m would lie between the muon shield and the downstream detector. The schematic view is seen in Fig.~\\ref{fig:exp}, and the similar design of the setup can be found in \\cite{Sakaki:2020mqb,Asai:2021ehn}. Here, we additionally assume a multi-layer tracker in the decay volume. We consider a time frame of 10-year run  for both ILC-250 and ILC-1000 where the  beam energy is 125~GeV and 500~GeV, respectively.  \n\n\nThe EM shower (photons, electrons, and positions) starts in the beam dump. The ILC beam dump experiment is unique compared to the past electron beam dump experiments with regards to the higher beam energy and the high intensity, i.e., the number of electrons on the beam dump  $N_{\\rm EOT}$ is about $4\\times 10^{21}$ per year\\footnote{In ILC-1000, we assume to be $N_{\\rm EOT}=4\\times 10^{21}$ per year in numerical calculations.}.\nThe energetic beam  creates photons at $\\cal O$(1-10)~GeV energy scale (Fig.~9 of \\cite{Asai:2021ehn}), and the secondary interaction between the photon and the nucleus can  produce light mesons ($\\pi$ and $K$), heavy mesons ($D$, $B$, and even $B_c$) and $\\tau$ lepton.\nThey loose the energy in the beam dump and finally decay, and some of them produce the HNLs.  The yield and energy spectrum of the SM particles at the decay is therefore important to study the sensitivity of the ILC beam dump experiment.\nWe show the evaluation in Sec.~\\ref{sec:spectrum}. \n\n\nThe secondary muons are stopped in the muon shield at ILC-250, but their penetration behind the shield cannot be neglected for $E_{\\rm beam}=500$~GeV at ILC-1000. In this case, an additional active muon shield behind the muon shield would be necessary, and we assume that the muon shield consists of the lead shield ($l^{\\rm lead}_{\\rm sh}=10$~m) and the active shield ($l^{\\rm active}_{\\rm sh}=70~{\\rm m}-l^{\\rm lead}_{\\rm sh}$). The HNL with a dominant mixing with the muon neutrino can be produced inside the muon shield by  scattering of the shower muon, and we approximate that the HNLs produced behind the lead shield do not contribute to the signal events. In Appendix~\\ref{app:muonshield}, we study how  different depth of the muon shield affects the sensitivity for the HNL at ILC-1000. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figs\/exp.pdf}\n\\caption{A setup for ILC beam dump experiments. It consists of the main beam dump, a muon shield, and a decay volume. We assume a multi-layer tracker is placed in the decay volume so that the charged tracks are measured.}\n\\label{fig:exp}\n\\end{figure}\n\n\n\\section{Meson and $\\tau$ lepton Spectra}\\label{sec:spectrum}\n\n\nThis section presents the meson and $\\tau$ lepton spectrum obtained by Monte-Carlo simulation at the ILC beam dump. We use PHITS~3.25~\\cite{Sato:2018imy} for production and transport of particles other than heavy mesons. PHITS (Particle and Heavy Ion Transport code System) is a general-purpose Monte Carlo particle transport simulation code developed under collaboration between JAEA, RIST, KEK, and several other institutes. PHITS can transport most particle species for a given geometry of the materials, and it is tested thoroughly by benchmarks studies~\\cite{iwamoto2017benchmark,matsuda2011benchmarking}. For heavy mesons production, we implement their differential production cross sections obtained by PYTHIA8.3~\\cite{Bierlich:2022pfr} into PHITS. More details are given in the following. \n\n\n\\subsection{Light Mesons}\n\n\n\\begin{figure*}\n\\centering \n\\includegraphics[width=0.49\\textwidth]{figs\/decay_125GeV.pdf}~~\n\\includegraphics[width=0.49\\textwidth]{figs\/decay_500GeV.pdf}\n\\caption{The kinetic energy distribution of light mesons when they decay, where $\\pi^\\pm$ and $K^\\pm$ indicates the sum of charged pions and kaons.\nWe consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively.}\n\\label{fig:decay}\n\\end{figure*}\n\n\nThe light mesons are mainly produced by the interaction of real photons in the electromagnetic shower with the nucleons in the beam dump. If the decay length of the produced light mesons is in the same order of magnitude of or greater than the mean free path in the material, the particles reduce their energy or change into different flavors by  (in-)elastic scattering or  multiple scattering. We use the following codes and models which are available in PHITS to simulate the electromagnetic shower and the production and transport of the light mesons. For the electromagnetic shower, the simulation is performed by EGS5~\\cite{Hirayama:2005zm}. For the light meson production and transport, the JAM~\\cite{Nara:1999dz} and JQMD~\\cite{PhysRevC.52.2620} models modified for photoproduction (photonuclear interaction) are used. In addition to these models, INCL4.6~\\cite{PhysRevC.87.014606} is also employed to calculate the interaction of the mesons with nuclei during transport. The energy loss of the charged particles due to multiple scattering is evaluated by ATIMA~\\cite{ATIMA}.\n\n\nFig.~\\ref{fig:decay} shows the kinetic energy distribution of light mesons when they decay. The decay energy distribution is more important than the production energy distribution because the kinematics of new particles is determined by the parent particle distribution at the decay. For reference, the production energy distribution of light mesons is shown in Fig.~\\ref{fig:production}.\n\n\n\n\n\n\\subsection{Heavy Mesons}\n\n\nWe use PYTHIA8 to calculate the differential cross sections of the  $\\gamma p(n)\\to B(D)+X$ process for the heavy mesons.\\footnote{We thank the Pythia team, especially to Ilkka Helenius for helping us understand the latest photoproduction feature of PYTHIA~8.3.}\nWe have checked that the sum of the direct and non-diffractve cross section of the $D$ meson production agrees very well with the photoproduction data~\\cite{SLACHybridFacilityPhoton:1983yfx,TaggedPhotonSpectrometer:1989bpi}, see Appendix~\\ref{app:Dmeson}. Therefore we regard the sum of the two cross sections as the total cross section, \n$\\sigma_{\\rm total}(\\gamma p(n))$\n$=\\sigma_\\text{non-diff}(\\gamma p(n))$\n$+ \\sigma_{\\rm direct}(\\gamma p(n))$. The total cross section of atomic neucleus is obtained by taking into account the shadowing effect \\cite{Caldwell:1978ik,Kopeliovich:2012kw}, \n\\begin{equation}\n\\sigma_{\\rm total}(\\gamma A)=  A^{l} \\left(\\frac{Z}{A}\\sigma_{\\rm total}(\\gamma p)+\\frac{A-Z}{A}\\sigma_{\\rm total}(\\gamma n)\\right) ,\n\\label{eq:atomic}\n\\end{equation}\nwhere $l=0.92$~\\cite{Kopeliovich:2012kw}. \n\n\nThe differential production cross sections for heavy meson photoproduction are obtained by PYTHIA8~\\cite{Bierlich:2022pfr} and implemented in PHITS. Since the decay length of the heavy mesons are much shorter than the mean free path in the material, the spectra at their production and decay are similar. In Fig.~\\ref{fig:production}, we show the production rate per electron injection in the beam dump for mesons and $\\tau$ lepton with respect to the kinetic energy at production, where the energy is normalized by the beam energy. The results for $\\pi^{\\pm}(\\pi^+, \\pi^-)$, $K(K^+, K^-, K_S^0, K_L^0)$, $D(D^+, D^-, D^0, \\overline{D}^0)$, $D_s(D_s^+, D_s^-)$, $B(B^+, B^-, B^0, \\overline{B}^0)$, $B_s(B_s^0, \\overline{B}_s^0)$, $B_c(B_c^+, B_c^-)$, and $\\tau(\\tau^+, \\tau^-)$ produced in the beam dump are shown, which represent the sum of the particles in the parenthesis. We consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively. The overall yield of the heavy meson production increases as the beam energy gets higher, and $B_c$ becomes accessible at ILC-1000. For the sake of comparison, we also include $\\pi$, $K$ distributions at production. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/production_125GeV.pdf}\n\\includegraphics[width=0.45\\textwidth]{figs\/production_500GeV.pdf}\n\\caption{The production rate per electron injection in the beam dump for mesons and $\\tau$ lepton with respect to the kinetic energy at production, where the energy is normalized by the beam energy.\nThe results for $\\pi^{\\pm}(\\pi^+, \\pi^-)$, $K(K^+, K^-, K_S^0, K_L^0)$, $D(D^+, D^-, D^0, \\overline{D}^0)$, $D_s(D_s^+, D_s^-)$, $B(B^+, B^-, B^0, \\overline{B}^0)$, $B_s(B_s^0, \\overline{B}_s^0)$, $B_c(B_c^+, B_c^-)$, and $\\tau(\\tau^+, \\tau^-)$ produced in the beam dump are shown, which represent the sum of the particles in the parenthesis.\nWe consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively.}\n\\label{fig:production}\n\\end{figure*}\n\n\n\\subsection{$\\tau$ lepton}\nAs discussed in the previous literature~\\cite{2018}, the primary source of $\\tau$ lepton is the $D_s$ decay with approximately 5\\% branching ratio. PHITS simulates $D_s$ meson propagation and decay, which accounts for the $\\tau$ lepton production. The sub-dominant source of $\\tau$ lepton is the $\\tau$ lepton pair production, $\\gamma + {\\rm nucleus\/nucleon} \\to \\tau^+ + \\tau^- + X$. We implement a complete differential cross section calculated with the Born approximation in QED in the PHITS code and generate events for the process~\\cite{Tsai:1973py,Sakaki:2020cux}. The form factors for coherent (nucleus elastic), quasi-elastic and inelastic interactions are included. The spectrum of $\\tau$ lepton is shown in Fig.~\\ref{fig:production}.\n\n\nWe find that the number of $\\tau$ leptons from the pair production is about 20 times smaller than those from the decay of $D_s$. However, the pair production process becomes dominant in the high-energy region where the kinetic energy of $\\tau$ lepton is above 65\\% of the beam energy. So the pair production process will be necessary when considering the physics of high-energy $\\tau$ leptons or $\\tau$ neutrinos at the ILC beam dump.\n\n\n\n\n\\section{Heavy Neutral Leptons}\\label{sec:HNL}\n\nIf gauge singlet fermions $N$ exist in the BSM sector, a renormalizable interaction with the SM sector is possible\n\\begin{align}\n{\\cal L} = -\\lambda_{\\ell I} (\\bar L_\\ell \\tilde{H}) N_I -\\frac{1}{2}M_{I} {\\bar{N}_I^c} N_I +\\rm h.c. \\label{eq:HNL}, \n\\end{align}\nwhere $L_\\ell$ is the SM lepton doublet of flavor $\\ell=e,\\mu,\\tau$, and $I$ is the index of $N$. \nIf $M_I\\gg \\lambda_{iI} v$, the standard seesaw mechanism   make the SM neutrino mass light, $m_{\\nu, \\ell\\ell'}\\sim  \\sum_I (\\lambda_{\\ell I}\\lambda_{\\ell'I}) v^2\/M_I$. For $\\mathcal{O}(1)$  Yukawa coupling, the singlet fermion $N$ has to be extremely heavy to satisfy the bounds on the neutrino mass, $m_{\\nu} \\gtrsim 0.05~{\\rm eV}$~\\cite{ParticleDataGroup:2020ssz}. On the other hand,  if the size of Yukawa coupling $\\lambda$ is small, $M$ can be in MeV-GeV mass scale to satisfy the same condition. Such particles can be searched directly in laboratories, and they are often called  heavy neutral leptons (HNLs) or sterile neutrinos. The active neutrinos $\\nu$ from $L$ doublet and the HNLs $N$ are almost in the mass eigenstates up to a small admixture between $\\nu_i$ and $N_I$ characterized by the mixing angle\n\\begin{align}\nU_{\\ell I} = \\frac{v \\lambda_{\\ell I}}{M_I}.  \\label{eq:mixing}\n\\end{align}\nSince the mixing and mass determine the HNL interactions with the SM particles, we use the mixing parameter $U_{\\ell I}$ instead of the Yukawa couplings to discuss the HNL phenomenology. \n\n\n\\begin{figure*}[tt]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/benchmarkplot.pdf}\n\\caption{\nThe target parameter space in a scenario of two degenerate HNLs with respect to the HNL mass and the sum of the mixing squared defined in Eq.\\eqref{eq:mixingsum}. The region between dashed (dotted) green lines is favored as the HNL can generate the baryon asymmetry of the universe~\\cite{Akhmedov:1998qx, Asaka:2005pn}, and the lines are adopted from Fig.~4.17 of \\cite{Alekhin:2015byh}. \nThe bottom shaded region cannot explain the neutrino oscillation data in Type-I seesaw mechanism with the two degenerate HNLs. Another shaded region with the BBN label is excluded by that the long-lived HNLs affect the successful big bang nucleosynthesis. The bound is obtained with respect to $U_e$, $U_\\mu$, or $U_\\tau$ in \\cite{Boyarsky:2020dzc}, and we take $U^2<\\min_{\\ell=e,\\mu,\\tau}[|U_{\\ell}^{(\\rm BBN)}|^{2}]$ for this plot. \n}\n\\label{fig:bench}\n\\end{figure*}\n\n\nThe HNLs in the GeV mass range have another exciting aspect other than explaining the mass of active neutrinos and testability. They can be responsible for the baryon asymmetry by leptogenesis via HNL oscillation~\\cite{Akhmedov:1998qx, Asaka:2005pn}. The effective leptogenesis occurs for fast $N_I$ oscillation, and therefore two degenerate HNLs are an excellent benchmark model to investigate. \n\n\nIn the minimalistic scenario with two quasi-degenerate HNLs, the target parameter space is well-defined by the baryon asymmetry and the neutrino mass. We schematically show it in Fig.~\\ref{fig:bench}. Note that the vertical axis is the sum of the mixing angles  over both the active neutrinos and the HNLs, \n\\begin{align}\nU^2\\equiv \\sum_{i=e,\\mu,\\tau \\ I=1,2} |U_{\\ell I}|^2 \n\\label{eq:mixingsum}\n\\end{align}\nWe also include a bound on  $U^2$ from Big Bang Nucleosynthesis (BBN). The small mixing angle is disfavored as the HNLs are sufficiently long-lived to decay during or slightly before BBN and  affect the ratio of the neutron and proton number densities~\\cite{Boyarsky:2020dzc}. \n\n\nIt is essential to probe all flavor mixings to cover the target parameter space for baryon asymmetry characterized by $U^2$. The sensitivity to $U_{e I}$ and $U_{\\mu I}$ of the current and proposed experiments is high, but the $\\tau$ neutrino mixing is poorly constrained. In the ILC beam dump experiment, the relevant region of $U_{\\tau I}$ can be probed because  $\\tau$ leptons can be copiously produced from $D_s$ meson decay thanks to the higher beam energy.\n\n\nIn many phenomenological studies of HNLs, one assumes a single HNL (say $N_1$) in the low energy  because having two HNLs will have little impact on  the search sensitivity as long as the two HNLs are degenerate. In the following we deal with one HNL for simplicity, and thus the index $I$ is omitted. Furthermore we turn on only one $U_\\ell$, a mixing with one of the active neutrinos in the flavor eigenstate ($\\nu_e, \\nu_\\mu, \\nu_\\tau$), at a time, which helps us to understand which underlying process matters. Under these assumptions, the phenomenology is well-described by the HNL mass and the single mixing in each benchmark model. Also, these benchmarks are used commonly in the literature, which allows us to compare our results with the previous works.  \n\n\n\\subsection{Sensitivities at ILC beam dump}\n\n\nIn this subsection we evaluate the sensitivity of the ILC beam dump experiment to the HNLs. We consider the following two production mechanisms of HNL:\n\\begin{enumerate}[(i)]\n \\item Productions from meson and $\\tau$ lepton decays; \n \\item Direct productions from electrons and muons in EM shower interacting with nucleons.\n\\end{enumerate}\nIn both cases, we consider the HNLs decaying inside the decay volume as the signal. We adopt the decay widths of HNL to the SM particles based on Ref.~\\cite{2018}, and most decay patterns of the HNL would leave multiple tracks, which is distinguishable from the background. Then, we assume zero background and account for all the HNL decays inside the decay volume as signal except for the invisible decay mode, $\\rm HNL\\to \\nu\\nu\\bar\\nu$. One of the possible backgrounds is hardons produced by high energy neutrinos, such as $K_{S,L}$ decaying to charged pions. This type of background was estimated with GIANT~4 in the SHiP setup~ \\cite{Bonivento:2013jag}. They found that   the relevant hadrons are always produced at the edge of the muon shield and can be significantly reduced by requiring the charged tracks consistent with the HNL kinematics. Note that PHITS does not take into account the high energy neutrinos interactions, which is, therefore, not included in our results. Although cosmic muons may produce beam-unrelated backgrounds, the deep underground location of the detector (100 m from the ground surface), direction of tracks, and coincidence time window based on the pulsed electron beam significantly reduce the backgrounds. The detector setup and event selections to reduce the background are beyond the scope of this paper.\n\n\nIn the HNL production process (i), evaluating the position and 4-momentum of the HNL is not straightforward because it involves various intermediate particles with possible higher multiplicity. Therefore, it is suitable to  estimate the sensitivity using Monte-Carlo simulation. We include the SM particle decays to an HNL  which are summarized in Appendix~\\ref{sec:Br} and Ref.~\\cite{2018}\n\n\nFor the other production (ii), associated physical processes are  tractable   without simulation. We therefore evaluate its sensitivity by the numerical integration and provide relevant formulae for it. To help the quantitative understanding, we also provide an approximate formula of the signal rate of the process (i) that can be integrated numerically. The signal rate of the numerical integration will be compared with the one obtained by the Monte-Carlo simulation. \n\n\nIn the following, we describe the detail of each method. \n\n\n\\subsubsection{Monte-Carlo method} \nWe simulate particle production and transport by PHITS with the help of PYTHIA8 for electron injection as described in Sec.~\\ref{sec:spectrum}. We also modify the decay tables of mesons and $\\tau$ lepton of PHITS  to include decay modes to an HNL as discussed in Appendix~\\ref{sec:Br}. It is essential to perform the Monte-Carlo simulation to track how the system evolves from the incident electron since the intermediate steps are very involved in the production (i). The result of the Monte-Carlo simulation also becomes the guiding post for the course-grained integration method which is described in Sec.~\\ref{sec:nummethod}.\n\n\nHowever, a naive Monte-Carlo simulation of the long-lived particle would suffer from technical challenges in obtaining sufficient statistics, in particular in the following cases:\n\\begin{enumerate}[(a)]\n \\item Small cross section of new physics process or photoproduction of mesons.\n \\item Small decay branching ratio from a SM particle to the new particle.\n \\item The decay length of the new particle much shorter than the shield length.\n \\item The decay length of the new particle much longer than the length of the experimental setup.\n\\end{enumerate}\n\n\nIn cases (a) and (b), the processes resulting in the signal process are so rare that it is difficult to obtain the sufficient number of events. The issue can be solved in PHITS-based simulation using the {\\it biasing technique}. In this technique, PHITS provides a biasing parameter for photoproduction. In this technique, the biased process occurs more frequently according to the biasing parameter, and an appropriate weight of the produced particle lower than that of the incident particle is assigned. We obtain the correct expected value of physical quantities by adding up the weights, rather than adding up the number of particles produced. \n\n\nIn case (c), the new particles generated by the simulation predominantly decay inside the shield, so  sufficient signal statistics are not easily obtained. This issue can be avoided by using the {\\it importance sampling technique} which PHITS supports. This technique allows us to assign an {\\it importance} to different regions of the simulation geometry. When a particle passes through the boundary between different regions with increasing importance, several copies of the particle are created in an event, and their weights are reduced depending on the importance ratio between the regions. We divide the region of the shield in the direction of the beam axis and increase their importance value exponentially, so that a short lifetime particle pass through the shield more efficiently without exponential loss. \n\n\nThe importance technique is also useful in the opposite case (d) as we can set a large importance value in the decay volume to increase the event sampling in the relevant region. In addition, {\\it forced-decay technique} is more useful when the decay length of the long-lived particle $X$ is extremely longer than the length of the shield and decay volume. In this technique, we introduce a {\\it maximum decay length}, $l_{\\rm max}$. When the decay length of $X$, $l_X$, is sufficiently longer than the typical length of experiment, $l_{\\rm exp} (\\sim l_{\\rm dump}+l_{\\rm sh}+l_{\\rm dec})$, the differential decay rate of $X$ with respect to the flight distance $z$ is\n\\begin{align}\n\\left. \\frac{dP}{dz}\\right\\vert_{l_X}\n= \\frac{1}{l_X} {\\rm e}^{-\\frac{z}{l_X}}\n\\simeq\n\\frac{1}{l_X},~~(z<l_{\\rm exp}).\n\\end{align}\nThis is a very small value, so it's hard to get enough Monte-Carlo statistics for the decay of $X$ in the decay volume.  To deal with this problem, we multiply the decay probability by $b = l_X \/ l_{\\rm max} (\\,>1)$ and the weight of $X$ by $1\/b$. The rescaled probability is\n\\begin{align}\n\\left. b \\frac{dP}{dz}\\right\\vert_{l_X}\n\\simeq\n\\frac{b}{l_X}\n\\simeq\n\\left. \\frac{dP}{dz}\\right\\vert_{l_{\\rm max}}.\n\\end{align}\nThis corresponds to the decay rate when the lifetime of $X$ is multiplied by $1\/b$. In summary, when $l_{\\rm max}<l_X$, multiplying the lifetime and weight of $X$ by $1\/b$ forces a higher probability of decay within the decay volume and reduces the computational cost by a factor $b$. This approach is an approximation, and the higher order correction to weight is $\\mathcal{O}(l_X^2\/l_{\\rm max}^2)$.  We typically use $l_{\\rm max}\\sim 10^{3}$~m, so that we can safely ignore the error. We count the number of the HNL decays to the visible SM particles in the decay region counting up all the weight factors appropriately.\n\n\n\n\n\n\\subsubsection{Coarse-grained integration method}\n\\label{sec:nummethod}\nIn this subsection, we describe the approximate estimation for the HNL productions (i) and (ii). Here, we apply suitable approximations to obtain simplified expressions involving multiple integrals for the signal yield, and then perform the integrals numerically. We refer this calculation scheme to as the course-grained integration (CGI) method. The same scheme was used in Ref.~\\cite{Sakaki:2020mqb,Asai:2021ehn} in which the ALPs and dark photon productions by the EM shower were studied.\n\n\n\nFor the production mechanism (i), we consider processes where a photon from the EM shower interacts with a nucleus in the beam dump and produces the SM particle $k$. \nThe number of signal events is schematically given by\n\\begin{align}\nN_{\\rm signal}^{\\rm (i)}&=N_{\\rm EOT}\\times\\sum_{k=\\pi, K, D, B,\\tau} \\left( l_{\\gamma}\\times n_{\\rm nucleus}\\times  \\sigma_{\\gamma~{\\rm nucleus}\\to k X}\\right)\n\\notag\n\\\\ \n& \\times{\\rm Br}(k \\to {\\rm HNL}~X) \\times {\\rm Br}_{\\rm vis}\\times {\\rm Acc}^{\\rm (i)}({\\rm HNL}), \\label{eq:sig}\n\\end{align}\nwhere  $l_{\\gamma}$ is  the photon track length in the beam dump, $n_{\\rm nucleus}$ is the number density of beam dump nucleus and the production cross section of particle $k$ is denoted by $\\sigma_{\\gamma~{\\rm nucleus}\\to k X}$. The branching ratio ${\\rm Br}(k \\to {\\rm HNL}~X)$ and ${\\rm Br}_{\\rm vis}={\\rm Br}({\\rm HNL} \\to {\\rm visible~SM})$ are summarized in Ref.~\\cite{2018} and also Appendix~\\ref{sec:Br}. ${\\rm Acc}^{\\rm (i,ii)}({\\rm HNL})$ denotes the detector acceptance depending on the production process. The distribution of photon track length  $l_{\\gamma}$ in the beam dump can be  obtained by the Monte-Carlo simulation, and the result is given in Ref.~\\cite{Asai:2021ehn} and also Appendix~\\ref{app:track}.\n\n\nIn the production mechanism (ii), an incoming lepton $\\ell=e,\\mu$ from the EM shower interacts with a nucleon and would produce an HNL. The number of signal events is schematically expressed as\n\\begin{align}\nN_{\\rm signal}^{\\rm (ii)}&=N_{\\rm EOT}\\times l_{{\\ell}^{\\pm}}\\times n_{N}^{\\rm dump\/shield}\\times \\sigma_{{\\ell}^{\\pm}~{\\rm nucleon}\\to {\\rm HNL}~ X}\\times{\\rm Br}_{\\rm vis}\\times {\\rm Acc}^{\\rm (ii)}({\\rm HNL}),\\label{eq:sig2}\n\\end{align}\nwhere the track length $l_{\\ell^{\\pm}}$ of the lepton in the beam dump is provided in Ref.~\\cite{Asai:2021ehn} and Appendix~\\ref{app:track}, and $n_{N}^{\\rm dump\/shield}$ is the number density of nucleon in the beam dump or muon shild. For the incoming muon, we ignore the HNL production in the beam dump. The production cross section of  HNL is denoted by $\\sigma_{{\\ell}^{\\pm}~{\\rm nucleon}\\to {\\rm HNL}~X}$, which is summarized in Appendix~\\ref{app:DIS}.\n\n\nWe elaborate the schematic formulae Eqs.~\\eqref{eq:sig} and \\eqref{eq:sig2} in the following. Given that the differential distribution of  mesons and $\\tau$ lepton were obtained by PHITS and PYTHIA8, as the results are showed in Sec.~\\ref{sec:spectrum}, we fit the differential distribution of the number of SM particle $N_k$ as the function of $E_k$. This quantity is expressed by the more fundamental quantity as follows,\n\\begin{align}\n\\frac{d N_k}{dE_k}=\\int dE_{\\gamma}\\frac{d l_{\\gamma}}{dE_{\\gamma}}\\times n_{\\rm nucleus}\\times \\int d \\Pi _X  \\frac{d\\sigma_{\\gamma~{\\rm nucleus}\\to k X}}\n{dE_k d \\Pi_X}.\n\\end{align}\nThis corresponds to the schematical terms in the parentheses of Eq.~\\eqref{eq:sig}. Note that ${d l_{\\gamma}}\/{dE_{\\gamma}}$ and $\\sigma_{\\gamma~{\\rm nucleus}\\to k X}$ is absorbed in ${d N_k}\/{dE_k}$.\n\n\nBy using $dN_k\/dE_k$ the production rate approximately expressed as follows, \n\\begin{align}\n    N^{\\rm (i)}_{\\rm signal}&\\sim N_{\\rm EOT}\\sum_{k=\\pi, K, D, B,\\tau}\\int d E_k \\frac{d N_k}{dE_k}\\cdot {\\rm Br}(k \\to {\\rm HNL}~X)\\cdot {\\rm Br}_{\\rm vis}\\cdot{\\rm Acc}_{\\rm dump}^{\\rm (i)}({\\rm HNL}), \\label{eq:N1approx}\n\\end{align}\nwhere we assume momentum of the SM particle $k$ is aligned to the electron beam direction. The HNL production via the SM particle decay is the leading production channel at the ILC beam dump experiment for the most accessible parameter space of mass and mixing angle. As we will see later, this production mode even contributes to the sensitivity of $U_\\tau$.\n\n\nAdditionally, the direct production initiated by the charged leptons of the EM shower takes an essential role to  explore the higher mass region of the HNL in the presence of $U_e$ or $U_\\mu$. The direct production from the electron and positron occurs inside the beam dump, while the one from the muon occurs dominantly in the muon shield. Several direct production processes contribute to the HNL productions~\\cite{Formaggio:2012cpf} such as quasi-elastic scattering, resonance production, and deep inelastic scattering~(DIS). Among those, DIS would be a significant contribution to the HNL productions around $E_{e^{\\pm},\\mu^{\\pm}}\\geq 10~{\\rm GeV}$ which is favored by the acceptance of the HNL production angle. Therefore, we focus on the DIS process, and the formulae of the number of signal events are expressed as follows, \n\\begin{align}\n    N^{{\\rm (ii)},e^{\\pm}}_{\\rm signal}&=N_{\\rm EOT}\\sum_{i=e^-,e^+}\\int d E_{i}\\frac{1}{2}\\frac{dl_{i}}{d E_{i}}\n      \\sum_{N=n,p}n_{N}^{\\rm dump}\n\\int dx \\int dy \\frac{d^2\\sigma (i N\\to {\\rm HNL}~X)}{dx dy}\n\\nonumber\\\\ &\\times\n{\\rm Br}_{\\rm vis}\\cdot {\\rm Acc}_{\\rm dump}^{{\\rm (ii)},e^{\\pm}}({\\rm HNL}),\\label{eq:DISel}\n    \\\\\n    N^{{\\rm (ii)},\\mu^{\\pm}}_{\\rm signal}&=N_{\\rm EOT}\\sum_{i=\\mu^-,\\mu^+}\\int d E_{i 0}\\frac{1}{2}\\frac{d Y_{i 0}}{dE_{i 0}}\\int dE_{i}\\frac{1}{2}\\frac{dl_{i}}{d E_{i}} \\int d\\theta_{\\rm MCS} \\frac{d P_{\\rm MCS}}{d\\theta_{\\rm MCS}} \\notag\n    \\\\\n&\\times \\sum_{N=n,p} n_{N}^{\\rm shield}\\int dx \\int dy \\frac{d^2\\sigma (i N\\to {\\rm HNL}~X)}{dx dy}\\cdot {\\rm Br}_{\\rm vis}\\cdot{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},\\mu^{\\pm}}({\\rm HNL}).\\label{eq:DISmu}\n\\end{align}\nThe energy spectrum of the muon yield per incident electron behind the beam dump $dY_{\\mu 0}\/d E_{\\mu 0}$ where $E_{\\mu^{\\pm}0}\\geq E_{\\mu^{\\pm}}$ is evaluated in Ref.~\\cite{Sakaki:2020cux}, {also see Eq.(A.3) of Ref.~\\cite{Sakaki:2020mqb}} and Appendix~\\ref{app:track}, and the track length of muons $l_{\\mu^{\\pm}}$ in the muon shield is provided in Ref.~\\cite{Sakaki:2020mqb} and Appendix~\\ref{app:track}. The angular distribution stemming from the multiple Coulomb scattering (MCS)  of the muon $dP_{\\rm MCS}\/d\\theta_{\\rm MCS}$ is given in Ref.~\\cite{Lynch:1990sq,ParticleDataGroup:2020ssz}. The integral variable $x\\equiv$ $Q^2\/[2M_N(E_{i}-E_{\\rm HNL})]$ denotes the Bjorken scaling, where $E_{i}$ is the incoming lepton energy at the HNL production, and $E_{\\rm HNL}$ is the HNL energy in the nucleon rest frame. The other integral variable $y\\equiv(E_{i}-E_{\\rm HNL})\/E_{i}$ is the fraction of the lepton energy loss in the rest frame. \n\n\nFinally, we give the formula we used to estimate the acceptance ${\\rm Acc}_{\\rm dump,shield}^{(\\rm i,ii)}({\\rm HNL})$ as follows,\n\\begin{align}\n{\\rm Acc}_{\\rm dump}^{{\\rm (i)}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm dump}}{dz}\\cdot \\int_0^{r_{\\rm det}\/(l_{\\rm dump}+l_{\\rm sh})}d\\theta_{\\rm HNL}^{\\rm dec} \\frac{dP_{\\rm ang}}{d\\theta_{\\rm HNL}^{\\rm dec}}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm dump}}^{{\\rm (i)}}\\right),\\label{eq:acci}\n\\\\\n{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},e^{\\pm}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm shield}}{dz}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm shield}}^{{\\rm (ii)},e^{\\pm}}\\right),\\label{eq:acciie}\n\\\\\n{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},\\mu^{\\pm}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm shield}}{dz}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm shield}}^{{\\rm (ii)},\\mu^{\\pm}}\\right)\\times\n\\begin{cases}\n\\Theta \\left(l_{\\rm sh}-\\delta_{\\mu}\\right)~~~{\\rm for}~{\\rm ILC\\text{-}250}\n\\\\\n\\Theta \\left(10~{\\rm m}-\\delta_{\\mu}\\right)~~~{\\rm for}~{\\rm ILC\\text{-}1000}\n\\end{cases},\\label{eq:accmuon}\n\\end{align}\nwhere, as in the Monte-Carlo simulation, all the HNL decays inside the decay volume are assumed to be the visible signal except for the ones to entirely invisible particles ${\\rm HNL}\\to \\nu\\nu\\bar{\\nu}$.\nHere $z$ is the decay position of HNL in the decay volume, and $P_{\\rm dec}$ denotes the decay probability of HNL given by\n\\begin{align}\n\\frac{d P_{\\rm dec}^{\\rm dump}}{dz}&=\\frac{1}{l^{({\\rm lab})}_{\\rm HNL}}{\\rm exp}\\left(-\\frac{l_{\\rm dump}+l_{\\rm sh}+z}{l^{({\\rm lab})}_{\\rm HNL}}\\right),\\label{eq:decP}\n\\\\\n\\frac{d P_{\\rm dec}^{\\rm shield}}{dz}&=\\frac{1}{l^{({\\rm lab})}_{\\rm HNL}}{\\rm exp}\\left(-\\frac{l_{\\rm sh}-\\delta_{\\mu}+z}{l^{({\\rm lab})}_{\\rm HNL}}\\right), \n\\end{align} \nwhere $l^{({\\rm lab})}_{\\rm HNL}$ denotes the lab-frame decay length of HNL given by\n\\begin{align}\nl^{(\\rm lab)}_{\\rm HNL}=\\frac{p_{\\rm HNL}}{m_{\\rm HNL}}\\frac{1}{\\Gamma_{\\rm HNL}}\n\\end{align}\nwith $p_{\\rm HNL}$ being the momentum, $m_{\\rm HNL}$ the mass, and $\\Gamma_{\\rm HNL}$ the total decay width of HNL evaluated in \\cite{2018}.\nThe angular distribution of the HNLs is denoted as $dP_{\\rm ang}\/d \\theta^{\\rm dec}_{\\rm HNL}$, where $\\theta^{\\rm dec}_{\\rm HNL}$ is the decay angle of HNL from the particle $k$ with respect to the direction of $k$, and we use an approximated formula as follows,\n\\begin{align}\n    \\frac{dP_{\\rm ang}}{d\\theta_{\\rm HNL}^{\\rm dec}}=\\sin \\theta_{\\rm HNL}^{\\rm dec} \\frac{1}{2}\\left(\\frac{m_k}{(E_k+m_k)-p_k\\cos\\theta_{\\rm HNL}^{\\rm dec}}\\right)^2,\n\\end{align}\nwhere $m_k$ is the mass of the SM particle $k$, and $p_k$ denotes the momentum of $k$ in the lab-frame.\nThen, the energy of the HNL is approximately given by\n\\begin{align}\n    E_{\\rm HNL}=\\frac{1}{2}\\frac{m_k^2}{(E_k+m_k)-p_k\\cos\\theta_{\\rm HNL}^{\\rm dec}}.\n\\end{align}\nThe distance that a muon passes through the muon shield before emitting the HNL is the function of $E_{\\mu0}$ and $E_{\\mu}$~\\cite{Sakaki:2020mqb}, \n\\begin{align}\n    \\delta_{\\mu}=\\frac{E_{\\mu 0}-E_{\\mu}}{{\\langle dE\/dx\\rangle}_{\\rm Lead}},\n\\end{align}\nwith the energy independent stopping power: ${\\langle dE\/dx\\rangle}_{\\rm Lead}=0.02~{\\rm GeV}\/{\\rm cm}$. \nFor $E_{\\rm beam}=500$~GeV, the additional active shield behind the muon shield may be necessary due to their strong penetrating power, and the muon's trajectory becomes non-trivial after the active shield. Then we account for only the HNLs produced before the active shield, which leads to  the Heaviside step function in Eq.~\\eqref{eq:accmuon}. Regarding the impact of varying the active shield length, we study how  different depth of the muon shield affects the sensitivity for the HNL at ILC-1000 in Appendix~\\ref{app:muonshield}. The angular acceptance is expressed by the Heaviside step function associated with the transverse radius $r_\\perp (z)$ in Eqs.~\\eqref{eq:acci}, \\eqref{eq:acciie}, and \\eqref{eq:accmuon}. The typical deviation of the visible SM particles emitted from HNL from the beam axis is estimated as\n\\begin{align}\nr_{\\perp,{\\rm dump}}^{\\rm (i)}&=|\\theta_{\\rm HNL}^{\\rm dec}|\\cdot (l_{\\rm dump}+l_{\\rm sh}+z),\\label{eq:ri}\n\\\\\nr_{\\perp,{\\rm dump}}^{{\\rm (ii)},e^{\\pm}}&=\\sqrt{(\\theta_{e}^{\\rm shower})^2\\cdot (l_{\\rm dump}+l_{\\rm sh}+z)^2+(\\theta_{\\rm HNL}^{\\rm prod})^2\\cdot (l_{\\rm dump}+l_{\\rm sh}+z)^2},\n\\\\\nr_{\\perp,{\\rm dump}}^{{\\rm (ii)},\\mu^{\\pm}}&=\\sqrt{(\\theta_{\\mu}^{\\rm shower})^2\\cdot (l_{\\rm sh}-\\delta_{\\mu}+z)^2+(\\theta_{\\rm HNL}^{\\rm prod})^2\\cdot (l_{\\rm sh}-\\delta_{\\mu}+z)^2},\n\\end{align}\nwhere $\\theta_e^{\\rm shower}$ ($\\theta_{\\mu}^{\\rm shower}$) is the angle of shower electrons and positrons (muons) with respect to the beam axis, and $\\theta_{\\rm HNL}^{\\rm prod}$ is the production angle of HNL.\nFor the DIS process, the production angle is expressed as\n\\begin{align}\n    \\cos\\theta_{\\rm HNL}^{\\rm prod}=\\frac{1}{|p_{e,\\mu}|\\cdot|p_{\\rm HNL}|}\\left(E_{e,\\mu}E_{\\rm HNL}-\\frac{1}{2}\\left(Q^2+m_{e,\\mu}^2+m_{\\rm HNL}^2\\right) \\right),\n\\end{align}\nwhere $p_{e,\\mu}$ is the momentum of incoming electron and muon, and $Q^2=2 M_N E_{e,\\mu} xy$ with the mass of nucleus. \nAs provided in Refs.~\\cite{Sakaki:2020mqb} and \\cite{Asai:2021ehn}, $\\theta_{e}^{\\rm shower}$ is estimated by the Monte-Carlo simulation, and we use the mean value $\\theta_{e}^{\\rm shower}=16~{\\rm mrad}\\cdot {\\rm GeV}\/E_{e^{\\pm}}$.\nFor the incoming muons, we use $\\theta_{\\mu}^{\\rm shower}=\\theta_{\\rm MCS}$. Although for the production mechanism (i), the angle of shower photon and the production angle of the particle $k$ are precisely calculated in the Monte-Carlo simulation, we omit them in the coarse-grained integration method. This is because the heavy mesons are produced by high-energy photons and the mean value of the angle of shower photons $\\theta^{\\rm shower}_{\\gamma}=8~{\\rm mrad}\\cdot {\\rm GeV}\/E_{\\gamma}$~\\cite{Sakaki:2020mqb,Asai:2021ehn} is suppressed.\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/Ue.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the electron neutrino in the mass and mixing plane, assuming 10 year run at ILC-250 (black solid) and ILC-1000 (red solid).  It requires more than three signal events in the detector assuming no background, corresponding to $95\\%$ C.L. sensitivity. The current exclusion bounds are shown in the gray region, see Sec.~\\ref{sec:bounds}. The darker grey region is from the laboratory bounds, and the lighter gray region is the BBN bound for the HNL, which is roughly $\\tau_{\\rm HNL}>0.02$~s~\\cite{Boyarsky:2020dzc}. Sensitivity reach through $10^ 9$ $Z$-decays  at ILC is shown as a blue solid line. See Sec.~\\ref{sec:ZatILC}. For a comparison, dashed lines show a sensitivity reach of the DUNE experiment~\\cite{Coloma:2020lgy} (brown), the FASER2 experiment~\\cite{Kling:2018wct} (purple), the NA62 experiment~\\cite{Beacham:2019nyx} (orange), and the SHiP experiment~\\cite{Alekhin:2015byh} (magenta), the  MATHUSLA experiment\\cite{Curtin:2018mvb} (green), and $10^{12}$ $Z$-decays that could be realized at the FCC-ee experiment~(cyan). \\label{fig:sensitivityUe}\n}\n\\end{figure*}\n\n\n\n\\subsubsection{Projected sensitivities}\nWe give the prospects of the ILC beam dump experiment in ILC-1000 and ILC-250 for 10 year run. We assume the parameter region with more than three signal events can be probed, corresponding to the expected $95\\%$ C.L. exclusion sensitivity. After independently evaluating the signal events by the productions (i) and (ii), we combine the plots of the sensitivity of ILC.\n\n\n \n\n\\subsection*{$U_e$ dominance}\n\nFig.~\\ref{fig:sensitivityUe} shows the sensitivity of ILC for the HNL whose mixing to active neutrinos is dominated by the $U_e$ mixing. The region above the red (black) curves is the region {expected $95\\%$ C.L. exclusion sensitivity} that can be searched by ILC-1000 (ILC-250) with 10-year statistics. The gray-shaded regions are constrained from past experiments which is discussed in Sec.~\\ref{sec:bounds}.\n\n\nLet us explore the results in Fig.~\\ref{fig:sensitivityUe}, highlighting each of the HNL production processes.\n\\begin{enumerate}[(a)]\n\\item Meson and $\\tau$ lepton decays\n\nIn the small $|U_e|^2$ regions where the lifetime of HNL becomes longer, the decay probability in Eq.~\\eqref{eq:decP} is approximately $d P_{\\rm dec}^{\\rm dump}\/dz \\simeq 1\/l^{\\rm (lab)}_{X}$. Depending on the mass of HNL, the HNLs are produced by the decay of mesons and $\\tau$ lepton, see Figs.~\\ref{fig:brUeli}, \\ref{fig:brUeD}, \\ref{fig:brUeB}, and \\ref{fig:brUetau} in Appendix~\\ref{sec:Br}.\nBelow the kaon mass $m_{K}\\simeq 0.5~{\\rm GeV}$, the HNLs are mainly produced by the kaon decay. For $m_{\\rm HNL}\\leq m_{D,\\tau}-m_e\\sim 2~{\\rm GeV}$, the $D(D_s)$ meson decay dominates the HNL productions. In the region of $1~{\\rm GeV}\\lesssim m_{\\rm HNL}\\lesssim 2~{\\rm GeV}$ there are many thresholds of production mode, such as $D\\to K+e+{\\rm HNL}$, $D_s\\to \\eta+e+{\\rm HNL}$, $D\\to \\pi+e+{\\rm HNL}$ and $\\tau\\to \\nu_{\\tau}+e+{\\rm HNL}$, see Fig.~\\ref{fig:brUeD}.   \n\n\nThe limit obtained by the Monte-Carlo simulation can be checked using the approximate formula Eq.~\\eqref{eq:N1approx}. As explained in Appendix \\ref{app:Com}, the approximate formula agrees well with the results of the Monte-Carlo simulation. For the electron beam energy $E_{\\rm beam}=500~{\\rm GeV}$, the number of events by the leptonic $D$ meson decays $D^\\pm\\to e^\\pm +{\\rm HNL}$ is given by \n\\begin{align}\nN_{\\rm signal}^{\\rm (i)}&\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)  \\left(\\frac{l_{\\rm dec}}{50~{\\rm m}}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2\\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2\\left(\\frac{|U_e|^2}{ 10^{-10}}\\right)^2  \\left(\\frac{m_{\\rm HNL}}{1~{\\rm GeV}}\\right)^4,\n\\end{align}\nwhere we used follwing approximations:\n\\begin{align}\n {\\rm Br}(D^{\\pm}\\to e^{\\pm}+{\\rm HNL})\\propto |U_e|^2 , \\quad   \n {\\rm Br}_{\\rm vis}\\simeq 1, \\quad \n (l^{\\rm lab}_X)^{-1}\\propto |U_e|^2  m_{\\rm HNL}^4\/E^{\\rm lab}_{\\rm HNL} \\ . \n\\end{align}\nNear the kinematic threshold, the number of events is rapidly decreased by a suppression of the branching ratio ${\\rm Br}(D^{\\pm}\\to e^{\\pm}+{\\rm HNL})$. For $m_D-m_e \\lesssim m_{\\rm HNL} \\lesssim m_B-m_e$, the $D$ meson decay channels are closed, and the $B$ meson decay channels become dominant in the HNL productions up to around $m_{\\rm HNL}\\sim 3$ GeV.\n\n\n\n\n\\item Direct production\n\nAbove $m_{\\rm HNL}\\sim 3$ GeV, the direct production channel with the incoming electrons and positrons become significant. \nIn particular, the production with the high-energy primary electron is essential to extend the mass reach of the $U_e$ dominant scenario.\nIn the small $|U_e|^2$ regions, the decay probability in Eq.~\\eqref{eq:decP} is approximately $d P_{\\rm dec}^{\\rm dump}\/dz \\simeq 1\/l^{\\rm (lab)}_{X}$, and the energy of the produced HNL is approximately $E^{\\rm lab}_{\\rm HNL}\\simeq E_{e^{\\pm}}$. Then, the number of events for $E_{\\rm beam}=500~{\\rm GeV}$ is given by\n\\begin{align}\nN_{\\rm signal}^{\\rm (ii),e^{\\pm}}\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)\\left(\\frac{l_{\\rm dec}}{50~{\\rm m}}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2\\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2\\left(\\frac{|U_e|^2}{10^{-10}}\\right)^2 \\left(\\frac{m_{\\rm HNL}}{10~{\\rm GeV}}\\right)^6,\n\\end{align}\nwhere we use, \n\\begin{align}\n    &dl_{e^{\\pm}}\/d E_{e^{\\pm}}\\simeq \\left(dl_{e^{\\pm}}\/d E_{e^{\\pm}}\\right)_{\\rm primary}\\propto 1\/E_{e^{\\pm}},\\quad\n    d^2\\sigma(e^{\\pm} N\\to {\\rm HNL})\/dx dy \\propto |U_e|^2 E_{e^{\\pm}},\n    \\nonumber\\\\\n    &(l^{(\\rm lab)}_X)^{-1}\\propto |U_e|^2 m_{\\rm HNL}^6\/E^{\\rm lab}_{\\rm HNL},\\quad\n    {\\rm Br}_{\\rm vis}\\simeq 1,\\quad\n    y\\lesssim x^{-1}E_{e^{\\pm}} r^2_{\\rm det}(l_{\\rm dump}+l_{\\rm sh})^{-2}.\n\\end{align}\n\n\nIn the larger $|U_e|^2$ regions, where $l^{\\rm (lab)}_{\\rm HNL}\\ll l_{\\rm dump}+l_{\\rm sh}$, the shape of the upper contour lines in Fig.~\\ref{fig:sensitivityUe} is determined by the probability to decay inside the decay volume given in Eq.~\\eqref{eq:decP}. \nThe contour is characterized by the exponent of the decay probability\n\\begin{align}\n\\frac{m_{\\rm HNL}\\Gamma_{\\rm HNL}}{E^{\\rm lab}_{\\rm HNL}}(l_{\\rm dump}+l_{\\rm sh})\\sim {\\rm const}.\\label{eq:diupp}\n\\end{align} \nCombining $E^{\\rm lab}_{\\rm HNL}\\simeq E_{\\rm beam}$ and $\\Gamma_{\\rm HNL}\\propto |U_e|^2\\cdot m_{\\rm HNL}^5$, Eq.~\\eqref{eq:diupp} becomes $|U_e|^2\\propto m_{\\rm HNL}^{-6}$ , which is consistent with the parameter dependence of Fig.~\\ref{fig:sensitivityUe}. \n\n\\end{enumerate}\n\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/Umu.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the mu-neutrino in the mass and mixing\nplane. For the description of this plot, see the caption of Fig.~\\ref{fig:sensitivityUe}.  \\label{fig:sensitivityUmu}\n}\n\\vspace{20pt}\n\\includegraphics[width=0.6\\textwidth]{figs\/Utau.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the tau-neutrino in the mass and mixing\nplane. For the description of this plot, see the caption of Fig.~\\ref{fig:sensitivityUe}. \n\\label{fig:sensitivityUtau}\n}\n\\end{figure*}\n\n\n\\subsection*{$U_\\mu$ dominance}\nFig.~\\ref{fig:sensitivityUmu} summarizes the prospects of the ILC beam dump experiment in ILC-1000 and ILC-250 for the HNL mixing dominantly with $\\nu_{\\mu}$.\nThe red (black) curves show the expected sensitivity for $95\\%$ C.L. exclusion of ILC-1000 (ILC-250) with 10-year statistics given by the meson and $\\tau$ lepton decays, and the DIS process for incoming muons from the EM shower. \nWe provide approximated formulae of the number of signal events for each of the HNL production mode for ease of understanding.\n\n\n\\begin{enumerate}[(a)]\n\\item Meson and $\\tau$ lepton decays\n\nThe production process from meson and $\\tau$ lepton decays of the HNL that mix dominantly with $\\nu_{\\mu}$ can be calculated in parallel with $U_e$ dominant case except for minor threshold difference.\nThe mass dependence of the dominant HNL production process is the same as that of $U_e$ dominant case with $e$ replaced with $\\mu$.\n\n\nAs shown in Figs.~\\ref{fig:brUmuD} and \\ref{fig:brUmuB}, for $m_D-m_{\\mu} \\lesssim m_{\\rm HNL} \\lesssim m_B-m_{\\mu}$, \nthe $B$ meson decay gives a significant contribution to the HNL production.\nThe zigzag curves near $m_{\\rm HNL}\\sim 3~{\\rm GeV}$ correspond to the threshold of $B\\to D+{\\mu}+{\\rm HNL}$.\nAbove $m_{\\rm HNL}\\simeq 3~{\\rm GeV}$, the leptonic $B$ meson decay such as $B^{\\pm}\\to \\mu^{\\pm} +{\\rm HNL}$ dominates the HNL productions.\n\n\n\\item Direct production\n\nAs shown in Fig.~\\ref{fig:exp}, incident real photons produce muon pairs in the beam dump through the electromagnetic interaction with the nucleus or nucleon~\\cite{Sakaki:2020cux}, and the HNL can be generated by the DIS process in the muon shield.\nThe muon pair production cross section is about $(m_{\\mu}\/m_e)^2\\simeq 10^{5}$ times smaller than that of the electron pair production. \n\n\nIn the small $|U_{\\mu}|^2$ regions, the projected sensitivity scales with the ratio of numbers of muon and electron in the \nshower, and the projected sensitivity via the DIS process  is less significant compared with the \n$U_e$ dominant case.\nOn the other hand in the large $|U_{\\mu}|^2$ regions, the number of signal events  is mostly determined by the muon shield length and  less sensitive to the number of muon pairs, because the lifetime is shorter.\nThe shape of the upper side of contours in Fig.~\\ref{fig:sensitivityUmu} is determined by the exponential factor in Eq.~\\eqref{eq:decP}, which is characrized by \n\\begin{align}\n    \\frac{m_{\\rm HNL} \\Gamma_{\\rm HNL}}{E^{\\rm lab}_{\\rm HNL}}(l_{\\rm sh}-\\delta_{\\mu})\\sim {\\rm const}.\\label{eq:larUmu}\n\\end{align}\nCombining $E^{\\rm lab}_{\\rm HNL}\\simeq E_{\\rm beam}$ and $\\Gamma_{\\rm HNL}\\propto |U_{\\mu}|^2\\cdot m^5_{\\rm HNL}$, Eq.~\\eqref{eq:larUmu} becomes $|U_{\\mu}|^2\\propto m^{-6}_{\\rm HNL}$. \nSince the DIS process with a muon happens in the muon shield,  the shorter distance from the HNL production point to the decay volume enhances the acceptance compared the DIS process in the beam dump.\n  \n\\end{enumerate}\n\n\n\n\\subsection*{$U_\\tau$ dominance}\n\nThe projected sensitivities in the $U_\\tau$ dominant scenario are shown in Fig.~\\ref{fig:sensitivityUtau} with a similar notation as in the $U_{e,\\mu}$ dominance.\nThe red (black) curves show the expected $95\\%$ C.L. exclusion sensitivity with 10-year statistics in ILC-1000 (ILC-250) by $B$ and $D_s$ mesons and $\\tau$ lepton decay production. The direct (DIS) production is absent in this case due to very limited flux of $\\tau$ leptons.\n\n\nAs shown in Figs.~\\ref{fig:production}, \\ref{fig:brUtauD}, \\ref{fig:brUtauB}, and \\ref{fig:brUtautau}, below the threshold of $m_{D_s}-m_{\\tau}\\simeq 0.2~{\\rm GeV}$,\nthe main HNL production channel is the $D_s\\to \\tau+{\\rm HNL}$ decay.\nFor $0.2~{\\rm GeV}\\lesssim m_{\\rm HNL}\\leq m_{\\tau}-m_e\\simeq 1.8~{\\rm GeV}$, the production of HNL is dominated by $\\tau$ lepton decay.\nMost of the $\\tau$ leptons are produced by $D_s$ meson decays, and the $\\tau$ pair production in the electromagnetic showers are subdominant, which therefore is dropped in the HNL study. \nThe branching ration of $D_s\\to \\tau +\\nu_{\\tau}$ is $5.48\\%$, and an energy dependence of $\\tau$ lepton spectra is determined by that of $D_s$ meson spectra in Fig.~\\ref{fig:production}.\nThe main decay channels of $\\tau$ are $\\tau\\to \\rho +{\\rm HNL}$, $\\tau\\to \\mu+\\nu+{\\rm HNL}$, $\\tau\\to e+\\nu+{\\rm HNL}$, $\\tau\\to \\pi+{\\rm HNL}$, $\\tau\\to K^{\\ast}+{\\rm HNL}$, and $\\tau\\to K+{\\rm HNL}$, see Fig~\\ref{fig:brUtautau}.\n\nIn the small $|U_{\\tau}|$ regions with the longer lifetime of HNL, the decay probability in Eq.~\\eqref{eq:decP} is approximated by $d P_{\\rm dec}^{\\rm dump} \/dz\\simeq 1\/l^{\\rm (lab)}_X$.\nFor the electron beam energy $E_{\\rm beam}=500~{\\rm GeV}$, the number of events by the $\\tau^{\\pm}\\to \\rho^{\\pm} +{\\rm HNL}$ is approximately given by\n\\begin{align}\n    N_{\\rm signal}^{\\rm (i)}&\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)\\left(\\frac{l_{\\rm dec}}{50~{\\rm }}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2 \\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2 \\left(\\frac{|U_{\\tau}|^2}{5\\times 10^{-9}}\\right)^2 \\left(\\frac{m_{\\rm HNL}}{0.5~{\\rm GeV}}\\right)^4, \n\\end{align}\nwhere we use\n\\begin{align}\n    {\\rm Br}(\\tau^{\\pm}\\to \\rho^{\\pm}+{\\rm HNL})\\propto |U_{\\tau}|^2,\\quad\n    {\\rm Br}_{\\rm vis}\\simeq 1,\\quad\n    (l^{\\rm lab}_X)^{-1}\\propto |U_{\\tau}|^2 \\cdot m^4_{\\rm HNL} \/E^{\\rm lab}_{\\rm HNL}.\n\\end{align}\nFor $1.8~{\\rm GeV}\\lesssim m_{\\rm HNL}$, the $B$ meson decays dominate the HNL productions, see Figs.~\\ref{fig:brUtauD}, \\ref{fig:brUtauB}, and \\ref{fig:brUtautau}.\n\n\n\n\\subsection{Sensitivities from $Z$~decays at ILC and FCC-ee}\\label{sec:ZatILC}\n\nThe future $e^+ e^-$ colliders can be seen as a $Z$ boson factory, such as Giga-$Z$ program of the ILC and Tera-$Z$ program of the CERN Future $e^+ e^-$ Circular Collider, dubbed FCC-ee. The HNLs can leave a clear signal with displaced tracks at $e^+ e^-$ colliders once  they are produced via $Z\\to N \\bar\\nu, \\bar N \\nu$. This type of signal was examined at DELPHI detector of the Large Electron-Positron collider (LEP) \\cite{DELPHI:1996qcc}. We briefly study the future sensitivities of the HNL search using the displaced tracks from the $Z$ decay, which is complementary to the sensitivities of the ILC beam dump experiment. \n\n\nThe proposed ILC detector \\cite{Behnke:2013lya} has three layers of vertex detector (VTX) starting from 1.5~cm to 19.5~cm surrounding the beam,  Time Projection Chambers (TPC) as the main tracking detector spreading from 33~cm to 170~cm, and  two layers of silicon strip detectors (SIT) arranged between TPC and VTX. We assume the ILC detector is sensitive to the HNL signal with negligible background if the HNL decays between 3~cm and 170~cm from the collision point. We adopt  the radius-dependent track-detection efficiency $\\epsilon_{\\rm trk}$ linearly decreasing from $r = 3$~cm ($\\epsilon_{\\rm trk}=100\\%$) to $r = 170$~cm ($\\epsilon_{\\rm trk}=0\\%$)~\\cite{Bertholet:2021hjl}. Then, we assume $10^9$ $Z$ decays and  zero background after requiring the displaced tracks and consider all the HNL decays except for the fully invisible mode, $N\\to \\nu\\nu\\bar\\nu$. The $95\\%$~CL future sensitivity corresponds to three HNL decays with the multi-displaced tracks in the fiducial volume, and the result is included in Figs.~\\ref{fig:sensitivityUe},~\\ref{fig:sensitivityUmu},~and~\\ref{fig:sensitivityUtau}.\n\n  \nSimilarly, we study the future projection at the FCC-ee. For the simplicity, we take the same detector setup of the ILC and rescale the statistics assuming $10^{12}$ $Z$ decays. This is also shown in Figs.~\\ref{fig:sensitivityUe},~\\ref{fig:sensitivityUmu},~and~\\ref{fig:sensitivityUtau}. Ref.~\\cite{Blondel:2014bra} studied also the projected sensitivities at the FCC-ee, but our estimate is different with respect to the detector coverage and the signal efficiency. \n\n\n\\subsection{Existing constraints on HNL and projected sensitivities at other experiments}\\label{sec:bounds}\n\nThe presence of HNL can have drastic consequences on early-universe observables and has been explored extensively in the literature. The HNL is thermally produced in the early universe, and its late decay can disrupt the standard  Big Bang Nucleosynthesis. The typical constraint is for $\\tau_{\\rm HNL}\\lesssim 1\\rm s$, see \\cite{Abazajian:2012ys} and references therein. In addition, Ref ~\\cite{Boyarsky:2020dzc}  studies that the pions from the HNL hadronic decay alter the decoupling of neutrons from the prediction of the standard BBN, which affects the ratio of neutrons to protons. The upper limit on the HNL lifetime is obtained from the $^4$He abundance as $\\tau_{\\rm HNL}\\lesssim 0.02$~s (see also other recent studies \\cite{Alonso-Alvarez:2022uxp, Bondarenko:2021cpc}).\n\n\nThe long-lived HNLs are also extensively searched in the accelerator-based experiments. The HNLs may be produced directly either in the beam-target collisions, or in the decays of secondary particles such as $B$, $D$ mesons and $\\tau$ leptons. The searches are carried out either by detecting the displaced decay of the HLN or the significant missing momentum of the events. In the following, we briefly list the existing constraints included in Fig.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau}. \n\n\n\\begin{itemize}\n\\item CHARM  proton beam dump experiment --- \nSearches for HNLs produced at a proton beam dump operating at the 400 GeV CERN SPS are  sensitive to all the scenarios we consider~\\cite{CHARM:1983ayi, CHARM:1985nku}. Recently, the result of the $U_\\tau$ dominant scenario was reanalyzed, and the bound was improved for  $ 0.3~{\\rm GeV}\\lesssim m_{\\rm HNL}\\lesssim 1.5~{\\rm GeV}$~\\cite{Boiarska:2021yho}. \n\n\n\\item Other beam damp\/neutrino beam experiments --- \nIn addition to the CHARM experiment, the HNL searches were conducted at a variety of fixed target and neutrino experiments, such as NuTeV~\\cite{NuTeV:1999kej}, BEBC~\\cite{WA66:1985mfx}, and PS191~\\cite{Bernardi:1987ek}. They are mostly sensitive to the $U_\\mu$ mixing, while PS191 placed a strong bound on the $U_e$ dominant scenario as well.\n \n\n\\item Long-baseline neutrino experiment --- \nAt the T2K experiment, the near detector ND280 was utilized to search for the HNL~\\cite{T2K:2019jwa}. The detection principle is similar to the one at  PS191. \nWe include the bounds of ``single-channel'' analysis where only one of  $U_\\ell$ is non-zero at a time. In the future, this type of search can be performed at the near detector of the DUNE experiment~\\cite{Coloma:2020lgy}.\n\n\n\\item Super-Kamiokande --- \nThe HNLs can be copiously produced from kaon and pion decays in atmospheric showers and decay in the Super-Kamiokande detector. In~\\cite{Coloma:2019htx}, the authors analyzed the Super-Kamiokande data~\\cite{Super-Kamiokande:2017yvm}, and obtained bounds on the HNL in all the scenarios. The bound on $U_{\\tau}$ is as strong as the T2K one. \n \n\n\\item Pion decay --- \nPrecision measurements of charge pions can test $\\pi^+ \\to \\ell^+ +\\rm HNL$  with no subsequent HNL decay which is sensitive to  the $U_e$ and $U_{\\mu}$ mixings \\cite{PIENU:2017wbj, PIENU:2019usb}. It gives an unique bound in the low mass region $m_{\\rm HNL}\\sim 0.1~{\\rm GeV}$ of the $U_{e}$ dominant scenario. \n\n\n\\item  Kaon decay ---  \nSimilar to the search with the charged pion, the HNL searches have been conducted by $K^+\\to \\ell^+ +\\rm HNL(invisible)$ at NA62~\\cite{NA62:2020mcv, NA62:2021bji}, E949~\\cite{E949:2014gsn}, and KEK\\cite{Yamazaki:1984sj}.  Typically they give the best limit near the threshold of $m_{\\rm HNL}\\lesssim m_{K^+}-m_{\\ell^+}$. \n\n\n\\item $B$ meson and $\\tau$ lepton decay --- \nThe heavier HNL can be looked for from the $B$ meson decays. The HNL production by $B\\to X \\ell +{\\rm HNL}\\ (\\ell=e,\\mu)$  with a subsequent displaced decay of ${\\rm HNL}\\to \\ell \\pi$ was examined using the Belle data in~\\cite{Belle:2013ytx}.  The $B$-factories also produce $\\tau$ leptons  copiously, and  the search of long-lived HNL from $\\tau$ lepton decays at Belle and Babar placed a bound on $U_{\\tau}.$ \\cite{Dib:2019tuj}\n\n\n\\item $Z$ boson decay --- \nThe search of $Z$ boson decaying into HNL using the DELPHI data at the LEP collider constraints all the scenarios. It sets strongest limit  in mass range around $m_{\\rm HNL}\\sim {\\cal O}(10)~\\rm GeV$ \\cite{DELPHI:1996qcc}. See Sec.~\\ref{sec:ZatILC} for more detail and  the  projection at the future $e^+e^-$ colliders.    \n\n\n\\item Large Hadron Collider --- \nAt the ATLAS and CMS detectors of LHC, the $W$ boson mediated process efficiently produces the HNL. Depending on the final states and decay length, different analyses were performed at ATLAS~\\cite{ATLAS:2019kpx, ATLAS:2022atq} and CMS~\\cite{CMS:2018jxx, CMS:2018iaf, CMS:2022fut}. In particular, the search for the displaced decays of HNL~\\cite{CMS:2022fut, ATLAS:2022atq} excludes the large parameter space. \n \n\n\\end{itemize}\n\n\nAmong studies of the future HNL searches, we include projected sensitivities at FASER2~\\cite{FASER:2018eoc}, NA62~\\cite{Beacham:2019nyx}, DUNE~\\cite{Coloma:2020lgy}, SHiP~\\cite{Alekhin:2015byh}, and MATHUSLA~\\cite{Curtin:2018mvb}  in Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau} to compare  to the sensitivities at the ILC beam dump experiment. Other experiments including LHCb~\\cite{Antusch:2017hhu, Cvetic:2019shl}, Codex-b~\\cite{Aielli:2019ivi}, Belle~II~\\cite{Dib:2019tuj}, Dark-Quest~\\cite{Batell:2020vqn}, IceCube~\\cite{Coloma:2017ppo},  ATLAS and CMS experiments~\\cite{ATLAS:2022atq, CMS:2022fut} at HL-LHC can also search for the HNLs beyond the current experimental constrains. \n\nIn Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau}, the estimated sensitivity contours of ILC is better than the other proposed searches. We remind here  that the contour is calculated  assuming zero background events, but it is very encouraging to explore the possibility further. The ILC sensitivity is very close to the one of SHiP in the low mass region ($m_{\\rm HNL}<2$~GeV). Above 2~GeV, the sensitivity of the ILC beam dump experiment is better than SHiP because the initial electron energy is high enough to produce $B$ mesons at a higher rate. Thanks to the higher HNL mass, the HNL decay products are more clearly separated from the background so that the neutrino background would be less critical in the region. \n\n\n\n\n\\section{Discussions}\\label{sec:discussion}\n\nThe ILC beam dump experiment is a seamless extension of the ILC program, which provides a unique opportunity to test feebly interacting light particles. The electron beam energy of ILC is much higher than the ones at the current and past electron beam dump experiments, and therefore, not only the light mesons but also the heavier SM particles, such as heavy flavor mesons and $\\tau$ lepton, can be produced in the beam dump. Moreover, a large number of electrons on target are expected thanks to the high-intensity beam. In many BSM scenarios, new particles can be efficiently produced by decays of the SM particles. Therefore, it is essential to estimate the production rate of the SM particles at the ILC beam dump experiment. In this paper, we evaluate the light and heavy mesons and $\\tau$ lepton spectra at the decay for the first time using  PHITS and PYTHIA8. PHITS is responsible for producing and transporting light SM particles, and we incorporate the heavy meson productions calculated by PYTHIA8 into PHITS. The main results are in Fig.~\\ref{fig:decay} for the light mesons and in Fig.~\\ref{fig:production} for the heavy mesons and $\\tau$ lepton. \n\n\nThe spectra of SM particles can be used to estimate the  yield of BSM particles from SM particle decay. As a demonstration, we studied the projected sensitivity of the heavy neutral leptons at the ILC beam dump experiment. Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau} show that  ILC would explore the HNLs with the heavier mass and small mixing and cover the large parameter space motivated by the baryon asymmetry of the Universe. We use the Monte-Carlo simulation to evaluate the HNL signal from the SM particle decay, and the results are well reproduced by the the course-grained integration method convolving the SM particle spectra given in Figs.~\\ref{fig:decay} and \\ref{fig:production}. \n\n\nApart from the SM particle decays, the HNLs can be produced through various processes at ILC. The EM shower can directly create an HNL via DIS, and we account for this production by the course-grained integration method. Moreover, copious $Z$ decays expected in the primary detector of ILC would realize  a different search method for the mildly long-lived HNLs.\n\n\nOther than the HNLs, many motivated long-lived particles can be predominantly produced by the SM particle decay. For example, dark scalars such as the QCD axion and the Higgs portal scalar can be efficiently produced via flavor-changing decays of meson (especially $B\\to K X$ and $K\\to \\pi X$ where $X$ denotes a long-lived particle).  The heavy mesons produced from the high energy electron beam allow us to probe long-lived particles heavier than several GeV. Another advantage of the abundant heavy mesons is that the experiment is sensitive to a new particle that preferably couples to the third generation fermions ($b$ quark or $\\tau$ lepton). One can estimate the sensitivity of these particles at the ILC beam dump experiment based on our results in Figs.~\\ref{fig:decay}, \\ref{fig:production} and the approximate formula like Eqs.~\\eqref{eq:sig}. \n\n\n\n\n\\section*{Note added}\\label{sec:note}\nWhile completing this work, we became aware of \\cite{Giffin:2022}, which considers related topics.\n\n\\section*{Acknowledgement}\\label{sec:ackn}\nMN is supported by \t Grant-in-Aid for Scientific Research on Innovative Areas(16H06492) JSPS KAKENHI 22K03629, YS is supported by JSPS KAKENHI JP21H05466.  KT is supported by in part the US Department of Energy grant DE-SC0010102 and JSPS KAKENHI 21H01086. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nRecent observational studies of nearby star-forming regions with the {\\em Herschel Space Observatory} have convincingly shown that stars are born in self-gravitating filaments \n \\citep[e.g., ][]{Andre+2010,Arzoumanian+2011}.  \nIn addition, the resultant mass function of star-forming dense cores are now explained by the mass distribution along filaments \\citep{Inutsuka2001,Andre+2014}. \nThis simplifies the question of the initial conditions of star formation, but poses the question of how such filamentary molecular clouds are created in the interstellar medium (ISM) prior to the star formation process. \nRecent high-resolution magneto-hydrodynamical simulations of two-fluid dynamics with cooling\/heating and thermal conduction by \\citet{InoueInutsuka2008,InoueInutsuka2009} have shown that the formation of molecular clouds requires multiple episodes of supersonic compression \\citep[see also][]{Heitsch+2009}. \n\\citet{InoueInutsuka2012} further investigated the formation of molecular clouds in the magnetized ISM \nand revealed the formation of a magnetized molecular cloud by the accretion of HI clouds created through thermal instability. \nSince the mean density of the initial multi-phase HI medium is an order of magnitude larger than the typical warm neutral medium (WNM) density, this formation timescale is shorter than that of molecular cloud formation solely through the accumulation of diffuse WNM \n\\citep[see, e.g.,][for the cases of WNM flows]{KoyamaInutsuka2002,Hennebelle+2008,HeitschHartmann2008,Banerjee+2009,Vazquez-Semadeni+2011}. \nThe resulting timescale of molecular cloud formation of $\\gtrsim$10 Myrs is consistent with the evolutionary timescale of molecular clouds in the LMC \\citep{Kawamura+2009}.\n\n\nWe have done numerical simulations of additional compression of already-formed but low-mass molecular clouds, and found interesting features associated with realistic evolution.\nFigure 1 shows a snapshot of the face-on view of the layer created by compressing a non-uniform molecular cloud with a shock wave propagating at 10 km\/s. The direction of the shock compression is perpendicular to the layer. The magnetic field lines are mainly in the dense layer of compressed gas. \nThe strength of the initial magnetic field prior to the shock compression is $20\\mu$Gauss and that of the dense region created after compression is about $200\\mu$Gauss on average. \nMany dense filaments are created with axes perpendicular to the mean magnetic field lines. \nWe can also see many faint filamentary structures that mimic ``striations'' observed in the Taurus Dark Cloud and are almost parallel to the mean magnetic field lines \\citep[][]{Goldsmith+2008}. \nIn our simulations, these faint filaments appear to be feeding gas onto dense filaments (similar to what is observed for local clouds by \\citet[e.g.,][]{Sugitani+2011,Palmeirim+2013,Kirk+2013}). \nOnce the line-mass of a dense filament exceeds the critical value ($2C_{\\rm s}^2\/G$), star formation is expected to start \\citep{InutsukaMiyama1992,InutsukaMiyama1997,Andre+2010}. \nThis threshold of line-mass for star formation is equivalent to the threshold of the column density of molecular gas $116M_\\sun {\\rm pc}^{-2}$ \\citep[][]{Lada+2010}, if the widths of filaments are all close to 0.1pc \\citep[][]{Arzoumanian+2011,Andre+2014}. \n\nAlthough further analysis is required for quantitative comparison between the results of simulation and observed structures, Figure 1 clearly shows that the structures created by multiple shock wave passages do match the characteristic structures observed in filamentary molecular clouds. \nThis motivates us to describe a basic scenario of molecular cloud formation.\nThe present paper is focused on the implications of this identification of the mechanism of molecular cloud formation.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{inutsuka_fig1.eps}\n\\caption{Face-on column density view of a shock-compressed dense layer of molecular clouds.  \nWe set up low-mass molecular clouds by the compression of two-phase HI clouds. \nThis snapshot shows the result of an additional compression of \nlow-mass\nmolecular clouds by a shock wave propagating at 10 km\/s. \nThe magnetic field lines are mainly in a dense sheet of a compressed gas. \nThe color scale for column density (in cm$^{-2}$) is shown on top. \nThe mean magnetic field is in the plane of the layer and its direction is shown by white bars.\nNote the formation of dense magnetized filaments whose axes are almost perpendicular to the mean magnetic field. \nFainter ``striation''-like filaments can also be seen, that are almost perpendicular to the dense filaments. \n\\label{fig1}}\n\\end{figure}\n\n\\section{A Scenario of Cloud Formation Driven by Expanding Bubbles}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{inutsuka_fig2.eps}\n\\caption{\nA schematic picture of sequential formation of molecular clouds by multiple compressions by overlapping dense shells driven by expanding bubbles. \nThe thick red circles correspond to magnetized dense multi-phase ISM where cold turbulent HI clouds are embedded in WNM.  \nMolecular clouds can be formed only in the limited regions where the compressional direction is almost parallel to the local mean magnetic field lines, or in regions experiencing an excessive number of compressions.  \nAn additional compression of a molecular cloud tends to create multiple filamentary molecular clouds. \nOnce the line-mass of a filament exceeds the critical value, even in a less massive molecular cloud, star formation starts. \nIn general, star formation in a cloud accelerates with the growth in total mass of the cloud. \nGiant molecular clouds collide with one another at limited frequency. \nThis produces very unstable molecular gas and may trigger very active star formation.  \n\\label{fig2}}\n\\end{figure}\n\nHI observations of our Galaxy reveal many shell-like structures near the galactic plane \\citep[e.g.,][]{HartmannBurton1997,Taylor+2003}. \nWe identify repeated interactions of expanding shock waves as a basic mechanism of molecular cloud formation, and we depict the overall scenario of cloud formation in our Galaxy as a schematic picture in Figure \\ref{fig2}. \nIn this picture, red circles correspond to the remnants of shock waves due to old and slow supernova remnants or  expanding HII regions. \nCold HI clouds embedded in WNM are almost ubiquitously found in the shells of these remnants  \n\\citep[e.g.,][]{HartmannBurton1997,Taylor+2003,2007ApJ...664..363H}. \nMolecular clouds are expected to be formed in limited regions where the mean magnetic field is parallel to the direction of shock wave propagation, or in regions where an excessive number of shock wave sweepings are experienced. \nTherefore, molecular clouds can be found only in limited regions in shells.  \nNote that the typical timescale of each shock wave is of order 1Myr, but the formation of molecular clouds requires many Myrs. \nSome bubbles become invisible as a supernova remnant or an HII region many million years after their birth. \nTherefore, this schematic picture corresponds to a ``very long exposure snapshot'' of the real structure of the ISM. \nEach molecular cloud may have random velocity depending on the location in the most recent bubble that interacts with the cloud. \nInterestingly, this multi-generation picture of the evolution of molecular clouds seems to agree with the observational findings of \\citet{Dawson+2011a,Dawson+2011b,Dawson+2015}, who investigated the transition of atomic gas to molecular gas in the wall of Galactic supershells.\nIn the case of LMC, \\citet{Dawson+2013} concluded that only $12\\sim25$\\% of the molecular mass can be apparently attributed to the formation due to presently visible shell activity.  \nThis may not be inconsistent with our scenario since Dawson et al (2013) only considered HI supergiant shells, whereas molecular clouds in our model can form at the interface of much smaller bubbles and shells (which observationally are more difficult to identify and characterize in the HI data) and the timescale for cloud forming shells to become invisible is much shorter than the growth timescale of molecular mass.\n\nA typical velocity of the shock wave due to an expanding ionization-dissociation front is 10km\/s, as shown by Hosokawa \\& Inutsuka (2006a), since it is essentially determined by the sound speed of ionized gas ($\\sim 10^4$K).  \nIwasaki et al. (2011b) have shown that if a molecular cloud is swept-up by shock wave of ~10km\/s, it moves with a velocity slightly less than the shock speed.  \nThus, the mean velocity of each molecular cloud should be somewhat smaller than that of the most recent shock wave. \nWhen the shock velocity of a supernova remnant is much higher than 10km\/s, the resultant interaction would result in the destruction of molecular clouds.  \nTherefore, the cloud-to-cloud velocity dispersion of molecular clouds should be similar to 10km\/s.\nAccording to this acquisition mechanism of random velocity, the velocity of a cloud is not expected to depend strongly on its mass.  \nIn other words, random velocities of molecular clouds of different masses are not expected to be in equipartition of energy ($M \\delta v^2\/2=$const.).   \nObservations by Stark \\& Lee (2005) have shown that the random velocities of low-mass molecular clouds ($< 2 \\times 10^5 M\\sun$) only vary by a few, with no dependence on cloud mass. \nThese observations are therefore more consistent with our picture than a model in which molecular clouds acquire their relative velocities via mutual gravitational interaction.\n\n\nIn limited circumstances, created molecular clouds collide with one another. \nThis produces highly gravitationally unstable molecular gas and may trigger very active star formation \\citep[e.g.,][]{Fukui+2014}.  \nInoue \\& Fukui (2013) have done magnetohydrodynamical simulations of a cloud-cloud collision and argue that it may lead to active formation of massive stars \\citep[see also][]{Vaidya+2013}. \nThis mode of massive star formation is not, however, a prerequisite of our model.\n\n\n\\subsection{Formation Timescale of Molecular Clouds}\nLet's first model the growth of molecular clouds. \n\\cite{InoueInutsuka2012} have shown that we need multiple episodes of compression of HI clouds to create molecular clouds. \nAccording to the standard picture of supernova-regulated ISM dynamics (e.g., McKee \\& Ostriker 1977), the typical timescale between consecutive compressions by supernova remnants is about 1Myr.  \nThe total creation rate of expanding bubbles is larger than the occurrence rate of supernova explosions, since the former can also be created by radiation from massive stars less massive than supernova progenitors.  \nTherefore, the actual timescale of compressions in ISM, $T_{\\rm exp}$, should be somewhat smaller than 1Myr if it is averaged over the Galactic thin disk.  \nObviously the compression timescale is smaller in the spiral arms and larger in inter-arm regions since star formation activity is concentrated in the spiral arms.\nThus, we have to consider the time evolution of cloud mass for much longer than 1 Myr. \n\nLet us estimate the typical timescale of molecular cloud growth. \n\\cite{InoueInutsuka2012} have shown that the angle between the converging flow direction and the average direction of the magnetic field should be less than a certain angle for molecular cloud formation.  \nAlthough Inoue \\& Inutsuka (2009) shows that this critical angle depends on the flow speed, we adopt a critical angle of 15 degrees (=0.26 radian) in the following discussion for simplicity. \nThis value is not so different from the angle ($\\sim 20$ degrees) for possible compression in the simpler one-dimensional model by \\cite{HennebellePerault2000}. \nFor simplicity we assume that magnetic field is uniform in the region we consider and the direction of compression is isotropic. \nThe solid angle spanned by the possible directions of compression resulting in the formation of a molecular cloud is $0.26^2 \\pi$.  \nThe anti-parallel directions are also possible.  \nTherefore, the probability, $p$, of successfully forming a molecular cloud in a single compression can be estimated by the ratio of solid angle over which compressions lead to molecular cloud formation to the solid angle of the whole sphere, i.e., $p=2 \\cdot 0.26^2 \\pi\/(4\\pi)=0.034$.\nNote that Figure 1 is not the snapshot just after the birth of molecular clouds, but the result of one additional compression of the molecular clouds in which the direction of compression is perpendicular to the mean direction of the magnetic field lines.\nWe also emphasize that since the formation of a GMC requires many episodes of compression, our model does not predict a strong correlation between the present-day magnetic field direction and the orientation of the GMC.\n\nAfter each compression a cloud may slightly expand because of the reduced pressure of the ambient medium, which may result in the loss of diffuse components of cloud mass. \nObservationally the average column densities of molecular clouds do not seem to change very much and always appear to correspond to a visual extinction of several.\nThis means that the mass of a cloud is proportional to its cross-section. \nSince the compressional formation of molecular material is expected to be proportional to the cross-section of the pre-existing cloud, we can model the rate of increase of molecular cloud mass as \n\\begin{equation}\n  \\frac{dM}{dt} = \\frac{M}{T_{\\rm f}} ,  \\label{eq:Eqformation}\n\\end{equation}\nwhere $T_{\\rm f}$ denotes the formation timescale.  \nThis equation shows that resultant mass of each molecular cloud grows exponentially with a long timescale $T_{\\rm f}$ if we average in time over a few Myr.\nIf self-gravity increases the accumulation rate of mass into the molecular cloud, the right-hand side of Equation (1) may have a stronger dependence on mass.  \nFor example, the so-called ``gravitational focusing factor'' increases the cross section of coalescence by a factor proportional to the square of mass for the large mass limit.  \nThis will produce a significantly steeper slope of the cloud mass function (see Section 4).   \nA linear dependence on mass in our formulation implicitly assumes that self-gravity of the whole molecular cloud does not significantly affect the cloud growth.\n\nBased on our investigation of molecular cloud formation described above,\nwe estimate the formation timescale as follows:\n\\begin{equation}\n  T_{\\rm f} = \\frac{1}{p} \\cdot T_{\\rm exp}.  \\label{eq:Tformation}\n\\end{equation}\nThe average value in spiral arm regions would be $T_{\\rm f} \\sim 10$ Myr, but can be factor of a few longer in the inter-arm regions. \nIn reality, many repeated compressions with large angles between the flow direction and mean magnetic field lines gradually increase the dense parts of clouds, and hence contribute to the formation of molecular clouds over a long timescale. \nThis may mean that the actual value of $T_{\\rm f}$ is somewhat smaller than the estimate of Equation (2). \n\nFukui et al. (2009) has shown that the clouds with masses (a few $\\times 10^5M_\\sun$) gain their mass at a rate 0.05 $M_\\sun$\/yr over a timescale 10Myr.  \nThis means that the mass of a cloud in their sample doubles in $\\sim 10$Myr, which is consistent with our choice of $T_{\\rm f}=10$Myr. \nNote, however, that Fukui et al. (2009) argued that the atomic gas accretion is driven by the self-gravity of a GMC, which is not included in the present modelling where we assume that gas accretion is essentially driven by the interaction with expanding bubbles.  \nIf the gravitational force is significant for the HI accretion onto GMC, it possibly enhances the growth rate of molecular cloud (i.e., smaller $T_{\\rm f}$).  \nFurther quantitative studies of the effect of self-gravity on the accretion of gas onto a GMC remain to be done.  \nIn the present paper we neglect this effect and do not distinguish self-gravitationally bound and pressure-confined clouds, for simplicity.\n\nIn regions where the number density of molecular clouds is very large, cloud-cloud collision may contribute to the increase of cloud mass, and hence, may also affect the mass function of molecular clouds.  \nThe detailed modelling of cloud-cloud collision will be given in our forthcoming paper.  \nHere we ignore the contribution of cloud-cloud collision to the change of mass function and simply use the constant value of $T_{\\rm f}$.\n\n\n\\section{Quenching of Star Formation in Molecular Clouds\nNext we consider the destruction of molecular clouds to determine how the star formation is quenched. \nDale et al. (2012, 2013) have done extensive three dimensional simulations of star cluster formation with ionization or stellar wind feedback and shown that the effects of photo-ionization and stellar winds are limited in quenching the star formation in massive molecular clouds \\citep[see also][]{Walch+2012}. \n\\citet{Diaz-Miller+1998} calculated the steady-state structures of HII regions and pointed out that the photodissociation of hydrogen molecules due to FUV photons is much more important than photoionization due to UV photons for the destruction of molecular clouds. \n\\cite{2005ApJ...623..917H,2006ApJ...646..240H,2006ApJ...648L.131H,2007ApJ...664..363H} actually included photodissociation in the detailed radiation hydrodynamical calculations of an expanding HII region in a non-magnetized ISM (by resolving photodissociative line radiation), and found the limited effect of ionizing radiation and essentially confirmed the critical importance of FUV radiation for the ambient molecular cloud. \n\n\\subsection{Expanding HII Regions in Magnetized ISM}\nIn the case of non-magnetized molecular gas of density $10^2{\\rm cm}^{-3}$ around a massive star larger than $\\sim 20 M_\\sun$, a large amount of gas ($\\sim 3\\times 10^4 M_\\sun$) is photodissociated and re-processed into molecular material in the dense shell around the expanding HII region within 5 Myrs \\citep{2006ApJ...648L.131H}. \nAccording to the series of papers by \\citet{InoueInutsuka2008,InoueInutsuka2009}, however, \nthe inclusion of magnetic field is expected to reduce the density of the swept-up shell substantially.   \nTherefore the magnetic field should affect significantly the actual structure of compressed shell and subsequent star formation process \\citep[c.f., 3D simulation by][]{Arthur+2011}. \n\nTo quantitatively analyze the consequence, we have done numerical magnetohydrodynamics simulations of an expanding bubble due to UV and FUV photons from the massive star. \nThe details of the method is the same as described in \n\\cite{2006ApJ...646..240H,2006ApJ...648L.131H} except for the inclusion of the magnetic field. \nSince the calculation assumes spherical symmetry, we include only the $13\\mu$Gauss magnetic field that is transverse to the radial direction as a simplification.  \nThe magnetic pressure due to transverse field is accounted for in the Riemann solver as in \\cite{Sano+1999} \\citep[see][]{SuzukiInutsuka2006,IwasakiInutsuka2011}. \n\nThe upper panel of Figure \\ref{fig3} shows the resultant masses of ionized gas in HII region and atomic gas in photo-dissociation region transformed from cold molecular gas around an expanding HII region at the termination time as a function of the mass of a central star ($M_*$). \nAlso plotted is the warm molecular gas in and outside the compressed shell around the HII region. \nThe temperature of the warm molecular gas exceeds 50K. \nIts column density is smaller than $10^{21} {\\rm cm}^{-2}$, and hence, dust shielding for CO molecule is not effective and all the CO molecules are photo-dissociated.\nThis warm molecular gas without CO (so-called, CO-dark H$_2$) is not expected to be gravitationally bound unless the mass of the parental molecular cloud is exceptionally large. \nTherefore the subsequent star formation in this warm molecular gas is not expected.  \nThe uppermost black solid line denotes the total mass ($M_{\\rm g}(M_*)$) of these non-star-forming gases. \n\nThe lower panel of Figure 3 shows gas mass in the upper panel multiplied by $M_*^{-1.3}$ (blue dashed curve), $M_*^{-1.5}$ (black solid curve), and $M_*^{-1.7}$ (red dotted curve).  \nThe areas under these curves are proportional to the mass affected by massive stars whose mass distribution follows \n$dn_*\/d(log M_*) \\propto M_*^{1.3}, M_*^{1.5}$, and $M_*^{1.7}$, respectively.  \nWe can see that the shape of the curve does not vary much, and stars with mass $20\\sim30 M_\\sun$ always dominate the disruption of the molecular cloud.  \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{inutsuka_fig3.eps}\n\\caption{\nUpper Panel: \nMasses in various phases transformed from cold molecular gas around an expanding HII region at the termination time as a function of the mass of a central star. \nThe red dot-dashed line, the blue dotted line, and purple dashed line correspond to \nionized hydrogen in the HII region, \nneutral hydrogen in the photodissociation region, \nand warm molecular hydrogen gas without CO, respectively.  \nThe uppermost black solid line denotes the total mass of these non-star-forming gases. \nLower Panel: \nThe IMF-weighted mass of non-star-forming gas transformed from molecular gas by a massive star of mass $M_*$. \nThe area under the curve is proportional to the mass generated by massive stars whose mass function follows $dn\/d(\\log M_*) \\propto M_*^{-\\beta +1}$. \nThe peak of the curve determines the inverse of star formation efficiency $\\epsilon_{\\rm SF}$ (see explanation below Equation \\ref{eq:SFE}). \n\\label{fig3}\n}\n\\end{figure}\n\n\nOur calculations include ionization, photodissociation, and magnetohydrodynamical shock waves, but are restricted to a spherical geometry.  \nTherefore we should investigate the dispersal process in more realistic three dimensional simulations.  \nHowever, the inclusion of photo-dissociation requires the numerical calculation of FUV line transfer and hence remains extremely difficult in multi-dimensional simulations.\n\n\\subsection{Star Formation Efficiency}\nHereafter we assume the power law exponent of mass in the initial mass function of stars for large mass ($M_* > 8 M_\\sun$) is $-\\beta$ and $2<\\beta<3$ \n($ dn_*\/dM_* \\propto M_*^{-\\beta} $). \nNow we calculate the total mass of non-star forming gas disrupted by new born stars in a cloud. \nOne might think that the total mass of non-star-forming mass in the stellar system can be calculated by \n$\n   M_{\\rm g,total} = \\int_0^\\infty M_{\\rm g}(M_*) (dn_*\/dM_*) dM_*\n$.\nHowever, this estimation is meaningful only in the case the number of massive stars in a cloud are very large, \n$\n   \\int_{20 M_\\sun}^\\infty (dn_*\/dM_*) dM_* \\gg 1. \n$ \nIn reality, the number of massive stars in a molecular cloud of intermediate mass is quite small, and even a single massive star can destruct the whole parental molecular cloud. \nThus, to analyze the quenching of star formation in molecular clouds, it is more appropriate to determine the most likely mass of the star that is responsible for the destruction of molecular clouds. \n\nSince we assume the large mass side of the stellar initial mass function can be approximated by the power law of the exponent $-\\beta$, we can express the mass function in logarithmic mass of massive stars created in a cloud as \n\\begin{equation}\n     \\frac{dn_*}{d\\log M_*} = M_* \\frac{dn_*}{dM_*} \n                         = N_* \\left( \\frac{M_*}{M_\\sun} \\right)^{-\\beta+1}\n                         ~~{\\rm for}~M_* > 8 M_\\sun\n                                                  \\label{eq:IMF} \n\\end{equation}\nNote that the pre-factor $N_*$ is defined for the mass distribution of stars in the individual cloud we are analyzing. \nFor convenience, we define the effective minimum mass ($M_{\\rm *m}$) of a star in the hypothetical power law mass function by the following formula for the total mass in the cloud,  \n\\begin{eqnarray}\n     M_{\\rm *,total} \n& = & \n       \\int_{0}        ^{\\infty} M_* \\frac{dn_*}{dM_*} dM_* \\nonumber \\\\ \n& \\equiv & \n       \\int_{M_{\\rm *m}}^{\\infty} \n          N_* \\left( \\frac{M_*}{M_\\sun} \\right)^{-\\beta+1} dM\n     = \\left( \\frac{N_*}{\\beta-2}     \\right)\n       \\left( \\frac{M_\\sun}{M_{\\rm *m}}\\right)^{\\beta-2} .\n                                                          \\label{eq:Matotal}\n\\end{eqnarray}\n\nSuppose that a single massive star more massive than $M_{\\rm *d}$ is created in the molecular cloud. \nThis condition can be expressed as \n\\begin{equation}\n    1 =\\int_{M_{\\rm *d}}^{\\infty} \\frac{dn_*}{dM_*} dM_* = \n       \\left( \\frac{N_*}{\\beta-1} \\right)\n       \\left( \\frac{M_\\sun}{M_{\\rm *d}} \\right)^{\\beta-1}   \n                         ~~{\\rm for}~M_* > 8 M_\\sun .\n                                                             \\label{eq:MdNa}\n\\end{equation}\nThis equation relates $M_{\\rm *d}$ and $N_*$. \nWe can express the total mass of stars in the cloud as a function of $M_{\\rm *d}$ by eliminating $N_*$ in equations (\\ref{eq:Matotal}) and (\\ref{eq:MdNa}), \n\\begin{equation}\n     M_{\\rm *,total} =\n       \\left( \\frac{\\beta-1  }{\\beta-2  } \\right)\n       \\left( \\frac{M_\\sun   }{M_{\\rm *m}} \\right)^{\\beta-2}  \n       \\left( \\frac{M_{\\rm *d}}{M_\\sun   } \\right)^{\\beta-1}   \n                         ~~{\\rm for}~M_* > 8 M_\\sun .\n                                                             \\label{eq:MtMd}\n\\end{equation}\nThus, $M_{\\rm *,total} \\propto M_{\\rm *d}^{\\beta-1}$ for $M_* > 8 M_\\sun$.\nNow we suppose that a molecular cloud of mass $M_{\\rm cl}$ is eventually destroyed by UV and FUV photons from a star of mass $M_{\\rm *d}$ born in the cloud, and hence, star formation in the cloud is quenched. \nThe condition for this to occur can be written as \n$\n    M_{\\rm cl} = M_{\\rm g}\n$\n and \n$\n    \\epsilon_{\\rm SF} M_{\\rm cl} = M_{\\rm *,total},  \n$\nwhere $\\epsilon_{\\rm SF}$ is the star formation efficiency (the ratio of the total mass of stars to the mass of the parental cloud).\n\nIf $\\epsilon_{\\rm SF}$ is smaller than the value that would satisfy the above condition, the cloud destruction is not sufficient and star formation continues using the remaining cold molecular material in the cloud, which in turn increases $\\epsilon_{\\rm SF}$. \nThus, we expect that the actual evolution of a molecular cloud finally satisifies the above condition when the star formation is eventually quenched.\nThis means that the star formation efficiency should be given by \n\\begin{equation}\n  \\epsilon_{\\rm SF} = \\frac{M_{\\rm *,total}}{M_{\\rm g}(M_{\\rm *d})} \n= \\left( \\frac{\\beta-1   }{\\beta-2   } \\right)\n  \\left( \\frac{M_\\sun    }{M_{\\rm *m}} \\right)^{\\beta-2}\n  \\left( \\frac{M_{\\rm *d}}{M_\\sun    } \\right)^{\\beta-1}\n  \\left( \\frac{M_{\\rm g} }{M_\\sun    } \\right)^{-1}.\n                        \\label{eq:SFE} \n\\end{equation}\nThe preceding argument suggests that the value of star formation efficiency should take the minimum value of the right hand side of this equation,  \ni.e., the maximum value of $M_{\\rm g} M_{\\rm *}^{-\\beta+1}$ where  $M_{\\rm g}$ is a function of $M_*$. \nFigure 3 shows that the maximum value of $M_{\\rm g} M_{\\rm *}^{-\\beta+1}$ is attained at $M_* \\sim 30M_\\sun$ where $M_{\\rm g}$ is about $10^5 M_\\sun$.  \nTherefore we can conclude that once a massive star of $M_{\\rm *d} = 30 \\pm 10 M_\\sun$ is created, the star formation is eventually quenched in a cloud of mass $M_{\\rm g}(M_{\\rm *d}) \\sim 10^5 M_\\sun$.  \nThis corresponds to $\\epsilon_{\\rm SF} \\sim 10^{-2}$, if we adopt $\\beta=2.5$ and $M_{\\rm *m} = 0.1M_\\sun$.\nThe dependence of $\\epsilon_{\\rm SF}$ on $M_{\\rm *m}$ is quite weak ($\\beta-2 \\sim 0.5$) as shown in Equation (7).   \nIt is also not sensitive on $\\beta$ in the limited range ($2.3 < \\beta < 2.7$) as shown in Figure 3.  \nThus, the authors think that this value of $\\epsilon_{\\rm SF} \\sim 10^{-2}$ is  robust in typical star forming regions in our Galaxy.  \nThis argument may explain the reason for the low star formation efficiency in molecular clouds observationally found many decades ago \\citep[e.g.,][]{ZuckermanEvans1974}. \n\nNote that a sharp increase of $M_{\\rm g} M_*^{-\\beta+1}$ is due to the sharp increase of UV\/FUV luminosity at $M_* \\sim 20 M_\\sun$. \nTherefore a star much smaller than $20M_\\sun$ is not expected to be the main disrupter of the molecular cloud.  \nFor example, the upper panel of Figure 3 shows that a $10M_\\sun$ star can quench $\\sim 10^3 M_\\sun$ of the surrounding molecular material. \nHowever a $10^3 M_\\sun$ molecular cloud is not likely to produce a $10M_\\sun$ star unless $\\epsilon_{\\rm SF} \\sim 1$ as can be seen in equation (\\ref{eq:MtMd}), and hence, the destruction of $10^3 M_\\sun$ cloud by a $10M_\\sun$ star is not expected, in general. \n\n\nIf the initial mass function does not depend on the parent cloud mass as we assume here, the star formation efficiency is not expected to depend on mass for a cloud larger than $\\sim 10^5 M_\\sun$. \nThis can be understood as follows. \nThe number of UV\/FUV photons is proportional to the number of massive stars, which increases with the mass of the cloud.  \nHowever, the required number of photons also increases with the mass of the cloud. \nFor example, a $10^6 M_\\sun$ cloud will produce 10 stars with mass $> 30M_\\sun$ \nif $\\epsilon_{\\rm SF} = 10^{-2}$, $\\beta=2.5$, and $M_{\\rm *m} = 0.1M_\\sun$.\nThen, these 10 massive stars will destroy $10 \\times 10^5 M_\\sun$ molecular gas.  \nTherefore star formation in the whole molecular cloud is quenched when $\\epsilon_{\\rm SF} = 10^{-2}$. \nTherefore we can conclude that the star formation efficiency does not depend on the mass of the cloud if the shape of the initial mass function does not depend on the mass of the cloud. \n\nNow we can estimate the timescale for the destruction of a molecular cloud. \nOur calculation of the expanding ionization\/dissociation front in the magnetized molecular cloud shows that a $\\sim 10^5 M_\\sun$ molecular cloud can be destroyed within 4 Myrs.\nThe actual destruction timescale, $T_{\\rm d}$, should be the sum of the timescale of formation of a massive star and expansion timescale of the HII region, i.e., $T_{\\rm d} \\approx T_* + 4$Myr, where $T_*$ denotes the timescale for a massive star to form once the cloud is created. \nAfter one cycle of molecular cloud destruction over a timescale $T_{\\rm d}$, only a fraction, $\\epsilon_{\\rm SF}$, of the molecular gas is transformed into stars. \nTherefore, the timescale to completely transform a molecular cloud to stars is $T_{\\rm d}\/\\epsilon_{\\rm SF} \\sim 1.4$Gyr for $T_* \\sim 10$Myrs and $T_{\\rm d} \\sim 14$Myrs. \nThis may explain the so-called ``depletion timescale'' of molecular clouds that manifests observationally in the Schmidt-Kennicutt Law \\citep[e.g.,][] {Bigiel+2011,Lada+2012,KennicuttEvans2012}. \n\n\\section{Mass Function of Molecular Clouds}\nIn order to describe the time evolution of the mass function of molecular clouds, $n_{\\rm cl}(M)=dN_{\\rm cl}\/dM$, over a timescale much longer than 1 Myr,\n we adopt coarse-graining of short-timescale growth and destruction of clouds, and describe the continuity equation of molecular clouds in mass space as \n\\begin{equation}\n  \\PD{n_{\\rm cl}}{t} + \\PD{}{M} \\left( n_{\\rm cl} \\frac{dM}{dt} \\right)\n               =  - \\frac{n_{\\rm cl}}{T_{\\rm d}} ,  \n\\end{equation}\nwhere \n$n_{\\rm cl}(dM\/dt)$ denotes the flux of mass function in mass space, \n$dM\/dt$ describes the growth rate of the molecular cloud as given in Equation (1).  \nThe sink term on the right hand side of this equation corresponds to the destruction rate of molecular clouds in the sense of ensemble average.  \nIf the dynamical effects such as shear and tidal stresses contribute to the cloud destruction \\citep[e.g.,][]{Koda+2009,DobbsPringle2013}, we should modify $T_{\\rm d}$ in this equation. \nSince the left hand side of this equation should be regarded as the ensemble average, the term $1\/T_{\\rm d}$ represents the sum of the destruction rate of all the possible processes. \nHere we simply assume that the resultant $T_{\\rm d}$ is not very different from our estimate of destruction due to radiation feedback from massive stars. \n\nAccording to the series of our work on the formation of molecular clouds, the molecular cloud as a whole is not necessarily created as a self-gravitationally bound object.  \nTherefore, our modelling of the mass function of molecular clouds is not restricted to the self-gravitationally bound clouds.\nHowever, our modelling is not intended to describe the spatially extended diffuse molecular clouds much larger than the typical size of the bubbles ($\\lesssim100$pc).\n\nA steady state solution of the above equation is \n\\begin{equation}\n  n_{\\rm cl}(M) = \\frac{N_0}{M_{\\sun}} \\left( \\frac{M}{M_{\\sun}} \\right)^{-\\alpha}, \n\\end{equation}\nwhere $N_0$ is a constant and \n\\begin{equation}\n  \\alpha = 1 + \\frac{T_{\\rm f}}{T_{\\rm d}} . \\label{eq:alpha}\n\\end{equation}\nFor conditions typical of spiral arm regions in our Galaxy, we expect $T_* \\sim T_{\\rm f}$ and thus $T_{\\rm f} \\la T_{\\rm d}$, which corresponds to $1 < \\alpha \\lesssim 2$.\nFor example, $T_{\\rm f}=T_*=10$Myrs corresponds to $\\alpha \\approx 1.7$, which agrees well with observations \\citep{Solomon+1987,Kramer+1998,Heyer+2001,RomanDuval+2010}. \n\nHowever, in a quiescent region away from spiral arms or in the outer disk, in which there is a very limited amount of dense material, $T_{\\rm f}$ is expected to be larger at least by a factor of a few than in spiral arms.  \nIn contrast, $T_{\\rm d}$ is not necessarily expected to be large even in such an environment, since the meaning of $T_{\\rm d}$ is the average timescale of cloud destruction that occurs after the cloud is created, and thus, it does not necessarily depend on the growth timescale of the cloud.  \nTherefore, we expect that $T_{\\rm d}$ can be smaller than $T_{\\rm f}$ in such an environment, which may produce $\\alpha = 1+ T_{\\rm f}\/T_{\\rm d} > 2$.  \nThis tendency is actually observed in Milky Way outer disk, LMC, M33, and M51 \\citep{Rosolowsky2005,Wong+2011,Gratier+2012,Hughes+2010,Colombo+2014} . \n\n\n\n\nThe total number of molecular clouds is calculated as \n\\begin{eqnarray}\n  N_{\\rm total} &=& \\int_{M_1}^{M_2} n(M) dM \n                 =  \\frac{N_0}{\\alpha-1} \n                    \\left[ \n                         \\left( \\frac{M_{\\sun}}{M_1} \\right)^{\\alpha-1}\n                       - \\left( \\frac{M_{\\sun}}{M_2} \\right)^{\\alpha-1}\n                    \\right]    \n                    \\nonumber\\\\ \n             &\\sim& \\frac{N_0}{\\alpha-1} \n                    \\left( \\frac{M_{\\sun}}{M_1} \\right)^{\\alpha-1}, \n\\end{eqnarray}\nwhere we used $M_2 \\gg M_1$ in the final estimate. \nThe total number of clouds is essentially determined by the lower limit of the mass of the cloud. \nLikewise, the total mass of the molecular clouds is \n\\begin{eqnarray}\n  M_{\\rm total} &=& \\int_{M_1}^{M_2} M n(M) dM \n                 =  \\frac{N_0 M_{\\sun}}{2-\\alpha} \n                    \\left[ \n                         \\left( \\frac{M_2}{M_{\\sun}} \\right)^{2-\\alpha}\n                       - \\left( \\frac{M_1}{M_{\\sun}} \\right)^{2-\\alpha}\n                    \\right]\n                    \\nonumber\\\\ \n             &\\sim& \\frac{N_0 M_{\\sun}}{2-\\alpha} \n                         \\left( \\frac{M_2}{M_{\\sun}} \\right)^{2-\\alpha}, \n\\end{eqnarray}\nwhere we used $M_2 \\gg M_1$ and $\\alpha < 2$ in the final estimate. \nThus, the total mass of molecular clouds is essentially determined by the upper limit of the mass of the cloud. \nLet us assume $M_{\\rm total} \\sim 10^9 M_{\\sun}$ in the Galaxy, then our simple choice of $M_1=10^2 M_{\\sun}$, $M_2=10^6 M_{\\sun}$, and $\\alpha=1.5$ corresponds to $N_{\\rm total}\\sim10^5$ and the average mass of molecular clouds is $M_{\\rm ave} \\equiv M_{\\rm total}\/N_{\\rm total} \\sim 10^4 M_{\\sun}$. \nNote that these numbers depend on our choice of $M_1$. \n\n\n\\section{Summary}\nIn general, dense molecular clouds cannot be created in shock waves propagating in magnetized WNM without cold HI clouds. \nIn this paper we identify repeated interactions of shock waves in dense ISM as a basic mechanism for creating filametary molecular clouds, which are ubiquitously observed in the nearby ISM \\citep{Andre+2014}. \nThis suggests an expanding-bubble-dominated picture of the formation of molecular clouds in our Galaxy, which enables us to envision an overall picture of the statistics of molecular clouds and resultant star formation.  \nTogether with the findings of our previous work, our conclusions are summarized as follows: \n\\begin{enumerate}\n\\item \nTurbulent cold HI clouds embedded in WNM can be readily created in the expanding shells of HII regions or in the very late phase of supernova remnants. \nIn contrast, the formation of molecular clouds in a magnetized ISM needs many compression events. \nOnce low-mass molecular clouds are formed, an additional compression creates many filamentary molecular clouds. \nOne compression corresponds to of order 1Myr on average in our Galaxy.\nThe timescale of cloud formation is a few times 10Myrs. \n\\item \nSince the galactic thin disk is occupied by many bubbles, molecular clouds are formed in the overlapping regions of (old and new) bubbles.  \nHowever, since the average lifetime of each bubble is shorter than the timescale of cloud formation, it is difficult to observationally identify the multiple bubbles that created the molecular clouds. \n\\item \nThe velocity dispersion of molecular clouds should originate in the expansion velocities of bubbles.  \nThis is estimated to be $\\lesssim$10km\/s and should not strongly depend on the mass of the molecular cloud. \n\\item \nTo describe the growth of molecular cloud mass we can temporally smooth out the evolution over timescales larger than $\\sim$ 1Myr. \nThe resultant mass (smoothed over time) of each molecular cloud is an almost exponentially increasing function of time. \n\n\\item \nThe destruction of a molecular cloud is mainly due to UV\/FUV radiation from massive stars more massive than $20M_\\sun$. \nThe probability of cloud destruction is not a sensitive function of the mass of molecular clouds.  \nIf the shape of the initial mass function does not vary much with the mass of parent molecular clouds, cloud destruction by $30 \\pm 10 M_\\sun$ stars results in a star formation efficiency of order 1\\%. \nThis property explains the observed constancy of the gas depletion timescale ($1 \\sim 2$ Gyr) of giant molecular clouds in the solar neighborhood and possibly in some external galaxies where the normalizations for the Schmidt-Kennicutt Law obtained by high-density tracers are shown to be similar.\n\n\\item \nThe steady state of the evolution of the cloud mass function corresponds to a power law with exponent $-n$ in the range $1 < n \\lesssim 2$ in the spiral arm regions of our Galaxy. \nHowever, a larger value of the exponent, such as $n > 2$, is possible in the inter-arm regions.  \n\\end{enumerate}\n\nNote that the first and third conclusions have partly shown in our previous investigations \\citep{InoueInutsuka2009}. \nIn addition we can suggest the following implications from these conclusions: \n\\begin{enumerate}\n\\setcounter{enumi}{6}\n\\item \nStar formation starts, even in small molecular clouds, once the line-mass of an internal self-gravitating filament exceeds the critical value \\citep{Andre+2010}.  \nOur analysis suggests that the mass of an individual molecular cloud increases roughly exponentially over $\\sim 10$ Myrs.\nAccording to the formation mechanism driven by repeated compressions, we expect that the total mass in filaments of sufficiently high line-mass increases with the number of compressional events. \nThis means that the mass of star-forming dense gas increases with the mass of the molecular cloud and the star formation should accelerate over many million years. \nThis conjecture may provide a clue in understanding the star formation histories found by \\citet{PallaStahler2010} in seven individual molecular clouds such as Taurus-Auriga and $\\rho$ Ophiuchi. \n\\item \nMolecular clouds may collide over a timescale of a few times 10 Myrs, depending on the relative locations in adjacent (almost invisible) bubbles. \nSuch a molecular cloud collision may result in active star formation in a giant molecular cloud \\citep[e.g.,][]{Fukui+2014}. \n\\end{enumerate}\n\nThese implications should be investigated in more detail by numerical simulations. \nOur radiation magnetohydrodynamics simulations of an expanding bubble due to UV and FUV photons from the massive star show that most of the material in molecular clouds become warm molecular clouds without CO molecules. \nAlthough we have to investigate the fate of the CO-dark gas in more detail, we expect that the total mass can be very large and may account for the dark gas indicated by various observations \\cite[e.g.,][]{Grenier+2005}. \n\nThere are many report that the Kennicutt-Schmidt correlation varies with some properties of galaxies \n\\citep[e.g., ][]{Saintonge+2011,Saintonge+2012,Meidt+2013,Davis+2014}. \nIn addition, a simple relation does not fit to the center of our Galaxy \\citep[e.g.,][]{Longmore+2013}.  \nThe reasons for these deviations remain to be studied.\n\n\n\n\\begin{acknowledgements}\nSI thanks Hiroshi Kobayashi and Jennifer M. Stone for useful discussions and comments. \nSI is supported by Grant-in-Aid for Scientific Research (23244027,23103005)\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe discrete memoryless interference channel (DM-IC) is the canonical model for studying the effect of interference in wireless systems. The capacity of this channel is only known in some special cases e.g. class of deterministic ICs \\cite{Gam82,Cho07}, strong interference conditions \\cite{Sat81,Cos87,Chu07}, degraded conditions \\cite{Ben79,Liu08} and a class of semideterministic ICs \\cite{Cho09}. Characterizing the capacity region in the general case has been one of the long standing open problems in information theory. The best known achievable rate region is the so-called Han-Kobayashi scheme, which can be achieved by using schemes that are based on the concepts of rate-splitting and superposition coding \\cite{Han81,Cho06}. Rate-splitting refers to the technique of splitting the message at a transmitter into a common and a private part, where the common part is decoded at all the receivers and the private part is decoded only at the intended receiver. The two parts of the message are then combined into a single signal using superposition coding, first introduced in \\cite{Cov72} in the context of the broadcast channel.\nIn all the special cases where the capacity is known, the Han-Kobayashi region equals the capacity region. However, it has been very recently shown that this inner bound is not tight in general \\cite{Nai15}.\n\nThe first result we present in this paper is to show that  the Han-Kobayashi region can be achieved by a multicoding scheme. This scheme does not involve any explicit rate-splitting. Instead, the codebook at each encoder is generated as a multicodebook, i.e. there are multiple codewords corresponding to each message. The auxiliary random variable in this scheme does not explicitly carry a part of the message, rather it implicitly carries \\emph{some} part of the message, and it is not required to specify which part.\\footnote{A similar idea, combined with block-Markov operation, has been recently used in \\cite{Lim14} to develop an achievability scheme called distributed-decode-forward for broadcast traffic on relay networks.} In this sense, it's role is different from that in the Han-Kobayashi scheme \\cite{Han81,Cho06}, and is reminiscent of the encoding for state-dependent channels in \\cite{Gel80}, and the alternative proof of Marton's achievable rate region for the broadcast channel given in \\cite{Gam81}. A key advantage of\nthe multicoding nature of the new scheme is that it can be easily extended to obtain simple achievability schemes for setups in which the canonical interference channel model is augmented to incorporate additional node capabilities such as cognition and state-dependence, while extending the original Han-Kobayashi scheme to such setups can quickly become highly involved. We demonstrate this by constructing schemes for settings which augment the canonical interference channel model in different ways.\n \n\nThe first setting we consider is when the interference channel is state-dependent and the state-information is available non-causally to one of the transmitters (cognitive transmitter). For simplicity, we focus on the case when the cross-link between the non-cognitive transmitter and its undesired receiver is weak enough to be ignored, giving rise to the so called $Z$-interference channel topology. We know that for a point-to-point state-dependent channel with non-causal state information at the encoder, the optimal achievability scheme due to Gelfand and Pinsker uses multicoding at the encoders. Hence, for state-dependent interference channels with noncausal state information at the encoders too, we would like to use the idea of multicoding. Since the new achievability scheme that we present for the canonical interference channel already involves multicoding, it requires almost no change to be applicable to the state-dependent setting. Apart from being simple, we are also able to prove its optimality for the case of the deterministic $Z$-interference channel.\n\nWe then specialize our capacity characterization for the state-dependent deterministic $Z$-interference channel to the case where the channels are governed by the linear deterministic model of \\cite{Ave11}. In the recent literature, this model has proven extremely useful for approximating the capacity of wireless networks and developing insights for the design of optimal communication strategies. We consider a linear deterministic Z-interference channel, in which the state of the channel denotes whether the interference link is present or not. When the transmitters are base-stations and the receivers are end-users, this can model the scenario where one of the transmitters is cognitive, for example it can be a central controller that knows when the other Tx-Rx pair will be scheduled to communicate on the same frequency band. When the two Tx-Rx pairs are scheduled to communicate on the same frequency band, this gives an interference channel; when they communicate on different frequency bands each pair gets a clean channel free of interference. Moreover, the cognitive transmitter can know the schedule ahead of time, i.e. the times at which its transmission will be interfering with the second Tx-Rx pair. For this special case, we identify auxiliary random variables and provide an explicit expression for the capacity region. This explicit capacity characterization allows us to identify interesting properties of the optimal strategy. In particular, with single bit level for the linear deterministic channels (which would imply low to moderate SNR for the corresponding Gaussian channels), the sum rate is maximized when the interfering transmitter remains silent (transmits $0$'s) at times when it interferes with the second transmission. It then treats these symbols as stuck to $0$ and performs Gelfand-Pinsker coding. The second transmitter observes a clean channel at all times and communicates at the maximal rate of $1$ bit per channel use. This capacity characterization also reveals that when all nodes are provided with the state information the sum-capacity cannot be further improved. Thus, for this\nchannel, the sum-capacity when all nodes have state information is the same as that when only the interfering encoder\nhas state information. \n\nMotivated by wireless applications, there has been significant recent interest in state-dependent interference channels (ICs), where the state information is known only to some of the transmitters. Given the inherent difficulty of the problem, many special cases have been considered  \\cite{Zha13,Goo13,Dua13a,Dua13b}, for which different coding schemes have been proposed. However, exact capacity characterizations have proven difficult. Another line of related work has been the study of cognitive state-dependent ICs  \\cite{Rin11,Som08,Dua12,Kaz13}. Here, the term ``cognitive'' is usually used to mean that the cognitive transmitters know not only the state of the channel but also messages of other transmitters. Note that this assumption is significantly stronger than assuming state information at the transmitter as we do here.\n\nThe second setting we consider is when one of the transmitters has the capability to overhear the signal transmitted by the other transmitter, which can be used to induce cooperation between the two transmitters. This is different from having orthogonal communication links (or conferencing) between the encoders, as studied in \\cite{Wan11b}. Instead, overhearing exploits the natural broadcasting nature of the wireless medium to establish cooperation without requiring any dedicated resources. A variety of different models have been used to capture overhearing \\cite{Pra11,Yan11,Car12}, and are known by different names such as cribbing, source cooperation, generalized feedback, cognition etc. We use \"partial cribbing\" to model the overhearing, in which some deterministic function of the signal transmitted by the non-cognitive transmitter is available at the cognitive transmitter in a strictly causal fashion. Again, for simplicity, we focus on the case of the $Z$-interference channel, where the cross-link between the non-cognitive transmitter and its undesired receiver is weak enough to be ignored. For this setting, we develop a simple achievability scheme by combining our multicoding-based scheme with block-Markov coding and show that it is optimal for deterministic configurations.\n\nFinally, to further illustrate the point that simple schemes can be obtained for augmented scenarios, we describe two extensions which introduce even more complexity in the model. In the first extension, a third message is introduced in  the state-dependent Z-interference channel, which is to be communicated from the interfering transmitter to the interfered receiver. The second extension combines the state-dependent Z-IC and the Z-IC with unidirectional partial cribbing. In both extensions, we are able to obtain simple optimal schemes by naturally extending the multicoding-based achievability schemes.\n\n\\subsection*{Organization}\nWe describe the models considered in this paper formally in Section~\\ref{sec:model}.\nThe alternate achievability scheme that achieves the Han-Kobayashi region is presented in Sections~\\ref{sec:outline}.  Section~\\ref{sec:state} describes the results concerning the state-dependent setup and section~\\ref{sec:cribbing} describes the results concerning the cribbing setup. The two extensions are described in Section~\\ref{sec:extensions} and we end the paper with a short discussion in Section~\\ref{sec:conclude}.\n\n\\section{Model}\\label{sec:model}\n\nCapital letters, small letters and capital calligraphic letters denote random variables, realizations and alphabets respectively. The tuple $(x(1),x(2),\\dots ,x(n))$ and the set $\\{a,{a+1},\\dots ,b\\}$ are denoted by $x^n$ and $[a:b]$ respectively, and $\\mc{T}_{\\epsilon}^{(n)}$ stands for the $\\epsilon$-strongly typical set of length-$n$ sequences.\n\nWe now describe the channel models considered in this paper.\n\n\\subsection{Canonical Interference Channel}\nThe two-user discrete memoryless interference channel $p_{Y_1,Y_2|X_1,X_2}(y_1,y_2|x_1,x_2)$ is depicted in Fig.~\\ref{fig:model}. Each sender $j\\in\\{1,2\\}$ wishes to communicate a message $M_j$ to the corresponding receiver.\n\nA $(n,2^{nR_1},2^{nR_2},\\epsilon)$ code for the above channel consists of the encoding and decoding functions:\n\\begin{IEEEeqnarray*}{rCl}\nf_{j,i} & : & [1:2^{nR_j}] \\rightarrow \\mc{X}_j, \\quad j\\in\\{1,2\\}, 1\\leq i\\leq n,\\\\ \ng_j & : & \\mc{Y}_j^n \\rightarrow [1:2^{nR_j}],\\quad j\\in\\{1,2\\},\n\\end{IEEEeqnarray*}\nsuch that \n$$\\text{Pr}\\left\\{g(Y_j^n)\\neq M_j\\right\\} \\leq \\epsilon,\\quad j\\in\\{1,2\\},$$\nwhere $M_1$ and $M_2$ are assumed to be distributed uniformly in $[1:2^{nR_1}]$ and $[1:2^{nR_2}]$ respectively. A rate pair $(R_1,R_2)$ is said to be \\emph{achievable} if for every $\\epsilon > 0,$ there exists a $(n,2^{nR_1},2^{nR_2},\\epsilon)$ code for sufficiently large $n$. The capacity region is defined to be the closure of the achievable rate region.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=1.5]{model.pdf}\n\\caption{Two-User Discrete Memoryless Interference Channel (DM-IC)}\n\\label{fig:model}\n\\end{figure}\n\n\n\\subsection{State-Dependent Z-Interference Channel}\\label{subsec:model_state}\n\nThe discrete memoryless Z-interference channel $p(y_1|x_1,s)p(y_2|x_1,x_2,s)$ with discrete memoryless state $p(s)$ is depicted in Fig.~\\ref{fig:model_gen}. The states are assumed to be known noncausally at encoder 1. Each sender $j\\in\\{1,2\\}$ wishes to communicate a message $M_j$ at rate $R_j$ to the corresponding receiver. For this setting, a $(n,2^{nR_1},2^{nR_2},\\epsilon)$ code consists of the encoding and decoding functions:\n\\begin{IEEEeqnarray*}{rCl}\nf_{1,i}& : &[1:2^{nR_1}]\\times \\mc{S}^n \\rightarrow \\mc{X}_1, \\quad  1\\leq i\\leq n,\\\\ \nf_{2,i}& :& [1:2^{nR_2}] \\rightarrow \\mc{X}_2, \\quad 1\\leq i\\leq n,\\\\ \ng_j &: & \\mc{Y}_j^n \\rightarrow [1:2^{nR_j}],\\quad j\\in\\{1,2\\},\n\\end{IEEEeqnarray*}\nsuch that \n$$\\text{Pr}\\left\\{g(Y_j^n)\\neq M_j\\right\\} \\leq \\epsilon,\\quad j\\in\\{1,2\\}.$$ The probability of error, achievable rate pairs $(R_1,R_2)$ and the capacity region are defined in a similar manner as before.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.5]{block_diag_gen.pdf}\n\\caption{The State-Dependent Z-Interference Channel (S-D Z-IC)}\n\\label{fig:model_gen}\\vspace{0mm}\n\\end{figure}\n\nThe deterministic S-D Z-IC is depicted in Fig.~\\ref{fig:model_state_det}. The channel output $Y_1$ is a deterministic function $y_1(X_1,S)$ of the channel input $X_1$ and the state $S$. At receiver 2, the channel output $Y_2$ is a deterministic function $y_2(X_2,T_1)$ of the channel input $X_2$ and the interference $T_1$, which is assumed to be a deterministic function $t_1(X_1,S)$.\nWe also assume that if $x_2$ is given, $y_2(x_2,t_1)$ is an injective function of $t_1$, i.e. there exists some function $g$ such that $t_1=g(y_2,x_2).$ \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.5]{block_diag_2.pdf}\n\\caption{The Injective Deterministic S-D Z-IC}\n\\label{fig:model_state_det}\n\\end{figure}\n\nWe consider a special case of the injective deterministic S-D Z-IC in detail, which is the modulo-additive S-D Z-IC, depicted in Fig.~\\ref{fig:model_modulo}. All channel inputs and outputs come from a finite alphabet $\\mc{X}=\\{0,1,\\dots ,|\\mc{X}|-1\\}$. The channel has two states. In state $S=0$, there is no interference while in state $S=1$, the cross-link is present. When the cross-link is present, the output at receiver~2 is the modulo-$\\mc{X}$ sum of $X_2$ and $X_1$. For all other cases, the output is equal to the input. We can describe this formally as: \n\\begin{equation*}\n\\begin{split}\nY_1 & = X_1,\\\\\nY_2 & = X_2 \\oplus (S\\cdot X_1).\n\\end{split}\n\\end{equation*}\nAssume that the state $S$ is i.i.d. Ber$(\\lambda)$. A generalization of this model that incorporates multiple levels is also considered subsequently.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.5]{modulo.pdf}\n\\caption{The Modulo-Additive S-D Z-IC. All channel inputs and outputs take values in the same finite alphabet $\\mc{X}$. The state $S$ is Ber$(\\lambda).$}\n\\label{fig:model_modulo}\n\\end{figure}\n\n\\subsection{Z-Interference Channel with Partial Cribbing}\\label{subsec:model_crib}\nThe discrete memoryless deterministic Z-interference channel is depicted in Fig.~\\ref{fig:model_crib}. The channel output $Y_1$ is a deterministic function $y_1(X_1)$ of the channel input $X_1$. At receiver 2, the channel output $Y_2$ is a deterministic function $y_2(X_2,T_1)$ of the channel input $X_2$ and the interference $T_1$, which is assumed to be a deterministic function $t_1(X_1)$. We also assume that if $x_2$ is given, $y_2(x_2,t_1)$ is an injective function of $t_1$, i.e. there exists some function $g$ such that $t_1=g(y_2,x_2).$ Each sender $j\\in\\{1,2\\}$ wishes to communicate a message $M_j$ at rate $R_j$ to the corresponding receiver. \n\nWe assume that encoder 1 can overhear the signal from transmitter 2 \\emph{strictly causally}, which is modeled as partial cribbing with a delay \\cite{Asn13}. The partial cribbing signal, which is a function of $X_2$ is denoted by $Z_2$. So $X_{1i}$ is a function of $(M_1,Z_2^{i-1})$ and $X_{2i}$ is a function of $M_2$.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=1.5]{block_diag.pdf}\n\\caption{Injective Deterministic Z-Interference Channel with Unidirectional Partial Cribbing}\n\\label{fig:model_crib}\n\\end{figure}\n\nA $(n,2^{nR_1},2^{nR_2},\\epsilon)$ code for this setting consists of \n\\begin{IEEEeqnarray*}{rCl}\nf_{1,i} & : & [1:2^{nR_1}]\\times \\mc{Z}_2^{i-1} \\rightarrow \\mc{X}_1, \\quad  1\\leq i\\leq n,\\\\ \nf_{2,i} & : & [1:2^{nR_2}] \\rightarrow \\mc{X}_2, \\quad 1\\leq i\\leq n,\\\\ \ng_j & : &  \\mc{Y}_j^n \\rightarrow [1:2^{nR_j}],\\quad j\\in\\{1,2\\},\n\\end{IEEEeqnarray*}\nsuch that \n$$\\text{Pr}\\left\\{g(Y_j^n)\\neq M_j\\right\\} \\leq \\epsilon,\\quad j\\in\\{1,2\\}.$$ The probability of error, achievable rate pairs $(R_1,R_2)$ and the capacity region are defined in a similar manner as before.\n\n\\section{Canonical Interference Channel}\\label{sec:outline}\n\n\\subsection{Preliminaries}\\label{sec:prelim}\nThe currently best known achievable rate region for the 2-user DM-IC was provided by Han and Kobayashi in \\cite{Han81}, using a scheme based on rate-splitting and superposition coding. An alternative achievable rate region that included the Han-Kobayashi rate region was proposed in \\cite{Cho06}, using another scheme that used rate-splitting and superposition coding. Using the terminology introduced in \\cite{Wan13}, the encoding in \\cite{Han81} can be described as employing \\emph{homogeneous} superposition coding, while that in \\cite{Cho06} can be described as employing \\emph{heterogeneous} superposition coding. It was then proved in \\cite{Cho08} that the two regions are, in fact, equivalent and given by the following compact representation (see also \\cite{Kra06,Kob07}).\n\n\\begin{thm}[Han-Kobayashi Region]\\label{thm:HK}\nA rate pair $(R_1,R_2)$ is achievable for the DM-IC $p(y_1,y_2|x_1,x_2)$ if\n\\begin{equation}\\label{eq:achreg_prelim}\n\\begin{split}\nR_1 & < I(X_1;Y_1|U_2,Q),\\\\\nR_2 & < I(X_2;Y_2|U_1,Q),\\\\\nR_1 + R_2 & < I(X_1;Y_1|U_1,U_2,Q) +I(X_2,U_1;Y_2|Q) ,\\\\\nR_1 + R_2 & < I(X_1,U_2;Y_1|U_1,Q) + I(X_2,U_1;Y_2|U_2,Q),\\\\\nR_1 + R_2 & < I(X_1,U_2;Y_1|Q) + I(X_2;Y_2|U_1,U_2,Q),\\\\\n2R_1 + R_2 & < I(X_1;Y_1|U_1,U_2,Q) + I(X_2,U_1;Y_2|U_2,Q) + I(X_1,U_2;Y_1|Q),\\\\\nR_1 + 2R_2 & < I(X_2;Y_2|U_1,U_2,Q) + I(X_1,U_2;Y_1|U_1,Q) + I(X_2,U_1;Y_2|Q),\n\\end{split}\n\\end{equation}\nfor some pmf $p(q)p(u_1,x_1|q)p(u_2,x_2|q),$ where ${|\\mc{U}_1|\\leq |\\mc{X}_1|+4}$, ${|\\mc{U}_2|\\leq |\\mc{X}_2|+4}$ and ${|\\mc{Q}|\\leq 4.}$\n\\end{thm}\n\n\\subsection{Outline of the new achievability scheme}\nWe first describe the alternative achievability scheme informally and discuss the similarities and differences with the existing achievability schemes. The later subsections describe and analyze the scheme formally.\n\nEncoder $j$, where $j\\in\\{1,2\\}$ prepares two codebooks: \n\\begin{itemize}\n\\item A transmission multicodebook\\footnote{The term ``multicodebook'' refers to the fact that there are multiple codewords corresponding to each message.}, which is a set of codewords $\\{x_j^n(\\cdot,\\cdot)\\}$ formed using the transmission random variable $X_j$. This set is partitioned into a number of bins (or subcodebooks), where the bin-index corresponds to the message, \n\\item A coordination codebook which is a set of codewords $\\{u_j^n(\\cdot)\\}$ formed using the auxiliary random variable $U_j$.\n\\end{itemize}\nGiven a message, one codeword $x_j^n$ from the corresponding bin in the transmission multicodebook is chosen so that it is jointly typical with some sequence $u_j^n$ in the coordination codebook. The codeword $x_j^n$ so chosen forms the transmission sequence.\n\nAt a decoder, the desired message is decoded by using joint typicality decoding, which uses the coordination codebook and the transmission multicodebook of the corresponding encoder and the coordination codebook of the other encoder. Thus, a receiver makes use of the interference via its knowledge of the coordination codebook at the interfering transmitter.\n\nFrom the above description, it can be seen that the coordination codebook does not carry any message. Its purpose is to ensure that the transmission sequence from a given bin is well-chosen, i.e. it is beneficial to the intended receiver and also the unintended receiver. To the best of our knowledge, this is the first time an auxiliary random variable (which is not the time-sharing random variable) appears in one of the best known achievability schemes without being explicitly associated with any message. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Achievability scheme}\\label{subsec:achHK}\nChoose a pmf $p(u_1,x_1)p(u_2,x_2)$ and $0<\\epsilon'<\\epsilon$.\n\\subsubsection*{Codebook Generation}\n\\begin{itemize}\n\\item Encoder 1 generates a coordination codebook consisting of $2^{nR_{1c}}$ codewords\\footnote{Though there is no notion of a common message or a private message in this achievability scheme, we use the subscripts $c$ and $p$ to convey if the corresponding random variables are used for decoding at all destinations or only the desired destination respectively.} $u_1^n(l_{1c}),\\;l_{1c}\\in[1:2^{nR_{1c}}]$ i.i.d. according to $\\prod_{i=1}^np(u_{1i})$. It also generates a transmission multicodebook consisting of $2^{n(R_1+R_{1p})}$ codewords $x_1^n(m_1,l_{1p}),\\;m_1\\in[1:2^{nR_1}],\\; l_{1p}\\in[1:2^{nR_{1p}}]$ i.i.d. according to $\\prod_{i=1}^np(x_{1i})$.\n\\item Similarly, encoder 2 generates a coordination codebook consisting of $2^{nR_{2c}}$ codewords $u_2^n(l_{2c}),\\;l_{2c}\\in[1:2^{nR_{2c}}]$ i.i.d. according to $\\prod_{i=1}^np(u_{2i})$. It also generates a transmission multicodebook consisting of $2^{n(R_2+R_{2p})}$ codewords $x_2^n(m_2,l_{2p}),\\;m_2\\in[1:2^{nR_2}],\\; l_{2p}\\in[1:2^{nR_{2p}}]$ i.i.d. according to $\\prod_{i=1}^np(x_{2i})$.\n\\end{itemize}\n\n\\subsubsection*{Encoding}\n\\begin{itemize}\n\\item To transmit message $m_1$, encoder 1 finds a pair $(l_{1c},l_{1p})$ such that $$(u_1^n(l_{1c}),x_1^n(m_1,l_{1p}))\\in\\mc{T}^{(n)}_{\\epsilon'}$$ and transmits $x_1^n(m_1,l_{1p})$. If it cannot find such a pair, it transmits $x_1^n(m_1,1)$.\n\\item Similarly, to transmit message $m_2$, encoder 2 finds a pair $(l_{2c},l_{2p})$ such that $$(u_2^n(l_{2c}),x_2^n(m_2,l_{2p}))\\in\\mc{T}^{(n)}_{\\epsilon'}$$ and transmits $x_2^n(m_2,l_{2p})$. If it cannot find such a pair, it transmits $x_2^n(m_2,1)$.\n\\end{itemize}\n\nThe codebook generation and encoding process are illustrated in Fig.~\\ref{fig:encoding}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1.5]{codebooks.pdf}\n\\vspace{1mm}\n\\caption{Codebook Generation and Encoding at Encoder~1. The independently generated $x_1^n$ sequences, lined up vertically in the figure, are binned into $2^{nR_1}$ bins. The independently generated coordination sequences $u_1^n$ are lined up horizontally. To transmit message $m_1$, a jointly typical pair $(x_1^n,u_1^n)$ is sought where $x_1^n$ falls into the $m_1$-th bin, and then $x_1^n$ is transmitted.}\n\\label{fig:encoding}\n\\end{figure}\n\n\\subsubsection*{Decoding}\n\\begin{itemize}\n\\item Decoder 1 finds the unique $\\hat{m}_1$ such that $$(u_1^n(l_{1c}),x_1^n(\\hat{m}_1,l_{1p}),u_2^n(l_{2c}),y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon}$$ for some $(l_{1c},l_{1p},l_{2c})$. If none or more than one such $\\hat{m}_1$ are found, then decoder 1 declares error.\n\\item Decoder 2 finds the unique $\\hat{m}_2$ such that $$(u_2^n(l_{2c}),x_2^n(\\hat{m}_2,l_{2p}),u_1^n(l_{1c}),y_2^n)\\in\\mc{T}^{(n)}_{\\epsilon}$$ for some $(l_{2c},l_{2p},l_{1c})$. If none or more than one such $\\hat{m}_2$ are found, then decoder 2 declares error.\n\\end{itemize}\n\n\\subsubsection*{Discussion}\nBefore providing the formal analysis of the probability of error to show that the coding scheme described above achieves the Han-Kobayashi region, we discuss the connection between the new scheme and the scheme from \\cite{Cho06} which motivates the equivalence of their rate regions.\n\nConsider the set of codewords used at encoder 1. While this set resembles a multicodebook, it can be reduced to a standard codebook (one codeword per message) by stripping away the codewords in each bin that are not jointly typical with any of the $u_1^n$ sequences, and therefore are never used by the transmitters. In other words, after we generate the multicodebook in Fig.~\\ref{fig:encoding}, we can form a smaller codebook by only keeping one codeword per message which is jointly typical with one of the $u_1^n$ sequences (i.e., those codewords highlighted in Fig.~\\ref{fig:encoding}). Note that this reduced codebook indeed has a superposition structure. Each of the $ 2^{nR_1} $ remaining codewords  $x_1^n$ is jointly typical with one of the $2^{nR_{1c}}$  $u_1^n$ codewords, and when $n$ is large there will be exactly $ 2^{n(R_1-R_{1c})} $   $x_1^n$ sequences that are typical with each $u_1^n$ sequence, i.e., these $ 2^{n(R_1-R_{1c})} $   $x_1^n$ sequences will look as if they were generated i.i.d. from $p(x_1|u_1)$. Therefore, the $u_1^n$ sequences can be indeed thought as the cloud centers in this superposition codebook and $x_1^n$'s as the satellite codewords. Therefore, our multicodebook construction can be viewed as an equivalent way to generate a superposition codebook as in \\cite{Cho08}. This reveals that both the codebook structure and the decoding in our scheme are similar to that in the Han-Kobayashi scheme and therefore the two achievable rate regions are, not surprisingly, equal.\n\nHowever, note that for broadcast channels, combining Marton coding (which employs multicoding) \\cite{Gam81} with Gelfand-Pinsker coding (which also employs multicoding) is more straightforward than combining superposition coding with Gelfand-Pinsker coding. The former has been shown to be optimal in some cases \\cite{Lap13}. Since our codebook construction for the interference channel also has the flavor of multicoding, extending this construction to setups where multicoding is required is also quite straightforward. As mentioned in the introduction, we exploit this to develop simple achievability schemes for more general setups described in later sections. \n\n\n\\subsubsection*{Probability of Error}\nDue to the symmetry of the code, the average probability of error $\\msf{P}(\\mc{E})$ is equal to $\\msf{P}(\\mc{E}|M_1,M_2)$, so we can assume $(M_1,M_2) = (1,1)$ and analyze $\\msf{P}(\\mc{E}|1,1)$. Let $(L_{1c},L_{1p},L_{2c},L_{2p})$ denote the indices chosen during encoding by encoder 1 and encoder 2. \n\nWe now define events that cover the event of error in decoding message $m_1$:\n\\begin{IEEEeqnarray*}{rCl}\n\\mc{E}_1 & \\triangleq & \\{(U_1^n(l_{1c}),X_1^n(1,l_{1p}))\\notin\\mc{T}^{(n)}_{\\epsilon'} \\;\\text{ for all } l_{1c}, l_{1p}\\}, \\\\\n\\mc{E}_2 & \\triangleq & \\{(U_1^n(L_{1c}),X_1^n(1,L_{1p}),U_2^n(L_{2c}),Y_1^n)\\notin\\mc{T}^{(n)}_{\\epsilon}\\}, \\\\\n\\mc{E}_3 & \\triangleq & \\{(U_1^n(L_{1c}),X_1^n(m_1,l_{1p}),U_2^n(L_{2c}),Y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon}\\text{ for some }m_1\\neq 1, \\text{ for some } l_{1p}\\}, \\\\\n\\mc{E}_4 & \\triangleq & \\{(U_1^n(L_{1c}),X_1^n(m_1,l_{1p}),U_2^n(l_{2c}),Y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon} \\text{ for some }m_1\\neq 1, \\text{ for some } l_{1p},l_{2c}\\} ,\\\\\n\\mc{E}_5 & \\triangleq & \\{(U_1^n(l_{1c}),X_1^n(m_1,l_{1p}),U_2^n(L_{2c}),Y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon} \\text{ for some }m_1\\neq 1, \\text{ for some } l_{1p},l_{1c}\\} ,\\\\\n\\mc{E}_6 & \\triangleq & \\{(U_1^n(l_{1c}),X_1^n(m_1,l_{1p}),U_2^n(l_{2c}),Y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon} \\text{ for some }m_1\\neq 1, \\text{ for some } l_{1c},l_{1p},l_{2c}\\}.\n\\end{IEEEeqnarray*}\n\nConsider also the event $\\mc{E}'_1$, analogous to $\\mc{E}_1$, which is defined as follows.\n\\begin{IEEEeqnarray*}{rCl}\n\\mc{E}'_1 & \\triangleq & \\{(U_2^n(l_{2c}),X_2^n(1,l_{2p}))\\notin\\mc{T}^{(n)}_{\\epsilon'} \\;\\text{ for all } l_{2c}, l_{2p}\\}. \\label{eq:E'1}\n\\end{IEEEeqnarray*}\n\nSince an error for $m_1$ occurs only if at least one of the above events occur, we use the union bound to get the following upper bound on the average probability of error in decoding $m_1$:\n$$ \\msf{P}(\\mc{E}_1) + \\msf{P}(\\mc{E}'_1)+ \\msf{P}(\\mc{E}_2\\cap\\mc{E}_1^c\\cap\\mc{E}'^{c}_1) + \\msf{P}(\\mc{E}_3) + \\msf{P}(\\mc{E}_4) + \\msf{P}(\\mc{E}_5) + \\msf{P}(\\mc{E}_6).$$\n\nBy the mutual covering lemma \\cite[Chap. 8]{Gam12}, $\\msf{P}(\\mc{E}_1)\\rightarrow 0$ as $n\\rightarrow\\infty$ if \n\\begin{IEEEeqnarray}{rCl}\nR_{1p} + R_{1c} & > & I(U_1;X_1) + \\delta(\\epsilon'),\\label{eq:ach1}\n\\end{IEEEeqnarray}\nwhere $\\delta(\\epsilon')\\rightarrow 0$ as $\\epsilon'\\rightarrow 0.$\n\nSimilarly, we get that \n$\\msf{P}(\\mc{E}'_1)\\rightarrow 0$ as $n\\rightarrow\\infty$ if \n\\begin{IEEEeqnarray}{rCl}\nR_{2p} + R_{2c} & > & I(U_2;X_2) + \\delta(\\epsilon').\\label{eq:ach1'}\n\\end{IEEEeqnarray}\n\nBy the conditional typicality lemma, $\\msf{P}(\\mc{E}_2\\cap\\mc{E}_1^c\\cap\\mc{E}'^{c}_1)$ tends to zero as $n\\rightarrow\\infty$.\n\nFor $\\msf{P}(\\mc{E}_3)\\rightarrow 0$, we can use the packing lemma from \\cite[Ch. 3]{Gam12} to get the condition\n\\begin{equation}\\label{eq:ach2}\nR_1 + R_{1p}  <  I(X_1;U_1,U_2,Y_1) - \\delta(\\epsilon),\n\\end{equation}where $\\delta(\\epsilon)\\rightarrow 0$ as $\\epsilon\\rightarrow 0.$\n\nFor $\\msf{P}(\\mc{E}_4)\\rightarrow 0$, we can again use the packing lemma to get the condition\n\\begin{equation}\\label{eq:ach3}\nR_1 + R_{1p} + R_{2c} < I(X_1,U_2;U_1,Y_1)- \\delta(\\epsilon).\n\\end{equation}\n\nFor $\\msf{P}(\\mc{E}_5)\\rightarrow 0$, we apply the multivariate packing lemma from the Appendix as shown in \\eqref{eq:multipack_2} to get the condition\n\\begin{IEEEeqnarray}{LCl}\nR_1 + R_{1p} + R_{1c} < I(U_1;X_1) + I(U_1,X_1;U_2,Y_1) - \\delta(\\epsilon).\\label{eq:ach4}\n\\end{IEEEeqnarray} \n\nFinally, for $\\msf{P}(\\mc{E}_6)\\rightarrow 0$ as $n\\rightarrow\\infty$, another application of the multivariate packing lemma as shown in \\eqref{eq:multipack_3} gives the condition\n\\begin{IEEEeqnarray}{rCl}\nR_1 + R_{1p} + R_{1c} + R_{2c} & < & I(U_1;X_1) + I(U_2;Y_1) + I(U_1,X_1;U_2,Y_1)-\\delta(\\epsilon).\\label{eq:ach5}\n\\end{IEEEeqnarray}\n\nA similar analysis leads to the following additional conditions for the probability of error in decoding $m_2$ to vanish as $n\\rightarrow\\infty$.\n\\begin{IEEEeqnarray}{rCl}\nR_2 + R_{2p} & < & I(X_2;U_2,U_1,Y_2) - \\delta(\\epsilon),\\label{eq:ach7}\\\\\nR_2 + R_{2p} + R_{1c} & < & I(X_2,U_1;U_2,Y_2)- \\delta(\\epsilon),\\label{eq:ach8}\\\\\nR_2 + R_{2p} + R_{2c} & < & I(U_2;X_2) + I(U_2,X_2;U_1,Y_2)- \\delta(\\epsilon),\\label{eq:ach9}\\\\\nR_2 + R_{2p} + R_{2c} + R_{1c} & < & I(U_2;X_2) + I(U_1;Y_2) + I(U_2,X_2;U_1,Y_2)-\\delta(\\epsilon).\\label{eq:ach10}\n\\end{IEEEeqnarray}\n\nHence the probability of error vanishes as $n\\rightarrow\\infty$ if the conditions \\eqref{eq:ach1}-\\eqref{eq:ach10} are satisfied.\nFor the sake of brevity, let us first denote the RHS of the conditions \\eqref{eq:ach1}-\\eqref{eq:ach10} by $a,b,c,d,e,f,g,h,i,j$ respectively (ignoring the $\\delta(\\epsilon')$ and $\\delta(\\epsilon)$ terms). \n\nWe then note the following relations among these terms which can be proved using the chain rule of mutual information, the Markov chains $U_1-X_1-(U_2,X_2,Y_1,Y_2)$ and $U_2-X_2-(U_1,X_1,Y_1,Y_2)$ and the independence of $(U_1,X_1)$ and $(U_2,X_2)$\n\\begin{equation}\\label{eq:relFM}\n\\begin{gathered}\ne-a  \\leq  \\min\\{c,d\\},\\\\\nf -a  \\leq  d \\leq f,\\\\\nc  \\leq  e  \\leq  f,\\\\\ni-b \\leq  \\min\\{g,h\\},\\\\\nj-b  \\leq  h \\leq j,\\\\\ng  \\leq  i  \\leq  j.\n\\end{gathered}\n\\end{equation}\n\nWe now employ Fourier-Motzkin elimination on the conditions \\eqref{eq:ach1}-\\eqref{eq:ach10} and $R_{1c},R_{1p},R_{2c},R_{2p}\\geq 0$ to eliminate $R_{1c},R_{1p},R_{2c},R_{2p}$. The set of relations \\eqref{eq:relFM} can be used to simplify this task by recognizing redundant constraints. At the end, we get the following achievable region:\n\\begin{equation}\\label{eq:achreg1}\n\\begin{split}\nR_1 & <  e-a,\\\\\nR_2 & <  i-b,\\\\\nR_1 + R_2 & <  c + j-a-b,\\\\\nR_1 + R_2 & <  d + h-a-b,\\\\\nR_1 + R_2 & <  f + g-a-b,\\\\\n2R_1 + R_2 & <  c+h+f-2a-b,\\\\\nR_1+2R_2 & < d +g+j-a-2b.\n\\end{split}\n\\end{equation}\n\nUsing the same facts as those used to prove \\eqref{eq:relFM}, we can show that the above region is the same as the Han-Kobayashi region. For the sake of completeness, we show this explicitly.\n\\begin{itemize}\n\\item Consider the upper bound on $R_1$:\n\\begin{IEEEeqnarray}{rCl}\ne-a & = & I(U_1,X_1;U_2,Y_1)\\nonumber\\\\\n& \\stackrel{(a)}{=} & I(X_1;U_2,Y_1)\\nonumber\\\\\n& \\stackrel{(b)}{=} & I(X_1;Y_1|U_2),\\label{eq:achreg2}\n\\end{IEEEeqnarray}where step $(a)$ follows since $U_1-X_1-(U_2,Y_1)$ is a Markov chain, and step $(b)$ follows since $X_1$ is independent of $U_2$.\n\\item Similarly, \\begin{equation}\\label{eq:achreg3}i-b= I(X_2;Y_2|U_1).\\end{equation}\n\\item Consider the first upper bound on the sum-rate ${c+j-a-b}$:\n\\begin{IEEEeqnarray}{lCl}\nc+j-a-b \\nonumber\\\\\n =  I(X_1;U_1,U_2,Y_1) + I(U_2;X_2) + I(U_1;Y_2)  \\nonumber\\\\\n \\quad\\quad +\\> I(U_2,X_2;U_1,Y_2)- I(U_2;X_2)-I(U_1;X_1)\\nonumber\\\\\n \\stackrel{(a)}{=}  I(X_1;U_2,Y_1|U_1) + I(U_1;Y_2) + I(U_2,X_2;U_1,Y_2)\\nonumber\\\\\n \\stackrel{(b)}{=}  I(X_1;U_2,Y_1|U_1) + I(U_1;Y_2) + I(X_2;U_1,Y_2)\\nonumber\\\\\n \\stackrel{(c)}{=}  I(X_1;U_2,Y_1|U_1) + I(U_1;Y_2) + I(X_2;Y_2|U_1)\\nonumber\\\\\n \\stackrel{(d)}{=}  I(X_1;Y_1|U_1,U_2) + I(X_2,U_1;Y_2),\\label{eq:achreg4}\n\\end{IEEEeqnarray}where step $(a)$ follows by the chain rule of mutual information, step $(b)$ follows by the Markov chain $U_2-X_2-(U_1,Y_2)$, step $(c)$ follows since $U_1$ and $X_2$ are independent and step $(d)$ follows by the independence of $U_2$ and $(U_1,X_1)$. \n\\item By similar steps, $f+g-a-b =$  \\begin{equation}\\label{eq:achreg5}I(X_1,U_2;Y_1) + I(X_2;Y_2|U_1,U_2).\\end{equation}\n\\item The remaining upper-bound on the sum-rate $d+h-a-b$ can be simplified as follows: \\begin{IEEEeqnarray}{lCl}\nd+h-a-b \\nonumber\\\\\n=  I(X_1,U_2;U_1,Y_1) + I(X_2,U_1;U_2,Y_2) \\nonumber\\\\\n\\quad -\\> I(U_1;X_1) - I(U_2;X_2)\\nonumber\\\\\n =  I(X_1,U_2;Y_1|U_1) + I(X_2,U_1;Y_2|U_2),\\label{eq:achreg6}\n\\end{IEEEeqnarray}which follows by the chain rule of mutual information and the independence of $(U_1,X_1)$ and $(U_2,X_2)$.\n\\item The upper bound on $2R_1+R_2$ can be simplified as follows:\n\\begin{IEEEeqnarray}{lCl}\nc+h+f-2a-b \\nonumber\\\\\n= I(X_1;U_1,U_2,Y_1) + I(X_2,U_1;U_2,Y_2) + I(U_1,X_1) \\nonumber\\\\\n\\quad  +\\> I(U_2;Y_1) + I(U_1,X_1;U_2,Y_1) - 2I(U_1;X_1) -  I(U_2;X_2)\\nonumber\\\\\n \\stackrel{(a)}{=}  I(X_1;U_2,Y_1|U_1) + I(X_2,U_1;Y_2|U_2) + I(U_2;Y_1) + I(U_1,X_1;U_2,Y_1)\\nonumber\\\\\n \\stackrel{(b)}{=}  I(X_1;U_2,Y_1|U_1) + I(X_2,U_1;Y_2|U_2) + I(U_2;Y_1) + I(X_1;Y_1|U_2)\\nonumber\\\\\n \\stackrel{(c)}{=}  I(X_1;Y_1|U_1,U_2) + I(X_2,U_1;Y_2|U_2) + I(X_1,U_2;Y_1),\\label{eq:achreg7}\n\\end{IEEEeqnarray}where step $(a)$ holds by the chain rule of mutual information and the independence of $U_1$ and $(U_2,X_2)$, step $(b)$ follows by $U_1-X_1-(U_2,Y_1)$ and the independence of $X_1$ and $U_2$, and step $(c)$ follows by the chain rule of mutual information and the independence of $U_2$ and $(U_1,X_1)$.\n\\item Finally, $d+g+j-a-2b$ can be similarly shown to be equal to \\begin{equation}\\label{eq:achreg8} I(X_2;Y_2|U_1,U_2) + I(X_1,U_2;Y_1|U_1) + I(X_2,U_1;Y_2).\\end{equation}\n\\end{itemize}\n\nFrom \\eqref{eq:achreg1}-\\eqref{eq:achreg8} and including a time-sharing random variable $Q$, we get that the following region is achievable:\n\\begin{equation}\\label{eq:achreg}\n\\begin{split}\nR_1 & < I(X_1;Y_1|U_2,Q),\\\\\nR_2 & < I(X_2;Y_2|U_1,Q),\\\\\nR_1 + R_2 & < I(X_1;Y_1|U_1,U_2,Q) +I(X_2,U_1;Y_2|Q) ,\\\\\nR_1 + R_2 & < I(X_1,U_2;Y_1|U_1,Q) + I(X_2,U_1;Y_2|U_2,Q),\\\\\nR_1 + R_2 & < I(X_1,U_2;Y_1|Q) + I(X_2;Y_2|U_1,U_2,Q),\\\\\n2R_1 + R_2 & < I(X_1;Y_1|U_1,U_2,Q) + I(X_2,U_1;Y_2|U_2,Q) + I(X_1,U_2;Y_1|Q),\\\\\nR_1 + 2R_2 & < I(X_2;Y_2|U_1,U_2,Q) + I(X_1,U_2;Y_1|U_1,Q) + I(X_2,U_1;Y_2|Q),\n\\end{split}\n\\end{equation}\nfor pmf $p(q)p(u_1,x_1|q)p(u_2,x_2|q).$ This region is identical to the region in \\eqref{eq:achreg_prelim}.\\hfill\\IEEEQED\n\n\n\n\\section{State-dependent Interference channels}\\label{sec:state}\nIn this section, we focus on the particular setup of the state-dependent Z-interference channel (S-D Z-IC) with noncausal state information at the interfering transmitter, as depicted in Fig.~\\ref{fig:model_gen}. We provide a simple achievability scheme for this setup, that is obtained from the alternative achievability scheme for the general interference channel. This scheme is shown to be optimal for the deterministic case. The auxiliary random variable used for encoding at the interfering transmitter now implicitly captures some part of the message as well as some part of the state sequence realization. \nThe achievability scheme can also be viewed as a generalization of the schemes presented in \\cite{Cad09} and \\cite{Dua13b}.\n\n\n\nAfter characterizing the capacity region of the deterministic S-D Z-IC, we investigate a special case in detail: the modulo-additive S-D Z-IC. The modulo-additive channel is motivated by the linear deterministic model which has gained popularity over the recent years for studying wireless networks \\cite{Ave11}. For this case (which can be thought of as a linear deterministic model with only one \\emph{bit level}), we obtain an explicit description of the capacity region and furthermore, show that the capacity region is also achieved by the standard Gelfand-Pinsker coding over the first link and treating interference as noise over the second link. Following this, the modulo-additive S-D Z-IC with multiple levels is considered and some discussion is provided about the capacity region and the performance of simple achievability schemes.\n\nTo summarize, this section contains the following contributions:\n\\begin{itemize}\n\\item An achievable rate region for the S-D Z-IC,\n\\item Capacity region of the injective deterministic S-D Z-IC,\n\\item Modulo-additive S-D Z-IC: optimality of treating interference-as-noise and other properties.\n\\end{itemize}\n\n\n\n\n\n\\subsection{Results for the State-Dependent Channel}\\label{subsec:main_res_state}\n\nThe following theorem provides an inner bound to the capacity region of the S-D Z-IC in Fig.~\\ref{fig:model_gen}.\n\\begin{thm}\\label{thm:gen_ach}\nA rate pair $(R_1,R_2)$ is achievable for the channel in Fig.~\\ref{fig:model_gen} if\n\\begin{equation}\\label{eq:gen_ach}\n\\begin{split}\nR_1 & < I(U;Y_1|Q)-I(U;S|Q),\\\\\nR_2 & < I(X_2;Y_2|V,Q),\\\\\nR_2 & < I(V,X_2;Y_2|Q) - I(V;S|Q),\\\\\nR_1 + R_2 & < I(U;Y_1|Q)+ I(V,X_2;Y_2|Q)\\\\\n & \\quad\\quad -I(U;S|Q)- I(U,S;V|Q),\n\\end{split}\n\\end{equation}for some pmf $p(q)p(u,v|s,q)p(x_1|u,v,s,q)p(x_2|q).$\n\\end{thm}\n\n\nFor the injective deterministic S-D Z-IC, we can identify natural choices for the auxiliary random variables in Theorem~\\ref{thm:gen_ach} that, in fact, yield the capacity region. This result is stated in the following theorem.\n\n\\begin{thm}\\label{thm:cap}\nThe capacity region of the injective deterministic S-D Z-IC in Fig.~\\ref{fig:model_state_det} is the set of rate pairs $(R_1,R_2)$ that satisfy\n\\begin{equation}\\label{eq:cap}\n\\begin{split}\nR_1 & \\leq H(Y_1|S,Q),\\\\\nR_2 & \\leq  H(Y_2|T_1,Q),\\\\\nR_2 & \\leq  H(Y_2|Q) - I(T_1;S|Q),\\\\\nR_1 + R_2 & \\leq H(Y_1|T_1,S,Q)+H(Y_2|Q)-I(T_1;S|Q),\n\\end{split}\n\\end{equation}\nfor some pmf $p(q)p(x_1|s,q)p(x_2|q),$ where $|\\mc{Q}|\\leq 4$.\n\\end{thm}\n\n\n\n\\begin{remark} Note that the capacity region remains unchanged even if the first receiver is provided with the state information. The proof of this theorem is presented in subsection~\\ref{subsec:proof_thm_cap}.\n\\end{remark}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:gen_ach}}\\label{subsec:proofthm1}\nFix $p(u,v|s)p(x_1|u,v,s)p(x_2)$ and choose $0<\\epsilon'<\\epsilon$. \n\n\\subsubsection*{Codebook Generation}\n\\begin{itemize}\n\\item Encoder 2 generates $2^{nR_2}$ codewords $x_2^n(m_2), m_2\\in[1:2^{nR_2}]$ i.i.d. according to $p(x_2)$.\n\\item Encoder 1 generates $2^{n(R_1+R_1')}$ codewords $u^n(m_1,l_1)$ i.i.d. according to $p(u)$, where $m_1\\in[1:2^{nR_1}]$ and $l_1\\in[1:2^{nR_1'}]$. Encoder 1 also generates $2^{nR_2'}$ codewords $v^n(l_2), l_2\\in[1:2^{nR_2'}]$ i.i.d. according to $p(v)$.\n\\end{itemize}\n\n\\subsubsection*{Encoding}\n\\begin{itemize}\n\\item To transmit message $m_2$, encoder~2 transmits $x_2^n(m_2)$.\n\\item Assume that the message to be transmitted by encoder~1 is $m_1$. After observing $s^n$, it finds a pair $(l_1,l_2)$ such that $(u^n(m_1,l_1),v^n(l_2),s^n)\\in\\mc{T}^{(n)}_{\\epsilon'}$. Then it transmits $x_1^n$, which is generated i.i.d. according to $p(x_1|u,v,s)$.\n\\end{itemize}\n\n\\subsubsection*{Decoding}\n\\begin{itemize}\n\\item Decoder 1 finds a unique $\\hat{m}_1$ such that $(u^n(\\hat{m}_1,l_1),y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon}$ for some $l_1$.\n\\item Decoder 2 finds a unique $\\hat{m}_2$ such that $(x_2^n(\\hat{m}_2),v^n(l_2),y_2^n)\\in\\mc{T}_\\epsilon^{(n)}$ for some $l_2$.\n\\end{itemize}\n\n\\subsubsection*{Probability of Error}\nDue to the symmetry of the code, the average probability of error $\\msf{P}(\\mc{E})$ is equal to $\\msf{P}(\\mc{E}|M_1,M_2)$, so we can assume $(M_1,M_2) = (1,1)$ and analyze $\\msf{P}(\\mc{E}|1,1)$. Let $(L_1,L_2)$ denote the pair of indices chosen by encoder 1 such that $(U^n(1,L_1),V^n(L_2),S^n)\\in\\mc{T}^n_{\\epsilon'}$. \n\nWe now define events that cover the error event:\n\\begin{IEEEeqnarray*}{rCl}\n\\mc{E}_1 & \\triangleq & \\{(U^n(1,l_1),V^n(l_2),S^n)\\notin\\mc{T}^{(n)}_{\\epsilon'} \\text{ for all } l_1, l_2\\}, \\label{eq:E1}\\\\\n\\mc{E}_2 & \\triangleq & \\{(U^n(1,L_1),Y_1^n)\\notin\\mc{T}^{(n)}_{\\epsilon}\\}, \\label{eq:E2}\\\\\n\\mc{E}_3 & \\triangleq & \\{(U^n(m_1,l_1),Y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon} \\text{ for some }m_1\\neq 1, l_1\\}, \\label{eq:E3}\\\\\n\\mc{E}_4 & \\triangleq & \\{(X_2^n(1),V^n(L_2),Y_2^n)\\notin\\mc{T}^{(n)}_{\\epsilon}\\} \\label{eq:E4},\\\\\n\\mc{E}_5 & \\triangleq & \\{(X_2^n(m_2),V^n(l_2),Y_2^n)\\in\\mc{T}^{(n)}_{\\epsilon} \\text{ for some }m_2\\neq 1, l_2\\} \\label{eq:E5}.\n\\end{IEEEeqnarray*}\n\nSince an error occurs only if at least one of the above events occur, we have the following upper bound on the average probability of error:\n$$\\msf{P}(\\mc{E}) \\leq \\msf{P}(\\mc{E}_1) + \\msf{P}(\\mc{E}_2\\cap\\mc{E}_1^c) + \\msf{P}(\\mc{E}_3) + \\msf{P}(\\mc{E}_4\\cap\\mc{E}_1^c) + \\msf{P}(\\mc{E}_5).$$\n\nSimilar to the proof of the mutual covering lemma \\cite[Ch. 8]{Gam12}, we can show that $\\msf{P}(\\mc{E}_1)\\rightarrow 0$ as $n\\rightarrow\\infty$ if \n\\begin{IEEEeqnarray}{rCl}\nR_1' & > & I(U;S) + \\delta(\\epsilon'),\\label{eq:ach1_state}\\\\\nR_2' & > & I(V;S) + \\delta(\\epsilon'),\\label{eq:ach2_state}\\\\\nR_1' + R_2' & > & I(U;S) + I(U,S;V) + \\delta(\\epsilon')\\label{eq:ach3_state},\n\\end{IEEEeqnarray}\nwhere $\\delta(\\epsilon')\\rightarrow 0$ as $\\epsilon'\\rightarrow 0.$\n\nBy the conditional typicality lemma \\cite[Ch. 2]{Gam12}, $\\msf{P}(\\mc{E}_2\\cap\\mc{E}_1^c)$ and $\\msf{P}(\\mc{E}_4\\cap\\mc{E}_1^c)$ both tend to zero as $n\\rightarrow\\infty$.\n\nBy the packing lemma \\cite[Ch. 3]{Gam12}, for $\\msf{P}(\\mc{E}_3)\\rightarrow 0$, we require\n\\begin{equation}\\label{eq:ach4_state} R_1+R_1' < I(U;Y_1) - \\delta(\\epsilon),\\end{equation}\nand for $\\msf{P}(\\mc{E}_5)\\rightarrow 0$, we require\n\\begin{IEEEeqnarray}{rCl}\nR_2 & < & I(X_2;Y_2|V) - \\delta(\\epsilon),\\label{eq:ach5_state}\\\\\nR_2 + R_2' & < & I(V,X_2;Y_2) - \\delta(\\epsilon),\\label{eq:ach6_state}\n\\end{IEEEeqnarray}\nwhere $\\delta(\\epsilon)\\rightarrow 0$ as $\\epsilon\\rightarrow 0.$ Hence, $\\msf{P}(\\mc{E})\\rightarrow 0$ if \\eqref{eq:ach1_state}, \\eqref{eq:ach2_state}, \\eqref{eq:ach3_state}, \\eqref{eq:ach4_state}, \\eqref{eq:ach5_state}, \\eqref{eq:ach6_state} are satisfied. \nAllowing coded-time sharing with a time-sharing random variable $Q$ and eliminating $R_1',R_2'$ via Fourier-Motzkin elimination, we obtain the region \\eqref{eq:gen_ach}.\\hfill\\IEEEQED\n\n\n\\subsection{Proof of Theorem~\\ref{thm:cap}}\\label{subsec:proof_thm_cap}\nAchievability follows from Theorem~\\ref{thm:gen_ach} by choosing $U=Y_1$ and $V=T_1$. These choices are valid since encoder 1 knows $(M_1,S^n)$, which determines $T_1^n$ and $Y_1^n$. We now prove the converse.\n\nGiven a sequence of codes that achieves reliable communication (i.e. $P_e^{(n)}\\rightarrow 0$ as $n\\rightarrow\\infty$) at rates $(R_1,R_2)$, we have, by Fano's inequality:\n\\begin{IEEEeqnarray*}{c}\nH(M_1|Y_1^n) \\leq n\\epsilon_n,\\\\\nH(M_2|Y_2^n) \\leq n\\epsilon_n,\n\\end{IEEEeqnarray*}\nwhere $\\epsilon_n\\rightarrow 0$ as $n\\rightarrow\\infty.$\n\nUsing these, we can establish an upper bound on $R_1$ as follows,\n\\begin{IEEEeqnarray}{rCl}\nnR_1 & = & H(M_1) \\nonumber\\\\\n& = & H(M_1|S^n) \\nonumber\\\\\n& \\leq & I(M_1;Y_1^n|S^n) + n\\epsilon_n \\nonumber\\\\\n& \\leq & H(Y_1^n|S^n) + n\\epsilon_n \\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{1i}|S_i) + n\\epsilon_n. \\nonumber\n\\end{IEEEeqnarray}\n\nA simple upper bound on $R_2$ is established in the following:\n\\begin{IEEEeqnarray}{rCl}\nnR_2 & = & H(M_2)\\nonumber\\\\\n& = & H(M_2|T_1^n)\\nonumber\\\\\n& \\leq & I(M_2;Y_2^n|T_1^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_2^n|T_1^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{2i}|T_{1i}) + n\\epsilon_n.\\nonumber\n\\end{IEEEeqnarray}\n\n\nFor the second upper bound on $R_2$, consider the following:\n\\begin{IEEEeqnarray}{rCl}\nnR_2 & = & H(M_2)\\nonumber\\\\\n& = & H(M_2) + H(Y_2^n|M_2) - H(Y_2^n|M_2) \\nonumber\\\\\n& = & H(Y_2^n) + H(M_2|Y_2^n) - H(Y_2^n|M_2) \\nonumber\\\\\n& \\leq & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - H(Y_2^n|M_2)\\nonumber\\\\\n& \\stackrel{(a)}{=} & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - H(T_1^n|M_2)\\nonumber\\\\\n& \\stackrel{(b)}{=} & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - H(T_1^n)\\nonumber\\\\\n& \\leq & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - I(T_1^n;S^n)\\nonumber\\\\\n& = & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - H(S^n) + H(T_1^n|S^n)\\nonumber\\\\\n& \\leq & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - H(S^n) + \\sum_{i=1}^nH(T_{1i}|S_i)\\nonumber\\\\\n& \\stackrel{(c)}{=} & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - \\sum_{i=1}^nH(S_i) + \\sum_{i=1}^nH(T_{1i}|S_i)\\nonumber\\\\\n& = & \\sum_{i=1}^nH(Y_{2i}) + n\\epsilon_n - \\sum_{i=1}^nI(T_{1i};S_i)\\nonumber\\\\\n\\end{IEEEeqnarray}\nwhere step $(a)$ follows by the injectivity property, step $(b)$ follows because $T_1^n$ is independent of $M_2$, and step $(c)$ follows because $S^n$ is an i.i.d. sequence.\n\n\nWe now establish an upper bound on the sum-rate\n\n \\begin{IEEEeqnarray}{rL}\nn(R_1+R_2) & = H(M_1|S^n) + H(M_2)\\nonumber\\\\\n& \\leq I(M_1;T_1^n,Y_1^n|S^n) + n\\epsilon_n + H(Y_2^n) + H(M_2|Y_2^n) - H(Y_2^n|M_2)\\nonumber\\\\\n& \\leq I(M_1;T_1^n,Y_1^n|S^n) + n\\epsilon_n + H(Y_2^n) + n\\epsilon_n - H(Y_2^n|M_2)\\nonumber\\\\\n& \\stackrel{(a)}{\\leq} H(T_1^n,Y_1^n|S^n) + H(Y_2^n) - H(T_1^n|M_2)+ 2n\\epsilon_n\\nonumber\\\\\n& \\stackrel{(b)}{=} H(T_1^n,Y_1^n|S^n) + H(Y_2^n) - H(T_1^n)+ 2n\\epsilon_n\\nonumber\\\\\n& = H(Y_1^n|S^n,T_1^n) + H(Y_2^n) - I(T_1^n;S^n)+ 2n\\epsilon_n\\nonumber\\\\\n& \\stackrel{(c)}{\\leq} \\sum_{i=1}^n H(Y_{1i}|S_i,T_{1i}) + \\sum_{i=1}^n H(Y_{2i}) - \\sum_{i=1}^nI(T_{1i};S_i)+ 2n\\epsilon_n\\nonumber\n\\end{IEEEeqnarray}\nwhere as before, steps $(a)$, $(b)$ and $(c)$ follow because of injectivity property, independence of $T_1^n$ and $M_2$, and i.i.d. state respectively.\n\n\n\nFrom the four bounds established in this section, we can complete the converse by introducing an independent time-sharing random variable $Q$ uniformly distributed on $[1:n]$ and defining $X_1$, $T_1$, $S$, $X_2$, $Y_1$, $Y_2$ to be $X_{1Q}$, $T_{1Q}$, $S_{Q}$, $X_{2Q}$, $Y_{1Q}$, $Y_{2Q}$ respectively. \\hfill\\IEEEQED\n\n\n\\subsection{Example: Modulo-Additive State-Dependent Z-Interference Channel}\\label{subsec:modulo}\n\n\\begin{thm}\\label{thm:modulo}\nThe capacity region of the modulo-additive S-D Z-IC in Fig.~\\ref{fig:model_modulo} is given by the convex closure of the rate pairs $(R_1,R_2)$ satisfying\n\\begin{equation}\\label{eq:cap_modulo}\n\\begin{split}\nR_1 & < (1-\\lambda)\\log |\\mc{X}| + \\lambda H(\\bm{p}),\\\\\nR_2 & < \\log |\\mc{X}| - H\\left(\\lambda \\bm{p} + (1-\\lambda)\\bm{\\delta}_0\\right),\n\\end{split}\n\\end{equation}\nfor some $\\bm{p}\\in\\mc{P}_{\\mc{X}}$, where $\\mc{P}_{\\mc{X}}$ denotes the probability simplex corresponding to $\\mc{X}$, $H(\\bm{p})$ stands for the entropy of the pmf $\\bm{p}$ and $\\bm{\\delta}_0$ denotes the pmf that has unit mass at $0$.\n\\end{thm}\n\nThe capacity region when $\\mc{X}=\\{0,1\\}$ and $S$ is i.i.d. Ber$\\left(\\frac{1}{2}\\right)$ is shown in Figure~\\ref{fig:modulo}.\n\n\n\n\\subsection*{Proof of Theorem~\\ref{thm:modulo}}\n\n\n\\begin{figure}[!t]\n\\centering\n\\input{modulo.tikz}\n\\caption{Capacity Region with $\\mc{X}=\\{0,1\\}$ and $S$ i.i.d. Ber$\\left(\\frac{1}{2}\\right).$ The dotted line shows the capacity region when all nodes have state information. Note that the maximal sum-rate of $1.5$ bits\/channel use is achievable with state information only at the interfering Tx.}\n\\label{fig:modulo}\n\\end{figure}\n\nConsider the capacity region stated in Theorem~\\ref{thm:cap}. Let $\\bm{p}_{1,0}$, $\\bm{p}_{1,1}$ and $\\bm{p}_{2}$, all in $\\mc{P}_{\\mc{X}}$, be used to denote the pmf's ${p(x_1|s=0,q)}$, $p(x_1|s=1,q)$ and $p(x_2|q)$ respectively. Evaluating each of the constraints in \\eqref{eq:cap} gives us the following expression for the capacity region:\n\\begin{equation}\\label{eq:cap_modulo_1}\n\\begin{split}\nR_1 & < (1-\\lambda)H(\\bm{p}_{1,0}) + \\lambda H(\\bm{p}_{1,1}),\\\\\nR_2 & < H(\\bm{p}_{2}),\\\\ \nR_2 & < H\\left((1-\\lambda)\\bm{p}_{2} + \\lambda\\widetilde{\\bm{p}}\\right) + \\lambda H(\\bm{p}_{1,1}) \\\\\n&\\quad\\quad - H\\left(\\lambda \\bm{p}_{1,1} + (1-\\lambda)\\bm{\\delta}_0\\right),\\\\\nR_1 + R_2 & < (1-\\lambda)H(\\bm{p}_{1,0})+H\\left((1-\\lambda)\\bm{p}_{2} + \\lambda\\widetilde{\\bm{p}}\\right)\\\\\n&\\quad\\quad + \\lambda H(\\bm{p}_{1,1}) - H\\left(\\lambda \\bm{p}_{1,1} + (1-\\lambda)\\bm{\\delta}_0\\right),\n\\end{split}\n\\end{equation}\nwhere $\\widetilde{\\bm{p}}\\in\\mc{P}_{\\mc{X}}$ is a pmf that is defined as \n$$\\widetilde{\\bm{p}}(k) = \\sum_{i=0}^{|\\mc{X}|-1} \\bm{p}_{1,1}(i)\\bm{p}_{2}(k-i),\\quad 0\\leq k\\leq |\\mc{X}|-1,$$ and $k-i$ should be understood to be $(k-i)\\text{ mod } |\\mc{X}|$.\n\nFirstly, we note that $\\bm{p}_{1,0}$ should be chosen as the pmf of the uniform distribution to maximize $H(\\bm{p}_{1,0})$, thus maximizing the RHS of the constraints in \\eqref{eq:cap_modulo_1}. Similarly, $\\bm{p}_2$ should also be chosen to be the pmf of the uniform distribution. Then, we can also remove the first constraint on $R_2$, since it is rendered redundant by the other constraint on $R_2$.\nThus, the capacity region is given by the convex closure of $(R_1,R_2)$ satisfying\n\\begin{equation}\\label{eq:cap_modulo_3}\n\\begin{split}\nR_1 & < (1-\\lambda)\\log(|\\mc{X}|) + \\lambda H(\\bm{p}_{1,1}),\\\\\nR_2 & < \\log(|\\mc{X}|) + \\lambda H(\\bm{p}_{1,1}) - H\\left(\\lambda \\bm{p}_{1,1} + (1-\\lambda)\\bm{\\delta}_0\\right),\\\\\nR_1 + R_2 & < (2-\\lambda)\\log(|\\mc{X}|)+ \\lambda H(\\bm{p}_{1,1})\\\\\n&\\quad\\quad\\quad\\quad - H\\left(\\lambda \\bm{p}_{1,1} + (1-\\lambda)\\bm{\\delta}_0\\right),\n\\end{split}\n\\end{equation} for $\\bm{p}_{1,1}\\in\\mc{P}_{\\mc{X}}.$\n\nFor any $\\bm{p}$, the region in \\eqref{eq:cap_modulo} is contained in the region in \\eqref{eq:cap_modulo_3} for $\\bm{p}_{1,1}=\\bm{p}$. Hence, the convex closure of \\eqref{eq:cap_modulo} is contained in the convex closure of \\eqref{eq:cap_modulo_3}.\n\nHowever, also note that the region in \\eqref{eq:cap_modulo_3} for any $\\bm{p}_{1,1}$ is contained in the convex hull of two regions, one obtained by setting $\\bm{p} = \\bm{p}_{1,1}$ in \\eqref{eq:cap_modulo} and the other obtained by setting $\\bm{p}=\\bm{\\delta}_0$ in \\eqref{eq:cap_modulo}. Hence, the convex closure of \\eqref{eq:cap_modulo_3} is also contained in the convex closure of \\eqref{eq:cap_modulo}. This concludes the proof of Theorem~\\ref{thm:modulo}.\\hfill\\IEEEQED \n\n\n\\begin{remark}\nThe optimal sum-rate $(2-\\lambda)\\log |\\mc{X}|$ is achieved by choosing $\\bm{p}=\\bm{\\delta}_0$. This corresponds to setting the transmitted symbols of the first transmitter to $0$ when $S=1$ so that it does not interfere with the second transmission. The first transmitter then treats these symbols as stuck to $0$ and performs Gelfand-Pinsker coding. The second transmitter transmits at rate $\\log(|\\mc{X}|)$ bits\/channel use. It can be easily verified that this is also the optimal sum-rate when all nodes are provided with the state~information. Thus, for this channel, the sum-capacity when all nodes have state information is the same as that when only encoder~1 has state information.\n\\end{remark}\n\n\\begin{remark}\nFinally, we note that there is also another way to achieve the capacity region of the modulo additive S-D Z-IC. For this, first recall that to get the capacity region expression in Theorem~\\ref{thm:cap}, we set the auxiliary random variables $U$ and $V$ in the expression in Theorem~\\ref{thm:gen_ach} to $Y_1$ and $T_1$ respectively. Another choice, which corresponds to standard Gelfand-Pinsker coding for the first transmitter-receiver pair and treating interference as noise at the second receiver is to choose $V=\\phi$ in Theorem~\\ref{thm:gen_ach}. This gives us the following achievable region:\n\\begin{equation}\\label{eq:int_noise}\n\\begin{split}\nR_1 & < I(U;Y_1|Q)-I(U;S|Q),\\\\\nR_2 & < I(X_2;Y_2|Q),\n\\end{split}\n\\end{equation} for some pmf $p(q)p(u|s,q)p(x_1|u,s,q)p(x_2|q)$. We can now see that for the modulo-additive S-D Z-IC, the capacity region is also achieved by making the following choices in the above region: $p(u|s=0)$ to be the uniform pmf over $\\mc{X}$, $p(u|s=1)$ to be $\\bm{p}$, $p(x_1|u,s)$ to be $\\bm{\\delta}_u$ (i.e. $X_1=U$) and $p(x_2)$ to be the uniform pmf over $\\mc{X}$. Thus, the capacity region of the modulo-additive S-D Z-IC can also be achieved by treating interference as noise at the second receiver.\n\\end{remark}\n\n\\subsection{Multiple-level modulo-additive S-D Z-IC}\\label{subsec:multiplelevel}\n\nThe linear deterministic model introduced in \\cite{Ave11} consists of multiple \\emph{bit levels} that roughly correspond to bits communicated at different power levels. The modulo-additive S-D Z-IC that we looked at in the previous subsection is a special case in which the number of levels is one. Extending the model to have multiple bit levels raises some interesting questions which we consider in this subsection. \n\nMore specifically, consider the model depicted in Fig.~\\ref{fig:model_modulo_multiple}, which can be thought of as three copies of the model in Fig.~\\ref{fig:model_modulo}, which are however related by the common state affecting them. For simplicity, we restrict attention to the case when the alphabet on each level, denoted by $\\mc{X}$, is the binary alphabet, i.e. $\\{0,1\\}$, and the state is Ber$(0.5).$ Let $L$ denote the number of bit levels.\n\n\\begin{figure}[!th]\n\\centering\n\\includegraphics[scale=1.5]{modulo_multiple_levels.pdf}\n\\caption{The Modulo-Additive S-D Z-IC wit multiple bit levels.}\n\\label{fig:model_modulo_multiple}\n\\end{figure}\n\n\\begin{figure}[!th]\n\\centering\n\\begin{subfigure}\n\\centering\n \\resizebox{.5\\linewidth}{!}{\\input{twolevel_binary_SD_ZIC.tikz}}\n\\caption{Comparison of the different rate regions for 2-level binary modulo-additive S-D Z-IC}\n\\label{fig:modulo_multiple_2}\n\\end{subfigure}\n\\vspace{4mm}\n\\begin{subfigure}\n\\centering\n\\resizebox{.5\\linewidth}{!}{\\input{threelevel_binary_SD_ZIC.tikz}}\n\\caption{Comparison of the different rate regions for 3-level binary modulo-additive S-D Z-IC}\n\\label{fig:modulo_multiple_3}\n\\end{subfigure}\n\\end{figure}\n\nThis model also falls under the injective-deterministic setup for which we have completely characterized the capacity region. So the capacity region can be easily computed, as we indeed do in the following. This evaluation also allows us to immediately compare the capacity region with the rates achieved by some straightforward achievability schemes that we can employ. In particular, consider the following two simple achievability schemes:\n\\begin{itemize}\n\\item ``Separation'': The simplest strategy one can employ is to separately consider each level and communicate over it independently of the other levels. This gives us that the rate pairs $(R_1,R_2)$ satisfying\n\\begin{equation}\\label{eq:ach_separation}\n\\begin{split}\nR_1 & < \\frac{L}{2} + \\sum_{i=1}^{L}\\frac{1}{2} H(\\bm{p}_i),\\\\\nR_2 & < L - \\sum_{i=1}^{L}H\\left( \\bm{p}_i + \\bm{\\delta}_0\\right),\n\\end{split}\n\\end{equation}\nfor some $\\bm{p}_1,\\bm{p}_2,\\dots,\\bm{p}_L\\in\\mc{P}_{\\mc{X}}$ are achievable.\n\\item ``Communicate state'': Alternatively, by noticing that strictly better rates could have been achieved if decoder~2 also had access to the state information, we can reserve one level to communicate the state from encoder~1 to decoder~2. This is done by ensuring that encoder~1 transmits a 1 on this reserved level whenever the state is 1, and encoder~2 constantly transmits a 0 on this level. The nodes communicate on the remaining levels keeping in mind that now decoder~2 also has state information. Note that while no communication can happen between encoder~2 and decoder~2 on the reserved level, encoder~1 can still communicate with decoder~1 at rate 0.5 on this level by treating it as a channel with stuck bits (bit equals 1 whenever state equals 1). This strategy provides us the following achievable region:\n\\begin{equation}\\label{eq:ach_state}\n\\begin{split}\nR_1 & < \\frac{L}{2} + \\frac{1}{2} H(\\bm{p}),\\\\\nR_2 & < L-1 - \\frac{1}{2}H\\left( \\bm{p}\\right),\n\\end{split}\n\\end{equation}\nfor some $\\bm{p}\\in\\mc{P}_{\\mc{X}^{L-1}}$.\n\\end{itemize}\n\nWe can expect that the suboptimality of reserving one level for communicating the state should become relatively small as the number of levels increases i.e. at high SNR. This is corroborated by the numerical analysis, shown in Figs.~\\ref{fig:modulo_multiple_2} and \\ref{fig:modulo_multiple_3}, in which we can see that there is a marked improvement in the rates achieved by this scheme relative to the capacity region as we increase the number of levels from 2 to 3. Indeed, since all the levels are affected by the same state, the entropy of the state becomes small compared to the communication rates as the SNR increases, so it is not a big overhead to explicitly communicate the state to decoder~2 at high SNR. However, at low SNR, the figures show that the overhead incurred is quite high due to which this approach is significantly suboptimal, while the simple scheme of treating the levels separately results in achieving very close to the entire capacity region.\n\n\n\n\n\n\\section{Interference Channels with Partial Cribbing}\\label{sec:cribbing}\n\n\nIn this section, we focus on deterministic Z-interference channels when the interfering transmitter can overhear the signal transmitted by the other transmitter after it passes through some channel. This channel is also modeled as a deterministic channel, dubbed as \\emph{partial cribbing} in \\cite{Asn13}. Deterministic models, in particular linear deterministic models \\cite{Ave11}, have gained popularity due to the observation that they are simpler to analyze and are provably close in performance to Gaussian models. \n\nThere have been quite a few very sophisticated achievability schemes designed for interference channels with causal cribbing encoders, however optimality of the achievable rate regions has not been addressed. In the most general interference channel model with causal cribbing \\cite{Yan11}, each encoder needs to split its message into four parts: a common part to be sent cooperatively, a common part to be sent non-cooperatively, a private part to be sent cooperatively and a private part to be sent non-cooperatively. Further, because of the causal nature of cribbing, achievability schemes usually involve block-Markov coding, so that each encoder also needs to consider the cooperative messages of both encoders from the previous block. Motivated by the alternative achievability scheme we have presented earlier for the general interference channel, we present a simple optimal achievability scheme that minimizes the rate-splitting that is required. Specifically, while encoder~2 only splits its message into a cooperative and non-cooperative private part, encoder~1 does not perform any rate-splitting at all. By focusing on the specific configuration of the Z-interference channel, we are able to prove the optimality of an achievability scheme that is simpler than the highly involved achievability schemes for the general case that are currently known.\n \n\n\n\n\n\n\n\n\\subsection{Result for Partial Cribbing}\\label{subsec:main_res_crib}\n\\begin{thm}\\label{thm:part_crib}\nThe capacity region of the injective deterministic Z-interference channel with unidirectional partial cribbing, depicted in Fig.~\\ref{fig:model_crib}, is given by the convex closure of $(R_1,R_2)$ satisfying \\begin{equation}\\label{eq:cap_part_crib}\n\\begin{split}\nR_1 & \\leq H(Y_1|W),\\\\\nR_2 & \\leq \\min\\Big(H(Y_2), H(Y_2,Z_2|T_1,W)\\Big),\\\\\nR_1 + R_2 & \\leq H(Y_1|T_1,W)+\\min\\Big(H(Y_2), H(Y_2,Z_2|W)\\Big),\n\\end{split}\n\\end{equation}\nfor $p(w)p(x_1|w)p(x_2|w),$ where $W$ is an auxiliary random variable whose cardinality can be bounded as\n$|\\mathcal{W}|\\leq |\\mathcal{Y}_2|+3.$\n\\end{thm}\n\nThe proof of this theorem is presented below.\n\n\\subsection{Proof of Theorem~\\ref{thm:part_crib}}\\label{subsec:proof_crib}\n\\emph{Achievability}\\\\\nChoose a pmf $p(w)p(u_d,u_c,x_1|w)p(x_2,z_2|w)$ and ${0<\\epsilon'<\\epsilon}$, where for the sake of generality, we use the auxiliary random variables $U_d$ and $U_c$. In the injective deterministic case at hand, they can be set to $Y_1$ and $T_1$ respectively.\n\\subsubsection*{Codebook Generation}\nThe communication time is divided into $B$ blocks, each containing $n$ channel uses, and an independent random code is generated for each block $b\\in[1:B]$. Whenever it is clear from the context, we suppress the dependence of codewords on $b$ to keep the notation simple. The messages in block $B$ are fixed apriori, so a total of $B-1$ messages are communicated over the $B$ blocks. The resulting rate loss can be made as negligible as desired by choosing a sufficiently large $B$.\n\nWe split $R_2$ as $R_2'+R_2''$, which corresponds to the split of message~2 into two parts, one that will be sent cooperatively by both transmitters to receiver~2 and the other non-cooperatively only by transmitter~2 to receiver~2. For each block $b$, let $m_{2,b}'\\in[1:2^{nR_2'}]$ and $m_{2,b}''\\in[1:2^{nR_2''}]$. For each block $b\\in [1:B]$, we generate $2^{nR_2'}$ sequences $w^n$ i.i.d. according to $p(w)$.\n\\begin{itemize}\n\\item For each $w^n$ in block $b$, we generate $2^{nR_{2}'}$ sequences $\\left\\{z_2^n(w^n,m'_{2,b})\\right\\}$ i.i.d. according to $p(z_2|w)$. Then for each $(w^n,z_2^n)$, we generate $2^{nR_2''}$ sequences $\\left\\{x_2^n(w^n,z_2^n,m''_{2,b})\\right\\}$ i.i.d. according to $p(x_2|z_2,w)$.\n\\item For each $w^n$ in block $b$, we generate $2^{nR_c}$ sequences $\\left\\{u_c^n(w^n,l_c)\\right\\}$ i.i.d. according to $p(u_c|w)$, where $l_c\\in[1:2^{nR_c}]$. We also generate $2^{n(R_1+R_d)}$ sequences $\\left\\{u_d^n(m_{1,b},l_d)\\right\\}$ i.i.d. according to $p(u_{d})$, where $m_{1,b}\\in[1:2^{nR_1}]$ and $l_d\\in[1:2^{nR_d}]$. \\footnote{Note that the $u_d^n$ sequences are generated independently of the $w^n$ sequences.}\n\\end{itemize}\n\n\\subsubsection*{Encoding}\nLet us assume for now that as a result of the cribbing, encoder 1 knows $m'_{2,b-1}$ at the end of block ${b-1}$. Then in block $b$, both encoders can encode $m'_{2,b-1}$ using $w^n(m'_{2,b-1})$ where $w^n$ is from the code for block $b$.\n\\begin{itemize}\n\\item To transmit message $m_{1,b}$, encoder 1 finds a pair $(l_{d},l_{c})$ such that $$(w^n(m'_{2,b-1}),u_c^n(w^n,l_c),u_d^n(m_{1,b},l_{d}))\\in\\mc{T}^{(n)}_{\\epsilon'}.$$ It transmits $x_1^n$ that is generated i.i.d. according to $p(x_1|w,u_d,u_c)$.\n\\item To transmit message $m_{2,b}=(m'_{2,b},m''_{2,b})$, encoder 2 encodes $m'_{2,b}$ as $z_2^n(w^n,m'_{2,b})$ and then transmits $x_2^n(w^n,z_2^n,m''_{2,b})$.\n\\end{itemize}\nWe fix apriori the messages in block $B$ to be $m_{1,B}=1$, $m'_{2,B}=1$ and $m''_{2,B}=1$. Also, to avoid mentioning edge cases explicitly, whenever $m_{1,0}$, $m'_{2,0}$ or $m''_{2,0}$ appear, we assume that all are fixed to 1.\n\n\\subsubsection*{Decoding}\n\\begin{itemize}\n\\item \\emph{Encoder~1:} At the end of block $b$, assuming it has already decoded $m'_{2,b-1}$ at the end of block $b-1$, encoder 1 decodes $m'_{2,b}$ by finding the unique $\\hat{m}'_{2,b}$ such that the sequence $z_2^n$ it has observed via cribbing is equal to $z_2^n(w^n,\\hat{m}'_{2,b})$.\n\\item \\emph{Decoder~1:} In each block $b$, decoder 1 finds the unique $\\hat{m}_{1,b}$ such that $(u_d^n(m_{1,b},l_{d}),y_1^n)\\in\\mc{T}^{(n)}_{\\epsilon}$ for some $l_{d}$.\n\\item \\emph{Decoder~2:} Decoder 2 performs backward decoding as follows: \n\\begin{itemize}\n\\item In block $B$, decoder 2 finds a unique $m'_{2,B-1}$ such that the condition \\eqref{eq:jtd_dec2_B} is satisfied for some $l_c.$\n\\begin{equation}\\label{eq:jtd_dec2_B}\n(w^n(\\hat{m}'_{2,B-1}),z_2^n(w^n,1),x_2^n(w^n,z_2^n,1),u_c^n(w^n,l_c),y_2^n)\\in\\mc{T}^{(n)}_{\\epsilon}\n\\end{equation}\n\\item In block $b$, assuming $m'_{2,b}$ has been decoded correctly, it finds the unique $(\\hat{m}'_{2,b-1}, \\hat{m}''_{2,b})$ such that  the condition \\eqref{eq:jtd_dec2} is satisfied for some $l_{c}$.\n\\begin{equation}\\label{eq:jtd_dec2}\n(w^n(\\hat{m}'_{2,b-1}),z_2^n(w^n,m'_{2,b}),x_2^n(w^n,z_2^n,\\hat{m}''_{2,b}),u_c^n(w^n,l_c),y_2^n)\\in\\mc{T}^{(n)}_{\\epsilon}\n\\end{equation}\n\\end{itemize}\n\\end{itemize}\n\n\n\n\n\\subsubsection*{Probability of Error}\n\nTo get a vanishing probability of error, we can impose the conditions described in the following list.\n\\begin{itemize}\n\\item\nSimilar to the proof of the mutual covering lemma \\cite[Ch. 8]{Gam12}, we can show that the following conditions are sufficient for the success of encoding at the first transmitter:\n\\begin{IEEEeqnarray}{rCl}\nR_d & > & I(U_d;W) +\\delta(\\epsilon'),\\label{eq:dec1}\\\\\nR_d + R_c & > & I(U_d;U_c,W)+\\delta(\\epsilon')\\label{eq:dec2}.\n\\end{IEEEeqnarray}\n\\item For the decoding at encoder 1 to succeed:\n\\begin{equation}\nR'_2 < H(Z_2|W)-\\delta(\\epsilon).\\label{eq:dec3}\n\\end{equation}\n\\item For decoding at decoder 1 to succeed:\n\\begin{equation}\nR_1 + R_d < I(U_d;Y_1)-\\delta(\\epsilon).\\label{eq:dec4}\n\\end{equation}\n\\item For the backward decoding at decoder 2 to succeed, it is sufficient that the following conditions are satisfied:\n\\begin{IEEEeqnarray}{rCl}\nR''_2 & < & I(X_2;Y_2|W,U_c,Z_2)-\\delta(\\epsilon),\\label{eq:dec5}\\\\\nR''_2+ R_c & < & I(U_c,X_2;Y_2|W,Z_2)-\\delta(\\epsilon),\\label{eq:dec6}\\\\\nR'_2 + R''_2 + R_c & < &  I(W,U_c,X_2;Y_2)-\\delta(\\epsilon).\\label{eq:dec7}\n\\end{IEEEeqnarray}\n\\end{itemize}\n\nNoting that $R'_2+R''_2=R_2$, eliminating $(R_d,R_c,R'_2,R''_2)$ from \\eqref{eq:dec1}-\\eqref{eq:dec7} via Fourier-Motzkin elimination, and substituting $U_d=Y_1$ and $U_c=T_1$, we get the achievable region in \\eqref{eq:cap_part_crib} with the following additional bound on $R_1$:\n$$R_1< H(Y_1|W,T_1) + H(Y_2|W,Z_2).$$\nTo conclude the proof of achievability, we show that this bound is rendered redundant by $R_1<H(Y_1|W)$ which can be proved by the following chain of inequalities:\n\\begin{IEEEeqnarray*}{rCl}\nH(Y_1|W,T_1) + H(Y_2|W,Z_2) & \\geq & H(Y_1|W,T_1) + H(Y_2|W,X_2)\\\\\n& = & H(Y_1|W,T_1) + H(T_1|W,X_2)\\\\\n& = & H(Y_1|W,T_1) + H(T_1|W)\\\\\n& = & H(Y_1,T_1|W)\\\\\n& \\geq & H(Y_1|W).\n\\end{IEEEeqnarray*}\n\n\\emph{Converse}\\\\\nWe now establish the converse. By Fano's inequality, we have the following two relations that are satisfied by any sequence of codes that achieve reliable communication:\n$$H(M_1|Y_1^n)\\leq n\\epsilon_n,\\quad H(M_2|Y_2^n)\\leq n\\epsilon_n,$$\nwhere $\\epsilon_n\\rightarrow 0$ as $n\\rightarrow\\infty.$\n\nFirst, an upper bound on $R_1$ is established in \\eqref{eq:conv1}.\n\\begin{IEEEeqnarray}{rCl}\nnR_1 & = & H(M_1)\\nonumber\\\\\n& = & H(M_1|Z_2^n)\\nonumber\\\\\n& \\leq & I(M_1;Y_1^n|Z_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_1^n|Z_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{1i}|Z_2^{i-1}) + n\\epsilon_n\\nonumber\\\\\n& = & \\sum_{i=1}^n H(Y_{1i}|W_i) + n\\epsilon_n,\\IEEEeqnarraynumspace\\label{eq:conv1}\n\\end{IEEEeqnarray}\nwhere $W_i\\triangleq Z_2^{i-1}$.\n\n\nNext, we establish two bounds on $R_2$, the first one in \\eqref{eq:conv2} as follows:\n\\begin{IEEEeqnarray}{rCl}\nnR_2 & = & H(M_2)\\nonumber\\\\\n& \\leq & I(M_2;Y_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{2i}) + n\\epsilon_n,\\IEEEeqnarraynumspace\\label{eq:conv2}\n\\end{IEEEeqnarray}\nand the second one in \\eqref{eq:conv3} below:\n\\begin{IEEEeqnarray}{rCl}\nnR_2 & = & H(M_2|M_1)\\nonumber\\\\\n& = & H(M_2,Z_2^n|M_1)\\nonumber\\\\\n& = & H(Z_2^n|M_1) + H(M_2|M_1,Z_2^n)\\nonumber\\\\\n& \\stackrel{(a)}{=} & H(Z_2^n|M_1) + H(M_2|M_1,Z_2^n,T_1^n)\\nonumber\\\\\n& \\leq & H(Z_2^n|M_1) + I(M_2;Y_2^n|M_1,Z_2^n,T_1^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Z_2^n) + H(Y_2^n|Z_2^n,T_1^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Z_{2i}|W_{i}) + \\sum_{i=1}^n H(Y_{2i}|W_i,T_{1i},Z_{2i}) + n\\epsilon_n,\\nonumber\\\\\n& \\stackrel{(b)}{=} & \\sum_{i=1}^n H(Z_{2i}|W_{i},T_{1i}) + \\sum_{i=1}^n H(Y_{2i}|W_i,T_{1i},Z_{2i}) + n\\epsilon_n,\\nonumber\\\\\n& = & \\sum_{i=1}^n H(Y_{2i},Z_{2i}|W_{i},T_{1i}) + n\\epsilon_n,\\IEEEeqnarraynumspace\\label{eq:conv3}\n\\end{IEEEeqnarray}\nwhere step $(a)$ follows because $T_1^n$ is a function of $X_1^n$ which is a function of $(M_1,Z_2^n)$, and step $(b)$ follows because $Z_{2i}-W_{i}-T_{1i}$.\n\nFinally, we establish two bounds on the sum-rate, the first one in \\eqref{eq:conv4} below:\n\\begin{IEEEeqnarray}{rRl}\n& n(R_1+R_2) \\nonumber\\\\\n& = & H(M_1|Z_1^n) + H(M_2,Z_2^n)\\nonumber\\\\\n& = & H(M_1,T_1^n|Z_2^n) + H(Z_2^n) + H(M_2|Z_2^n)\\nonumber\\\\\n& = & H(T_1^n|Z_2^n) + H(M_1|T_1^n,Z_2^n)+ H(Z_2^n)  + H(M_2|Z_2^n)\\nonumber\\\\\n& \\stackrel{(a)}{\\leq} & H(Y_2^n|X_2^n,Z_2^n) + I(M_1;Y_1^n|T_1^n,Z_2^n) + H(Z_2^n) + I(M_2;Y_2^n|Z_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_2^n|X_2^n,Z_2^n) + H(Y_1^n|T_1^n,Z_2^n) + H(Z_2^n)+ H(Y_2^n|Z_2^n) - H(Y_2^n|M_2,Z_2^n)+ n\\epsilon_n\\nonumber\\\\\n& = & H(Y_2^n|X_2^n,Z_2^n) + H(Y_1^n|T_1^n,Z_2^n) + H(Z_2^n)+ H(Y_2^n|Z_2^n) - H(Y_2^n|M_2,X_2^n,Z_2^n)+ n\\epsilon_n\\nonumber\\\\\n& \\stackrel{(b)}{=} & H(Y_2^n|X_2^n,Z_2^n) + H(Y_1^n|T_1^n,Z_2^n) + H(Z_2^n)+ H(Y_2^n|Z_2^n) - H(Y_2^n|X_2^n,Z_2^n)+ n\\epsilon_n\\nonumber\\\\\n& = & H(Y_1^n|T_1^n,Z_2^n) + H(Z_2^n)+ H(Y_2^n|Z_2^n) + n\\epsilon_n\\nonumber\\\\\n& = & H(Y_1^n|T_1^n,Z_2^n) + H(Y_2^n,Z_2^n)+ n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{1i}|T_{1i},W_i) + \\sum_{i=1}^n H(Y_{2i},Z_{2i}|W_i) + n\\epsilon_n,\\nonumber\\\\*\\label{eq:conv4}\n\\end{IEEEeqnarray}\nwhere step $(a)$ uses the fact that $H(T_1^n|Z_2^n)\\leq H(Y_2^n|X_2^n,Z_2^n)$, which is proved below: \\begin{IEEEeqnarray*}{rCl}\nH(T_1^n|Z_2^n) & \\leq & H(T_1^n)\\\\\n& = & H(Y_2^n|X_2^n)\\\\\n& = & H(Y_2^n|X_2^n,Z_2^n),\n\\end{IEEEeqnarray*}\nand step $(b)$ follows because $M_2 - (X_2^n,Z_2^n) - Y_2^n$.\n\nThe second bound on the sum-rate is established in \\eqref{eq:conv5} as follows:\n\\begin{IEEEeqnarray}{rRl}\n& n(R_1+R_2) \\nonumber\\\\\n& = & H(M_1,T_1^n|Z_2^n) + H(M_2)\\nonumber\\\\\n& \\leq & H(T_1^n|Z_2^n) + H(M_1|T_1^n,Z_2^n) + I(M_2;Y_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_2^n|X_2^n,Z_2^n) + I(M_1;Y_1^n|T_1^n,Z_2^n) + H(Y_2^n)- H(Y_2^n|M_2) + n\\epsilon_n\\nonumber\\\\\n& \\leq & H(Y_2^n|X_2^n,Z_2^n) + H(Y_1^n|T_1^n,Z_2^n) + H(Y_2^n)- H(Y_2^n|X_2^n,Z_2^n) + n\\epsilon_n\\nonumber\\\\\n& = &  H(Y_1^n|T_1^n,Z_2^n) + H(Y_2^n) + n\\epsilon_n\\nonumber\\\\\n& \\leq & \\sum_{i=1}^n H(Y_{1i}|T_{1i},W_i) + \\sum_{i=1}^n H(Y_{2i}) + n\\epsilon_n.\\IEEEeqnarraynumspace\\label{eq:conv5}\n\\end{IEEEeqnarray}\n\nIn \\eqref{eq:conv1}-\\eqref{eq:conv5}, we can introduce a time-sharing random variable $Q$ uniformly distributed on $[1:n]$. Defining $W$ to be $(W_Q,Q)$ and $(X_1,X_2,Y_1,Y_2)$ to be $(X_{1Q},X_{2Q},Y_{1Q},Y_{2Q})$, we get the required bounds on the rates.\n\nWe also require the Markov relationship $X_1-W-X_2$ to be satisfied. Since $W_i$ is chosen to be $Z_2^{i-1}$, we immediately have $X_{1i} - W_i - X_{2i}$ and hence $X_1-W-X_2$. Finally, the bound on the cardinality of $W$ can be established using the standard convex cover method.  \\hfill\\IEEEQED\n\n\n\\section{Extensions}\\label{sec:extensions}\n\n\\subsection{State-Dependent Z-channel}\n\nThe result in Theorem~\\ref{thm:cap} can be extended easily to Z-channels in which transmitter~1 also wishes to communicate a message $M_{21}$ at rate $R_{21}$ to receiver~2, as shown in Fig.~\\ref{fig:model_state_det_zc}. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.5]{block_diag_zc.pdf}\n\\caption{The Injective Deterministic S-D Z-C}\n\\label{fig:model_state_det_zc}\n\\end{figure}  \n\n\\begin{thm}\\label{thm:cap_z} The capacity region of the injective deterministic state-dependent Z-channel is the set of rate pairs $(R_{1},R_{21},R_{2})$ that satisfy\n\\begin{equation*}\\label{eq:cap_z}\n\\begin{split}\nR_{1} & \\leq H(Y_1|S,Q),\\\\\nR_{2} & \\leq  H(Y_2|T_1,Q),\\\\\nR_{21} & \\leq H(T_1|S,Q),\\\\\nR_{1} + R_{21} & \\leq H(T_1,Y_1|S,Q),\\\\\nR_{2} + R_{21} & \\leq  H(Y_2|Q) - I(T_1;S|Q),\\\\\nR_{1} + R_{2} + R_{21} & \\leq H(Y_1|T_1,S,Q)+H(Y_2|Q)-I(T_1;S|Q),\n\\end{split}\n\\end{equation*}\nfor some pmf $p(q)p(x_1|s,q)p(x_2|q),$ where $|\\mc{Q}|\\leq 6$.\n\\end{thm}\n\nThe achievability scheme for this case is similar to the achievability scheme for the Z-IC described in Section~\\ref{subsec:proof_thm_cap} except that now the $v^n$ sequences at encoder~1 are also binned, and this bin-index corresponds to the message $M_{21}$, that is to be communicated from transmitter~1 to receiver~2. The converse can be established by following similar steps as the converse for the Z-IC.\n\n\\begin{remark}\nSince broadcast channel and multiple-access channel are special cases of the Z-channel, Theorem~\\ref{thm:cap_z} also provides the capacity region for the deterministic broadcast channel and the injective deterministic multiple-access channel.\n\\end{remark}\n\n\\subsection{State-dependent injective deterministic Z-IC with unidirectional partial cribbing}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=1.5]{block_diag_combined.pdf}\n\\caption{State-Dependent Injective Deterministic Z-Interference Channel with Unidirectional Partial Cribbing}\n\\label{fig:model_state_crib}\n\\end{figure}\n\nTo illustrate further the advantages of the multicoding scheme, we consider a model that combines the state-dependent Z-IC and the Z-IC with unidirectional partial cribbing, as depicted in Fig.~\\ref{fig:model_state_crib}. We can combine the achievability schemes for the two component setups from Sections~\\ref{subsec:proof_thm_cap} and \\ref{subsec:proof_crib} in a straightforward manner to get an achievability scheme for this setup. It turns out that this is capacity-achieving, resulting in the following theorem. \n\\begin{thm} The capacity region of the channel in Fig.~\\ref{fig:model_state_crib} is given by the convex closure of rate pairs $(R_1,R_2)$ satisfying\n\\begin{IEEEeqnarray*}{rCl}\nR_1 & < & H(Y_1|W,S)\\\\\nR_2 & < & H(Y_2,Z_2|T_1,W)\\\\\nR_2 & < & \\min(H(Y_2),H(Y_2,Z_2|W)) - I(T_1;S|W)\\\\\nR_1 + R_2 & < & H(Y_1|W,T_1,S) + \\min(H(Y_2),H(Y_2,Z_2|W)) - I(T_1;S|W)\n\\end{IEEEeqnarray*}\nfor pmf of the form $p(w)p(x_2|w)p(x_1|w,s)$.\n\\end{thm}\n\nThe proof of the converse combines ideas from the converse proofs we have presented for the state-only case and the cribbing-only case.\n\n\\section{Discussion}\\label{sec:conclude}\nNote that there is one difference between the multicoding-based achievability scheme for the canonical interference channel and the achievability schemes for the settings in Sections~\\ref{sec:state} and \\ref{sec:cribbing}. For the former, the codebook associated with the auxiliary random variable $U$ is used at both receivers during decoding, whereas for the cribbing setup, the codebook associated with auxiliary random variable $U_d$ is only used at the desired receiver, while that associated with the auxiliary random variable $U_c$ is only used at the undesired receiver. One way to understand this dichotomy is to observe similarities with the inner bound for broadcast channel which combines Marton coding and superposition coding, given in \\cite[Proposition 8.1]{Gam12} which involves three auxiliary random variables $U_0,U_1,U_2$. Here, the random variable $U_0$ is used at both receivers during decoding, while $U_1$ and $U_2$ are used only at the respective receivers. Now for the deterministic cribbing setup that we have considered, we can think that in the optimal scheme, $U_0$ can be set to be the empty random variable $\\phi$, and $U_1$ and $U_2$ correspond to $U_d$ and $U_c$ respectively, i.e., there is no superposition coding, only Marton coding is used (with the distinction from usual Marton coding that the set of $U_c^n$ sequences is not binned). The situation is similar for deterministic and semideterministic broadcast channels where $U_0=\\phi$ is optimal too. On the other hand, the Han-Kobayashi scheme employing superposition coding can be thought of as setting $U_2$ to $\\phi$, and $U_0$ and $U_1$ correspond to $U$ and $X$ respectively (no Marton coding, only superposition coding). The key observation is that in the Han-Kobayashi scheme, it is not necessary to think of $U$ as encoding a part of the message explicitly, which can be exploited to view the superposition coding instead in a manner resembling Marton coding, as we have shown in Section~\\ref{sec:outline} (again with the distinction from usual Marton coding that the set of sequences corresponding to the auxiliary random variable is not binned). This clarifies the dichotomy mentioned at the beginning of this paragraph. Alternatively, we can understand both ways of decoding in a unified manner, as in \\cite{Ban12}. \n\n\n\n\n\n\\section*{Acknowledgment}\nWe gratefully acknowledge discussions with Young-Han Kim, Chandra Nair, Shlomo Shamai and Tsachy Weissman. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction to adjoint method}\nIn this section, we provide the general derivation of the adjoint method~\\cite{Pontryagin2018,Chen2018a} and apply it to the master equation governing the dynamics of a dissipative quantum system.\n\n\\subsection{Derivation of the adjoint method}\nConsider an initial value problem\n\\begin{equation}\n    \\frac{\\text{d}\\bm{v}}{\\text{d}t} = f(\\bm{v}, \\bm{\\theta})\n\\end{equation}\nwhere the state $\\bm{v}$ evolves from $t=0$ to $t=T$ and $\\bm{\\theta}$ are the time independent parameters for the ODE. Here we assume that there is a ``loss function'' $L(\\bm{v}(T))$ which only depends on the final state $\\bm{v}(T)$. This final state implicitly depends on both the initial state $\\bm{v}(0)$ and the parameters $\\bm \\theta$ that specify its time evolution through the integration of the ODE. Our goal is to compute the gradient of $L$ with respect to both the initial state ($\\partial L \/ \\partial \\bm{v}(0)$) and the ODE parameters ($\\partial L \/ \\partial \\bm{\\theta}$). Obtaining these gradients will enable optimization via gradient descent.\n\n\\subsubsection{Gradient with respect to the initial state}\nThe adjoint state is defined as $\\bm{a}(t) = \\partial L \/ \\partial \\bm{v}(t)$ and $\\bm{a}(T)$ can be computed directly as long as $L$ is differentiable while $\\bm{a}(0)$ is the gradient that we want. By the chain rule:\n\\begin{equation}\n    \\begin{split}\n        a_i(t) =& \\frac{\\partial L}{\\partial v_i(t)} = \\sum_j \\frac{\\partial L}{\\partial v_j(t+\\epsilon)} \\frac{\\partial v_j(t+\\epsilon)}{\\partial v_i(t)} \\\\\n        =& \\sum_j a_j(t+\\epsilon) \\frac{\\partial [v_j(t)+\\epsilon f_j(\\bm{v}(t), \\bm{\\theta})]}{\\partial v_i(t)} \\\\\n        =& a_i(t+\\epsilon) + \\epsilon \\sum_j a_j(t+\\epsilon) \\frac{\\partial  f_j(\\bm{v}(t), \\bm{\\theta})}{\\partial v_i(t)} .\n    \\end{split}\n\\end{equation}\nTherefore the adjoint state satisfies the differential equation:\n\\begin{equation}\n    \\dot{a}_i(t) = \\lim_{\\epsilon \\rightarrow 0} \\frac{a_i(t+\\epsilon)-a_i(t)}{\\epsilon} = -\\sum_j a_j(t) \\frac{\\partial f_j(\\bm{v}(t), \\bm{\\theta})}{\\partial v_i(t)}  .\n\\end{equation}\nMore compactly\n\\begin{equation}\n    \\dot{\\bm{a}}(t) = -C^T \\bm{a}(t) .\n\\end{equation}\nwhere $\\bm{a}$ is represented as a column vector and the matrix $C=\\frac{\\partial f(\\bm{v}(t), \\bm{\\theta})}{\\partial \\bm{v}(t)}$ is defined as $C_{ij} = \\frac{\\partial  f_i(\\bm{v}(t), \\bm{\\theta})}{\\partial v_j(t)}$. In other words, solving the above differential equation from $t=T$ to $t=0$ allows us to obtain $\\bm{a}(0) = \\partial L \/ \\partial \\bm{v}(0)$.\n\n\\subsubsection{Gradient with respect to ODE parameters}\nTo compute the gradient $\\partial L \/ \\partial \\bm{\\theta}$, we embed the parameters $\\bm\\theta$ into the state vector and consider a new ODE:\n\\begin{equation}\n    \\frac{\\text{d}}{\\text{d}t} \n    \\begin{pmatrix}\n    \\bm{v} \\\\\n    \\bm{\\theta}\n    \\end{pmatrix}\n    = \n    \\begin{pmatrix}\n    f(\\bm{v}, \\bm{\\theta}) \\\\\n    \\bm{0}\n    \\end{pmatrix} .\n\\end{equation}\nThis leads to a composite adjoint satisfying\n\\begin{equation}\n    \\frac{\\text{d}}{\\text{d}t} \n    \\begin{pmatrix}\n    \\bm{a}(t) \\\\\n    \\bm{a}_\\theta (t)\n    \\end{pmatrix}\n    = -\n    \\begin{pmatrix}\n    \\frac{\\partial f(\\bm{v}(t), \\bm{\\theta})}{\\partial \\bm{v}(t)} & \\frac{\\partial f(\\bm{v}(t), \\bm{\\theta})}{\\partial \\bm{\\theta}} \\\\\n    \\bm{0} & \\bm{0}\n    \\end{pmatrix} ^T\n    \\begin{pmatrix}\n    \\bm{a}(t) \\\\\n    \\bm{a}_\\theta (t)\n    \\end{pmatrix} .\n\\end{equation}\nDefine a matrix $D=\\frac{\\partial f(\\bm{v}(t), \\bm{\\theta})}{\\partial \\bm{\\theta}}$ as $D_{ij} = \\frac{\\partial  f_i(\\bm{v}(t), \\bm{\\theta})}{\\partial \\theta_j(t)}$, then we have\n\\begin{equation}\n    \\dot{\\bm{a}}_\\theta (t) = - D^T \\bm{a} (t) .\n\\end{equation}\nNotice that $\\bm{a}_\\theta (T) = 0$ since $L$ doesn't explicitly depend on $\\bm{\\theta}$.\n\n\\subsection{Adjoint method for the master equation}\nThe above derivation works for general ODEs and here we would like to adapt the results to a specific type of ODE, the quantum master equation.\n\n\\subsubsection{Gradient with respect to the initial state}\nIt's easier to take derivatives with respect to the initial density matrix by working with superoperators.\nConsider the linear transformation $\\ket{i}\\bra{j} \\rightarrow \\ket{i}\\otimes\\ket{j}$ that maps a density matrix $\\rho = \\sum \\rho_{ij}\\ket{i}\\bra{j}$ to a state vector $\\bar{\\rho} = \\sum \\rho_{ij}\\ket{i}\\otimes\\ket{j}$. By definition it's straightforward to check the following relation for left and right multiplication of an operator:\n\\begin{equation}\n    A\\rho B \\rightarrow (A\\otimes B^T) \\bar{\\rho} .\n\\end{equation}\nTherefore the master equation Eq.~(\\ref{eq:master}) is equivalent to\n\\begin{equation}\n    \\begin{split}\n        \\dot{\\bar{\\rho}} =& -i(H\\otimes I - I\\otimes H^T) \\bar{\\rho} + \\sum_{k=1}^{K} \\beta_k \\left(A_k\\otimes A_k^* \\vphantom{\\frac{1}{2}} \\right. \\\\\n        & \\left. - \\frac{1}{2} A_k^\\dagger A_k \\otimes I - \\frac{1}{2} I \\otimes (A_k^\\dagger A_k)^T \\right)\\bar{\\rho} \\\\\n        =& M \\bar{\\rho}.\n    \\end{split}\n\\end{equation}\nTo derive the adjoint equation in presence of complex numbers, we could simply separate the real part (label $x$) and imaginary part (label $y$). The master equation can be written as\n\\begin{equation}\\label{eq:super_real}\n    \\begin{split}\n        & \\frac{\\text{d}}{\\text{d} t}\n        \\begin{pmatrix}\n        \\bar{\\rho}_x \\\\\n        \\bar{\\rho}_y\n        \\end{pmatrix}\n        = (M_x + iM_y)(\\bar{\\rho}_x + i\\bar{\\rho}_y) \\\\\n        =&\n        \\begin{pmatrix}\n        M_x \\bar{\\rho}_x - M_y \\bar{\\rho}_y \\\\\n        M_x \\bar{\\rho}_y + M_y \\bar{\\rho}_x\n        \\end{pmatrix}\n        = \\begin{pmatrix}\n        M_x & -M_y  \\\\\n        M_y & M_x\n        \\end{pmatrix} \n        \\begin{pmatrix}\n        \\bar{\\rho}_x \\\\\n        \\bar{\\rho}_y\n        \\end{pmatrix}\\\\\n        =& \\bm{M} \n        \\begin{pmatrix}\n        \\bar{\\rho}_x \\\\\n        \\bar{\\rho}_y\n        \\end{pmatrix} .\n    \\end{split}\n\\end{equation}\nNow we could calculate the $C$ matrix by\n\\begin{equation}\n    C_{ij} = \\frac{\\partial [\\bm{M}\\rho(t)]_i}{\\partial \\rho_j(t)} = \\frac{\\partial \\sum_{k}M_{ik}\\rho_k(t)}{\\partial \\rho_j(t)} = M_{ij}\n\\end{equation}\nwhich gives $C = \\bm{M}$. Therefore the adjoint equation is given by\n\\begin{equation}\n    \\frac{\\text{d}}{\\text{d} t}\n    \\begin{pmatrix}\n    \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n    \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n    \\end{pmatrix}\n    = - \n    \\begin{pmatrix}\n    M_x^T & M_y^T \\\\\n    -M_y^T & M_x^T\n    \\end{pmatrix}\n    \\begin{pmatrix}\n    \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n    \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n    \\end{pmatrix} .\n\\end{equation}\nThis could be simplified by introducing the complex adjoint \n\\begin{equation}\n    \\bar{a} = \\frac{\\partial L}{\\partial \\bar{\\rho}_x} + i \\frac{\\partial L}{\\partial \\bar{\\rho}_y} = \\sum a_{ij} \\ket{i}\\otimes \\ket{j}, \\quad a_{ij} = \\frac{\\partial L}{\\partial \\rho_{ij}^x} + i \\frac{\\partial L}{\\partial \\rho_{ij}^y}\n\\end{equation}\nsuch that\n\\begin{equation}\n    \\dot{\\bar{a}} = -(M_x^T - i M_y^T) \\left( \\frac{\\partial L}{\\partial \\bar{\\rho}_x} + i \\frac{\\partial L}{\\partial \\bar{\\rho}_y} \\right) = -M^\\dagger \\bar{a} .\n\\end{equation}\nMore explicitly\n\\begin{equation}\n    \\begin{split}\n        \\dot{\\bar{a}} =& -M^\\dagger \\bar{a} = -i(H\\otimes I - I\\otimes H^T) \\bar{a} - \\sum_{k=1}^{K} \\beta_k \\left(A_k^\\dagger \\otimes A_k^T \\vphantom{\\frac{1}{2}} \\right. \\\\\n        & \\left. - \\frac{1}{2} A_k^\\dagger A_k \\otimes I - \\frac{1}{2} I \\otimes (A_k^\\dagger A_k)^T \\right)\\bar{a} .\n    \\end{split}\n\\end{equation}\nNow we inverse the mapping and define the adjoint matrix\n\\begin{equation}\n    \\bar{a} = \\sum a_{ij} \\ket{i}\\otimes \\ket{j} \\rightarrow a = \\sum a_{ij} \\ket{i} \\bra{j}\n\\end{equation}\nand the adjoint equation has a similar form as the original master equation:\n\\begin{equation}\\label{eq:state_adjoint}\n    \\dot{a}=-i[H, a] - \\sum_{k=1}^{K} \\beta_k \\left( A_k^\\dagger a A_{k} - \\frac{1}{2} \\left \\{ A_{k}^{\\dagger}A_{k}, a \\right\\}\\right) .\n\\end{equation}\nStarting from $a(T)$ and solving the adjoint equation backward in time to $t=0$ gives $a(0)$, which is the gradient with respect to the initial density matrix.\n\n\\subsubsection{Gradient with respect to system parameters}\nGoing back to the superoperator representation Eq.~(\\ref{eq:super_real}) and taking derivative with respect to some ODE parameter $\\theta$:\n\\begin{equation}\n    \\begin{split}\n        & \\dot{a}_\\theta (t) \\\\\n        =& - \\left[ \\partial_\\theta\\bm{M} \n        \\begin{pmatrix}\n        \\bar{\\rho}_x \\\\\n        \\bar{\\rho}_y\n        \\end{pmatrix} \\right]^T \n        \\begin{pmatrix}\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n        \\end{pmatrix}\n        = -(\\bar{\\rho}_x^T, \\bar{\\rho}_y^T) \\partial_\\theta\\bm{M}^T\n        \\begin{pmatrix}\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n        \\end{pmatrix} \\\\\n        =& -(\\bar{\\rho}_x^T, \\bar{\\rho}_y^T)\n        \\begin{pmatrix}\n        \\partial_\\theta M_x^T & \\partial_\\theta M_y^T \\\\\n        -\\partial_\\theta M_y^T & \\partial_\\theta M_x^T\n        \\end{pmatrix} \n        \\begin{pmatrix}\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n        \\end{pmatrix} \\\\\n        =& -(\\bar{\\rho}_x^T \\partial_\\theta M_x^T - \\bar{\\rho}_y^T \\partial_\\theta M_y^T, \\bar{\\rho}_x^T \\partial_\\theta M_y^T + \\bar{\\rho}_y^T \\partial_\\theta M_x^T)\n        \\begin{pmatrix}\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_x} \\\\\n        \\frac{\\partial L}{\\partial \\bar{\\rho}_y}\n        \\end{pmatrix} \\\\\n        =& - \\left[ (\\partial_\\theta M \\bar{\\rho})_x^T \\frac{\\partial L}{\\partial \\bar{\\rho}_x} + (\\partial_\\theta M \\bar{\\rho})_y^T \\frac{\\partial L}{\\partial \\bar{\\rho}_y} \\right] .\n    \\end{split}\n\\end{equation}\nDefine\n\\begin{equation}\n    \\bar{\\Delta}_\\theta = \\partial_\\theta M \\bar{\\rho} = \\sum \\Delta_{ij} \\ket{i} \\otimes \\ket{j} \\rightarrow \\Delta_\\theta = \\sum \\Delta_{ij} \\ket{i} \\bra{j}\n\\end{equation}\nthen the gradient could be simplified as\n\\begin{equation}\\label{eq:adjoint_theta}\n    \\dot{a}_\\theta (t) = -\\sum_{ij}(\\Delta_{ij}^x a_{ij}^x + \\Delta_{ij}^y a_{ij}^y) = -\\text{Re} \\tr{\\Delta_\\theta a^\\dagger(t)} .\n\\end{equation}\nFrom the master equation, it's not hard to see that\n\\begin{equation}\n    \\begin{split}\n        \\Delta_{\\alpha_j} =& -i[H_j, \\rho(t)] \\\\\n        \\Delta_{\\beta_k} =& A_k \\rho(t) A_{k}^{\\dagger} - \\frac{1}{2} \\left \\{ A_{k}^{\\dagger}A_{k}, \\rho(t) \\right\\} .\n    \\end{split}\n\\end{equation}\nTherefore after solving for $\\rho(t)$ from Eq.~(\\ref{eq:master}) and $a(t)$ from Eq.~(\\ref{eq:state_adjoint}), we could solve $a_\\theta(t)$ by integrating Eq.~(\\ref{eq:adjoint_theta}) backward in time starting from $a_{\\bm{\\theta}}(T)=0$.\n\n\n\\section{Average fidelity}\nWe could evaluate the integration over Bloch sphere in the definition of average Fidelity (main text Eq.(\\ref{eq:avg_F})) which leads to a simpler expression to work with\n\\begin{equation}\\label{eq:F_avg_eval}\n    \\begin{split}\n        \\bar{F} (t) =& \\tr{\\left(\\frac{1}{3} \\hat{\\rho}_{00} + \\frac{1}{6} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{00}(t)} \\\\\n        &+ \\tr{\\left(\\frac{1}{6} \\hat{\\rho}_{00} + \\frac{1}{3} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{11}(t)} \\\\\n        &+ \\text{Re} \\left\\{ \\tr{\\frac{1}{3} \\hat{\\rho}_{01} \\hat{\\rho}_{10}(t)} \\right\\} ,\n    \\end{split}\n\\end{equation}\nFor a single bosonic mode with $\\hat{H}=0$ and only photon loss error, the average fidelity for subspace of $\\ket{0}$ and $\\ket{1}$ is\n\\begin{equation}\n    \\bar{F} (t) = \\frac{1}{6} \\left( e^{-\\kappa t} + 2e^{-\\kappa t \/2} + 3 \\right)\n\\end{equation}\nwhere $\\kappa$ is the loss rate. This is our definition of break-even fidelity throughout the paper.\n\nOther fidelity definitions are also applicable as long as the gradients are computable. For example, we could learn AQEC with the entanglement fidelity~\\cite{Albert2018}\n\\begin{equation}\n    \\bar{F}(t) = \\frac{1}{4} \\tr{\\hat{\\rho}_{00} \\hat{\\rho}_{00}(t) + \\hat{\\rho}_{11} \\hat{\\rho}_{11}(t) + \\hat{\\rho}_{01} \\hat{\\rho}_{10}(t) + \\hat{\\rho}_{10} \\hat{\\rho}_{01}(t)} .\n\\end{equation}\nFrom our experience, we didn't observe any substantial differences in terms of training speed and optimization results when using entanglement fidelity compared to the average fidelity.\n\n\\subsection{Modified average fidelity}\nIn practice, we find out that there is a simple modification of the average fidelity that helps avoiding certain local minima during the training.\nInstead of learning an identity map on the Bloch sphere, we could aim at preserving the Bloch sphere up to an arbitrary rotation in the logical subspace.\nThis modification extends the set of Hamiltonians that protects a given QEC code, which could potentially accelerate the \\texttt{AutoQEC} searching.\n\nAnalytically deriving the modified average fidelity turns out to be challenging for general rotations on the Bloch sphere. We therefore restrict the allowed rotations to only along the $Z$ axis:\n\\begin{equation}\n    \\begin{split}\n        \\hat{U} \\ket{\\psi_0} =& \\ket{\\psi_0} \\\\\n        \\hat{U} \\ket{\\psi_1} =& e^{i\\varphi} \\ket{\\psi_1} .\n    \\end{split}\n\\end{equation}\nThe modified average fidelity becomes\n\\begin{equation}\\label{eq:modified_avg_F}\n    \\begin{split}\n        \\bar{F} (t) =& \\max_{\\hat{U}} \\tr{\\hat{U} \\left(\\frac{1}{3} \\hat{\\rho}_{00} + \\frac{1}{6} \\hat{\\rho}_{11}\\right) \\hat{U}^\\dagger \\hat{\\rho}_{00}(t)} \\\\\n        &\\qquad + \\tr{\\hat{U} \\left(\\frac{1}{6} \\hat{\\rho}_{00} + \\frac{1}{3} \\hat{\\rho}_{11}\\right) \\hat{U}^\\dagger \\hat{\\rho}_{11}(t)} \\\\\n        &\\qquad + \\text{Re} \\left\\{ \\tr{\\frac{1}{3} \\hat{U} \\hat{\\rho}_{01} \\hat{U}^\\dagger \\hat{\\rho}_{10}(t)} \\right\\} \\\\\n        =& \\max_{\\varphi} \\tr{\\left(\\frac{1}{3} \\hat{\\rho}_{00} + \\frac{1}{6} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{00}(t)} \\\\\n        &\\qquad + \\tr{\\left(\\frac{1}{6} \\hat{\\rho}_{00} + \\frac{1}{3} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{11}(t)} \\\\\n        &\\qquad + \\text{Re} \\left\\{ \\tr{\\frac{1}{3} e^{-i\\varphi} \\hat{\\rho}_{01} \\hat{\\rho}_{10}(t)} \\right\\} \\\\\n        =& \\tr{\\left(\\frac{1}{3} \\hat{\\rho}_{00} + \\frac{1}{6} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{00}(t)} \\\\\n        &+ \\tr{\\left(\\frac{1}{6} \\hat{\\rho}_{00} + \\frac{1}{3} \\hat{\\rho}_{11}\\right) \\hat{\\rho}_{11}(t)} \\\\\n        &+ \\left | \\tr{\\frac{1}{3} \\hat{\\rho}_{01} \\hat{\\rho}_{10}(t)} \\right | ,\n    \\end{split}\n\\end{equation}\nwhere the only change compared to Eq.~(\\ref{eq:F_avg_eval}) is to replace the real part of $\\tr{\\hat{\\rho}_{01} \\hat{\\rho}_{10}(t)}$ with its absolute value.\nThroughout this paper, we use this modified average fidelity (Eq.~(\\ref{eq:modified_avg_F})) as the objective function for \\texttt{AutoQEC}.\n\n\n\\section{Derivation of the $\\sqrt{3}$ code}\\label{appendix:b}\nWith $d=2$ Hamiltonian, \\texttt{AutoQEC} discovered an error correcting code which we were not able to find in literature. This simple ``$\\sqrt{3}$\" code warrants further investigation, and here we derive it analytically based on the Hamiltonian distance constraints and QEC properties. Notice that the AQEC Hamiltonian for a given code is not unique -- we therefore make some assumptions about the Hamiltonian structure to simplify the derivation.\n\n\\subsection{General results}\nConsider the problem of correcting a single photon loss error with a bosonic mode. The logical states are $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$. For simplicity, we assume the error states $\\ket{\\psi_2} \\propto \\hat a\\ket{\\psi_0}$ and $\\ket{\\psi_3} \\propto \\hat a\\ket{\\psi_1}$ are also mutually orthogonal to both logical states. Therefore $\\{\\ket{\\psi_0}, \\ket{\\psi_1}, \\ket{\\psi_2}, \\ket{\\psi_3}\\}$ forms the basis for a 4-dimensional subspace $\\mathcal{H}_1$. We choose the Hilbert space cutoff $\\ket{N}$ as the highest Fock level that has a non-zero overlap with either $\\ket{\\psi_0}$ or $\\ket{\\psi_1}$ (total Hilbert space dimension $N+1$). Now the Hilbert space is decomposed into $\\mathcal{H}_1$ and its orthogonal complement $\\mathcal{H}_2$ ($\\mathcal{H} = \\mathcal{H}_1 \\oplus \\mathcal{H}_2$) where a set of orthogonal basis for $\\mathcal{H}_2$ is $\\{\\ket{\\psi_4},...,\\ket{\\psi_N}\\}$.\n\nTo correct the single photon loss error autonomously, the Hamiltonian should include the following terms\n\\begin{equation}\\label{eq:QEC_1}\n    \\hat H = (\\ket{\\psi_0}\\bra{\\psi_2} + \\ket{\\psi_1}\\bra{\\psi_3}) \\otimes \\ket{e}\\bra{g} + \\mathrm{h.c.}\n\\end{equation}\nwhich basically maps the error states to the correct logical states and excited the qubit: $\\ket{\\psi_2,g} \\leftrightarrow \\ket{\\psi_0,e}$ and $\\ket{\\psi_3,g} \\leftrightarrow \\ket{\\psi_1,e}$. After relaxation of the ancilla qubit, the error states $\\ket{\\psi_2}$ and $\\ket{\\psi_3}$ are mapped back to the logical states $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$ while maintaining the relative phase between them due to the identical Rabi rate in the Hamiltonian.\nAdiabatically eliminating the ancilla qubit results in an effective dissipator $D[\\ket{\\psi_0}\\bra{\\psi_2} + \\ket{\\psi_1}\\bra{\\psi_3}]$, which provides an alternative way of understanding the reduced dynamics for the bosonic mode.\n\nThe Hamiltonian Eq.~(\\ref{eq:QEC_1}) is conceptually simple, but may be difficult to generate in experiment since it can be highly non-local in the Fock basis. In the case of the smallest binomial code $\\ket{\\psi_0} = \\frac{1}{\\sqrt{2}} (\\ket{0} + \\ket{4})$ and $\\ket{\\psi_1} = \\ket{2}$, realizing Eq.~(\\ref{eq:QEC_1}) would require a coupling term $\\ket{3,g}\\bra{0,e}$ which does not occur naturally~\\cite{Hu2019,Gertler2021}. Going to a higher order binomial code is even worse since it requires even more non-local interaction between Fock states.\n\nThis is where states in the orthogonal subspace $\\mathcal{H}_2$ could contribute. The basic idea is that those states won't change the QEC behavior, but their proper combinations could cancel certain non-local interactions in Eq.~(\\ref{eq:QEC_1}) and make the total Hamiltonian easier to implement. For this purpose, the general form of the QEC Hamiltonian is (we ignore all $\\ket{g}\\bra{g}$ and $\\ket{e}\\bra{e}$ terms since those won't solve the locality problem anyway)\n\\begin{equation}\\label{eq:QEC_full}\n    \\begin{split}\n        \\hat{H} =& \\tilde{H}^\\dagger \\otimes \\ket{e}\\bra{g} + \\tilde{H} \\otimes \\ket{g}\\bra{e} \\\\\n        \\tilde{H} =& \\ket{\\psi_2}\\bra{\\psi_0} + \\ket{\\psi_3}\\bra{\\psi_1} + \\sum_{i,j}^{N} \\beta_{ij} \\ket{\\psi_i} \\bra{\\psi_j}.\n    \\end{split}\n\\end{equation}\nWe impose a number of constraints on the coefficients $\\beta_{ij}$ such that the summation part of $\\tilde{H}$ does not modify the error correction dynamics. In particular, we require that $\\beta_{ij}=0$ for $i<4$ and $j<2$. The requirement $j<2$ removes overlap with the states $\\ket{\\psi_{0\\sim 1},e}$ -- we want only the first part of $\\tilde{H}$ to perform this correction function. Similarly, removing $i<4$ prevents overlap with states $\\ket{\\psi_{0\\sim 3},g}$ which is important for preventing the summation part of Hamiltonian from causing states to transition into the code and error subspaces.\n\nHaving set these conditions on $\\tilde{H}$, we now quantify the notion of locality for the Hamiltonian. We define the distance of a Hamiltonian $H$ to be $d$ if $\\tilde{H}_{mn}\\equiv\\mele{m}{\\tilde{H}}{n} = 0, \\forall |m-n|>d, 0\\leq m,n \\leq N$. From our numerical search (Fig.~\\ref{fig2}), we found that $d=1$ Hamiltonians generate only trivial error correcting codes. Therefore we will consider the $d=2$ case, which seems to be the minimal distance required for QEC exceeding break-even. More explicitly, a $d=2$ Hamiltonian satisfies\n\\begin{equation}\\label{eq:locality_constraints}\n    \\begin{pmatrix}\n    0 & \\tilde{H}_{03} & \\tilde{H}_{04} & ... & \\tilde{H}_{0N} \\\\\n    \\tilde{H}_{30} & 0 & \\tilde{H}_{14} & ... & \\tilde{H}_{1N} \\\\\n    \\tilde{H}_{40} & \\tilde{H}_{41} & 0 & ... & \\tilde{H}_{2N} \\\\\n    \\vdots & & & & \\tilde{H}_{N-3,N} \\\\\n    \\tilde{H}_{N0} & \\tilde{H}_{N1} & ... & \\tilde{H}_{N,N-3} & 0\n    \\end{pmatrix} =\\mathbf{0}.\n\\end{equation}\nThe goal here is to find $\\tilde{H}$, in other words we will solve for the coefficients $\\beta_{ij}$ as well as the logical states while satisfying these locality constraints.\n\n\\subsection{Example of the $\\sqrt{3}$ code}\nWe demonstrate how to solve this problem with a concrete example. Numerically the $\\sqrt{3}$ code we found with the $d=2$ Hamiltonian had logical states of the form\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_0} &= a_0 \\ket{0} + a_3 \\ket{3} \\\\\n        \\ket{\\psi_1} &= a_1 \\ket{1} + a_4 \\ket{4} + a_6 \\ket{6} .\n    \\end{split} \n\\end{equation}\nSince the two logical states don't share any Fock basis, we can always make all coefficients $a_0,a_3,a_1,a_4,a_6$ real by doing the basis transformation $\\ket{n} \\rightarrow e^{i\\theta_n} \\ket{n}$. The error states are\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_2} &= \\ket{2} \\propto \\hat a\\ket{\\psi_0} \\\\\n        \\ket{\\psi_3} &= \\mathcal{N}_1 ( a_1 \\ket{0} + 2 a_4 \\ket{3} + \\sqrt{6} a_6 \\ket{5} ) \\propto \\hat a\\ket{\\psi_1} .\n    \\end{split} \n\\end{equation}\nNotice that if Eq.~(\\ref{eq:locality_constraints}) does have a solution, the solution always exists no matter how we choose the basis for the orthogonal subspace $\\mathcal{H}_2$. In other words, we could always represent the new basis as linear combinations of the old basis and that together with the old solution $\\beta_{ij}$ gives the new solution $\\beta_{ij}'$. Therefore here we have complete freedom to select the basis $\\{ \\ket{\\psi_4},\\ket{\\psi_5},\\ket{\\psi_6} \\}$ for $\\mathcal{H}_2$ and for convenience of further analysis we make the following choice (notation $\\psi_i(n) = \\inp{n}{\\psi_i}$):\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_4} & = \\psi_4(1) \\ket{1} + \\psi_4(4) \\ket{4} + \\psi_4(6) \\ket{6} \\\\\n        \\ket{\\psi_5} & = \\psi_5(1) \\ket{1} + \\psi_5(4) \\ket{4} + \\psi_5(6) \\ket{6} \\\\\n        \\ket{\\psi_6} & = \\psi_6(0) \\ket{0} + \\psi_6(3) \\ket{3} + \\psi_6(5) \\ket{5} .\n    \\end{split} \n\\end{equation}\nWe can make all $\\psi_i(n)$ to be real here, which leads to all $\\beta_{ij}$ also being real.\nWith this basis choice, many constraints in Eq.~(\\ref{eq:locality_constraints}) can be easily satisfied either automatically or by setting certain $\\beta_{ij}=0$. More specifically, for any $|m-n|>2$ such that $\\mele{m}{(\\ket{\\psi_2}\\bra{\\psi_0} + \\ket{\\psi_3}\\bra{\\psi_1})}{n} = 0$, there are two different cases:\n\\begin{enumerate}\n    \\item $\\mele{m}{(\\ket{\\psi_i}\\bra{\\psi_j})}{n}=0, \\forall i,j$: in this case $\\tilde{H}_{mn}=0$ is already satisfied.\n    \\item there exists $i,j$ such that $\\mele{m}{(\\ket{\\psi_i}\\bra{\\psi_j})}{n}\\neq 0$: in this case we just set $\\beta_{ij}=0$.\n\\end{enumerate}\nTherefore the only non-trivial constraints from Eq.~(\\ref{eq:locality_constraints}) are those with $\\mele{m}{(\\ket{\\psi_2}\\bra{\\psi_0} + \\ket{\\psi_3}\\bra{\\psi_1})}{n} \\neq 0$ which are $\\tilde{H}_{04},\\tilde{H}_{06},\\tilde{H}_{36},\\tilde{H}_{51}$. It is easy to see that the only terms in Eq.~(\\ref{eq:QEC_full}) that will contribute to these matrix elements are $\\ket{\\psi_6}\\bra{\\psi_4}$ and $\\ket{\\psi_6}\\bra{\\psi_5}$. With these analysis, the ansatz Hamiltonian Eq.~(\\ref{eq:QEC_full}) can greatly simplified to the following:\n\\begin{equation}\\label{eq:H1}\n    \\tilde{H} = \\ket{\\psi_2}\\bra{\\psi_0} + \\ket{\\psi_3}\\bra{\\psi_1} + \\beta_1 \\ket{\\psi_6} \\bra{\\psi_4} + \\beta_2 \\ket{\\psi_6} \\bra{\\psi_5} ,\n\\end{equation}\nwhere the two free parameters $\\beta_1$ and $\\beta_2$ satisfy a set of linear equations\n\\begin{subequations}\n    \\begin{align}\n        \\tilde{H}_{04} &= \\psi_3(0) \\psi_1(4) + \\beta_1 \\psi_6(0) \\psi_4(4) + \\beta_2 \\psi_6(0) \\psi_5(4) = 0 \\label{eq:a} \\\\\n        \\tilde{H}_{06} &= \\psi_3(0) \\psi_1(6) + \\beta_1 \\psi_6(0) \\psi_4(6) + \\beta_2 \\psi_6(0) \\psi_5(6) = 0 \\label{eq:b} \\\\\n        \\tilde{H}_{36} &= \\psi_3(3) \\psi_1(6) + \\beta_1 \\psi_6(3) \\psi_4(6) + \\beta_2 \\psi_6(3) \\psi_5(6)  = 0 \\label{eq:c} \\\\\n        \\tilde{H}_{51} &= \\psi_3(5) \\psi_1(1) + \\beta_1 \\psi_6(5) \\psi_4(1) + \\beta_2 \\psi_6(5) \\psi_5(1) = 0 \\label{eq:d} .\n    \\end{align}\n\\end{subequations}\nThe crucial observation here is that the number of equations 4 is larger than the number of parameters 2, which means the coefficients must be linearly dependent. Since these coefficients are essentially functions of $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$, this eventually provides the extra constraints for determining the logical states. Here there should be $4-2=2$ constraints in total.\n\nBelow we will show in details how to obtain the two constraints and eventually the two logical states. Comparing Eq.~(\\ref{eq:b}) and Eq.~(\\ref{eq:c}), it's easy to see that the first constraint is\n\\begin{equation}\\label{eq:constraint1}\n    \\frac{\\psi_3(0)}{\\psi_3(3)} = \\frac{\\psi_6(0)}{\\psi_6(3)} .\n\\end{equation}\nTo get the second constraint, let us multiply Eq.~(\\ref{eq:a}) with $\\psi_1(4)$, multiply Eq.~(\\ref{eq:b}) with $\\psi_1(6)$, and then add them together:\n\\begin{equation}\n    \\begin{split}\n        & \\psi_3(0) ( [\\psi_1(4)]^2 + [\\psi_1(6)]^2 )  \\\\\n        & +\\beta_1 \\psi_6(0) [ \\psi_4(4) \\psi_1(4) + \\psi_4(6) \\psi_1(6) ] \\\\\n        & + \\beta_2 \\psi_6(0) [ \\psi_5(4) \\psi_1(4) + \\psi_5(6) \\psi_1(6) ] = 0 .\n    \\end{split}\n\\end{equation}\nUsing the fact that $\\ket{\\psi_1}$ is normalized and orthogonal to both $\\ket{\\psi_4}$ and $\\ket{\\psi_5}$, we have\n\\begin{equation}\n    \\begin{split}\n        & \\psi_3(0) ( 1 - [\\psi_1(1)]^2) + \\beta_1 \\psi_6(0) [ -\\psi_4(1) \\psi_1(1) ] \\\\\n        & + \\beta_2 \\psi_6(0) [ -\\psi_5(1) \\psi_1(1) ] = 0 \\\\\n        \\Rightarrow &  - \\psi_3(0) \\frac{1 - [\\psi_1(1)]^2}{\\psi_1(1)} + \\beta_1 \\psi_6(0) \\psi_4(1) \\\\\n        & + \\beta_2 \\psi_6(0) \\psi_5(1) = 0 .\n    \\end{split}\n\\end{equation}\nCompare this with Eq.~(\\ref{eq:d}), we immediately obtain the second constraint:\n\\begin{equation}\\label{eq:constraint2}\n    \\begin{split}\n        & - \\psi_3(0) \\frac{1 - [\\psi_1(1)]^2}{\\psi_1(1) \\psi_6(0)} = \\frac{\\psi_3(5) \\psi_1(1)}{\\psi_6(5)} \\\\\n        \\Rightarrow & \\psi_3(0) \\psi_6(5) (1 - [\\psi_1(1)]^2) + \\psi_3(5) \\psi_6(0) [\\psi_1(1)]^2 = 0 .\n    \\end{split}\n\\end{equation}\nLet us explicitly list all the relevant states here\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_0} &= a_0 \\ket{0} + a_3 \\ket{3} \\\\\n        \\ket{\\psi_1} &= a_1 \\ket{1} + a_4 \\ket{4} + a_6 \\ket{6} \\\\\n        \\ket{\\psi_3} &= \\mathcal{N}_1 ( a_1 \\ket{0} + 2 a_4 \\ket{3} + \\sqrt{6} a_6 \\ket{5} ) \\\\\n        \\ket{\\psi_6} &= \\mathcal{N}_2 ( a_1 \\ket{0} + 2 a_4 \\ket{3} + \\beta \\ket{5} ) ,\n    \\end{split} \n\\end{equation}\nwhere we have applied Eq.~(\\ref{eq:constraint1}) for $\\ket{\\psi_6}$ and $\\beta$ is another parameter. Combining the QEC criteria and Eq.~(\\ref{eq:constraint2}), we have\n\\begin{equation}\n    \\begin{split}\n        & a_0^2 + a_3^2 = 1 \\\\\n        & a_1^2 + a_4^2 + a_6^2 = 1 \\\\\n        & a_0 a_1 + 2 a_3 a_4 = 0 \\\\\n        & 3 a_3^2 = a_1^2 + 4 a_4^2 + 6 a_6^2 \\\\\n        & a_1^2 + 4 a_4^2 + \\sqrt{6}\\beta a_6 = 0 \\\\\n        & \\beta (1 - a_1^2) + \\sqrt{6} a_6 a_1^2 = 0 .\n    \\end{split}\n\\end{equation}\nWe have 6 equations and 6 parameters in total, and the solution is (there is some freedom to choose the signs which again is just a trivial basis transformation)\n\\begin{equation}\n    \\begin{split}\n        & a_0 = \\sqrt{1-\\frac{1}{\\sqrt{3}}}, a_3 = \\frac{1}{\\sqrt[4]{3}}, a_1 = \\sqrt{\\frac{2(6-\\sqrt{3})}{\\sqrt{3}+9}} \\\\\n        & a_4 = -\\sqrt{\\frac{(\\sqrt{3}-1)(6-\\sqrt{3})}{2(\\sqrt{3}+9)}}, a_6 = \\sqrt{\\frac{3-\\sqrt{3}}{2(\\sqrt{3}+9)}} .\n    \\end{split}\n\\end{equation}\nTherefore the logical states of the $\\sqrt{3}$ code are\n\\begin{equation}\\label{eq:logical_states}\n    \\begin{split}\n        \\ket{\\psi_0} =& \\sqrt{1-\\frac{1}{\\sqrt{3}}} \\ket{0} + \\frac{1}{\\sqrt[4]{3}} \\ket{3} \\\\\n        \\ket{\\psi_1} =& \\sqrt{\\frac{2(6-\\sqrt{3})}{\\sqrt{3}+9}} \\ket{1} - \\sqrt{\\frac{(\\sqrt{3}-1)(6-\\sqrt{3})}{2(\\sqrt{3}+9)}} \\ket{4} \\\\\n        & + \\sqrt{\\frac{3-\\sqrt{3}}{2(\\sqrt{3}+9)}} \\ket{6} .\n    \\end{split}\n\\end{equation}\nNotice that the average number of photons in the codewords is $3|a_3|^2 = \\sqrt{3}$.\n\nNow we could complete all basis states and the Hamiltonian Eq.~(\\ref{eq:H1}). The basis of $\\mathcal{H}_2$:\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_6} &= \\mathcal{N}_2 ( a_1 \\ket{0} + 2 a_4 \\ket{3} + \\beta \\ket{5} ) , \\quad \\beta = -\\frac{a_1^2+4a_4^2}{\\sqrt{6} a_6} \\\\\n        \\ket{\\psi_4} &= \\mathcal{N}_3 ( a_4 \\ket{1} - a_1 \\ket{4} ) \\\\\n        \\ket{\\psi_5} &= \\mathcal{N}_4 ( a_1 \\ket{1} + a_4 \\ket{4} + \\beta' \\ket{6} ) , \\quad \\beta' = -\\frac{a_1^2+a_4^2}{a_6} \n    \\end{split} \n\\end{equation}\nand the Hamiltonian parameters:\n\\begin{equation}\n    \\beta_2 = -\\frac{\\mathcal{N}_1 a_6}{\\mathcal{N}_2 \\mathcal{N}_4 \\beta'} \\qquad \\beta_1 = \\frac{\\mathcal{N}_1 a_4 (1-a_6\/\\beta')}{\\mathcal{N}_2 \\mathcal{N}_3 a_1} .\n\\end{equation}\n\nThere are some extra complexities in constructing the AQEC Hamiltonian and we actually need to keep more terms from the summation in Eq.~(\\ref{eq:QEC_full}) rather than just the $\\beta_1$ and $\\beta_2$ terms in Eq.~(\\ref{eq:H1}). To understand why this is required, let us study a simpler problem of stabilizing $\\ket{\\psi} = \\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{2})$ under photon loss error.\nEven though the Hamiltonian $\\hat{H} = (\\ket{0,e}+\\ket{2,e})\\bra{1,g} + \\text{h.c.}$ corrects the error after a single photon loss, it doesn't actually lead to state stabilization.\nThe reason is that when no photon loss happens, the state evolves within the subspace $\\{\\ket{0}, \\ket{2}\\}$ under the non-Hermitian Hamiltonian $\\hat{H}'=-i\\kappa \\hat{a}^\\dagger \\hat{a}\/2$ and eventually becomes $\\ket{0}$. This is because non-detection of a photon still provides us information about the state causing us to update it in a way that skews towards a lower number of photons. State-stabilization must undo this effect.\nWe protect $\\ket{\\psi}$ against $\\hat{H}'$ by engineering a large detuning for $\\ket{\\psi}$ within the subspace $\\{\\ket{0}, \\ket{2}\\}$. For example, adding extra terms such as $\\Omega \\ket{\\psi} \\bra{\\psi}$ or $\\Omega \\ket{0}\\bra{2}+\\text{h.c.}$ to $\\hat{H}$ will stabilize $\\ket{\\psi}$. This new interaction can be seen as rapidly repopulating the $\\ket{2}$ component of the wavevector as it decays through non-detection of photons.\n\nSimilarly, Eq.~(\\ref{eq:H1}) only protects the logical states against single photon loss error, but not the non-unitary dynamics under $\\hat{H}'$.\nFortunately, keeping a few extra terms from the summation in Eq.~(\\ref{eq:QEC_full}) is sufficient to generate the large detuning without changing the Hamiltonian distance as well as the above derivation.\nThe choices are not unique and one option is to add $\\ket{\\psi_4} \\bra{\\psi_2}$ as well as $(a_6 \\ket{4} - a_4 \\ket{6}) \\bra{5}$ in $\\tilde{H}$, which produces similar results compared to the discovered code in Fig.~\\ref{fig2}(c)i.\nOn the other hand, all these complications in constructing a proper AQEC Hamiltonian are automatically taken care of by \\texttt{AutoQEC} through  numerical optimization of the average fidelity.\n\n\n\n\\section{Minimizing the emission bandwidth $\\mathcal{B}$}\nHere we prove the claim in the main text that for a harmonic oscillator coupled to a three-level qubit with Hamiltonian $\\hat{H}_{ab}$ in Eq.~(\\ref{eq:H_ab}), the emission bandwidth $\\mathcal{B}$ is minimized when $g_1^2\/\\Delta_1 \\approx g_2^2\/\\Delta_2$.\n\n\\begin{proof}\nThe Hamiltonian can be written in the subspace of $\\{\\ket{n+2,g},\\ket{n+1,e},\\ket{n,f}\\}$ as a matrix\n\\begin{equation}\n    \\begin{pmatrix}\n    0 & \\sqrt{n+2} g_1 & 0 \\\\\n    \\sqrt{n+2} g_1 & \\Delta_1 & \\sqrt{n+1} g_2 \\\\\n    0 & \\sqrt{n+1} g_2 & \\Delta_2\n    \\end{pmatrix} \n\\end{equation}\nand the eigenvalues satisfy\n\\begin{equation}\n    \\begin{split}\n        & \\lambda^3 - (\\Delta_1+\\Delta_2) \\lambda^2 + \\left[\\Delta_1 \\Delta_2 - (n+2) g_1^2 - (n+1) g_2^2 \\right] \\lambda \\\\\n        & + (n+2) g_1^2 \\Delta_2 = 0 .\n    \\end{split}\n\\end{equation}\nIn the dispersive regime $\\Delta_{1,2} \\gg g_{1,2}$, the eigenvalues can be expanded perturbatively as\n\\begin{equation}\n    \\begin{split}\n        \\lambda =& \\lambda_0 + \\lambda_1 + \\lambda_2 + \\mathcal{O}\\left(\\frac{g^3}{\\Delta^3}\\right) g \\\\\n        \\lambda_1 =& \\mathcal{O}\\left(\\frac{g}{\\Delta}\\right) g, \\quad \\lambda_2 = \\mathcal{O}\\left(\\frac{g^2}{\\Delta^2}\\right) g .\n    \\end{split}\n\\end{equation}\nFor dressed eigenstates $\\ket{\\widetilde{n+2,g}}$, $\\lambda_0 = 0$ and\n\\begin{equation}\n    \\begin{split}\n        & \\left[\\Delta_1 \\Delta_2 - (n+2) g_1^2 - (n+1) g_2^2 \\right] \\lambda_1 + (n+2) g_1^2 \\Delta_2 = 0 \\\\ \n        \\Rightarrow & \\quad \\lambda_1 = -\\frac{(n+2) g_1^2}{\\Delta_1}\n    \\end{split}\n\\end{equation}\nwhich agrees with the dispersive coupling Hamiltonian and no level nonlinearity shows up at this order. To the next order,\n\\begin{equation}\n    \\begin{split}\n        & \\left[\\Delta_1 \\Delta_2 - (n+2) g_1^2 - (n+1) g_2^2 \\right] (\\lambda_1 + \\lambda_2) \\\\\n        & - (\\Delta_1+\\Delta_2) \\lambda_1^2  + (n+2) g_1^2 \\Delta_2 = 0\n    \\end{split}\n\\end{equation}\nwhich gives\n\\begin{equation}\n    \\lambda_2 = \\frac{(n+2)g_1^2}{\\Delta_1^2} \\left[ (n+2) \\frac{g_1^2}{\\Delta_1} - (n+1) \\frac{g_2^2}{\\Delta_2} \\right] .\n\\end{equation}\nNotice that in general $\\lambda_2$ will induce nonlinearity for the dressed states $\\ket{\\widetilde{n,g}}$ since it depends on $n^2$. However, when $g_1^2\/\\Delta_1 = g_2^2\/\\Delta_2$ the dependence on $n^2$ is completely removed which means the nonlinearity and therefore also the emission bandwidth $\\mathcal{B}$ is eliminated at this order.\n\\end{proof}\n\n\\subsection{Qubit choice for $\\hat{b}$}\nThe relevant dispersive coupling to the $e$ levels is\n\\begin{equation}\n    \\chi_e = \\frac{2g_1^2}{\\Delta_1} - \\frac{g_2^2}{\\Delta_2 - \\Delta_1}\n\\end{equation}\nand at the minimal nonlinearity point, we have\n\\begin{equation}\n    \\chi_e = \\frac{g_1^2}{\\Delta_1} \\frac{r-2}{r-1}\n\\end{equation}\nwhere $r=g_2^2\/g_1^2=\\Delta_2\/\\Delta_1$.\nIdeally, $\\chi_e$ should be as large as possible at this minimal nonlinearity point, such that we can selectively drive certain level transitions without introducing large $\\mathcal{B}$.\nFor a transmon qubit $r \\approx 2 \\Rightarrow \\chi_e \\approx 0$ and therefore cannot be used as qubit $\\hat{b}$.\nFortunately, other qubit designs could provide much more flexibility in engineering the coupling ratio $r$ and $r \\approx 1$ is favorable in terms of larger $\\chi_e$.\n\nIn this work, we choose a fluxonium type of Hamiltonian\n\\begin{equation}\n    \\hat{H} = 4 E_C \\hat{n}^2 - E_J \\cos (\\hat{\\phi} - \\phi_{\\text{ext}}) + \\frac{1}{2} E_L \\hat{\\phi}^2\n\\end{equation}\nfor qubit $\\hat{b}$. With realistic parameters $\\phi_{\\text{ext}}=0$, $E_C\/2\\pi=0.95~\\text{GHz}$, $E_J\/2\\pi=4.75~\\text{GHz}$ and $E_L\/2\\pi=0.65~\\text{GHz}$, the coupling ratio is $r =  g_2^2\/g_1^2 = |\\mele{f}{\\hat{n}}{\\hat{e}}|^2\/|\\mele{e}{\\hat{n}}{\\hat{g}}|^2 \\approx 1.2$ with $\\omega_{ge}\/2\\pi \\approx 5.43~\\text{GHz}$ and $\\omega_{ef}\/2\\pi \\approx 3.87~\\text{GHz}$.\n\n\n\\section{Full circuit design}\nIn this section, we provide details for the full circuit simulation in Fig.~\\ref{fig4}(e).\nThe AQEC Hamiltonian Eq.~(\\ref{eq:AQEC_H}) can be implemented with a more physical Hamiltonian\n\\begin{equation}\\label{eq:H_physical}\n    \\hat{H} = \\hat{H}_{ab} + \\left( f_1(t)\\hat{a}^\\dagger + f_2(t) \\hat{a} + f_3(t) \\hat{a}^2 \\right) \\hat{b}^\\dagger + f_4(t) \\hat{b}^\\dagger \\hat{c} + \\text{h.c.} \n\\end{equation}\nwhere\n\\begin{equation}\n    \\begin{split}\n        f_1(t) =& \\sum_n \\frac{\\alpha^{(1)}_n e^{-i(E_{n,e} - E_{n-1,g})t}}{\\mele{\\widetilde{n,e}}{\\hat{a}^\\dagger \\hat{b}^\\dagger}{\\widetilde{n-1,g}}} \\\\\n        f_2(t) =& \\sum_n \\frac{\\alpha^{(2)}_n e^{-i(E_{n,e} - E_{n+1,g})t}}{\\mele{\\widetilde{n,e}}{\\hat{a} \\hat{b}^\\dagger}{\\widetilde{n+1,g}}} \\\\\n        f_3(t) =& \\sum_n \\frac{\\alpha^{(3)}_n e^{-i(E_{n,e} - E_{n+2,g})t}}{\\mele{\\widetilde{n,e}}{\\hat{a}^2 \\hat{b}^\\dagger}{\\widetilde{n+2,g}}} \\\\\n        f_4(t) =& \\Omega \\sum_n \\frac{e^{-i(E_{n,e} - E_{n,g})t}}{\\mele{\\widetilde{n,e}}{\\hat{b}^\\dagger}{\\widetilde{n,g}}} \n    \\end{split}\n\\end{equation}\nand $\\ket{\\widetilde{n,g(e)}}$ are the dressed eigenstates of $\\hat{H}_{ab}$ with energies $E_{n,g(e)}$.\nNotice that the dressed states $\\ket{\\widetilde{n,g}}$ replace the bare Fock states $\\ket{n,g}$ in our definition of the logical basis in Eq.~(\\ref{eq:logical_states}) and the matrix elements such as $\\mele{\\widetilde{n,e}}{\\hat{a}^\\dagger \\hat{b}^\\dagger}{\\widetilde{n-1,g}}$ will be close but not equal to $\\sqrt{n}$.\nThe drivings $f_{1\\sim 4}(t)$ engineer couplings between dressed states rather than the bare Fock states.\n\nThe relevant dissipators are $\\{ \\sqrt{\\kappa} \\hat{a}, \\sqrt{\\kappa_q} \\hat{c} \\}$, but to be more realistic we also include an extra dissipator $\\sqrt{\\kappa} \\hat{b}$ in the simulation.\nThe coupling strength $\\Omega$ between $\\hat{b}$ and $\\hat{c}$ is chosen such that the effective decay rate for $\\hat{b}$ after adiabatically eliminating $\\hat{c}$~\\cite{Reiter2012a} is still $4\\Omega^2\/\\kappa_q=2\\pi \\times 20~\\text{MHz}$, same as the value used in the numerical optimization. We set the decay rate of $\\hat{c}$ as $\\kappa_q\/2\\pi=100~\\text{MHz}$.\n\nThe Hamiltonian Eq.~(\\ref{eq:H_physical}) can be furthermore implemented with a circuit model\n\\begin{equation}\\label{eq:H_circuit}\n    \\begin{split}\n        \\hat{H} =& \\omega_a \\hat{a}^\\dagger \\hat{a} + \\omega_{ge} \\ket{e} \\bra{e} + (\\omega_{ge} + \\omega_{ef}) \\ket{f} \\bra{f}+ \\omega_c \\hat{c}^\\dagger \\hat{c}  \\\\\n        &+ g_1 (\\hat{a}^\\dagger \\ket{g}\\bra{e} + \\hat{a}\\ket{e}\\bra{g}) + g_2 (\\hat{a}\\ket{f}\\bra{e} + \\hat{a}^\\dagger \\ket{e}\\bra{f}) \\\\\n        &+ \\varepsilon_1 (t) \\left[ g_{ab}^{(1)} \\cos \\left( \\varphi_a (\\hat{a}+\\hat{a}^\\dagger) + \\varphi_b (\\hat{b}+\\hat{b}^\\dagger) \\right) \\right. \\\\\n        &\\qquad \\qquad \\left. + g_{bc}^{(1)} \\cos \\left( \\varphi_b (\\hat{b}+\\hat{b}^\\dagger) + \\varphi_c (\\hat{c}+\\hat{c}^\\dagger) \\right) \\right] \\\\\n        &+ \\varepsilon_2 (t) \\left[ g_{ab}^{(2)} \\sin \\left( \\varphi_a (\\hat{a}+\\hat{a}^\\dagger) + \\varphi_b (\\hat{b}+\\hat{b}^\\dagger) \\right) \\right. \\\\\n        &\\qquad \\qquad \\left. + g_{bc}^{(2)} \\sin \\left( \\varphi_b (\\hat{b}+\\hat{b}^\\dagger) + \\varphi_c (\\hat{c}+\\hat{c}^\\dagger) \\right) \\right] ,\n    \\end{split}\n\\end{equation}\nwhere the drivings are given by\n\\begin{equation}\n    \\begin{split}\n        \\varepsilon_1 (t) =& - 2 \\text{Re} \\left\\{ \\frac{1}{\\varphi_a \\varphi_b g_{ab}^{(1)}} \\left[ e^{-2i\\omega_a t} f_1(t) + f_2(t)  \\right] \\right. \\\\\n        &\\left. + \\frac{1}{\\varphi_b \\varphi_c g_{bc}^{(1)}} e^{i(\\omega_c- \\omega_a) t} f_4(t) \\right\\} \\\\\n        \\varepsilon_2 (t) =& - 2 \\text{Re} \\left\\{ \\frac{2}{\\varphi_a^2 \\varphi_b g_{ab}^{(2)}} e^{i \\omega_a t} f_3(t) \\right\\} ,\n    \\end{split}\n\\end{equation}\nwhich are generated by the two independent flux pump through the larger and smaller loops~\\cite{Kapit2016a} in Fig.~\\ref{fig4}(c).\nAfter Taylor expanding the $\\cos$ and $\\sin$ interaction and dropping fast rotating terms in the rotating frame, we could show the equivalence of Eq.~(\\ref{eq:H_circuit}) to the Hamiltonian Eq.~(\\ref{eq:H_physical}) with $\\Delta_1 = \\omega_{ge}-\\omega_a$ and $\\Delta_2 = \\Delta_1 + \\omega_{ef}-\\omega_a$.\nTo ensure the validity of rotating wave approximation, we place the frequencies at $\\omega_a\/2\\pi=3.5~\\text{GHz}$ and $\\omega_c\/2\\pi=2.5~\\text{GHz}$ with qubit $\\hat{b}$ frequencies from the previous section. We also choose $\\varphi_a=\\varphi_b=\\varphi_c=0.1$ such that higher order terms in the $\\cos$ and $\\sin$ expansions can be safely dropped.\nAll AQEC Hamiltonian parameters as well as the logical basis states are directly imported from the \\texttt{AutoQEC} optimization result instead of using the analytical results in Appendix~\\ref{appendix:b}.\nWe use QuTiP~\\cite{Johansson2012,Johansson2013} for the full circuit simulation.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/SI_figure_1.pdf}\n    \\caption{(a-b) Wigner functions and photon number distributions for the discovered encoding in Fig.~\\ref{fig2}(c)ii. (c) Single state fidelity $F_{\\theta\\phi}$ at $t=10\\mu$s for the whole Bloch sphere. The white dashed line indicates the break-even fidelity. (d) $F_{\\theta\\phi}$ on the Bloch sphere for the $\\sqrt{3}$ code. For all Wigner funciton plots throughout this paper, the horizontal axis label is $x = \\ave{\\hat{a} + \\hat{a}^\\dagger}\/\\sqrt{2}$ and the vertical axis label is $p = i\\ave{\\hat{a}^\\dagger - \\hat{a}}\/\\sqrt{2}$.}\n    \\label{fig_SI_1}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/SI_figure_2.pdf}\n    \\caption{(a-b) Wigner functions and photon number distributions for another variant of the $\\sqrt{3}$ code discovered with $d=2$ Hamiltonian.}\n    \\label{fig_SI_2}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/SI_figure_3.pdf}\n    \\caption{(a) Learning curve for results in Fig.~\\ref{fig_SI_2}. (b) Wigner functions of $\\hat{\\rho}_{\\text{code}}$ at different iterations during training, which shows a relatively good convergence after a few thousand iterations.}\n    \\label{fig_SI_3}\n\\end{figure}\n\n\n\\section{Additional comments on the optimization results}\n\\subsection{Exceed break-even with partial protection}\nWe further investigate the result in Fig.~\\ref{fig2}(c)ii, which represents a class of optimization results that perform better than break-even fidelity but worse than full QEC codes.\nFig.~\\ref{fig_SI_1}(a) shows the Wigner functions for the code subspace as well as both logical states, and Fig.\\ref{fig_SI_1}(b) shows the photon number distribution for the logical states where $\\ket{\\psi_0} \\in \\{ \\ket{1}, \\ket{2}, \\ket{3} \\}$ only occupies low photon number states and $\\ket{\\psi_1} \\in \\{ \\ket{5}, \\ket{6}, \\ket{7} \\}$ only occupies high photon number states.\n\nTo understand how the logical subspace is preserved under the AQEC Hamiltonian, we plot the single state fidelity $F_{\\theta\\phi}$ over the Bloch sphere (Fig.~\\ref{fig_SI_1}(c)) at $t=10~\\mu$s. The logical state $\\ket{\\psi_0}$ is strongly stabilized by the AQEC Hamiltonian with a fidelity 0.985 and $\\ket{\\psi_1}$ is preserved with a lower fidelity 0.598.\nSome of their superposition states ($\\theta,\\phi$ in-between the white dashed line) have fidelities below break-even, but the average fidelity over the whole Bloch sphere still exceeds break-even (Fig.~\\ref{fig2}(e) red dashed line) due to the partial protection in the logical subspace.\nIn comparison, we also plot the single state fidelity for the $\\sqrt{3}$ code (Fig.~\\ref{fig2}(c)i) in Fig.~\\ref{fig_SI_1}(d) which shows a relatively uniform protection for any logical states.\n\nWe could study a simplified example to demonstrate that a partially protected logical subspace exceeds break-even.\nStabilizing Fock states $\\ket{0}$ and $\\ket{2}$ under photon loss error can be implemented with a distance 1 Hamiltonian $\\hat{H}=\\ket{2,e} \\bra{1,g}+\\ket{1,g} \\bra{2,e}$.\nAt long time, both logical states are stabilized with single state fidelities $F_{\\theta=0}(t) \\approx F_{\\theta=\\pi}(t) \\approx 1$ but any coherent superposition state becomes a complete mixture of $\\{\\ket{0}\\bra{0}, \\ket{2}\\bra{2}\\}$.\nThis leads to an average fidelity of $\\frac{2}{3}$ which is better than the break-even fidelity $\\frac{1}{2}$.\nIntuitively, stabilizing both $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$ preserves strictly more information compared to collapsing the whole Bloch sphere to $\\ket{\\psi_0}=\\ket{0}$.\n\n\\subsection{A different $\\sqrt{3}$ code}\nBesides the $\\sqrt{3}$ code explained in the main text, \\texttt{AutoQEC} also discovered another variant of $\\sqrt{3}$ code (Fig.~\\ref{fig_SI_2}(a)) protected by a distance 2 Hamiltonian.\nThe main difference is that $\\ket{\\psi_1} \\in \\{ \\ket{1}, \\ket{4}, \\ket{7} \\}$ instead of $\\{ \\ket{1}, \\ket{4}, \\ket{6} \\}$ (Fig.~\\ref{fig_SI_2}(b)). Following the same procedures as in Appendix~\\ref{appendix:b}, this new code can also be analytically derived as ($F \\approx 99.8\\%$ compared to the numerical results in Fig.~\\ref{fig_SI_2}(a))\n\\begin{equation}\n    \\begin{split}\n        \\ket{\\psi_0} =& \\sqrt{1-\\frac{1}{\\sqrt{3}}} \\ket{0} + \\frac{1}{\\sqrt[4]{3}} \\ket{3} \\\\\n        \\ket{\\psi_1} =& \\sqrt{\\frac{4(7-\\sqrt{3})}{3(7 + \\sqrt{3})}} \\ket{1} - \\sqrt{\\frac{(\\sqrt{3}-1)(7-\\sqrt{3})}{3(7 + \\sqrt{3})}} \\ket{4} \\\\\n        & + \\sqrt{\\frac{3-\\sqrt{3}}{3(7 + \\sqrt{3})}} \\ket{7} .\n    \\end{split}\n\\end{equation}\n\n\n\\section{Training details}\nWe use Adam optimizer~\\cite{Kingma2015} for the gradient based learning with a learning rate about 0.001. Usually after a few hundred iterations, we can tell whether the training is stuck at a bad local minimum below break-even or not and make a decision on early stops. The training often achieves good convergence after a few thousand iterations and we could lower the learning rate to about 0.0003 for the final learning stage.\n\nWe choose a Fock state cutoff of 20 for the bosonic mode with a total Hilbert space dimension of 40. At the beginning of each \\texttt{AutoQEC} run, the real and imaginary parts of the logical states $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$ as length 40 complex vectors are randomly initialized.\nDuring the optimization, in general $\\ket{\\psi_0}$ and $\\ket{\\psi_1}$ won't be perfectly orthogonal to each other after an Adam update step and therefore we choose to maintain their orthogonality by manually setting $\\ket{\\psi_1} \\rightarrow \\ket{\\psi_1} - \\frac{\\inp{\\psi_0}{\\psi_1}}{\\inp{\\psi_0}{\\psi_0}} \\ket{\\psi_0}$ after each iteration.\n\nFigure~\\ref{fig_SI_3} shows the learning curve for results in Fig.~\\ref{fig_SI_2} discovered with $d=2$ Hamiltonian. Similar learning curves occur frequently through many runs of \\texttt{AutoQEC}.\nRegarding computational cost, each iteration takes about 12 seconds on 3 CPUs (Intel Xeon CPU E5-2609 v4 @ 1.70GHz) for training with distance 2 Hamiltonians.\n\\texttt{AutoQEC} runs on 3 CPUs because $\\hat{\\rho}_{00}(t),\\hat{\\rho}_{11}(t),\\hat{\\rho}_{10}(t)$ in the definition of $\\bar{F}(t)$ can be evaluated in parallel with three independent master equation time evolutions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe goal for the construction that I describe in this note was to lift the stability result of \\cite{vineyards} to the setting when the simplicial filtrations are not necessarily defined on the same set. The ideas in this note were first presented at ATCMS in July 2012. A partial discussion appears in \\cite{facundo2012banff}. \n\nSection \\ref{sec:geodesic} proves that that this construction defines a geodesic metric on the collection of finite filtered spaces. There I give an improvement of the stability of persistence which uses these geodesics.\n\n\n\n\\section{Simplicial Homology}\nGiven a simplicial complex $L$ and simplices $\\sigma,\\tau\\in L$, we write $\\sigma\\subseteq \\tau$ whenever $\\sigma$ is a face of $\\tau$. For each integer $\\ell\\geq 0$ we denote by $L^{(\\ell)}$ the $\\ell$-skeleton of $L$.\n\nRecall that given two finite simplicial complexes $L$ and $S$, a \\emph{simplicial map} between them arises from any map $f:L^{(0)}\\rightarrow S^{(0)}$ with the property that whenever $p_0,p_1,\\ldots,p_k$ span a simplex in $L$, then $f(p_0),f(p_1),\\ldots,f(p_k)$ span a simplex of $S$. One does not require that the vertices $f(p_0),f(p_1),\\ldots,f(p_k)$ be all distinct.  Given a map $f:L^{(0)}\\rightarrow S^{(0)}$ between the vertex sets of the finite simplicial complexes $L$ and $S$, we let $\\overline{f}:L\\rightarrow S$ denote the induced \\emph{simplicial map}.\n\n\n\n\n\n\nWe will make use of the following theorem in the sequel.\n\\begin{theorem}[Quillen's Theorem A in the simplicial category, \\cite{quillen}] Let $\\zeta:S\\rightarrow L$ be a simplicial map between two finite complexes. Suppose that the preimage of each closed simplex of $L$ is contractible. Then $\\zeta$ is a  homotopy equivalence.\n\\end{theorem}\n\n\n\n\n \n\\begin{corollary}\\label{coro:eq-pd}\nLet $L$ be a finite simplicial complex and $\\varphi:Z\\rightarrow L^{(0)}$ be any  surjective map with finite domain $Z$. Let $S:=\\{\\tau\\subseteq Z|\\,\\varphi(\\tau)\\in L\\}$. Then $S$ is a simplicial complex and the induced simplicial map\n$\\overline{\\varphi}:S\\rightarrow L$ is an homotopy equivalence.\n\\end{corollary}\n\n\\begin{proof}\nNote that $S = \\bigcup_{\\sigma\\in L}\\{\\tau\\subseteq Z|\\,\\varphi(\\tau)=\\sigma\\}$ so it is clear that $S$ is a simplicial complex with vertex set $Z$. That the preimage of each $\\sigma\\in L$ is contractible is trivially true since those preimages are exactly the simplices in $S$. The conclusion follows directly from Quillen's Theorem A.\n\\end{proof}\n\nIn this paper we consider homology with coefficients in a field $\\mathbb{F}$ so that  given a simplicial complex $L$, then for each $k\\in\\mathbb{N}$, $H_k(L,\\mathbb{F})$ is a vector space. To simplify notation, we drop the argument $\\mathbb{F}$ from the list and only write $H_k(L)$ for the homology of $L$ with coefficients in $\\mathbb{F}$.\n\n\\section{Filtrations and Persistent Homology}\n\nLet $\\mathcal{F}$ denote the set of all finite \\emph{filtered spaces}: that is pairs $\\mathbf{X}=(X,F_X)$ where $X$ is a finite set and $F_X:\\mathrm{pow}(X)\\rightarrow \\mathbb{R}$ is a monotone function. Any such function is called a \\emph{filtration} over $X$. Monotonicity in this context refers to the condition that $F_X(\\sigma)\\geq F_X(\\tau)$ whenever $\\sigma \\supseteq \\tau.$ Given a finite set $X$, by $\\mathcal{F}(X)$ we denote the set of all possible filtrations $F_X:\\pow{X}\\rightarrow \\mathbb{R}$ on $X$. Given a filtered space $\\mathbf{X}=(X,F_X)\\in\\mathcal{F}$, for each $\\varepsilon\\in \\mathbb{R}$ define the simplicial complex \n$$L_\\varepsilon(\\mathbf{X}):=\\big\\{\\sigma\\subseteq X|\\,F_X(\\sigma)\\leq \\varepsilon\\big\\}.$$\nOne then considers the nested family of simplicial complexes\n\n\n$$L(\\mathbf{X}):=\\big\\{L_{\\varepsilon}(\\mathbf{X})\\subset L_{\\varepsilon'}(\\mathbf{X})\\}_{\\varepsilon\\leq \\varepsilon'}$$\n\n\nwhere each $L_{\\varepsilon}(\\mathbf{X})$ is, by construction, finite. At the level of homology, for each $k\\in \\mathbb{N}$ the above inclusions give rise to a system of vector spaces and linear maps\n\n$$\\mathbb{V}_k(\\mathbf{X}):=\\big\\{V_{\\varepsilon}(\\mathbf{X})\\stackrel{v_{\\varepsilon,\\varepsilon'}}{\\longrightarrow} V_{{\\varepsilon'}}(\\mathbf{X})\\big\\}_{\\varepsilon\\leq \\varepsilon'},$$\n\n\n\nwhich is called a \\emph{persistence vector space.} Note that each $V_{\\varepsilon}(\\mathbf{X})$ is finite dimensional.\n\n\nPersistence vector spaces admit a \\emph{classification up to isomorphism} in terms of collections of intervals so that to the persistence vector space  $\\mathbb{V}$ one assigns a multiset of intervals $I(\\mathbb{V})$ \\cite{zz}. These collections of intervals are sometimes referred to as \\emph{barcodes} or also \\emph{persistence diagrams}, depending on the graphical representation that is adopted \\cite{comptopo-herbert}. We denote by $\\mathcal{D}$ the collection of all finite persistence diagrams. An element $D\\in\\mathcal{D}$ is a \\emph{finite} multiset of points $$D= \\{(b_\\alpha,d_\\alpha),\\,0\\leq b_\\alpha\\leq d_\\alpha,\\,\\alpha\\in A\\}$$ for some (finite) index set $A$. Given $k\\in\\mathbb{N}$, to any filtered set $\\mathbf{X}\\in\\mathcal{F}$ one can attach a persistence diagram via \n$$\\mathbf{X}\\longmapsto L(\\mathbf{X}) \\longmapsto \\mathbb{V}_k(\\mathbf{X}) \\longmapsto I\\big(\\mathbb{V}_k(\\mathbf{X})\\big).$$\nWe denote by $\\dk{k}:\\mathcal{F}\\rightarrow \\mathcal{D}$ the resulting composite map. Given $\\mathbf{X}=(X,F_X)$,  we will sometimes write $\\dk{k}{(F_X)}$ to denote $\\dk{k}{(\\mathbf{X})}$.\n\n\n\\section{Stability of filtrations}\nThe \\emph{bottleneck distance} is a useful notion of distance between persistence diagrams and we recall it's definition next. We will follow the presentation on \\cite{carlsson_2014}. Let $\\Delta\\subset \\mathbb{R}^2_+$ be comprised of those points which sit above the diagonal: $\\Delta:=\\{(x,y)|\\,x\\leq y\\}.$ \n\n\n\nDefine the \\emph{persistence} of a point $P=(x_P,y_P)\\in\\Delta$ by  $\\pers{P}:=y_P-x_P$. \n\n Let $D_1=\\{P_\\alpha\\}_{\\alpha\\in A_1}$ and $D_2=\\{Q_\\alpha\\}_{\\alpha\\in A_2}$ be two persistence diagrams indexed over the finite index sets $A_1$ and $A_2$, respectively. Consider subsets $B_i\\subseteq A_i$ with $|B_1|=|B_2|$ and any bijection $\\varphi:B_1\\rightarrow B_2$, then define\n$$J(\\varphi):=\\max\\left(\\max_{\\beta\\in B_1}\\|P_\\beta-Q_{\\varphi(\\beta)}\\|_\\infty,\\max_{\\alpha\\in A_1\\backslash B_1}\\frac{1}{2}\\pers{P_\\alpha},\\max_{\\alpha\\in A_2\\backslash B_2}\\frac{1}{2}\\pers{P_\\alpha}\\right).$$\nFinally, one defines the bottleneck distance between $D_1$ and $D_2$ by $$d_{\\mathcal{D}}(D_1,D_2):=\\min_{(B_1,B_2,\\varphi)} J(\\varphi),$$\nwhere $(B_1,B_2,\\varphi)$ ranges over all $B_1\\subset A_1$, $B_2\\subset A_2$, and bijections $\\varphi:B_1\\rightarrow B_2$.\n\n\nOne of the standard results about the stability of persistent homology invariants, which is formulated in terms of the Bottleneck distance, is the proposition below which we state in  a weaker form that will suffice for our presentation:\\footnote{In \\cite{vineyards} the authors do not assume that the underlying simplicial complex is the full powerset.}\n \\begin{theorem}[\\cite{vineyards}]\\label{theo:stab-vineyards}\n For all finite sets $X$ and filtrations $F,G:\\mathrm{pow}(X)\\rightarrow \\mathbb{R}$, \n $$d_{\\mathcal{D}}(\\dk{k}(F),\\dk{k}(G))\\leq \\max_{\\sigma\\in\\mathrm{pow}(X)}|F(\\sigma)-G(\\sigma)|,$$\n for all $k\\in\\mathbb{N}.$\n \\end{theorem}\n\nThe proof of this theorem offered in \\cite{vineyards} is purely combinatorial and elementary. This result requires that the two filtrations be given on the \\emph{same} set. This restriction will be lifted using the ideas that follow.\n\n\n\\subsection{Filtrations defined over different sets}\n A \\emph{parametrization} of a finite set $X$ is any finite set $Z$ and a surjective map $\\varphi_X:Z\\rightarrow X$.\nConsider a filtered space $\\mathbf{X} = (X,F_X)\\in\\mathcal{F}$ and a parametrization $\\varphi_X:Z\\rightarrow X$ of $X$. By $\\varphi_X^\\ast F_X$ we denote the  \\emph{pullback filtration} induced by $F_X$ and the map $\\varphi_X$ on $Z$. This filtration is given by $\\tau\\mapsto F_X(\\varphi_X(\\tau))$ for all $\\tau\\in\\mathrm{pow}(Z).$ \n\n\n\nA useful corollary of the persistence homology isomorphism theorem \\cite[pp. 139]{comptopo-herbert} and Corollary \\ref{coro:eq-pd} is that the persistence diagrams of the original filtration and the pullback filtration are identical.\n\\begin{corollary}\\label{coro:same}\nLet $\\mathbf{X}=(X,F_X)\\in \\mathcal{F}$ and $\\varphi:Z\\twoheadrightarrow X$ a parametrization of $X$. Then, for all $k\\in\\mathbb{N}$, $\\dk{k}(\\varphi^\\ast F_X)=\\dk{k}(F_X).$\n\\end{corollary}\n\n\n\\subsubsection{Common parametrizations of two spaces: tripods.}\nNow, given $\\mathbf{X} = (X,F_X)$ and $\\mathbf{Y}=(Y,F_Y)$ in $\\mathcal{F}$, the main idea in comparing filtrations defined on different spaces is to consider parametrizations \n$\\varphi_X:Z\\twoheadrightarrow X$ and $\\varphi_Y:Z\\twoheadrightarrow Y$ of $X$ and $Y$ from a \\emph{common} parameter space $Z$, i.e. \\emph{tripods}:\n\n$$\\xymatrix{ & Z \\ar@{->>}[dl]_{{\\varphi_X}} \\ar@{->>}[dr]^{\\varphi_Y} &  \\\\ X\t& & Y}$$\n\nand compare the pullback filtrations  $\\varphi_X^*F_X$ and $\\varphi_Y^*F_Y$ on $Z$. Formally, define\n\n\n\n\n\n\n \\begin{multline}\n d_{\\mathcal{F}}\\big(\\mathbf{X},\\mathbf{Y}\\big):=\\\\\\inf\\left\\{\\max_{\\tau\\in\\mathrm{pow}(Z)}\\big|\\varphi^\\ast_X F_X(\\tau)-\\varphi^\\ast_Y F_Y(\\tau)\\big|;\\,\\varphi_X:Z\\twoheadrightarrow X,\\, \\varphi_Y:Z\\twoheadrightarrow Y\\,\\,\\mbox{parametrizations}\\right\\}.\n \\end{multline}\n\n\n\n\\begin{remark} \\label{rem:dist-F-simple} Notice that in case $X=\\{\\ast\\}$ and $F_{\\{\\ast\\}}(\\ast) = c\\in\\mathbb{R}$, then $d_{\\mathcal{F}}(X,Y)=\\max_{\\sigma\\subset Y}\\big|F_Y(\\sigma)-c\\big|,$ for any filtered space $Y$. If $c=0$,  $Y=\\{y_1,y_2\\}$ with $F_Y(y_1)=F_Y(y_2)=0$ and $F_{Y}(\\{y_1,y_2\\})=1$. Then, $d_{\\mathcal{F}}(X,Y)=1.$\n\nHowever, still with $c=0$ and $Y=\\{y_1,y_2\\}$,  but $F_Y(y_1)=F_Y(y_2)=F_{Y}(\\{y_1,y_2\\})=0$, one has $d_\\mathcal{F}(X,Y)=0$. This means that $d_{\\mathcal{F}}$ is at best a pseudometric on filtered spaces.\n\\end{remark}\n\n\n\\begin{proposition}\n$d_{\\mathcal{F}}$ is a pseudometric on $\\mathcal{F}$.\n\\end{proposition}\n\\begin{proof}\nSymmetry and non-negativity are clear. We need to prove the triangle inequality. Let $\\mathbf{X} = (X,F_X)$, $\\mathbf{Y} = (Y,F_Y)$, and $\\mathbf{W} = (W,F_W)$ in $\\mathcal{F}$ be non-empty and $\\eta_1,\\eta_2>0$ be s.t. \n$$d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y})<\\eta_1\\,\\,\\mbox{and}\\,\\,d_{\\mathcal{F}}(\\mathbf{Y},\\mathbf{W})<\\eta_2.$$ \nChoose, $\\psi_X:Z_1\\twoheadrightarrow X$, $\\psi_Y:Z_1\\twoheadrightarrow Y$, $\\zeta_Y:Z_2\\twoheadrightarrow Y$, and $\\zeta_W:Z_2\\twoheadrightarrow W$ surjective such that \n$$\\|F_X\\circ\\psi_X-F_Y\\circ\\psi_Y\\|_{\\ell^\\infty(\\pow{Z_1})}<\\eta_1$$\nand\n$$\\|F_Y\\circ\\zeta_Y-F_W\\circ\\zeta_W\\|_{\\ell^\\infty(\\pow{Z_2})}<\\eta_2.$$\nLet $Z\\subseteq Z_1\\times Z_2$ be defined by $Z:=\\{(z_1,z_2)\\in Z_1\\times Z_2|\\psi_Y(z_1)=\\zeta_Y(z_2)\\}$ and consider the following (pullback) diagram:\n\n$$\\xymatrix{& & Z \\ar[dl]_{{\\pi_1}} \\ar[dr]^{\\pi_2} & &  \\\\ & Z_1\\ar[dl]_{\\psi_X} \\ar[dr]^{\\psi_Y}& & Z_2\\ar[dl]_{\\zeta_Y}\\ar[dr]^{\\zeta_W} &\\\\ X & & Y & & W}.$$\n\nClearly, since $\\psi_Y$ and $\\zeta_Y$ are surjective, $Z$ is non-empty.\nNow, consider the following three maps with domain $Z$: $\\phi_X := \\psi_X\\circ\\pi_1$, $\\phi_Y := \\psi_Y\\circ\\pi_1=\\zeta_Y\\circ\\pi_2$, and  $\\phi_W:=\\zeta_W\\circ\\pi_2$. These three maps are surjective and therefore constitute parametrizations of $X$, $Y$, and $W$, respectively.  Then, since $\\pi_i:Z\\rightarrow Z_i$, $i=1,2$, are surjective and $\\psi_Y\\circ {\\pi_1} = \\zeta_Y\\circ\\pi_2$, we have\n\\begin{align*}\nd_{\\mathcal{F}}(\\mathbf{X},\\mathbf{W})&\\leq \\|F_X\\circ\\phi_X-F_W\\circ\\phi_W\\|_{\\ell^\\infty(\\pow{Z})}\\\\\n&\\leq \\|F_X\\circ\\phi_X - F_Y\\circ\\phi_Y\\|_{\\ell^\\infty(\\pow{Z})}+\\|F_Y\\circ\\phi_Y - F_W\\circ\\phi_W\\|_{\\ell^\\infty(\\pow{Z})}\\\\\n&= \\|F_X\\circ\\psi_X - F_Y\\circ\\psi_Y\\|_{\\ell^\\infty(\\pow{Z_1})}+\\|F_Y\\circ\\zeta_Y - F_W\\circ\\zeta_W\\|_{\\ell^\\infty(\\pow{Z_2})}\\\\\n&\\leq \\eta_1+\\eta_2.\n\\end{align*}\nThe conclusion follows by letting $\\eta_1\\searrow d_{\\mathcal{F}}(X,Y)$ and $\\eta_2\\searrow d_{\\mathcal{F}}(Y,W)$.\n\\end{proof}\n\nWe now obtain a lifted version of Theorem \\ref{theo:stab-vineyards}\n \\begin{theorem} \\label{theo:stab-pullback}\n For all finite filtered spaces $\\mathbf{X}  =(X,F_X)$ and $\\mathbf{Y} = (Y,F_Y)$, and all $k\\in\\mathbb{N}$ one has:\n $$d_{\\mathcal{D}}(\\dk{k}(\\mathbf{X}),\\dk{k}(\\mathbf{Y}))\\leq d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y}).$$\n \\end{theorem}\n\n\n\n \\begin{proof}[Proof of Theorem \\ref{theo:stab-pullback}]\n Assume $\\varepsilon>0$ is such that $d_{\\mathcal{F}}(F_X,F_Y)<\\varepsilon.$ Then, let $\\varphi_X:Z\\rightarrow X$ and $\\varphi_Y:Z\\rightarrow Y$ be surjective maps from the finite set $Z$ into $X$ and $Y$, respectively, such that $|\\varphi_X^\\ast F_X(\\tau) - \\varphi_Y^\\ast F_Y(\\tau)|<\\varepsilon$ for all $\\tau\\in\\mathrm{pow}(Z)$. Then, by Theorem \\ref{theo:stab-vineyards}, \n $$d_{\\mathcal{D}}(\\dk{k}(\\varphi_X^\\ast F_X),\\dk{k}(\\varphi_Y^\\ast F_Y))<\\varepsilon.$$\n for all $k\\in\\mathbb{N}.$ Now apply Corollary \\ref{coro:same} and conclude by letting $\\varepsilon$ approach $d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y})$.\n \\end{proof}\n\n\\begin{remark}\\label{rem:non-tight}\nConsider the case of $\\mathbf{Y}$ being the one point filtered space $\\{\\ast\\}$ such that $F_{\\{\\ast\\}}(\\{\\ast\\})=0$, and    $\\mathbf{X}$ such that $X=\\{x_2,x_2\\}$, and $F_X(\\{x_1\\})=F_X(\\{x_2\\})=0$, $F_X(\\{x_1,x_2\\})=1$. In this case $d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y})=1$. However, notice that for $k=0$ $\\mathrm{D}_0(\\mathbf{X}) = \\{[0,\\infty),[0,1)\\}$\n and $\\mathrm{D}_0(\\mathbf{Y}) = \\{[0,\\infty)\\}$. Additionaly, for all $k\\geq 1$ one has $\\mathrm{D}_k(\\mathbf{X}) = \\mathrm{D}_k(\\mathbf{Y}) = \\emptyset$. This means that the lower bound provided by Theorem \\ref{theo:stab-pullback} is equal to $\\frac{1}{2}<1=d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y})$.\n\\end{remark}\n\n\\section{Filtrations arising from metric spaces: Rips and \\v{C}ech}\nRecall \\cite{burago-book} that for two compact metric spaces $(X,d_X)$ and $(Y,d_Y)$, a correspondence between them is amy subset $R$ of $X\\times Y$ such that the natural projections $\\pi_X:X\\times Y\\rightarrow X$ and  $\\pi_Y:X\\times Y\\rightarrow Y$ are such that $\\pi_X(R)=X$ and $\\pi_Y(R)=Y$. The distortion of any such correspondence is given by \n$$\\mathrm{dis}(R):=\\sup_{(x,y),(x',y')\\in R}\\big|d_X(x,x')-d_Y(y,y')\\big|.$$\nThen, Gromov-Hausdorff distance between $(X,d_X)$ and $(Y,d_Y)$ is defined as \n$$\\dgro{X}{Y} := \\frac{1}{2}\\inf_{R}\\mathrm{dis}(R),$$\nwhere the infimum is taken over all correspondences $R$ between $X$ and $Y$.\n\n\\subsection{The Rips filtration}\nRecall the definition of the \\emph{Rips filtration} of a finite metric space $(X,d_X)$: for $\\sigma\\in \\pow{X}$,\n$$F^{\\mathrm{R}}_X(\\sigma)=\\diamms{\\sigma}{X}:=\\max_{x,x'\\in X}d_X(x,x').$$\n\nThe following theorem was first proved in \\cite{dgw-topo-pers}. A different proof (also applicable to compact metric spaces) relying on the interleaving distance and multivalued maps was given in \\cite{chazal-geom}. Yet another different proof avoiding multivalued maps is given in \\cite{dowker-ph}.\n\n\\begin{theorem}\\label{theo:stab-dD-R}\nFor all finite metric spaces $X$ and $Y$, and all $k\\in\\mathbb{N}$,\n$$d_{\\mathcal{D}}\\big(\\dk{k}(F^{\\mathrm{R}}_X),\\dk{k}(F^{\\mathrm{R}}_Y)\\big)\\leq 2\\,\\dgro{X}{Y}.$$\n\\end{theorem}\n\nA different proof of Theorem \\ref{theo:stab-dD-R} can be obtained by combining Theorem \\ref{theo:stab-pullback} and Proposition \\ref{prop:stab-R} below.\n\\begin{proposition}\\label{prop:stab-R}\nFor all finite metric spaces $X$ and $Y$, \n$$d_{\\mathcal{F}}\\big(F^{\\mathrm{R}}_X,F^{\\mathrm{R}}_Y\\big)\\leq 2\\,\\dgro{X}{Y}.$$\n\\end{proposition}\n\\begin{proof}[Proof of Proposition \\ref{prop:stab-R}]\nLet $X$ and $Y$ be s.t. $\\dgro{X}{Y}<\\eta$, and let $R\\subset X\\times Y$ be a surjective relation with $|d_X(x,x')-d_Y(y,y')|\\leq 2\\eta$ for all $(x,y),(x',y')\\in R$. Consider the parametrization $Z=R$, and $\\varphi_X=\\pi_1:Z\\rightarrow X$ and $\\varphi_Y=\\pi_2:Z\\rightarrow Y$, then \n\\beq{eq:param}\n|d_X(\\varphi_X(t),\\varphi_X(t'))-d_Y(\\varphi_Y(t),\\varphi_Y(t'))|\\leq 2\\eta\n\\end{equation}\nfor all $t,t'\\in Z$. Pick any $\\tau\\in Z$ and notice that\n\n$$\\varphi^*_XF_X^{\\mathrm{R}}(\\tau) = F_X^{\\mathrm{R}}(\\varphi_X(\\tau)) = \\max_{t,t'\\in \\tau}d_X(\\varphi_X(t),\\varphi_X(t')).$$\n\nNow, similarly, write \n\\begin{multline}\\varphi^*_YF_Y^{\\mathrm{R}}(\\tau) = \\max_{t,t'\\in \\tau}d_Y(\\varphi_Y(t),\\varphi_Y(t'))\\leq \\max_{t,t'\\in\\tau}d_X(\\varphi_X(t),\\varphi_X(t'))+2\\eta = \\varphi^*_XF_X^{\\mathrm{R}}(\\tau) + 2\\eta,\n\\end{multline}\n\nwhere the last inequality follows from \\refeq{eq:param}. The proof follows by interchanging the roles of $X$ and $Y$.\n\\end{proof}\n\n\n\\subsection{The \\v{C}ech filtration}\nAnother interesting and frequently used filtration is the \\emph{\\v{C}ech filtration}: for each $\\sigma\\in\\pow{X}$,  \n$$F^\\mathrm{C}_X(\\sigma) := \\mathbf{rad}_X(\\sigma)=\\min_{p\\in X}\\max_{x\\in \\sigma}d_X(x,p).$$ That is, the filtration value of each simplex corresponds to its \\emph{circumradius}.\n\\begin{proposition}\\label{prop:stab-C}\nFor all finite metric spaces $X$ and $Y$, \n$$d_{\\mathcal{F}}\\big(F^{W}_X,F^{W}_Y\\big)\\leq 2\\,\\dgro{X}{Y}.$$\n\\end{proposition}\n\nAgain, as a corollary of Theorem \\ref{theo:stab-pullback} and Proposition \\ref{prop:stab-C} we have the following \n\\begin{theorem}\\label{theo:stab-dD-C}\nFor all finite metric spaces $X$ and $Y$, and all $k\\in\\mathbb{N}$,\n$$d_{\\mathcal{D}}\\big(\\dk{k}(F^{\\mathrm{C}}_X),\\dk{k}(F^{\\mathrm{C}}_Y)\\big)\\leq 2\\,\\dgro{X}{Y}.$$\n\\end{theorem}\nA proof of this theorem via the interleaving distance and multi-valued maps has appeared in \\cite{chazal-geom}.\\footnote{The version in \\cite{chazal-geom} applies to compact metric spaces.} Another proof avoiding multivalued maps is given in \\cite{dowker-ph}.\n\n\\begin{proof}[Proof of Proposition \\ref{prop:stab-C}]\n\nThe proof is similar to that of Proposition \\ref{prop:stab-R}. Pick any $\\tau\\in Z$, then,\n\n$$\\varphi^*_XF_X^W(\\tau) = F_X^W(\\varphi_X(\\tau)) = \\min_{p\\in X}\\max_{t\\in \\tau}d_X(p,\\varphi_X(t))=\\max_{t\\in \\tau}d_X(p_\\tau,\\varphi_X(t))$$\nfor some $p_\\tau\\in X$. Let $t_\\tau\\in Z$ be s.t. $\\varphi_X(t_\\tau)=p_\\tau$, and from the above obtain\n$$\\varphi^*_XF_X^W(\\tau) = \\max_{t\\in \\tau}d_X(\\varphi_X(t_\\tau),\\varphi_X(t)).$$\n\nNow, similarly, write \n\\begin{multline}\\varphi^*_YF_Y^W(\\tau) = \\min_{q\\in Y}\\max_{t\\in \\tau}d_Y(q,\\varphi_Y(t))\\leq \\max_{t\\in\\tau}d_Y(\\varphi_Y(t_\\tau),\\varphi_Y(t))\\leq \\max_{t\\in\\tau}d_X(\\varphi_X(t_\\tau),\\varphi_X(t))+2\\eta\\\\ = \\varphi^*_XF_X^W(\\tau) + 2\\eta,\n\\end{multline}\n\nwhere the last inequality follows from \\refeq{eq:param}. The proof follows by interchanging the roles of $X$ and $Y$.\n\\end{proof}\n\n\n\\section{$d_{\\mathcal{F}}$ is geodesic}\\label{sec:geodesic}\nIn this section we construct geodesics between any pair $\\mathbf{X}$ and $\\mathbf{Y}$ of filtered spaces and obtain a strengthening of Theorem \\ref{theo:stab-pullback}.\n\n\n\\subsection{Geodesics}\nGiven $\\mathbf{X}$ and $\\mathbf{Y}$ in $\\mathcal{F}$ consider $\\mathcal{T}^\\mathrm{opt}(\\mathbf{X}.\\mathbf{Y})$ the set of all minimizing tripods: That is, for any $(Z,\\varphi_X,\\varphi_Y)\\in\\mathcal{T}(\\mathbf{X},\\mathbf{Y})$ we have $\\|\\varphi_X^\\ast F_X-\\varphi_Y^*F_Y\\|_{\\ell^\\infty(\\pow{Z}} = d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y}).$\n\nFor each minimizing tripod $T=(Z,\\varphi_X,\\varphi_Y)\\in\\mathcal{T}^\\mathrm{opt}(\\mathbf{X},\\mathbf{Y})$ consider the curve $$\\mbox{$\\gamma_T:[0,1]\\rightarrow \\mathcal{F}$ defined by $t\\mapsto \\mathbf{Z_t}:=(Z,F_t)$}$$ where \n $$F_t:=(1-t)\\cdot \\varphi_X^*F_X+t\\cdot \\varphi_Y^* F_Y.$$\n\n\n\\begin{theorem}\nFor each $T\\in\\mathcal{T}^\\mathrm{opt}(\\mathbf{X},\\mathbf{Y})$ the curve $\\gamma_T$ is a geodesic between $\\mathbf{X}$ and $\\mathbf{Y}$. Namely, for all $s,t\\in[0,1]$ one has:\n$$d_{\\mathcal{F}}(\\gamma_T(s),\\gamma_T(t))=|s-t|\\cdot d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y}).$$\n\\end{theorem}\n\\begin{proof}\nLet $\\eta = d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y})$. We check that $$(\\ast )\\,\\,\\,\\,d_{\\mathcal{F}}(\\gamma_T(s),\\gamma_T(t))\\leq |s-t|\\cdot \\eta$$ and notice that this is enough. Otherwise, let $s<t$ in $[0,1]$ be such that $d_{\\mathcal{F}}(\\gamma_T(s),\\gamma_T(t))\\leq (t-s)\\cdot \\eta$. Then,   by the triangle inequality for $d_{\\mathcal{F}}$ we would have\n$$\\eta \\leq d_{\\mathcal{F}}(\\gamma_T(0),\\gamma_T(s))+ d_{\\mathcal{F}}(\\gamma_T(s),\\gamma_T(t))+ d_{\\mathcal{F}}(\\gamma_T(t),\\gamma_T(1))$$\nwhich by $(*)$ and the definition of $s$ and $t$ would be \\emph{strictly} smaller than $\\eta\\cdot s+(t-s)\\cdot \\eta + (1-t)\\cdot \\eta = \\eta$, a contradiction. \n\nNow, in order to verify $(\\ast)$, we need to construct a tripod between $\\gamma_T(s) = (Z,F_s)$ and $\\gamma_T(t) = (Z,F_t)$. We consider the tripod $(Z,\\mathrm{id}_Z,\\mathrm{id}_Z)$ and notice that this tripod gives\n\\begin{align}\nd_{\\mathcal{F}}(\\gamma_T(s),\\gamma_T(t))&\\leq \\max_{\\tau\\in\\pow{Z}}\\big|F_s(\\tau)-F_t(\\tau)\\big|\\\\\n&= \\max_{\\tau\\in\\pow{Z}}\\bigg|\\big((1-s)-(1-t)\\big)\\cdot F_X(\\varphi_X(\\tau)) - (t-s)\\cdot F_Y(\\varphi_Y(\\tau))\\bigg|\\\\\n&=|t-s|\\cdot \\max_{\\tau\\in\\pow{Z}}| F_X(\\varphi_X(\\tau)) - F_Y(\\varphi_Y(\\tau))|\\\\\n&=|t-s|\\cdot \\eta.\n\\end{align}\n\\end{proof}\n\n\\subsection{A strengthening of Theorem \\ref{theo:stab-pullback}}\nRecall the definition of \\emph{length} in a metric space $(M,d_M)$. For a curve $\\alpha:[0,1]\\rightarrow M$, its length is\n$$\\mathrm{L}_M(\\alpha):= \\sup\\left\\{\\sum_{i=0}^{n-1}d_M(\\alpha(t_i),\\alpha(t_{i+1}))|\\,0=t_0<t_1<\\cdots t_n=1\\right\\}.$$\n\nWe now use the construction of geodesics above to strengthen Theorem \\ref{theo:stab-pullback}. \n\n\n\\begin{theorem}\\label{theo:strengthening}\nFor all $\\mathbf{X},\\mathbf{Y}\\in\\mathcal{F}$ and $k\\in\\mathbb{N}$ one has   \n$$\\sup_{T\\in\\mathcal{T}^\\mathrm{opt}} \\mathrm{L}_\\mathcal{D}(\\mathrm{D}_k(\\gamma_T)) \\leq d_\\mathcal{F}(\\mathbf{X},\\mathbf{Y}).$$\n\\end{theorem}\n\n\\begin{remark}[Strengthening] That this theorem is a strengthening can be argued as follows. Firstly, by definition of length  $\\mathrm{L}_\\mathcal{D}(\\mathrm{D}_k(\\gamma_T))\\geq d_\\mathcal{D}(\\mathrm{D}_k(\\gamma_T(0)),\\mathrm{D}_k(\\gamma_T(1))) = d_\\mathcal{D}(\\mathrm{D}_k(\\mathbf{X}),\\mathrm{D}_k(\\mathbf{Y})).$ Therefore the lower bound provided by Theorem \\ref{theo:strengthening} cannot be smaller than the one provided by Theorem \\ref{theo:stab-pullback}.\n\nNow, to argue about the improvement offered by the new lower bound consider the two spaces from Remark \\ref{rem:non-tight}. For those spaces, a minimizing tripod is $T=(X,\\mathrm{id}_X,\\varphi)$ where $\\varphi:X\\rightarrow \\{\\ast\\}$ is the unique map. In that case, one sees that $F_t = (1-t)\\cdot F_X$ and therefore $\\alpha_T(t):=\\mathrm{D}_0(\\gamma_T(t)) = \\{[0,\\infty),[0,(1-t))\\}.$ Notice that for ant two $t,s\\in[0,1]$ such that $|s-t|$ is sufficiently small, the bottleneck distance $d_{\\mathcal{D}}(\\alpha_T(t),\\alpha_T(s))$ equals $|t-s|.$ Then, $\\mathrm{L}_{\\mathcal{D}}(\\alpha) = 1=d_{\\mathcal{F}}(\\gamma_T(0),\\gamma_T(1)) >\\frac{1}{2}=d_{\\mathcal{D}}(\\alpha_T(0),\\alpha_T(1)),$ which is the bound provided by Theorem \\ref{theo:stab-pullback}.\n\n \\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{theo:strengthening}]\nLet $T\\in \\mathcal{T}^\\mathrm{opt}(\\mathbf{X},\\mathbf{Y})$ and let $0=t_0<\\cdots <t_n=1$ be any partition of $[0,1].$ Then, write\n\\begin{align*}\n\\sum_{i=0}^{n-1} d_\\mathcal{D}\\big(\\mathrm{D}_k(\\gamma_T(t_i)), \\mathrm{D}_k(\\gamma_T(t_{i+1}))\\big) &\\leq \\sum_{i=1}^{n-1} d_\\mathcal{F}\\big(\\gamma_T(t_i), \\gamma_T(t_{i+1})\\big)\\\\\n&=\\sum_{i=0}^{n-1} (t_{i+1}-t_i)\\cdot d_{\\mathcal{F}}(\\gamma_{T}(0),\\gamma_T(1))\\\\\n&=d_{\\mathcal{F}}(\\gamma_{T}(0),\\gamma_T(1))\\\\\n&= d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y}),\n\\end{align*}\nwhere the first inequality follows from Theorem \\ref{theo:stab-pullback}, and the equality immediately after it follows from the fact that $\\gamma_T$ is a geodesic.\n\nThus, we have for any partition $0=t_0<\\cdot <t_1=1$ that \n$$\\sum_{i=0}^{n-1} d_\\mathcal{D}\\big(\\mathrm{D}_k(\\gamma_T(t_i)), \\mathrm{D}_k(\\gamma_T(t_{i+1}))\\big)\\leq d_{\\mathcal{F}}(\\mathbf{X},\\mathbf{Y}).$$\nTaking supremum over all possible partitions yields the claim.\n\\end{proof}\n\n\\begin{remark}\nThe techniques of the above theorem and the results in \\cite{chowdhury2016constructing} imply similar strengthenings of Propositions \\ref{prop:stab-R} and \\ref{prop:stab-C}.\n\\end{remark}\n\n\n\n\\section{Discussion}\nIt seems possible to extend some of these ideas to the case of non necessarily finite filtered spaces.\n\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA quantum walk~\\cite{Kempe2003,SV,Kendon2006} is the quantum mechanical version of the classical walk and can either be discrete-time or continuous-time~\\cite{Childs2010}.  Quantum walks  provide a fascinating and versatile framework for studying a myriad of physical processes ranging from biological systems~\\cite{Mohseni2008} to satisfiability problems in computer science \\cite {HEN} to universal quantum computation \\cite{Child}.  A variety of experimental substrates have been used to implement proof-of-principle demonstrations of quantum walks; these include, single photons~\\cite{Schreiber2010,Broome2010,Peruzzo2010,Schreiber2012,Sansoni2012,Jeong2013}, trapped ions~\\cite{Zahringer2010,Schmitz2009}, and atoms in optical lattices~\\cite{Karski2009,Genske2013}. \n\nOne of the most fascinating aspects of quantum walks is related to the studying of topological phases in solid state physics~\\cite{Hasan2010,Qi2011}. It was shown~\\cite{Kit,Kitagawa2012,Kitagawa2012b} that the discrete version of quantum walks can be used to model and understand topological phases. This initial work has been theoretically extended~\\cite{Lindner2011,Obuse2011,Asboth2012,Asboth2013,Ramasesh2016,Rakovszky2015} and also experimentally demonstrated~\\cite{Cardano2016}.\n\nIn this work, we consider the continuous-time limit of discrete topological quantum walks. This is investigated with simple and split-step evolutions in which we identify their phases and boundaries. We confine ourselves to space-discrete and time-continuous walks which become relativistic in space limits~\\cite{Strauch}. One way to create non-trivial topological phases is to fashion the quantum walks in split steps using two coins with a requirement that the second coin's rotation is inhomogeneous~\\cite{Kit}. \n\nWe show that even with simple evolution one can have topologically bound states by considering an equivalent representation that splits the single coin's rotation in two steps. This is then generalized to split step with an additional coin providing a rich structure to the quantum walks adding new phases and boundaries. A key consideration in formulating continuous-time quantum walks from their discrete counterparts is to take the limit in the product topology of the spaces $\\theta\\in\\mathscr{R}\\times{R}\\ni{\\omega}$, where, $\\theta$ represents the rotation angle and $\\omega$ enters the limiting process as we study the systems in momentum space.\n\nOur discussion is organized as follows. In Section \\ref{sec:1}, we analyze the simple-step walk initially from a discrete-limit walk, and then from a continuous-time limit, perspective. In Section \\ref{sec:2}, we consider the case of split-step walk again with both a discrete-time walk and a continuous-time walk. We offer conclusions in Section \\ref{sec:3}.\n\n\n\\section{Simple-step Walk}\n\\label{sec:1}\n\nBefore we introduce the continuous-time limit for the simple-step walk, we rederive the discrete case. These dicsrete results were originally presented in~\\cite{Kit,Kitagawa2010,Obuse2015}. We do this for completeness for the reader and also to show that our results match, so we have a good limit in the continuum.\n\n\\subsection{Discrete-time walk}\nConsider a system consisting of a quantum coin and a quantum walker. The quantum coin is a qubit in $\\mathcal{H}_c$, whereas the Hilbert space of the quantum walker $\\mathcal{H}_w$ is infinite-dimensional. Let $\\{ |0\\rangle_c , |1\\rangle_c \\}$ and $\\{ |x\\rangle_w, x\\in\\mathbb{Z}\\}$ be orthonormal bases in $\\mathcal{H}_c$ and $\\mathcal{H}_w$, respectively. The integer $x$ can be thought of as the position of the (classical) walker. A general state can be written as\n\\begin{equation} |\\Psi\\rangle = \\sum_x \\left[ \\Psi_0(x) |0\\rangle_c + \\Psi_1(x) |1\\rangle_c \\right] |x\\rangle_w\n\\end{equation}\nIn momentum space, we define\n\\begin{equation} |k\\rangle_w = \\frac{1}{\\sqrt{2\\pi}} \\sum_x e^{-ikx} |x\\rangle_w \\ , \\ \\ k \\in (-\\pi, \\pi) \\end{equation}\nWe deduce\n\\begin{equation} \\langle k |k'\\rangle = \\delta (k-k') \\ , \\ \\ |x\\rangle = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\pi}^\\pi dk\\ e^{ikx} |k\\rangle \\end{equation}\nIntroduce operators which shift the position of the walker,\n\\begin{equation} L^\\pm |x\\rangle_w = |x\\pm 1\\rangle_w \\end{equation}\nand a unitary representing the flipping of the coin,\n\\begin{equation}\\label{eqTtheta} T(\\theta) = \\begin{pmatrix}\n\t\\cos\\theta & \\sin\\theta \\\\ \\sin\\theta & - \\cos\\theta\n\\end{pmatrix}\n= e^{-i\\theta Y} Z~.\n\\end{equation}\nSubsequently to coin flipping, the walker moves forward if the coin is in $|0\\rangle$, and backwards if the coin is in $|1\\rangle$. This is represented by the operator\n\\begin{equation} S = \\Pi_0 \\otimes L^+ + \\Pi_1 \\otimes L^- \\ , \\ \\ \\Pi_i = |i\\rangle\\langle i| \\ \\ \\ (i=0,1) \\end{equation}\nThus a step of the walker is represented by\n\\begin{equation} U = ST(\\theta) = S e^{-i\\theta Y}  Z~. \\end{equation}\nAfter $n$ steps, the initial state evolves with $U^n$.\nIn this chain, it is convenient to define the repeated block\n\\begin{equation}\\label{eq10a} U' = e^{ -i\\frac{\\theta}{2} Y}  Z  S e^{-i \\frac{\\theta}{2} Y} \\end{equation}\nThe advantage of working with $U'$ instead of $U$ is that $U'$ is of the form\n\\begin{equation}\\label{eq10} U' = iF X F^{-1} X \\end{equation}\nwhere\n\\begin{equation} F = e^{-i \\frac{\\theta}{2} Y} e^{-i\\frac{\\pi}{4} Z} S_-\n\\end{equation}\nand\n\\begin{equation} S_+ = \\Pi_0 \\otimes L^+ + \\Pi_1 \\otimes \\mathbb{I} \\ ,\\ \\\nS_- = \\Pi_0 \\otimes \\mathbb{I} + \\Pi_1 \\otimes L^- \\end{equation}\n$X$\nacts as a ``chiral\" transformation ($X e^{-i\\theta Y} X = e^{i\\theta Y}$, and $X Z X = -Z$).\n\nTo show \\eqref{eq10}, notice that $Z$ commutes with $S_\\pm$, and\n\\begin{equation} F^{-1} =  X S_+ X e^{i\\frac{\\pi}{4} Z}\ne^{i\\frac{\\theta}{2} Y}~, \\\nS = S_- S_+~. \\end{equation}\nTo find the eigenvalues and corresponding eigenstates, it is convenient to switch to the frame in which $X$ is diagonal. This can be accomplished with $e^{i\\frac{\\pi}{4} Y}$, since we have\n$e^{i\\frac{\\pi}{4} Y} X e^{-i\\frac{\\pi}{4} Y} = Z$. We will therefore work with\n\\begin{equation}\\label{eq13} W = e^{i\\frac{\\pi}{4} Y}  U'  e^{-i\\frac{\\pi}{4} Y} \\ , \\ \\ G = e^{i\\frac{\\pi}{4} Y}  F  e^{-i\\frac{\\pi}{4} Y} \\end{equation}\nTo solve the eigenvalue problem, notice that\nthe momentum eigenstates are eigenstates of the shift operators,\n\\begin{equation} L^\\pm |k\\rangle_w = e^{\\pm ik} |k\\rangle_w \\end{equation}\nTherefore,\n\\begin{equation}\\label{eq15} W |k\\rangle_w = W_c\n|k\\rangle_w \\ , \\ \\ G |k\\rangle_w = G_c\n|k\\rangle_w\\end{equation}\nwhere the matrices $W_c$ and $G_c$ act on the coin. We obtain\n\\begin{equation} W_c\n= \\begin{pmatrix}\n\ti\\sin k \\cos\\theta & -\\cos k -i \\sin k \\sin\\theta \\\\ -\\cos k +i \\sin k \\sin\\theta & i \\sin k \\cos\\theta\n\\end{pmatrix} \\end{equation}\nand\n\\begin{equation} G_c\n= e^{i\\frac{\\pi}{4} Y} \\begin{pmatrix}\n\te^{-i\\frac{\\pi}{4} } \\cos\\frac{\\theta}{2} & -e^{i\\frac{\\pi}{4} } e^{-ik} \\sin\\frac{\\theta}{2} \\\\ e^{-i\\frac{\\pi}{4} } \\sin\\frac{\\theta}{2} & e^{i\\frac{\\pi}{4} } e^{-ik} \\cos\\frac{\\theta}{2}\n\\end{pmatrix} e^{-i\\frac{\\pi}{4} Y} \\end{equation}\nThe eigenvalues of $W_c$ are\n\\begin{equation} e^{-i\\omega_\\pm (k)} =  i\\cos\\theta \\sin k \\mp  \\sqrt{1 - \\cos^2\\theta \\sin^2 k} \\end{equation}\nNote that $\\sin\\omega_\\pm = \\mp\\cos\\theta \\sin k$.\n\nWe introduce two topological invariants,\n\\begin{equation} \\nu_\\alpha = \\frac{1}{2\\pi i} \\int_{-\\pi}^{\\pi} dk \\frac{d}{dk} \\ln {}_c\\langle\\alpha|G_c|0\\rangle_c \\ , \\ \\ \\alpha = 0,1 \\end{equation}\nExplicitly,\n\\begin{equation} \\nu_0 =  \\int_{-\\pi}^{\\pi} \\frac{dk}{2\\pi} \\frac{1}{1 - i z e^{ik}} \\ , \\ \\\nz = \\frac{1+\\tan \\frac{\\theta}{2}}{1-\\tan \\frac{\\theta}{2}} \\end{equation}\nWe obtain\n\\begin{equation} \\nu_0 = \\left\\{ \\begin{array}{ccc}\n\t1 & , & \\theta > 0 \\\\ 0 & , & \\theta < 0\n\\end{array}\\right. \\end{equation}\nSimilarly,\n\\begin{equation} \\nu_1 =  \\int_{-\\pi}^{\\pi} \\frac{dk}{2\\pi} \\frac{1}{1 + i e^{ik} \/z} = \\left\\{ \\begin{array}{ccc}\n\t0 & , & \\theta > 0 \\\\ 1 & , & \\theta < 0\n\\end{array}\\right. \\end{equation}\nThus, we have two distinct topological phases with $\\nu_0 = 1$, $\\nu_1 = 0$ ($\\nu_0 = 0$, $\\nu_1 = 1$) for $\\theta > 0$ ($\\theta <0$).\n\nIn position space, the action of $W$ can be deduced directly from \\eqref{eq10a} and \\eqref{eq13}. We obtain\n\\begin{eqnarray}\\label{eq8} \\Psi_0 (x) &\\to& \\frac{\\cos\\theta}{2}    \\left[ \\Psi_0 (x-1) - \\Psi_0 (x+1) \\right] \\nonumber\\\\\n&& - \\frac{1 + \\sin\\theta}{2}  \\Psi_1 (x-1) - \\frac{1 -  \\sin\\theta}{2}  \\Psi_1 (x+1)\\nonumber\\\\\n\\Psi_1 (x) &\\to& \\frac{\\cos\\theta}{2}   \\left[ \\Psi_1 (x-1) - \\Psi_1 (x+1) \\right] \\nonumber\\\\\n&& - \\frac{1 - \\sin\\theta}{2}  \\Psi_0 (x-1) - \\frac{1 + \\sin\\theta}{2}  \\Psi_0 (x+1)\n\\nonumber\\\\\n\\end{eqnarray}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{QW-trapped.png}\n\t\\caption{Trapped state at the boundary of two topological phases of a discrete simple-step walk.}\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.4]{QW.png}\n\t\\caption{State being reflected at the boundary of two topological phases of a discrete simple-step walk.}\\label{fig:2}\n\\end{figure}\n\nIn Fig.~\\ref{fig:1}, we see that a discrete-time simple-step topological walk gives rise to bound states at $x=0$ and $x=-1$, which is chosen to be the boundary of two distinct topological phases. The initial state was centered around $x=0$ with a small spread $\\Delta x$, and $\\Psi_0(x) = \\Psi_1(x)$. As the system evolves in time, the parts of the state near the boundary diffuse ballistically away from the boundary, but the part at the boundary remains protected. In Fig.~\\ref{fig:2}, we choose the initial state around $x=50$, i.e., entirely within a single topological phase. In this case, we see that the quantum walk diffuses in both directions ballistically. The part of the walk that reaches the boundary of the other phase is reflected back and continues to diffuse away from the boundary ballistically without entering the region of the other topological phase.\n\n\\subsection{Continuous-time limit}\n\nTo go over to the continuous-time limit, we set $\\theta = \\frac{\\pi}{2} - \\epsilon$, and consider a scaling limit in which $\\epsilon \\to 0$ and $n\\to\\infty$, so that the product $n\\epsilon $ remains finite. In this limit, $\\omega_+ \\to \\pi$, and $\\omega_- \\to 0$.\nNotice that $W^2 = \\mathbb{I} + \\mathcal{O} (\\epsilon)$, which is not the case for $W$. We will therefore consider the limit of an \\emph{even} number of steps.\n\nSetting\n\\begin{equation}\\epsilon = \\gamma\\Delta t \\ , \\ \\ t = 2n\\Delta t \\end{equation}\nApplying \\eqref{eq8} twice, we obtain\n\\begin{eqnarray}\\label{eq8a} &&\\Psi_0 (x, n+2) - \\Psi_0 (x,n) \\nonumber\\\\\n&& = \\gamma \\Delta t \\left[  \\Psi_1 (x,n) - \\Psi_1 (x-2,n) \\right] \\nonumber\\\\\n&& \\Psi_1 (x, n+2) - \\Psi_1 (x,n) \\nonumber\\\\\n&&=  -\\gamma \\Delta t \\left[  \\Psi_0 (x,n) - \\Psi_0 (x+2,n)  \\right]\n\\end{eqnarray}\nfrom which we deduce the continuous-time quantum walk in the limit $\\Delta t\\to 0$,\n\\begin{eqnarray}\\label{eq8b}  \\frac{\\partial\\Psi_0 (x, t)}{\\partial t} &=& \\gamma  \\left[\n\\Psi_1 (x,t) -  \\Psi_1 (x-2,t) \\right] \\nonumber\\\\\n\\frac{\\partial\\Psi_1 (x, t)}{\\partial t} &=& \\gamma  \\left[\n- \\Psi_0 (x,t) +  \\Psi_0 (x+2,t) \\right]\n\\end{eqnarray}\nDefining\n\\begin{equation}\\label{eq27} \\Phi_\\pm (x) = \\pm \\Psi_0 (x) +  \\Psi_1 (x-1 ) \\end{equation}\nwe obtain the decoupled equations\n\\begin{equation}\\label{eq37} \\frac{\\partial\\Phi_\\pm (x, t)}{\\partial t} = \\pm \\gamma  \\left[\n\\Phi_\\pm (x+1,t) - \\Phi_\\pm (x-1,t) \\right]\n\\end{equation}\nrelated to each other by time reversal.\n\nWorking similarly in the other phase (with $\\nu_0 = 0$), we obtain in the continuous-time limit,\n\\begin{eqnarray}\\label{eq8bb}  \\frac{\\partial\\Psi_0 (x, t)}{\\partial t} &=& \\gamma  \\left[\n\\Psi_1 (x,t) -  \\Psi_1 (x+2,t) \\right] \\nonumber\\\\\n\\frac{\\partial\\Psi_1 (x, t)}{\\partial t} &=& \\gamma  \\left[\n- \\Psi_0 (x,t) +  \\Psi_0 (x-2,t) \\right]\n\\end{eqnarray}\nIt is easy to see that these are equivalent to the decoupled Eqs.\\ \\eqref{eq37} under the definition $\\Phi_\\pm (x) = \\mp \\Psi_0 (x) +  \\Psi_1 (x+1 )$ (\\emph{cf.}\\ with Eq.\\ \\eqref{eq27}).\n\nThe above results are no longer valid if the coin parameters are spatially dependent. In particular, we are interested in the case in which there are two regions in space which have different topological numbers. This can be achieved by having $\\theta > 0$ and $\\theta < 0$ in the respective regions.\nThen the above results are valid in the bulk of each region, but not along their boundaries.\n\nFor the continuous-time limit, we need to consider the limit of $4n$ steps as $n\\to\\infty$. This is because $W^2 \\ne \\mathbb{I} + \\mathcal{O} (\\epsilon)$, due to an obstruction at the boundary of the two regions, but we still have $W^4 = \\mathbb{I} + \\mathcal{O} (\\epsilon)$.\n\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{CTQW-trapped.png}\n\t\\caption{Trapped state at the boundary of two topological phases of a continuous simple-step walk.}\\label{fig:3}\n\\end{figure}\n\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.4]{CTQW.png}\n\t\\caption{State being reflected at the boundary of two topological phases of a continuous simple-step walk.}\\label{fig:4}\n\\end{figure}\nIn the continuous-time limit, away from the boundary the results match those of single topological phases. More precisely, for $x\\ge 2$, we recover Eq.\\ \\eqref{eq8bb} whereas for $x\\le -3$, we recover Eq.\\ \\eqref{eq8b}.\nNear the boundary (for $-3 < x < 2$), we obtain\n\\begin{eqnarray} \\frac{\\partial\\Psi_0(1)}{\\partial t}\n&=& \\gamma  \\Psi_1 (1) \\nonumber\n\\\\ \\frac{\\partial\\Psi_1 (1)}{\\partial t}  &=& - \\gamma \\left[  \\Psi_0 (1) -  \\Psi_0 (3) \\right]  \\nonumber\n\\\\ \\frac{\\partial\\Psi_0(0)}{\\partial t}\n&=& 0 \\nonumber\\\\\n\\frac{\\partial\\Psi_1 (0)}{\\partial t}  &=&  \\gamma  \\Psi_0 (2)  \\nonumber\n\\\\\n\\frac{\\partial\\Psi_0 (-1)}{\\partial t}  &=&  0\n\\\\\n\\frac{\\partial\\Psi_1(-1)}{\\partial t}\n&=& \\gamma  \\Psi_0 (-3)  \\nonumber \\nonumber\\\\\n\\frac{\\partial\\Psi_0 (-2)}{\\partial t}  &=& \\gamma  \\Psi_1 (-2)  \\nonumber\\\\\n\\frac{\\partial\\Psi_1(-2)}{\\partial t}\n&=& -\\gamma \\left[  \\Psi_0 (-2) -  \\Psi_0 (-4) \\right]\n\\end{eqnarray}\nNotice that $\\Psi_0 (0)$ and $\\Psi_1 (-1)$ decouple, so if initially $\\Psi_0(0) =1$, then the walk is trapped at $x=0$, and similarly for $\\Psi_1(-1)$. Moreover, if a walk starts entirely within one of the topologically distinct regions (or at the boundary, $x=0$), it remains in it. This matches the asymptotic behavior observed in the discrete-time case (cf. Fig.~\\ref{fig:1}).\n\nBy defining\n$\\Phi_\\pm$ as in \\eqref{eq27}, we recover the equations of motion for decoupled walks\n\\eqref{eq37} away from the boundary (for $|x|\\ge 2$). In the continuous-time limit of the simple-step quantum walk, we see the same behavior as we did for the discrete quantum walk. We have the same two bound states near the boundary at $x = 0,1$, respectively. This is shown in Fig.~\\ref{fig:3}. Away from the boundary, the system diffuses ballistically, and is reflected at the boundary, as shown in Fig.~\\ref{fig:4}.\n\n\\section{Split-step Walk}\n\\label{sec:2}\n\nAs with the previous section we rederive the discrete-time case here for completeness. The original results can be found in \\cite{Kit,Kitagawa2010,Obuse2015} along with our new results in the continuous-time limit following thereafter.\n\n\\subsection{Discrete-time walk}\nFor the split-step walk, we flip two coins, $T(\\theta_1)$ and $T(\\theta_2)$, where\n$T(\\theta)$ is defined in \\eqref{eqTtheta}.\nA step of the walker is represented by\n\\begin{equation} U = S_-T(\\theta_2)S_+ T(\\theta_1) \\end{equation}\nIt is convenient to define the repeated block\n\\begin{equation}\\label{eq10as} U' = e^{-i \\frac{\\theta_1}{2} Y} Z  S_- e^{-i \\theta_2 Y} Z S_+ e^{-i\\frac{\\theta_1}{2} Y} \\end{equation}\nThe advantage of working with $U'$ instead of $U$ is that $U'$ is of the form\n\\begin{equation}\\label{eq10s} U' = -F X F^{-1} X \\end{equation}\nwhere\n\\begin{equation}\\label{eqFsplit} F = e^{-i\\frac{\\theta_1}{2} Y} ZS_- e^{-i\\frac{\\theta_2}{2} Y}\n\\end{equation}\nAfter switching to a frame in which $X$ is diagonal, we arrive at $W$ given by \\eqref{eq13}, and $W_c$ acting on a coin (Eq.\\ \\eqref{eq15}) as\n\\begin{equation} W_c\n= \\begin{pmatrix}\n\t\\beta_0 & \\beta_1 \\\\ -\\beta_1^\\ast & \\beta_0\n\\end{pmatrix} \\end{equation}\nwhere\n\\begin{eqnarray} \\beta_0 &=& \\cos k \\cos\\theta_1\\cos\\theta_2 + \\sin\\theta_1\\sin\\theta_2 \\nonumber\\\\\n\\beta_1 &=& -(i\\sin k + \\cos k \\sin\\theta_1)\\cos\\theta_2 + \\cos\\theta_1\\sin\\theta_2\\ \\ \\\n\\end{eqnarray}\nThe eigenvalues are\n\\begin{equation} e^{-i\\omega_\\pm} = \\beta_0 \\mp i\\sqrt{1-\\beta_0^2}  \\end{equation}\nSimilarly, for $G_c$ defined by \\eqref{eq13} and \\eqref{eq15} with $F$ given by\n\\eqref{eqFsplit}, we obtain\n\\begin{equation} G_c\n= e^{-ik\/2}\\begin{pmatrix}\n\t\\gamma_0 & \\gamma_1 \\\\ \\gamma_1^\\ast & -\\gamma_0^\\ast\n\\end{pmatrix} \\end{equation}\nwhere\n\\begin{eqnarray} \\gamma_0 &=& \\cos \\frac{k}{2} \\cos\\frac{\\theta_-}{2} + i \\sin \\frac{k}{2} \\cos\\frac{\\theta_+}{2} \\nonumber\\\\\n\\gamma_1 &=& \\cos \\frac{k}{2} \\sin\\frac{\\theta_-}{2} - i \\sin \\frac{k}{2} \\cos\\frac{\\theta_+}{2}\\ \\ \\\n\\end{eqnarray}\nand we defined $\\theta_\\pm = \\theta_1 \\pm \\theta_2$.\n\nFor the two topological invariants, we obtain, respectively,\n\\begin{equation} \\nu_0 = \\int_{-\\pi}^{\\pi} \\frac{dk}{2\\pi}\\frac{1}{1-z_0 e^{ik} } \\ , \\ \\ z_0 = \\frac{\\cos\\frac{\\theta_+}{2} + \\sin\\frac{\\theta_-}{2}}{\\cos\\frac{\\theta_+}{2} - \\sin\\frac{\\theta_-}{2}} \\end{equation}\nand\n\\begin{equation} \\nu_1 = \\int_{-\\pi}^{\\pi} \\frac{dk}{2\\pi}\\frac{1}{1+ z_1 e^{ik} } \\ , \\ \\ z_1 = \\frac{\\cos\\frac{\\theta_-}{2} - \\sin\\frac{\\theta_+}{2}}{\\cos\\frac{\\theta_-}{2} + \\sin\\frac{\\theta_+}{2}}\n\\end{equation}\nIt is easy to see that\n\\begin{equation} \\nu_\\alpha = \\left\\{ \\begin{array}{ccc}\n\t1 & , & |z_\\alpha| < 1 \\\\ 0 & , & |z_\\alpha|>1\n\\end{array}\\right. \\ , \\ \\\n\\alpha = 0,1~.\n\\end{equation}\nThus we obtain four different topological phases,\n\\begin{align}\n\tI (\\nu_0,\\nu_1) &= (1, 1)\\\\\n\tII (\\nu_0,\\nu_1) &= (0, 0)\\\\\n\tIII (\\nu_0,\\nu_1)  &= (0,1)\\\\\n\tIV (\\nu_0,\\nu_1) &= (1,0)\n\\end{align}\nIn position space, the action of $W$ yields\n\\begin{widetext}\n\\begin{eqnarray}\\label{eq8split} \\Psi_0 (x) &\\to& \\frac{\\cos\\theta_1\\cos\\theta_2}{2}    \\left[ \\Psi_0 (x-1) + \\Psi_0 (x+1) \\right] + \\sin\\theta_1\\sin\\theta_2 \\Psi_0 (x)\\nonumber\\\\\n&& - \\frac{1 + \\sin\\theta_1}{2}\\cos\\theta_2  \\Psi_1 (x-1) + \\frac{1 -  \\sin\\theta_1}{2}\\cos\\theta_2  \\Psi_1 (x+1) + \\cos\\theta_1 \\sin\\theta_2 \\Psi_1 (x)\\nonumber\\\\\n\\Psi_1 (x) &\\to& \\frac{\\cos\\theta_1\\cos\\theta_2}{2}    \\left[ \\Psi_1 (x-1) + \\Psi_1 (x+1) \\right] + \\sin\\theta_1\\sin\\theta_2 \\Psi_1 (x) \\nonumber\\\\\n&& - \\frac{1 - \\sin\\theta_1}{2}\\cos\\theta_2  \\Psi_0 (x-1) + \\frac{1 +  \\sin\\theta_1}{2}\\cos\\theta_2  \\Psi_0 (x+1) - \\cos\\theta_1 \\sin\\theta_2 \\Psi_0 (x)\n\\end{eqnarray}\n\\end{widetext}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{SQW_type1-trapped.png}\n\t\\caption{Trapped state at the boundary of topological phases \\emph{I} and \\emph{II} of a continuous simple-step walk.}\\label{fig:5}\n\\end{figure}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{SQW1.png}\n\t\\caption{State being reflected at the boundary of topological phases \\emph{I} and \\emph{II} of a continuous simple-step walk.}\\label{fig:7}\n\\end{figure}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{SQW_type2-trapped.png}\n\t\\caption{Trapped state at the boundary of topological phases \\emph{I} and \\emph{III} of a continuous simple-step walk.}\\label{fig:6}\n\\end{figure}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{SQW2.png}\n\t\\caption{State being reflected at the boundary of topological phases \\emph{I} and \\emph{III} of a continuous simple-step walk.}\\label{fig:8}\n\\end{figure}\n\nFigures \\ref{fig:5} and \\ref{fig:7} show the behavior of a discrete split-step quantum walk with topological phases \\emph{III} for $x\\ge 0$ and \\emph{IV} for $x<0$. In Fig.~\\ref{fig:5}, we see the same two bound states near the boundary of the two phases as in the case of the simple-step walk. This is expected, because the boundary between phases \\emph{III} and \\emph{IV} is equivalent to the boundary between two topologically distinct phases of a simple-step quantum walk. This  behavior has been demonstrated experimentally by Kitagawa, \\emph{et al.}~\\cite{Kitagawa2012}. Figure \\ref{fig:7} shows the behavior of the split-step quantum walk away from the boundary. In particular, a system whose initial state is entirely within a single topological phase will never leave the region of that phase. When such a state comes in contact with the boundary between two phases, it is reflected back.\n\nFigures \\ref{fig:6} and \\ref{fig:8} show the behavior of a discrete split-step quantum walk with topological phases \\emph{I} for $x\\ge 0$ and \\emph{III} for $x<0$. Unlike in the previous case, there is a single bound state at the boundary between phases \\emph{I} and \\emph{III}, at $x=-1$. This has also been observed experimentally~\\cite{Kitagawa2012}. Away from the boundary, we obtain the same qualitative behavior as in the previous case, including reflection of the state at the boundary.\n\n\\subsection{Continuous-time limit}\n\nThe continuous-time limit is obtained as $\\theta_1 \\to\\pm \\frac{\\pi}{2}$, $\\theta_2\\to 0$, or $\\theta_2 \\to\\pm \\frac{\\pi}{2}$, $\\theta_1\\to 0$. They correspond to the four distinct topological phases listed above, respectively,\n\\begin{align}\n\tI (\\theta_1,\\theta_2) &= (0, \\frac{\\pi}{2})\\\\\n\tII (\\theta_1,\\theta_2) &= (0, -\\frac{\\pi}{2})\\\\\n\tIII (\\theta_1,\\theta_2) &= (\\frac{\\pi}{2},0)\\\\\n\tIV (\\theta_1,\\theta_2) &= (-\\frac{\\pi}{2},0)\n\\end{align}\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{CTSQW_type1-trapped.png}\n\t\\caption{Trapped state at the boundary of topological phases \\emph{I} and \\emph{II} of a continuous simple-step walk.}\\label{fig:9}\n\\end{figure}\n\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{CTSQW1.png}\n\t\\caption{State being reflected at the boundary of topological phases \\emph{I} and \\emph{II} of a continuous simple-step walk.}\\label{fig:10}\n\\end{figure}\nIn phase $I$, we set $\\theta_1 = \\epsilon_1$, $\\theta_2 = \\frac{\\pi}{2} - \\epsilon_2$, and consider the scaling limit in which $\\epsilon_{1,2}\\to 0$, $n\\to\\infty$, so that the products $n\\epsilon_{1,2}$ remain finite. In this limit, $\\omega_\\pm \\to \\pm \\frac{\\pi}{2}$. We have $W^2 = -\\mathbb{I} + \\mathcal{O} (\\epsilon_{1,2})$. We will therefore multiply the wavefunction by a phase $i$, and consider the limit of an even number of steps. Setting\n\\begin{equation} \\epsilon_{1,2} = \\gamma_{1,2} \\Delta t \\ , \\ \\ t = 2n\\Delta t \\end{equation}\nwe obtain from \\eqref{eq8split} the equations of motion in the limit $\\Delta t \\to 0$,\n\\begin{eqnarray}\\label{eqomI} \\frac{\\partial\\Psi_0 (x)}{\\partial t} &=& -2\\gamma_1 \\Psi_1 (x) - \\gamma_2 \\left[ \\Psi_1 (x-1)   + \\Psi_1 (x+1) \\right] \\nonumber\\\\\n\\frac{\\partial\\Psi_1(x)}{\\partial t} &=& 2\\gamma_1 \\Psi_0 (x) + \\gamma_2 \\left[  \\Psi_0 (x-1)    + \\Psi_0 (x+1) \\right]\\ \\ \\ \\\n\\end{eqnarray}\nWorking similarly, we obtain the same equations of motion \\eqref{eqomI} in the continuous-time limit in phase $II$.\n\nDefining\n\\begin{equation}\\label{eq47} \\Phi_\\pm (x) = \\pm i \\Psi_0 (x) + \\Psi_1 (x-1) \\end{equation}\nwe obtain the decoupled equations of motion,\n\\begin{equation}\\label{eq48} \\frac{\\partial\\Phi_\\pm (x)}{\\partial t} = \\pm 2i\\gamma_2 \\Phi_\\pm (x) \\pm i \\gamma_1 \\left[ \\Phi_\\pm (x-1)   + \\Phi_\\pm (x+1) \\right] \\end{equation}\nIn phase $III$, we obtain the equations of motion\n\\begin{eqnarray}\\label{eqomIII} \\frac{\\partial\\Psi_0 (x)}{\\partial t} &=&  \\gamma_1 \\left[ \\Psi_1 (x) +\\Psi_1 (x-2) \\right] +2\\gamma_2 \\Psi_1 (x-1)\\nonumber\\\\\n\\frac{\\partial\\Psi_1(x)}{\\partial t} &=&  - \\gamma_1 \\left[ \\Psi_0 (x) +\\Psi_0 (x+2)   \\right] - 2\\gamma_2 \\Psi_0 (x+1) \\ \\ \\ \\ \\\n\\end{eqnarray}\nand in phase $IV$,\n\\begin{eqnarray}\\label{eqomIV} \\frac{\\partial\\Psi_0 (x)}{\\partial t} &=&  \\gamma_1 \\left[ \\Psi_1 (x) +\\Psi_1 (x+2) \\right] +2\\gamma_2 \\Psi_1 (x+1)\\nonumber\\\\\n\\frac{\\partial\\Psi_1(x)}{\\partial t} &=&  - \\gamma_1 \\left[ \\Psi_0 (x) +\\Psi_0 (x-2)   \\right] - 2\\gamma_2 \\Psi_0 (x-1) \\ \\ \\ \\ \\\n\\end{eqnarray}\nThey can be put into the decoupled form \\eqref{eq48}, if we define $\\Phi_\\pm$ as in \\eqref{eq47} in phase $III$, and $\\Phi_\\pm (x) = \\pm i \\Psi_0 (x) + \\Psi_1 (x+1)$ in phase $IV$.\n\nThere are six different boundaries, but only two are qualitatively different. We proceed to consider a representative from each type.\n\nFor a system in phase $III$ for $x\\ge 0$, and phase $IV$ for $x<0$,\nworking as before, we obtain for $x\\le -3$  the equations of motion \\eqref{eqomIII}, and for $x\\ge 1$, the equations of motion \\eqref{eqomIV}. Near the boundary of the two phases, we have\n\t\\begin{eqnarray}\\label{eq53}\n\t\\frac{\\partial\\Psi_0 (-2)}{\\partial t} &=& \\gamma_1 \\Psi _1(-2) + 2\\gamma_2\\Psi _1(-1)\\nonumber\\\\\n\t\\frac{\\partial\\Psi_1 (-2)}{\\partial t} &=& -\\gamma_1\\left( \\Psi _0(-2) + \\Psi_0(-4) \\right) - 2\\gamma_2\\Psi _0(-3)\\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_0 (-1)}{\\partial t} &=& 0\\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_1 (-1)}{\\partial t} &=& -\\gamma_1\\Psi_0(-3) - 2\\gamma_2\\Psi_0(-2)\\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_0 (0)}{\\partial t} &=& 0 \\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_1 (0)}{\\partial t} &=& -\\gamma_1\\Psi _0(2) - 2\\gamma_2\\Psi_0(1)\\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_0(1)}{\\partial t} &=& \\gamma_1\\Psi_1(1) + 2\\gamma_2\\Psi_1(0) \\nonumber\\\\\n\t\t\\frac{\\partial\\Psi_1(1)}{\\partial t} &=& -\\gamma_1\\left( \\Psi_0(1) + \\Psi_0(3) \\right) - 2\\gamma_2\\Psi_0(2)\n\t\\end{eqnarray}\n\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=1.5]{CTSQW_type2-trapped.png}\n\t\\caption{Trapped state at the boundary of topological phases \\emph{I} and \\emph{III} of a continuous simple-step walk.}\\label{fig:11}\n\\end{figure}\n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{CTSQW2.png}\n\t\\caption{State being reflected at the boundary of topological phases \\emph{I} and \\emph{III} of a continuous simple-step walk.}\\label{fig:12}\n\\end{figure}\n\n\nSimilarly, for a system in phase $I$ for $x\\ge 0$, and phase $III$ for $x<0$,\nwe obtain for $x\\ge 1$, the equations of motion \\eqref{eqomI}, and for $x\\le -3$, the equations of motion \\eqref{eqomIII}.\nNear the boundary, we have\n\\begin{eqnarray}\\label{eq54}\n\\frac{\\partial\\Psi_0 (-2)}{\\partial t} &=& \\gamma_1\\left[\\Psi _1(-4)+\\Psi _1(-2)\\right] + 2\\gamma_2\\Psi _1(-3) \\nonumber\\\\\n\\frac{\\partial\\Psi_1 (-2)}{\\partial t} &=& -\\gamma_1\\Psi _0(-2)-2\\gamma_2 \\Psi _0(-1)\\nonumber\\\\\n\\frac{\\partial\\Psi_0 (-1)}{\\partial t} &=& \\gamma_1\\Psi _1(-3) +2\\gamma_2 \\Psi _1(-2)\\nonumber\\\\\n\\frac{\\partial\\Psi_1 (-1)}{\\partial t} &=& 0 \\nonumber\\\\\n\\frac{\\partial\\Psi_0 (0)}{\\partial t} &=& - \\gamma_2\\Psi _1(1)\\nonumber\\\\\n\\frac{\\partial\\Psi_1 (0)}{\\partial t} &=&  \\gamma _2\\Psi _0(1)\\nonumber\\\\\n\\end{eqnarray}\nFigures \\ref{fig:9}, \\ref{fig:10}, \\ref{fig:11}, and \\ref{fig:12} depict the continuous-time limit of the discrete quantum walks shown in Figs. \\ref{fig:5}, \\ref{fig:6}, \\ref{fig:7}, and \\ref{fig:8}, respectively. As expected, the observed behavior matches the asymptotic behavior at and away from the boundary in the discrete case. It should be noted that in the continuous-time limit, the bound states can be found analytically from \\eqref{eq53} for the boundary between \\emph{III} and \\emph{IV}, and \\ref{eq54} for the boundary between phases \\emph{I} and \\emph{III}. These bound states are all in agreement with the asymptotic results obtained above in the corresponding discrete cases, as well as experimental results \\cite{Kitagawa2012}.\n\n\\begin{widetext}\n\n\\begin{figure}[htp]\n    \\centering\n\t\\includegraphics[scale=1.0]{stability.png}\n\t\\caption{Bound states near the boundary between two phases. From left to right, the distributions of $|\\Psi_0(x)|^2, |\\Psi_1(x)|^2, |\\Psi_0(x)|^2 + |\\Psi_1(x)|^2$ after a short time $t=25$. In each plot, the value $R$ corresponds to the ratio between walk parameters $\\gamma_1, \\gamma_2$. (a) At the boundary between phases \\emph{I} and \\emph{II} with the initial state centered at $\\Psi_0(-1)\\ \\text{and}\\ \\Psi_0(0)$. (b) At the boundary between phases \\emph{I} and \\emph{III} with the initial state centered at $\\Psi_1(-1)$}\\label{fig:13}\n\\end{figure}\n\n\\end{widetext}\n\nAs discussed above, the split-step quantum walk with boundary between the topological phases \\emph{III} and \\emph{IV} gives rise to two topologically protected bound states at the boundary. In Fig.~\\ref{fig:13}, we show that these bound states are robust against small changes in the quantum walk parameters.\n\n\\begin{figure}[htp]\n    \\centering\n\t\\includegraphics[scale=0.75]{ballistic.png}\n\t\\caption{Ballistic diffusion of continuous walks without boundary. (a) In phase \\emph{III} with no boundary the quantum walk diffuses ballistically. This diffusion can occur to the right or to the left depending on the initial state. Here the initial state was centered at $\\Psi_0(0)$ and $\\Psi_1(0)$ with $\\Psi_1(0) = \\pm \\Psi_0(0)$. (b) In phase \\emph {I} with no boundary the quantum walk diffuses ballistically with the same behavior regardless of the initial state. }\\label{fig:14}\n\\end{figure}\n\nFigure \\ref{fig:14} shows that in the case of a single topological phase, the state ballistically diffuses away from its initial position. In topological phase \\emph{III}, one can choose the initial state in such a away that the diffusion only occurs in a single direction. Due to the linearity of the system, it follows that the initial state can be chosen so that the diffusion occurs in both directions. However, in phase \\emph{I}, regardless of the choice of initial state, the diffusion invariably occurs in both directions ballistically.\n\n\n\\section{Conclusions}\n\\label{sec:3}\n\nIn conclusion, we have investigated the continuous-time limit of discrete quantum walks with topological phases. In quantum walks, it is common to consider both discrete and continuous-times. In recent years much interest has been devoted to understanding how discrete-time quantum walks can simulator topological insulators. Here we have shown the existence of a continuous-time limit that preserves their topological phases. We considered both simple-step and split-step walks and derived analytically the equations of motion governing their behaviors. Through our analytical solutions we showed the existence of bound states at the boundary of two phases. We also solved the equations of motion numerically in the bulk. In terms of future work it would be interesting to consider the alternative continuous-limit approach given in \\cite{Dheeraj2015} to study topological properties of quantum walks.\n\n\\acknowledgments{We thank C.\\ M. Chandrashekar for illuminating discussions. D.\\ C.\\ and G.\\ S.\\ thank the Army Research Laboratory, where most of this work was performed, for its hospitality and financial support.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{Introduction}\n\nIn the past years measurements of the absolute sky emission at\nfrequency $\\nu \\sim 1$ GHz have been carried out to evaluate the\nbrightness temperature ($T_{cmb}$) of the Cosmic Microwave\nBackground (CMB). Besides the non-trivial problem of assuring an\naccurate absolute calibration of the measured signal, we need to\nremember that the sky emission is a superposition of different\ncontributions. After subtracting the local emissions (mainly due\nto atmosphere inside the main beam, ground and radio frequency\ninterferences in the far side-lobes) the sky brightness\ntemperature ($T_{sky}$) can be written as:\n\n\\begin{equation}\nT_{sky}(\\nu,\\alpha,\\delta) = T_{cmb}(\\nu) + T_{gal}(\\nu,\\alpha,\n\\delta) + T_{UERS}(\\nu)\n\\end{equation}\n\n\\noindent where $T_{gal}$ is the emission of our galaxy and\n$T_{UERS}(\\nu)$ the temperature of the unresolved extragalactic\nradio sources (UERS). In the present paper we evaluate the UERS\nbrightness temperature and its frequency dependence.\n\nThis paper follows a series of others describing the measurements\nof the sky brightness temperature at frequencies close to 1 GHz\ngathered by the TRIS experiment with an angular resolution\n$FWHM_{TRIS} \\sim 20$ deg (\\cite[]{TRIS-I}, \\cite[]{TRIS-II},\n\\cite[]{TRIS-III}). The results obtained in the present paper were\nused to disentangle the components of the sky brightness and to\nevaluate the CMB temperature at the frequencies $\\nu =$ 600, 820\nand 2500 MHz (\\cite[]{TRIS-II}). The aim of this work is to\nprovide a new estimate of the integrated contribution of UERS to\nthe diffuse brightness of the sky. An accurate estimate of\n$T_{UERS}(\\nu)$ is necessary for the TRIS experiment, but also for\nall the experiments aimed at the study of the spectral distortions\nin the Rayleigh-Jeans tail of the CMB spectrum. Deviations from\nthe black-body distribution can be present at low frequency, but\nthe amplitude of the distortions at frequencies around 1 GHz is\nnowadays constrained by past experiments at the level of few tens\nof mK \\cite[]{Fixen_96}.\n\nExperiments like TRIS \\cite[]{TRIS-I} can reach a control of\nsystematics at the level of $\\sim$50 mK, a remarkable improvement\nif compared to previous measurements at the same frequencies. On\nthe other hand, relying on the current knowledge of both amplitude\nand spectrum of the UERS signal \\cite[]{Longair_66}, we can\nestimate that at 600, 820, 1400 and 2500 MHz (where CMB\nobservations have been carried out in the past) the extra-galactic\ncontribution is respectively $810\\pm 180$ mK, $340\\pm 80$ mK,\n$79\\pm 19$ mK and $16\\pm 4$ mK (see for example \\cite{Sironi_90}).\nUsing the current 178 MHz normalization \\cite[]{Longair_66}, for\nstate-of-the-art experiments, this means that the uncertainty\nassociated with the UERS at the lowest frequencies (which are the\nmost interesting when looking for CMB spectral distortions), is\npotentially higher than instrumental systematics. In this paper we\nshow that by exploiting all the data available in literature we\ncan significantly improve the present status of our knowledge\nabout the UERS contribution, and that TRIS-like experiments are\nessentially limited by the current technology. New and updated\nestimates of the brightness temperature of UERS will be useful\nalso for feasibility studies of future experiments in this field.\n\nThis paper is organized as follows: Section \\ref{Sources}\ndiscusses the general properties of the UERS and the data in\nliterature. In Section \\ref{Fit} we describe the procedure to fit\nthe available number counts; in Section \\ref{Brightness} we\ncalculate the UERS sky brightness and its frequency dependence.\nFinally in Section \\ref{Discussion} we discuss the implications of\nthe results obtained for astrophysics and cosmology.\n\n\n\\section{The extragalactic radio sources}\\label{Sources}\n\n\\subsection{The population of sources}\n\nThe unresolved extragalactic radio sources contribute as a blend\nof point sources to the measurements of diffuse emission,\nespecially with poor angular resolution. Actually UERS are an\ninhomogeneous collection of quasars, radio galaxies, and other\nobjects. These can be both compact and extended sources with\ndifferent local radio luminosity function, lifetimes and cosmic\nevolution. An extensive discussion can be found in\n\\cite{Longair_78}.\n\nUsually we can distinguish two populations of radio sources: {\\it\nsteep spectrum} sources if $\\alpha > 0.5$, and {\\it flat spectrum}\nsources if $\\alpha < 0.5$, where $\\alpha$ is the spectral index of\nthe source spectrum ($S(\\nu)\\propto \\nu^{-\\alpha}$). Compact radio\nsources, like quasars, have mostly a flat spectrum ($\\alpha \\simeq\n0$) and are most commonly detected at higher frequencies. On the\nother hand, extended sources like radio-galaxies, have a steep\nspectrum ($\\alpha \\simeq 0.7-0.8$) and dominate low frequency\ncounts (see \\cite[]{Peacock_81a} and \\cite[]{Longair_78}).\n\n{\\it Steep spectrum} sources and {\\it flat spectrum} sources\ncontribute in different ways to the number counts. {\\it Flat\nspectrum} sources are important only at high frequency ($\\nu\n\\gtrsim 2$ GHz). In the same range, {\\it flat spectrum} source\ncounts seem to be comparable to {\\it steep spectrum} source counts\nfor high fluxes, but at low fluxes {\\it steep spectrum} sources\nare still dominating, as shown for example by \\cite{Condon_84a}\nand \\cite{Kellermann_87} The total number counts have been\nsuccessfully fitted using a luminosity evolution model by\n\\cite{Peacock_81a} and \\cite{Danese_87}.\n\n\\subsection{Isotropy}\n\nThe large scale isotropy of the extragalactic radio sources has\nbeen studied by \\cite{Webster_77}. He analyzed several samples\nof sources measured in a cube of 1 Gpc-side, getting an upper\nlimit $\\Delta N \/ N < 3 \\%$ on the fluctuation of the source\nnumber. This limit is set by the finite statistics of the sources\ncontained in the survey: $N \\sim 10^{4}$.\n\n\\cite{Franceschini_89} have evaluated the fluctuation of the\nsource counts assuming a poissonian distribution: at $\\nu = 5$ GHz\nthey found fluctuations of the antenna temperature $\\Delta T_A \/\nT_A < 10^{-4}$ over an angular scale $\\theta \\sim 5$ deg. This\nfluctuation rapidly decreases at larger scales. At lower\nfrequencies the fluctuations increase, but we have at most $\\Delta\nT_A \/ T_A < 10^{-2}$ for $\\nu=408$ MHz and $\\Delta T_A \/ T_A <\n10^{-4}$ for $\\nu=2.5$ GHz at an angular scale $\\theta \\sim 20$\ndeg.\n\nDue to these considerations, the\ncontribution of UERS to the sky brightness at large angular scale\nis assumed isotropic in the following discussion.\nMoreover the radio sources are supposed\nto be randomly distributed and the fluctuation in the number\ncounts is assumed to be poissonian. A possible anisotropic\ncontribution of UERS is in most cases negligible and limited to\nthe region of the super-galactic plane (see \\cite[]{Shaver_89}).\n\n\\subsection{The data set}\n\nMany source-number vs flux distributions have been produced in the\npast years: radio source counts at low frequency have been\npublished even since the Sixties, while deep surveys at higher\nfrequencies have been performed recently. Compilation of source\ncounts can be found in several papers (see for example\n\\cite[]{Condon_84a}, \\cite[]{Franceschini_89},\n\\cite[]{Toffolatti_98}, \\cite[]{Windhorst_93}). Most of the data\nwe used can be extracted from the references included in these\npapers.\n\nWe used the counts distributions at the frequencies between 150\nand 8000 MHz, spanning a frequency range larger than the one\ncovered by the TRIS experiment. In literature we found data at\neight frequencies: $\\nu=$ 151 MHz, 178 MHz, 408 MHz, 610 MHz, 1.4\nGHz, 2.7 GHz, 5.0 GHz, 8.44 GHz. The complete list of the papers\nwe examined is reported in Table \\ref{tab1}. Some of those\nmeasurements were performed at a frequency slightly different from\nthe nominal one. In this case, the original measurements were\nscaled to the nominal frequency by assuming a dependence of the\nsource flux $S(\\nu) \\sim \\nu^{-0.7}$. Usually the correction is\nnegligible. The number counts extracted from Table \\ref{tab1} is\nshown in Figure \\ref{fig1}.\n\n\n\\section{Number counts distribution fit}\\label{Fit}\n\nThe fit was performed on the differential number counts normalized\nto the Euclidean distribution of sources $E(S) = S^{5\/2}(dN\/dS)$,\nusing the following analytical expression:\n\n\\begin{equation}\nQ(S) = Q_1(S) + Q_2(S) = \\frac{1}{A_1 S^{\\varepsilon_1} + B_1\nS^{\\beta_1}} + \\frac{1}{A_2 S^{\\varepsilon_2} + B_2 S^{\\beta_2}}\n\\label{Fitequation}\n\\end{equation}\n\nThe best values of the fit parameters are summarized in Tables\n\\ref{tab2} and \\ref{tab3}. This analytical fit function is\nempirical. It reproduces the distribution of the experimental data\nwith some advantages: 1) it is a simple analytical function; 2) it\nallows to extend the extrapolation beyond the available data,\nbecause both the amplitude and slope of the tails are well\ndefined, both at low and high fluxes; 3) it is built as the sum of\ntwo \\textit{\"populations\"} of sources with different emission, but\nsimilar behavior and shape; 4) the fitting procedure is\nindependent on the source type and evolution but it is applied to\nthe source counts as currently observed. In addition, this\nanalytical form is simply a power law, with various indices at\ndifferent values of flux, exactly as the observations suggest.\n\n\\subsection{High flux distribution fit}\n\nTo get the fit we first considered the source distribution at high\nfluxes (i.e. we evaluated the parameters of the component\n$Q_1(S)$). For this high flux distribution we have data at all the\nconsidered frequencies. The best fit parameters are listed in\nTable \\ref{tab2}. We found that the values of the spectral indices\n$\\varepsilon_1$ and $\\beta_1$ obtained at all the frequencies are\nvery similar. We then decided to take a unique weighted average\nvalue for $\\varepsilon_1$ and $\\beta_1$. The parameter\n$\\varepsilon_1$ is particularly well constrained by the data at\n1400 MHz, where a large set of data with low scatter is available\n(see \\cite[]{White_97} and Figure \\ref{fig1}). Conversely\navailable data at 178 MHz do not span a flux range wide enough to\nconstrain the two slopes.\n\nSource counts at 2700 and 8440 MHz are the less accurate among the\nfull data set we used. At both frequencies there are sets of data\nnot completely overlapping and the statistics is poor. Therefore,\nto take into account these uncertainties, we assumed a uniform\ndistribution of the parameter $A_1$. At $\\nu=2700$ MHz we took\n$A_1 = (6 - 12) \\times 10^{-4}$, while at $\\nu=8440$ MHz we took\n$A_1 = (16 - 34) \\times 10^{-4}$ (see Table \\ref{tab2}). In this\nfitting procedure we used all the data collected from the papers\nlisted in the Table \\ref{tab1}. We excluded from the fit only the\nseven data points at lowest flux published by \\cite{White_97} at\n1400 MHz because these data present a roll-off, exactly in the\nregion where a changing of the slope is expected, but in the\nopposite direction. According to  the authors, this roll-off is an\nindication of the incompleteness of the survey at the faint-flux\nlimit.\n\n\\subsection{Low flux distribution fit}\n\nWe extended the fitting procedure to the counts at low flux, in\norder to constrain the distribution $Q_2(S)$. The two parameters\n$A_2$ and $\\varepsilon_2$ fixed amplitude and slope of the\nlow-flux tail of the distribution, while $B_2$ and $\\beta_2$ are\nneeded to fit the distribution in the region showing a change in\nthe slope. Deep counts are available only at 0.61, 1.4, 5 and 8.44\nGHz, but at 8.44 GHz the number of experimental points and their\nscatter do not allow to fit any parameter. In addition we\nconsidered the model of the low-flux tail published by\n\\cite{Franceschini_89} at 1.4 and 5 GHz. This source evolution\nmodel fits the data also after the addition of the most recent\nmeasurements both at 1.4 and 5 GHz. The low-flux tail of the model\ncompared with the experimental data and our fit $Q(S)$ are shown\nin Figure \\ref{fig2}. This evolution model is able to predict\naccurately the slope of the low-flux tail and we used it to get\nthe value of $\\varepsilon_2$, which is independent on the\nfrequency \\cite[]{Franceschini_07}. The values of $\\varepsilon_2$,\nobtained at 1.4 and 5 GHz, are fully compatible with the average\nvalue of $\\varepsilon_1$ previously evaluated. Combining the\nestimates at 1.4 and 5 GHz we get $\\varepsilon_2 = -0.856 \\pm\n0.021$.\n\nSince the model by \\cite{Franceschini_89} is not able to fix the\namplitude of the number counts, i.e. the parameter $A_2$, we\nevaluated this parameter by using the experimental data at 1.4 and\n5 GHz. We get: $A_2\/A_1 = 0.24 \\pm 0.02$ at 1.4 GHz; $A_2\/A_1 =\n0.30 \\pm 0.04$ at 5 GHz. The ratio $A_2\/A_1$ is almost independent\non the frequency. The small change is due to the different\ncontribution of the \\textit{flat-spectrum} sources in the total\ncounts at the two frequencies in the low flux region (below\n$10^{-5}$ Jy) and in the high flux region (10 mJy - 1 Jy).\n\nUsing the model of \\cite{Franceschini_89} and \\cite{Toffolatti_98}\nat 1.4, 5 and 8.4 GHz, we extrapolated the value of the ratio\n$A_2\/A_1$ also to the other frequencies. In order to evaluate\n$A_2\/A_1$ at lower frequencies we estimated the variation of\n$A_2\/A_1$ at 1.4 GHz excluding from the counts the\n\\textit{flat-spectrum} sources. This is the extreme condition,\nwhich holds at very low frequencies. In fact the other sources\ncontributing to the number counts do not change significantly with\nfrequency. We obtain in this situation that $0.23 \\leq A_2\/A_1\n\\leq 0.24$. The same result was obtained starting from data and\nmodel at 5 GHz. Therefore for the frequencies 151, 178, 408 and\n610 MHz we take the value obtained at 1.4 GHz, but associating a\nlarger error bar, due to the uncertainty in the extrapolation\nprocedure: $A_2\/A_1 = 0.24 \\pm 0.04$.\n\nWe then evaluated the contribution of the {\\it flat spectrum}\nsources in the total counts at 8.44 GHz (see \\cite{Toffolatti_98})\nin comparison with the counts at 5 GHz. In this way we estimate at\n8.44 GHz the value $A_2\/A_1 = 0.31 \\pm 0.04$. At 2.7 GHz we took a\nvalue constrained by the results obtained at 1.4 and 5 GHz:\n$A_2\/A_1 = 0.24 - 0.30$.\n\nWe finally estimated the $B_2$ and $\\beta_2$. They are important\njust to define the shape of the distribution in the region showing\na change in the slope. At 1.4 GHz data are accurate enough to\nconstrain both parameters, but $\\beta_2$ can not be constrained at\nthe other frequencies. Since the accuracy of these two parameters\nis not important for the calculation of the integrated brightness\ntemperature, we assumed for them the average value for all the\nfrequencies.\n\nThe summary of the best values of all the parameters of $Q_2(S)$\nis shown in Table \\ref{tab3}. The number counts and the function\nwhich fit them are shown in Figure \\ref{fig1}. In conclusion we\ncan note that: 1) $A_1$ and $B_1$, the two frequency dependent\nparameters of the fit, take different values at each frequency.\nThe same is true for the ratio $A_2\/A_1$. 2) The power law indices\n($\\varepsilon_1$, $\\beta_1$, $\\varepsilon_2$ and $\\beta_2$) are\nfrequency independent and we take a common value at all the\nfrequencies.\n\n\n\\section{The UERS contribution to the sky diffuse emission}\\label{Brightness}\n\n\\subsection{Evaluation of the diffuse emission}\n\nThe contribution of the UERS ($B_{UERS}(\\nu)$) to the sky\nbrightness is evaluated by integrating the function $S(dN\/dS)$\nfrom the largest flux ($S_{max}$) of the measured sources down to\nthe lowest fluxes ($S_{min}$) corresponding to the faintest\nsources:\n\n\\begin{equation}\nB_{UERS}(\\nu) = \\int^{S_{max}}_{S_{min}} \\frac{dN}{dS}(\\nu) \\cdot\nS \\ dS\n\\end{equation}\n\nThe brightness temperature $T_{UERS}(\\nu)$ is by definition:\n\n\\begin{equation}\nT_{UERS}(\\nu) = B_{UERS}(\\nu) \\frac{\\lambda^2}{2 \\ k_B},\n\\end{equation}\n\n\\noindent $k_B$ being the Boltzmann constant. The values of\n$T_{UERS}$ at the eight frequencies we considered are\nsummarized in Table \\ref{tab4}.\n\nFrom the observations we have $S_{max} \\sim 10^2$ Jy (as measured\nat 151, 408 and 1400 MHz) and $S_{min} \\sim 10^{-6}$ Jy (as\nmeasured in the deepest counts at 5 GHz). While sources at higher\nfluxes if present in the surveyed sky region can be easily\nmeasured, the limit at low flux is set by the confusion limit or\nby the observation completeness. In other words there is no sharp\nlimit at low flux in the population of the sources. We extended\nthe integration down to very faint limits, several orders of\nmagnitude below the faintest detected sources ($S_{min} \\sim\n10^{-6}$ Jy). When the integration is extended down to $S_{min}\n\\sim 10^{-12}$ Jy \\ the brightness increase by $3-4$ \\% and then\nthe value converges. This increment is comparable with the total\nuncertainty we get on the value of the brightness, as shown in\nFigure \\ref{fig3}.\n\nWe extended the integration also to higher values of the flux, in\norder to test also this integration limit. Increasing $S_{max}$\nwell beyond the flux of the strongest sources observed, the\nintegral change by less than $0.5\\%$ and quickly converges. This\nis a consequence of the very low statistics of sources at the\nhighest fluxes. This is a confirmation that the large scale\nbrightness is actually not sensitive to the upper limit of\nintegration.\n\n\\subsection{Evaluation of the uncertainty}\n\nThe error budget takes into account both the fit uncertainties and\nthe number counts fluctuations over the observed sky region. The\nfit uncertainties were evaluated by means of Monte Carlo\nsimulations. For each parameter we considered a gaussian\ndistribution with standard deviation as reported in Tables\n\\ref{tab2} and \\ref{tab3}. Only for the values of $A_1$ at 2.7 and\n8.44 GHz and for $A_2\/A_1$ at 2.7 GHz we assumed a uniform\ndistribution inside the interval reported in Tables \\ref{tab2} and\n\\ref{tab3}. The error bars of the parameters of the fit (in\nparticular $A_2\/A_1$ and $\\varepsilon_2$) include the uncertainty\non the extrapolation at the lowest fluxes for the various\nfrequencies. We underline that, as shown in Figure \\ref{fig3}, the\ncontribution to the brightness temperature of the low-flux tail\n(below $\\sim 10^{-6}$ Jy) is lower than the overall uncertainty\nreported in Table \\ref{tab4}.\n\nThe statistical fluctuations of the sources' number counts have no\neffect in a large part of the distribution because the number of\nsources is quite large. We concentrated on the effect of the\nfluctuation of the few sources with the highest flux. We\nconsidered these sources randomly distributed and therefore their\nfluctuation is Poissonian. We evaluated the fluctuation of the\nbrightness in a patch of the sky corresponding to the beam of\nTRIS: $\\Omega_{TRIS} \\sim 0.1$ sr. The upper limit of the\ncontribution to the temperature uncertainty due to the fluctuation\nin the number of sources is directly measured by the maximum of\nthe function\n\n\\begin{equation}\\label{}\nC(S_{min})=\\frac{\\lambda^2}{2k_B\\Omega_{TRIS}}\n\\frac{\\int_{S_{min}}^{100Jy}\\frac{dN}{dS}SdS}\n{\\int_{S_{min}}^{100Jy}\\frac{dN}{dS}dS}\n\\end{equation}\n\n\\noindent plotted in Figure \\ref{fig4} for the specific case of\nthe 1400 MHz data. For all the frequencies this maximum falls\naround $\\sim 5$ Jy, and its value is from 2 to 6 times smaller\nthan the corresponding values reported in Table \\ref{tab4}.\nTherefore for every frequency, and over the full flux range, the\nerror of the brightness temperature is dominated by the\nstatistical uncertainties of the fit parameters.\n\nThe relative error bar of the brightness temperature is $6-7\\%$ at\n151, 408, 610 and 1400 MHz increasing up to $9\\%$ at 5000 MHz. At\n178 MHz the available measurements are few and old and the\nobtained error bar is $13\\%$. At 2700 and 8440 MHz both the\nquantity and the quality of the data do not allow accurate\nestimates of the parameters of the fit and we get an uncertainty\nof $25-30\\%$.\n\n\\subsection{Frequency dependence}\n\nThe integrated contribution of the UERS to the brightness of the\nsky at the various frequencies is shown in Figure \\ref{fig5}. The\ndistribution can be fitted by a power law:\n\n\\begin{equation}\nT_{UERS}(\\nu) = T_0 \\Bigl( \\frac{\\nu}{\\nu_0}\\Bigr)^{\\gamma_0}\n\\end{equation}\n\n\\noindent Setting $\\nu_0 = 610$ MHz (chosen because it is\nclose to one of the channels of the TRIS experiment), we obtain\nthe best fit of $T_0$ and $\\gamma_0$ shown in Table \\ref{tab5}\n(\\textit{FIT1}). As shown in Figure \\ref{fig5}, in spite of the\nlarge error bars at 2700 and 8440 MHz, scatter of the data points is\nlimited. The fit of a single power law, done excluding the data with\nthe largest uncertainty (at $\\nu=178$, 2700 and 8440 MHz), gives\nthe values of $T_0$ and $\\gamma_0$ shown in Table \\ref{tab5}\n(\\textit{FIT2}). Now the scatter of the experimental data is much\nsmaller, as shown by the value of the reduced $\\chi^2$.\n\nIn both cases the results obtained are fully compatible with the\nslope of the {\\it steep spectrum} sources, which are therefore the\nmain contributors to the source counts. However it is interesting\nto check the contribution of the {\\it flat spectrum} sources\nespecially at high frequencies. We assumed a {\\it flat spectrum}\ncomponent with fixed slope $\\gamma_1 = -2.00$ and a  dominant {\\it\nsteep spectrum} component with slope $\\gamma_0 = -2.70$:\n\n\\begin{equation}\nT_{UERS}(\\nu) = T_0 \\Bigl( \\frac{\\nu}{\\nu_0}\\Bigr)^{\\gamma_0} +\nT_1 \\Bigl( \\frac{\\nu}{\\nu_0}\\Bigr)^{\\gamma_1}\n\\end{equation}\n\n\\noindent The results of the fit of $T_0$ and $T_1$ are shown in Table\n\\ref{tab5} (\\textit{FIT3}). Doing so we can get an estimate of\nthe contribution of the {\\it flat-spectrum} component in the\nnumber counts at the various frequencies. The value of\n$T_{UERS}(\\nu) (\\nu \/\\nu_0)^{2.70}$ \\ at the frequencies analyzed\nis shown in Figure \\ref{fig6}, together with the power law best\nfits. Figure \\ref{fig6} shows that the two last fits\n(\\textit{FIT2} and \\textit{FIT3}) are equivalent within the error\nbars.\n\n\\section{Discussion}\\label{Discussion}\n\n\\subsection{Analytical form of the fit function}\n\nAs discussed in Section \\ref{Fit}, the fit function $Q(S)$ is a\npower law distribution taking different slopes and amplitudes in\nthree different regions of the sources' flux. We prefer to deal\nwith a power law (like $Q(S)$) instead of a polynomial fit, as\nperformed by \\cite{Katgert_88} and \\cite{Hopkins_03}. A polynomial\nfit can be better adapted to the experimental points, but its\nvalidity range is restricted to the region covered by experimental\npoints. Therefore it is not possible to use a polynomial fit to\nextrapolate the distribution outside this range. Conversely our\nfit function can be extrapolated because amplitude and slope of\nthe tails are well defined and take into account the shapes\nexpected from the counts evolution models\n\\cite[]{Franceschini_89}.\n\nAccording to \\cite{Longair_78} we should expect a broadened\nmaximum in the distribution of the source counts with increasing\nfrequency. This indication seems to be confirmed looking at the\ndifferential counts shown in (\\cite[]{Kellermann_87} and\n\\cite[]{Condon_84a}). In spite of these expectations we have\nobtained a good fit of the counts at the various frequencies,\nusing the same function with the same slopes, as shown Figure\n\\ref{fig1}. Part of this effect is probably hidden by the larger\nscatter of the high frequency differential counts. In any case the\nbroadening of the maximum could be marginally observed above 1400\nMHz and the effect is not relevant for the calculation of the\ncontribution to the sky brightness.\n\n\\subsection{Frequency dependence}\n\nThe spectral dependence of the brightness temperature $T_{UERS}$\nfollows the expectations at low frequency where the number counts\nare dominated by the sources with steep spectrum: $\\alpha \\sim\n0.7$. At high frequency the situation is more complex. We could\nexpect a flattening, because at these frequencies the number\ncounts of {\\it flat-spectrum} sources begin to be important. It\nwill probably appear at frequencies higher than 1400 MHz.\nUnfortunately the available data are not accurate enough to\nconstrain the fit parameters (in particular $A_1$) at 2700 and\n8440 MHz. The values obtained at these two frequencies, lower than\nthe values expected looking at the other frequencies, could also\nbe an indication of incompleteness of the surveys (see Figure\n\\ref{fig6}). Conversely at 5000 MHz data are better and numerous.\nWe obtain a value for $T_{UERS}$ fully consistent with the slope\nof the data at low frequency, where {\\it steep-spectrum} sources\nare dominating, even if there is a marginal indication of a\nspectral flattening (see Figure \\ref{fig6}). In fact when we fit\nthe data adding the contribution of the {\\it flat-spectrum}\nsources we get an estimate of this contribution which is $T_1\/T_0\n\\simeq 2\\%$ at $\\nu=610$ MHz and $T_1\/T_0 \\simeq 9\\%$ at $\\nu=5$\nGHz (see \\textit{FIT3} in the Table \\ref{tab5}).\n\n\n\\subsection{Previous estimates of $T_{UERS}$}\n\nThe values of brightness temperature shown in Table \\ref{tab4}\nhave error bars of roughly 7\\% at most frequencies. The\nuncertainty is a bit larger at 178 MHz and even worse at 2700 and\n8440 MHz, because of the quality of the number counts data. So far\nvery few estimates of $T_{UERS}$ have been published (see\n\\cite[]{Longair_66}, \\cite[]{Wall_90} and \\cite[]{Burigana_04}),\nand the frequencies covered were limited and sometimes the\nuncertainty was not quoted. Our results are in agreement with the\nvalues previously estimated by \\cite{Longair_66}, $T_{UERS}(178 \\\nMHz)=23\\pm5$ K, and by \\cite{Wall_90}, $T_{UERS}(408 \\\nMHz)\\simeq2.6$ K, $T_{UERS}(1.4 \\ GHz)\\simeq0.09$ K, and\n$T_{UERS}(2.5 \\ GHz)\\simeq0.02$ K, but our error bars are\ndefinitely smaller. The accuracy of the estimated values of\n$T_{UERS}$ is particularly important if this contribution is to be\nsubtracted to calculate the value of the CMB temperature at low\nfrequency. Table \\ref{tab4} suggests that the value of $T_{UERS}$\nneeds to be accurately evaluated up to a frequency of several GHz,\nbecause its value is not negligible.\n\n\\section{Conclusions}\n\nWe used the source number - flux measurements in literature to\nevaluate the contribution of the Unresolved Extragalactic Radio\nSources to the diffuse brightness of the sky. We analyzed the\ncount distributions at eight frequencies between 150 and 8000 MHz,\nspanning over the frequency range partially covered by the TRIS\nexperiment (see \\cite[]{TRIS-I}): $\\nu=$ 151 MHz, 178 MHz, 408\nMHz, 610 MHz, 1.4 GHz, 2.7 GHz, 5.0 GHz, 8.44 GHz.\n\nWe optimized the fitting function of the experimental number\ncounts distribution. The differential number counts ($dN\/dS$) at\nthe various frequencies are well described by a multi power law\nempirical distribution $Q(S)$ (see Equation \\ref{Fitequation}).\nThe amplitudes ($A_1$ and $B_1$) are frequency dependent\nparameters of the fit and have different values at each frequency.\nConversely the power law indices ($\\varepsilon_1$, $\\beta_1$,\n$\\varepsilon_2$ and $\\beta_2$) have a common value at all the\nfrequencies.\n\nThe contribution of the UERS to the sky brightness was\nevaluated by integrating the function $S(dN\/dS)$ from the largest\nflux ($S_{max} = 10^{2}$) of the measured sources down to the\nlowest fluxes ($S_{min} = 10^{-12}$) corresponding to the expected\nfaintest sources. We got the brightness temperature with a\nrelative error bar of $\\delta T_{UERS}\/T_{UERS} \\simeq 6-7\\%$ at\n$\\nu =$ 151, 408, 610 and 1400 MHz, $\\delta T_{UERS}\/T_{UERS}\n\\simeq 9\\%$ at $\\nu =$ 5000 MHz, $\\delta T_{UERS}\/T_{UERS} \\simeq\n13\\%$ at $\\nu =$ 178 MHz and $\\delta T_{UERS}\/T_{UERS} \\simeq\n25-30\\%$ at $\\nu =$ 2700 and 8440 MHz.\n\nWe finally evaluated the spectral dependence of the point source\nintegrated brightness. As expected this dependence can be\ndescribed using a power law with a spectral index $\\gamma_0 \\simeq\n-2.7$, in agreement with the frequency dependence of the flux\nemitted by the {\\it steep-spectrum} sources. We have also tested\nthe contribution of the {\\it flat-spectrum} sources, adding a\nsecond component with the slope $\\gamma_1 = -2.0$. The\ncontribution of these sources starts to be relevant only at\nfrequencies above several GHz. In fact we estimated a contribution\nby {\\it flat-spectrum} sources $\\sim 2 \\%$ at 610 MHz and $\\sim 9\n\\%$ at 5 GHz.\n\nThe above results were used to evaluate the CMB temperature\nat frequencies close to 1 GHz from absolute measurements of the\nsky temperature made by our group (see \\cite[]{TRIS-I},\n\\cite[]{TRIS-II}, \\cite[]{TRIS-III}).\n\n\\acknowledgments  {\\bf Acknowledgements}: This work is part of the\nTRIS activity, which has been supported by MIUR (Italian Ministry\nof University and Research), CNR (Italian National Council of\nResearch) and the Universities of Milano and of Milano-Bicocca.\nThe authors acknowledge A. Franceschini for useful discussions and\nthe anonymous referee for helpfull comments to the draft.\n\n\\vfill \\eject\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOptimization problems are wide-spread in several domains of science and engineering. \nThe usual goal is to minimize or maximize some pre-defined objective(s). \nMost of the real-world scenarios place certain restrictions \non the variables of the problem i.e. the variables need to satisfy \ncertain pre-defined constraints to realize an acceptable solution.\n \nThe most general form of a constrained optimization problem (with inequality constraints,\nequality constraints and variable bounds) can be written as a nonlinear programming\n(NLP) problem: \n\n{\\small\\begin{eqnarray}\nMinimize\t &f(\\vec{x})\t\t&\t\\nonumber\\\\\nSubject to\t &g_j(\\vec{x}) \\ge 0,   & j= 1,...,J \\nonumber\\\\\n\t\t &h_k(\\vec{x})  =  0,\t& k= 1,...,K \\nonumber\\\\\n\t  \t & x_i^{(L)} \\le x_i \\le x_i^{(U)}, & i=1,...,n. \\label{eq:NLPprob}\n\\end{eqnarray}}\n\nThe NLP problem defined above contains $n$ decision variables (i.e. $\\vec{x}$ is a vector of size $n$),\n$J$ greater-than-equal-to type inequality constraints (less-than-equal-to can be expressed in this form by \nmultiplying both sides by $-1$), and $K$ equality-type constraints.\nThe problem variables $x_is$ are bounded by the lower ($x_i^{(L)}$) and upper ($x_i^{(U)}$)\nlimits. When only the variable bounds are specified then the \nconstraint-handling strategies are often termed as the boundary-handling methods.\n\\footnote{For the rest of the paper, by \\textit{constraint-handling} we imply\ntackling all of the following: variable bounds, inequality constraints and equality constraints. \nAnd, by a \\textit{feasible solution} it is implied that the solution satisfies all the variable bounds, inequality constraints,\nand equality constraints. The main contribution of the paper is to propose an efficient constraint-handling method\nthat operates and generates only feasible solutions during optimization.}\n\nIn classical optimization, the task of constraint-handling has been addressed\nin a variety of ways: (i) \\textit{using penalty approach} developed by Fiacoo and McCormick \\cite{jensen2003operations},\nwhich degrades the function value\nin the regions outside the feasible domain, (ii) \\textit{using barrier methods} which operate in a similar fashion \nbut strongly degrade the function values as the solution approaches a constraint boundary from \ninside the feasible space, (iii) \\textit{performing search in the feasible directions} using methods\nsuch gradient projection, reduced gradient and Zoutendijk's approach \\cite{zoutendyk1960methods}\n(iv) \\textit{using the augmented Lagrangian formulation} of the problem, as commonly \ndone in linear programming and sequential quadratic programming (SQP).\nFor a detailed account on these methods along with their implementation and \nconvergence characteristics the reader is referred to \\cite{rekl,debOptiBOOK,MichalewiczConstraint}.\nThe classical optimization methods reliably and effectively solve convex constrained optimization problems while ensuring convergence\nand therefore widely used in such scenarios.  However, same is not true in the presence of non-convexity.\nThe goal of this paper is to address the issue of constraint-handling for evolutionary algorithms in real-parameter optimization, \nwithout any limitations to convexity or a special form of constraints or objective functions. \n\nIn context to the evolutionary algorithms the constraint-handling has been addressed\nby a variety of methods; including borrowing of the ideas from the classical techniques. These include\n(i) \\textit{use of penalty functions} to degrade the fitness\nvalues of infeasible solutions such that the degraded solutions are given less emphasis during the evolutionary search. A\ncommon challenge in employing such penalty methods arises from\nchoosing an appropriate penalty parameter ($R$) that strikes the right balance between\nthe objective function value, the amount of constraint violation and the associated penalty. \nUsually, in EA studies, a trial-and-error method is employed to estimate $R$.\nA study \\cite{debpenalty} in 2000 suggested a parameter-less\napproach of implementing the penalty function concepts for population-based optimization method. \nA recent bi-objective method\n\\cite{deb-dutta} was reported to find the appropriate $R$ values adaptively during the optimization process.\nOther studies \\cite{wang2012dynamic,wang2012combining} have employed the concepts of multi-objective\noptimization by simultaneously considering the minimization of the constraint violation and optimization of the\nobjective function, (ii) \\textit{use of feasibility preserving operators},\nfor example, in \\cite{michalewicz1996genocop} specialized operators in the presence of linear constraints \nwere proposed to create new and feasible-only individuals from the feasible parents. In another example, \ngeneration of feasible child solutions within the variable bounds was achieved  \nthrough Simulated Binary Crossover (SBX) \\cite{debSBX} and polynomial mutation\noperators \\cite{debBookMO}. The explicit feasibility of child solutions was ensured by redistributing the probability distribution \nfunction in such a manner that the infeasible regions were assigned a zero probability \nfor child-creation \\cite{debpenalty}. Although explicit creation of feasible-only solutions during an EA search is an\nattractive proposition, but it may not be possible always since generic crossover or mutation operators\nor other standard EAs do not gaurantee creation of feasible-only solutions, (iii) \\textit{deployment of repair strategies}\nthat bring an infeasible solution back into the feasible domain.\nRecent studies \\cite{PadhyeBIC2012,Helwid-Constraing-handling-2013,amir,chu} investigated the \nissue of constraint-handling through repair techniques in context to PSO and DE, and   \nshowed that the repair mechanisms can introduce a bias in the search and \nhinder exploration. Several repair methods \nproposed in context PSO \\cite{padhyeCEC2009,Sabina,JonathanPareto} \nexploit the information about location of the optimum and fail to perform when the location of \noptimum changes \\cite{padhye2010}. These issues are universal and\noften encountered with all EAs (as shown in later sections).\nFurthermore, the choice of the evolutionary optimizer, the constraint-handling strategy, and \nthe location of the optima with respect to the search space, all play an important\nrole in the optimization task. To this end, authors have realized a need\nfor a reliable and effective repair-strategy that explicitly preserves feasibility.\nAn ideal evolutionary optimizer (evolutionary algorithm\nand its constrained-handling strategy) should be robust in terms of finding the \noptimum, irrespective of the location of the optimal location in the search space.   \nIn rest of the paper, the term constraint-handling strategy refers to explicit feasibility\npreserving repair techniques.  \n\nFirst we review the existing constraint-handling strategies \nand then propose two new constraint-handling schemes, namely, Inverse Parabolic Methods (IPMs). \nSeveral existing and newly proposed constrained-handling strategies are first tested on a class \nof benchmark unimodal problems with variable bound constraints.\nStudying the performance of constraint-handling strategies on problems with variable bounds\nallows us to gain better understanding into the operating principles in a simplistic manner. \nParticle Swarm Optimization, Differential Evolution and real-coded Genetic Algorithms are chosen \nas evolutionary optimizers to study the performance of different constraint-handling strategies. \nBy choosing different evolutionary optimizers, better understanding on the functioning\nof constraint-handlers embedded in the evolutionary frame-work can be gained. \nBoth, the search algorithm and constraint-handling strategy must operate efficiently and synergistically in\norder to successfully carry out the optimization task. It is shown that the constraint-handling\nmethods possessing inherent pre-disposition; in terms of bringing infeasible solutions back into the\nspecific regions of the feasible domain, perform poorly. Deterministic constraint-handling strategies such as \nthose setting the solutions on the constraint boundaries result in the loss of population diversity. \nOn the other hand, random methods of bringing the solutions back into the search space\narbitrarily; lead to complete loss of all useful information carried by the solutions. \nA balanced approach that utilizes the useful information from the solutions\nand brings them back into the search space in a meaningful way is desired. The newly proposed IPMs are motivated\nby these considerations.\nThe stochastic and adaptive components of IPMs (utilizing the information of the solution's\nfeasible and infeasible locations), and a user-defined parameter ($\\alpha$) render\nthem quite effective. \n  \nThe rest of the paper is organized as follows:\nSection~\\ref{sec:feasibility-preserving-existing} reviews existing constraint-handling techniques\ncommonly employed for problems with variable bounds.\nSection~\\ref{sec:IP} provides a detailed description on two newly proposed IPMs.\nSection~\\ref{sec:ResultsDiscussion} provides a description on the benchmark test problems and \nseveral simulations performed on PSO, GAs and DE with different constraint-handling techniques. \nSection~\\ref{sec:scale-up} considers \noptimization problems with larger number of variables. \nSection~\\ref{sec:Constraint-Programming} shows the extension and applicability of proposed\nIPMs for generic constrained problems.\nFinally, conclusions and scope for future work are discussed in Section~\\ref{sec:Conclusion}.\n    \n\n\n\n\n\\section{Feasibility Preserving Constraint-Handling Approaches for Optimization Problems with Variable Bounds}\n\\label{sec:feasibility-preserving-existing}\nSeveral constraint-handling strategies have been proposed to bring solutions back into the feasible region\nwhen constraints manifest as variable bounds. Some of these strategies can also be extended in \npresence of general constraints. An exhaustive recollection and \ncomparison of all the constraint-handling techniques is beyond the scope of this study. \nRather, we focus our discussions on the popular and representative constraint-handling techniques. \n\nThe existing constraint-handling methods for problems with variable bounds can be broadly categorized into two groups: \nGroup~$A$ techniques that perform feasibility check variable wise, and Group~$B$ techniques that perform feasibility\ncheck vector-wise. According to Group~$A$ techniques, for every solution, each variable is tested for its feasibility \nwith respect to its supplied bounds and made feasible if the corresponding bound is violated.\nHere, only the variables violating their corresponding bounds are altered, independently, and\nother variables are kept unchanged.  \nAccording to Group~$B$ techniques, if a solution (represented as a vector) \nis found to violate any of the variable bounds, it is brought back into\nthe search space along a vector direction into the feasible space. In\nsuch cases, the variables that explicitly do not violate their own\nbounds may also get modified.\n \nIt is speculated that for variable-wise separable problems, that is,\nproblems where variables are not linked to one another, \ntechniques belonging to Group~$A$ are likely to perform well. However, for the problems  \nwith high correlation amongst the variables (usually referred to as {\\em linked}-problems), Group~$B$ techniques are likely to be more useful. \nNext, we provide description of these constraint-handling methods in detail \\footnote{The implementation of several \nstrategies as C codes can be obtained by emailing npdhye@gmail.com or pulkitm.iitk@gmail.com}.\n\n\\subsection{Random Approach}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35]{Random.eps}    \n\\end{center}\n\\caption{Variable-wise random approach for handling bounds.}\n\\label{fig:RandomBH}\n\\end{figure}\nThis is one of the simplest and commonly used approaches for handling boundary\nviolations in EAs \\cite{chu}. This approach belongs to Group~$A$. \nEach variable is checked for a boundary violation and if the variable bound is violated\nby the current position, say $x_i^c$, then $x_i^c$ is replaced with a randomly chosen value\n$y_i$ in the range $[x_i^{(L)}, x_i^{(U)}]$, as follows:\n\\begin{equation}\ny_i = \\mbox{random} [x_i^{(L)}, x_i^{(U)}].\n\\end{equation}\nFigure~\\ref{fig:RandomBH} illustrates this\napproach. Due to the random choice of the feasible location, this approach explicitly maintains\ndiversity in the EA population. \n\n\\subsection{Periodic Approach}\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35]{Periodic.eps}    \n\\end{center}\n\\caption{Variable-wise periodic approach for handling bounds.}\n\\label{fig:PeriodicBH}\n\\end{figure}\nThis strategy assumes a periodic repetition of the objective function\nand constraints with a period $p=x_i^{(U)}-x_i^{(L)}$. This is carried out by\nmapping a violated variable $x_i^c$ in the range $[x_i^{(L)},\nx_i^{(U)}]$ to $y_i$, as follows:\n\\begin{equation}\ny_i  = \\left\\{ \n\\begin{array}{ll}\n         x_i^{(U)} - (x_i^{(L)}-x_i^c)\\%p, & \\quad \\text{if $x_i^c<x_i^{(L)}$},\\\\\n         x_i^{(L)} + (x_i^c-x_i^{(U)})\\%p, & \\quad \\text{if $x_i^c>x_i^{(U)}$}, \\\\\n\\end{array} \n\\right.\n\\end{equation}\nIn the above equation, \\% refers to the modulo operator.\nFigure~\\ref{fig:PeriodicBH} describes the periodic approach. \nThe above operation brings back an infeasible solution in a structured\nmanner to the feasible region. \nIn contrast to the random method, the periodic approach is too methodical and it is unclear \nwhether such a repair mechanism is supportive of\npreserving any meaningful information of the solutions that have created the\ninfeasible solution. This approach belongs to Group~$A$. \n\n\\subsection{SetOnBoundary Approach}\nAs the name suggests, according to this strategy a violated variable\nis reset on the \nbound of the variable which it violates.\n\\begin{equation}\ny_i  = \\left\\{ \n\\begin{array}{ll}\n         x_i^{(L)}, & \\quad \\text{if $x_i^c<x_i^{(L)}$},\\\\\n         x_i^{(U)}, & \\quad \\text{if $x_i^c>x_i^{(U)}$}.\\\\\n\\end{array} \\right.\n\\end{equation}\nClearly this approach forces all violated solutions to lie on the\nlower or on the upper boundaries, as the case may be. Intuitively, this approach will work\nwell on the problems when the optimum of the problem lies exactly on one of the variable\nboundaries. This approach belongs to Group~$A$.\n\n\\subsection{Exponentially Confined (Exp-C) Approach}\n\\begin{figure}[hbt]\n\\begin{minipage}{0.47\\linewidth}\n\\begin{center}\n\\includegraphics[scale=.35]{FieldSend.eps}    \n\\end{center}\n\\caption{Variable-wise exponentially approach (Exp-C) for handling\n  bounds.}\n\\label{fig:FieldSendDistBH}\n\\end{minipage}\\hfill\n\\begin{minipage}{0.47\\linewidth}\n\\begin{center}\n\\includegraphics[scale=.35]{expS.eps}    \n\\end{center}\n\\caption{Variable-wise exponentially approach (Exp-S) for handling bounds.}\n\\label{fig:expS}\n\\end{minipage}\n\\end{figure}\nThis method was proposed in \\cite{JonathanPareto}. According to this \napproach, a particle is brought back inside the feasible search space variable-wise in the region between \nits old position and the violated bound. The new location is created\nin such a manner that higher sampling probabilities are assigned to the regions\nnear the violated boundary. The developers suggested the use of an\nexponential probability distribution, shown in\nFigure~\\ref{fig:FieldSendDistBH}. \nThe motivation of this approach is based on the hypothesis \nthat a newly created infeasible point violates a particular variable\nboundary because the optimum solution lies closer to that variable\nboundary. Thus, this method will probabilistically \ncreate more solutions closer to the boundaries, unless the optimum lies well\ninside the restricted search space. This approach belongs to Group~$A$.\n\nAssuming that the exponential distribution is $p(x_i) =\nA\\exp(|x_i-x_i^p|)$, the value of $A$ can be obtained by integrating\nthe probability from $x_i=x_i^p$ to $x_i=x_i^{(B)}$ (where $B=L$ or\n$U$, as the case may be). Thus, the probability distribution is given\nas $p(x) = \\exp(|x_i-x_i^p|)\/(\\exp(|x_i^{(B)} - x_i^p|)-1)$. For any\nrandom number $r$ within $[0,1]$, the feasible solution is calculated as follows:\n\\begin{equation}\ny_i= \\left\\{\\begin{array}{ll}\nx_i^p - \\ln (1+r(\\exp (x_i^p-x_i^{(L)})-1))  &\\mbox{if $x_i<x_i^{(L)}$},\\\\\nx_i^p + \\ln (1+r(\\exp (x_i^{(U)}-x_i^p)-1)), & \\mbox{if\n  $x_i>x_i^{(U)}$}.\n\\end{array}\\right.\n\\label{eq:exp}\n\\end{equation}\n\n\n\\subsection{Exponential Spread (Exp-S) Approach}\nThis is a variation of the above approach, in which, instead of \nconfining the probability to lie between $x_i^p$ and the violated\nboundary, the exponential probability is spread over the entire\nfeasible region, that is, the probability is distributed from lower\nboundary to the upper boundary with an increasing probability towards\nthe violated boundary. This requires replacing $x_i^p$ with\n$x_i^{(U)}$ (when the lower boundary is violated) or $x_i^{(L)}$ \n(when the upper boundary is violated) in the Equation~\\ref{eq:exp} as follows: \n\\begin{equation}\ny_i= \\left\\{\\begin{array}{ll}\nx_i^{(U)} - \\ln (1+r(\\exp(x_i^{(U)}-x_i^{(L)})-1))  &\\mbox{if $x_i<x_i^{(L)}$},\\\\\nx_i^{(L)} + \\ln (1+r(\\exp(x_i^{(U)}-x_i^{(L)})-1)), & \\mbox{if\n  $x_i>x_i^{(U)}$}.\n\\end{array}\\right.\n\\end{equation}\nThe probability distribution is shown in Figure~\\ref{fig:expS}.\nThis approach also belongs to Group~$A$.\n\n\\subsection{Shrink Approach}\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35]{SHR.eps}    \n\\end{center}\n\\caption{Vector based {SHR.} strategy for handling bounds.}\n\\label{fig:SHRBH}\n\\end{figure}\nThis is a vector-wise approach and belongs to Group~$B$ in which the violated solution is\nset on the intersection point of the line joining the parent point\n($\\vec{x}_{not}$), child point ($\\vec{x}^c)$, and the violated boundary. Mathematically,\nthe mapped vector $\\vec{y}$ is created as follows:\n\\begin{equation}\n\\vec{y} = \\vec{x}_{not} +\\beta (\\vec{x}^c-\\vec{x}_{not}), \n\\end{equation}\nwhere $\\beta$ is computed as the minimum of all positive values of intercept\n$(x_i^{(L)}-x_{i,not})\/(x_i^c-x_{i,not})$ for a violated boundary\n$x_i^{(L)}$ and $(x_i^{(U)}-x_{i,not})\/(x_i^c-x_{i,not})$ for a violated boundary\n$x_i^{(U)}$.\nThis operation is shown in Figure~\\ref{fig:SHRBH}. In the case shown,\n$\\beta$ needs to be\ncomputed for variable bound $x_2^{(U)}$ only. \\\\ \\\\\n\n\n\\section{Proposed Inverse Parabolic (IP) Constraint-Handling Methods}\\label{sec:IP}\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.75]{ProposedDistDeb.eps}    \n\\end{center}\n\\caption{Vector based {Inverse Parabolic Methods}.}\n\\label{fig:ProbDistBH}\n\\end{figure}\nThe exponential probability distribution function described in the previous section brings \nviolated solutions back into the allowed range variable-wise, but \nignores the distance of the violated solution $x_i^c$ with respect to\nthe violated boundary. The distance from the violated boundary \ncarries useful information for remapping the violated solution into\nthe feasible region. One way to utilize this distance information is\nto bring solutions back into the allowed range with a higher\nprobability closer to the boundary, when the \\textit{fallen-out}\ndistance ($d_v$, as shown in Figure~\\ref{fig:ProbDistBH}) is small. \nIn situations, when points are too far outside the allowable range,\nthat is, the fallen-out\ndistance $d_v$ is large, particles are brought back more uniformly\ninside the feasible range. Importantly, when the fallen-out distance\n$d_v$ is small (meaning that the violated child solution is close to the\nvariable boundary), the repaired point is also close to the violated\nboundary but in the feasible side. Therefore, the nature of the exponential\ndistribution should become more and more like a uniform distribution\nas the fallen-out distance $d_v$ becomes large.\n\nLet us consider Figure~\\ref{fig:ProbDistBH} which shows a\nviolated solution $\\vec{x}^c$ and its parent solution $\\vec{x}^p$. Let\n$d_p=\\|\\vec{x}^c-\\vec{x}^p\\|$ denote the distance between the violated solution\nand the parent solution. Let $\\vec{v}$ and $\\vec{u}$ be the intersection points\nof the line joining $\\vec{x}^c$ and $\\vec{x}^p$ with the violated\nboundary and the non-violated boundary, respectively. The\ncorresponding distances of these two points from $\\vec{x}^c$ are $d_v$\nand $d_u$, respectively. Clearly, the violated distance is $d_v =\n\\|\\vec{x}^c-\\vec{v}\\|$.  We now define an inverse\nparabolic probability distribution function from $\\vec{x}^c$ along \nthe direction $(\\vec{x}^p-\\vec{x}^c)$ as:\n\\begin{equation}\np(d) = \\frac{A}{(d-d_v)^2+\\alpha^2d_v^2}, \\quad d_v \\leq d \\leq a,\n\\end{equation}\nwhere $a$ is the upper bound of $d$ allowed by the constraint-handling\nscheme (we define $a$ later) and $\\alpha$ is a pre-defined parameter. By calculating and\nequating the cumulative probability equal to one, we find:\n\\[A = \\frac{\\alpha d_v}{\\tan^{-1} \\frac{a-d_v}{\\alpha d_v}}.\\]\nThe probability is maximum at $d=d_v$ (at the violated boundary) and\nreduces as the solution enters the allowable range. Although this \ncharacteristic was also present in the exponential distribution, the\nabove probability distribution is also a function of violated distance\n$d_v$, which acts\nlike a variance to the probability distribution. If $d_v$ is\nsmall, then the variance of the distribution is small, thereby resulting in a\nlocalized effect of creating a mapped solution. \nFor a random number $r\\in [0,1]$, the distance of the mapped solution\nfrom $\\vec{x}^c$ in the allowable\nrange $[d_v,d_u]$ is given as follows:\n\\begin{equation}\nd' = d_v + \\alpha d_v \\tan \\left(r \\tan^{-1} \\frac{a-d_v}{\\alpha d_v}\\right).\n\\label{eq:s}\n\\end{equation}\nThe corresponding mapped solution is as follows:\n\\begin{equation}\n\\vec{y} = \\vec{x}^c + d' (\\vec{x}^p-\\vec{x}^c).\n\\label{eq:map}\n\\end{equation}\nNote that the IP method makes a vector-wise operation and is sensitive\nto the relative locations of the infeasible solution, the parent solution, and\nthe violated boundary. \n\nThe parameter $\\alpha$ has a direct {\\em external\\\/} effect of inducing small or large\nvariance to the above probability distribution. If $\\alpha$ is large,\nthe variance is large, thereby having uniform-like distribution. Later \nwe shall study the effect of the parameter $\\alpha$. A value $\\alpha$ $\\approx$ 1.2 is\nfound to work well in most of the problems and is recommended. \nNext, we describe two particular constraint-handling schemes employing this  \nprobability distribution. \n\n\\subsection{Inverse Parabolic Confined (IP-C) Method}\nIn this approach, the probability distribution is confined\nbetween $d \\in [d_v,d_p]$, thereby making $a=d_p$. Here, a mapped\nsolution $\\vec{y}$ lies strictly between violated boundary location\n($\\vec{v}$) and the parent ($\\vec{x}^p$). \n\n\\subsection{Inverse Parabolic Spread (IP-S) Method} \nHere, the mapped solution is allowed to lie in the entire\nfeasible range  between $\\vec{v}$ and\n$\\vec{u}$ along the vector $(\\vec{x}^p-\\vec{x}^c)$, but more emphasis is given on relocating the \nchild near the violated boundary. The solution can be found by using Equations~\\ref{eq:s} and\n\\ref{eq:map}, and by setting $a=d_u$. \n\n\\section{Results and Discussions}\\label{sec:ResultsDiscussion}\nIn this study, first we choose four standard scalable unimodal test functions (in presence of variable bounds): \nEllipsoidal ($F_{\\rm elp}$), Schwefel ($F_{\\rm sch}$), Ackley ($F_{\\rm sch}$),\nand Rosenbrock ($F_{\\rm ros}$), described as follows:\n\\begin{eqnarray}\nF_{\\rm elp} &=& \\sum_{i=1}^n ix_i^2 \\\\\nF_{\\rm sch} &=& \\sum_{i=1}^n\\left(\\sum_{j=1}^i x_j\\right)^2 \\\\\nF_{\\rm ack} &=& -20 exp \\left(-0.2 \\sqrt{ \\frac{1}{n} \\sum_{i=1}^{i=n} x_i^2}\\right) -exp\\left(\\frac{1}{n} \\sum_{i=1}^{n}cos(2\\pi x_i)\\right)  +20 + e\\\\\nF_{\\rm ros} &=& \\sum_{i=1}^{n-1} (100(x_i^2-x_{i+1})^2  + (x_i-1)^2) \n\\end{eqnarray}\n\nIn the unconstrained space, $F_{\\rm elp}$, $F_{\\rm sch}$ and $F_{\\rm ack}$ have a minimum\nat $x_i^{\\ast}=0$, whereas $F_{ros}$ has a minimum at $x_i^{\\ast}=1$.\nAll functions have minimum value $F^{\\ast}=0$.  \n$F_{\\rm elp}$ is the only variable separable problem.\n$F_{\\rm ros}$ is a challenging test problem that has a ridge which poses\ndifficulty for several optimizers. \nIn all the cases the number of variables is chosen to be $n=20$. \n\nFor each test problem three different scenarios corresponding to the\nrelative location of the \noptimum with respect to the allowable search range are considered. This is done \nby selecting different variable bounds, as follows:  \n \n\\begin{description}\n\\item[On the Boundary:] Optimum is exactly on one of the variable boundaries (for $F_{\\rm elp}$, $F_{\\rm sch}$ and $F_{\\rm ack}$ \n$x_i\\in [0, 10]$, and for $F_{\\rm ros}$, $x_i\\in [1, 10]$),\n\\item[At the Center:] Optimum is at the center of the allowable range (for $F_{\\rm elp}$, $F_{\\rm sch}$ and $F_{\\rm ack}$ \n$x_i\\in [-10, 10]$, and for $F_{\\rm ros}$, $x_i\\in [-8, 10]$), and \n\\item[Close to Boundary:] Optimum is near the variable boundary, \nbut not exactly on the boundary (for $F_{\\rm elp}$, $F_{\\rm sch}$ and $F_{\\rm ack}$ \n$x_i\\in [-1, 10]$, and for $F_{\\rm ros}$,  $x_i\\in [0, 10]$).\n\\end{description}\nThese three scenarios are shown in the Figure~\\ref{fig:3scenario} for a\ntwo-variable problem having variable bounds: $x_i^{(L)}=0$ and $x_i^{(U)}=10$.\nAlthough in practice, the optimum can lie anywhere in the allowable range,\nthe above three scenarios pose adequate representation of different\npossibilities that may exist in practice.  \n\n\\begin{figure}\n\\centering\n\\begin{subfigure}{}\n  \\centering\n  \\includegraphics[width=.65\\linewidth]{boundary-ellp-optima}\n\n  \\label{fig:boundary-ellp-optima}\n\\end{subfigure\n\n\\begin{subfigure}{}\n  \\centering\n  \\includegraphics[width=.65\\linewidth]{center-ellp-optima}\n \n  \\label{fig:center-ellp-optima}\n\\end{subfigure}\n\n\\begin{subfigure}{}\n  \\centering\n  \\includegraphics[width=.65\\linewidth]{close-to-boundary-ellp-optim}\n \n  \\label{fig:close-to-boundary-ellp-optim}\n\\end{subfigure}\n\n\\caption{Location of optimum for $F_{elp}$: (a) on the boundary (b) in the center and (c) close to the \nedge of the boundary by selecting different search domains.}\n\\label{fig:3scenario}\n\\end{figure}\n \nFor each test problem, the population is initialized uniformly in the\nallowable range. We count the number of function\nevaluations needed for the algorithm to find a solution close to the\nknown optimum solution and we call this our \nevaluation criterion $S$.  \nChoosing a high accuracy (i.e. small value of $S$)\nas the termination criteria minimizes the chances of locating the optimum due to \nrandom effects, and provides a better insight into the behavior of a constraint-handling mechanism.  \n\nTo eliminate the random effects and gather results of statistical importance, each algorithm \nis tested on a problem $50$ times (each run starting with a different\ninitial population). A particular run is terminated if the evaluation\ncriterion $S$ is met (noted as a successful run), \nor the number of function evaluations exceeds one million (noted as an\nunsuccessful run). If only a few out of the $50$ runs are\nsuccessful, then we report the number of successful runs in the\nbracket. In this case, the best, median and worst number of function\nevaluations are computed from the successful runs only. If none of the runs\nare successful, we denote this by marking \\textit{(DNC)} (Did Not\nConverge). In such cases, we report the best, median and worst\nattained function values of the best solution at the end of each\nrun.  To distinguish the unsuccessful results from successful\nones, we present the fitness value information of the unsuccessful\nruns in italics.\n\nAn in-depth study on the constraint-handling techniques is carried out\nin this paper. \nDifferent locations of the optimum are selected and systematic comparisons are \ncarried out for PSO, DE and GAs in Sections ~\\ref{subsec:psoresults}, \n~\\ref{subsec:DEresults} and ~\\ref{subsec:GAresults} , respectively. \n \n\\subsection{Results with Particle Swarm Optimization (PSO)}\\label{subsec:psoresults}\nIn PSO, decision variable and the velocity terms are updated\nindependently. Let us say, that the initial position is $\\vec{x}_{t}$, the newly created \nposition is infeasible and represented by $\\vec{x}_{t+1}$, and the repaired solution\nis denoted by  $\\vec{y}_{t}$. \n\nIf the velocity update is based on the infeasible solution as:\n\n\\begin{equation}\n\\label{eqn:velocity-standard}\n\\vec{v}_{t+1}=\\vec{x}_{t+1} - \\vec{x}_{t} \n\\end{equation}\n\nthen, we refer to this as ``Velocity Unchanged''. However, if the velocity update \nis based on the repaired location as: \n\n\\begin{equation}\n\\label{eqn:velocity-recomputed}\n\\vec{v}_{t+1}=\\vec{y}_{t} - \\vec{x}_{t},  \n\\end{equation}\n\nthen, we refer to this as ``Velocity Recomputed''. This terminology is used for rest of the paper. \nFor inverse parabolic (IP) and exponential (Exp) approaches, we use\n``Velocity Recomputed'' strategy only. We have performed ``Velocity\nUnchanged'' strategy with IP and exponential approaches, but the\nresults were not as good as compared to ``Velocity Recomputed'' strategy. \nFor the \\textit{SetOnBoundary} approach, we use the ``Velocity Recomputed''\nstrategy and two other strategies discussed as follows. \n\nAnother strategy named \n``Velocity Reflection'' is used, which simply implies \nthat if a particle is set on the $i$-th boundary, then $v_i^{t+1}$ \nis changed to $-v_i^{t+1}$. The goal of the velocity\nreflection is  to explicitly allow particles to move back into the search space.\nIn the ``Velocity Set to Zero'' strategy,  if a particle is set\non the $i$-th boundary, then the corresponding velocity component is set to zero i.e. $v_i^{t+1}=0$. \nFor the shrink approach, both ``Velocity Recomputed'' and ``Velocity\nSet to Zero'' strategies are used. \n\nFor PSO, a recently proposed {\\em hyperbolic} \\cite{Helwid-Constraing-handling-2013} constraint-handling approach is also\nincluded in this study. This strategy operates by first calculating velocity according to the standard mechanism ~\\ref{eqn:velocity-standard}, and \nin the case of violation a linear normalization is performed on the velocity to restrict the solution from jumping out of the constrained boundary as follows:\n\\begin{equation}\n\\label{eq:hyperbolic}\nv_{i,t+1} =  \\frac{v_{i,t+1}}{1+\\frac{|v_{i,t+1}|}{\\min(x_i^{(U)}-x_{i},x_{i,t}-x_{i}^{(L)})}}. \n\\end{equation}\nEssentially, the closer the particle gets to the boundary (e.g.,\n$x_{i,t}$ only slightly smaller than $x_{i}^{(U)}$), the more difficult it becomes to reach the boundary. In fact, the particle is never completely\nallowed to reach the boundary as the velocity tends to zero. We emphasize again that this strategy is only applicable to\nPSO. A standard PSO is employed in this study with a population size of 100. \nThe results for all the above scenarios with PSO are presented in\nTables~\\ref{tab:PSOEllp} to ~\\ref{tab:PSORos}.\n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\caption{Results on $F_{\\rm elp}$ with PSO for $10^{-10}$ termination criterion.}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\begin{tabular}{|lrrrr|} \\hline \n{Strategy}   & Velocity update             &  {Best} &\n Median &  Worst \\\\ \\hline \\hline \n\\multicolumn{5}{|c|}{\t{$F_{\\rm elp}$ in [0,10]: On the Boundary}} \\\\ \\hline\nIP Spread   & Recomputed          & 39,900  &    47,000    &\n67,000    \\\\ IP Confined & Recomputed            & 47,900 (49) &   88,600   &  140,800    \\\\ \nExp. Spread  & Recomputed     &\\textit{3.25e-01}    &       \\textit{5.02e-01}    &        \\textit{1.08e+00} \\\\\nExp. Confined  & Recomputed  & {\\bf 4,600}     &   {\\bf 5,900}   &\n{\\bf 7,500}         \\\\ \nPeriodic & Recomputed &\\textit{3.94e+02} \\textit{(DNC)}& \\textit{6.63e+02} & \\textit{1.17e+03} \\\\ \nPeriodic & Unchanged & \\textit{8.91e+02} \\textit{(DNC)} &\n\\textit{1.03e+03} &\\textit{1.34e+03} \\\\ \nRandom &  Recomputed &  \\textit{1.97e+01} \\textit{(DNC)}& \\textit{3.37e+01} & \\textit{8.10e+01}  \\\\ \nRandom &  Unchanged & \\textit{5.48e+02} \\textit{(DNC)}&\\textit{6.69e+02}  & \\textit{9.65e+02}   \\\\ \nSetOnBoundary & Recomputed      &  900 (44) & 1,300 & 5,100       \\\\\nSetOnBoundary & Reflected       & 242,100      & 387,100 &  811,400           \\\\   \nSetOnBoundary & Set to Zero      &  1,300 (48)      & 1,900 & 4,100                 \\\\ \nShrink & Recomputed       &  8,200 (49)      & 10,900 & 14,300                 \\\\ \nShrink & Set to Zero      &  33,000      & 40,700 & 53,900                 \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t  & 14,100 &15,100 & 16,500\t\t\t\\\\\t\\hline \\hline\n\\multicolumn{5}{|c|}{\t{$F_{\\rm elp}$ in [-10,10]: At the Center}}\t\\\\ \\hline\nIP Spread  & Recomputed            & 31,600   & 34,000   &37,900       \\\\\nIP Confined  & Recomputed         & 30,900   & 33,800   & 38,500     \\\\\nExp. Spread  & Recomputed      & 30,500   & 34,700  & 38,300          \\\\\nExp. Confined   & Recomputed    & 31,900   & 35,100  & 38,200          \\\\\nPeriodic & Recomputed    & 32,200   & 35,100  & 37,900    \\\\\nPeriodic & Unchanged      & 33,800   & 36,600  & 41,200          \\\\%\\hline\nRandom  & Recomputed        & 31,900   & 34,800 &  37,400      \\\\\nRandom & Unchanged          & 31,600   & 34,900    & 38,100             \\\\\nSetOnBoundary &Recomputed  & 31,900   & 35,500    & 40,500       \\\\   \nSetOnBoundary & Reflected   & 50,800 (38)  & 83,200  & 484,100        \\\\\nSetOnBoundary & Set to Zero    & 31,600   & 35,000     & 37,200        \\\\\nShrink &  Recomputed           &  32,000  & 34,400 & 48,200                 \\\\% \\hline \nShrink & Set to Zero             &  31,400  & 34,000 & 37,700                 \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\n& {\\bf 29,400} &  {\\bf 31,200}  & {\\bf 34,700}\t\t\t\\\\ \\hline \\hline\n\\multicolumn{5}{|c|}{\t  {$F_{\\rm elp}$ in [-1,10]: Close to Boundary}}        \\\\ \\hline\nIP Spread  & Recomputed            &  28,200       & 31,900    & 35,300    \\\\\nIP Confined  & Recomputed       &  28,300  & 32,900    & 44,600     \\\\\nExp. Spread  & Recomputed     &  28,300      &  30,700  & 33,200     \\\\% \\hline\nExp. Confined & Recomputed     &  29,500      &  33,000  & 44,700     \\\\\nPeriodic & Recomputed       &  \\textit{4.86e+01} \\textit{(DNC)}  &   \\textit{1.41e+02}  &  \n\\textit{4.28e+02}   \\\\\nPeriodic & Unchanged &  \\textit{2.84e+02}  \\textit{(DNC)} & \\textit{5.46e+02}  & \\textit{8.28e+02} \\\\\nRandom &  Recomputed         &  36,900      &  41,900  & 45,600     \\\\\nRandom & Unchanged &  \\textit{1.13e+02}  \\textit{(DNC)} & \\textit{2.26e+02} & \\textit{4.35e+02} \\\\ \nSetOnBoundary & Recomputed  &  \\textit{1.80e+01} \\textit{(DNC)} & \\textit{7.60e+01} &  \n\\textit{3.00e+02}  \\\\ \nSetOnBoundary & Reflected   &  \\textit{2.13e-01} \\textit{(DNC)}& \\textit{2.17e+01} &  \n\t\\textit{1.06e+02}  \\\\\nSetOnBoundary &  Set to Zero    &  31,700 (2)     &  31,700  & 32,600     \\\\\nShrink &  Recomputed           &  29,500 (6)      &  36,100  & 42,300     \\\\\nShrink &  Set to Zero             & 28,400 (36)      &  32,700  & 65,600     \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\n& {\\bf 25,900} & {\\bf 29,200} & {\\bf 31,000}\t\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:PSOEllp}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\\begin{table*}\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\caption{Results on $F_{\\rm sch}$ with PSO for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrrr|} \\hline \\hline\n{{Strategy}}  & Velocity   & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{5}{|c|}{{$F_{\\rm sch}$ in [0,10]: On the Boundary}} \\\\ \\hline\nIP Spread   &  Recomputed         &  67,200  & 257,800 & 970,400       \\\\% \\hline\nIP Confined   &  Recomputed          &  112,400 (6)  & 126,500 & 145,900       \\\\% \\hline\nExp. Spread    &  Recomputed     &  \\textit{3.79e+00} \\textit{(DNC)}& \\textit{8.37e+00} & \\textit{1.49e+01}  \\\\\nExp. Confined  &  Recomputed      & {\\bf  4,900} & {\\bf 6,100}   & {\\bf  13,500}        \\\\% \\hline\nPeriodic & Recomputed      &  \\textit{4.85e+03} \\textit{(DNC)}& \\textit{7.82e+03}&\\textit{1.34e+04}  \\\\\nPeriodic &Unchanged & \\textit{7.69e+03} \\textit{(DNC)}& \\textit{1.11e+04}& \\textit{1.51e+04}    \\\\% \\hline\nRandom &Recomputed    &  \\textit{2.61e+02} \\textit{(DNC)}& \\textit{5.44e+02}  &  \\textit{1.05e+03}       \\\\\n\nRandom &Unchanged   &  \\textit{5.30e+03} \\textit{(DNC)} & \\textit{7.60e+03} & \\textit{1.22e+04} \\\\% \\hline\n              \nSetOnBoundary &Recomputed  & 800 (30) & 1,100     &   3,900     \\\\ \nSetOnBoundary &Reflected    & 171,500  & 241,700   &   434,200     \\\\\nSetOnBoundary &Set to Zero   & 1,000 (40)      &  1,600      &  5,300              \\\\\nShrink &Recomputed\t        &   6,900    &  9,100     &     11,600                           \\\\%       \\hline\nShrink &Set to Zero             &  17,900     &   31,900    &    49,800                            \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t\t& 36,400 & 41,700 & 48,700\t\t\t\\\\   \\hline   \\hline\n\\multicolumn{5}{|c|}{{$F_{\\rm sch}$ in [-10,10]: At the Center}} \\\\ \\hline\nIP Spread  &  Recomputed            & 106,700 & 127,500 & 144,300       \\\\\nIP Confined  &  Recomputed          &  111,500  & 130,100    & 149,900     \\\\% \\hline\nExp. Spread  &  Recomputed      &  112,300  & 131,400   &  149,000         \\\\\nExp. Confined  &  Recomputed  &  116,400  & 131,300   &  148,200         \\\\\nPeriodic &Recomputed        &  113,400  & 130,900    &    150,600  \\\\%\t\\hline\nPeriodic &Unchanged         & 121,200   & 137,800       & 159,100          \\\\\nRandom &Recomputed          &  112,900  & 129,800     &  151,100       \\\\\nRandom &Unchanged           & 117,000  &   130,600    &  148,100        \\\\\nSetOnBoundary &Recomputed   &  118,500 (49)  & 132,300       & 161,100       \\\\\nSetOnBoundary &Reflected    & \\textit{3.30e-06} \\textit{(DNC)}& \\textit{8.32e+01}\\textit{(DNC)}&         \n\\textit{2.95e+02} \\textit{(DNC)}   \\\\% \\hline \nSetOnBoundary & Set to Zero    &  111,900  & 132,200      &   149,700               \\\\\nShrink.&Recomputed            & 111,800 (49)\t& 131,800\t& 183,500  \\\\\nShrink.&Set to Zero             & 108,400\t& 125,100\t& 143,600  \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t\t& {\\bf 101,300}\t& \t{\\bf  117,700} & {\\bf 129,700} \\\\\\hline \\hline\n\\multicolumn{5}{|c|}{{$F_{\\rm sch}$ in [-1,10]: Close to\n  Boundary}} \\\\ \\hline\nIP Spread   &  Recomputed           &  107,200         &  130,400       &  272,400      \\\\\nIP Confined  &  Recomputed         &  120,100 (44)        & 171,200        & 301,200    \\\\\nExp. Spread  &  Recomputed       &   92,800   & 109,200        &  126,400        \\\\\nExp. Confined &  Recomputed     &   110,200   & 127,400        &  256,100        \\\\\nPeriodic&Recomputed        &    \\textit{8.09e+02} \\textit{(DNC)}& \\textit{2.01e+03} \\textit{(DNC)}& \n\\textit{5.53e+03}\\textit{(DNC)}      \\\\\nPeriodic&Unchanged         &     \\textit{2.16e+03} \\textit{(DNC)}       &      \\textit{4.36e+03} \\textit{(DNC)}     &       \\textit{6.87e+03} \\textit{(DNC)}          \\\\\nRandom&Recomputed          & 123,300 & 165,600\t& 280,000         \\\\\nRandom&Unchanged           &   \\textit{8.17e+02} \\textit{(DNC)} & \\textit{1.96e+03} \\textit{(DNC)}  &   \n\\textit{2.68e+03} \\textit{(DNC)}               \\\\\n\t\t\nSetOnBoundary&Recomputed   &    \\textit{2.50e+00} \\textit{(DNC)}   &   \\textit{1.25e+01} \\textit{(DNC)}   &         \\textit{5.75e+02} \\textit{(DNC)}       \\\\\nSetOnBoundary&Reflected    &     \\textit{1.86e+00} \\textit{(DNC)}        &      \\textit{7.76e+00} \\textit{(DNC)}      &       \\textit{5.18e+01} \\textit{(DNC)}          \\\\\nSetOnBoundary&Set to Zero    &   \\textit{1.00e+00} \\textit{(DNC)}     &   \\textit{5.00e+00} \\textit{(DNC)}     &       \n\\textit{4.21e+02}  \\textit{(DNC)}              \\\\\nShrink & Recomputed            &      \\textit{5.00e-01} \\textit{(DNC)} &       \\textit{3.00e+00} \\textit{(DNC)}       &     \\textit{1.60e+01} \\textit{(DNC)}       \\\\\nShrink &Set to Zero             &   108,300 (8)   & 130,300        &  143,000        \\\\ \nHyperbolic \t& Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t& {\\bf 93,100} \t\t&{\\bf \t108,300}   &  {\\bf 119,000}\t\\\\ \\hline\n\\end{tabular}\n\\label{tab:PSOSch}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\caption{Results on $F_{\\rm ack}$ with PSO for $10^{-10}$ termination criterion.  }\n\\begin{center}\n\\begin{tabular}{|lrrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{5}{|c|}{{$F_{\\rm ack}$ in [0,10]: On the Boundary}} \\\\ \\hline\nIP Spread  &  Recomputed\t\t&\t150,600 (49)\t& 220,900\t & 328,000  \\\\\nIP Confined &  Recomputed\t& \\textit{4.17e+00} \\textit{(DNC)}  & \\textit{6.53e+00} & \\textit{8.79e+00}\\\\ \nExp. Spread &  Recomputed\t&\\textit{2.76e-01} \\textit{(DNC)}  & \\textit{9.62e-01}  & \n\\textit{2.50e+00} \\\\ \nExp. Confined &  Recomputed\t& 7,800\t& 9,600\t& 11,100 \\\\% \\hline \nPeriodic&Recomputed  \t&\\textit{6.17e+00} \\textit{(DNC)} & \\textit{6.89e+00}  & \n\\textit{9.22e+00}  \\\\\n\t\nPeriodic&Unchanged \t & \\textit{8.23e+00} \\textit{(DNC)}  & \\textit{9.10e+00} & \n\\textit{9.68e+00} \\\\\nRandom&Recomputed  \t& \\textit{3.29e+00} \\textit{(DNC)} & \\textit{3.40e+00} & \\textit{4.19e+00} \\\\\nRandom&Unchanged &\\textit{6.70e+00} \\textit{(DNC)}& \\textit{7.46e+00}& \\textit{8.57e+00} \n\\\\ \nSetOnBoundary&Recomputed  \t&{\\bf  800}\t& {\\bf 1,100}\t& {\\bf 2,100}  \\\\% \\hline \nSetOnBoundary&Reflected \t & 420,600\t& 598,600\t& 917,400  \\\\\nSetOnBoundary&Set to Zero\t & 1,100\t& 1,800\t& 3,100\t \\\\% \\hline \nShrink.&Recomputed  \t\t& 33,800 (5) & 263,100  & 690,400  \\\\% \\hline \nShrink.&Set Zero \t\t& \\textit{3.65e+00} \\textit{(DNC)}  & \\textit{6.28e+00}  & \\textit{8.35e+00} \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t& 24,600 (25) &26,100 &28,000 \\\\\t\\hline \\hline\n\\multicolumn{5}{|c|}{   {$F_{\\rm ack}$ in [-10,10]: At the Center}} \\\\ \\hline\nIP Spread &  Recomputed\t\t\t &\t53,900 (46)\t& 58,600\t& 66,500  \\\\% \\hline\nIP Confined &  Recomputed\t\t\t& 54,800 (49)\t& 59,200\t& 64,700  \\\\\nExp. Spread &  Recomputed\t\t& {\\bf 55,100}\t& {\\bf 59,300}\t& {\\bf 63,600} \\\\\nExp. Confined\t&  Recomputed\t &\t 56,800 & 59,600\t& 65,000  \\\\\nPeriodic&Recomputed \t\t & 55,700 (48)\t& 59,900\t& 64,700  \\\\\nPeriodic&Unchanged \t\t &\t57,900 (49)\t& 62,100\t& 66,700  \\\\% \\hline\nRandom&Recomputed  \t\t& 55,100 (47)\t& 59,400\t& 65,100  \\\\\nRandom&Unchanged \t\t&\t56,300\t & 59,700\t& 65,500  \\\\% \\hline\nSetOnBoundary&Recomputed \t & 55,100 (49)\t& 58,900\t& 65,400 \\\\\nSetOnBoundary&Reflected  \t& 86,900 (4)\t& 136,400\t& 927,600 \\\\% \\hline\nSetOnBoundary&Set to Zero \t& 53,900 (49)\t& 59,600\t& 67,700 \\\\\nShrink &Recomputed  \t\t& 55,800 (47)\t\t& 58,700\t& 65,800 \\\\\nShrink &Set to Zero \t\t\t& 55,700 (49)\t\t& 58,900\t& 62,000  \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t\t\t& 52,900 (49) & 56,200 & 64,400\t\t\\\\\\hline \\hline\n\\multicolumn{5}{|c|}{   {$F_{\\rm ack}$ in [-1,10]: Close to\n  Boundary}} \\\\ \\hline\nIP Spread &  Recomputed\t\t\t &\t54,600 (5)\t& 55,100\t& 56,600 \\\\\nIP Confined  &  Recomputed\t\t\t& 63,200 (1)\t& 63,200\t& 63,200  \\\\\nExp. Spread &  Recomputed\t\t& {\\bf 51,300}\t& {\\bf 55,200}\t& {\\bf 58,600} \\\\\nExp. Confined &  Recomputed & \\textit{1.42e+00} \\textit{(DNC)} & \\textit{2.17e+00} \n& \\textit{2.92e+00}  \\\\\nPeriodic&Recomputed  & \\textit{2.88e+00} \\textit{(DNC)}  & \\textit{4.03e+00} & \\textit{5.40e+00}  \\\\\nPeriodic&Unchanged  & \\textit{6.61e+00} \\textit{(DNC)}  & \\textit{7.46e+00}  & \\textit{8.37e+00} \\\\ \nRandom&Recomputed  & 60,300 (45) & 66,200 & 72,200 \\\\% \\hline\nRandom&Unchanged  \t& \\textit{4.21e+00} \\textit{(DNC)} &\n\\textit{4.93e+00}  & \\textit{6.11e+00} \\\\ \nSetOnBoundary&Recomputed & \\textit{2.74e+00} \\textit{(DNC)} & \\textit{3.16e+00} & \\textit{3.36e+00} \\\\\nSetOnBoundary&Reflected \t & 824,700 (1)\t& 824,700\t& 824,700 \\\\% \\hline\nSetOnBoundary&Set to Zero \t& \\textit{1.70e+00} \\textit{(DNC)} & \\textit{2.63e+00} \n& \\textit{3.26e+00}  \\\\ \nShrink&Recomputed  & \\textit{1.45e+00} \\textit{(DNC)} & \\textit{2.34e+00}  & \\textit{2.73e+00} \\\\ \nShrink&Set to Zero & \\textit{2.01e+00} \\textit{(DNC)} & \\textit{3.96e+00}  & \\textit{6.76e+00}  \\\\ \n\tHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic})) &\n        50,000 (39) &  53,500 & 58,100 \\\\ \\hline\n\\end{tabular}\n\\label{tab:PSOAck}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\caption{Results on $F_{\\rm ros}$ with PSO for $10^{-10}$ termination criterion.}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\begin{tabular}{|lrrrr|} \\hline \n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{5}{|c|}{   {$F_{\\rm ros}$ in [1,10]: On the Boundary}} \\\\ \\hline\nIP Spread  &  Recomputed                   & 89,800 & 195,900  & 243,300\t \\\\\nIP Confined &  Recomputed                  & 23,800 &164,300 & 209,300\t\\\\\nExp. Spread  &  Recomputed   & \\textit{9.55e-01} \\textit{(DNC)} & \\textit{2.58e+00}\n& \\textit{7.64e+00} \\\\ \nExp. Confined  &  Recomputed            & {\\bf 3,700} &\t{\\bf 128,100}\t& {\\bf 344,400}\t\\\\\nPeriodic&Recomputed                & \\textit{1.24e+04} \\textit{(DNC)}  & \\textit{2.35e+04}\n\t&\\textit{4.24e+04} \\\\\nPeriodic&Unchanged                 & \\textit{6.99e+04}  \\textit{(DNC)} & \\textit{1.01e+05} \n& \\textit{1.45e+05} \t\t\\\\\nRandom&Recomputed           & \\textit{6.00e+01} \\textit{(DNC)}  & \\textit{1.37e+02} & \n\t\\textit{4.42e+02}  \t\t  \\\\\nRandom&Unchanged          & \\textit{2.32e+04} \\textit{(DNC)}  & \\textit{3.90e+04} \n& \\textit{8.22e+04}  \\\\ \nSetOnBoundary&Recomputed    \t&  900 (45) &\t1,600 \t&89,800\t\t \\\\\nSetOnBoundary&Reflected     \t&  \\textit{2.14e-03} \\textit{(DNC)}  & \\textit{6.01e+02} \n& \\textit{5.10e+04} \t\t \\\\ \nSetOnBoundary&Set to Zero    \t& 1,400 (48) &\t3,000\t& 303,700 \t\t \\\\\nShrink.&Recomputed             \t& 3,900 (44) &\t5,100 \t& 406,000\t\t \\\\% \\hline\nShrink.&Set to Zero                     & 15,500 &136,200 & 193400\t\t \\\\\nHyperbolic    &   & 177,400 (45) & 714,300 & 987,500\\\\\t\t \\hline \\hline\n\\multicolumn{5}{|c|}{   {$F_{\\rm ros}$ in [-8,10]: Near the\n  Center}} \\\\ \\hline\nIP Spread   &  Recomputed                    & 302,300 (28) & 774,900 & 995,000 \t   \\\\% \\hline\nIP Confined  &  Recomputed                  & 296,600 (32) &729,000 &955,000 \t \\\\\nExp. Spread  &  Recomputed              &  208,800 (24) & 754,700 & 985,200  \t \\\\\nExp. Confined &  Recomputed              &  301,100 (33) & 801,400 & 961,800      \\\\\nPeriodic&Recomputed                &  26,200 (27)  & 705,100  & 986,200        \\\\% \\hline\nPeriodic&Unchanged                 &  247,300 (32) & 776,800 & 994,900  \t    \\\\% \\hline\nRandom&Recomputed                  &  311,200 (30)  &809,300 & 990,800        \\\\\nRandom&Unchanged                   & 380,100 (29) & 793,300 & 968,300\t \\\\%\t\\hline\nSetOnBoundary&Recomputed           & {\\bf 248,700} (35) & {\\bf 795,600} & {\\bf 973,900}  \\\\\nSetOnBoundary&Reflected            &  661,900 (01) & 661,900  & 661,900      \\\\% \\hline\nSetOnBoundary&Set to Zero           & 117,400 (25) &\t858,400 \t& 995,400              \\\\% \\hline\nShrink.&Recomputed                    & 347,900 (33) & 790,500& 996,300                \\\\% \\hline\nShrink.&Set to Zero                     & 353,300 (26) & 788,700 & 986,800              \\\\ \nHyperbolic   & Modified (Eq.~(\\ref{eq:hyperbolic}))\t           & \\textit{6.47e-08 (DNC)} & \\textit{1.27e-04 (DNC)}& \\textit{6.78e+00 (DNC)} \t\\\\\t\\hline \\hline\n\\multicolumn{5}{|c|}{   {$F_{\\rm ros}$ in [1,10]: Close to Boundary}}\n\\\\ \\hline\nIP Spread  &  Recomputed                    & 184,600 (47) &  442,200  & 767,500       \\\\% \\hline\nIP Confined &  Recomputed                   & 229,900 (40) & 457,600 & 899,200          \\\\% \\hline\nExp. Spread  &  Recomputed               &  19,400 (47) & 378,200 & 537,300       \\\\\nExp. Confined &  Recomputed             &  \\textit{6.79e-03} \\textit{(DNC)}   & \\textit{4.23e+00} & \n\\textit{6.73e+01}   \\\\ \nPeriodic&Recomputed                &  \\textit{1.51e-02} \\textit{(DNC)}  \t& \\textit{3.73e+00} \n& \\textit{5.17e+02}      \\\\\nPeriodic&Unchanged             &  \\textit{1.92e+04} \\textit{(DNC)}  & \\textit{2.86e+04} & \n\\textit{6.71e+04}       \\\\ \nRandom&Recomputed                  & {\\bf 103,800} \t& {\\bf 432,200}\t& {\\bf 527,200}      \\\\\nRandom&Unchanged                   &   \\textit{2.33e+02} \\textit{(DNC)}  & \\textit{1.47e+03}  \n& \\textit{4.23e+03}  \\\\  \nSetOnBoundary&Recomputed           & \\textit{1.71e+01} \\textit{(DNC)} & \\textit{1.87e+01}& \n\\textit{3.13e+02}  \t\\\\\nSetOnBoundary&Reflected            & \\textit{6.88e+00} \\textit{(DNC)}  \t& \\textit{5.52e+02}  & \n\t\\textit{2.14e+04}  \\\\\nSetOnBoundary&Set to Zero            & \\textit{6.23e+00} \\textit{(DNC)} & \\textit{1.80e+01}\n& \\textit{3.12e+02}          \\\\ \nShrink &Recomputed                    & 350,300 (3) \t& 350,900\t& 458,400               \\\\ \nShrink &Set to Zero                     & 163,700 (26) & 418,000  &531,900   \\\\ \nHyperbolic & Modified (Eq.~(\\ref{eq:hyperbolic}))\t & 920,900 (1) & 920,900 & 920,900     \t\\\\ \t\\hline\n\\end{tabular}\n\\label{tab:PSORos}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\nThe extensive simulation results are summarized using the following method. For each (say $j$) of the 14\napproaches, the corresponding number of the successful applications ($\\rho_j$) are recorded. Here,\nan application is considered to be successful if more than 45 runs out of 50\nruns are able to find the optimum within the specified accuracy. It is\nobserved that IP-S is successful in 10 out of 12 problem instances. Exponential confined approach (Exp-C) is successful in 9\ncases. To investigate the required number of function evaluations (FE) needed\nto find the optimum, by an approach (say $j$), we compute the average number of $\\bar{\\rm FE}_k^j$ \nneeded to solve a particular problem ($k$) and construct the following objective for\n$j$-th approach:\n\\begin{equation}\n\\mbox{FE-ratio}_j = \\frac{1}{\\rho_j}\\sum_{k=1}^{12} \\frac{\\mbox{FE}_k^j}{\\bar{\\rm FE}_k^j}, \n\\end{equation}\nwhere FE$_k^j$ is the FEs needed by the $j$-th approach to solve the\n$k$-th problem. Figure~\\ref{fig:pso_rank} shows the performance of\neach ($j$-th) of the 14 approaches on the two-axes plot ($\\rho_j$ and\n$\\mbox{FE-ratio}_j$). \n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=1.0]{pso_rank.eps}    \n\\end{center}\n\\caption{Performance comparison of 14 approaches for two metrics --\n  number of problems solved successfully and function evaluation ratio\n  -- with the PSO algorithm.}\n\\label{fig:pso_rank}\n\\end{figure}\n\nThe best approaches should have large $\\rho_j$ values and small $\\mbox{FE-ratio}_j$ values. \nThis results in a trade-off between three best approaches which are marked in filled circles. All other 11\napproaches are {\\em dominated\\\/} by these three approaches. The\n{\\it SetBound} (\\textit{SetOnBoundary}) with velocity set to zero performs in only six out of 12\nproblem instances. Thus, we ignore this approach. There is a clear\ntrade-off between IP-S and Exp-C approaches. IP-S solves one\nproblem more, but requires more FEs in comparison to Exp-C. Hence, we\nrecommend the use of both of these methods vis-a-vis all other methods used in this study. \n\nOther conclusions of this extensive study of PSO with different\nconstraint-handling methods are summarized as follows:\n\\begin{enumerate}\n\\item The constraint-handling methods show a large variation in the performance\ndepending on the choice of test problem and location of the optimum in\nthe allowable variable range.\n\n\\item When the optimum is on the variable boundary, periodic and\n  random allocation methods perform poorly. This\n  is expected intuitively. \n\n\\item When the optimum is on the variable boundary, methods that set infeasible\n  solutions on the violated boundary (\\textit{SetOnBoundary} methods) work very\n  well for obvious reasons, but these methods do not perform well for other cases. \n\n\\item When the optimum lies near the center of the allowable range,\n  most constraint-handling approaches work almost equally well. This can be understood intuitively from the fact that\ntendency of particles to fly out of the search space is small when the optimum is\nin the center of the allowable range. For example, the\nperiodic approaches fail in all the cases but are able to demonstrate\nsome convergence characteristics \nfor all test problems, when the optimum is at the center. When the optimum is on the boundary or close\nto the boundary, then the effect of the chosen \nconstraint-handling method becomes critical.\n\n\\item The shrink method (with ``Velocity Recomputed\" and ``Velocity Set\n  Zero'' strategies) succeeded in 10 of the 12 cases.\n\\end{enumerate}\n\n\\subsection{Parametric Study of $\\alpha$}\\label{sec:alpha-effect}\nThe proposed IP approaches involve a parameter $\\alpha$ affecting the\nvariance of the probability distribution for the mapped variable. In\nthis section, we perform a parametric study of $\\alpha$ to determine\nits effect on the performance of the IP-S approach.\n \nFollowing $\\alpha$ values are chosen: $0.1$, $1$, $10$, and\n$1,000$. To have an idea of the effect of $\\alpha$, we plot the\nprobability distribution of mapped values in the allowable range\n($[1,10]: On the Boundary$) for $d=1.0$ in Figure~\\ref{fig:Alpha-effect-figure}. It can\nbe seen that for $\\alpha=10$ and 1,000, the distribution is almost\nuniform. \n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.75]{alpha-effect-final.eps}    \n\\caption{Probability distribution function with different $\\alpha$\n  values. The $x$-axis denotes the increased distance from the\n  violated boundary. $x=1$ means the violated boundary. The child is\n  created at $x=0$.}\n\\label{fig:Alpha-effect-figure}\n\\end{center}\n\\end{figure}\n\nFigure ~\\ref{fig:Alpha-effect-result} shows the effect\nof $\\alpha$ on $F_{elp}$ problem. For the same termination criterion \nwe find that $\\alpha=0.1$ and $1.0$ perform better compared to other\nvalues. With\nlarger values of $\\alpha$ the IP-S method does not even find the\ndesired solution in all 50 runs. \n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=1.0]{Alpha-effect.eps}    \n\\caption{Performance of PSO algorithm with the IP-S approach with\n  different $\\alpha$ values on $F_{elp}$ problem.}\n\\label{fig:Alpha-effect-result}\n\\end{center}\n\\end{figure}\n\n\\subsection{Results with Differential Evolution (DE)}\\label{subsec:DEresults}\nDifferential evolution, originally proposed in \\cite{storn}, has gained \npopularity as an efficient evolutionary optimization algorithm. \nThe developers of DE proposed a total of ten different strategies \n\\cite{StornPriceBook}. In \\cite{DE-Boundary-Handling} it was shown that \nperformance of DE largely depended upon the choice of constraint-handling mechanism.\nWe use Strategy~1 (where the offspring is created around the population-best solution), which is\nmost suited for solving unimodal problems \\cite{padhyeJOGO2012}. \nA population size of $50$ was chosen with parameter values of  \n$CR=0.5$ and $F=0.7$. Other parameters are set the same as before. \nWe use $S=10^{-10}$ as our termination criterion. Results are tabulated \nin Tables~\\ref{tab:DEellp} to ~\\ref{tab:DEros}. \nFollowing two observations can be drawn:\n\\begin{enumerate}\n\\item For problems having optimum at one of the boundaries,\n  {\\it SetOnBoundary} approach performs the best. This is not a surprising\n  result.\n\\item However, for problems having the optimum near the center of the\n  allowable range, almost all eight algorithms perform in a similar\n  manner.\n\\item For problems having their optimum close to one of the boundaries,\nthe proposed IP and existing exponential approaches perform better than\nthe rest of the approaches with DE.\n\\end{enumerate}\nDespite the differences, somewhat similar performances of different constraint-handling\napproaches with DE indicates\nthat the DE is an efficient optimization algorithm and its performance\nis somewhat less dependent on the choice of constraint-handling scheme\ncompared to the PSO algorithm.\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results on $F_{\\rm elp}$ with DE for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrr|} \\hline \n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{$F_{\\rm elp}$ in [0,10]: On the Boundary} \\\\ \\hline\nIP Spread       &       25,600   & 26,850 &27,650  \\\\ \nIP Confined   &       22,400   &23,550  &24,200  \\\\ \nExp. Spread  &      38,350   & 39,800 &41,500  \\\\ \nExp. Confined &     19,200   & 20,700 & 21,900 \\\\ \nPeriodic                &       42,400   & 43,700 &       45,050   \\\\\nRandom                  &       40,650   & 43,050 & 44,250 \\\\ \nSetOnBoundary           & {\\bf 2,850}         &     {\\bf   3,350}    & {\\bf 3,900} \\\\ \nShrink                    & 4,050          & 4,900          & 5,850  \\\\       \\hline \\hline\n\\multicolumn{4}{|c|}{$F_{\\rm elp}$ in [-10,10]: At the Center} \\\\ \\hline  \nIP Spread       & 29,950 & 31,200 & 32,500     \\\\ \nIP Confined     & {\\bf 29,600}  & {\\bf 31,200} & {\\bf 32,400}      \\\\\nExp. Spread  & 29,950 &31,300 & 32,400      \\\\ \nExp. Confined &  30,500& 31,400       & 32,250    \\\\%  \\hline\nPeriodic                &   29,650 &     31,300 &         32,400       \\\\\nRandom                  &  30,000 &      31,200 &         31,250     \\\\ \nSetOnBoundary           & 29,850 & 31,200 & 32,700    \\\\\nShrink                     & 30,300 & 31,250  &32,750  \\\\        \\hline \\hline\n\\multicolumn{4}{|c|}{$F_{\\rm elp}$ in [-1,10]: Close to Boundary} \\\\ \\hline\nIP Spread       & 28,550 & 29,600 &       30,550   \\\\\nIP Confined    & 28,500 & 29,500 & 30,650   \\\\%  \\hline\nExp. Spread & {\\bf 28,050} & {\\bf 28,900} & {\\bf 29,850}   \\\\\nExp. Confined& 28,150 &      29,050   & 29,850   \\\\\nPeriodic                &  29,850 &      30,850 &         32,100      \\\\%  \\hline\nRandom                  &  28,900 &      30,200   & 31,000    \\\\\nSetOnBoundary           &  28,650 &      29,600   & 30,500  \\\\ \nShrink                     & 28,800 & 29,900 & 31,200  \\\\        \\hline \\hline\n\\end{tabular}\n\\label{tab:DEellp}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{footnotesize}\n\\caption{Results on $F_{\\rm sch}$ with DE for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrr|} \\hline \n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [0,10]: On the Boundary}} \\\\ \\hline\nIP Spread       &       26,600 &         27,400   & 28,000          \\\\\nIP Confined    &       22,450 &         23,350   & 24,300 \\\\ \nExp. Spread  &      40,500 &         42,050   & 43,200   \\\\\nExp. Confined&      19,650 &         20,350   & 22,050           \\\\ \nPeriodic                &     44,700   & 46,300           & 48,250   \\\\ \nRandom                  & 43,850       & 45,150           &       47,000     \\\\%  \\hline\nSetOnBoundary           &  {\\bf 2,100  }     &{\\bf  3,100  }          & {\\bf       3,750 } \\\\\nShrink                     & 3,450        & 4,400            &       5,100      \\\\     \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-10,10]: At the Center}} \\\\ \\hline\nIP Spread      &      {\\bf  258,750} & {\\bf 281,650}  & {\\bf 296,300}   \\\\\nIP Confined   &      268,150 & 283,050  & 300,450  \\\\\nExp. Spread &     266,850 & 283,950  & 304,500  \\\\\nExp. Confined &     266,450 & 283,700  & 305,550  \\\\\nPeriodic                &      269,700 & 284,100  & 310,100  \\\\\nRandom                  &      263,300 & 282,600  & 306,250         \\\\%  \\hline\nSetOnBoundary           &      267,750 & 284,550  & 298,850         \\\\\nShrink                     &      263,600 & 282,750  & 304,350         \\\\        \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-1,10]: Close to Boundary}} \\\\ \\hline\nIP Spread    &    {\\bf  228,950} & {\\bf  242,300}  & {\\bf 255,700}  \\\\%  \\hline\nIP Confined     &     232,200 &  243,900  & 263,400  \\\\\nExp. Spread  &    227,550 &  243,000  & 261,950   \\\\\nExp. Confined &   228,750  &  243,800  & 262,500    \\\\\nPeriodic                &    231,950   & 247,150  & 260,700    \\\\ \nRandom                  &    228,550   & 244,850  & 261,900     \\\\ \nSetOnBoundary           &    237,100  &  255,750  & 266,400     \\\\ \nShrink                     &    234,000  &  253,250  & 275,550     \\\\        \\hline\n\\end{tabular}\n\\label{tab:DEsch}\n\\end{footnotesize}\n\\end{center}\n\\end{table*}\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{footnotesize}\n\\caption{Results on $F_{\\rm ack}$ with DE for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrr|} \\hlin\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [0,10]: On the Boundary}} \\\\ \\hline\nIP Spread      & 43,400 &       44,950 & 45,950    \\\\ \nIP Confined    & 37,300 &       38,700 & 40,350    \\\\\nExp. Spread Dist. & 66,300 &      69,250 & 71,300      \\\\\nExp. Confined Dist& 32,750 &      34,600 & 36,200  \\\\\nPeriodic                &  72,500 &      74,250 & 75,900      \\\\\nRandom                  & 70,650  &      73,000 & 74,750       \\\\\nSetOnBoundary           & {\\bf  2,550} &    {\\bf    3,250}  &{\\bf  3,950 }      \\\\ \nShrink                     &  3,500 & 4,700        & 5,300       \\\\        \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-10,10]: At the Center}} \\\\ \\hline\nIP Spread     &     {\\bf   50,650}   &{\\bf  52,050} &     {\\bf   53,450}     \\\\\nIP Confined    &       51,050   & 52,200 &       53,800    \\\\ \nExp. Spread  &      51,200    &52,150  & 53,400    \\\\%  \\hline\nExp. Confined&      51,100   &52,300  & 53,850  \\\\ \nPeriodic                &       51,250   & 52,250  &53,500 \\\\ \nRandom                  &       50,950   & 52,200 & 53,450    \\\\\nSetOnBoundary           &       50,950   & 52,300 &       53,450  \\\\\nShrink                     &      50,450    & 52,300 &       53,550   \\\\        \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-1,10]: Close to Boundary}} \\\\ \\hline\nIP Spread      &       49,100   & 50,650 & 51,650      \\\\ \nIP Confined    &       48,650   & 50,400 & 52,100         \\\\%  \\hline\nExp. Spread &     {\\bf  48,300}    & {\\bf 49,900} & {\\bf 51,750}   \\\\\nExp. Confined&     48,900    & 50,000 & 51,250 \\\\ \nPeriodic                &      50,400    & 51,950 & 53,300 \\\\ \nRandom                  &      50,250     &51,200  &52,150   \\\\\nSetOnBoundary           &     49,900 (33) & 51,100 & 53,150  \\\\ \nShrink                     &     50,200     & 51,400 & 52,750  \\\\        \\hline\n\\end{tabular}\n\\label{tab:DEack}\n\\end{footnotesize}\n\\end{center}\n\\end{table*}\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{footnotesize}\n\\caption{Results on $F_{\\rm ros}$ with DE for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrr|} \\hline \n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [1,10]: On the Boundary}} \\\\ \\hline\nIP Spread       &       38,850   & 62,000 & 89,700 \\\\ \nIP Confined    &       24,850   & 45,700 & 73,400               \\\\ \nExp. Spread  &      57,100   & 86,800 & 118,600   \\\\\nExp. Confined &     16,600    & 21,400  &79,550 \\\\ \nPeriodic                &    69,550      & 93,500 & 18,1150        \\\\\nRandom                  &    65,850       &92,950  &157,600     \\\\%  \\hline\nSetOnBoundary           &   {\\bf  2,950}        & {\\bf 4,700} & {\\bf 30,450}   \\\\ \nShrink                     &    5,450       & 8,150   &55,550 \\\\        \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [-8,10]: At the Center}} \\\\ \\hline\nIP Spread      &       133,350 (41)      & 887,250&       995,700          \\\\\nIP Confined    &       712,500 (44)      & 854,800 &      991,400          \\\\\nExp. Spread   &      {390,700} (48)       & {866,150} &      {998,950}   \\\\\nExp. Confined &     138,550 (40)        &883,500 &      994,350  \\\\ \nPeriodic                &     764,650 (39) &      874,700 &        999,650      \\\\%  \\hline\nRandom                  &     {\\bf 699,400} (49)  &     {\\bf 885,450} &       {\\bf  999,600}   \\\\ \nSetOnBoundary           &     743,600 (38)  &     865,450& 995,500 \\\\\nShrink                     &    509,900 (40)  &      873,450 &        998,450 \\\\    \\hline    \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [0,10]: Close to Boundary}} \\\\ \\hline\nIP Spread      &      36,850    & 78,700 &       949,700          \\\\\nIP Confined    &       46,400 (46) &     95,900 &         891,450          \\\\ \nExp. Spread  &     49,550 (49)  &     85,900 &         968,200   \\\\%  \\hline\nExp. Confined &     87,300 (43)  &     829,200  & 973,350  \\\\ \nPeriodic                &     {\\bf  38,750}    &{\\bf  62,200}         & {\\bf 94,750}    \\\\\nRandom                  &      41,200    & 61,300 &       461,500  \\\\\nSetOnBoundary           &    8.23E+00 (DNC)    & 1.62E+01 &     1.89E+01        \\\\%  \\hline\nShrink                     &  252,650 (9) &  837,700  & 985,750         \\\\        \\hline\n\\end{tabular}\n\\label{tab:DEros}\n\\end{footnotesize}\n\\end{center}\n\\end{table*}\n\\subsection{Results with Real-Parameter Genetic Algorithms (RGAs)}\n\\label{subsec:GAresults}\nWe have used two real-parameter GAs in our study here:\n\\begin{enumerate}\n\\item {\\em Standard-RGA\\\/} using the simulated binary crossover (SBX) \\cite{debramb} operator\nand the polynomial mutation operator\n  \\cite{debBookMO}. In this approach, variables are expressed as\n  real numbers initialized within the allowable range of each\n  variable. The SBX and polynomial mutation operators can create\ninfeasible solutions. Violated boundary, if any, is handled using one\nof the approaches studied in this paper. Later we shall investigate a\nrigid boundary implementation of these operators which ensures\ncreation of feasible solutions in every recombination and mutation operations.\n\n\\item {\\em Elitist-RGA} in which two newly created offsprings are compared against the two\nparents, and the best two out of these four \nsolutions are retained as parents (thereby introducing elitism). Here, the \noffspring solutions are created using non-rigid\n versions of SBX and polynomial mutation operators. \nAs before, we test eight different constraint-handling approaches and, later\nexplore a rigid boundary implementation of the operators in presence of \nelite preservation. \n\\end{enumerate}\n\nParameters for RGAs are chosen as follows: population size of 100, \ncrossover probability $p_{c}$=0.9, mutation probability $p_{m}$=0.05, distribution index for crossover $n_{dist.,c}$=2, \ndistribution index for mutation $n_{dist.,m}$=100.\nThe results for the Standard-RGA are shown in Tables~\\ref{tab:RGAellp} to ~\\ref{tab:RGAros}\nfor four different test problems. Tables~\\ref{tab:GA-elitist-elp}\nto ~\\ref{tab:GA-elitist-ros} show results using the Elitist-RGA.\nFollowing key observations can be made:\n\\begin{enumerate}\n\\item For all the four test problems, Standard-RGA shows convergence\n  \\textit{only} in the situation when optima is on the boundary.  \n           \n\\item Elitist-RGA shows good convergence on $F_{elp}$ when the optimum is on the boundary and,\nonly some convergence is noted when the optima is at the other locations.\nFor other three problems, convergence is only obtained when optimum is present on the boundary.     \n\n\\item Overall, the performance of Elite-RGA is comparable or slightly better compared to Standard-RGA.\n\\end{enumerate}\n\nThe \\textit{Did Not Converge} cases can be explained on the fact that\nthe SBX operator has the property of creating solutions around\nthe parents; if parents are close to each other. This property is a likely cause of premature\nconvergence as the population gets closer to the optima. \nFurthermore the results suggest that the elitism implemented in this study\n(parent-child comparison) is not quite effective. \n \nAlthough RGAs are able to locate the optima,\nhowever, they are unable to fine-tune the optima due to undesired \nproperties of the generation scheme. This emphasizes the fact that \ngeneration scheme is primarily \nresponsible for creating good solutions, and\nthe constraint-handling methods cannot act as surrogate\nfor generating efficient solutions. \nEach step of the evolutionary \nsearch should be designed effectively in order to achieve overall success. \nOn the other hand one could argue that strategies such as increasing the mutation rate (in order to promote diversity so as to avoid\npre-mature convergence) should be tried, however, creation of good and meaningful solutions\nin the generation scheme is rather an important and a desired fundamental-feature.\n   \nAs expected, when the optima is on the boundary \\textit{SetOnBoundary} finds the optima most efficiently within a minimum number\nof function evaluations. Like in PSO the performance of \n\\textit{Exp. Confined} is better than \n\\textit{Exp. Spread}.\n\\textit{Periodic} and \\textit{Random} show comparable \nor slightly worse performances (these mechanisms don't have any preference\nof creating solutions close to the boundary and actually promote spread of \nthe population). \n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\caption{Results on $F_{\\rm elp}$ with Standard-RGA for termination criterion of $10^{-10}$.}\n\\begin{tabular}{|lrrr|} \\hline \n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [0,10]}} \\\\\nIP Spread                & 9,200 & 10,500  & 12,900                        \\\\% \\hline\nIP Confined            & 7,900 &  9,400 & 10,900     \\\\% \\hline\nExp. Spread        & 103,100 (6) & 718,900 & 931,200   \\\\\nExp. Confined        & 4,500 & 5,700 & 7,000                \\\\% \\hline\nPeriodic                         & 15,200 (1) & 15,200 & 15,200       \\\\\nRandom                           & 68,300 (12) & 314,700 & 939,800      \\\\\nSetOnBoundary                    & {\\bf 1,800} &  {\\bf 2,400} & {\\bf 2,800}               \\\\\nShrink                             & 3,700 & 5,100 & 6,600 \\\\   \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [-10,10]}} \\\\% \\hline\n\\multicolumn{4}{|c|}{   \t{2.60e-02 \\textit{(DNC)}}}            \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [-1,10]}} \\\\ \n\\multicolumn{4}{|c|}{\t{\\textit{1.02e-02 (DNC)}}}\t\t\\\\ \\hline\n\\end{tabular}\n\\label{tab:RGAellp}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\caption{Results on $F_{\\rm sch}$ with Standard-RGA for termination criterion of $10^{-10}$}\n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [0,10]}} \\\\% \\hline\nIP Spread               & 6,800         &9,800 & 11,800    \\\\% \\hline\nIP Confined           & 6,400         & 8,200  & 10,300     \\\\\nExp. Spread         & 21,200 (47)    & 180,000 &      772,200   \\\\% \\hline\nExp. Confined       & 4,300  &      5,500 &  6,300                 \\\\% \\hline\nPeriodic                         & 14,800 (26) &  143,500  & 499,400           \\\\\nRandom                           & 8,700 (43) &   195,200 &        979,300     \\\\% \\hline\nSetOnBoundary                    & {\\bf 1,800} &     {\\bf  2,300} &  {\\bf 2,900}               \\\\% \\hline\nShrink.                             & 3,600  &      4,600 &  5,500                    \\\\      \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-10,10]}} \\\\\n\\multicolumn{4}{|c|}{   {\t\\textit{1.20e-01 (DNC)}}}            \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-1,10]}} \\\\% \\hline\n\\multicolumn{4}{|c|}{   {\\textit{8.54e-02 (DNC)}}}            \\\\ \\hline\n\\end{tabular}\n\\label{tab:RGAsch}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\caption{Results on $F_{\\rm ack}$ with Standard-RGA for termination criterion of $10^{-10}$}\n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [0,10]}} \\\\\nIP Spread       & 12,100         &       22,600   & 43,400                 \\\\\nIP Confined    & 9,800          &       13,200   & 16,400                 \\\\ \nExp. Spread &      58,100 (29) &     355,900  & 994,000                \\\\  \nExp. Confined&    6,300      & 9,100        & 11,900                 \\\\  \nPeriodic                  & 19,600 (46)   & 122,300         &870,200                \\\\   \nRandom          &35,700 (38)      & 229,200        & 989,500                        \\\\    \nSetOnBoundary   & {\\bf 1,800} &       {\\bf  2,500}    & {\\bf 3,100}                                          \\\\%      \\hline\nShrink              &4,200  &       5,700 &  8,600                          \\\\ \\hline   \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-10,10]}} \\\\\n\\multicolumn{4}{|c|}{   { \\textit{7.76e-02(DNC)}}}               \\\\ \\hline \\hline\n \n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-1,10]}} \\\\% \\hline \\hline\n\\multicolumn{4}{|c|}{   {\\textit{4.00e-02 (DNC)}}}               \\\\ \\hline\n \n\\end{tabular}\n\\label{tab:RGAack}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\\begin{table*}\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\begin{center}\n\\caption{Results on $F_{\\rm ros}$ with Standard-RGA for termination criterion of $10^{-10}$}\n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [1,10]: On the Boundary}} \\\\\nIP Spread        & 12,400 (39)&   15,800&  20,000        \\\\\nIP Confined      & 9,400 (39)&    11,800&  13,600                \\\\% \\hline\nExp. Spread  &\\textit{9.73e+00} \\textit{(DNC)} & \\textit{1.83e+00}&  \\textit{2.43e+01}        \\\\% \\hline\nExp. Confined & 6,000       &       6,900&   8,200       \\\\\nPeriodic                  & \\textit{6.30E+01} \\textit{(DNC)}&  \\textit{4.92e+02}&   \\textit{5.27e+04}     \\\\\nRandom                    & \\textit{3.97e+02} \\textit{(DNC)}    &  \\textit{9.28e+02}&     \\textit{1.50e+03} \\\\% \\hline\nSetOnBoundary             &  {\\bf 1,900}     &    {\\bf   2,700} &   {\\bf 3,400}                   \\\\% \\hline\nShrink                       &  4,100      &      5,200 &  6,500                  \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [-8,10]}} \\\\% \\hline \n\\multicolumn{4}{|c|}{   { \\textit{3.64e+00 (DNC)}}} \\\\  \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [1,10]: On the Boundary}} \\\\\n\\multicolumn{4}{|c|}{   {\\textit{1.04e+01(DNC)}}} \\\\  \\hline\n\\end{tabular}\n\\label{tab:RGAros}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\n\n\n\n\\clearpage\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results on $F_{\\rm elp}$ with Elite-RGA for $10^{-10}$ termination criterion.}\n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [0,10]}} \\\\\nIP Spread       &  6,600 &      8,000    &  9,600                                  \\\\% \\hline\nIP Confined    &  6,300 & 8,100 & 9,800                                  \\\\\nExp. Spread   &     4,800 & 6,900  & 8,300                             \\\\\nExp. Confined &    4,600 & 5,800 & 6,700                                         \\\\% \\hline\nPeriodic                  &    6,500 & 8,800 & 11,500                                       \\\\\nRandom                    &     6,400 & 7,900 & 10,300                                  \\\\\nSetOnBoundary             &   {\\bf 2,200} & {\\bf 2,600} & {\\bf 3,500}                                       \\\\\nShrink                       &   4,000 & 5,200 & 6,700                                        \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [-10,10]}} \\\\\nIP Spread        &    980,200 (1) & 980,200 & 980,200               \\\\% \\hline\nIP Confined    &    479,000 (1) & 479,000 & 479,000                           \\\\% \\hline\nExp. Spread &   \\textit{2.06e-01} \\textit{(DNC)} & \\textit{4.53e-01} \n& \\textit{4.86e-01}                        \\\\\nExp. Confined &   954,400 (1)\t & 954,400 & 954,400                                   \\\\% \\hline\nPeriodic                  &   \\textit{1.55E-01} \\textit{(DNC)}& \\textit{2.48E-01} & \\textit{2.36E-01}  \\\\\nRandom                    &   \\textit{1.92E-01} \\textit{(DNC)}& \\textit{2.00E-01} & \\textit{2.46E-01}    \\\\\nSetOnBoundary             &   \\textit{2.11E-01} \\textit{(DNC)}&2.95E-01 & 1.94E-01                       \\\\% \\hline\nShrink                       &   {\\bf 530,900} (3) & {\\bf 654,000} & {\\bf 779,000}                \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm elp}$ in [-1,10]}} \\\\\nIP Spread        &   803,400 (5) & 886,100 & 947,600             \\\\\nIP Confined      &   643,300 (2) & 643,300 &963,000                          \\\\% \\hline\nExp. Spread  &  593,300 (3) & 628,900 & 863,500                       \\\\\nExp. Confined &  655,400 (3) & 940,500 & 946,700                                         \\\\% \\hline\nPeriodic                  &  653,800 (3) & 842,900 &843,100                                  \\\\\nRandom                    &  498,500 (2) & 498,500 &815,500                           \\\\\nSetOnBoundary             &  {\\bf 593,800} (5) & {\\bf 870,500} & {\\bf 993,500}                                \\\\\nShrink                       &  781,000 (2) & 781,000 &928,300                                   \\\\ \\hline\n\\end{tabular}\n\\label{tab:GA-elitist-elp}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results on $F_{\\rm sch}$ with Elite-RGA for termination criterion of $10^{-10}$}\n\\begin{tabular}{|lrrr|} \\hline \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [0,10]}} \\\\ \\hline\nIP Spread        &   5,000 &  6,500 & 7,900                     \\\\% \\hline\nIP Confined      &   4,900 & 6,500 & 7,900                          \\\\% \\hline\nExp. Spread  &  4,300 & 5,800 & 7,800                         \\\\\nExp. Confined &  4,300 & 4,900 & 5,600                                   \\\\\nPeriodic                  &  5,400 & 7,000 & 11,300                                 \\\\\nRandom                    &  5,300 & 6,600 & 8,500                            \\\\\nSetOnBoundary             &  {\\bf 1,600} & {\\bf 2,200} & {\\bf 2,600}                                   \\\\% \\hline\nShrink                       &  3,100 & 4,200 & 5,400              \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-10,10]}} \\\\\n\\multicolumn{4}{|c|}{   {\\textit{8.12e-05(DNC)}}}            \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm sch}$ in [-1,10]}} \\\\% \\hline\n\\multicolumn{4}{|c|}{   { \\textit{1.61e-01(DNC)}}}            \\\\ \\hline \n\\end{tabular}\n\\label{tab:GA-elitist-sch}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results on $F_{\\rm ack}$ with Elite-RGA for termination criterion of $10^-{10}$}\n\\begin{tabular}{|lrrr|} \\hline \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [0,10]}} \\\\ \\hline\nIP Spread       &    6,300 &8,700 &46,500                         \\\\% \\hline\nIP Confined     &    6,800 &9,200 &32,000                       \\\\\nExp. Spread  &   5,600 &6,800 &8,700                       \\\\\nExp. Confined &   5,200 &7,800 &9,900                               \\\\\nPeriodic                  &   6,300 &9,300 &12,200                              \\\\\nRandom                    &   6,200 &8,300 &53,700                        \\\\\nSetOnBoundary             &   {\\bf 1,900} &{\\bf 2,500} &{\\bf 4,000}                               \\\\% \\hline\nShrink                       &   3,900 &5,100 &7,700                           \\\\  \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-10,10]}} \\\\\n\\multicolumn{4}{|c|}{   {\\textit{1.03e-01(DNC)}}}            \\\\ \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ack}$ in [-1,10]}} \\\\\n\\multicolumn{4}{|c|}{   {\\textit{1.15e-00 (DNC)}}}            \\\\ \\hline\n\\end{tabular}\n\\label{tab:GA-elitist-ack}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n \n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results on $F_{\\rm ros}$ with Elite-RGA for termination criterion of $10^-{10}$}\n\\begin{tabular}{|lrrr|} \\hline \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [0,10]}} \\\\ \\hline\nIP Spread       &   9,900 (13) &12,500 &14,000                    \\\\ \nIP Confined      &   10,100 (12)& 12,100 &14,400                      \\\\ \nExp. Spread   &  8,500 (10) &11,000 &15,400                    \\\\ \nExp. Confined &  6,600 (30) &7,800 &8,900                       \\\\ \nPeriodic                  &  9,500 (10) &13,300 &16,800                          \\\\ \nRandom                    &  14,000 (3)& 15,300 &16,100                          \\\\ \nSetOnBoundary             &  {\\bf 2,300} (44) & {\\bf 3,200} & {\\bf 4,500}                              \\\\ \nShrink                       &  4,500 (32) &6,100 &8,100                           \\\\ \\hline \\hline\n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [-8,10]}} \\\\ \n\\multicolumn{4}{|c|}{   {\\textit{1.27e-00 (DNC)}}}            \\\\ \\hline \\hline\n \n\\multicolumn{4}{|c|}{   {$F_{\\rm ros}$ in [1,10]: On the Boundary}} \\\\\n\\multicolumn{4}{|c|}{   {\\textit{1.49e-00 (DNC)}}}            \\\\ \\hline \n\\end{tabular}\n\\label{tab:GA-elitist-ros}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n\n\\subsubsection{RGAs with Rigid Boundary}\\label{subsec:NonelitistGAresults}\n\n\\begin{table*}[htb]\n\\begin{footnotesize}\n\\caption{RGA with rigid boundary with termination criterion $10^{-10}$}\n{\\footnotesize\\begin{center}\n\\begin{tabular}{|lrrr|} \\hline \n\t\t\\multicolumn{4}{|c|}{ Optimum on the boundary   }\t\t\t\\\\\t\\hline\t\\hline\t\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n$F_{elp}$\t\t&\t8,100 & 8,500 & 8,800\t\t\\\\\n$F_{sch}$\t\t&\t7,800 & 8,100 & 8,300\t\t\\\\\n$F_{ack}$\t\t&\t9,500 & 10,100 & 10,800\t\t\\\\\n$F_{ros}$\t\t&\t10,100 (39) & 10,900 & 143,600\t\t\\\\\t\\hline\t\\hline \n\\multicolumn{4}{|c|}{ Optimum in the center}\t\t\\\\\n\\multicolumn{4}{|c|}{ {\\textit{3.88e-02 (DNC)}}}               \\\\\n\\hline \\hline   \n\\multicolumn{4}{|c|}{ Optimum close to the edge of the boundary  }      \\\\\t\n\\multicolumn{4}{|c|}{ {\\textit{9.44e-03 (DNC)}}}               \\\\\n\\hline \\hline   \n\\end{tabular}\n\\label{tab:rga-standard-rigid-boundary}\n\\end{center}}\n\\end{footnotesize}\n\\end{table*}\n\nWe also tested RGA (standard and its elite version) \nwith a rigid bound consideration in its operators. In this case, the\nprobability distributions of both\n{SBX} and polynomial mutation operator are changed in a way so as to\nalways create a feasible solution. \nIt is found that the Standard-RGA with rigid bounds shows \nconvergence only when optimum is on the boundary (Table~\\ref{tab:rga-standard-rigid-boundary}). \nThe performance of Elite-RGA with rigid bounds is slightly better \n(Table~\\ref{tab:rga-elite-rigid-boundary}).\nOverall, SBX operating within the rigid bounds is found to perform slightly better\ncompared to the RGAs employing boundary-handling mechanisms. \nHowever, as mentioned earlier, in the scenarios where the generation scheme cannot \nguarantee creation of feasible only solutions there is a necessary\nneed for constraint-handling strategies. \n\n\\begin{table*}[htb]\n\\begin{footnotesize}\n\\begin{minipage}[b]{1.0\\linewidth}\n\\caption{Elitist-RGA with rigid boundary with termination criterion $10^{-10}$.}\n\\begin{center}\n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}}     & {{Best}}& {{Median}}& {{Worst}} \\\\ \\hline \\hline\n                \\multicolumn{4}{|c|}{ Optimum on the boundary}                      \\\\      \n$F_{elp}$               & 7,300 & 7,900 \t& 8,400    \t\\\\   \n$F_{sch}$               & 6,500 & 6,900 & 7,500    \t \\\\   \n$F_{ack}$               & 9,400 & 10,400 & 12,200    \t\\\\    \n$F_{ros}$               & 11000 (10) & 12700 & 16400      \\\\      \\hline  \\hline\n\\multicolumn{4}{|c|}{ Optimum in the center }               \\\\  \n\\multicolumn{4}{|c|}{ {\\textit{1.24e-01(DNC)}}}               \\\\\n\\hline \\hline   \n\\multicolumn{4}{|c|}{ Optimum close to the boundary edge}      \\\\   \n$F_{elp}$               & 579,800 (3)  &885,900 & 908,600    \\\\     \n$F_{sch}$               & \\textit{2.73E-00} \\textit{DNC} & \\textit{6.18E-00} & \\textit{1.34E-00}   \\\\ \n$F_{ack}$               & \\textit{1.75E-01}  \\textit{DNC} & \\textit{8.38E-01} & \\textit{2.93E-00}  \\\\ \n$F_{ros}$               & \\textit{3.29E-00}  \\textit{DNC} & \\textit{4.91E+00} & \\textit{5.44E+00}  \\\\      \\hline\n\\end{tabular}\n\\label{tab:rga-elite-rigid-boundary}\n\\end{center}\n\\end{minipage}\n\\end{footnotesize}\n\\end{table*}\n\\section{Higher-Dimensional Problems}\\label{sec:scale-up}\nAs the dimensionality of the search space increases it becomes\ndifficult for a search algorithm to locate the optimum.\nConstraint-handling methods play even a more critical role\nin such cases.  \nSo far in this study, DE has been found to be the best algorithm. Next, we consider all\nfour unimodal test problems with an increasing problem size: $n=20$, $50$, $100$, $200$, $300$,\nand $500$. For all problems the variable bounds were chosen such that optima occured\nnear the center of the search space. No population scaling is used for DE. The DE parameters are chosen as $F=0.8$ and $CR=0.9$.\nFor $\\alpha = 1.2$ we were able to achieve a high degree of convergence and\nresults are shown in Figure~\\ref{fig:DE-scale-up}.\nAs seen from the figure, although it is expected that the required\nnumber of function evaluations would increase with the \nnumber of variables, the increase is sub-quadratic.\nEach case is run $20$ times and the termination criteria\nis set as $10^{-10}$. All 20 runs are found to be successful in each case,\ndemonstrating the robustness of the method in terms of finding the\noptimum with a high precision.\nParticularly problems with large variables, complex search spaces and \nhighly nonlinear constraints, such a methodology should be useful in terms of \napplying the method to different real-world problems.  \n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=1.0]{scale.eps}    \n\\end{center}\n\\caption{Scale-up study on four test problems for the DE algorithm with\n  the proposed IP-S approach shows sub-quadratic performance in all\n  problems. Slopes of a fitted polynomial line is marked within\n  brackets for each problem. Linear and quadratic limit lines are shown with dashed lines.}\n\\label{fig:DE-scale-up}\n\\end{figure}\n\nIt is worth mentioning that authors independently tested other scenarios for \nlarge scale problems with corresponding optimum on the boundary, and in order to achieve convergence\nwith IP methods we had to significantly reduce the \nvalues of $\\alpha$. Without lowering $\\alpha$, particularly, PSO did not show any convergence. \nAs expected in larger dimensions the probability to sample a point closer to the boundary\ndecreases and hence a steeper distribution is needed. However, this highlights the usefulness\nof the parameter $\\alpha$ to modify the behavior of the IPMs so as to yield the desired performance. \n\n\\section{General Purpose Constraint-Handling} \n\\label{sec:Constraint-Programming}\n\nSo far we have carried out simulations on problems where constraints have \nmanifested as the variable bounds. The IP methods proposed in this paper can be easily\nextended and employed for solving nonlinear constrained optimization problems (inclusive of \nvariable bounds).\\footnote{By \\textit{General Purpose Constraint-Handling} we imply\ntackling of all variable bounds, inequality constraints and equality constraints.}\n\nAs an example, let us consider a generic inequality constraint function: $g_j(\\vec{x})\n\\geq 0$ -- the $j$-th constraint in a set of $J$ inequality\nconstraints. In an optimization algorithm, every created (offspring) solution\n$\\vec{x}^c$ at an iteration must be\nchecked for its feasibility. If\n$\\vec{x}^c$ satisfies all $J$ inequality constraints, the solution is\nfeasible and the algorithm can proceed with the created solution. \nBut if $\\vec{x}^c$ does not\nsatisfy one or more of $J$ constraints, the solution is\ninfeasible and should be repaired before proceeding further. \n\nLet us\nillustrate the procedure using the inverse parabolic (IP) approach described in\nSection~\\ref{sec:IP}; though other constraint-handling\nmethods discussed before may also be used. The IP approaches require us to locate\nintersection points $\\vec{v}$ and $\\vec{u}$: two bounds in the\ndirection of ($\\vec{x}^p-\\vec{x}^c$), where $\\vec{x}^p$ is one of the\nparent solutions that created the offspring solution (see Figure~\\ref{fig:constr}). The critical intersection\npoint can be found by\nfinding multiple roots of the direction vector with each constraint\n$g_j(\\vec{x})$ and then choosing the\nsmallest root. \n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=1.0]{constr.eps}    \n\\caption{Extension of constraint-handling approaches to constrained optimization problems.}\n\\label{fig:constr}\n\\end{center}\n\\end{figure}\nWe define a parameter\n$\\alpha$ as the extent of a point from $\\vec{x}^c$, as\nfollows:\n\\begin{equation}\n\\vec{x}(\\alpha) = \\vec{x}^c + \\alpha\n\\frac{\\vec{x}^p-\\vec{x}^c}{\\|\\vec{x}^p-\\vec{x}^c\\|}.\n\\label{eq:mapping}\n\\end{equation}\n\nSubstituting above expression for $\\vec{x}(\\alpha)$ \\footnote{Note that $\\alpha$\nhere for calculating points should not be confused with parameter $\\alpha$ introduced in\nthe proposed constraint-handling methods.}in the $j$-th constraint function, we have the following\nroot-finding problem for the $j$-th constraint:\n\\begin{equation}\ng_j(\\vec{x}(\\alpha)) = 0.\n\\end{equation}\nLet us say the roots of the above equation are $\\alpha_k^j$ for\n$k=1,2\\ldots,K_j$. The above procedure can now be repeated for all $J$\ninequality constraints and corresponding roots can be found one at a time. \nSince the extent of $\\alpha$ to reach $\\vec{x}^p$ from $\\vec{x}^c$\nis given as \n\\[\\alpha^p = \\|\\vec{x}^p-\\vec{x}^c\\|,\\]\nwe are now ready to compute the two closest bounds (lower and upper bounds) on\n$\\alpha$ for our consideration, as\nfollows:\n\\begin{eqnarray}\n\\alpha^v &=& \\max \\{\\alpha_k^j | 0 \\leq \\alpha_k^j \\leq \\alpha^p,\n \\forall\nk, \\forall j\\}, \\\\ \n\\alpha^u &=& \\min \\{\\alpha_k^j | \\alpha_k^j \\geq \\alpha^p, \\forall\nk, \\forall j\\}.\n\\end{eqnarray}\nIP-S and IP-C approaches presented in Section~\\ref{sec:IP} now can be used to map the violated variable\nvalue $x_i^c$ into the feasible region using $d=\\alpha^v$ (the lower bound),\n$d_u=\\alpha^u$ (the upper bound) and $d^p=\\alpha^p$ (location of parent). \nIt is clear that the only difficult aspect of this method is to find\nmultiple intersection points in presence of nonlinear constraints. \n\nFor the sake of completeness, we show here that the two bounds for each variable: $x_i\\geq x_i^{(L)}$ and\n$x_i\\leq x_i^{(U)}$ used in previous sections can be also be treated\nuniformly using the above described approach. The two bounds can be written together as follows:\n\\begin{equation}\n\\left(x_i-x_i^{(L)}\\right)\\left(x_i^{(U)}-x_i\\right) \\geq 0.\n\\end{equation}\nNote that a simultaneous non-positive value of each of the bracketed terms\nis not possible, thus only way to satisfy the above left side is to\nmake each bracketed term non-negative. The above inequality can be\nconsidered as a quadratic constraint function, instead of two\nindependent variable bounds and treated as a single combined nonlinear\nconstraint and by finding both roots of the resulting quadratic root-finding equation.  \n\nFinally, the above procedure can also be extended to handle equality constraints ($h_k(\\vec{x})=0$) with a\nrelaxation as follows: $-\\epsilon_k \\leq h_k(\\vec{x}) \\leq\n\\epsilon_k$. Again, they can be combined together as follows:\n\\begin{equation}\n\\epsilon_k^2-(h_k(\\vec{x}))^2 \\geq 0.\n\\end{equation}\nAlternatively, the above can also be written as $\\epsilon_k -\n|h_k(\\vec{x})| \\geq 0$ and may be useful for non-gradient based\noptimization methods, such as evolutionary algorithms. We now show the\nworking of the above constraint handling procedure on a number of\nconstrained optimization problems. \n \n\\subsection{Illustrations on Nonlinear Constrained Optimization}\n\nFirst, we consider the three unconstrained problems used in previous\nsections as $f(\\vec{x})$, but now add an inequality constraint by imposing a\nquadratic constraint that makes the solutions fall within a radius of one-unit from\na chosen point $\\vec{o}$:\n\\begin{equation}\n\\begin{array}{rl}\n\\mbox{Minimize}  & f(\\vec{x}), \\\\\n\\mbox{subject to} & \\sum_{i=1}^{n} (x_i-o_i)^2 \\leq 1.\t\n\\end{array}\n\\label{eq:convex-opti-problem}\n\\end{equation}\nThere are no explicit variable bounds in the above problem. \nBy choosing different locations of the center\nof the hyper-sphere ($\\vec{o}$), we can have different scenarios of\nthe resulting constrained optimum. \nIf the minimum of the objective function (without constraints)\nlies at the origin, then setting $o_i=0$ the unconstrained minimum is also the solution to the\nconstrained problem, and this case is similar to the ``Optimum at the\nCenter'' (but in the context\nof constrained optimization now).\nThe optimal solution is at $x_i=0$ with $f^*=0$. DE with IP-S and previous parameter \nsettings is applied to this new constrained problem, and the results from 50 different runs for this case are shown in\nTable~\\ref{tab:convex-constraints-de-1}.\n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Results for test functions with DE for $o_i=0$ with\n  $S=10^{-10}$.}\n\\label{tab:convex-constraints-de-1}   \n\\begin{tabular}{|lrrr|} \\hline\n{{Strategy}} & {{Best}} & {{Median}} & {{Worst}}     \\\\\\hline \\hline\n$F_{\\rm elp}$             & 22,800 &\t23,750 &\t24,950 \t\t  \\\\ \n$F_{\\rm sch}$          &      183,750\t& 206,000 &\t229,150                          \\\\\n$F_{\\rm ack}$            & 42,800 &\t44,250\t& 45,500 \t                      \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n\nAs a next case, we consider $o_i = 2$. The constrained\nminimum is now different from that of the unconstrained problem, as\nthe original unconstrained minimum is no more feasible. This case is equivalent to ``Optima on \nthe Constraint Boundary''. Since the optimum value is not zero as before, instead of choosing\na termination criterion based on $S$ value, we allocate a maximum of\none million function evaluations for a run\nand record the obtained optimized solution. The {best fitness} values for \n$f(\\vec{x})$ as $f_{elp}$, $f_{sch}$ and $f_{ack}$ are shown in Table\n~\\ref{tab:convex-constraints-de-2}. For each function, we verified\nthat the obtained optimized solution satisfies the KKT optimality\nconditions \\cite{rekl,debOptiBOOK} suggesting that a truly optimum solution has\nbeen found by this procedure. \n\\begin{table*}[ht]\n\\begin{footnotesize}\n\\begin{center}\n\\caption{Best fitness results for test functions with DE for $o_i=2$.}\n\\begin{tabular}{|ccc|} \\hline\n$F_{\\rm elp}$             & $F_{\\rm sch}$& $F_{\\rm ack}$\t \t\t  \\\\ \\hline \\hline\n      640.93 $\\pm$ 0.00  & 8871.06 $\\pm$ 0.39\t& 6.56 $\\pm$ 0.00 \t                         \\\\ \\hline\n\\end{tabular}\n\\label{tab:convex-constraints-de-2}      \n\\end{center}\n\\end{footnotesize}\n\\end{table*}\n  \nNext, we consider two additional nonlinear constrained optimization\nproblems (TP5 and TP8) from \\cite{debpenalty} and a well-studied structural design and mechanics problem (`Weld') taken from \\cite{ragsdell1976optimal}.\nThe details on the mechanics of the welded structure and the beam deformation can be found in \\cite{shigley1963engineering,timoshenko1962theory}.\nThese problems have \nmultiple nonlinear inequality constraints and our goal is to demonstrate the performance of our proposed constraint-handling\nmethods. We used DE with\nIP-S method with the following \nparameter settings: $NP=50$, $F=0.7$, $CR=0.5$, and $\\alpha=1.2$ for all three problems.\nA maximum of 200,000 function evaluations were allowed and a termination criteria of $10^{-3}$\nfrom the known optima is chosen. \nThe problem definitions of TP5, TP8 and `Weld' are as follows:\\\\\n\n\\noindent \\textbf{TP5:}\n\\begin{equation}\n\\label{eq:TP5}\n\\begin{array}{rl}\n\\mbox{Minimize} & f(\\vec{x}) = (x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2  \\\\\t\n& \\qquad\t+ 10x_5^6+7x_6^2+x^4_7-4x_6x_7-10x_6-8x_7, \\\\\n\\mbox{subject to} & g_1(\\vec{x})  \\equiv 2x_1^2+3x_2^4+x_3+4x_4^2+5x_5 \\leq 127,\t\\\\\n& g_2(\\vec{x}) \\equiv   7x_1+3x_2+10x_3^2+x_4-x_5 \\leq 282, \\\\\t\n& g_3(\\vec{x})  \\equiv    23x_1+x_2^2+6x_6^2-8x_7 \\leq 196, \\\\\n& g_4(\\vec{x}) \\equiv 4x_1^2+x_2^2-3x_1x_2+2x_3^2 +5x_6-11x_7 \\leq 0,\\\\\t\t\n& -10 \\leq x_i \\leq 10,   \\quad  i = 1,\\ldots,7. \n\\end{array} \n\\end{equation}  \n\n\\noindent\\textbf{TP8:}\n\\begin{equation}\n\\label{eq:TP8}\n\\begin{array}{rl}\n\\mbox{Minimize} & f(\\vec{x}) = x_1^2+x_2^2+x_1x_2-14x_1-16x_2+2(x_9-10)^2\\\\\n & \\qquad + 2(x_6-1)^2 + 5x_7^2+7(x_8-11)^2+45+(x_{10}-7)^2 \\\\\n & \\qquad + (x_3-10)^2 + 4(x_4-5)^2 + (x_5-3)^2\\\\\n\\mbox{subject to} & g_1(\\vec{x})  \\equiv 4x_1+5x_2-3x_7+9x_8 \\leq 105,\\\\\n& g_2(\\vec{x})  \\equiv\t10x_1-8x_2-17x_7+2x_8 \\leq 0,\\\\\n& g_3(\\vec{x})  \\equiv -8x_1+2x_2+5x_9-2x_{10}  \\leq 12,\\\\\n& g_4(\\vec{x})\\equiv  3(x_1-2)^2 + 4(x_2-3)^2+2x_3^2-7x_4 \\leq 120,\\\\\n& g_5(\\vec{x})\t\\equiv 5x_1^2+8x_2+(x_3-6)^2-2x_4 \\leq 40,\t\\\\\n& g_6(\\vec{x})\t\\equiv\t x_1^2+2(x_2-2)^2-2x_1x_2+14x_5-6x_6 \\leq 0,\\\\\n& g_7(\\vec{x})\t \\equiv\t 0.5(x_1-8)^2+2(x_2-4)^2+3x_5^2-x_6  \\leq 30,\\\\\n& g_8(\\vec{x})\t \\equiv\t -3x_1+6x_2+12(x_9-8)^2-7x_{10} \\leq 0,\\\\\n& -10 \\leq x_i\\leq 10, \\quad i = 1,\\ldots,10. \n\\end{array} \n\\end{equation}\t \n\n\\noindent\\textbf{`Weld':}\n\\begin{equation}\n\\label{eq:weld-problem}\n\\begin{array}{rl}\n\\mbox{Minimize} & f(\\vec{x}) = 1.10471h^2l+0.04811tb(14.0+l),\t \\\\\t\n\\mbox{subject to} & g_1(\\vec{x}) \\equiv  \\tau(\\vec{x}) \\leq 13,600,   \\\\\n& g_2(\\vec{x})  \\equiv  \\sigma(\\vec{x}) \\leq 30,000,\t\\\\\n& g_3(\\vec{x})  \\equiv  h-b \\leq 0, \\\\\t \n& g_4(\\vec{x})  \\equiv  P_c(\\vec{x})  \\geq 6,000,\\\\\n& g_5(\\vec{x})  \\equiv  \\delta(\\vec{x}) \\leq 0.25,\t\t\\\\\n& 0.125\\leq (h,b) \\leq 5, \\mbox{ and } 0.1 \\leq (l,t) \\leq 10, \n\\end{array}\n\\end{equation}\nwhere,\t\t\t \n\\begin{eqnarray*}\n\\tau(\\vec{x})\t&=&  \\sqrt{ (\\tau')^2 + (\\tau'')^2 + (l\\tau'\\tau'')\/ \\sqrt{0.25(l^2+(h+t)^2)}}, \\\\\t\t\\\\\t\n\\tau' &=& \\frac{6,000}{\\sqrt{2}hl}, \\\\\n\\tau'' &=& \\frac{6,000(14+0.5l)\\sqrt{0.25*(l^2+(h+t)^2)}}{2[0.707hl(l^2\/12+0.25(h+t)^2)] }, \\\\\n\\sigma(\\vec{x}) &=& \\frac{504,000}{t^2b}, \\\\\n\\delta(\\vec{x}) &=& \\frac{2.1952}{t^3b}, \\\\\nP_c(\\vec{x}) &=& 64,746.022(1-0.0282346t)tb^3.   \n\\end{eqnarray*} \n \nThe feasible region in the above problems is quite complex, unlike hypercubes in case\nof problems with variable bounds, and since our methods require feasible initial population,\nwe first identified a single feasible-seed solution (known as the seed solution), and generated other several other \nrandom solutions. Amongst the several randomly generated solutions, those infeasible,\nwere brought into the feasible region using IP-S method and feasible-seed solution as the reference.\nThe optimum results for TP5, TP8 and `Weld', thus found, are shown in the\nTable~\\ref{tab:Non-linear-opti}. \n\n\\begin{table*}[htb]\n\\caption{Results from TP5, TP8 and `Weld' problem. For each problem, the\n  obtained solution also satisfies the KKT conditions.}\n\\label{tab:Non-linear-opti}\n\\begin{center}\n\\begin{footnotesize}\n\\begin{tabular}{|l|l|l|l|} \\hline\n\t \t&  \\multirow{2}{1.5cm}{Optimum Value ($f^*$)}& {Corresponding Solution Vector ($\\vec{x}^{\\ast}$)}\n                & \\multirow{2}{1.2cm}{Active Constraints} \\\\ \n & & &  \\\\ \\hline\n\\textbf{TP5}\t&    $680.63$\n&$(2.330,1.953,-0.473,4.362,-0.628,1.035,1.591)^T$\t&  $g_1$, $g_4$   \t\t\t\t\\\\      \\hline\n\\textbf{TP8}\t\t& $24.33$ &\n$(2.160,2.393,8.777,5.088,0.999,1.437,1.298,9.810,8.209,8.277)^T$&\n$g_1$ to $g_6$\\\\      \\hline  \n\\textbf{`Weld'}\t&\n$2.38$& $(0.244,6.219,8.291,0.244)^T$\t& $g_1$ to $g_4$  \\\\      \\hline  \n\\end{tabular}\n\\end{footnotesize}\n\\end{center}\n\\end{table*}\n\nTo verify the obtained optimality of our solutions, we employed MATLAB sequential\nquadratic programming (SQP) toolbox with a\ntermination criterion of $10^{-6}$ to solve each of the three problems. The solution obtained from SQP\nmethod matches with our reported solution indicating that our proposed constrained\nhandling procedure is successful\nin solving the above problems.\n\nFinally, the convergence plots of a sample run on these test\nproblems is shown in Figures~\\ref{fig:TP5}, \\ref{fig:TP8}, and \\ref{fig:WELD}, respectively.\nFrom the graphs it is clear that our proposed method is able to\nconverge to the final solution quite effectively. \n\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35,angle=-90]{TP5-sep-2014.eps}    \n\\end{center}\n\\caption{Convergence plot for TP5.}\n\\label{fig:TP5}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35,angle=-90]{TP8-sep-2014.eps}    \n\\end{center}\n\\caption{Convergence plot for TP8.}\n\\label{fig:TP8}\n\\end{figure}\n\n\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[scale=.35,angle=-90]{Weld-sep-2014.eps}    \n\\end{center}\n\\caption{Convergence plot for TP8.}\n\\label{fig:WELD}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:Conclusion}\nThe existing constraint-handling strategies that repair solutions by bringing\nthem back into the search spaces exhibit several\ninadequacies. This paper has addressed the task of studying, and proposing two new, explicit feasibility preserving\nconstraint-handling methods for real-parameter optimization using evolutionary algorithms. \nOur proposed single parameter Inverse Parabolic Methods with stochastic and adaptive components, \novercome limitations of existing methods and perform effectively\non several test problems.\nEmpirical comparisons on four different\nbenchmark test problems ($F_{elp}$, $F_{sch}$, $F_{ack}$, and $F_{ros}$) \nwith three different settings of the optimum relative to the\nvariable boundaries revealed key insights into the performance of PSO, GAs and DE\nin conjunction with different constraint-handling strategies.\nIt was noted that in addition to the critical role of constraint-handling strategy (which\nis primarily responsible for bringing the infeasible solutions back into the search space),\nthe generation scheme (child-creation step) in an evolutionary algorithm must create efficient solutions \nin order to proceed effectively.\nExponential and Inverse Parabolic Methods were most robust methods and never\nfailed to solve any problem. The other constraint-handling strategies\nwere either too deterministic and\/or operated without utilizing sufficient useful information from \nthe solutions. The probabilistic methods\nwere able to bring the solutions back into the\nfeasible part of the search space and showed a consistent\nperformance. In particular, scale-up studies on four problems, up to 500 variables, demonstrated\nsub-quadratic empirical run-time complexity with the proposed IP-S method.\n\nFinally, the success of the proposed IP-S scheme is demonstrated \non generalized nonlinear constrained optimization problems.   \nFor such problems, the IP-S method requires finding the lower and upper bounds\nfor feasible region along the direction of search by solving a series\nof root-finding problems. \nTo best of our knowledge, the\nproposed constraint-handling approach is a first explicit feasibility preserving method that has demonstrated\nsuccess on optimization problems with variable bounds and nonlinear constraints.\nThe approach is arguably general, and can be \napplied with complex real-parameter search spaces such as non-convex, discontinuous, etc., \nin addition to problems dealing with multiple conflicting objectives, multi-modalities, dynamic\nobjective functions, and other complexities.\nWe expect that proposed constraint-handling scheme will find its utility in \nsolving complex real-world constrained optimization problems using evolutionary algorithms. \nAn interesting direction for the future would be to carry out one-to-one comparison between\nevolutionary methods employing constrained handling strategies and the classical\nconstrained optimization algorithms. Such studies shall benefit the practitioners in optimization \nto compare and evaluate different algorithms in a unified frame-work.\nIn particular, further development of a robust, parameter-less\nand explicit feasibility preserving constraint-handling procedure can be attempted.\nOther probability distribution functions and utilizing of information \nfrom other solutions of the population can also be attempted. \n\\section*{Acknowledgments}\nNikhil Padhye acknowledges past discussions with Dr. C.K. Mohan on swarm intelligence. \nProfessor Kalyanmoy Deb acknowledges \nthe support provided by the J. C. Bose National\nfellowship generously provided by the Department of Science and\nTechnology (DST), Government of India. \n{\\footnotesize\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBootstrap percolation on a hypergraph with infection threshold $r\\geq 1$ is a deterministic infection process which evolves in rounds. In each round every vertex has exactly one of two possible states: it is either infected or uninfected. We denote the set of initially infected vertices by $\\mathcal{A}_r(0)$. We say that a vertex $u$ is a neighbour of $v$ if there exists an edge containing both $u$ and $v$. In each round of the process every uninfected vertex $v$ becomes infected if it has at least $r$ infected neighbours, otherwise it remains uninfected. Once a vertex has become infected it remains infected forever. The process stops once no more vertices become infected and we denote this time step by $T$. The final infected set is denoted by $\\mathcal{A}_r(T)$.\n\nBootstrap percolation was introduced by Chalupa, Leath, and Reich \\cite{bootstrapintr} in the context of\nmagnetic disordered systems. Since then bootstrap percolation processes (and extensions) have been used to describe several complex phenomena: from neuronal activity \\cite{MR2728841,inhbootstrap} to the dynamics of the Ising model at zero temperature \\cite{Fontes02stretchedexponential}.\n\nIn the context of social networks, bootstrap percolation provides a prototype model for the spread of ideas. In this setting infected vertices represent individuals who have already adopted a new belief and a person adopts a new belief if at least $r$ of his acquaintances have already adopted it.\n\nOn the $d$-dimensional grid $[n]^d$ bootstrap percolation has been studied by\nBalogh, Bollob{\\'a}s, Duminil-Copin, and Morris \\cite{MR2888224}, when the initial infected set contains every vertex independently with probability $p$. \nFor the size of the final infection set they showed the existence of a sharp threshold. More precisely, they established the threshold probability $p_\\mathrm{c}$, such that if $p\\leq (1-\\varepsilon )p_\\mathrm{c}$, then the probability that every vertex in $[n]^d$ becomes infected tends to 0, as $n\\rightarrow\\infty$, while if $p\\geq (1+\\varepsilon )p_\\mathrm{c}$, then the probability that every vertex in $[n]^d$ becomes infected tends to one, as $n\\rightarrow\\infty$.\n\nBootstrap percolation has also been studied for several random graph models. For instance Amini and Fountoulakis \\cite{bootpower} considered the Chung-Lu model \\cite{MR1955514} where the vertex weights follow a power law degree distribution and the presence of an edge $\\{u,v\\}$ is proportional to the product of the weights of $u$ and $v$. Taking into account that in this model a linear fraction of the vertices have degree less than $r$ and thus at most a linear fraction of the vertices can become infected the authors proved the size of the final infected set $\\mathcal{A}_r(T)$ exhibits a phase transition. \n\n\nJanson,  \\L uczak, Turova, and Vallier \\cite{MR3025687} analysed bootstrap percolation on the binomial random graph $G(n,p)$ where every edge appears independently with probability $p$. For $r\\geq 1$ and $n^{-1}\\ll p \\ll n^{-1\/r}$ they showed that there is a threshold such that if the initial number of infected vertices is below the threshold, then the process infects only a few additional vertices and if the initial number of infected vertices exceeds the threshold, then almost every vertex becomes infected.\n\nIn this paper we investigate the binomial random hypergraph $H_k(n,p)$, where every edge ($k$-tuple of vertices) is present independently with probability $p$. We choose the initial infected set uniformly at random and consider bootstrap percolation with infection threshold $r> 1$ in the regime $n^{-1}\\ll n^{k-2}p \\ll n^{-1\/r}$.\nThe main contribution of this paper are:\n\\begin{itemize}\n\\item strengthening of the result in \\cite{MR3025687}, by showing that the failure probability decreases {\\em exponentially} (Theorem~\\ref{thm:graph});\n\\item extension of the original results from graphs to hypergraphs (Theorem~\\ref{thm:hypergraph}).\n\\end{itemize}\n\n\n\\section{Main Results}\n\nWe extend the following result, which was originally proved in \\cite{MR3025687}, to $H_k(n,p)$: Consider bootstrap percolation with infection threshold $r$ on $G(n,p)$, where $n^{-1}\\ll p \\ll n^{-1\/r}$. There is a threshold $b_r=b_r(n,p)$ such that if $|\\mathcal{A}_r(0)|\\le(1-\\varepsilon)b_r$, then with probability tending to one as $n\\rightarrow \\infty$ (whp for short)  only a few additional vertices become infected, while if $|\\mathcal{A}_r(0)|\\ge(1+\\varepsilon)b_r$, then whp almost every vertex in the process becomes infected.\nFor integers $k\\geq 2$ and $r> 1$ set\n\\[\nb_{k,r}:=b_{k,r}(n,p)=\\left\\{ \\begin{array}{ll}\n\\left(1-\\frac{1}{r}\\right)\\left(\\frac{(r-1)!}{n\\left(\\binom{n}{k-2}p\\right)^r}\\right)^{1\/(r-1)} & \\mbox{if } r > 2 \\\\\n\\frac{1}{2(2k-3)}\\frac{1}{n\\left(\\binom{n}{k-2}p\\right)^2} & \\mbox{if } r = 2,\n\\end{array}\n\\right.\n\\]\nand note that the only difference for the $r=2$ case is a $1\/(2k-3)$ multiplier. Since $2k-3=1$ when $k=2$ this is consistent with the threshold in the graph case i.e.\\ $b_{2,r}=b_r$. \n\n\\begin{theorem}\\label{thm:hypergraph}\nFor $k\\geq 2$ consider bootstrap percolation with infection threshold $r> 1$ on $H_{k}(n,p)$ when $n^{-1}\\ll n^{k-2} p \\ll n^{-1\/r}$. Assume the initial infection set is chosen uniformly at random from all sets of vertices of size $a=a(n)$. Then for any fixed $\\varepsilon>0$ we have that\n\\begin{itemize}\n\\item if $a\\leq(1-\\varepsilon)b_{k,r}$ then whp $|\\mathcal{A}_r(T)|= O(b_{k,r})$;\n\\item if $a\\geq (1+\\varepsilon)b_{k,r}$ then whp $|\\mathcal{A}_r(T)|=(1+o(1))n$.\n\\end{itemize}\n\\end{theorem}\n\nUsing the methods developed for this result we also obtain a strengthened form of the result for $G(n,p)$ establishing exponentially small bounds on the failure probability.\n\n\\begin{theorem}\\label{thm:graph}\nConsider bootstrap percolation with infection threshold $r > 1$ on $G(n,p)$ when $n^{-1}\\ll p \\ll n^{-1\/r}$. Assume the initial infection set is chosen uniformly at random from the set of vertices of size $a=a(n)$. Then for any fixed $\\varepsilon>0$ the following holds with probability $1-\\exp(-\\Omega(b_{2,r}))$:\n\\begin{itemize}\n\\item if $a\\leq(1-\\varepsilon)b_{2,r}$, then $|\\mathcal{A}_r(T)|=O(b_{2,r})$;\n\\item if $a\\geq (1+\\varepsilon)b_{2,r}$, then $|\\mathcal{A}_r(T)|=(1+o(1))n$.\n\\end{itemize}\n\\end{theorem}\n\nThe proofs rely on surprisingly simple methods. When the number of vertices infected in the individual rounds is large, we apply Chebyshev's or Chernoff's inequality. However when the process dies out, these changes can become arbitrarily small. In this case we couple the infection process with a subcritical branching process which dies out very quickly.\n\n\n\n\n\\section{Proof outlines}\n\nWe first show the outline for the proof of Theorem~\\ref{thm:hypergraph}. For brevity we will only describe the $r>2$ case in detail and comment on the differences for $r=2$ at the end. \n\nStart with a given set of initially infected vertices $\\mathcal{A}_r(0)$ and consider the infection process round by round. At the end of round $t\\geq 1$ we partition the set of vertices into $\\mathcal{A}_0(t),\\mathcal{A}_1(t),...,\\mathcal{A}_r(t)$ where the set $\\mathcal{A}_i(t)$ consists of all the vertices which have exactly $i$ infected neighbours (these are vertices in $\\mathcal{A}_r(t-1)$), for $i<r$, and $\\mathcal{A}_r(t)$ consists of all the vertices which have at least $r$ infected neighbours.\n\nFor every $0\\leq i \\leq r$ we aim to define a sequence $\\{a_i(t)\\}_{t\\geq 0}$ in such a way that $|\\mathcal{A}_i(t)|\\approx a_i(t)$. We use the following initial values for the sequences $a_0(0)=n$, $a_r(0)=a$, and $a_i(0)=0$ for $0< i < r$.\n\nAs long as $|\\mathcal{A}_r(t)|=o\\left(\\left(n^{k-2}p\\right)^{-1}\\right)$ the expected number of infected neighbours of a vertex is $o(1)$ and thus the typical vertex that becomes infected in this round has exactly $r$ infected neighbours in $r$ different edges.\nIn round $t+1$  we determine for any uninfected vertex $v\\in V\\backslash \\mathcal{A}_r(t)$ whether it changes its partition class. Note that this only happens if it has at least one neighbour which became infected in round $t$. \nTherefore, for $0< i \\leq r$, the expected change $|\\mathcal{A}_i(t+1)\\backslash \\mathcal{A}_i(t)|$ of the size of the partition class can be approximated by\n\\begin{equation}\\label{eq:seqchange}\na_{i}(t+1)-a_i(t) \\approx \\sum_{j=1}^{i} \\left(\\frac{(a_r(t)-a_{r}(t-1))^j}{j!}a_{i-j}(t)\\right)\\left(\\binom{n}{k-2}p\\right)^{i-j}, \n\\end{equation}\nignoring any negative terms (which correspond to vertices leaving their partition class). Similarly we assume that the number of vertices without infected neighbours does not change significantly, i.e.\\ $a_0(t+1)=a_0(t)=n$.\nFrom \\eqref{eq:seqchange} we deduce\n\\begin{equation}\\label{eq:seq}\na_i(t+1)\\approx \\frac{a_r(t)^i}{i!}n\\left(\\binom{n}{k-2}p\\right)^i+a_i(0), \n\\end{equation}\nfor $0< i \\leq r$. The behaviour of the sequence depends on the size $a=|\\mathcal{A}_r(0)|$ of the initial set. We will show the following: in the subcritical regime, characterised by $a\\leq (1-\\varepsilon)b_{k,r}$, the sequence $\\{a_r(t)\\}_{t\\geq 0}$ converges to a value $a^*=O(b_{k,r})$ as $t\\rightarrow \\infty$. On the other hand, in the supercritical regime, $a\\geq (1+\\varepsilon)b_{k,r}$, the sequence $\\{a_r(t)\\}_{t\\geq 0}$ tends to infinity as $t\\rightarrow \\infty$. \n\nFirst consider the subcritical regime. Since in this case $a_r(t)$ converges we have that the differences $\\Delta(t):=a_{r}(t+1)-a_r(t)$ form a decreasing function in $t$ and show that, for any fixed $\\eta>0$, there exists a $\\tau$, which does not depend on $n$, such that $\\Delta(\\tau)\\leq \\eta b_{k,r}$.\nThe fact that $|\\mathcal{A}_i(t)|$ is concentrated around $a_i(t)$ for $t<\\tau$ follows from Chebyshev's inequality.\n\nSince we are in the subcritical regime the size of the individual generations will become small and the concentration will fail. In order to avoid this we attempt to analyse the remaining steps together. Consider the forest where every vertex in $\\mathcal{A}_r(\\tau+1)\\backslash \\mathcal{A}_r(\\tau)$ is a root.\nRecall that in order for a vertex to become infected in round $t+1$ it must have a neighbour that got infected in round $t$. The children of a vertex $v \\in \\mathcal{A}_r(t+1)\\backslash \\mathcal{A}_r(t)$ will be the vertices $u \\in \\mathcal{A}_r(t+2)\\backslash \\mathcal{A}_r(t+1)$ which lie in an edge containing $v$ and should this relation not be unique for some vertex $u$, $u$ is assigned arbitrarily to one of the candidates. Clearly every vertex of $A_r(T)\\setminus A_r(\\tau)$ is contained in the forest and thus the size of this forest matches the number vertices which got infected after round $\\tau$.\n\nNote that for every $\\delta>0$ there exists a $t_0$ such that $|\\mathcal{A}_i(t)|\\leq (1+\\delta)a_i(\\tau)$, for every $0\\leq i\\leq r$ and $\\tau<t \\leq t_0$. Also up until time $t_0+1$ we have an upper coupling by a Galton-Watson branching process with $|\\mathcal{A}_r(\\tau+1)\\backslash \\mathcal{A}_r(\\tau)|$ roots and offspring distribution $\\sum_{j=0}^{r-1}\\mathrm{Bin}((1+\\delta)a_{r-j}(\\tau),q_j)$ where\n\\[q_j=\\binom{n}{k-2}p \\frac{(\\delta a_r(\\tau))^{j-1}}{(j-1)!}\\left(\\binom{n}{k-2}p\\right)^{r-j-1}.\\]\nFor small enough $\\delta$ the expected number of offspring in one step is \\linebreak $\\sum_{j=0}^{r-1}(1+\\delta)a_{r-j}(\\tau) q_j<1$ and therefore this is a subcritical process, i.e.\\ it dies out with probability 1.\nFor every $t$ we have that $|\\mathcal{A}_r(t)|\\leq (1+\\delta)a_r(\\tau)$ implies $|\\mathcal{A}_i(t+1)|\\leq (1+\\delta)a_i(\\tau)$, for all $0\\leq i < r$, by \\eqref{eq:seq}, and thus it is enough to show that $|\\mathcal{A}_r(T)|\\leq (1+\\delta)a_r(\\tau)$. Due to the upper coupling with the branching process we have that the probability that $|\\mathcal{A}_r(T)|>(1+\\delta)a_r(\\tau)$ is dominated by the probability that the total size of the branching process exceeds $\\delta a_r(\\tau)$. However for properly chosen $\\eta,\\delta>0$ the probability that the total size of the branching process exceeds $\\delta a_r(\\tau)$ is sufficiently small. Therefore we have that there are at most $(1+\\delta)a_{r}(\\tau)$ infected vertices in total.\n\n\nNow for the supercritical case. Recall that \\eqref{eq:seqchange} and \\eqref{eq:seq} hold when \\linebreak[4] $a_r=o\\left(\\left(n^{k-2}p\\right)^{-1}\\right)$. Again we consider the differences $\\Delta(t)=a_{r}(t+1)-a_r(t)$. Although at the beginning of the process the values of $\\Delta(t)$ decrease there exists a value $t_1$ not depending on $n$ such that for every $t>t_1$ we have that $\\Delta(t+1)>\\Delta(t)$. In fact there exists a $t_2$ not depending on $n$ such that for $t\\geq t_2$ we have that $\\Delta(t+1)>2\\Delta(t)$.\nTherefore the probability of non-concentration is dominated by a geometric sequence and applying the union bound gives us concentration as long as $a_r(t)=o\\left(\\left({n}^{k-2}p\\right)^{-1}\\right)$.\nWhen $a_r(t)=\\Omega\\left(\\left({n}^{k-2}p\\right)^{-1}\\right)$ the expected number of neighbours is $\\Omega(1)$ and thus our approximation in \\eqref{eq:seqchange} does not hold any more. Refining these approximations shows that at most 2 rounds are required for almost every vertex to become infected, with $\\Theta(n)$ vertices becoming infected in every required step.\n\nRecall that for $r>2$ the typical vertex became infected when it was contained in $r$ different edges each containing a different infected vertex. When  $r=2$ this is equivalent to finding two intersecting edges each containing a different infected vertex. However unlike the $r>2$ case finding two such edges in step $t$ implies that every vertex in these edges is infected by step $t+1$. Two intersecting edges typically overlap in exactly one vertex and thus finding such an edge pair implies that $2k-3$ vertices will become infected, not just one. Taking this into account gives us the modified bound on the threshold.\n\nThe proof of Theorem~\\ref{thm:graph} is analogous. In the random graph case, in round $t$ of the process only those edges are examined which contain exactly one vertex from $\\mathcal{A}(t)\\backslash \\mathcal{A}(t-1)$ and no vertices from $\\mathcal{A}(t-1)$. Since each of these edges can contain at most one uninfected vertex the behaviour of the individual vertices is independent. Thus we can replace Chebyshev's inequality with Chernoff's inequality and achieve a stronger bound on the failure probability.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe large-scale distribution of dark matter halos is one of the key\ningredients of the theoretical description of large-scale structure (LSS).  \nSince most observed tracers of LSS, such as galaxies, reside in halos,\nthe statistics of halos determine those of galaxies on large scales.  \nSimilarly, the halo model description of the nonlinear matter density\nfield \\cite{cooray\/sheth} crucially relies on halo statistics.  In the context of perturbation\ntheory, the statistics of halos are written in terms of bias parameters \nmultiplying operators constructed out of the matter density field.  \nIn general, these operators consist of powers of the matter density\nand tidal field \\cite{fry\/gaztanaga,mcdonald\/roy}, as well as convective time derivatives of these quantities \\cite{MSZ,senatore:14}.  \nHowever, the most well-studied and phenomenologically most important\nbias parameters on large scales are those multiplying powers of\nthe matter density field, i.e. \n\\begin{equation}\n\\d_h(\\v{x},\\tau) \\supset b_1(\\tau) \\delta_\\rho(\\v{x},\\tau) + \\frac12 b_2(\\tau) \\delta_\\rho^2(\\v{x},\\tau)\n+ \\frac16 b_3(\\tau) \\delta_\\rho^3(\\v{x},\\tau) + \\cdots\\,,\n\\label{eq:localbias}\n\\end{equation}\nwhere $\\d_h$ is the fractional number density perturbation of a given\nhalo sample,\nwhile $\\delta_\\rho$ is the matter density perturbation.  More precisely,\nthe powers of $\\delta_\\rho$ should be understood as renormalized operators\n\\cite{mcdonald,assassi\/etal,PBSpaper}.  \nThe $b_n$ are commonly called (nonlinear) \\emph{local bias parameters}.  \nThe goal of this paper is to present precision measurements of\n$b_1,\\,b_2,\\,b_3$ using a novel technique, \\emph{separate universe simulations}.\n\nIn the separate universe approach \n\\citep{lemaitre:1933,sirko:2005,baldauf\/etal:2011,li\/hu\/takada:2014,Wagner:2014}, a long-wavelength density\nperturbation is included in an N-body simulation by changing the \ncosmological parameters, in particular $\\Omega_m,\\,\\Omega_\\Lambda,\\,\\Omega_K$ and $H_0$, \nfrom their fiducial values, and running the simulation to a different\nscale factor.  As argued in \\cite{baldauf\/etal:11,jeong\/etal,PBSpaper}, \nthe (renormalized) local bias parameters defined in \\refeq{localbias}\ncorrespond to the response of the halo abundance, $\\bar n_h$, to a long-wavelength\ndensity perturbation, equivalent to a change in the background density, $\\bar\\rho$,\n\\begin{equation}\nb_{n} = \\frac{\\bar\\rho^{\\hskip 1pt n}}{\\bar n_h} \\frac{\\partial^n \\bar n_h}{\\partial\\bar\\rho^{\\hskip 1pt n}}\\, .\n\\end{equation}\nThis can be understood as an exact formulation of the peak-background split (PBS) \\cite{kaiser:1984,mo\/white:1996}.  Thus, the $b_n$ can be measured through the mass function of halos in a suite\nof separate universe simulations.  This technique has several advantages:  \nfirst, it is guaranteed to recover the large-scale limit of the $b_n$, without\nscale-dependent or nonlinear corrections which affect measurements of the\nbias parameters from the halo power spectrum and bispectrum, or\nfrom the cross-correlation with smoothed fields.  Note that, starting \nat second order, ``nonlocal'' bias parameters such as those with respect\nto powers of the tidal field will enter in these latter measurements at\nthe same level as the $b_n$.  Second, \nwe can straightforwardly obtain measurements of higher order bias parameters\nsuch as $b_3$, which become cumbersome to measure using correlations.  Finally,\nby using the same initial phases for simulations with different density,\nwe can cancel to a large extent the cosmic variance contribution to the measurement error.  \n\nSeparate universe simulations are expected to estimate the same set of bias parameters as those obtained from matter-halo cross-correlations. We will thus compare the biases obtained from the separate universe simulations to\nthose determined by fitting to halo two- and three-point statistics.  \nWe also compare the results to biases derived from universal mass functions\nusing the classic peak-background split argument, and \nrecent theoretical predictions from\nthe excursion set-peaks (ESP) approach \\cite{Paranjape:2012,Paranjape:2013}, which incorporates some aspects\nof the Gaussian peaks model into the excursion set framework.  \n\nHigher order bias parameters have previously been measured in simulations\nby correlating the halo number with powers of the smoothed density field \nat the final time (Eulerian frame) \n\\cite{angulo\/baugh\/lacey:2008,manera\/gaztanaga:2011}\nor in the initial conditions \\cite{paranjape\/etal:2013}.  \nHowever, the bias parameters measured in this way depend on the smoothing\nscale adopted, while the local bias parameters that are relevant for perturbation theory predictions, and that we are interested in here, correspond to\na smoothing scale of infinity.  Further, all these references\nneglect the nonlocal bias terms mentioned above, \nwhich will affect the inferred values of $b_2$ and higher.  \nFor these reasons, it is difficult to directly compare our measurements of\nnonlinear bias parameters with these previous results (although we find\nbroad agreement).    \nWe stress again that in the separate universe approach we are guaranteed\nto obtain the local bias in the large-scale limit, without nonlinear\nor tidal corrections.  Moreover, we simultaneously obtain both\nthe Eulerian ($b_n$) and Lagrangian ($b_n^L$) bias parameters.  \n\nTwo related papers appeared on the preprint archive simultaneously\nto this paper.  Ref.~\\cite{li\/etal:15} measured the linear bias using\nseparate universe simulations through an abundance matching technique\nwhich yields the integrated halo bias above a mass threshold.  This\ntechnique reduces the shot noise in the bias measurement.  \nRef.~\\cite{baldauf\/etal:15} also measured the linear bias via\nthe mass function.  In addition, they present measurements of $b_2$\nthrough the response of the halo power spectrum to a long-wavelength mode\n(as done in \\cite{li\/hu\/takada:2014,Wagner:2015} for the matter power spectrum).  \nOur results are consistent with the findings of both of these references.  \nHowever, unlike these and any other previous published results, we\nuse the fully nonlinear separate universe approach to obtain accurate\nmeasurements of the \\emph{linear and nonlinear} local biases.\n\n\nIn this paper we adopt a flat $\\Lambda$CDM fiducial cosmology with $\\Omega_m=0.27$, $h=0.7$, $\\Omega_b h^2=0.023$ and $\\mathcal{A}_s=2.2\\cdot 10^{-9}$.  \nThe outline of the paper is as follows.  In \\refsec{theory}, we present\nthe theoretical predictions that we will compare our measurements with.  \\refSec{bsep}\ndescribes the technique of measuring bias parameters from separate universe\nsimulations, while \\refsec{bcorr} presents the estimators for $b_1$ and $b_2$\nusing the conventional approach of measuring halo correlations.  \nWe discuss the results in \\refsec{res}.  We conclude in \\refsec{concl}.  \nThe appendices contain more details on the ESP predictions as well as\nour bias measurements.  \n\n\\section{Theory predictions}\n\\label{sec:theory}\n\nIn this section we present several theoretical predictions for the large-scale bias from the literature.  We first recap the PBS argument in \\refsec{PBS} and briefly present the ESP formalism in \\refsec{ESP}.\n\nBefore jumping into details, we briefly explain the definitions of \\textit{Lagrangian} and \\textit{Eulerian} halo bias. The Lagrangian bias links the abundance of dark matter halos to the density perturbations in Lagrangian space, i.e. it describes the relation of proto-halos in the initial conditions that correspond to halos identified at redshift $z$ to the initial linear density perturbation field. On the other hand, the Eulerian bias relates the halos identified at redshift $z$ to the nonlinear density field, $\\delta_\\rho$, at redshift $z$.  \nIn the case of the local bias parameters considered here, there is\nan exact nonlinear mapping between the Lagrangian bias parameters $b_n^L$\nand their Eulerian counterparts $b_m$, see \\refapp{compbsep}.  We will\nmake use of this mapping both for the theory predictions and measurements.  \n\nIn the following, the top-hat filtered variance on a scale $R_{\\rm TH}$ \n(the Lagrangian radius of halos) is denoted as\n\\begin{equation}\n\\s{0}^2 \\equiv \\int {\\rm dln} k\\, \\Delta^2 (k) [W_{\\rm TH}(kR_{\\rm TH})]^2,\n\\label{eq:sigma0}\n\\end{equation}\nwhere $\\Delta^2(k) = k^3 P(k)\/2\\pi^2$ is the dimensionless linearly extrapolated matter power spectrum and \nthe top-hat filter in Fourier space $W_{\\rm TH}(k R_{\\rm TH})$ is given in \\refeq{WTH}.  \n\n\\subsection{Peak-background split bias}\n\\label{sec:PBS}\n\nWe briefly recap how the bias parameters can be derived from the differential halo mass function using the PBS argument,\nas initially proposed in \\cite{kaiser:1984,cole\/kaiser:1989,mo\/white:1996}.  \nFollowing the PBS argument, the effect of a long wavelength mode $\\d_0$ on the small scale formation can be seen as locally modulating the density threshold for halo formation, or barrier $B$, sending it to $B-\\d_0$ (here we denote the barrier as $B$ to emphasize that this argument is not restricted to the constant spherical collapse threshold $\\d_c$ and can be extended to barriers depending e.g. on the halo mass $M$ through $\\s{0}$). Note that, in the case where stochasticity should be introduced in the barrier, this shift does not modify the stochastic contribution to the barrier, which is supposed to capture the effect of small-scale modes. We define the differential mass function as  \n\\begin{equation}\nn_{\\rm}(\\nu_{\\rm B}) = \\frac{\\bar{\\rho}_m}{M}f_{\\rm}(\\nu_{\\rm B}) \\left|\\frac{{\\rm d ln}\\,\\s{0}}{{\\rm d ln }\\,M}\\right|,\n\\label{eq:nf}\n\\end{equation}\nwith $\\nu_{\\rm B} \\equiv B(\\s{0})\/\\s{0}$ (we reserve the notation $\\nu$ for $\\nu\\equiv\\d_c\/\\s{0}$), $M$ the corresponding mass and $f(\\nu_{\\rm B})$ the mass fraction contained in halos of mass $M$. The scale-independent large-scale Lagrangian bias parameters are then defined by the well known relation \n\\begin{equation}\nb^L_n(\\nu_{\\rm B}) = \\frac{1}{n(\\nu_{\\rm B})}\\frac{\\partial ^n n([B(\\s{0})-\\d_0]\/\\sigma_0)}{\\partial \\d_0^n}\\Bigg|_{\\d_0=0}.\n\\label{eq:biasPBS}\n\\end{equation}\nAs we have indicated, this also applies if the deterministic part of the barrier is mass-dependent. We will use \\refeq{biasPBS} both to derive the bias in the ESP model and from the fits to the mass function proposed in \\cite{Sheth:1999} and \\cite{Tinker:2008} (hereafter ST99 and T08 respectively). \n\n\\subsection{Excursion set peaks}\n\\label{sec:ESP}\n\nIn this section, we review the ESP formalism proposed in \\citep{Paranjape:2012} and \\citep{Paranjape:2013}.  The details of the calculation are relegated to \\refapp{ESP}.  All the results that we present here and in \\refapp{ESP} were already derived in these two references, but in a different way; here, we use the PBS argument to derive the bias parameters directly.  Further, the ESP predictions\nfor $b_3$ and $b_4$ are computed here for the first time.  \n\nThe ESP aims at unifying the peak model of Bardeen et al. in 1986 (hereafter BBKS) \\citep{Bardeen:1986} and the excursion set formalism of Bond et al. in 1991 \\citep{Bond:1991}. It can be seen either as addressing the cloud-in-cloud problem within the peak model, or as applying the excursion set formalism to a special subset of all possible positions (the peaks).  \nWe follow \\citep{Paranjape:2013}, who chose a top-hat filter for the excursion\nset part, and a Gaussian filter to identify peaks (in order to ensure finite moments\nof derivatives of the smoothed density field).  \n\nMore importantly, \\citep{Paranjape:2013} improved the model by adding a mass-dependent stochastic scatter to the threshold.  Specifically, the barrier is\ndefined as \\citep{Paranjape:2012}\n\\begin{equation} \nB(\\s{0}) = \\d_c + \\beta \\s{0}\\,.\n\\label{eq:barrier}\n\\end{equation}\nHere, $\\beta$ is a stochastic variable and \\cite{Paranjape:2013} chose its PDF $p(\\beta)$ to be lognormal with mean and variance corresponding to $\\<\\beta\\> = 0.5$ and ${\\rm Var}(\\beta)=0.25$.  This choice was made to match the peak height measured in simulations by \\cite{robertson\/etal}.   Hence $\\beta$ takes only positive values.  Note that \\refeq{barrier} then corresponds to a mass-dependent mean barrier $\\d_c + 0.5 \\s{0}$.  \n\nAs we show in \\refapp{ESP}, the Lagrangian bias parameters in the ESP\ncan be directly derived from \\refeq{biasPBS} by inserting the multiplicity\nfunction $f_{\\rm ESP}(\\nu)$ into \\refeq{nf}, and sending \n$\\nu = \\d_c\/\\s{0}$ to $\\nu_1 = \\nu\\left(1-\\d_0\/\\d_c\\right)$.\\footnote{Here one needs to take care not to shift one instance of $\\nu$ in the expression for $f_{\\rm ESP}(\\nu)$ that is actually unrelated to the barrier. See \\refapp{ESP}.}  \nOur results for the bias, \\refeq{btheo}, are identical to the large-scale bias parameters derived using\na different approach in \\citep{Paranjape:2012,Paranjape:2013}.  \nWe will see that the choice of barrier \\refeq{barrier} leads to significant differences from the standard PBS biases derived using $B = \\d_c$\nfrom the T08 and ST99 mass functions. \n\n\n\\section{Bias parameters from separate universe simulations}\n\\label{sec:bsep} \n\nOur results are based on the suite of separate universe simulations described in \\cite{Wagner:2014,Wagner:2015}, performed using the cosmological code GADGET-2 \\citep{Springel:2005}. The idea of the separate universe simulations is that a uniform matter overdensity $\\delta_\\rho$ of a scale larger than the simulation box can be absorbed in the background density $\\tilde{\\rho}_m$ of a modified cosmology simulation (throughout the whole paper, quantities in modified cosmologies will be denoted with a tilde), where\n\\begin{equation} \n\\tilde{\\rho}_m(t) = \\rho_m(t)\\left[1+\\delta_\\rho(t)\\right], \n\\label{eq:dr}\n\\end{equation}\nwith $\\rho_m$ the mean matter density in a simulation with no overdensity (which we call the fiducial cosmology).  Indeed, a uniform density can only be included in this way, since the Poisson equation for the potential enforces a vanishing mean density perturbation over the entire box. Thus one can see a simulation with a constant overdensity $\\delta_\\rho$ as a separate universe simulation with a properly modified cosmology. Qualitatively, a positive overdensity causes slower expansion and enhances the growth of  structure, i.e. more halos, whereas a negative one will have the opposite effect. The precise mapping of $\\delta_\\rho$ to modified cosmological parameters is described in \\cite{Wagner:2014}.  Crucially, we work to fully nonlinear order in $\\delta_\\rho(t)$.  \n\nWe use two sets of simulations denoted by ``lowres'' and ``highres'' throughout the paper. Both have a comoving box size of $500\\,h^{-1}{\\rm Mpc}$ in the fiducial cosmology.  The ``lowres'' set uses $256^3$ particles in each simulation, while ``highres'' employs $512^3$ particles.  For both sets, we run the fiducial cosmology, i.e. $\\delta_\\rho=0$,  and simulations with values of $\\delta_\\rho$ corresponding to $\\d_L$ = \\{$\\pm$0.5, $\\pm$0.4, $\\pm$0.3, $\\pm$0.2, $\\pm$0.1, $\\pm$0.07, $\\pm$0.05, $\\pm$0.02, $\\pm$0.01\\}, where $\\d_L$ is the present-day linearly extrapolated matter density contrast. \nIn addition, we simulate separate universe cosmologies corresponding to $\\d_L$ = 0.15, 0.25, and 0.35 for both resolutions. \nThis makes the sampling in the final, nonlinear $\\delta_\\rho$ more symmetric around 0 which should help diminish the covariance between the bias parameters.\\footnote{We have not performed a systematic study on the number of $\\d_L$ values that are necessary to derive accurate measurements of the $b_n$ up to a given order.  \nGiven the significant degeneracies between $b_n$ and $b_{n+2}$ we have found\n(\\refapp{cov}), this is a nontrivial question.}  \nThe comoving box size in the modified cosmology simulations is adjusted to match that in the fiducial cosmology, $L=500\\,h^{-1}{\\rm Mpc}$. \nHence, in the high redshift limit ($z\\rightarrow \\infty$ for which $\\delta_\\rho\\rightarrow 0$) the physical size of the box is the same for all simulations whereas at the present time ($z=0$ in the fiducial cosmology) the physical size of the simulation box varies with $\\delta_\\rho$. However, this choice of the box size has the advantage that the physical mass resolution is the same within each set of simulations regardless of the simulated overdensity $\\delta_\\rho$ (i.e. $\\tilde{m}_p = m_p$ where $m_p$ is the particle mass in the fiducial cosmology). \nSince the biases are determined by comparing halo abundances between different overdensities, this eliminates any possible systematic effects in the biases due to varying mass resolution. \nThe mass resolution is $m_p = 5.6\\cdot 10^{11}h^{-1} M_\\odot$ in the ``lowres'' set of simulations and $m_p=7\\cdot 10^{10}h^{-1} M_\\odot$ in the ``highres'' one. Furthermore, for the ``lowres'' set of simulation, we ran 64 realizations of the entire set of $\\d_L$ values. For the ``highres'' one we ran only 16 realizations of each $\\d_L$ value as they are more costly in terms of computation time.  \nEach simulation was initialized using 2LPT at $z_i$ = 49.  \nFor further details about the simulations, see \\citep{Wagner:2015}.\n\n\\subsection{Halo catalogs}\n\\label{sec:HC}\n\nThe halos were identified using the Amiga Halo Finder (hereafter AHF) \\cite{Gill:2004,Knollmann:2009}, which identifies halos with a spherical overdensity (SO) algorithm.  We identify halos at a fixed proper time corresponding to $z=0$ in the fiducial cosmology.  \nIn this paper, we only use the number of distinct halos and do not consider their sub-halos. \n\nThe key point in identifying halos with the spherical overdensity criterion is the setting of the density threshold.  We choose here a value of $\\Delta_{\\rm SO}=200$ times the background matter density in the \\emph{fiducial} cosmology.  \nThus, our measured bias parameters are valid for this specific halo definition.  \nFor the simulations with a different background density, the threshold must be rescaled in order to compare halos identified using the same physical density in each simulation. Specifically, we need to use\n\\begin{equation}\n\\Delta_{\\rm SO} = \\frac{200}{1+\\delta_\\rho}\\,.\n\\label{eq:DSO}\n\\end{equation}\nAnother point is the treatment of the particle unbinding in a halo.  AHF has the ability to remove unbound particles, i.e particles which are not gravitationally bound to the halo they are located in.  However, in order to avoid having to implement the complicated matching of the unbinding criterion between the modified and fiducial cosmologies, we have turned unbinding off in all halo catalogs.  \nNote that the effect of unbinding is very small (of order 1\\% on the mass function), and\nthat we consistently use the same halo catalogs for all measurements,\nso that this choice does not affect our comparison between different methods\nfor measuring bias.  \n\nWe count halos in top-hat bins given by \n\\begin{equation}\nW_n(M,M_{{\\rm center}})=\\begin{cases} 1 &\\mbox{if } \\left|{\\rm log}_{10}(M)-{\\rm log}_{10}(M_{{\\rm center}})\\right| \\leq 0.1 \\\\\n0 & \\mbox{otherwise, } \\end{cases}\n\\label{eq:bins}\n\\end{equation}\nwhere $M$ is the mass ($M_{{\\rm center}}$ corresponding to center of the bin).  \nFor the high resolution simulations, we count halos in 12 bins centered from ${\\rm log}_{10}\\left(M_{{\\rm center}}\\right) = 12.55$ to ${\\rm log}_{10}\\left(M_{{\\rm center}}\\right) = 14.75$, to ensure that we have enough halos in each bin. For the low resolution simulations, we have 7 bins from ${\\rm log}_{10}\\left(M_{{\\rm center}}\\right) = 13.55$ to ${\\rm log}_{10}\\left(M_{{\\rm center}}\\right) = 14.75$. With this binning choice, the lowest bin is centered around halos with 63 particles for the ``lowres'' set of simulations, with a lower limit at halos containing around 50 particles. For the ``highres'' set of simulations, the lowest mass bin is centered on halos with around 51 particles, with a lower limit around 40 particles. These numbers are quite low compared to more conservative values (e.g. 400 particles in T08). However $\\d_h$ is the \\emph{relative difference} of the number of halos between the fiducial and modified cosmology simulations (see \\refeq{DN_N} hereafter) and therefore that quantity should be less affected by resolution effects. For halos with a minimum number of 40 particles, we did not find any systematic difference between the bias parameters measured from the ``lowres'' and ``highres'' simulations. Thus, we present results for halos that are resolved by at least 40 particles. \n\n\\subsection{Eulerian biases}\n\\label{sec:BE}\n\nInstead of fitting the Eulerian bias parameters directly to the simulation results, we derive them from the measured Lagrangian biases for which the fitting is more robust, using the exact nonlinear evolution of $\\delta_\\rho$ (see \\refapp{compbsep} for the details of the mapping).  \nIn order to obtain the Lagrangian bias parameters, we compute $\\d_h(M,\\d_L)$ versus $\\d_L$ where $\\d_h(M,\\d_L)$ is the overdensity of halos in a bin of mass $M$ compared to the fiducial case $\\d_L=0$,\n\\begin{equation}\n\\delta_h (M,\\d_L) = \\frac{\\tilde{N}(M,\\d_L) - N(M)}{N(M)},\n\\label{eq:DN_N}\n\\end{equation}\nwith $\\tilde N(M,\\d_L)$ the number of halos in a bin centered around mass $M$ in the presence of the linear overdensity $\\d_L$ and $N(M)=\\tilde N(M,\\d_L=0)$. \nNote that $\\d_h(M,\\d_L)$ is the overdensity of halos in Lagrangian space as the physical volumes of the separate universe simulations only coincide at high redshift.\n\nIn order to obtain the Lagrangian bias parameters $b_n^L$, we then fit \\refeq{DN_N} by  \n\\begin{equation}\n\\d_h = \\sum_{n=1}^5  \\frac{1}{n!} b_n^L (\\d_L)^n\\,.\n\\label{eq:BiasExp}\n\\end{equation}\nAs indicated in \\refeq{BiasExp} we use a $5^{\\rm th}$ order polynomial in $\\d_L$ by default.  In \\refapp{deg} we study the effect of the degree of the polynomial on the results; as a rough rule, if one is interested in $b_n^L$, then one should fit a polynomial up to order $n+2$.  \n\nIn order to estimate the overall best-fit of and error bars on the bias parameters, we use a bootstrap technique. For each non zero $\\d_L$ value, we randomly produce $p$ resamples of the mass function. Each resample is composed of the same number of realizations as the original sample (i.e. 16 or 64) and we choose $p=100\\cdot 64$ ($100\\cdot 16$) for the low (high) resolution simulations. We then compute the average number of halos per mass bin for each resample. This gives us $p$ numbers $\\tilde{N}^i(M,\\d_L)$. \nFor a given $\\d_L$, we also create the same set of resamples for the fiducial cosmology  \nand again compute the average number of halos, i.e. $N^i(M)$.  We then compute $p$ times $\\d^i_h$ according to \\refeq{DN_N} for every $\\d_L$ value. \nSince we use the same resamples for the separate universe results, $\\tilde{N}^i(M,\\d_L)$, and the fiducial case, $N^i(M)$, the cosmic variance is canceled to leading order. The error on $\\d_h$ at fixed mass and $\\d_L$ is given by the sample variance and we use it as a weight for the fit. \nWe neglect, however, the covariance between $\\tilde{N}^i(M,\\d_L)$ for different $\\d_L$ values.\nWe then produce $p$ fits with a weighted least squares method. For every bias parameter, the value we report is the mean of the results of the $p$ fits while the corresponding error bar is given by the square root of the variance of the distribution.   Within the mass range common to both sets of simulations ``lowres'' and ``highres'', the measurements are consistent with each other and hence we perform a volume-weighted average of the biases from the two sets of simulations.\n\n\\section{Bias parameters from correlations}\n\\label{sec:bcorr}\n\nTraditionally bias parameters are used for and measured from $n$-point correlation functions or $n$-spectra. The $n$-th order bias parameters enter the tree-level calculation of the $n+1$-point functions. For instance, $b_1$ appears at the leading order in the large-scale behavior of the halo power spectrum, $b_2$ in the large-scale limit of the bispectrum and $b_3$ in the large-scale limit of the trispectrum. For the comparison to $n$-point functions, we will restrict ourselves to the power spectrum and bispectrum  at tree level here.  The bispectrum also contains nonlocal bias parameters, i.e. biases with respect to the tidal field, that arise from triaxial collapse and gravitational evolution. The estimation of the first and second order bias parameters closely follows the steps outlined in \\cite{Baldauf:2012} (see also \\cite{Saito:2014}), with the difference that we are performing a joint fit for all the bias parameters, instead of first fitting $b_1$ to the halo power spectrum and then using its value in the bispectrum analysis.\n\nLet us start by discussing the power spectrum. We measure the halo-matter cross power spectrum $P_\\text{hm}$, which at tree level (on large scales) is given by\n\\begin{equation}\nP_\\text{hm}(k)=b_1 P_\\text{mm}(k).\n\\end{equation}\nWe refrain from explicitly including the loop corrections, since they contain third order biases not present in the bispectrum as well as scale-dependent biases $\\propto k^2$ \\cite{assassi\/etal}.  \nThe advantage of the halo-matter cross power spectrum over the halo-halo power spectrum is that it is free of shot noise. To ensure that our measurements are not contaminated by higher order contributions or scale dependent bias, we will \nin fact fit $P_\\text{hm}(k)=(b_1 + b_{P,k^2} k^2 ) P_\\text{mm}(k)$ to the simulation\nresults, where $b_{P,k^2}$ is a free nuisance parameter.  This term absorbs the\nloop corrections in the large-scale limit.  \nWe measure the matter and halo power spectra in the same wavenumber bins in the simulation and take their ratio to cancel the leading cosmic variance, i.e. we define a quantity $q(k)=P_\\text{hm}(k)\/P_\\text{mm}(k)$ and the $\\chi^2$\n\\begin{equation}\n\\chi^2_P=\\sum_{k}^{k_\\text{max}} \\left(\\frac{q(k)-b_1-b_{P,k^2}k^2}{\\sigma[q(k)]}\\right)^2,\n\\end{equation}\nwhere the variance $\\sigma^2(q)$ is estimated from the box-to-box scatter between the simulation realizations. \n\nLet us now turn to the bispectrum. One can form three different bispectra containing the halo field, the halo-halo-halo, the halo-halo-matter and the halo-matter-matter bispectrum. We are using the latter, since it is the only bispectrum free of shot noise. Furthermore, we will employ the unsymmetrized bispectrum, where the halo mode is the one associated with the wavevector $\\vec k_3$. This unsymmetrized bispectrum measurement allows for a clear distinction of the second order local bias $b_2$ and tidal tensor bias $b_{s^2}$, once the matter bispectrum is subtracted out. The unsymmetrized tree-level bispectrum reads\n\\begin{equation}\nB_\\text{mmh}(k_1,k_2,k_3)=b_1 B_\\text{mmm}(k_1,k_2,k_3) + b_2 P(k_1)P(k_2)+2 b_{s^2} S_2(\\vec k_1,\\vec k_2) P(k_1)P(k_2)\\; ,\n\\end{equation}\nwhere $B_\\text{mmm}$ is the tree-level matter bispectrum (e.g., \\cite{Baldauf:2012}), and we employed the tidal operator $S_2$ defined as\n\\begin{equation}\nS_2(\\vec k_1,\\vec k_2)=\\left(\\frac{\\vec k_1\\cdot \\vec k_2}{k_1^2 k_2^2}-\\frac{1}{3}\\right).\n\\end{equation}\nSimilarly to the power spectrum defined above, this bispectrum does not include loop corrections or scale dependent biases. Thus, we again add a term of the form\n$b_{B,k^2} (k_1^2+k_2^2)P(k_1) P(k_2)$ with a free coefficient $b_{B,k^2}$,\ndesigned to absorb the loop corrections.  \nTo cancel cosmic variance, we define the ratio of bispectrum and power spectrum measurements\n\\begin{equation}\nQ(k_1,k_2,k_3;b_1)=\\frac{B_\\text{mmh}(k_1,k_2,k_3)-b_1 B_\\text{mmm}(k_1,k_2,k_3)}{P_\\text{mm}(k_1) P_\\text{mm}(k_2)},\n\\end{equation}\nand using this we define the corresponding $\\chi^2$\n\\begin{equation}\n\\chi^2_B=\\sum_{k_1,k_2,k_3}^{k_\\text{max}} \\left(\\frac{Q(k_1,k_2,k_3;b_1)-b_2  -2b_{s^2}S_2-b_{B,k^2}(k_1^2+k_2^2)}{\\sigma[Q(k_1,k_2,k_3;b_{1,\\text{fid}})]}\\right)^2\\; ,\n\\end{equation}\nwhere the variance of $Q$ is estimated from the box-to-box scatter between the simulation realizations for a fiducial $b_{1,\\text{fid}}$. Equivalent results could have been obtained using the estimator presented in \\cite{Schmittfull:2014tca}. We decided to stick with the more traditional bispectrum estimation for the following reasons: for their method the smoothing scale of the fields needs to be chosen before the simulation data is reduced, complicating convergence tests. Furthermore, \\cite{Schmittfull:2014tca} ignored two-loop corrections to their estimator and higher derivative terms, while we marginalize over an effective shape accounting for the onset of scale dependence. A detailed comparison of the two methods is however beyond the scope of this work.\n\nAll measurements are done on the ``lowres'' and ``highres'' sets of the fiducial cosmology.  \nWe find the best fit biases $b_1$ and $b_2$ by sampling the log-likelihood $\\ln \\mathcal{L}=-\\chi^2_\\text{tot}\/2$, where $\\chi^2_\\text{tot}=\\chi^2_P+\\chi^2_B$ using the Markov Chain code EMCEE \\cite{emcee}.  \nThe errors on the bias parameters are estimated from the posterior distribution of sampling points after marginalizing over the (for our purposes) nuisance parameters $b_{P,k^2}$,  $b_{B,k^2}$ and $b_s^2$.  \nWe have varied that maximum wavenumber $k_\\text{max}$ to ensure that we remain in the regime where the tree level bias parameters remain consistent with increasing $k_\\text{max}$.  Further, we demand that the total $\\chi^2$ per degree of freedom is approximately unity.  The results shown below use a conservative value of $k_\\text{max} = 0.06 \\; h \\, {\\rm Mpc}^{-1}$. This limits the number of modes to $\\mathcal{O}(100)$ and thus also the number of power and bispectrum configurations. Due to the cancellation of the leading order cosmic variance this is not of major concern. We have compared the clustering constraints with a larger $2400\\; h^{-1}\\text{Mpc}$ box providing a factor of 100 more modes to the same cutoff and found consistent results.\n\n\\section{Results}\n\\label{sec:res}\n\nThis section presents the results for the Eulerian bias parameters $b_1$ to $b_3$.  For completeness, we also present results for $b_4$, which is poorly constrained, in \\refapp{b4}.\n\nIn order to obtain a precise comparison between any theoretical prediction for the bias $b_n(M)$ (such as the ESP, \\refeq{btheo}) and our data points, we convolve the theoretical prediction with the mass bins used in the simulation (see \\refsec{bsep}).  I.e., the theory predictions we will show in the following are given by\n\\begin{equation}\nb_n^{\\rm conv}(M) = \\frac{\\int{W_n(M',M) n(M') b_n(M') {\\rm d} M'}}{\\int{ W_n(M',M) n(M'){\\rm d}M'}},\n\\label{eq:b1TbinConv}\n\\end{equation} \nwhere $W_n(M',M)$ is the window function of the mass bin given by \\refeq{bins}, and $n(M')$ is the differential halo mass function, parametrized by the fitting formula of Eq. (2) in T08.  \nIn this way, we obtain smooth curves for the theory prediction whose\nvalue at the center of a given mass bin can be compared directly to the simulation results.\n\n\\subsection{Linear bias}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.55]{b1_comp_deg5.pdf} \n\\caption{\\textbf{Top panel:} comparison between the linear halo bias from separate universe simulations (green dots), and from clustering (red crosses; displaced slightly horizontally for clarity). Error bars that are not visible are within the marker size. The solid black curve is the Tinker et al. (2010) best fit curve for $b_1$, while the dot-dashed green curve is the ESP prediction \\refeq{btheo}.  We also show the result obtained by applying the PBS argument [\\refeq{biasPBS}] to the T08 \nand ST99 mass functions (blue dashed curves). \\textbf{Bottom panel:} relative difference between the measurements and the Tinker et al. (2010) best fit.} \n\\label{fig:b1}\n\\end{figure}\n\n\\refFig{b1} presents the results for $b_1$. The green points show the results obtained from the separate universe simulations, while the red crosses show those from fitting $P_{\\text{hm}}$ and $B_{\\text{mmh}}$. The mutual agreement of the two measurements is very good (the only point with relative difference greater than the $1\\sigma$ uncertainty is at $\\log M=13.15$). The error bars of the separate universe measurements are significantly smaller. Note however that the effective volume used by these measurements is also larger, since the halo-matter power spectrum was only measured in the fiducial boxes. This is a first validation of the separate universe method and also proves its efficiency.\n\nThese results are consistent with the ones presented in \\cite{li\/etal:15} who derived the linear bias from abundance matching.  Since Ref.~\\cite{li\/etal:15} used a linearized implementation of separate universe simulations, they are restricted to small overdensities (they take $\\delta_\\rho= \\pm 0.01$), resulting in very small changes in the halo abundance.  For such small changes, abundance matching is much more efficient than binning halos into finite mass intervals.  We circumvent this issue by using fully nonlinear separate universe simulations which allow us to simulate arbitrary values of $\\delta_\\rho$.  \n\nWe also compare our data with several results from the literature. The solid black curve is the fit to $P_{\\text{hm}}$ measurements from Tinker et al. (2010) \\cite{Tinker:2010} [their Eq. (6)]. As shown in the lower panel of\n\\refFig{b1}, the agreement is better than 5\\%, the quoted accuracy of the\nfitting formula. Note that we do not remove unbound particles from our\nhalos, which we expect to lead to a slight underestimate of the bias at the few percent level at low masses.   \nNext, we turn to the ``standard'' peak-background split \nargument \\refeq{biasPBS} applied to the universal mass functions of ST99 \nand T08 (blue dashed curves).  At low masses, the T08 curve is at 1\\% level agreement but the ST99 prediction overestimates the bias by around 8\\%.  The agreement is worse at high mass where these two curves underestimate the bias by around 8\\%  and 11\\% respectively.\n\nThe green dot-dashed line finally shows the prediction from excursion set\npeaks \\refeq{btheo}.  The agreement at high masses is excellent, where\nthe ESP matches the measured $b_1$ to better than 2\\%.  The agreement is far less good at low masses where the ESP prediction overestimates the bias by roughly 10\\%.  Note that the assumption that halos correspond to peaks in the\ninitial density field is not expected to be accurate at low masses\n\\cite{ludlow\/porciani}. Part of the discrepancy might also come from the up-crossing criterion applied to derive the ESP prediction, which is only expected to be accurate at high masses \\cite{Musso:2013}. It is worth emphasizing that \\refeq{biasPBS} still \napplies in the case of the ESP.  That is, the large-scale bias can still be\nderived directly from the mass function.  The key difference to the\nPBS curves discussed previously is that, following \\cite{Paranjape:2013},\nwe employ a stochastic moving barrier, which changes the relation between\nmass function and bias.  This more realistic barrier leads to the\nsignificant improvement in the prediction of the bias for high-mass halos. \n\n\\subsection{Higher order biases}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.5]{b2_comp_deg5.pdf} \n\\caption{\\textbf{Top panel:} same as \\refFig{b1}, but for the quadratic bias $b_2$.  The color code is as in \\refFig{b1}. \\textbf{Bottom panel:} relative difference between measurements and the theoretical prediction of the ESP. In each panel, the clustering points have been horizontally displaced as in \\refFig{b1}.} \n\\label{fig:b2}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.55]{b3_comp_deg5.pdf} \n\\caption{As \\refFig{b2} but for $b_3$.} \n\\label{fig:b3}\n\\end{figure}\n\n\\refFigs{b2}{b3} present the analogous results of \\refFig{b1} for $b_2$ and $b_3$, respectively.  For $b_2$ at masses below $10^{13.5} h^{-1} M_\\odot$, there is some scatter in the separate universe results that is apparently larger than what is expected given the error bars (a hint of a similar effect can be seen in $b_1$ as well).  \nNote however that there is significant residual degeneracy\nbetween the $b_n$ for a given mass bin, so that a ``$\\chi$-by-eye''\ncan be misleading.  As an example, we show projections of the likelihood for one mass bin in \\refFig{contours}.  \nThe covariance between the bias parameters is further explored in \\refapp{cov}.  Covariance in the halo shot noise between different mass bins, which we do not take into account in the likelihood, could also contribute to the fluctuations in the bias parameters.\n\nIn the case of $b_2$, we can compare the separate universe results to the results of fitting to $P_{\\text{hm}}$ and $B_{\\text{mmh}}$.  Again, we find good agreement, with all points being within $2\\sigma$ from each other.  Note that $b_2$ is most difficult to constrain from correlations around its zero-crossing.   The difference in constraining power between the two methods is now even larger than in the case of $b_1$.  This is because, when using correlations, $b_2$ has to be measured from a higher order statistic which has lower signal-to-noise.  \nIn the case of $b_3$, a measurement from correlations would have to\nrely on the trispectrum and accurate subtraction of 1-loop contributions in \nperturbation theory.  We defer this significantly more involved measurement\nto future work.  \nAs discussed in the introduction, it is difficult to rigorously compare these\nmeasurements to previously published results, since those were measured\nat a fixed smoothing scale and did not take into account nonlocal bias\nterms.  Nevertheless, our results for $b_2$ and $b_3$ appear broadly consistent with those of \n\\cite{angulo\/baugh\/lacey:2008,paranjape\/etal:2013}\nand \\cite{angulo\/baugh\/lacey:2008}, respectively.\n\nWe again compare with the peak-background split results, now derived at\nsecond and third order from the ST99 and T08\nmass functions.  For $b_2$, at low mass, both predictions deviate from our measurements by about 50\\%. At high mass, the deviation is at most 25\\% for T08 and 40\\% for ST99. In the low mass range, this apparently big discrepancy is also due to the smallness of the absolute value of $b_2$. \nIn the case of $b_3$, the PBS predictions using either the T08 or ST99\nmass functions are in fact completely consistent with the measurements \nat masses $\\gtrsim 10^{12.7} h^{-1} M_\\odot$ and $10^{13.5} h^{-1} M_\\odot$, respectively.\n\nTurning to the ESP prediction, we again find very good agreement at\nhigh masses, although for $b_2$ and $b_3$ the performance is not significantly better than the\nPBS-derived biases from the T08 mass function.  At low masses,\nwe again find larger discrepancies, with the ESP now underpredicting\nthe magnitude of $b_2$ and $b_3$.  The same caveats regarding the relation of low-mass halos to peaks and the efficiency of the up-crossing condition apply here, i.e. we do not expect the ESP\nprediction to work well for those masses.\\\\  \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.37]{b2overb1_of_b1_zdep.pdf} \n\\includegraphics[scale=0.37]{b3overb1_of_b1_zdep.pdf}\n\\caption{$b_2$ and $b_3$ as a function of $b_1$ obtained from separate universe simulations and for different redshifts. The dashed curves present the third order best fit polynomial for each bias. See text for details about the fit.} \n\\label{fig:b2b3(b1)}\n\\end{figure}\n\nSo far, we have only shown results at redshift $0$.  \\refFig{b2b3(b1)}\nshows results from various redshifts by plotting $b_2,\\,b_3$ as functions of\n$b_1$.  If the bias parameters are uniquely determined by $\\sigma_0 = \\sigma(M)$, then this relation will be redshift-independent.  Indeed, we find no\nevidence for a redshift dependence over the range $z = 0 \\dots 2$ and\n$b_1 = 1 \\dots 10$.  Note that we have kept the overdensity criterion\n$\\Delta_{\\rm SO}=200$ fixed.  \nSince the separate universe simulation measurements of $b_2$ and $b_3$ \nare very accurate, we provide fitting formulas in \nthe form of $b_n(b_1)$ for convenience. \nGiven the consistency with a universal behavior, we perform a joint fit of results from all redshifts. We use a $3^{\\rm rd}$ order polynomial form for both $b_2$ and $b_3$. Again, we use a weighted least squares method for the fit but do not take into account the error on $b_1$ since it is much smaller than those in $b_2,\\,b_3$. We obtain\n\\begin{equation}\nb_2(b_1) = 0.412-2.143\\,b_1+0.929\\,b_1^2+0.008\\,b_1^3, \n\\label{eq:b2(b1)} \n\\end{equation}\nand\n\\begin{equation}\nb_3(b_1) = -1.028+7.646\\,b_1-6.227\\,b_1^2+0.912\\,b_1^3. \n\\label{eq:b3(b1)} \n\\end{equation}\nThe fits are shown as dashed lines in the two panels of \\refFig{b2b3(b1)}. Notice that we restricted ourselves to $b_1 < 9.6$ on these figures for clarity but we used the full range of results to produce the fits.\nNote that one should be careful when using these formulas outside the fitting range $1\\lesssim b_1\\lesssim 10$.\n\\refeqs{b2(b1)}{b3(b1)} are similar to the fitting formulas provided in \\cite{Hoffmann:2015} who fitted $2^{\\rm nd}$ and $3^{\\rm rd}$ order polynomials for\n$b_2(b_1)$ and $b_3(b_1)$, respectively, to PBS predictions, and found no redshift dependence of their results. Such universal relations became already apparent in \\cite{Saito:2014} (their figure 9). \n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have presented a new method to measure the large-scale, renormalized\nlocal density bias parameters $b_n$ of dark matter halos, with $n=1,2,3$,\nby running simulations which simulate an infinite-wavelength density\nperturbation of arbitrary amplitude.  This method can be seen as an \nexact implementation of the peak-background split.  \nThis method has several advantages, including a simple implementation applicable,\nin principle, to arbitrarily high $n$.  The most important advantage,\nhowever, is that the measured biases are not affected\nby the modeling of scale-dependent or nonlinear corrections, and there\nis no ambiguous choice of $k_{\\rm max}$, with the associated risk of \noverfitting, as when\nfitting halo $N$-point functions.  The most significant disadvantage\nof the method is that it needs a set of dedicated simulations with\nvarying cosmological parameters to generate a range of $\\d_L$ (note\nhowever that once the simulations are done, they can be used for\nvarious studies, such as for example the nonlinear power spectrum\nresponse \\cite{Wagner:2015}).  \n\nWe have compared our results for $b_1$ and $b_2$ to those measured\nfrom the halo-matter power spectrum and halo-matter-matter bispectrum,\nand find excellent agreement overall.  One necessary condition for this\nagreement is a careful fitting procedure of the halo statistics and choice of $k_{\\rm max}$.  \n\nWe also compared our results to predictions based on the analytical peak-background\nsplit.  Once a specific barrier $B$ is assumed, the PBS allows for\na derivation of all local bias parameters $b_n$ from a given halo mass\nfunction.  The simplest and most common choice is $B=\\d_c$, which we have\napplied to the ST99 and T08 mass function prescriptions.  \nWe found that even though the latter provides a very accurate mass function, the linear bias derived via the PBS and simple collapse threshold is only accurate at the $\\sim 10$\\% level, in agreement\nwith previous results \\cite{manera\/etal:2010}.  Things are even worse for $b_2$, with up to 50\\% discrepancy at low mass, although the absolute difference between the PBS predictions and the measurements is similar to that in $b_1$.  For $b_3$, the simple PBS predictions are consistent with the measurements (at least at high masses), but this is not a very strong statement given the large error bars on $b_3$.  \n\nWe also derived the biases predicted in the excursion set-peaks\napproach, which includes a stochastic moving barrier motivated by \nsimulation results.  At high mass, this performs much better, at least for $b_1$, showing\nthat the choice of barrier is a key ingredient in deriving accurate\nbias parameters.  In this context, it is important to note that previous\nresults on the inaccuracy of PBS bias parameters \\cite{manera\/etal:2010}\nrelied on the simple constant threshold $B=\\d_c$.  This shows that the cause of \ntheses inaccuracies is not the peak-background split itself.  The\ninaccuracy of the peak-background split thus depends on what one\ndefines PBS to mean, and can be summarized as follows:\n\\begin{itemize}\n\\item The PBS implemented via the separate universe approach is exact.\n\\item The PBS using a simulation-derived stochastic moving barrier \\cite{robertson\/etal,Paranjape:2013}, as in the ESP, is accurate to a few percent, at least at high masses.  The discrepancy found at low mass can be explained by the failure of the peak assumption at such masses, an issue unrelated to the choice of the barrier.\n\\item The PBS using the constant spherical collapse barrier is no better than 10\\%\\,.\n\\end{itemize}  \n\nWe also provide fitting formulas for $b_2,\\,b_3$ as a function of $b_1$\nwhich are valid over a range of redshifts and \ncan be useful for predictions and forecasts based on halo statistics,\nsuch as for example the halo model.  \n\nIn the future, we plan to extend our analysis to accurately measure\nassembly bias, i.e. the dependence of bias on halo properties beyond\nthe mass (e.g., \\cite{gao\/etal,wechsler\/etal,dalal\/etal}).  \nFurther, it will be interesting to extend this technique beyond the \ninfinite wavelength, spherically symmetric ``separate universe'' to allow for precision measurements of the \ntidal and scale-dependent biases. \n\n\\acknowledgments{We thank Aseem Paranjape, Marcello Musso and Vincent~Desjacques for useful discussion about the ESP. F.S.~acknowledges support from the Marie Curie Career Integration Grant  (FP7-PEOPLE-2013-CIG) ``FundPhysicsAndLSS''. T.B.~gratefully acknowledges support from the Institute for Advanced Study through a Corning Glass works foundation grant.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe promise of automating tedious tasks and keeping humans away from hazardous environments has driven the development of robotic systems to reach tremendous capabilities.\nWith the increased maturity of the single agent systems, the interest in teams of robots collaborating with each other has been steadily rising.\nThe use of such a team of robots collaborating towards a common goal promises to increase the efficiency and robustness of a mission by distributing the tasks among the participating agents and has been proposed for a variety of different tasks, such as in search-and-rescue missions \\cite{srr}, archaeological mapping \\cite{col_arch_map}, precision agriculture \\cite{agri-robots}, and surveillance \\cite{surveillance}.\nSharing information across the agents not only enables the robots to perform a task faster but also enables the individual agents to make better-informed decisions as they can profit from information beyond their own gathered experience, as shown in \\cite{bartolomei2020multi}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/expt_eurocv3.png}\n    \\caption{Collaborative SLAM estimate for 5 agents (EuRoC MH Dataset) running different VIO front-ends. The COVINS-G back-end does not require map points (shown here only for visualization) and thus, is compatible with any arbitrary VIO front-end.\n    }\n    \\label{fig:irchel_5ag}    \n    \\vspace{-20pt}\n\\end{figure}\n\nHowever, in order for the robots to work towards any higher-level goal, they need to be aware of their surroundings and their pose in their workspace.\nMoreover, for the robots to be able to collaborate, the knowledge of the pose of all other robots within the team is crucial.\nThe use of external sensors, such as GPS or motion capture systems can provide such a shared reference frame enabling the coordination of the robotic team, however, for many practical applications, such data is not reliable or simply not available in the first place.\nFor example, the uncertainty of GPS measurements can be in the order of tens of meters close to larger structures (e.g. within a city) or not available at all inside buildings or underground.\nIn order to remove dependencies on external sensors, research into Simultaneous Localization And Mapping (SLAM) has made significant progress.\nIn particular, the use of cameras and Inertial Measurement Units (IMUs) for visual-inertial SLAM has proven to provide robustness and accuracy, which led to their deployment onboard products in the market already.\nWith the increasing maturity of single-agent vision-based SLAM techniques, the extension towards multi-agent SLAM as a core enabler for robotic collaboration in real-world scenarios has been increasingly gaining interest, sparking a variety of works addressing multi-agent SLAM \\cite{multi-uav-slam, ccm-slam, cvi-slam, door-slam, kimera-multi, covins}. \nWhile the capabilities and robustness of the developed systems have been steadily increasing, due to their tailored architectures, their modularity is often sacrificed. \nAs a result, any exchange and modification of the front-end onboard such systems most often requires significant effort to adapt the back-end to the structure.\nIn this spirit, this work addresses this issue by proposing a generic back-end solution built on top of the architecture of \\cite{covins}.\nAs the proposed approach requires only 2D features, it is agnostic to the front-end running onboard each agent and even allows mixing different front-end algorithms on the different agents during the same mission as illustrated in Figure \\ref{fig:irchel_5ag}.\nIn summary, the contributions of this work are the following:\n\\begin{itemize}[leftmargin=10pt]\n    \\item a generalized collaborative SLAM back-end, which requires only 2D keypoints and a pose estimate to fuse the estimates from multiple agents, enabling the use of any arbitrary VIO and stereo front-ends onboard each agent, \n    \\item a publicly available codebase\\footnote{\\href{https:\/\/github.com\/VIS4ROB-lab\/covins}{https:\/\/github.com\/VIS4ROB-lab\/covins}}, which is integrated with the framework of \\cite{covins}. Furthermore, a front-end wrapper is provided, to support any off-the-shelf front-end, and\n    \\item an extensive evaluation of the proposed back-end on both the EuRoC dataset \\cite{euroc} as well as newly collected datasets. Our evaluation reveals the flexibility of the proposed approach, using and combining different types of front-ends onboard collaborating agents within the same mission.\n\\end{itemize}\n\n\\vspace{0pt}\n\\section{Related Work}\n\\label{sec:relatedwork}\nThe capability to process multiple trajectories sequentially ({\\it{aka}} the multi-session capability) can be seen as a special case of collaborative SLAM.\nRecent SLAM systems, such as ORB-SLAM3\\cite{ORBSLAM3_TRO} and VINS-mono \\cite{vins-mono} have such multi-session capabilities, which enables them to achieve joint pose and scene estimates similar to collaborative SLAM estimates.\nWhile these approaches achieve greater accuracy and robustness when compared to single-agent SLAM, they are not designed to be used in real-time applications where multiple agents are operating at the same time. \n\n\nIn multi-agent SLAM literature, the classification of the systems is generally made into decentralized and centralized architectures.\nOne of the first decentralized approaches to multi-agent SLAM is DDF-SAM \\cite{ddfsam}, which communicates and propagates condensed local graphs between the robots to distribute the information.\nCombining efficient decentralized place recognition \\cite{Cieslewski:Scaramuzza:MRS2017} and with a Gauss-Seidel based distributed pose graph optimization \\cite{Coudhary:etal:ICRA2016}, in \\cite{Cieslewski:etal:ICRA2018} a data-efficient and complete decentralized visual SLAM approach was proposed. \nIn \\cite{dist-coslam}, a monocular vision-only distributed SLAM for mapping large-scale environments is presented. \nThe recent works \\cite{door-slam, kimera-multi} both make use of distributed pose graph optimization schemes along with a robust mechanism for identifying and rejecting incorrect loop closures. \nWhile these distributed SLAM approaches have advantages in terms of scalability, in general, as the information is also distributed the extent of collaboration is limited in order to keep the communication requirements feasible.\n\nOn the other hand, in centralized systems, all relevant information passes through a central node.\nIn \\cite{PG-slam}, the authors present a back-end for real-time multi-robot collaborative SLAM where the server combines the local pose graphs obtained from different agents into a global pose graph.\nIn MOARSLAM \\cite{moarslam}, each agent runs a full SLAM pipeline onboard, while the server is used to store the agents' maps and perform map merges across them.\nAs the agents perform all the computationally expensive tasks, in particular global optimization, this approach is not well-suited for resource-constrained platforms.\nOn the other end of the spectrum is  C${}^2$TAM \\cite{riazuelo2014c}, which offloads all tasks onto a server platform, except pose tracking.\nWhile this allows for very limited computational load onboard the agents, it limits the autonomy of the agents, as a loss of connection to the server eventually causes the onboard tracking to fail.\nA middle ground between MOARSLAM and C${}^2$TAM was proposed in \\cite{multi-uav-slam}, which introduces a system architecture enabling the agents to function autonomously, but is still able to offload heavy computations to a server and crucially, enable two-way information flow between the agents and the server.\nThis work was extended in CCM-SLAM \\cite{ccm-slam}, shown to perform in real-time on real data, proposing redundancy detection that enabled scalability, which is key in large-scale missions.\nPushing for the incorporation of inertial cues onboard each agent, aside from the monocular cues, CVI-SLAM \\cite{cvi-slam} was demonstrated to achieve higher accuracy and metrically scaled SLAM estimates aligned with the gravity direction -- which are core to robotic autonomy in reality.\nMost recently, COVINS \\cite{covins} was proposed, revisiting the most important components for centralized collaborative SLAM, shown to achieve significant gains in accuracy, while pushing the scalability of the architecture to up to 12 participating agents.\n\nDespite the improved performance, however, one of the major limitations of COVINS is that this performance is highly dependent on the choice of the VIO front-end employed onboard the agents.\nFor example, using an ORB-SLAM3\\cite{ORBSLAM3_TRO} front-end with the COVINS back-end gives an exceptional performance, but the performance drops significantly when using an alternative state-of-the-art VIO front-end, such as VINS-mono \\cite{vins-mono}.\nThis is mainly caused due to the reliance of COVINS on large numbers of highly accurate map points for closing loops, which holds for ORB-SLAM3, for example, but not for VINS-mono. \nBy utilizing a generic multi-camera relative pose solver for map fusion and loop-closures we break the dependency on highly accurate map points and instead perform the operations using only 2D image observations.\nThis does not only enable COVINS-G to perform well with virtually any VIO frontend, but also permits the usage of more powerful image features, which in return, allows to handle drastic view-point changes where previous systems failed.\n\n\n\\section{Methodology}\n\\label{sec:methodology}\nIn this section, we first provide an overview of the overall architecture in Section \\ref{sec:architecture}.\nSince our system is closely inspired by the COVINS framework, we then focus on the individual modules of the whole architecture, which are directly impacted by the contributions of this work.\n\n\\subsection{System Overview}\n\\label{sec:architecture}\nA summary of our system architecture can be seen in Figure \\ref{fig:architecture}.\nThe system consists of $n$ individual agents, which all run their own local VIO independently and are able to communicate their Keyframe (KF) information to a central server.\nThe communication module is based on the approach of \\cite{covins}, and is shown to be resilient to potential message losses as well as network delays and occasional bottlenecks in the bandwidth.\nHowever, owing to the fact that each agent is running an independent VIO, even a complete loss in connection does not violate the autonomy of an agent.\n\nThe central server, on the other hand, is responsible for maintaining the data of the individual agents, fusing the information from the agent, and carrying out computationally expensive tasks, such as global optimization.\nAfter decoding the received KF information, for every KF a visual descriptor based on the Bag-Of-Words (BoW) approach gets computed and added to the KF Database containing the visual descriptor for all the collected KFs.\nFor every incoming KF, a place-recognition query is performed to find the closest visual matches (candidate KFs).\nBased on the visual similarity, it is attempted to compute the transformation between the newly received KF and the candidate KFs.\nUpon successful computation of the transformation between the KFs, it is used to either perform a loop closure within a map or if it is across multiple maps, these get fused based on the computed transformation.\n\nAt the start of a mission, each of the agents is initialized separately and has a dedicated map on the server.\nIn our system, the map closely resembles a Pose Graph, where nodes correspond to KFs and edges are based on the relative poses from the odometry and loop closures.\nAs soon as a loop is found across two agents, their maps get merged together by transforming the map of one agent into the coordinate frame of the other based on the estimated loop transformation.\nUpon detection of any loop, a Pose Graph Optimization (PGO) is carried out over all connected Maps.\n\n\nAs opposed to COVINS, which strongly relies on having a well-estimated and consistent set of map points, in our approach we only operate on 2D keypoint information.\nThis opens up the possibility to use and combine all sorts of front-ends, even if no map points are accessible at all (e.g. in the case of an off-the-shelf tracking sensor such as the T265).\nIn order to be able to compute the loop constraints without access to map points, we make use of a multi-camera relative pose estimation algorithm \\cite{17PT} by treating neighboring KFs as a multi-camera system.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{images\/covins_arch_v3.pdf}\n    \\caption{Overview of the COVINS-G system architecture.\n    }\n    \\label{fig:architecture}    \n    \\vspace{-19pt}\n\\end{figure}\n\n\n\\subsection{Loop Closure and Map Fusion}\n\\label{sec:loop_closure}\nThe process of closing loops consists of two main processes; first, suitable candidates for the current query KF are detected, and second, the candidates get geometrically verified by estimating the relative pose between the candidate and the query KF.\nThe first step is handled by the Place Recognition module, which queries the KF Database for similar KFs based on the BoW \\cite{dbow2} image descriptor.\n\nThe second step, the geometrical verification, is used as an additional check beside the visual appearance  and in order to obtain a transformation between the query KF $\\text{KF}_q$ and the candidate KF $\\text{KF}_c$.\nHence, with $q$ being the sensor coordinate frame of the $\\text{KF}_q$ and $c$ being the coordinate frame of the sensor for $\\text{KF}_c$, the goal is to find the transformation $T_{cq}$, describing the transformation from frame $q$ into frame $c$.\nThe standard way of computing this transformation is to establish 3D-2D correspondences between the query KF and the map points associated with the candidate KF. \nHowever, in COVINS-G the system does not have access to 3D map points, but only 2D keypoints.\nEstimating the transformation between $\\text{KF}_q$ and $\\text{KF}_c$ using standard 2D-2D relative pose estimation algorithms like the 5-point algorithm \\cite{5PT} would result in a scale ambiguity.\nInstead, we propose to not only use the query and candidate KFs but also use some of their neighboring KFs and the relative odometry transformations to form two sets of cameras which can be treated as a multi-camera system as illustrated in Figure \\ref{fig:17_PT}.\nUsing the 17-point algorithm \\cite{17PT}, the transformation between two such camera systems can be estimated at metric scale.\n\n\\subsubsection{The 17-point Algorithm}\n\\label{sec:seventeen_pt}\nThe 17-point (or 17-ray) algorithm \\cite{17PT} can be used to solve for the relative motion between two multi-camera systems. \nA multi-camera system consists of more than one camera where the relative transformations between the cameras are known. \nUsing 17 2D-2D point correspondences, the relative transformation between two viewpoints of a multi-camera system can be estimated. \nTo do so, the generalized epipolar constraints \\cite{generalized_epipolar} are used, which can be seen as an extension of the epipolar constraints to systems without a central point of projection.\nNote that in our setup we only have a monocular camera, however, we leverage the fact that we have a good estimate of the relative pose between adjacent KFs provided by the VIO front-end.\nHence for a $\\text{KF}_a$, we take the $m$ neighbors $\\text{KF}_{a_{n_i}}, i \\in [1, m]$ of $\\text{KF}_a$ and use the relative transformations $T_{aa_{n_{i}}}$ to build up a conceptual multi-camera system (Figure \\ref{fig:17_PT}).\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{images\/17PT_v5.png}\n    \\caption{The 17-point algorithm requires 17 2D point correspondences to estimate the relative transformation $T_{cq}$ between the multi-camera systems $S_{c}$ and $S_{q}$ (represented by blue dashed lines). }\n    \\label{fig:17_PT}\n    \\vspace{-15pt}\n\\end{figure}\n\n\n\\subsubsection{Implementation}\n\\label{sec:seventeen_pt_approach}\nTo estimate the transformation between the candidate KF ($\\text{KF}_c$) and the query KF ($\\text{KF}_q$), we extend the neighborhood to form two multi-camera systems $S_{q}$ and $S_{c}$ for the $\\text{KF}_q$ and the $\\text{KF}_c$, respectively.\nIn our setup the two sets comprised of one additional neighbor for $\\text{KF}_q$ and two additional neighbors for $\\text{KF}_c$, i.e. $S_{q} = \\{ \\text{KF}_q, \\text{KF}_{q_{n_1}} \\}$ and $S_{c} = \\{ \\text{KF}_c, \\text{KF}_{c_{n_1}}, \\text{KF}_{c_{n_2}}\\}$.\nIn order to establish the 2D-2D candidate correspondences, we first perform a brute force descriptor matching across all pairs of the two sets $S_{q}$ and $S_{c}$, leading to 6 sets of correspondences.\n\nInstead of using this set of correspondences directly with the 17-point algorithm inside a RANSAC framework, we first perform a pre-filtering, as a high outlier ratio would render a RANSAC with 17 correspondences infeasible.\nThe pre-filtering consists of performing a standard 2D-2D RANSAC on each of the 6 sets of correspondences.\nTo reject bad candidates early on, we require a minimum number of inliers (30) in each of the 6 sets.\nOn the remaining inlier correspondences, we carry out a 17-point RANSAC, where during the sampling we ensure that candidates from all sets are present in order to prevent degenerate configurations (e.g. all 17 correspondences from a single set).\nFor both the pre-filtering as well as the 17-point RANSAC, we utilize the OpenGV library \\cite{opengv}.\n\nFor accepting a transformation we require to have at least 100 inliers after RANSAC.\nTo quantify the uncertainty of the computed transformation we use a sampling-based approach to compute a covariance matrix.\nThe samples are obtained by repeatedly selecting 17 inliers and computing the transformation using the 17-point algorithm.\nWe favor the sampling approach over an analytical one as it results in a more consistent covariance estimate since unmodelled effects (e.g. the relative pose uncertainty of the odometry) are better reflected in the samples.\n\n\\subsection{Pose Graph Optimization}\n\\label{sec:pgo}\nThe Pose Graph Optimization (PGO) in COVINS-G is where the information from the different agents gets fused and the drift in the corresponding trajectories can be corrected.\nThe PGO step is triggered every time a new constraint is added to the graph as a result of the loop detection.\nThe state that is optimized in the PGO consists of all KF poses that are in the corresponding graph. \nIn the following, we denote the pose of a KF $k$ by a rotation $q_{ws_k}$ and a translation $p_{ws_k}$, where $w$ represents the coordinate frame in which the poses are expressed and $s_k$ denotes the device coordinate frame at time-point $k$.\nHence, the state in PGO can be defined as $\\mathcal{S} = \\{ q_{ws_1}, p_{ws_1}, \\cdots, q_{ws_n}, p_{ws_n} \\}$, with $n$ being the number of keyframes in the graph.\n\nIn the actual PGO, we optimize the following objective:\n\\begin{equation}\n\\label{eq:pg}\n   \\mathcal{S}^{*} = \\underset{\\mathcal{S}}{\\text{arg min}} \\{ \\sum_{k=1}^{n}  \\sum_{l=1}^{q} \\norm{e_{kk+l}}^2_{W_{kk+l}} + \\sum_{i,j \\in \\mathcal{L}} \\varrho (\\norm{e_{ij}}^2_{W_{ij}}) \\},\n\\end{equation}\nwhere $\\norm{e}^2_{W} = e^TWe$ is the squared Mahalanobis distance with the information matrix $W$, $\\mathcal{L}$ denotes the set of all loop-closure edges and $\\varrho(\\cdot)$ denotes the use of a robust loss function, here, the Cauchy loss function.\nThe parameter $q$ represents the number of neighbor KFs added as odometry constraints (e.g. $q=1$ would correspond to having an edge only with the subsequent KF). \nThis is used to approximate correlations between poses which are generally computed in a sliding window fashion and in our implementation is set $q=4$.\nThe error terms $e_{ij}$ are of the form\n\\begin{equation}\ne_{ij} = \\begin{bmatrix}\n\\left(p_{ij} - \\hat{p_{ij}} \\right)^T  & 2 \\cdot \\text{vec}(q_{ij}^{-1} \\cdot \\hat{q_{ij}})^T  \n\\end{bmatrix}^{T},\n\\end{equation}\nwhere $\\text{vec}(q)$ extracts the vector part of a quaternion. \nThe variables denoted with $\\hat{\\cdot}$ correspond to measurements, e.g. from a loop closure transformation or the delta pose estimated by the odometry front-end.\nWe use the simplified notation $p_{ij}, q_{ij}$ to denote the translation and rotation of the delta pose between the pose of KF $i$ and $j$ given by $T_{ij} = T_{ws_i}^{-1} T_{ws_j}$.\nFor the optimization, the Ceres's implementation of the Levenberg-Marquardt is used.\nThe information matrices $W$ for the loop constraints are obtained via the estimated covariance outlined in section \\ref{sec:seventeen_pt_approach}.\nThe ones corresponding to the odometry edges are obtained using the expected accuracy of the corresponding front-end.\n\n\n\\section{Experiments and Discussions}\n\\label{sec:experiments}\nIn our evaluation, we demonstrate the generic nature of our collaborative back-end and its capability to use and combine all sorts of VIO front-ends.\nIn Sections \\ref{sec:tracking_cam} and \\ref{sec:feature_support} we show two applications where our back-end enables a collaborative estimation, which previous approaches cannot support.\nAll our experiments are performed using pre-recorded datasets which we play back in real-time.\nThe agents' front-ends are run on Intel NUC 7i7BNH @3.5 GHz hardware, while the server is run on a laptop with a core i7-6700HQ @2.6 GHz.\nNote that in our setup the agents and the server are connected via a wireless network, in order for real communication to take place.\nThis setup allows us to have more comparable results over multiple runs while still making use of a real wireless network as would be the case during real-world deployment.\n\n\\subsection{Collaborative SLAM Estimation Accuracy}\n\\label{sec:euroc_eval}\nWe evaluate the accuracy of the collaborative SLAM estimate on the EuRoC Dataset\\cite{euroc} using the Machine Hall (MH) and the Vicon Room 1 (V1) sequences to establish a collaborative estimation scenario with three to five participating agents. \nWe use various combinations of front-ends with our back-end and compare the results against COVINS \\cite{covins} as well as the state-of-the-art multi-session capabilities of VINS-mono\\cite{vins-mono} and  ORB-SLAM3 \\cite{ORBSLAM3_TRO}.\nAs multi-session methods only support one agent at a time, we process the datasets sequentially with one agent at a time, in contrast to the collaborative SLAM approaches, where all agents' processes are run in parallel.\n\nTo showcase the flexibility of our approach, we perform the evaluation of our back-end with different front-end combinations.\nIn the first experimental setup, all agents are operated using the same front-end, which in our experiments is either the VINS-mono or the ORB-SLAM3 front-end.\nIn subsequent experiments, we test COVINS-G using different front-ends for the different agents, namely OpenVINS\\cite{Openvins}, VINS-mono, ORB-SLAM3 and SVO-pro\\cite{Forster17troSVO}.\nSince COVINS performs as a post-processing step a Global Bundle Adjustment (GBA) at the end of every run, we do not directly compare against it but rather include its performance here as a reference.\nFor more appropriate comparisons with COVINS-G, we include the COVINS result without the additional GBA step.\nThe averaged results over 5 runs are summarized in Table \\ref{tab:multi_ag}.\n\nUsing the ORB-SLAM3 front-end, we achieve on-par performance to the ORB-SLAM3 multi-session back-end, even though in our approach we do not perform map re-use as in ORB-SLAM3.\nAs ORB-SLAM3 performs a GBA upon every loop detected, it can potentially reach very high accuracy (as COVINS demonstrates with GBA enabled), however, as long as it is able to find sufficient correspondences in the re-localization, no GBA gets triggered.\nTherefore, the overall accuracy is coupled also to the time that the last GBA step was performed.\nCompared to COVINS without GBA, our approach reduces the error almost by a factor of two.\nThis can be explained by the fact that in COVINS-G we are able to close more loops, as also shown in Table \\ref{tab:computation}, and also because the covariance of the loop transformations gets explicitly estimated instead of using fixed heuristics.\n\nThe comparison with the VINS-mono front-end showcases a similar outcome that COVINS-G performs similarly to the VINS-mono multi-session back-end.\nThe small gap in performance can be explained in that VINS-mono incorporates a larger number of loops, whereas in COVINS-G we set a minimum number of KFs between consecutive loops in order to reduce the computational burden.\nCompared to COVINS with the VINS-mono frontend, COVINS-G shows a significant improvement with a factor of around 3.\nThis is because COVINS only detects and closes few loops because it's loop-closure pipeline is tailored to a high quality map with a large number of map points.\nDue to this weak connection within and across the trajectories, even the GBA is unable to reduce the error.\nWith COVINS-G, even with a mix of different front-ends, we can see that the performance is in a similar range as when using the VINS-mono front-end, demonstrating the effectiveness of our generic approach.\n\n\\begin{table*}[]\n\\caption{Evaluation of joint-trajectory estimates for different methods on the EuRoC Dataset\\cite{euroc} (lowest error in bold) reported as the average trajectory error over 5 runs each. $^\\#$For the heterogenous front-end, agents 1-5 utilize OpenVINS\\cite{Openvins}, VINS-mono\\cite{vins-mono}, ORB-SLAM3\\cite{ORBSLAM3_TRO}, SVO-Pro\\cite{Forster17troSVO} and VINS-mono\\cite{vins-mono} front-ends, respectively. As *COVINS\\cite{covins} performs a Global Bundle Adjustment (GBA) step at the end of the run it has been included here for reference only and is excluded from our comparison.\n}\n\\label{tab:multi_ag}\n\\centering\n\\def1.2{1.2}\n\n\\begin{tabular}{|cc|ccc|}\n\\hline\n\\multicolumn{2}{|c|}{\\textbf{Method}}                                                          & \\multicolumn{3}{c|}{\\textbf{Translational RMSE (m)}}                                                                                 \\\\ \\hline\n\\multicolumn{1}{|c|}{}                                 &                                       & \\multicolumn{1}{c|}{MH01-MH03}                    & \\multicolumn{1}{c|}{MH01-MH05}                    & V101-V103                    \\\\ \\cline{3-5} \n\\multicolumn{1}{|c|}{\\multirow{-2}{*}{Front-end}}       & \\multirow{-2}{*}{Back-end}             & \\multicolumn{1}{c|}{(3 Agents)}                   & \\multicolumn{1}{c|}{(5 Agents)}                   & (3 Agents)                   \\\\ \\hline\n\n\\multicolumn{1}{|c|}{ORB-SLAM3\\cite{ORBSLAM3_TRO}}                        & ORB-SLAM3 MS\\cite{ORBSLAM3_TRO}                          & \\multicolumn{1}{c|}{0.041}                        & \\multicolumn{1}{c|}{0.082}                        & \\textbf{0.048}               \\\\ \\hline\n\\multicolumn{1}{|c|}{ORB-SLAM3\\cite{ORBSLAM3_TRO}}                        & COVINS (No GBA) \\cite{covins}                      & \\multicolumn{1}{c|}{0.075}                        & \\multicolumn{1}{c|}{0.119}                        & 0.130                        \\\\ \\hline\n\\multicolumn{1}{|c|}{{\\color[HTML]{666666} ORB-SLAM3\\cite{ORBSLAM3_TRO}}} & {\\color[HTML]{666666} COVINS (GBA) \\cite{covins}*} & \\multicolumn{1}{c|}{{\\color[HTML]{666666} 0.024}} & \\multicolumn{1}{c|}{{\\color[HTML]{666666} 0.036}} & {\\color[HTML]{666666} 0.042} \\\\ \\hline\n\\multicolumn{1}{|c|}{ORB-SLAM3\\cite{ORBSLAM3_TRO}}                        & Ours                                  & \\multicolumn{1}{c|}{\\textbf{0.040}}               & \\multicolumn{1}{c|}{\\textbf{0.064}}               & 0.067                        \\\\ \\hline \\hline\n\n\\multicolumn{1}{|c|}{VINS-mono \\cite{vins-mono}}                        & VINS MS\\cite{vins-mono}                               & \\multicolumn{1}{c|}{\\textbf{0.062}}                        & \\multicolumn{1}{c|}{0.100}                        & \\textbf{0.076}                        \\\\ \\hline\n\\multicolumn{1}{|c|}{VINS-mono \\cite{vins-mono}}                        & COVINS (No GBA) \\cite{covins}                               & \\multicolumn{1}{c|}{0.259}                        & \\multicolumn{1}{c|}{0.305}                        & 0.183                       \\\\ \\hline\n\\multicolumn{1}{|c|}{{\\color[HTML]{666666} VINS-mono \\cite{vins-mono}}}                        & { \\color[HTML]{666666} COVINS (GBA) \\cite{covins}*}                               & \\multicolumn{1}{c|}{ {\\color[HTML]{666666} 0.261}}                        & \\multicolumn{1}{c|}{{\\color[HTML]{666666} 0.321}}                        & {\\color[HTML]{666666} 0.183}                       \\\\ \\hline\n\\multicolumn{1}{|c|}{VINS-mono\\cite{vins-mono}}                        & Ours                                  & \\multicolumn{1}{c|}{0.081}                        & \\multicolumn{1}{c|}{\\textbf{0.095}}                        & 0.090                        \\\\ \\hline \\hline\n\n\\multicolumn{1}{|c|}{Heterogenous$^\\#$}                   & Ours                                  & \\multicolumn{1}{c|}{0.081}                        & \\multicolumn{1}{c|}{0.090}                        & 0.088                        \\\\ \\hline\n\\end{tabular}\n\\vspace{-10pt}\n\\end{table*}\n\n\\subsection{Large-Scale Outdoor Experiments}\n\\label{sec:outdoor_expt}\nTo demonstrate the applicability of our back-end to large-scale scenarios, we captured a dataset consisting of 4 sequences using a hand-held setup with an Intel Realsense D455.\nThe dataset was recorded by walking on the sidewalks around the campus of ETH Zurich in the center of Zurich and with a combined trajectory length of around 2400$m$ covered an area of approximately 67500$m^2$.\nFor this experiment, the VINS-mono front-end was used on all sequences.\nAs we do not have a ground truth trajectory for the dataset, we superimposed the estimated combined trajectory on a satellite image of the area as illustrated in Figure \\ref{fig:outdoor_expt}. \nAs can be seen, the different trajectories are well aligned with the road structure. \nA complete visualization of the whole experiment can be found in the supplementary video.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/expt_outdoor_largescalev3.png}\n    \\caption{Joint trajectory estimates for 4 agents superimposed on the satellite image of our large-scale outdoor experiment with a combined trajectory length of 2400 $m$.}\n    \\label{fig:outdoor_expt}    \n    \\vspace{-12pt}\n\\end{figure}\n\n\\subsection{Tracking Camera Support}\n\\label{sec:tracking_cam}\nThis experiment aims at highlighting the generalization capabilities of our system with respect to the front-end that is used.\nTo do so, we captured a dataset with two different sensors, one Intel Realsense D455 camera, and one Intel Realsense T265 tracking camera inside an office floor.\nFor the D455 we used the VINS-mono as a front-end, whereas for the T265 we used the odometry estimate that is provided directly by the sensor.\nAs this sensor only provides its images and the corresponding poses but does not offer access to its internal state, we use our ROS-based front-end wrapper, which does a motion-based keyframe selection, detects feature points, and creates and communicates the KF messages to the back-end.\n\nThe outcome of the experiment is illustrated in Figure \\ref{fig:tracking}, where one can see an example image for each of the sensors and the combined trajectory overlaid with the floor plan of the office (the complete experiment is visualized in the supplementary video).\nAs it can be seen, the estimated trajectory fits the floor plan, indicating its consistency.\nIn addition to the different odometry sources, the two sensors have also two rather different lenses, rendering matching between the two systems more difficult, however, our proposed back-end is able to merge the estimates nonetheless.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/expt_trackingv3.png}\n    \\caption{Joint trajectory estimate for a heterogeneous system of 2 agents superimposed on the floor plan of the \n    building. Map points are shown for visualization purposes only}\n    \\label{fig:tracking}    \n    \\vspace{-9pt}\n\\end{figure}\n\n\\subsection{Alternative Feature Descriptor support}\n\\label{sec:feature_support}\nVisual features are designed to be tolerant to illumination changes, perspective distortions as well as viewpoint changes.\nWhile for runtime efficiency, real-time SLAM systems are mainly restricted to highly computational efficient binary features such as ORB \\cite{ORB} or BRISK \\cite{Leutenegger:etal:ICCV2011}.\nHowever, it is accepted that to date, binary features are not able to match the robustness of more expensive descriptors such as e.g. SIFT \\cite{SIFT}.\nAs our approach is largely decoupled from the front-end, we are able to make use of more expensive features, also because we do not require them to be computed at frame rate, thus allowing us to potentially match trajectories with much larger viewpoint differences.\n\nIn this experiment, we recorded another outdoor dataset with two hand-held trajectories walking in parallel on two sides of a street.\nWe modified our front-end wrapper to detect SIFT instead of ORB features and run the back-end using these SIFT features as well.\nWhile using the framework with ORB features no overlap can be detected to merge the trajectories, using the powerful SIFT features allowed us to successfully detect loops across the agents and align the trajectories.\nThe comparison of this experiment can be found in the accompanying video.\nSuch an improved tolerance to large view-point changes allows the system to be used in scenarios where larger view-point changes are inherent to the use case, for example, the collaboration between aerial and ground robots.\n\n\\subsection{Computational and Communication Requirements}\n\\label{sec:computation}\n\\begin{table}[]\n\\caption{Comparison of the runtime for the loop computation and the average communication traffic.}\n\\label{tab:computation}\n\\centering\n\\def1.2{1.2}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|l|}{\\textbf{}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}\\# Loops\\\\  detected\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Computation Time \\\\ per loop\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Network Traffic\\\\  per Agent\\end{tabular}} \\\\ \\hline\nOurs                            & 75                                                                    & 213 $\\pm$ 171 ms                                                                        & 179.74 kB\/s                                                                   \\\\ \\hline\nCOVINS \\cite{covins}                         & 57                                                                    & 35 $\\pm$ 23 ms                                                                         & 486.41 kB\/s                                                                   \\\\ \\hline\n\\end{tabular}\n\\vspace{-15pt}\n\\end{table}\n\nThe statistics for communication and loop transformation computation are generated and compared with COVINS for the experiment performed on EuRoC MH dataset with 5 agents, each running an ORB-SLAM3 front-end. \nThe computation time for estimating the loop transformation using our approach (including feature matching, 2D-2D pre-filtering, 17-point RANSAC and covariance matrix computation) is compared against the standard approach used in COVINS (feature matching + PnP RANSAC) and summarized in Table \\ref{tab:computation}. \nThough the computation time for our method is around an order of magnitude higher than for COVINS, the average computation time per loop is around $213 ms$ which still makes it possible to operate in real-time for up to 5 loop computations per second. \nFor this experiment, a total of 75 loops were detected for a mission time of around 135 seconds, resulting in 0.55 loops per second over all agents. \nThe network traffic for our approach is around three times lower than that of COVINS.\nThis is owed to the fact that in our back-end we require only 2D keypoints and their descriptors for each KF, unlike COVINS which also requires sending the map points as well as the data required to perform the IMU pre-integration.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, we present a front-end-agnostic back-end for collaborative visual-inertial SLAM.\nBy making use of a multi-camera relative pose estimator for estimating loop-closure transformations, our system is able to work using only 2D keypoint information.\nThis allows our collaborative back-end to be compatible with virtually any VIO front-end with minimal to no modifications to it.\nIn our experimental evaluation, we achieve at least on-par accuracy with state-of-the-art multi-session and collaborative SLAM systems, which use back-ends specifically designed for the respective front-end.\nOwed to the decoupled nature of the data used in our back-end, we can make use of more powerful keypoint descriptors like SIFT, allowing us to close loops and merge trajectories with drastic viewpoint changes, which cannot be handled by state-of-the-art systems.\nOur open-source implementation of the system enables users to unlock the capabilities of a collaborative SLAM system without the need to change their state estimation pipeline of choice.\n\nIn proceeding work, we would like to make more explicit use of the information coming from our VIO, in particular the gravity alignment, by replacing the 17-point algorithm with a minimal 4-point solver which leverages the gravity information \\cite{4PT_gravity}.\nThis allows to speed up the relative pose estimation step significantly, as it requires drawing fewer samples within the RANSAC computation.\nIn the future, we would also like to deploy this system on a team of robots for multi-robot applications, such as exploration and 3D reconstruction of large areas.\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nInteracting particle systems are relevant to wide-ranging phenomena in \nphysics, chemistry, biophysics, ecology, etc. The concept of\n'particles' is used in a broad sense, that is \n'particles' can be atoms, molecules, spins, individuals, etc, and\nwhilst attention is\ndrawn to the interactions among particles no attempt is made in order to\nachieve a detailed description (e. g. quantum mechanical) of the particle \nitself. Therefore, due to interactions, the occurrence of complex behavior, \nsuch as phase transitions, self-organization, chaos, bistability, etc, may \nbe observed \\cite{Ligget}. \n\nWithin this context, an active field of research is the study of\nfar-from-equilibrium reaction systems \\cite{ZGB,Eze1}. Irreversible\nphase \ntransitions (IPT) between active regimes (where reactions are\nsustained) and absorbing states (where \nreactions are no longer possible) have been reported in a great variety of \nmodels such as the Ziff, Gulari, and Barshad (ZGB) model for the\ncatalytic oxidation of CO \\cite{ZGB}, the \ndimer-dimer model \\cite{Eze2}, the contact process \\cite{Jens1},\nforest-fire models \\cite{Eze3}, etc (for a recent review see\ne. g. \\cite{Eze1}). According to the Janssen-Grassberger conjecture\n\\cite{Janss,Grass1}, irreversible reaction systems that exhibit a\nphase transition to a single absorbing state characterized by a scalar\norder parameter belong to the Directed Percolation (DP) universality\nclass. This conjecture, stated a long time ago for unique\nabsorbing states, has \nbeen further generalized for the cases where such states are\nnon unique \n\\cite{Jens2,Eze4}. A special case corresponds to non-equilibrium systems\nwhere, provided an IPT exists, there is in addition a local or global\nconservation of particles modulo two, such as the branching\nand annihilating random walks with an even number of offsprings\n\\cite{BARWO,Jens4}. In these cases a new universality class emerges, commonly called \nparity conserving (PC) class, which is due\nto the existence of two statistically equivalent absorbing states at\nthe critical point \\cite{Kolea}. However, global conservation of\nparticles of modulo two\nmay also lead to exponents in the PC class only when local spontaneous annihilation\n($1X \\rightarrow 0$) is highly inhibited. Then, at a coarse grained\nlevel, the relevant surviving processes are those conserving\nparity. In other words, parity conservation can \nbe restored at a coarse grained level \\cite{Haye,Roberto}. A nice example where global\nparity conservation still leads to DP exponents is given by Inui et. al. \\cite{Ponja}. \nIt is clear in this case that spontaneous annihilation must be taken\ninto account.\n\n\nIPT are studied largely by means of Monte Carlo simulations\nand mean-field approaches. Recent developments of field-theoretic\nrenormalization group techniques have provided a new theoretical framework\nwhere non-equilibrium phase transitions can be studied \\cite{Uwe}. These\ntechniques are able to identify the relevant processes in a given\nuniversality class although the quantitative predictions are still\npoor. \n\nSo far, most of the simulations have been performed using\n\\underline{discrete} lattices where each particle fills a single site\non the lattice and neighboring particles may react with a certain\nprobability. In contrast, our knowledge on the behavior of\nirreversible reaction systems in continuous media is rather poor. In\norder to stress some dramatic differences that may arise for a reaction system\nwhen it is simulated off lattice, let us consider the simplest case of\nthe $B + B \\rightarrow 0$ irreversible reaction which proceeds according\nto the Langmuir-Hinshelwood mechanism.\n\\begin{eqnarray}\n & B(g) + S \\rightarrow B(a) \\\\ \\nonumber  \n & B(g) + S \\rightarrow B(a)  \\\\   \n & B(a) + B(a) \\rightarrow 0 + 2 S \\nonumber  \n\\end{eqnarray}\nwhere $g$ and $a$ refer to the gas and adsorbed phases, respectively,\nwhile $S$ represents a site on the surface. At first, we assume that\n$B$-species adsorbed on nearest neighbor sites react with unitary\nprobability ($P_r = 1$). If we used a discrete lattice, reactions\nwould be sustained indefinitely, i. e. the system could not irreversibly\nevolve into an absorbing state. However, considering a continuous\nmedia, the random adsorption of $B$ particles of finite size $\\sigma$\ncauses the formation of several interparticle gaps of size smaller\nthan $\\sigma$. So, in the infinite time limit ($t \\rightarrow \\infty$) the\nsample becomes imperfectly covered by $B$-species separated by small\ninterparticle gaps. Reaction is no longer possible and the system\nbecomes irreversible trapped into an absorbing state (infinitely\ndegenerated). The maximum jamming coverage attained in one dimension\nis $\\Theta_j \\approx 0.74759$, which corresponds to the so called car\nparking problem \\cite{Jim,car}. \n\nIn this paper we show that by introducing the adsorption of species\nwith transient mobility in a continuous one-dimensional medium, it is\npossible to open a window where reactions are sustained. However, by \ntuning the external parameter which controls the transient mobility\nof the particles it is possible to irreversible drive the system into\nan absorbing state. \n\nIt should be mentioned that the study of reactions\nof atoms in the gas phase possessing thermal energy with adsorbed\natomic CO species on metal and \nsemiconductor surfaces is a topic of current interest. In contrast to\nthermally activated reactions among adsorbed species, i. e., the so\ncalled Langmuir-Hinshelwood mechanism, these kind of reactions take\nplace under far-from-equilibrium conditions. Consequently, the\ndetermination of the underlying mechanism as well as the understanding\nof the dynamic behavior is challenging work. Within this context,\nvery recently Kim {\\it et al} \\cite{Kim} have reported experimental studies\nof the reaction of hot $H$-atoms with adsorbed $D$-atoms (for further\nexperimental works see references in \\cite{Kim}).\n\nIt should be noted that from the theoretical point of view a number of \nrelated models for random sequential adsorption with diffusion\n\\cite{priv,eis} and desorption \\cite{stinc} have also been proposed and studied\n(for a review see also \\cite{Jim}). However, interest in\nsuch studies is addressed to the asymptotic approach to the jammed state.\nIn contrast, in this paper our interest is focused on the irreversible\ncritical behaviour of a reactive system.\n\nSo this work\nis devoted to the characterization of \nsuch IPT in continuous media and it is organized as follows: section\n2 gives the description of the model and the simulation technique. In\nsection 3 we discuss the results while conclusions and remarks are\npresented in section 4.\n\n\\section{The Model and the Monte Carlo simulation method}\n\nIn this paper, we study a 1D off-lattice\nadsorption-reaction model in which particles of size $\\sigma$ undergo a \nballistic flight just after depositing on the substrate.\nThe system evolves in time under the following local rules. (i) A position $x$\nis randomly selected on the substrate. If the interval $[x - \\sigma\/2, x + \\sigma\/2]$ is \nempty, then the adsorption trial is successful, otherwise it is rejected. So, it is\nclear that double occupancy of positions is forbidden. (ii) Right after a successful\nadsorption trial, a random direction is selected (left or right). Then, the particle \nundergoes a ballistic flight in the previously selected direction up to a distance $R$ \nfrom the adsorption position $x$, provided that no other already deposited particle is found \nalong the way. (iii) If during the flight one particle hits another\npreviously adsorbed particle which is \nalready at rest on the substrate, the following alternatives can occur: (1) the annihilation \n($B+B \\rightarrow 0$) occurs with probability $P_r$. Then, both particles react and leave \nthe system. (2) Particles do not react (with probability $(1- P_r)$), and the flying particle \nis frozen at the collision point.\n\nThe ballistic flight mimics 'hot monomer' adsorption, allowing the incoming particle to \ntransform its energy into degrees of freedom parallel to the\nsubstratum. The length of the flight $R$ is finite in order to account for\nfrictional dissipation. The model has two externally tunable\nparameters, namely $R$ and $P_r$. For $P_r = 0$ one recovers the 'hot\nmonomer' random sequential adsorption model \\cite{Daniel} while for\n$R = 0$ and $P_r = 0$ one has the 1d car parking problem \\cite{car}.\n\nIn order to simulate a continuous medium on a digital computer, one\nactually considers a discrete lattice. However, each site of size\n$\\sigma$ is subdivided into $2^{64}$ different adsorption\npositions. This high degree of discretization has provided excellent\nresults when compared with the exact analytic solution of a related\nproblem \\cite{Daniel}. \n\nPreliminary results show that the system can undergo continuous IPT\nbetween a stationary \nreactive state and an absorbing state without reactions when varying\nthe parameters. This can easily be tested by considering the case\n$P_r = 1$ and $R = 0$ ($R > 1$) which gives an absorbing (reactive)\nstate, respectively. \nIt should be pointed out that continuous IPT are dominated by\nfluctuations. Consequently, in a finite\nsystem and close to the critical point, the stationary state of the reactive\nphase can irreversibly evolve into the saturated state (absorbing state). Due\nto this circumstance, the precise determination of both critical points and\ncritical exponents is rather difficult. However, this shortcoming can be \navoided by performing an epidemic analysis. \nFor this purpose one starts, at $t=0$, with a a configuration close to\none of the absorbing states. It is clear that different absorbing states will normally differ\nin the density of monomers. It should be pointed out that the dynamical critical behavior\nof systems with infinitely many absorbing configurations is expected to depend upon the\ninitial density of inactive particles \\cite{Jens2,Mendes}. However, static critical behavior appears\nto be independent of it. \nFrom the set of possible initial densities, \na value $\\rho_n$ is particularly important, namely the stationary density of inactive particles\nwhich results after the natural evolution of the system in the subcritical region has finished.\nThe value $\\rho_n$ is relevant, since only for this value the natural dynamical critical \nbehavior emerges.  Preliminary simulation results show that $\\rho_n$ depends on the parameter \n$P_r$, but their values have not been included in this work for the sake of space.\nThe dependence of the critical behavior on the set of initial densities is \nthe subject of an ongoing investigation.\n\nConsequently, the initial state for the epidemic analysis is \ngenerated by the evolution of the system very close to the critical\npoint until poisoning is achieved. After generating this stationary\npoisoned state, we remove one or two particles \nfrom the middle of the system in order to create a small active\narea where adsorption is now possible. It should be noted that an\nempty area is considered to be active \nif it is longer than or equal to $\\sigma$. Then, the time evolution of\nthe system is analyzed by measuring the following properties: (i) the\naverage amount of active area at time $t$, $A(t)$; (ii) the survival\nprobability of the active area at time $t$, $P_s (t)$; and (iii) the average\ndistance over which the active area have spread at time $t$,\n$D(t)$. Finite size effects are absent because the system is taken\nlarge enough to avoid the presence of active area at the boundaries. \nFor this purpose a sample of $10^4 \\sigma$ is enough. Averages are taken over \n$10^5$ to $10^6$ different samples. Near the critical point, the\namount of active area is often very small. Then, we improve the\nefficiency of the algorithm by keeping a list of the positions where\nthere is active area. Time is incremented by $1\/a(t)$, where $a(t)$\nis the amount of active area at time $t$. The time evolution of the active\narea is monitored up to $t =10^5$. At criticality, the following\nscaling behavior holds: \n\\begin{equation}\n\\label{Eta}\nA(t) \\propto t^{\\eta},\n\\end{equation}\n\\begin{equation}\n\\label{Delta}\nP_s(t) \\propto t^{-\\delta},\n\\end{equation}\nand\n\\begin{equation}\n\\label{Zeta}\nD(t) \\propto t^{z\/2}\n\\end{equation}\nwhere $\\delta $, $\\eta $ and $z$ are \\underline{dynamic} exponents. \n\n\\section{Results and discussion}\nPreliminary simulations show that for $P_r = 1$ it is possible to\nachieve a stationary reactive state in the large $R$ limit (i. e. for\n$R \\ge 2$) while in the opposite limit ($R \\le 1.5$) the system\nbecomes irreversible saturated by $B$-species. In order to obtain a\nquantitative description of the IPT we have performed epidemic studies\naround the critical edge. Figure 1 (a - c) shows log-log plots of $A$, $P_s$\nand $D$ versus the time $t$, obtained for different parameter values. The \nthree plots exhibit a power law behavior which is the signature of a\ncritical state. Using smaller (greater) $R$-values we observe slight\nupwards (downwards) deviations in three plots which indicate\nsupercritical (subcritical) behavior (these results are not shown for\nthe sake of clarity). Then, the critical exponents obtained by\nregressions are:\n\\begin{equation}\n\\label{expon}\n\\eta =  0.308 \\pm 0.004 \\; \\; \\; \\delta = 0.165 \\pm 0.003 \\; \\;\n\\; z\/2 = 0.625 \\pm 0.002 \n\\end{equation}\nThese values are in excellent agreement with the exponents\ncorresponding to the DP universality class in $1+1$\ndimensions \\cite{Janss,Grass1}. Recently, extended series expansions calculations \\cite{series}\nhave provided very accurate values for the DP critical exponents, namely\n\\begin{equation}\n\\label{dpexpon}\n\\eta = 0.31368(4) \\; \\; \\; \\delta = 0.15947(3) \\; \\; \\; z\/2 = 0.63261(2)\n\\end{equation}\nTherefore we conclude that the studied adsorption-reaction model on a\n\\underline{continuous medium} belongs to the DP universality class\nlike many other systems already studied on \\underline{discrete\nlattices}. It should be noticed that the present model has infinitely\nmany absorbing states, so as in the case of the dimer-dimer model\n\\cite{Eze2} the DP conjecture holds for non-unique absorbing states\n\\cite{Janss} at least as long as the absorbing states can be solely \ncharacterized by the vanishing of a single scalar order parameter. \n\nWe have also studied the case of imperfect reaction, i. e. $P_r <\n1$. Figure 2 shows a plot of the phase diagram. The phase boundary\ncurve was determined by means of an epidemic analysis, as is shown\nin figure 1. The obtained critical exponents are\n\\begin{equation}\n\\label{imexpon}\n\\eta = 0.312 \\pm 0.004 \\; \\; \\; \\delta = 0.157 \\pm 0.003\\; \\; \\; z\/2 =\n0.631 \\pm 0.001 \n\\end{equation}\nOnce again, these exponents are in good agreement with those corresponding\nto DP. Scanning the whole critical curve we obtain second order IPT's\nthat belong to the DP universality class. However, the special case\n$R \\rightarrow \\infty$ merits further discussion. For $P_r = 1$ ($P_r =\n0$) the system evolves towards a reactive (absorbing) state,\nrespectively. Then, a transition is expected at some intermediate\n$P_r$ value. In this case, the time evolution of the active area can\nbe described by means of a mean field equation. In fact, the active\narea $A(t)$ will grow (decrease) proportionally to $A(t) P_r$ ($A(t) (1 - P_r)$),\nrespectively; so\n\\begin{equation}\n\\label{mfield1}\n\\frac{d A}{dt} = A(t) P_r - A(t) (1 - P_r)\n\\end{equation}\nwhich leads to\n\\begin{equation}\n\\label{mfield3}\nA(t) = A_{0} \\; e^{(2 P_r - 1)t} \n\\end{equation}\nTherefore, $P_r = 1\/2$ is a critical value such as for $P_r > 1\/2$\n($P_r < 1\/2$) the active area will increase (decrease) exponentially\nin time, while just at criticality $A(t)$ will remain constant ($A(t)\n= A_0$), which is consistent with a mean field exponent $\\eta_{MF} =\n0$. The predicted behavior is confirmed by means of simulations as it\nis shown in figure 3. By means of linear regressions the following\nexponents are obtained for $P_{r} = \\frac{1}{2}$:\n\\begin{equation}\n\\label{imexp}\n\\eta \\approx 0.0  \\; \\; \\; \\delta \\approx 1.0 \\; \\; \\; z\/2 \\approx 1.0\n\\end{equation}\nThen, our mean field estimate for $\\eta$ is in good agreement with the\nsimulation results. Regrettably, we were unable to derive the mean\nfield values for the remaining exponents.\n\nWe conclude that the particular point $P_r =\n1\/2$, $R \\rightarrow \\infty$ is a first order point (see figure 2)\nwhich is not in the DP class which characterizes the whole critical curve. \n\nIn the following, we give theoretical arguments by means of a coarse-grained \nLangevin description that support the result concerning the\nuniversality class of the model. First, note that the normalized variables\nneeded to characterize the configurations of the system are the amount\nof active area $a(x,t)$, the number of monomers in the system $n(x,t)$ and\nthe amount of inactive area $v(x,t)$. These variables are not\nindependent since we have the constrain $a(x,t) + n(x,t) + v(x,t) = 1$. It\nis clear from the above discussion that the time evolution of the\nsystem ends when $a(x,t) = 0$. Since $a(x) \\rightarrow 0$ at\ncriticality, this quantity can be chosen as the order parameter of\nthe system. Then, we will try to  describe the time\nevolution of the system near the critical point by means of two\ncoupled Langevin equations, for instance, one for $a(x,t)$ and the\nother for $n(x,t)$. Due to the nature of the absorbing configurations,\neach term of these equations must vanish when $a(x,t) \\rightarrow\n0$. \\\\ Let us consider the microscopic processes which are \nrelevant to characterize the critical behavior of the system. First of\nall, both diffusion of $a(x)$ and $n(x)$ can be interpreted as\nsuccessive adsorption-reaction processes. Within a one site\ndescription, both $n(x)$ and $a(x)$ will increase proportional to\n$a(x)$. The reaction processes will contribute to the equations with a\ncoupling term proportional to $a(x)n(x)$. It is also clear that monomer\nflights will introduce terms proportional to $a(x)^2$, $a(x)^3$,\netc. Since only the lower order terms are relevant for a\nrenormalization group treatment \\cite{cardyb}, we just keep the term\nproportional to $a(x)^2$. Then, we can write down the following\nLangevin equations: \n\\begin{equation}\n\\label{lang}\n\\partial n(x,t) \/ \\partial t = k_{1} \\nabla^{2} a(x,t) + k_{2} a(x,t)\n-k_{3} a(x,t)^{2} - k_{4} a(x,t) n(x,t) + \\eta_{1} (x,t) \n\\end{equation}\n\n\\begin{equation}\n\\label{lang1}\n\\partial a(x,t) \/ \\partial t = u_{1} \\nabla^{2} a(x,t) + u_{2} a(x,t)\n-u_{3} a(x,t)^{2} - u_{4} a(x,t) n(x,t) + \\eta_{2} (x,t) \n\\end{equation}\nwhere $\\eta_{1} (x,t)$ and $\\eta_{2} (x,t)$ are uncorrelated noises\nproportional to $\\sqrt{a(x,t)}$, $k_{i}$ and  $u_{i}$ are coefficients. \nThis system of coupled Langevin equations is similar to that obtained\nfor the `pair contact process' \\cite{Jens1,Jens2} which is one\nof the prototype systems with multiple absorbing states.  Mu\\~{n}oz\n{\\it et al} \\cite{munioz1} have shown that for large $t$ the equation\ncorresponding to the activity (equation (\\ref{lang1}) for the present model)\nreduces to the Langevin representation of DP. Then, our simulation\nresults are consistent with the above-presented theoretical\narguments. The same authors have also shown that systems with many\navailable absorbing configurations display strong memory effects\nthat may lead to anomalous scaling. In addition to this, Mendes {\\it et al} \\cite{Mendes} have \nproposed a generalized hyperscaling relation which has proved to be valid in systems with \nmultiple absorbing configurations.\nSimulation results on several\nlattice models with infinitely-many absorbing states support both\ntheoretical argument \\cite{Mendes,Jens2,dick}. The role that initial states play in \nthe temporal evolution of the present model is under investigation. \n\n\\section{Conclusions and final remarks}\nA model for the irreversible adsorption-reaction of a single species\non a continuous medium is studied by means of numerical\nsimulations. We would like to stress and comment upon the following\ninteresting features of the system: \n(i) in contrast to standard (reversible) transitions, non-equilibrium\nIPT can happen in one dimension. (ii) the studied adsorption-reaction model\nclearly shows interesting new effects that may arise when a\nprocess is modelled on a continuous medium. Since the system always reaches a\nstationary reactive state when simulated on a discrete lattice but a\nfinal poisoned state can be observed on a continuous one (e. g. for\n$P_r = 1$ and $R = 0$), one may expect a crossover behaviour when the\n'discretization degree' of the surface is tuned from one extreme to\nthe other. This can be achieved by considering the adsorption on\n\\underline{discrete lattices} of species of arbitrary length $r$,\ni. e. $r$-mers. We found that the reactive random sequential\nadsorption (RRSA) of dimers ($r = 2$) always\nleads to a reactive steady state, whose stationary coverage is\nclose to $\\Theta \\approx 0.5$, and no poisoning is observed. However,\nthe RRSA of trimers ($r = 3$) causes irreversible poisoning of the lattice with\nan average saturation coverage close to $\\Theta_{ST} \\approx 0.7$. In\nthe case of dimers, two absorbing states of the type \n\\begin{eqnarray}\n...BBVBBVBBVBB... \\\\ \\nonumber\n...VBBVBBVBBVBBV... \\nonumber\n\\end{eqnarray}\nwhere $V$ is an empty site, can be expected. So, during the RRSA the formation\nof several interfaces of the type $...BBVBBVVBBVBB...$ takes place. Due to\ncoarsening we expect that in the asymptotic limit ($t \\rightarrow\n\\infty$) both absorbing states will form two semi-infinite domains\nseparated by an interface. The competition between these domains will\nkeep the system in the reactive state for ever. Just by increasing \n$r = 2$ to $r = 3$, an infinite number of absorbing configurations\nappear in the system. So, the arguments developed above no longer hold\nand poisoning is observed. Consequently, an IPT is located between $r = 2$ and $r\n= 3$ and its precise location requires the study of the\nadsorption-reaction problem with particles of non-integer\nlength. However, adsorption of $r$-mers of length $r = 2 + \\epsilon$\nwould cause the argument of the two competing absorbing states to\nfail. Therefore, we expect that the critical size, i. e. 'the\ndiscretization degree' is $2$. (iii) To our best knowledge, this is the\nfirst off-lattice model which exhibits a second-order IPT in the DP\nuniversality class. Consequently, this results once again supports the\nDP conjecture \\cite{Janss} which can now be generalized to off-lattice\nmodels with infinitely many absorbing states.   \n\nWe expect that the interesting behavior of the present simple reaction\nmodel will stimulate further work on irreversible transitions and\ncritical phenomena on continuous media, a field that, to\nour best knowledge, remains almost unexplored. \n\\vskip 1.5 true cm\n{\\bf Acknowledgments}: This work was financially supported by CONICET,\nUNLP, CIC (Provincia Bs. As.), ANPCyT, the Fundaci\\'on Antorchas,\nand the Volkswagen Foundation (Germany).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tMulti-task representation learning~\\cite{caruana1997multitask} is an important problem which aims to learn a common low-dimensional representation from multiple related tasks. Representation learning has received extensive attention in both empirical applications~\\cite{ando2005framework,bengio2013representation,li2014joint} and theoretical study~\\cite{maurer2016benefit,du2020few,tripuraneni2021provable}.\n\n\n\t\n\tRecently, an emerging number of works~\\cite{yang2020impact,yang2022nearly,hu2021near,cella2022multi} investigate representation learning for sequential decision making, and show that if all tasks share a joint low-rank representation, then by leveraging such a joint representation,\n\tit is possible to learn faster than treating each task independently. \n\tDespite the accomplishments of these works, they mainly focus on the regret minimization setting, where the performance is measured by the cumulative reward gap between the optimal option and the actually chosen options. \n\t\n\n\t\n\tHowever, in real-world applications where obtaining a sample is expensive and time-consuming, e.g., clinical trails~\\cite{zhang2012multi},\n\n\tit is often desirable to identify the optimal option using as few samples as possible, i.e., we face the \\emph{pure exploration} scenario rather than regret minimization.\n\n\tMoreover, in many decision-making applications, we often need to tackle multiple related tasks, e.g., treatment planning for different diseases~\\cite{bragman2018uncertainty} and content optimization for multiple websites~\\cite{agarwal2009explore}, and there\n\tusually exists a common representation among these tasks, e.g., the features of drugs and the representations of website items. Thus,  we desire to exploit the shared representation among tasks to expedite learning.\n\tFor example, in clinical treatment planning, we want to identify the optimal treatment for multiple diseases, and there exists a joint representation of treatments. In this case, since conducting a clinical trial and collecting a sample is time-consuming, we desire to make use of the shared representation and reduce the number of samples required.  \\looseness=-1\n\t\n\tMotivated by the above fact, in this paper, we study representation learning for multi-task pure exploration in sequential decision making. Following prior works~\\cite{yang2020impact,yang2022nearly,hu2021near}, we consider the linear bandit setting, which is one of the most popular settings in sequential decision making and has various applications such as clinical trials and recommendation systems. \n\n\n\tSpecifically, we investigate two pure exploration problems, i.e., representation learning for best arm identification in linear bandits ($\\textup{RepBAI-LB}$) and best policy identification in contextual linear bandits ($\\textup{RepBPI-CLB}$). \n\t\n\tIn $\\textup{RepBAI-LB}$, an agent is given a confidence parameter $\\delta$, an arm set $\\mathcal{X}:=\\{\\boldsymbol{x}_1,\\dots,\\boldsymbol{x}_n\\} \\subseteq \\mathbb{R}^{d}$ and $M$ tasks. For each task $m \\in [M]$, the expected reward of each arm $\\boldsymbol{x} \\in \\mathcal{X}$ is generated by $\\boldsymbol{x}^\\top \\boldsymbol{\\theta}_m$, where $\\boldsymbol{\\theta}_m \\in \\mathbb{R}^{d}$ is an underlying reward parameter. There exists an unknown global feature extractor $\\boldsymbol{B} \\in \\mathbb{R}^{d \\times k}$ and an underlying prediction parameter $\\boldsymbol{w}_m$  such that $\\boldsymbol{\\theta}_m=\\boldsymbol{B} \\boldsymbol{w}_m$ \n\tfor any $m \\in [M]$, where $M \\gg d \\gg k$. We can understand the problem as that all tasks share a joint representation $\\boldsymbol{f}(\\boldsymbol{x}):= \\boldsymbol{B}^\\top \\boldsymbol{x}$ for arms, where the dimension of $\\boldsymbol{f}(\\boldsymbol{x})$ is much smaller than that of $\\boldsymbol{x}$.  The agent sequentially selects arms and tasks to sample, and observes noisy rewards. The goal of the agent is to identify the best arm with the maximum expected reward for each task with confidence $1-\\delta$, using as few samples as possible. \n\t\n\tThe $\\textup{RepBPI-CLB}$ problem is an extension of $\\textup{RepBAI-LB}$ to environments with random and varying contexts. \n\tIn $\\textup{RepBPI-CLB}$, there are a context space $\\mathcal{S}$, an action space $\\mathcal{A}$, a known feature mapping $\\boldsymbol{\\phi}:\\mathcal{S} \\times \\mathcal{A} \\mapsto \\mathbb{R}^d$ and an \\emph{unknown} context distribution $\\mathcal{D}$. \n\n\n\tFor each task $m \\in [M]$, the expected reward of each context-action pair $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$ is generated by $\\boldsymbol{\\phi}(s,a)^\\top \\boldsymbol{\\theta}_m$, where $\\boldsymbol{\\theta}_m=\\boldsymbol{B} \\boldsymbol{w}_m$.\n\n\tWe can similarly interpret the problem as that all tasks share a low-dimensional context-action representation $\\boldsymbol{B}^\\top \\boldsymbol{\\phi}(s,a) \\in \\mathbb{R}^k$.  At each timestep, the agent first observes a context drawn from $\\mathcal{D}$, and chooses an action and a task to sample, and then observes a random reward.\n\n\tGiven a confidence parameter $\\delta$ and an accuracy parameter $\\varepsilon$, the agent aims to identify an $\\varepsilon$-optimal policy (i.e., a mapping $\\mathcal{S} \\mapsto \\mathcal{A}$ that gives suboptimality within $\\varepsilon$)\n\n\tfor each task with confidence $1-\\delta$, while minimizing the number of samples used.\n\n\t\n\n\n\n\n\t\n\t\n\tIn contrast to existing representation learning works~\\cite{yang2020impact,yang2022nearly,hu2021near,cella2022multi}, we focus on the pure exploration scenario and face several unique challenges: \n\n\t(i) The sample complexity minimization objective  requires us to plan an optimal sample allocation for recovering the low-rank representation, in order to save samples to the highest degree. \n\t(ii) Unlike prior works which either assume that the arm set is an ellipsoid\/sphere~\\cite{yang2020impact,yang2022nearly} or are computationally inefficient~\\cite{hu2021near}, we allow an arbitrary arm set that spans $\\mathbb{R}^d$, which poses challenges on how to efficiently schedule samples according to the shapes of arms.\n\t(iii) Different from prior works~\\cite{huang2015efficient,li2022instance}, we do not assume prior knowledge of the context distribution. This imposes additional difficulties in sample allocation planning and estimator construction.\n\t%\n\tTo handle these challenges, we design computationally and sample efficient algorithms, which effectively estimate the context distribution and employ the experimental design approaches to plan samples.\n\n\t\n\tWe summarize our contributions in this paper as follows.\n\t\\begin{itemize}\n\t\t\\item We formulate the problems of multi-task representation learning for best arm identification in linear bandits ($\\textup{RepBAI-LB}$) and best policy identification in contextual linear bandits ($\\textup{RepBPI-CLB}$). To the best of our knowledge, this is the first work to study representation learning in the multi-task pure exploration scenario.\n\t\t\\item For $\\textup{RepBAI-LB}$, we propose an efficient algorithm $\\mathtt{DouExpDes}$ equipped with \\emph{double experimental designs}. The first design optimally schedules samples to learn the joint representation according to arm shapes, and the second design minimizes the estimation error for rewards using low-dimensional representations.\n\t\n\t\n\t\n\t\tFurthermore, we establish a sample complexity guarantee $\\tilde{O}(\\frac{Mk}{\\Delta_{\\min}^2})$, which shows superiority over the baseline result $\\tilde{O}(\\frac{Md}{\\Delta_{\\min}^2})$ (i.e., solving each task independently). Here $\\Delta_{\\min}$ denotes the minimum reward gap.\n\t\n\t\t\\item For $\\textup{RepBPI-CLB}$, we develop $\\mathtt{C \\hyphen DouExpDes}$, an algorithm which efficiently estimates the context distribution and conducts double experimental designs under the estimated context distribution to learn the global representation. A sample complexity result $\\tilde{O}(\\frac{Mk^2}{\\varepsilon^2})$ is also provided for $\\mathtt{C \\hyphen DouExpDes}$, which significantly outperforms the baseline result $\\tilde{O}(\\frac{Md^2}{\\varepsilon^2})$, and demonstrates the power of representation learning.\n\t\\end{itemize}\n\t\n\t\n\t\n\t\\section{Related Work}\n\t\n\tIn this section, we introduce two lines of related works, and defer a more complete literature review to Appendix~\\ref{apx:related_work}. \\looseness=-1\n\t\n\t\\textbf{Representation Learning.}\n\tThe study of representation learning has been initiated and developed in the supervised learning setting, e.g., \\cite{baxter2000model,ando2005framework,maurer2016benefit,du2020few,tripuraneni2021provable}. Recently, representation learning for sequential decision making has attracted extensive attention.\n\n\n\t\\citet{yang2020impact,yang2022nearly} study representation learning for linear bandits with the regret minimization objective, where they assume that the arm set is an ellipsoid or sphere. \\citet{hu2021near} relax this assumption and allow arbitrary arm sets, but their algorithms based on a multi-task joint least-square estimator are computationally inefficient. \\citet{cella2022meta,cella2022multi} design algorithms which do not need to know the dimension of the underlying representation. \n\n\tDifferent from the above works which consider regret minimization, we study representation learning for (contextual) linear bandits with the pure exploration objective, which brings unique challenges on how to optimally allocate samples to learn the feature extractor, and motivates us to design algorithms building upon double experimental designs. \\looseness=-1\n\t\n\t\\textbf{Pure Exploration in (Contextual) Linear Bandits.}\n\tMost existing linear bandit works focus on regret minimization, e.g.,~\\cite{dani2008stochastic,chu2011contextual,abbasi2011improved}. Recently, there has been a surge of interests in the pure exploration objective for (contextual) linear bandits.\n\n\tFor linear bandits, \\citet{soare2014best} firstly apply the experimental design approach to distinguish the optimal arm, and establish sample complexity that heavily depends on the minimum reward gap.\n\t\\citet{tao2018best} design a novel randomized estimator for the underlying reward parameter, and achieve tighter sample complexity which depends on the reward gaps of the best $d$ arms.\n\n\n\t\\citet{fiez2019sequential} provide the first near-optimal sample complexity upper and lower bounds for best arm identification in linear bandits. \n\n\tFor contextual linear bandits, \\citet{zanette2021design} develop a non-adaptive policy to collect data, from which a near-optimal policy can be computed. \\citet{li2022instance} build instance-optimal sample complexity for best policy identification in contextual linear bandits, with prior knowledge of the context distribution. By contrast, our work studies a multi-task setting where tasks share a common representation,\n\tand does not assume any prior knowledge of the context distribution.\n\t\n\t\n\t\\section{Problem Formulation}\n\t\n\tIn this section, we present the formal problem formulations of $\\textup{RepBAI-LB}$ and $\\textup{RepBPI-CLB}$. Before describing the formulations, we first introduce some useful notations.\n\t\n\t\\textbf{Notations.}\n\tWe use bold lower-case letters to denote vectors and bold upper-case letters to denote matrices.\n\tFor any matrix $\\boldsymbol{A}$,  $\\|\\boldsymbol{A}\\|$ denotes the spectral norm of $\\boldsymbol{A}$, and  $\\sigma_{\\min}(\\boldsymbol{A})$ denotes the minimum singular value of $\\boldsymbol{A}$. For any positive semi-definite matrix $\\boldsymbol{A} \\in \\mathbb{R}^{d' \\times d'}$ and vector $\\boldsymbol{x} \\in \\mathbb{R}^{d'}$,  $\\|\\boldsymbol{x}\\|_{\\boldsymbol{A}}:=\\sqrt{\\boldsymbol{x}^\\top \\boldsymbol{A} \\boldsymbol{x}}$. We use $\\textup{polylog}(\\cdot)$ to denote a polylogarithmic factor in given parameters, and $\\tilde{O}(\\cdot)$ to denote an expression that hides polylogarithmic factors in all problem parameters except $\\delta$ and $\\varepsilon$.\n\n\t\n\t\n\t\\textbf{Representation Learning for Best Arm Identification in Linear Bandits ($\\textup{RepBAI-LB}$).}\n\tAn agent is given a set of arms $\\mathcal{X}:=\\{\\boldsymbol{x}_1,\\dots,\\boldsymbol{x}_n\\} \\subseteq \\mathbb{R}^d$ and $M$ best arm identification tasks. Without loss of generality, we assume that $\\mathcal{X}$ spans $\\mathbb{R}^d$, as done in many prior works~\\cite{fiez2019sequential,katz2020empirical,degenne2020gamification}. For any $\\boldsymbol{x} \\in \\mathcal{X}$, $\\|\\boldsymbol{x}\\|\\leq L_{x}$ for some constant $L_{x}$. For each task $m \\in [M]$, the expected reward of each arm $\\boldsymbol{x} \\in \\mathcal{X}$ is $\\boldsymbol{x}^\\top \\boldsymbol{\\theta}_m$, where $\\boldsymbol{\\theta}_m  \\in \\mathbb{R}^d$ is an unknown reward parameter. Among all tasks, there exists a common underlying feature extractor $\\boldsymbol{B} \\in \\mathbb{R}^{d \\times k}$, which satisfies that for each task $m \\in [M]$, $\\boldsymbol{\\theta}_m=\\boldsymbol{B} \\boldsymbol{w}_m$. Here $\\boldsymbol{B}$ has orthonormal columns, $\\boldsymbol{w}_m \\in \\mathbb{R}^k$ is an unknown prediction parameter, and $M \\gg d \\gg k$. For any $m \\in [M]$, $\\|\\boldsymbol{w}_m\\|\\leq L_{w}$ for some constant $L_{w}$. \n\t\n\tAt each timestep $t$, the agent chooses an arm $\\boldsymbol{x} \\in \\mathcal{X}$ and a task $m \\in [M]$, to sample arm $\\boldsymbol{x}$ in task $m$. Then, she observes a random reward $r_t=\\boldsymbol{x}^\\top \\boldsymbol{\\theta}_m+\\eta_t=\\boldsymbol{x}^\\top \\boldsymbol{B} \\boldsymbol{w}_m+ \\eta_t$, where $\\eta_t$ is an independent, zero-mean and sub-Gaussian noise. For simplicity of analysis, we assume that $\\mathbb{E}[\\eta_t^2]=1$, which can be easily relaxed by using a more carefully-designed estimator in our algorithm.\n\tGiven a confidence parameter $\\delta \\in (0,1)$, the agent aims to identify the best arms $\\boldsymbol{x}^{*}_m:=\\operatornamewithlimits{argmax}_{\\boldsymbol{x} \\in \\mathcal{X}} \\boldsymbol{x}^\\top \\boldsymbol{\\theta}_m$ for all tasks $m \\in [M]$ with probability at least $1-\\delta$, using as few samples as possible. We define sample complexity as the total number of samples used over all tasks, which is the performance metric considered in our paper.\n\t\n\tTo efficiently learn the underlying low-dimensional representation,\n\twe make the following standard assumptions.\n\t\n\t\\begin{assumption}[Diverse Tasks] \\label{assumption:diverse_task}\n\t\tWe assume that $\\sigma_{\\min}(\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{w}_m \\boldsymbol{w}_m^\\top) = \\Omega(\\frac{1}{k})$.\n\t\\end{assumption}\n\t\n\tThis assumption indicates that the prediction parameters $\\boldsymbol{w}_1,\\dots,\\boldsymbol{w}_{M}$  are uniformly spread out in all directions of $\\mathbb{R}^k$, which was also assumed in~\\cite{du2020few,tripuraneni2021provable,yang2020impact}, and is necessary for recovering the feature extractor $\\boldsymbol{B}$.\n\t\n\tFor any distribution $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}$ and $\\boldsymbol{B} \\in \\mathbb{R}^{d \\times k}$, let $\\boldsymbol{A}(\\boldsymbol{\\lambda}, \\boldsymbol{B}):=\\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}$.\n\tFor any task $m \\in [M]$, let\n\t\\begin{align*}\n\t\t\\boldsymbol{\\lambda}^*_m := & \\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top (\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x}) \\|^2_{\\boldsymbol{A}(\\boldsymbol{\\lambda}, \\boldsymbol{B})^{-1}} }{ ((\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m)^2 } .\n\t\\end{align*}\n\tHere $\\boldsymbol{\\lambda}^*_m$ denotes the optimal sample allocation that minimizes prediction error of arms (i.e., the solution of G-optimal design~\\cite{pukelsheim2006optimal}) under the underlying low-dimensional representation.\n\t\n\t\\begin{assumption}[Eigenvalue of G-optimal Design Matrix] \\label{assumption:lambda^*_B_x_x_B_invertible}\n\t\tFor any task $m \\in [M]$, $\\sigma_{\\min}(\\boldsymbol{A}(\\boldsymbol{\\lambda}^*_m, \\boldsymbol{B})) \\geq \\omega$ for some constant $\\omega>0$.\n\t\\end{assumption}\n\t\n\tThis assumption implies that the covariance matrix $\\boldsymbol{A}(\\boldsymbol{\\lambda}^*_m, \\boldsymbol{B})$ under the optimal sample allocation is invertible, which is necessary for estimating $\\boldsymbol{w}_m$.  \\looseness=-1\n\t\n\t\\textbf{Representation Learning for Best Policy Identification in Contextual Linear Bandits ($\\textup{RepBPI-CLB}$).}\n\tIn this problem, there are  a context space $\\mathcal{S}$, an action space $\\mathcal{A}$, a feature mapping $\\boldsymbol{\\phi}(\\cdot,\\cdot):\\mathcal{S}\\times\\mathcal{A} \\mapsto \\mathbb{R}^d$ and an \\emph{unknown} context distribution $\\mathcal{D} \\in \\triangle_{\\mathcal{S}}$. For any $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$, $\\|\\boldsymbol{\\phi}(s,a)\\|\\leq L_{\\phi}$ for some constant $L_{\\phi}$.\n\tAn agent needs to solve $M$ best policy identification tasks. For each task $m \\in [M]$, the expected reward of each context-action pair $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$ is $\\boldsymbol{\\phi}(s,a)^\\top \\boldsymbol{\\theta}_m$, where $\\boldsymbol{\\theta}_m \\in \\mathbb{R}^d$ is an unknown reward parameter. Similar to $\\textup{RepBAI-LB}$, there exists a global feature extractor $\\boldsymbol{B} \\in \\mathbb{R}^{d \\times k}$ with orthonormal columns, such that for each task $m \\in [M]$, $\\boldsymbol{\\theta}_m=\\boldsymbol{B} \\boldsymbol{w}_m$. Here $\\boldsymbol{w}_m \\in \\mathbb{R}^k$ is an unknown prediction parameter, $\\|\\boldsymbol{w}_m\\|\\leq L_{w}$ for any $m \\in [M]$, and $M \\gg d \\gg k$.\n\t\n\tAt each timestep $t$, the agent first observes a random context $s_t$, which is i.i.d. drawn from $\\mathcal{D}$. Then, she selects an action $a_t \\in \\mathcal{A}$ and a task $m \\in [M]$, to sample action $a_t$ in context $s_t$ under task $m$. After sampling, she observes a random reward $r_t=\\boldsymbol{\\phi}(s_t,a_t)^\\top \\boldsymbol{\\theta}_m+\\eta_t=\\boldsymbol{\\phi}(s_t,a_t)^\\top \\boldsymbol{B} \\boldsymbol{w}_m+ \\eta_t$, where $\\eta_t$ is an independent, zero-mean and $1$-sub-Gaussian noise. \n\t\n\tWe define a policy $\\pi$ as a mapping from $\\mathcal{S}$ to $\\mathcal{A}$. For each task $m \\in [M]$, we say a policy $\\hat{\\pi}_m$ is $\\varepsilon$-optimal if\\looseness=-1\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\max_{a \\in \\mathcal{A}} \\sbr{\\boldsymbol{\\phi}(s,a) - \\boldsymbol{\\phi}(s,\\hat{\\pi}_m(s) }^\\top \\boldsymbol{\\theta}_m } \\leq \\varepsilon .\n\t\\end{align*}\n\tGiven a confidence parameter $\\delta \\in (0,1)$ and an accuracy parameter $\\varepsilon>0$, the goal of the agent is to identify an $\\varepsilon$-optimal policy $\\hat{\\pi}_m$ for each task $m \\in [M]$ with probability at least $1-\\delta$, and minimize the number of samples used, i.e., sample complexity.\n\t\n\tWe also make two standard assumptions for $\\textup{RepBPI-CLB}$:  Assumption~\\ref{assumption:diverse_task} and the following assumption on the context distribution and context-action features.\n\t\n\t\\begin{assumption}\\label{assumption:bpi_rho_E_D_is_finite}\n\t\tThere exists some $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{A}}$ such that  \n\t\t$$\n\t\t\\sigma_{\\min}\\sbr{\\sum_{a \\in \\mathcal{A}} \\lambda(a) \\mathbb{E}_{s \\sim \\mathcal{D}}\\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top }} \\geq \\nu \n\t\t\\vspace*{-1em}\n\t\t$$ \n\t\\end{assumption}\n\tfor some constant $\\nu>0$.\n\t\n\tAssumption~\\ref{assumption:bpi_rho_E_D_is_finite} manifests that there exists at least one sample allocation, under which the expected covariance matrix with respect to random contexts is invertible. This assumption enables one to reveal the feature extractor $\\boldsymbol{B}$, despite stochastic and varying contexts. \n\tNote that Assumption~\\ref{assumption:bpi_rho_E_D_is_finite} only assumes the existence of a feasible sample allocation, rather than the knowledge of this sample allocation. \n\n\t\n\t\n\tIt is worth mentioning that in this work, we do not assume that we can sample arbitrary vectors in an ellipsoid\/sphere as in \\cite{yang2020impact,yang2022nearly}, or assume that each arm (action) has zero mean and identity covariance as in \\cite{tripuraneni2021provable}. In contrast, we allow arbitrary shapes of arms (actions), and efficiently allocate samples according to their different shapes. Moreover, we do not assume prior knowledge of the context distribution as in \\cite{huang2015efficient,li2022instance}.\n\tInstead, we design an effective scheme to estimate the context distribution, and carefully bound the estimation error in our analysis. \n\t\n\tBelow we will introduce our algorithms and results. We defer all our proofs to Appendix due to space limit.\n\t\n\t\n\t\\section{Representation Learning for Best Arm Identification in Linear Bandits}\n\t\n\t\n\t\n\t\n\tIn this section, we design a computationally efficient algorithm $\\mathtt{DouExpDes}$ for $\\textup{RepBAI-LB}$, which performs double delicate experimental designs to recover the feature extractor and distinguish the best arms using low-rank representations. Furthermore, we provide sample complexity guarantees that mainly depend on the underlying low dimension.\n\t\n\n\tTo better describe our algorithm, we first introduce the notion of \\emph{experimental design}.\n\tExperimental design is an important problem in statistics~\\cite{pukelsheim2006optimal}. Consider a set of feature vectors and an unknown linear regression parameter. Sampling each feature vector will produce a noisy feedback of the inner-product of this feature vector and the unknown parameter.\n\tExperimental design investigates how to schedule samples to maximize the statistical power of estimating the unknown parameter. \n\n\tIn our algorithm, we mainly use two popular types of experimental design, i.e., \\emph{E-optimal design}, which minimizes the spectral norm of the inverse of sample covariance matrix, and \\emph{G-optimal design}, which minimizes the maximum prediction error for feature vectors.  \\looseness=-1\n\t\n\t\n\t\n\t\\subsection{Algorithm $\\mathtt{DouExpDes}$}\n\t\n\t\n\t\n\t\n\n\t\n\tNow we present our algorithm $\\mathtt{DouExpDes}$, whose pseudo-code is provided in Algorithm~\\ref{alg:repbailb}. $\\mathtt{DouExpDes}$ is a phased elimination algorithm, which first conducts the E-optimal design to optimally schedule samples for learning the feature extractor $\\boldsymbol{B}$, and then performs the G-optimal design with low-dimensional representations to eliminate suboptimal arms.\n\n\n\t\n\t$\\mathtt{DouExpDes}$ uses a \\emph{rounding procedure} $\\mathtt{ROUND}$~\\cite{allen2017near,fiez2019sequential}, which transforms a given continuous sample allocation (design) into a discrete sample sequence and maintains important properties (e.g., E-optimality and G-optimality) of the design. $\\mathtt{ROUND}(\\{(\\boldsymbol{q}_i,\\boldsymbol{Q}_i)\\}_{i=1}^{n'}, \\boldsymbol{\\lambda}, \\zeta, N)$ takes $n'$ arm-matrix pairs $(\\boldsymbol{q}_1,\\boldsymbol{Q}_1),\\dots,(\\boldsymbol{q}_{n'},\\boldsymbol{Q}_{n'}) \\in \\mathcal{X} \\times \\mathbb{R}^{{d'} \\times {d'}}$, a distribution $\\boldsymbol{\\lambda} \\in \\triangle_{\\{\\boldsymbol{q}_1,\\dots,\\boldsymbol{q}_{n'}\\}}$, a rounding approximation parameter $\\zeta>0$, and the number of samples $N$ such that $N \\geq \\frac{180d'}{\\zeta^2}$ as inputs. It will return a sample sequence $\\boldsymbol{s}_1, \\dots, \\boldsymbol{s}_N \\in \\mathcal{X}$, which correspond to feature matrices $\\boldsymbol{S}_1, \\dots, \\boldsymbol{S}_N \\in \\{\\mathcal{Q}_1,\\dots,\\mathcal{Q}_{n'}\\}$, and $\\sum_{j=1}^N \\boldsymbol{S}_j$ has similar properties as the covariance matrix of the inputted design $N \\sum_{i=1}^{n'} \\lambda(\\boldsymbol{q}_i) \\boldsymbol{Q}_i$ (see Appendix~\\ref{apx:rounding_procedure} for more details).\n\t\n\t\n\t\n\tThe procedure of $\\mathtt{DouExpDes}$ is as follows. At the beginning, $\\mathtt{DouExpDes}$ performs the E-optimal design with raw representations, to plan an optimal sample allocation $\\boldsymbol{\\lambda}^E$  for the purpose of recovering the feature extractor $\\boldsymbol{B}$ (Line~\\ref{line:bai_E_optimal_design}). \n\tThen, $\\mathtt{DouExpDes}$ calls $\\mathtt{ROUND}$ to convert the E-optimal sample allocation $\\boldsymbol{\\lambda}^E$ into a discrete sample batch $\\bar{\\boldsymbol{x}}_1,\\dots,\\bar{\\boldsymbol{x}}_p$, which satisfies that\n\t$$\n\t\\bigg\\| \\Big(\\sum_{j=1}^p \\bar{\\boldsymbol{x}}_j \\bar{\\boldsymbol{x}}_j^\\top \\Big)^{-1} \\bigg\\| \\leq (1+\\zeta) \\bigg\\| \\Big(p \\sum_{i=1}^n \\lambda^E(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\Big)^{-1} \\bigg\\| .\n\t$$\n\tNext, $\\mathtt{DouExpDes}$ enters multiple phases, and maintains a candidate arm set $\\hat{\\mathcal{X}}_{t,m}$ for each task. \n\tThe specific value of $T_t$ in Line~\\ref{line:bai_T_t} is presented in Eq.~\\eqref{eq:value_T_t} of Appendix~\\ref{apx:bai_feature_recover}\n\t\n\t\n\t\\begin{algorithm}[t]\n\t\t\\caption{$\\mathtt{DouExpDes}$ (Double Experimental Design)} \\label{alg:repbailb}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\STATE {\\bfseries Input:} $\\mathcal{X}$, $\\delta$, rounding procedure $\\mathtt{ROUND}$, rounding approximation parameter $\\zeta:=\\frac{1}{10}$, and the size of sample batch $p:= \\frac{180d}{\\zeta^2}$.\n\t\t\n\t\t\t\\STATE Let $\\boldsymbol{\\lambda}^{E}$ and $\\rho^{E}$ be the optimal solution and the optimal value of the E-optimal design optimization:\n\t\t\t$$\n\t\t\t\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\Big\\| \\big( \\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\big)^{-1} \\Big\\|\n\t\t\t\\vspace*{-0.5em}\n\t\t\t$$\\label{line:bai_E_optimal_design}\\\\\n\t\t\t\\STATE $\\bar{\\boldsymbol{x}}_1,\\dots,\\bar{\\boldsymbol{x}}_p \\leftarrow \\mathtt{ROUND}(\\{(\\boldsymbol{x}_i, \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top)\\}_{i=1}^{n}, \\boldsymbol{\\lambda}^{E}, \\zeta, p)$ \\label{line:bai_E_optimal_round}\n\t\t\t\\STATE $\\hat{\\mathcal{X}}_{1,m} \\leftarrow \\mathcal{X}$ for any $m \\in [M]$. $\\delta_t \\leftarrow \\frac{\\delta}{2 t^2}$ for any $t \\geq 1$\\; \n\t\t\t\\FOR{phase $t=1,2,\\dots$}\n\t\t\n\t\t\t\\STATE $T_t \\leftarrow \\lceil \\frac{c_1 \\sbr{1+\\zeta}^3 (\\rho^E)^2 k^4 L_{x}^4 L_{w}^4}{M} \\max\\{2^{2t},\\ \\frac{L_x^4}{\\omega^2}\\}\\cdot$\\\\$ \\textup{polylog}(\\zeta,\\rho^E,p,k,L_{x},L_{w},\\frac{1}{\\delta_t}, \\frac{1}{\\omega}) \\rceil$, where $c_1$ is an absolute constant \\label{line:bai_T_t}\n\t\t\t\\STATE $\\hat{\\boldsymbol{B}}_t \\leftarrow \\mathtt{FeatRecover}(T_t, \\{\\bar{\\boldsymbol{x}}_i\\}_{i \\in [p]})$\\;\n\t\t\n\t\t\t\\STATE $\\{\\hat{\\mathcal{X}}_{t+1,m}\\}_{m \\in [M]} \\leftarrow$\\\\$ \\mathtt{EliLowRep} (t, \\mathcal{X}, \\{\\hat{\\mathcal{X}}_{t,m}\\}_{m \\in [M]}, \\delta_t, \\mathtt{ROUND}, \\zeta, \\hat{\\boldsymbol{B}}_t)$\n\t\t\t\\IF{$|\\hat{\\mathcal{X}}_{t+1,m}|=1$, $\\forall m \\in [M]$}\n\t\t\t\\STATE {\\bfseries return}  $\\hat{\\mathcal{X}}_{t+1,m}$ for all tasks $m \\in [M]$\\;\n\t\t\t\\ENDIF\n\t\t\t\\ENDFOR\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\t\\begin{algorithm}[t] \n\t\t\\caption{$\\mathtt{FeatRecover}(T,  \\{\\bar{\\boldsymbol{x}}_i\\}_{i \\in [p]})$} \\label{alg:feat_recover}\n\t\t\\begin{algorithmic}[1]\n\t\t\n\t\t\t\\FOR{task $m \\in [M]$} \\label{line:bai_stage2_sample_start}\n\t\t\t\\FOR{round $j \\in [T]$ }\n\t\t\t\\FOR{arm $i \\in [p]$}\n\t\t\t\\STATE Sample $\\bar{\\boldsymbol{x}}_i$, and observe random reward $\\alpha_{m,j,i}$\\; \\label{line:bai_stage2_sample}\n\t\t\t\\ENDFOR\n\t\t\t\\STATE $\\tilde{\\boldsymbol{\\theta}}_{m,j} \\leftarrow (\\sum_{i=1}^{p} \\bar{\\boldsymbol{x}}_i \\bar{\\boldsymbol{x}}_i^\\top)^{-1} \\sum_{i=1}^{p} \\bar{\\boldsymbol{x}}_i \\alpha_{m,j,i}$\n\t\t\t\\ENDFOR\n\t\t\t\\ENDFOR \\label{line:bai_stage2_sample_end}\n\t\t\t\\STATE $\\boldsymbol{Z} \\leftarrow \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{\\theta}}_{m,j} (\\tilde{\\boldsymbol{\\theta}}_{m,j})^\\top - (\\sum_{i=1}^{p} \\bar{\\boldsymbol{x}}_i \\bar{\\boldsymbol{x}}_i^\\top)^{-1} $ \\label{line:bai_Z_t}\\;\n\t\t\t\\STATE Perform SVD decomposition on $\\boldsymbol{Z}$, and let $\\hat{\\boldsymbol{B}}$ be the top-$k$ left singular vectors of $\\boldsymbol{Z}$. {\\bfseries return} $\\hat{\\boldsymbol{B}}$ \\label{line:svd}\\;\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\t\n\t\\begin{algorithm}[t!] \n\t\t\\caption{$\\mathtt{EliLowRep}(t, \\mathcal{X}\\!,\\! \\{\\hat{\\mathcal{X}}_{m}\\}_{\\! m \\in [M]}, \\delta'\\!, \\mathtt{ROUND},\\! \\zeta,\\! \\hat{\\boldsymbol{B}})$} \\label{alg:elim_low_rep}\n\t\t\\begin{algorithmic}[1]\n\t\t\n\t\t\t\\FOR{task $m \\in [M]$}\n\t\t\t\\STATE Let $\\boldsymbol{\\lambda}^{G}_{m}$ and $\\rho^{G}_{m}$ be the optimal solution and the optimal value of the G-optimal design optimization:\n\t\t\t$$\n\t\t\t\\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x},\\boldsymbol{x}' \\in \\hat{\\mathcal{X}}_{m}} \\nbr{ \\hat{\\boldsymbol{B}}^\\top (\\boldsymbol{x}-\\boldsymbol{x}') }^2_{\\boldsymbol{A}(\\boldsymbol{\\lambda},\\hat{\\boldsymbol{B}})^{-1}}\n\t\t\t\\vspace*{-0.5em}\n\t\t\t$$ \\label{line:bai_G_optimal_design}\\\\\n\t\t\t\\STATE $N_{m} \\leftarrow \\lceil \\max \\{ 32 (1+\\zeta) 2^{2t}  \\rho^{G}_{m} \\log (\\frac{4n^2 M}{\\delta'}),$ $\\frac{180 k}{\\zeta^2} \\} \\rceil$\\;\n\t\t\t\\STATE $\\boldsymbol{z}_{m,1},\\dots,\\boldsymbol{z}_{m,N_{m}} \\leftarrow$\\\\$ \\mathtt{ROUND}(\\{(\\boldsymbol{x}_i, \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}})\\}_{i=1}^{n}, \\boldsymbol{\\lambda}^{G}_{m}, \\zeta, N_{m})$ \\label{line:bai_stage3_round}\\;\n\t\t\t\\STATE Sample the arms $\\boldsymbol{z}_{m,1},\\dots,\\boldsymbol{z}_{m,N_{m}} \\in \\mathcal{X}$, and observe random rewards $r_{m,1}, \\dots, r_{m,N_{m}}$\\; \\label{line:bai_stage3_sample}\n\t\t\t\\STATE Let $\\tilde{\\boldsymbol{z}}_{m,j} := \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{z}_{m,j}$ for any $j \\in [N_{m}]$\n\t\t\t\\STATE $\\hat{\\boldsymbol{w}}_{m} \\leftarrow ( \\sum_{j=1}^{N_{m}}  \\tilde{\\boldsymbol{z}}_{m,j} \\tilde{\\boldsymbol{z}}_{m,j}^\\top )^{-1}  \\sum_{j=1}^{N_{m}}  \\tilde{\\boldsymbol{z}}_{m,j} r_{m,j}$ \\label{line:bai_est_theta}\\;\n\t\t\t\\STATE $\\hat{\\boldsymbol{\\theta}}_{m} \\leftarrow \\hat{\\boldsymbol{B}} \\hat{\\boldsymbol{w}}_{m}$ \\label{line:bai_est_w}\\;\n\t\t\t\\STATE $\\hat{\\mathcal{X}}'_{m} \\leftarrow \\hat{\\mathcal{X}}_{m} \\setminus \\{ \\boldsymbol{x} \\in \\hat{\\mathcal{X}}_{m} \\ | \\ \\exists \\boldsymbol{x}' \\in \\hat{\\mathcal{X}}_{m}: (\\boldsymbol{x}'-\\boldsymbol{x})^\\top \\hat{\\boldsymbol{\\theta}}_{m} > 2^{-t} \\}$ \\label{line:bai_elimination}\\;\n\t\t\t\\ENDFOR\n\t\t\t\\STATE {\\bfseries return}  $\\{\\hat{\\mathcal{X}}'_{m}\\}_{m \\in [M]}$\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\tIn each phase $t$, \n\t$\\mathtt{DouExpDes}$ first calls subroutine $\\mathtt{FeatRecover}$ to recover the feature extractor $\\boldsymbol{B}$. In $\\mathtt{FeatRecover}$ (Algorithm~\\ref{alg:feat_recover}), we repeatedly sample $\\bar{\\boldsymbol{x}}_1,\\dots,\\bar{\\boldsymbol{x}}_p$ in all tasks, and construct an estimator $\\boldsymbol{Z}$ for  $\\frac{1}{M} \\sum_{i=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top$, which contains the information of underlying reward parameters (Line~\\ref{line:bai_Z_t}). Then, we perform SVD  on $\\boldsymbol{Z}$ and obtain the estimated feature extractor $\\hat{\\boldsymbol{B}}$ (Line~\\ref{line:svd}). \n\t\n\tThen, $\\mathtt{DouExpDes}$ calls subroutine $\\mathtt{EliLowRep}$ to eliminate suboptimal arms using low-dimensional representations.\n\tIn $\\mathtt{EliLowRep}$ (Algorithm~\\ref{alg:elim_low_rep}), we conduct the G-optimal design with the reduced-dimensional representations $\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x}$, and obtain sample allocation $\\boldsymbol{\\lambda}^{G}_{m}$ for each task (Line~\\ref{line:bai_G_optimal_design}). We further use $\\mathtt{ROUND}$ to transform $\\boldsymbol{\\lambda}^{G}_{m}$ into a sample sequence $\\boldsymbol{z}_{m,1},\\dots,\\boldsymbol{z}_{m,N_{m}}$, which satisfies that\n\t\\begin{align*}\n\t\t& \\max_{\\boldsymbol{x}, \\boldsymbol{x}' \\in \\hat{\\mathcal{X}}_{m}} \\nbr{\\boldsymbol{x}-\\boldsymbol{x}'}^2_{\\sbr{\\sum_{j=1}^{N_{m}} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{z}_{m,j} \\boldsymbol{z}_{m,j}^\\top \\hat{\\boldsymbol{B}}}^{-1}} \n\t\t\\\\\n\t\t\\leq & (1+\\zeta) \\max_{\\boldsymbol{x}, \\boldsymbol{x}' \\in \\hat{\\mathcal{X}}_{m}} \\nbr{\\boldsymbol{x}-\\boldsymbol{x}'}^2_{\\sbr{N_{m} \\sum_{i=1}^{n} \\lambda^{G}_{m}(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}}^{-1}} .\n\t\\end{align*}\n\tAfter sampling this sequence, we build estimators $\\hat{\\boldsymbol{w}}_{t,m}$ and $\\hat{\\boldsymbol{\\theta}}_{t,m}$ for the underlying prediction parameter $\\boldsymbol{w}_m$ and reward parameter $\\boldsymbol{\\theta}_m$, respectively (Lines~\\ref{line:bai_est_theta}-\\ref{line:bai_est_w}). Then, we discard the arms that show large gaps to the estimated optimal arm for each task (Line~\\ref{line:bai_elimination}). \n\t\n\t\n\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\n\t\n\t\\subsection{Theoretical Performance of $\\mathtt{DouExpDes}$}\n\t\n\tIn this subsection, we provide sample complexity guarantees for $\\mathtt{DouExpDes}$.\n\t%\n\tTo formally present our sample complexity,\n\twe first revisit existing results for conventional single-task best arm identification in linear bandits (BAI-LB).\n\t\n\tFor a single-task BAI-LB instance with arm set $\\mathcal{X} \\in \\mathbb{R}^d$ and underlying reward parameter $\\boldsymbol{\\theta} \\in \\mathbb{R}^{d}$, the instance-dependent hardness is defined as~\\cite{fiez2019sequential}\n\t\\begin{align*}\n\t\t\\rho^{S}(\\mathcal{X}, \\boldsymbol{\\theta}) := & \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}\\}} \\frac{\\| \\boldsymbol{x}^{*} - \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top }^{-1}} }{ ((\\boldsymbol{x}^{*} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta})^2 } ,\n\t\\end{align*}\n\tand the best known sample complexity result is $\\tilde{O}(\\rho^{S}(\\mathcal{X}, \\boldsymbol{\\theta}) \\log(\\frac{1}{\\delta}))=\\tilde{O}(\\frac{d}{(\\Delta^{S}_{\\min})^2} \\log(\\frac{1}{\\delta}))$~\\cite{fiez2019sequential}.\n\tHere $\\boldsymbol{x}^{*}:=\\operatornamewithlimits{argmax}_{\\boldsymbol{x} \\in \\mathcal{X}} \\boldsymbol{x}^\\top \\boldsymbol{\\theta}$ denotes the best arm, and $\\Delta^{S}_{\\min}:=\\min_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}\\}}(\\boldsymbol{x}^{*} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}$ refers to the minimum reward gap.\n\t\n\tIt can be seen that a naive algorithm for RepBAI-LB is to run an existing single-task BAI-LB algorithm~\\cite{fiez2019sequential,katz2020empirical} to solve $M$ tasks independently. Then, the sample complexity of such naive algorithm is\n\t\\begin{align}\n\t\t\\!\\!\\!\\! \\tilde{O} \\! \\sbr{ \\sum_{m=1}^{M} \\rho^{S}(\\mathcal{X}, \\boldsymbol{\\theta}_m) \\log\\sbr{ \\frac{1}{\\delta} } \\!} \\!\\!=\\! \\tilde{O}\\sbr{\\! \\frac{Md}{\\Delta_{\\min}^2} \\log\\sbr{ \\frac{1}{\\delta} } \\!} \\!, \\label{eq:bai_naive_sample_complexity}\n\t\\end{align}\n\twhere $\\Delta_{\\min}:=\\min_{m \\in [M], \\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_m\\}}(\\boldsymbol{x}^{*}_m - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m$ denotes the minimum reward gap among all tasks. In the following, we take Eq.~\\eqref{eq:bai_naive_sample_complexity} as the baseline to demonstrate the power of representation learning.\n\t\n\tNow we state the sample complexity for $\\mathtt{DouExpDes}$.\n\t\\begin{theorem} \\label{thm:bai_ub}\n\t\tWith probability at least $1-\\delta$, algorithm $\\mathtt{DouExpDes}$   returns the best arms $\\boldsymbol{x}^{*}_{m}$ for all tasks $m \\in [M]$, and the number of samples used is bounded by\n\t\t\\begin{align*}\n\t\t\t\\tilde{O}  \\bigg( & \\sum_{m=1}^{M} \\! \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\! \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\!\\!\\!\\!\\! \\frac{\\| \\boldsymbol{B}^\\top (\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x}) \\|^2_{\\boldsymbol{A}(\\boldsymbol{\\lambda},\\boldsymbol{B})^{-1}} }{ ((\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m)^2 } \\! \\log\\Big(\\frac{1}{\\delta}\\Big)\n\t\t\t\\\\& \n\t\t\t+  (\\rho^E)^2 d k^4 L_{x}^2 L_{w}^2 D \\log^4 \\Big(\\frac{1}{\\delta}\\Big) \\bigg)\n\t\t\t\\\\\n\t\t\t= & \\ \\tilde{O} \\bigg(  \\frac{M k}{\\Delta_{\\min}^2}  \\log\\Big(\\frac{1}{\\delta}\\Big) + (\\rho^E)^2 d k^4  L_{x}^2 L_{w}^2  D \\log^4 \\Big(\\frac{1}{\\delta}\\Big)  \\bigg) ,\n\t\t\\end{align*}\n\t\twhere\n\t\t$D:=\\max\\{ \\frac{1}{\\Delta_{\\min}^2} ,\\ \\frac{L_{x}^4}{\\omega^2} \\}$.\n\t\\end{theorem}\n\t\n\t\\textbf{Remark 1.} In the above theorem, the first term, $\\sum_{m=1}^{M}  \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top (\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x}) \\|^2_{\\boldsymbol{A}(\\boldsymbol{\\lambda},\\boldsymbol{B})^{-1}} }{ ((\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m)^2 }=O(\\frac{M k}{\\Delta_{\\min}^2})$, is the instance-dependent hardness of $M$ $k$-dimensional linear bandit problems with arm set $\\tilde{\\mathcal{X}}:=\\{\\boldsymbol{B}^\\top \\boldsymbol{x}: \\boldsymbol{x} \\in \\mathcal{X}\\} \\subseteq \\mathbb{R}^k$ and underlying reward parameters $\\boldsymbol{w}_1,\\dots,\\boldsymbol{w}_{M} \\in \\mathbb{R}^k$. This term only depends on the reduced dimension $k$, instead of $d$. \n\tIn other words, it is an essential price that is needed for solving $M$ low-dimensional tasks, even if one knows the feature extractor $\\boldsymbol{B}$. \n\tThe second term $(\\rho^E)^2 d k^4 L_{x}^2 L_{w}^2 D$, which depends on the raw dimension $d$, is a cost paid for learning the feature extractor. Note that since this term does not contain $M$, the cost for learning the underlying features is paid only once, rather than for all tasks. \\looseness=-1\n\t\n\tWhen $M \\gg d \\gg k$, the first term dominates the bound. This indicates that algorithm $\\mathtt{DouExpDes}$ effectively learns the low-dimensional representation, and exploits the intrinsic problem structure to\n\n\treduce the sample complexity from $\\tilde{O}(\\frac{M d}{\\Delta_{\\min}^2} \\log (\\frac{1}{\\delta}))$ (i.e., learning each task independently)\n\tto only $\\tilde{O}(\\frac{M k}{\\Delta_{\\min}^2} \\log (\\frac{1}{\\delta}))$. Our result corroborates the benefits of representation learning for multi-task pure exploration.\n\t\n\t\\textbf{Analytical Novelty.}\n\tWe highlight several novelties in the analysis of Theorem~\\ref{thm:bai_ub} as follows. (i) We develop a delicate  concentration inequality for $\\|\\boldsymbol{Z} - \\frac{1}{M} \\sum_{i=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top\\|$, by bounding $\\|(\\sum_{i=1}^{p} \\bar{\\boldsymbol{x}}_i \\bar{\\boldsymbol{x}}_i^{\\top})^{-1}\\|$ via the E-optimality of the sample batch $\\bar{\\boldsymbol{x}}_1,\\dots,\\bar{\\boldsymbol{x}}_p$ and applying the matrix Bernstern inequality with truncated noises. Then, we employ the Davis-Kahan sin $\\theta$ Theorem~\\cite{bhatia2013matrix} to bound the estimation error of $\\hat{\\boldsymbol{B}}$\n\n\tincurred from the SVD decomposition on $\\boldsymbol{Z}$.\n\t(ii) Furthermore, we decompose the prediction error $\\boldsymbol{x}^\\top (\\hat{\\boldsymbol{\\theta}}_{m}-\\boldsymbol{\\theta}_m)$ into three parts, i.e., the sampling variance and bias for learning $\\hat{\\boldsymbol{w}}_{m}$ with estimated representation $\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x}$, and the estimation error of $\\hat{\\boldsymbol{B}}$. (iii) To interpret our sample complexity by the essential instance-dependent hardness with true representation $\\boldsymbol{B}^\\top \\boldsymbol{x}$, we establish a connection between the actual effective dimension $\\| \\boldsymbol{B}^\\top \\boldsymbol{x} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} }$ and our estimated effective dimension $\\| \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}} }^{-1} }$.\n\t\n\t\n\t\\vspace*{-0.5em}\n\t\\section{Representation Learning for Best Policy Identification in Contextual Linear Bandits}\n\t\n\t\n\tIn this section, we turn to contextual linear bandits.\n\tDifferent from prior contextual linear bandit works, e.g.,~\\cite{huang2015efficient,li2022instance}, here we do not assume any knowledge of context distribution. \n\tAs a result, our $\\textup{RepBPI-CLB}$ problem faces several unique challenges: (i) how to plan an efficient sample allocation for recovering the feature extractor in advance under an \\emph{unknown} context distribution, and (ii) how to construct an estimator for the feature extractor with a partially observed context space.\n\t\n\n\t\n\n\tWe propose algorithm $\\mathtt{C \\hyphen DouExpDes}$, which first (i) efficiently estimates the context distribution and conducts experimental designs under the estimated context distribution, and then (ii) builds a delicate estimator for the feature extractor using instantaneous contexts. \n\tMoreover, we also establish a sample complexity guarantee for $\\mathtt{C \\hyphen DouExpDes}$, which mainly depends on the low dimension of the common representation among tasks.\n\t\n\t\\vspace*{-0.5em}\n\t\\subsection{Algorithm $\\mathtt{C \\hyphen DouExpDes}$}\n\t\n\t\\begin{algorithm}[t]\n\t\t\\caption{$\\mathtt{C \\hyphen DouExpDes}$ (Contextual Double Experimental Design)} \\label{alg:repbpiclb}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\STATE {\\bfseries Input:} $\\delta$, $\\varepsilon$, $\\boldsymbol{\\phi}(\\cdot,\\cdot)$, regularization parameter $\\gamma \\geq 1$, rounding procedure $\\mathtt{ROUND}$, rounding approximation parameter $\\zeta:=\\frac{1}{10}$, and the size of sample batch $p:= \\lceil \\frac{c_2 (1+\\zeta)^2 L_{\\phi}^4}{\\nu^2} \\textup{polylog}(\\zeta,M,d,k,L_{\\phi},L_{w},\\gamma,\\frac{1}{\\nu},\\frac{1}{\\delta},\\frac{1}{\\varepsilon}) \\rceil$, where $c_2$ is an absolute constant. \\label{line:bpi_value_p}\n\t\t\n\t\t\n\t\t\t\\STATE $T_0 \\leftarrow \\lceil \\frac{32^2 (1+\\zeta)^2 L_{\\phi}^4}{\\nu^2} \\log^2 (\\frac{20d |\\mathcal{A}|}{\\delta}) \\rceil$. $\\hat{\\mathcal{D}} \\leftarrow \\emptyset$\n\t\t\n\t\t\t\\FOR{$\\tau \\in [T_0]$}  \n\t\t\t\\STATE \\hspace*{-0.4em} Observe context $s_{\\tau}$, and randomly sample an action \\label{line:bpi_estimate_context_dis}\\;\n\t\t\t\\STATE $\\hat{\\mathcal{D}} \\leftarrow \\hat{\\mathcal{D}} \\cup \\{s_{\\tau}\\}$\\;\n\t\t\t\\ENDFOR\n\t\t\t\\STATE Let $\\boldsymbol{\\lambda}^{E}_{\\hat{\\mathcal{D}}}$ and $\\rho^{E}_{\\hat{\\mathcal{D}}}$ be the optimal solution and the optimal value of the E-optimal design optimization: \n\t\t\t$$\n\t\t\t\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{A}}} \\Big\\| \\big( \\sum_{a \\in \\mathcal{A}} \\lambda(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}}\\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} \\big)^{-1} \\Big\\|\n\t\t\t\\vspace*{-1em}\n\t\t\t$$ \\label{line:bpi_E_optimal_design}\\\\\n\t\t\t\\STATE $\\{\\bar{a}_i\\}_{i \\in [p]} \\!\\! \\leftarrow \\!\\! \\mathtt{ROUND}(\\{(a, \\mathbb{E}_{\\! s \\sim \\hat{\\mathcal{D}} \\!\\!}\\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top})\\}_{a \\in \\mathcal{A}},$\\\\$ \\boldsymbol{\\lambda}^{E}_{\\hat{\\mathcal{D}}}, \\zeta, p)$ \\label{line:bpi_E_design_round}\\; \n\t\t\n\t\t\n\t\t\t\\STATE $T \\leftarrow \\lceil \\frac{c_3 (1+\\zeta)^2 k^4 L_{\\phi}^4 L_{w}^4 }{ M \\nu^2 \\varepsilon^2 } \\textup{polylog}(\\zeta,d,k,L_{\\phi},L_{w},\\gamma,\\frac{1}{\\nu},$\\\\$\\frac{1}{\\delta},\\frac{1}{\\varepsilon}) \\rceil$, where $c_3$ is an absolute constant \\label{line:bpi_value_T}\n\t\t\t\\STATE $\\hat{\\boldsymbol{B}} \\leftarrow \\mathtt{C \\hyphen FeatRecover}(T, \\{\\bar{a}_i\\}_{i \\in [p]})$\n\t\t\t\\STATE $N \\leftarrow \\lceil \\frac{ (k^2 + \\gamma k L_{w}^2) }{\\varepsilon^2} \\log^4 ({\\frac{  \\gamma k  L_{w} }{\\varepsilon \\delta}} ) \\rceil$\\;\n\t\t\n\t\t\t\\STATE $\\{\\hat{\\boldsymbol{\\theta}}_{m,N}\\}_{m \\in [M]} \\leftarrow \\mathtt{EstLowRep}(N, \\gamma, \\hat{\\boldsymbol{B}})$\n\t\t\t\\STATE {\\bfseries return} $\\hat{\\pi}_m(\\cdot):=\\operatornamewithlimits{argmax}_{a \\in \\mathcal{A}} \\boldsymbol{\\phi}(\\cdot,a)^\\top \\hat{\\boldsymbol{\\theta}}_{m,N}$ for all tasks $m \\in [M]$ \\label{line:bpi_return}\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\t\n\t\\begin{algorithm}[t]\n\t\t\\caption{$\\mathtt{C \\hyphen FeatRecover}(T, \\{\\bar{a}_i\\}_{i \\in [p]})$} \\label{alg:con_feat_recover}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\FOR{task $m \\in [M]$} \\label{line:bpi_stage2_sample_start}\n\t\t\t\\FOR{round $j \\in [T]$}\n\t\t\n\t\t\t\\FOR{arm $i \\in [p]$}\n\t\t\t\\STATE Observe context $s_{m,j,i}^{(1)}$, sample action $\\bar{a}_i$ in task $m$, and observe reward $\\alpha^{(1)}_{m,j,i}$\\; \\label{line:bpi_stage2_sample1}\n\t\t\t\\STATE Observe context $s_{m,j,i}^{(2)}$, sample action $\\bar{a}_i$ in task $m$, and observe reward $\\alpha^{(2)}_{m,j,i}$\\; \\label{line:bpi_stage2_sample2}\n\t\t\t\\ENDFOR\n\t\t\t\\STATE Let $\\boldsymbol{\\phi}^{(\\ell)}_{m,j,i}\\!:=\\!\\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)$, $\\forall i \\!\\in\\! [p]$, $\\forall \\ell \\!\\in\\! \\{1,2\\}$\\;\n\t\t\t\\STATE $\\tilde{\\boldsymbol{\\theta}}^{(\\ell)}_{m,j} \\! \\leftarrow \\!\\! ( \\sum_{i=1}^{p} \\! \\boldsymbol{\\phi}^{(\\ell)}_{m,j,i} {\\boldsymbol{\\phi}^{(\\ell)}_{m,j,i}}^{\\!\\!\\!\\!\\top} \\!)^{-1} \\! \\sum_{i=1}^{p} \\! \\boldsymbol{\\phi}^{(\\ell)}_{m,j,i} \\alpha^{(\\ell)}_{m,j,i}$, $\\forall \\ell \\in \\{1,2\\}$\\;\n\t\t\t\\ENDFOR\n\t\t\n\t\t\t\\ENDFOR  \\label{line:bpi_stage2_sample_end}\n\t\t\t\\STATE $\\boldsymbol{Z} \\leftarrow \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{\\theta}}^{(1)}_{m,j} (\\tilde{\\boldsymbol{\\theta}}^{(2)}_{m,j})^\\top$ \\label{line:bpi_Z}\\;\n\t\t\t\\STATE Perform SVD decomposition on $\\boldsymbol{Z}$, and let $\\hat{\\boldsymbol{B}}$ be the top-$k$ left singular vectors. {\\bfseries return} $\\hat{\\boldsymbol{B}}$\\; \\label{line:bpi_svd}\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\t\n\t\\begin{algorithm}[t]\n\t\t\\caption{$\\mathtt{EstLowRep}(N, \\gamma, \\hat{\\boldsymbol{B}})$} \\label{alg:est_low_rep}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\STATE $\\boldsymbol{\\Sigma}_{m,0} \\leftarrow \\gamma I$ for any $m \\in [M]$\\;\n\t\t\n\t\t\t\\FOR{task $m \\in [M]$}\n\t\t\t\\FOR{timestep $t \\in [N]$}\n\t\t\t\\STATE Observe context $s_{m,t}$\\; \\label{line:bpi_stage3_context}\n\t\t\t\\STATE $a_{m,t} \\leftarrow \\operatornamewithlimits{argmax}_{a \\in \\mathcal{A}} \\| \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,t},a) \\|_{\\boldsymbol{\\Sigma}_{m,t-1}^{-1}}$\\;\n\t\t\t\\STATE Sample action $a_{m,t}$, and observe reward $r_{m,t}$\\; \\label{line:bpi_stage3_sample}\n\t\t\t\\STATE $\\boldsymbol{\\Sigma}_{m,t} \\leftarrow \\boldsymbol{\\Sigma}_{m,t-1} +$\\\\$ \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,t},a_{m,t}) \\boldsymbol{\\phi}(s_{m,t},a_{m,t})^\\top \\hat{\\boldsymbol{B}}$\\;\n\t\t\t\\STATE $\\hat{\\boldsymbol{w}}_{m,t} \\leftarrow \\boldsymbol{\\Sigma}_{m,t}^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) r_{m,\\tau}$ \\label{line:bpi_est_w}\\;\n\t\t\t\\STATE $\\hat{\\boldsymbol{\\theta}}_{m,t} \\leftarrow \\hat{\\boldsymbol{B}} \\hat{\\boldsymbol{w}}_{m,t}$ \\label{line:bpi_est_theta}\\;\n\t\t\t\\ENDFOR\n\t\t\t\\ENDFOR\n\t\t\t\\STATE {\\bfseries return} $\\{\\hat{\\boldsymbol{\\theta}}_{m,N}\\}_{m \\in [M]}$\\;\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\t\n\tAlgorithm~\\ref{alg:repbpiclb} presents the pseudo-code of $\\mathtt{C \\hyphen DouExpDes}$.\n\n\tAt the beginning, $\\mathtt{C \\hyphen DouExpDes}$ uses $T_0$ samples to estimate the context distribution $\\mathcal{D}$ (Line~\\ref{line:bpi_estimate_context_dis}). Then, it performs the E-optimal design under the estimated context distribution $\\hat{\\mathcal{D}}$, and obtains an efficient sample allocation $\\boldsymbol{\\lambda}^{E}_{\\hat{\\mathcal{D}}}$ for the purpose of recovering the feature extractor $\\boldsymbol{B}$ (Line~\\ref{line:bpi_E_optimal_design}). Further, $\\mathtt{C \\hyphen DouExpDes}$ calls the rounding procedure $\\mathtt{ROUND}$ to transform $\\boldsymbol{\\lambda}^{E}_{\\hat{\\mathcal{D}}}$ into a sample batch $\\bar{a}_1,\\dots,\\bar{a}_p$, such tha\n\t\\begin{align*}\n\t\t& \\Big\\| \\big(\\sum_{j=1}^p \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}}\\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_j) \\boldsymbol{\\phi}(s,\\bar{a}_j)^\\top} \\big)^{-1} \\Big\\| \n\t\t\\\\ \n\t\t\\leq & (1+\\zeta) \\Big\\| \\big(p \\sum_{a \\in \\mathcal{A}} \\lambda^{E}_{\\hat{\\mathcal{D}}}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}}\\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} \\big)^{-1} \\Big\\| .\n\t\\end{align*}\n\tThe specific values of $p$ and $T$ in Lines~\\ref{line:bpi_value_p},~\\ref{line:bpi_value_T} are provided in Eq.~\\eqref{eq:value_p} of Appendix~\\ref{apx:bpi_sample_batch_planning} and Eq.~\\eqref{eq:value_T} of Appendix~\\ref{apx:bpi_feat_recover}, respectively.\\looseness=-1\n\t\n\tNext, $\\mathtt{C \\hyphen DouExpDes}$ runs subroutine $\\mathtt{C \\hyphen FeatRecover}$ to estimate the feature extractor $\\boldsymbol{B}$ using the sample batch $\\bar{a}_1,\\dots,\\bar{a}_p$.\n\tIn $\\mathtt{C \\hyphen FeatRecover}$ (Algorithm~\\ref{alg:con_feat_recover}), we repeatedly sample $\\bar{a}_1,\\dots,\\bar{a}_p$ in all tasks with random contexts. \n\tIn Lines~\\ref{line:bpi_stage2_sample1}-\\ref{line:bpi_stage2_sample2}, we sample this batch twice, and the superscripts $(1)$ and $(2)$ denotes the first and second samples, respectively.\n\tAfter sampling, we carefully establish an estimator $\\boldsymbol{Z}$ for the reward parameter related matrix $\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top$, using instantaneous context-action features $\\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_{i})^\\top$.\n\tWe then perform SVD decomposition on $\\boldsymbol{Z}$ to obtain the estimated feature extractor $\\hat{\\boldsymbol{B}}$ (Lines~\\ref{line:bpi_Z}-\\ref{line:bpi_svd}). \n\t\n\n\tThen, $\\mathtt{C \\hyphen DouExpDes}$ calls subroutine $\\mathtt{EstLowRep}$, which adapts existing reward-free-exploration algorithm in \\cite{zanette2021design} with low-rank representations to estimate $\\boldsymbol{\\theta}_m$.\n\tIn $\\mathtt{EstLowRep}$ (Algorithm~\\ref{alg:est_low_rep}), we employ the estimated representation $\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)$  to sample the actions with the maximum uncertainty under the observed contexts.\n\tAfter that, we construct estimators $\\hat{\\boldsymbol{w}}_{m,t}$ and $\\hat{\\boldsymbol{\\theta}}_{m,t}$ for the prediction parameter $\\hat{\\boldsymbol{w}}_m$ and reward parameter $\\hat{\\boldsymbol{\\theta}}_m$ (Lines~\\ref{line:bpi_est_w}-\\ref{line:bpi_est_theta}). \n\tAt last, $\\mathtt{C \\hyphen DouExpDes}$ returns the greedy policy with respect to the estimated reward parameter $\\hat{\\boldsymbol{\\theta}}_{m,N}$ for each task. \n\t\n\t\n\t\n\t\\subsection{Theoretical Performance of $\\mathtt{C \\hyphen DouExpDes}$}\n\t\n\tNext, we establish sample complexity guarantees for algorithm $\\mathtt{C \\hyphen DouExpDes}$. In order to illustrate the advantages of representation learning, we first review existing results for traditional single-task best policy identification in contextual linear bandits (BPI-CLB).\n\tFor a single  BPI-CLB instance with context-action features $\\boldsymbol{\\phi}(s,a) \\in \\mathbb{R}^d$ and reward parameter $\\boldsymbol{\\theta} \\in \\mathbb{R}^{d}$, the  best known sample complexity is $\\tilde{O}(\\frac{d^2}{\\varepsilon^2}\\log(\\frac{1}{\\delta}))$~\\cite{zanette2021design,li2022instance}.\n\t\n\tApparently, if one naively solves the $\\textup{RepBPI-CLB}$ problem by running single-task BPI-CLB algorithms to tackle $M$ tasks independently, one will have a  sample complexity \n\t\\begin{align*}\n\t\t\\vspace*{-0.5em}\n\t\t\\tilde{O}\\bigg( \\frac{Md^2}{\\varepsilon^2} \\log \\Big(\\frac{1}{\\delta}\\Big) \\bigg) ,\n\t\t\\vspace*{-0.5em}\n\t\\end{align*}\n\twhich heavily depends on the raw dimension $d$ of context-action features.\n\tThe goal of representation learning is to leverage the common representation among tasks to alleviate the dependency of dimension and save samples.\n\t\n\tNow we present the sample complexity for $\\mathtt{C \\hyphen DouExpDes}$. \\looseness=-1\n\t\n\t\\begin{theorem} \\label{thm:bpi_ub}\n\t\tWith probability at least $1-\\delta$,  $\\mathtt{C \\hyphen DouExpDes}$  returns an $\\varepsilon$-optimal policy $\\hat{\\pi}_m$ such that\n\t\t$\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}}  [ \\max_{a \\in \\mathcal{A}} (\\boldsymbol{\\phi}(s,a) - \\boldsymbol{\\phi}(s,\\hat{\\pi}_m(s) )^\\top \\boldsymbol{\\theta}_m ] \\leq \\varepsilon\n\t\t$ \n\t\tfor each task $m \\in [M]$, and the number of samples used is \n\t\t\\begin{align*}\n\t\t\t\\tilde{O} \\sbr{ \\frac{ M \\sbr{k^2 + \\gamma k L_{w}^2 } }{\\varepsilon^2} + \\frac{ k^4 L_{\\phi}^8 L_{w}^4 }{ \\nu^4 \\varepsilon^2 } } .\n\t\t\\end{align*}\n\t\\end{theorem}\n\t\\vspace*{-1em}\n\t\n\t\\textbf{Remark 2.} The first term $\\frac{ M (k^2 + \\gamma k L_{w}^2) }{\\varepsilon^2}$ is a cost of identifying optimal policies for $M$ tasks with underlying $k$-dimensional representation $\\boldsymbol{B}^\\top \\boldsymbol{\\phi}(s,a)$. The second term $\\frac{ k^4 L_{\\phi}^8 L_{w}^4 }{ \\nu^4 \\varepsilon^2 }$ is a price paid for learning global feature extractor $\\boldsymbol{B}$ and does not depend on $M$, which indicates that we only need to pay this price once and then enjoy the benefits of dimension reduction for all $M$ tasks.\n\tTheorem~\\ref{thm:bpi_ub} shows that when $M \\gg \\frac{1}{\\nu} \\gg k$, $\\mathtt{C \\hyphen DouExpDes}$ \n\n\tperforms almost as well as an oracle  that knows inherent low-dimensional features $\\boldsymbol{B}^\\top \\boldsymbol{\\phi}(s,a)$. This sample complexity significantly outperforms the baseline result $\\tilde{O}(\\frac{M d^2}{\\varepsilon^2})$ (i.e., solving $M$ tasks independently), and demonstrates the power of representation learning.\n\t\n\t\\textbf{Innovations in Analysis.}\n\tThe proof of Theorem~\\ref{thm:bpi_ub} has several innovations.  (i) We carefully bound the deviation between the estimated context distribution $\\hat{\\mathcal{D}}$ and the true context distribution $\\mathcal{D}$, and utilize the E-optimality of sample batch $\\bar{a}_1,\\dots,\\bar{a}_p$\n\n\tto bound $\\|(\\sum_{i=1}^{p} \\boldsymbol{\\phi}^{(\\ell)}_{m,j,i} {\\boldsymbol{\\phi}^{(\\ell)}_{m,j,i}}^\\top)^{-1}\\|$. (ii) A concentration inequality for $\\|\\boldsymbol{Z}-\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top\\|$ is established, using the bounded $\\|(\\sum_{i=1}^{p} \\boldsymbol{\\phi}^{(\\ell)}_{m,j,i} {\\boldsymbol{\\phi}^{(\\ell)}_{m,j,i}}^\\top)^{-1}\\|$ and matrix Bernstern inequality with truncated noises. Furthermore, an estimation error guarantee for $\\hat{\\boldsymbol{B}}$ is derived by applying the SVD perturbation analysis. (iii) We provide a decomposition of the prediction error $\\boldsymbol{\\phi}(s,a)^\\top (\\hat{\\boldsymbol{\\theta}}_{m,t} - \\boldsymbol{\\theta}_m)$ into three components, including the variance and bias incurred in the sampling for learning $\\hat{\\boldsymbol{w}}_{m,t}$, and the estimation error of $\\hat{\\boldsymbol{B}}$. Then, we bound them via self-normalized concentration inequalities with reduced dimension $k$.\n\t\n\t\n\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\n\t\n\t\\section{Experiments}\n\t\n\t\n\t\n\tIn this section, we present experiments to evaluate the empirical performance of our algorithms.\n\t\n\n\tIn our experiments, we set $\\delta=0.005$, $d=5$, $k=2$ and $M \\in [50,230]$, where $k$ divides $M$. In $\\textup{RepBAI-LB}$, $\\mathcal{X}$ is the canonical basis of $\\mathbb{R}^{d}$. \n\tIn $\\textup{RepBPI-CLB}$, we set $\\varepsilon=0.1$, $|\\mathcal{S}|=5$ and $|\\mathcal{A}|=5$.\n\t$\\mathcal{D}$ is the uniform distribution on $\\mathcal{S}$. For any $s \\in \\mathcal{S}$, $\\{\\boldsymbol{\\phi}(s,a)\\}_{a \\in \\mathcal{A}}$ is the canonical basis of $\\mathbb{R}^d$.  \n\tIn both problems, $\\boldsymbol{B}=[I_k;\\boldsymbol{0}]$, where $I_k$ denotes the $k \\times k$ identity matrix. \n\t$\\boldsymbol{w}_1,\\dots,\\boldsymbol{w}_M$ are divided into $k$ groups, with $\\frac{M}{k}$ same members in each group. \n\tThe members in the $i$-th group ($i \\in [k]$), i.e., $\\boldsymbol{w}_{(M\/k)\\times(i-1)+1},\\dots,\\boldsymbol{w}_{(M\/k)\\times i}$, have $1$ in the $i$-th coordinate and $0$ in all other coordinates. \n\n\n\tFor any $m \\in [M]$, $\\boldsymbol{\\theta}_m=\\boldsymbol{B} \\boldsymbol{w}_m$. We vary $M$ and perform $50$ independent runs  to report the average sample complexity across runs.\n\t\n\t\n\t\\begin{figure}[t]\n\t\t\\centering     \n\t\t\\subfigure[$\\textup{RepBAI-LB}$] { \n\t\t\t\\label{fig:bai}        \n\t\t\t\\includegraphics[width=0.45\\columnwidth]{fig\/BAI.pdf} \n\t\t}    \n\t\t\\subfigure[$\\textup{RepBPI-CLB}$] { \n\t\t\t\\label{fig:bpi}       \n\t\t\t\\includegraphics[width=0.45\\columnwidth]{fig\/BPI.pdf}\n\t\t}\n\t\t\\vspace*{-1em}\n\t\t\\caption{Experimental results for $\\textup{RepBAI-LB}$ and $\\textup{RepBPI-CLB}$. The two figures compare the sample complexities of our algorithms with the naive algorithms which treat each task independently.\n\t\t}\n\t\t\\label{fig:experiments}\n\t\\end{figure} \n\t\n\tFor $\\textup{RepBAI-LB}$, we compare algorithm $\\mathtt{DouExpDes}$ with the baseline $\\mathtt{IndRAGE}$ which runs the state-of-the-art single-task BAI-LB algorithm $\\mathtt{RAGE}$~\\cite{fiez2019sequential} to solve $M$ tasks independently.\n\tFigure~\\ref{fig:bai} shows the empirical results for $\\textup{RepBAI-LB}$. From Figure~\\ref{fig:bai}, we can see that $\\mathtt{DouExpDes}$ has a better sample complexity than $\\mathtt{IndRAGE}$, and as the number of tasks $M$ increases, the sample complexity of $\\mathtt{DouExpDes}$ increases at a lower rate than that of $\\mathtt{IndRAGE}$. This demonstrates that $\\mathtt{DouExpDes}$ effectively utilize the shared representation among tasks to reduce the number of samples needed for multi-task learning.\n\t\n\tFor $\\textup{RepBPI-CLB}$, our algorithm $\\mathtt{C \\hyphen DouExpDes}$ is compared with the baseline $\\mathtt{IndRFLinUCB}$, which tackles $M$ tasks independently by calling the state-of-the-art single-task BPI-CLB algorithm $\\mathtt{Reward \\hyphen free\\ LinUCB}$~\\cite{zanette2021design}. As presented in Figure~\\ref{fig:bpi}, $\\mathtt{C \\hyphen DouExpDes}$ achieves a significantly lower sample complexity than $\\mathtt{IndRFLinUCB}$. In addition, the slope of the sample complexity curve of $\\mathtt{C \\hyphen DouExpDes}$ with respect to $M$ is much smaller than that of $\\mathtt{IndRFLinUCB}$, which validates that $\\mathtt{C \\hyphen DouExpDes}$ enjoys a lighter dependency on dimension in multi-task learning. These empirical results match our theoretical bounds, and corroborate the power of representation learning. \n\t\n\t\\section{Conclusion and Future Work}\n\t\n\tIn this paper, we investigate representation learning for pure exploration in multi-task (contextual) linear bandits. We propose two efficient algorithms which conduct double experimental designs to optimally allocate samples for learning the low-rank representation. The sample complexities of our algorithms mainly depend on the low dimension of the underlying joint representation among tasks, instead of the raw high dimension. Our theoretical and experimental results demonstrate the benefit of representation learning for pure exploration in multi-task bandits.\n\n\tThere are many interesting directions for further exploration. One direction is to establish lower bounds to validate the optimality of our algorithms. Another direction is to extend this work to more complex (nonlinear) representation settings.\n\t\n\t\n\t\n\t\n\t\n\\section{Related Work} \\label{apx:related_work}\n\nIn this section, we present a full literature review for two lines of related works, i.e., representation learning and pure exploration in (contextual) linear bandit.\n\n\\textbf{Representation Learning.}\nThe study of representation learning has been initiated and developed in the supervised learning setting, e.g., \\cite{baxter2000model,ben2003exploiting,ando2005framework,maurer2006bounds,cavallanti2010linear,maurer2016benefit,du2020few,tripuraneni2021provable}. Recently, representation learning for sequential decision making has attracted extensive attention, e.g., ~\\cite{yang2020impact,yang2022nearly,hu2021near,cella2022meta,cella2022multi,qin2022non}.\n\\citet{tripuraneni2021provable} propose a method-of-moments estimator for recovering the feature extractor, and establish error guarantees for transferring the learned representation from past tasks to a new task.\n\\citet{yang2020impact,yang2022nearly} study representation learning for linear bandit with the regret minimization objective, and assume that the action set at each timestep is an ellipsoid or sphere. \\citet{hu2021near} further relax this assumption and allow arbitrary action sets, but their algorithms equipped with a multi-task joint least-square estimator are computationally inefficient. \\citet{cella2022meta,cella2022multi} also investigate the problem in \\cite{yang2020impact} and propose algorithms which do not need to know the dimension of the underlying representation. \n\\citet{qin2022non} study representation learning for linear bandit under non-stationary environments, and develop algorithms that learn and transfer non-stationary representations adaptively.\nDifferent from the above works which consider regret minimization, we study representation learning for (contextual) linear bandit with the pure exploration objective, which imposes unique challenges in how to optimally allocate samples to learn the feature extractor, and motivates us to design algorithms building upon double experimental designs.\n\n\\textbf{Pure Exploration in (Contextual) Linear Bandit.}\nMost linear bandit studies consider regret minimization, e.g.,~\\cite{dani2008stochastic,rusmevichientong2010linearly,chu2011contextual,abbasi2011improved}. Recently, there is a surge of interests in pure exploration for (contextual) linear bandit, e.g., \\cite{soare2014best,tao2018best,xu2018fully,fiez2019sequential,katz2020empirical,degenne2020gamification,jedra2020optimal,du2021combinatorial,zanette2021design,li2022instance}. \nFor linear bandit, \\citet{soare2014best} firstly apply the G-optimal design to identify the best arm, and  provide a sample complexity result that heavily depends on the minimum reward gap.\n\\citet{tao2018best} design a novel randomized estimator for the underlying reward parameter, and achieve tighter sample complexity which depends on the reward gaps of the best $d$ arms. \\citet{du2021combinatorial} further extend the algorithm in \\citep{tao2018best} to develop a polynomial-time algorithm for combinatorially large arm sets. \\citet{xu2018fully} propose a fully-adaptive algorithm which changes the arm selection strategy at each timestep. \\citet{fiez2019sequential} establish the first near-optimal sample complexity upper and lower bounds for best arm identification in linear bandit. \\citet{katz2020empirical} further extend the algorithm in \\cite{fiez2019sequential} and use empirical processes to avoid an explicit union bound over the number of arms. \\citet{degenne2020gamification,jedra2020optimal} develop asymptotically optimal algorithms using the track-and-stop approaches. For contextual linear bandit, \\citet{zanette2021design} design a single non-adaptive policy to collect a dataset, from which a near-optimal policy can be computed. \\citet{li2022instance} build the first instance-dependent upper and lower bounds for best policy identification in contextual linear bandit, with the prior knowledge of the context distribution. By contrast, our work studies multi-task best arm\/policy identification in (contextual) linear bandit with a shared representation among tasks, and does not assume any prior knowledge of the context distribution.\n\n\n\\section{Rounding Procedure} \\label{apx:rounding_procedure}\n\nIn this section, we introduce the rounding procedure $\\mathtt{ROUND}$ in detail.\n\nLet $\\mathcal{X}^{+}:=\\mathcal{X} \\cup \\mathcal{A}$ denote the union space of arm set $\\mathcal{X}$ and action space $\\mathcal{A}$. There are $n$ arms or actions $p_1,\\dots,p_n \\in \\mathcal{X}^{+}$ and $n$ positive semi-definite matrices $\\boldsymbol{Q}_1,\\dots,\\boldsymbol{Q}_n \\in \\mathbb{S}^{d}_{+}$, where $\\boldsymbol{Q}_i$ represents the feature of arm or action $p_i$ for any $i \\in [n]$. Denote $\\mathcal{P}:=\\{p_1,\\dots,p_n\\}$ and $\\mathcal{Q}:=\\{\\boldsymbol{Q}_1,\\dots,\\boldsymbol{Q}_n\\}$.\n\nThe rounding procedure $\\mathtt{ROUND}(\\{(p_i,\\boldsymbol{Q}_i)\\}_{i=1}^{n}, \\boldsymbol{\\lambda}, \\zeta, N)$~\\cite{allen2017near,fiez2019sequential} takes $n$ arm-matrix or action-matrix pairs $(p_1,\\boldsymbol{Q}_1),\\dots,(p_n,\\boldsymbol{Q}_n) \\in \\mathcal{X}^{+} \\times \\mathbb{S}^{d}_{+}$, a distribution $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{P}}$ (or equivalently, $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{Q}}$), an approximation parameter $\\zeta>0$, and the number of samples $N$ which satisfies that $N \\geq \\frac{180d}{\\zeta^2}$ as inputs. \nRoughly speaking, it will find a $N$-length discrete arm or action sequence whose associated feature matrices maintain the similar property (e.g., G-optimality and E-optimality) as the continuous sample allocation $\\boldsymbol{\\lambda}$. \n\nFormally, $\\mathtt{ROUND}(\\{(p_i,\\boldsymbol{Q}_i)\\}_{i=1}^{n}, \\lambda, \\zeta, N)$ returns a discrete sample sequence $s_1, \\dots, s_N \\in \\mathcal{P}^{N}$ associated with feature matrices $\\boldsymbol{S}_1, \\dots, \\boldsymbol{S}_N \\in \\mathcal{Q}^{N}$, which satisfy the following properties: \n\n(i) If $\\boldsymbol{\\lambda}$ is an E-optimal design, i.e., $\\boldsymbol{\\lambda}$ is the optimal solution of the optimization\n$$\n\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{Q}}} \\nbr{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i}^{-1}} ,\n$$\nthen $\\boldsymbol{S}_1, \\dots, \\boldsymbol{S}_N$ satisfy that\n\\begin{align*}\n\t\\nbr{\\sbr{\\sum_{j=1}^N \\boldsymbol{S}_j }^{-1}} \\leq (1+\\zeta) \\nbr{\\sbr{N \\sum_{i=1}^n \\lambda(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i }^{-1}} .\n\\end{align*}\n\n(ii) If $\\boldsymbol{\\lambda}$ is a G-optimal design, i.e., for a given prediction set $\\mathcal{Y} \\subseteq \\mathbb{R}^d$, $\\lambda$ is the optimal solution of the optimization\n$$\n\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{Q}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}} \\nbr{\\boldsymbol{y}}^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i}^{-1}} ,\n$$\nthen $\\boldsymbol{S}_1, \\dots, \\boldsymbol{S}_N$ satisfy that\n\\begin{align*}\n\t\\max_{\\boldsymbol{y} \\in \\mathcal{Y}} \\nbr{\\boldsymbol{y}}^2_{\\sbr{\\sum_{j=1}^{N} \\boldsymbol{S}_j}^{-1}} \\leq (1+\\zeta) \\max_{\\boldsymbol{y} \\in \\mathcal{Y}} \\nbr{\\boldsymbol{y}}^2_{\\sbr{N \\sum_{i=1}^{n} \\lambda(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i}^{-1}} .\n\\end{align*}\n\n\nWe implement $\\mathtt{ROUND}$ by setting $\\boldsymbol{\\pi}^*=N\\boldsymbol{\\lambda}$, $k=r=N$ and $\\boldsymbol{x}_i \\boldsymbol{x}_i^\\top = (\\sum_{i=1}^{n} \\pi^*(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i)^{-\\frac{1}{2}} \\boldsymbol{Q}_i (\\sum_{i=1}^{n} \\pi^*(\\boldsymbol{Q}_i) \\boldsymbol{Q}_i)^{-\\frac{1}{2}}$ for any $i \\in [n]$ in Algorithm 1 of \\cite{allen2017near}. \nNote that Algorithm 1 in \\cite{allen2017near} only needs to access the feature matrix $\\boldsymbol{x}_i \\boldsymbol{x}_i^\\top$ rather than the separate feature vector $\\boldsymbol{x}_i$, which allows us to apply it to our problem. We refer interested readers to \\cite{allen2017near} and Appendix B in \\cite{fiez2019sequential} for more implementation details of this rounding procedure.\n\n\n\\section{Proofs for Algorithm $\\mathtt{DouExpDes}$}\n\nIn this section, we provide the proofs for Algorithm~$\\mathtt{DouExpDes}$. \n\nThroughout our proofs, we use $L_{\\theta}$ to denote the upper bound of $\\|\\boldsymbol{\\theta}_m\\|$ for any $m \\in [M]$. \nSince $\\boldsymbol{\\theta}_m=\\boldsymbol{B}\\boldsymbol{w}_m$ for any $m \\in [M]$, we have that $\\|\\boldsymbol{\\theta}_m\\| \\leq \\|\\boldsymbol{B}\\| \\|\\boldsymbol{w}_m\\| \\leq \\|\\boldsymbol{w}_m\\| \\leq L_{w}$, and thus $L_{\\theta} \\leq L_{w}$.\n\n\\subsection{Sample Batch Planning}\n\nRecall that\n\\begin{align*}\n\t\\boldsymbol{\\lambda}^{E} := & \\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\nbr{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} \n\\end{align*}\nand\n\\begin{align*}\n\t\\rho^{E} := & \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\nbr{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} \n\\end{align*}\nare the optimal solution and the optimal value of the E-optimal design optimization, respectively (Line~\\ref{line:bai_E_optimal_design} in Algorithm~\\ref{alg:repbailb}). $\\bar{\\boldsymbol{x}}_1,\\dots,\\bar{\\boldsymbol{x}}_p$ is an arm sequence generated according to sample allocation $\\boldsymbol{\\lambda}^{E}$ via rounding procedure $\\mathtt{ROUND}$ (Line~\\ref{line:bai_E_optimal_round} in Algorithm~\\ref{alg:repbailb}).\n\nLet \n$$\n\\boldsymbol{X}_{\\textup{batch}}:=\n\\begin{bmatrix}\n\t\\bar{\\boldsymbol{x}}_1^\\top\n\t\\\\\n\t\\dots\n\t\\\\\n\t\\bar{\\boldsymbol{x}}_p^\\top\n\\end{bmatrix} ,\n$$\nand \n$$\n\\boldsymbol{X}_{\\textup{batch}}^{+} := (\\boldsymbol{X}_{\\textup{batch}}^\\top \\boldsymbol{X}_{\\textup{batch}})^{-1} \\boldsymbol{X}_{\\textup{batch}}^{\\top} .\n$$\nAccording to the fact that $\\mathcal{X}$ spans $\\mathbb{R}^d$, the definition of E-optimal design and the guarantee of $\\mathtt{ROUND}$, we have that $\\boldsymbol{X}_{\\textup{batch}}^\\top \\boldsymbol{X}_{\\textup{batch}}$ is invertible.\n\nNow, we first give an upper bound of $\\|\\boldsymbol{X}_{\\textup{batch}}^{+}\\|$.\n\n\\begin{lemma} \\label{lemma:X_batch_ub}\n\n\tIt holds that\n\t\\begin{align*}\n\t\t\\|\\boldsymbol{X}_{\\textup{batch}}^{+}\\| \\leq \\sqrt{\\frac{(1+\\zeta) \\rho^{E}}{p}} .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:X_batch_ub}]\n\tWe have\n\t\\begin{align*}\n\t\t\\|\\boldsymbol{X}_{\\textup{batch}}^{+}\\| = & \\nbr{\\sbr{\\boldsymbol{X}_{\\textup{batch}}^{\\top} \\boldsymbol{X}_{\\textup{batch}}}^{-1} \\boldsymbol{X}_{\\textup{batch}}^{\\top}}\n\t\t\\\\\n\t\t= & \\sqrt{ \\nbr{\\sbr{\\boldsymbol{X}_{\\textup{batch}}^{\\top} \\boldsymbol{X}_{\\textup{batch}}}^{-1} \\boldsymbol{X}_{\\textup{batch}}^{\\top} \\boldsymbol{X}_{\\textup{batch}} \\sbr{\\boldsymbol{X}_{\\textup{batch}}^{\\top} \\boldsymbol{X}_{\\textup{batch}}}^{-1}} }\n\t\t\\\\\n\t\t= & \\sqrt{ \\nbr{\\sbr{\\boldsymbol{X}_{\\textup{batch}}^{\\top} \\boldsymbol{X}_{\\textup{batch}}}^{-1}} }\n\t\t\\\\\n\t\t= & \\sqrt{ \\nbr{\\sbr{\\sum_{i=1}^{p} \\bar{\\boldsymbol{x}}_i \\bar{\\boldsymbol{x}}_i^{\\top}}^{-1}} }\n\t\t\\\\\n\t\t\\leq & \\sqrt{(1+\\zeta) \\nbr{\\sbr{p \\sum_{i=1}^{n} \\lambda^{E}(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^{\\top}}^{-1}} }\n\t\t\\\\\n\t\t= & \\sqrt{\\frac{(1+\\zeta) \\rho^{E}}{p}} .\n\t\\end{align*}\n\\end{proof}\n\n\n\\subsection{Global Feature Extractor Recovery} \\label{apx:bai_feature_recover}\n\nFor clarity of notation, we add subscript $t$ to the notations in subroutine $\\mathtt{FeatRecover}$ to denote the quantities generated in phase $t$. Specifically, we use $\\alpha_{t,m,j,i}$, $\\tilde{\\boldsymbol{\\theta}}_{t,m,j}$, $\\boldsymbol{Z}_t$ and $\\hat{\\boldsymbol{B}}_t$ to denote the random reward, estimator of reward parameter, estimator of $\\frac{1}{M} \\sum_{i=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top$ and estimator of feature extractor in phase $t$, respectively.\n\n\nFor any phase $t>0$, task $m \\in [M]$, round $j \\in [T_t]$ and arm $i \\in [p]$, let $\\eta_{t,m,j,i}$ denote the noise of the sample on arm $\\bar{\\boldsymbol{x}}_i$ in the $j$-th round for task $m$, during the execution of $\\mathtt{FeatRecover}$ in phase $t$ (Line~\\ref{line:bai_stage2_sample} in Algorithm~\\ref{alg:feat_recover}). The noise $\\eta_{t,m,j,i}$ is zero-mean and sub-Gaussian, and has variance $1$. $\\eta_{t,m,j,i}$ is independent for different $t,m,j,i$.\n\nFor any phase $t>0$, task $m \\in [M]$, round $j \\in [T_t]$, let $\\boldsymbol{\\alpha}_{t,m,j}:=[\\alpha_{t,m,j,1},\\dots,\\alpha_{t,m,j,p}]^\\top$.\nThen, we have that\n$$\n\\tilde{\\boldsymbol{\\theta}}_{t,m,j} = \\boldsymbol{X}_{\\textup{batch}}^{+} \\boldsymbol{\\alpha}_{t,m,j} ,\n$$\nand\n$$\n\\boldsymbol{Z}_t = \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\tilde{\\boldsymbol{\\theta}}_{t,m,j} (\\tilde{\\boldsymbol{\\theta}}_{t,m,j})^\\top - \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top .\n$$\n\n\n\\begin{lemma}[Expectation of $\\boldsymbol{Z}_t$] \\label{lemma:bai_expectation_Z_t}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\mathbb{E} \\mbr{ \\boldsymbol{Z}_t } = \\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:bai_expectation_Z_t}]\n\t$\\boldsymbol{Z}_t$ can be written as\n\t\\begin{align}\n\t\t\\boldsymbol{Z}_t = & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\tilde{\\boldsymbol{\\theta}}_{t,m,j} (\\tilde{\\boldsymbol{\\theta}}_{t,m,j})^\\top - \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top \n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \\begin{bmatrix}\n\t\t\t\\alpha_{t,m,j,1}\n\t\t\t\\\\\n\t\t\t\\vdots\n\t\t\t\\\\\n\t\t\t\\alpha_{t,m,j,p} \n\t\t\\end{bmatrix}  \n\t\t[\\alpha_{t,m,j,1}, \\dots, \\alpha_{t,m,j,p}]^\\top (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t- \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top \n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \\begin{bmatrix}\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,1}\n\t\t\t\\\\\n\t\t\t\\vdots\n\t\t\t\\\\\n\t\t\t\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,p} \n\t\t\\end{bmatrix}  \n\t\t[\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,1}, \\dots,\n\t\t\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,p} ]^\\top (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t- \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top \n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\begin{bmatrix}\n\t\t\t(\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,1})^2 & \\!\\!\\!\\!\\!\\cdots\\!\\!\\!\\!\\! & (\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,1}) (\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,p}) \\\\\n\t\t\t\\cdots\t& \\!\\!\\!\\!\\!\\cdots\\!\\!\\!\\!\\! & \\cdots \\\\\n\t\t\t(\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,p})(\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,1}) & \\!\\!\\!\\!\\!\\cdots\\!\\!\\!\\!\\! & (\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m + \\eta_{t,m,j,p})^2\n\t\t\\end{bmatrix}\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t\\nonumber\\\\& - \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top \n\t\t\\nonumber\\\\\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\n\t\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\Bigg(\n\t\t\\begin{bmatrix}\n\t\t\t(\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m)^2 & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m & \\cdots & (\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m)^2 \n\t\t\\end{bmatrix}\n\t\t\\nonumber\\\\& +\n\t\t\\begin{bmatrix}\n\t\t\t2 \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & 2 \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p}\n\t\t\\end{bmatrix}\n\t\t\\nonumber\\\\& +\n\t\t\\begin{bmatrix}\n\t\t\t(\\eta_{t,m,j,1})^2 & \\cdots & \\eta_{t,m,j,1} \\eta_{t,m,j,p} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\eta_{t,m,j,1} \\eta_{t,m,j,p} & \\cdots & (\\eta_{t,m,j,p})^2\n\t\t\\end{bmatrix}\n\t\t\\Bigg)\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t- \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top . \\label{eq:Z_t_decompose}\n\t\\end{align}\n\t\n\n\tThen, taking the expectation on $\\boldsymbol{Z}_t$, we have\n\t\\begin{align*}\n\t\t\\mathbb{E} [\\boldsymbol{Z}_t] = & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\Bigg( \\begin{bmatrix}\n\t\t\t(\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m)^2 & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m  & \\cdots & (\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m)^2\n\t\t\\end{bmatrix} + \\boldsymbol{I}_d \\Bigg)\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t- \\boldsymbol{X}_{\\textup{batch}}^{+} (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\begin{bmatrix}\n\t\t\t(\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m)^2 & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m  & \\cdots & (\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m)^2\n\t\t\\end{bmatrix} \n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\begin{bmatrix}\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m\n\t\t\t\\\\\n\t\t\t\\vdots\n\t\t\t\\\\\n\t\t\t\\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m\n\t\t\\end{bmatrix} [\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m, \\dots, \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m]^\\top\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\boldsymbol{X}_{\\textup{batch}} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top \\boldsymbol{X}_{\\textup{batch}}^\\top\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top\n\t\t\\\\\n\t\t= & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top \n\t\t\\\\\n\t\t= & \\frac{1}{M} \\sum_{m=1}^{M}  \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top .\n\t\\end{align*}\n\\end{proof}\n\nRecall that for any $t>0$, $\\delta_t:=\\frac{\\delta}{2t^2}$.\n\nFor any phase $t>0$, define events \n\\begin{align*}\n\t\\mathcal{E}_t:= \\lbr{ \\nbr{\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t]} \n\t\t\\leq \\frac{ 96 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 p L_{x} L_{\\theta} \\log\\sbr{\\frac{16p}{\\delta_t}} }{\\sqrt{MT_t}} \\log \\sbr{\\frac{16pMT_t}{\\delta_t}} } ,\n\\end{align*}\nand\n\\begin{align*}\n\t\\mathcal{E}:=\\cap_{t=1}^{\\infty} \\mathcal{E}_t .\n\\end{align*}\n\n\\begin{lemma}[Concentration of $\\boldsymbol{Z}_t$] \\label{lemma:Z_t_est_error}\n\tIt holds that\n\n\t\n\t\n\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{E}} \\geq \\frac{\\delta}{2} .  \n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:Z_t_est_error}]\n\tAccording to Eq.~\\eqref{eq:Z_t_decompose}, we have\n\t\\begin{align*}\n\t\t\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t] = & \\frac{1}{M T_t} \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{X}_{\\textup{batch}}^{+} \n\t\t\\Bigg(\n\t\t\\begin{bmatrix}\n\t\t\t2 \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & 2 \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p}\n\t\t\\end{bmatrix}\n\t\t\\\\& -\n\t\t\\mathbb{E}\n\t\t\\begin{bmatrix}\n\t\t\t2 \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & 2 \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p}\n\t\t\\end{bmatrix}\n\t\t\\nonumber\\\\& +\n\t\t\\begin{bmatrix}\n\t\t\t(\\eta_{t,m,j,1})^2 & \\cdots & \\eta_{t,m,j,1} \\eta_{t,m,j,p} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\eta_{t,m,j,1} \\eta_{t,m,j,p} & \\cdots & (\\eta_{t,m,j,p})^2\n\t\t\\end{bmatrix}\n\t\t-\n\t\t\\mathbb{E}\n\t\t\\begin{bmatrix}\n\t\t\t(\\eta_{t,m,j,1})^2 & \\cdots & \\eta_{t,m,j,1} \\eta_{t,m,j,p} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\eta_{t,m,j,1} \\eta_{t,m,j,p} & \\cdots & (\\eta_{t,m,j,p})^2\n\t\t\\end{bmatrix}\n\t\t\\Bigg)\n\t\t(\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top .\n\t\\end{align*}\n\t\n\tDefine the following matrices:\n\t\\begin{align*}\n\t\t\\boldsymbol{A}_{t,m,j} &:= \\frac{1}{M T_t} \\begin{bmatrix}\n\t\t\t2 \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,1} & \\cdots & 2 \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\eta_{t,m,j,p} \n\t\t\\end{bmatrix} ,\n\t\t\\\\\n\t\t\\boldsymbol{A}_t &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{A}_{t,m,j} ,\n\t\t\\\\ \n\t\t\\boldsymbol{C}_{t,m,j} &:= \\frac{1}{M T_t} \\begin{bmatrix}\n\t\t\t(\\eta_{t,m,j,1})^2 & \\cdots & \\eta_{t,m,j,1} \\eta_{t,m,j,p} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\eta_{t,m,j,1} \\eta_{t,m,j,p} & \\cdots & (\\eta_{t,m,j,p})^2 \n\t\t\\end{bmatrix} ,\n\t\t\\\\\n\t\t\\boldsymbol{C}_t &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\boldsymbol{C}_{t,m,j} .\n\t\\end{align*}\n\t\n\tThen, we can write $\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t]$ as\n\t\\begin{align*}\n\t\t\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t] = \\boldsymbol{X}_{\\textup{batch}}^{+} \\sbr{ \\boldsymbol{A}_t - \\mathbb{E}[\\boldsymbol{A}_t] + \\boldsymbol{C}_t - \\mathbb{E}[\\boldsymbol{C}_t] } (\\boldsymbol{X}_{\\textup{batch}}^{+})^\\top ,\n\t\\end{align*}\n\tand thus,\n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t]} \\leq \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 \\sbr{ \\nbr{\\boldsymbol{A}_t - \\mathbb{E}[\\boldsymbol{A}_t]} + \\nbr{\\boldsymbol{C}_t - \\mathbb{E}[\\boldsymbol{C}_t]} } . \\label{eq:Z_t_expressed_by_A}\n\t\\end{align}\n\t\n\tNext, we analyze $\\|\\boldsymbol{A}_t - \\mathbb{E}[\\boldsymbol{A}_t]\\|$ and $\\|\\boldsymbol{C}_t - \\mathbb{E}[\\boldsymbol{C}_t]\\|$. In order to use the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}), we define the truncated noise and truncated matrices as follows.\n\t\n\tLet $R>0$ be a truncation level of noises, which will be chosen later.  For any $t>0$, $m \\in [M]$, $j \\in [T_t]$ and $i \\in [p]$, let $\\tilde{\\eta}_{t,m,j,i}=\\eta_{t,m,j,i} \\mathbbm{1}\\{|\\eta_{t,m,j,i}| \\leq R\\}$ denote the truncated noise. Then, we define the following truncated matrices:\n\t\\begin{align}\n\t\t\\tilde{\\boldsymbol{A}}_{t,m,j} &:= \\frac{1}{M T_t} \\begin{bmatrix}\n\t\t\t2 \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,1} & \\cdots & \\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,1} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\bar{\\boldsymbol{x}}_1^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,p} + \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,1} & \\cdots & 2 \\bar{\\boldsymbol{x}}_p^\\top \\boldsymbol{\\theta}_m \\tilde{\\eta}_{t,m,j,p}\n\t\t\\end{bmatrix} \n\t\t\\nonumber\\\\\n\t\t\\tilde{\\boldsymbol{A}}_t &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\tilde{\\boldsymbol{A}}_{t,m,j} ,\n\t\t\\nonumber\\\\\n\t\t\\tilde{\\boldsymbol{C}}_{t,m,j} &:= \\frac{1}{M T_t} \\begin{bmatrix}\n\t\t\t(\\tilde{\\eta}_{t,m,j,1})^2 & \\cdots & \\tilde{\\eta}_{t,m,j,1} \\tilde{\\eta}_{t,m,j,p} \\\\\n\t\t\t\\cdots\t& \\cdots & \\cdots \\\\\n\t\t\t\\tilde{\\eta}_{t,m,j,1} \\tilde{\\eta}_{t,m,j,p} & \\cdots & (\\tilde{\\eta}_{t,m,j,p})^2 \n\t\t\\end{bmatrix}  \\label{eq:tilde_C_t_m_j}\n\t\t\\\\\n\t\t\\tilde{\\boldsymbol{C}}_t &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T_t} \\tilde{\\boldsymbol{C}}_{t,m,j}  \\nonumber\n\t\\end{align}\n\t\n\tFirst, we bound $\\|\\boldsymbol{A}_t-\\mathbb{E}[\\boldsymbol{A}_t]\\|$. \n\tSince for any $t>0$, $m \\in [M]$, $j \\in [T_t]$ and $i \\in [p]$, $|\\tilde{\\eta}_{t,m,j,i}| \\leq R$ and $|\\bar{\\boldsymbol{x}}_i^\\top \\boldsymbol{\\theta}_m| \\leq L_{x} L_{\\theta}$, we have $\\|\\tilde{\\boldsymbol{A}}_{t,m,j}\\| \\leq \\frac{1}{M T_t} \\cdot 2p L_{x} L_{\\theta} R$.\n\t\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\t\n\t\n\n\n\t\n\t\n\n\n\t\n\t\n\t\n\tRecall that for any $t>0$, $m \\in [M]$, $j \\in [T_t]$ and $i \\in [p]$, $\\eta_{t,m,j,i}$ is 1-sub-Gaussian. Using a union bound over $i \\in [p]$, we have that for any $t>0$, $m \\in [M]$, $j \\in [T_t]$, with probability at least $1-2p\\exp(-\\frac{R^2}{2})$, $|\\eta_{t,m,j,i}| \\leq R$ for all $i \\in [p]$. Thus, with probability at least $1-2p\\exp(-\\frac{R^2}{2})$, $\\|\\boldsymbol{A}_{t,m,j}\\| \\leq \\frac{1}{M T_t} \\cdot 2p L_{x} L_{\\theta} R$. \n\t\n\tThen, we have\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}[\\boldsymbol{A}_{t,m,j}]-\\mathbb{E}[\\tilde{\\boldsymbol{A}}_{t,m,j}] } \\leq & \\nbr{ \\mathbb{E} \\mbr{\\boldsymbol{A}_{t,m,j} \\cdot \\indicator{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\geq \\frac{2p L_{x} L_{\\theta} R}{M T_t} }} }\n\t\t\\\\\n\t\t\\leq & \\mathbb{E} \\mbr{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\cdot \\indicator{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\geq \\frac{2p L_{x} L_{\\theta} R}{M T_t} } } \n\t\t\\\\\n\t\t= & \\mathbb{E}\\mbr{ \\frac{2p L_{x} L_{\\theta} R}{M T_t} \\cdot \\indicator{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\geq \\frac{2p L_{x} L_{\\theta} R}{M T_t} }} \\\\& + \\mbr{\\sbr{ \\nbr{\\boldsymbol{A}_{t,m,j}} - \\frac{2p L_{x} L_{\\theta} R}{M T_t}} \\cdot \\indicator{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\geq \\frac{2p L_{x} L_{\\theta} R}{M T_t} } } \n\t\t\\\\\n\t\t= & \\frac{2p L_{x} L_{\\theta} R}{M T_t} \\cdot \\Pr\\mbr{ \\nbr{\\boldsymbol{A}_{t,m,j}} \\geq \\frac{2p L_{x} L_{\\theta} R}{M T_t} } + \\int_0^{\\infty}  \\Pr \\mbr{ \\nbr{\\boldsymbol{A}_{t,m,j}} - \\frac{2p L_{x} L_{\\theta} R}{M T_t} > x}  dx  \n\t\t\\\\\n\t\t\\leq & \\frac{2p L_{x} L_{\\theta} R}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{x} L_{\\theta}}{M T_t}  \\int_R^{\\infty} \\Pr \\mbr{ \\nbr{\\boldsymbol{A}_{t,m,j}} > \\frac{2p L_{x} L_{\\theta} y}{M T_t} }  dy  \n\t\t\\\\\n\t\t\\leq & \\frac{2p L_{x} L_{\\theta} R}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{x} L_{\\theta}}{M T_t}  \\int_R^{\\infty} 2p \\exp\\sbr{-\\frac{y^2}{2}}  dy\n\t\t\\\\\n\t\t\\leq & \\frac{2p L_{x} L_{\\theta} R}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{x} L_{\\theta}}{M T_t} \\cdot 2p \\cdot \\frac{1}{R} \\cdot \\exp\\sbr{-\\frac{R^2}{2}} \n\t\t\\\\\n\t\t= & \\frac{2p L_{x} L_{\\theta}}{M T_t} \\cdot 2p \\cdot \\sbr{R+\\frac{1}{R}} \\exp\\sbr{-\\frac{R^2}{2}} .\n\t\\end{align*}\n\t\n\tLet $\\delta' \\in (0,1)$ be a confidence parameter which will be chosen later.\n\tUsing the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) with $n=MT_t$, $R=\\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}}$, $n \\Pr[\\|\\boldsymbol{A}_{t,m,j}\\| \\geq \\frac{1}{M T_t} \\cdot 2p L_{x} L_{\\theta} R] \\leq \\delta'$, $U=\\frac{2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}}}{MT_t}$, $\\sigma^2=M T_t U^2$, $\\tau=\\frac{4\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} + \\frac{4\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{M T_t}$ and $\\Delta=\\frac{2p L_{x} L_{\\theta} \\cdot 2 \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}}}{M T_t} \\cdot \\frac{\\delta'}{M T_t}$, we have that with probability at least $1-2\\delta'$, \n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{A}_t - \\mathbb{E}[\\boldsymbol{A}_t]} \\leq & \\frac{4\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} + \\frac{4\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{M T_t} \n\t\t\\nonumber\\\\ \n\t\t\\leq & \\frac{8\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} .\n\t\t\\label{eq:A_t_concentration}\n\t\\end{align}\n\t\n\tNow we investigate $\\|\\boldsymbol{C}_t-\\mathbb{E}[\\boldsymbol{C}_t]\\|$. Recall that in Eq.~\\eqref{eq:tilde_C_t_m_j}, for any $t>0$, $m \\in [M]$, $j \\in [T_t]$ and $i \\in [p]$, $|\\tilde{\\eta}_{t,m,j,i}| \\leq R$. Then, we have $\\|\\tilde{\\boldsymbol{C}}_{t,m,j}\\| \\leq \\frac{1}{M T_t} \\cdot pR^2$.\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\t\n\t\n\n\n\t\n\t\n\t\n\tRecall that for any $t>0$, $m \\in [M]$ and $j \\in [T_t]$, with probability at least $1-2p\\exp(-\\frac{R^2}{2})$, $|\\eta_{t,m,j,i}| \\leq R$ for all $i \\in [p]$. Thus, with probability at least $1-2p\\exp(-\\frac{R^2}{2})$, $\\|\\boldsymbol{C}_{t,m,j}\\| \\leq \\frac{1}{M T_t} \\cdot pR^2$. Then, we have\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}[\\boldsymbol{C}_{t,m,j}]-\\mathbb{E}[\\tilde{\\boldsymbol{C}}_{t,m,j}] } \\leq & \\nbr{ \\mathbb{E} \\mbr{\\boldsymbol{C}_{t,m,j} \\cdot \\indicator{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\geq \\frac{pR^2}{M T_t} }} }\n\t\t\\\\\n\t\t\\leq & \\mathbb{E} \\mbr{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\cdot \\indicator{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\geq \\frac{pR^2}{M T_t} } } \n\t\t\\\\\n\t\t= & \\mathbb{E}\\mbr{ \\frac{pR^2}{M T_t} \\cdot \\indicator{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\geq \\frac{pR^2}{M T_t} }} + \\mbr{\\sbr{ \\nbr{\\boldsymbol{C}_{t,m,j}} - \\frac{pR^2}{M T_t}} \\cdot \\indicator{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\geq \\frac{pR^2}{M T_t} } } \n\t\t\\\\\n\t\t= & \\frac{pR^2}{M T_t} \\cdot \\Pr\\mbr{ \\nbr{\\boldsymbol{C}_{t,m,j}} \\geq \\frac{pR^2}{M T_t} } + \\int_0^{\\infty}  \\Pr \\mbr{ \\nbr{\\boldsymbol{C}_{t,m,j}} - \\frac{pR^2}{M T_t} > x}  dx  \n\t\t\\\\\n\t\t\\leq & \\frac{pR^2}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p}{M T_t}  \\int_R^{\\infty} \\boldsymbol{y} \\cdot \\Pr \\mbr{ \\nbr{\\boldsymbol{C}_{t,m,j}} > \\frac{dy^2}{M T_t} } dy  \n\t\t\\\\\n\t\t\\leq & \\frac{pR^2}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p}{M T_t}  \\int_R^{\\infty}  \\boldsymbol{y} \\cdot 2p \\exp\\sbr{-\\frac{y^2}{2}} dy\n\t\t\\\\\n\t\t\\leq & \\frac{pR^2}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p}{M T_t} \\cdot 2p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} \n\t\t\\\\\n\t\t= & \\frac{p}{M T_t} \\cdot 2p \\cdot \\sbr{R^2+2} \\exp\\sbr{-\\frac{R^2}{2}} .\n\t\\end{align*}\n\t\n\tUsing the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) with $n=MT_t$, $R=\\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}}$, $n \\Pr[\\|\\boldsymbol{C}_{t,m,j}\\| \\geq \\frac{1}{M T_t} \\cdot pR^2] \\leq \\delta'$, $U=\\frac{p \\cdot 2\\log \\sbr{\\frac{2pMT_t}{\\delta'}} }{MT_t}$, $\\sigma^2=\\frac{32p}{M T_t}$, $\\tau=\\frac{4\\cdot p \\cdot 2\\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} + \\frac{4\\cdot p \\cdot 2\\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log\\sbr{\\frac{2p}{\\delta'}}}{M T_t}$ and $\\Delta=\\frac{p \\cdot 2 \\cdot 2\\log \\sbr{\\frac{2pMT_t}{\\delta'}} }{M T_t} \\cdot \\frac{\\delta'}{M T_t}$, we have that with probability at least $1-2\\delta'$, \n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{C}_t - \\mathbb{E}\\mbr{\\boldsymbol{C}_t}} \\leq & \\frac{4\\cdot 2p \\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} +  \\frac{4 \\cdot 2p \\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log \\sbr{\\frac{2p}{\\delta'}}}{M T_t} \n\t\t\\nonumber\\\\\n\t\t\\leq & \\frac{8\\cdot 2p \\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}}\n\t\t\\label{eq:C_t_concentration}\n\t\\end{align}\n\t\n\tPlugging Eqs.~\\eqref{eq:A_t_concentration} and \\eqref{eq:C_t_concentration} into Eq.~\\eqref{eq:Z_t_expressed_by_A}, we have that with probability at least $1-4\\delta'$,\n\t\\begin{align*}\n\t\t\\nbr{\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t]} \\leq & \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 \\sbr{\\nbr{\\boldsymbol{A}_t - \\mathbb{E} \\mbr{\\boldsymbol{A}_t}} + \\nbr{\\boldsymbol{C}_t  - \\mathbb{E} \\mbr{\\boldsymbol{C}_t}} }\n\t\t\\\\\n\t\t\\leq & \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 \\sbr{ \\frac{8\\cdot 2p L_{x} L_{\\theta} \\sqrt{2\\log \\sbr{\\frac{2pMT_t}{\\delta'}}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} + \\frac{8\\cdot 2p \\log \\sbr{\\frac{2pMT_t}{\\delta'}} \\log\\sbr{\\frac{2p}{\\delta'}}}{\\sqrt{M T_t}} }\n\t\t\\\\\n\t\t\\leq & \\frac{ 96 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 p L_{x} L_{\\theta} \\log\\sbr{\\frac{2p}{\\delta'}} }{\\sqrt{MT_t}} \\log \\sbr{\\frac{2pMT_t}{\\delta'}} .\n\t\\end{align*}\n\tLet $\\delta'=\\frac{\\delta_t}{8}$. Then, we obtain that with probability at least $1-\\frac{\\delta_t}{2}$,\n\t\\begin{align*}\n\t\t\\nbr{\\boldsymbol{Z}_t - \\mathbb{E} [\\boldsymbol{Z}_t]} \\leq & \\frac{ 96 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 p L_{x} L_{\\theta} \\log\\sbr{\\frac{16p}{\\delta_t}} }{\\sqrt{MT_t}} \\log \\sbr{\\frac{16pMT_t}{\\delta_t}} , \n\t\\end{align*}\n\twhich implies that\n\t$\\Pr \\mbr{\\mathcal{E}_t} \\geq 1-\\frac{\\delta_t}{2}$.\n\t\n\tTaking a union bound over all phases $t\\geq 1$ and recalling $\\delta_t:=\\frac{\\delta}{2t^2}$, we obtain\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{E}} \n\t\t\\geq & 1- \\sum_{t=1}^{\\infty} \\Pr \\mbr{\\bar{\\mathcal{E}_t}}\n\t\t\\\\\n\t\t\\geq & 1- \\sum_{t=1}^{\\infty} \\frac{\\delta_t}{2}\n\t\t\\\\\n\t\t= & 1- \\sum_{t=1}^{\\infty} \\frac{\\delta}{4t^2}\n\t\t\\\\\n\t\t\\geq & 1-\\frac{\\delta}{2} .\n\t\\end{align*}\n\\end{proof}\n\nFor any matrix $\\boldsymbol{A} \\in \\mathbb{R}^{m \\times n}$ with $m \\geq n$, let $\\sigma_{\\max}(\\boldsymbol{A})$ and $\\sigma_{\\min}(\\boldsymbol{A})$ denote the maximum and minimum singular values of $\\boldsymbol{A}$, respectively. For any $i \\in [m]$, let $\\sigma_i(\\boldsymbol{A})$ denote the $i$-th singular value of $\\boldsymbol{A}$.\n\nFor any matrix $\\boldsymbol{A} \\in \\mathbb{R}^{m \\times n}$ with $m \\geq n$, let $\\boldsymbol{A}_{\\bot}$ denote the orthogonal complement matrix of $\\boldsymbol{A}$, where the columns of $\\boldsymbol{A}_{\\bot}$ are the orthogonal complement of those of $\\boldsymbol{A}$. Then, it holds that $\\boldsymbol{A} \\boldsymbol{A}^\\top + \\boldsymbol{A}_{\\bot} \\boldsymbol{A}_{\\bot}^\\top = \\boldsymbol{I}_m$, where $\\boldsymbol{I}_m$ is the $m \\times m$ identity matrix.\n\nAccording to Assumption~\\ref{assumption:diverse_task}, there exists an absolute constant $c_0$ which satisfies that $\\sigma_{\\min}(\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{w}_m \\boldsymbol{w}_m^\\top) = \\sigma_{\\min}(\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top) \\geq \\frac{c_0}{k}$.\n\n\n\\begin{lemma}[Concentration of $\\hat{\\boldsymbol{B}}_t$] \\label{lemma:concentration_B_hat_t}\n\tSuppose that event $\\mathcal{E}$ holds. Then, for any phase $t>0$,\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{ 192 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 k p L_{x} L_{\\theta} \\log\\sbr{\\frac{16p}{\\delta_t}} }{\\sqrt{MT_t}} \\log \\sbr{\\frac{16pMT_t}{\\delta_t}} .\n\t\\end{align*}\n\tFurthermore, for any phase $t>0$, if \n\t\\begin{align}\n\t\n\t\t%\n\t\tT_t = \\Bigg \\lceil & \\frac{68 \\cdot 192^2 \\cdot 8^2 \\sbr{1+\\zeta}^3 (\\rho^E)^2 k^4 L_{x}^4 L_{\\theta}^2 L_{w}^2}{c_0^2 M} \\cdot \\max\\lbr{2^{2t},\\ \\frac{L_x^4}{\\omega^2}} \\cdot \\log^2 \\sbr{\\frac{16p}{\\delta_t}} \\nonumber\\\\& \\log^2 \\sbr{ \\frac{192 \\cdot 16 \\cdot 8  \\sbr{1+\\zeta}^{\\frac{3}{2}} \\rho^E k^2 p L_{x}^2 L_{\\theta} L_{w}}{c_0} \\cdot \\max\\lbr{2^t,\\ \\frac{L_x^2}{\\omega}} \\cdot \\frac{1}{\\delta_t} \\cdot \\log\\sbr{\\frac{16p}{\\delta_t}} } \\Bigg \\rceil , \\label{eq:value_T_t}\n\t\\end{align}\n\tthen\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\min\\lbr{\\frac{1}{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta}},\\ \\frac{\\omega}{6 L_x^2}} .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:concentration_B_hat_t}]\n\tFrom Assumption~\\ref{assumption:diverse_task}, $\\sigma_{k}(\\mathbb{E}[\\boldsymbol{Z}_t]) - \\sigma_{k+1}(\\mathbb{E}[\\boldsymbol{Z}_t])=\\sigma_{\\min}( \\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top ) \\geq \\frac{c_0}{k}$. Using the Davis-Kahan sin $\\theta$ Theorem~\\cite{bhatia2013matrix} and letting $T_t$ be large enough to satisfy $\\nbr{\\boldsymbol{Z}_t - \\mathbb{E}[\\boldsymbol{Z}_t]} \\leq \\frac{c_0}{2k}$, we have\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq & \\frac{ \\nbr{\\boldsymbol{Z}_t - \\mathbb{E}[\\boldsymbol{Z}_t]} }{ \\sigma_{k}(\\mathbb{E}[\\boldsymbol{Z}_t]) - \\sigma_{k+1}(\\mathbb{E}[\\boldsymbol{Z}_t]) - \\nbr{\\boldsymbol{Z}_t - \\mathbb{E}[\\boldsymbol{Z}_t]} }\n\t\t\\\\\n\t\t\\leq & \\frac{2k}{c_0} \\nbr{\\boldsymbol{Z}_t - \\mathbb{E}[\\boldsymbol{Z}_t]} \n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\frac{ 192 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 k p L_{x} L_{\\theta} \\log\\sbr{\\frac{16p}{\\delta_t}} }{c_0 \\sqrt{MT_t}} \\log \\sbr{\\frac{16pMT_t}{\\delta_t}} ,.\n\t\\end{align*}\n\twhere inequality (a) uses the definition of event $\\mathcal{E}$.\n\t\n\tUsing Lemma~\\ref{lemma:technical_tool_bai_stage2} with $A= \\frac{192 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 k p L_{x} L_{\\theta}}{c_0} \\log\\sbr{\\frac{16p}{\\delta_t}} $, $B=\\frac{16p}{\\delta_t}$ and $\\kappa=\\min\\{\\frac{1}{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta}}, \\frac{\\omega}{6 L_x^2}\\}$, we have that if \n\t\\begin{align*}\n\t\tM T_t \\geq & 68 \\sbr{ \\frac{192 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 k p L_{x} L_{\\theta}}{c_0} \\log\\sbr{\\frac{16p}{\\delta_t}}}^2 \\cdot \\max\\lbr{ \\sbr{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta} }^2 ,\\ \\frac{6^2 L_x^4}{\\omega^2} } \\cdot \\\\& \\log^2 \\sbr{ \\frac{192 \\nbr{\\boldsymbol{X}_{\\textup{batch}}^{+}}^2 k p L_{x} L_{\\theta}}{c_0} \\log\\sbr{\\frac{16p}{\\delta_t}} \\cdot \\frac{16p}{\\delta_t} \\cdot \\max\\lbr{ 8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta} ,\\ \\frac{6 L_x^2}{\\omega} } } ,\n\t\n\t\n\t\n\t\\end{align*}\n\tthen $\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\min\\lbr{\\frac{1}{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta}},\\ \\frac{\\omega}{6 L_x^2}}$.\n\t\n\n\tAccording to Lemma~\\ref{lemma:X_batch_ub}, we have $\\|\\boldsymbol{X}_{\\textup{batch}}^{+}\\| \\leq \\sqrt{\\frac{(1+\\zeta) \\rho^{E}}{p}}$.\n\t\n\tThen, further enlarging $M T_t$, we have that if\n\t\\begin{align*}\n\t\tM T_t \\geq & \\frac{68 \\cdot 192^2 \\cdot 8^2 \\sbr{1+\\zeta}^3 (\\rho^E)^2 k^4 L_{x}^4 L_{\\theta}^2 L_{w}^2}{c_0^2} \\cdot \\max\\lbr{2^{2t},\\ \\frac{L_x^4}{\\omega^2}} \\cdot \\log^2 \\sbr{\\frac{16p}{\\delta_t}} \\\\& \\log^2 \\sbr{ \\frac{192 \\cdot 16 \\cdot 8  \\sbr{1+\\zeta}^{\\frac{3}{2}} \\rho^E k^2 p L_{x}^2 L_{\\theta} L_{w}}{c_0} \\cdot \\max\\lbr{2^t,\\ \\frac{L_x^2}{\\omega}} \\cdot \\frac{1}{\\delta_t} \\cdot \\log\\sbr{\\frac{16p}{\\delta_t}} } ,\n\t\\end{align*}\n\tthen\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\min\\lbr{\\frac{1}{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta}},\\ \\frac{\\omega}{6 L_x^2}} .\n\t\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Elimination with Low-dimensional Representations}\n\nFor clarity of notation, we also add subscript $t$ to the notations in subroutine $\\mathtt{EliLowRep}$ to denote the quantities generated in phase $t$. Specifically, we use the notations $\\hat{\\boldsymbol{B}}_t$, $\\hat{\\mathcal{X}}_{t,m}$, $\\boldsymbol{\\lambda}^{G}_{t,m}$, $\\rho^{G}_{t,m}$, $N_{t,m}$, $\\{\\boldsymbol{z}_{t,m,i}\\}_{i \\in [N_{t,m}]}$, $\\{r_{t,m,i}\\}_{i \\in [N_{t,m}]}$, $\\hat{\\boldsymbol{w}}_{t,m}$ and $\\hat{\\boldsymbol{\\theta}}_{t,m}$ to denote the corresponding quantities used in $\\mathtt{EliLowRep}$ in phase $t$.\n\n\nBefore analyzing the sample complexity of $\\mathtt{EliLowRep}$, we first prove that there exists a sample allocation $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}$ such that $\\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t$ is invertible, i.e., the G-optimal design optimization with $\\hat{\\boldsymbol{B}}_t$ is non-vacuous (Line~\\ref{line:bai_G_optimal_design} in Algorithm~\\ref{alg:elim_low_rep}).\n\n\nFor any task $m \\in [M]$, let\n\\begin{align*}\n\t\\boldsymbol{\\lambda}^*_m := & \\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{x}^{*}_{m} - \\boldsymbol{B}^\\top \\boldsymbol{x} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} } }{\\sbr{ ({\\boldsymbol{x}^{*}_{m}} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m }^2} .\n\\end{align*}\n$\\boldsymbol{\\lambda}^*_m$ is the optimal solution of the G-optimal design optimization with true feature extractor $\\boldsymbol{B}$.\n\n\n\\begin{lemma} \\label{lemma:invertible_under_hat_B}\n\tFor any phase $t>0$ and task $m \\in [M]$, if $\\|\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}\\| \\leq \\frac{\\omega}{6 L_x^2}$, we have\n\t\\begin{align*}\n\t\t\\sigma_{\\min}\\sbr{\\sum_{i=1}^{n} \\lambda^{*}_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t} > 0 .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:invertible_under_hat_B}]\n\tFor any task $m \\in [M]$, let $\\boldsymbol{A}_m:=\\sum_{i=1}^{n} \\lambda^{*}_m(\\boldsymbol{x}_i) \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top$.\n\tThen, for any phase $t>0$ and task $m \\in [M]$, we have\n\t\\begin{align*}\n\t\t\\sum_{i=1}^{n} \\lambda^{*}_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t = & \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{A}_m \\hat{\\boldsymbol{B}}_t\n\t\t\\\\\n\t\t= & \\hat{\\boldsymbol{B}}_t^\\top \\sbr{ \\boldsymbol{B} \\boldsymbol{B}^\\top + \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top } \\boldsymbol{A}_m \\sbr{ \\boldsymbol{B} \\boldsymbol{B}^\\top + \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top } \\hat{\\boldsymbol{B}}_t\n\t\t\\\\\n\t\t= & \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t \n\t\t\\\\& + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{A}_m \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{A}_m \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t .\n\t\\end{align*}\n\tHence, we have\n\t\\begin{align*}\n\t\t\\sigma_{\\min}\\sbr{\\sum_{i=1}^{n} \\lambda^{*}_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t} \\geq & \\sigma_{\\min}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t } - \\sigma_{\\max}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t }\n\t\t\\\\& - \\sigma_{\\max}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{A}_m \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t } - \\sigma_{\\max}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{A}_m \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t }\n\t\t\\\\\n\t\t\\geq & \\sigma_{\\min}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} } \\sigma_{\\min}\\sbr{ \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B} } \\sigma_{\\min}\\sbr{ \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t } -  \\nbr{ \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t } \\nbr{\\boldsymbol{A}_m}\n\t\t\\\\& - \\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} } \\nbr{\\boldsymbol{A}_m} - \\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} } \\nbr{\\boldsymbol{A}_m}\n\t\t\\\\\n\t\t\\geq & \\sigma^2_{\\min}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} } \\sigma_{\\min}\\sbr{ \\boldsymbol{B}^\\top \\boldsymbol{A}_m \\boldsymbol{B} } - 3\\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} } L_x^2\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\geq} & \\sbr{ 1 - \\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} }^2 } \\omega - 3\\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} } L_x^2 ,\n\t\\end{align*}\n\twhere inequality (a) uses the fact that $\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t = \\hat{\\boldsymbol{B}}_t^\\top (B \\boldsymbol{B}^\\top + \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top) \\hat{\\boldsymbol{B}}_t =\\hat{\\boldsymbol{B}}_t^\\top \\hat{\\boldsymbol{B}}_t=\\boldsymbol{I}_k$, and thus, $\\sigma^2_{\\min}( \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} )=1 - \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\|^2$.\n\t\n\t\n\t\n\tLet $\\|\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}\\| \\leq \\frac{\\omega}{6 L_x^2}$. Then, we have\n\t\\begin{align*}\n\t\t\\sigma_{\\min}\\sbr{\\sum_{i=1}^{n} \\lambda^{*}_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t} \\geq & \\sbr{ 1 - \\frac{\\omega^2}{36 L_x^4} } \\omega - \\frac{\\omega}{2} \n\t\t\\\\\n\t\t= & \\frac{\\omega}{2} - \\frac{\\omega^3}{36 L_x^4} \n\t\t\\\\\n\t\t> & 0 ,\n\t\\end{align*}\n\twhere the last inequality is due to $\\omega \\leq L_x^2 < \\sqrt{18} L_x^2$.\n\\end{proof}\n\n\nNext, we bound the optimal value $\\rho^{G}_{t,m}$ of the G-optimal design optimization with the estimated feature extractor $\\hat{\\boldsymbol{B}}_t$.\n\nFor any $\\mathcal{Z} \\subseteq \\mathcal{X}$, let $\\mathcal{Y}(\\mathcal{Z}):=\\{ \\boldsymbol{x}-\\boldsymbol{x}':\\ \\forall \\boldsymbol{x},\\boldsymbol{x}' \\in \\mathcal{Z},\\ \\boldsymbol{x} \\neq \\boldsymbol{x}' \\}$.\nRecall that in Line~\\ref{line:bai_G_optimal_design} of Algorithm~\\ref{alg:elim_low_rep}, for any phase $t>0$ and task $m \\in [M]$, \n$$\n\\rho^{G}_{t,m}:=\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} .\n$$\n\n\n\\begin{lemma} \\label{lemma:rho_t_leq_4k}\n\tFor any phase $t>0$ and task $m \\in [M]$, \n\t\\begin{align*}\n\t\t\\rho^{G}_{t,m} \\leq 4k .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:rho_t_leq_4k}]\n\tFor any phase $t>0$ and task $m \\in [M]$, we have that $\\hat{\\mathcal{X}}_{t,m} \\subseteq \\mathcal{X}$ and $ \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m}) \\subseteq \\mathcal{Y}(\\mathcal{X})$.\n\t\n\tFor any fixed $\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}$, \n\t\\begin{align*}\n\t\t\\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} \\leq & \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\mathcal{X})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t= & \\| \\hat{\\boldsymbol{B}}_t^\\top (\\boldsymbol{x}'_1-\\boldsymbol{x}'_2) \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t\\leq & \\sbr{\\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}'_1 \\|_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}+\\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}'_2 \\|_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}^2\n\t\t\\\\\n\t\t\\leq & 2\\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}'_1 \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} + 2\\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}'_2 \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t\\leq & 4 \\max_{\\boldsymbol{x} \\in \\mathcal{X}} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} , \n\t\\end{align*}\n\twhere $\\boldsymbol{x}'_1$ and $\\boldsymbol{x}'_2$ are the arms which satisfy that $\\boldsymbol{y}=\\boldsymbol{x}'_1-\\boldsymbol{x}'_2$ achieves the maximum value $\\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\mathcal{X})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}$.\n\t\n\tSince $\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\in \\mathbb{R}^k$, according to the Equivalence Theorem in \\cite{kiefer1960equivalence}, we have \n\t\\begin{align*}\n\t\t\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X}} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} = k .\n\t\\end{align*}\n\t\n\tTherefore, we have\n\t\\begin{align*}\n\t\t4k = & 4 \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X}} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} \\\\\n\t\t= & 4 \\max_{\\boldsymbol{x} \\in \\mathcal{X}} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda'(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t\\geq & \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda'(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t\\geq & \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\t\\\\\n\t\t= & \\rho^{G}_{t,m} ,\n\t\\end{align*}\n\twhere $\\boldsymbol{\\lambda}':=\\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X}} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}$.\n\\end{proof}\n\nNow we analyze the estimation error of the estimated reward parameter $\\hat{\\boldsymbol{\\theta}}_{t,m}=\\hat{\\boldsymbol{B}}_t \\hat{\\boldsymbol{w}}_{t,m}$ in $\\mathtt{EliLowRep}$.\n\nFor any phase $t>0$, task $m \\in [M]$ and arm $j \\in [N_{t,m}]$, let $\\xi_{t,m,j}$ denote the noise of the sample on arm $\\boldsymbol{z}_{t,m,j}$ for task $m$, during the execution of $\\mathtt{EliLowRep}$ in phase $t$ (Line~\\ref{line:bai_stage3_sample} in Algorithm~\\ref{alg:elim_low_rep}).\n\nFor any phase $t>0$, define events\n\\begin{align}\n\t\\mathcal{F}_t := \\Bigg \\{ &\n\t\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j} \n\t\\nonumber\\\\\n\t& \\leq \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}  \\sqrt{2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}}}, \\ \\forall m \\in [M] ,\\ \\forall \\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m}) \\Bigg\\}, \\label{eq:definition_cF}\n\\end{align}\nand \n\\begin{align*}\n\t\\mathcal{F} := \\cap_{t=1}^{\\infty} \\mathcal{F}_t .\n\\end{align*}\n\n\\begin{lemma}[Concentration of the Variance Term]\\label{lemma:concentration_variance_bai}\n\tIt holds that\n\n\t\n\t\n\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{F}} \\geq 1-\\frac{\\delta}{2} .  \n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:concentration_variance_bai}]\n\tLet $\\boldsymbol{\\Sigma}_{t,m}:=\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t$.\n\tThen, we can write\n\t\\begin{align*}\n\t\t\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j} =\\sum_{j=1}^{N_{t,m}} \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1}  \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j} .\n\t\\end{align*}\n\t\n\tFor any phase $t>0$, task $m \\in [M]$ and arm $j \\in [N_{t,m}]$, $\\hat{\\boldsymbol{B}}_t$, $\\boldsymbol{\\Sigma}_{t,m}$ and $\\{\\boldsymbol{z}_{t,m,j}\\}_{j=1}^{N_{t,m}}$ are fixed before the sampling in $\\mathtt{EliLowRep}$, and the noise $\\xi_{t,m,j}$ is 1-sub-Gaussian (Line~\\ref{line:bai_stage3_sample} in Algorithm~\\ref{alg:elim_low_rep}).\n\tThus, we have that for any $t>0$, $m \\in [M]$ and $j \\in [N_{t,m}]$, $\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j} $ is $(\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j})$-sub-Gaussian.\n\t\n\tUsing Hoeffding's inequality and taking a union bound over all $m \\in [M]$ and $\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})$, we have that with probability at least $1-\\frac{\\delta_t}{2}$,\n\t\\begin{align*}\n\t\n\t\n\t\t& \\sum_{j=1}^{N_{t,m}} \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1}  \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j}\n\t\t\\\\\n\t\t\\leq & \\sqrt{2 \\sum_{j=1}^{N_{t,m}} \\sbr{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j}}^2 \\cdot \\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} \n\t\t\\\\\n\t\t= & \\sqrt{2 \\sum_{j=1}^{N_{t,m}} \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\cdot \\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} \n\t\t\\\\\n\t\t= & \\sqrt{2 \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\boldsymbol{\\Sigma}_{t,m}^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\sum_{j=1}^{N_{t,m}} \\boldsymbol{z}_{t,m,j} \\cdot {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}  \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\cdot \\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} \n\t\t\\\\\n\t\t= & \\sqrt{2 \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\boldsymbol{\\Sigma}_{t,m}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\cdot \\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} \n\t\t\\\\\n\t\t= & \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\boldsymbol{\\Sigma}_{t,m}^{-1}} \\sqrt{2 \\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} ,\n\t\\end{align*}\n\twhich implies that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\mathcal{F}_t } \\geq 1 - \\frac{\\delta_t}{2} .\n\t\\end{align*}\n\t\n\tTaking a union bound over all phases $t \\geq 1$ and recalling $\\delta_t:=\\frac{\\delta}{2t^2}$, we obtain\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{F}} \n\t\t\\geq & 1- \\sum_{t=1}^{\\infty} \\Pr \\mbr{\\bar{\\mathcal{F}_t}}\n\t\t\\\\\n\t\t\\geq & 1- \\sum_{t=1}^{\\infty} \\frac{\\delta_t}{2}\n\t\t\\\\\n\t\t= & 1- \\sum_{t=1}^{\\infty} \\frac{\\delta}{4t^2}\n\t\t\\\\\n\t\t\\geq & 1-\\frac{\\delta}{2} .\n\t\\end{align*}\n\\end{proof}\n\n\n\\begin{lemma}[Concentration of $\\hat{\\boldsymbol{\\theta}}_{t,m}$] \\label{lemma:bai_concentration_theta_t_m}\n\tSuppose that event $\\mathcal{E} \\cap \\mathcal{F}$ holds. Then, for any phase $t>0$, task $m \\in [M]$ and $\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})$,\n\t\\begin{align*}\n\t\t\\abr{\\boldsymbol{y}^\\top \\sbr{\\hat{\\boldsymbol{\\theta}}_{t,m}-\\boldsymbol{\\theta}_m}} \\leq \\frac{1}{2^t} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:bai_concentration_theta_t_m}]\n\t\n\tFor any phase $t>0$, task $m \\in [M]$ and $\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})$,\n\t\\begin{align}\n\t\t\\boldsymbol{y}^\\top \\sbr{\\hat{\\boldsymbol{\\theta}}_{t,m}- \\boldsymbol{\\theta}_m} = & \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\hat{\\boldsymbol{w}}_{t,m}-\\boldsymbol{y}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t \\hat{\\boldsymbol{B}}_t^\\top + \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top } \\boldsymbol{\\theta}_m\n\t\t\\nonumber\\\\\n\t\t= & \\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\sbr{\\hat{\\boldsymbol{w}}_{t,m} - \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{\\theta}_m } -\\boldsymbol{y}^\\top  \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{\\theta}_m . \\label{eq:y_theta_est_err_decompose}\n\t\\end{align}\n\t\n\t\n\tHere, $\\hat{\\boldsymbol{w}}_{t,m}$ can be written as\n\t\\begin{align}\n\t\t\\hat{\\boldsymbol{w}}_{t,m} = & \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot r_{t,m,j}\n\t\t\\nonumber\\\\\n\t\t= & \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\sbr{{\\boldsymbol{z}_{t,m,j}}^\\top \\boldsymbol{\\theta}_m + \\xi_{t,m,j}}\n\t\t\\nonumber\\\\\n\t\t= & \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\sbr{{\\boldsymbol{z}_{t,m,j}}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t \\hat{\\boldsymbol{B}}_t^\\top + \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top } \\boldsymbol{\\theta}_m + \\xi_{t,m,j}}\n\t\t\\nonumber\\\\\n\t\t= & \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{\\theta}_m + \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{\\theta}_m \\nonumber\\\\& + \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j} . \\label{eq:hat_w_decompose}\n\t\\end{align}\n\t\n\tPlugging Eq.~\\eqref{eq:hat_w_decompose} into Eq.~\\eqref{eq:y_theta_est_err_decompose}, we can decompose the estimation error of $\\hat{\\boldsymbol{\\theta}}_{t,m}$ in $\\mathtt{EliLowRep}$ into three parts as\n\t\\begin{align*}\n\t\t\\boldsymbol{y}^\\top \\sbr{\\hat{\\boldsymbol{\\theta}}_{t,m}- \\boldsymbol{\\theta}_m} = & \n\t\t\\underbrace{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}_{\\textup{Bias}}  \\\\& + \\underbrace{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_t \\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1} \\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot \\xi_{t,m,j}}_{\\textup{Variance}} - \\underbrace{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}_{\\textup{Estimation error of $\\hat{\\boldsymbol{B}}_t$}} .\n\t\\end{align*}\n\t\n\t\n\tTaking the absolute value on both sides, and using the Cauchy\u2013Schwarz inequality and definition of event $\\mathcal{F}$ (Eq.~\\eqref{eq:definition_cF}), we have\n\t\\begin{align*}\n\t\t& \\abr{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{\\theta}}_{t,m}-\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m} \n\t\t\\\\\n\t\t\\leq & \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} \\cdot  \\nbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} \\cdot {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}_{\\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}  \\\\& + \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}  \\sqrt{2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} + \\abr{\\boldsymbol{y}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\frac{ \\sqrt{1+\\zeta}  \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}{\\sqrt{N_{t,m}}} \\cdot  L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}   \\cdot \\sum_{j=1}^{N_{t,m}}  \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j}}_{\\sbr{\\sum_{j=1}^{N_{t,m}} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{z}_{t,m,j} {\\boldsymbol{z}_{t,m,j}}^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}  \\\\& + \\frac{ \\sqrt{1+\\zeta} \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}{\\sqrt{N_{t,m}}} \\cdot \\sqrt{2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}}}  +  2 L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \n\t\t\\\\\n\t\t\\overset{\\textup{(b)}}{\\leq} & \\frac{ \\sqrt{1+\\zeta}  \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}{\\sqrt{N_{t,m}}} \\cdot  L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\cdot \\sqrt{k N_{t,m}}  \\\\& + \\frac{ \\sqrt{1+\\zeta} \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}{\\sqrt{N_{t,m}}} \\cdot \\sqrt{2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}}}  +  2 L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \n\t\t\\\\\n\t\t\\leq & \\sqrt{1+\\zeta} \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}} \\cdot  L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\cdot \\sqrt{k}  \\\\& + \\frac{ \\sqrt{1+\\zeta}  \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\sbr{\\sum_{i=1}^{n} \\lambda^{G}_{t,m}(\\boldsymbol{x}_i) \\cdot \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}}{\\sqrt{N_{t,m}}} \\cdot \\sqrt{2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}}} + 2 L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \n\t\t\\\\\n\t\t\\leq &  \\sqrt{(1+\\zeta) \\cdot k \\cdot \\rho^{G}_{t,m}} \\cdot  L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}  + \\frac{ \\sqrt{(1+\\zeta) \\cdot \\rho^{G}_{t,m} \\cdot 2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}} } }{\\sqrt{N_{t,m}}} +  2 L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}\n\t\t\\\\\n\t\t\\overset{\\textup{(c)}}{\\leq} &  \\sqrt{(1+\\zeta) \\cdot 4 k^2} \\cdot  L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}  + \\frac{ \\sqrt{(1+\\zeta) \\cdot \\rho^{G}_{t,m} \\cdot 2\\log \\sbr{\\frac{4n^2 M}{\\delta_t}} } }{\\sqrt{N_{t,m}}} +  2 L_{x} L_{w} \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}\n\t\t\\\\\n\t\t\\overset{\\textup{(d)}}{\\leq} & \\sqrt{(1+\\zeta) \\cdot 4 k^2} \\cdot L_{x} L_{w} \\cdot \\frac{1}{8 k L_{x} L_{w} \\cdot 2^t \\sqrt{1+\\zeta}}  + \\frac{1}{4 \\cdot 2^t} +  2 L_{x} L_{w} \\cdot \\frac{1}{8 k L_{x} L_{w}  \\cdot 2^t \\sqrt{1+\\zeta}}\n\t\t\\\\\n\t\t\\leq & \\frac{1}{4 \\cdot 2^t}  + \\frac{1}{4 \\cdot 2^t} +  \\frac{1}{4 \\cdot 2^t}\n\t\t\\\\\n\t\t\\leq & \\frac{1}{2^t} .\n\t\\end{align*}\n\tHere inequality (a) is due to the guarantee of rounding procedure $\\mathtt{ROUND}$ and the triangle inequality. Inequality (b) uses Lemma~\\ref{lemma:sqrt_n_k}, and inequality (c) follows from Lemma~\\ref{lemma:rho_t_leq_4k}. Inequality (d) comes from Lemma~\\ref{lemma:concentration_B_hat_t} and $N_{t,m}:=\\max \\{ \\lceil  32 \\cdot 2^{2t} (1+\\zeta) \\rho^{G}_{t,m} \\log (\\frac{4n^2 M}{\\delta_t}) \\rceil,\\ \\frac{180k}{\\zeta^2} \\}$.\n\\end{proof}\n\n\n\nFor any task $m \\in [M]$ and arm $\\boldsymbol{x} \\in \\mathcal{X}$, let $\\Delta_m(\\boldsymbol{x}):=({\\boldsymbol{x}^{*}_{m}} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m$ denote the reward gap between the optimal arm $\\boldsymbol{x}^{*}_{m}$ and arm $\\boldsymbol{x}$ in task $m$.\nFor any phase $t>0$ and task $m \\in [M]$, let $\\mathcal{Z}_{t,m}:=\\{ \\boldsymbol{x} \\in \\mathcal{X} : \\Delta_{m}(\\boldsymbol{x}) \\leq 4 \\cdot 2^{-t} \\}$.\n\n\\begin{lemma} \\label{lemma:cX_subseteq_cS}\n\tSuppose that event $\\mathcal{E} \\cap \\mathcal{F}$ holds. For any phase $t>0$ and task $m \\in [M]$,\n\t\\begin{align*}\n\t\t\\boldsymbol{x}^{*}_{m} \\in \\hat{\\mathcal{X}}_{t,m} ,\n\t\\end{align*}\n\tand for any phase $t\\geq2$ and task $m \\in [M]$,\n\t\\begin{align*}\n\t\t\\hat{\\mathcal{X}}_{t,m} \\subseteq \\mathcal{Z}_{t,m} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:cX_subseteq_cS}]\n\tThis proof follows a similar analytical procedure as that of Lemma~2 in \\cite{fiez2019sequential}.\n\t\n\tFirst, we prove $\\boldsymbol{x}^{*}_{m} \\in \\hat{\\mathcal{X}}_{t,m}$ for any phase $t>0$ and task $m \\in [M]$ by contradiction.\n\t\n\tSuppose that for some $t>0$ and some $m \\in [M]$, $\\boldsymbol{x}^{*}_{m}$ is eliminated from $\\hat{\\mathcal{X}}_{t,m}$ in phase $t$.\n\tThen, we have that there exists some $\\boldsymbol{x}' \\in \\hat{\\mathcal{X}}_{t,m}$ such that\n\t\\begin{align*}\n\t\t(\\boldsymbol{x}'-\\boldsymbol{x}^{*}_{m})^\\top \\hat{\\boldsymbol{\\theta}}_{t,m} > 2^{-t} .\n\t\\end{align*}\n\tThen, we have\n\t\\begin{align*}\n\t\t(\\boldsymbol{x}'-\\boldsymbol{x}^{*}_{m})^\\top \\boldsymbol{\\theta}_m = & (\\boldsymbol{x}'-\\boldsymbol{x}^{*}_{m})^\\top \\hat{\\boldsymbol{\\theta}}_{t,m} - (\\boldsymbol{x}'-\\boldsymbol{x}^{*}_{m})^\\top \\sbr{\\hat{\\boldsymbol{\\theta}}_{t,m} - \\boldsymbol{\\theta}_m} \n\t\t\\\\\n\t\t\\geq & (\\boldsymbol{x}'-\\boldsymbol{x}^{*}_{m})^\\top \\hat{\\boldsymbol{\\theta}}_{t,m} - 2^{-t}\n\t\t\\\\\n\t\t> & 2^{-t} - 2^{-t}\n\t\t\\\\\n\t\t= & 0 ,\n\t\\end{align*}\n\twhich contradicts the definition of $\\boldsymbol{x}^{*}_{m}$. Thus, we obtain that $\\boldsymbol{x}^{*}_{m} \\in \\hat{\\mathcal{X}}_{t,m}$ for any phase $t>0$ and task $m \\in [M]$.\n\t\n\tNext, we prove $\\hat{\\mathcal{X}}_{t,m} \\subseteq \\mathcal{Z}_{t,m}$ for any phase $t\\geq2$ and task $m \\in [M]$, i.e., each $\\boldsymbol{x} \\in \\hat{\\mathcal{X}}_{t,m}$ satisfies that $\\Delta_m(\\boldsymbol{x}) \\leq 4 \\cdot 2^{-t}$. \n\t\n\tSuppose that there exists some phase $t$, some task $m$ and some $\\boldsymbol{x} \\in \\hat{\\mathcal{X}}_{t,m}$ such that $\\Delta_m(\\boldsymbol{x}) > 4 \\cdot 2^{-t}$. Then, in phase $t-1 \\geq 1$, we have\n\t\\begin{align*}\n\t\t(\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x})^\\top \\hat{\\boldsymbol{\\theta}}_{t-1,m} = & (\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m - (\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x})^\\top \\sbr{\\boldsymbol{\\theta}_m-\\hat{\\boldsymbol{\\theta}}_{t-1,m}} \n\t\t\\\\\n\t\t\\geq & (\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m - 2^{-(t-1)}\n\t\t\\\\\n\t\t> & 4 \\cdot 2^{-t} - 2^{-(t-1)}\n\t\t\\\\\n\t\t= & 2^{-(t-1)} , \n\t\\end{align*}\n\twhich implies that $x$ should have been eliminated from $\\hat{\\mathcal{X}}_{t,m}$ in phase $t-1$, and contradicts our supposition. Thus, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:bai_ub}}\n\nBefore proving Theorem~\\ref{thm:bai_ub}, we first introduce a useful lemma.\n\nFor any task $m \\in [M]$, let\n\\begin{align*}\n\t\\boldsymbol{\\lambda}^*_m := & \\operatornamewithlimits{argmin}_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{x}^{*}_{m} - \\boldsymbol{B}^\\top \\boldsymbol{x} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} } }{ \\sbr{({\\boldsymbol{x}^{*}_{m}} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m}^2 } ,\n\\end{align*}\nand\n\\begin{align*}\n\t\\rho^*_m := & \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{x}^{*}_{m} - \\boldsymbol{B}^\\top \\boldsymbol{x} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} } }{ \\sbr{({\\boldsymbol{x}^{*}_{m}} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m}^2 } .\n\\end{align*}\n$\\boldsymbol{\\lambda}^*_m$ and $\\rho^*_m$ are the optimal solution and the optimal value of the G-optimal design optimization with true feature extractor $\\boldsymbol{B}$, respectively.\n\n\n\\begin{lemma}\\label{lemma:connection_ub_lb}\n\tSuppose that event $\\mathcal{E} \\cap \\mathcal{F}$ holds. For any task $m \\in [M]$ and $\\boldsymbol{y} \\in \\mathbb{R}^d$,\n\t\\begin{align*}\n\t\t\\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t }^{-1} } \\leq \\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} } + \\frac{ 11 L_{x}^4 }{ k \\omega^2 \\cdot 2^t } .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:connection_ub_lb}]\n\t\n\tWe first handle the term $( \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t )^{-1}$. \n\t\n\tFor any task $m \\in [M]$, we have\n\t\\begin{align*}\n\t\t& \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t \n\t\t\\\\\n\t\t= & \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i}^\\top\n\t\t\\\\\n\t\t= & \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\Bigg( \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top + \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i}^\\top \n\t\t\\\\\n\t\t& + \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top + \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i}^\\top \\Bigg)\n\t\t\\\\\n\t\t= & \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top\n\t\t+ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\Bigg( \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i}^\\top \n\t\t\\\\& + \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top + \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i}^\\top \\Bigg) .\n\t\\end{align*}\n\t\n\tLet $\\boldsymbol{P}_t:=\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i) \\cdot (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i)^\\top$.\n\tLet $\\boldsymbol{Q}_t:=\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) ( (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i) \\cdot (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i)^\\top \n\t+ (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i) \\cdot (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i)^\\top + (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i) \\cdot (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{x}_i)^\\top )$.\n\tThen, we have $\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t = \\boldsymbol{P}_t+\\boldsymbol{Q}_t$.\n\t\n\tFrom Assumption~\\ref{assumption:lambda^*_B_x_x_B_invertible}, we have that for any task $m \\in [M]$, $\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}$ is invertible.\n\tSince $\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}$ is also invertible, we have that $\\boldsymbol{P}_t$ is invertible. \n\tAccording to Lemmas~\\ref{lemma:concentration_B_hat_t} and \\ref{lemma:invertible_under_hat_B}, we have that $\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t$ is also invertible.\n\tThus, we can write $(\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t)^{-1}$ as follows.\n\t\\begin{align*}\n\t\t\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t }^{-1} = & \\boldsymbol{P}_t^{-1}-\\sbr{\\boldsymbol{P}_t + \\boldsymbol{Q}_t}^{-1} \\boldsymbol{Q}_t \\boldsymbol{P}_t^{-1} \n\t\\end{align*}\n\t\n\t\n\tHence, for any task $m \\in [M]$ and $\\boldsymbol{y} \\in \\mathbb{R}^d$, we have\n\t\\begin{align}\n\t\t\\|\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}\\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t }^{-1}} = &\n\t\t\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t }^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \n\t\t\\nonumber\\\\\n\t\t= & \\underbrace{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\boldsymbol{P}_t^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\textup{Term 1}} - \\underbrace{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\sbr{\\boldsymbol{P}_t + \\boldsymbol{Q}_t}^{-1} \\boldsymbol{Q}_t \\boldsymbol{P}_t^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}_{\\textup{Term 2}} . \\label{eq:norm_B_hat_y_decompose}\n\t\\end{align}\n\t\n\t\n\tFrom Lemma~\\ref{lemma:concentration_B_hat_t}, we have\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\min\\lbr{\\frac{1}{8 k \\cdot 2^t \\sqrt{1+\\zeta}},\\ \\frac{\\omega}{6 L_x^2}} \\leq \\min\\lbr{\\frac{1}{8 k \\cdot 2^t},\\ \\frac{\\omega}{6 L_x^2}} .\n\t\\end{align*}\n\t\n\t\n\tSince $\\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_{t} \\hat{\\boldsymbol{B}}_{t}^\\top \\boldsymbol{B} + \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} = \\boldsymbol{B}^\\top (\\hat{\\boldsymbol{B}}_{t} \\hat{\\boldsymbol{B}}_{t}^\\top + \\hat{\\boldsymbol{B}}_{t,\\bot} \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top) \\boldsymbol{B} = \\boldsymbol{B}^\\top \\boldsymbol{B} =\\boldsymbol{I}_k$, we have $\\sigma^2_{\\min} ( \\hat{\\boldsymbol{B}}_{t}^\\top \\boldsymbol{B} )=1 - \\| \\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B} \\|^2$.\n\t\n\tThus, we have\n\t\\begin{align*}\n\t\t\\sigma_{\\min}(\\hat{\\boldsymbol{B}}_{t}^\\top \\boldsymbol{B}) = \\sqrt{1 - \\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}}^2} \\geq \\sqrt{1 - \\min\\lbr{\\frac{1}{64 k^2 \\cdot 2^{2t}},\\ \\frac{\\omega^2}{36 L_x^4}} } > 0 ,\n\t\\end{align*}\n\twhich implies that $\\hat{\\boldsymbol{B}}_{t}^\\top \\boldsymbol{B}$ is invertible.\n\t\n\n\t\n\tNow, we first analyze Term 1 in Eq.~\\eqref{eq:norm_B_hat_y_decompose}.\n\t\n\t\\begin{align*}\n\t\t\\textup{Term 1} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\boldsymbol{P}_t^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y} + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} }^\\top \\boldsymbol{P}_t^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y} + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} }\n\t\t\\\\\n\t\t\\\\\n\t\t= & \\underbrace{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\boldsymbol{P}_t^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}}_{\\textup{Term 1-1}} + \\underbrace{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\boldsymbol{P}_t^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} } }_{\\textup{Term 1-2}}\n\t\t\\\\\n\t\t& + \\underbrace{\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} }^\\top \\boldsymbol{P}_t^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}}_{\\textup{Term 1-3}} + \\underbrace{\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top B_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} }^\\top \\boldsymbol{P}_t^{-1} \\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} }}_{\\textup{Term 1-4}} .\n\t\\end{align*}\n\t\n\t\n\tIn the following, we bound Terms 1-1, 1-2, 1-3 and 1-4, respectively.\n\t\n\tFirst, we have\n\t\\begin{align*}\n\t\t\\textup{Term 1-1} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B}}  \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^\\top }^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} }^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\boldsymbol{B}^\\top \\boldsymbol{y} \n\t\t\\\\\n\t\t= & \\nbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1}} .\n\t\\end{align*}\n\t\n\tWe note that since $\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\hat{\\boldsymbol{B}}_t + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\hat{\\boldsymbol{B}}_t = \\hat{\\boldsymbol{B}}_t^\\top (B \\boldsymbol{B}^\\top + \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top) \\hat{\\boldsymbol{B}}_t = \\hat{\\boldsymbol{B}}_t^\\top \\hat{\\boldsymbol{B}}_t =\\boldsymbol{I}_k$, $\\sigma^2_{\\min}\\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} }=1 - \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}}^2$.\n\tIn addition, $\\nbr{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1}}=\\frac{1}{ \\sigma_{\\min}(\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}) }=\\frac{1}{ \\sqrt{1 - \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}}^2} }$.\n\t\n\tThen, second, we have\n\t\\begin{align*}\n\t\t\\textup{Term 1-2} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B}}  \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^\\top }^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} }^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}  \n\t\t\\\\\n\t\t\\leq & 2 L_{x} \\cdot \\frac{1}{\\omega} \\cdot \\frac{1}{ \\sqrt{1-\\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}}^2 } } \\cdot \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}} \\cdot 2 L_{x}\n\t\t\\\\\n\t\t\\leq & 4 L_{x}^2 \\cdot \\frac{1}{\\omega} \\cdot \\frac{1}{\\sqrt{ 1-\\sbr{\\frac{1}{8k \\cdot 2^t}}^2 }} \\cdot \\frac{1}{8k \\cdot 2^t}\n\t\t\\\\\n\t\t\\leq & 4 L_{x}^2 \\cdot \\frac{1}{\\omega} \\cdot \\frac{1}{\\sqrt{ 1-\\frac{3}{4} }} \\cdot \\frac{1}{8k \\cdot 2^t}\n\t\t\\\\\n\t\t= & \\frac{ L_{x}^2 }{ k \\omega \\cdot 2^t} .\n\t\\end{align*}\n\t\n\tThird, we have\n\t\\begin{align*}\n\t\t\\textup{Term 1-3} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B}}  \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^\\top }^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} }^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{ \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\boldsymbol{B}^\\top \\boldsymbol{y} ,\n\t\t\\\\\n\t\t\\leq & \\frac{1}{ \\sqrt{1-\\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}}^2 } } \\cdot \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}} \\cdot 2 L_{x} \\cdot \\frac{1}{\\omega} \\cdot 2 L_{x} \n\t\t\\\\\n\t\t\\leq & 4 L_{x}^2 \\cdot \\frac{1}{\\omega} \\cdot \\frac{1}{\\sqrt{ 1-\\sbr{\\frac{1}{8k \\cdot 2^t}}^2 }} \\cdot \\frac{1}{8k \\cdot 2^t}\n\t\t\\\\\n\t\t\\leq & 4 L_{x}^2 \\cdot \\frac{1}{\\omega} \\cdot \\frac{1}{\\sqrt{ 1-\\frac{3}{4} }} \\cdot \\frac{1}{8k \\cdot 2^t}\n\t\t\\\\\n\t\t= & \\frac{ L_{x}^2 }{k \\omega \\cdot 2^t} .\n\t\\end{align*}\n\t\n\tFinally, we have\n\t\\begin{align*}\n\t\t\\textup{Term 1-4} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i} \\cdot \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\boldsymbol{B}^\\top \\boldsymbol{x}_i}^\\top}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} \\sbr{\\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B}}  \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^\\top }^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{\\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} }^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}\n\t\t\\\\\n\t\t= & \\sbr{ \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y}}^\\top \\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1} \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}}^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} \\boldsymbol{B}_{\\bot}^\\top \\boldsymbol{y} ,\n\t\t\\\\\n\t\t\\leq & \\sbr{\\frac{1}{ \\sqrt{1-\\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}}^2 } } \\cdot \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}} \\cdot 2 L_{x} }^2 \\cdot \\frac{1}{\\omega}\n\t\t\\\\\n\t\t\\leq & \\sbr{ 2 L_{x} \\cdot \\frac{1}{\\sqrt{ 1-\\sbr{\\frac{1}{8k \\cdot 2^t}}^2 }} \\cdot \\frac{1}{8k \\cdot 2^t} }^2 \\cdot \\frac{1}{\\omega}\n\t\t\\\\\n\t\t\\leq & \\sbr{ 2 L_{x} \\cdot \\frac{1}{\\sqrt{ 1-\\frac{3}{4} }} \\cdot \\frac{1}{8k \\cdot 2^t} }^2 \\cdot \\frac{1}{\\omega}\n\t\t\\\\\n\t\t= & \\frac{L_{x}^2}{4k^2 \\omega \\cdot 2^{2t}} .\n\t\\end{align*}\n\t\n\tThus, we have\n\t\\begin{align}\n\t\t\\textup{Term 1} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\boldsymbol{P}_t^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}\n\t\t\\nonumber\\\\\n\t\t\\leq & \\nbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1}} + \\frac{ 2L_{x}^2 }{k \\omega \\cdot 2^t} + \\frac{L_{x}^2}{4k^2 \\omega \\cdot 2^{2t}} \n\t\t\\nonumber\\\\\n\t\t\\leq & \\nbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1}} + \\frac{ 3L_{x}^2 }{k \\omega \\cdot 2^t} . \\label{eq:term1_ub}\n\t\\end{align}\n\t\n\t\n\tNext, we investigate Term 2. In order to bound Term 2, we first bound the minimum singular value of $\\boldsymbol{P}_t$ and the maximum singular value of $\\boldsymbol{Q}_t$.\n\t\n\t\n\tSince $\\boldsymbol{P}_t=\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} (\\sum_{i=1}^{n} \\lambda^*(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}) (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B})^\\top$, we have\n\t\\begin{align*}\n\t\t\\sigma_{\\min}(\\boldsymbol{P}_t) \\geq & \\sigma^2_{\\min}(\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}) \\cdot \\omega\n\t\t\\\\\n\t\t= & \\sbr{1-\\|\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot}\\|^2 } \\omega \n\t\t\\\\\n\t\t\\geq & \\sbr{ 1-\\frac{1}{8^2 k^2 \\cdot 2^{2t}} } \\omega \n\t\t\\\\\n\t\t\\geq & \\frac{3}{4} \\omega .\n\t\\end{align*}\n\t\n\n\tSince $\\boldsymbol{Q}_t=\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B} (\\sum_{i=1}^{n} \\lambda^*(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}_{\\bot} ) (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot})^\\top \n\t+ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} (\\sum_{i=1}^{n} \\lambda^*(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top_{\\bot} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B} ) (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B})^\\top + \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} (\\sum_{i=1}^{n} \\lambda^*(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top_{\\bot} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}_{\\bot} ) (\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot})^\\top $, we have \n\t\\begin{align*}\n\t\t\\sigma_{\\max}(\\boldsymbol{Q}_t) \\leq & 3 L_{x}^2 \\nbr{ \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{B}_{\\bot} } \n\t\t\\\\\n\t\t\\leq & \\min \\lbr{ \\frac{3 L_{x}^2}{8k \\cdot 2^t} , \\ \\frac{ \\omega }{2} } .\n\t\\end{align*}\n\t\n\tThen, we can bound Term 2 as \n\t\\begin{align}\n\t\t\\textup{Term 2} = & \\sbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^\\top \\sbr{\\boldsymbol{P}_t + \\boldsymbol{Q}_t}^{-1} \\boldsymbol{Q}_t \\boldsymbol{P}_t^{-1} \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}\n\t\t\\nonumber\\\\\n\t\t\\leq & \\nbr{\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}}^2 \\cdot \\nbr{\\sbr{\\boldsymbol{P}_t + \\boldsymbol{Q}_t}^{-1}} \\cdot \\nbr{\\boldsymbol{Q}_t} \\cdot \\nbr{\\boldsymbol{P}_t^{-1}}\n\t\t\\nonumber\\\\\n\t\t\\leq & \\frac{ 4 L_{x}^2 \\cdot \\sigma_{\\max}\\sbr{\\boldsymbol{Q}_t} }{ \\sigma_{\\min}\\sbr{\\boldsymbol{P}_t + \\boldsymbol{Q}_t} \\cdot \\sigma_{\\min}\\sbr{\\boldsymbol{P}_t} }\n\t\t\\nonumber\\\\\n\t\t\\leq & \\frac{ 4 L_{x}^2 \\cdot \\sigma_{\\max}\\sbr{\\boldsymbol{Q}_t} }{ \\sbr{\\sigma_{\\min}\\sbr{\\boldsymbol{P}_t} - \\sigma_{\\max}\\sbr{\\boldsymbol{Q}_t}} \\cdot \\sigma_{\\min}\\sbr{\\boldsymbol{P}_t} }\n\t\t\\nonumber\\\\\n\t\n\t\n\t\t\\leq & \\frac{ 4 L_{x}^2 \\cdot \\frac{3 L_{x}^2}{8k \\cdot 2^t} }{ \\sbr{ \\frac{3}{4} \\omega - \\frac{ 1 }{2} \\omega } \\cdot \\frac{3}{4} \\omega }\n\t\t\\nonumber\\\\\n\t\t= & \\frac{ 8 L_{x}^4 }{ k \\omega^2 \\cdot 2^t } . \\label{eq:term2_ub}\n\t\\end{align}\n\t\n\t\n\tPlugging Eqs.~\\eqref{eq:term1_ub} and \\eqref{eq:term2_ub} into Eq.~\\eqref{eq:norm_B_hat_y_decompose}, we have\n\t\\begin{align*}\n\t\t\\|\\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y}\\|^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t }^{-1}} \\leq & \\nbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1}} + \\frac{ 3L_{x}^2 }{k \\omega \\cdot 2^t} + \\frac{ 8 L_{x}^4 }{ k \\omega^2 \\cdot 2^t } .\n\t\t\\\\\n\t\t\\leq & \\nbr{\\boldsymbol{B}^\\top \\boldsymbol{y}}^2_{\\sbr{ \\sum_{i=1}^{n} \\lambda^*_m(\\boldsymbol{x}_i)  \\boldsymbol{B}^\\top \\boldsymbol{x}_i  \\boldsymbol{x}_i^\\top \\boldsymbol{B} }^{-1}} + \\frac{ 11 L_{x}^4 }{ k \\omega^2 \\cdot 2^t } .\n\t\\end{align*}\n\t\n\\end{proof}\n\nBelow we prove the sample complexity for algorithm $\\mathtt{DouExpDes}$ (Theorem~\\ref{thm:bai_ub}).\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bai_ub}]\n\tAccording to Lemmas~\\ref{lemma:Z_t_est_error} and \\ref{lemma:concentration_variance_bai}, we have $\\Pr [\\mathcal{E} \\cap \\mathcal{F}] \\geq 1-\\delta$. Below, supposing that event $\\mathcal{E} \\cap \\mathcal{F}$ holds, we prove the correctness and sample complexity.\n\t\n\tWe first prove the correctness.\n\t\n\tFor any task $m \\in [M]$, let $t^{*}_m$ denote the first phase which satisfies $|\\hat{\\mathcal{X}}_{t,m}|=1$. Let $t_*=\\max_{m \\in [M]} t^{*}_m$ denote the total number of phases used.\n\tFor any task $m \\in [M]$, let $\\Delta_{m,\\min}:=\\min_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} (\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m$ denote the minimum reward gap for task $m$. Let $\\Delta_{\\min}:=\\min_{m \\in [M]} \\Delta_{\\min,m}$ denote the minimum reward gap among all tasks.\n\t\n\tFrom Lemma~\\ref{lemma:cX_subseteq_cS}, we can obtain the following facts: (i) For any task $m \\in [M]$, the optimal arm $\\boldsymbol{x}^{*}_{m}$ will never be eliminated. (ii) $t^{*}_m \\leq \\lceil \\log(\\frac{4}{\\Delta_{m,\\min}}) \\rceil +1$, and thus, $t_*\\leq \\lceil \\log(\\frac{4}{\\Delta_{\\min}}) \\rceil +1$. Therefore, after at most $\\lceil \\log(\\frac{4}{\\Delta_{\\min}}) \\rceil +1$ phases, algorithm $\\mathtt{DouExpDes}$ will return the optimal arms $\\boldsymbol{x}^{*}_{m}$ for all tasks $m \\in [M]$.\n\t\n\tNow we prove the sample complexity. \n\tIn the following, we first prove that the sample complexity of algorithm $\\mathtt{DouExpDes}$ is bounded by $\\tilde{O} ( \\frac{M k}{\\Delta_{\\min}^2}  \\log(\\delta^{-1}) + (\\rho^E)^2 d k^4  L_{x}^2 L_{w}^2  D \\log^4(\\delta^{-1}) )$.\n\t\n\tRecall that $p=\\frac{180d}{\\zeta^2}$ and $\\zeta=\\frac{1}{10}$. Then, summing the number of samples used in subroutines $\\mathtt{FeatRecover}$ and $\\mathtt{EliLowRep}$ in all phases (Line~\\ref{line:bai_stage2_sample} in Algorithm~\\ref{alg:feat_recover}, Line~\\ref{line:bai_stage3_sample} in Algorithm~\\ref{alg:elim_low_rep}), we have that the total number of samples is\n\t\\begin{align}\n\t\t& \\sum_{t=1}^{t^*} p M T_t + \\sum_{m=1}^{M} \\sum_{t=1}^{t^{*}_m} N_{t,m}\n\t\t\\nonumber\\\\\n\t\t= & \\sum_{t=1}^{t^*} p \\cdot O \\Bigg( \\sbr{1+\\zeta}^3 (\\rho^E)^2 k^4 L_{x}^2 L_{\\theta}^2  \\max\\lbr{2^{2t},\\ \\frac{L_{x}^4}{\\omega^2}}  \\log^2 \\sbr{\\frac{p}{\\delta_t}} \\cdot \\nonumber\\\\& \\hspace*{5em} \\log^2 \\sbr{ \\sbr{1+\\zeta} \\rho^E k p L_{x} L_{\\theta}  \\max\\lbr{2^t,\\ \\frac{L_{x}}{\\omega}}  \\frac{1}{\\delta_t} \\log\\sbr{\\frac{p}{\\delta_t}} } \\Bigg)\n\t\t\\nonumber\\\\\n\t\t& + \\sum_{m=1}^{M} \\sum_{t=1}^{t^{*}_m} O\\sbr{ 2^{2t} (1+\\zeta) \\rho^{G}_{t,m} \\log \\sbr{\\frac{n^2 M}{\\delta_t}} + \\frac{k}{\\zeta^2} }\n\t\t\\nonumber\\\\\n\t\t= & \\sum_{t=1}^{O(\\log(\\Delta_{\\min}^{-1}))} O \\Bigg( (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{2^{2t},\\ \\frac{L_{x}^4}{\\omega^2}}  \\log^2 \\sbr{\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}} \\cdot \\nonumber\\\\& \\hspace*{5em} \\log^2 \\sbr{ \\rho^E k d L_{x} L_{\\theta}  \\max\\lbr{\\Delta_{\\min}^{-1},\\ \\frac{L_{x}}{\\omega}}  \\frac{\\log(\\Delta_{\\min}^{-1})}{\\delta} \\log\\sbr{\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}} } \\Bigg)\n\t\t\\nonumber\\\\\n\t\t& + \\sum_{m=1}^{M} \\sum_{t=1}^{O(\\log(\\Delta_{m,\\min}^{-1}))} O\\sbr{ 2^{2t} \\rho^{G}_{t,m} \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{m,\\min}^{-1})  }{\\delta}}  + k}\n\t\t\\label{eq:bai_sample_complexity} \\\\\n\t\t= & O \\Bigg( (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{ \\Delta_{\\min}^{-2} ,\\ \\frac{ L_{x}^4 \\log(\\Delta_{\\min}^{-1}) }{\\omega^2} }  \\log^2 \\sbr{\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}} \\cdot \\nonumber\\\\& \\hspace*{5em} \\log^2 \\sbr{ \\rho^E k d L_{x} L_{\\theta}  \\max\\lbr{\\Delta_{\\min}^{-1},\\ \\frac{L_{x}}{\\omega}}  \\frac{\\log(\\Delta_{\\min}^{-1})}{\\delta} \\log\\sbr{\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}} } \\Bigg)\n\t\t\\nonumber\\\\\n\t\t& + O\\sbr{ M k \\Delta_{\\min}^{-2} \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{\\min}^{-1})}{\\delta}} } . \\nonumber\n\t\\end{align}\n\\end{proof}\n\n\nNext, we prove that the sample complexity of algorithm $\\mathtt{DouExpDes}$ is bounded by \n$\n\\tilde{O} ( \\sum_{m=1}^{M}  \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{x} \\in \\mathcal{X} \\setminus \\{\\boldsymbol{x}^{*}_{m}\\}} \\frac{\\| \\boldsymbol{B}^\\top (\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x}) \\|^2_{\\boldsymbol{A}(\\boldsymbol{\\lambda})^{-1}} }{ ((\\boldsymbol{x}^{*}_{m} - \\boldsymbol{x})^\\top \\boldsymbol{\\theta}_m)^2 } \\log (\\delta^{-1}) +  (\\rho^E)^2 d k^4 L_{x}^2 L_{w}^2 D \\log^4 (\\delta^{-1}) ) .\n$\n\n\nFrom Eq.~\\eqref{eq:bai_sample_complexity}, we have that with probability $1-\\delta$, the number of samples used by algorithm $\\mathtt{DouExpDes}$ is bounded by\n\\begin{align}\n\t\\tilde{O} \\Bigg( \\sum_{t=1}^{\\log(\\Delta_{\\min}^{-1})} (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{2^{2t},\\ \\frac{L_{x}^4}{\\omega^2}} + \\sum_{m=1}^{M} \\sum_{t=1}^{\\log(\\Delta_{m,\\min}^{-1})} 2^{2t} \\rho^{G}_{t,m} + Mk \\Bigg) .\t\\label{eq:bai_sample_complexity_tilde}\n\\end{align}\n\n\n\n\n\nFor any $\\mathcal{Z} \\subseteq \\mathcal{X}$, $\\mathcal{Y}(\\mathcal{Z}):=\\{ \\boldsymbol{x}-\\boldsymbol{x}':\\ \\forall \\boldsymbol{x},\\boldsymbol{x}' \\in \\mathcal{Z},\\ \\boldsymbol{x} \\neq \\boldsymbol{x}' \\}$ and $\\mathcal{Y}^*_m(\\mathcal{Z}):=\\{ \\boldsymbol{x}^{*}_{m}-\\boldsymbol{x}:\\ \\forall \\boldsymbol{x} \\in \\mathcal{Z},\\ \\boldsymbol{x} \\neq \\boldsymbol{x}^{*}_{m} \\}$.\nThen, we have that for any task $m \\in [M]$ and phase $t\\geq 2$,\n\n\\begin{align}\n\t\\sbr{2^t}^2 \\rho^{G}_{t,m} \n\t= & \\sbr{2^t}^2 \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\\nonumber\\\\\n\t\\leq & \\sbr{2^t}^2 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\hat{\\mathcal{X}}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\\nonumber\\\\\n\t\\overset{\\textup{(a)}}{\\leq} & \\sbr{2^t}^2 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}(\\mathcal{Z}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\\nonumber\\\\\n\t\\overset{\\textup{(b)}}{\\leq} & 4 \\sbr{2^t}^2 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{Z}_{t,m})} \\| \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\hat{\\boldsymbol{B}}_t^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\hat{\\boldsymbol{B}}_t}^{-1}}\n\t\\nonumber\\\\\n\t\\overset{\\textup{(c)}}{\\leq} & 4 \\sbr{2^t}^2 \\sbr{\\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{Z}_{t,m})} \\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}} + \\frac{ 11 L_{x}^4 }{ \\omega^2 k \\cdot 2^t } }\n\t\\nonumber\\\\\n\t= & 4 \\sbr{ \\frac{16 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{Z}_{t,m})} \\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{\\sbr{4 \\cdot 2^{-t}}^2} +  \\frac{ 11 L_{x}^4 \\cdot 2^t }{ \\omega^2 k } }\n\t\\nonumber\\\\\n\t\\overset{\\textup{(d)}}{\\leq} & 4 \\sbr{ 16 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{Z}_{t,m})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2} + \\frac{ 11 L_{x}^4 \\cdot 2^t }{ \\omega^2 k } }\n\t\\nonumber\\\\\n\t\\leq & 4 \\sbr{ 16 \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{X})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda^*_m(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2} + \\frac{ 11 L_{x}^4 \\cdot 2^t }{ \\omega^2 k } }\n\t\\nonumber\\\\\n\t\\overset{\\textup{(e)}}{=} & 4 \\sbr{ 16 \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{X})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2} + \\frac{ 11 L_{x}^4 \\cdot 2^t }{ \\omega^2 k } } . \\label{eq:connection_ub_lb}\n\\end{align}\nHere inequality (a) is due to $\\hat{\\mathcal{X}}_{t,m} \\subseteq \\mathcal{Z}_{t,m}$ (from Lemma~\\ref{lemma:cX_subseteq_cS}). Inequality (b) uses the fact that for any $\\boldsymbol{y}=\\boldsymbol{x}_i-\\boldsymbol{x}_j \\in \\mathcal{Y}(\\mathcal{Z}_{t,m})$, we can write $\\boldsymbol{y}=(\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x}_j)-(\\boldsymbol{x}^{*}_{m}-\\boldsymbol{x}_i)$, and the triangle inequality. Inequality (c) follows from Lemma~\\ref{lemma:connection_ub_lb}, and inequality (d) is due to that for any  $\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{Z}_{t,m})$, $\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m \\leq 4 \\cdot 2^{-t}$ (from the definition of $\\mathcal{Z}_{t,m}$). Equality (e) comes from the definition of $\\boldsymbol{\\lambda}^*_m$.\n\nLet $L:=\\log^2 (\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}) \\cdot \\log^2 (\\rho^E k d L_{x} L_{\\theta}  \\max \\{\\Delta_{\\min}^{-1},\\ \\frac{L_{x}}{\\omega}\\}  \\frac{\\log(\\Delta_{\\min}^{-1})}{\\delta} \\log (\\frac{d \\log(\\Delta_{\\min}^{-1})}{\\delta}) ) $.\nPlugging Eq.~\\eqref{eq:connection_ub_lb} into Eq.~\\eqref{eq:bai_sample_complexity_tilde}, we have that with probability $1-\\delta$, the number of samples used by algorithm $\\mathtt{DouExpDes}$ is bounded by\n\\begin{align*}\n\n\n\n\n\n\t& O \\sbr{ \\sum_{m=1}^{M} \\!\\!\\! \\sum_{t=1}^{\\log(\\Delta_{m,\\min}^{-1})} \\!\\!\\!\\!\\!\\! 2^{2t} \\rho^{G}_{t,m}  \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{m,\\min}^{-1})  }{\\delta}} + Mk \\log(\\Delta_{\\min}^{-1}) + \\sum_{t=1}^{\\log(\\Delta_{\\min}^{-1})} (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{2^{2t},\\ \\frac{L_{x}^4}{\\omega^2}}  L } \n\t\\\\\n\t= & O \\Bigg( \\sum_{m=1}^{M} \\sum_{t=2}^{\\log(\\Delta_{m,\\min}^{-1})} \\Bigg( \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{X})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2} + \\frac{ L_{x}^4 \\cdot 2^t }{ \\omega^2 k }  \\Bigg) \\cdot \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{m,\\min}^{-1})  }{\\delta}}  \\\\& + \\sum_{m=1}^{M} \\rho^*_{1,m} \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{m,\\min}^{-1})  }{\\delta}} + Mk \\log(\\Delta_{\\min}^{-1}) + \\sum_{t=1}^{\\log(\\Delta_{\\min}^{-1})} (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{2^{2t},\\ \\frac{L_{x}^4}{\\omega^2}} \\cdot L \\Bigg) \n\t\\\\\n\t\\overset{\\textup{(a)}}{=} & O \\Bigg( \\sum_{m=1}^{M}  \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{X})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2} \\cdot \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{m,\\min}^{-1})  }{\\delta}} \\cdot \\log(\\Delta_{m,\\min}^{-1}) \\\\& + \\frac{ M L_{x}^4 }{ \\omega^2 k \\cdot \\Delta_{\\min} } \\cdot \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{\\min}^{-1})  }{\\delta}} + Mk \\cdot \\log \\sbr{\\frac{n^2 M \\log(\\Delta_{\\min}^{-1})  }{\\delta}} \\cdot \\log(\\Delta_{\\min}^{-1}) \\\\& \\qquad +  (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{ \\Delta_{\\min}^{-2} ,\\ \\frac{L_{x}^4 \\cdot \\log(\\Delta_{\\min}^{-1})}{\\omega^2}} \\cdot L  \\Bigg) ,\n\\end{align*}\nwhere equality (a) uses Lemma~\\ref{lemma:rho_t_leq_4k}.\n\nWhen $L_x=\\omega=\\Theta(1)$, we have that with probability $1-\\delta$, the sample complexity of algorithm $\\mathtt{DouExpDes}$ is bounded by\n\\begin{align*}\n\t\\tilde{O} \\Bigg( \\sum_{m=1}^{M}  \\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{X}}} \\max_{\\boldsymbol{y} \\in \\mathcal{Y}^*_m(\\mathcal{X})}  \\frac{\\| \\boldsymbol{B}^\\top \\boldsymbol{y} \\|^2_{\\sbr{\\sum_{i=1}^n \\lambda(\\boldsymbol{x}_i) \\boldsymbol{B}^\\top \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top \\boldsymbol{B}}^{-1}}}{ \\sbr{\\boldsymbol{y}^\\top \\boldsymbol{\\theta}_m}^2}  \\log\\sbr{\\frac{1}{\\delta}} +  (\\rho^E)^2 k^4 d L_{x}^2 L_{\\theta}^2  \\max\\lbr{ \\Delta_{\\min}^{-2} ,\\ \\frac{L_{x}^4}{\\omega^2}}  \\log^4\\sbr{\\frac{1}{\\delta}}  \\Bigg) .\n\\end{align*}\n\n\n\\section{Proofs for Algorithm~$\\mathtt{C \\hyphen DouExpDes}$}\n\nIn this section, we present the proofs for Algorithm~$\\mathtt{C \\hyphen DouExpDes}$.\n\n\\subsection{Context Distribution Estimation and Sample Batch Planning} \\label{apx:bpi_sample_batch_planning}\n\nDefine $\\lambda_{\\mathcal{D}}^{E}$ and $\\rho_{\\mathcal{D}}^{E}$ as the optimal solution and the optimal value of the following E-optimal  design optimization:\n\\begin{align}\n\t\\min_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{A}}} \\nbr{ \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} }^{-1} } . \\label{eq:E_optimal_design}\n\\end{align}\n\n\\begin{lemma}\\label{lemma:bound_rho_E_cD}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\rho_{\\mathcal{D}}^{E} \\leq \\frac{1}{\\nu} .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:bound_rho_E_cD}]\n\tThe optimization in Eq.~\\eqref{eq:E_optimal_design} is equivalent to maximize the minimum singular value of the matrix $\\sum_{a \\in \\mathcal{A}} \\lambda(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top}$.\n\t\n\tThus, $\\lambda_{\\mathcal{D}}^{E}$ is the optimal solution of the following optimization:\n\t\\begin{align*}\n\t\t\\max_{\\boldsymbol{\\lambda} \\in \\triangle_{\\mathcal{A}}} \\sigma_{\\min} \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } .\n\t\\end{align*}\n\t\n\tUsing Assumption~\\ref{assumption:bpi_rho_E_D_is_finite}, we have \n\t\\begin{align*}\n\t\t\\sigma_{\\min} \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda_{\\mathcal{D}}^{E} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } \\geq \\nu .\n\t\\end{align*}\n\t\n\tThen, we have\n\t\\begin{align*}\n\t\t\\rho_{\\mathcal{D}}^{E} = & \\nbr{ \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda_{\\mathcal{D}}^{E} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} }^{-1} }\n\t\t\\\\\n\t\t= & \\frac{1}{\\sigma_{\\min} \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda_{\\mathcal{D}}^{E} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} }}\n\t\t\\\\\n\t\t\\leq & \\frac{1}{\\nu} .\n\t\\end{align*}\n\\end{proof}\n\nDefine event\n\\begin{align*}\n\t\\mathcal{K}:= \\lbr{ \\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } } \\leq \\frac{ 8 L_{\\phi}^2 \\log \\sbr{\\frac{20d |\\mathcal{A}|}{\\delta}} }{ \\sqrt{T_0} } , \\ \\forall a \\in \\mathcal{A} } .\n\\end{align*}\n\n\\begin{lemma} \\label{lemma:number_of_samples_T0}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{K}} \\geq 1-\\frac{\\delta}{5} .\n\t\\end{align*}\n\tFurthermore, if event $\\mathcal{K}$ holds and \n\t\\begin{align*}\n\t\n\t\t%\n\t\tT_0 = \\left \\lceil \\frac{32^2 (1+\\zeta)^2 L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{20d |\\mathcal{A}|}{\\delta}} \\right \\rceil , \n\t\n\t\\end{align*}\n\twe have that for any $a \\in \\mathcal{A}$,\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } } \\leq & \\frac{\\nu}{ 4(1+\\zeta) } .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:number_of_samples_T0}]\n\tFor any $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$, $\\| \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top \\| \\leq L_{\\phi}^2$. Then, using the matrix Bernstern inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) and a union bound over $a \\in \\mathcal{A}$, we have that with probability $1-\\frac{\\delta}{5}$, for any $a \\in \\mathcal{A}$,\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } } \\leq & 4 L_{\\phi}^2 \\sqrt{ \\frac{ \\log \\sbr{\\frac{10 \\cdot 2d |\\mathcal{A}|}{\\delta}} }{ T_0 } } + \\frac{ 4 L_{\\phi}^2 \\log \\sbr{\\frac{10 \\cdot 2d |\\mathcal{A}|}{\\delta}} }{ T_0 } \n\t\t\\\\\n\t\t\\leq & \\frac{ 8 L_{\\phi}^2 \\log \\sbr{\\frac{20d |\\mathcal{A}|}{\\delta}} }{ \\sqrt{T_0} } .\n\t\\end{align*} \n\tIf $T_0 \\geq 32^2 (1+\\zeta)^2 \\nu^{-2} L_{\\phi}^4 \\log^2 \\sbr{\\frac{20d |\\mathcal{A}|}{\\delta}}$, we have\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } } \\leq \\frac{\\nu}{ 4(1+\\zeta) } ,\n\t\\end{align*} \n\twhich completes the proof.\n\\end{proof}\n\nDefine event\n\\begin{align*}\n\t\\mathcal{L}:= \\Bigg\\{ & \\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\leq 8 L_{\\phi}^2 \\sqrt{ p } \\log \\sbr{\\frac{40dMT}{\\delta}} , \\\\& \\forall m \\in [M], \\ \\forall j \\in [T], \\ \\forall \\ell \\in \\{1,2\\} \\Bigg\\} .\n\\end{align*}\n\n\\begin{lemma} \\label{lemma:number_of_samples_p}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{L}} \\geq 1 - \\frac{\\delta}{5} .\n\t\\end{align*}\n\tFurthermore, if event $\\mathcal{L}$ holds and \n\t\\begin{align}\n\t\tp = \\left \\lceil \\frac{32^2 (1+\\zeta)^2 L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{40dMT}{\\delta}} \\right \\rceil , \\label{eq:value_p}\n\t\\end{align}\n\twe have that for any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$,\n\t\\begin{align*}\n\t\t\\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\leq & \\frac{p \\nu}{ 4(1+\\zeta) } .\n\t\\end{align*}\n\tHere, the value of $T$ is specified in Eq.~\\eqref{eq:value_T}.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:number_of_samples_p}]\n\tFor any $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$, $\\| \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top \\| \\leq L_{\\phi}^2$. Then, using the matrix Bernstern inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) and a union bound over $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$, we have that with probability $1-\\frac{\\delta}{5}$, for any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$,\n\t\\begin{align*}\n\t\t\\nbr{ \\sum_{i=1}^{p} \\! \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top \\!\\!-\\!\\! \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\leq & 4 L_{\\phi}^2 \\sqrt{ p \\log \\sbr{\\frac{10 \\cdot 4dMT}{\\delta}} } \\!+\\! 4 L_{\\phi}^2 \\log \\sbr{\\frac{10 \\cdot 4dMT}{\\delta}}\n\t\t\\\\\n\t\t\\leq & 8 L_{\\phi}^2 \\sqrt{ p } \\log \\sbr{\\frac{40dMT}{\\delta}} .\n\t\\end{align*}\n\tIn addition, if $p \\geq 32^2 (1+\\zeta)^2 \\nu^{-2} L_{\\phi}^4 \\log^2 \\sbr{\\frac{40dMT}{\\delta}}$, we have that \n\t\\begin{align*}\n\t\t8 L_{\\phi}^2 \\sqrt{ p } \\log \\sbr{\\frac{40dMT}{\\delta}} \\leq \\frac{p \\nu}{ 4(1+\\zeta) }\n\t\\end{align*}\n\tand thus,\n\t\\begin{align*}\n\t\t\\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\leq \\frac{p \\nu}{ 4(1+\\zeta) } ,\n\t\\end{align*} \n\twhich completes the proof.\n\\end{proof}\n\nFor any task $m \\in [M]$, round $j \\in [T]$ and $\\ell \\in \\{1,2\\}$, let\n$$\n\\boldsymbol{\\Phi}_{m,j}^{(\\ell)} = \n\\begin{bmatrix}\n\t\\boldsymbol{\\phi}(s_{m,j,1}^{(\\ell)},\\bar{a}_1)^\\top\n\t\\\\\n\t\\dots\n\t\\\\\n\t\\boldsymbol{\\phi}(s_{m,j,p}^{(\\ell)},\\bar{a}_p)^\\top\n\\end{bmatrix} ,\n$$ \nand\n$$\n(\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+} = ((\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{-1} (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} .\n$$\n\n\\begin{lemma} \\label{lemma:norm_Phi_plus}\n\tSuppose that event $\\mathcal{K} \\cap \\mathcal{L}$ holds. Then, for any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$,\n\t\\begin{align*}\n\t\t\\nbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+} } \n\t\t\\leq 2 \\sqrt{\\frac{(1+\\zeta)}{p \\nu}} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:norm_Phi_plus}]\n\n\tWe first assume that $(\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)}$ is invertible. In our later analysis, we will prove that as long as $T_0$ and $p$ are large enough, $(\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)}$ is invertible.\n\t\n\tFor any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$, we have\n\t\\begin{align}\n\t\t\\nbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+} } = & \\nbr{ ((\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{-1} (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} } \n\t\t\\nonumber\\\\\n\t\t= & \\sqrt{ \\nbr{ ((\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{-1} (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)} ((\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{-1} } }\n\t\t\\nonumber\\\\\n\t\t= & \\sqrt{ \\nbr{ ((\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{-1} } }\n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{ \\sqrt{ \\sigma_{\\min} \\sbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)} } } } . \\label{eq:norm_Phi_plus}\n\t\\end{align}\n\t\n\tIn addition, we have\n\t\\begin{align}\n\t\t& \\sigma_{\\min} \\sbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)} } \n\t\t\\nonumber\\\\ \n\t\t= & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top }\n\t\t\\nonumber\\\\\n\t\t= & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} + \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} }\n\t\t\\nonumber\\\\\n\t\t\\geq & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } - \\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \n\t\t\\nonumber\\\\\n\t\t= & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} + \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\nonumber\\\\& - \\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \n\t\t\\nonumber\\\\\n\t\t\\geq & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } - \\nbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\nonumber\\\\& - \\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} }\n\t\t\\nonumber\\\\\n\t\t\\geq & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } -  \\sum_{i=1}^{p} \\nbr{ \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} - \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\nonumber\\\\& - \\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \n\t\t\\nonumber\\\\\n\t\t\\geq & \\sigma_{\\min} \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } - \\frac{p \\nu}{ 4(1+\\zeta) } - \\frac{p \\nu}{ 4(1+\\zeta) }\n\t\t, \\label{eq:sigma_min_Phi_top_times_Phi}\n\t\\end{align}\n\twhere the last inequality uses Lemmas~\\ref{lemma:number_of_samples_T0} and \\ref{lemma:number_of_samples_p}.\n\t\n\tIn the following, we analyze $\\sigma_{\\min}  (\\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top})$.\n\tAccording to the guarantee of the rounding procedure $\\mathtt{ROUND}$, we have\n\t\\begin{align*}\n\t\t\\nbr{ \\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} }^{-1} } \\leq & (1+\\zeta) \\nbr{ \\sbr{ p \\sum_{a \\in \\mathcal{A}} \\lambda^{E}_{\\hat{\\mathcal{D}}}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} }^{-1} }\n\t\t\\\\\n\t\t\\leq & (1+\\zeta) \\nbr{ \\sbr{ p \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} }^{-1} } ,\n\t\\end{align*}\n\twhich implies that\n\t\\begin{align}\n\t\t& \\sigma_{\\min}\\sbr{ \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \n\t\t\\nonumber\\\\\n\t\t\\geq & \\frac{p}{1+\\zeta} \\sigma_{\\min} \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } \n\t\t\\nonumber\\\\\n\t\t\\geq & \\frac{p}{1+\\zeta} \\sigma_{\\min} \\Bigg( \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} \\nonumber\\\\& + \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} - \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} \\Bigg)\n\t\t\\nonumber\\\\\n\t\t\\geq & \\frac{p}{1+\\zeta} \\Bigg( \\sigma_{\\min} \\sbr{ \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } \\nonumber\\\\& - \\nbr{ \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} - \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } \\Bigg)\n\t\t\\nonumber\\\\\n\t\t\\geq & \\frac{p}{1+\\zeta} \\Bigg( \\frac{1}{\\rho_{\\mathcal{D}}^{E}} - \\sum_{a \\in \\mathcal{A}} \\lambda^{E}(a) \\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top} } \\Bigg)\n\t\t\\nonumber\\\\\n\t\t\\overset{\\textup{(a)}}{\\geq} & \\frac{p}{1+\\zeta} \\sbr{ \\nu - \\frac{\\nu}{ 4(1+\\zeta) } }\n\t\t\\nonumber\\\\\n\t\t\\geq & \\frac{3p \\nu}{ 4(1+\\zeta) } , \\label{eq:sigma_min_batch_under_hat_cD}\n\t\\end{align}\n\twhere inequality (a) uses Lemmas~\\ref{lemma:bound_rho_E_cD} and \\ref{lemma:number_of_samples_T0}. \n\n\t\n\tPlugging Eq.~\\eqref{eq:sigma_min_batch_under_hat_cD} into Eq.~\\eqref{eq:sigma_min_Phi_top_times_Phi}, we have\n\t\\begin{align}\n\t\t\\sigma_{\\min} \\sbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)} } \\geq & \\frac{3p \\nu}{ 4(1+\\zeta) } - \\frac{p \\nu}{ 4(1+\\zeta) } - \\frac{p \\nu}{ 4(1+\\zeta) }\n\t\t\\nonumber\\\\\n\t\t= & \\frac{p \\nu}{ 4(1+\\zeta) } . \\label{eq:sigma_min_Phi_top_Phi_value}\n\t\\end{align}\n\tEquations~\\eqref{eq:sigma_min_Phi_top_times_Phi} and \\eqref{eq:sigma_min_Phi_top_Phi_value} show that if $T_0$ and $p$ are large enough to satisfy that $\\nbr{ \\mathbb{E}_{s \\sim \\hat{\\mathcal{D}}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } - \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{ \\boldsymbol{\\phi}(s,a) \\boldsymbol{\\phi}(s,a)^\\top } } \\leq  \\frac{\\nu}{ 4(1+\\zeta) }$ for any $a \\in \\mathcal{A}$ and $\\nbr{ \\sum_{i=1}^{p} \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i) \\boldsymbol{\\phi}(s_{m,j,i}^{(\\ell)},\\bar{a}_i)^\\top - \\sum_{i=1}^{p} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\boldsymbol{\\phi}(s,\\bar{a}_i) \\boldsymbol{\\phi}(s,\\bar{a}_i)^\\top} } \\leq \\frac{p \\nu}{ 4(1+\\zeta) }$ for any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$, respectively, then we have that $ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(\\ell)} $ is invertible.\n\t\n\tContinuing with Eq.~\\eqref{eq:norm_Phi_plus}, we have\n\t\\begin{align*}\n\t\t\\nbr{ (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+} } \n\t\t\\leq 2 \\sqrt{\\frac{(1+\\zeta)}{p \\nu}} .\n\t\\end{align*}\n\t\n\\end{proof}\n\n\n\\subsection{Global Feature Extractor Recovery with Stochastic Contexts} \\label{apx:bpi_feat_recover}\n\nIn subroutine $\\mathtt{C \\hyphen FeatRecover}$, for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, let $s^{(\\ell)}_{m,j,i}$ and $\\eta^{(\\ell)}_{m,j,i}$ denote the random context and noise of the $\\ell$-th sample on action $\\bar{a}_i$ in the $j$-th round for task $m$, respectively. Here, the superscript $\\ell \\in \\{1,2\\}$ refers to the first sample (Line~\\ref{line:bpi_stage2_sample1} in Algorithm~\\ref{alg:con_feat_recover}) or the second sample (Line~\\ref{line:bpi_stage2_sample2} in Algorithm~\\ref{alg:con_feat_recover}) on an action $\\bar{a}_i$.\n\nIn $\\mathtt{C \\hyphen FeatRecover}$, for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, let $\\boldsymbol{\\alpha}^{(\\ell)}_{m,j} \\leftarrow [\\alpha^{(\\ell)}_{m,j,1}, \\dots, \\alpha^{(\\ell)}_{m,j,p}]^\\top$, and then,  $\\tilde{\\boldsymbol{\\theta}}^{(\\ell)}_{m,j} = (\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+} \\boldsymbol{\\alpha}^{(\\ell)}_{m,j}$. Recall that $\\boldsymbol{Z} = \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{\\theta}}^{(1)}_{m,j} (\\tilde{\\boldsymbol{\\theta}}^{(2)}_{m,j})^\\top$.\n\n\\begin{lemma}[Expectation of $\\boldsymbol{Z}$] \\label{lemma:bpi_expectation_Z}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\mathbb{E} \\mbr{ \\boldsymbol{Z} } = \\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:bpi_expectation_Z}]\n\t$\\boldsymbol{Z}$ can be written as\n\t\\begin{align}\n\t\t\\boldsymbol{Z} = & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{\\theta}}^{(1)}_{m,j} (\\tilde{\\boldsymbol{\\theta}}^{(2)}_{m,j})^\\top  \n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\begin{bmatrix}\n\t\t\t\\alpha^{(1)}_{m,j,1}\n\t\t\t\\\\\n\t\t\t\\vdots\n\t\t\t\\\\\n\t\t\t\\alpha^{(1)}_{m,j,p} \n\t\t\\end{bmatrix}\n\t\t\\mbr{\\alpha^{(2)}_{m,j,1}, \\dots, \\alpha^{(2)}_{m,j,p}} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^{\\top}\n\t\t\\nonumber\\\\\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\n\t\t= & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\Bigg( \\!\\!\n\t\t\\begin{bmatrix}\n\t\t\t&\\!\\!\\!\\!\\!\\! \\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\!\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\!\\! &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\dots &\\!\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\!\\! &\\dots\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\!\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\!\\! &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\\end{bmatrix} \n\t\t\\nonumber\\\\& \\hspace*{-2em} +\n\t\t\\begin{bmatrix}\n\t\t\t&\\!\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,1} + \\eta^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,p} + \\eta^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\dots &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\dots\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,1} + \\eta^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m  & \\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,p} + \\eta^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\\end{bmatrix} \n\t\t\\nonumber\\\\& \\hspace*{-2em} +\n\t\t\\begin{bmatrix}\n\t\t\t&\\!\\!\\!\\!\\!\\! \\eta^{(1)}_{m,j,1} \\cdot \\eta^{(2)}_{m,j,1} &\\dots & \\eta^{(1)}_{m,j,1} \\cdot \\eta^{(2)}_{m,j,p}\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\dots &\\dots &\\dots\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\! \\eta^{(1)}_{m,j,p} \\cdot \\eta^{(2)}_{m,j,1} &\\dots &\\eta^{(1)}_{m,j,p} \\cdot \\eta^{(2)}_{m,j,p}\n\t\t\\end{bmatrix} \n\t\t\\Bigg)\n\t\t((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^{\\top} \\label{eq:decompose_Z} .\n\t\\end{align}\n\t\n\tFor any task $m \\in [M]$, $j \\in [T]$, $i \\in [p]$, the sample on action $a_i$ in the first round (i.e., $s^{(1)}_{m,j,i}$ and $\\eta^{(1)}_{m,j,i}$) is independent of that in the second round (i.e., $s^{(2)}_{m,j,i}$ and $\\eta^{(2)}_{m,j,i}$).  \n\tHence, taking the expectation on $\\boldsymbol{Z}$, we obtain\n\t\n\t\\begin{align*}\n\t\t\\mathbb{E}[\\boldsymbol{Z}] = & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E} \\Bigg[ (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\cdot\n\t\t\\\\\n\t\t&\n\t\t\\begin{bmatrix}\n\t\t\t&\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\t\\\\\n\t\t\t&\\dots &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\dots\n\t\t\t\\\\\n\t\t\t&\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\\end{bmatrix}\n\t\t((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^{\\top} \\Bigg]\n\t\t\\\\\n\t\t= & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E} \\Bigg[ ((\\boldsymbol{\\Phi}_{m,j}^{(1)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(1)})^{-1} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{\\top}  \n\t\t\\begin{bmatrix}\n\t\t\t\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m\n\t\t\t\\\\\n\t\t\t\\vdots\n\t\t\t\\\\\n\t\t\t\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\\end{bmatrix}\n\t\t\\mbr{ \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m, \\dots, \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m }\n\t\t\\cdot\n\t\t\\\\\n\t\t& \\boldsymbol{\\Phi}_{m,j}^{(2)} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(2)})^{-1} \\Bigg]\n\t\t\\\\\n\t\t= & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E} \\mbr{ ((\\boldsymbol{\\Phi}_{m,j}^{(1)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(1)})^{-1} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{\\top} \\cdot \\boldsymbol{\\Phi}_{m,j}^{(1)} \\boldsymbol{\\theta}_m\n\t\t\t(\\boldsymbol{\\theta}_m)^{\\top} (\\boldsymbol{\\Phi}_{m,j}^{(2)})^{\\top} \\cdot\n\t\t\t\\boldsymbol{\\Phi}_{m,j}^{(2)} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{\\top} \\boldsymbol{\\Phi}_{m,j}^{(2)})^{-1} }\n\t\t\\\\\n\t\t= & \\frac{1}{M T} \\sum_{m=1}^{M} \\sum_{j=1}^{T}  \\boldsymbol{\\theta}_m (\\boldsymbol{\\theta}_m)^\\top\n\t\t\\\\\n\t\t= & \\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top .\n\t\\end{align*}\n\t\n\\end{proof}\n\nDefine event \n\\begin{align*}\n\t\\mathcal{G}:= \\lbr{ \\nbr{\\boldsymbol{Z} - \\mathbb{E} [\\boldsymbol{Z}]} \\leq \\frac{ 256 (1+\\zeta) L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{\\nu \\sqrt{MT}} \\log \\sbr{\\frac{100pMT}{\\delta}} } .\n\\end{align*}\n\n\\begin{lemma}[Concentration of $Z$] \\label{lemma:Z_est_error}\n\tSuppose that $\\mathcal{K} \\cap \\mathcal{L}$ holds. Then, it holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{G}} \\geq 1-\\frac{\\delta}{5} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:Z_est_error}]\n\tDefine the following matrices:\n\t\\begin{align*}\n\t\t\\boldsymbol{D}_{m,j} &:= \\frac{1}{M T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\cdot\n\t\t\\\\\n\t\t&\n\t\t\\begin{bmatrix}\n\t\t\t&\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\dots &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\t\\\\\n\t\t\t&\\dots &\\dots &\\dots\n\t\t\t\\\\\n\t\t\t&\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m} &\\dots &\\sbr{\\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m} \\sbr{\\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m}\n\t\t\\end{bmatrix} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^\\top\n\t\t\\\\\n\t\t\\boldsymbol{D} &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\boldsymbol{D}_{m,j}\n\t\t\\\\\n\t\t\\boldsymbol{E}_{m,j} &:= \\frac{1}{M T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\cdot\n\t\t\\\\\n\t\t&\n\t\t\\hspace*{-1.8em}\n\t\t\\begin{bmatrix}\n\t\t\t&\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,1} + \\eta^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,p} + \\eta^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\dots &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\dots\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,1} + \\eta^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m  & \\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\eta^{(2)}_{m,j,p} + \\eta^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\\end{bmatrix} \\! \\cdot \n\t\t\\\\\n\t\t& ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^\\top\n\t\t\\\\\n\t\t\\boldsymbol{E} &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\boldsymbol{E}_{m,j}\n\t\t\\\\\n\t\t\\boldsymbol{F}_{m,j} &:= \\frac{1}{M T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+}\n\t\t\\begin{bmatrix}\n\t\t\t&\\eta^{(1)}_{m,j,1} \\cdot \\eta^{(2)}_{m,j,1} &\\dots & \\eta^{(1)}_{m,j,1} \\cdot \\eta^{(2)}_{m,j,p}\n\t\t\t\\\\\n\t\t\t&\\dots &\\dots &\\dots\n\t\t\t\\\\\n\t\t\t&\\eta^{(1)}_{m,j,p} \\cdot \\eta^{(2)}_{m,j,1} &\\dots &\\eta^{(1)}_{m,j,p} \\cdot \\eta^{(2)}_{m,j,p}\n\t\t\\end{bmatrix} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^\\top\n\t\t\\\\\n\t\t\\boldsymbol{F} &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\boldsymbol{F}_{m,j}\n\t\\end{align*}\n\t\n\t\n\tFrom Eq.~\\eqref{eq:decompose_Z}, we can bound $\\|\\boldsymbol{Z} - \\mathbb{E} [\\boldsymbol{Z}]\\|$ as\n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{Z} - \\mathbb{E} [\\boldsymbol{Z}]} \\leq & \\nbr{\\boldsymbol{D} - \\mathbb{E}[\\boldsymbol{D}]} + \\nbr{\\boldsymbol{E} - \\mathbb{E}[\\boldsymbol{E}]} + \\nbr{\\boldsymbol{F} - \\mathbb{E}[\\boldsymbol{F}]} . \\label{eq:decompose_Z_est_err}\n\t\\end{align}\n\t\n\tSimilar to the proof of Lemma~\\ref{lemma:Z_t_est_error}, in order to use the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}), we define the truncated noise and some truncated matrices as follows.\n\t\n\tLet $R>0$ be a truncation parameter of noises which will be chosen later.  For any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, let $\\tilde{\\eta}^{(\\ell)}_{m,j,i}=\\eta^{(\\ell)}_{m,j,i} \\mathbbm{1}\\{|\\eta^{(\\ell)}_{m,j,i}| \\leq R\\}$ denote the truncated noise. Furthermore, we define the following matrices with truncated noises:\n\t\\begin{align*}\n\t\t\\tilde{\\boldsymbol{E}}_{m,j} &:= \\frac{1}{M T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+} \\cdot\n\t\t\\\\\n\t\t&\n\t\t\\hspace*{-1.8em} \n\t\t\\begin{bmatrix}\n\t\t\t&\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\tilde{\\eta}^{(2)}_{m,j,1} + \\tilde{\\eta}^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,1},\\bar{a}_1)^\\top  \\boldsymbol{\\theta}_m \\cdot \\tilde{\\eta}^{(2)}_{m,j,p} + \\tilde{\\eta}^{(1)}_{m,j,1} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\!\\dots &\\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! &\\dots\n\t\t\t\\\\\n\t\t\t&\\!\\!\\!\\!\\! \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\tilde{\\eta}^{(2)}_{m,j,1} + \\tilde{\\eta}^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,1},\\bar{a}_1)^\\top \\boldsymbol{\\theta}_m  & \\!\\!\\!\\!\\!\\dots\\!\\!\\!\\!\\! & \\boldsymbol{\\phi}(s^{(1)}_{m,j,p},\\bar{a}_p)^\\top  \\boldsymbol{\\theta}_m \\cdot \\tilde{\\eta}^{(2)}_{m,j,p} + \\tilde{\\eta}^{(1)}_{m,j,p} \\cdot \\boldsymbol{\\phi}(s^{(2)}_{m,j,p},\\bar{a}_p)^\\top \\boldsymbol{\\theta}_m\n\t\t\\end{bmatrix} \\! \\cdot\n\t\t\\\\\n\t\t& ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^\\top\n\t\t\\\\\n\t\t\\tilde{\\boldsymbol{E}} &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{E}}_{m,j}\n\t\t\\\\\n\t\t\\tilde{\\boldsymbol{F}}_{m,j} &:= \\frac{1}{M T} (\\boldsymbol{\\Phi}_{m,j}^{(1)})^{+}\n\t\t\\begin{bmatrix}\n\t\t\t&\\tilde{\\eta}^{(1)}_{m,j,1} \\cdot \\tilde{\\eta}^{(2)}_{m,j,1} &\\dots & \\tilde{\\eta}^{(1)}_{m,j,1} \\cdot \\tilde{\\eta}^{(2)}_{m,j,p}\n\t\t\t\\\\\n\t\t\t&\\dots &\\dots &\\dots\n\t\t\t\\\\\n\t\t\t&\\tilde{\\eta}^{(1)}_{m,j,p} \\cdot \\tilde{\\eta}^{(2)}_{m,j,1} &\\dots &\\tilde{\\eta}^{(1)}_{m,j,p} \\cdot \\tilde{\\eta}^{(2)}_{m,j,p}\n\t\t\\end{bmatrix} ((\\boldsymbol{\\Phi}_{m,j}^{(2)})^{+})^\\top\n\t\t\\\\\n\t\t\\tilde{\\boldsymbol{F}} &:= \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\tilde{\\boldsymbol{F}}_{m,j}\n\t\\end{align*}\n\t\n\t\n\tRecall that from Lemma~\\ref{lemma:norm_Phi_plus}, we have that for any $m \\in [M]$, $j \\in [T]$ and $\\ell \\in \\{1,2\\}$, $\\|(\\boldsymbol{\\Phi}_{m,j}^{(\\ell)})^{+}\\| \\leq 2 \\sqrt{\\frac{(1+\\zeta)}{p \\nu}}$. Let $B_{\\Phi}:=2 \\sqrt{\\frac{(1+\\zeta)}{p \\nu}}$.\n\t\n\tWe first analyze $\\|\\boldsymbol{D}-\\mathbb{E}[\\boldsymbol{D}]\\|$. Since  $|\\boldsymbol{\\phi}(s^{(\\ell)}_{m,j,i},\\bar{a}_i)^\\top \\boldsymbol{\\theta}_m| \\leq L_{\\phi}L_{\\theta}$ for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, we have that $\\|\\boldsymbol{D}_{m,j}\\| \\leq \\frac{1}{MT} \\cdot p L_{\\phi} L_{\\theta} B_{\\Phi}^2$ and $\\| \\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E}[\\boldsymbol{D}^2_{m,j}]\\| \\leq MT \\cdot \\frac{1}{M^2 T^2} \\cdot p^2 L_{\\phi}^2 L_{\\theta}^2 B_{\\Phi}^4 = \\frac{1}{M T} \\cdot p^2 L_{\\phi}^2 L_{\\theta}^2 B_{\\Phi}^4$ for any $m \\in [M]$ and $j \\in [T]$. \n\t\n\tLet $\\delta' \\in (0,1)$ be a confidence parameter which will be chosen later.\n\tUsing the matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}), we have that with probability at least $1-\\delta'$,\n\t\n\t\\begin{align}\n\t\t\\nbr{ \\boldsymbol{D} - \\mathbb{E}[\\boldsymbol{D}] } \\leq & 4 \\sqrt{ \\frac{p^2 L_{\\phi}^2 L_{\\theta}^2 B_{\\Phi}^4 \\log \\sbr{\\frac{2d}{\\delta'}}}{M T} } +  \\frac{4 p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\log \\sbr{\\frac{2d}{\\delta'}}}{MT} \n\t\t\\nonumber\\\\\n\t\t\\leq &  \\frac{8 \\cdot 4 p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\log \\sbr{\\frac{2d}{\\delta'}}}{\\sqrt{MT}}  \\label{eq:D_est_err}.\n\t\\end{align}\n\t\n\tNext, we bound $\\|\\boldsymbol{E}-\\mathbb{E}[\\boldsymbol{E}]\\|$. \n\tSince $|\\boldsymbol{\\phi}(s^{(\\ell)}_{m,j,i},\\bar{a}_i)^\\top  \\boldsymbol{\\theta}_m| \\leq L_{\\phi} L_{\\theta}$ and $|\\tilde{\\eta}^{(\\ell)}_{m,j,i}| \\leq R$ for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, we have that  $\\|\\tilde{\\boldsymbol{E}}_{m,j}\\| \\leq \\frac{1}{M T} \\cdot 2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2$ and $\\nbr{\\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E} [\\tilde{\\boldsymbol{E}}_{m,j}^2]} \\leq \\frac{1}{M T} \\cdot 4p^2R^2 L_{\\phi}^2 L_{\\theta}^2 B_{\\Phi}^4$ for any $m \\in [M]$ and $j \\in [T]$.\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\tSince $\\eta^{(\\ell)}_{m,j,i}$ is 1-sub-Gaussian for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, using a union bound over $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, we have that for any $m \\in [M]$ and $j \\in [T]$, with probability at least $1-4p\\exp(-\\frac{R^2}{2})$, $|\\eta^{(\\ell)}_{m,j,i}| \\leq R$ for all $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, and thus, $\\|\\boldsymbol{E}_{m,j}\\| \\leq \\frac{1}{M T} \\cdot 2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2$. Then, we have\n\t\\begin{align*}\n\t\t& \\nbr{ \\mathbb{E}[\\boldsymbol{E}_{m,j}]-\\mathbb{E}[\\tilde{\\boldsymbol{E}}_{m,j}] } \n\t\t\\nonumber\\\\\n\t\t\\leq & \\nbr{ \\mathbb{E} \\mbr{\\boldsymbol{E}_{m,j} \\cdot \\indicator{ \\nbr{\\boldsymbol{E}_{m,j}} \\geq \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} }} }\n\t\t\\\\\n\t\t\\leq & \\mathbb{E} \\mbr{ \\nbr{\\boldsymbol{E}_{m,j}} \\cdot \\indicator{ \\nbr{\\boldsymbol{E}_{m,j}} \\geq \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} } } \n\t\t\\\\\n\t\t= & \\mathbb{E}\\mbr{ \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\!\\cdot\\! \\indicator{ \\nbr{\\boldsymbol{E}_{m,j}} \\!\\geq\\! \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} }} \\!+\\! \\mathbb{E} \\mbr{\\sbr{ \\nbr{\\boldsymbol{E}_{m,j}} \\!-\\! \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T}} \\!\\cdot\\! \\indicator{ \\nbr{\\boldsymbol{E}_{m,j}} \\!\\geq\\! \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} } } \n\t\t\\\\\n\t\t= & \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot \\Pr\\mbr{ \\nbr{\\boldsymbol{E}_{m,j}} \\geq \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} } + \\int_0^{\\infty}  \\Pr \\mbr{ \\nbr{\\boldsymbol{E}_{m,j}} - \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} > x}  dx  \n\t\t\\\\\n\t\t\\leq & \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T}  \\int_R^{\\infty} \\Pr \\mbr{ \\nbr{\\boldsymbol{E}_{m,j}} > \\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 y}{M T} }  dy  \n\t\t\\\\\n\t\t\\leq & \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T}  \\int_R^{\\infty} 4p \\exp\\sbr{-\\frac{y^2}{2}}  dy\n\t\t\\\\\n\t\t\\leq & \\frac{2pR L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\frac{1}{R} \\cdot \\exp\\sbr{-\\frac{R^2}{2}} \n\t\t\\\\\n\t\t= & \\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\sbr{R+\\frac{1}{R}} \\exp\\sbr{-\\frac{R^2}{2}} .\n\t\\end{align*}\n\t\n\t\n\tUsing the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) with $n=MT$, $R=\\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}}$, $U=\\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}}}{MT}$, $\\sigma^2=\\frac{(2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}})^2}{MT}$, $\\tau=4\\sqrt{\\frac{ (2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}})^2 \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{MT}}+\\frac{4 \\cdot 2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}} \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{M T}$ and $\\Delta=\\frac{2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\cdot 2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}}}{M T} \\cdot \\frac{\\delta'}{M T}$, we have that with probability at least $1-2\\delta'$, \n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{E} - \\mathbb{E}[\\boldsymbol{E}]} \\leq  \\frac{8 \\cdot 2p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}} \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{\\sqrt{M T}} . \\label{eq:E_est_err}\n\t\\end{align}\n\t\n\tNow we investigate $\\nbr{\\boldsymbol{F} - \\mathbb{E}[\\boldsymbol{F}]}$. Since $|\\tilde{\\eta}^{(\\ell)}_{m,j,i}| \\leq R$ for any $m \\in [M]$, $j \\in [T]$, $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, we have that $\\|\\tilde{\\boldsymbol{F}}_{m,j}\\| \\leq \\frac{1}{M T} \\cdot pR^2 B_{\\Phi}^2$ and $\\nbr{\\sum_{m=1}^{M} \\sum_{j=1}^{T} \\mathbb{E} \\mbr{\\tilde{\\boldsymbol{F}}_{m,j}^2} } \\leq \\frac{1}{M T} \\cdot p^2 R^4 B_{\\Phi}^4$.\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\t\n\tRecall that for any $m \\in [M]$ and $j \\in [T]$, with probability at least $1-4p\\exp(-\\frac{R^2}{2})$, $|\\eta^{(\\ell)}_{m,j,i}| \\leq R$ for all $i \\in [p]$ and $\\ell \\in \\{1,2\\}$, and thus, $\\|\\boldsymbol{F}_{m,j}\\| \\leq \\frac{1}{M T} \\cdot p B_{\\Phi}^2 R^2 $. Then, we have\n\t\\begin{align*}\n\t\t\\nbr{ \\mathbb{E}[\\boldsymbol{F}_{m,j}]-\\mathbb{E}[\\tilde{\\boldsymbol{F}}_{m,j}] } \\leq & \\nbr{ \\mathbb{E} \\mbr{\\boldsymbol{F}_{m,j} \\cdot \\indicator{ \\nbr{\\boldsymbol{F}_{m,j}} \\geq \\frac{p B_{\\Phi}^2 R^2}{M T} }} }\n\t\t\\\\\n\t\t\\leq & \\mathbb{E} \\mbr{ \\nbr{\\boldsymbol{F}_{m,j}} \\cdot \\indicator{ \\nbr{\\boldsymbol{F}_{m,j}} \\geq \\frac{p B_{\\Phi}^2 R^2}{M T} } } \n\t\t\\\\\n\t\t= & \\mathbb{E}\\mbr{ \\frac{p B_{\\Phi}^2 R^2}{M T} \\cdot \\indicator{ \\nbr{\\boldsymbol{F}_{m,j}} \\geq \\frac{p B_{\\Phi}^2 R^2}{M T} }} + \\mbr{\\sbr{ \\nbr{\\boldsymbol{F}_{m,j}} - \\frac{p B_{\\Phi}^2 R^2}{M T}} \\cdot \\indicator{ \\nbr{\\boldsymbol{F}_{m,j}} \\geq \\frac{p B_{\\Phi}^2 R^2}{M T} } } \n\t\t\\\\\n\t\t= & \\frac{p B_{\\Phi}^2 R^2}{M T} \\cdot \\Pr\\mbr{ \\nbr{\\boldsymbol{F}_{m,j}} \\geq \\frac{p B_{\\Phi}^2 R^2}{M T} } + \\int_0^{\\infty}  \\Pr \\mbr{ \\nbr{\\boldsymbol{F}_{m,j}} - \\frac{p B_{\\Phi}^2 R^2}{M T} > x}  dx  \n\t\t\\\\\n\t\t\\leq & \\frac{p B_{\\Phi}^2 R^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p B_{\\Phi}^2}{M T}  \\int_R^{\\infty} \\boldsymbol{y} \\cdot \\Pr \\mbr{ \\nbr{\\boldsymbol{F}_{m,j}} > \\frac{p B_{\\Phi}^2 y^2}{M T} } dy  \n\t\t\\\\\n\t\t\\leq & \\frac{p B_{\\Phi}^2 R^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p B_{\\Phi}^2}{M T}  \\int_R^{\\infty}  \\boldsymbol{y} \\cdot 4p \\exp\\sbr{-\\frac{y^2}{2}} dy\n\t\t\\\\\n\t\t\\leq & \\frac{p B_{\\Phi}^2 R^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} + \\frac{2p B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\exp\\sbr{-\\frac{R^2}{2}} \n\t\t\\\\\n\t\t= & \\frac{p B_{\\Phi}^2}{M T} \\cdot 4p \\cdot \\sbr{R^2+2} \\exp\\sbr{-\\frac{R^2}{2}} .\n\t\\end{align*}\n\t\n\t\n\tUsing the truncated matrix Bernstein inequality (Lemma~\\ref{lemma:matrix_bernstein_tau}) with $n=MT$, $R=\\sqrt{2\\log \\sbr{\\frac{4pMT}{\\delta'}}}$, $U=\\frac{p B_{\\Phi}^2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}}}{MT}$, $\\sigma^2=\\frac{(p B_{\\Phi}^2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}})^2}{MT}$, $\\tau=4\\sqrt{\\frac{ (p B_{\\Phi}^2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}})^2 \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{MT}}+\\frac{4\\cdot p B_{\\Phi}^2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}} \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{M T}$ and $\\Delta=\\frac{p B_{\\Phi}^2 \\cdot 2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}} }{M T} \\cdot \\frac{\\delta'}{M T}$, we have that with probability at least $1-2\\delta'$, \n\t\\begin{align}\n\t\t\\nbr{\\boldsymbol{F} - \\mathbb{E}\\mbr{\\boldsymbol{F}}} \\leq \\frac{8 \\cdot p B_{\\Phi}^2 \\cdot 2\\log \\sbr{\\frac{4pMT}{\\delta'}} \\cdot \\log\\sbr{\\frac{2d}{\\delta'}}}{\\sqrt{MT}} \\label{eq:F_est_err} .\n\t\\end{align}\n\t\n\tPlugging Eqs.~\\eqref{eq:D_est_err}-\\eqref{eq:F_est_err} into Eq.~\\eqref{eq:decompose_Z_est_err}, we have that with probability at least $1-5\\delta'$,\n\t\\begin{align*}\n\t\t\\nbr{\\boldsymbol{Z} - \\mathbb{E} [\\boldsymbol{Z}]} \\leq & \\nbr{\\boldsymbol{D} - \\mathbb{E} \\mbr{\\boldsymbol{D}}} + \\nbr{\\boldsymbol{E} - \\mathbb{E} \\mbr{\\boldsymbol{E}}} + \\nbr{\\boldsymbol{F} - \\mathbb{E} \\mbr{\\boldsymbol{F}}}\n\t\t\\\\\n\t\t\\leq & \\frac{64 p L_{\\phi} L_{\\theta} B_{\\Phi}^2 \\log \\sbr{\\frac{4pMT}{\\delta'}} \\log\\sbr{\\frac{2d}{\\delta'}}}{\\sqrt{MT}} .\n\t\\end{align*}\n\t\n\tLet $\\delta'=\\frac{\\delta}{25}$. Recall that $B_{\\Phi}:=2 \\sqrt{\\frac{(1+\\zeta)}{p \\nu}}$. Then, we obtain that with probability at least $1-\\frac{\\delta}{5}$,\n\t\\begin{align*}\n\t\t\\nbr{\\boldsymbol{Z} - \\mathbb{E} [\\boldsymbol{Z}]} \\leq \\frac{ 256 (1+\\zeta) L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{\\nu \\sqrt{MT}} \\log \\sbr{\\frac{100pMT}{\\delta}} ,\n\t\\end{align*}\n\twhich implies that\n\t$\\Pr[\\mathcal{G}]\\geq 1-\\frac{\\delta}{5}$.\n\\end{proof}\n\nAccording to Assumption~\\ref{assumption:diverse_task}, there exists an absolute constant $c_0$ which satisfies that $\\sigma_{\\min}(\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{w}_m \\boldsymbol{w}_m^\\top) = \\sigma_{\\min}(\\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top) \\geq \\frac{c_0}{k}$.\n\n\\begin{lemma}[Concentration of $\\hat{\\boldsymbol{B}}$] \\label{lemma:concentration_hat_B_clb}\n\tSuppose that event $\\mathcal{G}$ holds. Then, \n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{ 2048 (1+\\zeta) k L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{c_0 \\nu \\sqrt{MT}} \\log \\sbr{\\frac{ 135 (1+\\zeta) d L_{\\phi} M T}{\\nu \\delta} } .\n\t\\end{align*}\n\tFurthermore, if\n\t\\begin{align}\n\t\n\t\t%\n\t\tT = \\left \\lceil \\frac{68 \\cdot 2048^2 \\cdot 96^2 (1+\\zeta)^2 k^4 L_{\\phi}^4 L_{\\theta}^2 L_{w}^2 }{ c_0^2 \\nu^2 \\varepsilon^2 M } \\log^6 \\sbr{ \\frac{ 2048 \\cdot 135 \\cdot 96 \\cdot 50 \\cdot 5 (1+\\zeta)^2 k^2 d^2 L_{\\phi}^3 L_{\\theta} L_{w} N }{c_0 \\nu^2 \\delta^3 \\varepsilon} } \\right \\rceil , \\label{eq:value_T}\n\t\\end{align}\n\twe have \n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{\\varepsilon}{ 96 k \\log \\sbr{\\frac{5N}{\\delta}} L_{\\phi} L_{w} } .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:concentration_hat_B_clb}]\n\tFirst, we have that $\\sigma_{k}(\\mathbb{E}[\\boldsymbol{Z}]) - \\sigma_{k+1}(\\mathbb{E}[\\boldsymbol{Z}])=\\sigma_{\\min}( \\frac{1}{M} \\sum_{m=1}^{M} \\boldsymbol{\\theta}_m \\boldsymbol{\\theta}_m^\\top )\\geq \\frac{c_0}{k}$. Let $p := \\lceil 32^2 (1+\\zeta)^2 \\nu^{-2} L_{\\phi}^4 \\log^2 \\sbr{\\frac{40dMT}{\\delta}} \\rceil$.\n\tThen, using the Davis-Kahan sin $\\theta$ Theorem~\\cite{bhatia2013matrix} and letting $T_t$ be large enough to satisfy that $\\nbr{\\boldsymbol{Z} - \\mathbb{E}[\\boldsymbol{Z}]} \\leq \\frac{c_0}{2k}$, we have\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq & \\frac{ \\nbr{\\boldsymbol{Z} - \\mathbb{E}[\\boldsymbol{Z}]} }{ \\sigma_{k}(\\mathbb{E}[\\boldsymbol{Z}]) - \\sigma_{k+1}(\\mathbb{E}[\\boldsymbol{Z}]) - \\nbr{\\boldsymbol{Z} - \\mathbb{E}[\\boldsymbol{Z}]} }\n\t\t\\\\\n\t\t\\leq & \\frac{2k}{c_0} \\nbr{\\boldsymbol{Z} - \\mathbb{E}[\\boldsymbol{Z}]} \n\t\t\\\\\n\t\t\\leq & \\frac{ 512 (1+\\zeta) k L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{c_0 \\nu \\sqrt{MT}} \\log \\sbr{\\frac{100pMT}{\\delta}} \n\t\t\\\\\n\t\t\\leq & \\frac{ 512 (1+\\zeta) k L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{c_0 \\nu \\sqrt{MT}} \\log \\sbr{\\frac{100MT}{\\delta} \\cdot \\frac{2 \\cdot 32^2 (1+\\zeta)^2 L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{40dMT}{\\delta}}} \n\t\t\\\\\n\t\t\\leq & \\frac{ 512 (1+\\zeta) k L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{c_0 \\nu \\sqrt{MT}} \\log \\sbr{\\frac{ 2 \\cdot 100 \\cdot 32^2 \\cdot 40^2 (1+\\zeta)^2 d^2 L_{\\phi}^4 M^3 T^3}{\\nu^2 \\delta^3} } \n\t\t\\\\\n\t\t\\leq & \\frac{ 2048 (1+\\zeta) k L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}} }{c_0 \\nu \\sqrt{MT}} \\log \\sbr{\\frac{ 135 (1+\\zeta) d L_{\\phi} M T}{\\nu \\delta} } .\n\t\\end{align*}\n\t\n\tUsing Lemma~\\ref{lemma:technical_tool_bai_stage2} with $A=2048 (1+\\zeta) k c_0^{-1} \\nu^{-1} L_{\\phi} L_{\\theta} \\log\\sbr{\\frac{50d}{\\delta}}$, $B=\\frac{ 135 (1+\\zeta) d L_{\\phi} }{\\nu \\delta}$ and $\\kappa=\\frac{\\varepsilon}{ 96 k \\log \\sbr{\\frac{5N}{\\delta}} L_{\\phi} L_{w}}$, we have that if \n\t\\begin{align*}\n\t\tM T \\geq & \\frac{68 \\cdot 2048^2 \\cdot 96^2 (1+\\zeta)^2 k^4 L_{\\phi}^4 L_{\\theta}^2 L_{w}^2 }{ c_0^2 \\nu^2 \\varepsilon^2 } \\cdot \\\\& \\log^2 \\sbr{\\frac{50d}{\\delta}} \\log^2 \\sbr{\\frac{5N}{\\delta}}  \\log^2 \\sbr{ \\frac{ 2048 \\cdot 135 \\cdot 96 (1+\\zeta)^2 k^2 d L_{\\phi}^3 L_{\\theta} L_{w} }{c_0 \\nu^2 \\delta \\varepsilon} \\log\\sbr{\\frac{50d}{\\delta}} \\log \\sbr{\\frac{5N}{\\delta}} } ,\n\t\\end{align*}\n\tthen $\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{\\varepsilon}{ 96 k \\log \\sbr{\\frac{5N}{\\delta}} L_{\\phi} L_{w} }$.\n\t\n\tFurther enlarging $MT$, if \n\t\\begin{align*}\n\t\tM T \\geq \\frac{68 \\cdot 2048^2 \\cdot 96^2 (1+\\zeta)^2 k^4 L_{\\phi}^4 L_{\\theta}^2 L_{w}^2 }{ c_0^2 \\nu^2 \\varepsilon^2 } \\log^6 \\sbr{ \\frac{ 2048 \\cdot 135 \\cdot 96 \\cdot 50 \\cdot 5 (1+\\zeta)^2 k^2 d^2 L_{\\phi}^3 L_{\\theta} L_{w} N }{c_0 \\nu^2 \\delta^3 \\varepsilon} } ,\n\t\\end{align*}\n\tthen\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{\\varepsilon}{ 96 k \\log \\sbr{\\frac{5N}{\\delta}} L_{\\phi} L_{w} } .\n\t\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Estimation with Low-dimensional Representations}\n\n\\begin{lemma} \\label{lemma:computation_logdet}\n\tIn subroutine $\\mathtt{EstLowRep}$ (Algorithm~\\ref{alg:est_low_rep}), for any $m \\in [M]$ and $t > 0$, we have \n\t\\begin{align*}\n\t\t\\log \\sbr{ \\frac{\\det\\sbr{ \\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} } }{ \\det \\sbr{ \\gamma I} } } \\leq k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:computation_logdet}]\n\tThis proof uses a similar idea as Lemma 11 in \\cite{abbasi2011improved}.\n\t\n\tIt holds that\n\t\\begin{align*}\n\t\t& \\log \\sbr{ \\frac{\\det\\sbr{ \\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} } }{ \\det \\sbr{ \\gamma I} } } \n\t\t\\\\\n\t\t\\leq & \\log \\sbr{ \\frac{\\sbr{ \\frac{ \\textup{Trace} \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}} }{k}}^{k} }{  \\gamma^{k} } } \n\t\t\\\\\n\t\t= & k \\log \\sbr{ \\frac{ \\textup{Trace} \\sbr{\\gamma I} + \\sum_{\\tau=1}^{t} \\textup{Trace} \\sbr{ \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}} }{\\gamma k} } \n\t\t\\\\\n\t\t= & k \\log \\sbr{ \\frac{ \\gamma k  + \\sum_{\\tau=1}^{t} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})}^2  }{\\gamma k} } \n\t\t\\\\\n\t\t\\leq & k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } .\n\t\\end{align*}\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:decreasing_uncertainty}\n\tIn subroutine $\\mathtt{EstLowRep}$ (Algorithm~\\ref{alg:est_low_rep}), for any $m \\in [M]$ and $t \\geq 0$, we have \n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} } \\geq \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t+1}^{-1}} } .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:decreasing_uncertainty}]\n\tThis proof is similar to that of Lemma 6 in \\cite{zanette2021design}.\n\t\n\tFor any $m \\in [M]$ and $t \\geq 0$, since $\\boldsymbol{\\Sigma}_{m,t+1} \\succeq \\boldsymbol{\\Sigma}_{m,t}$, we have $\\boldsymbol{\\Sigma}_{m,t}^{-1} \\succeq \\boldsymbol{\\Sigma}_{m,t+1}^{-1}$. Hence, for any $m \\in [M]$, $t \\geq 0$, $s \\in \\mathcal{S}$ and $a \\in \\mathcal{A}$, we have\n\t\\begin{align*}\n\t\t\\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}} \\boldsymbol{\\Sigma}_{m,t}^{-1} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a) \\geq \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}} \\boldsymbol{\\Sigma}_{m,t+1}^{-1} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a) ,\n\t\\end{align*}\n\twhich implies that\n\t\\begin{align*}\n\t\t\\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\geq \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t+1}^{-1}} .\n\t\\end{align*}\n\t\n\tTherefore, for any $m \\in [M]$ and $t \\geq 0$, we have\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} } \\geq \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t+1}^{-1}} } .\n\t\\end{align*}\n\\end{proof}\n\n\nIn subroutine $\\mathtt{EstLowRep}$, for any $m \\in [M]$ and $t>0$, let $\\xi_{m,t}$ denote the noise of the sample at timestep $t$ for task $m$ (Line~\\ref{line:bpi_stage3_sample} in Algorithm~\\ref{alg:est_low_rep}).\n\n\nDefine event \n\\begin{align*}\n\t\\mathcal{H} := \\Bigg\\{ &\n\t\\nbr{\\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\xi_{m,\\tau}}_{\\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1}} \\leq \\\\&  k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}, \\ \\forall m \\in [M],\\ \\forall t>0 \\Bigg\\} . \n\\end{align*}\n\n\\begin{lemma}[Martingale Concentration of the Variance Term] \\label{lemma:martingale_concentration_variance}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{H}} \\geq 1-\\frac{\\delta}{5} .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:martingale_concentration_variance}]\n\tLet $\\delta'$ be a confidence parameter which will be chosen later.\n\tSince $\\hat{\\boldsymbol{B}}$ is fixed before sampling $(s_{m,\\tau},a_{m,\\tau})$ for all $m \\in [M]$ and $\\tau>0$, using Lemma~\\ref{lemma:self-normalized_vector_concentration}, we have that with probability at least $1-\\delta'$, for any task $m \\in [M]$ and $t>0$,\n\t\\begin{align*}\n\t\t& \\nbr{\\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\xi_{m,j}}_{\\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1}} \n\t\t\\\\\n\t\t\\leq & 2 \\log \\sbr{ \\frac{\\det\\sbr{ \\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{\\frac{1}{2}} }{ \\det \\sbr{ \\gamma I}^{\\frac{1}{2}} \\cdot \\delta'} } \n\t\t\\\\\n\t\t\\leq & \\log \\sbr{ \\frac{\\det\\sbr{ \\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} } }{ \\det \\sbr{ \\gamma I} } } + 2 \\log\\sbr{\\frac{1}{\\delta'}}\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{1}{\\delta'}} ,\n\t\\end{align*}\n\twhere inequality (a) uses Lemma~\\ref{lemma:computation_logdet}.\n\t\n\tLetting $\\delta'=\\frac{\\delta}{5}$, we obtain this lemma.\n\\end{proof}\n\nDefine event \n\\begin{align*}\n\t\\mathcal{J} := \\Bigg\\{ & \\sum_{t=1}^{N} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} } \\leq \\\\& \\frac{1}{4} \\sbr{ 2\\sqrt{\\log \\sbr{\\frac{5}{\\delta}}} + \\sqrt{ 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\sbr{ \\sum_{t=1}^{N} \\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_t,a) }_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} + 2 \\log \\sbr{\\frac{5}{\\delta}} } } }^2 \\Bigg\\} .\n\\end{align*}\n\n\\begin{lemma} \\label{lemma:reverse_bernstein_uncertainty}\n\tIt holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{\\mathcal{J}} \\geq 1-\\frac{\\delta}{5} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:reverse_bernstein_uncertainty}]\n\tUsing Lemma~\\ref{lemma:reverse_bernstein}, we can obtain this lemma.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:number_of_samples_N}\n\tSuppose that event $\\mathcal{K} \\cap \\mathcal{L} \\cap \\mathcal{G} \\cap \\mathcal{H} \\cap \\mathcal{J}$ holds. For any task $m \\in [M]$, we have\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} & \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \\leq  \\sbr{2\\sqrt{ \\frac{2 k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }}{N} } + \\frac{8 \\log \\sbr{\\frac{5}{\\delta}} }{N}} \\cdot \\\\& \\sbr{ \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} \\sqrt{Nk} +  \\sqrt{k \\log \\sbr{ 1 + \\frac{ N  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}} + \\sqrt{\\gamma} } + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} .\n\t\\end{align*}\n\tFurthermore, if \n\t\\begin{align*}\n\t\n\t\t%\n\t\tN = \\left \\lceil \\frac{4^2 \\cdot 26^4 \\cdot 24^2 \\cdot 2 \\sbr{k^2 + k \\gamma L_{\\theta}^2 } \\log^4 \\big({\\frac{240 ( k + \\sqrt{k \\gamma} L_{\\theta}) }{\\varepsilon \\delta}} \\big) }{\\varepsilon^2} \\right \\rceil , \n\t\n\t\\end{align*}\n\tthen\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \\leq \\frac{\\varepsilon}{2}\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:number_of_samples_N}]\n\tFor any task $m \\in [M]$ and $t \\in [N]$,\n\t\\begin{align*}\n\t\t\\hat{\\boldsymbol{w}}_{m,t} = & \\boldsymbol{\\Sigma}_{m,t}^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) r_{m,\\tau}\n\t\t\\\\\n\t\t= & \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\sbr{ \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\boldsymbol{\\theta}_m + \\xi_{m,j} }\n\t\t\\\\\n\t\t= & \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\cdot \\\\& \\sbr{ \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m + \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}_{\\perp} \\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{\\theta}_m + \\xi_{m,j} }\n\t\t\\\\& + \\gamma \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m \n\t\t\\\\& - \\gamma \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m\n\t\t\\\\\n\t\t= & \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m + \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\cdot\n\t\t\\\\&\n\t\t\\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})  \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}_{\\perp} \\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B} \\boldsymbol{w}_m\n\t\t\\\\&\n\t\t+ \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\xi_{m,j}\n\t\t\\\\& \n\t\t- \\gamma \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m .\n\t\\end{align*}\n\t\n\t\n\tHence, for any task $m \\in [M]$, $t \\in [N]$ and $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$, \n\t\\begin{align*}\n\t\t\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,t} - \\boldsymbol{\\theta}_m} = & \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}} \\hat{\\boldsymbol{w}}_{m,t} - \\boldsymbol{\\phi}(s,a)^\\top \\sbr{\\hat{\\boldsymbol{B}}\\hat{\\boldsymbol{B}}^\\top+\\hat{\\boldsymbol{B}}_{\\bot}\\hat{\\boldsymbol{B}}_{\\bot}^\\top} \\boldsymbol{\\theta}_m\n\t\t\\\\\n\t\t= & \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}} \\sbr{ \\hat{\\boldsymbol{w}}_{m,t} - \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m } - \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}}_{\\bot}\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{\\theta}_m\n\t\t\\\\\n\t\t= & \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}} \n\t\t\\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\cdot \\\\& \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})  \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}_{\\perp} \\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B} \\boldsymbol{w}_m\n\t\t\\\\&\n\t\t+ \\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}}  \\sbr{\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}} }^{-1} \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\xi_{m,j}\n\t\t\\\\& \n\t\t-\\! \\gamma \\boldsymbol{\\phi}(s,a)^{\\!\\top} \\! \\hat{\\boldsymbol{B}}  \\sbr{ \\! \\gamma I \\!+\\! \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^{\\!\\top} \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^{\\!\\top} \\hat{\\boldsymbol{B}} \\! }^{\\!\\!-1} \\!\\!\\!\\!\\!\\! \\hat{\\boldsymbol{B}}^\\top \\! \\boldsymbol{\\theta}_m  \\!-\\! \\boldsymbol{\\phi}(s,a)^{\\!\\top} \\! \\hat{\\boldsymbol{B}}_{\\bot}\\hat{\\boldsymbol{B}}_{\\bot}^{\\!\\top} \\boldsymbol{B} \\boldsymbol{w}_m .\n\t\\end{align*}\n\t\n\tFor any $m \\in [M]$, let $\\boldsymbol{\\Sigma}_{m,0}:=\\gamma I$. For any $m \\in [M]$ and $t \\geq 1$, let $\\boldsymbol{\\Sigma}_{m,t}:=\\gamma I + \\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}$.\n\t\n\tTaking the absolute value on both sides and using the Cauchy\u2013Schwarz inequality, we obtain that for any $m \\in [M]$, $t \\in [N]$ and $(s,a) \\in \\mathcal{S} \\times \\mathcal{A}$, \n\t\\begin{align*}\n\t\t& \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,t} - \\boldsymbol{\\theta}_m}} \n\t\t\\\\\n\t\t\\leq & \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\nbr{\\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})  \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}_{\\perp} \\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}\n\t\t\\\\&\n\t\t+ \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}  \\nbr{\\sum_{\\tau=1}^{t} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\xi_{m,j}}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}\n\t\t\\\\& \n\t\t+ \\gamma \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}  \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \n\t\t\\\\& \n\t\t+ \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\hat{\\boldsymbol{B}}_{\\bot}\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B} \\boldsymbol{w}_m}\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\sum_{\\tau=1}^{t} \\abr{ \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}_{\\perp} \\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B} \\boldsymbol{w}_m } \\cdot \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}\n\t\t\\\\&\n\t\t+ \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}  \\sqrt{k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}}\n\t\t\\\\& \n\t\t+ \\gamma \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\cdot \\frac{1}{\\sqrt{\\gamma}} \\cdot \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\theta}_m} + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w}\n\t\t\\\\\n\t\t\\leq & \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\cdot \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\cdot \\sum_{\\tau=1}^{t}  \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}\n\t\t\\\\&\n\t\t+ \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}  \\sqrt{k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}}\n\t\t\\\\& \n\t\t+ \\sqrt{\\gamma} L_{\\theta} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w}\n\t\t\\\\\n\t\t\\overset{\\textup{(b)}}{\\leq} & \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\cdot \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\cdot \\sqrt{tk}\n\t\t\\\\&\n\t\t+ \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}}  \\sqrt{k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}}\n\t\t\\\\& \n\t\t+ \\sqrt{\\gamma} L_{\\theta} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w}\n\t\t\\\\\n\t\t= & \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{m,t}^{-1}} \\sbr{ \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\cdot \\sqrt{tk} + \\sqrt{k \\log \\sbr{ 1 + \\frac{ t  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}} + \\sqrt{\\gamma} L_{\\theta} } \\\\& + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} ,\n\t\\end{align*}\n\twhere inequality (a) uses the triangle inequality and the definition of event $\\mathcal{H}$, and inequality (b) is due to Lemma~\\ref{lemma:sqrt_n_k_gamma}.\n\t\n\t\n\tTaking the maximum over $a \\in \\mathcal{A}$ and taking the expectation on $s \\sim \\mathcal{D}$, we have that for any task $m \\in [M]$,\n\t\\begin{align}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \\leq & \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_N^{-1}} } \\cdot\n\t\t\\nonumber\\\\&\n\t\t\\sbr{ \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\cdot \\sqrt{Nk} + \\sqrt{k \\log \\sbr{ 1 + \\frac{ N  }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}} + \\sqrt{\\gamma} L_{\\theta} } \n\t\t\\nonumber\\\\& \n\t\t+ \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} . \\label{eq:ex_max_phi_theta}\n\t\\end{align}\n\t\n\t\n\tAccording to Lemma~\\ref{lemma:decreasing_uncertainty}, $\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_t^{-1}} }$ is non-increasing with respect to $t$. Hence, we have\n\t\\begin{align}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{N}^{-1}} } \\leq & \\frac{1}{N} \\sum_{t=1}^{N} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{t}^{-1}} }\n\t\t\\nonumber\\\\\n\t\t\\leq & \\frac{1}{N} \\sum_{t=1}^{N} \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} } \n\t\t\\nonumber\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\frac{1}{4N} \\Bigg( 2\\sqrt{\\log \\sbr{\\frac{5}{\\delta}}} \\nonumber\\\\& + \\sqrt{ 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\sbr{ \\sum_{t=1}^{N} \\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_t,a) }_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} + 2 \\log \\sbr{\\frac{5}{\\delta}} } } \\Bigg)^2 \n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{4N} \\sbr{ 2\\sqrt{\\log \\sbr{\\frac{5}{\\delta}}} + \\sqrt{ 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\sbr{ \\sum_{t=1}^{N} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_t,a_t)}_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} + 2 \\log \\sbr{\\frac{5}{\\delta}} } } }^{\\!\\!\\!2} \\!\\! , \\label{eq:ex_max_B_phi}\n\t\\end{align}\n\twhere inequality (a) is due to the definition of event $\\mathcal{J}$.\n\t\n\tIn addition, we have \n\t\\begin{align}\n\t\t\\sum_{t=1}^{N} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_t,a_t)}_{\\boldsymbol{\\Sigma}_{t-1}^{-1}} \\leq & \\sqrt{N \\cdot \\sum_{t=1}^{N} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_t,a_t)}^2_{\\boldsymbol{\\Sigma}_{t-1}^{-1}}}\n\t\t\\nonumber\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\sqrt{2N \\log \\sbr{\\frac{\\det \\sbr{\\gamma I + \\sum_{\\tau=1}^{N} \\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau}) \\boldsymbol{\\phi}(s_{m,\\tau},a_{m,\\tau})^\\top \\hat{\\boldsymbol{B}}}}{\\det \\sbr{\\gamma I}}}}\n\t\t\\nonumber\\\\\n\t\t\\overset{\\textup{(b)}}{\\leq} & \\sqrt{2 N k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }} , \\label{eq:B_phi_leq_k_log}\n\t\\end{align}\n\twhere inequality (a) uses Lemma~\\ref{lemma:elliptical_potential}, and inequality (b) is due to Lemma~\\ref{lemma:computation_logdet}.\n\t\n\t\n\tCombining Eqs.~\\eqref{eq:ex_max_B_phi} and \\eqref{eq:B_phi_leq_k_log}, we have\n\t\\begin{align}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\nbr{\\hat{\\boldsymbol{B}}^\\top \\boldsymbol{\\phi}(s,a)}_{\\boldsymbol{\\Sigma}_{N}^{-1}} } \\leq &\n\t\t\\frac{1}{4N} \\sbr{ 2\\sqrt{\\log \\sbr{\\frac{5}{\\delta}}} + \\sqrt{ 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\sbr{ \\sqrt{2 N k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }} + 2 \\log \\sbr{\\frac{5}{\\delta}} } } }^2\n\t\t\\nonumber\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\frac{1}{2N} \\sbr{ 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\log \\sbr{\\frac{5}{\\delta}} + 4 \\sbr{ \\sqrt{2 N k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }} + 2 \\log \\sbr{\\frac{5}{\\delta}} } }\n\t\t\\nonumber\\\\\n\t\t= & \\frac{1}{N} \\sbr{ 2\\sqrt{2 N k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }} + 8 \\log \\sbr{\\frac{5}{\\delta}} }\n\t\t\\nonumber\\\\\n\t\t= &  2\\sqrt{ \\frac{2 k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }}{N} } + \\frac{8 \\log \\sbr{\\frac{5}{\\delta}} }{N} , \\label{eq:sum_ex_max_B_phi_final_bound}\n\t\\end{align}\n\twhere inequality (a) uses the Cauchy\u2013Schwarz inequality.\n\t\n\tFurthermore, plugging Eq.~\\eqref{eq:sum_ex_max_B_phi_final_bound} into Eq.~\\eqref{eq:ex_max_phi_theta} and using $\\gamma \\geq 1$, we have that for $N \\geq 1$ and $\\sqrt{k}\\log(2N) \\geq 1$,\n\t\n\t\n\t\\begin{align}\n\t\t& \\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \n\t\t\\\\\n\t\t\\leq & \\sbr{2\\sqrt{ \\frac{2 k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} }}{N} } + \\frac{8 \\log \\sbr{\\frac{5}{\\delta}} }{N}} \\cdot \\nonumber\\\\& \\sbr{ \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\sqrt{N k} + \\sqrt{k \\log \\sbr{ 1 + \\frac{ N }{\\gamma k} } + 2 \\log\\sbr{\\frac{5}{\\delta}}} + \\sqrt{\\gamma} L_{\\theta} } + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w}\n\t\t\\nonumber\\\\\n\t\t\\leq & \\frac{ 12 \\sqrt{k} \\log \\sbr{\\frac{5N}{\\delta}} }{\\sqrt{N}} \\sbr{ \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} \\sqrt{Nk} +  2\\sqrt{k} \\log\\sbr{\\frac{5N}{\\delta}} + \\sqrt{\\gamma} L_{\\theta} } + \\nbr{\\hat{\\boldsymbol{B}}_{\\bot}^\\top \\boldsymbol{B}} L_{\\phi} L_{w}\n\t\t\\nonumber\\\\\n\t\t\\leq &  \\frac{ \\sbr{24 k + 12 \\sqrt{k \\gamma} L_{\\theta} } \\log^2 \\sbr{\\frac{5N}{\\delta}} }{\\sqrt{N}} +  24 k \\log \\sbr{\\frac{5N}{\\delta}} \\nbr{\\hat{\\boldsymbol{B}}_{\\perp}^\\top \\boldsymbol{B}} L_{\\phi} L_{w} . \\label{eq:ex_max_phi_theta_B_hat_times_B}\n\t\n\t\n\t\n\t\n\t\\end{align}\n\t\n\t\n\tUsing Lemma~\\ref{lemma:technical_tool_log_N_square} with $A=24 k + 12 \\sqrt{k \\gamma} L_{\\theta}$, $B=\\frac{5}{\\delta}$ and $\\kappa=\\frac{\\varepsilon}{4}$, we have that if\n\t\\begin{align*}\n\t\tN \\geq \\frac{26^4 \\sbr{24 k + 12 \\sqrt{k \\gamma} L_{\\theta}}^2 \\log^4 \\big({\\frac{2 \\cdot 5 (24 k + 12 \\sqrt{k \\gamma} L_{\\theta}) }{\\varepsilon \\delta}} \\big) }{\\sbr{\\frac{\\varepsilon}{4}}^2} ,\n\t\\end{align*} \n\tthen $\\frac{ \\sbr{24 k + 12 \\sqrt{k \\gamma} L_{\\theta} } \\log^2 \\sbr{\\frac{5N}{\\delta}} }{\\sqrt{N}} \\leq \\frac{\\varepsilon}{4}$.\n\t\n\tFurther enlarging $N$, if \n\t\\begin{align}\n\t\tN \\geq \\frac{4^2 \\cdot 26^4 \\cdot 24^2 \\cdot 2 \\sbr{k^2 + k \\gamma L_{\\theta}^2 } \\log^4 \\big({\\frac{240 ( k + \\sqrt{k \\gamma} L_{\\theta}) }{\\varepsilon \\delta}} \\big) }{\\varepsilon^2} , \\label{eq:number_of_samples_N}\n\t\\end{align}\n\tthen\n\t\\begin{align*}\n\t\t\\frac{ \\sbr{24 k + 12 \\sqrt{k \\gamma} L_{\\theta} } \\log^2 \\sbr{\\frac{5N}{\\delta}} }{\\sqrt{N}} \\leq \\frac{\\varepsilon}{4} .\n\t\\end{align*}\n\t\n\tAccording to Lemma~\\ref{lemma:concentration_hat_B_clb}, we have $\\nbr{\\hat{\\boldsymbol{B}}_{t,\\bot}^\\top \\boldsymbol{B}} \\leq \\frac{\\varepsilon}{ 96 k \\log \\sbr{\\frac{5N}{\\delta}} L_{\\phi} L_{w} }$.\n\t\n\tThus, setting $N$ as the value in Eq.~\\eqref{eq:number_of_samples_N}, and continuing with Eq.~\\eqref{eq:ex_max_phi_theta_B_hat_times_B}, we have\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \\leq  \\frac{\\varepsilon}{4} + \\frac{\\varepsilon}{4}\n\t\t=  \\frac{\\varepsilon}{2} .\n\t\\end{align*}\n\t\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:bpi_ub}}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bpi_ub}]\n\tCombining Lemmas~\\ref{lemma:number_of_samples_T0}, \\ref{lemma:number_of_samples_p}, \\ref{lemma:Z_est_error}, \\ref{lemma:martingale_concentration_variance} and \\ref{lemma:reverse_bernstein_uncertainty}, we have that $Pr[\\mathcal{K} \\cap \\mathcal{L} \\cap \\mathcal{G} \\cap \\mathcal{H} \\cap \\mathcal{J}] \\geq 1-\\delta$.\n\tSuppose that event $\\mathcal{K} \\cap \\mathcal{L} \\cap \\mathcal{G} \\cap \\mathcal{H} \\cap \\mathcal{J}$ holds.\n\t\n\tFirst, we uses a similar analytical procedure as that\n\n\tin \\cite{zanette2021design} to prove the correctness.\n\t\n\tUsing Lemma~\\ref{lemma:number_of_samples_N}, we have that for any task $m \\in [M]$,\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s \\sim \\mathcal{D}} \\mbr{\\max_{a \\in \\mathcal{A}} \\abr{\\boldsymbol{\\phi}(s,a)^\\top \\sbr{ \\hat{\\boldsymbol{\\theta}}_{m,N} - \\boldsymbol{\\theta}_m}}} \\leq \\frac{\\varepsilon}{2} .\n\t\\end{align*}\n\t\n\tFor any $m \\in [M]$ and $s \\in \\mathcal{S}$, let $\\beta_m(s):=\\max_{a\\in\\mathcal{A}}|\\boldsymbol{\\phi}(s,a)^\\top (\\hat{\\boldsymbol{\\theta}}_{m,N}-\\boldsymbol{\\theta}_m)|$ and $\\pi^*_m(s):=\\operatornamewithlimits{argmax}_{a \\in \\mathcal{A}} \\boldsymbol{\\phi}(s,a)^\\top \\boldsymbol{\\theta}_m$.\n\t\n\tFor any $m \\in [M]$ and $s \\in \\mathcal{S}$, we have\n\t\\begin{align*}\n\t\t\\boldsymbol{\\phi}(s,\\hat{\\pi}_m(s))^\\top \\boldsymbol{\\theta}_m \\geq & \\boldsymbol{\\phi}(s,\\hat{\\pi}_m(s))^\\top \\hat{\\boldsymbol{\\theta}}_{m,N} - \\beta_m(s)\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\geq} & \\boldsymbol{\\phi}(s,\\pi^*_m(s))^\\top \\hat{\\boldsymbol{\\theta}}_{m,N} - \\beta_m(s)\n\t\t\\\\\n\t\t\\geq & \\boldsymbol{\\phi}(s,\\pi^*_m(s))^\\top \\boldsymbol{\\theta}_m - 2\\beta_m(s) ,\n\t\\end{align*}\n\twhere inequality (a) is due to that $\\hat{\\pi}_m(s)$ is greedy with respect to $\\hat{\\boldsymbol{\\theta}}_{m,N}$.\n\t\n\tRearranging the above equation and taking the expectation of $s$ on both sides, we have\n\t\\begin{align*}\n\t\t\\mathbb{E}_{s\\sim\\mathcal{D}}\\mbr{\\max_{a \\in \\mathcal{A}} \\sbr{\\boldsymbol{\\phi}(s,a) - \\boldsymbol{\\phi}(s,\\hat{\\pi}_m(s))}^\\top \\boldsymbol{\\theta}_m }\n\t\t\\leq  2 \\mathbb{E}_{s\\sim\\mathcal{D}}[\\beta_m(s)] \\leq \\varepsilon .\n\t\\end{align*}\n\t\n\t\n\tNow we prove the sample complexity.\n\tSumming the number of samples used in the main algorithm of $\\mathtt{C \\hyphen DouExpDes}$ and subroutines $\\mathtt{C \\hyphen FeatRecover}$ and $\\mathtt{EstLowRep}$ (Line~\\ref{line:bpi_estimate_context_dis} in Algorithm~\\ref{alg:repbpiclb}, Lines~\\ref{line:bpi_stage2_sample1}-\\ref{line:bpi_stage2_sample2} in Algorithm~\\ref{alg:con_feat_recover} and Line~\\ref{line:bpi_stage3_sample} in Algorithm~\\ref{alg:est_low_rep}), we have that the total number of samples is bounded by\n\t\\begin{align*}\n\t\t& T_0+2MTp+MN \n\t\t\\\\\n\t\t= & O \\Bigg( \\frac{L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{d |\\mathcal{A}|}{\\delta}} + \\frac{ k^4 L_{\\phi}^4 L_{\\theta}^2 L_{w}^2 }{ \\nu^2 \\varepsilon^2 } \\log^6 \\sbr{ \\frac{ k d L_{\\phi} L_{\\theta} L_{w} N }{\\nu \\delta \\varepsilon} } \\cdot \\frac{L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{dMT}{\\delta}} \\\\&+ M \\cdot \\frac{\\sbr{k^2 + k \\gamma L_{\\theta}^2 } \\log^4 \\big({\\frac{ k + \\sqrt{k \\gamma} L_{\\theta} }{\\varepsilon \\delta}} \\big) }{\\varepsilon^2} \\Bigg)\n\t\t\\\\\n\t\t= & O \\Bigg( \\frac{ k^4 L_{\\phi}^4 L_{\\theta}^2 L_{w}^2 }{ \\nu^2 \\varepsilon^2 } \\log^6 \\sbr{ \\frac{ |\\mathcal{A}| k d L_{\\phi} L_{\\theta} L_{w} N }{\\nu \\delta \\varepsilon} } \\cdot \\frac{L_{\\phi}^4}{\\nu^2} \\log^2 \\sbr{\\frac{dMT}{\\delta}} \\\\&+ M \\cdot \\frac{\\sbr{k^2 + k \\gamma L_{\\theta}^2 } \\log^4 \\big({\\frac{ k + \\sqrt{k \\gamma} L_{\\theta} }{\\varepsilon \\delta}} \\big) }{\\varepsilon^2} \\Bigg)\n\t\t\\\\\n\t\t= & \\tilde{O} \\Bigg( \\frac{ k^4 L_{\\phi}^8 L_{\\theta}^2 L_{w}^2 }{ \\nu^4 \\varepsilon^2 } + \\frac{ M \\sbr{k^2 + k \\gamma L_{\\theta}^2 } }{\\varepsilon^2} \\Bigg) .\n\t\\end{align*}\n\\end{proof}\n\n\n\\section{Technical Tools}\n\nIn this section, we provide some useful technical tools.\n\n\\begin{lemma}[Matrix Bernstern Inequality - Average, Lemma 31 in \\cite{tripuraneni2021provable}] \\label{lemma:matrix_bernstein_tripuraneni2021}\n\tConsider a truncation level $U>0$. If $\\{\\boldsymbol{Z}_1,\\dots,\\boldsymbol{Z}_n\\}$ is a sequence of $d_1 \\times d_2$ independent random matrices and \n\t$\\boldsymbol{Z}'_i = \\boldsymbol{Z}_i \\cdot \\indicator{\\|\\boldsymbol{Z}_i\\| \\leq U}$ for any $i \\in [n]$, then\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\nbr{ \\frac{1}{n} \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq t } \\leq \n\t\t\\Pr \\mbr{ \\nbr{ \\frac{1}{n} \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}'_i - \\mathbb{E}[\\boldsymbol{Z}'_i] } } \\geq t - \\Delta } + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } ,\n\t\\end{align*}\n\twhere $\\Delta \\geq \\|\\mathbb{E}[\\boldsymbol{Z}_i]-\\mathbb{E}[\\boldsymbol{Z}'_i]\\|$ for any $i \\in [n]$. \n\t\n\tIn addition, for $t \\geq \\Delta$, we have\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\nbr{ \\frac{1}{n} \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}'_i - \\mathbb{E}[\\boldsymbol{Z}'_i] } } \\geq t - \\Delta } \\leq (d_1 + d_2) \\exp \\sbr{- \\frac{n^2 (t-\\Delta)^2}{2\\sigma^2 + \\frac{2Un(t-\\Delta)}{3}}} ,\n\t\\end{align*}\n\twhere\n\t\\begin{align*}\n\t\t\\sigma^2 = & \\max \\lbr{ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[(\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])^\\top (\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])]},\\ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[(\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i]) (\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])^\\top]} } \n\t\t\\\\\n\t\t\\leq & \\max \\lbr{ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[{\\boldsymbol{Z}'_i}^\\top \\boldsymbol{Z}'_i]},\\ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[\\boldsymbol{Z}'_i {\\boldsymbol{Z}'_i}^\\top]} } .\n\t\n\t\n\t\\end{align*} \n\\end{lemma}\n\nLemma 31 in \\cite{tripuraneni2021provable} gives a truncated matrix Bernstern inequality for symmetric random matrices. Here we extend it to general random matrices.\n\nLemma~\\ref{lemma:matrix_bernstein_tripuraneni2021} can be obtained by combining the truncation argument in the proof of Lemma 31 in \\cite{tripuraneni2021provable} and Theorem 6.1.1 in \\cite{tropp2015introduction} (classic matrix Bernstern inequality for general random matrices).\n\n\\begin{lemma}[Matrix Bernstern Inequality - Summation] \\label{lemma:matrix_bernstein_tau}\n\tConsider a truncation level $U>0$. If $\\{\\boldsymbol{Z}_1,\\dots,\\boldsymbol{Z}_n\\}$ is a sequence of $d_1 \\times d_2$ independent random matrices, and $\\boldsymbol{Z}'_i = \\boldsymbol{Z}_i \\cdot \\indicator{\\|\\boldsymbol{Z}_i\\| \\leq U}$ and $\\Delta \\geq \\|\\mathbb{E}[\\boldsymbol{Z}_i]-\\mathbb{E}[\\boldsymbol{Z}'_i]\\|$ for any $i \\in [n]$, then for $\\tau \\geq 2 n\\Delta$,\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\nbr{ \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq \\tau } \\leq \n\t\t(d_1 + d_2) \\exp \\sbr{- \\frac{1}{4} \\cdot \\frac{ \\tau^2}{2\\sigma^2 + \\frac{U \\tau}{3}}} + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } ,\n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t\t\\sigma^2 = & \\max \\lbr{ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[(\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])^\\top (\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])]},\\ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[(\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i]) (\\boldsymbol{Z}'_i-\\mathbb{E}[\\boldsymbol{Z}'_i])^\\top]} } \n\t\t\\\\\n\t\t\\leq & \\max \\lbr{ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[{\\boldsymbol{Z}'_i}^\\top \\boldsymbol{Z}'_i]},\\ \\nbr{\\sum_{i=1}^{n} \\mathbb{E}[\\boldsymbol{Z}'_i {\\boldsymbol{Z}'_i}^\\top]} } .\n\t\n\t\n\t\\end{align*}\n\t\n\tFurthermore, we have\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\nbr{ \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq 4 \\sqrt{ \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } + 4 U \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } \\leq \n\t\t\\delta + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:matrix_bernstein_tau}]\n\tUsing Lemma~\\ref{lemma:matrix_bernstein_tripuraneni2021} and defining $\\tau:=nt$, we have that for $\\tau>n\\Delta$,\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\nbr{ \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq \\tau } \\leq & \n\t\t(d_1 + d_2) \\exp \\sbr{- \\frac{ (\\tau-n \\Delta)^2}{2\\sigma^2 + \\frac{2U(\\tau-n \\Delta)}{3}}} + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } .\n\t\\end{align*}\n\tIf $\\tau>2n\\Delta$, then $\\tau-n\\Delta>\\frac{1}{2} \\tau$ and we have\n\t\\begin{align}\n\t\t\\Pr \\mbr{ \\nbr{ \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq \\tau } \n\t\t\\leq & (d_1 + d_2) \\exp \\sbr{- \\frac{ \\sbr{\\frac{1}{2} \\tau}^2}{2\\sigma^2 + \\frac{2U \\sbr{\\frac{1}{2} \\tau}}{3} } } + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } \n\t\t\\nonumber\\\\\n\t\t\\leq & (d_1 + d_2) \\exp \\sbr{- \\frac{1}{4} \\cdot \\frac{ \\tau^2 }{2\\sigma^2 + \\frac{U \\tau}{3} }} + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } . \\label{eq:matrix_bernstein_error}\n\t\\end{align} \n\tPlugging $\\tau=4 \\sqrt{ \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } + 4 U \\log \\sbr{\\frac{d_1 + d_2}{\\delta}}$ into Eq.~\\eqref{eq:matrix_bernstein_error}, we have\n\t\\begin{align*}\n\t\t& \\Pr \\mbr{ \\nbr{ \\sum_{i=1}^{n} \\sbr{ \\boldsymbol{Z}_i - \\mathbb{E}[\\boldsymbol{Z}_i] } } \\geq 4 \\sqrt{ \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } + 4 U \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } \n\t\t\\\\\n\t\t\\leq & (d_1 + d_2) \\exp \\sbr{- \\frac{1}{4} \\cdot \\frac{ 16 \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} + 16 U^2 \\log^2 \\sbr{\\frac{d_1 + d_2}{\\delta}} + 32 U \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} \\sqrt{ \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } }{2\\sigma^2 + \\frac{1}{3} \\sbr{4U \\sqrt{ \\sigma^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } + 4 U^2 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} }} }  \\\\& + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U}\n\t\t\\\\\n\t\t\\leq & (d_1 + d_2) \\exp \\sbr{- \\frac{1}{4} \\cdot 4 \\log \\sbr{\\frac{d_1 + d_2}{\\delta}} } + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U }\n\t\t\\\\\n\t\t= & \\delta + n \\Pr \\mbr{ \\|\\boldsymbol{Z}_i\\| \\geq U } .\n\t\\end{align*}\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:technical_tool_bai_stage2}\n\tFor any $A,B>1$, $\\kappa \\in (0,1)$ and $T>0$ such that $\\log\\sbr{\\frac{AB}{\\kappa}}>1$ and $\\log(BT)>2$, if \n\t$$\n\tT \\geq \\frac{68 A^2 \\log^2 \\sbr{\\frac{AB}{\\kappa}} }{\\kappa^2} ,\n\t$$\n\tthen\n\t$$\n\t\\frac{A}{\\sqrt{T}} \\log(B T) \\leq \\kappa .\n\t$$\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:technical_tool_bai_stage2}]\n\tIf $T = \\frac{68 A^2 \\log^2 \\sbr{\\frac{AB}{\\kappa}} }{\\kappa^2}$, we have\n\t\\begin{align*}\n\t\t\\frac{A}{\\sqrt{T}} \\log(B T) = & \\frac{A \\kappa}{ \\sqrt{68} A \\log \\sbr{\\frac{AB}{\\kappa}} } \\log \\sbr{ \\frac{68 A^2 B \\log^2 \\sbr{\\frac{AB}{\\kappa}} }{\\kappa^2}}\n\t\t\\\\\n\t\t= & \\frac{\\kappa}{ \\sqrt{68} \\log \\sbr{\\frac{AB}{\\kappa}} } \\sbr{ \\log \\sbr{68} + \\log \\sbr{ \\frac{A^2 B}{\\kappa^2}} + \\log \\sbr{ \\log^2 \\sbr{\\frac{AB}{\\kappa}} } }\n\t\t\\\\\n\t\t\\leq & \\frac{\\kappa}{ \\sqrt{68} \\log \\sbr{\\frac{AB}{\\kappa}} } \\sbr{ \\log \\sbr{68} + 2 \\log \\sbr{ \\frac{A B}{\\kappa}} + 2 \\log \\sbr{ \\frac{AB}{\\kappa} } }\n\t\t\\\\\n\t\t\\leq & \\frac{\\kappa}{ \\sqrt{68} \\log \\sbr{\\frac{AB}{\\kappa}} } \\sbr{ \\log \\sbr{68} \\log \\sbr{ \\frac{A B}{\\kappa}} + 4 \\log \\sbr{ \\frac{A B}{\\kappa}} }\n\t\t\\\\\n\t\t\\leq & \\kappa .\n\t\\end{align*}\n\t\n\tLet $f(T)=\\frac{A}{\\sqrt{T}} \\log(B T)$. Then, the derivative of $f(T)$ is\n\t\\begin{align*}\n\t\tf'(T)= \\frac{2A-A\\log(BT)}{2T\\sqrt{T}} .\n\t\\end{align*}\n\tIf $\\log(BT)>2$, then $f'(T)<0$, and thus $f(T)$ is decreasing with respect to $T$.\n\t\n\tTherefore, if $T \\geq \\frac{68 A^2 \\log^2 \\sbr{\\frac{AB}{\\kappa}} }{\\kappa^2}$, we have\n\t\\begin{align*}\n\t\t\\frac{A}{\\sqrt{T}} \\log(B T) \\leq \\kappa.\n\t\\end{align*}\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:technical_tool_log_N_square}\n\tFor any $A,B>1$ and $\\kappa \\in (0,1)$ such that $\\log (\\frac{AB}{\\kappa})>1$ and $\\log(BN)>4$, if \n\t$$\n\tN \\geq \\frac{26^4 A^2 \\log^4 (\\frac{AB}{\\kappa}) }{\\kappa^2} ,\n\t$$\n\tthen\n\t$$\n\t\\frac{A \\log^2 \\sbr{BN}}{\\sqrt{N}} \\leq \\kappa .\n\t$$\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:technical_tool_log_N_square}]\n\tIf $N = \\frac{26^4 A^2 \\log^4 (\\frac{AB}{\\kappa}) }{\\kappa^2}$, we have\n\t$\n\t\\kappa \\sqrt{N} = 26^2 A \\log^2 (\\frac{AB}{\\kappa}) \n\t$,\n\tand\n\t\\begin{align*}\n\t\tA \\log^2 \\sbr{BN} = & A \\log^2 \\sbr{ \\frac{26^4 A^2 B \\log^4 (\\frac{AB}{\\kappa}) }{\\kappa^2} }\n\t\t\\\\\n\t\t\\leq & A \\log^2 \\sbr{ \\frac{26^4 A^2 B }{\\kappa^2} \\cdot \\frac{A^4 B^4}{\\kappa^4}  }\n\t\t\\\\\n\t\t\\leq & 36 A \\log^2 \\sbr{ \\frac{26 A B }{\\kappa} }\n\t\t\\\\\n\t\t= & 36 A \\sbr{\\log \\sbr{26} + \\log \\sbr{ \\frac{A B }{\\kappa} }}^2\n\t\t\\\\\n\t\t\\leq & 36 A \\sbr{\\log \\sbr{26} \\log \\sbr{ \\frac{A B }{\\kappa} } + \\log \\sbr{ \\frac{A B }{\\kappa} }}^2\n\t\t\\\\\n\t\t= & 36 \\sbr{\\log \\sbr{26}+1}^2 A \\log^2 \\sbr{ \\frac{A B }{\\kappa} }\n\t\t\\\\\n\t\t\\leq & 26^2 A \\log^2 \\sbr{ \\frac{A B }{\\kappa} }\n\t\t\\\\\n\t\t= & \\kappa \\sqrt{N} ,\n\t\\end{align*}\n\tand thus $\\frac{A \\log^2 \\sbr{BN}}{\\sqrt{N}} \\leq \\kappa$.\n\t\n\tLet $f(N)=\\frac{A \\log^2 \\sbr{BN}}{\\sqrt{N}}$. Then, the derivative function of $f(N)$ is\n\t\\begin{align*}\n\t\tf'(N)= \\frac{4A\\log(BN)-A\\log^2(BN)}{2N\\sqrt{N}} = \\frac{A\\log(BN) \\cdot (4-\\log(BN))}{2N\\sqrt{N}} .\n\t\\end{align*}\n\tIf $\\log(BN)>4$, then $f'(N)<0$, and thus $f(N)$ is decreasing with respect to $N$.\n\t\n\tTherefore, if $N \\geq \\frac{26^4 A^2 \\log^4 (\\frac{AB}{\\kappa}) }{\\kappa^2}$, we have\n\t$\\frac{A \\log^2 \\sbr{BN}}{\\sqrt{N}} \\leq \\kappa$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:sqrt_n_k}\n\tFor any $\\boldsymbol{x}_1,\\dots,\\boldsymbol{x}_n \\in \\mathbb{R}^k$, we have\n\t\\begin{align*}\n\t\t\\sum_{j=1}^{n} \\|\\boldsymbol{x}_j\\|_{\\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} \\leq \\sqrt{nk} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:sqrt_n_k}]\n\tIt holds that\n\t\\begin{align*}\n\t\t\\sum_{j=1}^{n} \\|\\boldsymbol{x}_j\\|_{\\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} = & \\sum_{j=1}^{n} \\sqrt{\\boldsymbol{x}_j^\\top \\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}\n\t\t\\\\\n\t\t\\leq & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\boldsymbol{x}_j^\\top \\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}\n\t\t\\\\\n\t\t\\leq & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\textup{Trace} \\sbr{ \\boldsymbol{x}_j^\\top \\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\textup{Trace} \\sbr{ \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} }}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\sum_{j=1}^{n} \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} }}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\boldsymbol{I}_k } }\n\t\t\\\\\n\t\t= & \\sqrt{ n k }\n\t\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:sqrt_n_k_gamma}\n\tFor any $\\boldsymbol{x}_1,\\dots,\\boldsymbol{x}_n \\in \\mathbb{R}^k$ and $\\gamma>0$, we have\n\t\\begin{align*}\n\t\t\\sum_{j=1}^{n} \\|\\boldsymbol{x}_j\\|_{\\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} \\leq \\sqrt{nk} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:sqrt_n_k_gamma}]\n\tIt holds that\n\t\\begin{align*}\n\t\t\\sum_{j=1}^{n} \\|\\boldsymbol{x}_j\\|_{\\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1}} = & \\sum_{j=1}^{n} \\sqrt{\\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}\n\t\t\\\\\n\t\t\\leq & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\textup{Trace} \\sbr{ \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} \\boldsymbol{x}_j}}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot  \\sum_{j=1}^{n} \\textup{Trace} \\sbr{ \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} }}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\sum_{j=1}^{n} \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} }}\n\t\t\\\\\n\t\t\\overset{\\textup{(a)}}{\\leq} & \\sqrt{ n \\!\\cdot\\! \\sbr{\\textup{Trace} \\sbr{ \\sum_{j=1}^{n} \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{\\!\\!\\!-1} } \\!+\\! \\textup{Trace} \\sbr{ \\gamma \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{\\!\\!\\!-1} } } }\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\sum_{j=1}^{n} \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} + \\gamma \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} }}\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\sbr{\\gamma I + \\sum_{j=1}^{n} \\boldsymbol{x}_j \\boldsymbol{x}_j^\\top} \\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}^{-1} } }\n\t\t\\\\\n\t\t= & \\sqrt{ n \\cdot   \\textup{Trace} \\sbr{ \\boldsymbol{I}_k } }\n\t\t\\\\\n\t\t= & \\sqrt{ n k } ,\n\t\\end{align*}\n\twhere inequality (a) is due to that $\\sbr{\\gamma I + \\sum_{i=1}^{n} \\boldsymbol{x}_i \\boldsymbol{x}_i^\\top}$ is a positive definite matrix.\n\\end{proof}\n\n\n\n\\begin{lemma}[Self-normalized Concentration for Martingales, Theorem 1 in \\cite{abbasi2011improved}]\\label{lemma:self-normalized_vector_concentration}\n\tLet $\\{\\mathcal{F}_t\\}_{t=0}^{\\infty}$ be a filtration such that for any $t \\geq 1$, the selected action $\\boldsymbol{X}_t \\in \\mathbb{R}^k$ is $\\mathcal{F}_{t-1}$-measurable, the noise $\\eta_t \\in \\mathbb{R}$ is $\\mathcal{F}_{t}$-measurable, and conditioning on $\\mathcal{F}_{t-1}$, $\\eta_t$ is zero-mean and $R$-sub-Gaussian. Let $\\boldsymbol{V}_0 \\in \\mathbb{R}^{k \\times k}$ be a positive definite matrix and let $\\boldsymbol{V}_t=\\sum_{i=1}^{t} \\boldsymbol{X}_i \\boldsymbol{X}_i^\\top$ for any $t \\geq 1$. Then, for any $\\delta>0$, with probability at least $1-\\delta$, for all $t \\geq 1$,\n\t\\begin{align*}\n\t\t\\nbr{\\sum_{i=1}^{t} \\boldsymbol{X}_i \\cdot  \\eta_i}^2_{\\sbr{\\boldsymbol{V}_0+\\boldsymbol{V}_t}^{-1}} \\leq 2R^2 \\log \\sbr{ \\frac{\\det(\\boldsymbol{V}_t)^{\\frac{1}{2}} }{ \\det(\\boldsymbol{V}_0)^{\\frac{1}{2}} \\cdot \\delta} } .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{lemma}[Reverse Bernstein Inequality for Martingales, Theorem 3 in \\cite{zanette2021design}] \\label{lemma:reverse_bernstein}\n\tLet $(\\boldsymbol{\\Sigma},\\mathcal{F},\\Pr[\\cdot])$ be a probability space and consider the stochastic process $\\{\\boldsymbol{X}_t\\}$ adapted to the filtration $\\{\\mathcal{F}_t\\}$. Let $\\mathbb{E}_t [\\boldsymbol{X}_t]:=\\mathbb{E}[\\boldsymbol{X}_t|\\mathcal{F}_{t-1}]$ be the conditional expectation of $\\boldsymbol{X}_t$ given $\\mathcal{F}_{t-1}$. If $0 \\leq \\boldsymbol{X}_t \\leq 1$ then it holds that\n\t\\begin{align*}\n\t\t\\Pr \\mbr{ \\sum_{t=1}^{T} \\mathbb{E}_t [\\boldsymbol{X}_t] \\geq \\frac{1}{4} \\sbr{ 2\\sqrt{\\log \\sbr{\\frac{1}{\\delta}}} + \\sqrt{ 4 \\log \\sbr{\\frac{1}{\\delta}} + 4 \\sbr{ \\sum_{t=1}^{T} \\boldsymbol{X}_t + 2 \\log \\sbr{\\frac{1}{\\delta}} } } }^2 } \\leq \\delta .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{lemma}[Elliptical Potential Lemma, Lemma 11 in \\cite{abbasi2011improved}] \\label{lemma:elliptical_potential}\n\tLet $\\{\\boldsymbol{X}_t\\}_{t=1}^{\\infty}$ be a sequence in $\\mathbb{R}^k$. Let $\\boldsymbol{V}_0$ be a $k \\times k$ positive definite matrix and let $\\boldsymbol{V}_t = \\boldsymbol{V}_0 + \\sum_{i=1}^{t} \\boldsymbol{X}_i \\boldsymbol{X}_i^\\top$ such that for any $t \\geq 1$, $\\|\\boldsymbol{X}_t\\|^2_{\\boldsymbol{V}_{t-1}^{-1}} \\leq 1$. Then, we have that\n\t\\begin{align*}\n\t\t\\sum_{t=1}^{n} \\nbr{ \\boldsymbol{X}_t }^2_{\\boldsymbol{V}_{t-1}^{-1}} \\leq 2 \\log \\frac{\\det(\\boldsymbol{V}_n)}{\\det(\\boldsymbol{V}_0)} .\n\t\\end{align*}\n\\end{lemma}\n\n\n\n\n\n\\begin{lemma}[Moments of Sub-Gaussian Random Variables, Proposition 3.2 in \\cite{subgaussian_note}] \\label{lemma:subgaussian_moment}\n\tFor a $\\sigma^2$-sub-Gaussian random variable $\\boldsymbol{X}$ which satisfies\n\t\\begin{align*}\n\t\t\\mathbb{E} \\mbr{ \\exp \\sbr{\\mu \\boldsymbol{X} } } \\leq \\exp \\sbr{ \\frac{\\sigma^2 \\mu^2}{2} }, \\ \\forall \\mu \\in \\mathbb{R} ,\n\t\\end{align*}\n\twe have that for any integer $n \\geq 1$,\n\t\\begin{align*}\n\t\t\\mathbb{E} [|\\boldsymbol{X}|^{n}] \\leq \\sbr{2\\sigma^2}^{\\frac{n}{2}} n \\cdot \\Gamma\\sbr{\\frac{n}{2}} ,\n\t\\end{align*}\n\twhere $\\Gamma(n):=(n-1)!$ for any integer $n\\geq1$.\n\\end{lemma}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\vskip4mm Let $R$ be a ring with an identity. Idempotents, units\nand nilpotent elements play important rules in ring theory. The\nmotivation of this paper is to investigate the structures of\nvarious rings involving such special elements. An element $a$ in a\nring is called very idempotent if $a$ or $-a$ is an idempotent. An\nelement $a\\in R$ is called (weakly) clean if there exists a (very)\nidempotent $e\\in R$ and a unit such that $a=e+u$~\\cite{A}. An\nelement $a\\in R$ is (weakly) nil-clean provided that there exists\na (very) idempotent $e\\in R$ and a nilpotent $w\\in R$ such that\n$a=e+w$~\\cite{B} and ~\\cite{D}. These inspire us introduce two\nconcepts. We call an element $a\\in R$ is (weakly) precious if\nthere exists a (very) idempotent $e\\in R$, a unit $u\\in R$ and a\nnilpotent $w\\in R$ such that $a=e+u+w$. A ring $R$ is called a\nweakly clean (weakly precious, nil-clean, precious) ring if every\nelement in $R$ is weakly clean (weakly precious, nil-clean,\nprecious). Many fundamental properties about commutative weakly\nclean rings were obtained in ~\\cite{Ah} and ~\\cite{A}, and that\nweakly nil-clean rings were comprehensive studied by Breaz et al.\nin ~\\cite{B}.\n\nIn this paper, we shall explore the structures of these rings. In\nSection 2, we prove that the direct product $R=\\prod R_{i}$ of\nrings $R_i$ is weakly precious if and only if each $R_{i}$ is\nweakly precious and at most one is not precious. Furthermore, we\nshow that the precious property is invariant for any Morita\ncontext. In Section 3, we are concern on weakly clean rings and\nnil-clean rings. Let $R$ be a commutative ring with at most three\nmaximal ideals. If $2\\in U(R)$ and $J(R)$ is nil, we prove that\n$R$ is weakly clean. This provides a new type of weakly clean\nrings. A ring $R$ is abelian if every idempotent is central. We\nshow that if $R$ is abelian then $M_n(R)$ is nil-clean if and only\nif $R\/J(R)$ is Boolean and $M_n(J(R))$ is nil. This extend the\nmain results of Breaz et al. ~\\cite{BGDT} and that of Ko\\c{s}an et\nal.~\\cite{KLZ}. In the last section, we investigate when a ring\nconsists entirely of very idempotent, units, and nilpotent\nelements. We prove that a ring consists entirely of very\nidempotents, units and nilpotent elements if and only if $R$ is\nisomorphic to one of the following: a Boolean ring; ${\\Bbb\nZ}_3\\bigoplus {\\Bbb Z}_3$; ${\\Bbb Z}_3\\oplus B$ where $B$ is a\nBoolean ring; local ring with a nil Jacobson radical;\n$M_2\\big({\\Bbb Z}_2\\big)$ or $M_2\\big({\\Bbb Z}_3\\big)$; or the\nring of a Morita context with zero pairings where the underlying\nrings are ${\\Bbb Z}_2$ or ${\\Bbb Z}_3$. The structure of such\nrings is thereby completely determined.\n\nThroughout, all rings are associative with an identity. $M_n(R)$\nand $T_n(R)$ will denote the ring of all $n\\times n$ full matrices\nand triangular matrices over $R$, respectively. $J(R)$ and $P(R)$\nstand for the Jacobson radical and prime radical of $R$. $Id(R)=\\{\ne\\in R~|~e^2=e\\in R\\},-Id(R)=\\{ e\\in R~|~e^2=-e\\in R\\}$, $U(R)$ is\nthe set of all units in $R$, and $N(R)$ is the set of all\nnilpotent elements in $R$.\n\n\\section{Weakly Previous Rings}\n\n\\vskip4mm We start this section by indicating that \"weakly\ncleanness\" and \"weakly preciousness\", \"nil-cleanness\" and\n\"preciousness\" are not the same for elements in a ring.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Example 2.1.}\\ \\ {\\it $(1)$ Every\nweakly clean element in a ring is weakly previous, but the\nconverse is not true.}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(2)] {\\it Every nil-clean element in a ring is previous, but the converse is not true.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $(1)$ Obviously, every\nweakly clean element in a ring is weakly previous. But the\nconverse is not true. Consider the matrix $A=\\left(\n\\begin{array}{cc}\n3&9\\\\\n-7&-2\n\\end{array}\n\\right)\\in M_2({\\Bbb Z})$. By ~\\cite[Theorem 4]{An}, $A$ is not\nclean. Thus, $(A,-A)\\in M_2({\\Bbb Z})\\times M_2({\\Bbb Z})$ is not\nweakly clean. But $(A,-A)$ is previous, as it has the previous\ndecomposition\n$$\\big(\\left(\n\\begin{array}{cc}\n1&0\\\\\n7&0\n\\end{array}\n\\right),\\left(\n\\begin{array}{cc}\n1&0\\\\\n6&0\n\\end{array}\n\\right)\\big)+\\big(\\left(\n\\begin{array}{cc}\n-1&0\\\\\n-13&1\n\\end{array}\n\\right),\\left(\n\\begin{array}{cc}\n-1&0\\\\\n0&-1\n\\end{array}\n\\right)\\big)+\\big(\\left(\n\\begin{array}{cc}\n3&9\\\\\n-1&-3\n\\end{array}\n\\right),\\left(\n\\begin{array}{cc}\n-3&-9\\\\\n1&3\n\\end{array}\n\\right)\\big).$$ Thus, $(A,-A)$ is weakly precious.\n\n$(2)$ Let $a\\in R$ be nil-clean. Then there exists an idempotent\n$e\\in R$ and a nilpotent $w\\in R$ such that $a=e+w$, and so\n$a=(1-e)+(2e-1)+w$. As $(2e-1)^2=1$, we see that $a\\in R$ is\nprecious. Thus, every nil-clean element in a ring is previous. The\nconverse is not true. For instance, $-1\\in {\\Bbb Z}_3$ is not\nnil-clean, but it is precious.\\hfill$\\Box$\n\n\\vskip4mm Example 2.1 shows that $\\{~\\mbox{weakly clean\nelements}~\\}\\subsetneq $ $\\{~\\mbox{weakly precious elements}\\}$\nand $\\{~\\mbox{nil-clean elements}~\\}\\subsetneq \\{~\\mbox{precious\nelements}~\\}.$ Though weakly precious rings are rich, but there\nindeed exist rings which are not weakly precious. Since\n$Id\\big({\\Bbb Z}\\big)=\\{ 0,1\\}, U\\big({\\Bbb Z}\\big)=\\{ 1,-1\\}$ and\n${\\Bbb Z}$ has no nonzero nilpotent element, we easily check that\n$5\\in {\\Bbb Z}$ is not weakly precious. Therefore, the ring ${\\Bbb\nZ}$ of all integers is not weakly previous. The purpose of this\nsection is to investigate when a ring is weakly previous or\nprevious. Clearly, every homomorphic image of weakly precious\nrings is weakly precious. Further, we derive\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 2.2.}\\ \\ {\\it Let $I$ be a\nnil ideal of a ring $R$. Then $R$ is weakly precious if and only\nif $R\/I$ is weakly precious.} \\vskip2mm\\hspace{-1.8em} {\\it\nProof.}\\ \\ Let $R$ be a weakly precious ring. Then $R\/I$ is weakly\nprecious. Now assume that $R\/I$ is weakly precious. Let $a\\in R$.\nThen $\\overline{a}= a+I=\\overline{e} +\\overline{u}+\\overline{w}$\nor $\\overline{a}= a+I=\\overline{-e} +\\overline{u}+\\overline{w}$\nfor an idempotent $\\overline{e}\\in R\/I$, a unit $\\overline{u}\\in\nR\/I$ and a nilpotent $\\overline{w}\\in R\/I$. As $I$ is a nil ideal\nof $R$, we easily check that $u$ is a unit element in $R$ and $w$\nis a nilpotent element. Since every idempotent lifts modulo $I$,\nwe can find an idempotent $e\\in R$ such that\n$\\overline{e}=\\overline{f}$. Hence, $a= e+u+(w+c)$ or $a=\n-e+u+(w+c)$ for some $c\\in I$. Write $w^m=0~(m\\geq 1)$. Then\n$(w+c)^m\\in I$, and so $w+c\\in R$ is a nilpotent element.\nTherefore $a\\in R$ is weakly precious, as required.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.3.}\\ \\ {\\it Let $R$ be a\nring. Then $R$ is weakly precious if and only if $R[[x]]\/(x^n)\n(n\\in {\\Bbb N})$ is weakly precious.} \\vskip2mm\\hspace{-1.8em}\n{\\it Proof.}\\ \\ Clearly, $R[[x]]\/(x^n)=\\{ a_0+a_1x+\\cdots\n+a_{n-1}x^{n-1}~|~a_0,\\cdots ,a_{n-1}\\in R\\}$. Let $\\alpha :\nR[[X]]\/(x^n)\\longrightarrow R$ be a morphism such that\n$\\alpha(f)=f(0)$. It is easy to check that $\\alpha$ is an\n$R$-epimorphism and $ker\\alpha$ is a nil ideal of $R$, and\ntherefore the result follows from Lemma 2.2.\\hfill$\\Box$\n\n\\vskip4mm An element $a\\in R$ is strongly nilpotent if for every\nsequence $a_0,a_1,\\cdots ,a_i,\\cdots$ such that $a_0 =a$ and\n$a_{i+1}\\in a_iRa_i$, there exists an $n$ with $a_n=0$. The prime\nradical (i.e., the intersection of all prime ideals) of a ring is\nexactly the set of all its strongly nilpotent elements.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 2.4.}\\ \\ {\\it Let $R$ be a\nring. Then the following are equivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it $R$ is weakly precious.}\n\\vspace{-.5mm} \\item [(2)] {\\it $R\/P(R)$ is weakly\nprecious.}\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ This is\nobvious by Lemma 2.2, as $P(R)$ is nil.\\hfill$\\Box$\n\n\\vskip4mm Recall that a ring $R$ is 2-primal provided that every\nnilpotent element of $R$ is strongly nilpotent~\\cite{MMZ}.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.5.}\\ \\ {\\it Let $R$ be\n2-primal. Then the following are equivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it $R$ is weakly precious;}\n\\vspace{-.5mm} \\item [(2)] {\\it $R$ is weakly clean.}\n\\vspace{-.5mm} \\item [(3)] {\\it $R\/P(R)$ is weakly\nclean.}\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\\n$(1)\\Rightarrow (2)$ Let $a\\in R$. As $R$ is weakly precious,\n$a=e+u+w$ or $a=-e+u+w$ for an idempotent $e\\in R$, a unit $u\\in\nR$ and a nilpotent $w\\in R$. This shows that $a=\ne+u\\big(1+u^{-1}w)$ or $a=-e+u\\big(1+u^{-1}w)$. As $R$ is a\n2-primal ring, we get $w\\in P(R)$. Since $P(R)$ is a nil ideal of\n$R$, $1+u^{-1}w\\in U(R)$, and therefore $R$ is weakly clean.\n\n$(2)\\Rightarrow (3)$ is clear.\n\n$(3)\\Rightarrow (1)$ Since $R\/P(R)$ is weakly clean, it is weakly\nprecious. Therefore we complete the proof, by Lemma\n2.4.\\hfill$\\Box$\n\n\\vskip4mm A ring $R$ is called nil-semicommutative  if $ab=0$ in\n$R$ implies that $aRb=0$ for every $a, b \\in N(R)$ (see\n\\cite{MMZ}). For instance, every semicommutative ring (i.e.,\n$ab=0$ in $R$ implies that $aRb=0$) is nil-semicommutative.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 2.6.}\\ \\ {\\it Let $R$ be\nnil-semicommutative. Then the following are\nequivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it $R$ is weakly precious;}\n\\vspace{-.5mm} \\item [(2)] {\\it $R$ is weakly clean.}\n\\vspace{-.5mm} \\item [(3)] {\\it $R\/P(R)$ is weakly\nclean.}\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\  Let $R$ be\na nil-semicommutative ring. Then, by ~\\cite[Lemma 2.7]{MMZ}, $R$\nis 2-primal, so the result follows from Theorem 2.5.\\hfill$\\Box$\n\n\\vskip4mm A ring $R$ a right (left) quasi-duo ring if every\nmaximal right (left) ideal of $R$ is an ideal. For instance, local\nrings, duo rings and weakly right (left) duo rings are all right\n(left) quasi-duo rings. We now derive\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Proposition 2.7.}\\ \\ {\\it Let $R$\nbe a right (left) quasi-duo ring. Then the following are\nequivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it $R$ is weakly precious.}\n\\vspace{-.5mm} \\item [(2)] {\\it $R$ is weakly clean.}\n\\vspace{-.5mm} \\item [(3)] {\\it $R\/P(R)$ is weakly\nclean.}\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\\n$(1)\\Rightarrow (2)$ Let $a\\in R$. Then there exists a very\nidempotent $e\\in R$, a unit $u\\in R$ and a nilpotent element $w\\in\nR$ such that $a=e+u+w$. As $R$ is right (left) quasi-duo, it\nfollows from ~\\cite[Lemma 2.3]{Yu} that $w\\in J(R)$. Thus,\n$a=e+u\\big(1+u^{-1}w\\big)$; hence, $a\\in R$ is weakly clean.\n\n$(2)\\Rightarrow (3)$ This is obvious.\n\n$(3)\\Rightarrow (1)$ Clearly, $R\/P(R)$ is weakly precious, and\ntherefore the result follows, by Lemma 2.4.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 2.8.}\\ \\ {\\it Let $R$ be\nweakly precious and $S$ be precious. Then $R\\oplus S$ is weakly\nprecious.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Set $A=R\\oplus\nS$. Let $(a,b)\\in A$. Then there exists an idempotent $e\\in R$, a\nunit $u\\in R$ and a nilpotent $v\\in R$ such that $a=e+u+v$ or\n$a=-e+u+v$.\n\nCase I. $a=e+u+v$. Then we have an idempotent $f\\in S$, a unit\n$s\\in S$ and a nilpotent $w\\in S$ such that $b=f+v+w$. Thus,\n$(a,b)=(e,f)+(u,s)+(v,w)$, where $(e,f)\\in A$ is an idempotent,\n$(u,s)\\in A$ is a unit and $(v,w)\\in A$ is nilpotent.\n\nCase II. $a=-e+u+v$. Then we have an idempotent $f\\in S$, a unit\n$s\\in S$ and a nilpotent $w\\in S$ such that $-b=f+s+w$. Thus,\n$(a,b)=-(e,f)+(u,-s)+(v,-w)$, where $(e,f)\\in A$ is an idempotent,\n$(u,-s)\\in A$ is a unit and $(v,-w)\\in A$ is nilpotent.\n\nTherefore we conclude that $(a,b)$ is the sum of a very\nidempotent, a unit and a nilpotent element in $A$, hence that\nresult.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.9.}\\ \\ {\\it Let\n$\\{R_{i}\\}$ be a family of rings. Then the direct product $R=\\prod\nR_{i}$ of rings $R_i$ is weakly precious if and only if each\n$R_{i}$ is weakly precious and at most one is not precious.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ $\\Longrightarrow $\nObviously, each $R_i$ is weakly precious. Suppose $R_{i_1}$ and\n$R_{i_2} (i_1\\neq i_2)$ are not precious. Then there exist some\n$x_{i_j}\\in R_{i_j} (j=1,2)$ such that $x_{i_1}\\in R_{i_1}$ and\n$-x_{i_2}\\in R_{i_2}$ are not precious. Choose $x=(x_i)$ where\n$x_i=0$ whenever $i\\neq i_j (j=1,2)$. Then $x\\pm e$ is not the sum\nof a unit and a nilpotent for all idempotents $e\\in R$. This gives\na contradiction. Therefore each $R_{i}$ is a weakly precious and\nat most one is not precious.\n\n$\\Longleftarrow$ Suppose that $R_{i_0}$ is weakly precious and all\nthe others $R_i$ are precious. Then $\\prod\\limits_{i\\neq\ni_{0}}R_{i_0}$ is precious. In light of Lemma 2.8, We conclude\nthat $R$ is weakly precious.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 2.10.}\\ \\ {\\it Let\n$L=\\prod\\limits_{i\\in I}R_{i}$ be the direct product of rings\n$R_i\\cong R$ and $|I|\\geq 2$. Then $L$ is weakly precious if and\nonly if $R$ is precious if and only if $L$ is precious.}\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 2.11.}\\ \\ {\\it Let $e=e^2 \\in\nR$ be such that $eRe$ is weakly precious and $(1-e)R(1-e)$ is\nprecious. Then $R$ is weakly precious.} \\vskip2mm\\hspace{-1.8em}\n{\\it Proof.}  Let $e\\in R$ be an idempotent, we have $R\\cong\n\\left(\n\\begin{array}{cc}\neRe&eR(1-e)\\\\\n(1-e)Re&(1-e)R(1-e)\n\\end{array}\n\\right)$. Now suppose that $A=\\left(\n\\begin{array}{cc}\na&b\\\\\nc&d\n\\end{array}\n\\right)$ be an element of $R$. As $eRe$ is weakly precious and\n$(1-e)R(1-e)$ is precious, $a=f+u+w$ or $a=-f+u+w$ for some\nidempotent $f\\in eRe$, $u\\in U(eRe)$ and a nilpotent $w\\in\\ eRe$.\n\nCase I. Let $a=f+u+w$. Then $d-cu^{-1}b \\in (1-e)R(1-e)$, and so\n$d-cu^{-1}b=g+v+z$ for some idempotent $g$, unit $v$ and nilpotent\n$z\\in (1-e)R(1-e)$. Now $$A= \\left(\n\\begin{array}{cc}\nf+u+w&b\\\\\nc&g+v+z+cu^{-1}b\n\\end{array}\n\\right) =\\left(\n\\begin{array}{cc}\nf&0\\\\\n0&g\n\\end{array}\n\\right)+  \\left(\n\\begin{array}{cc}\nu&b\\\\\nc&v+cu^{-1}b\n\\end{array}\n\\right)+ \\left(\n\\begin{array}{cc}\nw&0\\\\\n0&z\n\\end{array}\n\\right).$$ It is clear that $\\left(\n\\begin{array}{cc}\nf&0\\\\\n0&g\n\\end{array}\n\\right)$ is an idempotent element of $R$ and  $\\left(\n\\begin{array}{cc}\nw&0\\\\\n0&z\n\\end{array}\n\\right) $ is a nilpotent element of $R$, so we need only to show\nthat $\\left(\n\\begin{array}{cc}\nu&b\\\\\nc&v+cu^{-1}b\n\\end{array}\n\\right)$ is a unit of $R$. One easily checks that\n$$\\left(\n\\begin{array}{cc}\ne&0\\\\\n-cu^{-1}&1-e\n\\end{array}\n\\right)\\left(\n\\begin{array}{cc}\nu&b\\\\\nc&v+cu^{-1}b\n\\end{array}\n\\right)\\left(\n\\begin{array}{cc}\ne&-u^{-1}\\\\\n0&1-e\n\\end{array}\n\\right)\\\\\n=\\left(\n\\begin{array}{cc}\nu&0\\\\\n0&v\n\\end{array}\n\\right).$$ Hence, $\\left(\n\\begin{array}{cc}\nu&b\\\\\nc&v+cu^{-1}b\n\\end{array}\n\\right)$ is invertible. Thus, $A$ is precious.\n\nCase II. Let $a=-f+u+w$. Then $-a=f-u-w$. By the similar way of\nCase I, we see that $-A$ is precious, as required.\\hfill$\\Box$\n\n\\vskip4mm A Morita context $(R,S,M,N,\\psi,\\varphi)$ consists of\ntwo rings $R$ and $S$, two bimodules $_RN_S$ and $_SM_R$, and a\npair of bimodule homomorphisms $\\psi : N\\bigotimes\\limits_{S}M\\to\nR$ and $\\varphi: M\\bigotimes\\limits_{R} N\\to S$ which satisfy the\nfollowing associativity: $\\psi \\big(n\\bigotimes m\\big)n'=n\\varphi\n\\big(m\\bigotimes n'\\big)$ and $\\varphi \\big(m\\bigotimes\nn\\big)m'=m\\psi \\big(n\\bigotimes m'\\big)$ for any $m,m'\\in M,\nn,n'\\in N$. The ring $T=\\{ \\left(\n\\begin{array}{cc}\nr&m\\\\\nn&s \\end{array} \\right)~|~r\\in R,s\\in S,m\\in M,n\\in N\\}$ is called\nthe ring of the Morita context $(R,S,M,N,\\psi,\\varphi)$.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.12.}\\ \\ {\\it Let $T$ be\nthe ring of the Morita context $(R,S,M,N,\\varphi,\\phi)$. If $R$ is\nweakly precious and $S$ is precious, then $T$ is weakly precious.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $R$ be weakly\nprecious and $S$ be precious, and let $e=diag(1_{R},0)$. Since\n$eTe\\cong R$ and $(1_{T}-e)T(1_{T}-e)\\cong S$, it follows by Lemma\n2.11 that $T$ is a weakly precious ring, as asserted.\\hfill$\\Box$\n\n\\vskip4mm Many properties of weakly precious rings can be extended\nto precious rings. For instance, $R$ is precious if and only if\n$R[[x]]\/(x^n) (n\\in {\\Bbb N})$ is precious. The direct product\n$\\prod R_{i}$ of rings $R_i$ is precious if and only if each\n$R_{i}$ is precious. But the subdirect product of (weakly)\nprecious rings is not necessarily (weakly) precious. For instance,\n${\\Bbb Z}$ is a subdirect product of rings $\\lbrace{\\Bbb Z_n},\nn\\geq 2\\rbrace$, where each ${\\Bbb Z_n} (n\\geq 2)$ is precious,\nbut ${\\Bbb Z}$ is not.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 2.13.}\\ \\ {\\it Let $e=e^2 \\in\nR$ be such that $eRe$ and $(1-e)R(1-e)$ are precious. Then $R$ is\nprecious.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $e\\in R$\nbe an idempotent element, we have $R\\cong \\left(\n\\begin{array}{cc}\neRe&eR(1-e)\\\\\n(1-e)Re&(1-e)R(1-e)\n\\end{array}\n\\right)$. Now suppose that  $A=\\left(\n\\begin{array}{cc}\na&b\\\\\nc&d\n\\end{array}\n\\right)$ be an element of $R$. Since $eRe$ and $(1-e)R(1-e)$ are\nprecious rings, $a=f+u+w$ for some idempotent $f\\in eRe$, $u\\in\nU(eRe)$ and a nilpotent $w\\in\\ eRe$. Let $u^{-1}$ be the inverse\nof $u$. Then $d-cu^{-1}b\\in (1-e)R(1-e)$, so $d-cu^{-1}b=g+v+z$\nfor some idempotent $g$, unit $v$ and nilpotent $z\\in\n(1-e)R(1-e)$. Now $A= \\left(\n\\begin{array}{cc}\nf+u+w&b\\\\\nc&g+v+z+cu^{-1}b\n\\end{array}\n\\right) =\\left(\n\\begin{array}{cc}\nf&0\\\\\n0&g\n\\end{array}\n\\right)+\\left(\n\\begin{array}{cc}\nu&0\\\\\n0&v+cu^{-1}b\n\\end{array}\n\\right)+ \\left(\n\\begin{array}{cc}\nw&0\\\\\n0&z\n\\end{array}\n\\right)$. As in the proof of Lemma 2.11, we easily checks that\n$\\left(\n\\begin{array}{cc}\nu&b\\\\\nc&v+cu^{-1}b\n\\end{array}\n\\right)$ is a unit of $R$, and the the result follows.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.14.}\\ \\ {\\it Let $T$ be\nthe ring of the Morita context $(R,S,M,N,\\varphi,\\phi)$. If $R$\nand $S$ are precious, then $T$ is precious.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $R,S$ be precious\nrings and let $e=diag(1_{R},0)$. Then $eTe\\cong R$ and\n$(1_{T}-e)T(1_{T}-e)\\cong S$. By virtue of Lemma 2.13, $T$ is a\nprecious ring. \\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 2.15.}\\ \\ {\\it Let $R$ be\nprecious. Then $M_n(R)$ is precious.} \\vskip2mm\\hspace{-1.8em}\n{\\it Proof.}\\ \\ If $n=2$. Then the result follows by Theorem 2.14.\nSuppose that the result holds for $n\\leq k ( k\\geq 2)$. Then $R$\nand $M_k(R)$ are both precious. In light of Theorem 2.14, the ring\n$\\left(\n\\begin{array}{cc}\nR&M\\\\\nN&M_k(R)\n\\end{array}\n\\right)$ is precious, where $M=\\{\\left(\n\\begin{array}{ccc}\nb_{1}&\\cdots &b_k\n\\end{array}\n\\right)~|~b_1,\\cdots ,b_k\\in R\\}$ and $N=\\{\\left(\n\\begin{array}{c}\nc_{1}\\\\\n\\vdots\\\\\nc_k\n\\end{array}\n\\right)~|~c_1,\\cdots ,c_k\\in R\\}$. This completes the proof by\ninduction.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 2.16.}\\ \\ {\\it Let $R$ be a\nring. Then the following are equivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it $R$ is precious.}\n\\vspace{-.5mm}\n\\item [(2)] {\\it $T_n(R)$ is precious for all $n\\in {\\Bbb N}$.}\n\\vspace{-.5mm}\n\\item [(3)] {\\it\n$T_n(R)$ is precious for some $n\\in {\\Bbb N}$.} \\vspace{-.5mm}\n\\item [(4)] {\\it\n$T_n(R)$ is weakly precious for some $n\\geq 2$.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $(1)\\Rightarrow\n(2)$  The result holds for $n=2$ by Theorem 2.14. Assume that the\nresult holds for $n\\leq k$ $(k\\geq 2)$. Let $n=k+1$. Then\n$T_n(R)\\cong \\left(\n\\begin{array}{cc} R&M\\\\\n0&T_{k}(R) \\end{array} \\right)$, where $M =\\{ ( c_{1}, \\cdots ,\nc_{k})\\mid c_1,\\cdots ,c_{k}\\in R\\}$. In light of Theorem 2.14,\n$T_{k+1}(R)$ is precious. Then, proving $(2)$, by induction.\n\n$(2)\\Rightarrow (3)$ is trivial.\n\n$(3)\\Rightarrow (1)$ This is obvious, as a triangular matrix over\n$R$ is an idempotent (unit, nilpotent matrix) if and only if every\nits diagonal entry is an idempotent (unit, nilpotent matrix).\n\n$(2)\\Rightarrow (4)$ is trivial.\n\n$(4)\\Rightarrow (1)$ Let $a\\in R$. Choose $A=\\left(\n\\begin{array}{cccccc}\na&&&&&\\\\\n&-a&&&&\\\\\n&&0&&&\\\\\n&&&\\ddots&&\\\\\n&&&&&0\n\\end{array}\n\\right)\\in T_n(R)$. By hypothesis, we can find an idempotent\n$\\left(\n\\begin{array}{ccccc}\ne_1&&&&\\\\\n&e_2&&&\\\\\n&&\\ddots&&\\\\\n&&&&e_n\n\\end{array}\n\\right)$, a unit $\\left(\n\\begin{array}{ccccc}\nu_1&&&&\\\\\n&u_2&&&\\\\\n&&\\ddots&&\\\\\n&&&&u_n\n\\end{array}\n\\right)$ and a nilpotent $\\left(\n\\begin{array}{ccccc}\nw_1&&&&\\\\\n&w_2&&&\\\\\n&&\\ddots&\\\\\n&&&&w_n\n\\end{array}\n\\right)$ such that\n$$A=\\pm\n\\left(\n\\begin{array}{ccccc}\ne_1&&&&\\\\\n&e_2&&&\\\\\n&&\\ddots&&\\\\\n&&&&e_n\n\\end{array}\n\\right)+\\left(\n\\begin{array}{ccccc}\nu_1&&&&\\\\\n&u_2&&&\\\\\n&&\\ddots&&\\\\\n&&&&u_n\n\\end{array}\n\\right)+\\left(\n\\begin{array}{ccccc}\nw_1&&&&\\\\\n&w_2&&&\\\\\n&&\\ddots&&\\\\\n&&&&w_n\n\\end{array}\n\\right).$$ It follows that $a=e_1+u_1+w_1$ or $a=e_2-u_2-w_2$.\nClearly, $e_1,e_2$ are idempotents, $u_1, u_2$ are units and\n$w_1,w_2$ are nilpotent. Therefore proving $(1)$.\\hfill$\\Box$\n\n\\vskip4mm A ring $R$ is weakly periodic provided that for any\n$a\\in R$ there exists some $p=p^m(m\\geq 2)$ such that $a-p\\in R$\nis nilpotent. For instance, every periodic ring is weakly\nperiodic.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 2.17.}\\ \\ {\\it Let $R$ be\na weakly periodic ring. Then $M_n(R)$ and $T_n(R)$ are precious\nfor all $n\\in {\\Bbb N}$.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\\nFor any $a\\in R$, there exists a $p=p^{k+1} (k\\in {\\Bbb N})$ and a\nnilpotent $w\\in R$ such that $a=p+w$. If $k=1$, then $p\\in R$ is\nan idempotent, and so $a\\in R$ is precious. Suppose that $k\\geq\n2$. Set $e=1-p^k, u=p-1+p^k$. Then $e=e^2\\in\nR,u^{-1}=p^{k-1}-1+p^k$, and that $p=e+u$. Thus, $a=e+u+w$, and\nthen $R$ is precious. Therefore we complete the proof, by\nCorollary 2.15 and Theorem 2.16.\\hfill$\\Box$\n\n\\vskip4mm Let $R$ be a ring and $M$ an $R$-$R$-bimodule. The\ntrivial extension of $R$ by $M$,\n$$R\\propto M=\\{ \\left(\n\\begin{array}{cc}\nr&m\\\\\n&r\n\\end{array}\n\\right)~|~r\\in R,m\\in M\\}$$ is (weakly) precious if and only if\n$R$ is (weakly) precious.\n\n\\section{Certain Subclasses}\n\n\\vskip4mm Weakly clean rings and nil-clean rings forms main types\nof subclasses of weakly precious rings and precious rings,\nrespectively. The purpose of this section is to off new types of\nsuch rings. In ~\\cite{A}, Anderson and Camillo proved that if a\nring $R$ has at most two maximal ideals and $2\\in U(R)$ then $R$\nis weakly clean. We extend this result as follows.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 3.1.}\\ \\ {\\it Let $R$ be a\ncommutative ring with at most three maximal ideals. If $2\\in U(R)$\nand $J(R)$ is nil, then $R$ is weakly clean.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Case I. $R$ has only one\nmaximal ideal. Then $R$ is local, and so it is clean.\n\nCase II. $R$ has only two maximal ideals. Then $R$ is weakly\nclean, by ~\\cite[Proposition 16]{A}.\n\nCase III. $R$ has three maximal ideals $M_1,M_2$ and $M_2$. Let\n$a\\in R$. If $a\\not\\in M_1,M_2,M_3$, then $aR=R$; hence, $a\\in\nU(R)$. So we may assume that $a\\in M_1$. If $1-a\\not\\in\nM_1,M_2,M_3$, then $1-a\\in U(R)$, and so $a\\in R$ is clean. If\n$1-a\\in M_1$, then $1=a+(1-a)\\in M_1$, a contradiction. Thus,\n$1-a\\in M_2$ or $1-a\\in M_3$. If $1+a\\not\\in M_1,M_2,M_3$, then\n$1+a\\in U(R)$; hence, $a\\in R$ is weakly clean. If $1+a\\in M_1$,\nthen $1=(1+a)-a\\in M_1$, a contradiction. Thus, $1+a\\in M_2$ or\n$1+a\\in M_3$. There are only two possible cases: $1-a\\in M_2,\n1+a\\in M_3$ or $1-a\\in M_3, 1+a\\in M_2$, as $2\\in U(R)$. Thus, we\nmay assume that $1-a\\in M_2, 1+a\\in M_3$. Hence, $a(1-a)(1+a)\\in\nM_1M_2M_3\\subseteq J(R)$. Thus, $\\overline{a}=\\overline{a}^3\\in\nR\/J(R)$. Set $\\overline{e}=\\overline{1-a^2}$ and\n$\\overline{u}=\\overline{a^2+a-1}$. Then $\\overline{e}\\in R\/J(R)$\nis an idempotent and $\\overline{u}^2=\\overline{1}$. Further, we\nhave $\\overline{a}=\\overline{e}+\\overline{u}$. By hypothesis,\n$J(R)$ is nil, and then every unit and every idempotent lift\nmodulo $J(R)$. So we may assume that $e\\in R$ is an idempotent and\n$u\\in U(R)$. Set $w:=a-e-u$. Then $a=e+u+w$ where $w\\in J(R)$.\nClearly, $u+w=u(1+u^{-1}w)\\in U(R)$, and so $a\\in R$ is clean.\nTherefore $R$ is weakly clean.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Example 3.2.}\\ \\ {\\it Let\n$R=k[[x,y,z]]$ where $k$ is a field with $char(k)\\neq 2$. Let\n$S=R-(x)\\cup (y)\\cup (z)$. Then the ring $R_S\/J^2(R_S)$ is weakly\nclean.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Since\n$k[[x,y,z]]\/(x)\\cong k[[y,z]]$ is an integral domain, we see that\n$(x)$ is a prime ideal of $k[[x,y,z]]$. Likewise, $(y)$ and $(z)$\nare prime ideals of $R$. Let $S=R-(x)\\bigcup (y)\\bigcup (z)$. Then\n$S$ is a multiplicative closed subset of $R$. Let $P$ be a maximal\nideal of $R_S$. Then we have an ideal $Q$ of $R$ such that $P=Q_S$\nsuch that $Q\\bigcap S=\\emptyset$. Thus, $Q\\subseteq (x)\\bigcup\n(y)\\bigcup (z)$. Assume that $Q\\nsubseteq (x), Q\\nsubseteq (y)$\nand $Q\\nsubseteq (z)$. Then we have some $b,c,d\\in Q$, but\n$b\\not\\in (x)$, $c\\not\\in (y)$ and $d\\not\\in (z)$. Choose\n$a=b+c+d$. Then $a\\in Q$. If $a\\in (x)$, then $c+d\\in (x)$. If\n$c\\not\\in (x)$, then $c\\in (z)$. This implies that $d\\in (x)$.\nHence, $c\\in (z)\\bigcap (x)=0$. This gives a contradiction. If\n$c\\in (x)$, then $d\\in (x)$; hence that $b=a-(c+d)\\in (x)$, a\ncontradiction. Hence, $a\\not\\in (x)$. Likewise, $a\\not\\in (y)$ and\n$a\\not\\in (z)$. Thus, $a\\not\\in (x)\\bigcup (y)\\bigcup (z)$, a\ncontradiction. We infer that $Q\\subseteq (x)$, or $Q\\subseteq (y)$\nor $Q\\subseteq (z)$. Hence, $Q_S\\subseteq (x)_S$, or $Q_S\\subseteq\n(y)_S$, or $Q_S\\subseteq (z)_S$. By the maximality of $P$, we get\n$P=(x)_S$, or $(y)_S$, or $(z)_S$. Thus, $R_S$ has exactly three\nmaximal ideals $(x)_S$, $(y)_S$ and $(z)_S$. Therefore $R$ has at\nmost three maximal ideals. Since $char(k)\\neq 2$, we see that\n$2\\in U(R_S)$.\n\nSet $A=R_S\/J^2(R_S)$. Then $A$ has at most three maximal ideals\nand $2\\in U(A)$. If $\\overline{x}\\in J(A)$, then\n$\\overline{1-xr}\\in U(A)$ for any $r\\in R_S$. Hence, $1-xr\\in\nU(R_S)$. This implies that $x\\in J(R_S)$, and so\n$\\overline{x^2}=\\overline{0}$. That is, $\\overline{x}$ is\nnilpotent. So, $J(A)$ is nil. Therefore we complete the proof, in\nterms of Theorem 3.1.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Example 3.3.}\\ \\ {\\it Let $R=\\{\n\\frac{m}{n}~|~(m,n)=1, m,n\\in {\\Bbb Z}~\\mbox{and}~3,5,7~\\nmid\nn\\}$. Then the ring $R\/J^2(R)$ is weakly clean.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $M$ be an ideal of\n$R$ such that $3R\\subsetneq M\\subseteq R$. Choose $\\frac{m}{n}\\in\nM$ while $\\frac{m}{n}\\not\\in 3R$. Then $3\\nmid m$, and so\n$(3,m)=1$. So $3k+lm=1$ for some $k,l\\in {\\Bbb Z}$. This shows\nthat $\\frac{1}{1}=3\\cdot \\frac{k}{1}+l\\frac{m}{n}\\cdot\n\\frac{n}{1}\\in M$, i.e., $M=R$. Thus, $3R$ is a maximal ideal of\n$R$. Likewise, $5R$ and $7R$ are maximal ideals of $R$. For any\n$\\frac{m}{n}\\in 3R\\bigcap 5R\\bigcap 7R$ and $\\frac{a}{b}\\in R$,\nthen $\\frac{1}{1}-\\frac{m}{n}\\frac{a}{b}=\\frac{nb-ma}{nb}$. Write\n$\\frac{m}{n}=\\frac{3s}{t}$. Then $3sn=mt$, and so $3~|~mt$. Since\n$3\\nmid t$, we get $3~|~m$. Obviously, $3\\nmid nb$; hence, $3\\nmid\n(nb-ma)$. Similarly, $5,7\\nmid (nb-ma)$. It follows that\n$\\frac{nb}{nb-ma}\\in U(R)$. We infer that $\\frac{m}{n}\\in J(R)$.\nTherefore $3R\\bigcap 5R\\bigcap 7R\\subseteq J(R)$. Let $M$ be a\nmaximal ideal of $R$ and $M\\neq 3R, 5R, 7R$. Then $3R+M=R, 5R+M=R$\nand $7R+M=R$. Thus, $R=(3R+M)(5R+M)(7R+M)\\subseteq 3R\\bigcap\n5R\\bigcap 7R+M=J(R)+M\\subseteq M$, hence, $R=M$, an absurd. We\ninfer that $R$ is a commutative ring with exactly three maximal\nideals. Obviously $2\\in R$ is invertible. Therefore $A:=R\/J^2(R)$\nis a commutative ring with exactly three maximal ideals. Obviously\n$2\\in A$ is invertible. As in the proof of Example 3.2, $A$ has\nthe nil Jacobson radical. We conclude that $A$ is weakly clean, by\nTheorem 3.1.\\hfill$\\Box$\n\n\\vskip4mm In ~\\cite[Question 3]{D}, Diesl asked: Let $R$ be a nil\nclean ring, and let $n$ be a positive integer. Is $M_n(R)$ nil\nclean?  In ~\\cite[Theorem 3]{BGDT}, Breaz et al. proved that their\nmain theorem: for a field $K$, $M_n(K)$ is nil-clean if and only\nif $K\\cong {\\Bbb Z}_2$. They also asked if this result could be\nextended to division rings. As a main result in ~\\cite{KLZ},\nKo\\c{s}an et al. gave a positive answer to this problem. They\nshowed that the preceding equivalence holds for any division ring.\nWe shall extend ~\\cite[Theorem 3]{BGDT} and ~\\cite[Theorem 3]{KLZ}\nto an arbitrary abelian ring.\n\n\\vskip4mm Recall that a ring $R$ is an exchange ring if for every\n$a\\in R$ there exists an idempotent $e\\in aR$ such that $1-e\\in\n(1-a)R$. Clearly, every nil-clean ring is an exchange ring.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 3.4.}\\ \\ {\\it Let $R$ be an\nabelian exchange ring, and let $x\\in R$. Then $RxR=R$ if and only\nif $x\\in U(R)$.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ If $x\\in\nU(R)$, then $RxR=R$. Conversely, assume that $RxR=R$. As in the\nproof of ~\\cite[Proposition 17.1.9]{CH}, there exists an\nidempotent $e\\in R$ such that $e\\in xR$ such that $ReR=R$. This\nimplies that $e=1$. Write $xy=1$. Then $yx=y(xy)x=(yx)^2$. Hence,\n$yx=y(yx)x$. Therefore $1=x(yx)y=xy(yx)xy=yx$, and so $x\\in U(R)$.\nThis completes the proof.\\hfill$\\Box$\n\n\\vskip4mm Set $J^*(R)=\\bigcap \\{ P~|~P~\\mbox{is a maximal ideal\nof}~R\\}.$ We will see that $J(R)\\subseteq J^*(R)$. In general,\nthey are not the same. For instance, $J(R)=0$ and $J^*(R)=\\{ x\\in\nR~|~dim_F(xV)<\\infty\\}$, where $R=End_F(V)$ and $V$ is an\ninfinite-dimensional vector space over a field $F$.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 3.5.}\\ \\ {\\it Let $R$ be an\nabelian exchange ring. Then $J^*(R)=J(R)$.}\n\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $M$ be a maximal\nideal of $R$. If $J(R)\\nsubseteq M$, then $J(R)+M=R$. Write\n$x+y=1$ with $x\\in J(R),y\\in M$. Then $y=1-x\\in U(R)$, an absurd.\nHence, $J(R)\\subseteq M$. This implies that $J(R)\\subseteq\nJ^*(R)$. Let $x\\in J^*(R)$, and let $r\\in R$. If $R(1-xr)R\\neq R$,\nthen we can find a maximal ideal $M$ of $R$ such that\n$R(1-xr)R\\subseteq M$, and so $1-xr\\in M$. It follows that\n$1=xr+(1-xr)\\in M$, which is imposable. Therefore $R(1-xr)R=R$. In\nlight of Lemma 3.4, $1-xr\\in U(R)$, and then $x\\in J(R)$. This\ncompletes the proof.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 3.6.}\\ \\ {\\it Let $R$ be a\nring with no non-trivial idempotents, and let $n\\in {\\Bbb N}$.\nThen the following are equivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)]{\\it $M_n(R)$ is nil-clean.}\n\\item [(2)]{\\it $R\/J(R)\\cong {\\Bbb Z}_2$ and $M_n(J(R))$ is nil.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $(1)\\Rightarrow (2)$ In view of ~\\cite[Proposition 3.16]{D}, $J(M_n(R))$ is\nnil, and then so is $M_n(J(R))$.\n\nLet $a\\in R$. By hypothesis, $M_n(R)$ is nil-clean. If $n=1$, then\n$R$ is nil-clean. Hence, $a\\in N(R)$ or $a-1\\in N(R)$. This shows\nthat $a\\in U(R)$ or $1-a\\in U(R)$, and so $R$ is local. That is,\n$R\/J(R)$ is a division ring. As $R\/J(R)$ is nil-clean, it follows\nfrom ~\\cite[Theorem 3]{BGDT} that $R\/J(R)\\cong {\\Bbb Z}_2$. We now\nassume that $n\\geq 2$. Then there exists an idempotent $E\\in\nM_n(R)$ and a nilpotent $W\\in GL_n(R)$ such that $I_n+\\left(\n\\begin{array}{cccc}\na&&\\\\\n&0&\\\\\n&&\\ddots&\\\\\n&&&0\n\\end{array}\n\\right)=E+W$. Set $U=-I_n+W$. Then $U\\in GL_n(R)$. Hence,\n$$U^{-1}\\left(\n\\begin{array}{cccc}\na&&\\\\\n&0&\\\\\n&&\\ddots&\\\\\n&&&0\n\\end{array}\n\\right)=U^{-1}E+I_n=\\big(U^{-1}EU\\big)U^{-1}+I_n.$$ Set\n$F=U^{-1}EU$. Then $F=F^2\\in M_n(R)$, and that\n$$(I_n-F)U^{-1}\\left(\n\\begin{array}{cccc}\na&&\\\\\n&0&\\\\\n&&\\ddots&\\\\\n&&&0\n\\end{array}\n\\right)=I_n-F.$$ Write $I_n-F=\\left(\n\\begin{array}{cccc}\ne&0&\\\\\n*&0&\\\\\n\\vdots&&\\ddots&\\\\\n*&0&&0\n\\end{array}\n\\right).$ By hypothesis, $e=0$ or $1$. If $e=0$, then $I_n-F=0$,\nand so $E=I_n$. This shows that $\\left(\n\\begin{array}{cccc}\na&&\\\\\n&0&\\\\\n&&\\ddots&\\\\\n&&&0\n\\end{array}\n\\right)=W$ is nilpotent; hence that $a\\in R$ is nilpotent. Thus,\n$1-a\\in U(R)$.\n\nIf $e=1$, then $F=\\left(\n\\begin{array}{cccc}\n0&0&\\\\\n*&1&\\\\\n\\vdots&&\\ddots&\\\\\n*&0&&1\n\\end{array}\n\\right).$ Write $U^{-1}=\\left(\n\\begin{array}{cc}\n\\alpha&\\beta\\\\\n\\gamma&\\delta\n\\end{array}\n\\right),$ where $\\alpha\\in R, \\beta\\in M_{1\\times (n-1)}(R),$\n$\\gamma\\in M_{(n-1)\\times 1}(R)$ and $\\delta\\in M_{(n-1)\\times\n(n-1)}(R)$. Then $$\\left(\n\\begin{array}{cc}\n\\alpha&\\beta\\\\\n\\gamma&\\delta\n\\end{array}\n\\right)\\left(\n\\begin{array}{cccc}\na&&\\\\\n&0&\\\\\n&&\\ddots&\\\\\n&&&0\n\\end{array}\n\\right)=\\left(\n\\begin{array}{cc}\n0&0\\\\\nx&I_{n-1}\n\\end{array}\n\\right)\\left(\n\\begin{array}{cc}\n\\alpha&\\beta\\\\\n\\gamma&\\delta\n\\end{array}\n\\right)+I_n,$$ where $x\\in M_{(n-1)\\times 1}(R)$. Thus, we get\n$$\\begin{array}{c}\n\\alpha a=1, \\gamma a=x\\alpha+\\gamma, 0=x\\beta+\\delta+I_{n-1}.\n\\end{array}$$ One easily checks that\n$$\\left(\n\\begin{array}{cc}\n1&\\beta\\\\\n0&I_{n-1}\n\\end{array}\n\\right)\\left(\n\\begin{array}{cc}\n1&0\\\\\nx&I_{n-1}\n\\end{array}\n\\right)U^{-1}\\left(\n\\begin{array}{cc}\n1&0\\\\\n\\gamma a&I_{n-1}\n\\end{array}\n\\right)=\\left(\n\\begin{array}{cc}\n\\alpha+\\beta\\gamma a&0\\\\\n0&-I_{n-1}\n\\end{array}\n\\right).$$ This implies that $u:=\\alpha+\\beta\\gamma a\\in U(R)$.\nHence, $\\alpha=u-\\beta\\gamma a$. It follows from $\\alpha a=1$ that\n$(u-\\beta\\gamma a)a=1$. Since $R$ has only trivial idempotents, we\nget $a(u-\\beta\\gamma a)=1$, and so $a\\in U(R)$. This shows that\n$a\\in U(R)$ or $1-a\\in U(R)$. Therefore $R$ is local, and then\n$R\/J(R)$ is a division ring. Since $M_n(R)$ is nil-clean, we see\nthat so is $M_n(R\/J(R))$. In light of ~\\cite[Theorem 3]{BGDT},\n$R\/J(R)\\cong {\\Bbb Z}_2$, as desired.\n\n$(2)\\Rightarrow (1)$ By virtue of ~\\cite[Theorem 3]{BGDT},\n$M_n(R\/J(R))$ is\n nil-clean. Since $M_n(R)\/J(M_n(R))\\cong M_n(R\/J(R))$ and\n$J\\big(M_n(R)\\big)=M_n(J(R))$ is nil, it follows from ~\\cite[Lemma\n4]{BGDT} that $M_n(R)$ is nil-clean, as asserted. \\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Example 3.7.}\\ \\ Let $K$ be a\nfield, and let $R=K[x,y]\/(x,y)^2$. Then $M_n(R)$ is nil-clean if\nand only if $K\\cong {\\Bbb Z}_2$. Clearly, $J(R)=(x,y)\/(x,y)^2$,\nand so $R\/J(R)\\cong K$. Thus, $R$ is a local ring with a nilpotent\nJacobson radical. Hence, $R$ has no non-trivial idempotents. Thus,\nwe are done by Lemma 3.6.\n\n\\vskip4mm We are now ready to prove:\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 3.8.}\\ \\ {\\it Let $R$ be\nabelian, and let $n\\in {\\Bbb N}$. Then the following are\nequivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)]{\\it $M_n(R)$ is nil-clean.}\n\\item [(2)]{\\it $R\/J(R)$ is Boolean and $M_n(J(R))$ is nil.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $(1)\\Rightarrow (2)$ Clearly, $M_n(J(R))$ is nil. Let $M$ be a maximal ideal of\n$R$, and let $\\varphi_M: R\\to R\/M.$ Since $M_n(R)$ is nil-clean,\nthen so is $M_n(R\/M)$. Hence, $R\/M$ is an exchange ring with all\nidempotents central. In view of ~\\cite[Lemma 17.2.5]{CH}, $R\/M$ is\nlocal, and so $R\/M$ has only trivial idempotents. It follows from\nLemma 3.6 that $R\/M\/J(R\/M)\\cong {\\Bbb Z}_2$. Write $J(R\/M)=K\/M$.\nThen $K$ is a maximal ideal of $R$, and that $M\\subseteq K$. This\nimplies that $M=K$; hence, $R\/M\\cong {\\Bbb Z}_2$. Construct a map\n$\\varphi_M: R\/J^*(R)\\to R\/M, r+J^*(R)\\mapsto r+M$. Here, $J^*(R)$\nis the intersection of all maximal two-sided ideal of $R$. Then\n$\\bigcap\\limits_{M}Ker\\varphi_M=\\bigcap\\limits_{M}\\{\nr+J^*(R)~|~r\\in M\\}=0$. Therefore $R\/J^*(R)$ is isomorphic to a\nsubdirect product of some ${\\Bbb Z}_2$. Hence, $R\/J^*(R)$ is\nBoolean. In light of Lemma 3.5, $R\/J(R)$ is Boolean, as desired.\n\n$(2)\\Rightarrow (1)$ Since $R\/J(R)$ is Boolean, it follows by\n~\\cite[Corollary 6]{BGDT} that $M_n(R\/J(R))$ is nil-clean. That\nis, $M_n(R)\/J(M_n(R))$ is nil-clean. But $J(M_n(R))=M_n(J(R))$ is\nnil. Therefore we complete the proof, in terms of Lemma\n3.6.\\hfill$\\Box$\n\n\\vskip4mm We note that the \"$(2)\\Rightarrow (1)$\" in Theorem 3.8\nalways holds, but \"abelian\" condition is necessary in\n\"$(1)\\Rightarrow (2)$\". Let $R=M_n({\\Bbb Z}_2) (n\\geq 2)$. Then\n$R$ is nil-clean. But $R\/J(R)$ is not Boolean. Here, $R$ is not\nabelian.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 3.9.}\\ \\ {\\it Let $R$ be\ncommutative, and let $n\\in {\\Bbb N}$. Then the following are\nequivalent:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)]{\\it $M_n(R)$ is nil-clean.}\n\\item [(2)]{\\it $R\/J(R)$ is Boolean and $J(R)$ is nil.}\n\\item [(3)]{\\it For any $a\\in R$, $a-a^2\\in R$ is nilpotent.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $(1)\\Rightarrow (3)$ Let $a\\in R$. In view of Theorem 3.8,\n$a-a^2\\in J(R)$. Since $R$ is commutative, we see that $J(R)$ is\nnil if and only if $J(M_n(R))$ is nil. Therefore $a-a^2\\in R$ is\nnilpotent.\n\n$(3)\\Rightarrow (2)$ Clearly, $R\/J(R)$ is Boolean. For any $a\\in\nJ(R)$, we have $(a-a^2)^n=0$ for some $n\\geq 1$. Hence,\n$a^n(1-a)^n=0$, and so $a^n=0$. This implies that $J(R)$ is nil.\n\n$(2)\\Rightarrow (1)$ As $R$ is commutative, we see that\n$M_n(J(R))$ is nil. This completes the proof, by Theorem 3.8.\n\\hfill$\\Box$\n\n\\vskip4mm Furthermore, we observe that the converse of\n~\\cite[Corollary 7]{BGDT} is true as the following shows.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 3.10.}\\ \\ {\\it A\ncommutative ring $R$ is nil-clean if and only if $M_n(R)$ is\nnil-clean.} \\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ One direction\nis obvious by ~\\cite[Corollary 7]{BGDT}. Suppose that $M_n(R)$ is\nnil-clean. In view of Corollary 3.9, $R\/J(R)\\cong {\\Bbb Z}_2$ is\nnil-clean, and that $J(R)$ is nil. Therefore $R$ is nil-clean, by\n~\\cite[Lemma 4]{BGDT}.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Corollary 3.11.}\\ \\ Let $m,n\\in\n{\\Bbb N}$. Then $M_n\\big({\\Bbb Z}_m\\big)$ is nil-clean if and only\nif $m=2^r$ for some $r\\in {\\Bbb N}$. Write $m=p_1^{r_1}\\cdots\np_s^{r_s} (p_1,\\cdots ,p_s~\\mbox{are distinct primes}, r_1,\\cdots\n,r_s\\in {\\Bbb N}$). Then $Z_m\\cong Z_{p_1^{r_1}}\\oplus \\cdots\n\\oplus Z_{p_m^{r_s}}$. In light of Corollary 3.10, $M_n\\big({\\Bbb\nZ}_m\\big)$ is nil-clean if and only if ${\\Bbb Z}_m$ is nil-clean,\nif and only if $s=1$ and $Z_{p_1^{r_1}}$ is nil-clean. Therefore\nwe are done by Lemma 3.6.\n\n\\section{A Special Case}\n\n\\vskip4mm A natural problem is asked when a ring consists entirely\nof very idempotents, units, and nilpotent elements. We will extend\nthe study of the rings consisting entirely of some special\nelements in ~\\cite{I}, and explore such type rings. Surprisingly,\nour case will be involved in both Boolean rings and the ring\n${\\Bbb Z}_3$ of integers modulo $3$. Their structures will be\nthereby completely determined. The following is a generalization\nof ~\\cite[Corollary 2.29]{Ah} which is for a commutative ring.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 4.1.}\\ \\ {\\it Let $R$ be a\nring. Then $R=U(R)\\bigcup Id(R)\\bigcup -Id(R)$ if and only if $R$\nis isomorphic to one of the following:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it a Boolean ring;} \\vspace{-.5mm}\n\\item [(2)] {\\it a division ring;}\n\\vspace{-.5mm}\n\\item [(3)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3$; or} \\vspace{-.5mm}\n\\item [(4)] {\\it\n${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $\\Longrightarrow $\nIt is easy to check that $R$ is reduced; hence, it is abelian.\n\nCase I. $R$ is indecomposable. Then $R$ is a division ring.\n\nCase II. $R$ is decomposable. Then $R=A\\oplus B$ where $A,B\\neq\n0$. If $0\\neq x\\in A$, then $(x,0)\\in R$ is a very idempotent.\nHence, $x\\in R$ is very idempotent. Hence, $A=Id(A)\\bigcup\n-Id(A)$. Likewise, $B=Id(B)\\bigcup -Id(B)$. In view of\n~\\cite[Theorem 1.12]{Ah}, $A$ and $B$ are isomorphic to one of the\nfollowing:\n\\begin{enumerate}\n\\item [(1)] {\\it ${\\Bbb Z}_3$;}\n\\vspace{-.5mm} \\item [(2)] {\\it a Boolean ring;} \\vspace{-.5mm}\n\\item [(3)] {\\it ${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring.}\n\\end{enumerate}\n\nThus, $R$ is isomorphic to one of the following:\n\\begin{enumerate}\n\\item [(a)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3$;}\n\\vspace{-.5mm} \\item [(b)] {\\it ${\\Bbb Z}_3\\oplus B$ where $B$ is\na Boolean ring;} \\vspace{-.5mm}\n\\item [(c)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus B$ where $B$ is\na Boolean ring;} \\vspace{-.5mm}\n\\item [(d)] {\\it a Boolean ring;}\\end{enumerate}\n\nCase (c). $(1,-1,0)\\not\\in U(R)\\bigcup Id(R)\\bigcup -Id(R)$, an\nabsurd.\n\nTherefore we conclude that $R$ is one of Cases (a), (b) and (d),\nas desired.\n\n$\\Longleftarrow $ Case (1). $R=Id(R)$. Case (2). $R=U(R)\\bigcup\nId(R)$. Case (3). $U(R)=\\{ (1,1),(1,-1),$ $(-1,1),$ $(-1,-1)$,\n$Id(R)=\\{ (0,0),(0,1),(1,0)\\}$ and $-Id(R)=\\{ (0,0),(0,-1),$ $\n(-1,0),(-1,-1)\\}$. Thus, $R=U(R)\\bigcup Id(R)\\bigcup -Id(R)$. Case\n(4). $Id(R)=\\{ (0,x),(1,x) ~|~x\\in B\\}$ and $-Id(R)=\\{\n(0,x),(-1,x)~|~x\\in B$. Therefore $R=Id(R)\\bigcup -Id(R)$, as\ndesired.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 4.2.}\\ \\ {\\it Let $R$ be a\ndecomposable ring. Then $R$ consists entirely of very idempotents,\nunits, and nilpotent elements if and only if $R$ is isomorphic to\none of the following:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it a Boolean ring;}\n\\vspace{-.5mm}\n\\item [(2)] {\\it ${\\Bbb Z}_3\\times {\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(3)] {\\it\n${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $\\Longrightarrow$ Write $R=A\\oplus B$\nwith $A,B\\neq 0$. Then $A$ and $B$ are rings that consisting\nentirely of very idempotents, units, and nilpotent elements. If\n$0\\neq x\\in N(A)$, then $(x,1)\\not\\in Id(R)\\bigcup -Id(R)\\bigcup\nU(R)\\bigcup N(R)$. This shows that $A=U(A)\\bigcup Id(A)\\bigcup\n-Id(A)$. Likewise, $B=U(B)\\bigcup Id(B)\\bigcup -Id(B)$. In light\nof Lemma 4.1, $R$ is one of the following:\n\\begin{enumerate}\n\\item [(a)] {\\it a Boolean ring;} \\vspace{-.5mm}\n\\item [(b)] {\\it $B\\oplus D$ where $B$ is a Boolean ring and $D$ is a division ring;}\n\\vspace{-.5mm}\n\\item [(c)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus B$ where $B$ is\na Boolean ring;} \\vspace{-.5mm}\n\\item [(d)] {\\it ${\\Bbb Z}_3\\oplus B$ where $B$ is\na Boolean ring;} \\vspace{-.5mm}\n\\item [(e)] {\\it $D\\oplus D'$ where $D$ and $D'$ are division rings; or}\n\\vspace{-.5mm}\n\\item [(f)] {\\it ${\\Bbb Z}_3\\oplus B\\oplus D$ where $B$ is a Boolean ring and $D$ is a division ring.}\n\\vspace{-.5mm}\n\\item [(g)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus D$;}\\vspace{-.5mm}\n\\item [(h)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus {\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(i)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus B$ where $B$ is adivision ring;}\\vspace{-.5mm}\n\\item [(j)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus {\\Bbb Z}_3\\oplus {\\Bbb Z}_3$;}\\end{enumerate}\n\nIn Case (b). If $0,\\pm 1\\neq x\\in D$, then $(0,x)\\not\\in\nU(R)\\bigcup Id(R)\\bigcup -Id(R)$. This forces $D\\cong {\\Bbb Z}_2,\n{\\Bbb Z}_3$. Hence, $(b)$ forces $R$ is in $(1)$ or $(3)$. $(c)$\ndoes not occur. $(e)$ forces $D,D'\\cong {\\Bbb Z_2}$ or ${\\Bbb\nZ}_3$. Hence, $R$ is in $(1)-(3)$. $(f)$ does not occur except\n$D\\cong {\\Bbb Z}_2$. Thus, $R$ is in $(1)-(3)$. Cases (g)-(j) do\nnot occur as $(1,-1,0), (1,-1,0,0)\\not\\in I(R)\\bigcup\n-Id(R)\\bigcup N(R)$, as desired.\n\n$\\Longleftarrow$ This is clear.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 4.3.}\\ \\ {\\it Let $R$ be an\nabelian ring. Then $R$ consists entirely of very idempotents,\nunits, and nilpotent elements if and only if $R$ is isomorphic to\none of the following:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it ${\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(2)] {\\it a Boolean ring;}\n\\vspace{-.5mm}\n\\item [(3)] {\\it ${\\Bbb Z}_3\\oplus {\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(4)] {\\it\n${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring; or}\n\\item [(5)] {\\it local ring with nil Jacobson radical.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $\\Longrightarrow$\nCase I. $R$ is indecomposable. Then $R=U(R)\\bigcup N(R)$. This\nshows that $R$ is local. Let $x\\in J(R)$, then $x\\in N(R)$, and so\n$J(R)$ is nil.\n\nCase II. $R$ is decomposable. In view of Lemma 4.2, $R$ is\nisomorphic to one of the following: \\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it a Boolean ring;}\n\\vspace{-.5mm}\n\\item [(2)] {\\it ${\\Bbb Z}_3\\times {\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(3)] {\\it\n${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring.}\n\\end{enumerate} This shows that $R$ is isomorphic to one of\n$(1)-(5)$, as desired.\n\n$\\Longleftarrow$ This is obvious.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 4.4.}\\ \\ {\\it Let $R$ be any\nring that consists entirely of very idempotents, units and\nnilpotent elements. Then $eRe$ is a division ring for any\nnoncentral idempotent $e\\in R$.}\\vskip2mm\\hspace{-1.8em} {\\it\nProof.}\\ \\ Let $e\\in R$ be a noncentral idempotent, and let\n$f=1-e$. Then $R\\cong \\left(\n\\begin{array}{cc}\neRe&eRf\\\\\nfRe&fRf\n\\end{array}\n\\right)$. The subring $\\left(\n\\begin{array}{cc}\neRe&0\\\\\n0&fRf\n\\end{array}\n\\right)$ consists entirely of very idempotents, units and\nnilpotent elements. That is, $eRe\\times fRf$ consists entirely of\nvery idempotents, units and nilpotent elements. Set $A=eRe$ and\n$B=fRf$. Similarly to Lemma 4.2, $A=U(A)\\bigcup Id(A)\\bigcup\n-Id(A)$. In view of Lemma 4.1, $A$ is isomorphic to one of the\nfollowing:\n\\begin{enumerate}\n\\item [(1)] {\\it ${\\Bbb Z}_3$;}\n\\vspace{-.5mm} \\item [(2)] {\\it a Boolean ring;} \\vspace{-.5mm}\n\\item [(3)] {\\it a division ring;}\n\\vspace{-.5mm}\n\\item [(4)] {\\it ${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring.}\n\\end{enumerate}\nThat is, $A$ is a division ring or a ring in which every element\nis very idempotent. Suppose that $eRe$ is not a division ring.\nThen $eRe$ must contains a nontrivial idempotent, say $a\\in R$.\nLet $b=e-a$. Let $x\\in eRf$ and $y\\in fRe$. Choose\n$$X_1=\\left(\n\\begin{array}{cc}\na&x\\\\\n0&0\n\\end{array}\n\\right),X_2=\\left(\n\\begin{array}{cc}\nb&x\\\\\n0&0\n\\end{array}\n\\right),Y_1=\\left(\n\\begin{array}{cc}\na&0\\\\\ny&0\n\\end{array}\n\\right),Y_2=\\left(\n\\begin{array}{cc}\nb&0\\\\\ny&0\n\\end{array}\n\\right).$$ Then $X_1,X_2,Y_1,Y_2$ are not invertible. As $a,b\\in\neRe$ is nontrivial idempotents, we see that $X_1,X_2,Y_1,Y_2$ are\nall not nilpotent matrices. This shows that $X_1$ and $X_2$ are\nboth very idempotents. It follows that $X_1=\\pm X_2^2$ or\n$X_2^2=\\pm X_2$. As $x\\in eRf, y\\in fRe$, we have $ex=x$ and\n$fy=y$.\n\nCase I. $X_1=X_1^2, X_2=X_2^2$. Then $ax=x, bx=x$, and so\n$x=ex=2x$; hence, $x=0$.\n\nCase II. $X_1=X_1^2, X_2=-X_2^2$. Then $ax=x, bx=-x$, and so\n$x=ex=0$.\n\nCase III. $X_1=-X_1^2, X_2=X_2^2$. Then $ax=-x, bx=x$, and so\n$x=ex=0$.\n\nCase IV. $X_1=-X_1^2, X_2=-X_2^2$. Then $a=-a^2, ax=-x, b=-b^2$\nand $bx=-x$. Hence, $(e-a)x=-x$, and so $x=ex=-2x$, hence, $3x=0$.\nAs $a\\in R$ is an idempotent, we see that $a=a^2$, hence, $a=-a$,\nand so $2a=0$. It follows that $x=-ax=(2a)x-(3x)a=0$.\n\nThus, $x=0$ in any case. We infer that $eRf=0$. Likewise, $fRe=0$.\nHence, $e\\in R$ is central, an absurd. This completes the\nproof.\\hfill$\\Box$\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Lemma 4.5.}\\ \\ {\\it Let $R$ be any\nring that consists entirely of very idempotents, units and\nnilpotent elements. Then $eRe$ is isomorphic to ${\\Bbb Z}\/2{\\Bbb\nZ}$ or ${\\Bbb Z}\/3{\\Bbb Z}$ for any noncentral idempotent $e\\in\nR$.}\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Let $e\\in R$ be a\nnoncentral idempotent. In view of Lemma 4.4, $eRe$ is a division\nring. Set $f=1-e$. For any $u\\in eRe$, we assume that $u\\neq 0, e,\n-e$, then the matrix\n$$X= \\left(\n\\begin{array}{cc}\nu&0\\\\\n0&0\n\\end{array}\n\\right)\\in \\left(\n\\begin{array}{cc}\neRe&eRf\\\\\nfRe&fRf\n\\end{array}\n\\right)$$ is not be a unit, a very idempotent, or a nilpotent\nelement. This gives a contradiction. Therefore $u=0,e$ or $-e$, as\ndesired.\\hfill$\\Box$\n\n\\vskip4mm Recall that a ring $R$ is semiprime if it has no nonzero\nnilpotent ideals. Furthermore, we derive\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 4.6.}\\ \\ {\\it Let $R$ be\nany nonabelian ring that consists entirely of units, very\nidempotents, and nilpotent elements. If $R$ is semiprime, then it\nis isomorphic to $M_2({\\Bbb Z}_2)$ or $M_2({\\Bbb\nZ}_3)$.}\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Suppose that $R$\nis semiprime. In view of Lemma 4.4, $eRe$ is a division ring for\nany noncentral idempotent $e\\in R$. It follows by ~\\cite[Lemma\n21]{Du} that $R$ is isomorphic to $M_2(D)$ for a division ring\n$D$. Choose $E_{11}=\\left(\n\\begin{array}{cc}\n1&0\\\\\n0&0\n\\end{array}\n\\right)\\in M_2(D)$. Then $E_{11}$ is a noncentral idempotent.\nAccording to Lemma 4.5, $R\\cong {\\Bbb Z}\/2{\\Bbb Z}$ or ${\\Bbb\nZ}\/3{\\Bbb Z}$, as asserted.\\hfill$\\Box$\n\n\\vskip4mm Recall that a ring $R$ is a NJ-ring provided that for\nany $a\\in R$, either $a\\in R$ is regular or $1-a\\in R$ is a\nunit~\\cite{N}. Clearly, all rings in which every elements consists\nentirely of units, very idempotents, and nilpotent elements are\nNJ-rings.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 4.7.}\\ \\ {\\it Let $R$ be\nany nonabelian ring that consists entirely of very idempotents,\nunits and nilpotent elements. If $R$ is not semiprime, then it is\nisomorphic to the ring of a Morita context with zero pairings\nwhere the underlying rings are ${\\Bbb Z}_2$ or ${\\Bbb\nZ}_3$.}\\vskip2mm\\hspace{-1.8em} {\\it Proof.}\\ \\ Suppose that $R$\nis not semiprime. Clearly, $R$ is a NJ-ring. In view of\n~\\cite[Theorem 2]{N}, $R$ must be a regular ring, a local ring or\nisomorphic to the ring of a Morita context with zero pairings\nwhere the underlying rings are both division ring. If $R$ is\nregular, it is semiprime, a contradiction. If $R$ is local, it is\nabelian, a contradiction. Therefore, $R$ is isomorphic to the ring\nof a Morita context $T=(A,B,M,N,\\varphi,\\psi)$ with zero pairings\n$\\varphi,\\psi$ where the underlying rings are division rings $A$\nand $B$. Choose $E=\\left(\n\\begin{array}{cc}\n1_A&0\\\\\n0&0\n\\end{array}\n\\right)\\in T$. Then $E\\in T$ is a noncentral idempotent. In light\nof Lemma 4.5, $A\\cong ETE\\cong {\\Bbb Z}_2$ or ${\\Bbb Z}_3$.\nLikewise, $B\\cong  {\\Bbb Z}_2$ or ${\\Bbb Z}_3$. This completes the\nproof.\\hfill$\\Box$\n\n\\vskip4mm With these information we completely determine the\nstructure of rings that consist entirely of very idempotents,\nunits and nilpotent elements.\n\n\\vskip4mm \\hspace{-1.8em} {\\bf Theorem 4.8.}\\ \\ {\\it Let $R$ be a\nring. Then $R$ consists entirely of very idempotents, units and\nnilpotent elements if and only if $R$ is isomorphic to one of the\nfollowing:}\\vspace{-.5mm}\n\\begin{enumerate}\n\\item [(1)] {\\it a Boolean ring;}\n\\vspace{-.5mm}\n\\item [(2)] {\\it ${\\Bbb Z}_3\\times {\\Bbb Z}_3$;}\n\\vspace{-.5mm}\n\\item [(3)] {\\it\n${\\Bbb Z}_3\\oplus B$ where $B$ is a Boolean ring;}\n\\item [(4)] {\\it local ring with a nil Jacobson radical;}\n\\vspace{-.5mm}\n\\item [(5)] {\\it $M_2\\big({\\Bbb Z}_2\\big)$ or $M_2\\big({\\Bbb Z}_3\\big)$; or} \\vspace{-.5mm}\n\\item [(6)] {\\it the ring of a Morita context\nwith zero pairings where the underlying rings are ${\\Bbb Z}_2$ or\n${\\Bbb Z}_3$.}\n\\end{enumerate} \\vspace{-.5mm} {\\it Proof.}\\ \\ $\\Longrightarrow$\nThis is obvious by Theorem 4.3, Theorem 4.6 and Theorem 4.7.\n\n$\\Longleftarrow$ Cases (1)-(4) are easy. Case (5)-(6) are verified\nby checking all possible (generalized) matrices over ${\\Bbb Z}_2$\nand ${\\Bbb Z}_3$.\\hfill$\\Box$\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the Euclidean case, we can roughly distinguish three types of Improved Sobolev inequalities following the method used in their proof and the parameter's range defining the functional spaces. Let us recall these inequalities (for a precise definition of the functional spaces used below, please refer to section \\ref{espa}).\\\\\n\nHistorically the first method, due to P. G\\'erard, F. Oru and Y. Meyer \\cite{GMO}, is based on a Littlewood-Paley decomposition and interpolation results applied to dyadic blocks. For a function $f$ such that $f\\in \\dot{W}^{s_1,p}(\\mathbb{R}^n)$ and $f\\in \\dot{B}^{-\\beta,\\infty}_{\\infty}(\\mathbb{R}^n)$, the inequality obtained reads as follows:\n\\begin{equation}\\label{ISI1}\n\\|f\\|_{\\dot{W}^{s,q}}\\leq C \\|f\\|_{\\dot{W}^{s_1,p}}^\\theta\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-\\theta}\n\\end{equation}\nwhere $1<p<q<+\\infty$, $\\theta=p\/q$, $s=\\theta s_1 -(1-\\theta)\\beta$ and $-\\beta<s<s_1$. Let us stress that the value $p=1$ is forbidden here. We write $\\dot{W}^{s,p}$ for homogeneous $(s,p)$-Sobolev spaces and  $\\dot{B}^{-\\beta,\\infty}_{\\infty}$ for homogeneous $(-\\beta,\\infty, \\infty)$-Besov spaces.\\\\\n\nThe second method, studied by M. Ledoux in \\cite{Ledoux}, use semi-group properties related to Laplacian and heat kernel and allows us to treat the case $p=1$. If  $\\nabla f\\in L^{p}(\\mathbb{R}^n)$ and $f\\in \\dot{B}^{-\\beta,\\infty}_{\\infty}(\\mathbb{R}^n)$, we have\n\\begin{equation}\\label{ISI2}\n\\|f\\|_{L^q}\\leq C \\|\\nabla f\\|_{L^p}^\\theta\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-\\theta}\n\\end{equation}\nwith $1\\leq p < q<+\\infty$, $\\theta=p\/q$ and $\\beta=\\theta\/(1-\\theta)$.\\\\ \n\nFinally, the third method proposed by A. Cohen, W. Dahmen, I. Daubechies \\& R. De Vore in \\cite{Cohen2} use a BV-norm weak estimation using wavelet coefficients and isoperimetric inequalities and gives, for a function $f$ such that $f\\in BV(\\mathbb{R}^n)$ and $f\\in \\dot{B}^{-\\beta,\\infty}_{\\infty}(\\mathbb{R}^n)$, the estimation below:\n\\begin{equation}\\label{ISI3}\n\\|f\\|_{\\dot{W}^{s,q}}\\leq C \\| f\\|_{BV}^{1\/q}\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-1\/q}\n\\end{equation}\nwhere $1<q\\leq 2$, $0\\leq s<1\/q$ and $\\beta=(1-sq)\/(q-1)$. When $s=0$, this last result implies (\\ref{ISI2}) with $p=1$, but is limited by the fact that $1<q\\leq 2$.\\\\\n\nIn this paper we will study these inequalities using as a framework stratified Lie groups which are a natural generalization of $\\mathbb{R}^n$ when modifying dilations (see (\\ref{DefDilat}) below for a definition). However, this seemingly simple modification induces some serious technical problems at many levels since the whole group structure is changed: for example, the underlying geometry is totally different (see \\cite{Strichartz} and \\cite{Garofalo}) which makes the techniques used in \\cite{Cohen2} hardly transposable to this setting; observe also that the use of Fourier transform, and the classical associated tools, is not as straightforward as in $\\mathbb{R}^n$.\\\\\n\nFor the Heisenberg group, inequalities of type (\\ref{ISI1}) have been carried out in \\cite{Bahouri} and with the work realized in \\cite{Furioli2} we can deduce these inequalities for stratified Lie groups in a unweighted setting. Note that these authors develop systematically in each case a Littlewood-Paley decomposition in order to obtain these estimates, we will show here how to treat these inequalities in a far more direct and simpler way using maximal functions.\\\\ \n\nThis is one of the main novelties of this paper, however, our principal aim is to generalize inequality (\\ref{ISI2}) to stratified Lie groups using weighted spaces and to give a new weak-type estimation which lies, roughly speaking, between (\\ref{ISI2}) and (\\ref{ISI3}). In order to achieve this, we will develop some techniques using properties associated to the sub-Laplacian spectral decomposition. \\\\\n\nWe will consider throughout this paper weighted functional spaces with weights $\\omega$ belonging to the Muckenhoupt classes $A_{p}$ for $1\\leq p<+\\infty$\\footnote{When $p=+\\infty$, we do not consider them, since in this case weighted spaces $X^{\\infty}(\\omega)$ coincide with traditional ones $X^{\\infty}$ where $X$ is a Lebesgue, Sobolev or Besov space.}. The main reason for considering these weights lies in their connection with maximal functions which will lead us to a painless proof for inequalities of type (\\ref{ISI1}).\\\\\n\nOur principal theorem treats the critical case $p=1$ of improved Sobolev inequalities: \n\\begin{Theoreme}\\label{smile00} \nLet $\\mathbb{G}$ be a stratified Lie group and $\\omega$ a weight in the Muckenhoupt class $A_{1}$.  If $\\nabla f\\in L^{1}(\\mathbb{G},\\omega)$ and $f\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$, then we have the following inequalities:\n\\begin{enumerate}\n\\item[$\\bullet$] \\textbf{\\emph{[Strong inequalities]}}\n\\begin{equation}\\label{smile0} \n\\|f\\|_ {L^{q}(\\omega)}\\leq C \\|\\nabla f \\|_ {L^{1}(\\omega)}^{\\theta} \\|f\\|_{\\dot{B}^{-\\beta, \\infty}_{\\infty}}^{1-\\theta} \n\\end{equation} \nwhere $1<q<+\\infty$, $\\theta = 1\/q$ and $\\beta=\\theta\/(1-\\theta)$.  \n\n\\item[$\\bullet$] \\textbf{\\emph{[Weak inequalities]}} \n\\begin{equation}\\label{smile1} \n\\|f\\|_ {\\dot{W}^{s, q}_{\\infty}(\\omega)}\\leq C \\|\\nabla f\\|_{L^1(\\omega)}^{\\theta}\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-\\theta} \n\\end{equation} \nwhere $1<q<+\\infty$, $0<s<1\/q<1$, $\\theta=1\/q$ and $\\beta=\\frac{1-sq}{q-1}$.\nHere $\\|\\cdot\\|_{\\dot{W}^{s, q}_{\\infty}(\\omega)}$ characterizes the weighted weak homogenous Sobolev space which definition is given by formula (\\ref{DefDebilSoboLe}) in section \\ref{espa}.\n\\end{enumerate}\n\\end{Theoreme} \n\nIt is possible to see inequality (\\ref{smile1}) as a weak-type improvement of (\\ref{ISI3}). Indeed, the general Sobolev-like inequality obtained in \\cite{Cohen2} uses in fact a Besov space in the left-hand side: \n$$\\|f\\|_{\\dot{B}^{s,q}_q}\\leq C \\| f\\|_{BV}^{1\/q}\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-1\/q}$$\nwhich turns to be (\\ref{ISI3}) only if $1<q\\leq 2$ since in this case we have $\\dot{B}^{s,q}_q\\subset \\dot{W}^{s,q}$. The restriction $q\\in ]1,2]$ is a serious limitation as, for $q>2$, this Besov-Sobolev spaces embbeding is reversed. It is then important to observe that our weak inequality does not have this restriction since we allow $q$ to be in the interval $]1, +\\infty[$. \\\\\n\nNote also that in the context of stratified Lie groups, the weak inequality (\\ref{smile1}) is the sharpest result available.\\\\\n\nOur second result provides the main tool for proving theorem \\ref{smile00}:\n\\begin{Theoreme}[Modified pseudo-inequality of Poincar\\'e]\\label{CHAME} \nLet $\\mathbb{G}$ be a stratified Lie group, $\\omega \\in A_{1}$ and $\\nabla f\\in L^{1}(\\mathbb{G}, \\omega)$.  We have the following estimate for $0\\leq s<1$ and for $t>0$:  \n\\begin{equation}\\label{CHAME1} \n\\|\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}f \\|_ { L^{1}(\\omega)}\\leq C \\; t^{\\frac{1-s}{2 } } \\|\\nabla f\\|_ { L^{1}(\\omega) }.  \n\\end{equation}\nWhere $\\mathcal{J}$ is a sub-Laplacian on $\\mathbb{G}$ invariant with respect to the family of dilation, $H_{t}$ stands for the associated heat semi-group and the constant $C=C(s)$ depends on the group $\\mathbb{G}$.\n\\end{Theoreme} \nThis estimate will be a consequence of the sub-Laplacian $\\mathcal{J}$ several spectral properties and we will specially use operators of type $m(\\mathcal{J})$ where $m$ is a well suited Borel function (see section \\ref{spectre} for the details). \\\\\n\nFinally, we will prove a in very straightforward way the next theorem for a weighted functional setting: \n\\begin{Theoreme}\\label{PGLR}\nLet $\\mathbb{G}$ be a stratified Lie group and $\\omega \\in A_{p}$ with $1<p<+\\infty$. If $f\\in \\dot{W}^{s_1,p}(\\mathbb{G},\\omega)$ and $f\\in \\dot{B}^{-\\beta,\\infty}_{\\infty}(\\mathbb{G})$ then\n\\begin{equation}\\label{PGLR1}\n\\|f\\|_{\\dot{W}^{s,q}(\\omega)}\\leq C \\|f\\|_{\\dot{W}^{s_1,p}(\\omega)}^\\theta\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}^{1-\\theta}\n\\end{equation}\nwhere $1<p<q<+\\infty$, $\\theta=p\/q$, $s=\\theta s_1 -(1-\\theta)\\beta$ and $-\\beta<s<s_1$.\n\\end{Theoreme} \nIn the proof of this theorem we will show as annouced how to pass over the Littlewood-Paley theory.\\\\\n\n\nThe plan of the article is the following: in section \\ref{def} we make a short presentation of stratified Lie groups; in sections \\ref{Herbata} and \\ref{espa} we define maximal functions, Muckenhoupt weights and weighted functional spaces respectively; we detail the necessary results concerning spectral resolution of the sub-Laplacian in section \\ref{spectre}; and finally, in section \\ref{demos}, we give the proof of theorems \\ref{smile00}, \\ref{CHAME} and \\ref{PGLR}.  \n\n\\section{Notation and preliminaries}\\label{def} \n\nIn this section we recall some basic facts about stratified Lie groups, for further information see \\cite{Folland2}, \\cite{Varopoulos},\\cite{Stein2}  and the references given therein. \\\\\n\nA \\textit{homogeneous group} $\\mathbb{G}$ is the data of $\\mathbb{R}^{n}$ equipped with a structure of Lie group and with a family of dilations \nwhich are group automorphisms. We will always suppose that the origin is the identity. For \\textit{dilations}, we define them by fixing integers $(a_{i})_{1\\leq i\\leq n}$ such that $1=a_{1}\\leq... \\leq a_{n}$ and by writing:  \n\\begin{eqnarray}\\label{DefDilat}\n\\delta_{\\alpha }:  \\mathbb{R}^{n } & \\longrightarrow & \\mathbb{R}^{n } \\label{dilat} \\\\\nx &\\longmapsto & \\delta_{\\alpha}[x]=(\\alpha^{a_{1}}x_{1},...,\\alpha^{a_{n}}x_{n})\\nonumber \n\\end{eqnarray} \nWe will often note $\\alpha x$ instead of $\\delta_{\\alpha}[x]$ and $\\alpha$ will always indicate a strictly positive real number. \\\\  \n\nThe reader can be easily convinced that the Euclidean space $\\mathbb{R}^{n}$ with its group structure and provided with its usual dilations \n(i.e. $a_{i}=1$, for $i=1,...,n$) is a homogeneous group. Here is another example: if $x=(x_{1}, x_{2}, x_{3})$ is an element of $\\mathbb{R}^{3}$, we can fix a dilation by writing \n$\\delta_{\\alpha}[x]=(\\alpha x_{1 }, \\alpha x_{2}, \\alpha^{2 } x_{3})$ for $\\alpha>0$. Then, the well suited group law with respect to this dilation is given by  \n\\begin{equation*}\n x\\cdot y=(x_{1}, x_{2}, x_{3})\\cdot(y_{1}, y_{2}, y_{3})=(x_{1}+y_{1}, x_{2}+y_{2}, x_{3}+y_{3}+\\frac{1}{2}(x_{1}y_{2}-y_{1}x_{2 })). \n\\end{equation*} \nThis triplet $(\\mathbb{R}^{3}, \\cdot,\\delta )$ corresponds to the Heisenberg group $\\mathbb{H}^{1}$ which is the first non-trivial example of a homogeneous group.\\\\  \n\nThe \\emph{homogeneous dimension} with respect to dilation (\\ref{dilat}) is given by the sum of the exponents of dilation:  \n\\begin{equation*} N=\\sum_{1\\leq i\\leq n}a_{i }.  \n\\end{equation*} \nWe observe that it is always larger than the topological dimension $n$ since integers $a_{i}$ verifies $a_{i}\\geq 1$ for all $i=1,...,n$. \nFor instance, in the Heisenberg group $\\mathbb{H}^{1}$ we have $N=4$ and $n=3$ while in the Euclidean case these two concepts coincide. \\\\  \n\nWe will say that a function on $\\mathbb{G}\\setminus \\{0\\}$ is homogeneous of degree $\\lambda \\in \\mathbb{R}$ if $f(\\delta_{\\alpha}[x])=\\alpha^{\\lambda}f(x)$ \nfor all $\\alpha>0$.  \nIn the same way, we will say that a differential operator $D$ is homogeneous of degree $\\lambda$ if \n$$D(f(\\delta_{\\alpha}[x]))=\\alpha^{\\lambda}(Df)(\\delta_{\\alpha}[x])$$ for all $f$ in operator's domain.  \nIn particular, if $f$ is homogeneous of degree $\\lambda$  and if $D$ is a differential operator of degree $\\mu$, then $Df$ is homogeneous of degree $\\lambda-\\mu$. \\\\ \n\nFrom the point of view of measure theory, homogeneous groups behave in a traditional way since Lebesgue measure $dx$ is bi-invariant and coincides with the Haar measure. For any subset $E$ of $\\mathbb{G}$ we will note its measure as $|E|$. The convolution of two functions $f$ and $g$ on $\\mathbb{G}$ is defined by\n\\begin{equation*} \nf\\ast g(x)=\\int_{\\mathbb{G}}f(y)g(y^{-1}\\cdot x)dy=\\int_{\\mathbb{G}}f(x\\cdot y^{-1})g(y)dy,  \\quad x\\in \\mathbb{G}.\n\\end{equation*}\nWe also have the useful Young's inequalities:\n\\begin{Lemme}\\label{LemYoungG} \nIf $1\\leq p, q, r\\leq +\\infty$ such that $1+\\frac{1}{r}=\\frac{1}{p}+\\frac{1}{q}$.  \nIf $f\\in L^{p}(\\mathbb{G})$ and $g\\in L^{q}(\\mathbb{G})$, then $f\\ast g \\in L^{r}(\\mathbb{G})$ and \n\\begin{equation}\\label{Young} \n\\|f\\ast g\\|_{L^r}\\leq \\|f\\|_{L^p}\\|g\\|_{L^q}. \n\\end{equation} \n\\end{Lemme} \nA proof is given in \\cite{Folland2}.\\\\\n\nFor a homogeneous group $\\mathbb{G}=(\\mathbb{R}^{n}, \\cdot, \\delta)$ we consider now its Lie algebra $\\mathfrak{g}$ whose elements can be conceived in two different ways: as \\textit{left}-invariant vector fields or as \\textit{right}-invariant vector fields. The left-invariant vectors fields $(X_j)_{1\\leq j\\leq n}$ are determined by the formula\n\\begin{equation}\\label{loiG} \n(X_{j}f)(x)=\\left.\\frac{\\partial f(x\\cdot y)}{\\partial y_{j}}\\right|_{y=0}=\\frac{\\partial f}{\\partial x_{j}}+\\sum_{j<k}q^{k}_{j}(x)\\frac{\\partial f}{\\partial x_{k}} \n\\end{equation} \nwhere $q^{k}_{j}(x)$ is a homogeneous polynomial of degree $a_{k}-a_{j}$ and $f$ is a smooth function on $\\mathbb{G}$. By this formula one deduces easily that these vectors fields are homogeneous of degree $a_{j}$:  \n\\begin{equation*} \nX_{j}\\left(f(\\alpha x)\\right)=\\alpha^{a_{j}}(X_{j}f)(\\alpha x).  \n\\end{equation*} \nWe will note $(Y_{j})_{1\\leq j\\leq n}$ the right invariant vector fields defined in a totally similar way:\n$$\n(Y_{j}f)(x)=\\left.\\frac{\\partial f(y\\cdot x)}{\\partial y_{j}}\\right|_{y=0}\n$$\n\nA homogeneous group $\\mathbb{G}$ is  \\emph{stratified} if its Lie algebra $\\mathfrak{g}$ breaks up into a sum of linear subspaces  \n$\\mathfrak{g}=\\bigoplus_{1\\leq j\\leq k } E_{j}$ such that $E_{1}$ generates the algebra $\\mathfrak{g}$ and $[E_{1}, E_{j}]=E_{j+1}$ for $1\\leq j < k$ and $[E_{1}, E_{k}]=\\{0\\}$ and $E_{k}\\neq\\{0\\}$, but $E_{j}=\\{0\\}$ if $j>k$. Here $[E_{1}, E_{j}]$ indicates the subspace of $\\mathfrak{g}$ generated by the elements $[U, V]=UV-VU$ with $U\\in E_{1}$ and $V\\in E_{j}$. The integer $k$ is called the \\emph{degree} of stratification of $\\mathfrak{g}$. For example, on Heisenberg group $\\mathbb{H}^1$, we have $k=2$ while in the Euclidean case $k=1$.  \\\\\n\nWe will suppose henceforth that $\\mathbb{G}$ is \\textbf{stratified}. Within this framework, if we fix the vectors fields $X_{1},...,X_{m}$ such that $a_{1}=a_{2}=\\ldots=a_{m}=1$ $(m<n)$, then the family $(X_{j})_{1\\leq j\\leq m}$ is a base of $E_{1}$ and generates the Lie algebra of $\\mathfrak{g}$, which is precisely the H\\\"ormander's condition (see \\cite{Folland2} and \\cite{Varopoulos}).\\\\\n\nTo the family $(X_{j})_{1\\leq j\\leq m}$ is associated the Carnot-Carath\\'eodory distance $d$ which is left-invariant and compatible with the topology on $\\mathbb{G}$ (see \\cite{Varopoulos} for more details). For any $x\\in \\mathbb{G}$ we will note $|x|=d(x,e)$ and for $r>0$ we form the balls by writing $B(x,r)=\\{y\\in \\mathbb{G}: d(x,y)<r\\}$.\\\\\n\nThe main tools of this paper depends on the properties of the gradient, the sub-Laplacian and the associated heat kernel. Before introducing them, we make here three remarks on general vectors fields $X_{j}$ and $Y_{j}$. Let us fix some notation. For any multi-index $I=(i_{1},...,i_{n})\\in \\mathbb{N}^{n}$, one defines $X^{I}$ by $X^{I}=X_{1}^{i_{1}}\\dots X_{n}^{i_{n}}$ and $Y^{I}$ by $Y^{I}=Y_{1}^{i_{1}}\\dots Y_{n}^{i_{n}}$. We note $|I|= i_{1}+\\ldots+i_{n}$ the order of the derivation $X^I$ or $Y^I$ and $d(I)=a_{1}i_{1}+\\ldots+a_{n}i_{n}$ the homogeneous degree of this one. \\\\  \n\nFirstly, for $\\varphi, \\psi \\in \\mathcal{C}^{\\infty}_{0}(\\mathbb{G})$ we have the equality \n\\begin{equation*} \n\\int_{\\mathbb{G}}\\varphi(x)(X^{I}\\psi)(x)dx=(-1)^{|I|}\\int_{\\mathbb{G}}(X^{I}\\varphi)(x)\\psi(x)dx.  \n\\end{equation*} \nSecondly, interaction of operators $X^{I}$ and $Y^{I}$ with convolutions is clarified by the following identities:  \n\\begin{equation}\\label{izder}\nX^{I}(f*g)=f*(X^{I}g), \\qquad Y^{I}(f*g)=(Y^{I}f)*g, \\qquad (X^{I}f)*g=f*(Y^{I}g). \n\\end{equation} \nFinally, one will say that a function $f \\in \\mathcal{C}^{\\infty}(\\mathbb{G})$ belongs to the Schwartz class $\\mathcal{S}(\\mathbb{G})$ if the following semi-norms are bounded for all $k\\in \\mathbb{N}$ and any multi-index $I$:  $N_{k,I}(f)=\\underset{x\\in \\mathbb{G}}{\\sup } \\, (1+|x|)^{k }|X^{I}f(x)|$. \n\\begin{Remarque}\n\\emph{To characterize the Schwartz class $\\mathcal{S}(\\mathbb{G})$ we can replace vector fields $X^I$ in the semi-norms $N_{k,I}$ above by right-invariant vector fields $Y^I$.}\n\\end{Remarque}\nFor a proof of these facts and for further details see \\cite{Folland2} and \\cite{Furioli2}.\\\\\n\nWe define now the \\textit{gradient} on $\\mathbb{G}$ from vectors fields of homogeneity degree equal to one by fixing $\\nabla = (X_{1},...,X_{m })$.  This operator is of course left invariant and homogeneous of degree $1$. The length of the gradient is given by the formula $|\\nabla f|= \\left((X_{1}f)^{2}+... +(X_{m}f)^{2 } \\right)^{1\/2}$.\\\\\n \nLet us notice that there is not a single way to build a \\textit{sub-Laplacian}, see for example \\cite{Furioli2} and \\cite{Chame}. In this article, we will work with the following sub-Laplacian:\n\\begin{equation}\\label{laplaciano} \n\\mathcal{J}=\\nabla^{*}\\nabla=-\\sum_{j=1}^{m}X^{2}_{j} \n\\end{equation} \nwhich is a positive self-adjoint, hypo-elliptic operator (since $(X_j)_{1\\leq j\\leq m}$ satisfies the H\\\"ormander's condition), having as domain of definition $L^2(\\mathbb{G})$. Its associated \\textit{heat operator} on $\\mathbb{G}\\times]0, +\\infty[$ is given by $\\partial_{t}+\\mathcal{J}$.\\\\\n\nWe recall now some well-known properties of this operator. \n \n\\begin{Theoreme}\\label{fol} There exists a unique family of continuous linear operators $(H_{t})_{t>0}$ defined on $L^{1}+L^{\\infty}(\\mathbb{G})$ with the semi-group property $H_{t+s}=H_{t}H_{s}$ for all $t, s>0$ and $H_{0}=Id$, such that: \\\\  \n\\begin{enumerate} \n\\item[1)] the sub-Laplacian $\\mathcal{J}$ is the infinitesimal generator of the semi-group  $H_{t}=e^{-t\\mathcal{J}}$; \\\\  \n\\item[2)] $H_{t}$ is a contraction operator on $L^{p}(\\mathbb{G})$ for $1\\leq p\\leq +\\infty$ and for $t>0$; \\\\  \n\\item[3)] the semi-group $H_t$ admits a convolution kernel $H_{t}f=f\\ast h_{t}$ where $h_{t}(x)=h(x, t) \\in \\mathcal{C}^{\\infty}(\\mathbb{G}\\times]0, +\\infty[)$ is the heat kernel which satisfies the following points:  \n\\begin{enumerate} \n\\item $(\\partial_{t}+\\mathcal{J})h_{t}=0$ on $\\mathbb{G}\\times]0, +\\infty[$, \\\\ \n\\item $h(x, t)=h(x^{-1}, t)$, $h(x, t)\\geq 0$ and $\\int_{\\mathbb{G}}h(x, t)dx=1$, \\\\ \n\\item $h_{t}$ has the semi-group property:  $h_{t}\\ast h_{s}=h_{t+s}$ for $t, s>0$, \\\\ \n\\item $h(\\delta_{\\alpha}[x], \\alpha^{2}t)=\\alpha^{-N}h(x, t)$, \\\\ \n\\item For every $t>0$, $x\\mapsto h(x, t)$ belong to the Schwartz class in $\\mathbb{G}$. \\\\  \n\\end{enumerate} \n\\item[4)] $ \\|H_{t}f-f \\|_ {L^p}\\to 0$ if $t\\to 0$ for $f \\in L^{p}(\\mathbb{G})$ and $1\\leq p < +\\infty$; \\\\  \n\\item[5)] If $f\\in L^{p}(\\mathbb{G})$, $1\\leq p\\leq +\\infty$, then the function $u(x, t)=H_{t}f(x)\\in \\mathcal{C}^{\\infty}(\\mathbb{G}\\times \\mathbb{R}^{+})$ is a solution of the heat equation: \n$$ \\qquad\\left \\lbrace\\begin{array}{l}(\\frac{\\partial}{\\partial t}+\\mathcal{J})u(x, t)=0\\quad \\mbox{for } \\quad x\\in \\mathbb{G } \\quad \\mbox{and}\\quad t>0\\, ;\\\\[8mm] \nu(x, 0)=f(x)\\qquad\\quad \\mbox{for } \\quad x\\in \\mathbb{G }.\\\\  \n\\end{array}\\right. $$ \n\\end{enumerate} \n\\end{Theoreme} \n\nFor a detailed proof of these and other important facts concerning the heat semi-group see \\cite{Folland2} and \\cite{Saka}.\\\\\n\nTo close this section we recall the definition of the sub-Laplacian fractional powers $\\mathcal{J}^s$ with $s>0$.\\\\ We write:\n$$\\mathcal{J}^s f(x)=\\underset{\\varepsilon \\to 0}{\\lim}\\frac{1}{\\Gamma(k-s)}\\int_{\\varepsilon}^{+\\infty}t^{k-s-1}\\mathcal{J}^k H_tf(x)dt$$\nfor all $f\\in \\mathcal{C}^{\\infty}(\\mathbb{G})$ with $k$ the smallest integer greater than $s$.\n\n\\section{Maximal functions and Muckenhoupt weights}\\label{Herbata}\nThere are several ways of defining maximal functions in stratified Lie groups and our principal reference is \\cite{Folland2}. In this article we will mainly work with the following function  \n\\begin{Definition} Let $f\\in \\mathcal{S}'(\\mathbb{G})$ and $\\varphi\\in\\mathcal{S}(\\mathbb{G})$.  \nThe maximal function $\\mathcal{M}_{\\varphi}$ is given by the expression \n\\begin{equation*} \n\\mathcal{M}_{\\varphi}f(x)=\\underset{0<t<+\\infty}{\\sup}\\{|f\\ast\\varphi_{t}(x)|\\} \n\\end{equation*} with $\\varphi_{t}(x)=t^{-N\/2}\\varphi(t^{-1\/2}x)$.  \n\\end{Definition} \nThis definition still has a sense if $f$ and $\\varphi$ are two distributions such that $(x, t)\\longmapsto f\\ast\\varphi_{t}(x)$ is a continuous function on $\\mathbb{G}\\times]0, +\\infty[$:  for example, if $f\\in L^{p}(\\mathbb{G})$ and $\\varphi\\in L^{q}(\\mathbb{G})$ where $1\\leq p\\leq +\\infty$ and $1\/p+1\/q=1$. \\\\ \n\nAn important special case is given by the Hardy-Littlewood function which consists in taking as function $\\varphi$ the characteristic function of the unit ball:  \n$$\\mathcal{M}_{B}f(x)=\\underset{B \\ni x}{\\sup } \\;\\frac{1}{|B|}\\int_{B }|f(y)|dy.$$ \nThe next lemma explain the relationship between these maximal functions.\n\\begin{Lemme}\\label{Led11} \nLet $\\varphi$ a function on $\\mathbb{G}$ such that $|\\varphi(x)|\\leq C(1+|x|)^{-N-\\varepsilon}$ for some $\\varepsilon>0$, then\n\\begin{equation}\\label{Wilma} \n\\mathcal{M}_{\\varphi}f(x)\\leq C \\mathcal{M}_{B}f(x). \n\\end{equation} \n\\end{Lemme} \nWe will use this property in the sequel and we request the reader to consult the proof in \\cite{Folland2}. \\\\  \n\nThe reader can consult \\cite{Folland2}, \\cite{Grafakos} and \\cite{Garcia} for a more detailed study of these important functions. \nFor our part, we will be interested in the relationship existing between these functions and weights. A \\emph{weight} $\\omega$ is, in a very general way, a locally integrable function on $\\mathbb{G}$ with values in $]0,+\\infty[$.  \nFor a given weight $\\omega$ and a measurable set $E\\subset\\mathbb{G}$ we use the following notation:  \n\\begin{equation*} \n\\omega(E)=\\int_{E}\\omega(x)dx.  \n\\end{equation*} \nWe will define thus, for $1\\leq p<+\\infty$, weighted Lebesgue spaces by the norm \n\\begin{equation}\\label{pico1}\n \\|f\\|_ {L^{p}(\\omega)}=\\left(\\int_{\\mathbb{G}}|f(x)|^{p}\\omega(x)dx\\right)^{1\/p }\n\\end{equation}\nHistorically, the characterization of Muckenhoupt weights comes from the following problem: for a fixed $p\\in ]1, +\\infty[$ we want to know for which functions \n$\\omega$ one has the strong estimate\n\\begin{equation}\\label{MaximalProper}\n\\int_{\\mathbb{G}}\\mathcal{M}_{B}f(x)^{p}\\omega(x)dx\\leq C \\int_{\\mathbb{G}}|f(x)|^{p}\\omega(x)dx \\qquad (f\\in L^{p}(\\mathbb{G},\\omega)).  \n\\end{equation} \nIt follows the condition below and the next definition (see \\cite{Grafakos}): \n\\begin{equation}\\label{brown} \n\\underset{B}{\\sup}\\left(\\frac{1}{|B|}\\int_{B}\\omega(x)dx\\right)\\left(\\frac{1}{|B|}\\int_{B}\\omega(x)^{-\\frac{1}{p-1}}dx\\right)^{p-1}<+\\infty. \n\\end{equation} \n\n\\begin{Definition} \nLet $\\mathbb{G}$ a stratified Lie group and let $1<p<+\\infty$.  We will say that a weight $\\omega$ belongs to the Muckenhoupt class $A_{p}$ if it satisfies condition (\\ref{brown}). Moreover, we will define weights in the class $A_{1}$ by:  \n\\begin{equation}\\label{soledad} \n\\mathcal{M}_{B}\\;\\omega(x)\\leq C \\;\\omega(x) \\qquad (\\forall x\\in \\mathbb{G}).  \n\\end{equation} \n\\end{Definition}\n\nHere are some traditional examples:  the trivial weight $\\omega(x) \\equiv 1$ for all $x\\in \\mathbb{G}$ is a $A_p$ weight for $1\\leq p<+\\infty$ and the function $|x|^{\\alpha}$ is in $A_{p}$ if and only if  $-N<\\alpha<N(p-1)$, where $N$ is the homogeneous dimension. For $p=1$, the function $|x|^{\\alpha}$ belongs to $A_{1}$ if and only if $-N<\\alpha\\leq 0$. \\\\  \n\nLet us finally say that we have following inclusion:\n\\begin{Proposition}\\label{soledad1bis} \nIf $1< p<q<+\\infty$, then $A_{1}\\subset A_{p}\\subset A_{q}$.\n\\end{Proposition} \nWe request the reader to consult the proof of this result in \\cite{Folland2} or \\cite{Grafakos}.  \n\\section{Weigthed spaces}\\label{espa} \n\nWe give in this section the precise definition of weighted functional spaces involved in theorems \\ref{smile00}, \\ref{CHAME} and \\ref{PGLR}. In a general way, given a norm $\\|\\cdot\\|_{X(\\omega)}$, we will define the corresponding weighted functional space $X(\\mathbb{G}, \\omega)$ by $\\{f\\in \\mathcal{S}'(\\mathbb{G}): \\|f\\|_{X(\\omega)}<+\\infty\\}$ where $\\omega$ is a Muckenhoupt weight belonging to a certain class $A_p$.\\\\\n\n\\begin{enumerate}\n\\item[$\\bullet$] \\textbf{Lebesgue spaces} $L^p(\\mathbb{G}, \\omega)$. We have already considered how to define weigthed Lebesgue spaces with the formula (\\ref{pico1}). Let us notice that we also have a characterization with the distribution function:\n\\begin{equation*} \n\\|f\\|^{p}_{L^{p}(\\omega)}=\\int_{0}^{+\\infty}p\\sigma^{p-1 } \\omega(\\{x\\in \\mathbb{G}:|f(x)|> \\sigma\\})d\\sigma.  \n\\end{equation*} \n\\item[$\\bullet$] \\textbf{weak-$L^{p}$ spaces} or Lorentz spaces $L^{p,\\infty}(\\mathbb{G}, \\omega)$. We define them by\n\\begin{equation*}\n\\|f\\|_ { L^{p, \\infty}(\\omega)}=\\underset{\\sigma>0}{\\sup}\\{\\sigma \\; \\omega(\\{x\\in \\mathbb{G}:|f(x)|> \\sigma\\})^{1\/p}\\}.\n\\end{equation*} \n\\item[$\\bullet$]\\textbf{Sobolev spaces} $\\dot{W}^{s,p}(\\mathbb{G}, \\omega)$. For $\\omega\\in A_{p}$ we write:  \n\\begin{equation*} \n\\|f\\|_ {\\dot{W}^{s, p}(\\omega)} =\\|\\mathcal{J}^{s\/2}f\\|_{L^{p}(\\omega)} \\qquad (1<p<+\\infty) \n\\end{equation*}\nand when $p=s=1$ we will note\n\\begin{equation*}\n\\|f\\|_ {\\dot{W}^{1,1}(\\omega)} =\\|\\nabla f\\|_{L^{1}(\\omega)}.\n\\end{equation*} \n\\item[$\\bullet$]\\textbf{weak Sobolev spaces} $\\dot{W}^{s,p}_\\infty(\\mathbb{G}, \\omega)$: \n\\begin{equation} \\label{DefDebilSoboLe}\n\\|f\\|_ {\\dot{W}^{s, p}(\\omega)} =\\|\\mathcal{J}^{s\/2}f\\|_{L^{p,\\infty}(\\omega)} \\qquad (1<p<+\\infty) \n\\end{equation}\n\\item[$\\bullet$] \\textbf{Besov spaces} $\\dot{B}^{s,q}_p(\\mathbb{G}, \\omega)$. We define them in the following way: \n\\begin{equation*} \n\\|f\\|_{\\dot{B}^{s, q}_{p}(\\omega)}=\\left[\\int_{0}^{+\\infty}t^{(m-s\/2)q}\\left \\|\\frac{\\partial^{m}H_{t}f}{\\partial t^{m}}(\\cdot)\\right\\|^{q}_{L^{p}(\\omega)}\\frac{dt}{t}\n\\right]^{1\/q} \n\\end{equation*} \nfor $1\\leq p, q\\leq +\\infty, s>0$ and $m$ an integer such that $m>s\/2$. \\\\ \n \nFinally, for Besov spaces of indices $(-\\beta, \\infty, \\infty)$ which appear in all the Improved Sobolev inequalities we have:\n\\begin{equation}\\label{BesovChaleurGhomo}\n\\|f\\|_{\\dot{B}^{-\\beta,\\infty}_{\\infty}}=\\underset{t>0}{\\sup}\\;\\;t^{\\beta\/2 } \\|H_{t}f\\|_ {L^\\infty} \n\\end{equation} \n\\end{enumerate}\n\\section{Spectral resolution of the sub-Laplacian}\\label{spectre}\n\nThe use in this article of spectral resolution for the sub-Laplacian consists roughly in expressing this operator by the formula $\\mathcal{J}=\\int_{0}^{+\\infty}\\lambda \\;dE_{\\lambda}$ and, by means of this characterization, build a family of new operators $m(\\mathcal{J})$ associated to a Borel function $m$. This kind of operators have some nice properties as shown in the next propositions.\n\\begin{Proposition} \nIf $m$ is a bounded Borel function on $]0,+\\infty[$ then the operator $m(\\mathcal{J})$ fixed by\n\\begin{equation}\\label{little} \nm(\\mathcal{J})=\\int_{0}^{+\\infty}m(\\lambda) \\;dE_{\\lambda }, \n\\end{equation} \nis bounded on $L^{2}(\\mathbb{G})$  and admits a convolution kernel $M$ i.e.: $m(\\mathcal{J})(f)=f\\ast M \\qquad (\\forall f\\in L^{2}(\\mathbb{G}))$. \n\\end{Proposition} \nSee \\cite{Folland2} and the references given therein for a proof. For our purposes, it will be particularly interesting to combine this result with the structure of dilation:  \n\\begin{Lemme}\\label{madera} \nLet $m$ be a bounded function on $]0,+\\infty[$ and let $M$ be the kernel of the operator $m(\\mathcal{J})$.  \nThen, for all $t>0$ we can build a bounded operator on $L^{2}(\\mathbb{G})$ by writing $m_{t}(\\mathcal{J})=m(t\\mathcal{J})$ with an associated kernel given by \n\\begin{equation*} \nM_{t}(x)=t^{-N\/2}M(t^{-1\/2}x).  \n\\end{equation*} \n\\end{Lemme} \n\nFollowing \\cite{Hulanicki} and \\cite{Furioli2} we can improve the conclusion of the above proposition. Let $k\\in\\mathbb{N}$ and $m$ be a function of class $\\mathcal{C}^{k}(\\mathbb{R}^{+})$, we write \n\\begin{equation*} \n\\|m \\|_ {(k)}=\\underset{\\underset{\\lambda>0}{1\\leq r\\leq k}}{\\sup } (1+\\lambda)^{k }|m^{(r)}(\\lambda)|.\n\\end{equation*}\nThis formula gives us a necessary condition to obtain certain properties of the operators defined by (\\ref{little}):\n\\begin{Proposition}\\label{toge}\nLet $\\alpha\\in \\mathbb{N}$, $I=(i_{1},...,i_{n})$ be a multi-index and $p\\in [1, +\\infty]$.  \nThere is a constant $C>0$ and an integer $k$ such that, for any function $m\\in \\mathcal{C}^{k}(\\mathbb{R}^{+})$ with $ \\|m\\|_{(k)}<+\\infty$, the kernel $M_{t}$ associated to the operator $m(t\\mathcal{J})$, $t>0$, satisfies \n \\begin{equation*} \n\\|(1+|\\cdot|)^{\\alpha}X^{I}M_{t}(\\cdot)\\|_{L^p}\\leq C(1+\\sqrt{t})^{\\alpha}t^{-(\\frac{N}{2p'}+\\frac{d(I)}{2})}\\|m\\|_{(k)}. \n\\end{equation*} \nwhere $\\frac{1}{p}+\\frac{1}{p'}=1$.\n\\end{Proposition} \n\\begin{Corollaire}\\label{black} Let $t>0$.\n\\begin{enumerate}\n\\item[1)] Let $m$ be the restriction on $\\mathbb{R}^{+}$ of a function defined on $\\mathcal{S}(\\mathbb{R})$.  Then, the kernel $M$ \nof the operator $m(\\mathcal{J})$ is in $\\mathcal{S}(\\mathbb{G})$.  \n\\item[2)] If $m$ is as above; and if it is vanishing at all orders near of the origin, then the kernel $M$ belongs to the \nspace $\\mathcal{S}_{0}(\\mathbb{G})$ formed by the functions of the Schwartz class which every moment is null.  \n\\end{enumerate} \n\\end{Corollaire}\nFor more details and proofs see \\cite{Folland2}, \\cite{Furioli2} and \\cite{Hulanicki}.\n\\section{Improved Sobolev Inequalities on stratified groups: the proofs}\\label{demos} \n\nAs said in the introduction, inequalities given in theorem \\ref{smile00} depends on the theorem \\ref{CHAME}. We will thus begin proving this result in the following lines and we will continue our study by treating separately weak inequalities (\\ref{smile1}) and strong inequalities (\\ref{smile0}).\n\n\\subsection{The modified pseudo-inequality of Poincar\\'e}\\label{ISPGH} \n\nUnder the hypothesis of theorem \\ref{CHAME}, we have to prove the inequality  \n\\begin{equation*} \n\\|\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}f\\|_{L^{1}(\\omega)}\\leq C \\; t^{\\frac{1-s}{2}}\\|\\nabla f\\|_{ L^{1}(\\omega)}.  \n\\end{equation*} \nTo begin the proof, we observe that the following identity occurs: \n\\begin{equation*} \n(\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}f)(x)=\\left(\\int_{0}^{+\\infty}m(t\\lambda)dE_{\\lambda}\\right)t^{1-s\/2}\\mathcal{J} f(x), \n\\end{equation*} \nwhere we noted $m(\\lambda)=\\lambda^{s\/2-1}(1-e^{-\\lambda }$) for $\\lambda>0$, note that $m$ is a bounded function which tends to $0$ at infinity since $s\/2-1<0$. We break up this function by writing:  \n$$m(\\lambda)=m_{0}(\\lambda)+m_{1}(\\lambda)=m(\\lambda)\\theta_{0}(\\lambda)+m(\\lambda)\\theta_{1}(\\lambda)$$ \nwhere we chose the auxiliary functions $\\theta_{0}(\\lambda), \\theta_{1}(\\lambda)\\in \\mathcal{C}^{\\infty}(\\mathbb{R}^{+})$ defined by:  \n\\begin{eqnarray*} \n\\bullet\\quad \\theta_{0}(\\lambda) = 1 \\quad\\mbox{on } \\quad]0, 1\/2 ] \\quad\\mbox{and}\\quad 0 \\quad \\mbox{on}\\quad]1, +\\infty[, \\\\[5mm ] \n\\bullet\\quad \\theta_{1}(\\lambda) = 0 \\quad\\mbox{on} \\quad]0, 1\/2 ] \\quad\\mbox{and}\\quad 1 \\quad \\mbox{on }\\quad]1, +\\infty[, \n\\end{eqnarray*} \nso that $\\theta_{0}(\\lambda)+\\theta_{1}(\\lambda)\\equiv 1$. Then, we obtain the formula: \n\\begin{equation*} \n(\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}f)(x)=\\left(\\int_{0}^{+\\infty}m_{0}(t\\lambda)dE_{\\lambda}\\right)t^{1-s\/2}\\mathcal{J}f(x)+\n\\left(\\int_{0}^{+\\infty}m_{1}(t\\lambda)dE_{\\lambda}\\right)t^{1-s\/2}\\mathcal{J}f(x).  \n\\end{equation*} \nIf we note $M^{(i)}_{t}$ the kernel of the operator fixed by $\\int_{0}^{+\\infty}m_{i}(t\\lambda)dE_{\\lambda}$ for $i=0,1$, we have:  \n\\begin{equation*} \n(\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}f)(x)=t^{1-s\/2}\\mathcal{J}f\\ast M^{(0)}_{t}(x)+t^{1-s\/2}\\mathcal{J}f\\ast M^{(1)}_{t}(x).  \n\\end{equation*} \nWe now multiply the above equality by a weight $\\omega\\in A_1$ to obtain the inequality\n\\begin{equation}\\label{poids1} \n\\int_{\\mathbb{G}}\\left|\\mathcal{J}^{s\/2}f-H_{t}\\mathcal{J}^{s\/2}\\right|\\omega(x)dx \\leq \\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(0)}_{t}(x)\\right|\\omega(x)dx +\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(1)}_{t}(x)\\right|\\omega(x)dx.\n\\end{equation} \nWe will now estimate the right side of the above inequality by the two following propositions:  \n\\begin{Proposition}\\label{Sir} \nFor the first integral in the right-hand side of (\\ref{poids1}) we have the inequality:  \n\\begin{equation*} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(0)}_{t}(x)\\right|\\omega(x)dx\\leq Ct^{\\frac{1-s}{2}}\\|\\nabla f\\|_{L^{1}(\\omega)} \n\\end{equation*}\n\\end{Proposition} \n\\emph{\\textbf{Proof}}. The function $m_{0}$ is the restriction on $\\mathbb{R}^{+}$ of a function belonging to the Schwartz class. This function satisfies the assumptions of corollary \\ref{black} which we apply after having noticed the identity\n\\begin{equation*} \nI=\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(0)}_{t}(x)\\right|\\omega(x)dx=\n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\nabla f\\ast\\tilde{\\nabla}M^{(0)}_{t}(x)\\right|\\omega(x)dx \n\\end{equation*} \nwhere we noted $\\tilde{\\nabla}$ the gradient formed by vectors fields $(Y_{j})_{1\\leq j\\leq m}$. We have then\n\\begin{equation*} \nI\\leq \\int_{\\mathbb{G}}\\int_{\\mathbb{G}}t^{1-s\/2}|\\nabla f(y)||\\tilde{\\nabla} M^{(0)}_{t}(y^{-1}\\cdot x)|\\omega(x)dxdy.  \n\\end{equation*} \nBy corollary \\ref{black}, one has $M^{(0)}_{t}\\in \\mathcal{S}(\\mathbb{G})$ and, since $M_{t}^{(0)}(x)=t^{-N\/2}M^{(0)}(t^{-1\/2}x)$, we can write \n$$K_{t}(x)=t^{1\/2 }|\\tilde{\\nabla}M^{(0)}_{t}(x^{-1 })|\\in L^{1}(\\mathbb{G}).$$ \nOne obtains\n\\begin{equation*} \nI\\leq \\int_{\\mathbb{G}}\\int_{\\mathbb{G}}t^{\\frac{1-s}{2}}|\\nabla f(y)|\\, K_{t}(x\\cdot y^{-1})\\omega(x)dxdy=\\int_{\\mathbb{G}}t^{\\frac{1-s}{2}}|\\nabla f(y)|\\; \n\\omega\\ast K_{t}(y)dy.  \n\\end{equation*} \nBy definition of maximal functions and by the estimate (\\ref{Wilma}), we have the inequality:  \n$$\\underset{t>0}{\\sup } \\;\\omega\\ast K_{t}(y)\\leq C\\, \\left(\\mathcal{M}_{B}\\;\\omega\\right)(y), $$ \nhence,   \n\\begin{equation*}\n I\\leq Ct^{\\frac{1-s}{2}}\\int_{\\mathbb{G}}|\\nabla f(y)|\\mathcal{M}_{B}\\, \\omega(y)dy.  \n\\end{equation*} \nIt remains to notice that, by assumption, $\\omega \\in A_{1}$ if and only if $(\\mathcal{M}_{B}\\, \\omega)(\\cdot)\\leq C \\;\\omega(\\cdot).$ \nWe obtain then the desired estimation:  \n\\begin{equation*} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(0)}_{t}(x)\\right|\\omega(x)dx\\leq Ct^{\\frac{1-s}{2}}\\int_{\\mathbb{G}}|\\nabla f(y)|\\omega(y)dy. \n\\end{equation*} \n\\begin{flushright}{$\\blacksquare$}\\end{flushright}\n\\begin{Proposition}\\label{PropoChame2}\nFor the last integral of (\\ref{poids1}) we have the inequality\n\\begin{equation*} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(1)}_{t}(x)\\right|\\omega(x)dx\\leq Ct^{\\frac{1-s}{2}}\\|\\nabla f\\|_{L^{1}(\\omega)} \n\\end{equation*}\n\\end{Proposition} \n\\emph{\\textbf{Proof}}. Here, it is necessary to make an additional step. We cut out the function $m_{1}$ in the following way:  \n\\begin{equation*}\nm_{1}(\\lambda)=\\left(\\frac{1-e^{-\\lambda}}{\\lambda}\\right)\\theta_{1}(\\lambda)=m_{a}(\\lambda)-m_{b}(\\lambda) \n\\end{equation*} \nwhere $m_{a}(\\lambda)=\\frac{1}{\\lambda}\\theta_{1}(\\lambda)$ and $m_{b}(\\lambda)=\\frac{e^{-\\lambda}}{\\lambda}\\theta_{1}(\\lambda)$. We will note $M^{(a)}_{t}$ and $M^{(b)}_{t}$ the associated kernels of these two operators. We obtain thus the estimate \n\\begin{equation}\\label{Rolling1} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(1)}_{t}(x)\\right|\\omega(x)dx\\leq \\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(a)}_{t}(x)\\right|\\omega(x)dx+ \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(b)}_{t}(x)\\right|\\omega(x)dx \n\\end{equation} \nObserve that $m_{b}\\in \\mathcal{S}(\\mathbb{R}^{+})$ and then $M^{(b)}_{t}\\in \\mathcal{S}(\\mathbb{G})$.  We have the next lemma for the last integral in (\\ref{Rolling1}).\n\\begin{Lemme}\\label{BishopLem0}\n\\begin{equation*} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(b)}_{t}(x)\\right|\\omega(x)dx\\leq Ct^{\\frac{1-s}{2}}\\|\\nabla f\\|_{L^{1}(\\omega)}.\n\\end{equation*} \n\\end{Lemme} \n\\emph{\\textbf{Proof}}. The proof is straightforward and follows the same steps as those of the preceding proposition \\ref{Sir}.  \n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \nWe treat the other part of (\\ref{Rolling1}) with the following lemma:  \n\\begin{Lemme} \\label{BishopLem}\n\\begin{equation}\\label{Bishop} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(a)}_{t}(x)\\right|\\omega(x)dx\\leq Ct^{\\frac{1-s}{2}}\\|\\nabla f\\|_{L^{1}(\\omega)} \n\\end{equation} \\end{Lemme} \n\\emph{\\textbf{Proof}}. We consider the auxiliary function \n$$\\psi(\\lambda)=\\theta_{0}(\\lambda\/2)-\\theta_{0}(\\lambda)=\\theta_{1}(\\lambda)-\\theta_{1}(\\lambda\/2)$$ \nin order to obtain the identity $$\\sum_{j=0}^{+\\infty}\\psi(2^{-j}\\lambda)=\\theta_{1}(\\lambda).$$  \nWe have then \n$$m_{a}(t\\lambda)=\\frac{1}{t\\lambda}\\sum_{j=0}^{+\\infty}\\psi(2^{-j}t\\lambda)=\\sum_{j=0}^{+\\infty}2^{-j}\\tilde{\\psi}(2^{-j}t\\lambda)$$ \nwhere $\\tilde{\\psi}(\\lambda)=\\frac{\\psi(\\lambda)}{\\lambda}$ is a function in $\\mathcal{C}^{\\infty}_{0}(\\mathbb{R}^{+})$. By corollary \\ref{black}, the kernel $\\tilde{K}$ associated with the function $\\tilde{\\psi}$ belongs to $\\mathcal{S}_{0}(\\mathbb{G})$. Then, from the point of view of operators, one has:  \n\\begin{equation}\\label{Stone} \nM^{(a)}_{t}(x)=\\sum^{+\\infty}_{j=0}2^{-j}\\tilde{K}_{j, t}(x) \n\\end{equation} \nwhere $\\tilde{K}_{j,t}(x)=2^{N\/2}t^{-N\/2}\\tilde{K}(2^{j\/2}t^{-1\/2}x)$. With formula (\\ref{Stone}) we return to the left side of (\\ref{Bishop}):  \n$$\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(a)}_{t}(x)\\right|\\omega(x)dx\\leq\\sum^{+\\infty}_{j=0}2^{-j}\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast \n\\tilde{K}_{j,t}(x)\\right|\\omega(x)dx.$$\nUsing the sub-Laplacian definition and vector fields properties, we have\n\\begin{equation}\\label{Chicago}\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(a)}_{t}(x)\\right|\\omega(x)dx\\leq \\sum^{+\\infty}_{j=0}2^{-j}t^{1-s\/2}\\int_{\\mathbb{G}}\n\\int_{\\mathbb{G } }|\\nabla f(y)| |\\tilde{\\nabla}\\tilde{K}_{j, t}(y^{-1}\\cdot x)|\\omega(x)dxdy.\n\\end{equation} \nWe note this time $K_{j, t}(x)=2^{-j\/2}t^{1\/2}|\\tilde{\\nabla}\\tilde{K}_{j, k}(x^{-1})|$ to obtain the following formula for the right side of (\\ref{Chicago}):\n$$\\sum^{+\\infty}_{j=0}2^{-j\/2}t^{\\frac{1-s}{2}}\\int_{\\mathbb{G}}\\int_{\\mathbb{G}}|\\nabla f(y)|K_{j,t}(x\\cdot y^{-1})\\omega(x)dxdy = \n\\sum^{+\\infty}_{j=0}2^{-j\/2}t^{\\frac{1-s}{2}}\\int_{\\mathbb{G}}|\\nabla f(y)|\\;\\omega\\ast K_{j, t}(y)dy.$$ \nIt remains to apply the same arguments used in proposition \\ref{Sir}, namely the assumption $\\omega \\in A_{1}$ and, for $K_{j,t}$, the estimations $\\underset{j, t>0}{\\sup } \\;\\omega \\ast K_{j, t}(y)\\leq C\\, (\\mathcal{M}_{B } \\; \\omega)(y)\\leq C\\, \\omega(y).$ \nThen, we finally get the inequality  \n\\begin{equation*} \n\\int_{\\mathbb{G}}\\left|t^{1-s\/2}\\mathcal{J}f\\ast M^{(a)}_{t}(x)\\right|\\omega(x)dx\\leq C\\, t^{\\frac{1-s}{2}}\\sum_{j=0}^{+\\infty}2^{-j\/2}\\int_{\\mathbb{G}}|\\nabla f|\n(y)\\omega(y)dy = C\\, t^{\\frac{1-s}{2} } \\|\\nabla f\\|_{L^{1}(\\omega)}.  \n\\end{equation*} \nWhich ends the proof of the lemma \\ref{BishopLem}.\n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \nWith these two last lemmas we conclude the proof of the proposition \\ref{PropoChame2}. Now, getting back to the formula (\\ref{poids1}), with propositions \\ref{Sir} and \\ref{PropoChame2} we finally finish the proof of theorem \\ref{CHAME}.\n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \n\\subsection{Weak inequalities}\n\nTo begin the proof notice that operator $\\mathcal{J}^{s\/2}$ carries out an isomorphism between the spaces $\\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$ and $\\dot{B}^{-\\beta-s, \\infty}_{\\infty}(\\mathbb{G})$ (see \\cite{Saka}). Thus inequality (\\ref{smile1}) rewrites as:  \n\\begin{equation}\\label{ForIsoLap}\n\\|\\mathcal{J}^{s\/2}f\\|_{L^{q, \\infty}(\\omega)}\\leq C \\|\\nabla f\\|_{L^{1}(\\omega)}^{\\theta}\\|\\mathcal{J}^{s\/2}f\\|_{\\dot{B}^{-\\beta-s, \\infty}_{\\infty}}^{1-\\theta} \n\\end{equation} \nBy homogeneity, we can suppose that the norm $\\|\\mathcal{J}^{s\/2}f \\|_{\\dot{B}^{-\\beta-s,\\infty}_{\\infty}}$ is bounded by $1$; then we have to show \n\\begin{equation}\\label{CHAMEFAIBLE2} \n\\|\\mathcal{J}^{s\/2}f \\|_{L^{q, \\infty}(\\omega)}\\leq C \\|\\nabla f \\|_{L^{1}(\\omega)}^{\\theta}. \n\\end{equation} \nWe have thus to evaluate the expression $\\omega\\left(\\{x\\in \\mathbb{G}:|\\mathcal{J}^{s\/2}f(x)|> 2\\alpha\\}\\right)$ for all $\\alpha>0$. If we use the thermic definition of the Besov space (\\ref{BesovChaleurGhomo}), we have \n$$\\|\\mathcal{J}^{s\/2}f\\|_{\\dot{B}^{-\\beta-s,\\infty}_{\\infty}}\\leq 1 \\iff \\underset{t>0}{\\sup}\\left\\{t^{\\frac{\\beta+s}{2}}\\|H_{t}\\mathcal{J}^{s\/2}f\\|_{L^\\infty}\\right\\} \\leq 1.$$ \nBut, if one fixes $t_{\\alpha}=\\alpha^{-\\left(\\frac{2}{\\beta+s}\\right)}$, we obtain $\\|H_{t_{\\alpha}}\\mathcal{J}^{s\/2}f \\|_{L^\\infty}\\leq \\alpha $. \nNote also that with the definition of parameter $\\beta$ one has $t_{\\alpha}=\\alpha^{-\\frac{2(q-1)}{(1-s)}}$. Therefore, since we have the following set inclusion\n$$\\left\\{x\\in \\mathbb{G}: |\\mathcal{J}^{s\/2}f(x)|> 2\\alpha\\right\\}\\subset \\left\\{x\\in \\mathbb{G}: |\\mathcal{J}^{s\/2}f(x)-H_{t_{\\alpha}}\\mathcal{J}^{s\/2}f(x)|> \\alpha\\right\\}, $$ \nthe Tchebytchev inequality implies \n\\begin{equation*} \n\\alpha^{q}\\omega\\left(\\{x\\in \\mathbb{G}: |\\mathcal{J}^{s\/2}f(x)|> 2\\alpha\\}\\right)\\leq \\alpha^{q-1}\\int_{\\mathbb{G } }|\\mathcal{J}^{s\/2}f(x)-H_{t_{\\alpha}}\\mathcal{J}^{s\/2}f(x)|\\omega(x)dx.  \n\\end{equation*} \nAt this point, we use the modified Poincar\\'e pseudo-inequality, given by theorem \\ref{CHAME}, to estimate the right side of the preceding inequality:  \n\\begin{equation}\\label{T2g} \n\\alpha^{q}\\omega\\left(\\{x\\in \\mathbb{G}:|\\mathcal{J}^{s\/2}f(x)|> 2\\alpha\\}\\right)\\leq C \\alpha^{q-1}\\;t_{\\alpha}^{\\frac{1-s}{2}}\\int_{\\mathbb{G}}|\\nabla f(x)|\\omega(x)dx.  \n\\end{equation} \nBut, by the choice of $t_{\\alpha}$, one has $\\alpha^{q-1}\\alpha^{-\\frac{2(q-1)}{(1-s)}\\frac{(1-s)}{2}}=1 $. Then (\\ref{T2g}) implies the inequality \n$$\\qquad\\alpha^{q}\\omega\\left(\\{x\\in \\mathbb{G}:|\\mathcal{J}^{s\/2}f(x)|> 2\\alpha\\}\\right)\\leq C \\|\\nabla f \\|_ { L^{1}(\\omega) } \\;; $$  \nand, finally, using definition (\\ref{DefDebilSoboLe}) of weak Sobolev spaces it comes \n$$\\qquad \\qquad \\|\\mathcal{J}^{s\/2}f \\|^{q}_{L^{q, \\infty}(\\omega)}\\leq  C \\|\\nabla f \\|_{L^{1}(\\omega)}$$ \nwhich is the desired result.\\begin{flushright}{$\\blacksquare$}\\end{flushright} \n\\subsection{Strong inequalities}\nWhen $s=0$ in the weak inequalities it is possible to obtain stronger estimations. To achieve this, we will need an intermediate step:  \n\\begin{Proposition}\\label{escalera2} \nLet $1<q<+\\infty$, $\\theta=\\frac{1}{q}$ and $\\beta=\\theta\/(1-\\theta)$.  Then we have\n$$\\|f\\|_ { L^{q}(\\omega)}\\leq C \\|\\nabla f\\|_{L^{1}(\\omega)}^{\\theta } \\|f\\|^{1-\\theta}_{\\dot{B}^{-\\beta, \\infty}_{\\infty}}$$ \nwhen the three norms in this inequality are bounded.  \n\\end{Proposition} \n\\textit{\\textbf{Proof}.}  We will follow closely \\cite{Ledoux}. Just as in the preceding theorem, we will start by supposing that $\\|f\\|_{\\dot{B}^{-\\beta, \\infty}_{\\infty}}\\leq 1$.  Thus, we must show the estimate \n\\begin{equation}\\label{ForteG} \n\\|f\\|_{L^{q}(\\omega)}\\leq C \\|\\nabla f\\|_{L^{1}(\\omega)}^{\\theta}. \n\\end{equation} \nLet us fix $t$ in the following way:  $t_{\\alpha}=\\alpha^{-2(q-1)\/q}$ where $\\alpha>0$.  We have then, by the thermic definition of Besov spaces, the estimate $\\|H_{t}f \\|_{L^\\infty}\\leq \\alpha$. We use now the characterization of Lebesgue space given by the distribution function:  \n\\begin{equation}\\label{formulaG} \n\\frac{1}{5^{q }} \\|f\\|_{L^{q}(\\omega)}^{q}=\\int_{0}^{+\\infty}\\omega\\left(\\{x\\in \\mathbb{G}: |f(x)|> 5\\alpha\\}\\right)d(\\alpha^{q}).  \n\\end{equation} \nIt now remains to estimate $\\omega(\\{x\\in \\mathbb{G}: |f(x)|> 5\\alpha\\})$ and for this we introduce the following thresholding function:  \n\\begin{equation*}\\label{teta}\n\\Theta_{\\alpha}(t)=\\left\\lbrace\\begin{array}{l} \\Theta_{\\alpha}(-t)=-\\Theta_{\\alpha}(t)\\\\[4mm] \n0 \\qquad \\qquad\\qquad\\mbox{ if }\\qquad 0\\leq T \\leq \\alpha\\\\[4mm]\nt-\\alpha \\qquad\\qquad\\mbox{ if } \\qquad\\alpha\\leq T \\leq M\\alpha \\\\[4mm]\n(M-1)\\alpha \\qquad\\mbox{ if } \\qquad T > M\\alpha \n\\end{array}\\right.\\\\[3mm ] \n\\end{equation*}\nHere, $M$ is a parameter which depends on $q$ and which we will suppose for the moment larger than 10.  \\\\\n\nThis cut-off function enables us to define a new function $f_{\\alpha}=\\Theta_{\\alpha}(f)$. We write in the next lemma some significant properties of this function $f_{\\alpha}$: \n\\begin{Lemme}\\label{Taboo} \n\\begin{enumerate} \n\\item[]\n\\item[1)] the set defined by $\\{x\\in \\mathbb{G}: |f(x)|> 5\\alpha\\}$ is included in the set $\\{x\\in \\mathbb{G}: |f_{\\alpha }(x)|> 4\\alpha\\}$. \\\\  \n\\item[2)] On the set $\\{x\\in \\mathbb{G}: |f(x)|\\leq M\\alpha\\}$ one has the estimate $|f-f_{\\alpha }|\\leq \\alpha$. \\\\  \n\\item[3)] If $f\\in \\mathcal{C}^{1}(\\mathbb{G})$, one has the equality $\\nabla f_{\\alpha}=(\\nabla f)\\mathds{1}_{\\{\\alpha\\leq|f|\\leq M\\alpha\\}}$ \nalmost everywhere.  \n\\end{enumerate} \n\\end{Lemme} \nFor a proof see \\cite{Ledoux}. \\\\  \n\nLet us return now to (\\ref{formulaG}). By the first point of the lemma above we have\n\\begin{equation}\\label{cray} \n\\int_{0}^{+\\infty}\\omega\\left(\\left\\{x\\in \\mathbb{G}: |f(x)|> 5\\alpha\\right\\}\\right)d(\\alpha^{q})\\leq \\int_{0}^{+\\infty } \\omega\\left(\\{x\\in \\mathbb{G}: |f_{\\alpha }(x)|> 4\\alpha\\}\\right)d(\\alpha^{q})=I. \n\\end{equation} \nWe note $A_{\\alpha}=\\{x\\in \\mathbb{G}: |f_{\\alpha }(x)|> 4\\alpha\\}$, $B_{\\alpha}=\\{x\\in \\mathbb{G}: |f_{\\alpha}(x)-H_{t_{\\alpha}}(f_{\\alpha})(x)|> \\alpha\\}$ and $C_{\\alpha}=\\{x\\in \\mathbb{G}: |H_{t_{\\alpha}}(f_{\\alpha}-f)(x)|> 2\\alpha\\}$. Now, by linearity of $H_{t}$ we can write: $f_{\\alpha}=f_{\\alpha}-H_{t_{\\alpha}}(f_{\\alpha})+H_{t_{\\alpha}}(f_{\\alpha}-f)+H_{t_{\\alpha}}(f)$. Then, holding in account the fact $\\|H_{t}f\\|_{L^\\infty}\\leq \\alpha$, we obtain $A_{\\alpha}\\subset B_{\\alpha}\\cup C_{\\alpha}$. Returning to (\\ref{cray}), this set inclusion gives us the following inequality \n\\begin{equation}\\label{dosG} \nI\\leq \\int_{0}^{+\\infty}\\omega\\left(B_{\\alpha}\\right)d(\\alpha^{q})+\\int_{0}^{+\\infty}\\omega\\left(C_{ \\alpha}\\right)d(\\alpha^{q }) \n\\end{equation} \nWe will study and estimate these two integrals, which we will call $I_{1}$ and $I_{2}$ respectively, by the two following lemmas:  \n\\begin{Lemme} For the first integral of (\\ref{dosG}) we have the estimate:  \n\\begin{equation}\\label{ARG} \nI_{1 } = \\int_{0}^{+\\infty}\\omega\\left(B_{\\alpha}\\right)d(\\alpha^{q})\\leq C\\, q\\log(M) \\|\\nabla f\\|_ { L^{1}(\\omega) } \n\\end{equation} \n\\end{Lemme} \n\\textbf{\\emph{Proof}.} Tchebytchev's inequality implies \n\\begin{equation*} \n\\omega\\left(B_{\\alpha}\\right)\\leq \\alpha^{-1}\\int_{\\mathbb{G } }|f_{\\alpha}(x)-H_{t_{\\alpha}}(f_{\\alpha })(x)|\\omega(x)dx.  \n\\end{equation*} \nUsing the modified Poincar\\'e pseudo-inequality (\\ref{CHAME1}) with $s=0$ in the above integral we obtain:  \n$$\\omega\\left(B_{\\alpha}\\right)\\leq C \\, \\alpha^{-1 } \\, t_{\\alpha}^{1\/2}\\int_{\\mathbb{G } }|\\nabla f_{\\alpha }(x)|\\omega(x)dx$$ \nRemark that the choice of $t_{\\alpha}$ fixed before gives $t_{\\alpha}^{1\/2}=\\alpha^{1-q}$, then we have\n$$\\omega\\left(B_{\\alpha}\\right)\\leq C \\, \\alpha^{-q}\\int_{\\{\\alpha\\leq|f|\\leq M\\alpha \\} }|\\nabla f(x)|\\omega(x)dx.$$ \nWe integrate now the preceding expression with respect to $d(\\alpha^{q})$:  \n$$I_{1}\\leq C\\int_{0}^{+\\infty}\\alpha^{-q}\\left(\\int_{\\{\\alpha\\leq|f|\\leq M\\alpha \\} }|\\nabla f(x)|\\omega(x)dx\\right)d(\\alpha^{q }) = C\\;q\\int_{\\mathbb{G } }|\\nabla f(x)|\\left(\\int_{\\frac{|f|}{M}}^{|f|}\\frac{d\\alpha}{\\alpha}\\right)\\omega(x)dx$$ \nIt follows then $I_{1}\\leq C\\, q\\, \\log(M) \\|\\nabla f \\|_ { L^{1}(\\omega)}$ and one obtains the estimation needed for the first integral.\n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \n\\begin{Lemme}\\label{papas}\nFor the second integral of (\\ref{dosG}) one has the following result:  \n\\begin{equation*}\nI_{2}=\\int_{0}^{+\\infty}\\omega(C_{\\alpha})d(\\alpha^{q})\\leq \\frac{q}{q-1}\\;\\frac{1}{M^{q-1 } } \\|f\\|_ {L^q}^{q } \n\\end{equation*} \n\\end{Lemme} \n\\textbf{\\emph{Proof}.} For the proof of this lemma, we write:  $$|f-f_{\\alpha }|=|f-f_{\\alpha }|\\mathds{1}_{\\{|f|\\leq M \\alpha\\}}+|f-f_{\\alpha }|\\mathds{1}_{\\{|f|> M\\alpha\\}}.$$ \nAs the distance between $f$ and $f_{\\alpha}$ is lower than $\\alpha$ on the set $ \\{x\\in \\mathbb{G}: |f(x)|\\leq M \\alpha\\}$, one has the inequality \n$$|f-f_{\\alpha }|\\leq\\alpha+|f|\\mathds{1}_{\\{|f|> M \\alpha\\}}$$ \nBy applying the heat semi-group to both sides of this inequality we obtain $H_{t_{\\alpha}}(|f-f_{\\alpha }|)\\leq \\alpha + H_{t_{\\alpha}}(|f|\\mathds{1}_{\\{|f|> M \\alpha\\}})$ and we have then the following set inclusion $C_{\\alpha}\\subset \\left\\{x\\in \\mathbb{G}: H_{t_{\\alpha}}(|f|\\mathds{1}_{\\{|f|> M \\alpha\\}})>\\alpha\\right\\}$.\nThus, considering the measure of these sets and integrating with respect to $d(\\alpha^{q})$, it comes \n\\begin{equation*} \nI_{2}=\\int_{0}^{+\\infty}\\omega\\left(C_{\\alpha}\\right)d(\\alpha^{q})\\leq\\int_{0}^{+\\infty}\\omega\\bigg(\\{H_{t_{\\alpha}}(|f|\\mathds{1}_{\\{|f|> M\\alpha\\}})>\\alpha\\}\\bigg)d(\\alpha^{q}) \n\\end{equation*} \nWe obtain now, by applying Tchebytchev inequality, the estimate \n$$I_{2}\\leq\\int_{0}^{+\\infty}\\alpha^{-1}\\bigg(\\int_{\\mathbb{G}}H_{t_{\\alpha}}\\left(|f|\\mathds{1}_{\\{|f|> M\\alpha\\}}\\right)\\omega(x)dx\\bigg)d(\\alpha^{q}),$$ \nthen by Fubini's theorem we have\n$$I_{2}\\leq q\\int_{\\mathbb{G}}|f(x)|\\bigg(\\int_{0}^{+\\infty}\\mathds{1}_{\\{|f|> M\\alpha\\}}\\alpha^{q-2}d\\alpha\\bigg)\\omega(x)dx=\\frac{q}{q-1}\\int_{\\mathbb{G}}|f(x)|\\frac{|f(x)|^{q-1}}{M^{q-1}}\\omega(x)dx= \\frac{q}{q-1}\\frac{1}{M^{q-1 }}\\|f\\|_{L^{q}(\\omega)}^{q}.$$ \nAnd this concludes the proof of this lemma.\n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \nWe finish the proof of proposition \\ref{escalera2} by connecting together these two lemmas \\textit{i.e.}:\n$$\\frac{1}{5^{q}} \\|f\\|_ {L^q(\\omega)}^{q}\\leq Cq\\, \\log(M) \\|\\nabla f\\|_ {L^1(\\omega)}+\\frac{q}{q-1}\\frac{1}{M^{q-1 } } \\|f\\|_{L^q(\\omega)}^{q}$$ \nSince we supposed all the norms bounded and  $M\\gg 1$, we finally have\n$$\\left(\\frac{1}{5^{q}}-\\frac{q}{q-1}\\frac{1}{M^{q-1}}\\right) \\|f\\|_ {L^q(\\omega)}^{q}\\leq C q\\, \\log(M) \\|\\nabla f\\|_{L^1(\\omega)}$$ \n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \nThe proof of the theorem \\ref{smile00} is not yet completely finished. The last step is provided by the \n\\begin{Proposition}\\label{escalera3} \nIn proposition \\ref{escalera2} it is possible to consider only the two assumptions \n$\\nabla f\\in L^{1}(\\mathbb{G},\\omega)$ and $f\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$.  \n\\end{Proposition}\n\\textbf{\\emph{Proof}.} For the proof of this proposition we will build a homogeneous Littlewood-Paley like approximation of $f$ writing:  \n\\begin{equation*} \nf_{j}=\\left(\\int_{0}^{+\\infty}\\left(\\varphi(2^{-2j}\\lambda)-\\varphi(2^{2j}\\lambda)\\right)dE_{\\lambda}\\right)(f) \\qquad (j\\in \\mathbb{N})\n\\end{equation*} \nwhere $\\varphi$ is a $\\mathcal{C}^{\\infty}(\\mathbb{R}^+)$ function such that $\\varphi=1$ on $]0,1\/4[$ and $\\varphi=0$ on $[1, +\\infty[$.\n\\begin{Lemme} \nIf $q>1$, if $\\nabla f\\in L^{1}(\\mathbb{G}, \\omega)$ and if $f\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$ then $\\nabla f_j\\in L^{1}(\\mathbb{G}, \\omega)$, $f_j\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$ and $f_{j}\\in L^{q}(\\mathbb{G},\\omega)$.  \n\\end{Lemme} \n\\textbf{\\emph{Proof}.} The fact that $\\nabla f_j\\in L^{1}(\\mathbb{G}, \\omega)$ and $f_j\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$ is an easy consequence of the definition of $f_j$. For $f_{j}\\in L^{q}(\\mathbb{G},\\omega)$ the starting point is given by the relation:  \n\\begin{equation*}\nf_{j}=\\left(\\int_{0}^{+\\infty}m(2^{-2j}\\lambda)\\, dE_{\\lambda}\\right)2^{-2j}\\mathcal{J}(f), \n\\end{equation*} \nwhere we noted $$m(2^{-2j}\\lambda)=\\frac{\\varphi(2^{-2j}\\lambda)-\\varphi(2^{2j}\\lambda)}{2^{-2j}\\lambda}.$$ \nObserve that the function $m$ vanishes near of the origin and satisfies the assumptions of corollary \\ref{black}. We obtain then the following identity where $M_{j}\\in \\mathcal{S}(\\mathbb{G})$ is the kernel of the operator $m(2^{-2j}\\mathcal{J})$:  \n$$f_{j}=2^{-2j}\\mathcal{J}f\\ast M_{j}=2^{-2j}\\nabla f\\ast \\tilde{\\nabla}M_{j},$$ \nwhere we denoted $\\tilde{\\nabla}$ the gradient of the right invariant vectors fields and used the property (\\ref{izder}). Let us now calculate the norm $L^{q}(\\mathbb{G},\\omega)$ in the preceding identity:\n$$\\|f_{j}\\|_{L^{q}(\\omega)}=\\|2^{-2j}\\nabla f\\ast\\tilde{\\nabla}M_{j}\\|_{L^{q}(\\omega)}\\leq 2^{-2j}\\|\\nabla f\\|_{L^{1}(\\omega)}\\|\\tilde{\\nabla}M_{j}\\|_{L^{q}(\\omega)}.$$ \nFinally, we obtain:  \n$$\\|f_{j } \\|_ { L^{q}(\\omega)}\\leq C\\, 2^{j(N(1-\\frac{1}{q})-1) } \\|\\nabla f \\|_ { L^{1}(\\omega)}<+\\infty$$ \n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \n\nThanks to this estimate, we can apply the proposition \\ref{escalera2} to $f_{j}$ whose $L^{q}(\\mathbb{G}, \\omega)$ norm is bounded, and we obtain:  \n$$\\|f_{j } \\|_ { L^{q}(\\omega)}\\leq C \\|\\nabla f_j\\|_ { L^{1}(\\omega)}^{\\theta } \\|f_j\\|_ { \\dot{B}^{-\\beta, \\infty}_{\\infty}}^{1-\\theta}.$$ \nNow, since $f\\in \\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$, we have $f_{j } \\rightharpoonup f$ in the sense of distributions. It follows  \n$$\\|f\\|_{ L^{q}(\\omega)}\\leq \\underset{j \\to +\\infty}{\\lim \\inf } \\|f_{j } \\|_ { L^{q}(\\omega)}\\leq C \\|\\nabla f\\|_ { L^{1}(\\omega)}^{\\theta }\n \\|f\\|_ { \\dot{B}^{-\\beta, \\infty}_{\\infty}}^{1-\\theta}.$$ \nWe restricted ourselves to the two initial assumptions, namely $\\nabla f\\in L^{1}(\\mathbb{G}, \\omega)$ and $f\\in\\dot{B}^{-\\beta, \\infty}_{\\infty}(\\mathbb{G})$. The strong inequalities (\\ref{smile0}) are now completely proved for stratified groups.  \n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \n\n\\subsection{Maximal function and Improved Sobolev inequalities}\nWe will study now theorem \\ref{PGLR}. Just as for weak inequalities (\\ref{ForIsoLap}), we can rewrite (\\ref{PGLR1}) in the following way \n\\begin{equation*}\n\\| \\mathcal{J}^{\\frac{s-s_1}{2}}f\\|_{L^q(\\omega)}\\leq C \\|f\\|_{L^p(\\omega)}^\\theta\\|f\\|_{\\dot{B}^{-\\beta-s_1,\\infty}_{\\infty}}^{1-\\theta}\n\\end{equation*}\nwhere $1<p<q<+\\infty$, $\\theta=p\/q$, $s=\\theta s_1 -(1-\\theta)\\beta$ and $-\\beta<s<s_1$. Using the sub-Laplacian fractional powers characterization  we have the identity\n\\begin{equation}\\label{Kashmor}\n\\mathcal{J}^{\\frac{-\\alpha}{2}}f(x)=\\frac{1}{\\Gamma(\\frac{\\alpha}{2})}\\int_{0}^{+\\infty}t^{\\frac{\\alpha}{2}-1}H_t f(x)dt=\\frac{1}{\\Gamma(\\frac{\\alpha}{2})}\\left(\\int_{0}^{T}t^{\\frac{\\alpha}{2}-1}H_t f(x)dt+\\int_{T}^{+\\infty}t^{\\frac{\\alpha}{2}-1}H_t f(x)dt\\right)\n\\end{equation}\nwhere $\\alpha=s_1-s>0$ and $T$ will be fixed in the sequel.\\\\\n\nFor studying each one of these integrals we will use the estimates\n\\begin{enumerate}\n\\item[$\\bullet$] $|H_tf(x)|\\leq C \\mathcal{M}_B f(x)$ \\qquad\\qquad\\qquad\\;(by lemma \\ref{Led11})\\\\\n\n\\item[$\\bullet$] $|H_tf(x)|\\leq C t^{\\frac{-\\beta-s_1}{2}}\\|f\\|_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}$ \\qquad (by the thermic definition of Besov spaces (\\ref{BesovChaleurGhomo}))\\\\\n\\end{enumerate}\nThen, applying these inequalities in (\\ref{Kashmor}) we obtain\n$$|\\mathcal{J}^{\\frac{-\\alpha}{2}}f(x)|\\leq \\frac{c_1}{\\Gamma(\\frac{\\alpha}{2})}T^{\\frac{\\alpha}{2}} \\mathcal{M}_B f(x)+ \\frac{c_2}{\\Gamma(\\frac{\\alpha}{2})}T^{\\frac{\\alpha-\\beta-s}{2}}\\|f\\|_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}.$$\nWe fix now $$T=\\left(\\frac{\\|f\\|_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}}{\\mathcal{M}_B f(x)}\\right)^{ \\frac{2}{\\beta+s_1}}$$\nand we get \n$$|\\mathcal{J}^{\\frac{-\\alpha}{2}}f(x)|\\leq \\frac{c_1}{\\Gamma(\\frac{\\alpha}{2})}\\mathcal{M}_B f(x)^{1-\\frac{\\alpha}{\\beta+s_1}}+ \\frac{c_2}{\\Gamma(\\frac{\\alpha}{2})}\\mathcal{M}_B f(x)^{1-\\frac{\\alpha}{\\beta+s_1}}\\|f\\|^{\\frac{\\alpha}{\\beta+s_1}}_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}.$$\nSince $\\frac{\\alpha}{\\beta+s_1}=1-\\theta$, we have\n$$|\\mathcal{J}^{\\frac{-\\alpha}{2}}f(x)|\\leq \\frac{c}{\\Gamma(\\frac{\\alpha}{2})}\\mathcal{M}_B f(x)^{\\theta}\\|f\\|^{1-\\theta}_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}.$$\nMultiplying this inequality by a $A_p$ weight $\\omega$, using the fact that $A_p\\subset A_q$ if $p<q$, and the property of maximal function (\\ref{MaximalProper}) we obtain\n$$\\|\\mathcal{J}^{\\frac{-\\alpha}{2}}f(x)\\|_{L^q(\\omega)}\\leq c\\|f\\|_{L^p(\\omega)}^{\\theta}\\|f\\|^{1-\\theta}_{\\dot{B}^{-\\beta-s_1, \\infty}_{\\infty}}$$\nand we are done. \n\\begin{flushright}{$\\blacksquare$}\\end{flushright} \nIt is worth noting the simplicity of this proof: the arguments used are classical tools from harmonic analysis. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nRecent experiments have shown that it is possible\nto study unitary nonequilibrium dynamics\nin quantum systems on long time scales \\cite{experiments}.\nOne important result of these investigations has been the\nobserved lack of thermalization.\nThe reason for this non-thermal behaviour is\nattributed to the near integrability of the\nsystem. This observation\nhas motivated recent studies of\nthe non-equilibrium properties of\nintegrable quantum model Hamiltonians \\cite{theo}.\nThe interest in this topic is also motivated by its\nrelevance to a variety of experimental situations,\nincluding cold atoms \\cite{BlDaZw}, Penning traps \\cite{CiMaTo}\nand Josephson-junction arrays \\cite{MaScSh}.\n\nThere are many ways in which it is possible\nto drive a system out of equilibrium. Quantum quenches,\nand coupling to baths with different temperatures (or potentials)\nare the most common protocols. While in the quench\nscenario the focus is on the unitary dynamics of the full system,\nin the second case one usually deals with an effective\ndescription of the dynamics of the subsystem only.\nDifferent out-of-equilibrium dynamics have nevertheless\nsimilar characteristics, for example the presence of\ncurrents (of particles, energy, or heat).\n\nIn this work, we study an Ising-like model system\ndriven out of equilibrium with a quantum quench in the\npresence of an energy current. In the usual\nquench protocol the system is prepared in the ground-state\nin the absence of any current,\nand subsequently the dynamics drives it to some excited\nstate. The model Hamiltonian we consider here allows for the\nstudy of two different new situations: a quench from the ground-state\nin the presence of an energy current, and a quench from\nan initial excited state in the absence of an energy current.\nThe two scenarios are associated with different Hamiltonians\nwhich, in a sense, are dual to each other.\nTo characterize the system's behaviour\nwe focus our attention on the Entanglement Entropy (EE), as measured by the Von Neumann entropy,\nwhich is a central quantity in the characterization of nonequilibrium\nquantum dynamics \\cite{CaCa}.\n\nThe structure of the paper is the following: in Sec.II we introduce and describe the\nproperties of the model Hamiltonian, in Sec.III we consider the static properties of EE for this model,\nand in Sec.IV we study the dynamics of\nentanglement following a quench. Finally, we states our conclusions.\nIn the Appendix the reader can find more details of the calculations.\n\n\\section{The model}\nWe consider an Ising spin chain in transverse field $ H_I $,\nwith an additional Dzyaloshinskii-Moriya (DM)\ninteraction $ H_{DM}$. The total Hamiltonian $ H_I + H_{DM} $ is defined as follows\n\\begin{equation}\nH=-\\sum_{j}\\left[\\frac{1}{2}\\sigma_{j}^{x}\\sigma_{j+1}^{x}+\\frac{h}{2}\\sigma_{j}^{z}+\\frac{\\zeta}{8}\\left(\\sigma_{j}^{x}\\sigma_{j+1}^{y}-\\sigma_{j}^{y}\\sigma_{j+1}^{x}\\right)\\right],\n\\label{Eq:ham}\n\\end{equation}\nwhere $ h $ is the external magnetic field, and $ \\zeta $ is the coupling parameter determining the strength\nof the DM interaction. Such an anisotropic interaction is present in many low-dimensional\nmaterials with the necessary crystal symmetry,\nand it originates from spin-orbit coupling \\cite{materials,dm}.\nFurthermore, the DM interaction is of relevance in quantum information theory,\nsince it plays an important role in the\nphysics of quantum dots \\cite{ChFrJo},\nand in fault-tolerant quantum computation \\cite{WuLi}.\n\nAdding the DM term to $ H_I $\ndoes not affect the solvability of the model \\cite{SiCaGa}, and interestingly\nenough it provides the system with a richer phase diagram.\nThese features have been used in \\cite{AnRaSa} to study\nthe effective out-of-equilibrium quantum dynamics of the model.\n$ H_{DM} $ can be viewed as a current term.\nThe reason for this is the following.\nThe equation of motion for the local energy density of $ H_I $,\ndefined by\n$\n\\epsilon_j = \\frac{1}{2}\\sigma_{j}^{x}\\sigma_{j+1}^{x}+\\frac{h}{2}\\sigma_{j}^{z},\n$\n\\begin{equation}\n\\dot{\\epsilon_j} = \\frac{i}{\\hbar}\\left[H_I,\\epsilon_j \\right].\n\\end{equation}\nOne can write the time derivative of the energy current as the divergence of the energy current\n\\begin{equation}\n\\dot{\\epsilon_j}=C_j-C_{j+1},\n\\end{equation}\nwith\n\\begin{equation}\nC_j \\propto \\sigma^{y}_{j} \\left( \\sigma^{x}_{j-1} - \\sigma^{x}_{j+1} \\right).\n\\end{equation}\nIt thus follows that $ H_{DM} $ is precisely the sum over all sites of the\nlocal currents\n $ \\sum_j C_j $.\nTherefore, the ground-state expectation value of $ H_{DM} $\n\\begin{equation}\nJ \\equiv\n\\langle \\sum_j \\frac{\\zeta}{8}\n\\left(\\sigma_{j}^{x}\\sigma_{j+1}^{y}-\\sigma_{j}^{y}\\sigma_{j+1}^{x}\\right)\\rangle,\n\\end{equation}\nbecomes an order parameter indicating the presence of an energy current.\nOnce the total Hamiltonian has been diagonalized, which can be done\nwith the usual Jordan-Wigner and Bogoliubov transformations\n\\begin{equation}\nH=\\sum_q \\Lambda_q b^\\dagger_q b_q,\n\\end{equation}\nthe effect of the DM interaction is clearly observed at the single-particle level.\nThe single-particle spectrum is given by\n\\begin{equation}\n\\Lambda_q=\\sqrt{1+h^2+2h\\cos q}+\\zeta \\sin q,\n\\label{Eq:spectrum}\n\\end{equation}\nwith  $q\\in [-\\pi,\\pi)$ the momentum of the quasi-particle.\nAs can be seen in the above expression and in Fig.\\ref{Fig:spectrum}, the DM interaction\nmakes the spectrum non-symmetric with respect to $q = 0$.\nNote that the ground state of the Hamiltonian including the DM interaction\nis the same for all values of $ \\zeta $ in the interval $ [0,1] $. In particular this\nmeans  that, within this range of values, the Ising model in a transverse field $ H_I $\nand the system described by Eq.\\ref{Eq:ham} have the same ground state.\nBeyond $ \\zeta=1 $ (with $h\\leq\\zeta$) the Fermi sea starts to be populated\nby the modes in between the zeros of\nthe single particle spectrum (i.e. between $ q_+ $ and $ q_- $ in Fig.\\ref{Fig:spectrum}).\nThis implies that the ground state is not anymore that of $ H_I $. Furthermore,\nsince the DM term commutes with the rest of the Hamiltonian,\nat the many-body level when $ \\zeta=1 $ we must have a level crossing\nbetween the ground state of $ H_I $ and some previously excited Hamiltonian eigenstates.\nFig.\\ref{Fig:phase} shows the phase diagram of the model \\cite{AnRaSa}.\nThere are three regions:\nferromagnetic, polarized paramagnetic, and the so-called\ncurrent phase, characterized by $ J \\neq 0 $.\nThe current phase is gapless, and the two-point correlation functions\nshow a power-law behaviour with an oscillatory amplitude:\n$\n\\langle\n\\sigma^x_l\\sigma^x_{l+n} \\rangle_{gs}\n\\sim\\frac{Q(h,\\zeta)}{\\sqrt n}\\cos (kn),\n$\nwhere $Q$ is a non-universal function and\n$k\\equiv\\arccos{\\frac{1}{\\zeta}}$  \\cite{AnRaSa}.\n\nIn the following, we study the entanglement properties of this model,\nand subsequently we analyze new quench protocols for this spin system.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{fig1.eps}\n\\caption{(Color online) Spectrum of the Hamiltonian in Eq.\\ref{Eq:ham}. We show the spectrum for 4 different values of $\\zeta$, while keeping $h=0.5$ fixed.\nSee also figures in \\cite{AnRaSa}.}\n\\label{Fig:spectrum}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{fig2}\n\\caption{(Color online) Phase diagram in the $h-\\zeta$ plane of the model in Eq.\\ref{Eq:ham}.\nThe dotted line is a critical line where the model shows the same universal properties of the quantum Ising model in transverse field.\nSee also figures in \\cite{AnRaSa}.}\n\\label{Fig:phase}\n\\end{figure}\n\n\\section{Static entanglement entropy and phase diagram}\n\\label{sec:static}\nIn this section we show how entanglement can be used to characterize\nthe different phases of the model.\nTo measure the EE, we consider a bipartition\nof the spin chain into two subsystems A and B.\nFor this setup a good\nmeasure of EE between the two partitions\nis given by the von Neumann entropy\n$ S_A \\equiv -Tr \\rho_L \\ln \\rho_L $, where $ \\rho_L $ is the ground-state\nreduced density matrix of the subsystem A with $ L $ spins.\n\nIt is known that for critical one-dimensional systems\nthe EE scales logarithmically in the subsystem size,\nwith a prefactor given by the central charge of the associated Conformal Field Theory (CFT),\n\\begin{equation}\n\\label{Eq:critScaling}\nS_L=\\frac{c}{3}\\ln L+S_0,\n\\end{equation}\nwhere $ c $ is the central charge and $ S_0 $ is a non-universal constant \\cite{ViLaRi,AmFaOs}.\nOn the other hand, in the non-critical region of the phase diagram, the entanglement entropy\nsaturates to a value which depends on the correlation length $ \\xi $,\n\\begin{equation}\n\\label{Eq:noncritScaling}\nS_L\\propto \\frac{c}{3}\\ln \\xi.\n\\end{equation}\nBoth Eq.\\ref{Eq:critScaling} and Eq.\\ref{Eq:noncritScaling} characterize\nthe ground-state properties of EE for one-dimensional systems.\n\nApart from the ground state it is also of interest to investigate\nentanglement properties of excited states. Recently, two works have appeared\non this topic. In \\cite{AlFaCa} it has been shown that\nthere are excited states for which\nthe logarithmic scaling of EE can have prefactors different from\nthe ground state, and that for some excited states the\nscaling can be extensive in the subsystem size, instead of logarithmic.\nIn \\cite{AlBeSi} the authors have studied\nthe connection\nbetween EE for excited states and\nproperties of the associated CFT not contained in the central charge.\nThe EE of excited states is of interest also in the context of quantum quenches\nsince in this setting the system is unitarily driven from the initial ground state\nto an excited state.\n\nThe Hamiltonian in Eq.\\ref{Eq:ham} naturally fits in this set of problems.\nFollowing the discussion from the previous section\nthe model we are considering allows us to study the EE\nof some excited states of $ H_I $, simply\nby tuning the coupling constant associated with the DM term.\nThe excitations we can consider in this way\nare characterized by the modes in between the zeros of\nthe single particle spectrum that get populated when $ \\zeta > 1 $ and $h < \\zeta$ (\\ref{Eq:spectrum}).\nFor these states the EE can be evaluated\nanalogously to the ground state of $ H_I $\n(see also \\cite{KaZi} for related analytical study).\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{fig3.eps}\n\\caption{(Color online) $S_L$ vs $L$ at $h=1.0$ for different $\\zeta$. \nThe scaling behavior changes from $S_L\\sim\\frac{1}{6}\\ln L$ \non the critical line separating the ordered and disordered phase to \n$S_L\\sim\\frac{1}{3}\\ln L$ inside the current-carrying phase.}\n\\label{Fig:scaling}\n\\end{figure}\n\nLet us consider in detail the entanglement properties of the different phases\nshown in Fig.\\ref{Fig:phase}.\nFirst, we compare the scaling of the\nentanglement in the non-current-carrying critical regions and in\nthe region where an energy current is present.\nFig.\\ref{Fig:scaling} shows the result of the simulations for the scaling\nbehaviour of the ground-state EE with $\\zeta<1$ and $h=1$.\nFor all values of $\\zeta \\in [0,1]$\none observes the same scaling result. In the\nnon-current-carrying region\ncritical states are present only on\nthe $h=1$ line of the phase diagram.\nOn this line, separating the ferromagnetic and polarized\nparamagnetic phases,\nthe ground state of the system is the same as in $ H_I $.\nThis implies that the EE scaling is logarithmic with\na prefactor of $c\/3$, and $c=1\/2$.\nNote that also the entire current-carrying\nphase ($\\zeta>1$ and $h<\\zeta $) is gapless, and in this sense critical. At any point\nin this phase we observe logarithmic scaling of the EE in the subsystem size.\nThis is consistent with the discussion in the previous section\non the algebraic decay of the two point correlation function.\nInterestingly, the prefactor of the logarithmic scaling of EE in the\ncurrent phase is twice as large as the prefactor in the non-current phase.\nThis doubling reflects the increased number of zeros\nin the single-particle spectrum, consistent with\nthe results of \\cite{KaZi}.\nIn fact, when $\\zeta>1$ the ground state of Eq.\\ref{Eq:ham} is\nthe filled Fermi sea of modes in between $q_-$ and $q_+$ (see Fig.\\ref{Fig:spectrum}).\nSince the ground-state in the current phase is effectively\nan excited eigenstate of $ H_I $, we could expect a scaling of EE\nthat is extensive in the system size, as shown in \\cite{AlFaCa} for excited states.\nThe reason why this is not the case is due to the\nnature of the single-particle spectrum, which at most can have two zeros (see Fig.\\ref{Fig:spectrum}),\nand thus does not satisfy the requirements found in \\cite{AlFaCa} for an extensive scaling of EE.\n\nThe DM interaction in the Hamiltonian affects also the sub-leading term\nin the scaling of EE\n$ S_L=\\frac{1}{3}\\ln{L} +S_0(h,\\zeta) $.\nDeriving the analytical form of the sub-leading order term $S_0(h,\\zeta)$ is\ncomplicated. Nonetheless,\none can investigate this term numerically.\nIn Fig.\\ref{Fig:sublead} we see that $S_0$ is\nconstant on the critical line $h=1$ with $\\zeta\\leq 1$.\nAs soon as $\\zeta >1$ and $h<\\zeta$, $S_0$ increases, but becomes almost\nconstant for large $\\zeta$. Also $S_0$ is maximum at $h=0$,\nand $S_0$ is minimum at the critical point $h=\\zeta$. From the behaviour of\n$S_0$ we can conclude that a given block has the highest\nentanglement when all the negative modes ($q\\in [-\\pi,0)$) in the Fermi sea are filled.\nConsequently EE increases with higher values of the energy current.\n\nWe now consider the differences between the critical lines shown in\nthe phase diagram (Fig.\\ref{Fig:phase}),\nseparating different phases.\nThe only second-order quantum phase transition is found along\nat the $h=1$ line (with $\\zeta \\leq 1$), which corresponds to the\nIsing quantum phase transition\n(see Fig. \\ref{Fig:IIqpt}). On the hand the\nboundaries of the current-carrying phase with\nboth the paramagnetic and the ferromagnetic phases\nare characterized by a level crossing.\nThis translates into a sudden jump in EE (see Fig.\\ref{Fig:IIqpt} and Fig.\\ref{Fig:Iqpt}).\nThe value of EE is always higher in the current\ncarrying phase because of the presence of long-range correlations\nthat decay algebraically.\nThe plots in Fig.\\ref{Fig:IIqpt} and Fig.\\ref{Fig:Iqpt} show that controlling\nthe DM term can be used as an entanglement switch.\nThe amount of entanglement can be driven by the\nDM coupling term or the magnetic field,\nwhich are controllable parameters in\noptical lattices \\cite{optical}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig4.eps}\n\\caption{(Color online) Non-universal nature of $S_0$. (upper panel) $S_0$ vs. $\\zeta$ at different $h$;\n(lower panel) $S_0$ vs. $h$ at\ndifferent $\\zeta$.}\n\\label{Fig:sublead}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig5.eps}\n\\caption{(Color online) $S_L$ vs. $h$ at $L=60$ for different $\\zeta$.\n(upper panel) Static entanglement along the transition between the ordered ferromagnetic and the disordered paramagnetic phase. The peak signals\nthe presence of long-range correlations at the critical point, which is a signature of a second-order quantum phase transition. (lower panel) Static entanglement along the transition between the disordered paramagnetic\nand the current-carrying phase. The sudden change in entanglement followed by the absence of a peak at the\ncritical point is a signature of the first-order quantum phase transition.}\n\\label{Fig:IIqpt}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig6.eps}\n\\caption{(Color online) $S_L$ vs. $\\zeta$ at $L=60$ for different $h$.\nThe plots for $h=0.5$ corresponds to the static entanglement\nalong the transition between the ordered ferromagnetic and the current-carrying phase. This is a first order\nquantum phase transition which occurs at $\\zeta=1$ (analogously for $h=1$).  \nThe plots for $h=2$ and $h=3$ correspond to the static entanglement along the transition between the disordered and\nthe current-carrying phase. This is a first order QPT which occurs at $\\zeta=h$.}\n\\label{Fig:Iqpt}\n\\end{figure}\n\n\\section{Entanglement dynamics following a quench}\nIn this section, we focus on the quench dynamics\nof the EE. Quenching provides a way to excite a system, initially\nprepared in the ground state, and to subsequently study\nthe non-equilibrium dynamics of the model\n(in the following, we denote\nwith a subscript $0$ the value of the parameters\ndescribing the initial Hamiltonian).\nAs stated previously, the model in Eq.\\ref{Eq:ham}\nis of interest because it combines two different\nmechanisms typically used to drive a system out of equilibrium:\nquantum quenching, and the coupling to a field originating a current\nin the system. Furthermore the inclusion of the DM term allows us to\nstudy a model Hamiltonian where the energy current can be controlled\nand used in the quench protocol.\n\nIn our setup, the quench can either involve\nthe magnetic field $h$, the DM coupling $\\zeta$ or a combination of the two.\nSince the DM term commutes with the Hamiltonian, a quench\nin $ \\zeta $ leaves the system in one of its eigenstates,\nproviding a trivial evolution of the EE.\nOn the other hand, quenches in the magnetic field give more interesting behaviours.\nIf the quench is done with the initial state prepared in a region\nwith no current\nthe results are similar to those found in \\cite{CaCa}, where quenches for the\n$ H_I $ were considered.\nThis is due to the fact that, in the absence of\na current, the ground-state wave function initially is identical\nto that of $ H_I $, and the time evolution\nis not affected by the presence of the DM term\n(see the derivation of Eq.\\ref{Eq:timeCorrelation} in the Appendix for a proof of this).\nMore interestingly, if the quench involves an initial state inside\nthe current phase, new non-trivial behaviours can be expected, since the\nground state now is radically different.\nIt is important to notice that the DM coupling\nenters only in the specification of the initial state,\nwhereas the evolution can be effectively described by\nthe Hamiltonian without the DM term. The calculations\nshowing that this is in fact the case can\nbe found in the Appendix.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig7.eps}\n\\caption{(Color online) Quenches from the current carrying phase. \n$S_L(t)$ vs. the time steps, with $L=60$, $h_0=4.0$, $\\zeta_0=\\zeta=5.0$ for different $h$.\nNote that the extent of the initial linear regime depends on the particular evolving Hamiltonian.}\n\\label{Fig:QuInCurr1}\n\\end{figure}\nWe first compare the evolution of the EE for different\nquenches inside the current-carrying phase. Fixing the\ncoupling constant of the DM term, and quenching only the\nexternal magnetic field we obtain the results shown in\nFig.\\ref{Fig:QuInCurr1}.\nOne always has an initial ballistic\nevolution of the EE, which grows linearly in time (measured in units where the speed of the elementary excitation is unity)\nand saturates at some point. Quite interestingly,\nthe saturation time (hence also the rate at\nwhich entanglement is initially building up) depends on the\nparticular evolving Hamiltonian.\nThis way we can control the time needed to\ngenerate the maximal asymptotic amount of entanglement.\nThis property is relevant also from a computational point of view.\nIn fact, DMRG-like schemes, used for the simulation of the time evolution\nof quantum systems, can take advantage of the lower rate at\nwhich entanglement is generated. Knowing the regions in the\nphase diagram where such rates are lower can provide more\nefficient time simulations.\nAs far as we know this is a new feature that is\nnot present in other quench protocols considered so far\nin the literature.\nThe other aspect that is important to notice in Fig.\\ref{Fig:QuInCurr1}\nis the special role played by the line $h=1$ in the phase diagram,\nwhich turns out to provide the maximum asymptotic EE\nfor different quench parameters.\nThis can be understood by mapping the quench for\n$ H_I+H_{DM} $\nto a quench protocol for $ H_I $ only.\nAs stated in the previous section,\nthe entanglement evolution with respect to $H(h,\\zeta)$\nis identical to the evolution with respect to $H(h,0)$.\nFurthermore, the ground state of $H(h_0,\\zeta_0)$, the initial Hamiltonian in the quench protocol\nis also an excited eigenstate of $H(h_0,0)$, because of the commutativity\nof the DM term with the total Hamiltonian.\nFrom this dual perspective\nthe effect of the current is that of effectively\nquenching an excited eigenstate without the current term.\nFor the Ising model, a quench from $h_0\\neq1$ yields the maximum value\nof $S_L(\\infty)$ when quenched to $h=1$, because the energy gap closes\nat $h=1$, and hence a large number of zero energy excitations can be produced.\nWhile the asymptotic value\nof EE depends on the particular\nexcited state at the beginning of the quench.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig8.eps}\n\\caption{(Color online) Quenches from the current carrying-phase with different values of the current driving field\n$\\zeta$. $S_L(t)$ vs. the time steps with $L=60$, $h_0=3.0$ to $h=1$ for different $\\zeta_0=\\zeta$.}\n\\label{Fig:QuInCurr2}\n\\end{figure}\n\nFig.\\ref{Fig:QuInCurr2} shows results of simulations for quenches\nwith increasing values of the DM field in the current-carrying phase.\nThe asymptotic value of the EE decreases with\nincreasing $\\zeta$. This is consistent\nwith the phenomenological picture provided in \\cite{CaCa}, and\nwith the fact that if the system starts in an\nexcited state, the available number of unoccupied modes that\ncan be occupied after the quench is smaller than in the case\nof having the ground state as an initial state.\nFurthermore, Fig.\\ref{Fig:QuInCurr2} shows that\nthe time at which the EE saturates does not depend on\n$ \\zeta $, and consequently does not depend on\nthe particular initial Hamiltonian eigenstates\n(as long as it is not an eigenstate of the evolving Hamiltonian).\nThe line with $ \\zeta=100 $ in Fig.\\ref{Fig:QuInCurr2} shows that very deep into\nthe current phase quenching does not create\nentanglement. In fact, when $ \\zeta \\gg h_0 $\nquenching the magnetic field is just a small perturbation\nto the Hamiltonian, which then approximately stays in\nthe ground state.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.40]{fig9.eps}\n\\caption{(Color online) $S_L(t)$ vs. the time steps \ninside the current-carrying phase for different block sizes $L$. \nQuenching is done from $h_0=2.0$ to $h=1.0$ with $\\zeta_0=\\zeta=3.0$.}\n\\label{Fig:L}\n\\end{figure}\n\nFinally, we verify that the presence of\nan energy current does not affect the extensive nature of the\nasymptotic value of EE (Fig.\\ref{Fig:L}), and\nits proportionality with the quench size (Fig.\\ref{Fig:quenchsize}).\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{fig10.eps}\n\\caption{(Color online) $S_{60}(t)$ vs. the time steps \nfor quenches to $h=1$ from various $h_0$ inside the current-carrying phase at $\\zeta=\\zeta_0=4$.}\n\\label{Fig:quenchsize}\n\\end{figure}\n\n\n\n\\section{Conclusions}\nWe have studied the static and dynamic properties\nof the entanglement entropy in the Ising spin chain with a transverse field and\na Dzyaloshinskii-Moriya interaction. The model is characterized\nby the presence of an energy current for certain regions of the phase diagram.\n\nConcerning the static properties we have analyzed\nthe transitions between phases with no energy current and the\nphase where an energy current is present.\nThe transition is captured by a discontinuity of EE as a function of the parameters,\nand by a distinguishable scaling behaviour in the current-carrying and non-current-carrying regions.\nIn particular, the leading logarithmic term of the\nEE scaling with respect to the system size\nhas a prefactor in the current-carrying region which\nis twice as large compared to the second order Ising critical line.\n\nConcerning the behaviour of the entanglement evolution\nfollowing a quench, the model in Eq.\\ref{Eq:ham}\nallows us to study new quench protocols.\nThe usual schemes consider quenches from an initial ground state.\nThis scenario, for the model in Eq. \\ref{Eq:ham},\neffectively corresponds to a quench from an initially excited state\nof the Ising spin chain in transverse field (without DM interaction).\nThe main result of this analysis shows that\nthe ballistic picture presented in \\cite{CaCa} is still valid,\nalthough with a significantly different aspect.\nIn particular the entanglement saturation time in the current-carrying\nphase depends on the\ndetails of the evolving Hamiltonian.\nThis is an indication of the role played by the evolving \nHamiltonian on the propagation of\nexcitations. This result is of relevance in\ntuning the dynamics of the system in regions with a\ndifferent rate for the propagation of entanglement.\nFurthermore it also provides\na characterization of the regions in the phase diagram that can be\nsimulated more efficiently with DMRG-like techniques.\n\nFrom a general point of view, the model in Eq. \\ref{Eq:ham} also\nsuggests a simple way to study the quench dynamics of initial excited\nstates in integrable systems. The addition of a commuting term\nin the Hamiltonian causes a reshuffling of the spectrum that,\nwithout changing the integrability of the model, allows us to\nobtain non-trivial results about the excitations in the original model.\nThe same trick can be applied to other systems of interest.\n\nWe thank Letian Ding, Zoltan R\\'acz, and Paolo Zanardi for useful comments.\nThis work has been supported by NSF grants PHY-803304, and DMR-0804914.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA long time ago Nielsen and Olesen \\cite{Nielsen:1973cs}\ndiscussed the vortex solution\nof the (3+1) dimensional Abelian Higgs model as a possible model\nfor strings. In the same publication the authors discuss the\neffective action for the transverse collective oscillations of the \nvortex which they find to be equivalent to the\nNambu-Goto action of string theory (see e.g. \\cite{Green:1987sp}).\nTheir derivation is based on a general consideration of Lorentz\ntransformation properties of such oscillations.\nWe here derive the nonrelativistic limit of \nthis effective action from an analysis of\nthe fluctuations in the underlying quantum field theory.\n\nFor quantum kinks\n(see, e.g., \\cite{Rajaraman:1982is,Coleman85})\n the collective motion is related to the\n translation mode. It is generated by infinitesimal\ndisplacements of the classical kink solution\n and is a zero mode of the \nfluctuation operator. Its quantization requires a special\napproach, by which the zero mode is found to carry\n the kinetic energy of the collective\nmotion of the kink.\n\nIn the Abelian Higgs model in two dimensions  one finds an instanton\nsolution which describes topological transitions\n\\cite{Kripfganz:1989vm}. The\nfluctuations around the classical solution display two\nzero modes which again are related to translation invariance.\nThe one-loop prefactor for the semiclassical transition rate\nis related to the functional determinant of the fluctuation operator.\nThe zero modes would cause this determinant to be infinite and have\nto be eliminated. If handled properly, this elimination \nproduces the correct dimension for the transition rate\n\\cite{Baacke:1994bk,Baacke:2008zx}.\n\nThe instanton of the Abelian Higgs model reappears in the\n$(3+1)$ dimensional version of the model as the\nvortex solution which we will consider here. The vortex solution is identical\nto the instanton solution in the transversal $x$ and $y$ coordinates,\n and it is independent on $z$ and of time. The translation of the\nclassical solution now becomes local, dependent on $z$.\nInstead of a collective motion of a quantum kink we have\ncollective oscillations of the vortex.\nThe pole at $p_\\perp=0$ in the Euclidian Green's function of the \ntwo-dimensional model becomes a cut in the Green' s function of the\nmodel in four dimensions. In the computation  of one-loop\ncorrections to the string tension \\cite{Baacke:2008sq}\nthese modes can be included in the same\nway as all other fluctuations, in contrast to the\ntwo-dimensional case. Indeed, for the renormalization of\nthese corrections it is necessary to include the\nzero mode contribution. \nStill the translation modes play a special r\\^ole, \nand in the present work we will discuss these particular aspects.\n\nThe text is organized as follows:\nIn Sec. \\ref{sec:basics} we present the model, the classical vortex solution\nand the classical string tension.\nIn Sec. \\ref{sec:flucsandtension} \nwe define the fluctuation operator and relate it to the one-loop \ncorrection to the string tension. Based on the derivation of the\ntranslation mode wave functions in Appendix \\ref{app:translationmode}\nand using a virial theorem proven in Appendix \\ref{app:virialtheorem}\nwe discuss in Sec. \\ref{sec:collectivestringoscillations}\nthe collective string oscillations and their effective action.\nWe discuss in Sec. \\ref{sec:fluctuationenergy} the r\\^ole of the \nzero modes in the computation of the renormalized the string tension. \nConclusions are presented in Sec. \\ref{sec:conclusions}.\n\n \n \n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Basic relations}\n\\setcounter{equation}{0}\n\\par\n\\label{sec:basics}\nThe Abelian Higgs model in (3+1) dimensions is\ndefined by the Lagrange density \n\\begin{equation}\n  {\\cal L}=-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n+\\frac{1}{2}(D_\\mu\\phi)^*D^\\mu\\phi-\\frac{\\lambda}\n{4}\\left(|\\phi|^2-v^2\\right)^2\\; .  \n\\end{equation}\nHere $\\phi$ is a complex scalar field and\n\\begin{eqnarray}\nF_{\\mu\\nu}&=&\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu\\; , \\\\\nD_\\mu&=&\\partial_\\mu-igA_\\mu \n\\; .\\end{eqnarray} \nThe particle spectrum consists of Higgs bosons of mass\n$m_H^2=2\\lambda v^2$ and vector bosons of mass $m_W^2=g^2v^2$.\nThe model allows for vortex type solutions, representing\nstrings with a magnetic flux, the Nielsen-Olesen vortices \n\\cite{Abrikosov:1957,Nielsen:1973cs,deVega:1976mi}.\nThe cylindrically symmetric\nansatz for this solution is given, in the singular gauge, by \n\\footnote{ We use Euclidean notation for the transverse components, so\n$A^\\perp_1\\equiv A^1=-A_1$ etc. } \n\\begin{eqnarray}\nA^{{\\rm cl}\\perp}_i (x,y,z)&=&\\frac{\\varepsilon_{ij}x^\\perp_j} \n{gr^2}\\left[A(r)+1\\right] \\;\\;\\; i=1,2 \\\\\n\\phi^{cl}(x,y,z)&=&vf(r) \\; .\n\\end{eqnarray}\nwhere $r=\\sqrt{x^2+y^2}$ and $\\varphi$ is the polar angle. \nFurthermore $A_3^{\\rm cl}=A_0^{\\rm cl}=0$. \nWith this ansatz the energy per unit length, or string\ntension $\\sigma$ takes the form\n\\begin{eqnarray}\n\\sigma_{\\rm cl}&=&\\pi v^2\n \\int^{\\infty}_{0}\\!\\!\\! dr \\left\\{\\frac{1}{rm_W^2}\\left[\n\\frac{dA(r)}{dr}\\right]^2\\!\\!+r\\left[\\frac{df(r)}{dr}\\right]^2\\!\\!\n+\\frac{f^2(r)}{r}\\left[A(r)+1\\right]^2\\!\\! \\right. \\nonumber \\\\ \n&+&\\left. \\frac{rm^2_H}{4}\\left[\nf^2(r)-1\\right]^2\\!\\right\\}\\; .\n\\label{eq:classtens} \\end{eqnarray}\nThe classical equations of motion are given by\n\\begin{eqnarray}\n\\left\\{\\frac{\\partial^2}{\\partial r^2}+\n\\frac{1}{r}\\frac{\\partial}{\\partial r}-\\frac{\\left[A(r)+1\\right]^2}{r^2}-\n\\frac{m^2_H}{2}\\left[f^2(r)-1\\right]\n\\right\\}f(r)&=&0 \\; , \\\\\n\\left\\{\\frac{\\partial^2}\n{\\partial r^2}-\\frac{1}{r}\n\\frac{\\partial}{\\partial r}-m^2_W f^2(r)\\right\\}\n\\left[A(r)+1\\right]&=&0\n\\; ,\\end{eqnarray}\nwhich are to be solved numerically with\n\\begin{equation}  \\label{rb}\n\\begin{array}{rcccccr}\nA(r)&\\stackrel{\\scriptscriptstyle{r\\to 0}}\n{\\longrightarrow}& {\\rm const.}\\cdot r^2                       \n&,&A(r)&\\stackrel{\\scriptscriptstyle{r\\to\\infty}}\n{\\longrightarrow}&-1 \\; , \\\\ \nf(r)&\\stackrel{\\scriptscriptstyle{r\\to 0}}\n{\\longrightarrow}&{\\rm const.}\\cdot r \n&,&f(r)&\\stackrel{\\scriptscriptstyle{r\\to\\infty}}\n{\\longrightarrow}&1\\; , \n\\end{array}\n\\; .\\end{equation}\n\n\n\\section{Fluctuation operator and one-loop  string tension}\n\\label{sec:flucsandtension}\nExpanding the gauge and Higgs fields as\n\\begin{eqnarray}\n\\phi &=& \\phi_i^{\\rm cl} + \\varphi_1 + i \\varphi_2\n\\\\\nA_\\mu&= &A_\\mu^{\\rm cl} + a_\\mu\n\\end{eqnarray}\nthe dynamics of the fluctuations is described by the second\norder gauge fixed Lagrangian \\cite{Baacke:1994bk}\n\\begin{eqnarray} \\label{lag2}\n{\\cal L}^{II}&=&\n-a_\\mu\\frac{1}{2}\\left(-\\Box-g^2\\phi^2\\right)\na^\\mu\\nonumber \\\\\n&&+\\varphi_1\\frac{1}{2}\\left[-\\Box+g^2A_\\mu A^\\mu-\n\\lambda\\left(3\\phi^2-\nv^2\\right)\\right]\\varphi_1  \\nonumber \\\\  \n&&+\\,\\varphi_2\\frac{1}{2}\\left[-\\Box+g^2A_\\mu A^\\mu-g^2\\phi^2-\n\\lambda\\left(\\phi^2-v^2\\right)\\right]\\varphi_2 \\nonumber \\\\\n&&+\\,\\varphi_2(gA^\\mu\\partial_\\mu)\\varphi_1\n+\\varphi_1(-gA^\\mu\\partial_\\mu)\\varphi_2  \\\\\n&&+\\,a^\\mu(2g^2A_\\mu\\phi)\\varphi_1\n+\\,a^\\mu(2g\\partial_\\mu\\phi)\\varphi_2 \\nonumber \\\\\n&&+\\eta_1\\frac{1}{2}\\left(-\\Box- g^2\\phi^2\\right)\\eta_1\n+\\eta_2\\frac{1}{2}\\left(-\\Box-g^2\\phi^2\\right)\\eta_2\n\\nonumber  \n\\; ,\\end{eqnarray}\nHere $\\varphi_1$ and $\\varphi_2$ denote the real and imaginary part of\nthe Higgs field fluctuations, $\\eta_i$ are the Faddeev-Popov ghosts, and\nwe have chosen the 't Hooft-Feynman background gauge.\nIn compact notation this may be written as\n \\begin{equation}\n{\\cal L}^{II} =  \\frac{1}{2}  \\psi^*_i {\\cal M}_{ij}  \\psi_j \\; .\n\\end{equation}\nThe fields $\\psi_i$ denote the ensemble of gauge, Higgs and\nFaddeev-Popov fields and  the\nfluctuation operator ${\\cal M}_{ij}$ is defined by this and the\nprevious equation.\nIn terms of the fluctuation operators\n${\\cal M}$ on the vortex and ${\\cal M}^0$ for the\nvacuum background fields, the effective action\nis defined as\n\\begin{equation}\nS_{eff} = \\frac{i}{2} \\ln \\left\\{ \\frac{\\det {\\cal M}+i\\epsilon}\n{\\det {\\cal M}^0+i\\epsilon} \\right\\} \\; .\n\\end{equation}\nAs the background field is time-independent and also independent of\n$z$ the fluctuation operators take the form\n\\begin{equation}\\label{flucsep}\n{\\cal M}_{ij} = (\\partial_0^2-\\partial_3^2)\\delta_{ij} +{\\cal M}_{\\perp ij}\n\\; , \\end{equation}\nwhere ${\\cal M}_\\perp$ is a positive-definite operator describing the transversal\nfluctuations.\nIt is identical for the longitudinal and timelike gauge fields and for\nthe Faddeev-Popov ghosts, so these contributions to the effective\naction cancel. The remaining degrees of freedom form a\ncoupled system of four fields $\\psi_i$: the real and imaginary part of \nthe Higgs field fluctuations $\\varphi_1,\\varphi_2$ and the transverse \ngauge field fluctuations $a_1,a_2$.\n \nAs is well known the  logarithm of the determinant can be written as the\ntrace  of the logarithm. One can do the trace over $k_0$, the\nmomentum associated with the time variable, by integrating over\n$T\\int dk_0\/2\\pi$, where $T$ is the lapse of time.\nOne then obtains\n\\begin{equation}\nS_{\\rm eff} = -T \\frac{1}{2} \\sum \\left[E_\\alpha-E_\\alpha^{(0)}\\right]\\; ,\n\\end{equation}\nwhere $E_\\alpha$ are square roots of  the eigenvalues of the \npositive definite operator\n\\begin{equation}\n-\\partial_3^2+{\\cal M}_\\perp\\; ,\n \\end{equation}\nand likewise $E_\\alpha^{(0)}$ are those of the analogous operator\nin the vacuum\n\\begin{equation}\n-\\partial_3^2+{\\cal M}_\\perp^0=-\\partial_3^2-\\vec \\nabla_\\perp^2 + {\\bf m^2}\n\\; . \\end{equation}\nHere ${\\bf m}^2={\\rm diag}(m_1^2,\\dots,m_n^2)$ is the diagonal mass squared \noperator for the various fluctuations.\n\n  So the effective action is  equal to the sum of differences between \nthe zero point energies of the quantum\nfluctuations around the vortex and the ones of the\nquantum fluctuations in the vacuum, multiplied by $-T$.\nFurther, we can  do the trace over the variable $k_3$\nby integrating over $L\\int dk_3\/2\\pi$. We then obtain\n\\begin{equation}\nS_{\\rm eff}= -TL\\sum_\\alpha\\int\\frac{dk_3}{2\\pi}\n\\frac{1}{2}\n\\left[\\sqrt{k_3^2+\\mu_\\alpha^2}-\\sqrt{k_3^2+\\mu_\\alpha^{(0)~ 2}}\\right]\n\\; , \\end{equation}\nwhere $\\mu_\\alpha^2$ are the eigenvalues of the operator ${\\cal M}^\\perp$ and\n$\\mu_\\alpha^{(0)~ 2}$ those of $-\\vec \\nabla_\\perp^2+{\\bf m}^2$. \nIn the same way the classical action becomes\n\\begin{equation}\nS_{\\rm cl}=-TL\\sigma_{\\rm cl}\n\\end{equation}\nwhere $\\sigma_{\\rm cl}$ is the classical string tension.\nThe fluctuation part of the string tension is \ngiven by\n\\begin{equation} \\label{fluctuationstringtension}\n\\sigma_{\\rm fl}=\\sum_\\alpha\\int\\frac{dk_3}{2\\pi}\n\\frac{1}{2}\\left[\\sqrt{k_3^2+\\mu_\\alpha^2}-\\sqrt{k_3^2+\\mu_\\alpha^{(0)~ 2}}\\right]\n\\; . \\end{equation}\nOf course all expressions are formal, the integrals do not exist before\na suitable regularization. \n\nA way of computing $\\sigma_{\\rm fl}$ has been formulated in Ref.\n\\cite{Baacke:2008sq}: one computes the matrix valued \nEuclidian Green' s function of the\nfluctuation operator defined by\n\\begin{equation}\n(p^2{\\bf 1} +{\\cal M}_\\perp) {\\cal G}({\\bf x}_\\perp,{\\bf x}'_\\perp,p) =\n{\\bf 1}\\delta^2({\\bf x}_\\perp-{\\bf x}'_\\perp)\n\\end{equation}\nand similarly for ${\\cal M}_0$.\nThen with\n\\begin{equation} \\label{fpdef}\nF(p)=\\int d^2 x_\\perp {\\rm Tr} \\left[{\\cal G}({\\bf x}_\\perp,{\\bf x}_\\perp,p)-\n{\\cal G}_0({\\bf x}_\\perp,{\\bf x}_\\perp,p)\\right]\n\\end{equation}\nthe fluctuation correction to the string tension is given by\n\\begin{equation}\\label{sigmanum}\n\\sigma_{\\rm fl}=-\\int_0^\\infty \\frac{p^3~dp}{4\\pi} F(p)\n\\end{equation}\nThe function $F(p)$ has been computed in Ref. \\cite{Baacke:2008sq}.\nFor small $p$ it behaves as $2\/p^2$, as expected for \ntwo translation modes which yield  poles at $p=0$.\nThis is displayed in Fig. \\ref{fig:zeromode} for\n$\\xi= m_H\/m_W = 1$. For large momenta $F(p)$ again behaves\nas $p^{-2}$, but with a different coefficient.\nThe contribution of the translation modes, \nas well as the entire fluctuation correction\nare quadratically divergent. The renormalization has been\ndiscussed previously \\cite{Baacke:2008sq}.\nWe come back to the fluctuation correction below,\nbut at first we discuss  explicitly the translation mode\nwhich describes collective oscillations of the vortex. \n\n\n\n\\section{Collective string oscillations}\n\\label{sec:collectivestringoscillations}\n\\setcounter{equation}{0}\n\nFor quantum kinks $\\phi(x)$ the translation mode is proportional to the\nderivative of the classical solution, $\\psi_0= N d \\phi_{\\rm cl}(x)\/dx$.\nIt is an eigenstate of the fluctuation operator with eigenvalue\nzero, and it leads to a pole in the Green's function of the fluctuation\noperator at energy $\\omega=0$. For the quantum kink the translation mode is \nrelated to the collective motion of the entire kink, and\nits contribution to the quantum corrections is the kinetic energy.\nHere we are considering local transverse displacements of the vortex.\nEach slice between $z$ and $z+\\Delta z$ is moving separately and the\nresulting motions of the vortex can be described in terms of\nwaves propagating along the string. In the Green's function\nof the complete fluctuation operator ${\\cal M}$ the pole appears as a cut,\nstarting at $\\omega=0$.\n\nOf course we again expect the zero modes to be related to the derivatives\nof the classical solution $\\nabla_i \\phi_{\\rm cl}$, but the fluctuation \noperator is matrix valued and therefore we have to determine all\nfour components of the eigenvector. This is discussed in Appendix\n\\ref{app:translationmode}, in a cylindrical basis of modes for which\nthe fluctuation operator was derived in Refs. \n\\cite{Baacke:1994bk,Baacke:2008sq}. \nThe wave functions of the zero modes arising from local translation invariance\nare derived in Appendix \\ref{app:translationmode}.\nCombining the modes with azimutal quantum numbers $m=\\pm 1$\nproportional to $\\exp{\\pm i\\varphi}$ one finds that an infinitesimal\nshift in the $x$ direction generates a four component wave function\n\\begin{eqnarray}\n\\varphi^x_1(r,\\varphi)&=&v f'(r) \\cos \\varphi \\; ,\n\\\\\n\\varphi^x_2(r,\\varphi)&=&-v \\frac{A(r)+1}{r}f(r)\\sin \\varphi\\; ,\n\\\\\n{\\bf a}^x(r,\\varphi)&=& \\left(\\begin{array}{c}0\\\\-1\\end{array}\\right)\n\\frac{A'(r)}{gr}\\; ,\n\\end{eqnarray}\nand a shift in the $y$ direction leads to\n\\begin{eqnarray}\n\\varphi^y_1(r,\\varphi)&=&v f'(r) \\sin \\varphi\\; ,\n\\\\\n\\varphi^y_2(r,\\varphi)&=&v  \\frac{A(r)+1}{r}f(r) \\cos \\varphi\\; ,\n\\\\\n{\\bf a}^y(r,\\varphi)&=& \\left(\\begin{array}{c}1\\\\0\\end{array}\\right)\n\\frac{A'(r)}{gr}\\; .\n\\end{eqnarray}\nThe norm of these wave functions is given by\n\\begin{eqnarray} \\nonumber\n||\\psi_t^x||^2&=&\\int r~dr~\\int d\\varphi\n\\left\\{(a^x_1)^2+(a^x_2)^2+(\\varphi^x_1)^2+(\\varphi^x_2)^2\\right\\}\n\\\\\\nonumber&=&2 \\pi v^2\\int r~dr~\\left\\{\\frac{1}{m_W^2}\\frac{A'{}^2(r)}{r^2}+\n\\frac{1}{2}\\frac{(A(r)+1)^2}{r^2}f^2(r)+\\frac{1}{2}f'{}^2(r)\\right\\}\n\\; ,\\end{eqnarray}\nand analogously for the $y$ mode. Using the virial theorem proven in\nAppendix \\ref{app:virialtheorem} it takes the value\n\\begin{equation}\n||\\psi_t^x||^2 = \\sigma_{\\rm cl}\n\\; ,\n\\end{equation}\nwhere $\\sigma_{\\rm cl}$ is the \nclassical string tension. \n\nIn the mode expansion of the quantum fields these modes appear\nin the form\n\\begin{eqnarray} \\label{fieldexpansion1}\n\\phi_i(r,\\varphi,z,t)=\\phi^{\\rm cl}(r,\\varphi)\\delta_{i1}+ \n{\\bf X}(z,t)\\varphi^x_i(r,\\varphi)\n+ {\\bf Y}(z,t)\\varphi^y_i(r,\\varphi) + \\dots &&\n\\\\\\label{fieldexpansion2}\nA_i(r,\\varphi,z,t)=A_i^{\\rm cl}(r, \\varphi)+\n{\\bf X}(z,t)a^x_i(r,\\varphi)\n+{\\bf Y}(z,t)a^y_i(r,\\varphi) + \\dots&&\n\\; ,\\end{eqnarray}\nwhere the dots indicate the contributions of all other\neigenfunctions of the fluctuation operator. While these have,\nin general, complex wave functions we have written the contributions\nof the translation modes, which are real, in a suggestive form\nusing operators ${\\bf X}(z,t)$ and ${\\bf Y}(z,t)$. \nThe canonical momenta of the field operators are given by\n\\begin{eqnarray}\n\\Pi^\\Phi_i(r,\\varphi,z,t)=\n\\dot\\Phi_i(r,\\varphi,z,t = \\dot{\\bf X}(z,t)\\phi^x_i(r,\\varphi) + \n\\dot{\\bf Y}(z,t)\\phi^y_i(r,\\varphi) + \\dots && \n\\\\\n\\Pi^A_i(r,\\varphi,z,t)=\\dot A_i(r,\\varphi,z,t)=\\dot{\\bf X}(z,t)a^x_i(r,\\varphi)\n+\\dot{\\bf Y}(z,t)a^y_i(r,\\varphi) + \\dots &&\n\\; ,\\end{eqnarray}\n The relation of the operator ${\\bf X}$ to the\nusual creation and annihilation operators is given by\n\\begin{eqnarray}\\label{xcrelation}\n{\\bf X}(z,t)&=&\\frac{1}{\\sqrt{\\sigma_{\\rm cl}}}\n\\int \\frac{dk}{2\\pi 2|k|}\\left(c_x(k)e^{i(kz-|k|t)}+\nc_x^\\dagger(k)e^{-i(kx-|k|t)}\\right) \\; ,\n\\\\\\label{pcrelation}\n{\\bf P}_x(z,t)&=&\\sqrt{\\sigma_{\\rm cl}} \\int \\frac{dk}{2\\pi 2i}\\left(c_x(k)e^{i(kz-|k|t)}-\nc_x^\\dagger(k)e^{-i(kx-|k|t)}\\right)\n\\; ,\\end{eqnarray}\nand analogously for ${\\bf Y}$. Here we have used the fact that for the zero \nmodes we have $\\omega=|k|$. \nThe operators $c_\\alpha(k)$ satify the commutation relations\n\\begin{equation}\n\\left[c_\\alpha(k),c_\\beta^\\dagger(k')\\right]=\n2\\pi 2|k|\\delta_{\\alpha\\beta}\\delta(k-k')\n\\; ,\\end{equation}\nand the commutation relation between ${\\bf X}$ and ${\\bf P}_x$ is\n\\begin{equation}\n\\left[{\\bf X}(z,t),{\\bf P}_x(z',t)\\right]= i \\delta(z-z')\n\\; .\\end{equation}\n The normalization factors $\\sqrt\\sigma_{\\rm cl}$\nin Eqs. \\eqn{xcrelation} and \\eqn{pcrelation} are determined by \nthe requirement that in the field \nexpansion  the operators $c_x(k), c_x^\\dagger(k)$ have to appear multiplied\nby wave functions normalized to unity. \nThis is necessary for obtaining the \ncanonical equal time commutation relations for the fields,\n\\begin{eqnarray}\n\\left[\\Phi_i(x,y,z,t),\\dot \\Phi_j(x',y',z',t)\\right]=i\\delta_{ij}\n\\delta^3({\\bf x}-{\\bf x}')\n\\\\\n\\left[A_i(x,y,z,t),\\dot A_j(x',y',z',t)\\right]=i\\delta_{ij}\n\\delta^3({\\bf x}-{\\bf x}')\n\\end{eqnarray}\nvia the completeness relation of the wave functions. Of course, this \ncompleteness relation requires of the inclusion of all\neigenfunctions of the fluctuation operator, which \nabove are indicated by dots.\n\nThe second order \nHamilton operator corresponding to the Lagrangian\n\\eqn{lag2} can be written in the form\n\\begin{equation}\n{\\bf H}^{II}=\\int d^2 x_\\perp \\int dz\n\\frac{1}{2}\\left\\{\\sum_i \\dot \\psi_i^2+\\sum_i\\left[\\frac {d\\psi_i}{dz}\\right]^2\n+\\sum_{ij}\\psi_i {\\cal M}_{\\perp ij}\\psi_j\\right\\}\n\\; .\\end{equation}\nHere we are interested only in the contribution of the zero modes\nof ${\\cal M}_\\perp$.\nIf we insert the fluctuation fields of the\nfield expansion Eqs. \\eqn{fieldexpansion1} and \\eqn{fieldexpansion2}\nthe operator ${\\cal M}_{\\perp ij}$ does not contribute. Using further the\nnorm of the translation modes in order to do the integration\nover $d^2 x_\\perp$ we obtain\n\\begin{equation}\n{\\bf H}^{II}_{\\rm transl.}=\n\\sigma_{\\rm cl}\\int dz \\frac{1}{2}\\left\\{\n\\dot {\\bf X}^2+ {\\bf X}'{}^2+\\dot {\\bf Y}^2+ {\\bf Y}'{}^2\\right\\}\n\\; . \\end{equation}\nIncluding the classical string tension we find\n\\begin{equation} \\label{Heffnonrel}\n{\\bf H}=\\sigma_{\\rm cl} \\int dz \\left[1+\\frac{1}{2}\\left(\n\\dot {\\bf X}^2+ {\\bf X}'{}^2+\\dot {\\bf Y}^2+ {\\bf Y}'{}^2\\right)\\right\\}+\\dots\n\\; ,\\end{equation}\nwhere the dots indicate the contributions of higher modes.\nThis looks analogous to the result for the quantization of kinks\n\\begin{equation}\nH= M+\\frac{1}{2M}{\\bf P}^2+\\dots= M(1+\\frac{1}{2}\\dot{\\bf X}^2)+\\dots\n\\; .\\end{equation}\nThere we know that the complete result, which only appears if higher loops\nare included, must be Lorentz covariant:\n\\begin{equation}\nH= \\frac{M}{\\sqrt{1-\\dot{\\bf X}^2}}+\\dots\n\\; ,\\end{equation}\ncorresponding to an action\n\\begin{equation}\nS=-M\\int dt \\sqrt{1-\\dot {\\bf X}^2}\n\\; .\n\\end{equation}\nFor the case of the Nielsen-Olesen vortex the action\n\\begin{equation} \\label{Seffrel}\nS=-\\sigma_{\\rm cl}\\int dt\\int dz\\sqrt{1-\\dot {\\bf X}^2-\\dot {\\bf Y}^2+\n{\\bf X}'{}^2+{\\bf Y}'{}^2}\n\\; .\\end{equation}\nimplies the string Hamiltonian\n\\begin{equation} \\label{Heffrel}\nH=\\sigma_{\\rm cl} \\int dz \\frac{1+{\\bf X}'{}^2+{\\bf Y}'{}^2}\n{\\sqrt{1-\\dot {\\bf X}^2-\\dot {\\bf Y}^2+\n{\\bf X}'{}^2+{\\bf Y}'{}^2}}\n\\end{equation}\nwhich in the nonrelativistic limit leads to Eq. \\eqn{Heffnonrel}.\nIn this limit these results are in agreement with Ref. \\cite{Nielsen:1973cs}\n\n\n\\section{Energy of collective fluctuations and renormalization}\n\\label{sec:fluctuationenergy}\n\\setcounter{equation}{0}\nThe Hamiltonian for collective oscillations of the vortex\nprimarily describes excitations of the string, here:\ntransversal waves that propagate along the $z$ axis.\nThe zero point energies associated with these degrees of freedom\ncan be absorbed, in the string picture,\n into the redefinition of the string tension.\nIn quantum field theory they are absorbed, as all other divergences,\nby counter terms local in the fields, the string tension does not appear\nin the basic Lagrangian and there is no related counter term either.\nThe difference between the two approaches appears in a similar\nway in the case of the Casimir effect \\cite{Baacke:1985fh,Graham:2002fi}.\n We would like to discuss this in some detail. \n\nThe trace of the  Euclidian  Green' s function \n$F(p)$ for the gauge-Higgs sector is displayed in Fig.\n\\ref{fig:zeromode}. We have mentioned\nalready that at low momenta it behaves as $2\/p^2$ which is the\nreflection of the two zero modes. At high momenta it behaves\nas $a\/p^2+b\/p^4+ O\\left(p^{-6}\\right)$ where the coefficients\n$a$ and $b$ are determined  by the lowest orders \nof perturbation theory. A contribution to the Green' s function \nwhich at high momenta is proportional to $1\/p^2$ is converted, \nvia Eq. \\eqn{sigmanum} into a quadratic divergence for the \none-loop string tension. \nIf we consider the complete Green' s function then this quadratic\ndivergence, as well as the subleading logarithmic one,\ncan be handled \\cite{Baacke:2008sq} by subtracting \nthe leading orders perturbation theory\nanalytically and by regularizing and renormalizing them in the\nusual way. The subtracted function, whose integral is finite,\nbehaves as $p^{-6}$; it is plotted in Fig. \\ref{fig:zeromode}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{zeromode.eps}\n\\end{center}\n\\vspace{4mm}\n\\caption{\\label{fig:zeromode}\nThe integrand function $F(p)$ defined in Eq.\n\\eqn{fpdef}: circles: the unsubtracted function;\ndashed line: asymptotic behaviour $ a\/p^2$;\nsolid line: zero mode contribution $2\/p^2$;\ndiamonds: the subtracted function; dotted line: asymptotic\nbehaviour $\\propto 1\/p^6$ of the subtracted function. }\n\\end{figure}\nThe zero mode poles are part of the asymptotic\nGreen' s function; if they were removed, the asymptotic behaviour\nof the subtracted Green's function would be $-2\/p^2$ and its\nintegral would again be divergent. \nWe neither have a prescription to handle this divergence\nnor the one of the separated zero mode pole. \nSo within usual renormalized perturbation\ntheory there is no obvious way to quantify the contribution of the\ncollective string oscillations to the total one-loop fluctuation\nenergy: the renormalization of the contribution of collective\nfluctuations to the string tension is embedded into\nthe renormalization of the entire one-loop contribution within the\nframework of renormalized quantum field theory. It is not necessary to\ninvoke a mathematical definition of divergent sums like the\nzeta function regularization.\n\nThere is further a conceptual difference between the zero point energy\nof the collective fluctuations in a string picture\n and their contribution to the one-loop corrections\nin quantum field theory:\nIn a pure string picture the presence of fluctuations\ntrivially requires the presence of the string. Once included\ntheir zero point energies are added to the string tension. \nSo, if their contribution were be finite it would positive.\nIn quantum field theory the fluctuations of the field are present\neven in the absence of the vortex. The vortex generates an\nattractive potential. The presence of the\nzero modes implies that levels of a\ncontinuum which starts at energies larger than $m_W$ or $m_H$\nare pulled down such that at least in\none channel the continuum starts at energy zero. So we expect\na negative contribution to the string tension. Indeed the\nunsubtracted  function $F(p)$ is positive and if the\nintegral of Eq. \\eqn{sigmanum} were finite, $\\sigma_{\\rm fl}$\nwould be negative. This simple\nfeature gets obscured in the process of regularization and \nrenormalization.\n\n\n\n\n\\section{Conclusions}\n\\setcounter{equation}{0}\n\\label{sec:conclusions}\nWe have considered here a particular aspect of the one-loop\ncorrections to the string tension of the Nielsen-Olesen vortex,\nthe r\\^ole of the translation modes. We have identified their wave functions\nand derived their contribution to the string tension. This contribution\ndescribes the energy of transversal waves propagating along the\ndirection of the string. These can be considered\nas collective fluctuations of the classical vortex, in the same way\nas the translation modes of quantum kinks describe the collective\nmotion of the kink. This relation is made precise, in both cases,\nby virial theorems. The effective action for the\nfluctuations is found to be the nonrelativistic limit\nof the Nambu-Goto action.\n\nFor the handling of the divergent one-loop corrections\nwe have discussed conceptual differences between standard\nstring theory and the vortex of the Abelian Higgs model.\nIn the latter case the divergences associated \nwith the the zero point energies of collective\nfluctuations are treated along with those of other\nfluctuations within the standard framework\nof renormalized quantum field theory.\n\nThe approach described here only pertains to a\nstraight line vortex of infinite length. It can be expected\nto hold for more general string configurations as long as the\ncurvature radii and a possible finite length are large\ncompared with the transverse extension of the string. The \nconceptual differences in renormalization between \nthe idealized string model\nand the vortex of a quantum field theory are of course\nof an entirely general nature. Indeed they are more general than\nthe special model considered here. E.g., if we elevate a kink\nof an $1+1$ dimensional quantum field theory to a domain wall in $3+1$ \ndimensions, its translation mode reappears in the form of collective\nsurface oscillations and the renormalization of this degree of freedom\nis again embedded into the renormalization of the energy\nof all quantum fluctuations.\n\n\n\\section*{Acknowledgements}\nThe author has pleasure in thanking Nina Kevlishvili for\nuseful discussions and comments.\n\n\n\n\n\\newpage\n\\noindent\n{\\LARGE \\bf Appendix}\n\\setcounter{equation}{0}\n\n\\begin{appendix}\n\n\n\\section{The translation mode}\n\\setcounter{equation}{0}\n\\label{app:translationmode}\nThe fluctuation operator  for the coupled system\nof transverse gauge and Higgs fields\nwas derived in Ref. \\cite{Baacke:2008sq}\nin a basis of partial waves with ``magnetic'' quantum numbers\n$m$, proportional to $\\exp(im\\varphi)$. We refer to\nthis reference for details. Essentially, the amplitudes\n$F_4^m$ corresponds to the real part of the Higgs field\n$\\varphi_1$, $F_3^m$ to the imaginary part of the\nHiggs field $\\varphi_2$, and the amplitudes $F_1^m$ and $F_2^m$ to\ncombinations of the transverse gauge fields $a_1,a_2$. The\nbasis was chosen in such a way that the fluctuation operator\nbecomes symmetric, and the amplitudes are real relative to each other.\nThe amplitudes $F_i$ for $m= 1$ satisfy the following \ncoupled system of linear\ndifferential equations:\n\\begin{eqnarray}\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}+m_W^2 f^2\\right\\}\nF^1_1+\\sqrt{2}m_W f' F^1_3+\\sqrt{2}m_W\\frac{A+1}{r}f F^1_4=0&&\n\\\\\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}+\\frac{4}{r^2}+m_W^2 f^2\\right\\}\nF^1_2+\\sqrt{2}m_W f' F^1_3-\\sqrt{2}m_W\\frac{A+1}{r}f F^1_4=0&& \n\\\\\\nonumber\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}+\\frac{1}{r^2}+\n\\frac{(A+1)^2}{r^2}+m_W^2 f^2+\\frac{m_H^2}{2}\\left(f^2-1\\right)\\right\\}\nF^1_3&&\n\\\\+\\sqrt{2}m_W f'(F^1_1+F_2)-2\\frac{A+1}{r} F^1_4=0&& \n\\\\\n\\nonumber\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}+\\frac{1}{r^2}+\n\\frac{(A+1)^2}{r^2}+\\frac{m_H^2}{2}\\left(3f^2-1\\right)\\right\\}\nF^1_4&&\n\\\\+\\sqrt{2}m_W\\frac{A+1}{r}(F^1_1-F^1_2)-2\\frac{A+1}{r} F^1_3=0&& \n\\; .\\end{eqnarray}\nThe last equation corresponds to the real part of the fluctuations\nof the field $\\phi$, and the translation mode is obtained\nas $\\nabla \\phi_{cl}=\\hat x v f'$. We therefore start with the ansatz\n\\begin{equation}\nF^1_4= c f'\n\\; ,\\end{equation}\nwhere the coefficient $c$ is a prefactor that will be fixed\nlater. Applying the derivative $d\/dr$ to the equation of motion\nfor $F_4^1$ we find\n\\begin{eqnarray}\\nonumber\n\\frac{d}{dr}\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{\\left[A(r)+1\\right]^2}{r^2}+\\frac{m^2_H}{2}\\left[f^2(r)-1\\right]\n\\right\\}f(r)&=&\n\\\\\\nonumber\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{\\left[A(r)+1\\right]^2}{r^2}+\\frac{m^2_H}{2}\\left[3f^2(r)-1\\right]\n\\right\\}f'&&\n\\\\\n-2\\frac{(A+1)^2}{r^3}f+2\\frac{A+1}{r^2}fA'&=&0\n\\; .\\end{eqnarray}\nThis is the  equation of motion for $F_1$ if the choose\n\\begin{eqnarray}\nF_3^1=c\\frac{A+1}{r}f\n\\\\\nF_1^1-F_2^1=c \\frac{\\sqrt{2}}{m_W}\\frac{A'}{r}\n\\; .\n\\end{eqnarray}\nThe assignment has to be checked for consistency with the\nremaining equations of motion. Multiplying the equation of motion of \n$f(r)$ with $(A+1)\/r$ and commuting this factor with \nthe derivatives we find\n\\begin{eqnarray}\\nonumber\n\\frac{A+1}{r}\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{\\left[A(r)+1\\right]^2}{r^2}+\\frac{m^2_H}{2}\\left[f^2(r)-1\\right]\n\\right\\}f(r)&=&\n\\\\\\nonumber\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{1}{r^2}+\\frac{\\left[A(r)+1\\right]^2}{r^2}+\\frac{m^2_H}{2}\\left[f^2(r)-1\\right] + m_W^2 f^2\n\\right\\}\\frac{A+1}{r}f(r)&&\n\\\\\n-2\\frac{(A+1)^2}{r^2}f'+2f'\\frac{A'}{r}=0&&\n\\; .\\end{eqnarray}\nIn the intermediate steps we have used the equation of motion\nfor $A(r)$ in order to replace a second derivative $A\"$.\nThe result is consistent with the previous assignement if we\nchoose\n\\begin{equation}\nF_1^1+F_2^2=c \\frac{\\sqrt{2}}{m_W}\\frac{A'}{r}\n\\; .\\end{equation}\nSo we find\n\\begin{eqnarray}\nF_2^1&=&0\n\\\\\nF_1^1&=&c \\frac {\\sqrt{2}}{m_W}\\frac{A'}{r}\n\\; .\\end{eqnarray} \nThis has to be verified by deriving the equation of motion for\n$A'\/r$. We obtain it by applying $1\/r~d\/dr$ to the \nclassical equation of motion for $A(r)$:\n\\begin{eqnarray}\\nonumber\n\\frac{1}{r}\\frac{d}{dr}\\left\\{\n-r \\frac{d}{dr}\\frac{1}{r}\\frac{d}{dr}\n+m^2_W f^2\\right\\}\n\\left[A+1\\right]&=&\n\\\\\n\\left\\{-\\frac{1}{r}\\frac{d}{dr}r \\frac{d}{dr}+m_W^2 f^2\\right\\}\n\\frac{A'}{r}+2m_W^2\\frac{A+1}{r} ff'=0&&\n\\; .\n\\end{eqnarray}\nThis is consistent with the previous assigments and the equation\nof motion for $F_1^1$.\n\nSo we have derived that the wave function of the translation\nmode with $m=1$ is given by\n\\begin{equation}\n\\left(\\begin{array}{c}F_1^1\\\\F_2^1\\\\F_3^1\\\\F_4^1\\end{array}\n\\right)=c\\left(\\begin{array}{c}\n\\sqrt{2}\/m_W~A'\/r\\\\\n0\\\\\n(A+1)f\/r\\\\\nf'\\end{array}\\right)\n\\end{equation}\nWe have started this derivation by considering the gradient\napplied to the scalar field $\\phi=v f(r)$, but this has fixed\nonly the component $F_4$. The component $F_3$ is easily seen\nas arising from replacing the derivatives $\\nabla_i$ by the\ncovariant derivatives $\\nabla_i-ig A_i$. The wave function for the \nvector potential is not given by an infinitesimal shift of the\nclassical potential, but it correctly describes the shift in\nthe classical magnetic field.\n\nThere is of course a second translation mode, with $m=-1$. Its wave function\nis given by\n\\begin{equation}\n\\left(\\begin{array}{c}F_1^{-1}\\\\F_2^{-1}\\\\F_3^{-1}\\\\F_4^{-1}\\end{array}\n\\right)=c'\\left(\\begin{array}{c}\n0\\\\\n\\sqrt{2}\/m_W~A'\/r\\\\\n-(A+1)f\/r\\\\\nf'\\end{array}\\right)\n\\end{equation}\nFrom these wave functions in the azimutal basis and in terms\nof the amplitudes $F_i^{\\pm 1}$ we go back to the physical basis\n$\\varphi_1,\\varphi_2,a_1,a_2$.\nThese are related to the functions $F_i^1$\nvia \\footnote{The prefactors appearing in the definitions\nof $F_1$ and $F_2$, Eq. (4.2) of Ref. \\cite{Baacke:2008sq}\nshould be $1\/\\sqrt 2$, not $1\/2$. This misprint also appears\nin Refs. \\cite{Baacke:1994bk} and \\cite{Baacke:2008zx}}\n\\begin{equation}\n\\left(\\begin{array}{c}a_1\\\\a_2\\\\\\varphi_1\\\\\\varphi_2\\end{array}\n\\right)=\\frac{1}{\\sqrt{2\\pi}}\\left(\\begin{array}{c}\n(F_1^1+F_1^{1*})\/\\sqrt{2}\\\\\ni(F_1^1-F_1^{1*}\/\\sqrt{2}\\\\\n-i[F_4^1\\exp(i\\varphi)-F_4^{1*}\\exp(-i\\varphi)]\\\\\nF_3^1\\exp(i\\varphi)+F_3^{1*}\\exp(-i\\varphi)\n\\end{array}\\right)\n\\end{equation}\nHere we have used reality constraints for the fields in order\nto eliminate the functions $F_i^{-1}$.\nThe wave functions corresponding to infinitesimal translations\ninto the $x$ and $y$ directions are obtained by\nchoosing the free coefficient $c$ is such a way that the real part of the\nHiggs field fluctuation $\\varphi_1$ is given by $d \\phi_{\\rm cl}\/dx$\nand $d \\phi_{\\rm cl}\/dy$, respectively. Explicitly, \n$c_x=iv\\sqrt{\\pi\/2}$ and $c_y=v \\sqrt{\\pi\/2}$.\nThe complete wave functions  are \ngiven in section \\ref{sec:collectivestringoscillations}.\n\n\n\n\n\\section{The virial theorem}\n\\label{app:virialtheorem}\n\\setcounter{equation}{0}\nAs we have discussed in section \\ref{sec:collectivestringoscillations}\nthe normalization of the translation mode derived from the\ndeformation of the classical solution\nis given by\n\\begin{equation}\n|\\psi_t|^2=\\pi v^2\\int r~dr~\\left\\{\\frac{2}{m_W^2}\\frac{A'{}^2(r)}{r^2}+\n\\frac{(A(r)+1)^2}{r^2}f^2(r)+f'{}^2(r)\\right\\}\n\\end{equation}\nThe classical string tension is given by Eq. \\eqn{eq:classtens},\ni.e., \n\\begin{eqnarray}\\nonumber\n\\sigma_{\\rm cl}&=&\\pi v^2\\int r~dr~\\left\\{\\frac{1}{m_W^2}\\frac{A'{}^2(r)}{r^2}+\n\\frac{(A(r)+1)^2}{r^2}f^2(r)+f'{}^2(r)\\right.\n\\\\\n&&\\left.\\hspace{25mm} +\\frac{m_H^2}{4}\\left[f^2(r)-1\\right]^2\\right\\}\n\\end{eqnarray}\nIn analogy with the virial theorem for quantum kinks, where the normalization\nof the translation mode is equal to the classical mass,\nwe expect a virial theorem $|\\psi_t|^2=\\sigma_{\\rm cl}$ which\nreduces to the identity\n\\begin{equation}\n\\int r~dr~\\frac{1}{m_W^2}\\frac{A'{}^2(r)}{r^2}\n=\\int r~dr~\\frac{m_H^2}{4}\\left(f^2(r)-1\\right)^2\n\\; .\\end{equation}\nIt can readily be  verified numerically. In order\nto derive the relation analytically we use the following weighted integrals \nover the classical equations of motion\n\\begin{eqnarray}\n&&I_1=\\int r~dr~f\\left\\{\n-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{\\left[A+1\\right]^2}{r^2}+\n\\frac{m^2_H}{2}\\left(f^2-1\\right)\n\\right\\}f=0\n\\\\\n&&I_2=\\int r~dr~f r\\frac{d}{dr}\\left\\{\n-\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr}\n+\\frac{\\left[A+1\\right]^2}{r^2}+\n\\frac{m^2_H}{2}\\left(f^2-1\\right)\n\\right\\}f=0\n\\\\\n&&I_3=\\int dr~(A+1)\\frac{d}{dr}\n\\left\\{-r\\frac{d}{dr}\\frac{1}{r}\\frac{d}{dr}+m^2_W f^2\\right\\}\n\\left(A+1\\right)=0 \\hspace{25mm}\n\\end{eqnarray}\nIntegrating by parts these take the form\n\\begin{eqnarray} \nI_1 &=&\\int r~dr~\\left\\{\\left(\\frac{df}{dr}\\right)^2\n+\\frac{(A+1)^2}{r^2}f^2+\\frac{m_H^2}{2}f^2(f^2-1)\\right\\}=0 \n\\\\\\nonumber\nI_2 &=&\\int r~dr~\\left\\{-2\\left(\\frac{df}{dr}\\right)^2\n-2\\frac{(A+1)^2}{r^2}f^2-m_H^2f^2(f^2-1)\\right.\n\\\\\n&&\\left.\\hspace{25mm}+\\frac{m_H^2}{4}(f^2-1)^2-\\frac{(A+1)^2}{r}ff'\\right\\}=0 \n\\\\\nI_3 &=& \\int r~dr~\\left\\{-\\frac{A'{}^2}{r^2}\n+m_W^2\\frac{(A+1)^2}{r}ff'\\right\\}\n\\end{eqnarray}\nCombining the first and second integral we find\n\\begin{equation}\nI_2+2I_1=\\int r~dr~\\left\\{\\frac{m_H^2}{4}(f^2-1)^2-\n\\frac{(A+1)^2}{r}ff'\\right\\}=0\n\\end{equation}\nAdding the third integral we obtain the relation\n\\begin{equation}\nI_2+2I_1+\\frac{1}{m_W^2}I_3=\\int r~dr~\\left\\{\n-\\frac{1}{m_W^2}\\frac{A'{}^2}{r^2}+\n\\frac{m_H^2}{4}\\left(f^2-1\\right)^2\\right\\}=0\n\\end{equation}\nwhich is the expected result.\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nMagnetic reconnection in the lower solar atmosphere manifests itself through dynamical events such as jets \\citep[e.g.,][]{shibata:2007, Tian:2014:IRIS}, explosive events such as Ellerman bombs \\citep[][]{Ellerman:1917, PFChen:2001, CNelson:2013, HYang:2013, Peter:2014:IRIS}, and possibly Type II spicules \\citep{depontieu:2007}.  Reconnection events contribute to the heating of the chromosphere \\citep[e.g.,][]{jess:2014}, and the contribution of jets and Type II spicules to the mass and energy budgets of the corona and solar wind is under active investigation \\citep{depontieu:2011, cranmer:2010, madjarska:2011}.  \\cite{arge:1998} proposed that chromospheric reconnection could cause the elemental fractionation that is responsible for the first ionization potential (FIP) effect \\citep[see also][]{sturrock:1999A, feldman:2002}.  Partial ionization effects are important not just in the chromosphere but also in molecular clouds, protoplanetary disks, the neutral phases of the interstellar medium, some exoplanetary atmospheres \\citep{koskinen:2010}, Earth's ionosphere \\citep{leake:2014}, the edge of tokamaks \\citep{mladams:thesis}, and dedicated reconnection experiments \\citep[e.g.,][]{lawrence:2013}.\n\nTheories and simulations of reconnection have generally assumed that the plasma is fully ionized.  This approximation is valid for the fully ionized solar corona but invalid for weakly ionized plasmas such as the solar chromosphere which has ionization fractions ranging from {$\\lesssim$}$0.01$ to {$\\sim$}$0.5$.  Partial ionization effects modify the dynamics of reconnection in several ways \\citep{zweibel:1989, zweibel:2011}.  Before the onset of reconnection, current sheets thin significantly due to ambipolar diffusion \\citep{brandenburg:1994, brandenburg:1995}.  When there is strong coupling between ions and neutrals (e.g., on length scales longer than the neutral-ion mean free path, $\\lambda_{ni}$), the effective ion mass is increased by the ratio of the total mass density $\\rho$ to the ion mass density $\\rho_i$.  This decreases the bulk Alfv\\'en speed and consequently the predicted reconnection rate \\citep{zweibel:1989}.  The plasma resistivity has contributions from both electron-ion and electron-neutral collisions \\cite[e.g.,][]{piddington:1954, cowling:1956, ni:2007}.  \n\nThe Hall effect is expected to be important on scales comparable to or less than the ion inertial length $d_i$.  However, the effective ion inertial length is predicted to be enhanced in plasmas with strong ion-neutral coupling: $d_i' = d_i\\sqrt{\\rho\/\\rho_i}$ \\citep[e.g.,][]{pandey:2008a}.  Consequently, it has been proposed that Hall reconnection may occur at longer length scales or larger densities than would be predicted from the ion density alone \\citep{malyshkin:2011, vekstein:2013}.  \\citet{malyshkin:2011} predict that enhancement of the Hall effect will lead to fast reconnection in molecular clouds and protoplanetary disks, but that this transition is less likely to be important in the solar chromosphere.  In general, two regimes may be considered.  \nWhen $\\lambda_{ni} \\ll d_i$, the Hall effect will be enhanced because of strong coupling and fast reconnection is predicted to occur.  When $d_i \\lesssim \\lambda_{ni} \\lesssim d_i'$, some enhancement of the Hall effect due to ion-neutral coupling is also expected. However, as shown previously by \\citet[hereafter, \\citetalias{leake:partial1}]{leake:partial1} and \\citet[hereafter, \\citetalias{leake:partial2}]{leake:partial2}, the ion density (and therefore also the local values for $d_i$, $d_i'$, and $\\lambda_{ni}$)  may vary by more than an order of magnitude between the reconnection current sheet and the ambient plasma, which makes it difficult to \\emph{a priori} evaluate the importance of the Hall effect in determining the rate of reconnection and the structure of the reconnection region in this parameter regime.\n\nThe behavior of weakly ionized plasmas can be modeled using either single-fluid or multi-fluid formulations.  A single-fluid approach incorporates the ambipolar diffusion term into the generalized Ohm's law to account for relative drift between ions and neutrals \\citep[e.g.,][]{brandenburg:1994, brandenburg:1995}.  This approach is less computationally expensive but implicitly assumes that the ions and neutrals are in ionization equilibrium.  Multi-fluid models evolve each of the ion, neutral, and sometimes electron components separately to allow relative drifts between each of these populations and the fluid to be out of ionization equilibrium \\citep[e.g.,][]{meier:2012A}.\n\nIn recent years, considerable progress has been made in modeling magnetic reconnection in the lower solar atmosphere. \\citet{sakai:2006}, \\citet{smith:2008}, \\citet{sakai:2008}, and \\citet{sakai:2009} performed two-fluid (ion-neutral) simulations of coalescing current loops in the solar chromosphere.  \\citet{smith:2008} found that the rate of reconnection in simulations of the upper chromosphere was about $\\sim${$20$} times greater than in the lower chromosphere which has a considerably lower ionization fraction.  \\cite{sakai:2008} and \\citet{sakai:2009} investigated the role of reconnection in penumbra filaments.  These models assume fixed ionization and recombination rates rather than assuming a dependence on temperature and density.  However, it is essential to incorporate rates that depend on physical conditions in order to accurately capture the important role of recombination during chromospheric reconnection.\n\n\\citetalias{leake:partial1} and \\citetalias{leake:partial2} used the plasma-neutral module of the HiFi framework \\citep{lukin:thesis, meier:thesis} to model chromospheric reconnection.  The simulations showed a strong enhancement of ion density inside the current sheet as a result of ions being preferentially dragged along by the reconnecting magnetic field.  Recombination and ion outflow were of comparable importance in the ion continuity equation for removing ions from the current sheet.  The ions and neutral inflows were decoupled, but the ion and neutral outflow jets were strongly coupled.  Late in time, the simulations by \\citetalias{leake:partial1} and \\citetalias{leake:partial2} showed the development of the secondary tearing instability known as the plasmoid instability \\citep[][]{loureiro:2007}.  The increase in the ion and electron densities in the plasmoids led to an increase in the recombination rate that allowed further contraction of the magnetic islands on timescales comparable to the advection time of islands out of the current sheet.  \\citet{ni:2015} present one-fluid simulations of the plasmoid instability in the solar chromosphere using the NIRVANA code with and without guide fields.  They find that the onset of the plasmoid instability leads to fast reconnection, and that slow shocks develop near the X-points.  Cases without guide fields show rapid thinning of the current sheet due to ambipolar diffusion and radiative cooling, but these effects are not significant during guide field simulations.\n\nPrior models of magnetic reconnection in weakly ionized plasmas have generally assumed symmetric inflow.  In general, however, there will be some asymmetry in the upstream magnetic field strengths, temperatures, and densities.  The standard model for anenome jets predicts that chromospheric reconnection occurs when newly emerged flux interacts with pre-existing, overlying flux \\citep[e.g.,][]{shibata:2007}.  It is reasonable to expect that these different plasma domains will have different plasma parameters, so this configuration naturally leads to asymmetric reconnection.  Because the chromosphere is a dynamic magnetized environment \\citep{leenaarts:2007}, asymmetry in the reconnection process is likely to be the norm.\n\nThe physics of asymmetric inflow reconnection has been investigated in detail for fully ionized plasmas.  One of the principal applications of this work has been Earth's dayside magnetopause. Asymmetric inflow reconnection has also been investigated in the context of Earth's magnetotail and elsewhere in the magnetosphere \\citep{oieroset:2004, muzamil:2014}, the solar atmosphere \\citep{NNakamura:2012, murphy:double, YingnaSu:2013:prominence, YangSu:2013}, laboratory experiments \\citep{yamada:2007, yoo:2014, murphy:mrx}, and plasma turbulence \\citep{servidio:2009, servidio:2010}.  \\citet{cassak:asym} performed a scaling analysis for asymmetric inflow reconnection.  They found that the outflow is governed by a hybrid upstream Alfv\\'en speed that is a function of the magnetic field strength and density in both upstream regions \\citep[see also][]{birn:2010} and that the flow stagnation point in the simulation frame and magnetic field null were not colocated.  The structure and dynamics of asymmetric reconnection in fully ionized collisionless and two-fluid plasmas have been studied previously in several works \\citep[e.g.,][]{cassak:hall, cassak:dissipation, malakit:2010, malakit:2013, pritchett:2008:asym, pritchett:2009, mozer:2008, mozer:2009, swisdak:2003, aunai:2013b, aunai:2013a, murphy:mrx}. Simulations of the plasmoid instability during reconnection with asymmetric upstream magnetic fields by \\citet{murphy:plasmoidasym} showed that the resultant magnetic islands developed primarily into the weak field upstream region.  Because the reconnection jets impacted the islands obliquely rather than directly, the islands developed net vorticity.  In addition to asymmetric inflow reconnection, several groups have investigated asymmetric outflow reconnection \\citep[e.g.,][]{oka:2008, murphy:asym, murphy:retreat} and reconnection with three-dimensional asymmetry \\citep[e.g.,][]{AlHachami:2010, wyper:2013}.  \n\nObservational investigations of partially ionized reconnection in the chromosphere are challenging because the dissipation length scales are significantly shorter than can be resolved with current instrumentation.  Diagnosing the chromospheric magnetic field is an important but very difficult problem that requires inversion of spectropolarimetric data \\citep[e.g.,][]{kleint:2012}, and this will be even more challenging for short-lived dynamical events.  Comparisons between simulations and observations should therefore concentrate on the large-scale consequences of chromospheric reconnection which depend on small-scale processes.  A complementary approach to solar observations is to study partially ionized reconnection in the laboratory.  \\citet{lawrence:2013} have performed such studies at the Magnetic Reconnection Experiment \\citep[MRX;][]{yamada:1997a}.\n\nOur motivation is to investigate the role of asymmetry on the small-scale physics of reconnection in partially ionized chromospheric plasmas.  While there are similarities to symmetric cases, there are also physical effects such as neutral flows through the current sheet that do not occur in cases with symmetric inflow.  We use the plasma-neutral module of the HiFi modeling framework \\citep{lukin:thesis} to perform simulations of magnetic reconnection with asymmetric upstream magnetic field strengths, temperatures, and\/or densities in the weakly ionized solar chromosphere.  In Section \\ref{numerical}, we describe the numerical method and problem setup (see also \\citetalias{leake:partial1} and \\citetalias{leake:partial2}). In Section \\ref{global}, we describe the global dynamics of reconnection, the structure of the current sheet, the role of the Hall effect, the dynamics of the plasmoid instability, the motion of the X-point, and ion\/neutral flows at the X-point.  We discuss our results in Section \\ref{discussion}.\n\n\\section{NUMERICAL MODEL AND PROBLEM SETUP\\label{numerical}}\n\nThe HiFi framework\\footnote{See \\begin{tt}http:\/\/faculty.washington.edu\/vlukin\/HiFi\\_Framework.html\\end{tt}}\n\\citep{lukin:thesis, glasser:2004} uses a spectral element spatial representation and implicit time advance to solve systems of partial differential equations.  The modular approach makes new physical models straightforward to implement \\citep[e.g.,][]{gray:2010B, lukin:2011, ohia:2012, le:2013, stanier:2013, browning:2014, elee:2014}.  In this paper, we use the module for partially ionized, reacting plasmas \\citep[][\\citetalias{leake:partial1}, \\citetalias{leake:partial2}]{meier:thesis}.  These plasmas consist of neutral and ionized hydrogen and electrons.  A multi-fluid approach allows the plasma and neutral components to be modeled separately.  We summarize the equations used for our 2.5D simulations in this section, but direct the reader to \\citetalias{leake:partial1} and \\citetalias{leake:partial2} for further detail.  In general, we use the notation and conventions from \\citetalias{leake:partial1} and \\citetalias{leake:partial2}\\@.  The subscripts `n', `i', and `e' refer to neutrals, ions, and electrons, respectively.  \n\n\\subsection{Normalizations\\label{normalizations}}\n\nFollowing \\citetalias{leake:partial1} and \\citetalias{leake:partial2}, we normalize the fluid equations to values characteristic of the lower chromosphere.  We choose a characteristic length scale of $L_\\star\\equiv 1\\times 10^4$ m, a characteristic number density of $n_\\star \\equiv 3\\times 10^{16}$ m$^{-3}$, and a characteristic magnetic field strength of $B_\\star \\equiv 1\\times 10^{-3}$ T\\@.  From these quantities, we derive additional normalizing values to be $V_\\star\\equiv B_\\star \/ \\sqrt{\\mu_0 m_p n_\\star} = 1.26\\times 10^5$ m\\,s$^{-1}$ for velocity, $t_\\star \\equiv L_\\star \/ V_\\star = 0.0794$ s for time, $T_\\star \\equiv B_\\star^2 \/ k_B \\mu_0 n_\\star = 1.92 \\times 10^6$ K for temperature, $P_\\star \\equiv B_\\star^2\/\\mu_0 = 0.796$ Pa for pressure, $J_\\star \\equiv B_\\star\/\\mu_0 L_\\star = 7.96\\times 10^{-2}$ A\\,$\\cdot$\\,m$^{-2}$ for current density, and $\\eta_\\star \\equiv \\mu_0 L_\\star V_\\star = 1.58 \\times 10^3$ $\\Omega$\\,{$\\cdot$}\\,m for resistivity.  Unless otherwise indicated (e.g., Sections \\ref{irce} and \\ref{initial}), the equations presented for the simulation will be in dimensionless units according to these normalizations.  \n\n\\subsection{Ionization and Recombination\\label{irce}}\n\nThe HiFi module for partially ionized plasmas includes both ionization and recombination of hydrogen to allow departures from ionization equilibrium \\citepalias{leake:partial1, leake:partial2}.  The ionization and recombination rates are given by\n\\begin{eqnarray}\n  \\Gamma^{ion}_n & \\equiv & -n_n \\nu^{ion}, \\label{ionrate}  \\\\\n  \\Gamma^{rec}_i & \\equiv & -n_i \\nu^{rec}, \\label{recrate}\n\\end{eqnarray}\nwith $\\Gamma^{ion}_i = -\\Gamma^{ion}_n$ and $\\Gamma^{rec}_i =\n-\\Gamma^{rec}_n$.  The ionization frequency of hydrogen is approximated to be\n\\begin{equation}\n  \\nu^{ion} = \\frac{n_e A}{X + \\phi_{ion}\/T_e^*}\n  \\left(\\frac{\\phi_{ion}}{T_e^*}\\right)^K \n  \\exp\\left(-\\frac{\\phi_{ion}}{T_e^*}\\right)\n  \\mbox{~m}^3\\mbox{\\,s}^{-1}, \\label{ionfreq}\n\\end{equation}\nwith $A = 2.91\\times 10^{-14}$, $K=0.39$, $X=0.232$, and $\\phi_{ion} = 13.6$ eV \\citep{voronov:1997}. Here, $T_e^*$ is the electron temperature in eV\\@. \\citet{smirnov:2003} approximates the recombination frequency of hydrogen to be\n\\begin{equation}\n  \\nu^{rec} = 2.6 \\times 10^{-19} \\frac{n_e}{\\sqrt{T_e^*}} \n    \\mbox{~m}^3\\mbox{\\,s}^{-1}\n    \\label{recfreq}.\n\\end{equation}\nAt a characteristic temperature of $9000$ K, this corresponds to an equilibrium ionization fraction of $4.1\\times 10^{-4}$.  These expressions do not include the consequences of non-local thermodynamic equilibrium radiative transfer.  For example, if Lyman $\\alpha$ is optically thick, then there will be more neutral hydrogen in the $n=2$ state than predicted from these expressions which would allow for enhanced ionization. At lower temperatures, ionization from low-FIP elements may contribute more to the electron density than hydrogen.\n\n\\subsection{Multi-fluid Equations\\label{fluidequations}}\n\nIn this section, we summarize the equations solved by the plasma-neutral module of HiFi.  The equations are described more thoroughly by \\citetalias{leake:partial1} and \\citetalias{leake:partial2} \\citep[see also][]{meier:thesis, meier:2012A}.  Our simulations include the modifications and extensions to the model described by \\citetalias{leake:partial2}.  \n\nThe ion and neutral continuity equations are\n\\begin{eqnarray}\n  \\frac{\\partial n_i}{\\partial t} + \\nabla\\cdot\n  \\left(n_i\\ensuremath{\\mathbf{V}_i}\\right) & = & \\Gamma^{rec}_i + \\Gamma^{ion}_i,\n  \\\\ \n  \\frac{\\partial n_n}{\\partial t} + \\nabla\\cdot\n  \\left(n_n\\ensuremath{\\mathbf{V}_n}\\right) & = & \\Gamma^{rec}_n + \\Gamma^{ion}_n.\n\\end{eqnarray}\nThese equations include ionization and radiative recombination as described in Section \\ref{irce}, \\citetalias{leake:partial1}, and \\citetalias{leake:partial2}.  We assume quasineutrality such that $n_i=n_e$.\n\nThe ion and neutral momentum equations are given by\n\\begin{eqnarray}\n  \\frac{\\partial }{\\partial t}\n  \\left(m_in_i\\ensuremath{\\mathbf{V}_i}\\right) + \n  \\nabla \\cdot \\left( m_in_i\\ensuremath{\\mathbf{V}_i}\\vi + \\ensuremath{\\mathsf{P}_i} + \\ensuremath{\\mathsf{P}_e} \\right) = \n  \\hspace{6cm}\\nonumber\\\\ \\hspace{2cm}\n  \\ensuremath{\\mathbf{J}} \\times \\ensuremath{\\mathbf{B}} + \\ensuremath{\\mathbf{R}^{in}_i} + \\Gamma^{ion}_im_i\\ensuremath{\\mathbf{V}_n} - \\Gamma^{rec}_nm_i\\ensuremath{\\mathbf{V}_i} \n  + \\Gamma^{cx}m_i\\left(\\ensuremath{\\mathbf{V}_n}-\\ensuremath{\\mathbf{V}_i}\\right) +\\ensuremath{\\mathbf{R}^{cx}_{in}} - \\ensuremath{\\mathbf{R}^{cx}_{ni}}, \\label{momentum_plasma}\n  \\\\\n  \\frac{\\partial }{\\partial t}\n  \\left(m_in_n\\ensuremath{\\mathbf{V}_n}\\right) + \n  \\nabla \\cdot \\left( m_in_n\\ensuremath{\\mathbf{V}_n}\\vn + \\ensuremath{\\mathsf{P}_n} \\right) = \n  \\hspace{6cm}\\nonumber\\\\ \\hspace{2cm}\n  -\\ensuremath{\\mathbf{R}^{in}_i} + \\Gamma^{rec}_nm_i\\ensuremath{\\mathbf{V}_i} - \\Gamma^{ion}_im_i\\ensuremath{\\mathbf{V}_n}\n  + \\Gamma^{cx} m_i\\left(\\ensuremath{\\mathbf{V}_i}-\\ensuremath{\\mathbf{V}_n}\\right)\n  -\\ensuremath{\\mathbf{R}^{cx}_{in}} + \\ensuremath{\\mathbf{R}^{cx}_{ni}} \n  . \\label{momentum_neutrals}\n\\end{eqnarray}\nThe Lorentz force acts directly on the plasma but not the neutrals, while the neutral pressure gradient acts directly on the neutrals but not the plasma.  Coupling between the plasma and neutrals is achieved though many of the terms on the right hand side of Eqs.\\ \\ref{momentum_plasma} and \\ref{momentum_neutrals}. The term $\\ensuremath{\\mathbf{R}^{in}_i}$ represents momentum transfer from neutrals to ions due to identity preserving collisions and is given by\n\\begin{equation}\n    \\ensuremath{\\mathbf{R}^{in}_i} = m_{in} n_i \\nu_{in} \\left(\\ensuremath{\\mathbf{V}_n}-\\ensuremath{\\mathbf{V}_i}\\right), \\label{rinidef}\n\\end{equation}\nwhere $m_{in} = m_i m_n \/(m_i+m_n)$ and the collision frequency $\\nu_{in}$ is given by Eq.\\ 7 of \\citetalias{leake:partial2}.  The pressure tensor for species $\\alpha$ is\n\\begin{equation}\n    \\mathsf{P}_\\alpha=P_\\alpha\\mathsf{I} + \\pi_\\alpha\n    \\label{ptensdef},\n\\end{equation}\nwhere $P_\\alpha$ is the scalar pressure and $\\mathsf{I}$ is the identity tensor.  The viscous stress tensor is then\n\\begin{equation}\n    \\pi_\\alpha = - \\xi_\\alpha\\left[\\nabla\\mathbf{V}_\\alpha + \\left(\\nabla\\mathbf{V}_\\alpha\\right)^\\top\\right],\n\\end{equation}\nwith $\\xi_\\alpha$ as the isotropic dynamic viscosity coefficient.  The terms that depend on the ionization and recombination rates represent momentum transfer by particles that change identity from neutrals to ions and vice-versa. We follow \\citetalias{leake:partial2} and include charge exchange. Here, $\\Gamma^{cx}$ is the charge exchange reaction rate and $\\mathbf{R}^{cx}_{\\alpha\\beta}$ is the momentum transfer due to charge exchange rections from species $\\beta$ to species $\\alpha$.  Charge exchange leads to increased momentum and energy transfer between species by about a factor of two, and therefore more effective coupling.  We use the formulation for charge exchange given by Eqs.\\ 4--6 of \\citetalias{leake:partial2} \\citep[see also][]{meier:thesis, meier:2012A, leake:partial1, Barnett:1990}.\n\nWe adjust the collisional cross sections to be $\\Sigma_{in}=\\Sigma_{ni}=5\\times 10^{-19}$ m$^2$ (used in Eq.\\ 7 of \\citetalias{leake:partial2} to calculate $\\nu_{in}$) and $\\Sigma_{nn}=5\\times 10^{-19}$ m$^2$ (used in Eq.\\ 9 of \\citetalias{leake:partial2} to calculate $\\nu_{nn}$).  These values differ from \\citetalias{leake:partial1} and \\citetalias{leake:partial2}, but the value for $\\Sigma_{in}$ is consistent with \\citet{khomenko:2012} and \\citet{ni:2015}.  These cross sections are a function of energy, but we use constant values appropriate for the chromosphere as a simplifying assumption \\citep[see also][]{draine:1983}.  The neutral and ion viscosity coefficients $\\xi_n$ and $\\xi_i$ are set by Eq.\\ 8 of \\citetalias{leake:partial2} using the revised value of $\\Sigma_{nn}$.  While $\\xi_n$ is a function of local physical conditions, $\\xi_i$ and $\\xi_e$ are constant throughout the domain and correspond to the parallel component of the Braginskii viscosity for each species computed using the mean of the asymptotic upstream densities and temperatures.  We use $\\xi_i=2.8\\times 10^{-6}$ and $\\xi_e=4.6\\times 10^{-14}$ with a normalization of $\\xi_\\star=m_p n_\\star L_\\star V_\\star$.  \n\nThe neutral-ion mean free path\n\\begin{equation}\n    \\lambda_{ni} = \\frac{V_{T,n}}{\\nu_{ni}^\\dagger}  \\label{lambdanidef}\n\\end{equation}\ngoverns the lengths scales above which the neutrals and ions are coupled to each other.  Here, the neutral thermal velocity is $V_{T,n}=\\sqrt{2 k_B T_n\/m_n}$ where $k_B$ is the Boltzmann constant, the neutral temperature is $T_n$, and the neutral mass is $m_n$.  The frequency $\\nu_{ni}^\\dagger = \\nu_{ni} + \\nu_{ni}^{CX}$ includes contributions from the neutral-ion collision frequency $\\nu_{ni}$ (see Eq.\\ 16 of \\citetalias{leake:partial1}) and the neutral-ion charge exchange frequency $\\nu_{ni}^{CX}$ (see also Eqs.\\ 4--6 of \\citetalias{leake:partial2}).  The ions and neutrals will be coupled on length scales much longer than $\\lambda_{ni}$, and decoupled on scales much shorter than $\\lambda_{ni}$.\n\nThe energy equations for the plasma and neutral components are given by Eqs.\\ 19 and 20 of \\citetalias{leake:partial1}.  Both equations include frictional heating due to identity preserving collisions, thermal transfer due to changes in identity, and the effects of charge exchange.  The energy equation for the plasma component combines the electron and ion energy equations and includes Ohmic heating, optically thin radiative losses, and anisotropic thermal conduction \\citepalias[Eq.\\ 21 of][]{leake:partial1}. The neutral energy equation includes isotropic thermal conduction that depends on plasma parameters and is set by Eq.\\ 10 of \\citetalias{leake:partial2}.  Neutral thermal conduction dominates thermal diffusion, in part due to rapid thermal transfer between neutrals and ions.  For characteristic values of $T=9000$ K and $n_n=7.6\\times 10^{18}$ m$^{-3}$, the neutral thermal conductivity is $\\kappa_n=3.05\\times 10^{22}$ m$^{-1}$\\,s$^{-1}$. We assume that the ion and electron temperatures are equal, but the neutral temperature is evolved separately.  \n\nThe generalized Ohm's law for this paper is given by\n\\begin{equation}\n    \\ensuremath{\\mathbf{E}} + \\ensuremath{\\mathbf{V}_i}\\times\\ensuremath{\\mathbf{B}} \n    =\n    \\eta\\ensuremath{\\mathbf{J}}\n    + \\frac{\\ensuremath{\\mathbf{J}}\\times\\ensuremath{\\mathbf{B}}}{e n_e} \n    - \\frac{\\nabla P_e}{e n_e}\n    - \\frac{m_e \\nu_{en}}{e} \\left(\\ensuremath{\\mathbf{V}_i}-\\ensuremath{\\mathbf{V}_n}\\right) \\label{genohms}\n\\end{equation}\nThe resistivity includes both electron-ion and electron-neutral collisions and is given by\n\\begin{equation}\n    \\eta = \\frac{m_e n_e \\left(\\nu_{ei} + \\nu_{en}\\right)}{\\left( e n_e \\right)^2},\n\\end{equation}\nwhere the electron-ion collision frequency $\\nu_{ei}$ and the electron-neutral collision frequency $\\nu_{en}$ are functions of number density and temperature and are given by Eq.\\ 13 of \\citetalias{leake:partial2} with $\\Sigma_{en}=\\Sigma_{ne}=1\\times 10^{-19}$ m$^{2}$ as used previously in \\citetalias{leake:partial1}, \\citetalias{leake:partial2}, \\citet{khomenko:2012}, and \\citet{ni:2015}.  The resistivity therefore depends on plasma parameters and is not a constant as in \\citetalias{leake:partial1}.  We include the Hall term and evolve scalar plasma and neutral pressures. \n\n\\subsection{Initial Conditions\\label{initial}}\n\nHere we describe the procedure to establish an approximate initial equilibrium with asymmetric upstream magnetic field strengths, densities, temperatures, and ionization fractions.  Before proceeding, we define several parameters to quantify asymmetries in different fields.  The magnetic, temperature, number density (of ions and neutrals), and ionization fraction asymmetries are defined as\n\\begin{eqnarray}\n  \\ensuremath{\\mathcal{B}} & \\equiv & \\frac{B_{2}}{B_{1}}, \\label{basym}\n  \\\\  \n  \\ensuremath{\\mathcal{T}} & \\equiv & \\frac{T_{2}}{T_{1}}, \\label{tasym}\n  \\\\\n  \\ensuremath{\\mathcal{N}} & \\equiv &\n  \\frac{n_{i,2}+n_{n,2}}{n_{i,1}+n_{n,1}}, \\label{nasym}\n  \\\\\n  \\ensuremath{\\mathcal{F}} & \\equiv & \\frac{f_2}{f_1}, \\label{fasym}\n\\end{eqnarray}\nwhere the subscripts `1' and `2' correspond to the asymptotic upstream magnitudes of each field for $y>0$ and $y<0$, respectively, and the ionization fraction is defined as\n\\begin{equation}\n  f \\equiv \\frac{n_i}{n_i + n_n}. \\label{ionfrac}\n\\end{equation}\nAll of these quantities are functions of time, so the subscript `0' in expressions below indicates correspondence to the initial conditions.\n\nThe equilibrium magnetic field is specified as a modified Harris sheet,\n\\begin{equation}\n  B_{x0}\\left(y\\right) = B_{1,0} \\left[ \\frac{\\tanh \\left(\n      \\frac{y}{\\ensuremath{\\lambda_\\psi}} - b\\right) + b}{1+b}\n    \\right],  \\label{initial_b}\n\\end{equation}\nwhere $\\ensuremath{\\lambda_\\psi}$ is the initial thickness of the current sheet \\citep[see also][]{birn:2008, birn:2010, murphy:double, murphy:plasmoidasym}.  The initial magnetic asymmetry is given by $\\ensuremath{\\mathcal{B}}_0 = (1-b)\/(1+b)$ with the convention that $0 \\leq b < 1$ so that $B_{2,0}\\leq B_{1,0}$.  We describe the in-plane magnetic field using the magnetic flux, $A_z$, such that \n\\begin{equation}\n  \\mathbf{B} = \\nabla\\times \\left(A_z\\ensuremath{\\hat{\\mathbf{z}}}\\right) + B_z\\ensuremath{\\hat{\\mathbf{z}}}. \\label{fluxdef}\n\\end{equation}\nThe flux corresponding to Eq.\\ \\ref{initial_b} is \n\\begin{equation}\n  A_{z0}(y) = \\frac{B_{1,0}}{1+b}\n           \\left[\n             \\ensuremath{\\lambda_\\psi}\n             \\ln\\cosh\\left(\\frac{y}{\\ensuremath{\\lambda_\\psi}}-b\\right)\n             + b y\n           \\right].\n\\end{equation}\nThe out-of-plane current density is then\n\\begin{equation}\n  J_{z0}(y) = - \\frac{B_{1,0}}{\\mu_0\\ensuremath{\\lambda_\\psi}}\n  \\left[\n  \\frac{\n    \\mathrm{sech}^2{\\left(\\frac{y}{\\ensuremath{\\lambda_\\psi}} - b\n      \\right)}}{1+b} \\right]. \\label{initial_j}\n\\end{equation}\n\nThe initial temperature is set using the relations\n\\begin{eqnarray}\n  T_0(y) & = & T_{1,0} \n  \\left[ 1 + \n  \\left( \\ensuremath{\\mathcal{T}}_0 - 1 \\right)\n  \\left( 1 - \\zeta^2   \\right)\n  \\right], \\label{tempinit}\n  \\\\\n  \\zeta &=& \\frac{1}{2} \n  \\left[ 1 + \\tanh\n     \\left(\n         \\frac{y}{\\ensuremath{\\lambda_\\psi}} - b\n     \\right) \n  \\right] . \\label{zetadef}\n\\end{eqnarray}\nThe initial ionization fraction is found numerically by equating the ionization rate $\\Gamma^{ion}_i$ with the recombination rate $\\Gamma^{rec}_i$ using Eqs.\\ \\ref{ionrate}--\\ref{recfreq}.  The initial conditions are therefore in ionization equilibrium.  If the initial temperature is non-uniform, then the initial ionization fraction will also be non-uniform: if $\\ensuremath{\\mathcal{T}}_0 \\ne 1$, then $\\ensuremath{\\mathcal{F}}_0 \\ne 1$. This is in contrast to \\citetalias{leake:partial1} and \\citetalias{leake:partial2} which both assume that the temperature and ionization fraction are both initially uniform.\n\nWe define $\\ensuremath{P_\\mathrm{tot}}$ to be the sum of the plasma and neutral pressures and the plasma pressure $P_p$ to be the sum of the electron an ion pressures,\n\\begin{eqnarray}\n  \\ensuremath{P_\\mathrm{tot}} & \\equiv & P_n + P_p,\n  \\\\\n  P_p  & \\equiv & P_i + P_e. \n\\end{eqnarray}\nFor equal temperatures, the neutral and plasma pressures are related to the ionization fraction by\n\\begin{eqnarray}\n  P_n &=& \\ensuremath{P_\\mathrm{tot}} \\left(\\frac{1 - f}{1 + f}\\right),\n  \\\\\n  P_p &=& \\ensuremath{P_\\mathrm{tot}} \\left(\\frac{ 2 f }{1 + f}\\right),\n\\end{eqnarray}\nwhere we recall that the total number of particles depends on the ionization fraction and that electrons are included in the plasma pressure.  We calculate ${\\ensuremath{P_\\mathrm{tot}}}_{,0}$ to balance the Lorentz force associated with the magnetic field profile given by Eq.\\ \\ref{initial_b},\n\\begin{equation}\n  P_{\\mathrm{tot},0}(y) = \\left( 1 + \\beta_{1,0} \\right)\n  \\frac{B_{1,0}^2}{2\\mu_0} - \\frac{B_{x0}(y)^2}{2\\mu_0}, \\label{ptot}\n\\end{equation}\nwhere $\\beta_{1,0}$ is the ratio of ${\\ensuremath{P_\\mathrm{tot}}}_{,0}$ to the magnetic pressure for $y \\gg 0$.  The number densities are then given by\n\\begin{eqnarray}\n  n_n &=& \\frac{P_n}{k_B T},\n  \\\\ \n  n_i &=& \\frac{P_p}{2 k_B T}.\n\\end{eqnarray} \nWe assume that the ions and electrons have equal temperatures and number densities.\n\nThe above magnetic and pressure profiles satisfy the relation\n\\begin{equation}\n  0 = - \\nabla \\left( P_p + P_n \\right) + \\ensuremath{\\mathbf{J}} \\times \\ensuremath{\\mathbf{B}}.\n\\end{equation}\nHowever, there must also be a relative velocity between the plasma and neutrals to allow coupling via the momentum equations of the different species through collisions and charge exchange.  In the absence of charge exchange, the steady state momentum equations for the plasma and neutral components are\n\\begin{eqnarray}\n  0 &=& -\\nabla P_p + \\ensuremath{\\mathbf{J}}\\times\\ensuremath{\\mathbf{B}} + \\ensuremath{\\mathbf{R}^{in}_i},\n  \\\\\n  0 &=& -\\nabla P_n - \\ensuremath{\\mathbf{R}^{in}_i},\n\\end{eqnarray}\nrespectively, where the frictional force $\\ensuremath{\\mathbf{R}^{in}_i}$ is defined in Eq.\\ \\ref{rinidef}. In this approximation, the initial ion and neutral velocities along the inflow direction can be chosen so that the initial configuration contains no net force on either of the plasma and neutral components,\n\\begin{eqnarray}\nV_{iy0}-V_{ny0} = \\frac{\\partial P_n\/\\partial y}{m_{in}n_i\\nu_{in}}, \\label{viy0def}\n\\end{eqnarray}\nwhere the neutral pressure gradient is calculated analytically using the expression\n\\begin{equation}\n  \\frac{\\partial P_n}{\\partial y} = -\n  \\left( \\frac{1-f_0}{1+f_0} \\right)\n  \\left( \\frac{B_{1,0}}{\\mu_0 \\lambda_\\psi \\left(1+b\\right)} \\right)\n  B_{x0}(y)\\,\n  \\mathrm{sech}^2\\left( \\frac{y}{\\lambda_\\psi} - b\\right) .\n\\end{equation}\nBecause our simulations include charge exchange, momentum transfer between the ions and neutrals is of order twice as effective as the case with frictional coupling alone.  In lieu of an exact solution to the steady state momentum equation, we reduce the initial ion-neutral drift velocity given in Eq.\\ \\ref{viy0def} by a factor of two so that the initial conditions for both ions and neutrals are in approximate but not exact force balance.  We set $V_{iy0}$ and $V_{ny0}$ so that they have the same magnitude but opposite sign.  Example initial conditions are shown in Fig.\\ \\ref{ICplot}\\@.\n\n\\begin{figure}[t!]\n  \\begin{center}\n    \\includegraphics{f1.pdf}\n  \\end{center}\n  \\caption{The initial conditions for case E with $\\ensuremath{\\mathcal{B}}_0=1$, $\\ensuremath{\\mathcal{T}}_0=0.95$, $\\ensuremath{\\mathcal{N}}_0=1.47$, and $\\ensuremath{\\mathcal{F}}_0=0.35$.\n    \\label{ICplot}}\n\\end{figure}\n\nThe initial state does not represent an exact equilibrium, which leads to an outward moving pulse along the inflow direction that propagates from the initial current sheet.  This pulse is mostly damped by placing a layer of significantly enhanced viscosity near the conducting walls at $y=\\pm 1$.  \n\n\\subsection{Boundary Conditions\\label{BCs}}\n\nWe simulate a half-domain that extends from $0\\leq x\\leq L_x$ and from $-L_y \\leq y \\leq L_y$, where $x$ is the outflow direction and $y$ is the inflow direction.  We assume that the fields $A_z$, $n_i$, $n_n$, $p_i$, $p_n$, $V_{iy}$, $V_{ny}$, $V_{iz}$, $V_{nz}$, and $J_z$ are symmetric about $x=0$ [e.g., $A_z(x,y)=A_z(-x,y)$], and that the fields $V_x$ and $B_z$ are antisymmetric about $x=0$ [e.g., $V_x(x,y)=-V_x(-x,y)$].  We assume that all fields are periodic along the $x$ direction with a period of $2L_x$.  We apply perfectly conducting, zero-flux boundary conditions along the boundaries at $y=\\pm L_y$.  \n\nThe choice of initial and boundary conditions enforces that the reconnection outflow is symmetric about $x=0$.  The development of the plasmoid instability under these conditions often leads to the formation of a magnetic island centered about $x=0$ that cannot be advected out of the system.  In general, there will be some degree of asymmetry along the outflow direction that can modify the internal structure of the reconnection layer and the dynamics of reconnection \\citep{oka:2008, murphy:asym, murphy:retreat}.  This can be achieved by simulating the whole domain of interest and allowing the driving term, the outflow boundaries, or the initial perturbations to be asymmetric \\citep[e.g.,][]{murphy:plasmoidasym}.  \n\n\\subsection{Initializing Reconnection\\label{initialization}}\n\nTo initialize reconnection, we apply a source function to the evolution equation for magnetic flux of the form\n\\begin{equation}\n    \\frac{\\dif A_z}{\\dif t} = \n    \\epsilon \\lambda_\\psi \n    \\Lambda\\left(x,y\\right)\n    \\Omega\\left(x,y\\right)\n    \\Theta\\left(t\\right)\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\Lambda\\left(x,y\\right) &=&\n    \\exp\\left[\n        - \\left( \\frac{x}{h_x} \\right)^2\n        - \\left( \\frac{y}{h_y} \\right)^2\n    \\right]\n    \\left\\{ 1 - \\frac{1}{2} \\exp\n        \\left[\n            - 3 \\left( \\frac{y}{h_y} \\right)^2\n        \\right] \n    \\right\\},\n    \\\\\n    \\Omega\\left(x,y\\right) &=&\n        \\left[ 1 - \\left( \\frac{x}{L_x} \\right)^{16} \\right]\n        \\left[ 1 - \\left( \\frac{y}{L_y} \\right)^{16} \\right],\n    \\\\  \n    \\Theta\\left(t\\right) &=&\n    \\frac{1}{2}\n    \\left[\n        1 - \\cos\\left( \\frac{2\\pi t}{t_{D}}\\right)\n    \\right].\n\\end{eqnarray}\nThe spatial length scales for this source function are given by $h_x = 4\\lambda_\\psi$ and $h_y = \\lambda_\\psi$.  The function $\\Lambda(x,y)$ localizes the source function in a form akin to a tearing mode eigenfunction to create an X-point at the field reversal along $x=0$.  The function $\\Omega(x,y)$ ensures that the pulse goes to zero along the outer boundaries.  The pulse is applied between $0\\leq t \\leq t_{D}$ with a waveform given by $\\Theta\\left(t\\right)$.  We use $\\epsilon=0.05$ and $t_{D}=5$.  The application of this electric field ceases before the reconnection layer has had a chance to develop.  In contrast, \\citetalias{leake:partial1} applied a small, localized perturbation to the magnetic flux while \\citetalias{leake:partial2} applied a small amplitude localized rotational flow perturbation to initialize reconnection.\n\n\\section{LOCAL AND GLOBAL DYNAMICS OF RECONNECTION\\label{global}}\n\n\\begin{deluxetable}{ccccccccccccc}\n\\tablecolumns{7}\n\\tablewidth{0in}\n\\tablecaption{Simulation initial parameters\\label{table_simpars}}\n\\tablehead{\n\\colhead{Case} &\n\\colhead{$B_{1,0}$} &\n\\colhead{$B_{2,0}$} & \n\\colhead{$T_{1,0}$} &\n\\colhead{$T_{2,0}$} &\n\\colhead{$n_{1,0}$} &\n\\colhead{$n_{2,0}$} &\n\\colhead{$f_{1,0}$} & \n\\colhead{$f_{2,0}$} &\n\\colhead{$\\beta_{1,0}$} &\n\\colhead{$\\beta_{2,0}$} & \n\\colhead{$\\lambda_{ni,1,0}$} &\n\\colhead{$\\lambda_{ni,2,0}$}\n}\n\\startdata\nA & $7.9$ & $7.9$ & $9000$ & $9000$ & $7.6$ & $7.6$ & $0.00041$ & $0.00041$ & $3.8$ & $3.8$ & $220$ & $220$ \\\\\nB & $7.9$ & $7.9$ & $8750$ & $9250$ & $7.8$ & $7.4$ & $0.00024$ & $0.00068$ & $3.8$ & $3.8$ & $366$ & $136$ \\\\\nC & $10$ & $5$    & $9000$ & $9000$ & $8.8$ & $6.4$ & $0.00041$ & $0.00041$ & $11.0$ & $2.0$ & $261$ & $190$ \\\\\nD & $10$ & $5$    & $8750$ & $9250$ & $6.6$ & $8.6$ & $0.00024$ & $0.00068$ & $2.0$  & $11.0$ & $432$ &$117$ \\\\\nE & $10$ & $5$    & $9250$ & $8750$ & $6.2$ & $9.1$ & $0.00068$ & $0.00024$ & $2.0$ & $11.0$& $163$ & $314$ \\\\\n\\enddata\n\\vspace{-0.8cm}\n\\tablecomments{The units are G for magnetic field, K for temperature, $10^{18}$ m$^{-3}$ for number density (including both neutrals and ions), and m for the neutral-ion mean free paths. The neutral-ion mean free path $\\lambda_{ni}$ is calculated using Eq.\\ \\ref{lambdanidef} and includes charge exchange reactions.  When calculating the charge exchange cross section using Eqs.\\ 5--6 of \\citetalias{leake:partial2}, we use that the relative velocity between ions and neutrals is much less than the neutral and ion thermal speeds for the initial conditions.\n}\n\\end{deluxetable}\n\n\\afterpage{\\clearpage}\n\nWe present five simulations to investigate the impact of asymmetry during magnetic reconnection in partially ionized chromospheric plasmas.  The simulation parameters are shown in Table \\ref{table_simpars}.  Case A is the symmetric test case.  Case B has asymmetric upstream temperatures, and consequently asymmetric upstream densities and ionization fractions, but symmetric upstream magnetic field strengths.  Case C has symmetric upstream temperatures but a factor of two difference in the upstream magnetic field strengths.  Cases D and E have asymmetric temperatures and magnetic field strengths.  We keep three parameters constant between runs: the sum of the initial magnetic pressure, neutral pressure, and plasma pressure which equals $1.495$ ($1.19$ Pa); the mean of the initial asymptotic upstream magnetic energy densities which is given by  $\\frac{1}{2}\\left( B_{1,0}^2\/{2} + B_{2,0}^2\/{2}\\right) = 0.3125$ ($0.249$ J\\,m$^{-3}$); and the mean of the initial asymptotic upstream temperatures which is $(T_{1,0}+T_{2,0})\/2 = 4.69\\times 10^{-3}$ ($9000$ K).  \n\nThe domain size for all simulations is $(L_x,L_y)=(2,1)$.  The resolution in Case A is $m_x=256$ elements along the outflow direction and $m_y=128$ elements along the inflow direction.  The resolution in Cases B--E is $m_x=m_y=256$.  We use sixth order basis functions for all simulations, resulting in effective total resolution of $(M_x,M_y)=6(m_x,m_y)$.  Grid packing is used to concentrate mesh in the reconnection region.  In Case A, the current sheet does not move from $y=0$ so the mesh packing along the inflow direction is concentrated to a thin region near $y=0$.  In Cases B--D, the current sheet drifts slowly in the $-\\ensuremath{\\hat{\\mathbf{y}}}$ direction so the highest resolution is needed over a much longer distance to resolve the dynamics \\citep[see also][]{murphy:double, murphy:plasmoidasym}.  High resolution along the outflow direction helps capture the front end of the reconnection jet as well as the dynamics of the plasmoid instability.  The resolution along the outflow direction from $0\\leq x \\lesssim 1.2$ and $1.95\\lesssim x \\leq 2$ is approximately twice as high as from $1.2\\lesssim x \\lesssim 1.95$.  \n\n\\subsection{Structure of Reconnection Region\\label{structure}}\n\nAll five simulations show broadly similar evolution.  The electric field application from $t=0$ to $t=5$ described in Section \\ref{initialization} allows the inflow\/outflow pattern associated with two-dimensional reconnection to develop in the portion of the current sheet near the origin.  The outflow jet lengthens as it plows into a region of enhanced ion density associated with current sheet thinning outside of the reconnection region.  By $t\\sim 25$, laminar reconnection is well-established.  The current sheet has thinned from its initial thickness of ${0.1}$ to a thickness of $\\delta\\sim 1.0\\times 10^{-3}$ ({$\\sim$}$10$ m).  This is comparable to the value of the neutral-ion mean free path evaluated inside the current sheet: $\\lambda_{ni} \\gtrsim 6.0\\times 10^{-4}$ to $1.0\\times 10^{-3}$ ({$\\gtrsim$}$6$ to $10$ m).  The ionization fraction inside each current sheet is of order $0.01$ at this time.  The structure of the reconnection region for all simulations at $t=25$ is shown in Figures \\ref{jzvxplot}, \\ref{flowplot}, \\ref{bzniplot}, and \\ref{inflowslice}.  At $t=25$, the current sheets have thinned to {$\\sim$}$\\lambda_{ni}$, laminar reconnection is well-established, and plasmoid formation has not yet begun.  The plasmoid instability onsets during all of these simulations.  The simulations end when structures develop on scales comparable to the resolution scale as a result of the plasmoid instability.\n\nFigure \\ref{jzvxplot} shows the out-of-plane current density and the ion outflow.  In Case B, the current sheet is slightly arched so that the X-point is closer to the low temperature upstream region ($y>0$) than the ends of the current sheet.  In the cases with magnetic asymmetry, the current sheet is arched so that the X-point is closer to the strong field upstream region than the ends of the outflow jets (see also Fig. \\ref{inflowslice}b).  Case E, which has the strong field side coincident with the high temperature side, shows the fastest development of these three cases and the strongest current density.\n\n\\begin{figure}[tp]\n    \\begin{center}\n    \\vspace{-7mm}\n    \\includegraphics[height=7.5in]{f2_midres.pdf}\n    \\end{center}\n    \\vspace{-7mm}\n    \\caption{The out-of-plane current density $J_z$ (left) and the ion outflow (right) at $t=25$ for Cases A--E\\@.  The right half of each plot shows color contours of the outflow component of ion velocity $V_{ix}$, ion velocity vectors, and contours of magnetic flux $A_z$.  This image is scaled significantly along the $y$ direction which exaggerates the inflow $y$-component of velocity.\n        \\label{jzvxplot}}\n\\end{figure}\n\n\\begin{figure}[tp]\n    \\begin{center}\n    \\vspace{-7mm}\n    \\includegraphics[height=7.5in]{f3_midres.pdf}\n    \\end{center}\n    \\vspace{-7mm}\n    \\caption{The inflow components of the ion velocity $V_{iy}$ (left) and the neutral velocity $V_{ny}$ (right) in the reconnection region at $t=25$ for Cases A--E\\@.  The solid black contours represent the magnetic flux $A_z$.  The dashed green contour indicates the locations where the inflow component of velocity equals zero.  \n    \\label{flowplot}}\n\\end{figure}\n\n\\begin{figure}[tp]\n    \\begin{center}\n    \\vspace{-7mm}\n    \\includegraphics[height=7.5in]{f4_midres.pdf}\n    \\end{center}\n    \\vspace{-7mm}\n    \\caption{The logarithm of the ion number density $\\log_{10}\\,{n_i}$ (left) and the out-of-plane magnetic field $B_z$ (right) at $t=25$ for Cases A--E\\@.  The solid black contours represent the magnetic flux $A_z$.\\label{bzniplot}}\n\\end{figure}\n\n\\begin{figure}[tp]\n    \\begin{center}\n    \\includegraphics[width=3.3in]{f5.pdf}\n    \\end{center}\n    \\caption{\n        The structure of the current sheet along $x=0$ at $t=25$ very near the current sheet relative to the X-point.  Shown are (a) the reconnecting component of the magnetic field $B_x$, (b) the out-of-plane current density $J_z$, (c) the logarithm of ion density $\\log_{10}\\,n_i$, (d) the inflow component of ion velocity $V_{iy}$, (e) the inflow component of neutral velocity $V_{ny}$, and (f) the neutral pressure.\n    \\label{inflowslice}}   \n\\end{figure}\n\nAs in \\citetalias{leake:partial1} and \\citetalias{leake:partial2}, the ion and neutral outflows are tightly coupled.  The right half of each panel in Figure \\ref{jzvxplot} only show pseudocolor maps of $V_{ix}$ because $V_{nx}$ is very similar in the outflow jet.  The strong coupling of the ion and neutral outflow jets occurs because the mean free path for neutral-ion collisions is comparable to the current sheet thickness which is significantly shorter than the length of the current sheet.  \n\nFigure \\ref{flowplot} shows the inflow components of the ion velocity on the left side of each panel and the neutral inflow speed on the right side (see also Figs.\\ \\ref{inflowslice}d and \\ref{inflowslice}e).  As in \\citetalias{leake:partial1} and \\citetalias{leake:partial2}, the inflow velocities are decoupled. In Cases B--E, the decoupling is asymmetric in part because $\\lambda_{ni}$ is different in each upstream region.  The neutrals and the ions at the X-point are moving in opposite directions.  For simulations with magnetic asymmetry (Cases C, D, and E), there is neutral flow through the current sheet from the weak magnetic field side toward the strong magnetic field side.  This corresponds to a neutral pressure gradient that is pushing neutrals from the weak field side into the strong field side (see Fig.\\ \\ref{inflowslice}f).  \n\nThe ion density is strongly peaked within the current sheet for all cases, as shown on the left side of each panel in Fig.\\ \\ref{bzniplot} \\citepalias[see also][]{leake:partial1, leake:partial2}.  The decoupling of ions and neutrals on scales below the neutral-ion mean free path allows the ions to be swept into the current sheet by the magnetic field.  Neutrals are dragged along by collisions with ions, though less effectively on these short length scales.  The current sheet is therefore out of ionization equilibrium: the ionization fraction is much higher than the equilibrium value.  As a result, recombination becomes of comparable importance to the outflow in the ion continuity equation.\n\nNext we consider the consequences of changing the temperature asymmetry while maintaining the same magnetic asymmetry.  As we go from D to C to E, the magnetic asymmetry remains constant ($\\ensuremath{\\mathcal{B}}=0.5$) but the temperature asymmetry goes from $\\ensuremath{\\mathcal{T}}=1.06$ to $1$ to $0.95$.  Case D has higher temperature in the weak field upstream region; Case C has initially uniform temperature; and Case E has higher temperature in the strong field upstream region.  \n\n\\begin{figure}[bt]\n    \\begin{center}\n        \\includegraphics{f6.pdf}\n    \\end{center}\n    \\caption{The logarithm of the neutral-ion mean free path $\\lambda_{ni}=V_{T,n}\/\\nu_{ni}^\\dagger$ along $x=0$ at $t=25$. The minimum values of $\\lambda_{ni}$ in this plot are $6.5\\times 10^{-4}$, $6.3\\times 10^{-4}$, $7.9\\times 10^{-4}$, $7.2\\times 10^{-4}$, and $8.3\\times 10^{-4}$ for cases A through E, respectively.\n        \\label{lambda_ni}}\n\\end{figure}\n\nSwitching the temperature asymmetry while maintaining the same magnetic asymmetry has a significant impact on how quickly reconnection develops, the reconnection rate at a given time, and the onset time and mode structure of the plasmoid instability.  Reconnection develops more quickly and occurs at a faster rate in Case E where the strong field upstream region has a higher temperature than the weak field upstream region. As we proceed from Case D to C to E, the relative velocity between the ions and neutrals increases in magnitude on the weak field side but does not change substantially on the strong field side (see, e.g., Fig.\\ \\ref{inflowslice}d while noting that the $V_{ny}$ profiles in Fig.\\ \\ref{inflowslice}e are nearly identical between these three runs).  The neutral-ion mean free path $\\lambda_{ni}$ along $x=0$ at $t=25$ is shown in Fig.\\ \\ref{lambda_ni}.  In this figure, the minimum value of $\\lambda_{ni}$ increases by 10\\% from Case D to C and another 5\\% from Case C to E\\@.  In contrast, the values of $\\lambda_{ni}$ increase by {$\\sim$}$30$\\% to {$\\sim$}$60$\\% from D to C to E in both near-upstream regions.  The reconnection rate at $t=25$ measured as $\\dif A_z\/\\dif t$ evaluated at the X-point along $x=0$ is $3.4\\times 10^{-4}$, $3.8\\times 10^{-4}$, and $4.5\\times 10^{-4}$ in Cases D, C, and E, respectively, which suggests that differences in ion-neutral coupling inside and around the current sheet are largely responsible for the observed differences between simulations.  In particular, the reconnection rate in these simulations is greater when there is weaker coupling between neutrals and ions.  The $V_{ny}$ profile in Fig.\\ \\ref{inflowslice}d remains nearly unchanged between simulations with the same magnetic asymmetry, which suggests that the neutral flow across the current sheet depends strongly on magnetic asymmetry but only weakly on temperature asymmetry.\n\n\\subsection{Role of the Hall Effect\\label{halleffect}}\n\nThe Hall effect is not expected to be significant in the solar chromosphere during magnetic reconnection unless structures develop on scales comparable to or below the ion inertial length $d_i$ \\citep{malyshkin:2011}.  In the case of strong coupling between ions and neutrals, it has been proposed that the ion inertial length may be enhanced because each ion will be dragging along a much larger mass \\citep{pandey:2008a, malyshkin:2011}; consequently the effective ion inertial length is proposed to be $d_i' = d_i \\sqrt{\\rho\/\\rho_i}$.  We therefore include the Hall effect in these simulations. \n\nThe characteristic thickness of the current sheet is {$\\sim$}$10^{-3}$ ($10$ m; see Fig.\\ \\ref{inflowslice}). The ion inertial length in the upstream regions is initially of order $d_i\\sim 4\\times 10^{-4}$ ($4$ m; using upstream parameters from Case A), while $d_i'\\sim 0.02$ ($200$ m).  In the current sheet, the pileup of ions leads to a local ion inertial length of $d_i\\approx 8\\times 10^{-5}$ ($0.8$ m; based on an ion density of {$\\sim$}$3n_0$), which corresponds to $d_i' \\sim 8\\times 10^{-4}$ ($8$ m).  These lengths scales can also be compared to $\\lambda_{ni}$ which is of order {$\\sim$}$0.01$ to $0.03$ in the regions just upstream of the current sheet and {$\\sim$}$10^{-3}$ in the current sheet.\n\nThe right side of Figure \\ref{bzniplot} shows the quadrupole structure of the out-of-plane magnetic field $B_z$ that forms as a result of the Hall effect during laminar reconnection.  The quadrupole field strength remains about an order of magnitude smaller than the asymptotic values of $B_x$, but $|B_z|\/\\sqrt{B_x^2+B_y^2}$ locally reaches {$\\sim$}$0.5$ near the location where $|B_z|$ is largest on the weak field upstream region in Cases C--E\\@.  This occurs in part because the in-plane field is weaker in the near upstream regions than in the far upstream regions.  Despite the locally high value of the ratio of $|B_z|$ to the in-plane field, these cases show neither the significant enhancement of the reconnection rate nor the development of the low aspect ratio (X-point) geometry expected during Hall-mediated reconnection in fully ionized plasmas \\citep[e.g.,][]{biskamp:1997}.\n\nThe structure of the quadrupole field is modified by both temperature and magnetic asymmetries.  When the temperature or magnetic asymmetries are changed, this also impacts the neutral density, ion density, and ionization fraction asymmetries and therefore $\\lambda_{ni}$.  In Case B, the quadrupole lobe on the low ion density side ($y<0$) has a greater spatial extent and higher magnitude than the high ion density side.  In Cases C--E with magnetic asymmetry, the quadrupole lobe on the strong magnetic field side is localized near the current sheet while the quadrupole lobe on the weak magnetic field side has a greater spatial extent.  This asymmetry of the quadrupole field is qualitatively similar with previous fully kinetic simulations \\citep[see, e.g., Fig.\\ 8c of][]{pritchett:2008:asym}.  As we proceed from Case D to C to E at $t=25$, the quadrupole field on the strong field side increases in strength while the quadrupole field on the weak field side decreases in magnitude but increases in spatial extent.  \n\nEven though the ion inertial scale is shorter than the current sheet thickness throughout these simulations, this might not be the case during the long-term evolution of the plasmoid instability.  While simulations of fully ionized plasmas have shown that the plasmoid instability allows reconnection to occur at a rate that is roughly independent of Lundquist number, \\citet{Daughton:2009} and \\citet{shepherd:2010} have proposed that the most important role of the plasmoid instability might actually be to allow structure to develop on scales smaller than the ion inertial length which would then allow fast collisionless reconnection to take place.  This proposed mechanism has not yet been tested in weakly ionized plasmas, which would require capturing highly nonlinear behavior at later times and will be explored in future work.  \n\n\\subsection{Plasmoid Formation\\label{plasmoidformation}}\n\n\\begin{figure}[tp]\n    \\vspace{-7mm}\n    \\begin{center}\n        \\includegraphics[height=7.5in]{f7_midres.pdf}\n    \\end{center}\n    \\vspace{-7mm}\n    \\caption{The ion density $n_i$ (left) and current density $J_z$ (right) for Cases A--E near the end of each simulation after plasmoids form.\n        \\label{plasmoidplot}}\n\\end{figure}\n\nEach simulation shows the formation of plasmoids after the current sheets have thinned sufficiently.  Figure \\ref{plasmoidplot} shows the ion density, out-of-plane current density, and magnetic flux during the early nonlinear evolution of the plasmoid instability for each case.  In contrast to Figures \\ref{jzvxplot}--\\ref{lambda_ni}, the times for each panel were chosen to be near the end of each simulation but before the plasma substructures associated with the cascading nonlinear evolution of the plasmoids reached the grid scale. The appearance of additional in-plane null points occurs at $t=27.9$ for Case A, $t=29.8$ for Case B, $t=29.6$ for Case C, $t=31.1$ for Case D, and $t=25.9$ for Case E\\@.  In all cases, the secondary islands develop in the central portion of the current sheet: within $x\\lesssim 0.16$ compared to a current sheet length of {$\\sim$}$0.8$.  The out-of-plane current density has a local extremum near each X-point.\n\nWe first consider the effect of introducing a temperature asymmetry by comparing Cases A and B\\@.  The plasmoid instability takes longer to develop in Case B which has temperature asymmetry, but the mode structure does not change substantially. Both cases show a central O-point with two X-points on each side with an additional X-point\/O-point pair further out.  The symmetry about $x=0$ allows a pitchfork bifurcation to change the central X-point into an O-point.  The resulting island then grows due to reconnection and is unable to be advected out of the reconnection region by the outflow.  Prior simulations of the plasmoid instability in resistive MHD have avoided this situation by simulating the entire domain and including a slight asymmetry along the outflow direction so that any central islands that form do not remain in the current sheet indefinitely.\n\nThe structure of the current sheet and the resultant plasmoid instability are more directly modified by magnetic asymmetry, including when it is coupled with temperature asymmetry.  While Case D also undergoes a pitchfork bifurcation to change the central X-point into an O-point, Cases C and E maintain a central X-point and develop multiple alternating X-points and magnetic islands.  As in prior simulations using the resistive MHD approximation \\citep{murphy:plasmoidasym}, the resulting islands preferentially grow into the weak field upstream region.  \n\nWe now compare our simulations with the complementary simulations of the plasmoid instability during partially ionized chromospheric reconnection by \\citet{ni:2015}.  Ni et al.\\ capture the long term nonlinear evolution of the plasmoid instability using a single fluid approach that incorporates ambipolar diffusion and assumes ionization equilibrium.  The parameter regimes for the two sets of simulations differ.  For example, Ni et al.\\ assume an initial upstream $\\beta$ of $0.1$ which allows heating up to {$\\sim$}$8\\times 10^4$ K, while our simulations have high $\\beta$ and thus do not result in significant heating because less magnetic free energy is available.  Ni et al.\\ present current sheets with lengths of order {$\\sim$}$1$ Mm which is a significant fraction of the size of the chromosphere, in contrast to lengths of {$\\sim$}$10$ km in our simulations.  The thicknesses of the Ni et al.\\ current sheets are generally much larger than $\\lambda_{ni}$ which suggests that the neutrals and ions are strongly coupled in their regime.  While recombination plays an important role in the simulations presented by \\citetalias{leake:partial1}, \\citetalias{leake:partial2}, and this work, the high temperatures found by Ni et al.\\ make it less likely that net recombination will play an important role in the plasma continuity equation.  Physical parameters such as magnetic field strength, temperature, density, and ionization fraction vary significantly within the chromosphere even at a single height, so it is likely that there are regions where each parameter regime is valid.  These differences highlight the need for detailed parameter studies on how partial ionization impacts the reconnection process including the plasmoid instability at various locations in the chromosphere.\n\n\\subsection{Reconnection Rate\\label{reconrate}}\n\nThe reconnection rate for all five simulations is shown in Fig.\\ \\ref{ratefig}.  This rate was measured as the maximum of the time derivatives of $A_z$ at all X-points within the reconnection region.  A common alternative method for measuring the reconnection rate is to measure the inflow velocity divided by the outflow velocity; however, during asymmetric reconnection there is ambiguity in how to define the inflow velocity.  The reconnection rate before $t=5$ is not shown because an electric field was applied to initialize reconnection at early times.  When interpreting this figure, it is important to recall that the initial total pressure and average upstream magnetic energy density are constant between runs.  With this convention, magnetic and temperature asymmetry reduce the reconnection rate for all cases studied.\n\n\\begin{figure}[tb]\n    \\begin{center}\n    \\includegraphics[width=8.5cm]{f8.pdf}\n    \\end{center}\n    \\caption{\n        The reconnection rate as a function of time as measured by the maximum change in magnetic flux among all X-points in the current sheet, $\\dif A_z\/\\dif t$.  This rate is presented in dimensionless units according to Section \\ref{normalizations}; consequently, this rate has not been renormalized to values immediately upstream of the reconnection layer.\n    \\label{ratefig}}\n\\end{figure}\n\nFrom early times until around when plasmoids form, the reconnection rate is highest at the X-point along $x=0$.  The reconnection rate increases as the current sheet thins.  Around the time that plasmoids form, the reconnection rate at this X-point decreases.  For some cases, this X-point bifurcates into an O-point.  The peak reconnection rate then occurs at one of the nearby X-points that is not located along $x=0$.  Cases A, B, and D show a considerable increase in the reconnection rate after the formation of plasmoids.  It is likely that Cases C and E would also show a similar increase in the reconnection rate; however, the simulations ended before this occurred.  At even later times, structure on small enough scales could develop so that Hall reconnection could become important \\citep[see also][]{Daughton:2009, shepherd:2010}.\n\n\\subsection{X-Point Motion and Flows Across the X-Point\\label{nullflows}}\n\n\\begin{figure*}[t!]\n    \\begin{center}\n    \\includegraphics[width=6.5in]{f9.pdf}\n    \\end{center}\n    \\caption{A comparison between the X-point velocity $\\frac{\\dif y_n}{\\dif t}$, the ion flow at the X-point along the inflow direction $V_{iy}(y_n)$, and the neutral flow at the X-point $V_{ny}(y_n)$ for Cases B--E\\@.  We consider only the X-point along $x=0$ and times after the early electric field application and before the onset of the plasmoid instability.  Case A is not shown because all of these velocities equal zero due to symmetry.\n    \\label{nullflowfig}}  \n\\end{figure*}\n\nWe consider three velocities that are relevant to the small-scale physics near the X-point located along the symmetry axis at $\\mathbf{x}_n = (0,y_n)$.  The first velocity is the rate of motion of the X-point along the inflow direction: $\\frac{\\dif y_n}{\\dif t}$.  This velocity is not a fluid velocity, but rather the velocity of a topologically stable magnetic feature \\citep{murphy:retreat}.  The second is the ion flow at the X-point along the inflow direction: $V_{iy}(y_n)$.  The third is the neutral flow at the X-point along the inflow direction: $V_{ny}(y_n)$.  Differences between $\\frac{\\dif y_n}{\\dif t}$ and $V_{iy}(y_n)$ must be due to non-ideal behavior such as resistivity or the Hall effect.  Differences between $V_{iy}(y_n)$ and $V_{ny}(y_n)$ correspond to momentum transfer between ions and neutrals (e.g., Eq.\\ \\ref{rinidef}).\n\nFigure \\ref{nullflowfig} shows $\\frac{\\dif y_n}{\\dif t}$, $V_{iy}(y_n)$, and $V_{ny}(y_n)$ as a function of time for all of the cases with asymmetric inflow.  Case A is not shown because these velocities equal zero due to symmetry.  The bump in velocity around $t=16$ is a consequence of the initial conditions being slightly out of equilibrium; the resulting pulse is mostly but not entirely damped by a viscous layer at the boundary and the reflected pulse returns to the current sheet region at this time.  The magnitude of this pulse is weak compared to the ion inflow speeds, but is the same order of magnitude as the velocities associated with each null point.  For the rest of this section, we consider $t\\gtrsim 20$ for which reconnection is well established.  \n\nIn general, there is a small but nonzero difference between $\\frac{\\dif y_n}{\\dif t}$ and $V_{iy}(y_n)$.  This similarity indicates that the magnetic field is predominantly carried by the ion flow.  The residual difference in flow results from resistive diffusion of the magnetic field and the Hall effect.  The difference between either of these velocities and the neutral flow at the X-point $V_{ny}(y_n)$ is greater, which is consistent with decoupling between the ion and neutral inflows.  The $V_{ny}$ profile along the inflow direction remains roughly constant between Cases A and B which both have symmetric magnetic fields but different temperature profiles, and also between Cases C--E which have the same asymmetric magnetic field configuration but differing initial temperature profiles. Case B has symmetric upstream magnetic fields but the $y<0$ upstream region is warmer and less dense than the $y>0$ upstream region.  That $\\frac{\\dif y_n}{\\dif t} \\approx V_{iy}(y_n) < 0$ indicates that the X-point is being carried primarily by the ions in the direction of the warmer upstream region.  \n\nThe simulations with magnetic asymmetry (Cases C, D, and E) show neutral flow from the weak field upstream region into the strong field upstream region.  This flow results from a neutral pressure gradient that is pushing the neutrals from the weak field side into the strong field side where the neutral pressure is lower.  This flow is able to occur because of imperfect coupling between species on scales below or comparable to the neutral-ion mean free path.  Neutrals swept along with the outflow will have preferentially originated from the weak field upstream region.  This flow may have important observational consequences because the abundances of neutrals swept along with the outflow are likely to better match the weak field upstream region.  In contrast, there is net ion flow inside the current sheet from the strong field upstream region into the weak field upstream region.  While the neutrals are pushed by a neutral pressure gradient, the dominant force on the ions is a magnetic pressure gradient.  \n\n\\section{DISCUSSION\\label{discussion}}\n\nIn this paper we perform simulations of asymmetric magnetic reconnection in partially ionized chromospheric plasmas using the plasma-neutral module of the HiFi framework.  We include cases with symmetric or asymmetric upstream temperatures and magnetic field strengths.  The plasma and neutrals are modeled as separate fluids with momentum and energy transfer between species due to collisions and charge exchange.  These simulations self-consistently include ionization and recombination without assuming ionization equilibrium.  \n\nSeveral of the properties of asymmetric partially ionized reconnection are qualitatively similar to the symmetric cases \\citepalias[Case A;][]{leake:partial1, leake:partial2}.  The current sheet rapidly thins so that its thickness becomes comparable to the neutral-ion mean free path evaluated inside the current sheet.  The ion and neutral outflows are strongly coupled, but the ion and neutral inflows are decoupled.  The reconnecting magnetic field drags ions into the current sheet which leads to a significant enhancement of ion density so that the current sheet is out of ionization equilibrium.  Recombination becomes of comparable importance to the outflow in the equation for ion conservation of mass.  In this paper, we show that these properties are modified by the temperature and magnetic field asymmetries.\n\nDuring our simulations of asymmetric reconnection, the ion and neutral inflows remain decoupled, but the decoupling is asymmetric.  When there is magnetic asymmetry, there is net neutral flow through the current sheet from the weak magnetic field upstream region into the strong field upstream region.  This neutral flow results from a large scale neutral pressure gradient and imperfect coupling between ions and neutrals along the inflow direction.  Similarly, the greater Lorentz force acting on the ions from the strong field upstream region leads to the ions pulling the X-point into the weak field upstream region.  These effects are not present during symmetric simulations.  An observational consequence of these neutral flows through the current sheet is that most of the neutrals swept along with the outflow are more likely to have originated from the weak magnetic field upstream region.  This may be especially important if there are different elemental abundances in each upstream region (e.g., different FIP enhancements).  \n\nPrior simulations of asymmetric reconnection in fully ionized plasmas have shown non-ideal flows through null points \\citep[e.g.,][]{oka:2008, murphy:retreat, murphy:double}.  These flows result from non-ideal effects such as resistivity and the Hall effect.  In resistive MHD, for example, the motion of null points results from a combination of resistive diffusion of the magnetic field and advection by the bulk plasma flow \\citep{murphy:retreat}.  During asymmetric reconnection in partially ionized plasmas, an additional contributor to non-ideal plasma flows across null points is the electric field contribution from electron-neutral drag (see the fourth term on the right hand side of Eq.\\ \\ref{genohms}).  Neutral flow through the current sheet is a new effect that is not present in fully ionized situations.\n\nWe include the Hall effect in our simulations to test whether or not a transition to Hall reconnection can develop in this regime.  We find that the out-of-plane quadrupole magnetic field associated with antiparallel Hall reconnection does develop.  The out-of-plane magnetic field strength is a significant fraction of the in-plane field strengths just upstream of the current sheet.  However, the characteristic low aspect ratio geometry and high reconnection rate expected during Hall reconnection do not develop over the course of these simulations, including during the early nonlinear evolution of the plasmoid instability.  The transition to Hall reconnection in a partially ionized plasma has been investigated analytically by \\citet{malyshkin:2011}, and will continue to be investigated by ongoing simulation efforts to determine the conditions under which such a transition is likely to occur.\n\nAfter an initial phase, each of our simulations show the development of the plasmoid instability.  These simulations capture the early nonlinear evolution until structures develop on scales comparable to the resolution scale.  In cases with magnetic asymmetry, the plasmoids develop preferentially into the weak field upstream region, which is similar to fully ionized simulations. Recombination due to the enhanced ion density remains an important loss term in the plasma continuity equation.  We anticipate that secondary merging of plasmoids will be further modified by these asymmetries.  Our simulations complement the work of \\citet{ni:2015} who use a single fluid framework to investigate the long term nonlinear evolution of the plasmoid instability in a different parameter regime.\n\nWe investigate the coupling between magnetic and temperature asymmetries by comparing cases with the same magnetic asymmetry but different temperature asymmetries.  We find that reconnection develops more quickly, occurs at a faster rate, and becomes unstable to plasmoid formation as we increase the temperature in the strong field upstream region while decreasing the temperature in the weak field upstream region.  This change corresponds to $\\lambda_{ni}$ increasing in both upstream regions.  The neutral flow through the current sheet does not noticeably change and therefore depends more strongly on the magnetic asymmetry than the temperature asymmetry.\n\nThere are many important remaining directions for modeling work to understand how reconnection occurs in the solar chromosphere and other weakly ionized plasmas.  The simulations of \\citetalias{leake:partial1}, \\citetalias{leake:partial2}, and this paper have all simulated two-dimensional, antiparallel reconnection.  The simulations presented in these works have modeled the early nonlinear evolution of the plasmoid instability in a weakly ionized plasma, and to model the later evolution will likely require a combination of increased resolution (including along the outflow direction to capture plasmoid merging) and increased diffusion.  An alternative strategy would be to evolve the logarithm of density instead of the density itself \\citep[e.g.,][]{elee:2014} which precludes the possibility of the number density becoming negative.  \\citet{ni:2015} adopt adaptive mesh refinement to ensure adequate resolution.  The presence of a guide field can suppress the thinning of current sheets due to ambipolar diffusion \\citep{brandenburg:1994, brandenburg:1995}, and thus should be considered in future efforts \\citep[see also][]{ni:2015}.  Physical conditions in the chromosphere vary significantly even at a single height, so a detailed examination of parameter space will be necessary to characterize the different regimes of partially ionized reconnection in the chromosphere.  The parameter studies may also include regimes relevant to partially ionized plasmas in the laboratory and in astrophysics.  Future studies should investigate the impact of a realistic geometry and the interplay between small-scale physics and global dynamics (e.g., how small and large scales feed back on each other).  Finally, much remains to be understood about the role of three-dimensional effects during weakly ionized reconnection and the behavior of current sheet thinning in cases with and without null points. \n\nIn addition to the modeling efforts, much work remains to compare the simulation results to observations of chromospheric reconnection and to validate the results against laboratory experiments on partially ionized reconnection.  A challenge in comparing simulations against observations is the disparity in length scales.  The current sheets in these simulations have characteristic thicknesses of $\\lesssim 10^2$~m and lengths of $\\lesssim 10$~km, while the spatial resolution of \\emph{IRIS} observations is of order $200$~km, so direct diagnostics of the reconnection layer itself remain extremely challenging.  However, the simulations could be used to predict velocities, spectra (including the charge state distributions of minor ions), densities, and morphologies that could then be compared against observation.  The experiments that have recently been performed at MRX provide an opportunity to directly validate HiFi's plasma-neutral module against laboratory measurements where key quantities can be observed using \\emph{in situ} probes.  Using a code that has been validated against experimental data will improve confidence that it is able to accurately capture the relevant physical processes in the lower solar atmosphere and in astrophysical plasmas.  \n\n\\acknowledgments\n\nThe authors thank H.\\ Ji, J.\\ Leake, J.\\ Lin, L.\\ Ni, N.\\ Nishizuka, J.\\ Raymond, K.\\ Reeves, C.\\ Shen, H.\\ Tian, and E.\\ Zweibel for useful discussions and an anonymous referee for useful comments that helped to improve this paper.  N.A.M.\\ acknowledges support from NASA grants NNX12AB25G, NNX15AF43G, \\& NNX11AB61G; NSF SHINE grants AGS-1156076 and AGS-1358342; contract 8100002705 from LMSAL; and NASA contract NNM07AB07C to the Smithsonian Astrophysical Observatory (SAO). V.S.L.\\ acknowledges support from the NASA LWS \\& Solar and Heliospheric Physics programs, as well as the National Science Foundation. Resources supporting this work were provided by the NASA High-End Computing Program through the NASA Advanced Supercomputing Division at Ames Research Center.  We thank J.\\ Sattelberger formerly from SAO and J.\\ Chang from NASA for technical support.  This work has benefited from the use of NASA's Astrophysics Data System.\n\n\n\\bibliographystyle{apj}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{M}{ulti-carrier} technology has shown great potentials in improving the transmission efficiency and reliability in fading channels \\cite{Cimini1985}. The multi-carrier based multi-access schemes, such as orthogonal frequency division multi-access (OFDMA) and single-carrier frequency division multi-access (SC-FDMA)\\footnote{SC-FDMA can be seen as a coded OFDMA scheme, which is discussed in Section \\ref{subsec:coding_schemes}.} etc., have been widely adopt by IEEE 802.16 and 3GPP LTE-A standards \\cite{IEEE802.16-2009,3GPPLTE}. Both academia and industry believe that the multi-carrier multi-access technology will be playing a more and more important role in future wireless communication systems \\cite{Yang2009}.\n\nIt is a key advantage that the multi-carrier multi-access channel provides great flexibility and adaptability in sharing the channel resources among multiple users, because the channel resources are composed by plenty of small \\emph{resource blocks} (RBs) in time-frequency domain \\cite{Bahai04}. In this context, a RB corresponds to a subcarrier in the duration of a timeslot. Previous works for channel resources allocation can be roughly divided into two categories: RB based scheme, and chunk based scheme. The RB based scheme implies that the based station (BS) directly allocates RBs, which are normally not adjacent in time-frequency domain, to every user. The interleaved subcarrier allocation in IEEE 802.16 and 3GPP LTE-A is a typical RB based allocation scheme, where the uniformly-spaced subcarriers are allocated to each user \\cite{IEEE802.16-2009,3GPPLTE}. In \\cite{Wong1999}, Wong, Cheng, Letaief, and Murch proposed an OFDMA system and studied the adaptive subcarrier, bit, and power allocation problem. In \\cite{Hoo2004}, Hoo, Halder, Tellado, and Cioffi considered allocating subcarriers and power in multi-carrier broadcast channels to maximize the weighted sum rate. In \\cite{Song2005}, Song and Li proposed a cross-layer optimization framework for subcarrier and power allocation in OFDM wireless networks. In contrast to RB based schemes, the chunk based scheme means that multiple RBs are bundled together as a chunk. The BS then allocates these chunks to every user. In IEEE 802.16 and 3GPP LTE-A, a typical chunk based allocation scheme is referred to as the localized allocation, where the adjacent subcarriers in the frequency domain are allocated to each user \\cite{IEEE802.16-2009,3GPPLTE}. In \\cite{Shen2005}, Shen, Li, and Liu studied the performance of a chunk based scheme, where the chunks are allocated to users according to their average SNR within each chunk. Zhu and Wang developed this idea and proposed a joint chunk, power, and bit allocation scheme for OFDMA systems in \\cite{Zhu2012}. The basic principle behind these schemes is to maximize the system capacity or some utility function by dynamically allocating RBs\/chunks and power under the constraints of total power, fairness, rate, and latency \\cite{Stuber05}. The used mathematical tools are mainly based on convex optimization and integral programming \\cite{Liu2005,Hanzo06}.\n\nAlthough many iterative or heuristic algorithms are derived from the aforementioned works to improve the transmission efficiency or enhance the reliability, it is not trivial to conduct a thorough analysis on the fundamental tradeoff between them. Since there lacks an analytic framework with a comprehensive performance metric, the relationship among the efficiency-reliability tradeoffs of different users is also still unknown. Moreover, these proposed allocation schemes usually require perfect channel state information (CSI) at the BS. Based on these considerations, an analytic coded $f$-matching framework is proposed for channel resources allocation problem in multi-carrier multi-access channels. The outage exponent region (OER), which is the region bounded by outage exponents of all the users at given target rates and SNR, is first defined as a fundamental performance metric. In this context, the outage exponent, proposed in our previous work \\cite{Bai2011c}, characterizes the relationship among the outage probability, the number of channels, and SNR at the same time. As a matter of fact, the region bounded by diversity-multiplexing tradeoffs (DMTs) of every user, referred to as the diversity-multiplexing region (DMR), is actually the asymptotic OER when SNR tends to infinity. Similar to the capacity region, the proposed OER and DMR comprehensively illustrate the relationship among the achieved outage performances of all the users for a given resources allocation scheme in slow fading multi-access channels. To achieve the optimal OER and DMR, the random bipartite graph (RBG) \\cite{Bollobas01} approach, which only depends on $\\unit[1]{bit}$ CSI per coherence bandwidth per user, is then applied to formulate the channel resources allocation problem in multi-carrier multi-access channels. The frequency-domain coding based maximum $f$-matching method, which can be applied to both RB and chunk based allocation schemes, is then proposed to allocate the channel resources to all of the users according to their different target rates.\n\nIn contrast to previous works, the outage probability, OER, and DMR of the coded $f$-matching framework can be analyzed in closed-form formulas. By applying the saddle-point approximation method, which has a good balance between the approximation accuracy and complexity \\cite{Butler2007}, the tight upper bound has been first derived for the outage probability of the optimal coding scheme with $\\unit[1]{bit}$ CSI feedback per coherence bandwidth. The combinatorial structure of the RBG based maximum $f$-matching is then studied by using the results in matching theory \\cite{Lovasz1986}. Based on these results, the approximation formulas for the outage probability of each user are obtained for RB and chunk based coded $f$-matching schemes in the high and low SNR regimes, respectively. The best achievable OER and DMR for RB and chunk based coded $f$-matching schemes are then obtained accordingly. Although there are serious conflicts in channel resources allocation, the proposed coded $f$-matching framework is still capable of achieving the optimal OER and DMR with multi-user diversity. As a result, all of the users share the total multiplexing gain according to their target rates, while achieving the full frequency diversity simultaneously.\n\nThe low-complexity channel resources allocation algorithm and coding scheme are also very important for practical multi-carrier multi-access systems. By applying the principle of parallel computations \\cite{Karpinski1998}, we propose the parallel vertices expansion \\& random rotation based Hopcroft-Karp (PVER\\textsuperscript{2}HK) algorithm. It can be shown that the time complexity is only $\\mathcal{O}\\left(\\log^{2\\eta}N\\right)$, where $N$ is the number of resource blocks and $\\eta$ is a constant. The proposed PVER\\textsuperscript{2}HK algorithm is even faster than FFT, $\\mathcal{O}\\left(N\\log N\\right)$. To achieve the optimal DMR, two practical DMT optimal coding approaches are discussed: rotated lattices code \\cite{Oggier2004}, and permutation code \\cite{Tavildar2006}. From the simulation examples, the proposed closed-form formulas for outage probabilities are nearly identical with the simulation curves in both high and low SNR regimes. The results also show that for zero multiplexing gain (i.e., with a fixed target rate), the RB based coded $f$-matching scheme has $\\unit[2]{dB}$ SNR gains compared to the interleaved or TDMA based allocation in IEEE 802.16 and 3GPP LTE-A. The SNR gain also becomes greater as the multiplexing gain increases. For the chunk based coded $f$-matching scheme, it achieves a much greater diversity gain than the localized allocation scheme. Therefore, the proposed framework cannot only be applied to analyze the performance of existing multi-carrier multi-access systems but also provide powerful tools for designing practical channel resources allocation schemes for future multi-carrier multi-access systems.\n\nThe rest of this paper is organized as follows. Section \\ref{sec:system_model} presents the system model and precise problem formulation. The RBG based maximum $f$-matching framework and the requirements on coding schemes are presented in Section \\ref{sec:coded_f_matching}. Section \\ref{sec:oer_dmr_analysis} presents the closed-form formulas for outage probabilities, OER, and DMR of the proposed framework. The practical issues, such as the optimal parallel channel resources allocation algorithm and asymptotic optimal coding schemes, are studied in Section \\ref{sec:practical_issues}. Section \\ref{sec:simulation_results} presents the simulation results to verify our theoretical results. Finally, Section \\ref{sec:conclusions} concludes this paper.\n\n\n\\section{System Model and Problem Formulation}\\label{sec:system_model}\n\n\\subsection{Multi-Carrier Multi-Access Channel Model}\\label{subsec:channel_model}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tikzpicture}[>=stealth]\n\t\\draw[thick] (0,2.8) rectangle (1.6,3.6);\n\t\\node at (0.8,3.2) {BS};\n\t\\draw[thick] (1.6,3.2) -- (2,3.2) -- (2,3.6);\n\t\\draw[thick] (2,3.6) -- (2.3,3.9) -- (1.7,3.9) -- (2,3.6);\n\t\n\t\\draw[thick] (9,0) rectangle (7.4,0.8);\n\t\\node at (8.2,0.4) {User $u_{N}$};\n\t\\draw[thick] (7.4,0.4) -- (7,0.4) -- (7,0.8);\n\t\\draw[thick] (7,0.8) -- (7.3,1.1) -- (6.7,1.1) -- (7,0.8);\n\t\n\t\\fill\n\t(8.2,1.5) circle (1.5pt)\n\t(8.2,1.8) circle (1.5pt)\n\t(8.2,2.1) circle (1.5pt);\n\t\n\t\\draw[thick] (9,2.8) rectangle (7.4,3.6);\n\t\\node at (8.2,3.2) {User $u_{2}$};\n\t\\draw[thick] (7.4,3.2) -- (7,3.2) -- (7,3.6);\n\t\\draw[thick] (7,3.6) -- (7.3,3.9) -- (6.7,3.9) -- (7,3.6);\n\t\n\t\\draw[thick] (9,5.1) rectangle (7.4,5.9);\n\t\\node at (8.2,5.5) {User $u_{1}$};\n\t\\draw[thick] (7.4,5.5) -- (7,5.5) -- (7,5.9);\n\t\\draw[thick] (7,5.9) -- (7.3,6.2) -- (6.7,6.2) -- (7,5.9);\n\t\n\t\\draw[<->,very thick] (2.4,3.8) -- (6.6,1.1);\n\t\\draw[<->,very thick] (2.4,3.9) -- (6.6,3.9);\n\t\\draw[<->,very thick] (2.4,4) -- (6.6,6.2);\n\t\n\n\t\\filldraw[fill=black!70] (-7,3.9) rectangle (-5.75,4.15);\n\t\\filldraw[fill=black!70] (-7,5.15) rectangle (-5.75,5.4);\n\t\\filldraw[fill=black!70] (-5.75,4.9) rectangle (-4.5,5.15);\n\t\\filldraw[fill=black!70] (-5.75,5.65) rectangle (-4.5,5.9);\n\t\\filldraw[fill=black!70] (-4.5,4.15) rectangle (-3.25,4.65);\n\t\\filldraw[fill=black!70] (-4.5,5.4) rectangle (-3.25,5.65);\n\t\\filldraw[fill=black!70] (-3.25,5.4) rectangle (-2,5.65);\n\n\t\\filldraw[fill=black!40] (-7,4.15) rectangle (-5.75,5.15);\n\t\\filldraw[fill=black!40] (-5.75,5.4) rectangle (-4.5,5.65);\n\t\\filldraw[fill=black!40] (-4.5,3.9) rectangle (-3.25,4.15);\n\t\\filldraw[fill=black!40] (-3.25,5.65) rectangle (-2,5.9);\n\n\t\\filldraw[fill=black!10] (-7,5.4) rectangle (-5.75,5.65);\n\t\\filldraw[fill=black!10] (-5.75,5.15) rectangle (-4.5,5.4);\n\t\\filldraw[fill=black!10] (-5.75,3.9) rectangle (-4.5,4.15);\n\t\\filldraw[fill=black!10] (-5.75,4.4) rectangle (-4.5,4.65);\n\t\\filldraw[fill=black!10] (-4.5,4.9) rectangle (-3.25,5.15);\n\t\\filldraw[fill=black!10] (-3.25,4.4) rectangle (-2,4.9);\n\t\\filldraw[fill=black!10] (-3.25,5.15) rectangle (-2,5.4);\n\t\n\t\\foreach \\x in {-7,-5.75,-4.5,-3.25}\n\t\t\\foreach \\y in {3.9,4.15,4.4,4.65,4.9,5.15,5.4,5.65}\n\t\t{\n\t\t\t\\draw (\\x,\\y) +(0,0) rectangle ++(1.25,.25);\n\t\t}\n\t\n\t\\draw[->,very thick] (-7,3.9) -- (-1.5,3.9) node[right] {Time};\n\t\\draw[->,very thick] (-7,3.9) -- (-7,6.3) node[above] {Frequency};\n\t\\node at (-4.5,3.6) {RB based Allocation Scheme};\n\t\\node at (-4,6.3) {$1$ Subchannel $=1$ RB};\n\n\t\\node at (-7.3,5.65) {$\\mathcal{S}_{4}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,5.15) {$\\mathcal{S}_{3}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,4.65) {$\\mathcal{S}_{2}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,4.15) {$\\mathcal{S}_{1}^{\\mathrm{c}}\\,\\big\\{$};\n\n\t\t\t\t\n\t\\filldraw[fill=black!70] (-7,0.3) rectangle (-2,0.8);\n\t\\filldraw[fill=black!40] (-7,0.8) rectangle (-2,1.3);\n\t\\filldraw[fill=black!10] (-7,1.3) rectangle (-2,1.8);\n\t\t\t\n\t\\foreach \\x in {-7,-5.75,-4.5,-3.25}\n\t\t\\foreach \\y in {0.3,0.55,0.8,1.05,1.3,1.55,1.8,2.05}\n\t\t{\n\t\t\t\\draw (\\x,\\y) +(0,0) rectangle ++(1.25,.25);\n\t\t}\n\n\t\\draw[->,very thick] (-7,0.3) -- (-1.5,0.3) node[right] {Time};\n\t\\draw[->,very thick] (-7,0.3) -- (-7,2.7) node[above] {Frequency};\n\t\\node at (-4.5,0) {Chunk based Allocation Scheme};\n\t\\node at (-4,2.7) {$1$ Subchannel $=8$ RBs};\n\t\n\t\\node at (-7.3,2.05) {$\\mathcal{S}_{4}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.55) {$\\mathcal{S}_{3}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.05) {$\\mathcal{S}_{2}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,0.55) {$\\mathcal{S}_{1}^{\\mathrm{c}}\\,\\big\\{$};\n\t\n\t\n\t\\filldraw[fill=black!70] (0,0.8) rectangle (1.25,1.05);\n\t\\node at (1.9,0.925) {User $u_{1}$};\n\t\\filldraw[fill=black!40] (0,0.3) rectangle (1.25,0.55);\n\t\\node at (1.9,0.425) {User $u_{2}$};\n\t\\filldraw[fill=black!10] (2.75,0.8) rectangle (4,1.05);\n\t\\node at (4.7,0.925) {User $u_{N}$};\n\t\\draw (2.75,0.3) rectangle (4,0.55);\n\t\\node at (4.92,0.425) {Unallocated};\n\\end{tikzpicture}\n\\caption{A multi-carrier multi-access system with one BS and $M\\geq2$ users over the frequency-selective Rayleigh slow fading channel. BS allocates the subchannels in the set $\\mathcal{S}_{m}$ to the user $u_{m}$ according to $\\unit[1]{bit}$ CSI feedback.}\\label{fig:system_desc}\n\\end{figure*}\n\nConsider a multi-carrier multi-access channel, as shown in Fig. \\ref{fig:system_desc}, where the basic resource unit allocated to each user is referred to as the \\emph{subchannel} in order to construct the unified analytic framework. Clearly, the subchannel is the aforementioned RB or chunk in the RB or chunk based allocation schemes, respectively.\\footnote{The term ``RB'' or ``chunk'' will be used, if the term ``subchannel'' indicates one of them specifically.} In the considered multi-carrier multi-access channel, $M$ users communicate with a BS through $N$ subchannels with $N\\geq M$. The user set is defined as $\\mathcal{U}=\\left\\{u_{1},\\ldots,u_{M}\\right\\}=\\left\\{u_{m}\\right\\}_{m=1}^{M}$, where $u_{m}$ denotes the $m$th user.\\footnote{The script symbol $\\mathcal{X}$ denotes a set, whose cardinality is denoted by $\\left|\\mathcal{X}\\right|$.} The subchannel set in a given frame is defined as $\\mathcal{S}=\\left\\{s_{n}\\right\\}_{n=1}^{N}$, where $s_{n}$ denotes the $n$th subchannel. The channel is assumed to be under-spread, i.e., the coherence time $T^{\\mathrm{c}}$ is much larger than the multi-path delay spread $T^{\\mathrm{d}}$ and also the length of each fame \\cite{Proakis2007}. The signals of each user are assumed to undergo independent $L$-path frequency-selective fading, i.e., the channel contains $L$ coherence bandwidths. Let $\\mathcal{S}_{l}^{\\mathrm{c}},\\,l=1,\\ldots,L$ denote the set of subchannels in the $l$th coherence bandwidth, then $\\left|\\mathcal{S}_{l}^{\\mathrm{c}}\\right|=N_{\\mathrm{c}}=\\frac{N}{L}$, where the non-integer case is omitted since $N$ can be exactly divided by $L$ in the chunk based scheme or $N\\gg L$ in the RB based scheme \\cite{IEEE802.16-2009}. For convenience, the coherence bandwidth is used to normalize the total bandwidth, and the length of each frame is normalized as $1$.\n\nAccording to the results in \\cite{Wong1999}, it is quite reasonable to assume that the channel gains of the subchannels in one coherence bandwidth are the same, whereas they are independent with one another for different coherence bandwidths. Let $g_{ml}$ denote the channel gain of the $l$th coherence bandwidth for the user $u_{m}$. Then, $g_{ml}$, for every $m=1,\\ldots,M$ and $l=1,\\ldots,L$, are independent with the identical distribution of $\\mathcal{CN}\\left(0,1\\right)$.\\footnote{$\\mathcal{CN}\\left(\\bm{\\mu},\\bm{C}\\right)$ denotes a circularly symmetric Gaussian distribution with mean vector $\\bm{\\mu}$ and covariance matrix $\\bm{C}$.} The channel gain on each subchannel, denoted by $h_{mn}$, can then be defined as $h_{mn}=g_{ml},\\,\\forall s_{n}\\in\\mathcal{S}_{l}^{\\mathrm{c}}$. The mutual information of a subchannel $s_{n}\\in\\mathcal{S}_{l}^{\\mathrm{c}}$ for the user $u_{m}$ is then given by\n\\begin{equation}\\label{eq:subchannel_capacity}\n\tI_{mn}=\\frac{1}{N_{\\mathrm{c}}}\\ln\\left(1+\\left|h_{mn}\\right|^{2}\\gamma\\right)=\\frac{1}{N_{\\mathrm{c}}}\\ln\\left(1+\\left|g_{ml}\\right|^{2}\\gamma\\right),\n\\end{equation}\nwhere $\\gamma$ is the average SNR at the receiver. Throughout the paper, the natural logarithm function is used, and the unit of information is ``$\\mathrm{nat}$''.\n\nBecause the subchannel must be allocated at the beginning of each frame in real-time, each user is only allowed to feedback $\\unit[1]{bit}$ CSI for each coherence bandwidth so as to reduce the complexity in both signaling and subchannel allocation processes. Let $\\bm{Q}$ denote the quantized CSI matrix at BS, where the element at the $m$th row and the $l$th column $\\left[\\bm{Q}\\right]_{ml}$ is an $\\unit[1]{bit}$ quantized value of $g_{ml}$. A subchannel allocation scheme, denoted by $\\mathscr{S}$, can be seen as a mapping from $\\bm{Q}$ to a family of subchannel allocation sets, that is\n\\begin{equation}\\label{eq:subchannel_allocation}\n\t\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}=\\mathscr{S}\\left(\\bm{Q}\\right),\n\\end{equation}\nwhere $\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}=\\left(\\mathcal{S}_{1},\\ldots,\\mathcal{S}_{M}\\right)$ is a vector of sets, and $\\mathcal{S}_{m}$ is the set of subchannels allocated to the user $u_{m}$, which satisfies $\\mathcal{S}_{m}\\cap\\mathcal{S}_{m'}=\\emptyset,\\,\\forall m'\\neq m$, and $\\mathcal{S}_{\\mathrm{alloc}}=\\bigcup_{u_{m}\\in\\mathcal{U}}\\mathcal{S}_{m}\\subseteq\\mathcal{S}$. Over the time duration of one frame, the received signal of each symbol $\\bm{y}\\in\\mathbb{C}^{N}$ in the frequency domain is given by\n\\begin{equation}\\label{eq:channel_model}\n\t\\bm{y}=\\bm{H}\\bm{x}+\\bm{w},\n\\end{equation}\nwhere $\\bm{w}\\in\\mathbb{C}^{N}$ is the additive white Gaussian noise with the distribution of $\\mathcal{CN}\\left(\\bm{0},\\bm{I}\\right)$. The channel gain $\\bm{H}$ is a diagonal matrix and $\\left[\\bm{H}\\right]_{nn}=h_{mn}$, if and only if $s_{n}\\in\\mathcal{S}_{m}$. The transmission symbol of all the users is denoted by the vector $\\bm{x}=\\left(x_{n}\\right)_{n=1}^{N}$, where $\\bm{x}_{m}=\\left(x_{n}\\right)_{s_{n}\\in\\mathcal{S}_{m}}$ is the symbol vector transmitted by the user $u_{m}$. Let $\\mathscr{C}$ denote the coding scheme in the frequency domain, then $\\bm{x}_{m}=\\mathscr{C}\\left(\\bm{b}_{m}\\right)$, where $\\bm{b}_{m}$ is the uncoded-symbol vector for user $u_{m}$.\n\nThe objective of this paper is to propose a unified analytic framework for designing the  $\\mathscr{S}$ and choosing $\\mathscr{C}$ for subchannel allocation in multi-carrier multi-access systems. In the following, we will first assume $\\mathscr{C}$ has been properly chosen and focus on designing the optimal $\\mathscr{S}$. Then, the coding scheme, which can meet the requirement of the optimal $\\mathscr{S}$, will be discussed.\n\n\n\\subsection{Outage Exponent Region \\& Diversity-Multiplexing Region}\nAs defined in \\cite{Ozarow1994}, if the instantaneous mutual information is smaller than a target transmission rate, this channel is in outage. The results in compound channels showed that the reliable communication can be achieved if the channel is not in outage \\cite{Csiczar1981}. Therefore, the outage probability, which establishes the relationship between the reliability and efficiency, is a key performance metric in slow fading channels \\cite{Zheng2003}.\n\nAccording to the QoS requirement of the considered multi-carrier multi-access channel, each user has a specific target rate, which is denoted by $R_{m}$. A given vector of target rates $\\bm{R}=\\left(R_{m}\\right)_{m=1}^{M}$ will be referred to as an \\emph{operating point} of the system. If $\\bm{Q}$ is already known at the BS, $\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}$ can be generated by a given subchannel allocation scheme $\\mathscr{S}$. The coding scheme $\\mathscr{C}$ is assumed to be properly chosen such that the sum mutual information of the subchannels allocated to each user can be achieved. For a given operating point $\\bm{R}$, therefore, the outage probability of a user $u_{m}$ is defined by\n\\begin{equation}\\label{eq:outage_definition}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)=\\Pr\\left\\{\\left.\\sum_{s_{n}\\in\\mathcal{S}_{m}}I_{mn}<R_{m}\\right|\\mathcal{S}_{m}\\right\\}.\n\\end{equation}\nEq. \\eqref{eq:outage_definition} implies that different users may have different outage probabilities. To evaluate the comprehensive performance of a subchannel allocation scheme, we define the \\emph{outage exponent region} (OER), which specifies the set of outage exponents that are simultaneously achievable by the users for a fixed operating point. Similar to the error exponent region proposed in \\cite{Weng2008}, the following definition describes this problem formally.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[>=stealth]\n\t\\fill[domain=0:4,gray!50] plot (\\x,{2*sqrt(1-pow(\\x,2)\/pow(4,2))}) -- (0,0) -- (0,2);\n\t\\draw[domain=0:4,thick] plot (\\x,{2*sqrt(1-pow(\\x,2)\/pow(4,2))});\n\t\\draw[domain=0:4.5,thick,dashed] plot (\\x,{2.5*sqrt(1-pow(\\x,2)\/pow(4.5,2))});\n\t\\node at (1.8,0.9) {$\\mathcal{R}_\\mathrm{oer}\\left(R_{1},R_{2};\\gamma\\right)$};\n\t\\node at (3.5,2.7) {Boundary of $\\mathcal{R}_\\mathrm{dmr}\\left(r_{1},r_{2}\\right)$};\n\t\\node at (1.3,2.1) {$\\gamma\\rightarrow\\infty$};\n\n\t\\draw[->] (3.5,2.5) -- (3,1.8634);\n\t\\draw[->] (1.8,1.7861) -- (2.1,2.2111);\n\n\t\\draw[->,very thick] (0,0) -- (5,0) node[right] {$E_{1}\\left(R_{1}\\right)$};\n\t\\draw[->,very thick] (0,0) -- (0,3) node[above] {$E_{2}\\left(R_{2}\\right)$};\n\\end{tikzpicture}\n\\caption{The OER $\\mathcal{R}_{\\mathrm{oer}}\\left(R_{1},R_{2};\\gamma\\right)$ and DMR $\\mathcal{R}_{\\mathrm{dmr}}\\left(r_{1},r_{2}\\right)$ for an operating point $\\left(R_{1},R_{2}\\right)$.}\\label{fig:region_example}\n\\end{figure}\n\n\\medskip\n\\begin{defn}\\label{def:OER}\nFor a given operating point $\\bm{R}$, the OER, denoted by $\\mathcal{R}_{\\mathrm{oer}}\\left(\\bm{R};\\gamma\\right)$, consists of all vectors of outage exponents $\\left(E_{m}\\left(R_{m},\\gamma\\right)\\right)_{m=1}^{M}$, which can be achieved by at least one subchannel allocation scheme $\\mathscr{S}$. In this context, the outage exponent is defined as\n\\begin{equation}\\label{eq:outage_exponent}\n\tE_{m}\\left(R_{m},\\gamma\\right)=-\\frac{\\partial\\ln p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)}{\\partial\\ln\\gamma}.\n\\end{equation}\n\\end{defn}\n\\medskip\n\nFig. \\ref{fig:region_example} shows an example of the OER for a subchannel allocation scheme operated at point $\\left(R_{1},R_{2}\\right)$, which is a two-dimensional region depending on the operating point. It should be noted that there is only one capacity region for a given system, but there is one OER for every operating point inside the capacity region. Clearly, a larger OER implies a better performance for a given operating point.\n\nAccording to the results in \\cite{Bai2011c}, the diversity-multiplexing tradeoff (DMT) for a user $u_{m}$ is closely related to the outage exponent as follows:\n\\begin{equation}\\label{eq:diversity_gain}\n\t\\lim_{\\mathsf{\\gamma\\rightarrow\\infty}}E_{m}\\left(R_{m},\\gamma\\right)=d_{m}\\left(r_{m}\\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\tr_{m}=\\lim_{\\mathsf{\\gamma\\rightarrow\\infty}}\\frac{R_{m}}{\\ln\\left(1+\\gamma\\right)}\n\\end{equation}\nis known as the multiplexing gain. As shown in Fig. \\ref{fig:region_example}, the \\emph{diversity-multiplexing region} (DMR), defined in the following, is the asymptotic case of the OER when $\\gamma$ tends to infinity.\n\n\\medskip\n\\begin{defn}\\label{def:DMR}\nFor a given operating point $\\bm{r}=\\left(r_{m}\\right)_{m=1}^{M}$, the DMR, denoted by $\\mathcal{R}_{\\mathrm{dmr}}\\left(\\bm{r}\\right)$, consists of all vectors of diversity gains $\\left(d_{m}\\left(r_{m}\\right)\\right)_{m=1}^{M}$, which can be achieved by at least one subchannel allocation scheme $\\mathscr{S}$.\n\\end{defn}\n\\medskip\n\nThe optimal DMT for parallel fading channels is given by $d^{*}\\left(r\\right)=L\\left(1-r\/L\\right)$ \\cite{Bai2011c}. Therefore, for the DMR of the considered multi-carrier multi-access channel, the element of the optimal DMT vector should be given by\n\\begin{equation}\\label{eq:opt_dmt}\n\td_{m}^{*}\\left(r_{m}\\right)=L\\left(1-\\frac{r_{m}}{r_{m}^{*}}\\right),\\quad m=1,\\ldots,M,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:opt_dmt_cond}\n\\left\\{\\begin{aligned}\n\t& \\frac{r_{1}^{*}}{R_{1}}=\\cdots=\\frac{r_{M}^{*}}{R_{M}},\\\\\n\t& \\sum_{u_{m}\\in\\mathcal{U}}r_{m}^{*}=L.\n\\end{aligned}\\right.\n\\end{equation}\nIn fact, Eqs. \\eqref{eq:opt_dmt} \\eqref{eq:opt_dmt_cond} indicate that each user shares the total multiplexing gain according to the operating point, while all of them achieve the full frequency diversity. Thus, a subchannel allocation scheme is referred to be \\emph{asymptotic optimal}, denoted by $\\mathscr{S}^{*}$, if and only if Eqs. \\eqref{eq:opt_dmt} \\eqref{eq:opt_dmt_cond} are achieved.\n\n\n\\section{RBG based Coded $f$-Matching Framework}\\label{sec:coded_f_matching}\n\nIn this section, the maximum $f$-matching approach will first be proposed for designing the asymptotic optimal subchannel allocation schemes $\\mathscr{S}^{*}$. The coding scheme $\\mathscr{C}$ which fulfills the requirement of the proposed framework, denoted by $\\mathscr{C}^{*}$, will also be discussed.\n\n\n\\subsection{Random Bipartite Graph Formulation}\\label{subsec:RBG_Formulation}\n\nBased on the aforementioned $\\bm{Q}$ matrix, the considered multi-carrier multi-access channel can be formulated as a \\emph{random bipartite graph} (RBG) model. In order to satisfy Eq. \\eqref{eq:opt_dmt_cond}, it requires every subchannel to contribute the same amount of efficiency and reliability to each user. This requirement implies that all of the elements in $\\bm{Q}$ should be independent and follows the identical distribution. The subchannel set $\\mathcal{S}$ will then be shared by all of the users according to the operating point of the considered multi-carrier multi-access channel. Define\n\\begin{equation}\\label{eq:subchannel_rate}\n\tR_{\\mathrm{s}}=\\frac{1}{N}\\sum_{u_{m}\\in\\mathcal{U}}R_{m}.\n\\end{equation}\nIf $I_{mn}<R_{\\mathrm{s}}$, we let $\\left[\\bm{Q}\\right]_{ml}=0,\\,\\forall s_{n}\\in\\mathcal{S}_{l}^{\\mathrm{c}}$, and refer this  subchannel as an outage subchannel for user $u_{m}$. Otherwise, $\\left[\\bm{Q}\\right]_{ml}=1$. According to Eq. \\eqref{eq:subchannel_capacity}, the subchannel outage probability is given by\n\\begin{equation}\\label{eq:subchannel_outage}\tp_{\\mathrm{s}}=\\Pr\\left\\{\\left[\\bm{Q}\\right]_{ml}=0\\right\\}=1-\\exp\\left(-\\frac{e^{N_{\\mathrm{c}}R_{\\mathrm{s}}}-1}{\\gamma}\\right).\n\\end{equation}\nThe non-outage probability of the subchannel is given by\n\\begin{equation}\n\tq_{\\mathrm{s}}=1-p_{\\mathrm{s}}.\n\\end{equation}\nIt can be seen that the subchannel outage probability is determined by $N_{\\mathrm{c}}R_{\\mathrm{s}}$, which is the target rate normalized by the coherence bandwidth. Since this parameter is very important in this paper, we let $R_{\\mathrm{c}}=N_{\\mathrm{c}}R_{\\mathrm{s}}$ for convenience.\n\nNote that $\\bm{Q}$ denotes the independent outage state of each subchannel for each user at the target rate $R_{\\mathrm{s}}$. In this context, each subchannel provides the same contribution of the rate and diversity to each user. Therefore, the user $u_{m}$ will be allocated\n\\begin{equation}\n\t\\widetilde{K}_{m}=\\frac{R_{m}}{R_{\\mathrm{s}}}\n\\end{equation}\nsubchannels to guarantee the target rate. Recalling Eq. \\eqref{eq:opt_dmt_cond}, we have $r_{m}^{*}=\\frac{\\widetilde{K}_{m}}{N_{\\mathrm{c}}}$.\n\nBefore describing the RBG formulation in detail, we need to introduce some concepts in graph theory \\cite{Bollobas1998}. A bipartite graph is a graph on which the set of vertices $\\mathcal{V}$ admits a partition into two classes such that every edge has its ends in different class. A bipartite graph is often denoted by $\\mathcal{G}\\left(\\mathcal{V}_{1}\\cup\\mathcal{V}_{2},\\mathcal{E}\\right)$, where $\\mathcal{E}$ is the set of edges, $\\mathcal{V}_{1}$ and $\\mathcal{V}_{2}$ are the sets of vertices in different partition classes with $\\mathcal{V}=\\mathcal{V}_{1}\\cup\\mathcal{V}_{2}$ and $\\mathcal{V}_{1}\\cap\\mathcal{V}_{2}=\\emptyset$. A bipartite graph is called complete if every two vertices from different partition classes are adjacent. The complete bipartite graph with $M$ and $N$ vertices in different partition classes is denoted by $\\mathcal{K}_{MN}$. Define a probability space $\\left(\\Omega,\\mathscr{A},\\mathsf{P}\\right)$, in which the sample space contains all the bipartite graphs with $M$ vertices and $N$ vertices in different partition classes, $\\mathscr{A}$ is the $\\sigma$-field on $\\Omega$, and $\\mathsf{P}$ is the probability measure on $\\mathscr{A}$. It is easy to verify that the sample space has $2^{MN}$ bipartite graphs. Similar to \\cite{Bollobas2001}, the probability space of a RBG is denoted by $\\mathscr{G}\\left\\{ \\mathcal{K}_{MN};\\mathsf{P}\\right\\}$ which means that every edge of the sample bipartite graph is chosen from $\\mathcal{K}_{MN}$ with the probability measure $\\mathsf{P}$. Clearly, a sample of $\\mathscr{G}\\left\\{ \\mathcal{K}_{MN};\\mathsf{P}\\right\\}$ is a bipartite graph $\\mathcal{G}\\left(\\mathcal{V}_{1}\\cup\\mathcal{V}_{2},\\mathcal{E}\\right)$. We also introduce some terms, which will be used in the following part. A vertex $v'$ is called a neighbor of vertex $v$, if they are adjacent. All the neighboring vertices of a given vertex $v$ are denoted by set $\\mathcal{N}\\left(v\\right)\\subset\\mathcal{V}$, and the cardinality of $\\mathcal{N}\\left(v\\right)$ is referred to as the degree of the vertex $v$, and denoted by $\\deg_{\\mathcal{G}}\\left(v\\right)$.\n\nFor the considered multi-carrier multi-access channel, the RBG model can be constructed as follows. First, all of the vertices in one partition class are used to represent all of the users in $\\mathcal{U}$, and all of the vertices in the other partition class are used to represent all of the subchannels in $\\mathcal{S}$, i.e., $\\mathcal{V}_{1}=\\mathcal{U}$ and $\\mathcal{V}_{2}=\\mathcal{S}$. If a subchannel $s_{n}\\in\\mathcal{S}$ is not in outage for a user $u_{m}\\in\\mathcal{U}$ according to the $\\bm{Q}$ matrix, we join the vertices $u_{m}$ and $s_{n}$ with an edge $e_{mn}=u_{m}s_{n}$ in the RBG. Otherwise, if $s_{n}$ is in outage for $u_{m}$, there will be no edge between them. Thus, $\\mathcal{N}\\left(u_{m}\\right)$ contains all of the subchannels that are not in outage for the user $u_{m}$. According to the construction method of $\\bm{Q}$, the probability measure $\\mathsf{P}$ can be defined as follows:\n\\begin{enumerate}\n\t\\item Let $s_{n}\\in\\mathcal{S}_{l}^{\\mathrm{c}}$ and $s_{n'}\\in\\mathcal{S}_{l'}^{\\mathrm{c}}$, the edges $e_{mn}$ and $e_{m'n'}$ are chosen from $\\mathcal{K}_{MN}$ independently with the identical probability $q_{\\mathrm{s}}$, if and only if $\\mathcal{S}_{l}^{\\mathrm{c}}\\neq\\mathcal{S}_{l'}^{\\mathrm{c}}$.\n\t\\item The edges $e_{mn}$ and $e_{m'n'}$ are chosen from $\\mathcal{K}_{MN}$ simultaneously, if and only if $\\mathcal{S}_{l}^{\\mathrm{c}}=\\mathcal{S}_{l'}^{\\mathrm{c}}$.\n\\end{enumerate}\nTherefore, a sample of $\\mathscr{G}\\left\\{\\mathcal{K}_{MN};\\mathsf{P}\\right\\}$ corresponds to a sample of $\\bm{Q}$ and vice verse. A sample of $\\mathscr{G}\\left\\{\\mathcal{K}_{3,6};\\mathsf{P}\\right\\}$ is shown in Fig. \\ref{fig:RBG_example}.\n\n\n\\subsection{Maximum $f$-Matching based Subchannel Allocation}\\label{sec:max_f_matching}\n\nGiven a sample of $\\mathscr{G}\\left\\{\\mathcal{K}_{MN};\\mathsf{P}\\right\\}$, BS should allocate $\\widetilde{K}_{m}$ subchannels to the user $u_{m}$ so as to guarantee Eqs. \\eqref{eq:opt_dmt} \\eqref{eq:opt_dmt_cond}. For designing the optimal subchannel allocation scheme, we introduce the concept of $f$-matching and $f$-factor in matching or factorization theory \\cite{Lovasz1986}. Define a non-negative integer function $f\\left(v\\right)$ on vertex set $\\mathcal{V}=\\mathcal{U}\\cup\\mathcal{S}$, and let $w\\left(e\\right),\\:\\forall e\\in\\mathcal{E}$ denote an integral weight over edge $e$. Let $e_{v}$ denote the edge that is incident with the vertex $v$, the $\\mathcal{M}_{f}$-degree of vertex $v$ is then given by\n\\begin{equation}\n\t\\deg_{\\mathcal{M}_{f}}\\left(v\\right)=\\sum_{e_{v}\\in\\mathcal{M}_{f}}w\\left(e_{v}\\right).\n\\end{equation}\nTherefore, the $f$-matching can be defined as follows.\n\n\\medskip\n\\begin{defn}\\label{def:f_matching}\nAn $f$-matching $\\mathcal{M}_{f}$ is a subset of $\\mathcal{E}$ which satisfies\n\\begin{equation}\\label{eq:f_matching}\n\t\\deg_{\\mathcal{M}_{f}}\\left(v\\right)\\leq f\\left(v\\right),\\quad\\forall v\\in\\mathcal{V}.\n\\end{equation}\nIf $\\deg_{\\mathcal{M}_{f}}\\left(v\\right)=f\\left(v\\right)$, the vertex $v$ is referred to as the $\\mathcal{M}_{f}$-saturated. The $f$-matching with the maximum number of edges is known as the maximum $f$-matching, denoted by $\\mathcal{M}_{f}^{\\mathrm{m}}$. Especially, the maximum $f$-matching is called the perfect $f$-matching or $f$-factor, denoted by $\\mathcal{M}_{f}^{\\mathrm{p}}$, if the equality of Eq. \\eqref{eq:f_matching} holds for every vertex $v\\in\\mathcal{V}$.\n\\end{defn}\n\\medskip\n\nConsider now the subchannel allocation problem in the considered multi-carrier multi-access channel. Let $w\\left(e\\right)=1,\\,\\forall e\\in\\mathcal{E}$, and define $f\\left(v\\right)$ as\n\\begin{equation}\\label{eq:f-function}\n\tf\\left(v\\right)=\n\t\\left\\{\\begin{aligned}\n\t\t& K_{m}, & v=u_{m}\\in\\mathcal{U};\\\\\n\t\t& 1, & v=s_{n}\\in\\mathcal{S}.\n\t\\end{aligned}\\right.\n\\end{equation}\nIn order to guarantee Eq. \\eqref{eq:opt_dmt_cond}, the maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$ should be generated to satisfy the following conditions:\n\\begin{equation}\\label{eq:opt_cond_f-function}\n\t\\frac{K_{1}}{\\widetilde{K}_{1}}=\\cdots=\\frac{K_{M}}{\\widetilde{K}_{M}}.\n\\end{equation}\nFor a generated maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$, the number of non-outage subchannels allocated to the user $u_{m}$ is given by $k_{m}=\\deg_{\\mathcal{M}_{f}^{\\mathrm{m}}}\\left(u_{m}\\right)$. Clearly, the following condition must also be guaranteed by the generated $\\mathcal{M}_{f}^{\\mathrm{m}}$:\n\\begin{equation}\\label{eq:opt_cond_fmatching}\n\t\\frac{k_{1}}{\\widetilde{K}_{1}}=\\cdots=\\frac{k_{M}}{\\widetilde{K}_{M}}.\n\\end{equation}\nHere, we let $(\\widetilde{K}_{\\left(m\\right)})_{m=1}^{M}$, $(K_{\\left(m\\right)})_{m=1}^{M}$, and $(k_{\\left(m\\right)})_{m=1}^{M}$ be the ordered vectors which are from the minimum value to the maximum one. For convenience, we also define\n\\begin{equation}\\label{eq:K_sum}\n\\left\\{\\begin{aligned}\n\t& \\widetilde{K}^{\\mathrm{sum}}=\\sum_{u_{m}\\in\\mathcal{U}}\\widetilde{K}_{m}=N,\\\\\n\t& K^{\\mathrm{sum}}=\\sum_{u_{m}\\in\\mathcal{U}}K_{m},\\\\\n\t& k^{\\mathrm{sum}}=\\sum_{u_{m}\\in\\mathcal{U}}k_{m};\n\\end{aligned}\\right.\n\\end{equation}\nand\n\\begin{equation}\n\\left\\{\\begin{aligned}\n\t& \\widetilde{K}_{m}^{\\mathrm{sum}}=\\widetilde{K}^{\\mathrm{sum}}-\\widetilde{K}_{m},\\\\\n\t& K_{m}^{\\mathrm{sum}}=K^{\\mathrm{sum}}-K_{m},\\\\\n\t& k_{m}^{\\mathrm{sum}}=k^{\\mathrm{sum}}-k_{m}.\n\\end{aligned}\\right.\n\\end{equation}\nIf $k_{m}=K_{m}=\\widetilde{K}_{m}$ for any $u_{m}\\in\\mathcal{U}$, all of the user are $\\mathcal{M}_{f}$-saturated, i.e., no users are in outage. The maximum $f$-matching satisfying this condition will be referred to as the $\\mathcal{U}$-perfect $f$-matching, denoted by $\\mathcal{M}^{\\mathcal{U}}_{f}$. Therefore, the optimal subchannel allocation scheme has two steps:\n\\begin{enumerate}\n\t\\item BS generates the maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$ that satisfies Eq. \\eqref{eq:opt_cond_fmatching} on each sample bipartite graph of $\\mathscr{G}\\left\\{ \\mathcal{K}_{MN};\\mathsf{P}\\right\\}$ so as to allocate $k_{m}$ non-outage subchannels to the user $u_{m}$.\n\t\\item If $\\mathcal{M}_{f}^{\\mathrm{m}}\\neq\\mathcal{M}^{\\mathcal{U}}_{f}$, BS randomly allocates $\\widetilde{K}_{m}-k_{m}$ subchannels from the set of outage subchannels to the user $u_{m}\\in\\mathcal{U}$.\n\\end{enumerate}\nAccording to Eq. \\eqref{eq:f-function}, if a $\\mathcal{U}$-perfect $f$-matching $\\mathcal{M}^{\\mathcal{U}}_{f}$ is generated in a bipartite graph, then the target rates of every user can be guaranteed. The $\\mathcal{U}$-perfect $f$-matching, however, may not always be exist for every sample of $\\mathscr{G}\\left\\{\\mathcal{K}_{MN};\\mathsf{P}\\right\\} $. For the case of $k_{m}<\\widetilde{K}_{m}$, whether the user $u_{m}$ is in outage or not also depends on the applied coding scheme.\n\nAn example of the maximum $f$-matching based subchannel allocation is shown in Fig. \\ref{fig:RBG_example}. The maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}=\\left\\{u_{1}s_{1},u_{1}s_{4},u_{2}s_{5},u_{3}s_{2},u_{3}s_{6}\\right\\}$ are denoted by thick lines. It can be seen that the users $u_{1}$ and $u_{3}$ are $\\mathcal{M}_{f}^{\\mathrm{m}}$-saturated, and thus are not in outage. Since the user $u_{2}$ is allocated one non-outage subchannel and one outage subchannel, its outage state will then be determined by the sum mutual information achieved by the applied coding scheme.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\fill\n\t\t(-2,5) circle (3pt)\n\t\t(-2,3) circle (3pt)\n\t\t(-2,1) circle (3pt);\n\t\t\n\t\t\\node at (-2.5,5) {$u_{1}$};\n\t\t\\node at (-2.5,3) {$u_{2}$};\n\t\t\\node at (-2.5,1) {$u_{3}$};\n\t\t\n\t\t\\draw[line width=0.5pt,style=dashed] (-2.8,0.5) rectangle (-1.7,5.5);\n\t\t\\node at (-2.25,5.8) {Users};\n\t\t\t\t\n\t\t\\fill\n\t\t(2,5.5) circle (3pt)\n\t\t(2,4.5) circle (3pt)\n\t\t(2,3.5) circle (3pt)\n\t\t(2,2.5) circle (3pt)\n\t\t(2,1.5) circle (3pt)\n\t\t(2,0.5) circle (3pt);\t\t\n\n\t\t\\node at (2.5,5.5) {$s_{1}$};\n\t\t\\node at (2.5,4.5) {$s_{2}$};\n\t\t\\node at (2.5,3.5) {$s_{3}$};\n\t\t\\node at (2.5,2.5) {$s_{4}$};\n\t\t\\node at (2.5,1.5) {$s_{5}$};\n\t\t\\node at (2.5,0.5) {$s_{6}$};\n\t\t\n\t\t\\draw[line width=0.5pt,style=dashed] (1.7,0) rectangle (2.8,6);\n\t\t\\node at (2.25,6.3) {Subchannels};\n\t\t\n\t\t\\draw [line width=2pt]\n\t\t(-2,5) -- (2,5.5)\n\t\t(-2,5) -- (2,2.5)\n\t\t(-2,3) -- (2,1.5)\n\t\t(-2,1) -- (2,4.5)\n\t\t(-2,1) -- (2,0.5);\n\t\t\n\t\t\\draw [line width=0.8pt]\n\t\t(-2,5) -- (2,4.5)\n\t\t(-2,5) -- (2,3.5)\n\t\t(-2,5) -- (2,1.5)\n\t\t(-2,5) -- (2,0.5)\n\t\t(-2,3) -- (2,5.5)\n\t\t(-2,3) -- (2,4.5)\n\t\t(-2,3) -- (2,3.5)\n\t\t(-2,3) -- (2,2.5)\n\t\t(-2,3) -- (2,0.5)\n\t\t(-2,1) -- (2,5.5)\n\t\t(-2,1) -- (2,3.5)\n\t\t(-2,1) -- (2,2.5)\n\t\t(-2,1) -- (2,1.5);\n\t\\end{tikzpicture}\n\t\\caption{A sample of $\\mathscr{G}\\left\\{\\mathcal{K}_{3,6};\\mathsf{P}\\right\\}$ with $K_{m}=\\widetilde{K}_{m}=2,\\,m=1,2,3$, where the outage probability of each edge is given by Eq. \\eqref{eq:subchannel_outage}. The thick lines are the edges in the maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$.}\\label{fig:RBG_example}\n\\end{figure}\n\nClearly, the RBG based $f$-matching approach can be directly applied to both RB and chunk based allocation schemes, only if we see the RB or chunk as the subchannel.\n\n\n\\subsection{Requirements on Coding Schemes}\\label{sec:coding}\n\nAccording to the proposed maximum $f$-matching method, if the frequency-domain coding scheme $\\mathscr{C}$ is not used, the outage state of a user is only determined by the relationship between $\\widetilde{K}_{m}$ and $k_{m}$. In this context, the outage probability should be given by\n\\begin{equation}\n\t\\Pr\\left\\{\\left.k_{m}<\\widetilde{K}_{m}\\right|\\mathcal{S}_{m}\\right\\},\n\\end{equation}\nwhich is apparently greater than $p_{m}^{\\mathrm{out}}$ defined in Eq. \\eqref{eq:outage_definition}. Therefore, the maximum $f$-matching should be generated with the help of coding schemes to achieve the optimal outage performance.\n\nIt is well known that the minimum product distance between two codewords determines the capability of this coding scheme to combat the channel fading \\cite{Tse2005}, which will determine the outage probability defined in Eq. \\eqref{eq:outage_definition}. In the signal space of user $u_{m}$, each codeword can be seen as a vector or point in the $\\widetilde{K}_{m}$-dimension space, i.e., $\\widetilde{K}_{m}$-dimension constellation. If we choose any $s_{n}\\in\\mathcal{S}_{m}$, then the $n$th axis of the $\\widetilde{K}_{m}$-dimension constellation corresponds to the channel gain $h_{mn}$. For convenience, let\n\\begin{equation}\n\\left\\{\\begin{aligned}\n\t& \\bm{x}_{m}^{A}=\\left(x_{mn}^{A}\\right)_{s_{n}\\in\\mathcal{S}_{m}},\\\\\n\t& \\bm{x}_{m}^{B}=\\left(x_{mn}^{B}\\right)_{s_{n}\\in\\mathcal{S}_{m}},\n\\end{aligned}\\right.\n\\end{equation}\ndenote the codewords of the user $u_{m}$ for the information $A$ and $B$, respectively. The normalized product distance of $\\bm{x}_{m}^{A}$ and $\\bm{x}_{m}^{B}$ is then defined by\n\\begin{equation}\\label{eq:product_distance}\n\tD_{m}=\\frac{1}{\\sqrt{\\gamma}}\\prod_{s_{n}\\in\\mathcal{S}_{m}}\\left|x_{mn}^{A}-x_{mn}^{B}\\right|.\n\\end{equation}\n\nThe coding scheme must be designed to maximize the minimum value of $D_{m}$, denoted by $D_{m}^{\\mathrm{min}}$, so as to optimize the outage performance. If $D_{m}^{\\mathrm{min}}=0$, for example $x_{n}^{A}=x_{n}^{B}$ for some $n$, there must be a hyperplane which is orthogonal with the $n$th axis such that $x_{n}^{A}-x_{n}^{B}$ is parallel with this hyperplane. Clearly, if all the channel gains on that hyperplane are very small, the receiver cannot distinguish $\\bm{x}_{m}^{A}$ from $\\bm{x}_{m}^{B}$. In other words, we lose one dimension, i.e., the $n$th subchannel, to combat the channel fading. Therefore, the number of terms that $x_{n}^{A}\\neq x_{n}^{B},\\,s_{n}\\in\\mathcal{S}_{m}$ in Eq. \\eqref{eq:product_distance} is the diversity order achieved by this signal constellation \\cite{Oggier2004}. This diversity order is another understanding, from the perspective of coding scheme, of the DMR in Definition \\ref{def:DMR}. Therefore, the optimal coding scheme should be capable of achieving the optimal DMT in point-to-point parallel fading channels.\n\nIn this context, the coded $f$-matching based subchannel allocation scheme can be summarized as follows:\n\\begin{enumerate}\n\t\\item Allocate subchannels to each user by using the maximum $f$-matching method in Section \\ref{sec:max_f_matching};\n\t\\item Apply the optimal coding scheme on the subchannel set $\\mathcal{S}_{m}$, if $k_{m}<\\widetilde{K}_{m}$ for the user $u_{m}$.\n\\end{enumerate}\n\n\n\\section{OER \\& DMR Analysis for the Coded $f$-Matching Framework}\\label{sec:oer_dmr_analysis}\n\n\\begin{table*}\n  \\centering\n  \\caption{Summarization of the Main Results}\\label{tab:comparison}\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n    \\hline\n    \\multicolumn{2}{|c|}{Allocation Schemes} & OER & DMR & \\begin{tabular}{c} Multi-User\\\\ Diversity \\end{tabular} & \\begin{tabular}{c} Allocation\\\\ Complexity \\end{tabular} & \\begin{tabular}{c} Coding\\\\ Length \\end{tabular} & CSI & FOE\\tabularnewline\n    \\hline\n    \\multirow{3}{*}{\\begin{tabular}{c} RB\\\\ Based \\end{tabular}} & Coded $f$-Matching & Eqs. \\eqref{eq:outage_RB_high} \\eqref{eq:outage_RB_LowSNR} & {\\begin{tabular}{c} Eq. \\eqref{eq:DMR_RB}\\\\ Optimal\\end{tabular}} & Yes & $\\mathcal{O}\\left(\\log^{2\\eta}N\\right)$ & $\\frac{\\widetilde{K}_{m}}{N_{\\mathrm{c}}}$ & $\\unit[1]{bit}$ & High\\tabularnewline\n    \\cline{2-9}\n    & Interleaved & Eq. \\eqref{eq:outage_exponent_interleaved} & {\\begin{tabular}{c} Eq. \\eqref{eq:DMR_RB}\\\\ Optimal\\end{tabular}} & No & $\\mathcal{O}\\left(1\\right)$ & $L$ & --- & High\\tabularnewline\n    \\hline\n    \\multirow{5}{*}{\\begin{tabular}{c} Chunk\\\\ Based\\end{tabular}} & Coded $f$-Matching & Eqs. \\eqref{eq:outage_chunk_1} \\eqref{eq:outage_RB_LowSNR} & {\\begin{tabular}{c} Eq. \\eqref{eq:DMR_chunk}\\\\ Suboptimal\\end{tabular}} & Yes & $\\mathcal{O}\\left(\\log^{2\\eta}L\\right)$ & $K_{m}$ & $\\unit[1]{bit}$ & Middle\\tabularnewline\n    \\cline{2-9}\n    & Localized & Eq. \\eqref{eq:outage_exponent_interleaved} & {\\begin{tabular}{c} Eq. \\eqref{eq:DMT_Lower}\\\\ Worst\\end{tabular}} & No & $\\mathcal{O}\\left(1\\right)$ & $K_{m}$ & --- & Low\\tabularnewline\n    \\cline{2-9}\n    & TDMA & Eq. \\eqref{eq:outage_exponent_interleaved} & {\\begin{tabular}{c} Eq. \\eqref{eq:DMR_RB}\\\\ Optimal\\end{tabular}} & No & $\\mathcal{O}\\left(1\\right)$ & $L$ & --- & Middle\\tabularnewline\n    \\hline\n  \\end{tabular}\n\\end{table*}\n\nIn this section, the OER and DMR are analyzed for the proposed coded $f$-matching framework. The tight upper bound of the conditional outage probability is first presented to illustrate the performance of the optimal coding schemes with $\\unit[1]{bit}$ CSI feedback per coherence bandwidth. The properties of the maximum $f$-matching method are then studied in detail. With these results, the outage probability, OER and DMR of the proposed coded $f$-matching framework are presented for both RB and chunk based allocation schemes. The main results can be briefly summarized in Table \\ref{tab:comparison}, where the complexity of frequency offset estimation (FOE) is also listed \\cite{Beek1997} .\n\n\n\\subsection{Tight Upper Bound of the Conditional Outage Probability}\n\nLet the optimal coding scheme $\\mathcal{C}^{*}$ be used such that $D_{p}^{\\mathrm{min}}$ has been achieved. Focus on the user $u_{m}$ with a generated maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$, then the \\emph{conditional outage probability} (COP) is given by\n\\begin{equation}\\label{eq:con_outage_probability}\n\tp_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)=\\Pr\\left\\{\\left.\\sum_{n=1}^{K_{m}}I_{mn}<R_{m}\\right|\\mathcal{D}_{K_{m}k_{m}}\\right\\},\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:state_space}\n\\begin{aligned}\n\t\\mathcal{D}_{K_{m}k_{m}}= & \\left\\{I_{m1},\\ldots,I_{mK_{m}}\\left|I_{m1}\\geq R_{\\mathrm{s}},\\ldots,I_{mk_{m}}\\geq R_{\\mathrm{s}},\\right.\\right.\\\\\n \t& \\left.I_{m,k_{m}+1}<R_{\\mathrm{s}},\\ldots,I_{mK_{m}}<R_{\\mathrm{s}}\\right\\}.\n\\end{aligned}\n\\end{equation}\nAccording to the proposed coded $f$-matching framework, it is only needed to consider the case that different subchannel in $\\mathcal{S}_{m}$ belongs to different coherence bandwidth. This condition is true because only the scheme of coding across coherence bandwidths is capable of providing diversity gains. Even under this condition, it is still not trivial to obtain the exact closed-form formula for Eq. \\eqref{eq:con_outage_probability}. In this paper, the saddle-point approximation method, which has a good balance between accuracy and complexity \\cite{Butler2007}, is applied to derive a tight upper bound of Eq. \\eqref{eq:con_outage_probability}. Let $\\beta_{m}=1-\\frac{k_{m}}{K_{m}}$, and define a symbol ``$\\lesssim$'' as\n\\begin{equation}\\label{eq:symbol}\n\tf\\left(x\\right)\\lesssim g\\left(x\\right)\\Leftrightarrow\n\t\\left\\{\\begin{aligned}\n\t\t& f\\left(x\\right)\\leq g\\left(x\\right);\\\\\n\t\t& \\lim_{x\\rightarrow\\infty}\\frac{f\\left(x\\right)}{g\\left(x\\right)}=1.\n\t\\end{aligned}\\right.\n\\end{equation}\nThe tight upper bound of Eq. \\eqref{eq:con_outage_probability} can then be summarized in the following theorem.\n\n\\medskip\n\\begin{thm}\\label{thm:con_outage_exponent}\nThe exponentially tight upper bound of the conditional outage probability in Eq. \\eqref{eq:con_outage_probability} is given by\n\\begin{equation}\\label{eq:con_outage_exponent}\n\t\\begin{aligned}\n\t\t& p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)\\lesssim p_{m}^{\\mathrm{upper}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)\\\\\n\t\t& =\\psi \\exp\\left(-K_{m}\\left[\\left(\\ln\\gamma-R_{\\mathrm{c}}\\right)J_{2}\\left(\\gamma\\right)+J_{1}\\left(\\gamma\\right)-J_{0}\\left(\\gamma\\right)\\right]\\right),\n\t\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:exponent_part1}\n\t\\psi=\\frac{1}{\\sqrt{2\\pi K_{m}\\sigma^{2}}\\lambda^{*}},\n\\end{equation}\n\\begin{equation}\\label{eq:exponent_part2}\n\t\\left\\{\\begin{aligned}\n\t\t& J_{2}\\left(\\gamma\\right)=\\lambda^{*};\\\\\n\t\t& J_{1}\\left(\\gamma\\right)=\\ln p_{\\mathrm{s}}^{\\beta_{m}}q_{\\mathrm{s}}^{\\left(1-\\beta_{m}\\right)};\\\\\n\t\t& J_{0}\\left(\\gamma\\right)=\\frac{1}{\\gamma}+\\left(1-\\beta_{m}\\right)\\ln\\Gamma\\left(1-\\lambda^{*},e^{R_{\\mathrm{c}}}\\gamma^{-1}\\right)\\\\\n\t\t& +\\beta_{m}\\ln\\left(\\Gamma\\left(1-\\lambda^{*},\\gamma^{-1}\\right)-\\Gamma\\left(1-\\lambda^{*},e^{R_{\\mathrm{c}}}\\gamma^{-1}\\right)\\right).\n\t\\end{aligned}\\right.\n\\end{equation}\nIn Eqs. \\eqref{eq:exponent_part1} \\eqref{eq:exponent_part2}, $\\lambda^{*}$ and $\\sigma^{2}$ satisfy Eq. \\eqref{eq:exponent_equation} and Eq. \\eqref{eq:exponent_parameter}, respectively. In these equations,\n\\begin{equation}\n\t\\Gamma\\left(z,\\alpha\\right)=\\int_{\\alpha}^{\\infty}e^{-t}t^{z-1}dt\n\\end{equation}\nis the incomplete Gamma function, and\n\\begin{equation}\n\\begin{aligned}\n\t& G_{p,q}^{m,n}\\left(z\\left|\n\t\\begin{aligned}\n\t\ta_{1}, & \\ldots,a_{p}\\\\\n\t\tb_{1}, & \\ldots,b_{q}\n\t\\end{aligned}\\right.\\right)=\\\\\n\t & \\frac{1}{2\\pi i}\\oint_{\\mathcal{L}}\\frac{\\prod_{j=1}^{m}\\Gamma\\left(b_{j}-s\\right)\\prod_{j=1}^{n}\\Gamma\\left(1-a_{j}+s\\right)}{\\prod_{j=m+1}^{q}\\Gamma\\left(1-b_{j}+s\\right)\\prod_{j=n+1}^{p}\\Gamma\\left(a_{j}-s\\right)}z^{s}ds,\n\\end{aligned}\n\\end{equation}\nis the Meijer's $G$-function \\cite{Gradshteyn2007}.\n\\end{thm}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:con_outage_exponent}.\n\\end{IEEEproof}\n\\medskip\n\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{equation}\\label{eq:exponent_equation}\n\t\\begin{aligned}\n\t\tR_{\\mathrm{c}}= & \\ln\\gamma+(1-\\beta_{m})\\ln\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}+(1-\\beta_{m})\\frac{G_{2,3}^{3,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\\\\n \t\t& +\\beta_{m}\\frac{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)\\ln\\frac{1}{\\gamma}-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\ln\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}+G_{2,3}^{3,0}\\left(\\frac{1}{\\gamma}\\left|\n\t \t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)-G_{2,3}^{3,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}.\n\t\\end{aligned}\n\\end{equation}}\n\\rule[0.5ex]{1\\textwidth}{0.5pt}\n\\end{figure*}\n\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{equation}\\label{eq:exponent_parameter}\n\t\\begin{aligned}\n\t\t\\sigma^{2}= & \\beta_{m}\\frac{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)\\left(\\ln\\frac{1}{\\gamma}\\right)^{2}-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\left(\\ln\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)^{2}+2G_{2,3}^{3,0}\\left(\\frac{1}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)\\ln\\frac{1}{\\gamma}}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\\\\n\t\t& +\\beta_{m}\\frac{-2G_{2,3}^{3,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)\\ln\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}+2G_{3,4}^{4,0}\\left(\\frac{1}{\\gamma}\\left|\n\t\t\\begin{array}{cccc}\n\t\t\t1, & 1, & 1, & -\\\\\n\t\t\t0, & 0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)-2G_{3,4}^{4,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{cccc}\n\t\t\t1, & 1, & 1, & -\\\\\n\t\t\t0, & 0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\\\\n\t\t& -\\beta_{m}\\left(\\frac{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)\\ln\\frac{1}{\\gamma}-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\ln\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}+G_{2,3}^{3,0}\\left(\\frac{1}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)-G_{2,3}^{3,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\right)^{2}\\\\\n\t\t& +(1-\\beta_{m})\\frac{2G_{3,4}^{4,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{cccc}\n\t\t\t1, & 1, & 1, & -\\\\\n\t\t\t0, & 0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}-(1-\\beta_{m})\\left(\\frac{G_{2,3}^{3,0}\\left(\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\left|\n\t\t\\begin{array}{ccc}\n\t\t\t1, & 1, & -\\\\\n\t\t\t0, & 0, & 1-\\lambda^{*}\n\t\t\\end{array}\\right.\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\right)^{2}.\n\t\\end{aligned}\n\\end{equation}}\n\\rule[0.5ex]{1\\textwidth}{0.5pt}\n\\end{figure*}\n\nRecalling the definition in Eq. \\eqref{eq:diversity_gain}, the conditional DMT, which illustrates the contribution of the coding scheme in the asymptotic case, can be summarized as follows.\n\n\\medskip\n\\begin{cor}\\label{cor:CDMT}\nFor the optimal coding scheme with $\\unit[1]{bit}$ CSI feedback per coherence bandwidth, the conditional DMT for the user $u_{m}$ is given by\n\\begin{equation}\\label{eq:dmt}\n\\begin{aligned}\n\td_{m}^{\\mathrm{con}}\\left(r_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right) & =k_{m}\\left(1-\\frac{r_{m}}{K_{m}}\\right)\\\\\n\t& =K_{m}\\left(1-\\beta_{m}\\right)\\left(1-\\frac{r_{m}}{K_{m}}\\right).\n\\end{aligned}\n\\end{equation}\n\\end{cor}\n\\medskip\n\n\\begin{rmk}\nCorollary \\ref{cor:CDMT} shows that the diversity gain will be decreased to $K_{m}\\left(1-\\beta_{m}\\right)$, when the transmitter knows that $K_{m}\\left(1-\\beta_{m}\\right)$ subchannels are not in outage. This phenomenon is quite counter-intuitive that the conventional point of view thinks CSI feedback will increase the communication performance. How to explain this? As a matter of fact, the observation of the channel in the transmitter will result in the state space collapse of the channel gains. If the transmitter does not observe the channel, we have no choice but evaluate the outage probability in the full state space, i.e., the prior probability space. When the transmitter knows CSI, the outage probability will be evaluated in a restricted state space, i.e., the posterior probability space. For the partial CSI feedback, the sate space will be collapsed to $\\mathcal{D}_{K_{m}k_{m}}$ as defined in Eq. \\eqref{eq:state_space}. With perfect CSI feedback, the state space will be reduced to one element, i.e., whether the channel is in outage or not is a deterministic event. Therefore, the conditional outage performance can be seen as a ``local'' performance metric in the time domain. Let $I^{t}_{mn}$ and $\\mathcal{D}^{t}_{K_{m}k_{m}}$ denote the mutual information and CSI feedback at timeslot $t$, respectively. In the long run, the outage probability of the system, which is a ``global'' performance metric, can be given by\n\\begin{equation}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)=\\frac{1}{T}\\sum_{t=1}^{T}\\Pr\\left\\{\\left.\\sum_{n=1}^{K_{m}}I^{t}_{mn}<R_{m}\\right|\\mathcal{D}^{t}_{K_{m}k_{m}}\\right\\}.\n\\end{equation}\nAccording to the law of large numbers, we have\n\\begin{equation}\n\\begin{aligned}\n\t& \\begin{aligned}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)= & \\sum_{\\mathcal{D}_{K_{m}k_{m}}\\in\\mathscr{D}_{m}}\\Pr\\left\\{\\mathcal{D}_{K_{m}k_{m}}\\right\\}\\cdot\\\\\n\t& \\Pr\\left\\{\\left.\\sum_{n=1}^{\\widetilde{K}_{m}}I_{mn}<R_{m}\\right|\\mathcal{D}_{K_{m}k_{m}}\\right\\}\n\t\\end{aligned}\\\\\n\t& = \\sum_{\\mathcal{D}_{K_{m}k_{m}}\\in\\mathscr{D}_{K_{m}k_{m}}}p_{m}^{\\mathrm{cop}}\\left(R_{m}|\\mathcal{D}_{K_{m}k_{m}}\\right)\\Pr\\left\\{\\mathcal{D}_{K_{m}k_{m}}\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathscr{D}_{K_{m}k_{m}}$ is the space constructed by all possible $\\mathcal{D}_{K_{m}k_{m}}$. Since\n\\begin{equation}\n\t\\Pr\\left\\{\\mathcal{D}_{K_{m}k_{m}}\\right\\}=p_{\\mathrm{s}}^{K_{m}\\beta_{m}}q_{\\mathrm{s}}^{K_{m}\\left(1-\\beta_{m}\\right)},\n\\end{equation}\nthen in the high SNR regime the ``global'' outage probability will converge to\n\\begin{equation}\n\\begin{aligned}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right) & \\approx\\gamma^{-K_{m}\\left(1-\\beta_{m}\\right)\\left(1-\\frac{r_{m}}{K_{m}}\\right)}p_{\\mathrm{s}}^{K_{m}\\beta_{m}}\\\\\n\t& \\approx\\gamma^{K_{m}\\left(1-\\frac{r_{m}}{K_{m}}\\right)},\n\\end{aligned}\n\\end{equation}\nwhich is just the ``global'' DMT in \\cite{Bai2011c}.\n\\end{rmk}\n\n\n\\subsection{Properties of the Maximum $f$-Matching}\n\nTheorem \\ref{thm:con_outage_exponent} and Corollary \\ref{cor:CDMT} presents the outage performance achieved by the optimal coding scheme with $\\unit[1]{bit}$ CSI feedback per coherence bandwidth. To analyze the outage performance of the proposed coded $f$-matching framework, we still need to study the properties of the maximum $f$-matching on the RBG formulation of the considered multi-carrier multi-access channel. As a powerful tool for studying these properties, the \\emph{max-flow min-cut} theorem is listed as Lemma \\ref{lem:max_flow_min_cut} in the following \\cite{Bollobas1998}.\n\\medskip\n\\begin{lem}\\label{lem:max_flow_min_cut}\nFor a directed graph with source $\\theta$ and sink $\\zeta$, the maximum flow value from $\\theta$ to $\\zeta$ is equal to the minimum of the capacities of cuts separating $\\theta$ from $\\zeta$.\n\\end{lem}\n\\medskip\n\nBased on the max-flow min-cut theorem in Lemma \\ref{lem:max_flow_min_cut}, the following lemma can be obtained.\n\n\\medskip\n\\begin{lem}\\label{lem:f_matching}\nLet $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ be a bipartite graph with a non-negative integer function $f\\left(v\\right)$ defined on $\\mathcal{U}\\cup\\mathcal{S}$. Then, the bipartite graph $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ has a maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$ which cannot saturate a given vertex $u_{m}$, if and only if there is a subset $\\mathcal{X}$ of $\\mathcal{U}$ with $u_{m}\\in\\mathcal{X}$ satisfying\n\\begin{equation}\\label{eq:f_matching_cond}\n\t\\sum_{u_{i}\\in\\mathcal{X}}f\\left(u_{i}\\right)>\\sum_{s_{n}\\in\\mathcal{N}\\left(\\mathcal{X}\\right)}f\\left(s_{n}\\right),\n\\end{equation}\nwhere $\\mathcal{N}\\left(\\mathcal{X}\\right)\\subseteq\\mathcal{S}$ is the adjacent vertex set of $\\mathcal{X}$.\n\\end{lem}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:f_matching}.\n\\end{IEEEproof}\n\\medskip\n\nFrom this lemma, the sufficient and necessary conditions for the existence of a specific maximum $f$-matching can be summarized as follows, where $K_{m}^{\\mathrm{th}}$ is an integer such that\n\\begin{equation}\n\tK_{m}^{\\mathrm{th}}=N+1-\\left\\lceil\\frac{M}{M-1}K_{m}^{\\mathrm{sum}}\\right\\rceil.\n\\end{equation}\n\n\\medskip\n\\begin{lem}\\label{lem:matching_edges}\nLet $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ be a bipartite graph with $\\left|\\mathcal{U}\\right|=M$, $\\left|\\mathcal{S}\\right|=N$ and $2\\leq M\\leq N$. Define a non-negative integer function $f\\left(v\\right)$ as Eq. \\eqref{eq:f-function}, then for any edge set $\\mathcal{E}$ if and only if\n\\begin{enumerate}\n   \\item $|\\mathcal{E}|\\geq\\left(M-1\\right)N+K_{m}$ for the case of $K_{m}=1,\\ldots,K_{m}^{\\mathrm{th}}$,\n   \\item or $|\\mathcal{E}|\\geq M\\left(K^{\\mathrm{sum}}-1\\right)+1$ for the case of $K_{m}=K_{m}^{\\mathrm{th}}+1,\\ldots,\\widetilde{K}_{m}$;\n\\end{enumerate}\nthere must be a maximum $f$-matching in $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ which can saturate the vertices $(u_{\\left(i\\right)})_{i=1}^{m}$, where $u_{\\left(i\\right)}$ has the same order as $K_{\\left(i\\right)}$, and $K^{\\mathrm{sum}}$ is defined in Eq. \\eqref{eq:K_sum}.\n\\end{lem}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:matching_edges}.\n\\end{IEEEproof}\n\\medskip\n\n\n\\subsection{OER \\& DMR for RB based Coded $f$-Matching Scheme}\n\nAs shown in Fig. \\ref{fig:system_desc}, the subchannels, i.e., RBs themselves, are directly allocated to each user in the RB based coded $f$-matching allocation scheme. According to the channel model, the number of outage RBs is the integral multiples of the number of RBs in one coherence bandwidth, i.e., the integral multiples of $N_{\\mathrm{c}}$. Moreover, each user will be allocated $\\widetilde{K}_{m}$ RBs in this scheme so as to achieve the optimal performance. The exact performance analysis is very difficult for the coded $f$-matching approach, we will focus on the first order approximations in the high and low SNR regimes. Based on Theorem \\ref{thm:con_outage_exponent} and Lemma \\ref{lem:matching_edges}, the main results are summarized in the following theorem.\n\n\\medskip\n\\begin{thm}\\label{thm:outage_RB}\nFor the considered multi-carrier multi-access channel with $M$ users ($M\\geq2$), $L$ coherence bandwidths, and $N$ RBs, if the RB based coded $f$-matching allocation scheme is applied, the achieved outage probability for the user $u_{m}$, in the high SNR regime, is given by\n\\begin{equation}\\label{eq:outage_RB_high}\n\\begin{aligned}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)= & \\sum_{\\kappa=L-\\left\\lceil\\frac{\\widetilde{K}_{m}}{N_{\\mathrm{c}}}\\right\\rceil+1}^{L}\\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{\\widetilde{K}_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}}{\\kappa!\\left(L-\\kappa\\right)!}\\\\\n\t& +\\mathsf{O}\\left(p_{\\mathrm{s}}^{L}\\right),\n\\end{aligned}\n\\end{equation}\n\\end{thm}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:outage_RB}.\n\\medskip\n\nAccording to Definition \\ref{def:DMR} and Corollary \\ref{cor:CDMT}, the achieved DMR of the RB based coded $f$-matching allocation scheme is obtained from Theorem \\ref{thm:outage_RB}.\n\n\\medskip\n\\begin{thm}\\label{thm:DMR_RB}\nFor an operating point $\\bm{r}=\\left(r_{m}\\right)_{m=1}^{M}$, the best achievable bound of the DMR $\\mathcal{R}_{\\mathrm{dmr}}\\left(\\bm{r}\\right)$ is given by the vector $\\left(d_{m}^{*}\\left(r_{m}\\right)\\right)_{m=1}^{M}$ which satisfies\n\\begin{equation}\\label{eq:DMR_RB}\n\td_{m}^{*}\\left(r_{m}\\right)=L\\left(1-\\frac{N_{\\mathrm{c}}r_{m}}{\\widetilde{K}_{m}}\\right),\n\\end{equation}\nfor $0<r_{m}\\leq\\widetilde{K}_{m}$.\n\\end{thm}\n\\medskip\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{DMTCurve}\n\t\\caption{DMT curves of the user $u_{m}$ for different allocation schemes with $M=2$, $L=6$, and $N=12$.}\\label{fig:DMTCurve}\n\\end{figure}\n\nThe DMT curve of the user $u_{m}$ and the DMR for the RB based coded $f$-matching allocation scheme are shown in Fig. \\ref{fig:DMTCurve} and Fig. \\ref{fig:OER}, respectively. Recalling Definition \\ref{def:DMR} and Eqs. \\eqref{eq:opt_dmt} \\eqref{eq:opt_dmt_cond}, the RB based coded $f$-matching allocation scheme achieves the optimal DMR, such that all of the users share the multiplexing gain according to their target rates, and also achieve the full frequency diversity.\n\nThe previous results show the outage performance in the high SNR regime. A long distance between the user and the BS will lead to the low SNR regime. According to Eq. \\eqref{eq:subchannel_outage}, we have $p_{\\mathrm{s}}\\rightarrow1$ as $\\gamma\\rightarrow0$, which indicates that almost all the RBs are in outage. Thus, the outage state caused by two users competing for one RB will nearly not happen. As a result, the outage state in the low SNR regime is mainly determined by the RB outage. This intuitive thinking is proved to be true by the following theorem.\n\n\\medskip\n\\begin{thm}\\label{thm:outage_RB_LowSNR}\nFor the considered multi-carrier multi-access channel with $M$ users and $L$ coherence bandwidths, if the RB based coded $f$-matching allocation scheme is applied, the achieved outage probability for the user $u_{m}$, in the low SNR regime, is given by\n\\begin{equation}\\label{eq:outage_RB_LowSNR}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)=p_{\\mathrm{s}}^{L}+Lp_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},1}\\right.\\right)p_{\\mathrm{s}}^{L-1}q_{\\mathrm{s}}+\\mathsf{O}\\left(q_{\\mathrm{s}}\\right).\n\\end{equation}\n\\end{thm}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:outage_RB_LowSNR}.\n\\end{IEEEproof}\n\\medskip\n\nBased on Theorem \\ref{thm:outage_RB} and Theorem \\ref{thm:outage_RB_LowSNR}, the OER of the RB based coded $f$-matching allocation scheme is summarized as the following theorem, which illustrate the comprehensive performance of the proposed scheme in the full SNR range.\n\n\\medskip\n\\begin{thm}\\label{thm:OER_RB}\nFor an operating point $\\bm{R}=\\left(R_{m}\\right)_{m=1}^{M}$, the best achievable bound of the OER $\\mathcal{R}_{\\mathrm{oer}}\\left(\\bm{R};\\gamma\\right)$ is given by the vector $\\left(E_{m}\\left(R_{m},\\gamma\\right)\\right)_{m=1}^{M}$, which can be directly obtained by plugging Eq. \\eqref{eq:outage_RB_high} and Eq. \\eqref{eq:outage_RB_LowSNR} into Eq. \\eqref{eq:outage_exponent}.\n\\end{thm}\n\\medskip\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{OER}\n\t\\caption{OER $\\mathcal{R}_{\\mathrm{oer}}\\left(\\bm{R};\\gamma\\right)$ and DMR $\\mathcal{R}_{\\mathrm{dmr}}\\left(\\bm{r}\\right)$ for the RB based coded $f$-matching allocation scheme at operating point $\\bm{r}=\\left(0.9,0.9\\right)$ with $M=2$, $L=6$, $N=12$.}\\label{fig:OER}\n\\end{figure}\n\nThe DMR for the RB based coded $f$-matching allocation scheme at different SNRs are shown in Fig. \\ref{fig:OER}. It can be seen that the OER approaches the DMR as SNR tends to infinity.\n\n\\begin{rmk}\nThere is a so called \\emph{interleaved} allocation scheme in both IEEE 802.16 and LTE-A standards \\cite{IEEE802.16-2009,3GPPLTE}. As shown in Fig. \\ref{fig:interleaved}, one subcarrier in the duration of one frame is seen as one RB in this scheme, and the BS allocates RBs in different coherence bandwidths to users in an interleaved way.  As a matter of fact, the interleaved scheme is a special case of the RB based coded $f$-matching allocation scheme, whose OER and DMR performance can also be analysed by the code $f$-matching approach. For the user $u_{m}$, BS will allocate $\\widetilde{K}_{m}$ RBs to it without observing the outage state of these RBs. According to the saddle-point approximation in the proof of Theorem \\ref{thm:con_outage_exponent}, the exponentially tight upper bound of the outage probability $p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)$ is given by\n\\begin{equation}\\label{eq:outage_exponent_interleaved}\n\t\\begin{aligned}\n\t\t& p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)\\lesssim p_{m}^{\\mathrm{upper}}\\left(R_{m}\\right)\\\\\n\t\t& =\\psi \\exp\\left(-\\widetilde{K}_{m}\\left[\\left(\\ln\\gamma-R_{\\mathrm{c}}\\right)J_{2}\\left(\\gamma\\right)-J_{0}\\left(\\gamma\\right)\\right]\\right),\n\t\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\psi=\\frac{1}{\\sqrt{2\\pi\\widetilde{K}_{m}\\sigma^{2}}\\lambda^{*}},\n\\end{equation}\n\\begin{equation}\n\t\\left\\{\\begin{aligned}\n\t\t& J_{2}\\left(\\gamma\\right)=\\lambda^{*};\\\\\n\t\t& J_{0}\\left(\\gamma\\right)=\\frac{1}{\\gamma}+\\ln\\Gamma\\left(1-\\lambda^{*},\\gamma^{-1}\\right).\n\t\\end{aligned}\\right.\n\\end{equation}\nThe parameter $\\lambda^{*}$ is the solution of the following equation:\n\\begin{equation}\n\tR_{\\mathrm{c}}-\\frac{1}{\\Gamma\\left(1-\\lambda^{*},\\gamma^{-1}\\right)}G_{2,3}^{3,0}\\left(\\frac{1}{\\gamma}\\left|\n\t\\begin{aligned} & 1,1\\\\\n \t\t\t       & 0,0,1-\\lambda^{*}\n\t\\end{aligned}\\right.\\right)=0,\n\\end{equation}\nwhile $\\sigma^{2}$ is given by\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\sigma^{2}= & \\frac{2}{\\Gamma\\left(1-\\lambda^{*},\\gamma^{-1}\\right)}G_{3,4}^{4,0}\\left(\\frac{1}{\\gamma}\\left|\\begin{aligned}\n\t\t     \t& 1,1,1\\\\\n\t\t\t& 0,0,0,1-\\lambda^{*}\n\t\t     \\end{aligned}\\right.\\right)-\\\\\n\t & \\left[\\frac{1}{\\Gamma\\left(1-\\lambda^{*},\\gamma^{-1}\\right)}G_{2,3}^{3,0}\\left(\\frac{1}{\\gamma}\\left|\\begin{aligned}\n\t& 1,1\\\\\n\t& 0,0,1-\\lambda^{*}\n\\end{aligned}\\right.\\right)\\right]^{2}.\n\t\\end{aligned}\n\\end{equation}\nIt can be shown that the outage probability in Eq. \\eqref{eq:outage_RB_high} is smaller than Eq, \\eqref{eq:outage_exponent_interleaved}. Moreover, the achieved DMT of the interleaved allocation scheme is also shown to be given by Eq. \\eqref{eq:DMR_RB}, if and only if $\\widetilde{K}_{m}$ RBs are uniformly distributed in $L$ coherence bandwidths. The DMT curve of the user $u_{m}$ is plotted in Fig. \\ref{fig:DMTCurve} for comparison.\n\\end{rmk}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[>=stealth]\n\t\\filldraw[fill=black!70] (-7,0.3) rectangle (-2,0.55);\n\t\\filldraw[fill=black!10] (-7,0.55) rectangle (-2,0.8);\n\t\\filldraw[fill=black!70] (-7,0.8) rectangle (-2,1.05);\n\t\\filldraw[fill=black!10] (-7,1.05) rectangle (-2,1.3);\n\t\\filldraw[fill=black!70] (-7,1.3) rectangle (-2,1.55);\n\t\\filldraw[fill=black!10] (-7,1.55) rectangle (-2,1.8);\n\t\\filldraw[fill=black!70] (-7,1.8) rectangle (-2,2.05);\n\t\\filldraw[fill=black!10] (-7,2.05) rectangle (-2,2.3);\n\t\t\t\n\t\\foreach \\x in {-7}\n\t\t\\foreach \\y in {0.3,0.55,0.8,1.05,1.3,1.55,1.8,2.05}\n\t\t{\n\t\t\t\\draw (\\x,\\y) +(0,0) rectangle ++(5,.25);\n\t\t}\n\n\t\\draw[->,very thick] (-7,0.3) -- (-1.5,0.3) node[right] {Time};\n\t\\draw[->,very thick] (-7,0.3) -- (-7,2.7) node[above] {Frequency};\n\t\\node at (-4.5,0) {Interleaved Allocation Scheme};\n\t\\node at (-4,2.7) {$1$ Subchannel $=1$ RB};\n\t\n\t\\node at (-7.3,2.05) {$\\mathcal{S}_{4}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.55) {$\\mathcal{S}_{3}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.05) {$\\mathcal{S}_{2}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,0.55) {$\\mathcal{S}_{1}^{\\mathrm{c}}\\,\\big\\{$};\n\t\n\t\\filldraw[fill=black!70] (-7,-0.7) rectangle (-5.75,-0.45);\n\t\\node at (-5.1,-0.625) {User $u_{1}$};\n\t\\filldraw[fill=black!10] (-4.25,-0.7) rectangle (-3,-0.45);\n\t\\node at (-2.3,-0.625) {User $u_{N}$};\n\\end{tikzpicture}\n\\caption{The interleaved allocation scheme in IEEE 802.16 and LTE-A, where the RBs in different coherence bandwidths are allocated to different users in an interleaved way.}\\label{fig:interleaved}\n\\end{figure}\n\n\\subsection{OER \\& DMR for Chunk based Coded $f$-Matching Scheme}\n\nThe chunk based coded $f$-matching allocation scheme is considered in this subsection, where we assume $L>M$. The case of $L\\leq M$ has been shown to achieve the optimal OER and DMR without requiring the frequency-domain coding scheme in \\cite{Bai2011b}. As shown in Fig. \\ref{fig:system_desc}, all of the RBs in each coherence bandwidth will be bundled as one chunk, i.e., $N=L$, because:  1) dividing each coherence bandwidth into more chunks can be seen as a special case of RB based coded $f$-matching allocation scheme; 2) bundling all of the RBs in multiple coherence bandwidths as one chunk must be worse than allocating more chunks in the proposed bundling method. Due to the difficulty in calculating the exact formula of the outage probability, we still focus on the first order approximations in the high and low SNR regimes. Based on Theorem \\ref{thm:con_outage_exponent} and Lemma \\ref{lem:matching_edges}, the main results are summarized in the following theorem.\n\n\\medskip\n\\begin{thm}\\label{thm:outage_chunk}\nFor the considered multi-carrier multi-access channel with $M$ users and $L$ coherence bandwidths ($M\\leq L$), if the chunk based coded $f$-matching allocation scheme is applied, the achieved outage probability for the user $u_{m}$, in the high SNR regime, is given by\n\\begin{equation}\\label{eq:outage_chunk_1}\n\\begin{aligned}\n\tp_{m}^{\\mathrm{out}}\\left(R_{m}\\right)= & \\sum_{\\kappa=L-K_{m}+1}^{L}\\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}}{\\kappa!\\left(L-\\kappa\\right)!}\\\\\n\t& +\\mathsf{O}\\left(p_{\\mathrm{s}}^{L}\\right),\n\\end{aligned}\n\\end{equation}\nfor the case of $K_{m}=1,\\ldots,K_{m}^{\\mathrm{th}}$; or\n\\begin{equation}\\label{eq:outage_chunk_2}\n\\begin{aligned}\n   p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)= & \\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)}}{M\\left(L-K^{\\mathrm{sum}}+1\\right)!\\left(K^{\\mathrm{sum}}-1\\right)!}\\\\\n   & +\\mathsf{O}\\left(p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)+K_{m}-1}\\right),\n\\end{aligned}\n\\end{equation}\nfor the case of $K_{m}=K_{m}^{\\mathrm{th}}+1,\\ldots,\\widetilde{K}_{m}$. For the case of $K_{m}=K_{m}^{\\mathrm{th}}$ and $\\left(M-1\\right)|MK_{m}^{\\mathrm{sum}}$, the outage probability of the user $u_{m}$ is the summation of Eq. \\eqref{eq:outage_chunk_1} and Eq. \\eqref{eq:outage_chunk_2}.\n\\end{thm}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:outage_chunk}.\n\\end{IEEEproof}\n\\medskip\n\nSimilar to Theorem \\ref{thm:DMR_RB}, it is not difficult to obtain the following corollary from Theorem \\ref{thm:outage_chunk} and Corollary \\ref{cor:CDMT}.\n\n\\medskip\n\\begin{cor}\\label{cor:DMT_chunk}\nFor the operating point $\\left(r_{m}\\right)_{m=1}^{M}$, the achieved DMT for the user $u_{m}$ is given by\n\\begin{equation}\\label{eq:DMT_Lower}\n\td_{m}\\left(r_{m}\\right)=L\\left(1-\\frac{r_{m}}{K_{m}}\\right),\n\\end{equation}\nfor the case of $K_{m}=1,\\ldots,K_{m}^{\\mathrm{th}}$; and\n\\begin{equation}\\label{eq:DMT_Upper}\n\td_{m}\\left(r_{m}\\right)=\\left[M\\left(L-K^{\\mathrm{sum}}+1\\right)+K_{m}-1\\right]\\left(1-\\frac{r_{m}}{K_{m}}\\right),\n\\end{equation}\nfor the case of $K_{m}=K_{m}^{\\mathrm{th}}+1,\\ldots,\\widetilde{K}_{m}$.\n\\end{cor}\n\\medskip\n\nBoth Theorem \\ref{thm:outage_chunk} and Corollary \\ref{cor:DMT_chunk} show that the chunk based coded $f$-matching allocation will not be optimal in OER and DMR. For a given user $u_{m}$, the best achievable DMT is a piece-wised linear function. In Eq. \\eqref{eq:DMT_Lower}, the slope $-\\frac{L}{K_{m}}$ increases as $K_{m}$ increasing from $1$ to $K_{m}^{\\mathrm{th}}$, therefore\n\\begin{equation}\\label{eq:OptDMT_Lower}\n\td_{m}^{*}\\left(r_{m}\\right)=L\\left(1-\\frac{r_{m}}{K_{m}^{\\mathrm{th}}}\\right)\n\\end{equation}\noutperforms the cases of $K_{m}=1,\\ldots,K_{m}^{\\mathrm{th}}-1$. In Eq. \\eqref{eq:DMT_Upper}, similarly, the slope\n\\begin{equation}\n\\begin{aligned}\n\t& -\\frac{M\\left(L-K^{\\mathrm{sum}}+1\\right)+K_{m}-1}{K_{m}}\\\\\n\t= & -\\frac{M\\left(L-K_{m}^{\\mathrm{sum}}+1\\right)-1}{K_{m}}+M-1\n\\end{aligned}\n\\end{equation}\nincreases as $K_{m}$ increasing from $K_{m}^{\\mathrm{th}}+1$ to $\\widetilde{K}_{m}$. Thus,\n\\begin{equation}\\label{eq:OptDMT_Upper}\n\td_{m}^{*}\\left(r_{m}\\right)=\\left[M\\left(L-K^{\\mathrm{sum}}+1\\right)+\\widetilde{K}_{m}-1\\right]\\left(1-\\frac{r_{m}}{\\widetilde{K}_{m}}\\right)\n\\end{equation}\noutperforms the cases of $K_{m}=K_{m}^{\\mathrm{th}}+1,\\ldots,\\widetilde{K}_{m}$. Since $L>M\\left(L-K^{\\mathrm{sum}}+1\\right)+\\widetilde{K}_{m}-1$, a threshold of $r_{m}$ can be obtained by solving the following equation\n\\begin{equation}\n\\begin{aligned}\n\t& L\\left(1-\\frac{r_{m}}{K_{m}^{\\mathrm{th}}}\\right)\\\\\n\t= & \\left[M\\left(L-K^{\\mathrm{sum}}+1\\right)+\\widetilde{K}_{m}-1\\right]\\left(1-\\frac{r_{m}}{\\widetilde{K}_{m}}\\right).\n\\end{aligned}\n\\end{equation}\nClearly, we have\n\\begin{equation}\n\tr_{m}^{\\mathrm{th}}=\\frac{K_{m}^{\\mathrm{th}}\\widetilde{K}_{m}\\left[L-\\widetilde{K}_{m}+1-M\\left(L-K^{\\mathrm{sum}}+1\\right)\\right]}{\\widetilde{K}_{m}\\left(L-K_{m}^{\\mathrm{th}}\\right)+K_{m}^{\\mathrm{th}}\\left[1-M\\left(L-K^{\\mathrm{sum}}+1\\right)\\right]}.\n\\end{equation}\nTherefore, the optimal chunk based coded $f$-matching allocation scheme within the specific bundle method can be described as follows. If the operating point for the user $u_{m}$ satisfies $r_{m}\\leq r_{m}^{\\mathrm{th}}$, the proposed coded $f$-matching method will allocate $K_{m}^{\\mathrm{th}}$ chunks to this user; whereas if $r_{m}> r_{m}^{\\mathrm{th}}$, the proposed method will allocate $\\widetilde{K}_{m}$ chunks to this user. The best achievable DMT of the user $u_{m}$ is the piece-wised linear function joined by Eq. \\eqref{eq:OptDMT_Lower} and Eq. \\eqref{eq:OptDMT_Upper}. This result is summarized as follows.\n\n\\medskip\n\\begin{thm}\\label{thm:DMR_chunk}\nFor an operating point $\\bm{r}=\\left(r_{m}\\right)_{m=1}^{M}$, the best achievable bound of the DMR $\\mathcal{R}_{\\mathrm{dmr}}\\left(\\bm{r}\\right)$ is given by the vector $\\left(d_{m}^{*}\\left(r_{m}\\right)\\right)_{m=1}^{M}$ which satisfies\n\\begin{equation}\\label{eq:DMR_chunk}\n\td_{m}^{*}\\left(r_{m}\\right)=L\\left(1-\\frac{r_{m}}{K_{m}^{\\mathrm{th}}}\\right),\n\\end{equation}\nfor $0<r_{m}\\leq r_{m}^{\\mathrm{th}}$; or\n\\begin{equation}\n\td_{m}^{*}\\left(r_{m}\\right)=\\left[M\\left(L-K^{\\mathrm{sum}}+1\\right)+\\widetilde{K}_{m}-1\\right]\\left(1-\\frac{r_{m}}{\\widetilde{K}_{m}}\\right),\n\\end{equation}\nfor $r_{m}^{\\mathrm{th}}<r_{m}<\\widetilde{K}_{m}$.\n\\end{thm}\n\\medskip\n\nThe DMT curve of the user $u_{m}$ and the DMR for the chunk based coded $f$-matching allocation scheme are shown in Fig. \\ref{fig:DMTCurve} and Fig. \\ref{fig:OER}, respectively. It can be seen that the maximum achievable DMT for this scheme is not optimal. However, this scheme provides the multi-user diversity, and enjoys the low complexity of resource allocation algorithm and low requirement on frequency shift estimation.\n\nThe previous results show the outage performance in the high SNR regime. Consider now the low SNR regime, where almost all of the chunks are in outage. Similar to the RB based coded $f$-matching allocation scheme, the outage state in the low SNR regime is mainly determined by the chunk outage. Therefore, Theorem \\ref{thm:outage_RB_LowSNR} can still be established for the chunk based coded $f$-matching allocation scheme.\n\nAccording to Theorem \\ref{thm:con_outage_exponent}, $p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)$ is a decreasing function of $K_{m}$. Therefore, the aforementioned analysis for the best achievable DMR is also applicable for OER. Let $\\mathbf{1}_{\\mathcal{X}}$ denote the indicative function of the event $\\mathcal{X}$, then the result can be summarized as follows.\n\n\\medskip\n\\begin{thm}\\label{thm:OER_chunk}\nFor an operating point $\\bm{R}=\\left(R_{m}\\right)_{m=1}^{M}$, the best achievable bound of the OER $\\mathcal{R}_{\\mathrm{oer}}\\left(\\bm{R};\\gamma\\right)$ is given by the vector $\\left(E_{m},\\gamma\\left(R_{m}\\right)\\right)_{m=1}^{M}$, which can be directly obtained by plugging\n\\begin{equation}\\label{eq:Opt_outage_chunk_1}\n\\begin{aligned}\n\t& p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)=\\sum_{\\kappa=L-K_{m}^{\\mathrm{th}}+1}^{L}\\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}}{\\kappa!\\left(L-\\kappa\\right)!}\\\\\n\t& +\\mathbf{1}_{\\left\\{\\left(M-1\\right)\\left|MK_{m}^{\\mathrm{th}}\\right.\\right\\}}\\frac{L!}{M\\left(L-K^{\\mathrm{sum}}+1\\right)!\\left(K^{\\mathrm{sum}}-1\\right)!}\\cdot\\\\\n\t& p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}^{\\mathrm{th}}-1}\\right.\\right)p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)}+\\mathsf{O}\\left(p_{\\mathrm{s}}^{L}\\right),\n\\end{aligned}\n\\end{equation}\ninto Eq. \\eqref{eq:outage_exponent} for $0<r_{m}\\leq r_{m}^{\\mathrm{th}}$ in the high SNR regime; or\n\\begin{equation}\\label{eq:Opt_outage_chunk_2}\n\\begin{aligned}\n   p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)= & \\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},\\widetilde{K}_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)}}{M\\left(L-K^{\\mathrm{sum}}+1\\right)!\\left(K^{\\mathrm{sum}}-1\\right)!}\\\\\n   & +\\mathsf{O}\\left(p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)+\\widetilde{K}_{m}-1}\\right),\n\\end{aligned}\n\\end{equation}\ninto Eq. \\eqref{eq:outage_exponent} for $r_{m}^{\\mathrm{th}}<r_{m}<\\widetilde{K}_{m}$ in the high SNR regime. In the low SNR regime, the OER can be obtained by plugging Eq. \\eqref{eq:outage_RB_LowSNR} into Eq. \\eqref{eq:outage_exponent}.\n\\end{thm}\n\\medskip\n\nThe DMR for the chunk based coded $f$-matching allocation scheme at different SNRs are similar to the regions in Fig. \\ref{fig:OER}. Also, the OER approaches the DMR as SNR tends to infinity.\n\n\\begin{rmk}\nThere is another allocation scheme in both IEEE 802.16 and LTE-A standards, which is referred to as the \\emph{localized} scheme \\cite{IEEE802.16-2009,3GPPLTE}. As a matter of fact, the localized scheme is a fixed version of the chunk based coded $f$-matching allocation scheme in this subsection. Thus, the OER and DMR performance can also be analyzed by the proposed code $f$-matching approach. For the user $u_{m}$, BS will allocate $\\widetilde{K}_{m}$ chunks to it without observing the outage state of these chunks. The outage probability $p_{m}^{\\mathrm{out}}\\left(R_{m}\\right)$ is then exponentially tight upper bounded by Eq. \\eqref{eq:outage_exponent_interleaved} with $\\widetilde{K}_{m}$ being replaced by $\\frac{\\widetilde{K}_{m}}{N_{\\mathrm{c}}}$ in the equation. The DMT curve of the user $u_{m}$ for the localized allocation scheme is plotted in Fig. \\ref{fig:DMTCurve}. Compared to other allocation schemes, the localized allocation scheme can only achieve the worst DMT performance.\n\\end{rmk}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[>=stealth]\n\t\\filldraw[fill=black!70] (-7,0.3) rectangle (-5.75,2.3);\n\t\\filldraw[fill=black!40] (-5.75,0.3) rectangle (-4.5,2.3);\n\t\\filldraw[fill=black!10] (-4.5,0.3) rectangle (-3.25,2.3);\n\t\t\t\n\t\\foreach \\x in {-7,-5.75,-4.5,-3.25}\n\t\t\\foreach \\y in {0.3,0.55,0.8,1.05,1.3,1.55,1.8,2.05}\n\t\t{\n\t\t\t\\draw (\\x,\\y) +(0,0) rectangle ++(1.25,.25);\n\t\t}\n\n\t\\draw[->,very thick] (-7,0.3) -- (-1.5,0.3) node[right] {Time};\n\t\\draw[->,very thick] (-7,0.3) -- (-7,2.7) node[above] {Frequency};\n\t\\node at (-4.5,0) {TDMA based Allocation Scheme};\n\t\\node at (-4,2.7) {$1$ Subchannel $=8$ RBs};\n\t\n\t\\node at (-7.3,2.05) {$\\mathcal{S}_{4}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.55) {$\\mathcal{S}_{3}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,1.05) {$\\mathcal{S}_{2}^{\\mathrm{c}}\\,\\big\\{$};\n\t\\node at (-7.3,0.55) {$\\mathcal{S}_{1}^{\\mathrm{c}}\\,\\big\\{$};\n\t\n\t\\filldraw[fill=black!70] (-7,-0.7) rectangle (-5.75,-0.45);\n\t\\node at (-5.1,-0.625) {User $u_{1}$};\n\t\\filldraw[fill=black!40] (-7,-1.25) rectangle (-5.75,-1);\n\t\\node at (-5.1,-1.125) {User $u_{2}$};\n\t\\filldraw[fill=black!10] (-4.25,-0.7) rectangle (-3,-0.45);\n\t\\node at (-2.3,-0.625) {User $u_{N}$};\n\t\\draw (-4.25,-1.25) rectangle (-3,-1);\n\t\\node at (-2.08,-1.125) {Unallocated};\n\\end{tikzpicture}\n\\caption{TDMA is also a special case of the chunk based coded $f$-matching allocation scheme, where all of the RBs in one timeslot are bundled as one chunk.}\\label{fig:TDMA}\n\\end{figure}\n\n\\begin{rmk}\nBesides the bundling method shown in Fig. \\ref{fig:system_desc}, another method, often referred to as TDMA, is shown in Fig. \\ref{fig:TDMA}, where the RBs in one timeslot are bundled as one chunk. The OER and DMR performance can also be analyzed by the coded $f$-matching approach. For the user $u_{m}$, BS will allocate $\\widetilde{K}_{m}$ chunks to it without observing the outage state of these chunks. Since the frame is in one coherence time, the outage probabilities are the same for each user and is exponentially tight upper bounded by Eq. \\eqref{eq:outage_exponent_interleaved} with $\\widetilde{K}_{m}$ being replaced by $L$ in the equation. The DMT curve of the user $u_{m}$ for TDMA scheme is plotted in Fig. \\ref{fig:DMTCurve}. It can be seen that the optimal DMT can also be achieved by this scheme. However, this scheme has a high requirement on frequency shift estimation, and no multi-user diversity.\n\\end{rmk}\n\n\n\\section{Practical Issues}\\label{sec:practical_issues}\n\nThe complexity of the proposed framework is a key issue for practical multi-carrier multi-access systems. In this section, the parallel maximum $f$-matching algorithm for subchannel allocation with log-polynomial complexity is discussed. The asymptotic optimal coding scheme, which is easy to implement, is also presented.\n\n\n\\subsection{Parallel Subchannel Allocation Algorithm}\n\nThe Hopcroft-Karp algorithm is well known for solving the maximum matching problem in bipartite graphs \\cite{Hopcroft1973}. According to the matching theory \\cite{Lovasz1986}, a matching $\\mathcal{M}$ is maximum if and only if there is no augmenting path with respect to this matching. Consider a bipartite graph $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$, an augmenting path is an alternating path that ends in an unsaturated vertex of $\\mathcal{S}$. Here, an alternating path with respect to $\\mathcal{M}$ is defined as a path in $\\mathcal{G}$ which starts in $\\mathcal{U}$ at an unsaturated vertex and then contains, alternately, edges from $\\mathcal{E}\\setminus\\mathcal{M}$ and from $\\mathcal{M}$. Therefore, the basic principle of the Hopcroft-Karp algorithm is to search the shortest augmenting path from all of the unsaturated vertices in $\\mathcal{S}$ simultaneously \\cite{Hopcroft1973}. If there are augmenting paths, a new matching with more saturated vertices can be obtained by computing the symmetric difference between the previous matching and the augmenting paths. It can be shown that the Hopcroft-Karp algorithm is capable of finding a maximum matching for any bipartite graph with the time complexity of $\\mathcal{O}\\left(N^{2.5}\\right)$, which is the fastest deterministic sequential matching algorithm ever known.\n\nThe Hopcroft-Karp algorithm, however, can only compute the maximum matching. It is still needed to be generalized for generating the maximum $f$-matching. For a sample of the RBG formulation of the considered multi-carrier multi-access channel, say $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$, a new bipartite graph $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$ will then be constructed as follows: For the user $u_{m}\\in\\mathcal{U}$, let $\\mathcal{X}_{u_{m}}$ be a set of $K_{m}$ elements such that $\\mathcal{X}_{u_{m}}$ and $\\mathcal{X}_{u_{m'}}$ are disjointed if $m\\neq m'$, i.e., expand vertex $u_{m}$ to a set of $K_{m}$ vertices. Let\n\\begin{equation}\\label{eq:vertex_extension}\n\t\\widetilde{\\mathcal{U}}=\\bigcup_{u_{m}\\in\\mathcal{U}}\\mathcal{X}_{u_{m}},\n\\end{equation}\nthen connect each element in $\\mathcal{X}_{u_{m}}$ to each element of $\\mathcal{S}$, whenever $u_{m}$ and $s_{n}$ are adjacent in $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$. The induced new edge set is denoted by $\\widetilde{\\mathcal{E}}$\n\n\\medskip\n\\begin{lem}\\label{lem:perfect_f_matching}\nThe bipartite graph $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$ has a perfect matching if and only if $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ has a perfect $f$-matching.\n\\end{lem}\n\\begin{IEEEproof}\nSee Appendix \\ref{app:perfect_f_matching}.\n\\end{IEEEproof}\n\\medskip\n\nFrom Lemma \\ref{lem:perfect_f_matching}, we only need to extend the user vertices as Eq. \\eqref{eq:vertex_extension}. The maximum $f$-matching can then be generated by using Hopcroft-Karp algorithm. However, the maximum $f$-matching method also requires Eq. \\eqref{eq:opt_cond_fmatching} to be hold for achieving the optimality of the framework. Moreover, Hopcroft-Karp algorithm will always saturate the user vertices in sequence, which is not fair for the users in subchannel allocation. Therefore, the vertices in the extended set $\\widetilde{\\mathcal{U}}$ should be grouped and random rotated with the following rules: The vertex set $\\widetilde{\\mathcal{U}}$ will be divided into $K_{(1)}$ subsets, where the $i$th subset is denoted by $\\widetilde{\\mathcal{U}}_{i},\\,i=1,\\ldots,K_{(1)}$. All of the vertices in the set $\\mathcal{X}_{u_{m}}$ will be distributed uniformly into these subsets. Thus, the subset $\\widetilde{\\mathcal{U}}_{i}$ has $\\left\\lceil\\frac{K_{m}}{K_{(1)}}\\right\\rceil$ vertices from the set $\\mathcal{X}_{u_{m}}$ for any $u_{m}\\in\\mathcal{U}$. Clearly, $\\widetilde{\\mathcal{U}}_{i}\\cap\\widetilde{\\mathcal{U}}_{i'}=\\emptyset,\\,\\forall i'\\neq i$, and $\\bigcup_{i=1}^{K_{(1)}}\\widetilde{\\mathcal{U}}_{i}=\\widetilde{\\mathcal{U}}$. In the ideal case, all of the subsets $\\widetilde{\\mathcal{U}}_{i}$ have the identical number of elements from the same set $\\mathcal{X}_{u_{m}}$. To guarantee the fairness among all of the users, the family of the subsets $\\widetilde{\\mathcal{U}}_{i},\\,i=1,\\ldots,K_{(1)}$ will be randomly rotated, and all of the vertices in each subset $\\widetilde{\\mathcal{U}}_{i}$ will also be randomly rotated.\n\nWith the development of multi-core processors, the parallel algorithms, which can execute on multiple processors simultaneously, attract much more attentions from both industry and academia. In order to fulfill the stringent computation time requirement of the next generation communication systems, the parallel implementation of Hopcroft-Karp algorithm will also be applied in the considered multi-carrier multi-access. Hence, the proposed algorithm will be referred to as the \\emph{parallel vertices extension \\& random rotation based Hopcroft-Karp} (PVER\\textsuperscript{2}HK) algorithm.\n\n\\begin{algorithm}[t]\n\\begin{algorithmic}[1]\n\t\\STATE Input $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ and let $l^{*}\\leftarrow2\\log^{\\eta}N+1$.\n\t\\STATE Expand the bipartite graph $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ to $\\widetilde{\\mathcal{G}}(\\widetilde{\\mathcal{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$.\n\t\\STATE Generated $(\\widetilde{\\mathcal{U}}_{i})_{i=1}^{K_{(1)}}$ from $\\widetilde{\\mathcal{U}}$.\n\t\\STATE Randomly rotate the vertex sets $(\\widetilde{\\mathcal{U}}_{i})_{i=1}^{K_{(1)}}$ and the elements in them.\n\t\\STATE Let $\\widetilde{\\mathcal{M}}\\leftarrow\\emptyset$ and $l\\leftarrow0$.\n\t\\REPEAT\n\t\t\\STATE $\\mathcal{G}_{\\mathrm{d}}\\leftarrow\\mathrm{D}(\\widetilde{\\mathcal{G}},\\widetilde{\\mathcal{M}})$: Construct a digraph $\\mathcal{G}_{\\mathrm{d}}$ form $\\widetilde{\\mathcal{G}}$ with $\\widetilde{\\mathcal{M}}$, and define $\\widetilde{\\mathcal{U}}'$ and $\\mathcal{S}'$ as the sets of free vertices in $\\widetilde{\\mathcal{U}}$ and $\\mathcal{S}$, respectively.\n\t\t\\STATE $\\left(\\mathcal{L},l\\right)\\leftarrow\\mathrm{PBFS}(\\widetilde{\\mathcal{U}}',\\mathcal{S}',\\mathcal{G}_{\\mathrm{d}},l^{*})$: Execute a parallel breadth-first searching to find the $l$-layer graph $\\mathcal{L}$ with $l<l^{*}$. If $\\mathcal{L}$ has more layers than $l^{*}$, we let $l=\\infty$.\n\t\t\\STATE $\\mathcal{P}\\leftarrow\\mathrm{PVDP}\\left(\\mathcal{L},l\\right)$: Search in parallel a maximal set of vertex-disjoint paths $\\mathcal{P}$ from $\\widetilde{\\mathcal{U}}'$ to $\\mathcal{S}'$ of length $l$ in the layer graph $\\mathcal{L}$.\n\t\t\\STATE $\\mathcal{M}\\leftarrow\\mathrm{PA}(\\mathcal{P},\\widetilde{\\mathcal{M}})$: Execute the parallel augmenting operation to get the next matching.\n\t\\UNTIL{$\\mathcal{P}=\\emptyset$ or $l>\\eta$.}\n\t\\STATE Map $\\widetilde{\\mathcal{M}}$ to $\\mathcal{M}_{f}^{\\mathrm{m}}$ by applying the reversion of the vertex expansion and random rotation process.\n\t\\STATE Construct $\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}$ from $\\mathcal{M}_{f}^{\\mathrm{m}}$, and compute $\\mathcal{S}_{\\mathrm{alloc}}^{\\mathrm{c}}=\\mathcal{S}\\setminus\\bigcup_{u_{m}\\in\\mathcal{U}}\\mathcal{S}_{m}$.\n\t\\STATE $\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}\\leftarrow\\mathrm{RA}\\left(\\mathcal{S}_{\\mathrm{alloc}}^{\\mathrm{c}}\\right)$: Randomly allocate $\\widetilde{K}_{m}-k_{m}$ outage subchannels in $\\mathcal{S}_{\\mathrm{alloc}}^{\\mathrm{c}}$ to the user $u_{m}$.\n\t\\STATE Output subchannel allocation results $\\left(\\mathcal{S}_{m}\\right)_{m=1}^{M}$, then stop.\n\\end{algorithmic}\n\\caption{PVER\\textsuperscript{2}HK}\\label{alg:PVER2HK}\n\\end{algorithm}\n\nIn \\cite{Karpinski1998}, a parallel implementation of Hopcroft-Karp algorithm is studied, which can find the maximum matching with a poly-logarithmic complexity within a given approximation error. A matching $\\mathcal{M}$ is said to approximate a maximum matching $\\mathcal{M}^{\\mathrm{m}}$ with an approximation factor $\\epsilon$ if and only if\n\\begin{equation}\n\t\\left|\\mathcal{M}\\right|\\geq\\left(1-\\epsilon\\right)\\left|\\mathcal{M}^{\\mathrm{m}}\\right|.\n\\end{equation}\nThe detailed description of PVER\\textsuperscript{2}HK is listed in Algorithm \\ref{alg:PVER2HK}. For the operation $\\mathrm{D}(\\widetilde{\\mathcal{G}},\\mathcal{M})$, we orient all the matching edges in $\\mathcal{M}$ from their ends in $\\widetilde{\\mathcal{U}}$ to the ends in $\\mathcal{S}$, while the other edges have an opposite direction, i.e., from $\\mathcal{S}$ to $\\widetilde{\\mathcal{U}}$. The operation $\\mathrm{PBFS}(\\widetilde{\\mathcal{U}},\\mathcal{S},\\mathcal{G}_{\\mathrm{d}},l^{*})$ will start at all the vertices in $\\widetilde{\\mathcal{U}}$ and search along the directed edges in $\\mathcal{G}_{\\mathrm{d}}$, and end when some of them first hit on the vertices in $\\mathcal{S}$. Then, the layer graph $\\mathcal{L}$ can be generated with $l<l^{*}$. The $\\mathrm{PVDP}\\left(\\mathcal{L},l\\right)$ operation executes the parallel maximal matching algorithm between two vertex layers in order to search all of the vertex-disjoint paths, which are just all the augmenting paths with length $l$. The operation $\\mathrm{PA}\\left(\\mathcal{P},\\mathcal{M}\\right)$ will execute the augmenting operation on $\\mathcal{M}$, which yields a larger maximum matching. All of the parallel operations will work on multiple processors simultaneously. The performance of this parallel implementation is summarized in the following theorem.\n\n\\medskip\n\\begin{thm}\nAlgorithm \\ref{alg:PVER2HK} generates a matching in bipartite graphs with an approximation factor $\\log^{-\\eta}N$ for some constant $\\eta$. The time complexity is $\\mathcal{O}\\left(\\log^{2\\eta}N\\right)$.\n\\end{thm}\n\\medskip\n\nThis assertion is the direct consequence of Lemma 12.1.1 and Theorem 12.3.2 in \\cite{Karpinski1998}. It can be seen that the proposed algorithm is even faster than FFT, $\\mathcal{O}\\left(N\\log N\\right)$, which is essentially a parallel implementation of discrete Fourier transform (DFT).\n\n\n\\subsection{Asymptotic Optimal Coding Schemes}\\label{subsec:coding_schemes}\n\nAs stated before, the optimal outage performance can only be achieved by using the maximum $f$-matching method with the help of coding schemes. To the best of our knowledge, however, the optimal coding scheme discussed in Section \\ref{sec:coding} is a NP-Hard problem and still unknown by present. However, there are two main practical approaches to maximize the minimum product distance. The induced coding schemes are capable of achieving the optimal DMT in parallel fading channels, i.e., the asymptotic optimal coding schemes.\n\nThe first class is the rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices code for the user $u_{m}$, which is based on the algebraic number theory and lattices theory. The basic idea is to construct the $\\widetilde{K}_{m}$-dimension constellation with the size $2^{R_{m}}$ based on $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices. Then, the constellation will be rotated to an appropriate angle such that $D_{p}^{\\min}$ is maximized. According to the lattices theory, $D_{p}^{\\min}$ can be calculated in theory. One can refer to \\cite{Wang2003,Oggier2004} and its references for detailed discussions on this coding scheme. Another advantage of the rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices code lies in the fact that it can reduce the peak-to-average power ratio (PAPR) \\cite{Henkel2000}. As a matter of fact, SC-FDMA can be seen as a special case of rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices codes, where the rotated angle is determined by the DFT matrix. This rotation increases $D_{p}^{\\mathrm{min}}$ or the distance between two points with zero product distance. Therefore, compared to uncoded OFDMA systems, SC-FDMA has a better performance in form of decoding error probability and PAPR.\n\nAnother approach is the permutation code which was proposed to achieve the optimal DMT \\cite{Tse2005,Tavildar2006}. The basic idea is to use the constellation of size $2^{R_{m}}$ on a complex plane for each subchannel. The points in the constellation for each subchannel are permuted, so that the product distance is maximized. The permutation operation on each subchannel can be seen as that of the original information times a specific matrix, which is referred to as the universal decodable matrix (UDM). Based on the Pascal triangle, the UDMs can be constructed directly for every subchannel \\cite{Ganesan2007}. From the perspective of rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices code, the permutation code can be seen as a way to construct a $\\widetilde{K}_{m}$-dimension constellation with size $2^{\\widetilde{K}_{m}R_{m}}$. The permutation rules imply that we should choose $2^{R_{m}}$ points from $2^{\\widetilde{K}_{m}R_{m}}$, so that the product distance is maximized. It is not trivial to evaluate the decoding error probability of the permutation code, because the analytic formula for $D_{p}^{\\min}$ has not been obtained. According to the construction process of the permutation code, the PAPR performance could be worse than the rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices code for the same average power. Another important difference is that the constellation size of each subchannel is $2^{R_{\\mathrm{s}}}=2^{R_{m}\/\\widetilde{K}_{m}}$ for the rotated $\\mathbb{Z}^{\\widetilde{K}_{m}}$-lattices code, while it is $2^{R_{m}}$ for the permutation code.\n\n\n\\section{Simulation Results}\\label{sec:simulation_results}\n\nIn this section, some simulation examples are presented to verify the theoretical derivations and the proposed algorithm. In the first group of simulation results, the tight upper bound of the conditional outage probability is verified, which is the key fundamental of the proposed code $f$-matching framework. The second group of simulation results verify the outage performance of the proposed framework.\n\nFor the first group of simulation results, a large number of samples are generated at each SNR value according to the channel model in Section \\ref{subsec:channel_model}. The number of outage events can be obtained under the condition that the transmitter knows $\\unit[1]{bit}$ CSI by computing the instantaneous channel capacity. The conditional outage probability $p^{\\mathrm{cop}}_{m}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)$ can then be calculated accordingly. The theoretical approximations are calculated by applying the results in Theorem \\ref{thm:con_outage_exponent}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{ConOutExp}\n\t\\caption{The tight upper bound of the conditional outage probability for the user $u_{m}$ with $K_{m}=4$, $k_{m}=2$, and $r_{m}=1.2$.}\\label{fig:cond_outage_exponent}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{FiniteSNRDMT}\n\t\\caption{The conditional outage exponent and conditional DMT for the user $u_{m}$ with $K_{m}=4$, $k_{m}=2$, and $r_{m}=1.2$.}\\label{fig:diversity_gain}\n\\end{figure}\n\nFig. \\ref{fig:cond_outage_exponent} compares the simulation results and the theoretical curves of the conditional outage probability. The user $u_{m}$ has $K_{m}=4$ coherence bandwidths with $k_{m}=2$ non-outage ones, and the multiplexing gain $r$ is set to $1.2$. It can be seen that the proposed tight upper bound is nearly identical with the simulation results in the realistic SNR range. For comparison, the curve of non-conditional outage probability is also plotted. The outage performance with $\\unit[1]{bit}$ CSI is worse than the corresponding one without CSI in this situation. In the high SNR regime, the slope of the conditional outage probability is determined by Eq. \\eqref{eq:dmt}, while the slope of the outage probability is determined by $K_{m}\\left(1-\\frac{r_{m}}{K_{m}}\\right)$. Fig. \\ref{fig:diversity_gain} presents the curves of conditional outage exponent and conditional DMT at a given multiplexing gain. Clearly, the conditional outage exponent is an increasing function of SNR for a fixed multiplexing gain. The proposed theoretical curve is nearly the same as the simulation one. In Fig. \\ref{fig:diversity_gain}, the conditional DMT is plotted as a constant which is much larger than the outage exponent at realistic SNRs. However, the conditional outage exponent is approaching the conditional DMT as SNR tends to infinity. Therefore, the proposed conditional outage exponent can be used to estimate the decreasing slope of conditional outage probabilities for the considered multi-carrier multi-access with $\\unit[1]{bit}$ CSI.\n\nFor the second group of simulation results, a large number of samples are generated according to the RBG formulation $\\mathscr{G}\\left\\{\\mathcal{K}_{MN},\\mathsf{P}\\right\\}$ of the considered multi-carrier multi-access in Section \\ref{subsec:RBG_Formulation}. The subchannels are then allocated by the proposed PVER\\textsuperscript{2}HK algorithm to each user for RB and chunk based coded $f$-matching allocation schemes, respectively. The simulation values of the outage probability can then be obtained by counting the number of outage events of each user in the total number of simulations. In proposed allocation schemes, the first order approximation of the outage probabilities are calculated by the formulas in Theorem \\ref{thm:outage_RB} and Theorem \\ref{thm:outage_chunk}, respectively. Besides, the interleaved allocation scheme, TDMA based scheme, and the localized allocation scheme have been simulated so as to illustrate the performance gain of the coded $f$-matching approach. The asymptotic line $\\mathsf{SNR}^{d_{m}^{*}\\left(r_{m}\\right)}$ is also plotted to compare the slope of these curves in high SNRs.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{RB_1}\n\t\\caption{The outage probability of the RB based coded $f$-matching allocation scheme for the user $u_{m}$ with $K_{m}=3$ and $R_{m}=1$ ($r_{m}=0$).}\\label{fig:RB_1}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{RB_2}\n\t\\caption{The outage probability of the RB based coded $f$-matching allocation scheme for the user $u_{m}$ with $K_{m}=3$ and $r_{m}=0.9$.}\\label{fig:RB_2}\n\\end{figure}\n\nFigs. \\ref{fig:RB_1} and \\ref{fig:RB_2} compare the simulation results and the theoretical curves of the RB based coded $f$-matching allocation scheme. These simulation examples consider the multi-carrier multi-access channel with $M=2$ users and $L=6$ coherence bandwidths, where each coherence bandwidth contains $N_{\\mathrm{c}}=M$ RBs. The fixed target rate schemes, i.e., $R=1$, are plotted in Fig. \\ref{fig:RB_1}. It can be seen that both PVER\\textsuperscript{2}HK algorithm and the interleaved \\& TDMA based allocation scheme have the same slope in the high SNR regime, which means that they have the same diversity order. The proposed algorithm, however, has $\\unit[2]{dB}$ SNR gain. The theoretical approximation in Eq. \\eqref{eq:outage_RB_high} approaches the simulation results as SNR increases. In Fig. \\ref{fig:RB_2}, the dynamic rate scenario is considered, where the multiplexing gain $r$ is set to $0.9$. The theoretical approximation is computed by Eq. \\eqref{eq:outage_RB_high} and Eq. \\eqref{eq:outage_RB_LowSNR} in high and low SNR regimes, respectively. It can be seen that the theoretical approximation is nearly identical with the simulation results of the proposed PVER\\textsuperscript{2}HK algorithm. Compared with the interleaved \\& TDMA based allocation schemes, the proposed algorithm has $\\unit[4]{dB}$ performance gain. Therefore, the performance gain of the proposed algorithm increases as the multiplexing gain increases.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{Chunk_1}\n\t\\caption{The outage probability of the chunk based coded $f$-matching allocation scheme for the user $u_{m}$ with $K_{m}=1$ and $R_{m}=1$ ($r_{m}=0$).}\\label{fig:chunk_1}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.6in]{Chunk_2}\n\t\\caption{The outage probability of the chunk based coded $f$-matching allocation scheme for the user $u_{m}$ with $K_{m}=2$ and $r_{m}=0.6$.}\\label{fig:chunk_2}\n\\end{figure}\n\nFigs. \\ref{fig:chunk_1} and \\ref{fig:chunk_2} compare the simulation results and the theoretical curves of chunk based coded $f$-matching allocation schemes. These simulation examples consider the multi-carrier multi-access channel with $M=3$ users and $L=6$ coherence bandwidths, where $N_{\\mathrm{c}}$ RBs in one coherence bandwidth is bundled as one chunk. The fixed target rate schemes, i.e., $R=1$, are plotted in Fig. \\ref{fig:chunk_1}. The PVER\\textsuperscript{2}HK algorithm has the full diversity order, whereas the diversity gain of the localized allocation scheme is $2$. Therefore, the SNR gain of the proposed algorithm increases as SNR increases. The theoretical approximation in Eq. \\eqref{eq:outage_chunk_1} is nearly identical with the simulation results of the proposed algorithm. The dynamic rate scenario is presented in Fig. \\ref{fig:chunk_2}, where the multiplexing gain $r$ is set to $0.6$. The theoretical approximation is computed by Eq. \\eqref{eq:outage_chunk_2}. Once again, the theoretical approximation is nearly identical with the simulation result of the proposed PVER\\textsuperscript{2}HK algorithm. Compared with the fixed rate scenario, the proposed algorithm has greater SNR gains, i.e., the performance gain of the proposed algorithm increases as the multiplexing gain increases.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\nThis paper proposed an analytic coded $f$-matching framework for channel resources allocation in multi-carrier multi-access channels, where the OER and DMR were defined as the comprehensive performance metrics. Compared to conventional approaches, the proposed framework sufficiently combined the advantages of optimal frequency-domain coding scheme and RBG based maximum $f$-matching approach, which can fully explore the combinatorial structure of the channel allocation problem. It was shown in theory that the outage probability, OER and DMR are optimal such that all of the users share the total multiplexing gain while achieving the full frequency diversity. The low complexity optimal allocation algorithm is a natural result of the proposed framework, which only depends on $\\unit[1]{bit}$ CSI feedback. The simulation results did not only verify the theoretical derivations, but also showed the significant performance gains in both diversity and SNR. Therefore, the proposed framework provides powerful tools for designing future multi-carrier multi-access systems, and has the potential to generalized to other wireless communication systems.\n\n\n\n\n\n\n\n\n\\appendices\n\n\\section{Proof of Theorem \\ref{thm:con_outage_exponent}}\\label{app:con_outage_exponent}\n\nDefine a sequence of random variables $\\left\\{Z_{mn}\\right\\}_{n=1}^{K_{m}}$ with the following distribution\n\\begin{equation}\n\\begin{aligned}\n\t& F_{Z_{mn}}\\left(z\\right)=\\Pr\\left\\{\\left.I_{mn}<z\\right|\\mathcal{D}_{K_{m}k_{m}}\\right\\}\\\\\n \t& =\\left\\{\n \t\\begin{aligned}\n\t\t& 1-\\frac{1}{q_{\\mathrm{s}}}\\exp\\left(-\\frac{e^{N_{\\mathrm{c}}z}-1}{\\gamma}\\right), & n=1,\\ldots,k_{m};\\\\\n\t\t& \\frac{1}{p_{\\mathrm{s}}}\\left[1-\\exp\\left(-\\frac{e^{N_{\\mathrm{c}}z}-1}{\\gamma}\\right)\\right], & n=k_{m}+1,\\ldots,K_{m}.\n\t\\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\nLet $X_{n}=R_{\\mathrm{s}}-Z_{mn}$ and $Y_{K_{m}}=\\sum_{n=1}^{K_{m}}X_{n}$, then the conditional outage probability defined in Eq. \\eqref{eq:con_outage_probability} is given by\n\\begin{equation}\n\\begin{aligned}\n\tp_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right) & =\\Pr\\left\\{\\left.\\sum_{n=1}^{K_{m}}I_{mn}<R_{m}\\right|\\mathcal{D}_{K_{m}k_{m}}\\right\\}\\\\\n\t& =\\Pr\\left\\{Y_{K_{m}}>0\\right\\}.\n\\end{aligned}\n\\end{equation}\n\nAccording to the considered condition, the elements of $\\left\\{h_{mn}\\right\\}_{n=1}^{K_{m}}$ are independent with the identical distribution of $\\mathcal{CN}\\left(0,1\\right)$. Let $1\\leq n_{1}\\leq k_{m}$, $k_{m}+1\\leq n_{2}\\leq K_{m}$, the cumulant-generating function of $Y_{K_{m}}$ is then given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\Phi_{K_{m}}\\left(\\lambda\\right)=\\ln\\mathsf{E}\\left\\{e^{\\lambda}Y_{K_{m}}\\right\\}=K_{m}R_{\\mathrm{s}}\\lambda\\\\\n\t& +\\ln\\left(\\mathsf{E}\\left\\{e^{-\\lambda Z_{mn_{2}}}\\right\\}\\right)^{\\beta_{m}K_{m}}+\\ln\\left(\\mathsf{E}\\left\\{e^{-\\lambda Z_{mn_{1}}}\\right\\}\\right)^{\\left(1-\\beta_{m}\\right)K_{m}}\\\\\n\t& =K_{m}\\left[R_{\\mathrm{s}}\\lambda+\\beta_{m}\\ln\\int_0^{R_{\\mathrm{s}}}e^{-\\lambda z}dF_{Z_{mn_{2}}}\\left(z\\right)\\right.\\\\\n\t& \\left.+\\left(1-\\beta_{m}\\right)\\ln\\int_{R_{\\mathrm{s}}}^{\\infty}e^{-\\lambda z}dF_{Z_{mn_{1}}}\\left(z\\right)\\right]\\\\\n\t& =K_{m}\\left[\\beta_{m}\\ln\\int_0^{R_{\\mathrm{s}}}\\frac{N_{\\mathrm{c}}e^{-\\lambda z}}{p_\\mathrm{s}\\gamma}\\exp\\left(-\\frac{e^{N_{\\mathrm{c}}z}-1}{\\gamma}+N_{\\mathrm{c}}z\\right)dz\\right.\\\\\n\t& \\left.+\\left(1-\\beta_{m}\\right)\\ln\\int_{R_{\\mathrm{s}}}^{\\infty}\\frac{N_{\\mathrm{c}}e^{-\\lambda z}}{q_{\\mathrm{s}}\\gamma}\\exp\\left(-\\frac{e^{N_{\\mathrm{c}}z}-1}{\\gamma}+N_{\\mathrm{c}}z\\right)dz\\right]\\\\\n    &+K_{m}R_{\\mathrm{s}}\\lambda\\\\\n    & =K_{m}\\left[R_{\\mathrm{s}}\\lambda+\\beta_{m}\\ln\\frac{e^{\\frac{1}{\\gamma}}}{p_\\mathrm{s}\\gamma^{\\frac{\\lambda}{N_{\\mathrm{c}}}}}+\\beta_{m}\\ln\\int_{\\frac{1}{\\gamma}}^{\\frac{e^{R_\\mathrm{c}}}{\\gamma}}t^{-\\frac{\\lambda}{N_{\\mathrm{c}}}}e^{-t}dt\\right.\\\\\n    &\\left.+\\left(1-\\beta_{m}\\right)\\ln\\frac{e^{\\frac{1}{\\gamma}}}{q_\\mathrm{s}\\gamma^{\\frac{\\lambda}{N_{\\mathrm{c}}}}}+\\left(1-\\beta_{m}\\right)\\ln\\int_{\\frac{e^{R_{\\mathrm{c}}}}{\\gamma }}^{\\infty }t^{-\\frac{\\lambda}{N_{\\mathrm{c}}}}e^{-t}dt\\right]\n\\end{aligned}\n\\end{equation}\n\\begin{equation*}\n\\begin{aligned}\n\t& =K_{m}\\left[\\left(R_{\\mathrm{c}}-\\ln\\gamma\\right)\\frac{\\lambda}{N_{\\mathrm{c}}}+\\frac{1}{\\gamma}-\\ln p_{\\mathrm{s}}^{\\beta_{m}}q_{\\mathrm{s}}^{1-\\beta_{m}}\\right.\\\\\n\t& +\\beta_{m}\\ln\\left(\\Gamma\\left(1-\\frac{\\lambda}{N_{\\mathrm{c}}},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\frac{\\lambda}{N_{\\mathrm{c}}},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\right)\\\\\n\t& \\left.+\\left(1-\\beta_{m}\\right)\\ln\\Gamma\\left(1-\\frac{\\lambda}{N_{\\mathrm{c}}},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\right],\n\\end{aligned}\n\\end{equation*}\nwhere $\\beta_{m}=1-\\frac{k_{m}}{K_{m}}$. Therefore, we have\n\\begin{equation}\n\\begin{aligned}\n\t\\Lambda\\left(\\lambda\\right)=&\\lim_{K_{m}\\rightarrow\\infty}\\frac{1}{K_{m}}\\Phi_{K_{m}}\\left(\\lambda\\right)\\\\\n\t= & \\left(R_{\\mathrm{c}}-\\ln\\gamma\\right)\\lambda+\\frac{1}{\\gamma}-\\ln p_{\\mathrm{s}}^{\\beta_{m}}q_{\\mathrm{s}}^{1-\\beta_{m}}\\\\\n\t& +\\beta_{m}\\ln\\left(\\Gamma\\left(1-\\lambda,\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)\\right)\\\\\n\t& +\\left(1-\\beta_{m}\\right)\\ln\\Gamma\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right).\n\\end{aligned}\n\\end{equation}\nConsidering the relationship between the cumulant-generating function and the characteristic function, the characteristic function of $Y_{K_{m}}$ can then by given by\n\\begin{equation}\n\t\\Phi_{K_{m}}\\left(i\\lambda\\right)=K_{m}\\Lambda\\left(i\\lambda\\right).\n\\end{equation}\n\nDefine $G\\left(y\\right)=\\Pr\\left\\{Y_{K_{m}}>y\\right\\}$, then according to L\\'evy's theorem \\cite{Shiryaev1996}, we have\n\\begin{equation}\n\t\\begin{aligned}\n\t\tG\\left(y\\right) & =\\frac{1}{2\\pi i}\\int_{-\\infty}^{+\\infty}\\frac{1}{\\xi}\\exp\\left(K_{m}\\Lambda\\left(i\\xi\\right)-i\\xi y\\right)d\\xi\\\\\n\t\t& =\\frac{1}{2\\pi i}\\int_{\\lambda-i\\infty}^{\\lambda+i\\infty}\\frac{1}{z}\\exp\\left(K_{m}\\Lambda\\left(z\\right)-zy\\right)dz,\n\t\\end{aligned}\n\\end{equation}\nwhere $z=\\lambda+i\\xi$, and $\\lambda$ is chosen from the convergence region of this integral. According to the saddle-point approximation method \\cite{Butler2007}, if we let $z^{*}$ with $\\Re\\left(z^{*}\\right)=\\lambda^{*}$ be a solution of the saddle-point equation\n\\begin{equation}\n\t\\Lambda'\\left(z\\right)=\\frac{y}{K_{m}},\n\\end{equation}\nthen\n\\begin{equation}\n\t\\begin{aligned}\n\t\t& \\frac{1}{2\\pi i}\\int_{\\lambda^{*}-i\\infty}^{\\lambda^{*}+i\\infty}\\frac{1}{z}\\exp\\left(K_{m}\\Lambda\\left(z\\right)-zy\\right)dz\\lesssim\\\\\n\t\t& \\frac{1}{\\sqrt{2\\pi K_{m}\\Lambda''\\left(\\lambda^{*}\\right)}\\lambda^{*}}\\exp\\left(K_{m}\\Lambda\\left(z^{*}\\right)-z^{*}y\\right).\n\t\\end{aligned}\n\\end{equation}\nSince $G\\left(y\\right)$ is a real function, we can always choose a real $z^{*}$ only if the cumulant-generating function exists. Moreover, $\\frac{\\lambda y}{K_{m}}-\\Lambda\\left(\\lambda\\right)$ is a conjugate function when $\\lambda\\in\\mathbb{R}$, then $\\lambda^{*}$ must be unique. Therefore, the exponentially tight upper bound of the conditional outage probability $p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)$ is the tail distribution of $G\\left(y\\right)$ with $y>\\mathsf{E}\\left\\{Y_{K_{m}}\\right\\}$, which is given by\n\\begin{equation}\\label{eq:saddle-point}\n\\begin{aligned}\n\t& p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right)=G\\left(0\\right)\\\\\n\t\\lesssim & \\frac{1}{\\sqrt{2\\pi K_{m}\\sigma^{2}}\\lambda^{*}}\\exp\\left(K_{m}\\Lambda\\left(\\lambda^{*}\\right)\\right)=p_{m}^{\\mathrm{upper}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m}k_{m}}\\right.\\right),\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda^{*}$ is the solution of $\\Lambda'\\left(\\lambda\\right)=0$, and $\\sigma^{2}=\\Lambda''\\left(\\lambda^{*}\\right)$. Clearly, we have\n\\begin{equation}\\label{eq:CGF_derivative}\n\\begin{aligned}\n\t\\Lambda'\\left(\\lambda\\right)= & \\left(R_{\\mathrm{c}}-\\ln\\gamma\\right)+\\left(1-\\beta_{m}\\right)\\frac{\\Gamma'\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\\\\n\t& +\\beta_{m}\\frac{\\Gamma'\\left(1-\\lambda,\\frac{1}{\\gamma}\\right)-\\Gamma'\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda,\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda,\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}=0.\n\\end{aligned}\n\\end{equation}\nAccording to the properties of Gamma function and Meijer's $G$-function, we have\n\\begin{equation}\\label{eq:gamma_derivative}\n\\begin{aligned}\n\t\\Gamma'\\left(1-\\lambda,z\\right)= & -\\Gamma\\left(1-\\lambda,z\\right)\\ln z\\\\\n\t& -G_{2,3}^{3,0}\\left(z\\left|\n\t\\begin{array}{ccc}\n\t\t1, & 1, & -\\\\\n\t\t0, & 0, & 1-\\lambda\n\t\\end{array}\\right.\n\t\\right).\n\\end{aligned}\n\\end{equation}\nTherefore, Eq. \\eqref{eq:exponent_equation} can be obtained by plugging Eq. \\eqref{eq:gamma_derivative} into Eq. \\eqref{eq:CGF_derivative}. For $\\sigma^{2}$, we have\n\\begin{equation}\n\\begin{aligned}\\label{eq:CGF_parameter}\n\t& \\sigma^{2}=\\Lambda''\\left(\\lambda^{*}\\right)\\\\\n\t=&\\left(1-\\beta_{m}\\right)\\left[\\frac{\\Gamma''\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}-\\left(\\frac{\\Gamma'\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\right)^{2}\\right]\\\\\n\t& +\\beta_{m}\\frac{\\Gamma''\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma''\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\\\\n\t& -\\beta_{m}\\left(\\frac{\\Gamma'\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma'\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}{\\Gamma\\left(1-\\lambda^{*},\\frac{1}{\\gamma}\\right)-\\Gamma\\left(1-\\lambda^{*},\\frac{e^{R_{\\mathrm{c}}}}{\\gamma}\\right)}\\right)^{2}.\n\\end{aligned}\n\\end{equation}\nAccording to the properties of Meijer's $G$-function, we have\n\\begin{equation}\\label{eq:Meijer_derivative}\n\\begin{aligned}\n\t& \\frac{\\partial}{\\partial x}G_{2,3}^{3,0}\\left(z\\left|\n\t\\begin{array}{ccc}\n\t\t1, & 1, & - \\\\\n\t\t0, & 0, & 1-\\lambda^{*}\\\\\n\t\\end{array}\\right.\\right)\\\\\n\t= & G_{2,3}^{3,0}\\left(z\\left|\n\t\\begin{array}{ccc}\n\t\t1, & 1, & - \\\\\n\t\t0, & 0, & 1-\\lambda^{*}\\\\\n\t\\end{array}\\right.\\right)\\ln z\\\\\n\t&+2G_{3,4}^{4,0}\\left(z\\left|\n\t\\begin{array}{cccc}\n\t\t1, & 1, & 1, & -\\\\\n\t\t0, & 0, & 0, & 1-\\lambda^{*}\\\\\n\t\\end{array}\\right.\\right)\n\\end{aligned}\n\\end{equation}\nTherefore, Eq. \\eqref{eq:exponent_parameter} can be obtained by plugging Eq. \\eqref{eq:Meijer_derivative} into Eq. \\eqref{eq:CGF_parameter}.\n\nFor the tail distribution with $y<\\mathsf{E}\\left\\{Y_{K_{m}}\\right\\}$, a similar result can be obtained by applying the same process. Therefore, Theorem \\ref{thm:con_outage_exponent} has been established.\n\n\n\\section{Proof of Lemma \\ref{lem:f_matching}}\\label{app:f_matching}\n\n($\\Rightarrow$). Suppose $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ has a maximum $f$-matching $\\mathcal{M}_{f}^{\\mathrm{m}}$, which cannot saturates the vertex $u_{m}$. Then, there must be conflicts between $u_{m}$ and other vertices in $\\mathcal{U}$. Therefore, we have a subset $\\mathcal{X}\\subseteq\\mathcal{U}$ with $u_{m}\\in\\mathcal{X}$ satisfying\n\\begin{equation}\n\t\\sum_{u_{i}\\in\\mathcal{X}}f\\left(u_{i}\\right)=\\sum_{u_{i}\\in\\mathcal{X}}K_{i}>\\sum_{s_{n}\\in\\mathcal{N} \\left(\\mathcal{X}\\right)}f\\left(s_{n}\\right).\n\\end{equation}\n\n($\\Leftarrow$). Orient all edges of $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$ from $\\mathcal{U}$ to $\\mathcal{S}$. Add a source $\\theta$ joined all vertices in $\\mathcal{U}$ and a sink $\\zeta$ to which every vertex in $\\mathcal{S}$ is joined. This induced directed graph is denoted by $\\mathcal{G}_{\\mathrm{d}}$. Assign capacities to all of the edges as follows:\n\\begin{equation}\n\t\\left\\{\\begin{aligned}\n\t\t& c\\left(e\\right)=f\\left(u_{i}\\right), & \\textrm{if }e\\textrm{ is from }\\theta\\textrm{ to }u_{i}\\in\\mathcal{U};\\\\\n\t\t& c\\left(e\\right)=1, & \\textrm{on all the other edges}.\n\t\\end{aligned}\\right.\n\\end{equation}\nChoose a subset $\\mathcal{X}$ of $\\mathcal{U}$ with $u_{m}\\in\\mathcal{X}$ such that Eq. \\eqref{eq:f_matching_cond} holds. Let $\\mathcal{C}=\\mathcal{X}\\cup\\left\\{\\theta\\right\\}$, which is clearly a cut set separating $\\theta$ from $\\zeta$. The capacity of $\\mathcal{C}$ is then given by\n\\begin{equation}\n\t\\sum_{u_{i}\\in\\mathcal{U}-\\mathcal{X}}f\\left(u_{i}\\right)+\\sum_{s_{n}\\in\\mathcal{N}\\left(\\mathcal{X}\\right)}f\\left(s_{n}\\right)<\\sum_{u_{i}\\in\\mathcal{U}}f\\left(u_{i}\\right).\n\\end{equation}\nAccording to max-flow min-cut theorem in Lemma \\ref{lem:max_flow_min_cut}, the maximum flow $\\mathcal{F}^{*}$ from $\\theta$ to $\\zeta$ of $\\mathcal{G}_{\\mathrm{d}}$ must be smaller than $\\sum_{u_{i}\\in\\mathcal{U}}f\\left(u_{i}\\right)$. Moreover, since $u_{m}\\in\\mathcal{X}\\subseteq\\mathcal{C}$, the maximum flow $\\mathcal{F}^{*}$ cannot achieve the capacities of all the edges from $\\theta$ to vertices in $\\mathcal{X}$. Thus, greater flow rate can be allocated to the edge from $\\theta$ to $u_{i}$ in $\\mathcal{U}$ than the edge from $\\theta$ to $u_{m}$ when they are conflicts. Therefore, the generated maximum $f$-matching cannot saturate $u_{m}$.\n\n\n\\section{Proof of Lemma \\ref{lem:matching_edges}}\\label{app:matching_edges}\n\nLet $\\mathcal{G}(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E})$ satisfy the conditions in this lemma, and we assume there is an edge set $\\mathcal{E}_{0}$ that contains at least one maximum $f$-matching which cannot saturate $u_{m}$. According to Lemma \\ref{lem:f_matching}, there must be a subset $\\mathcal{X}\\subseteq\\mathcal{U}$ with the maximum number of elements, which satisfies\n\\begin{equation}\n   \\left|\\mathcal{N}\\left(\\mathcal{X}\\right)\\right|<K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)=\\sum_{u_{i}\\in\\mathcal{X}}K_{i},\n\\end{equation}\nIt can be seen that there are $\\left|\\mathcal{X}\\right|\\left(K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)-1\\right)$ edges from $\\mathcal{X}$ to $\\mathcal{S}$ at most.\n\nFor any $\\mathcal{X}$ with $|\\mathcal{X}|=1,\\ldots,M$, there are at most\n\\begin{equation}\n\t\\left|\\mathcal{X}\\right|\\left(K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)-1\\right)+\\left(M-\\left|\\mathcal{X}\\right|\\right)N\n\\end{equation}\nedges. Therefore, $\\left|\\mathcal{E}\\right|$ must be equal or greater than\n\\begin{equation}\\label{eq:edge_number}\n\\begin{aligned}\n\t& \\max_{|\\mathcal{X}|=1,\\ldots,M}\\left\\{\\left|\\mathcal{X}\\right|\\left(K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)-1\\right)+\\left(M-\\left|\\mathcal{X}\\right|\\right)N+1\\right\\}\\\\\n\t= & \\max_{|\\mathcal{X}|=1,\\ldots,M}\\left\\{\\left|\\mathcal{X}\\right|\\left(K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)-N-1\\right)+MN+1\\right\\}.\n\\end{aligned}\n\\end{equation}\nSince $K^{\\mathrm{sum}}\\left(\\mathcal{X}\\right)$ is an increasing function of $\\left|\\mathcal{X}\\right|$, the maximum value can only be achieved for $\\left|\\mathcal{X}\\right|=1$ or $\\left|\\mathcal{X}\\right|=M$.\n\nConsider first $\\mathcal{X}$ has only one vertex such that $\\mathcal{X}=\\left\\{u_{m}\\right\\}$. There will be at most $\\left(M-1\\right)N+K_{m}-1$ edges. Thus, if $\\left|\\mathcal{E}\\right|\\geq\\left(M-1\\right)N+K_{m}$ there must be a maximum $f$-matching which can saturate $u_{m}$ for any $\\mathcal{E}$. Because $\\left(M-1\\right)N+K_{m}>\\left(M-1\\right)N+K_{i}$ with $K_{\\left(i\\right)}<K_{\\left(m\\right)}$, all of the vertices $(u_{\\left(i\\right)})_{i=1}^{m}$ can be saturated by the maximum $f$-matching for any $\\mathcal{E}$. Consider then the case that $\\mathcal{X}=\\mathcal{U}$. There will be at most $M\\left(K^{\\mathrm{sum}}-1\\right)$ edges. Thus, if $\\left|\\mathcal{E}\\right|\\geq M\\left(K^{\\mathrm{sum}}-1\\right)+1$ there must be a maximum $f$-matching which can saturate $u_{m}$ for any $\\mathcal{E}$. By solving the inequality that\n\\begin{equation}\n\t\\left(M-1\\right)N+K_{m}\\geq M\\left(K^{\\mathrm{sum}}-1\\right)+1,\n\\end{equation}\nwe can get\n\\begin{equation}\\label{eq:one}\n\tK_{m}\\leq N+1-\\frac{M}{M-1}K_{m}^{\\mathrm{sum}}.\n\\end{equation}\nIf Eq. \\eqref{eq:one} holds, the bipartite graph with $\\left|\\mathcal{X}\\right|=1$ has the maximum number of edges. Otherwise, the bipartite graph with $\\left|\\mathcal{X}\\right|=M$ has the maximum number of edges. Therefore, Lemma \\ref{lem:matching_edges} has been established.\n\n\n\\section{Proof of Theorem \\ref{thm:outage_RB}}\\label{app:outage_RB}\nAccording to the second part of Lemma \\ref{lem:matching_edges}, $MN-(M\\widetilde{K}^{\\mathrm{sum}}-M)=M$ indicates the case that there is one isolated coherence bandwidth in $\\mathcal{S}$. Recalling Eq. \\eqref{eq:opt_cond_fmatching} in the coded $f$-matching approach, thus any user $u_{m}$ will be allocated $\\frac{\\widetilde{K}_{m}}{L}$ RBs in each coherence bandwidth. In this context, the outage probability of the user $u_{m}$ has the term\n\\begin{equation}\n\\begin{aligned}\n\t& \\frac{1}{M}\\binom{L}{L-1}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{L,L-1}\\right.\\right)p_{\\mathrm{s}}^{M}q_{\\mathrm{s}}^{M\\left(L-1\\right)}\\\\\n\t= & \\frac{Lp_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{L,L-1}\\right.\\right)p_{\\mathrm{s}}^{M}}{M}+\\mathsf{O}\\left(p_{\\mathrm{s}}^{M+L-1}\\right).\n\\end{aligned}\n\\end{equation}\nClearly, the lowest order of $p_{\\mathrm{s}}$ is given by $M+L-1>L$, since $M\\geq2$ in the considered multi-carrier multi-access channel. Therefore, this case is the higher order terms in the formula of the outage probability.\n\nAccording to the first part of Lemma \\ref{lem:matching_edges}, there must be $\\kappa$ outage coherence bandwidths for the user $u_{m}$, with $\\kappa=L-\\left\\lceil\\frac{\\widetilde{K}_{m}}{N_{\\mathrm{c}}}\\right\\rceil+1,\\ldots,L$, if this user cannot be saturated by the generated maximum $f$-matching. Recalling the definition of maximum $f$-matching, thus all of the RBs in $L-\\kappa$ coherence bandwidths must be allocated to the user $u_{m}$. According to Theorem \\ref{thm:con_outage_exponent}, therefore, the outage probability of the user $u_{m}$ for a given $\\kappa$ is given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\binom{L}{\\kappa}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{\\widetilde{K}_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}q_{\\mathrm{s}}^{ML-\\kappa}\\\\\n\t= & \\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{\\widetilde{K}_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}}{\\kappa!\\left(L-\\kappa\\right)!}+\\mathsf{O}\\left(p_{\\mathrm{s}}^{L}\\right).\n\\end{aligned}\n\\end{equation}\nHence, according to the law of total probability, the first order approximation of the user $u_{m}$ is given by Eq. \\eqref{eq:outage_RB_high}.\n\\end{IEEEproof}\n\n\n\\section{Proof of Theorem \\ref{thm:outage_RB_LowSNR}}\\label{app:outage_RB_LowSNR}\nConsider the low SNR regime such that the sample of $\\mathscr{G}\\left(\\mathcal{K}_{MN};\\mathsf{P}\\right)$ only has a few edges. For a user $u_{m}$ in the set $\\mathcal{U}$, there are two cases that make $u_{m}$ unsaturated: 1) There are no $K_{m}$ non-outage RBs in $\\mathcal{S}$ for $u_{m}$; and 2) There are other users competing for the same RB with $u_{m}$ and it is not saturated by the maximum $f$-matching.\n\nIn the first case, the occurrence probability of this event is given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\sum_{\\kappa=L-K_{m}+1}^{L}\\binom{L}{\\kappa}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},L-\\kappa}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}q_{\\mathrm{s}}^{L-\\kappa}\\\\\n\t= & p_{\\mathrm{s}}^{L}+Lp_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},1}\\right.\\right)p_{\\mathrm{s}}^{L-1}q_{\\mathrm{s}}+\\mathsf{O}\\left(q_{\\mathrm{s}}\\right),\n\\end{aligned}\n\\end{equation}\nfor $\\gamma\\rightarrow0$. In the second case, there will be at least two edges in the bipartite graph $\\mathcal{G}\\left(\\mathcal{U}\\cup\\mathcal{S},\\mathcal{E}\\right)$. One is $u_{m}s_{n}$, and the other one is $u_{m'}s_{n}$. Assuming that there are only two edges in the sample of this random bipartite graph, i.e., $\\mathcal{E}=\\left\\{u_{m}s_{n},u_{m'}s_{n}\\right\\}$. In the maximum $f$-matching, $u_{m}$ or $u_{m'}$ is chosen with equal probability. The outage probability of $u_{m}$ must then have the term\n\\begin{equation}\n\t\\frac{1}{2}\\binom{M}{2}\\binom{L}{1}p_{\\mathrm{s}}^{ML-2}q_{\\mathrm{s}}^{2}=\\mathsf{O}\\left(q_{\\mathrm{s}}\\right),\\quad\\gamma\\rightarrow0.\n\\end{equation}\n\nThus, if there are more than two edges in this random bipartite graph, the outage probability of $u_{m}$ must have a factor $q_{\\mathrm{s}}^{x}$ with $x\\geq3$. Hence, Eq. \\eqref{eq:outage_RB_LowSNR} has then been established.\n\n\n\\section{Proof of Theorem \\ref{thm:outage_chunk}}\\label{app:outage_chunk}\nWe first consider the case of $K_{m}=1,\\ldots,K_{m}^{\\mathrm{th}}$. According to Lemma \\ref{lem:matching_edges}, if the user $u_{m}$ is not saturated by the generated maximum $f$-matching, there are at least $L-K_{m}+1$ chunks in outage state. Let $\\kappa$ be the number of outage chunks. Since the first order approximation is considered in this paper, $\\kappa$ must be in the range from $L-K_{m}+1$ to $L$. According to the proof of Lemma \\ref{lem:matching_edges}, the $L-K_{m}+1$ outage chunks are only outage for the user $u_{m}$. In this specific condition, the outage probability of the user $u_{m}$ is given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\binom{L}{L-K_{m}+1}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{L-K_{m}+1}\\cdot\\\\\n\t& q_{\\mathrm{s}}^{ML-L+K_{m}-1}=\\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{L-K_{m}+1}}{\\left(L-K_{m}+1\\right)!\\left(K_{m}-1\\right)!}\\\\\n\t& +\\mathsf{O}\\left(p_{\\mathrm{s}}^{L}\\right),\n\\end{aligned}\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, where the lowest order of $p_{\\mathrm{s}}$ is $K_{m}-1+L-K_{m}+1=L$. For $\\kappa=L-K_{m}+2,\\ldots,L$, the $\\kappa$ outage chunks must also be in outage only for the user $u_{m}$. If this is not true, for example, $\\Delta_{1}$ outage chunks for $u_{m_{1}}$ and $\\Delta_{2}$ outage chunks for $u_{m_{2}}$ with $\\kappa=\\Delta_{1}+\\Delta_{2}$, the approximation with the lowest order of the outage probability for the user $u_{m_{i}},\\,i=1,2$ is given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\binom{L}{\\Delta_{1}}\\binom{L-\\Delta_{1}}{\\Delta_{2}}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},L-\\Delta_{i}}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}q_{\\mathrm{s}}^{ML-\\kappa}\\\\\n\t& = \\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},L-\\Delta_{i}}\\right.\\right)p_{\\mathrm{s}}^{\\kappa}}{\\Delta_{1}!\\Delta_{2}!\\left(L-\\Delta_{1}-\\Delta_{2}\\right)!}+\\mathsf{O}\\left(p_{\\mathrm{s}}^{L-\\Delta_{i}+\\kappa}\\right).\n\\end{aligned}\n\\end{equation}\nTherefore, the lowest order of $p_{\\mathrm{s}}$ is given by $L-\\Delta_{i}+\\kappa>L$. In other words, this situation results in a higher order term. Hence, according to the law of total probability, the first order approximation of the user $u_{m}$ is given by Eq. \\eqref{eq:outage_chunk_1}.\n\nConsider now the case of $K_{m}=K_{m}^{\\mathrm{th}}+1,\\ldots,\\widetilde{K}_{m}$. According to Lemma \\ref{lem:matching_edges}, if the user $u_{m}$ is not saturated by the generated maximum $f$-matching, there are at least $ML-M\\left(K^{\\mathrm{sum}}-1\\right)=M\\left(L-K^{\\mathrm{sum}}+1\\right)$ chunks in outage state. According to the proof of Lemma \\ref{lem:matching_edges}, the only outage user $u_{m}$ has $K_{m}-1$ non-outage chunks. Therefore, the first order approximation of the outage probability for the user $u_{m}$ is given by\n\\begin{equation}\n\\begin{aligned}\n\t& \\frac{\\binom{L}{K^{\\mathrm{sum}}-1}}{M}p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)}q_{\\mathrm{s}}^{M\\left(K^{\\mathrm{sum}}-1\\right)}\\\\\n   \t& =\\frac{L!p_{m}^{\\mathrm{cop}}\\left(R_{m}\\left|\\mathcal{D}_{K_{m},K_{m}-1}\\right.\\right)p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)}}{M\\left(L-K^{\\mathrm{sum}}+1\\right)!\\left(K^{\\mathrm{sum}}-1\\right)!}\\\\\n   \t& +\\mathsf{O}\\left(p_{\\mathrm{s}}^{M\\left(L-K^{\\mathrm{sum}}+1\\right)+K_{m}-1}\\right),\n\\end{aligned}\n\\end{equation}\nwhen $\\gamma\\rightarrow\\infty$.\n\nIf it happens that $K_{m}=K_{m}^{\\mathrm{th}}$ and $\\left(M-1\\right)|MK_{m}^{\\mathrm{sum}}$, the outage probability of the user $u_{m}$ is clearly the summation of Eq. \\eqref{eq:outage_chunk_1} and Eq. \\eqref{eq:outage_chunk_2}.\n\n\n\\section{Proof of Lemma \\ref{lem:perfect_f_matching}}\\label{app:perfect_f_matching}\nSuppose $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$ has a perfect matching $\\widetilde{\\mathcal{M}}$. For each edge $u_{m}s_{n}\\in\\mathcal{E}$, let $l\\left(u_{m}s_{n}\\right)$ denote the number of edges of $\\widetilde{\\mathcal{M}}$ connecting $\\mathcal{X}_{u_{m}}$ and $s_{n}\\in\\mathcal{S}$. Then the number of edges of $\\widetilde{\\mathcal{M}}$ incident with $\\mathcal{X}_{u_{m}}$ is exactly $\\sum_{s_{n}\\in\\mathcal{S}}l\\left(u_{m}s_{n}\\right)$, which is also equal to $\\left|\\mathcal{X}_{u_{m}}\\right|=K_{m}$. Since $\\widetilde{\\mathcal{M}}$ is a perfect matching, then\n\\begin{equation}\n\t|\\widetilde{\\mathcal{M}}|=\\sum_{u_{m}\\in\\mathcal{U}}\\sum_{s_{n}\\in\\mathcal{S}}l\\left(u_{m}s_{n}\\right)=MN\n\\end{equation}\nis the maximum number of edges in any matching of $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$. This fact shows that if $\\mathcal{X}_{u_{m}}$ is seen as $u_{m}$, this is a perfect $f$-matching $\\mathcal{M}_{f}^{\\mathrm{p}}$. If this is not true, there will be more edges in $\\mathcal{M}_{f}^{\\mathrm{p}}$ than in $\\widetilde{\\mathcal{M}}$. In fact, when constructing $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$, we can dispatch one edge incident with $u_{m}$ in $\\mathcal{M}_{f}^{\\mathrm{p}}$ to one vertex in $\\mathcal{X}_{u_{m}}$. As a result, a new matching which contains more edges than $\\widetilde{\\mathcal{M}}$ can be obtained. This contradicts the fact that $\\widetilde{\\mathcal{M}}$ is a perfect matching of $\\mathcal{G}(\\mathcal{\\widetilde{U}}\\cup\\mathcal{S},\\widetilde{\\mathcal{E}})$.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The goal of the\npaper is to prove local boundedness and other regularity of weak\nsolutions to the next two parabolic equations.\n\n\\[\n\\Delta u(x, t) - b(x, t) \\nabla u(x, t) - \\partial_t u(x, t)=0,\n\\quad (x, t) \\in \\Omega \\subset {\\bf R}^n \\times {\\bf R},\n\\leqno(1.1)\n\\]\n\\[\n\\begin{cases}\n\\Delta u(x, t) - b(x, t)  \\nabla u(x, t) + \\nabla P(x, t) -\n\\partial_t\nu(x, t)=0, \\quad (x, t) \\in \\Omega \\subset {\\bf R}^3 \\times {\\bf R},\\\\\ndiv u = 0, \\ div b=0, \\ b(\\cdot, t) \\in L^2_{loc}.\n\\end{cases}\n\\leqno(1.2)\n\\]Here $\\Delta$ is the standard Laplacian and $b=b(x, t)$ is a  given\n$L^2_{loc}$ singular vector field  to be specified later. $\\Omega$\nis a domain.\n\nThere has been a mature theory of existence and regularity for\nequation (1.1) (see [LSU], [Lieb] e.g.). For instance when\n$b=b(x)$ and $|b| \\in L^p_{loc}({\\bf R}^n)$, $p>n$, weak solutions\nto (1.1) are locally bounded and H\\\"older continuous. This\ncondition is sharp in general. Here is an example (see [HL] p108).\nThe function $u=\\ln \\ln |x|^{-1} - \\ln \\ln R^{-1}$ is an unbounded\nweak solution of\n\\[\n\\Delta u + b \\nabla u =0\n\\]\nin the ball $B(0, R)$ in ${\\bf R}^2$, $R<1$. Here $b = \\nabla u =\n- \\frac{\\nabla |x|}{|x| \\ln |x|^{-1}}$ and hence $b \\in L^2_{loc}$\nwith $n=2$.\n\nThe first goal of the paper is to show that the simple condition\n$div b \\le 0$ will ensure that weak solutions of (1.1) are locally\nbounded when the data $b$ is  almost twice as singular as before.\nThis will be made precise in Theorem 1.1 and Remark 1.1 below.\nThus one has achieved a leap in boundedness condition rather than\na marginal improvement.\n\n\nClearly  a strong impetus still exists for the study of parabolic\nequations with very singular coefficients. In the study of\nnonlinear equations with gradient structure such as the\nNavier-Stokes equations and harmonic maps, highly singular\nfunctions occur naturally. So, it is very important investigate a\npossible gain of regularity in the presence of singular drift term\n$b$. This line of research has been followed in the papers [St],\n[KS], [Os], [CrZ], [ChZ], [CS] and [Se]. Under the condition $|b|\n\\in L^n({\\bf R}^n)$, Stampacchia [St] proved that bounded\nsolutions of $\\Delta u + b \\nabla u=0$ are H\\\"older continuous. In\nthe paper [CrZ], Cranston and Zhao proved that solutions to this\nequation are continuous when $b$ is in a suitable Kato class i.e.\n$\\lim_{r \\to 0} \\sup_x \\int_{|x-y| \\le r}\n\\frac{|b(y)|}{|x-y|^{n-1}} dy =0$. In the paper [KS] Kovalenko and\nSemenov proved the H\\\"older continuity of solutions to (1.1), when\n$| b|^2$ is independent of time and  is sufficiently small in the\nform sense, i.e., for a sufficiently small $\\epsilon>0$,\n\\[\n\\int_{{\\bf R}^n}  |b|^2(x)  \\phi^2(x) dx \\le \\epsilon\n\\int_{\\mathbf{R}^n} |\\nabla \\phi|^2(x) dx, \\quad \\phi\\in\nC^\\infty_0(\\mathbf{R}^n).\n\\]It is a well known fact that form boundedness condition provides\na more general class of singular functions than corresponding\n$L^p$ class, Morrey-Campanato class and Kato class functions.\n  This result was  generalized in [Se] to\nequations with leading term in divergence form. In [Os], Osada\nproved, among other things, that the fundamental solution of (1.1)\nhas global Gaussian upper and lower bound when $\\bf b$ is the\nderivative of bounded functions (in distribution sense) and $div\nb=0$. Recently in the paper [LZ], H\\\"older continuity of solutions\nto (1.1) was established when $b=b(x)$, $|b|^2$ is form bounded\nand $div {b} =0$. Most recently, in [Z2], we considered (1.1) with\ntime-dependent functions $b=b(x, t)$.  It was proven that weak\nsolutions to (1.1) are locally bounded provided that $div b =0$\nand for a fixed $m>1$, $|b|^m$ is form bounded. That is for any\n$\\phi \\in C^\\infty({\\bf R}^n \\times (0, \\infty) )$ with compact\nsupport in the spatial direction,\n\\[\n\\int \\int_{{\\bf R}^n} |b| ^m \\phi^2 dxdt \\le k \\int \\int_{{\\bf\nR}^n} |\\nabla \\phi|^2 dxdt\n\\]where $k$ is independent of $\\phi$. Note the key improvement\nover previous result is that the power on $b$ drops from $2$ to\nany number greater than $1$.\n\nIt is interesting to note that this class of data $b$ contains the\nvelocity function in the $3$ dimensional Navier-Stokes equations.\nAs a result we gave a different proof of the {\\it local}\nboundedness of velocity in $2$ dimensional case. Moreover assuming\na {\\it local}  bound in the pressure, we prove boundedness of\nvelocity in $3$ the dimensional case.\n\nThe first goal of the paper is to treat the end point case of the\nabove condition i.e. $m=1$. We will prove that weak solutions to\n(1.1) are locally bounded provided that $|b| [\\ln (1+ |b|)]^2$ is\nform bounded and $div b \\le 0$.\n\nMany authors have also studied the regularity property of the\nrelated heat equation $ \\Delta u + V u - u_t =0$. Here $V$ is s\nsingular potential. We refer the reader to the papers by Aizenman\nand Simon [AS], Simon [S] and the reference therein. The function\n$V$ is allowed in the Kato class which is a little more singular\nthan the corresponding $L^p$ class. It remains a  challenging\nproblem is to push this theory to broader class of functions.\n\n\nIn this paper we use the following definition of weak\n solutions.\n\n\n{\\it {\\bf Definition 1.1} Let $D \\subseteq {\\bf R}^n$ be a domain\nand $T \\in (0, \\infty]$. A function $u$ such that $u, |\\nabla u|\n\\in L^2_{loc}(D \\times [0, T])$ is a weak solution to (1.1) if:\nfor any $\\phi \\in C^{\\infty}_0(D \\times (-T, T))$, there holds\n\\[\n\\int^T_0\\int_D ( u \\partial_t \\phi - \\nabla u \\nabla \\phi)\n dxdt - \\int^T_0\\int_D b \\nabla u \\  \\phi \\  dxdt\n = - \\int_D u_0(x) \\phi(x, 0) dx.\n\\]}\n\n\n\n\n\n\\begin{theorem}\n\\label{th:1.1} Suppose $div b \\le 0$ in the weak sense, $b \\in\nL^2_{loc}$,  and that $|b| [\\ln (1 + |b|)]^2$ is form bounded i.e.\nthere exists $k>0$ such that for any $\\phi \\in C^\\infty({\\bf R}^n\n\\times (0, \\infty) )$ with compact support in the spatial\ndirection,\n\\[\n\\int \\int_{{\\bf R}^n} |b| [\\ln (1 + |b|)]^2 \\phi^2 dxdt \\le k \\int\n\\int_{{\\bf R}^n} |\\nabla \\phi|^2 dxdt. \\leqno(1.3)\n\\]\nThen weak solutions to equation (1.1) are locally bounded.\n\\end{theorem}\n\\medskip\n\n{\\it Remark 1.1.} In the special case that $b$ is independent of\ntime and  $b \\in L^p({\\bf R}^n)$ with $p>n\/2$, then it is easy to\ncheck that (1.3) is satisfied. Recall that the standard theory\nessentially only allows functions in $L^p$ with $p\n> n$. The strength of the theorem comes from the fact that weak\nsolutions are locally bounded in any domain regardless of its\nvalue on the parabolic boundary. If the domain is ${\\bf R}^n\n\\times (0, \\infty)$ or if the initial Dirichlet boundary condition\nis imposed, then using a Nash type estimate, one can show that\nsolutions are locally bounded when $t>0$ as long as the\nfundamental solution is well defined. In this case one can choose\n$b$ to be as singular as any $L^2$ functions. Actually the\npresence of $b$ is totally irrelevant except for the purpose of\nmaking the integrals in the definition of a weak solution finite.\nHere is a sketch of the proof. Let u solves\n\\[\n\\begin{cases}\n \\Delta u - b \\nabla u - u_t =0,  \\qquad \\text{in} \\\n \\text D \\times (0, \\infty),\\\\\nu(x, t) = 0, \\qquad (x, t) \\in \\partial D \\times (0, \\infty)\\\\\n  u(x, 0) = u_0(x).\n\\end{cases}\n  \\]\nLet $G(x, t; y, t)$ be the fundamental solution with initial\nDirichlet boundary condition. If $div b=0$, differentiating in\ntime shows that $\\int_D G(x, t; y, 0) dy \\le 1$. By Nash\ninequality, one has\n\\[\n\\frac{d}{dt} \\int_D G(x, t; y, t)^2 dy  = - 2 \\int_D |\\nabla G(x,\nt; y, t)|^2 dy  \\le - c \\frac{ \\big{(} \\int_D G(x, t; y, t)^2 dy\n\\big{)}^{1+(2\/n)}}{\\big{(} \\int_D G(x, t; y, 0) dy \\big{)}^{4\/n}}.\n\\]\nHence\n\\[\nG(x, t; y, 0) \\le c\/t^{n\/2}.\n\\]Therefore\n\\[\nu(x, t) = \\int_D G(x, t; y) u_0(y) dy\n\\] is bounded as soon as $t>0$\nand $u_0$ is in $L^1(D)$.\n\n However this does not imply local\nboundedness of weak solutions unconditionally. It would be\ninteresting to establish existence and more regularity result for\n(1.1) with the singular data in Theorem 1.1.\n\nWe should mention that in the time independent elliptic case,\nthere was already strong indication that standard regularity\ntheory can be improved in the presence of divergence free data. In\nthe important papers [FR1-2], local boundedness of Green's\nfunction (away from the singularity) of the operator $ -\\Delta  -\nb \\nabla $ with Dirichlet boundary conditions was proved, under\nthe conditions:  $n=5$, $|\\nabla^2 b| \\in L^{4\/3}$, $b=0$ on the\nboundary and $div b =0$ ( c.f. Lemma 1.48 and Lemma1.11 [FR1]).\nThe upshot of the result is that the bounds on the Green's\nfunction is independent of the norm $|\\nabla^2 b| \\in L^{4\/3}$.\nThe proof uses essentially the fact that the Green's function\nvanishes on the boundary. So the drift term is integrated out. In\ncontrast we do not have the benefit of a zero boundary.\n\n\n\n\nIn the three dimension case, we derive a further regularity\nresult:\n\n\\begin{corollary}\n Assume $|b| \\in L^{\\infty}([0, T], L^2({\\bf R}^3))\n \\cap L^q(\n {\\bf R}^3 \\times [0, T])$ with $q>3$ and $div b =0$. Suppose $u$ is a weak solution of (1.1) in ${\\bf R}^3\n\\times [0, T]$ with $\\int_{{\\bf R}^3} u^2(x, 0) dx < \\infty$. Then\n$u$ is locally bounded and for almost every $t$, $u(\\cdot, t)$ are\nH\\\"older continuous. \\proof\n\\end{corollary}\n\nAccording to Corollary 1 in [Z2], $b $ satisfies the conditions of\nTheorem 1.1. For completeness we provide the proof here.\n\n Let us take $m=4\/3$ and\n$p=2\/m=3\/2$. Then, by H\\\"older's inequality,\n\\[\n\\aligned\n \\int^T_0 \\int_{{\\bf R}^3}& |b |^{4\/3} \\phi^2 dxdt \\le \\int^T_0\n\\bigg{(} \\int_{{\\bf R}^3} |b |^{mp} dx \\bigg{)}^{1\/p}  \\ \\bigg{(}\n\\int_{{\\bf R}^3}\n\\phi^{2p\/(p-1)} dx \\bigg{)}^{(p-1)\/p} dt\\\\\n&=\\int^T_0 \\bigg{(} \\int_{{\\bf R}^3} |b |^2 dx \\bigg{)}^{2\/3} \\\n\\bigg{(} \\int_{{\\bf R}^3}\n\\phi^6 dx \\bigg{)}^{1\/3} dt\\\\\n&\\le \\sup_{t \\in [0, T]} \\bigg{(} \\int_{{\\bf R}^3} |b |^2(x, t) dx\n\\bigg{)}^{2\/3}\n\\ \\int^T_0 \\bigg{(} \\int_{{\\bf R}^3} \\phi^6 dx \\bigg{)}^{1\/3} dt\\\\\n&\\le C \\sup_{t \\in [0, T]} \\bigg{(} \\int_{{\\bf R}^3} |b |^2(x, t)\ndx \\bigg{)}^{2\/3} \\ \\int^T_0 \\int_{{\\bf R}^3} |\\nabla \\phi|^2 dx\ndt.\n\\endaligned\n\\]The last step is by Sobolev imbedding. This shows that condition\n(1.3) holds.\n\nHence Theorem 1.1 implies that $u$ is locally bounded.  Notice the\nfact that\n\\[\n\\int_{{\\bf R}^3} u^2(x, t) dx\n\\]is non-increasing in time since $b$ is divergence free.\nBy this and Theorem 1.1, we know that $u \\in L^{\\infty}({\\bf R}^3\n\\times [t_0, T])$ for any $t_0>0$.\n\nDenote by $G_0$ the Gaussian heat kernel of the heat equation.\nThen, for $t>t_0$,\n\\[\nu(x, t) = \\int_{{\\bf R}^3} G_0(x, t; y, t_0) u(y, t_0) dy -\n\\int^t_{t_0} \\int_{{\\bf R}^3} G_0(x, t; y, s) b  \\nabla u(y, s)\ndyds.\n\\]Since $b$ is divergence free, we have\n\\[\nu(x, t) = \\int_{{\\bf R}^3} G_0(x, t; y, t_0) u(y, t_0) dy +\n\\int^t_{t_0} \\int_{{\\bf R}^3} \\nabla_y G_0(x, t; y, s) b u(y, s)\ndyds.\n\\]Therefore, in the weak sense,\n\\[\n\\aligned\n \\nabla_x u(x, t) &= \\int_{{\\bf R}^3} \\nabla_x G_0(x, t; y,\nt_0) u(y, t_0) dy + \\int^t_{t_0} \\int_{{\\bf R}^3} \\nabla_x\n\\nabla_y G_0(x, t; y, s) b u(y, s) dyds\\\\\n& \\equiv I_1(x, t) + I_2(x, t).\n\\endaligned\n\\]It is well known that\n\\[\n\\nabla_x \\nabla_y G_0(x, t; y, t_0) =-\\nabla_x \\nabla_x G_0(x, t;\ny, t_0)\n\\]is a parabolic Calderon-Zygmond kernel (see [Lie] e.g.).\nHence by our assumption on $b$ and the fact that $u$ is bounded in\n${\\bf R}^3 \\times [t_0, T]$, the second term $I_2$  in the last\nintegral is in $L^q({\\bf R}^3 \\times [t_0, T])$, $q>3$. It follows\nthat\n\\[\n|I_2(\\cdot, t) | \\in L^q({\\bf R}^3), \\qquad q>3.\n\\]for a.e. $t$. Sobolev imbedding theorem then shows that\n $u (\\cdot, t)$ is\nH\\\"older continuous for a.e. $t$. \\qed\n\n\n\\medskip\n\n Next we turn to equation (1.2), which is the first step in\ntackling the full Navier-Stokes equations. When $b=0$, (1.2) is\njust the Stokes equations which has been studied for long time.\nOur focus is on how to allow $b$ as singular as possible while\nretaining the boundedness of weak solutions. As far as equation\n(1.2) is concerned, our result does not improve the standard\ntheory as dramatically as for equation (1.1). We have to restrict\nthe data $b$ in a suitable Kato class for (1.2). Nevertheless,\nTheorem 1.2 still generalizes the key part of the important work\n[AS], [CrZ] on the elliptic equations to the case of linearized\nNavier-Stokes system. Moreover, we even obtain gradient estimates\nfor solutions of (1.2) while only continuity was expected.\n\nAs pointed out in the papers [AS] and [Si], Kato class functions\nare quite natural objects in studying elliptic and parabolic\nequations with singular lower order terms. Roughly speaking, a\nfunction is in a Kato class with respect to an equation if its\nconvolutions with certain kernel functions are small in some\nsense. The kernel function usually is related to the fundamental\nsolution of the principal term of the equation. For instance, for\nthe equation $\\Delta u(x) + V(x) u(x) =0$ in ${\\bf R}^n$, $n \\ge\n3$, the function $V$ is in Kato class if\n\\[\n\\lim_{r \\to 0} \\sup_{x} \\int_{B(x, r)} \\frac{|V(y)|}{|x-y|^{n-2}}\ndy = 0. \\leqno(1.4)\n\\]In [AS], it is proven that weak solutions to $\\Delta u + V u=0$\nare continuous and satisfy a Harnack inequality when $V$ is in the\nabove Kato class. Numerous papers have been written on this\nsubject in the last thirty years, mainly in the context of\nelliptic and heat equations.\n\nIn the context of Navier-Stokes equations, the corresponding time\ndependent Kato class was defined recently in [Z3], which mirrors\nthose for the heat equation [Z]. Normally, with data in the Kato\nclass, weak solutions of elliptic equations are just continuous as\nproven in [AS], [CrZ]. It was proved  in [Z3] that weak solutions\nto (1.2) are bounded when $b$ is in the Kato class. Here we prove\nthat the {\\it spatial gradient} of solutions to (1.2) are bounded\nprovided that $b$ is in the Kato class locally. Let us mention\nthat one can use the idea of Kato class to recover some (but not\nall) the decay estimate in the interesting papers [Scho1-2] and to\nprove some pointwise decay estimate (see [Z3]).\n\nIn order to make our statement precise, we introduce a number of\nnotations. Throughout the paper, we write\n\\[\nK_1(x, t; y, s) = \\begin{cases} \\frac{1}{[ \\ |x-y| + \\sqrt{t-s} \\\n]^{n+1}}, \\quad t \\ge  s, x \\neq y\\\\\n0, s<t.\n\\end{cases}\n\\leqno (1.5)\n\\]We write $Q_r(x, t) = B(x, r) \\times [t-r^2, t]$.\n\n\\medskip\n\n{\\it {\\bf Definition 1.2}. A vector valued function $b=b(x, t) \\in\nL^1_{loc}({\\bf R}^{n+1})$ is in class $K_1$ if it satisfies the\nfollowing  condition:\n\\[\n\\lim_{h \\to 0 } \\sup_{(x, t) \\in {\\bf R}^{n+1}} \\int^{t}_{t-h}\n\\int_{{\\bf R}^n} [K_1(x, t; y, s) + K_1(x, s; y, t-h) ] |b(y, s)|\ndyds =0, \\leqno (1.6)\n\\]}\n\\medskip\n\nFor clarity of presentation, given $t>l$,  we introduce the\nquantity\n\\[\nB(b, l, t) = \\sup_{x} \\int^{t}_l \\int_{{\\bf R}^n} [K_1(x, t; y, s) +\nK_1(x, s; y, l) ] |b(y, s)| dyds. \\leqno(1.6')\n\\]\n\nBy the example in Remark 1.2 in [Z3], we see that the function\nclass $K_1$ permits solutions which  are very singular. In case\nthe spatial dimension is $3$,  a function in this class can have\nan apparent singularity of certain type that is not $L^p_{loc}$\nfor any $p>1$ and of dimension $1$. One can also construct time\ndependent functions in $K_1$ with quite singular behaviors. The\nclass $K_1$ also contains the space $L^{p,q}$ with $n\/p + 2\/q<1$,\nwhich sometimes is referred to as the Prodi-Serrin class. For the\nnonlinear Navier-Stokes equation, if a weak solution is known to\nbe in this class, then it is actually smooth. As for the\nlinearized equation (1.2), following the argument in [Ser], it is\nclear that weak solutions are bounded if $b$ is in the above\n$L^{p, q}$ class. Now we are able to prove that the spatial\ngradient of weak solutions are bounded {\\it automatically},\nwithout resorting to the nonlinear structure. Using H\\\"older's\ninequality, one can see that the class $K_1$ also contains the\nMorrey type space introduced in [O] by O' Leary, where boundedness\nof weak solutions in Morrey space are proven.\n\nOne can also define a slightly bigger Kato class by requiring the\nlimit in (1.6) to be a small positive number rather than $0$. We\nwill not seek such generality this time. The appearance of two\nkernel functions is due to the asymmetry of the equation in time\ndirection.\n\n\n\\medskip\n\nLet $D$ for a domain in ${\\bf R}^3$ and $T>0$. Following standard\npractice, we will use this definition for solutions of (1.2)\nthroughout the paper.\n\n\\medskip\n\n\\noindent{\\it  {\\bf Definition 1.3.} A divergence free vector\nfield $u \\in L^{\\infty}(0, T; L^2(D)) \\cap L^2(0, T; W^{1, 2}(D))$\nis called a\n (weak) solution of (1.2) if:\n\nfor any vector valued $\\phi \\in C^{\\infty}(D \\times [0, T])$ with\n$ div \\phi =0$ and $\\phi=0$ on $\\partial D \\times [0, T]$, $u$\nsatisfies\n\\[\n\\int^{t_2}_{t_1}\\int_{{\\bf R}^n} <u, \\partial_t \\phi + \\Delta\n\\phi>\n dxdt - \\int^{t_2}_{t_1}\\int_{{\\bf R}^n} <b \\nabla u,  \\phi>  dxdt\n = - \\int_{{\\bf R}^n} <u(x, t), \\phi(x, t)> |^{t_2}_{t_1} dx.\n\\]}\n\n\n\nNext we state the theorem on equation (1.2), the linearized\nNavier-Stokes equations.\n\n\n\\bigskip\n\n\\begin{theorem}\n\\label{th:1.2}\n\nLet $u$ be a solution of (1.2) in a domain $\\Omega \\subset {\\bf\nR}^3 \\times {\\bf R}$.  Suppose $Q_{4r}(x, t) \\subset \\Omega$, $div\nb =0$ and that $b |_{Q_{2r}(x, t)}$ is in class $K_1$ and $b \\in\nL^2_{loc}$. Then both $u$ and $|\\nabla u|$ are bounded functions\nin $Q_{2r}(x, t)$.\n\nMoreover, for some positive constants $C=C(b)$ and $r_0$,\ndepending on the size of the Kato norm of $b$,   there hold, when\n$0<r<r_0$,\n\\[\n\\aligned\n |u(x, t)| &\\le  \\frac{C}{r^5} \\int_{Q_{2r}(x, t)} |u(y,\ns)| dyds,\\\\\n|\\nabla u(x, t)| &\\le \\frac{C}{r^5} \\int_{Q_{2r}(x, t)} |\\nabla\nu(y, s)| dyds + \\frac{C}{r^6} \\int_{Q_{2r}(x, t)} |u(y,\ns)-\\overline{u}_{Q_{2r}}| dyds.\n\\endaligned\n\\leqno(1.7)\n\\]Here $\\overline{u}_{Q_{2r}}$ is the average of $u$ in $Q_{2r}(x, t)$.\n\nIf in addition that $u(\\cdot, t) \\in W^{1, 2}_0(B(x, 2r))$ for\na.e. $t$, then\n\\[\n|\\nabla u(x, t)| \\le \\frac{C}{r^5} \\int_{Q_{2r}(x, t)} |\\nabla\nu(y, s)| dyds. \\leqno(1.8)\n\\]\n\\end{theorem}\n\n\\medskip\n\n{\\it Remark 1.2.} One may think that the last term on the right\nhand side of the gradient estimate (1.7) is too singular to be\nnatural, especially when $r \\to 0$. However, both (1.7) and\nequation (1.2) are actually scaling invariant under the scaling:\n$u_\\lambda = u(\\lambda x)$, $b_\\lambda = \\lambda b(\\lambda x)$. So\n(1.7) is in the right setting. Moreover, the gradient estimate in\n(1.7) immediately simplifies to (1.8) when $u$ vanishes on the\nlateral boundary or when $u$ enjoys a suitable extension property\nin $W^{1, 2}$ space.\n\n\n\n{\\it Remark 1.3.} The reader may wonder whether any extra\nregularity in the time direction is possible. The answer is no as\nindicated in the example in [Ser], restated in [O]. Let $\\phi$ be\na harmonic function in ${\\bf R}^3$ and $a=a(t)$ be an integrable\nfunction. Then $u= a \\nabla \\phi$ is a weak solution of the\nNavier-Stokes equation. Obviously in the time direction $u$ is no\nmore regular than $a$.\n\n\\medskip\n\n\n\nTheorem 1.1 and 1.2 will be proven in section 3 and 4\nrespectively. The idea for the proof of Theorem 1.2 is to combine\na recent localization argument in [O] with a refined iteration.\nUsing the idea in the proof of Theorem 1.1,  in Section 4  we\nintroduce a sufficient condition on the velocity that implies\nboundedness of weak solutions of 3 dimensional Navier-Stokes\nequations. The main improvement is that no absolute value of the\nvelocity is involved.\n\n\\section{Proof of Theorem 1.1}\n\n\nSince the drift term $b$ in (1.1) can be much more singular than\nthose allowed by the standard theory, the existence and uniqueness\nof  weak solutions of (1.1) can not be taken for granted. In order\nto proceed first we need  some approximation results whose proof\ncan be found in [Z2], with only small modifications.\n\n\\begin{proposition}\nSuppose $u$ is a weak solution of  equation (1.1) in the cube $Q =\nD \\times [0, T]$, where $b$ satisfies the condition in Theorem\n1.1, part (a). Here $D$ is a domain in ${\\bf R}^n$.  Then $u$ is\nthe $L^1_{loc}$ limit of functions $\\{ u_k \\}$. Here $\\{ u_k \\}$\nis a weak solution of (1.1) in which $b$ is replaced by smooth\n$b_k$  such that $div b_k \\le 0$ and  $b_k \\to b$ strongly in\n$L^2(Q)$, $k \\to \\infty$. \\proof\n\\end{proposition}\n\nThe proof is almost identical to that of Proposition 2.4 in [Z2].\nThe only difference is that we are assuming $div b \\le 0$ instead\nof $div b =0$. Let $G=G(x,t; y, s)$ be the fundamental solution of\n(1.1) with $div b \\le 0$ and $b$ smooth. Then it is easy to show\nby differentiation that\n\\[\n\\int_{{\\bf R}^n} G(x, t; y, s) dx \\le 1.\n\\]The rest of the proof is identical to that of Proposition 2.4\n[Z2]. \\qed\n\n\n\n\n\n\n\n\\medskip\n\nNow we are ready to give a\n\n\n{\\bf Proof of Theorem 1.1.}\n\n\\medskip\n\n{\\bf Step 1.} $L^2$ gradient estimate. \\medskip\n\nBy the above approximation result, we can and do assume that the\nvector $b$ is smooth and we will differentiate freely as we wish.\n\nIn this section, we actually will prove the following local \"mean\nvalue\" property.\n\nLet $u$ be a  nonnegative solution of (1.1) in the parabolic cube\n$Q_{\\sigma_0 r} =B(x, \\sigma_0 r) \\times [t-(\\sigma_0 r)^2, t]$.\nHere $x \\in {\\bf R}^n$,\n  $r>0$, $t>0$ and $\\sigma_0$ is a suitable number greater than $1$.\n  Suppose $b$ satisfies Condition A in\n  $Q_{\\sigma_0 r}$. Then there exist $C=C(r, b)>0$ such that\n\\[\n\\sup_{Q_r} u^2 \\le C(r, b) \\frac{1}{|Q_{\\sigma_0 r}|} \\int_{Q_{\n\\sigma_0 r}} u^2 dyds.\n\\]\n\n\n\nPick a solution $u$ of (1.1) in the parabolic cube $Q_{\\sigma r}\n=B(x, \\sigma r) \\times [t-(\\sigma r)^2, t]$, where $x \\in {\\bf\nR}^n$, $\\sigma>1$,  $r>0$ and $t>0$. By direct computation, for\nany rational number $p \\ge 1$, which can be written as quotient of\ntwo integers with the denominator being odd, one has\n\\[\n\\Delta u^p - b \\nabla u^p  - \\partial_t u^p = p(p-1) |\\nabla u|^2\nu^{p-2}. \\leqno (2.1)\n\\]Here the condition on $p$ is to ensure that $u^p$ makes sense\nwhen $u$ changes sign. Actually $u^2$ is a sub-solution to (1.1).\nHence one can also assume that $u$ is a non-negative sub-solution to\n(1.1) by working with $u^2$.\n\n\n\n\nChoose $\\psi=\\phi(y) \\eta(s)$ to be a refined cut-off function\nsatisfying\n\\[\nsupp \\  \\eta \\subset [t- (\\sigma r)^2, t]; \\quad \\eta(s)=1, \\quad\ns \\in [t- r^2, t]; \\quad |\\eta'| \\le 2\/( (\\sigma-1) r)^2; \\quad  0\n\\le \\eta \\le 1;\n\\]\n\\[\nsupp \\  \\phi \\subset B(x, \\sigma r); \\quad \\phi(y)=1, \\quad y \\in\nB(x, r);  \\quad 0 \\le \\phi \\le 1;\n\\]\n\\[\n\\frac{|\\nabla \\phi|}{\\phi} \\le  \\frac{A}{(\\sigma-1) r} |\\ln \\phi\n|^{3\/2}, \\qquad A>0.\n\\]\nBy modifying the following function\n\\[\n\\exp \\big{(} -\\frac{ \\sigma^2}{\\sigma^2 - |x-y|^2} \\big{)}^k\n\\]\nand scaling, it is easy to show that such a function $\\phi$\nexists. Here $k$ is a sufficiently large number.\n\n\n\nDenoting $w=u^p$ and using $w \\psi^2$ as a test function on (2.1),\none obtains\n\\[\n\\int_{Q_{\\sigma r}} (\\Delta w - b \\nabla w  -\n\\partial_s w) w \\psi^2 dyds  = p(p-1) \\int_{Q_{\\sigma r}}\n|\\nabla u|^2 w^2 u^{-2} \\ge 0.\n\\]Using integration by parts, one deduces\n\\[\n\\int_{Q_{\\sigma r}} \\nabla(w \\psi^2) \\nabla w dyds \\le -\n\\int_{Q_{\\sigma r}} b \\nabla w (w \\psi^2) dyds  - \\int_{Q_{\\sigma\nr}} (\\partial_s w) w \\psi^2 dyds. \\leqno (2.2)\n\\]By direct calculation,\n\\[\n\\aligned &\\int_{Q_{\\sigma r}} \\nabla(w \\psi^2)  \\nabla w dyds =\n\\int_{Q_{\\sigma r}} \\nabla [(w \\psi )  \\psi] \\nabla w dyds\\\\\n&=\\int_{Q_{\\sigma r}} [ \\ \\nabla (w \\psi ) ( \\ \\nabla (w \\psi ) -\n(\\nabla \\psi) w ) + w \\psi \\nabla \\psi \\nabla w ] dyds\\\\\n&=\\int_{Q_{\\sigma r}} [\\  |\\nabla (w \\psi )|^2 -|\\nabla \\psi |^2\nw^2 \\ ]dyds.\n\\endaligned\n\\]Substituting this to (2.2), we obtain\n\\[\n\\aligned\n\\int_{Q_{\\sigma r}} &|\\nabla (w \\psi )|^2 dyds \\\\\n&\\le - \\int_{Q_{\\sigma r}} b \\nabla w (w \\psi^2) dyds  -\n\\int_{Q_{\\sigma r}} (\\partial_s w) w \\psi^2 dyds + \\int_{Q_{\\sigma\nr}} |\\nabla \\psi |^2 w^2 dyds.\n\\endaligned \\leqno (2.3)\n\\]\n\nNext, notice that\n\\[\n\\aligned \\int_{Q_{\\sigma r}}& (\\partial_s w) w \\psi^2 dyds =\n\\frac{1}{2}\n\\int_{Q_{\\sigma r}} (\\partial_s w^2) \\psi^2 dyds\\\\\n&=-\\int_{Q_{\\sigma r}} w^2 \\phi^2 \\eta \\partial_s \\eta dyds +\n\\frac{1}{2} \\int_{B(x, \\sigma r)} w^2(y, t) \\phi^2(y) dy.\n\\endaligned\n\\]Combining this with (2.3), we see that\n\\[\n\\aligned & \\int_{Q_{\\sigma r}} |\\nabla (w \\psi )|^2 dyds +\n\\frac{1}{2} \\int_{B(x, \\sigma r)} w^2(y, t) \\phi^2(y) dy \\\\\n&\\le \\int_{Q_{\\sigma r}} (|\\nabla \\psi |^2 + \\eta \\partial_s \\eta)\n\\ w^2 dyds -\n\\int_{Q_{\\sigma r}} b (\\nabla w) (w \\psi^2) dyds  \\\\\n&\\equiv T_1+T_2.\n\\endaligned\n\\leqno(2.4)\n\\]The first term on the righthand side of (2.4) is\nalready in good shape. So let us estimate the second term as\nfollows.\n\\[\n\\aligned\nT_2 &= - \\int_{Q_{\\sigma r}} b (\\nabla w) (w \\psi^2) dyds\\\\\n&= - \\frac{1}{2} \\int_{Q_{\\sigma r}} b \\psi^2 \\nabla w^2 dyds =\n\\frac{1}{2} \\int_{Q_{\\sigma r}} div (b \\psi^2)  w^2 dyds\\\\\n&=\\frac{1}{2} \\int_{Q_{\\sigma r}} div b (\\psi  w) ^2 dyds +\n\\frac{1}{2}\n\\int_{Q_{\\sigma r}} b \\nabla (\\psi^2) w^2 dyds\\\\\n&=\\frac{1}{2} \\int_{Q_{\\sigma r}} div b (\\psi  w) ^2 dyds +\n\\int_{Q_{\\sigma r}} b (\\nabla \\psi) \\psi w^2 dyds\\\\\n&\\le   \\int_{Q_{\\sigma r}} b (\\nabla \\psi) \\psi w^2 dyds.\n\\endaligned\n\\]Here we just used the assumption that $div b \\le 0$.\n\n\n\n\n\nThe next paragraph contains a key argument of the paper.\n\nLet  $D>0$ be a number to be chosen later\n\\[\n\\aligned T_2 &\\le  \\int_{Q_{\\sigma r}} |b| \\  |\\nabla \\psi| \\psi\nw^2\ndyds  \\\\\n&\\le \\int_{|b| \\ge D} |b| \\  |\\nabla \\psi| \\psi w^2 dyds\n+\\int_{|b| \\le D} |b| \\  |\\nabla \\psi| \\psi w^2 dyds\n\\\\\n&\\le \\int_{|b| \\ge D} |b| \\  |\\nabla \\psi| \\psi w^2 dyds + \\frac{C\nD }{(\\sigma - 1) r} \\int_{Q_{\\sigma r}}  w^2 dyds.\n\\endaligned\n\\]Using\nthe property that $\\psi = \\phi \\eta$ and\n\\[\n|\\nabla \\phi| \\le  \\frac{A}{(\\sigma -1) r} \\phi | \\ln \\phi\n|^{3\/2},\n\\]\nwe have\n\\[\n\\aligned T_2& \\le \\int \\int_{|b| \\ge 1\/\\phi, |b| \\ge D} |b|\n|\\nabla \\phi| \\phi w^2  dy \\eta^2 ds  + \\int \\int_{|b|\n\\le 1\/\\phi}  |b| |\\nabla \\phi| \\phi w^2 dy \\eta^2 ds\\\\\n&\\qquad \\qquad \\qquad+\n\\frac{C D }{(\\sigma - 1) r} \\int_{Q_{\\sigma r}}  w^2 dyds\\\\\n&\\le \\frac{A}{(\\sigma -1) r} \\int \\int_{|b| \\ge 1\/\\phi, |b| \\ge D}\n|b| \\ |\\ln \\phi|^{3\/2} (\\phi w)^2 dy \\eta^2 ds \\\\\n&\\qquad \\qquad + \\frac{A}{(\\sigma -1) r} \\int \\int_{B(x, \\sigma\nr), |b| \\le 1\/\\phi} |b| \\ |\\ln \\phi|^{3\/2} (\\phi w)^2  dy \\eta^2\nds\n\\\\\n&\\qquad \\qquad \\qquad+\n\\frac{C D }{(\\sigma - 1) r} \\int_{Q_{\\sigma r}}  w^2 dyds\\\\\n&\\le \\frac{A}{(\\sigma -1) r} \\int_{|b| \\ge D} |b| (\\ln |b|)^{3\/2}\n(\\psi w)^2 dyds  + \\frac{A}{(\\sigma -1) r} \\int_{Q_{\\sigma r}}\n\\frac{1}{\\phi} |\\ln \\phi|^{3\/2} (\\phi\nw)^2 \\eta^2 dyds\\\\\n&\\qquad \\qquad \\qquad +\\frac{C D }{(\\sigma - 1) r} \\int_{Q_{\\sigma\nr}}  w^2 dyds\\\\\n&\\le \\frac{A}{(\\sigma -1) r (\\ln D)^{1\/2}} \\int_{Q_{\\sigma r}} |b|\n(\\ln |b|)^2 (\\psi w)^2 dyds  + \\frac{A}{(\\sigma -1) r}\n\\int_{Q_{\\sigma r}} \\phi |\\ln \\phi|^{3\/2}\nw^2 \\eta^2 dyds\\\\\n&\\qquad \\qquad \\qquad +\\frac{C D }{(\\sigma - 1) r} \\int_{Q_{\\sigma\nr}}  w^2 dyds.\\\\\n\\endaligned\n\\]\n\n\nBy our assumptions on $b$,\n\\[\n\\int_{Q_{\\sigma r}} |b| (\\ln |b|)^2 (\\psi  w)^2 dyds \\le k\n\\int_{Q_{\\sigma r}} |\\nabla (\\psi  w)|^2 dyds,\n\\]and the fact that $\\phi |\\ln \\phi|^{3\/2}$ is a bounded function, we\ndeduce\n\\[\nT_2 \\le \\frac{A}{(\\sigma -1) r (\\ln D)^{1\/2}} \\int_{Q_{\\sigma r}}\n|\\nabla (\\psi w)|^2 dyds + \\frac{C_1 D }{(\\sigma - 1) r}\n\\int_{Q_{\\sigma r}} w^2 dyds.\n\\]Now we choose $D$ so that $\\frac{A}{(\\sigma -1) r (\\ln D)^{1\/2}}\n= \\frac{1}{2}$, i.e. $D= e^{( 2A\/ [(\\sigma-1) r])^2}$, then\n\\[\nT_2 \\le  \\frac{1}{2} \\int_{Q_{\\sigma r}} |\\nabla (\\psi  w)|^2 dyds\n+ c_0 e^{( c_1\/ [(\\sigma-1) r])^2} \\int_{Q_{\\sigma r}}  w^2 dyds.\n\\leqno(2.6)\n\\]Here $c_0$ and $c_1$ are positive constants independent of $r$\nand $\\sigma$.\n\n\n\nCombining  (2.4) with (2.6), we reach\n\\[\n\\int_{Q_{\\sigma r}} |\\nabla (w \\psi )|^2 dyds +  \\int_{B(x, \\sigma\nr)} w^2(y, t) \\phi^2(y) dy  \\le c_0 e^{( c_1\/ [(\\sigma-1) r])^2}\n\\int_{Q_{\\sigma r}}  w^2 dyds. \\leqno(2.7)\n\\]\n\n\n\n\\medskip\n\n{\\bf step 2.} $L^2 - L^\\infty$ bounds.\n\\medskip\n\nBy modifying Moser's iteration moderately,  we deduce from (2.7)\nthe following $L^2-L^\\infty$ estimate.\n\\[\n\\sup_{Q_r} u^2 \\le C(r, b) \\frac{1}{|Q_{\\sigma_0 r}|}\n\\int_{Q_{\\sigma_0 r}} u^2 dyds. \\leqno(2.8)\n\\]Indeed, by H\\\"older's inequality,\n\\[\n\\aligned\n \\int_{{\\bf R}^n} &{(\\phi w)}^{2(1+(2\/n))} =\n \\int_{{\\bf R}^n} (\\phi w)^2 \\ (\\phi w)^{4\/n}\\\\\n &\\le \\big{(}\n\\int_{{\\bf R}^n} {(\\phi w)}^{2n\/(n-2)} \\big{)}^{(n-2)\/n}\n             \\big{(}\\int_{{\\bf R}^n} {(\\phi w)}^2 \\big{)}^{2\/n}.\n\\endaligned\n\\]Using  Sobolev inequality, one obtains\n\\[\n\\int_{{\\bf R}^n} {(\\phi w)}^{2(1+(2\/n))} \\le\n            C \\big{(}\\int {(\\phi w)}^2 \\big{)}^{2\/n}\n             \\big{(}\\int_{{\\bf R}^n}\n            |\\nabla (\\phi w)|^2 \\big{)}.\n\\]The last inequality, together with (2.7) implies, for some\n$C_1>0$,\n\\[\n\\int_{Q_{\\sigma' r }(x, t)}  u^{2p\\theta} \\le  \\big{(} c_0 e^{(\nc_1\/ [(\\sigma-\\sigma') r])^2} \\int_{Q_{\\sigma r}(x, t)}\nu^{2p}\\big{)}^{\\theta},\n\\]where $\\theta = 1+ (2\/n)$ and $\\sigma'<\\sigma$.\n\n\nTake a  number $\\rho>1$ so that $\\rho^2< \\theta$.\n  We  set\n$\\tau_i=\\rho^{-i}$, $\\sigma_0=1\/(1-\\rho^{-1})$,\n$\\sigma_i=\\sigma_{i-1}-\\tau_i= \\sigma_0-\\Sigma^{i}_1 \\tau_j$,\n$p=\\theta^i$, $i=1, 2, ...$. The above then yields, for some $c_2,\nc_3>0$,\n\\[\n\\int_{Q_{\\sigma_{i+1} r}(x, t)}  u^{2 \\theta^{i+1} } \\le c_3\n\\big{(} c^{i+1}_2 e^{ c^2_1 r^{-2} \\rho^{2 i} } \\int_{Q_{\\sigma_i\nr }(x, t)} u^{2 \\theta^i}\\big{)}^{\\theta}.\n\\]After iterations the above implies, for some $c_4 >0$,\n\\[\n\\big{(}\\int_{Q_{\\sigma_{i+1} r }(x, t)}\n        u^{2 \\theta^{i+1}} \\big{)}^{\\theta^{-i-1}}\n\\le \\exp (c_4 \\Sigma^i_1 j \\theta^{-j}) \\ \\exp (c_1 r^{-2}\n\\Sigma^i_1 \\rho^{2j} \\theta^{-j})\n    \\int_{Q_{\\sigma_0 r}(x, t)} u^2.\n\\]Observe that $\\rho^2\/\\theta<1$. Letting $i \\rightarrow \\infty$ and\nobserving that $\\sigma_i \\to 1$ as $i \\to \\infty$, we obtain\n\\[\n\\sup_{Q_r} u^2 \\le C(r, b) \\int_{Q_{\\sigma_0 r}} u^2.\n\\]\n This completes\nproof the theorem. \\qed\n\n\\medskip\n\n\n\n\n\\section{Proof of Theorem 1.2}\n\nFirst we will need a short lemma concerning kernel function $K_1$\ndefined in (1.5). It was proved, among others things, in [Z3]. We\ngive a proof for completeness.\n\n\\begin{lemma}\nThe following inequality holds for all $x, y, z \\in {\\bf R}^n$ and\n$t> \\tau >0$.\n\\[\n\\aligned K_1*b K_1 &\\equiv \\int^t_0\\int_{\\bf R^n} \\frac{1}{(\n|x-z|+\\sqrt{t-\\tau} )^{n+1}} \\frac{|b(z, \\tau)|}{(\n|z-y|+\\sqrt{\\tau} )^{n+1}} dzd\\tau \\\\\n&\\le C B(b, 0, t) K_1(x, t; y, 0). \\endaligned \\leqno(3.1)\n\\]Here, recalling from (1.6'),\n\\[\nB(b, 0, t) \\equiv \\sup_{x \\in {\\bf R}^n} \\int^{t}_0 \\int_{{\\bf\nR}^n} [K_1(x, t; y, s) + K_1(x, s; y, 0) ] |b(y, s)| dyds. \\leqno\n(3.2)\n\\]\n \\proof\n\\end{lemma}\n\nSince\n\\[\n|x-z| + \\sqrt{t-\\tau} + |y-z| + \\sqrt{\\tau} \\ge |x-y| + \\sqrt{t},\n\\]we have, either\n\\[\n|x-z| + \\sqrt{t-\\tau} \\ge \\frac{1}{2} (|x-y| + \\sqrt{t}),\n\\leqno(3.3)\n\\]or\n\\[\n|z-y| + \\sqrt{\\tau} \\ge \\frac{1}{2} (|x-y| + \\sqrt{t}).\n\\leqno(3.4)\n\\]\n\nSuppose (3.3) holds then\n\\[\nK_1*b K_1 \\le \\frac{ 2^n}{(|x-y| + \\sqrt{t})^{n+1}}\n\\int^t_0\\int_{\\bf R^n} \\frac{|b(z, \\tau)|}{(|z-y|+\\sqrt{\\tau}\n)^{n+1}} dzd\\tau.\n\\]That is\n\\[\nK_0*b K_1 \\le \\frac{ 2^n  B(b, 0, t)}{(|x-y| + \\sqrt{t})^{n+1}}.\n\\leqno (3.5)\n\\]\n\nSuppose (3.4) holds but (3.3) fails, then\n\\[\n|z-y| + \\sqrt{\\tau} \\ge \\frac{1}{2} (|x-y| + \\sqrt{t}) \\ge |x-z| +\n\\sqrt{t-\\tau}.\n\\]This shows\n\\[\n\\frac{1}{( |x-z|+\\sqrt{t-\\tau} )^{n+1} \\ ( |z-y|+\\sqrt{\\tau}\n)^{n+1}} \\le \\frac{2^n}{( |x-z|+\\sqrt{t-\\tau} )^{n+1} (\n|x-y|+\\sqrt{t} )^{n+1}}.\n\\]Substituting this to (3.1), we obtain\n\\[\nK_1*b K_1 \\le \\frac{ 2^n}{(|x-y| + \\sqrt{t})^{n+1}}\n\\int^t_0\\int_{\\bf R^n} \\frac{|b(z, \\tau)|}{(|x-z|+\\sqrt{t-\\tau}\n)^{n+1}} dzd\\tau.\n\\]That is\n\\[\nK_1*b K_1 \\le \\frac{ 2^n  B(b, 0, t)}{(|x-y| + \\sqrt{t})^n}.\n\\]\n\nClearly the only remaining case to consider is when both (3.3) and\n(3.4) holds. However this case is already covered by (3.5). Thus\n(3.1) is proven.  \\qed\n\n\\medskip\n\n\n\n\nNext we state and prove a representation formula for solutions of\n(1.2) and their spatial gradient, following and extending the idea\nin [O]. The formula for solutions is contained in [O]. However, we\nwill outline the proof since it is useful in the proof of the\nformula for the gradient, which is a new contribution of this\npaper.\n\n\n\n\\medskip\n\n{\\bf Remark 3.1.}  Let us note that at this moment, the\nrepresentation formula (3.6') below for the gradient is understood\nas a comparison of two $L^1_{loc}$ functions in space-time. This\nis legal for two reasons. First we assumed that $\\nabla u$ is a\n$L^2$ function a priori. Second, it is easy to check\n $K_1(\\cdot,\n\\cdot; y, s)$ is $L^1_{loc}$ and $b \\nabla u$ is $L^1_{loc}$ by\nthe assumption that $b$ is $L^2$. Therefore the last function on\nthe righthand side of (3.6') is a $L^1_{loc}$ function.\n\n\\medskip\n\n\\begin{lemma} (mean value inequality)\n\n(a). Let $u$ be a solution of (1.2) in the region $\\Omega$.\nSuppose $Q_{2r}(x, t) \\subset \\Omega$. Then there exists a\nconstant $\\lambda$ such that\n\\[\n\\aligned\n |u(x, t)| &\\le \\lambda \\frac{1}{r^5} \\int_{Q_r(x,\nt)-Q_{r\/2}(x, t)} |u(y, s)| dyds \\\\\n&\\qquad + \\lambda \\int_{Q_r(x, t)} K_1(x, t; y, s) |b(y, s)| \\\n|u(y, s)| dyds. \\endaligned\n \\leqno(3.6)\n\\]\n\n(b). Under the same assumption as (a), there exists a constant\n$\\lambda$ such that\n\\[\n\\aligned |\\nabla u(x, t)| &\\le  \\frac{\\lambda}{r^5} \\int_{Q_r(x,\nt)-Q_{r\/2}(x, t)} | \\nabla u(y, s)|  dyds\\\\\n&\\qquad  + \\frac{\\lambda}{r^6}\n\\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)| dyds\\\\\n&\\qquad  + \\lambda \\int_{Q_r(x, t)} K_1(x, t; y, s) |b(y, s)| \\\n|\\nabla u(y, s)| dyds. \\endaligned \\leqno(3.6')\n\\]\n\n\\proof of (a).\n\\end{lemma}\n\nLet $E=E(x, t; y, s)$ be the fundamental solution (matrix) of the\nStokes system in ${\\bf R}^3 \\times (0, \\infty)$ and $E_k$ be the\n$k$-th column of $E$. This function has been studied for a long\ntime. All of its basic properties we are using below can be found\nin [So] and [FJR].  Fixing $(x, t)$, we construct a standard\ncut-off function $\\eta$ such that $\\eta(y, s)=1$ in $Q_{r\/2}(x,\nt)$, $\\eta(y, s)=0$ outside of $Q_r(x, t)$, $0 \\le \\eta \\le 1$ and\n$|\\nabla \\eta|^2 + |\\Delta \\eta| + |\\partial_s \\eta| \\le c\/r^2$.\n\nDefine a vector valued function\n\\[\n\\Phi_k=\\Phi^{(x, t)}_k(y, s) = \\frac{1}{4 \\pi} curl \\big{(} \\eta\n(y, s) \\int_{{\\bf R}^3} \\frac{ curl E_k(x, t; z, s)}{|z-y|} dz\n\\big{)}. \\leqno(3.7)\n\\]It is clear that when $t>s$,  $\\Phi_k$ is a valid test function\nfor equation (1.2) since $\\Phi_k$ is smooth, compactly supported\nand divergence free. Using $\\Phi_k$ as a test function on (1.2),\nby Definition (1.2), we obtain\n\\[\n\\int^t_0\\int u(y, s) \\big{(} \\Delta \\Phi_k + b \\nabla \\Phi_k +\n\\partial_s \\Phi_k \\big{)} dyds = \\lim_{s \\to t} \\int u(y, s)\n\\Phi_k(y, s) dy.\n\\]Here and later,we will suppress the superscript $(x, t)$ on\n$\\Phi$, unless there is a confusion.\n\nSince $E_k$ is divergence free $curl \\ curl E_k = - \\Delta E_k$.\nThus\n\\[\n\\aligned \\Phi_k(y, s) &= \\eta(y, s) E_k(x, t; y, s) + \\frac{1}{4\n\\pi} \\nabla \\eta(y, s) \\times \\int_{{\\bf R}^3} \\frac{ curl E_k(x,\nt; z, s)}{|z-y|} dz\\\\\n&\\equiv \\eta E_k + \\overrightarrow{Z}.\n\\endaligned\n\\leqno(3.8)\n\\]\nUsing  the property of the fundamental matrix $E$ and the fact\nthat $\\overrightarrow{Z}$ is a lower order term, it is easy to see\nthat\n\\[\n\\lim_{s \\to t} \\int u(y, s) \\Phi_k(y, s) dy = \\eta (y, t) u_k(x,\nt) = u_k(x, t)\n\\]where $u_k$ is the k-th component of $u$. Hence\n\\[\n\\aligned u_k(x, t) &= \\int_{Q_r(x, t)} u [ \\Delta (\\eta E_k) +\n\\partial_s (\\eta\nE_k)]dyds  + \\int_{Q_r(x, t)} u [\\Delta \\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}] dyds\\\\\n&\\qquad \\qquad +   \\int_{Q_r(x, t)} u b \\nabla \\Phi_k dyds.\n\\endaligned\n\\]Here and later $u b \\nabla \\Phi_k = u \\cdot \\sum^3_{i=1} b_i\n\\partial_i \\Phi_k$.\n\nNote that $\\Delta E_k + \\partial_s E_k=0$ when $s<t$. The above\nimplies\n\\[\n\\aligned u_k&(x, t) \\\\\n&= \\int_{Q_r(x, t)} u(y, s) [ E_k(x, t; y, s) (\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y E_k(x, t; y, s)]dyds \\\\\n&\\qquad + \\int_{Q_r(x, t)} u(y, s) [\\Delta \\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}](y, s) dyds\\\\\n&\\qquad \\qquad+ \\int_{Q_r(x, t)} u b \\nabla_y \\Phi_k(y, s) dyds\\\\\n&\\equiv J_1+J_2+J_3.\n\\endaligned\n\\leqno(3.9)\n\\]\n\n\nWe are going to estimate $J_1, J_2, J_3$ separately. By well known\nestimates on $E$ (see [So] or [FJR] for example), for $(y, s) \\in\nQ_r(x, t)- Q_{r\/2}(x, t)$,\n\\[\n|E_k(x, t; y, s)| \\le \\frac{c}{(|x-y| + \\sqrt{t-s})^3} \\le\n\\frac{c}{r^3},\n\\]\n\\[\n|\\nabla_y E_k(x, t; y, s)| \\le \\frac{c}{(|x-y| + \\sqrt{t-s})^4}\n\\le \\frac{c}{r^4}.\n\\]Using these and the property of $\\eta$, we see that\n\\[\n|J_1| \\le \\frac{C}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)|\ndyds. \\leqno(3.10).\n\\]\n\nNext we give an estimate $J_2$, which follows verbatim from $[O]$\np626. Recall that\n\\[\nE_k(x, t; y, s) =  G(x, t; y, s) e_k + \\frac{1}{4 \\pi} \\nabla\n\\partial_k \\int_{{\\bf R}^3} \\frac{G(x, t; z, s)}{|y-z|} dz.\n\\]Here $G$ is the fundamental solution of the heat equation and\n$\\{ e_1, e_2, e_3 \\}$ is the standard orthonormal basis of ${\\bf\nR}^3$. Since $curl \\nabla =0$, from (3.8), $\\overrightarrow{Z}$\ndefined in (3.8) takes a very simple form\n\\[\n\\overrightarrow{Z}= \\frac{1}{ 4 \\pi} \\nabla_y \\eta(y, s) \\times\n\\bigg{(} \\nabla_y \\big{[}\\int_{{\\bf R}^3} \\frac{ G(x, t; z,\ns)}{|z-y|} dz \\big{]} \\times e_k \\bigg{)}.\n\\]Hence\n\\[\nJ_2 = \\frac{1}{ 4 \\pi} \\int_{Q_r(x, t)} u(y, s) [\\Delta  +\n\\partial_s ] \\bigg{[} \\nabla_y \\eta(y, s) \\times\n\\bigg{(} \\nabla_y \\big{[}\\int_{{\\bf R}^3} \\frac{ G(x, t; z,\ns)}{|z-y|} dz \\big{]} \\times e_k \\bigg{)} \\bigg{]} dyds.\n\\leqno(3.11)\n\\]From here direct computation, using the estimates ([So], Chapter\n2, Section 5),\n\\[\n|D^l_s D^m_y \\int_{{\\bf R}^3} \\frac{ G(x, t; z, s)}{|z-y|} dz| \\le\n\\frac{C_{m, l}}{(|x-y| + \\sqrt{t-s})^{1+m+2l}}, \\leqno(3.12)\n\\] and the fact that $|x-y| \\ge r\/2$ here, it is easy to show that\n\\[\n|J_2| \\le  \\frac{C}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)|\ndyds. \\leqno(3.13)\n\\]\nFinally direct  computation using (3.12) shows that\n\\[\n|J_3| \\le \\lambda \\int_{Q_r(x, t)} K_1(x, t; y, s) |b(y, s)| \\\n|u(y, s)| dyds. \\leqno(3.14)\n\\]Substituting (3.10), (3.13) and (3.14) to (3.9), we finish the\nproof of part (a) of the lemma.\n\\medskip\n\n{\\bf Proof of part (b).}\n\n\nOur next task is to prove the representation formula for $\\nabla\nu$. In (3.9) the cut-off function apparently depends on $(x, t)$.\nIn order to prove the gradient estimate, we need modify it a\nlittle. For $(w, l) \\in Q_{r\/4}(x, t)$, we take\n\\[\n\\Phi_k=\\Phi^{(w, l)}_k(y, s) = \\frac{1}{4 \\pi} curl \\big{(} \\eta\n(y, s) \\int_{{\\bf R}^3} \\frac{ curl E_k(w, l; z, s)}{|z-y|} dz\n\\big{)}.\n\\]as a text function for (1.2). Since $\\eta(w, l)=1$ in this case,\nfollowing the computation before (3.9), we obtain\n\\[\n\\aligned u_k&(w, l) \\\\\n&= \\int_{Q_r(x, t)} u(y, s) [ E_k(w, l; y, s) (\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y E_k(w, l; y, s)]dyds \\\\\n&\\qquad + \\int_{Q_r(x, t)} u(y, s) [\\Delta \\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}](y, s) dyds\\\\\n&\\qquad \\qquad + \\int_{Q_r(x, t)} u b \\nabla_y \\Phi^{(w, l)}_k(y,\ns) dyds.\n\\endaligned\n\\leqno(3.9')\n\\]Here $\\overrightarrow{Z}=\\overrightarrow{Z}(w, l; y, s)$ is defined\n as $\\overrightarrow{Z}$ in (3.8) except that $E_k(x, t; y, s)$ is\nreplaced by $E_k(w, l; y, s)$. i.e.\n\\[\n\\overrightarrow{Z} =\\frac{1}{4 \\pi} \\nabla \\eta(y, s) \\times\n\\int_{{\\bf R}^3} \\frac{ curl E_k(w, l; z, s)}{|z-y|} dz.\n\\]We would like to differentiate (3.9) in the spatial variables.\nHowever, since $\\nabla u$ is only known as a $L^2$ function, we\nhave to consider the weak derivatives.\n\nLet $\\rho = \\rho(w)$ be a smooth cut-off function supported in\n$B(x, r\/4)$. Then (3.9') implies, for $i=1, 2, 3$, and a.e. $l$,\n\\[\n\\aligned\n&\\int_{{\\bf R}^3} u_k(w, l) \\partial_{w_i} \\rho(w) dw  \\\\\n&= - \\int_{{\\bf R}^3} {\\bf \\bigg{(}}  \\int_{Q_r(x, t)} u(y, s)\n\\partial_{w_i}[ E_k(w, l; y, s)\n(\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y E_k(w, l; y, s)]dyds \\\\\n&\\qquad + \\int_{Q_r(x, t)} u(y, s) \\partial_{w_i}[\\Delta\n\\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}](y, s) dyds\\\\\n&\\qquad \\qquad +  \\int_{Q_r(x, t)} u b \\nabla_y\n\\partial_{w_i} \\Phi^{(w, l)}_k(y, s) dyds {\\bf \\bigg{ )}} \\rho(w) dw\\\\\n&\\equiv - \\int_{{\\bf R}^3} \\bigg{(} M_1 + M_2 + M_3 \\bigg{)}\n\\rho(w) dw.\n\\endaligned\n\\leqno(3.15)\n\\]Here we just used integration by parts which is legitimate since\nwe will show that $M_1, M_2$ are bounded functions and $M_3$ is\n$L^1_{loc}$. Note also we should have integrated in the time\ndirection as well since $\\nabla u$ is only known to $L^2$ in space\ntime. However, since all the estimates below are uniform for $l\n\\in [t-(r\/4)^2, t]$, our estimates are valid.\n\n\nLet us estimate $M_1$ first. Noting that $\\partial_{w_i} E_k(w, l;\ny, s) = -\\partial_{y_i} E_k(w, l; y, s)$, we deduce\n\\[\nM_1 = - \\int_{Q_r(x, t)} u(y, s) [ \\partial_{y_i} E_k(w, l; y, s)\n(\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y \\partial_{y_i} E_k(w, l; y, s)] dyds.\n\\]Using integration by parts, we obtain\n\\[\n\\aligned &M_1 =\\\\\n&  \\int_{Q_r(x, t)} \\partial_{y_i} u(y, s) [ E_k(w, l; y, s)\n(\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y  E_k(w, l; y, s)] dyds\\\\\n&+ \\int_{Q_r(x, t)} u(y, s) [  E_k(w, l; y, s)\n\\partial_{y_i} (\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 (\\partial_{y_i} \\nabla \\eta(y, s)) \\nabla_y  E_k(w, l; y, s)]\ndyds.\n\\endaligned\n\\]By standard properties of  $E$ and the bounds on $\\eta$ and\nits derivatives, we have\n\\[\n|M_1| \\le \\frac{c}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |\\nabla\nu(y, s)| dyds +\\frac{c}{r^6} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y,\ns)| dyds. \\leqno(3.16)\n\\]Here we also utilize the fact that the arguments $(w, l)$ and $(y, s)$\nhave a parabolic distance of at least $r\/4$\n\nNext we need to find an upper bound for $M_2$. Recall from the\nformula before (3.11) that\n\\[\n\\aligned \\overrightarrow{Z}&= \\frac{1}{ 4 \\pi} \\nabla_y \\eta(y, s)\n\\times \\bigg{(} \\nabla_y \\big{[}\\int_{{\\bf R}^3} \\frac{ G(w, l; z,\ns)}{|z-y|} dz\n\\big{]} \\times e_k \\bigg{)}\\\\\n&=\\frac{1}{ 4 \\pi} \\nabla_y \\eta(y, s) \\times \\bigg{(} \\nabla_y\n\\big{[}\\int_{{\\bf R}^3} \\frac{ G(w, l; z+y, s)}{|z|} dz \\big{]}\n\\times e_k \\bigg{)}.\n\\endaligned\n\\]Using the vector identity $F \\times (G \\times H) =\n(F \\cdot H) G - H (F \\cdot G)$, we have\n\\[\n\\overrightarrow{Z}= \\frac{1}{ 4 \\pi} \\bigg{[} \\partial_{y_k} \\eta\n\\nabla_y \\int_{{\\bf R}^3} \\frac{ G(w, l; z+y, s)}{|z|} dz -\n\\big{(} \\sum^3_{j=1}\n\\partial_{y_j} \\eta \\partial_{y_j} \\int_{{\\bf\nR}^3} \\frac{ G(w, l; z+y, s)}{|z|} dz  \\big{)} e_k \\bigg{]}\n\\]Since $\\partial_{w_i} G(w, l; y+z, s) =\n-\\partial_{y_i} G(w, l; y+z, s)$, the above shows\n\\[\n\\aligned\n&\\partial_{w_i} \\overrightarrow{Z}\\\\\n&=\\frac{1}{ 4 \\pi} \\bigg{[}\n\\partial_{y_k} \\eta \\nabla_y \\int_{{\\bf R}^3} \\frac{\n\\partial_{w_i} G(w, l; z+y, s)}{|z|} dz - \\big{(} \\sum^3_{j=1}\n\\partial_{y_j} \\eta \\partial_{y_j} \\int_{{\\bf\nR}^3} \\frac{\\partial_{w_i} G(w, l; z+y, s)}{|z|} dz  \\big{)} e_k\n\\bigg{]}. \\endaligned \\]Hence\n\\[\n\\aligned\n &\\partial_{w_i} \\overrightarrow{Z}\n  =-\\frac{1}{ 4 \\pi} \\bigg{[}\n\\partial_{y_k} \\eta \\nabla_y \\int_{{\\bf R}^3} \\frac{\n\\partial_{y_i} G(w, l; z+y, s)}{|z|} dz \\\\\n&\\qquad \\qquad - \\big{(} \\sum^3_{j=1}\n\\partial_{y_j} \\eta \\partial_{y_j} \\int_{{\\bf\nR}^3} \\frac{\\partial_{y_i} G(w, l; z+y, s)}{|z|} dz  \\big{)} e_k\n\\bigg{]}.\n\\endaligned\n\\leqno(3.16')\n\\]Substituting the above into the defining formula for $M_2$\n((3.15)) and use integration by parts, we obtain\n\\[\n\\aligned M_2&= \\int_{Q_r(x, t)} \\partial_{y_i} u(y, s) [\\Delta\n\\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}](y, s) dyds\\\\\n&\\qquad + \\frac{1}{ 4 \\pi} \\int_{Q_r(x, t)} u(y, s) (\\Delta  +\n\\partial_s) [ \\partial_{y_i}\n\\partial_{y_k} \\eta \\nabla_y \\int_{{\\bf\nR}^3} \\frac{ G(w, l; z+y, s)}{|z|}] dz dyds\\\\\n&\\qquad + \\frac{1}{ 4 \\pi} \\int_{Q_r(x, t)} u(y, s) (\\Delta  +\n\\partial_s) [ \\sum^3_{j=1} \\partial_{y_i}\n(\\partial_{y_j} \\eta)   \\partial_{y_j} \\int_{{\\bf R}^3} \\frac{\nG(w, l; z+y, s)}{|z|} dz] dyds\n\\endaligned\n\\]Just like the estimate of $J_2$ in part (a), we have\n\\[\n|\\Delta \\overrightarrow{Z} +\\partial_s \\overrightarrow{Z}| \\le\nc\/r^5.\n\\]Here, as before, we used the fact that the parabolic\ndistance between $(w, l)$ and $(y, s)$ is a least $r\/4$.\n\nFor the rest of the terms, using (3.12) and the same argument as\nin the estimate of $K_1$, we deduce\n\\[\n|M_2| \\le \\frac{c}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |\\nabla\nu(y, s)| dyds +\\frac{c}{r^6} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y,\ns)| dyds. \\leqno(3.17)\n\\]\n\n\\medskip\n\n\nFinally we bound $M_3$. Using integration by parts on the $y$\nvariable and using the assumption that $div \\ b=0$, we can write\n\\[\n\\aligned M_3&=  \\int_{Q_r(x, t)} u b \\nabla_y\n\\partial_{w_i} \\Phi^{(w, l)}_k(y, s) dyds \\\\\n&= - \\int_{Q_r(x, t)} \\partial_{w_i} \\Phi^{(w, l)}_k(y, s)  b\n\\nabla_y u dyds \\\\\n&= - \\int_{Q_r(x, t)} (\\partial_{w_i} (\\eta E_k +\n\\overrightarrow{Z}) )  b \\nabla u dyds.\n\\endaligned\n\\]The last step is by (3.8), with $(w, l)$ replacing $(x, t)$ there. From the well known property of the Stokes system\n\\[\n|\\partial_{w_i} E_k(w, l; y, s) | \\le C K_1(w, l; y, s).\n\\]This together with (3.16') and (3.12) yield\n\\[\n|\\partial_{w_i} (\\eta E_k + \\overrightarrow{Z})| \\le C K_1(w, l;\ny, s)\n\\]\n\n\nTherefore\n\\[\n|M_3| \\le C \\int_{Q_r(x, t)} K_1(w, l; y, s)  |b \\nabla u| dyds.\n\\leqno(3.18)\n\\]Recall (see Remark 3.1 just before Lemma 3.2),  the integral on\nthe righthand side of (3.18) is a $L^1_{loc}$ function of space\ntime.\n\n\nCombining (3.15)-(3.18), we have, for $(w, l) \\in Q_{r\/4}(x, t)$,\na.e.\n\\[\n\\aligned\n| \\int_{{\\bf R}^3}& u(w, l) \\partial_{w_i} \\rho(w) dw |\\\\\n &\\le \\frac{c}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |\\nabla u(y,\ns)| dyds +\\frac{c}{r^6}\n\\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)| dyds\\\\\n&\\qquad + c \\int_{{\\bf R}^3} \\int_{Q_r(x, t)} K_1(w, l; y, s)  |b|\n\\ |\\nabla u| dyds \\ \\rho(w) dw.\n\\endaligned\n\\]Since $\\rho$ is arbitrary and $\\nabla u$ and $M_3$ is\n$L^1_{loc}$, we deduce, for $(w, l) \\in Q_{r\/4}(x, t)$,\n\\[\n\\aligned\n|\\partial_{w_i} &u(w, l)|\\\\\n &\\le \\frac{c}{r^5} \\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |\\nabla u(y,\ns)| dyds +\\frac{c}{r^6}\n\\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)| dyds\\\\\n&\\qquad +  \\int_{Q_r(x, t)} K_1(w, l; y, s)  |b| \\ |\\nabla u|\ndyds.\n\\endaligned\n\\]\nThis proves the lemma. \\qed\n\n\n\\medskip\n\nNow we are ready to give the\n\n{\\bf Proof of Theorem 1.2.}\n\\medskip\n\n\n\n{\\it (a) We will  prove the first formula, i.e. the bound on $u(x,\nt)$.}\n\n The idea of the\nproof is to iterate (3.6) in a special manner since simple\niteration will double the domain of integration in each step and\nwill not yield a local formula. The key is to cut in half the size\nof the cube in (3.6) after each iteration. Here are the details.\n\nIn order to simplify the presentation, we will use capital letters\nto denote points in space-time. For example $X=(x, t), Y=(y, s),\nZ=(z, \\tau)$, etc. From (3.6), we have\n\\[\n|u(X)| \\le \\lambda \\frac{1}{r^5} \\int_{Q_r(X)} |u(Y)| dY + \\lambda\n\\int_{Q_r(X)} K_1(X; Y) |b(Y)| \\ |u(Y)| dY. \\leqno(3.19)\n\\]For $Y \\in Q_r(X)$, we apply the representation formula for\ncubes of half of the previous size to get\n\\[\n|u(Y)| \\le \\lambda \\frac{2^5}{r^5} \\int_{Q_{r\/2}(Y)} |u(Z)| dZ +\n\\lambda \\int_{Q_{r\/2}(Y)} K_1(Y; Z) |b(Z)| \\ |u(Z)| dZ.\n\\]Combining the above two inequalities, we obtain\n\\[\n\\aligned |u(X)| &\\le \\lambda \\frac{1}{r^5} \\int_{Q_r(X)} |u(Y)| dY\n+ \\lambda \\int_{Q_r(X)} K_1(X; Y) |b(Y)| \\ \\lambda \\frac{2^5}{r^5}\n\\int_{Q_{r\/2}(Y)} |u(Z)| dZ dY \\\\\n&\\qquad + \\lambda \\int_{Q_r(X)} K_1(X; Y) |b(Y)| \\lambda\n\\int_{Q_{r\/2}(Y)} K_1(Y; Z) |b(Z)| |u(Z)| \\  dZ dY.\n\\endaligned\n\\]Notice that $Q_{r\/2}(Y) \\subset Q_{3r\/2}(X)$ when $Y \\in\nQ_r(X)$. The above thus shows\n\\[\n\\aligned |u(X)| &\\le \\lambda \\frac{1}{r^5} \\int_{Q_r(X)} |u(Y)| dY\n+ \\lambda^2 \\frac{2^5}{r^5} \\Vert u \\Vert_{L^1(Q_{2r}(X))}\n\\int_{Q_r(X)} K_1(X; Y) |b(Y)| dY \\\\\n&\\qquad + \\lambda^2 \\int_{Q_{3r\/2}(X)} \\ \\int_{Q_r(X)} K_1(X; Y)\n|b(Y)|  K_1(Y; Z) dY |b(Z)| |u(Z)| dZ.\n\\endaligned\n\\]\n\nApplying Lemma 3.1, we deduce\n\\[\n\\aligned\n |u(X)| &\\le \\lambda \\frac{1}{r^5}  \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} + \\lambda^2 \\frac{2^5}{r^5} \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} B(b) \\\\\n&\\qquad + \\lambda^2  c B(b) \\int_{Q_{3r\/2}(X)} K_1(X; Z) |b(Z)|\n|u(Z)| dZ,\n\\endaligned\n\\]here and throughout this section\n$ B(b)=B(b, t-(4r)^2, t)$ defined in (1.6').\n\nFor $u(Z)$, we use the representation formula in the cube\n$Q_{r\/4}(Z)$:\n\\[\n|u(Z)| \\le \\lambda \\frac{4^5}{r^5} \\int_{Q_{r\/4}(Z)} |u(W)| dW +\n\\lambda \\int_{Q_{r\/4}(W)} K_1(Z; W) |b(W)| \\ |u(W)| dW.\n\\]Note that for $Z \\in Q_{3r\/2}(X)$,  we have\n$Q_{r\/4}(Z) \\subset Q_{7r\/4}(X) \\subset Q_{2r}(X)$. Hence\n\\[\n\\aligned |u(X)| &\\le \\lambda \\frac{1}{r^5}  \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} +  \\lambda^2 \\frac{2^5}{r^5} \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} B(b)  \\\\\n&+ \\lambda^2 \\frac{4^5}{r^5} c B(b)\n\\int_{Q_{3r\/2}(X)}  K_1(X; Z) |b(Z)|dZ \\,  \\Vert u \\Vert_{L^1(Q_{2r}(X))} \\\\\n&+ \\lambda^3  c B(b) \\int_{Q_{3r\/2}(X)}  K_1(X; Z) |b(Z)|\n\\int_{Q_{r\/4}(Z)} K_1(Z; W) |b(W)| |u(W)| dW dZ.\n\\endaligned\n\\]Exchanging the integrals and using Lemma 3.1 again, we deduce\n\\[\n\\aligned |u(X)| &\\le \\lambda \\frac{1}{r^5}  \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} + \\lambda^2 \\frac{2^5}{r^5} \\Vert u\n\\Vert_{L^1(Q_{2r}(X))} B(b) + \\lambda^2 \\frac{4^5}{r^5} c B(b)^2 \\Vert u \\Vert_{L^1(Q_{2r}(X))}\n \\\\\n&+ \\lambda^3  c^2 B(b)^2 \\int_{Q_{7r\/4}(X)}  K_1(X; W)  |b(W)|\n|u(W)| dW dZ.\n\\endaligned\n\\]\n\nWe iterate the above process, halving the size of each cube. By\ninduction, it is clear that for some $C, c_1>0$,\n\\[\n|u(X)| \\le \\frac{C}{r^5}  \\Vert u \\Vert_{L^1(Q_{2r}(X))}\n\\Sigma^\\infty_{j=1} [2^5 c_1  B(b)]^j.\n\\]When $2^5 c_1 B(b)<1$ the above series converges to yield the mean\nvalue inequality for $u$. Since $b$ is in the Kato class defined\nin Definition 1.2, we know that $2^5 c_1 B(b)= 2^5 c_1 B(b,\nt-(4r)^2, t)<1$ when $r$ is sufficiently small. This proves the\nbound on $u$.\n\n\\medskip\n\n{\\it (b) We prove the gradient bound.}\n\\medskip\n\n\nSince $u+c$ is also a solution of (1.2) for any constant $c$, we\nwill assume that $\\overline{u}_{Q_{2r}}$, the average of $u$ in\n$Q_{2r}(x, t)$ is $0$.\n\nThe idea is to iterate (3.6') in the above manner. From (3.6'),\nusing the same notation as in part (a), we have\n\\[\n |\\nabla u(X)| \\le  m(X, r)\n + \\lambda \\int_{Q_r(X)} K_1(X; Y) |b(Y)| \\ |\\nabla\nu(Y)| dY, \\leqno(3.20)\n\\]where\n\\[\nm(X, r) \\equiv\n \\frac{\\lambda}{r^5} \\int_{Q_r(X)} | \\nabla u(Y)|  dyds\n + \\frac{\\lambda}{r^6} \\int_{Q_r(X)} |u(Y)| dY.\n\\]Here, we remind the reader that both sides of (3.20) are\n$L^1_{loc}$ functions and hence finite almost everywhere (see\nRemark 3.1 just before Lemma 3.2).\n\nApplying (3.20) to $\\nabla u(Y)$ and $Q_{r\/2}(Y)$, we obtain\n\\[\n|\\nabla u(Y)| \\le  m(Y, r\/2)\n + \\lambda \\int_{Q_{r\/2}(Y)} K_1(Y; Z) |b(Z)| \\ |\\nabla\nu(Z)| dZ.\n\\]For $Y \\in Q_r(X)$, it is clear that there exists a $\\mu >0$,\nindependent of $r$ such that\n\\[\nm(Y, r\/2) \\le \\mu m(X, 2 r).\n\\]Hence\n\\[\n|\\nabla u(Y)| \\le  \\mu m(X, 2 r)\n + \\lambda \\int_{Q_{r\/2}(Y)} K_1(Y; Z) |b(Z)| \\ |\\nabla\nu(Z)| dZ. \\leqno(3.21)\n\\]Substituting (3.21) to (3.20) we have\n\\[\n\\aligned |\\nabla u(X)| &\\le  m(X, r) + \\mu \\ m(X, 2r) \\lambda\n\\int_{Q_r(X)} K_1(X; Y) |b(Y)| dy \\\\\n&\\qquad + \\lambda \\int_{Q_r(X)} K_1(X; Y) |b(Y)| \\lambda\n\\int_{Q_{r\/2}(Y)} K_1(Y; Z) |b(Z)| \\ |\\nabla u(Z)| dZ dY.\n\\endaligned\n\\]Therefore\n\\[\n\\aligned |\\nabla u(X)| &\\le  m(X, r) + \\mu \\lambda \\ m(X, 2r) B(b) \\\\\n&\\qquad + \\lambda^2 \\int_{Q_{3r\/2}(X)} \\int_{Q_{r\/2}(Z)} K_1(X; Y)\n|b(Y)| K_1(Y; Z) dZ |b(Z)| \\ |\\nabla u(Z)| dZ dY.\n\\endaligned\n\\]Lemma 3.1 then implies\n\\[\n|\\nabla u(X)| \\le  m(X, r) + \\mu \\lambda \\ m(X, 2r) B(b) \\\\\n + c \\lambda^2 B(b) \\int_{Q_{3r\/2}(X)} K_1(X,Z) |b(Z)| \\ |\\nabla\nu(Z)| dZ.\n\\]Now, using (3.20) on $|\\nabla\nu(Z)|$ and the cube $Q_{r\/4}(Z)$ and repeat the above argument,\nhalving the size of the cube in each step, we finally reach\n\\[\n|\\nabla u(X)| \\le  C  \\ m(X, 2r) \\Sigma^{\\infty}_{k=1}\n\\lambda^{k+1} \\mu^{k+1} B(b)^k.\n\\]As before this implies the desired gradient bound when $B(b)$ is\nsmall.\n\n\nThe last statement of Theorem 1.2 is a simple consequence of the\ngradient estimate and Poincar\\`e inequality. \\qed\n\n\n\n\n\n\nWe end the section by showing that if the sum of the entries of\nthe drift term $b$ is zero, then (1.2) in torus has bounded\nsolutions for many initial values, regardless of the singularity\nof $b$. To state the result rigourously, we will assume that $b$\nis bounded. However all coefficients are independent of the bounds\nof $b$.\n\n\\medskip\n\n\\begin{proposition} Given bounded vector fields $b=(b_1(x, t), ...,\nb_n(x, t))$, consider the linearized Navier-Stokes equation with\nperiodic boundary condition i.e. in a torus.\n\\[\n\\begin{cases}\n\\Delta u(x, t) - b(x, t)  \\nabla u(x, t) + \\nabla P(x, t) -\n\\partial_t\nu(x, t)=0, \\quad (x, t) \\in D  \\times {\\bf R},\\\\\ndiv u = 0, \\ div b=0, \\ b(\\cdot, t) \\in L^\\infty \\\\\nu(x, 0) = u_0(x).\n\\end{cases}\n\\leqno(3.22)\n\\]Here $D = [0, 2 \\pi]^n$. The functions $b(\\cdot, t)$,\n$u_0(\\cdot)$ and $u(\\cdot, t)$ have period $2 \\pi$.\n\n\nSuppose $\\sum^n_{j=1} b_j(x, t) = \\lambda$, a constant and $u_0$\nis any finite linear combination of\n\\[\n\\int_{D} e^{i k \\sum^n_{j=1}(x_j-y_j)} f(y) dy,\n\\]where $k$ is a positive integer and $f$ is a bounded,\ndivergence free vector field with period $2 \\pi$. Then there\nexists a constant $c$ independent of $b$ and $\\lambda$ such that\n\\[\nu(x, t) \\le c e^{-t} C(\\Vert u_0 \\Vert_{\\infty}).\n\\]\n\\proof\n\\end{proposition}\n\nUnder the assumption that $b$ is bounded, the existence of\nsolutions to (3.22) follows from the standard theory. Let $E=E(x,\nt; y, s)$ be the fundamental solution of (3.22). The existence of\n$E$ is also standard.\n\n First we\njust assume that $u_0$ has  one term, i.e.\n\\[\nu_0(x) = \\int_{D} e^{i k \\sum^n_{j=1}(x_j-y_j)} f(y) dy,\n\\]Fixing $(x, t)$, consider\n\\[\nI(s) \\equiv \\int_D E(x, t; y, s) u_0(y) dy. \\leqno(3.23)\n\\]By the fact that the rows of $E$ satisfies the conjugate\nequation of (3.22), we have\n\\[\n\\aligned I'(s) & = \\int_{D} \\frac{d}{ds} E(x, t; y, s) u_0(y) dy\\\\\n&= \\int_{D} \\big{[} - \\Delta_y E(x, t; y, s) - b(y, s) \\nabla_y\nE(x, t; y, s) + \\nabla Y(y, s)] u_0(y) dy. \\endaligned\n\\]Here\n\\[\n\\nabla Y =\\left( \\begin{array}{cc} \\partial_1 P_1 \\ ...\\\n\\partial_n P_1\\\\... \\ ... \\ ... \\\\ \\partial_1 P_1\\ ... \\\n\\partial_n P_n\n\\end{array} \\right)\n\\]with $P_1, ..., P_n$ being scalar functions. Using integration\nby parts and the divergence free property of $b$, we deduce\n\\[\nI'(s) = - \\int_{D}   E(x, t; y, s) \\Delta u_0(y) dy  + \\int_{D}\nE(x, t; y, s) \\sum^n_{j=1} b_j \\partial_{y_j} u_0(y) dy.\n\\]Noticing that $\\Delta u_0 = - n k^2 u_0$ and $\\partial_{y_j} u_0(y)\n= i k u_0$, we have\n\\[\nI'(s) =  n k^2 \\int_{D}   E(x, t; y, s) u_0(y) dy + i k \\int_{D}\n(\\sum^n_{j=1} b_j) E(x, t; y, s)  u_0(y) dy.\n\\]By our assumption that $\\sum^n_{j=1} b_j = \\lambda$, the above\nshows\n\\[\nI'(s) =  (n k^2 + i k \\lambda) I(s).\n\\]Hence\n\\[\nI(s) = e^{(n k^2 + i k \\lambda) s} I(0).\n\\]From (3.23) and the fact that $E$ is the fundamental solution to\n(3.22), we have $I(0) = u(x, t)$ and $I(t) = u_0(x)$. This shows\n\\[\nu(x, t) = e^{- (n k^2 + i k \\lambda) t} u_0(x). \\leqno(3.24)\n\\]\n\nNow let $S$ be a set of finite positive integers and\n\\[\nu_0(x) = \\sum_{l \\in S} c_l \\int_{D} e^{i k_l\n\\sum^n_{j=1}(x_j-y_j)} f_l(y) dy.\n\\]Here $k_l$ is a positive integer and $f_l$ is a bounded,\ndivergence free vector field with period $2 \\pi$. By (3.24) one\nhas\n\\[\nu(x, t) = \\sum_{l \\in S} e^{- (n k^2_l + i k_l \\lambda) t} c_l\n\\int_{D} e^{i k_l \\sum^n_{j=1}(x_j-y_j)} f_l(y) dy.\n\\]\\qed\n\n\n\n\\section{A regularity condition for Navier-Stokes equations not involving\nabsolute values}\n\nIn this section we introduce another sufficient condition on the\nvelocity for boundedness of weak solutions of 3 dimensional\nNavier-Stokes equations. The novelty is that no absolute value of\n$u$ is involved. This is useful since it allows more cancellation\neffect to be taken into account. Throughout the years, various\nconditions on $u$ that imply regularity have been proposed. One of\nthem is the Prodi-Serrin condition which requires that $u \\in\nL^{p, q}$ with $\\frac{3}{p}+\\frac{2}{q} \\le 1$ for some $3<p \\le\n\\infty$ and $q \\ge 2$. See ([Ser], [Str] e.g.) Recently the\nauthors in [ESS] showed that the condition $p=3$ and $q=\\infty$\nalso implies regularity. In another development the author of [Mo]\nimproved the Prodi-Serrin condition by a log factor, i.e. by\nrequiring\n\\[\n\\int^T_0 \\frac{\\Vert u(., t) \\Vert^q_p}{1+ \\log^+ \\Vert u(.,t)\n\\Vert_p} dt < \\infty,\n\\]where $3\/p + 2\/q = 1$ and $3<p<\\infty$, $2<q<\\infty$\n\nMost recently in [Z3], a form boundedness condition on velocity\nwas introduced, which will imply the boundedness of weak\nsolutions. The form boundedness condition,  with its root in the\nperturbation theory of elliptic operators and mathematical\nphysics, seems to be different from all the previous conditions.\nIt seems to be one of the most general condition under the\navailable tools. This fact has been well documented in the theory\nof linear elliptic equations. See [Si] e.g.  Moreover, as\nindicated in [Z3],  it contains the Prodi-Serrin condition except\nwhen $p$ or $q$ are infinite. It also includes suitable\nMorrey-Compamato type spaces. However we are not sure this\ncondition contains the one in [Mo].\n\nMore precisely we proved\n\\medskip\n\n\\noindent {\\it {\\bf Theorem 5.1 ([Z3])}\nLet $u$ be a Leray-Hopf solution to the 3 dimensional\nNavier-Stokes equation  in ${\\bf R}^3 \\times (0, \\infty)$.\n\\[\n\\begin{cases}\n\\Delta u(x, t) - u(x, t)  \\nabla u(x, t) + \\nabla P(x, t) -\n\\partial_t\nu(x, t)=0, \\quad (x, t) \\in \\Omega \\subset {\\bf R}^3 \\times {\\bf R},\\\\\ndiv u = 0.\n\\end{cases}\n\\leqno(4.1)\n\\]\n Suppose\nfor every $(x_0, t_0) \\in {\\bf R}^3 \\times (0, \\infty)$, there\nexists a cube $Q_r=B(x_0, r) \\times [t_0-r^2, t_0]$ such that  $u$\nsatisfies the form bounded condition\n\\[\n\\int_{Q_r} |u|^2 \\phi^2 dyds \\le \\frac{1}{24} \\bigg{(} \\int_{Q_r}\n|\\nabla \\phi|^2 dyds + \\sup_{s \\in [t_0-r^2, t_0]} \\int_{B(x_0,\nr)} \\phi^2(y, s) dy \\bigg{)} + B(\\Vert \\phi \\Vert_{L^2(Q_r)}).\n\\leqno(4.2)\n\\]Here $\\phi$ is any smooth function vanishing on the parabolic side of\n$Q_r$ and $B=B(t)$ is any given locally bounded function of $t \\in\n{\\bf R}^1$. Then $u$ is a classical solution when $t>0$.}\n\\medskip\n\n\nIn this section we are able to extend the form boundedness\ncondition further. The main improvement is that our new condition\nin $u$ ((4.3)) below does not involve the absolute value of $u$.\nThis differs significantly from known conditions on $u$ so far\nwhere $|u|$ is always present. By a simple integration by parts\nargument, it is clear that Condition (4.3) is more general than\nCondition (4.2).\n\n\\medskip\n\n\\begin{theorem}\nLet $u$ be a Leray-Hopf solution to the 3 dimensional\nNavier-Stokes equation  in ${\\bf R}^3 \\times (0, \\infty)$.\n\n Suppose\nfor every $(x_0, t_0) \\in {\\bf R}^3 \\times (0, \\infty)$, there\nexists a cube $Q_r=B(x_0, r) \\times [t_0-r^2, t_0]$ such that  $u$\nsatisfies the form bounded condition: for a given $\\delta>0$,\n\\[\n\\int_{Q_r} \\phi \\nabla u \\cdot \\phi dyds \\le \\frac{1-\\delta}{2}\n\\bigg{(} \\int_{Q_r} |\\nabla \\phi|^2 dyds + \\frac{1}{2} \\sup_{s \\in\n[t_0-r^2, t_0]} \\int_{B(x_0, r)} \\phi^2(y, s) dy \\bigg{)} +\nB(\\Vert \\phi \\Vert_{L^2(Q_r)}). \\leqno(4.3)\n\\]Here $\\phi$ is any smooth vector field vanishing on parabolic the side of\n$Q_r$ and $B=B(t)$ is any given locally bounded function of $t \\in\n{\\bf R}^1$. Then $u$ is a classical solution when $t>0$.\n\\end{theorem}\n\\medskip\n\n{\\bf Remark 4.1.} Condition 4.3 is actually a condition on the\nstrain tensor $\\nabla u + (\\nabla u)^T$. Theorem 4.1 immediately\nimplies that weak solutions to the 3 dimensional Navier-Stokes\nequations are locally bounded in any open subset of the region\nwhere the strain tensor is negative definite.\n\\medskip\n\n\n{\\bf Proof of Theorem 4.1.} Let $t_0$ be the first moment of\nsingularity formation. We will reach a contradiction. It is clear\nthat we only need to prove that $u$ is bounded in\n$Q_{r\/8}=Q_{r\/8}(x_0, t_0)$ for some $r>0$. In fact the number $8$\nis not essential. Any number greater than $1$ would work.\n\n\n Consider the equation for\nvorticity $w = \\nabla \\times u$. It is well known that, in the\ninterior of $Q_r$,  $w$ is a classical solution to the parabolic\nsystem with singular coefficients\n\\[\n\\Delta w - u \\nabla w + w \\nabla u - w_t = 0. \\leqno(4.4)\n\\]Let $\\psi=\\psi(y, s)$ be the refined cut-off function defined right after\n(2.1) such that $\\psi=1$ in $Q_{r\/2}$, $\\psi=0$ in $Q^c_r$ and\nsuch that $0 \\le \\psi \\le 1$, $|\\nabla \\psi| \\le C\/r$ and\n$|\\psi_t| \\le C\/r^2$. We can use $w \\psi^2$ as a test function on\n(4.4) to obtain\n\\[\n\\aligned\n \\int_{Q_r} | &\\nabla (w \\psi) |^2 dyds + \\frac{1}{2}\n \\int_{B(x_0, r)} |w \\psi|^2(y, t_0) dy\\\\\n&\\le \\frac{C}{r^2} \\int_{Q_r} |w|^2 dyds - \\int_{Q_r} u \\nabla w\n\\cdot w \\psi^2 dyds  + \\int_{Q_r} w \\nabla u \\cdot w \\psi^2\ndyds \\\\\n&\\equiv I_1 + I_2 + I_3.\n\\endaligned\n\\leqno(4.5)\n\\]\n\nThe term $I_1$ is already in good shape. Next, using integration\nby parts and the divergence free condition on $u$, we have\n\\[\nI_2 = \\frac{1}{2} \\int_{Q_r} u \\cdot \\nabla \\psi \\psi |w|^2 dyds.\n\\]Since $\\nabla u \\in L^2_{loc}({\\bf R}^3 \\times [0, \\infty))$\nand $\\Vert u( \\cdot, t)\\Vert_{{\\bf R}^3}$ is non-increasing in\ntime, it is easy to prove by Sobolev imbedding and H\\\"older's\ninequality that $u|_{Q_r}$ satisfies the form boundedness\ncondition (1.3). In fact this has been proven in Corollary 2 in\n[Z2]. Hence we can bound $I_2$ in exactly the same way as $T_2$ in\n(2.4) with $b$ being chosen as $u$ here. Following the argument\nbetween (2.4) and (2.6), we obtain, for any given $\\delta>0$,\n\\[\nI_2 \\le  \\frac{\\delta}{4} \\int_{Q_{r}} |\\nabla (\\psi  w)|^2 dyds +\nc_\\delta e^{( c_1\/ r)^2} \\int_{Q_r}  w^2 dyds. \\leqno(4.6)\n\\]Note that in (2.6), $\\delta$ was chosen as $2$. However, a\ncloser look at the proof shows that (4.6) is true.\n\n\n\nNext we estimate $I_3$. From Condition (4.3)\n\\[\nI_3 \\le \\frac{1-\\delta}{2} \\bigg{(}  \\int_{Q_r} |\\nabla (w \\psi)\n|^2 dyds + \\frac{1}{2} \\sup_{s \\in [t_0-r^2, t_0]} \\int_{B(x_0,\nr)} (w \\psi)^2(y, s) dy \\bigg{)} + B(\\Vert w \\psi\n\\Vert_{L^2(Q_r)}). \\leqno(4.7)\n\\]\n\n\nSubstituting (4.6)-(4.7) to (4.5) we obtain,\n\\[\n\\aligned\n \\int_{Q_r}& | \\nabla (w \\psi) |^2 dyds + \\frac{1}{2}\n\\int_{B(x_0, r)} |w \\psi|^2(y, t_0) dy \\\\\n&\\le \\frac{\\delta}{4} \\int_{Q_{r}} |\\nabla (\\psi  w)|^2 dyds +\nc_\\delta e^{( c_1\/ r)^2}\n\\int_{Q_r}  w^2 dyds \\\\\n&+ \\frac{1-\\delta}{2} \\bigg{(}  \\int_{Q_r} |\\nabla (w \\psi) |^2\ndyds + \\frac{1}{2} \\sup_{s \\in [t_0-r^2, t_0]} \\int_{B(x_0, r)} (w\n\\psi)^2(y, s) dy \\bigg{)}\n + C B(\\Vert w\n\\Vert_{L^2(Q_r)}).\n\\endaligned\n\\]Repeating the above process, but restrict the integrals to $Q_r\n\\cap \\{ (y, s) \\ | s<t \\}$ with $t<t_0$, we obtain, for any $t \\in\n[t_0-r^2, t_0]$,\n\\[\n\\aligned &\\frac{1}{2}\n\\int_{B(x_0, r)} |w \\psi|^2(y, t) dy \\\\\n&\\le \\frac{\\delta}{4} \\int_{Q_{r}} |\\nabla (\\psi  w)|^2 dyds +\nc_\\delta e^{( c_1\/ r)^2}\n\\int_{Q_r}  w^2 dyds \\\\\n&+ \\frac{1-\\delta}{2} \\bigg{(}  \\int_{Q_r} |\\nabla (w \\psi) |^2\ndyds + \\frac{1}{2} \\sup_{s \\in [t_0-r^2, t_0]} \\int_{B(x_0, r)} (w\n\\psi)^2(y, s) dy \\bigg{)}\n + C B(\\Vert w\n\\Vert_{L^2(Q_r)}).\n\\endaligned\n\\]\n\n\nCombining the last two estimates, we deduce,\n\\[\n\\aligned\n &\\int_{Q_r} | \\nabla (w \\psi) |^2 dyds + sup_{t_0-r^2 \\le\ns \\le t_0} \\int_{B(x_0, r)} |w \\psi|^2(y, s) dy \\\\\n&\\le \\frac{C_1}{r^2} \\Vert w \\Vert_{L^2(Q_r)} + B_1(\\Vert w\n\\Vert_{L^2(Q_r)}). \\endaligned\n \\leqno(4.8)\n\\]Here $B_1$ is a locally bounded, one variable function.\n\n\nUsing standard results, we know that (4.8) implies that $u$ is\nregular. Here is the proof.\n\nFrom (4.8), it is clear that $\\int_{Q_r} | curl \\ (w \\psi) |^2\ndyds \\le C.$ Hence, since $\\psi =1$ in $Q_{r\/2}$,\n\\[\n\\int_{Q_{r\/2}} | \\Delta  u|^2 dyds \\le C. \\leqno(4.9)\n\\]Let $\\eta =\\eta(y)$ be a cut-off function such that $\\eta=1$ in\n$B(x_0, r\/4)$ and $\\eta=0$ in $B(x_0, r\/2)^c$. Then for each $s\n\\in [t_0-(r\/4)^2, t_0]$, we have, in the weak sense\n\\[\n\\Delta (u \\eta) = \\eta \\Delta u + 2 \\nabla u \\nabla \\eta + u\n\\Delta \\eta \\equiv f,\n\\]in $Q_{r\/2}$. By standard elliptic estimates, using the fact\nthat $u \\eta=0$ on the boundary,\n\\[\n\\Vert D^2 u( \\cdot, s) \\Vert_{L^2(B(x_0, r\/4))} \\le C \\Vert f(\n\\cdot, s) \\Vert_{L^2(B(x_0, r\/2))}.\n\\]This shows that\n\\[\n\\Vert D^2 u \\Vert_{L^2(Q_{r\/4})} \\le C \\Vert \\Delta u\n\\Vert_{L^2(Q_{r\/2})} + C \\Vert u \\Vert_{L^2(Q_{r\/2})}.\n\\]By Sobolev imbedding\n\\[\n\\nabla u \\in L^{6, 2}(Q_{r\/4}). \\leqno(4.10)\n\\]\n\nNext, from (1.22) on p316 [Te],\n\\[\n\\Vert u( \\cdot, s) \\eta \\Vert_{W^{1, 2}} \\le C \\big{(} \\Vert u(\n\\cdot, s) \\eta \\Vert_{L^2} + \\Vert div (u \\eta) ( \\cdot, s)\n\\Vert_{L^2} + \\Vert curl (u( \\cdot, s) \\eta) \\Vert_{L^2} \\big{)}.\n\\]Here all norms are over the ball $B(x_0, r\/2)$. Therefore\n\\[\n\\Vert u \\eta ( \\cdot, s)  \\Vert_{W^{1, 2}} \\le C \\big{(} \\Vert u(\n\\cdot, s) \\eta \\Vert_{L^2} + \\Vert u \\nabla \\eta ( \\cdot, s)\n\\Vert_{L^2} + \\Vert w \\eta ( \\cdot, s)  \\Vert_{L^2} + \\Vert |u(\n\\cdot, s)| \\ |\\nabla \\eta| \\Vert_{L^2} \\big{)}.\n\\]It follows\n\\[\n\\Vert u( \\cdot, s)  \\Vert_{W^{1, 2}(B(x_0, r\/4))} \\le C.\n\\]From Sobolev imbedding we know that\n\\[\nu \\in L^{6, \\infty}(Q_{r\/4}). \\leqno(4.11)\n\\]We treat $u$ and $\\nabla u$ as coefficients in the vortex equation (4.4).\nBy (4.10) and (4.11), the standard parabolic theory (see [Lieb]\ne.g.), shows that $w$ is bounded and  H\\\"older continuous in\n$Q_{r\/8}$. Here the bound depends only on the $L^2$ norm of $w$ in\n$Q_r$ and $r$. This is so because of the relation $ 3\/6 + 2\/\\infty\n< 1$ for the norm of $u$ and $3\/6 + 2\/2 < 2$ for the norm of\n$\\nabla u$. Now a standard bootstrapping argument shows that $u$\nis smooth. \\qed\n\n\n\n\\medskip\n\n\n\n\n\n\n\\medskip\n\n{\\bf Acknowledgement} I thank Professors Z. Grujic, I. Kukavica\nand V. Liskevich for helpful discussions.\n\n\\medskip\n\n\\section{Addendum May 2019} \n\nThe purpose of the addendum is to fill in a missing term in Theorem 1.7 and Lemma 3.3 in the current paper that is published in Pacific Journal of Mathenatics, Vol. 223, No.2, 2006, which will be cited as \\cite{Z:4}.  This is due to the use of a formula in a cited reference, which omitted a term.\nThe main conclusion that local solutions of certain linearized Navier-Stokes equation have bounded spatial gradient is intact. This includes bounded local Leray-Hopf solutions to the Navier Stokes equation without condition on the pressure.\n\nLet us recall the basic set up of \\cite{Z:4}. The equation is: for $(x, t) \\in \\Omega \\subset {\\bf R}^3 \\times {\\bf R}$,\n\\be\n\\lab{lns}\n\\begin{cases}\n\\Delta u(x, t) - b(x, t)  \\nabla u(x, t) + \\nabla P(x, t) -\n\\partial_t\nu(x, t)=0, \\\\\ndiv \\, u = 0, \\ div \\, b=0, \\ b(\\cdot, t) \\in L^2_{loc}.\n\\end{cases}\n\\ee\nHere $\\Delta$ is the standard Laplacian and $b=b(x, t)$ is a  given\n$L^2_{loc}$  vector field  to be specified later. $\\Omega$\nis a domain.\nThe parabolic Kato norm is:\n\\be\nB(b, l, t) = \\sup_{x} \\int^{t}_l \\int_{{\\bf R}^n} [K_1(x, t; y, s) +\nK_1(x, s; y, l) ] |b(y, s)| dyds.\n\\lab{(1.6')}\n\\ee We say $b$ is in class $K_1$ if $\\lim_{l \\to t} B(b, l, t)=0$. $K_1$ class in a bounded space time domain contains the usual $L^{p, q}$ functions with $(3\/p)+(2\/q)<1.$\n\n\n\nLet $D$ for a domain in ${\\bf R}^3$ and $T>0$, we will use this definition for solutions of (\\ref{lns}).\n\n\n\n\\begin{definition}\n\\lab{defsol}\n A divergence free vector\nfield $u \\in L^{\\infty}(0, T; L^2(D)) \\cap L^2(0, T; W^{1, 2}(D))$\nis called a\n (weak) solution of (\\ref{lns}) if:\nfor any vector valued $\\phi \\in C^{\\infty}(D \\times [0, T])$ with\n$ div \\, \\phi =0$ and $\\phi=0$ on $\\partial D \\times [0, T]$, $u$\nsatisfies\n\\[\n\\int^{t_2}_{t_1}\\int_{{\\bf R}^n} <u, \\partial_t \\phi + \\Delta\n\\phi>\n dxdt - \\int^{t_2}_{t_1}\\int_{{\\bf R}^n} <b \\nabla u,  \\phi>  dxdt\n = - \\int_{{\\bf R}^n} <u, \\phi>(x, t) |^{t_2}_{t_1} dx.\n\\]\n\\end{definition}\n\n\n\nThe corrected version of Theorem 1.7 in \\cite{Z:4} is:\n\n\\begin{theorem}\n\\label{th:1.2}\nLet $u$ be a solution of (\\ref{lns}) in a domain $\\Omega \\subset {\\bf\nR}^3 \\times {\\bf R}$.  Suppose $Q_{4r}(x, t) \\subset \\Omega$, $div\nb =0$ and that $b |_{Q_{2r}(x, t)}$ is in class $K_1$ and $b \\in\nL^2_{loc}$. Then both $u$ and $|\\nabla u|$ are bounded functions\nin $Q_{2r}=Q_{2r}(x, t)$, the standard parabolic cube of size $2r$.\n\nMoreover, for some positive constants $C=C(b)$ and $r_0$,\ndepending on the size of the Kato norm of $b$,   there hold, when\n$0<r<r_0$,\n\\[\n\\aligned\n |u(x, t)| &\\le  \\frac{C}{r^3} \\sup_{s \\in [t-(2 r)^2, t]} \\int_{B_{r}(x)} |u(y,\ns)| dy,\\\\\n|\\nabla u(x, t)| &\\le \\frac{C}{r^5} \\Vert \\nabla\nu \\Vert_{L^1(Q_{2r})}\n + \\frac{C}{r^4} \\sup_{s \\in [t-(2 r)^2, t]} \\int_{B_{r}(x)} |u(y,\ns)| dy.\n\\endaligned\n\\]\n\\end{theorem}\n\n\n\n\n\nThe proof of the theorem is through a scaled iteration process starting with Lemma 3.3 in \\cite{Z:4}, which contains mean value inequalities for: (a) solutions of\n(\\ref{lns}); (b)  spatial gradient of solutions. Inequality (a) is first obtained in \\cite{O:1}, which is also the basis for\ninequality (b).  Recently Professor Hongjie Dong kindly informed us that inequality (a) and hence (b) misses a term.  The corrected version of Lemma 3.3 in \\cite{Z:4} is the following. In our opinion the main localization idea in \\cite{O:1} is still nice.\n\n\n\n\n\n\\begin{lemma} (mean value inequalities, replacing Lemma 3.3 in \\cite{Z:4})\n\n(a). Let $u$ be a solution of (\\ref{lns}) in the region $\\Omega$.\nSuppose $Q_{2r}(x, t) \\subset \\Omega$. Then there exists a\nconstant $\\lambda$ such that\n\\[\n\\aligned\n |u(x, t)| &\\le  \\frac{\\lambda}{r^5} \\int_{Q_r(x,\nt)-Q_{r\/2}(x, t)} |u(y, s)| dyds + \\underbrace{\\frac{\\lambda}{r^3} \\int_{B_r(x)-B_{r\/2}(x)} |u(y, t)| dy}\\\\\n&\\qquad + \\lambda \\int_{Q_r(x, t)} K_1(x, t; y, s) |b(y, s)| \\\n|u(y, s)| dyds. \\endaligned\n\\]\n\n(b). Under the same assumption as (a), there exists a constant\n$\\lambda$ such that\n\\[\n\\aligned\n&|\\nabla u(x, t)| \\le  \\frac{\\lambda}{r^5} \\int_{Q_r(x,\nt)-Q_{r\/2}(x, t)} | \\nabla u(y, s)|  dyds\n + \\frac{\\lambda}{r^6}\n\\int_{Q_r(x, t)-Q_{r\/2}(x, t)} |u(y, s)| dyds \\\\\n&\\quad + \\underbrace{\\frac{\\lambda}{r^4} \\int_{B_r(x)-B_{r\/2}(x)} |u(y, t)| dy}\n + \\lambda \\int_{Q_r(x, t)} K_1(x, t; y, s) |b(y, s)| \\\n|\\nabla u(y, s)| dyds. \\endaligned\n\\]\n\\end{lemma}\n\n\\proof of (a). The under-braced terms were missing. We indicate the changes needed for the proof,  dividing into 3 steps.\n\n{\\it Step 1.}\nLet $E=E(x, t; y, s)$ be the fundamental solution (matrix) of the\nStokes system in ${\\bf R}^3 \\times (0, \\infty)$ and $E_k$ be the\n$k$-th column of $E$. Then\n\\be\n\\lab{Ekform}\nE_k(x, t; y, s) =  G(x, t; y, s) e_k + \\frac{1}{4 \\pi} \\nabla\n\\partial_k \\int_{{\\bf R}^3} \\frac{G(x, t; z, s)}{|y-z|} dz\n\\ee where $G$ is the fundamental solution of the heat equation.   Fixing $(x, t)$, we construct a standard\ncut-off function $\\eta$ such that $\\eta(y, s)=1$ in $Q_{r\/2}(x,\nt)$, $\\eta(y, s)=0$ outside of $Q_r(x, t)$, $0 \\le \\eta \\le 1$ and\n$|\\nabla \\eta|^2 + |\\Delta \\eta| + |\\partial_s \\eta| \\le c\/r^2$.\n\nDefine for $k=1, 2, 3$, after \\cite{O:1},  a vector valued function\n\\be\n\\lab{(3.7)phik}\n\\Phi_k=\\Phi^{(x, t)}_k(y, s) = \\frac{1}{4 \\pi} curl \\big{(} \\eta\n(y, s) \\int_{{\\bf R}^3} \\frac{ curl E_k(x, t; z, s)}{|z-y|} dz\n\\big{)}.\n\\ee It is clear that when $t>s$,  $\\Phi_k$ is a valid test function\nfor equation (\\ref{lns}) since $\\Phi_k$ is smooth, compactly supported\nand divergence free. Using $\\Phi_k$ as a test function on (\\ref{lns}),\nby Definition \\ref{defsol}, we obtain\n\\be\n\\lab{uphi1}\n\\int^t_0\\int u(y, s) \\big{(} \\Delta \\Phi_k + b \\nabla \\Phi_k +\n\\partial_s \\Phi_k \\big{)} dyds = \\lim_{s \\to t} \\int u(y, s)\n\\Phi_k(y, s) dy.\n\\ee Here,we will suppress the superscript $(x, t)$ on\n$\\Phi_k$, unless there is a confusion. Also $u$ is regarded as a row vector so that $u \\Phi_k$ etc is\na scalar.\n\nSince $E_k$ is divergence free $curl \\ curl E_k = - \\Delta E_k$.\nThus\n\\be\n\\lab{(3.8)}\n\\aligned \\Phi_k(y, s) &= \\eta(y, s) E_k(x, t; y, s) + \\frac{1}{4\n\\pi} \\nabla \\eta(y, s) \\times \\int_{{\\bf R}^3} \\frac{ curl E_k(x,\nt; z, s)}{|z-y|} dz\\\\\n&\\equiv \\eta E_k + \\overrightarrow{Z}.\n\\endaligned\n\\ee\n\n{\\it Step 2.}\nUsing formula (\\ref{Ekform}) of the fundamental matrix $E$ and\nthat of $\\overrightarrow{Z}$ in (\\ref{(3.8)}), direct computation shows\nthat\n\\be\n\\lab{uphi2}\n\\al\n&\\lim_{s \\to t} \\int u(y, s) \\Phi_k(y, s) dy = \\eta  u_k(x,t)\\\\\n& + \\int \\left[\n\\partial_{y_k} \\Gamma (x, y) (\\nabla \\eta \\cdot u)(y, t)\n+\\partial_{y_k} \\eta (y, t) (\\nabla_y \\Gamma(x, y) \\cdot u(y, t) )\n- (\\nabla \\eta \\cdot \\nabla_y \\Gamma(x, y)) u_k(y, t) \\right] dy.\n\\eal\n\\ee\nwhere $u_k$ is the k-th component of $u$ and $\\Gamma=\\frac{1}{4 \\pi |x-y|}$\nis the Green's function on ${\\bf R}^3$.\n\nAlternatively, from (\\ref{(3.8)}),\n\\be\n\\lab{i+ii}\n\\al\n\\lim_{s \\to t} \\int u(y, s) \\Phi_k(y, s)  dy\n&=\\lim_{s \\to t} \\int \\eta u(y, s) E_k(x, t; y, s) dy + \\lim_{s \\to t} \\int u(y, s) \\vec{Z} dy\\\\\n&\\equiv I + II.\n\\eal\n\\ee First we treat $I$. Consider the Helmholtz decomposition\n\\[\n\\eta u = \\nabla f + X,\n\\]where $\\Delta f = 0$, $ div \\, X = 0$. Since $\\Delta f = \\nabla \\eta \\cdot u$ and\n$\\eta(\\cdot, s)$ is compactly supported, we can take\n$\nf(y, s) = -\\int \\Gamma(y, z) ( \\nabla \\eta \\cdot u)(z, s) dz.\n$ Then\n\\be\n\\lab{x=}\nX=\\eta u - \\nabla f = \\eta u(y, s) + \\int \\nabla_y \\Gamma(y, z) (\\nabla \\eta \\cdot u)(z, s) dz.\n\\ee\nUsing the decay estimates\n$\n|\\nabla_y E_k(x, t; y, s)| \\le \\frac{c}{(|x-y| + \\sqrt{t-s})^4},\n $ $|f(y, s)| \\le \\frac{c}{|y|}$ and the fact that $E_k$ is a divergence vector field of variable $y$, we can integrate by parts to deduce that\n$\n\\int \\nabla f E_k(x, t; y, s) dy=0.\n$ Therefore\n\\be\n\\lab{i=}\n\\al\nI&=\\lim_{s \\to t} \\int (\\nabla f + X) E_k(x, t; y, s) dy = X_k(x, t)\\\\\n&= \\eta u_k(x, t) + \\int \\partial_{y_k} \\Gamma(x, y) (\\nabla \\eta \\cdot u)(y, t) dy.\n\\eal\n\\ee\n\nIn order to compute $II$, we see from (\\ref{Ekform}) and (\\ref{(3.8)}) that\n\\[\n\\vec{Z}\n= \\frac{1}{ 4 \\pi} \\nabla_y \\eta(y, s) \\times\n\\bigg{(} \\nabla_y \\big{[}\\int_{{\\bf R}^3} \\frac{ G(x, t; z,\ns)}{|z-y|} dz \\big{]} \\times e_k \\bigg{)}.\n\\]This implies\n\\be\n\\lab{ii=}\n\\al\nII=\\lim_{s \\to t} \\int u(y, s) \\vec{Z} dy\n=\\int  u(y, t) \\cdot \\left[ \\nabla_y \\eta \\times (\\nabla_y \\Gamma(x, y) \\times e_k )\\right]  dy.\n\\eal\n\\ee\n\nA combination of (\\ref{i=}), (\\ref{ii=}) and (\\ref{i+ii}) proves (\\ref{uphi2}). By (\\ref{uphi2}) and (\\ref{uphi1}):\n\n\\[\n\\aligned\n&u_k(x, t) = \\int_{Q_r(x, t)} u [ \\Delta (\\eta E_k) +\n\\partial_s (\\eta\nE_k)]dyds  + \\int_{Q_r(x, t)} u [\\Delta \\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}] dyds\\\\\n& + \\int_{B_r(x)} \\left[(\\nabla \\eta \\cdot \\nabla_y \\Gamma) u_k -\n\\partial_{y_k} \\Gamma  (\\nabla \\eta \\cdot u)\n-\\partial_{y_k} \\eta  (\\nabla_y \\Gamma \\cdot u )\n \\right](y, t) dy  +\\int_{Q_r(x, t)} u b \\nabla \\Phi_k dyds.\n\\endaligned\n\\]Here and later $u b \\nabla \\Phi_k = u \\cdot \\sum^3_{i=1} b_i\n\\partial_i \\Phi_k$.\n\nNote that $\\Delta E_k + \\partial_s E_k=0$ when $s<t$. The above\ninfers, with $\\Gamma=\\Gamma(x, y)$, a local representation formula which may be of independent interest:\n\\be\n\\aligned\n&u_k(x, t)\n= \\int_{Q_r(x, t)} u(y, s) [ E_k(x, t; y, s) (\\Delta \\eta +\n\\partial_s \\eta)(y, s) +\n2 \\nabla \\eta(y, s) \\nabla_y E_k(x, t; y, s)]dyds \\\\\n&\\qquad + \\int_{Q_r(x, t)} u(y, s) [\\Delta \\overrightarrow{Z} +\n\\partial_s \\overrightarrow{Z}](y, s) dyds + \\int_{Q_r(x, t)} u b \\nabla_y \\Phi_k(y, s) dyds\\\\\n&\\qquad + \\int_{B_r(x)} \\left[(\\nabla \\eta \\cdot \\nabla_y \\Gamma) u_k -\n\\partial_{y_k} \\Gamma  (\\nabla \\eta \\cdot u)\n-\\partial_{y_k} \\eta  (\\nabla_y \\Gamma \\cdot u )\n \\right](y, t) dy \\\\\n&\\equiv J_1+J_2+J_3+J_4.\n\\endaligned\n\\lab{(3.9)}\n\\ee\n\n\n{\\it Step 3.} It is clear that\n$\n|J_4| \\le \\frac{C}{r^3} \\int_{B_r(x)-B_{r\/2}(x)} |u(y, t)| dy.\n$\nThe estimates $J_1, J_2, J_3$ are done on p381 of \\cite{Z:4} following \\cite{O:1}. This\nproves of part (a) of the lemma.\n\n{\\it Proof of part (b).}\n Given $(w, l) \\in Q_{r\/4}(x, t)$, (\\ref{(3.9)}) still holds when $(x, t)$ is replaced by $(w, l)$.  It is clear that, for $(w, l) \\in Q_{r\/4}(x, t)$,\n\\[\n|\\nabla_w J_4|  \\le \\frac{C}{r^4} \\int_{B_r(x)-B_{r\/2}(x)} |u(y, t)| dy.\n\\] The rest of the proof of the lemma is the same as that on p382-285 of \\cite{Z:4} by taking the gradient of $J_1, J_2,  J_3$. \\qed\n\n\n\n\n\n\nWith the lemma in hand, the proof of Theorem \\ref{th:1.2} is then the same as the scaled iteration on p385-388 of \\cite{Z:4} with the additional mild integral terms $J_4$ and $|\\nabla J_4|$. Note these additional terms are spatial integrals only. These account for the difference\nbetween the result of Navier Stokes equation and related result for the heat equation.\n\n\n\n{\\bf Acknowledgement} We wish to sincerely thank Professor Hongjie Dong for pointing out  the issue of the theorem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Dataset}\n\\label{appendix-dataset}\n\n\\subsection{Dataset requirements}\n\nOur dataset $\\mathcal{D}$ is composed of triples of the form\n$(A, B, A \\vDash{} B)$,\nwhere $A \\vDash{} B$ is 1 if $A$ entails\\footnote{Throughout, we focus on \\emph{classical} propositional logic, and do not consider e.g., intuitionistic entailment.} $B$, and 0 otherwise.\nFor example:\n\\begin{eqnarray*}\n(p \\wedge q, q, 1) \\\\\n(q \\vee r, r, 0)\n\\end{eqnarray*}\n\nWe wanted to ensure that simple baseline models are unable to exploit simple statistical regularities to perform well in this task.\nWe define a series of baseline models which, due to their structure or the information they have access to, should not be able to solve the entailment recognition problem described in this paper. We distinguish baselines for which we believe there is little chance of them detecting entailment, from those for which there categorically cannot be true modelling of entailment.\nThe baselines which categorically cannot detect entailment are encoding models which only observe one side of the sequent:\n\\[\nP(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(f(A))\\right) \\quad \\textrm{or} \\quad P(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(f(B))\\right)\n\\]\nwhere $f$ is a linear bag of words encoder, an MLP bag of words encoder, or a TreeNet.\n\nBecause the dataset contains a roughly balanced number of positive and negative examples, it follows that we should expect any model which only sees part of the sequent to perform in line with a random classifier. If they outperform a random baseline on test, there is a structural or symbolic regularity on one side (or both) which is sufficient to identify some subset of positive or negative examples. \nWe use these baselines to verify the soundness of the generation process.\n\nLet $\\mathcal{D}^+$ and $\\mathcal{D}^-$ be the positive and negative entailments:\n\\begin{eqnarray*}\n\\mathcal{D}^+ = \\{ (A, B) \\; | \\; (A, B, 1) \\in \\mathcal{D} \\} \\\\\n\\mathcal{D}^- = \\{ (A, B) \\; | \\; (A, B, 0) \\in \\mathcal{D} \\}\n\\end{eqnarray*}\n\n\nWe impose various requirements on the dataset, to rule out  superficial syntactic differences between  $\\mathcal{D}^+$ and  $\\mathcal{D}^-$ that can be easily exploited by the simple baselines described above.\nWe require that our classes are balanced:\n\\begin{eqnarray*}\n|\\mathcal{D}^+| & = & |\\mathcal{D}^-|\n\\end{eqnarray*}\nWe do not want there to be any obvious difference in the length of formulas in $\\mathcal{D}^+$ and  $\\mathcal{D}^-$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} length(A) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} length(A) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} length(B) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} length(B) \\\\\n\\end{eqnarray*}\nWe want there to be the same number of new free variables (variables appearing in B that do not appear in A) in both $\\mathcal{D}^+$ and  $\\mathcal{D}^-$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} |vars(B)-vars(A)| & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} |vars(B)-vars(A)|\n\\end{eqnarray*}\n\n\n\nLet $num(A, op)$ be the number of occurrences of operator $op$ in formula $A$. So, for example, $num(\\neg (p \\wedge \\neg q), \\neg) = 2$. \nWe impose the constraint that for each operator $op \\in \\{\\neg, \\wedge, \\vee, \\rightarrow\\}$, that\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num(A, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num(A, op) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num(B, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num(B, op)\n\\end{eqnarray*}\nFurthermore, we require that the number of occurrences of an operator \\emph{at each level in the abstract syntax tree} is the same in $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nIt would not be acceptable if, for example, a typical $B^+$ from $\\mathcal{D}^+$ had more disjunctions at the top of the syntax tree than $B^-$ from $\\mathcal{D}^-$.\nLet $num\\_at(B, level, op)$ be the number of occurrences of operator $op$ at $level$ in the syntax tree for $B$. \nWe also require that, for each $op$ and $level$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num\\_at(A, level, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num\\_at(A, level, op) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num\\_at(B, level, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num\\_at(B, level, op)\n\\end{eqnarray*}\n\n\\subsection{Dataset generation}\n\n\\subsubsection{A naive approach to dataset generation}\n\nA simple way to generate an entailment dataset would be to alternate between first sampling formulas $A^+$ and $B^+$ such that $A^+ \\vDash{} B^+$, and second sampling formulas $A^-$ and $B^-$ such that $A^- \\nvDash{} B^-$.\nSince we are alternating between $\\vDash{}$ and $\\nvDash{}$, we are guaranteed to produce balanced classes.\nUnfortunately, this straightforward approach generates datasets that violate most of our requirements above. \nSee Table \\ref{dataset-naive} for the details.\n\nIn particular, the mean number of negations, conjunctions, and disjunctions at the top of the syntax tree ($num\\_at(\\cdot, 0, op)$) is markedly different.\n$A^+$ has significantly more conjunctions at the top of the syntax tree than $A^-$, while $B^+$ has significantly fewer than $B^-$.\nConversely, $A^+$ has significantly fewer disjunctions at the top of the syntax tree than $A^-$, while $B^+$ has significantly more than $B^-$.\n\nThe mean number of satisfying truth-value assignments ($sat(\\cdot)$) is also markedly different: $A^+$ is true in on average $3.7$ truth-value assignments (i.e. it is a very specific formula which is only true under very particular circumstances), while $A^-$ is true in $10.3$ truth-value assignments (i.e. it is true in a wider range of circumstances).\n\nIf we look at the mean number of variables appearing in $B$ that do not appear in $A$, there is also a striking difference between $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nThe mean number of new variables in $vars(B^+) - vars(A^+)$ is $0.80$ while the mean number of new variables in $vars(B^-) - vars(A^-)$ is $1.39$ with a $\\chi^2$ of $3308.1$ and $8$ degrees of freedom. \n\nWe can use these statistics to develop simple heuristic baselines that will be unreasonably effective on the dataset described above:\nwe can estimate whether $A \\vDash{} B$ by comparing the lengths of $A$ and $B$, or by looking at the number of variables in $B$ that do not appear in $A$, or by looking at the topmost connective in $A$ and $B$.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Requirement violations in the naive approach, with $|\\mathcal{D}| = 50,000$}\n\\label{dataset-naive}\n\\begin{tabular}{lllllllll}\n\\hline\n& {\\bf $A^+$} & {\\bf $A^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df & {\\bf $B^+$} & {\\bf $B^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df \\\\ \\hline\n{$length(.)$} & 6.62 & 6.45 & 70.6 & 9 & 8.33 & 8.28 & 304.9 & 16 \\\\\n{\\bf $num(\\cdot, \\neg)$} & 1.47 & 1.33 & 309.4 & 8 & 1.77 & 1.91 & 139.0 & 9 \\\\\n{\\bf $num(\\cdot, \\wedge)$} & 1.52 & 1.33 & 308.6 & 8 & 1.70 & 1.94 & 134.0 & 11 \\\\\n{\\bf $num(\\cdot, \\vee)$} & 1.30 & 1.40 & 86.9 & 8 & 1.95 & 1.69 & 127.0 & 10 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\neg)$} & 0.31 & 0.22 & 532.4 & 1 & 0.18 & 0.30 &350.9 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\neg)$} & 0.32 & 0.31 & 7.5 & 2 & 0.39 & 0.41 & 3.2 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\neg)$} & 0.31 & 0.31 & 8.8 & 4 & 0.56 & 0.54 & 5.3 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\wedge)$} & {\\color{red} 0.35} & {\\color{red} 0.2} & 1382.9 & 1 & {\\color{red} 0.13} & {\\color{red} 0.33} & 1076.4 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\wedge)$} & 0.32 & 0.31 & 36.5 & 2 & 0.39 & 0.40 & 6.5 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\wedge)$} & 0.31 & 0.32 & 3.2 & 4 & 0.56 & 0.53 & 16.5 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\vee)$} & {\\color{red} 0.16} & {\\color{red} 0.28} & 1070.3 & 1 & {\\color{red} 0.34} & {\\color{red} 0.16} & 752.4 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\vee)$} & 0.30 & 0.32 & 66.0 & 2 & 0.42 & 0.34 & 141.1 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\vee)$} & 0.32 & 0.31 & 12.9 & 4 & 0.57 & 0.52 & 39.7 & 4 \\\\\n{\\bf $\\#sat(\\cdot)$} & {\\color{red} 3.7} & {\\color{red} 10.3} & 11265 & 174 & {\\color{red} 22.1} & {\\color{red} 11.7} & 3702.8 & 241 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{Our preferred approach to dataset generation}\n\nIn order to satisfy our requirements above, we took a different approach to dataset generation.\nIn order to ensure that there are no crude statistical measurements that can detect differences between $\\mathcal{D}^+$ and $\\mathcal{D}^-$, we change the generation procedure so that every formula appears in both $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nWe sample 4-tuples of formulas $(A_1, B_1, A_2, B_2)$ such that:\n\\begin{eqnarray*}\nA_1 & \\vDash{} & B_1 \\\\\nA_2 & \\vDash{} & B_2 \\\\\nA_1 & \\nvDash{} & B_2 \\\\\nA_2 & \\nvDash{} & B_1\n\\end{eqnarray*}\nHere, each of the four formulas appears in one positive entailment and one negative entailment\\footnote{One consequence of this method is that it rules out $A_1$ from being impossible (if it was impossible, we would not have $A_1 \\nvDash{} B_2$) and $B_1$ from being a tautology (if it was a tautology, we would not have $A_2 \\nvDash{} B_1)$.}.\n\nUsing this alternative approach, we are able to satisfy the requirements above.\nBy construction, the mean length, number of operators at a certain level in the syntax tree, and the number of satisfying truth-value assignments is \\emph{exactly the same} for $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nSee Table \\ref{dataset-preferred}.\n\nThe only crude difference remaining is in the number of new variables.\nIf we look at the number of variables appearing in $B$ that do not appear in $A$, there is a noticeable difference between $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nThe mean number of new variables in $vars(B^+) - vars(A^+)$ is $1.25$ while the mean number of new variables in $vars(B^-) - vars(A^-)$ is $1.60$ with a $\\chi^2$ of $922.1$ and $8$ degrees of freedom. \n\n\\begin{table}[ht]\n\\centering\n\\caption{Statistics for the preferred approach that generates 4-tuples, with $|\\mathcal{D}| = 50,000$}\n\\label{dataset-preferred}\n\\begin{tabular}{lllllllll}\n\\hline\n& {\\bf $A^+$} & {\\bf $A^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df & {\\bf $B^+$} & {\\bf $B^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df \\\\ \\hline\n{$length(.)$} & 6.33 & 6.33 & 0.0 & 9 & 6.38 & 6.38 & 0.0 & 16 \\\\\n{\\bf $num(\\cdot, \\neg)$} & 1.42 & 1.42 & 0.0 & 9 & 1.26 & 1.26 & 0.0 & 8 \\\\\n{\\bf $num(\\cdot, \\wedge)$} & 1.63 & 1.63 & 0.0 & 7 & 1.16 & 1.16 & 0.0 & 7 \\\\\n{\\bf $num(\\cdot, \\vee)$} & 1.14 & 1.14 & 0.0 & 7 & 1.53 & 1.53 & 0.0 & 8 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\neg)$} & 0.33 & 0.33 & 0.0 & 1 & 0.16 & 0.16 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\neg)$} & 0.29 & 0.29 & 0.0 & 2 & 0.32 & 0.32 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\neg)$} & 0.30 & 0.30 & 0.0 & 3 & 0.31 & 0.31 & 0.0 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\wedge)$} & 0.49 & 0.49 & 0.0 & 1 & 0.1 & 0.1 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\wedge)$} & 0.34 & 0.34 & 0.0 & 2 & 0.31 & 0.31 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\wedge)$} & 0.30 & 0.30 & 0.0 & 4 & 0.30 & 0.30 & 0.0 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\vee)$} & 0.08 & 0.08 & 0.0 & 1 & 0.39 & 0.39 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\vee)$} & 0.27 & 0.27 & 0.0 & 2 & 0.35 & 0.35 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\vee)$} & 0.29 & 0.29 & 0.0 & 3 & 0.29 & 0.29 & 0.0 & 3 \\\\\n{\\bf $\\#sat(\\cdot)$} & 3.86 & 3.86 & 0.0 & 86 & 14.42 & 14.42 & 0.0 & 157 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Dataset Example}\n\nOur method generates 4-tuples such as the following:\n\\begin{eqnarray*}\np \\vee p & \\vDash{} & (r \\rightarrow c) \\rightarrow ((r \\rightarrow v) \\vee p) \\\\\n((g \\vee p) \\vee s) \\rightarrow (g \\rightarrow g) \\wedge r & \\vDash{} & r \\wedge (r \\rightarrow r) \\\\\np \\vee p & \\nvDash{} & r \\wedge (r \\rightarrow r) \\\\\n((g \\vee p) \\vee s) \\rightarrow (g \\rightarrow g) \\wedge r & \\nvDash{} & (r \\rightarrow c) \\rightarrow ((r \\rightarrow v) \\vee p)\n\\end{eqnarray*}\n\n\\section{The Dataset}\n\\label{appendix-dataset}\n\n\\subsection{Dataset requirements}\n\nOur dataset $\\mathcal{D}$ is composed of triples of the form\n$(A, B, A \\vDash{} B)$,\nwhere $A \\vDash{} B$ is 1 if $A$ entails\\footnote{Throughout, we focus on \\emph{classical} propositional logic, and do not consider e.g., intuitionistic entailment.} $B$, and 0 otherwise.\nFor example:\n\\begin{eqnarray*}\n(p \\wedge q, q, 1) \\\\\n(q \\vee r, r, 0)\n\\end{eqnarray*}\n\nWe wanted to ensure that simple baseline models are unable to exploit simple statistical regularities to perform well in this task.\nWe define a series of baseline models which, due to their structure or the information they have access to, should not be able to solve the entailment recognition problem described in this paper. We distinguish baselines for which we believe there is little chance of them detecting entailment, from those for which there categorically cannot be true modelling of entailment.\nThe baselines which categorically cannot detect entailment are encoding models which only observe one side of the sequent:\n\\[\nP(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(f(A))\\right) \\quad \\textrm{or} \\quad P(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(f(B))\\right)\n\\]\nwhere $f$ is a linear bag of words encoder, an MLP bag of words encoder, or a TreeNet.\n\nBecause the dataset contains a roughly balanced number of positive and negative examples, it follows that we should expect any model which only sees part of the sequent to perform in line with a random classifier. If they outperform a random baseline on test, there is a structural or symbolic regularity on one side (or both) which is sufficient to identify some subset of positive or negative examples. \nWe use these baselines to verify the soundness of the generation process.\n\nLet $\\mathcal{D}^+$ and $\\mathcal{D}^-$ be the positive and negative entailments:\n\\begin{eqnarray*}\n\\mathcal{D}^+ = \\{ (A, B) \\; | \\; (A, B, 1) \\in \\mathcal{D} \\} \\\\\n\\mathcal{D}^- = \\{ (A, B) \\; | \\; (A, B, 0) \\in \\mathcal{D} \\}\n\\end{eqnarray*}\n\n\nWe impose various requirements on the dataset, to rule out  superficial syntactic differences between  $\\mathcal{D}^+$ and  $\\mathcal{D}^-$ that can be easily exploited by the simple baselines described above.\nWe require that our classes are balanced:\n\\begin{eqnarray*}\n|\\mathcal{D}^+| & = & |\\mathcal{D}^-|\n\\end{eqnarray*}\nWe do not want there to be any obvious difference in the length of formulas in $\\mathcal{D}^+$ and  $\\mathcal{D}^-$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} length(A) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} length(A) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} length(B) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} length(B) \\\\\n\\end{eqnarray*}\nWe want there to be the same number of new free variables (variables appearing in B that do not appear in A) in both $\\mathcal{D}^+$ and  $\\mathcal{D}^-$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+} |vars(B)-vars(A)| & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-} |vars(B)-vars(A)|\n\\end{eqnarray*}\n\n\n\nLet $num(A, op)$ be the number of occurrences of operator $op$ in formula $A$. So, for example, $num(\\neg (p \\wedge \\neg q), \\neg) = 2$. \nWe impose the constraint that for each operator $op \\in \\{\\neg, \\wedge, \\vee, \\rightarrow\\}$, that\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num(A, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num(A, op) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num(B, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num(B, op)\n\\end{eqnarray*}\nFurthermore, we require that the number of occurrences of an operator \\emph{at each level in the abstract syntax tree} is the same in $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nIt would not be acceptable if, for example, a typical $B^+$ from $\\mathcal{D}^+$ had more disjunctions at the top of the syntax tree than $B^-$ from $\\mathcal{D}^-$.\nLet $num\\_at(B, level, op)$ be the number of occurrences of operator $op$ at $level$ in the syntax tree for $B$. \nWe also require that, for each $op$ and $level$:\n\\begin{eqnarray*}\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num\\_at(A, level, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num\\_at(A, level, op) \\\\\n\\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^+}num\\_at(B, level, op) & = & \\mathop{\\mathbb{E}}_{(A,B) \\sim \\mathcal{D}^-}num\\_at(B, level, op)\n\\end{eqnarray*}\n\n\\subsection{Dataset generation}\n\n\\subsubsection{A naive approach to dataset generation}\n\nA simple way to generate an entailment dataset would be to alternate between first sampling formulas $A^+$ and $B^+$ such that $A^+ \\vDash{} B^+$, and second sampling formulas $A^-$ and $B^-$ such that $A^- \\nvDash{} B^-$.\nSince we are alternating between $\\vDash{}$ and $\\nvDash{}$, we are guaranteed to produce balanced classes.\nUnfortunately, this straightforward approach generates datasets that violate most of our requirements above. \nSee Table \\ref{dataset-naive} for the details.\n\nIn particular, the mean number of negations, conjunctions, and disjunctions at the top of the syntax tree ($num\\_at(\\cdot, 0, op)$) is markedly different.\n$A^+$ has significantly more conjunctions at the top of the syntax tree than $A^-$, while $B^+$ has significantly fewer than $B^-$.\nConversely, $A^+$ has significantly fewer disjunctions at the top of the syntax tree than $A^-$, while $B^+$ has significantly more than $B^-$.\n\nThe mean number of satisfying truth-value assignments ($sat(\\cdot)$) is also markedly different: $A^+$ is true in on average $3.7$ truth-value assignments (i.e. it is a very specific formula which is only true under very particular circumstances), while $A^-$ is true in $10.3$ truth-value assignments (i.e. it is true in a wider range of circumstances).\n\nIf we look at the mean number of variables appearing in $B$ that do not appear in $A$, there is also a striking difference between $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nThe mean number of new variables in $vars(B^+) - vars(A^+)$ is $0.80$ while the mean number of new variables in $vars(B^-) - vars(A^-)$ is $1.39$ with a $\\chi^2$ of $3308.1$ and $8$ degrees of freedom. \n\nWe can use these statistics to develop simple heuristic baselines that will be unreasonably effective on the dataset described above:\nwe can estimate whether $A \\vDash{} B$ by comparing the lengths of $A$ and $B$, or by looking at the number of variables in $B$ that do not appear in $A$, or by looking at the topmost connective in $A$ and $B$.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Requirement violations in the naive approach, with $|\\mathcal{D}| = 50,000$}\n\\label{dataset-naive}\n\\begin{tabular}{lllllllll}\n\\hline\n& {\\bf $A^+$} & {\\bf $A^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df & {\\bf $B^+$} & {\\bf $B^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df \\\\ \\hline\n{$length(.)$} & 6.62 & 6.45 & 70.6 & 9 & 8.33 & 8.28 & 304.9 & 16 \\\\\n{\\bf $num(\\cdot, \\neg)$} & 1.47 & 1.33 & 309.4 & 8 & 1.77 & 1.91 & 139.0 & 9 \\\\\n{\\bf $num(\\cdot, \\wedge)$} & 1.52 & 1.33 & 308.6 & 8 & 1.70 & 1.94 & 134.0 & 11 \\\\\n{\\bf $num(\\cdot, \\vee)$} & 1.30 & 1.40 & 86.9 & 8 & 1.95 & 1.69 & 127.0 & 10 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\neg)$} & 0.31 & 0.22 & 532.4 & 1 & 0.18 & 0.30 &350.9 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\neg)$} & 0.32 & 0.31 & 7.5 & 2 & 0.39 & 0.41 & 3.2 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\neg)$} & 0.31 & 0.31 & 8.8 & 4 & 0.56 & 0.54 & 5.3 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\wedge)$} & {\\color{red} 0.35} & {\\color{red} 0.2} & 1382.9 & 1 & {\\color{red} 0.13} & {\\color{red} 0.33} & 1076.4 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\wedge)$} & 0.32 & 0.31 & 36.5 & 2 & 0.39 & 0.40 & 6.5 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\wedge)$} & 0.31 & 0.32 & 3.2 & 4 & 0.56 & 0.53 & 16.5 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\vee)$} & {\\color{red} 0.16} & {\\color{red} 0.28} & 1070.3 & 1 & {\\color{red} 0.34} & {\\color{red} 0.16} & 752.4 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\vee)$} & 0.30 & 0.32 & 66.0 & 2 & 0.42 & 0.34 & 141.1 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\vee)$} & 0.32 & 0.31 & 12.9 & 4 & 0.57 & 0.52 & 39.7 & 4 \\\\\n{\\bf $\\#sat(\\cdot)$} & {\\color{red} 3.7} & {\\color{red} 10.3} & 11265 & 174 & {\\color{red} 22.1} & {\\color{red} 11.7} & 3702.8 & 241 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{Our preferred approach to dataset generation}\n\nIn order to satisfy our requirements above, we took a different approach to dataset generation.\nIn order to ensure that there are no crude statistical measurements that can detect differences between $\\mathcal{D}^+$ and $\\mathcal{D}^-$, we change the generation procedure so that every formula appears in both $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nWe sample 4-tuples of formulas $(A_1, B_1, A_2, B_2)$ such that:\n\\begin{eqnarray*}\nA_1 & \\vDash{} & B_1 \\\\\nA_2 & \\vDash{} & B_2 \\\\\nA_1 & \\nvDash{} & B_2 \\\\\nA_2 & \\nvDash{} & B_1\n\\end{eqnarray*}\nHere, each of the four formulas appears in one positive entailment and one negative entailment\\footnote{One consequence of this method is that it rules out $A_1$ from being impossible (if it was impossible, we would not have $A_1 \\nvDash{} B_2$) and $B_1$ from being a tautology (if it was a tautology, we would not have $A_2 \\nvDash{} B_1)$.}.\n\nUsing this alternative approach, we are able to satisfy the requirements above.\nBy construction, the mean length, number of operators at a certain level in the syntax tree, and the number of satisfying truth-value assignments is \\emph{exactly the same} for $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nSee Table \\ref{dataset-preferred}.\n\nThe only crude difference remaining is in the number of new variables.\nIf we look at the number of variables appearing in $B$ that do not appear in $A$, there is a noticeable difference between $\\mathcal{D}^+$ and $\\mathcal{D}^-$.\nThe mean number of new variables in $vars(B^+) - vars(A^+)$ is $1.25$ while the mean number of new variables in $vars(B^-) - vars(A^-)$ is $1.60$ with a $\\chi^2$ of $922.1$ and $8$ degrees of freedom. \n\n\\begin{table}[ht]\n\\centering\n\\caption{Statistics for the preferred approach that generates 4-tuples, with $|\\mathcal{D}| = 50,000$}\n\\label{dataset-preferred}\n\\begin{tabular}{lllllllll}\n\\hline\n& {\\bf $A^+$} & {\\bf $A^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df & {\\bf $B^+$} & {\\bf $B^-$} & {\\bf $\\chi^2$} & $\\chi^2$ df \\\\ \\hline\n{$length(.)$} & 6.33 & 6.33 & 0.0 & 9 & 6.38 & 6.38 & 0.0 & 16 \\\\\n{\\bf $num(\\cdot, \\neg)$} & 1.42 & 1.42 & 0.0 & 9 & 1.26 & 1.26 & 0.0 & 8 \\\\\n{\\bf $num(\\cdot, \\wedge)$} & 1.63 & 1.63 & 0.0 & 7 & 1.16 & 1.16 & 0.0 & 7 \\\\\n{\\bf $num(\\cdot, \\vee)$} & 1.14 & 1.14 & 0.0 & 7 & 1.53 & 1.53 & 0.0 & 8 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\neg)$} & 0.33 & 0.33 & 0.0 & 1 & 0.16 & 0.16 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\neg)$} & 0.29 & 0.29 & 0.0 & 2 & 0.32 & 0.32 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\neg)$} & 0.30 & 0.30 & 0.0 & 3 & 0.31 & 0.31 & 0.0 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\wedge)$} & 0.49 & 0.49 & 0.0 & 1 & 0.1 & 0.1 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\wedge)$} & 0.34 & 0.34 & 0.0 & 2 & 0.31 & 0.31 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\wedge)$} & 0.30 & 0.30 & 0.0 & 4 & 0.30 & 0.30 & 0.0 & 4 \\\\\n{\\bf $num\\_at(\\cdot, 0, \\vee)$} & 0.08 & 0.08 & 0.0 & 1 & 0.39 & 0.39 & 0.0 & 1 \\\\\n{\\bf $num\\_at(\\cdot, 1, \\vee)$} & 0.27 & 0.27 & 0.0 & 2 & 0.35 & 0.35 & 0.0 & 2 \\\\\n{\\bf $num\\_at(\\cdot, 2, \\vee)$} & 0.29 & 0.29 & 0.0 & 3 & 0.29 & 0.29 & 0.0 & 3 \\\\\n{\\bf $\\#sat(\\cdot)$} & 3.86 & 3.86 & 0.0 & 86 & 14.42 & 14.42 & 0.0 & 157 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Dataset Example}\n\nOur method generates 4-tuples such as the following:\n\\begin{eqnarray*}\np \\vee p & \\vDash{} & (r \\rightarrow c) \\rightarrow ((r \\rightarrow v) \\vee p) \\\\\n((g \\vee p) \\vee s) \\rightarrow (g \\rightarrow g) \\wedge r & \\vDash{} & r \\wedge (r \\rightarrow r) \\\\\np \\vee p & \\nvDash{} & r \\wedge (r \\rightarrow r) \\\\\n((g \\vee p) \\vee s) \\rightarrow (g \\rightarrow g) \\wedge r & \\nvDash{} & (r \\rightarrow c) \\rightarrow ((r \\rightarrow v) \\vee p)\n\\end{eqnarray*}\n\n\\section{Introduction}\n\nThis paper seeks to answer two questions: ``Can neural networks understand logical formulae well enough to detect entailment?'', and, more generally, ``Which architectures are best at inferring, encoding, and relating features in a purely structural sequence-based problem?''. In answering these questions, we aim to better understand the inductive biases of popular architectures with regard to structure and abstraction in sequence data. Such understanding would help pave the road to agents and classifiers that reason structurally, in addition to reasoning on the basis of essentially semantic representations. In this paper, we provide a testbed for evaluating some aspects of neural networks' ability to reason structurally and abstractly. We use it to compare a variety of popular network architectures and a new model we introduce, called PossibleWorldNet.\n\nNeural network architectures lie at the heart of a variety of applications. They are practically ubiquitous across vision tasks~\\citep{lecun1995convolutional,krizhevsky2012imagenet,simonyan2014very} and natural language understanding, from machine translation~\\citep{kalchbrenner2013recurrent,sutskever2014sequence,bahdanau2014neural} to textual entailment~\\citep{bowman2015large,rocktaschel2015reasoning} via sentiment analysis~\\citep{socher2013recursive,kalchbrenner2014convolutional} and reading comprehension~\\citep{hermann2015teaching,hill2015goldilocks,rajpurkar2016squad}. They have been used to synthesise programs~\\citep{ling2016latent,parisotto2016neuro,devlin2017robustfill} or internalise algorithms~\\citep{graves2016hybrid,grefenstette2015learning,joulin2015inferring,kaiser2015neural,reed2015neural}. They form the basis of reinforcement learning agents capable of playing video games~\\citep{mnih2015human}, difficult perfect information games~\\citep{silver2016mastering,tian2015better}, and navigating complex environments from raw pixels~\\citep{mirowski2016learning}. An important question in this context is to find the inductive and generalisation properties of different neural architectures, particularly towards the ability to capture structure present in the input, an ability that might be important for many language and reasoning tasks. However, there is little work on studying these inductive biases in isolation by running these models on tasks that are primarily or purely \\emph{about} sequence structure, which we intend to address.\n\nThe paper's contribution is three-fold. First, we introduce a new dataset for training and evaluating models. Second, we provide a thorough evaluation of the existing neural models on this dataset.\nThird, inspired by the semantic (model-theoretic) definition of entailment, we propose a variant of the TreeNet that evaluates the formulas in multiple different ``possible worlds'', and which significantly outperforms the benchmarks.\nThe structure of this paper is as follows. In Section~\\ref{sec:dataset_creation}, we introduce the new dataset and describe a generic data generation process for entailment datasets, which offers certain guarantees against the presence of superficial exploitable biases. In Section~\\ref{sec:models}, we describe a series of baseline models used to validate the dataset, benchmarks from which we will derive our analyses of popular model architectures, and also introduce our new neural model, the {PossibleWorldNet}. In Section~\\ref{sec:exp_setup}, we describe the structure of experiments, from which we obtained the results presented and discussed in Section~\\ref{sec:results}. We offer a brief survey of related work in Section~\\ref{sec:related}, before making concluding remarks in Section~\\ref{sec:conclusion}.\n\n\\section{Dataset Creation}\n\\label{sec:dataset_creation}\n\nFormal logics provide a symbolic toolkit for encoding and examining patterns of reasoning. They are structural calculi aiming to codify the norms of correct thought. The meanings of such statements are invariant to what the particular propositions stand for: to understand the entailment $(p \\wedge q) \\vDash{} q$, we only need to understand the semantics of---or related syntactic rules governing---a finite set of logical connectives, while $p$ and $q$ are meaningless arbitrary symbols selected to stand for distinct propositions. In other words, the problem of determining whether an entailment holds is a purely structural sequence-based problem: to evaluate whether an entailment is true, only the meaning of---or inference rules governing---the connectives is relevant. Everything else only has meaning via its place in the structure specified by an expression. These qualities suggest that detecting logical entailment is an excellent task for measuring the ability of models to capture, understand, or exploit structure. We present in this paper a generic process for generating entailment datasets, explained in detail in Appendix~\\ref{appendix-dataset}, for any given logical system. In the specific dataset---generated through this process---presented in this section, we will focus on propositional logic, which is decidable but requires a worst case of $O(2^n)$ operations (e.g.~resolution steps, truth table rows), where $n$ is the number of unique propositional variables, to verify entailment.\n\nOur dataset\\footnote{We aim to release the dataset used for experiments, and the code used to generate it according to the constraints discussed in this paper, upon publication of the paper.} $\\mathcal{D}$ is composed of triples of the form\n$(A, B, A \\vDash{} B)$,\nwhere $A$ and $B$ are formulas of propositional logic, and $A \\vDash{} B$ is 1 if $A$ entails $B$, and 0 otherwise. For example, the data point $(p \\wedge q, q, 1)$ is positive because $p \\wedge q$ entails $q$, whereas $(q \\vee r, r, 0)$ is negative because $q \\vee r$ does not entail $r$.\nEntailment is primarily a semantic notion: $A$ entails $B$ if every model in which $A$ is true is also a model in which $B$ is true.\n\nWe impose various requirements on the dataset, to rule out superficial structural differences between  $\\mathcal{D}^+$ and  $\\mathcal{D}^-$ that can be easily exploited by ``trivial'' baselines\\footnote{E.g., bag-of-words baselines that cannot look at the structure of $A$ and $B$, or baselines that ignore $A$ and only look at $B$.}.\nWe impose the following high level constraints on our data through the generative process, explained in detail in Appendix~\\ref{appendix-dataset}: our classes must be balanced, and formulas in positive and negative examples must have the same distribution over length. Furthermore, we attempt to ensure that there are no recognisable differences in the distributions of lexical or syntactic features between the positive and negative examples. It would not be acceptable, for example, if a typical $B$ formula in a positive entailment $(A, B, 1)$ had more disjunctions than a $B'$ formula in a negative entailment $(A', B', 0)$.\n\nIf we simply sample formulas $A$ and $B$ and evaluate whether $A \\vDash{} B$, there are significant differences between the distributions of formulas for the positive and negative examples, which models can learn to exploit without needing to understand the structure of the problem. To avoid these issues, we use a different approach, that satisfies the above requirements. We sample 4-tuples of formulas $(A_1, B_1, A_2, B_2)$ such that:\n\\[\n A_1  \\vDash{} B_1 \\qquad\n A_2  \\vDash{} B_2 \\qquad\n A_1 \\nvDash{} B_2 \\qquad\n A_2  \\nvDash{} B_1\n\\]\nHere, each of the four formulas appears in one positive entailment and one negative entailment.\nThis way, we minimise crude structural differences between the positive and negative examples.\nHere is a simple example (although the actual dataset has much longer formulas) of such a 4-tuple of datapoints:\n\\[\n p \\vDash{}  p \\vee q \\qquad\n \\neg p \\wedge \\neg q  \\vDash{}  \\neg q \\qquad\n p  \\nvDash{}  \\neg q \\qquad\n \\neg p \\wedge \\neg q \\nvDash{}  p \\vee q\n\\]\nTo generate these 4-tuples, we first generate pairs $(A, B)$ such that $A \\vDash{} B$.\n(To test if $A \\vDash{} B$, we test whether $A \\wedge \\neg B$ is satisfiable, using {\\bf minisat} \\citep{sorensson2005minisat}).\nThen we search through the set of pairs, looking for pairs of pairs, $(A_1, B_1)$ and $(A_2, B_2)$, such that $A_1 \\nvDash{} B_2$ and $A_2 \\nvDash{} B_1$. We present, in Appendix \\ref{appendix-dataset}, the full details of this generative process, its constraints and guarantees, and how we used particular baselines to validate the data.\n\n\\subsection{Splitting the Dataset}\n\nWe produced train, validation, and test (easy) by generating one large set of 4-tuples, and splitting them into groups of sizes 100000, 5000, and 5000. The difficulty of evaluating an entailment depends on the number of propositional variables and the number of operators in the two formulas. In training, validation, and test (easy), we sample the number of propositional variables uniformly between 1 and 10 (there are 26 propositional variables in total: $a$ to $z$). In test (hard), we sample uniformly between 5 and 10. Our formula sampling method takes a parameter specifying the desired number of operators in the formula. In training, validation, and test (easy), the number of operators in a formula is sampled uniformly between 1 and 10. In our hard test set, the number of operators in a formula is sampled uniformly between 15 and 20.\n\nFor the test (big) dataset, we sampled formulas using between 1 and 20 variables (uniformly), and between 10 and 30 operators (again, uniformly).\nFor test (massive), we used a different generating mechanism.\nWe first sampled pairs of formulas $A$, $B$ such that $A \\models B$.\nThese had between 20 and 26 variables, and between 20 and 30 operators each.\nThen we generated a $B^*$ by mutating $B$ and checking that $A \\nvDash B^*$.\nSee Table \\ref{dataset-statistics} for detailed statistics of the dataset sections, including the average difficulty (based on a complexity of $\\mathcal{O}(2^{\\textbf{\\# Vars}})$) of sequents in each fold.\n\nThe test (exam) dataset was assembled from 100 examples of logical entailment \"in the wild\".\nWe looked through various logic textbooks for classic examples of entailments.\nFrom these textbooks, we extracted true entailment triples $(A, B, 1)$ where $A \\models B$.\nWe added false triples $(A, B^*, 0)$, by mutating $B$ into $B^*$ and checking that $A \\nvDash B^*$.\n\n\\begin{table}[tb]\n\\centering\n\\caption{Dataset Statistics}\n\\label{dataset-statistics}\n\\begin{tabular}{lrrrrr}\n\\toprule\n& {\\bf Size} & {\\bf \\shortstack[c]{Mean\\\\ \\# Vars}} & {\\bf \\shortstack[c]{Mean\\\\ \\# Ops}} & {\\bf \\shortstack[c]{Mean\\\\ Length}} & {\\bf \\shortstack[c]{Mean \\\\ $2^{\\textbf{\\# Vars}}$}} \\\\\n\\midrule\n{\\bf Train} & 100,000 & 4.5 & 5.3 & 11.3 & 52.2 \\\\\n{\\bf Validate} & 5,000 & 5.1 & 6.8 & 13.0 & 75.7 \\\\\n{\\bf Test (easy)} & 5,000 & 5.2 & 6.9 & 13.1 & 81.0 \\\\\n{\\bf Test (hard)} & 5,000 & 5.8 & 17.4 & 31.5 & 184.4 \\\\\n{\\bf Test (big)} & 5,000 & 8.0 & 20.9 & 38.7 & 3310.8 \\\\\n{\\bf Test (massive)} & 2,230 & 18.4 & 49.4 & 88.8 & 848,570.0 \\\\\n{\\bf Test (exam)} & 100 & 2.4 & 3.9 & 8.6 & 5.8 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nIn order to test models' ability to generalise to new unseen formulas,\nwe pruned out cases where formulas seen in validation and test were $\\alpha$-equivalent (equivalent up to renaming of symbols) to formulas seen in training.\nSo, for example, if it had seen $p \\models (\\neg q \\wedge p)$ in training, we did not want $r \\models (\\neg s \\wedge r)$ to appear in either the test or validation sets.\nTo do this, we converted all formulas to de-Bruijn form (see~\\cite{pierce2002types}, Chapter 6), and filtered out formulas in validation and test whose de-Bruijn form was identical to one of those in training.\nThis prevents the system from being able to simply memorise examples it has seen in training.\n\n\\subsection{Data Augmentation through Symbolic Vocabulary Permutation}\n\\label{ssec:data_aug}\n\nAs discussed above, the logical connectives ($\\lor$, $\\land$, \\ldots) are the only elements of the language in each dataset that have consistent implicit semantics across expressions.\nIn this sense, two entailments $p \\land q \\vDash{} q$ and $a \\land b \\vDash{} b$ should ideally be treated as identical by the model. To encourage models to capture this invariance, we add an optional data processing layer during training (not testing) whereby symbols are consistently replaced by other symbols of the same type within individual entailments before being input to the network according to the process described below. This is achieved by randomly sampling a permutation of $a, \\ldots, z$ (the propositional variables used) for every training example, and applying this permutation to the left and right sequents. This process is analogous to augmenting image classification training with random reflections and crops.\n\n\\section{Models}\n\\label{sec:models}\n\nIn this section, we first describe a couple of baseline models that verify the basic difficulty of the dataset, followed by a description of benchmark models which are commonly used (with some variation) in a variety of problems, and finally by a description of our new model, PossibleWorldNet.\n\n\\subsection{Baselines}\n\n\n\nThe classes in the dataset are balanced in training, validation, and both test sets, so a random baseline (and a constant, majority-class predicting baseline) will obtain an accuracy of 50\\% on the test sets.\n\nWe define two neural baselines which, we believe, should not be able to perform competitively on this task, but may do better than random. The first is a linear bag of words (Linear BoW) model which embeds each symbol to a vector, and averages them, to produce a representation of each side of the sequent. These representations are then passed through a linear layer:\n\\[\nP(A \\vDash{} B) = \\sigma \\left(W \\cdot \\text{concat} \\left(g(A), g(B)\\right) + b \\right) \\quad \\textrm{where} \\quad g(X) =  \\frac{1}{|X|} \\sum_{x \\in X} \\text{embed}(x)\n\\]\nThe second is a similar architecture, where the final linear layer is replaced with a multi-layer perceptron (MLP BoW):\n\\[\nP(A \\vDash{} B) = \\sigma (\\text{MLP}(\\text{concat}\\left(g(A), g(B)\\right))) \\quad \\textrm{where} \\quad g(X) = \\frac{1}{|X|} \\sum_{x \\in X} \\text{embed}(x)\n\\]\nIn both of these cases, the baselines are expected to have limited performance since they can only capture entailment by modelling the contribution of symbols individually, rather than by modelling structure, since the summation in $g$ destroys all structural information (including word order). We use these results to provide an indication of the difficulty of the dataset.\n\n\\subsection{Benchmarks}\n\\label{subsec:benchmarks}\n\nWe present here a series of benchmark models, not only to serve the purpose of being grounds for comparison for new models tested against this dataset, but also to compare and contrast the performance of fairly ubiquitous model architectures on this purely syntactic problem.\n\nWe distinguish two categories of models: encoding models and relational models. Encoding models, with exceptions specified below, jointly learn an encoding function $f$ and an MLP, such that given a sequent $A \\vDash{} B$, the model expresses\n\\[\nP(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(\\text{concat}(f(A), f(B)))\\right).\n\\]\nIn this sense $f$ produces a representation of each side of the sequent which contains all the information needed for the MLP to decide on entailment. In contrast, relational models will observe the pair of expressions and make a decision, perhaps by traversing both expressions, or by relating sub-structure of one expression to that of the other. These models express a more general formulation\n\\[\nP(A \\vDash{} B) = \\sigma \\left(f(A, B)\\right).\n\\]\n\n\\subsubsection{Encoder benchmarks}\n\nThe first encoder benchmark implemented is a Deep Convolutional Network Encoder (ConvNet Encoders), akin to architectures described in the convolutional networks for text literature~\\citep{kalchbrenner2014convolutional,zhang2015character,kim2016character}. Here, the encoder function $f$ is a stack of one dimensional convolutions over sequence symbols embedded by an embedding operation \\mbox{embedSeq}, interleaved with max pooling layers every $k$ layers (which is a model hyperparameter), followed by $n$ (also a hyperparameter) fully connected layers:\n\\[f(X) = \\text{MLP}(\\text{Conv1D}_n(\\ldots \\text{maxPool}(\\text{Conv1D}_k(\\ldots \\text{Conv1D}_1(\\text{embedSeq}(X)) \\ldots)) \\ldots))\\]\n\nThe second and third encoder benchmarks are an LSTM~\\citep{hochreiter1997long} encoder network (LSTM Encoders), and its bidirectional LSTM variant (BiDirLSTM Encoders). For the LSTM encoder, we embed the sequence symbols, and run an LSTM RNN over them, ignoring the output until the final state:\n\\[\nf(X) = h_\\text{final} \\quad \\textrm{where} \\quad h_\\text{final} = \\text{LSTM}(\\text{embedSeq}(X))\n\\]\nFor the bidirectional variant, two separate LSTM RNNs $\\text{LSTM}^{\\leftarrow}$ and $\\text{LSTM}^{\\rightarrow}$ are run over the sequence in opposite directions. Their respective final states are concatenated to form a representation of the expression:\n\\begin{align*}\n    f(X) = \\text{concat}(h_\\text{final}^{\\leftarrow}, h_\\text{final}^\\rightarrow)  \\quad  \\textrm{where} \\quad & h_\\text{final}^{\\leftarrow} = \\text{LSTM}^{\\leftarrow}(\\text{embedSeq}(X))\\\\\n     \\textrm{and} \\quad & h_\\text{final}^{\\rightarrow} = \\text{LSTM}^\\rightarrow(\\text{embedSeq}(X))\n\\end{align*}\n\nThe benchmarks described thus far do not explicitly condition on structure, even when it is known, as they are designed to traverse a sequence from left to right and model dependencies in the data implicitly. In contrast, we now consider encoder benchmarks which rely on the provision of the syntactic structure of the sequence they encode, and exploit it to determine the order of composition. This inductive bias, which may be incorrect in certain domains (e.g., where no syntax is defined) or difficult to achieve in domains such as natural language text (where syntactic structure is latent and ambiguous), is easy to achieve for logic (where the syntax is known). The experiments below will seek to demonstrate whether is a helpful inductive architectural bias.\n\nThe fourth and fifth encoding benchmarks are (tree) recursive neural networks \\citep{tai2015improved, le2015compositional, zhu2015long, allamanis2016learning}, also known as TreeRNNs.\nThese recursively encode the logical expression using the parse structure\\footnote{Completely accurate parses of logical expressions are trivial to obtain, and these are provided to the model rather than learned.}, where leaf nodes of the tree (propositional variables) are embedded as learnable vectors, and each logical operator then combines one or more of these embedded values to produce a new embedding.\nFor example, the expression $(\\neg a) \\lor b$ is parsed as the tree with leaves $a$ and $b$, a unary node $\\neg$ (with input the embedding of $a$), and a binary node $\\lor$ (with inputs the embeddings of $\\neg a$ and $b$).\nFollowing~\\cite{allamanis2016learning}, the fourth encoding benchmark is a simple TreeRNN (TreeNet Encoders), where each operator `op' concatenates its inputs to a vector $x$, and produces the output\n\\[\np = \\frac{h}{\\lVert h \\rVert_2} \\quad \\textrm{where} \\quad h = W_1^\\text{op} x + W_2^\\text{op} \\sigma( W_3^\\text{op} x + b_3^\\text{op} ) + b_1^\\text{op}.\n\\]\nThe fifth and final encoding benchmark (TreeLSTM Encoders) is a variant of TreeRNNs which adapts LSTM cell updates. This helps capture long range dependencies and propagate gradient within the tree. Our implementation follows \\cite{tai2015improved}, modified to have per-op parameters as per TreeRNNs (see, also, the work by~\\cite{le2015compositional} and~\\cite{zhu2015long}).\n\n\\subsubsection{Relational benchmarks}\n\nIn addition to these encoding benchmarks, we define a pair of relational benchmarks, following~\\cite{rocktaschel2015reasoning}. We will traverse the entire sequent with LSTM RNNs or bidirectional LSTM RNNs but concatenating the left hand side and right hand side sequences into a single sequence separated by a held-out symbol (effectively standing for $\\vDash{}$). For the LSTM variant (LSTM Traversal), the model is:\n\\[\nP(A \\vDash{} B) = \\sigma(\\text{MLP}(h_\\text{final})) \\quad \\textrm{where} \\quad h_\\text{final} = \\text{LSTM}(\\text{embedSeq}(\\text{join}(A, \\textrm{``}\\vDash{}\\textrm{''}, B)))\n\\]\nFor the bidirectional case (BiDirLSTM Traversal), the extension is\n\\begin{align*}\n     P(A \\vDash{} B) = \\sigma(\\text{MLP}(h_\\text{final}^{\\leftrightarrow})) \\quad  \\textrm{where} \\quad & h_\\text{final}^{\\leftrightarrow} = \\text{concat}(h_\\text{final}^{\\leftarrow}, h_\\text{final}^{\\rightarrow}) \\\\\n    \\textrm{with} \\quad & h_\\text{final}^{\\leftarrow} = \\text{LSTM}^{\\leftarrow}(\\text{embedSeq}(X))\\\\\n     \\textrm{and} \\quad & h_\\text{final}^{\\rightarrow} = \\text{LSTM}^{\\rightarrow}(\\text{embedSeq}(X))\n\\end{align*}\n\n\\subsubsection{The Transformer benchmark}\n\nWe also benchmark the Transformer model, also known as \\textit{Attention Is All You Need}~\\citep{vaswani2017attention}, which is a sequence-to-sequence model achieving state-of-the-art results in machine translation. As in the relational LSTM models, we concatenate and embed the sequents, but instead of separating the sequents by a held-out symbol, we add a learnable bias to the right sequent in this embedding. This augments the Transformer's method of adding timing signals to distinguishing symbols at different positions. We then decode a sequence of length 1 and apply a linear transformation to get the final entailment prediction logits.\n\n\\subsection{The PossibleWorldNet}\n\nIn this section, we introduce our new model.\nInspired by the semantic (model-theoretic) definition of entailment, we propose a variant on TreeNets that evaluates the pair of formulas in different ``possible worlds''.\n\nEntailment is, first and foremost, a \\emph{semantic} notion.\nGiven a set $\\mathcal{W}$ of worlds,\n\\[\nA \\models B \\; \\mbox{iff for every world} \\; w \\in \\mathcal{W}, \\; sat(w,A) \\; \\mbox{implies} \\; sat(w, B)\n\\]\nHere $sat : World \\times Formula \\rightarrow Bool$ indicates whether a formula is satisfied in a particular world.\n\nWe shall first define a variant of $sat$ that produces \\emph{integers}, and then define another variant that operates on real values.\nFirst, define $sat_2 : World \\times Formula \\rightarrow \\{0, 1\\}$:\n\\[\nsat_2(w, A) = \\mathbbm{1} (sat(w,A))\n\\]\nUsing $sat_2$, we can redefine entailment as:\n\\[\nA \\models B \\; \\mbox{iff} \\; \\forall w \\in \\mathcal{W} \\; sat_2(w, A) \\leq sat_2(w, B)\n\\]\nAssume we have a finite set of worlds $\\mathcal{W} = \\{w_1, ..., w_n\\}$; then we can recast as:\n\\begin{eqnarray}\n\\label{key}\nP(A \\models B) = \\prod_{i=1}^n \\mathbbm{1} (sat_2(w_i, A) \\leq sat_2(w_i, B))\n\\end{eqnarray}\nWe are going to produce a relaxation of Proposition \\ref{key} by replacing $sat_2$ and $\\leq$ with continuous functions.\nAssume we have a variant of $sat_2$ that produces vectors of real values:\n\\[\nsat_3 : World \\times Formula \\rightarrow \\mathbb{R}^d\n\\]\nAssume we have a function $f : \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow [0, 1]$ that generalises $\\leq$ to vectors of real values.\nNow we can rewrite as:\n\\begin{eqnarray}\n\\label{key2}\nP(A \\models B) = \\prod_{i=1}^n f(sat_3(w_i, A), sat_3(w_i, B))\n\\end{eqnarray}\nIn our neural model, $f$ is implemented by a simple linear layer using learnable weights $W_f$ and $b_f$:\n\\[\nf(x, y) = \\sigma(W_f \\cdot \\text{concat}(x, y) + b_f)\n\\]\n\nWe use a set of random vectors to represent our worlds $\\{w_1, ..., w_n\\}$, where $w_i \\in \\mathbb{R}^k$ is a vector of length $k$ of values drawn uniformly randomly.\nWe implement $sat_3$ using a simplified TreeNN (see Section \\ref{subsec:benchmarks}) as described below.\nSince $sat_3$ depends on the particular world $w_i$ we are currently evaluating, we add an additional parameter to the TreeNN so that the embedder has access to the current world $w_i$.\nWe add an additional weight matrix $W_4^{op}$ so that propositional variables can learn which aspect of the current world to focus on.\nIf the formula is of the form $op(l, r)$, where $op$ is nullary (a propositional variable), unary (e.g., negation), or binary (e.g., conjunction), and $l$ and $r$ are the embeddings of the constituents of the expression, then\n\\[\nsat_3(w_i, op(l, r)) = \\frac{h}{\\lVert h \\rVert_2} \\quad \\textrm{where} \\quad h =\n\\left\\{\n\\begin{tabular}{l l}\n    $W_4^{op} w_i$ & where $op$ is nullary (leaf)\\\\\n    $W_1^{op} x + b_1^{op}$ & otherwise\n\\end{tabular}\n\\right.\n\\]\n where $x = \\text{concat}(l, r)$.\n\nTo evaluate whether $A \\vDash{} B$, the PossibleWorldNet generates a set of imagined ``worlds'', and then evaluates $A$ and $B$ in each of those worlds.\nIt is a form of ``convolution over possible worlds''.\nAs we will see in Section \\ref{sec:results}, the quality of the model increases steadily as we increase the number of imagined worlds.\n\nThis architecture was inspired by semantic (model-theoretic) approaches to detecting entailment, but it does not encode any constraint on propositional logic \\emph{in particular} or formal logic in general.\nThe procedure of evaluating sentences in multiple worlds, and combining those evaluations in one product, is just what ``entailment'' means; so we speculate that an architecture like this should, in principle, be equally applicable to other logics (e.g., intuitionistic logic, modal logics, first-order logic) and also to non-formal entailments in natural language sentences\n\nAbstracting away from the particular interpretation of these vectors as ``worlds'', this method generates $n$ copies of the model with shared weights, one for each vector $w_i$; each nullary operator learns a different projection on $w_i$. It makes predictions via a linear layer combining two representations, and then takes the product of the predictions as the overall prediction.\n\n\\section{Experimental Setup}\n\\label{sec:exp_setup}\n\nFor each encoder benchmark architecture, the parameters of the encoders for the left and right hand sides of the sequent are shared. The MLP which performs binary classification to detect entailment based on the expression representations produced by the encoders is model-specific (re-initialised for each model) and jointly trained. Symbol embedding matrices are also model-specific, shared across encoders, and jointly trained.\n\nWe implemented all architectures in TensorFlow~\\citep{abadi2016tensorflow}. We optimised all models with Adam~\\citep{kingma2014adam}. We grid searched across learning rates in [\\num{1e-5}, \\num{1e-4}, \\num{1e-3}], minibatch sizes in $[64, 128]$, and trained each model thrice with different random seeds. Per architecture, we grid-searched across specific hyperparameters as follows. We searched across 2 and 3 layer MLPs wherever an MLP existed in a benchmark, and across layer sizes in $[32, 64]$ for MLP hidden layers, embedding sizes, and RNN cell size (where applicable). Additionally for convolutional networks, we searched across a number of convolutional layers in $[4, 6, 8]$, across kernel size in $[5, 7, 9]$, across number of channels in $[32, 64]$, and across pooling interval in $[0, 5, 3, 1]$ (where $0$ indicates no pooling). For the Transformer model, we searched across the number of encoder and decoder layers in the range $[6, 8, 10]$, dropout probability in the range $[0, 0.1, 0.5]$, and filter size in the range $[128, 256, 384]$. Finally, for all models, we ran them with and without the symbol permutation data augmentation technique described in Section~\\ref{ssec:data_aug}.\n\nAs a result of the grid search, we selected the best model for each architecture against validation results, and record training, validation, and all test accuracies for the associated time step, which we present below.\n\n\\section{Results and Discussion}\n\\label{sec:results}\n\nExperimental results are shown in Table~\\ref{tab:prop_logic_results}. The test scores of the best performing overall model are indicated in bold. The test scores of the best performing model which does not have privileged access to the syntax or semantics of the logic (i.e.~excluding TreeRNN-based models) are italicised. The best benchmark test results are underlined.\n\n\\begin{table}[htb]\n    \\centering\n    \\caption{Propositional Logic Model Accuracy.}\n    \\begin{tabular}{llllllll}\n        \\toprule\n        & \\textbf{model} & \\textbf{valid} & \\textbf{\\shortstack[l]{test\\\\ (easy)}} & \\textbf{\\shortstack[l]{test\\\\ (hard)}} & \\textbf{\\shortstack[l]{test\\\\ (big)}} & \\textbf{\\shortstack[l]{test\\\\ (massive)}} & \\textbf{\\shortstack[l]{test\\\\ (exam)}}\\\\\n        \\midrule\n        \\multirow{2}{*}{\\textbf{baselines}} & Linear BoW  & 52.6 & 51.4 & 50.0 & 49.7 & 50.0 & 52.0 \\\\\n        & MLP BoW & 57.8 & 57.1 & 51.0 & 55.8 & 49.9 & 56.0 \\\\\n        \\midrule\n        \\multirow{7}{*}{\\textbf{\\shortstack[l]{benchmark\\\\ models}}} & Transformer & 57.1 & 56.8 & 50.8 & 51.2 & 50.3 & 46.9 \\\\\n        & ConvNet Encoders & 59.3 & 59.7 & 52.6 & 54.9 & 50.4 & 54.0 \\\\\n        & \\emph{LSTM Encoders} & 68.3 & 68.3 & 58.1 & 61.1 & 52.7 & 70.0 \\\\\n        & BiDirLSTM Encoders  & 66.6 & 65.8 & 58.2 & 61.5 & 51.6 & 78.0 \\\\\n        & TreeNet Encoders & 72.7 & 72.2 & 69.7 & 67.9 & 56.6 & \\underline{85.0} \\\\\n        & \\underline{TreeLSTM Encoders}  & 79.1 & \\underline{77.8} & \\underline{74.2} & \\underline{74.2} & \\underline{59.3} & 75.0 \\\\\n        & LSTM Traversal & 62.5 & 61.8 & 56.2 & 57.3 & 50.6 & 61.0 \\\\\n        & BiDirLSTM Traversal & 63.3 & 64.0 & 55.0 & 57.9 & 50.5 & 66.0 \\\\\n        \\midrule\n       \n        \\textbf{new model} & \\textbf{PossibleWorldNet} & {98.7} & \\textbf{98.6} & \\textbf{96.7} & \\textbf{93.9} & \\textbf{73.4} & \\textbf{96.0} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\label{tab:prop_logic_results}\n\\end{table}\n\nWe observe that the baselines are doing better than random (8.2 points above for the easy test set, for the MLP BoW, and 2.6 above random for the hard test set). This indicates that there are some small number of exploitable regularities at the symbolic level in this dataset, but that they do not provide significant information.\n\nThe baseline results show that convolution networks and BiDirLSTMs encoders obtain relatively mediocre results compared to other models, as do LSTM and BiDirLSTM Traversal models. LSTM encoders is the best performing model which does not have privileged access to the syntax trees. Their success relative to BiDirLSTMs Encoders could be due to their reduced number of parameters guarding against overfitting, and rendering them easier to optimise, but it is plausible BiDirLSTMs Encoders would perform similarly with a more fine-grained grid search. Both tree-based models take the lead amongst the benchmarks, with the TreeLSTM being the best performing benchmark overall on both test sets. For most models except baselines, the symbol permutation data augmentation yielded 2--3 point increase in accuracy on weaker models (BiDirLSTM encoders and traversals, an convolutional networks) and between 7--15 point increases for the Tree-based models. This indicates that this data augmentation strategy is particularly well fitted for letting structure-aware models capture, at the representational level, the arbitrariness of symbols indicating unbound variables.\n\nOverall, these results show clearly that models that exploit structure in problems where it is provided, unambiguous, and a central feature of the task, outperform models which must implicitly model the structure of sequences. LSTM-based encoders provide robust and competitive results, although bidirectionality is not necessarily always the obvious choice due to optimisation and overfitting problems. Perhaps counter-intuitively, given the results of~\\cite{rocktaschel2015reasoning}, traversal models do not outperform encoding models in this pair-of-sequences traversal problem, indicating that they may be better at capturing the sort of long-range dependencies need to recognise textual entailment better than they are at capturing structure in general.\n\nWe conclude, from these benchmark results, that tree structured networks may be a better choice for domains with unambiguous syntax, such as analysing formal languages or programs. For domains such as natural language understanding, both convolutional and recurrent network architectures have had some success, but our experiments indicate that this may be due to the fact that existing tasks favour models which capture representational or semantic regularities, and do not adequately test for structural or syntactic reasoning. In particular, the poor performance of convolutional nets on this task serves as a useful indicator that while they present the right inductive bias for capturing structure in images, where topological proximity usually indicates a joint semantic contribution (pixels close by are likely to contribute to the same ``part'' of an image, such as an edge or pattern), this inductive bias does not carry over to sequences particularly well (where dependencies may be significantly more sparse, structured, and distant)\\footnote{Related to this point, \\cite{kim2016character} show that convolutional networks make for good character-level encoders, to produce word representations, which are in turn better exploited by RNNs. This is consistent with our interpretation of our results, since at the character level, topological distance is---like for images---a good indicator of semantic grouping (characters that are close are usually part of the same word or n-gram).}. The results for the transformer benchmark indicate that while this architecture can capture sufficient structure for machine translation, allowing for the appropriate word order in the output, and accounting for disambiguation or relational information where it exists within sentences, it does not capture with sufficient precision the more hierarchical structure which exists in logical expressions.\n\nThe best performing model overall is the PossibleWorldNet, which achieves significantly higher results than the other models, with\n$99.3\\%$ accuracy on test (easy), and $97.3\\%$ accuracy on test (hard).\nThis is as to be expected, as it has the strongest inductive bias.\nThis inductive bias has two components.\nFirst, the model has knowledge of the syntactic structure of the expression, since it is a variant of a TreeNet.\nSecond, inspired by the definition of semantic (model-theoretic) entailment in general, the model evaluates the pair of formulas in lots of different situations (``possible worlds'') and combines the various results together in a product\\footnote{See Formula~\\ref{key2} above. This general notion of entailment as truth-in-all-worlds is not dependent on any particular formal logic, and applies to entailment in both formal logics and natural languages.}.\n\nThe quality of the PossibleWorldNet depends directly on the number of ``possible worlds'' it considers (see Figure \\ref{fig:pwn}).\nAs we increase the number of possible worlds, the validation error rate goes down steadily.\nNote that the data-efficiency also increases as we increase the number of  worlds.\nThis is because adding worlds to the model does not increase the number of model parameters---it just increases the number of different ``possibilities'' that are considered.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{pwn.png}\n\\caption{The quality of the PossibleWorldNet as we vary the number of possible worlds}\n\\label{fig:pwn}\n\\end{figure}\n\nIn propositional logic, of course, if we are allowed to generate \\emph{every single} truth-value assignment, then it is trivial to detect entailment by checking each one. In our big test set, there are on average more than 3,000 possible truth-value assignments.\nIn our massive test set, there are on average over 800,000 possible assignments.\n(See Table \\ref{dataset-statistics}).\nThe PossibleWorldNet considers at most 256 different worlds, which is only 7\\% of the expected total number of rows needed in the big test set, and only 0.03\\% of the expected number of rows needed for the massive test set.\n\nTo understand this result, we sample 32, 64, 128 and 256 truth table rows (variable truth-value assignments) for each pair of formulas in Test (hard), and reject entailment if a single evaluation for the formulas amongst these finds the left hand side to be true while the right hand side is false. This gives us an estimate of the accuracy of sampling a number of truth table rows equal to the number of possible worlds in our model. We estimate that these statistical methods have 75.9\\%, 86.5\\%, 93.4\\% and 97.2\\% chance of finding a countermodel, respectively. This seems to indicate that PossibleWorldNet is capable of exploiting repeated computation across projections of random noise in order to learn, solely based on the label likelihood objective, something akin to a model-based solution to entailment by treating the random-noise as variable valuations.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\cite{zaremba2014learning} show how a neural architecture can be used to optimise matrix expressions.\nThey generate all expressions up to a certain depth, group them into equivalence classes, and train a recursive neural network classifier to detect whether two expressions are in the same equivalence class.\nThey use a recursive neural network~\\citep{socher2012semantic} to guide the search for an optimised equivalent expression.\nThere are two major differences between this work and ours.\nFirst, the classifier is predicting whether two matrix expressions (e.g. $A$ and $(A^T)^T$) compute the same values; this is an equivalence relation, while entailment is a \\emph{partial order}.\nSecond, their dataset consists of matrix expressions containing at most one variable, while our formulas contain many variables.\n\n\\cite{allamanis2016learning} use a recursive neural network to learn whether two expressions are equivalent.\nThey tested on two datasets: propositional logic and polynomials.\nThere are two main differences between their approach and ours.\nFirst, we consider \\emph{entailment} while they consider \\emph{equivalence}; equivalence is a symmetric relation, while entailment is not symmetric.\nSecond, we consider entailment as a \\emph{relational} classification problem: given a pair of expressions $A$ and $B$, predict whether $A$ entails $B$.\nIn their paper, by contrast, they generate a set of $k$ equivalence-classes of formulas with the same truth-conditions, and ask the network to predict which of these $k$ classes a \\emph{single} formula falls into.\nTheir task is more specific: their network is only able to classify a formula from a new equivalence class that has not been seen during training if it has additional auxiliary information about that class (e.g.~exemplar members of the class).\n\nRecognizing textual entailment (RTE) between \\emph{natural language sentences} is a central task in natural language processing. (See \\cite{dagan2006pascal};\nfor a recent dataset, see \\cite{bowman2015large}).\nSome approaches (e.g., \\cite{wang2015learning} and \\cite{rocktaschel2015reasoning}) use LSTMs with attention, while others (e.g., \\cite{yin2015abcnn}) use a convolutional neural network with attention.\nOf course, recognizing entailment between natural language sentences is a very different task from recognizing entailment between logical formulas.\nEvaluating an entailment between natural language sentences requires understanding the meaning of the non-logical terms in the sentence.\nFor example, the inference from\n\\emph{``An ice skating rink placed outdoors is full of people''} to\n\\emph{``A lot of people are in an ice skating park''}\nrequires knowing the non-logical semantic information that an outdoors ice skating rink is also an ice skating park.\n\nCurrent neural models do not always understand the structure of the sentences they are evaluating.\nIn \\cite{bowman2015large}, all the neural models they considered wrongly claimed that \\emph{``A man wearing padded arm protection\nis being bitten by a German shepherd dog''}\nentails \\emph{``A man bit a dog''}.\nWe believe that isolating the purely structural sub-problem will be useful because only networks that can reliably predict entailment in a purely formal setting, such as propositional (or first-order) logic, will be capable of getting these sorts of examples consistently correct.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we have introduced a new process for generating datasets for the purpose of recognising logical entailment. This was used to compare benchmarks and a new model on a task which is primarily about understanding and exploiting structure. We have established two clear results on the basis of this task. First, and perhaps most intuitively, architectures which make explicit use of structure will perform significantly better than those which must implicitly capture it. Second, the best model is the one that has a strong architectural bias towards capturing the possible world semantics of entailment. In addition to these two points, experimental results also shed some light on the relative abilities of implicit structure models---namely LSTM and Convolution network-based architectures---to capture structure, showing that convolutional networks may not present the right inductive bias to capture and exploit the heterogeneous and deeply structured syntax in certain sequence-based problems, both for formal and natural languages.\n\nThis conclusion is to be expected: the most successful models are those with the most prior knowledge about the generic structure of the task at hand. But our dataset throws new light on this unsurprising thought, by providing a new data-point on which to evaluate neural models' ability to understand structural sequence problems. Logical entailment, unlike textual entailment, depends \\emph{only} on the meaning of the logical operators, and of the place particular arbitrarily-named variables hold within a structure. Here, we have a task in which a network's understanding of structure can be disentangled from its understanding of the meaning of words.\n\n\\subsubsection*{Acknowledgments}\nWe thank our colleagues at DeepMind for their insightful comments during the preparation of this paper, and in particular Yujia Li, Chris Dyer, and Alex Graves.\n\n\n\n\\section{Introduction}\n\nThis paper seeks to answer two questions: ``Can neural networks understand logical formulae well enough to detect entailment?'', and, more generally, ``Which architectures are best at inferring, encoding, and relating features in a purely structural sequence-based problem?''. In answering these questions, we aim to better understand the inductive biases of popular architectures with regard to structure and abstraction in sequence data. Such understanding would help pave the road to agents and classifiers that reason structurally, in addition to reasoning on the basis of essentially semantic representations. In this paper, we provide a testbed for evaluating some aspects of neural networks' ability to reason structurally and abstractly. We use it to compare a variety of popular network architectures and a new model we introduce, called PossibleWorldNet.\n\nNeural network architectures lie at the heart of a variety of applications. They are practically ubiquitous across vision tasks~\\citep{lecun1995convolutional,krizhevsky2012imagenet,simonyan2014very} and natural language understanding, from machine translation~\\citep{kalchbrenner2013recurrent,sutskever2014sequence,bahdanau2014neural} to textual entailment~\\citep{bowman2015large,rocktaschel2015reasoning} via sentiment analysis~\\citep{socher2013recursive,kalchbrenner2014convolutional} and reading comprehension~\\citep{hermann2015teaching,hill2015goldilocks,rajpurkar2016squad}. They have been used to synthesise programs~\\citep{ling2016latent,parisotto2016neuro,devlin2017robustfill} or internalise algorithms~\\citep{graves2016hybrid,grefenstette2015learning,joulin2015inferring,kaiser2015neural,reed2015neural}. They form the basis of reinforcement learning agents capable of playing video games~\\citep{mnih2015human}, difficult perfect information games~\\citep{silver2016mastering,tian2015better}, and navigating complex environments from raw pixels~\\citep{mirowski2016learning}. An important question in this context is to find the inductive and generalisation properties of different neural architectures, particularly towards the ability to capture structure present in the input, an ability that might be important for many language and reasoning tasks. However, there is little work on studying these inductive biases in isolation by running these models on tasks that are primarily or purely \\emph{about} sequence structure, which we intend to address.\n\nThe paper's contribution is three-fold. First, we introduce a new dataset for training and evaluating models. Second, we provide a thorough evaluation of the existing neural models on this dataset.\nThird, inspired by the semantic (model-theoretic) definition of entailment, we propose a variant of the TreeNet that evaluates the formulas in multiple different ``possible worlds'', and which significantly outperforms the benchmarks.\nThe structure of this paper is as follows. In Section~\\ref{sec:dataset_creation}, we introduce the new dataset and describe a generic data generation process for entailment datasets, which offers certain guarantees against the presence of superficial exploitable biases. In Section~\\ref{sec:models}, we describe a series of baseline models used to validate the dataset, benchmarks from which we will derive our analyses of popular model architectures, and also introduce our new neural model, the {PossibleWorldNet}. In Section~\\ref{sec:exp_setup}, we describe the structure of experiments, from which we obtained the results presented and discussed in Section~\\ref{sec:results}. We offer a brief survey of related work in Section~\\ref{sec:related}, before making concluding remarks in Section~\\ref{sec:conclusion}.\n\n\\section{Dataset Creation}\n\\label{sec:dataset_creation}\n\nFormal logics provide a symbolic toolkit for encoding and examining patterns of reasoning. They are structural calculi aiming to codify the norms of correct thought. The meanings of such statements are invariant to what the particular propositions stand for: to understand the entailment $(p \\wedge q) \\vDash{} q$, we only need to understand the semantics of---or related syntactic rules governing---a finite set of logical connectives, while $p$ and $q$ are meaningless arbitrary symbols selected to stand for distinct propositions. In other words, the problem of determining whether an entailment holds is a purely structural sequence-based problem: to evaluate whether an entailment is true, only the meaning of---or inference rules governing---the connectives is relevant. Everything else only has meaning via its place in the structure specified by an expression. These qualities suggest that detecting logical entailment is an excellent task for measuring the ability of models to capture, understand, or exploit structure. We present in this paper a generic process for generating entailment datasets, explained in detail in Appendix~\\ref{appendix-dataset}, for any given logical system. In the specific dataset---generated through this process---presented in this section, we will focus on propositional logic, which is decidable but requires a worst case of $O(2^n)$ operations (e.g.~resolution steps, truth table rows), where $n$ is the number of unique propositional variables, to verify entailment.\n\nOur dataset\\footnote{We aim to release the dataset used for experiments, and the code used to generate it according to the constraints discussed in this paper, upon publication of the paper.} $\\mathcal{D}$ is composed of triples of the form\n$(A, B, A \\vDash{} B)$,\nwhere $A$ and $B$ are formulas of propositional logic, and $A \\vDash{} B$ is 1 if $A$ entails $B$, and 0 otherwise. For example, the data point $(p \\wedge q, q, 1)$ is positive because $p \\wedge q$ entails $q$, whereas $(q \\vee r, r, 0)$ is negative because $q \\vee r$ does not entail $r$.\nEntailment is primarily a semantic notion: $A$ entails $B$ if every model in which $A$ is true is also a model in which $B$ is true.\n\nWe impose various requirements on the dataset, to rule out superficial structural differences between  $\\mathcal{D}^+$ and  $\\mathcal{D}^-$ that can be easily exploited by ``trivial'' baselines\\footnote{E.g., bag-of-words baselines that cannot look at the structure of $A$ and $B$, or baselines that ignore $A$ and only look at $B$.}.\nWe impose the following high level constraints on our data through the generative process, explained in detail in Appendix~\\ref{appendix-dataset}: our classes must be balanced, and formulas in positive and negative examples must have the same distribution over length. Furthermore, we attempt to ensure that there are no recognisable differences in the distributions of lexical or syntactic features between the positive and negative examples. It would not be acceptable, for example, if a typical $B$ formula in a positive entailment $(A, B, 1)$ had more disjunctions than a $B'$ formula in a negative entailment $(A', B', 0)$.\n\nIf we simply sample formulas $A$ and $B$ and evaluate whether $A \\vDash{} B$, there are significant differences between the distributions of formulas for the positive and negative examples, which models can learn to exploit without needing to understand the structure of the problem. To avoid these issues, we use a different approach, that satisfies the above requirements. We sample 4-tuples of formulas $(A_1, B_1, A_2, B_2)$ such that:\n\\[\n A_1  \\vDash{} B_1 \\qquad\n A_2  \\vDash{} B_2 \\qquad\n A_1 \\nvDash{} B_2 \\qquad\n A_2  \\nvDash{} B_1\n\\]\nHere, each of the four formulas appears in one positive entailment and one negative entailment.\nThis way, we minimise crude structural differences between the positive and negative examples.\nHere is a simple example (although the actual dataset has much longer formulas) of such a 4-tuple of datapoints:\n\\[\n p \\vDash{}  p \\vee q \\qquad\n \\neg p \\wedge \\neg q  \\vDash{}  \\neg q \\qquad\n p  \\nvDash{}  \\neg q \\qquad\n \\neg p \\wedge \\neg q \\nvDash{}  p \\vee q\n\\]\nTo generate these 4-tuples, we first generate pairs $(A, B)$ such that $A \\vDash{} B$.\n(To test if $A \\vDash{} B$, we test whether $A \\wedge \\neg B$ is satisfiable, using {\\bf minisat} \\citep{sorensson2005minisat}).\nThen we search through the set of pairs, looking for pairs of pairs, $(A_1, B_1)$ and $(A_2, B_2)$, such that $A_1 \\nvDash{} B_2$ and $A_2 \\nvDash{} B_1$. We present, in Appendix \\ref{appendix-dataset}, the full details of this generative process, its constraints and guarantees, and how we used particular baselines to validate the data.\n\n\\subsection{Splitting the Dataset}\n\nWe produced train, validation, and test (easy) by generating one large set of 4-tuples, and splitting them into groups of sizes 100000, 5000, and 5000. The difficulty of evaluating an entailment depends on the number of propositional variables and the number of operators in the two formulas. In training, validation, and test (easy), we sample the number of propositional variables uniformly between 1 and 10 (there are 26 propositional variables in total: $a$ to $z$). In test (hard), we sample uniformly between 5 and 10. Our formula sampling method takes a parameter specifying the desired number of operators in the formula. In training, validation, and test (easy), the number of operators in a formula is sampled uniformly between 1 and 10. In our hard test set, the number of operators in a formula is sampled uniformly between 15 and 20.\n\nFor the test (big) dataset, we sampled formulas using between 1 and 20 variables (uniformly), and between 10 and 30 operators (again, uniformly).\nFor test (massive), we used a different generating mechanism.\nWe first sampled pairs of formulas $A$, $B$ such that $A \\models B$.\nThese had between 20 and 26 variables, and between 20 and 30 operators each.\nThen we generated a $B^*$ by mutating $B$ and checking that $A \\nvDash B^*$.\nSee Table \\ref{dataset-statistics} for detailed statistics of the dataset sections, including the average difficulty (based on a complexity of $\\mathcal{O}(2^{\\textbf{\\# Vars}})$) of sequents in each fold.\n\nThe test (exam) dataset was assembled from 100 examples of logical entailment \"in the wild\".\nWe looked through various logic textbooks for classic examples of entailments.\nFrom these textbooks, we extracted true entailment triples $(A, B, 1)$ where $A \\models B$.\nWe added false triples $(A, B^*, 0)$, by mutating $B$ into $B^*$ and checking that $A \\nvDash B^*$.\n\n\\begin{table}[tb]\n\\centering\n\\caption{Dataset Statistics}\n\\label{dataset-statistics}\n\\begin{tabular}{lrrrrr}\n\\toprule\n& {\\bf Size} & {\\bf \\shortstack[c]{Mean\\\\ \\# Vars}} & {\\bf \\shortstack[c]{Mean\\\\ \\# Ops}} & {\\bf \\shortstack[c]{Mean\\\\ Length}} & {\\bf \\shortstack[c]{Mean \\\\ $2^{\\textbf{\\# Vars}}$}} \\\\\n\\midrule\n{\\bf Train} & 100,000 & 4.5 & 5.3 & 11.3 & 52.2 \\\\\n{\\bf Validate} & 5,000 & 5.1 & 6.8 & 13.0 & 75.7 \\\\\n{\\bf Test (easy)} & 5,000 & 5.2 & 6.9 & 13.1 & 81.0 \\\\\n{\\bf Test (hard)} & 5,000 & 5.8 & 17.4 & 31.5 & 184.4 \\\\\n{\\bf Test (big)} & 5,000 & 8.0 & 20.9 & 38.7 & 3310.8 \\\\\n{\\bf Test (massive)} & 2,230 & 18.4 & 49.4 & 88.8 & 848,570.0 \\\\\n{\\bf Test (exam)} & 100 & 2.4 & 3.9 & 8.6 & 5.8 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nIn order to test models' ability to generalise to new unseen formulas,\nwe pruned out cases where formulas seen in validation and test were $\\alpha$-equivalent (equivalent up to renaming of symbols) to formulas seen in training.\nSo, for example, if it had seen $p \\models (\\neg q \\wedge p)$ in training, we did not want $r \\models (\\neg s \\wedge r)$ to appear in either the test or validation sets.\nTo do this, we converted all formulas to de-Bruijn form (see~\\cite{pierce2002types}, Chapter 6), and filtered out formulas in validation and test whose de-Bruijn form was identical to one of those in training.\nThis prevents the system from being able to simply memorise examples it has seen in training.\n\n\\subsection{Data Augmentation through Symbolic Vocabulary Permutation}\n\\label{ssec:data_aug}\n\nAs discussed above, the logical connectives ($\\lor$, $\\land$, \\ldots) are the only elements of the language in each dataset that have consistent implicit semantics across expressions.\nIn this sense, two entailments $p \\land q \\vDash{} q$ and $a \\land b \\vDash{} b$ should ideally be treated as identical by the model. To encourage models to capture this invariance, we add an optional data processing layer during training (not testing) whereby symbols are consistently replaced by other symbols of the same type within individual entailments before being input to the network according to the process described below. This is achieved by randomly sampling a permutation of $a, \\ldots, z$ (the propositional variables used) for every training example, and applying this permutation to the left and right sequents. This process is analogous to augmenting image classification training with random reflections and crops.\n\n\\section{Models}\n\\label{sec:models}\n\nIn this section, we first describe a couple of baseline models that verify the basic difficulty of the dataset, followed by a description of benchmark models which are commonly used (with some variation) in a variety of problems, and finally by a description of our new model, PossibleWorldNet.\n\n\\subsection{Baselines}\n\n\n\nThe classes in the dataset are balanced in training, validation, and both test sets, so a random baseline (and a constant, majority-class predicting baseline) will obtain an accuracy of 50\\% on the test sets.\n\nWe define two neural baselines which, we believe, should not be able to perform competitively on this task, but may do better than random. The first is a linear bag of words (Linear BoW) model which embeds each symbol to a vector, and averages them, to produce a representation of each side of the sequent. These representations are then passed through a linear layer:\n\\[\nP(A \\vDash{} B) = \\sigma \\left(W \\cdot \\text{concat} \\left(g(A), g(B)\\right) + b \\right) \\quad \\textrm{where} \\quad g(X) =  \\frac{1}{|X|} \\sum_{x \\in X} \\text{embed}(x)\n\\]\nThe second is a similar architecture, where the final linear layer is replaced with a multi-layer perceptron (MLP BoW):\n\\[\nP(A \\vDash{} B) = \\sigma (\\text{MLP}(\\text{concat}\\left(g(A), g(B)\\right))) \\quad \\textrm{where} \\quad g(X) = \\frac{1}{|X|} \\sum_{x \\in X} \\text{embed}(x)\n\\]\nIn both of these cases, the baselines are expected to have limited performance since they can only capture entailment by modelling the contribution of symbols individually, rather than by modelling structure, since the summation in $g$ destroys all structural information (including word order). We use these results to provide an indication of the difficulty of the dataset.\n\n\\subsection{Benchmarks}\n\\label{subsec:benchmarks}\n\nWe present here a series of benchmark models, not only to serve the purpose of being grounds for comparison for new models tested against this dataset, but also to compare and contrast the performance of fairly ubiquitous model architectures on this purely syntactic problem.\n\nWe distinguish two categories of models: encoding models and relational models. Encoding models, with exceptions specified below, jointly learn an encoding function $f$ and an MLP, such that given a sequent $A \\vDash{} B$, the model expresses\n\\[\nP(A \\vDash{} B) = \\sigma \\left(\\text{MLP}(\\text{concat}(f(A), f(B)))\\right).\n\\]\nIn this sense $f$ produces a representation of each side of the sequent which contains all the information needed for the MLP to decide on entailment. In contrast, relational models will observe the pair of expressions and make a decision, perhaps by traversing both expressions, or by relating sub-structure of one expression to that of the other. These models express a more general formulation\n\\[\nP(A \\vDash{} B) = \\sigma \\left(f(A, B)\\right).\n\\]\n\n\\subsubsection{Encoder benchmarks}\n\nThe first encoder benchmark implemented is a Deep Convolutional Network Encoder (ConvNet Encoders), akin to architectures described in the convolutional networks for text literature~\\citep{kalchbrenner2014convolutional,zhang2015character,kim2016character}. Here, the encoder function $f$ is a stack of one dimensional convolutions over sequence symbols embedded by an embedding operation \\mbox{embedSeq}, interleaved with max pooling layers every $k$ layers (which is a model hyperparameter), followed by $n$ (also a hyperparameter) fully connected layers:\n\\[f(X) = \\text{MLP}(\\text{Conv1D}_n(\\ldots \\text{maxPool}(\\text{Conv1D}_k(\\ldots \\text{Conv1D}_1(\\text{embedSeq}(X)) \\ldots)) \\ldots))\\]\n\nThe second and third encoder benchmarks are an LSTM~\\citep{hochreiter1997long} encoder network (LSTM Encoders), and its bidirectional LSTM variant (BiDirLSTM Encoders). For the LSTM encoder, we embed the sequence symbols, and run an LSTM RNN over them, ignoring the output until the final state:\n\\[\nf(X) = h_\\text{final} \\quad \\textrm{where} \\quad h_\\text{final} = \\text{LSTM}(\\text{embedSeq}(X))\n\\]\nFor the bidirectional variant, two separate LSTM RNNs $\\text{LSTM}^{\\leftarrow}$ and $\\text{LSTM}^{\\rightarrow}$ are run over the sequence in opposite directions. Their respective final states are concatenated to form a representation of the expression:\n\\begin{align*}\n    f(X) = \\text{concat}(h_\\text{final}^{\\leftarrow}, h_\\text{final}^\\rightarrow)  \\quad  \\textrm{where} \\quad & h_\\text{final}^{\\leftarrow} = \\text{LSTM}^{\\leftarrow}(\\text{embedSeq}(X))\\\\\n     \\textrm{and} \\quad & h_\\text{final}^{\\rightarrow} = \\text{LSTM}^\\rightarrow(\\text{embedSeq}(X))\n\\end{align*}\n\nThe benchmarks described thus far do not explicitly condition on structure, even when it is known, as they are designed to traverse a sequence from left to right and model dependencies in the data implicitly. In contrast, we now consider encoder benchmarks which rely on the provision of the syntactic structure of the sequence they encode, and exploit it to determine the order of composition. This inductive bias, which may be incorrect in certain domains (e.g., where no syntax is defined) or difficult to achieve in domains such as natural language text (where syntactic structure is latent and ambiguous), is easy to achieve for logic (where the syntax is known). The experiments below will seek to demonstrate whether is a helpful inductive architectural bias.\n\nThe fourth and fifth encoding benchmarks are (tree) recursive neural networks \\citep{tai2015improved, le2015compositional, zhu2015long, allamanis2016learning}, also known as TreeRNNs.\nThese recursively encode the logical expression using the parse structure\\footnote{Completely accurate parses of logical expressions are trivial to obtain, and these are provided to the model rather than learned.}, where leaf nodes of the tree (propositional variables) are embedded as learnable vectors, and each logical operator then combines one or more of these embedded values to produce a new embedding.\nFor example, the expression $(\\neg a) \\lor b$ is parsed as the tree with leaves $a$ and $b$, a unary node $\\neg$ (with input the embedding of $a$), and a binary node $\\lor$ (with inputs the embeddings of $\\neg a$ and $b$).\nFollowing~\\cite{allamanis2016learning}, the fourth encoding benchmark is a simple TreeRNN (TreeNet Encoders), where each operator `op' concatenates its inputs to a vector $x$, and produces the output\n\\[\np = \\frac{h}{\\lVert h \\rVert_2} \\quad \\textrm{where} \\quad h = W_1^\\text{op} x + W_2^\\text{op} \\sigma( W_3^\\text{op} x + b_3^\\text{op} ) + b_1^\\text{op}.\n\\]\nThe fifth and final encoding benchmark (TreeLSTM Encoders) is a variant of TreeRNNs which adapts LSTM cell updates. This helps capture long range dependencies and propagate gradient within the tree. Our implementation follows \\cite{tai2015improved}, modified to have per-op parameters as per TreeRNNs (see, also, the work by~\\cite{le2015compositional} and~\\cite{zhu2015long}).\n\n\\subsubsection{Relational benchmarks}\n\nIn addition to these encoding benchmarks, we define a pair of relational benchmarks, following~\\cite{rocktaschel2015reasoning}. We will traverse the entire sequent with LSTM RNNs or bidirectional LSTM RNNs but concatenating the left hand side and right hand side sequences into a single sequence separated by a held-out symbol (effectively standing for $\\vDash{}$). For the LSTM variant (LSTM Traversal), the model is:\n\\[\nP(A \\vDash{} B) = \\sigma(\\text{MLP}(h_\\text{final})) \\quad \\textrm{where} \\quad h_\\text{final} = \\text{LSTM}(\\text{embedSeq}(\\text{join}(A, \\textrm{``}\\vDash{}\\textrm{''}, B)))\n\\]\nFor the bidirectional case (BiDirLSTM Traversal), the extension is\n\\begin{align*}\n     P(A \\vDash{} B) = \\sigma(\\text{MLP}(h_\\text{final}^{\\leftrightarrow})) \\quad  \\textrm{where} \\quad & h_\\text{final}^{\\leftrightarrow} = \\text{concat}(h_\\text{final}^{\\leftarrow}, h_\\text{final}^{\\rightarrow}) \\\\\n    \\textrm{with} \\quad & h_\\text{final}^{\\leftarrow} = \\text{LSTM}^{\\leftarrow}(\\text{embedSeq}(X))\\\\\n     \\textrm{and} \\quad & h_\\text{final}^{\\rightarrow} = \\text{LSTM}^{\\rightarrow}(\\text{embedSeq}(X))\n\\end{align*}\n\n\\subsubsection{The Transformer benchmark}\n\nWe also benchmark the Transformer model, also known as \\textit{Attention Is All You Need}~\\citep{vaswani2017attention}, which is a sequence-to-sequence model achieving state-of-the-art results in machine translation. As in the relational LSTM models, we concatenate and embed the sequents, but instead of separating the sequents by a held-out symbol, we add a learnable bias to the right sequent in this embedding. This augments the Transformer's method of adding timing signals to distinguishing symbols at different positions. We then decode a sequence of length 1 and apply a linear transformation to get the final entailment prediction logits.\n\n\\subsection{The PossibleWorldNet}\n\nIn this section, we introduce our new model.\nInspired by the semantic (model-theoretic) definition of entailment, we propose a variant on TreeNets that evaluates the pair of formulas in different ``possible worlds''.\n\nEntailment is, first and foremost, a \\emph{semantic} notion.\nGiven a set $\\mathcal{W}$ of worlds,\n\\[\nA \\models B \\; \\mbox{iff for every world} \\; w \\in \\mathcal{W}, \\; sat(w,A) \\; \\mbox{implies} \\; sat(w, B)\n\\]\nHere $sat : World \\times Formula \\rightarrow Bool$ indicates whether a formula is satisfied in a particular world.\n\nWe shall first define a variant of $sat$ that produces \\emph{integers}, and then define another variant that operates on real values.\nFirst, define $sat_2 : World \\times Formula \\rightarrow \\{0, 1\\}$:\n\\[\nsat_2(w, A) = \\mathbbm{1} (sat(w,A))\n\\]\nUsing $sat_2$, we can redefine entailment as:\n\\[\nA \\models B \\; \\mbox{iff} \\; \\forall w \\in \\mathcal{W} \\; sat_2(w, A) \\leq sat_2(w, B)\n\\]\nAssume we have a finite set of worlds $\\mathcal{W} = \\{w_1, ..., w_n\\}$; then we can recast as:\n\\begin{eqnarray}\n\\label{key}\nP(A \\models B) = \\prod_{i=1}^n \\mathbbm{1} (sat_2(w_i, A) \\leq sat_2(w_i, B))\n\\end{eqnarray}\nWe are going to produce a relaxation of Proposition \\ref{key} by replacing $sat_2$ and $\\leq$ with continuous functions.\nAssume we have a variant of $sat_2$ that produces vectors of real values:\n\\[\nsat_3 : World \\times Formula \\rightarrow \\mathbb{R}^d\n\\]\nAssume we have a function $f : \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow [0, 1]$ that generalises $\\leq$ to vectors of real values.\nNow we can rewrite as:\n\\begin{eqnarray}\n\\label{key2}\nP(A \\models B) = \\prod_{i=1}^n f(sat_3(w_i, A), sat_3(w_i, B))\n\\end{eqnarray}\nIn our neural model, $f$ is implemented by a simple linear layer using learnable weights $W_f$ and $b_f$:\n\\[\nf(x, y) = \\sigma(W_f \\cdot \\text{concat}(x, y) + b_f)\n\\]\n\nWe use a set of random vectors to represent our worlds $\\{w_1, ..., w_n\\}$, where $w_i \\in \\mathbb{R}^k$ is a vector of length $k$ of values drawn uniformly randomly.\nWe implement $sat_3$ using a simplified TreeNN (see Section \\ref{subsec:benchmarks}) as described below.\nSince $sat_3$ depends on the particular world $w_i$ we are currently evaluating, we add an additional parameter to the TreeNN so that the embedder has access to the current world $w_i$.\nWe add an additional weight matrix $W_4^{op}$ so that propositional variables can learn which aspect of the current world to focus on.\nIf the formula is of the form $op(l, r)$, where $op$ is nullary (a propositional variable), unary (e.g., negation), or binary (e.g., conjunction), and $l$ and $r$ are the embeddings of the constituents of the expression, then\n\\[\nsat_3(w_i, op(l, r)) = \\frac{h}{\\lVert h \\rVert_2} \\quad \\textrm{where} \\quad h =\n\\left\\{\n\\begin{tabular}{l l}\n    $W_4^{op} w_i$ & where $op$ is nullary (leaf)\\\\\n    $W_1^{op} x + b_1^{op}$ & otherwise\n\\end{tabular}\n\\right.\n\\]\n where $x = \\text{concat}(l, r)$.\n\nTo evaluate whether $A \\vDash{} B$, the PossibleWorldNet generates a set of imagined ``worlds'', and then evaluates $A$ and $B$ in each of those worlds.\nIt is a form of ``convolution over possible worlds''.\nAs we will see in Section \\ref{sec:results}, the quality of the model increases steadily as we increase the number of imagined worlds.\n\nThis architecture was inspired by semantic (model-theoretic) approaches to detecting entailment, but it does not encode any constraint on propositional logic \\emph{in particular} or formal logic in general.\nThe procedure of evaluating sentences in multiple worlds, and combining those evaluations in one product, is just what ``entailment'' means; so we speculate that an architecture like this should, in principle, be equally applicable to other logics (e.g., intuitionistic logic, modal logics, first-order logic) and also to non-formal entailments in natural language sentences\n\nAbstracting away from the particular interpretation of these vectors as ``worlds'', this method generates $n$ copies of the model with shared weights, one for each vector $w_i$; each nullary operator learns a different projection on $w_i$. It makes predictions via a linear layer combining two representations, and then takes the product of the predictions as the overall prediction.\n\n\\section{Experimental Setup}\n\\label{sec:exp_setup}\n\nFor each encoder benchmark architecture, the parameters of the encoders for the left and right hand sides of the sequent are shared. The MLP which performs binary classification to detect entailment based on the expression representations produced by the encoders is model-specific (re-initialised for each model) and jointly trained. Symbol embedding matrices are also model-specific, shared across encoders, and jointly trained.\n\nWe implemented all architectures in TensorFlow~\\citep{abadi2016tensorflow}. We optimised all models with Adam~\\citep{kingma2014adam}. We grid searched across learning rates in [\\num{1e-5}, \\num{1e-4}, \\num{1e-3}], minibatch sizes in $[64, 128]$, and trained each model thrice with different random seeds. Per architecture, we grid-searched across specific hyperparameters as follows. We searched across 2 and 3 layer MLPs wherever an MLP existed in a benchmark, and across layer sizes in $[32, 64]$ for MLP hidden layers, embedding sizes, and RNN cell size (where applicable). Additionally for convolutional networks, we searched across a number of convolutional layers in $[4, 6, 8]$, across kernel size in $[5, 7, 9]$, across number of channels in $[32, 64]$, and across pooling interval in $[0, 5, 3, 1]$ (where $0$ indicates no pooling). For the Transformer model, we searched across the number of encoder and decoder layers in the range $[6, 8, 10]$, dropout probability in the range $[0, 0.1, 0.5]$, and filter size in the range $[128, 256, 384]$. Finally, for all models, we ran them with and without the symbol permutation data augmentation technique described in Section~\\ref{ssec:data_aug}.\n\nAs a result of the grid search, we selected the best model for each architecture against validation results, and record training, validation, and all test accuracies for the associated time step, which we present below.\n\n\\section{Results and Discussion}\n\\label{sec:results}\n\nExperimental results are shown in Table~\\ref{tab:prop_logic_results}. The test scores of the best performing overall model are indicated in bold. The test scores of the best performing model which does not have privileged access to the syntax or semantics of the logic (i.e.~excluding TreeRNN-based models) are italicised. The best benchmark test results are underlined.\n\n\\begin{table}[htb]\n    \\centering\n    \\caption{Propositional Logic Model Accuracy.}\n    \\begin{tabular}{llllllll}\n        \\toprule\n        & \\textbf{model} & \\textbf{valid} & \\textbf{\\shortstack[l]{test\\\\ (easy)}} & \\textbf{\\shortstack[l]{test\\\\ (hard)}} & \\textbf{\\shortstack[l]{test\\\\ (big)}} & \\textbf{\\shortstack[l]{test\\\\ (massive)}} & \\textbf{\\shortstack[l]{test\\\\ (exam)}}\\\\\n        \\midrule\n        \\multirow{2}{*}{\\textbf{baselines}} & Linear BoW  & 52.6 & 51.4 & 50.0 & 49.7 & 50.0 & 52.0 \\\\\n        & MLP BoW & 57.8 & 57.1 & 51.0 & 55.8 & 49.9 & 56.0 \\\\\n        \\midrule\n        \\multirow{7}{*}{\\textbf{\\shortstack[l]{benchmark\\\\ models}}} & Transformer & 57.1 & 56.8 & 50.8 & 51.2 & 50.3 & 46.9 \\\\\n        & ConvNet Encoders & 59.3 & 59.7 & 52.6 & 54.9 & 50.4 & 54.0 \\\\\n        & \\emph{LSTM Encoders} & 68.3 & 68.3 & 58.1 & 61.1 & 52.7 & 70.0 \\\\\n        & BiDirLSTM Encoders  & 66.6 & 65.8 & 58.2 & 61.5 & 51.6 & 78.0 \\\\\n        & TreeNet Encoders & 72.7 & 72.2 & 69.7 & 67.9 & 56.6 & \\underline{85.0} \\\\\n        & \\underline{TreeLSTM Encoders}  & 79.1 & \\underline{77.8} & \\underline{74.2} & \\underline{74.2} & \\underline{59.3} & 75.0 \\\\\n        & LSTM Traversal & 62.5 & 61.8 & 56.2 & 57.3 & 50.6 & 61.0 \\\\\n        & BiDirLSTM Traversal & 63.3 & 64.0 & 55.0 & 57.9 & 50.5 & 66.0 \\\\\n        \\midrule\n       \n        \\textbf{new model} & \\textbf{PossibleWorldNet} & {98.7} & \\textbf{98.6} & \\textbf{96.7} & \\textbf{93.9} & \\textbf{73.4} & \\textbf{96.0} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\label{tab:prop_logic_results}\n\\end{table}\n\nWe observe that the baselines are doing better than random (8.2 points above for the easy test set, for the MLP BoW, and 2.6 above random for the hard test set). This indicates that there are some small number of exploitable regularities at the symbolic level in this dataset, but that they do not provide significant information.\n\nThe baseline results show that convolution networks and BiDirLSTMs encoders obtain relatively mediocre results compared to other models, as do LSTM and BiDirLSTM Traversal models. LSTM encoders is the best performing model which does not have privileged access to the syntax trees. Their success relative to BiDirLSTMs Encoders could be due to their reduced number of parameters guarding against overfitting, and rendering them easier to optimise, but it is plausible BiDirLSTMs Encoders would perform similarly with a more fine-grained grid search. Both tree-based models take the lead amongst the benchmarks, with the TreeLSTM being the best performing benchmark overall on both test sets. For most models except baselines, the symbol permutation data augmentation yielded 2--3 point increase in accuracy on weaker models (BiDirLSTM encoders and traversals, an convolutional networks) and between 7--15 point increases for the Tree-based models. This indicates that this data augmentation strategy is particularly well fitted for letting structure-aware models capture, at the representational level, the arbitrariness of symbols indicating unbound variables.\n\nOverall, these results show clearly that models that exploit structure in problems where it is provided, unambiguous, and a central feature of the task, outperform models which must implicitly model the structure of sequences. LSTM-based encoders provide robust and competitive results, although bidirectionality is not necessarily always the obvious choice due to optimisation and overfitting problems. Perhaps counter-intuitively, given the results of~\\cite{rocktaschel2015reasoning}, traversal models do not outperform encoding models in this pair-of-sequences traversal problem, indicating that they may be better at capturing the sort of long-range dependencies need to recognise textual entailment better than they are at capturing structure in general.\n\nWe conclude, from these benchmark results, that tree structured networks may be a better choice for domains with unambiguous syntax, such as analysing formal languages or programs. For domains such as natural language understanding, both convolutional and recurrent network architectures have had some success, but our experiments indicate that this may be due to the fact that existing tasks favour models which capture representational or semantic regularities, and do not adequately test for structural or syntactic reasoning. In particular, the poor performance of convolutional nets on this task serves as a useful indicator that while they present the right inductive bias for capturing structure in images, where topological proximity usually indicates a joint semantic contribution (pixels close by are likely to contribute to the same ``part'' of an image, such as an edge or pattern), this inductive bias does not carry over to sequences particularly well (where dependencies may be significantly more sparse, structured, and distant)\\footnote{Related to this point, \\cite{kim2016character} show that convolutional networks make for good character-level encoders, to produce word representations, which are in turn better exploited by RNNs. This is consistent with our interpretation of our results, since at the character level, topological distance is---like for images---a good indicator of semantic grouping (characters that are close are usually part of the same word or n-gram).}. The results for the transformer benchmark indicate that while this architecture can capture sufficient structure for machine translation, allowing for the appropriate word order in the output, and accounting for disambiguation or relational information where it exists within sentences, it does not capture with sufficient precision the more hierarchical structure which exists in logical expressions.\n\nThe best performing model overall is the PossibleWorldNet, which achieves significantly higher results than the other models, with\n$99.3\\%$ accuracy on test (easy), and $97.3\\%$ accuracy on test (hard).\nThis is as to be expected, as it has the strongest inductive bias.\nThis inductive bias has two components.\nFirst, the model has knowledge of the syntactic structure of the expression, since it is a variant of a TreeNet.\nSecond, inspired by the definition of semantic (model-theoretic) entailment in general, the model evaluates the pair of formulas in lots of different situations (``possible worlds'') and combines the various results together in a product\\footnote{See Formula~\\ref{key2} above. This general notion of entailment as truth-in-all-worlds is not dependent on any particular formal logic, and applies to entailment in both formal logics and natural languages.}.\n\nThe quality of the PossibleWorldNet depends directly on the number of ``possible worlds'' it considers (see Figure \\ref{fig:pwn}).\nAs we increase the number of possible worlds, the validation error rate goes down steadily.\nNote that the data-efficiency also increases as we increase the number of  worlds.\nThis is because adding worlds to the model does not increase the number of model parameters---it just increases the number of different ``possibilities'' that are considered.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{pwn.png}\n\\caption{The quality of the PossibleWorldNet as we vary the number of possible worlds}\n\\label{fig:pwn}\n\\end{figure}\n\nIn propositional logic, of course, if we are allowed to generate \\emph{every single} truth-value assignment, then it is trivial to detect entailment by checking each one. In our big test set, there are on average more than 3,000 possible truth-value assignments.\nIn our massive test set, there are on average over 800,000 possible assignments.\n(See Table \\ref{dataset-statistics}).\nThe PossibleWorldNet considers at most 256 different worlds, which is only 7\\% of the expected total number of rows needed in the big test set, and only 0.03\\% of the expected number of rows needed for the massive test set.\n\nTo understand this result, we sample 32, 64, 128 and 256 truth table rows (variable truth-value assignments) for each pair of formulas in Test (hard), and reject entailment if a single evaluation for the formulas amongst these finds the left hand side to be true while the right hand side is false. This gives us an estimate of the accuracy of sampling a number of truth table rows equal to the number of possible worlds in our model. We estimate that these statistical methods have 75.9\\%, 86.5\\%, 93.4\\% and 97.2\\% chance of finding a countermodel, respectively. This seems to indicate that PossibleWorldNet is capable of exploiting repeated computation across projections of random noise in order to learn, solely based on the label likelihood objective, something akin to a model-based solution to entailment by treating the random-noise as variable valuations.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\cite{zaremba2014learning} show how a neural architecture can be used to optimise matrix expressions.\nThey generate all expressions up to a certain depth, group them into equivalence classes, and train a recursive neural network classifier to detect whether two expressions are in the same equivalence class.\nThey use a recursive neural network~\\citep{socher2012semantic} to guide the search for an optimised equivalent expression.\nThere are two major differences between this work and ours.\nFirst, the classifier is predicting whether two matrix expressions (e.g. $A$ and $(A^T)^T$) compute the same values; this is an equivalence relation, while entailment is a \\emph{partial order}.\nSecond, their dataset consists of matrix expressions containing at most one variable, while our formulas contain many variables.\n\n\\cite{allamanis2016learning} use a recursive neural network to learn whether two expressions are equivalent.\nThey tested on two datasets: propositional logic and polynomials.\nThere are two main differences between their approach and ours.\nFirst, we consider \\emph{entailment} while they consider \\emph{equivalence}; equivalence is a symmetric relation, while entailment is not symmetric.\nSecond, we consider entailment as a \\emph{relational} classification problem: given a pair of expressions $A$ and $B$, predict whether $A$ entails $B$.\nIn their paper, by contrast, they generate a set of $k$ equivalence-classes of formulas with the same truth-conditions, and ask the network to predict which of these $k$ classes a \\emph{single} formula falls into.\nTheir task is more specific: their network is only able to classify a formula from a new equivalence class that has not been seen during training if it has additional auxiliary information about that class (e.g.~exemplar members of the class).\n\nRecognizing textual entailment (RTE) between \\emph{natural language sentences} is a central task in natural language processing. (See \\cite{dagan2006pascal};\nfor a recent dataset, see \\cite{bowman2015large}).\nSome approaches (e.g., \\cite{wang2015learning} and \\cite{rocktaschel2015reasoning}) use LSTMs with attention, while others (e.g., \\cite{yin2015abcnn}) use a convolutional neural network with attention.\nOf course, recognizing entailment between natural language sentences is a very different task from recognizing entailment between logical formulas.\nEvaluating an entailment between natural language sentences requires understanding the meaning of the non-logical terms in the sentence.\nFor example, the inference from\n\\emph{``An ice skating rink placed outdoors is full of people''} to\n\\emph{``A lot of people are in an ice skating park''}\nrequires knowing the non-logical semantic information that an outdoors ice skating rink is also an ice skating park.\n\nCurrent neural models do not always understand the structure of the sentences they are evaluating.\nIn \\cite{bowman2015large}, all the neural models they considered wrongly claimed that \\emph{``A man wearing padded arm protection\nis being bitten by a German shepherd dog''}\nentails \\emph{``A man bit a dog''}.\nWe believe that isolating the purely structural sub-problem will be useful because only networks that can reliably predict entailment in a purely formal setting, such as propositional (or first-order) logic, will be capable of getting these sorts of examples consistently correct.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we have introduced a new process for generating datasets for the purpose of recognising logical entailment. This was used to compare benchmarks and a new model on a task which is primarily about understanding and exploiting structure. We have established two clear results on the basis of this task. First, and perhaps most intuitively, architectures which make explicit use of structure will perform significantly better than those which must implicitly capture it. Second, the best model is the one that has a strong architectural bias towards capturing the possible world semantics of entailment. In addition to these two points, experimental results also shed some light on the relative abilities of implicit structure models---namely LSTM and Convolution network-based architectures---to capture structure, showing that convolutional networks may not present the right inductive bias to capture and exploit the heterogeneous and deeply structured syntax in certain sequence-based problems, both for formal and natural languages.\n\nThis conclusion is to be expected: the most successful models are those with the most prior knowledge about the generic structure of the task at hand. But our dataset throws new light on this unsurprising thought, by providing a new data-point on which to evaluate neural models' ability to understand structural sequence problems. Logical entailment, unlike textual entailment, depends \\emph{only} on the meaning of the logical operators, and of the place particular arbitrarily-named variables hold within a structure. Here, we have a task in which a network's understanding of structure can be disentangled from its understanding of the meaning of words.\n\n\\subsubsection*{Acknowledgments}\nWe thank our colleagues at DeepMind for their insightful comments during the preparation of this paper, and in particular Yujia Li, Chris Dyer, and Alex Graves.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Chernoff bound for bisections}\n\n\\begin{lemma}\n\\label{lem:bisect-chernoff}\nSuppose $S$ is a fixed subset of $[m']$, and $(A,B)$ a random partition of $[m']$, chosen uniformly among all partitions where $|A| = |B| = m'\/2$. Then\n$$ \\Pr\\left[ ||S \\cap A| - |S \\cap B|| > \\beta m'] \\right] < 4 e^{-\\beta^2 m' \/ 2}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe use the fact that $A$ has distribution very close to a uniformly random subset of $[m']$ (where elements appear independently with probability $1\/2$). More precisely, we couple the two distributions as follows. Let $A$ be a random set of size $m'\/2$, $B$ its complement, and let $X$ be a binomial random variable $Bi(m',1\/2)$. Let $R$ be a random set chosen as follows: if $X \\leq m'\/2$, take a random subset of $A$ of size $X$. If $X > m'\/2$, take the union of $A$ and $X-m'\/2$ random elements from $B$. This defines a set $R$ which is uniformly random. Hence, by the Chernoff bound (see e.g. \\cite[Theorem A.1.16]{AlonSpencer}),\n$$ \\Pr[|R \\Delta A| > \\alpha m'] = \\Pr[Bi(m',1\/2) \\notin [m'\/2-\\alpha m', m'\/2+\\alpha m']] < 2 e^{-2 \\alpha^2 m'}.$$\nSimilarly, $S \\Delta R$ has the distribution of a uniformly random set (because $S$ is fixed), and hence\n$$ \\Pr[|S \\Delta R| \\notin [m'\/2 - \\alpha m', m'\/2 + \\alpha m']] < 2 e^{-2 \\alpha^2 m'}.$$\nUsing the triangle inequality $|S \\Delta A| \\leq |S \\Delta R| + |R \\Delta A|$, we get\n$$ \\Pr[|S \\Delta A| \\notin [m'\/2 - 2 \\alpha m', m'\/2 + 2 \\alpha m']] < 4 e^{-2 \\alpha^2 m'}.$$\nThe lemma follows by taking $\\alpha = \\beta\/2$, since $|S \\cap A| - |S \\cap B| = |A| - |S \\Delta A| = \\frac{m'}{2} - |S \\Delta A|.$\n\\end{proof}\n\n\n\\section{Product composition of submodular functions}\n\n\\begin{lemma}\n\\label{lem:submod-product}\nLet $f_1,f_2:2^M \\rightarrow [0,1]$ be monotone submodular. Then\n$$ f(S) = 1 - (1-f_1(S)) (1-f_2(S)) $$\nis also monotone submodular.\n\\end{lemma}\n\n\\begin{proof}\nLet $g_1(S) = 1-f_1(S)$, $g_2(S) = 1-f_2(S)$; these are non-negative non-increasing supermodular functions. Clearly, $g(S) = g_1(S) g_2(S)$ is also non-increasing. Our goal is to prove that $g(S) = g_1(S) g_2(S)$ is supermodular, which implies the claim. By the properties of $g_1, g_2$, we get for any $i,j \\notin S$\n$$ g_1(S) (g_2(S) - g_2(S+i)) \\geq g_1(S) (g_2(S+j) - g_2(S+i+j)) \\geq g_1(S+j) (g_2(S+j) - g_2(S+i+j)) $$\nand\n$$ (g_1(S) - g_1(S+i)) g_2(S+i) \\geq (g_1(S+j) - g_1(S+i+j)) g_2(S+i) \\geq (g_1(S+j) - g_1(S+i+j)) g_2(S+i+j).$$\nAdding up these two inequalities, we get the condition of supermodularity for $g(S) = g_1(S) g_2(S)$:\n$$ g_1(S) g_2(S) - g_1(S+i) g_2(S+i) \\geq g_1(S+j) g_2(S+j) - g_1(S+i+j) g_2(S+i+j).$$\n\\end{proof}\n\n\n\n\n\\section{Proof of hardness for combinatorial auctions}\n\\label{sec:CA-proof}\n\nIn this section, we present the proof of Theorem~\\ref{thm:CA-hardness}.\n\n\\subsection{The basic random instance}\n\\label{sec:basic-inst}\n\nWe choose a parameter $\\ell \\geq 1$ and construct instances with $|N| = n = 2^\\ell$ players and $|M| = m = 400^\\ell$ items. We define \"polar valuations\" as in \\cite{Dobzin11}.\n\n\\begin{definition}\nGiven a set of items $A \\subset M$ and a parameter $\\omega>0$, the polar valuation $v^*_A$ associated with $A$ is defined by \n$$ v^*_A(S) = |A \\cap S| + \\omega |S \\setminus A|.$$\n\\end{definition}\n\nOur \"basic instance\" is an instance where each player has a polar valuation associated with a random set of size $m\/n$.\n\n\\begin{definition}\nIn the basic instance, player $i$ has valuation $v^*_i = v^*_{A^{(0)}_i}$ where $A^{(0)}_i$ is a uniformly random set of size $m\/n$, chosen independently for each player.\n\\end{definition}\n\nNext, we prove that for some player, his allocation overlaps significantly with his desired set.\n\n\\begin{lemma}\n\\label{lem:basic-inst}\nFor any $c$-approximation mechanism applied to the random basic instance, there is a player $i$ and sets $A^{(0)}_j, j \\neq i$, such that conditioned on the desired sets for players $j \\neq i$ being $A^{(0)}_j$, player $i$ gets allocated a random set $R^{(0)}_i$ such that\n$$ \\E[|R^{(0)}_i \\cap A^{(0)}_i|] > (c\/4-\\omega) \\E[|R^{(0)}_i \\cup A^{(0)}_i|].$$\n\\end{lemma}\n\n\\begin{proof}\nFirst, let us estimate the optimal social welfare that the basic instance admits in expectation. Given $(A^{(0)}_1,\\ldots,A^{(0)}_n)$, each item in $\\bigcup_{i=1}^{n} A^{(0)}_i$ can be allocated to some player so that it brings value $1$. We ignore the remaining items. Observe that a fixed item $j$ appears in each $A^{(0)}_i$ independently with probability $1\/n$, therefore $Pr[j \\in \\bigcup_{i=1}^{n} A^{(0)}_i] = 1 - (1-1\/n)^n \\geq 1-1\/e > 1\/2$. Hence,\n$$ \\E[OPT] \\geq \\E[|\\bigcup_{i=1}^{n} A^{(0)}_i|] > \\frac{m}{2}.$$\nWe remind the reader that the expectation is over the random choices of $(A^{(0)}_1,\\ldots,A^{(0)}_n)$. A $c$-approximate mechanism should provide at least $c \\cdot OPT$ in expectation for every particular instance. Hence also in expectation over the random choice of $(A^{(0)}_1,\\ldots,A^{(0)}_n)$. If $(R_1,\\ldots,R_n)$ is the allocation provided by the mechanism, this means\n$$ \\sum_{i=1}^{n} \\E[|R_i \\cap A^{(0)}_i| + \\omega|R_i \\setminus A^{(0)}_i|] \\geq c \\cdot OPT > \\frac{c m}{2}.$$\nSince each of $A^{(0)}_1,\\ldots,A^{(0)}_n$ has size $m\/n$ and the sizes of $R_1,\\ldots,R_n$ add up to at most $m$,\nwe can write\n$$ \\sum_{i=1}^{n} \\E[|R_i \\cap A^{(0)}_i| + \\omega|R_i \\setminus A^{(0)}_i|] > \\frac{cm}{2} \\geq \\frac{c}{4} \\sum_{i=1}^{n} \\E[|R_i| + |A^{(0)}_i|] \\geq \\frac{c}{4} \\sum_{i=1}^{n} \\E[|R_i \\cup A^{(0)}_i|].$$\nBy an averaging argument, there must be $i$ such that\n$$ \\E[|R_i \\cap A^{(0)}_i| + \\omega|R_i \\setminus A^{(0)}_i|] > \\frac{c}{4} \\E[|R_i \\cup A^{(0)}_i|] $$\nand therefore\n$$ \\E[|R_i \\cap A^{(0)}_i|] > (c\/4 - \\omega) \\E[|R_i \\cup A^{(0)}_i|].$$\nThis holds in expectation over the choices of $(A^{(0)}_j: j \\neq i)$, and again by an averaging argument it also holds conditioned on some particular choice of $(A^{(0)}_j: j \\neq i)$. We call the random set allocated to player $i$ under this conditioning $R^{(0)}_i$.\n\\end{proof}\n\n\n\\subsection{Setup for higher-level valuations}\n\\label{sec:prep}\n\nIn the following, the valuations of all players except $i$ are fixed to be $v^*_j = v^*_{A^{(0)}_j}$ for some choice of sets $(A^{(0)}_j: j \\neq i)$. Now we will vary the valuation of player $i$ in order to be able to apply the symmetry gap argument as before. Since we work only with player $i$, we call him the \"special player\" and we drop the index $i$ in the following.\n\nRecall Definition~\\ref{def:rnd-bisect}, the definition of a random bisection sequence. We will use the same concept here, where at level $j$ we have a random set $A^{(j)}$ partitioned randomly into $A^{(j-1)} \\cup B^{(j-1)}$. These sets have sizes $|A^{(j-1)}| = |B^{(j-1)}| = 2^{j-1-\\ell} m$.\nWe will use valuation functions associated with each pair of sets. We denote these valuation functions by $v_{A^{(j-1)},B^{(j-1)}}$ at level $j$. In particular, this valuation function depends only on the elements of $A^{(j)} = A^{(j-1)} \\cup B^{(j-1)}$. In other words, $A^{(j-1)}$ and $B^{(j-1)}$ are the desired sets of items at level $j$. \n\n\\paragraph{Distribution menu.}\nFor each particular choice of a valuation function $v_{A^{(j)},B^{(j)}}$ at a certain level, the mechanism needs to produce a distribution over item sets for the special player, along with a certain price. Recall that due to the definition of (approximate) truthfulness in expectation, this choice should give (approximately) the optimal utility for the special player among all possible choices given the other valuations $v^*_{-i}$. After fixing a set of valuation functions $v_{A^{(j)},B^{(j)}}$ for each pair $(A^{(j)},B^{(j)})$, the output distribution will depend only on $(A^{(j)},B^{(j)})$ and hence we denote the respective random set by $R(A^{(j)},B^{(j)})$; we also denote the associated price by $P(A^{(j)},B^{(j)})$. Thus the mechanisms assigns distributions over sets and prices to all pairs of sets. As before, we distill the important information from the distribution into a random variable $X_j$. There is some additional information now expressed by the price; we associate the price with a separate random variable $P_j$. The possible choices of distributions for $(X_j,P_j)$ are what we call a {\\em distribution menu} at level $j$.\n\n\\begin{definition}\n\\label{def:menu}\nGiven a mechanism and a special player with other valuations fixed, the \"distribution menu at level $j$\", $\\cM_j$, is the set of all probability distributions of a pair of variables $(X_j,P_j)$ that arise as follows: There exist valuations $v_{A^{(j-1)},B^{(j-1)}}$ such that when declaring  $v_{A^{(j-1)},B^{(j-1)}}$, the special player receives a random set $R(A^{(j-1)},B^{(j-1)})$ at a price $P(A^{(j-1)},B^{(j-1)})$. Then, for $A^{(j)} = A^{(j-1)} \\cup B^{(j-1)}$ chosen as the $(j-1)$-th level of a random bisection sequence, i.e. a random pair of disjoint sets of size $2^{j-1-\\ell} m$, we have\n$$ X_j = \\frac{|A^{(j)} \\cap R(A^{(j-1)},B^{(j-1)})|}{|A^{(j)}|}, $$\n$$ P_j = P(A^{(j-1)},B^{(j-1)}). $$\n\\end{definition}\n\nIn other words, $X_j$ encodes the (random) fraction of the relevant items that the special player receives at level $j$, and $P_j$ is the respective (random) price.\nNote that there are two sources of randomness in $(X_j,P_j)$: one is the random choice of $(A^{(j-1)},B^{(j-1)})$, and one arises from the randomness of the mechanism for fixed $(A^{(j-1)},B^{(j-1)})$.\n\n\\paragraph{Closure of a distribution menu.}\nFurthermore, it will be convenient to make the menu closed and convex as follows.\n\n\\begin{definition}\n\\label{def:menu-hull}\nWe define $\\overline{\\cM_j}$, the closure of the distribution menu at level $j$, to be the topological closure of the set of all convex combinations of distributions from the menu $\\cM_j$. \n\\end{definition}\n\nI.e., we take the convex hull of the menu and then its topological closure. \nWe emphasize that the convex hull is generated by averaging distributions, and not the values of $(X_j,P_j)$. In other words, a  distribution of $(X'_j,P'_j)$ is in $\\overline{\\cM_j}$ if its distribution can be approximated arbitrarily closely by some convex combination of distributions in $\\cM_j$. It is important that we keep all the randomness present in $(X_j,P_j)$ and do not take expectations until the end.\n\n\n\\subsection{Symmetry gap revisited}\n\nRecall Lemma~\\ref{lem:sym-gap} which was proved using the symmetry gap argument and played an important role in our proof for the CPP problem.  We still want to use this lemma; however, the difference now is the presence of prices.  In order to deal with prices, we need to introduce a parameter $\\lambda$ which acts as a conversion factor between values and prices. For that purpose, we prove the following slight variation of Lemma~\\ref{lem:sym-gap}.\n\n\\begin{lemma}\n\\label{lem:sym-gap-2}\nLet $\\phi:[0,1] \\rightarrow [0,1]$ be a non-decreasing concave function and $\\lambda \\geq 0$ any constant.\nLet $A^{(j+1)}$ be a fixed set of size $2^{j+1-\\ell} m \\geq m\/n$ and $(A^{(j)}, B^{(j)})$ a random partition of $A^{(j+1)}$ into two sets of equal size.\nThen there is a monotone submodular function $\\tilde{v}_{A^{(j)}, B^{(j)}}$ for each partition $(A^{(j)}, B^{(j)})$ such that\n\\begin{compactitem}\n\\item For any distribution of a random set $R^{(j)}$ (possibly correlated with $A^{(j)}$) and the associated random variable $X_{j} = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$, we have\n$$ \\E[\\tilde{v}_{A^{(j)}, B^{(j)}}(R^{(j)})] \\geq \\lambda \\E[\\phi(X_j - n m^{-1\/2})]. $$\n\\item Any mechanism that uses $poly(n)$ value queries, when applied to the random input $\\tilde{v}_{A^{(j)}, B^{(j)}}$ will return a random set $R^{(j+1)}$ such that for the random variable $X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$,\n$$ \\E[\\tilde{v}_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\leq \\lambda \\E[1 - (1-\\phi(X_{j+1}))^2 + e^{-\\Omega(n)}].$$\n\\end{compactitem}\nThe expectations are over both $(A^{(j)}, B^{(j)})$ and $R^{(j)}$ or $R^{(j+1)}$ respectively.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is easily obtained from the proof of Lemma~\\ref{lem:sym-gap}. The only difference is the scaling by $\\lambda \\geq 0$. (For $\\lambda=0$ the statement is trivial.) Given $\\phi:[0,1] \\rightarrow [0,1]$ and $\\lambda \\geq 0$, we take the function $\\tilde{f}_{A^{(j)}, B^{(j)}}$ provided by Lemma~\\ref{lem:sym-gap} and scale it by $\\lambda$:\n$$ \\tilde{v}_{A^{(j)}, B^{(j)}}(S) = \\lambda \\tilde{f}_{A^{(j)}, B^{(j)}}(S).$$\nRandomizing over $(A^{(j)}, B^{(j)})$, the same proof shows that any mechanism will return a random set $R^{(j+1)}$ with high probability balanced with respect to $(A^{(j)}, B^{(j)})$. Hence we obtain the same bounds as in Lemma~\\ref{lem:sym-gap} with the right-hand side scaled by $\\lambda$.\n\\end{proof}\n\n\nApplying the assumption of approximate truthfulness, we obtain the following.\n\n\\begin{lemma}\n\\label{lem:lambda-gap}\nConsider a $(1-\\epsilon)$-approximately truthful-in-expectation mechanism for combinatorial auctions with $n=2^\\ell$ players and $m=400^\\ell$ items. Let $\\phi:[0,1] \\rightarrow [0,1]$ be a non-decreasing concave function. If $(X_j,P_j)$ has a distribution in the closure of the level-$j$ menu $\\overline{\\cM_j}$, then for any $\\lambda',\\lambda'' \\geq 0$ there is $(X_{j+1},P_{j+1})$ with a distribution in the closure of the level-$(j+1)$ menu $\\overline{\\cM_{j+1}}$ such that \n$$ \\lambda' \\E[1 - (1-\\phi(X_{j+1}))^2] - \\lambda'' \\E[P_{j+1}] \\geq \\lambda' \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}] - \\lambda'' \\E[P_j].$$\n\\end{lemma}\n\n\\begin{proof}\nFirst let us assume that $\\lambda''>0$ and set $\\lambda = \\lambda'\/\\lambda''$. If $(X_j,P_j)$ is on the menu $\\cM_j$, it means that the mechanism under certain valuations depending on the (random) set $A^{(j)}$ allocates to the special player a random set $R^{(j)}$ (at some price $P_j$) such that $X_{j} = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$. Given $\\phi$, by Lemma~\\ref{lem:sym-gap-2} there are valuation functions $\\tilde{v}_{A^{(j)}, B^{(j)}}$ such that \n$$ \\E[\\tilde{v}_{A^{(j)}, B^{(j)}}(R^{(j)})] \\geq \\lambda \\E[\\phi(X_j - n m^{-1\/2})] \n$$\nand on the other hand, the mechanism executed on this random input allocates a random set $R^{(j+1)}$ such that with $X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$,\n$$ \\E[\\tilde{v}_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\leq \\lambda \\E[1 - (1-\\phi(X_{j+1}))^2 + e^{-\\Omega(n)}]\n.$$\nLet us assume that the mechanism allocates this distribution at a price $P_{j+1}$. \nDue to the assumption of $(1-\\epsilon)$-truthfulness, the utility provided by the mechanism must be approximately maximized for the true valuation. Hence, we must have\n\\begin{equation}\n\\label{eq:lambda-bound}\n \\lambda \\E[1 - (1-\\phi(X_{j+1}))^2 + e^{-\\Omega(n)}] - \\E[P_{j+1}] \\geq (1-\\epsilon) \\lambda \\E[\\phi(X_j-n m^{-1\/2})] - \\E[P_j].\n\\end{equation}\nGiven our parameters $m=400^\\ell, n=2^\\ell$, we have $n m^{-1\/2} = 10^{-\\ell}$ and $e^{-\\Omega(n)} = e^{-\\Omega(2^\\ell)} << 10^{-\\ell}$, therefore\n$$ \\lambda \\E[1 - (1-\\phi(X_{j+1}))^2] - \\E[P_{j+1}] \\geq \\lambda \\E[(1-\\epsilon) \\phi(X_j - 10^{-\\ell}) - 10^{-\\ell}] - \\E[P_j].$$\nSince this inequality is preserved under convex combinations and limits of the distributions of $(X_j,P_j)$ and $(X_{j+1},P_{j+1})$ (the non-linearity of $\\phi$ is irrelevant here!), the same holds for the closures $\\overline{\\cM_j}, \\overline{\\cM_{j+1}}$: For any $(X_j,P_j)$ with a distribution in $\\overline{\\cM_j}$ and $\\lambda>0$, there exists $(X_{j+1},P_{j+1})$ with a distribution in $\\overline{\\cM_{j+1}}$ such that $(\\ref{eq:lambda-bound})$ holds. This proves the statement of the lemma when $\\lambda''>0$.\n\nWhen $\\lambda''=0$, the statement claims that given $(X_j,P_j) \\in \\overline{\\cM_j}$, there is $(X_{j+1},P_{j+1}) \\in \\overline{\\cM_{j+1}}$, such that $\\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}]$,\nwithout regard to prices. This can be obtained from the previous discussion as follows. Let us assume that $p^*$  is an absolute lower bound on the expected price $\\E[P_{j+1}]$. (If the mechanism possibly pays arbitrarily large amounts on the menu of the special player, then given a zero valuation it cannot maximize utility over the menu.)\nGiven $(X_j,P_j)$ in $\\overline{\\cM_j}$, let $\\lambda = 10^{\\ell+1} (\\E[P_j] - p^*)$; there must be a pair $(X_{j+1},P_{j+1})$ in $\\overline{\\cM_{j+1}}$ satisfying\n (\\ref{eq:lambda-bound}). Using the (still very crude) estimate $e^{-\\Omega(n)} << 10^{-\\ell-1}$, (\\ref{eq:lambda-bound}) implies\n\\begin{eqnarray*}\n\\E[1 - (1-\\phi(X_{j+1}))^2] & \\geq & \\E[(1-\\epsilon) \\phi(X_j-n m^{-1\/2}) - e^{-\\Omega(n)}] - \\frac{\\E[P_j] - p^*}{\\lambda} \\\\\n& \\geq & \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell-1}] - 10^{-\\ell-1} \\\\\n& \\geq & \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}].\n\\end{eqnarray*}\n\\end{proof}\n\n\\subsection{The convex separation argument}\n\nNext, we use a geometric argument, essentially Farkas' lemma in 2 dimensions, which shows that the bounds for varying multipliers $\\lambda', \\lambda'' \\geq 0$ allow us to obtain separate bounds on value and price.\n\n\\begin{lemma}\n\\label{lem:convex}\nConsider a $(1-\\epsilon)$-approximately truthful-in-expectation mechanism for combinatorial auctions with $n=2^\\ell$ players and $m=400^\\ell$ items. Let $\\phi:[0,1] \\rightarrow [0,1]$ be a non-decreasing concave function. If $(X_j,P_j)$ has a distribution in the closure of the level-$j$ menu $\\overline{\\cM_j}$, then there is $(X_{j+1},P_{j+1})$ with a distribution in the closure of the level-$(j+1)$ menu $\\overline{\\cM_{j+1}}$ such that\n$$\\E[P_{j+1}] \\leq \\E[P_j] $$ and\n$$\\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}].$$\n\\end{lemma}\n\n\\begin{figure}[!h]\n\\label{fig:covex-sep}\n\\centering\n\\input{convex-sep.tex}\n\\caption{The convex separation argument.}\n\\label{fig:convex-sep}\n\\end{figure}\n\n\n\\begin{proof}\nDenote $q_j = \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}]$ and $q_{j+1} = \\E[1 - (1-\\phi(X_{j+1}))^2]$. Set also $p_{j} = \\E[P_{j}]$ and $p_{j+1} = \\E[P_{j+1}]$. \nBy Lemma~\\ref{lem:lambda-gap}, for any $(X_{j},P_{j}) \\in \\overline{\\cM_j}$ and any $\\lambda',\\lambda'' \\geq 0$, there is $(X_{j+1},P_{j+1}) \\in \\overline{\\cM_{j+1}}$  such that $\\lambda' q_{j+1} - \\lambda'' p_{j+1} \\geq \\lambda' q_j - \\lambda'' p_j$.\nUsing these transformations, let us map the set $\\overline{\\cM_j}$ to \n$$ \\cQ_j = \\{ (q_j, p_j): (X_j, P_j) \\in \\overline{\\cM_j} \\}.$$\nand map $\\overline{\\cM_{j+1}}$ to\n$$ \\cQ_{j+1} = \\{ (q_{j+1}, p_{j+1}): (X_{j+1}, P_{j+1}) \\in \\overline{\\cM_{j+1}} \\}. $$\nBoth $\\cQ_j$ and $\\cQ_{j+1}$ are closed convex sets, because they are the images of the closed convex sets $\\overline{\\cM_j}, \\overline{\\cM_{j+1}}$ under a linear map (the map being the expectation of a certain function over a distribution; this is linear\nas a function of the distribution even though the function is non-linear).\n\nBy this transformation, we have reduced the proof to a geometric question in the plane (see Figure~\\ref{fig:convex-sep}): Given $(q_j,p_j)$, assume that for any $\\lambda', \\lambda'' \\geq 0$, there is $(q_{j+1}, p_{j+1}) \\in \\cQ_{j+1}$ such that $\\lambda' q_{j+1} - \\lambda'' p_{j+1} \\geq \\lambda' q_j - \\lambda'' p_j$.\nIs it possible that there is no point $(q_{j+1}, p_{j+1}) \\in \\cQ_{j+1}$ such that $q_{j+1} \\geq q_j$ and $p_{j+1} \\leq p_j$?\n\nSuppose that there is no such point in $\\cQ_{j+1}$.\nThis means that $\\cQ_{j+1}$ and $\\{(q,p): q \\geq q_j, p \\leq p_j\\}$ are disjoint.\nSince these are closed convex sets, they can be separated by a line.\nThis line cannot have a negative slope, otherwise it would intersect the quadrant $\\{(q,p): q \\geq q_j, p \\leq p_j\\}$. Such a separating line gives $\\lambda', \\lambda'' \\geq 0$ such that $\\lambda' q_{j+1} - \\lambda'' p_{j+1} < \\lambda' q_j - \\lambda'' p_j$ for all $(q_{j+1}, p_{j+1}) \\in \\cQ_{j+1}$. However, this contradicts the assumption above. Hence there is a point $(q_{j+1}, p_{j+1}) \\in \\cQ_{j+1}$ such that $q_{j+1} \\geq q_j$ and\n $p_{j+1} \\leq p_j$.\n\\end{proof}\n\n\n\\subsection{Putting it all together}\n\nIn this section, we finish the proof of our main hardness result for combinatorial auctions.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:CA-hardness}]\nLet $\\epsilon>0$ and $\\delta>0$ be the constants provided by Lemma~\\ref{lem:level-bound}. Let the number of players be $n=2^\\ell$ and the number of items $m = 400^\\ell$. Suppose there is a $(1-\\epsilon)$-approximately truthful-in-expectation mechanism that provides a $c$-approximation in social welfare, where $c = 1\/n^\\gamma = 2^{-\\gamma \\ell}$ for some constant $\\gamma>0$.\n\nConsider the basic instance (Section~\\ref{sec:basic-inst}). Choose a special player and fix the remaining valuations, based on Lemma~\\ref{lem:basic-inst}. Let $R^{(0)}$ be the random set allocated to the special player, $P_0$ the respective price, $A^{(0)}$ his desired set, $X_0 = \\frac{|R^{(0)} \\cap A^{(0)}|}{|A^{(0)}|}$, $c_0 = \\E[X_0]$ and $p_0 = \\E[P_0]$. \nLemma~\\ref{lem:basic-inst} implies $c_0 \\geq c\/4 - \\omega$. We set $\\omega = c\/8$. Then\n $c_0 \\geq c\/8 = 1\/(8 n^\\gamma) = 2^{-\\gamma \\ell - 3}$.\n\nNow consider the distribution menus at different levels and their closures $\\overline{\\cM_j}$ (Section~\\ref{sec:prep}).\nLet us define $\\cX_j$ to be the collection of random variables $X_j$ such that $(X_j,P_j)$ is in $\\overline{\\cM_j}$ for some price $P_j$ such that $\\E[P_j] \\leq p_0$. As discussed above, we have $X_0$ in $\\cX_0$ such that $\\E[X_0] = c_0 \\geq 2^{-\\gamma \\ell - 3} \\geq 2^{-\\ell}$ for $\\ell$ sufficiently large. Also, Lemma~\\ref{lem:convex} says that for any $X_j$ in $\\cX_j$ and any non-decreasing concave $\\phi:[0,1] \\rightarrow [0,1]$, we have $X_{j+1}$ in $\\cX_{j+1}$ such that\n$$\\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[(1-\\epsilon) \\phi(X_j-10^{-\\ell}) - 10^{-\\ell}].$$\nIn other words, the collections $\\cX_0, \\cX_1, \\ldots, \\cX_\\ell$ satisfy the assumptions of Lemma~\\ref{lem:level-bound}.\nHence, by Lemma~\\ref{lem:level-bound} for $j=\\ell$, there is $X_\\ell$ in $\\cX_\\ell$ and $\\alpha_\\ell \\in [0,1]$ such that\n$$ \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell c_0^{1+\\delta}.$$\nRecall that $\\phi_{\\alpha_\\ell}(t) = \\min \\{ \\frac{t}{\\alpha_\\ell}, 1\\}$. Therefore, we have \n$$  \\E[X_\\ell] \\geq \\alpha_\\ell \\E[\\phi_{\\alpha_\\ell}(X_\\ell)] \\geq  \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell c_0^{1+\\delta} \\geq 2^{\\delta^2 \\ell - \\ell} c_0^{1+\\delta}.$$\nSince $X_\\ell$ is in $\\cX_\\ell$, the respective price is bounded by $\\E[P_\\ell] \\leq p_0$. \nNow consider an expression related to the utility the special player would derive in the basic instance:\n$$ \\E[(1-\\epsilon) \\omega m X_\\ell - P_\\ell] \\geq\n (1-\\epsilon) \\omega m 2^{\\delta^2 \\ell - \\ell} c_0^{1+\\delta} - p_0.$$\nThe distribution of $(X_{\\ell}, P_\\ell)$ might not be on the actual menu $\\cM_\\ell$ of the special player; however, since it is in the closure of its convex hull, there exists a pair $(\\tilde{X}_{\\ell}, \\tilde{P}_\\ell)$ with a distribution in $\\cM_\\ell$ such that\n$$ \\E[(1-\\epsilon) \\omega m \\tilde{X}_\\ell - \\tilde{P}_\\ell]\n > (1-2\\epsilon) \\omega m 2^{\\delta^2 \\ell - \\ell} c_0^{1+\\delta} - p_0.$$\n(If not, we get a contradiction since if the reverse inequality holds for $\\cM_\\ell$, it holds also for the closure $\\overline{M}_\\ell$.) \nThe random variable $\\tilde{X}_\\ell$ represents a random set $\\tilde{R}^{(\\ell)}$, possibly allocated to the special player at price $\\tilde{P}_\\ell$: $\\tilde{X}_\\ell = \\frac{|\\tilde{R}^{(\\ell)}|}{|A^{(\\ell)}|} = \\frac{1}{m} |\\tilde{R}^{(\\ell)}|$. \n\nNow let us go back to the basic instance. Considering that the valuation of the special player in the basic instance satisfies $v^*_i(S) \\geq \\omega|S|$, we obtain\n$$ \\E[(1-\\epsilon) v^*_i(\\tilde{R}^{(\\ell)}) - \\tilde{P}_\\ell]\n \\geq \\E[(1-\\epsilon) \\omega m \\tilde{X}_{\\ell} - \\tilde{P}_\\ell]\n > (1-2\\epsilon) \\omega m 2^{\\delta^2 \\ell - \\ell} c_0^{1+\\delta} - p_0.$$ \nUsing $c_0 \\geq c\/8 = \\omega = 2^{-\\gamma \\ell - 3}$, we get\n\\begin{equation}\n\\label{eq:menu-item}\n\\E[(1-\\epsilon) v^*_i(\\tilde{R}^{(\\ell)}) - \\tilde{P}_\\ell]\n> (1-2\\epsilon) \\left( \\frac{c}{8} \\right)^{1+\\delta} 2^{\\delta^2 \\ell - \\ell} m c_0 - p_0\n= (1-2\\epsilon) 2^{\\delta^2 \\ell - (1+\\delta) (\\gamma \\ell + 3) - \\ell} m c_0 - p_0.\n\\end{equation}\nOn the other hand, the set $R^{(0)}$ actually allocated under declared valuation $v^*_i$ gives\n$$ \\E[v^*_i(R^{(0)})] = \\E[|R^{(0)} \\cap A^{(0)}|] + \\omega \\E[|R^{(0)} \\setminus A^{(0)}|]\n \\leq \\frac{m}{n} \\E[X_0] + \\omega \\E[|R^{(0)}|] \\leq \\frac{2m}{n} \\E[X_0] $$ \nusing again Lemma~\\ref{lem:basic-inst} to say that $\\frac{m}{n} \\E[X_0] = \\E[|R^{(0)} \\cap A^{(0)}|] \\geq (c\/4-\\omega) \\E[|R^{(0)}|] = \\omega \\E[|R^{(0)}|] $. Therefore, since $\\E[X_0] = c_0$ and $\\E[P_0] = p_0$,\n\\begin{equation}\n\\label{eq:true-item}\n\\E[v^*_i(R^{(0)}) - P_0] \\leq \\frac{2m}{n} \\E[X_0] - \\E[P_0] = 2^{1-\\ell} m c_0 - p_0.\n\\end{equation}\nSince $\\tilde{R}^{(\\ell)}$ is a random set the special player could receive at price $\\tilde{P}_\\ell$ if he had declared a suitable valuation, $(1-\\epsilon)$-approximate truthfulness implies that\n$$ \\E[v^*_i(R^{(0)}) - P_0] \\geq \\E[(1-\\epsilon) v^*_i(\\tilde{R}^{(\\ell)}) - \\tilde{P}_\\ell]. $$\nConsidering (\\ref{eq:menu-item}) and (\\ref{eq:true-item}), this implies\n$$ 2^{1-\\ell} > (1-2\\epsilon) 2^{\\delta^2 \\ell - (1+\\delta)(\\gamma \\ell+3)-\\ell}.$$\nWe conclude that $\\gamma \\geq \\frac{\\delta^2}{1+\\delta}$, otherwise we get a contradiction for a large enough $\\ell$.\n\\end{proof}\n\n\n\\section{Hardness for combinatorial auctions}\n\\label{sec:auctions-hardness}\n\nThe following is our main result for combinatorial auctions.\n\n\\begin{theorem}\n\\label{thm:CA-hardness}\nThere are absolute constants $\\epsilon, \\gamma > 0$ such that there is no $(1-\\epsilon)$-approximately truthful-in-expectation mechanism for combinatorial auctions with monotone submodular valuation functions in the value oracle model, achieving a better than $1\/n^\\gamma$-approximation in expectation in terms of social welfare, where the number of players is $n$ and the number of items is $m=\\poly(n)$. \n\\end{theorem}\n\n\\iffalse\n\\paragraph{Approximate truthfulness.}\nOur main reason for considering the notion of approximate truthfulness is that the mechanisms of \\cite{DRY11,Dughmi11}, if implemented in the value oracle model, may at best be approximately truthful-in-expectation (for an arbitrarily small $\\epsilon>0$). The value oracle model seems too weak to make the mechanisms of \\cite{DRY11,Dughmi11} exactly truthful-in-expectation; however, \\cite{DRY11,Dughmi11}  make it quite conceivable that there might be an approximately truthful-in-expectation mechanism for combinatorial auctions and combinatorial public projects, both with submodular valuations. We note that the construction of \\cite{Dobzin11} is not robust with respect to approximate truthfulness and rules out only exactly (universally) truthful mechanisms. Our hardness result rules out both approximate truthfulness and truthfulness in expectation simultaneously.\n\\fi\n\n\\paragraph{Discussion.}\nThis theorem extends previous negative results for combinatorial auctions with submodular valuation functions, which were known in the cases of deterministic truthful and randomized universally truthful mechanisms \\cite{Dobzin11}. Also, it appears that as stated these results do not rule out approximately truthful mechanisms.\n\nWe remark that there is still the possibility of a truthful-in-expectation mechanism in the ``lottery-value\" oracle model which was introduced in \\cite{DRY11}. Here, a player is able to provide the {\\em exact} expectation $\\E[v_i(\\hat{\\bx})]$ for a product distribution given by $\\bx$. Since the exact expectations $\\E[v_i(\\hat{\\bx})]$ are hard to compute even in very special cases like the budget-additive case, this is a severe limitation. Our hardness result does not apply directly to this stronger oracle model. However, what our result implies is that if a truthful-in-expectation mechanism exists in the lottery-value model, then it must be very sensitive to the accuracy of the oracle's answers, and does not remain even approximately truthful-in-expectation if the oracle's answers involve some small noise. This is because if we had a mechanism in the lottery-value oracle model, which remains approximately t.i.e. under small noise in the oracle and provides a good approximation, then we could simulate this mechanism in the value oracle model (by sample-average approximation). Thus we would obtain an approximately t.i.e. mechanism contradicting Theorem~\\ref{thm:CA-hardness}. \n\n\\paragraph{Proof strategy.} Our hardness result for combinatorial public projects (Section~\\ref{sec:CPP-hardness}) can be adapted to show that there is no (approximately) MIDR mechanism for submodular combinatorial auctions that guarantees a good approximation ratio. However, unlike in CPP, we are unable to prove that truthful-in-expectation mechanisms and MIDR algorithms are equivalent in power (even in the approximate sense). This is not surprising, since randomized truthful mechanisms that are not maximal-in-distributional-range have been designed for combinatorial auctions (see for example \\cite{D07}). Therefore, additional ideas are needed to rule out all truthful-in-expectation mechanisms.\nSuch ideas have been recently put forth in a paper by Dobzinski \\cite{Dobzin11}. The {\\em direct hardness} approach of \\cite{Dobzin11} provides a way to avoid the characterization step and instead attack the truthful mechanism directly. This idea applies to truthful-in-expectation mechanisms as well. \n\nThe main idea of the direct hardness approach can be stated as follows. If we identify a special player whose range of possible allocations is sufficiently ``rich\" when the valuations of other players are fixed to particular functions, then we can work with the special player directly using the \\emph{taxation principle}: There is a fixed price for each distribution over allocations in the ``range\" of the mechanism as the special player varies his valuation, and  the mechanism outputs the distribution in this range that maximizes the player's utility (his expected value for the distribution on allocations less the price of that distribution). Thus, our symmetry gap techniques from Section~\\ref{sec:CPP-hardness} apply here quite naturally, though the presence of payments poses an additional technical challenge that was not present for CPP.\nNext, we present a sketch of our proof. The complete proof is presented in Appendix~\\ref{sec:CA-proof}.\n\n\\iffalse\n\\paragraph{Proof overview.}\n\\begin{itemize}\n\\item We start from a ``basic instance\" where each player chooses independently a random set whose items have value $1$ and other items are worth only some small $\\omega>0$ (as in \\cite{Dobzin11}). We call these valuations $v^*_i$.\n\\item If the mechanism provides a good approximation in social welfare, there must be a player $i$ whose allocated set $R^{(0)}_i$ overlaps significantly with his desired set, for certain valuations $v^*_{-i}$ of the other players. In the following, we fix $v^*_{-i}$ and work with player $i$ only.\n\\item Considering other valuations $v^{(j)}_i$ for player $i$ and $1 \\leq j \\leq \\ell$, we prove that there must be other distributions possibly allocated to him, depending on $v^{(j)}_i$. Using the assumption of truthfulness and a symmetry gap argument, we prove that when declaring $v^{(j+1)}_i$, he must obtain a larger (random) set than when declaring $v^{(j)}_i$, at a comparable price.\n\\item Eventually, we prove that player $i$ with valuation $v^{(\\ell)}_i$ gets allocated a set $R^{(\\ell)}_i$ which is very large in expectation, for a low price. This set is so large that in terms of the initial valuation $v^*_i$, $R^{(\\ell)}_i$ brings more utility to player $i$ than $R^{(0)}_i$. Therefore, assuming the true valuation of player $i$ is $v^*_i$, he would be better off declaring $v^{(\\ell)}_i$, which is a contradiction to the assumption of truthfulness.\n\\end{itemize}\n\\fi\n\n\\paragraph{The basic instance.}\nWe start from the following ``basic instance\".\nFor an integer $\\ell$, we construct instances with $|N| = n = 2^\\ell$ players and $|M| = m = \\poly(n)$ items.\nEach player has a ``polar valuation\" $v^*_i$ (as in \\cite{Dobzin11}), where items in a certain set $A^{(0)}_i$ have value $1$ for player $i$ and other items have (small) value $\\omega>0$. The sets $A^{(0)}_i$ are chosen independently at random, under the constraint that $|A^{(0)}_i| = m\/n$.\n\nA counting argument shows that if a mechanism provides a $c$-approximation in social welfare,\nthen there must be a player whose allocated set $R^{(0)}_i$ overlaps significantly with his desired set $A^{(0)}_i$:\n\\begin{equation*}\n \\E[|R^{(0)}_i \\cap A^{(0)}_i|] \\geq (c\/4-\\omega) \\E[|R^{(0)}_i \\cup A^{(0)}_i|].\n\\end{equation*}\n(See Lemma~\\ref{lem:basic-inst}.)\nBy an averaging argument, this is also true for a certain fixed choice of the other players' valuations. In the following, we fix that choice and consider varying valuations for player $i$ only, who we refer to as the ``special player\". We also drop the index $i$, since we do not consider the other players anymore.\n\nIn the following, we set $\\omega = c\/8$, so that $\\E[|R^{(0)} \\cap A^{(0)}|] \\geq \\omega \\E[|R^{(0)} \\cup A^{(0)}|]$. Hence we can estimate the expected value received by the special player as follows: \n$$\\E[v^*(R^{(0)})] = \\E[|R^{(0)} \\cap A^{(0)}|] + \\omega \\E[|R^{(0)} \\setminus A^{(0)}|] \\leq 2 \\E[|R^{(0)} \\cap A^{(0)}|].$$\nDenoting $X_0 = \\frac{|R^{(0)} \\cap A^{(0)}|}{|A^{(0)}|}$, we have  $\\E[v^*(R^{(0)})] \\leq \\frac{2m}{n} \\E[X_0]$.\nAlso, $\\E[v^*(R^{(0)})] \\geq \\E[|R^{(0)} \\cap A^{(0)}|] = \\frac{m}{n} \\E[X_0]$. \nSo the special player's utility in the basic instance (in expectation over the random instances) is $\\frac{m}{n} \\E[X_0]$, up to a factor of $2$.\n\n\\paragraph{Symmetry gap again.}\nWe consider valuations for the special player at $\\ell$ levels, in the same form that we considered in the case of combinatorial public projects. The difference now is that the mechanism is not necessarily maximal-in-distributional-range. Instead, we use the definition of truthfulness in expectation directly. The same symmetry gap argument as in Section~\\ref{sec:CPP-hardness} gives the following:\nIf there is a random set $R^{(j)}$ possibly allocated at level $j$ at a price $P_j$, and $X_j = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$, then there is a random set $R^{(j+1)}$ possibly allocated at level $j+1$ at a price $P_{j+1}$, and\n$X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$, so that\n$$  \\E[1 - (1-\\phi(X_{j+1}))^2] - \\E[P_{j+1}] \\geq \\E[\\phi(X_j)] - \\E[P_j].$$\nAgain, we are ignoring certain error terms and we are also ignoring the issue of approximate truthfulness.\nUsing the fact that the valuation functions can be scaled arbitrarily and the mechanism must still be truthful in expectation,\nwe obtain that for any $\\lambda',\\lambda'' \\geq 0$, there is distribution possibly allocated at level $j+1$ such that\n\\begin{equation}\n\\label{eq:sym-lam}\n \\lambda' \\E[1 - (1-\\phi(X_{j+1}))^2] - \\lambda'' \\E[P_{j+1}] \\geq \\lambda' \\E[\\phi(X_j)] - \\lambda'' \\E[P_j].\n\\end{equation}\n\n\\paragraph{Convex hulls and the separation argument.}\nOur goal is to eliminate the prices from the picture, so that we can use arguments similar to Section~\\ref{sec:CPP-hardness}. For that purpose, it is convenient to pass to convex hulls as follows. We define the {\\em distribution menu} $\\cM_j$ at level $j$ to consist of all distributions of pairs of random variables $(X_j,P_j)$, such that $X_j = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$ for some random set $R^{(j)}$ allocated for a level-$j$ valuation at a price $P_j$. Then we define the {\\em closure} of a distribution menu, $\\overline{\\cM}_j$, to be the topological closure of the convex hull of $\\cM_j$ (in the sense of taking convex combinations of distributions). By convexity, (\\ref{eq:sym-lam}) still holds in the sense that for any $(X_j,P_j)$ with a distribution in $\\overline{\\cM}_j$ and any $\\lambda',\\lambda'' \\geq 0$, there is $(X_{j+1},P_{j+1})$ with a distribution in $\\overline{\\cM_{j+1}}$ such that (\\ref{eq:sym-lam}) holds.\n\nA convex separation argument, essentially Farkas' lemma in 2 dimensions, actually implies the following.\nFor any $(X_j,P_j)$ with a distribution in $\\overline{\\cM}_j$, there is $(X_{j+1},P_{j+1})$ with a distribution in $\\overline{\\cM_{j+1}}$ such that $\\E[P_{j+1}] \\leq \\E[P_j]$ and\n\\begin{equation*}\n \\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[\\phi(X_j)].\n\\end{equation*}\nIn other words, there is a distribution in the closure of the menu at level $j+1$ at a price no higher than the price we had at level $j$, and the respective random variables $X_j, X_{j+1}$ satisfy the same relationship (\\ref{eq:sym-simple}) that we had in Section~\\ref{sec:CPP-hardness}.\nThe rest of the proof goes exactly as in Section~\\ref{sec:CPP-hardness}, using (\\ref{eq:ampl}) and eventually producing a distribution represented by $(X_\\ell,P_\\ell)$ in $\\overline{\\cM_\\ell}$ such that $\\E[P_\\ell] \\leq \\E[P_0]$ and\n$$ \\E[X_\\ell] \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell (\\E[X_0])^{1+\\delta}. $$\nHere, $(X_0,P_0)$ represents the distribution and price allocated in the basic instance.\nNow we consider the utility that the distribution represented by $(X_\\ell,P_\\ell)$ would provide in the basic instance: since every element has value at least $\\omega$ there, the utility would be\n\\begin{equation}\n\\label{eq:ell-value}\n \\E[v^*(R^{(\\ell)}) - P_\\ell] \\geq \\E[\\omega m X_\\ell - P_\\ell] \\geq \\omega m \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell (\\E[X_0])^{1+\\delta} - \\E[P_0].\n\\end{equation}\nRecall that the distribution of $(X_\\ell,P_\\ell)$ is not on the menu $\\cM_\\ell$ but rather in its convex hull. However, by using the properties of the convex hull, there must be a distribution on the actual menu $\\cM_\\ell$ that satisfies the same linear inequality. So we can assume without loss of generality that the distribution of $(X_\\ell, P_\\ell)$ is on the actual menu at level $\\ell$, and $R^{(\\ell)}$ is the respective random set that would be allocated to the special player if he declared a level-$\\ell$ valuation.\n\nRecall that in the basic instance, the value received by the special player is at most $\\frac{2m}{n} \\E[X_0]$, and the respective utility is at most $\\frac{2m}{n} \\E[X_0] - \\E[P_0]$. We also have $n = 2^\\ell$ and $\\E[X_0] \\geq c\/4 - \\omega = c\/8$.\nIf $c = 8\\omega \\geq n^{-\\gamma}$ for a suitable constant $\\gamma>0$, we would obtain from (\\ref{eq:ell-value}) that the special player could substantially improve his utility in the basic instance by declaring a level-$\\ell$ valuation instead. We conclude that this would contradict the property of truthfulness in expectation.\n\nThe complete proof appears in Appendix~\\ref{sec:CA-proof}.\n\n\n\n\n\n\\section{Proof of hardness for combinatorial public projects}\n\\label{sec:CPP-proof}\n\nIn this section, we present the proof of Theorem~\\ref{thm:CPP-hardness}.\n\n\\subsection{The basic setup}\n\\label{sec:basics}\n\nWe consider a ground set of $|M|=m=400^\\ell$ items for some $\\ell \\geq 1$. We set the cardinality bound to be $k = 200^\\ell = m\/n$ where $n = 2^\\ell$. (We note that $n$ is just a parameter unrelated to the number of players, which is 1 in this case. This parameter will however denote the number of players in Section~\\ref{sec:auctions-hardness}.)\nAn important object in the following will be a {\\em random bisection sequence}.\n\n\\begin{definition}\n\\label{def:rnd-bisect}\nA random bisection sequence is a random sequence of pairs of sets $(A^{(0)},B^{(0)})$, $(A^{(1)},B^{(1)})$,\n$\\ldots$, $(A^{(\\ell)}, B^{(\\ell)})$ generated as follows. We define $A^{(\\ell)} = B^{(\\ell)} = M$. Given $A^{(j)}$ for $0 < j \\leq \\ell$, we pick $(A^{(j-1)}, B^{(j-1)})$ uniformly among all partitions of $A^{(j)}$ into two parts of size $\\frac12 |A^{(j)}|$.\n\\end{definition}\n\nI.e., $|A^{(j)}| = |B^{(j)}| = 2^{j-\\ell} m$.\nWe refer to $A^{(j)} = A^{(j-1)} \\cup B^{(j-1)}$ as the $j$-th level of the bisection sequence. Observe that the distribution of $(A^{(j-1)},B^{(j-1)})$ is uniform among all pairs of disjoint sets of size $2^{j-1-\\ell} m$.\nWe will use valuation functions associated with each level of a bisection sequence. We denote these valuation functions at level $j$ by $f_{A^{(j-1)},B^{(j-1)}}$. In particular, this valuation function depends only on the elements of $A^{(j)} = A^{(j-1)} \\cup B^{(j-1)}$.\n\nThe bisection sequence is generated at random and unknown to the mechanism. For each particular choice of a valuation function at a certain level, the mechanism needs to produce a probability distribution over feasible sets, which is purportedly the (approximately) optimal one over a certain fixed range of distributions $\\cR$. The distribution will depend on the choice of a valuation function, in particular on the relevant set of items $A^{(j)}$. A function assigning a distribution over sets to every set $A^{(j)}$ is a complicated object; in order to be able to argue about all possible such functions, we distill the important information into a single random variable for each level $j$.\n\n\\begin{definition}\n\\label{def:density}\nWe say that a random variable $X_j$ is constructible by a range $\\cR$ at level $j$, if there is a distribution $D(A^{(j)}) \\in \\cR$ for each set $A^{(j)}$ of size $2^{j-\\ell} m$ such that if a random set $R$ is generated by first choosing $A^{(j)}$ uniformly among all sets of size $2^{j-\\ell} m$ and then sampling $R$ from the distribution $D(A^{(j)})$, then\n$$ X_j = \\frac{|R \\cap A^{(j)}|}{|A^{(j)}|}. $$\n\\end{definition}\n\nNote that the normalization is chosen so that we have $X_j \\in [0,1]$. There are two sources of randomness in defining $X_j$: one is the randomness in $A^{(j)}$, and one arises from the probability distribution $D(A^{(j)})$.\n\nThe first useful fact is the following (easy) lemma.\n\n\\begin{lemma}\n\\label{lem:base-case}\nConsider a mechanism returning distributions from a range $\\cR$ that achieves a $c$-approximation for the problem $\\max \\{f(S): |S| \\leq k \\}$ for $f$ monotone submodular. Then there is a random variable $X_0$ constructible by $\\cR$ at level $0$ such that\n$$ \\E[X_0] \\geq c.$$\n\\end{lemma}\n\n\\begin{proof}\nConsider the valuation function $f_{A^{(0)}}(S) = \\frac{|S \\cap A^{(0)}|}{|A^{(0)}|}$. Let $X_0 = \\frac{|R^{(0)} \\cap A^{(0)}|}{|A^{(0)}|}$ where $R^{(0)}$ is the random set returned by the mechanism, given valuation $f_{A^{(0)}}$ for $A^{(0)}$ chosen randomly among all sets of size $2^{-\\ell} m$. By Definition~\\ref{def:density}, $X_0$ is a random variable constructible by the range $\\cR$ at level $0$. \n\nSince the optimum under valuation  $f_{A^{(0)}}$ is $1$ (achieved by $A^{(0)}$ itself), the mechanism should return expected value at least $c$. The value returned by the mechanism is exactly the random variable $X_0$, hence $\\E[X_0] \\geq c$.\n\\end{proof}\n\nWe remark that Lemma~\\ref{lem:base-case} is the only place where we use the assumption of $c$-approximation.\nIn the following, our goal is to use the MIDR property to argue about distributions that must be in the range at higher levels, and prove successive bounds on the random variables $X_1, X_2, \\ldots$.\n\n\n\\subsection{The symmetry gap argument}\n\nThe main building block of our proof is a symmetry gap argument whose goal is to show the following. If the mechanism optimizes over a certain fixed range $\\cR$ which supports distributions of ``high density\" at level $j$, then $\\cR$ must support distributions of even higher density (when properly scaled) at level $j+1$. However, the way we measure density is quite intricate. Recall the random variables $X_0,X_1,\\ldots,X_\\ell$ that encode certain distributions in the range at each level. It would be nice to say that for any $X_j$ constructible at level $j$, there must be $X_{j+1}$ constructible at level $j+1$ such that $\\E[X_{j+1}] > \\frac{1+\\delta}{2} \\E[X_j]$ (which would correspond to sets of larger cardinality at level $j+1$ than $j$). But this is not true - the distributions of $X_j, X_{j+1}$ also matter and we cannot get a guaranteed boost just in terms of expectation. Instead, we define a measure of density using a test function $\\phi$ that we specify later. \nThe symmetry gap argument allows us to prove the following.\n\n\\begin{lemma}\n\\label{lem:sym-gap}\nLet $\\phi:[0,1] \\rightarrow [0,1]$ be a non-decreasing concave function. Fix $j \\in \\{0,1,\\ldots,\\ell-1\\}$ and \na set $A^{(j+1)}$ of size $2^{j+1-\\ell} m \\geq m\/n$. Let $(A^{(j)}, B^{(j)})$ be a random partition of $A^{(j+1)}$ into two sets of equal size.\nThen there is a monotone submodular function $\\tilde{f}_{A^{(j)}, B^{(j)}}$ for each partition $(A^{(j)}, B^{(j)})$ such that\n\\begin{compactitem}\n\\item For any distribution of a random set $R^{(j)}$ (possibly correlated with $A^{(j)}$) and the associated random variable $X_{j} = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$, we have\n$$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j)})] \\geq \\E[\\phi(X_j - n m^{-1\/2})]. $$\n\\item Any mechanism that uses $poly(n)$ value queries, when applied to the random input $\\tilde{f}_{A^{(j)}, B^{(j)}}$ will return a random set $R^{(j+1)}$ such that for the random variable $X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$,\n$$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\leq \\E[1 - (1-\\phi(X_{j+1}))^2] + e^{-\\Omega(n)}.$$\n\\end{compactitem}\nThe expectations are over both $(A^{(j)}, B^{(j)})$ and $R^{(j)}$ or $R^{(j+1)}$ respectively.\n\\end{lemma}\n\nThe key point here is that the performance of a mechanism depends only on the fraction of elements taken from $A^{(j+1)}$, and not on the partition $(A^{(j)}, B^{(j)})$. While there might be a ``good distribution\" $R^{(j)}$ in the range which is correlated with $A^{(j)}$, the mechanism cannot find such a distribution and must compensate for it by returning larger sets. This will be important later.\n\n\n\\iffalse\nTo get an intuitive feeling for the bounds, ignore the error terms $n m^{-1\/2}, e^{-\\Omega(n)}$ and assume for now that $X_j, X_{j+1}$ are deterministic and moreover $\\phi$ is linear.\nIf $X_{j+1} \\simeq \\frac12 X_j$ and the value of $\\phi(X_{j+1})$ is very close to $0$, then the right-hand side expectations would be roughly equal. If this happens, the expected values (in terms of $f$) of the solutions $R^{(j)}, R^{(j+1)}$ could be very close, i.e. we do not get any contradiction with the assumption that the mechanism optimizes over $\\cR$. Also,\nwe are not making any progress, since the mechanism can return a set $R^{(j+1)}$ of roughly the same size as $R^{(j)}$. (Recall the normalization of $X_j, X_{j+1}$.) However, if $\\phi$ is suitably chosen, then $R^{(j+1)}$ is forced to be larger than $R^{(j)}$ due to the quadratic term. We use this intuition later.\n\\fi\n\nThe proof of Lemma~\\ref{lem:sym-gap} relies on the notion of symmetry gap developed in \\cite{FMV07,MSV08,V09}. Since what we need here is a special case where the construction can be carried out explicitly quite easily, we present a self-contained proof here instead of referring to the general framework of \\cite{V09}.\n\n\\begin{proof\nConsider a pair of sets  $(A^{(j)}, B^{(j)})$.\nGiven a non-decreasing concave function $\\phi:[0,1] \\rightarrow [0,1]$ ,\nwe define the valuation function $f_{A^{(j)},B^{(j)}}$ as follows:\n$$ f_{A^{(j)},B^{(j)}}(S) = 1 - \\left( 1 - \\phi\\left(\\frac{|S \\cap A^{(j)}|}{|A^{(j)}|}\\right) \\right)\n \\left( 1 - \\phi\\left(\\frac{|S \\cap B^{(j)}|}{|B^{(j)}|}\\right) \\right). $$\nThe function depends only on how many elements we take from $A^{(j)}$ and how many from $B^{(j)}$. Moreover, the two sets play the same role in $f_{A^{(j)},B^{(j)}}$; i.e., all elements in $A^{(j+1)} = A^{(j)} \\cup B^{(j)}$ contribute equivalently to $f_{A^{(j)},B^{(j)}}$. This is the kind of situation where we can apply a symmetry gap argument.\n\nLet us simplify the notation and write $$\\psi(x,y) = 1 - (1-\\phi(x))(1-\\phi(y)),$$ where $x,y \\in [0,1]$; i.e. $f_{A^{(j)},B^{(j)}}(S) = \\psi\\left(\\frac{|S \\cap A^{(j)}|}{|A^{(j)}|}, \\frac{|S \\cap B^{(j)}|}{|B^{(j)}|}\\right)$. It is elementary to verify that since $\\phi$ is non-decreasing concave, the first partial derivatives of $\\psi$ are non-negative and non-increasing with respect to both coordinates. Now we replace $\\psi$ by a modified function $\\tilde{\\psi}$ which has the property that if $|x-y|$ is very small, the function value depends only on $x+y$. This can be accomplished explicitly as follows: For some $\\beta>0$, let\n\n\\\n\n\\begin{compactitem}\n\\item $\\tilde{\\psi}(x,y) = \\psi(\\frac12(x+y), \\frac12(x+y))$ if $|x-y| \\leq \\beta$.\n\\item $\\tilde{\\psi}(x,y) = \\psi(x-\\frac12 \\beta, y+\\frac12 \\beta)$ if $x-y > \\beta$.\n\\item $\\tilde{\\psi}(x,y) = \\psi(x+\\frac12 \\beta, y-\\frac12 \\beta)$ if $y-x > \\beta$.\n\\end{compactitem}\n\n\\\n\n\\begin{figure}[!ht]\n\\centering\n\\input{sym-gap.tex}\n\\caption{Construction of $\\tilde{\\psi}(x,y)$ from $\\psi(x,y)$, assuming that $\\psi(x,y) = 1-(1-x)(1-y)$. The solid lines denote the diagonal $x=y$ and the shifted diagonals $x-y = \\pm \\beta$. The gray lines are the level sets of $\\psi(x,y)$ and $\\tilde{\\psi}(x,y)$.}\n\\label{fig:sym-gap}\n\\end{figure}\n\n\n\nGeometrically, this construction can be seen as taking the graph of $\\psi(x,y)$, pulling it away from the diagonal $x=y$ on both sides, and patching the area close to the diagonal with a function which depends only on $x+y$ and is equal to the function on the diagonal. Using the properties of $\\psi$, one can check that again the first partial derivatives of $\\tilde{\\psi}$ are non-negative and non-increasing with respect to both coordinates. We define the function promised by the lemma as\n$$\\tilde{f}_{A^{(j)},B^{(j)}}(S) = \\tilde{\\psi}\\left(\\frac{|S \\cap A^{(j)}|}{|A^{(j)}|},\\frac{|S \\cap B^{(j)}|}{|B^{(j)}|} \\right).$$\nThe properties of $\\tilde{\\psi}$ imply that $\\tilde{f}_{A^{(j)},B^{(j)}}$ is a monotone submodular function\n(see e.g.~\\cite{MSV08,V09}).\n\nWe observe the following (which is the case in all proofs using the symmetry gap). For a ``typical query\" $S$, oblivious to the random partition $(A^{(j)}, B^{(j)})$, with high probability $S$ will contain approximately the same number of elements from these two sets. (Recall that $|A^{(j)}| = |B^{(j)}|$.) We call a query $S$ balanced if the parameters $x = \\frac{|S \\cap A^{(j)}|}{|A^{(j)}|}$ and $y = \\frac{|S \\cap B^{(j)}|}{|B^{(j)}|}$ are in the range where $|x-y| \\leq \\beta$, and hence $\\tilde{f}_{A^{(j)},B^{(j)}}(S) = \\tilde{\\psi}(x,y) = \\psi(\\frac12 (x+y), \\frac12(x+y))$  is independent of the particular partition $(A^{(j)}, B^{(j)})$. By Lemma~\\ref{lem:bisect-chernoff} (applied to the ground set $A^{(j+1)}$), the probability that any fixed query $S$ is unbalanced is exponentially small:\n $$ \\Pr[|x-y| > \\beta] = \\Pr\\left[||S \\cap A^{(j)}| - |S \\cap B^{(j)}|| > \\beta |A^{(j)}|\\right] \\leq e^{-\\Omega(\\beta^2 |A^{(j+1)}|)}.$$\nRecall that $|A^{(j+1)}| = 2^{j+1-\\ell} m \\geq m\/n$. Therefore, if we pick $\\beta = n m^{-1\/2}$, the probability is $e^{-\\Omega(n)}$. \nLet us fix for now the random coin flips of the mechanism. As long as all query answers are independent of the partition $(A^{(j)}, B^{(j)})$, the mechanism will follow the same computation path, independent of $(A^{(j)}, B^{(j)})$, and we can use a union bound over its $poly(n)$ queries. Hence, the probability that a mechanism ever makes a query such that $|x-y| > \\beta$ is $poly(n) e^{-\\Omega(n)} = e^{-\\Omega(n)}$. \nThis is still true if we average over the random coin flips of the algorithm.\nTherefore, the output of the mechanism will be independent of $(A^{(j)}, B^{(j)})$ with probability $1-e^{-\\Omega(n)}$.\n\nTo summarize, the output of the mechanism, $R^{(j+1)}$, is with high probability independent of $(A^{(j)}, B^{(j)})$ and again by Lemma~\\ref{lem:bisect-chernoff} with high probability balanced with respect to $(A^{(j)}, B^{(j)})$. Given the definition of the random variable $X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$, this means the output random set contains an $X_{j+1}$-fraction of the set $A^{(j+1)}$, approximately balanced between its two halves. For some $|\\beta'| \\leq \\frac12 \\beta$, the value of such a set is\n$$ \\tilde{\\psi}(X_{j+1}+\\beta',X_{j+1}- \\beta') = \\psi(X_{j+1},X_{j+1}) = 1 - (1-\\phi(X_{j+1}))^2.$$\nThus the expected value of this solution is  $\\E[f_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\leq \\E[1 - (1-\\phi(X_{j+1}))^2] + e^{-\\Omega(n)}$ (where $e^{-\\Omega(n)}$ accounts for the small probability of finding an unbalanced solution, whose value could be up to $1$). This proves the second statement of the lemma.\n\nFinally, consider any random set $R^{(j)}$ and the associated random variable $X_{j} = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$. We have \n$$ \\tilde{f}_{A^{(j)},B^{(j)}}(R^{(j)})) \\geq  \\tilde{f}_{A^{(j)},B^{(j)}}(R^{(j)} \\cap A^{(j)}) = \\tilde{\\psi}(X_j,0)\n \\geq \\psi\\left(X_j- \\beta, 0 \\right) = \\phi\\left(X_j - \\beta \\right).$$\nTherefore $\\E[\\tilde{f}_{A^{(j)},B^{(j)}}(R^{(j)}))] \\geq \\E[\\phi(X_j - \\beta)]$. Recall that $\\beta = n m^{-1\/2}$, so this proves the first statement of the lemma.\n\\end{proof}\n\nConsidering the setup of random variables $X_0,X_1,\\ldots,X_\\ell$ constructible by $\\cR$ at different levels\n(Section~\\ref{sec:basics}), we obtain the following.\n\n\\begin{lemma}\n\\label{lem:level-gap}\nConsider a mechanism of polynomial query-complexity that $(1-\\epsilon)$-approximately maximizes over a range of distributions $\\cR$ for the problem $\\max \\{f(S): |S| \\leq k\\}$ for $f$ monotone submodular, ground set of size $m=400^\\ell$ and $k = 2^{-\\ell} m$. Let $\\phi:[0,1] \\rightarrow [0,1]$ be a non-decreasing concave function. If a random variable $X_j$ is constructible by $\\cR$ at level $j$, then there is a random variable $X_{j+1}$ constructible by $\\cR$ at level $j+1$ such that\n$$\\E[1 - (1-\\phi(X_{j+1}))^2] \\geq (1-\\epsilon) \\E[\\phi(X_j - 10^{-\\ell})] - 10^{-\\ell}.$$\n\\end{lemma}\n\n\n\\begin{proof}\nGiven $\\phi$, let $\\tilde{f}_{A^{(j)}, B^{(j)}}$ be the valuation function provided by Lemma~\\ref{lem:sym-gap}. Consider $A^{(j+1)}$ uniformly random among sets of size $2^{j+1-\\ell} m$, bisected randomly into $A^{(j)} \\cup B^{(j)}$. If $X_j$ is constructible by the range $\\cR$ at level $j$, it means that for each $A^{(j)}$ there is a distribution $D(A^{(j)})$ in $\\cR$ such that $X_j = \\frac{|R^{(j)} \\cap A^{(j)}|}{|A^{(j)}|}$ where $A^{(j)}$ is random and $X^{(j)}$ is sampled from $D(A^{(j)})$. \nBy Lemma~\\ref{lem:sym-gap}, conditioned on any $A^{(j+1)}$ and taking expectation over the random partition $(A^{(j)}, B^{(j)})$,\n$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j)}) \\mid A^{(j+1)}] \\geq \\E[\\phi(X_j  - n m^{-1\/2}) \\mid A^{(j+1)}]. $\nTherefore the same holds also without the conditioning. Recall that we have $n m^{-1\/2} = 2^\\ell 400^{-\\ell\/2} = 10^{-\\ell}$. So we get\n$$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j)})] \\geq \\E[\\phi(X_j  - n m^{-1\/2})] =  \\E[\\phi(X_j  - 10^{-\\ell})].$$\n\nNow let us run the mechanism on the same random instance and denote the output random set by $R^{(j+1)}$.\nBy Lemma~\\ref{lem:sym-gap},\n$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j+1)}) \\mid A^{(j+1)}]\n \\leq \\E[1 - (1-\\phi(X_{j+1}))^2 \\mid A^{(j+1)}] + e^{-\\Omega(n)}$, where $X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j)}|}$.\nHence this holds also without the conditioning: \n$$ \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\leq \\E[1 - (1-\\phi(X_{j+1}))^2] + e^{-\\Omega(n)}\n \\leq \\E[1 - (1-\\phi(X_{j+1}))^2] + 10^{-\\ell} $$\n and by definition $X_{j+1}$ is constructible by $\\cR$ at level $j+1$.\n\nTo conclude, if the mechanism maximizes $(1-\\epsilon)$-approximately over $\\cR$, then the expected value of $R^{(j+1)}$ conditioned on $(A^{(j)}, B^{(j)})$ must be at least $(1-\\epsilon) \\times$ that provided by $R^{(j)}$. Therefore, the same holds in expectation over $(A^{(j)}, B^{(j)})$, which means $\\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j+1)})] \\geq (1-\\epsilon) \\E[\\tilde{f}_{A^{(j)}, B^{(j)}}(R^{(j)})]$ and the lemma follows.\n\\end{proof}\n\n\n\\subsection{The gap amplification argument}\n\nIn this section, we develop an inductive argument based on Lemma~\\ref{lem:base-case} and Lemma~\\ref{lem:level-gap}, which proves that a certain notion of density of the distributions at level $j$ increases exponentially in $j$. \nBy Lemma~\\ref{lem:level-gap}, for any $X_j$ constructible at level $j$ there is $X_{j+1}$ constructible at level $j+1$ such that\n$$ \\E[1 - (1-\\phi(X_{j+1}))^2] \\geq (1-\\epsilon) \\E[\\phi(X_j - 10^{-\\ell})] - 10^{-\\ell}.$$\nWe\nwant to prove that $X_{j+1}$ is in some sense ``significantly larger\" than $\\frac12 X_j$.\nOur main technical lemma formalizing this intuition is the following.\n\n\\begin{lemma}\n\\label{lem:level-bound}\nThere are absolute constants $\\epsilon,\\delta > 0$ such that the following holds for any sufficiently large $\\ell \\in \\NN$. If $\\cX_0,\\ldots,\\cX_\\ell$ are collections of random variables in $[0,1]$ such that\n\\begin{itemize}\n\\item there is $X_0$ in $\\cX_0$ such that $\\E[X_0] \\geq c$ for some $c \\geq 2^{-\\ell}$, and\n\\item for every $X_j$ in $\\cX_j$ and every non-decreasing concave function $\\phi:[0,1] \\rightarrow [0,1]$,\nthere is $X_{j+1}$ in $\\cX_{j+1}$ such that\n$$ \\E[1 - (1-\\phi(X_{j+1}))^2] \\geq (1-\\epsilon) \\E[\\phi(X_j - 10^{-\\ell})] - 10^{-\\ell}$$\n\\end{itemize}\nthen there is a sequence of variables $X_j$ in $\\cX_j$ and parameters $1 = \\alpha_0 \\geq \\alpha_1 \\geq \\ldots \\alpha_\\ell > 0$\nsuch that if we define\n$\\phi_\\alpha(t) = \\min \\left\\{ \\frac{t}{\\alpha}, 1 \\right\\} $ then\n$$ \\alpha_j (\\E[\\phi_{\\alpha_j}(X_j)])^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^j c^{1+\\delta}.$$\n\\end{lemma}\n\nThe use of $1+\\delta$ in the exponent is crucial here; note that it makes the statement stronger, but this is what makes the inductive proof work.\nThe intuitive meaning of this lemma is as follows: there exist random variables $X_j$ constructible at different levels that, when measured by suitable test functions, decrease roughly as $\\left(\\frac{1+\\delta^2}{2}\\right)^j$, rather than $\\frac{1}{2^j}$. In terms of the cardinality of the returned sets, this means they increase by a factor of $(1+\\delta^2)$ at each level. This gives the exponential amplification that we need.\n\n\\begin{proof}\nThe base case $j=0$ holds trivially with $\\alpha_0 = 1$ and $\\phi_{\\alpha_0}(t) = t$.\nTo prove the inductive step, suppose that there is $\\alpha_j \\in [0,1]$ that satisfies the statement of the lemma for $X_j$. Let us define $\\xi_j = \\E[\\phi_{\\alpha_j}(X_j)]$; then the inductive statement reads\n\\begin{equation}\n\\label{eq:hypo}\n\\alpha_j \\xi_j^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^j c^{1+\\delta}.\n\\end{equation}\nOur goal is to prove that $\\alpha_{j+1} \\xi_{j+1}^{1+\\delta} \\geq \\frac{1+\\delta^2}{2} \\alpha_j \\xi_j^{1+\\delta}$, which implies the inductive statement for $j+1$.\n\nBy assumption, for the non-decreasing concave function $\\phi_{\\alpha_j}$, we get\n$$ \\E[1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2] \\geq (1-\\epsilon) \\E[\\phi_{\\alpha_j}(X_j - 10^{-\\ell})] - 10^{-\\ell}.$$\nFirst, we simplify the error terms on the right-hand side. Let us keep in mind that $\\epsilon,\\delta>0$ are (small) absolute constants which will be suitably chosen at the end of the proof. \nRecall that $\\phi_{\\alpha_j}(t) = \\min \\{ \\frac{t}{\\alpha_j},1\\}$. Therefore,\n $(1-\\epsilon) \\E[\\phi_{\\alpha_j}(X_j - 10^{-\\ell})] \\geq (1-\\epsilon) \\E[\\phi_{\\alpha_j}(X_j)] - \\frac{1}{\\alpha_j 10^\\ell}\n  = (1-\\epsilon) \\xi_j - \\frac{1}{\\alpha_j 10^\\ell}.$\nRecall the inductive hypothesis (\\ref{eq:hypo}).\nSince $\\alpha_j, \\xi_j \\in [0,1]$, and $c \\geq 2^{-\\ell}$, this means in particular that $\\alpha_j \\xi_j \\geq  2^{-j} c^{1+\\delta} \\geq 2^{-3 \\ell}$. Also, $\\xi_j \\geq 2^{-j} c \\geq 2^{-2 \\ell}$. Hence, we can estimate \n\\begin{equation}\n\\label{eq:quad}\n\\E[1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2] \\geq (1-\\epsilon) \\xi_j - \\frac{1}{\\alpha_j 10^\\ell} - \\frac{1}{10^\\ell} \n\\geq \\left(1-\\epsilon - \\frac{2^{3\\ell}}{10^\\ell} - \\frac{2^{2\\ell}}{10^\\ell} \\right) \\xi_j\n \\geq (1-2\\epsilon) \\xi_j\n\\end{equation}\nfor $\\ell$ sufficiently large.\n \nNow we come to the meat of the inductive argument. Instead of the expression $\\E[1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2]$, we would like to estimate $\\xi_{j+1} = \\E[\\phi_{\\alpha_{j+1}}(X_{j+1})]$ for a suitable value of $\\alpha_{j+1}$. The reason why the values of $\\alpha_j$ are not specified by the lemma is that their choice depends on the particular distributions of $X_j$ over which we have no control. For example, $\\alpha_{j+1} = \\frac12 \\alpha_j$ is a natural choice which works for some distributions of $X_{j+1}$ but not always. In the following, we split the analysis into 2 cases.\n \n\\paragraph{Case 1:} $\\Pr[\\frac{X_{j+1}}{\\alpha_j} > \\sqrt{\\delta}] > 2 \\delta \\xi_j$. \\\\\nIn this case, $X_{j+1}$ is with non-negligible probability quite large, in the region where $1 - (1-X_{j+1}\/\\alpha_j)^2$ is significantly smaller than $2 X_{j+1} \/ \\alpha_j$ . In this case, we can gain by making $\\alpha_{j+1}$ slightly larger than $\\frac12 \\alpha_j$, specifically $\\alpha_{j+1} = \\frac12 (1+\\delta) \\alpha_j$.\nWe obtain:\n\\begin{eqnarray*}\n\\xi_{j+1} = \\E[\\phi_{\\alpha_{j+1}}(X_{j+1})] = \\E\\left[ \\min \\left\\{ \\frac{X_{j+1}}{\\alpha_{j+1}}, 1 \\right\\}\\right]\n = \\E\\left[ \\min \\left\\{ \\frac{2 X_{j+1}}{(1+\\delta) \\alpha_j}, 1 \\right\\} \\right] \n  = \\frac{1}{1+\\delta} \\E\\left[ \\min \\left\\{ \\frac{2 X_{j+1}}{\\alpha_j}, 1+\\delta \\right\\}\\right].\n\\end{eqnarray*}\nObserve the following: $\\min \\{ \\frac{2 X_{j+1}}{\\alpha_j}, 1+\\delta \\} \\geq \\min \\{ \\frac{2 X_{j+1}}{\\alpha_j}, 1 \\}\n \\geq 1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2$ for all $X_{j+1} \\geq 0$. Moreover, \nif $X_{j+1} > \\sqrt{\\delta} \\alpha_j$, we gain an additional $\\delta$, because then\n$$ \\min \\left\\{ \\frac{2 X_{j+1}}{\\alpha_j}, 1+\\delta \\right\\}\n \\geq 1 - \\left(1-\\min \\left\\{ \\frac{X_{j+1}}{\\alpha_j}, 1 \\right\\} \\right)^2 + \\delta\n = 1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2 + \\delta $$\n(with equality for $X_{j+1} = \\sqrt{\\delta} \\alpha_j$ and $X_{j+1} \\geq \\alpha_j$; the best way to verify this is to ponder the graph\nin Figure~\\ref{fig:phi-graph}).\n\\begin{figure}[!ht]\n\\centering\n\\input{phi-graph.tex}\n\\caption{Comparison of the 3 relevant functions for Case 1: Note that for $x \\geq \\sqrt{\\delta} \\alpha_j$, the top two functions differ by at least $\\delta$; i.e, $\\min \\{\\frac{2x}{\\alpha_j}, 1+\\delta\\} \\geq 1 - (1 - \\min \\{\\frac{x}{\\alpha_j}, 1\\})^2 + \\delta$.}\n\\label{fig:phi-graph}\n\\end{figure}\nTherefore,\n\\begin{eqnarray*}\n\\xi_{j+1} = \\frac{1}{1+\\delta} \\E\\left[ \\min \\left\\{ \\frac{2 X_{j+1}}{\\alpha_j}, 1+\\delta \\right\\} \\right]\n \\geq \\frac{1}{1+\\delta}\\E[1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2] + \\frac{\\delta}{1+\\delta} \\Pr[X_{j+1} > \\sqrt{\\delta} \\alpha_j].\n\\end{eqnarray*}\nUsing (\\ref{eq:quad}) and $\\Pr[X_{j+1} > \\sqrt{\\delta} \\alpha_j] > 2 \\delta \\xi_j$, we get\n$$ \\xi_{j+1} \\geq \n \\frac{1-2\\epsilon}{1+\\delta} \\xi_j + \\frac{2 \\delta^2}{1+\\delta} \\xi_j =\n\\frac{1+2\\delta^2-2\\epsilon}{1+\\delta} \\xi_j. $$\nSince $\\alpha_{j+1} = \\frac{1+\\delta}{2} \\alpha_j$, we get\n$$ \\alpha_{j+1} \\xi_{j+1}^{1+\\delta}\n\\geq \\frac{1+\\delta}{2} \\alpha_j \\left( \\frac{1+2\\delta^2-2\\epsilon}{1+\\delta}\\right)^{1+\\delta} \\xi_j^{1+\\delta} \n= \\frac{(1+2\\delta^2-2\\epsilon)^{1+\\delta}}{2(1+\\delta)^\\delta} \\alpha_j \\xi_j^{1+\\delta}.$$\nWe choose $\\epsilon = \\delta^4$, so that $(1+2\\delta^2-2\\epsilon)^{1+\\delta} = (1+2\\delta^2-2\\delta^4)^{1+\\delta}\\geq 1 + 2\\delta^2 + \\delta^4$ (it can be verified that this holds for $\\delta \\in [0,\\frac12]$). We also use $(1+\\delta)^\\delta \\leq 1+\\delta^2$ (which holds for $\\delta \\in [0,1]$). This implies the inductive statement:\n$$ \\alpha_{j+1} \\xi_{j+1}^{1+\\delta} \\geq \\frac{1+2\\delta^2+\\delta^4}{2(1+\\delta^2)} \\alpha_j \\xi_j^{1+\\delta}\n= \\frac{1+\\delta^2}{2} \\alpha_j \\xi_j^{1+\\delta}.$$ \n\n\\paragraph{Case 2:} $\\Pr[\\frac{X_{j+1}}{\\alpha_j} > \\sqrt{\\delta}] \\leq 2 \\delta \\xi_j$. \\\\\nIn this case, $X_{j+1}$ is almost always very small compared to $\\alpha_j$. Then we can gain by making $\\alpha_{j+1}$ much smaller than $\\alpha_j$; we let $\\alpha_{j+1} = \\sqrt{\\delta} \\alpha_j$. We have\n\\begin{eqnarray*}\n\\xi_{j+1} = \\E[\\phi_{\\alpha_{j+1}}(X_{j+1})] & = & \\E\\left[ \\min \\left\\{ \\frac{X_{j+1}}{\\alpha_j \\sqrt{\\delta}}, 1 \\right\\} \\right]\n = \\frac{1}{\\sqrt{\\delta}} \\, \\E\\left[ \\min \\left\\{ \\frac{X_{j+1}}{\\alpha_j}, \\sqrt{\\delta} \\right\\} \\right] \\\\\n & \\geq & \\frac{1}{\\sqrt{\\delta}} \\left( \\E\\left[ \\min \\left\\{ \\frac{X_{j+1}}{\\alpha_j}, 1 \\right\\}\\right] - (1-\\sqrt{\\delta}) \\Pr\\left[\\frac{X_{j+1}}{\\alpha_j} > \\sqrt{\\delta}\\right] \\right) \\\\\n& \\geq & \\frac{1}{\\sqrt{\\delta}} \\left( \\E[\\phi_{\\alpha_j}(X_{j+1})] - (1-\\sqrt{\\delta}) \\cdot 2 \\delta \\xi_j \\right)\n \\end{eqnarray*}\nAn elementary bound together with (\\ref{eq:quad}) gives\n$$ \\E[\\phi_{\\alpha_j}(X_{j+1})] \\geq \\frac12 \\E[1 - (1-\\phi_{\\alpha_j}(X_{j+1}))^2]\n \\geq \\frac12 (1-2\\epsilon) \\xi_j.$$\nTherefore, using our choice of $\\epsilon = \\delta^4$,\n$$ \\xi_{j+1} = \\E[\\phi_{\\alpha_{j+1}}(X_{j+1})]\n \\geq \\frac{1}{\\sqrt{\\delta}} \\left(\\frac12 (1-2\\epsilon) \\xi_j - 2 (1-\\sqrt{\\delta}) \\delta \\xi_j\\right)\n  = \\frac{1 - 2\\delta^4 - 4 \\delta + 4\\delta^{3\/2}}{2 \\sqrt{\\delta}} \\xi_j \\geq \\frac{1-4\\delta+2 \\delta^{3\/2}}{2\\sqrt{\\delta}} \\xi_j. $$\nFrom here, using $\\alpha_{j+1} = \\sqrt{\\delta} \\alpha_j$ and $(1-4\\delta+2\\delta^{3\/2})^{1+\\delta}\n \\geq 1 - 4\\delta$ (which holds for any $\\delta \\in [0,\\frac14]$),\n$$ \\alpha_{j+1} \\xi_{j+1}^{1+\\delta} \\geq \\sqrt{\\delta} \\alpha_j \\left(\\frac{1-4\\delta+2\\delta^{3\/2}}{2\\sqrt{\\delta}}\\right)^{1+\\delta} \\xi_j^{1+\\delta} \\geq \\frac{1-4\\delta}{2^{1+\\delta} \\delta^{\\delta\/2}} \\alpha_j \\xi_j^{1+\\delta}.$$\nWe choose $\\delta = e^{-10}$ so that $\\delta^{\\delta\/2} = e^{-5\\delta}$. Then,\n$$ \\alpha_{j+1} \\xi_{j+1}^{1+\\delta} \\geq \\frac{1-4\\delta}{2^{1+\\delta}} e^{5\\delta} \\alpha_j \\xi_j^{1+\\delta}\n \\geq \\frac{1+\\delta^2}{2} \\alpha_j \\xi_j^{1+\\delta} $$\nwhich finishes the inductive step.\n\\end{proof}\n\nPutting together Lemma~\\ref{lem:level-bound} and the cardinality bound which applies to every feasible solution, we complete our hardness result for combinatorial public projects.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:CPP-hardness}]\nLet $\\epsilon>0$ and $\\delta>0$ be the constants provided by Lemma~\\ref{lem:level-bound}. Let $n=2^\\ell$ and $m = 400^\\ell$. Suppose there is a mechanism for the problem $\\max \\{f(S): |S| \\leq m\/n\\}$ that maximizes $(1-\\epsilon)$-approximately over a distributional range $\\cR$ and provides a $c$-approximation, where $c \\geq 1\/n$. By Lemma~\\ref{lem:level-gap} and Lemma~\\ref{lem:base-case}, there are collections of random variables $\\cX_0, \\cX_1, \\ldots, \\cX_\\ell$ constructible at the respective levels by $\\cR$, satisfying the conditions of Lemma~\\ref{lem:level-bound}. Hence,\nby Lemma~\\ref{lem:level-bound} for $j=\\ell$, there is $X_\\ell$ constructible by $\\cR$ at level $\\ell$ such that\n$$ \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\geq c^{1+\\delta} \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell  = \\frac{c^{1+\\delta}}{n} (1+\\delta^2)^\\ell.$$\nRecall that $\\phi_{\\alpha_\\ell}(t) = \\min \\{ \\frac{t}{\\alpha_\\ell}, 1\\}$. Therefore, we have \n$$ \\frac{c^{1+\\delta}}{n} (1+\\delta^2)^\\ell \\leq \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\leq \n\\alpha_\\ell \\E[\\phi_{\\alpha_\\ell}(X_\\ell)] \\leq \\E[X_\\ell]. $$\nWe have $X_\\ell = \\frac{|R|}{|M|}$ where $R$ is a random set sampled according to some distribution in the range $\\cR$. \nAll distributions in the range must be feasible in expectation, otherwise the mechanism cannot possibly maximize over them and return a feasible solution. Therefore, \n$ \\E[X_\\ell] \\leq \\frac{1}{n} $\nwhich implies that\n$$ c \\leq (1+\\delta^2)^{-\\frac{\\ell}{1+\\delta}} < 2^{-\\delta^2 \\ell} = n^{-\\delta^2}. $$\nTherefore, there is no $(1-\\epsilon)$-approximately MIDR mechanism providing an $n^{-\\delta^2}$-approximation in the objective function.\nAlso, we have $m = 400^\\ell = \\poly(n)$, so the approximation cannot be better than $m^{-\\gamma}$ for some constant $\\gamma>0$.\nThe only bound we have used on the mechanism was that the number of value queries is polynomial in $n$, or equivalently\npolynomial in $m$.\n\\end{proof}\n\n\\section{Hardness for combinatorial public projects}\n\\label{sec:CPP-hardness}\n\n\n\\iffalse\nWe start by defining the combinatorial public project problem. We restrict our attention to the case with submodular valuation functions. Two variants have been considered.\n\n\\paragraph{Exact Submodular CPP.}\n{\\em Given $n$ players with monotone submodular functions $f_i:2^M \\rightarrow \\RR_+$, find a solution $S \\subset M$ of size $|S| = k$ maximizing $\\sum_{i=1}^{n} f_i(S)$.}\n\n\\paragraph{Flexible Submodular CPP.}\n{\\em Given $n$ players with monotone submodular functions $f_i:2^M \\rightarrow \\RR_+$, find a solution $S \\subset M$ of size $|S| \\leq k$ maximizing $\\sum_{i=1}^{n} f_i(S)$.}\n\n\\\n\nThe two variants are equivalent from the optimization point of view -\nin the absence of truthfulness issues, this is monotone submodular maximization subject to a cardinality constraint, which admits a $(1-1\/e)$-approximation. However, the requirement of truthfulness can make the two problems potentially different.\n\\fi\n\nWe start with the combinatorial public project problem.\nThe (exact) combinatorial public project problem was introduced in \\cite{PSS08} as a model problem for the study of truthful approximation mechanisms. This problem is better understood than combinatorial auctions, in the sense that a useful characterization of all deterministic truthful mechanisms is known: every truthful mechanism for 2 players is an \\emph{affine maximizer} --- a weighted generalization of maximal-in-range mechanisms \\cite{PSS08}. \nUsing this characterization, it was proved in \\cite{PSS08} that the exact submodular CPP problem does not admit any (deterministic) truthful $m^{\\epsilon-1\/2}$-approximation using a subexponential amount of communication, and moreover there is no  $m^{\\epsilon-1\/2}$-approximation even for a certain class of succintly represented submodular valuations unless $NP \\subseteq BPP$. In contrast, the simple greedy algorithm is a non-truthful $(1-1\/e)$-approximation algorithm for this problem \\cite{NWF78}. This was the first example of such a dramatic gap in approximability between truthful mechanisms and non-truthful algorithms.\n\nIn follow-up work, a simpler\ncharacterization-type statement for CPP was shown in \\cite{BSS10}: Every truthful mechanism for a single player with a coverage valuation can, via a non-uniform polynomial time reduction, be converted to a truthful maximal-in-range mechanism without degrading its approximation ratio. Since every truthful mechanism for $n$ players must embed a truthful mechanism for a single player, this allowed the authors to restrict attention to maximal-in-range mechanisms for a single player in proving an $m^{\\epsilon-1\/2}$-approximation threshold for CPP with coverage valuations, assuming that $NP \\not\\sse P\/poly$. The following easy converse of their characterization is notable: A maximal-in-range mechanism for CPP with a single player can directly be used as a maximal-in-range mechanism for any number of players\n\n\n\n\\iffalse\nHere, we consider randomized truthful-in-expectation mechanisms. \n\n\\begin{definition}\n\\label{def:CPP-TIE-mech}\nA mechanism for the exact\/flexible CPP problem (for 1 player) is truthful-in-expectation, if given a declared valuation $f:2^M \\rightarrow \\RR_+$, it outputs a random set $R(f)$ and charges a random price $p(f)$, such that for any other valuation $f'$,\n$$ \\E[f(R(f)) - p(f)] \\geq \\E[f(R(f')) - p(f')]. $$\nA mechanism is $(1-\\epsilon)$-approximately truthful, if given $f$ it outputs a random set $R(f)$ and charges a random price $p(f)$ such that for any other valuation $f'$,\n$$ \\E[f(R(f)) - p(f)] \\geq \\E[(1-\\epsilon) f(R(f')) - p(f')].$$\n\\end{definition}\n\nNote that in the case of approximate truthfulness, the approximation factor is in front of the valuation function and not the price. We discuss this definition in Section~\\ref{sec:auctions-hardness}.\nA natural extension of the notion of a maximal-in-range mechanism is that of a {\\em maximal-in-distributional-range} mechanism.\n\n\\begin{definition}\n\\label{def:MIDR}\nA mechanism for the problem $\\max \\{ f(S): |S| \\leq k \\}$ is {\\em maximal-in-distributional-range}, if there is a range $\\cR$ of probability distributions over $2^M$ such that the mechanism for any function $f$ returns a distribution $D \\in \\cR$  maximizing $\\E_{R \\sim D}[f(R)]$ over $D \\in \\cR$. \nA mechanism is $(1-\\epsilon)$-approximately maximal-in-distributional-range, if it maximizes $\\E_{R \\sim D}[f(R)]$ within a factor of $(1-\\epsilon)$ over $D \\in \\cR$.\n\\end{definition}\n\\fi\n\nRecently, it was proved by Dobzinski \\cite{Dobzin11} that the exact variant of the submodular CPP problem\n(under the constraint $|S|=k$) does not admit a truthful-in-expectation $m^{\\epsilon-1\/2}$-approximation in the value oracle model. However, as noted in \\cite{Dobzin11},  the flexible variant of CPP (under the constraint $|S| \\leq k$) \nis arguably  more natural in the strategic setting. For the flexible variant of CPP, \\cite{Dobzin11} proves that there is no universally truthful $m^{\\epsilon-1\/2}$-approximation, but leaves open the possibility of a better truthful-in-expectation mechanism.  Problems that have a packing structure like flexible CPP have historically proven to be easier to approximate using truthful-in-expectation mechanisms \\cite{LS05,DD09,DR10,DRY11}. Flexible CPP has exhibited a similar pattern; Dughmi \\cite{Dughmi11} recently designed a truthful-in-expectation $(1-1\/e)$-approximation mechanism for CPP when players have explicit coverage valuations (which is optimal regardless of strategic issues \\cite{Feige98}), and more generally when players have matroid rank sum valuations that support a certain randomized variant of value queries. \n\n\\paragraph{Transformation to MIDR mechanisms.}\nWhile deterministic truthful mechanisms for the CPP problem are no more powerful in terms of approximation than maximal-in-range mechanisms \\cite{PSS08,BSS10}, the situation is slightly more complicated for randomized  mechanisms. It is not clear whether truthful-in-expectation mechanisms are equivalent to maximal-in-distributional-range mechanisms. Nonetheless, we prove the following.\n\n\\begin{theorem}\n\\label{thm:TIE=MIDR}\nFor every $\\epsilon \\geq 0$ and $c(m)>0$ the following holds. If there is a $(1-\\epsilon)$-approximately truthful-in-expectation mechanism $\\cM$ for the (exact or flexible) CPP problem that achieves a $c(m)$-approximation for submodular valuations on $m$ elements, then for any $\\delta>0$ there is a non-uniform  $(1-3 \\epsilon - \\delta)$-approximately maximal-in-distributional-range mechanism $\\cM'$ that achieves a $c(m)$-approximation for submodular valuations on $m$ elements and uses at most $m$ more value queries than $\\cM$.\n\\end{theorem}\n\n\nBy a non-uniform mechanism, we mean a separate fixed mechanism for each input size $m$; i.e., the size of the program can depend arbitrarily on $m$. The only bound on the non-uniform mechanism is the number of value queries used.  The main idea is that the although the range of prices offered by a truthful-in-expectation mechanism can be unbounded, the mechanism can be made MIDR ``in the limit\", when the input valuation is scaled by a sufficiently large constant. This constant can be fixed for each input size $m$ and acts as an ``advice string'' to the mechanism. We present the proof in Appendix~\\ref{sec:TIE-MIDR}.\n\n\\paragraph{Hardness for MIDR mechanisms.}\nOur hardness result for flexible submodular CPP rules out mechanisms purely based on the number of value queries used,\nand hence it rules out even the non-uniform mechanisms mentioned in Theorem~\\ref{thm:TIE=MIDR}.\n\n\\begin{theorem}\n\\label{thm:CPP-hardness}\nThere are absolute constants $\\epsilon, \\gamma > 0$ such that there is no $(1-\\epsilon)$-approximately maximal-in-distributional-range mechanism for the flexible submodular CPP problem with 1 player in the value oracle model, $\\max \\{f(S): |S| \\leq k\\}$, achieving a better than $1\/m^\\gamma$-approximation in expectation in the objective function, where $m$ is the size of the ground set. This holds even for non-uniform mechanisms of arbitrary computational complexity, as long as the number of value queries is bounded by $\\poly(m)$.\n\\end{theorem}\n\nIn the following, we present a sketch of the proof of this theorem. The full proof appears in Appendix~\\ref{sec:CPP-proof}.\n\n\\paragraph{Proof strategy.}\nWe assume that a mechanism optimizes over a range of distributions $\\cR$. (We assume for simplicity that the mechanism is MIDR rather than approximately MIDR.) We emphasize that the range $\\cR$ is fixed beforehand, and the mechanism must optimize over $\\cR$ for any particular submodular function $f$. This gives us a lot of flexibility in arguing about the properties of $\\cR$. \n\n\nSuppose that the size of the ground set is $m = 2^{O(\\ell)}$ and the cardinality bound is $k = m \/ 2^\\ell$. We consider $\\ell+1$ different ``levels\" of valuation functions. (See Figure~\\ref{fig:bisect-seq}.) At level $0$, we have a set $A^{(0)}$ of $m\/2^\\ell$ items, where the valuation function is nonzero and additive. Assuming that the mechanism achieves a $c$-approximation, there must be a distribution $D_0 \\in \\cR$ which allocates at least a $c$-fraction of $A^{(0)}$ in expectation to player $i$. This must be true for every set $A^{(0)}$ of size $m\/2^\\ell$. It will be useful to think of this set as random (and hidden from the mechanism.)\n\n\\begin{figure}[!ht]\n\\centering\n\\input{bisect-seq.tex}\n\\caption{A bisection sequence $(A^{(j)}, B^{(j)})$, with the distributions $D_j$ returned by the mechanism at level $j$. The density of $D_j$ increases in a certain technical sense exponentially in $j$, although much slower than $2^j$.}\n\\label{fig:bisect-seq}\n\\end{figure}\n\n\nAt level $j$, $1 \\leq j \\leq \\ell$, we have a (random) set $A^{(j)}$ of $m \/ 2^{\\ell-j}$ items, which is partitioned randomly into two sets $A^{(j-1)} \\cup B^{(j-1)}$ of equal size; these are level-$(j-1)$ sets.\nThe valuation function at level $j$ will be as in Section~\\ref{sec:exp-gap} but restricted to the set $A^{(j)} = A^{(j-1)} \\cup B^{(j-1)}$ (the two parts play the role of $M_1,M_2$ from Section~\\ref{sec:exp-gap}). The mechanism can detect the set $A^{(j)}$; however, the partition of $A^{(j)}$ into $A^{(j-1)} \\cup B^{(j-1)}$ remains hidden. By the symmetry gap argument, the mechanism cannot learn what the partition is, and hence any distribution $D_j$ returned by the algorithm will be with high probability balanced with respect to $(A^{(j-1)}, B^{(j-1)})$. The MIDR property implies that this distribution must be ``dense\" enough in order to beat the distribution $D_{j-1}$ guaranteed by the previous level, which is sensitive to the partition $(A^{(j-1)}, B^{(j-1)})$. (By density, we mean a certain notion of average size for sets sampled from $D_j$.) Since distributions concentrated inside $A^{(j-1)}$ or $B^{(j-1)}$ are more profitable than distributions balanced between $(A^{(j-1)}, B^{(j-1)})$, we will ideally obtain a constant-factor boost in density at each level. As $\\ell$ grows, this will eventually contradict the fact that the mechanism cannot choose more than $k$ items. \n\nFinding the right definition of density that yields a constant-factor boost at each level is the main technical difficulty. The most natural definition of density seems to be the expected size of the set returned by the mechanism. However, this notion does not yield the desired boost. (This is related to the fact that we cannot get any contradiction for coverage functions.) The notion of density that turns out to be useful is more complicated; it is derived from functions that exhibit non-concave behavior of the extension $F^{exp}$. \nThis strategy will be made more explicit in the following.\n\n\n\\paragraph{The symmetry gap.}\nAt level $j+1$, we consider valuation functions of the form\n$$ f_{A^{(j)},B^{(j)}}(S) = 1 - \\left( 1 - \\phi\\left(\\frac{|S \\cap A^{(j)}|}{|A^{(j)}|}\\right) \\right)\n \\left( 1 - \\phi\\left(\\frac{|S \\cap B^{(j)}|}{|B^{(j)}|}\\right) \\right) $$\nwhere $\\phi:[0,1] \\rightarrow [0,1]$ is a suitable non-decreasing concave function. Note that under this valuation function, the value of a (random) set $R$ depends only on how many elements it takes from $A^{(j)}$ and $B^{(j)}$.\nIn particular, if we denote $X_j = \\frac{|R \\cap A^{(j)}|}{|A^{(j)}|}$,\n$Y_j = \\frac{|R \\cap B^{(j)}|}{|B^{(j)}|}$, then we have\n$$ \\E[f_{A^{(j)},B^{(j)}}(R)] = \\E[1 - (1-\\phi(X_j))(1-\\phi(Y_j))].$$\nSince the expected value depends only on $X_j,Y_j$, we say that the random variables $X_j, Y_j$ represent the distribution of $R$.\n\nBy the symmetry gap argument (if the valuation function is suitably perturbed and the partition $(A^{(j)},B^{(j)})$ is random), then the mechanism with high probability returns a solution $R^{(j+1)}$ independent of the partition and hence symmetric with respect to it. Denoting\n$X_{j+1} = \\frac{|R^{(j+1)} \\cap A^{(j+1)}|}{|A^{(j+1)}|}$, we obtain that the mechanism returns expected value\n$$ \\E[f_{A^{(j)},B^{(j)}}(R^{(j+1)})] = \\E[1 - (1-\\phi(X_{j+1}))^2] $$\nwhich is typically less than $\\E[1 - (1-\\phi(X_j))(1-\\phi(Y_j))]$ if $X_{j+1} = \\frac12 (X_j + Y_j)$\nand $X_j \\neq Y_j$.\n(There are certain error terms arising from the symmetry gap argument but let us ignore them for now.)\n\nThe main point here is that if the mechanism is MIDR, then the expected value of the returned random set $\\E[f_{A^{(j)},B^{(j)}}(R^{(j+1)})]$ must be at least that of any other random set whose distribution is in the range - in particular, the random set $R^{(j)}$ whose presence in the range we prove at the previous level. If this random set $R^{(j)}$ is represented by the random variables $X_j, Y_j$, then the mechanism must return a distribution represented by $X_{j+1}$ such that\n$$ \\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[1 - (1-\\phi(X_j))(1-\\phi(Y_j))].$$\nWe in fact ignore the contribution of $Y_j$ and use the weaker inequality\n\\begin{equation}\n\\label{eq:sym-simple}\n\\E[1 - (1-\\phi(X_{j+1}))^2] \\geq \\E[\\phi(X_j)].\n\\end{equation}\nHence, the existence of certain distributions in the range forces the existence of other distributions, satisfying the bound (\\ref{eq:sym-simple}). \n\n\\paragraph{Gap amplification.}\nNow we would like to say that if two distributions represented by $X_j$ and $X_{j+1}$ satisfy (\\ref{eq:sym-simple}),\nthen the distribution at level $j+1$ is ``more dense\" than the one at level $j$.\nConsidering the scaling at different levels, we want to prove that $X_{j+1}$ is ``significantly larger\" than $\\frac12 X_j$.\nThis is intuitive, since $X_{j+1} = \\frac12 X_j$ is not enough to satisfy (\\ref{eq:sym-simple}), for example when $\\phi$ is linear.\nUnfortunately,  (\\ref{eq:sym-simple}) does not imply any useful relationship between the expectations $\\E[X_j]$, $\\E[X_{j+1}]$, beyond $\\E[X_{j+1}] \\geq \\frac12 \\E[X_j]$.\nFor example, we could have $X_j = 1$ with probability $\\xi - \\xi^2$ and $0$ otherwise. Then $X_{j+1} = \\frac12 \\xi$ satisfies (\\ref{eq:sym-simple}) for any concave function $\\phi:[0,1] \\rightarrow [0,1]$. This does not provide a constant-factor improvement over $\\frac12 \\E[X_j]$.\n\nWe still want to prove that $X_{j+1}$ is in some sense ``significantly larger\" than $\\frac12 X_j$.\nOur main technical inequality formalizing this intuition is the following: \nDefine $\\phi_\\alpha(t) = \\min \\left\\{ \\frac{t}{\\alpha}, 1 \\right\\}$.\nThen for any distribution in the range represented by $X_j$ at level $j$\nand any $\\alpha_j \\in [0,1]$, there is a distribution in the range represented by $X_{j+1}$ at level $j+1$, and $\\alpha_{j+1} \\in [0,1]$\nsuch that\n\\begin{equation}\n\\label{eq:ampl}\n\\alpha_{j+1} (\\E[\\phi_{\\alpha_{j+1}}(X_{j+1})])^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right) \\alpha_{j} (\\E[\\phi_{\\alpha_{j}}(X_{j})])^{1+\\delta} .\n\\end{equation}\nwhere $\\delta>0$ is some (small) absolute constant. The use of $1+\\delta$ in the exponent is crucial here.\nWe remark that formulating and proving this inequality was the most challenging part of the proof.\n(The precise statement with a proof appears as Lemma~\\ref{lem:level-bound} in the appendix.)\n\n\\paragraph{The contradiction.}\nUsing this bound, we arrive at a contradiction as follows. As we already mentioned, assuming that an MIDR mechanism provides a $c$-approximation for the CPP problem, then for any feasible set $A^{(0)}$ there must be a distribution $D_0$ in its range such that\n$$ \\E[X_0] = \\E_{R \\sim D_0}\\left[\\frac{|R \\cap A^{(0)}|}{|A^{(0)}|}\\right] \\geq c.$$\nNow we apply the symmetry gap argument and the gap amplification technique to random pairs of sets $(A^{(j)}, B^{(j)})$\nat each level $j$. Starting from $\\E[X_0] \\geq c$ and $\\alpha_0=1$, by repeated use of (\\ref{eq:ampl})\nwe obtain that there is $\\alpha_\\ell \\in [0,1]$ and a distribution at level $\\ell$ represented by $X_\\ell$ such that\n$$ \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell c^{1+\\delta}. $$\nNote that $ \\alpha_\\ell (\\E[\\phi_{\\alpha_\\ell}(X_\\ell)])^{1+\\delta} \\leq \\alpha_\\ell \\E[\\phi_{\\alpha_\\ell}(X_\\ell)] \n = \\E[\\min \\{X_\\ell, \\alpha_\\ell\\}] \\leq \\E[X_\\ell]$. So in fact\n$$ \\E[X_\\ell] \\geq \\left( \\frac{1+\\delta^2}{2} \\right)^\\ell c^{1+\\delta} > \\frac{2^{\\delta^2 \\ell}}{2^\\ell} c^{1+\\delta}. $$\nThe meaning of $X_\\ell$\nis simply the fraction of the ground set that the mechanism returns at level $\\ell$. Since $m = 2^{O(\\ell)}$, we have $2^{\\delta^2 \\ell} \\geq m^{(1+\\delta)\\gamma}$ for some constant $\\gamma>0$.\nIf the approximation factor is $c \\geq m^{-\\gamma}$, then we get $\\E[X_\\ell] > 2^{-\\ell}$,\nwhich would violate the cardinality constraint of the CPP problem.\n\nAs we mentioned, the full proof appears in Appendix~\\ref{sec:CPP-proof}.\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe design of incentive-compatible mechanisms for welfare maximization in combinatorial auctions is a central problem of algorithmic mechanism design. In a combinatorial auction, there are $n$ players and a set $M$ of $m$ items. Player $i$ has a (private) valuation function $v_i:2^M \\rightarrow \\RR_+$ which is assumed to be monotone ($v_i(S) \\leq v_i(T)$ whenever $S \\subset T$) and normalized ($v_i(\\emptyset) = 0$). The goal is to design a computationally efficient mechanism that yields an allocation of items $(S_1,\\ldots,S_n)$ to the players along with payments $(p_1,\\ldots,p_n)$ so that (a) the {\\em social welfare} $\\sum_{i=1}^{n} v_i(S_i)$ is approximately maximized, and (b) the mechanism is {\\em incentive-compatible}, or {\\em truthful}, meaning that each player maximizes his \\emph{utility} $v_i(S_i) - p_i$ by reporting his true valuation $v_i$.\n\nThis problem has been studied extensively in both strategic and non-strategic settings. Various strategic solution concepts have been considered, including deterministic truthfulness, universal truthfulness, and truthfulness in expectation. Moreover, both strategic and non-strategic formulations of the problem have been studied for various restricted classes of valuations, as well as under various assumptions on how valuations are accessed or represented. Absent assumptions on the class of valuations, the welfare maximization problem is very hard to approximate even by non-truthful algorithms (NP-hardness of  $m^{\\epsilon-1\/2}$-approximation follows from the set packing problem). Better approximation ratios are possible for valuation classes that restrict complementarity between items. The most prominent such class of valuations is \\emph{submodular functions}: functions $v_i$ where the marginal value $v_i(S \\cup \\{j\\}) - v_i(S)$ for a each fixed item $j$ is non-increasing in $S$. It is known that the welfare maximization problem with submodular valuation functions admits a (non-truthful) $(1-1\/e)$-approximation algorithm \\cite{V08}, and this is optimal  assuming $P \\neq NP$ \\cite{KLMM05}. The hardness result of \\cite{KLMM05} holds even in the special case of coverage valuations; the algorithmic result of \\cite{V08} holds in the {\\em value oracle model}, where each $v_i$ can be queried only through an oracle returning $v_i(S)$ for a given query $S$. \nIn the value oracle model, it is known that any $(1-1\/e+\\epsilon)$-approximation for combinatorial auctions with submodular valuations would require an exponential number of queries \\cite{MSV08}. This is also the model we consider in this paper.\n\nThe classical VCG mechanism is incentive compatible and maximizes welfare in combinatorial auctions. Unfortunately, however, VCG can not be implemented in polynomial time even for very special classes of valuation functions, including submodular functions. Combining computational efficiency and truthfulness for combinatorial auctions appears difficult. A series of works have provided evidence that computational efficiency and truthfulness are in conflict: (deterministic) VCG-type mechanisms have been ruled out for submodular combinatorial auctions in the communication complexity model \\cite{DN07}, and even for explicitly given budget-additive valuations \\cite{BDFKMPSSU10}. Recently,  Dobzinski \\cite{Dobzin11} proved that there is no deterministic truthful or even randomized universally truthful mechanism for submodular combinatorial auctions in the value oracle model, achieving an approximation ratio better than $m^{\\epsilon-1\/2}$.\n\n\nTherefore, it came as a surprise when a $(1-1\/e)$-approximate randomized mechanism was discovered by Dughmi, Roughgarden and Yan \\cite{DRY11} for a large subclass of submodular valuations. Their mechanism is {\\em truthful in expectation} --- a weaker notion than truthfulness in the universal sense --- and applies to explicitly represented coverage functions. More generally, their mechanism applies to ``black-box'' valuations that are expressible as weighted sums of matroid rank functions, provided they support ``lottery-value queries\" (what is the expected value $\\E[v_i(\\hat{\\bx})]$ for a given product distribution $\\hat{\\bx}$). \nThe mechanism can be also implemented in the value oracle model, \nat the cost of relaxing the solution concept to approximate truthfulness in expectation \\cite{DRVY11}.\n\nThis  development  raises a natural question: Could truthfulness-in-expectation be the cure for combinatorial auctions, perhaps providing an optimal $(1-1\/e)$-approximation for all submodular valuations? Given that a $(1-1\/e)$-approximation for welfare maximization in combinatorial auctions (without truthfulness) was also discovered first for coverage functions \\cite{DS06}, then for weighted sums of matroid rank functions \\cite{CCPV07} and later extended to monotone submodular functions \\cite{V08}, it seems reasonable to conjecture that the same might happen for truthful-in-expectation mechanisms.\n\n\n\\paragraph{Our results.}\nWe prove that this is not the case,  and there is a significant separation between the class of coverage functions and general monotone submodular functions. More precisely, there is no truthful-in-expectation mechanism (even $(1-\\epsilon)$-approximately truthful-in-expectation) for submodular combinatorial auctions in the value oracle model, guaranteeing an approximation better than $1\/m^\\gamma$ for some fixed $\\epsilon,\\gamma>0$ (Theorem~\\ref{thm:CA-hardness}). In particular, the results of \\cite{DRY11} cannot be extended to all monotone submodular functions.\n\n\nWe also prove a similar result for the \\emph{flexible submodular combinatorial public projects} problem (see Section \\ref{sec:CPP-hardness} for a history of this problem): there is no $(1-\\epsilon)$-approximately truthful-in-expectation mechanism providing approximation better than $1\/m^\\gamma$ for some $\\gamma>0$. This is true even in the case of a single player. The combinatorial public projects problem admits a simpler structure than combinatorial auctions,  and hence we use it as a warm-up to demonstrate our approach.\n\n\n\\begin{figure}[here]\n$\\begin{array}{|| c || c | c | c ||} \\hline\n\\mbox{Class of valuations} & \\mbox{Approximation} & \\mbox{Universally truthful} & \\mbox{Truthful-in-expectation}  \\\\\n\\hline \\hline\n\\mbox{submodular \/ value oracle} & 1-1\/e & m^{-1\/2} \\mid m^{\\epsilon-1\/2} & m^{-1\/2} \\mid {m^{-\\gamma}} \\mbox{\\bf [new]} \\\\\n\\hline\n\\mbox{coverage, matroid rank sums} & 1-1\/e & m^{-1\/2} \\mid 1-1\/e & 1-1\/e \\\\\n\\hline\n\\mbox{budget-additive} & \\frac{3}{4} \\mid \\frac{15}{16} & m^{-1\/2} \\mid \\frac{15}{16} & m^{-1\/2} \\mid \\frac{15}{16} \\\\\n\\hline\n\\mbox{submodular \/ demand oracle} & 1-1\/e+\\epsilon \\mid \\frac{15}{16} & \\Omega(1\/\\log m \\log \\log m) \\mid \\frac{15}{16} & \\Omega(1\/\\log m \\log \\log m) \\mid \\frac{15}{16} \\\\\n\\hline\n\\end{array} $\n\\caption{\\small Currently known results for combinatorial auctions: approximation $|$ inapproximability.\nIf only one result is given, it is known to be optimal. For randomized maximal-in-range (universally truthful) mechanisms, it is known that it is hard to achieve a better than $1\/n$-approximation for coverage valuations; however, other universally truthful mechanisms might exist. No non-trivial hardness was previously known for truthful-in-expectation combinatorial auctions, even when restricted to maximal-in-distributional-range mechanisms.}\n\\end{figure}\n\n\\paragraph{Our techniques.}\nOur hardness results are obtained by combining two recently developed techniques: the {\\em symmetry gap} technique for submodular functions \\cite{V09}, and the {\\em direct hardness} approach for combinatorial auctions \\cite{Dobzin11}.\n\nFirst, we consider the possibility of maximal-in-distributional range (MIDR) mechanisms.\nWe endeavor to explain why the approach of \\cite{DRY11} breaks down when applied to monotone submodular functions. The answer lies in a certain convexity phenomenon that can be exploited in a symmetry gap argument. The symmetry gap argument on its own rules out the approach of \\cite{DRY11}. Furthermore, it is possible to generalize the argument to an arbitrary MIDR mechanism, and moreover amplify the gap to some constant power of $m$. In fact our approach rules out even non-uniform approximately-MIDR mechanisms.\n\nIn the case of combinatorial public projects (CPP), we prove that if non-uniformity is allowed, then  approximately truthful-in-expectation mechanisms are no more powerful --- in terms of approximating combinatorial auctions using a polynomial number of value queries --- than  MIDR mechanisms. Therefore, by ruling out MIDR mechanisms, we also rule out  truthful-in-expectation mechanisms.\nIn the case of combinatorial auctions, no such  equivalence in power between truthful-in-expectation and MIDR mechanisms is known. Instead, we apply the direct hardness approach of Dobzinski \\cite{Dobzin11} to identify a single player for whom the allocation problem in some sense mimics the CPP problem. Again, the symmetry gap argument can be used here, though payments complicate  the picture. We address this difficulty by employing a scaling argument and invoking the separating hyperplane theorem --- this allows us to essentially get rid of the payments and use the same gap amplification technique we used for the CPP problem to obtain a hardness of $m^{-\\gamma}$-approximation.\n\n\\paragraph{Organization of the paper.}\nAfter the necessary preliminaries (Section~\\ref{sec:prelims}), we present our intuition on the separation between coverage and submodular functions in Section~\\ref{sec:intuition}. In Section~\\ref{sec:CPP-hardness}, we present an overview of the proof of hardness for combinatorial public projects, and in Section~\\ref{sec:auctions-hardness} an overview of the proof for combinatorial auctions. The complete proofs are deferred to the appendices.\n\n\n\n\\section{Intuition - what fails for submodular valuations}\n\\label{sec:intuition}\n\\label{sec:exp-gap}\n\nThe main obstacle in proving our hardness result for submodular functions is the fact that the natural subclass of coverage functions {\\em does} admit a truthful-in-expectation $(1-1\/e)$-approximation \\cite{DRY11}. In the absence of strategic considerations, coverage functions capture the full difficulty of submodular functions in the context of welfare maximization, in the sense that they exhibit the same hardness threshold of $1-1\/e$. Hence, it is not immediately clear where the dramatic jump in hardness should come from.\n\n\nLet us recall the main idea of \\cite{DRY11}: Let $f:2^M \\to \\RRp$ be a submodular set function. Given $\\bx \\in [0,1]^M$, the expected value of $f(S)$ when $S$ includes each item $j$ independently with probability $x_j$ is measured by the {\\em multilinear extension} $F(\\bx)$, which has been previously used in work on submodular maximization \\cite{CCPV07,V08,KST09,V09,OV11}. $F$ is an \\emph{extension} of $f$, in the sense that it agrees with $f$ on integer points, and therefore maximizing $F(\\bx)$ over fractional allocations would yield an optimal algorithm. However, $F(\\bx)$ is not a concave function and can be maximized only approximately. \nInstead, the authors of \\cite{DRY11} consider a different rounding process --- which they call the \\emph{Poisson rounding scheme} --- that includes each $j$ in $S$ with probability $1-e^{-x_j}$ instead. The expected value of applying the Poisson rounding rounding scheme to a point $\\bx$ is measured by a modified function $F^{exp}(x_1,\\ldots,x_m) = F(1-e^{-x_1},\\ldots,1-e^{-x_m})$, which fortuitously {\\em turns out to be concave} for a subclass of submodular functions, including coverage functions and weighted sums of matroid rank functions. In this case,  $F^{exp}(\\bx)$ can be maximized exactly, and yields a maximal-in-distributional-range algorithm whose range is the image of the Poisson rounding scheme.\nSince the ratio between $F(\\bx)$ and $F^{exp}(\\bx)$ is bounded by $1-1\/e$, this leads to a truthful-in-expectation $(1-1\/e)$-approximation.\n\nThe first question is whether $F^{exp}$ can be maximized for any monotone submodular function. \nIt was observed by the authors of \\cite{DRY11} that $F^{exp}$ is not concave for every submodular function:\none example is the budget-additive function $f(S) = \\min \\{ \\sum_{i \\in S} w_i, 2 \\}$ where $w_1=w_2=w_3=1$ and $w_4=2$. Hence convex optimization techniques cannot be used for $F^{exp}(\\bx)$ directly; still, perhaps $F^{exp}(\\bx)$ could be maximized for a different reason.\nWe prove that this is impossible, using a {\\em symmetry gap} argument \\cite{FMV07,MSV08,V09}. \n\nThe budget-additive function above does not lend itself well to the symmetry gap argument, because there is a clear asymmetry between the elements of weight 1 and the element of weight 2. Instead, we construct an example where $F^{exp}$ is not concave and all elements are in some sense ``equivalent\". For this purpose, we use the following construction: If $f_1,f_2:2^M \\rightarrow [0,1]$ are monotone submodular functions, then \n$$f(S) = 1 - (1-f_1(S))(1-f_2(S))$$\nis also a monotone submodular function (see Lemma~\\ref{lem:submod-product}).\nIn particular, let $M = M_1 \\cup M_2$, $|M_1| = |M_2| = m$,\n$|M|=2m$, and let $f_i(S) = \\min \\{ \\frac{1}{\\alpha m} |S \\cap M_i|, 1 \\}$ for some $\\alpha > 0$. These are budget-additive and hence monotone submodular functions. Then we set \n$$ f(S) = 1 - (1-f_1(S))(1-f_2(S)) = 1 - \\left(1 - \\frac{1}{\\alpha m} |S \\cap M_1|\\right)_+ \\left(1 - \\frac{1}{\\alpha m} |S \\cap M_2| \\right)_+.$$\nHere, $(y)_+ = \\max \\{ y, 0\\}$ denotes the positive part of a number. By Lemma~\\ref{lem:submod-product}, $f(S)$ is a monotone submodular function.\nLet's consider the function $F^{exp}(x_1,\\ldots,x_{2m}) = F(1-e^{-x_1},\\ldots,1-e^{-x_{2m}})$. If $m \\rightarrow \\infty$, a random set obtained by sampling with probabilities $1-e^{-x_i}$ will have cardinality very close to $\\sum (1-e^{-x_i})$.\nWe obtain\n$$ F^{exp}(\\bx) \\simeq 1 - \\left(1 - \\frac{1}{\\alpha m} \\sum_{i \\in M_1} (1-e^{-x_i}) \\right)_+\n \\left(1 - \\frac{1}{\\alpha m} \\sum_{j \\in M_2} (1-e^{-x_j}) \\right)_+.$$\nThe reader can verify that this function is concave for $\\alpha=1$.\nBut this is a very special coincidence. (The reason is that $f$ for $\\alpha=1$ can be represented as a coverage function.)\nAny smaller value of $\\alpha$, for instance $\\alpha = 1\/2$, gives a non-concave function $F^{exp}$,\nas can be seen by checking $\\bx = \\b1_{M_1}$, $\\bx = \\b1_{M_2}$ and $\\bx = \\frac12 \\b1_M$:\n$ F^{exp}(\\b1_{M_1}) = F^{exp}_2(\\b1_{M_2}) = 1 - (-1 + 2 e^{-1})_+ = 1 $\n(note that $-1 + 2e^{-1} < 0$), while the value at the midpoint is\n$ F^{exp}\\left(\\frac12 \\b1_M \\right) \\simeq 1 - (-1 + 2 e^{-1\/2})^2 = 4 e^{-1\/2} - 4 e^{-1} \\simeq 0.955.$\nTherefore, we have an example where $F^{exp}(\\bx)$ is not concave and moreover, all elements play the same symmetric role in $f$.\n(Formally, $f$ has an element-transitive group of symmetries.)\nFunctions of this type will play a crucial role in our proof.\n\n\\paragraph{The symmetry gap argument.}\nThe symmetry gap argument from \\cite{V09}, building up on previous work \\cite{FMV07, MSV08}, shows the following:\nInstances exhibiting some kind of symmetry can be blown up and modified in such a way that the only solutions that an algorithm can find (using a polynomial number of value queries) are symmetric with respect to the same notion of symmetry.\nThus the gap between symmetric and asymmetric solutions implies an inapproximability threshold.\nWe use this argument here as follows. The instance above (for $\\alpha=1\/2$) can be slightly modified as in \\cite{FMV07,MSV08,V09}, in such a way that it is impossible to find any solution that is asymmetric with respect to $M_1, M_2$. \nConsider the optimization problem\n$$ \\max \\{F^{exp}(\\bx): \\sum x_i \\leq m \\}.$$\nThe best symmetric solution is $F^{exp}(\\frac12 \\b1_M) \\simeq 0.955$, while the optimum is $F^{exp}(\\b1_{M_1}) = 1$. The only solutions found by a polynomial number of value queries are the symmetric ones, and hence we cannot solve the optimization problem within a factor better than $0.955$. A similar argument shows that we cannot solve the welfare maximization problem (for 2 players) with respect to $F^{exp}(\\bx)$ within a factor better than $0.955$.\n\n\\iffalse\n\\paragraph{Towards arbitrary distribution ranges.}\nIn the following, we harness this construction towards showing that there can be no good maximum-in-distributional range mechanism, and eventually, no good truthful-in-expectation mechanism. This will be much more technical; however, the main idea is as follows: If the valuation function is as above and there is a distribution possibly returned by a mechanism that is concentrated on $M_1$ or $M_2$, then it provides more value than a similar distribution which is symmetric with respect to $(M_1,M_2)$. However, the only distributions that can be efficiently found are symmetric with respect to $(M_1,M_2)$. Therefore, in order to beat the asymmetric distribution, a mechanism needs to compensate for this by returning a (in some technical sense) ``denser\" distribution symmetric with respect to $(M_1,M_2)$. By using this argument recursively, we reach a contradiction with the fact that the mechanism provides a good approximation. \nNext, we show how this can be applied to the CPP problem.\n\\fi\n\nIn the following, we harness this construction towards showing that there can be no good maximum-in-distributional-range mechanism, and eventually, no good truthful-in-expectation mechanism. \n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:prelims}\n\n\\subsection{Mechanism Design Basics}\\label{sec:MD}\n\n\\paragraph{Mechanism Design Problems.} We consider mechanism design problems where there are $n$ players, and a set $\\Omega$ of feasible solutions. Each player $i$ has a non-negative \\emph{valuation function} $v_i: \\Omega \\to \\RRp$. We are concerned with \\emph{welfare maximization} problems, where the objective is $\\sum_{i=1}^n v_i(\\omega)$.\n\n\\paragraph{Mechanisms.} We consider direct-revelation mechanisms for mechanism design problems.  Such a mechanism  comprises an {\\em\n  allocation rule} $\\A$, which is a function from (hopefully\ntruthfully) reported valuation functions $v=(v_1,\\ldots,v_n)$ to an outcome\n$\\A(v) \\in \\Omega$, and a {\\em payment rule} $p$, which is a function from\nreported valuation functions to a required payment $p_i(v)$ from each player $i$.\nWe allow the allocation and payment rules to be randomized. We restrict our attention to mechanisms that are individually rational in expectation --- i.e. $\\E[v_i(\\A(v)) - p_i(v)] \\geq 0 $ --- and  the payments are non-negative in expectation --- i.e. $\\E[p_i(v)] \\geq 0$ --- for each player $i$ and each input $v=(v_1,\\ldots,v_n)$, when the expectations are over the random coins of the mechanism. \n\n\\paragraph{Truthfulness.} A mechanism with allocation and payment rules $\\A$ and $p$ is {\\em\n  truthful-in-expectation} if every player always maximizes its expected\n  payoff by truthfully reporting its valuation function, meaning that\n\\begin{equation}\\label{eq:truthful}\n\\E[v_i(\\A(v)) - p_i(v)] \\geq \\E[ v_i(\\A(v'_i,v_{-i})) -\n  p_i(v'_i,v_{-i})]\n\\end{equation}\nfor every player~$i$, (true) valuation function~$v_i$, (reported)\nvaluation function~$v'_i$, and (reported) valuation functions~$v_{-i}$ of\nthe other players.\nThe expectation in~\\eqref{eq:truthful} is over the coin flips of the\nmechanism.  If \\eqref{eq:truthful} holds for every flip of the coins, rather than merely in expectation, we call the mechanism \\emph{universally truthful}.\n\n\n\\paragraph{VCG-Based Mechanisms.} Mechanisms for welfare maximization problems are often variants of the classical VCG mechanism.  \nRecall that the {\\em VCG\n  mechanism} is defined by the (generally intractable)\nallocation rule that selects the welfare-maximizing outcome with\nrespect to the reported valuation functions, and the payment rule that\ncharges each player~$i$ a bid-independent ``pivot term'' minus the\nreported welfare earned by other players in the selected outcome.  This\n(deterministic) mechanism is truthful; see e.g.~\\cite{Nis07}.\n\nLet $dist(\\Omega)$ denote the probability distributions over the set of \nfeasible solutions $\\Omega$, and let $\\cR \\sse dist(\\Omega)$ be a compact subset of\nthem.  The corresponding {\\em Maximal in Distributional Range (MIDR)}\nallocation rule is defined as follows: given reported valuation\nfunctions $v_1,\\ldots,v_n$, return an outcome that is sampled randomly\nfrom a distribution $D^* \\in \\cR$ that maximizes the expected welfare\n$\\E_{\\omega \\sim D}[\\sum_i v_i(\\omega)]$ over all distributions $D \\in \\cR$.\nAnalogous to the VCG mechanism, there is a (randomized) payment rule\nthat can be coupled with this allocation rule to yield a\ntruthful-in-expectation mechanism (see~\\cite{DD09}). We note that deterministic MIDR allocation rules\n --- i.e. those where $\\cR$ is a set of point distributions --- are called \\emph{maximal-in-range (MIR)}.\n\n\\paragraph{Approximate Truthfulness.}\nFor $\\epsilon\\geq 0$, a  mechanism with allocation and payment rules $\\A$ and $p$ is {\\em $(1-\\epsilon)$-approximately   truthful-in-expectation} if \n\\begin{equation}\\label{eq:approx-truthful}\n\\E[v_i(\\A(v)) - p_i(v)] \\geq (1-\\epsilon) \\E[ v_i(\\A(v'_i,v_{-i})) -\n  p_i(v'_i,v_{-i})]\n\\end{equation}\nfor every player~$i$, (true) valuation function~$v_i$, (reported)\nvaluation function~$v'_i$, and (reported) valuation functions~$v_{-i}$ of\nthe other players.\nThe expectation in~\\eqref{eq:approx-truthful} is over the coin flips of the\nmechanism.  \nUsing the fact that payments are non-negative in expectation, a $(1-\\epsilon)$-approximately truthful-in-expectation mechanism also satisfies the following weaker condition. (This condition is sufficient for our hardness results.)\n\\begin{equation}\\label{eq:approx-truthful-relaxed}\n\\E[v_i(\\A(v)) - p_i(v)] \\geq  \\E[ (1-\\epsilon) v_i(\\A(v'_i,v_{-i})) - p_i(v'_i,v_{-i})]\n\\end{equation}\nApproximately truthful mechanisms are related to \\emph{approximately maximal-in-distributional-range} allocation rules. An allocation rule $\\A: \\V \\to \\Omega$ is $(1-\\epsilon)$-approximately maximal-in-distributional range if it fixes a $\\cR \\sse dist(\\Omega)$, and returns an outcome that is sampled from $D^* \\in \\cR$ that $(1-\\epsilon)$-approximately maximizes the expected welfare\n$\\E_{\\omega \\sim D}[\\sum_i v_i(\\omega)]$ over all distributions $D \\in \\cR$.  We show in Appendix  \\ref{sec:TIE-MIDR} a sense in which approximately maximal-in-distributional-range allocation rules are no less powerful -- in terms of approximating the social welfare -- than approximately truthful-in-expectation mechanisms.\n\nOur main reason for considering the notion of approximate truthfulness is that the mechanisms of \\cite{DRY11,Dughmi11}, if implemented in the value oracle model, are only approximately truthful-in-expectation (for an arbitrarily small $\\epsilon>0$) \\cite{DRVY11}. The value oracle model seems too weak to make the mechanisms of \\cite{DRY11,Dughmi11} exactly truthful-in-expectation; however, \\cite{DRVY11}  makes it quite conceivable that there might be an approximately truthful-in-expectation mechanism for combinatorial auctions and combinatorial public projects, both with submodular valuations.\n\n\n\\subsection{Combinatorial Auctions}\n\n\nIn \\emph{Combinatorial Auctions} there is a set $M$ of $m$ items, and a set of $n$ players. Each player $i$ has a valuation\nfunction $v_i:2^M\\rightarrow \\RRp$ that is normalized\n($v_i(\\emptyset) = 0$) and monotone ($v_i(A) \\leq v_i(B)$ whenever\n$A \\sse B$). A feasible solution is an \\emph{allocation}\n$(S_1,\\ldots,S_n)$, where $S_i$ denotes the items assigned to player $i$,\nand $\\set{S_i}_i$ are mutually disjoint subsets of $M$.\nPlayer $i$'s value for outcome $(S_1,\\ldots,S_n)$ is equal to $v_i(S_i)$.  The\ngoal is to choose an allocation maximizing \\emph{social welfare}: $\\sum_i v_i(S_i)$.  \n\n\n\n\\subsection{Combinatorial Public Projects}\n\n\nIn \\emph{Combinatorial Public Projects} there is a set\n$[m]= \\set{1,\\ldots,m}$ of \\emph{projects}, a cardinality bound $k$ such that $0 \\leq k \\leq m$, and a set\n$[n]=\\set{1,\\ldots,n}$ of \\emph{players}. Each player $i$ has a valuation\nfunction $v_i:2^{[m]}\\rightarrow \\RRp$ that is normalized\n($v_i(\\emptyset) = 0$) and monotone ($v_i(A) \\leq v_i(B)$ whenever\n$A \\sse B$). In this paper, we focus on the \\emph{flexible} variant of combinatorial public projects: a feasible solution is a set $S \\sse [m]$ of projects with $|S| \\leq k$. Player $i$'s value for outcome $S$ is equal to $v_i(S)$. Prior work \\cite{PSS08,BSS10,Dobzin11} has also considered the \\emph{exact} variant, where a feasible solution is a set $S \\sse [m]$ with $|S| = k$. In both variants, the\ngoal is to choose a feasible set $S$ maximizing \\emph{social welfare}: $\\sum_i v_i(S)$.  \n\n\n\n\n\n\n\n\n\\section{Transforming an approximately truthful-in-expectation mechanism into approximately maximal-in-distribution range}\n\\label{sec:TIE-MIDR}\nIn this section, we prove Theorem \\ref{thm:TIE=MIDR}.\n\nLet $\\Omega= \\set{ S \\sse [m] : |S| \\leq k}$ be the set of outcomes of CPP. Let $\\Delta(\\Omega)$ denote the simplex in $\\RR^\\Omega$, representing the set of distributions over $\\Omega$. Let $\\V$ denote the set of submodular valuations on $[m]$. We think of $\\V$ as a subset of $\\RRp^\\Omega$ --- specifically, each $v \\in \\V$ is a vector in $\\RRp^\\Omega$, where $v_S$ is the value of outcome $S$ for a player with valuation $v$. We note that for each $v \\in \\V$,  the infinity norm $||v||_\\infty$  is equal to the value of the optimum solution.\n\nFix $\\epsilon$, $c$, $\\cM$, and $\\delta$ as in the statement of the theorem. Let $\\A: \\V \\to \\Delta(\\Omega)$ be the allocation rule of $\\cM$ when there is a single player. By assumption, $\\A$ is a $c$-approximation for $c>0$ -- specifically, $\\frac{v^T \\A(v)}{||v||_\\infty} \\geq c$.  The following is an approximate variant of \\emph{weak monotonicity} \\cite{LMN03}, and follows from the fact that $\\A$ is the allocation rule of a  $(1-\\epsilon)$-approximately truthful mechanism.\n\n\\begin{fact}[Similar to \\cite{LMN03}]\\label{fact:wmon}\n  For any $u,v \\in \\V$,\n\\[ v^T \\A(v) - (1-\\epsilon) u^T \\A(v) \\geq (1-\\epsilon) v^T \\A(u) - u^T \\A(u).  \\]\n\\end{fact}\n\nWe prove Theorem \\ref{thm:TIE=MIDR} by showing that there is a black-box reduction  that converts $\\A$ to a new allocation rule $\\B$ that is $(1- 3\\epsilon - \\delta)$-MIDR. The reduction will be non-uniform -- specifically, $\\B$ will utilize an advice string that depends on $m$, but is independent of the input valuation $v \\in \\V$. The length of the advice string will not be bounded, polynomially or otherwise --- this is OK, since we are only interested in preserving value oracle lower-bounds. $\\B$ preserves  the approximation ratio of $\\A$, and moreover makes only $m$ more value queries than does $\\A$.\n\n\n\n\nThe proof consists of two main steps. First, we show that $\\A$ tends to a $(1-\\epsilon)$-approximately maximal-in-distributional-range allocation rule ``in the limit'' as we scale up the valuations.  Then, we use this fact to construct, via a non-uniform black box reduction,  an allocation rule $\\B$ that approximates the limit behavior of $\\A$,  in the sense that it $(1-3\\epsilon -\\delta)$-approximately maximizes over the range of $\\A$. \n\n\\begin{remark}\nWe note that the proofs of this section apply more generally than CPP with submodular valuations. In particular, the only properties of this problem that are used in the proofs are: (1) The multiple-player allocation problem is algorithmically equivalent to the single player allocation problem (2) The set $\\Omega$ of outcomes is finite, (3) The set of valuations $\\V \\sse \\RRp^\\Omega$ is closed under scaling by a non-negative constant, and (4) There is a deterministic algorithm $s: \\V \\to \\RRp$ that runs in finite time, makes a polynomial number of value queries, and returns a ``weak approximation'' to the optimal value -- i.e. we only require that $s(v) > 0$ when $||v||_\\infty >0$. When a welfare-maximization mechanism design problem satisfies these four conditions, as do all variants of CPP and other ``public-project''-type problems in the literature, then the analogue of Theorem \\ref{thm:TIE=MIDR} holds for that problem.\n\\end{remark}\n\n\\subsection{Limit behavior of truthful in expectation mechanisms}\n\n\n\n\n\n\n\n\nWe will show that $\\A$ is $(1-\\epsilon)$-approximately MIDR in the limit as we scale up the valuations. Recall that a mechanism $\\B: \\V \\to \\Delta (\\Omega)$ is $(1-\\epsilon)$-approximately MIDR if $v^T\\B(v) \\geq (1-\\epsilon) \\sup_{w \\in \\V} v^T \\B(w)$ for all $v \\in \\V$. The following statement is analogous.\n\n\n\\begin{proposition}\\label{prop:lim_midr}\n$\\liminf_{\\alpha \\to \\infty} v^T\\A(\\alpha v) \\geq (1-\\epsilon) \\sup_{w \\in \\V} v^T\\A(w)$ for all $v \\in \\V$.\n\\end{proposition}\n\\begin{proof}\nLet $\\alpha,\\beta\\in \\RRp$, and let $w,v \\in \\V$. By Fact \\ref{fact:wmon}:\n\\begin{align*}\n \\alpha v^T \\A(\\alpha v) -   (1-\\epsilon)w^T \\A(\\alpha v) &\\geq (1-\\epsilon) \\alpha v^T \\A( w) -   w^T \\A( w).\n\\end{align*}\nDividing the expression by $\\alpha$, we get:\n\\begin{align*}\n v^T \\A(\\alpha v) -  \\frac{ (1-\\epsilon) w^T \\A(\\alpha v)}{\\alpha} &\\geq  (1-\\epsilon) v^T \\A( w) -  \\frac{ w^T \\A( w)}{\\alpha}\n\\end{align*}\nTaking the limit infimum as $\\alpha$ goes to infinity,\n\\begin{align*}\n\\liminf_{\\alpha \\to \\infty} v^T \\A(\\alpha v)  &\\geq  (1-\\epsilon) v^T \\A( w) .\n\\end{align*}\nNow, taking the supremum over $w$\n\\begin{align*}\n\\liminf_{\\alpha \\to \\infty} v^T \\A(\\alpha v)  &\\geq (1-\\epsilon) \\sup_{w \\in \\V}  v^T \\A(w) \n\\end{align*}\n This  completes the proof.\n\\end{proof}\n\n\n \n\n\n\\subsection{Approximating the limit behavior of a mechanism}\nIdeally, we would transform $\\A$ to an allocation rule that behaves as $\\A$ does in the limit -- by the results of the previous sub-section, such a ``limit allocation rule'' of $\\A$ would be $(1-\\epsilon)$-MIDR. However, since our reduction must take finite time, we must settle for approximating the limit behavior of $\\A$. Unfortunately, even that is non-trivial: given $v$, the ratio $\\alpha$ by which we would need to scale $v$ before coming close to the ``limit'' of $\\A(\\alpha v)$ is a complete mystery, and may be arbitrarily large. Therefore, we need to utilize some non-uniform advice to deduce that order of magnitude of the necessary scaling factor. An additional difficulty is that this advice must be independent of $v$ -- specifically,  the advice may depend only on the number of items $m$.\n\n\nFor each $\\delta' >0$ and $v \\in \\V$, we define a threshold  $t(\\delta',v)$. Roughly speaking, $t(\\delta',v)$ is the ``scale''  at which  $\\A$ is guaranteed to be within $(1-\\delta')$ of its limit behavior when given input in the direction of $v$.  Proposition \\ref{prop:lim_midr} guarantees that threshold $t(\\delta',v)$ exists for each $v\\in \\V$ and $\\delta' >0$.\n\\begin{equation}\n  \\label{eq:2}\n  t(\\delta',v) = \\sup \\left\\{t :  v^T \\A\\left(t \\frac{v}{||v||_\\infty}\\right) \\leq (1- \\epsilon-\\delta') \\sup_{w \\in \\V} v^T \\A(w)\\right\\} +1\n\\end{equation}\n We note that the motivation for adding $1$ (any arbitrary positive number would do) to the expression is to guarantee that $v^T \\A\\left(t(\\delta',v) \\frac{v}{||v||_\\infty}\\right) \\geq (1-\\epsilon - \\delta') \\sup_{w \\in \\V} v^T \\A(w)$, which may not be guaranteed by the supremum\n\nAssume that, for some $\\delta'>0$, we have some upper-bound $\\tau$ on $\\set{t(\\delta',v): v\\in \\V}$, and moreover we have a procedure $s(v)$ to estimate a non-zero lower bound on $||v||_\\infty$ for each $v \\in \\V$. Then, the following procedure is evidently a $c$-approximate and $(1-\\epsilon - \\delta')$-MIDR allocation rule: Given input $v \\in \\V$, output  $\\A(\\frac{\\tau}{s(v)} \\cdot v)$. The procedure $s(v)$ is easy to implement using only $m$ value queries --- indeed, we can take $s(v) = \\max_{j \\in [m]} v(\\set{j})$. It is not clear, however, that the  upper-bound $\\tau$ can be computed effectively. Even worse, it is not clear that such an upper-bound even exists: $\\V$ is infinite, and $t(\\delta',v)$ is not necessarily a continuous function of $v$!\n\nWe remedy this as follows. We will show that there exists such an upper-bound when $\\delta'$ is sufficiently large relative to $\\epsilon$. Specifically, we show that there exists an  upper-bound $\\tau$ on  $\\set{t(\\delta + 2\\epsilon,v) : v \\in \\V}$. However, computing such an upper-bound in finite time may be impossible in general, since the scale of valuations at which $\\A$ approaches its  limit behavior may be arbitrary. Instead, we take $\\tau$ as advice to our non-uniform reduction. By the discussion in the previous paragraph, showing that the upper-bound $\\tau$ exists  yields a non-uniform allocation rule that is $(1-3 \\epsilon -\\delta)$-MIDR, and makes at most $m$ more value queries than $\\A$, completing the proof of Theorem \\ref{thm:TIE=MIDR}. \n\nAs a tool  for proving that the upper-bound $\\tau$ exists, we define a finite net of $\\V$. Since $\\V$ is a  cone  in finite-dimensional euclidean space, its intersection with the infinity-norm unit ball admits a \\emph{$\\sigma$-net} in the infinity-norm for any $\\sigma>0$  --- specifically, a finite set  $\\U \\sse \\V$ such that\n\\begin{enumerate}\n\\item $||u||_\\infty =1$ for all $u \\in \\U$\n\\item  $\\forall v \\in \\V \\  \\exists u \\in \\U \\ \\left|\\left|\\frac{v}{||v||_\\infty} - u\\right|\\right|_\\infty \\leq \\sigma $\n\\end{enumerate}\n\n\nLet $\\sigma = c \\delta \/ 4$ and let $\\U$ be a $\\sigma$-net of $\\V$. Now let $\\beta= \\max_{u \\in \\U} t(\\frac{\\delta}{4},u)$, and let  $\\tau= 4 \\beta \/ \\delta$. It suffices to show that $v^T \\A ( \\tau v) \\geq (1-3\\epsilon -\\delta) \\sup_{w \\in \\V} v^T \\A (w)$ for each  $v \\in \\V$ with $||v||_\\infty = 1$. Let $v \\in \\V$ be such that $||v||_\\infty = 1$, and let $u$ be a point in the $\\sigma$-net $\\U$ such that $||v- u||_\\infty \\leq \\sigma$.  By Fact \\ref{fact:wmon}, we have\n\n\\begin{align}\\label{eq:wmon_tau}\n \\tau v^T \\A(\\tau v ) - (1-\\epsilon) \\beta u^T \\A(\\tau v) &\\geq (1-\\epsilon)\\tau v^T \\A (\\beta u)  - \\beta u^T \\A( \\beta u)    \n\\end{align}\nWe now use inequality \\eqref{eq:wmon_tau} to lower-bound $v^T \\A(\\tau v)$:\n\\begin{align*}\n  v^T \\A(\\tau v)  &\\geq  (1-\\epsilon) v^T \\A (\\beta u )  - \\frac{ \\beta}{\\tau} u^T \\A( \\beta u)    &&\\text{Dividing \\eqref{eq:wmon_tau} by $\\tau$ and loosening the inequality}\\\\\n   &=  (1-\\epsilon) v^T \\A (\\beta u)  - \\frac{\\delta}{4} u^T \\A( \\beta u)    &&\\text{By definition of $\\tau$}\\\\\n   &=  \\left(1  - \\epsilon - \\frac{\\delta}{4}\\right) u^T \\A( \\beta u) - (1-\\epsilon) (u-v)^T \\A(\\beta u)    &&\\\\\n   &\\geq  \\left(1- \\epsilon - \\frac{\\delta}{4}\\right) u^T \\A( \\beta u) - ||u-v||_\\infty     &&\\text{Since $||\\A(\\beta u)||_1 = 1$} \\\\\n   &\\geq  \\left(1  - \\epsilon-\\frac{\\delta}{4}\\right) u^T \\A( \\beta u) - \\sigma     &&\\text{By proximity of $u$ and $v$} \\\\\n   &\\geq  \\left(1 - \\epsilon -\\frac{\\delta}{2}\\right) u^T \\A( \\beta u)     &&\\text{By  $c \\leq u^T \\A(\\beta u)$ and definition of $\\sigma$} \\\\\n  &\\geq  \\left(1-2\\epsilon -\\frac{3}{4} \\delta\\right) \\sup_{w \\in \\V} u^T \\A (w)      &&\\text{By definition of $\\beta$}\\\\\n  &\\geq  \\left(1-2 \\epsilon -\\frac{3}{4} \\delta\\right) \\liminf_{\\alpha \\to \\infty} u^T \\A (\\alpha v)      &&\\\\\n  &=  \\left(1-2 \\epsilon -\\frac{3}{4} \\delta\\right) \\liminf_{\\alpha\\to \\infty} (v^T \\A (\\alpha v) - (v -u)^T \\A(\\alpha v) )      &&\\\\\n  &\\geq  \\left(1- 2 \\epsilon -\\frac{3}{4} \\delta\\right) \\liminf_{\\alpha \\to \\infty} (v^T \\A (\\alpha v) - ||v -u||_\\infty  )      &&\\text{Since $||\\A(\\alpha v)||_1 = 1$}\\\\\n  &\\geq  \\left(1-2 \\epsilon - \\frac{3}{4} \\delta\\right) \\liminf_{\\alpha \\to \\infty} (v^T \\A (\\alpha v) - \\sigma  )      &&\\text{By proximity of $u$ and $v$}\\\\\n  &\\geq  \\left(1- 2 \\epsilon - \\frac{3}{4}\\delta\\right) \\liminf_{\\alpha \\to \\infty} (v^T \\A (\\alpha v) - \\frac{\\delta}{4} v^T \\A(\\alpha v)  )      &&\\text{By  $c \\leq v^T \\A(\\alpha v)$ and definition of $\\sigma$}\\\\\n  &\\geq  \\left(1- 2 \\epsilon - \\delta\\right) \\liminf_{\\alpha \\to \\infty} v^T \\A (\\alpha v)  &&\\\\\n  &\\geq  \\left(1- 3 \\epsilon - \\delta\\right) \\sup_{w \\in \\V} v^T \\A (w)  &&\\text{By Proposition \\ref{prop:lim_midr}}\n\\end{align*}\nBy the previous discussion, this completes the proof of Theorem \\ref{thm:TIE=MIDR}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nLet $C$ be a non-hyperelliptic, smooth, projective curve \nof genus $g$ defined over\n$\\mathbb C$ and let\n$\\mathcal{R}_{C}:=\\oplus_{i=0}^{\\infty}H^{0}(C,\\omega_{C}^{\\otimes i})$ be its \ncanonical ring.\nIt is well-known that $C$ is isomorphic to the canonical curve $\\proj(\\mathcal{R}_{C})$, \nwhich embeds in \n$ \\mathbb P^{g-1}$ as a projectively normal variety\nand the homogeneous ideal ${\\mathcal{I}}_{C}$ of $C$ in $\\mathbb P^{g-1}$ \nis generated in degree\n$2$ unless $C$ is trigonal or a smooth plane quintic, see \n\\cite{ACGH}. \n\n\nSince Green's\nseminal papers, \\cite{g1}, \\cite{g3} and \\cite{g2}, the syzygies of $\\mathcal{R}_{C}$ have been studied\ndeeply by several authors; here we can quote, for instance, \\cite{Sch}, \\cite{mv}, \\cite{v}. \n\n\nIn this paper we follow an approach we learned from \\cite{IR}. \nWe put $ \\mathbb P^{g-1}:= \\proj(\\mathbb C[\\partial_{0},\\dotsc,\n\\partial_{g-1}])$ where $\\mathbb C[\\partial_{0},\\dotsc,\n\\partial_{g-1}]$ is the polynomial ring generated by the natural\nderivations over $\\mathbb C[x_{0},\\dotsc,\nx_{g-1}]$. We consider $\\eta_{1},\\eta_{2}$ two general\nlinear forms in $\\mathbb C[\\partial_{0},\\dotsc,\n\\partial_{g-1}]$ which can be assumed to be $\\eta_{1}=\\partial_{g-1}$ and\n$\\eta_{2}=\\partial_{g-2}$ and we construct the ring $A:=\n\\frac{\\mathcal{R}_{C}}{\\gen{\\eta_{1},\\eta_{2}}}$. An \neasy computation shows that $A$ is an Artinian graded Gorenstein ring of\nsocle degree $3$. Then, by a result of Macaulay, it can be\nrealised as $A=\\frac{\\mathbb C[\\partial_{0},\\dotsc,\n\\partial_{g-3}]}{F_{\\eta_{1},\\eta_{2}}^\\perp}$\nwhere $F_{\\eta_{1},\\eta_{2}}\\in \\mathbb C[x_{0},\\dotsc, x_{g-3}]$ \nis a cubic homogeneous\npolynomial and $F_{\\eta_{1},\\eta_{2}}^\\perp:=\n\\{ D\\in \\mathbb C[\\partial_{0},\\dotsc, \n\\partial_{g-3}]\\vert D(F_{\\eta_{1},\\eta_{2}})=0 \\}$.\n\nIn this way, it remains defined a rational map: \n\\begin{equation}\\label{eq:ac} \n\\alpha_C\\colon\\mathrm{Gr}(g-2,g)\\dashrightarrow H_{g-3,3}\n\\end{equation}\nwhere $\\mathrm{Gr}(g-2,g)$ is the Grassmannian of $(g-3)$-planes in $\\mathbb{P}^{g-1}$ and $H_{g-3,3}$ is the space of hypercubics \nin $\\check\\mathbb{P}^{g-3}$ modulo the action of $\\PGL(g-2,{\\mathbb C})$. \n\n\nWe will analyse this map when $C$ has a $g^1_n$, that is, it is an $n:1$ covering of \nthe projective line. In particular, \nthe above construction applied to the case of trigonal curves, \\textit{i.e. } $n=3$, gives a\nnice correspondence to the Fermat cubics in $g-2$ variables, and vice versa:\n\n\\begin{ta}\\label{Theorem A}\nA canonical curve $C$ is either trigonal or isomorphic \nto a smooth plane quintic \nif and only if $F_{\\eta_{1},\\eta_{2}}\\in \\mathbb C[x_{0},\\dotsc, x_{g-3}]$ \nis a Fermat cubic, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general $1$-forms. \n\\end{ta}\n\n\nTheorem A confirms the intuition that the more special is\nthe curve, the more special is the cubic; in other words, that the\nimage of the map $\\alpha_C$ depends on the\ngeometry of $C$.\n\nIn general, our method to estimate the \\emph{Waring number} of $F_{\\eta_1,\\eta_2}$ requires \nto find the degree of \na surface $S$ such that $C\\subset S\\subset X$, where $X$ is a rational normal \nscroll of dimension $n-1$. \n\nFor\ntetragonal curves we \nshow that $S$ can be obtained as \na rational surface such \nthat the $g^{1}_{4}$ is\ncut by a pencil of conics; \nmore precisely, \n\n\n\\begin{tb}\nIf a canonical curve $C$ has a $g^1_4$, then \n$F_{\\eta_{1},\\eta_{2}}$ is a sum of \ncubes of at most $\\lceil\\frac{3}{2}g-\\frac{7}{2}\\rceil$ \nlinear forms, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general. \n\\end{tb}\n\nWe show that the above bound is sharp for small genera and it is always better than the \none in \\cite{IR}; we analyse in particular the case of genus $7$ (Subsection~\\ref{g7}), showing that, for \nthe general tetragonal curve $[C]\\in \\mathcal{T}_7$ \n($\\mathcal{T}_7 \\subset\\mathcal M_7$ \nis the \\emph{tetragonal locus}, which is irreducible \nof dimension $17$), the corresponding $F_{\\eta_1,\\eta_2}$ is the sum of exactly \n$7$ cubes, and, for $[C]\\in \\mathcal{T}_7^{(3)}$ \n($\\mathcal{T}_7^{(3)}\\subset \\mathcal{T}_7$  \nis the locus of the curves  carrying exactly $3$ linear series $g^1_4$'s; it has dimension $16$), \n$F_{\\eta_1,\\eta_2}$ is the sum of exactly \n$6$ cubes, in Proposition~\\ref{prop:7}. \n\nThen, in Subsection~\\ref{ssec:stc}, \nwe have extended the method we used in Subsection~\\ref{g7} to some classes of tetragonal curves contained in \n\\emph{balanced} rational normal scrolls, see \nPropositions~\\ref{4gon}, \\ref{4gon-1}, and \\ref{4gon-2}, finding---for these curves---better estimates \nthan the one in Theorem~B. We have been able to extend the\nmethod to produce surfaces of the desired degree also for curves contained in non-balanced scrolls, \nsee Proposition~\\ref{specis}; \nbut the essential result given in Lemma~\\ref{lem:acm} is\nnot easy to generalise to the obtained surfaces.\n\n \nThen, in Subsection~\\ref{sec:agg}, we show that this construction can be\nextended to a class of special curves of every gonality thanks to a referee's suggestion.\nNevertheless the degree of such surfaces is, in general,\nrather high---at least higher than the one of \nIliev and Ranestad (or of Ciliberto and Harris \\cite{ch}). \n\n\nWe could not find a generalisation of Theorem~B to higher gonality, since in the proof of it  \nwe used the fact that the rational normal scroll $X$ which contains the tetragonal \n$C$ has dimension three, and therefore the surface $S$ such that $C\\subset S\\subset X$ gives a divisor in $X$, \nwhile for higher gonality the rational normal scroll $X$ \nhas higher dimension (more precisely, $\\dim(X)=n-1$ for the $n$-gonal \ncurve), and therefore $S$ has higher codimension.  \n\nMoreover, we observe that the obstruction to obtain \nthe vice versa of Theorem B (or even for the class of curves that we have \nstudied) is \nthat the geometry of surfaces in $\\mathbb P^{g-1}$ of degree $> g$ is\nnot well understood.\n\n\n\n\n\nWe think that the following problem has its own interest: \n\\begin{pb}\nFind a bound for the Waring number of the polar hypercubic associated to a general $n$-gonal curve. \n\\end{pb}\n\n\n\\begin{acks}\nWe would like to thank K. Ranestad, E. Mezzetti, G. Sacchiero and M. Brundu for interesting discussions and suggestions, \nG. Casnati and E. Ballico for the help to improve our work and\nfor pointing out some inaccuracies and valuable comments.  \nWe would like also to thank the anonymous referee for important remarks and advices. \n\\end{acks}\n\n\n\\section{Apolarity and hypercubics}\n\n\n\n\\subsection{Apolarity}\nLet $S:={\\mathbb C}[x_0,\\dotsc, x_N]$ be the polynomial ring in $(N+1)$-variables. \nThe algebra of the partial derivatives on $S$, \n\\begin{equation*}\nT:={\\mathbb C}[\\partial_0,\\dotsc,\\partial_N],\\qquad \\partial_i:=\\frac{\\partial}{\\partial_{x_i}},\n\\end{equation*}\nacts on the monomials in the following way\n\\begin{equation*}\n\\partial^a \\cdot x^b=\n\\begin{cases}a!\\binom{b}{a}x^{b-a} & \\text{if $b\\ge a$}\\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\end{equation*}\nwhere $a,b$ are multiindices $\\binom{b}{a}=\\prod_i\\binom{a_i}{b_i}$, etc. \n\nObviously, we can think of $S$ as the algebra of partial derivatives on $T$ \nby defining\n\\begin{equation*}\nx^a \\cdot \\partial^b=\n\\begin{cases}a!\\binom{b}{a}\\partial^{b-a} & \\text{if $b\\ge a$}\\\\\n0 & \\text{otherwise.}\n\\end{cases}\n\\end{equation*}\n\nThese actions define a perfect paring between the \nhomogeneous \nforms in degree $d$ in $S$ \nand $T$:  \n\\begin{equation*}\nS_d\\times T_d\\xrightarrow{\\cdot} {\\mathbb C}.\n\\end{equation*}\nIndeed, this is nothing but the extension of the duality between vector spaces: \nif $V:=S_1$, then $T_1=V^*$. \n\nMoreover, the perfect \nparing shows the natural duality between $\\mathbb{P}^N:=\\proj(S)$ and \n$\\check\\mathbb{P}^N=\\proj(T)$. \nMore precisely, if $c=: (c_0,\\dotsc, c_N)\\in\\check\\mathbb{P}^N$, this gives \n$f_c:=\\sum_i c_i x_i\\in S_1$, and if $D\\in T_a$, \n\\begin{equation*}\nD \\cdot f_c^b=\n\\begin{cases}a!\\binom{b}{a}D(c)f_c^{b-a} & \\text{if $b\\ge a$}\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation*}\nin particular, if $b\\ge a$\n\\begin{equation*}\\label{eq:lin}\n0=D \\cdot f_c^b \\iff D(c)=0.\n\\end{equation*}\n\n\\begin{df}\nWe say that two forms, $f\\in S$ and $g\\in T$ are \\emph{apolar} if \n\\begin{equation*}\ng \\cdot f =f\\cdot g =0.\n\\end{equation*}\n\\end{df}\n\nLet $f\\in S_d$ and $F:=V(f)\\subset\\mathbb{P}^N$ the corresponding hypersurface; \nlet us now define \n\\begin{equation*}\nF^\\perp :=\\{D\\in T\\mid D\\cdot f=0 \\}\n\\end{equation*}\nand\n\\begin{equation*}\nA^F:=\\frac{T}{F^\\perp}.\n\\end{equation*}\n\n\\begin{lem}\nThe ring $A^F$ is Artinian Gorenstein of socle \nof \ndimension one and degree $d$. \n\\end{lem}\n\\begin{proof}\nSee \\cite[\\S 2.3 page 67]{ik}.\n\\end{proof}\n\n\\begin{df}\n$A^F$ is called the \\emph{apolar} Artinian Gorenstein ring of $F$.\n\\end{df}\n\nIt holds the \\emph{Macaulay Lemma}, that is \n\\begin{lem}\nThe map \n\\begin{equation*}\nF\\mapsto A^F\n\\end{equation*}\nis a bijection between the hypersurfaces $F\\subset\\mathbb{P}^N$ of degree \n$d$ and graded Artinian Gorenstein quotient rings \n\\begin{equation*}\nA:=\\frac{T}{I}\n\\end{equation*}\nof $T$ with socledegree $d$. \n\\end{lem}\n\\begin{proof}\nSee \\cite[Lemma 2.12 page 67]{ik}.\n\\end{proof}\n\n\n\\subsubsection{Varieties of sum of powers}\n\nConsider a hypersurface $F=V(f)\\subset\\mathbb{P}^N$ of degree $d$. \n\\begin{df}\nA subscheme $\\Gamma\\subset\\check\\mathbb{P}^N$ is said to be \\emph{apolar to $F$} if\n\\begin{equation*}\n\\mathcal{I}(\\Gamma)\\subset F^\\perp.\n\\end{equation*}\n\\end{df}\n\nIt holds the \\emph{Apolarity Lemma}:\n\n\\begin{lem}\\label{lem:apo}\nLet us consider the linear forms $\\ell_1,\\dotsc,\\ell_s\\in S_1$ and \nlet us denote by \n$L_1,\\dotsc,L_s\\in\\check\\mathbb{P}^N$ the corresponding points in the dual space. \nThen \n\\begin{equation*}\n\\Gamma\\ \\textup{is apolar to}\\ \nF \n= \nV(f),\n\\iff \\exists \\lambda_1,\\dotsc,\\lambda_s \\in {\\mathbb C}^*\\ \\textup{such that}\\ f=\\lambda_1\\ell_1^d+\\dotsc+\\lambda_s\\ell_s^d\n\\end{equation*}\nwhere $\\Gamma:=\\{L_1,\\dotsc,L_s\\}\\subset \\check\\mathbb{P}^N$. If $s$ is minimal, then it is called the \n\\emph{Waring number} of $F$. \n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Lemma 1.15 page 12]{ik}.\n\\end{proof}\n\nBy this lemma, it is natural to define the \\emph{variety of apolar subschemes}\n\\begin{equation*}\n\\VPS(F,s):=\\overline{\\{\\Gamma\\in\\Hilb_s(\\check\\mathbb{P}^N)\\mid \n\\mathcal{I}(\\Gamma)\\subset F^\\perp \\}},\n\\end{equation*}\nwhere $\\Hilb_s(\\check\\mathbb{P}^N)$ is the Hilbert scheme of length $s$ \nzero-dimensional subschemes in $\\check\\mathbb{P}^N$. \n\n\n\n\\subsection{Hypercubics and canonical sections}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)=\\check\\mathbb{P}^{g-1}$ be a canonical curve. \nIt is a well-known fact that $C$ is \\emph{arithmetically Gorenstein} \n\\textit{i.e. } its homogeneous coordinate ring, $\\mathcal{R}_C$, is Gorenstein. \nTherefore, if we take two general linear forms \n$\\eta_1,\\eta_2\\in ({\\mathcal{R}_C})_1=H^0(\\omega_C)$, then \n\\begin{align*}\nT:&= \\frac{\\mathcal{R}_C}{\\gen{\\eta_1,\\eta_2}}\\\\\n&= \\mathcal{S}ym\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*\n\\end{align*}\nis Artinian Gorenstein, and its values of the Hilbert function are \n$1, g-2, g-2, 1$. Therefore, the socledegree of $T$ is \n$3$, and by the Macaulay Lemma, this defines a hypercubic in \n$\\proj(S)=\\mathbb{P}\\left(\n\\frac{ H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)$, $S:=T^*$. \n\nThus, we have the rational map \n$\\alpha_C\\colon\\mathrm{Gr}(g-2,g)\\dashrightarrow H_{g-3,3}$ introduced in \\eqref{eq:ac}. \n\n\n\n\n\n\n\n\n\\subsubsection{Gonality}\n\n\nIn the following sections we will study the image of the map \n$\\alpha_C$.\nWe will show that this \nis related to study the gonality of $C$. \n\n\n\n\\section{Trigonal curves} \n\nIn this section we will prove Theorem A:\n\n\\begin{thm}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a canonical curve. \nThen $C$ is trigonal or isomorphic to a smooth plane quintic \nif and only if for general $\\eta_1,\\eta_2\\in H^0(\\omega_C)$, \nthe image of the map \n$f=\n\\alpha_C\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*$, \ndefined in \\eqref{eq:ac}, is a \\emph{Fermat cubic}, \n\\textit{i.e. } it is the sum of $g-2$ cubes. \n\\end{thm}\n\n\\begin{proof}\nAssume that $C$ is trigonal or isomorphic to a smooth plane quintic. \nThen $\\mathcal{I}(C)$ is not generated by \nquadrics by the Enriques-Petri Theorem (see for instance \\cite{ACGH}). \nIn particular---again by Enriques-Petri---the quadrics determine a surface $S$, \nwhich is a rational normal scroll \n(or the Veronese surface, in the case of the plane quintic). \nThen, $\\mathcal{I}(S)\\subset\\mathcal{I}(C)$ and $\\mathcal{I}(S)_2=\\mathcal{I}(C)_2$. \nThis implies that the ideal $\\mathcal{I}:=(\\mathcal{I}(S),\\eta_1,\\eta_2)$ gives a \nzero-dimensional scheme $\\Gamma$ \nof length the degree of $S$ and $\\mathcal{I}(\\Gamma)=\\mathcal{I}$ since $S$ is arithmetically Cohen-Macaulay. Since $S$ is a surface of \nminimal degree in $\\check \\mathbb{P}^{g-1}$, then $\\deg(S)=g-2$. \n\nBy hypothesis, $\\eta_1,\\eta_2$\nare general, then $\\mathcal{I}$ gives $g-2$ points in $\\check \\mathbb{P}^{g-3}$, and by Lemma~\\ref{lem:apo}, \nthese determine $g-2$ linear forms in $\\mathbb{P}^{g-3}$, \n$\\ell_1,\\dotsc,\\ell_{g-2}$ \nsuch that\n$f=\\lambda_1\\ell_1^3+\\dotsc+\\lambda_{g-2}\\ell_s^3$.\n\n\nConversely, if the image of $\\alpha_C$ is a Fermat cubic for \na particular choice of $\\eta_1,\\eta_2\\in H^0(\\omega_C)$, \n\\textit{i.e. } for \n\\begin{equation*}\n\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*\\in \\mathrm{Gr}(g-2,g),\n\\end{equation*} \nwe can fix coordinates\n$(\\partial_0,\\dotsc,\\partial_{g-3})$ on \n$\\check\\mathbb{P}^{g-3}=\\proj(T)\n=\\mathbb{P}\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*$ \nand so coordinates \n$(x_0,\\dotsc,x_{g-3})$ on $\\mathbb{P}^{g-3}=\\proj(S)\n=\\mathbb{P}\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)$; therefore let us \nsuppose that \n\\begin{equation*}\\label{eq:F}\n\\alpha_C\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*=\n[x_0^3+\\dotsb+x_{g-3}^3]. \n\\end{equation*}\nWe can also suppose that the coordinates on the projective space $\\mathbb{P}(H^0(\\omega_C)^*)$ are \n$(\\partial_0,\\dotsc,\\partial_{g-3},\\partial_{g-2},\\partial_{g-1})$, \\textit{i.e. } we can think of $\\eta_1$ and \n$\\eta_2$ as the hyperplanes $\\{\\partial_{g-2}=0\\}$ and $\\{\\partial_{g-1}=0\\}$, \nrespectively. \n\n\nLetting $f:=x_0^3+\\dotsb+x_{g-3}^3$, we only need to find $f^\\perp$. \nIt is easy to see that \n\\begin{equation*}\nf^\\perp=(\\partial_i\\partial_j, \\partial_i^3-\\partial_j^3),\n\\quad i,j\\in\\{0,\\dotsc,g-3\\},\\quad i\\neq j. \n\\end{equation*}\nThen, the quadrics of $\\mathcal{I}(C)$ are of the form \n\\begin{equation*}\nQ_{i,j}:=\\partial_i\\partial_j+\\partial_{g-2}L_{i,j}+\\partial_{g-1}M_{i,j}, \n\\end{equation*}\nwhere $L_{i,j}$ and $M_{i,j}$ are linear forms on $\\check\\mathbb{P}^{g-1}$. \nBy the Enriques-Petri Theorem, \n$(Q_{i,j})\\subsetneq \\mathcal{I}(C)$ if and \nonly if $C$ is trigonal or isomorphic to a smooth plane quintic. Now, \n$(\\partial_i\\partial_j)\\subsetneq f^\\perp$, since for example \n$\\partial_0^3-\\partial_1^3$ is not contained \nin the ideal $(\\partial_0\\partial_1)$,\nso $(Q_{i,j})\\subsetneq \\mathcal{I}(C)$: in fact, if it were $(Q_{i,j})= \\mathcal{I}(C)$ this would imply \n$(\\partial_i\\partial_j)= f^\\perp$. \n\n\nWe have just proven that, if for a particular choice of $\\eta_1,\\eta_2$ the \nimage of $\\alpha_C$ is a Fermat cubic, then $C$ is trigonal or isomorphic to \nthe plane quintic, while we have seen before that if $C$ is trigonal or isomorphic to \nthe plane quintic for general $\\eta_1,\\eta_2$ the image of \n$\\alpha_C$ is a Fermat cubic. Then, if $\\eta_1,\\eta_2$ are general, $f=\n\\alpha_C\\left(\\frac{H^0(\\omega_C)}{\\gen{\\eta_1,\\eta_2}}\\right)^*$, is a Fermat cubic. \n\\end{proof}\n\n\n\n\n\n\\section{The tetragonal case}\n\nIn order to analyse $F_{\\eta_{1},\\eta_{2}}$ when $C$ is an $n$-gonal curve, \nwe recall some basic well-known facts about rational \nnormal scrolls.\n\n\n\n\\subsection{Rational normal scrolls}\nBy definition, a \\emph{rational normal scroll} (RNS for short, in the following) \nof type $(a_1,\\dotsc,a_k)$, $S_{a_1,\\dotsc,a_k}$, is the \nimage of the $\\mathbb{P}^{k-1}$-bundle $\\mathbb{P}(E)=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(a_1)\\oplus\\dotsb\\oplus\\mathcal{O}_{\\mathbb{P}^1}(a_k))$, \n$\\pi\\colon\\mathbb{P}(E)\\to\\mathbb{P}^1$, via \nthe map $j$ given by $\\mathcal{O}_{\\mathbb{P}(E)}(1)$ in $\\mathbb{P}^N$, $N=\\sum a_i+k-1$. \nEquivalently, one takes $k$ disjoint \nprojective spaces of dimension $a_i$, $\\mathbb{P}^{a_i}$, and $k$ rational normal curves $C_i\\subset\\mathbb{P}^{a_i}$, \ntogether with isomorphisms $\\phi_i\\colon\\mathbb{P}^1\\to C_i$ (if $a_i\\neq 0$, constant maps otherwise); \nthen\n\\begin{equation*}\nS_{a_1,\\dotsc,a_k}=\\bigcup_{P\\in\\mathbb{P}^1}\\gen{\\phi_1(P),\\dotsc,\\phi_k(P)}.\n\\end{equation*}\nWe have also that\n\\begin{align*}\n\\deg (S_{a_1,\\dotsc,a_k}) &=\\sum a_i\\\\\n&=N-k+1,\n\\end{align*}\nand $\\dim (S_{a_1,\\dotsc,a_k})=k$. \nFrom the second description, \nit is an easy exercise to show that, if $P\\in C_i$, then the projection of $S_{a_1,\\dotsc,a_k}$ from \n$P$ is a rational normal scroll of type $(a_1,\\dotsc,a_{i-1},a_i-1,a_{i+1},\\dotsc,a_k)$, with \nthe convention $$(a_1,\\dotsc,a_{i-1},-1,a_{i+1},\\dotsc,a_k)=(a_1,\\dotsc,a_{i-1},a_{i+1},\\dotsc,a_k).$$\n\nWe note that $j\\colon \\mathbb{P}(E) \\to S_{a_1,\\dotsc,a_k} \\subset\\mathbb{P}^N$ is an isomorphism (and $S_{a_1,\\dotsc,a_k}$ \nis smooth) if (and only if) $a_i>0$, for all $i$. \n\nThe \\emph{Picard group} of $\\mathbb{P}(E)$ is generated by the \\emph{hyperplane class}, defined by $H:=[j^*\\mathcal{O}_{\\mathbb{P}^N}(1)]$, \nand by its \\emph{ruling}, \\textit{i.e. } $F:=[\\pi^*\\mathcal{O}_{\\mathbb{P}^1}(1)]$, and the intersection product is given by  \n\\begin{equation*}\nH^k=N-k+1, \\qquad H^{k-1}F=1, \\qquad F^2=0. \n\\end{equation*}\nFinally, we recall that the canonical class of $\\mathbb{P}(E)$ is \n\\begin{equation*}\nK_{\\mathbb{P}(E)}=-kH+(N-k-1)F. \n\\end{equation*}\n\n\nThe following well-known theorem (due to A. Maroni, \\cite{Ma}) relates the $n$-gonal curves with the RNS: \n\n\\begin{thm}\\label{thm:sac}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a canonical curve and $n$ is an integer $n\\ge 4$. Then $C$ \nhas a $g^1_n$ if and only \nif it is contained in a\nrational normal scroll of dimension $n-1$ (and so \nof degree $g-n+1$). Moreover, the $\\check\\mathbb{P}^{n-2}$'s which are the fibres of the \nscroll, cut on $C$ precisely the $g^1_n$. \n\\end{thm}\n\n\\begin{rmk}\nWe put $n\\ge 4$ only to avoid the case of the plane quintic, which has a \n$g^2_5$ instead of a $g^1_3$. Of course it has a $g^1_4$ and it is contained \nin a rational normal cubic threefold in $\\check\\mathbb{P}^5$. \n\\end{rmk}\n\n\n\\begin{proof}\nBy the geometric version of the Riemann-Roch Theorem, a divisor $D\\in g^1_n$ \ngenerates a $\\check\\mathbb{P}^{n-2}$ in $\\check\\mathbb{P}^{g-1}$. The union of these $\\check\\mathbb{P}^{n-2}$'s\ngenerates \na rational scroll $X$ of dimension $n-1$. Therefore it is a RNS or \na projection of a RNS. \nIt is not a projection since $C$ is projectively normal (and hence linearly normal). \n\n\nFor the vice versa, if $C$ is contained in the scroll $X$, then the $\\check\\mathbb{P}^{n-2}$'s of $X$ \ndetermine a linear system $\\abs{D}$ of dimension at least one; \nthen, again by the geometric version of the Riemann-Roch Theorem,\n$\\deg(D)\\le n$, and the theorem is proved. \n\n\\end{proof}\n\n\\begin{rmk}\nA classical result (due to B. Segre, see \\cite{BSeg}) assure us that the above $\\check\\mathbb{P}^{n-2}$'s in the proof of the \ntheorem \nare in general positions for the general curve, and therefore $X$ is smooth if the curve $C$ is general. \n\\end{rmk}\n\n\nTo ease exposition, we give the following: \n\n\\begin{df}\nWe say that a RNS $X$ as in Theorem~\\ref{thm:sac} is a \\emph{balanced scroll}, if \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(a_1)\\oplus\\dotsb\\oplus\\mathcal{O}_{\\mathbb{P}^1}(a_{n-1}))$ is \nsuch that \n$a=\\lfloor\\frac{g}{n-1}\\rfloor-1$, where $a:=\\min_{i\\in\\{1,\\dotsc,n-1\\}}\\{a_i\\}$. \n\\end{df}\n\nBy a moduli computation \nwe note that a general $n$-gonal curve is contained in a balanced scroll (see for example \\cite{DG}, \\cite{Cha}, or \\cite[Corollary 3.3]{GV}). \n\n\n\n\n\\subsection{Theorem B}\nWe analyse a canonical curve \n$C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ having a $g^1_4$. \nHere, the intersection of the scroll $X$, given by Theorem~\\ref{thm:sac}, \nwith the $\\check\\mathbb{P}^{g-3}$, is \nno longer given by a finite number of points, but it is a (rational normal) \ncurve. \nThen, we want to find a surface $S$ such that $C\\subset S\\subset X$, and if $\\eta_1$ and $\\eta_2$ are \ntwo $1$-forms on $C$, then, letting $\\Gamma:=V(\\mathcal{I}(S),\\eta_1,\\eta_2)$, it holds that \n$\\mathcal{I}(\\Gamma)$ is contained in $(\\mathcal{I}(C),\\eta_1,\\eta_2)=F^\\perp$. \n\n\\begin{lem}\\label{lem:acm}\nLet $\\phi_{\\abs{K_S+C}}\\colon S\\to \\mathbb{P}^{g-1}$ be a generically 1--1 map, where $S$ is a rational (smooth) surface and \n$C$ is a smooth curve of genus $g\\ge 4$. Let us call $Y(\\subset \\mathbb{P}^{g-1})$ the image of this map. \nMoreover, let $\\eta_1,\\eta_2\\in H^0(\\mathcal{O}_{\\mathbb{P}^{g-1}}(1))$ be general sections and $\\Gamma:=V(\\mathcal{I}(Y),\\eta_1,\\eta_2)$ and  \nfinally, we suppose that $(\\mathcal{I}(C),\\eta_1,\\eta_2)=F^\\perp$, where \n$F$ is a cubic polynomial in the dual coordinates. Then \n$\\mathcal{I}(\\Gamma)\\subset (\\mathcal{I}(C),\\eta_1,\\eta_2)$. \n\\end{lem}\n\n\n\\begin{proof}\n Since $\\mathcal{I}(\\Gamma)$ is generated in degree $2$ and $3$,\n by the hypothesis on $F$, it is sufficient to prove that \n$H^0(\\mathbb{P}^{g-1},\\mathcal{I}_{\\Gamma\/\\mathbb{P}^{g-1}}(j))\\subset (F^\\perp)_j$, $j=2,3$. \n\nLet $D_1, D_2\\in \\abs{K_S+C}$ be such that $D_i:=\\{\\phi^*(\\eta_i) =0)\\}$, $i=1,2$. \nBy the cohomology of the standard resolution of \n$\\mathcal{I}_{\\Gamma\/S}:=\\phi^*(\\mathcal{I}_{\\Gamma\/Y})$, we obtain that \n$H^0(S,\\mathcal{I}_{\\Gamma\/S}(2))=\\phi^*(\\eta_1) H^0(S,K_S+C)+ \\phi^*(\\eta_2) H^0(S,K_S+C)$ since $S$ is \nregular. \n\nSince $H^1(S,K_S+C)=0$, then $H^0(S,\\mathcal{I}_{\\Gamma\/S}(3))=\\phi^*(\\eta_1) H^0(S,2(K_S+C))+ \\phi^*(\\eta_2) H^0(S,2(K_S+C))$. \n\nNow, consider the standard exact sequence of ideal sheaves:\n\\begin{equation*}\n0\\to\\mathcal{I}_{Y\/\\mathbb{P}^{g-1}}(j)\\to \\mathcal{I}_{\\Gamma\/\\mathbb{P}^{g-1}}(j)\\to \\mathcal{I}_{\\Gamma\/Y}(j)\\to 0; \n\\end{equation*}\nLet $Q\\in H^0(\\mathbb{P}^{g-1},\\mathcal{I}_{\\Gamma\/\\mathbb{P}^{g-1}}(j))$, $j=2,3$. If $Q\\in H^0(\\mathbb{P}^{g-1},\\mathcal{I}_{Y\/\\mathbb{P}^{g-1}}(j))$, we are done; otherwise, we obtain \na $q\\in H^0(\\mathbb{P}^{g-1},\\mathcal{I}_{\\Gamma\/Y}(j))\\cong H^0(S,\\mathcal{I}_{\\Gamma\/S}(j))$, and we can conclude by what we have proven above. \n\n\n\\end{proof}\n\n\\begin{rmk}Notice that the conclusion of Lemma~\\ref{lem:acm} holds if the surface $Y$ \nis arithmetically Cohen-Macaulay. In the paper \\cite{IR}, they use \nthe fact that their surfaces are cones over arithmetically \nCohen-Macaulay curves, hence any general linear section of these \ncones is aCM.\n\\end{rmk}\n\n\n\nLet us now prove Theorem B:\n\n\\begin{thm}\\label{thm:tetra}\nIf a canonical curve $C\\subset\\check\\mathbb{P}^{g-1}$ has a $g^1_4$, then \n$F_{\\eta_{1},\\eta_{2}}$ is a sum of \ncubes of at most $\\lceil\\frac{3}{2}g-\\frac{7}{2}\\rceil$ \nlinear forms, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general $1$-forms. \n\\end{thm}\n\n\n\\begin{proof}\nBy Theorem~\\ref{thm:sac}, $C\\subset X$, where $X\\subset\\check\\mathbb{P}^{g-1}$ is a rational normal smooth \nthreefold of degree $g-3$.  \n In the Chow ring of $X$ the curve $C$ is \n\\begin{equation*}\n[C]\\sim 4 H^2+2(5-g)HF \n\\end{equation*}\n(see \\cite[ Theorem 3.1]{GV}). By \\cite[Corollary~(4.4)]{Sch} \n(see also \\S 6 there), $C$ is \nthe complete intersection of two irreducible surfaces of type\n\\begin{equation*}\n[Y_1]\\sim 2H-b_1 F,\\ [Y_2]\\sim 2H-b_2 F, \\qquad \\textup{with}\\ b_1+b_2=g-5.  \n\\end{equation*}\nIt follows: \n\\begin{align}\n\\deg Y_i&=(2H-b_i F)\\cdot H^2\\\\\n&=2g-6-b_i,\\label{bi}. \n\\end{align}\nTherefore, if $b_1\\ge b_2$, then $\\deg(Y_2)\\ge\\deg(Y_1)$\nand, since $b_2\\ge \\lfloor\\frac{g-5}{2}\\rfloor$,\nwe deduce \n\\begin{align*}\n\\deg(Y_2)&\\le 2g-6-\\left \\lfloor \\frac{g-5}{2}\\right\\rfloor\\\\\n&= \\left\\lceil \\frac{3g-7}{2}\\right\\rceil.\n\\end{align*}\n\nWe note that $Y_{i}$ restricts to a quadric \non each fibre $\\Pi$ of the scroll $X$. \nThe intersection $Y_{1}\\cap Y_{2}\\cap\\Pi$ gives the four points \nof $C$ which are a divisor of the $g^1_4$. Again by \\cite[\\S 6]{Sch} \nat least $Y_{2}\\cap \\Pi$ is generically irreducible.  \nMoreover, $Y_2$ satisfies the hypothesis of \nLemma~\\ref{lem:acm} (see for example \\S 5 of \\cite{Sch}). Then, the \ntheorem is proved. \n\\end{proof}\n\n\n\\begin{rmk}\\label{rmk:sch}\nA more precise analysis of the proof of the preceding Theorem can be done if we distinguish the cases in which $g$ is \nodd or even: if $g=2k+1$ is odd, it is immediate that the degree of both $Y_1$ and $Y_2$ is bounded by \n$2g-6-(k-2)=3k-2=\\frac{3g-7}{2}$; instead, if $g=2k$ is even, we deduce that the degree of $Y_1$ is bounded by \n$2g-6-(k-2)=3k-4=\\frac{3}{2}g-4$, which is strictly less than $2g-6-(k-3)=3k-3=\\frac{3}{2}g-3$, the bound for $Y_2$. \nThe problem is that in this situation $Y_1$ could not be rational: in this case the $g^1_4$ is composed by an elliptic or \nhyperelliptic involution: see \\cite[\\S 6.5]{Sch}. \n\\end{rmk}\n\nThe surfaces $Y_i$ of the proof of the above theorem has been studied before (independently to us) \nand more deeply in \n\\cite{bs} to give a stratification of the moduli space of the tetragonal curves. \n\n\n\n\n\\subsection{Low genus cases}\n\nWe recall that every curve of genus $g\\le 5$ has a $g^1_4$ (see  \\cite[IV.5.5.1]{H}); \nso let us start to analyse the cases of low genus:\n\n\\subsubsection{g=6}\nIn this case, $C\\subset\\check\\mathbb{P}^5$ and $\\deg(C)=10$. If \n$C$ is tetragonal, it is contained in a smooth cubic threefold and we \nfind a sharp estimate: \nTheorem~\\ref{thm:tetra} says that there is a surface of \ndegree at most $6$ containing $C$ and contained in $X$. \nIndeed, from Remark~\\ref{rmk:sch} we can see that $C$ \nis contained in a quartic surface; but this is a surface of \nminimal degree, and it is either a RNS, in which case \n$C$ is trigonal, or it is a Veronese surface, in \nwhich case the $g^1_4$ is induced by the $g^2_5$\nwhich corresponds to the conics of the Veronese surface, in accordance \nwith Enriques-Petri Theorem. \n\n\n\\subsubsection{g=7}\\label{g7}\nIn this case, $C\\subset\\check\\mathbb{P}^6$ and $\\deg(C)=12$; if $C$ is tetragonal it is contained in a smooth quartic threefold.\nTheorem~\\ref{thm:tetra} says that there is a surface of degree at most $7$ containing $C$ and contained in $X$. \nFrom---for example---\\cite[\\S 6]{Mu}, \nwe see that $7$ is the correct estimate for a general tetragonal \ncurve. But it is also shown, again in \\cite[\\S 6]{Mu}, \nthat for the special ones, \\textit{i.e. } if $C$ possesses a $g^2_6$, \nthen $C$ is contained in a \n(possibly singular) \\emph{sextic del Pezzo} surface. \nWe give here an alternative proof of the existence of this surface. \n\nFirst, we note that the quartic scroll $X$  which contains $C$ is of type \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(2))$.\nLet us consider our surface $S$ as in the proof of \nTheorem~\\ref{thm:tetra}, \\textit{i.e. } $[S]=2H-bF$ as a divisor on $X$. \n\n\nThen,  \nwe project the $C$ to $\\check\\mathbb{P}^2$ by choosing the centre of projection \n$\\pi\\colon\\check\\mathbb{P}^6\\dashrightarrow\\check\\mathbb{P}^2$ a $\\check\\mathbb{P}^3$\ngenerated by a general plane of the scroll and a general point $P$ of the curve. \nSo, \nwe obtain a singular \nplane curve $Z$ of degree $7$. \nLet $V_i$'s be the singular points of $Z$. We can suppose that the \npoints are of simple multiplicity $m_i$. \n\n\nLet us now \\emph{suppose} that the surface $S$ containing $C$ is  \nthe blowing-up of $\\check\\mathbb{P}^2$ in the points $V_i$'s: $\\pi_{\\{V_i\\}}\\colon S \\to \\check\\mathbb{P}^2$. \nThis means that the projection \n\\begin{equation*}\n\\pi\\mid_{S}\\colon S\\dashrightarrow \\check\\mathbb{P}^2\n\\end{equation*}\nis generically 1--1. \n\nLet us denote by $H:=\\pi_{\\{V_i\\}}^*\\mathcal{O}_{\\check\\mathbb{P}^2}(1)$ the hyperplane section of $S$ \nand by $E_i$'s the $(-1)$-curves on $S$ which correspond to the $V_i$'s. \nThe complete linear system $\\abs{4H+\\sum_i(1-m_i)E_i}$ gives a generically 1--1 map \n$\\phi\\colon\\ S\\to\\check\\mathbb{P}^6$ such that $C=\\phi(\\pi_{\\{V_i\\}}^{-1}(Z))$. \nIn fact, we can write, by adjunction\n\\begin{align*}\nC&\\in\\abs{7H-\\sum_i m_i E_i}\\\\\nK_S&=-3H+\\sum_i E_i\\\\\n\\omega_C&=(4H+\\sum_i(1-m_i)E_i)_{\\mid_C}.\n\\end{align*}\nThen, the adjunction formula of $C$ on $S$ yields\n\\begin{equation*}\n(7H-\\sum_i m_iE_i)\\cdot (4H+\\sum_i(1-m_i)E_i)=12\n\\end{equation*}\nwhich means, \n\\begin{equation}\\label{adj4}\n\\sum_i m_i(m_i-1)=16. \n\\end{equation}\n\nIf we take a general plane $\\Pi_t$, $t\\in\\mathbb{P}^1$ of the ruling of the scroll \n$\\rho\\colon X\\to\\mathbb{P}^1$, then $\\pi\\mid_{\\Pi_t}\\colon\\Pi_t\\to\\check\\mathbb{P}^2$ \nis a birational isomorphism. Now, through the \nfour points of $C\\cap \\Pi_t$ there passes a pencil of conics $\\Lambda_{\\Pi_t}$. \nDenote by $Q_{\\Pi_t}$ the generic conic of $\\Lambda_{\\Pi_t}$. \nLet us denote by \n$Q_{\\Pi_t}':=\\pi(Q_{\\Pi_t})$. We want to show that there exists a pencil \n$\\Lambda:=\\{Q_t'\\}_{t\\in\\mathbb{P}^1}$ in $\\check\\mathbb{P}^2$ such that $Q_t'$ comes from \na general specialisation of $Q_{\\Pi_t}$. In fact, if we consider the projection of $X$ from $P$, which we can \nsuppose, by its generality, it is a general point of $C_2$, \\textit{i.e. } \na unisecant conic of the scroll, \nwe obtain that $X$ is mapped to a balanced scroll $\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(1))$, and \none unisecant line gives in fact a fifth point in each plane which determines a conic in each plane of the ruling and \nthen a pencil of conics when we project further to $\\check\\mathbb{P}^2$. \n\nSo, $\\Lambda$ cuts the $g^1_4$ on $Z$. Let $A_1,\\dotsc,A_4$ be the base points of the\npencil; by construction, we have\n\\begin{equation*}\nQ'_t\\mid_Z=\\sum_{i=1}^4 (n_iA_i+P'_{it}),\n\\end{equation*}\nwhere $\\{P_{1t},\\dotsb,P_{4t}\\}=\\Pi_t\\cap C$ and $P'_{it}=\\pi(P_{it})$. \nWithout loss of generality, by Bez\\'out, we can assume $V_i=A_i$, and by the generality of $Q'_t$, $n_i=m_i$. \nBut then\n\\begin{align*}\n\\deg Q'_t\\mid_Z &=2\\deg Z\\\\\n&=14\\\\\n&=4+\\sum_{i=1}^4 n_i, \n\\end{align*}\nwhich means \n\\begin{equation}\\label{qat}\n\\sum_{i=1}^4 n_i=10,\n\\end{equation}\nfrom which we deduce \n\\begin{equation*}\n\\sum_{i=1}^4 n_i^2\\ge 25,\n\\end{equation*}\nwhich implies \n\\begin{equation*}\n\\sum_{i=1}^4 n_i^2-n_i\\ge 15. \n\\end{equation*}\nNow, $\\sum_{i=1}^rm_i^2-m_i=16>15$, therefore $r=4$, and from \\eqref{qat}, we deduce \n\\begin{equation}\\label{3k-34}\n\\sum_{i=1}^4 (m_i-1)=6. \n\\end{equation}\nFrom this, \n\\begin{align*}\n\\deg(S)&=(4H+\\sum_i(1-m_i)E_i)^2\\\\\n&= 16-\\sum_i(m_i-1)^2,\n\\end{align*}\nbut then, by \\eqref{adj4} and \\eqref{3k-34}\n\\begin{align*}\n\\sum_i(m_i-1)^2&=\\sum_i (m_i(m_i-1)-(m_i-1))\\\\\n&=16-6\\\\\n&=10\n\\end{align*}\nwe obtain that \n\\begin{equation*}\n\\deg(S)=6. \n\\end{equation*}\nTherefore having assumed that $S$ is the blowing-up of \n$\\check\\mathbb{P}^{2}$, we have that $\\deg(S)=6$. In particular \n$[S]=2H-2F$ as a divisor on $X$, by \\eqref{bi}. \n\nBy the above proof, if the singular points \n$V_1,\\dotsc,V_4$ are distinct, then their multiplicities must be \n$m_1=m_2=3$ and $m_3=m_4=2$, and therefore we have three $g^1_4$ on $C$: \nthe first one given by the pencil \nof conics $\\Lambda$, \nand the other two correspond \nto the pencils of lines through $V_1$ and $V_2$. \n\nNow we show the main result of this subsection\nthat the canonical tetragonal curves \nof genus $7$ with a $g^2_6$ can be realised exactly\nas the above blowing-up of a curve $Z$. \n\nLet us see this. \nFirst of all, we recall that, in \ngeneral, the $n$-gonal locus is irreducible in the (coarse) \nmoduli space of algebraic curves \nof genus $g$, $\\mathcal{M}_g$; this follows for example from \\cite{F}. \n\nIn the case $g=7$, \nmuch more can be said: see for example \\cite[Table 1]{Mu}, \nwhere it is explained the stratification of this \nmoduli space. In particular, inside $\\mathcal{M}_7$, \nwhich has dimension $18$, there is \nthe codimension one \\emph{tetragonal locus},  $\\mathcal{T}_7$. \nThe general curve in it has only one $g^1_4$. \nInside the tetragonal locus, there is \nthe locus of curves possessing a $g^2_6$, $\\mathcal{G}^2_6$. \n$\\mathcal{G}^2_6$ has codimension one in $\\mathcal{T}_7$ so, \n$\\dim\\mathcal{G}^2_6=16$. \nThe general curve in $\\mathcal{G}^2_6$ has exactly two $g^2_6$. \nMoreover it can have one, two or three $g^1_4$. \n\n\nThis case has been analysed \ndeeply in \\cite{Cas}. First of all, one can observe that \n$\\mathcal{T}_7=\\mathcal{T}_7^{(1)}\\cup\\mathcal{T}_7^{(2)}\\cup\\mathcal{T}_7^{(3)}\\cup\\mathcal{M}_7^{be}$, where \n$\\mathcal{T}_7^{(i)}$  is the locus of the curves  carrying exactly $i$ linear series $g^1_4$'s, and \n$\\mathcal{M}_7^{be}$ is the bielliptic locus. \nOne of the main results of  \\cite{Cas} is that the loci  $\\mathcal{T}_7^{(2)}$ and $\\mathcal{T}_7^{(3)}$ are \n(irreducible) rational subvarieties of dimensions $15$ and $16$, respectively, of $\\mathcal{M}_7$. \nIn particular, the general element of $\\mathcal{G}^2_6$ \nis in $\\mathcal{T}_7^{(3)}$, \\textit{i.e. } it has three $g^1_4$'s. \n\nPrecisely, in \\cite[\\S 2]{Cas} it is shown why $\\mathcal{T}_7^{(3)}$ is rational of dimension $16$, with the \nfollowing geometric \nargument. Let $C$ be our general tetragonal curve of genus $g = 7$ carrying three $g^1_4$'s. \nThen, $C$ has a sextic plane model $\\tilde C \\subset\\check\\mathbb{P}^3$ with three non\u2013collinear nodes, which can be assumed \nto be $P_1=(1:0:0)$, $P_2=(0:1:0)$ and $P_3=(0:0:1)$. Clearly, such a model depends on the choice of the $g^2_6$.\nSet $X := \\{P_{1} , P_{2} , P_{3} \\}$. If $\\varphi\\colon C \\to C'$ is an isomorphism, \nthen, in particular, it sends the $g^r_n$'s on $C$ into $g^r_n$'s on $C'$, thus it induces\na birational automorphism $\\phi\\in \\bir_X (\\check\\mathbb{P}^2 )$, defined on the whole of $\\check\\mathbb{P}^2 \\setminus X$, leaving $X$ fixed \nand sending $\\tilde C$ to $\\tilde C'$. \n$\\bir_X(\\check\\mathbb{P}^2)$ is generated by the torus of diagonal matrices $\\mathbb{P} T\\subset \\mathbb{P} \\Gl(3)$, with $\\dim(\\mathbb{P} T)=2$, \nby the standard quadratic Cremona transformation $\\mu(x:y:z) = (yz:xz:xy)$, which \npermutes the two $g^2_6$'s on $C$ and by the group of permutations of the $P_i$'s. \n\nThe subspace $W\\subset {\\mathbb C}[x, y, z]$ of forms of degree $6$ representing\nplane curves having singularities at the points $P_i$'s has dimension $19$. Moreover, \nthe action of $\\bir_X(\\check\\mathbb{P}^2 )$ on $\\check\\mathbb{P}^2$ induces a linear action on $\\abs{W}$, \\textit{i.e. } \n$\\bir_X (\\check\\mathbb{P}^2)$ can be realised as a subgroup of $\\mathbb{P} \\Gl(W )$. \n\nConsider then the natural map  $p\\colon \\Gl(W ) \\to \\mathbb{P} \\Gl(W )$ and  let us define   \n$G := p^{-1} (\\bir_X (\\check\\mathbb{P}^2 )) \\subset\\Gl(W )$. Then, they notice that there exists \na dominant rational map $W\\to \\mathcal T_7^{(3)}$, whose\nfibres are the $G$-orbits of $W$, and therefore \n$\\mathcal T_7^{(3)}$ is irreducible of dimension $16$. \n\nLet us show now that there is a map which associates our $7$-tic $Z\\subset\\check\\mathbb{P}^2$ to one of their sextics $\\tilde C$. \nWe can suppose that the $P_1=(1:0:0)$ and $P_2=(0:1:0)$ are the triple points of $Z$, and $P_3=(0:0:1)$ and $P_4=(1:1:1)$ \nare its nodes. The two $g^2_6$ on $Z$ are given by the conics passing through $P_3$ and $P_4$, and \nthrough one of the two double points, \\textit{i.e. } $\\abs{D_i}:=\\abs{2H-P_1-P_2-P_i}$, $i=3,4$. Therefore, a map of the type \n\\begin{equation*}\n\\phi_{D_4}\\colon Z\\dashrightarrow \\tilde C\n\\end{equation*}\nis what we were looking for. In fact, $\\tilde C:=\\overline{\\phi_{D_4}(D_4)}$ is a sextic which has $P_1$, $P_2$ and \n$P_3$ as its singular points, and they are nodes. \n\nNow, it is not difficult to show that we can come back \nfrom $\\tilde C$ to one of our $Z$. \nThen even our construction \ngives all the curves of genus $7$ with a $g^2_6$.\nInstead, we give a computation with the moduli (and this is sufficient, \nsince the moduli spaces we are considering are all irreducible). \nAs above,  if $\\varphi\\colon C \\to C'$ ($C$ and $C'$ canonical curves, \nnormalisation of the two septics $Z$ and $Z'$) \nis an isomorphism, then in particular, it sends the $g^r_n$'s on $C$ \ninto $g^r_n$'s on $C'$, and induces \na birational automorphism $\\phi\\in \\bir_Y (\\check\\mathbb{P}^2 )$ \ndefined in $\\check\\mathbb{P}^2 \\setminus Y$, where $Y:=\\{P_1 , P_2 , P_3, P_4 \\}$. \n\nNow, the main difference with the case of the sextics is \nthat for $Z$ and $Z'$ the three $g^1_4$ are given by the lines \nthrough $P_1$ and $P_2$, and the conics through $Y$, \ntherefore in $\\bir_Y (\\check\\mathbb{P}^2 )$ we have also the Cremona \ntransformations of the plane which send the lines \nthrough $P_1$ (or $P_2$) to the conics through $Y$. \nThese transformations form a group of dimension one, and  $\\bir_Y (\\check\\mathbb{P}^2 )$ is generated by these transformations plus the \nmaps which change $P_1$ with $P_2$ and $P_3$ with $P_4$. \n\nNow, the subspace $W'\\subset {\\mathbb C}[x, y, z]$ of forms of degree $7$ \nrepresenting\nplane curves having singularities at the points $P_i$'s has dimension \n$9\\cdot 4-2\\cdot(3+6)=18$; therefore the family of \ncurves we have found has dimension $16$ in $\\mathcal T_7^{(3)}$, and \ntherefore it coincides with the whole \n$\\mathcal T_7^{(3)}$. \n\nSo, we can summarise what we have proven in the following:\n\n\\begin{prop}\\label{prop:7}\nThe general element of $\\mathcal T_7^{(3)}$ can be obtained from a (tetragonal)\ncurve \n$C\\subset S_{1,1,2} \\subset \\check\\mathbb{P}^6$ contained in  \na sextic rational surface $S$ of type $[S]=2H-2F$ in the rational normal scroll  \n$S_{1,1,2}$. \n\nThe projection $\\pi\\colon\\check\\mathbb{P}^6\\dashrightarrow\\check\\mathbb{P}^2$\nfrom the $\\check\\mathbb{P}^3$ generated by a general plane of  $S_{1,1,2}$ \nand a general point of $C$, \nrestricted to $S$, is generically 1--1 (\\textit{i.e. } $S$ is the blowing-up of this \n$\\check\\mathbb{P}^2$). \n\nThe curve $Z:=\\pi(C)\\subset\\check\\mathbb{P}^2$ has degree $7$ and has, as singular locus, four points in general position, two of them\nare nodes and two are triple points. \n\nMoreover, for the general tetragonal canonical curve of genus $7$, \nthe corresponding cubic $F_{\\eta_1,\\eta_2}$ is the sum of \nexactly $7$ cubes, while \nif a tetragonal canonical curve of genus $7$ is a \ngeneral element of $\\mathcal T_7^{(3)}$ (\\textit{i.e. } it has two $g^2_6$'s), \nthen  $F_{\\eta_1,\\eta_2}$ is the sum of exactly $6$ cubes. \n\\end{prop}\n\n\\begin{proof}\nIt remains to prove that $F_{\\eta_1,\\eta_2}$ is the sum of exactly $6$ cubes. \nWe refer to \\cite{Fu} for general facts about low degree varieties.  \nSince $C$ is neither trigonal nor degenerate, \nthe degree of such a surface $S$ cannot be $\\le 5$, \nand $S$ cannot have sectional geometric \ngenus zero (\\textit{i.e. } it cannot be a projection of a rational normal surface).  \n\nIf $C$ is general, it cannot be contained in a surface of \ndegree $6$, since this cannot have sectional geometric genus one; \nin fact, in this case, $S$ is either a Del Pezzo surface, and \nthis would imply that there is a $g^2_6$ on $C$, cut out by  \nrational normal cubic curves of $S$, \nor a cone over an elliptic curve, and therefore projecting \nfrom the vertex of the cone, we would see that $C$ is bielliptic.  \n\\end{proof}\n\n\n\n\\subsection{Special tetragonal curves}\\label{ssec:stc} \n\nIn this subsection we want to generalise \nthe construction obtained in Subsection~\\ref{g7}. \n\n\nIn the following Propositions~\\ref{4gon}, \\ref{4gon-1}, and \\ref{4gon-2},\nthe surfaces we consider are as in Lemma~\\ref{lem:acm}, so we can \napply them to the Waring problem. \n\n\n\nWe study first the case with $g=3k$, since we can find explicitly a rational smooth \nsurface $S$ of degree $\\frac{4}{3}g-3$, such that $C\\subset S\\subset X$: \n\n\\begin{prop}\\label{4gon}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a tetragonal canonical curve\nof genus $g=3k$, where $k\\geq 2$, contained in a rational surface $S$ of type $[S]=2H-bF$ in a balanced scroll \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1))$. \n\nLet us suppose that the projection $\\pi\\colon\\check\\mathbb{P}^{g-1}\\dashrightarrow\\check\\mathbb{P}^2$\nfrom $k-1$ general planes of the scroll restricted to $S$ is generically 1--1 (\\textit{i.e. } $S$ is the blowing-up of this \n$\\check\\mathbb{P}^2$). \nThen,  \n\\begin{equation*}\n\\deg(S)=\\frac{4}{3}g-3,  \n\\end{equation*}\nor, equivalently, $b=\\frac{2}{3}g-3$. \n\nIn particular, $F_{\\eta_1,\\eta_2}$ is a sum of \ncubes of at most  $\\frac{4}{3}g-3$ \nlinear forms, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general. \n\\end{prop}\n\n\n\\begin{proof}\n\n\nThe proof follows the idea behind our construction of \n$\\mathcal{T}^{(3)}_{7}$. The image of $C$ under $\\pi$ is \na singular \nplane curve $Z$ of degree $2k+2$. \nLet $V_i$'s be the singular points of $Z$. We can suppose that the \npoints are of simple multiplicity $m_i$. \n\n\nBy the hypothesis, we can think of the surface $S$ containing $C$ as \nthe blowing-up of $\\check\\mathbb{P}^2$ in the points $V_i$'s: $\\pi_{\\{V_i\\}}\\colon S \\to \\check\\mathbb{P}^2$.\nLet us denote by $H:=\\pi_{\\{V_i\\}}^*\\mathcal{O}_{\\check\\mathbb{P}^2}(1)$ the hyperplane section of $S$ \nand by $E_i$'s the $(-1)$-curves on $S$ which correspond to the $V_i$'s. \nThe complete linear system $\\abs{(2k-1)H+\\sum_i(1-m_i)E_i}$ gives a generically 1--1 map  \n$\\phi\\colon\\ S\\to\\check\\mathbb{P}^{g-1}$ such that $C=\\phi(\\pi_{\\{V_i\\}}^{-1}(Z))$. \nIn fact, we can write, by adjunction\n\\begin{align*}\nC&\\in\\abs{(2k+2)H-\\sum_i m_i E_i}\\\\\nK_S&=-3H+\\sum_i E_i\\\\\n\\omega_C&=((2k-1)H+\\sum_i(1-m_i)E_i)_{\\mid_C}.\n\\end{align*}\nThen, the adjunction formula of $C$ on $S$ yields\n\\begin{equation*}\n((2k+2)H-\\sum_i m_iE_i)\\cdot ((2k-1)H+\\sum_i(1-m_i)E_i)=6k-2\n\\end{equation*}\nwhich means, \n\\begin{equation}\\label{adj}\n\\sum_i m_i(m_i-1)=4k(k-1). \n\\end{equation}\n\nIf we take a general plane $\\Pi_t$, $t\\in\\mathbb{P}^1$ of the ruling of the scroll $\\rho\\colon X\\to\\mathbb{P}^1$, then $\\pi\\mid_{\\Pi_t}\\colon\\Pi_t\\to\\check\\mathbb{P}^2$ \nis an isomorphism. Now, through the \nfour points of $C\\cap \\Pi_t$ there passes a pencil of conics $\\Lambda_{\\Pi_t}$. Denote by $Q_{\\Pi_t}$ the generic conic of $\\Lambda_{\\Pi_t}$. \nLet us denote by \n$Q_{\\Pi_t}':=\\pi(Q_{\\Pi_t})$. We want to show that there exists a pencil $\\Lambda:=\\{Q_t'\\}_{t\\in\\mathbb{P}^1}$ in $\\check\\mathbb{P}^2$ such that $Q_t'$ comes from \na general specialisation of $Q_{\\Pi_t}$. In fact, if we consider the projection of $X$ from $k-2$ planes instead of \n$k-1$, we obtain that $X$ is mapped to a balanced scroll $\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(1))$, and \none unisecant line gives in fact a fifth point in each plane which determines a conic in each plane of the ruling and \nthen a pencil of conics when we project further to $\\check\\mathbb{P}^2$. \n\nSo, $\\Lambda$ cuts the $g^1_4$ on $Z$. Let $A_1,\\dotsc,A_4$ be the base points of the\npencil; by construction, we have\n\\begin{equation*}\nQ'_t\\mid_Z=\\sum_{i=1}^4 (n_iA_i+P'_{it}),\n\\end{equation*}\nwhere $\\{P_{1t},\\dotsb,P_{4t}\\}=\\Pi_t\\cap C$ and $P'_{it}=\\pi(P_{it})$. \nWithout loss of generality, by Bez\\'out, we can assume $V_i=A_i$, and by the generality of $Q'_t$, $n_i=m_i$. \nBut then\n\\begin{align*}\n\\deg Q'_t\\mid_Z &=2\\deg Z\\\\\n&=4k+4\\\\\n&=4+\\sum_{i=1}^4 n_i, \n\\end{align*}\nwhich means \n\\begin{equation}\\label{equat}\n\\sum_{i=1}^4 n_i=4k,\n\\end{equation}\nfrom which we deduce \n\\begin{equation}\\label{ineq}\n\\sum_{i=1}^4 n_i^2\\ge 4k^2,\n\\end{equation}\nand the equality holds iff $n_i=k$, $\\forall i$. \nThen \n\\begin{equation*}\\label{eq:ni}\n\\sum_{i=1}^4 n_i(n_i-1)\\ge 4k(k-1),\n\\end{equation*}\nwhich implies, by Equation \\eqref{adj}, $m_i=n_i=k$ for $1\\le i\\le 4$ and $m_j=0$ for $j\\ge 5$. \n\n\nThen, \n\\begin{equation}\\label{3k-3}\n\\sum_i (m_i-1)=4k-4. \n\\end{equation}\n\nNow, \n\\begin{align*}\n\\deg(S)&=((2k-1)H+\\sum_i(1-m_i)E_i)^2\\\\\n&= (2k-1)^2-\\sum_i(m_i-1)^2,\n\\end{align*}\nbut then, by \\eqref{adj} and \\eqref{3k-3}\n\\begin{align*}\n\\sum_i(m_i-1)^2&=\\sum_i (m_i(m_i-1)-(m_i-1))\\\\\n&=4k(k-1)-(4k-4)\\\\\n&=4k^2-8k+4\n\\end{align*}\nwe obtain that \n\\begin{align*}\n\\deg(S)&=4k-3\\\\\n&=\\frac{4}{3}g-3. \n\\end{align*}\n\\end{proof}\n\n\\begin{rmk}\nWe can show that the canonical curves and surfaces of the preceding proposition actually exist, in the following way. \nFrom the proof of the proposition, and with the same notation, we deduce that $S$ is the image of \nthe blowing-up of $\\check\\mathbb{P}^2$ in the points $P_1,\\dotsc,P_4$ by \nthe linear system $\\abs{(2k-1)H-(k-1)\\sum_{i=1}^4P_i}$. \nNow, it is immediate to see that the dimension of $\\abs{(2k-1)H-(k-1)\\sum_{i=1}^4P_i}$ is at least $3k+1$,\nand therefore, \nin order to show that $S$ exists, it is sufficient to show that this linear system  \nis ample. This fact can be obtained by the Nakai-Moishezon Criterion, \\cite[Theorem V.1.10]{H}: in fact, \nfirst of all, \n\\begin{equation*}\n((2k-1)H-(k-1)\\sum_{i=1}^4P_i)^2= 4k-3>0. \n\\end{equation*}\nThen, if $D$ is an irreducible curve in the blowing up of  $\\check\\mathbb{P}^2$ in the points $P_1,\\dotsc,P_4$, \nwe can think of it as $D\\in\\abs{aH-\\sum_{i=1}^4b_iP_i}$, \nwith $a>0$, and $b_i\\ge 0$, $\\forall i$; \nif, by contradiction, we suppose that \n$L\\cdot D \\le 0$, with $L\\in\\abs{(2k-1)H-(k-1)\\sum_{i=1}^4P_i}$, then, we deduce that \n\\begin{equation*}\n(2k-1)a-k\\sum_{i=1}^4b_i\\le 0, \n\\end{equation*}\nwhich means\n\\begin{align}\n\\sum_{i=1}^4b_i &\\ge \\frac{(2k-2)a+a}{k-1}\\\\\n&= 2a+\\frac{a}{k-1}\\\\\n&> 2a. \\label{bii}\n\\end{align}\nFrom this, since we can write $\\sum_{i=1}^4b_i>4\\frac{a}{2}$, we obtain \n\\begin{align}\n\\sum_{i=1}^4b_i^2&>4\\frac{a^2}{4}\\\\\n&=a^2.\\label{bi2}\n\\end{align}\nNow, since $D$ is irreducible, we infer, by the Clebsch formula,  \n\\begin{equation}\\label{eq:con}\n\\binom{a-1}{2}-\\sum_{i=1}^4\\frac{b_i(b_i+1)}{2}\\ge 0,\n\\end{equation}\nwhere $p_a(D)=\\binom{a-1}{2}$ is the arithmetic genus of $D$. \nBut, by \\eqref{bii} and \\eqref{bi2}, \n\\begin{align*}\n\\binom{a-1}{2}-\\sum_{i=1}^4\\frac{b_i(b_i+1)}{2}&<\\frac{a^2-3a+2}{2}-\\frac{a^2}{2}-a\\\\\n&=-\\frac{5}{2}a+1<0,\n\\end{align*}\nwhich contradicts \\eqref{eq:con}. \n\nAnalogously, we can deduce that there exists canonical curves $C$ contained in $S$. In fact $C$, in $S$, is \n\\begin{equation*}\nC\\in\\abs{(2k+2)H-k\\sum_{i=1}^4P_i}. \n\\end{equation*}\nNow, it is immediate to see that the dimension of $\\abs{(2k+2)H-k\\sum_{i=1}^4P_i}$ is at least $5k+5$, and therefore, \nin order to prove our claim, it is sufficient to show that the general element in this linear system  \nis irreducible. This fact can be obtained again by the Nakai-Moishezon Criterion: in fact, \nfirst of all, \n\\begin{align*}\nC^2&=((2k+2)H-k\\sum_{i=1}^4P_i)^2\\\\\n&= 8k+4>0. \n\\end{align*}\nThen, if $D\\in\\abs{aH-\\sum_{i=1}^4b_iP_i}$ is an irreducible curve in $S$, as above, \nwe deduce that \n\\begin{equation*}\n(2k+2)a-k\\sum_{i=1}^4b_i\\le 0, \n\\end{equation*}\nwhich means\n\\begin{align*}\n\\sum_{i=1}^4b_i &\\ge \\frac{(2k+2)a}{k}\\\\\n&= 2a+\\frac{2a}{k}\\\\\n&> 2a, \n\\end{align*}\nand we can conclude as above. \n\nClearly, the same kind of results (with the same proofs) \ncan be obtained in the cases of the following Propositions~\\ref{4gon-1} and \\ref{4gon-2}. \n\\end{rmk}\n\nWe then pass to case $g=3k-1$: \n\n\\begin{prop}\\label{4gon-1}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a tetragonal canonical curve\nof genus $g=3k-1$, where $k\\geq 2$ contained in a rational surface $S$ of type $[S]=2H-bF$ in a balanced scroll \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(k-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1))$. \n\nLet us suppose that the projection $\\pi\\colon\\check\\mathbb{P}^{g-1}\\dashrightarrow\\check\\mathbb{P}^2$\nfrom the linear space generated by $2$ general points of $C$ and $k-2$ general planes of the scroll, \nrestricted to $S$, is generically 1--1 (\\textit{i.e. } $S$ is the blowing-up of this \n$\\check\\mathbb{P}^2$). \n\nThen,  \n$\\deg(S)\\le \\frac{4}{3}g-\\frac{5}{3}$ (or $b\\ge \\frac{2}{3}g-\\frac{13}{3}$). \n\nIn particular, $F_{\\eta_1,\\eta_2}$ is a sum of \ncubes of at most  $\\frac{4}{3}g-\\frac{5}{3}$ \nlinear forms, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general. \n\\end{prop}\n\n\n\\begin{proof}\nWe can follow the proof of the preceding result: \nwe have to consider  \na plane curve $Z$ of degree $2k+2$. \nLet $V_i$'s be the singular points of $Z$, \nand we can suppose that the points are of simple multiplicity $m_i$. \n\n\nWe can think of the surface $S$ containing $C$ as \nthe blowing-up of $\\check\\mathbb{P}^2$ in the points $V_i$'s: $\\pi_{\\{V_i\\}}\\colon S \\to \\check\\mathbb{P}^2$.\nLet us denote by $H:=\\pi_{\\{V_i\\}}^*\\mathcal{O}_{\\check\\mathbb{P}^2}(1)$ the hyperplane divisor of $S$ \nand by $E_i$'s the $(-1)$-curves on $S$ which correspond to the $V_i$'s. \nThe complete linear system $\\abs{(2k-1)H-\\sum_i (1-m_i)E_i}$ gives a generically 1--1 map  \n$\\phi\\colon\\ S\\to\\check\\mathbb{P}^{g-1}$ such that $C=\\phi(\\pi_{\\{V_i\\}}^{-1}(Z))$. \nThen, the adjunction formula of $C$ on $S$ in this case yields\n\\begin{equation*}\n((2k+2)H-\\sum_i m_iE_i)\\cdot ((2k-1)H-\\sum_i(1-m_i)E_i)=6k-4\n\\end{equation*}\nwhich means, \n\\begin{equation*}\\label{adj-1}\n\\sum_i m_i(m_i-1)=2(2k^2-2k+1). \n\\end{equation*}\nAgain, the $g^1_4$, is cut out on $Z$ by a pencil of conics, \nand as in the proof of the preceding proposition, with\nthe same notations, we deduce \n\\begin{equation*}\n\\sum_{i=1}^4 n_i=4k\n\\end{equation*}\nand \n\\begin{equation*}\n\\sum_{i=1}^4 n^2_i\\ge 4k^2,\n\\end{equation*}\nso that \n\\begin{equation*}\n\\sum_{i=1}^4 n_i^2-n_i\\ge 4k^2-4k.\n\\end{equation*}\nThe situation here is a little more complicated, since \n$\\sum_{i=1}^r m_i(m_i-1)=4k^2-4k+2> 4k^2-4k$, so we have \ntwo possibilities. The first one is that $r>4$, which means \nthat $r=5$, $m_1=n_1=\\dotsb m_4=n_4=k$ and $m_5=2$, or $r=4$, \nbut then $\\sum_{i=1}^4 n^2_i= 4k^2+2$ and $\\sum_i m_i=4k$. \n\n\nNow, \n\\begin{align*}\n\\deg(S)&=((2k-1)H-\\sum_i(1-m_i) E_i)^2\\\\\n&= (2k-1)^2-\\sum_i(m_i-1)^2\\\\\n&=\\sum_i(m_i-1)-1,\n\\end{align*}\nbut then, \nif $r=5$, we deduce $\\sum_i (m_i-1)=4k-2$ and then \nwe obtain that \n\\begin{align*}\n\\deg(S)&=4k-3\\\\\n&=\\frac{4}{3}g-\\frac{5}{3}; \n\\end{align*}\nif instead $r=4$, \n\\begin{align*}\n\\deg(S)&=4k-5\\\\\n&=\\frac{4}{3}g-\\frac{11}{3}. \n\\end{align*}\n\n\\end{proof}\n\n\nFinally, we analyse the case $g=3k+1$: \n\n\\begin{prop}\\label{4gon-2}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a tetragonal canonical curve\nof genus $g=3k+1$, where $k\\geq 2$, contained in a rational surface $S$ of type $[S]=2H-bF$ in a balanced scroll \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k))$. \n\nLet us suppose that the projection $\\pi\\colon\\check\\mathbb{P}^{g-1}\\dashrightarrow\\check\\mathbb{P}^2$\nfrom the linear space generated by a general point of $C$ and $k-1$ general planes of the scroll, \nrestricted to $S$, is generically 1--1 (\\textit{i.e. } $S$ is the blowing-up of this \n$\\check\\mathbb{P}^2$). \n\nThen, \n\\begin{equation*}\n\\deg(S)=\\frac{4}{3}g-\\frac{10}{3}, \n\\end{equation*} \nor, equivalently, $b=\\frac{2}{3}g-\\frac{8}{3}$. \n\nIn particular, $F_{\\eta_1,\\eta_2}$ is a sum of \ncubes of at most  $\\frac{4}{3}g-\\frac{10}{3}$ \nlinear forms, where \n$\\eta_1,\\eta_2\\in H^{0}(C,\\omega_{C})$ are general. \n\\end{prop}\n\n\n\\begin{proof}\nAs above \nwe obtain \na plane curve $Z$ of degree $2k+3$. \nLet $V_i$'s be the singular points of $Z$, \nand we can suppose that the points are of simple multiplicity $m_i$. \n\n\nWe can think of the surface $S$ containing $C$ as \nthe blowing-up of $\\check\\mathbb{P}^2$ in the points $V_i$'s: $\\pi_{\\{V_i\\}}\\colon S \\to \\check\\mathbb{P}^2$.\nLet us denote by $H:=\\pi_{\\{V_i\\}}^*\\mathcal{O}_{\\check\\mathbb{P}^2}(1)$ the hyperplane divisor of $S$ \nand by $E_i$'s the $(-1)$-curves on $S$ which correspond to the $V_i$'s. \nThe complete linear system $\\abs{2kH-\\sum_i(1-m_i) E_i}$ gives a generically 1--1 map  \n$\\phi\\colon\\ S\\to\\check\\mathbb{P}^{g-1}$ such that $C=\\phi(\\pi_{\\{V_i\\}}^{-1}(Z))$. \nThen, the adjunction formula of $C$ on $S$ in this case yields\n\\begin{equation*}\n((2k+3)H-\\sum_i m_iE_i)\\cdot (2kH-\\sum_i(1-m_i)E_i)=6k\n\\end{equation*}\nwhich means, \n\\begin{equation*}\\label{adj-2}\n\\sum_i m_i(m_i-1)=4k^2. \n\\end{equation*}\nAgain, the $g^1_4$, is cut out on $Z$ by a pencil of conics, \nand as in the proof of the preceding results, with\nthe same notations, we deduce \n\\begin{equation*}\n\\sum_{i=1}^4 n_i=4k+2 \n\\end{equation*}\nand \n\\begin{equation*}\n\\sum_{i=1}^4 n^2_i\\ge 4k^2+4k+1,\n\\end{equation*}\nso that \n\\begin{equation*}\n\\sum_{i=1}^4 n_i^2-n_i\\ge 4k^2-1.\n\\end{equation*}\nThe situation here is that \n$\\sum_{i=1}^r m_i(m_i-1)=4k^2> 4k^2-1$, and we can consider \nonly the case $r=4$, for which $\\sum_i (m_i-1)=4k-2$. \nNow, \n\\begin{align*}\n\\deg(S)&=(2kH-\\sum_i(1-m_i) E_i)^2\\\\\n&= 4k^2-\\sum_i(m_i-1)^2\\\\\n&=\\sum_i(m_i-1)\\\\\n&=4k-2\\\\\n&=\\frac{4}{3}g-\\frac{10}{3}. \n\\end{align*}\n\n\\end{proof}\n\n\n\n\n\nThe worst estimate of the preceding proposition is $\\frac{4}{3}g-\\frac{5}{3}$, which is the estimate reported in the \nabstract.  \n\n\n\n\n\\subsubsection{A generalisation}\\label{sec:special}\nIn this subsection we will show that the bounds for the surfaces \ngiven in the three propositions of Subsection~\\ref{ssec:stc} \nextends also to the case where $C$ is contained in a non-balanced scroll $X$. \n\nInstead of a check case by case, where we expect even better \nestimates, we concern ourselves only\nto extend the bounds  to these special cases.\nTo obtain this result we consider the tetragonal loci \n${\\mathcal{T}}_{g}\\subset{\\mathcal{M}}_{g}$ inside the moduli space of\nsmooth curves of genus $g$. Let \n${\\mathcal{T}}^{s}_g\\subset{\\mathcal{T}}_{g}$ be the loci\nof \\emph{special} tetragonal curves, \\textit{i.e. } not contained in a balanced scroll. It can be\nshown that  ${\\mathcal{T}}^{s}_g$ is a proper subscheme of \n${\\mathcal{T}}_{g}$  and then, given a \ncurve $C$ such that $[C]\\in {\\mathcal{T}}^{s}_g$, there exists an open\nset $U\\subset {\\mathcal{M}}_{g}$, the local universal family\n$\\rho\\colon{\\mathcal{U}}\\to U$ and a nontrivial morphism\n$\\Delta\\to U$, where $\\Delta\\subset {\\mathbb C}$ is the unitary \ndisk, such that the pull-back family \n$\\pi\\colon{\\mathcal{C}}\\rightarrow\\Delta$ has the central fibre\n$C_{0}=C$ \nwhich is a special\ntetragonal curve and the general one is in \n${\\mathcal{T}}_{g}\\setminus {\\mathcal{T}}^{s}_g$. More formally, we\nrecall that by the Hurwitz formula $\\omega_{C}\\simeq\nf^{\\star}(\\omega_{ {\\mathbb P}^{1} })\\otimes R$, where $R$ is the\nramification divisor of the morphism $f\\colon C\\to \\mathbb{P}^{1}$ which gives the $g^{1}_{4}$. Since $C$ is general in \n${\\mathcal{T}}^{s}_g$, we can assume \n$2p_{\\infty}^{1}+p_{\\infty}^{2}+p_{\\infty}^{3}=f^{\\star}(\\infty)$ and \n$2p_{0}^{1}+p_{0}^{2}+p_{0}^{3}=f^{\\star}(0)$. In particular, we\nidentify a meromorphic $1$-form\n$\\frac{df}{f}\\in\nH^{0}(C,\\omega(\\sum_{i=1}^{3}p_{\\infty}^{i}+\\sum_{i=1}^{3}p_{0}^{i}))$. \nInside the vector space $H^{1}(C, T_{C}(-\\sum_{i=1}^{3}p_{\\infty}^{i}-\n\\sum_{i=1}^{3}p_{0}^{i}))$ which parametrises the first order\ndeformations of $(C, p_{\\infty}^{1},\\dotsc ,p_{0}^{3})$, we want to\nidentify the subspace ${\\mathcal{T}}_{f}$\nof the first order deformations which extends\nthe $g^{1}_{4}$. So, consider $\\pi\\colon{\\mathcal{C}}_{\\epsilon}\n\\rightarrow{\\spec}\\,{{\\mathbb C}}\\frac{[\\epsilon]}{(\\epsilon^{2})}$ an \ninfinitesimal deformation given by $\\eta\\in H^{1}(C,\nT_{C}(-\\sum_{i=1}^{3}p_{\\infty}^{i}-\n\\sum_{i=1}^{3}p_{0}^{i}))$. Let $P_{\\infty}^{i}$, $P_{0}^{i}$ be the\ninfinitesimal sections which extend $p_{\\infty}^{i}$, $p_{0}^{i}$,\nrespectively, where $i=1,2,3.$ It is a trivial remark that\nthe extension \n\\begin{equation}\\label{vedid}\n0\\rightarrow \\mathcal{O}_{C}\\rightarrow\n\\Omega^{1}_{\\mathcal{C}_{\\epsilon}}({\\log}(\\sum_{i=1}^{3}P_{\\infty}^{i}+\\sum_{i=1}^{3}P_{0}^{i}))_{|C}\\rightarrow\n\\omega_{C}(\\sum_{i=1}^{3}p_{\\infty}^{i}+\\sum_{i=1}^{3}p_{0}^{i})\\rightarrow 0\n\\end{equation} \nrepresents the class \n\\begin{align*}\n\\eta&\\in{\\Ext}^{1}( \\omega_{C}(\\sum_{i=1}^{3}p_{\\infty}^{i}+\\sum_{i=1}^{3}p_{0}^{i},\\mathcal{O}_{C}))\\\\\n&\\simeq H^{1}(C,T_{C}(-\\sum_{i=1}^{3}p_{\\infty}^{i}-\\sum_{i=1}^{3}p_{0}^{i})). \n\\end{align*}\nThe fact that $f$ extends translates to\nthe fact that $\\frac{df}{f}$ extends to a meromorphic form\n$\\frac{dF}{F}$ on $\\mathcal{C}_{\\epsilon}$. This in turns means that \n$\\frac{df}{f}$ belongs to the kernel of the co-boundary operator in\nthe long sequence of cohomology given by the sequence \\eqref{vedid}:\n\n\\begin{equation*}\n    \\partial_{f}\\colon \n    H^{0}(C,\\omega(\\sum_{i=1}^{3}p_{\\infty}^{i}+\\sum_{i=1}^{3}p_{0}^{i}))\n    \\rightarrow H^{1}(C, \\mathcal{O}_{C}).\n\\end{equation*}\nThen ${\\mathcal{T}}_{f}$ is contained in the orthogonal subspace of\n$\\gen{\\frac{df}{f}}$ with respect to the cup product\n\n\\begin{equation*}\\label{vedidtris}\n    H^{1}(C,T_{C}(-\\sum_{i=1}^{3}p_{\\infty}^{i}-\\sum_{i=1}^{3}p_{0}^{i}))\\otimes\n    H^{0}(C,\\omega(\\sum_{i=1}^{3}p_{\\infty}^{i}+\\sum_{i=1}^{3}p_{0}^{i}))\n    \\rightarrow H^{1}(C, \\mathcal{O}_{C}).\n\\end{equation*}\nSince ${\\dim}_{{\\mathbb C}}{\\mathcal{T}}_{g}=2g+3$, \n${\\mathcal{T}}_{f}\\subset\\gen{\\frac{df}{f}}^{\\perp}$ and \n${\\dim}_{{\\mathbb C}}\\gen{\\frac{df}{f}}^{\\perp}=2g+3$ then\nit follows ${\\mathcal{T}}_{f}=\\gen{\\frac{df}{f}}^{\\perp}$ and \nthat ${\\mathcal{T}}_{g}$ is  smooth around its general\nspecial points. Now we turn to the pull-back family \n$\\pi\\colon{\\mathcal{C}}\\to\\Delta$ of the universal family \n$\\rho\\colon{\\mathcal{U}}\\to U$. \nBy flatness of $\\rho\\colon{\\mathcal{U}}\\to U$\nwe also have the relative canonical fibration $\\pi'\\colon \\mathbb{P}\\to\\Delta$ and inside it a relative fibration \n$\\pi'\\colon{\\mathcal{X}}\\to\\Delta$ where the general fibre is\na balanced scroll $X_{t}$.\n\\begin{prop}\\label{specis}\nLet $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ be a tetragonal canonical curve\nof genus $g$. Assume that $C$ is in the closure in\n$\\mathcal T_g$ of the class of the curves studied in\nSubsection~\\ref{ssec:stc}; \nthen it is contained in a surface $S$ such that: \n\\begin{equation*}\n\\deg(S)\\leq \\frac{4}{3}g-\\frac{5}{3}.\n\\end{equation*} \n\\end{prop}\n\\begin{proof} We assume that $g=3k-1$ and we will use Proposition~\\ref{4gon-1}.\nThe other cases are similar and give better bounds. Consider \nthe relative canonical fibration $\\pi'\\colon \\mathbb{P}\\to\\Delta$ \nand inside it a relative fibration \n$\\pi'\\colon{\\mathcal{X}}\\to\\Delta$ where the general fibre is\na balanced scroll $X_{t}$ as we have done above.\nUp to restrict $\\Delta$ if\nnecessary, we can construct a fibered surface $\\tau\\colon\\mathbb{P}^{1}_{\\Delta}\\to\\Delta$ whose\nfibre is a general section for the scroll $X_{t}$. By generality\nof the section $\\tau^{-1}(t)$ for the scroll $X_{t}$, it follows\nthat $\\tau^{-1}(0)$ is a general section of $X_{0}$. Then, by the\ngenerality of all the projections involved in our method, taking a \n$k-2$-multisection of $\\tau\\colon\\mathbb{P}^{1}_{\\Delta}\\to\\Delta$ which gives exactly $k-2$ points\non the fibre $\\tau^{-1}(0)$, we can perform relative projections\nwhich send the general fibre of $\\pi'\\colon{\\mathcal{X}}\n\\to\\Delta$ on a $\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^{1}}\\oplus\\mathcal{O}_{\\mathbb{P}^{1}}(1)\\oplus\\mathcal{O}_{\\mathbb{P}^{1}}(1))\\subset\\check\\mathbb{P}^{4}$\nand the\nspecial fibre $X_{0}$ on a suitable $3$-fold.\nThen, up to restrict $\\Delta$ if\nnecessary again, we can construct a $2$-section of \n$\\pi\\colon{\\mathcal{C}}\\to\\Delta$ in order to relative project to \n$\\check\\mathbb{P}^{2}$. By \\cite[Proposition\nIII.9.8]{H} we can\nconstruct a flat \nfamily of surfaces $\\pi'' \\colon{\\mathcal S}\\to\n\\Delta$ whose general fibre is embedded into the general fibre of \nthe relative canonical fibration $\\pi'\\colon \\mathbb{P}\\to\\Delta$. These surfaces are obtained by the blowing up of \n$\\check\\mathbb{P}^{2}\\times\\Delta$ along a $4$ or $5$ multisection \nby (the proof of) Proposition~\\ref{4gon-1}. Since the general fibre of $\\pi'' $ \nhas degree $=\\frac{4}{3}g-\\frac{5}{3}$\nthen the limiting surface $S_{0}$ still satisfies this bound.\n\\end{proof}\n\n\n\n\n\n\\subsection{Higher order gonality}\\label{sec:agg}\n\nNow we consider an $n$-gonal curve. Following a referee's comment \nwe will show that \nit is possible to construct surfaces as in Subsection~\\ref{ssec:stc}, \nbut it turns out that the degree of them is too big.\nFor simplicity, we show this in the easiest case only, \n\\textit{i.e. } the case of the $n$-gonal curve such that $n$ \ndivides the genus $g$ of $C$.\n\nMore precisely, we can write $g=(n-1)k$, and we can proceed as for Proposition~\\ref{4gon}: \nso we \\emph{suppose} that $C\\subset\\mathbb{P}(H^0(\\omega_C)^*)$ is an $n$-gonal canonical curve\nof genus $g=(n-1)k$, where $k\\geq 2$, contained in a rational surface $S$ in a balanced scroll \n$X=\\mathbb{P}(\\mathcal{O}_{\\mathbb{P}^1}(k-1)\\oplus\\dotsb\\oplus\\mathcal{O}_{\\mathbb{P}^1}(k-1))$ of dimension $n-1$. \n\nLet us \\emph{suppose also} that the projection $\\pi\\colon\\check\\mathbb{P}^{g-1}\\dashrightarrow\\check\\mathbb{P}^2$\nfrom the linear space generated by $(k-1)$ $\\check\\mathbb{P}^{n-2}$'s general fibres of the scroll and $(n-4)$ points of $C$ \nis generically 1--1 (\\textit{i.e. } $S$ is the blowing-up of this \n$\\check\\mathbb{P}^2$). \n\nLet us find now the degree of $S$. \n\nIndeed, if we project from $(k-1)$ $\\check\\mathbb{P}^{n-2}$'s general fibres of the scroll,  we arrive to a $\\check\\mathbb{P}^{n-2}$\nwhich contains the \nimage of the curve, $Z'$. We have \n\\begin{align*}\n\\deg(Z')&=2k(n-1)-2-(k-1)n\\\\\n&=(k+1)(n-2).\n\\end{align*}\nIf then we choose $(n-4)$ points on $Z'$ and we project again from these points, \nwe find a plane curve $Z$ of degree $k(n-2)+2$. \nAs above, the $V_i$'s are the singular points of $Z$, of \nsimple multiplicity $m_i$. \nIn this case, the adjunction formula gives, in the same way as \nwe obtained Formula~\\eqref{adj}\n\\begin{equation*}\\label{adjgen}\n\\sum_i m_i(m_i-1)=nk(nk-1)-4k^2(n-1). \n\\end{equation*}\n\nIf we take a general $(n-2)$-plane $\\Pi_t$, $t\\in\\mathbb{P}^1$, of the ruling of \nthe scroll $\\rho\\colon X\\to\\mathbb{P}^1$, then $C\\cap \\Pi_t$ \nis given by $n$ points $\\{P_{1t},\\dotsc,P_{nt}\\}$. Now \ntake a general section of the scroll. \nWe have now $(n+1)$ (general) points in  $\\check\\mathbb{P}^{n-2}\\cong \\Pi_t$, and \nthrough them there pass only one rational normal curve (of degree $(n-2)$). \nIf we project these curves to $\\check\\mathbb{P}^2$, we see that \nthere exists a pencil $\\Lambda$ of rational curves of degree $(n-2)$ which \ncuts the $g^1_n$ on $Z$. \nLet $A_1,\\dotsc,A_{(n-2)^2}$ be the base points of the\npencil; as above, we write:\n\\begin{equation}\\label{prima}\nQ'_t\\mid_Z=\\sum_{i=1}^{(n-2)^2} n_iA_i+ \\sum_{i=1}^{n}P'_{it},\n\\end{equation}\nwhere $Q'_t\\in\\Lambda$ and the $P'_{it}$'s are the projection\nof the points $P_{it}$'s, and $t\\in\\mathbb{P}^1$, \nas in the proof of Proposition~\\ref{4gon}.\nCalculating the degree, we obtain, as in Formula~\\eqref{equat}\n\\begin{equation}\\label{seconda}\n\\sum_{i=1}^{(n-2)^2} n_i=(n-2)^2k+n-4.\n\\end{equation}\nWe note that we can write an inequality as in Formula~\\eqref{ineq}.\nIt is not restrictive to suppose that $V_i=A_i$ \n(for the first $(n-2)^2$ $V_i$'s) and that $m_i=n_i$; then, by Formulas~\\eqref{prima} \nand \\eqref{seconda} we deduce\n\\begin{align*}\n\\deg(S)&=(k(n-2)-1)^2-\\sum_i(m_i-1)^2\\\\\n&=kn^2-5kn+8k-n^2+5n-7+\\sum_{i>(n-2)^2}(m_i-1), \n\\end{align*}\nwhich unfortunately is greater than $2g-3$, which is the estimate of \nIliev-Ranestad and Ciliberto-Harris, if $n\\gg 0$. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n\\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix: Renormalizing Higher-Order Tails}\n\nThe CTP effective actions in eqs.~\\eqref{unrenctpactiontail}, \\eqref{unrenctpactionTT}, \n\\eqref{unrenctpactionTTT} contain DimReg poles, and go through a renormalization. \nAs we illustrate below, there is an interplay among lower-order DimReg zeros and higher-order \nDimReg poles, similar to that in purely conservative effective potentials as of the N$^3$LO \nsectors. \nThe renormalization we apply is essentially similar to that in \\cite{Goldberger:2009qd}, where the \nquadrupole moment gets renormalized and displays an RG flow as \na Wilson coefficient of the EFT at the radiation scale.\nHere we shall demonstrate the renormalization needed for the extraction of dissipative \nphysics, which was discussed in the above. \n\nFirst, we note that the CTP effective action of radiation-reaction actually contains a piece \nproportional to a simple DimReg zero beyond the leading expression presented in \neq.~\\eqref{ctpactionRR}:\n\\begin{align}\n\\label{dimregzeroRR}\n\\Delta S_{\\text{RR}}\\Big|_{\\ensuremath{\\epsilon_d}^1}=&\\ensuremath{\\epsilon_d} \\frac{G_N}{20} \\int \\frac{d\\omega}{2\\pi} \\omega^5 \n\\Big[ -\\pi \\sgn\\omega \\big(\\kappa_{+-}(\\omega) + \\kappa_{-+}(\\omega) \\big) \\nonumber\\\\\n&+i \\bigg(\\frac{9}{10}-\\gamma_E+ \\log\\pi-\\log\\frac{\\omega^2}{\\mu_0^2}\\bigg)\\nonumber\\\\\n&\\big(\\kappa_{+-}(\\omega) - \\kappa_{-+}(\\omega) \\big)\\Big].\n\\end{align}\nIn the effective\naction of the tail, eq.~\\eqref{unrenctpactiontail}, the DimReg pole\n(and corresponding logarithm) is purely in the conservative part of\nthe effective action, so it does not affect dissipative observables.\n\nThe first dissipative DimReg pole occurs in the TT effective action,\neq.~\\eqref{unrenctpactionTT}.\nFollowing textbook renormalization procedures, (and Ref.~\\cite{Goldberger:2009qd}'s \napplication in a similar context)\nwe introduce a renormalized coupling to the quadrupoles:\n\\begin{equation}\n\\label{rgflow}\n\\kappa_{ij} \\to \\overline{\\kappa}_{ij} \\equiv \n\\kappa_{ij} \\Bigl( 1+\\frac{214}{105} \\ensuremath{\\epsilon_d}^{-1} \\, G_N^2E^2\\omega^2 \\Bigr). \n\\end{equation}\nWith this substitution in eq.~\\eqref{ctpactionRR} we find that\n\\begin{equation}\n\\overline{S}_{TT} \\equiv \\left(S_{RR} + S_{T} +  S_{TT}\\right) \\Big|_{\\kappa_{ij} \n\t\\to \\overline{\\kappa}_{ij}}\n\\end{equation}\nis free of dissipative DimReg poles through $\\mathcal{O}(G_N^3)$.\nExtracting the $\\mathcal{O}(G_n^3)$ contribution from\n$\\overline{S}_{TT}$ defines the\nrenormalized TT effective action:\n\\begin{align}\n  S_{\\text{TT}}^{\\text{Ren}}\n  &= \n\\frac{G_N^3E^2}{10} \\int \\frac{d\\omega}{2\\pi} \\omega^7\n\\kappa_{-+}(\\omega)\n\\Bigg[\\frac{428}{105}\\Big[\\pi\\text{sgn}\\omega + \\nonumber\\\\\n&i\\Big(\\gamma_E-\\text{log}\\pi+\\text{log}\\frac{\\omega^2}{\\mu_0^2}\\Big)\\Big]\n  -i\\Bigg(\\frac{634913}{22050}+16\\zeta_2 \\Bigg)\\Bigg].\n\\end{align}\nWhile the TTT effective action, eq. \\eqref{unrenctpactionTTT},\ncontains higher-order DimReg poles, the dissipative part only\ncontains a simple pole.  \nAs such, the same renormalization in eq.~\\eqref{rgflow}\napplied to the $\\mathcal{O}(\\ensuremath{\\epsilon_d}^1)$ part of the tail is\nsufficient to remove the dissipative TTT pole and obtain the\nrenormalized action $\\overline{S}_{TTT}$.  \nWe defer a discussion of the full renormalization including the conservative sector \nto future work.\n\nWith renormalized couplings, we also expect an RG flow of the\nquadrupoles (or equivalently $\\kappa_{ij}$).  The flow equation can\nbe found by allowing $\\overline{\\kappa}$ to depend on the log scale\n$\\mu_0$ then demanding that $\\overline{S}_{TTT}$ does not depend on $\\mu_0$.  \nDoing so, we find\n\\begin{equation}\n\\label{eq:rgflow}\n\\frac{d}{d \\log \\mu} \\overline{\\kappa} = - \\frac{428}{105}(G_N \\omega E)^2 \\overline{\\kappa},\n\\end{equation}\nin exact agreement with \\cite{Goldberger:2009qd} (noting that\n  $\\frac{d}{d \\mu} \\kappa \\sim 2 \\frac{d}{d \\mu} I_{ij}$).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper we consider quasi-linear elliptic partial differential equations over a polygonal\/polyhedral domain $\\Omega\\subset \\mathbb R^d$ ($d=2,3$)  of the form\n\\begin{equation}\\label{E:ley de conservacion}\n\\left\\{\n\\begin{aligned}\n-\\nabla\\cdot \\big[\\alpha(\\,\\cdot\\,,|\\nabla u|^2)\\nabla u\\big]&= f\\qquad & & \\text{in}\\,\\Omega\\\\\nu&= 0\\qquad & &\\text{on}\\,\\partial \\Omega,\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\alpha:\\Omega\\times \\mathbb R_+\\to \\mathbb R_+$ is a function whose properties will be stated in Section~\\ref{S:setting} below, and $f\\in L^2(\\Omega)$ is given.\nThese equations describe stationary conservation laws which frequently arise in problems of mathematical physics~\\cite{Zeidler}. \nFor example, in hydrodynamics and gas dynamics (subsonic and supersonic flows), electrostatics, magnetostatics, heat conduction, elasticity and plasticity (e.g., plastic torsion of bars), etc. Some of these examples are better modeled by variational inequalities (see~\\cite{HJS} and the references therein), but these fall beyond the scope of this article, which attempts to set a first step towards understanding the convergence and optimality of inexact Ka\\v canov-type iterations, in the context of adaptive finite element methods.\n\nFor a summary of convergence and optimality results of AFEM we refer the  reader to the survey~\\cite{NSV-survey}, and the references therein. We restrict ourselves to those references strictly related to our work.\n\n\\emph{Inexact} adaptive finite element methods have been considered for Stokes problem in~\\cite{BMN,KS} using Uzawa's algorithm. Briefly, a Richardson iteration is applied to the infinite-dimensional Schur complement operator and in each iteration, the elliptic problem is solved up to a decreasing tolerance. In~\\cite{BMN} linear convergence is proved and in~\\cite{KS} the optimality of the method in terms of degrees of freedom is proved, after adding some new refinement steps to the algorithm.\n\nIn~\\cite{DK}, a contraction property is proved for an adaptive algorithm based on D\\\"orfler's marking strategy~\\cite{Doerfler} for nonlinear problems of the type~\\eqref{E:ley de conservacion}, using Orlicz norms to cope with the very mild assumptions on the nonlinear term $\\alpha$.\n\nIn this work we impose stronger assumptions on $\\alpha$, which guarantee the convergence of Ka\\v canov's iteration~\\cite{HJS}. More precisely, we assume that $\\alpha(\\cdot,\\cdot)$ is decreasing with respect to its second variable (cf.~\\eqref{alpha-decreasing}), and fulfills condition~\\eqref{E:betasegunda acotada} below; which are the same assumptions stated in~\\cite{HJS}, where they consider the iteration on a fixed space, either finite- or infinite-dimensional. Our focus is the convergence analysis of the adaptive method that results from performing one mesh adaptation in each iteration of the non-linear solver. This turns out to be a very realistic and efficient method, based on the sole assumption that a \\emph{linear} system is exactly solved in each iteration step.\n\nThis paper is organized as follows. In Section~\\ref{S:setting} we present the  class of specific nonlinear problems that we study, and some of its properties. In Section~\\ref{S:adaptive loop}, we present the inexact adaptive Ka\\v canov algorithm and in Section~\\ref{S:convergence} we state and prove the main result of this article, namely the convergence of the discrete solutions produced by the algorithm to the exact solution of the nonlinear problem. Finally, in Section~\\ref{S:numerical experiments}, we present some numerical experiments which illustrate the theory, and explore the practical boundaries of applicability of the algorithm.\n\n\\section{Setting}\\label{S:setting}\n\nLet $\\Omega\\subset\\mathbb R^d$ be a bounded polygonal ($d = 2$) or polyhedral ($d = 3$) domain with Lipschitz boundary. A weak formulation of~\\eqref{E:ley de conservacion} consists in finding $u\\in H^1_0(\\Omega)$ such that\n\\begin{equation}\\label{E:cont_prob_nolineal}\na(u;u,v)=L(v),\\qquad\\forall\\,v\\in H^1_0(\\Omega),\n\\end{equation}\nwhere\n\\begin{equation*\na(w;u,v)=\\int_\\Omega \\alpha(\\,\\cdot\\,,|\\nabla w|^2)\\nabla u\\cdot \\nabla v,\\qquad\\forall\\,w,u,v\\in H^1_0(\\Omega),\n\\end{equation*}\nand\n\\begin{equation*\nL(v)=\\int_\\Omega fv,\\qquad\\forall\\,v\\in H^1_0(\\Omega).\n\\end{equation*}\nWe require that $\\alpha$ is locally Lipschitz in its first variable, uniformly with respect to its second variable (cf.~\\eqref{E:alpha localmente lipschitz} below). On the other hand, we assume that $\\alpha$ is $\\mathcal{C}^1$ as a function of its second variable and there exist positive constants $c_a$ and $C_a$ such that\n\\begin{equation}\\label{E:betasegunda acotada} \nc_a\\le \\alpha(x,t^2)+2t^2D_2\\alpha(x,t^2)\\le C_a,\\qquad\\forall\\,x\\in\\Omega,\\,t>0,\n\\end{equation} \nwhere $D_2\\alpha$ denotes the partial derivative of $\\alpha$ with respect to its second variable.\nThe last condition is related to the well-posedness of problem~\\eqref{E:cont_prob_nolineal} as we will show below. Also, it is possible to prove that~\\eqref{E:betasegunda acotada} implies that\n\\begin{equation}\\label{E:acotacion de alfa}\nc_a\\le\\alpha(x,t)\\le C_a,\\qquad\\forall\\,x\\in\\Omega,\\,t>0.\n\\end{equation}\nThese same assumptions are stated in~\\cite{HJS}, where several interesting applied problems are shown to satisfy them.\n\nAdditionally, it is easy to check that the form $a$ is linear and symmetric in its second and third variables. Also, from~\\eqref{E:acotacion de alfa} it follows that $a$ is bounded,\n\\begin{equation}\\label{E:a_acotada}\n|a(w;u,v)|\\le C_a \\|\\nabla u\\|_\\Omega\\|\\nabla v\\|_\\Omega,\\qquad\\forall\\,w,u,v\\in H^1_0(\\Omega),\n\\end{equation}\nand coercive,\n\\begin{equation}\\label{E:a_coercitiva}\n c_a\\|\\nabla u\\|_\\Omega^2\\le a(w;u,u),\\qquad\\forall\\,w,u\\in H^1_0(\\Omega).\n\\end{equation}\n\nIf we define $A:H^1_0(\\Omega)\\to H^{-1}(\\Omega)$ as the nonlinear operator given by\n\\begin{equation*}\n\\langle Au,v\\rangle := a(u;u,v),\\qquad\\forall\\,u,v\\in H^1_0(\\Omega),\n\\end{equation*}\nthen problem~\\eqref{E:cont_prob_nolineal} is equivalent to the equation\n$$Au=L,$$\nwhere $L \\in H^{-1}(\\Omega)$ is given. Assumption~\\eqref{E:betasegunda acotada} implies that $A$ is Lipschitz and strongly monotone (see~\\cite{Zeidler}), i.e., there exist positive constants $C_A$ and $c_A$ such that\n\\begin{equation}\\label{E:A Lipschitz}\n\\|Au-Av\\|_{H^{-1}(\\Omega)}\\le C_A \\|\\nabla (u -v)\\|_\\Omega,\\qquad \\forall\\,u,v\\in H^1_0(\\Omega),\n\\end{equation}\nand\n\\begin{equation}\\label{E:A fuertemente monotono}\n\\langle Au-Av,u-v\\rangle\\ge c_A\\|\\nabla (u -v)\\|_\\Omega^2,\\qquad \\forall\\,u,v\\in H^1_0(\\Omega).\n\\end{equation}\nAs a consequence of~\\eqref{E:A Lipschitz} and~\\eqref{E:A fuertemente monotono}, problem~\\eqref{E:cont_prob_nolineal} has a unique solution and is stable~\\cite{Zarantonello,Zeidler}.\n\n\n\\section\nAdaptive algorithm}\\label{S:adaptive loop}\n\nIn order to define discrete adaptive approximations to problem~\\eqref{E:cont_prob_nolineal} we will consider \\emph{triangulations} of the domain $\\Omega$. Let $\\Tau_0$ be an\ninitial conforming triangulation of $\\Omega$, that is, a partition\nof $\\Omega$ into $d$-simplices such that if two elements intersect,\nthey do so at a full vertex\/edge\/face of both elements. Let\n$\\TT$ denote the set of all conforming triangulations of\n$\\Omega$ obtained from $\\Tau_0$ by refinement using the bisection procedures presented by Stevenson~\\cite{Stevenson-refine}. These coincide (after some re-labeling) with the so-called \\textit{newest vertex} bisection procedure in two dimensions and the bisection procedure of Kossaczk\\'y in three\ndimensions~\\cite{Alberta}.\n\nDue to the processes of refinement used, the family $\\TT$ is shape regular, i.e., \n$$\\sup_{\\Tau\\in\\TT}\\,  \\sup_{T \\in \\Tau} \\frac{\\diam(T)}{\\rho_T} =: \\kappa_\\TT <\\infty,$$\nwhere $\\diam(T)$ is the diameter of $T$, and $\\rho_T$ is the radius of the largest ball contained in it. Throughout this article, we only consider meshes $\\Tau$ that belong to the family $\\TT$, so the shape regularity of all of them is bounded by the uniform constant $\\kappa_\\TT$ which only depends on the initial triangulation $\\Tau_0$~\\cite{Alberta}. Also, the diameter of any element $T\\in\\Tau$ is equivalent to the local mesh-size $H_T:=|T|^{1\/d}$, which in turn defines the global mesh-size $H_\\Tau := \\displaystyle\\max_{T\\in\\Tau} H_T$.\n\nFor the discretization we consider the Lagrange finite element spaces consisting of continuous functions vanishing on $\\partial \\Omega$ which are piecewise linear over a mesh $\\Tau\\in\\TT$, i.e.,\n\\begin{equation}\\label{E:V_Tau}\n\\VV_{\\Tau}:=\\{v\\in H^1_0(\\Omega)\\mid\\quad v_{|_T}\\in \\PP_1(T),\\quad \\forall~T\\in\\Tau\\}.\n\\end{equation}\nWe are now in position to state the adaptive loop to approximate the solution $u$ of the problem~\\eqref{E:cont_prob_nolineal}.\n\n\\medskip\n\\begin{center}\n\\doublebox{\n\\begin{minipage}{.9\\textwidth}\n\\textbf{Adaptive Algorithm.}\n\\tt\nLet $\\Tau_0$ be an initial conforming mesh of $\\Omega$ and $u_0\\in\\VV_{\\Tau_0}$ be an initial approximation. Let $\\Tau_1=\\Tau_0$ and $k=1$.\n\\begin{enumerate}\n\\vspace{0.15cm}\n \\item [1.] $u_k:= \\textsf{SOLVE}(u_{k-1},\\Tau_k)$.\n\\vspace{0.15cm}\n\\item [2.] $\\{\\eta_k(T)\\}_{T\\in\\Tau_k}:= \\textsf{ESTIMATE}(u_{k-1},u_k,\\Tau_k)$.\n\\vspace{0.15cm}\n\\item [3.] $\\MM_k:= \\textsf{MARK}(\\{\\eta_k(T)\\}_{T\\in\\Tau_k},\\Tau_k)$.\n\\vspace{0.15cm}\n\\item [4.] $\\Tau_{k+1}:= \\textsf{REFINE}(\\Tau_k,\\MM_k,n)$.\n\\vspace{0.15cm}\n\\item [5.] Increment $k$ and go back to step 1.\n\\end{enumerate}\n\\end{minipage}\n}\n\\end{center}\n\nWe now explain each module of the last algorithm in detail.\n\n\\begin{description}\n\\item[\\textbf{The module \\textsf{SOLVE}.}]\n Given the conforming triangulation $\\Tau_k$ of $\\Omega$, and the solution $u_{k-1}$ from the previous iteration, the module \\textsf{SOLVE} outputs the solution $u_k\\in\\VV_k:= \\VV_{\\Tau_k}$ of the \\emph{linear} problem\n\\begin{equation}\\label{E:kdisc-problem-nolineal inexacto}\na(u_{k-1};u_k,v_k)=L(v_k),\\qquad \\forall~v_k\\in  \\VV_k.\n\\end{equation}\n\\end{description}\n\n\\begin{remark}[Linear vs.\\ nonlinear] \nNotice that while \\textsf{SOLVE} requires the solution of a linear system, the usual discretization of~\\eqref{E:cont_prob_nolineal} in $\\VV_k$ consists in finding $u_k\\in \\VV_k$ such that\n\\begin{equation}\\label{E:kdisc-problem-nolineal}\na(u_k;u_k,v_k)=L(v_k),\\qquad \\forall~v_k\\in  \\VV_k,\n\\end{equation}\nwhich is nonlinear. We propose here to make only one step of a fixed point iterative method to solve the nonlinear problem, and proceed to the mesh adaptation, whereas the usual approach would be to approximate the discrete nonlinear problem up to a very fine accuracy (pretending to have it exactly solved) before proceeding to mesh adaptation~\\cite{GMZ-optimality-nonlinear}. Each iteration of the adaptive loop is thus much cheaper in our approach. Our theory guarantees convergence of this algorithm, and the numerical experiments of Section~\\ref{S:numerical experiments} show that the convergence is quasi-optimal, although the latter is not yet rigorously proved.\n\n\n\\end{remark}\n\n\n\n\n\\begin{remark}[Ka\\v{c}anov's Method]\nLet us consider problem~\\eqref{E:cont_prob_nolineal} (resp.\\ problem~\\eqref{E:kdisc-problem-nolineal} with $k\\in\\NN$ fixed). We denote the space $H^1_0(\\Omega)$ (resp.\\ $\\VV_k$) by $\\VV$, and the solution $u$ (resp.\\  $u_k$) by $U$. Given an initial approximation $U_0\\in\\VV$ of the solution $U$, we consider the sequence $\\{U_i\\}_{i\\in\\NN_0}$ where $U_i\\in\\VV$ is the solution of the \\emph{linear} problem\n\\begin{equation*\na(U_{i-1};U_i,v)=L(v),\\qquad\\,\\forall\\,v\\in\\VV,\\quad i \\in \\NN.\n\\end{equation*}\n\nThis is known as Ka\\v{c}anov's Method, and it follows~\\cite{HJS} that the sequence $\\{U_i\\}_{i\\in\\NN_0}$ converges to the solution $U$, provided $D_2\\alpha(x,t)\\le 0$ for all $x\\in\\Omega$ and $t>0$.\n\nNotice that our algorithm consists in performing \\emph{only one} step of Ka\\v{c}anov iteration in each step of the adaptive loop.\n\\end{remark}\n\n\\begin{description}\n\\item[\\textbf{The module \\textsf{ESTIMATE}.}] \nGiven $\\Tau_k$, $u_{k-1}$ and the corresponding output $u_k$ of \\textsf{SOLVE}, the module \\textsf{ESTIMATE} computes and outputs the local error estimators $\\{\\eta_k(T)\\}_{T\\in\\Tau_k}$ given by\n\\begin{equation}\\label{E:estimadores kakanov def}\n\\eta_k^2(T):= H_T^2\\normT{R_k}^2 + H_T\\normbT{J_k}^2,\n\\end{equation}\nfor all $T\\in\\Tau_k$.\nHere $R_k$ denotes the \\emph{element residual} given by \\begin{equation*\n{R_k}_{|_{T}}:= -\\nabla \\cdot \\big[\\alpha(\\,\\cdot \\,,|\\nabla u_{k-1}|^2)\\nabla u_k\\big]-f,\\qquad\\forall\\,T\\in\\Tau_k,\n\\end{equation*}\nand $J_k$ denotes the \\emph{jump residual} given by\n\\begin{equation*\n{J_k}_{|_{S}}:=\\frac12\\left[(\\alpha(\\,\\cdot \\,,|\\nabla u_{k-1}|^2)\\nabla u_k)_{|_{T}}\\cdot \\vec{n}+(\\alpha(\\,\\cdot \\,,|\\nabla u_{k-1}|^2)\\nabla u_k)_{|_{T'}} \\cdot\\vec{n}' \\right],\n\\end{equation*}\nfor each interior side $S$, and ${J_k}_{|_{S}}:= 0$, if $S$ is a side lying on the boundary of $\\Omega$. Here, $T$ and $T'$ denote the elements of $\\Tau_k$ sharing $S$, and $\\vec{n}$, $\\vec{n}'$ are the outward unit normals of $T$, $T'$ on $S$, respectively.\n\\end{description}\n\nThe squared sum of the a posteriori error estimators is an upper bound for the \\emph{residual} $\\RRR(u_k) \\in H^{-1}(\\Omega)$ of $u_k$ which is defined as\n\\begin{equation*}\n\\langle \\RRR(u_k),v\\rangle:= a(u_{k-1};u_k,v)-L(v)=\\int_\\Omega \\left(\\alpha(\\,\\cdot\\,,|\\nabla u_{k-1}|^2)\\nabla u_k\\cdot \\nabla v- fv\\right),\n\\end{equation*}\nfor $v\\in H^1_0(\\Omega)$.\nIn fact, integrating by parts on each $T\\in\\Tau_k$ we have that\n\\begin{equation}\\label{E:relacion fundamental}\n\\langle\\RRR(u_k),v\\rangle=\\sum_{T\\in\\Tau_k} \\left(\\int_T R_kv +\\int_{\\partial T} J_kv\\right),\n\\end{equation}\nand since $\\langle\\RRR(u_k),v_k\\rangle =0$ for $v_k\\in\\VV_k$, using~\\eqref{E:relacion fundamental} and interpolation estimates, we arrive at the following upper bound\n\\footnote{From now on, we will write $a \\lesssim b$ to indicate that $a \\le C b$ with $C>0$ a constant depending on the data of the problem and possibly on shape regularity $\\kappa_\\TT$ of the meshes.\n}\n\\begin{equation}\\label{E:local-residual-upperbound}\n|\\langle\\RRR(u_k),v\\rangle|\\lesssim\\sum_{T\\in\\Tau_k} \\eta_k(T)\\|\\nabla v\\|_{\\omega_k(T)},\\quad\\forall\\,v\\in H^1_0(\\Omega),\n\\end{equation} \nwhere $\\omega_k(T)$ is the union of $T$ and its neighbors in $\\Tau_k$. \n\nThe next result is some kind of \\emph{stability} result for the estimators, which is a property somewhat weaker than the usual \\emph{efficiency}, but sufficient to guarantee convergence of our adaptive algorithm (see also~\\cite{Siebert-convergence,GM-09}). In order to prove it, we assume that $\\alpha$ is locally Lipschitz in its first variable, uniformly with respect to its second variable, as we mentioned before. More precisely, we assume that there exists a constant $C_\\alpha>0$ such that\n\\begin{equation}\\label{E:alpha localmente lipschitz}\n|\\alpha(x,t)-\\alpha(y,t)|\\le C_\\alpha |x-y|,\\qquad\\forall\\,x,y\\in T,\\,t>0,\n\\end{equation}\nfor each $T\\in\\Tau_0$.\n\n\n\\begin{proposition}[Stability of the local error estimators]\\label{P:estabilidad kakanov}\nLet $\\{u_k\\}_{k\\in\\NN}$ be the sequence of discrete solutions computed with the Adaptive Algorithm. Then, the local error estimators given by~\\eqref{E:estimadores kakanov def} are stable. More precisely, there holds $$\\eta_k(T)\\lesssim\\|\\nabla u_k\\|_{\\omega_k(T)}+\\|f\\|_{T},\\qquad\\forall\\, T\\in\\Tau_k,$$\nfor all $k\\in\\NN$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\{u_k\\}_{k\\in\\NN}$ be the sequence computed with the Adaptive Algorithm. Let $k\\in\\NN$ and $T\\in\\Tau_k$ be fixed. On the one hand, using that ${u_k}_{|T}$ is linear, and thus $\\Delta u_k = 0$ inside $T$, we have that\n\\begin{align*}\n\\normT{R_k} &= \\normT{-\\nabla \\cdot \\big[\\alpha(\\cdot,|\\nabla u_{k-1}|^2)\\nabla u_k\\big]-f } \n\\\\\n&\\le \\normT{\\nabla \\big[\\alpha(\\cdot,|\\nabla u_{k-1}|^2)\\big]\\cdot\\nabla u_k}+\\normT{f}.\n\\end{align*}\nSince $\\alpha$ is locally Lipschitz in its first variable (cf.~\\eqref{E:alpha localmente lipschitz}), it follows that if $\\xi:=\\nabla u_{k-1}$ (constant over $T$),\n\\begin{align*}\n\\left|\\frac{\\partial}{\\partial x_i}\\alpha(x,|\\xi|^2)\\right|&=\\left|\\frac{\\partial\\alpha}{\\partial x_i}(x,|\\xi|^2)\\right|\\le C_\\alpha,\\qquad \\forall\\, x\\in T,\\quad 1\\le i\\le d,\n\\end{align*}\nand thus,\n\\begin{align*}\n\\normT{R_k} &\\lesssim  \\normT{\\nabla u_k}+\\normT{f}.\n\\end{align*}\n\nOn the other hand, if $S$ is a side of $\\Tau_k$ shared by the elements $T,T' \\in \\Tau_k$,\n\\begin{align*}\n\\normS{J_k} &\\le \n\\normS{ (\\alpha(\\,\\cdot \\,,|\\nabla u_{k-1}|^2)\\nabla u_k)_{|_{T}}}\n+\n\\normS{(\\alpha(\\,\\cdot \\,,|\\nabla u_{k-1}|^2)\\nabla u_k)_{|_{T'}} }\n\\\\\n&\\lesssim \\normS{{\\nabla u_k}_{|_{T}}}+\\normS{{\\nabla u_k}_{|_{T'}}}\n\\\\\n&\\lesssim  H_T^{-1\/2}\\|\\nabla u_k\\|_{T}+H_{T'}^{-1\/2}\\|\\nabla u_k\\|_{T'}\\lesssim H_T^{-1\/2}\\|\\nabla u_k\\|_{T\\cup T'},\n\\end{align*}\nwhere we have used~\\eqref{E:acotacion de alfa} and a scaled trace theorem. Therefore,\n\\begin{equation*}\nH_T^{1\/2}\\normbT{J_k}\\lesssim\\|\\nabla u_k\\|_{\\omega_k(T)},\n\\end{equation*}\nwhich completes the proof.\n\\end{proof}\n\n\n\\begin{description}\n\\item[\\textbf{The module \\textsf{MARK}.}] Based on the local error indicators, the module \\textsf{MARK} selects a subset $\\MM_k$ of $\\Tau_k$, using any of the so-called \\emph{reasonable} marking strategies, such as the \\emph{maximum strategy}, the \\emph{equidistribution strategy}, or \\emph{D\\\"orfler's strategy}~\\cite{Alberta}. More precisely, we only require that the set of marked elements $\\MM_k$ has at least one element of $\\Tau_k$ holding the largest local estimator. That is, there exists an element $T_k^{\\max} \\in \\MM_k$ such that\n\\begin{equation}\\label{E:estrategia razonable}\n \\eta_k(T_k^{\\max}) = \\max_{T \\in \\Tau_k} \\eta_k(T).\n\\end{equation}\nThis is what practitioners usually do in order to maximize the error reduction with a minimum computational effort.\n\n\\item[\\textbf{The module \\textsf{REFINE}.}] Finally, the module \\textsf{REFINE} takes the mesh $\\Tau_k$ and the subset $\\MM_k\\subset \\Tau_k$ as inputs. By using the bisection rule described by Stevenson in~\\cite{Stevenson-refine}, this module refines (bisects) each element in $\\MM_k$ at least $n$ times (where $n \\ge 1$ is fixed), in order to obtain a new conforming triangulation $\\Tau_{k+1}$ of $\\Omega$, which is a refinement of $\\Tau_k$ and the output of this module. By definition, $\\Tau_{k} \\in \\TT$ for all $k \\in \\NN$ and the family of meshes obtained by the Adaptive Algorithm is shape-regular. Finally, it is worth observing that the resulting spaces are nested, i.e., $\\VV_{k} \\subset \\VV_{k+1}$; this fact will be used in the proof of Lemma~\\ref{L:sucesivas a cero} below.\n\\end{description}\n\n\n\n\\begin{remark}[The adaptive sequence $\\{u_k\\}_{k\\in\\NN_0}$ is bounded]\\label{R:uk acotada no lineal}\nBy the coercivity of $a$ given by~\\eqref{E:a_coercitiva}, and using the definition~\\eqref{E:kdisc-problem-nolineal inexacto} of $u_k$ we have that\n$$\\|\\nabla u_k\\|_\\Omega^2\\le\\frac{1}{c_a}a(u_{k-1};u_k,u_k)=\\frac{1}{c_a}L(u_k)\\le \\frac{\\|L\\|_{H^{-1}(\\Omega)}}{c_a}\\|\\nabla u_k\\|_\\Omega,$$\nfor all $k\\in\\NN$, and then\n\\begin{equation*}\n\\|\\nabla u_k\\|_\\Omega\\le\\frac{\\|L\\|_{H^{-1}(\\Omega)}}{c_a}.\n\\end{equation*}\nTherefore, $\\{u_k\\}_{k\\in\\NN_0}$ is bounded in $H^1_0(\\Omega)$.\n\\end{remark}\n\n\\section{Convergence Analysis}\\label{S:convergence}\n\nIn this section we prove the convergence of the sequence computed with the Adaptive Algorithm described in the previous section. We combine the ideas of the proof of convergence of adaptive algorithms for linear problems given in~\\cite{MSV-convergence} and~\\cite{Siebert-convergence} with new techniques needed to overcome the difficulties arisen by the nonlinear nature of the problem treated in this paper, adapting some ideas from~\\cite{GMZ-08,GM-09}. \n\nAs a first step to prove the convergence of the $\\{u_k\\}_{k\\in\\NN_0}$ to the exact solution $u$, we prove that $\\|\\nabla (u_k-u_{k+1})\\|_\\Omega\\to 0$, as $k$ tends to infinity, for which we need the following auxiliary lemma.\n\nFrom now on we assume that \n\\begin{equation}\\label{alpha-decreasing}\n\\alpha(x,t_1) \\ge \\alpha(x, t_2)  \\text{ whenever } 0 \\le t_1 \\le t_2 \\text{ and } x\\in\\Omega.\n\\end{equation}\n\n\\begin{lemma}\\label{L:key property}\nLet us assume that $\\alpha(x,\\cdot)$ is a monotone decreasing function for all $x \\in \\Omega$, i.e.,~\\eqref{alpha-decreasing} holds. Then\n$$\\JJ(v)-\\JJ(w)\\le\\frac12 \\big(a(w;v,v)-a(w;w,w)\\big),\\qquad\\forall\\,v,w\\in H^1_0(\\Omega),$$\nwhere $\\JJ(v)= \\int_0^1 a(sv;sv,v)~ds$.\n\\end{lemma}\n\n\\begin{proof}\nLet $v,w\\in H^1_0(\\Omega)$. Then, a change of variables in the integral defining $\\JJ(\\cdot)$ yields\n\\begin{align*}\n\\JJ(v)-\\JJ(w)&=\\frac12\\int_\\Omega \\left(\\int_0^{|\\nabla v|^2} \\alpha(x,t)~dt-\\int_0^{|\\nabla w|^2}\\alpha(x,t)~dt\\right)~dx\n\\\\\n&=\\frac12\\int_\\Omega \\int_{|\\nabla w|^2}^{|\\nabla v|^2} \\alpha(x,t)~dt~dx.\n\\end{align*}\nSince $\\alpha(x,\\cdot)$ is a decreasing function for all $x \\in \\Omega$,\n\\begin{align*}\n\\frac12\\int_\\Omega \\int_{|\\nabla w|^2}^{|\\nabla v|^2} \\alpha(x,t)~dt~dx\n&\\le\\frac12\\int_\\Omega \\alpha(x,|\\nabla w|^2)\\left(|\\nabla v|^2-|\\nabla w|^2\\right)~dx\n\\\\\n&=\\frac12\\big(a(w;v,v)-a(w;w,w)\\big),\n\\end{align*}\nand the assertion of the lemma follows.\n\\end{proof}\n\n\n\\begin{lemma}\\label{L:sucesivas a cero}\nLet $\\{u_k\\}_{k\\in\\NN_0}$ denote the sequence of discrete solutions computed with the Adaptive Algorithm. Then,\n$$\\lim_{k\\to\\infty} \\|\\nabla (u_k-u_{k+1})\\|_\\Omega=0.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\{u_k\\}_{k\\in\\NN_0}$ be the sequence obtained with the Adaptive Algorithm. Since $a$ is coercive (cf.~\\eqref{E:a_coercitiva}), and linear and symmetric in its second and third variables, we have that\n\\begin{multline}\\label{E:sucesivas a cero aux}\nc_a\\|\\nabla (u_k-u_{k+1})\\|_\\Omega^2\\le a(u_k;u_k-u_{k+1},u_k-u_{k+1}) \\\\\n=a(u_k;u_k,u_k)-2a(u_k;u_{k+1},u_k)+a(u_k;u_{k+1},u_{k+1}).\n\\end{multline}\nFrom~\\eqref{E:kdisc-problem-nolineal inexacto}, since $u_k \\in \\VV_{k+1}$, it follows that $a(u_k;u_{k+1},u_k-u_{k+1})=L(u_k-u_{k+1})$ and thus\n$$a(u_k;u_{k+1},u_k)=L(u_k-u_{k+1})+a(u_k;u_{k+1},u_{k+1}).$$\nReplacing this equality in~\\eqref{E:sucesivas a cero aux} and taking into account Lemma~\\ref{L:key property} we obtain\n\\begin{equation}\\label{E:sucesivas a cero aux 2}\n\\begin{aligned}\n c_a\\|\\nabla (u_k-u_{k+1})\\|_\\Omega^2&\\le a(u_k;u_k,u_k)-2L(u_k-u_{k+1})-a(u_k;u_{k+1},u_{k+1})\\\\\n&\\le 2\\JJ(u_k)-2\\JJ(u_{k+1})-2L(u_k)+2L(u_{k+1})\\\\\n&=2(\\FF(u_k)-\\FF(u_{k+1})),\n\\end{aligned}\n\\end{equation}\nwhere $\\FF:= \\JJ-L$, and therefore, $\\{\\FF(u_k)\\}_{k\\in\\NN_0}$ is a monotone decreasing sequence.\n\nOn the other hand, $\\{\\FF(u_k)\\}_{k\\in\\NN_0}$ is bounded below since\n\\begin{align*}\n\\FF(u_k)&=\\int_0^1 sa(su_k;u_k,u_k)~ds-L(u_k)\n\\\\\n&\\ge \\frac12 c_a\\|\\nabla u_k\\|_\\Omega^2-\\|L\\|_{H^{-1}(\\Omega)}\\|\\nabla u_k\\|_\\Omega\\ge -\\frac{\\|L\\|_{H^{-1}(\\Omega)}^2}{2c_a}.\n\\end{align*}\n\nFinally, from the last two assertions it follows that $\\{\\FF(u_k)\\}_{k\\in\\NN_0}$ is convergent. Considering~\\eqref{E:sucesivas a cero aux 2}, we conclude the proof of this lemma.\n\\end{proof}\n\n\n\nWe show now that the sequence obtained with the Adaptive Algorithm is convergent, and more precisely, that it converges to a function in the limiting space $\\VV_\\infty:= \\overline{\\cup \\VV_k}^{H^1_0(\\Omega)}$. Note that $\\VV_\\infty$ is a Hilbert space with the inner product inherited from $H^1_0(\\Omega)$.\n\n\\begin{theorem}[The adaptive sequence is convergent]\\label{T:convergencia a un limite}\nLet $\\{u_k\\}_{k\\in\\NN_0}$ be the sequence obtained with the Adaptive Algorithm. Let $u_\\infty\\in\\VV_\\infty$ be the only solution to \n\\begin{equation}\\label{E:cont_prob_vinfty inexacto}\nu_\\infty \\in \\VV_\\infty : \\quad a(u_\\infty;u_\\infty,v)=L(v),\\qquad \\forall~v\\in  \\VV_\\infty.\n\\end{equation}\nThen\n$$u_k \\longrightarrow u_\\infty \\quad \\text{in } H^1_0(\\Omega).$$\n\\end{theorem}\n\n\\begin{remark} Notice that~\\eqref{E:cont_prob_vinfty inexacto} always has a solution because~\\eqref{E:A Lipschitz} and~\\eqref{E:A fuertemente monotono} hold on $\\VV_\\infty$, which is itself a Hilbert space.\n\\end{remark}\n\n\\begin{proof}\nLet $\\{u_k\\}_{k\\in\\NN_0}$ be the sequence obtained with the Adaptive Algorithm. Let $u_\\infty\\in\\VV_\\infty$ denote the solution of~\\eqref{E:cont_prob_vinfty inexacto} and $\\PP_{k+1}:H^1_0(\\Omega)\\to\\VV_{k+1}$ be the orthogonal projection onto $\\VV_{k+1}$. Since $A$ is strongly monotone (cf.~\\eqref{E:A fuertemente monotono}), using~\\eqref{E:cont_prob_vinfty inexacto} and~\\eqref{E:kdisc-problem-nolineal inexacto} we have that\n\\begin{align*}\nc_A\\|\\nabla (u_k-u_\\infty)\\|_\\Omega^2&\\le \\langle Au_k-Au_\\infty,u_k-u_\\infty\\rangle\\\\\n&=\\langle Au_k,u_k-u_\\infty\\rangle-L(u_k-u_\\infty)\\\\\n&=\\langle Au_k,u_k-\\PP_{k+1}u_\\infty\\rangle+\\langle Au_k,\\PP_{k+1}u_\\infty-u_\\infty\\rangle\\\\\n&\\quad-L(u_k-\\PP_{k+1} u_\\infty)-L(\\PP_{k+1} u_\\infty-u_\\infty)\\\\\n&=a(u_k;u_k-u_{k+1},u_k-\\PP_{k+1}u_\\infty)+\\langle Au_k,\\PP_{k+1}u_\\infty-u_\\infty\\rangle\n\\\\\n&\\quad -L(\\PP_{k+1} u_\\infty-u_\\infty) \\\\\n&\\le C_a\\|\\nabla (u_k-u_{k+1})\\|_\\Omega(\\|\\nabla u_k\\|_\\Omega+\\|\\nabla u_\\infty\\|_\\Omega) \\\\\n&\\quad+(C_a\\|\\nabla u_k\\|_\\Omega+\\|L\\|_{H^{-1}(\\Omega)})\\|\\nabla (\\PP_{k+1}u_\\infty-u_\\infty)\\|_\\Omega,\n\\end{align*}\nfor all $k\\in\\NN_0$, where in the last inequality we have used~\\eqref{E:a_acotada}. From Remark~\\ref{R:uk acotada no lineal} it follows that $\\{u_k\\}_{k\\in\\NN_0}$ is bounded in $H^1_0(\\Omega)$, and using Lemma~\\ref{L:sucesivas a cero}, together with the fact that the spaces $\\{\\VV_k\\}_{k\\in\\NN_0}$ are nested and $\\cup_{k\\in\\NN_0} \\VV_k$ is dense in $\\VV_\\infty$, we conclude that $u_k\\to u_\\infty$ in $H^1_0(\\Omega)$. \n\\end{proof}\n\nIn order to show that the limiting function $u_\\infty$ is, in fact, the solution of the problem~\\eqref{E:cont_prob_nolineal} and thereby conclude that the adaptive sequence converges to the solution of this problem, we establish first two auxiliary results (see Lemma~\\ref{L:marcados tienden a cero} and Theorem~\\ref{T:convergencia debil del residuo}). We need the following\n\n\\begin{definition}\\label{D:splitting}\nGiven any sequence of meshes $\\{\\Tau_k\\}_{k\\in\\NN_0}\\subset\\TT$, with $\\Tau_{k+1}$ a refinement of $\\Tau_k$, for each $k \\in \\NN_0$, we define\n$$\\Tau_k^+:= \\{T\\in\\Tau_k \\mid T\\in\\Tau_m,\\quad \\forall\\,m\\ge k\\},\\qquad\\qquad\\Tau_k^0:= \\Tau_k \\setminus \\Tau_k^+,$$\nand\n$$\\qquad\\qquad\\Omega_k^+:= \\bigcup\\limits_{T\\in\\Tau_k^+} \\omega_k(T),\\qquad\\qquad\\qquad\\qquad\\Omega_k^0:= \\bigcup\\limits_{T\\in\\Tau_k^0} \\omega_k(T).$$\nIn words, $\\Tau_k^+$ is the subset of the elements of $\\Tau_k$ which are never refined in the adaptive process, and $\\Tau_k^0$ consists of the elements which are eventually refined.\n\\end{definition}\n\n\nIt can be proved~\\cite{MSV-convergence} that if $\\chi_{\\Omega_k^0}$ denotes the characteristic function of $\\Omega_k^0$, then\n\\begin{equation}\\label{E:hktendstozero}\n \\lim_{k\\rightarrow\\infty} \\|h_k \\chi_{\\Omega_k^0}\\|_{L^\\infty(\\Omega)}=0,\n\\end{equation}\nwhere $h_k\\in L^\\infty(\\Omega)$ denotes the piecewise constant mesh-size function satisfying ${h_k}_{|_T}\\definedasH_T$, for all $T\\in\\Tau_k$.\n\nSince the error estimators are stable (cf. Proposition~\\ref{P:estabilidad kakanov}), using the convergence proved in the last theorem and~\\eqref{E:hktendstozero} we can establish the following\n\n\\begin{lemma}[Estimator on marked elements]\\label{L:marcados tienden a cero} \nLet $\\left\\{\\{\\eta_k(T)\\}_{T\\in\\Tau_k}\\right\\}_{k\\in\\NN_0}$ be the sequence of local error estimators computed with the Adaptive Algorithm, and let $\\{\\MM_k\\}_{k\\in\\NN_0}$ be the sequence of subsets of marked elements over each mesh. Then, $$\\lim_{k\\to\\infty} \\max_{T\\in\\MM_k} \\eta_k(T)=0.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\left\\{\\{\\eta_k(T)\\}_{T\\in\\Tau_k}\\right\\}_{k\\in\\NN_0}$ and $\\{\\MM_k\\}_{k\\in\\NN_0}$ be as in the assumptions. For each $k\\in\\NN_0$, we select $T_k\\in\\MM_k$ such that $\\eta_k(T_k)=\\max_{T\\in\\MM_k} \\eta_k(T)$. Using Proposition~\\ref{P:estabilidad kakanov} we have that\n\\begin{equation}\\label{E:marcados tienden a cero aux1}\n\\eta_k(T_k)\\lesssim \\|\\nabla u_k\\|_{\\omega_k(T_k)}+\\|f\\|_{T_k}\\lesssim \\|\\nabla u_k-\\nabla u_\\infty\\|_{\\Omega}+\\|\\nabla u_\\infty\\|_{\\omega_k(T_k)}+\\|f\\|_{T_k},\n\\end{equation}\nwhere $u_\\infty\\in\\VV_\\infty$ is the solution of~\\eqref{E:cont_prob_vinfty inexacto}. On the one hand, the first term in the right hand side of~\\eqref{E:marcados tienden a cero aux1} tends to zero due to Theorem~\\ref{T:convergencia a un limite}. On the other hand, since $T_k\\in\\MM_k\\subset \\Tau_k^0$, from~\\eqref{E:hktendstozero} it follows that\n\\begin{equation*}\n|T_k|\\le|\\omega_k(T_k)|\\lesssim H_{T_k}^d\\le \\|h_k\\chi_{\\Omega_k^0}\\|_{L^\\infty(\\Omega)}^d\\rightarrow 0, \\quad\\text{as } k\\to\\infty,\n\\end{equation*}\nand the last two terms in the right hand side of~\\eqref{E:marcados tienden a cero aux1} also tend to zero.\n\\end{proof}\n\nUsing the upper bound~\\eqref{E:local-residual-upperbound}, the stability of the estimators (Proposition~\\ref{P:estabilidad kakanov}), the facts that $\\{u_k\\}_{k\\in\\NN_0}$ is bounded (cf. Remark~\\ref{R:uk acotada no lineal}) and the marking strategy is reasonable (cf.~\\eqref{E:estrategia razonable}), and Lemma~\\ref{L:marcados tienden a cero}, we now prove the following important result.\n\n\\begin{theorem}[Weak convergence of the residual]\\label{T:convergencia debil del residuo}\nIf $\\{u_k\\}_{k\\in\\NN_0}$ denotes the sequence of discrete solutions computed with the Adaptive Algorithm, then\n$$\n\\lim_{k\\to\\infty} \\langle\\RRR(u_k),v\\rangle=0,\n\\quad\\text{for all $v\\in H^1_0(\\Omega)$}.\n$$\n\\end{theorem}\n\n\\begin{proof}\nWe prove first the result for $v\\in H^2(\\Omega)\\cap H^1_0(\\Omega)$, and then extend it to $H^1_0(\\Omega)$ by density. Let $p\\in\\NN$ and $k>p$. By Definition~\\ref{D:splitting} we have that $\\Tau_p^+\\subset\\Tau_k^+\\subset\\Tau_k$.  Let $v_k\\in \\VV_k$ be the Lagrange's interpolant of $v$. Since $\\langle \\RRR(u_k),v_k\\rangle=0$, using~\\eqref{E:local-residual-upperbound}, and Cauchy-Schwartz's inequality we have that\n\\begin{align*}\n|\\langle\\RRR(u_k),v\\rangle|&=|\\langle\\RRR(u_k),v-v_k\\rangle|\n\\lesssim \\sum_{T\\in\\Tau_k} \\eta_k(T)\\|\\nabla (v-v_k)\\|_{\\omega_k(T)}\\\\\n&= \\sum_{T\\in\\Tau_p^+} \\eta_k(T)\\|\\nabla(v-v_k)\\|_{\\omega_k(T)}+\\sum_{T\\in\\Tau_k\\setminus\\Tau_p^+} \\eta_k(T)\\|\\nabla(v-v_k)\\|_{\\omega_k(T)}\\\\\n&\\lesssim \\eta_k(\\Tau_p^+)\\|\\nabla(v-v_k)\\|_{\\Omega_p^+}+\\eta_k(\\Tau_k\\setminus\\Tau_p^+)\\|\\nabla(v-v_k)\\|_{\\Omega_p^0},\n\\end{align*}\nwhere, for any $\\Xi\\subset\\Tau_k$ we hereafter denote $\\left(\\sum_{T\\in\\Xi}\\eta_k^2(T)\\right)^{\\frac12}$ by $\\eta_k(\\Xi)$.\nTaking into account Proposition~\\ref{P:estabilidad kakanov} and the boundedness of the discrete solutions (cf. Remark~\\ref{R:uk acotada no lineal}) we have that $\\eta_k(\\Tau_k\\setminus\\Tau_p^+)\\le\\eta_k(\\Tau_k)\\lesssim 1$, and therefore,\n\\[\n|\\langle\\RRR(u_k),v\\rangle|\\lesssim \\left(\\eta_k(\\Tau_p^+)+\\|h_p \\chi_{\\Omega_p^0}\\|_{L^\\infty(\\Omega)}\\right)|v|_{H^2(\\Omega)},\n\\]\ndue to interpolation estimates.\n\nIn order to prove that $\\langle\\RRR(u_k),v\\rangle \\to 0$ as $k\\to\\infty$ we now let $\\varepsilon>0$ be arbitrary. Due to~\\eqref{E:hktendstozero}, there exists $p\\in\\NN$ such that\n$$ \\|h_p \\chi_{\\Omega_p^0}\\|_{L^\\infty(\\Omega)}<\\varepsilon.$$\nOn the other hand, since $\\Tau_p^+\\subset\\Tau_k^+\\subset\\Tau_k$ and the marking strategy is reasonable (cf.~\\eqref{E:estrategia razonable}), \n$$\\eta_k(\\Tau_p^+)\\le (\\# \\Tau_p^+)^{1\/2} \\max_{T\\in\\Tau_p^+}\\eta_k(T)\\le (\\# \\Tau_p^+)^{1\/2} \\max_{T\\in\\MM_k}\\eta_k(T).$$\nNow, by Lemma~\\ref{L:marcados tienden a cero}, we can select $K>p$ such that\n$\\eta_k(\\Tau_p^+)<\\varepsilon$,\nfor all $k>K$.\n\nSummarizing, we have proved that\n\\[\n\\lim_{k\\to\\infty} \\langle\\RRR(u_k),v\\rangle=0,\\qquad \\text{for all } v\\in H^2(\\Omega)\\cap H^1_0(\\Omega).\n\\]\nFinally, since $H^{2}(\\Omega)\\cap H^1_0(\\Omega)$ is dense in $H^1_0(\\Omega)$, this limit is also zero for all $v\\in H^1_0(\\Omega)$.\n\\end{proof}\n\nAs a consequence of Theorem~\\ref{T:convergencia debil del residuo} we now prove that $u_\\infty$ is the solution of problem~\\eqref{E:cont_prob_nolineal}.\n\n\\begin{theorem}[The limiting function is the solution]\\label{T:u_infty es solucion inexacto}\nIf $u_\\infty$ denotes the solution of~\\eqref{E:cont_prob_vinfty inexacto}, then $u_\\infty$ is the solution of problem~\\eqref{E:cont_prob_nolineal}, i.e.,\n\\begin{equation*}\na(u_\\infty;u_\\infty,v)=L(v),\\qquad\\forall\\,v\\in H^1_0(\\Omega).\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nLet $u_\\infty$ be the solution of~\\eqref{E:cont_prob_vinfty inexacto}. If $v\\in H^1_0(\\Omega)$, and $\\{u_k\\}_{k\\in\\NN_0}$ denotes the sequence of discrete solutions computed with the Adaptive Algorithm, then\n\\begin{align*}\n|a(u_\\infty;u_\\infty,v)-L(v)|&=|a(u_\\infty;u_\\infty,v)-L(v)-a(u_k;u_{k+1},v)+a(u_k;u_{k+1},v)|\\\\\n&\\le |a(u_\\infty;u_\\infty,v)-a(u_k;u_{k+1},v)|+|\\langle \\RRR(u_{k+1}),v\\rangle|\\\\\n&\\le|a(u_\\infty;u_\\infty,v)-a(u_k;u_k,v)|\n\\\\\n&\\quad +|a(u_k;u_k-u_{k+1},v)|+|\\langle \\RRR(u_{k+1}),v\\rangle|\\\\\n&\\le \\|Au_\\infty-Au_k\\|_{H^{-1}(\\Omega)}\\|\\nabla v\\|_\\Omega\n\\\\\n&\\quad+C_a\\|\\nabla (u_k-u_{k+1})\\|_\\Omega\\|\\nabla v\\|_\\Omega+|\\langle \\RRR(u_k),v\\rangle|\\\\\n&\\le  C_A\\|\\nabla (u_\\infty-u_k)\\|_\\Omega\\|\\nabla v\\|_\\Omega\n\\\\\n&\\quad+C_a\\|\\nabla (u_k-u_{k+1})\\|_\\Omega\\|\\nabla v\\|_\\Omega+|\\langle \\RRR(u_k),v\\rangle|,\n\\end{align*}\nwhere we have used that $A$ is Lipschitz (cf.~\\eqref{E:A Lipschitz}) and $a$ is bounded (cf.~\\eqref{E:a_acotada}). Using  Theorem~\\ref{T:convergencia a un limite}, Lemma~\\ref{L:sucesivas a cero} and Theorem~\\ref{T:convergencia debil del residuo} it follows that the right-hand side in the last inequality tends to zero as $k$ tends to infinity\n\\end{proof}\n\nAs an immediate consequence of Theorems~\\ref{T:convergencia a un limite} and~\\ref{T:u_infty es solucion inexacto} we finally obtain the main result of this article.\n\n\\begin{theorem}[Main result]\\label{T:main}\nLet $\\{u_k\\}_{k\\in\\NN_0}$ denote the sequence of discrete solutions computed with the Adaptive Algorithm. If $\\alpha(\\cdot,\\cdot)$ satisfies assumptions~\\eqref{E:betasegunda acotada} and~\\eqref{alpha-decreasing}, then $\\{u_k\\}_{k\\in\\NN_0}$ converges to the solution $u$ of  problem~\\eqref{E:cont_prob_nolineal}.\n\\end{theorem}\n\nWe conclude this section with a couple of remarks.\n\n\\begin{remark}\nThe problem given by~\\eqref{E:ley de conservacion} is a particular case of the more general problem:\n\\begin{equation*\n\\left\\{\n\\begin{aligned}\n-\\nabla\\cdot \\big[\\alpha(\\,\\cdot\\,,|\\nabla u|_\\AAA^2)\\AAA\\nabla u\\big]&= f\\qquad & & \\text{in}\\,\\Omega\\\\\nu&= 0\\qquad & &\\text{on}\\,\\partial \\Omega,\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $\\alpha:\\Omega\\times \\mathbb R_+\\to \\mathbb R_+$ and $f\\in L^2(\\Omega)$ satisfy the properties assumed in the previous sections, and $\\AAA:\\Omega\\to \\mathbb R^{d\\times d}$ is\nsymmetric for all $x\\in\\Omega$, and uniformly positive definite, i.e., there exist constants $\\underline{a},\\overline{a}>0$ such that\n\\begin{equation*\n\\underline{a}|\\xi|^2\\leq \\AAA(x)\\xi\\cdot \\xi\\leq \\overline{a}|\\xi|^2,\\qquad \\forall~x\\in\\Omega,\\,\\xi\\in\\mathbb R^d.\n\\end{equation*}\n If $\\AAA$ is piecewise Lipschitz over an initial conforming mesh $\\Tau_0$ of $\\Omega$, i.e., there exists $C_\\AAA>0$ such that\n\\begin{equation*\n\\|\\AAA(x)-\\AAA(y)\\|_2\\le C_\\AAA|x-y|,\\qquad\\forall\\,x,y\\in T,\\quad\\forall\\,T\\in\\Tau_0,\n\\end{equation*}\nthen the convergence results previously presented also hold for this problem.\n\\end{remark}\n\n\n\\begin{remark}\\label{R:polynomial degree}\nWe have assumed the use of \\emph{linear} finite elements for the discretization (see~\\eqref{E:V_Tau}). It is important to notice that the only place where we used this is in Proposition~\\ref{P:estabilidad kakanov}.  The rest of the steps of the proof hold regardless of the degree of the finite element space. The use of linear finite elements is customary in nonlinear problems, because they greatly simplify the analysis. The numerical experiments of the next section show a competitive performance of the adaptive method for any tested polynomial degree (up to four).\n\\end{remark}\n\n\\section{Numerical Experiments}\\label{S:numerical experiments}\n\nWe conclude this article reporting on the behavior of the adaptive algorithm for some particular nonlinear problems. \nIn the first subsection we study the convergence rate in terms of degrees of freedom for an exact solution and different functions $\\alpha(\\cdot,\\cdot)$. \nIn the second subsection we show the performance of the algorithm when approximating an unknown solution of a prescribed curvature equation. \n\n\\subsection{Exact solution}\n\nLet us consider the problem\n\\begin{equation}\\label{E:problema general}\n\\left\\{\n\\begin{aligned}\n-\\nabla\\cdot \\big[\\alpha(|\\nabla u|^2)\\nabla u\\big]&= f& &\\qquad \\text{in}\\,\\Omega\\\\\nu&= g& &\\qquad\\text{on}\\,\\partial \\Omega,\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\Omega\\subset\\mathbb R^2$ is the $L$-shaped domain given in Figure~\\ref{f:solucion}. In the following examples, in order to study experimentally the behavior of the Adaptive Algorithm, we consider~\\eqref{E:problema general} with different choices of the function $\\alpha$, defining $f$ and $g$ in each case so that the solution of the problem is the function $u$ depicted in Figure~\\ref{f:solucion}, given in polar coordinates by\n\\begin{equation}\\label{E:u polar}\nu(r,\\varphi)=r^{\\frac{2}{3}}\\sin\\left(\\frac{2}{3}\\varphi\\right).\n\\end{equation}\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{W}{\\small$\\Omega$}\n\\psfrag{1}{\\tiny $1$}\n\\psfrag{-1}{\\tiny $-1$}\n\\hfil\n\\includegraphics[width=.30\\textwidth]{figuras\/L-shape}\n\\hfil\n\\includegraphics[width=.34\\textwidth]{figuras\/funcionsingular1}\n\\caption{The domain $\\Omega$ where the problem~\\eqref{E:problema general} is posed and the function $u$ which is the solution of the problem in each example.}\\label{f:solucion}\n\\end{center}\n\\end{figure}\n\nWe consider the Adaptive Algorithm using different marking strategies, namely, \\emph{global refinement}, \\emph{maximum strategy} with $\\theta=0.7$ and \\emph{D\\\"orfler's strategy} with $\\theta=0.5$ (see~\\cite{Alberta}).\n\nWe implemented the Adaptive Algorithm using the finite element toolbox ALBERTA~\\cite{Alberta}. We iterated the algorithm until the global error estimator was below $10^{-6}$ or the number of degrees of freedom exceeded $5\\times 10^5$. We tested the limits of our theory by trying with some functions $\\alpha$ which did not satisfy all the assumptions of the theoretical results above.\n\n\n\\begin{example}[Optimal rate of convergence when $\\alpha$ satisfies the hypotheses]\\label{Ex:todo_ok}\nAs a first example, in order to study experimentally the rate of convergence of the Adaptive Algorithm, we consider\n$$\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{2},\\qquad t>0,$$\nwhich satisfies the hypotheses to guarantee the convergence (see Figure~\\ref{f:todo_ok}), i.e., $\\alpha$ is a $\\CCC^1$-function, and there exist positive constant $c_a$ and $C_a$ such that\n\\begin{equation}\\label{E:Hip beta segunda}\nc_a\\le\\alpha(t^2)+2t^2\\alpha'(t^2)\\le C_a,\\qquad\\forall\\,t>0,\n\\end{equation}\nand\n\\begin{equation}\\label{E:Hip alpha decreciente}\n\\text{$\\alpha$ is monotone decreasing, i.e., $\\alpha'(t)\\le 0$ for all $t>0$.}\n\\end{equation}\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{alpha}{\\hspace{-3pt}\\tiny $\\alpha(t)$}\n\\psfrag{betasegundabetasegundabetasegunda}{\\tiny$\\alpha(t^2)+2t^2\\alpha'(t^2)$}\n\\psfrag{t}{\\tiny $t$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/alpha}\n\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/betasegunda}\n\\caption{The function $\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{2}$, of Example~\\ref{Ex:todo_ok} satisfies the properties we require to guarantee the convergence.}\\label{f:todo_ok}\n\\end{center}\n\\end{figure}\n\nIn Figure~\\ref{f:todo_ok_error} we plot the $H^1(\\Omega)$-error versus the number of degrees of freedom (DOFs), for finite elements of degree $\\ell=1,2,3,4$. In this case, the rate of the convergence is optimal for adaptive strategies, that is, $\\| u - u_k \\|_{H^1(\\Omega)} =O(\\text{DOFs}_k^{-\\ell\/2})$. For global refinement, the observed order of convergence is $\\text{DOFs}_k^{-1\/3}$ for all tested polynomial degrees, due to the fact that the solution $u$ belongs to $H^{1+\\delta}(\\Omega)$, for all $0<\\delta<\\frac{2}{3}$, and does not belong to $H^{1+2\/3}(\\Omega)$. \n\nNote that, although the theory only guarantees the plain convergence for linear elements (cf. Theorem~\\ref{T:main}), the numerical results suggest that the method works for any polynomial degree (see Remark~\\ref{R:polynomial degree}), and the convergence rate is optimal.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{Refinamiento Global}{\\tiny Global refinement}\n\\psfrag{Estrategia del Maximo}{\\tiny Maximum Strategy}\n\\psfrag{Estrategia de Dorfler}{\\tiny D\\\"orfler Strategy}\n\\psfrag{Pendiente Optima}{\\tiny Optimal slope}\n\\psfrag{Error}{\\hspace{-10pt}\\tiny $H^1$-error}\n\\psfrag{DOFs}{\\tiny DOFs}\n\\psfrag{Aproximacion con polinomios de grado 1}{ \\tiny Polynomial degree $\\ell=1$}\n\\psfrag{Aproximacion con polinomios de grado 2}{ \\tiny Polynomial degree $\\ell=2$}\n\\psfrag{Aproximacion con polinomios de grado 3}{ \\tiny Polynomial degree $\\ell=3$}\n\\psfrag{Aproximacion con polinomios de grado 4}{ \\tiny Polynomial degree $\\ell=4$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/error1}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/error2}\n\n\\bigskip\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/error3}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo1\/error4}\n\\caption{\nError versus DOFs for Example~\\ref{Ex:todo_ok}.\nWe present the $H^1(\\Omega)$-error between the exact solution and discrete solutions, versus the number of degrees of freedom (DOFs) used to represent each of them. We note that the convergence rate is optimal for the considered adaptive strategies, but not for global refinement, due to the fact that the solution $u$ is not sufficiently smooth. In this case, $\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{2}$ satisfies all the properties established to guarantee the convergence with linear finite elements. The numerical experiments suggest that the method converges with optimal rate for any polynomial degree.}\\label{f:todo_ok_error}\n\\end{center}\n\\end{figure}\n\\end{example}\n\n\n\\begin{example}[About the hypothesis~\\eqref{E:Hip beta segunda}]\\label{Ex:betasegundabad}\nWe consider the function\n$$\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{10},\\qquad t>0,$$\nwhich is monotone decreasing, i.e., satisfies~\\eqref{E:Hip alpha decreciente}, but not~\\eqref{E:Hip beta segunda}, as it is shown in Figure~\\ref{f:betasegundabad}. Since~\\eqref{E:Hip beta segunda} guarantees the well-posedness of problem~\\eqref{E:problema general} (uniqueness and stability), we could be facing an example with multiple solutions.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{alpha}{\\hspace{-3pt}\\tiny $\\alpha(t)$}\n\\psfrag{betasegundabetasegundabetasegunda}{\\tiny$\\alpha(t^2)+2t^2\\alpha'(t^2)$}\n\\psfrag{t}{\\tiny $t$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/alpha}\n\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/betasegunda}\n\\caption{The function $\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{10}$ of Example~\\ref{Ex:betasegundabad} does not satisfy $\\alpha(t^2)+2t^2\\alpha'(t^2)>0$ for all $t>0$.}\\label{f:betasegundabad}\n\\end{center}\n\\end{figure}\n\nIn Figure~\\ref{f:betasegundabad_error} we plot the $H^1(\\Omega)$-error versus the number degrees of freedom, for different polynomial degrees. For $\\ell=3$ and $\\ell=4$ the algorithm stopped with an estimator below the desired tolerance $10^{-6}$, although the error is around $10^{-2}$ in all cases. On the other hand, as we can see in Figure~\\ref{f:betasegundabad_estimador}, the global error estimator decreases with optimal rate for the adaptive strategies, indicating that the adaptive algorithm may be converging to another solution of the nonlinear problem, different from the one given by~\\eqref{E:u polar}.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{Refinamiento Global}{\\tiny Global refinement}\n\\psfrag{Estrategia del Maximo}{\\tiny Maximum Strategy}\n\\psfrag{Estrategia de Dorfler}{\\tiny D\\\"orfler Strategy}\n\\psfrag{Pendiente Optima}{\\tiny Optimal slope}\n\\psfrag{Error}{\\hspace{-10pt}\\tiny $H^1$-error}\n\\psfrag{DOFs}{\\tiny DOFs}\n\\psfrag{Aproximacion con polinomios de grado 1}{ \\tiny Polynomial degree $\\ell=1$}\n\\psfrag{Aproximacion con polinomios de grado 2}{ \\tiny Polynomial degree $\\ell=2$}\n\\psfrag{Aproximacion con polinomios de grado 3}{ \\tiny Polynomial degree $\\ell=3$}\n\\psfrag{Aproximacion con polinomios de grado 4}{ \\tiny Polynomial degree $\\ell=4$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/error1}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/error2}\n\n\\bigskip\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/error3}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/error4}\n\\caption{\nError versus DOFs for Example~\\ref{Ex:betasegundabad}.\nWe present the $H^1(\\Omega)$-error between the exact and discrete solutions, versus DOFs. We observe that the method does not converge, but the error stagnates around $10^{-2}$.\nSince $\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{10}$ does not satisfy the condition which guarantees uniqueness of solutions, and based on the fact that the a posteriori error estimator does tend to zero (see Figure~\\ref{f:betasegundabad_estimador}) we conclude that the method converges to \\emph{another solution} of the same problem, different from the one given by~\\eqref{E:u polar}.\n}\\label{f:betasegundabad_error}\n\\end{center}\n\\end{figure}\n\n\n\nBased on this remark, it seems that the adaptive algorithm converges to a solution $u_1$ such that $\\|u-u_1\\|_{H^1(\\Omega)}\\approx 10^{-2}$. We recall that $\\alpha$ does not satisfy the condition~\\eqref{E:Hip beta segunda} which guarantees the uniqueness of solutions.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{Refinamiento Global}{\\tiny Global refinement}\n\\psfrag{Estrategia del Maximo}{\\tiny Maximum Strategy}\n\\psfrag{Estrategia de Dorfler}{\\tiny D\\\"orfler Strategy}\n\\psfrag{Pendiente Optima}{\\tiny Optimal slope}\n\\psfrag{Estimador global}{\\hspace{-10pt}\\tiny global error estimator}\n\\psfrag{DOFs}{\\tiny DOFs}\n\\psfrag{Aproximacion con polinomios de grado 1}{ \\tiny Polynomial degree $\\ell=1$}\n\\psfrag{Aproximacion con polinomios de grado 2}{ \\tiny Polynomial degree $\\ell=2$}\n\\psfrag{Aproximacion con polinomios de grado 3}{ \\tiny Polynomial degree $\\ell=3$}\n\\psfrag{Aproximacion con polinomios de grado 4}{ \\tiny Polynomial degree $\\ell=4$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/estimadores\/estimador1}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/estimadores\/estimador2}\n\n\\bigskip\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/estimadores\/estimador3}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo2\/estimadores\/estimador4}\n\\caption{\nEstimator versus DOFs for Example~\\ref{Ex:betasegundabad}.\nWe present the global error estimator $\\eta_k(\\Tau_k)$ versus DOFs used to represent each discrete solution. We note that for the adaptive strategies the global error estimator decreases with the optimal rate for the $H^1$-error, although the error does not tend to zero (see Figure~\\ref{f:betasegundabad_error}). In this case, $\\alpha(t)=\\frac{1}{1+t}+\\frac{1}{10}$ does not satisfy the condition which guarantees uniqueness of solutions. It seems that the method converges to \\emph{another solution} of the problem\n}\\label{f:betasegundabad_estimador}\n\\end{center}\n\\end{figure}\n\n\\end{example}\n\n\n\\begin{example}[About the hypothesis~\\eqref{E:Hip alpha decreciente}]\\label{Ex:creciente}\nWe consider\n$$\\alpha(t)=-\\frac12\\e^{-\\frac{3}{2}t}+1,\\qquad t>0,$$\nwhich satisfies the hypothesis~\\eqref{E:Hip beta segunda} related to the  well-posedness of the problem but not~\\eqref{E:Hip alpha decreciente}, because $\\alpha$ is monotone increasing (see Figure~\\ref{f:creciente}).\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{alpha}{\\hspace{-3pt}\\tiny $\\alpha(t)$}\n\\psfrag{betasegundabetasegundabetasegunda}{\\tiny$\\alpha(t^2)+2t^2\\alpha'(t^2)$}\n\\psfrag{t}{\\tiny $t$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/alpha}\n\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/betasegunda}\n\\caption{The function $\\alpha(t)=-\\frac12\\e^{-\\frac{3}{2}t}+1$ of Example~\\ref{Ex:creciente} satisfies the hypothesis~\\eqref{E:Hip beta segunda} but is monotone increasing.}\\label{f:creciente}\n\\end{center}\n\\end{figure}\n\nIn Figure~\\ref{f:creciente_error} we plot $H^1(\\Omega)$-error versus the number of degrees of freedom, for finite elements of degrees $\\ell = 1, 2, 3, 4$. Note that in this case the optimal convergence rate $\\text{DOFs}^{-\\ell\/2}$ is still observed for the adaptive algorithm. This is an indication that the assumption~\\eqref{E:Hip alpha decreciente} about $\\alpha$ being monotone decreasing can be superfluous, and only an artificial requirement for the presented proof (see Lemma~\\ref{L:key property}). A more detailed study about this hypothesis is subject of future research, and beyond the scope of this article.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{Refinamiento Global}{\\tiny Global refinement}\n\\psfrag{Estrategia del Maximo}{\\tiny Maximum Strategy}\n\\psfrag{Estrategia de Dorfler}{\\tiny D\\\"orfler Strategy}\n\\psfrag{Pendiente Optima}{\\tiny Optimal slope}\n\\psfrag{Error}{\\hspace{-10pt}\\tiny $H^1$-error}\n\\psfrag{DOFs}{\\tiny DOFs}\n\\psfrag{Aproximacion con polinomios de grado 1}{ \\tiny Polynomial degree $\\ell=1$}\n\\psfrag{Aproximacion con polinomios de grado 2}{ \\tiny Polynomial degree $\\ell=2$}\n\\psfrag{Aproximacion con polinomios de grado 3}{ \\tiny Polynomial degree $\\ell=3$}\n\\psfrag{Aproximacion con polinomios de grado 4}{ \\tiny Polynomial degree $\\ell=4$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/error1}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/error2}\n\n\\bigskip\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/error3}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo3\/error4}\n\\caption{\nError versus DOFs for Example~\\ref{Ex:creciente}.\nWe present the $H^1(\\Omega)$-error between the exact and discrete solutions, versus DOFs. We note that the convergence rate is optimal for the considered adaptive strategies, although the function $\\alpha(t)=-\\frac12\\e^{-\\frac{3}{2}t}+1$ is not monotone decreasing. This could mean that this hypothesis is not necessary for the convergence of the Adaptive Algorithm, which performs better than expected by our theory.\n}\\label{f:creciente_error}\n\\end{center}\n\\end{figure}\n\\end{example}\n\n\n\n\n\\begin{example}\\label{Ex:derivadainfinita}\nFinally, we consider an extreme case, with\n$$\\alpha(t)=2-\\frac{\\sqrt{t}}{1+\\sqrt{t}},\\qquad t>0,$$\nwhich satisfies~\\eqref{E:Hip beta segunda} and~\\eqref{E:Hip alpha decreciente}, but $\\lim_{t\\to 0^+} \\alpha'(t)=-\\infty$, as can be observed in Figure~\\ref{f:derivadainfinita}. This means that $\\alpha$ is not Lipschitz continuous. Since we only require that $|t\\alpha'(t)|$ is bounded (cf.~\\eqref{E:Hip beta segunda}), it still satisfies the assumptions of the convergence theory, and optimality is observed regardless of the fact that $\\lim_{t\\to 0^+} \\alpha'(t)=-\\infty$.\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{alpha}{\\hspace{-3pt}\\tiny $\\alpha(t)$}\n\\psfrag{betasegundabetasegundabetasegunda}{\\tiny$\\alpha(t^2)+2t^2\\alpha'(t^2)$}\n\\psfrag{t}{\\tiny $t$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/alpha}\n\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/betasegunda}\n\\caption{The function $\\alpha(t)=2-\\frac{\\sqrt{t}}{1+\\sqrt{t}}$ of Example~\\ref{Ex:derivadainfinita} satisfies our assumptions which do not require that $\\alpha$ is Lipschitz continuous.}\\label{f:derivadainfinita}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{f:derivadainfinita_error} shows $H^1(\\Omega)$-error versus DOFs, for polynomial degrees $\\ell = 1, 2, 3, 4$. We obtain optimal convergence rate $\\text{DOFs}^{-\\ell\/2}$ for the adaptive strategies. Thus, even when $\\alpha$ is not Lipschitz, the convergence rate is optimal. As a consequence we conclude that it should not be necessary to make additional assumptions in order to prove optimality.\n\n\n\\begin{figure}[h!tbp]\n\\begin{center}\n\\psfrag{Refinamiento Global}{\\tiny Global refinement}\n\\psfrag{Estrategia del Maximo}{\\tiny Maximum Strategy}\n\\psfrag{Estrategia de Dorfler}{\\tiny D\\\"orfler Strategy}\n\\psfrag{Pendiente Optima}{\\tiny Optimal slope}\n\\psfrag{Error}{\\hspace{-10pt}\\tiny $H^1$-error}\n\\psfrag{DOFs}{\\tiny DOFs}\n\\psfrag{Aproximacion con polinomios de grado 1}{ \\tiny Polynomial degree $\\ell=1$}\n\\psfrag{Aproximacion con polinomios de grado 2}{ \\tiny Polynomial degree $\\ell=2$}\n\\psfrag{Aproximacion con polinomios de grado 3}{ \\tiny Polynomial degree $\\ell=3$}\n\\psfrag{Aproximacion con polinomios de grado 4}{ \\tiny Polynomial degree $\\ell=4$}\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/error1}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/error2}\n\n\\bigskip\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/error3}\\hfil\n\\includegraphics[width=.40\\textwidth]{figuras\/ejemplo5\/error4}\n\\caption{\nError versus DOFs for Example~\\ref{Ex:derivadainfinita}.\nWe present the $H^1(\\Omega)$-error between the exact and discrete solutions, versus DOFs. We note that the convergence rate is optimal for the considered adaptive strategies, even though the function $\\alpha(t)=2-\\frac{\\sqrt{t}}{1+\\sqrt{t}}$ has an infinite derivative at $0$. This example falls inside the theory, and the fact that $\\alpha$ is not Lipschitz continuous does not destroy the optimality of the sequence of discrete solutions.\n}\\label{f:derivadainfinita_error}\n\\end{center}\n\\end{figure}\n\\end{example}\n\n\\subsection{Unknown solution}\n\nIn this section we use the Adaptive Algorithm to approximate a solution to a prescribed mean curvature problem. We consider the problem\n\\begin{equation}\\label{E:curvature}\n\\left\\{\\begin{aligned}\n-\\nabla \\cdot \\left[ \\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}} \\right] &= f \\quad & &\\text{in }\\Omega = (-1,1)\\times(-1,1) \n\\\\\nu &= 0 \\quad& &\\text{on }\\partial\\Omega,\n\\end{aligned}\\right.\n\\end{equation}\nwith \n\\[\nf(x) = \\begin{cases}\n5 \\quad &\\text{if } |x| \\le 1\/3,\n\\\\\n-3 \\quad &\\text{if } 1\/3 < |x| \\le 2\/3,\n\\\\\n0 \\quad&\\text{otherwise}.\n\\end{cases}\n\\]\nThe function $\\alpha(t) = 1\/\\sqrt{1+t}$ corresponding to this equation does not fulfill~\\eqref{E:Hip beta segunda} because $\\alpha(t),\\alpha'(t) \\to 0$ as $t\\to\\infty$. Nevertheless, for many right-hand side functions $f$, as the one stated above, the solution belongs to $W^{1,\\infty}(\\Omega)$. This implies that $|\\nabla u|$ is bounded and $\\alpha$ could be replaced by a function satisfying~\\eqref{E:Hip beta segunda} without changing the solution. This is not needed in practice, but is rather a theoretical tool for proving that the assumptions hold in some sense.\n\n We experimented with several right-hand sides $f$ and observed that the method performs robustly whenever $u \\in W^{1,\\infty}(\\Omega)$. When the solution has an unbounded gradient, the method produces a sequence with a maximum value increasing to infinity. This is a drawback of our method, since it cannot be used to approximate singular solutions (not belonging to $W^{1,\\infty}(\\Omega)$), if $\\alpha$ does not satisfy~\\eqref{E:Hip beta segunda}. It is worth recalling that adaptivity is still a good tool for approximating regular solutions, specially taking into account the fact that in this context \\emph{regularity} is a concept relative to the polynomial degree. Using adaptivity with higher order finite elements can improve drastically the performance even when the solution is in $H^2(\\Omega)\\cap W^{1,\\infty}(\\Omega)$.\n\nWe present in Figure~\\ref{F:curvature} a picture of the solution obtained with our method and several meshes at different iteration steps, for polynomial degrees 1, 2 and 3. It is worth observing the stronger grading obtained for higher polynomial degrees. This obeys the fact that we mention above, that the singularity of the solution is relative to the polynomial degree of the finite element space. Recall that in practice adaptive methods with polynomials of degree $\\ell$ on domains in $\\mathbb R^d$ converge with order DOF$^{-\\ell\/d}$. In order to obtain this rate with uniform meshes, we would need the solution to belong to $H^{\\ell+1}(\\Omega)$.\n\n\\begin{figure}[h!tbp]\n\n\\hfil\n\\includegraphics[width=.6\\textwidth]{figuras\/curvature\/curvature20}\n\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-1\/mesh-00010}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-1\/mesh-00015}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-1\/mesh-00020}\n\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-2\/mesh-00010}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-2\/mesh-00015}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-2\/mesh-00020}\n\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-3\/mesh-00010}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-3\/mesh-00015}\n\\hfil\n\\includegraphics[width=.32\\textwidth]{figuras\/curvature\/meshes-degree-3\/mesh-00020}\n\\caption{\\label{F:curvature}Adaptive meshes obtained when solving the prescribed mean curvature equation~\\eqref{E:curvature} with polynomials of degree 1 (top), 2 (middle), 3 (bottom). The meshes correspond to iteration count 10 (left), 15 (middle) and 20 (right). It is worth observing the higher grading presented by the meshes corresponding to higher polynomial degree.}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThanks to the Internet, we are now living in the global village where emails from the U.S. can be transmitted to China within a tenth of second, and real-time teleconferences connect people all over the world. The  Internet of Things (IoT) goes even further beyond, not only affecting the way we exchange data, but also touching the physical world.  Fig. \\ref{fig1-iot} shows some scenarios where devices connected to IoT has changed our living:  the smart household appliances in our homes, the wearable gadgets accompany us everyday, the autonomous vehicles, and the industrial control system. In the not too distant future, it would be almost impossible to buy new devices that are not connected to the IoT. And it is estimated that IoT technologies will have an  impact of several trillions to the global economy by 2020 \\cite{IEC2016}.\n\nHowever, the security  and privacy concerns on IoT are always   clouds hanging upon us. As pointed out by Bruce Schneier \\cite{Schneier2017}, a security technologist at Harvard University and the chief technology officer of IBM Resilient, the  IoT  companies  are rushing to make their products cheaper and smarter, but without much care about security. The Ukraine power grid cyberattacks remind us that the smart IoT can help us control our light-bulbs, but if under attacks it might also take us into darkness.\nNowadays, many literatures have tried to address the concerns on IoT security  \\cite{Fernandes2017}, but few of them take into consideration the sever threat to IoT coming from the advances of quantum computing.\n\n\\begin{figure}[!htbp]\n\\centering\n \\includegraphics[width=.7\\textwidth]{p1.eps}\n\\caption{Illustration of smart IoT applications.}\n \\label{fig1-iot}\n\\end{figure}\n\n\n\n\nAlthough quantum computers bear some debates over scientists, with the ever-looming breakthroughs of quantum computing, many researchers are becoming more and more positive about the future of large-scale quantum computers. In March 2017, IBM launched an industry-first initiative, called the IBM Q system, to build a commercially available universal quantum computing system for business and science applications. The publicly available universal quantum processor consists of 15 qubits and their commercially available 17-qubit processor is claimed to be at least twice as powerful.\n\nThe quantum threats to cryptography apply equally, or even to a greater extent, to smart objects extensively used in smart IoT services since they involve platforms and systems which are difficult to update. For example, embedded devices in wearables and furnitures are difficult to update  and the scalability issue in IoT devices further complicates the problem. Therefore, we should  taken into consideration post-quantum security   when designing secure architectures and systems for smart IoT, \\emph{now}.\n\nRecently, Cheng et al. has called the attention to using post-quantum cryptography (PQC) to secure IoT \\cite{Cheng2017}. As a promising candidate for the future PQC standard, lattice-based cryptography enjoys the advantages of strong security guarantees and high efficiency, which make  it extremely suitable for IoT applications. In this paper, we focus on introducing the advantages of lattice-based cryptography  and the state of art of  their implementations for IoT devices.\n\n In the following, we first give a brief introduction to cryptography and the impact of quantum computers. Then we explain why lattice-based cryptography is a proper choice for smart IoT. Next we give detailed discussions on the state-of-art implementations of lattice-based cryptography on constrained devices, following a high-level overview of lattice based cryptography. Finally we share our opinion on current challenges and directions for future explorations regarding the application of lattice-based cryptography in IoT systems.\n\n\n\n\\section{Cryptography and Quantum Computers }\nBeneath all security protocols, cryptography is used as a fundamental building block.  The canonical implication of security is \\emph{confidentiality}, which requires that sensitive information can not be learned by unauthorized party. Symmetric encryption is the simplest and the most popular way of achieving confidentiality. Two communicating parties, Alice and Bob, share a common secret key which is used for both encryption and decryption. Without the knowledge of the secret key, a third party can not learn the encrypted information from the ciphertext.\n\n\nSymmetric encryption requires a shared common key between two parties, which belongs to the area of symmetric-key cryptography. One drawback of symmetric-key cryptography is the difficulty of establishing secret keys. This is usually done via some costly secure channels such as face-to-face meeting, use of trusted courier or even quantum key distribution. These methods are highly difficult and expensive. Asymmetric-key cryptography (aka public-key cryptography) can be used to overcome this difficulty as it provides a mechanism to distribute cryptographic keys over insecure channel. In public-key cryptography, Alice has a pair of related keys: one is the private key and the other is called the public key.  The private key, as suggested by its name, is kept private to Alice herself while her public key is known to everyone.\n\nUsing public-key encryption algorithms, everyone can encrypt message and send it to Alice using her public key. But only Alice who has the private key is able to decrypt. This feature allows Bob to encrypt a secret session key of a symmetric encryption scheme such as AES and transmit it to Alice. After decrypting, Alice gets the key for AES and can now establish a secure channel with Bob via AES using the session key. This is called hybrid encryption and is used in many security protocols such as Transport Layer Security (TLS) protocol. Another method known as the key exchange protocol allows Alice and Bob to negotiate session key over an  insecure channel.\n\nYet another problem arises. How can Bob, or anyone, make sure that the claimed public key for Alice indeed belongs to Alice but not Eve? This involves the notion of \\emph{trust} in cryptography. Generally two solutions are available. One is to use the Public Key Infrastructure (PKI) and the other is to use Identity Based Encryption (IBE).\n\nPKI is a mechanism that can bind the public keys with the identities of their owners. A trusted certificate authority (CA) can issue certificate to an entity to prove that the public key indeed belongs to this entity. Informally a certificate can be viewed as a digital signature made by the CA (using its private key) on the message that ``This public key belongs to Alice''. A digital signature of a message is a digital counterpart to the hand-written signature which assures that the message is generated by the signer (this relates to \\emph{authentication} in cryptography). Everyone can use the CA's public key to verify the validity of the CA's signature so as to verify the certificate. Of course as CA is trusted, its public key must be well known. This can be easily achieved since trusted CAs (like government agencies or global organizations) usually have large influence and rich resources to distribute their public keys to the public.\n\nThe other method of using IBE also requires a trusted authority to generate the public and private key pair of an entity. But no certificate is needed. In an IBE system, the public key of an entity can be anything so an entity can use its identity, such as name of an organization, email address of a person,  as its public key which can be easily verified by others. The PKI mechanism requires users to verify each certificate issued by CAs. Thus heavy public-key operations are needed in PKI, which are obviously not friendly to IoT applications. IBE can efficiently reduce the cost to verify the correctness of public keys, which turns to be favorable in the scenario of IoT.\n\nModern cryptography bases its security on rigorous proofs for assuring security in extreme adversarial situations. The acknowledged security of essentially all provably secure cryptographic primitives is reduced to the confidence on well-established hardness of some  mathematical problems.  The integer factorization problem and (Elliptic Curve) discrete logarithm  problem are two famous problems of this kind. They are the bases for RSA, Diffie-Hellman and Elliptic Curve Cryptography (ECC), which are widely used in today's cryptography. The best known classical algorithms (on Turing machines) for solving factorization and discrete logarithm problem work with sub-exponential time complexity. But Shor's quantum algorithm can solve both within polynomial time. A direct consequence is that once large-scale quantum computers are available our current public-key cryptography system such as RSA and ECC, would be \\emph{completely broken}. Hence, it is of high priority that we explore alternative problems which are intractable for both classical computers and quantum computers.\n\nAnother mild yet universally influential impact of quantum computing techniques comes from Grover's algorithm which presents a quadratic speedup for searching problems over classical algorithms. Grover's algorithm can be used for many cryptanalysis methods which require some sort of brute force. For example guessing the secret key of AES can be accelerated using Grover's algorithm. Generally speaking, one can simply double the length of the key to achieve the same post-quantum security level regarding the impact of Grover's algorithm.\n\nThe  quantum threat has been well recognized by government agencies, large corporations and academic researchers all over the world. The alternative solution called PQC, which aims to provide cryptographic solutions those remain secure even the adversary has access to large-scale quantum computers, is now a hot and steadily growing topic. National Security Agency (NSA) announced, in 2015, their preliminary plans for transitioning to quantum resistant cryptography for protecting classified information. In December 2016, National Institute of Standards and Technology (NIST) issued an open call for standardization consideration of post quantum cryptographic algorithms. At the time of writing (December 2017), the open call is finished. NIST is arranging the first PQC standardization conference, to be held in April, 2018, for the submitters to present and discuss their submissions. Currently Google is experimenting post-quantum cryptography in its web browser Chrome. The Tor (a software which protects its users against Internet surveillance) project is also trying to implement lattice-based key exchange protocols to achieve post-quantum security.\n\n\n\n\n\n\n\n\n\\section{Why Lattice-Based Cryptography?}\nDifferent proposals have been proposed to achieve post-quantum security including hash-based signatures, code-based cryptography, multi-variate polynomial-based cryptography, and lattice-based cryptography. We focus on lattice-based cryptography in this article. In our opinion, lattice-based cryptography is highly suitable for smart IoT applications. Firstly, the strong security guarantees and high efficiency shown by lattice-based cryptography make it extremely suitable for IoT applications. Secondly, the wide applicability of lattice-based cryptography can accommodate further advances of smart IoT services. Last but not least, lattice-based cryptography receives the most intensive attention among all subfields of post-quantum cryptography. The recent NIST call has received 82 submissions for post-quantum cryptographic algorithms and 28 of them are based on lattice, taking the lead.\n\nLattice-based cryptography has strong security guarantees. The underlying hard problems  are intensively studied for decades but no efficient algorithm, both classically and quantumly, is known for those problems. Moreover, lattice-based cryptography enjoys worst-case to average-case reduction. Cryptography inherently requires average-case intractability considering the  requirement of random keys. The worst-case to average-case reduction essentially guarantees that lattice-based cryptography is secure on average unless every instance of the underlying lattice problem is easy. From the practical aspect, this worst-case reduction makes parameter selection and key generation much easier in lattice-based cryptography. For example, the RSA cryptosystem is based on the hardness of integer factorization. But this is an worst-case problem. It is known that if the primes have certain number-theoretic properties, the problem turns out to be essentially easy. Hence it is important to \\emph{avoid such structures} in key generation for RSA. Unfortunately, we do not know whether such structures have been fully explored. In contrast, lattice-based cryptography is based on average-case hard problems. When generating keys for lattice-based cryptography, one only needs to select proper \\emph{parameter size} and then generate keys uniformly.\n\nLattice-based cryptographic algorithms operate over relatively smaller integers, compared with large integers  used in RSA. The computations involved in the state of art of lattice-based algorithms mainly consists of simple operations between matrices and vectors in some rings or fileds of small order.    Actually lattice-based cryptography runs faster than RSA   and it can be implemented on low-power devices with 8-bit microcontrollers. Recent implementations of lattice-based cryptography have been already  an order of magnitude faster than the corresponding\nRSA implementations. For example, the current state-of-art implementation of R-LWE based encryption on 8-bit AVR microconstroller can finish an encryption within 2 million cycles, while the RSA-1024 (has a lower level of security and no post-quantum security) implementations on comparable devices need more than 23 million cycles for the same task \\cite{Liu2017}.\n\nOther candidates of post-quantum cryptography, for example the code-based cryptography, may present even better performance regarding computational efficiency but inevitably require larger sizes for keys and ciphertexts. We stress that it is the balance  among performance metrics, such as key size, ciphertext and signature lengths, computational efficiency and confidence of security, that make lattice-based cryptography a well fit for IoT applications.\n\n\\begin{figure}\n\\centering\n        \\includegraphics[width=.6\\textwidth]{p2.eps}\n\\caption{Illustration of a 2-Dimensional lattice}\n\\label{fig2-svp}\n \\end{figure}\n\n\\section{An Introduction to Lattice Based Cryptography}\nSince the impact of quantum algorithms is mild for symmetric-key cryptography but devastating for current public-key cryptography, many efforts have been put forth in the research of PQC to seek for replacement of universally used cryptosystems such as RSA and ECC. Among them, lattice-based cryptography is a very promising candidate. See Peikert \\cite{Peikert2016} for a comprehensive survey on lattice-based cryptography.\n\nLattice-based cryptography is based on the   hardness of solving some   geometric problems over high-dimensional lattices, such as the shortest vector problem (SVP) and the closest vector problem (CVP). But these problems are easy to solve if one has a good basis. A good basis consists of short vectors which are nearly orthogonal, while a bad basis consists of long vectors which generally point in the same direction. In Fig. \\ref{fig2-svp}, the points  coloured in blue  are lattice vectors which are integer  combinations of the basis  $\\mathbf{B} = \\{\\mathbf{v}_1, \\mathbf{v}_2\\}$. The  basis $\\mathbf{B}$, colored in red, is a bad basis and the  basis $\\mathbf{B}'$, colored in green, is a good basis consisting of almost orthogonal vectors. The SVP   is to find a shortest nonzero vector such as the vector $\\mathbf{v}'_1$. For an arbitrary vector in the space such as the purple point $\\mathbf{t}$, the CVP is to find a lattice vector $\\mathbf{t}'$ which is the closest to $\\mathbf{t}$. Note that we use 2-dimensional (2D) examples for easy visualization but SVP and CVP are easy for 2D case. They get (exponentially) harder as the dimension of lattice grows.\n\n\\begin{figure}\n\\centering\n        \\includegraphics[width=.6\\textwidth]{p3.eps}\n\\caption{Landscape of hard problems in lattice-based cryptography. }\n\\label{fig3-lattice}\n \\end{figure}\n\n\nIn Fig. \\ref{fig3-lattice}, we give a landscape of the hard problems in lattice-based cryptography. The above mentioned SVP and CVP are called \\emph{worst-case} hard problems. This type of problems only guarantee that there exist instances that are intractable. But the problem might be easy on average. One big advantage of lattice-based cryptography is that there are many \\emph{average-case} problems such as the short integer solution (SIS) and learning with errors (LWE) problems. These are well encapsulated average problems enjoying a worst-case to average-case reduction which states that SIS and LWE  are hard on average (for a random instance) unless the related problems on lattices are easy for \\emph{all} instances.  The worst-case to average-case reduction gives the lattice-based cryptography a easy way to  construct  cryptographic schemes and prove their security. That is, one can work on the conceptually simple average-case problems to construct cryptographic primitives while at the same time gains confidence about the constructions from the low-level hard lattice problems in the worst case.\n\nThe SIS and LWE problems can be easily described in the form of solving linear equation systems. The SIS  problem is to find short nontrivial (excluding the all-zero solutions) integer solutions to an homogeneous linear system  with the coefficient matrix uniformly randomly generated. The   LWE problem asks to  find the  the secret vector $\\textbf{s}$, given polynomially many samples $A\\textbf{s}+\\textbf{e}$, where $A$ is a uniformly generated matrix  and    $\\textbf{e} $ represents the noises  selected from some specified error distribution.  More compact ring versions of SIS and LWE are called Ring-SIS (R-SIS) and Ring-LWE (R-LWE).  These ring variants can  reduce memory requirement and computational costs for the cryptographic schemes.\n \nIn practice, besides the primitives based on (R-)SIS and (R-)LWE, NTRUEncrypt is a lattice-based encryption scheme standardized by the  industry. The security of NTRUEncrypt is based on  hard problems over the so called NTRU lattices. Although there is no worst-case reduction for standard NTRUEncrypt, it has withstood attacks for 20 years (with update in the parameters). Another problem called learning with rounding (LWR) is a derandomization of LWE, where no error distribution is needed  to boost the  performance of related schemes.\n\n\n\n\n\\section{Implementations of Lattice-Based Cryptography for Resource-Constrained Devices}\n\\label{section-recomm}\nThe ambition of connecting everything inherently raises the challenge of implementing cryptographic algorithms on resource-constrained devices such as sensors and actuators. In this section we review the state of art of implementations of lattice-based cryptography on constrained devices, which can be divided into software implementations on microcontrollers and hardware implementations on FPGAs.\n\nMain operations of lattice-based cryptographic constructions involve matrix-vector multiplication (schemes based on SIS and LWE) and polynomial multiplications (schemes based on R-SIS, R-LWE, and NTRU).   Polynomial multiplication used in R-LWE based schemes can be optimized via the number theoretic transform (NTT) method, which is a  discrete Fourier transform (DFT) variant. In NTT, the primitive root  of unity modulo an integer $N$ is used instead of complex roots of unity used in DFT. The NTT can transform the classic polynomial multiplication to point-wise multiplication to reduce the complexity from quadratic to quasi-linear. The underlying algorithm can be adjusted to reduce the number of necessary NTT transformations as suggested by Roy et al. for the R-LWE based encryption \\cite{Roy2014}. Optimization methods for FFT can also be borrowed.\n\nMany lattice-based cryptographic constructions (LWE and R-LWE based) require sampling from discrete Gaussian distribution \\cite{Howe2016}. There are many methods to implement the discrete Gaussian sampler including rejection sampling, cumulative distribution table (CDT) sampling, Bernoulli sampling and Knuth-Yao sampling. Sampling from a discrete Gaussian is difficult due to the fact that it requires high precision computation of the exponential function or large precomputed tables.\n\n\n\n\nIn the following we discuss implementations of  lattice-based proposals for public-key encryption, digital signatures and key exchange protocols. We summarize the  implementations  on low-cost microcontrollers in Table \\ref{tab1}, in which the `$\/$' within a cell is used to separate figures for two different operations in the underlying algorithm. The two operations are encryption and decryption for public-key encryption schemes; signing and verification for signature schemes; sever-side computation and client-side computation for key exchange protocols. The `ROM' presents the memory used by the implementation. The hardware implementations of lattice-based cryptography on FPGAs is summarized  in Table \\ref{tab2}.\n\\begin{table}[!htb]\n\\centering\n\\caption{Software implementations of lattice-based cryptography on low-cost microcontrollers.}\n\\label{tab1}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n  \\mr{Schemes} &  \\mr{Bit Security} & \\multicolumn{3}{c|}{Platform} & \\mr{Cycles} & \\mr{Time   (ms)} & \\mr{ROM (KB)}  \\\\ \\cline{3-5}\n  &    & Device & CPU & MHz &   &    & \\\\\n  \\hline\n  NTRUEncrypt \\cite{Guillen2017}                  & 128 (pre)                                                      & \\begin{tabular}[c]{@{}c@{}}Cortex-M0\\\\ (XMC1100)\\end{tabular}       & 32-bit & 32  &   588,044 \/ 950,371      &  18.4 \/ 29.7                 & 9                         \\\\ \\hline\n  R-LWEenc \\cite{Liu2017}                     & \\begin{tabular}[c]{@{}c@{}}106 (pre) \\\\ 46 (post)\\end{tabular} & ATxmega128                                                           & 8-bit  & 32  &    796,872 \/ 215,031   &  24.9 \/ 6.7            & 6.5                       \\\\ \\hline\n  \\multirow{2}{*}{R-BIN-LWEenc \\cite{Buchmann2016}} & \\multirow{2}{*}{94 (pre)}                                      & ATxmega128                                                          & 8-bit  & 32  & 1,573,000 \/ 740,000      &  49.2 \/ 23.1                  & 2.7                       \\\\ \\cline{3-8}\n                                &                                                                & Cortex-M0                                                           & 32-bit & 32  & 999,000 \/ 437,000         &  31.2 \/ 13.7                & 5.6                       \\\\ \\hline\n  \\multirow{2}{*}{IBE \\cite{Guneysu2017}}          & \\multirow{2}{*}{80 (pre)}                                      & Cortex-M0                                                           & 32-bit & 32  &  3,297,380 \/ 1,155,000    &  103.0 \/ 36.1               & 17                        \\\\ \\cline{3-8}\n                                &                                                                & Cortex-M4                                                           & 32-bit & 168 & 972,744 \/ 318,539        &  5.8 \/ 1.9                  & 18.7                      \\\\ \\hline\n  BLISS \\cite{Liu2017}                         & 128 (pre)                                                      & ATxmega128                                                          & 8-bit  & 32  &  10,156,247 \/ 2,760,244    &  317.4 \/ 86.3               & 18.4                      \\\\ \\hline\n  \\multirow{2}{*}{NewHope \\cite{Alkim2016}}      & \\multirow{2}{*}{128 (post)}                                    & \\begin{tabular}[c]{@{}c@{}}Cortex-M0\\\\ (STM32F051R8T6)\\end{tabular} & 32-bit & 48  & 1,467,101 \/ 1,738,922   & 30.6 \/ 54.3             & 30.2                      \\\\ \\cline{3-8}\n                                &                                                                & \\begin{tabular}[c]{@{}c@{}}Cortex-M4\\\\ (STM32F407VGT6)\\end{tabular} & 32-bit & 168 &  860,388 \/ 984,761       &  5.1 \/ 5.9                  & 22.8                      \\\\ \\hline\n  \\end{tabular}\n  \\end{table}\n\nAs the first lattice-based encryption scheme, the NTRUEncrypt encryption has been accepted by the IEEE P1363.1 standard. The NTRU cryptosystem was patented by its inventors along with a variant using `product-from keys' for efficient implementation. But recently (March, 2017) they announced their decision for placing all of its NTRUEncrypt patents in the public domain. Recently Guillen et al. \\cite{Guillen2017} explored the feasibility of employing NTRUEncrypt in constrained devices (a Cortex-M0 based microcontroller).\n\n The R-LWE based encryption scheme is similar to NTRUEncrypt regarding communication and computational costs. One advantage of R-LWE based encryption is its provable security but it comes at the price of a high-precision Gaussian sampler. \tLiu et al. \\cite{Liu2017} presented a constant-time implementation of the R-LWE based encryption scheme with 46-bit post-quantum security level on an 8-bit ATxmega128 microcontroller (32 MHz, 128 KB flesh memory, 8 KB RAM) with encryption and decryption time of 24.9 ms and 6.7 ms, respectively. Buchmann et al. \\cite{Buchmann2016} proposed an encryption scheme by replacing the Gaussian noise in R-LWE with a binary distribution and implemented the scheme (R-BIN-LWEenc) on low-cost microcontrollers.\n\n\n\\begin{table}[!htb]\n\\centering\n\\caption{Hardware implementations of lattice-based cryptography on FPGAs. }\n\\label{tab2}\n\\begin{tabular}{  |c|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Schemes}                           & \\multirow{2}{*}{Bit Security} & \\multirow{2}{*}{Devices} & \\multirow{2}{*}{MHz} & \\multirow{2}{*}{Cycles} & \\multirow{2}{*}{Time($\\mu$s)} \\\\\n   & & & & & \\\\  \\hline\nR-LWEenc \\cite{Roy2014}                         & 128 (pre)                    & V6LX75T                  & 313                 & 6,300 \/ 2,800             & 20.1 \/ 9.1                       \\\\ \\hline\n\\multirow{2}{*}{R-LWEenc \\cite{Poppelmann2014}} & \\multirow{2}{*}{128 (pre)}    & \\multirow{2}{*}{S6LX9}  & 144                    & 136,986                 & 946                            \\\\ \\cline{4-6}\n                                                  &                               &                         & 189                    & 66,338                  & 351                            \\\\ \\hline\n\\multirow{2}{*}{BLISS \\cite{Poppelmann2014b}}   & \\multirow{2}{*}{128 (pre)}    & \\multirow{2}{*}{S6LX25} & 129                & 16,210                  & 126.6                          \\\\ \\cline{4-6}\n                                                  &                               &                         & 142                & 9,835                   & 69.3                           \\\\ \\hline\nIBE \\cite{Guneysu2017}                          & 80 (pre)                      & S6LX25                  & 174                     & 13,958 \/ 9,530            & 80.2 \/ 54.8                      \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n G\\\"{u}neysu and Oder \\cite{Guneysu2017} demonstrated that IBE has become practical even for embedded devices such as Cortex-M microcontrollers and FPGAs by implementing a R-LWE based IBE scheme. Many FPGA implementations of R-LWEenc exist, we introduce two of them. Roy et al. \\cite{Roy2014} presented an FPGA implementation of R-LWEenc optimized for throughput on V6LX75T. P\\\"{o}ppelmann and G\\\"{uneysu} \\cite{Poppelmann2014} presented an area-optimized implementation of R-LWEenc on S6LX9 by carefully choosing parameters to eliminate modular reduction operations.\n\n\nWe refer the readers to Howe et al. \\cite{Howe2015} for a comprehensive discussion on lattice-based signatures. Original proposals for lattice-based signature schemes such as the GGH signature and NTRUSign, which utilize the hardness of CVP, have been broken. The current state-of-art signature scheme  is BLISS, which is based on R-LWE and has been proven secure in the random oracle model. Discrete Gaussian sampling accounts for more budget in signature schemes than that in encryption schemes because the deviation used in signature schemes is much larger.   State-of-art software implementation and hardware implementation of BLISS are Liu et al. \\cite{Liu2017} and P\\\"{o}ppelmann et al. \\cite{Poppelmann2014b}.  Ducas proposed a variant called BLISS-B which can reduce the repetition rate and in turn speeding up the key generation by a factor of 5 to 10.\n\n\n    NewHope is a post-quantum key exchange protocol based on R-LWE and has been used by Google's post-quantum security experiments within Chrome. Alkim et al. \\cite{Alkim2016} presented a software implementation of NewHope for Cortex-M family of 32-bit microcontrollers. With various generic and platform-specific optimizations, their implementation demonstrated that lattice-based key exchange protocols are indeed promising candidates for post-quantum IoT security. The Cortex-M0 implementation requires about 1.5 million cycles for server side computation and 1.8 million for client side. On the more powerful M4 platform the corresponding cycles are 0.8 million and 9.8 million.\n\n\n\n\n \\section{Challenges and Future Research Directions}\nVarious implementation results have demonstrated that lattice-based cryptography is practical even for resource-constrained devices. Regarding computational speed, lattice-based cryptography is already faster than traditional public-key cryptography such as RSA or even ECC. But one can not draw the conclusion that the former performs better in practice since lattice based cryptography usually requires more communication cost which is much more resource-consuming. Further improvement of lattice-based cryptography remains a challenge. The implementation optimization is of course necessary, but theoretical improvement for reducing ciphertext and signature size might be more promising. Further directions include tighter security proof, efficient construction, reducing the use of discrete Gaussian noise and efficient encoding techniques.\n\nMost of implementations we discussed do not provide protection against side-channel attacks (SCAs). It is of prominent significance to provide side-channel-attack-resistant implementations for smart IoT applications since they are more venerable to SCAs.\n\nThe provable security of most lattice-based cryptography does not guarantee security in practice and might even cause overlook on practical security. Choosing appropriate parameters for lattice-based schemes is another challenge. Many incomparable algorithms for analyzing lattice problems are available and the performance for some of them are not well understood. A unified model for evaluating security level of lattice-based cryptography is highly desirable. In R-LWE based constructions, the parameters are also constrained by the requirement of NTT-friendly choices which may lead to gaps in security level.\n\nAnalyzing the security of lattice-based cryptography against fully quantum attacks is more than a theoretical interest. Current PQC only considers the impact of quantum computers (or quantum algorithms). However, in the quantum world the attacker is able to have quantum interaction with the cryptosystem. The quantum random oracle model in the literature is one kind of this directions.\n\n\n\n\n\n\\section*{Acknowledgments}\n The work presented in this paper was supported in part by the National Natural Science Foundation of China under Grant no. 61672029.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nCrystal structure prediction is increasingly becoming one of the most effective approaches for discovering new functional materials \\cite{oganov2019structure} due to the ease to obtain new compositions either by enumeration \\cite{davies2016computational}, heuristic knowledge, or the latest deep learning-based generative machine learning models \\cite{dan2020generative}. While the peer protein structure prediction problem has been recently almost solved by the deep learning-based AlphaFold and RossettaFold algorithms, the crystal structure prediction problem remains elusive for a majority of categories of compositions. There are mainly three types of crystal structure prediction approaches including the ab initio based global optimization \\cite{glass2006uspex, wang2015materials,liang2020cryspnet, demir2021ffcasp,falls2020xtalopt}, machine learning-based prediction \\cite{hu2021alphacrystal}, and template-based elemental substitution \\cite{hautier2011data}. The first approach instead depends on computationally expensive DFT calculations and is applicable to only small chemical systems. The second approach is inspired by the AlphaFold family of deep learning algorithms \\cite{senior2020improved,jumper2021highly}, but is only at the early stage of development. The last template-based CSP methods are the most widely used and easiest to implement. Even though this method cannot predict crystal structures of new prototypes, recent deep generative models can discover new prototype materials which can partially address this issue \\cite{zhao2021high}. In a pioneering work \\cite{hautier2011data}, Hautier et al. proposed a data mining-based approach to identify the probabilities for different pairs of ionic substitutions, which can be applied to any prototype structures to generate new structures or used to select templates for template-based crystal structure prediction. However, despite the wide usage of template-based CSP methods, there are many different ways to implement, and there is no working web app\/server that is user-friendly to make it accessible to all materials scientists (The structure predictor of the Materials Project \\cite{ceder2010materials} website was available before but is not functional now). \n\nHere we propose a fast and user-friendly template-based crystal structure prediction algorithm and related companion web server for broad adoption of crystal structure prediction in the daily life of materials science. Our algorithm TCSP is based on the careful selection of template structures based on chemical formula similarity and the match of oxidation states using an exhaustive enumeration strategy. Our predicted structures can be optimized by DFT or machine learning-based structure relaxation. By using seven case studies, we have shown that our user-friendly and fast crystal structure prediction web server has high prediction performance when appropriate templates are available. We also apply our TCSP algorithm to predict the structures of all 98,290 formulas using leave-one-out evaluation and have achieved good performance for a large portion of the targets: more than 13,145 target structures have been found with maximum RMSD less than 0.1. The good performance of this high-throughput crystal structure prediction shows that the template\/prototype-based element substitution CSP approach has big potential in exploratory materials discovery. With the development of large scale prototype databases \\cite{su2017construction,hicks2021aflow} and their applications in the generative design of new crystals \\cite{bushlanov2019topology,sorkun2020artificial}, the performance of our template-based crystal structure prediction algorithm can be further improved.\n\n\n\\section{Method}\n\\label{sec:headings}\n\n\n\\subsection{Template based crystal structure prediction algorithm}\n\nOur template-based crystal structure prediction algorithm is illustrated in Figure~\\ref{fig:predict_algorithm}. Given an input formula (e.g., SrTiO\\textsubscript{3}) with which can be specified by the user or predicted by algorithms \\cite{zhao2020machine,li2021composition} or without space group. We then search structure templates with the same prototype (sometimes called an anonymous formula)  (e.g., ABC\\textsubscript{3}) and the same space group if specified. This step may retrieve too many matched templates, so we use Module A1, an Element's mover distance, to measure the composition similarity between the query formula and the compositions of all the template structures, which are then ranked by ascending order. We then pick the top K structures as template candidates with the smallest composition distances. For each of the candidate templates, we use the Pymatgen package to estimate its oxidation states and compare them to those of the query formula. If we find templates with identical oxidation states, we then add them to the final template list. If no such templates are found, we then neglect the oxidation match requirements and directly add them as final templates. The next step is to determine all the possible element substitution pairs between the query formula and the template formula using the algorithm described in Algorithm 2. Next, we will pick the template structure files and replace the elements according to the pair arrangements found by Algorithm 2. A replacement quality score is also calculated for each such element substitution arrangement using the procedure as described in Module 3. The resulting structures will then be subject to DFT or machine learning-based structure relaxation, which can be further used to calculate formation energy, e-above-hull energy, and phonon dispersion for validation. \n\n\n\n\n\\paragraph{Module A1: Element mover's distance for formula similarity calculation}\nWe use the Element's Mover Distance measure $ElMD$ \\cite{hargreaves2020earth} to select most similar template structures. ElMD is a metric that allows measuring the chemical similarity of two formulas in an explainable fashion. The EMD is computed between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale (several other element similarities can also be used such as Mendeleev, Petti, Atomic, Mod\\_petti, Oliynyk, Oliynyk\\_sc, Jarvis, Jvarvis\\_sc, magpie,magpie\\_sc, CGCNN, Elemnet, mat2vec, Matscholar, megnet16, random). This metric shows clear strength at distinguishing compounds. It is shown that ElMD distances have greater alignment with chemical understanding than the Euclidean distance. The ElMD is defined in formula (1) below:\n\n\n\\begin{equation}\nElM D(X, Y)=\\min \\sum_{i=1}^{m} \\sum_{j=1}^{n} q_{i j}\\left|p_{i}-p_{j}\\right|\n\\textrm{, subject to} \\quad q_{i j} \\geq 0 \\quad \\textrm{for ${\\forall}$ } i, j\n\\end{equation}\n\n\n\\begin{equation}\n\\sum_{j=1}^{n} q_{i j} \\leq x_{i}  \\textrm{, for ${\\forall}$ } 1 \\leq i \\leq m\n\\end{equation}\n\n\\begin{equation}\n\\sum_{i=1}^{m} q_{i j} \\leq y_{j} \\textrm{,  for ${\\forall}$ } 1 \\leq j \\leq n\n\\end{equation}\n\n\\begin{equation}\n\\sum_{i=1}^{m} \\sum_{j=1}^{n} q_{i j}=1\n\\end{equation}\n\nwhere the distance is first calculated by matching and pairing each of the $m$ elements in a vector, $X$, to its most similar unmatched partner in the $n$ elements of a second vector $Y$, until all have been paired. The quantity matched, $q$, from the $i$-th element of $X$, to the $j$-th element of $Y$, is given by $q_{ij}$.\n\n\\paragraph{Algorithm A2: element replacement pair enumeration algorithm:} \n\nThis algorithm is used to enumerate all possible element replacement strategies between two pairs of formulas. \n\n\\begin{algorithm}[h]\n    \\caption*{Algorithm A2: Element replacement pair enumeration algorithm}\n    \\begin{algorithmic}[1]\n        \\State Given two formula X,Y, calculate theirs oxidation states then get the statelist and elementlist.\n        \n        \\If{$stateList_x == stateList_y$}\n        \n        {Replace element pairs}\n        \\Else{\\State Create elementGroupList to represent and distinguish elements of equal state\n            \\For {$i = 0,1,\\ldots$}\n                \\If{$i == 0$}\n                    \\State $elementGroup = elementList_y[i]$;\n                \\ElsIf {($stateList_y[i] == stateList_y[i-1]$)}\n                \\State append $elementList_y[i]$ to $elementGroup$;\n                \\Else\n                    \\State append $elementList_y[i]$ to $elementGroup$;\n                \\EndIf\n            \\EndFor\n            \\State Create $pre\\_patterns$ to represent permutation and combination of elements in $elementGroupList$\n            \n            \\For{$element =0,1,\\ldots,j$}\n                \\If{$j = 0$}\n                    \\State $pre\\_patterns$ = permutations of $elementGroupList[j]$;\n                \\Else\n                    \\State $patterns$ =permutations of $elementGroupList[j]$;\n                    \\For{$p1 = 0,1,\\ldots,m$} \n                        \\For{$p2 = 0,1,\\ldots,n$}\n                            \\State append $p1,p2$ to $new\\_patterns$;\n                        \\EndFor\n                        \\State $pre\\_patterns = new_patterns$;\n                    \\EndFor\n                \\EndIf\n            \\EndFor\n          \n            \\State If the element in $pre\\_patterns$ is different from the element in $elementList_x$, then replace the element\n            \n            \n        \\EndIf\n}\n    \\end{algorithmic} \n\\end{algorithm}\n\n\n\\paragraph{Module A3: element substitution scoring function:}\nTo differentiate the resulting structures from different element substitution arrangements between the query formula and the template structure and rank the final output structures from different templates, we use the ElMD distance in Module A1 to calculate the similarity score between each pair of substitution elements for a given query formula and the final structure. Then we sum up these similarity scores for all the element substitution pairs and use them to calculate the quality scores of final structures. \n\n\\begin{equation}\nS_{es}= ElMD(e1,e2), S_{r}= \\sum_{i=1}^{p}  S^{i}_{es}\n\\end{equation}\n\n\nWhere $S_{es}$ measures the element matching quality between two substitution elements $e1$ and $e2$. $S_{r}$ is the replacement distance score for a given element substitution arrangement of a given query formula and template structure, which is equal to the sum of all the element similarity scores of all substitution element pairs in the element substitution arrangement.\n\nThe final quality of the generated structures is then measured by the $S_r$ scores, with lower scores corresponding to higher quality. \n\n\n\n\\paragraph{DFT or Genetic algorithm based structure relaxation:}\n\n\nAs with all predicted crystal structures, they usually need a fine-tuning or relaxation step to adjust the local atomic coordinates either using the DFT-based structural relaxation method or the recently developed machine learning and optimization-based relaxation approach \\cite{zuo2021accelerating} which is much faster than the DFT approach. In this study, we used the DFT approach for evaluation purposes.\n\n \n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=0.45\\linewidth]{predict_template.pdf}\n  \\caption{Flowchart of our TCSP, a template based crystal structure prediction algorithm. The space group specification is optional. }\n  \\label{fig:predict_algorithm}\n\\end{figure}\n\n\\FloatBarrier\n\n\\subsection{User interface of our web server}\n\nOur template-based crystal structure prediction server has a user-friendly web interface, as shown in Figure\\ref{fig:ui}. Each time, a user can just put in a formula\/composition, and then the target space group number from 1 to 230 can be set or just assigned to 0 to allow a template with any space group. Then the user types in their email for receiving the job completion notification email with a downloadable URL link for the predicted structures. After a few minutes, an email will be sent to the user with the download URL link for the predicted structures. After the zipped result file is downloaded and unzipped, the user can go into the folder and click the Name column to sort the files by filename. Then it shows several key files: (1) $results.txt$, which shows the template similarity scores, the templates with compatible oxidation states, and the element replace pairs for each template. (2) $similar\\_formulas.csv$ file shows the distance scores of all templates to the query formula; (3) $TemplateCandiates.csv$ shows the Materials Project IDs of the selected templates. (4) all the remaining $cif$ files are predicted, which are sorted by their replacement quality score (the number before \\_mp of the filename), which is better when the number is smaller. However, it is strongly suggested to validate a couple of top-scored candidate structures as the candidate structure with the top quality score is not always the best one.\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=0.75\\linewidth]{TCSP_UI.png}\n  \\caption{User interface of our TCSP web app for crystal structure prediction.}\n  \\label{fig:ui}\n\\end{figure}\n\n\n\\subsection{DFT validation of predicted structures}\n\nThe first principle calculations based on the density functional theory (DFT) are carried out using the  Vienna \\textit{ab initio} simulation package (VASP) \\cite{Vasp1,Vasp2,Vasp3,Vasp4}.  The projected augmented wave (PAW) pseudopotentials, where 520 eV plane-wave cutoff energy, were used to treat the electron-ion interactions \\cite{PAW1, PAW2}. The exchange-correlation functional was considered with the generalized gradient approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) method \\cite{GGA1, GGA2}. The energy convergence criterion was set as 10$^{-5}$ eV, while the atomic positions were optimized with the force convergence criterion of 10$^{-2}$ eV\/{\\AA}. The Brillouin zone integration for the unit cells was computed using the $\\Gamma$-centered  Monkhorst-Pack $k$-meshes. The Formation energies (in eV\/atom) of several materials were determined based on the expression in  Eq.~\\ref{eq:form}, where $E[\\mathrm{Material}]$ is the total energy per unit formula of the considered structure, $E[\\textrm{A}_i]$ is the energy of $i^\\mathrm{th}$ element of the material, $x_i$ indicates the number of A$_i$ atoms in a unit formula, and $n$ is the total number of atoms in a unit formula($n=\\sum_i x_i$).\n\n\\begin{equation}\n    E_{\\mathrm{form}} =\\frac{1}{n}(E[\\mathrm{Material}] - \\sum_i x_i E[\\textrm{A}_i])\n    \\label{eq:form}\n\\end{equation}\n\n\n\n\n\n\\subsection{Evaluation criteria}\n\n\nTo evaluate the reconstruction performance of  algorithm, we define the root mean square distance (RMSD) and mean absolute error (MAE) of two structures as below:\n\\begin{equation}\n    \\begin{aligned}\n\\mathrm{RMSD}(\\mathbf{v}, \\mathbf{w}) &=\\sqrt{\\frac{1}{n} \\sum_{i=1}^{n}\\left\\|v_{i}-w_{i}\\right\\|^{2}} \\\\\n&=\\sqrt{\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\left(v_{i x}-w_{i x}\\right)^{2}+\\left(v_{i y}-w_{i y}\\right)^{2}+\\left(v_{i z}-w_{i z}\\right)^{2}\\right)}\n\\end{aligned}\n\\end{equation}\n\n\n\n\\begin{equation}\n        \\begin{aligned}\n\\mathrm{MAE}(\\mathbf{v}, \\mathbf{w}) &=\\frac{1}{n} \\sum_{i=1}^{n}\\left\\|v_{i}-w_{i}\\right\\| \\\\\n&=\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\|v_{i x}-w_{i x}\\|+\\|v_{i y}-w_{i y}\\|+\\|v_{i z}-w_{i z}\\|\\right)\n\\end{aligned}\n\\end{equation}\n\nwhere $n$ is the number of independent atoms in the target crystal structure. For symmetrized CIF structures, $n$ is the number of independent atoms of the set of Wyckoff equivalent positions. For regular CIF structures, it is the total number of atoms in the compared structure. $v_i$ and $w_i$ are the corresponding atoms in the predicted crystal and the target crystal structure. \n\n\n\\section{Results}\n\\label{sec:others}\n\n\n\\subsection{Dataset}\n\nWe used more than 130,000 crystal structures deposited in the Materials Project database as our template sources. We picked seven test materials including SrTiO\\textsubscript{3} (mp-5229), Ni\\textsubscript{3}S\\textsubscript{4} (mp-1050), NiS\\textsubscript{2} (mp-849059), GaBN\\textsubscript{2} (mp-1007823), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740), GaB\\textsubscript{2}N\\textsubscript{3} (mp-1245554), and Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743). We then ran our algorithm and checked if it can predict the correct structures that match the target structures. \n\n\n\\subsection{ Performance of template based crystal structure prediction}\n\nWe selected seven formulas of target structures from materials project database for evaluating the capability of our TCSP algorithm for structure prediction. The first test target formula is SrTiO\\textsubscript{3}, which has three different phases corresponding to space groups of 140, 149, 221. Its most famous structure is the cubic perovskite structure as shown in Figure \\ref{fig:SrTiO3_target}. Our algorithm identified thousands of compatible templates and picked top 10 as templates including BaZrO\\textsubscript{3} (mp-3834), MgTiO\\textsubscript{3} (mp-1016830), CaZrO\\textsubscript{3} (mp-542112), MgZrO\\textsubscript{3} (mp-1017000), BaTiO\\textsubscript{3} (mp-504715), BaTiO\\textsubscript{3} (mp-5020), SrZrO\\textsubscript{3} (mp-613402), CaTiO\\textsubscript{3} (mp-5827), BaTiO\\textsubscript{3} (mp-2998), SrHfO\\textsubscript{3} (mp-4551), among which all are cubic templates except BaTiO\\textsubscript{3}(mp-5020), which is a trigonal structure with space group 160. The top four predicted structures all have a zero rmsd error compared to the perovskite target structure: they all have the identical fractional coordinates as the target structure except that the cubic lengths are different (the predicted cubic structure in Figure \\ref{fig:SrTiO3_predict} has a lattice length of 4.256 \u00c5 while the target structure has a lattice length of 3.945 \u00c5), which may be fine-tuned using DFT based relaxation.\n\nThe second test sample is Ni\\textsubscript{3}S\\textsubscript{4}, which only has one cubic phase with the space group of 227. The structure is shown in Figure  \\ref{fig:Ni3S4_target}. Our algorithm found top four templates including Co\\textsubscript{3}S\\textsubscript{4} (mp-943), Co\\textsubscript{3}Se\\textsubscript{4} (mp-20456), Ni\\textsubscript{3}Se\\textsubscript{4} (mp-1120781), Co\\textsubscript{3}O\\textsubscript{4} (mp-18748), all of them are cubic structures with the space group of 227. The predicted structure with the lowest rmsd is 0.000714  which is predicted by our algorithm using Co\\textsubscript{3}S\\textsubscript{4} as the template. The rmsd errors of the structures from the other three templates are much larger, all around 0.288 . We can see that the predicted structure in Figure \\ref{fig:Ni3S4_predict} is very close to the target structure in Figure \\ref{fig:Ni3S4_target}. We also found that the predicted structure of NiS\\textsubscript{2} in Figure \\ref{fig:NiS2_predict} also matches well with the target structure in Figure \\ref{fig:NiS2_target}, which has the small rmsd error of 0.004918 .\n\nWe also tested four formulas of the chemical system Ga-B-N including GaBN\\textsubscript{2}, GaB\\textsubscript{3}N\\textsubscript{4}, Ga\\textsubscript{2}BN\\textsubscript{3}, and Ga\\textsubscript{3}BN\\textsubscript{4}. For the GaBN\\textsubscript{2} with the space group of 115, the best eight templates are found by our algorithm. Five of them including AlBN\\textsubscript{2} (mp-1008557)  ,AlBN\\textsubscript{2} (mp-1008557), AlGaP\\textsubscript{2} (mp-1228888), AlGaN\\textsubscript{2} (mp-1008556), B\\textsubscript{2}AsP(mp-1008528) have the same tetragonal crystal system as the target structure with the space group of 115. The remaining are trigonal with the space group of 166. The predicted structures AlBN\\textsubscript{2} (mp-1008557) and AlBN\\textsubscript{2} (mp-1008557) have the same lowest rmsd error of 0.003889 . However they have different structure patterns as shown in Figure \\ref{fig:GaBN2_predict1} and Figure \\ref{fig:GaBN2_predict2}. For the same template, our algorithm suggests two element replacement strategies. In the first strategy, the element Ga in the test formula is used to replace the element Al in the template AlBN\\textsubscript{2}; in the second strategy, the element Ga in the test formula GaBN\\textsubscript{2} replaces the element B in the same template AlBN\\textsubscript{2}. For the formula GaB\\textsubscript{3}N\\textsubscript{4} with the space group of 215, TCSP finds top 9 most similar templates AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), CrGa\\textsubscript{3}P\\textsubscript{4} (mp-985440), AlGa\\textsubscript{3}N\\textsubscript{4} (mp-1019508), Al\\textsubscript{3}GaN\\textsubscript{4} (mp-1019378), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743), Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743) of the same space group 215 as well. The lowest rmsd error with different structure templates AlB\\textsubscript{3}N\\textsubscript{4} and AlB\\textsubscript{3}N\\textsubscript{4} is 0.002336  by using two element replacement strategies. The element Ga in the first strategy is used to replace Al in the template as shown in Figure \\ref{fig:GaB3N4_predict1} and\nas shown in Figure \\ref{fig:GaB3N4_predict2}, Ga replaces B in  AlB\\textsubscript{3}N\\textsubscript{4} in the second strategy. For the formula GaB\\textsubscript{2}N\\textsubscript{3} (mp-1245554) in Figure \\ref{fig:GaB2N3_target} which has monoclinic structure with the space group 15, TCSP only find two templates AuC2N3 (mp-1245653) and AuC2N3 (mp-1245653) with the same space group. The lowest rmsd is 0.24746 for these two predicted structures. As shown in Figure \\ref{fig:GaB2N3_predict1}, our algorithm uses the first strategy,  TCSP uses Ga in the test formula GaB\\textsubscript{2}N\\textsubscript{3} to replace Au in the first template AuC\\textsubscript{2}N\\textsubscript{3}. In the second strategy, Ga replaces B in the second template as shown in Figure \\ref{fig:GaB2N3_predict2}. For the formula Ga\\textsubscript{3}BN\\textsubscript{4}(mp-1019743) with the space group 215, TCSP finds 9 most similar templates Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}GaN\\textsubscript{4} (mp-1019378), AlGa\\textsubscript{3}N\\textsubscript{4} (mp-1019508), CrGa\\textsubscript{3}P\\textsubscript{4} (mp-985440), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740) with the same space group 215. The templates Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380) and Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380) with different structure patterns have the same lowest rmsd error of 0.004017781. For Al\\textsubscript{3}BN\\textsubscript{4}, AlB\\textsubscript{3}N\\textsubscript{4} and GaB\\textsubscript{3}N\\textsubscript{4}, they all have different structure patterns by using the two elements replacement strategies.\n\n\n\n\n\n\n\n\\begin{figure}[ht!] \n \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{SrTiO3.png}\n        \\caption{SrTiO\\textsubscript{3}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:SrTiO3_target}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{Ni3S4.png}\n        \\caption{Ni\\textsubscript{3}S\\textsubscript{4}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:Ni3S4_target}\n    \\end{subfigure}    \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{NiS2.png}\n        \\caption{NiS\\textsubscript{2}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:NiS2_target}\n    \\end{subfigure}    \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{2.0_mp-3834_SrTiO3_predict.png}\n        \\caption{SrTiO\\textsubscript{3}(Predicted)}\n        \\vspace{-3pt}\n        \\label{fig:SrTiO3_predict}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{3.0_mp-943_Ni3S4_predict.png}\n        \\caption{Ni\\textsubscript{3}S\\textsubscript{4}(Predicted)}\n        \\vspace{-3pt}\n        \\label{fig:Ni3S4_predict}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{3.0_mp-849086_NiS2_predict.png}\n        \\caption{NiS\\textsubscript{2}(Predicted)}\n        \\vspace{-3pt}\n        \\label{fig:NiS2_predict}\n    \\end{subfigure}\\hfill\n \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{GaBN2.png}\n        \\caption{GaBN\\textsubscript{2}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:GaBN2_target}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{1.0_mp-1008557_GaBN2_predict.png}\n        \\caption{GaBN\\textsubscript{2}(Predicted1)}\n        \\vspace{-3pt}\n        \\label{fig:GaBN2_predict1}\n    \\end{subfigure}\\hfill    \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=0.9\\textwidth]{3.0_mp-1008557_GaBN2_predict.png}\n        \\caption{GaBN\\textsubscript{2}(Predicted2)}\n        \\vspace{-3pt}\n        \\label{fig:GaBN2_predict2}\n    \\end{subfigure}\\hfill    \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{GaB3N4.png}\n        \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:GaB3N4_target}\n    \\end{subfigure}    \n        \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{1.0_mp-1019379_GaB3N4_predict.png}\n        \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Predicted1)}\n        \\vspace{-3pt}\n        \\label{fig:GaB3N4_predict1}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{3.0_mp-1019379_GaB3N4_predict.png}\n        \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Predicted2)}\n        \\vspace{-3pt}\n        \\label{fig:GaB3N4_predict2}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{GaB2N3.png}\n        \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Target)}\n        \\vspace{-3pt}\n        \\label{fig:GaB2N3_target}\n    \\end{subfigure}          \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{11.0_mp-1245653_GaB2N3_presict.png}\n        \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Predicted1)}\n        \\vspace{-3pt}\n        \\label{fig:GaB2N3_predict1}\n    \\end{subfigure} \n    \\begin{subfigure}[t]{0.33\\textwidth}\n        \\includegraphics[width=\\textwidth]{7.0_mp-1245653_GaB2N3_presict.png}\n        \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Predicted2)}\n        \\vspace{-3pt}\n        \\label{fig:GaB2N3_predict2}\n    \\end{subfigure}     \n\n   \\caption{predicted structures by TCSP compared to targets.}\n  \\label{fig:predictedstructures}\n\\end{figure}\n\n\n\n\n\nOur TCSP algorithm can output multiple predictions using different templates. To understand this capability, Table~\\ref{tab:results_materials} shows the RMSD and quality scores of the top 10 predictions for each input formula. For SrTiO\\textsubscript{3}, the top 4 structures all have zero RMSD errors for their fractional coordinates with their replacement distance scores ranging from 2 to 3. The MAE errors of these four structures are also 0. We do find their lattice length is different from the target structures, which, however, can be tuned by DFT-based structure relaxation. For Ni\\textsubscript{3}S\\textsubscript{4}, only the top 1 result is very close to the target structure with three much worse results. For NiS\\textsubscript{2}, the top 2 predicted structures have RMSD of 0.0049  and 0.0124. Both are good compared to the target structures. For GaBN\\textsubscript{2}, the top four results all have small RMSD errors ranging from 0.0039 to 0.0209. The same is true for the predicted structures of GaB\\textsubscript{3}N\\textsubscript{4}. The worst prediction performance is on the formula GAB\\textsubscript{2}N\\textsubscript{3}, which can only find two compatible templates, both leading to very different structures from the target structure. Their MAE errors are 0.1853 . For GA\\textsubscript{3}BN\\textsubscript{4}, the top three predictions are all of high quality with RMSD of only 0.004\/0.004\/0.0206  respectively. In terms of the quality score distribution, we found that low replacement distance scores indicate predicted structures with good quality: for example, predictions with replacement distance scores greater than 5 are all low-quality results. However, low replacement distance scores do not always mean their structures are of high quality. For example, in the case of SrTiO3, the structures with high RMSD have lower replacement distance scores than those top 4 results.  \n\n\\FloatBarrier\n\n\\begin{table}[th]\n\n\\centering\n\\caption{Prediction performance (rmsd error) of top 10 results for each sample in the benchmark set by TCSP\n}\n\\label{tab:results_materials}\n\\begin{tabular}{|llrrrrrrrrll|}\n\\hline\n\\rowcolor[HTML]{FFFFFF} \n\\textbf{Formula}                                              & \\textbf{Metric}                & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top1}}    & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top2}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top3}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top4}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top5}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top6}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top7}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top8}} & \\textbf{Top9}                        & \\textbf{Top10}                       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}SrTiO\\textsubscript{3}} & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0}      & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0}                           & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0}                           & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0}                           & \\multicolumn{1}{r|}{0.1667}                      & \\multicolumn{1}{r|}{0.2832}                      & \\multicolumn{1}{r|}{0.4082}                      & \\multicolumn{1}{r|}{0.4410}                      & \\multicolumn{1}{r|}{0.4410} & \\multicolumn{1}{r|}{0.4410} \\\\ \\hline\n\\multicolumn{1}{|l|}{}                               & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{2}                              & \\multicolumn{1}{r|}{2}                           & \\multicolumn{1}{r|}{2}                           & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}      & \\multicolumn{1}{r|}{2}      \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}Ni\\textsubscript{3}S\\textsubscript{4}}  & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0007} & \\multicolumn{1}{r|}{0.2885}                      & \\multicolumn{1}{r|}{0.2888}                      & \\multicolumn{1}{r|}{0.2897}                      & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}}       & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3}                              & \\multicolumn{1}{r|}{4}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{4}                           & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}NiS\\textsubscript{2}}   & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0049} & \\multicolumn{1}{r|}{0.0124}                      & \\multicolumn{1}{r|}{0.0777}                      & \\multicolumn{1}{r|}{0.2282}                      & \\multicolumn{1}{r|}{0.2424}                      & \\multicolumn{1}{r|}{0.2846}                      & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}}       & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3}                              & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{2}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{2}                           & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}GaBN\\textsubscript{2}}  & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0039} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0039}                      & \\multicolumn{1}{r|}{0.0174}                      & \\multicolumn{1}{r|}{0.0209}                      & \\multicolumn{1}{r|}{0.3410}                      & \\multicolumn{1}{r|}{0.3412}                      & \\multicolumn{1}{r|}{0.3412}                      & \\multicolumn{1}{r|}{0.3886}                      & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}}       & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3}                              & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{2}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{9}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{5}                           & \\multicolumn{1}{r|}{31}                          & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{GaB\\textsubscript{3}N\\textsubscript{4}}                         & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0023} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0023}                      & \\multicolumn{1}{r|}{0.0152}                      & \\multicolumn{1}{r|}{0.0162}                      & \\multicolumn{1}{r|}{0.3335}                      & \\multicolumn{1}{r|}{0.3344}                      & \\multicolumn{1}{r|}{0.3344}                      & \\multicolumn{1}{r|}{0.3347}                      & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{}                               & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{1}                              & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{22}                          & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{4}                           & \\multicolumn{1}{r|}{0}      & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}GaB\\textsubscript{2}N\\textsubscript{3}} & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.2475} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.2475}                      & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}}       & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{11}                             & \\multicolumn{1}{r|}{7}                           & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}                            & \\multicolumn{1}{l|}{}       & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}Ga\\textsubscript{3}BN\\textsubscript{4}} & \\multicolumn{1}{l|}{rmsd}  & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0040} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0040}                      & \\multicolumn{1}{r|}{0.0206}                      & \\multicolumn{1}{r|}{0.3336}                      & \\multicolumn{1}{r|}{0.3337}                      & \\multicolumn{1}{r|}{0.3345}                      & \\multicolumn{1}{r|}{0.3345}                      & \\multicolumn{1}{r|}{0.3347}                      & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}}       & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{1}                              & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{22}                          & \\multicolumn{1}{r|}{1}                           & \\multicolumn{1}{r|}{3}                           & \\multicolumn{1}{r|}{0}                           & \\multicolumn{1}{r|}{4}      & \\multicolumn{1}{l|}{}       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nTo further evaluate the performance of our TCSP algorithm, we conduct comprehensive predictions of all 98,290 formulas in the Materials Project database using the leave-one-out evaluation approach. For each formula, we predict its structure using existing templates that do not have the same formula. Here we use the strict mode for finding templates: only top 10 templates with the same prototype and compatible oxidation states are used to predict new structures. For each formula, we first identify all its corresponding mp-ids with corresponding structures and then for each of these target structures, we pick the structure with the lowest rmsd error out of all the predicted structures and we show the distribution of these rmsd errors to see how our template based TCSP algorithm can recover the structures in the MP database. The result is shown in Figure \\ref{fig:histogram_rmsd}. We find that for 34,569 MP structures, our algorithm has identified templates for structure prediction. Out of these target structures, TCSP has predicted hypothetical structures with a maximum RMSD less than 0.01 for 11,764 MP structures, or with RMDS less than 0.1 for 13,145 MP structures. We also find that for 10,433 structures, the algorithm could not find the correct templates that generates the same number of atoms in the unit cell for which we set the RMSD error at 1.0.\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=0.55\\linewidth]{histogram_tcsp.png}\n  \\caption{Distribution of rmsd errors of all Materials Project structures as predicted by our TCSP  using leave-one-out evaluation.}\n  \\label{fig:histogram_rmsd}\n\\end{figure}\n\n\\FloatBarrier\n\n\\subsection{Discovery of new materials and DFT validation of the predict structures}\n\nWe are interested in how our TCSP algorithm can help to discover novel stable materials. We started with the Ga-B-N chemical system, for which we found four materials in the materials project database as shown in our test set discussed in the previous section. According to the materials project database, those available Ga-B-N structures have non-zero energy-above-hull values indicating those materials are thermodynamically unstable. We wonder if there exist thermodynamically and dynamically stable materials of this chemical system. We use the composition enumeration tool from our MaterialsAtlas.com toolbox to identify new Ga-B-N formulas prototypes and their predicted formation energy. We picked the top 41 formulas that do not exist in the MP database and used our TCSP to predict a set of candidate structures for each formula. We then conducted DFT relaxation and formation energy, e-above-hull energy, and phonon dispersion calculations to verify their thermodynamical and dynamical stability.\n\nOur calculations (Table 2) show that almost all candidate structures found by our TCSP have negative formation energies. This is immensely helpful for discovering new materials using the first-principles calculations. If most of the candidate structures of a selected composition have positive formation energies, we have to waste computational hours to find the structures with negative formation energies. However, our ML model is able to filter out the unsuitable candidate structures to reduce the computational burden.\n\nWe further calculated the e-above-hull to investigate the stability against the Ga-B-N competing phases. As given in the MP database, GaN, BN, Ga, B, and N2 are the stable competing phases available. Total energy calculations of competing phases were done with the same VASP settings used for Ga-B-N systems to determine the e-above-hull using the Pymatgen code. Our calculations suggest that 4 out of 41 materials exhibit zero e-bove-hull, indicating they are thermodynamically stable (Figure \\ref{fig:struct}). Those materials and their candidate structures are shown in Table~2. We further carried out phonon calculations for the candidate structures with the lowest formation energies for those four materials. It is clear from Fig.~\\ref{fig:phonons} that GaB$_2$N$_4$ material with mp-780282 template structure is dynamically stable at 0K temperature. It has an interesting layered honeycomb structure.\n\n\\begin{table}[]\n\\caption{Formation energy and corresponding templates of top 10 predictions for each of the four new materials}\n\\label{tab:my-table}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|}\n\\hline\n\\multicolumn{2}{|c|}{GaB$_2$N$_4$}                                 & \\multicolumn{2}{c|}{GaB$_4$N$_3$}                                   & \\multicolumn{2}{c|}{Ga$_2$BN$_4$}                                  & \\multicolumn{2}{c|}{GaBN$_4$}                                    \\\\ \\hline\n\\multicolumn{1}{|c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$(eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$ (eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$(eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$ (eV)} \\\\ \\hline\nmp-780282                   & -3.2957                        & mp-1224009                 & -2.4471                          & mp-532446                  & -2.9443                         & mp-30979                   & -3.29                            \\\\ \\hline\nmp-778103                   & -3.2414                        & mp-1225800                 & -2.2103                          & mp-698589                  & -2.8493                         & mp-20790                   & -2.9865                          \\\\ \\hline\nmp-13335                    & -3.2224                        & mp-1228436                 & -2.1092                          & mp-761314                  & -2.7696                         & mp-1224951                 & -2.7224                          \\\\ \\hline\nmp-780395                   & -3.1119                        & mp-1120750                 & -1.838                           & mp-5712                    & -2.5712                         & mp-1224810                 & -2.5462                          \\\\ \\hline\nmp-30161                    & -2.9124                        & mp-29672                   & -1.69                            & mp-1212041                 & -2.5703                         & mp-555538                  & -2.4165                          \\\\ \\hline\nmp-1194477                  & -2.4454                        & mp-1019378                 & -1.6865                          & mp-1255006                 & -2.5669                         & mp-1071955                 & -2.4012                          \\\\ \\hline\nmp-756317                   & -2.2645                        & mp-1228943                 & -0.3167                          & mp-765466                  & -2.5623                         & mp-1102285                 & -2.3984                          \\\\ \\hline\nmp-36866                    & -2.2639                        & mp-29672                   & -0.2741                          & mp-753397                  & -2.5622                         & mp-1071955                 & -2.398                           \\\\ \\hline\nmp-1208866                  & -2.1376                        & mp-1019508                 & -0.1409                          & mp-756649                  & -2.5621                         & mp-27462                   & -2.3971                          \\\\ \\hline\nN\/A                            & N\/A                                & mp-1223879                 & -0.0784                          & mp-1178203                 & -2.5621                         & mp-1102285                 & -2.3967                          \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/Structures.pdf}\n  \\caption{Candidate new structures with zero e-above-hull for GaB$_2$N$_4$ (using template mp-780282), GaB$_4$N$_3$ (using template mp-1224009), GaBN$_4$ (using template mp-30979), and Ga$_2$BN$_4$ (using template mp-698589).}\n  \\label{fig:struct}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nCrystal structure prediction plays a key role in new materials discovery \\cite{oganov2019structure}. However, large-scale fast prediction of crystal structures is challenging, and user-friendly web apps are missing for such an important function despite the availability of a few public software that needs expensive high-performance computing (HPC) resources and expertise of computational materials. We believe such fast crystal structure prediction web apps are critical to the materials science community, as demonstrated by the bioinformatics field, which has more than 9000 web servers \\cite{hu2021materialsatlas}. Here we propose a template-based crystal structure prediction algorithm, TCSP, and its companion web server for fast and quick crystal structure prediction. Due to the widely observed structure similarity across many materials families such as perovskites in the materials database, TCSP achieves strong prediction performance as benchmarked on the whole Materials Project structure using leave-one-out evaluation due to its flexible template selection algorithm using prototype and oxidation information. To our knowledge, this is the largest experiments for crystal structure prediction. We believe our web-based TCSP algorithm will be of great interest to materials scientists for exploratory materials discovery. \n\n\\begin{figure}[htb]\n  \\centering\n   \\begin{subfigure}[t]{0.49\\textwidth}\n        \\includegraphics[width=\\textwidth]{stableGaB2N4filling.png}\n        \\caption{predicted structure of GaB$_2$N$_4$ }\n        \\vspace{-3pt}\n        \\label{fig:GaB2N3_predict1}\n    \\end{subfigure} \n     \\begin{subfigure}[t]{0.5\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/phonons.pdf}\n          \\caption{Phonon dispersion of GaB$_2$N$_4$ with mp-780282 template structure.}\n          \\label{fig:phonons}\n    \\end{subfigure} \n    \\caption{New material GaB$_2$N$_4$ discovered by our TCSP with zero e-above-hull energy, negative formation energy (-3.2957 eV) and dynamical stability}\n\\end{figure}\n\n\\section{Data Availability}\n\nDataset is downloaded from the Materials Project database website. \n\n\\section{Contribution}\nConceptualization, J.H.; methodology,J.H., L.W., W.Y., E.S., N.F., S.O., R.D.; software, J.H., L.W.,W.Y., N.F. ; resources, J.H.; writing--original draft preparation, J.H., L.W., E.S.,N.F.,R.X.; writing--review and editing,  J.H, L.W.; visualization, J.H., L.W., E.S.; supervision, J.H.;  funding acquisition, J.H.\n\n\\section*{Acknowledgement}\nThe research reported in this work was supported in part by National Science Foundation under the grant and 1940099 and 1905775. The views, perspectives, and content do not necessarily represent the official views of the NSF. We thank Andrew Hughes for his help in proofreading the manuscript. \n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently there was suggestion to extend the Universe beyond the Big Bang using the analytic continuation of the radiation-dominated epoch across the singularity\\cite{Turok2018,Turok2018b}. In this analytic continuation, at which the scale factor $a(\\tau)$ changes sign at $\\tau=0$, the gravitational tetrads also change sign giving rise to what is called the antispacetime. This means that the Universe on the other side of the Big Bang is the mirror image of the Universe on our side of the Big Bang.\n\nDifferent types of the antispacetime obtained by the space reversal $P$ and time reversal $T$ operations  were considered ealier, including those where the determinant of the tetrads $e$ changes sign.\\cite{Diakonov2011,Diakonov2012,Rovelli2012b,Rovelli2012a,NissinenVolovik2018,Volovik2019b,Vergeles2019}\nLater the consideration has been extended to the thermal states, where the possible analytic continuation of the temperature across the transition from spacetime to antispacetime has been considered \\cite{Volovik2019}. Here we discuss two different scenarios of analytic contunuation across the Big Bang.\n\n\n\n\n\n\n\\section{CPT-symmetric Universe and negative temperature}\n\nThe CPT-symmetric Universe has been obtained in the conformal time frame\\cite{Turok2018}, where the metric in the spatially flat radiation-dominated era is:\n\\begin{equation}\ng_{\\mu\\nu}=a^2(\\tau)\\eta_{\\mu\\nu}\\,,\n\\label{metric}\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}$ is the flat Minkowski metric; $a(\\tau)$ is the scale factor and $\\tau$ is the conformal time. \nSince the scale factor $a(\\tau)\\propto\\tau$, it was suggested that it can be analytically transformed to the region $\\tau<0$ before the Big Bang. Then the terad fields, which are proporional to  $a(\\tau)$, also change sign under this analytic continuation,\n$e^a_\\mu(-\\tau)=-e^a_\\mu(\\tau)$. \n\nNow let us go further and extend the analytical continuation to the temperature of the system.\nIn the spatially flat radiation-dominated era one has\n\\begin{equation}\nT(\\tau)\\propto \\frac{1}{N(\\tau)}= \\frac{1}{e_{00}(\\tau)}  \\,\\,, {\\rm or} \\,\\, \\beta(\\tau)\\propto\\tau\\,.\n\\label{TinCPT}\n\\end{equation}\nThe analytic continuation of $\\beta(\\tau)$ suggests that in the Universe at $\\tau<0$ the temperature is negative.\nThe transition between the states with positive and negative temperatures occurs via the infinite $T$ at $\\tau=0$ ($\\beta(0)=0$).\nThe states with negative temperature are typically unstable thermodynamically. This means that if the analytic continuation is really valid, then one of the two states (before or after Big Bang) is thermodynamically unstable.\nIt is more natural to assume, that the evolution of the system from $\\tau = -\\infty$ to $\\tau=0$ was equilibrium and corresponded to the positive temperature, $T(\\tau<0)>0$. This is because the system had enough time to equilibrate before the collapse. So we should have the following analytic time dependence of temperature:\n\\begin{equation}\nT(\\tau)\\propto -\\frac{1}{e_{00}(\\tau)} \\,\\,, {\\rm or} \\,\\, \\beta(\\tau)\\propto -\\tau\\,,\n\\label{TinCPTopposite}\n\\end{equation}\nand thus on our side of the Big Bang the temperature is negative, $T(\\tau>0)<0$. Of course, this may happen only at the first stage of the evolution of our Universe, i.e. immediately after the Big Bang. After some time the equilibration occurs, and the system returns again to the evolution with the equilibrium positive temperature,  $T(\\tau>0)>0$.\nThe proper justification of this suggestion is beyond this comment.\n\n\\section{Euclidean signature Universe and spontaneously broken CPT symmetry}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{Bifurcation.pdf}\n\\caption{ Bifurcation at Big Bang. Analytic continuation in physical proper time $t$: the scale factor\n$a(t)\\propto \\sqrt{t}$ changes sign around the Big Bang point, transforming spacetime into antispacetime.\nAt $t<0$ the scale factor is imaginary, which corresponds to the metric with Euclidean  signature.\nCrossing the Big Bang from the $t<0$ semiaxis, the Universe approaches either spacetime or antispacetime. In this scenario, the Big Bang represents the bifurcation point at which the $Z_2$ symmetry \nbetween the spacetime and antispacetime is spontaneously broken. The quantum superposition of the two states is not allowed  in the macroscopic system.\n}\n\\label{TwoRoads_Fig}\n\\end{figure}\n\n\nLet us consider the analytic continuation in terms of the physical proper time $t$. The metric for the radiation-dominated Universe is:\n\\begin{equation}\ng_{\\mu\\nu}=dt^2 -a^2(t)d{\\bf r}^2\\,\\,,\\, a^2(t)  \\propto t \\,.\n\\label{metric2}\n\\end{equation}\nlet us now assume that the scale factor $a(t) \\sim \\sqrt{t}$ can be analytically extended around the singularity at $t=0$. Then the spacetime (with positive spatial components of tetrads) and antispacetime\n (with negative spatial components of tetrads) can be connected by analytic continuation: by $2\\pi$ rotation about $t=0$. As distinct from scenario in \\cite{Turok2018}, where the two Univereses live together,\nin the proper time scenario the spacetime and antispacetime represent two different realizations of the Universe at $t>0$, which exclude each other. \n\nThe Universe at $t>0$ can be obtained by analytic continuation from the negative $t$ region, where the metric has  Euclidean signature. In this scenario, the Big Bang (at $t=0$) represents the point of \nthe phase transition from the Euclidean to Minkowski signature, which is similar to that, say,  in \\cite{Steinacker2018,Stern2018,Sakharov1991,GibbonsHartle1990} (an example of such transition in condensed matter can be found e.g. in Ref. \\cite{NissinenVolovik2017b}). At the Big Bang, the state with Euclidean signature transforms either to the spacetime or to antispacetime, but not to the quantum superposition of the two Universes. The latter is not allowed for the macroscopic systems, which experience the spontaneous symmetry breaking.\\cite{Grady1994}\nIn principle, immediately after the Big Bang the Universe may evolve as the quantum superposition of these two states, but due to rapid decoherence only one of the two states survives. Thus in this analytic continuation the $Z_2$ ($CPT$) symmetry is spontaneously broken, as distinct from the scenario in \\cite{Turok2018}, which\ndescribes the creation of two Universes both evolving to the future with the conservation of the  $CPT$ symmetry. \n\n\n\n\\section{Discussion}\n\nIn conclusion, we extended the analytic continuation across the Big Bang proposed in Ref.\\cite{Turok2018}in two different ways. The analytic continuation in the conformal time frame $\\tau$ was extended to the temperature of the system. This extension suggests that if the analytic continuation of the terad is valid, then it is possible that\nthe initial stage of the evolution of our Universe after the Big Bang was characterized by the negative temperature. The analytic continuation in the proper time $t$  suggests that the  Big Bang is the bifurcation point of the second order quantum transition from the Euclidean to Minkowski signature, at which the symmetry between the spacetime and antispacetime is spontaneously broken.\n\n{\\bf Acknowledgements}. \n I thank A. Starobinsky for criticism. This work has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 694248).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHopf proved that the (singular) cohomology of a real connected compact Lie group $\\mathcal{G}$ is an exterior algebra on $\\mathrm{rank}(\\mathcal{G})$ generators of odd degree~\\cite{hopf1964topologie}.  Its Poincar\\'e series is therefore given by \\[\\mathrm{Hilb}(H^*(\\mathcal{G});q) = \\prod_{i=1}^n (1+q^{2e_i+1}).\\]\nChevalley presented these $e_i$ for the exceptional simple Lie algebras in his 1950 address at the International Congress of Mathematicians~\\cite{chevalley1950betti}, and Coxeter recognized them from previous work with real reflection groups~\\cite{coxeter1951product}.  This observation has led to deep relationships between the cohomology of $G$, and the invariant theory of the corresponding Weyl group $W=N_\\mathcal{G}(T)\/T$~\\cite{reeder1995cohomology,reiner2019invariant}---notably, $H^*(\\mathcal{G}) \\simeq \\left(H^*(\\mathcal{G}\/T) \\times H^*(T)\\right)^W \\simeq \udbff\udc01\\left(S\/S_+^W \\otimes \\bigwedge V^*\\right)^W.$\nFor more information, we refer the reader to the wonderful survey~\\cite{barcelo1994combinatorial}.\n\n\nIt turns out that one method to compute the $e_i$ is the generating function for the dimension of the fixed space ${\\mathrm{fix}}(w)=\\dim(\\ker(1-g))$ over the Weyl group:\n\\[\\sum_{w \\in W} q^{{\\mathrm{fix}}(w)} = \\prod_{i=1}^r (q+e_i).\\]\n\n\nShephard and Todd~\\cite{shephard1954finite} verified case-by-case that the same sum still factors when $W$ is replaced by a complex reflection group $G$. Let $G$ be a finite complex reflection group, acting by reflections on $V$.  The $e_i$ are now determined by the degrees $d_i$ of the fundamental invariants of $G$ on $V$ as $e_i=d_i-1$.  A case-free proof of this result was given by Solomon~\\cite{solomon1963invariants}, mirroring Hopf's result: writing $S=\\mathrm{Sym}(V^*)$ and $\\Lambda=\\bigwedge(V^*)$, $(\\left(S \\otimes \\Lambda\\right)^G$ is a free exterior algebra over the ring of $G$-invariant polynomials, which gives a factorization of the Poincar\\'e series \n\n\\begin{equation}\n\\label{eq:solomon}\n\\mathrm{Hilb}(\\left(S \\otimes \\Lambda \\right)^G;q,u) = \\prod_{i=1}^r \\frac{1+u q^{e_i}}{1-q^{d_i}}.\n\\end{equation}\nComputing the trace on $S \\otimes \\Lambda$ of the projection to the $G$-invariants $\\frac{1}{G}\\sum_{g\\in G} g$, specializing to $u=q(1-x)-1$, and taking the limit as $x \\to 1$ gives the Shephard-Todd result:\n\n\\begin{theorem}[{\\cite{shephard1954finite,solomon1963invariants}}]\\label{thm:OS-untwisted}\nFor any complex reflection group $G$,\n\\begin{equation}\n\\label{eq:shephard_todd}\n\\sum_{g \\in G} q^{{\\mathrm{fix}}(g)} = \\prod_{i=1}^r (q+e_i).\n\\end{equation}\n\\end{theorem}\n\nMore generally, define the \\defn{fake degree} of an $m$-dimensional (simple) $G$-module $M$ to be the polynomial encoding the degrees in which $M$ occurs in the coinvariant ring $\\mathbb{C}[V]_G$: $f_M(q)=\\sum_{i} (\\mathbb{C}[V]_G^i,M)q^i =\\sum_{i=1}^m q^{e_i(M)}.$\n\n\\medskip\nLet $G \\subset \\mathrm{GL}(V)$ be a complex reflection group.  We say that $N \\triangleleft G$ is a \\defn{normal reflection subgroup} of $G$ if it is a normal subgroup of $G$ that is generated by reflections.  For Weyl groups, nontrivial normal reflection subgroups can be constructed using root lengths.   More generally, normal reflection subgroups are constructed by taking the union of conjugacy classes of reflections.    We give the classification of normal reflection subgroups in~\\Cref{sec:classification}, and tie our work with previous work on their numerology in~\\Cref{sec:reflexponents}.\n\nThe following theorem is a special case of results in~\\cite{bessis2002quotients} (where the authors consider the more general notion of ``bon sous-groupe distingu\\'e'' in lieu of our normal reflection subgroup $N$ of $G$). We emphasize that our proof of this result in \\Cref{sec:quotients} follows the main ideas in~\\cite{bessis2002quotients}, specialized to our more restricted setting.\n\n\\begin{theorem}\n\\label{thm:reflection_action}\nLet $N$ be a normal reflection subgroup of a complex reflection group $G$ acting on $V$.  Then the quotient group $H=G\/N$ acts as a reflection group on the vector space ${E}=V\/N$.\\end{theorem}\n\n\nIn \\Cref{sec:quotients} we will build on our proof of \\Cref{thm:reflection_action} to prove the following numerological identities.\n\n\\begin{theorem}\n\\label{thm:numbers}\nLet $G,N,H$ be as in~\\Cref{thm:reflection_action}. For a suitable choice of indexing of degrees and fake degrees, we have the following equalities:\n\\begin{align*}\ne_i^N(V) {+} e_i^G({E}) &= e_i^G(V) \\\\\nd_i^N \\cdot d_i^H & = d_i^G \\\\\nd_i^N \\cdot e_i^H({E}) & = e_i^G({E})\n\\end{align*}\n\\end{theorem}\n\n\\begin{example}\nTake $G=W(F_4)=G_{28}$ and $N$ to be the normal subgroup generated by the reflections corresponding to short roots.  Then $N\\simeq W(D_4)$, $G\/N\\simeq W(A_2)=\\mathfrak{S}_3$ acting by reflections on $\\mathbb{C} \\oplus \\mathbb{C} \\oplus \\mathbb{C}^2$ (trivially on $\\mathbb{C}\\oplus\\mathbb{C}$), so that \\Cref{thm:numbers} gives the equations\n\\begin{align*}\n(1,5,3,3) {+} (0,0,4,8) &= (1,5,7,11) \\\\\n(2,6,4,4) \\cdot (1,1,2,3) & = (2,6,8,12) \\\\\n(2,6,4,4) \\cdot (0,0,1,2) & = (0,0,4,8).\n\\end{align*}\n\\end{example}\n\nThe following result simultaneously generalizes the results of~\\cite{williams2019reflexponents} and the Shephard-Todd formula~\\Cref{eq:shephard_todd} from \\Cref{thm:OS-untwisted}. We state a generalized version that incorporates Galois twists in \\Cref{sec:galois}.\n\n\\begin{theorem}\n\\label{thm:main_theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Then\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).\\]\n\\end{theorem}\n\n\\begin{example}\n\\label{ex:c2}\nThe dihedral group $G=G(2,1,2)=\\left\\langle s,t | s^2=t^2=(st)^4=1\\right\\rangle$ acts as a reflection group on $V=\\mathbb{C}^2$ by $s=\\left[\\begin{smallmatrix} -1 & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right] \\text{ and } t=\\left[\\begin{smallmatrix} 0 & 1 \\\\ 1 & 0 \\end{smallmatrix} \\right].$  Take $N$ to be the normal subgroup generated by the reflections conjugate to $s$.   Then $N \\simeq C_2 \\times C_2$ is a normal reflection subgroup, isomorphic to the direct product of the cyclic group of order two with itself, with invariants $N_1=x_1^2$ and $N_2=x_2^2$, and $G$ acts on ${E}$ dual to ${E^*}=\\mathrm{span}_\\mathbb{C}\\{N_1,N_2\\}$ as the quotient group $G\/N\\simeq C_2$ by $s=\\left[\\begin{smallmatrix} 1 & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right] \\text{ and } t=\\left[\\begin{smallmatrix} 0 & 1 \\\\ 1 & 0 \\end{smallmatrix} \\right].$  In this case, \\Cref{thm:main_theorem} expresses the equality\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = q^2t^2+2qt^2+2qt+2t+t^2 =(qt+t)(qt+t+2).\\]\n\\end{example}\n\n\\noindent\n{\\bf Acknowledgements.} We thank Theo Douvropoulos for many helpful comments and suggestions. The first author was partially supported by NSF grant CCF-1815108. The second author was partially supported by Simons Foundation award number 585380. \n\n\\section{Quotients by Normal Reflection Subgroups}\\label{sec:quotients} \n\nLet $V$ be a complex vector space of dimension $r$. A \\defn{reflection} is an element of $\\mathrm{GL}(V)$ that fixes some hyperplane pointwise. A \\defn{complex reflection group} $G$ is a finite subgroup of $\\mathrm{GL}(V)$ that is generated by reflections. A complex reflection group $G$ is called \\defn{irreducible} if $V$ is a simple $G$-module; $V$ is then called the \\defn{reflection representation} of $G$. A \\defn{(normal) reflection subgroup} of $G$ is a (normal) subgroup of $G$ that is generated by reflections.\n\nLet $S(V^*)$ be the symmetric algebra on the dual vector space $V^*$, and write $S(V^*)^G$ for its $G$-invariant subring.  By a classical theorem of Shephard-Todd~\\cite{shephard1954finite} and Chevalley~\\cite{chevalley1955invariants}, a subgroup $G$ of $\\mathrm{GL}(V)$ is a complex reflection group if and only if $S(V^*)^G$ is a polynomial ring, generated by $r$ algebraically independent homogeneous $G$-invariant polynomials---the \\defn{degrees} $d_1\\leq \\cdots \\leq d_r$ of these polynomials are invariants of $G$.\n\n\\begin{theorem}[\\cite{shephard1954finite,chevalley1955invariants}]\n\\label{thm:shephard_todd}\n  Let $G \\leq \\mathrm{GL}(V)$ be finite.  Then $G$ is a complex reflection group if and only if there exist $r$ homogeneous algebraically independent polynomials ${G}_1,\\ldots,{G}_r\\in S(V^*)^G$ such that $S(V^*)^G=\\mathbb{C}[{G}_1,\\ldots,{G}_r]$.  In this case, $|G|=\\prod_{i=1}^r d_i$, where $d_i=\\mathrm{deg}({G}_i)$.\n\\end{theorem}\n\nAlthough \\Cref{thm:reflection_action} is a special case of results in~\\cite{bessis2002quotients} (where they consider the more general notion of ``bon sous-groupe distingu\\'e''), the proof is more straightforward in our restricted setting, and also leads directly to a proof of \\Cref{thm:numbers}.\n\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:reflection_action}}\n\\begin{theorem}\nLet $N$ be a normal reflection subgroup of a complex reflection group $G$ acting on $V$.  Then the quotient group $H=G\/N$ acts as a reflection group on the vector space ${E}=V\/N$.\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\n\\begin{proof} We claim that there exist homogeneous generators ${N}_1,\\dots,{N}_r$ of $S(V^*)^N$ such that ${E}^*=\\mathrm{span}_\\mathbb{C}\\{{N}_1,\\ldots,{N}_r\\}$ is $H$-stable. By Theorem~\\ref{thm:shephard_todd}, $S(V^*)^N=\\mathbb{C}[\\tilde{{N}}_1,\\dots,\\tilde{{N}}_r]$ for some homogeneous algebraically independent $\\tilde{{N}}_i$. Let $I_+\\subset S(V^*)^N$ be the ideal generated by homogeneous elements of positive degree. Then both $I_+$ and $I_+^2$ are $H$-stable homogeneous ideals, and therefore the algebraic tangent space $I_+\/I_+^2$ to ${E}=V\/N$ at $0$ inherits a graded action of $H$ that is compatible with the (graded) quotient map $\\pi:I_+\\twoheadrightarrow I_+\/I_+^2$. Hence there exists a graded $H$-equivariant section $\\varphi:I_+\/I_+^2\\rightarrow I_+$. Letting ${N}_i=\\varphi\\circ\\pi(\\tilde{{N}}_i)$ we see that ${N}_1,\\dots,{N}_r$ are still homogeneous algebraically independent generators for $S(V^*)^N$ with $\\mathrm{deg}({N}_i)=\\mathrm{deg}(\\tilde{{N}}_i)$ and such that ${E}^*={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ is $H$-stable, as claimed.\n\n\nWrite ${G}_1, \\ldots, {G}_r$ for the homogeneous generators of $S(V^*)^G$, again as in Theorem~\\ref{thm:shephard_todd}. Consider the action of $H$ on ${E}^*$ defined by $(gN){N}_i\\coloneqq  g{N}_i.$ Since $S(V^*)^G= (S(V^*)^N)^H=S(E^*)^H$, there exist polynomials ${H}_1,\\ldots,{H}_r \\in \\mathbb{C}[\\mathbf{{N}}]$ such that ${H}_i(\\mathbf{{N}})={G}_i(\\mathbf{x})$, where $\\mathbf{{N}}=\\{{N}_1,\\dots,{N}_r\\}$ and $\\mathbf{x}=\\{x_1,\\dots,x_r\\}$ denote dual bases for $E$ and $V$, respectively.\n\nSince any algebraic relation $f({H}_1,\\ldots,{H}_r)=0$ would result in an algebraic relation $f({G}_1,\\ldots,{G}_r)=0$, the ${H}_i$ must be algebraically independent. By \\Cref{thm:reflection_action}, $H$ is a complex reflection group.\\end{proof}\n\n\\begin{remark}\nNote that the quotient group $H=G\/N$ does not necessarily lift to a reflection subgroup of $G$ nor even a subgroup of $G$. A counterexample is given by $G(4,2,2)=N \\triangleleft G=G_8$, so that $G\/N \\simeq \\mathfrak{S}_3$.\n\\end{remark}\n\n\\begin{remark}\n\\label{rem:indexing}\nIn the course of the proof of \\Cref{thm:reflection_action} we showed that the vector space ${E}=V\/N$ on which $H$ acts by reflections is dual to ${E^*}:={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ for a certain choice of fundamental $N$-invariants ${N}_1,\\dots{N}_r\\in S(V^*)^N$ such that ${E^*}$ is $G$-stable. The resulting action of $H$ on ${E}$ respects the $\\mathbf{x}$-grading on the $N$-invariants ${N}_i(\\mathbf{x})$, and therefore there is a choice of fundamental $H$-invariants ${H}_1,\\dots,{H}_r\\in S({E^*})^H=S(V^*)^G$ such that each ${H}_i(\\mathbf{N})={G}_i(\\mathbf{x})$ is both $\\mathbf{x}$-homogeneous of $\\mathbf{x}$-degree $=:d_i^G$ and $\\mathbf{N}$-homogeneous of $\\mathbf{N}$-degree $=:d_i^H$. Since the action of $H$ on ${E^*}$ respects the homogeneous decomposition of ${E^*}$ according to $\\mathbf{x}$-degree, the ${H}_i(\\mathbf{N})$ may be chosen such that the $N$-invariants ${N}_j(\\mathbf{x})\\in\\mathbf{N}$ occurring non-trivially in ${H}_i(\\mathbf{N})$ are all of the same $\\mathbf{x}$-degree $=:d_i^N$. This relationship between the fundamental invariants for $N$, $G$, and $H$ acting on their corresponding reflection representations leads to interesting numerological identities.\n\\end{remark}\n\nThe following result motivates some of the theoretical ingredients in our proofs.\n\n\\begin{theorem}[{\\cite{solomon1963invariants}}]\n\\label{thm:solomon} If $G\\subset\\mathrm{GL}(V)$ is a complex reflection group, then the ring $\\left(S(V^*)\\otimes \\bigwedge V^*\\right)^G$ is a free exterior algebra over the ring of $G$-invariant polynomials:\n\\[\\left(S(V^*)\\otimes \\bigwedge V^*\\right)^G \\simeq S(V^*)^G \\otimes \\bigwedge U_G,\\] where $U_G={\\mathrm{span}}_\\mathbb{C}\\left\\{d{G}_1,\\ldots,d{G}_s\\right\\}$ and $d{G}_i=\\sum_{j=1}^r \\frac{\\partial {G}_i}{\\partial x_j}  \\otimes x_j$.\\end{theorem}\n\n\\begin{comment}\n\\begin{proof}\nLet $u_i:=d{N}_i$. Consider the multiplication map $\\mu:{U^N}\\to {E}^*$ given by \\[\\mu(u_i)=\\sum_{j=1}^rx_j\\frac{\\partial {N}_i}{\\partial x_j},\\] and note that $\\mu$ is $H$-equivariant of degree $+1$ and $\\mu\\circ d{N}_i=d_i^N{N}_i$.\n\\end{proof}\n\\end{comment}\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:numbers}}\n\\begin{theorem}\nLet $G,N,H$ be as in~\\Cref{thm:reflection_action}. For a suitable choice of indexing of degrees and fake degrees, we have the following equalities:\n\\begin{align*}\ne_i^N(V) {+} e_i^G({E}) &= e_i^G(V) \\\\\nd_i^N \\cdot d_i^H & = d_i^G \\\\\nd_i^N \\cdot e_i^H({E}) & = e_i^G({E})\n\\end{align*}\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\n\\begin{proof} Having chosen fundamental $N$-invariants ${N}_1,\\dots,{N}_r\\in S(V^*)^N$ such that ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ is $G$-stable as in the proof of \\Cref{thm:reflection_action} and \\Cref{rem:indexing}, we have fundamental $G$-invariants ${G}_i(\\mathbf{x})={H}_i(\\mathbf{N})$ that are $\\mathbf{x}$-homogeneous of $\\mathbf{x}$-degree $d_i^G$ and $\\mathbf{N}$-homogeneous of $\\mathbf{N}$-degree $d_i^H$, and where the ${N}_j\\in\\mathbf{N}$ occurring non-trivially in ${H}_i(\\mathbf{N})$ are all of the same $\\mathbf{x}$-degree $d_i^N$. The equality $d_i^Nd_i^H=d_i^G$ is immediate.\n\nLet us show that this same choice of indexing of fundamental invariants for $N$, $G$, and $H$ results in the other two equalities. We begin by comparing $\\mathbf{x}$-degrees in \\[d{G}_i=\\sum_{j=1}^r\\frac{\\partial{G}_i}{\\partial x_j}\\otimes x_j=\\sum_{k=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\cdot d{N}_k=\\sum_{k=1}^r\\sum_{j=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\cdot\\frac{\\partial{N}_k}{\\partial x_j}\\otimes x_j.\\] Recall that $e_i^G(V)=d_i^G-1=\\mathrm{deg}_\\mathbf{x}(d{G}_i)$ and $e_i^N(V)=d_i^N-1=\\mathrm{deg}_\\mathbf{x}(d{N}_i)$. Similarly, $e_i^H(E)=d_i^H-1=\\mathrm{deg}_\\mathbf{N}(d{H}_i)$, where this time $d{H}_i=\\sum_{k=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\otimes{N}_k\\in  (S({E^*})\\otimes {E^*})^H$. Since $\\frac{\\partial {H}_i}{\\partial {N}_k}=0$ whenever $\\mathrm{deg}_\\mathbf{x}({N}_k)\\neq d_i^N$, it follows that \\[e_i^G(V)=e_i^N(V)+d_i^N\\cdot (d_i^H-1)=e_i^N(V)+d_i^N\\cdot e_i^H(E).\\] It remains to show that $d_i^Ne_i^H(E)=e_i^G({E})$.\n\nThe $e_i^G({E})$ are known to coincide with the $\\mathbf{x}$-degrees of any set of homogeneous generators for $(S(V^*)\\otimes{E^*})^G$ as a free $S(V^*)^G$-module. Since ${E^*}$ consists of $N$-invariants, \\[(S(V^*)\\otimes{E^*})^G=((S(V^*)\\otimes{E^*})^N)^H=(S({E^*})\\otimes{E^*})^H\\simeq S({E^*})^H\\otimes U_H=S(V^*)^G\\otimes U_H,\\] where again $U_H:={\\mathrm{span}}_\\mathbb{C}\\{d{H}_1,\\dots,d{H}_r\\}$ and the non-trivial isomorphism comes from \\Cref{thm:solomon} applied to the reflection representation ${E}$ of $H$. Hence $(S(V^*)\\otimes{E^*})^G$ is generated by $d{H}_i$ as a free $S(V^*)^G$-module, whence $e_i^G({E})=\\mathrm{deg}_\\mathbf{x}(d{H}_i)=d_i^Ne_i^H({E})$.\\end{proof}\n\n\\begin{remark}\\label{rem:bigraded_solomon}\nThe same argument used in the proof of \\Cref{thm:numbers} shows more generally: \\[(S(V^*)\\otimes\\bigwedge {E^*})^G=((S(V^*)\\otimes \\bigwedge {E^*})^N)^H=(S({E^*})\\otimes\\bigwedge {E^*})^H\\simeq S(V^*)^G\\otimes\\bigwedge U_H.\\] \n\\end{remark}\n\n\\section{Poincar\\'e Series and Specializations}\\label{sec:poincare}\n\nOur goal in this section is to prove our main result:\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:main_theorem}}\n\\begin{theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Then\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).\\]\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\nWe refer to the left-hand side of~\\Cref{thm:main_theorem} as the \\defn{sum side}, and to the right-hand side as the \\defn{product side}.  \nWe prove \\Cref{thm:main_theorem} by computing the Poincar\\'e series for $(S(V^*)\\otimes\\bigwedge {E^*})^G$ in two different (and standard) ways, keeping track of the supplemental grading afforded by the $\\mathbf{x}$-degrees of $N$-invariants in ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$: one way corresponds to the product side (\\Cref{sec:prod}), and the other to the sum side (\\Cref{sec:sum}).  A subtlety arises when trying to compute the term-by-term specialization for the sum side, which is dealt with in~\\Cref{sec:coseti,sec:cosetii}.\n\nA more general version of~\\Cref{thm:main_theorem} that incorporates  Galois twists is stated in~\\Cref{sec:galois}. For technical reasons that arise in that generalization, we will define the shifted homogeneous decomposition ${E}^*_m:={\\mathrm{span}}_\\mathbb{C}\\{{N}_i \\ | \\ \\mathrm{deg}_\\mathbf{x}({N}_i)=m+1\\}$, and similarly\n\\[({\\textstyle\\bigwedge^p}{E^*})_m:={\\mathrm{span}}_\\mathbb{C}\\{ {N}_{i_1}\\wedge\\dots\\wedge{N}_{i_p}\\in{\\textstyle\\bigwedge^p}{E^*} \\ | \\ {\\textstyle\\sum_{j=1}^p}\\mathrm{deg}_\\mathbf{x}({N}_{i_j})=m+p\\}.\\]\nWriting $S(V^*)_\\ell$ for the homogeneous component of $\\mathbf{x}$-degree $\\ell$, we define the Poincar\\'e series\n\\begin{equation}\\label{eq:p-series-defn}\n\\mathcal{P}(x,y,u):=\\sum_{\\ell,m,p\\geq 0}\\mathrm{dim}_\\mathbb{C}((S(V^*)_\\ell\\otimes ({\\textstyle\\bigwedge^p}{E^*})_m)^G)x^\\ell y^m u^p.\n\\end{equation}\nWe write ${\\mathcal{E}^N_\\sigma}=\\{e_1^N({V^\\sigma}),\\dots,e_r^N({V^\\sigma})\\}$ for the set of fake degrees of ${V^\\sigma}$ as an $N$-representation.\n\n\\subsection{Product Side}\n\\label{sec:prod}\nWe first obtain the following product formula for the Poincar\\'e series $\\mathcal{P}(x,y,u)$ defined in \\Cref{eq:p-series-defn} from \\Cref{thm:numbers} and \\Cref{rem:bigraded_solomon}, since $d{H}_i\\in S(V^*)_{e_i^G(E)}\\otimes {E}^*_{e_i^N(V)}$.\n\n\\begin{lemma}\n\\label{lem:product_side}\n$\\mathcal{P}(x,y,u)=\\displaystyle\\prod\\limits_{i=1}\\limits^r\\frac{1+x^{e_i^G({E})}y^{e_i^N(V)}u}{1-x_i^{d_i^G}}.$\n\\end{lemma}\n\n\\begin{corollary}\n\\label{cor:prod_side_specialization}\n$\\lim\\limits_{x\\to1}\\mathcal{P}\\left(x,x^t,qt(1-x)-1\\right)=\\displaystyle\\prod\\limits_{i=1}\\limits^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).$\n\\end{corollary}\n\n\n\\subsection{Sum Side}\n\\label{sec:sum}\nBy taking traces, we now compute a formula for the Poincar\\'e series $\\mathcal{P}(x,y,u)$ defined in \\Cref{eq:p-series-defn} as a sum over elements of $G$. To simplify notation, we denote by ${E}_m$ the homogeneous component of ${E}$ corresponding to the dual of ${E}^*_m$.\n\n\n\\begin{lemma} \\label{thm:5.2} \n$\n\\mathcal{P}(x,y,u)=\n\\frac{1}{|G|} \\displaystyle\\sum\\limits_{g \\in G} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g|_{{E}_m})}{\\det(1-xg|_{V})}.$\n\\end{lemma}\n\n\\begin{proof}\nSince ${(U^N_\\sigma)^*}\\simeq \\bigoplus_{m\\in{\\mathcal{E}^N_\\sigma}} {E}^*_m$, we have that\n$\\bigwedge {(U^N_\\sigma)^*}\\simeq \\bigotimes_{m\\in{\\mathcal{E}^N_\\sigma}}\\bigwedge {E}^*_m$ as $G$-modules. hence, for each $g\\in G$, \\[\\sum_{m,p\\geq 0} \\mathrm{tr}(g|({\\textstyle\\bigwedge^p}{(U^N_\\sigma)^*})_{m})y^mu^p= \\prod_{m\\in{\\mathcal{E}_N}}\\left(\\sum_{p\\geq 0}\\mathrm{tr}(g|{\\textstyle\\bigwedge^p} {E}^*_m )y^{pm}u^p\\right).\\]\n\nFor each $m\\in{\\mathcal{E}^N_\\sigma}$ we have $\\displaystyle\\sum\\limits_{p\\geq 0}\\left(\\mathrm{tr}(g|{\\textstyle\\bigwedge^p}{E}^*_m)y^{pm}u^p\\right)=\\mathrm{det}(1+y^{m}ug|_{{E}^*_{m}}).$\n Therefore,\n\\begin{align*}\n\\mathcal{P}(x,y,u)&=\\left(\\sum_{\\ell\\geq 0}\\mathrm{tr}(g|(S({V^*})_\\ell)x^\\ell\\right)\\left(\\sum_{m,p\\geq 0}\\mathrm{tr}(g|({\\textstyle \\bigwedge^p}{(U^N_\\sigma)^*})_m)y^mu^p\\right)\\\\\n&=\\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g^{-1}|_{{U^N_\\sigma}_m})}{\\det(1-xg^{-1}|_{V})}.\\end{align*}\nThe result follows after taking the average over $G$ on each side.\\qedhere\n\\end{proof}\n\nThe story is not quite so simple as just setting the sum over $G$ from~\\Cref{thm:5.2} equal to the product, and then specializing.  The trouble is that in this specialized sum over $G$ from~\\Cref{thm:5.2}, each element of $G$ \\emph{does not} necessarily contribute the ``correct amount'' specified by the sum side of~\\Cref{thm:main_theorem}---in particular, $(ng)|_{{E}}$ often has larger fixed space than $(ng)|_{{V}}$, which causes many terms in the term-by-term limit to be zero.  It turns out, as we will now show, that the contributions \\emph{are} correct when taken coset-by-coset.\n\n\n\n\\subsection{Sum Side, Coset-by-Coset I}\n\\label{sec:cosetii}\nFix some $g \\in G$.  We find a product formula for \\Cref{thm:5.2} restricted to the coset $gN$. Define \\[\\mathcal{P}_{gN}(x,y,u):=\\frac{1}{|N|}\\sum_{n \\in N} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}} \\det\\left(1+u y^{m} (ng)|_{{E}^*_{m}}\\right) }{\\det(1-x(ng)|_{V^*})}.\\]\nGiven $g\\in G$, we can choose the fundamental $N$-invariants ${N}_1,\\dots,{N}_r\\in S(V^*)^G$ to also form a $g$-eigenbasis for ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$, since this space is $G$-stable and $g$ has finite order.  For $g \\in G$, let $\\epsilon_1^g,\\dots,\\epsilon_r^g$ denote the eigenvalues of $g$ on ${E^*}$, so that $g{N}_i=\\epsilon_i^g{N}_i$. \n\n\\begin{proposition}\n\\label{prop:twisted_graded}\n$\\mathcal{P}_{gN}(x,y,u) = \\displaystyle\\prod\\limits_{i=1}^r \\frac{1+ \\epsilon_i^g u y^{e_i^N(V)}}{1-\\epsilon_i^g x^{d_i^N}}.$\n\\end{proposition}\n\n\\begin{proof}\nFirst, $\\displaystyle\\prod\\limits_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}(ng)|_{{E}^*_{m}})=\\displaystyle\\prod\\limits_{i=1}\\limits^r(1+\\epsilon_i^guy^{e_i^N({V^\\sigma})})$ uniformly for any $n\\in N$, since ${E}^*_m$ is $N$-invariant. It remains to show that \\[{\\frac{1}{|N|}\\sum_{n\\in N}\\frac{1}{\\mathrm{det}(1-x(ng)|_{{V^*}})}=\\prod_{i=1}^r\\frac{1}{1-\\epsilon_i^gx^{d_i^N}}}.\\]\n\nSince $S({E}^*)\\simeq\\bigotimes_{m\\in\\mathcal{E}_N}\\mathrm{Sym}({E}^*_m),$ where ${E}^*_m$ denotes the span of fundamental $N$-invariants having $\\mathbf{x}$-degree $m+1$ and $\\mathrm{Sym}({E}^*_m)$ denotes its symmetric algebra, we have\n\\[\\sum_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr)x^\\ell = \\prod_{m\\in\\mathcal{E}_N}\\Bigl(\\sum_{\\ell\\geq 0}\\bigl(\\mathrm{tr}(g|\\mathrm{Sym}^\\ell({E}^*_m))(x^{m+1})^\\ell\\bigr), \\] where $S({E}^*)_\\ell:=S({E}^*)\\cap S(V^*)_\\ell$ and $S(V^*)_\\ell$ as before denotes the homogeneous subspace of polynomials of $\\mathbf{x}$-degree $\\ell$. On the other hand,\n\\[\\prod_{m\\in\\mathcal{E}_N}\\Bigl(\\sum_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|\\mathrm{Sym}^\\ell({E}^*_{m})\\bigr)(x^{m+1})^\\ell\\Bigr)=\\prod_{m\\in\\mathcal{E}_N}\\frac{1}{\\mathrm{det}(1-x^{m+1}(g|_{{E}^*_{m}}))}=\\prod_{i=1}^r\\frac{1}{1-\\epsilon_i^gx_i^{d_i^N}},\\]\nsince $d_i^N=e_i^N(V)$. Therefore, $\\displaystyle\\sum\\limits_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr)x^\\ell=\\displaystyle\\prod\\limits_{i=1}^r\\frac{1}{1-\\epsilon_i^gx_i^{d_i^N}}.$ Since for each $n\\in N$ we have $\\displaystyle\\sum\\limits_{\\ell\\geq 0}\\mathrm{tr}\\bigl(ng|S(V^*)_\\ell\\bigr)x^\\ell = \\frac{1}{\\mathrm{det}(1-x(ng|_{V^*}))},$ it remains to show that, for each $\\ell\\geq 0$, \\[\\frac{1}{|N|}\\sum_{n\\in N}\\Bigl(\\mathrm{tr}\\bigl(ng|S(V^*)_\\ell\\bigr)\\Bigr)=\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr).\\]\nTo see this, note that the operator $\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n\\in N}ng=g\\cdot\\left(\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n\\in N}n\\right)=g\\circ\\mathrm{pr}^N_\\ell,$ where $\\mathrm{pr}^N_\\ell=\\frac{1}{|N|}\\sum_{n\\in N} n$ is the projection from $S(V^*)_\\ell$ onto its $g$-stable subspace $S(V^*)_\\ell^N=S(E^*)_\\ell$, whence $\\mathrm{tr}\\bigl((g\\circ\\mathrm{pr}^N_\\ell)|S(V^*)_\\ell\\bigr)=\\mathrm{tr}\\bigl(g|S(E^*)_\\ell\\bigr)$. \\end{proof}\n\n\\subsection{Sum Side, Coset-by-Coset II}\n\\label{sec:coseti}\nWe next specialize some results of \\cite{bonnafe2006twisted} to the case when $N$ is a normal reflection subgroup of a complex reflection group $G$. They consider the more general situation when $N$ is translated by an arbitrary element in the normalizer of $N$ in $\\mathrm{GL}(V)$.  Continue to fix some $g \\in G$. \n\n\\begin{proposition}[{\\cite{bonnafe2006twisted}}]\n\\label{thm:twisted_os}\n\\[\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n \\in N} \\frac{\\det(1+u (ng)|_{V^*})}{\\det(1-x (ng)|_{V^*})} = \\displaystyle\\prod\\limits_{i=1}^r \\frac{1+ \\epsilon_i^g u x^{e_i^N(V)}}{1-\\epsilon_i^g x^{d_i^N}} \\left( = \\mathcal{P}_{gN}(x,x,u)\\right).\\]\n\\end{proposition}\n\nSpecializing both sides of~\\Cref{thm:twisted_os} to $u=q(1-x)-1$ and then taking the limit $x \\to 1$ yields the following simple formula for sums over cosets.\n\n\\begin{corollary}[{\\cite{bonnafe2006twisted}}]\n\\label{cor:twisted_os} \n\\[\\displaystyle\\sum\\limits_{n \\in N} q^{{\\mathrm{fix}}_V(ng)} = \\left(\\displaystyle\\prod\\limits_{\\epsilon^g_i =1} q+e_i^N(V)\\right) \\left(\\displaystyle\\prod\\limits_{\\epsilon_i^g \\neq 1} d_i^N\\right).\\]\n\\end{corollary}\n\nUsing~\\Cref{cor:twisted_os}, we obtain the following crucial specialization of \\Cref{prop:twisted_graded}, exploiting the fact that \\Cref{prop:twisted_graded} gives the series $\\mathcal{P}_{gN}(x,y,u)$, while \\Cref{thm:twisted_os} gives the series $\\mathcal{P}_{gN}(x,x,u)$.\n\n\\begin{corollary}\n\\label{cor:twisted_graded}\n$\\lim_{x\\to 1} \\mathcal{P}_{gN}(x,x^t,qt(1-x)-1) = t^{{\\mathrm{fix}}_E g}\\sum_{n \\in N} q^{{\\mathrm{fix}}_V (ng)}.$\n\\end{corollary}\n\n\\subsection{Proof of Theorem \\ref{thm:main_theorem}}  We now prove our main theorem.\n\\begin{proof}[Proof of~\\Cref{thm:main_theorem}] \n  Equating the formulas from~\\Cref{lem:product_side,thm:5.2} gives\n\\begin{align}\\label{eq:sum_eq_product}\n\\mathcal{P}(x,y,u)=\\sum_{g \\in G} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g|_{{E}_m})}{\\det(1-xg|_{V})} = |G|\\prod_{i=1}^r\\frac{1+x^{e_i^G({U^N_\\sigma})}y^{e_i^N(V)}u}{1-x_i^{d_i^G}}. \\end{align}\n\nLet $\\{g_j\\}_{j=1}^{|H|}$ be a set of coset representatives for $N$ in $G$. By~\\Cref{cor:twisted_graded}, splitting the sum side of~\\Cref{eq:sum_eq_product} into a sum over the cosets of $N$ and specializing gives\n\\begin{align*}\n\\lim_{x\\to 1} \\mathcal{P}(x,x^t,qt(1-x)-1)&= \\sum_{j=1}^{|H|}\\lim_{x\\to 1} \\mathcal{P}_{gN}(x,x^t,qt(1-x)-1) \\\\ &= \\sum_{j=1}^{|H|} t^{{\\mathrm{fix}}_{E} g_j}\\sum_{n \\in N} q^{{\\mathrm{fix}}_V ng_j} = \\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{E} g}.\n\\end{align*}\n\nThe result now follows from~\\Cref{eq:sum_eq_product} by equating this specialization of the sum side with the same specialization of the product side from~\\Cref{cor:prod_side_specialization}.\n\\end{proof}\n\n\\section{Classification of Normal Reflection Subgroups}\n\\label{sec:classification}\nIn the interest of space, we restrict our classification of normal reflection subgroups to $\\mathrm{rank} \\geq 3$.  In rank 2, there are two connected posets of imprimitive complex reflection groups ordered by normality: one has maximal element $G_{11}$ and minimal elements $G(4,2,2)$, $G_4$, and $G_{12}$, while the other has maximal element $G_{19}$ and minimal elements $G_{16}$, $G_{20}$, and $G_{22}$.\n\n\n\\begin{theorem}[{\\cite[Corollary 2.18]{lehrer2009unitary}}]\nFor $r\\geq 3$, the normal reflection subgroups of $G(ab,b,r)$ are $(C_d)^r$ and  $G(ab,db,r)$ for $d|a$, giving quotients $G(ab,b,r)\/(C_d)^r = G((a\/d) b,b,r)$ and $G(ab,b,r)\/G(ab,db,r) = C_d$.\n\\end{theorem}\n\nAs $G_{26}$ and $G_{28}$ are the only exceptional reflection groups with more than a single orbit of reflections, there are three nontrivial exceptional (that is, not imprimitive) examples of normal reflection subgroups in rank greater than two: $G(3,3,3) \\triangleleft G_{26}$, with quotient $G_4$; $G_{25} \\triangleleft G_{26}$, with quotient $C_2$;  and $G(2,2,4) \\triangleleft G_{28} \\simeq W(F_4)$, with quotient $\\mathfrak{S}_3$.\n\n\n\\section{Reflexponents}\n\\label{sec:reflexponents}\nFix $G$ a complex reflection group of rank $r$ with reflection representation $V$.  Call an $r$-dimensional representation $M$ of $G$ \\defn{factorizing} if $M$ has dimension $r$ and\n\\[\\sum_{g \\in G} q^{\\mathrm{fix}_V(g)} t^{\\mathrm{fix}_M(g)} = \\prod_{i=1}^r \\Big(qt+(e_i^G(V)-m_i)t+ m_i\\Big),\\] for some nonnegative integers $m_1,\\ldots,m_r$.  More generally, call a representation $M$ of $G$ of dimension $\\dim M \\leq r$ factorizing if it is factorizing in the above sense after adding in $r-\\dim M$ copies of the trivial representation.\n\nWe can now give a uniform explanation for certain ad-hoc identities from~\\cite{williams2019reflexponents}.  Let $\\mathcal{H}$ be an orbit of reflecting hyperplanes, write $\\mathcal{R}_\\mathcal{H}$ for the set of reflections fixing some $L \\in \\mathcal{H}$, and let $N_\\mathcal{H} = \\left\\langle \\mathcal{R}_\\mathcal{H} \\right\\rangle$ be the subgroup generated by reflections around hyperplanes in $\\mathcal{H}$.  Since these reflections form a conjugacy class in $G$, $N_\\mathcal{H}$ is a normal reflection subgroup of $G$. Furthermore: the quotient $G\/N_\\mathcal{H}$ acts as a reflection group on the $N_\\mathcal{H}$-invariants of $V$; and this action gives a $G$-representation $M_\\mathcal{H}$ that is factorizing by~\\Cref{thm:main_theorem}.\n\n\\section{Galois Twists}\\label{sec:galois}\n\n\nLet $V$ be an $r$-dimensional complex vector space and $G\\subset\\mathrm{GL}(V)$ be a complex reflection group. It is known that $G$ can be realized over $\\mathbb{Q}(\\zeta_G)$, where $\\zeta_G$ is a primitive $|G|$-th root of unity, in the sense that there is a basis for $V$ with respect to which $G\\subset \\mathrm{GL}_r(\\mathbb{Q}(\\zeta_G))$. For $\\sigma\\in\\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$, the \\defn{Galois twist} $V^\\sigma$ is the representation of $G$ on the same underlying vector space $V$ obtained by applying $\\sigma$ to the matrix entries of $g\\in\\mathrm{GL}(\\mathbb{Q}(\\zeta_G))$. Orlik and Solomon found a beautiful generalization of \\Cref{eq:shephard_todd} that takes into account these Galois twists. Below we write $\\lambda_1(g),\\dots,\\lambda_r(g)$ for the eigenvalues of $g\\in G$ on $V$.\n\n\\begin{theorem}[{\\cite{orlik1980unitary}}]\n\\label{thm:orlik_solomon}\nFix $G$ a reflection group and $\\sigma \\in \\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$.  Then \n\\[\\sum_{g \\in G} \\left(\\prod_{\\lambda_i(g) \\neq 1} \\frac{1-\\lambda_i(g)^\\sigma}{1-\\lambda_i(g)}\\right) q^{{\\mathrm{fix}}_V g} = \\prod_{i=1}^r \\left(q+e_i(V^\\sigma) \\right).\\]\n\\end{theorem}\n\n\\begin{definition}\nLet $I^G_+\\subset S(V^*)$ be the ideal generated by homogeneous $G$-invariant polynomials of positive degree, and let $\\mathcal{C}_G$ be a $G$-stable homogeneous subspace of $S(V^*)$ such that $S(V^*)\\simeq I^G_+\\oplus \\mathcal{C}_G$ as $G$-modules. For $\\sigma\\in\\mathrm{Gal}(\\zeta_G)$, define the \\defn{Orlik-Solomon space} $U_G^\\sigma:=(\\mathcal{C}_G\\otimes ({V}^\\sigma)^*)^G={\\mathrm{span}}_\\mathbb{C}\\{u_1^G,\\dots,u_r^G\\}$, where the $u_i^G$ are homogeneous with $\\mathrm{deg}(u_i^G)=e_i^G(V^\\sigma)$ and such that $\\bigl(S(V^*)\\otimes (V^\\sigma)^*\\bigr)^G\\simeq S(V^*)^G\\otimes U_G^\\sigma$.\n\\label{def:osspace}\n\\end{definition}\n\n We can now state the general form of our main theorem.\n\n\\renewcommand{\\thetheorem}{\\ref{thm:main_theorem}}\n\\begin{theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Let $\\sigma\\in\\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$ and define the Orlik-Solomon space $U_N^\\sigma$ as in \\Cref{def:osspace}. Then\n\\[\\sum_{g \\in G}\\left( \\prod_{\\lambda_i(g) \\neq 1} \\frac{1-\\lambda_i(g)^\\sigma}{1-\\lambda_i(g)}\\right) q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{E} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V^\\sigma) t + e_i^G((U_N^\\sigma)^*)\\right).\\]\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n\n\\begin{remark}\nThe Orlik-Solomon space $U_N^\\sigma$ playes the role of ${E}^*$ in this generalization of~\\Cref{thm:main_theorem}. When $\\sigma=1$, a straightforward argument yields a graded $G$-module homomorphism (of degree $-1$) ${E^*}\\simeq U_N$. This is why we shifted the obvious $\\mathbf{x}$-grading of ${E}^*$ by $-1$ in \\Cref{sec:poincare}. However, it is not true in general that $({E}^\\sigma)^*\\simeq U_N^\\sigma$ when $\\sigma\\neq 1$.\n\\end{remark}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\IEEEPARstart{B}{ounds} on the performance of electrically small antennas provide meaningful estimates to the limiting trends in key metrics for wireless systems.  Bounds on Q-factor\\footnote{Readers are directed to \\cite{Schab2017energy} for a detailed discussion of Q-factor and its various definitions for radiating systems.} (closely related to impedance bandwidth), and radiation efficiency are two of the most studied parameters, with mature methodologies available for their computation on a variety of objects.  From analytic methods on spheres \\cite{Chu1948,Pfeiffer_FundamentalEfficiencyLimtisForESA} to advanced numerical procedures for arbitrary geometries \\cite{capek2017minimization,gustafsson2019trade}, the general trends are that both Q-factor and efficiency are dramatically constrained as the size of an electrically small antenna is reduced.\n\nOne possibility to circumvent these bounds is through the use of non-linear or time-varying (non-LTI) antennas, which are not necessarily constrained in the same ways as their linear time-invariant (LTI) counterparts.  Several promising approaches have been reported using switched matching networks \\cite{Galejs1963,Vallese1972,Gamble1973,Xu2006,Zhu2014,manteghi2017,schab2019pulse,manteghi2019}, switched DC supplies \\cite{Merenda2006,Manteghi2016}, or time-varying tuning and matching elements on an antenna itself \\cite{Keller2010,Salehi2013} to successfully transmit broadband signals from electrically small antennas .  However, lacking classically defined impedances or Q-factors, these time-varying systems cannot be directly compared to analogous LTI systems by the common practice of comparing impedance bandwidth and continuous wave (CW) efficiency.  This presents a major obstacle in the fair, replicable evaluation of non-LTI devices and their relative performance tradeoffs compared to conventional systems. \n\nThe goal of this work is to reformulate well-known behavioral trends of LTI electrically small dipole antennas into forms compatible with time domain metrics applicable to non-LTI systems.  These metrics are based heavily on those developed for assessing signal distortion (or equivalently, fidelity) in ultra-wideband (UWB) transmit systems \\cite{farr1992,Lamensdorf1994} which deal directly with the signals radiated by an arbitrary source.  By moving to this methodology, we remove all dependence of assumptions on the LTI \/ non-LTI nature of a transmitter.\n\nWe begin by summarizing distortion, as used in this paper, in Section~\\ref{sec:dist-and-eff}.  This is followed in Section~\\ref{sec:mappings} by development of field-circuit mappings necessary for the study of distortion due to both spherical TM$_{01}$ radiators (exact, specific) and single resonance electrically small dipole type antennas (approximate, general).  Several numerical examples studying the accuracy of the approximate single resonance model and comparison of various simulated electrically small antennas are presented in Section~\\ref{sec:examples}.  Finally we apply the developed methodology to assess the performance of measured conventional (LTI) and direct antenna modulation (DAM, non-LTI) systems reported in \\cite{schab2019pulse}.\n\n\n\n\n\\section{Distortion}\n\\label{sec:dist-and-eff}\nThe narrow impedance bandwidth of an LTI electrically small antenna distorts a transmitted or received signal, reducing the efficacy of information transfer.  Typically, this distortion is not considered directly, but rather by proxy through the antenna's impedance bandwidth.  In an effort to study electrically small antennas in ways applicable both to LTI (conventional) and non-LTI (time-varying systems for which impedance bandwidth is not defined), we revert to the study of signals produced by a radiating antenna directly in the time domain. \n\nLet the distortion measured between an ideal signal $x_\\mathrm{i}$ and a realized signal $x$ be defined as \\cite{Lamensdorf1994}\n\\begin{equation}\n\\label{eq:d-td}\nd(x_\\mathrm{i},x) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\hat{x}_\\mathrm{i}(t) - \\hat{x}(t-\\tau) \\right|^2\\mathrm{d}t.\n\\end{equation}\nwhere $\\hat{\\cdot}$ denotes the normalization\n\\begin{equation}\n\t\\hat{f}(t) =\\frac{f}{\\left[\\int_{-\\infty}^{\\infty} |f|^2\\mathrm{d}t\\right]^{1\/2}}.\n\\end{equation}\nNote the optimization over time delay $\\tau$, with the true distortion defined by the minimum possible value.  Parseval's Theorem allows this to be alternatively written in the frequency domain as \n\\begin{equation}\n\\label{eq:d-fd}\nd(X_\\mathrm{i},X) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\hat{X}_\\mathrm{i}(\\omega) - \\hat{X}(\\omega)\\mathrm{e}^{-\\mathrm{j}\\omega\\tau} \\right|^2\\mathrm{d}\\omega.\n\\end{equation}\nFrom distortion, fidelity $F$ may also be calculated, as the two metrics are closely related by\n\\begin{equation}\n\\label{eq:fid}\n    F= 1 -\\frac{d}{2}\n\\end{equation}\nwhere the minimization over time delay $\\tau$ is implicitly taken on whichever quantity is directly computed \\cite{Lamensdorf1994}.  By these conventions, a linear system with a non-zero transfer function produces zero distortion (unity fidelity) under CW excitation. \n\nIf there exist multiple degrees of freedom in each observable, for example, two polarizations of a radiated field, the integrand in \\eqref{eq:d-td} and \\eqref{eq:d-fd} can be generalized to the norm of the difference between the ideal and realized signals in vector form, i.e.,\n\\begin{equation}\n\\label{eq:d-td-gen}\nd(\\boldsymbol{x}_\\mathrm{i},\\boldsymbol{x}) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\boldsymbol{x}_\\mathrm{i}(t) - \\boldsymbol{x}(t-\\tau) \\right|^2\\mathrm{d}t.\n\\end{equation}\n\nIf measurement distortion (e.g., channel distortion, limited receiver bandwidth) or preconditioning (e.g., polarization selection) is known and representable by the linear operator $\\mathcal{L}$, the same formulation for distortion can be applied using \\cite{Lamensdorf1994}\n\\begin{equation}\n\\label{eq:op-transform}\n\tx = \\mathcal{L}y,\\quad x_\\mathrm{i} = \\mathcal{L}y_\\mathrm{i}\n\\end{equation}\nwhere $y$ and $y_\\mathrm{i}$ are the realized and ideal transmit signals, and now $x$ and $x_\\mathrm{i}$ are the corresponding observables. \n\nThough distortion is used extensively in the characterization of UWB transmitters, the impact of the bandwidth and efficiency limitations of small antennas on distortion bounds has not been previously described.  Our goal here is to do so in order to provide a benchmark against which the distortion properties of non-LTI systems can be compared.  We proceed by developing operational relationships between ideal signals (e.g., input voltage to an antenna) and some produced output (e.g., a normalized radiated field) in order to calculate distortion in terms of parameters bounded for LTI systems.  In this way, we project the effect of an LTI system's Q-factor and efficiency onto the distortion metric which is compatible for comparison with non-LTI systems for which Q-factor, impedance bandwidth, and other frequency domain parameters are not defined.  The rest of this paper uses distortion as the metric of interest, though all data and results can be recast in terms of fidelity using \\eqref{eq:fid}\n\n\\section{Radiated Fields from Electrically Small TM$_{01}$ Antennas}\n\\label{sec:mappings}\nThe goal of this section is to derive expressions for calculating the fields radiated by a small TM$_{01}$ antenna under specific (LTI) matching conditions.  These in turn can be used to calculate distortion as defined in Sec.~\\ref{sec:dist-and-eff} under arbitrary excitation.  A spherical radiator with a well-defined equivalent circuit is first studied before applying approximations to generalize the calculations to the study of any structure (antenna or current distribution) with a Q-factor and efficiency defined at a single frequency.\n\n\\subsection{Spherical Shell}\nIn a seminal paper studying the physical bounds of omnidirectional antennas, Chu established an equivalent circuit to model the impedance and radiation behavior of a dipole-like (TM$_{01}$) spherical antenna \\cite{Chu1948}.  In terms of the input current $I$ to this equivalent circuit, the radiated electric field $\\boldsymbol{E}$ is\n\\begin{equation}\n    \\boldsymbol{E} = \\hat{\\boldsymbol{\\theta}}\\frac{1}{\\rho\\mathrm{h}^{(2)}_1}\\sqrt{\\frac{3\\eta_0}{8\\pi}}\\sin\\theta \\frac{\\mathrm{e}^{-\\mathrm{j} kr}}{r} I.\n\\end{equation}\nHere $\\rho$ denotes the electrical size $ka = 2\\pi a\/\\lambda$ and $\\mathrm{h}^{(2)}_1$ is the first order spherical Hankel function of the second kind evaluated with argument $\\rho$.  Without loss of generality, we consider only broadside ($\\theta=\\pi\/2$) radiated fields, drop the fixed constant $\\sqrt{3\\eta_0\/(8\\pi)}$, suppress the far field propagation term, and consider only the scalar magnitude of the $\\theta$-polarized field, i.e.,\n\\begin{equation}\n    E = \\frac{I}{\\rho\\mathrm{h}^{(2)}_1}.\n    \\label{eq:sphere-ei}\n\\end{equation}\nThe equivalent antenna input impedance of the TM$_{01}$ radiator is defined as\n\\begin{equation}\n    Z_\\mathrm{a} = \\frac{\\mathrm{j}(\\rho\\mathrm{h}^{(2)}_1)'}{\\rho\\mathrm{h}^{(2)}_1} = \\frac{\\rho^4 - \\mathrm{j}\\rho}{\\rho^4+\\rho^2}\n\\label{eq:z1}\n\\end{equation}\nand can be readily implemented using a three element RLC circuit \\cite{Chu1948}.  Here $(\\cdot)'$ denotes a derivative taken with respect to electrical size $\\rho$.  Note that this impedance represents only an outward going wave impedance.  Internal fields can easily be included using parallel networks \\cite{thal1978exact} with the effect of increased Q-factors in the calculation of electrically small spheres in Sec.~\\ref{sec:examples}.\n\nSuppose the antenna (equivalent circuit) is tuned to resonance at $\\rho_0$ by a series inductor $L$.  Further, a series resistance $R_\\Omega$ is introduced to represent ohmic losses.  This resistance can be determined based on material parameters and electrical size by various models.  Because of the intricate nature of its definition and scaling (e.g., skin depth model or surface resistance) we leave this resistance as a free parameter to tune at will. With these alterations,  the tuned, lossy impedance model is written as \n\\begin{equation}\n    \\tilde{Z}_\\mathrm{a} = Z_\\mathrm{a} + \\mathrm{j}\\rho X_0\/\\rho_0 + R_\\Omega,\n\\end{equation}\nwhere the tuning inductance is scaled in terms of its reactance \\mbox{$X_0 = \\mathrm{Im}~\\tilde{Z}_\\mathrm{a}(\\rho_0)$} at the tuned frequency.  A non-dispersive source with impedance \\mbox{$R_0 = \\mathrm{Re}~\\tilde{Z}_\\mathrm{a}(\\rho_0)$} is connected to the antenna circuit.  For a given excitation $V_\\mathrm{s}$, the radiated field is then given by\n\\begin{equation}\n    E = \\frac{V_\\mathrm{s}}{\\rho \\mathrm{h}^{(2)}_1(R_0+\\tilde{Z}_\\mathrm{a})}.\n    \\label{eq:sphere-ev}\n\\end{equation}\nThe above expression is general in the sense that no approximations have been made regarding the electrical size of the spherical radiator, only that it radiates solely in the TM$_{10}$ mode.  Using some form of this electric field as an observable quantity, the relation in \\eqref{eq:sphere-ev} can be applied to calculate distortion from a sphere of arbitrary size ($\\rho$), arbitrary loss parameters ($R_\\Omega$), and under arbitrary excitation ($V_\\mathrm{s}$); where the ideal observable $x_\\mathrm{i}$ may be the excitation itself.\n\n\n\\subsection{Arbitrary Electrically Small TM$_{01}$ Radiators}\nSevere Q-factor and efficiency limits arise in the electrically small limit ($\\rho\\rightarrow 0$).  Here we simplify \\eqref{eq:sphere-ev} in this limit and make narrowband, stationary approximations to allow for the calculation of the radiated field $E$ with knowledge only of a radiator's tuned Q-factor and efficiency at a given operating frequency.\n\nThe current-field relation in \\eqref{eq:sphere-ei} simplifies to the expected time-derivative form in the electrically small limit, i.e.,\n\\begin{equation}\nE = -\\mathrm{j} \\rho I.\n\\end{equation}\nUsing this simplification to represent the general behavior of all small dipole-like radiators and the definition of the reflection coefficient \n\\begin{equation}\n    \\varGamma = \\frac{\\tilde{Z}_\\mathrm{a} - R_0}{\\tilde{Z}_\\mathrm{a} + R_0},\n    \\label{eq:gamma}\n\\end{equation}\nthe radiated field may be written as \n\\begin{equation}\n    E = \\frac{\\mathrm{j} \\rho V_\\mathrm{s}}{2R_0}\\left(1-\\varGamma\\right).\n    \\label{eq:esa-ev}\n\\end{equation}\nAssuming that the resonant system can be approximated locally in frequency as a series RLC circuit \\cite{Yaghjian2005}, the reflection coefficient can be rewritten using \\eqref{eq:gamma} as a function parameterized by the system's resonant radiation Q-factor $Q$ and efficiency $\\eta$\n\\begin{equation}\n    \\varGamma = \\frac{\\mathrm{j} Q\\eta (\\rho-\\rho_0) \/ \\rho_0}{1+\\mathrm{j} Q\\eta (\\rho-\\rho_0) \/ \\rho_0}.\n    \\label{eq:gamma}\n\\end{equation}\nHere $Q$ denotes only the radiation Q-factor where as $Q\\eta$ is the total system Q-factor including loss.  In this RLC approximation, the ohmic and radiation losses are implicitly assumed to be stationary in frequency.\n\nThe expressions in \\eqref{eq:esa-ev} and \\eqref{eq:gamma} represent a framework for calculating the frequency domain radiated fields for an electrically small dipole-like antenna specified by a radiation Q-factor and efficiency at a single frequency.  Arbitrary excitation $V_\\mathrm{s}$ may be applied and loss models may be parameterized by the total system efficiency $\\eta$.  Given that closed form expressions are available for the radiation Q-factor and efficiency of a TM$_{01}$ spherical current distribution, the strength of the stationary narrowband approximations can be easily assessed.  Similarly, many techniques allow for the calculation of optimal Q-factor and optimal efficiency currents on arbitrarily shaped objects (e.g., \\cite{capek2017minimization,gustafsson2019trade}), meaning that \\eqref{eq:esa-ev} and \\eqref{eq:gamma}, in conjunction with the metrics defined in Sec.~\\ref{sec:dist-and-eff}, can be applied to the study of the optimal distortion and efficiency achievable by arbitrarily shaped electrically small dipole-like radiators.\n\n\\subsection{Removal of Time Derivative Term}\nThe expressions \\eqref{eq:sphere-ev} and \\eqref{eq:esa-ev} suggest that the source voltage and radiated electric field may be candidates for the ideal and measured signals used in computing distortion via \\eqref{eq:d-td}.  However, we note that even in the case of perfect broadband matching, there exists a time derivative term between these two quantities.  In an effort to obtain a comparison that reduces to zero distortion in the ideally matched case, either the source voltage must be multiplied by this derivative term $\\mathrm{j}\\rho$ or the radiated field must be integrated to remove this term.  We opt for the latter in this study to avoid issues related to the unbounded nature of the differentiation operator.  Thus in applying equation \\eqref{eq:d-td} to calculate distortion, we take for the ideal signal $x_\\mathrm{i} = \\mathcal{F}^{-1}\\left[V_\\mathrm{s}\\right]$ and the measured signal $x= \\mathcal{F}^{-1}\\left[E\/(\\mathrm{j}\\rho)\\right]$ where $E$ is the radiated field calculated either by \\eqref{eq:sphere-ev} (exact sphere) or \\eqref{eq:esa-ev} (approximate single resonance model).  This is similar to the treatment discussed in \\cite{farr1992}.\n\n\n\\subsection{Discussion of Parameter Dependencies}\nThe rigorous calculation of the electric field produced by a spherical shell \\eqref{eq:sphere-ev} depends on many parameters: input signal type, antenna materials, antenna efficiency $\\eta$, and electrical size $\\rho$.  By contrast, it can be observed that, for signals classified by some nominal bandwidth $B$ (e.g., $99\\%$ power occupied bandwidth), all of these parameters can be collapsed into a single product $Q\\eta B$ using the single resonance model in \\eqref{eq:esa-ev}.  Note that here $B$ represents the bandwidth of the input signal (not the antenna bandwidth). Thus, the complete term $Q\\eta B$ is proportional to the quotient of the signal and antenna impedance bandwidths \\cite{Yaghjian2005}. \n\nThis implies that all small antennas which fit the assumptions of the single resonance dipole model transmitting a signal of bandwidth $B$ can be classified by a single number $Q\\eta B$ regardless of their specific Q-factor, efficiency, or the bandwidth of the signal.  Thus, the distortion for a specific signal type and bit sequence can be plotted as a function of this single composite variable.  The resulting curve contains the distortion performance of any and all single resonance antennas transmitting that particular message.  The shape of this curve depends on the exact message being transmitted for short signals but converges as the message length is increased and the power spectral density of the input signal approaches that of an infinite sequence of the chosen modulation type.\n\nAt this point it is necessary to highlight a few of the practical advantages of collapsing several antenna metrics into the combined parameter $Q\\eta B$.  First, to predict realized distortion under a given signal bandwidth $B$, there is no need to evaluate an antenna's radiation Q-factor or efficiency, rather only the product $Q\\eta$ which can be easily estimated from an analytic, numerical, or measured antenna's input reflection coefficient \\cite{Yaghjian2005}.  Second, the effect of broadbanding an antenna via resistive loading can quickly be assessed by calculating the associated shift in $Q\\eta B$ and moving along the single resonance model curve.  Finally, if a non-LTI antenna is capable of operating in an LTI mode with no changes to hardware (e.g., with a dynamic switch held static as done in \\cite{schab2019pulse}), then its effective value of $Q\\eta B$ can be determined.  The realized distortion in both LTI and non-LTI modes of such a transmitter can then be measured directly in the time domain and compared to the single resonance model presented here.  Alternative methods for comparing non-LTI data using these metrics are discussed in Sec.~\\ref{sec:dam}.\n\n\n\n\\section{Example Calculations on LTI Systems}\n\\label{sec:examples}\nIn this section, we apply distortion calculations to assess LTI structures.  In anticipating the analysis of datasets collected from non-LTI transmitters in \\cite{schab2019pulse}, we calculate distortion using pseudorandom bit sequence (PRBS) on-off-keyed (OOK) excitations where the data rate is defined by the number $N$ of carrier periods per symbol.  The nominal bandwidth of these signals is defined here as $B = 1 \/ N$.  Additional examples using differentiated Gaussian and modulated Gaussian pulses are also included.  Throughout this section, $ka$ denotes the electrical size at the tuned frequency, i.e., $\\rho_0$.\n\n\\subsection{Assessment of Stationary, Narrowband Approximations}\nHere we compare the distortion resulting from the rigorous expressions derived for a spherical shell with those from the approximate single resonance model.  A 128-bit PRBS sequence is used to excite spherical radiators of varying electrical size, with the radiated field calculated via \\eqref{eq:sphere-ev}.  Frequency dependent losses were not considered, instead $R_\\Omega$ was selected to enforce a CW radiation efficiency of $\\eta = 10\\%$ at the tuned carrier frequency for each electrical size.  At each frequency, the tuned factor $Q\\eta$ was calculated from the differential impedance bandwidth \\cite{Yaghjian2005}.  Data from this calculation are overlaid with a curve of distortion values obtained by sweeping $Q\\eta$ in the single resonance model \\eqref{eq:esa-ev} in Fig.~\\ref{fig:sphere-comparison}.  Excellent agreement is observed for all tested electrical sizes.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/1-sphere_points_processed.pdf}\n    \\caption{Comparison of rigorous sphere distortion calculations (points) with those based on the single resonance model (solid line).  Bracketed points are generated for a given electrical size $ka$, while opacity denotes the number of cycles $N$ per bit in the input 128-bit OOK PRBS.}\n    \\label{fig:sphere-comparison}\n\\end{figure}\n\nMany time domain signals may yield the same distortion values.  The level of tolerable distortion in a communications link would depend on many parameters related to the receiver architecture.  Making judgements on what is or is not an acceptable level of distortion is beyond the scope of this work, however we do note that link resilience and capacity would nearly always improve with lowered distortion.  For reference, in Fig.~\\ref{fig:sphere-td} we plot time domain signals produced by a small selection of the data points in Fig.~\\ref{fig:sphere-comparison} along with their corresponding values of distortion.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/2-sphere_signals_processed.pdf}\n    \\caption{Normalized time-domain signals $\\hat{x}$ for points corresponding to \\mbox{$N = 20$} in Fig.~\\ref{fig:sphere-comparison}.  The ideal signal $\\hat{x}_\\mathrm{i}$ is shown in the top panel.  Time axis normalized to the symbol length $T_\\mathrm{s}$.}\n    \\label{fig:sphere-td}\n\\end{figure}\n\nAdditionally, we can examine the validity of the single resonance model under extremely broadband conditions using traditional UWB pulses.  Rather than the OOK PRBS signal used throughout the rest of this paper, Fig.~\\ref{fig:sphere-pulses} shows the same comparison as Fig.~\\ref{fig:sphere-comparison} using differentiated Gaussian \n\\begin{equation}\n    v_\\mathrm{s} = -\\frac{t}{T_\\mathrm{p}}\\mathrm{e}^{-\\frac{1}{2}\\left(t\/T_\\mathrm{p}\\right)^2}\n\\end{equation}\nand modulated Gaussian \n\\begin{equation}\n    v_\\mathrm{s} = \\sin\\left(2\\pi f_\\mathrm{c} t\\right)\\mathrm{e}^{-\\frac{1}{2}\\left(tf_\\mathrm{c}\/N\\right)^2}\n\\end{equation}\npulses \\cite{jin2015theory}.  The width of the differentiated Gaussian pulses $T_\\mathrm{p}$ corresponds to a peak frequency component located at the tuned frequency of a spherical shell with swept electrical size.  The pulse repetition period is $2048T_\\mathrm{p}$.  In the case of the modulated Gaussian pulses, the pulse width is written in terms of the carrier period and is varied by the parameter $N$ and the pulse repetition rate is $2048\/f_\\mathrm{c}$, where $f_\\mathrm{c}$ corresponds to the modulation frequency and the tuned frequency of a spherical shell of size $ka = 0.05$.  In each of these analyses, $\\eta = 10\\%$.  Excellent agreement is observed.  The nominal signal bandwidth is normalized to $B = 1$ for the differentiated Gaussian and $B = 1\/N$ for the modulated Gaussian pulses.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/3-sphere_pulses_points_processed.pdf}\n    \\caption{Pulse distortion from a sphere via rigorous calculation (points) and single resonance model (curve).  Differentiated Gaussian pulses for varying sphere sizes (top) and modulated Gaussian pulses for fixed sphere size and varying pulse width (bottom) are shown.}\n    \\label{fig:sphere-pulses}\n\\end{figure}\n\n\\subsection{Analysis of Optimal Currents on Arbitrary Objects}\n\nWith the single resonance model, distortion may be predicted for any system with a Q-factor and efficiency defined at a single frequency.  This includes non-driven current distributions on arbitrary structures, extending the general approach used for spheres to the analysis of any arbitrary geometry.  Currents optimal in radiation Q-factor and efficiency are readily obtained using a method of moments formulation \\cite{capek2017minimization}.  In many ways, these optimal, non-driven current distributions are exactly analogous to the spherical harmonics used by Chu, though their behavior is not obtained via analytic expansion, but rather by deterministic convex optimization methods.\n\nThe tradeoff between optimal radiation Q-factor ($Q$) and optimal efficiency may be calculated via a Pareto analysis~\\cite{gustafsson2019trade}.  Here we conduct this analysis for an L-shaped plate\\footnote{Aspect ratio $2:1$, notch extends exactly halfway into each dimension.} with electrical size $ka = 0.5$, homogeneous surface resistivity \\mbox{$R_\\mathrm{s} = 0.1~\\Omega$}, and without the explicit constraint of self resonance (i.e., lossless tuning implied for non-resonant solutions).  The results are plotted in the inset of Fig.~\\ref{fig:ell} as the tradeoff between $(Q\\eta)^{-1}$ (proportional to antenna bandwidth $B_\\mathrm{a}$) and efficiency $\\eta$.  Three points are selected at various positions along the tradeoff curve and the properties of the associated optimal current distributions are used to calculate distortion of a 128-bit PRBS OOK signal with $N=10$.  The resulting points are shown along side these current distributions in Fig.~\\ref{fig:ell}.  Note that, unlike the preceding analysis of a sphere, these data points are computed directly from the single resonance model and lie exactly on the single resonance model curve, drawn in black.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/4-ell_points_processed.pdf}\n    \\caption{Distortion calculated for Pareto-optimal currents on an L-shaped plate$^\\mathrm{1}$.  Three points along the bandwidth-efficiency Pareto front (inset) are selected and the corresponding distortion values are calculated using the single resonance model.}\n    \\label{fig:ell}\n\\end{figure}\n\nThis analysis demonstrates how distortion analysis may be combined with the study of optimal currents to predict \\textit{a priori} the ideal tradeoff between message distortion and radiator efficiency.  This is particularly relevant for the analysis of non-LTI systems where it is prudent to assess how improvements in signal fidelity compare versus the gains that could have been achieved by resistively broadbanding a comparable LTI transmitter.\n\n\n\n\\subsection{Analysis of Driven Antennas}\n\nIn the preceding examples, non-driven optimal current distributions were assessed using hypothetical equivalent circuit models.  By contrast, here we demonstrate the calculation of distortion from several driven antennas evaluated by full wave simulations and compare the obtained data to curves predicted by the single resonance model.\n\nWe begin with a thin wire dipole, depicted in Fig.~\\ref{fig:dipoles}.  The dipole is simulated using NEC2++ \\cite{nec}, a method of moments solver, to obtain broadband input impedance and radiation data.  These data are then used to numerically tune the dipole to a carrier frequency represented by an electrical size $ka$.  In each case considered here, the dipole is tuned below its natural resonance using a series inductance.  The tuned system is driven by a matched source using the same class of OOK signals as in the previous examples with bandwidths classified by the number of cycles per symbol $N$.  For each signal considered, the broadside co-polarized field is computed, integrated, and compared to the driving voltage via calculation of distortion.  Additionally, at each tuned frequency, the parameter $Q\\eta$ is estimated from simulated impedance data using the model in \\cite{Yaghjian2005}.  This factor, combined with the nominal signal bandwidth yields the coordinate $Q\\eta B$ with which the simulated distortion can be compared to the single resonance distortion model. Fig.~\\ref{fig:dipoles} shows excellent agreement between the fullwave dipole simulation and single resonance distortion model over three electrical sizes and five data rates, with each parameter spanning at least one order of magnitude.\n\n\\begin{figure}\n    \\centering  \\includegraphics[width=3.25in]{figures\/5-dipole_points_processed.pdf}\n    \\caption{Comparison of full wave distortion calculations on a copper dipole simulated using the method of moments (points) with those based on the single resonance model (solid line).}\n    \\label{fig:dipoles}\n\\end{figure}\n\nSeveral self resonant structures\\footnote{Dimensions and materials are listed in Appendix A.} are subjected to the same simulation and distortion calculation procedure, with results shown in Fig.~\\ref{fig:sr-antennas}.  A spherical helix, a quarterwave patch on a finite dielectric substrate, and a capacitively coupled circular patch \\cite{vanniekerk2013analysis} are examined.  The latter two antennas were simulated over an infinite perfect electric conductor (PEC) ground plane using CST Microwave Studio.  As in the previous example, the factor $Q\\eta$ was again estimated from simulated impedance data, though here each antenna is driven with a nominal $50~\\Omega$ source.  All transmitted fields were calculated in the polarization and direction of maximum gain at the self resonance frequency.  Despite the relatively high complexity of these antennas, we observe excellent agreement between the simulated distortion and the predicted single resonance curve.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/6-full_points_processed.pdf}\n    \\caption{Comparison of full wave distortion calculations using three self-resonant antennas$^\\mathrm{2}$ (points) with those based on the single resonance model (solid line).}\n    \\label{fig:sr-antennas}\n\\end{figure}\n\n\n\\section{Assessment of a non-LTI Transmitter}\n\\label{sec:dam}\nWith the accuracy of the single resonance model verified in previous examples, we move to our primary motivation for studying distortion: assessing the relative performance of LTI and non-LTI transmitters.\n\nAs stated in Sec.~\\ref{sec:intro}, while previous efforts to demonstrate non-LTI transmitters' ability to outperform comparable LTI transmitters have provided promising results, reporting of the realized benefits of non-LTI systems has been largely visual or qualitative.  This is due in no small part to the fact that frequency domain quantities, such as impedance bandwidth, are not directly defined for non-LTI systems and thus cannot be used as performance metrics as is done with LTI systems.  Here we use distortion, which can be measured directly in the time domain for either LTI or non-LTI systems, to compare the performance of the non-LTI OOK transmitter reported in \\cite{schab2019pulse}.\n\nThe transmitter studied in \\cite{schab2019pulse} is based on the switched matching network architecture described in \\cite{Galejs1963}.  As such, it is capable of operating in conventional (LTI) and direct antenna modulation (DAM, non-LTI) modes without changes to the physical hardware.  Thus, the transmitter can be characterized by a value of the parameter $Q\\eta$ measured when operating in LTI modes.  Note that not all non-LTI systems may be characterized in this way, but as we shall point out later this does not prevent the use of distortion analysis.\n\nRaw over the air time domain data from the experiments conducted in \\cite{schab2019pulse} were used to calculate distortion for 64-bit PRBS OOK signals in conventional and DAM modes.  Channel inversion was used to isolate distortion due to transmitter mismatch, see \\cite{schab2019pulse} for details.  The resulting data are plotted in Fig.~\\ref{fig:dam}, where good agreement between the conventional transmitter mode and the single resonance model is observed.  For all three transmissions using the DAM transmitter mode, the distortion is clearly reduced below that of the single resonance model, in line with the visual assessment of corresponding eye diagrams made in \\cite{schab2019pulse}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/7-dam_points_processed.pdf}\n    \\caption{Comparison of OOK LTI distortion bounds with measured non-LTI transmitter performance.  Here $N\\sim 1\/B$ is a metric of symbol length.  The same 64-bit OOK sequence was used for the analytic calculation and the measured transmissions. }\n    \\label{fig:dam}\n\\end{figure}\n\nBeyond qualitative comparison, the use of distortion analysis enables the quantification of gains afforded by this non-LTI transmitter.  Fig.~\\ref{fig:dam-detail} shows a detail view of the self resonance curve and measured distortion from the conventional and DAM transmitter modes with $N=3$.  Starting from the indicated LTI measurement, we observe that the non-LTI system provides a four-fold reduction in distortion (from $0.36$ to $0.09$).  To achieve equivalent distortion for the same transmission, the LTI system would require resistive broadbanding amounting to a reduction in efficiency (or gain, assuming constant directivity) by a multiplicative factor of $0.17$ or equivalently $-7.6~\\mathrm{dB}$.  Conversely, since the LTI and non-LTI systems have the same efficiency and the signal bandwidths are identical in both measurements, the non-LTI system can be interpreted as having an ``effective Q-factor'' a multiplicative factor of $0.17$ lower than that of the LTI system, or equivalently an ``effective bandwidth'' that is a multiplicative factor of $5.7$ larger.  \n\nNeither gain nor Q-factor is necessarily defined for the non-LTI transmitter. However, this example demonstrates how distortion may be used as an intermediate quantity to determine the non-LTI transmitter's potential benefits in terms of these critical system parameters.  Additionally, while there is no non-trivial lower bound on distortion for non-LTI systems, the curve containing the distortion properties of all single resonance LTI antennas helps illustrate the relative gains and performance parameters afforded by the use of any particular non-LTI transmitter implementation.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.25in]{figures\/8-dam_detail.pdf}\n    \\caption{Comparison of OOK LTI distortion bounds with measured non-LTI transmitter performance.  The same 64-bit OOK sequence was used for the analytic calculation and the measured transmissions. }\n    \\label{fig:dam-detail}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nElectric field relations for a spherical shell and single resonance dipole-like antennas are cast in forms compatible with the calculation of distortion.  We have provided several examples justifying the approximation that all LTI, electrically small, dipole-like radiators are well characterized by a single curve of distortion versus the composite Q-factor, efficiency, and signal bandwidth parameter $Q\\eta B$.  This finding is used to compare the performance of a non-LTI transmitter by obtaining quantitative measures of its behavior relative to the bandwidth and \/ or efficiency of a comparable LTI system.  \n\nGoing forward, further studies into quantifying the efficiency and directive nature (intentional or unintentional) of non-LTI radiators in order to complete a framework for assessing their integration into communications or sensing systems.  Furthermore, the procedure presented in this paper will allow for the more rigorous analysis of previously proposed theoretical and experimental non-LTI transmitters.\n\n\\appendices\n\\section{Dimensions of Example Self-Resonant Antennas}\n\n\\subsection{Spherical Helix}\nThe five-turn spherical helix is simulated using NEC2++, a method of moments based solver.  The simulated helix is made of a cylindrical wire of radius $r_\\mathrm{w} = 0.1~\\mathrm{mm}$ with conductivity $\\sigma = 5.8\\cdot10^{7}~\\mathrm{S\/m}$.  The radius of the bounding sphere is \\mbox{$r = 1~\\mathrm{m}$} and the helix is described by the curve $(x(t),~y(t),~z(t))$ with coordinates\n\\begin{equation}\n    z = -r + 2rt,\\quad\\quad\n    \\rho = \\sqrt{r^2-z^2}\n\\end{equation}\n\\begin{equation}\n    x = \\rho \\cos (10\\pi t),\\quad\\quad\n    y = \\frac{z}{|z|}\\rho \\sin (10\\pi t)\n\\end{equation}\nwhere $t \\in [0,1]$.  The structure is discretized into 175 segments and fed at the segment crossing $t=0.5~(z = 0~\\mathrm{m})$.  The helix is resonant near $f = 6.6~\\mathrm{MHz}~(ka = 0.14)$.\n\n\\subsection{Quarterwave Patch}\nThe square quarterwave patch is simulated using the finite element method in CST.  The ground plane and patch are made of PEC and the ground plane is of infinite extent. There is a shorting wall the entire extent of the patch on the side closest to the feed point.  Parameters for this antenna are given in Table~\\ref{tab:diel}.\n\n\\subsection{Dual Reactively Loaded Monopole (DRLM)}\nThe DRLM is based on the horizontally meandered PIFA from \\cite{vanniekerk2013analysis}, where detailed dimensions may be obtained.  The circumscribing sphere for this antenna (and its image below the infinite ground) has a radius of $22~\\mathrm{mm}$.  Simulations of this antenna are carried out in CST with an infinite PEC ground plane.  The self resonant frequency is near $810~\\mathrm{MHz}~(ka = 0.37)$. \n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{l|c}\n    \\textit{Parameter} & \\textit{Value}\\\\ \\hline\n        Substrate dimensions & $9$~mm \\\\\n        Patch dimensions & $8$~mm \\\\\n        Shorting wall width & $8$~mm\\\\\n        Feed distance from shorting wall & $3.4$~mm \\\\\n        Substrate relative permittivity & $9.8$ \\\\\n        Feed type & SMA coaxial \\\\ \n        Self resonant frequency & $2.978~\\mathrm{GHz}$\\\\\n    \\end{tabular}\n    \\caption{Quarterwave patch parameters.}\n    \\label{tab:diel}\n\\end{table}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe cosmic infrared background (CIB or CIRB) contains a considerable fraction of the collective radiation emitted by stars through the cosmic time. Stars in the Epoch of Reionization (EoR) have the bulk of their radiation redshifted into the near-infrared band ($\\sim$0.7 - 5 $\\mu$m) and the CIB measured in this band is specifically named ``near-infrared background\" (NIRB). \nAs such, the NIRB offers\na unique opportunity to study faint high-$z$ galaxies that remain largely undetected in deep galaxy surveys (see e.g. \\citealt{2006MNRAS.367L..11S, 2006ApJ...646..703F,2010ApJ...710.1089F,2012ApJ...750...20F,2013ApJ...764...56F,2013MNRAS.433.2047F}). This is particularly important as these objects are commonly believed to provide most of the ionizing power that drives cosmic reionization \\citep{2007MNRAS.380L...6C,2011MNRAS.410..775R,2011MNRAS.414..847S}. Besides, the NIRB might also help characterizing the stellar populations of the first cosmic systems \n\\citep{2003MNRAS.339..973S,2006MNRAS.368L...6S,2002MNRAS.336.1082S,2004ApJ...608....1K,2005Natur.438...45K,2003MNRAS.342L..25M,2004MNRAS.351L..71C,2004ApJ...606..611C}. \n\nMost recent studies have converged on the prediction that on scales of $\\approx 1000''$ the fluctuation level from normal star-forming galaxies at $z \\lower.5ex\\hbox{\\gtsima} 5$ is $\\approx 10^{-3}$~nWm$^{-2}$sr$^{-1}$ at 3.6~$\\mu$m \\citep{2012ApJ...756...92C,2013MNRAS.431..383Y,2016MNRAS.455..282H}. However, extracting such signal from available data has been so far very challenging,  as even when the deepest galaxy\nsubtraction from NIRB maps is applied, the remaining flux fluctuations\\footnote{NIRB studies usually concentrate on  fluctuations rather than absolute flux, as the latter is difficult to measure due to the presence of an overwhelming foreground. However, as the foreground is rather smooth on scales at which the NIRB is measured, the fluctuations analysis is more robust -- see e.g. \\citealt{2007ApJ...657..669T,2011ApJ...742..124M,2012ApJ...753...63K,2012Natur.490..514C}.} still cannot be ascribed to the known high-$z$ galaxy population. On small angular scales, most of the signal arises from unresolved low-$z$ galaxies (see the first analysis by \\citealt{2002ApJ...579L..53K}).\nOn larger scales the measured power  (see e.g. \\citealt{2004ApJ...608....1K,2005Natur.438...45K,2007ApJ...654L...1K,2007ApJ...654L...5K,2012ApJ...753...63K,2005ApJ...626...31M, 2011ApJ...742..124M,2015arXiv150405681S,2007ApJ...659L..91C,2012Natur.490..514C}) is $\\lower.5ex\\hbox{\\gtsima} 100$ times larger than the low-$z$ galaxies \\citep{2012ApJ...752..113H}, and $\\lower.5ex\\hbox{\\gtsima} 1000$ times larger than that expected from high-$z$ normal star-forming galaxies and first stars \\citep{2012ApJ...756...92C,2013MNRAS.431..383Y}. Therefore it must be attributed to some, yet unknown, alternative sources. Basically, two different explanations have been proposed  for the origin of such large scale ($\\sim1000''$) ``power excess\". They involve either early accreting black holes \\citep{2013MNRAS.433.1556Y,2014MNRAS.440.1263Y} which could explain the detected NIRB-cosmic X-ray background coherence \\citep{2013ApJ...769...68C}, or ``intrahalo light'' that radiated by stars ejected from their parent galaxies during merger events \\citep{2012Natur.490..514C,2014Sci...346..732Z}. \n\nAt present it is unclear which hypothesis should be preferred. Besides, it is also possible that first stars or black holes are much more abundant than in the standard theoretical framework \\--- for example if the slope of the density fluctuations power spectrum sightly deviates from the standard one, the number of small halos (thus stars or black holes therein) could be boosted exponentially \\--- they can then provide sufficient radiation power. To be consistent with electron scattering reionization bounds, at the same time the ionizing photon escape probability must be rather low (for a detailed discussion see \\citealt{2016MNRAS.455..282H}). To further complicate the interpretation, recent observations \\citep{2014Sci...346..732Z} show that, on large scales and at least for the 1.1 and 1.6~$\\mu$m bands, diffuse Galactic light (DGL) might provide a non-negligible flux contribution. However \\citet{2007ApJ...654L...5K} (see \\citealt{2010ApJS..186...10A} as well) found that at the 3.6 and 4.5 $\\mu$m bands the DGL or the Galactic cirrus component is largely subdominant. \n\nFor the above reasons, it is urgent to devise new strategies that put our understanding on firmer grounds. In order to isolate the signal from increasingly high redshifts, with sufficient depth and angular resolution one can in principle remove foreground galaxies down to extremely faint levels. This is challenging for current instrumentation, and also not an easy task for the {\\tt JWST}, as the signal from high-$z$ galaxies is expected to be subdominant even at $\\sim$32 AB mag \\citep{2016MNRAS.455..282H}.  \nAlternatively, cross-correlation studies seem promising. The NIRB-HI 21cm line cross-correlation \\citep{2014MNRAS.440..298F,2014ApJ...790..148M} has the advantage that it selectively picks up the signal from reionization sources.  Also, the NIRB, if produced by sources in the EoR, would be cross-correlated with the CMB through the thermal Sunyaev-Zeldovich effect. This correlation could be seen in the forthcoming Euclid-based all-sky CIB maps \\citep{2014ApJ...797L..26A}.  In addition to cross-correlation, a recent work \\citep{2015ApJ...804...99K} proposed the use of Lyman-break tomography to constrain the NIRB contribution from sources above a certain redshift.\n\n\\citet{2007ApJ...666L...1K} analyzed the cross-correlation between the ACS-detected faint sources and the source-subtracted NIRB. They found significant correlation on small scales, implying that the faint ACS galaxies do contribute to NIRB fluctuations. However, these sources (or objects associated with them) cannot account for the large scale clustering as the measured correlation is negligible. Their analysis did not pay particular attention to the high-$z$ sources, therefore the information on high-$z$ galaxies cannot be directly derived from there.\nIt is surprising that so far little attention has been devoted to the search for high-$z$ normal star-forming galaxy signatures in the NIRB, given that deep galaxy surveys have made tremendous progresses in obtaining the UV luminosity functions (LF) up to $z=10$ \\citep{2015ApJ...803...34B}, and the detection limits of Lyman break galaxy (LBG) surveys carried out by {\\tt HST} have already reached $H\\sim 29 - 31$ \\citep{2013ApJS..209....6I,2011ApJ...737...90B}.  \nMeasuring their intensity fluctuations present (though tiny) in the NIRB is essentially one of the most exciting perspectives as it might convincingly show that NIRB fluctuation experiments can be used to study the first galaxies. \nSuch an experiment is also complementary to more traditional galaxy surveys that derive the LF of the brightest galaxies among early populations. \n\nIn summary, it is mandatory to show that the NIRB power spectrum signal of already known high-$z$ galaxies can be recovered robustly. Demonstrating a successful strategy will represent a major step forward in the methodology and allows to obtain, in addition to the clustering signal, other key quantities, as e.g. the colors of high-$z$ galaxies beyond the observed $H$ band. Colors, in turn, provide potentially information on stellar ages and initial mass function of the stars harbored by galaxies in the EoR.  \n\nThe idea we propose here is to isolate the targeted LBG signal and show the feasibility of statistically detecting reionization sources by cross-correlating deep LBG surveys with NIRB maps. To this aim, we: (a) construct large scale mock maps of the source-subtracted NIRB and LBG catalogs using semi-numerical simulations; (b) perform a cross-correlation analysis between the two data sets to extract the contribution of high-$z$ galaxies in the NIRB. The paper is organized as follows. In Sec. \\ref{method} we describe the steps to construct the mock maps. In Sec. \\ref{results} we present the analysis about the correlation coefficient and the colors. Conclusions and discussions are presented in Sec. \\ref{conclusions}. Throughout this paper we use {\\tt Planck} cosmological parameters: $\\Omega_{\\rm m} = 0.31, \\Omega_\\Lambda = 0.69, \\Omega_{\\rm b} = 0.048, n_s = 0.96, \\sigma_8 = 0.82$ and $h = 0.68$ \\citep{2014A&A...571A..16P}. All magnitudes are in the AB-system \\citep{1983ApJ...266..713O}.\n\n\\section{Construction of mock maps}\\label{method}\n\n\\subsection{ High-$z$ galaxies }\nUsing the code {\\tt DexM}\\footnote{\\url{http:\/\/homepage.sns.it\/mesinger\/DexM___21cmFAST.html}}   \\citep{2007ApJ...669..663M} we carry out semi-numerical simulations to get catalogs of halos with mass $M_{\\rm h}\\lower.5ex\\hbox{\\gtsima} 5\\times10^8~M_\\odot$ from $z = 5$ to 10 for every $\\Delta z = 0.1$. We adopt a 400~Mpc box size that corresponds  to an angular size  of $\\approx2.4$~deg at $z = 10$. We construct a cuboid by replicating the output boxes along the line-of-sight, adding random translations, rotations and reflections \\citep{2005MNRAS.360..159B}. This is our light-cone since we assume that all lines-of-sight are parallel \\--- an assumption that is convenient and safe enough when $z \\lower.5ex\\hbox{\\gtsima} 5$. \n\n   \nTo construct flux maps from the light-cone, we link galaxy luminosities to halo mass $M_{\\rm h}$. The observed  LFs could be reproduced exactly if we derive the $L_{\\rm UV} - M_{\\rm h}$ relation through abundance matching, i.e. we force the number density of galaxies with luminosity $>L_{\\rm UV}$ to match the number density of halos with  mass $>M_{\\rm h}$. Formally, \n\\begin{equation}\n\\int_{M_{\\rm UV}} \\Phi(M^\\prime_{\\rm UV},z) dM^\\prime_{\\rm UV} = \\int_{M_h} \\frac{dn}{dM^\\prime_h}dM^\\prime_h,\n\\end{equation}\nwhere $\\Phi$ is the UV LF at 1600~\\AA~and we use the Schechter parameterization with the redshift-dependent fitting parameters given in \\citet{2015ApJ...803...34B}. As a reference, in the derived $L_{\\rm UV} - M_{\\rm h}$ relations \nour minimum mass $5\\times10^8~M_\\odot$ corresponds  to an absolute magnitude $M_{\\rm UV} = -10.5, -12.1, -13.0$ at $z=5, 8, 10$ respectively. Luminosity at other UV wavelengths is obtained through the luminosity-dependent Spectral Energy Distribution (SED) slope $\\beta$ (i.e., $f_\\lambda \\propto \\lambda^{\\beta}$) in \\citet{2014ApJ...793..115B}\\footnote{$\\beta=\\beta_{-19.5}+\\frac{d\\beta}{dM_{\\rm UV}}(M_{\\rm UV}+19.5)$, the values of $\\beta_{-19.5}$ and $\\frac{d\\beta}{dM_{\\rm UV}}$ at $z= 4, 5, 6$ and 7 could be found in \\citet{2014ApJ...793..115B}. For convenience of using this form at in-between redshifts, we fit $z$-dependent forms \\citep{2015MNRAS.450.3829Y}: $\\beta_{-19.5}=-1.97-0.06(z-6)$ and $\\frac{d\\beta}{dM_{\\rm UV}}=-0.18-0.03(z-6)$. We use these fittings anyway when $5 < z < 10$.}. However, generally speaking this power-law only holds at $\\lambda \\lower.5ex\\hbox{\\ltsima} 2000 - 3000$~\\AA, while we need luminosities at least until $4.5\/(1+z)~\\mu$m, say 7500~\\AA~when $z=5$. Therefore at $\\lambda>$2000~\\AA~we use the SED template from {\\tt Starburst99}\\footnote{http:\/\/www.stsci.edu\/science\/starburst99\/docs\/default.htm} \\citep{1999ApJS..123....3L,2005ApJ...621..695V,2010ApJS..189..309L}, adopting a continuous star formation mode, with metallicity 0.1~$Z_\\odot$ and 200~Myr stellar age. The SB99 SEDs are normalized to match the power-law form at 2000~\\AA. \n\n\nThe flux received in each pixel in the map is the sum of radiation from all galaxies seen by the pixel,\n\\begin{equation}\nF(\\nu_0)=\\nu_0\\frac{1}{(\\theta_{\\rm pix})^2}\\sum_j \\frac{L^j(\\nu)(1+z_j)}{4\\pi r_j^2(1+z_j)^2},\n\\end{equation}\nwhere $\\nu=\\nu_0(1+z_j)$, $z_j$ is the redshift of the $j$-th halos\\footnote{We do not model photometric redshift uncertainties, because the redshift range considered here $z = 5-10$ is much larger than the uncertainties.} in the solid angle $(\\theta_{\\rm pix})^2$ (we adopt $\\theta_{\\rm pix}=3.6''$ for all mock maps in this work),\n $r_j$ is its comoving distance.\nIn Fig. \\ref{maps} we show the 3.6~$\\mu$m flux map (bottom left panel) from all galaxies between $z = 5 -10$ \n(flux from galaxies at $z > 10$ is negligible compared with galaxies at lower redshift, we ignore it here).  \n  \n\\begin{figure*}\n\\centering{\n\\subfigure{\\includegraphics[scale=0.4]{fig1a.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1b.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1c.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1d.eps}}\n \\caption{{\\it Upper:} The 1.6~$\\mu$m flux map constructed from resolved LBGs with $H_{\\rm lim} = 25$ (left) and $H_{\\rm lim} = 27$ (right) respectively. {\\it Lower:} Map of the 3.6~$\\mu$m flux from galaxies with $5 < z < 10$ (left) and map of contamination at the same wavelength (right). The mean flux of the latter is 1 nWm$^{-2}$sr$^{-1}$.}\n\\label{maps}\n}\n\\end{figure*}\n\nFinally, we construct the flux map of LBGs at $5<z<10$. To take into account selection effects, we assume a completeness function of the form\n\\begin{equation}\nf(m)=0.5[1-{\\rm erf}(m-m_{\\rm lim})],\n\\end{equation}\nwhere erf is the error function, and $m_{\\rm lim}$ is the limiting magnitude. When constructing the flux map, for each LBG with apparent magnitude $m$, we generate an uniformly distributed number $x_r$. Flux from these galaxies is added to the map only if $x_r \\le f(m)$.\nIn Fig. \\ref{maps} we show the 1.6~$\\mu$m\\footnote{In this paper we only discuss the 1.6~$\\mu$m flux maps of LBGs, because we consider a redshift range from $z = 5$ to 10. For shorter wavelengths all procedures (and conclusions) are similar, with the only exception of a slightly smaller signal due to the Lyman dropout of $z\\gtrsim 8$ galaxies.}\nflux map constructed from the LBGs for $H$-band limiting magnitudes $H_{\\rm lim} = 25$ (top left) and $H_{\\rm lim} = 27$ (top right) respectively.\n\n\\subsection{NIRB contamination maps}\n\nIn addition to the flux from high-$z$ ($z > 5$) galaxies, the observed NIRB also contains radiation from unresolved, low-$z$ galaxies, and an excess radiation from unknown sources \\citep{2013MNRAS.433.1556Y,2014MNRAS.440.1263Y,2012Natur.490..514C,2014Sci...346..732Z}. We collectively refer to these two components as {\\it contamination}, since in this work the targeted signal is the flux from high-$z$ galaxies. To model such contaminating signal we construct random maps with mean flux 1.0 (0.7) nWm$^{-2}$sr$^{-1}$ at 3.6 (4.5) $\\mu$m and reproduce the sum of (i) the angular power spectrum of the power {\\it excess}  (see \\citealt{2013MNRAS.433.1556Y}) matching available observations \\citep{2012Natur.490..514C,2012ApJ...753...63K}, and (ii) the angular power spectrum of low-$z$ galaxies \\citep{2012ApJ...752..113H} producing shot noise level matching \n\\citealt{2012ApJ...753...63K} ($P_{\\rm SN} = 4.8\\times10^{-11}$nW$^{2}$m$^{-4}$sr$^{-1}$ at 3.6~$\\mu$m and $P_{\\rm SN} = 2.2\\times10^{-11}$nW$^{2}$m$^{-4}$sr$^{-1}$ at 4.5~$\\mu$m, the corresponding subtraction magnitude is $\\sim25$ mag). \nThe steps are:\n\\begin{itemize}\n\\item A white noise map, i.e. a Gaussian random field, is generated.\n\\item This map is then transformed into spatial-frequency space by FFT.\n\\item For each complex number in frequency space, its modulus is rescaled to be $\\sqrt{P(q)}$, where $P$ is the given power spectrum and $q$ is the spatial-frequency. The zero-frequency ($q = 0$) element is set to be the mean flux.\n\\item The above map is then transformed back into real space by inverse FFT, resulting in a synthetic image with the same 2-point clustering properties as the measured $P(q)$.\n\\end{itemize}\nIn Fig. \\ref{maps} we plot a single realization of the contamination map at 3.6~$\\mu$m as an example (bottom right). The contamination is not correlated with the high-$z$ galaxy component; however, it adds noise to the cross-correlation signal. To account for the statistical variance of the contamination, we make 30 independent realizations. In Fig. \\ref{APS} we show the angular power spectrum of bottom panels in Fig. \\ref{maps}. All maps are convolved with a circular symmetric {\\tt Spitzer} PSF before further analysis. \n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig2.eps}\n \\caption{Angular power spectrum of the 3.6~$\\mu$m flux map for $5 < z < 10$ galaxies (triangles) and contamination (squares). Errorbars show uncertainties due to limited number of Fourier modes in each $q$ bin. As a comparison, in the same panel we also plot  the \\citet{2012ApJ...752..113H} model for $z < 5$ galaxies (solid line), and the observational points in \\citet{2012ApJ...753...63K} (diamonds) and \\citet{2012Natur.490..514C} (hourglasses).  \nOn large scales ($\\theta \\lower.5ex\\hbox{\\gtsima} 100''$) the angular power spectrum of our co-added map (quite similar to squares in the panel) is consistent with both observations. Compared with K12, at $\\theta \\lower.5ex\\hbox{\\ltsima} 100''$ our prediction slightly falls short, probably because the non-linear clustering of low-$z$ galaxies is not modeled here. The C12 observations have a shallower (i.e. $\\sim24$ mag) source subtraction, hence a higher shot-noise level.\n}\n\\label{APS}\n}\n\\end{figure}\n \n\\section{Cross-correlation of LBG\\lowercase{s} and NIRB}\\label{results}\n\n\\subsection{the correlation coefficient}\nWe first analyze the correlation coefficient between the LBG flux map and the NIRB map. It is defined as\n \\begin{equation}\n {\\cal R}=\\frac{\\mean{\\delta F_{1.6} \\delta F_{\\lambda_0}}}{\\sqrt{ \\mean{\\delta F^2_{1.6}} \\mean{  \\delta F^2_{\\lambda_0}}    }  },\n \\end{equation}\n where $\\lambda_0$ refers to either 3.6 $\\mu$m or 4.5 $\\mu$m, $\\delta F_{1.6}$ and $\\delta F_{\\lambda_0}$ are the flux fluctuations of the same pixel (zero lag) at those two wavelengths. The brackets refer to the pixel-averaged fluctuations. The correlation coefficient indicates the fraction of sources contributing to common signals.  Before calculating the correlation coefficient we smooth both the NIRB maps and the LBG flux maps \nby a real space top-hat window function with diameter $\\theta=10''$. This is to suppress the instrumental noise since in the measured NIRB maps \\citep{2012ApJ...753...63K} the instrumental noise is negligible at $\\theta \\lower.5ex\\hbox{\\gtsima} 10''$.\nThe smoothing is not related to the LBG detection limits since it is performed on maps constructed from LBGs already present in catalogs.\nTo mimic surveys with different areas, we cut out sub-maps with different areas from the full map. We choose three areas:\n$(0.036)^2$, $(0.3)^2$ and $(1.2)^2$~deg$^2$, representing a survey region similar to HUDF\/XDF, UDS, and an hypothetical larger field, respectively. \n \nBefore calculating the correlation coefficient, in both maps we mask pixels containing galaxies brighter than mag $\\sim25$ at {\\it either} 3.6~$\\mu$m or 4.5~$\\mu$m. From this procedure we obtain the source-subtracted NIRB map. The correlation coefficient vs. LBG limiting magnitude is shown in Fig. \\ref{correlation_coefficient}. The filled regions  contain 68.3\\% probability of all sub-map samples. \nNote that we have 30 random realizations for each full map ((2.4)$^2$ deg$^2$), so even for the $(1.2)^2$~deg$^2$ case there are 120 samples, sufficient to compute the correlation coefficient variance.  \n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{.\/fig3.eps}\n\\caption{\nCorrelation coefficient between the 3.6~$\\mu$m source-subtracted NIRB and LBG flux maps vs. $H$-band limiting magnitudes for three different map areas. Filled regions bracket the 68.3\\% probability ranges around the peaks.\nAll fields are smoothed on scale $\\theta=10''$. }\t\n\\label{correlation_coefficient}\n}\n\\end{figure}\n\nFig. \\ref{correlation_coefficient} shows that it is indeed feasible to detect the cross-correlation signal from the mock maps, even for a relatively shallow survey with $H_{\\rm lim}\\sim25$, for which ${\\cal R}\\approx 0.04$. By pushing the limiting magnitude to fainter values, the correlation coefficient rapidly increases by a factor $\\sim2$, and then slowly approaches $\\approx 0.09$ at $H_{\\rm lim}\\sim29$. It is worth noting that in small fields the measured correlation coefficient has a $\\sim 30 - 80\\%$ relative field-by-field scatter for $H_{\\rm lim} \\lower.5ex\\hbox{\\gtsima} 26$, and even larger scatter at $H_{\\rm lim} < 26$. In some extreme cases there would be no or negative cross-correlation detected, due to the small number of LBGs contained in the fields.\n\nTo elucidate the differential contribution of LBGs, for a $(0.3)^2$ deg$^2$ field, we further show in Fig. \\ref{correlation_coefficient_z} the correlation coefficient from LBGs at $>z$ for $H_{\\rm lim} = 25, 26$ and 27, respectively. For example, LBGs at $z>8$ contribute ${\\cal R}\\approx 0.01$ at $H\\lower.5ex\\hbox{\\ltsima} 27$.\n\nOur results are for NIRB maps with a fixed shot noise level matching \\citet{2012ApJ...753...63K} measurements.  Since the shot noise is mainly from low-$z$ galaxies, an increased subtraction depth implies a relatively larger number of low-$z$ galaxies (with respect to the high-$z$ ones) are resolved and removed. As a result we expect a higher correlation coefficient. On the other hand, if we resolve more LBGs, the cross-power signal also becomes stronger. Thus, the strength of the cross-power signal vs. different LBGs detection depths can be used to determine the differential contribution of early galaxies.  \n \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig4.eps}\n\\caption{ \nContribution to correlation coefficient from LBGs with  $>z$. The 68.3\\% probability ranges are plotted by errorbars. For displaying purpose we slightly shift the x-positions of some curves.}\t\n\\label{correlation_coefficient_z}\n}\n\\end{figure}\n\n\\subsection{Detectability vs. scales}\n\nSo far we have investigated the behavior of the correlation coefficient by at a specific smoothing scale $\\theta=10''$. We now examine its dependence on angular scale. To this aim we re-define the correlation coefficient in the spatial-frequency domain via the power spectrum\n\\begin{equation} \\label{eqn_coherence}\n{\\cal R}_q(\\theta) = \\frac{P_{\\rm IR\\times G}(\\theta)}{  \\sqrt{P_{\\rm IR}(\\theta)\\times P_{\\rm G}(\\theta)} },\n\\end{equation}\nwhere $P_{\\rm 1\\times 2}(\\theta)$ is the cross-power spectrum and $P_{\\rm 1}$,$P_{\\rm 2}$ are the auto-power spectra. They are all calculated using the 2D FFT. Since $\\delta F^2 \\simeq q^2P\/2\\pi$, ${\\cal R}_q \\sim {\\cal R}$ for the same $\\theta$. \nFirst we derive the shot noise due to LBGs fainter than a given H-magnitude (Fig. \\ref{PSN}). It is the value of $P_{\\rm G}$ taken on a sufficiently small scale $\\theta \\approx 14''$ (a stability check of the results for scales  up to $\\approx40''$ has been performed). As we will point out in the following,  shot noise  is the most important term contributing to the  cross-correlation. The shot noise level shown here is a useful reference for possible follow-up work on real data.  \n \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig5.eps}\n\\caption{Shot noise due to LBGs with magnitude $>H$ and at $5 < z < 10$.}\t\n\\label{PSN}\n}\n\\end{figure}\n \n \nThe ${\\cal R}_q(\\theta)$ between the source-subtracted NIRB at $3.6~\\mu$m and the 1.6~$\\mu$m flux map of LBGs with $H_{\\rm lim} =  25, 26$ and 27 is shown by Fig. \\ref{correlation_coefficient_q}. The errorbars are the r.m.s of 30 samples each one using  different random contamination realizations. \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig6.eps}\n\\caption{The ${\\cal R}_q$ between the source-subtracted NIRB and the flux maps constructed from LBGs with different $H_{\\rm lim}$. The errorbars here show the r.m.s of 30 random realizations. For displaying purpose we slightly shift the x-positions of $H_{\\rm lim} = 26, 27$ curves.}\n\\label{correlation_coefficient_q}\n}\n\\end{figure}\n\nThe cross-power spectrum includes both the shot noise and clustering terms. Shot noise is from sources common to both NIRB and LBG flux, and it is dominant on scales $ \\theta \\lower.5ex\\hbox{\\ltsima} 100''$. On larger scales, the clustering term progressively takes over. This term arises from all sources sharing the same large scale structure, including galaxies even fainter than the LBG limiting magnitude. As a consequence, in principle the clustering term could allow the detection of fainter galaxies in the source-subtracted NIRB through the cross-correlation with relatively bright LBGs. However, as shown by Fig. \\ref{correlation_coefficient_q} the cross-correlation is more easily detected on small scales. Although the clustering term may contain more information, the main difficulty to be overcome in order to efficiently use this strategy is that even for a relatively large survey area of  $(2.4)^2$~deg$^2$ (i.e. our full map), the signal-to-noise ratio never exceeds $\\sim 3$ at $\\theta \\gsim300''$ for $H_{\\rm lim} < 27$.  While increasing the limiting magnitude to $>27$ does not help much to this aim, the noise could be reduced by using larger survey areas, as expected with, e.g. EUCLID and WFIRST. \n\n\n\\subsection{Colors}\n\nFrom our mock maps it is also possible to characterize the colors, i.e. the ratio between flux in two bands, of unresolved galaxies in NIRB observations. With the assumption \n\\begin{equation}\n\\frac{F_{4.5}}{F_{3.6}}\\approx \\frac{\\mean{ (\\delta F_{4.5} \\delta F_{1.6})_\\theta  }} { \\mean{ (\\delta F_{3.6} \\delta F_{1.6})_\\theta }  },\n\\end{equation} \nwe derive the magnitude difference as \n\\begin{align}\n[3.6]-[4.5] &= 2.5{\\rm log}\\left( \\frac{4.5F_{4.5}}{3.6F_{3.6}}  \\right) \\nonumber \\\\\n&\\approx2.5{\\rm log}\\left( \\frac{4.5}{3.6}  \\frac{\\mean{ (\\delta F_{4.5} \\delta F_{1.6} )_\\theta }} { \\mean{ (\\delta F_{3.6} \\delta F_{1.6} )_\\theta}  }   \\right)  \n\\end{align} \nSuch magnitude difference, which might considerably vary for individual galaxies, represents the combined and weighted color of the unresolved galaxy population in these bands. In practice the PSF adds a bias to the measured flux ratio. Therefore before calculating the magnitude difference it would be necessary to first deconvolve the PSF from each map. We skip this step here and directly calculate the magnitude difference on maps without PSF convolution. Again we specify $\\theta=10''$. The predicted color as a function of $H_{\\rm lim}$ is reported in Fig. \\ref{color}, allowing us to conclude that the magnitude difference which is $\\sim$-0.13 mag, could be detected by cross-correlating survey maps with area $\\lower.5ex\\hbox{\\gtsima} (0.3)^2$ deg$^2$. The figure reiterates that smaller area fields would be affected by bias effects.\n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig7.eps}\n\\caption{The magnitude difference, $[3.6]-[4.5]$, of fluctuations due to unresolved galaxies in the source-subtracted NIRB maps as a function of the survey limiting magnitude. Filled regions indicate 68.3\\% probability ranges.}\t\n\\label{color}\n}\n\\end{figure}\n \n \\section{Conclusions}\\label{conclusions} \n\nMotivated by the fact that LBG surveys carried out by the {\\tt HST} have reached detection limits much deeper than the NIRB measured by {\\tt Spitzer} at longer wavelengths, in this paper we investigated the feasibility to pick the resolved LBGs component out of the NIRB by a cross-correlation analysis. Our investigations were based on mock maps constructed from semi-numerical simulations of halo formation, and empirically determined $L_{\\rm UV} - M_{\\rm h}$ relations.\n \nWe found that in the NIRB observed at 3.6 and 4.5~$\\mu$m and with sources subtracted down to $\\sim$25 apparent magnitude, at smallest scales where the shot noise dominates, about 10\\%  of the flux fluctuations arises from LBGs in $5 < z < 10$ and with $H \\lower.5ex\\hbox{\\ltsima} 29$. Such faint galaxies have already been resolved in the existing deep surveys.  However, this fractional contribution, if measured from narrow fields with area $\\sim (0.036)^2$~deg$^2$ (as the HUDF\/XDF), could vary from $\\sim3\\%$ to $\\sim16\\%$. \nIf the limiting magnitude is decreased to $H \\sim 27$, the fractional contribution decreases to about 8\\%. If in this case we consider a larger field, e.g. with area similar to the UDS ($\\sim (0.3)^2$~deg$^2$), then the correlation coefficient  varies in the narrower range $\\sim 6\\%-9\\%$. \nWe remind that the variance at hand here is due to both the large-scale inhomogeneity of the signal and the contamination: it is the field-to-field variance of the correlation itself. We do not model errors introduced by data analysis procedures, as for example mask effects. However, at least theoretically we have shown that the contribution from the faintest galaxies could be isolated from the NIRB through the cross-correlation analysis. \n\nWe pointed out that it is still challenging to use the cross-correlation arising from the clustering term of the cross-power spectrum to study galaxies unresolved not only in NIRB observations, but also in LBG surveys. This term is dominant at $\\lower.5ex\\hbox{\\gtsima} 200''$, but even for survey areas as large as $(2.4)^2$~deg$^2$, the signal  has a small significance, with a S\/N ratio $\\lsim3$.   Although very unlikely, if the NIRB clustering excess originates from LBGs we would detect a correlation coefficient close to 1 at scales where the clustering term dominates. Stated differently, a measured correlation coefficient $ {\\cal R} \\ll 1$ would actually rule out this scenario.\n\nOur results have interesting implications also for the colors of high-$z$ galaxy population. The colors of LBGs detected at wavelengths $< 1.6~\\mu$m but too faint to be individually resolved by existing telescopes at  3.6 and 4.5~$\\mu$m, can be obtained by using the cross-correlation with the NIRB in these two longer wavelength bands. \n\n \nIt is worth noting that the predictions presented in this paper assume a contaminating signal in the form of NIRB fluctuation excess which originates  from direct collapse black holes at high $z$ \\citep{2013MNRAS.433.1556Y}. If however the excess is found to arise more locally, we might be able to model or subtract it more accurately, thereby making the signal calculated in this paper more easily detectable, i.e. ${\\mean {\\cal R}} \\lower.5ex\\hbox{\\gtsima} 0.1$. \n \nImportantly, the study proposed here can be further developed to infer the properties of even fainter high-$z$ galaxy populations that are currently inaccessible to direct telescopic detection. This could, for example, be accomplished by a Lyman-break tomography study designed to isolate high-$z$ populations via multi-band cross-correlations without requiring prior LBG detections \\citep{2015ApJ...804...99K}. \n\n\\section*{ACKNOWLEDGMENTS}\n\nWe thank A. Kashlinsky for valuable comments on the manuscript. \t\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nRecent investigations seem to show that soft $\\gamma$-ray repeaters and some\nanomalous X-ray pulsars are neutron stars which may have surface magnetic fields larger that\n$10^{15}$ G~\\cite{duncan,usov,pacz}, the so called magnetars.  Until recently, the strongest \nestimated magnetic field is of the order of $B=2\\times 10^{15}$ G and was detected \nin a quite young star, SGR 1806-20~\\cite{sgr}. According to~\\cite{kouve} a fraction as high as \n$10\\%$ of the neutron star population could be magnetars. \n\nThe effect of the strong magnetic fields on the equation of state (EOS) of\nstellar matter in neutron stars has been\nstudied both at low densities below neutron drip, of interest for the study of \nthe outer crust of neutron stars~\\cite{low}, and at  high densities, of interest for the\nstudy of the interior of compact stars~\\cite{chakrabarty96,broderick}. In\nthis last case field-theoretical descriptions based on the non-linear \nWalecka model (NLWM)~\\cite{bb}  were used and several\nparametrizations compared. It was shown that they have an overall similar\nbehavior.\nIt was recently shown in~\\cite{aziz08} that the EOS at subsaturation densities, including densities\nof the order of the densities at the\ninner edge of the crust of a compact star, was particularly affected by fields of\nthe order of 10$^{18}$ G. \n\nAn important characteristic of the nuclear matter is the appearance of a\nliquid-gas phase transition at subsaturation densities. The role of the\nisospin is of particular importance. Indeed, since nuclear matter is composed\nof two different fluids, namely protons and neutrons, the liquid-gas phase\ntransition can lead to an isospin distillation phenomenon~\\cite{chomaz}. \n The region of \ninstability is determined by the spinodal curve. Due to the symmetry energy, the EOS of $\\beta$-equilibrium of magnetic free nuclear matter is thermodynamically stable. The stability of the EOS is determined from the curvature of the free-energy: a positive curvature corresponds to thermodynamically stable matter.\n\nIf we\nconsider stellar matter at very low densities, nuclei in matter are expected\nto form a Coulomb lattice embedded in the neutron-electron sea that\nminimizes the Coulomb interaction energy. With an increase of the density,\nnuclear \"pasta\" structures emerge~\\cite{rpw83}. The existence of pasta phases\nmay modify some important processes by\nchanging the hydrodynamic properties and the neutrino opacity in supernova\nmatter and in the matter of newly born neutron stars~\\cite{hor04}. Also, the\npasta phases may influence neutron star quakes and pulsar glitches via the\nchange of mechanical properties of the crust matter~\\cite{quakes}. It is\ntherefore important to study how the magnetic field could affect the extension\nof the pasta phase and the isospin distillation effect.\n\n\n\n In fact, sufficiently strong magnetic field affect the extension of the unstable region.\nIn order to\nhave a better understanding of the effect of the magnetic field on the\ninstabilities of nuclear matter at subsaturation densities we study in the\npresent work the effect of a strong magnetic field on the thermodynamical\nspinodal instabilities obtained from the free energy curvature matrix~\\cite{umodes06,abmp06}.\nRecently, it was shown that the magnetic field and Joule heating have \nthe important effect of maintaining compact stars warm for a\nlonger time~\\cite{pons08}. This kind of simulations need the EOS of the crust. It is,\ntherefore, important to make a study that shows when should the magnetic field \nbe taken explicitly into account in the EOS of the crust. An unstable\nregion in a wider density range  will correspond to a larger crust and the\nproperties of the star depending on the crust will be affected. \n\nIn the present paper, we will consider two  relativistic effective approaches: a\nNLWM, TM1~\\cite{tm1}, with constant coupling parameters,\nand a density dependent relativistic hadronic (DDRH) model TW~\\cite{tw} \nwith density-dependent coupling parameters. DDRH models  seem to give results closer Skyrme interactions  than NLWM,\n at subsaturation densities~\\cite{camille08}.\n\nIn section II we make a brief review of the models, the EOS under the effect of\na magnetic field and  the stability conditions.  Results are discussed in section III and conclusions are drawn\nin the last section. \n\n\\section{The formalism}\n\n\\subsection{EOS of nuclear matter under a strong magnetic field}\n\nIn the present section we make a short review of\nthe field-theoretical approach used to obtain the EOS of nuclear matter. Within this approach, the baryons\ninteract via the exchange of $\\sigma$, $\\omega$ and $\\rho$ mesons in the presence of a\nuniform magnetic field $B$ along the $z$-axis.  We start from the Lagrangian\ndensity of TW~\\cite{fuchs,tw} model\n\\begin{equation}\n{\\cal L}= \\sum_{b=n, p}{\\cal L}_{b} + {\\cal L}_{m}.\n\\label{lan}\n\\end{equation}\nThe baryon ($b$=$n$, $p$) and meson ($\\sigma$, $\\omega$ and $\\rho$) Lagrangians are given by\n\\begin{widetext}\n\\begin{eqnarray}\n{\\cal L}_{b}&=&\\bar{\\Psi}_{b}\\left(i\\gamma_{\\mu}\\partial^{\\mu}-q_{b}\\gamma_{\\mu}A^{\\mu}- \nm_{b}+\\Gamma_{\\sigma}\\sigma\n-\\Gamma_{\\omega}\\gamma_{\\mu}\\omega^{\\mu}-\\frac{1}{2}\\Gamma_{\\rho}\\tau_{3 b}\\gamma_{\\mu}\\rho^{\\mu}\n-\\frac{1}{2}\\mu_{N}\\kappa_{b}\\sigma_{\\mu \\nu} F^{\\mu \\nu}\\right )\\Psi_{b}, \\cr\n{\\cal L}_{m}&=&\\frac{1}{2}\\partial_{\\mu}\\sigma \\partial^{\\mu}\\sigma\n-\\frac{1}{2}m^{2}_{\\sigma}\\sigma^{2}\n+\\frac{1}{2}m^{2}_{\\omega}\\omega_{\\mu}\\omega^{\\mu}\n-\\frac{1}{4}\\Omega^{\\mu \\nu} \\Omega_{\\mu \\nu}  \\cr\n&-&\\frac{1}{4} F^{\\mu \\nu}F_{\\mu \\nu}\n+\\frac{1}{2}m^{2}_{\\rho}\\rho_{\\mu}\\rho^{\\mu}-\\frac{1}{4}  P^{\\mu \\nu}P_{\\mu \\nu},\n\\label{lagran}\n\\end{eqnarray}\n\\end{widetext}\nrespectively, where $\\Psi_{b}$ are the baryon Dirac fields. The nucleon mass and isospin projection for the protons and neutrons are denoted by $m_{b}$  and $\\tau_{3 b}=\\pm 1$, respectively. The mesonic and electromagnetic field strength tensors are given by their usual expressions: $\\Omega_{\\mu \\nu}=\\partial_{\\mu}\\omega_{\\nu}-\\partial_{\\nu}\\omega_{\\mu}$, $P_{\\mu \n\\nu}=\\partial_{\\mu}\\rho_{\\nu}-\\partial_{\\nu}\\rho_{\\mu}$, and  $F_{\\mu\n\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$. The nucleon anomalous\nmagnetic moments are introduced via the coupling of the baryons to the\nelectromagnetic field tensor with $\\sigma_{\\mu\n  \\nu}=\\frac{i}{2}\\left[\\gamma_{\\mu},  \\gamma_{\\nu}\\right] $ and strength\n$\\kappa_{b}$ with $\\kappa_{n}=g_n\/2=-1.91315$ for the neutron and\n$\\kappa_{p}=(g_p\/2-1)=1.79285$ for the proton, respectively, and where $g_i$\nare the Land\\'e $g$ factors for the particle $i$ ($g_p=5.5856912$ and\n$g_n=-3.8260837$), and $\\mu_N$ is the nuclear magneton. The electromagnetic field is assumed to be externally generated (and thus has no associated field equation), and only frozen-field configurations will be considered.  The electromagnetic couplings are denoted by $q$.\nThe parameters of the model are the nucleon mass $m_b=939$~MeV, the\nmasses of mesons $m_\\sigma$, $m_\\omega$ and $m_\\rho$ and the density\ndependent coupling parameters $\\Gamma$ which are adjusted in order to\nreproduce some of the nuclear matter bulk properties and  DBHF (Dirac\nBrueckner Hartree-Fock) calculations~\\cite{dbhf}, using the following parametrization\n\\begin{equation}\n\\Gamma_{i}(\\rho)=\\Gamma_{i}(\\rho_{sat})f_{i}(x),\\quad i=\\sigma, \\omega, \\rho\n\\label{gam1}\n\\end{equation}\nwhere $x={\\rho}\/{\\rho_{sat}}$, with\n\\begin{equation}\nf_{i}(x)=a_{i}\\frac{1+b_{i}\\left(x+d_{i}\\right)^{2}}{1+c_{i}\\left(x+d_{i}\\right)^{2}},\n\\quad i=\\sigma, \\omega,\n\\label{gam2}\n\\end{equation}\nand,\n\\begin{equation}\nf_{\\rho}(x)=\\exp\\left[ -a_{\\rho}(x-1)\\right], \n\\end{equation}\nwith the values of the parameters $m_i$, $\\Gamma_{i}$, $a_{i}$, $b_{i}$,\n$c_{i}$ and $d_{i}$, $i=\\sigma, \\omega, \\rho$ \ngiven in Table~\\ref{table1} for TW model~\\cite{tw}. Other possibilities for these parameters are also found in the\nliterature~\\cite{hbt}. \n\nThe symmetry energy and its first and second derivatives are important to understand the instability \nregion.\nNLWM models become very stiff above saturation densities while DDRH models have \na softer behavior. On the other hand, at subsaturation densities DDRH models\nhave larger symmetry energies and  a larger extension of the\ninstability region for very asymmetric matter. In Fig. \\ref{esym} we compare\nthe symmetry energy of the models we will consider in the present study: TM1\nand TW. As expected TM1 has a smaller symmetry energy at subsaturation\ndensities and a larger one above the saturation density.\n\n\\begin{table}[Htb]\n\\begin{tabular}{ccccccc}\n\\hline\n\\hline\ni &$m_{i}$(MeV)& $\\Gamma_{i}$ & $a_{i}$ & $b_{i}$ & $c_{i}$ & $d_{i}$  \\\\\n\\hline\n$\\sigma$   & 550 &10.72854 &1.365469 &0.226061 &0.409704&0.901995\\\\\n$\\omega$ &783 &13.29015 & 1.402488&0.172577 &0.344293&0.983955 \\\\\n$\\rho$       &763 & 7.32196 & 0.515 & & & \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Parameters of the TW model.}\n\\label{table1}\n\\end{table} \n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure1.eps}\n\\caption{Symmetry energy for all models used in the present work. }\n\\label{esym}\n\\end{figure}\n\nNotice that in the DDRHM  the nonlinear meson terms are not present, in contrast\nwith the usual NLWM. For TM1 model we add\nto the Langrangian density, Eq. (\\ref{lagran}) , with $g_i=\\Gamma_i$,\n$${\\cal L}=\\frac{1}{3}bm_n(g_{\\sigma}\\sigma)^3+\\frac{1}{4}c(g_{\\sigma}\\sigma)^4+\\frac{1}{4!}\\xi g^4_\\omega(\\omega_{\\mu}\\omega^{\\mu} )^2. $$\nThe coupling parameters are constant and given in~\\cite{tm1}.\n\n\nFrom now we take the standard mean-field theory (MFT) approach and display\nonly some of the equations. A complete set of equations and description of the\nmethod can be found in the literature (\\textit{e.g.}, Ref.~\\cite{yuan,\n  broderick}). For  the description of the system, we need the baryonic\ndensity, the energy density of nuclear  matter, and the pressure. The energy density of nuclear\nmatter is given by \n\\begin{equation}\n\\varepsilon=\\sum_{b=p,n} \\varepsilon_{b}+\\frac{1}\n{2}m^{2}_{\\sigma}\\sigma^{2}+\\frac{1}{2}m^{2}_{\\omega}\\omega^{2}_{0}+\\frac{1}{2}m^{2}_{\\rho}\\rho^{2}_{0}\n\\label{ener}\n\\end{equation} \nwhere the energy densities of nucleons have the following forms\n\\begin{eqnarray}\n\\varepsilon_{p}&=&\\frac{q_{p}B}{4\\pi^ {2}}\\sum_{\\nu, s}\\left[k^{p}_{F,\\nu,s}E^{p}_{F}\n+\\left(\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B\\right) ^{2} \n\\ln\\left|\\frac{k^{p}_{F,\\nu,s}+E^{p}_{F}}{\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B} \\right|\\right] , \\cr\n\\varepsilon_{n}&=&\\frac{1}{4\\pi^ {2}}\\sum_{s}\\bigg[\\frac{1}{2}k^{n}_{F, s}E^{n 3}_{F}-\\frac{2}\n{3}s\\mu_{N}\\kappa_{n} B E^{n 3}_{F}\\left(\\arcsin\\left(\\frac{\\bar{m}_{n}}{E^{n}_{F}} \\right)-\\frac{\\pi}\n{2}\\right)-\\left(\\frac{1}{3}s\\mu_{N}\\kappa_{n} B +\\frac{1}{4}\\bar{m}_{n}\\right) \\cr\n&&\\left(\\bar{m}_{n}k^{n}_{F, s}E^{n}_{F}+\\bar{m}^{3}_{n}\\ln\\left|\\frac{k^{n}_{F,s}+E^{n}_{F}}{\\bar{m}_{n}} \n\\right|\\right) \\bigg].\n\\end{eqnarray}\nFor the neutrons we have introduced\n\\begin{equation}\n\\bar{m}_{n}=m^{*}-s\\mu_{N}\\kappa_{n}B,\n\\label{barm}\n\\end{equation}\nwhere the effective baryon masses are given by \n\\begin{equation}\nm^{*}=m-\\Gamma_{\\sigma}\\sigma.\n\\end{equation} \n\nThe pressure of the system is obtained from the expression\n\\begin{equation}\nP_{m}=\\sum_{b=n,p}\\mu_{b}\\rho_{b}-\\varepsilon.\n\\label{press}\n\\end{equation}\n  \nThe energy spectra for protons are neutrons are given by\n\\begin{eqnarray}\nE^{p}_{\\nu, s}&=& \\sqrt{k^{2}_{z}+\\left(\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B \\right) \n^{2}}+\\Gamma_{\\omega} \\omega^{0}+\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}+\\Sigma^{R}_{0}, \\label{enspc1}\\\\\nE^{n}_{s}&=& \\sqrt{k^{2}_{z}+\\left(\\sqrt{m^{* 2}+k^{2}_{x}+k^{2}_{y}}-s\\mu_{N}\\kappa_{n}B \n\\right)^{2}}+\\Gamma_{\\omega} \\omega^{0}-\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}+\\Sigma^{R}_{0},\n\\label{enspc2} \n\\end{eqnarray}\nrespectively, where $\\nu=n+\\frac{1}{2}-sign(q)\\frac{s}{2}=0, 1, 2, \\ldots$ enumerates the Landau levels of the fermions with electric charge $q$, the quantum number $s$ is $+1$ for spin up and $-1$ for spin down\ncases, and the rearrangement term is given by \n\\begin{equation}\n\\Sigma^{R}_{0}=\\frac{\\partial \\Gamma_{\\omega}}{\\partial \\rho}\\rho_b\\omega_{0}+\\frac{\\partial \\Gamma_{\\rho}}{\\partial \n\\rho}\\rho_{3}\\frac{\\rho_{0}}{2}-\\frac{\\partial \\Gamma_{\\sigma}}{\\partial \\rho}\\rho^s\\sigma,\n\\end{equation}\nwhere $\\rho^s=\\rho^s_p+\\rho^s_n$ and $\\rho_b=\\rho_p+\\rho_n$, with the expressions of the scalar and vector densities for protons and neutrons given by~\\cite{broderick}\n\\begin{eqnarray}\n\\rho^{s}_{p}&=&\\frac{q_{p}Bm^{*}}{2\\pi^{2}}\\sum_{\\nu=0}^{\\nu_{\\mbox{\\small max}}}\\sum_{s}\\frac{\\sqrt{m^{* 2}+2\\nu \nq_{p}B}-s\\mu_{N}\\kappa_{p}B}{\\sqrt{m^{* 2}+2\\nu q_{p}B}}\\ln\\left|\\frac{k^{p}_{F,\\nu,s}+E^{p}_{F}}\n{\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B} \\right|, \\cr\n\\rho^{s}_{n}&=&\\frac{m^{*}}{4\\pi^{2}}\\sum_{s} \\left[E^ {n}_{F}k^{n}_{F, s}-\\bar{m}^{2}_{n}\\ln\\left|\n\\frac{k^{n}_{F,s}+E^{n}_{F}}{\\bar{m}_{n}} \\right|\\right],  \\cr\n\\rho_{p}&=&\\frac{q_{p}B}{2\\pi^{2}}\\sum_{\\nu, s}k^{p}_{F,\\nu,s},  \\cr\n\\rho_{n}&=&\\frac{1}{2\\pi^{2}}\\sum_{s}\\left[ \\frac{1}{3}\\left(k^{n}_{F, s}\\right) ^{3}-\\frac{1}\n{2}s\\mu_{N}\\kappa_{n}B\\left(\\bar{m}_{n}k^{n}_{F,s}+E^{n 2}_{F}\\left(\\arcsin\\left( \\frac{\\bar{m}_{n}}\n{E^{n}_{F}}\\right) -\\frac{\\pi}{2} \\right)  \\right) \\right] \n\\end{eqnarray}\nwhere $k^{p}_{F, \\nu, s}$ and $ k^{n}_{F, s}$ are the Fermi momenta of protons and neutrons which are related to the Fermi energies $E^{p}_{F}$ and $E^{n}_{F}$ through\n\\begin{eqnarray}\nk^{p 2}_{F,\\nu,s}&=&E^{p 2}_{F}-\\left[\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B\\right] ^{2} \\cr\nk^{n 2}_{F,s}&=&E^{n 2}_{F}-\\bar{m}^{2}_{n}.\n\\end{eqnarray}\n\nThe chemical potentials of nucleons within TW are given  by\n\\begin{eqnarray}\n\\mu_{p}&=& E^{p}_{F}+\\Gamma_{\\omega}\\omega^{0}+\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}\n+\\Sigma^{R}_{0} \\cr\n\\mu_{n}&=& E^{n}_{F}+\\Gamma_{\\omega}\\omega^{0}-\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}\n+\\Sigma^{R}_{0}.\n\\label{mu}\n\\end{eqnarray}\nFor TM1 we have similar expressions with the last term, the rearrangement\nterm, equal to zero. \n\n\\subsection{Stability conditions for nuclear matter}\n\nAt subsaturation densities nuclear matter has a liquid-gas phase transition\nand homogeneous matter is not stable within a given range of densities. \nThe stability conditions for asymmetric nuclear matter, keeping volume and temperature constant, \nare obtained from the free energy density $\\cal F$, imposing that this\nfunction is a convex function of the densities $\\rho_p$ and $\\rho_n$. For\nstable homogeneous matter, the symmetric matrix with the elements~\\cite{Bar03,  marg03}, \n\\begin{equation}\n{\\cal F}_{ij}=\\left( \\frac{\\partial^{2} {\\cal F}}{\\partial \\rho_{i}\\partial\\rho_{j}}\\right) _{T},\n\\end{equation}\nknown as stability matrix,  must be positive. This corresponds to imposing that the trace and the\ndeterminant of ${\\cal F}_{ij}$ are positive. In terms of the proton and\nneutron chemical potentials $\\mu_i$, the stability matrix is given by\n\\begin{equation}\n{\\cal F}=\n\\begin{pmatrix}\n\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{n}} \n&\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{p}} \\\\\n\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{n}}\n&\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{p}}\n\\label{cur}\n\\end{pmatrix},\n\\end{equation}\nwith $\\mu_{i}=\\frac{\\partial{\\cal F}}{\\partial \\rho_{i}}|_{T,\\rho_{j\\neq i}} $.\n\nFor nuclear matter,\nthe largest eigenvalue of the stability matrix is always positive and the\nother becomes negative at subsaturation densities.\nWe define the thermodynamical spinodal at T=0 as the curve on the  $\\rho_n$,\n$\\rho_p$ plane defined by the points  for which the determinant of $ {\\cal  F}_{ij}$ is zero; \nthat is, the smallest  eigenvalue is zero. Inside the region limited by the thermodynamical\nspinodal  the smallest eigenvalue of  ${\\cal F}_{ij}$ is negative and nuclear matter is \nunstable~\\cite{marg03}. At $T=0$, $\\cal F$ is equal to the energy density defined \nby Eq. (\\ref{ener}). The eigenvalues of the stability matrix are given by\n\\begin{equation}\n\\lambda_{\\pm}=\\frac{1}{2}\\left(\\hbox{Tr}({\\cal F})\\pm\\sqrt{\\hbox{Tr}({\\cal F})^2-4 \\hbox{Det}({\\cal F})}\\right).\n\\label{lambd}\n\\end{equation}\nThe stability condition requires that they are both positive. When one curvature becomes negative the system is thermodynamically unstable and can decrease its free energy by going in the instability direction \\cite{marg03}, defined by the direction of the eigenvector associated to the negative eigenvalue.\nThe eigenvectors $\\delta {\\bf \\rho}_{\\pm}$ of the stability matrix are given by \n\\begin{equation}\n\\frac{\\delta \\rho^{\\pm}_{i}}{\\delta \\rho^{\\pm}_{j}}=\\frac{\\lambda_{\\pm}-{\\cal F}_{jj}}{{\\cal F}_{ji}}, \\: i, j = p, n.\n\\end{equation}\nIn the following we study the direction of instability inside the spinodal\nsection for both  models considered. \n\n\n\\subsection{Stability conditions  for $npe$ matter}\nStellar matter at low densities is formed by protons, neutrons and\nelectrons in equilibrium with respect to weak interaction processes. Until now we have considered the subsaturation instability region\nof neutron-proton ($np$) nuclear matter.  In this section we investigate the\neffect of the inclusion of electrons on the stability conditions of nuclear matter when electrons are\nincluded. In particular, we will calculate the thermodynamical spinodal\nsections for $npe$ (neutron-proton-electron) neutral matter. Since matter is neutral the proton and electron densities must be equal, i.e. $\\rho_p=\\rho_e$.\n\nElectrons are included in a minimal way in the system, and are   described by the following the Lagrangian density \n\\begin{equation}\n{\\cal L}_{e} = \\bar{\\psi}_{e}\\left(i\\gamma_{\\mu}\\partial^{\\mu}-q_{e}\\gamma_{\\mu}A^{\\mu}\n-m_{e}\\right )\\psi_{e},\n\\label{lagran}\n\\end{equation}\nwhere $\\psi_{l} $ are the electron Dirac fields and $m_e=0.511 \\hbox{ MeV}$. \n\nIncluding the electrons, the stability matrix  (\\ref{cur}) becomes\n\\begin{equation}\n{\\cal F}=\n\\begin{pmatrix}\n\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{n}} &\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{p}} \\\\\n\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{n}}\n&\\displaystyle\\frac{\\partial\\left(\\mu_{p}+\\mu_{e}\\right)}{\\partial \\rho_{p}}\n\\end{pmatrix},\n\\label{cur1}\n\\end{equation}\nand the stability conditions are equivalent to the ones indicated in the previous subsection: the trace and the determinant of ${\\cal F}$ must be positive.\n\nThe electron density is given by\n\\begin{equation}\n\\rho_e=\\frac{|q_{e}|B}{2\\pi^{2}}\\sum_{\\nu, s}k^{e}_{F, \\nu, s}\n\\end{equation}\nwhere $k^{e}_{F, \\nu, s}$ is  the electron Fermi momentum related to the Fermi energy $E^{e}_{F}$ by\n\\begin{equation}\nk^{e 2}_{F,\\nu,s}=E^{e 2}_{F}-\\left(m^{2}_{e}+2\\nu |q_{e}| B\\right).\n\\end{equation}\nFor $npe$ neutral matter\n$${\\cal F}=\\epsilon+ \\epsilon_e$$\nwhere $\\epsilon$ was defined in (\\ref{ener}) and\nthe electron contribution is given by\n\\begin{equation}\n\\varepsilon_{e}=\\frac{|q_{e}|B}{4\\pi^ {2}}\\sum_{\\nu, s}\\left[k^{l}_{F,\\nu,s}E^{e}_{F}\n+\\left(m^{2}_{e}+2\\nu |q_{e}|B\\right) \n\\ln\\left|\\frac{k^{e}_{F,\\nu,s}+E^{e}_{F}}{\\sqrt{m^{2}_{e}+2\\nu |q_{e}| B}} \\right|\\right] .\n\\end{equation}\nWe next discuss $np$  nuclear matter and  $npe$ neutral matter. $\\beta$-equilibrium stellar matter is a particular case of  $npe$ neutral matter, with the proton and electron fractions defined by chemical equilibrium conditions, namely\n$$\\mu_p=\\mu_n-\\mu_e,$$\nwith $\\mu_p$ and $\\mu_n$ defined in (\\ref{mu}) and $\\mu_e=E^{e }_{F}$.\n \n\\section{Results and discussion}\n\nIn the present section we first show the dependence of the spinodal section on  the magnetic field, \nboth for neutron-proton  ($np$) and neutron-proton-electron ($npe$) matter. \nFrom the crossing of the  $\\beta$-equilibrium  EOS of stellar matter with the thermodynamical spinodal we make a prevision of the transition density and transition pressure at the inner edge of the crust of a compact star. \n\nFor a zero magnetic field,  the direction of instability of nuclear matter gives rise to a \ndistillation effect, corresponding to the formation of droplets of  dense matter with low isospin asymmetry\nin a background of a neutron gas with a small fraction of protons. This effect has been observed experimentally in heavy-ion reactions \\cite{chomaz}. Therefore, \nwe also discuss the effect of the magnetic field on the direction of instability, namely in which way it affects the distillation effect. \n\n\n\\subsection{Spinodal section $np$ matter}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure2.eps}\n\\caption{(Color online) Spinodal section on the  $\\rho_p$, $\\rho_n$\n  plane for TM1 at $T = 0 \\hbox{ MeV}$ and for severals values of\n  magnetic fields, $B=B^*\\, B_e$, a)  without  and b) with  AMM. For $B^*=10^5$ the spinodal section is formed by two separate regions.}\n\\label{spztm1npn}\n\\end{figure}\n\nWe will first consider $np$ nuclear  matter and determine the instability region limited by the spinodal surface.\nIn Fig.~\\ref{spztm1npn} and Fig.~\\ref{spztwnp} we show the spinodal sections obtained making $\\lambda_-=0$, where $\\lambda_-$ was defined in Eq.  (\\ref{lambd}),  on\nthe ($\\rho_p$, $\\rho_n$) plane for TM1  and TW and  for several values of the\nmagnetic field.\nWe define the magnetic field in units of the critical  field $B^c_e=4.414 \\times 10^{13}$~G, so that  $B=B^* \\, B^c_e$. For a field with the intensity $B^c_e$ the electron cyclotron energy is equal to the electron mass. \n\n We present the numerical results both not including and including the contribution of the anomalous magnetic moment (AMM).  In all figures where both cases are considered we \nshow on the left panel the results without the magnetic field   and on the right panel the results\nincluding  the AMM. \n\nThe magnetic\nfield has a strong effect not only on the size, giving rise to larger\ninstability regions, but also on the shape of the\ninstability zones which is not symmetric with respect to the $\\rho_p=\\rho_n$ line, contrary to $np$ matter for $B=0$. \nThe increase of the instability region is due to \nLandau quantization which softens the EOS. \n For magnetic fields $B^{*}>2\\times10^{5}$, the\nprotons are totally polarized for all the values of the densities\nconsidered and the size of the spinodal zone is larger than\nthe one obtained for $B=0$.\nIncluding AMM decreases the spinodal region with respect to the results\nwithout AMM for all the values of the magnetic field considered.\nThis behavior is explained by the extra stiffness that  \nthe inclusion of AMM brings  into  the system due to neutron spin polarization.\n\\bigskip\n\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure3.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus $\\rho_n$\nfor TW at $T = 0 \\hbox{ MeV}$ and for severals values of magnetic fields\na) without  and b) with  AMM. For $B^*=10^5$ and $2\\times 10^5$ the spinodal section has respectively three and two separated parts.}\n\\label{spztwnp}\n\\end{figure}\n\nThe effect of the Landau quantization on the spinodal section  is explicitly seen\n in the spinodal for $B^{*}=10^{5}$ for TM1 [dashed line\nFig. \\ref{spztm1npn} a)]. The\nspinodal region consists of two separate zones each one corresponding to one\nLandau level, the one corresponding to the first Landau level extends to\nlarger neutron densities than the one corresponding to the second level.\nIn order to understand this effect,  we plot in Fig. \\ref{chem2} the proton chemical potential for $B^*=0$, \n$B^{*}=10^{5}$ and $B^{*}=3\\times 10^{5}$. We have  identified the\ninstable regions with thick lines. It is seen that for the larger\nfield the proton chemical potential changes smoothly  because for all\nthe densities shown only the first Landau level (LL) is occupied. At\nlow densities the chemical potential decreases with density and only\nabove 0.025 fm$^{-3}$ it starts increasing with density. For\n$B^{*}=10^{5}$ the proton chemical potential shows a cusp\ncorresponding to the end of the first LL and beginning of the second LL. The unstable regions, in this case, correspond to the beginning of each LL when the slope of the chemical potential is smaller. The smaller the magnetic field the larger the number of LL occupied at subsaturation densities and the larger of independent sections which make up the whole spinodal section. For reference we include the chemical potential at $B=0$: it increases smoothly with density with a quite constant slope.\n\n\nIf the AMM is included the instabilities regions are smaller, as discussed above.\nThe structure (bump) appearing at $\\rho_n\\sim 0.05$ fm$^{-3}$ is due to the neutron\npolarization: below that value of the density the neutrons are totally polarized. \n\n\nFor the TW model with $B^{*}=2 \\times 10^{5}$  and  $10^{5}$, Fig. \\ref{spztwnp},  we also get a \nspinodal region formed by several bands, respectively two and three bands. \nAn interesting feature of this model, is that the band with the largest Landau level may\nextend to larger neutron densities than lower levels. This does not occur for NLWM and \nhas to do with the behavior of the symmetry energy which increases in a smoother \nway for DDRHM than for NLWM above $\\rho=0.1$ fm$^{-3}$. As a result the slope of the chemical potentials is not so large.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure4.eps}\n\\caption{(Color online) The proton chemical potential for $B=0$ and for two magnetic field  intensities, $B^*=10^5$ and  $B^*=2\\times 10^5$, obtained within the TM1\nmodel for the neutron density $\\rho_n=0.05$ fm$^{-3}$ and excluding the\nAMM. The thick lines represent the regions of instability.}\n\\label{chem2}\n\\end{figure}\n \nAnother feature of the spinodal regions with magnetic field and without AMM, that is\npresent in both models we have studied, is the extension of the spinodal for\nzero proton fraction: in the absence of the magnetic field there is no\ninstability but the inclusion of the magnetic field changes this behavior: the instability region \nat $\\rho_p=0$ extends until a finite  $\\rho_n$ value, model dependent, but independent of\n$B$. This value is $\\sim 0.15$ fm$^{-3}$ for TW and  $\\sim 0.186$ fm$^{-3}$ for TM1. \n\nTo understand this behavior seen at zero proton fraction, we consider the TM1 model.  For $\\rho_p=0$, we obtain the\ncorresponding finite value of $\\displaystyle\\rho_n=\\frac{k^{n 3}_ {F}}{3\\pi^2}$ from the Fermi neutron momenta $k^{n}_ {F}$ solution of the equation $\\hbox{Det}({\\cal F})=0$ with $\\rho_p=0$. Explicitly, the latter equation can be written as follows \n\n\\begin{eqnarray}\n\\left({\\cal A}_{+}-\\frac{\\cal C}{m^{* 2}} -{\\cal D} \\right) \\left(\\frac{\\pi^{2}}{E^{n}_{F} k^{n}_{F}}+{\\cal A}_{+}-\\frac{\\cal C}{E^{n 2}_{F}}-{\\cal D} \\right) -\\left({\\cal A}_{-}-\\frac{\\cal C}{m^{*} E^{n }_{F}}-{\\cal D} \\right)^{2} = 0\n\\end{eqnarray}\nwhere $\\displaystyle{\\cal A}_{\\pm}=\\left(\\frac{g_{\\omega}}{m_{\\omega}}\\right)^{2}\n\\pm\\frac{1}{4}\\left(\\frac{g_{\\rho}}{m_{\\rho}}\\right)^{2}$, $E^{n}_{F}=\\sqrt{k^{n 2}_ {F}+m^{* 2}}$ , and $\\displaystyle {\\cal C}=\\left(\\frac{g_\\sigma}{m_{\\sigma}}\\right)^2 \\frac{m^{* 2}}{\\cal K}$ with,\n\\begin{eqnarray}\n{\\cal K}=1+\\left(\\frac{g_{\\sigma}}{m_{\\sigma}}\\right)^{2}\\left[2 b (g_{\\sigma}\\sigma)+3c(g_{\\sigma}\\sigma)^2+\\frac{1}{2 \\pi^ 2 E^{n}_{F}}\\left(k^{n 3}_ {F}+3 m^{* 2} k^{n}_ {F}-3 m^{* 2} E^{n}_{F}\\log \\left|\\frac{k^{n}_ {F}+E^{n}_{F}}{m^{*}}\\right|\\right) \\right] \n\\end{eqnarray}\nand $\\displaystyle{\\cal D}=\\frac{\\frac{1}{2}\\xi\\left( \\frac{g_{\\omega}}{m_{\\omega}}\\right)^{4}\\left(g_{\\omega}\\omega^0 \\right)^2}{1+\\frac{1}{2}\\xi\\left( \\frac{g_{\\omega}}{m_{\\omega}}\\right)^{2}\\left(g_{\\omega}\\omega^0 \\right)^2}.$ \\\\\nThis equation is independent of the magnetic field and, therefore, all the\nspinodal regions for different magnetic fields, without AMM, in\nFigs.~\\ref{spztm1npn} and~\\ref{spztwnp} have\nthe same value of $\\rho_n$ for $\\rho_p=0$. The inclusion of AMM changes this\nfeature: $\\rho_n$ is still finite for $\\rho_p=0$ but its value is $B$ dependent.\n\nFor models with density dependent couplings the spinodal region \nextends to larger  densities for the larger proton densities\nwhen compared with NLWM.\nThis is due to the behavior of the symmetry energy: while for\nNLWM the symmetry energy increases quite steeply for densities above\nsaturation densities, DDRH models have a much smoother behavior and the\nsymmetry energy of these models take much smaller values than NLWM for\ndensities above $\\rho=0.15$ fm$^{-3}$. \n\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure5.eps}\n\\caption{(Color online) Proton and neutron chemical potentials  as function of\nthe proton density for $\\rho_n = 0.02$  $\\hbox{ fm}^{-3}$ and for $B^*=10^5$\na) without and b) with  AMM for TM1 (dashed line)  and TW (full line).\nThe thick segments of each curve represent the regions of instability.}\n\\label{chem}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure6.eps}\n\\caption{(Color online) Spinodal section in the  $\\rho_p$,  $\\rho_n$ plane for\nTM1 at $T = 0 \\hbox{ MeV}$ and for severals values of the magnetic field without\nAMM. For each value of the magnetic field, it also plotted the EOS of\nstellar matter in $\\beta$-equilibrium  (thin lines). The crossing of the EOS\nwith the respective spinodal, large dot in each spinodal, represent the\ntransition density to continuous matter. }\n\\label{spztm1npd}\n\\end{figure}\n\n\n\n\n In Fig. \\ref{chem} we plot $\\mu_p$ and  $\\mu_n$ for $B^*=10^5$ with\nthe models TM1 (dashed lines) and TW (full lines). The thick segments of the curves lines identify\nthe instability regions defined by the two (TM1) and three (TW) bands which form the spinodal.  These curves are obtained for a fixed\nneutron density, $\\rho_n=0.02$ fm$^{-3}$. It is seen that above  $\\rho_p=0.1$\nfm$^{-3}$ the proton chemical potential within TW is much softer and this\nseems to be the reason for the appearance of the third band in this model. \n\n\nFor each model TM1 and TW we  identified the crossing density,\nand corresponding pressure, of the EOS of $\\beta$-equilibrium stellar\nmatter and the corresponding spinodal for each value of the magnetic\nfield considered.  The EOS of state was obtained considering neutrons,\nprotons and electrons in $\\beta$-equilibrium. In order to illustrate\nwhat was done we represent in Fig.~\\ref{spztm1npd} the spinodal\nsections obtained within TM1 for $B^*=0, 10^5$ and $5\\times10^5$\nrespectively by full, dashed and dotted thick lines. We include in the\nsame figure, using thin lines with the same type of curve for each $B$ value, the corresponding EOS of  $\\beta$-equilibrium stellar matter. The crossing spinodal-EOS is identified by a big dot. Both the spinodal and the EOS are plotted in the $\\rho_p, \\rho_n$ plane. \n\nThe crossing density of the EOS with the thermodynamical spinodal gives a \nprevision of the transition density \\cite{bao, pasta1} to an homogeneous phase, \nand is always larger than the one obtained from the crossing of the EOS with the\ndynamical spinodal for $npe$ matter, which includes the Coulomb\ninteraction. In \\cite{link99} the authors have shown how the transition density\nand respective pressure were related to the fraction of the star's\nmoment of inertia contained in the solid crust, and obtained\n a relation between the radius and mass of compact stars. \n\nIn Tables~\\ref{table4} and \\ref{table5} the values of the crossing density and\nrespective pressure are given for stellar matter under different magnetic\nfields, respectively without and with the AMM.\nThe values of the crossing density for $B=0$, 0.069 fm$^{-3}$ for TM1 and  0.085 fm$^{-3}$  for TW,  can be compared with the\ncorresponding ones obtained from the crossing of the dynamical spinodal with\nthe EOS Ref. \\cite{camille08,pasta1}, respectively 0.06 fm$^{-3}$ for TM1  and 0.075 fm$^{-3}$ for TW. As expected they are a\nbit larger, with TW model having a larger crossing density than the other.\nThe effect of the magnetic field is to increase the values of the crossing\ndensity: at $B^*=10^5$ both models have similar transition densities of the\norder of $\\sim 0.1$ fm$^{-3}$ corresponding to a much larger pressure for \nTM1 than TW. For  $B^*=3\\times 10^5$ the transition densities increase to\n$\\rho\\sim 0.14-0.15$ fm$^{-3}$.\n\n\\begin{table*}[t]\n\\caption{\nPredicted density, proton fraction and pressure at the inner edge of the\ncrust of a compact star at zero temperature, as defined by the crossing\nbetween the thermodynamical instability region of $np$ matter and the\n$\\beta$-equilibrium EOS for homogeneous, neutrino-free stellar\nmatter in the $\\rho_p,\\rho_n$ plane. The AMM is not included.} \n\\label{table4}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n$B^{*}$  & Models & $\\rho^{\\hbox{cross}}_b (\\hbox{ fm}^{-3}) $ & $Y_p$ & $P_{m}(\\hbox{ MeV}\\hbox{ fm}^{-3}) $ \\\\\n\\hline\n  $0$            & TM1 & 0.069509 & 0.024713 & 0.50288 \\\\\n                                & TW  & 0.084955 & 0.036690  & 0.52246 \\\\\n$10^{5}$         & TM1  & 0.097030 & 0.14645 & 0.95944  \\\\\n                                       &  TW  & 0.10099   & 0.14641   & 0.67321  \\\\\n$2\\times10^{5}$    & TM1 & 0.12266 & 0.24283 & 1.4008  \\\\\n                                             & TW  & 0.12786  & 0.23599 & 1.0156  \\\\\n$3\\times10^{5}$    & TM1 & 0.14085 & 0.31304 & 1.5944 \\\\\n                                             & TW  & 0.15159 & 0.30219 & 1.3795 \\\\\n$5\\times10^{5}$    & TM1 & 0.16783 & 0.40921  & 1.6324 \\\\\n                                             & TW &0.19784 & 0.40194 & 2.3310 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nIn Table~\\ref{table5} we show the same data given in Table~\\ref{table4} but\nincluding the AMM in the calculation. The conclusions are similar: the\ntransition density increases with the increase of the magnitude of the magnetic field but not\nso fast. However, the corresponding pressures are larger than before. We conclude that the existence of a strong magnetic field at the crust gives rise to a larger crust.\n\n\\begin{table*}[t]\n\\caption{\nPredicted density, proton fraction and pressure at the inner edge of the crust of a compact star at zero temperature, as defined by the crossing between the thermodynamical instability region of $np$ matter and the $\\beta$-equilibrium condition for homogeneous, neutrino-free stellar matter. The case where the AMM is  included.}  \n\\begin{ruledtabular}\n\\label{table5}\n\\begin{tabular}{c c c c c}\n$B^{*}$ & Models & $\\rho^{\\hbox{cross}}_b (\\hbox{ fm}^{-3}) $ & $Y_p$ & $P_{m}(\\hbox{ MeV}\\hbox{ fm}^{-3}) $ \\\\\n\\hline\n           $10^{5}$             & TM1  & 0.086942 & 0.17670  & 1.3801  \\\\\n                                           &TW  & 0.091391 & 0.17829 & 1.2809 \\\\\n $2\\times10^{5}$ &TM1 & 0.096337 & 0.29468 & 1.5373 \\\\\n                                           &TW & 0.092438 & 0.30512 & 1.3549  \\\\\n$3\\times10^{5}$ &TM1 & 0.11046 & 0.37135 & 1.7091 \\\\\n                                           &TW  & 0.11251 & 0.38035 & 1.8047 \\\\\n $5\\times10^{5}$ & TM1  & 0.12822 & 0.47093 & 1.6616 \\\\\n                                            & TW    & 0.14880 & 0.48803 & 2.7717 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\subsection{Spinodal section $npe$ matter}\n\nWe have studied the effect of the magnetic field on the instability region of\n$np$ matter in the previous section.\nFor $npe$ matter without magnetic field, NLWM models still present a small\nthermodynamical instability region but for DDRHM models there is no instability region \\cite{abmp06}. The\nincompressibility of the free electron gas is so high that the spinodal\ndisappears or almost disappears.     \n\n In Fig. \\ref{spin_npe_nlw} the spinodals for $npe$ matter are shown\n for TM1 and different magnetic fields. \nIn fact,  although including electrons, the instability region can become almost as\nlarge as the $B=0$ $np$-spinodal.\nThis is due both to the Landau quantization of the orbital motion of protons\nand electrons: the incompressibility of the electron gas is smaller than the\none of a magnetic free electron gas.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure7.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus $\\rho_n$ \nfor TM1 for $npe$ neutral matter at $T = 0 \\hbox{ MeV}$ and for severals values of magnetic \nfields (a) without  and (b) with  AMM.}\n\\label{spin_npe_nlw}\n\\end{figure}\n\nFor TW, contrary to the $B=0$ case,  the inclusion of the magnetic field gives\nrise to a spinodal region as seen in  Fig. \\ref{spin_npe_tw}.\nThe behavior of this model with the magnetic field  is similar to\nTM1. We also point out that the\n inclusion of the AMM has a strong effect on the spinodal part corresponding to the first LL: it is drastically reduced or even disappears.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure8.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus\n  $\\rho_n$ for TW for $npe$ neutral matter at $T = 0  \\hbox{ MeV}$ and\n  for severals values of magnetic fields (a) without  and  (b) with  AMM.}\n\\label{spin_npe_tw}\n\\end{figure}\n\n\n\\subsection{Direction of instability}\n\nThe eigenvector associated with the negative eigenvalue of the free energy curvature\nmatrix  defines the direction of the instability and tells us how does the system\nseparate into a dense liquid and a gas phase. It was shown in \\cite{abmp06,camille08} that\nin the absence of the magnetic field the direction of instability favors the\nreduction of the isospin asymmetry of the dense clusters of the system, and\nincreases the isospin asymmetry of the gas surrounding the clusters, the so\ncalled distillation effect. This effect is represented in Fig.~\\ref{spdtm2}\nwhere it is seen that for  the $B=0$ curve (thick full line)  the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\nis larger than $\\rho_p\/\\rho_n$ below $y_p=0.5$ and the other way round above.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure9.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ plotted as a\nfunction of the proton fraction with $T = 0  \\hbox{ MeV}$ and $\\rho= 0.06\n\\hbox{ fm}^{-3}$  for the TM1 model and several values of the magnetic\nfields  without  AMM. The fraction $\\rho_p\/\\rho_n$ is given by the thin dotted line.}\n\\label{spdtm2}\n\\end{figure}\n\nIn Fig. \\ref{spdtm2} we plot, for TM1,  the ratio  $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$  for $\\rho=0.06$ fm$^{-3}$ as a function of the proton fraction. Several results for different values of the magnetic field are shown by the thick lines. The thin lines represent the ratio $\\rho_p\/\\rho_n$ for reference and $y_p=0.5$ points corresponding to symmetric matter, as well as   $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}=1$, which is the ratio of density fluctuations for symmetric matter with no field. For the largest field considered the spinodal region contains a single\nLandau level and the curve varies smoothly starting at  $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}\\sim1.5$. We\npoint out the very large value of this fraction, always above 1, for\n$y_p<0.5$.  The magnetic field favors an increase of the proton fraction\nquite above the symmetric matter value. For $B^*=10^5$ the spinodal has two bands, see Fig.~\\ref{spztm1npn} and \\ref{spztwnp}\n, corresponding to the occupation of the\nfirst two Landau levels. The transition from one to the other is clearly seen\nwith  a large discontinuity of $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ at $y_p\\sim 0.7$. Above this $y_p$ value the\ncurve behaves like the previous ones. However for $y_p<0.7$ the behavior is\nquite different: the curve decreases from the value at $y_p$=0, which is\nindependent of the magnitude of the  magnetic field, to a value much smaller\nthan the corresponding value of the  fraction $\\rho_p\/\\rho_n$. The fluctuations will not drive the system\nout of the first Landau level and therefore the larger the proton fraction, the\ncloser the system comes to the top of the band and the smaller are the allowed\nproton fluctuations. For $y_p>0.7$ or for the larger magnetic fields the\nLandau levels are only partially filled and the fluctuations will never drive\nthe system out of the corresponding Landau level. \n\n\\begin{figure}[htb]\n\\begin{tabular}{c}\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure10.eps}\\vspace{1.5cm}\\\\\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure11.eps}\n\\end{tabular}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ plotted as a\nfunction of the proton fraction with $T = 0  \\hbox{ MeV}$ and $\\rho= 0.06\n\\hbox{ fm}^{-3}$  for the TM1 (top) and TW (bottom) and for severals values of the magnetic\nfields  without (a) and (c) and with (b) and (d) AMM. The fraction $\\rho_p\/\\rho_n$ is given by the thin dotted line.}\n\\label{spdtm1}\n\\end{figure}\n\nSimilar features are obtained for TW and\/or including the AMM. In Fig. \\ref{spdtm1}\n  we show, respectively for TM1 (top)  and TW (bottom), the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ as a function of $y_p$ for a fixed baryonic\ndensity, $\\rho= 0.06 \\hbox{ fm}^{-3}$, chosen inside the instability region.\nFor $y_p>0.5$, TM1 and TW behave in a similar way, while below this value the main \ndifference is the smaller $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ for TW, corresponding to a smaller distillation effect. This behavior is also present for $B=0$ and it was shown that this was due to the presence of the rearrangement term. The inclusion of the AMM favors larger proton fractions because neutron polarization stiffens the EOS.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure12.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\n  plotted as a function of the density with $T = 0  \\hbox{ MeV}$ and\n  $y_p = 0.2$ with (thin lines) and without (thick lines) electrons\n  for the TM1 and TW models and for severals values of the magnetic\n  fields (a) and (c) without  and (b) and (d) with AMM.}\n\\label{drpdrnyp02}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure13.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\n  plotted as a function of the density with $T = 0  \\hbox{ MeV}$ and\n  $y_p = 0.4$ with (thin lines) and without (thick lines) electrons\n  for the TM1 and TW models and for severals values of the magnetic\n  fields a)  without  and b)  with  AMM.}\n\\label{drpdrnyp04}\n\\end{figure}\n\nIn Figs.~\\ref{drpdrnyp02} and~\\ref{drpdrnyp04} we represent the fraction\n$\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ as a function of density\nrespectively for two\nvalues of $y_p$, 0.2 and 0.4, for $np$ matter (thick lines) and $npe$ matter\n(thin lines). We consider TM1 and TW.\n Both models have\na very similar behavior for finite values of $B$ although for $y_p=0.2$ and $B=0$ they differ: for\nTM1, $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$   increases with density while,\nfor TW, this fraction decreases for $\\rho>0.02$ fm$^{-3}$. This effect is not\nso strong for $y_p=0.4$ and for $npe$ matter the fraction is always quite small\ndue to  the presence of electrons which prevents large proton variations.\n\nFor $B=10^5$,  $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ decreases with\ndensity while for $B=5\\times 10^5$ the opposite occurs. In both cases only the first\nLandau level is occupied, however for the lower field the first Landau level is\nalmost full and the density\nfluctuations will occur in such a way that the system stays in the same Landau\nlevel: the larger the total density the smaller the fluctuations. For the\nlarger field the first Landau level is only partially filled, far the top of the band. For the same nuclear density, the larger the proton fraction\nthe lower the system energy and therefore the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ increases with density.\n\n\nIn Fig.~\\ref{drpdrnyp04} we give the same information with $y_p=0.4$. While for\n$B^* =5\\times 10^5$ for the range of densities considered, matter occupies only\none Landau level, for $B^* =10^5$ we may have two (TM1) or three (TW) Landau\nlevels, see Figs.~\\ref{spztm1npn} and~\\ref{spztwnp}.  This explains the\ndiscontinuities occurring for $\\sim 0.1$ fm$^{-3}$. For the highest magnetic\nfield only one Landau level partially filled comes into play and\ntherefore the\nfraction $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ \nincreases with density because that is favored energetically. For $B^* =10^5$\nthe presence of almost filled Landau levels prevents the existence of large\nproton density fluctuations. \n\n\n\\section{Conclusions and outlook}\n\nIn the present work we have studied the instabilities of $np$ matter and $npe$ neutral matter under\nvery strong magnetic fields. The fields considered are much stronger than the\nstrongest field measured until now at the surface of a magnetar which is $B^*\\sim 10^2$\nfor SGR 1806-20 \\cite{sgr}. However, it is expected that  fields in the interior of neutrons\nstars will be much larger. The present work shows how fields of the order\nof $B=5\\times 10^{18}$ G could affect the inner crust of a compact star.\n\nWe have considered  two relativistic nuclear models: one NLW model (TM1)  and one\nDDRH model (TW). For both models, we have determined the spinodal surface, from the curvature\nmatrix of the free energy, for different magnitudes of the magnetic field. It\nwas shown that the instability region could be divided into several bands\naccording to the magnitude of the magnetic field and the number of  the Landau\nlevels occupied.  The presence of the magnetic field will generally  increase the\ninstability region. Making a crude estimation of the transition density at the\ninner crust of a compact star under a strong magnetic field from the crossing\nof the EOS of $\\beta$-equilibrium stellar matter with the thermodynamical spinodal, it was shown that the transition\ndensity and associated pressure increases with the magnetic field. This will\naffect the structure of the star increasing the fraction of mass and of the\nstar' s moment of inertia  concentrated at the crust. These effects will be\nnoticeable if, for densities of the order of 0.1 fm$^{-3}$, the magnetic field\nis  of the order of $B^*=10^4$ or larger.\n\nThe TW model has larger instability regions than  the TM1 model for the larger\nproton densities. A smoother increase of the proton chemical potential for the\nfirst model justifies this behavior. This behavior of the symmetry energy may  even give rise to a larger\nnumber of bands in the spinodal of TW than the spinodal of TM1 for the same magnetic field.\n\nWe have also investigated the direction of instability. It was  shown\nthat if the Landau level is only partially occupied the density fluctuations\nare such that the system evolves for a state with dense clusters very proton\nrich immersed in a proton poor gas. A larger proton fraction is favored\nenergetically due to the degeneracy of the Landau levels. If on the other hand,\nwe study the fluctuation of particles occupying an almost complete Landau level, proton fluctuations\ncannot be so large and it may even occur an anti-distillation effect with a\ndecrease of the proton fraction of the dense clusters. This is due to the fact\nthat these fluctuations will keep the system in the same Landau level. \n   \n\n\\begin{acknowledgments}\n\nThis work was partially supported by FEDER and FCT (Portugal)\nunder the grant SFRH\/BPD\/14831\/2003, and projects  PTDC\/FP\/64707\/2006 and  POCI\/FP\/81923\/2007. \n\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nA highlight in the theory of Drinfeld-Jimbo quantum groups is the construction of canonical bases by Lusztig and Kashiwara; cf.  \\cite{Lu93}. Let $(\\mbf U, {\\mbf U}^\\imath)$ be a quantum symmetric pair of finite type \\cite{Le99} (also cf. \\cite{BK15}).  A general theory of ($\\imath$-)canonical bases arising from quantum symmetric pairs of finite type was developed recently by Bao and the first author in \\cite{BW16}, where $\\imath$-canonical bases on based $\\mbf U$-modules and on the modified form of ${\\mbf U}^\\imath$ were constructed.\n\nOne notable feature of the definition of quantum symmetric pairs is its dependence on parameters; see \\cite{Le99, BK15, BW16} where various conditions on parameters are imposed at different levels of generalities for various constructions. In \\cite{BWW18} it is shown that the unequal parameters of type B Hecke algebras correspond under the $\\imath$-Schur duality to certain specializations of parameters in the type AIII\/AIV quantum symmetric pairs. \n\n\n\\vspace{3mm}\n\nFor the remainder of the paper we let $\\mbf U$ be the quantum group of $\\mathfrak{sl}_2$ over $\\mathbb Q(q)$ with generators $E, F, K^{\\pm 1}$. Let ${\\mbf U}^\\imath$ be the $\\mathbb Q(q)$-subalgebra of $\\mbf U$ (with a parameter $\\kappa$), which is a polynomial algebra in  $t$, where \n\\[\nt = F+ q^{-1}EK^{-1}  +\\kappa K^{-1}. \n\\]\nThen ${\\mbf U}^\\imath$ is a coideal subalgebra, and $(\\mbf U, {\\mbf U}^\\imath)$ is an example of quantum symmetric pairs; cf. \\cite{Ko93}. For the consideration of $\\imath$-canonical bases (which will be referred to as $\\imath$-divided powers from now on) the parameter $\\kappa$ is taken to be a bar invariant element in $\\mathcal A =\\mathbb Z[q,q^{-1}]$; cf. \\cite{BW16}. \n\nIn contrast to the usual quantum $\\mathfrak{sl}_2$ case where the divided powers are simple to define, finding explicit formulae for the $\\imath$-divided powers is a highly nontrivial problem. Closed formulae for the $\\imath$-divided powers in ${\\mbf U}^\\imath$ were known earlier only in two distinguished cases when $\\kappa=0$ or $1$ (cf. \\cite{BeW18}); this verified a conjecture in \\cite{BW13} when $\\kappa=1$.\n\n\nThe goal of this paper is to establish closed formulae  for the $\\imath$-divided powers in ${\\mbf U}^\\imath$  as polynomials in $t$ and also viewed as elements in $\\mbf U$ (via the embedding of ${\\mbf U}^\\imath$ to $\\mbf U$), when $\\kappa$ is an arbitrary $q$-integer which is clearly bar invariant. For arbitrary $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, we are able to present a closed formula only for the {\\em second} $\\imath$-divided power (note the first $\\imath$-divided power is simply $t$ itself). \n\n\n\n\\vspace{3mm}\n\nIn contrast to the quantum group setting, there are {\\em two} $\\mathcal A$-forms, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$, for ${\\mbf U}^\\imath$ corresponding to the parities $\\{\\rm ev, \\rm odd\\}$ of highest weights of finite-dimensional simple $\\mbf U$-modules \\cite{BW13, BW16}. As a very special case of a main theorem in \\cite{BW16}, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ (and respectively, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$) admits  $\\imath$-canonical bases ($= \\imath$-divided powers) for an arbitrary parameter $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, which are invariant with respect to a bar map (which fixes $t$ and hence is not a restriction of Lusztig's bar map on $\\mbf U$ to ${\\mbf U}^\\imath$) and satisfy an asympototic compatibility with the $\\imath$-canonical bases on finite-dimensional simple $\\mbf U$-modules. \n\nComputations by hand and by Mathematica have\nled us to make an ansatz for the formulae for the $\\imath$-divided powers as polynomials in $t$ when $\\kappa$ is an arbitrary $q$-integer. In further discussions we need to separate the cases when $\\kappa$ is an even or odd $q$-integer, and let us restrict ourselves to the case for the $q$-integer $\\kappa$ being even in the remainder of the Introduction. Our ansatz is that the $\\imath$-divided powers $\\dvev{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and $\\dv{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ are given as follows: for $a\\in \\mathbb N$, \n\\begin{align}\n\\label{def:idp:evevKa}\n\\dvev{n} = \n\\begin{cases}\n\\frac{t}{[2a]!}  (t  - [-2a+2])(t -[-2a+4]) \\cdots (t -[2a-4]) (t  - [2a-2]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{1}{[2a+1]!} (t  - [-2a])(t -[-2a+2]) \\cdots (t -[2a-2]) (t  - [2a]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align}\n\\begin{align}\n\\label{def:dv:evoddK}\n{\\small \\dv{n} = \n\\begin{cases}\n\\frac{1}{[2a]!}   (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{t}{[2a+1]!}     (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), &\\text{if } n=2a+1.\n\\end{cases}\n}\n\\end{align}\nThat is, the $\\imath$-divided power formulas as polynomials in $t$ for $\\kappa$ being even $q$-integers are the same as for $\\kappa=0$  \\cite{BW13, BeW18}. \n\nTo verify the above formulae \\eqref{def:idp:evevKa}-\\eqref{def:dv:evoddK} are indeed the $\\imath$-canonical bases as defined and established in \\cite{BW16}, we need to verify 2 properties: (1) these polynomials in $t$ lie in the corresponding $\\mathcal A$-forms of ${\\mbf U}^\\imath$; (2) they satisfy the asymptotic compatibility with $\\imath$-canonical bases on finite-dimensional simple $\\mbf U$-modules. \n\nTo that end, we find the expansions for the polynomials in $t$ defined in \\eqref{def:idp:evevKa}-\\eqref{def:dv:evoddK}  in terms of $E, F, K^{\\pm 1}$ of $\\mbf U$ (via the embedding ${\\mbf U}^\\imath \\to \\mbf U$); they are given by explicit triple sum formulae. Once these explicit formulae are found, they are verified by lengthy inductions. In the formulation of the expansion formulae, a sequence of degree $n$ polynomials $p^{(n)} (x) =p_n(x)\/ [n]!$ which are defined recursively arise naturally; see \\S \\ref{subsec:pn}. The presence of $p^{(n)} (x)$ makes the formulation of the expansion formula and its proof much more difficult than the distinguished cases when $\\kappa=0$ or $1$ treated in \\cite{BeW18}. \n\nThe polynomials  $p^{(n)} (x)$ satisfy a crucial integrality property in the sense that $p^{(n)} (\\kappa)\\in \\mathcal A$ which leads to the integral property (1) above. To prove such an integrality we express these polynomials in terms of another sequence of polynomials (which are a reincarnation of the $\\imath$-divided powers viewed as polynomials) with some $q^2$-binomial coefficients. Note that $p_0(0)=1$ and $p_n(0)=0$ for $n\\ge 1$, our triple sum expansion formula reduces at $\\kappa=0$ to a double sum formula in \\cite{BeW18}. \n\nWhen applying the expansion formulas for the $\\imath$-divided powers to the highest weight vectors of the finite-dimensional simple $\\mbf U$-modules, we establish the asymptotic compatibility property (2) above explicitly. Note this form of compatibility cannot be made as strong as the compatibility between $\\imath$-divided powers on ${\\mbf U}^\\imath$ and simple $\\mbf U$-modules when $\\kappa=0$ or $1$ (as expected in \\cite{BW16}).\n\n\nWith Huanchen Bao's help, we compute the closed formulae for the second $\\imath$-divided power for an arbitrary parameter $\\overline{\\kappa} =\\kappa \\in \\mathcal A$; see Appendix~\\ref{sec:genK}. It is an interesting open problem to find closed formulae for higher $\\imath$-divided power with such an arbitrary parameter $\\kappa$. \n \n %\n\\vspace{3mm}\nThe paper is organized as follows. There are 4 sections depending on the parities of the weights and of the parameter $\\kappa$. \n\nIn Section~\\ref{sec:evevK}, we recall the basics of the quantum symmetric pair $(\\mbf U, {\\mbf U}^\\imath)$. Throughout the section we take $\\kappa$ to be an arbitrary even $q$-integer. We establish a key integral property of the polynomials  $p^{(n)} (x)$, and use it to formulate and establish the expansion formula for the $\\imath$-divided powers $\\dvev{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$. We prove that $\\{\\dvev{n} | n\\ge 0 \\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ by showing $\\dvev{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda)$, for integers $\\lambda \\gg n$. \n\nIn Section~\\ref{sec:oddevK}, letting $\\kappa$ be an arbitrary even $q$-integer, we establish the expansion formula for the $\\imath$-divided powers $\\dv{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$. We prove that $\\{\\dv{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ by showing $\\dv{n}v^+_{2\\la+1} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda+1)$, for integers $\\lambda \\gg n$. \n\nIn Section~\\ref{sec:evoddK}, we take $\\kappa$ to be an arbitrary odd $q$-integer. We establish a key integral property of another sequence of polynomials  $\\mathfrak p^{(n)} (x)$, and use it to establish the expansion formula for the $\\imath$-divided powers $\\dvp{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$. We show that $\\{\\dvp{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and $\\dvp{n}v^+_{2\\la} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda)$, for integers $\\lambda \\gg n$.\n\nIn Section~\\ref{sec:oddoddK}, we take $\\kappa$ to be an arbitrary odd $q$-integer. We establish the expansion formula for the $\\imath$-divided powers $\\dvd{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$. We show that $\\{\\dvd{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ and $\\dvd{n}v^+_{2\\la+1} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda+1)$, for integers $\\lambda \\gg n$.\n\n\nIn Appendix~\\ref{sec:genK}, for arbitrary $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, we present closed formulae for the second $\\imath$-divided powers in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$.\n\n\n\\vspace{.3cm}\n\n{\\bf Acknowledgement.} \nWW thanks Huanchen Bao for his insightful collaboration. The formula in the Appendix for the second $\\imath$-divided power with arbitrary parameter $\\kappa$ (which was obtained with help from Huanchen) was crucial to this project, and to a large extent this paper grows by exploring for what values for the parameter $\\kappa$ reasonable formulae for higher divided powers can be obtained. The research of WW  and the undergraduate research of CB are partially supported by a grant from National Science Foundation. Mathematica was used intensively in this work.\n\n\n\\section{The $\\imath$-divided powers $\\dvev{n}$ for even weights and even $\\kappa$}\n  \\label{sec:evevK}\n\n\\subsection{The quantum $\\mathfrak{sl}_2$}\n\nRecall  the quantum group $\\mbf U=\\mbf U_q(\\mathfrak{sl}_2)$ is the $\\mathbb Q(q)$-algebra generated by $F, E, K, K^{-1}$, subject to the relations:  \n\\[\nKK^{-1}=K^{-1}K=1,\n \\;\n EF -FE =\\frac{K-K^{-1}}{q-q^{-1}}, \n \\;\n K E =q^2 E K, \n \\;\n K F=q^{-2} FK.\n\\]\nThere is an anti-involution $\\varsigma$ of the $\\mathbb Q$-algebra $\\mbf U$:\n\\begin{align}\n \\label{eq:vs}\n\\varsigma: \\mbf U \\longrightarrow \\mbf U,\n\\qquad E\\mapsto E, \\quad F\\mapsto F, \\quad K\\mapsto K, \\quad q\\mapsto q^{-1}.\n\\end{align}\n\n Let $\\mathcal A =\\mathbb Z[q,q^{-1}]$ and $\\kappa \\in \\mathcal A$.  Set\n\\begin{equation}  \\label{eq:Y}\n\\check{E} :=q^{-1}EK^{-1}, \n\\qquad\nh :=\\frac{K^{-2}-1}{q^2-1}.\n\\end{equation} \n\n\n  Define, for $a\\in \\mathbb Z, n\\ge 0$, \n\\begin{equation}  \\label{kbinom}\n\\qbinom{h;a}{n} =\\prod_{i=1}^n \\frac{q^{4a+4i-4} K^{-2} -1}{q^{4i} -1},\n\\qquad \n[h;a]= \\qbinom{h;a}{1}.\n\\end{equation}\nThen we have, for $a\\in \\mathbb Z, n\\in \\mathbb N$, \n\\begin{align}\n  \\label{FEk}\n  \\begin{split}\nF \\check{E}  - & q^{-2} \\check{E}  F =h,\n\\\\\n\\qbinom{h;a}{n} F = F \\qbinom{h;a+1}{n}, &\n\\qquad \\qbinom{h;a}{n} \\check{E}  =\\check{E}  \\qbinom{h;a-1}{n}.\n\\end{split}\n\\end{align}\nIt follows by definition that\n\\begin{equation}  \\label{kbinom2}\n\\qbinom{h;a}{n} =\\qbinom{h;1+a}{n}\n -q^{4a}K^{-2} \\qbinom{h;1+a}{n-1}.\n\\end{equation}\n\nThe following formula holds for $n\\ge 0$:  \n\\begin{align}   \\label{FYn}\nF \\check{E}^{(n)} &=q^{-2n} \\check{E}^{(n)} F +  \\check{E}^{(n-1)} \\frac{q^{3-3n} K^{-2} - q^{1-n} }{q^2-1}.\n\\end{align} \n\n\n\n\nLet \n\\[\n\\check{E}^{(n)} =\\check{E}^n\/[n]!, \\quad\nF^{(n)} =F^n\/[n]!,\\quad \\text{ for } n\\ge 1.\n\\] \nThen $\\check{E}^{(n)}= q^{-n^2} E^{(n)} K^{-n}$. It is understood that $\\check{E}^{(n)}=0$ for $n<0$ and $\\check{E}^{(0)}=1$.\n\n\n\\subsection{The coideal subalgebra ${\\mbf U}^\\imath$}\n\nRecall $\\check{E}$ from \\eqref{eq:Y}. Let \n\\begin{align}\n  \\label{eq:t}\nt &=F + \\check{E} +\\kappa K^{-1}. \n\\end{align}\nDenote by ${\\mbf U}^\\imath$ the $\\mathbb Q(q)$-subalgebra of $\\mbf U$ generated by $t$. Then ${\\mbf U}^\\imath$ is a coideal subalgebra of $\\mbf U$ and $(\\mbf U, {\\mbf U}^\\imath)$ forms a quantum symmetric pair \\cite{Ko93, Le99}; also cf. \\cite{BW16}.\n\nDenote by $\\dot{\\mbf U}$ the modified quantum group of $\\mathfrak{sl}_2$ \\cite{Lu93}, which is the $\\mathbb Q(q)$-algebra generated by $E\\mathbf 1_\\lambda, F\\mathbf 1_\\lambda$ and the  idempotents $\\mathbf 1_\\lambda$, for $\\lambda\\in \\mathbb Z$. Let ${}_\\A{\\dot{\\mbf U}}$ (respectively, ${}_\\A{\\dot{\\mbf U}}_{\\rm{ev}}$, ${}_\\A{\\dot{\\mbf U}}_{\\text{odd}}$) be the $\\mathcal A$-subalgebra of $\\dot{\\mbf U}$  generated by $E^{(n)} \\mathbf 1_\\lambda, F^{(n)} \\mathbf 1_\\lambda, \\mathbf 1_\\lambda$, for all $n\\ge 0$ and for $\\lambda\\in \\mathbb Z$ (respectively, for $\\lambda$ even, for $\\lambda$ odd). Note ${}_\\A{\\dot{\\mbf U}} ={}_\\A{\\dot{\\mbf U}}_{\\rm{ev}} \\oplus {}_\\A{\\dot{\\mbf U}}_{\\text{odd}}$.\nThere is a natural left action of $\\mbf U$ (and hence ${\\mbf U}^\\imath$) on $\\dot{\\mbf U}$ such that $K \\mathbf 1_\\lambda =q^\\lambda \\mathbf 1_\\lambda$. \nFor $\\mu\\in \\mathbb N$, denote by $L(\\mu)$ the finite-dimensional simple $\\mbf U$-module of highest weight $\\mu$, and denote by $L(\\mu)_\\mathcal A$ its Lusztig $\\mathcal A$-form. \n\nFollowing \\cite{BW16}, we introduce the following $\\mathcal A$-subalgebras of ${\\mbf U}^\\imath$, for ${\\rm p}\\in \\{ {\\rm ev}, {\\rm odd} \\}$: \n\\begin{align*}\n{}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}\n&=\\{x\\in {\\mbf U}^\\imath~|~x u \\in {}_\\A{\\dot{\\mbf U}}_{\\rm p}, \\forall u\\in {}_\\A{\\dot{\\mbf U}}_{\\rm p} \\}\n\\\\\n&=\\big \\{x\\in {\\mbf U}^\\imath~|~x v \\in L(\\mu)_\\mathcal A, \\forall v\\in L(\\mu)_\\mathcal A, \\forall {\\rm p}\\equiv\\mu\\pmod{2} \\big \\}. \n\\end{align*}\nBy \\cite{BW16}, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}$ is a free $\\mathcal A$-submodule of ${\\mbf U}^\\imath$ such that ${\\mbf U}^\\imath =\\mathbb Q(q) \\otimes_\\mathcal A {}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}$, for ${\\rm p}\\in \\{ {\\rm ev}, {\\rm odd} \\}$. \n\n\\subsection{Definition of $\\dvev{n}$ for even $\\kappa$}\n\nIn this section~\\ref{sec:evevK} we shall always take $\\kappa$ to be an even $q$-integer, i.e., \n\\begin{align}  \n   \\label{eq:evenK}\n\\kappa & =[2\\ell], \\quad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{align}\nWe shall take the following as a definition of the $\\imath$-divided powers $\\dvev{n}$. \n\n\\begin{definition}\nSet $\\dvev{1} =t = F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dvev{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt}\n\\begin{split}\nt \\cdot \\dvev{2a-1} &=[2a] \\dvev{2a},\n\\\\\nt \\cdot \\dvev{2a} &=  [2a+1] \\dvev{2a+1} +   [2a] \\dvev{2a-1}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \nEquivalently, $\\dvev{n}$ is defined by the closed formula  \\eqref{def:idp:evevKa}. \n\\end{definition}\nThe bar involution $\\psi_\\imath$ on ${\\mbf U}^\\imath$, which fixed $t$ and sends $q\\mapsto q^{-1}$, clearly fixes the above $\\imath$-divided powers. \n\nThe following is a variant of \\cite[Lemma~2.2]{BeW18} (where $\\kappa=0$) with the same proof.\n\n\\begin{lem} \n  \\label{lem:anti}\nThe anti-involution $\\varsigma$ on $\\mbf U$\nsends $\nF\\mapsto F,  \\check{E} \\mapsto \\check{E},  K^{-1} \\mapsto K^{-1},  q \\mapsto q^{-1}$;  in addition, $\\varsigma$ sends \n\\[\nh\\mapsto -q^{2} h, \n\\quad\n\\dvev{n} \\mapsto \\dvev{n}, \n\\quad\n\\qbinom{h;a}{n}\\mapsto (-1)^n q^{2n(n+1)} \\qbinom{h;1-a-n}{n}, \\; \\forall a\\in\\mathbb Z, n\\in\\mathbb N.\n\\] \n\\end{lem}\n\n\\subsection{Polynomials $p_n(x)$ and $p^{(n)}(x)$}\n \\label{subsec:pn}\n \n\\begin{definition}\nFor $n\\in \\mathbb N$, the monic polynomial $p_n(x)$ in $x$ of degree $n$ is defined as\n\\begin{align}\n \\label{def:pn}\np_{n+1} =x p_n  + q^{1-2n} [n] [n-1] p_{n-1}, \\qquad  p_0 =1.\n\\end{align}\n\\end{definition}\nAlso set $p_n =0$ for all $n<0.$\nNote that $p_n$ is an odd polynomial for $n$ odd while it is an even polynomial for $n$ even. These polynomials $p_n$ will appear in the expansion formula for the $\\imath$-divided powers in $\\mbf U$. \n\n\\begin{example}\nHere are $p_n(x)$, for the first few $n\\ge 0$:\n\\begin{align*}\np_0& =1, \\qquad p_1 =x,\n\\qquad\np_2=x^2,\n\\qquad\np_3=x^3 +(q^{-4} +q^{-2})x,\n\\\\\np_4&=x^4 + (q^{-4} +q^{-2})(q^{-4} +q^{-2}+2)x^2,\n\\\\\np_5&=x^5 + (q^{-4} +q^{-2}) (q^{-8}+q^{-6}+3q^{-4} +2q^{-2}+3)x^3\n\\\\\n& \\qquad \\qquad + (q^{-4} +q^{-2})^2 (q^{-8}+q^{-6}+2q^{-4} +q^{-2}+1)x.\n\\end{align*}\n\\end{example}\n\nIntroduce the monic polynomial of degree $n$:\n\\begin{align*}\ng_n(x) =\n\\begin{cases}\n\\prod_{i=0}^{m-1} (x^2-[2i]^2), & \\text{ if } n=2m \\text{ is even};\n\\\\\nx \\prod_{i=1}^{m} (x^2-[2i]^2), & \\text{ if } n=2m+1 \\text{ is odd}.\n\\end{cases}\n\\end{align*}\nDefine \n\\begin{equation}  \\label{eq:divpg}\np^{(n)}(x) =p_n(x)\/[n]!, \\qquad\ng^{(n)}(x) =g_n(x)\/[n]!. \n\\end{equation} \nThe recursion for $p_n$ can be rewritten as\n\\begin{equation}   \\label{eq:f(n)}\n[n+1]p^{(n+1)} =x p^{(n)}  + q^{1-2n} [n-1] p^{(n-1)}.\n\\end{equation}\n\nOur next goal is to show that $p^{(n)}([2\\ell]) \\in \\mathcal A$ for all $\\ell \\in \\mathbb Z$; cf. Proposition~\\ref{prop:fg:ev}. This is carried out by relating $p^{(n)}$ to $g^{(n)}$ for various $n$, and showing that $g^{(n)}([2\\ell]) \\in \\mathcal A$ for all $\\ell \\in \\mathbb Z$.\n\n\n\\begin{lem} \\label{lem:ev:g-integral}\nFor $\\kappa =[2\\ell]$ with $\\ell \\in \\mathbb Z$ (see \\eqref{eq:evenK}), we have $g^{(n)}(\\kappa) \\in \\mathcal A$, for all $n\\ge 0$. \n\\end{lem}\n\n\\begin{proof} \nWe separate the cases for $n=2m+1$ and $n=2m$. \nNoting that\n\\begin{equation}  \\label{eq:A}\n\\kappa -[2i] =[2\\ell]-[2i] = [\\ell-i] (q^{\\ell+i} +q^{-\\ell-i}),\n\\end{equation} \nwe have\n \\begin{align*}\n g^{(2m+1)}(\\kappa) &= \\frac1{[2m+1]!} \\prod_{i=-m}^{m} (\\kappa-[2i])\n =\n \\qbinom{\\ell+m}{2m+1} \\prod_{i=-m}^{m}(q^{\\ell+i} +q^{-\\ell-i} ) \\in \\mathcal A.\n \\end{align*}\n\n\nSimilarly, we have\n \\begin{align*}\n g^{(2m)}(\\kappa) &= \\frac1{[2m]!} \\kappa \\prod_{i=1-m}^{m-1} (\\kappa-[2i])\n \\\\\n &= \\frac1{[2m]!} \\prod_{i=1-m}^{m} (\\kappa-[2i]) + \\frac1{[2m-1]!}  \\prod_{i=1-m}^{m-1} (\\kappa-[2i])\n \\\\\n &=\\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i} +q^{-\\ell-i} ) \n  + g^{(2m-1)}(\\kappa)  \\in \\mathcal A.\n  \\end{align*}\n  The lemma is proved. \n\\end{proof}\n\n\\begin{prop}\n \\label{prop:fg:ev}\n For $m \\in \\mathbb Z_{\\ge 1}$, we have\n\\begin{align*}\np^{(2m)} &= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)},\n\\\\\np^{(2m-1)} &= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}.\n\\end{align*}\nIn particular, we have $p^{(n)} (\\kappa) \\in \\mathcal A$, for all $\\kappa =[2\\ell]$ with $\\ell \\in \\mathbb Z$ and all $n\\ge 0$. \n\\end{prop}\n\n\\begin{proof}  \n It follows from  these formulae for $p^{(n)}$ and Lemma~\\ref{lem:ev:g-integral} that $p^{(n)}([2\\ell]) \\in \\mathcal A$.\n\nIt remains to prove these formulae. Recall\n\\begin{align}  \\label{recursive:f}\n x \\cdot p^{(n)} = [n+1] p^{(n+1)}  -  q^{1-2n} [n-1] p^{(n-1)}, \\quad \\text{ for } n \\ge 1.\n\\end{align}\nAlso recall\n\\begin{align}  \\label{recursive:g}\n\\begin{split}\nx \\cdot g^{(2a-1)} &=[2a] g^{(2a)},\n\\\\\nx \\cdot g^{(2a)} &=  [2a+1] g^{(2a+1)} +   [2a] g^{(2a-1)}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}\n\nWe prove the formulae by induction on $m$, with the base cases for $p^{(1)}$ and $p^{(2)}$ being clear. We separate in two cases. \n\n(1). Let us prove the formula for $p^{(2m+1)}$, assuming the formulae for $p^{(2m-1)}$ and $p^{(2m)}$.\nBy \\eqref{recursive:f}--\\eqref{recursive:g} we have\n\\begin{align*}\n[2m+1]p^{(2m+1)} \n&= x p^{(2m)} +q^{1-4m} [2m-1] p^{(2m-1)}  \\\\\n&= \\sum_{a=0}^{m-1}q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} x \\cdot g^{(2m-2a)} \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(3-2m)a+1-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}\n \\\\\n&= \\sum_{a=0}^{m-1}q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2}\n      \\left( [2m-2a+1]g^{(2m-2a+1)} +[2m-2a]g^{(2m-2a-1)}\n    \\right) \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(3-2m)a+1-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}.\n\\end{align*}\nBy combining the like terms for $g^{(2m-2a-1)}$ above and using the identity\n\\[\nq^{(1-2m)a} [2m-2a] +q^{(3-2m)a+1-4m} [2m-1] =q^{(1-2m)(a+1)} [4m-2a-1], \n\\]\nwe obtain\n\\begin{align*}\n[2m+1]p^{(2m+1)} \n&= \\sum_{a=0}^{m-1}q^{(1-2m)a}  [2m-2a+1] \\qbinom{m-1}{a}_{q^2}  g^{(2m-2a+1)} \n     \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(1-2m)(a+1)} [4m-2a-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}\n  \\\\ %\n&= [2m+1]\\sum_{a=0}^{m}q^{(1-2m)a} \\qbinom{m}{a}_{q^2} g^{(2m-2a+1)}, \n\\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and using\nthe identity\n\\[\n[4m-2a+1] \\qbinom{m-1}{a}_{q^2} +q^{(1-2m)a} [4m-2a+1] \\qbinom{m-1}{a-1}_{q^2}\n=[2m+1] \\qbinom{m}{a}_{q^2}. \n\\]\n\n(2) We shall prove the formula for $p^{(2m+1)}$, assuming the formulae for $p^{(2m-1)}$ and $p^{(2m)}$.\nBy \\eqref{recursive:f}--\\eqref{recursive:g} we have\n\\begin{align*}\n&[2m+2]p^{(2m+2)} \n\\\\\n&= x p^{(2m+1)} +q^{-1-4m} [2m] p^{(2m)}  \\\\\n&= \\sum_{a=0}^{m}q^{(1-2m)a} \\qbinom{m}{a}_{q^2} x \\cdot g^{(2m-2a+1)}\n +\\sum_{a=0}^{m-1} q^{(1-2m)a-1-4m} [2m] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)}\n \\\\\n %\n&= \\sum_{a=0}^{m} q^{(1-2m)a} [2m-2a+2] \\qbinom{m}{a}_{q^2}  g^{(2m-2a+2)}\n +\\sum_{a=0}^{m-1} q^{(1-2m)a-1-4m} [2m] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)}\n \\\\\n &= [2m+2] \\sum_{a=0}^{m} q^{(-1-2m)a} \\qbinom{m}{a}_{q^2} g^{(2m-2a+2)},\n \\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and \n using the following identity\n \\[\n [2m-2a+2] \\qbinom{m}{a}_{q^2} \n+ q^{-2-2m} [2m] \\qbinom{m-1}{a-1}_{q^2}\n=  q^{-2a} [2m+2] \\qbinom{m}{a}_{q^2}. \n \\]\nThe proposition is proved.\n\\end{proof}\n\n\n\\begin{rem}\n\\label{rem:positive-p}\nNote that $p_n(x) \\in \\mathbb N [q^{-1}] [x]$ for all $n$. Therefore, for all non-negative even $q$-integer $\\kappa$, we have $p_n (\\kappa) \\in \\mathbb N [q,q^{-1}]$. \n\\end{rem}\n\n\\subsection{Formulae for $\\dvev{n}$ with even $\\kappa$}\n\nRecall $p^{(n)}(x)$ from \\eqref{def:pn}--\\eqref{eq:divpg}. \n\\begin{thm}   \n \\label{thm:dvev:evKappa}\nAssume $\\kappa$ is an even $q$-integer as in \\eqref{eq:evenK}. Then we have, for $m\\ge 1$,  \n\\begin{align}\n\\dvev{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evev2} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvev{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n p^{(2m-b-2c-1)}(\\kappa) \n\\label{t2m-1:evev2}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\nThe proof of Theorem~\\ref{thm:dvev:evKappa} will be given in \\S\\ref{subsec:proof:dvev:evKappa} below. \nLet us list some equivalent formulae here. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvev:evKappa} into the following.\n\n\\begin{thm}   \n \\label{thm:dvev:evKappa2}\nAssume $\\kappa$ is an even $q$-integer as in \\eqref{eq:evenK}. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvev{2m} &= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{3c+b(2m-b-2c) +a(b-a)}  p^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evevFhE} \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  \\check{E}^{(a)},\n\\notag\n\\\\ %\n\\dvev{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c-1) +a(b-a)}\n p^{(2m-b-2c-1)}(\\kappa) \n\\label{t2m-1:evevFhE}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c+1}  \\qbinom{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:anti}, the anti-involution $\\varsigma$ sends \n$\\qbinom{h;1-m}{c}\\mapsto (-1)^c q^{2c(c+1)} \\qbinom{h;m-c}{c}$, \n$q\\mapsto q^{-1}$, while fixing $\\kappa, K, \\check{E}^{(a)}, F^{(a)}$ and $\\dvev{n}$. The formulae \\eqref{t2m:evevFhE}--\\eqref{t2m-1:evevFhE} now follows by applying $\\varsigma$ to the formulae  \\eqref{t2m:evev2}--\\eqref{t2m-1:evev2} in Theorem~\\ref{thm:dvev:evKappa}. \n\\end{proof}\nFor $c\\in \\mathbb N, m\\in\\mathbb Z$, we define\n\\begin{align}   \\label{cbinom}\n  \\cbinom{m}{c} &= \\prod_{i=1}^c \\frac{q^{4(m+i-1)} -1}{q^{-4i}-1}  \\in \\mathcal A,\n  \\qquad \\cbinom{m}{0}=1.\n\\end{align}\n Note that $\\cbinom{m}{c}$ are $q^2$-binomial coefficients up to some $q$-powers.\nLet $\\lambda, m \\in \\mathbb Z$ with $m\\ge 1$.  We note that\n\\begin{align}\n  \\label{eq:qbinom-hc}\n\\qbinom{h;m-c}{c} \\mathbf 1_{2\\lambda} \n&= \\prod_{i=1}^c \\frac{q^{4(m-\\lambda-c+i-1)} -1}{q^{4i} -1}  \\mathbf 1_{2\\lambda}\n= (-1)^c q^{-2c(c+1)} \\cbinom{m-\\lambda-c}{c}  \\mathbf 1_{2\\lambda}.\n\\end{align}\n\nThe following corollary is immediate from \\eqref{eq:qbinom-hc}, Proposition~\\ref{prop:fg:ev}, and Theorem~\\ref{thm:dvev:evKappa}. \n\\begin{cor}  \nWe have $\\dvev{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$, for all $n$. \n\\end{cor}\n\n\\begin{rem}   \nNote $p_n(0)=0$ for $n>0$, and $p_0(0)=1$. \nThe formulae in Theorems~\\ref{thm:dvev:evKappa} and  \\ref{thm:dvev:evKappa2} in the special case for $\\kappa=0$ recover the formulae in \\cite[Theorem~2.5, Proposition~2.7]{BeW18}.\n\\end{rem}\n\n\n\n\nFor $n\\in \\mathbb N$, we denote\n\\begin{align}  \\label{eq:divB}\nb^{(n)}  = \\sum_{a=0}^n  q^{-a(n-a)}  \\check{E}^{(a)}F^{(n-a)}.\n\\end{align}\n\n\\begin{example}\nThe formulae for $\\dvev{n}$ in Theorem~\\ref{thm:dvev:evKappa}, for $1\\le n \\le 3$, read as follows:\n\\begin{align*}\n\\dvev{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\\n\\dvev{2} \n&= \\check{E}^{(2)} +q^{-1} \\check{E} F + F^{(2)}  + q [h;0] \n + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  + \\kappa^2 \\frac{K^{-2}}{[2]},\n\\\\\n\\dvev{3}  &   = b^{(3)} + q^3[h;-1]F+ q^3 \\check{E} [h;-1]\n\\\\\n&\\qquad \n+ \\big(q^{-2} \\check{E}^{(2)}K^{-1} +q^{-3} \\check{E} K^{-1}F + q^{-2} K^{-1}F^{(2)} \n+ q^3 [h;-1] K^{-1} \\big) \\kappa\n\\\\\n&\\qquad + \\frac{q^{-2}}{[2]} (\\check{E} K^{-2} +K^{-2}F) \\kappa^2\n+\\frac{\\kappa^3 + (q^{-4} +q^{-2})\\kappa}{[3]!} K^{-3}. \n\\end{align*}\n\\end{example}\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}^\\imath_{\\rm{ev}}$ for even $\\kappa$}\n  \\label{sec:iCB:ev}\n  \nDenote by $L(\\mu)$ the \nsimple $\\mbf U$-module of highest weight $\\mu \\in \\mathbb N$, with highest weight vector $v^+_\\mu$.\nThen $L(\\mu)$ admits a canonical basis $\\{ F^{(a)} v^+_\\mu\\mid 0\\le a \\le \\mu\\}$. \nFollowing \\cite{BW13,BW16}, there exists a new bar involution $\\psi_\\imath$ on $L(\\mu)$, which, in our current rank one setting, can be defined simply by requiring $\\dvev{n} v^+_\\mu$ to be $\\psi_\\imath$-invariant for all $n$. \nAs a very special case of the general results in \\cite[Corollary~F]{BW16} (also cf. \\cite{BW13}), we know that\n the $\\imath$-canonical basis $\\{b^{\\mu}_{a} \\}_{0\\le a \\le \\mu}$ of $L(\\mu)$ exists and is characterized by the following 2 properties: \n\n(iCB1) $b^{\\mu}_{a}$ is $\\psi_\\imath$-invariant; \\qquad \n\n(iCB2) $b^{\\mu}_{a} \\in F^{(a)} v^+_\\mu + \\sum_{0\\le r< a} q^{-1} \\mathbb Z[q^{-1}] F^{(r)} v^+_\\mu$. \n\n \n \\begin{thm}  \\label{thm:iCB:ev}\n   $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvev{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element for $L(2\\lambda)$. \n \n \\item\nThe set $\\{\\dvev{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$). \n\\end{enumerate}\n \\end{thm}\n \n \\begin{proof}\nLet $\\lambda, m \\in \\mathbb N$ with $m\\ge 1$.  \nWe compute by Theorem~\\ref{thm:dvev:evKappa2}(1) and \\eqref{eq:qbinom-hc} that \n\\begin{align} \n\\dvev{2m} v^+_{2\\la}  \n&= \\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c q^{3c+b(2m-b-2c)}  p^{(2m-b-2c)}(\\kappa) \nF^{(b)} K^{b-2m+2c} \\qbinom{h;m-c}{c} v^+_{2\\la}  \n\\notag \\\\\n&= \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{(b-2\\lambda)(2m-b-2c)-2c^2+c}  p^{(2m-b-2c)}(\\kappa) \n\\cbinom{m-\\lambda-c}{c}F^{(b)}   v^+_{2\\la}  \n  \\label{dvev:2m-mod}.\n\\end{align}\nSimilarly we compute by Theorem~\\ref{thm:dvev:evKappa2}(2) that \n\\begin{align} \n\\dvev{2m-1} v^+_{2\\la}  \n&= \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c-1)}\n p^{(2m-b-2c-1)}(\\kappa) \nF^{(b)} K^{b-2m+2c+1}  \\qbinom{h;m-c}{c}  v^+_{2\\la} \n\\notag\n\\\\\n&= \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0}  q^{(b-2\\lambda)(2m-b-2c-1)-2c^2-c}\n p^{(2m-b-2c-1)}(\\kappa) \n\\cbinom{m-\\lambda-c}{c} F^{(b)}  v^+_{2\\la} . \n \\label{dvev:2m-1-mod}\n\\end{align}\n\nNote that\n\\begin{equation}\n  \\label{eq:q-1}\n\\cbinom{m-\\lambda-c}{c} \\in \\mathbb N[q^{-1}], \\qquad \\text{ for }\\lambda \\ge m, c\\ge 0.\n\\end{equation}  \n\nLet $n\\in \\mathbb N$. Recall $\\kappa =[2\\ell]$, for $\\ell \\in \\mathbb Z$.  \nOne computes that $\\deg_q p^{(n)} (\\kappa) =(2\\ell-1)n - {n \\choose 2}$ and hence $q^{(b-2\\lambda)(n-b-2c)-2c^2 \\pm c}\n p^{(n-b-2c)}(\\kappa) \\in q^{-1} \\mathbb N[q^{-1}]$ when $\\lambda \\gg n>b+2c$. It follows by \\eqref{dvev:2m-mod}, \\eqref{dvev:2m-1-mod} and \\eqref{eq:q-1} that \n \\begin{align}\n   \\label{eq:lattice}\n \\dvev{n} v^+_{2\\la}   & \\in F^{(n)}  v^+_{2\\la}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align}\n Therefore, the first statement follows by the characterization properties (iCB1)--(iCB2) of the $\\imath$-canonical basis for $L(2\\lambda)$.\n \n The second statement follows now from the definition of  the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda) \\}_{\\lambda\\in \\mathbb N}$ with ${\\mbf U}^\\imath$-homomorphisms $L(2\\lambda+2) \\rightarrow L(2\\lambda)$; cf. \\cite[\\S6]{BW16}; this set $\\{\\dvev{n} \\mid n \\in \\mathbb N \\}$ forms an $\\mathcal A$-basis for  ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$. \nThe theorem is proved. \n \\end{proof}\n \n \\begin{rem}\nIt follows by \\eqref{dvev:2m-mod} that \n\\[\n\\dvev{2m} v^+_{2\\la}    \\in F^{(2m)}  v^+_{2\\la}  +  q^{2m-2\\lambda-1} \\kappa F^{(2m-1)}  v^+_{2\\la}  +\\sum_{k\\ge 2} \\mathcal A \\cdot F^{(2m-k)}  v^+_{2\\la} .\n\\]\nRecall $\\kappa=[2\\ell]$. When $\\ell -1 \\ge \\lambda -m \\ge 0$, we have $q^{2m-2\\lambda-1} \\kappa \\not \\in q^{-1} \\mathbb Z[q^{-1}]$, and therefore, the nonzero element $\\dvev{2m} v^+_{2\\la} $ is not an $\\imath$-canonical basis element in $L(2\\lambda)$. So we cannot expect to have a stronger form of Theorem~\\ref{thm:iCB:ev} as \\cite[Theorem 2.10(1)]{BeW18} in  the case when $\\kappa=0$ (i.e., $\\ell=0$). \n \\end{rem}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvev:evKappa} }\n\\label{subsec:proof:dvev:evKappa}\nWe prove the formulae for $\\dvev{n}$ by induction on $n$, in two steps (1)--(2) below.  The base cases when $n=1,2$ are clear. \n\n(1) We shall prove the formula \\eqref{t2m:evev2} for $\\dvev{2m}$, assuming the formula \\eqref{t2m-1:evev2} for $\\dvev{2m-1}$.\n\nRecall $[2m] \\dvev{2m} = t \\cdot \\dvev{2m-1}$, and $t= F+ \\check{E}  +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dvev{2m-1},\n\\qquad\nII=F \\dvev{2m-1},\n\\qquad\nIII=\\kappa K^{-1} \\dvev{2m-1}.\n\\]\nFirst we have\n\\begin{align*}\nI\n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n    [a+1] p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -(b-1)(2m-b-2c) -(a-1)(b-a)}\n    [a] p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$ (and then adding some zero terms to make the bounds of the summations uniform throughout).\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n   p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n   \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)} =II^1 +II^2,\n\\end{align*}\nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$ and then adding some zero terms, we further have\n\\begin{align*}\nII^1&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c) -a(b-a-1)-2a+2b+4c-4m}\n   [b-a] p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad \\qquad \\cdot  \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2\n &=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c) - (a+1)(b-a)-2}\n  p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad \\qquad \\cdot  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{recursive:f} we also compute\n\\begin{align*}\nIII\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a}\n    \\kappa \\cdot p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a}\n    [2m-b-2c] p^{(2m-b-2c)}(\\kappa)  \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n  \\\\\n & \\quad+\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a -4m}\n    [2m-b-2c] p^{(2m-b-2c)}(\\kappa)  \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dvev{2m-1} = \n \\sum_{0 \\le a \\le b \\le 2m}  \\sum_{c\\ge 0}  p^{(2m-b-2c)}(\\kappa)  \\check{E}^{(a)} H_{a,b,c}  K^{b-2m+2c} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\nH_{a,b,c} & :=\n q^{\\binom{2c}{2}+2c -(b-1)(2m-b-2c) -(a-1)(b-a)} [a]  \\qbinom{h;1-m}{c} \n  \\\\\n& + q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c) -a(b-a-1)-2a+2b+4c-4m} [b-a]  \\qbinom{h;-m}{c} \n\\\\\n& + q^{\\binom{2c}{2}-2c+1 - (b+1)(2m-b-2c) - (a+1)(b-a)}\n  \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1}    \n\\\\\n&+ q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a} [2m-b-2c] \\qbinom{h;1-m}{c}  \n  \\\\\n& - q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a -4m} [2m-b-2c] \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nRecall $[2m] \\dvev{2m} = t \\cdot \\dvev{2m-1}$. To prove the formula \\eqref{t2m:evev2},  by the PBW basis theorem it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H}\nH_{a,b,c} = q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} [2m]  \\qbinom{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a} [a] -[2m] =q^{-a} [a-2m]$, we can combine the RHS with the first summand of LHS \\eqref{eq:H}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)-a}$ on both sides, we see that \\eqref{eq:H} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD}\nA+B+C+D_1 +D_2=0,\n\\end{equation}\nwhere   \n\\begin{align*}\nA &= [a-2m]  \\qbinom{h;1-m}{c},\n  \\\\\nB &=    q^{4c-2m+b} [b-a] \\qbinom{h;-m}{c},\n\\\\\nC   &=  q^{2a-2m+1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1},\n\\\\\nD_1 &=  q^{2c+b-a} [2m-b-2c] \\qbinom{h; 1-m}{c},\n   \\\\\nD_2 &= -  q^{2c-4m+b-a} [2m-b-2c] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD}. Using \\eqref{kbinom2}, we can write $B=B_1+B_2$, where \n\\begin{align*}\nB_1 &=  q^{4c-2m+b} [b-a]  \\qbinom{h; 1-m}{c},\n   \\quad\nB_2 = -  q^{4c-6m+b} [b-a] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{2m-2c-2a-1} [2m-2c-a],\n \\] \nwe rewrite $C=C_1+C_2$, where\n\\begin{align*}\nC_1 &= q^{-2c+2m-a} [2c] \\qbinom{h; 1-m}{c},\n   \\quad\nC_2 =   q^{-2c} [2m-2c-a]  \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\nA+B_1+C_1+D_1 \n&= (q-q^{-1})  [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c},\n\\\\\nC_2 +(B_2+D_2)  \n&= C_2 -  q^{2c-4m} [2m-2c-a] \\qbinom{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (q-q^{-1}) [2c] [2m-2c-a] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $A+B+C+D_1+D_2=0,$ whence \\eqref{ABCD}, completing Step~(1). \n\n\\vspace{3mm}\n\n(2) Assuming the formulae for $\\dvev{n}$ with $n\\le 2m$, we shall now prove the following formula for $\\dvev{2m+1}$ (obtained from \\eqref{t2m-1:evev2} with $m$ replaced by $m+1$):\n\\begin{align}\n\\dvev{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:evev2}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dvev{2m} =[2m+1] \\dvev{2m+1} +[2m] \\dvev{2m-1}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dvev{2m}$ from \\eqref{t2m:evev2}. We shall compute \n\\[\n\\texttt I=\\check{E}  \\dvev{2m},\n\\qquad\n\\texttt{II}=F \\dvev{2m},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvev{2m}, \n\\]\nrespectively. First we have\n\\begin{align*}\n\\texttt{I}\n&= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \n [a+1] p^{(2m-b-2c)}(\\kappa)  \n \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\\\\\n&= \n \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a] p^{(2m-b-2c+1)}(\\kappa) \\\\\n&\\qquad\\qquad\\qquad\\qquad \n\\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$. We also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\\\\n& \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n   \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)} \n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{II}_1&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -a(b-a-1)-2a +2(b+2c-2m-1)} \n[b-a] p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad \\qquad \\cdot \n  \\check{E}^{(a)}   \n      \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2} -(b+1)(2m-b-2c+1) -(a+1)(b-a)} \np^{(2m-b-2c+1)}(\\kappa) \n\\\\\n& \\qquad\\qquad \\qquad \\cdot \n  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n   \\qbinom{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{recursive:f} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a} \n[2m-b-2c+1]  \n\\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n& \\quad + \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a-4m} \n[2m-b-2c+1]  \n\\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dvev{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0} p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} L_{a,b,c} K^{b-2m+2c-1} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\nL_{a,b,c} := &\nq^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -a(b-a-1)-2a +2(b+2c-2m-1)} [b-a]  \\qbinom{h;-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2}-4c+3 -(b+1)(2m-b-2c+1) -(a+1)(b-a)} \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}   \\qbinom{h;1-m}{c-1}   \\\\\n&+ q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a} [2m-b-2c+1]  \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a-4m} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nOn the other hand, using \\eqref{t2m-1:evev2} (with an index shift $c\\to c-1$) and \\eqref{t2m+1:evev2} we write \n\\begin{align*}\n[2m+1] & \\dvev{2m+1} +[2m] \\dvev{2m-1} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+1} \\sum_{c \\geq 0} p^{(2m-b-2c+1)}(\\kappa) \n\\check{E}^{(a)} R_{a,b,c} K^{b-2m+2c-1} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\nR_{a,b,c} :=&\nq^{\\binom{2c}{2}+2c -b(2m-b-2c+1) -a(b-a)}   [2m+1] \\qbinom{h;-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2}-2c+1 -b(2m-b-2c+1) -a(b-a)} [2m]  \n \\qbinom{h;1-m}{c-1}.  \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+1:evev2} for $\\dvev{2m+1}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R}\nL_{a,b,c}=R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}+2bc-b(2m-b+1)-a(b-a)}$ on both sides, we see that the identity \\eqref{L=R} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r}\n \\begin{split}\nq^{2m-2c-a+1} [a] \\qbinom{h;1-m}{c}  \n&+ q^{2c-2m+b-a-1} [b-a] \\qbinom{h;-m}{c}  \n\\\\\n&+ q^{-2m-2c+a+2} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}   \\qbinom{h;1-m}{c-1}    \\\\\n&+ q^{b -2a} [2m-b-2c+1]  \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{b -2a -4m} [2m-b-2c+1] \\qbinom{h;1-m}{c-1} K^{-2}\n\\\\\n=& q^{2c} [2m+1] \\qbinom{h;-m}{c}  \n + q^{-2c+1} [2m] \\qbinom{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ}\nW+X+Y+Z_1+Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\nW&=  [a]  \\qbinom{h;1-m}{c},\n\\\\\nX&= q^{4c-2m+b-1} [-2m+b-a-1] \\qbinom{h;-m}{c},\n\\\\\nY&= \\frac{q^{-4m-a+1}K^{-2}-q^{a+1}}{q^2-1}   \\qbinom{h;1-m}{c-1},  \\\\\nZ_1&=  q^{2c-2m+b -a-1} [2m-b-2c+1] \\qbinom{h;1-m}{c},\n  \\\\\nZ_2&=  q^{2c-6m+b-a-1} [2m-b-2c+1] \\qbinom{h;1-m}{c-1} K^{-2}.  \n\\end{align*}\n\nLet us prove the identity \\eqref{WXYZ}. Using \\eqref{kbinom2}, we can write $X=X_1+X_2$, where \n\\begin{align*}\nX_1 &=q^{4c-2m+b-1} [-2m+b-a-1] \\qbinom{h; 1-m}{c},\n   \\\\\nX_2 &= - q^{4c-6m+b-1} [-2m+b-a-1] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-4m-a+1}K^{-2}-q^{a+1}}{q^2-1}\n= q^{-4c-a+1}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c} [-2c-a],\n \\] \nwe rewrite $Y=Y_1+Y_2$, where\n\\begin{align*}\nY_1 &=q^{-2c-a} [2c] \\qbinom{h; 1-m}{c},\n   \\qquad\nY_2 = q^{-2c} [-2c-a] \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\nW+X_1+Y_1+Z_1 &=\n- (q-q^{-1}) [2c+a] [2c]  \\qbinom{h; 1-m}{c},\n\\\\\n(X_2+Z_2) +Y_2 \n&= q^{2c-4m} [2c+a] \\qbinom{h; 1-m}{c-1} K^{-2} +Y_2\n\\\\\n&= (q-q^{-1}) [2c+a] [2c] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $W+X+Y+Z_1+Z_2=0$, whence \\eqref{WXYZ}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvev:evKappa} is completed. \\qed\n\n\n\n  \n\\section{The $\\imath$-divided powers $\\dv{n}$ for odd weights and even $\\kappa$}\n  \\label{sec:oddevK}\n\nIn this section we shall always take $\\kappa$ to be an even $q$-integer, i.e., \n\\[\n\\kappa =[2\\ell], \\qquad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\]  \n\n\\subsection{Definition of $\\dv{n}$ for even $\\kappa$}\n\n\nWe introduce some notations. Set, for $n\\ge 1, a\\in \\mathbb Z$, \n\\begin{equation}   \\label{brace}\n\\LR{h;a}{0}=1, \n\\qquad\n\\LR{h;a}{n}= \\prod_{i=1}^n \\frac{q^{4a+4i-4} K^{-2}-q^2}{q^{4i}-1}, \n\\qquad \\llbracket h;a\\rrbracket = \\LR{h;a}{1}.\n\\end{equation}\nIt follows from \\eqref{brace} that, for $n\\ge 0$ and $a\\in \\mathbb Z$, \n\\begin{align}\n \\label{eq:commLR}\n \\begin{split}\n\\LR{h;a}{n} F &= F  \\LR{h;a+1}{n},\n\\qquad\n\\LR{h;a}{n} \\check{E} =\\check{E} \\LR{h;a-1}{n},\n\\\\\n\\LR{h;a}{n} &=\\LR{h;1+a}{n} -q^{4a}K^{-2} \\LR{h;1+a}{n-1}.  \n  \\end{split}\n\\end{align}\n\nWe shall take the following as a definition of the $\\imath$-divided powers $\\dv{n}$. \n\n\\begin{definition}\nSet $\\dv{1} =t= F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dv{n}$, for $n\\ge 1$, are defined by the  recursive relations: \n\\begin{align}   \n  \\label{eq:tt:oddevKa}\n\\begin{split}\nt \\cdot \\dv{2a-1} &=[2a] \\dv{2a}+   [2a-1] \\dv{2a-2},\n\\\\\nt \\cdot \\dv{2a} &=  [2a+1] \\dv{2a+1} , \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \nEquivalently, $\\dv{n}$ is defined by the closed formula \\eqref{def:dv:evoddK}.\n\\end{definition}\n\nThe following lemma is a variant of \\cite[Lemma~3.3]{BeW18} (where $\\kappa=0$) with the same proof.\n\n\\begin{lem}   \n  \\label{lem:anti2}\nThe anti-involution $\\varsigma$ on $\\mbf U$ fixes $F,  \\check{E},  K,  \\dvd{n}$, respectively. Moreover, $\\varsigma$ sends \n$q \\mapsto q^{-1}$,  and \n\\[\n\\LR{h;a}{n}\\mapsto (-1)^n q^{2n(n-1)}  \\LR{h;2-a-n}{n}, \\quad \\forall a \\in \\mathbb Z,\\; n\\in \\mathbb N. \n\\]\n\\end{lem}\n\n\\subsection{Formulae of $\\dv{n}$ with even $\\kappa$}\n\nRecall the polynomials $p^{(n)}$, for $n\\ge 0$, from \\S\\ref{subsec:pn}.\n\n\\begin{thm}  \n  \\label{thm:dv:evenKappa}\nLet $\\kappa$ be an even $q$-integer. Then we have, for $m\\ge 0$,  \n\\begin{align}\n\\dv{2m} &= \n\\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:oddev} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\%\n\\dv{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:oddev}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\n\nThe proof of Theorem~\\ref{thm:dv:evenKappa} will be given in \\S\\ref{subsec:proof:dv:evKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dv:evenKappa} into the following.\n\n\\begin{thm}  \n  \\label{thm:dv:evenKappa2}\nLet $\\kappa$ be an even $q$-integer. Then we have, for $m\\ge 0$,  \n\\begin{align}\n\\dv{2m} &= \n\\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{-c +b(2m-b-2c) +a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:oddevFhE}  \\\\\n&\\qquad \\qquad \\qquad\\qquad \\cdot F^{(b-a)} K^{b-2m+2c} \\LR{h;1+m-c}{c}  \\check{E}^{(a)}\n\\notag\n\\\\\n\\dv{2m+1} &=  \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c+1) +a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:oddevFhE}  \\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot F^{(b-a)} K^{b-2m+2c-1} \\LR{h;1+m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nRecall from Lemma~\\ref{lem:anti2} that the anti-involution $\\varsigma$ on $\\mbf U$   \nfixes $F,  \\check{E},  K,  \\dvd{n}, \\kappa$ while sending\n$q \\mapsto q^{-1}$, \n$\\LR{h;1-m}{c}\\mapsto (-1)^c q^{2c(c-1)}  \\LR{h;1+m-c}{c}$.\nThe formulae  \\eqref{t2m:oddevFhE}--\\eqref{t2m+1:oddevFhE} now follows from \\eqref{t2m:oddev}--\\eqref{t2m+1:oddev}. \n\n\n\n\n\\end{proof}\n\n  Let $\\lambda, m \\in \\mathbb Z$ and $c\\in \\mathbb N$.  Recall $\\cbinom{m}{c}$ from \\eqref{cbinom}.\n We note that\n\\begin{align}\n  \\label{eq:LRhc}\n\\LR{h; 1+m-c}{c} \\mathbf 1_{2\\lambda+1}  \n&= \\prod_{i=1}^c \\frac{q^{4m-4c+4i} K^{-2} -q^2}{q^{4i} -1} \\mathbf 1_{2\\lambda+1}   \n= (-1)^c q^{-2c^2} \\cbinom{m-\\lambda-c}{c} \\mathbf 1_{2\\lambda+1}.\n\\end{align}\n\nThe following corollary is immediate from \\eqref{eq:LRhc}, Proposition~\\ref{prop:fg:ev}, and Theorem~\\ref{thm:dv:evenKappa}. \n\\begin{cor}  \nWe have $\\dv{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$, for all $n$. \n\\end{cor}\n\n\n\n\\begin{example}\nThe formulae of $\\dv{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dv:evenKappa} reads as follows. \n\\begin{align*}\n\\dv{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\%\n\\dv{2} \n&=  b^{(2)}  + q\\llbracket h;0 \\rrbracket + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  +  \\frac{\\kappa^{2}}{[2]} K^{-2},  \n \\\\%\n \\dv{3}  &   = b^{(3)} + q^{-1} \\llbracket h;0 \\rrbracket F + q^{-1} \\check{E} \\llbracket h;0 \\rrbracket\n\\\\\n & \\qquad +(q^{-2} \\check{E}^{(2)} K^{-1} + q^{-3} \\check{E} K^{-1} F + q^{-2} K^{-1} F^{(2)} + q^{-1} \\llbracket h;0 \\rrbracket K^{-1})\\kappa\n\\\\\n & \\qquad + \\frac{q^{-2}}{[2]}(\\check{E} K^{-2} + K^{-2} F)\\kappa^2\n + \\frac{K^{-3}}{[3]!}\\big( \\kappa^3 + (q^{-4} + q^{-2})\\kappa \\big).\n\\end{align*}\n\\end{example}\n\n\\begin{rem}\nThe formula for $\\dv{2m}$ is formally obtained from $\\dvev{2m}$ in Theorem~\\ref{thm:dvev:evKappa},\nwith $\\qbinom{h;a}{c}$ replaced by $\\LR{h;a}{c}$.\nThe formula for $\\dv{2m-1}$ is formally obtained from $\\dvev{2m-1}$ in Theorem~\\ref{thm:dvev:evKappa},\nwith $\\qbinom{h;a}{c}$ replaced by $q^{-4c}\\LR{h;a+1}{c}$.\n\\end{rem}\n\n\\begin{rem}   \nNote $p_n(0)=0$ for $n>0$, and $p_0(0)=1$. \nThe formulae in Theorems~\\ref{thm:dv:evenKappa} and  \\ref{thm:dv:evenKappa2} in the special case for $\\kappa=0$ recover the formulae in \\cite[Theorem~3.1, Proposition~3.4]{BeW18}.\n\\end{rem}\n\n\n\n\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\text{odd}}$ for even $\\kappa$}\n  \\label{sec:iCB:odd}\n  \nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:odd}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dv{n} v^+_{2\\la+1} $ is an $\\imath$-canonical basis element for $L(2\\lambda+1)$. \n \n \\item\nThe set $\\{\\dv{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\n  Recall $\\cbinom{m}{c}$ from \\eqref{cbinom} and $\\LR{h; a}{c}$ from \\eqref{brace}.  Let $\\lambda, m \\in \\mathbb N$.  \nIt follows by a direct computation using Theorem~\\ref{thm:dv:evenKappa2} and \\eqref{eq:LRhc} that \n\\begin{align}\n\\dv{2m} v^+_{2\\la+1}  &= \n\\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c q^{-c +b(2m-b-2c)} \np^{(2m-b-2c)}(\\kappa) \n\\notag\n \\\\\n&\\qquad \\qquad \\qquad\\qquad \\cdot F^{(b)} K^{b-2m+2c} \\LR{h;1+m-c}{c}  v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda-1)(2m-b-2c)}  p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la+1} .\n\\label{t2m:oddevF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dv:evenKappa2} we have \n\\begin{align}\n\\dv{2m+1} v^+_{2\\la+1}  &=  \n\\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c+1)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\notag  \\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot F^{(b)} K^{b-2m+2c-1} \\LR{h;1+m-c}{c} v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda-1)(2m-b-2c+1)} p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la+1} . \n\\label{t2m+1:oddevF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:oddevF}--\\eqref{t2m+1:oddevF}  we obtain\n \\begin{align*}\n \\dv{n} v^+_{2\\la+1}   & \\in F^{(n)}  v^+_{2\\la+1}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la+1} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda+1) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dv:evenKappa} }\n   \\label{subsec:proof:dv:evKappa}\n\n \nWe prove the formulae for $\\dv{n}$ by induction on $n$, in two steps (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n(1) We shall prove the formula \\eqref{t2m+1:oddev} for $\\dv{2m+1}$, assuming the formula \\eqref{t2m:oddev} for $\\dv{2m}$.\n\nRecall $[2m+1] \\dvev{2m+1} = t \\cdot \\dv{2m}$, and $t= F +\\check{E} +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dv{2m},\n\\qquad\nII=F \\dv{2m},\n\\qquad\nIII=\\kappa K^{-1} \\dv{2m}.\n\\]\n\nFirst we have\n\\begin{align*}\nI\n&= \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)} [a+1] p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\%\n&= \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$ (and then adding some zero terms to make the bounds of the summations uniform throughout).\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=   \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)}  p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n    \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\end{align*}  \nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$, we further have\n\\begin{align*}\nII^1\n&=\\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \n       \\LR{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+1) -(a+1)(b-a)}  p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n    \\LR{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}  \nUsing \\eqref{recursive:f}\nwe also compute\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n\\kappa \\cdot p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n \\\\%%\n&= \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n[2m-b-2c+1]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%\n&\\quad \n- \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+2) -a(b-a)-2a+1} \n[2m-b-2c+1] \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dv{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} \n \\mathcal H_{a,b,c}  K^{b-2m+2c-1} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\n\\mathcal H_{a,b,c}  :=\n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] \\LR{h;1-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2} - (b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\LR{h;-m}{c} \n\\\\\n&+ q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+1) -(a+1)(b-a)}\n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1} \n\\\\\n&+ q^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n[2m-b-2c+1]  \\LR{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+2) -a(b-a)-2a+1} \n[2m-b-2c+1] \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\nRecall $[2m+1] \\dv{2m+1} = t \\cdot \\dv{2m}$. To prove the formula \\eqref{t2m+1:oddev} for $\\dv{2m+1}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H3}\n\\mathcal H_{a,b,c} = q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)} [2m+1] \\LR{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a+1} [a] -[2m+1] = q^{-a} [a-2m-1]$, we can combine the RHS with the first summand of LHS \\eqref{eq:H3}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a) -a}$\non both sides, we see that \\eqref{eq:H3} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD3}\n\\mathcal A+\\mathcal B+\\mathcal C+\\mathcal D_1 +\\mathcal D_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathcal A &= [a-2m-1] \\LR{h;1-m}{c},\n\\\\\n\\mathcal B &= q^{4c-2m+b-1} [b-a]  \\LR{h;-m}{c},   \n\\\\\n\\mathcal C &= q^{2a-2m+2} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1},   \n\\\\\n\\mathcal D_1 &= q^{2c+b-a} [2m-b-2c+1]  \\LR{h;1-m}{c},\n\\\\\n\\mathcal D_2 &= - q^{2c-4m+b-a} [2m-b-2c+1] \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD3}. Using \\eqref{eq:commLR}, we can write $\\mathcal B=\\mathcal B_1+\\mathcal B_2$, where \n\\begin{align*}\n\\mathcal B_1 &=  q^{4c-2m+b-1} [b-a]  \\LR{h;-m}{c},\n   \\quad\n\\mathcal B_2 = -  q^{4c-6m+b-1} [b-a]  \\LR{h;-m}{c}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a}\\frac{( q^{4c-4m}K^{-2}-q^2)}{q^2-1}\n +  q^{2m-2c-2a} [2m-2c-a+1],\n \\] \nwe rewrite $\\mathcal C=\\mathcal C_1+\\mathcal C_2$, where\n\\begin{align*}\n\\mathcal C_1 &= q^{2m-2c-a+1} [2c] \\LR{h; 1-m}{c},\n   \\quad\n\\mathcal C_2 =   q^{2-2c} [2m-2c-a+1]  \\LR{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\n\\mathcal A+\\mathcal B_1+\\mathcal C_1+\\mathcal D_1 \n&= (q-q^{-1}) [2c] [2m-2c-a+1]  \\LR{h; 1-m}{c},\n\\\\\n\\mathcal C_2 +(\\mathcal B_2+\\mathcal D_2)  \n&= \\mathcal C_2 -  q^{2c-4m} [2m-2c-a+1] \\LR{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (q-q^{-1}) [2c] [2m-2c-a+1]  \\LR{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathcal A+\\mathcal B+\\mathcal C+\\mathcal D_1+\\mathcal D_2=0,$ whence \\eqref{ABCD3}, completing Step~(1). \n\n\\vspace{3mm}\n(2) Assuming the formulae for $\\dv{n}$ with $n\\le 2m+1$, we shall now prove the following formula \\eqref{t2m+2:oddev} for $\\dv{2m+2}$ (obtained with $m$ replaced by $m+1$ in \\eqref{t2m:oddev}):\n\\begin{align}\n\\dv{2m+2} &= \n\\sum_{b=0}^{2m+2}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+2) -a(b-a)} \np^{(2m-b-2c+2)}(\\kappa) \n  \\label{t2m+2:oddev}\n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dv{2m+1} =[2m+2] \\dv{2m+2} +[2m+1] \\dv{2m}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dv{2m+1}$ from \\eqref{t2m+1:oddev}. We shall compute \n\\[\n\\texttt{I}=\\check{E}  \\dv{2m+1},\n\\qquad\n\\texttt{II}=F \\dv{2m+1},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dv{2m+1}, \n\\]\nrespectively. First by by shifting indices $a\\to a-1$ and $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n[a+1] p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)}\n[a] p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{FYn} we   also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{II}_1 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c - (b+1)(2m-b-2c+2) -a(b-a-1)-2a}\n[b-a] p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n    \\check{E}^{(a)} \n    \\LR{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c - (b+1)(2m-b-2c+2) -(a+1)(b-a)+2}\n p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\LR{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n \\end{align*}\n\nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n\\kappa \\cdot p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\\n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\%\n& - \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c - (b+2)(2m-b-2c+3) -a(b-a)-2a+3}\n[2m-b-2c+2] \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c-1} K^{b-2m+2c-4} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dv{2m+1} = \n \\sum_{0 \\le a \\le b \\le 2m+2}  \\sum_{c\\ge 0} p^{(2m-b-2c+2)}(\\kappa)  \n  \\check{E}^{(a)} \\mathcal L_{a,b,c} K^{b-2m+2c-2} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\n\\mathcal L_{a,b,c} := \n& \\; q^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)} \n[a] \\LR{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c}{2}-2c - (b+1)(2m-b-2c+2) -a(b-a-1)-2a} \n[b-a]  \\LR{h;-m}{c} \n\\\\\n&+q^{\\binom{2c-2}{2}-2c - (b+1)(2m-b-2c+2) -(a+1)(b-a)+2}\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \\LR{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2}-2c - (b+2)(2m-b-2c+3) -a(b-a)-2a+3}\n[2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}. \n\\end{align*}\n\nOn the other hand, using \\eqref{t2m:oddev} (with an index shift $c\\to c-1$) and \\eqref{t2m+2:oddev} we write \n\\begin{align*}\n [2m+2] & \\dv{2m+2} +[2m+1] \\dv{2m} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+2} \\sum_{c \\geq 0} \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)} \\mathcal R_{a,b,c} K^{b-2m+2c-2} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathcal R_{a,b,c} \n&:= q^{\\binom{2c}{2} - b(2m-b-2c+2) -a(b-a)} \n[2m+2]  \\LR{h;-m}{c}  \n\\\\\n&\\qquad + q^{\\binom{2c-2}{2} - b(2m-b-2c+2) -a(b-a)} \n[2m+1]  \\LR{h;1-m}{c-1}.    \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+2:oddev} for $\\dv{2m+2}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R3}\n\\mathcal L_{a,b,c}= \\mathcal R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)}$ on both sides, we see that the identity \\eqref{L=R3} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r3}\n \\begin{split}\n[a] \\LR{h;1-m}{c}  \n&+q^{4c-4m+b-4}  [b-a]  \\LR{h;-m}{c}   \n\\\\\n&+q^{2a-4m+1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1}   \n\\\\\n&+q^{2c-2m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c}\n\\\\\n&- q^{2c-6m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}\n\\\\\n&= q^{4c-2m+a-2}  [2m+2]  \\LR{h;-m}{c}  \n+ q^{a-2m+1}  [2m+1]  \\LR{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r3} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ3}\n\\mathcal W+\\mathcal X+\\mathcal Y+\\mathcal Z_1+\\mathcal Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\n\\mathcal W & =[a] \\LR{h;1-m}{c},\n\\\\\n\\mathcal X & =q^{4c-2m+b-2}  [b-a-2m-2]  \\LR{h;-m}{c},   \n\\\\\n\\mathcal Y & =  \\frac{q^{-a-4m+1}K^{-2}-q^{a+3}}{q^2-1} \\LR{h;1-m}{c-1},    \n\\\\\n\\mathcal Z_1 & =q^{2c-2m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c},\n\\\\\n\\mathcal Z_2 & = - q^{2c-6m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us finally prove the identity \\eqref{WXYZ3}. Using \\eqref{eq:commLR}, we can write $\\mathcal X=\\mathcal X_1+\\mathcal X_2$, where \n\\begin{align*}\n\\mathcal X_1 &=q^{4c-2m+b-2}  [b-a-2m-2]  \\LR{h; 1-m}{c},\n   \\\\\n\\mathcal X_2 &= - q^{4c-6m+b-2}  [b-a-2m-2]  \\LR{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-a-4m+1}K^{-2}-q^{a+3}}{q^2-1}  \n= q^{-4c-a+1}\\frac{( q^{4c-4m}K^{-2}-q^2)}{q^2-1}\n +  q^{2-2c} [-2c-a],\n \\] \nwe rewrite $\\mathcal Y=\\mathcal Y_1+\\mathcal Y_2$, where\n\\begin{align*}\n\\mathcal Y_1 &=q^{-2c-a} [2c] \\LR{h; 1-m}{c},\n   \\qquad\n\\mathcal Y_2 = q^{2-2c} [-2c-a] \\LR{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathcal W+\\mathcal X_1+\\mathcal Y_1+\\mathcal Z_1 &=\n- (q-q^{-1}) [2c+a] [2c]  \\LR{h; 1-m}{c},\n\\\\\n(\\mathcal X_2+\\mathcal Z_2) +\\mathcal Y_2 \n&= q^{2c-4m} [2c+a] \\LR{h; 1-m}{c-1} K^{-2} +\\mathcal Y_2\n\\\\\n&= (q-q^{-1}) [2c+a] [2c] \\LR{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathcal W+\\mathcal X+\\mathcal Y+\\mathcal Z_1+\\mathcal Z_2=0$, whence \\eqref{WXYZ3}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dv:evenKappa} is completed. \\qed\n\n\n\\section{The $\\imath$-divided powers  $\\dvp{n}$ for even weights and odd $\\kappa$}\n  \\label{sec:evoddK}\n\nIn this section~\\ref{sec:evoddK} we shall always take $\\kappa$ to be an odd $q$-integer, i.e., \n\\begin{align}  \n   \\label{eq:oddK}\n\\kappa & =[2\\ell-1], \\quad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{align}\n\\subsection{Definition of $\\dvp{n}$ for odd $\\kappa$}\n\nWe shall take the following as a definition of the $\\imath$-divided powers  $\\dvp{n}$ with odd $\\kappa$, which is formally identical to the formulae for $\\dv{n}$ with even $\\kappa$. \n\n\\begin{definition}\nSet $\\dvp{1} = t =  F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dvp{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt:oddevKa}\n\\begin{split}\nt \\cdot \\dvp{2a-1} &=[2a] \\dvp{2a}+   [2a-1] \\dvp{2a-2},\n\\\\\nt \\cdot \\dvp{2a} &=  [2a+1] \\dvp{2a+1} , \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \n\\end{definition}\nEquivalently, we have the following closed formula for $\\dvp{n}$, for $a\\in\\mathbb N$:\n\\begin{align*}\n\\dvp{n} = \n\\begin{cases}\n\\frac{1}{[2a]!}  (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{t}{[2a+1]!}  (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align*}\nNote the above formulae are formally the same as \\eqref{def:dv:evoddK} for $\\dv{n} \\in {}_\\mathcal A{\\mbf U}^\\imath_{\\rm odd}$ for  $\\kappa$ an even $q$-integer.\n\n\n\\subsection{Polynomials $\\mathfrak p_n(x)$  and $\\mathfrak p^{(n)}(x)$}\n  \\label{subsec:pn2}\n\nWe define a sequence of polynomials $\\mathfrak p_n(x)$ in a variable $x$, for $n\\in \\mathbb N$,  by letting\n\\begin{align}\n  \\label{eq:pn2}\n\\mathfrak p_{n+1} =x \\mathfrak p_n  + q^{2-2n} [n] [n-2] \\mathfrak p_{n-1}, \\qquad \\mathfrak p_0=1.\n\\end{align}\nNote $\\mathfrak p_n$ is a monic polynomial in $x$ of degree $n$. Also set $\\mathfrak p_n=0$ for $n<0$. These polynomials $\\mathfrak p_n$ will appear in the expansion formula for the $\\imath$-divided powers in $\\mbf U$. \n\n\n\\begin{example}\nHere are the polynomials $\\mathfrak p_n(x)$, for $1\\le n \\le 5$:\n\\begin{align*}\n& \\mathfrak p_1(x)  =x,\n\\qquad\n\\mathfrak p_2(x) =x^2-1,\n\\qquad\n\\mathfrak p_3(x) =x^3-x,\n\\\\\n\\mathfrak p_4(x) &=x^4 +(q^{-4}[3]-1)x^2 -q^{-4}[3],  \n\\quad\n\\mathfrak p_5(x) = x (x^2 - 1) (x^2+q^{-5}[3]! + q^{-6}[5]).\n\\end{align*}\n\\end{example}\nA simple induction using the recursive formula \\eqref{eq:pn2} shows that $\\mathfrak p_n\/(x-1) \\in \\mathbb N[q^{-1}][x]$, for $n\\ge 2$. In particular,  for $\\kappa$ any positive odd $q$-integer, we always have $\\mathfrak p_n (\\kappa) \\in \\mathbb N[q,q^{-1}]$. \n\nIntroduce the monic polynomials $\\mathfrak g_n(x)$ of degree $n$:\n\\begin{align}   \\label{eq:gn-ood}\n\\mathfrak g_n(x) =\n\\begin{cases}\n\\prod_{i=1}^{m} (x^2-[2i-1]^2), & \\text{ if } n=2m \\text{ is even};\n\\\\\nx \\prod_{i=1}^{m} (x^2-[2i-1]^2), & \\text{ if } n=2m+1 \\text{ is odd}.\n\\end{cases}\n\\end{align}\n\nDefine \n\\[\n\\mathfrak p^{(n)}(x) =\\mathfrak p_n(x)\/[n]!, \n\\qquad\n\\mathfrak g^{(n)}(x) =\\mathfrak g_n(x)\/[n]!. \n\\]\nThen Equation \\eqref{eq:pn2} implies that\n\\begin{align}\n  \\label{eq:pn2div}\nx \\cdot \\mathfrak p^{(n)}  =[n+1] \\mathfrak p^{(n+1)}-q^{2-2n} [n-2] \\mathfrak p^{(n-1)} \\quad (n\\ge 1).\n\\end{align}\n\n\nOur next goal is to prove that $\\mathfrak p^{(n)}(\\kappa)$ are integral, i.e., $\\mathfrak p^{(n)}([2\\ell-1]) \\in \\mathcal A$, for all $n, \\ell \\in \\mathbb Z$; see Proposition~\\ref{prop:fg:odd}. This is achieved by relating $\\mathfrak p^{(n)}$ to $\\mathfrak g^{(n)}$ for varied $n$. \n\n\\begin{lem} \\label{lem:odd:g-integral}\nFor $\\kappa =[2\\ell-1]$ as in \\eqref{eq:oddK}, we have $\\mathfrak g^{(n)} (\\kappa) \\in \\mathcal A$, for all $n\\in \\mathbb N$. \n\\end{lem}\n\n\\begin{proof}\nWe separate the cases for $n=2m+1$ and $n= 2m$. \nNoting that\n\\begin{equation}  \\label{eq:A2}\n\\kappa -[2i-1] =[2\\ell-1]-[2i-1] = [\\ell-i] (q^{\\ell+i-2} +q^{-\\ell-i+2}),\n\\end{equation} \nwe have\n \\begin{align*}\n \\mathfrak g^{(2m)}(\\kappa) &= \\frac1{[2m]!} \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n =  \n \\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \\in \\mathcal A.\n \\end{align*}\n \nSimilarly, using \\eqref{eq:A2} we have\n \\begin{align*}\n \\mathfrak g^{(2m+1)}(\\kappa) &= \\frac1{[2m+1]!} \\kappa \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n \\\\\n &= \\frac1{[2m+1]!} \\prod_{i=-m}^{m} (\\kappa-[2i-1]) - \\frac1{[2m]!}  \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n \\\\\n &=\\qbinom{\\ell+m}{2m+1} \\prod_{i=-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \n  + \\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \\in \\mathcal A.\n  \\end{align*}\n  The lemma is proved. \n\\end{proof}\n\n\\begin{prop}\n \\label{prop:fg:odd}\nFor $m \\in \\mathbb N$, we have\n\\begin{align*}\n\\mathfrak p^{(2m)} &= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)},\n\\\\\n\\mathfrak p^{(2m+1)} &= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+1)}.\n\\end{align*}\nIn particular, we have $\\mathfrak p^{(n)} (\\kappa) \\in \\mathcal A$, for all $n\\in \\mathbb N$ and all $\\kappa =[2\\ell-1]$ with $\\ell \\in \\mathbb Z$. \n\\end{prop}\n\n\\begin{proof}\n It follows from  these formulae for $\\mathfrak p^{(n)}$ and Lemma~\\ref{lem:odd:g-integral} that $\\mathfrak p^{(n)} ([2\\ell-1]) \\in \\mathcal A$.\n\nLet us prove these formulae. Note that \n\\begin{align}  \\label{recursive:g2}\n\\begin{split}\nx \\cdot \\mathfrak g^{(2a-1)} &=[2a] \\mathfrak g^{(2a)} +   [2a-1] \\mathfrak g^{(2a-2)},\n\\\\\nx \\cdot \\mathfrak g^{(2a)} &=  [2a+1] \\mathfrak g^{(2a+1)}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}\n\nWe prove by induction on $m$, with the base cases for $\\mathfrak p^{(1)}$ and $\\mathfrak p^{(2)}$ being clear. We separate in two cases. \n\n(1) Let us prove the formula for $\\mathfrak p^{(2m+1)}$, assuming the formulae for $\\mathfrak p^{(2m-1)}$ and $\\mathfrak p^{(2m)}$.\nBy \\eqref{eq:pn2div} and \\eqref{recursive:g2} we have \n\\begin{align*} \n[2m+1] \\mathfrak p^{(2m+1)} \n&= x \\mathfrak p^{(2m)} +q^{2-4m} [2m-2] \\mathfrak p^{(2m-1)}  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} x \\cdot \\mathfrak g^{(2m-2a)}\n\\\\&\\qquad \n  +\\sum_{a=0}^{m-2} q^{(3-2m)a+2-4m}  [2m-2] \\qbinom{m-2}{a}_{q^2} \\mathfrak g^{(2m-2a-1)}\n  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2}   \\mathfrak g^{(2m-2a+1)}\n\\\\&\\qquad \n  +\\sum_{a=0}^{m-2} q^{(3-2m)a+2-4m}  [2m-2] \\qbinom{m-2}{a}_{q^2} \\mathfrak g^{(2m-2a-1)}\n  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2}   \\mathfrak g^{(2m-2a+1)}\n\\\\&\\qquad \n  +\\sum_{a=1}^{m-1} q^{(3-2m)(a-1)+2-4m}  [2m-2] \\qbinom{m-2}{a-1}_{q^2} \\mathfrak g^{(2m-2a+1)}\n  \\\\\n  &=[2m+1] \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+1)},\n\\end{align*}\nwhere the last equation results from shifting the second summation by $a\\to a-1$ and using\nthe identity \n\\[\nq^{2a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2} +q^{2a-2m-1}  [2m-2] \\qbinom{m-2}{a-1}_{q^2}\n=[2m+1] \\qbinom{m-1}{a}_{q^2}. \n\\]\n\\vspace{3mm} \n\n(2) We shall prove the formula for $\\mathfrak p^{(2m+2)}$, assuming the formulae for $\\mathfrak p^{(2m+1)}$ and $\\mathfrak p^{(2m)}$.\nBy \\eqref{eq:pn2div} and \\eqref{recursive:g2} we have  \n\\begin{align*} \n&[2m+2] \\mathfrak p^{(2m+2)} \n\\\\\n&= x \\mathfrak p^{(2m+1)} +q^{-4m} [2m-1] \\mathfrak p^{(2m)} \n \\\\\n&= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} x\\cdot \\mathfrak g^{(2m-2a+1)}\n+ \\sum_{a=0}^{m-1} q^{(3-2m)a-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}\n\\\\%\n&= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \n\\left( [2m-2a+2] \\mathfrak g^{(2m-2a+2)} +[2m-2a+1]  \\mathfrak g^{(2m-2a)} \\right)\n\\\\& \\qquad \n+ \\sum_{a=0}^{m-1} q^{(3-2m)a-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}. \n\\end{align*}\nBy combining the like terms for $\\mathfrak g^{(2m-2a)}$ and using the identity\n\\[\nq^{(1-2m)a} [2m-2a+1] + q^{(3-2m)a-4m} [2m-1]\n=q^{(1-2m)(a+1)} [4m-2a],\n\\]\nwe have\n\\begin{align*} \n[2m+2] \\mathfrak p^{(2m+2)} \n&= \\sum_{a=0}^{m-1} q^{(1-2m)a}  [2m-2a+2] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+2)}  \n\\\\& \\qquad \n+ \\sum_{a=0}^{m-1} q^{(1-2m)(a+1)} [4m-2a] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}\n\\\\%\n&= [2m+2]  \\sum_{a=0}^{m} q^{(1-2m)a} \\qbinom{m}{a}_{q^2} \\mathfrak g^{(2m-2a+2)},\n\\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and using the identity\n\\[\n[2m-2a+2] \\qbinom{m-1}{a}_{q^2} + [4m-2a+2] \\qbinom{m-1}{a-1}_{q^2} \n= [2m+2] \\qbinom{m}{a}_{q^2}.\n\\]\nThe proposition is proved.\n\\end{proof}\n\n\n\\subsection{Formulae for $\\dvp{n}$ with odd $\\kappa$}\n\n\\begin{thm}   \n  \\label{thm:dvp:oddKappa}\nLet $\\kappa$ be an odd $q$-integer. Then, for $m\\ge 0$, we have\n\\begin{align}\n\\dvp{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evodd} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvp{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n\\label{t2m+1:evodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\nThe proof of Theorem~\\ref{thm:dvp:oddKappa} will be given in \\S\\ref{subsec:proof:dvp:oddKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvp:oddKappa} as follows.\n\n\\begin{thm}  \n  \\label{thm:dvp:oddKappa2}\nLet $\\kappa$ be an odd $q$-integer. Then we have, for $m\\ge 1$,  \n\\begin{align}\n\\dvp{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \n(-1)^c q^{c +b(2m-b-2c) +a(b-a)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\label{t2m:oddevKa}  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  \\check{E}^{(a)}\n\\notag\n\\\\\n\\dvp{2m+1}   &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \n(-1)^c q^{3c +b(2m-b-2c+1) +a(b-a)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n \\label{t2m+1:oddevKa} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c-1} \\qbinom{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:anti}, the anti-involution $\\varsigma$ sends \n$\\qbinom{h;1-m}{c}\\mapsto (-1)^c q^{2c(c+1)} \\qbinom{h;m-c}{c}$, \n$q\\mapsto q^{-1}$, while fixing $\\kappa, K, \\check{E}^{(a)}, F^{(a)}$ and $\\dvev{n}$. The formulae \\eqref{t2m:oddevKa}--\\eqref{t2m+1:oddevKa}  now follows from \\eqref{t2m:evodd}--\\eqref{t2m+1:evodd} in Theorem~\\ref{thm:dvp:oddKappa}. \n\\end{proof}\n\nThe following corollary is immediate from \\eqref{eq:qbinom-hc}, Proposition~\\ref{prop:fg:odd}, and Theorem~\\ref{thm:dvp:oddKappa}. \n\\begin{cor}  \nWe have $\\dvp{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$, for all $n$. \n\\end{cor}\n\n\\begin{rem}   \nNote $\\mathfrak p_n(1)=0$ for $n\\ge 2$, and $\\mathfrak p_0(1)=\\mathfrak p_1(1)=1$. \nThe formulae in Theorems~\\ref{thm:dvp:oddKappa} and  \\ref{thm:dvp:oddKappa2} in the special case for $\\kappa=1$ recover the formulae in \\cite[Theorem~4.1, Proposition~4.3]{BeW18}.\n\\end{rem}\n\n\n\\begin{example}\nThe formulae of $\\dvp{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dvp:oddKappa} reads as follows. \n\\begin{align*}\n\\dvp{1} \n&= F+ \\check{E} + K^{-1},\n\\\\\n\\dvp{2} \n&= b^{(2)}  + q^3 [h;0]+\\kappa( q^{-1}K^{-1}F +q^{-1}\\check{E} K^{-1} ) +  \\frac{\\kappa^{2}-1}{[2]} K^{-2},\n\\\\\n\\dvp{3} \n&=  b^{(3)} + q \\check{E} [h;0] + q [h;0] F \n+ \\kappa( q^{-2} \\check{E}^{(2)} K^{-1} + q^{-3} \\check{E} K^{-1} F + q^{-2} K^{-1} F^{(2)} + q [h;0] K^{-1})\n\\\\\n & \\qquad + q^{-2}\\frac{\\kappa^{2}-1}{[2]}(\\check{E} K^{-2} + K^{-2} F)  \n + \\frac{\\kappa^3 -\\kappa}{[3]!} K^{-3}.\n\\end{align*}\n\\end{example}\n \n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\rm{ev}}$ with odd $\\kappa$}\n  \\label{sec:iCB:evwt-oddK}\n  \n\nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:evwt-oddK}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvp{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element for $L(2\\lambda)$. \n \n \\item\nThe set $\\{\\dvp{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\nLet $\\lambda, m \\in \\mathbb N$. Recall $\\cbinom{m}{c}$ from \\eqref{cbinom} and $\\qbinom{h; a}{c}$ from \\eqref{kbinom}. It follows by a direct computation using Theorem~\\ref{thm:dvp:oddKappa2} and \\eqref{eq:qbinom-hc} that \n\\begin{align}\n\\dvp{2m} v^+_{2\\la}  &= \n\\sum_{b=0}^{2m} \\sum_{c\\geq 0} \n(-1)^c q^{c +b(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) F^{(b)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  v^+_{2\\la} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda)(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la} .\n\\label{t2m:evwt-oF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dvp:oddKappa2} we have \n\\begin{align}\n\\dvp{2m+1} v^+_{2\\la}  &=  \n\\sum_{b=0}^{2m+1} \\sum_{c\\geq 0} \n(-1)^c q^{3c +b(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n F^{(b)} K^{b-2m+2c-1} \\qbinom{h;m-c}{c} v^+_{2\\la} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda)(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la} . \n\\label{t2m+1:evwt-oF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:evwt-oF}--\\eqref{t2m+1:evwt-oF}   we obtain\n \\begin{align*}\n \\dvp{n} v^+_{2\\la}   & \\in F^{(n)}  v^+_{2\\la}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvp:oddKappa} }\n  \\label{subsec:proof:dvp:oddKappa}\nWe prove the formulae for $\\dvp{n}$ by induction on $n$, in two steps (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n\n(1) We shall prove the formula \\eqref{t2m+1:evodd} for $\\dvp{2m+1}$, assuming the formula \\eqref{t2m:evodd} for $\\dvp{2m}$.\n\nRecall $[2m+1] \\dvp{2m+1} = t \\cdot \\dvp{2m}$, and $t= F + \\check{E}  +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dvp{2m},\n\\qquad\nII=F \\dvp{2m},\n\\qquad\nIII=\\kappa K^{-1} \\dvp{2m}.\n\\]\n\nFirst we have\n\\begin{align*}\nI\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n[a+1] \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\\\%\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$.\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=  \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n& \\qquad\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n\\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n=II^1 +II^2, \n\\\\\n\\end{align*}  \nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$, we further have\n\\begin{align*}\nII^1\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c -(b+1)(2m-b-2c+1) -a(b-a+1)} \n[b-a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n   \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c+1) -(a+1)(b-a)-2} \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\qbinom{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}  \nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a} \n\\kappa \\cdot \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%%\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a}  [2m-b-2c+1]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%\n&\\quad - \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c-2}{2}+2c - (b+2) (2m-b-2c+2) -a(b-a)-2a}  [2m-b-2c]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot\n\\mathfrak p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dvp{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  \\mathfrak p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} \n \\mathscr H_{a,b,c}  K^{b-2m+2c-1} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\n\\mathscr H_{a,b,c} := \n& \\; q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a]  \\qbinom{h;1-m}{c} \n\\\\\n&+q^{\\binom{2c}{2}+2c -(b+1)(2m-b-2c+1) -a(b-a+1)} \n[b-a] \\qbinom{h;-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c+1) -(a+1)(b-a)-2} \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a}  \n[2m-b-2c+1]    \\qbinom{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}+2c - (b+2) (2m-b-2c+2) -a(b-a)-2a}  \n[2m-b-2c]   \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\nRecall $[2m+1] \\dv{2m+1} = t \\cdot \\dvp{2m}$. To prove the formula \\eqref{t2m+1:evodd} for $\\dv{2m+1}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H4}\n\\mathscr H_{a,b,c} = q^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)}\n [2m+1] \\qbinom{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a+1} [a] -[2m+1] = q^{-a} [a-2m-1]$, we can combine the RHS\\eqref{eq:H4} with the first summand of LHS\\eqref{eq:H4}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}- b(2m-b-2c+1) -a(b-a) -a}$\non both sides, we see that \\eqref{eq:H4} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD4}\n\\mathscr A+\\mathscr B+\\mathscr C+\\mathscr D_1 +\\mathscr D_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathscr A &= [a-2m-1] \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr B &= q^{4c-2m+b-1} [b-a]  \\qbinom{h;-m}{c},   \n\\\\\n\\mathscr C &= q^{2a-2m} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1},   \n\\\\\n\\mathscr D_1 &= q^{2c+b-a} [2m-b-2c+1]  \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr D_2 &= - q^{2c-4m+b-a-1} [2m-b-2c] \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD4}. Using \\eqref{kbinom2}, we can write $\\mathscr B=\\mathscr B_1+\\mathscr B_2$, where \n\\begin{align*}\n\\mathscr B_1 &=  q^{4c-2m+b-1} [b-a]  \\qbinom{h;-m}{c},\n   \\quad\n\\mathscr B_2 = -  q^{4c-6m+b-1} [b-a]  \\qbinom{h;-m}{c}.\n\\end{align*}\nNoting that  \n\\[\nq^{2a-2m}\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{2m-4c-a}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c-1} [2m-2c-a],\n \\] \nwe rewrite $\\mathscr C=\\mathscr C_1+\\mathscr C_2$, where\n\\begin{align*}\n\\mathscr C_1 &= q^{2m-2c-a-1} [2c] \\qbinom{h; 1-m}{c},\n   \\quad\n\\mathscr C_2 =   q^{-2c-1} [2m-2c-a]  \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\n\\mathscr A+\\mathscr B_1+\\mathscr C_1+\\mathscr D_1 \n&=(1-q^{-2}) [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c},\n\\\\\n\\mathscr C_2 +(\\mathscr B_2+\\mathscr D_2)  \n&= \\mathscr C_2 -  q^{2c-4m-1} [2m-2c-a] \\qbinom{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (1-q^{-2}) [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathscr A+\\mathscr B+\\mathscr C+\\mathscr D_1+\\mathscr D_2=0,$ whence \\eqref{ABCD4}, completing Step~(1). \n\n\\vspace{3mm}\n\n(2) Assuming the formulae for $\\dvp{n}$ with $n\\le 2m+1$, we shall now prove the following formula \\eqref{t2m+2:evodd} for $\\dvp{2m+2}$ (obtained with $m$ replaced by $m+1$ in \\eqref{t2m:evodd}):\n\\begin{align}\n\\dvp{2m+2} &= \n\\sum_{b=0}^{2m+2}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} +2c- b(2m-b-2c+2) -a(b-a)} \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n  \\label{t2m+2:evodd}\n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dvp{2m+1} =[2m+2] \\dvp{2m+2} +[2m+1] \\dvp{2m}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dvp{2m+1}$ from \\eqref{t2m+1:evodd}. We shall compute \n\\[\n\\texttt{I}=\\check{E}  \\dvp{2m+1},\n\\qquad\n\\texttt{II}=F \\dvp{2m+1},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvp{2m+1}, \n\\]\nrespectively. First by by shifting indices $a\\to a-1$ and $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)}\n[a+1] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)}\n[a] \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{FYn} we   also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we further have\n\\begin{align*}\n\\texttt{II}_1 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b+1)(2m-b-2c+2) -a(b-a-1)-2a}\n[b-a] \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n    \\check{E}^{(a)} \n    \\qbinom{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+2) -(a+1)(b-a)}\n \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\qbinom{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n \\end{align*}\n\nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n\\kappa \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\\n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\%\n& - \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+3) -a(b-a)-2a+2}\n[2m-b-2c+1] \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-4} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dvp{2m+1} = \n \\sum_{0 \\le a \\le b \\le 2m+2}  \\sum_{c\\ge 0} \\mathfrak p^{(2m-b-2c+2)}(\\kappa)  \n  \\check{E}^{(a)} \\mathscr L_{a,b,c} K^{b-2m+2c-2} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\n\\mathscr L_{a,b,c} := \n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)} \n[a] \\qbinom{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c}{2} - (b+1)(2m-b-2c+2) -a(b-a-1)-2a} \n[b-a]  \\qbinom{h;-m}{c} \n\\\\\n&+q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+2) -(a+1)(b-a)}\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \\qbinom{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+3) -a(b-a)-2a+2}\n[2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}. \n\\end{align*}\n\nOn the other hand, using \\eqref{t2m:evodd} (with an index shift $c\\to c-1$) and \\eqref{t2m+2:evodd} we write \n\\begin{align*}\n [2m+2] & \\dvp{2m+2} +[2m+1] \\dvp{2m} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+2} \\sum_{c \\geq 0} \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)} \\mathscr R_{a,b,c} K^{b-2m+2c-2} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathscr R_{a,b,c} \n&:= q^{\\binom{2c}{2} +2c- b(2m-b-2c+2) -a(b-a)} \n[2m+2]  \\qbinom{h;-m}{c}  \n\\\\\n&\\qquad + q^{\\binom{2c-2}{2} +2c- b(2m-b-2c+2) -a(b-a)-2} \n[2m+1]  \\qbinom{h;1-m}{c-1}.    \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+2:oddev} for $\\dvp{2m+2}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R4}\n\\mathscr L_{a,b,c}= \\mathscr R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)}$ on both sides, we see that the identity \\eqref{L=R4} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r4}\n \\begin{split}\n[a] \\qbinom{h;1-m}{c}  \n&+q^{4c-4m+b-4}  [b-a]  \\qbinom{h;-m}{c}   \n\\\\\n&+q^{2a-4m-1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1}   \n\\\\\n&+q^{2c-2m+b-a-2} [2m-b-2c+2]  \\qbinom{h;1-m}{c}\n\\\\\n&- q^{2c-6m+b-a-3} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}\n\\\\\n&= q^{4c-2m+a-2}  [2m+2]  \\qbinom{h;-m}{c}  \n+ q^{a-2m-1}  [2m+1]  \\qbinom{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r4} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ4}\n\\mathscr W+\\mathscr X+\\mathscr Y+\\mathscr Z_1+\\mathscr Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\n\\mathscr W & =[a] \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr X & =q^{4c-2m+b-2}  [b-a-2m-2]  \\qbinom{h;-m}{c},   \n\\\\\n\\mathscr Y & =  \\frac{q^{-a-4m-1}K^{-2}-q^{a+1}}{q^2-1} \\qbinom{h;1-m}{c-1},    \n\\\\\n\\mathscr Z_1 & =q^{2c-2m+b-a-2} [2m-b-2c+2]  \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr Z_2 & = - q^{2c-6m+b-a-3} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us finally prove the identity \\eqref{WXYZ4}. Using \\eqref{kbinom2}, we can write $\\mathscr X=\\mathscr X_1+\\mathscr X_2$, where \n\\begin{align*}\n\\mathscr X_1 &=q^{4c-2m+b-2}  [b-a-2m-2]  \\qbinom{h; 1-m}{c},\n   \\\\\n\\mathscr X_2 &= - q^{4c-6m+b-2}  [b-a-2m-2]  \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-a-4m-1}K^{-2}-q^{a+1}}{q^2-1}  \n= q^{-4c-a-1}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c-1} [-2c-a-1],\n \\] \nwe rewrite $\\mathscr Y=\\mathscr Y_1+\\mathscr Y_2$, where\n\\begin{align*}\n\\mathscr Y_1 &=q^{-2c-a-2} [2c] \\qbinom{h; 1-m}{c},\n   \\qquad\n\\mathscr Y_2 = -q^{-2c-1} [2c+a+1] \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathscr W+\\mathscr X_1+\\mathscr Y_1+\\mathscr Z_1 &=\n- (1-q^{-2}) [2c+a+1] [2c] \\qbinom{h; 1-m}{c},\n\\\\\n(\\mathscr X_2+\\mathscr Z_2) +\\mathscr Y_2  \n&= q^{2c-4m-1} [2c+a+1] \\qbinom{h; 1-m}{c-1} K^{-2} +\\mathscr Y_2\n\\\\\n&= (1-q^{-2}) [2c+a+1] [2c] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathscr W+\\mathscr X+\\mathscr Y+\\mathscr Z_1+\\mathscr Z_2=0$, whence \\eqref{WXYZ4}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvp:oddKappa} is completed. \\qed\n\n\n\n\n\n\n\\section{The $\\imath$-divided powers $\\dvd{n}$ for odd weights and odd $\\kappa$}\n  \\label{sec:oddoddK}\n  \nIn this section~\\ref{sec:oddoddK} we always take $\\kappa$ to be an odd $q$-integer, i.e., \n\\begin{equation*}\n\\kappa =[2\\ell-1], \\qquad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{equation*} \n\n\\subsection{Definition of $\\dvd{n}$ for odd $\\kappa$}\n\n\n\\begin{definition}\nSet $\\dvd{1} =t= F +\\check{E} +\\kappa K^{-1}$. \nThe divided powers $\\dvd{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt:evoddKa}\n\\begin{split}\nt \\cdot \\dvd{2a-1} &=[2a] \\dvd{2a},\n\\\\\nt \\cdot \\dvd{2a} &=  [2a+1] \\dvd{2a+1} +   [2a] \\dvd{2a-1}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \n\\end{definition}\nEquivalently, we have the following closed formula for $\\dvd{n}$:\n\\begin{align}\n\\label{def:dvd:evoddKa}\n\\dvd{n} = \n\\begin{cases}\n\\frac{t}{[2a]!} (t  - [-2a+2])(t -[-2a+4]) \\cdots (t -[2a-4]) (t  - [2a-2]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{1}{[2a+1]!} (t  - [-2a])(t -[-2a+2]) \\cdots (t -[2a-2]) (t  - [2a]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align}\nNote the formulae for $\\dvd{n}$ with odd $\\kappa$ is formally identical to the formulae for $\\dvev{n}$ with even $\\kappa$. \n\n\\subsection{Formulae for $\\dvd{n}$ with odd $\\kappa$}\n\nRecall the polynomials $\\mathfrak p^{(n)}$, for $n\\ge 0$, from \\S\\ref{subsec:pn2}.\n\n\\begin{thm}  \n   \\label{thm:dvd:oddKappa}\nAssume $\\kappa$ is an odd $q$-integer. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvd{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa)\n\\label{t2moddodd} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvd{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\label{t2m-1oddodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\nThe proof of Theorem~\\ref{thm:dvd:oddKappa} will be given in \\S\\ref{subsec:proof:dvd:oddKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvd:oddKappa} as follows.\n\n\\begin{thm}   \n \\label{thm:dvd:oddKappa2}\nAssume $\\kappa$ is an odd $q$-integer. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvd{2m} \n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n(-1)^c  q^{c +b(2m-b-2c) +a(b-a)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \n\\label{t2moddodd2} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c} \\LR{h;m-c}{c}  \\check{E}^{(a)}, \n\\notag\n\\\\%\n\\dvd{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n(-1)^c q^{-c+ b(2m-b-2c-1) +a(b-a)}\n \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\label{t2m-1oddodd2} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c+1}   \\LR{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nRecall from Lemma~\\ref{lem:anti2} that the anti-involution $\\varsigma$ on $\\mbf U$   \nfixes $F,  \\check{E},  K,  \\dvd{n}, \\kappa$ while sending\n$q \\mapsto q^{-1}$, \n$\\LR{h;2-m}{c}\\mapsto (-1)^c q^{2c(c-1)}  \\LR{h;m-c}{c}$.\nThe formulae \\eqref{t2moddodd2}--\\eqref{t2m-1oddodd2} now follow by applying $\\varsigma$ to the formulae \\eqref{t2moddodd}--\\eqref{t2m-1oddodd} in Theorem~\\ref{thm:dvd:oddKappa}. \n\\end{proof}\n\nThe following corollary is immediate from \\eqref{eq:LRhc}, Proposition~\\ref{prop:fg:odd}, and Theorem~\\ref{thm:dvd:oddKappa}. \n\n\\begin{cor}  \nWe have $\\dvd{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$, for all $n$. \n\\end{cor}\n\n\n\\begin{rem}   \nNote $\\mathfrak p_n(1)=0$ for $n\\ge 2$, and $\\mathfrak p_0(1)=\\mathfrak p_1(1)=1$. \nThe formulae in Theorems~\\ref{thm:dvd:oddKappa} and  \\ref{thm:dvd:oddKappa2} in the special case for $\\kappa=1$ recover the formulae in \\cite[Theorem~5.1, Proposition~5.3]{BeW18}.\n\\end{rem}\n\n\n\\begin{example}\nThe formulae of $\\dvd{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dvd:oddKappa} read as follows. \n\\begin{align*}\n\\dvd{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\\n\\dvd{2} \n&= \\check{E}^{(2)} +q^{-1} \\check{E} F + F^{(2)}  + q^{-1} [[h;1]] \n + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  +  \\frac{\\kappa^2-1}{[2]} K^{-2},\n\\\\\n\\dvd{3}  &   = b^{(3)} + q[[h;0]]F+ q \\check{E} [[h;0]]\n\\\\\n&\\qquad \n+ \\big(q^{-2} \\check{E}^{(2)}K^{-1} +q^{-3} \\check{E} K^{-1}F + q^{-2} K^{-1}F^{(2)} \n+ q [[h;0]] K^{-1} \\big) \\kappa\n\\\\\n&\\qquad + \\frac{(\\kappa^2-1)}{[2]} (q^{-2}\\check{E} K^{-2} +q^{-2}K^{-2}F) \n+\\frac{\\kappa^3 -\\kappa}{[3]!} K^{-3}.\n\\end{align*}\n\\end{example}\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\text{odd}}$ with odd $\\kappa$}\n  \\label{sec:iCB:evwt-oddK}\n  \n\nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:evwt-oddK}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvd{n} v^+_{2\\la+1} $ is an $\\imath$-canonical basis element for $L(2\\lambda+1)$. \n \n \\item\nThe set $\\{\\dvd{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\n Let $\\lambda, m \\in \\mathbb N$. Recall $\\cbinom{m}{c}$ from \\eqref{cbinom}. It follows by a direct computation using Theorem~\\ref{thm:dvd:oddKappa2} and \\eqref{eq:LRhc} that \n\\begin{align}\n\\dvd{2m} v^+_{2\\la+1}  &= \n\\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c  q^{c +b(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \nF^{(b)} K^{b-2m+2c} \\LR{h;m-c}{c}  v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda)(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la+1} .\n\\label{t2m:oddoF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dvd:oddKappa2} we have \n\\begin{align}\n\\dvd{2m-1} v^+_{2\\la+1}  &=  \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} \n(-1)^c q^{-c+ b(2m-b-2c-1)} \\mathfrak p^{(2m-b-2c-1)}(\\kappa) F^{(b)} K^{b-2m+2c+1}   \\LR{h;m-c}{c} v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda)(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la+1} . \n\\label{t2m-1:oddoF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:oddoF}--\\eqref{t2m-1:oddoF}  we obtain\n \\begin{align*}\n \\dvd{n} v^+_{2\\la+1}   & \\in F^{(n)}  v^+_{2\\la+1}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la+1} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda+1) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvd:oddKappa} }\n  \\label{subsec:proof:dvd:oddKappa}\n  \nWe prove the formulae for $\\dvd{n}$ by induction on $n$, in two separate cases (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n\n(1) We shall prove the formula \\eqref{t2moddodd} for $\\dvd{2m}$, assuming the formula \\eqref{t2m-1oddodd} for $\\dvd{2m-1}$.\n\nRecall $[2m] \\dvd{2m} = t \\cdot \\dvd{2m-1}$, and $t= F +\\check{E} +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E} \\dvd{2m-1},\n\\qquad\nII=F \\dvd{2m-1},\n\\qquad\nIII=\\kappa K^{-1} \\dvd{2m-1}.\n\\]\n\nWe compute\n\\begin{align*}\nI\n&= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)} [a+1] \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n\\\\\n &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c) -(a-1)(b-a)} [a] \\mathfrak p^{(2m-b-2c)}(\\kappa)\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}. \n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$.\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)}\\mathfrak p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n    \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n   =II^1 +II^2,\n\\end{align*}\nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$ and then adding some zero terms, we obtain\n\\begin{align*}\nII^1\n\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - (b+1)(2m-b-2c) -a(b-a+1)} [b-a] \\mathfrak p^{(2m-b-2c)}(\\kappa)  \\\\\n&\\qquad\\qquad\\qquad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and then adding some zero terms, we also obtain\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c) -(a+1)(b-a)}\\mathfrak p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad\\qquad\\qquad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n    \\LR{h;2-m}{c-1} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy the identity \\eqref{eq:pn2div} we have\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}\n \\kappa \\cdot \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\\\ \n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\%%\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}\n[2m-b-2c]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c)} \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\\n&\\quad - \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n q^{\\binom{2c-2}{2} -(b+2)(2m-b-2c+1) -a(b-a)-2a +2}\n[2m-b-2c-1]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c)} \\check{E}^{(a)}  \\LR{h;2-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2, III$ gives us\n\\[\nt \\cdot \\dvd{2m-1} = \\sum_{0 \\le a \\le b \\le 2m}  \\sum_{c\\ge 0} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \\check{E}^{(a)} \\mathfrak H_{a,b,c}  K^{b-2m+2c} F^{(b-a)}, \n\\]\nwhere \n\\begin{align*}\n\\mathfrak H_{a,b,c} := \n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c) -(a-1)(b-a)} [a]  \\LR{h;2-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2} - (b+1)(2m-b-2c) -a(b-a+1)} [b-a] \\LR{h;1-m}{c} \n\\\\\n&+ q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c) -(a+1)(b-a)} %\n   \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}  \n\\\\%\n&+ q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}[2m-b-2c] \\LR{h;2-m}{c} \n\\\\\n&- q^{\\binom{2c-2}{2} -(b+2)(2m-b-2c+1) -a(b-a)-2a +2}\n[2m-b-2c-1] \\LR{h;2-m}{c-1} K^{-2}, \n\\end{align*}\n\nRecall $[2m] \\dvd{2m} = t \\cdot \\dvd{2m-1}$. To prove the formula \\eqref{t2moddodd} for $\\dvd{2m}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:Hoo}\n\\mathfrak H_{a,b,c} = q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} [2m] \\LR{h;2-m}{c}.\n\\end{align}  \nThanks to $q^{2m-a} [a] -[2m] =q^{-a} [a-2m]$, we can combine the RHS\\eqref{eq:Hoo} with the first summand of the LHS\\eqref{eq:Hoo}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-a}$ on both sides, we see that \\eqref{eq:Hoo} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD2}\n\\mathfrak A+\\mathfrak B+\\mathfrak C+\\mathfrak D_1 +\\mathfrak D_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}  \n\\mathfrak A &=  [a-2m] \\LR{h;2-m}{c},\n  \\\\\n\\mathfrak B &=  q^{b+4c-2m} [b-a] \\LR{h;1-m}{c},\n\\\\\n\\mathfrak C  &=  q^{2a-2m+3} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1},\n\\\\\n\\mathfrak D_1 &=  q^{2c+b-a} [2m-b-2c] \\LR{h;2-m}{c},\n   \\\\\n\\mathfrak D_2 &= - q^{2c-4m+b-a+3} [2m-b-2c-1] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD2}. Using \\eqref{eq:commLR}, \nwe can write $\\mathfrak B=\\mathfrak B_1+\\mathfrak B_2$, where \n\\begin{align*}\n\\mathfrak B_1 &=  q^{b+4c-2m} [b-a] \\LR{h;2-m}{c},\n   \\quad\n\\mathfrak B_2 = - q^{b+4c-6m+4} [b-a] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\n Noting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a-4}\\frac{( q^{4c-4m+4}K^{-2}-q^2)}{q^2-1}\n +  q^{2m-2c-2a-2} [2m-2c-a-1],\n \\] \nwe rewrite $\\mathfrak C=\\mathfrak C_1+\\mathfrak C_2$, where \n\\begin{align*}\n\\mathfrak C_1 &= q^{2m-a-2c-2} [2c] \\LR{h;2-m}{c},\n   \\qquad\n\\mathfrak C_2 =   q^{1-2c} [2m-2c-a-1] \\LR{h;2-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us  \n\\begin{align*}\n\\mathfrak A+\\mathfrak B_1+\\mathfrak C_1+\\mathfrak D_1 \n&=  (1-q^{-2}) [2c] [2m-2c-a-1] p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c},\n\\\\\n\\mathfrak C_2 +(\\mathfrak B_2+\\mathfrak D_2)  \n&= \\mathfrak C_2 -  q^{2c-4m+3} [2m-2c-a-1]  p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c-1} K^{-2}\n\\\\\n&=  - (1-q^{-2}) [2c] [2m-2c-a-1] p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathfrak A+\\mathfrak B+\\mathfrak C+\\mathfrak D_1+\\mathfrak D_2=0,$ whence \\eqref{ABCD2}, completing Step~(1). \n\n\\vspace{3mm}\n\n\n\n(2) Assuming the formulae for $\\dvd{n}$ with $n\\le 2m$, we shall now prove the following formula for $\\dvd{2m+1}$ (obtained from \\eqref{t2m-1oddodd} with $m$ replaced by $m+1$):\n\\begin{align}\n\\dvd{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)} p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1oddodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\nWe first need to compute $t \\cdot \\dvd{2m}$, where we recall $t= F +\\check{E}  +\\kappa K^{-1}$ and $\\dvd{2m}$ from \\eqref{t2moddodd}. We shall compute \n\\[\n\\texttt I=\\check{E}  \\dvd{2m},\n\\qquad\n\\texttt{II}=F \\dvd{2m},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvd{2m}, \n\\]\nrespectively. First we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \n[a+1] \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\ %\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}  [a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting the indices $a\\to a-1$, $b\\to b-1$. We also have\n\\begin{align*}\n\\texttt{II} \n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n&\\qquad    \\cdot \n\\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we further have\n\\begin{align*}\n\\texttt{II}_1\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -(b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n\\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot\n \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c -(b+1)(2m-b-2c+1)-(a+1)(b-a)+2} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n  \\\\\n&\\qquad    \\cdot  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\LR{h;2-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nUsing the identity \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n \\texttt{III} =\n\n& \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} \\kappa \\cdot \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n& \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n  \\\\%\n= & \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} [2m-b-2c+1]  \n  \\\\\n& \\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n& -  \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2}-2c -(b+2)(2m-b-2c+2)-a(b-a)-2a+4} \n [2m-b-2c]  \n  \\\\\n& \\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;2-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2, \\texttt{III}$, we obtain an expression of the form\n\\[\nt \\cdot \\dvd{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  \n \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)} \\mathfrak L_{a,b,c} K^{b-2m+2c-1} F^{(b-a)},\n\\]\nwhere\n\\begin{align*}\n\\mathfrak L_{a,b,c} :=& \\;\nq^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}  [a]  \\LR{h;2-m}{c}\n\\\\\n&+q^{\\binom{2c}{2}-2c -(b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\LR{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}-2c -(b+1)(2m-b-2c+1)-(a+1)(b-a)+2}  \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}\n\\\\\n&+q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} [2m-b-2c+1] \\LR{h;2-m}{c}\n\\\\\n&-q^{\\binom{2c-2}{2}-2c -(b+2)(2m-b-2c+2)-a(b-a)-2a+4} \n [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\nOn the other hand, using \\eqref{t2m-1oddodd} (with an index shift $c\\to c-1$) and \\eqref{t2m+1oddodd} we write \n\\begin{align*}\n[2m+1] & \\dvd{2m+1} +[2m] \\dvd{2m-1} \n\\\\\n& =\\sum_{0 \\le a \\le b \\le 2m+1} \\sum_{c \\geq 0} \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)} \\mathfrak R_{a,b,c} K^{b-2m+2c-1} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathfrak R_{a,b,c} :=&\nq^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)} [2m+1]  \\LR{h;1-m}{c}  \n\\\\\n&\\quad + q^{\\binom{2c-2}{2} - b(2m-b-2c+1) -a(b-a)} [2m]  \\LR{h;2-m}{c-1}.\n\\end{align*}\n\n\nTo prove the formula \\eqref{t2m+1oddodd} for $\\dvd{2m+1}$, it suffices to show that, for all $a,b,c$, \n\\begin{equation}\n  \\label{L=R2}\n \\mathfrak L_{a,b,c}=\\mathfrak R_{a,b,c}.\n \\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}$  on both sides, we see that the identity \\eqref{L=R2} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n  \\label{l=r2}\n \\begin{split}\n[a]  \\LR{h;2-m}{c} \n&+q^{4c-4m+b-2} [b-a] \\LR{h;1-m}{c}\n\\\\\n&+q^{2a-4m+3} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}\n\\\\\n&+q^{2c-2m+b-a-1} [2m-b-2c+1] \\LR{h;2-m}{c}\n\\\\\n&-q^{2c-6m+b-a+2} [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}\n \\\\\n =& \\; q^{4c-2m+a-1} [2m+1]  \\LR{h;1-m}{c}  \n+ q^{a-2m+2} [2m]   \\LR{h;2-m}{c-1}.\n \\end{split}\n\\end{align}\n\nBy combining the second summand of the LHS\\eqref{l=r2} with the first summand of the RHS\\eqref{l=r2} as well as combining the third summand of the LHS with the second summand of the RHS, the identity \\eqref{l=r2} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ2}\n\\mathfrak W+ \\mathfrak X+\\mathfrak Y+\\mathfrak Z_1+\\mathfrak Z_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathfrak W&= [a]  \\LR{h;2-m}{c},\n\\\\\n\\mathfrak X&= q^{4c-2m+b-1} [b-a-2m-1] \\LR{h;1-m}{c},\n\\\\\n\\mathfrak Y&=  \\frac{q^{3-a-4m}K^{-2}-q^{3+a}}{q^2-1} \\LR{h;2-m}{c-1},  \n\\\\\n\\mathfrak Z_1&= q^{2c-2m+b-a-1} [2m-b-2c+1] \\LR{h;2-m}{c},\n\\\\\n\\mathfrak Z_2&= -q^{2c-6m+b-a+2} [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\nLet us finally prove the identity \\eqref{WXYZ2}. Using \\eqref{eq:commLR}, we can write $\\mathfrak X=\\mathfrak X_1+\\mathfrak X_2$, where \n\\begin{align*}\n\\mathfrak X_1 &=q^{4c-2m+b-1} [b-a-2m-1] \\LR{h; 2-m}{c},\n   \\\\\n\\mathfrak X_2 &= - q^{4c-6m+b+3} [b-a-2m-1] \\LR{h; 2-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{3-a-4m}K^{-2}-q^{3+a}}{q^2-1}\n= q^{-4c-a-1}\\frac{( q^{4c-4m+4}K^{-2}-q^2)}{q^2-1}\n +  q^{1-2c} [-2c-a-1],\n \\] \nwe rewrite $\\mathfrak Y=\\mathfrak Y_1+\\mathfrak Y_2$, where\n\\begin{align*}\n\\mathfrak Y_1 &=q^{-2c-a-2} [2c] \\LR{h; 2-m}{c},\n   \\qquad\n\\mathfrak Y_2 = -q^{1-2c} [2c+a+1] \\LR{h; 2-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathfrak W+\\mathfrak X_1+\\mathfrak Y_1+\\mathfrak Z_1 &=\n-(1-q^{-2}) [2c+a+1] [2c]  \\LR{h; 2-m}{c},\n\\\\\n(\\mathfrak X_2+\\mathfrak Z_2) +\\mathfrak Y_2 \n&= q^{-2m-a+1} [2c+a] \\LR{h; 2-m}{c-1} K^{-2} +\\mathfrak Y_2\n\\\\\n&= (1-q^{-2}) [2c+a+1] [2c]  \\LR{h; 2-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathfrak W+\\mathfrak X+\\mathfrak Y+\\mathfrak Z_1+\\mathfrak Z_2=0$, whence \\eqref{WXYZ2}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvd:oddKappa} is completed. \\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe upcoming measurements at LHC are renewing considerable interest \nregarding predictions for total \ncross-sections. The model \\cite{corsetti,ggps}\n we shall describe in the following, \nattempts to \nlink the rate with which total cross-sections rise, to   \nthe infrared behaviour of the strong coupling constant \n$\\alpha_s$  and to QCD hard parton-parton scattering, using \n known phenomenological entities such as the available QCD parton density \nfunctions (PDFs). \n\n\\section{The model}\nThe energy behaviour of the total cross-section exhibits the following \nproperties \\cite{PDG}\n\\begin{itemize}\n\\item an initial decrease\n\\item a sharp change in curvature occurring somewhere between  \n20 and 50 GeV in the c.m. of the scattering hadrons\n\\item a smooth rise which asymptotically follows a   \n$\\ln s$ or $\\ln^2 {s}$ type \nincrease in consonance with the Froissart bound \\cite{froissart,blockfroissart}\n\\end{itemize}\nThe model we use is based on\n\\begin{enumerate}\n\\item hard component of scattering responsible for the rise of the total \ncross-section \\cite{allhalzen,minijets}\n\\item soft gluon emission from scattering particles which softens \nthe rise \\cite{corsetti}\n\\item eikonal transformation which implies \nmultiple scattering and requires \nimpact parameter distributions inside scattering particles and basic \nscattering cross-sections \\cite{durand}\n\\end{enumerate}\n\nAccording to our model, soft gluon emission is responsible for  the initial \ndecrease in $p p $, as well as for the transformation of the sharp rise due\nto the increase in gluon-gluon interactions\ninto \na smooth behavior.\nThus soft gluon emission plays a crucial role, with care taken to \nextend resummation to the zero energy modes, in complete analogy for what \nis  required by the Bloch-Nordsieck \ntheorem for QED\\cite{BN}. \nThe model can then be referred to as the BN model, for reasons which will \nalso be clearer in the following.\n\n\\subsection{Details of the BN model} \nWe use the following eikonal expression for the total inelastic \ncross-section:\n\\begin{equation}\n\\label{siginel}\n\\sigma_{inel}=\\int d^2 {\\vec b} [1-e^{-n(b,s)}]\n\\end{equation}\nwhere $n(b,s)$ corresponds to the average number of inelastic \ncollisions at  any given value of the impact parameter b. Neglecting the\nreal part of the eikonal, we then calculate the total cross-section as\n\\begin{equation}\n\\label{sigtot}\n\\sigma_{total}=2 \\int d^2 {\\vec b} [1-e^{-n(b,s)\/2}]\n\\end{equation}\nIn our model \n$n(b,s)$\nis split as\n\\begin{equation}\nn(b,s)=n_{soft}(b,s)+n_{hard}(b,s)\n\\end{equation}\nwhere we postulate the following factorization \n\\begin{equation}\nn_{soft\/hard}(b,s)=A_{BN}^{soft\/hard}(b,s) \\sigma_{soft\/hard}(s)\n\\end{equation}\nwith \n\\begin{equation}\nA_{BN}(b,s)=N \\int d^2 K_{\\perp}\\  e^{-iK_\\perp\\cdot b}\n {{d^2P(K_\\perp)}\\over{d^2 K_\\perp}} \n\\end{equation}\nwhere N is a normalization factor such that $\\int d^2 {\\vec b} A(b) =1$  and\n\\begin{equation}\n\\label{dpdk}\n {{d^2P(K_\\perp)}\\over{d^2K_\\perp}} = {{1}\\over{(2\\pi)^2}}\\int d^2 {\\vec b}\\ \ne^{iK_\\perp\\cdot b -\\int_0^{q_{max}} d^3{\\bar n}(k)[1-e^{-ik_t\\cdot b}]}\n\\end{equation}\nis the transverse momentum distribution \nof initial state soft gluons emitted in the  \nparton-parton collisions\nand where, for simplicity, \n$k_\\perp\\cdot b = {\\vec k}_\\perp\\cdot {\\vec b}$. In Equation (\\ref{dpdk})\n$q_{max}$ is the maximum transverse momentum allowed by kinematics to\nsingle soft gluon emission in a given hard collision, \naveraged over the parton \ndensities. \nAccording to the basic\nansatz of the Eikonal Minijet Model (EMM), \n\\begin{equation}\n\\label{sigmajet}\n\\sigma_{hard}\\equiv \\sigma^{AB}_{\\rm jet} (s) = \n\\int_{p_{tmin}}^{\\sqrt{s\/2}} d p_t \\int_{4\np_t^2\/s}^1 d x_1 \\int_{4 p_t^2\/(x_1 s)}^1 d x_2 \\sum_{i,j,k,l}\nf_{i|A}(x_1) f_{j|B}(x_2)~~\n \\frac { d \\hat{\\sigma}_{ij}^{ kl}(\\hat{s})} {d p_t}.\n\\end{equation}\n Here   $A$ and $B$ denote particles ($\\gamma, \\ p,\n\\dots$), $i, \\ j, \\ k, \\ l$ are parton types and $x_1,x_2$ the fractions\nof the parent particle momentum carried by the parton. $\\hat{s} =\nx_1 x_2 s$  and $\\hat{ \\sigma}$ are hard parton scattering\ncross--sections. As discussed in \\cite{corsetti}, kinematical considerations\nsuggest \\cite{greco}\n\\begin{equation}\n\\label{qmaxav}\nq_{max}(s)={{\\sqrt{s}} \n\\over{2}}{{ \\sum_{i,j}\\int {{dx_1}\\over{ x_1}}\nf_{i|A}(x_1)\\int {{dx_2}\\over{x_2}}f_{j|B}(x_2)\\sqrt{x_1x_2} \\int_{z_{min}}^1\n dz (1 - z)}\n\\over{\\sum_{i,j}\\int {dx_1\\over x_1}\nf_{i|A}(x_1)\\int {{dx_2}\\over{x_2}}f_{j|B}(x_2) \\int_{z_{min}}^1 (dz)}}\n\\end{equation}\nwith $z_{min}=4p_{tmin}^2\/(sx_1x_2)$ and $f_{i\/a}$ the valence \nquark densities used in the jet cross-section calculation.   \nThe steps we  follow to compare the model with data  are then the following:\n\\begin{enumerate}\n\\item choose the parameters for the hard scattering part, namely\n\\begin{description}\n\\item{(i)} parton densities (PDF), \n$p_{tmin}$ and $\\Lambda_{QCD}$ for the\nchosen PDF set in Equation (\\ref{sigmajet}) \n\\item{(ii)} model for $\\alpha_s$ in the infrared region and relevant \nparameters in Equation (\\ref{dpdk})\n\\end{description}\n\\item calculate $q_{max}(s,p_{tmin})$ for the given densities and\n  $p_{tmin}$\nusing Equation (\\ref{qmaxav})\n\\item calculate $n_{hard}(b,s)= A_{BN}^{hard}(b,s)\n  \\sigma_{jet}(s,p_{tmin})$\n \\item choose the parameters for the low energy part, namely\n\\begin{description}\n\\item{(i)}  the constant low energy cross-section $\\sigma_0$\n\\item {(ii)}  values for  $q_{max}^{soft}$\n\\end{description}\n\\item  calculate $n_{soft}(b,s)=\n A_{BN}^{soft}(b,s) \\sigma_0(1+\\epsilon {{2}\\over{\\sqrt{s}}})$ with\n $\\epsilon=0, 1$ depending upon the process being $p p$ or $p \n\\bar{p}$\n\\item calculate $n(b,s)$ and thus $\\sigma_{tot}$ \n\\item choose the parameter set which gives the best description of the\n  total cross-section up to the Tevatron data \\cite{e710,cdf,e811}\n\\end{enumerate}\nNotice that once a good set of parameters is found, one can \nuse $n(b,s)$ with fitted parameters to calculate survival \nprobabilities or diffractive Higgs production.\n\\subsection{Application to  total cross-section data}\nWe show in this section the application of the model to  the\ntotal \ncross-section for different PDFs. We find that our model is \nflexible enough to be able to reproduce the present data for \n$\\sigma_{tot}$ using all presently available PDFs. \nIn particular, all GRV \\cite{GRV,GRV94,GRV98} and MRST \\cite{MRST}\n densities give a good description  using \nthe singular $\\alpha_s$ model described in \\cite{corsetti}, \nwhile CTEQ densities \n\\cite{CTEQ} give an acceptable \nfit  up to Tevatron data, but  then fail to rise further. \n\nWe present these results by following the previously listed steps. We start\nby choosing the parameters for the hard scattering, and, following our\nprevious results \\cite{ggps}, we fix $p_{tmin}=1.15 \\ GeV$ in the jet\ncross-sections and calculate   $q_{max}$  for different PDF sets.\n We show the result in Fig.(1).\n\\begin{figure}\n\\label{qmax}\n\\includegraphics[width=\\textwidth,height=\\textwidth\n]{qmax_115_band.eps}\n\\caption{Average value of the maximum transverse momentum allowed to single\n  gluon emission, according to the model in {\\protect \\cite{ggps}}.}\n\\end{figure}\nWe notice the following:\\begin{itemize}\n\\item GRV densities are of two types, GRV98\\cite{GRV98}  \nfor which $q_{max}$ keeps on increasing \nlogarithmically and the older ones \n\\cite{GRV} for  which \n$q_{max}$ slows down past the TeV region, albeit still slowly increasing\n\\item CTEQ densities give values for $q_{max}$ which  increase more\n  rapidly than $\\ln {s}$ after typical Tevatron energies \n\\item MRST densities indicate a behaviour opposite to CTEQ, since they give\n$q_{max}$ values decreasing after the TeV cross mark.\n\\end{itemize} \nWe now turn to the jet cross-sections and examine\nthe  growth with energy of \n $\\sigma_{jet}$   for different PDFs. \n In Fig.(2)\nwe plot\nthese cross-sections for the same set of densities used to calculate\n$q_{max}$\nand for  $p_{tmin}=1.15 \\ GeV$.\n\\begin{figure}\n\\label{sigjets}\n\\includegraphics[width=\\textwidth,height=\\textwidth\n]{sigjet_115_band.eps}\n\\caption{Mini-jet QCD cross-sections for different PDFs as indicated in the\n  figure}\n\\end{figure}\nFrom this figure we notice that :\n\\begin{itemize}\n\\item the jet cross-sections  for GRV densities\n  increase faster than all the others\n\\item the jet cross-sections for CTEQ increase more or less similarly to\n  those for the MRST group\n\\end{itemize}\nThe implications are that $\\sigma_{jet}$ with GRV densities, \nwhich increase faster than $\\sigma_{jet}$ with MRST,\nneed more softening, $\\sigma_{jet}$ with CTEQ, which increase \nless than with GRV, should not be\nsmeared that much.\n\nTo proceed further, we now need to calculate the $b-distribution$ and \nfix the low-energy parameters, like $\\sigma_0$. The $b-distribution$ requires\nto input the behaviour of $\\alpha_s$ in the infrared region. We have shown\nin \\cite{ggps} the need to use a singular but integrable expression for \n$\\alpha_s$ in order to reproduce both the sudden rise and the subsequent\nsoftening of the total cross-section. Our choice is an expression like\n\\begin{equation}\n\\label{alphaRich}\n\\alpha_s(k_t)={{12 \\pi }\\over{(33-2N_f)}}{{p}\\over{\\ln[1+p({{k_t}\n\\over{\\Lambda}})^{2p}]}}\n\\end{equation}\nwhich depends on the singularity parameter $p$, in addition to the scale\n$\\Lambda$. In \\cite{ggps} we have chosen the value $p=0.75$ and \n$\\Lambda=100\\ MeV$, other choices\nare also possible \\cite{pramana}. Turning now to the low-energy part, $n_{soft}(b,s)$, we\nchoose $\\sigma_0=48\\ mb$ and use for $A_{BN}^{soft}$ a set of \n$q_{max}$ values which\nreproduce the low energy behaviour, and which appear in Fig. (1).  \n\\section{Comparison with data and expectations at LHC densities}\nWe can now input all the above in the eikonal representation for the total\ncross-section and obtain\nthe results shown in Fig.(\\ref{sigtotsing})\n\n\\begin{figure}[t]\n\\label{sigtotsing}\n\\includegraphics[width=\\textwidth,height=\\textwidth]{eiksig_115_band_pp.eps}\n\\caption{Data for total cross-sections for $pp$ and $p{\\bar p}$ \ncompared with model predictions at LHC for $p p$ scattering,\n using different PDFs.}\n\\end{figure}\n for the singular $\\alpha_s$ case\n and GRV, MRST and CTEQ \ndensities. For the sake of clarity, we only plot curves for \n$p p$ scattering, referring the reader to \n\\cite{ggps,pramana} for the curves for $p{\\bar p}$ and for a different\n parameter set,  or for predictions from \nother models \\cite{blockfroissart,dl,bghp,cudell,luna}. \nFrom the figure we see that the calculation with CTEQ densities \nappears very unlikely. The effect is due to the fact that the jet\ncross-sections in the CTEQ case do not rise as much as the others \nwhile the softening\neffect is stronger, as it is driven by $q_{max}$, which is strongly \nincreasing for these densities. As a\n result, the cross-section starts decreasing. \nNotice that while the behaviour of $\\sigma_{jet}$ is dominated by the \ngluon densities, that of $q_{max}$ is determined only by the valence \nquarks, as we assume this to be the leading order effect. \n\nThe curves shown in Fig.(\\ref{sigtotsing}) indicate that the coming\nmeasurement at $\\sqrt{s}=900\\ GeV$ will be very important in determining\nwhich of these curves  best describes the data. It   can then be used to \nselect  the parameter set, basically PDF's and $p_{tmin}$, \nfor a prediction at the  project LHC energy, $\\sqrt{s}=14\\ TeV$.   \nIf the UA5 \\cite{ua5} value  at $\\sqrt{s}=900\\ GeV$ \nis confirmed with a comparable error, \nthen, for the set of parameters discussed in this note,  at\n$\\sqrt{s}=14\\ TeV$ our model gives\n$\\sigma_{total}^{GRV98}=90.2\\  mb$, $\\sigma_{total}^{GRV}=100.2\\  mb$\nand $\\sigma_{total}^{MRST76}=103.4\\  mb$. As shown in \\cite{pramana}, \nchanging the parameter set, namely $\\sigma_0$, $p_{tmin}$ or the singularity \nindex $p$, give values in the range $88\\div 111\\ mb$.\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n\n\n\n\\section{Conclusions}\nWe have presented a version of the eikonal minijet model which allows a\ngood description of total cross-sections at asymptotic energies and\ndiscussed its connection with the small x-behaviour of various types of\nparton densities.\nThis model is based on a softening of the\nmini-jet cross-sections due to  an s-dependent \nb-distribution in\nthe proton, which we calculate using a soft gluon resummation model down to\nzero energy of the soft gluons.\n\\section*{Acknowledgments}\nWe acknowledge  partial support from EU CT-2002-0311. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe breakthrough discovery of Witten-Kontsevich (\\cite{Witten,Kon}) established an intimate link between mathematical physics and enumerative geometry. From a general perspective, one aims at studying the Gromov-Witten invariants of a smooth projective variety $X$  in terms of suitable integrable hierarchies. From this point of view, the Witten-Kontsevich case corresponds to the situation when $X$ is a point and it was shown that the exponential of the generating function of intersection numbers on the moduli space of curves was a common solution of the Virasoro constraints and of the KdV hierarchy. Therefore, following the generalization for the case of the projective space proposed in \\cite{EHX}, on the one hand one wonders if the generating function fulfills a generalization of the Virasoro constraints. On the other hand, one also wants to know if the generating function is given by (the logarithm of) a particular tau-function of an integrable hierarchy. Nowadays, the case of varieties with semisimple quantum cohomology is well understood and the answer to both questions is affirmative (\\cite{Teleman} and \\cite{DubrovinZ2}; see also \\cite{DubrovinZ, Getzler,Givental, LiuTian, OK, OK-Toda}). \n\n\nIt is worth pointing out that, for each $X$, the explicit Virasoro operators as well as the relevant integrable hierarchy may vary; for instance, the $2$-Toda hierarchy appears when dealing with the equivariant GW invariants of ${\\mathbb P}^1$ (\\cite{OK-Toda}). Nevertheless, one recognizes some common features that arise among these results. Let us mention some of them. In \\cite{GiventalAn}, Givental studied a case in which the total descendent potential is a $\\tau$-function for the $n$KdV-hierarchy by using $n-1$ copies of the KdV. Thus,  the total descendent potential of a semisimple Frobenius manifold was defined in \\cite{Givental} as a product of $n$ copies of Witten-Kontsevich $\\tau$-functions. Dealing with a case of orbifold quantum cohomology, it has been proved in \\cite{JarvisKimura} that the Virasoro constraints decomposed as $n$ copies of (half of) the Virasoro algebra, that their solution was the product of Witten-Kontsevich $\\tau$-functions, and that the relevant integrable hierarchy consisted of $n$ commuting copies of the KdV hierarchy. Finally, in \\cite{DVV, KacSchwarz} it was shown that  the solution of the Virasoro constraints in the case of Witten-Kontsevich is unique  (up to a constant factor) and this uniqueness also holds in other setups (e.g. \\cite{LiuUnique}).\n\nThis paper, making use of the representation theoretic properties of the Virasoro algebra, \noffers new insights and results on these properties and provides evidences that the above mentioned properties rely heavily on the structure of the Virasoro algebra and its representations. Our study of explicit expressions for Virasoro representations (see  \\S2) is general enough to encode many of the known representations within the framework of Virasoro constraints. Further, it allows us to determine whether a representation is the tensor product of Lie algebra representations and if a solution factorizes as a product of solutions of those representations. An explicit realization of these ideas is carried out in \\S3 for the case of  smooth projective varieties with trivial odd cohomology and vanishing first Chern class. Thus, we think that our approach may help in determining the explicit expression of the Virasoro operators as well as the corresponding integrable hierarchies for other types of varieties $X$ (see \\S\\ref{subsec:final}).  Now, let us be more precise and explain the contents of the paper.\n\n\nWe begin by fixing a pair $(A,(\\, ,\\,))$ consisting of a finite dimensional vector space and a non-degenerate bilinear form. Associated to this data, we consider a Heisenberg algebra ${\\mathbb H}(A)$ and its universal enveloping algebra $\\uh[(A)]$. Let us denote by ${\\mathcal W}_>$ the positive half of the Virasoro algebra and recall that it contains $\\sl(2)$ canonically. Section 2 is entirely devoted to the study of Lie algebra maps ${\\mathcal W}_>\\to \\uh[(A)]$. To begin with, we show that, under some homogeneity condition, there is a canonical bijection between $\\Hom_{\\text{Lie-alg}}( {\\mathcal W}_>, \\uh[(A)])$ and $\\Hom_{\\text{Lie-alg}}( \\sl(2), \\uh[(A)]$ (Theorem~\\ref{thm:W^+-sl2}). This is highly non-trivial since, in general, the problem of extending a map defined on $\\sl(2)$ to ${\\mathcal W}_>$ involves  infinitely many conditions (see~\\cite{PT}). Accordingly, it is natural to expect that many properties of a map ${\\mathcal W}_> \\to \\uh[(A)]$ can be stated in terms of its restriction to $\\sl(2)$. Actually, we prove that such a map decomposes as tensor product of Lie algebra representations if and only if its restriction does. Moreover, this factorization is possible only if $A$ decomposes as the orthogonal sum of  two subspaces (Theorem~\\ref{thm:rhodecomp}). We finish this section by showing that the fact that ${\\mathcal W}_>$ admits no non-trivial finite dimensional representations has important consequences for the structure of the solutions of the equations $(\\rho_1\\otimes 1 + 1\\otimes \\rho_2)(L)(\\sum_i f_i\\otimes g_i)=0$, where $L\\in {\\mathcal W}_>$ (see Theorem~\\ref{thm:solutionrhoi}). That is, decompositions of the representation and of the solutions depend strongly on the structure of ${\\mathcal W}_>$ and of $(A,(\\, ,\\,))$. \n\nAlthough the previous results are interesting on their own, \\S3 explores their application to concrete situations; for instance, relations with integrable hierarchies (e.g. multicomponent KdV). The case we have chosen to illustrate this issue is that of the differential operators appearing in the Virasoro conjecture when $X$ has trivial odd cohomology (for instance, whenever $X$ has semisimple even quantum cohomology) and its first Chern class vanishes. Then, Theorem~\\ref{thm:Lbar=Lhat} shows explicitly how to obtain these operators as the images of the generators $L_k\\in{\\mathcal W}_>$ by:\n\t{\\small $$\n\t\\hat\\rho\\,:\\, {\\mathcal W}_>\\, \\overset{\\rho}\\longrightarrow\\,  \\uh[(A)] \\, \\overset{\\widehat{\\,}}\\longrightarrow\\, \\End\\big({\\mathbb C}[[\\{t_{i,\\alpha}\\vert 1\\leq \\alpha\\leq \\dim(A) , i=1,3,\\ldots\\}]]\\big)\n\t$$}for $A=H^*(X,{\\mathbb C})$ endowed with the Poincar\\'e pairing. Then, our results of \\S2 imply that $\\hat\\rho$ decomposes as the tensor product of Lie algebra representations associated to data $({\\mathbb C},(\\, , \\,))$; i.e. the $1$-dimensional case. The detailed study of the $1$-dimensional case carried out in \\S\\ref{subsec:solutions1dim} shows that, up to re-scaling the variables, the corresponding operators \\emph{always }come from a representation:\n\t$$\n\t\\sigma: {\\mathcal W}_> \\,\\longrightarrow \\, \\operatorname{Diff}^1({\\mathbb C}((z)))\n\t$$\nwhich means that we can profit from \\cite{KacSchwarz, Plaza-AlgSol} to build the unique solution in terms of a $\\tau$-function of the KdV hierarchy. Putting everything together, we have the main results of this section. First, in the case of $\\dim A=1$:\n\n\\begin{thm*}[see Theorem~\\ref{thm:solutionWscaleKdV}]\nLet $\\rho \\in \\homlie({\\mathcal W}_>,\\End({\\mathbb C}[[t_1,t_3,\\ldots]]))$ be such that $\\rho(L_k)$  is of type $k$ for $k\\geq -1$ and that all coefficients of $\\rho(L_{-1})$ are non zero. \n\nThen, there exists a unique $\\tau(t)\\in{\\mathbb C}[[t_1,t_3,\\ldots]]$, with $\\tau(0)=1$, such that:\n\t$$\n\t\\rho(L_k)(\\tau(t))\\,=\\,0\\qquad k\\geq -1\n\t$$\nFurther, the solution $\\tau(t)$ is a $\\tau$-function of the scaled KdV hierarchy.\n\\end{thm*}\\noindent and, for $\\dim A=N \\geq 2$:\n\n\n\\begin{thm*}[see Theorem~\\ref{thm:solutionproductKdV}]\nLet $\\rho:{\\mathcal W}_>\\to \\uh[(A)]$ be as in \\S\\ref{subsec:baby}.\n\nThere exist $S\\in \\Gl(A)$ and functions $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)\\in {\\mathbb C}[[t_{1,\\alpha},t_{3,\\alpha},\\ldots]]$ such that:\n\t$$\n\t\\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big)\\,=\\, 0\n\t$$\n\nFurther, $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)$ are $\\tau$-functions of the scaled KdV hierarchy.\\end{thm*}\n\nWe hope that our methods shed some light on the explicit expressions of the Virasoro operators and of the relevant integrable hierarchies that appear when studying the Virasoro conjecture.  We also think that the techniques presented here can be applied to many instances of representations of ${\\mathcal W}_>$ which appear in a variety or problems such as recursion relations, Hurwitz numbers, knot theory, etc.~. We sketch some ideas in \\S\\ref{subsec:final} although all of them deserve further research.  \n\n\\emph{Acknowledgements.} We thank A. Givental and G. Borot for explaining us some facts on their papers. The first author  wish to express his gratitude to the Max Planck Institute f\\\"ur Mathematik (Bonn) for the invitation in fall 2014. \n\n\n\n\n\n\n\\section{Lie algebras}\\label{sec:ReprW>}\n\nLet ${\\mathcal W}$ be the Witt algebra; that is, the ${\\mathbb C}$-vector space\nwith basis $\\{L_k\\}_{k\\in{\\mathbb Z}}$ endowed with the Lie bracket\n$[L_i,L_j]=(i-j) L_{i+j}$, and let ${\\mathcal W}_>$ be the subalgebra\ngenerated by $\\{L_k\\}_{k\\geq -1}$. It contains a copy of $\\sl(2)$ via $\\sl(2)=<\\{L_{-1}, L_0, L_1\\}> \\subset {\\mathcal W}_>$. Recall that ${\\mathcal W}_>$ is also called the \\emph{positive half of the centereless Virasoro algebra}. \n\nIn this section, we study certain maps from  $\\sl(2)$ and their extensions to  ${\\mathcal W}_>$. These results will eventually allow us to relate the representation theories of ${\\mathcal W}_>$ and $\\sl(2)$. A further consequence is that, in order to construct the operators $L_0, L_1, L_2, \\ldots$ one only has to start with $L_{-1}$ and follow some simple procedures and choices. \n\nIt is worth mentioning that a study of the representation theory of ${\\mathcal W}_>$ in terms of the representation theory of its subalgebra $\\sl(2) \\subset {\\mathcal W}_>$ has been carried out in \\cite{PT} in full generality. \n\t\n\\subsection{Preliminaries}\\label{subsec:prelim-beta-Diff-to-C[[t]]}\n\n\nLet us be more precise. Let $(A, (\\, ,\\,))$ be given, where $A$ is a finite dimensional ${\\mathbb C}$-vector space and $(\\,,\\,)$ is a non-degenerated bilinear pairing. For a basis $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots, n\\}$ of $A$, let $\\eta=(\\eta_{\\alpha\\beta})$ denote the matrix associated to the given bilinear product; that is, $\\eta_{\\alpha\\beta}:=(a_{\\alpha}, a_{\\beta})$. The inverse will be denoted with superindexes; i.e. $\\eta^{\\alpha\\beta}:=(\\eta^{-1})_{\\alpha\\beta}$. \n\nLet us consider unknowns $\\{p_i,q_i\\vert i\\geq 1\\}$ and introduce $p_{i,\\alpha}:=p_i\\otimes a_{\\alpha}$ and $q_{i,\\alpha}:=q_i\\otimes a_{\\alpha}$. Let ${\\mathbb H}(A)$ be the Heisenberg algebra generated by $\\{1, p_{i,\\alpha},q_{i,\\alpha}\\vert i\\geq 1, \\alpha=1,\\ldots, n\\}$, whose elements will be called \\emph{operators},  endowed with the Lie bracket: \n    \\begin{equation}\\label{eq:pqcommutation}\n    \\begin{gathered}\n    \\, [p_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \\delta_{i,j} i \\eta^{\\alpha\\beta}\\cdot 1\n    \\\\\n   \\,  [p_{i,\\alpha}, p_{j,\\beta}] \\,=\\,[q_{i,\\alpha}, q_{j,\\beta}] \\,=\\, 0\n    \\\\\n   \\,  [p_{i,\\alpha}, 1] \\,=\\,[q_{i,\\alpha}, 1] \\,=\\, 0\n    \\end{gathered}\n    \\end{equation}\nWe define their degree by $\\deg(q_{i,\\alpha})=i$, $\\deg(p_{i,\\alpha})=-i$ and $\\deg(1)=0$. \n\nAlthough the definition of the Heisenberg algebra depends on the pair $(A, (\\, ,\\,))$, it will be simply denoted by ${\\mathbb H}$ if no confusion arises.\n\nFor ${\\mathbb H}$ as above, let us define $\\uh$ the universal enveloping algebra of ${\\mathbb H}$, which is the quotient of the tensor algebra of  ${\\mathbb H}$ by the two-sided ideal generated by the relations $u\\otimes v- v\\otimes u-[u,v]$. \n\nMotivated by the explicit forms of the Virasoro operators considered in the literature (\\cite{DVV, DubrovinZ,EO, Getzler, Givental,KSU, KazarianZograf}), we introduce the following notion. The ultimate meaning of this notion is unveiled in Lemma~\\ref{lem:Diff1scaleW}.\n \n\n\\begin{defn}\\label{defn:type}\nAn operator $T\\in \\uh$ is \\emph{of type $i\\geq -1$} if it is a linear combination of $p_{2i+3,\\alpha}$ and double products of degree $-2i$; i.e. $p_{j,\\alpha}p_{2i-j,\\beta}$, $q_{j,\\alpha}p_{2i+j,\\beta}$ and $q_{j,\\alpha}q_{-j-2i,\\beta}$. If $i=0$ we also allow a constant term.\n\nThe subset consisting of operators of type $i\\geq -1$ will be denoted by $\\uh[(A)]_i$ (or, simply, $\\uh_i$). \n\\end{defn}\n\nThis section deals with the study of homomorphisms of Lie algebras:\n\t$$\n\t\\rho:{\\mathcal W}_> \\, \\longrightarrow\\, \\uh \\qquad \\text{s.t. } \\rho(L_i)\\in \\uh_i\n\t$$\n\t\nLet us illustrate the previous definition. From now on, according to Einstein convention, summation over repeated indices will be understood. For instance, an operator of type $-1$ is of the form:\n    \\begin{equation}\\label{eq:type-1}\n    b_{-1}^{0,1} p_1 +  q_1 a_{-1}^{1,1}  q_1^T  + q_{i+2} b_{-1}^{i+2,i}  p_i \\, \\in\\, \\uh_{-1} \n    \\end{equation}\n(the sum  runs over the set of odd positive integers $i$),  $p_i$ is the column vector $(p_{i,1}\\ldots,p_{i,n})^T$, $q_i$ is the row vector $(q_{i,1}\\ldots,q_{i,n})$,  $b_{-1}^{0,1}$ is a row vector, $a_{-1}^{1,1}$  and $b_{-1}^{i+2,i}$ are $n\\times n$ matrices. For brevity, we set $a:=a_{-1}^{1,1} $. \n\nSimilarly, an type $0$ operator can be expressed as:\n    \\begin{equation}\\label{eq:type0}\n   b_{0}^{0,3} p_3 + b_0^{0,0}  + q_{i} b_{0}^{i,i}  p_i \\, \\in\\, \\uh_{0}\n    \\end{equation}\nwhile an operator of type $i\\geq 1$ is of the form:\n\t\\begin{equation}\\label{eq:type}\n\tb_i^{0,2i+3} p_{2i+3} + p_{j}^T c_i^{j,2i-j} p_{2i-j} +  q_j b_i^{j,2i+j} p_{2i+j}\n\t\\,\\in\\,\\uh_i \\qquad i\\geq 1\n\t\\end{equation}\nfor a row vector $b_i^{0,2i+3}$ and $n\\times n$-matrices $b_i^{j,2i+j}$ and $c_i^{j,2i-j}$, where  $c_i^{j,2i-j}=(c_i^{2i-j,j})^T$ and the sum runs over $j$ odd.\n\nIt is convenient to offer an interpretation of these matrices. Recall that $q_i$ is the row vector $(q_{i,1},\\ldots, q_{i,n})$, which can be thought as an ${\\mathbb H}$-valued vector of $A$. A similar argument holds for the column vector $p_i$. Thus, under a basis change in $A$, the matrix $b$ in $q_i\\cdot b\\cdot p_i$ behaves as a bilinear form on $A$. The same fact applies to all $a$, $b$ and $c$ matrices.  Similarly, column vectors $b_i^{0,2i+3}$ are understood as vectors on $A$ while row vectors are like linear forms. \n\nIt is worth noticing how these operators behave w.r.t. the Lie bracket. Indeed, the computations given in subsection~\\S\\ref{subsect:CR} and the linearity of the bracket show that it is compatible with the degree:\n\t\\begin{equation}\\label{eq:LieT}\n\t[\\,\\, , \\,\\, ] \\,:\\, \\uh_i \\times \\uh_j \\, \\longrightarrow\\, \\uh_{i+j}\n\t\\end{equation}\nwhere $i,j,i+j\\geq -1$. In particular, it follows that $\\oplus_{i\\geq -1} \\uh_i$ is a partial Lie algebra. \n\t\n\n\n\n\\subsection{Maps from $\\sl(2)$ to Heisenberg}\\label{subsect:sl2}\n\n\n\n\nLet $\\sl(2)$ be the Lie algebra of $\\operatorname{Sl}(2,{\\mathbb C})$. We fix a  basis $\\{e,f,h\\}$  of $\\sl(2)$  satisfying the relations:\n    $$\n    [e,f]=h\\; , \\qquad\n    [h,e]=2e \\; , \\qquad\n    [h,f]=-2f\\; .\n    $$\nIn particular, the previous choice yields a natural embedding:\n    \\begin{equation}\\label{eq:emb-sl2-W+}\n    \\iota:\\sl(2)\\,\\hookrightarrow\\, {\\mathcal W}_>\n    \\end{equation}\nby mapping $f$ to $L_{-1}$, $h$ to $-2L_0$, and $e$ to $-L_1$. \n\n\n\\begin{lem}\\label{lem:FyieldsH}\nLet $F\\in  \\uh_{-1}$ be as in \\eqref{eq:type-1}. \nAssume that\n$b_{-1}^{i+2,i}$ is invertible for all $i$. It holds that:\n    {\\small $$\n    \\left\\{\n    \\begin{gathered}\n    H\\in\\uh_0 \\text{ s.t.}\n    \\\\\n    [H,F]\\,=\\, - 2F\n    \\end{gathered}\n    \\right\\}\n    \\,\\simeq\\,\n    \\left\\{\n    \\begin{gathered}\n    (b,B)\\in {\\mathbb C}\\times \\operatorname{Mat}_{n\\times n}({\\mathbb C})\n    \\text{ s.t.}\n    \\\\ (B \\eta^{-1} + \\operatorname{Id}) (a +a^T) +  (a +a^T)(B \\eta^{-1} +\\operatorname{Id})^T = 0\n    \\end{gathered}\n    \\right\\}\n    $$}\n\\end{lem}\n\n\\begin{proof}\nOur task consists of computing the  bracket $[H,F]$ explicitly. Recall that, for simplicity, we have set $a=a_{-1}^{1,1}$. Since $H\\in\\uh_0$, it must be of the form $H:=  b_0^{0,3} p_3 +  b_{0}^{0,0}   +  q_{i}  b_{0}^{i,i} p_i $ where $b_0^{0,3}$ is a row vector, $b_{0}^{0,0}$ is an homothety, and $b_{0}^{i,i}$ are $n\\times n$ matrices.\n\nHaving in mind the commutation relations of \\S\\ref{subsect:CR}, the bracket $[H,F]$ is a linear combination of $p_{1}$, $q_{1\\alpha}q_{1\\beta}$ and $q_{i+2,\\alpha}p_{i,\\beta}$. Therefore, the expression $[H,F]=-2F$ is equivalent to the following identities:\n\t$$\n\t    \\begin{aligned}\n    \\big( 3 b_0^{0,3} \\eta^{-1}  b_{-1}^{3,1} - b_{-1}^{0,1}\\eta^{-1}  b_0^{1,1} \\big) p_1 \\, & =\\, -2 b_{-1}^{0,1} p_1\n    \\\\\n     q_1 b_0^{1,1}\\eta^{-1}  \\big( a  + a^T   \\big) q_1^T   \\, & =\\,  -2 q_1 a   q_1^T\n    \\\\\n    q_{i+2}\\big( (i+2) b_0^{i+2,i+2}\\eta^{-1}  b_{-1}^{i+2,i}  - i  b_{-1}^{i+2,i}\\eta^{-1}  b_0^{i,i}\\big) p_i\n    \\, & =\\,  - 2 q_{i+2} b_{-1}^{i+2,i} p_i\n    \\qquad\\forall i \\geq 1\n    \\end{aligned}\n\t$$\nObserve that $q_1 A q_1^T = q_1 B q_1^T$ if and only if $A+A^T=B+B^T$.  Hence, \nthe above system is equivalent to the following equations:\n        \\begin{subequations}\\label{eq:FyieldsH}\n    \\begin{align}\n    \\label{eq:FyieldsH:1}\n    3 b_0^{0,3} \\eta^{-1} b_{-1}^{3,1} - b_{-1}^{0,1}\\eta^{-1} b_0^{1,1} \\, & =\\, -2 b_{-1}^{0,1}\n    \\\\\n     \\label{eq:FyieldsH:2}\n    b_0^{1,1} \\eta^{-1} (a +a^T)+  (a +a^T)( b_0^{1,1}\\eta^{-1} )^T  \n     \\, & =\\,  - 2 (a +a^T)\n         \\\\\n    \\label{eq:FyieldsH:3}\n    (i+2) b_0^{i+2,i+2}\\eta^{-1} b_{-1}^{i+2,i}  - i b_{-1}^{i+2,i} \\eta^{-1}  b_0^{i,i}\n    \\, & =\\,  - 2 b_{-1}^{i+2,i}\n    \\qquad\\forall i\\geq 1 \n    \\end{align}\n        \\end{subequations}\n\nNote that, since $b_{-1}^{i+2,i}$ and $\\eta$ are invertible,  given a pair $(b,B)$ as in the statement, this system has a unique solution for $b_0^{0,0}=b$ and $b_0^{1,1}=B$; namely,\n\t\\begin{equation}\\label{eq:coefH}\n\t\\begin{aligned}\n\tb_0^{0,3}\\, &=\\, \\frac13 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) (\\eta^{-1} b_{-1}^{3,1})^{-1} \n\t\\\\\n\tb_0^{i+2,i+2} \\,&=\\, \\frac1{i+2}  b_{-1}^{i+2,i} (i \\eta^{-1} b_0^{i,i}-2 )(\\eta^{-1} b_{-1}^{i+2,i})^{-1} \\qquad\\forall i\\geq 1\n\t\\end{aligned}\n\t\\end{equation}\nThe converse is straightforward.\n\\end{proof}\n\n\n\\begin{exam} \nSet  $F= b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  \\frac{i+2}2 q_{i+2} p_i$ and $b_0^{1,1}=-\\frac12$, then  $H=-2 b_{-1}^{0,1} p_3 +   b_{0}^{0,0}   + i q_i p_i$.  Note that $i q_{i} p_i$ is the \\emph{degree operator}.\n\\end{exam}\n\n\\begin{exam} \nLet us consider the case where the chosen basis in $A$ is orthonormal; i.e. $\\eta$ is the identity matrix, and suppose that:\n    $$\n    F\\,=\\, b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  q_{i+2} p_i\\,\\in\\,\\uh_{-1}\n    $$\nThen, operators $H$ given by Lemma~(\\ref{lem:FyieldsH}) acquire the form:\n    $$\n    H \\,=\\,    \\frac13 b_{-1}^{0,1}( b_0^{1,1}-2)   p_3 \n    +   b_{0}^{0,0}   +  \\frac1{i} q_i  ( b_0^{1,1}-(i-1)) p_i,\\in\\,\\uh_0\n    $$\nwhere $ b_{0}^{0,0} \\in {\\mathbb C}$ and $b_0^{1,1}$ verifies $(b_0^{1,1} +\\operatorname{Id}) a  +  a (b_0^{1,1} +\\operatorname{Id})^T =0$. \n\\end{exam}\n\n\\begin{exam}\\label{exam:dim1}\nFinally, let $\\dim A=1$ , $a, \\eta\\in{\\mathbb C}^{\\ast}$ and $F = b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  q_{i+2} \\eta p_i$. Then, $ b_0^{i,i}=-\\eta$ for all $i$ and $H=-  b_{-1}^{0,1}  p_3     +   b_{0}^{0,0}   -   q_i  \\eta  p_i$. \n\\end{exam}\n\n\n\n\n\n\\begin{lem}\\label{lem:T'T'bracket}\nLet $H$ be as in equation~\\eqref{eq:type0} and $\\uh'_i$ be the subspace:\n\t$$\n\t\\uh'_i\\,:=\\, \\{ T\\in\\uh_i  \\text{ s.t. } [H,T]= 2i T \\}\n\t$$\nThen, it holds that $[ \\uh'_i ,  \\uh'_j ]\\subseteq \\uh'_{i+j} $.\n\\end{lem}\n\n\\begin{proof}\nThe claim follows easily from~\\eqref{eq:LieT} and the Jacobi identity.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:sl2-symmmatrices}\nLet $F$ and $H$ be as in equations \\eqref{eq:type-1} and \\eqref{eq:type0} respectively. \n\nThere is a surjective map:\n    {\\small\n    $$\n    \\left\\{\n        \\begin{gathered}\n            c \\in M_{n\\times n}({\\mathbb C})\\text{ such that }\n            \\\\\n             b_0^{0,0} =   \\operatorname{Tr}( c \\eta^{-1}  (a+a^T)  (\\eta^{-1})^T\n            \\\\\n            \\text{and equation~\\eqref{eq:HF-2F:2} below}\n        \\end{gathered}\n    \\right\\} \n    \\, \\longrightarrow\\,\n    \\left\\{\n        \\begin{gathered}\n        \\sigma \\in \\homlie(\\sl(2),\\uh) \\\\\n         \\text{such that } \\sigma(f)=F, \\\\\n         \\sigma(h)=H  \\text{ and }\\sigma(e)\\in\\uh_1\n        \\end{gathered}\n    \\right\\}\n    $$}\nMoreover, $c_1$ and $c_2$ have the same image iff $c_1+c_1^T= c_2+c_2^T$. Thus, the restriction of the above map to symmetric matrices yields a bijection. \n\\end{thm}\n\n\\begin{proof}\nGiving a map $\\sigma$ as in the r.h.s. is equivalent to set an operator $E\\in\\uh_1$, such that $[E,F] = H $ and $ [H,F]= -2F$. Consider:\n     \\begin{equation}\\label{eq:E=type1}\n     E\\,=\\,  b_1^{0,5} p_5 + p_1^T c_1^{1,1}p_1  + q_i  b_1^{i,i+2}  p_{i+2}\\,\\in\\, \\uh_1\n     \\end{equation}\nwhere, for simplicity, we will set $c= c_1^{1,1}$. The identity $[H,E]= 2E$, expressed in terms of the coefficients of the operators, is equivalent to the following equations (thanks to the computations of \\S\\ref{subsect:CR}):\n    \\begin{subequations}\n    \\label{eq:HF-2F}\n    \\begin{align}\n     3 b_{0}^{0,3}\\eta^{-1} b_1^{3,5} - 5 b_{1}^{0,5}\\eta^{-1} b_0^{5,5}\n     \\, & =\\,  2 b_1^{0,5}\\label{eq:HF-2F:1}\n    \\\\\n    - (c^T + c) \\eta^{-1}b_0^{1,1} - (\\eta^{-1}b_0^{1,1})^T  (c^T + c)  \n    \\, & =\\, 2 (c^T + c) \\label{eq:HF-2F:2}\n  \n    \\\\\n    r b_0^{r,r}  \\eta^{-1}  b_1^{r,r+2} - (r+2) b_1^{r,r+2} \\eta^{-1}  b_0^{r+2,r+2}\n    \\,&=\\, 2 b_1^{r,r+2} \\label{eq:HF-2F:3}\n    \\end{align}\n    \\end{subequations}\n\n\nAnalogously, expanding the relation $[E,F]=H$ with the help of \\S\\ref{subsect:CR} yields the system:\n    \\begin{subequations}\n    \\label{eq:EFH}\n    \\begin{align}\n    \\label{eq:EFH:1}\n    \\operatorname{Tr}( c \\eta^{-1} (a+a^T) (\\eta^{-1})^T ) \\, & = \\,  b_0^{0,0}\n    \\\\\n    \\label{eq:EFH:2}\n    - b_{-1}^{0,1}\\eta^{-1}  b_1^{1,3} + 5  b_1^{0,5} \\eta^{-1} b_{-1}^{5,3} \\, & = b_0^{0,3}\n    \\\\\n    \\label{eq:EFH:3}\n    3 b_1^{1,3}\\eta^{-1} b_{-1}^{3,1}   +  (a+a^T) (  \\eta^{-1})^T  (c+c^T )  \\, & = \\,  b_0^{1,1}\n    \\\\\n    \\label{eq:EFH:4}\n    (r+2) b_1^{r,r+2} \\eta^{-1} b_{-1}^{r+2,r} - (r-2) b_{-1}^{r,r-2} \\eta^{-1}   b_1^{r-2,r} \\, & = \\, b_0^{r,r} \\quad\\forall r>2\n    \\end{align}\n    \\end{subequations}\nHaving in mind the properties of the trace, one observe that these equations only depends on $c+c^T$.\n\n\nIt remains to show that equations \\eqref{eq:HF-2F} and \\eqref{eq:EFH} are equivalent to the conditions of the claim; that is, that they can be reduced to \\eqref{eq:HF-2F:2} and \\eqref{eq:EFH:1}. \n\nAssuming \\eqref{eq:HF-2F:2} and \\eqref{eq:EFH:1}, one gets $b_1^{1,3}$ from \\eqref{eq:EFH:3}; then, $b_1^{0,5}$ is determined by \\eqref{eq:EFH:2}; and, $b_1^{r,r+2}$ is obtained from \\eqref{eq:EFH:4}. We claim that \\eqref{eq:HF-2F:1} is fulfilled too. Indeed, a long but straightforward computation shows that \\eqref{eq:HF-2F:1} is derived from \\eqref{eq:coefH}, \\eqref{eq:HF-2F:2} together with  the case $r=3$ of \\eqref{eq:EFH:4}. Similarly,  \\eqref{eq:HF-2F:3} follows from  \\eqref{eq:coefH}, \\eqref{eq:FyieldsH:3}  and \\eqref{eq:EFH:4}.\n\\end{proof}\n\n\\subsection{Extending to ${\\mathcal W}_>$}\\label{subsect:W>}\n\nIn order to extend a map defined on $\\sl(2)$ to one on ${\\mathcal W}_>$, one should choose an endomorphism $T$, define $\\rho(L_i)$ by equations~\\eqref{eq:rho(L_2)def} and \\eqref{eq:rho(L_i)def} and check infinitely many constraints (see~\\cite{PT}). However, in our situation the following Lemma simplifies that approach drastically; there will exist a unique $T$ satisfying all the requirements. \n\n\\begin{lem}\\label{lem:adFinjective}\nLet $F,H$ be as in equations~\\eqref{eq:type-1}   and~\\eqref{eq:type0}. The map:\n\t$$\\ad(F):\\uh'_i \\overset{\\sim}\\longrightarrow \\uh'_{i-1}\n\t$$\nis an isomorphism for $i\\geq 2$.\n\\end{lem}\n\n\\begin{proof}\nFirst, one has to prove that given an operator:\n\t$$\n\tS\\,:=\\, b_{i-1}^{0,2i+1} p_{2i+1} \\, +\\, p_j^T c_{i-1}^{j,2i-j-2} p_{2i-j-2} \\, +\\, q_j b_{i-1}^{j,j+2i-2} p_{j+2i-2}\\,\\in\\,\\uh_{i-1}\n\t$$\nof type $i-1\\geq 1$, there is exactly one operator:\n\t$$\n\tT\\,:=\\, b_i^{0,2i+3} p_{2i+3} \\, +\\, p_j^T c_i^{j,2i-j} p_{2i-j} \\, +\\, q_j b_i^{j,j+2i} p_{j+2i}\\,\\in\\,\\uh_i\n\t$$\nof type $i$ satisfying $\\ad(F)(T)=S$ where  $\\ad$ denotes the adjoint representation and $F$ is given by equation~\\eqref{eq:type-1}.\n\nNow, one proceeds as in the proof of Lemma~\\ref{lem:FyieldsH} and shows that  $\\ad(F)(T)=[F,T]=S$ has exactly one solution. \n\n\nFinally, let us check that if $S\\in\\uh'_{i-1}$  and $\\ad(F)(T)=S$, then $T\\in\\uh'_{i}$. Using the injectivity of $\\ad(F)$ and the relation:\n    $$\n    \\begin{aligned}\n    \\ad(F)(\\ad(H)(T))\n    \\,& =\\,\\ad(H)(\\ad(F)(T)) + \\ad([F,H])(T)\n    \\,= \\\\ & =\\, \\ad(H)(S) + \\ad(2F)(T)\n    \\,=\\, 2(i-1)S + 2S\\,=\\, 2 i S\n    \\end{aligned}\n    $$\none obtains $\\ad(H)(T)=2iT$, as we wanted.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:W^+-sl2}\nLet $F$ be as in \\eqref{eq:type-1} where  $a $ is symmetric and $b_{-1}^{i,i-2}$ are invertible.\n\nThen, the map $\\iota^*$ of \\eqref{eq:emb-sl2-W+} yields a bijection:\n    {\\small\n    $$\n    \\left\\{\n        \\begin{gathered}\n        \\rho \\in \\homlie({\\mathcal W}_>,\\uh)\n        \\\\\n         \\text{such that }\n        \\rho(L_{-1})=F\n        \\\\\n        \\text{ and }\n       \n        \\rho(L_i) \\in\\uh_i \\text{ for }i\\geq 0\n        \\end{gathered}\n    \\right\\}\n    \\, \\overset{\\sim}\\longrightarrow\\,\n    \\left\\{\n        \\begin{gathered}\n        \\sigma \\in \\homlie(\\sl(2),\\uh) \\\\\n         \\text{such that }\n        \\sigma(f)=F\\, ,\n        \\\\\n        \\sigma(h) \\in\\uh_0 \\text{ and }\\sigma(e)\\in\\uh_1\n        \\end{gathered}\n    \\right\\}\n    $$}\n\\end{thm}\n\n\\begin{proof}\nGiven $\\rho$, we define $\\sigma:=\\iota^*(\\rho)$ and, therefore, $\\sigma(f)=\\rho(\\iota(f))=\\rho(L_{-1})$, $\\sigma(h)=\\rho(-2L_0)$ and $\\sigma(e)=\\rho(-L_1)$.\n\nFor the converse, one requires several steps and the previous Lemmas.\n\n\\emph{Step 1.} Let  $\\sigma$ be given. There exists a ${\\mathbb C}$-linear homomorphism $\\rho:{\\mathcal W}_>\\to \\uh$ such that $\\sigma=\\iota^*(\\rho)$. First, we set:\n    $$\n    \\rho(L_{-1}):= \\sigma(f)=F\n    \\; , \\quad\n    \\rho(L_0):= -\\frac12 \\sigma(h)\n    \\; , \\quad\n    \\rho(L_1):= -\\sigma(e)\n\t$$\nThe fact that $\\sigma$ is a map of Lie algebras and Lemma~\\ref{lem:FyieldsH} imply that:\n\t$$\n\t\\rho(L_0)\\,=\\, -\\frac12H\n\t$$\nwhere $H$ is as in equation~\\eqref{eq:type0}. Furthermore, it holds that $\\rho(L_{i})\\in\\uh'_i$ for $i=-1,1$. Having in mind Lemma~\\ref{lem:adFinjective} we obtain that there is a unique $T\\in\\uh'_2$ such that:\t\n\t$$\n\t\\operatorname{ad}(\\rho(L_{-1}))(T) \\,=\\, \\rho(L_1)\n\t$$\n\nThen, we define:\n\t\\begin{equation}\\label{eq:rho(L_2)def}\n\t\\rho(L_2)\\,:= \\, -3 T \\,\\in \\, \\uh'_2\n\t\\end{equation}\nand, recursively,\n    \\begin{equation}\\label{eq:rho(L_i)def}\n    \\rho(L_i):= \\frac1{i-2}[\\sigma(e),\\rho(L_{i-1})]\n    \\qquad \\text{for }i>2\n    \\;.\n    \\end{equation}\n\n\n\\emph{Step 2.} It holds that $ [\\rho(L_{0}),\\rho(L_j)] = -j\\rho(L_{j})$ for $j\\geq -1$. This is equivalent to show that\n$\\rho(L_j) \\in  \\uh'_j$ for all $j\\geq 1$.  Bearing in mind that $\\rho(L_1)\\in\\uh'_1$ and Lemma~\\ref{lem:T'T'bracket}, the conclusion follows.\n\n\\emph{Step 3.} It holds that $ [\\rho(L_{-1}),\\rho(L_j)] = -(1+j)\\rho(L_{j-1})$ for $j\\geq -1$. The cases $j\\leq 1$ follow from the fact that $\\sigma$ is a homomorphism of Lie algebras.  The choice of $T$ implies the case $j=2$. Let us proceed by induction on $j$.  For $j\\geq 3$, the definition of $\\rho(L_i)$, the Jacobi identity and the induction hypothesis yield:\n    {\\small $$\n    \\begin{aligned}\\;\n    [\\rho(L_{-1}), & \\rho(L_j)]\n    \\,  =\\,\n    [\\rho(L_{-1}),\\; -\\frac1{j-2}[\\rho(L_{1}),\\rho(L_{j-1})]]\n    \\, = \\\\ & = \\,\n    \\frac1{j-2}\\Big( [\\rho(L_{1}),\\; [\\rho(L_{j-1}), \\rho(L_{-1})]] \\;+\\;\n    [\\rho(L_{j-1}),\\; [\\rho(L_{-1}), \\rho(L_{1})]]\\Big)\n    \\, = \\\\ & = \\,\n    \\frac1{j-2}\\Big( [\\rho(L_{1}),\\; j \\rho(L_{j-2})] \\;+\\;\n     [\\rho(L_{j-1}),\\; (-2) \\rho(L_{0})]\\Big)\n     \\, = \\\\ & = \\,\n    \\frac1{j-2}\\big( j(3-j) \\rho(L_{j-1}) -2 (j-1)\\rho(L_{j-1})\\big)\n    \\,=\\, (-1-j)\\rho(L_{j-1})\n    \\end{aligned}\n    $$}\n\n\n\\emph{Step 4.} The identity:\n    \\begin{equation}\\label{eq:braket-hom}\n    [\\rho(L_i),\\rho(L_j)]\\,-\\,(i-j)\\rho(L_{i+j}) \\,=\\,0\n    \\end{equation}\nholds for $i,j\\geq 1$. We proceed by induction on $n=i+j$. The case $n=4$ (i.e. $i,j\\geq 1$ and $i+j=4$) holds by the very definition of $\\rho(L_4)$. Now, let us assume that it holds true up to  $n-1=i+j-1$ and let us prove the case $n=i+j>4$. Observe that, by Step 2, the l.h.s of the equation \\eqref{eq:braket-hom} lies in $\\uh_{i+j}'$. By Lemma~\\ref{lem:adFinjective}, it suffices to show that its image under $\\ad(F)=\\ad(\\rho(L_{-1}))$ vanishes. In fact, the Jacobi identity, the Step 3 and the induction hypothesis show that:\n\t{\\small $$\n\t\\begin{aligned}\n\t\\ad & (\\rho( L_{-1}))\\big(   [\\rho(L_i),\\rho(L_j)]\\,-\\,(i-j)\\rho(L_{i+j})\\big) \\, = \\\\ & = \\,\n\t [[\\rho(L_{-1}), \\rho(L_i)],\\rho(L_j)] \\,+\\,  [\\rho(L_i), [\\rho(L_{-1}),\\rho(L_j)]]\n\t \\,-\\, (i-j)[\\rho(L_{-1}),\\rho(L_{i+j})]\n\t \\, = \\\\ & = \\,\n\t [-(1+i)\\rho(L_{i-1}),\\rho(L_j)] \\,+\\, [\\rho(L_i), -(1+j)\\rho(L_{j-1})] \\,+\\, (i-j)(1+i+j) \\rho(L_{i+j-1})\n\t \\, = \\\\ & = \\,\n\t - \\big( (1+i)(i-j-1) \\,+\\, (1+j)(i-j+1)  \\,-\\, (i-j)(1+i+j) \\big) \\rho(L_{i+j-1})\n\t \\, = \\\\ & = \\,\n\t0\n\t \\end{aligned}\n\t $$}\n\n\\emph{Step 5.} $\\rho$ is a Lie algebra homomorphism. This follows from the properties of $\\sigma$ and Steps 2, 3, 4. \t\n\t\n\\end{proof}\n\n\n\\subsection{Factorization as a product}\n\nIt is remarkable that if the vector space $(A, (\\, ,\\,))$ decomposes as $A_1\\perp A_2$ (i.e. $A=A_1\\oplus A_2$ and $(a_1,a_2)=0$ for all $a_i\\in A_i$), then the very definition of the associated Heisenberg algebra implies that:\n\t$$\n\t{\\mathbb H}(A) \\,\\simeq\\, {\\mathbb H}(A_1)  \\widehat\\otimes_{{\\mathbb C}} {\\mathbb H}(A_2) \n\t$$\nas Lie algebras and ${\\mathbb H}(A_i)$ is a subalgebra of ${\\mathbb H}(A)$. So, we may wonder under which circumstances a morphism\n$\\rho :{\\mathcal W}_> \\to \\uh[(A)]$ would decompose accordingly. The following Theorem provides an answer in terms of the restriction $\\rho\\vert_{\\sl(2)}$. For this goal, recall that matrices $a$, $b$ and $c$'s behave as bilinear forms on $A$ (w.r.t. the action of $\\Gl(A)$).\n\n\\begin{thm}\\label{thm:rhodecomp}\nLet $F, H, E$ be as in~\\eqref{eq:type-1}, \\eqref{eq:type0} and \\eqref{eq:E=type1}. Let $\\rho :{\\mathcal W}_> \\to \\uh[(A)]$ satisfy $\\rho(L_{-1})=F$, $\\rho(L_0)=-\\frac12 H$ and $\\rho(L_1)=-E$. \n\nIf the vector space $A$ decomposes as $A_1\\perp A_2$ w.r.t. $\\eta$ and this decomposition is compatible with the action of $F$ and with the bilinear forms  $b_0^{1,1}$ and  $c_1^{1,1}$, then there are Lie algebra maps $\\rho_{i} :{\\mathcal W}_> \\to \\uh[(A_i)]$ for $i=1,2$ such that:\n\t$$\n\t\\rho \\,=\\, \\rho_1\\otimes 1 + 1\\otimes \\rho_2\n\t$$\n\nIf this is the case, and $\\rho(L_k)\\in \\uh[(A)]_k'$ for all $k\\geq -1$, then $\\rho_{i}(L_k)\\in \\uh[(A_{i})]_k'$ for all $k\\geq -1$ and $i=1,2$. \n\\end{thm}\n\n\n\\begin{proof}\n\\emph{Step 1}. The case of $\\rho(L_{-1})$. The hypothesis says that we can find $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots,n\\}$, a basis of $A$, and an index $m$ such that, for $1\\leq i <m \\leq j\\leq n$, the vectors $a_i$ and $a_j$ are orthogonal w.r.t. to the bilinear form defined by $a $.  Equivalently, w.r.t. the splitting $A_1\\oplus A_2$ the matrix of this bilinear form acquires a block decomposition as follows:\n\t$$\n\ta  \\,=\\, \n\t\\begin{pmatrix} \\ast  & 0 \\\\ 0 & \\ast \\end{pmatrix}\n\t$$\nIt is now straightforward that the terms of the operator $F$ (as given in equation~\\eqref{eq:type-1}) can be grouped in two sets, the first one involving $p_{i,\\alpha}$ and $q_{i,\\alpha}$ for $i\\in{\\mathbb N}$ and $1\\leq \\alpha <m$, and the second one depending only on $p_{i,\\alpha}$ and $q_{i,\\alpha}$ for $i\\in{\\mathbb N}$ and $m\\leq \\alpha \\leq n$. Denote these operators as $\\bar L_{-1,1}$ and $\\bar L_{-1,2}$ respectively. One checks that:\n\t\\begin{equation}\\label{eq:barL_-1}\n\t\\begin{gathered}\n\t\\rho( L_{-1}) = \\bar L_{-1,1}\\otimes 1 + 1\\otimes  \\bar L_{-1,2}\n\t\\\\\n\t\\bar L_{-1,\\alpha}\\,\\in \\, \\uh[(A_{\\alpha})]_{-1}\\qquad\\alpha=1,2.\n\t\\end{gathered}\n\t\\end{equation}\n\t\n\\emph{Step 2}. The case of $\\rho(L_{0})$. Bearing in mind that it is defined as $-\\frac12H$ and that the coefficients of the latter fulfill the relations \\eqref{eq:FyieldsH}, one can proceed as in the previous case. More precisely, considering the following block decompositions:\n\t$$\n\t\t\\eta  \\,=\\, \n\t\\begin{pmatrix} \\eta_1  & 0 \\\\ 0 & \\eta_2 \\end{pmatrix}\n\t\\qquad\n\t\ta  \\,=\\, \n\t\\begin{pmatrix} a_1  & 0 \\\\ 0 & a_2 \\end{pmatrix}\n\t\\qquad\n\t\tc_{1}^{1,1} \\,=\\, \n\t\\begin{pmatrix} c_1  & 0 \\\\ 0 & c_2 \\end{pmatrix}\n\t$$\none may use the following identity as a defining relation for $\\bar L_{0,\\alpha} \\in \\uh[(A_{\\alpha})]_0$:\n\t$$\n\t\\begin{aligned} \n\t\\rho(L_0)-b_0^{0,0} \n\t\\, & =\\, \\big( \\bar L_{0,1}- 2\\Tr( c_{1}\\eta_1^{-1}( a_{1}+a_1^T)(\\eta_1^T)^{-1} )\\big) \n\t\\, + \\\\ & \\qquad +\\, \n\t \\big( \\bar L_{0,2}- 2\\Tr( c_{2}\\eta_2^{-1} (a_{2}+a_2^T)(\\eta_2^T)^{-1})\\big)  \n\t\\end{aligned}\n\t$$\n\n\\emph{Step 3}. The case of $\\rho(L_{k})$ for $k\\geq 1$.  Recall from the proof of Theorem~\\ref{thm:sl2-symmmatrices} that the coefficients $b_1^{r,r+2}$ of $\\rho(L_{1})$ can be expressed in terms of $a $, $b_0^{1,1}$ and $c_1^{1,1}$ and that a close look of these expressions shows that $b_1^{r,r+2}$ are compatible w.r.t. to the splitting of $A$. Thus, we can express $\\rho(L_1)$ as the sum of two factors; namely, $\\bar L_{1,\\alpha}$ for $\\alpha=1,2$ which consists of the terms of $\\rho(L_{1})$ in   $p_{i,\\alpha}, q_{i,\\alpha}$ for  $1\\leq \\alpha <m$ and  for $m\\leq \\alpha \\leq n$, respectively. Now, we proceed as above. \n\nFor the case of $\\rho(L_k)$ for $k\\geq 2$ one proceeds recursively (using the expressions of the proof of Lemma~\\ref{lem:adFinjective}). \n\n\n\n\\emph{Step 4}. \n$[ \\bar L_{k,\\alpha} ,  \\bar L_{l,\\beta}]=0$ for $k,l \\geq -1$ and  $\\alpha\\neq\\beta$, since these two operators involve disjoint sets of variables. \n\n\\emph{Step 5}. The maps $\\rho_{\\alpha}$. Consider:\n\t$$\n\t\\rho_{\\alpha}(L_k)\\,:=\\,\n\t\\bar L_{k,\\alpha} \n\t\\qquad\\text{for $k\\geq -1$ and $\\alpha=1,2$}\n\t$$\nThe previous steps show that $\\rho = \\rho_1\\otimes 1 + 1\\otimes \\rho_2$.\n\nIt remains to check that $\\rho_{\\alpha}$ are morphisms of Lie algebras. For this goal we will expand both sides of the identity $[\\rho(L_k),\\rho(L_l)]=(k-l)\\rho(L_{k+l})$ using the above facts. The l.h.s. is:\n\t{$$\n\t\\, [\\rho(L_k),\\rho(L_l)]\\,=\\, \n\t[\\bar L_{k,1}+\\bar L_{k,2} , \\bar L_{l,1}+\\bar L_{l,2}]\n\t\\,=\\, \n\t[\\bar L_{k,1} , \\bar L_{k,1}]  + [\\bar L_{k,2} , \\bar L_{l,2}]\n\t$$}\nwhile the r.h.s. reads:\n\t{$$\n\t(k-l)\\rho(L_{k+l}) \\,=\\, \n\t(k-l)( \\bar L_{k+l,1}+\\bar L_{k+l,2})\n\t$$}\nComparing both expressions and having in mind the separation of variables, it follows that:\n\t{$$\n\t[\\bar L_{k,\\alpha} , \\bar L_{k,\\alpha }]  \\,=\\, \n\t(k-l) \\bar L_{k+l,\\alpha}\n\t$$}\n\\noindent and we conclude that $\\rho_{\\alpha}$ is a map of Lie algebras ${\\mathcal W}_> \\to \\uh[(A_{\\alpha})]$.\n\n\\emph{Step 6}. Type of the operators. In order to show that $\\rho(L_k)\\in \\uh[(A)]_k'$ implies that  $\\rho_{\\alpha}(L_k)\\in \\uh[(A_{\\alpha})]_k'$, it suffices to expand the Lie bracket $[ \\rho(L_0), \\rho(L_k)]$ using  $ \\rho(L_k)= \\rho_1(L_k) \\otimes 1 + 1\\otimes \\rho_2(L_k)$. \n\\end{proof}\n\n\\begin{rem}\nIt is worth noticing that if a decomposition is compatible with $a $, it does not need to be compatible with $b_0^{1,1}$. Indeed, for $A={\\mathbb C}^2$, $\\eta=a  = \\tiny{\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} }$, the general form of $b_0^{1,1}$ is given by\n$\\tiny{\\begin{pmatrix} -1  & \\lambda \\\\ \\lambda  & -1 \\end{pmatrix} }$.\n\\end{rem}\n\n\nFor later use, the following general result will be required.\n\n\\begin{thm}\\label{thm:solutionrhoi}\nLet $\\rho_i:{\\mathcal W}_> \\to \\End V_i$, $i=1,2$, be two representations of the Lie algebra ${\\mathcal W}_>$. And let us consider the product representation:\n\t$$\n\t\\rho=\\rho_1\\otimes 1 +1\\otimes \\rho_2\\, : \\, {\\mathcal W}_> \\longrightarrow \\End (V_1\\otimes V_2)$$\n\nLet $\\sum_{i=1}^r f_{1,i}\\otimes f_{2,i} \\in V_1 \\otimes V_2$.  Assume that $f_{i,1},\\ldots,f_{i,r}$ are linearly independent (for $i=1,2$).\n\nIt then holds that:\n\t$$\n\t\\rho(L_k)(\\sum_{i}f_{1,i}\\otimes f_{2,i})\\,=\\, 0 \\quad \\forall k\\geq -1\n\t$$\nif and only if:\n\t$$\n\t\\rho_{i}(L_k)(f_{i,j})\\,=0\\, \\text{ for all $i,j$ and $k\\geq -1$ } \n\t$$\n\\end{thm}\n\n\\begin{proof}\nLet us prove the converse. Bearing in mind that $\\rho =\\rho_1\\otimes 1 + 1\\otimes \\rho_2$, one can do the following computation:\n\t{\\small $$\n\t\\begin{aligned}\n\t\\rho(L_k)(\\sum_i f_{1,i} & \\otimes f_{2,i} )\\, \n\t=\\, \n\t(\\rho_1\\otimes 1 + 1 \\otimes \\rho_2) (L_k)(\\sum_i f_{1,i}\\otimes f_{2,i} )\n\t\\, \\\\ & =\\, \n\t \\sum_i  \\rho_1(L_k)(f_{1,i}) \\otimes f_{2,i} \\, +\\, \n\t \\sum_i f_{1,i}\\otimes \\rho_2 (L_k)(f_{2,i}  )\n\t\\,=\\, 0\n\t\\end{aligned}\n\t$$}\nand the conclusion follows. \n\nThe direct implication is more subtle. The hypothesis and the decomposition of $\\rho$ yield:\n\t{\\small $$\n\t\\begin{aligned}\n\t0\\,=\\, \\rho(L_k)(\\sum_i f_{1,i} & \\otimes f_{2,i} )\\, \n\t=\\, \n\t(\\rho_1\\otimes 1 + 1 \\otimes \\rho_2) (L_k)(\\sum_i f_{1,i}\\otimes f_{2,i} )\n\t\\, \\\\ & =\\, \n\t \\sum_i  \\rho_1(L_k)(f_{1,i}) \\otimes f_{2,i} \\, +\\, \n\t \\sum_i f_{1,i}\\otimes \\rho_2 (L_k)(f_{2,i}  )\n\t\\end{aligned}\n\t$$}\nLet $E$ be  the vector space generated by $\\{f_{1,1},\\ldots, f_{1,r}\\}\\subset V_1$. Suppose that there exists $l$ such that $\\rho_1(f_{1,l})$ does not belong to $E$. Then, let $\\chi: V_1 \\to {\\mathbb C}$ be a linear form such that $\\chi(f_{1,i})=0$ for all $i$ and $\\chi( \\rho_1(f_{1,l}))\\neq 0$. Applying $\\chi$ to the above equation, one obtains:\n\t$$\n\t0\\,=\\, \\sum_i \\chi\\big( \\rho_1(L_k)(f_{1,i}) \\big) f_{2,i} \\,\\in\\, V_2\n\t$$\nwhich contradicts the fact that $f_{2,1},\\ldots,f_{2,r}$ are linearly independent. Therefore, it follows that $\\rho_1(f_{1,l})$ belongs to $E$ for all $l$ or, equivalently, \n\t$$\n\t\\begin{aligned}\n\t\\rho_{1,E}:{\\mathcal W}_> \\,&\\longrightarrow \\,\\End(E)\n\t\\\\\n\tL_k \\quad &\\longmapsto \\, \\rho_{1,E}(L_k):= \\rho_1(L_k)\\vert_E\n\t\\end{aligned}\n\t$$\nis a Lie algebra homomorphism. Recall that, being ${\\mathcal W}_>$ simple, the non-trivial representations of ${\\mathcal W}_>$ are faithful. Since $E$ is finite dimensional, $\\rho_{1,E}$ must be trivial; that is, $\\rho_{1,E}(L_k)=0$ for all $k$. In particular, \n\t$$\n\t0\\,=\\, \\rho_{1,E}(L_k)(f_{1,j}) \\,=\\,  \\rho_1(L_k)(f_{1,j}) \\qquad \\forall j\n\t$$\nThe identities $\\rho_2(L_k)(f_{2,j})=0$ are proven similarly. \n\\end{proof}\n\t\n\n\n\n\\subsection{Commutation Relations}\\label{subsect:CR}\n\nThis subsection only collects the explicit computations of some Lie brackets used  previously and, thus, the reader can skip it. From a formal point of view,  we are dealing with generators of $\\uh$, $ 1, q_{i,\\alpha}, p_{i,\\alpha}$,  with $i=1,2,\\ldots$ and $\\alpha=1,\\ldots, n$, that satisfy the following relations:\n    $$\n    \\begin{gathered}\n    \\, [p_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \\delta_{i,j} i \\eta^{\\alpha\\beta}\\cdot 1 \n    \\\\\n   \\,  [p_{i,\\alpha}, p_{j,\\beta}] \\,=\\,[q_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \n    [p_{i,\\alpha}, 1] \\,=\\,[q_{i,\\alpha}, 1] \\,=\\,  0\n    \\end{gathered}\n    $$\nand, because of the associativity of composition, we will also use:\n\t$$\n\t[a,bc]=\\, [a,b] c \\,+\\, b[a,c]\n\t$$\n\nWe will use the Einstein convention; that is, repeated subindices of the variables $p,q$'s imply the summation is to be done. Recall that $b_i^{0,2i+3}$  and $q_i:=(q_{i,1},\\ldots, q_{i,n})$ denote row vectors, $p_i:=(p_{i,1},\\ldots, p_{i,n})^T$ are column vectors (the superscript $T$ denotes the transpose), and $a , b_i^{j,2i+j}, c_i^{j,2i-j}$ are $n\\times n$ square matrices.\n\nLet us compute some Lie brackets. For instance,\n    $$\n    \\begin{aligned}\n    \\, & [ b_i^{0,2i+3} p_{2i+3} , b_{j}^{0,2j+3} p_{2j+3}]\n    \\,=\\\\ & \\qquad  =\\,\n    [ (b_i^{0,2i+3})_{\\alpha} (p_{2i+3})_{\\alpha}\t\\, ,\\, (b_{j}^{0,2j+3})_{\\beta} (p_{2j+3})_{\\beta} ]\n    \\,=\\\\ & \\qquad  =\\,\n    (b_i^{0,2i+3})_{\\alpha}  [ (p_{2i+3})_{\\alpha}\t\\, ,\\,(p_{2j+3})_{\\beta} ]  (b_{j}^{0,2j+3})_{\\beta} \\,=\\, 0\n    \\end{aligned}\n    $$\nwhere subindices $\\alpha,\\beta$ denote the corresponding entries of the vectors. Analogously, we have the following identities:\n    \\begin{subequations}\n    \\begin{align*}\n\t& [ q_1 a  q_1^T ,  b_i^{0,2i+3} p_{2i+3} ] \\,=\\, 0 \\qquad \\forall i\\geq 0\n\t\\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  b_j^{0,2j+3} p_{2j+3}]\n    \\,=\\\\ & \\qquad  =\\,\n    - [b_j^{0,2j+3} p_{2j+3} ,  (q_{r})_{\\alpha}] (b_{i}^{r,r+2i} p_{r+2i})_{\\alpha}\n    \\\\ & \\qquad \\quad \n    -\\, (q_{r} b_{i}^{r,r+2i})_{\\alpha} [  b_j^{0,2j+3} p_{2j+3}, (p_{r+2i})_{\\alpha}]\n    \\,=\\\\ & \\qquad  =\\,\n    - (b_j^{0,2j+3})_{\\beta} [ (p_{2j+3})_{\\beta}, (q_{r})_{\\alpha}]\n    (b_{i}^{r,r+2i})_{\\alpha\\gamma} (p_{r+2i} )_{\\gamma}\n    \\,=\\\\ & \\qquad  =\\,\n    - (2j+3) (b_j^{0,2j+3})_{\\beta} \\eta^{\\beta\\alpha} (b_{i}^{2j+3,2j+3+2i})_{\\alpha\\gamma} (p_{2j+3+2i} )_{\\gamma}\n    \\,=\\\\ & \\qquad  =\\,\n    - (2j+3) b_j^{0,2j+3} \\eta^{-1} b_{i}^{2j+3,2j+3+2i} p_{2j+3+2i}\n     \\\\ & \n     [ p_{r} c_{i}^{r,2i-r} p_{2i-r} ,  b_j^{0,2j+3} p_{2j+3}]\n    \\,=\\, 0 \n    \\\\ & \n    [ q_1 a  q_1^T , q_{r} b_{j}^{r,r+2j} p_{r+2j} ]\n    \\,=\\, 0 \\qquad\\forall j\\geq 1\n       \\\\ &\n    [ q_1 a  q_1^T , q_{r} b_{0}^{r,r} p_{r} ]\n    \\,=\\\\ & \\qquad  =\\,\n   (  q_{1} b_{0}^{1,1} )_{\\alpha} [ q_1 a  q_1^T , ( p_{1})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n\t- (  q_{1} b_{0}^{1,1} )_{\\alpha} \\big(\n\t[( p_{1})_{\\alpha} , ( q_{1})_{\\beta}] (a  q_1^T)_{\\beta} +\n\t(q_1 a )_{\\beta} [( p_{1})_{\\alpha} , ( q_{1})_{\\beta}]\\big)\n    \\,=\\\\ & \\qquad  =\\,\n    - q_{1} b_{0}^{1,1}\\eta^{-1}  \\big( a  +a^T \\big) q_1^T\n\\\\ &\n    [ q_1 a  q_1^T , p_{r}^T c_{j}^{r,2j-r} p_{2j-r} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [ q_1 a  q_1^T , (p_{r})_{\\alpha}] (c_{j}^{r,2j-r} p_{2j-r})_{\\alpha}\n    \\,+\\,\n      (p_{r}^T c_{j}^{r,2j-r})_{\\alpha}  [ q_1 a  q_1^T ,  (p_{2j-r} )_{\\alpha}]\n    \\,=\\\\ & \\qquad  =\\,\n    -\\big( [(p_1)_{\\alpha}, (q_1)_{\\beta}] (a  q_1^T)_{\\beta} + (q_1 a )_{\\beta}[(p_1)_{\\alpha}, (q_1)_{\\beta}]\\big)\n    (c_{j}^{1,2j-1} p_{2j-1})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad - \\, (p_{2j-1}^T c_{j}^{2j-1,1})_{\\alpha} \\big([(p_1)_{\\alpha},(q_1)_{\\beta}] (a  q_1^T)_{\\beta} + ( q_1 a )_{\\beta} [(p_1)_{\\alpha},(q_1)_{\\beta}]\\big)\n    \\,=\\\\ & \\qquad  =\\,\n    -q_1 \\big(  a    + ( a )^T\\big)(\\eta^{-1})^T  c_{j}^{1,2j-1} p_{2j-1}     \n    \\,- \\, p_{2j-1}^T c_{j}^{2j-1,1} \\eta^{-1} \\big(   a   + (a   )^T\\big) q_1^T\n    \\,=\\\\ & \\qquad  =\\,\n    -q_1 \\big(  a    + ( a )^T\\big)(\\eta^{-1})^T   \n    \\big(  c_{j}^{1,2j-1}  +    (c_{j}^{2j-1,1})^T \\big) p_{2j-1} -\n    \\\\  \n    & \\qquad \n    -  \\delta_{j1}  \\operatorname{Tr}\\big(c_{1}^{1,1} \\eta^{-1} ( a   + a^T)(\\eta^{-1})^T\\big)\n\\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i}   , p_{s}^T c_{j}^{s,2j-s} p_{2j-s} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [  q_{r} b_{i}^{r,r+2i} p_{r+2i} , (p_{s})_{\\alpha} ] (c_{j}^{s,2j-s} p_{2j-s})_{\\alpha}\n    \\,+\n    \\\\ & \\qquad\\quad + \\,\n      (p_{s}^T c_{j}^{s,2j-s} )_{\\alpha} [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  (p_{2j-s})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n    - [ (p_{s})_{\\alpha} ,  (q_{r})_{\\beta}  ] (b_{i}^{r,r+2i} p_{r+2i} )_{\\beta} (c_{j}^{s,2j-s} p_{2j-s})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad -\\,\n      (p_{s}^T c_{j}^{s,2j-s} )_{\\alpha}  [ (p_{2j-s})_{\\alpha},  (q_{r})_{\\beta}   ](b_{i}^{r,r+2i} p_{r+2i})_{\\beta}\n    \\,=\\\\ & \\qquad  =\\,\n    - r p_{2j-r}^T (c_{j}^{r,2j-r})^T \\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n    \\,-\\, r  p_{2j-r}^T c_{j}^{2j-r,r} \\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n    \\,=\\\\ & \\qquad  =\\,\n    - r p_{2j-r}^T \\big( (c_{j}^{r,2j-r})^T + c_{j}^{2j-r,r}\\big)\\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n\n  \n      \\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i}   , q_{s} b_{j}^{s,s+2j} p_{s+2j} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [  q_{r} b_{i}^{r,r+2i} p_{r+2i} , (q_{s})_{\\alpha} ] (b_{j}^{s,s+2j} p_{s+2j})_{\\alpha}\n    \\,+\\\n     \\\\ & \\qquad \\qquad +\\,\n      (q_{s} b_{j}^{s,s+2j} )_{\\alpha} [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  (p_{s+2j})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n    - (q_{r} b_{i}^{r,r+2i})_{\\beta}[    (q_{s})_{\\alpha}, (p_{r+2i})_{\\beta} ] (b_{j}^{s,s+2j} p_{s+2j})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad -\\,\n      (q_{s} b_{j}^{s,s+2j} )_{\\alpha} [ (p_{s+2j})_{\\alpha}, (q_{r} )_{\\beta} ](b_{i}^{r,r+2i} p_{r+2i})_{\\beta}\n    \\,=\\\\ & \\qquad  =\\,\n    (r+2i) q_{r} b_{i}^{r,r+2i}  \\eta^{-1}   b_{j}^{r+2i,r+2i+2j} p_{r+2i+2j}\n    \\,-\\,\n      r q_{r-2j} b_{j}^{r-2j,r}\\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n   \\end{align*}\n   \\end{subequations}\n\n\n\\section{An Application}\n\nAs an application of the previous sections, we offer here an example that illustrates how our results can be used for studying the representation of ${\\mathcal W}_>$ appearing in the study of the Virasoro conjecture. Regarding the Virasoro conjecture our main references are the works of  Dubrovin-Zhang, Eguchi-Hori-Xiong, Getzler, Givental and Liu-Tian (\\cite{DubrovinZ, EHX, Getzler, Givental, LiuTian}). \n\nIn our example will consider $(A, (,))$ to be the cohomology ring of a smooth projective variety $X$ with $c_1(X)=0$ and trivial odd cohomology groups. Recall that the hypothesis on the first Chern class is equivalent to the vanishing of the operator $R$ in \\cite{DubrovinZ, Getzler}; however, it does not seem difficult to extend the results of \\S\\ref{sec:ReprW>} to include this case. On the other hand, the hypothesis on the odd cohomology groups is fulfilled if $X$ has generically semisimple even quantum cohomology (\\cite{HMT}). It seems to be very hard to weaken this assumption.  \n\n\n\\subsection{Preliminaries}\n\nLet $A$ be a $n$ dimensional vector space over ${\\mathbb C}$ endowed with a bilinear form $(\\, , \\, )$. Let $\\{a_1,\\ldots,a_n\\}$ be a basis and $\\eta$ be the  matrix associated to the pairing,  $\\eta_{\\alpha\\beta}:=(a_{\\alpha},a_{\\beta})$. Let us  consider  the subspace ${\\mathbb C}[[t_1,t_3,t_5,\\ldots]]$ of the the boson Fock space ${\\mathbb C}[[t_1,t_2,\\ldots]]$ and the subalgebra of $ {\\mathbb C}[[t_1,\\ldots]] \\hat\\otimes_{{\\mathbb C}} S^{\\bullet}A$ generated by $t_{i,\\alpha}:= t_i\\otimes a_{\\alpha} $ with $i$ odd:\n\t{\\small\n\t\\begin{equation}\n\t\\label{eq:Vkdv}\n\t\\Vkdv(A)\\,:={\\mathbb C}[[\\{t_{i,\\alpha} \\vert 1\\leq \\alpha\\leq n, i \\text{ odd}\\} ]]\n\t\\,\\subseteq \\,  {\\mathbb C}[[t_1,\\ldots]] \\hat\\otimes_{{\\mathbb C}} S^{\\bullet}A\n\t\\end{equation}}\nIf no confusion arises, we will simply write $\\Vkdv$. \n\nNow we study a distinguished representation of ${\\mathcal W}_>$ in $\\Vkdv$; eventually, we will see that it is the representation coming from the action of the Heisenberg algebra via Givental's quantization (\\cite{Givental}). More precisely, we will combine the chain of inclusions of Lie algebras:\n\t$$\n\t\\sl(2)\\,\\hookrightarrow\\, {\\mathcal W}_> \\, \\hookrightarrow \\, \\uh\n\t$$\nwhich has been studied in the previous section, with a map:\n\t$$\n\t\\begin{aligned}\n\t\\widehat{\\,}\\,: \\uh[(A)] &\\, \\longrightarrow \\End_{{\\mathbb C}}(\\Vkdv(A))\n\t\\\\\n\tP &\\, \\longmapsto \\, \\hat P\n\t\\end{aligned}\n\t$$\nwhose obstruction to be compatible with the Lie brackets is governed by a cocycle. This map is defined following  the results of Dubrovin-Zhang, Givental and Kazarian (\\cite{DubrovinZ, Givental, Kaz}); namely, we set:\n\t\\begin{equation}\\label{eq:quant}\n\\hat 1 \\,=\\, 1 \\, , \\qquad \n\\hat p_{i,\\alpha} \\,=\\, \\eta^{\\alpha\\beta}\\frac{\\partial}{\\partial t_{i,\\beta}}   \\, ,\\qquad\n\\hat q_{i,\\alpha} \\,=\\,  i t_{i,\\alpha}\n\t\\end{equation}\n(recall that $i$ is a positive odd integer number).\n\n\\begin{rem}\nGivental has developed a beautiful formalism for this construction in terms of quantization of quadratic hamiltonians (\\cite{Givental}). An alternative approach, originated in the Japanese school and strongly linked to the Sato grassmannian, can be found in \\cite{KSU}. The forthcoming section (\\S\\ref{subsec:solutions1dim}) is deeply inspired by the latter. \n\\end{rem}\n\n\\subsection{The representation}\\label{subsec:representation}\n\nBearing in mind the results of \\S\\ref{subsect:sl2}, we know that the operator:\n\t$$\n\t    F\\,  :=\\, b_{-1}^{0,1} p_1 + q_1\\eta q_1^T + q_{i+2} \\eta p_i \n\t$$\ntogether with the data:\n\t\\begin{itemize}\n\t\\item  $b_{-1}^{0,1}$ arbitrary,\n\t\\item $b_0^{1,1}$ such that \\eqref{eq:FyieldsH:2} holds, and\n\t\\item  $c_1^{1,1} := \\frac1{16} \\eta^T b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) $,\n\t\\end{itemize}\ndetermine a map  $\\sigma:\\sl(2)\\to \\uh$. Indeed, equations~\\eqref{eq:FyieldsH}, \\eqref{eq:EFH} and \\eqref{eq:HF-2F} allow us to obtain the explicit expressions for  $H$ and $F$:\n\t\\begin{equation}\\label{eq:HEparticular}\n\t\\begin{aligned}\n\tH\\, & = \\, \n\t\\frac13 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) p_3 +\n\t \\frac1i q_i (b_0^{1,1}-(i-1)\\eta) p_i +\n\t \\\\ & \\qquad + \n\t \\frac1{16} \\Tr(b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n\t\\\\\n\tE \\, &=\\, \\frac1{5!!} b_{-1}^{0,1} \n\t\\big(2\\eta^{-1}b_0^{1,1} - 2- (\\eta^{-1}+(\\eta^{-1})^T )(c_1^{1,1}+(c_1^{1,1})^T)\\big) p_5 \n\t+ \\\\\n\t & \\qquad +\n\t \\frac1{16} p_1^T \\eta^T b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2)  p_1 - \\\\\n\t & \\qquad - \\frac1{4 i (i+2) } q_{i} (  b_0^{1,1} \\eta^{-1}- (i-1) ) (  b_0^{1,1} \\eta^{-1}-(i+1) ) \\eta  p_{i+2}\n\t\\end{aligned}\n\t\\end{equation}\n\nNow, by  Theorem \\ref{thm:W^+-sl2}, the map $\\sigma$ extends uniquely to an homomorphism $\\rho:{\\mathcal W}_>\\to \\uh$. And one can now compute the induced action on $\\Vkdv$. Let us write down the first operators:\n    \\begin{subequations}\n    \\begin{align*}\n    \\hat{L}_{-1}\\, & :=\\, (\\rho(L_{-1}))^{\\hat\\,} \\, =\\, \\hat{F}\\, =\\, b_{-1}^{0,1} \\eta^{-1}\\frac{\\partial}{\\partial t_{1}}\n    \t + t_{1} \\eta t_{1}^T + (i+2) t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n    \\\\\n     \\hat{L}_{0}\\, & :=\\, (\\rho(L_{0}))^{\\hat\\,} \\, =\\, -\\frac12 \\hat{H}\\, =\\, \n     - \\frac16 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) \\eta^{-1} \\frac{\\partial}{\\partial t_{3}}  -\n      \\\\ & \\qquad \t-\\frac12 t_i (b_0^{1,1}\\eta^{-1} -(i-1)) \\frac{\\partial}{\\partial t_{i}}\n      -\\frac1{32} \\Tr(b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n     \\\\\n     \\hat{L}_{1}\\, & :=\\, (\\rho(L_{1}))^{\\hat\\,} \\, =\\, - \\hat{E}\\, = \\\\ & \\qquad\n     - \\frac1{5!!} b_{-1}^{0,1} \n\t\\big(2\\eta^{-1}b_0^{1,1} - 2- (\\eta^{-1}+(\\eta^{-1})^T )(c_1^{1,1}+(c_1^{1,1})^T)\\big) \\eta^{-1}\\frac{\\partial}{\\partial t_{5}} - \\\\ & \\qquad\n     -  \\frac1{16} (\\frac{\\partial}{\\partial t_{1}})^T \\eta^T  b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2)   \\frac{\\partial}{\\partial t_{1}} + \\\\\n\t & \\qquad + \\frac1{4 (i+2)} t_{i} (  b_0^{1,1} \\eta^{-1}- (i-1) ) (  b_0^{1,1} \\eta^{-1}-(i+1) )  \\frac{\\partial}{\\partial t_{i+2}}\n     \\end{align*}\n     \\end{subequations}\nwhere, as usual, we write $t_i$ for the row vector $(t_{i,1},\\ldots, t_{i,n})$ and $\\frac{\\partial}{\\partial t_{i}}$ for the column vector $(\\frac{\\partial}{\\partial t_{i,1}},\\ldots, \\frac{\\partial}{\\partial t_{i,n}})^T$. \n     \n     \n\n\\subsection{The operators of the Virasoro Conjecture: a baby model}\\label{subsec:baby}\n\nNow, we are ready to show how the operators appearing in the Virasoro conjecture agree with our approach for the case of manifold with trivial odd cohomology and whose first Chern class vanishes. \n\nFrom now on, we suppose we are given $X$, whose first Chern class is zero, and with trivial odd cohomology. Under this hypothesis, the Poincar\\'e pairing defines on $A:= H^{\\bullet}(X,{\\mathbb C})$  a symmetric non-degenerated bilinear form:\n\t$$\n\t( a, b ) \\,=\\, \\int_X a\\cup b \\quad \\text{ for }a,b\\in A ,\n\t$$\nLet $r:=\\operatorname{dim}(X)$ and fix a basis $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots, n \\}$ of $A$,  with $a_1=1\\in H^0(X,{\\mathbb C})$, such that it is homogeneous w.r.t. the Hodge decomposition; that is, $a_{\\alpha}\\in H^{p_{\\alpha},q_{\\alpha}}(X)$ for certain $p_{\\alpha}, q_{\\alpha}$.  Let $\\bar \\eta$ the matrix associated to the Poincar\\'e pairing w.r.t. the chosen basis and let us define  $\\mu_{\\alpha}:= p_{\\alpha}-\\frac{r}2$ and $\\mu$ the matrix with $\\mu_1,\\ldots, \\mu_r$ along its diagonal and $0$ elsewhere. Observe that the compatibility of the Poincar\\'e pairing w.r.t. the Hodge decomposition yields:\n\t\\begin{equation}\\label{eq:mua+mb=0}\n\t\\bar\\eta_{\\alpha\\beta}\\neq 0 \\qquad \\implies \\qquad \\mu_{\\alpha} + \\mu_{\\beta}=0\n\t\\end{equation}\n\n\nThe operators appearing in the Virasoro conjecture when the first Chern class vanishes (\\cite[Equation (1.2)]{Getzler}) are as follows:\n\t{\\small\n\t{\\begin{equation}\\label{eq:Getzler-101}\n\t\\begin{aligned}\n\t\\bar L_{-1}\\,& :=\\,\n\t- \\frac{\\partial}{\\partial \\bar t_{0,1}}  + \\frac{1}{2\\hbar} \\bar t_{0} \\bar \\eta \\bar t_0^T + \\bar t_{i+1} \\frac{\\partial}{\\partial \\bar t_{i}} \n\t\\\\\n\t\\bar L_0 \\,& :=\\, -\\frac{3-r}2\\frac{\\partial}{\\partial  \\bar t_{1,1}}  + (\\mu_{\\alpha}+i+ \\frac{1}2) \\bar t_{i,\\alpha}  \\frac{\\partial}{\\partial \\bar t_{i,\\alpha}}  +  \n\t\\frac{1}{48}(3-r)\\int_X c_r(X)\n\t\\end{aligned}\n\t\\end{equation}}}\nand, for $k\\geq 1$, as:\n\t{\\small\n\t\\begin{equation}\\label{eq:GetzlerLk}\n\t\\begin{aligned}\n\t\\bar L_k \\,& :=\\, -\\frac{\\Gamma(k+\\frac{5-r}2)}{\\Gamma(\\frac{3-r}2)} \n\t\\frac{\\partial}{\\partial \\bar t_{k+1,1}} +\n\t\t \\frac{\\Gamma(\\mu_{\\alpha}+i+k+\\frac{3}2)}{\\Gamma(\\mu_{\\alpha}+i+\\frac{1}2)} \\bar t_{i,\\alpha }   \\frac{\\partial}{\\partial \\bar t_{k+i,\\alpha}} \n\t\t\\\\ & \\qquad\n\t\t+ \\frac{\\hbar}2 (-1)^{i} \\frac{\\Gamma(\\mu_{\\alpha}+i+k+\\frac{3}2)}{\\Gamma(\\mu_{\\alpha}+i+\\frac{1}2)} \\bar \\eta^{\\alpha\\beta} \\frac{\\partial}{\\partial \\bar t_{-1-i,\\alpha}}  \\frac{\\partial}{\\partial \\bar t_{k+i,\\beta}} \n\t\\end{aligned}\n\t\\end{equation}}\nwhere $c_r(X)$ is the $r$-th Chern class and we have used variables $\\bar t_{i,\\alpha}$ with $\\alpha=1,\\ldots, n$ and $i=0,1,2,\\ldots$.\n\nSimilarly to the case of $\\uh$, we say that a second order differential operator in $\\{\\bar t_{i,\\alpha}\\}$ is of type $i$ if it is a linear combination of $\\frac{\\partial}{\\partial \\bar t_{i+1,\\alpha}}$ and the following terms \n$\\frac{\\partial}{\\partial \\bar t_{j-1,\\alpha}}\\frac{\\partial}{\\partial \\bar t_{i-j,\\beta}}$, $\\bar t_{j,\\alpha} \\frac{\\partial}{\\partial \\bar t_{j+i,\\beta}}$ and $\\bar t_{j-1,\\alpha}\\bar t_{-i-j,\\beta}$ and, if $i=0$, a constant term. Observe that $\\bar L_k$ is of type $k$. Now, we offer a simple proof of a folk statement.\n\n\\begin{prop}\nThe operators $\\{\\bar L_k\\vert k\\geq 2\\}$ are uniquely determined by $\\{\\bar L_{-1},\\bar L_0, \\bar L_1\\}$ and the condition that $\\bar L_k$ is of type $k$ for all $k\\geq -1$. \n\\end{prop}\n\n\\begin{proof}\nUnder the change of variables $\\bar t_i:= \\sqrt{2\\hbar} (2i+1)!! t_{2i+1}$, it is clear that a second order differential operator in $\\bar t_i$'s  is of type $k$ if and only if is equal to $\\hat T$ for $T\\in \\uh_{k}$. Now, it is easy to check that the hypothesis of Theorem \\ref{thm:W^+-sl2} hold; namely, $\\hat F=\\bar L_{-1}$ and $\\bar L_k$ are of type $k$ for $k=0,1$. The conclusion follows. \n\\end{proof}\n\n\\begin{thm}\\label{thm:Lbar=Lhat}\nIt holds:\n\t$$\n\t\\bar L_i \\,=\\, \\hat L_i \\qquad i=-1,0,1,\\ldots\n\t$$\nfor the choice $\\bar t_i:= \\sqrt{2\\hbar} (2i+1)!! t_{2i+1}$, $\\eta=\\bar \\eta$, $b_{-1}^{0,1}= (0,\\ldots, 0, \\frac{-1}{\\sqrt{2\\hbar}})$ and:\n\t$$\n\tb_0^{1,1}\\,:=\\, -(2 \\mu+1) \\eta \\,=\\, -{\\small \\begin{pmatrix} 0 & & 2\\mu_1+1 \\\\ & \\iddots & \\\\ 2\\mu_n+1 & & 0\\end{pmatrix}}$$\n\\end{thm}\n\n\\begin{proof}\nTheorem \\ref{thm:W^+-sl2} implies that it suffices to show that $\\bar L_i \\,=\\, \\hat L_i$ for $i=-1,0,1$. Indeed, this fact follows from the explicit substitution of $t_i$, $\\eta$, etc. as in the statement in the operators $ \\hat L_i$. The only identity which is not obvious is the one corresponding to the constant term of $ \\hat L_0$. Bearing in mind the definitions and the fact that $\\eta$ is symmetric, this term is:\n\t{\\small $$\n\t\\begin{aligned}\n\t -\\frac1{32} & \\Tr (b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n\t \\,= \\\\ & = \\,\n\t -\\frac1{16}\n\t \\sum_{\\alpha=1}^n (2 p_{\\alpha} -r+1) (2 p_{\\alpha}-r -1)  \n\t  \\,=\\,\n\t \\frac14 \\sum_{p,q} h^{p,q} (\\frac{r+1}2 - p ) ( p-\\frac{r-1}2)\n\t \\end{aligned}\n\t $$}\nwhere $h^{p,q}=\\dim H^p(X,\\Omega^q)$.\n\nNow, observe that the Libgober-Wood identity (\\cite[Proposition~2.3]{LW}) can be stated as:\n\t{\\small $$\n\t\\sum_{p,q} (-1)^{p+q}h^{p,q}  (\\frac{r+1}2 - p ) ( p-\\frac{r-1}2) \n\t\\,=\\, \\frac16\\int_X \\big(\\frac{3-r}2 c_r(X)- c_1(X)c_{r-1}(X)\\big)\n\t$$}\nRecalling that we are assuming that that it has trivial odd cohomology, the constant term equals:\n\t$$\n\t \\frac1{48}\\int_X \\big( (3-r) c_r(X)- 2 c_1(X)c_{r-1}(X)\\big)\n\t$$\nwhich agrees with the free term of $\\bar L_0$ (see \\eqref{eq:Getzler-101}) since $c_1(X)=0$. \n\\end{proof}\n\n\\begin{rem}\nIt is worth noticing that up to rescaling the variables and a Dilaton shift, these operators coincide with those of \\cite[Equation (3.5)]{DVV} and \\cite[\\S3]{Givental} (for $b_1=0$) and with those of \\cite[Equation (7.33)]{Dijkgraaf} and \\cite[Equation (2.59)]{Witten} (for $b_1=-\\frac13\\sqrt{\\frac{\\eta}{2\\hbar}}$). \n\\end{rem}\n\n\nNow, we will go one step further in the study of the above representation. Recall that in \\S\\ref{subsec:prelim-beta-Diff-to-C[[t]]} it was stated that matrices $a$, $b_{i}^{j-2i,j}$ and $c_i^{j,2i-j}$ behave as bilinear forms under the action of $\\Gl(A)$. \\emph{A fundamental observation is that all results and equations above are invariant under the action of the general linear group} (acting as base changes on the given basis $\\{a_1,\\ldots, a_n\\}$). Let us briefly discuss this statement. For instance, let $S\\in \\Gl(A)$, then the row vector $q_i=(q_{i,1},\\ldots, q_{i,n})$ is transformed to $q_i S^T$, accordingly the column vector $p_i$ goes to $S p_i$. The action of $S$ sends the bilinear form $\\eta$  to $(S^{-1})^T \\eta S^{-1}$ and analogously with $a$, etc.~. Note that, since $\\eta$ and $b_0^{1,1}$ behave as bilinear forms, $\\eta^{-1}b_0^{1,1}$ defines an endomorphism of $A$.  Finally, the Heisenberg algebra is also affected. \n\n\n\\begin{defn}\nLet ${\\mathbb H}^{\\eta}$ be the Heisenberg algebra defined in \\eqref{eq:pqcommutation}. Given a map of Lie algebras $\\rho:{\\mathcal W}_>\\to {\\mathcal U}({\\mathbb H}^{\\eta})$ and  $S\\in\\Gl(A)$,  we denote by $\\rho^S$ the map of Lie algebras:\n\t$$\n\t{\\mathcal W}_>\\, \\overset{\\rho}\\longrightarrow \\,  {\\mathcal U}({\\mathbb H}^{\\eta})\n\t\\,\\overset{\\sim}\\longrightarrow\\,  {\\mathcal U}({\\mathbb H}^{(S^{-1})^T \\eta S^{-1}})\n\t$$\nwhere the last map sends \t$q_{i}$ to $q_{i} S^T$ and $p_i$ to $S p_i$. \n\\end{defn}\n\nWith the hypothesis and choices of above, we have the following,\n\n\n\\begin{thm}\\label{thm:Virasorodecomp}\nLet $\\rho:{\\mathcal W}_>\\to {\\mathcal U}({\\mathbb H}^{\\eta})$ be as above; i.e. $\\hat\\rho$ defines  the Virasoro constraints,  \\eqref{eq:Getzler-101} and \\eqref{eq:GetzlerLk},  of a smooth projective variety with trivial odd cohomology and vanishing first Chern class. \n\nThen  there exists $S\\in \\Gl(A)$ such that  $\\rho^S$ decomposes as the product of $n$ representations of dimension $1$; that is, there exist $\\rho_i: {\\mathcal W}_>\\to \\uh[({\\mathbb C})]$ such that:\n\t\\begin{equation}\\label{eq:rho=sumrhoalpha}\n\t\\rho^S \\,=\\, \n\t\\rho_1 \\otimes 1\\otimes\\ldots \\otimes 1 \\, +\\, \\ldots\\, +\\,\n\t1\\otimes\\ldots \\otimes 1 \\otimes \\rho_n\n\t\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet us consider a basis  which is orthonormal for $\\eta$. Let $S\\in\\Gl(A)$ be the matrix associated to this change of basis. Due to the choices of $a$, $b_{-1}^{i+2,i}, b_0^{1,1}$ and $c_1^{1,1}$, it is trivial that $S$ also bring them into diagonal form; or, equivalently, there is a common orthogonal basis for all these bilinear pairings. Applying Theorem \\ref{thm:rhodecomp}, one concludes.\n\\end{proof}\n\nIn this situation, for each $\\alpha=1,\\ldots,n$, one obtains a one dimensional representation $\\rho_{\\alpha}$ or, what is tantamount, our study essentially reduces to the case of Example~\\ref{exam:dim1}. That is,  $\\dim A=1$ , $a= \\eta\\in{\\mathbb C}^{\\ast}$ and, thus, $b_0^{1,1}=-\\eta$. Setting $b_0:= b_{0}^{0,0}$, one has that \\eqref{eq:HEparticular} gives:\n\t$$\n\t\\begin{aligned}\n\tF \\,& =\\,  b_{-1}  p_1 +   q_1  \\eta  q_1  +  q_{i+2}\\eta p_i\n\t\\\\\n\tH\\, &=\\,  -  b_{-1}   p_3   -  q_i  \\eta  p_i  - \\frac18 \n    \t\\\\\n\tE\\,&=\\, - \\frac1{4} b_{-1}  p_5 - \\frac1{4} p_1 \\eta  p_1 -  \\frac1{4}  q_{i}  \\eta  p_{i+2}\n\t\\end{aligned}\n\t$$\nwhere $b_{-1} $ and $\\eta$ are computed from the $n$-dimensional setup \\eqref{eq:Getzler-101}.\n\nThese three operators determine $\\rho$ completely and, according to the map \\eqref{eq:quant} and Theorem~\\ref{thm:Lbar=Lhat}, one has:\n\t{\\small\n\t{\\begin{equation}\\label{eq:barL}\n\t\t\\begin{aligned}\n\t\\bar L_{-1}\\,& :=\\, \n\tb_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{0}}  \n\t+ \\frac{1}{2\\hbar} \\eta \\bar t_{0}^2\n\t+ \\bar t_{i+1} \\frac{\\partial}{\\partial \\bar t_{i}} \n\t\\\\\n\t\\bar L_0 \\,& :=\\, \\frac32  b_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{1}}  \n\t+ (i+\\frac12) \\bar t_{i}  \\frac{\\partial}{\\partial \\bar t_{i}}  + \\frac{1}{16} \n\t\\\\\n\t\\bar L_1 \\,& :=\\, \\frac{5!!}4  b_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{2}}  \n\t+ \\frac{\\hbar}2\\eta^{-1} \\frac{\\partial}{\\partial \\bar t_0} \\frac{\\partial}{\\partial \\bar t_0} \n\t+ (i+\\frac12)(i+\\frac32) \\bar t_{i } \\frac{\\partial}{\\partial \\bar t_{i+1}} \n\t\\end{aligned}\n\t\\end{equation}}}\n\n\n\n\\subsection{On the solutions for the $1$-dimensional situation}\\label{subsec:solutions1dim}\n Once the representation has been decomposed in terms of $1$-dimensional parts, we wonder if one could deduce some properties of the solutions of the Virasoro constraints. Our approach follows closely our previous work \\cite{Plaza-AlgSol} which is inspired in \\cite{KacSchwarz}. Briefly, the idea is to show that each of our representations $\\rho_i$ come from an action of ${\\mathcal W}_>$ on the Sato Grassmannian and that that they admit exactly one solution $\\tau_i$, which are $\\tau$-functions for the KP hierarchy and, then, conclude that the product $\\tau_1\\cdot\\ldots\\cdot \\tau_n$ is a solution for $\\rho^S$. \n\nLet us begin recalling that the Sato Grassmannian is the set of subspaces $U\\subset {\\mathbb C}((z))$ such that the kernel and cokernel of $\\pi_U:U\\to {\\mathbb C}((z))\/{\\mathbb C}[[z]]$ are finite dimensional (\\cite{Sato, SW}). Actually, it is an infinite dimensional scheme (\\cite{AMP}) and carries a distinguished line bundle, the determinant line bundle ${\\mathbb D}$. Each integer $n$ correspondes to a connected component, $\\operatorname{Gr}^n$; namely, those subspaces $U$ such that $\\dim\\ker\\pi_U-\\dim\\coker\\pi_U=n$. Sato-Sato's achievement was to show that there was a bijection between the set of those $U$ s.t. $\\pi_U$ is an isomorphism and the set of functions $\\tau(t)\\in{\\mathbb C}[t_1,t_2,\\ldots]]$ with $\\tau(0)=1$ and fulfilling the KP hierarchy (thus, each $U$ has a $\\tau$-function; see \\cite{Sato,SW,AMP} for details). The same holds for the Sato grassmannian of ${\\mathbb C}((z))^{\\oplus n}$ and the $n$-multicomponent KP hierarchy.\n\nThe fact that the space of global section of ${\\mathbb D}^*$ is isomorphic to the semi-infinite wedge product or Fermion Fock space:\n\t{\\small $$\n\tH^0(\\operatorname{Gr}^n,{\\mathbb D}^*)\\simeq \\wedge^{\\frac{\\infty}2}{\\mathbb C}((z)) \\, =\\,\n\t<\\left\\{ \n\t\\begin{gathered}\n\tz^{i_1}\\wedge z^{i_2}\\wedge\\ldots \\text{ s.t. } i_1<i_2<\\ldots \\\\\n\t \\text{ and } i_k=k+n \\,\\,  \\forall k>>0\n\t \\end{gathered}\\right\\}>\n\t $$}\nhave allowed its extensive use in CFT's (in particular, by the Japanese school, see \\cite{KSU} and references therein). Recall that the boson-fermion correspondence is the isomorphism (we restrict us to $\\operatorname{Gr}^0$; that is, the charge $0$ sector):\n\t$$\n\t \\wedge^{\\frac{\\infty}2}{\\mathbb C}((z)) \\, \\simeq \\, {\\mathbb C}[[t_1,t_2,\\ldots]]\n\t $$\nthat maps $z^{i_1}\\wedge z^{i_2}\\wedge\\ldots$ to the Schur polynomial associated with the partition $1-i_1\\geq 2-i_2\\geq \\ldots$. Similarly, the space of global sections of ${\\mathbb D}^*$ over the Sato grassmannian of ${\\mathbb C}((z))^{\\oplus n}$ is isomorphic to ${\\mathbb C}[[\\{t_{i,\\alpha} \\vert \\alpha=1,\\ldots, n , i=1,2,\\ldots \\}]]$. \n\n\t\nGiven a subgroup of the restricted linear group of ${\\mathbb C}((z))$ (see \\cite{SW}), one has an induced action on $\\operatorname{Gr}^n({\\mathbb C}((z)))$. Moreover, if the action preserves the determinant bundle, it will yield a projective action on the space of global sections. In fact, an analogous statement holds for the case of Lie algebras. Let us illustrate this issue with the case of the Lie algebra $\\operatorname{Diff}^1({\\mathbb C}((z)))$ of first order differential operators on ${\\mathbb C}((z))$. An operator $D\\in\\operatorname{Diff}^1({\\mathbb C}((z)))$ acts on sections as follows. If the matrix $(d_{ij})$ corresponding to $D$ w.r.t. the basis $\\{z^i\\}$ has no non-trivial diagonal elements, then:\n    $$\n    D ( z^{i_1}\\wedge z^{i_2}\\wedge\\ldots)\n    \\,:=\\,\n     D ( z^{i_1})\\wedge z^{i_2}\\wedge\\ldots +\n     z^{i_1}\\wedge  D ( z^{i_2})\\wedge\\ldots + \\ldots\n     $$\nIf the matrix $(d_{ij})$ is diagonal, then:\n    $$\n    D ( z^{i_1}\\wedge z^{i_2}\\wedge\\ldots)\n    \\,:=\\,\n     \\sum_{j=1}^{\\infty} (d_{i_j i_j}- d_{jj})  z^{i_1}\\wedge z^{i_2}\\wedge\\ldots\n     $$\nHaving in mind the boson-fermion correspondence, the above construction gives rise to a linear map:\n    \\begin{equation}\\label{eq:betaDiffEnd}\n    \\begin{aligned}\n    \\operatorname{Diff}^1({\\mathbb C}((z))) \\,&\n    \\overset{\\beta}\\longrightarrow\n    \\End({\\mathbb C}[[t_1,t_2,\\ldots]])\n    \\\\\n    D\\,& \\longmapsto \\, \\beta( D)\n    \\end{aligned}\n    \\end{equation}\nwhich defines a projective representation. Note, nevertheless, that if we are given a map of Lie algebras $\\sigma:{\\mathcal W}_>\\to\\operatorname{Diff}^1({\\mathbb C}((z)))$, then, $\\beta\\circ\\sigma$ can be canonically promoted to a linear representation since ${\\mathcal W}_>$ has no non-trivial central extensions. Indeed, for this goal, if suffices to add a constant to $\\beta\\circ\\sigma(L_0)$.\n\nThe following results will show that the operators of \\S\\ref{subsec:baby} arise from the previous setup. \n\n\\begin{lem}\\label{lem:DinDiffistypei}\nLet $D\\in\\operatorname{Diff}^1({\\mathbb C}((z)))$. Then, $\\beta(D)$  is of type $i$ if and only if $D$ is a linear combination of $ 1$, $ z^{-(2i+3)} $ and $  z^{-2i}(z\\partial_z+\\frac{1-2i}2)$. \n\\end{lem}\n\n \\begin{proof}\nRecall that $\\operatorname{Diff}^1({\\mathbb C}((z)))$ is generated as ${\\mathbb C}$-vector space by $1$, $z^m$ for $m\\in{\\mathbb Z}$ acting as an homothety and $ z^{m}(z\\partial_z+\\frac{m+1}2)$ for $m\\in{\\mathbb Z}$. Let us recall from~\\cite[Table 1]{Kaz} the description of  the operators induced by them via the boson-fermion correspondence: \n    $$\n    \\beta(z^m) \\,=\\,\n        \\begin{cases}\n        m t_m & \\text{ for } m >0 \\\\\n        0 & \\text{ for } m=0 \\\\\n        \\frac{\\partial}{\\partial t_{-m}} &\\text{ for } m<0\n        \\end{cases}\n    $$\nand, for $m>0$, \n    {\\small $$\n    \\beta\\big(z^m(z\\partial_z+\\frac{1+m}2)\\big)\n    \\,  = \\,\n    \\frac12 \\sum_{j=1}^{m-1} j(m-j) t_j t_{m-j} +\n    \\sum_{j=1}^{\\infty} (j+m) t_{m+j} \\frac{\\partial}{\\partial t_{j}}\n    $$}\nAnalogously, the\naction of $z^{-m}(z\\partial_z+\\frac{1-m}2)$ on ${\\mathbb C}((z))$\ncorresponds to the action of:\n    {\\small $$\n\t\\beta\\big(z^{-m}(z\\partial_z+\\frac{1-m}2))\\big)\n    \\, = \\,\n    \\sum_{j=1}^{\\infty} j t_j  \\frac{\\partial}{\\partial t_{m+j}}  +\n    \\frac12\\sum_{j=1}^{m-1} \\frac{\\partial}{\\partial t_{j}} \\frac{\\partial}{\\partial t_{m-j}}\n    $$}\nFinally, recall that the case $m=0$ is regularized as follows:\n    {\\small $$\n    \\beta\\big(z^{-m}(z\\partial_z+\\frac{1-m}2))\\big)    \\, := \\,  \\sum_{j=1}^{\\infty} j t_j \\frac{\\partial}{\\partial t_{-j}}\n    $$}\nChecking the degrees, the conclusion follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem:sigma}\nLet $\\sigma\\in\\homlie({\\mathcal W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))$. Recall that $\\Vkdv = {\\mathbb C}[[t_1,t_3,\\ldots]]$. \n\n$\\beta({\\sigma(L_i)}) \\vert_{\\Vkdv}$ takes values in $\\Vkdv$ and it is of type $i$ for all $i$, if and only if there exists $s,  t\\in {\\mathbb C}$ such that:\n    $$\n    \\sigma(L_i)\\,=\\,\n    t^i\\Big(\n        \\frac12 z^{-2i}(z\\partial_z + \\frac{1-2i}{2})+ s z^{-2i-3}\\Big)\n        \\qquad \\forall i\\geq -1\n    $$\n\\end{lem}\n\n\\begin{proof}\nThe ``if'' part follows from Lemma \\ref{lem:DinDiffistypei} and the fact that $\\sigma$ as in the statement defines a map of Lie algebras. Let us now deal with the ``only if'' part.\n\n\nWe know from~\\cite[\\S2]{Plaza-AlgSol} (see also~\\cite{Plaza-Opers}) that there is a 1-1-correspondence:\n    {\\footnotesize\n    $$\n    \\left\\{\\begin{gathered}\n    \\sigma\\in\\Hom_{\\text{Lie-alg}}({\\mathcal\n    W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))\n   \\\\\n    \\text{such that }\\sigma\\neq 0\n    \\end{gathered}\\right\\}\n    \\, \\overset{1-1}\\longleftrightarrow \\,\n    \\left\\{\\begin{gathered}\n    \\text{triples }(h(z),c,b(z))\\text{ such that}\n    \\\\\n    h'(z)\\in{\\mathbb C}((z))^*,\\; c\\in{\\mathbb C},\\; b(z)\\in{\\mathbb C}((z))\n    \\end{gathered}\\right\\}\n    $$\n    }\nwhich is explicitly given by:\n    \\begin{equation}\\label{eq:explicitexpreL_idiffC((z))}\n    \\sigma(L_i) \\,=\\,\n    \\frac{-h(z)^{i+1}}{h'(z)} \\partial_z  -\n    (i+1) c \\cdot h(z)^i + \\frac{h(z)^{i+1}}{h'(z)}b(z)\n    \\end{equation}\n\nOn the other hand, due to Lemma~\\ref{lem:DinDiffistypei}, the fact that $\\sigma(L_i)$ is of type $i$ implies that there exist $r_i,s_i,t_i\\in{\\mathbb C}$ satisfying:\n    \\begin{equation}\\label{eq:explicitexpreL_ilinearcomb}\n    \\sigma(L_i) \\,=\\,\n    r_i\\cdot 1 + s_i\\cdot z^{-(2i+3)} + t_i\\cdot z^{-2i}(z\\partial_z+\\frac{1-2i}2)\n    \\end{equation}\n\nComparing the coefficients of $\\partial_z$ in the previous identities, it follows that $h(z)=\\frac{t_i}{t_{i-1}} z^{-2}$. Hence, the quotients $\\frac{t_i}{t_{i-1}}$ are all equal to a constant, say $t$. Hence $t_i=t^i t_0$  and $h(z)=t z^{-2}$. Further, the case $i=0$ yields $t_0=\\frac12$.\n\nPluging this in equations~\\eqref{eq:explicitexpreL_idiffC((z))} and~\\eqref{eq:explicitexpreL_ilinearcomb}, one gets:\n    {\\small\n    $$\n    -(i+1)c(tz^{-2})^i\n    -\\frac{(tz^{-2})^{i+1}}{2 t z^{-3}} b(z)\n    \\,=\\,\n    r_i + s_i z^{-(2i+3)} + \\frac12 t^i z^{-2i}(\\frac{1-2i}2)\n    $$}\nand, thus:\n    $$\n    b(z)\\,=\\,\n    -2(i+1)c z^{-1}\n    -  2 t^{-i} r_i z^{2i-1} - 2 s_i t^{-i} z^{-4} - \\frac12  z^{-1}(1-2i)\n    $$\nObserve that the l.h.s. does not depend on $i$, one gets many conditions. First, for $i\\neq 0$ the term $z^{2i-1}$ is an odd power of $z$ different from $z^{-1}$ that can not be cancelled with any other term; consequently, $r_i=0$ for $i\\neq 0$. Further, since the coefficient of $z^{-4}$ in $b(z)$ has to be independent of $i$, it follows that $t^{-i} s_i$ is a constant independent of $i$; and, thus, equal to $s_0$. Finally, the coefficient of $z^{-1}$ in $b(z)$ is:\n    $$\n    -2(i+1)c - 2 r_0 \\delta_{i,0} - \\frac12 (1-2i)\n    $$\nSince it has to be independent of $i$, it follows that $c=\\frac12 $, $r_0=0$ and, thus:\n    $$\n    b(z)\\,=\\, -\\frac32  z^{-1} - 2 s_0 z^{-4}\n    $$\nSubstituting $h(z), c, b(z)$ into expression~\\eqref{eq:explicitexpreL_idiffC((z))} and setting $s=s_0$, one obtains the result.\n\\end{proof}\n\nLet us recall that the rescaling of the variables yields an action on the boson Fock space. More precisely, $\\lambda =\\{\\lambda_i \\} \\in\\prod_{i \\text{ odd}}{\\mathbb C}^*$ maps $t_i$ to $\\lambda_i t_i$. Accordingly, it acts on $ \\homlie({\\mathcal W}_>,\\End(\\Vkdv))$ and sends $\\rho$ to $\\rho^{\\lambda}:=\\lambda\\circ\\rho\\circ \\lambda^{-1}$. \n\n\n\\begin{defn}\nThe $\\lambda$-scaled KP hierarchy is the hierarchy obtained by replacing $t_i$ by $\\lambda_i t_i$ in the KP hierarchy (for given  $\\lambda=(\\lambda_i ) \\in\\prod_{i \\in{\\mathbb N}}{\\mathbb C}^*$).\nA function $\\tau_1(t)\\in {\\mathbb C}[[t_1, t_2,\\ldots]]$ is called $\\tau$-function of the $\\lambda$-scaled KP hierarchy if $\\tau(\\lambda^{-1}t):=\\tau(\\lambda_1^{-1}t_1, \\lambda_2^{-1} t_2,\\ldots)$ is a $\\tau$-function of the KP hierarchy. For brevity, we simply say scaled KP. We do similarly for KdV, multicomponent KP. \n\\end{defn}\n\nNote that the $\\lambda$-scaled KP hierarchy for $\\lambda=(\\mu^i)$ for $\\mu\\in{\\mathbb C}^*$ coincides with the KP hierarchy. However, this does not happen in general. \n\nThe following Lemma is the key point to go from Virasoro to KdV. \n\n\\begin{lem}\\label{lem:Diff1scaleW}\nThe map $\\beta$ of \\eqref{eq:betaDiffEnd} induces a bijection between:\n\\begin{itemize}\n\t\\item the set of $ \\sigma\\in\\Hom_{\\text{Lie-alg}}({\\mathcal  W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))$ such that there exist $s\\in {\\mathbb C}$\n\tsatisfying:\n\t$$\n\t \\sigma(L_i)= \\frac12 z^{-2i}\\big(z \\partial_z + \\frac{1-2i}{2}\\big)+ s z^{-2i-3}  \n\t     $$\n\t\\item the set of scale equivalence classes of $\\rho \\in \\homlie({\\mathcal W}_>,\\End(\\Vkdv))$ whose coefficients of quadratic terms in $\\rho(L_{-1})$ do not vanish and such that  $\\rho(L_i)$  is of type $i$ for $i\\geq -1$.\n\t\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nFirst, we prove the statement with no reference to $r(z)$ on the first item and with no mention to a linear function on the second item. Under these circumstances,  given $\\sigma$ as in the statement, Lemma~\\ref{lem:DinDiffistypei} shows that $(\\beta\\circ\\sigma)(L_i) \\vert_{\\Vkdv}$ takes values in $\\Vkdv$ and it is of type $i$ for all $i$. An explicit computation yields:\n\t$$\n\t\\begin{aligned}\n    (\\beta\\circ\\sigma)(L_{-1})  \\,& = \\,\n    s \\frac{\\partial}{\\partial t_{1}}+\n    \\frac14 t_1^2 +\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j+2} \\frac{\\partial}{\\partial t_{j}}\n\\\\\n    (\\beta\\circ\\sigma)(L_{0}) & \\,= \\,\n     s \\frac{\\partial}{\\partial t_{3}} +\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j} \\frac{\\partial}{\\partial t_{j}}\n    \\\\\n    (\\beta\\circ\\sigma)(L_{i}) & \\,= \\,\n    s \\frac{\\partial}{\\partial t_{2i+3}} +\n    \\frac14  \\sum_{j=1}^{2i-1} \\frac{\\partial}{\\partial t_{j}} \\frac{\\partial}{\\partial t_{2i-j}}+\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j} \\frac{\\partial}{\\partial t_{2i-j}}\n        \\end{aligned}\n    $$\n(where $j$, as usual, is odd) and thus:\n    $$\n    [(\\beta\\circ\\sigma)(L_{-1}), (\\beta\\circ\\sigma)(L_{1}) ]\n    \\; - \\beta({[\\sigma(L_{-1}), \\sigma(L_{1})]})\n    \\,=\\,\n    -\\frac18\n    $$\nwhich implies that we have a map of Lie algebras defined by:\n    \\begin{equation}\\label{eq:rhoWK}\n    \\rho(L_i)\\,:=\\, (\\beta\\circ\\sigma)(L_i)+ \\frac1{16}\\delta_{i,0}\n    \\end{equation}\n\nConversely, let us start with $\\rho$ as in the second set of the statement. The assumptions yield the following expression:\n\t$$\n\t\\rho(L_{-1})\\, = \\,   b_{-1}^{0,1} \\frac{\\partial}{\\partial t_{1}} +  a  t_1^2   + b_{-1}^{i+2,i}  t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n\t$$\nwith $ a, b_{-1}^{i+2,i}  \\neq 0$. Considering the action of $\\prod_{i \\text{ odd}}{\\mathbb C}^*$ by conjugation, one finds $\\lambda = \\{\\lambda_i \\in{\\mathbb C}^* \\vert i\\text{ odd}\\}$ and $s\\in {\\mathbb C}$ such that:\n\t$$\n\t\\rho^{\\lambda}(L_{-1}) \\, =\\, \\lambda \\circ \\rho(L_{-1}) \\circ \\lambda^{-1}\n\t\\,=\\,\n\ts \\frac{\\partial}{\\partial t_{1}} +  \\frac14 t_1^2   + (\\frac{i+2}2) t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n\t$$\n\nLemma \\ref{lem:sigma} and the previous discussion show that $\\rho^{\\lambda}$ is the representation associated to the map  $\\sigma : {\\mathcal W}_>\\to \\operatorname{Diff}^1({\\mathbb C}((z)))$ defined by:\n\t$$\n\t\\sigma(L_i)= \\frac12 z^{-2i}(z\\partial_z + \\frac{1-2i}{2})+ s z^{-2i-3}\n\t\\qquad \\forall i\\geq -1\n\t$$\n\\end{proof}\n\n\\begin{rem}\\label{rem:VirtoKdV}\nThe statement can be generalized. On the one hand, we may consider the conjugation of $\\sigma$ by an operator of the type  $\\exp(r(z))$ while, on the other hand, we replace $\\rho$ by its conjugate by $\\exp(\\beta(r(z)))$. For instance, for $r(z)\\in {\\mathbb C}[[z^2]]$, one has that $\\beta(r(z))$ is a linear function on $t_1,t_3,\\ldots$. Thus, the first representation is:\n \t$$\n\t \\sigma(L_i)= \\frac12 z^{-2i}\\big(z(- r(z)+\\partial_z) + \\frac{1-2i}{2}\\big)+ s z^{-2i-3}  \n\t     $$\nwhile  $\\rho$ is as in the statement up to a linear function on $t_i$'s. \n\\end{rem}\n\n\\begin{rem}\nIt is worth noticing that the Virasoro operators studied by Witten (\\cite{Witten}) correspond to the case $s=-\\frac12$, $r(z)=0$. Kac-Schwarz (\\cite{KacSchwarz}), using the fact the these operators come from a representation in $\\operatorname{Diff}^1({\\mathbb C}((z)))$, proved that there is a point in the Sato Grassmannian whose $\\tau$-function is a solution of these equations and, hence, is a solution of KdV hierarchy too.  A study of common solutions of Virasoro-like constraints and KdV has been carried out in \\cite{Plaza-AlgSol}. \n\\end{rem}\n\n\\begin{lem}\\label{lem:uniquenesTauVirasoro}\nLet $\\rho$ be as in Lemma~\\ref{lem:Diff1scaleW} and let $\\tau(t)\\in \\Vkdv={\\mathbb C}[[t_1,t_3,\\ldots]]$. Then, the Virasoro constraints:\n\t$$\n\t\\rho(L_k)(\\tau(t)) \\,=\\, 0\\qquad k\\geq -1\n\t$$\nwith the initial condition $\\tau(0)=1$ admits no solution for $s=0$ and at most one solution for $s\\neq 0$. \n\\end{lem}\n\n\\begin{proof}\nSince $\\tau(0)=1$, let us consider the problem in terms of a formal function $F(t)\\in \\Vkdv= {\\mathbb C}[[t_1,t_3,\\ldots]]$ with $F(0)=0$ and $\\tau(t)=\\exp(F(t))$. The function $F(t)$ has a series expansion:\n    \\begin{equation}\\label{eq:F=generating}\n    F(t)\\,=\\, \\sum_{\\bf n} f_{\\bf n} {\\bf t}^{\\bf n}\n        \\end{equation}\nwhere ${\\bf n}:=\\{n_1,n_3,\\ldots\\}$ is a sequence of non-negative integers such that $n_i=0$ for all $i\\gg 0$, $f_{\\bf n}\\in {\\mathbb C}$ and ${\\bf t}^{\\bf n}:=\\prod_{i\\geq 1} t_i^{n_i}$. Further, the topology of $\\Vkdv={\\mathbb C}[[t_1,t_3,\\ldots]]$ comes from the definition $\\deg(t_i)=i$. In particular, the degree of ${\\bf t}^{\\bf n}$ is given by $\\vert {\\bf n}\\vert:= \\sum_{i\\geq 0} i n_i$.\n\nFor the sake of brevity, let us denote by  $f_{n_1 n_3\\ldots n_k} = f_{\\bf n}$ for ${\\bf n}=\\{n_1,n_3,\\ldots\\}$ with $n_k\\neq 0$ and $n_i=0$ for all $i>k$ and we set $f_0=F(0)=0$. As a brief summary, let us write down the monomials and their coefficients up to degree $5$:\n    {\\small\n    $$\n    \\begin{array}{ccccccc}\n    \\text{degree} & 0 & 1 & 2 & 3 & 4 & 5\n    \\\\\n    \\text{monomials} & 1 & t_1 & t_1^2 & t_1^3 , t_3 & t_1^4 , t_1t_3 &\n    t_1^5 , t_1^2 t_3 , t_5\n    \\\\\n    \\text{coefficient} & f_0 & f_1 & f_2 & f_3, f_{01} & f_4, f_{11} &\n    f_5, f_{21}, f_{001}\n    \\end{array}\n    $$}\n\nAfter rescaling $t_i$'s and conjugation by an exponential, if needed, we may assume that $\\rho$ is given by \\eqref{eq:rhoWK}. The hypothesis $\\rho(L_k)(\\tau(t)) =0$ is equivalent to the vanishing of the corresponding homogeneous parts of degree $i$ for $i=0,1,2,\\ldots$. An explicit computation for low values of $k$ and $i$ yields:\n\t    $$\n    \\begin{array}{ccc}\n    k &\\quad  i \\quad& \\text{part of degree $i$ in }\\rho(L_k)(\\tau(t))\n    \\\\\n    -1 & 0 & s f_1 \n    \\\\\n    -1 & 1 & 2s f_2 t_1 \n    \\\\\n    -1 & 2 & 3s f_3 t_1^2 + \\frac12 t_1^2 \n    \\\\\n    -1 & 3 & s\\big(4 f_4 t_1^3 +f_{13} t_3\\big) + t_3 f_1  \n    \\\\\n    0 & 0 & s f_{01}+\\frac1{16} \n    \\\\\n    0 & 1 &  s f_{11} t_1 + t_1 f_1  \n    \\\\\n    0 & 2 & s f_{21} t_1^2 +2 f_2 t_1^2  \n    \\\\\n    1 & 0 & s f_{001} + \\frac12 f_2 +\\frac14 f_1^2  \n    \\\\\n    1 & 1 & s f_{101} + \\frac32 f_3 t_1 + f_{11} t_1  \n    \\end{array}\n    $$\nThus, it is clear that if a solution $F$ does exist, then $s\\neq 0$. In this case,  the vanishing of the above polynomials implies that  $f_1=0$, $f_2=0$, $f_3=-\\frac1{3! s}$, $f_{01}=-\\frac1{16 s}$, $f_{11}= 0$, $f_4=0$, $f_{11}= 0$, etc.~. Writing down the general expression for the homogeneous part of degree $i$ of $\\rho(L_k)(\\tau(t))$, one observes that it allows us to determine $f_{\\bf n}$ with $\\vert n\\vert =i$ and $n_k\\neq 0$ in terms of $f_{\\bf n}$ with $\\vert n\\vert \\leq i-2$. Thus, if a solution $F$ exists, the coefficients $f_{\\bf n}$ can be recursively determined.\n\n\n\\end{proof}\n\n\n\n\n\n\n\\begin{thm}\\label{thm:solutionWscaleKdV}\nLet $\\rho \\in \\homlie({\\mathcal W}_>,\\End({\\mathbb C}[[t_1,t_3,\\ldots]]))$ be such that $\\rho(L_k)$  is of type $k$ for $k\\geq -1$ and that all coefficients of $\\rho(L_{-1})$ are non zero. \n\nThen, there exists a unique $\\tau(t)\\in{\\mathbb C}[[t_1,t_3,\\ldots]]$, with $\\tau(0)=1$, such that:\n\t$$\n\t\\rho(L_k)(\\tau(t))\\,=\\,0\\qquad k\\geq -1\n\t$$\nFurther, the solution $\\tau(t)$ is a $\\tau$-function of the scaled KdV hierarchy.\n\\end{thm}\n\n\\begin{proof}\nLemma~\\ref{lem:Diff1scaleW} implies that there is $\\lambda$ and $\\sigma:{\\mathcal W}_>\\to \\operatorname{Diff}^1({\\mathbb C}((z)))$ such that $\\rho^{\\lambda}=\\beta_*(\\sigma)$.  Recalling  Theorem~3.12  of~\\cite{Plaza-AlgSol}, one knows that there is a function $\\tau_0(t)$ which satisfies that $\\rho^{\\lambda}(L_n)(\\tau_0(t))=0$ and that it is a $\\tau$-function of the KdV hierarchy. Then, $\\tau(t):=\\tau_0(\\lambda t)$ fulfills the requirements. Since Lemma~\\ref{lem:uniquenesTauVirasoro} implies the uniqueness of the solution, the conclusion follows. \n\n\\end{proof}\n\n\n\\begin{rem}\nLet us make two comments on the solutions. First, an instance of the notion of scaled KdV appears already in Kontsevich's Theorem when it is claimed that the exponential of the generating function in variables $T_{2i+1}:=t_i\/(2i+1)!!$ is a $\\tau$-function for the KdV hierarchy (\\cite[Theorem 1.2]{Kon}). On the other hand, although the  dilation shift $\\bar t_{i}\\mapsto \\bar t_{i}-\\delta_{i,0}$ transforms the operators $\\rho(L_k)$, it should be noted that it does not induce an automorphism of the algebra ${\\mathbb C}[[\\bar t_0,\\bar t_1,\\ldots]]$.\n\\end{rem}\n\n\n\n\n\n\\subsection{On the solutions for the $n$-dimensional situation}\n\nLet us now focus in the $n$-dimensional situation. That is, we aim at studying the interplay between Virasoro representations and multicomponent KP hierarchy. Special attention will be paid at their common solutions. \n\nRecall that $\\Vkdv(A)$ is the subalgebra of ${\\mathbb C}[[t_1,t_3,\\ldots]]\\widehat\\otimes_{{\\mathbb C}} S^\\bullet A$ generated by $t_i\\otimes a$. Then, $S\\in\\Gl(A)$ acts on it by the automorphism of algebras $t_i\\otimes a\\mapsto t_i\\otimes S(a)$. \n\n\\begin{thm}\\label{thm:solutionproductKdV}\nLet $\\rho:{\\mathcal W}_>\\to \\uh[(A)]$ be as in \\S\\ref{subsec:baby}.\n\nThere exist $S\\in \\Gl(A)$ and functions $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)\\in {\\mathbb C}[[t_{1,\\alpha},t_{3,\\alpha},\\ldots]]$ such that:\n\t\\begin{equation}\\label{eq:scaleequivmultiKP}\n\t\\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big)\\,=\\, 0\n\t\\end{equation}\n\nFurther, $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)$ are $\\tau$-functions of the scaled KdV hierarchy.\n \\end{thm}\n\n\\begin{proof}\nTheorem~\\ref{thm:Virasorodecomp} shows that there is $S\\in\\Gl(A)$ such that $\\rho^S$ decomposes as the tensor product of $n$ $1$-dimensional Lie algebra representations of ${\\mathcal W}_>$. More precisely,  if $\\{a_\\alpha\\}$ is the chosen basis for $A$, then $\\{S(a_\\alpha)\\}$ is a orthogonal basis for $\\eta$. Consequently, there are $\\rho_\\alpha: {\\mathcal W}_>\\to \\uh[(<S(a_\\alpha)>)]$ such that \\eqref{eq:rho=sumrhoalpha} holds. \n\nNow, apply the results of \\S\\ref{subsec:solutions1dim} on the $1$-dimensional case. Indeed, since $\\eta$ is non-degenerated and $\\{S(a_\\alpha)\\}$ is a orthogonal basis,  from Theorem~\\ref{thm:solutionWscaleKdV} one obtains functions $\\tau_\\alpha(t_\\alpha)$, such that $\\tau_\\alpha(0)=1$,  $\\rho_\\alpha(L_k)(\\tau_\\alpha)=0$ for all $\\alpha,k$ and they are  $\\tau$-functions for the scaled KdV hierarchy. \n\nObserve that \\eqref{eq:scaleequivmultiKP} holds if and only if $\\hat\\rho^S(L_k)( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))$ vanishes. Applying the converse of Theorem~\\ref{thm:solutionrhoi} one concludes. \n\\end{proof} \n\n\n\n\\begin{rem}\nThe previous Theorem means that, assuming the uniqueness of the solution (\\cite[Theorem 3.10.20]{DubrovinZ}), the solution of the Virasoro constraints has to be of the above form; that is, an operator acting on a product of Witten-Kontsevich $\\tau$-functions. Thus, it agrees with the results of Givental (\\cite{Givental}) for the total descendent potential. It would be interesting to relate both expressions explicitly (see also \\cite{FvLS, GiventalAn,Lee}). Alternatively, one could combine Teleman's classification of semi simple cohomological field theories (\\cite{Teleman}) with Givental's results to deduce that this is the right expression for the solution. Nevertheless our result can be applied on other frameworks, as it will mentioned in \\S\\ref{subsec:final}).\n\\end{rem}\n\n\n\n\n\n\\begin{cor}\nLet $\\rho$ be as in the Theorem~\\ref{thm:solutionproductKdV}. \n\nIf $S, \\tau_\\alpha$ satisfy \\eqref{eq:scaleequivmultiKP}, then $\\rho^S=\\rho_1+\\ldots+\\rho_n$ and $\\hat\\rho_\\alpha(L_k)(\\tau_\\alpha)=0$. \n\nThe matrix $S$ is unique up to an ortogonal matrix. \n\\end{cor}\n\n\\begin{proof}\nIf $S$ and $\\tau_\\alpha$ are such that \\eqref{eq:scaleequivmultiKP} vanishes, then the following expression also vanishes:\n\t$$\n\t0\\,=\\, \n\t{\\exp(-\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha))} S^{-1} \\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big) \n\t\\,=\\, \n\t\\frac{ \\hat \\rho^S(L_k)(\\exp(\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha)))} {\\exp(\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha))}\n\t$$\nRecall that an operator $\\rho^S(L_k)$ of type \\eqref{eq:type} is the same as $\\rho(L_k)$  where the matrix $a$  has been replaced by $(S^{-1})^T a S^{-1}$ (and, accordingly, $b_k$, $c_k$, etc.). Expanding the case $k=-1$ of the last identity, one obtains that $(S^{-1})^T \\eta S^{-1}$ is diagonal. Then, Theorem~\\ref{thm:rhodecomp}, implies that $\\rho^S$ decomposes as a sum $\\rho_1+\\ldots+\\rho_n$ and Theorem~\\ref{thm:solutionrhoi} implies that $\\hat\\rho_\\alpha(L_k)(\\tau_\\alpha(t_\\alpha))=0$. \n\nIt is straightforward that $S$ is unique up to an ortogonal matrix.\n\\end{proof} \n\n\n\n\\begin{cor}\nLet $\\rho$ be as in the Theorem~\\ref{thm:solutionproductKdV}.  If either $S$ is diagonal or $\\tau_\\alpha$ are $\\tau$-functions of the same scaled hierarchy, then the solution is a $\\tau$-function for the scaled multicomponent KP hierarchy. \n\\end{cor}\n\n\\begin{proof}\nIn particular, the product $S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))$ is uniquely determined by $\\rho$. Each function $\\tau_\\alpha( t_\\alpha)$ satisfies the scaled KdV and, thus, there are $\\lambda_\\alpha:=(\\lambda_{i,\\alpha})\\in\\prod_{i\\text{ odd}}{\\mathbb C}^*$ such that $\\tau_\\alpha( \\lambda_\\alpha^{-1} t_\\alpha)$ defines a point $U_\\alpha\\in \\operatorname{Gr}({\\mathbb C}((z)))$. If $\\rho$ is expressed w.r.t. a basis $\\{a_1,\\ldots , a_n\\}$, then $S$ determines a second basis $\\{S(a_1),\\ldots, S(a_n)\\}$ or, equivalently, an isomorphism  ${\\mathbb C}\\oplus\\ldots \\oplus{\\mathbb C}\\simeq A$. This isomorphism induces:\n\t{\\small $$\n\t\\operatorname{Gr}({\\mathbb C}((z)))\\times \\ldots\\times \\operatorname{Gr}({\\mathbb C}((z))) \\hookrightarrow \n\t\\operatorname{Gr}({\\mathbb C}((z))\\oplus\\ldots\\oplus{\\mathbb C}((z))) \n\t\\simeq \\operatorname{Gr}(A\\otimes {\\mathbb C}((z)))\n\t$$}\nSince $\\tau$-function of the image of $(U_1,\\ldots, U_n)$, which is $U_1\\oplus\\ldots\\oplus U_n$, is given by $\\prod_\\alpha \\tau_\\alpha(\\lambda_\\alpha^{-1} t_\\alpha)$ it follows that $S \\prod_\\alpha \\tau_\\alpha(t_\\alpha)$ is a $\\tau$-function of the scaled multicomponent KP in the two cases of the statement.\n\\end{proof}\n\n\n\\begin{rem}\nRecalling Remark~\\ref{rem:VirtoKdV}, we observe that Theorems~\\ref{thm:solutionWscaleKdV} and~\\ref{thm:solutionproductKdV} could be weaken and stated for representations satisfying the hypothesis up to a linear function on $t_i$'s.\n\\end{rem}\n\n\n\n\\subsection{Final Comments}\\label{subsec:final}\n\nLet us finish with some brief comments. From a general perspective, we hope that our methods shed some light on the explicit expressions of the Virasoro operators and of the relevant integrable hierarchies that appear in   the Virasoro conjecture.  Furthermore, they can also be applied to many instances of representations of ${\\mathcal W}_>$ such as recursion relations, Hurwitz numbers, knot theory, etc.~. \n\nAs an illustration, let us point out the results of \\cite{AmbjornChekhov,KazarianZograf} on Hurwitz numbers. In both cases, the authors study the generating functions of the number of coverings of ${\\mathbb P}_1\\setminus\\{0,1,\\infty\\}$ with some properties. It is shown that these functions satisfy Virasoro constraints, KP hierarchy and topological recursion (of the Eynard-Orantin type \\cite{EO}). It is remarkable that the Virasoro constraints are explicitly expressed as differential operators of the form considered in \\S\\ref{sec:ReprW>} for the case $A={\\mathbb C}$. Thus, the results of \\S\\ref{subsec:solutions1dim} can be directly applied to conclude that Virasoro constraints imply the scaled KP hierarchy. \n\n\nOur results could also be of interest within the context of Eynard-Orantin topological recursion (\\cite{EO}). Indeed,  we learn from \\cite{MulaseSafnuk} that Mirzakhani's recursion formula for the Weil-Petersson volumes (\\cite{Mir}) is indeed a Virasoro constraint imposed on a generating function of these volumes and that this function satisfies the KdV hierarchy. It is worth pointing out some recent results on the relation of topological recursion and Virasoro constraints (\\cite{Eynard, Milanov}). On the one hand, it has been shown in \\cite{Eynard} that these Virasoro constraints are actually equivalent to Eynard-Orantin topological recursion for some spectral curve. On the other hand, one knows from \\cite{Milanov}  that the correlation functions of a semisimple cohomological field theory satisfy the Eynard-Orantin topological recursion and that these recursion formulas are equivalent to $n$ copies of the Virasoro constraints for the ancestor potential. Therefore, two problems can be faced with our techniques. First, we think that Theorem~\\ref{thm:solutionproductKdV} should imply  some bilinear relations of Hirota type for the solution of the Eynard-Orantin topological recursion. Second, due to the uniqueness of the solution and the fact that the solution satisfies the KP hierarchy, there must be a relation of the Eynard-Orantin spectral curve and the Krichever construction.  \n\nSimilarly, it would be interesting to interpret the recent papers \\cite{Da,DOSS} from our perspective. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\paragraph{}We have proposed to augment the standard model by a ``knot algebra\" that will define new ``elementary particles\".\n\nDirac \\cite{Dirac1} and Schwinger \\cite{Schwinger1} have both proposed adding hypothetical magnetic poles to the Quantized Maxwell Theory. We have shown how magnetic poles might be expected already in the context of the ``Knot Algebra\", where the field operators of the standard model acquire the factors $D^j_{mm'}(q,z)$, which are the irreducible representations of the quantum group $SL_q(2)$. This extension of the standard model is topological and could unlock magnetic poles. At the same time, because of advances in observational astronomy, the quantization of the gravitational field itself has become a topological challenge, ie, in most of the already explored universe, the influence of the gravitational field has been believed until recently to be too weak to be topologically significant. There is now, however, a growing opinion that different topologies may describe the consequences of earlier and major astronomical events, including gravitational collapse and nuclear explosions, as illustrated by the Gamow ``Big Bang\" \\cite{Hawking}. With the recent discovery of gravitational waves, physical spacetime may have become a field of speculation about its topological structure.\n\nSome of these topological possibilities are discussed by van der Bij\\cite{Bij1} in his review of recent experimental results at CERN.  It is also there pointed out that physical spacetime at earlier times may have been 3-dimensional\\cite{carlip} before entering its present phase of physical expansion. Both the new topologies and the new possibilities offered by a hypothetical earlier 2-d space were discussed by us in earlier works describing the knot model\\cite{Fink1}. \n\n\n\\section{Knot Model \\cite{Fink1} \\cite{finkelstein14a} }\n\\paragraph{} The original purpose of the knot algebra was to provide additional degrees of freedom to the Standard Model. Since the Knot Model associates a classical knot (N, w, r) with every quantum ``knot state\" $(j, m, m')$ of the standard model, it is necessary to add an odd number, o, to a defining equation of the ``knot algebra\" as follows:\n\\begin{equation}\n    \\boxed{(j, m, m')_q = \\frac{1}{2}(N, w, r+o)}\n    \\label{knotAssociation}\n\\end{equation}\nwhere w and r are of opposite parity while $m$ and $m'$ are of the same parity. Here q is the deformation parameter of $SL_q(2)$\n\nWe postulate that the quantum states of the ``new standard model\" $(j, m, m')_q$ are restricted by \\ref{knotAssociation}.\n\nThe Noether charges of the associated simplest classical knots are then\n\\begin{empheq}{align}\n    Q_w \\equiv -k_wm \\equiv -k_w\\frac{w}{2} \\label{qwdef} \\\\\n    Q_r \\equiv -k_rm' \\equiv -k_r\\frac{r+1}{2}  \\label{qrdef}\n\\end{empheq}\nwhere we set o = 1 for the simplest knots. The $k_w$ and $k_r$ are themselves undetermined constants that determine the writhe and rotation charges. The total Noether charge of the simplest knots satisfying (2.2) and (2.3) is then\n\\begin{equation}\n    Q = -k_w\\frac{w}{2}-k_r\\frac{r+1}{2}\n\\end{equation}\n\nThe association of classical knots with leptons and quarks is somewhat similar to the first association of classical knots by Bohr with the limiting states of the quantized Hydrogen atom. \n\nThere is in addition an empirical relation between the isotopic spin $(t, t_3, t_0)$ of elementary fermions and the knot characterization of the trefoils $(N, w, r)$, as shown in Table 1.\n\n\\renewcommand{\\arraystretch}{2}\n\n\\begin{figure}[H]\n\\begin{tabular}{r c r r r | p{14mm} c r r c | r}\n\\multicolumn{10}{c}{\\textbf{Table 1:} Empirical Support for $6(t,-t_3,-t_0) = (N,w,r+1)$} \\\\ \n\\hline \\hline\n & Elementary Fermions & $t$ & $t_3$ & $t_0$ & Classical Trefoil & $N$ & $w$ & $r$ & $r+1$ & $D^{N\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$ \\\\[0.2cm]\n\\hline\n\\multirow{2}{*}{\\hspace{-4pt}leptons $\\Bigg \\{$} & $(e, \\mu, \\tau)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & & 3 & 3 & 2 & 3  & $D^{3\/2}_{\\frac{3}{2}\\frac{3}{2}}$\\\\\n& $(\\nu_e, \\nu_{\\mu}, \\nu_{\\tau})_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & & 3 & $-3$ & 2 & 3 & $D^{3\/2}_{-\\frac{3}{2}\\frac{3}{2}}$\\\\\n\\multirow{2}{*}{quarks $\\Bigg \\{$} & $(d, s, b)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{6}$ & & 3 & 3 & $-2$ & $-1$ & $D^{3\/2}_{\\frac{3}{2}-\\frac{1}{2}}$\\\\\n& $(\\Bar{u}, \\Bar{c}, \\Bar{t})_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{6}$ & & 3 & $-3$ & $-2$ & $-1$ & $D^{3\/2}_{-\\frac{3}{2}-\\frac{1}{2}}$\\\\ [0.2cm]\n\\hline\n\\end{tabular}\n\n\\caption*{{The symbols $(\\quad)_L$ designate the left chiral states in the usual notation. The topological labels $(N,w,r)$ on the right side of the table provide a natural way to label the same chiral states. Note that $D^{3\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$, which labels the chiral states, correlates with the elementary fermions on the left.}}\n\\end{figure}\n\nThe chiral states $(N, w, r+1)$ which may be described as 2d projections of 3d trefoils, while the corresponding $(t, -t_3, -t_0)$ states, which may be described as 2d projections of quark states of spin, together suggest the possible relevance of an early two dimensional space (for which there may be independent evidence\\cite{carlip}).\n\nThere is also the empirical relation \\ref{empirical}, which is shown in Tables 1 and 2.\n\n\\begin{equation}\n    (j, m, m')_q = 3(t, -t_3, -t_0) \\label{empirical}\n\\end{equation}\n\nOnly for the particular row-to-row correspondences shown in Table 1 does \\ref{empirical} hold, i.e., \\emph{each of the four families of fermions labelled by $(t_3, t_0)$ is uniquely correlated with a specific $(w,r)$ classical knot, and therefore with a specific state $D^{3\/2}_{\\frac{w}{2}\\frac{r+1}{2}}(q)$ of the quantum knot.}\n\nNote in Table 1 that the $t_3$ doublets of the standard model now correspond to the writhe doublets ($w = \\pm 3$). Note also that with this same correspondence the leptons and quarks form a knot rotation doublet ($r = \\pm 2$); the lepton-quark relation depends on $(w,r)$.\n\nRetaining the row to row correspondence described in the Tables, it is then possible to compare in Table 2 the electroweak charges, $Q_e$, \\emph{of the most elementary fermions with the total Noether charges, $Q_w + Q_r$, of the simplest quantum knots, which are the quantum trefoils.}\n\n\\begin{center}\n\\begin{tabular} {c r r r c | c l c c c}\n\\multicolumn{10}{c}{{\\textbf{Table 2:} Electric Charges of Leptons, Quarks, and Quantum Trefoils}} \\\\\n\\hline \\hline\n\\multicolumn{5}{c |}{{Standard Model}} & \\multicolumn{5}{c}{{Quantum Trefoil Model}} \\\\\n\\hline\n{$(f_1, f_2, f_3)$} & {$t$} & {$t_3$} &{$t_0$} & {$Q_e$} & {$(N,w,r)$} & {$D^{N\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$} & {$Q_w$} & {$Q_r$} & {$Q_w + Q_r$} \\\\ [0.2cm]\n\\hline\n$(e, \\mu, \\tau)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-e$ & $(3,3,2)$ & $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}}$ & $-k_w \\left( \\frac{3}{2} \\right)$ & $-k_r \\left( \\frac{3}{2} \\right)$ & $-\\frac{3}{2}(k_r + k_w)$ \\\\\n$(\\nu_e, \\nu_{\\mu}, \\nu_{\\tau})_L \\hspace{-10pt}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $0$ & $(3,-3,2)$ & $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}}$ & $-k_w \\left( -\\frac{3}{2} \\right)$ & $-k_r \\left( \\frac{3}{2} \\right)$ & $\\frac{3}{2}(k_w - k_r) $ \\\\\n$(d,s,b)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{6}$ & $-\\frac{1}{3}e$ & $(3,3,-2)$ & $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}}$ & $-k_w \\left( \\frac{3}{2} \\right)$ & $-k_r \\left( -\\frac{1}{2} \\right)$ & $\\frac{1}{2}(k_r - 3k_w)$ \\\\\n$(u,c,t)_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{6}$ & $\\frac{2}{3}e$ & $(3,-3, -2)$ & $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\hspace{-5pt}$ & $-k_w \\left( -\\frac{3}{2} \\right)$ & $-k_r \\left( -\\frac{1}{2} \\right)$ & $\\frac{1}{2}(k_r + 3k_w)$ \\\\\n& & \\multicolumn{3}{c |}{$ \\hspace{-8pt} Q_e = e(t_3+t_0) \\hspace{-10pt}$} & \\multicolumn{2}{c}{\\normalsize $(j, m, m') = \\frac{1}{2}(N, w, r + 1)$} & $Q_w = -k_w \\frac{w}{2} \\hspace{-5pt}$ & $Q_r = -k_r \\frac{r+1}{2} \\hspace{-15pt}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\n\\emph{One sees that \\boxed{Q_w + Q_r = Q_e} is satisfied for charged leptons, neutrinos and for both up and down quarks with only a single value of k as follows:}\n\\begin{equation}\n\\boxed{k_r = k_w (= k)=\\frac{e}{3}} \\label{3k}\n\\end{equation}\nand also that $t_3$ isospin and $t_0$ hypercharge then respectively measure the writhe and rotation charges of the associated classical knot:\n\\begin{equation}\nQ_w = et_3 \\label{qwet3}\n\\end{equation}\n\\begin{equation}\nQ_r = e t_0 \\label{qret0}\n\\end{equation}\nThen $Q_w + Q_r = Q_e$ becomes by \\eqref{qwet3} and \\eqref{qret0} an alternative statement of\n\\begin{equation}\nQ_e = e(t_3 + t_0)\n\\end{equation}\nof the standard model. It is important to note that this classification of fundamental particles in Tables 1 and 2 is made by \\emph{charge and not by mass}\n\nIn SLq(2) measure $Q_e = Q_w + Q_r$ is by \\eqref{qwdef} and \\eqref{qrdef}:\n\\begin{equation}\n\\boxed{Q_e = -\\frac{e}{3}(m+m'), \\label{qeem}}\n\\end{equation}\nor by (2.3)\n\\begin{equation}\nQ_e = - \\frac{e}{6}(w+r+1). \\label{qeewr}\n\\end{equation}\nfor the quantum trefoils, that represent the elementary fermions.\n\n\nWe can denote the fundamental two dimensional representation of $(j, m, m')_q$ by\n\n\\begin{equation}\nD^{\\frac{1}{2}}_{mm'}(q) = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\label{jhalfrep}\n\\end{equation}\n\nIn the physical model with the \\eqref{jhalfrep} coupling we interpret $(a,b,c,d)$ as creation operators for $(a,b,c,d)$ particles, which we term preons. Then we assume that $D^j_{mm'}(a,b,c,d)$ is the creation operator for the state representing the superposition of $(n_a, n_b, n_c, n_d)$ preons. Since we shall regard the preons as fermions, they will also carry an anti-symmetrizing index to satisfy the Pauli principle. \n\nHere we may additionally generalize by describing some aspects of a generic field theory where the field quanta have two couplings that may be expressed in the coupling matrix\n\\begin{equation}\n\\boxed{\n\\varepsilon_q = \\begin{pmatrix}\n0 & \\alpha_2 \\\\\n-\\alpha_1 & 0\n\\end{pmatrix}. \\cite{FinkSchwing}} \\label{couplingmatrix}\n\\end{equation}\nwhere the couplings $\\alpha_1$ and $\\alpha_2$ are assumed to be dimensionless and real and may be written as\n\\begin{equation}\n(\\alpha_1, \\alpha_2) \\text{ or } (\\alpha_2, \\alpha_1) = \\left( \\frac{e}{\\sqrt{\\hbar c}}, \\frac{g}{\\sqrt{\\hbar c}} \\right) \\label{couplingconstants}\n\\end{equation}\nwhere $e$ and $g$ refer to a specific two charge model and have dimensions of an electric charge. We assume that e and g may be energy dependent and normalized at relevant energies. The reference charge is the universal constant $\\sqrt{\\hbar c}$. \n\nThe fundamental assumption that we make on this coupling matrix is that it is invariant under SLq(2) as follows\n\\begin{equation}\nT \\varepsilon_q T^t = T^t \\varepsilon_q T = \\varepsilon_q \\label{slq2invariant}\n\\end{equation}\nwhere $t$ means transpose and $T$ is a two dimensional representation of SLq(2):\n\\begin{equation}\nT = \\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} . \\label{2drepslq2}\n\\end{equation}\nwhere the elements of $T$ obey the knot algebra:\n\\begin{equation}\n\\begin{split}\nab = qba \\qquad bd = qdb \\qquad ad-qbc = 1 \\qquad bc &= cb \\\\\nac = qca \\qquad cd = qdc \\qquad da -q_1cb = 1 \\qquad q_1 &\\equiv q^{-1}\n\\end{split}\n\\tag{A}\n\\end{equation}\nand\n\\begin{equation}\n\\boxed{ q = \\frac{\\alpha_1}{\\alpha_2}}\n\\end{equation}\nso that the two physical couplings fix the algebra through their ratio.\n\nIf also\n\\begin{equation}\n\\text{det} \\hspace{2pt} \\varepsilon_q = 1 \\label{couplingdet}\n\\end{equation}\none has\n\\begin{equation}\n\\alpha_1 \\alpha_2 = 1\n\\end{equation}\nIf the two couplings $(\\alpha_1, \\alpha_2)$ are given by \\eqref{couplingconstants}, where $e$ and $g$ are the electroweak and ``gluon''-like couplings, or electric and magnetic couplings respectively, then\n\\begin{equation}\n\\boxed{eg = \\hbar c} \\label{quantize}\n\\end{equation}\n\nSince $g$ represents magnetic charge, \\eqref{quantize} copies the Dirac proposal according to which the magnetic charge is very much stronger than the electric charge. If the magnetic pole is very much heavier as well, it may be observable only at early and not at current cosmological temperatures or at currently achievable accelerator energies.\n\nWe may alternatively assume that q-magnetic poles do exist and shall study the possible extension of q-knot symmetry to magnetic charges, in particular as sources of the gravitational field, with \\emph{strength} q.\n\\begin{equation}\n    \\boxed{g = qe}    \n\\end{equation}\nby (2.17), where q is the deformation parameter of $SL_q(2)$, so that one may assume either (2.20) or (2.21), and $g\/e$ may be either $\\hbar c \/ e^2$ or $q$ or both: then $q = \\hbar c \/ e^2$. This new expression for q would directly relate Gamow cosmology to Bohr quantum mechanics, and would be required to be experimentally realized.\n\n\\section{Graphical Representation of Corresponding Classical Structures}\nThe representation of the four classical trefoils as composed of three overlapping preon loops is shown in Figure 1. In interpreting Figure 1, note that the two lobes of all the preon loops make opposite contributions to the rotation, $r$, so that the total rotation of each preon loop vanishes. When the three $a$-preons and $c$-preons are combined to form charged leptons and neutrinos, respectively, as suggested by Harari\\cite{harari79}, Shupe\\cite{Schupe1}, and Raitio\\cite{Raitio1} each of the three labelled circuits is counterclockwise and contributes $+1$ to the rotation while the single unlabeled and shared (overlapping) circuit is clockwise and contributes $-1$ to the rotation so that the total $r$ for both charged leptons and neutrinos is $+2$. In this way the leptons, neutrinos, up and down quarks may be considered composite massive particles that are \\emph{sources of the gravitational field}, and are here abreviated by a, b, c, d, in accord with the fundamental representation of $SL_q(2)$. \\vspace{-0.3em}\n\n\\newgeometry{bottom=2.5cm}\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c | c} \n         \\multicolumn{2}{c}{} \\\\ [-0.54cm]\n\t\\multicolumn{2}{c}{\\textbf{Figure 1:} Preonic Structure of Elementary Fermions} \\\\ [-0.49cm]\n\t\\multicolumn{2}{c}{$Q = -\\frac{e}{6}(w+r+o)$, and $(j, m, m') = \\frac{1}{2} (N, w, r+o)$} \\\\ [-0.25cm]\n\t\\multicolumn{2}{c}{$D^j_{mm'} = D^{\\frac{N}{2}}_{\\frac{w}{2} \\frac{r+o}{2}}$}\\\\\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Charged Leptons, $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}} \\sim a^3$} \\\\ [0.2cm]\n\t\t \\Large{$\\varepsilon^{ijk}$} & $\\hspace{17pt} a_j$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} a_i$ & & $\\hspace{-58pt} a_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{ \\xoverh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu }\n[l(2)]\n!{\\vcap[2] =>}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xunderh }\n[d(0.5)]\n!{ \\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl- }\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5] =<}\n[u(1.75)] [l(2)]\n!{ \\xoverv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[u(1)] [l(1)]\n!{ \\hcap[-2]=>}\n[d(1)]\n!{ \\xbendl \\color{black}}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{ \\hspace{33pt} \\textcolor{red}{\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =>}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =>}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =>}\n}}}\n \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(3,2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{\\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{$a$-preons, $D^{1\/2}_{\\frac{1}{2} \\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $a_i$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=>}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.2cm]\n\\multicolumn{3}{l}{\\hspace{39pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(0.75)] [l(0.75)]\n!{\\xcaph[-1]=>}\n}\n}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(1,0,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{Neutrinos, $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}} \\sim c^3$} \\\\ [0.2cm]\n\t\t \\Large{$\\varepsilon^{ijk}$}& $\\hspace{17pt} c_j$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} c_i$ & & $\\hspace{-58pt} c_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{ \\vcap[2]=>}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xoverh}\n[d(0.5)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{ \\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl \\color{black}}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\textcolor{red} {\\hspace{32pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =<}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =<}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-3,2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$c$-preons, $D^{1\/2}_{-\\frac{1}{2} \\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $c_i$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=>}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.2cm]\n\\multicolumn{3}{l}{\\textcolor{red}{\\hspace{39pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[d(0.75)] [l(0.75)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-1,0,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-quarks, $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}} \\sim ab^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} b$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} a_i$ & & $\\hspace{-56pt} b$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{ \\xoverv}\n[u(0.5)] [r(1)]\n!{ \\xbendr }\n[u(2)]\n!{\\hcap[2]=<}\n[l(1)]\n!{\\xbendl- }\n[u(0.5)] [l(3)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr- }\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl  \\color{black}}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\[-1.6cm]\n\\multicolumn{3}{l}{\\hspace{42pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=<}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=<}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$b$-preons, $D^{1\/2}_{\\frac{1}{2} -\\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $b$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\hspace{40pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=>}\n}}} \\\\ [-0.1cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(1,0,-1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$u$-quarks, $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\sim cd^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} d$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} c_i$ & & $\\hspace{-56pt} d$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{\\vcap[2] =<}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xunderv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2] =<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl \\color{black}}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\[-1.75cm]\n\\multicolumn{3}{l}{\\hspace{44pt}\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=>}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=>}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-preons, $D^{1\/2}_{-\\frac{1}{2} -\\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $d$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\hspace{39pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=<}\n}}} \\\\ [-0.1cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-1,0,-1)$} \\\\\t\t \n\t\\end{tabular} \\\\ [-0.43cm]\n\t\\multicolumn{2}{c}{The clockwise and counterclockwise arrows are given opposite weights $(\\mp 1)$ respectively.} \\\\ [-0.51cm]\n\t\\multicolumn{2}{c}{The (rotation\/writhe charge) is measured by the sum of the weighted (black\/red) arrows.} \\\\\n\t[-0.51cm]\n\t\\multicolumn{2}{c}{The central loops of the trefoils contribute oppositely to the rotation of the complete trefoil.}\n\n\t\\end{tabular}\n\\end{center}\n\n\\restoregeometry\n\n\\setlength{\\baselineskip}{1.6\\baselineskip}\n\nFor quarks the three labelled loops contribute $-1$ and the shared loop $+1$ so that $r=-2$, as required.\n\nIn each case the three preons that form a lepton trefoil contribute their three negative rotation charges. The geometric and charge profile of the lepton trefoil is thus similar to the geometric and charge profile of a triatomic molecule composed of neutral atoms since the valence electronic charges of the atoms, which cancel the nuclear electronic charges of the atoms, are shared among the atoms to create the chemical binding of the molecule just as the negative rotation charges which cancel the positive rotation charges of the preons are shared among the preons to create the preon binding of the trefoils. There is a similar correspondence between quarks and antimolecules.\n\nWe next display the general representation $D^j_{m m'}(q)$ of the algebra $SL_q(2)$ in the Weyl monomial basis just as in the current standard model.\n\\begin{equation}\n\\boxed{ {D^j_{mm'}(q) = \\sum_{n_a, n_b, n_c, n_d} A(q | n_a, n_b, n_c, n_d) a^{n_a} b^{n_b} c^{n_c} d^{n_d} \\label{long}}}\n\\end{equation}\n\nwhere $a, b, c, d$ satisfy the algebra (A) and the sum  on $n_a, n_b, n_c, n_d$ is over all positive integers and zero that satisfy the following equations: \\cite{finkelstein14a}\n\\begin{empheq}[box=\\fbox]{align}\n    n_a + n_b + n_c + n_d &= 2j \\label{n2j}\\\\\n    n_a + n_b - n_c - n_d &= 2m \\label{n2m}\\\\\n    n_a - n_b + n_c - n_d &= 2m' \\label{n2m'}\n\\end{empheq}\n\nand\n\n\\begin{equation}\nA (q \\vert n_a n_b n_c n_d) = \\left [ \\frac{ \\langle n_+ ' \\rangle_1 ! \\langle n_- ' \\rangle_1 ! }{\\langle n_+ \\rangle_1 ! \\langle n_- \\rangle_1 !} \\right ]^{\\frac{1}{2}} \\frac{ \\langle n_+ \\rangle_1 !}{\\langle n_a \\rangle_1 ! \\langle n_b \\rangle_1 !} \\frac{\\langle n_- \\rangle_1 !}{\\langle n_c \\rangle_1 ! \\langle n_d \\rangle_1 !}.\n\\end{equation}\nwhere $n_\\pm = j \\pm m$, $n ' _\\pm = j \\pm m '$, and $\\langle n \\rangle _1 = \\frac{q_1 ^n - 1}{q_1 ^1 - 1}$ and $q_1 = q^{-1}$\n\nThe two dimensional representation, $T$, introduced by \\eqref{2drepslq2} now reappears as the $j = \\frac{1}{2}$ fundamental representation\n\\begin{equation}\n\\boxed{D^{\\frac{1}{2}}_{mm'}(q) = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\label{}}\n\\end{equation}\nIn the physical model with the \\eqref{couplingmatrix} coupling we interpret $(a,b,c,d)$ in \\eqref{long} as creation operators for $(a,b,c,d)$ particles, which we have termed preons. Then $D^j_{mm'}(a,b,c,d)$ is the creation operator for the state representing the superposition of $(n_a, n_b, n_c, n_d)$ gravitational preons. Since we shall regard the preons as fermions, they will also carry an anti-symmetrizing index to satisfy the Pauli principle.\n\n\\emph{In proceeding to the quantization of the gravitational field we represent its sources as Standard Model x Knot Model sources ($SL_q(2)$) where the particles of the Knot Model are the preons (a, b, c, d) described in Fig. 1 as $D^{3\/2}_{m, m'}(q)$ and also represented as twisted loops as the auxiliary knot particles}. The physical particles are here graphically represented by closed loops of energy-momentum with $n \\frac{e}{3}$ charge where n is the number of net positive turns of the tangent in one transit of the loop.\n\n\n\\section{Presentation of the Model in the Preon Representation\\cite{fink2018}\\cite{FinkSchwing}} \n\nThe particles $(a,b,c,d)$ described in the following sections are assumed to be either e or g preons and carry both e and g charges. The knot representation (3.1) of $D^j_{mm'}$ as a function of $(a,b,c,d)$ and $(n_a,n_b,n_c,n_d)$ implies the following constraints on the exponents:\n\n\\begin{IEEEeqnarray*}{+rCl+x*}\nn_a+n_b+n_c+n_d & = & 2j & (4.1) \\\\\nn_a+n_b - n_c - n_d & =  & 2m & (4.2) \\\\\nn_a-n_b+n_c-n_d & = & 2m' . & (4.3)\n\\end{IEEEeqnarray*}\nThe two relations defining the quantum kinematics and giving physical meaning to $D^j_{mm'}$, namely the postulated \\eqref{knotAssociation}:\n\\begin{equation*}\n(j,m,m')_q = \\frac{1}{2}(N,w, r+o) \\quad \\text{field (flux loop) description} \\tag{4.4}\n\\end{equation*}\nand the semi-empirical \\eqref{empirical} shown in the tables.\n\\begin{equation*}\n(j,m,m')_q = 3(t, -t_3, -t_0)_L \\quad \\text{particle description} \\tag{4.5}\n\\end{equation*}\nimply two complementary interpretations of the relations (4.1) -- (4.3). By (4.4) and these equations, one has a \\textit{field} description $(N, w, \\tilde{r})$ of the quantum state $(j, m, m')_q$ as follows\n\\begin{equation}\n\\left.\n{\\arraycolsep=1.2pt\n\\begin{array}{rl}\nN &= n_a + n_b + n_c + n_d \\\\\nw &= n_a + n_b - n_c - n_d  \\\\\n\\tilde{r} \\equiv r+o &= n_a - n_b + n_c - n_d\n\\end{array}\n}\n\\quad \\right\\} \\text{field (flux loop) (N, w, $\\tilde{r}$) description} \\label{field (flux loop) description}\n\\tag{4.6}\n\\end{equation}\nIn the last line of \\eqref{field (flux loop) description}, where $\\tilde{r} \\equiv r+o$ and $o$ is a parity index, $\\tilde{r}$ has been termed ``the quantum rotation,\" and $o$ the ``zero-point rotation.\" \n\nBy \\eqref{empirical} one has a ``\\textit{particle} description\" $(t, t_3, t_0)$ of the same quantum state $(j, m, m')$.\n\\begin{comment}\n\\begin{align}\nt &= \\frac{1}{6} (n_a + n_b +n_c +n_d) \\\\\nt_3 &= -\\frac{1}{6}(n_a + n_b - n_c - n_d) \\quad \\text{particle description} \\multirow{3}{*}{\\}} \\\\ \nt_0 &= -\\frac{1}{6}(n_a - n_b + n_c - n_d)\n\\end{align}\n\\end{comment}\n\\begin{equation}\n\\left.\n{\\arraycolsep=1.2pt\n\\begin{array}{rl}\nt &= \\frac{1}{6} (n_a + n_b +n_c +n_d)  \\\\\nt_3 &= -\\frac{1}{6}(n_a + n_b - n_c - n_d)  \\\\\nt_0 &= -\\frac{1}{6}(n_a - n_b + n_c - n_d) \n\\end{array}\n}\n\\quad \\right\\} \\text{particle ($t$, $t_3$, $t_0$) description} \\label{particle_description}\n\\tag{4.7}\n\\end{equation}\n\\emph{In \\eqref{particle_description}, $(t, t_3, t_0)$ are to be read as SLq(2) preon indices agreeing with standard SU(2) $\\times$ U(1) notation} only \\emph{at $j = \\frac{3}{2}$. In general, however, we do not assume $SU(2)\\times U(1)$ and instead may assume that $t_3$ measures writhe charge, $t_0$ measures rotation hypercharge and $t$ measures the total preon population or the total number of crossings of the associated classical knot.}\n\nThe attempt to invert (4.4) as (4.6) is quite successful but the corresponding attempt to rewrite the semi-empirical (2.5) as $t, t_3, t_0$ is not. We interpret this difference in favor of (4.4) or as a weakness of (4.5).\n\n\n\\section{Interpretation of the Complementary Equations}\nThere is also an alternative particle interpretation of the flux loop equations \\eqref{field (flux loop) description}\n\\begin{align*}\nN = n_a + n_b + n_c + n_d \\tag{5.1$N$} \\\\\nw = n_a + n_b - n_c - n_d \\tag{5.1$w$} \\\\\n\\tilde{r} = n_a - n_b + n_c - n_d \\tag{5.1$\\tilde{r}$}\n\\end{align*}\n\nHere the left-hand side with coordinates $(N,w,\\tilde{r})$ label a 2d-projected knot, and the right-hand side describes the preon population of the corresponding quantum state. \n\n\\emph{Equation (5.1$N$) states that the number of crossings, $N$, equals the total number of preons, $N'$, as given by the right side of this equation. Since we assume that the preons are fermions, the knot describes a fermion or a boson depending on whether the number of crossings is odd or even.} Viewed as a knot, a fermion becomes a boson when the number of crossings is changed by attaching or removing a geometric curl\n\\begin{turn}{90}\n\\xygraph{\n    !{0;\/r0.75pc\/:}\n    !{\\xunderh}\n    [u l(0.75)]!{\\xbendu}\n    [l (1.5)]!{\\vcap[1.5]}\n    !{\\xbendd-}}\n\\end{turn}\n. This picture is consistent with the view of a geometric curl as an opened preon loop, in turn viewed as a twisted loop\n\\begin{turn}{90}\n\\xygraph{\n    !{0;\/r0.75pc\/:}\n    !{\\xunderh}\n    [u l(0.75)]!{\\xbendu}\n    [l (1.5)]!{\\vcap[1.5]}\n    !{\\xbendd-}\n    [l] [d(0.5)]!{\\xbendu-}\n    [ld]!{\\vcap[-1.5]}\n    [u] [r(0.5)]!{\\xbendd}\n}\n\\end{turn}. Each counterclockwise or clockwise classical curl corresponds to a preon creation operator or antipreon creation operator respectively.\n\\newline\n\n\\subsection{\\emph{Gravitational} Preon Numbers}\n\nSince $a$ and $d$ are creation operators for antiparticles with opposite charge and hypercharge, while $b$ and $c$ are neutral antiparticles with opposite values of the hypercharge, we may introduce the \\emph{gravitational preon numbers}\n\\begin{align}\n\\nu_a &= n_a -n_d \\\\\n\\nu_b &= n_b - n_c\n\\end{align}\nThen (5.1$w$) and (5.1$\\tilde{r}$) may be rewritten in terms of gravitational preon numbers as\n\\begin{align}\n&\\nu_a + \\nu_b = w \\hspace{3pt} (= -6t_3) \\\\\n&\\nu_a - \\nu_b = \\tilde{r} \\hspace{3pt} (=-6t_0)\n\\end{align}\nBy (5.3) and (5.4) the conservation of the preon numbers and of the charge and hypercharge is equivalent to the conservation of the writhe and rotation, which are topologically conserved at the 2d-classical level. In this respect, these quantum conservation laws for preon numbers correspond to the classical conservation laws for writhe and rotation.\n\nEqns. $(5.1N) - (5.1\\tilde{r})$may also be interpreted directly in terms of Fig. 2 by describing the right-hand side of these equations as the possible populations of the conjectured preons at these crossings of Fig. 2 and interpreting the left-hand side as parameters of the binding field that links the 3 conjectured preons.\n\\newline\n\n\n\\section{Summary on the Measure of Charge by SU(2) $\\times$ U(1) and by SLq(2)} \nThe SU(2)$\\times$U(1) measure of charge requires the assumption of fractional charges for the quarks. The SLq(2) measure requires the replacement of the fundamental charge $(e)$ for charged leptons by a new fundamental charge $(e\/3)$ or $(g\/3)$ for charged preons but then does not require fractional charges for quarks\\cite{Schupe1} if quark charge is expressed in terms of the new ``fundamental or minimal charges\", i.e. $(e\/3)$ or $q(e\/3)$.\n\nThe SLq(2), or $(j,m,m')_q$ measure, has a direct preon interpretation since $2j$ is the total number of preonic sources, while $2m$ and $2m'$ respectively measure the numbers of writhe and rotation sources of preonic charge.\\cite{Fink2} Since $N$, $w$, and $r$ all measure the handedness of the source, charge is measured by the chirality of the source. \\emph{The electric charge of the resultant trefoil or of any composite of preons would then be a measure of the chirality generated by the knotting of an original unknotted flux loop of energy-momentum.}\n\nIf neutral unknotted flux tubes predated the particles, and the particles were initially formed by the knotting of unknotted flux tubes of energy-momentum, then the simplest fermions that could have formed with topological stability must have had three 2d crossings and therefore three preons, but the topological stability of this trefoil could be protected only as long as spacetime remains 2-dimensional as suggested by the comments on Tables 1 and 2.\n\nThe total SLq(2) charge sums the signed two dimensional clockwise and counterclockwise turns that any energy-momentum current makes both at the crossings and in making a single circuit of the 2d-projected knot. This measure of charge, ``knot charge\", which is suggested by the leptons and quarks, appears more fundamental than the electroweak isotopic measure that originated in the neutron-proton system, since it reduces the concept of charge to the chirality of the corresponding energy-momentum curve which may also be described as a SLq(2) chirality and in this way reduces charge to a topological concept similar to the way energy-momentum is geometrized by the curvature of spacetime in the Einstein-Hilbert equations or in lattice methods of Regge for solving the differential equations. In this way the energy-momentum conservation may also be formulated graphically.\\cite{Regge}\n\n\n\\section{Other Possible Physical Interpretations of Corresponding Quantum States}\n\n\\begin{comment}\nThe point particle $(N', \\nu_a, \\nu_b)$ representation and the flux loop $(N, w, \\tilde{r})$ complementary representation are related by the $\\delta$-function transform:\n\\begin{align}\n\\tilde{D}^{N'}_{\\nu_a, \\nu_b}(q, a, b, c, d) = \\sum_{N,w,r} \\delta(N',N)\\delta(\\nu_a+\\nu_b, w)\\delta(\\nu_a - \\nu_b, \\tilde{r}) D^{N\/2}_{\\frac{w}{2} \\frac{\\tilde{r}}{2}} (q, a, b, c, d)\n\\end{align}\n\\end{comment}\n\nSince one may interpret the elements $(a,b,c,d)$ of the SLq(2) algebra as creation operators for either preonic particles or current loops, the $D^j_{mp}(q)$ may be interpreted as a creation operator for a composite quantum particle composed of either preonic particles $(N', \\nu_a, \\nu_b)$ or current loops $(N, w, \\tilde{r})$ . \\emph{These two complementary views of the same particle may be reconciled as describing $N$-preon systems bound by a knotted field having $N$-crossings with the preons at the crossings as illustrated in Figure 2 for $N=3$.} In the limit where the three outside lobes become small or infinitesimal compared to the central circuit, the resultant structure will resemble a three particle system tied together by a string.\n\\begin{figure}[!tb]\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c | c}\n\t\\multicolumn{2}{c}{\\textbf{Figure 2:} Leptons and Quarks Pictured as Three Preons Bound by a Trefoil Field} \\\\ [-0.0cm]\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Neutrinos, $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}} \\sim c^3$} \\\\ [0.2cm]\n\t\t & $\\hspace{17pt} c_j$ & \\\\ [-.22 cm]\n\t\t \\multicolumn{3}{c}{\\hspace{-1.655 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} c_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm} & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} c_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=>}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xoverh}\n[d(0.5)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl}}} \\\\[-1.5cm]\n\\multicolumn{3}{l}{\\textcolor{red} {\\hspace{32pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =<}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =<}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-3,2,1)$}\n\t\t \n\t\\end{tabular}\n&\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Charged Leptons, $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}} \\sim a^3$} \\\\ [0.2cm]\n\t\t & $\\hspace{17pt} a_j$ & \\\\ [-.22 cm]\n\t\t \\multicolumn{3}{c}{\\hspace{-1.58 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} a_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-58pt} a_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=>}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xunderh}\n[d(0.5)]\n!{\\xbendr }\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{\\hspace{33pt} \\textcolor{red}{\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =>}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =>}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(3,2,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-quarks, $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}} \\sim ab^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} b$ & \\\\ [-.22 cm]\n\t\t  \\multicolumn{3}{c}{\\hspace{-1.52 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} a_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} b$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xoverv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2] =<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{\\hspace{42pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=<}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=<}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$u$-quarks, $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\sim cd^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} d$ & \\\\ [-.22 cm]\n\t\t  \\multicolumn{3}{c}{\\hspace{-1.75 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} c_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.9cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} d$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xunderv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\ [-1.75cm]\n\\multicolumn{3}{l}{\\hspace{44pt}\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=>}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=>}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-3,-2,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.2cm]\n\\multicolumn{2}{c}{The preons conjectured to be present at the crossings are suggested by the blue dots at the crossings} \\\\ [-0.4cm]\n\\multicolumn{2}{c}{ of the lepton-quark diagrams, or at the crossings of any diagram with more crossings.}\n\\end{tabular}\n\\end{center}\n\\end{figure}\n\\emph{The physical models suggested by Fig. 2 may be further studied with the aid of preon Lagrangians similar to that given in reference 3}. The Hamiltonians of these three body systems may be parametrized by degrees of freedom characterizing both the preons and the binding field that come from the \\emph{form factors required by SLq(2) invariance}. The masses of the leptons, quarks, and binding quanta are determined by the eigenvalues of this Hamiltonian in terms of the parameters describing the constituent preons and energy-momentum flux loops. There is currently no experimental guidance at these conjectured energies. These three body systems are, however, familiar in different contexts, namely\n\\begin{center}\nH$^3$ composed of one proton and two neutrons: $PN^2$ \\\\\n$P$ composed of one down and two up quarks: $DU^2$ \\\\\n$N$ composed of one up and two down quarks: $UD^2$ \\\\\n\\end{center}\nwhich are similar to\n$U$ as $cd^2$ and $D$ as $ab^2$,\nwhere $U$ and $D$ are up and down quarks, presented as three binding preon states. These different realizations of energy-momentum and charge represent different expressions of curvature and chirality, and in particular as displayed by a closed loop of energy-momentum.\n\n\\section{Alternate Interpretation}\nIn the model suggested by Fig. 2 the parameters of the preons and the parameters of the current loops are to be understood as codetermined. On the other hand, in an alternative interpretation of complementarity, the hypothetical preons conjectured to be present in Figure 2 may carry no independent degrees of freedom and may simply describe \\emph{concentrations of energy and momentum at the crossings of the energy-momentum tube}. In this interpretation of complementarity, $(t, t_3, t_0)$ and $(N,w, \\tilde{r})$ are just two ways of describing the same quantum trefoil of field. \\emph{In this picture the preons are bound}, i.e. they do not appear as free particles. This view of the elementary particles as either non-singular lumps of field or as solitons has also been described as a unitary field theory\\cite{Raitio1}.\n\n\\section{Lower Representations}\nWe have so far considered the states $j = 3, \\frac{3}{2}, \\frac{1}{2}$ representing electroweak vectors, leptons and quarks, and preons, respectively. We finally consider the states $j=1$ and $j=0$. Here we shall not examine the higher $j$ states.\n\nIn the adjoint representation $j=1$, the particles are the vector bosons by which the $j=\\frac{1}{2}$ preons interact and there are two crossings. These vectors are different from the $j=3$ vectors by which the $j=\\frac{3}{2}$ leptons and the $j=\\frac{3}{2}$ quarks interact.\n\\bigskip\n\nIf $j=0$, the indices of the quantum knot are\n\\begin{align}\n(j,m,m')_q = (0,0,0)\n\\end{align}\nand by the basic rule \\eqref{knotAssociation} for interpreting the knot indices on the left chiral fields\n\\begin{align}\n\\frac{1}{2}(N,w,\\tilde{r}) = (j, m, m')_q &= (0, 0, 0) \n\\end{align}\nThen the $j=0$ quantum states correspond to classical loops with no crossings $(N=0)$ just as preon states correspond to classical twisted loops with one crossing. Since $N=0$, the $j=0$ states also have no preonic sources of charge and therefore no electroweak interaction.  \\emph{It is possible that these }$j=0$ \\emph{hypothetical quantum states are realized as (electroweak non-interacting) loops of field flux with} $w=0$, $\\tilde{r}= r+o = 0$\\emph{, and }$r = \\pm1$, $o =\\mp1$ \\emph{ i.e. with the topological rotation }$r=\\pm1$. The two states $(r, o) = (+1, -1)$ and $(-1, +1)$ are to be understood as quantum mechanically coupled.\n\nIf, as we are assuming, the leptons and quarks with $j= \\frac{3}{2}$ correspond to 2d projections of knots with three crossings, and if the heavier preons with $j= \\frac{1}{2}$ correspond to 2d projections of twisted loops with one crossing, then if the $j=0$ states correspond to 2d projections of simple loops with no crossings, one might ask if these particles with no electroweak interactions and which are smaller and heavier than the preons, are among the candidates for ``dark matter.\" If these $j=0$ particles predate the $j=\\frac{1}{2}$ preons, one may refer to them as ``yons\" as suggested by the term ``ylem\" for primordial matter.\n\n\\section{Speculations about an earlier universe and dark matter} \\vspace{-0.8em}\nOne may speculate about an earlier universe before leptons and quarks had appeared, when there was no charge, and when energy and momentum existed only in the SLq(2) $j=0$ neutral state as simple loop currents of gravitational energy-momentum. Then the gravitational attraction would bring some pairs of opposing loops close enough to permit the transition from two $j=0$ loops into two opposing $j=\\frac{1}{2}$ twisted loops. A possible geometric scenario for the transformation of two simple loops of current (yons) with opposite rotations into two $j=\\frac{1}{2}$ twisted loops of current (preons) is suggested in Fig. 3. Without attempting to formally implement this scenario, one notes according to Fig. 3 that the fusion of two yons may result in a doublet of preons as twisted loops, which might also qualify as Higgs particles.\n\n\\newpage\n\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c  c  c}\n\\multicolumn{3}{c}{ \\textbf{Figure 3:} Creation of Preons as Twisted Loops} \\\\\n\\vspace{-3pt}\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\hcap[2]=>}\n!{\\hcap[-2]}} \\hspace{3pt} \n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\hcap[2]}\n!{ \\hcap[-2]=< }} &\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1.25)]\n!{\\xcaph[3]=<@(0)}} & \\hspace{-4pt}\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1)]\n!{\\color{red} \\vcap[2]=> \\color{blue}}\n!{\\vcap[-2]}} \\hspace{-7pt} \n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1)]\n!{\\color{blue} \\vcap[2]=<}\n!{\\color{red} \\vcap[-2]=< \\color{black}}}\\\\ [-1.7cm]\n\\hspace{4pt}\\scriptsize{$r=1 \\hspace{20pt}+ \\hspace{16pt}r=-1$} & & \\scriptsize{$\\hspace{3pt}r=1 \\hspace{8pt} r=-1$} \\\\ [-0.7cm]\n\\hspace{0pt}\\scriptsize{$\\tilde{r}=0 \\hspace{32pt} \\hspace{16pt}\\tilde{r}=0$} & & \\scriptsize{$\\hspace{0pt}\\tilde{r}=0 \\hspace{10pt} \\tilde{r}=0$} \\\\\nTwo $j=0$ neutral loops & gravitational attraction & interaction causing the crossing or  \\\\ [-0.55cm]\n with opposite topological & & redirection of neutral current flux \\\\ [-0.55cm]\n  rotation & & shown below \\\\ [-0.55cm]\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{c c}\n\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=>}\n}} &\n\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=<}\n}} \\\\ [-1cm]\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}} &\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}} \\\\ [-1.1cm]\n\\multicolumn{2}{c}{+} \\\\ [-0.95cm]\n\\hspace{-95pt} & \\hspace{-95pt} \\\\ [-0.9cm]\na preon & \\hspace{-1pt} c preon \\\\ [-0.4cm]\n\\hspace{2pt}$r=0$ & \\hspace{2pt}$r=0$ \\\\ [-0.4cm]\n$w_a = +1$ & $w_c=-1$ \\\\ [-0.4cm]\n$ Q_a = -\\frac{e}{3}$ & $ Q_c = 0$  \\\\\n\\end{tabular}\n\\end{center}\n\nIn the scenario suggested by Figure 3 the opposing states are quantum mechanically entangled and may undergo gravitational exchange scattering. \n\nThe $\\binom{c}{a}$ doublet of Fig. 3 is similar to the Higgs doublet which is independently required by the mass term of the Lagrangian described in reference 3 to be a SLq(2) singlet $(j=0)$ and a SU(2) charge doublet $(t=\\frac{1}{2})$. The matrix elements connecting the preon a and c states in Fig. 3 is obviously fundamental for the model. Since the Higgs mass contributes to the inertial mass, one expects a fundamental connection with the gravitational field at this point. \n\n\\begin{comment}\nA fraction of the preons $(j= \\frac{1}{2})$ produced by the fusion of two yons might in turn combine to form two preon $(j=1)$ and then three preon $(j = \\frac{3}{2})$ states with two and three crossings respectively. The $j = \\frac{3}{2}$ states would be recognized in the present universe as leptons and quarks. Since, however, the $j = \\frac{1}{2}$ and $j=1$ particles with one and two crossings, respectively, are not topologically stable in three dimensions and can relapse into a $j=0$ state with no crossings, the building up process from yons would not produce topologically stable particles before the $j=\\frac{3}{2}$ leptons and quarks with three crossings are reached. These remarks apply equally to the e and g sectors.\n\\end{comment}\n \nIf at an early cosmological time, only a fraction of the initial gas of quantum loops was converted to preons and these in turn led to a still smaller number of leptons and quarks, then most of the mass and energy of the universe would at the present time still reside in the dark loops while charge and current and visible mass would be confined to structures composed of leptons and quarks. \\emph{In making experimental tests for particles of dark matter one might expect the SLq(2) $j=0$ dark loops to be different in mass than the dark neutrino trefoils where $j= \\frac{3}{2}$, although both $j=0$ and $j=\\frac{3}{2}$ would contribute to the dark matter.}\n\n\\bigskip\n\n\\section*{Acknowledgements}\nI thank E. Abers, C. Cadavid, J. Smit, and S. Mackie for comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\nAmyotrophic lateral sclerosis (ALS) is a rare progressive neurological disorder resulting in the degeneration of both upper motor neurons of the cerebral cortex and lower motor neurons of the spinal cord and peripheral nervous system, with a very poor prognosis.  Currently, there is no cure for ALS and clinical care is generally limited to treating secondary infections and palliative care, such as surgically inserting a percutaneous endogastrostomy (PEG) tube to provide enteral nutrition for individuals having difficulty swallowing \\citep{procaccini2008}. Our objective is to assess the effect of inserting a PEG feeding tube on preventing weight loss. PEG insertion is an individual decision and one that must be made while the individual is strong enough to proceed with surgery.  Hence, a randomized controlled trial to study the effect of PEG would be implausible.  We develop new methods to evaluate PEG using data from the Emory ALS Clinic registry.\n\nLet $T$ denote the continuously-defined time of PEG insertion for a randomly selected patient from the population.  The observed outcome $Y$ is collected at or just after a fixed point at time $L$, which consequently restricts the time of PEG insertion. If subjects were randomly assigned to receive PEG prior to $L$ and randomly assigned to treatment times, then both treatment effect and dose-response curve could be estimated using standard methods. However, treatment assignment depends on patient characteristics and confounds the effect of treatment on outcome.  To remove confounding associated with covariate imbalance among treatment levels, we rely on the general concept of the propensity score \\citep{rosenbaumandrubin1983,rubin1996}.\n\nWhen treatment assignment is binary, the propensity score \\citep{rosenbaumandrubin1983} is defined as probability of receiving a treatment given a set of observed variables. Generalizations of the propensity score as a balancing score have been investigated in various settings \\citep{hirano2004,imaiandvandyk2004,hansen2008prognostic,allen2011control,hu2014estimation}. For continuously-defined treatment levels, \\citet{hirano2004} proposed a direct translation of the propensity score by replacing the conditional probability mass function with the conditional density function of treatment assignment given covariates, known as a generalized propensity score (GPS); while this approach leads to as many propensity scores as there are levels of the treatment it uses only one single score at a time. Although \\citet{imaiandvandyk2004} similarly found that the conditional density function of treatment assignment given covariates could serve as a propensity score, they noted potential limitations of this approach and suggested instead using the linear predictor in regression models or other summary statistic that are of finite dimension. When treatment assignment occurs over time as in the case where an individual chooses to receive PEG insertion or not, we must allow for the possibility of time-dependent confounding.  To this end,\nlet $X_t$ denote a set of $p$-dimensional time-dependent covariates at time $t$ and ${\\ensuremath{\\mathcal X}}_t=\\left\\{X_s,~0\\le s\\le t\\right\\}$ denote the history of covariates up to time $t$. Then, the probability of treatment assignment at time $t$ given the covariate history up to time $t$ is\n\\begin{equation}\\label{eq:density.tz}\nf(t\\mid{\\ensuremath{\\mathcal X}}_t) = \\lim_{\\epsilon\\rightarrow 0} \\epsilon^{-1} P\\left(t\\le T < t+\\epsilon \\mid {\\ensuremath{\\mathcal X}}_t\\right),\n\\end{equation}\nwhere $f(t\\mid {\\ensuremath{\\mathcal X}}_t) = h(t\\mid X_t)\\exp\\left\\{-\\int_0^t h(s\\mid X_s)\\, ds\\right\\}$\nand the hazard function is\n\\begin{equation}\\label{eq:haz.tz}\nh(t\\mid X_t) = \\lim_{\\epsilon\\rightarrow 0} \\epsilon^{-1} P\\left(t\\le T < t+\\epsilon \\mid T\\ge t, X_t\\right).\n\\end{equation}\nBecause $h(t\\mid X_t)$ uniquely parameterises $f(t\\mid \\cal{X}_t)$, either model~\\eqref{eq:density.tz} or model~\\eqref{eq:haz.tz} may be regarded as a legitimate treatment assignment model for continuous treatment with time-independent or time-dependent confounding \\citep{li2001,lu2005}.\n\nOf note, $f(t\\mid \\cal{X}_t)$ is a function of the entire covariate history ${\\ensuremath{\\mathcal X}}_t$, whereas the hazard function $h(t\\mid X_t)$ is a function of $X_t$ only.  This subtle, yet important difference can lead to difficulties when extending methods proposed by \\cite{imaiandvandyk2004} and \\citet{hirano2004} to time-dependent confounding via standard hazard modeling. In addition, both \\citet{li2001} and \\citet{lu2005} used the hazard function $h(t\\mid X_t)$ as a GPS for matching which allows for balancing $X_t$ at the time of treatment in a matched set. However, they did not establish the strong ignorability of treatment assignment given their time-dependent GPS; this property does not hold if $Y$ is associated with ${\\ensuremath{\\mathcal X}}_t$ rather than just $X_t$, in which case their proposed procedures may not lead to valid causal inference. Additionally, their proposed methods are only applicable to studies with data routinely collected at regular intervals, which is often not true in clinical registries.\n\n\n\nWe propose the propensity process to correct for confounding in observational studies by balancing the covariate history ${\\ensuremath{\\mathcal X}}_t$. After the propensity process is estimated,\nbias-corrected data analyses can be achieved through matching or stratification \\citep{rosenbaumandrubin1983}. Establishing formally the theoretical properties of the propensity process for time-independent confounding requires different arguments than those presented in \\citet{imaiandvandyk2004}.\n\n\n\n\n\\section{Methods}\\label{sec:PS}\n\n\\subsection{Notation and Assumptions}\n\nOur framework is constructed through potential outcomes \\citep{rubin2005causal}.  For $t\\in [0,L)$, we define $U_t=T\\wedge t$ as the treatment time restricted to time $t$ and $U=T\\wedge L$ as the treatment time restricted to time $L$, where $a \\wedge b$ denotes the minimum of $a$ and $b$. Let ${\\ensuremath{\\mathcal T}}_t=\\left\\{[0,t),t+\\right\\}$ define the set of potential treatment times restricted to $t$, $t\\in[0,L)$, where $t+$ means that a patient did not receive PEG treatment before $t$. Let $Y^*_t$ be the potential outcome if a subject received PEG treatment at time $t$, $t\\in[0,L)$, and $Y^*_{t+}$ the potential outcome if a subject did not receive PEG treatment in the interval $[0,t)$. It follows that $Y^*_{L+}$ denotes the potential outcome if a subject did not receive PEG treatment in the interval $[0,L)$.  We also define the treatment-free potential covariate process ${\\ensuremath{\\mathcal X}}^*_t,~t\\le L$. Then, the set of potential outcomes and treatment-free potential covariate process for a randomly selected subject from the population is $\\{Y^*_s, {\\ensuremath{\\mathcal X}}^*_s,~s\\in {\\ensuremath{\\mathcal T}}_t\\}$ when treatment time is restricted at $t$, $t\\in[0,L)$. In contrast, the observed data are $(Y, U, {\\ensuremath{\\mathcal X}}_{U})$, where the observed outcome $Y=Y^{*}_{U}$, and the observed covariate history ${\\ensuremath{\\mathcal X}}_{U}={\\ensuremath{\\mathcal X}}^*_{U}$.\n\nGiven $\\theta_t=h(t\\mid X^*_t)$, we define the propensity process as the sample path of the hazard function from baseline to time $t$, i.e.,\n\\begin{equation}\\label{eq:pp}\n\\Theta_t = \\left\\{ \\theta_s=h(s\\mid X^*_s),~0\\le s\\le t \\right\\},\n\\end{equation}\nnoting that $\\Theta_t$ is dependent on ${\\ensuremath{\\mathcal X}}^*_t$. As  ${\\ensuremath{\\mathcal X}}^*_t$ is observable only up to $U$,  $\\Theta_t$ is estimable only up to $U$. While this concept seems similar to the propensity function \\citep{imaiandvandyk2004},  the distinguishing factor of the propensity process is that $\\Theta_t$ depends on $t$ and is of infinite dimension and $\\Theta_L$ cannot be fully estimated for subjects receiving PEG before $L$, whereas the propensity function in \\citet{imaiandvandyk2004} only allows for incorporation of time-independent covariates and can be estimated for all subjects.\n\nIn our framework, we make two assumptions.\n\\begin{assump} [Stable unit treatment value assumption] The distributions of potential outcomes for different subjects are independent of one another.\n\\end{assump}\n\\begin{assump}[Strong Ignorability]  For every $t\\in [0, L)$, $\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid Y^*_s,{\\ensuremath{\\mathcal X}}^*_t)=\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid {\\ensuremath{\\mathcal X}}^*_t)$ and $\\mbox{pr}\\left(U_t\\in{\\ensuremath{\\mathcal A}}\\mid {\\ensuremath{\\mathcal X}}^*_t\\right)>0$ for all $s\\in {\\ensuremath{\\mathcal T}}_t$, ${\\ensuremath{\\mathcal X}}^*_t$, and  ${\\ensuremath{\\mathcal A}}\\subseteq{\\ensuremath{\\mathcal T}}_t$.\n\\end{assump}\nAssumption~1 is a common assumption in causal inference. However, our Assumption~2 is defined for each time point $t$ and differs from the standard strong ignorability of treatment assignment assumption used in earlier work for balancing scores. One implication of Assumption~2 is that, conditional on the treatment-free history ${\\ensuremath{\\mathcal X}}^*_t$, receiving treatment at $t$ or not is independent of the set of potential outcomes, allowing us to model treatment assignment without conditioning on potential outcomes. \n\n\\subsection{Main results}\\label{theory}\n\nWe establish the large-sample results of the propensity process assuming that the true propensity process is known along the lines of \\citet{rosenbaumandrubin1983} and \\citet{imaiandvandyk2004}.\n\\begin{prop}\n\\label{prop1}\n$U$ is conditionally independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_L$ given $\\Theta_L$, where ${\\ensuremath{\\mathcal X}}^*_L$ and $\\Theta_L$ are the entire treatment-free covariate history and propensity process, respectively.\n\\end{prop}\nProposition 1 establishes $\\Theta_L$ as a balancing functional that balances the entire covariate history. Proposition 1  requires that  $\\Theta_L$ is known or can be estimated in the entire domain $[0,L)$.  In practice, however, we can only observe the covariate process ${\\ensuremath{\\mathcal X}}^*_{U}$ and hence estimate $\\Theta_{U}$. Proposition 2 establishes the balancing property for every given time point $t$ in $[0,L)$.\n\n\\begin{prop}\\label{prop2}\nFor every $t\\in[0,L)$, $U_t$ is conditionally independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_t$ given $\\Theta_t$, where ${\\ensuremath{\\mathcal X}}^*_t$ and $\\Theta_t$ are the treatment-free covariate history and propensity process through time $t$, respectively.\n\\end{prop}\nWhen $t=U$ in Proposition~\\ref{prop2}, we have that  $U$ is independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_U$ given $\\Theta_U$, where ${\\ensuremath{\\mathcal X}}^*_U={\\ensuremath{\\mathcal X}}_U$ is observable and hence $\\Theta_U$ is estimable.\n\\begin{thm}\\label{Thm1} For every $t\\in [0, L)$, $\\mbox{pr} \\left(U_t\\in{\\ensuremath{\\mathcal A}}\\mid Y^*_s,\\Theta_t\\right)=\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid \\Theta_t)$ for all $s\\in {\\ensuremath{\\mathcal T}}_t$, $\\Theta_t$, and  ${\\ensuremath{\\mathcal A}}\\subseteq{\\ensuremath{\\mathcal T}}_t$.\n\\end{thm}\nWhen $t=U$ in Theorem~\\ref{Thm1}, we have that  $U$ is independent of potential outcomes given $\\Theta_U$, where $\\Theta_U$ is estimable. Several remarks are in order. First, in \\S~\\ref{ssec:pp} we suggest modeling the hazard function in~\\eqref{eq:haz.tz} through the proportional hazards model ~\\eqref{eq:PH}; one could also use other model formulations for~\\eqref{eq:haz.tz} and the results in Propositions~1--2 and Theorem~1 would still apply. Second, Proposition~\\ref{prop2} and Theorem~\\ref{Thm1} provide justifications for matching a subject treated at $t$ with an eligible control subject untreated at $t$ based on the propensity process up to $t$. It follows that each matched pair would have the same distribution for the covariate process up to $t$ and their potential outcomes are independent of their treatment assignments, allowing for valid causal inference. Third, our Proposition 2 is similar in spirit to Proposition 1 in \\citet{lu2005} but is more general in the sense that the propensity process balances the entire covariate history up to $t$ not just the covariates measured at $t$. In addition, \\citet{lu2005} did not establish the strong ignorability of treatment assignment given propensity scores similar to our Theorem 1. Proofs for Propositions~1--2 and Theorem~1 are given in the Appendix.\n\n\\section{Implementation and Practical Considerations} \\label{methods}\n\n\\subsection{Interpolated Propensity Processes}\\label{ssec:pp}\n\nIn practice, the propensity process $\\Theta_{U}$ must be estimated from the observed data. The challenge for estimating the propensity process is that we may not observe the complete treatment-free covariate process ${\\ensuremath{\\mathcal X}}^*_U$ on $[0,U]$; rather, we only get to observe the covariate process at a coarse set of discrete time points as is the case in the motivating ALS study.  Here, we propose to borrow strength across subjects in the study sample by modeling each time-dependent covariate as a random curve over time via nonlinear mixed effects models. This allows a predictive curve to be estimated for the entire treatment-free covariate process for each subject.\n\nFirst, suppose we parameterize the hazard function in \\eqref{eq:haz.tz} through Cox's proportional hazards model and define the propensity process\nthrough the linear predictor,\n\\begin{align}\\label{eq:PH}\n&h(t\\mid X_t;\\beta)=h_{0}(t)\\exp(\\beta^{\\rm T}X_t),\n&&\\Theta_t = \\left\\{ \\theta_s=\\beta^{\\rm T}X^*_{s},~0\\le s\\le t \\right\\},\n\\end{align}\nwhere $h_{0}(t)$ is the unspecified baseline hazard function.  Next, write the observed treatment-free covariate history for the $i$-th subject and $k$-th covariate as ${\\ensuremath{\\mathcal X}}_{ik}=\\left(X_{i1k},\\ldots,X_{im_ik}\\right)$, with time-dependent covariate $X_{ijk}$ measured at time $t_{ij}$. We note that the observation times $(t_{ij},~j=1,\\ldots,m_i)$ may be different for each subject but are assumed to be the same for all covariates within a subject. Then, for each time-dependent covariate, we fit the model,\n\\begin{eqnarray}\nX_{ijk} &=& b_k^{\\rm T}(t_{ij})\\gamma_k + b_k^{\\rm T}(t_{ij})\\alpha_{ik} + \\epsilon_{ijk},~(i=1,\\ldots,n;~j=1,\\ldots,m_i;k=1,\\ldots,p),\\label{eq:mm}\n\\end{eqnarray}\nwhere $\\epsilon_{ijk}$ are independent, mean-zero random errors. To provide greater flexibility in modeling the covariate process over time, we use spline-type models \\citep{ruppert2003semiparametric} in \\eqref{eq:mm} where $b(\\cdot)$ denotes a set of basis functions and $\\gamma_k$ and $\\alpha_{ik}$ are regression coefficients corresponding to the basis functions for the fixed and random effects, respectively. The interpolated treatment-free $\\widehat{{\\ensuremath{\\mathcal X}}}_t$ can be obtained from model~\\eqref{eq:mm} by replacing regression coefficients $\\gamma_k$ and $\\alpha_{ik}$ with their estimates $\\widehat\\gamma_k$ and $\\widehat\\alpha_{ik}$, respectively. Then the estimated propensity process $\\widehat\\Theta_{U}$ can be obtained from~\\eqref{eq:PH} by plugging in the interpolated $\\widehat{{\\ensuremath{\\mathcal X}}}_U$ and $\\widehat\\beta$, where $\\widehat\\beta$ is the estimated regression coefficient vector in the Cox proportional hazards model.\n\n\n\\subsection{Matching}\\label{subsec:matching}\n\n\nThe use of matched analyses based on propensity scores for testing causal null hypotheses has been advocated by several other authors; for example, see  \\citet{rosenbaumandrubin1983}, \\citet{li2001} and \\citet{lu2005} and references therein. Matching can be performed by minimizing the integrated squared error between the estimated propensity process $\\widehat\\Theta_{t}$ of a subject who received PEG treatment at time $t$ and that of each eligible control with $U>t$.  To accomplish this task, we implement a sequential matching algorithm. We start by ordering chronologically subjects according to their time of PEG treatment or censoring, namely $U$.  Set the matched pair counter to $m=1$ and select the subject with the smallest time to PEG treatment, say subject $i_1$.  Define the integrated squared difference in interpolated propensity processes between $i_1$ and $l$ as $Q(i_1,l) = I(T_{i_1}\\le L) \\int_0^{T_{i_{1}}} (\\widehat\\theta_{i_{1},t}-\\widehat\\theta_{l,t})^2\\, dt,$ for all subjects $l$ in the set of $n-1$ eligible controls $\\mathcal{C}_1=\\{l\\mid l=1,\\ldots,n,~l\\ne i\\}$.  The matched control for $i_1$ is the nearest neighbor in interpolated propensity processes among eligible controls, i.e., $\\mbox{argmin}_{l\\in\\mathcal{C}_1} Q(i_1,l)$. Increment the matched pair counter by one to $m=2$ and select the subject with the smallest time to PEG treatment, say $i_2$, excluding the two subjects in the first matched pair.  Therefore, the set of eligible controls, say $\\mathcal{C}_2$, contains $n-3$ subjects: all $n$ subjects less the two subjects in the first matched pair and $i_2$. The matched control for $i_2$ is the nearest neighbor in interpolated propensity processes among the set of eligible controls, $\\mbox{argmin}_{l\\in\\mathcal{C}_2} Q(i_2,l)$.  Increment the matched pair counter by one and continue until all treated individuals are matched or until there are no suitable controls available for matching.\n\n\\section{Analysis of the ALS Registry Data} \\label{results}\n\nUsing a data set from the Emory ALS Registry, we assess the association of PEG treatment with the change in body mass index (BMI) from baseline to 18 months, i.e., $L$ = 18 months. The data set includes 240 patients who survived past $L$ and had at least one clinic visit between baseline and $L$. The patients who received PEG did so after their first clinic visit. The timing of recommending PEG by the physician involved many factors and the final decision to have PEG was made by each patient. We model treatment\nassignment through the proportional hazards model \\eqref{eq:PH} including the following covariates. The baseline risk factors are age at diagnosis, sex, site of onset of disease, negative inspiratory force, and time from diagnosis to the first clinic visit.  Two time-varying covariates are forced vital capacity and body mass index, which may not be measured at every clinic visit for every patient. Each time-varying covariate is modeled over time using the mixed model~\\eqref{eq:mm}, where polynomial spline basis functions are used. The estimated curves are used to interpolate the covariate values needed for estimating the propensity process based on \\eqref{eq:PH}.\n\nWe compare three alternative approaches to the proposed propensity process. First, a na\\\"{i}ve analysis compares all treated individuals to those who are untreated prior to $L$. The second approach is the propensity function \\citep{imaiandvandyk2004} that uses baseline risk factors $X_0$ only in the treatment assignment model~\\eqref{eq:PH}, where $\\theta_0=\\beta^{\\rm T}X_0$ defines the propensity function. The third approach is the interpolated generalized propensity score, which uses the interpolated treatment-free $\\widehat{X}_t$ defined in \\S~\\ref{ssec:pp} to obtain the GPS for each subject in the spirit of \\citet{lu2005}, noting that $X_t$ may not be observed at time $U$ for a subject and its eligible controls as defined in \\S~\\ref{subsec:matching}. The same sequential matching algorithm in \\S~\\ref{subsec:matching} is used for all propensity score methods.  Our matching algorithm resulted in $M=74$ pairs for the analysis using the propensity function and $M=76$ pairs for both analyses using the generalized propensity score and propensity process.\n\nFollowing \\citet{li2001} and \\citet{lu2005}, we assess balance of covariates by examining Type I errors from a log-rank test of the effect of the covariate on time to treatment, one covariate at a time. In the matched analyses, this model is stratified by the $M$ matched pairs. As shown in Table 1, prior to matching, balance is not achieved. While other methods improve covariate balance,  they do not balance all covariates. However, matching using the propensity process results in balance across all covariates. This indicates that the propensity process outperforms the baseline propensity function or interpolated GPS in terms of balancing covariates and there may be residual confounding after matching by the other propensity score methods.\n\n\\begin{table}[h]\\label{tab:balance}\n  \\centering\n  \\caption{Covariate balance before and after matching}\n    \\begin{tabular}{lcccc}\n \\hline\n             & Prior to  & Propensity   & Generalized  & Propensity      \\\\\n    Covariate &         Matching        & Function & Propensity Score & Process \\\\\\hline\n    Body mass index & 0.277 & 0.245 & 0.986 & 0.991 \\\\\n    Forced vital capacity & 0.764 & 0.539 & 0.201 & 0.317 \\\\\n    Negative inspiratory force & 0.151 & 0.022 & 0.016 & 0.704 \\\\\n    Age & 0.162 & 0.718 & 0.378 & 0.195 \\\\\n    Sex & 0.577 & 0.695 & 0.002 & 0.706 \\\\\n    Site & 0.001 & 0.003 & 1.000 & 0.341 \\\\\n    Time from diagnosis & 0.676 & 0.633 & 0.033 & 0.854\\\\\n\\hline\n    \\end{tabular}\n\\end{table}\n\nAfter matching, we test the causal null hypothesis that the mean potential outcome is the same whether a patient received PEG treatment at time $t$ versus PEG treatment at some time after $t$ or untreated by $L$, which can be written as $H_0: E(Y^*_t)=E(Y^*_s)$ for all $t<s\\leq L$. We test this hypothesis by a Wilcoxon signed rank test on matched pairs for all the matched analyses.  The Wilcoxon rank sum test is used for hypothesis testing in the na\\\"ive analysis.  Table 2 presents the median difference in BMI change at 18 months and p-value of the Wilcoxon test for each approach. The propensity process matched analysis suggests a protective effect of PEG on BMI, whereas the other three methods all show effects that are attenuated towards 0 and are not statistically significant.\n\n\\begin{table}[h]\\label{tab:outcome}\n  \\centering\n  \\caption{Results in the data analysis}\n    \\begin{tabular}{lcc}\n    \\hline\n       & Median Difference&  P-value\\\\\n    \\hline\n    Na\\\"ive & 0.035 & 0.673 \\\\\n    Propensity Function& 0.030 & 0.466 \\\\\n    Generalized Propensity Score& 0.360 & 0.453 \\\\\n    Propensity Process& 0.830 & 0.022 \\\\\n    \\hline\n    \\end{tabular}\n\\end{table}\n\n\\section{Discussion} \\label{discussion}\nCompared to the existing propensity score methods, the propensity process offers the advantage of balancing time-varying covariate history from baseline to time of treatment. A key  component of this approach is the interpolation of covariate curves. We propose to use nonlinear mixed models to provide flexibility for modeling covariate history, though there must be enough individual longitudinal data collected to estimate these curves, which is a potential limitation in settings with sparsely collected longitudinal data. However, data interpolation may not be needed in settings such as critical care in intensive care units where time series data including heart rate and blood pressure are continuously recorded \\citep{lehman2013tracking}.\n\n\nIn our data analysis, we use a straightforward approach for hypothesis testing after matching. Future extensions may include conditional likelihood methods for estimating treatment effects based on matched pairs\/sets and methods for stratification and covariate adjustment using the propensity process. Additionally, our analysis excludes individuals who died prior to $L$ in order to avoid complications due to censoring by death \\citep{rubin2006,zhang2003estimation}, which could be addressed in future extensions such that no such exclusion is necessary.\n\n\n\\section*{Acknowledgements}\nThe authors thank Dr. Jonathan Glass and Ms. Meraida Polak at the Emory ALS Center for providing the ALS data and Dr. Xin Qi at the Georgia State University for helpful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\input{sections\/introduction.tex}\n\n\\input{sections\/related.tex}\n\n\n\\section{Problem Statement and Approach}\n\\input{sections\/problem.tex}\n\n\n\n\\section{Trajectory Planning}\n\\input{sections\/trajectory_optimization.tex}\n\n\n\\section{Local Feedback Control}\n\\input{sections\/pwa.tex}\n\n\n\\section{Experiment}\n\\input{sections\/experiment.tex}\n\n\\section{Discussion and Conclusion} \n\\label{sec:conclusion}\n\\input{sections\/conclusion.tex}\n\n\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\\iffalse\n    \\section*{APPENDIX}\n\\fi\n\n\\iffalse \n\\section*{ACKNOWLEDGMENT}\nThis work was supported by Air Force\/Lincoln Laboratory Award No. PO\\# 7000374874, and Lockheed Martin Corporation Award No. RPP2016-002.\n\n\\fi \n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\n\\input{sections\/introduction.tex}\n\n\n\n\\subsection*{Notation}\n\n\\section{Problem Statement and Approach}\n\\input{sections\/problem.tex}\n\n\n\n\\section{Hierarchical path planning}\n\\input{sections\/trajectory_optimization.tex}\n\n\n\\section{Local Multi-Contact Dynamics: Piecewise Affine Models}\n\\input{sections\/pwa.tex}\n\n\n\\section{Stabilizing Controller}\n\n\\section{Experiments}\n\\input{sections\/experiment.tex}\n\n\\section{Conclusion} \n\\label{sec:conclusion}\n\nThe conclusion goes here.\n\n\n\n\\bibliographystyle{plainnat}\n\n\n\n\\subsection{Balancing Bar with Two Fingers}\nWe consider a horizontal bar in 2D that is subject to gravity. Two point fingers are used to balance the bar. The 10D state is $\\textbf{x}=(x,y,\\theta,x_1,y_1,x_2,y_2,\\dot{x},\\dot{y},\\dot{\\theta})$, where $x,y,\\theta$ are the 3D pose of the bar, and $(x_i,y_i), i=1,2$ are the 2D position of the each finger. The fingers are driven by velocity commands and have first order dynamics. The contact between each finger and the bar has 4 possible modes: sticking, sliding left, sliding right, and no contact. 16 possible contact combinations are possible. \n\nThe task is to steer the bar into the equilibrium point $\\textbf{x}_{\\text{goal}}=(0,0,0,1,0,-1,0,0,0,0)$ - each finger carrying half of the bar weight from $\\textbf{x}=(0,0,0,1,-0.2,-1,-0.3,1,0.5,0)$ toward goal - the bar has initial velocities in the $x$ and $y$ direction and the fingers are not initially in contact. Trajectory synthesis for such a problem with a large number of contact modes is hard, so we make the build the MLD model around the equilibrium to be able to optimize a trajectory using MICP.  First, we solve a MICP problem with horizon $T=30$, $h=0.03$ for time discretization. We observe that the fingers approach the bar, tilt it toward the left, and maintain contact to counteract the initial velocity to the right balance it. Snapshots are shown in Fig. \\ref{fig:balance}. \n\nNext, we make 30 MLD systems with each environment being a point of the nominal trajectory, and find a funnel from a nearby random point to connect to the central funnel. The projection of corresponding zonotopic funnels into $x-y$, $x-\\theta$, and $x-\\dot{x}$ are shown in Fig. \\ref{fig:balance_funnels}. Using the sampling-based procedure outlined in \\cite{sadraddini2018sampling}, these funnels can be even grown further to gain more coverage of the state-space. Obtaining the corresponding hybrid control law is straightforward. \n\n\\begin{figure}\n    \\includegraphics[width=0.48\\linewidth]{sections\/Figures\/bar_0.png}\n    \\includegraphics[width=0.48\\linewidth]{sections\/Figures\/bar_5.png}\n    \\includegraphics[width=0.48\\linewidth]{sections\/Figures\/bar_10.png}\n    \\includegraphics[width=0.48\\linewidth]{sections\/Figures\/bar_29.png}\n    \\caption{Balancing the bar: snapshots of a trajectory}\n    \\label{fig:balance}\n\\end{figure}\n\n\n\\begin{figure}\n    \\includegraphics[width=0.32\\linewidth]{sections\/Figures\/funnel_0_1.png}\n    \\includegraphics[width=0.32\\linewidth]{sections\/Figures\/funnel_0_2.png}\n    \\includegraphics[width=0.32\\linewidth]{sections\/Figures\/funnel_0_7.png}\n    \\caption{Projections of the caterpillar funnels corresponding to a trajectory steering the system to the equilibrium. The central funnel is shown in red with $T=30$. The connecting funnels, each with $T=3$, are shown in green.}\n    \\label{fig:balance_funnels}\n\\end{figure}\n\\fi \n\n\n\\label{sec:experiment}\n\\begin{figure*}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_0_degree_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_90_degrees_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_80_degrees_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_135_degrees_200x267.png}\n   \n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_180_degrees_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_human_disturbance_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_disturbance_before_opening_gripper_200x267.png}\n    \\includegraphics[width=0.12\\linewidth]{sections\/Figures\/series\/carrot_disturbance_after_opening_gripper_200x267.png}\n    \n    \\quad\\quad\\ (A) \\quad\\ \\  \n    \\quad\\quad\\quad (B) \\quad\\  \\  \n    \\quad\\quad\\quad (C) \\quad\\  \\  \n    \\quad\\quad\\quad (D) \\quad\\  \\  \n    \\quad\\quad\\quad (E) \\quad\\  \\  \n    \\quad\\quad\\quad (F) \\quad\\  \\  \n    \\quad\\quad\\quad (G) \\quad\\  \\  \n    \\quad\\quad\\quad (H) \\quad\n    \\caption{Flipping the half-cylinder}\n    \\label{fig:flipping_motions}\n\\end{figure*}\n\nWe carried out the experiment on a Kuka robot with a two-finger Schunk gripper. \nThe half-cylinder representing the carrot is 0.11 m long and its radius is 0.036 m.\nWe put an AR-tag \\cite{fiala2005artag} with side width 3 cm on a side of the half-cylinder and use a Kinect to track the pose of the half-cylinder.\nWe can get in real time the pose of the gripper relative to the Kuka base and hence the pose of the gripper relative to the table. \nOnce we compute the initial pose of the half-cylinder relative to the table, we can track the pose of the half-cylinder relative to the gripper in real time.\n{We determine whether the gripper is in contact of the half-cylinder or not by relative poses, since we do not have any force or tactile sensors.}\n\nThe goal is to flip the half-cylinder 180 degrees, i.e., to manipulate the half-cylinder with the flat surface facing upwards to the pose where the flat surface is facing downwards to the table. \nWe use our algorithm to design a trajectory that flips the half-cylinder 90 degrees so that the grippers are holding the half-cylinder.\nAfter that, a manually-designed (open-loop) controller would transport the half-cylinder and flip another 90 degrees.\nWe mainly focus on the first 90-degree rotation for several reasons. First, it is most challenging in the entire manipulation process, and manually designed open-loop controllers usually fail in this phase.\nEven an experienced human operator tele-operating the robot cannot accomplish this task in one or two attempts and the failures are often in the first 90-degree phase (see the accompanying video).\nSecond, in the next 90-degree rotation, the dynamics is different from that in the first 90-degree rotation, so one needs to design a different trajectory separately, instead of simply extending the first trajectory.\nBesides, it is very simple to design a good open-loop controller for the next 90-degree rotation (see the accompanying video).\n\nThe state is $\\textbf{x} = [x,y,\\theta, \\dot{x},\\dot{y},\\dot{\\theta}, \\varphi, w]$, where $x$ and $y$ are the coordinates of CoM of the half-cylinder, $\\theta$ is the angle between the flat surface of the half-cylinder and the horizontal axis, $\\dot{x},\\dot{y},\\dot{\\theta}$ are the first-order time derivatives of $x,y,\\theta$, respectively, $\\varphi$ is the angle between the finger of the gripper and the horizontal axis, and $w$ is the separation between two fingers.\nThe control input is $\\textbf{u} = [F_N,F_t, F_1,F_{1t},F_2,F_{2t}, \\dot{\\varphi}, \\dot{w}]$, where $F_N,F_t$ are the normal force and the friction between the half-cylinder and the ground, and similar definitions for $F_1, F_{1t}, F_2, F_{2t}$ (Figure \\ref{fig:contact_modes}).\nWe set up trajectory optimization with time horizon $N = 100$ and sampling time $dt = 0.01$ s.\n{We constrain that there is always contact between the left finger and the flat surface of the half-cylinder. The friction coefficients are chosen to be between 0.2 and 0.5.}\nThe goal state region of the trajectory optimization is $X_G = \\{\\textbf{x}: \\varphi = \\theta = 90^\\circ \\}$.\n{The objective is to minimize $\\sum_{t=0}^N |\\varphi_t - \\theta_t|$.}\nIt took IPOPT about 2 seconds to solve the trajectory optimization on an Intel i7 3.3\nGHz, 32 GB RAM machine\\footnote{By varying the time horizon or other constraints, the solving time of IPOPT varies from 1 to 25 seconds or IPOPT cannot find a feasible solution. We found that on solving this particular problem with varying constraints, IPOPT generally behaves better than SNOPT \\cite{gill2005snopt}.}.\n\n\\begin{figure} \n    \\centering\n    \\includegraphics[width=0.23\\textwidth]{sections\/Figures\/closest_nominal_index.pdf}\n    \\includegraphics[width=0.23\\textwidth]{sections\/Figures\/contact_modes_plot2.pdf}\n    \\caption{The figure on the left plots the index of the closest nominal trajectory state for the current state \\textit{before} reaching the goal region $\\widetilde{X}_G$ vs. time steps for four typical trajectories. Trajectory 1 can be thought of as the open loop trajectory and is just for reference. Trajectory 2, 3, and 4 are typical trajectories for the closed-loop controller, the closed-loop controller under the first type of disturbance, and the closed-loop controller under the second type of disturbance, respectively. They terminate early, because $\\widetilde{X}_G$ is strictly larger than ${X}_G$. The figure on the right plots the contact modes for 3 typical trajectories. From top to bottom are typical contact modes for the closed-loop controller under the first type of disturbance, the closed-loop controller under the second type of disturbance, and the close-loop controller under the second type of disturbance where the gripper opens 4 cm.}\n    \\label{fig:two_plots}\n\\end{figure}\n\nWe design controllers in 2D, and in execution the half-cylinder is 3D, so we always operate at the center section of the half-cylinder.\nDuring execution, we let the goal region be $\\widetilde{X}_G = \\{\\textbf{x}: \\varphi = \\theta, w \\leq c \\} \\supseteq X_G$ for some constant $c$. Once the state is in the goal region, the gripper is in grasp of the half-cylinder, ready for the remaining operations. \nThe open-loop controller obtained from trajectory optimization brings the half-cylinder from 0 degree (Fig \\ref{fig:flipping_motions}.A) to 90 degrees (Fig \\ref{fig:flipping_motions}.B).\nThen a manually-designed controller flips the half-cylinder completely (Fig \\ref{fig:flipping_motions}.DE).\nThe closed-loop controller stops early once the goal region has been reached (Fig \\ref{fig:flipping_motions}.C), followed by the same manually-designed controller.\n\nThe open-loop controller is already very robust during execution, so we tested some larger disturbances to show the robustness of the closed-loop controller.\nWe experimented two types of disturbances. One is to force the half-cylinder to rotate clockwise using a human hand (Fig \\ref{fig:flipping_motions}.F). \nWe can easily fail the open-loop controller by holding the half-cylinder long enough so that the controller finishes open-loop execution.\nWe found that the closed-loop controller always recovers when the disturbances are no more than 30 degrees.\nThe success rate is 100\\% (20 out of 20).\nWe observe that the contact modes do not change under small disturbances (Fig \\ref{fig:two_plots} Right).\nWe also observe that the controller can sometimes recover from very large disturbances that violate the assumptions we made when designing the trajectory, e.g., the disturbance is more than 30 degrees and the right finger is on the flat surface of the half-cylinder.\nThe other type of disturbance is to ``accidentally\" open the gripper for 2 cm at a certain time step (before opening: Fig \\ref{fig:flipping_motions}.G, after opening: Fig \\ref{fig:flipping_motions}.H).\nThis tests the local multi-contact stabilizing controller.\nThe success rate is also 100\\% (20 out of 20).\nThe contact modes change when the disturbance happens (Fig \\ref{fig:two_plots} Right).\nThe controller can also recover from large disturbances that violate our design assumptions, e.g., the gripper opens 4 cm at a certain time step, {in which case the left finger moves a long distance along the flat surface of the half-cylinder, or even breaks contact with the half-cylinder.}\n\n\\begin{table}\n\\centering \n\\begin{tabular}{|c|c|c|c|\n \\hline\n \\multicolumn{2}{|c|}{Disturbance 1} & \\multicolumn{2}{|c|}{Disturbance 2}\\\\\n \\hline\n $\\approx 15^\\circ$ & $\\approx 30^\\circ$ & at time step 7 & at time step 14\\\\\n \\hline\n 10\/10 & 10\/10 & 10\/10 & 10\/10\\\\\n \\hline \n\\end{tabular}\n\\caption{Success rate for recovery from two types of disturbances.}\n\\end{table}\n\n\n\\subsection{Local Multi-Contact Dynamics}\n\\label{sec:pwa}\n\\begin{figure}[t]\n \n  \\begin{minipage}{0.49\\linewidth}\n  \\centering \n  \\includegraphics[width=0.7\\linewidth]{sections\/Figures\/carrot_mode1.pdf}\\\\ \n  \\small (A)\n  \\end{minipage}\\ \n  \\begin{minipage}{0.49\\linewidth}\n  \\centering \n  \\includegraphics[width=0.7\\linewidth]{sections\/Figures\/carrot_mode2.pdf} \\\\\n  \\small (B)\n  \\end{minipage}\n  \\caption{Two contact modes. (A) Right finger in contact with carrot. (B) Right finger not in contact with carrot.}\n  \\label{fig:contact_modes}\n\\end{figure}\n\nFrom the nonlinear trajectory optimization, we obtain a nominal trajectory $\\{\\bar{x}_0,\\bar{u}_0,\\bar{x}_1,\\bar{u}_1,\\ldots,\\bar{x}_N\\}$, where $\\bar{x}_N \\in X_G$.\nAt each nominal point $(\\bar{x}_i,\\bar{u}_i)$, where $i = 0,\\ldots, N$, we linearize the dynamics $\\dot{x} = {f}(x,u)$ as $\\dot{x} =  {A}_i(x-\\bar{x}_i) + {B}_i (u-\\bar{u}_i) + {c}_i$, where ${A}_i = {\\partial {f} \\over \\partial x}(\\bar{x}_i,\\bar{u}_i), {B}_i = {\\partial {f} \\over \\partial u}(\\bar{x}_i,\\bar{u}_i)$, and ${c}_i = {f}(\\bar{x}_i, \\bar{u}_i)$.\nThis continuous time affine system can be discretized as $x^+ =  \\tilde{{A}}_i x + \\tilde{{B}}_i u + \\tilde{{c}}_i$.\nThis can equivalently be obtained by linearizing the discretized system $x^+ = \\phi(x,u)$.\nThe corresponding state space $X_{i}^1$ is obtained by linearizing all constraints ${g}(x,u) \\leq {0}$ at $(\\bar{x}_i,\\bar{u}_i)$.\n\nNow we consider different contact modes due to making or breaking contacts between an object and the manipulator.\nWe fix a nominal point $(\\bar{x}_i,\\bar{u}_i)$.\nSuppose there are $p\\in \\mathbb{N}$ contact locations that may make or break contacts.\nEach contact mode corresponds to a distinct dynamics $\\dot{x} = {f}_j (x,u)$ and constraints ${g}_j (x,u) \\leq {0}$, where $j = 1,\\ldots,s := 2^p$, ${f}_1 = {f}$, and ${g}_1 = {g}$.\nFor the half-cylinder example, the right finger can be touching or not touching the half-cylinder, giving two contact modes with distinct dynamics and distinct state space regions (Figure \\ref{fig:contact_modes}).\nWe call the contact mode in which the nominal trajectory is computed the \\textit{nominal mode} or Mode 1.\nWe linearize the dynamics and the constraints for modes other than the nominal mode and evaluate at $(\\bar{x}_i,\\bar{u}_i)$.\nSince making and breaking contacts can happen when the system state makes very small changes, the linearization is in the vicinity of the nominal point and hence is valid.\nIn fact, the nominal points can be on the boundaries of state space cells of a few modes.\nThus we obtain a piecewise affine system $x^+ =  {{A}}_{i,j}x + {{B}}_{i,j} u + {{c}}_{i,j} = : {h}_{i,j}(x,u)$ with state space $X_i^j$, $j = 1,\\ldots,s$, around each nominal point $(\\bar{x}_i,\\bar{u}_i)$, $i = 1,\\ldots,N$.\nWe call $x^+ = {h}_{i,1}(x,u)$ the \\textit{nominal linearization}.\n\n\n\\label{sec:control}\n\n\\subsection{Polytopic Funnel around Nominal Trajectory}\nAfter we get a nominal trajectory, we are going to build a funnel around the trajectory so that if the system state is inside the funnel, it will always stay inside the funnel until reaching the target region.\nFunnels can be sum-of-squares (SOS) \\cite{tedrake2010lqr,majumdar2017funnel} or polytopic  \\cite{sadraddini2019sampling}.\nWe use the latter, because numerical computations involving polytopes requires solving LP or quadratic program (QP), which are more reliable than solving SOS programs.\n\nHere we briefly review the polytopic tree method in \\cite{sadraddini2019sampling}. \nSuppose the system is a time-varying affine system\n\\[\nx_{t+1} = A_t x_t + B_t u_t + c_t.\n\\]\nGiven a target region $X_G$ and time horizon $N$, the method computes a trajectory $\\{\\bar{x}_i,\\bar{u}_i\\}_{i=0}^N$ alongside with polytopes \n\\begin{align}\n    Y_i = \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P} \\label{poly_Y}\n\\end{align}\nand a control law \n\\begin{align}\n    u_i(x) = \\bar{u}_i + \\theta_i p(x) \\label{poly_u}\n\\end{align}\naround each point $(\\bar{x}_i,\\bar{u}_i)$ on the trajectory by solving a main LP.\nHere $\\oplus$ represents Minkowski sum, $G_i$ and $\\theta_i$ are decision matrices the main LP searches over, $G_{i+1} = A_iG_i+B_i\\theta_i$ captures the evolution of the polytopes over time with respect to the system dynamics, $\\mathbb{P}$ is the hyper-cube $[-1,1]^n$, and $p(x) \\in \\mathbb{P}$ satisfies \n\\begin{align}\n    x = \\bar{x}_i + G_i p(x). \\label{poly_x}\n\\end{align}\nThe main LP to be solved offline encodes the state space constraints in polytopic containment forms, including the final polytopic containment constraint $Y_N \\subseteq X_G$, as well as trying to maximize the volumes of the polytopes.\nDuring online execution, if the current state $x$ is in some polytope $Y_i$, then $p(x)$ can be found by solving an LP through Equation (\\ref{poly_x}) and $u_i(x)$ can be calculated by Equation (\\ref{poly_u}). \nBy following the control law $u_i(x)$, the system is guaranteed to land inside the next polytope $Y_{i+1}$ and hence eventually it will reach $Y_N\\subseteq X_G$.\n\nWe use the method to build polytopes for the nominal linearization $x^+ = h_{i,1}(x,u)$ around the nominal trajectory $(\\bar{x}_i,\\bar{u}_i)$.\nWhile \\cite{sadraddini2019sampling} deals with PWA systems, we show that the polytopic tree method can be extended to non-smooth nonlinear systems.\n\\begin{proposition}\\label{prop1}\nIf $x^+ = \\phi(x,u)$ and $\\phi$ is Lipschitz continuous, and if the polytopic tree method finds the polytopes $Y_i$ as in Equation (\\ref{poly_Y}) for the nominal linearization $x^+ = h_{i,1}(x,u)$ at the nominal trajectory, with $\\mathbb{P} = [-1,1]^n$ and $G_i$ full rank $\\forall i$, then there exists $\\mathbb{P}_i = [-a_i,a_i]^n, 0 < a_0 \\leq a_1 \\leq \\cdots \\leq a_N = 1$ such that if $x \\in \\tilde{Y}_i :=  \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}_i$, then by following $u$ as in Equation (\\ref{poly_u}), $\\phi(x,u) \\in \\tilde{Y}_{i+1}$. So $x$ will eventually land in $\\tilde{Y}_N = Y_N \\subseteq X_G$.\n\\end{proposition}\n\\begin{proof}[Proof Sketch]\nIf the system state $x \\in \\tilde{Y}_{N-1} =  \\{\\bar{x}_{N-1}\\} \\oplus G_{N-1} \\mathbb{P}_{N-1}$, by following $u_{N-1}$, $x^+ \\in \\{\\bar{x}_{N} + e_{N-1}\\} \\oplus G_{N} \\mathbb{P}_{N-1}$, where $e_{N-1}$ is the residual error induced by the linearization.\nSince the dynamics $\\phi$ is Lipschitz, $e_{N-1}$ goes to 0 as $(x,u)$ goes to $(\\bar{x}_{N-1},\\bar{u}_{N-1})$. \nGiven small $\\varepsilon > 0$, we can find $\\delta > 0$ such that if $||(x,u)-(\\bar{x}_{N-1},\\bar{u}_{N-1})||_2 < \\delta$, then $||e_{N-1}||_2 < \\varepsilon$.\nWe can choose $\\varepsilon$ and $a_{N-1}$ such that any $(x,u)$ satisfying $x \\in \\tilde{Y}_{N-1}$ and Equation (\\ref{poly_u}) also satisfies $||(x,u)-(\\bar{x}_{N-1},\\bar{u}_{N-1})||_2 < \\delta$ and such that $e_{N-1} \\oplus G_N \\mathbb{P}_{N-1} \\subseteq G_N \\mathbb{P}_{N}$.\nThen $x^+ \\in \\tilde{Y}_N$.\nThe proof is complete by repeating the procedure for $i=N-2,\\ldots,0$.\n\\end{proof}\nThe Proposition says that if the system state is close enough to the nominal trajectory and if the nonlinear dynamics is Lipschitz, then we can use the {affine} control law (\\ref{poly_u}) for the nominal linearization of the trajectory to steer the nonlinear system to the target region for sure.\n{\nThis gives us stability guarantees near the vicinity of the nominal trajectory.\nIn practice, for any arbitrary system, we do not know how large its polytopic funnel can be, or how likely it is for the state to fall into the funnel.}\nWe want to be able to handle larger external disturbances during online execution.\nThis is what we are going to {address} in the next section.\n\n\\section{Online Execution}\nThe polytopic tree method in \\cite{sadraddini2019sampling} hopes to probabilistically cover the state space by polytopes by growing a single polytopic trajectory to an existing polytopic tree, similar in methodology to the growth of the LQR-trees \\cite{tedrake2010lqr}.\nThe method samples points in the state space, steers sample points to the current polytopic tree as well as building polytopes along the way by solving mixed-integer linear program (MILP), and enlarges the current tree by adding the new polytopes to the tree.\n\nIn practice, for example for the half-cylinder flipping experiment, there are several problems with the polytopic tree method.\nFirst, the volumes of the polytopes {can be} very small, and hence a state {may} never fall into any polytope.\nSecond, the polytopic tree method requires checking the closest polytope online, which is computationally inefficient in the naive implementation when there are large number of polytopes.\nThird, the polytopic tree method deals with PWA systems, but our system is nonlinear and computing trajectory from a sample point to the current tree is potentially an expensive nonlinear trajectory optimization problem which cannot be carried out online.\n\nTherefore, we propose the practical improvement of the polytopic tree method, with the sacrifice of stability guarantees when there are large deviations to the nominal trajectory.\nWe only keep one nominal trajectory, which is computed in Section \\ref{sec:trajectory}.\nWe build polytopes $Y_i = \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}, i=0,\\ldots,N$ around the nominal linearization of the trajectory.\nThis amounts to solving an LP.\nDuring online execution, we compute the closest nominal state $\\bar{x}_i$ to the current state $x$ (with respect to some weighted $L_2$ norm) and determine the current contact mode $j$.\nIf $x$ is inside the polytope $\\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}$, the we use the corresponding control law $u_i(x) = \\bar{u}_i + \\theta_i p(x)$, where $x = \\bar{x}_i + G_i p(x)$. \nOtherwise we let the target index be $v = \\min\\{i+1,N\\}$ and solve the following LP to get control $u$:\n\\begin{align}\\label{lp1}\n    &\\quad \\ \\min_{\\gamma,p,\\delta,u} \\ \\alpha^\\top \\gamma \\\\\n    &\\text{subject to} \\ x_{v} + G_{v} p = h_{i,j}(x,u) + \\delta \\nonumber\\\\\n    & \\quad \\quad \\quad \\quad \\ p \\in \\mathbb{P},  |\\delta_k| \\leq \\gamma_k, k=1,\\ldots,n  \\nonumber\n\\end{align}\nwhere $\\alpha$ is some weight or cost vector.\nThis LP means we want the state to get to the polytope with index $v$ as close as possible.\nWhen, for example in the half-cylinder flipping experiment, the volumes of the polytopes are very small, we can directly solve the LP\n\\begin{align}\\label{lp2}\n    &\\quad \\quad\\min_{\\gamma,\\delta,u} \\ \\alpha^\\top \\gamma \\\\\n    &\\text{subject to} \\ x_{v} = h_{i,j}(x,u) + \\delta \\nonumber\\\\\n    & \\quad \\quad \\quad \\quad \\  |\\delta_k| \\leq \\gamma_k, k=1,\\ldots,n  \\nonumber\n\\end{align}\nwhich means we want the state to get to the nominal state with index $v$ as close as possible.\n\n\\begin{algorithm}\n\\caption{Stabilizing controller around nominal trajectory}\\label{alg:stabilizing_controller}\n\\hspace*{\\algorithmicindent} \\textbf{Input} Current state $x \\not\\in X_G$ \\\\\n\\hspace*{\\algorithmicindent} \\textbf{Output} Control $u$\n\\begin{algorithmic}[1] \n    \\If {$x \\in \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}, \\forall i \\in I$, for some index set $I$,} \\Return $u = \\bar{u}_{i_0} + \\theta_{i_0} p(x)$, where $x = \\bar{x}_{i_0} + G_{i_0} p(x)$, and ${i_0}$ is the largest element in $I$.\n    \\EndIf\n    \\State Find the closest nominal state $\\bar{x}_i$ to $x$, w.r.t. some weighted $L_2$ norm.\n    \\State Determine the current contact mode $j$.\n    \\State Let the target index be $v = \\min\\{i+1,N\\}$.\n    \\State Solve LP (\\ref{lp1}) or (\\ref{lp2}), \\Return $u$.\n\\end{algorithmic}\n\\end{algorithm}\n\nThe procedure is summarized in Algorithm \\ref{alg:stabilizing_controller}.\nThere can be many variants to Steps 4 and 5.\nFor example, one might use MPC-style planning based on local PWA linearization.\nDuring offline phase, one samples states $\\tilde{x}_k$ not in the nominal mode and solve MICP to get to some target points $\\bar{x}_{v(k)}$ on the nominal trajectory, hence storing a list of samples $\\{($state $\\tilde{x}_k$, mode sequence to get to target $\\bar{x}_{v(k)})\\}_{k=1}^M$.\nThe target index $v(k)$ for each sample state $\\tilde{x}_k$ can be chosen by comparing the cost to get to all nominal states $\\bar{x}_i, i=0,\\ldots,N$.\nDuring online execution, one finds the closest sample $\\tilde{x}_k$ to the current state $x$ and solve QP or LP to get to $\\bar{x}_{v(k)}$ fixing the mode sequence as stored.\n\nWe find empirically that for the half-cylinder flipping experiment, solving LP like (\\ref{lp1}) or (\\ref{lp2}) to directly go to the nominal trajectory is more efficient than MPC-style planning which plans multiple steps to reach the nominal trajectory.\nThis might be because of our assumptions that the change of contact modes is only caused by the manipulator making and breaking contacts with the object and that the manipulator is fully actuated.\nSo instructing the manipulator to directly go back to the desired position works.\nAlso MPC-style planning on linearized PWA systems may accumulate linearization errors.\n\nDuring the online execution, we use $\\mathbb{P} = [-1,1]^n$ instead of $\\mathbb{P}_i$ in Proposition \\ref{prop1}, because it is more computationally efficient to use $[-1,1]^n$ and it is hard to compute $\\mathbb{P}_i$.\nSince $\\mathbb{P}_i \\subseteq \\mathbb{P}$, we know once $x$ happens to fall into $\\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}_i$, then the system is guaranteed to reach the target region.\n\n\\iffalse \nIn our hardware experiment, in order to simplify the linearized piecewise affine system, we assume there is no sliding.\nWe only distinguish between two objects in contact or not in contact. \nWhen in contact, the two objects are sticking and not sliding.\nThis assumption reduces the dimension of the states, and simplifies the linearized PWA system, making the system easier to control.\nIn practice this simplification works better.\n\\fi \n\\section{Related Work}\n\\textit{Model-based feedback control for manipulation}:\nLynch's group used feedback control for sliding \\cite{shi2017dynamic}, rolling \\cite{ryu2013control}, vibratory manipulation \\cite{umbanhowar2012effect,vose2012manipulation}, and hybrid manipulation using motion primitives \\cite{woodruff2017planning,dafle2014extrinsic}.\nThey designed specific manipulators for specific tasks.\nIn contrast, our algorithm is more general and can be carried out on common robot platforms.\nRodriguez's group used feedback control for planar pushing \\cite{hogan2018reactive}.\nThey linearize the nonlinear system and solve linear model predictive control (MPC) online to plan the trajectory.\n\n\\textit{Path planning}:\nThere are generally two categories of approaches to path planning.\nOne is motion planning algorithms \\cite{lavalle2006planning}. \nFor discretized configuration space, grid search algorithms such as A$^*$ and its variants \\cite{likhachev2005anytime} are widely used.\nSampling-based algorithms such as rapidly exploring random trees \\cite{shkolnik2010sample}  are common approaches for continuous configuration space. \nThe other category is the trajectory optimization algorithms.\nA common approach in this category would be to formulate the problem as nonlinear optimization programs and solve it using off-the-shelf numerical solvers \\cite{posa2014direct,dai2014whole,mordatch2012discovery}.\nOther approaches in this category include augmented Lagrangian \\cite{toussaint2014novel}, mixed-integer convex optimization (MICP) \\cite{valenzuela2016mixed}, differential dynamic programming (DDP) \\cite{tassa2014control}, and iterative linear quadratic Gaussian (iLQG) \\cite{tassa2012synthesis}. \nA combination of these two methods have proved even more useful \\cite{dolgov2010path,zhang2018autonomous}. \nIn this work, we use trajectory optimization and use off-the-shelf numerical solvers to solve a nonlinear optimization program.\n\n\\textit{Local feedback controllers}: In control literature, it is quite standard to track a system trajectory using linear quadratic servo (LQ servo) or time-varying linear quadratic regulator (TVLQR) based on linearization of the system trajectory around nominal states \\cite{anderson2007optimal}.\nIn reinforcement learning, it is common to learn local linear models of the system and linear feedback control gains \\cite{levine2015learning,kumar2016optimal}.\nHowever, the local linear model does not fully capture the contact-rich nature of manipulation. \nRobot fingers may make and break multiple contacts with the object due to small disturbances. \nWhen contact modes change, the dynamics may change.\nSo it is more natural to model the local dynamics as a PWA system, i.e., a system whose state-input space is partitioned into several polytopic regions, with each region\nassociated with a different affine dynamics equation.\n\nHowever, stabilizing PWA systems alone is not an easy problem. \nExplicit solutions of optimal control for PWA systems can be computed offline by multi-parametric  programming \\cite{baoti2006constrained, christophersen2005optimal,  borrelli2005dynamic, baric2008efficient, bemporad2000optimal,bemporad2000piecewise,bemporad2002optimal,mayne2002optimal}.\nThe computational complexity of these methods grows exponentially with respect to the number of time steps.\nLyapunov-based approaches \\cite{rodrigues2002dynamic,lazar2006model,han2017feedback} and occupation measure approaches \\cite{zhao2017optimal,han2019controller} do not depend on the number of time steps, but are quite conservative and may not always find solutions.\nSampling-based methods \\cite{marcucci2017approximate,sadraddini2019sampling} suffer from the issue of scalability.\nIn this work, we consider manipulation problems in which there is only one rigid object, the manipulators are fully actuated, and the PWA dynamics is caused by manipulators making and breaking contacts with the object.\nWe track the system trajectory by solving LP online.\n\\subsection{Trajectory Optimization}\nWe plan the path using trajectory optimization methods.\nIn particular, we use direct transcription as in \\cite{dai2014whole}.\nThe continuous-time dynamics $\\dot{x} = f(x,u)$ is discretized into a discrete-time system $x^+ = \\phi(x,u)$ with sampling time $dt$.\nThe trajectory is discretized into $N$ time steps with $N\\cdot dt = T$, where $T$ is the time horizon:\n\\begin{align*}\n    \\text{minimize}\\ & \\sum_{t = 0}^T L(x[t],u[t])\\\\\n    \\text{subject to} \\ & m \\ddot{{r}}[t] = m{g} + \\sum_j {F}_j[t] \\\\\n    & I \\ddot{\\theta}[t] = \\sum_j ({c}_j[k]-{r}[k])\\times {F}_j[t] \\\\\n    & \\text{friction cone constraints}, \\text{contact constraints}, \\\\\n    & \\text{kinematics constraints}, \\text{time integration constraints}\n\\end{align*}\nwhere ${r}[t]$ is the position of the center of mass, ${F}_j[t]$ are forces, ${c}_j[t]$ is the the contact position of the $j$-th force for each $j$, $L$ is the loss function, and the decision variables include states $x[t]$, controls $u[t]$, and variables in the contact constraints.\nThe variables ${r}[t]$ and ${c}_j[t]$ are part of the states $x[t]$, and the forces ${F}_j[t]$ are part of the controls $u[t]$.\n\nThe first two constraints are Newton-Euler equations.\nThe friction cone constraints for planar systems are\n    $-\\mu F_{j,n} \\leq F_{j,t} \\leq \\mu F_{j,n}$,\nwhere $F_{j,n}$ is the normal force and $F_{j,t}$ is the frictional force.\nFor 3-dimensional systems, we can use a polyhedral cone to approximate the friction cone \\cite{stewart2000implicit}:\n    ${F}_j[t] = \\sum_{i} \\beta_{ij} {w}_{ij}, \\beta_{ij} \\geq 0$,\nwhere ${w}_{ij}$'s are the spanning vectors of the polyhedral cone.\nFor some contacts, we formulate the contact constraints as linear complementarity problems (LCP)  \\cite{posa2014direct} to fully characterize all possible contact modes -- sticking, sliding, or breaking contacts.\nWe use IPOPT \\cite{wachter2006implementation} to solve the trajectory optimization offline.\nSince IPOPT is an interior point method solver, the LCP constraints $P(x)^\\top Q(x) = 0, P(x) \\geq 0,Q(x) \\geq 0$, can be replaced by the equivalent constraints $P(x)^\\top Q(x) \\leq 0, P(x) \\geq 0, Q(x) \\geq 0$, and further be relaxed as $P(x)^\\top Q(x) \\leq \\epsilon, P(x) \\geq 0, Q(x) \\geq 0$ for small $\\epsilon > 0$.\n\n\n\\subsection{Force as Control Input}\nIn the trajectory planning described in the previous subsection, we borrowed the idea of zero-moment point (ZMP) for bipedal footstep planning in the humanoid robot literature \\cite{kajita2014introduction, kajita2003biped}. \nFor a bipedal robot walking on the ground, ZMP is by definition a point on the ground where the sum of all the tangential moments equals zero.\nAlthough many humanoid robots have pressure sensors on the feet, because of lack of reliability (in the case of Atlas), researchers do not measure ZMP directly. What they do is to plan a joint ZMP and center of mass (CoM) trajectory, and then only track the CoM trajectory during online execution \\cite{kuindersma2016optimization}.\n\nSimilarly, in the trajectory optimization formulation, we use forces as part of the control input. \nIn the manipulation context, the forces are those between the gripper and the object, and those between the object and the environment, e.g. the table.\nAlthough the hardware we are using cannot directly measure the force it applies to the object, we still use forces as control variables to help plan the CoM trajectory of the object as well as the pose trajectory of the gripper.\nDuring online execution, we only track the trajectory of the CoM of the object and that of the gripper.\nWe find in practice this approach works well.\n\n\\iffalse \n\\subsection{Path Planning on Simplified States}\nWhen the state space is very complicated, for example due to the existence of obstacles or due to the dramatic change in dynamics, directly formulating and solving the trajectory optimization can be hard.\nHere we describe a practical extension to the single trajectory planning.\n\nThe idea is to concatenate several trajectories and finally reach the goal state.\nThe user can define some waypoints in the state space, and compute trajectories between each waypoints.\nHowever, specifying a point in the entire state space might be tricky, so we introduce the \\textit{simplified states} or \\textit{equilibrium states}, where the velocities and the accelerations are all 0, and everything is static.\nMore formally, if the state is $[\\mathbf{q}_1, \\dot{\\mathbf{q}}_1,\\mathbf{q}_2]$, where $\\mathbf{q}_1, \\mathbf{q}_2$ are vectors, then the simplified state representation is a vector consisting of some entries in the vector $[\\mathbf{q}_1, \\mathbf{q}_2]$. \nFor example, if the carrot state is $[x,y,\\theta,\\dot{x},\\dot{y},\\dot{\\theta}]$, then the simplified state can be $[x,y,\\theta]$ or even just $\\theta$, while $\\dot{x},\\dot{y},\\dot{\\theta}$ are assumed to be 0.\n\nIt is much more computationally efficient and more user-friendly to specify waypoints or do RRT-style search on simplified states.\nFor example, if we want to flip the carrot $90$ degrees, we can plan to first rotate $45$ degrees, and then another $45$ degrees without caring about the pose of the gripper as long as it is feasible.\nIf we want to push a block through a maze, we specify static waypoints as 2D coordinates in the plane, without worrying about whether the pusher is sticking or sliding on the surface of the block at those equilibrium states.\nThis idea of planning in simplified states is also exploited in autonomous car path planning \\cite{zhang2018autonomous,dolgov2010path}. \n\\fi \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMisinformation is the act of accidentally spreading false or inaccurate information \\cite{online2019}. Some cases of Misinformation can be false rumors and pranks. Contrary to this, Disinformation describes the dissemination of malicious content, like, but not limited to, hoaxes, spear-phishing, and propaganda \\cite{wooley}.\n\nThe problem of the quality of information and how it reaches the European citizens has become enormous nowadays. It is important to note that a false expressed opinion by anybody, even without an intention to manipulate facts, could potentially fuel disinformation. An even more significant fact, is that the structure of the internet allows for a snowballing effect, potentially reaching a huge audience \\cite{renda}.\n\nThe phenomenon of Misinformation is emerging across Europe and has many manifestations. An insight into whether and to what extent European citizens trust their national information network comes from the fact that 23 out of the 28 EU states (82\\%) have at least a medium level of trust in the information provided by their national media \\cite{ebu}. Another statistic that should worry us regarding the confidence of European citizens in the media is that Social networks are the least trusted media in 29 out of 33 of the countries surveyed (88\\%). Citizens of the EU are more likely to trust radio and television over the internet and social media \\cite{ebu}. \n\nNot enough in-depth research has been conducted on networking patterns, which are a very valuable tool \\cite{koulas2}. On a more specific level, few studies have been carried out about Misinformation and fake news spreading. Still, they primarily measured the level of the phenomenon and the corresponding impact of it on the political and social environment. In the domain our research was held, there aren't any relevant studies dealing with the correlation among the websites containing misinformation content. \n\nNonetheless, few studies have focused on the analysis of Misinformation through Europe. A limited number of systematic reviews have examined the key actors that cause and encourage the spread of Misinformation. This study provides insights about the role of the European Governmental Organizations (GOs), Non-Governmental Organizations (NGOs), International Organizations (IOs), and International Institutes (IIs) in the spreading of that phenomenon. The method we used in order to construct our website collection sheet was by searching some key phrases about Misinformation on the search engines Google and Yahoo. On Bing, we faced some difficulties due to the lack of websites when we searched about specific key phrases. We collected and analyzed 49 seed websites with the criterion of the high ranking in the search engine. \n\nOur study provides a detailed webometric network analysis, based on the seed websites enlisted as Misinformation in Europe. It fills a gap in the literature for reviews of the correlation between sites. This study aims to empirically and methodologically assist in combating Misinformation both in Europe and the Global level. \n\nThe remainder of the paper is structured as follows. In Section 2, we perform an extensive literature review, comprising of the misinformation and disinformation effects in Europe, related webometric studies and action taken by European stakeholder on the issue. In Section 3, we formulate our research questions and analyze the procedure we follow regarding the selection of the seed sites and their analysis. Results of the webometrics analysis follow at Section 4. Finally, in Section 5 we discuss our findings and we provide our concluding remarks in Section 6.\n\n\\section{Literature Review}\n\n\\subsection{Misinformation and Disinformation effects in Europe}\nMisinformation and Disinformation are both phenomena that physically exist in everyday life almost since always. It is their digital nature that is novel nowadays and occurred a few years ago. \n\nRecently, the misinformation effect has been the source of significant concern in the Member States and has provoked discussions and investigations on this issue. The European Parliament, in June 2017, required, the European Commission analyzed thoroughly misinformation. Furthermore, it enquired from the Commission to formulating strategies for the effective mitigation of the problem. The Parliament even considered the possibility of legislative intervention to accomplish that mitigation, using as justification that fake news and Disinformation consist a considerable threat to the freedom of opinion, expression and democracy, which are of paramount importance under the European Union's Charter of Fundamental Rights. In a 2018 research conducted by the Robert Schuman Foundation, 68\\% of European citizens claim they encounter fake news at least once a week, while at the same time, 37\\% claim they encounter fake news daily \\cite{schuman}.\n\nThe European Commission, in order to address the issue, on the 26th of April 2018, proceeded with the publication of a communication titled \"Communication on tackling Disinformation: A European approach\" \\cite{commission}. Within this communication self-regulatory tools, which constitute the first step into countering effectively online disinformation in Europe, are contained.\n\nPresident Juncker, in his speech on the State of Union, presented in detail the actions that will take place based on this Communication and noted that he would do everything within his power to protect the civil rights and democracy. The purpose of this communication is to ensure that all citizens can have access to objective, quality information.\n\nThe Communications work plan was a series of actions that lead to the creation of a robust mechanism that prevents fake news and wrong information to spread. A team of fact-checkers was created, who reviewed information coming from reputable public sources and evaluated them. Another task was to promote non-toxic and quality journalism and punish any news media channel that doesn't provide valid information to the public. \n\nAt the same time, actions to educate people about choosing the right online and media information sources were taken in order to raise awareness for the issues of Misinformation and Disinformation and how these can affect the people. \n\nThe most significant success of this Communication was the acceptance and adaptation of the Code of good practice to fight online disinformation \\cite{commission1}, on the 26th of September 2018, which represents all the points and goals of the Communication. This Code successfully classifies all the prerequisites for a trustworthy only campaign, while at the same time enhancing fundamental principles, like the freedom of expression and media pluralism.\n\nMisinformation can be tackled, and its effects can be mitigated through ICT and monitored via the use of the internet. Internet is a useful tool to accomplish that feat, and ICT is the medium that gives the capability to collect, accumulate, interpret, and show data in order to make crucial decisions and formulate strategies. And as the initiator of Europe's democratic system Robert Schuman has said: \"Technology is changing, but our fundamental values remain. A citizen that is equipped with the necessary skills and that can listen, watch, and read critically is a prior condition for the success of these values\".\n\n\\subsection{Webometrics related studies}\nThe application of bibliometric and infometric approaches to study the web, its information resources, structures, and technologies, is known as webometrics. The term webometrics is a coinage from the English word \"web\" and the andicet Greek \"metric\", which means to measure. Since the term was coined in 1997 by Tomas Almind and Peter Ingwersen, the value of webometrics quickly became established through the Web Impact Factor, the critical metric for measuring and analyzing website hyperlinks \\cite{thelwall12}. If we would like to specify that methodology, an excellent definition to show would be the one by Mike Thelwall in 2004 \"\u2026 the study of web-based content with primarily quantitative methods for social science research goals using techniques that are not specific to one field of study\". The purpose of this alternative definition was not to replace the description within Information Science \\cite{thelwall09}. The actual use of the first definition is to support the publishing of appropriate methods out of the scope of Information Science \\cite{kunosic}.\n\nSince the emergence of webometrics, this tool has become a useful methodology that applies in many fields such as the ranking of universities and scientometric evaluations or investigations of research areas. In order for the effective analysis of data for webometrics usage, it is of paramount importance to use known credible sources, for every category of webometrics.\n\nThe study from Roghaye Tafaroji, Iman Tahamtan, Masoud Roudbari, and Shahram Sedghi, which was conducted in 2012, aimed to present the findings of a webometric analysis of web sites of medical universities of Iran. They tried to investigate the Webometric ranking of Iranian Universities of Medical Sciences. In comparison to other similar studies that were conducted before and used inlinks and size as the main webometric criteria. The authors of this study examined rich text format files (doc, pdf, etc.) as a webometric indicator, and measured the impact of this new indicator on webometric ranks. The main findings indicate that Iranian Medical Universities performed poorly in regard to number of webpages, external links, and rich files. This observation is very useful because it can explain the anemic presence of these universities on the web  \\cite{tafaroji}.\n\nAnother study presented a ranking of Alternative Search Engines (ASEs). Using indicators to evaluate a large amount of data that can be retrieved effortlessly and effectively, Bernd Markscheffel and Bastian Eine managed to create a picture of the most popular ASEs currently available. This approach allows further investigation for other studies, exploring the dynamics of the search engine market, while at the same time assessing the categories of ASEs \\cite{markscheffel}.\n\nA recent study conducted by The University of Burdwan measured and gave a clear idea about the information provided by the websites of the 25 High courts using this time just Google Search engine in contrast with the previous studies. This paper highlighted the various web impact factors, scores, and ranking of the websites of high courts in India and the final results that gave did open the door to further studies of other new areas of the webometric analysis \\cite{mahji}.\n\nAnother study also examined information originating from the website of 71 universities in Bangladesh. The results given indicate that the universities of Bangladesh do not have greater web visibility and, it is clear that these universities need to focus on several issues to increase the visibility of their websites \\cite{islam}. \n\nFurthermore, Webometrics Analysis was also used to measure Web Impact Factor (WIF) of 8 National Libraries' websites in South Asian Countries. WIF provides tools for quantitative research for several categories, like ranking, evaluation, categorization and comparison of websites, both on top-level and sub-level domains. The results visualized that the National Library of India leads with the highest Domain Authority and Page Authority, and it is followed by the National Library of Sri Lanka and National Library of Bhutan among the other National Libraries websites. Users visit the websites of the National Libraries for their information needs \\cite{verma}. \n\nLast but not least, one of the significant studies regarding webometrics is \"Open Data in Nepal: A Webometric Analysis\", which measures the impact of Open Data in the Nepalese cyber domain \\cite{acharya}. Acharya and Park's study serves as a guide for this study.\n\nTaking in mind the related studies above it is clear that Webometrics is a tool used in many studies to examine the World Wide Web and give specific results about the construction and use of information resources. This is the reason why we decided to use Webometrics analysis in order to search the web and examine Misinformation in Europe.\n\n\\subsection{Anti-misinformation stakeholders inside and outside Europe}\nAs the volume of information flowing on the internet snowballs, the phenomenon of Misinformation is becoming more and more intense. For this reason, in recent years, many Governmental Organizations (GOs), Non-Governmental Organizations (NGOs) and International Organizations (IOs), inside and outside of the European border, have been mobilized to deal with Misinformation.\n\n\\subsubsection{European Level}\nThe anti-misinformation concern of the European Commission increased after some cases of intense manipulation of the public opinion on political issues. These phenomena occurred during the U.S and French presidential elections (2016-2017), as well as the Italian constitutional referendum (2016) \\cite{service}. After a two-day conference in Brussels, the European Commission concluded that expert's help is vital in order to combat Misinformation \\cite{commission17}. \n\nFinally, a High-Level Expert Group (HELG) on Artificial Intelligence was formed in January 2018 to reduce Misinformation, fake news, and Disinformation at any level within Europe. A report containing the best strategies and solutions about every disinformation issue depended on the same set of fundamental principles, which was the main deliverable of the HELG \\cite{connect}.\n\n\\subsubsection{Member states of European Union}\n\nBesides the European Commission, there are also many mechanisms within the states of Europe, that try to combat Misinformation.\n\nIn Germany and Croatia, strict laws about Misinformation and hate speech were applied. Websites that would not comply with the law within a specific period after the warning would pay a considerable fine \\cite{bbc1, vintof}. \n\nIn Italy, a portal where people could report to the authorities, fake news occurrences, was set, but unfortunately, it didn't work rationally because of the lack of knowledge about fake news \\cite{commissario}. However, when a man was sent to prison for using a false identity in TripAdvisor reviews, the government's intentions were very clear. After that, the country's communications authority released a report on Misinformation \\cite{agcom}.\n\nSweden, Denmark, Belgium, and the Netherlands launched campaigns on websites and social media in order to inform the citizens about Misinformation and fake news, on the initiative of their governments (2018-2019) \\cite{funke}. \n\nIn Spain, when Russia was blamed for spreading Misinformation concerning the Catalan referendum by national authorities, an agreement was signed between the two of them, to create together a cybersecurity team to prevent Misinformation \\cite{funke}. Moreover, a task force of about 100 officials was activated during the 2019 elections, with the aim to combat fake political posts, especially on social media.\n\nIn France, a very innovative law was set on the press, which gives the power to the authorities to do whatever they must do with sites that illegally use fake news and Misinformation, enabling them to shut them down.  However, this measure was criticized especially from opposition senators and journalists because according to what they say, it is against the principle of proportional justice and press freedom \\cite{ricci}.\n\nIn Greece, there are many NGOs nowadays, dealing with the refugees coming from regions where the situation is turbulent. Because of the significant number of migrants, this situation has become very sensitive. As a result, Misinformation and fake news are spread both from people and the media very fast. In order to help the migrant, many mechanisms collaborated to protect refugees' rights and fight Misinformation about this topic. They also take care of their housing and education.  These mechanisms are the \"United Nations High Commissioner for Refugees (UNHCR)\", together with the \"Emergency Support to Integration and Accommodation (ESTIA)\" program,  funded by the European Union Civil Protection and Humanitarian Aid, and some other local NGOs.\n\nIn the United Kingdom, a parliamentary report was published with the purpose to enforce citizens to avoid fake news and misinformation spreading, because the country suffers from a democracy crisis since the idea of Brexit came to the surface \\cite{waterson}. Furthermore, the National Security Communications Unit was launched with the task to fight Disinformation from authority people and others, after Russia got involved in Brexit by spreading fake news \\cite{bbc2}.\n\n\\subsubsection{NGOs}\nAt the same time, even though the mobilization of governmental organizations is essential, the contribution of non-governmental organizations to combating misinformation is just as remarkable. An international NGO, \"Reporters Without Borders (RSF)\", launched an innovative media to deal with Disinformation online \\cite{boulay}. It is called the \"Journalism Trust Initiative (JTI)\" and is designed to encourage journalism by applying some agreed transparency standards to protect information and combat misinformation.\n\nMoreover, charity NGOs, who are dealing either with refugees or with citizens that need help, are those who try besides their actions to protect people's rights and to publish only the real thing about the issues in which they are involved. \"ActionAid\" is one fascinating example of international NGO which has already offered very much in this sector.\n\n\\subsubsection{Anti-misinformation stakeholders outside Europe}\n\nIn addition to the efforts made to combat Misinformation within the European Union, efforts are also being made from countries outside of Europe. In some cases, Europe is firmly connected and immediately affected by the efforts made to fight Misinformation outside of Europe.\n\nRussia has been one very important player in spreading fake news in recent years across Europe and the whole world, at both a political and social level. Especially in the US presidential elections, many campaigns where bots were used, were set in order to manipulate the result. For this reason, media continuously publishing fake news and misinformation are punished. For a start with a fine, if they would not conform with the law the people accountable for this would go to prison, and if that was not enough, their website would shut down \\cite{stanglin}.\n\nIn the Americas, fines are the most widely used method of combating fake news and misinformation. In the USA, Brazil, and Chile, when someone is found to be disseminating fake news, the responsible party will be fined, and may even face prison time. This applies to everyone whether they are citizens writing on the net, whether they are journalists, webpages or even politicians who spread fake news \\cite{funke}.\n\nIn Asia, many countries have adopted strict laws to deal with misinformation. In China and Malaysia spreading fake news is considered as a crime. Those who rumor fake news that can be harmful in public order are punished by the law with up to seven years in prison or public surveillance \\cite{zhang, leong}. Moreover, south Asia countries, Thailand and Indonesia have also enforced laws in cases of misinformation detection. Many people were arrested for fabricating fake news, especially on social media, and many others paid huge fines \\cite{funke}.\n\n\n\\section{Methods}\n\\subsection{Goals and Research Questions}\nThe rise of the internet and computational power has allowed for the disproportional growth of misinformation phenomena in the last years. In this study we want to discuss the measures taken by stakeholders in Europe to tackle those incidents and assess their effectiveness. For this, we formulated two research questions: (RQ1) Which are the key stakeholders and how do they fight the phenomenon of misinformation? (RQ2) Do the actions of the key stakeholders have a palpable impact? The first question aims to mapping the key stakeholders, as well as, assess their actions and cooperation. The second question tries to showcase the impact, if any, those actions have.\n\n\\subsection{Data Collection}\nOur method for gathering information and seed sites regarding Misinformation was Google's search engine, as well as, Bing and Yahoo search engines. Trying to have a variety in our results and have the whole idea of Misinformation, we used different keyword combinations to get more accurate results. These are the search queries we used: \n\n\\begin{itemize}\n\\item Misinformation in Europe\n\\item Role of NGOs in Europe to tackle Misinformation\n\\item Governmental Organizations tackling Misinformation in the Europe Area\n\\item Expanding Misinformation in society\n\\item The Consequences of Misinformation\n\\item Fighting Misinformation\n\\item Misinformation in Belgium and\n\\item Misinformation Tackling\n\\end{itemize}\n\nThe search results included most of European and Global NGOs, IOs, GOs, and IIs. The homepages of these organizations were visited and read individually to assess the importance of researching the phenomenon of Misinformation in Europe. We opted to include Belgium as a separate search query, due to the fact that the European Union's present there has led to the creation of a variety of think tanks, NGOs and corporations.\n\nSince the exposure to a large scale of disinformation is proliferating, fighting misinformation is a top priority for the European Commission. Therefore we emphasize identifying the key factors that encourage the spread of this phenomenon throughout Europe. It is highly essential to understand the role of all the European Stakeholders and Institutions, such as the European Commission, Council of Europe, etc., that are responsible for measuring all the appropriate criteria that give us a more analytic point of view about the issue. \n\nFurthermore, it is significant to measure the impact and the consequences of Misinformation to find new and more efficient ways to counter misleading information in the European Area. Subsequently, we should emphasize on social-economic elements that will be able to lead us to determine a general framework for the protection of information throughout Europe, in the same way, the General Data Protection Regulation has been defined.\n\nWe choose to collect data from three Top Level Domain Categories and the specific eight Country Code Top Level Domains. In this context, to make it more understandable, we must observe the following hierarchy tree: \n\n\\begin{figure}[h]\n\\caption{Hierarchy tree}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{tree.png}\n\\end{figure}\n\nFrom the hierarchy tree in Figure 1, we notice the Top-Level Domain categories that there will be shown in the webometrics analysis and the following Country Code Top Level Domains. It's time to perceive the provenance of each domain. Firstly, .com derives from commercial, indicating its original intended purpose for domains registered by commercial organizations. Secondly, .org is truncated from organizations. Following the same way of thinking, we recognize that all Country Code Top Level Domains reserved for specific countries. In our case, we ended up with seedsites having the country codes of the European Union, Germany, Belgium, and Luxembourg. Finally, it's meaningful to recognize the role of Sponsored Top Level Domains. This category of the domain name is supported by a community or organization and considered to have the strictest application policies of all TLDs, as it implies that the holder is a subject of international law.\n\nTable 2 lists the organizations, their established date, the sector in which they belong, their website address, and their URL.\n\n\\subsection{Process and Assessment}\nThe above websites are analyzed using Webometric Analyst 2.0 (http:\/\/lexiurl. wlv.ac.uk) and the Bing Application Program Interface, which is capable of carrying out advanced Boolean searches and tracking external websites linked to URL citations under study. Thus, the lists of external sites corresponding to the base query, i.e., the websites mentioned above, were obtained. \n\nInterlinkage and co-mention, as explained in Table 1, will be used for the data analysis.\n\n\\begin{table}[]\n\\caption{Analytical techniques and concepts of Webometrics\\cite{acharya}}\n\\label{tab:tools}\n\\begin{tabular}{p{0.2\\textwidth}p{0.8\\textwidth}}\n\\hline\nInter-mention network analysis & Network diagrams illustrate the accompanying networks of the communication strength of the provided pairs of websites. It is the indicators based on asymmetric (directed) inter-mention counts between a pair of websites. A diagram illustrates the pattern of interconnectivity between collections of sites. This analysis gives a proxy for the hyperlinks between the websites under study \\\\\n\\\\\nCo-mention network analysis            & Network diagrams and their indicators based on the number of external sites referring to a pair of target sites. The co-mentions show something important in common but are not directly related to each other. The competitors who are also considered as stakeholders show a different pattern in the webometric analysis. Co-mention does not have a direction                                \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFor the assessment we are going to use the inlink degree, the outlink degree and the betweenness centrality. The inlink degree shows how many links from the network are inbound for each specific node, while the outdegree shows the outbound links towards other nodes of the network. Betweenness centrality shows the sum of times any particular node is found on the shortest path between different nodes of the network \\cite{disney, valente, friedkin}. The higher the metrics, the more influential a node, thus an organization, is in the network. \n\\section{Results}\n Figure 2 and Tables 2 and 3 answer our first research question. Figure 3 and Table 4 answer our second research question.  Figure 2 depicts a network diagram that demonstrates the inter-mention between websites conclusively. The red nodes and arrows show the linkage between the websites, whereas the green nodes indicate that there is no connectivity with any of the seed sites.\n \nEvery website domain and URL was converted accordingly to meet the requirements for Bing classification so that Bing can make the connections between the URLs and seed websites. According to the results below, websites are firmly connected and are the core of our findings, www.theverge.com, www.nytimes.com, www.cnet.com, www.washingtonpost.com, www.reuters.com, and www.bloomberg.com. As you see, the core URLs are oriented to the private company sector, and they have .com TLD. Most of the government based websites are connected, having a substantial presence on the web, for example, www.coe.int and www.poynter.org. \n\n\\begin{figure}[h]\n\\caption{Inter-mention Network Diagram}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{mininfo-interlinks.png}\n\\end{figure}\n\nWe also observe that www.goethe.de (which is a Nonprofit organization), www.zeit.de, and www.welt.de, which are both POs, are strongly connected to www.dw.com which is the most centralized website from all NPO's. Another seed site that has many connections to the central nodes of the diagram and vice versa is www.technologyreview.com and www.newscientist.com; both of them have a .com TLD. Additionally, we see that from websites with TLD .be and .lu only today.rtl.lu is not connected with any website. On the contrary, www.lesoir.be and www.knack.be are connected between them and also to the most central websites.\n\nFurthermore, the interlinkage was investigated for the purpose of analyzing the online networking patterns in different networking scenarios. The values that we have observed are the networking density value for the directed network, which is 0,0791, and the value for the undirected network which is 0,1182, increased in comparison to the direct network. Density is calculated by diving the number of relations by the maximum number of possible relations. Density means an average value of entire cell blocks, when we refer to a network matrix that is weighted and valued. Next, 'degree' and 'betweenness' network centrality values are calculated. The term degree centrality refers to the amount of ties that are immediately connected to a node (i.e., website), rather than indirect ties to all others in the network\\cite{acharya}. The two-degree centrality, specifically indegree, and outdegree, is calculated by the direction of the connection between two nodes. On the other hand, betweenness centrality measures how important a node is in the network. This is calculated by the effectiveness that a specific node plays as a broker, while connecting a pair of nodes. In this instance, the number of the shortest paths via the node is considered. Our network metrics were calculated utilizing built-in functions within Webometric Analyst.\n\n\\begingroup\n\\footnotesize\n\\begin{longtable}{p{0.1\\textwidth}p{0.3\\textwidth}p{0.1\\textwidth}p{0.1\\textwidth}p{0.3\\textwidth}}\n\\caption{Description of the Selected Seed Sites}\n\\label{tab:seedsites}\\\\\n\\hline\nNo. & Organization Name                                                        & Est. date & Sector                                 & URL                                     \\\\ \\hline\n\\endfirsthead\n\\multicolumn{5}{c}%\n{{\\bfseries Table \\thetable\\ continued from previous page}} \\\\\n\\hline\nNo. & Organization Name                                                        & Est. date & Sector                                 & URL                                     \\\\ \\hline\n\\endhead\n\\hline\n\\endfoot\n\\endlastfoot\n1   & Council of   Europe                                                      & 1999      & GOV                                    & coe.int                    \\\\\n2   & European Commission                                                      & 1958      & GOV                                    & ec.europa.eu                   \\\\\n3   & Public Data Lab                                                          & 2017      & IO                                     & publicdatalab.org               \\\\\n4   & Reporters Without Borders                                                & 1985      & IO                                     & rsf.org                        \\\\\n5   & European   External Action Service's East StratCom Task Force            & 2015      & NGO                                    & euvsdisinfo.eu                 \\\\\n6   & Center for   Strategic and International Studies (CSIS)                  & 1962      & Non-profit Organization                 & csis.org                   \\\\\n7   & Investigate Europe                                                       & 2014      & NGO                                    & investigate-europe.eu      \\\\\n8   & Journalismfund.eu                                                        & 2008      & NGO                                    & journalismfund.eu          \\\\\n9   & CNN Digital                                                              & 1980      & Private Company                        & edition.cnn.com                \\\\\n10  & European Data   Journalism Network (EDJNet)                              & 2017      & Private Organization                   & europeandatajournalism.eu   \\\\\n11  & European Centre   for Press and Media Freedom (ECPMF)                    & 2009      & NGO                                    & ecpmf.eu                       \\\\\n12  & Poynter Institute                                                        & 1975      & GOV                                    & poynter.org                \\\\\n13  & Social   Observatory for Disinformation and Social Media Analysis (SOMA) & 2018      & Project                                & disinfobservatory.org      \\\\\n14  & Media Freedom Resource Centre                                            & 2015      & NGO                                    & rcmediafreedom.eu          \\\\\n15  & WAN-IFRA - World   Association of News Publishers                        & 1948      & ORG                                    & wan-ifra.org               \\\\\n16  & Parliament of Europe                                                     & 1952      & Int'l Institution              & europarl.europa.eu         \\\\\n17  & EU Open Data Portal                                                      & 2012      & Portal                                 & data.europa.eu                  \\\\\n18  & Euractiv                                                                 & 1999      & Network of Media                       & www.euractiv.com               \\\\\n19  & Fandango Project                                                         & 2018      & Project                                & fandango-project.eu            \\\\\n20  & The Guardian                                                             & 2011      & Private Company                        & www.theguardian.com            \\\\\n21  & Sunlight Foundation                                                      & 2006      & Non-profit Organization                 & sunlightfoundation.com         \\\\\n22  & New Scientist                                                            & 1956      & NGO                                    & newscientist.com           \\\\\n23  & Fipp                                                                     & 1920      & Private Company                        & fipp.com                   \\\\\n24  & The New York Times                                                       & 1851      & Private Company                        & nytimes.com                \\\\\n25  & Washington Post                                                          & 1877      & Private Company                        & washingtonpost.com         \\\\\n26  & Euronews                                                                 & 1993      & Portal                                 & euronews.com               \\\\\n27  & MIT Technology Review                                                    & 1899      & Private Company                        & technologyreview.com       \\\\\n28  & Media Frenzy Global                                                      & 2006      & Private Company                        & mediafrenzyglobal.com      \\\\\n29  & Singularity University                                                   & 2008      & Univer-sity                             & singularityhub.com             \\\\\n30  & The Atlantic                                                             & 1857      & Private Organization                   & theatlantic.com            \\\\\n31  & The Verge                                                                & 2011      & NGO                                    & theverge.com               \\\\\n32  & Civil Beat                                                               & 2010      & NGO & https:\/\/www.civilbeat.org\/              \\\\\n33  & The World Bank Group                                                     & 1944      & NGO                                    & blogs.worldbank.org            \\\\\n34  & The Social   Science Research Council (SSRC)                             & 1923      & NGO                                    & ssrc.org                   \\\\\n35  & Bloomberg                                                                & 1981      & Private Company                        & bloomberg.com\/europe        \\\\\n36  & Star Tribune                                                             & 1897      & Private Organization                   & startribune.com             \\\\\n37  & Science Direct                                                           & 1997      & Private Organization                   & sciencedirect.com          \\\\\n38  & NetGov                                                                   & 2008      & GOV                                    & nextgov.com                \\\\\n39  & Lesoir                                                                   & 1928      & Private Organization                   & lesoir.be                   \\\\\n40  & Knack                                                                    & 1971      & Private Organization                   & knack.be                    \\\\\n41  & Deutsche Welle                                                           & 1953      & Non-profit Organization                 & dw.com                      \\\\\n42  & Zeit                                                                     & 1946      & Private Organization                   & zeit.de                     \\\\\n43  & Welt                                                                     & 1946      & Private Organization                   & welt.de                     \\\\\n44  & Goethe Institute                                                         & 1951      & Non-profit                             & goethe.de                   \\\\\n45  & RTL                                                                      & 1929      & Private Organization                   & today.rtl.lu                    \\\\\n46  & American Press Institute                                                 & 1946      & Institute                              & americanpressinstitute.org \\\\\n47  & Reuters                                                                  & 1851      & Private Organization                   & reuters.com                \\\\\n48  & Computer Network                                                         & 1994      & Private Organization                   & cnet.com                   \\\\\n49  & Pew Research Center                                                      & 2004      & Research Center                        & pewresearch.org            \\\\ \\hline\n\\end{longtable}\n\\endgroup\n\nParticularly, the 14 websites of our seed sites with the highest indegree and outdegree centrality are presented in Table 3. The Reuters (www.reuters.com) has the highest indegree centrality (74), and a Private Company 'The Guardian' (www.theguardian.com) has the highest outdegree centrality (66).  The balance between big organizations, like the European Parliament, and case specific seed sites like the EUvsDisinfo, ensure that the metrics are accurately depicting connectivity around misinformation.\n\n\\begin{table}[]\n\\caption{Top 14 websites with the highest indegree and outdegree centralities.}\n\\label{tab:centralities}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lllllll}\n\\cline{1-7}\nOrganization            & Sector                  & Indegree &  & Organization            & Sector                  & Outdegree \\\\ \\hline\nReuters                 & Private Company         & 74       &  & The Guardian            & Private Company         & 66        \\\\\nWashington Post         & Private Company         & 54       &  & The New York Times      & Private Company         & 60        \\\\\nThe Atlantic            & Private Organisation    & 50       &  & Washington Post         & Private Company         & 56        \\\\\nThe Guardian            & Private Company         & 44       &  & The Atlantic            & Private Organization    & 28        \\\\\nBloomberg               & Private Company         & 36       &  & Reuters news agency     & Private Company         & 24        \\\\\nThe New York Times      & Private Company         & 26       &  & European Commission     & GOV                     & 24        \\\\\nThe Verge               & NGO                     & 24       &  & Science Direct          & Private Organisation    & 24        \\\\\nPew Research Center     & Research Center         & 20       &  & The Verge               & NGO                     & 20        \\\\\nDeutsche Welle          & Non-profit Organisation & 16       &  & CNET (Computer Network) & Private Company         & 20        \\\\\nCNET (Computer Network) & Private Company         & 14       &  & Deutsche Welle          & Non-profit Organisation & 14        \\\\\nMIT Technology Review   & Private Company         & 14       &  & Poynter                 & GOV                     & 14        \\\\\nNew Scientist           & NGO                     & 14       &  & CNN World News          & Private Company         & 14        \\\\\nThe Star Tribune        & Online Media Company    & 10       &  & MIT Technology Review   & Private Company         & 12        \\\\\nEuronews                & Portal                  & 10       &  & The Star Tribune        & Online media Company    & 10        \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\nConcerning the Private Organizations, like theguardian.com, theatlantic.com, washingtonpost.com, and reuters.com, they have high betweenness centrality (248, 185, 184, 180, 137). The local NGO www.theverge.com has the highest betweenness centrality among all NGOs (11,217). The Government Organization ec.europa.eu has a high betweenness centrality (113,867). The Nonprofit Organization Deutsche Welle (www.dw.com) have 42,35 betweenness centrality.  The betweenness centrality of a website shows the amount of control that this website exerts over the interactions of other websites in the network. The Pew Research Center is a research center that has the minimum betweenness centrality (0,2). Thus, it is noticeable that the Private Organizations have the highest betweenness centrality from NGOs.\nIn addition to the above conclusions, we see many websites with a weak connection between them such as www.disinfobservatory.org which is a technology-based company, also www.publicdatalab.org the only IO which has no connection. We have 6 NGO's websites with a minor presence on the web and no connection at all with the rest of the websites.  Finally, www.fipp.com and www.mediafrenzyglobal.com are two private companies with no connection between them.\n\nFigure 3 shows the co-mention links of the websites. All of the nodes are colored red because all websites have at least one co-mention with another website. Each line's width is proportionately calculated and drawn based on the number of co-linking websites.\n\nThe paramount importance of the issues related to Misinformation is highlighted by the vast co-mentions among the websites analyzed. It is observed that the European Commission spearheads the efforts for tackling Misinformation.\n\nThe European Commission's role in promoting effectively European initiatives for tackling Misinformation can be shown by the fact that ec.europa.eu and www.disinfobservatory.org are co-mentioned 15 times, and ec.europa.eu and euvsdisinfo.eu are co-mentioned 185 times.\n\n\\begin{figure}[h]\n\\caption{Co-mention Network Diagram}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{misinfo_comention.png}\n\\end{figure}\n\n\\begin{table}[]\n\\caption{Seed site calculation (N=49). Data values are defined as the total number of counts of TLDs citing seed sites.}\n\\label{tab:domains}\n\\begin{tabular}{llll}\n\\hline\nNo & Seed site TLDs\/GTLDs\/CCTLDs & Total Number of Seed Sites & Percentage (\\%) \\\\ \\hline\n1  & .com                        & 21                         & 42.85\\%         \\\\\n2  & .org                        & 11                         & 22.44\\%         \\\\\n3  & .eu                         & 10                         & 20.40\\%         \\\\\n4  & .de                         & 3                          & 6.12\\%          \\\\\n5  & .be                         & 2                          & 4.08\\%          \\\\\n6  & .int                        & 1                          & 2.04\\%          \\\\\n7  & .lu                         & 1                          & 2.04\\%          \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nAt the same time, the Commission is used as a reference alongside the world's most influential think tanks, as it is observed that ec.europa.eu and www.theatlantic.com are co-mentioned 305 times; ec.europa.eu and www.csis.org are co-mentioned 359 times; ec.europa.eu and pewresearch.org are co-mentioned 541 times.\n\nLastly, the Commission is used to set a paradigm with other major organizations since ec.europa.eu and www.wan-ifra.org are co-mentioned 180 times; ec.europa.eu and www.rsf.org are co-mentioned 479 times, and ec.europa.eu and www.coe.int are co-mentioned 721 times.\n\nFurthermore, it is observed that resource produced by European initiatives is often used alongside resources produced by leading think tanks. For example, euvsdisinfo.eu and www.csis.org are co-mentioned 49 times; euvsdisinfo.eu and www.pewresearch.org are co-mentioned 81 times, and euvsdisinfo.eu and www.theatlantic.com are co-mentioned 178 times. \n\nThere are, however, initiatives that fall behind when it comes to European Projects. For instance, www.disinfobservatory.org and www.theatlantic.com are co-mentioned once, while www.disinfobservatory.org and www.csis.org are co-mentioned once, and www.disinfobservatory.org and pewresearch.org are co-mentioned twice.\n\nLast but not least, European NGOs play an important role in the efforts to tackle misinformation since www.investigate-europe.eu and ec.europa.eu are co-mentioned 33 times; www.investigate-europe.eu and www.theatlantic.com are co-mentioned 661 times; www.investigate-europe.eu and www.csis.org are co-mentioned 841 times; www.investigate-europe.eu and www.pewresearch.org are co-mentioned 845 times; www.investigate-europe.eu and www.rsf.org are co-mentioned 849 times; and www.investigate-europe.eu and fandango-project.eu are co-mentioned 979 times.\n\nAll those findings show the massive effort, and the resources pooled from various stakeholders to tackle Misinformation.\n\n\\section{Discussion}\n\nMisinformation is a constantly evolving threat that requires rigorous checks and balances in order to address it. The results can be categorized as those that stem from the theoretical part of the paper and those that derive from the webometrics analysis.\n\nOn the theoretical part, it is an interesting fact that in many cases, Webometrics is used as an evaluation system of a wide range of universities in the world. This system is known as a \"ranking\" system, where ranking describes a process where the position of the elements in a group regarding its entirety is defined by the relevance between the elements. The ranking process appears in many areas besides academics. For example, there was also a study on the ranking of Alternative Search Engines (ASEs) \\cite{markscheffel}. \n\nWebometrics is a tool used in many studies to examine the World Wide Web for different reasons. In this project, we chose to use this tool to explore a totally different issue that also has an enormous impact on people and tries to interpret the data exported from the analysis. \n\nFrom the research on combating Misinformation at the state level, it can be concluded that the use of strict laws or regulations, to punish the people accountable for this phenomenon , is not an effective strategy. Inform people how to detect Misinformation, thus preventing them from reproducing it, is much more effective. Democratic societies ought to help their citizens learn how to acquire their information only from proven reputable sources, question what they read, examine its accuracy, avoid reading only the headlines of articles, and in case something is fake to avoid sharing it.\n\nLast but not least, we observe the involvement of International Organizations in the effort to tackle Misinformation and Disinformation whose original mandate was totally irrelevant. For instance, the Council of Europe is heavily investing in the cyberspace domain, to remain relevant in a shifting world \\cite{koulas}, while this change in perspective can foster meaningful collaborations \\cite{brass}.\n\nIn the webometrics area, this study aims to investigate on a webometric dimension the role of all public and private sectors in Europe, as well as on the international level, that have taken actions as stakeholders to tackle the misinformation effect in Europe. We explore the structure of these stakeholders' portals and websites, their source, the organization's vision, methods of gathering and crosschecking data and information, and actions on the matter of Misinformation in Europe. We searched the seed sites countrywide and on an international level, for URL- based hyperlinks, title mentions and external links that refer to the seed sites of the stakeholders. According to the results, European governmental websites and portals are cooperating with the concerned NGOs inside Europe, but not as much as IOs are among themselves. In the co-mention network, all European portals show a strong connection with international websites and the other way around. Also, news organizations among them and other organizations of the same work- nature appear to have a very strong co-mention relationship. Besides, diverse organizations are also well co-mentioned. This could be because of the severe and sensitive nature of the issue and the urgency to counter it. Overall, European IOs and Governmental portals seem to have the most interlinkage and co-mentions as legal bodies that officially take actions to counter the Misinformation in Europe on a national governmental level with legislation, new laws and various efforts to raise awareness.\n\nIn our study we have identified 49 different stakeholder that took action in the fight against misinformation in Europe, as it is shown in Table 2. We found out that these efforts had some success, in terms of networking patterns. The limitations we faced are similar to those Acharya and Park \\cite{acharya} face. Firstly, from a total of 71 possible websites, we manually selected the ones we deemed more important and more relevant to our study and used them as seed sites. Early tries to use all 71 of the original findings resulted in messy diagrams and due to the high number of irrelevant sites, the different metrics, especially the betweenness centrality, were boosted, without actually ensuring the misinformation focus we opted for. Secondly, many of our seed sites are not solely focused on Misinformation and Misinformation tackling alone but have made contributions towards that field. Thirdly, the seed site analysis was conducted with third party software and the search engine www.bing.com.\n\n\\section{Conclusions}\nMisinformation is one of the significant challenges that modern societies need to address effectively because it severely impacts multiple aspects of our lives. We observe that there have been attempts to tackle Misinformation, with mixed results. The European Union spearheads these efforts, in cooperation with other organizations, but there it is possible to further enhance and improve those efforts. The ease world wide web provides for the spreading of Misinformation is a significant factor that increases the complexity of the situation. Further study should focus on the limitations of this study, as well as the usage of new technologies, like artificial intelligence, in order to process more data and yield better results.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nWhen an individual has to pay for the consumption of goods or services then there is a natural incentive to moderate this consumption. But there are cases when the relationship between the consumption and the emerging cost is more subtle because both actions are made by a group of persons. For example, in a university dormitory or a shared household each member can use electricity or water freely, but all members share the related bills equally. This collective responsibility may induce less trustworthy individual consumption because members may easily consume more than they would if the personal consumption is recorded and the related bill is paid separately. As a result, they can easily become careless because the extra use is paid collectively. Notably, the reverse is also true: an individual saving of usage would result in just a modest decrease in an individual bill because others will also benefit from a responsible act. On the other side, if all participants behave trustworthily and consume just the necessary resource then they would pay the minimal cost. In parallel, they could still enjoy the benefit of a collective venture, like maintaining a less expensive joint infrastructure. This establishes a dilemma between individual and collective interests when players are motivated to defect and consume more than they would individually. A similar dilemma occurs at a larger scale when countries use the same natural resources, like the oceans or the atmosphere~\\cite{milinski_pnas08,hilbe_prsb10,pacheco_plrev14,sun_ww_is21}.\n\nThe conflict of individual and collective interests, often called as a social dilemma, is the central problem of several game-theoretical models, including prisoner's dilemma game \\cite{szabo1998evolutionary,takeshue_epl19,amaral_pre21,zhu_pc_epjb21}, snowdrift game \\cite{hauert2004spatial,chen_xj_epl10,graeser_njp11}, stag-hunt game \\cite{skyrms2004stag,starnini_jsm11,deng_ys_epjb22}, ultimatum game \\cite{page2000spatial,sinatra_jstat09,szolnoki_prl12,chen_w_pa19}, trust game \\cite{berg1995trust,chica_srep19,zheng_lp_pa21}, donation game \\cite{nowak2006evolutionary,li_xy_epjb20,yang_hx_pa20}, or recently proposed involution game \\cite{wang2021replicator,wang2022modeling,wang_cq_amc22}, including the $N$-person versions of these games \\cite{zheng2007cooperative,souza2009evolution,li_k_csf21,pacheco2009evolutionary,luo2021evolutionary,chen_w_srep17,chica_cnsns19}. The common feature of all related situations, which are captured by the mentioned mathematical games, is that cooperation would provide the highest collective benefit, but players can gain more individually if they decide to defect and exploit others \\cite{sigmund2010calculus,hilbe_n18,couto_njp22}. However, from the viewpoint of our present work, the most important version is the so-called public goods game (PGG) played by several players simultaneously \\cite{perc2013evolutionary,alvarez2021evolutionary,lv_amc22,kang_hw_pla21,shen_y_pla22,liu_lj_rspa22}. In this framework, group members decide independently whether they want to contribute to a common pool or not. The accumulated contributions are then enhanced by an $r>1$ productivity factor which expresses the synergy among collaborators: their collective efforts are more effective than a simple sum of their contributions. Finally, the enhanced sum is distributed among all group members equally regardless of whether they contributed to the common pool earlier. It is worth noting that the original public good dilemma can be framed from different aspects: we may establish or maintain a common pool where both goals generate the same individual dilemma \\cite{gaechter17}.\n\nIn the original PGG, the contribution, which involves direct negative income to a participant, is optional, while the benefit is universal for all group members. From this viewpoint, the situation we described earlier is the opposite: consumption, which means a positive personal income, is optional while the contribution to the resulting cost is mandatory. Put differently, in a PGG the negative part is optional and the positive part is universal. In contrast, in our present case the positive part is optional and the negative part is universal. This is why we can call this setup a {\\it reversed} public goods game (R-PGG) where cooperation means to consume the minimal from the common resource, while a defector chooses to consume the pool extravagantly and wastefully. \n\nIn this work, we explore the proposed R-PGG model and reveal its relation to the original PGG. We therefore study the equilibrium and evolutionary behavior of competing strategies and investigate how it is possible to reduce the undesired overuse of common resources. Evidently, this is the core question of evolutionary game theory which targets to understand the evolution of cooperation among self-interested competitors \\cite{nowak2004emergence,quan_j_pa21,su2018understanding,fu_mj_pa21,ohdaira_srep22}.\nBesides a well-mixed population, studied by replicator dynamics \\cite{schuster1983replicator,duong_dga20,wang_q_amc18,liang_rh_pre22}, we also consider structured populations, which could be a decisive factor in how evolution proceeds \\cite{nowak1992evolutionary,ohtsuki2006simple,perc2017statistical,allen2017evolutionary,flores_jtb21,quan_j_c19,wei_x_epjb21}. Notably, we not only check the consequences when players are arranged on a lattice topology, but more realistic topologies, including small-world and scale-free features are also considered.\n\nMoreover, we also assume more complex conditions, including the heterogeneity of key parameters, and explore their impacts on the evolutionary outcome. Indeed, heterogeneity in various forms is not simply a more realistic approach, but could also be a decisive factor in how cooperation evolves \\cite{wang_cq_amc22,perc_pre08,santos_n08}. For the public goods game, the heterogeneity of allocation \\cite{lei2010heterogeneity,szolnoki_amc20,lee_hw_pa21}, productivity \\cite{liu_jz_epjb21,rong_zh_c19,deng_ys_pa21}, input \\cite{yuan2014role,huang2015effect,weng2021heterogeneous,ma2021effect}, and their combinations have been studied intensively \\cite{zhang2017impact,fan2017promotion,liu2021effects}. In particular, Hauser {\\it et al.} \\cite{hauser2019social} systematically analyzed public goods games among unequals by equilibrium calculation, evolutionary analysis, and human experiments, where heterogeneous game parameters depict unequal individuals. While heterogeneity can sometimes be a cooperator-supporting condition, it could also lead to inequality \\cite{mcavoy2020social}.\n\nThe remaining of our paper is organized as follows. In the next section we define the R-PGG model and discuss its extension to heterogeneous situations. Section~\\ref{equivalence} compares the original and reverse homogeneous models for well-mixed and spatial structured populations where their equivalence is demonstrated. In Sec.~\\ref{difference} we present the heterogeneous case where the surprising difference between the models is revealed. We here also present the mechanisms which explain the equivalence and difference between their behaviors. In the last section we discuss our findings and provide some potential applications to alternative real-life situations. Our arguments are greatly supported by an intensive Appendix where all complementary observations are presented.\n\n\\section{Model}\n\\label{model}\n\nWe here define the R-PGG model in analogy with the original PGG. First, we consider a specific case, where players are homogeneous (Sec.~\\ref{homomodel}), then we extend our model to a more general case, where the players are not necessarily equal (Sec.~\\ref{hetemodel}).\n\n\\subsection{The underlying model}\n\\label{homomodel}\n\nSimilarly to PGG, in an R-PGG a group is formed by $g$ members. A player is free to require goods, but all involved members should cover the emerging cost of the goods equally. For simplicity, we assume binary strategies: (i) not requiring goods (cooperation, $C$); (ii) requiring goods (defection, $D$). Later we explain why these strategies can be considered as cooperation or defection. If a player defects and requires goods, she receives goods valued $m$ ($m>0$). Otherwise, she receives nothing. When all group members made their independent decisions about their consumptions, each player equally pays a cost to cover the goods required by the whole group. The payoff $\\pi_C$ and $\\pi_D$ for a cooperative or for a defective player can be calculated as \n\\begin{subequations}\\label{payoffCD}\n\t\\begin{align}\n\t\t\\pi_C&=-\\frac{g_D m\\lambda}{g} \\label{payoffC}\\\\\n\t\t\\pi_D&=m-\\frac{(g_D+1) m\\lambda}{g} \\label{payoffD}\\,,\n\t\\end{align}\n\\end{subequations}\nwhere there are $g_D$ other defectors among the neighbors of the focal player. Here $\\lambda > 1$ parameter denotes the waste factor associated with requiring public goods. The larger the waste factor, the greater the loss caused by the consumption. If  $\\lambda<g$ then requiring goods always brings positive income to a defective individual. Therefore, to meet a proper dilemma, we assume that $1<\\lambda<g$. \n\n\\subsection{The extension to heterogeneous players}\n\\label{hetemodel}\n\nMore generally, we can consider heterogeneous $m$ and $\\lambda$ game parameters for different players. A player $i$ ($i=1,2,\\dots,g$) in a group may require goods valued $m_i$ ($m_i>0$), with its waste factor $\\lambda _i$. In the extended version, we introduce a continuous strategy space, hence the strategy of player $i$ can be denoted by a proportion $y_i$ ($0\\leq y_i\\leq 1$) of goods $m_i$, where $y_i=0$ means full cooperation and $y_i=1$ is full defection. If not mentioned otherwise, we use the pure strategy set, $y_i \\in \\{ 0,1\\}$, in this work. The payoff $\\pi_i$ of player $i$ can be calculated as\n\\begin{equation} \\label{unequalpayoff}\n\t\\pi_i=y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\,,\n\\end{equation}\nwhere $j$ goes through every player in the group, including player $i$. \n\nIn this extension, $m_i$ measures the full consumption capacity of player $i$. In real-world scenarios, an individual's needs and capacity may vary. Secondly, $\\lambda_i$ characterizes the ``effectiveness\" of player $i$ in spoiling the required public goods. Note that players with the same consumption capacity and strategy may generate different final cost values due to individual efficiency.\n\n\\section{Equivalence between R-PGG \\& PGG}\n\\label{equivalence}\n\nWe first demonstrate that the proposed R-PGG model is equivalent to the PGG in many aspects. The static analysis shows the equivalence in general homogeneous cases. From the perspective of evolutionary dynamics, the equivalence of the underlying homogeneous model also holds both in well-mixed and structured populations.\n\n\\subsection{Static analysis among unequals}\n\\label{unequals}\n\nHauser {\\it et al.}~\\cite{hauser2019social} proposed a general framework for the PGG among unequal participants (i.e., players with heterogeneous game parameters). We here show that our R-PGG model fits this general framework. In the following, we perform an equilibrium analysis.\n\nAccording to the general heterogeneous case defined by Eq.~(\\ref{unequalpayoff}), we denote the requirement vector by $\\mathbf{y}=(y_1, y_2, ..., y_g)$ in a group of $g$ players, and assume that $\\mathbf{y}>\\mathbf{y}'$ if $y_i>y'_i$ for all $i \\in \\{1,\\dots g\\}$ group members. We also denote the full consumption or desire profile by vector $\\mathbf{m}=(m_1, m_2,\\dots, m_g)$ and the waste vector $\\boldsymbol{\\lambda}=(\\lambda_1, \\lambda_2,\\dots, \\lambda_g)$ for the actual group. These $\\mathbf{m}$, $\\boldsymbol{\\lambda}$, and $\\mathbf{y}$ vectors determine the payoff values $\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\in \\mathbb R^g$ \nfor all group members. \nIn addition, we denote the accumulated group desire or maximal group consumption by $M\\coloneqq \\sum_{i=1}^g m_i$, and the total payoff of all involved players by $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\coloneqq \\sum_{i=1}^g \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})$.\n\nIn Ref.~\\cite{hauser2019social}, the authors introduced a so-called ``cooperation vector\", denoted by $\\mathbf{x}=(x_1, x_2,\\dots, x_g)$, which characterizes the cooperation profile of all involved participants. In our R-PGG model, consumption is the focus, therefore it is appropriate to use a ``defection vector\" which practically determines the individual aims to consume their maximal capacities via $\\mathbf{y}=(y_1, y_2, \\dots, y_g)$ requirement vector. Since cooperation and defection complement each other, the sum of these vectors is always constant and gives a unit value for every $i$ component. Namely, $\\mathbf{y}+\\mathbf{x}=\\mathbf{1}$.\nIn strong analogy with the heterogeneous PGG model~\\cite{hauser2019social}, the following properties are valid here, which describe the conflict between the collective and individual incentives on how to utilize public resources:\n\\begin{itemize}\n\t\\item\n\t\\textbf{Continuity:} \n\tThe payoff function $\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})$ is continuous in both the arguments $\\mathbf{m}$ and $\\mathbf{y}$.\n\t\\item \n\t\\textbf{Negative externalities:}\n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requirement vectors such that $y_i=y'_i$ and $y_j\\geq y'_j$ for $\\forall j\\neq i$, then $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\leq \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ for $\\forall \\mathbf{m}$. The inequality $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})< \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff $\\exists j$ such that $m_j>0, y_j>y'_j$.\n\t\\item \n\t\\textbf{Incentives to free-require:} \n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requirement vectors such that $y_i>y'_i$ and $y_j=y'_j$ for $\\forall j\\neq i$, then $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\geq \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ for $\\forall \\mathbf{m}$. The inequality $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})> \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff the required goods are positive, $m_i>0$.\n\t\\item \n\t\\textbf{Non-optimality of defection:}\n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requiring vectors such that $\\mathbf{y}\\leq \\mathbf{y}'$, then $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\geq U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$. The inequality $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})> U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff $\\exists i$ such that $m_i>0, y_{i}<y'_i$.\t\n\\end{itemize}\n\nThe so-called Grim player $i$ cooperates in the initial round ($y_i=0$). Then, if all players cooperate as well, player $i$ keeps cooperating. Otherwise, if any player defects in previous rounds, player $i$ defects and keep defecting in the future ($y_i=1$) \\cite{sigmund2010calculus}. In anology to Ref.~\\cite{hauser2019social}, when all players are Grim players, it is a subgame perfect equilibrium for the given requirement distribution, which means that full cooperation can be sustained among Grim players. Given that all players follow Grim strategy, the following condition must hold for all players $i$ to sustain cooperation:\n\\begin{equation}\\label{grimcondi}\n\t\\delta(\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{0}_{+i})-\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{1}))\\geq \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{0}_{+i})-\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{0})\\,,\n\\end{equation}\nwhere $\\delta$ is the probability of another round. Here vector $\\mathbf{1}$ means all players defect, $\\mathbf{0}$ means all players cooperate, and $\\mathbf{0}_{+i}$ means that all players cooperate but player $i$ defects. According to Eq.~(\\ref{grimcondi}), the expected benefit from the future cooperation of others must exceed the incentive to defect in the current round.\n\nBased on Eq.~(\\ref{unequalpayoff}), the individual payoff value is\n\\begin{eqnarray}\\label{rpgggrim}\n\t\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})&=-\\frac{1}{g}\\sum_{j} y_j m_j \\lambda_j+m_i y_i \\nonumber\\\\\n\t&=-\\frac{1}{g}\\sum_{j\\neq i}y_j m_j \\lambda_j +m_i(1-\\frac{\\lambda_i}{g})y_i \\,.\n\\end{eqnarray}\nTherefore, \n\\begin{align}\\label{rpgguvalue}\n\\begin{cases} \n\t\\displaystyle{\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{1})\\,\\,\\,\\,\\,=-\\frac{1}{g}\\sum_{j\\neq i}m_j\\lambda_j +m_i(1-\\frac{\\lambda_i}{g})}\\\\\n\t\\displaystyle{\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{0})\\,\\,\\,\\,\\,=0}\\\\\n\t\\displaystyle{\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{0}_{+i})=m_i(1-\\frac{\\lambda_i}{g})}\\,.\n\\end{cases}\n\\end{align}\nBy substituting Eqs.~(\\ref{rpgguvalue}) into Eq.~(\\ref{grimcondi}), the necessary and sufficient condition for the feasibility of cooperation for all involved player $i$ is\n\\begin{equation}\\label{rpgggrimcondi}\n\t\\frac{\\delta}{g}\\sum_{j\\neq i}m_j \\lambda_j \\geq m_i (1-\\frac{\\lambda_i}{g})\\,.\n\\end{equation}\n\nLike the PGG, repeated games (indicated by a larger $\\delta$) promote cooperation. For example, according to Eq.~(\\ref{rpgggrimcondi}), given equal desire $\\mathbf{m}=(M\/g,\\dots,M\/g)$, full cooperation is feasible iff\n\\begin{equation}\\label{rpggdeltacondi}\n\t\\delta\\geq \\frac{g-\\min{(\\lambda_j)}}{\\sum_{j=1}^{g}\\lambda_j-\\min{(\\lambda_j})}\\,.\n\\end{equation}\n\nWe can deduce the optimal requirement vector $\\mathbf{m}$, maximally cooperative when requiring the lowest continuation probability $\\delta$ for cooperation to be feasible. In particular, if we consider only two players, then full cooperation is feasible iff  \n\\begin{equation}\\label{rpggtwoplayercondi}\n\t\\delta\\geq \\max{\\left(\\frac{m_1(2-\\lambda_1)}{m_2\\lambda_2},\\frac{m_2(2-\\lambda_2)}{m_1\\lambda_1}\\right)}\\,.\n\\end{equation}\nTo minimize the right-hand side of Eq.~(\\ref{rpggtwoplayercondi}), we have the condition\n\\begin{equation}\\label{rpggmaxcondi}\n\t\\frac{m_1}{m_2}=\\sqrt{\\frac{\\lambda_2(2-\\lambda_2)}{\\lambda_1(2-\\lambda_1)}}\\,,\n\\end{equation}\nwhich maximizes cooperation.\n\nWe can see that in the conditions of Eq.~(\\ref{rpgggrimcondi}), Eq.~(\\ref{rpggdeltacondi}), and Eq.~(\\ref{rpggmaxcondi}), the parameter $\\lambda$ plays the same role as $r$ in PGG, and parameter $m$ plays the same role as $c$ in PGG (see Ref.~\\cite{hauser2019social} for the corresponding conditions in PGG and note that they denote the contribution $c$ by endowment $e$). In other words, R-PGG is equivalent to PGG in the framework of social dilemmas among unequals.\n\n\\subsection{Evolutionary dynamics in a well-mixed population}\n\\label{unstruc}\n\nThe evolutionary game theory focuses on how cooperation can evolve via a dynamic process. Unlike the traditional game theory, discussed in Sec.~\\ref{unequals}, here we adopt the simplest strategy set (a player employs unconditional cooperation or defection) \\cite{nowak1992evolutionary}.\nIn particular, according to microscopic dynamics, a strategy with a higher payoff value can be reproduced with a higher probability via a certain dynamical rule, including, but not limited to death-birth \\cite{ohtsuki2006simple}, birth-death \\cite{ohtsuki2006simple}, pairwise comparison \\cite{szabo1998evolutionary}, and imitation \\cite{ohtsuki2006simple} rules. For simplicity, we here apply the pairwise comparison dynamics with Fermi-function-type probability. Accordingly, player $i$ with payoff $\\pi_i$ adopts the strategy of player $i'$ who has payoff $\\pi_{i'}$ with the probability\n\\begin{equation}\\label{fermi}\n\tQ(\\pi_i \\gets \\pi_{i'})=\\dfrac{1}{1+\\e^{-\\omega(\\pi_{i'}-\\pi_i)}}\\,.\n\\end{equation}\nHere, $\\omega$ ($\\omega>0$) is considered as the strength of selection. A stronger selection strength $\\omega$ means that players are more sensitive to the payoff difference. The higher the payoff of player ${i'}$, the higher probability is that player $i$ imitates the strategy of player ${i'}$.\n\nWithout loss of generality, we first study the evolutionary dynamics of R-PGG in a well-mixed population of homogeneous players. \nIn Appendix~A1 we first apply deterministic dynamics and find two solutions to the replicator equation. They are full defector and full cooperator states. Their stability depends on the sign of the payoff difference between the competing strategies. To compare the results with the PGG case, we obtain equivalent expressions if we replace the original $r$ and $c$ parameters with $\\lambda$ and $m$. Hence the homogeneous R-PGG is equivalent to homogeneous PGG in the framework of replicator dynamics in an infinite and unstructured population.\n\nWe can also consider stochastic dynamics in a finite population of size $N$, as specified in Appendix~A2. By calculating the expected payoff values for both strategies, the transition matrix of the Markov process can be determined. Similarly to the above-discussed approach, the process has two absorbing states. In the $\\omega \\to 0$ weak selection limit, the fixation probability agrees with the one obtained for the original PGG if we replace $\\lambda$ and $m$ parameters with $r$ and $c$. Therefore, we can conclude that homogeneous R-PGG \\& PGG are also equivalent in the framework of stochastic dynamics in a finite and unstructured population.\n\n\\subsection{Evolutionary dynamics in structured populations (the homogeneous case)}\n\\label{struc}\n\nNow, we consider R-PGG on a graph $\\mathcal{G}=(\\mathcal{N},\\mathcal{L})$ of a finite and structured population $N$. The graph $\\mathcal{G}$ is unweighted and undirected. $\\mathcal{N}$ is the node set of the graph, $\\mathcal{N}=\\{1,2,\\dots,N\\}$ and each node represents a player. $\\mathcal{L}$ is the link set of the graph, $\\mathcal{L}=\\{l_1,l_2,\\dots,l_{|\\mathcal{L}|}\\}$. We denote by $l_{i_1 i_2}$ a link with order 2, connecting two nodes $i_1$ and $i_2$. \nThe size of $\\mathcal{L}$ varies with specific graphs (e.g., for a complete graph, the size of $\\mathcal{L}$ is $|\\mathcal{L}|=N(N-1)\/2$). The degree of node $i$ (i.e., the number of link(s) containing $i$) is denoted by $k_i$. Self-loop is not allowed.\n\nAt each elementary Monte Carlo (MC) step, a random node $i\\in \\mathcal{N}$ is selected. If $k_i=0$, nothing happens. If $k_i\\neq 0$, then a random link $l_{ii'}$ is selected between the focal player $i$ and a neighbor $i'$. We then calculate their payoff values. Player $i$ participates in $k_i+1$ R-PGGs centered on herself and all her neighbor(s). First, the contribution is calculated from the $g_i=k_i+1$-size group centered on player $i$. After, we calculate the income from each group centered on an $i$'s neighbor(s). Last, player $i$ accumulates these  payoff values to reach the total sum\n\\begin{align}\\label{graphpayoff}\n\t\\pi_i=&~\\frac{1}{k_i+1}\\left[ y_im_i-\\frac{1}{g_i}\\left(y_im_i\\lambda_i+\\sum_{l_{ij_1}\\in\\mathcal{L}}y_{j_1}m_{j_1}\\lambda_{j_1}\\right)\\right.\\nonumber\n\t\\\\\n\t&\\left.+\\sum_{l_{ij_1}\\in \\mathcal{L}}\\left(y_i m_i-\\frac{1}{g_{j_1}}\\left(y_im_i\\lambda_i+\\sum_{l_{j_1j_2}\\in \\mathcal{L}} y_{j_2} m_{j_2}\\lambda_{j_2}\\right)\\right)\\right]\\,.\n\\end{align}\n\nThe $\\pi_{i'}$  payoff of player $i'$ can be calculated in a similar way. Then, player $i$ adopts the strategy of model player $i'$ with the probability $Q(\\pi_{i}\\gets\\pi_{i'})$ defined by Eq.~\\ref{fermi}. Each full time step contains $N$ elementary MC steps specified above.\n\nIn the weak selection limit, we can determine the $\\lambda^\\star$ threshold value for the success of cooperation through the identity-by-descent method introduced by Allen and Nowak \\cite{allen2014games}. The details can be found in \nAppendix~B. This calculation suggests that the threshold $\\lambda^\\star$ in R-PGG is exactly the same as $r^\\star$ for the original PGG (see Refs.~\\cite{su2018understanding,su2019spatial} for the deduction of $r^\\star$ in the original PGG). In other words, homogeneous R-PGG \\& PGG predict equivalent threshold values for the emergence of cooperation in the weak selection limit.\n\nFor non-transitive graphs, which may lead to varying $k_i$ degree among different players $i$, the explicit threshold of cooperation emergence in multiplayer games remains unsolved (only the results in two-player games have been solved \\cite{allen2017evolutionary}). In this case, we may use the average degree $\\langle k\\rangle=(\\sum_{i=1}^N k_i)\/N$ to estimate the threshold value. In this case, the average number of players in each group is $\\langle g\\rangle =\\langle k\\rangle+1$.\n\n\\begin{itemize}\n\t\\item \n\t\\textbf{Monte Carlo simulations}\n\\end{itemize}\n\nTo extend our study, we also apply numerical simulations which can be done for arbitrary selection strength. Now we choose $\\omega=10$ (i.e., $1\/\\omega=0.1$) and $N=40000$. This intermediate level of selection strength was frequently applied by previous works \\cite{lv_amc22,kang_hw_pla21,shen_y_pla22,deng_ys_pa21,szolnoki_pre09c,gracia-lazaro_pre14,yang_hx_pa19,meloni_rsos17,szolnoki_epl10}.\nAt the beginning, we randomly assign each agent's strategy by cooperation or defection. Then, we simulate for $15000$ MC steps, and measure the  $\\langle x \\rangle$ stationary portion of cooperators. This can be done by averaging $x_i$ values over the last $5000$ steps of the simulations.\n\nTo cover all significantly different interaction topologies, we apply both regular and heterogeneous graphs. In the former case we use an $L \\times L$ square lattice with periodic boundary conditions where players have four nearest neighbors, hence they form 5-member groups. The typical linear system size was $L=200$. Furthermore, we also use  Watts-Strogatz-type (WS) small-world topology where the graph is generated from a ring of $N$ nodes having $k=4$ bonds for each player \\cite{watts1998collective}. Then, each link is rewired randomly with probability $p=0.5$, where self-loop and double links are forbidden. Finally, we also use Barab{\\'a}si-Albert-type (BA) scale-free graphs where the graph is generated starting from a 3-node complete graph and new nodes are attached to the network with two new links \\cite{barabasi1999emergence}. In this way we keep $\\langle k\\rangle=4$ average degree, which makes the results comparable for all cases.\n\nOur observations are summarized in Fig.~\\ref{graph r} where we present the $\\langle x\\rangle$ cooperation level both for PGG and R-PGG models. Evidently, as we pointed out earlier, parameters $r$ and $\\lambda$ serve as control parameters in the former and latter cases, respectively. The overlap between the results of the models is convincing. Evidently, there are differences between the applied topologies, but both PGG and R-PGG change identically. A similar conclusion can be drawn about the influence of cost $c$ and consumption $m$ parameters. If one can require more goods (by increasing $m$), then the cooperation proportion is more sensitive to the waste factor (i.e., increasing with $\\lambda$ more gradually). On the one hand, this leads to the emergence of cooperation, $\\langle x\\rangle>0$, at a smaller $\\lambda$ value. On the other hand, it also leads to the dominance of cooperation, $\\langle x\\rangle=1$, at a greater $\\lambda$. The latter effect is more evident on the small-world graph than on the square lattice graph, and is most evident on the scale-free graph.\n\\begin{figure}[h!]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\\includegraphics[width=14.5cm]{Fig1.eps}}\\\\\n\t\\caption{Cooperation level as a function of normalized synergy factor $r\/\\langle g\\rangle$ and waste factor $\\lambda\/\\langle g\\rangle$. Panels show the results on different graphs, as indicated in the titles. In addition, we mark by arrows the threshold level of the control parameter favoring cooperation under weak selection.}\\label{graph r}\n\\end{figure}\n\nAccording to Eq.~(\\ref{graphpayoff}), $m$ can be extracted as a common factor in every $\\pi_i$ for the homogeneous case. Then, $m$ plays the same role as $\\omega$ in the Fermi-function defined by Eq.~(\\ref{fermi}). In other words, the selection strength is also directly measured by $m$. The same inference holds in PGG, where the selection strength can be measured by $c$. Taking $c=m=10^{-2}$, we have the results valid under a weak selection in numerical simulations. Using $N=40000$ and $k=4$ in Eq.~(\\ref{emergelimit}), we have $\\lambda^\\star\\approx 5$ or $\\lambda^\\star\/\\langle g\\rangle \\approx 1$. This weak selection threshold is consistent with the cases of $c=m=10^{-2}$ in Fig.~\\ref{graph r}.\n\nFor completeness, we also measured how the cooperation level changes by varying the $c$ contribution (in PGG) or $m$ requirement (in R-PGG). Our results are shown in Fig.~\\ref{graph c} in Appendix~C for different topologies. The comparison supports our original observation about the equivalence of homogeneous PGG and R-PGG, no matter whether we consider spatially structured populations.\n\nSumming up, both $r$ $\\&$ $\\lambda$ and $c$ $\\&$ $m$ dependencies demonstrate nicely that R-PGG and PGG models are equivalent if applying the appropriate $\\lambda-m$ or $r-c$ parameter pairs, no matter we have unstructured or structured populations with different interaction topologies or how intensive the selection strength is. The only crucial condition, as we will demonstrate in the next section, is that the population should be homogeneous where all players can be characterized by the same parameter value of the game.\n\n\\section{Difference between R-PGG \\& PGG}\n\\label{difference}\n\nOur abovementioned conclusion becomes invalid if the population is more realistic and players are not uniform anymore. Because of realistic conditions, we here focus on structured populations where interactions are limited and fixedpermanent. The introduction of heterogeneity reveals the difference between PGG and R-PGG models, as we illustrate in Sec.~\\ref{hetestruc}. By means of the translation of payoff functions we will argue that R-PGG is not a simple transformation of  the original PGG, but instead a reversed form of it. \n\n\\subsection{Evolutionary dynamics in structured populations (the heterogeneous case)}\n\\label{hetestruc}\n\nTo generalize our model, we here assume that players can be heterogeneous hence they may not participate in the same way in the game. This can be done by introducing player-specific parameter values. Accordingly, we assume uniform random distribution for each player's parameters $\\lambda_i$ and $m_i$. For each player $i$, we randomly generate two numbers between 0 and 1: $\\chi_i^{(\\lambda)}\\in [0,1)$, $\\chi_i^{(m)}\\in [0,1)$. Then, we set the parameters $\\lambda_i$ and $m_i$ for player $i$ as\n\\begin{subequations}\\label{etalambda}\n\t\\begin{align}\n\t\\lambda_i&=\\lambda+(-2\\chi_i^{(\\lambda)}+1)\\eta_{\\lambda}\\,,\\\\\n\tm_i&=m+(-2\\chi_i^{(m)}+1)\\eta_m\\,,\n\t\\end{align}\n\\end{subequations}\nwhere $\\eta_{\\lambda}$ ($\\eta_{\\lambda}\\geq 0$) measures the heterogeneity of the waste factor $\\lambda_i$, and $\\eta_m$ ($\\eta_m\\geq 0$) measures the heterogeneity of the requirement $m_i$. In this way, $\\lambda_i$ and $m_i$ are selected randomly from $[\\lambda-\\eta_{\\lambda},\\lambda+\\eta_{\\lambda})$ and $[m-\\eta_m,m+\\eta_m)$ intervals where $\\lambda$ and $m$ values act as the ``baselines\": the expected waste factor and requirement over all players are still $\\int_{0}^{1}\\lambda_i~\\mathrm{d}\\chi_i^{(\\lambda)}=\\lambda$ and $\\int_{0}^{1}m_i~\\mathrm{d}\\chi_i^{(m)}=m$. In addition, by taking $\\eta_{\\lambda}=0$, $\\eta_m=0$, we return to the homogeneous case where $\\lambda_i=\\lambda$ and $m_i=m$ for all players.\n\nFor PGG, we introduce the heterogeneity in the same way. Here, the two key parameters are $\\eta_r$ and $\\eta_c$, which make $r$ and $c$ player-specific in the extended case. As we demonstrated in the previous section, $\\lambda$ \\& $r$, furthermore $m$ \\& $c$ parameters are identical in the original and reversed games, therefore the $\\eta_\\lambda$ \\& $\\eta_r$ and $\\eta_m$ \\& $\\eta_c$ amplitudes have similar roles in characterizing heterogeneity in the extended versions. \n\nOur key findings are summarized in Fig.~\\ref{graph hetero r}, where we plot the $\\langle x\\rangle$ cooperation level in dependence of $\\eta_r$ or $\\eta_\\lambda$. While they are still equivalent when $\\eta_\\lambda=\\eta_r=0$ (in the homogeneous case), their difference becomes striking as we increase the heterogeneity of the systems. In agreement with previous observations about the principal role of heterogeneity \\cite{perc_pre08,santos_n08}, the cooperation level increases as we increase $\\eta_r$ in PGG for all interaction graphs. But the opposite is true for R-PGG where the extension to a heterogeneous population has the reversed consequence for the cooperation level. While increasing the heterogeneity in the synergy factor usually promotes cooperation in PGG, the opposite is observed for unequal waste factors in R-PGG. This is generally true independently of the applied topology or the dilemma strength.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=14.5cm]{Fig2.eps}\\\\\n\t\\caption{The cooperation level in dependence on the heterogeneity of the synergy factor or the waste factor in structured populations. Panels show different topologies, as indicated. When $\\eta_r=0$ \\& $\\eta_\\lambda=0$, PGG \\& R-PGG are equivalent. But their difference becomes crucial as we increase $\\eta_r$ and in parallel $\\eta_\\lambda$. We fix $c=m=1$.}\\label{graph hetero r}\n\\end{figure}\n\nOur observations can be explained as follows. In PGG, those with a higher synergy factor tend to have a higher payoff, for themselves and their co-players, which leads to an environment beneficial to cooperation. The strategy of cooperation, hence, is favored to reproduce. In R-PGG, however, if we transform the payoff function (see Eq.~(\\ref{rpgg1}) for details), we can observe a redundant item $-m\\lambda$ compared with the payoff function of PGG. Originally, those with a higher waste factor in R-PGG could have the chance of a higher payoff. Nevertheless, the item $-m\\lambda$ tends to bring them a lower payoff. As a result, the same effect in PGG is inhibited in R-PGG, and the heterogeneity of the waste factor usually hinders cooperation in R-PGG.\n\nEvidently, we can introduce heterogeneity in an alternative way when a cooperator's contribution (in PGG) or a defector's requirement (in R-PGG) becomes player-specific. Accordingly, we use individual $c_i$ or $m_i$ values as it is introduced in Eq.~\\ref{etalambda}b. Our results are summarized in Fig.~\\ref{graph hetero c} of Appendix~D. The comparison of PGG and R-PGG confirms what we found previously. Namely, these systems show similar cooperation levels only if the heterogeneity is mild. But their difference becomes striking if we increase the amplitude of $\\eta_c$ and $\\eta_m$.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13.5cm]{Fig3.eps}\\\\\n\t\\caption{(a) The heat map of cooperation level on the parameter plane of normalized synergy factor $r\/g$ and heterogeneous $r_i$. (b) The same plot in dependence of normalized waste factor $\\lambda\/g$ and heterogeneous $\\lambda_i$. The difference between PGG and R-PGG is more striking at low $r$ and low $\\lambda$ region when there is a proper social dilemma and less noticeable for high $r$ and high $\\lambda$ values. But it still exists if the individual heterogeneity is large enough. The simulations are performed on square lattice topology. We fix $c=m=1$.}\\label{graph r hetero r}\n\\end{figure}\n\nA more comprehensive overview of the system behavior can be obtained if we plot the cooperation level not just at a specific $r$ or $\\lambda$, but for arbitrarily large average values. For simplicity, we only show the results obtained on square lattice topology. The results can be seen in Fig.~\\ref{graph r hetero r} where we present the cooperation level as a heat map on a two-dimensional parameter plane. Notably, Fig.~\\ref{graph hetero r} can be considered a cross-section of this new diagram. Our first qualitative observation is the ``red'' area, which represents the highest cooperation level in the population, is significantly larger for PGG than for R-PGG, which highlights that heterogeneity is generally more beneficial for the PGG system. If we compare the heat maps more carefully then we can see that the difference between PGG and R-PGG is more striking in the low $r$ and low $\\lambda$ region when there is a proper dilemma situation. In this parameter region, even minimal cooperation is challenging for the R-PGG model if the average $\\lambda$ value remains below $\\lambda\/g \\approx 0.75$. For large $r$ and large $\\lambda$ values the difference is less visible between the models. But it still exists if the amplitude of individual heterogeneity is large enough. Hence, we can generally conclude that the heterogeneous synergy factor in PGG can promote cooperation while the heterogeneous waste factor in R-PGG cannot.\n\nWe have also compared the cooperation levels by means of heat maps when heterogeneity was introduced via individual $c_i$ and $m_i$ values. The results are shown in Fig.~\\ref{graph r hetero c} in Appendix~D. The difference between PGG and R-PGG can also be revealed, showing that heterogeneity is a general condition to make a difference between these games. Summing up, we can conclude that introducing the same level of heterogeneity has significantly different consequences on PGG and R-PGG systems. This effect is robust and remains valid no matter how we introduce the distinction among players.\n\n\\subsection{Discussion on the translation of payoff function}\n\\label{translation}\n\nTo explain why R-PGG and PGG become different when heterogeneous players are present, we study the translation of payoff functions. \nAs Equation~(\\ref{payoffCD}) describes, a focal player's income from a group venture can always be expressed as a function of $g_C$ which is the number of cooperators among group neighbors. In the following, we consider four different translations of PGG. The details of these games are as follows.\n\\begin{itemize}\n\\item (i) {\\bf PGG}\nIn the most well-known and widely accepted version of PGG a cooperator contributes a $c$ amount to the common pool, while a defector player does not. The sum of contributions is enlarged and distributed among all group members. The corresponding payoff values of the focal player playing either $C$ or $D$ in the group are\n\\begin{align}\\label{pgg1}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\\pi_C=\\frac{(g_C+1) rc}{g}-c}\\\\\n\t\t\\displaystyle{\n\t\t\\pi_D=\\frac{g_C rc}{g} } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (ii) {\\bf Alternative PGG}\nIn an alternative form, all players have an initial endowment $c$ \\cite{hauser2019social}. A cooperator player invests this whole amount to the common pool, while a defector player keeps it. As previously, the contributions are summed, enhanced and distributed among all competitors. The resulting payoff values are \n\\begin{align}\\label{pgg2}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=\\frac{(g_C+1) rc}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=\\frac{g_C rc}{g}+c } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (iii) {\\bf R-PGG}\nAccording to our proposal, in an R-PGG a defector requires $m$ from a common resource, while a cooperator player does not. The sum of required goods is wasted at a certain rate, which means an enlarged, and equally shared cost to everyone in the group, yielding the payoff values \n\\begin{align}\\label{rpgg1}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=-\\frac{g_D m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m-\\frac{(g_D+1) m\\lambda}{g} }\n\t\\end{cases}\n\t\t\\stackrel{g_C+g_D=g-1}{\\Longleftrightarrow}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=-m\\lambda +\\frac{(g_C+1) m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=-m\\lambda +m+\\frac{g_C m\\lambda}{g} } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (iv) {\\bf Alternative R-PGG}\nIn the alternative version, each player has an initial $m\\lambda$ deposit. Additionally, a defector requires $m$ from a common resource, and related (enlarged) cost is shared among everyone in the group. Therefore, the payoff values are \n\\begin{align}\\label{rpgg2}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=m\\lambda-\\frac{g_D m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m\\lambda+m-\\frac{(g_D+1) m\\lambda}{g} }\n\t\\end{cases}\n\t\\stackrel{g_C+g_D=g-1}{\\Longleftrightarrow}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=\\frac{(g_C+1) m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m+\\frac{g_C m\\lambda}{g} }\\,.\n\t\\end{cases}\n\\end{align}\n\\end{itemize}\nNote that in Eq.~(\\ref{rpgg1}) and in Eq.~(\\ref{rpgg2}) we have used $g_D=g-g_C-1$ notation to obtain a consistent $g_C$-dependent form of payoff values for all cases. The connections between the above-specified cases are summarized in Fig.~\\ref{figureasymmetric}, where the curves represent the payoff of a focal player employing either cooperation or defection.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13cm]{Fig4.eps}\\\\\n\t\\caption{Payoff value for a focal cooperator or defector player as a function of $g_C$, denoting the number of cooperators among neighbors. The four panels show the cases as indicated in the titles. They are equivalent in the payoff gap because the translational items in $\\pi_D-\\pi_C$ can be subtracted. They are different in $\\pi_C(g_C=0)$, $\\pi_C(g_C=g-1)$, $\\pi_D(g_C=0)$, and $\\pi_D(g_C=g-1)$, which means that their translational items in $\\pi_D-\\pi_C$ cannot be subtracted when parameters become heterogeneous. The green arrows indicate that the given game models are equivalent in the homogeneous case. The red arrows mean the two models are different in the heterogeneous case. The yellow arrow means the payoff functions of the models are seemingly equivalent, but later we will show they are actually different. $g=5$, $r=\\lambda=1.5$, and $c=m=1$ parameters were used.}\\label{figureasymmetric}\n\\end{figure}\n\nThe key assumption, which explains the equivalence, is the linear dependence on the $g_C$ value. We can see that with the same parameters, the slope of payoff functions is constant everywhere. Also, the difference between $\\pi_D$ and $\\pi_C$ (i.e., $\\pi_D-\\pi_C$) is constant at any $g_C$ value. Therefore, despite different interpretations, we can say that all four translations expressed in panels~(a1),~(a2),~(b1), and (b2) represent conceptually similar public goods games. \n\nThe equivalence in the homogeneous case is clear because the results depend only on $\\pi_{i'}-\\pi_i$. Here the translational items in $\\pi_{i'}-\\pi_i$ can be subtracted, which leads to the equivalence among the four interpretations. \nHowever, the absolute values of payoff functions vary from case to case, as indicated by the value of $\\pi_C(g_C=0)$, $\\pi_C(g_C=g-1)$, $\\pi_D(g_C=0)$, and $\\pi_D(g_C=g-1)$ in Fig.~\\ref{figureasymmetric}. This leads to a significant difference when introducing heterogeneity. As an example, take alternative PGG and original R-PGG versions. We can see that R-PGG has a redundant item $-m\\lambda$ compared with alternative PGG. As mentioned earlier, in a homogeneous case, such an item can be subtracted when calculating $\\pi_{i'}-\\pi_i$ for strategy update. However, in a heterogeneous case, such an item varies from player to player, $-m_i\\lambda_i\\neq -m_{i'}\\lambda_{i'}$, which cannot be subtracted anymore. \n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13cm]{Fig5.eps}\\\\\n\t\\caption{Payoff values for a cooperator (open symbols) and for a defector (solid symbols) player in dependence on the number of other cooperative players in the group. Panels show different models, as indicated. While $g=5$, $c=m=1$ values are fixed, the three curves for each $\\pi_C$ or $\\pi_D$ are plotted by taking $r_i=\\lambda_i=1.5-2$, $1.5$, $1.5+2$ ($r=\\lambda=1.5$, $\\eta_r=\\eta_{\\lambda}=2$). The shaded areas mark the traces of curves when $r_i$ or $\\lambda_i$ changes in the mentioned interval bordered by $r\\pm\\eta_r$ ($\\lambda\\pm\\eta_{\\lambda}$). These areas are marked by different colors for competing strategies (green for $D$ and yellow for $C$). The meaning of color arrows are similar to those we used in Fig.~\\ref{figureasymmetric}.}\\label{figureasymmetric heter}\n\\end{figure}\n\nTo visualize this effect in a more intuitive way, we present Fig.~\\ref{figureasymmetric heter}  which shows $\\pi_C$ and $\\pi_D$ as a function of $g_C$ in the mentioned cases when heterogeneous $r$ or $\\lambda$ are introduced. The curves represent three specific average values of key parameters, but we also mark the location of payoff values for intermediate values by shaded areas. In this way the conceptual difference between different cases becomes clear. Staying at Fig.~\\ref{figureasymmetric heter}, the shaded areas in panel~(a1) and in panel~(b1) are completely different because the difference in the payoff values leads to unequal outcomes in the presence of heterogeneity. \n\nNotably, the shape of the shaded area remains intact if we change PGG, shown in panel Fig.~\\ref{figureasymmetric heter}(a1), to the alternative PGG, presented in panel Fig.~\\ref{figureasymmetric heter}(a2). This is because their coefficients in front of $r$ are equal: $(g_C+1)c\/g$ in $\\pi_C$, and $g_C c\/g$ in $\\pi_D$. Therefore, the same change in $r$ leads to the same change in the payoff function. Although they are still different in their vertical positions, the identical shape confirms their equivalence in the stationary cooperation level (the reader may compare Fig.~\\ref{figurealter r}(a) in Appendix~D and Fig.~\\ref{graph r hetero r}(a)). Furthermore, if we compare the case of alternative PGG (Fig.~\\ref{figureasymmetric heter}(a2)) and the case of alternative R-PGG (Fig.~\\ref{figureasymmetric heter}(b2)) then we can see that they show not only the same shape of shaded areas, but also the same absolute payoff values. Their cooperation levels, however, as we  show in Fig.~\\ref{figurealter r} in Appendix~D, are different. This fact is a direct consequence of the conceptual difference between PGG $\\&$ R-PGG, as we will discuss later.\n\nFor completeness, we also studied the consequence of heterogeneity in $c$ or $m$ parameter on the payoff functions. The results are presented in Fig.~\\ref{figureasymmetric hetec} in Appendix~D. It shows that the same level of change in individual $c$ leads to a varying change in their payoff functions, resulting in different shapes of shaded areas. We, however, stress that alone the payoff functions are not able to reveal all differences among the cases we discussed above. For example, similar shapes of shaded areas can produce unequal heat maps of cooperation levels on the parameter plane, as shown in Appendix~D.\n\nTo give a deeper insight into the origin of varying behavior in heterogeneous games, let us turn back to the payoff functions. When we used the term $g_C=g-g_D-1$ to replace $g_D$ with $g_C$ in R-PGG (as well as in alternative R-PGG), we implicitly assumed homogeneous payoff functions. Therefore, the individual parameter values have no importance. However, if we assume distinct players and consider player-specific payoff values via Eq.~(\\ref{unequalpayoff}), then we can rewrite the payoff function for the heterogeneous cases. For example, the payoff for the original PGG can be written as\n\\begin{equation}\n\t\\pi_i=\\frac{1}{g}\\sum_{j=1}^g x_j c_j r_j-x_i c_i\\,,\n\\end{equation}\nand for the original R-PGG it is\n\\begin{align} \\label{transRPGG}\n\t\\pi_i&=y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\nonumber\\\\\n\t&=(1-x_i) m_i-\\frac{1}{g}\\sum_{j=1}^g (1-x_j) m_j\\lambda_j\\nonumber\\\\\n\t&=-\\frac{\\sum_{j=1}^g m_j\\lambda_j}{g}+\\frac{1}{g}\\sum_{j=1}^g x_j m_j\\lambda_j+(1-x_i) m_i\\,.\n\\end{align}\nNow we can see that the term $-\\sum_{j=1}^g m_j\\lambda_j\/g$ is related to the waste factor of all players in the group. Importantly, we can write the payoff for alternative R-PGG as\n\\begin{align} \\label{transalterRPGG}\n\t\\pi_i&=m_i\\lambda_i+y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\nonumber\\\\\n\t&=m_i\\lambda_i+(1-x_i) m_i-\\frac{1}{g}\\sum_{j=1}^g (1-x_j) m_j\\lambda_j\\nonumber\\\\\n\t&=m_i\\lambda_i-\\frac{\\sum_{j=1}^g m_j\\lambda_j}{g}+\\frac{1}{g}\\sum_{j=1}^g x_j m_j\\lambda_j+(1-x_i) m_i\\,,\n\\end{align}\nwhile for alternative PGG it is\n\\begin{equation} \\label{alterPGG}\n\t\\pi_i=\\frac{1}{g}\\sum_{j=1}^g x_j c_j r_j+(1-x_i) c_i\\,.\n\\end{equation}\n\nBy comparing Eq.~(\\ref{transalterRPGG}) and Eq.~(\\ref{alterPGG}), we can identify a non-zero redundant item $m_i\\lambda_i-(\\sum_{j=1}^g m_j\\lambda_j)\/g$ in R-PGG. This is the reason behind the observed difference in cooperation levels when we change the independent variable from $g_D$ to $g_C$. In particular, if we apply identical extension to PGG and alternative PGG, then we have no such difficulties because cooperators take the key role hence crucial game parameters belonging to this strategy in both model versions. A similar argument can be raised when we compare R-PGG and alternative R-PGG, because the decisive game parameters determine the payoff values that belong to defectors in both models. This explains why we cannot transform the family of PGGs into the family of R-PGGs by simply switching cooperators \\& defectors. \n\nThe conceptual difference between the models can be termed a ``reversed effect'' and its essence is explained in the following way. The game parameters working in PGG \\& R-PGG belong to different strategies. In PGG, the personal features of cooperator players count: if a player $i$ cooperates, she contributes $c_i$ and enlarges the contribution by $r_i$; meanwhile, a defective player $j$ has no chance to ``activate'' her specific $c_j$ and $r_j$ values. But she only receives the public goods of $r_i c_i$ produced by a cooperator. On the contrary, in R-PGG, the diversity of defectors players becomes essential: if a player $j$ defects, she receives $m_j$ which is multiplied by $\\lambda_j$ factor as the public cost. Meanwhile, a cooperative player $i$ never has a chance to validate her $m_i$ and $\\lambda_i$ parameter values. Her role is limited to sharing the public cost of $m_j \\lambda_j$ of a defector player.\n\nIn this way, the role of defectors is essential in R-PGG which is irreplaceable in a more complex environment. Hence, we can conclude that R-PGG is not a simple translational transformation of the original PGG, but rather a reversed form of it.\n \n\\section{Conclusion}\n\\label{conclusion}\n\nGenerally, people make logical, and frequently economical decisions when their consumption and its cost directly relate. However, if this connection is less clear, because the emerging cost is shared among a group of people, then we face a dilemma situation: we may require more goods from a common source than it is absolutely necessary because the related extra expense will be covered collectively. The reverse is also true: our economic behavior by consuming fewer resources will not necessarily be awarded by a proportionally smaller bill in the mentioned situation because all the others will also benefit from it. One might think this social dilemma is just a simple transformation of the original PGG where players contribute freely to a common pool and receive goods equally. In the reversed case described above, players receive goods from a common resource freely and should contribute equally. But the relation between the original PGG and our present R-PGG is more subtle.\n\nIn the case of a homogeneous population, where potential contributions and demands are constant, R-PGG is equivalent to PGG. This fact can be supported by means of a static calculation, deterministic and stochastic evolutionary analysis in a well-mixed population, and Monte Carlo simulations in structured populations. In evolutionary game dynamics, the requirement $m$ parameter in R-PGG plays the same role as the cooperator contribution $c$ does in PGG. Furthermore, the waste factor $\\lambda$ in R-PGG has the same duty as the synergy factor $r$ has in PGG. By transforming these parameters between R-PGG and PGG, we can obtain exactly the same system behaviors.\n\nThe network reciprocity reveals the following conclusions in R-PGG. If requiring goods leads to more cost to the group, then players tend to require fewer goods. Intriguingly, if players are allowed to require more goods, then fewer players may do it. The opposite behavior, which is consistent with our intuition, can only be detected clearly on small-world and scale-free graphs under the weak dilemma limit.\n\nAlthough the underlying R-PGG model seems equivalent to PGG, the agreement diminishes when we introduce more realistic conditions, like supposing heterogeneous players with diverse contribution capacities or unequal requirement levels for common resources. In the mentioned cases, R-PGG and PGG show major differences and produce highly different cooperation levels even if we apply the same average values of key parameters with equally strong amplitudes of diversity. The key observation is that while heterogeneity can generally support cooperation in PGG, the mentioned condition impedes the evolution of cooperation in R-PGG.\n\nTo understand the deeper origin of these diverse behaviors, we studied how payoff functions vary in response to the heterogeneity condition for different cases. In the case of equivalent models, the mentioned function should vary similarly if we apply varying average or growing diversity of key parameters. This requirement is justified between alternative forms of the original PGG where cooperators play the decisive role in collective income. This is also true between the alternative versions of R-PGG where the individual defector's profile becomes crucial. In this way, the equivalence cannot be held between PGG and R-PGG where the heterogeneity reveals the conceptual difference between the model families. Hence, we can say that R-PGG is not a trivial transformation of the original PGG, but can be considered a reversed version of it.\n\nOur present work focused on the difference in structured populations, but of course, similar study can also be done in a well-mixed system. Hopefully, our observations will stimulate forthcoming research. When the applied microscopic rule leads to the heterogeneity of game parameters by any means, one can study the consequence of extended parameters on PGGs and R-PGGs separately. The different results under the same topic (e.g., environmental protection) can reveal how the introduced rule works in PGGs (e.g., players donate to environmental protection), and how it works in corresponding R-PGGs (e.g., players litter for convenience). Aside from heterogeneity, the potential difference between R-PGG and PGG can be further investigated.\n\n\\section*{Acknowledgements}\nA.S. was supported by the National Research, Development and Innovation Office (NKFIH) under Grant No. K142948.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\nThe problem of characterizing the capacity region of a broadcast channel\\footnote{Throughout this article we restrict attention to memoryless channel. Please refer to \\cite{1998ADITI_Mas} or \\cite{2006EIT_CovTho} for a definition.} (BC) was proposed by Cover \\cite{197201TIT_Cov} in 1972, and he introduced a novel coding technique to derive achievable rate regions for particular degraded BCs. In a seminal work aimed at deriving an achievable rate region for the general degraded BC,\nBergmans \\cite{197303TIT_Ber} generalized Cover's technique into what is currently referred to as superposition coding. Gallager \\cite{197403PPI_Gal} and Bergmans \\cite{197403TIT_Ber} concurrently and independently proved optimality of\nsuperposition coding for the class of degraded BCs. This in particular yielded capacity region for\nthe scalar additive Gaussian BC. However, the case of general discrete BC (DBC) remained open. This (problem) led to the discovery of another ingenious coding technique\\footnote{We remark that general DBC is richer in terms of the strategies it permits.}. Gelfand \\cite{197707PIT_Gel} proposed an ingenious coding technique for a particular 2-DBC. In 1979, Marton \\cite{197905TIT_Mar} proposed the technique of binning. In conjunction with superposition, she derived the largest known achievable rate region for the general two user DBC (2-DBC). A generalization \\cite[p.391 Problem 10(c)]{CK-IT2011} of superposition and binning to incorporate\na common message is the largest known achievable rate region for the general 2-DBC and it's capacity is yet unknown.\\footnote{It is of interest to note that though superposition and binning were known in particular settings \\cite{197201TIT_Cov}, \\cite{197707PIT_Gel}, it's generalization led to fundamentally new ideas. For example, the description of a rate region using an auxiliary random variable \\cite{197303TIT_Ber} and the technique of binning have proved to be invaluable in\nderiving information theoretic achievable rate regions.}\n\nThough the capacity region has been found for many interesting classes of BCs  \\cite{197201TIT_Cov,197507TIT_Cov,197303TIT_Ber,197403TIT_Ber, 197511TIT_AhlKor,197701TIT_KorMar,197903TIT_Elg, \n1977MMISIT_Mar,197804PPI_Pin,197905TIT_Mar,1980MMPPI_GelPin,198001TIT_Elg,198101TIT_GamMeu,197503TIT_Meu,197901TIT_HajPur}, the question of whether the\nrate region derived by Marton is optimal for the general DBC has remained open for over thirty years. Following a period of reduced activity, there has been renewed interest \\cite{201006ISIT_GenNaiShaWan,200910TIT_NaiElg} in settling this question. Gohari and Anantharam \\cite{201202TIT_GohAna} have proved computability of Marton's rate region. This enabled them identify a class of binary 2-DBCs for which Marton's \\cite{197905TIT_Mar} rate region when computed is strictly smaller than the tightest known outer bound \\cite{197805TIT_Sat,200701TIT_NaiGam}, which is due to Nair\nand El Gamal. On the other hand, Weingarten, Steinberg and Shamai \\cite{200609TIT_WeiSteSha} have proved Marton's binning (also referred to, in the Gaussian setting, as Costa's dirty paper coding \\cite{198305TIT_Cos}) to be optimal for Gaussian MIMO BC, and thereby characterized capacity region for the particular class of Gaussian vector BCs. We remark \\cite{200609TIT_WeiSteSha} proves optimality of Marton's binning technique for Gaussian vector BCs with arbitrary number of receivers.\n\nIn this article, (1) we propose a coding technique based on nested coset codes, an ensemble of codes endowed with algebraic structure, that enables us (2) present an achievable rate region for the general three user DBC (3-DBC), and thereby (3) strictly enlarge the current known largest achievable rate region\\footnote{The largest known achievable rate region for the general 3-DBC is the natural extension of Marton's rate region for the 2-DBC. We henceforth refer to this as Marton's rate region for 3-DBC.}. We characterize \\3To1BC DBCs, a class of 3-DBCs for which Marton's coding technique is proved to be strictly sub-optimal. In particular, we identify a novel \\3To1BC DBC for which the proposed coding technique yields achievable, a rate triple not contained within Marton's rate region. We remark that the \\3To1BC DBC presented herein is the first known BC for which the natural generalization of Marton's coding technique is strictly sub-optimal.  Indeed, one of the key elements of our work is an analytical proof of sub-optimality of Marton's rate region for this 3-DBC. Our findings emphasize the need for codes endowed with algebraic structure for communicating over a general multi-terminal communication setting.\n\nWhy do codes endowed with algebraic structure outperform traditional independent unstructured codes for a BC? The central aspect of a coding technique designed for a BC is interference management. Marton's coding incorporates two techniques - superposition and binning - for tackling interference. Superposition enables each user decode a \\textit{univariate} component of the other user's signal and thus subtract it off. Binning enables the encoder counter the component of each user's interfering signal not decoded by the other, by precoding for the same. Except for particular cases, the most popular being dirty paper coding, precoding results in a rate loss, and is therefore less efficient than decoding the interfering signal at the decoder. The presence of a rate loss motivates each decoder to decode as large a part of interference as possible.\\footnote{For the Gaussian case, there is no rate loss. Thus the encoder can precode all the interference. Indeed, the optimal strategy does not require any user to decode a part of signal not intended for\nit. This explains why lattices are not necessary to achieve capacity of Gaussian vector BC.} However decoding a large part of the interference constrains the individual rates. In a three user BC, each user's reception is plagued by interference caused by signals intended for the other two users. The interference is in general a bi-variate function of signals intended for the other users. If the signals of the two users are endowed with structure that can help compress the range of this bi-variate function when applied to all possible signals, then the receivers can decode a large part of the interfering signal. This minimizes the component of the interference precoded, and therefore the rate loss.\\footnote{For the Gaussian case, precoding suffers no rate loss and hence no part of the interference needs to be decoded. Thus constraining interference patterns is superfluous. } This is where codebooks endowed with algebraic structure outperform unstructured independent codebooks. Indeed, linear codes constrain the interference pattern to an affine subspace if the interference is the sum of user 2 and 3's signals. It is our belief that additional degrees of freedom prevalent in a three user information theoretic problem can be harnessed with codebooks endowed with algebraic structure. Whether structure in codebooks can be exploited for a two user problem remains open.\n\nThis article is organized as follows. We begin with preliminaries in section \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}. In particular, we provide relevant definition for a DBC and state Marton's achievable rate region. In section \\ref{Sec:3To1BCAndTheNeedForStructuredCodes}, describe a 3-DBC for which Marton's coding technique is strictly suboptimal and generalize this example into a class of 3-DBCs called \\3To1BC DBCs. We present our achievable rate region for the general 3-DBC in three steps. In the first two steps, presented in section \\ref{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}, we present achievable rate regions for a \\3To1BC DBC. The final step, where we derive an achievable rate region for a general 3-DBC, is presented in section \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes}. The proof of strict sub-optimality of Marton's coding technique for the \\3To1BC DBC is detailed in section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}, \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}. In section \\ref{Sec:EnlargingMarton'sRateRegionUsingNestedCosetCodes}, we indicate structure of a coding technique that yields the largest known achievable rate region for the 3-DBC which subsumes all of the known achievable rate regions. We conclude in section \\ref{Sec:ConcludingRemarks} by pointing at fundamental connection between the several layers of coding in a three user communication problem and common parts of a triple of random variables, thereby arguing the inherent role played by structured codes in information theory. \n\nThe reader solely interested in proof of strict sub-optimality of Marton's coding technique may read through sections \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}, \\ref{SubSec:LimitationsOfUnstructuredCodes}, \\ref{SubSec:TheThreeUserBroadcastChannel} and skip over to section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique} without missing out on any necessary elements. However, we encourage the reader to peruse the proposed coding technique based on nested coset codes.\n\nWhile sections \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}-\\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique} contain all the technical details presented in this article, we encourage the reader to read through section \\ref{Sec:RoleOfStructuredCodesInNetworkInformationTheory} for an account of the role of structured codes in network information theory.\n\n\\section{The role of structured codes in network information theory}\n\\label{Sec:RoleOfStructuredCodesInNetworkInformationTheory}\n\nIn this article, we demonstrate the role of algebraic structure through the simplest example of structured codes - linear codes - that contain a bivariate function - sum - of transmitted codewords to an affine space, thereby leaving sufficient room for generalization using other algebraic objects. In particular, we employ the ensemble of nested coset codes \\cite{201108ISIT_PadPra} to derive a new achievable rate region for the \\textit{general} 3-DBC. Indeed, analyzing the proposed coding technique for a \\textit{general} 3-DBC requires joint typical encoding and decoding. One of the key elements of our study is therefore the interplay of joint typical encoding and decoding of statistically correlated codebooks resulting in new proof techniques. While we do not provide proofs in this article, we refer the curious reader to \\cite[proof of theorem 2]{201301arXivMACDSTx_PadPra}, \\cite[proof of theorem 1]{201108ISIT_PadPra} that contain several of these techniques.\n\nThe astute reader will question the case when the\nbi-variate function is not a sum. Towards answering this question, we consider a natural\ngeneralization of linear codes to sets with looser algebraic structure such as groups. Our\ninvestigation of group codes, kernels of group homomorphisms, to improve achievable rate\nregions for information theoretic problems has been pursued in a concurrent research\nthread \\cite{201108ISIT_SahPra}.\\footnote{We also bring to the attention of the interested\nreader, our investigation \\cite{201207ISIT_SahPra} of pseudo group codes. While linear\ncodes are completely {\\it compressive} under the operation of addition, and unstructured\nindependent codes are completely {\\it explosive}, pseudo group codes lie in between. In\nother words, when two pseudo group codes of rate $R$ are operated under the group\noperation, the range of the resulting codebook lies between $R$ and $2R$. Pseudo group\ncodes are of interest since they outperform group codes for point to point communication.} Containing the sum of transmitted codewords using linear codes is just the first step, and we envision an achievable rate region involving a union over all algebraic objects pertaining to the several bivariate functions.\n\nThe role of structured codes for improving information theoretic rate regions began with the ingenious technique of K\\\"orner and Marton \\cite{197903TIT_KorMar} proposed for the source coding problem of reconstructing modulo$-2$ sum of distributed binary sources. Han and Kobayashi \\cite{198701TIT_HanKob} categorized a class of function reconstruction problems for which K\\\"orner and Marton's technique provided strict gains over the largest known rate regions using unstructured codes. Ahlswede and Han \\cite{198305TIT_AhlHan} proposed a universal coding technique that brings together coding techniques based on unstructured and structured codes\\footnote{Indeed, the coding techniques based on   structured codes do not substitute for coding techniques based on unstructured codes. For example, lossless reconstruction of a pair of sources with two source codes that are partitioned using a common channel code can be strictly sub optimal. Similarly, the technique of partitioning independent source codes using independent channel codes is sub optimal for the problem of lossless reconstruction of modulo$-2$ sum of a class of binary sources.}. More recently, there is a wider interest \\cite{200710TIT_NazGas,200912arXiv_CadJaf} in developing coding techniques for particular problem instances that perform better than the best known techniques based on unstructured codes. Philosof and Zamir \\cite{200906TIT_PhiZam} employ nested linear codes to communicate over a particular binary doubly dirty multiple access channel (MAC) and analytically prove strict sub-optimality of best known coding technique based on unstructured independent codes. Bresler, Parekh and Tse \\cite{201009TIT_BreParTse} study the problem of characterizing approximate capacity of many-to-one Gaussian interference channels and demonstrate the need for lattice codes. Sridharan et. al. \\cite{200809Allerton_SriJafVisJafSha} employ lattice based scheme to prove achievability of rates not contained within the largest known achievable rate region using unstructured independent codes.\n\nIt was shown in \\cite{198701TIT_HanKob}, in the setting of distributed source coding that for every any non-trivial and truly bi-variate function, there exists at least one source distribution for which linear codes outperform random codes,  Even then, it was largely believed that codebooks possessing algebraic structure can be exploited only for modulo additive channel and source coding problems. Indeed, linear codes were known to be sub optimal for communicating over arbitrary point to point channels \\cite{197102TAMS_Ahl} (and similarly for lossy compression of sources subject to an arbitrary distortion), and therefore, the basic building block in the coding scheme for any multi-terminal communication problem could not be filled by linear codes. For over thirty years, since the work of K\\\"orner and Marton came to light, neither did we know of a coding technique based on unstructured codes that performed as well, nor did we know of a framework for coding based on structured codes for which the above findings was a particular case.\n\nKrithivasan and Pradhan \\cite{201103TIT_KriPra} have proposed the ensemble of nested coset codes as the basic building block of algebraic codes for compressing sources subject to any arbitrary distortion. They employ this ensemble to propose a framework for communicating information from distributed encoders observing correlated sources to a centralized decoder. Firstly, this framework generalizes the technique proposed by K\\\"orner and Marton for the general problem of distributed function computation, joint quantization of distributed sources etc. Secondly, in conjunction with the Berger Tung \\cite{Berger-MSC} technique this strictly enlarges the largest known achievable rate region for the problem of distributed function computation. In the same spirit, we proposed the ensemble of nested coset codes \\cite{201108ISIT_PadPra} as an ensemble of codes possessing algebraic structure that achieves capacity of arbitrary point to point channels. We employed this ensemble to (i) elevate the technique proposed by Philosof and Zamir to derive a new achievable rate region \\cite{201301arXivMACDSTx_PadPra} for an arbitrary discrete multiple access channel with distributed states (ii) derive a new achievable rate region for the general discrete three user interference channel \\cite{201207ISIT_PadSahPra}.\n\n\\section{Broadcast channel: definitions and Marton's rate region}\n\\label{Sec:BroadcastChannelDefinitionsMartonRateRegion}\nWe begin with remarks on notation in section \\ref{SubSec:Notation}. In section \\ref{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}, we state relevant definitions - code, achievability and capacity - with respect to a 3-DBC. In section \\ref{SubSec:MartonRateRegionFor2BC}, we briefly discuss Marton's coding technique and state Marton's rate region for a 2-DBC. In section \\ref{SubSec:NaturalExtensionOfMartonTo3BC}, we state a natural extension of Marton's coding technique for the three user case and provide a description of the corresponding achievable rate region.\n\\subsection{Notation}\n\\label{SubSec:Notation}\nWe employ notation that has now been widely accepted in the information theory literature\nsupplemented with the following. For a set $A \\subseteq \\reals^{k}$, $\\cocl \\left( A \\right)$ denotes closure of convex hull of $A$. Throughout this article, $\\log$ and $\\exp$ functions are taken with respect to the base $2$. Hence, units for information theoretic quantities such as entropy and mutual information are expressed in bits\/symbol. Let $h_{b}(x) \\define -x\\log_{2}x - (1-x)\\log_{2}(1-x)$ denote binary entropy function. Let $a * b \\define a(1-b)+(1-a)b$ denote binary convolution. For $K \\in \\naturals$, we let $[K]\\define \\left\\{ 1,2\\cdots,K \\right\\}$. We let $\\fieldq$ denote the finite field of cardinality $q$. While $+$ denotes addition in $\\reals$, we let $\\oplus$\ndenote addition in a finite field. The particular finite field,\nwhich is uniquely determined (up-to an isomorphism) by it's cardinality, is clear from\ncontext. When ambiguous, or to enhance clarity, we specify addition in $\\fieldq$ using\n$\\oplus_{q}$. For elements $a,b$, in a finite field, $a \\ominus b\n\\define a \\oplus (-b)$, where $(-b)$ is the additive inverse of $b$.\\begin{comment}{ In sections \\ref{Sec:3To1BCAndTheNeedForStructuredCodes} and \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, $\\oplus$ denotes addition in the binary field $\\BinaryField$.}\\end{comment} In this article, we will need to define multiple objects, mostly triples, of the same type. In order to reduce clutter, we use an \\underline{underline} to\ndenote aggregates of objects of similar type. For example, (i) if\n$\\OutputAlphabet_{1},\\OutputAlphabet_{2}, \\OutputAlphabet_{3}$ denote (finite) sets, we\nlet $\\underlineSetY$ either denote the Cartesian product\n$\\OutputAlphabet_{1} \\times \\OutputAlphabet_{2} \\times \\OutputAlphabet_{3}$ or\nabbreviate the collection $(\\OutputAlphabet_{1},\\OutputAlphabet_{2},\\OutputAlphabet_{3})$\nof sets, the particular reference being clear from context, (ii) if $y_{k} \\in \\OutputAlphabet_{k}:k=1,2,3$, we let $\\underliney \\in\n\\underlineSetY$ abbreviate $(y_{1},y_{2},y_{3}) \\in \\OutputAlphabet$ (iii) if\n$\\mathscr{M}_{1},\\mathscr{M}_{2}, \\mathscr{M}_{3}$ denote cardinalities of finite sets\n$\\MessageSetM_{1},\\MessageSetM_{2},\\MessageSetM_{3}$ respectively, then we let\n$\\underlineCardinalityMessageSet$ abbreviate the collection\n$\\mathscr{M}_{1},\\mathscr{M}_{2}, \\mathscr{M}_{3}$, (iv) if\n$d_{k}:\\OutputAlphabet_{k}^{n} \\rightarrow \\MessageSetM_{k}:k=1,2,3$ denote (decoding)\nmaps, then we let $\\underlined(\\underliney^{n})$ denote $(d_{1}(y_{1}^{n}),\nd_{2}(y_{2}^{n}), d_{3}(y_{3}^{n}))$.\n\\subsection{Definitions: Broadcast channel, code, achievability and capacity}\n\\label{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}\nA 3-DBC consists of a finite input alphabet\nset $\\InputAlphabet$ and three finite output alphabet sets $\\OutputAlphabet_{1},\n\\OutputAlphabet_{2}, \\OutputAlphabet_{3}$. The discrete time channel is (i) time\ninvariant, i.e., the pmf of $\\underlineY_{t}=(Y_{1t},Y_{2t},Y_{3t})$, the output at time\n$t$, conditioned on $X_{t}$, the input at time $t$, is invariant with $t$, (ii)\nmemoryless, i.e., conditioned on present input $X_{t}$, the present output $\\underlineY_{t}$ is\nindependent of past inputs $X_{1},\\cdots, X_{t-1}$ and (iii) used without feedback, i.e.,\nthe encoder has no information of the symbols received by the decoder. Let\n$W_{\\underlineY|X}(\\underliney|x)=W_{Y_{1}Y_{2}Y_{3}|X}(y_{1},y_{2},y_{3}|x)$ denote\nprobability of observing $\\underliney \\in \\underlineSetY$ at the respective outputs\nconditioned on $x \\in \\InputAlphabet$ being input. Input is constrained with respect\nto a cost function $\\kappa : \\InputAlphabet \\rightarrow [0, \\infty)$. The cost function\nis assumed additive, i.e., cost of transmitting the vector $x^{n} \\in \\InputAlphabet^{n}$\nis $\\bar{\\kappa}^{n}(x^{n}) \\define \\sum_{i=1}^{n}\\kappa(x_{i})$. We refer to this 3-DBC as\n$(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$.\n\nIn this article, we restrict attention to communicating private messages to the three users. The focus of this article therefore is the (private message) capacity region of a 3-DBC, and in particular corresponding achievable rate regions. The following definitions make the relevant notions precise.\n\n\\begin{definition}\n \\label{Defn:3-BCCode}\n A 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ consist of (i) index\nsets $\\MessageSetM_{1}, \\MessageSetM_{2},\\MessageSetM_{3}$ of messages, of cardinality\n$\\mathscr{M}_{1},\\mathscr{M}_{2},\\mathscr{M}_{3}$ respectively, (ii) encoder map\n$e:\\underlineMessageSetM \\rightarrow\n\\InputAlphabet^{n}$, and (iii) three decoder maps $d_{k} : \\OutputAlphabet_{k}^{n}\n\\rightarrow \\MessageSetM_{k}:k=1,2,3$.\n\\end{definition}\n\n\\begin{definition}\n \\label{Defn:BCCodeErrorProbability}\n The error probability of a 3-DBC code\n$(n,\\underlineCardinalityMessageSet,e,\\underline{d})$ conditioned on message triple\n$(m_{1},m_{2},m_{3}) \\in \\underlineMessageSetM$ is\n\\begin{equation}\n \\label{Eqn:ErrorProbabilityOf3-BCCode}\n \\xi(e,\\underlined|\\underlinem) \\define 1-\\sum_{\\underliney^{n} : \\underlined\n(\\underliney^{n}) = \\underlinem}\nW_{\\underlineY|X}(\\underliney^{n}|e(\\underlinem)).\\nonumber\n\\end{equation}\nThe average error probability of\na 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ is\n$\\bar{\\xi}(e,\\underlined)\\define \\sum_{\\underlinem\n \\in \\underlineMessageSetM}\n\\frac{1}{\\mathscr{M}_{1}\\mathscr{M}_{2}\\mathscr{M}_{3}}\n\\xi(e,\\underlined|\\underlinem)$. Cost of transmitting message $\\underlinem \\in\n\\underlineMessageSetM$ per symbol is $\\tau(e|\\underlinem) \\define\n\\frac{1}{n}\\bar{\\kappa}^{n}(e(\\underlinem))$ and\naverage cost of 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ is $\\tau(e)\n\\define \\frac{1}{\\mathscr{M}_{1}\\mathscr{M}_{2}\\mathscr{M}_{3}}\\sum_{\\underlinem\n \\in \\underlineMessageSetM}\\tau(e|\\underlinem)$.\n\\end{definition}\n\n\\begin{definition}\n\\label{Defn:3-BCAchievabilityAndCapacity}\nA rate-cost quadruple $(R_{1},R_{2},R_{3},\\tau)\\in [0,\\infty)^{4}$ is\nachievable if for every $\\eta > 0$, there exists $N(\\eta)\\in \\naturals$ such that for all\n$n > N(\\eta)$, there exists a 3-DBC code $(n, \\underlineCardinalityMessageSet^{(n)},\ne^{(n)},\\underlined^{(n)})$ such that (i)\n$\\frac{\\log_{2}\\mathscr{M}_{k}^{(n)}}{n} \\geq R_{k}-\\eta:k=1,2,3$, (ii)\n$\\bar{\\xi}(e^{(n)},\\underlined^{(n)})\n\\leq \\eta$, and (iii) average cost $\\tau(e^{(n)}) \\leq \\tau+\\eta$. The capacity region is\n$\\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau) \\define \\cl{\\left\\{ \\underlineR \\in \\reals^{3}:\n(\\underlineR,\\tau)\\mbox{ is achievable} \\right\\}}$.\\begin{comment}{\nA sequence of 3-DBC codes $(n, \\underlineCardinalityMessageSet^{(n)},\ne^{(n)},\\underlined^{(n)}):n \\geq 1$ achieves $(\\underlineR,\\tau)$ if\n$\\lim_{n \\rightarrow \\infty } \\frac{\\log_{2}\\mathscr{M}_{k}^{(n)}}{n} \\geq\nR_{k}:k=1,2,3$ and $\\lim_{n \\rightarrow\n\\infty }\\bar{\\xi}(e^{(n)},\\underlined^{(n)}) = 0$ and $\\lim_{n \\rightarrow\n\\infty } \\tau(e^{(n)}) \\leq \\tau$.}\\end{comment}\n\\end{definition}\n\nThe currently known largest achievable rate region for 3-DBC is obtained by a natural extension of Marton's coding technique proposed for the 2-DBC. We begin with a brief review of Marton's coding technique for the 2-DBC in section \\ref{SubSec:MartonRateRegionFor2BC} and state it's natural extension to a general 3-DBC in section \\ref{SubSec:NaturalExtensionOfMartonTo3BC}.\n\n\\subsection{Marton's rate region}\n\\label{SubSec:MartonRateRegionFor2BC}\n\\begin{comment}{\nWe begin with a brief review of Marton's coding technique followed by a characterization of the corresponding achievable rate region.\n}\\end{comment}\nMarton's coding incorporates two fundamental coding techniques - superposition and precoding. Superposition involves each user decode a part of the signal carrying the other user's information and thereby enhance it's ability to decode the intended signals. The technique of jointly choosing each user's message bearing signal to contain mutual interference is precoding. Superposition coding is accomplished using a two layer coding scheme. First layer, which is public, contains a codebook over $\\PublicRVSet$. Second layer is private and contains two codebooks one each on $\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$. Precoding is accomplished by setting aside a \\textit{bin} of codewords for each private message, thus enabling the encoder to choose a compatible pair of codewords in the indexed bins. User $j$th message is split into two parts - public and private. The public parts together index a codeword in $\\PublicRVSet-$codebook and the private part of user $j$th message index a codeword in $\\PrivateRVSet_{j}-$codebook. Both users decode from the public codebook and their respective private codebooks.\n                                                                                                                                                                                                                                                                                                           \nDefinition \\ref{Defn:MartonTestChannels} and theorem \\ref{Thm:MartonRateRegionFor2-BC}\nprovide a characterization of rate pairs achievable using Marton's coding technique.\nWe omit restating the definitions analogous to definitions \\ref{Defn:3-BCCode},\n\\ref{Defn:BCCodeErrorProbability}, \\ref{Defn:3-BCAchievabilityAndCapacity} for a 2-BC.\n\n\\begin{definition}\n \\label{Defn:MartonTestChannels}\nLet $\\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of distributions\n$p_{\\TimeSharingRV\\PublicRV \\PrivateRV_{1} \\PrivateRV_{2}XY_{1}Y_{2}}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times\n\\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times \\InputAlphabet \\times\n\\OutputAlphabet_{1} \\times \\OutputAlphabet_{2}$, where (i) $\\TimeSharingRVSet$, $\\PublicRVSet$,\n$\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$ are finite sets of cardinality at most\n$|\\InputAlphabet|+4$, $|\\InputAlphabet|+4$, $|\\InputAlphabet|+1$ and $|\\InputAlphabet|+1$ respectively, (ii)\n$p_{\\underlineY|X \\underlinePrivateRV \\PublicRV \\TimeSharingRV}=p_{\\underlineY|X} = W_{\\underlineY|X}$,\n(iii) $\\Expectation\\left\\{\\kappa(X)\\right\\} \\leq \\tau$. For $p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY} \\in \\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)$,\nlet $\\alpha_{M}(p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY})$ denote the set of $(R_{1},R_{2}) \\in \\reals^{2}$ that satisfy\n\\begin{comment}{\n\\begin{flalign}\n \\label{Eqn:MartonAchievableRateRegionForParticularTestChannel}\n0 \\leq R_{k} &\\leq I(\\PublicRV\n\\PrivateRV_{k};Y_{k}|\\TimeSharingRV):k=1,2,\\nonumber\\\\ R_{1}+R_{2} &\\leq\\! I(W \\PrivateRV_{1};Y_{1}|\\TimeSharingRV)+\nI(\\PrivateRV_{2};Y_{2}|\\PublicRV,\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV) \\nonumber\\\\\nR_{1}+R_{2} &\\leq\\! I(\\PrivateRV_{1};Y_{1}|W,\\TimeSharingRV)+\nI(\\PublicRV\\PrivateRV_{2};Y_{2}|\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV)\n\\nonumber\n\\end{flalign}\n}\\end{comment}\n\\begin{flalign}\n \\label{Eqn:MartonAchievableRateRegionForParticularTestChannel}\n0 \\leq R_{k} &\\leq I(\\PublicRV\n\\PrivateRV_{k};Y_{k}|\\TimeSharingRV):k=1,2,\\nonumber\\\\ \nR_{1}+R_{2} &\\leq\\! \\min\\left\\{ I(W;Y_{1}|\\TimeSharingRV),I(W;Y_{2}|\\TimeSharingRV)  \\right\\}+I(\\PrivateRV_{1};Y_{1}|\\TimeSharingRV W )+\nI(\\PrivateRV_{2};Y_{2}|\\PublicRV,\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV) \\nonumber\n\\end{flalign}\nand\n\\begin{equation}\n \\label{Eqn:MartonRateRegionAsAUnionOfTestChannelRateRegions}\n \\alpha_{M}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\alpha_{M}(p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY})\\right)\\nonumber\n\\end{equation}\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:MartonRateRegionFor2-BC}\nFor 2-DBC $(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$,\n$\\alpha(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\alpha(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\n\\begin{remark}\n \\label{Rem:CardinalityBoundsDerivedByGohariAndAnantharam}\n The bounds on cardinality of $\\PublicRVSet,\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$\nwere derived by Gohari and Anantharam in \\cite{201202TIT_GohAna}.\n\\end{remark}\nWe refer the reader to \\cite{197905TIT_Mar} for a proof of achievability. El Gamal and Meulen\n\\cite{198101TIT_GamMeu} provide a simplified proof using the method of second moment.\\begin{comment}{ We make a\ncouple of observations that lead us to an alternate view of the coding technique. This\nwill motivate the use of structured codes in deriving a larger rate region.\n\nThe encoder splits each message into two parts - public and private. The public parts are\nused to index the codebook over $\\PublicRVSet$. Each user decodes both public parts and\nit's private part. We make a couple of observations. If we let $\\bar{V_{j}} \\define\n(W,V_{j})$ to be the pair of random variables decoded by user $j$, then $\\PublicRV$ is\nthe common part \\cite{1972MMPCT_GacKor} of $\\bar{V_{1}},\\bar{V_{2}}$. \\textit{When the\ndecoded random variables share a common part, this common information is best encoded\nand decoded in a separate layer.} Since the common part of the pair\n$\\bar{V}_{1},\\bar{V}_{2}$ is identified through univariate functions\n$f_{j}:\\PublicRVSet \\times \\PrivateRVSet_{j}  \\rightarrow \\PublicRVSet:j=1,2$ such that\n$P(W=f_{1}(\\bar{V}_{1})=f_{2}(\\bar{V}_{2}))=1$, \\textit{each user decodes a function of\nthe random variable decoded by the other user.} The latter observation is central to\nMarton's coding technique. Indeed, Nair \\cite{} has proved the strategy of decoding a\nunivariate function of the other user's codeword subsumes that of Marton. It is natural\nto expect these observations to carry over for the natural extension of Marton to the\n3-BC.}\\end{comment}\n\\subsection{Natural extension of Marton's rate region for the 3-BC}\n\\label{SubSec:NaturalExtensionOfMartonTo3BC}\nThe natural extension to the case of 3 users will involve a 3 layer coding scheme. For simplicity, we describe the coding technique without referring to the time sharing random variable and employ the same in describing the achievable rate region. User $j$th message $M_{j}$ is split into four parts - two semi-private parts, and one, private and public parts each. We let message (i) $M^{W}_{j} \\in \\MessageSetM^{W}_{j}$ of rate $K_{j}$ denote it's public part (ii) $M^{U}_{i\\msout{j}} \\in \\MessageSetM^{U}_{i\\msout{j}},M^{U}_{\\msout{j}k} \\in \\MessageSetM^{U}_{\\msout{j}k}$ of rates $L_{ij},K_{jk}$ respectively, denote it's semi-private parts, where $(i,j,k)$ is an appropriate triple in $\\left\\{(1,2,3),(2,3,1),(3,1,2)  \\right\\}$, and (iii) $M^{V}_{j} \\in \\MessageSetM^{V}_{j}$ of rate $T_{j}$ denote it's private part.\n\nThe first layer is public with a single codebook $(w^{n}(\\underline{m}^{W}):\\underline{m}^{W} \\in \\underlineMessageSetM^{W})$ of rate $K_{1}+K_{2}+K_{3}$ over $\\PublicRVSet$. $\\underline{M}^{W}$ indexes a codeword in $\\PublicRVSet-$codebook and each user decodes from\n$\\PublicRVSet-$codebook. \n\nEach codeword in $\\PublicRVSet-$codebook is linked to a triple of codebooks - one each on $\\SemiPrivateRVSet_{ij}:(i,j) \\in \\left\\{ (1,2),(2,3),(3,1) \\right\\}$- in the second layer. The second layer is semi-private. Each of the three semi-private codebooks is composed of \\textit{bins}, wherein each bin comprises a collection of codewords. For each pair $(i,j) \\in \\left\\{ (1,2),(2,3),(3,1) \\right\\}$ the following hold. $M^{U}_{i\\msout{j}}$ and $M^{U}_{\\msout{i}j}$ together index a bin in $\\SemiPrivateRVSet_{ij}-$codebook. Each bin in $\\SemiPrivateRVSet_{ij}-$codebook is of rate $S_{ij}$. Let $(u_{ij}^{n}(\\underlinem^{W},m_{\\msout{i}j}^{U},m_{i\\msout{j}}^{U},s_{ij}):s_{ij} \\in [\\exp\\{ nS_{ij} \\}])$ denote bin corresponding to semi-private messages $\\underline{m^{U}_{ij}}\\define (m_{\\msout{i}j}^{U},m_{i\\msout{j}}^{U}) \\comment{ \\in \\underline{\\MessageSetM_{ij}}^{U}}$ in the $\\SemiPrivateRVSet_{ij}-$codebook linked to public message $\\underlinem^{W}$. Users $i,j$ decode from $\\SemiPrivateRVSet_{ij}-$codebook and it maybe verified that $\\SemiPrivateRVSet_{ij}-$codebook is of rate $K_{ij}+L_{ij}+S_{ij}$.\n\nLet $(i,j)$ and $(j,k)$ be distinct pairs in $\\left\\{ (1,2),(2,3),(3,1) \\right\\}$. Every pair of codewords in $\\SemiPrivateRVSet_{ij}-$ and $\\SemiPrivateRVSet_{jk}-$codebooks is linked to a codebook on $\\PrivateRVSet_{j}$. The codebooks over $\\PrivateRVSet_{j}:j=1,2,3$ comprise the third layer which is private. $M_{j}^{V}$ indexes a bin in $\\PrivateRVSet_{j}-$codebook, each of which is of rate $S_{j}$, and thus $\\PrivateRVSet_{j}-$codebook is of rate $T_{j}+S_{j}$. Let $(v_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}):s_{j} \\in [\\exp\\{ nS_{j} \\}])$ denote bin corresponding to private message $m_{j}^{V}$ in the $\\PrivateRVSet_{j}-$codebook linked to codeword pair $(u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}))$. User $j$ decodes from the private codebook over $\\PrivateRVSet_{j}$.\n\nHow does the encoder map messages to a codeword?\nLet $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}$ be a distribution on\n$\\PublicRVSet \\times \\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet$ such that $\\Expectation\\left\\{ \\kappa(\\InputRV) \\right\\} \\leq \\tau$. The encoder looks for $(s_{12},s_{23},s_{31},s_{1},s_{2},s_{3})$ such that the septuple\n\\begin{eqnarray}\n \\label{Eqn:JointlyTypicalSeptupleOfCodewords}\n\\left(\\substack{w^{n}(\\underlineM^{W}),u_{ij}^{n}(\\underlineM^{W},\\underline{M^{U}_{ij}},s_{ij}):(i,j)=(1,2),(2,3),(3,1),\\\\v_{j}^{n}(\\underlineM^{W},\\underline{M^{U}_{ij}},s_{ij},\\underline{M^{U}_{jk}},s_{jk},M_{j}^{V},s_{j}):(i,j,k)=(1,2,3),(2,3,1),(3,1,2)}\\right)\\nonumber\n\\end{eqnarray}\nof codewords is jointly typical with respect to $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV}$. If such a septuple is found, this is mapped to a codeword on $\\InputAlphabet^{n}$ which is input to the channel. If it does not find any such septuple, an error is declared. \n\nDecoder $j$ looks for all quadruples $(\\underline{\\hatm}^{W},\\underline{\\hatm_{ij}}^{U},\\underline{\\hatm_{jk}}^{U},\\hatm_{j}^{V})$ such that\n\\begin{equation}\n \\label{Eqn:DecodingRuleOfDecoderj}\n\\left(   w^{n}(\\underline{\\hatm}^{W}),u_{ij}^{n}(\\underline{\\hatm}^{W},\\underline{\\hatm^{U}_{ij}},s_{ij}),u_{jk}^{n}(\\underline{\\hatm}^{W},\\underline{\\hatm^{U}_{jk}},s_{jk}),v_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}),Y_{j}^{n} \\right) \\nonumber\n\\end{equation}\nis jointly typical with respect to $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV\\underlineY}\\define p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}W_{\\underline{Y}|X}$, where (i) $(i,j,k)$ is the appropriate triple in $\\left\\{ (1,2,3),(2,3,1),(3,1,2) \\right\\}$ and (ii) $Y_{j}^{n}$ is the received vector. If there is a unique such quadruple, it declares $\\hatm_{j} \\define (\\hatm_{j}^{W},\\hatm_{i\\msout{j}}^{U},\\hatm_{\\msout{j}k}^{U},\\hatm^{V}_{j})$ as user $j$th message. Otherwise, i.e., none or more than one such quadruple is found, it declares an error.\n\nAs is typical in information theory, we average error probability over the entire ensemble of codebooks and upper bound the same. Moreover, we incorporate the time sharing random variable in the above coding technique using the standard approach. Let $\\TimeSharingRV$, taking values over the finite alphabet $\\TimeSharingRVSet$, denote the time sharing random variable. Let $p_{\\TimeSharingRV}$ be a pmf on $\\TimeSharingRVSet$ and $q^{n} \\in \\TimeSharingRVSet^{n}$ denote a sequence picked according to $\\prod_{t=1}^{n}p_{Q}$. $q^{n}$ is revealed to the encoder and all decoders. The codewords in $\\PublicRVSet-$codebook are identically and independently distributed according to $\\prod_{t=1}^{n}p_{W|Q}(\\cdot|q_{t})$. Conditioned on entire public codebook $(W^{n}(\\underline{m}^{W})=w^{n}(\\underline{m}^{W}):\\underline{m}^{W} \\in \\underlineMessageSetM^{W})$ and the time sharing sequence $q^{n}$, each of the codewords $U_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}):(\\underline{m^{U}_{ij}},s_{ij}) \\in \\underline{\\MessageSetM_{ij}}^{U} \\times [\\exp\\{ nS_{ij} \\}]$ are independent and identically distributed according to $\\prod_{t=1}^{n}p_{U_{ij}|WQ}(\\cdot|w_{t}(\\underline{m}^{W}),q_{t})$. Conditioned on a realization of the entire collection of public and semi-private codebooks, the private codewords\n\\comment{\\begin{eqnarray}\n\\left(\\substack{W^{n}(\\underline{m}^{W}),U_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),\\\\U_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}),U_{ki}^{n}(\\underlinem^{W},\\underline{m^{U}_{ki}},s_{ki})}\\right)=\\nonumber\\\\\\left(\\substack{w^{n}(\\underline{m}^{W}),u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),\\\\u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}),u_{ki}^{n}(\\underlinem^{W},\\underline{m^{U}_{ki}},s_{ki})}\\right):\\left(\\substack{\\underlinem^{W},\\underlinem^{U}\\\\s_{ij},s_{jk},s_{ki}}\\right) \\in \\substack{\\underline{\\MessageSetM}^{W} \\times \\underline{\\MessageSetM}^{U}},\\nonumber                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \\end{eqnarray}\neach of the codewords }$(V_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}):s_{j} \\in [\\exp\\{ nS_{j} \\}])$ are independent and identically distributed according to\n\\begin{eqnarray}\n \\label{Eqn:DistributionOfVCodebookConditionedOnPublicAndSemiPrivate}\n\\prod_{t=1}^{n}p_{V_{j}|U_{ij}U_{jk}WQ}\\left(\\cdot|w_{t}(\\underline{m}^{W}),(u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}))_{t},(u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}))_{t},q_{t}\\right).\\nonumber\n\\end{eqnarray}\nWe now average error probability over the ensemble of codebooks. An upper bound on the error event at the encoder is derived using the method of second moment \\cite{198101TIT_GamMeu}. The probability of the error event at the encoder decays exponentially with $n$ if for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $\n\\begin{eqnarray}\n \\label{Eqn:3BCSourceCodingBoundNonnegativity}\nS_{i}\\!\\!\\!\\!&>&\\!\\!\\!\\! 0\\\\\n\\label{Eqn:3BCSourceCodingPairwiseBound}\nS_{ij}+S_{jk}\\!\\!\\!\\! &>& \\!\\!\\!\\!I(U_{ij};U_{jk}|WQ) \\\\\n\\label{Eqn:3BCSourceCodingTripleBound}\nS_{ij}+S_{jk}+S_{ki} \\!\\!\\!\\!&>&\\!\\!\\!\\! I(U_{ij};U_{jk};U_{ki}|WQ)\\footnotemark\\\\\n\\label{Eqn:3BCSourceCodingQuadrapleBound}\nS_{i}+S_{ij}+S_{jk}+S_{ki}\\!\\!\\!\\!&>&\\!\\!\\!\\! I(U_{ij};U_{jk} ;U_{ki}|WQ)\n +I(V_{i};U_{jk}|U_{ij},U_{ki},WQ)\\\\\nS_{i}+S_{j}+S_{ij}+S_{jk}+S_{ki} \\!\\!\\!\\!&>& \\!\\!\\!\\!I(V_{i};U_{jk}|U_{ij},U_{ki},WQ)+I(V_{j};U_{ki}|U_{ij},U_{jk},WQ)\\nonumber\\\\\n\\label{Eqn:3BCSourceCodingPentaBound}\n&&\\!\\!\\!\\!+I(U_{ij};U_{jk};U_{ki}|WQ)+I(V_{i};V_{j}|U_{jk},U_{ij},U_{ki},WQ)\\\\\nS_{1}+S_{2}+S_{3}+S_{12}+S_{23}+S_{31} \\!\\!\\!\\!&>& \\!\\!\\!\\!I(V_{1};U_{23}|U_{12},U_{31},WQ)+I(V_{2};U_{31}|U_{12},U_{23},WQ)+I(V_{1};V_{2};V_{3}|\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV)\\nonumber\\\\\n\\label{Eqn:3BCSourceCodingSextupleBound}\n&&\\!\\!\\!\\!+I(U_{12};U_{23};U_{31}|WQ)+I(V_{3};U_{12}|U_{23},U_{31},WQ).\n\\end{eqnarray}\n\\footnotetext{For three random variables, $A,B,C, I(A;B;C)=I(A;B)+I(AB;C)$.}\nThe probability of error event at the decoders decays exponentially if for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $\n\\begin{flalign}\n\\label{Eqn:3BCChannelCodingSingleBound}\nI(V_i;Y_i|QWU_{ij}U_{ki}) &\\geq T_i+S_i  \\\\\n\\label{Eqn:3BCChannelCodingDoubleBound}\nI(U_{ij}V_i;Y_i|QWU_{ki})+I(U_{ij};U_{ki}|QW)  &\\geq\nK_{ij}+L_{ij}+S_{ij}+T_i+S_i \\\\\n\\label{Eqn:3BCChannelCodingDoubleSecondBound}\nI(U_{ki}V_i;Y_i|QWU_{ij})+I(U_{ij};U_{ki}|QW)  &\\geq\nK_{ki}+L_{ki}+S_{ki}+T_i+S_i \\\\\n\\label{Eqn:3BCChannelCodingTripleBound}\nI(U_{ij}U_{ki}V_i;Y_i|QW)+I(U_{ij};U_{ki}|QW) &\\geq\nK_{ij}+L_{ij}+S_{ij}+K_{ki}+L_{ki}+S_{ki}+T_i+S_i \\\\ \n\\label{Eqn:3BCChannelCodingSextupleBound}\nI(WU_{ij}U_{ki}V_i;Y_i|Q)+I(U_{ij};U_{ki}|QW) &\\geq\nK_i+K_j+K_k+K_{ij}+L_{ij}+S_{ij}+K_{ki}+L_{ki}+S_{ki}+T_i+S_i\n\\end{flalign}\n\\begin{comment}{\\begin{flalign}\n\\label{Eqn:3BCChannelCodingSingleBound}\nI(V_i;Y_i|QWU_{ij}U_{ki}) &\\geq \\bold{T_i} \\\\\n\\label{Eqn:3BCChannelCodingDoubleBound}\nI(U_{ij}V_i;Y_i|QWU_{ki})+I(U_{ij};U_{ki}|QW)  &\\geq\n\\bold{K_{ij}}+\\bold{T_i} \\\\\n\\label{Eqn:3BCChannelCodingDoubleSecondBound}\nI(U_{ki}V_i;Y_i|QWU_{ij})+I(U_{ij};U_{ki}|QW)  \\geq&\n\\bold{K_{ki}}+\\bold{T_i}  \\\\\n\\label{Eqn:3BCChannelCodingTripleBound}\nI(U_{ij}U_{ki}V_i;Y_i|QW)+I(U_{ij};U_{ki}|QW) \\geq\n\\bold{K_{ij}}&+\\bold{K_{ki}}+\\bold{T_i} \\\\ \nI(WU_{ij}U_{ki}V_i;Y_i|Q)+I(U_{ij};U_{ki}|QW) &\\geq\nK_i+K_j+K_k\\nonumber\\\\\n\\label{Eqn:3BCChannelCodingSextupleBound}+\\bold{K_{ij}}+\\bold{K_{ki}}+&\\bold{T_i},\n\\end{flalign}\n}\\end{comment}\nFor each pmf $p_{\\TimeSharingRV W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}W_{\\underlineY|\\InputRV}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet \\times \\underline{\\OutputAlphabet}$, let $\\alpha_{NEM}(p_{QW\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV\\underlineY})$ denote the set of all triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty)^{4}$ such that (i) there exists non-negative real numbers $K_{ij},L_{ij},S_{ij},K_{j},T_{j},S_{j}$ that satisfies (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for each pair $(i,j) \\in \\left\\{ (1,2),(2,3),(3,1)  \\right\\}$ and (ii) $R_{j}=T_{j}+K_{jk}+L_{ij}+K_{j}$ for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $. Natural extension of Marton's rate region for the 3-DBC with private messages is\n\\begin{equation}\n \\label{Eqn:3BCMartonRateRegionAsAUnionOfTestChannelRateRegions}\n \\alpha_{NEM}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV\n\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\n\\underlinePrivateRV X \\underlineY})\\right),\\nonumber\n\\end{equation}\nwhere $\\SetOfDistributions_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of distributions\n$p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV \\underlineY}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times\n\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet \\times\n\\underlineSetY$, where (i) $\\TimeSharingRVSet,\\PublicRVSet,\n\\underlineSemiPrivateRVSet, \\underlinePrivateRVSet$ are finite sets, (ii)\n$p_{\\underlineY|X \\underlinePrivateRV \\underlineSemiPrivateRV \\PublicRV\\TimeSharingRV}=p_{\\underlineY|X} = W_{\\underlineY|X}$,\n(iii) $\\Expectation\\left\\{\\kappa(X)\\right\\} \\leq \\tau$.\n\\section{\\3To1BC broadcast channels and the need for structured codes}\n\\label{Sec:3To1BCAndTheNeedForStructuredCodes}\n\nIn this section, we define \\3To1BC DBC, a class of BCs for which a natural extension of Marton's coding technique is strictly suboptimal. In particular, in section \\ref{SubSec:TheThreeUserBroadcastChannel} we present a novel 3-DBC for which the natural extension of Marton's coding technique is strictly suboptimal. The class of \\3To1BC DBCs, defined in section \\ref{SubSec:ThreeToOneBroadcastChannels}, is obtained by a natural generalization of this example. \\3To1BC DBCs also being the simplest class of DBCs for which the limitations of traditional unstructured independent codes and the need for structure can be easily demonstrated provides a pedagogical step for describing our findings. We begin by a discussion on the limitations of traditional unstructured independent codes.\n\n\\subsection{Limitations of unstructured codes}\n\\label{SubSec:LimitationsOfUnstructuredCodes}\nThe central aspect of any coding technique designed for a BC is \\textit{interference management}. Marton's coding incorporates two fundamental coding strategies for interference management - superposition and binning. In this discussion, we restrict attention to the strategy of superposition in Marton's coding. Transmission of information bearing signal of each user causes interference to the other user. As against to being completely oblivious to the other user's transmissions and therefore treating it as noise, superposition enables each user to decode a component of the other user's signal. Traditional unstructured independent codebooks facilitates this by building a two layer - cloud center and satellite - code. The cloud center code contains components of each user's signal that is decoded by the other user. Over a degraded BC, for which superposition is optimal, the more capable receiver decodes the entire signal carrying the less capable user's information.\n\nOver a general non-degraded channel what does it mean to say a \\textit{component} of the other users' signal? The coding technique as proposed by Marton does not provide an obvious answer to this question. However, a recent interpretation by Nair \\cite{2009MMICWCSP_Nai} makes this quite explicit. In particular, Nair proves superposition in Marton's coding is tantamount to each user decoding a (univariate) function of the signal carrying the other users' information. Due to the presence of just one interferer in a 2-BC this is a reasonable strategy.\n\nHow does one tackle interference in a 3-BC? The signal input to this channel is a function of three signals, one intended for each user. \\textit{The reception at each receiver is therefore plagued, in general, by a bivariate function of signals intended for the other users. It is therefore natural to enable each user decode the relevant bivariate interfering component, in addition to a univariate component of the other two user's signal.} Does the natural extension of Marton's coding `truly' decode a bivariate interference component of the other users' signals and if yes, how? Traditional unstructured independent coding, on which Marton's technique is based does not enable decoding a bivariate component of signals without decoding the arguments in their entirety. The latter strategy is in general inefficient. Before we provide a formal proof of this claim in section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we lend credence by referring to the gamut of related problems.\n\nAptly describing this phenomenon, the problem of reconstructing modulo-2 sum of distributed binary sources \\cite{197903TIT_KorMar} brings to light the limitations of traditional independent unstructured coding. Even after three decades, we are unaware of a coding technique based on traditional unstructured independent coding that achieves rates promised by K\\\"orner and Marton. More recently, other problem instances \\cite{200906TIT_PhiZam}, \\cite{200710TIT_NazGas}, \\cite{201103TIT_KriPra}, \\cite{200809Allerton_SriJafVisJafSha},\\cite{201207ISIT_PadSahPra} have been identified, wherein the need to decode a bivariate function has been efficiently met by employing structured codebooks. While all of the above works propose ingenious coding techniques based on structured codes, the example studied in \\cite[Section V]{201207ISIT_PadSahPra} closely illustrates the role played by structured codes in decoding bivariate interference component for the problem in context.\\footnote{That all of the above works are centered around an additive problem - reconstruction of modulo-2 sum of binary sources or communicating over additive channels corroborates the phenomenon under question. Decoding a bivariate function without decoding it's arguments is more efficient than decoding the individual arguments if the bivariate function is compressive, i.e., the entropy of the function is significantly lower when compared to the joint entropy of the arguments. Indeed, the simplest such bivariate function of binary sources is an addition. Furthermore, a good understanding of linear codes that facilitated decoding the sum has constrained much of the attention thus far to additive problems. It is our belief that these examples, being only the tip of an iceberg, point to more efficient encoding and decoding techniques for multi-terminal communication problems based on structured codebooks.}\n\nHow do structured codes enable decode the bivariate interference component more efficiently? This is best described by the use of linear codes in decoding the sum interference component. Let us assume that codebooks of users $2$ and $3$ are built over the the binary field $\\BinaryField$ and the sum of user $2$ and $3$ codewords is the interference component at receiver $1$. If user $2$ and $3$ build independent codebooks of rate $R_{2}$ and $R_{3}$ respectively, the range of interference patterns has rate $R_{2}+R_{3}$. Enabling user $1$ decode the sum interference pattern constrains the sum $R_{2}+R_{3}$. Instead if codebooks of users $2$ and $3$ are sub-codes of a common linear code, then enabling user $1$ to decode the sum will only constrain $\\max\\left\\{ R_{2},R_{3}\\right\\}$.\n\n\\subsection{A three user broadcast channel}\n\\label{SubSec:TheThreeUserBroadcastChannel}\nIn this section, we present a novel 3-DBC that makes concrete our remarks in section \\ref{SubSec:LimitationsOfUnstructuredCodes}. In section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove strict sub-optimality of Marton's coding technique for the 3-DBC presented herein. The novelty of this 3-DBC lies in our ability to prove strict containment of Marton's rate region even without a compact description of the latter. \n\\begin{example}\n \\label{Ex:3-BCExample}\nConsider the 3-DBC depicted in figure \\ref{Fig:TheThreeUserBroadcastChannel}. Let the input alphabet $\\InputAlphabet = \\InputAlphabet_{1} \\times \\InputAlphabet_{2} \\times \\InputAlphabet_{3}$ be a triple Cartesian product of the binary field $\\InputAlphabet_{1}=\\InputAlphabet_{2} = \\InputAlphabet_{3} = \\BinaryField$ and the output alphabets $\\OutputAlphabet_{1}=\\OutputAlphabet_{2}=\\OutputAlphabet_{3}=\\BinaryField$ be binary fields. If $X=X_{1}X_{2}X_{3}$ denote the three binary digits input to the channel, then the outputs are $Y_{1}=X_{1}\\oplus X_{2}\\oplus X_{3} \\oplus N_{1}$, $Y_{2} =X_{2} \\oplus N_{2}$ and $Y_{3} = X_{3} \\oplus N_{3}$, where (i) $N_{1}, N_{2}, N_{3}$ are independent binary random variables with $P(N_{j}=1)=\\delta_{j} \\in (0,\\frac{1}{2})$ and (ii) $(N_{1},N_{2},N_{3})$ is independent of the input $X$. The binary digit $X_{1}$ is constrained to an average Hamming weight of $\\tau \\in (0,\\frac{1}{2})$. In other words, $\\kappa (x_{1}x_{2}x_{3}) = 1_{\\left\\{x_{1}=1\\right\\}}$ and the average cost of input is constrained to $\\tau\\in (0,\\frac{1}{2})$. For the sake of clarity, we provide a formal description of this channel in terms of section \\ref{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}. This 3-DBC maybe referred to as $(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$ where $\\InputAlphabet \\define \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\} \\times \\left\\{ 0,1 \\right\\}, \\OutputAlphabet_{1}=\\OutputAlphabet_{2}=\\OutputAlphabet_{3}=\\left\\{ 0,1 \\right\\}, W_{\\underlineY|X}(y_{1},y_{2},y_{3}|x_{1}x_{2}x_{3})=BSC_{\\delta_{1}}(y_{1}|x_{1}\\oplus x_{2}\\oplus x_{3})BSC_{\\delta_{2}}(y_{2}|x_{2})BSC_{\\delta_{3}}(y_{3}|x_{3})$, where $\\delta_{j} \\in (0,\\frac{1}{2}):j=1,2,3$, $BSC_{\\eta}(1|0)=BSC_{\\eta}(0|1)=1-BSC_{\\eta}(0|0)=1-BSC_{\\eta}(1|1)=\\eta$ for any $\\eta \\in (0,\\frac{1}{2})$ and the cost function $\\kappa (x_{1}x_{2}x_{3}) = 1_{\\left\\{x_{1}=1\\right\\}}$.\n\\end{example}\n\\begin{figure}\n\\centering\n\\includegraphics[height=1.8in,width=1.9in]{binex}\n\\caption{A 3-DBC with  octonary input and  binary outputs described in example \\ref{Ex:3-BCExample}.}\n\\label{Fig:TheThreeUserBroadcastChannel}\n\\end{figure}\nWe begin with some observations for the above channel. Users $2$ and $3$ see \\textit{interference free point to point} links from the input. It is therefore possible to communicate to them simultaneously at their point to point capacities using any point to point channel codes achieving their respective capacities. For the purpose of this discussion, let us assume $\\delta \\define \\delta_{2}=\\delta_{3}$. This enables us employ the same capacity achieving code of rate $1-h_{b}(\\delta)$ for both users $2$ and $3$. What about user $1$? Three observations are in order. Firstly, if users $2$ and $3$ are being fed at their respective point to point capacities, then information can be pumped to user $1$ only through the first binary digit, henceforth referred to as $X_{1}$. In this case, we recognize that the sum of user $2$ and $3$'s transmissions interferes at receiver $1$. Thirdly, the first binary digit $X_{1}$ is costed, and therefore cannot cancel the interference caused by users $2$ and $3$.\n\nSince average Hamming weight of $X_{1}$ is restricted to $\\tau$, $X_{1}\\oplus N_{1}$ is restricted to an average Hamming weight of $\\tau * \\delta_{1}$. If the rates of users $2$ and $3$ are sufficiently small, receiver $1$ can attempt to decode codewords transmitted to users $2$ and $3$, cancel the interference and decode the desired codeword. This will require $2-2h_{b}(\\delta) \\leq 1- h_{b}(\\delta_{1} * \\tau)$ or equivalently $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} \\leq h_{b}(\\delta)$. What if this were not the case?\n\nIn the case $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} > h_{b}(\\delta)$, we are left with two choices. The first choice is to enable decoder $1$ decode as large a part of the interference as possible and precode for the rest of the uncertainty.\\footnote{Since $X_{1}$ is costed, precoding results in a rate loss, i.e., in terms of rate achieved, the technique of precoding is in general inferior to the technique of decoding interference. This motivates a preference for decoding the interference as against to precoding. However, for the Gaussian case, precoding suffers \\textit{no} rate loss. This is the precise reason for dirty paper coding being optimal for vector Gaussian BCs \\cite{200609TIT_WeiSteSha}.} The second choice is to attempt decoding the sum of user $2$ and $3$'s codewords, instead of the pair. In the sequel, we pursue the second choice using linear codes. In section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove Marton's coding technique is forced to take the first choice which results in it's sub-optimality.\n\nSince linear codes achieve capacity of binary symmetric channels, there exists a single linear code, or a coset thereof, of rate $1-h_{b}(\\delta)$ that achieves capacity of both user $2$ and $3$ channels. Let us employ this linear code for communicating to users $2$ and $3$. The code being linear or affine, the collection of sums of all possible pairs of codewords is restricted to a coset of rate $1-h_{b}(\\delta)$. This suggests that decoder $1$ decode the sum of user $2$ and $3$ codewords. Indeed, if $1-h_{b}(\\delta) \\leq 1-h_{b}(\\tau * \\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta$, then user $1$ can first decode the interference, peel it off, and then go on to decode the desired signal. Under this case, a rate $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ is achievable for user $1$ even while communicating independent information at rate $1-h_{b}(\\delta)$ for both users $2$ and $3$. We have therefore proposed a coding technique based on linear codes that achieves the rate triple $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta))$ if $\\tau*\\delta_{1} \\leq \\delta=\\delta_{2}=\\delta_{3}$. \n\nLet us now consider the general case with respect to $\\delta_{2}, \\delta_{3}$. Without loss of generality we may assume $\\delta_{2} \\leq \\delta_{3}$. We employ a capacity achieving linear code to communicate to user $2$. This code is sub sampled (uniformly and randomly) to yield a capacity achieving code for user $3$. This construction ensures the sum of all pairs of user $2$ and $3$ codewords to lie within user $2$'s linear code, or a coset thereof, of rate $1-h_{b}(\\delta_{2})$. If $1-h_{b}(\\delta_{2}) \\leq 1-h_{b}(\\tau*\\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta_{2}$, then decoder $1$ can decode the sum of user $2$ and $3$'s codewords, i.e., the interfering signal, peel it off and decode the desired message at rate $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$. If $\\delta_{3} \\leq \\delta_{2}$, then user $2$'s code is obtained by sub-sampling a capacity achieving linear code provided to user $3$. In this case, user $1$ can be fed at rate of $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ if $1-h_{b}(\\delta_{3}) \\leq 1-h_{b}(\\tau*\\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta_{3}$\nThe above arguments are summarized in the following lemma.\n\n\\begin{lemma}\n \\label{Lem:RateTripleAchievableUsingLinearCodes}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample}. If $\\tau*\\delta_{1} \\leq \\min\\left\\{\\delta_{2},\\delta_{3}\\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\mathbb{C}(\\tau)$.\n\\end{lemma}\n\nIn section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove that if $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\notin \\alpha_{NEM}(\\tau)$. We therefore conclude in corollary \\ref{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal} that if $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ and $\\min\\left\\{ \\delta_{2},\\delta_{3}\\right\\}\\geq \\delta_{1} * \\tau$, then Marton's coding technique is strictly suboptimal for the 3-DBC presented in example \\ref{Ex:3-BCExample}. In particular, if $\\tau, \\delta_{1}, \\delta=\\delta_{2}=\\delta_{3}$ are such that $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} > h_{b}(\\delta) \\geq h_{b}(\\delta_{1} * \\tau)$, then Marton's coding technique is strictly suboptimal for the 3-DBC presented in example \\ref{Ex:3-BCExample}. While the proof of this statement is long, the curious reader may sample our conclusion in theorem \\ref{Thm:ConditionsUnderWhichMartonCodingTechniqueDoesNotAchieveRateTriple} and corollary \\ref{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal}.\n\\subsection{\\3To1BC broadcast channels}\n\\label{SubSec:ThreeToOneBroadcastChannels}\n\nIn this section, we characterize \\3To1BC DBCs - a class of 3-DBCs for which the 3-DBC presented in section \\ref{SubSec:TheThreeUserBroadcastChannel} is an example. This charcterization resembles the notion of \\3To1BC interference channel studied in \\cite{200912arXiv_CadJaf,201009TIT_BreParTse}, \\cite{201207ISIT_PadSahPra}. Being the first known class of BCs for which Marton's coding technique is suboptimal, it's importance cannot be overemphasized. In addition, it provides a pedagogical step in deriving an achievable rate region for the 3-DBC using structured codes. Section \\ref{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes} is dedicated to deriving achievable rate regions for \\3To1BC DBC using nested coset codes \\cite{201108ISIT_PadPra}.\n\nA 3-DBC\n$(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X})$\nis a 3-to-1 DBC if $\\mathcal{X}=\\mathcal{X}_{1}\\times \\mathcal{X}_{2} \\times\n\\mathcal{X}_{3}$ is a Cartesian product of three alphabet sets such that\n$W_{Y_{2}|X}(y_{2} |\n(x_{1},x_{2},x_{3}))=W_{Y_{2}|X_{2}}(y_{2}|x_{2})$ and $W_{Y_{3}|X}(y_{3} |\n(x_{1},x_{2},x_{3}))=W_{Y_{3}|X_{3}}(y_{3}|x_{3})$.  Note that transition probabilities\n$W_{Y_1,Y_2,Y_3|X}$ of a 3-to-1 DBC can be denoted as\n$W_{Y_1,Y_2,Y_3|X_{1}X_{2}X_{3}}$. Since users $2$ and $3$ enjoy interference free point to point links, the corresponding receivers need to decode only signals intended for them. Therefore, there is no need for a (i) public codebook that each of the users decode into and (ii) a codebook that both users $2$ and $3$ decode into.\n\n\\section{Achievable rate regions for \\3To1BC broadcast channels using nested coset codes}\n\\label{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}\n\nIn this section, we present the first step in deriving an achievable rate region for the general 3-DBC using nested coset codes \\cite{201108ISIT_PadPra}. In particular, we restrict attention to \\3To1BC DBCs and derive an achievable rate region for this class using the ensemble of nested coset codes. The coding technique we propose is a generalization of the simple linear coding strategy proposed for example \\ref{Ex:3-BCExample}. The reader may wish to revisit the same.\n\n\\subsection{Decoding sum of codewords using nested coset codes}\n\\label{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}\nThe essential aspect of the linear coding strategy proposed for example \\ref{Ex:3-BCExample} is that users $2$ and $3$ employ a code that is closed under addition, the linear code being the simplest such example. Since linear codes only achieve symmetric capacity, we are forced to bin codewords from a larger linear code in order to find codewords that are typical with respect to a nonuniform distribution. This is akin to binning for channels with state information, wherein $\\exp\\left\\{ nI(U;S) \\right\\}$ codewords, each picked according to $\\prod_{t=1}^{n}p_{T}$, are chosen for each message in order to find a codeword in $T_{\\delta}(U|s^{n})$ jointly typical with state sequence $s^{n}$.\n\nWe now generalize the coding technique proposed for example \\ref{Ex:3-BCExample}. Consider auxiliary alphabet sets $\\PrivateRVSet_{1},\\SemiPrivateRVSet_{21},\\SemiPrivateRVSet_{31}$ where $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31} =\\fieldq$ be the finite field of cardinality $q$ and let $p_{\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31} X \\underlineY}$ be a pmf on $\\PrivateRVSet_{1} \\times \\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{31} \\times \\InputAlphabet \\times \\underlineSetY$.\\footnote{The seemingly queer notation employed herein is in anticipation of a generalization in section \\ref{SubSec:AnEnlargedAchievableRateRegion}. As earlier $\\PrivateRV$ stands for a private codebook and $\\SemiPrivateRV$ stands for a semi-private codebook. This will be evident from the coding technique described further.} For $j=2,3$, let $\\lambda_{j} \\subseteq \\SemiPrivateRVSet_{j1}^{n}$ be coset of a linear code $\\overline{\\lambda_{j}}\\subseteq\\fieldq^{n}$ of rate $S_{j1}+T_{j1}$. The linear codes are contained in one another, i.e., if $S_{j_{1}1}+T_{j_{1}1} \\leq S_{j_{2}2}+T_{j_{2}2}$, then $\\overline{\\lambda_{j_{1}}} \\subseteq \\overline{\\lambda_{j_{2}}}$. Codewords of $\\lambda_{j}$ are partitioned independently and uniformly into $\\exp \\left\\{ nT_{j} \\right\\}$ bins. A codebook $\\mathcal{C}_{1}$ of rate $K_{1}+L_{1}$ is built over $\\PrivateRVSet_{1}$. The codewords of $\\mathcal{C}_{1}$ are independently and uniformly partitioned into $\\exp \\left\\{ nL_{1} \\right\\}$ bins. Messages of users $1,2,3$ at rates $L_{1},T_{21},T_{31}$ is used to index a bins in $\\mathcal{C}_{1},\\lambda_{2},\\lambda_{3}$ respectively. The encoder looks for a jointly typical triple, with respect to $p_{\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}$, of codewords in the indexed triple of bins. Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical triple if\n\\begin{eqnarray}\n \\label{Eqn:SingleLayerSourceCodingBounds}\nS_{j1} \\geq \\log_{2} q - H(U_{j1}):j=2,3,~~~ K_{1} \\geq 0,~~~\nS_{21}+S_{31} \\geq 2\\log_{2} q - H(U_{21})-H(U_{31})+I(U_{21};U_{31})\\\\\nS_{j1}+K_{1} \\geq \\log_{2} q - H(U_{j1})+I(U_{j1};V_{1}):j=2,3,\n\\sum_{j=2}^{3}S_{j1}+K_{1} \\geq 2\\log_{2} q - \\sum_{j=2}^{3}H(U_{j1})+I(U_{21};U_{31};V_{1}).\n\\end{eqnarray}\nHaving chosen one such jointly typical triple, say $V_{1}^{n},U_{21}^{n},U_{31}^{n}$, it generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}(\\cdot | \\PrivateRV_{1}^{n},\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n})$ and inputs the same on the channel.\n\nDecoders $2$ and $3$ perform a standard point to point channel decoding. For example, decoder $2$ receives $Y_{2}^{n}$ and looks for all codewords in $\\lambda_{2}$ that are jointly typical with $Y_{2}^{n}$. If it finds all such codewords in a unique bin it declares the corresponding bin index as the decoded message. It can be proved by following the technique similar to \\cite[Proof of Theorem 1]{201108ISIT_PadPra} that if\n\\begin{equation}\n \\label{Eqn:BoundsOnUser2And3Rates}\nS_{j1}+T_{j1} \\leq \\log_{2} q - H(\\SemiPrivateRV_{j1}|Y_{j}):j=2,3 \n\\end{equation}\nthen probability of decoding error at decoders $2$ and $3$ can be made arbitrarily small for sufficiently large $n$.\n\nHaving received $Y_{1}^{n}$, decoder $1$ looks for all codewords $v_{1}^{n} \\in \\mathcal{C}_{1}$ for which there exists a codeword $u_{2 \\oplus 3}^{n} \\in (\\lambda_{2} \\oplus \\lambda_{3})$ such that $(v_{1}^{n},u_{2 \\oplus 3}^{n},Y_{1}^{n})$ are jointly typical with respect to $p_{\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{3},\\PrivateRV_{1},Y_{1}}$. Here $(\\lambda_{2}\\oplus \\lambda_{3}) \\define \\left\\{ v_{2}^{n}\\oplus v_{3}^{n}: v_{j}^{n} \\in \\lambda_{j}^{n}:j=2,3 \\right\\}$.\\footnote{Recall that structure of $\\lambda_{2},\\lambda_{3}$ contains cardinality of $(\\lambda_{2}\\oplus \\lambda_{3})$. In particular $|(\\lambda_{2}\\oplus \\lambda_{3})|\\leq \\exp \\left\\{ \\min \\left\\{ S_{21}+T_{21},S_{31}+T_{31} \\right\\} \\right\\}$} If all such codewords in $\\mathcal{C}_{1}$ belong to a unique bin, the corresponding bin index is declared as the decoded message. Again following the technique similar to \\cite[Proof of Theorem 1]{201108ISIT_PadPra}, it can be proved, that if\n\\begin{flalign}\n \\label{Eqn:BoundsOnUser1Rates}\nK_{1}\\!+\\!L_{1} \\!\\leq \\!H(V_{1})\\! - \\!H(\\PrivateRV_{1}|\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{31},Y_{1}) ,~~~\nK_{1}\\!+\\!L_{1}\\!+\\!S_{j1}\\!+\\!T_{j1} \\leq \\log_{2} q +H(V_{1})\\!-\\! H(\\PrivateRV_{1},\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{31}|Y_{1})\n\\end{flalign}\nfor $j=2,3$, then probability of decoding error at decoder $1$ falls exponentially with $n$.\n\nSince $L_{1},T_{21},T_{31}$ denotes rate achievable by users $1,2,3$ respectively, eliminating $K_{1},S_{21},S_{31}$ from the set of equations (\\ref{Eqn:SingleLayerSourceCodingBounds})-(\\ref{Eqn:BoundsOnUser1Rates}) yields an achievable rate region. The following definition and theorem provide a precise mathematical characterization of this achievable rate region.\n\\begin{definition}\n \\label{Defn:TestChannelsForSingleLayerWithOneUserDecodingInterference}\nLet $\\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of pmf's $p_{\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}\\PrivateRV_{1} X \\underlineY}$ defined on $\\SemiPrivateRVSet_{21}\\times \\SemiPrivateRVSet_{31}\\times \\PrivateRVSet_{1} \\times \\InputAlphabet \\times \\underlineSetY$, where (i) $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31}=\\fieldq$ is the finite field of cardinality $q$, $\\PrivateRVSet_{1}$ is a finite set, (ii) $p_{\\underlineY|X\\PrivateRV_{1}\\underlineSemiPrivateRV}=p_{\\underlineY|X}=W_{\\underlineY|X}$, and (iii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$. For $p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY} \\in \\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})$ be defined as the set of triples $(R_{1},R_{2},R_{3})$ that satisfy\n\\begin{flalign}\n\\label{Eqn:OneLayerRateRegionForSpecificTestChannel}\n0 \\leq R_{1} &\\leq I(V_{1};U_{21} \\oplus U_{31},Y_{1}),~~~~R_{1}+R_{j} \\leq H(V_{1}U_{j1})-H(U_{j1}|Y_{j})-H(V_{1}|U_{21}\\oplus U_{31},Y_{1}):j=2,3\\nonumber\\\\\n0 &\\leq R_{j} \\leq I(U_{j1};Y_{j}),~~~~R_{1}+R_{j} \\leq H(V_{1}U_{j1})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1}):j=2,3\\nonumber\\\\\nR_{2}+R_{3} &\\leq I(U_{21};Y_{2})+I(U_{31};Y_{3})-I(U_{21};U_{31}),\\nonumber\\\\ \n\\sum_{k=1}^{3}R_{k} &\\leq H(U_{21}U_{31}V_{1})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1})-\\max \\left\\{H(U_{21}|Y_{2}),H(U_{31}|Y_{3})  \\right\\}\\nonumber\\\\\n\\sum_{k=1}^{3}R_{k}&\\leq H( U_{21}U_{31}V_{1})\\!-\\!H(V_{1}|U_{21}\\!\\oplus\\! U_{31},Y_{1})-\\sum_{k=2}^{3}H(U_{k1}|Y_{k})\\nonumber\\\\\nR_{1}+\\sum_{k=1}^{3}R_{k} &\\leq H(V_{1})+H(U_{21}U_{31}V_{1})\\!-\\!2H(V_{1},U_{21}\\!\\oplus\\! U_{31}|Y_{1}) \\nonumber\\\\\nR_{j}+\\sum_{k=1}^{3}R_{k} &\\leq H(V_{1}U_{j1})+H(U_{21}U_{31})-2H(U_{j1}|Y_{j})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1}):j=2,3\\nonumber\n\\end{flalign}\nand\n\\begin{eqnarray}\n \\label{Eqn:OneLayerAchievableRateRegionWithOneUserDecodingInterference}\n \\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY} \\\\\\in \\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:OneLayerAchievableRateRegionWithOneUserDecodingInterference}\nFor a 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nThe proof is based on upper bounding the average probability of error for the coding scheme described earlier. The codewords in user $1$'s codebook, the $\\mathcal{V}_{1}-$codebook, are picked independently according to $\\prod_{t=1}^{n}p_{V_{1}}$. Assuming $S_{j_{1}1}+T_{j_{1}1} \\leq S_{j_{2}1}+T_{j_{2}1}$, we first pick generator matrix of $\\overline{\\lambda_{j_{1}}}$ by picking each entry independently and uniformly over $\\mathcal{U}_{j_{1}1}=\\mathcal{F}_{q}$. We append this with $n\\left[S_{j_{2}1}+T_{j_{2}1}-(S_{j_{1}1}+T_{j_{1}1})\\right]$ rows, again picking each new entry independently and uniformly over $\\mathcal{U}_{j_{2}1}=\\mathcal{F}_{q}$ to yield generator matrix for $\\overline{\\lambda_{j_{2}}}$. The row space of these generator matrices are shifted by uniformly distributed independent bias vectors $B_{j_{1}1}^{n}:j=2,3$ and the resulting collection of codewords in uniformly and independently partitioned into $\\exp\\left\\{nT_{j_{1}1}\\right\\}$ bins. This induces a distribution over the collection of all triple codebooks and as is typical in information theory, we upper bound the average probability of error. The key element is the use of joint typical encoding and decoding. This enables analyze decoding sum of codewords over arbitrary channels. While we do not provide details in this version of the article, we refer the diligent reader to \\cite[Proof of theorem 2]{201301arXivMACDSTx_PadPra} where a similar decoding rule is analyzed. In the interest of brevity, we omit a proof. The new elements in the proof is the interplay of joint typical encoding and decoding with correlated codebooks.\\footnote{While particular decoding rules such as syndrome decoding of linear codes can achieve capacity of particular channels such as binary symmetric, we will need to employ typical decoding to achieve capacity of arbitrary channels.} Indeed, codebooks being cosets of a common linear code are correlated and moreover, it's codewords are correlated. The analysis of error events contain several new elements.\n\nFor example \\ref{Ex:3-BCExample}, if $\\tau * \\delta_{1} \\leq \\min \\left\\{ \\delta_{2},\\delta_{3} \\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau)$. Indeed, it can be verified that if $\\tau * \\delta_{1} \\leq \\min \\left\\{ \\delta_{2},\\delta_{3} \\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})$, where $p_{\\underlineSemiPrivateRV \\PrivateRV_{1}X}=p_{V_{1}}p_{U_{21}}p_{U_{31}}1_{\\left\\{X_{1}=V_{1}\\right\\}}1_{\\left\\{X_{2}=U_{21}\\right\\}}1_{\\left\\{X_{3}=U_{31}\\right\\}}$, $p_{U_{21}}(1)=p_{U_{31}}(1)=\\frac{1}{2}$ and $p_{V_{1}}(1)=\\tau$.\n\\subsection{An enlarged achievable rate region}\n \\label{SubSec:AnEnlargedAchievableRateRegion}\nWe revisit the coding technique proposed in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}. Observe that (i) user $1$ decodes a sum of the entire codewords\/signals transmitted to users $2$ and $3$ and (ii) users $2$ and $3$ decode only their respective codewords. This technique may be enhanced in the following way. User $1$ can decode the sum of \\textit{one component} of user $2$ and $3$ signals. In other words, we may include private codebooks for users $2$ and $3$.\n\nWe begin with a description of the coding technique. In addition to the codebooks $\\mathcal{C}_{1},\\lambda_{2},\\lambda_{3}$ described in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}, we incorporate private layer codebooks for users $2$ and $3$. Specifically, in addition to auxiliary alphabet sets $\\PrivateRVSet_{1},\\SemiPrivateRVSet_{21},\\SemiPrivateRVSet_{31}$ introduced in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}, let $\\PrivateRVSet_{2},\\PrivateRVSet_{3}$ denote arbitrary finite sets and $p_{U_{21}U_{31}\\underlinePrivateRV}$ denote a pmf on $\\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{31} \\times \\underlinePrivateRVSet$. Two codebooks $\\mathcal{C}_{2} \\subseteq \\PrivateRVSet_{2}^{n},\\mathcal{C}_{3} \\subseteq \\PrivateRVSet_{3}^{n}$ of rates $K_{2}+L_{2}$, $K_{3}+L_{3}$ are partitioned independently and uniformly into $\\exp \\left\\{ nL_{2} \\right\\}, \\exp \\left\\{ nL_{3} \\right\\}$ bins respectively. Messages of users' $2$ and $3$ are split into two parts each. One part of user $2$'s ($3$'s) message, of rate $T_{21}$ ($T_{31}$), index a bin in $\\lambda_{2}$ ($\\lambda_{3}$), and the other part, of rate $L_{2}$ ($L_{3}$), index a bin in $\\mathcal{C}_{2}$ ($\\mathcal{C}_{3}$). The sole part of user $1$'s message indexes a bin in $\\mathcal{C}_{1}$. The encoder looks for a quintuple of jointly typical codewords with respect to $p_{U_{21}U_{31}\\underlinePrivateRV}$, in the quintuple of indexed bins. Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical triple if\n\\begin{equation}\n \\label{Eqn:ManyToOneSourceCodingBounds}\nS_{A}+K_{B} \\geq |A|\\log_{2} q + \\sum_{b \\in B} H(V_{b}) - H(U_{A},V_{B})\n\\end{equation}\nfor all $A \\subseteq \\left\\{21,31\\right\\}, B \\subseteq \\left\\{ 1,2,3 \\right\\}$, where $S_{A} = \\sum _{j1 \\in A}S_{j1}$, $K_{B} = \\sum_{b \\in B}K_{b}$, $U_{A} = (U_{j1}:j1 \\in A)$ and $V_{B}=(V_{b}:b \\in B)$. Having chosen one such jointly typical quintuple, say $(\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n},\\underlinePrivateRV^{n})$, the encoder generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\underlinePrivateRV\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}(\\cdot | \\underlinePrivateRV^{n},\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n})$ and inputs the same on the channel.\n\nThe operations of decoders $2$ and $3$ are identical and we describe one of them. Decoder $3$ receives $Y_{3}^{n}$ and looks for all pairs of codewords in the Cartesian product $\\lambda_{3} \\times \\mathcal{C}_{3}$ that are jointly typical with $Y_{3}^{n}$ with respect to $p_{U_{31}V_{3}Y_{3}}$. If all such pairs belong to a unique pair of bins, the corresponding pair of bin indices is declared as the decoded message of user $3$. Else an error is declared. It can be proved that if\n\\begin{eqnarray}\n \\label{Eqn:UpperboundOnPublicCodebook}\nS_{j1}+T_{j1} \\leq \\log_{2} q-H(U_{j1}|V_{j},Y_{j}),&&\nK_{j}+L_{j} \\leq H(V_{j})-H(V_{j}|Y_{j},U_{j1})\\\\\n\\label{Eqn:UpperboundOnPublicAndPrivateCodebook}\nS_{j1}+T_{j1}+K_{j}+L_{j} &\\leq& \\log_{2} q+H(V_{j})-H(V_{j},U_{j1}|Y_{j})\n\\end{eqnarray}\nfor $j=2,3$, then probability of users $2$ or $3$ decoding into an incorrect message falls exponentially with $n$.\n\nOperation of decoder $1$ is identical to that described in \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}. If (\\ref{Eqn:BoundsOnUser1Rates}) holds, then probability of error at decoder 1 falls exponentially with $n$. Substituting $R_{1}=K_{1}, R_{2}=T_{21}+L_{2}, R_{3}=T_{31}+L_{3}$ and eliminating $S_{21},S_{31},K_{1},K_{2},K_{3}$ in (\\ref{Eqn:BoundsOnUser1Rates})-(\\ref{Eqn:UpperboundOnPublicAndPrivateCodebook}) yields an achievable rate region. We provide a mathematical characterization of this achievable rate region.\n\\begin{definition}\n \\label{Defn:TestChannelsForTwoLayersWithOneUserDecodingInterference}\nLet $\\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of pmf's $p_{\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}\\PrivateRV_{1}\\PrivateRV_{2}\\PrivateRV_{3} X \\underlineY}$ defined on $\\SemiPrivateRVSet_{21}\\times \\SemiPrivateRVSet_{31}\\times \\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times\\PrivateRVSet_{3} \\times \\InputAlphabet \\times \\underlineSetY$, where (i) $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31}=\\fieldq$ is the finite field of cardinality $q$, $\\PrivateRVSet_{1},\\PrivateRVSet_{2}, \\PrivateRVSet_{3}$ are finite sets, (ii) $p_{\\underlineY|X\\underlinePrivateRV\\underlineSemiPrivateRV}=p_{\\underlineY|X}=W_{\\underlineY|X}$, and (iii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$. For $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in \\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{2}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})$ be defined as the set of triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty)^{3}$ for which there exists nonnegative numbers $S_{21},T_{21},S_{31},T_{31},K_{j},L_{j}:j=1,2,3$ such that $R_{1}=K_{1}, R_{2}=T_{21}+L_{2}, R_{3}=T_{31}+L_{3}$ and (\\ref{Eqn:BoundsOnUser1Rates}),(\\ref{Eqn:ManyToOneSourceCodingBounds}),(\\ref{Eqn:UpperboundOnPublicCodebook}),(\\ref{Eqn:UpperboundOnPublicAndPrivateCodebook}) hold. Let\n\\begin{eqnarray}\n \\label{Eqn:TwoLayerAchievableRateRegionWithOneUserDecodingInterference}\n \\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{2}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}\nFor a 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nThe proof is similar to that of theorem \\ref{Thm:OneLayerAchievableRateRegionWithOneUserDecodingInterference}. The only differences being (i) the encoder looks for a quintuple of codewords instead of a triple, and (ii) decoders $2$ and $3$ decode from a pair of codebooks. From  \\cite[Proof of theorem 1]{201108ISIT_PadPra}, the informed reader can see why the second difference can be easily handled. Indeed, in \\cite[Theorem 1]{201108ISIT_PadPra}, we prove nested coset codes achieve capacity of arbitrary point to point channels. This indicates that for $j=2,3$, $\\SemiPrivateRVSet_{j1}-$codebook can be used to communicate at rate $I(\\SemiPrivateRV_{j1};Y_{j})$ and the private layer $\\PrivateRVSet_{j}-$codebook can be used to communicate at rate $I(\\SemiPrivateRV_{j}:Y_{j}|\\SemiPrivateRV_{j1})$, thereby satisfying user $j$'s rate. This leaves us to argue only the first difference pointed above. Employ a second moment method similar to that employed in \\cite{198101TIT_GamMeu}, \\cite{201108ISIT_PadPra}, it can be shown that probability of encoder not finding a jointly typical quintuple decays exponentially if (\\ref{Eqn:ManyToOneSourceCodingBounds}) holds. We defer a detailed proof of theorem \\ref{Thm:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes} to a subsequent enlarged version of this article.\n\\section{Achievable rate region for $3-$user broadcast channel using nested coset codes}\n\\label{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes}\nIn this section, we derive an achievable rate region for the general 3-DBC. This is obtained by a simple extension of the coding technique proposed for the \\3To1BC DBC. In section \\ref{SubSec:AnEnlargedAchievableRateRegion}, only user $1$ decoded a bivariate interference component. Users $2$ and $3$ only decoded from their respective codebooks. This suffices for a \\3To1BC DBC since receivers $2$ and $3$ enjoyed interference free point to point links. In general, each user would attempt to decode a bivariate interference component of the other users' signals. In the sequel, we propose a simple extension of the technique presented in section \\ref{SubSec:AnEnlargedAchievableRateRegion} to enable each user decode a bivariate interference component.\n\nUser $j$ splits it's message $M_{j}$ into three parts $(M_{ji}^{U},M_{jk}^{U},M_{j}^{V})$, where $i,j,k$ are distinct indices in $\\left\\{ 1,2,3 \\right\\}$. Let $\\SemiPrivateRVSet_{ji}=\\fieldqi, \\SemiPrivateRVSet_{jk}=\\fieldqk$ be finite fields and $\\PrivateRVSet_{j}$ be an arbitrary finite set. Let $\\lambda_{ji} \\subseteq \\SemiPrivateRVSet_{ji}^{n}$, $\\lambda_{jk} \\subseteq \\SemiPrivateRVSet_{jk}^{n}$ denote cosets of linear codes $\\overline{\\lambda_{ji}}, \\overline{\\lambda_{jk}}$ of rates $S_{ji}+T_{ji}, S_{jk}+T_{jk}$ respectively. Observe that cosets $\\lambda_{ji}$ and $\\lambda_{ki}$ are built over the same finite field $\\fieldqi$. To enable contain the range the sum of these cosets, the larger of $\\lambda_{ji}$, $\\lambda_{ki}$ contains the other. Codewords of $\\lambda_{ji}$ and $\\lambda_{jk}$ are independently and uniformly partitioned into $\\exp\\left\\{nT_{ji}\\right\\}$ and $\\exp\\left\\{nT_{jk}\\right\\}$ bins respectively. A codebook $\\mathcal{C}_{j}$ of rate $K_{j}+L_{j}$ is built over $\\PrivateRVSet_{j}$. Codewords of $\\mathcal{C}_{j}$ are partitioned into $\\exp\\left\\{ nL_{j}\\right\\}$ bins uniformly and independently at random. \n\n\n$M_{ji}^{U}$,$M_{jk}^{U}$ and $M_{j}^{V}$ index bins in $\\lambda_{ji}$, $\\lambda_{jk}$ and $\\mathcal{C}_{j}$ respectively. The encoder looks for a collection of $9$ codewords from the indexed bins that are jointly typical with respect to a pmf $p_{\\underlineSemiPrivateRV \\underlinePrivateRV}$ defined on $\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet$.\\footnote{$\\underlineSemiPrivateRV$ abbreviates $\\SemiPrivateRV_{12}\\SemiPrivateRV_{13}\\SemiPrivateRV_{21}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}\\SemiPrivateRV_{32}$.} Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical collection if\n\\begin{equation}\n \\label{Eqn:ManyToManySourceCodingBounds}\nS_{A}+K_{B} \\geq \\sum_{a \\in A}\\log |\\mathcal{U}_{a}| + \\sum_{b \\in B} H(V_{b}) - H(U_{A},V_{B})\n\\end{equation}\nfor all $A \\subseteq \\left\\{12,13,21,23,31,32\\right\\}, B \\subseteq \\left\\{ 1,2,3 \\right\\}$, where $S_{A} = \\sum _{jk \\in A}S_{jk}$, $K_{B} = \\sum_{b \\in B}K_{b}$, $U_{A} = (U_{jk}:jk \\in A)$ and $V_{B}=(V_{b}:b \\in B)$. Having chosen one such jointly typical collection, say $(\\underlineSemiPrivateRV^{n},\\underlinePrivateRV^{n})$, the encoder generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\underlineSemiPrivateRV\\underlinePrivateRV}(\\cdot | \\underlineSemiPrivateRV^{n},\\underlinePrivateRV^{n})$ and inputs the same on the channel.\n\nDecoder $j$ receives $Y_{j}^{n}$ and looks for all triples $(u_{ji}^{n},u_{jk}^{n},v_{j}^{n})$ of codewords in $\\lambda_{ji} \\times \\lambda_{jk} \\times \\mathcal{C}_{j}$ such that there exists a $u^{n}_{ij\\oplus kj} \\in (\\lambda_{ij} \\oplus \\lambda_{kj})$ such that $(u_{ij\\oplus kj}^{n},u_{ji}^{n},u_{jk}^{n},v_{j}^{n},Y_{j}^{n})$ are jointly typical with respect to $p_{U_{ij}\\oplus U_{kj},U_{ji},U_{jk},V_{j},Y_{j}}$. If it finds all such triples in a unique triple of bins, the corresponding triple of bin indices is declared as decoded message of user $j$. Else an error is declared. The probability of error at decoder $j$ can be made arbitrarily small for sufficiently large block length if\n\\begin{equation}\n\\begin{aligned}\n\\label{Eqn:ManyToManyBCChannelCodingBounds}\n\\lefteqn{S_{A_{j}}+T_{A_{j}} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| - H(U_{A_{j}}|U_{A_{j}^{c}},U_{ij}\\oplus U_{kj},V_{j},Y_{j})}\n\\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+S_{ij}+T_{ij} \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j} - H(U_{A_{j}},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},V_{j},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+S_{kj}+T_{kj} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j} - H(U_{A_{j}},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},V_{j},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}|+H(V_{j}) - H(U_{A_{j}},V_{j}|U_{A_{j}^{c}},U_{ij}\\oplus\nU_{kj},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j}+S_{ij}+T_{ij} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j}+H(V_{j}) -\nH(U_{A_{j}},V_{j},U_{ij}\\oplus U_{kj}|U_{A_{j}^{c}},Y_{j})} \\\\\n&S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j}+S_{kj}+T_{kj} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j}+H(V_{j}) - H(U_{A_{j}},V_{j},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},Y_{j}), \n\\end{aligned}\n\\end{equation}\nwhere for every $A_{j} \\subseteq \\left\\{ ji,jk\\right\\}$ with distinct indices $i,j,k$ in $\\left\\{ 1,2,3 \\right\\}$, \n\\begin{equation}\n\\label{Eqn:DefinitionOfS_A_jAndT_A_j}\nS_{A_{j}} \\define \\left\\{  \\begin{array}{ll} S_{ji}&\\mbox{ if }S_{A_{j}}=\\left\\{ ji \\right\\} \\\\  S_{jk}&\\mbox{ if }S_{A_{j}}=\\left\\{ jk \\right\\}  \\\\ S_{ji}+S_{jk}&\\mbox{ if }S_{A_{j}}=\\left\\{ ji,jk \\right\\} \\end{array} \\right. \nT_{A_{j}} \\define \\left\\{  \\begin{array}{ll} T_{ji}&\\mbox{ if }T_{A_{j}}=\\left\\{ ji \\right\\} \\\\  T_{jk}&\\mbox{ if }T_{A_{j}}=\\left\\{ jk \\right\\}  \\\\ T_{ji}+T_{jk}&\\mbox{ if }T_{A_{j}}=\\left\\{ ji,jk \\right\\} \\end{array} \\right.  .\n\\end{equation}\nRecognize that user $j$'s rate $R_{j}=T_{ji}+T_{jk}+L_{j}$. We are now equipped to state an achievable rate region for the general 3-DBC using nested coset codes.\n\n\\begin{definition}\n \\label{Defn:CollectionOfTestChannelsForCommunicatingOverGeneral3BCUsingNCC}\nLet $\\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of probability mass functions $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY}$ defined on $\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\mathcal{X} \\times \\underlineSetY$, where $\\underlineSemiPrivateRV\\define (\\SemiPrivateRV_{12},\\SemiPrivateRV_{13},\\SemiPrivateRV_{21},\\SemiPrivateRV_{23},\\SemiPrivateRV_{31},\\SemiPrivateRV_{32})$, $\\underlinePrivateRV\\define (\\PrivateRV_{1},\\PrivateRV_{2},\\PrivateRV_{3})$, $\\underlineSemiPrivateRVSet \\define \\SemiPrivateRVSet_{12} \\times \\SemiPrivateRVSet_{13} \\times \\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{23} \\times \\SemiPrivateRVSet_{31} \\times \\SemiPrivateRVSet_{32},\\underlinePrivateRVSet \\define \\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times \\PrivateRVSet_{3}, \\SemiPrivateRVSet_{ij}=\\fieldqj$, the finite field of cardinality $q_{j}$ for each $1\\leq  i,j\\leq3$, $\\SemiPrivateRVSet_{i}$ is an arbitrary finite set such that (i) $p_{\\underlineY| X \\underlinePrivateRV \\underlineSemiPrivateRV}=p_{\\underlineY |X}=W_{\\underlineY |X}$, (ii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$.\n\nFor $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in \\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{f}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})$ is defined as the set of rate triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty]^{3}$ for which there exists nonnegative numbers $S_{ij}:ij \\in \\left\\{12,13,21,23,31,32 \\right\\}, T_{jk}:jk \\in \\left\\{12,13,21,23,31,32 \\right\\}, K_{j}:j \\left\\{ 1,2,3\\right\\}, L_{j}:\\left\\{1,2,3 \\right\\}$ that satisfy (\\ref{Eqn:ManyToManySourceCodingBounds})-(\\ref{Eqn:DefinitionOfS_A_jAndT_A_j}) and $R_{1}=T_{12}+T_{13}+L_{1}, R_{2}=T_{21}+T_{23}+L_{2}, R_{3}=T_{31}+T_{32}+L_{3}$. Let\n\\begin{eqnarray}\n \\label{Eqn:AchievableRateRegionFor3BCUsingNestedCosetCodes}\n \\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in\\\\ \\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{f}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:AchievableRateRegionFor3BCUsingNestedCosetCodes}\nFor 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nWe defer the proof of this theorem to a subsequent enlarged version of this article.\n\\begin{comment}{\n\\begin{eqnarray}\n \\label{Eqn:ManyToManyBCChannelCodingBounds}\n\\lefteqn{ST_{A} \\leq |A|\\log_{2} q - H(U_{A}|U_{A^{c}},U_{ij}\\oplus U_{kj},V_{j},Y_{j})}\n\\nonumber\\\\\n\\lefteqn{ST_{A}+ST_{ij} \\leq (|A|+1)\\log_{2} q - H(U_{A},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},V_{j},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+ST_{kj} \\leq (|A|+1)\\log_{2} q - H(U_{A},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},V_{j},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+KL_{j} \\leq |A|\\log_{2} q+H(V_{j}) - H(U_{A},V_{j}|U_{A^{c}},U_{ij}\\oplus\nU_{kj},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+KL_{j}+ST_{ij} \\leq (|A|+1)\\log_{2} q+H(V_{j}) -\nH(U_{A},V_{j},U_{ij}\\oplus U_{kj}|U_{A^{c}},Y_{j})} \\nonumber\\\\\n&\\!\\!\\!\\!\\!\\!ST_{A}+KL_{j}+ST_{kj} (\\leq |A|+1)\\log_{2} q+H(V_{j}) -\nH(U_{A},V_{j},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},Y_{j}) \\nonumber\n\\end{eqnarray}\nwhere $ST_{A} \\define S_{A}+T_{A}$, $ST_{ij}\\define S_{ij}+T_{ij}$, $ST_{kj}\n=S_{kj}+T_{kj}$, $KL_{j}=K_{j}+L_{j}$\n}\\end{comment}\n\n\\section{Enlarging Marton's rate region using nested coset codes}\n\\label{Sec:EnlargingMarton'sRateRegionUsingNestedCosetCodes}\nThe natural question that arises is whether the achievable rate region using nested coset codes $\\beta(W_{\\underlineY|X},\\kappa,\\tau)$ contains Marton's rate region $\\alpha_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)$. It is our belief that coding techniques based on structured codes do not substitute their counterparts based on traditional unstructured independent codes, but enhance the same. Indeed, the technique proposed by K\\\"orner and Marton \\cite{197903TIT_KorMar} is strictly suboptimal to that studied by Berger and Tung \\cite{Berger-MSC} if the function is not sufficiently compressive, i.e., entropy of the sum is larger than one half of the joint entropy of the sources.\\footnote{If $X$ and $Y$ are the distributed binary sources whose modulo$-2$ sum is to be reconstructed at the decoder, then K\\\"orner and Marton technique is strictly suboptimal if $H(X \\oplus Y) > \\frac{H(X,Y)}{2}$.} The penalty paid in terms of the binning rate for endowing structure is not sufficiently compensated for by the function. This was (recognized)\/(hinted at) by Ahlswede and Han \\cite[Section VI]{198305TIT_AhlHan} for the problem studied by K\\\"orner and Marton.\n\nWe follow the approach of Ahlswede and Han \\cite[Section VI]{198305TIT_AhlHan} to build upon Marton's rate region by gluing to it the coding technique proposed herein. In essence the coding techniques studied in section \\ref{SubSec:NaturalExtensionOfMartonTo3BC} and \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes} are glued together.\\footnote{This is akin to the use of superposition and binning in Marton's coding.} Indeed, a description of the resulting rate region is quite involved and we spare the reader of these details. The resulting coding technique will involve each user split it's message into six parts - one public and private part each, two semi-private and \\textit{bivariate} parts each. This can be understood by splitting the message as proposed in sections \\ref{SubSec:NaturalExtensionOfMartonTo3BC} and \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes} and identifying the private parts. In essence each user decodes a univariate component of every other user's transmission particularly set apart for it, and furthermore decodes a bivariate component of the other two user's transmissions.\\footnote{An informed and inquisitive reader may begin to see a relationship emerge between the several layers of coding and common parts of a collection of random variables. Please refer to section \\ref{Sec:ConcludingRemarks} for a discussion.} A mathematical characterization of the resulting achievable rate region will be provided in a subsequent enlarged version of this article. For now, we conclude by stating that the resulting achievable rate region 1) contains and strictly enlarges Marton's rate region for the general 3-DBC and 2) is therefore the currently known largest achievable rate region for the same.\n\n\\section{Concluding remarks : Common parts of random variables and the need for structure}\n\\label{Sec:ConcludingRemarks}\nLet us revisit Marton's coding technique for 2-BC. Define the pair $\\overline{\\PrivateRV_{j}}\\define (\\PublicRV, \\PrivateRV_{j}):j=1,2$ of random variables decoded by the two users and let $\\overline{\\PrivateRVSet_{j}}\\define \\PublicRVSet \\times \\PrivateRVSet_{j}:j=1,2$. Let us stack the collection of compatible codewords over $\\overline{\\PrivateRVSet_{1}}^{n} \\times \\overline{\\PrivateRVSet_{2}}^{n}$. The encoder can work with this stack, being  oblivious to the distinction between $\\PublicRVSet$ and $\\PrivateRVSet_{j}:j=1,2$. In other words, it does not recognize that a symbol over $\\overline{V_{j}}$ is indeed a pair of symbols. A few key observations of this stack of codewords is in order. Recognize that many pairs of compatible codewords agree in their `$\\PublicRVSet-$coordinate'. In other words, they share the same codeword on the $\\PublicRVSet-$codebook. $\\PublicRV$ is a common part \\cite{1972MMPCT_GacKor} of the pair $(\\overline{\\PrivateRV_{1}},\\overline{\\PrivateRV_{2}})$. Being a common part, it can be realized through univariate functions. Let us say $W=f_{1}(V_{1})=f_{2}(V_{2})$. This indicates, \\textit{$\\PublicRVSet-$codebook is built such that, the range of these univariate functions when applied on the collection of codewords in this stack, is contained.}\n\nHow did Marton accomplish this containment? Marton proposed building the $\\PublicRV-$codebook first, followed by conditional codebooks over $\\PrivateRV_{1},\\PrivateRV_{2}$. Conditional coding with a careful choice of order therefore contained the range under the action of univariate function. How is all of this related to the need for containing bivariate functions of a pair of random variables. The fundamental underlying thread is the notion of common part \\cite{1972MMPCT_GacKor}. What are the common parts of a triple of random variables? Clearly, one can simply extend the notion of common part defined for a pair of random variables. This yields four common parts - one part that is simultaneously to common to all three random variables and one common part each, corresponding to each pair in the triple.\nIndeed, if $\\overline{V_{1}}=(W,U_{12},U_{31},V_{1}),\\overline{V_{2}}=(W,U_{12},U_{23},V_{2}),\\overline{V_{3}}=(W,U_{23},U_{31},V_{3})$, then $W$ is the part simultaneously to common to $\\overline{V_{1}},\\overline{V_{2}},\\overline{V_{3}}$ and $U_{ij}:ij\\in\\left\\{12,23,31\\right\\}$ are the pairwise common parts. A simple extension of Marton's coding suggests a way to handle these common parts.\n\nThis does not yet answer the need for containment under bivariate function. We envision a fundamentally richer notion of common part for a triple of random variables. Indeed, three nontrivial binary random variables $X,Y, Z=X\\oplus Y$ have no common parts as defined earlier, since each pair has no common part and the triple does not admit a simultaneous common part. Yet, the degeneracy in the joint probability matrix hints at a common part. Indeed, they possess a \\textit{conferencing} common part. For example, the pair $(X,Y),Z$ have a common part. In other words, there exists a \\textit{bivariate} function of $X,Y$ and a univariate function of $Z$ that agree with probability $1$. Containment of this bivariate function brings in the need for structured codes. Indeed, the resemblance to the problem studied by K\\\"orner and Marton \\cite{197903TIT_KorMar} is striking. We therefore believe the need for structured codes for three (multi) user communication problems is closely linked to the notion of common parts of a triple (collection) of random variables. Analogous to conditional coding that contained univariate functions, endowing codebooks with structure is an inherent need to carefully handle additional degrees of freedom prevalent in larger dimensions.\n\n\\section{Strict sub-optimality of Marton's coding technique}\n\\label{Sec:StrictSubOptimalityOfMartonCodingTechnique}\n\nIn this section, we prove strict sub-optimality of Marton's coding technique for the 3-DBC presented in example \\ref{Ex:3-BCExample}. In particular, we prove that if parameters $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ and $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(\\tau)$, then $R_{1} < h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$.\n\nWhy is Marton's coding technique suboptimal for the case described above. As mentioned in section \\ref{SubSec:TheThreeUserBroadcastChannel}, in this case, receiver $1$ is unable to decode the pair of codewords transmitted to users $2$ and $3$. Furthermore, based on unstructured independent coding, it does not attempt to decode a function of transmitted codewords - in this case the modulo$-2$ sum. This forces decoder $1$ to be content by decoding only individual components of user $2$ and $3$'s transmissions, leaving residual uncertainty in the interference. The encoder helps out by precoding for this residual uncertainty. However, as a consequence of the cost constraint on $X_{1}$, it is forced to live with a rate loss.\n\nSince our proof traces through the above arguments in three stages, it is instructive. In the first stage, we characterize all test channels $p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X \\underlineY}$ for which $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X \\underlineY})$. This stage enables us identify `active' codebooks, their corresponding rates and characterize two upper bounds on $R_{1}$. One of these contains the rate loss due to precoding. In the second stage, we therefore characterize the condition under which there is no rate loss. As expected, it turns out that there is no rate loss only if decoder $1$ has decoded codewords of users $2$ and $3$. This gets us to the third stage, where we conclude that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ precludes this possibility. The first stage is presented in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}, second stage is stated in lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss} and proved in appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}, \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}. Third stage can be found in arguments following lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}.\n\nWe begin with a characterization of a test channel $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY}$ for which $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY})$. Since independent information needs to be communicated to users $2$ and $3$ at their respective point to point capacities, it is expected that their codebooks are not precoded for each other's signal, and moreover none of users $2$ and $3$ decode a part of the other users' signal. The following lemma establishes this. We remind the reader that $X_{1}X_{2}X_{3}=X$ denote the three binary digits at the input, where $Y_{2}$, the output at receiver $2$ is obtained by passing $X_{2}$ through a BSC with cross over probability $\\delta_{2}$, $Y_{3}$, the output at receiver $3$ is obtained by passing $X_{3}$ through a BSC with cross over probability $\\delta_{3}$ and $Y_{1}$ is obtained by passing $X_{1}\\oplus X_{2}\\oplus X_{3}$ through a BSC with cross over probability $\\delta_{1}$. Moreover, the binary symmetric channels (BSC's) are independent. Input symbol $X_{1}$ is constrained with respect to a Hamming cost function and the constraint on the average cost per symbol is $\\tau$. Formally, $\\kappa(x_{1}x_{2}x_{3})=1_{\\left\\{ x_{1}=1 \\right\\}}$ is the cost function and the average cost per symbol is not to exceed $\\tau$.\n\n\\begin{lemma}\n \\label{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}\nIf there exists a test channel $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY} \\in \\SetOfDistributions_{NEM}(\\tau)$ and nonnegative numbers $K_{i},S_{ij},K_{ij},L_{ij},S_{i},T_{i}$ that satisfy (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2) \\right\\}$ such that $R_{2}=K_{2}+K_{23}+L_{12}+T_{2}=1-h_{b}(\\delta_{2}),R_{3}=K_{3}+K_{31}+L_{23}+T_{3}=1-h_{b}(\\delta_{3})$, then\n\\begin{enumerate}\n \\item \\label{Item:RatesOfPublicAndSemiPrivateCodebooks}$K_1=K_2=K_3=K_{23}=L_{23}=K_{12}=L_{31}=S_2=S_3=0$ and $I(\\SemiPrivateRV_{31} V_{1}V_{3};Y_{2}|\\TimeSharingRV WU_{23}U_{12}V_{2})=0$,\n\\item \\label{Item:BinningRatesOfSemiPrivateCodebooks}$S_{31}=I(U_{31};U_{23}|\\TimeSharingRV W),S_{12}=I(U_{12};U_{23}|\\TimeSharingRV W)$, $S_{23}=I(U_{12};U_{31}|\\TimeSharingRV WU_{23})=0$,\n\\item \\label{Item:WU23IsConditionallyIndependentOfY2Y3GivenQ}$I(V_{2}U_{12};V_{3}U_{31}|\\TimeSharingRV WU_{23})=0$, $I(WU_{23};Y_{j}|\\TimeSharingRV )=0:j=2,3,I(V_{2}U_{12};Y_{2}|\\TimeSharingRV W U_{23})=1-h_{b}(\\delta_{2})$ and $I(V_{3}U_{31};Y_{3}|\\TimeSharingRV W U_{23})=1-h_{b}(\\delta_{3})$, \n\\item \\label{ItemNumber:MarkovChains} $(V_3,X_3,V_1,U_{31}) - (\\TimeSharingRV WU_{23}U_{12}V_2) - (X_2,Y_2) $ and $(V_2,X_2,V_1,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - (X_3,Y_3)$ are Markov chains,\n\\item \\label{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}$X_{2} - \\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12} \\SemiPrivateRV_{23} \\SemiPrivateRV_{31} - X_{3}$ is a Markov chain,\n\\item \\label{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23}$\\SemiPrivateRV_{12}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{31}-X_{3}$ and $\\SemiPrivateRV_{31}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{12}-X_{2}$ are Markov chains.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Substituting (i) $(2,3,1)$ for $(i,j,k)$ in (\\ref{Eqn:3BCChannelCodingSextupleBound}), (ii) $(1,2,3)$ for $(i,j,k)$ in (\\ref{Eqn:3BCSourceCodingPairwiseBound}) and combining the resulting bounds yields \n\\begin{flalign}\n\\label{Eqn:UpperBoundOnR_2PlusABunchOfTerms}\nI(WU_{23}U_{12}V_{2};Y_{2}|\\TimeSharingRV) \\geq R_{2}+K_{3}+K_{1}+L_{23}+K_{12}+S_{2}\\geq R_{2}=1-h_{b}(\\delta_{2}),\n\\end{flalign}\nwhere the second inequality follows from non-negativity of $K_{3},K_{1},L_{23},K_{12},S_{2}$. Moreover, \n\\begin{eqnarray}\n\\label{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}\n1-h_{b}(\\delta_{2}) &\\geq&  I(X_{2};Y_{2}) = I(\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV\\underlinePrivateRV X_{1}Y_{1}X_{3}Y_{3}X_{2};Y_{2}) \\geq I(W \\SemiPrivateRV_{23} \\SemiPrivateRV_{12} \\PrivateRV_{2} ; Y_{2}|\\TimeSharingRV) \\\\\n\\label{Eqn:SubstitutingUpperBoundOnR_2}\n&\\geq& R_{2}\\!+\\! K_{3}\\!+\\! K_{1}\\!+\\! L_{23}\\!+\\! K_{12}\\!+\\! S_{2} \\geq R_{2}=1-h_{b}(\\delta_{2}),\n\\end{eqnarray}\nwhere (i) equality in (\\ref{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}) follows from Markov chain $\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X_{1}Y_{1}X_{3}Y_{3}- X_{2} -Y_{2}$, and (ii) (\\ref{Eqn:SubstitutingUpperBoundOnR_2}) follows from substituting (\\ref{Eqn:UpperBoundOnR_2PlusABunchOfTerms}). Since all the terms involved are non-negative, equality holds through the above chain of inequalities to yield\n\\begin{eqnarray}\n\\label{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}\n&S_{12}+S_{23}=I(U_{12};U_{23}|\\TimeSharingRV W),K_{1}\\!=\\!K_{3}\\!=\\!L_{23}\\!=\\!K_{12}\\!=\\!S_{2}\\!=\\!\nI(\\TimeSharingRV;Y_{2})\\!=\\!0\\\\\n\\label{Eqn:K3K1L23K12S2AreZero}\n&I(U_{31}V_{1}X_{1}Y_{1}V_{3}X_{3}Y_{3}X_{2};Y_{2}|QWU_{12}U_{23}V_{2})\\!=\\!0\\!\\\\\n\\label{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}\n&\\mbox{ and therefore }\n(V_1,V_3,X_3,U_{31}) - (\\TimeSharingRV WU_{12}U_{23}V_2) - Y_2\\mbox{ is a Markov chain}\n\\end{eqnarray}\nwhere the first equality in (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}) follows from condition for equality in the first inequality of (\\ref{Eqn:UpperBoundOnR_2PlusABunchOfTerms}). The above sequence of steps are repeated by substituting (i) $(3,1,2)$ for $(i,j,k)$ in (\\ref{Eqn:3BCChannelCodingSextupleBound}), (ii) $(2,3,1)$ for $(i,j,k)$ in (\\ref{Eqn:3BCSourceCodingPairwiseBound}). It can be verified that\n\\begin{eqnarray} \n\\label{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}\n&S_{31}+S_{23}=I(U_{31};U_{23}|\\TimeSharingRV W), K_{1}\\!=\\!K_{2}\\!=\\!L_{31}\\!=\\!K_{23}\\!=\\!S_{3}\\!=\\!I(\\TimeSharingRV; Y_{3})\\!=\\!0,\\\\\n\\label{Eqn:K2K1L31K23S3AreZero}\n&I(\\SemiPrivateRV_{12} V_{1}X_{1}Y_{1}V_{2}X_{2}Y_{2}X_{3};Y_{3}|\\TimeSharingRV WU_{23}U_{31}V_{3})\\!=\\!0\\!\\\\\n\\label{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}\n&\\mbox{ and therefore }\n(V_1,V_2,X_2,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - Y_3\\mbox{ is a Markov chain}.\n\\end{eqnarray}\nThe second set of equalities in (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}), (\\ref{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}) lets us conclude\n\\begin{equation}\n \\label{Eqn:RatesOfUsers12And3InTermsOfSimplifiedCodebooks}\nR_{1}=T_{1},R_{2}=L_{12}+T_{2} \\mbox{ and }R_{3}=K_{31}+T_{3}.\n\\end{equation}\nFrom $I(U_{12};U_{23}|\\TimeSharingRV W) + I(U_{31};U_{23}|\\TimeSharingRV W) = S_{12}+S_{23}+S_{31}+S_{23}$, and (\\ref{Eqn:3BCSourceCodingTripleBound}), we have $I(U_{12};U_{23}|\\TimeSharingRV W) + I(U_{31};U_{23}|\\TimeSharingRV W) \\geq I(U_{12};U_{23};U_{31}|\\TimeSharingRV W)+S_{23}$. The non-negativity of $S_{23}$ (\\ref{Eqn:3BCSourceCodingBoundNonnegativity}) implies $S_{23}=0$ and $I(U_{31};U_{12}|\\TimeSharingRV WU_{23})=0$. We therefore conclude\n\\begin{eqnarray}\n\\label{Eqn:SemiPrivateLayerBinningRates}\nS_{12}=I(U_{12};U_{23}|\\TimeSharingRV W),S_{31}=I(U_{31};U_{23}|\\TimeSharingRV W),S_{23}=0,\nI(U_{31};U_{12}|\\TimeSharingRV W U_{23})=0\n\\end{eqnarray}\nSubstituting (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}), (\\ref{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}), (\\ref{Eqn:SemiPrivateLayerBinningRates}) in (\\ref{Eqn:3BCSourceCodingQuadrapleBound}) for $(i,j,k)=(2,3,1)$ and $(i,j,k)=(3,1,2)$ and (\\ref{Eqn:3BCSourceCodingPentaBound}) for $(i,j,k)=(2,3,1)$, we obtain\n\\begin{eqnarray}\n\\label{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}\nI(V_{2};U_{31}|\\TimeSharingRV WU_{12}U_{23})=I(V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})\n=I(V_{2};V_{3}|\\TimeSharingRV WU_{12}U_{23}U_{31})=0.\n\\end{eqnarray}\n(\\ref{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}) and last equality in (\\ref{Eqn:SemiPrivateLayerBinningRates}) yield \n\\begin{equation}\n\\label{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}\nI(V_{2}U_{12};V_{3}U_{31}|\\TimeSharingRV WU_{23})=0.\n\\end{equation}\nSubstituting (\\ref{Eqn:RatesOfUsers12And3InTermsOfSimplifiedCodebooks}), (\\ref{Eqn:SemiPrivateLayerBinningRates}) in (\\ref{Eqn:3BCChannelCodingDoubleBound})\\begin{comment}{ and (\\ref{Eqn:3BCChannelCodingSextupleBound})}\\end{comment} with $(i,j,k)=(2,3,1)$ yields the upper bound $R_2 \\leq I(U_{12}V_2;Y_2|\\TimeSharingRV WU_{23})$. Since\n\\begin{eqnarray}\n&\\!\\!\\!\\!1-h_{b}(\\delta_{2}) \\!=\\! R_{2} \\!\\leq\\! I(\\SemiPrivateRV_{12} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23}) \\!\\leq\\! I(\\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV) \\leq 1-h_{b}(\\delta_{2}),\\nonumber\n\\end{eqnarray}\nwhere the last inequality follows from (\\ref{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}), equality holds in all of the above inequalities to yield $I(W\\SemiPrivateRV_{23};Y_{2}|\\TimeSharingRV )=0$ and $I(\\SemiPrivateRV_{12} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=1-h_{b}(\\delta_{2})$. A similar argument proves $I(W\\SemiPrivateRV_{23};Y_{3}|\\TimeSharingRV )=0$ and $I(\\SemiPrivateRV_{31} \\PrivateRV_{3};Y_{3}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=1-h_{b}(\\delta_{3})$.\n\nWe have proved the Markov chains in (\\ref{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}), (\\ref{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}). In order to prove Markov chains in item \\ref{ItemNumber:MarkovChains}, we prove the following lemma.\n\\begin{lemma}\n\\label{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}\nIf $A,B,X,Y$ are discrete random variables such that (i) $X,Y$ take values in $\\left\\{0,1\\right\\}$ with $P(Y=0|X=1)=P(Y=1|X=0)=\\eta \\in (0,\\frac{1}{2})$, (ii) $A-B-Y$ and $AB-X-Y$ are Markov chains, then $A-B-XY$ is also a Markov chain.\n\\end{lemma}\nPlease refer to appendix \\ref{AppSec:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} for a proof. Markov chains in (\\ref{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}), (\\ref{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}) in conjunction with lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} establishes Markov chains in item \\ref{ItemNumber:MarkovChains}.\n\n(\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}) and (\\ref{Eqn:K3K1L23K12S2AreZero}) imply $I(\\SemiPrivateRV_{31}\\PrivateRV_{3};\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=0$. This in conjunction with (\\ref{Eqn:K2K1L31K23S3AreZero}) implies\n\\begin{equation}\n \\label{Eqn:U31V3Y3AndU12V2Y2AreIndependentConditionedOnWU23}\nI(\\SemiPrivateRV_{31}\\PrivateRV_{3}Y_{3};\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}|\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23})=0\\mbox{ and thus }\\SemiPrivateRV_{31}\\PrivateRV_{3}Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}-\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}\\mbox{ is a Markov chain.}\n\\end{equation}\n(\\ref{Eqn:U31V3Y3AndU12V2Y2AreIndependentConditionedOnWU23}) implies $\\SemiPrivateRV_{31}Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}-\\SemiPrivateRV_{12}Y_{2}$ is a Markov chain, and therefore $Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-Y_{2}$ is a Markov chain. Employing lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} twice we observe $Y_{3}X_{3}-\\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-X_{2}Y_{2}$ is a Markov chain and furthermore $X_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-X_{2}$ is a Markov chain, thus proving item \\ref{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}.\n\nFinally, we prove Markov chains in item \\ref{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23}. From Markov chain $(V_3,X_3,V_1,U_{31}) - (\\TimeSharingRV WU_{23}U_{12}V_2) - (X_2,Y_2)$ proved in item \\ref{ItemNumber:MarkovChains}, we have $I(X_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12}V_{2})=0$. From (\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}), we have $I(V_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$. Summing these two, we have $I(X_{2}V_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$ and therefore $I(X_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$ implying the Markov chain $X_{2}-\\TimeSharingRV WU_{23}U_{12}-U_{31}$. Similarly, Markov chain $(V_2,X_2,V_1,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - (X_3,Y_3)$ proved in item \\ref{ItemNumber:MarkovChains} implies $I(X_{3};U_{12}|WU_{23}U_{31}V_{3}\\TimeSharingRV )=0$. From (\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}), we have $I(V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$. Summing these two, we have $I(X_{3}V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$ and therefore $I(X_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$ implying the Markov chain $X_{3}-\\TimeSharingRV WU_{23}U_{31}-U_{12}$.\n\\end{proof}\nLemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} enables us simplify the bounds (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for the particular test channel under consideration. Substituting (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks})-(\\ref{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}) in (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) and employing statements of lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}, we conclude that if $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY})$, then there exists nonnegative numbers $S_{1},T_{1},L_{12},K_{31}$ that satisfy $R_{1}=T_{1},R_{2}=L_{12}+T_{2}=1-h_{b}(\\delta_{2}),R_{3}=K_{31}+T_{3}=1-h_{b}(\\delta_{3})$,\n\\begin{eqnarray}\n \\label{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}\n&S_1  \\geq I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})\\begin{comment}{\\define a_{1}}\\end{comment},~~~T_1+S_1\\leq I(V_1;Y_1|\\TimeSharingRV WU_{12}U_{31})\\begin{comment}{\\define a_{2}}\\end{comment} \\\\\n\\label{Eqn:BoundsOnRatesOfUser123CodebooksAsAResultOfUser1Decoding}\n&L_{12}+K_{31}+T_1+S_1 \\leq I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)+I(V_1U_{12}U_{31};Y_1|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\begin{comment}{\\define a_{5}}\\end{comment}\\\\\n\\label{Eqn:BoundsOnRatesOfUser2CodebookAsAResultOfUser2Decoding}\n&0 \\leq T_{2} \\leq I(V_2;Y_2|\\TimeSharingRV WU_{12}U_{23})\\begin{comment}{\\define a_{6}}\\end{comment},~~~1-h_{b}(\\delta_{2})=T_{2}+L_{12} = I(U_{12}V_2;Y_2|\\TimeSharingRV WU_{23})\\begin{comment}{\\define a_{7}}\\end{comment}\\\\\n\\label{Eqn:BoundsOnRatesOfUser3CodebookAsAResultOfUser3Decoding}\n&0 \\leq T_{3}\\leq I(V_3;Y_3|\\TimeSharingRV WU_{31}U_{23})\\begin{comment}{\\define a_{8}}\\end{comment},~~~1-h_{b}(\\delta_{3})=T_{3}+K_{31}= I(U_{31}V_3;Y_3|\\TimeSharingRV WU_{23})\\begin{comment}{\\define a_{9}}\\end{comment}.\n\\end{eqnarray}\n(\\ref{Eqn:BoundsOnRatesOfUser2CodebookAsAResultOfUser2Decoding}), (\\ref{Eqn:BoundsOnRatesOfUser3CodebookAsAResultOfUser3Decoding}) imply\n\\begin{equation}\n \\label{Eqn:LowerBoundsOnRateOfCodebooksSharedWithUser1}\nL_{12}\\geq I(U_{12};Y_{2}|\\TimeSharingRV \\PublicRV U_{23}),~~~~~~~~~~~ K_{31} \\geq I(U_{31};Y_3|\\TimeSharingRV WU_{23}),\n\\end{equation}\n(\\ref{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}) implies\n\\begin{eqnarray}\nT_{1}\\!\\!\\!\\!&=&\\!\\!\\!\\!R_1 \\leq I(V_1;Y_1|\\TimeSharingRV W\\SemiPrivateRV_{12}\\SemiPrivateRV_{31})\n -I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31}), \\nonumber\\\\\n\\label{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}\n&\\leq&\\!\\!\\!\\! I(V_1;Y_1U_{23}|\\TimeSharingRV W\\SemiPrivateRV_{12}\\SemiPrivateRV_{31})\n -I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})=I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV)\n -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV),\n\\end{eqnarray}\nand (\\ref{Eqn:BoundsOnRatesOfUser123CodebooksAsAResultOfUser1Decoding}) in conjunction with (\\ref{Eqn:LowerBoundsOnRateOfCodebooksSharedWithUser1}), and lower bound on $S_{1}$ in (\\ref{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}) imply\n\\begin{eqnarray}\nR_1 \\!\\!\\!\\!&\\leq&\\!\\!\\!\\! I(U_{12}U_{31}V_1;Y_1|\\TimeSharingRV W)\n-I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})-I(U_{12};Y_2|\\TimeSharingRV WU_{23})-I(U_{31};Y_3|\\TimeSharingRV WU_{23}) \\nonumber\\\\\n&&\\!\\!\\!\\!+I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\nonumber\\\\\n\\!\\!\\!\\!&\\leq&\\!\\!\\!\\! I(U_{12}U_{31}V_{1};Y_1 U_{23}|\\TimeSharingRV W)\n-I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})-I(U_{12};Y_2|\\TimeSharingRV WU_{23})-I(U_{31};Y_3|\\TimeSharingRV WU_{23})\\nonumber\\\\\n&&\\!\\!\\!\\!+I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\nonumber\\\\\\label{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}\n\\!\\!\\!\\!&=&\\!\\!\\!\\!I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) \\!-\\!I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV)\\! +\\!I(U_{12}U_{31};Y_1|\\TimeSharingRV W U_{23}) \\!-\\!I(U_{12};Y_2|\\TimeSharingRV WU_{23})\\!-\\!I(U_{31};Y_3|\\TimeSharingRV WU_{23}),\n\\end{eqnarray}\nwhere (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}) follows from the last equality in (\\ref{Eqn:SemiPrivateLayerBinningRates}). We have thus obtained (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}), two upper bounds on $R_{1}$ we were seeking, and this concludes the first stage of our proof. In the sequel, we prove the minimum of the above upper bounds on $R_{1}$ is strictly lesser than $h_{b}(\\tau * \\delta_{1})-h_{b}(\\delta_{1})$. Towards, that end, note that upper bound (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) contains the rate loss due to precoding. In the second stage, we work on (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and derive conditions under which there is \\textit{no} rate loss.\n\nMarkov chains of lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{ItemNumber:MarkovChains} imply $V_{1}-QW\\underlineSemiPrivateRV V_{2}V_{3}-X_{2}$ and $V_{1}-QW\\underlineSemiPrivateRV V_{2}V_{3}X_{2}-X_{3}$ are Markov chains. Therefore, $I(V_{1};X_{2}|QW\\underlineSemiPrivateRV V_{2}V_{3})=0$ and $I(V_{1};X_{3}|QW\\underlineSemiPrivateRV V_{2}V_{3}X_{2})=0$. Summing these, we have $I(V_{1};X_{2}X_{3}|QW\\underlineSemiPrivateRV V_{2}V_{3})=0$. Employing this in (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}),\\begin{comment}{ abbreviating $\\tildeQRV \\define \\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}$,\\footnote{This notation is employed for the rest of this section.} $\\tildeQRVSet \\define \\TimeSharingRVSet \\times \\PublicRVSet \\times \\SemiPrivateRVSet_{23} $,\\addtocounter{footnote}{-1}\\footnotemark and letting $\\tildeq \\define (q,w,u_{23})$\\addtocounter{footnote}{-1}\\footnotemark denote a generic element in $\\tildeQRVSet$,}\\end{comment} we note\n\\begin{eqnarray}\n\\label{eq:inequality1}\nR_1 &\\leq& I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV) =I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3X_2X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\\\\n&\\leq& I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) - I(V_1;X_2,X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\leq I(V_1;Y_1 |\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;X_2\\oplus X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\label{eq:inequality3}\\\\\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[  I(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\right]\\label{eq:inequality4}\\\\\n&\\leq&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) I(X_{1}X_{2}X_{3}V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\nonumber\\\\\n&\\leq&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[H(Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))-H(Y_1|X_{1}X_{2}X_{3}V_{1},(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\right]\\nonumber\\\\\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[H(X_{1}\\oplus N_{1}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))-h_{b}(\\delta_{1})\\right]\\nonumber\\\\\n\\label{Eqn:TheStepJustBeforeApplicationOfJensenInequality}\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}} p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}), \\mbox{ where }\\tau_{q,w,\\underline{u}}=p_{X_{1}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})\\\\\n\\begin{comment}{\n&=&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\sum_{\\substack{(u_{12},u_{31}) \\in\\\\  \\SemiPrivateRVSet_{12} \\times \\SemiPrivateRVSet_{31}}}p_{ U_{12} U_{31}|\\tildeQRV}(u_{12},u_{31}|\\tildeq) h_{b}(\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1})-h_{b}(\\delta_{1}), \\nonumber\\\\\n&=&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\Expectation_{U_{12}U_{31}|\\tildeQRV=\\tildeq} \\left\\{h_{b}(\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1})\\right\\}-h_{b}(\\delta_{1}), \\nonumber\\\\\n\\label{Eqn:ApplicationOfJensensInequalityFirstInstance}\n&\\leq&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)h_{b}(\\Expectation_{U_{12}U_{31}|\\tildeQRV=\\tildeq} \\left\\{\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1}\\right\\})-h_{b}(\\delta_{1})=\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\left\\{h_{b}(\\tau_{\\tildeq}*\\delta_{1})-h_{b}(\\delta_{1})\\right\\}, \\\\\n}\\end{comment}\n\\label{Eqn:ApplicationOfJensensInequality}\n&=&\\Expectation_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV} \\left\\{h_{b}(\\tau_{q,w,\\underline{u}} * \\delta_{1})\\right\\}-h_{b}(\\delta_{1})\\leq h_{b}(\\Expectation_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV} \\left\\{ \\tau_{q,w,\\underline{u}} * \\delta_{1}\\right\\})-h_{b}(\\delta_{1})\\leq h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})\n\\end{eqnarray}\nwhere (\\ref{Eqn:ApplicationOfJensensInequality}) follows from application of Jensen's inequality to the strictly concave function $h_{b}(\\cdot)$, and second inequality in (\\ref{Eqn:ApplicationOfJensensInequality}) follows from $\\delta \\in (0,\\frac{1}{2})$. We conclude that $R_{1}=h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ if and only if equality holds in the above chain of inequalities, and in particular, equality holds in (\\ref{Eqn:ApplicationOfJensensInequality}), which by the condition for equality in Jensen's inequality implies $\\tau_{q,w,\\underline{u}}=\\tau $ for every $(q,w,\\underline{u}) \\in \\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV$ that satisfies $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0$. This in conjunction with\n\\begin{equation}\nI(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\leq h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}) \\nonumber\n\\end{equation}\nwhich follows from the chain of inequalities from (\\ref{eq:inequality4}) through (\\ref{Eqn:TheStepJustBeforeApplicationOfJensenInequality}) implies\n\\begin{equation}\n \\label{Eqn:AfterProcessingTheRateLossUpperBoundAndFitForApplicationOfRateLossLemma}\nI(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\leq h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})\n\\end{equation}\nwith equality if and only if\n\\begin{eqnarray}\n\\label{Eqn:EqualityInJensenImpliesChannelMustOperateAtUniformCost}\n&\\mbox{for every }(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRV \\mbox{ that satisfies }p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0, p_{X_{1}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})=\\tau_{q,w,\\underline{u}}=\\tau,~~~~\\\\\n\\label{Eqn:NoRateLossForEveryConditionalTestChannel}\n&\\mbox{ and }I(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) = h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}).\n\\end{eqnarray}\n\nAn informed reader, by now must have made the connection to capacity of the point to point channel with non-causal state\n\\cite{1980MMPCT_GelPin}. We develop this connection in appendix \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}. For now, we provide a characterization for (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) to hold. This will require us to define a few mathematical objects that may initially seem unrelated to a reader unaware of findings in \\cite{1980MMPCT_GelPin}. Very soon, we argue the relevance. An informed reader will find the following development natural.\n\nLet $\\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)$ denote the collection of all probability mass functions $p_{\\GPV\\GPS\\GPX\\GPY}$ defined on $\\GPSetV \\times \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\}$, where $\\GPSetV$ is an arbitrary finite set such that (i) $p_{\\GPY|\\GPX\\GPS\\GPV}(x\\oplus s|x,s,v)=p_{\\GPY|\\GPX\\GPS}(x\\oplus s|x,s)=1-\\delta$, where $\\delta \\in (0,\\frac{1}{2})$, (ii) $p_{\\GPS}(1)=\\epsilon \\in [0,1]$, and (iii) $p_{\\GPX}(1)\\leq \\tau \\in (0,\\frac{1}{2})$. For $p_{\\GPV\\GPS\\GPX\\GPY} \\in \\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{T}(p_{\\GPV\\GPS\\GPX\\GPY})=I(\\GPV;\\GPY)-I(\\GPV;\\GPS)$ and $\\alpha_{T}(\\tau,\\delta,\\epsilon)=\\sup_{p_{\\GPV\\GPS\\GPX\\GPY} \\in \\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)}\\alpha_{T}(p_{\\GPV\\GPS\\GPX\\GPY})$.\n\n\nFor every $(q,w,\\underline{u}) \\in \\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV$ that satisfies $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0$, we note $p_{Y_{1}|X_{1},X_{2}\\oplus X_{3}\\PrivateRV_{1}\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(x_{1}\\oplus x_{2}\\oplus x_{3}|x_{1},x_{2}\\oplus x_{3},v_{1},q,w,\\underline{u})=p_{Y_{1}|X_{1},X_{2}\\oplus X_{3}\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(x_{1}\\oplus x_{2}\\oplus x_{3}|x_{1},x_{2}\\oplus x_{3},q,w,\\underline{u})=1-\\delta_{1}$. In other words, conditioned on the event $\\left\\{ (\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)= (q,w,\\underline{u})\\right\\}$, $V_{1}-X_{1},X_{2}\\oplus X_{3}-Y_{1}$ is a Markov chain. We conclude $p_{V_{1}X_{2}\\oplus X_{3}X_{1}Y_{1}|QW\\underlineSemiPrivateRV}(\\cdots|q,w,\\underline{u}) \\in \\mathbb{D}_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}})$, where $\\epsilon_{q,w,\\underline{u}}=p_{X_{2}\\oplus X_{3}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})$, and hence\n\\begin{equation}\n \\label{Eqn:CharacterizationOfAchievableRateForUser1InTermsOfGelfandPinskerCapacity}\nI(V_1;Y_1|(\\TimeSharingRV ,\\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV ,\\PublicRV ,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\leq \\alpha_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}}). \\nonumber\n\\end{equation}\nTherefore, (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) holds only if $\\alpha_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}}) = h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1})$, where $\\tau_{q,w,\\underline{u}}=\\tau \\in (0,\\frac{1}{2})$. The following lemma characterizes conditions under which this is the case. Please refer to appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx},\\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} for a proof.\n\\begin{lemma} \n\\label{lem:GP}\nIf $\\tau ,\\delta \\in (0,\\frac{1}{2})$ and $\\epsilon \\in (0,1)$, then $\\alpha_{T}(\\tau,\\delta,\\epsilon) < h_b(\\tau * \\delta)-h_b(\\delta)$. Alternatively, if $\\tau ,\\delta \\in (0,\\frac{1}{2})$ and $\\epsilon \\in [0,1]$, then either $\\alpha_{T}(\\tau,\\delta,\\epsilon) < h_b(\\tau * \\delta)-h_b(\\delta)$ or $\\epsilon \\in \\left\\{ 0,1\\right\\}$.\\end{lemma}\nRecall that arguments in relation to (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) imply that if for any $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet$ that satisfies $P((\\TimeSharingRV, \\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u})) >0$, \n\\begin{equation}\n\\label{Eqn:NoRateLossForEveryConditionalTestChannelRestated}\n I(V_1;Y_1|(\\TimeSharingRV ,\\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV ,\\PublicRV ,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))<h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1})\\nonumber\n\\end{equation}\nwhere $\\tau_{q,w,\\underline{u}}=p_{X_{1}|\\TimeSharingRV, \\PublicRV, \\underlineSemiPrivateRV}(1|q,w,\\underline{u})$, then $R_{1} < h_{b}(\\tau* \\delta_{1})-h_{b}(\\delta_{1})$ and we have proved strict sub-optimality of Marton's coding technique. We therefore assume (\\ref{Eqn:EqualityInJensenImpliesChannelMustOperateAtUniformCost}), (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) hold for every $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet$ that satisfies $P((\\TimeSharingRV, \\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u})) >0$. From lemma \\ref{lem:GP}, we conclude for every such $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet$, $\\epsilon_{q,w,\\underline{u}}=p_{X_{2}\\oplus X_{3}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u}) \\in \\left\\{ 0,1 \\right\\}$. We therefore assume \n\\begin{equation}\n \\label{Eqn:CaseOfNoRateLoss}\nI(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV)\n-I(V_1;X_2\\oplus X_3|\\TimeSharingRV W\\underlineSemiPrivateRV)  = h_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})\\mbox{ and }H(X_2 \\oplus  X_3|\\TimeSharingRV W\\underlineSemiPrivateRV)=0.\n\\end{equation}\nThis has got us to the third and final stage. Here we argue (\\ref{Eqn:CaseOfNoRateLoss}) implies RHS of (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}) is strictly smaller than $h_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})$. Towards that end, note that Markov chain $X_{2}-\\TimeSharingRV WU_{23}U_{12}U_{31}-X_{3}$ proved in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}\nand (\\ref{Eqn:CaseOfNoRateLoss}) imply $H(X_{2}|\\TimeSharingRV W\\underlineSemiPrivateRV)=H(X_{3}|\\TimeSharingRV W\\underlineSemiPrivateRV)=0$.\\footnote{Indeed, for any $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underline{\\SemiPrivateRVSet}$ that satisfies $P((\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))>0$, if $P(X_{j}=1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))=\\alpha_{j}:j=2,3$, then $0=H(X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))=h_{b}(\\alpha_{2}*\\alpha_{3}) \\geq \\alpha_{2}h_{b}(\\alpha_{3})+(1-\\alpha_{2})h_{b}(1-\\alpha_{3})=\\alpha_{2}h_{b}(\\alpha_{3})+(1-\\alpha_{2})h_{b}(\\alpha_{3})=h_{b}(\\alpha_{3})\\geq 0$, where the first inequality follows from concavity of binary entropy function, and similarly, interchanging the roles of $\\alpha_{2}, \\alpha_{3}$, we obtain $0=H(X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\geq h_{b}(\\alpha_{2})\\geq 0$.} Furthermore, Markov chains $\\SemiPrivateRV_{12}-\\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{31}-X_{3}$ and $\\SemiPrivateRV_{31}-\\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{12}-X_{2}$ proved in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23} imply\n\\begin{equation}\n\\label{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}\nH(X_2|WU_{23}U_{12})=H(X_3|WU_{23}U_{31})=0.\n\\end{equation}\nObserve that\n\\begin{eqnarray}\n\\label{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}\nh_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})\\!\\!&=&\\!\\!I(V_1;Y_1|W\\underlineSemiPrivateRV) -I(V_1;X_2\\oplus X_3 W\\underlineSemiPrivateRV) = I(V_1;Y_1| W \\underlineSemiPrivateRV ) =I(V_1;Y_1| W \\underlineSemiPrivateRV ,X_2,X_3) \\\\\n\\!\\!&=&\\!\\! H(Y_{1}|W\\underlineSemiPrivateRV X_{2}X_{3})-H(Y_{1}|W\\underlineSemiPrivateRV V_{1}X_{2}X_{3})\n\\leq H(Y_{1}|W\\underlineSemiPrivateRV X_{2}X_{3})-H(Y_{1}|W\\underlineSemiPrivateRV V_{1}X_{1}X_{2}X_{3})\\nonumber\\\\\n\\label{Eqn:EntropyOfY1ConditionedOnX2AndX3}\n\\!\\!&=&\\!\\!H(Y_{1}| W \\underlineSemiPrivateRV ,X_2,X_3)-h_b(\\delta_{1})\n\\end{eqnarray}\nwhere the first two equalities in (\\ref{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}) follows from (\\ref{Eqn:CaseOfNoRateLoss}) and the last equality follows from (\\ref{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}). (\\ref{Eqn:EntropyOfY1ConditionedOnX2AndX3}) and first equality in (\\ref{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}) enables us conclude\n\\begin{equation}\n \\label{Eqn:LowerBoundOnEntropyOfY1ConditionedOnX2AndX3}\nH(Y_{1}| W \\underlineSemiPrivateRV ,X_2,X_3) \\geq h_b(\\tau * \\delta_{1})\n\\end{equation}\nWe now upper bound RHS of (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}). Note that it suffices to prove $I(U_{12}U_{31};Y_1|WU_{23})-I(U_{12};Y_2|WU_{23}) -I(U_{31};Y_3|WU_{23})$ is negative. Observe that\n\\begin{eqnarray}\n\\lefteqn{I(U_{12}U_{31};Y_1|WU_{23})-I(U_{12};Y_2|WU_{23})\n-I(U_{31};Y_3|WU_{23})}\\nonumber\\\\ \n&&=H(Y_{1}|WU_{23})-H(Y_{1}|W\\underlineSemiPrivateRV)-H(Y_{2}|WU_{23})+H(Y_{2}|WU_{23}U_{12})-H(Y_{3}|WU_{23})+H(Y_{3}|WU_{23}U_{31})\n\\nonumber\\\\\n&&=H(Y_{1}|WU_{23})-H(Y_{1}|W\\underlineSemiPrivateRV)-H(Y_{2})+H(Y_{2}|WU_{23}U_{12})-H(Y_{3})+H(Y_{3}|WU_{23}U_{31})\n\\nonumber\\\\\n\\label{Eqn:X2IsAFunctionOfWU23U12AndX3IsAFunctionOfWU23U31}\n&&=H(Y_{1}|WU_{23})-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-H(Y_{2})+H(Y_{2}|WU_{23}U_{12}X_{2})-H(Y_{3})+H(Y_{3}|WU_{23}U_{31}X_{3})\n\\\\\n&&= H(Y_{1}|WU_{23})-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-2+h_{b}(\\delta_{2})+h_{b}(\\delta_{3})\n\\nonumber\\\\\n\\label{Eqn:SubstitutingLowerBoundOnEntropyOfY1}\n&&\\leq 1-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-2+h_{b}(\\delta_{2})+h_{b}(\\delta_{3})\n\\leq h_{b}(\\delta_{2})+h_{b}(\\delta_{3})-h_b(\\delta_{1}*\\tau)-1\n\\end{eqnarray}\nwhere (\\ref{Eqn:X2IsAFunctionOfWU23U12AndX3IsAFunctionOfWU23U31}) follows from (\\ref{Eqn:CaseOfNoRateLoss}) and (\\ref{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}), second inequality in (\\ref{Eqn:SubstitutingLowerBoundOnEntropyOfY1}) follows from (\\ref{Eqn:LowerBoundOnEntropyOfY1ConditionedOnX2AndX3}).\nIf $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $h_b(\\delta_{2})+h_{b}(\\delta_{3}) < 1+h_b(\\delta_{1}*\\tau)\n$, then $R_1 < h_b(\\tau *\\delta_{1})-h_b(\\delta_{1})$ and RHS of (\\ref{Eqn:SubstitutingLowerBoundOnEntropyOfY1}) is negative. We summarize our findings in the following theorem and corollary.\n\n\\begin{thm}\n \\label{Thm:ConditionsUnderWhichMartonCodingTechniqueDoesNotAchieveRateTriple}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample}. If $h_b(\\delta_{2})+h_{b}(\\delta_{3}) < 1+h_b(\\delta_{1}*\\tau)\n$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\notin \\alpha_{NEM}(\\tau)$.\n\\end{thm}\n\n\\begin{corollary}\n\\label{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample} with $\\delta=\\delta_{2}=\\delta_{3}$. If $h_{b}(\\tau*\\delta_{1}) \\leq h_b(\\delta) < \\frac{1+h_b(\\delta_{1}*\\tau)}{2}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta)) \\notin \\alpha_{NEM}(\\tau)$ but $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta)) \\in \\mathbb{C}(\\tau)$ and thus $\\alpha_{NEM}(\\tau) \\neq \\mathbb{C}(\\tau)$. In particular, if $\\delta_{1}=0.01$ and $\\delta_{2} \\in (0.1325,0.21)$, then $\\alpha_{NEM}(\\frac{1}{8}) \\neq \\mathbb{C}(\\frac{1}{8})$.\n\\end{corollary}\n\n\\appendices\n\\section{Characterization for no rate loss in point to point channels with channel state information}\n\\label{AppSec:CharacterizationForNoRateLossInPTP-STx}\n\nWe now develop the connection between upper bound (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and the capacity of a point to point channel with non-causal state \\cite{1980MMPCT_GelPin}. We only describe the relevant additive channel herein and refer the interested reader to either to \\cite{1980MMPCT_GelPin} or \\cite[Chapter 7]{201201NIT_ElgKim} for a detailed study. The notation employed in this section and appendix \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} is specific to these sections.\n\nConsider a point to point channel with binary input and output alphabets $\\mathcal{X}=\\mathcal{Y}=\\left\\{0,1\\right\\}$. The channel transition probabilities depend on a random parameter, called state that takes values in the binary alphabet $\\mathcal{S}=\\left\\{ 0,1 \\right\\}$. The discrete time channel is time-invariant, memoryless and used without feedback. The channel is additive, i.e., if $S,X$ and $Y$ denote channel state, input and output respectively, then $P(Y=x \\oplus s | X=x, S=s)=1-\\delta$, where $\\oplus$ denotes addition in binary field and $\\delta \\in (0,\\frac{1}{2})$. The state is independent and identically distributed across time with $P(S=1)=\\epsilon \\in (0,1)$.\\footnote{Through appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx},\\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} we prove if $\\delta,\\tau \\in (0,\\frac{1}{2})$ and $\\epsilon \\in (0,1)$, then $\\alpha_{T}(\\tau,\\eta,\\epsilon) < h_b(\\tau * \\eta)-h_b(\\eta)$. This implies statement of lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}.} The input is constrained by an additive Hamming cost, i.e., the cost of transmitting $x^{n} \\in \\InputAlphabet^{n}$ is $\\sum_{t=1}^{n}1_{\\left\\{x_{t}=1\\right\\}}$ and average cost of input per symbol is constrained to be $\\tau \\in (0,\\frac{1}{2})$. \n\nThe quantities of interest - left and right hand sides of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) - are related to two scenarios with regard to knowledge of state for the above channel. In the first scenario we assume the state sequence is available to the encoder non-causally and the decoder has no knowledge of the same. In the second scenario, we assume knowledge of state is available to both the encoder and decoder non-causally. Let $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon),\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$ denote the capacity of the channel in the first and second scenarios respectively. It turns out, the left hand side of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) is upper bounded by $\\mathcal{C}(\\tau,\\delta,\\epsilon)$ and the right hand side of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) is $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. A necessary condition for (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) to hold, is therefore $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. For the point to point channel with non-causal state, this equality is popularly referred to as \\textit{no rate loss}. We therefore seek the condition for no rate loss.\n\nThe objective of this section and appendix \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} is to study the condition under which $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. In this section, we characterize each of these quantities, in the standard information theoretic way, in terms of a maximization of an objective function over a particular collection of probability mass functions.\n\nWe begin with a characterization of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ and $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\n\\begin{definition}\n \\label{Defn:TestChannelsForPTP-STx}\nLet $\\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ denote the set of all probability mass functions $p_{USXY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ that satisfy (i) $p_{S}(1) = \\epsilon$, (ii) $p_{Y|XSU}(x\\oplus s|x,s,u)=p_{Y|XS}(x\\oplus s | x,s)=1-\\delta$, (iii) $P(X=1)\\leq \\tau$. For $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{T}(p_{USXY}) = I(U;Y)-I(U;S)$ and $\\alpha_{T}(\\tau,\\delta,\\epsilon) = \\underset{p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)}{\\sup} \\alpha_{T}(p_{USXY})$.\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:CapacityOfBinaryAdditivePTP-STx}\n$\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) = \\alpha_{T}(\\tau,\\delta,\\epsilon)$\n\\end{thm}\nThis is a well known result in information theory and we refer the reader to \\cite{1980MMPCT_GelPin} or \\cite[Section 7.6, Theorem 7.3]{201201NIT_ElgKim} for a proof.\n\\begin{definition}\n \\label{Defn:TestChannelsForPTP-STxRx}\nLet $\\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ denote the set of all probability mass functions $p_{SXY}$ defined on $\\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ that satisfy (i) $p_{S}(1) = \\epsilon$, (ii) $p_{Y|XS}(x\\oplus s|x,s)=1-\\delta$, (iii) $P(X=1)\\leq \\tau$. For $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{TR}(p_{SXY}) = I(X;Y|S)$ and $\\alpha_{TR}(\\tau,\\delta,\\epsilon) = \\underset{p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)}{\\sup} \\alpha_{TR}(p_{SXY})$.\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:CapacityOfBinaryAdditivePTP-STxRx}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) = \\alpha_{TR}(\\tau,\\delta,\\epsilon)$\n\\end{thm}\nThis can be argued using Shannon's characterization of point to point channel capacity \\cite{194807BSTJ_Sha} and we refer the reader to \\cite[Section 7.4.1]{201201NIT_ElgKim} for a proof.\n\\begin{remark}\n\\label{Rem:SideInformationAtBothEncoderAndDecoderResultsInLargerCapacity}\n From the definition of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ and $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, it is obvious that $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) \\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, we provide an alternative argument based on theorems \\ref{Thm:CapacityOfBinaryAdditivePTP-STx}, \\ref{Thm:CapacityOfBinaryAdditivePTP-STxRx}. For any $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, it is easy to verify the corresponding marginal $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ and moreover $\\alpha_{T}(p_{USXY})=I(U;Y)-I(U;S) \\leq I(U;YS)-I(U;S) = I(U;Y|S)=H(Y|S)-H(Y|US) \\leq H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})\\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Since this this true for every $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, we have $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) \\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{remark}\n\nWe provide an alternate characterization for $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Lem:AlternateCharacterizationForCapacityOfBinaryAdditivePTP-STxRx}\nFor $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, let $\\beta_{TR}(p_{USXY}) = I(U;Y|S)$ and $\\beta_{TR}(\\tau,\\delta,\\epsilon) = \\underset{p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)}{\\sup} \\beta_{TR}(p_{USXY})$. Then $\\beta_{TR}(\\tau,\\delta,\\epsilon) = \\alpha_{TR}(\\tau,\\delta,\\epsilon) = \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{lemma}\n\\begin{proof}\n We first prove $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\leq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$. Note that for any $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, the corresponding marginal $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$. Moreover, $\\beta_{TR}(p_{USXY})=I(U;Y|S) = H(Y|S)-H(Y|US) \\leq H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Therefore, $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\leq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$. Conversely, given $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, define $\\AuxiliaryAlphabet=\\left\\{  0,1\\right\\}$ and a probability mass function $q_{USXY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ as $q_{USXY}(u,s,x,y)=p_{SXY}(s,x,y)1_{\\left\\{ u=x \\right\\}}$. Clearly $q_{SXY}=p_{SXY}$ and hence (i) and (iii) of definition \\ref{Defn:TestChannelsForPTP-STx} are satisfied. Note that $q_{USX}(x,s,x)=p_{SX}(s,x)$, and hence $q_{Y|XSU}(y|x,s,x)=p_{Y|XS}(y|x,s)=W_{Y|XS}(y|x,s)$. Hence $q_{USXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$. It is easy to verify $\\beta_{TR}(q_{USXY}) = \\alpha_{TR}(p_{SXY})$ and therefore $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\geq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{proof}\nWe now derive a characterization of the condition under which $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$. Towards that end, we first prove uniqueness of the pmf that achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTP-STxRx}\nSuppose $p_{SXY},q_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ are such that $\\alpha_{TR}(p_{SXY})=\\alpha_{TR}(q_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, then $p_{SXY}=q_{SXY}$. Moreover, if $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, then $p_{SX}=p_{S}p_{X}$, i.e., $S$ and $X$ are independent.\n\\end{lemma}\n\\begin{proof}\nClearly, if $q_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ satisfies $q_{SX}=q_{S}q_{X}$ with $q_{X}(1)=\\tau$, then $\\alpha_{TR}(q_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$ and since $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) \\leq h_{b}(\\tau * \\delta)-h_{b}(\\delta)$,\\footnote{This can be easily verified using standard information theoretic arguments.} we have $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) = h_{b}(\\tau * \\delta)-h_{b}(\\delta)$. Let $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ be another pmf for which $\\alpha_{TR}(p_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$. Let $\\chi_{0}\\define p_{X|S}(1|0)$ and $\\chi_{1}\\define p_{X|S}(1|1)$. $\\alpha_{TR}(p_{SXY})=I(X;Y|S)=H(Y|S)-H(Y|X,S)=H(X\\oplus S \\oplus N|S)-h_{b}(\\delta)$. We focus on the first term\n\\begin{eqnarray}\n\\lefteqn{H(X\\oplus S \\oplus N|S) = (1-\\epsilon)H(X\\oplus 0 \\oplus N|S=0)+\\epsilon H(X\\oplus 1 \\oplus N|S=1)}\\nonumber\\\\\n&=&(1-\\epsilon)h_{b}(\\chi_{0}(1-\\delta)+(1-\\chi_{0})\\delta)+\\epsilon h_{b}(\\chi_{1} (1-\\delta)+(1-\\chi_{1})\\delta)\n \\nonumber\\\\\n\\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}\n&\\leq& h_{b}((1-\\epsilon)\\chi_{0}(1-\\delta)+(1-\\epsilon)(1-\\chi_{0})\\delta+\\epsilon\\chi_{1} (1-\\delta)+\\epsilon(1-\\chi_{1})\\delta)\\\\\n\\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}\n&=&h_{b}(p_{X}(1)(1-\\delta)+(1-p_{X}(1))\\delta)=h_{b}(\\delta+p_{X}(1)(1-2\\delta)) \\leq h_{b}(\\delta+\\tau(1-2\\delta))=h_{b}(\\tau * \\delta)\n\\end{eqnarray}\nwhere (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}) follows from concavity of binary entropy function $h_{b}(\\cdot)$ and inequality in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}) follows from $\\delta \\in (0,\\frac{1}{2})$. We therefore have $\\alpha_{TR}(p_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$ if and only if equality holds in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}), (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}). $h_{b}(\\cdot)$ being strictly concave, equality holds in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}) if and only if $\\epsilon \\in \\left\\{ 0,1\\right\\}$ or $\\chi_{0}=\\chi_{1}$. The range of $\\epsilon$ precludes the former and therefore $\\chi_{0}=\\chi_{1}$. This proves $p_{SX}=p_{S}p_{X}$ and $p_{X}(1)=\\tau$. Given $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, these constrains completely determine $p_{SXY}$ and we have $p_{SXY}=q_{SXY}$.\n\\end{proof}\nFollowing is the main result of this section.\n\\begin{lemma}\n \\label{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ if and only if there exists a pmf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ such that\n\\begin{enumerate}\n \\item the corresponding marginal achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$,\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe first prove the reverse implication, i.e., the if statement. Note that $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\alpha_{TR}(p_{SXY})=I(X;Y|S)=H(Y|S)-H(Y|XS)\\overset{(a)}{=}H(Y|S)-H(Y|XSU)\\overset{(b)}{=}H(Y|S)-H(Y|US)=I(U;Y|S)=I(U;YS)-I(U;S)\\overset{(c)}{=}I(U;Y)-I(U;S) \\leq \\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, where (a) follows from (ii) of definition \\ref{Defn:TestChannelsForPTP-STx}, (b) follows from hypothesis 3) and (c) follows from hypothesis 2). We therefore have $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)\\leq \\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, and the reverse inequality follows from remark \\ref{Rem:SideInformationAtBothEncoderAndDecoderResultsInLargerCapacity}.\n\nConversely, let $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ achieve $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{T}(p_{USXY})=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$. We have $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\alpha_{T}(p_{USXY}) = I(U;Y)-I(U;S) \\overset{(b)}{\\leq} I(U;YS)-I(U;S) = I(U;Y|S)=H(Y|S)-H(Y|US) \\overset{(c)}{\\leq} H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})\\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Equality of $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon),\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ implies equality in (b), (c) and thus $I(U;S|Y)=0$ and $H(Y|US)=H(Y|USX)$ and moreover $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{proof}\n\nFor the particular binary additive point to point channel with state, we strengthen the condition for no rate loss in the following lemma.\n\\begin{lemma}\n \\label{Lem:XShouldBeAFunctionOfUAndS}\nIf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ satisfies\n\\begin{enumerate}\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain.\n\\end{enumerate}\nthen $H(X|U,S)=0$, or in other words, there exists a function $f : \\AuxiliaryAlphabet \\times \\StateAlphabet \\rightarrow \\InputAlphabet$ such that $P(X=f(U,S))=1$.\n\\end{lemma}\n\\begin{proof}\nWe prove this by contradiction. In particular, we prove $H(X|U,S) > 0$ violates Markov chain $X-(U,S)-Y$.\n\nIf $H(X|U,S) > 0$, then $H(X \\oplus S |U,S) >0$. Indeed, $0 < H(X|U,S)\\leq H(X,S|U,S)=H(X\\oplus S, S |U,S) = H(S|U,S)+H(X\\oplus S | U,S)=H(X \\oplus S | U, S)$. Since $(U,S,X)$ is independent of $X\\oplus S \\oplus Y$ and in particular, $(U,S,S\\oplus X)$ is independent of $X\\oplus S \\oplus Y$, we have $H((X\\oplus S)\\oplus(X\\oplus S \\oplus Y)|U,S)> H(X\\oplus S \\oplus Y|U,S)=h_{b}(\\delta)=H(Y|U,S,X)$, where the first inequality follows from concavity of binary entropy function. But $(X\\oplus S)\\oplus(X\\oplus S \\oplus Y)=Y$ and we have therefore proved $H(Y|U,S)>H(Y|U,S,X)$ contradicting Markov chain $X-(U,S)-Y$.\n\\end{proof}\nWe summarize the conditions for no rate loss below.\n\\begin{thm}\n \\label{Thm:CharacterizationOfConditionThatEnsuresNoRateLossForBinaryAdditive}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ if and only if there exists a pmf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ such that\n\\begin{enumerate}\n \\item the corresponding marginal achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, and in particular $S$ and $X$ are independent,\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain,\n\\item $H(X|U,S)=0$, or in other words, there exists a function $f : \\AuxiliaryAlphabet \\times \\StateAlphabet \\rightarrow \\InputAlphabet$ such that $P(X=f(U,S))=1$.\n\\end{enumerate}\n\\end{thm}\n\\section{The binary additive dirty point to point channel suffers a rate loss}\n\\label{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}\nThis section is dedicated to proving proposition \\ref{Prop:PropositionRegardindRateLoss}. We begin with an upper bound on cardinality of auxiliary set involved in characterization of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Lem:CardinalityBoundOnCostConstrainedGelfandPinskerAuxiliaryRV}\nConsider a point to point channel with state information available at transmitter. Let $\\mathcal{S}, \\mathcal{X}$ and $\\mathcal{Y}$ denote state, input and output alphabets respectively. Let $W_{S},W_{Y|XS}$ denote pmf of state, channel transition probabilities respectively. The input is constrained with respect to a cost function $\\kappa : \\mathcal{X} \\times \\mathcal{S} \\rightarrow [0,\\infty)$. Let $\\mathbb{D}_{T}(\\tau)$ denote the collection of all probability mass functions $p_{UXSY}$ defined on $\\mathcal{U} \\times \\mathcal{X} \\times \\mathcal{S} \\times \\mathcal{Y}$, where $\\mathcal{U}$ is an arbitrary set, such that (i) $p_{S}=W_{S}$, (ii) $p_{Y|XSU}=p_{Y|XS}=W_{Y|XS}$ and (iii) $\\Expectation \\left\\{ \\kappa(X,S) \\right\\} \\leq \\tau$. Moreover, let \\[\\overline{\\mathbb{D}_{T}}(\\tau) = \\left\\{ p_{UXSY} \\in \\mathbb{D}_{T}(\\tau): |\\mathcal{U}| \\leq \\min \\left\\{ \\substack{|\\mathcal{X}|\\cdot |\\mathcal{S}|, \\\\|\\mathcal{X}|+|\\mathcal{S}|+|\\mathcal{Y}|-2 }\\right\\} \\right\\}.\\] For $p_{UXSY} \\in \\mathbb{D}_{T}(\\tau)$, let $\\alpha(p_{UXSY})=I(U;Y)-I(U;S)$. Let \n\\begin{eqnarray} \n \\alpha_{T}(\\tau)=\\sup_{p_{UXSY} \\in \\mathbb{D}_{T}(\\tau)}\\alpha(p_{UXSY}) ,~~~~ \\overline{\\alpha_{T}}(\\tau)=\\sup_{p_{UXSY} \\in \\overline{\\mathbb{D}_{T}}(\\tau)}\\alpha(p_{UXSY}). \\nonumber\n\\end{eqnarray}\nThen $\\alpha_{T}(\\tau)=\\overline{\\alpha_{T}}(\\tau)$.\n\\end{lemma}\n\\begin{proof}\n The proof is based on Fenchel-Eggelston-Carath\\'eodory \\cite{Egg-Convexity1958}, \\cite[Appendix C]{201201NIT_ElgKim} theorem which is stated here for ease of reference.\n\\begin{lemma}\n\\label{Lem:FenchelEgglestonCaratheodory}\n let $\\mathcal{A}$ be a finite set and $\\mathcal{Q}$ be an arbitrary set. Let $\\mathcal{P}$ be a connected compact subset of pmf's on $\\mathcal{A}$ and $p_{A|Q}(\\cdot | q) \\in \\mathcal{P}$ for each $q \\in \\mathcal{Q}$. For $j=1,2,\\cdots,d$ let $g_{j}: \\mathcal{P} \\rightarrow \\reals$ be continuous functions. Then for every $Q \\sim F_{Q}$ defined on $\\mathcal{Q}$, there exist a random variable $\\overline{Q} \\sim p_{\\overline{Q}}$ with $|\\overline{\\mathcal{Q}}| \\leq d$ and a collection of pmf's $p_{A|\\overline{Q}}(\\cdot|\\overline{q}) \\in \\mathcal{P}$, one for each $\\overline{q} \\in \\overline{\\mathcal{Q}}$, such that\n\\begin{eqnarray}\n \\int_{\\mathcal{Q}}g_{j}(p_{A|Q}(a|q))dF_{Q}(q) = \\sum_{\\overline{q} \\in \\overline{\\mathcal{Q}}}g_{j}(p_{A|\\overline{Q}}(a|\\overline{q}))p_{\\overline{Q}}(\\overline{q}).\\nonumber\n\\end{eqnarray}\n\\end{lemma}\nThe proof involves identifying $g_{j}:j=1,2\\cdots,d$ such that rate achievable and cost expended are preserved. We first prove the bound $|\\mathcal{U}| \\leq |\\mathcal{X}|\\cdot|\\mathcal{S}|$.\n\nSet $\\mathcal{Q}=\\mathcal{U}$ and $\\mathcal{A}=\\mathcal{X} \\times \\mathcal{S}$ and $\\mathcal{P}$ denote the connected compact subset of pmf's on $\\mathcal{X} \\times \\mathcal{S}$. Without loss of generality, let $\\mathcal{X} = \\left\\{1,2,\\cdots, |\\mathcal{X}|  \\right\\}$ and $\\mathcal{S} = \\left\\{1,2,\\cdots, |\\mathcal{S}|  \\right\\}$. For $i=1,2, \\cdots, |\\mathcal{X}|$ and $k=1,2,\\cdots,|\\mathcal{S}|-1$, let $g_{i,k}(\\pi_{X,S})=\\pi_{X,S}(i,k)$ and $g_{l,|\\mathcal{S}|}(\\pi_{X,S})=\\pi_{X,S}(l,|\\mathcal{S}|)$ for $l=1,2,\\cdots, |\\mathcal{X}|-1$. Let $g_{|\\mathcal{X}|\\cdot |\\mathcal{S}|}(\\pi_{X,S})=H(S)-H(Y)$. It can be verified that\n\\begin{flalign}\n g_{|\\mathcal{X}|\\cdot |\\mathcal{S}|}(\\pi_{X,S}) =& -\\sum_{s \\in \\mathcal{S}} (\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))\\log_{2}(\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))+ \\sum_{y \\in \\mathcal{Y}} \\theta(y)\\log_{2}(\\theta(y)),\\mbox{ where }\\nonumber\\\\\n\\label{Eqn:ThetaOfy}\n\\!\\!\\!\\!\\!\\!\\!\\theta(y)=&\\sum_{(x,s) \\in \\mathcal{X}\\times \\mathcal{S}}\\pi_{X,S}(x,s)W_{Y|XS}(y|x,s)\n\\end{flalign}\nwhere, is continuous. An application of lemma \\ref{Lem:FenchelEgglestonCaratheodory} using the above set of functions, the upper bound $|\\mathcal{X}|\\cdot|\\mathcal{S}|$ on $|\\mathcal{U}|$ can be verified.\n\nWe now outline proof of upper bound $|\\mathcal{X}|+|\\mathcal{S}|+|\\mathcal{Y}|-2$ on $|\\mathcal{U}|$. Without loss of generality, we assume $\\mathcal{X}=\\left\\{ 1,\\cdots, |\\mathcal{X}| \\right\\}$, $\\mathcal{S}=\\left\\{ 1,\\cdots, |\\mathcal{S}| \\right\\}$ and $\\mathcal{Y}=\\left\\{ 1,\\cdots, |\\mathcal{Y}| \\right\\}$. As earlier, set $\\mathcal{Q}=\\mathcal{U}$ and $\\mathcal{A}=\\mathcal{X} \\times \\mathcal{S}$ and $\\mathcal{P}$ denote the connected compact subset of pmf's on $\\mathcal{X} \\times \\mathcal{S}$. For $j=1,\\cdots,|\\mathcal{S}|-1$, let $g_{j}(\\pi_{X,S})=\\sum_{x \\in \\mathcal{X}} \\pi_{X,S}(x,j)$. For $j=|\\mathcal{S}|, \\cdots, |\\mathcal{S}|+|\\mathcal{Y}|-2$, let $g_{j}(\\pi_{X,S})=\\sum_{(x,s) \\in \\mathcal{X} \\times \\mathcal{S}}\\pi_{X,S}(x,s)W_{Y|X,S}(j-|\\mathcal{S}|+1|x,s)$. For $j=|\\mathcal{S}|+|\\mathcal{Y}|-1, \\cdots, |\\mathcal{S}|+|\\mathcal{Y}|+|\\mathcal{X}|-3$, let $g_{j}(\\pi_{X,S})=\\sum_{s \\in \\mathcal{S}}\\pi_{X,S}(j-|\\mathcal{S}|-|\\mathcal{Y}|+2,s)$. Let $g_{t}(\\pi_{X,S})=H(S)-H(Y)$, i.e.,\n\\begin{eqnarray}\ng_{t}(\\pi_{X,S}) =  -\\sum_{s \\in \\mathcal{S}} (\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))\\log_{2}(\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))+ \\sum_{y \\in \\mathcal{Y}} \\theta(y)\\log_{2}(\\theta(y)),\\nonumber\n\\end{eqnarray}\nwhere $t=|\\mathcal{S}|+|\\mathcal{Y}|+|\\mathcal{X}|-2$, and $\\theta(y)$ as is in (\\ref{Eqn:ThetaOfy}). The rest of the proof follows by simple verification.\n\\end{proof}\n\n\\begin{prop}\n \\label{Prop:PropositionRegardindRateLoss}\nThere exists no probability mass function $p_{UXSY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ where $\\AuxiliaryAlphabet = \\left\\{  0,1,2,3 \\right\\}, \\InputAlphabet = \\StateAlphabet = \\OutputAlphabet = \\left\\{ 0,1 \\right\\}$, such that\n\\begin{enumerate}\n \\item $X$ and $S$ are independent with $P(S=1)=\\epsilon$, $P(X=1)=\\tau$, where $\\epsilon \\in (0,1)$, $\\tau \\in (0,\\frac{1}{2})$,\n\\item $p_{Y|X,S,U}(x \\oplus s|x,s,u)=p_{Y|X,S}(x\\oplus s |x,s)=1-\\delta$ for every $(u,x,s,y) \\in \\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$, where $\\delta \\in (0,\\frac{1}{2})$,\n\\item $U-Y-S$ and $X-(U,S)-Y$ are Markov chains, and\n\\item $p_{X|US}(x|u,s) \\in \\left\\{ 0,1 \\right\\}$ for each $(u,s,x) \\in \\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n The proof is by contradiction. If there exists such a pmf $p_{USXY}$ then conditions 1) and 2) completely specify it's marginal on $\\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ and it maybe verified that $p_{SY}(0,0)=(1-\\epsilon)(1-\\theta), p_{SY}(0,1)=(1-\\epsilon)\\theta, p_{SY}(1,0)=\\epsilon\\theta,p_{SY}(1,1)=\\epsilon(1-\\theta)$, where $\\theta \\define \\delta(1-\\tau)+(1-\\delta)\\tau$ takes a value in $(0,1)$. Since $\\epsilon \\in (0,1)$, $p_{SY}(s,y) \\in (0,1)$ for each $(s,y) \\in \\StateAlphabet \\times \\OutputAlphabet$.\nIf we let $\\beta_{i}\\define p_{U|Y}(i|0) :i=0,1,2,3$ and $\\gamma_{j}\\define p_{U|Y}(j|1):j=0,1,2,3$, then Markov chain $U-Y-S$ implies $p_{USY}$ is as in table \\ref{Table:p_USY}.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUSY & $p_{USY}$&USY & $p_{USY}$\\\\\n\\hline\n000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&200 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n001 & $(1-\\epsilon)\\theta\\gamma_{0}$&201 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n010 & $\\epsilon\\theta\\beta_{0}$&210 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n011 & $\\epsilon(1-\\theta) \\gamma_{0}$ &211 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n100 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &300 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n101 & $(1-\\epsilon)\\theta \\gamma_{1}$ &301 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n110 & $\\epsilon\\theta\\beta_{1}$ &310 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n111 & $\\epsilon(1-\\theta) \\gamma_{1}$ &311 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{USY}$} \\label{Table:p_USY} \n\\end{table}  \nSince $X$ is a function of $(U,S)$\\footnote{With probability $1$}, there exist $z_{i} \\in \\left\\{ 0,1 \\right\\}:i=0,1,\\cdots,7$ such that entries of table \\ref{Table:p_USX} hold true.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n$p_{USX}(0,0,0) =p_{US}(0,0)z_{0}$&$p_{USX}(0,1,0) =p_{US}(0,1)z_{4}$\\\\\n\\hline\n$p_{USX}(1,0,0) =p_{US}(1,0)z_{1}$&$p_{USX}(1,1,0) =p_{US}(1,1)z_{5}$\\\\\n\\hline\n$p_{USX}(2,0,0) =p_{US}(2,0)z_{2}$&$p_{USX}(2,1,0) =p_{US}(2,1)z_{6}$\\\\\n\\hline\n$p_{USX}(3,0,0) =p_{US}(3,0)z_{3}$&$p_{USX}(3,1,0) =p_{US}(3,1)z_{7}$\\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{USX}$} \\label{Table:p_USX} \n\\end{table}  \nMoreover, condition 4) and Markov chain $X-(U,S)-Y$ implies $p_{USXY}$ is completely determined in terms of entries of table \\ref{Table:p_USY} and $z_{i}:i=0,1,\\cdots,7$. For example $p_{USXY}(3,0,1,1)=p_{USY}(3,0,1)(1-z_{3})$. This enables us compute marginal $p_{SXY}$ in terms of entries of table \\ref{Table:p_USY} and $z_{i}:i=0,1,\\cdots,7$. This marginal must satisfy conditions 1) and 2) which implies\/is equivalent to the last two columns of table \\ref{Table:P_SXY} being equal.\n\\begin{eqnarray}\n\\label{Eqn:P_SYXOf000}\np_{SYX}(0,0,0)&=&(1-\\epsilon)(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2  z_2+\\beta_{3}z_{3} \\right] =(1-\\tau)(1-\\epsilon)(1-\\delta)\\\\\n\\label{Eqn:P_SYXOf001}\np_{SYX}(0,0,1)&=& (1-\\epsilon)(1-\\theta)\\left[1-\\beta_0z_0-\\beta_1 z_1-\\beta_2  z_2-\\beta_{3}z_{3} \\right] = \\tau(1-\\epsilon)\\delta\\nonumber\\\\\n\\label{Eqn:P_SYXOf010}\np_{SYX}(0,1,0)&=& (1-\\epsilon)\\theta \\left[\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2  z_2 +\\gamma_{3} z_{3} \\right]= (1-\\tau)(1-\\epsilon)\\delta\\\\\n\\label{Eqn:P_SYXOf011}\np_{SYX}(0,1,1)&=& (1-\\epsilon)\\theta\\left[1-\\gamma_0z_0-\\gamma_1 z_1-\\gamma_2  z_2-\\gamma_{3} z_{3} \\right]\n= \\tau(1-\\epsilon)(1-\\delta)\\nonumber\\\\\n\\label{Eqn:P_SYXOf100}\np_{SYX}(1,0,0)&=& \\epsilon \\theta\\left[\\beta_0z_4+\\beta_1 z_5+\\beta_2  z_6+\\beta_{3}z_{7} \\right] = (1-\\tau)\\epsilon \\delta\\\\\n\\label{Eqn:P_SYXOf101}\np_{SYX}(1,0,1)&=& \\epsilon \\theta\\left[1-\\beta_0z_4-\\beta_1 z_5-\\beta_2 z_6-\\beta_{3}z_{7} \\right] = \\tau \\epsilon(1-\\delta)\\nonumber\\\\\n\\label{Eqn:P_SYXOf110}\np_{SYX}(1,1,0)&=& \\epsilon(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2  z_6+\\gamma_{3}z_{7} \\right] = (1-\\tau) \\epsilon (1-\\delta)\\\\\n\\label{Eqn:P_SYXOf111}\np_{SYX}(1,1,1)&=& \\epsilon(1-\\theta) \\left[1-\\gamma_0z_4-\\gamma_1 z_5-\\gamma_2  z_6-\\gamma_{3}z_{7} \\right]= \\tau \\epsilon \\delta\\nonumber\n\\end{eqnarray}\nSince $\\epsilon \\notin \\left\\{ 0,1\\right\\}$, (\\ref{Eqn:P_SYXOf000}),(\\ref{Eqn:P_SYXOf110}) imply \n\\begin{eqnarray}\n\\label{Eqn:EquatingP_SYX000AndP_SYX110}\n(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2\n  z_2+\\beta_{3}z_{3} \\right]=(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n  z_6+\\gamma_{3}z_{7} \\right]\\nonumber\n\\end{eqnarray}\nwhich further implies\n\\begin{eqnarray}\n \\label{Eqn:ConsequenceEquatingP_SYX000AndP_SYX110}\n\\beta_0z_0+\\beta_1 z_1+\\beta_2\n  z_2+\\beta_{3}z_{3} =\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n  z_6+\\gamma_{3}z_{7}=:\\psi_{1}\\nonumber\n\\end{eqnarray}\nSimilarly (\\ref{Eqn:P_SYXOf010}),(\\ref{Eqn:P_SYXOf100}) imply\n\\begin{eqnarray}\n \\label{Eqn:ConsequenceEquatingP_SYX010AndP_SYX100}\n\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2\n  z_2 +\\gamma_{3} z_{3} =\\beta_0z_4+\\beta_1 z_5+\\beta_2\n  z_6+\\beta_{3}z_{7}=:\\psi_{2}\\nonumber\n\\end{eqnarray}\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nSYX & $p_{SYX}$ &\\\\\n\\hline\n000 & $(1-\\epsilon)(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2\n  z_2+\\beta_{3}z_{3} \\right]$ & $(1-\\tau)(1-\\epsilon)(1-\\delta)$ \\\\\n\\hline\n001 & $(1-\\epsilon)(1-\\theta)\\left[1-\\beta_0z_0-\\beta_1 z_1-\\beta_2\n  z_2-\\beta_{3}z_{3} \\right] $ & $\\tau(1-\\epsilon)\\delta$\\\\\n\\hline\n010 & $(1-\\epsilon)\\theta \\left[\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2\n  z_2 +\\gamma_{3} z_{3} \\right]$ & $(1-\\tau)(1-\\epsilon)\\delta$\\\\\n\\hline\n011 & $(1-\\epsilon)\\theta\\left[1-\\gamma_0z_0-\\gamma_1 z_1-\\gamma_2\n  z_2-\\gamma_{3} z_{3} \\right]$ & $\\tau(1-\\epsilon)(1-\\delta)$\\\\\n\\hline\n100 & $ \\epsilon \\theta\\left[\\beta_0z_4+\\beta_1 z_5+\\beta_2\n  z_6+\\beta_{3}z_{7} \\right] $ & $(1-\\tau)\\epsilon \\delta$\\\\\n\\hline\n101 & $\\epsilon \\theta\\left[1-\\beta_0z_4-\\beta_1 z_5-\\beta_2\n  z_6-\\beta_{3}z_{7} \\right]$ & $\\tau \\epsilon(1-\\delta)$\\\\\n\\hline\n110 & $\\epsilon(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n  z_6+\\gamma_{3}z_{7} \\right]$ & $(1-\\tau) \\epsilon (1-\\delta)$\\\\\n\\hline\n111 & $\\epsilon(1-\\theta) \\left[1-\\gamma_0z_4-\\gamma_1 z_5-\\gamma_2\n  z_6-\\gamma_{3}z_{7} \\right]$ & $\\tau \\epsilon \\delta$\\\\\n\\hline\n\\end{tabular} \\end{center}\n\\caption{Enforcing conditions 1) and 2) for $p_{SXY}$} \\label{Table:P_SXY}\n\\end{table}\nWe now argue there exists no choice of values for $z_{i}:i=0,1\\cdots,7$. Towards that end, we make a couple of observations. Firstly, we argue $\\psi_{1} \\neq \\psi_{2}$. Since $\\epsilon \\neq 1$ and $\\theta \\in (0,1)$, we have $\\psi_{1}=\\frac{(1-\\tau)(1-\\delta)}{(1-\\theta)}$ and $\\psi_{2} = \\frac{(1-\\tau)\\delta}{\\theta}$ from (\\ref{Eqn:P_SYXOf000}) and (\\ref{Eqn:P_SYXOf010}) respectively. Equating $\\psi_{1}$ and $\\psi_{2}$, we obtain either $\\tau=1$ or $\\tau=0$ or $\\delta=\\frac{1}{2}$. Since none of the latter conditions hold, we conclude $\\psi_{1}\\neq \\psi_{2}$. Secondly, one can verify $\\psi_{1}+\\psi_{2}-1=\\frac{\\delta (1-\\delta)(1-2\\tau)}{\\theta(1-\\theta)}$. Since $\\delta \\in (0,\\frac{1}{2}),\\theta \\in (0,1)$ and $\\tau \\in (0,\\frac{1}{2})$, $\\psi_{1}+\\psi_{2} > 1$.\nWe now eliminate the possible choices for $z_{i}:i=0,1\\cdots,7$ through the following cases. let $m \\define \\left| \\left\\{ i \\in \\left\\{ 0,1,2,3 \\right\\} :z_{i}=1\\right\\} \\right|$ and $l \\define \\left| \\left\\{ i \\in \\left\\{ 4,5,6,7 \\right\\} :z_{i}=1\\right\\} \\right|$.\n\n\\noindent \\textit{Case 1:} All of $z_{0},z_{1},z_{2},z_{3}$ or all of $z_{4},z_{5},z_{6},z_{7}$ are equal to 0, i.e., $m=0$ or $l=0$. This implies $\\psi_{1}=\\psi_{2}=0$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 2:} All of $z_{0},z_{1},z_{2},z_{3}$ or all of $z_{4},z_{5},z_{6},z_{7}$ are equal to 1, i.e., $m=4$ or $l=4$. This implies $\\psi_{1}=\\psi_{2}=1$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nCases 1 and 2 imply $m,l \\in \\left\\{1,2,3\\right\\}$.\n\n\\noindent \\textit{Case 3: $m=l=3$}. If $i_{1},i_{2},i_{3}$ are distinct indices in $\\left\\{ 0,1,2,3 \\right\\}$ such that $z_{i_{1}}=z_{i_{2}}=z_{i_{3}}=1$, then one among $z_{i_{1}+4},z_{i_{2}+4},z_{i_{3}+4}$ has to be $0$. Else $\\psi_{1}=\\beta_{i_{1}}+\\beta_{i_{2}}+\\beta_{i_{3}}$ and $\\psi_{2}=\\beta_{i_{1}}z_{i_{1}+4}+\\beta_{i_{2}}z_{i_{2}+4}+\\beta_{i_{3}}z_{i_{3}+4}=\\beta_{i_{1}}+\\beta_{i_{2}}+\\beta_{i_{3}}=\\psi_{1}$ contradicting $\\psi_{1} \\neq \\psi_{2}$. Let us consider the case $z_{0}=z_{1}=z_{2}=1$, $z_{3}=z_{4}=0$ and $z_{5}=z_{6}=z_{7}=1$. Table \\ref{Table:p_UXSY} tabulates $p_{USXY}$ for this case.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2000 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\beta_{3}$&2001 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\beta_{3}$ &2011 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\beta_{0}$ \\\\\n\\hline\n1010 & $\\epsilon\\theta\\beta_{1}$ &3010 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1011 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3011 & $\\epsilon(1-\\theta)\\beta_{0}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSY} \n\\end{table}\nWe have $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{1}+\\gamma_{2}+\\gamma_{3}$ or equivalently $\\psi_{1}=1-\\beta_{3}=1-\\gamma_{0}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{3}=\\beta_{1}+\\beta_{2}+\\beta_{3}$ or equivalently $\\psi_{2}=1-\\gamma_{3}=1-\\beta_{0}$. These imply $\\gamma_{3}=\\beta_{0}$, $\\gamma_{0}=\\beta_{3}$ which further imply $\\gamma_{1}+\\gamma_{2}=\\beta_{1}+\\beta_{2}$ (since $1=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}+\\gamma_{3}=\\beta_{0}+\\beta_{1}+\\beta_{2+}\\beta_{3}$). From table \\ref{Table:p_UXSY}, one can verify \n\\begin{eqnarray}\n\\label{Eqn:P_UGivenXSYOf0Given001}\n&p_{U|XSY}(0|0,0,1)=\\frac{\\beta_{3}(1-\\epsilon)\\theta}{(1-\\epsilon)\\theta(\\beta_{3}+\\gamma_{1}+\\gamma_{2})} = \\frac{\\beta_{3}}{\\beta_{1}+\\beta_{2}+\\beta_{3}},\n\\nonumber\\\\\n\\lefteqn{p_{U|XS}(0|0,0)=\\frac{(1-\\theta)\\beta_{0}+\\theta\\beta_{3}}{(1-\\theta)(\\beta_{0}+\\beta_{1}+\\beta_{2})+\\theta(\\beta_{3}+\\gamma_{1}+\\gamma_{2})}}.\\nonumber\n\\end{eqnarray}\nThe Markov chain $U-(X,S)-Y$ implies $p_{U|XSY}(0|0,0,1)=p_{U|XS}(0|0,0)$. Equating the right hand sides of the above equations, we obtain $(1-\\theta)(\\beta_{0}-\\beta_{3})(\\beta_{1}+\\beta_{2})=0$. Since $\\theta \\neq 0$, $\\beta_{1}+\\beta_{2}=0$ or $\\beta_{0}=\\beta_{3}$. If $\\beta_{0}=\\beta_{3}$, then $1-\\beta_{3}=\\psi_{1}=\\psi_{2}=1-\\beta_{0}$ thus contradicting $\\psi_{1} \\neq \\psi_{2}$. If $\\beta_{1}+\\beta_{2}=0$, then $\\beta_{0}+\\beta_{3}=1$ implying $\\psi_{1}+\\psi_{2}=1$ contradicting $\\psi_{1}+\\psi_{2}>1$.\n\n\\noindent \\textit{Case 4: $m=3, l=2$}. Let us assume $z_{0}=z_{1}=z_{2}=z_{6}=z_{7}=1,z_{3}=z_{4}=z_{5}=0$. We then have $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{2}+\\gamma_{3}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{2}+\\beta_{3}$. Since $\\beta_{0}+\\beta_{1}+\\beta_{2}=1-\\beta_{3}$ and $\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=1-\\gamma_{3}$, we have $\\gamma_{2}+\\gamma_{3}=1-\\beta_{3}$ and $\\beta_{2}+\\beta_{3}=1-\\gamma_{3}$ and therefore $\\gamma_{2}=\\beta_{2}$.Table \\ref{Table:p_UXSYCasemIs3lIs2} tabulates $p_{USXY}$ for this case.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2000 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\gamma_{0}$&2001 & $(1-\\epsilon)\\theta \\beta_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\gamma_{0}$ &2011 & $\\epsilon(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n1110 & $\\epsilon\\theta\\beta_{1}$ &3010 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1111 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3011 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSYCasemIs3lIs2} \n\\end{table}\nFrom table \\ref{Table:p_UXSYCasemIs3lIs2}, one can verify \n\\begin{eqnarray}\n\\label{Eqn:P_UGivenXSYOf0Given001}\n&p_{U|XSY}(2|0,0,1)=\\frac{\\beta_{2}(1-\\epsilon)\\theta}{(1-\\epsilon)\\theta(\\beta_{2}+\\gamma_{0}+\\gamma_{1})} = \\frac{\\beta_{2}}{\\beta_{2}+\\gamma_{0}+\\gamma_{1}},\n\\nonumber\\\\\n\\lefteqn{p_{U|XS}(2|0,0)=\\frac{\\beta_{2}}{(1-\\theta)(\\beta_{0}+\\beta_{1})+\\theta(\\gamma_{0}+\\gamma_{1})+\\beta_{2}}}.\\nonumber\n\\end{eqnarray}\nThe Markov chain $U-(X,S)-Y$ implies $p_{U|XSY}(2|0,0,1)=p_{U|XS}(2|0,0)$. Equating the RHS of the above equations, we obtain $\\beta_{0}+\\beta_{1}=\\gamma_{0}+\\gamma_{1}$. This implies $\\beta_{2}+\\beta_{3}=\\gamma_{2}+\\gamma_{3}$. However $\\psi_{1}=\\beta_{2}+\\beta_{3}$ and $\\psi_{2}=\\gamma_{2}+\\gamma_{3}$, this contradicting $\\psi \\neq \\psi_{2}$.\n\nLet us assume $z_{0}=z_{1}=z_{2}=z_{5}=z_{6}=1$ and $z_{3}=z_{4}=z_{7}=0$. It can be verified that $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{1}+\\gamma_{2}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{1}+\\beta_{2}$. This implies $\\psi_{1}-\\psi_{2}=\\beta_{0}=-\\gamma_{0}$. Since $\\beta_{0}$ and $\\gamma_{0}$ are non-negative, $\\beta_{0}=\\gamma_{0}=0$ implying $\\psi_{1}-\\psi_{2}=0$, contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 5: $m=3, l=1$}. Assume $z_{0}=z_{1}=z_{2}=z_{4}=1$, $z_{3}=z_{5}=z_{6}=z_{7}=0$. It can be verified that $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{0}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{0}$. Therefore $\\psi_{1}-\\psi_{2}=\\beta_{1}+\\beta_{2}$ and $\\psi_{2}-\\psi_{1}=\\gamma_{1}+\\gamma_{2}$. Since $\\beta_{i},\\gamma_{i}:i \\in \\left\\{ 0,1,2,3 \\right\\}$ are non-negative, $\\psi_{1}-\\psi_{2} \\geq 0$ and $\\psi_{2}-\\psi_{1} \\geq 0$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{2}=z_{7}=1$ and $z_{3}=z_{4}=z_{5}=z_{6}=0$. In this case, $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{3}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=1-\\gamma_{3}$. We have $\\psi_{1}+\\psi_{2}=1$ contradicting $\\psi_{1}+\\psi_{2}>1$.\n\n\\noindent \\textit{Case 6: $m=2, l=2$}. Assume $z_{0}=z_{1}=z_{4}=z_{5}=1$, $z_{2}=z_{3}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{0}=\\gamma_{1}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{0}+\\beta_{1}$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{6}=z_{7}=1$, $z_{2}=z_{3}=z_{4}=z_{5}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{2}+\\gamma_{3}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{2}+\\beta_{3}$ contradicting $\\psi_{1} + \\psi_{2}>1$.\n\nAssume $z_{0}=z_{1}=z_{5}=z_{6}=1$, $z_{2}=z_{3}=z_{4}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{1}+\\gamma_{2}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{1}+\\beta_{2}$ and therefore $\\beta_{2}+\\beta_{3}=\\gamma_{0}+\\gamma_{3}$ and $\\beta_{0}+\\beta_{3}=\\gamma_{2}+\\gamma_{3}$. We observe\n\\begin{equation}\n \\label{Eqn:CasemIs2lIs2Psi1-Psi2}\n\\psi_{1}-\\psi_{2}= \\beta_{0}-\\beta_{2}=\\gamma_{2}-\\gamma_{0}\n\\end{equation}\n\nPMF $p_{UXSY}$ is tabulated in \\ref{Table:p_UXSYCasemIs2lIs2} for this case. Table \\ref{Table:p_UXSYCasemIs2lIs2} enables us compute conditional pmf $p_{U|XSY}$ which is tabulated in table \\ref{Table:p_UGivenXSYCasemIs2lIs2}.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2100 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\gamma_{0}$&2101 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\gamma_{0}$ &2011 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n1010 & $\\epsilon\\theta\\beta_{1}$ &3110 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1011 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3111 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSYCasemIs2lIs2} \n\\end{table}\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{U|XSY}$&UXSY & $p_{U|XSY}$\\\\\n\\hline\n0000 & $\\frac{\\beta_{0}}{\\beta_{0}+\\beta_{1}}$&0001 & $\\frac{\\gamma_{0}}{\\gamma_{0}+\\gamma_{1}}$ \\\\\n\\hline\n0110 & $\\frac{\\beta_{0}}{\\beta_{0}+\\beta_{3}}$&0111 & $\\frac{\\gamma_{0}}{\\gamma_{0}+\\gamma_{3}}$ \\\\\n\\hline\n1000 & $\\frac{\\beta_{1}}{\\beta_{0}+\\beta_{1}}$ &1001 & $\\frac{\\gamma_{1}}{\\gamma_{0}+\\gamma_{1}}$ \\\\\n\\hline\n1010 & $\\frac{\\beta_{1}}{\\beta_{1}+\\beta_{2}}$ &1011 & $\\frac{\\gamma_{1}}{\\gamma_{1}+\\gamma_{2}}$ \\\\\n\\hline\n 2100& $\\frac{\\beta_{2}}{\\beta_{2}+\\beta_{3}}$&2101 & $\\frac{\\gamma_{2}}{\\gamma_{2}+\\gamma_{3}}$ \\\\\n\\hline\n 2010& $\\frac{\\beta_{2}}{\\beta_{1}+\\beta_{2}}$ &2011 & $\\frac{\\gamma_{2}}{\\gamma_{1}+\\gamma_{2}}$ \\\\\n\\hline\n 3100& $\\frac{\\beta_{3}}{\\beta_{2}+\\beta_{3}}$ &3101 & $\\frac{\\gamma_{3}}{\\gamma_{2}+\\gamma_{3}}$ \\\\\n\\hline\n3110 & $\\frac{\\beta_{3}}{\\beta_{0}+\\beta_{3}}$ &3111 & $\\frac{\\gamma_{3}}{\\gamma_{0}+\\gamma_{3}}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{U|XSY}$} \\label{Table:p_UGivenXSYCasemIs2lIs2} \n \\end{table}\nMarkov chain $U-(X,S)-Y$ implies columns 2 and 4 of table \\ref{Table:p_UGivenXSYCasemIs2lIs2} are identical. This implies\n\\begin{eqnarray}\n \\label{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}\n\\frac{\\beta_{0}}{\\gamma_{0}}\\overset{(a)}{=}\\frac{\\beta_{0}+\\beta_{1}}{\\gamma_{0}+\\gamma_{1}}\\overset{(b)}{=}\\frac{\\beta_{1}}{\\gamma_{1}}, \\frac{\\beta_{2}}{\\gamma_{2}}\\overset{(c)}{=}\\frac{\\beta_{2}+\\beta_{3}}{\\gamma_{2}+\\gamma_{3}}\\overset{(d)}{=}\\frac{\\beta_{3}}{\\gamma_{3}}, ~~\\mbox{ and }~~ \\frac{\\beta_{0}}{\\gamma_{0}}\\overset{(e)}{=}\\frac{\\beta_{0}+\\beta_{3}}{\\gamma_{0}+\\gamma_{3}}\\overset{(f)}{=}\\frac{\\beta_{3}}{\\gamma_{3}},\n\\end{eqnarray}\nwhere (a),(b),(c),(d) in (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}) is obtained by equating rows 1, 3, 5, 7 of columns 2 and 4 respectively and (e) and (f) in (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}) are obtained by equating rows 2 and 8 of columns 2 and 4 respectively. (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}), enables us conclude\n\\begin{equation}\n \\label{Eqn:BetaIsEqualToGamma}\n\\frac{\\beta_{0}}{\\gamma_{0}}=\\frac{\\beta_{1}}{\\gamma_{1}}=\\frac{\\beta_{2}}{\\gamma_{2}}=\\frac{\\beta_{3}}{\\gamma_{3}}. \\nonumber\n\\end{equation}\nSince $\\beta_{0}+\\beta_{1}+\\beta_{2}+\\beta_{3}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}+\\gamma_{3}=1$, we have $\\beta_{i}=\\gamma_{i}$ for each $i \\in \\left\\{ 0,1,2,3\\right\\}$ which yields $\\psi_{1}=\\psi_{2}$ in (\\ref{Eqn:CasemIs2lIs2Psi1-Psi2}) contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 7: $m=2, l=1$}. Assume $z_{0}=z_{1}=z_{4}=1, z_{2}=z_{3}=z_{5}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{0}, \\psi_{2}= \\gamma_{0}+\\gamma_{1}=\\beta_{0}$ and hence $\\psi_{1}-\\psi_{2}=\\beta_{1}$ and $\\psi_{2}-\\psi_{1}=\\gamma_{1}$. Since $\\gamma_{1}$ and $\\beta_{1}$ are non-negative, we have $\\psi_{1}=\\psi_{2}$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{7}=1, z_{2}=z_{3}=z_{4}=z_{5}=z_{6}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{3}, \\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{3}$ and hence $\\psi_{1}+\\psi_{2} = \\beta_{0}+\\beta_{1}+\\beta_{3} \\leq 1$ contradicting $\\psi_{1}+\\psi_{2} >1$.\n\n\\noindent \\textit{Case 6: $m=1, l=1$}. Assume $z_{0}=z_{4}=1, z_{1}=z_{2}=z_{3}=z_{5}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=beta_{0}=\\gamma_{0},\\psi_{2}=\\gamma_{0}=\\beta_{0}$, thus contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{5}=1, z_{1}=z_{2}=z_{3}=z_{4}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}=\\gamma_{1},\\psi_{2}=\\gamma_{0}=\\beta_{1}$, and hence $\\psi_{1}+\\psi_{2} = \\beta_{0}+\\beta_{1} \\leq 1$, thus contradicting $\\psi_{1} + \\psi_{2} >1$.\n\\end{proof}\n\n\\section{Proof of lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}}\n\\label{AppSec:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}\nSince $A-B-Y$ and $AB-X-Y$ are Markov chains, to prove $A-B-XY$ is a Markov chain, it suffices to prove $A-B-X$ is a Markov chain. We therefore need to prove $p_{XA|B}(x_{k},a_{i}|b_{j})=p_{X|B}(x_{k}|b_{j})p_{A|B}(a_{i}|b_{j})$ for every $(x_{k},a_{i},b_{j}) \\in \\left\\{ 0,1\\right\\}\\times \\mathcal{A} \\times \\mathcal{B}$ such that $p_{B}(b_{j})>0$. It suffices to prove $p_{XA|B}(0,a_{i}|b_{j})=p_{X|B}(0|b_{j})p_{A|B}(a_{i}|b_{j})$ for every $(a_{i},b_{j}) \\in \\mathcal{A} \\times \\mathcal{B}$ such that $p_{B}(b_{j})>0$.\\footnote{Indeed, $p_{XA|B}(1,a_{i}|b_{j})=p_{A|B}(a_{i}|b_{j})-p_{XA|B}(0,a_{i}|b_{j})=p_{A|B}(a_{i}|b_{j})(1-p_{X|B}(0|b_{j}))=p_{A|B}(a_{i}|b_{j})p_{X|B}(1|b_{j})$.}\n\nFix a $b_{j}$ for which $p_{B}(b_{j})>0$. Let $p_{A|B}(a_{i}|b_{j})=\\alpha_{i}$ for each $i \\in \\naturals$ and $p_{XA|B}(0,a_{i}|b_{j})=\\chi_{i}$ for each $(i,j) \\in \\naturals\\times \\naturals$. It can be verified $p_{XYA|B}(\\cdot,\\cdot,\\cdot|b_{j})$ is as in table \\ref{Table:p_XYAGivenB}. From table \\ref{Table:p_XYAGivenB}, we infer $p_{AY|B}(a_{i}0|b_{j})=\\chi_{i}(1-\\eta)+(\\alpha_{i}-\\chi_{i})\\eta=\\alpha_{i}\\eta+\\chi_{i}(1-2\\eta)$. From the Markov chain $A-B-Y$, we have $p_{AY|B}(a_{i}0|b_{j})=p_{A|B}(a_{i}|b_{j})p_{Y|B}(0|b_{j})= \\alpha_{i}p_{Y|B}(0|b_{j})$. Therefore, $ \\alpha_{i}p_{Y|B}(0|b_{j})=\\alpha_{i}\\eta+\\chi_{i}(1-2\\eta)$. Since $1-2\\eta \\neq 0$, we substitute for $\\chi_{i}$ and $\\alpha_{i}$ in terms of their definitions to conclude\n\\begin{equation}\n \\label{Eqn:ProbOfXAGivenB}\np_{XA|B}(0,a_{i}|b_{j})=\\chi_{i}= \\alpha_{i}\\cdot \\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}=p_{A|B}(a_{i}|b_{j})\\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}.\\nonumber\n\\end{equation}\nSince $\\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}$ is independent of $i$ and $b_{j}$ was an arbitrary element in $\\mathcal{B}$ that satisfies $p_{B}(b_{j})>0$, we have established Markov chain $A-B-X$.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \\hline\nAXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$\\\\\n\\hline\n$a_{i}00$& $\\chi_{i}(1-\\eta)$&$a_{i}01$ & $\\chi_{i}\\eta$&$a_{i}10$ & $(\\alpha_{i}-\\chi_{i})\\eta$&$a_{i}11$ & $(\\alpha_{i}-\\chi_{i})(1-\\eta)=$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$} \\label{Table:p_XYAGivenB} \n\\end{table}\n\\section*{Acknowledgment}\nThis work was supported by NSF grant CCF-1116021.\n\\bibliographystyle{..\/sty\/IEEEtran}\n{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\nAutonomous vehicles have ceased to be only a vision but are rapidly becoming a reality as cities around the world such as Pittsburgh, San Francisco, and Singapore have begun investigating and testing autonomous mobility concepts~\\cite{henderson2016autonomous}.\nThe planned introduction of thousands of autonomous taxis, as currently planned in Singapore~\\cite{henderson2016autonomous}, poses a challenge not only to the in-car systems of the \\ac{AV}, but also to the traffic system itself.\n\nBesides the deployment of more efficient ride sharing systems and the reduction of the total number of vehicles on the road, \\acp{AV} can  traverse a road faster while using less space.\nTo achieve the maximum benefit in terms of traffic speeds and congestion reduction, however, mixing of \\acp{AV} and \\acp{CV} should be avoided~\\cite{litman2016autonomous}.\nOne method to achieve this is the introduction of dedicated \\ac{AV} lanes on highways to allow \\acp {AV} to operate more efficiently due to the absence of unpredictable random behaviour introduced by humans and the use of communication capabilities, e.g. platoon organization~\\cite{segata2014supporting}.\n\nBlocking certain lanes for \\acp{CV}, and thereby limiting the overall road capacity for human drivers is certainly a step that can have considerable ramifications, depending on the portion of \\acp{AV} in the system.\nWhile at the early stages, where only few \\acp{AV} are on the road, it would constitute an incentive to obtain an \\ac{AV}, it could also possibly generate traffic congestion and increase travel times for other vehicles ~\\cite{ivanchev2017critical}.\nAs the level of \\ac{AV} penetration in the road transportation system is increased, the total congestion level would likely drop, however, the advantage of using \\acp{AV} over \\acp{CV} would gradually be diminished as well. \n\nLastly, converting highways into \\ac{AHS} could also affect the rest of the road network, as drivers of \\acp{CV} may then choose a different route, caused by the changed capacity of the highway, which can lead to a mismatch between road network and traffic demand ~\\cite{7313331}.\n\nIn this paper, we take a closer look at the benefits and drawbacks of introducing a dedicated \\ac{AV} lane on major highways Singapore.\nFocusing particularly on the effect of varying percentages of \\acp{AV} in the system, we study the differences in terms of capacity, travel time, fuel consumption, and impact on other roads compared to a setting where no dedicated \\ac{AV} lane is assigned.\nIn short, our two main contributions are:\n\\begin{itemize}\n\\item We present an analytical evaluation of the expected benefit from the introduction of dedicated \\ac{AV} lanes.\n\\item Using a macroscopic traffic simulation of the city-state of Singapore and based on realistic travel demand, we show the impact of vehicle automation with and without \\ac{AV} lanes.\n\\end{itemize}\n\nThe remainder of this paper is organized as follows:\nIn \\cref{sec:related} we discuss related work on \\ac{AHS}.\n\\cref{sec:system} explains the system model used in this study.\nIn \\cref{sec:analysis} we present our analytical evaluation of a synthetic scenario, followed by \\cref{sec:case} in which results for the entire city of Singapore are presented and discussed.\n\\cref{sec:conclusion} concludes our article.\n\n\n\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\ac{AHS} and their implications have received wide attention from both researchers and industry around the world.\nInvestigations include general AHS policies and concepts ~\\cite{litman2016autonomous,kanaris1997spacing}, effects on travel times and capacity ~\\cite{cohen2011impact,carbaugh1998safety,tsao1994capacity, harwood2014modelling,michael1998capacity,kanaris1997spacing, hall2005vehicle,van2006impact}, traffic safety~\\cite{carbaugh1998safety,kanaris1997spacing, godbole2000safety, van2006impact}, and interactions between conventional human-driven vehicles and autonomous vehicles~\\cite{dresner2007sharing}.\n\nSeveral \\ac{AHS} studies and field trials were conducted as part of the California \\ac{PATH} program:\nTsao et al.~discuss the relationship between lane changing manoeuvres and the overall throughput of the highway~\\cite{tsao1994capacity}.\nTheir analytical and simulation results indicate a direct trade-off between the two.\nLateral movement decreases the traffic flow, and higher traffic flow leads to longer lane change times.\nIn this work, we ignore decreased throughput caused by lane changing and focus on a best-case city-wide benefits of dedicated \\ac{AV} lanes.\nGodbole and Lygeros evaluate the increased capacity by the introduction of fully automated highways by treating the highway as a single-lane \\ac{AHS} pipe~\\cite{godbole2000safety}.\nSimilar to the work of Harwood and Reed~\\cite{harwood2014modelling}, they study different platoon sizes and speeds but do not consider separated \\ac{AV} and \\ac{CV} lanes.\n\n\\ac{AHS} pipeline capacity was also studied by Michael et al.~\\cite{michael1998capacity}.\nTheir results show that longer platoons of \\acp{AV} are favourable as they increase the capacity of the road due to lower inter-platoon distances.\nA high mixture of different vehicle classes, however, leads to a lower capacity.\nLastly, the presented analytical model shows that as highway speed increases, the capacity reaches a saturation point after which the speed decreases again.\nThis is inline with the findings presented in this paper when studying the throughput on only the \\ac{AV} lane.\n\nSimilar to the work presented in this article, Cohen and Princeton investigated the impact of dedicated lanes on the road capacity~\\cite{cohen2011impact}.\nThey find that assigning a dedicated lane can create bottlenecks due to increased lane changing manoeuvres of departing unauthorized vehicles.\nWhen the capacity of the remaining lanes is exceeded, it may even be impossible for allowed vehicles to access the dedicated lane.\nIn our macroscopic study, we abstract away from these problems to answer the more general question whether the introduction of dedicated \\ac{AV} lanes is a feasible approach.\n\nIn summary, it appears that while \\ac{AHS} seem to be a well-studied subject, a general evaluation of the introduction of \\ac{AV} lanes is still missing.\nFurthermore, automated highways are mostly investigated in an isolated manner (and often also simplified to a 1-lane road) without taking into considering the rest of the road network.\nUsing the city-state  of Singapore as a case study, we show how travel times of both \\acp{AV} and \\acp{CV} are affected and that the introduction of dedicated AV lanes has a considerable effect on the entire road network by changing the distribution of traffic demand within the transportation system. \n\n\n\n\n\\section{System Model}\n\\label{sec:system}\n\nThe goal of this study is to evaluate the allocation of one lane on every highway road to be used exclusively by autonomous vehicles.\nWe investigate an increasing percentage of \\acp{AV} in the system under two scenarios: with and without dedicated \\ac{AV} lanes on highways.\nIn the former scenario, all roads that are not highways will exhibit normal traffic conditions as there will be a mixture of human drivers and AVs.\nAll lanes on the highway roads will be accessible to \\acp{AV}, while \\acp{CV} will be able to utilize all lanes except one lane, which will be allocated exclusively for \\acp{AV} usage.\n\nIn the second scenario, all vehicles will share all lanes on all roads.\n\nWe model the different behaviour of \\acp{AV} and \\acp{CV} by means of smaller headway time, that is the time gap to the vehicle in front.\nWe assume that \\acp{AV} can afford a much smaller headway than normal vehicles since their reaction time is orders of magnitude smaller than that of humans~\\cite{ioannou1993autonomous}.\nA direct consequence of this is that effectively the capacity of the road is increased, as \\acp{AV} need less space.\nThe capacity (in cars per hour per lane) can then be calculated as follows:\n\n\\begin{equation}\nC=\\dfrac{3600}{h_{av}p_{av}+h_{cv}\\left(1-p_{av}\\right)}\\label{capacity_equation}\n\\end{equation}\n\nwhere $h_{av}$ is the headway time for AVs (values may vary between $0.5$ and $1$ second, depending on level of comfort~\\cite{van2006impact}), $p_{av}$ is the percentage of AVs on the road segment, $h_{cv}$ is the headway time for conventional vehicles set to $1.8$ seconds.\nEquation \\ref{capacity_equation} is based on \\cite{yokota1998evaluation}, where it was derived from collected data on highway roads in Japan for varying percentages of vehicles with \\ac{AHS}.\n\nFigure \\ref{capacity} illustrates the change of capacity as a function of the AV percentage for different AV headway values.\nIt can be observed that the capacity exponentially increases with the portion of AVs on the road.\nThis means that when there is a small portion of AVs on a road, their impact is only marginal, however, when the majority of vehicles are AVs there is a significant increase of the capacity of the road. The capacity for the same AV percentage also increases exponentially as the headway time decreases.\nFor the remainder of the paper we will assume a conservative \\ac{AV} headway of $h_{av} = 1s$.\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\t\n\t\n\t{\\includegraphics[width=0.6\\columnwidth, trim=50mm 0mm 78mm 0mm, clip]{lane_capacity.pdf}}\n\t\\caption{Changes of road capacity as a function of AV percentage and headway time}\n\t\\label{capacity}\n\\end{center}\n\\end{figure}\n\n\nThe primary measure we use to evaluate the impact on traffic caused by the introduction of an \\ac{AV} lane is the travel time of cars.\nThe travel time $T = \\sum_i^n t_i$ of a vehicle is determined by the traverse times of all $n$ segments (or links) included in its route.\nThe traverse time $t_i$ of a segment $i$ can be computed using the Bureau of Public Roads (BPR) function \\cite{dafermos1969traffic}:\\\\\n\\begin{equation}\nt_i=\\dfrac{l_i}{{v}_i}\\left(1+\\alpha_i\\left(\\dfrac{F_i}{C_iw_it}\\right)^{\\beta_i}\\right)\\label{eq:bpr}\n\\end{equation}\nwhere  $l_i$ is length of the road segment, ${v}_i$ is the free flow velocity of the segment, $F_i$ is flow, $w_i$ is the number of lanes, $t$ is time duration of the simulated period, $C_i$ is the capacity of road segment $i$, $\\alpha_i$ and $\\beta_i$ are parameters from the BPR function.\nFree flow velocities $\\hat{v}$ are extracted from historical GPS tracking data~\\cite{sindhwani2010singapore}.\nParameters $\\alpha_i$ and $\\beta_i$ are calibrated for different classes of roads depending on their speed limits using both GPS tracking data and a travel time distribution of the population for certain periods of the day.\nFor a more detailed description of the calibration and validation procedures we refer the reader to \\cite{t0zBWhqE4mBmv0Rm}.\n\n\n\n\n\n\n\\section{Analytical Evaluation}\n\\label{sec:analysis}\n\nIn this section, we will illustrate how to analytically determine the travel times of vehicles on highways in both cases with and without dedicated \\ac{AV} lanes.\nWe examine a simplistic scenario, where the system consists of a single road with two lanes.\nLet the overall percentage of AVs be $p$ and the percentage of AVs on the first and second lane respectively be $p_1$ and $p_2$. Let the number of vehicles on the respective lanes be $f_1$ and $f_2$ such that:\n\\begin{equation}\nf_1+f_2=F \\label{all_cars}\n\\end{equation}\nwhere $F$ is the total number of vehicles in the system.\nTherefore, we know that:\n\n\\begin{equation}\np_1f_1+p_2f_2=pF \\label{probs}\n\\end{equation}\nUnder the assumption of \\ac{UE}~\\cite{fisk1980some} we also know that:\n\\begin{equation}\nt(f_1)=t(f_2) \\label{equality}\n\\end{equation}\nwhere $t(f)$ represent the travel time on the road for flow $f$.\n\\ac{UE} is achieved when given the current traffic situation, a user would not change their route, i.e. the user is already on the shortest route in terms of travel time.\n\nSubstituting $t(f)$ for the BPR function (Equation \\ref{eq:bpr}) we get:\n\\begin{equation}\nt(f) = \\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{f}{C}\\right)^{\\beta}\\right)\n\\end{equation}\nWhere $\\alpha$ and $\\beta$ are parameters of the BPR function, $v$ and $l$ are the free flow velocity, and the length of the road and $C$ is the capacity of the road. \nSubstituting the expression for capacity from Equation \\ref{capacity_equation} we get for the BPR function:\n\\begin{equation}\nt(f)=\\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{f\\left(h_{av}p_{av}+h_{cv}\\left(1-p_{av}\\right)\\right)}{3600}\\right)^{\\beta}\\right)\n\\end{equation}\nExpanding Equation \\ref{equality} we get:\n\n\\begin{eqnarray}\n\\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{f_1\\left(h_{av}p_1+h_{cv}\\left(1-p_1\\right)\\right)}{3600}\\right)^{\\beta}\\right)=\\nonumber\\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{f_2\\left(h_{av}p_2+h_{cv}\\left(1-p_2\\right)\\right)}{3600}\\right)^{\\beta}\\right) \n\\end{eqnarray}\n\nRemoving common terms, simplifying and using Equations \\ref{all_cars} and \\ref{probs} we get:\n\n\\begin{eqnarray}\nf_1\\left(h_{av}p_1+h_{cv}\\left(1-p_1\\right)\\right)=\\nonumber\\left(h_{av}\\left(pF-p_1f_1\\right)+h_{cv}\\left(F-f_1-\\left(pF-p_1f_1\\right)\\right)\\right)\n\\end{eqnarray}\n\nLet's examine the two scenarios in this simplistic example: In scenario $1$ a lane is exclusively reserved for AVs ($p_1=1$) and in scenario $2$ all cars are allowed to use all lanes and therefore get distributed equally on the two lanes ($f_1=f_2=\\frac{F}{2}$ and $p_1=p_2=p$). For scenario $1$ we get:\n\n\\begin{equation}\nf_1h_{av}=h_{av}\\left(pF-f_1\\right)+h_{cv}F\\left(1-p\\right)\n\\end{equation}\n\nAfter simplifying we obtain:\n\n\\begin{equation}\nf_1h_{av}=\\dfrac{F}{2}\\left(h_{av}p+h_{cv}\\left(1-p\\right)\\right)\\label{equal}\n\\end{equation}\n\nUnder the assumption of user equilibrium it follows that for scenario $1$ the travel time for both lanes is the same:\n\n\\begin{equation}\nt_1=\\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{f_1h_{av}}{3600}\\right)^{\\beta}\\right)\\label{scenario_1}\n\\end{equation}\n\nThe travel time for a single vehicle in scenario $2$ is:\n\n\\begin{equation}\nt_2=\\dfrac{l}{v}\\left(1+\\alpha\\left(\\dfrac{\\dfrac{F}{2}\\left(h_{av}p+h_{cv}\\left(1-p\\right)\\right)}{3600}\\right)^{\\beta}\\right)\\label{scenario_2}\n\\end{equation}\n\nIt can be observed that because of Equation \\ref{equal} the expression in Equation \\ref{scenario_1} is equal to the expression in Equation \\ref{scenario_2}.\nThis means that in the simple examined case, regardless whether a dedicated AV lane is introduced or not, the resulting travel time for all commuters is going to be the same after the point where there are enough AVs to ensure equilibrium can be reached.\n\nThe number of vehicles on the first lane in scenario $1$ is bounded from above by the total number of autonomous vehicles: $f_1 \\leq pF$.\nIn the case where $f_1$ computed according to Equation \\ref{equal} is smaller than $pF$ equilibrium cannot be achieved.\nIt is trivial to demonstrate that the overall travel time under scenario $1$ is bigger than the travel time under scenario $2$ when $f_1 \\leq pF$.\nThe minimum number of AVs in terms of the AV percentage $p_{\\tau}$ (the threshold percentage) in order for the solutions of the two scenarios to coincide is:\n\n\\begin{eqnarray}\np_{\\tau}Fh_{av}&=&\\dfrac{F}{2}\\left(h_{av}p_{\\tau}+h_{cv}\\left(1-p_{\\tau}\\right)\\right)\\\\\np_{\\tau}&=&\\dfrac{h_{cv}}{h_{av}+h_{cv}}\\label{eq:tau}\n\\end{eqnarray}\n\nThe percentage $p_{\\tau}$ describes the point where for an additional AV it would be equally attractive to use the AV lane or the normal lane. This is the moment where the AV lane reaches saturation.\nBefore that percentage is reached, the AVs have a shorter commuting time than the CVs, however, the system is performing worse in terms of overall congestion compared to the case without a dedicated AV lane.\nEquation \\ref{eq:tau} can be easily extended for an arbitrary number of lanes in our minimalistic example.\nAssume we have $1$ dedicated AV lane and $N$ normal lanes.\nThen the percentage of AVs after which the dedicated lane is saturated becomes:\n\n\\begin{equation}\np_{\\tau}=\\dfrac{h_{cv}}{Nh_{av}+h_{cv}}\\label{threshold}\n\\end{equation}\n\nAssuming headway times of $h_{cv}=1.8s$ and $h_{av}=1s$, the percentage $p_\\tau$ of AVs on the road for saturation of the AV lane is $64.3\\%$, $47.4\\%$, $37.5\\%$, $31\\%$, $26.5\\%$ for respectively $2$, $3$, $4$, $5$, $6$ lane roads with one allocated AV lane. \n\n\n\\section{City-wide Simulation Study}\n\\label{sec:case}\n\nWe conduct a macroscopic city-wide simulation study of the city-state of Singapore to better understand the effects to be expected in a complex environment.\nWe take a look at the traffic conditions on the highway, but also closely investigate the effect dedicated \\ac{AV} lanes have on the rest of the road network. \n\n\\subsection{Methodology}\n\nIn order to evaluate the scenarios of AV introduction in a road transportation system, we make use of an agent-based macroscopic simulation approach.\nThe simulation consists of three steps: 1) Agent generation, 2) route computation and  3) travel time estimation.\n\nThe underlying road network on which the traffic assignment is performed is modelled by means of a uni-directional graph, where each edge represents a road segment and nodes represent decision points at which a road may split or merge.\n\nTo introduce dedicated \\ac{AV} lanes, we alter the original graph by duplicating start and stop nodes of highway segments and creating a new edge to represent the \\ac{AV} lane, while removing one of the normal lanes from the original segment.\nFurthermore, costless connections are added between the start and end nodes and their respective copies so that vehicles can change lanes at the start\/end of highway road segments.\n\n\n\nEvery agent is generated with an origin and destination sampled from an OD matrix representing the travel demand of the system.\nRealistic traffic volume is modelled by synthesizing a sufficiently large vehicle population according to the available survey data. \n\nThe routes of the agents are computed using an incremental user equilibrium approach~\\cite{fisk1980some} aiming at representing reality in the sense that every driver is satisfied with their route and would not choose a different one given the current traffic situation.\nWe assume a driver is satisfied when they are on the shortest route from origin to destination in terms of travel time.\nRouting is performed on the road network graph, where each edge has an attached weight representing the current traverse time of this road segment.\nWeights are updated after every batch route computation.\nTo disallow conventional vehicles from using the \\ac{AV} lane, the weight of the edges representing these lanes are set to infinity when the route of a \\ac{CV} is computed and set back to their traverse time values for \\ac{AV} route computation.\n\n\nFurther assumptions regarding to the macroscopic level of the simulation include:\n\\begin{itemize}\n\t\\item Agents want to minimize travel time and all perceive the traffic situation in the same way (there is no noise in the observations of the traffic states).\n\t\\item Traffic is spread homogeneously in time during the simulation.\n\tIn order for the BPR function (Equation \\ref{eq:bpr}) to give a reasonable estimation of the traffic conditions, the flow $F$ during the simulation time $t$ needs to be spread homogeneously in time.\n\tIf many vehicles try to enter the road at the same time for example, the traffic congestion generated will likely produce a much higher traverse time for the road than the expected one.\n\tIn order to ensure as much as possible that traffic demand stays static throughout the simulation period, both the period $t$ and the time of day, which is simulated, have to be chosen with care.\n\t\\item Agents do not reroute while on their trip.\n\tIn reality drivers may change their trip plan dynamically according to unexpected events or observed traffic conditions ahead.\n\tAs our traffic assignment provides information to the drivers about expected traffic conditions for the whole network, we assume that such events will not occur as drivers are given all the information they need prior to their trip.\n\\end{itemize}\n\n\n\\subsection{Data and Scenario Description}\n\n\nWe examine the city-state of Singapore with a total population of $5.4$ million and around $1$ million registered vehicles including taxis, delivery vans and public transportation vehicles.\nThe fact that Singapore is situated on an island simplifies our scenario as the examined system is relatively closed with only two expressways leading out of the country.\nThere are $3495$ km or roads of which $652$ km are major arterial roads and $161$ km are expressways (these are the roads on which we introduce the dedicated AV lane indicated on \\cref{fig:highways}) spreading over $715$ squared km of land area.\nWe have used publicly available data to acquire a unidirectional graph of Singapore, that comprises of $240,000$ links and $160,000$ nodes representing the road network of the city.\nThe number of lanes, speed limit and length of every link is available allowing us to extract information about its capacity.\n\nFor the purposes of our model we make use of two separate data sets.\nThe first data set consists of GPS trajectories of a $20,000$ vehicle fleet for the duration of one month, providing information about recorded velocities on the road network during different times of the day~\\cite{sindhwani2010singapore}.\nThe second one is the Household Interview Travel Survey (HITS) conducted in $2012$ in the city of Singapore, which studies the traffic habits of the population.\nInformation about the origin destination pairs, their temporal nature, and commuting time distribution during rush hour periods is extracted from it.\n\n\n \nIn order to achieve realistic traffic conditions, we estimated the number of agents based on the \nSingapore HITS data.\nWe extracted the ratio of people who actively create traffic on the streets (cab drivers, personal vehicle drivers, lorry drivers) and the total number of people interviewed.\n\nWe chose the morning commute hours (7:30am - 8:30am) as the period which seems to be the most stable in terms of traffic volume and estimated the traffic demand consisting of $309,000$ agents.\n\n                                                               \n\n                                                                        \n\nSimilar to the simplistic example the vehicle population is split into two parts: \\ac{AV} and \\ac{CV}.\nThe percentage of \\ac{AV} is varied in order to observe the effects in the initial stages of \\ac{AV} introduction, as well as the possible traffic situation if all vehicles were autonomous.\n \n\nIn order to have a benchmark for measuring the efficiency of the suggested policy, we also evaluate the scenario where no \\ac{AV} lane is introduced and all vehicles can access all lanes on all roads.\nThe analytical solution acquired for the simplistic example in the previous section indicates that the scenario with no \\ac{AV} lane will produce better or at least the same system performance as the introduction of the \\ac{AV} lane. \nOur case study will test our analytical results for a more realistic transportation system environment.\n\n\\subsection{Effects on Average Travel Time}\n\nWe evaluated the change in average commute time based on the percentage of \\acp{AV} in the system.\nWe compare two different settings: with and without a dedicated \\ac{AV} lane.\nAs a baseline we use the average commute time without the existence of \\acp{AV} (and thus no exclusive lanes).\nThis time was found to be approx. 18.5 minutes and can be seen as the status quo.\n\nWe simulated both settings with the exact same travel demand, that is, identical origin-destination pairs for all vehicles.\nWe then gradually increased the percentage of \\acp{AV} in the system.\nThe number of vehicles in the simulation was invariant; a higher percentage of \\acp{AV} means that \\acp{CV} were replaced with autonomous vehicles.\n\n\n\\cref{fig:travel_time} shows our result.\n\n\\begin{figure*}[h]         \n\t\\centering\n\t\n\t\\subfloat[Travel times with dedicated AV lane]{\n\t\t\\includegraphics[trim={150 0 150 0}, clip,width=.45\\textwidth]{with_lane_time_2.pdf}\\label{fig:with_lane}\n\t}\n\t\\qquad\n\t\\subfloat[Comparison for all vehicles]{\n\t\t\\includegraphics[trim={150 0 150 0}, clip,width=.45\\textwidth]{compare_scenarios_2.pdf}\\label{fig:comparison}\n\t\t\n\t}\n\t\n\t\\caption{Average travel time of whole population, AVs and CVs as a function of AV percentage}\n\t\\label{fig:travel_time}\n\\end{figure*}\n\nThe black dashed line serves as an illustration of the status quo.\nIt can be observed that when introducing a lane exclusively for \\acp{AV} and thereby taking away capacity from conventional vehicles (\\cref{fig:with_lane}), the travel time for \\acp{CV} increases, whereas the low percentage of \\acp{AV} (driving on almost empty lane) travel considerably faster.\nThis could be used as an incentive by policy-makers to increase the share of \\acp{AV} in the transportation system.\nWith an increasing percentage of \\acp{AV} we observe that travel times for \\acp{CV} decrease due to the fact that effectively vehicles move away from the common lane to the AV lane, which in turn increases the travel time on the AV lane.\nThis behaviour can be observed until the AV lane is saturated (somewhere between 40\\% and 50\\% AV percentage).\nFrom this point on, the choice of lane makes no difference for newly added \\acp{AV} as they experience equal travel times regardless whether they choose to take the \\ac{AV} lane or the mixed lane. \nTherefore, the difference between average travel time of \\ac{AV} and \\ac{CV} decreases. \nTravel time of both \\acp{AV} and \\acp{CV} still decreases with the increase of the percentage of \\acp{AV} since the capacity of the road network is effectively increased.\n\nThe average number of lanes in the Singapore road network is $3.14$.\nIf we apply Equation \\ref{threshold} from our analytical evaluation, we yield a saturation point of $45.2\\%$ AV penetration, which seems to be in agreement with our simulation results.\n\nWe conclude that computing the threshold percentage is useful in terms of AV lane introduction planning.\n\n\n\nFigure \\ref{fig:comparison} shows the comparison between the average travel times for the entire vehicle population with and without the introduction of an AV lane. \nIt can be observed that the setting without an AV lane is always performing better than the AV lane one. \nAfter saturation of the \\ac{AV} lane the difference between the two curves becomes marginal.\nThis finding is in complete agreement with the results of our analysis in \\cref{sec:analysis}. \nBefore saturation of the \\ac{AV} lane, the capacity of the highway is not fully utilized and that, on average, the smaller travel times of the \\acp{AV} cannot make up for the introduced delays for the \\acp{CV}.\nWe conclude that adding an \\ac{AV} lane, while initially being an incentive for early adopters, will noticeably penalize drivers of conventional vehicles at the early stages of \\ac{AV} introduction.\nAdditional delays will be introduced by lane changing manoeuvres and other microscopic effects not considered in this macroscopic study~\\cite{tsao1994capacity,cohen2011impact}.\n\n\n\\subsection{Analysis of Effect of Headway Time}\n\nThe headway time of vehicles is a crucial parameter for the computation of the road capacity (see Equation \\ref{capacity_equation}) and therefore an important input for the travel time computation using the BPR function (see Equation \\ref{eq:bpr}).\n\nAlthough, AVs can afford to have smaller headways, this might negatively affect the comfort of the passengers, as small distances at high velocities can induce anxiety~\\cite{van2006impact}. \nTo this end, we examine the effect of varying the headway time (from $0.5$s to $1$s) for the city-scale simulation scenario with a dedicated AV lane on highways. \nIt is therefore useful to evaluate the amount of overall travel time decrease as a function of the headway time. \n\n\\begin{figure}[h]\n\t\\begin{center}\n\t{\\includegraphics[width=0.6\\columnwidth, trim=150 0 150 0, clip]{vary_headway_2.pdf}}\n\t\\caption{Average travel time curves for varying headway time}\n\t\\label{fig:headway}\n\t\\end{center}\n\\end{figure}\n\nIn \\cref{fig:headway} we observed that, depending on the headway, the improvement of overall travel time can vary between $20\\%$ and $26\\%$. \nThe difference between the improvement in the case of all AVs seems is bigger between headway $0.5$s and $0.75$s than between $1$s and $0.75$s.\n\nThis is due to the non-linear nature of the BPR function.\nAs vehicles approach free flow velocities due to the increased capacity, any further improvement in capacity plays a smaller role and therefore the gains become less significant. \nIt must be noted, that this happens only when traffic congestion is such that vehicles travel at free flow velocities.\n\nIf the system was more congested, we would not be able to observe the decrease of travel time improvement for smaller headway time, since the BPR function will still be in its non-linear part thus providing significant changes of travel time as the capacity is altered.\n\n\n\\subsection{Effects on Fuel Consumption}\n\nWe have demonstrated that the introduction of AVs decreases the overall travel time in the system. \nThe next question we are trying to answer is by how much this saved time reduces fuel consumption (or energy in the case of electric vehicles). \nWe deploy the Elemental fuel consumption model~\\cite{faris2011vehicle} to evaluate the average fuel consumption of the vehicle population and subgroups.\n\nThe effects of introducing an AV lane on fuel consumption are shown in \\cref{fig:fuel_consumption}\n\n\\begin{figure}[h]\n\t\n\t\\begin{center}\n\t\t{\\includegraphics[width=0.6\\columnwidth, trim=145 0 150 0, clip]{with_lane_fuel_2.pdf}}\n\t\t\\caption{Average fuel consumption of whole population, AVs and regular vehicles as a function of AV percentage}\n\t\t\\label{fig:fuel_consumption}\n\t\\end{center}\n\t\n\\end{figure}\n\n\nAs expected, we observe a positive effect on fuel consumption.\nTo provide an idea of magnitude, the fuel saved at 20\\% AVs is about $9.3$ tons for the duration of one Singapore morning rush hour and approx.\\ $45$ tons when every vehicle is autonomous.\nAs for the travel time, the combination of one dedicated lane and a low percentage of \\acp{AV} penalizes \\acp{CV}.\n\nThe introduction of more AVs, however, decreases both the overall travel time and fuel consumption. \nIt can also be noted that the relative effect of the fuel consumption is smaller than the effect on travel time, which means that fuel consumption is less sensitive with respect to changes in the traffic states for the examined scenario.\nPlease note that we do not consider fuel or energy saving resulting from platoon organization~\\cite{hall2005vehicle} and other AV related technology but only seek to provide a order of magnitude with our macroscopic evaluation.\n\n\n\n\\subsection{Effects on Road Network Throughput and Traffic Distribution}\n\n\\begin{figure}[!t]         \n\t\\centering\n\t\\subfloat[Singapore road network with highways featuring dedicated AV lanes in bold]{\n\t\t\\includegraphics[trim={150 0 150 0}, clip,width=0.5\\columnwidth]{highways.png}\\label{fig:highways}\n\t}\\qquad    \t\n\t\\subfloat[Change of road throughput at 50\\% AVs (with AV lane)]{\n\t\t\\includegraphics[trim={150 0 150 0}, clip,width=0.5\\columnwidth]{diff_with_pic0.png}\\label{fig:with_through}\n\t}\\qquad\n\t\\subfloat[Change of road throughput at 50\\% AVs (without AV lane)]{\n\t\t\\includegraphics[trim={150 0 150 0}, clip,width=0.5\\columnwidth]{diff_no_pic0.png}\\label{fig:no_through}\n\t} \t\n\t\\caption{Road throughput change caused by 50\\% AVs compared to 0\\% AVs. Colours from the blue gamma represent higher throughput; colours in the red gamma represent lower throughput}\n\t\t\\label{throuput_change}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[trim={120 0 120 0}, clip,width=0.6\\columnwidth]{hist_throughput_2.pdf}\n\\caption{Change of road throughput for different road types}\n\\label{fig:hist}\n\\end{figure}\n\n\nLastly, we examine the traffic distribution on the entire road network to gain a better understanding of the changes occurring in traffic conditions as a results of the introduction of dedicated AV lanes.\nTo the best our knowledge, existing literature focuses on the traffic changes on the highways and ramps only, and such a study has not been conducted before.\n\\cref{fig:highways} shows the Singaporean road network with highways drawn in bold.\nAV lanes are added only to these highways, the rest of the network remained unchanged.\n\n\n\nAs a first step, we provide a qualitative measurement by visualizing the impact an introduction of $50\\%$ \\acp{AV} has on the road network (compared to $0\\%$ AVs).\nTo this end, we draw roads experiencing higher throughput in blue colours and roads experiencing lower throughput in red colours.\nWe show our results for both settings, i.e., with AV lane (\\cref{fig:with_through}) and without AV lane (\\cref{fig:no_through}).\n\n\n\nIt can be observed that in both cases the highways exhibit a considerably higher throughput of vehicles, which is expected since the capacity of the highway is technically increased by the introduction of the autonomous vehicles.\nOther roads exhibit a slight decrease of throughput, which may be due to the changes in routing that are triggered by the AVs, which have a strong preference towards the highways. \nIn other words, the AVs are, in a way, attracted to the highways, since they can traverse fast there and the capacity is sufficient. \nThis, however, makes them willing to pass through more congested roads in order to get to the highway, which creates additional time losses for the regular vehicles. \nThis argument is further strengthened by the fact that the roads leading to the highways do not exhibit increased throughput, although they exhibit higher demand since more vehicles want to use the highways. \nThis means that the level of congestion on those roads is too high to allow an increase of throughput.\n\n\n\n\n\nComparing the throughput changes in the two scenarios, it can be observed that the increase of throughput on the highways for the case with no AV lane is smaller. \nTherefore, the change of routing triggered by the introduction of AVs is qualitatively the same but quantitatively different for the two examined scenarios.\nThis can be observed in Fig. \\ref{fig:hist}.\nThe relative increase of throughput for highways is almost three times higher for the dedicated AV lane case.\nHigher level of throughput increase on highways leads to higher level of throughput decrease on the major roads, which represent alternative routes.\nAs mentioned before, if highways become too attractive for AVs, there can be negative effects on traffic conditions stemming from overly utilized roads which lead to the highways. \nThe more balanced distribution of traffic in the no AV lane scenario could be the reason for its slight, however, consistent superiority over the dedicated AV lane case in terms of average travel time observed in \\cref{fig:comparison}.\n\n\nFinally, we investigate the change in demand for different types of roads.\nWe define demand for a road as the number of vehicles that have this particular road in their route under user equilibrium traffic assignment.\nTaking a closer look at the difference in travel demand between the scenario with AV lane and the scenario without the introduction of AV lane, we measure the relative difference of travel demand between the scenarios.\nFormally, let the demand for road $i$ for a given AV percentage $k$ be $D_{i,k}^{a}$ for the AV lane scenario and $D_{i,k}^{b}$ for the benchmark scenario without an AV lane.\nThen, for the different classes of road (highways, major roads, other roads), we compute the difference in demand between the two examined scenarios relative to the AV lane scenario (a negative value therefore means that this particular type of road experiences less demand when AV lanes are introduced compared to not introducing AV lanes): \n\n\\begin{equation}\n\\Delta_{D,k}=\\dfrac{\\sum\\limits_{i}D_{i,k}^a-\\sum\\limits_{i}D_{i,k}^b}{D_{i,k}^a}\n\\end{equation}\n\n\n\n\n\n\\cref{fig:demand} shows our results for highways, major roads, and other roads. \nFor highways, we observe that initially there is lower demand.\nThis is due to the lower road capacity resulting from the dedicated lane, causing more \\acp{CV} to avoid the highways.\nWith more \\acp{AV} in the system, the demand for the highways increases until the saturation point of the AV lane is reached.\nFrom this point on, the difference between the both settings becomes smaller, eventually converging to zero as the addition of more \\acp{AV} cause the common lanes to achieve capacity values close to the one of the AV lane.\n\nFor the major roads we observe increased demand as \\acp{CV} will favour these roads as an alternative to the more congested highways caused by the dedicated AV lane.\nThis effect decreases until the saturation point of the AV lane.\n\nAt this moment the highways reach their maximum demand difference and therefore a smaller portion of the population takes the alternative routes utilizing major roads.\nFollowing the negative peak, the demand difference for major roads also converges to zero as the two scenarios become identical.\n\nThe difference for smaller roads is less pronounced and can only be observed at a low percentage of \\acp{AV} in the system.\nThe higher utilization of major roads will also increase traverse time of these roads.\nSome vehicles will therefore choose to use minor roads as a third alternative.\n\nIt is interesting to note that after the saturation point of the AV lane the difference in travel times between the two scenarios is almost negligible, however, the actual assignment of traffic, as can be observed in \\cref{fig:demand}, is qualitatively different.\nThis finding indicates that the introduction of an AV lane would not just affect the travel time of the population but also shift the route preferences of commuters.\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[trim={120 0 150 0}, clip,width=0.6\\columnwidth]{relative_difference_demand_2.pdf}\n\\caption{Demand difference between AV lane setting and no AV lane setting for varying percentage of AVs}\n\\label{fig:demand}\n\\end{figure}\n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec:conclusion}\n\n\nIn this article we demonstrated the effect of assigning one lane on highways exclusively for \\acp{AV}.\nWe showed that for lower percentages of \\acp{AV}, or more precisely, before the dedicated lane is saturated, travel times for \\acp{AV} can be significantly shorter, while at the same time \\acp{CV} are delayed due to the reduced capacity of the highway.\n\n\nLooking at the entire road network, we observe that also non-highways are affected as \\acp{CV} will effectively be drawn away from the highways onto the major roads.\nThis effect is especially pronounced at early stages of AV adaptation where the AV lane will remain mostly empty.\nRegardless of an introduction of the \\ac{AV} lane, we confirmed earlier findings that a larger number of \\acp{AV} will have a positive impact on travel times and fuel consumption for all vehicles.\nThe macroscopic simulation study confirmed our analytical evaluation where we showed how to compute the saturation point of the \\ac{AV} lane and illustrated that after this point is reached, the benefits for \\ac{AV} users become negligible.\nWe further compared the scenario with \\ac{AV} lane introduction to a baseline scenario where no changes to traffic regulations are made.\nThe latter scenario outperforms the former one over the whole range of \\ac{AV} percentages, however, the difference is of considerable amount only before the saturation point is reached.\nThis finding coincides with our analytical evaluation of the two scenarios.\n\nFuture work includes micro (and submicroscopic) studies to further provide insights on the effects of the introduction of AV lanes.\nThis includes benefits due to smart platooning strategies but also turbulences caused by lateral vehicle movement, e.g., from the on-ramp towards the AV lane.\nFurthermore, the authors would like to exchange the \\ac{UE} traffic assignment of the \\ac{AV} group of vehicles with the BISOS algorithm ~\\cite{7795902}, which looks for a system optimum assignment, in order to check whether the adoption of an \\ac{AV} lane would become more beneficial in this case.\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work was financially supported by the Singapore National Research Foundation under its Campus for Research Excellence And Technological Enterprise (CREATE) programme.\n\n\\vspace{-1em}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nIn advanced-retarded (A-R) differential equations, or mixed functional differential equations, the derivative of the associated function explicitly depends on itself evaluated at different advanced-retarded values of the variable \\cite{mfde1,mfde2,mfde3,mfde4,mfde5}.  In order to solve such A-R equations either analytically or numerically, we require the knowledge of the solution history out of the domain of the equation. In many scientific disciplines A-R differential equations are used to describe phenomena containing feedback and feedforward interactions in their evolution \\cite{aero,eco,ner}. In physics, for instance, A-R equations can be used to model dynamical systems exhibiting  certain symmetry in the evolution. As a prominent examples one may mention the application of A-R equations \\cite{wefe1} in Wheeler-Feynman absorber theory \\cite{wefe2,wefe3} and in the propagation of waves in discrete spatial systems \\cite{mapa}. \n\nIn the context of Quantum Mechanics, the implementation of feedback is more intricate than in the classical case due to the sensitivity of quantum systems to measurements. In this regard, a set of techniques has been developed for the realization of feedback-dependent systems, each of them employing different resources such as dynamical delays \\cite{grim,whalen,fede}, machine learning optimization \\cite{realt}, weak measurements \\cite{wk1,wk2}, including quantum memristors \\cite{qm1,qm2}, and projective measurements for digital feedback \\cite{l1}, among others. Certainly, the inclusion of feedback or memory effects in quantum dynamical systems has extended the scope of quantum protocols, and it has allowed for the study and reproduction of more complex phenomena. Therefore, devising schemes for engineering Hamiltonians that display advanced-retarded dynamics is of great relevance. Along these lines, the field of non-Markovian quantum dynamics focuses on the study of effective equations that govern the evolution of systems interacting with environments depending on previous times \\cite{nmk1,nmk2,nmk3,nmk4}. As a result, estimation of non-Markovianity sheds light on the memory content of the systems under study.\n\nIn this article we show that photonic lattices can be used to effectively tailor the dynamics of classical and quantum light fields in an advanced-retarded fashion. Our strategy is to exploit the duality between light propagation in space and time evolution \\cite{toni}. Our A-R photonic approach exploits the isomorphism existing between the steady state of judiciously-designed photonic waveguide circuits and solutions of A-R differential equations. We foresee that the inherent versatility of the proposed system will make the implementation of feedback and feedforward noticeably simple, in both quantum and classical frameworks, and thus may pave the way to interesting applications in integrated quantum technologies. \n\n\\section*{Results}\n\n\\subsection*{Advanced-Retarded Analogy}\n\\label{sec:ara}\n\nIn order to introduce our protocol, we start by considering the following first order linear and non-autonomous A-R differential equation\n\\begin{equation}\ni\\frac{dx(t)}{dt}=\\beta(t)x(t)+\\kappa^{-}(t)x(t-\\tau)+\\kappa^{+}(t)x(t+\\tau),\n\\label{are}\n\\end{equation}\nwith $\\kappa^+ (t)=\\kappa^- (t+\\tau)$, and boundary conditions $\\kappa^- (0< t < \\tau) = \\kappa^+ ((N-1)\\tau < t < N\\tau)=0$. Associating the functions $x(t)$, $x(t\\pm\\tau)$ with the mode amplitude of monochromatic waves traversing the $j$-th and $j\\pm 1$-th waveguides, $a_{j}(z)$ and $a_{j\\pm 1}(z)$, of an array of $N$ evanescently coupled waveguides, each supporting a single mode and having a ``time\" dependent propagation constant $\\beta(z)$, we obtain a system governed by a set of $N$ differential equations  \n\\begin{equation}\ni\\frac{da_j}{dz}=\\beta(z) a_j (z) + \\kappa_{j, j+1} (z) a_{j+1} (z) + \\kappa_{j,j-1} (z) a_{j-1} (z),\n\\label{chip}\n\\end{equation}\nwith $j\\in [1,N]$. In the quantum optics regime, the dynamics of single photons traversing this type of devices is governed by a set of Heisenberg equations that are isomorphic to Eq.~\\eqref{chip}. The only difference is that in the quantum case, the mode amplitude $a_{j}$ is replaced by the creation operators $a_{j}^{\\dagger}$ \\cite{Lai}. In order to make Eq.~\\eqref{chip} isomorphic to Eq.~\\eqref{are}, we must impose a continuity condition $a_j (0)=a_{j-1}(\\tau)$ within the interval $j \\in [2, N-1]$. Physically, this condition implies that the mode field $a_{j-1}$, at the propagation distance of $(j-1)\\tau$ ($\\tau$ is the length of the waveguides), has to be fed back into the input of the $j$-th waveguide. Furthermore, the finiteness of the waveguide array imposes the boundary conditions $\\kappa_{1,0}=\\kappa_{N,N+1}=0$. Additionally, we establish the aforementioned mapping of the independent variable $t$ into the spatial coordinate $z$. Therefore, our protocol requires the implementation of waveguide lattices endowed with input-output connections as illustrated in Fig.~\\ref{fig:fig1}. We point out that the phase introduced via the fiber propagation can be traced and disregarded given that it can be made equal for all fiber connections, being therefore a global phase.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.5\\linewidth]{Fig1}\n\\caption{{\\bf Scheme of the proposed implementation.} Illustration of the chip for implementing the photonic simulator in Eq.~\\eqref{chip}, where the arrows represent the input and output ports and the lines inside and outside the chip represent the waveguides and fiber connections respectively.}\n\\label{fig:fig1}\n\\end{figure}\n\nThe analogy we present here can be exploited with classical electromagnetic fields or single photons. The motivation for doing a quantum simulation is the possibility to embed the advanced-retarded dynamics in more general quantum protocols allowing the implementation of quantum feedback and feedforward. In the classical case, the initial condition $a_1 (0)$ is implemented by continuously injecting light into the system. This condition is crucial to establish the isomorphism between the light dynamics in the waveguide array and the solution of Eq.~\\eqref{are}. As a result, the solution of Eq.~\\eqref{are} is obtained in the stationary regime of our photonic system. Once the intensity is measured, the modulus square of the solution is obtained by merging the intensity evolution of each waveguide in a single variable. See Fig.~\\ref{fig:fig2} for a demonstration of the potentiality of our protocol, in which we analyze the setup depicted in Fig.~\\ref{fig:fig1}. The light dynamics occurring in such an array is governed by Eq.~\\eqref{chip}. \n\n\\begin{figure}[h!!]\n\\includegraphics[width=0.5\\textwidth, trim= 0cm 5cm 0cm 5cm]{Fig2}\n\\caption{{\\bf Numerical simulations of the dynamics.} (a) Intensity evolution for an array having $N=6$ waveguides and constant lattice parameters $\\beta~=~1$, $\\kappa~=~\\sqrt{\\beta+N}$ and $\\tau=1$. (b) Intensity of all the waveguides concatenated in a single curve, which represents the absolute square of the solution of Eq.~\\eqref{chip}.}\n\\label{fig:fig2}\n\\end{figure}\n\nAn interesting point to highlight here is the existence of a $z$ reversal symmetry in the simulation with respect to the central point of the evolution, $z_c=N\\tau \/2$, for constant lattice parameters. This relation holds for the modulus square of the solution, $a a^{*}(z_c +z)=a a^{*} (z_c -z)$. Consequently, after the system reaches the steady state, it simultaneously fulfills the periodic boundary condition, $a(0)=a(N\\tau)$, where we have neglected a global phase factor. This property combined with the space-time analogy opens a possible framework for the study and implementation of Closed and Open Timelike Curve gates \\cite{iqp1,iqp2,ctc1,ctc2,ctc3,ctc4,ctc5}.\n \n\\paragraph{Non-Autonomous Equations} A more general scenario arises when considering space-dependent parameters in Eq.~\\eqref{chip}, which in the context of photonic lattices, means that the system becomes dynamic. The dynamic character is achieved with modulations of the refractive index of individual waveguides. We consider an implementable system conformed by a periodic variation $\\beta(z)=\\beta_{0}+\\epsilon\\cos(\\omega z)$, where $\\beta_{0}$ is a constant, $\\epsilon$ is the modulation amplitude, and $\\omega$ stands for the modulation frequency along $z$. For illustration purposes, we provide a numerical calculation of this particular photonics simulator in the Supplementary Fig. 1. Notice that in the context of advanced-retarded equations the time dependence allows us to encode non-autonomous equations, which are hard to compute in general.\n\n\n\\paragraph{Systems of Equations} We now turn our attention to demonstrate how our protocol can be extended to provide solutions of systems of A-R equations. To this end, we encode every unknown function in a waveguide array, and place all the involved arrays close to each other in such a way that light fields traversing the system can tunnel from array to array. In this manner, all functions are self-coupled and coupled to others, enriching the dynamics of the systems. In order to explain the operating principle of the protocol, we focus on the simplest case of two variables. Note that in this case, we can relate each array to a component of a qubit to be implemented in the dual-rail encoding with a single photon. As a first possibility, we can put together two lattices in which the feedback takes place as depicted in Fig.~\\ref{fig:fig3}a. In this scenario, the light can hop to neighboring sites as well as to the sites of the adjacent array. This arrangement enables the time evolution simulation of a single qubit Hamiltonian combined with two terms corresponding to advanced and retarded couplings \n\\begin{equation}\n\\label{qubit}\ni|\\dot{\\psi}(t)\\rangle=H(t) |\\psi(t)\\rangle + H^+ (t)|\\psi(t+\\tau)\\rangle + H^- (t) |\\psi(t-\\tau)\\rangle\n\\end{equation}\n\\begin{eqnarray}\n\\label{hamiltonian}\nH(t)=\\left( \\begin{array}{cc} \\beta_x (t)&q(t)\\\\q(t)&\\beta_y (t) \\end{array} \\right), \\qquad  H^+(t)=\\left( \\begin{array}{cc} \\kappa^+_x (t)&d^+_{xy}(t)\\\\d^+_{yx}(t)&\\kappa^+_y (t) \\end{array} \\right), \\qquad H^-(t)=\\left( \\begin{array}{cc} \\kappa^-_x (t)&d^-_{xy}(t)\\\\d^-_{yx}(t)&\\kappa^-_y (t) \\end{array} \\right).\n\\end{eqnarray}\n\nA second configuration arises by mixing the connectors as shown in Fig.~\\ref{fig:fig3}b. In this case we flip the qubit in the advanced and retarded times. Even though the equation exhibits the same structure as Eq.~\\eqref{qubit}, the forward and backward Hamiltonians, $H^+$ and $H^-$ are the same matrices given in Eq.~\\eqref{hamiltonian} multiplied to the left by $\\sigma_x$. Here $H(t)$, $H^+ (t)$ and $H^- (t)$ depend on the propagation constants, on the coupling between waveguides belonging to different arrays and the coupling between the guides of the same array. The vertical coupling is refereed to as $q$ while the transverse coupling constants are represented by $\\kappa$ and the labels $(x,y)$ correspond to the plane where the arrays are located. Moreover, $d$ accounts for the diagonal coupling. In addition to the conditions for each array in \\eqref{are}, the following consistency and boundary conditions are fulfilled, \n\\begin{eqnarray}\n\\nonumber H^+ ((n-1)\\tau < t < n\\tau)= H^- (0< t < \\tau)=0, \\qquad  d^+_{yx}(t)=d^-_{xy}(t+\\delta), \\qquad d^+_{xy}(t)=d^-_{yx}(t+\\delta).\n\\end{eqnarray}\nIn Supplementary Fig. 2 a numerical calculation of Eq.~\\eqref{qubit} is provided.\nEven though our formalism is based mainly on classical optics, one can emulate a quantum system (qubit) via the encoding of multiple A-R equations, as shown in this section. Moreover, one may also include quantum effects via the loading of the chip with genuinely quantum photonic states, e.g., two-photon or few-photon Fock states, which will introduce quantum effects as photon bunching, in combination with advanced-retarded physics. A thorough analysis of this last possibility is, however, outside of the scope of this article.\n\n\\begin{figure*}[]\n\\includegraphics[width=15cm, trim= 0cm 0cm 0cm 0cm]{Fig3}\n\\caption{{\\bf Systems of equations.} Scheme of the chip in a perpendicular plane showing the input output connections and the parameters of the simulation. Here $\\beta$ is the propagation constant, $q$ is the vertical coupling constant, $\\kappa$ is the horizontal coupling constant and $d$ the diagonal coupling constant. a) The scheme in which each plane is associated with a component of the qubit simulates Eq.~\\eqref{qubit}. b) The crossed links allow for a stronger temporal mixing of the qubit components in the derivative. This situation corresponds to the second example of Eq.~\\eqref{qubit}.}\n\\label{fig:fig3}\n\\end{figure*}\n\n\\paragraph{Multiple Delays} We next consider a variant of Eq.~\\eqref{are} with multiple delays. This configuration arises when we allow each waveguide to couple to multiple neighbors and reordering the feedback connectors. The first non-trivial example is a two-time A-R equation, Eq.~\\eqref{are2}. \n\\begin{eqnarray}\n\\label{are2}\ni\\dot{x}(t)= \\beta(t)x(t)+\\kappa^+ (t) x(t+\\tau)+\\kappa^{++} (t) x(t+2\\tau)+ \\kappa^{-} (t) x(t-\\tau) + \\kappa^{--} (t) x(t-2 \\tau).\n\\end{eqnarray}\nExperimentally, the arrangement can be engineered by fabricating the waveguides in a zig-zag configuration. The resulting equation shares the structure of an A-R differential equation with additional feedback and feedforward terms. For this particular system the coupling coefficients are related as\n\\begin{equation}\nk^{++}(t)=k^{--}(t+2\\tau), \\qquad k^{+}(t)=k^{-}(t+\\tau),\n\\end{equation}\nwith the appropriate boundary conditions. See Supplementary Fig. 3 for a numerical simulation of Eq.~\\eqref{are2}.\n\n\\paragraph{Higher Order Equations} One last generalization of the A-R simulator consists in introducing complex dynamical parameters. This can be achieved by combining our feedback technique with Bloch oscillator lattices \\cite{bend1, bend2, bend3}. These types of arrays can be implemented by including a transverse ramping on the potential of the waveguides or by curving the waveguide arrays. Provided the evolution equations for the Bloch oscillator array, $i \\dot{a_n}+ n\\beta{z}a_{n} + \\kappa^+ a_{n+1} + \\kappa^- a_{n-1}=0$, and making the formal transformation $a_n=\\tilde{a}_{n}(z)\\exp(in\\phi(z))$, with $\\phi(z)=\\int_{0}^z\\beta(z')dz'$, one can show that it is formally equivalent to a system endowed with complex coefficients, $i\\dot{\\tilde{a}}_n + \\exp(in\\phi(z))\\kappa^+ \\tilde{a}_{n+1} + \\exp(-in\\phi(z))\\kappa^-\\tilde{a}_{n-1}$. The inclusion of arbitrary complex parameters could be used to enhance the versatility of the protocol. Furthermore, a complete control of the coupling constants would allow to simulate higher order equations via systems of first order equations.  \n\nEven though the toolbox introduced here is valid for simulating diverse physical configurations, it can be generalized by an appropriate mathematical treatment. Let $M(t)$ be the matrix containing the information about the propagation constant and couplings among the waveguides defined by $i \\dot{a}(t)=M(t) a(t)$, $F$ the matrix encoding the input-output connections, and $\\alpha$ the initial state independent of the feedback and feedforward mechanism, such that $a(0)=F a(\\tau) + \\alpha$. Consider now that the dynamical equation is solvable in terms of the evolution operator $U(t)$, $a(t)=U(t) a(0)$. Our goal is to determine the complete initial condition $a(0)$ in terms of the independent initial condition $\\alpha$, evolution operator $U(t)$ and input-output matrix $F$. The consistency relations at $a(\\tau)$ impose a set of equations that $a(0)$ has to fulfill, $a(\\tau)=U(\\tau) a(0)$, \n\\begin{eqnarray}\na(0)=F a(\\tau) + \\alpha \\Rightarrow a(0)=\\left( \\mathbb{1}-FU(\\tau) \\right)^{-1} \\alpha, \\qquad a(t)=U(t)( \\mathbb{1}-FU(\\tau))^{-1}\\alpha.\n\\end{eqnarray}   \n\nNotice that this relation holds for any $\\alpha$, allowing the input of quantum states superposed in more than one waveguide, and is also valid for different configurations of couplings $U$ and connections $F$, limited by the existence of the inverse of $(\\mathbb{1}-F U(\\tau))$. Moreover, we can think of different experimental conditions, in which the connections happen at distinct evolution times $\\tau_i$, $a(0)=\\sum F_i a(\\tau_i)+\\alpha$, resulting in $a(t)=U(t)( \\mathbb{1}-\\sum F_i U(\\tau_i))^{-1}\\alpha$.  \n\n\\subsection*{Implementation Analysis}\n\nWe analyze now the main limitations of the protocol regarding its interpretation and experimental realization. \n\nRegarding the quantum equation, although the probability is not normalized during the dynamics, it is normalized in the input and output points. Therefore, initialization and measurements retrieve the correct interpretation, even if the particle undergoes forward and backward jumps on time, an effect that could be useful for the simulation of absorbing potentials \\cite{ile}. \n\n\\paragraph*{Stationary State Proximity}\nA natural source of errors is given by the waveguide losses damaging the quantum state in the time elapsed until the photon escapes from the chip. This time is related with the population in the solution, which is unknown before the experiment is realized. Identifying the relation between the resonances and the dynamical parameters could be helpful for estimating the population in the stationary state, and therefore the total experimental time for achieving the stationary state in the classical case. See Fig. \\ref{fig:fig4} for a scheme of the error depending on the distance from the stationary state.   \n\n\\begin{figure}[t]\n\\includegraphics[width=0.5 \\textwidth, trim= 0cm 5cm 0cm 5cm]{Fig4}\n\\caption{{\\bf Error analysis.} We depict the decimal logarithm of the error as a function of time for three runs of the simulation with different distance with respect to the stationary state. The fact that the effective interaction between photons is null makes possible the analogy between the stationary state solution and the accumulation of solutions for an initial excitation combined until the initial population has escaped from the output port. Therefore, the distance is calculated as the norm of the population that remains in the chip. The dynamical constants of the system are equivalents to the ones in Fig. \\ref{fig:fig2}.}\n\\vspace{-0.5cm}\n\\label{fig:fig4}\n\\end{figure}\n\n\\paragraph*{Experimental Errors}\nWe have to take into account that the losses introduced by the fiber connections will break the continuity condition allowing us to simulate Eq.~\\eqref{are} in terms of Eq.~\\eqref{chip}. The length and propagation constant of this fibers have to be tuned so that no phase is introduced in the evolution. Additionally, the space dependence of propagation and coupling constants is limited by the experimentally realizable functions. The degrees of freedom to be considered are the writing precision for modifying the propagation constants and the spatial path of each waveguide for modifying the coupling constants. \n\n\\section*{Discussion}\n\nIn conclusion, we have developed a flexible but realistic toolbox for implementing advanced-retarded differential equations in integrated quantum photonics circuits. We have shown that our analogy enables the simulation of time dependent systems of advanced-retarded equations, which in the context of quantum information can be employed to realize feedback and feedforward driven dynamics. Therefore, we consider that our work enhances the versatility of quantum simulators in the abstract mathematical direction and in terms of applications for retrospective and prospective quantum memory.\n\n\n\\section*{Supplemental Information}\nSee the included Supplemental Information for supporting content.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFace recognition algorithms are good examples of recent advances in Artificial Intelligence (AI). The performance of automatic face recognition has been boosted during the last decade, achieving very competitive accuracies in the most challenging scenarios \\cite{1}. These improvements have been possible due to improved machine learning approaches (e.g., deep learning), powerful computation (e.g., GPUs), and larger databases (e.g., at scale of millions of images). However, the recognition accuracy is not the only aspect to consider when designing biometric systems. Algorithms have an increasingly important role in the decision-making of several processes involving humans. These decisions have therefore increasing effects in our lives. Thus, there is currently a growing need for studying AI behavior to better understand its impact in our society \\cite{31}.\n\nFace recognition systems are especially sensitive due to the personal information present in face images (e.g., identity, gender, ethnicity, and age). Previous works suggested that face recognition accuracy is affected by demographic covariates. In \\cite{6,5}, authors demonstrated that the performance of commercial face recognition systems varies according to demographic attributes. In \\cite{7,8}, the authors evaluated how covariates affect the performance of face recognition systems based on deep neural network models. Among the different covariates, the skin color is repetitively remarked as a factor with high impact in the performance \\cite{6,7}. However, ethnic face attributes are beyond skin color. The shape and size of facial features are partially defined by the ancestry origin. These differences can be used to accurate classify subjects according to their ancestry origin \\cite{8}. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics{AAAI\/figura1.pdf}\n\\caption{Face recognition block diagrams. The screener is an algorithm that given two face images decides if they belong to the same person. The trainer is an algorithm that generates the best data representation for the screener.}\n\\label{Figure1}\n\\end{figure*}\n\nThe number of published works pointing out the biases in the results of face detection \\cite{3} and recognition algorithms is large \\cite{5,8,4,6,7,32}. Yet, only a limited number of works analyze how biases affect the learning process of these algorithms. The aim of this work is to analyze face recognition models using a discrimination-aware perspective. Previous studies have demonstrated that ethnicity and gender affect the performance of face recognition models \\cite{gong2019debface}. However, there is a lack of understanding regarding how this demographic information affects the model beyond the performance. The main contributions of this work are: \n\\begin{quote}\n\\begin{itemize}\n\\item A general formulation of algorithmic discrimination for machine learning tasks. In this work, we apply this formulation in the context of face recognition.\n\\item Discrimination-aware performance analysis based on a new dataset \\cite{9}, with 24K identities equally distributed between six demographic groups.\n\\item Study of the effects of gender and ethnicity in the feature representation of deep models.\n\\item Analysis of the demographic diversity present in some of the most popular face databases.\n\\end{itemize}\n\\end{quote}\n\nThe rest of the paper is structured as follows: Section 2 presents our general formulation of algorithmic discrimination. Section 3 analyzes some of the most popular face recognition architectures and the experimental protocol followed in this work. Section 4 evaluates the causes and effects of biased learning in face recognition algorithms.  Finally, Section 5 summarizes the main conclusions.\n\n\n\\section{Formulation of Algorithmic Discrimination}\nDiscrimination is defined by the Cambridge Dictionary as treating a person or particular group of people differently, especially in a worse way than the way in which you treat other people, because of their skin color, sex, sexuality, etc.\n\nFor the purpose of studying discrimination in artificial intelligence at large, we now formulate mathematically algorithmic discrimination based on the previous dictionary definition. Even though similar ideas as the ones embedded in our formulation can be found elsewhere \\cite{23,22}, we didn't find this kind of formulation in related works. We hope that formalizing these concepts can be beneficial to foster further research and discussion in this hot topic.\n\nLet's begin with notation and preliminary definitions. Assume $\\textbf{x}_s^i$ is a learned representation of individual $i$ (out of $I$ different individuals) corresponding to an input sample $s$ (out of $S$ samples) of that particular subject. That representation $\\textbf{x}$ is assumed to be useful for task $T$, e.g., face authentication or emotion recognition. That representation $\\textbf{x}$ is learned using an artificial intelligence approach with parameters $\\theta$. We also assume that there is a goodness criterion $G$ on that task maximizing some performance real-valued function $f$ in a given dataset $\\mathcal{D}$ (collection of multiple samples) in the form:\n\n\\begin{equation}\n\\label{eqn:goodnes_criterion}\n    \\textit{G}(\\mathcal{D}) = \\max_{\\theta}\\textit{f}(\\mathcal{D},\\theta)\n\\end{equation}\n\nThe most popular form of the previous expression minimizes a loss function $\\mathcal{L}$ over a set of training samples $\\mathcal{D}$ in the form:\n\n\\begin{equation}\n\\label{eqn:learning_strategy}\n    \\theta^*=\\arg\\min_{\\theta}{\\sum_{\\textbf{x}_s^i\\in \\mathcal{D}}\\mathcal{L}(\\textit{O}(\\textbf{x}_s^i|\\theta),T^i_s)} \n\\end{equation}\nwhere \\textit{O} is the output of the learning algorithm that we seek to bring closer to the target function (or groundtruth) \\textit{T} defined by the task at hand. On the other hand, the \\textit{I} individuals can be classified according to \\textit{D} demographic criteria $\\textit{C}_d$, with $d = 1,..., D$, which can be the source for discrimination, e.g., $\\textit{C}_1 = \\textit{Gender} = \\{\\textit{Male, Female}\\}$ (demographic criterion $1 = \\textit{Gender}$ has two classes in this example). The particular class $k=1,...,K$ for a given demographic criterion $d$ and a given sample is noted as $\\textit{C}_d (\\textbf{x}_s^i)$, e.g., $\\textit{C}_1 (\\textbf{x}_s^i)=\\textit{Male}$. We assume that all classes are well represented in dataset $\\mathcal{D}$, i.e., the number of samples for each class in all criteria in $\\mathcal{D}$ is significant. $\\mathcal{D}_d^k \\in \\mathcal{D}$ represents all the samples corresponding to class $k$ of demographic criterion $d$.\n\nFinally, \\textbf{our definition of algorithmic discrimination}: an algorithm discriminates the group of people represented with class $k$ (e.g., \\textit{Female}) when performing the task \\textit{T} (e.g., face verification, or emotion recognition), if the goodness \\textit{G} in that task when considering the full set of data $\\mathcal{D}$ (including multiple samples from multiple individuals), is significantly larger than the goodness $\\textit{G}(\\mathcal{D}_d^k)$ in the subset of data corresponding to class $k$ of the demographic criterion $d$.\n\nThe representation $\\textbf{x}$ and the model parameters $\\theta$ will typically be real-valued vectors, but they can be any set of features combining real and discrete values. Note that the previous formulation can be easily extended to the case of varying number of samples $S_i$  for different subjects, which is a usual case; or to classes \\textit{K} that are not disjoint. Note also that the previous formulation is based on average performances over groups of individuals. Different performance across specific individuals is usual in many artificial intelligence tasks due to diverse reasons, e.g., specific users who were not sensed properly \\cite{24}, even for algorithms that on average may perform similarly for the different classes that can be the source of discrimination.\n\n\n\\section{Face Recognition Algorithms}\nA face recognition algorithm, as other machine learning systems, can be divided into two different algorithms: screener and trainer. Both algorithms are used for a different aim and therefore should be studied with a different perspective \\cite{33}.\n\nThe screener (see Fig. \\ref{Figure1}) is an algorithm that given two face images generates an output associated to the probability that they belong to the same person. This probability is obtained comparing the two learned representations obtained from a face model defined by the parameters $\\theta$. These parameters are trained previously based on a training dataset $\\mathcal{D}$ and the goodness criterion \\textit{G} (see Fig. \\ref{Figure1}). If trained properly, the output of the trainer would be a model with parameters $\\theta^*$ capable of representing the input data (e.g., face images) in a highly discriminant feature space $\\textbf{x}$.\n\nThe most popular architecture used to model face attributes is the Convolutional Neural Network (CNN). This type of network has drastically reduced the error rates of face recognition algorithms in the last decade \\cite{28} by learning highly discriminative features from large-scale databases. In our experiments we consider two popular face recognition pre-trained models: VGG-Face and ResNet-50. These models have been tested on competitive evaluations and public benchmarks \\cite{13,12}. \n\nVGG-Face is a model based on the VGG-Very-Deep-16 CNN architecture trained on the VGGFace dataset \\cite{13}. ResNet-50 is a CNN model with 50 layers and 41M parameters initially proposed for general purpose image recognition tasks \\cite{29}. The main difference between ResNet architecture and traditional convolutional neural networks is the inclusion of residual connections to allow information to skip layers and improve gradient flow. \n\nBefore applying the face models, we cropped the face images using the algorithm proposed in \\cite{26}. The pre-trained models are used as embedding extractor where $\\textbf{x}$ is a $l_2$-normalised learned representation of a face image. The similarity between two face descriptors $\\textbf{x}_r$ and $\\textbf{x}_s$ is calculated as the Euclidean distance $||\\textbf{x}_r-\\textbf{x}_s||$. Two faces are assigned to the same identity if their distance is smaller than a threshold $\\tau$. The recognition accuracy is obtained by comparing distances between positive matches (i.e., $\\textbf{x}_r$  and $\\textbf{x}_s$ belong to the same person) and negative matches (i.e., $\\textbf{x}_r$  and $\\textbf{x}_s$ belong to different persons).\n\nThe two face models considered in our experiments were trained with the VGGFace2 dataset according to the details provided in \\cite{12}. As we will show in Section \\ref{Bias in face databases}, databases used to train these two models are highly biased. Therefore, it is expected that the recognition models trained with this dataset present algorithmic discrimination.\n\n\\subsection{Experimental protocol}\n\nLabeled Faces in the Wild (LFW) is a database for research on unconstrained face recognition \\cite{20}. The database contains more than 13K images of faces collected from the web. In this study we consider the aligned images from the test set provided with view 1 and its associated evaluation protocol. This database is composed by images acquired in the wild, with large pose variations, and varying face expressions, image quality, illuminations, and background clutter among other variations. The performance achieved by the VGG-Face and ResNet-50 models for the LFW database is $4.1\\%$ and $1.7\\%$ Equal Error Rate respectively. These performances serve as a baseline for the  models and the rest of experiments. We can observe the superior performance of the ResNet-50 model, with a performance ca. 3 times better than the VGG-Face model. \n\nThe experiments with DiveFace will be carried out following a cross-validation methodology using three images for each of the 4K identities from each of the six classes available in DiveFace (72K face images in total). This results in 72K genuine comparisons and near 3M impostor comparisons.\n\n\\subsection{DiveFace database: an annotation dataset for face recognition trained on diversity}\n\nDiveFace was generated using the Megaface MF2 training dataset \\cite{11}. MF2 is part of the publicly available Megaface dataset with 4.7 million faces from 672K identities and it includes their respective bounding boxes. All images in the Megaface dataset were obtained from Flickr Yahoo's dataset \\cite{27}. \n\nDiveFace contains annotations equally distributed among six classes related to gender and ethnicity (see Fig. \\ref{Figure4} for example images). Gender and ethnicity have been annotated following a semi-automatic process. There are 24K identities (4K for class). The average number of images per identity is 5.5 with a minimum number of 3 for a total number of images greater than 120K. Users are grouped according to their gender (male or female) and three categories related with ethnic physical characteristics:\n\n\\begin{quote}\n\\begin{itemize}\n\\item \\textbf{Group 1}: people with ancestral origins in Europe, North-America, and Latin-America (with European origin).\n\\item \\textbf{Group 2}: people with ancestral origins in Sub-Saharan Africa, India, Bangladesh, Bhutan, among others.\n\\item \\textbf{Group 3}: people with ancestral origin in Japan, China, Korea, and other countries in that region.\n\\end{itemize}\n\\end{quote}\n\nWe are aware of the limitations of grouping all human ethnic origins into only three categories. According to studies, there are more than 5K ethnic groups in the world. We categorized according to only three groups in order to maximize differences among classes. Automatic classification algorithms based on these three categories show performances of up to 98\\% accuracy \\cite{9}.\n\n\\section{Causes and Effects of Biased Learning in Face Recognition Algorithms}\n\\subsection{Performance of face recognition: role of demographic information}\n\n\\begin{table*}[ht]\n\\centering\n\\caption{Performance (False Match Rate in \\% @ False Non-Match Rate = 0.1\\%) of Face Recognition Models on the DiveFace dataset. We show in brackets the relative error growth rates with respect to the best class (\\textit{Group 1 Male}).}\\smallskip\n\\begin{tabular}{@{}>{\\centering}p{2cm}>{\\centering}p{2.2cm}>{\\centering}p{2.2cm}\n                 >{\\centering}p{2.2cm}>{\\centering}p{2.2cm}>{\\centering}p{2.2cm}c@{}}\n        \\cmidrule(l){1-7}\n   \\multirow{2}{*}{\\textbf{Model}} & \\multicolumn{2}{c}{\\textcolor{mi_yellow}{\\textbf{Group 1}}}\n                                        & \\multicolumn{2}{c}{\\textcolor{mi_pink}{\\textbf{Group 2}}}\n                                            & \\multicolumn{2}{c}{\\textcolor{blue}{\\textbf{Group 3}}}\\\\\n        \\cmidrule(lr){2-3}\\cmidrule(lr){4-5}\\cmidrule(l){6-7}\n    & \\textbf{Male} & \\textcolor{gray}{\\textbf{Female}} & \\textbf{Male} \n                & \\textcolor{gray}{\\textbf{Female}} & \\textbf{Male} & \\textcolor{gray}{\\textbf{Female}} \\\\ \n        \\cmidrule(r){1-1}\n        \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\n        \\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\n        \\cmidrule(lr){6-6}\\cmidrule(l){7-7}\n    VGG-Face & 7.99  & 9.38 ($\\uparrow$17\\%) & 12.03 ($\\uparrow$50\\%) & 13.95 ($\\uparrow$76\\%) & 18.43 ($\\uparrow$131\\%) & 23.66 ($\\uparrow$196\\%) \\\\\n    ResNet-50 & 1.60  & 1.96 ($\\uparrow$22\\%) & 2.15 ($\\uparrow$34\\%) & 3.61 ($\\uparrow$126\\%) & 3.25 ($\\uparrow$103\\%) & 5.07 ($\\uparrow$217\\%) \\\\\n\\end{tabular}\n\\label{table1}\n\\end{table*}\n\n\nThis section explores the effects of biased models in the performance of face recognition algorithms. Table \\ref{table1} shows the performances obtained for each demographic group present in DiveFace.  Traditional face recognition benchmarks usually do not explore this kind of demographic covariates. Results reported in Table \\ref{table1} exhibit large gaps between performances obtained by different demographic groups, suggesting that both gender and ethnicity significantly affect the performance of biased models. These effects are particularly high for ethnicity, with a very large degradation of the results for the class less represented in the training data (\\textit{Group} 3 \\textit{Female}). This degradation produces a relative increment of the Equal Error Rate (EER) of 196\\% and 217\\% for VGG-Face and ResNet-50, respectively, with regard to the best class (\\textit{Group} 1 \\textit{Male}). These differences are important as they mark the percentage of faces successfully matched and faces incorrectly matched. These results suggest that your ethnic origin can highly affect your possibilities to be incorrectly matched (false positives).\n\n\\subsection{Understanding biased performances}\nThe relatively low performance in \\textit{Group} 3 seems to be originated by a limited ability to capture the best discriminant features for the groups underrepresented in the training databases. The results suggest that features capable of reaching high accuracy for a specific demographic group may be less competitive in others. Let's analyze the causes behind these degradations. Fig. \\ref{Figure2} represents the probability distributions of genuine and impostor scores for \\textit{Group} 1 \\textit{Male} (the best group) and \\textit{Group 3 Female} (the worst group). The comparison between genuine and impostor distributions reveals large differences for the impostor's ones. The genuine distribution (intra-class variability) between \\textit{Group} 3 and \\textit{Group} 1 is similar, but the impostor distribution (inter-class variability) is significantly different. The model has difficulties to differentiate face attributes from different subjects.\n\n\\textbf{Algorithmic discrimination implications:} define the performance function $f$ as the accuracy of the face recognition model, and $\\textit{G}(\\mathcal{D}_d^k)=f(\\mathcal{D}_d^k,\\theta^* )$ the goodness considering all the samples corresponding to class $k$ of the demographic criterion $d$, for an algorithm $\\theta^*$ trained on the full set of data $\\mathcal{D}$ (as described in Eq. \\ref{eqn:goodnes_criterion}). Results suggest large differences between the goodness $\\textit{G}(\\mathcal{D}_d^k)$ for different classes, especially for classes $k=\\textit{Group} \\, 1,\\textit{Group} \\, 2,\\textit{Group} \\, 3$.\n\n\\subsection{Bias in face databases}\\label{Bias in face databases}\n\n\\begin{figure}\n\\includegraphics[width=85mm]{AAAI\/figura2.png}\n\\caption{ResNet-50 face recognition score distributions for Group 3 females and Group 1 males.}\n\\label{Figure2}\n\\end{figure}\n\nBias and discrimination concepts are related to each other, but they are not necessarily the same thing. Bias is traditionally associated with unequal representation of classes in a dataset. The history of automatic face recognition has been linked to the history of the databases used for algorithm training during the last two decades. The number of publicly available databases is high, and they allow training models using millions of face images. Fig. \\ref{Figure3} summarizes the demographic statistics of some of the most cited face databases. Each of these databases is characterized by its own biases (e.g. image quality, pose, backgrounds, and aging). In this work, we highlight the unequal representation of demographic information in very popular face recognition databases. As it can be seen, the differences between ethnic groups are severe. Even though the people in \\textit{Group} 3 are more than 35\\% of the world's population, they represent only 9\\% of the users in those popular face recognition databases.\n\nBiased databases imply a double penalty for underrepresented classes. On the one hand, models are trained according to non-representative diversity. On the other hand, benchmark accuracies are reported over privileged classes and overestimate the real performance over a diverse society.\n\nRecently, diverse and discrimination-aware databases have been proposed in \\cite{3,25,wang2019mitigate}. These databases are valuable resources to explore how diversity can be used to improve face biometrics. However, some of these databases do not include identities \\cite{3,25}, and face images cannot be matched to other images. Therefore, these databases do not allow to properly train or test face recognition algorithms.\n\n\\textbf{Algorithmic discrimination implications}: classes $k$ are unequally represented in the most popular face databases $\\mathcal{D}$.\n\n\\begin{figure}[t]\n\\includegraphics{AAAI\/figura3.pdf}\n\\caption{Demographic statistics of the 12 most cited face databases available in the literature. BioSecure \\cite{21}, YouTubeFaces \\cite{14}, PubFig \\cite{17}, CasiaFace \\cite{15}, VGGFace \\cite{13}, CelebA \\cite{16}, MS-Celeb-1M \\cite{10}, Megaface \\cite{11}, LFW \\cite{20}, UTKface \\cite{19}, VGGFace2 \\cite{12}, IJB-C \\cite{18}, DiveFace \\cite{9}.}\n\\label{Figure3}\n\\end{figure}\n\n\\subsection{Biased embedding space of deep models}\n\nWe now analyze the effects of ethnicity and gender attributes in the embedding space generated by VGG-Face and ResNet-50 models. CNNs are composed of a large number of stacked filters. These filters are trained to extract the richest information for a pre-defined task (e.g. face recognition). As face recognition models are trained to identify individuals, it is reasonable to think that the response of the models can slightly vary from one person to another. In order to visualize the response of the model to different faces, we consider the specific Class Activation MAP (CAM) proposed in \\cite{30}, named Grad-CAM. This visualization technique uses the gradients of any target concept, flowing into the selected convolutional layer to produce a coarse localization map. The resulting heat map highlights the activated regions in the image for the mentioned target (e.g. an individual identity in our case). Fig. \\ref{Figure4} represents the heat maps obtained by the ResNet-50 model for faces from different demographic groups. Additionally, we include the heat map obtained after averaging results from 120 different individuals from the six demographic groups included in DiveFace. The activation maps show clear differences between ethnic groups with the highest activation for \\textit{Group} 1 and the lowest for \\textit{Group} 3. These differences suggest that features extracted by the model are, at least, partially affected by the ethnic attributes.\n\n\\begin{figure}[t]\n\\includegraphics{AAAI\/figura4.pdf}\n\\caption{Examples of the six classes available in the DiveFace database (columns 1 to 4).  Column 5 shows the averaged Class Activation MAP (first filter of the third convolutional block of ResNet-50) obtained from 20 random face images from each of the classes. Columns 1-4 show Class Activation MAPs for each of the face images. Maximum and minimum activations are represented by red and blue colors respectively. Average pixel value of the activation maps generated for the six classes (\\textit{Groups} 1 to 3, and \\textit{Male}\/\\textit{Female}): G1M=0.23, G1F=0.19, G2M=0.21, G2F=0.18, G3M=0.12, G3F=0.13. (This is a colored image, see the digital version for a better quality.)}\n\\label{Figure4}\n\\end{figure}\n\nOn a different front, we applied a popular data visualization algorithm to better understand the importance of ethnic features in the embedding space generated by deep models. t-SNE is an algorithm to visualize high-dimensional data. This algorithm minimizes the Kullback-Leibler divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data. Fig. \\ref{Figure5} shows the projection of each face into a 2D space generated from ResNet-50 embeddings and the t-SNE algorithm. Additionally, we have colored each point according to its ethnic attribute. As we can see, the resulting face representation results in three clusters highly correlated with the ethnicity attributes. Note that ResNet-50 has been trained for face recognition, not ethnicity detection. However, the ethnicity information is highly embedded in the feature space and a simple t-SNE algorithm reveals the presence of this information.\n\nThese two simple experiments illustrate the presence and importance of ethnic attributes in the feature space generated by face deep models.\n\n\\textbf{Algorithmic discrimination implications}: popular deep models trained for task \\textit{T} on biased databases (i.e., unequally represented classes $k$ for a given demographic criterion $d$ such as gender) result in feature spaces (corresponding to the solution $\\theta^*$ of the Eq. \\ref{eqn:goodnes_criterion}) that introduce strong differentiation between classes $k$. This differentiation affects the representation $\\textbf{x}$ and enables classifying between classes $k$ using $\\textbf{x}$, even though \\textbf{x} was trained for solving a different task ${T}$.\n\n\n\\section{Conclusions}\n\nThis work has presented a comprehensive analysis of face recognition models according to a new discrimination-aware perspective. This work presents a new general formulation of algorithmic discrimination with application to face recognition. We have shown the high bias introduced when training the deep models with the most popular databases employed in the literature, and testing with the DiveFace dataset with well balanced data across demographic groups\\footnote{Available at GitHub: https:\/\/github.com\/BiDAlab\/DiveFace}. We have evaluated two popular models according to the proposed formulation. Biased models based on competitive deep learning algorithms have been shown to be very sensitive to gender and ethnicity attributes. This sensitivity results in different feature representations and a large gap between performances depending on the ethnic origin. This gap between performances reached up to 200\\% of relative error degradation between the best class (\\textit{Group} 1 \\textit{Male}) and the worst (\\textit{Group} 3 \\textit{Female}). These results suggest that false positives are 200\\% more likely in \\textit{Group} 3 \\textit{Female} than in \\textit{Group} 1 \\textit{Male} for the models evaluated in this work. These results encourage training more diverse models and developing methods capable to deal with the differences inherent to demographic groups. Future work will go in line with this approach, as authors do in \\cite{wang2019mitigate}.\n\n\\begin{figure}[t]\n\\includegraphics[width=80mm]{AAAI\/figura5.png}\n\\caption{Projections of the ResNet-50 embeddings into the 2D space generated with t-SNE.}\n\\label{Figure5}\n\\end{figure}\n\n\\section*{Acknowledgments}\n\nThis work has been supported by projects: BIBECA (RTI2018-101248-B-I00 MINECO\/FEDER), Bio-Guard (Ayudas Fundacion BBVA a Equipos de Investigacion Cientifica 2017).\n\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThis work describes a linear gyrokinetic analysis of the electromagnetic universal instability in a collisionless slab plasma.\nThe mode we term a ``universal instability\" in this research differs slightly from the the most restrictive usage of that term, and might equally well be labeled an entropy mode. Historically, the names ``entropy mode\"~\\cite{kadomtsev1960convective} and ``universal instability\"~\\cite{galeev1963universal} were first introduced in 1960s.\nThe former refers to an ion Larmor scale thermal instability driven by density and temperature gradients, resulting a perturbed density and temperature while the plasma kinetic pressure remains undisturbed. In certain magnetic configurations (e.g., Z-pinch~\\cite{ricci2006gyrokinetic} and dipole~\\cite{simakov2001kinetic}), it is believed to play an essential role in particle and heat transport when ideal MHD instabilities are suppressed.\nFor instance, it is found that the entropy mode is responsible for the observed particle pinch (i.e., transport of particles along the direction of the density gradient) in the dipole configuration with both local gyrokinetic~\\cite{kobayashi2009gyrokinetic} and global fluid~\\cite{ou2020turbulent} simulations.\nThe universal mode (instability) was first named to refer to the electrostatic instability predicted in low $\\beta$ magnetized plasma occurring due to non-uniform densities -- a pervasive characteristic amongst almost all magnetized plasmas, and hence the term \\textit{universal}. This name was challenged later as subsequent work focusing on the long wavelength limit ($k_\\perp\\rho_i\\ll 1$) found that this instability is not truly ``universal\" as it can be stabilized in complex geometries\\cite{krall1965universal} or in the simple sheared slab in the absence of additional instability drivers (i.e., temperature gradients, parallel current or magnetic curvatures).~\\cite{antonsen1978stability,ross1978drift,tsang1978absolute}\nHowever, recent studies suggest that the universal instability could still exist at sub-ion Larmor scales $k_\\perp\\rho_i \\geq 1$ in a sheared slab~\\cite{smolyakov2002short,landreman2015universal} or in more general geometries~\\cite{helander2015universal}.\nTo date, most research into entropy modes and the universal instability has been carried out in the electrostatic limit.\n\nPreviously, we performed a local linear electromagnetic gyrokinetic analysis of a shearless, collisionless slab plasma with the constraint of MHD equilibrium, i.e., equilibrium pressure balance $p+B^2\/(8\\pi)=\\text{constant}$, and discovered an instability driven by density and temperature gradients.~\\cite{rogers2018gyrokinetic} Allowing electromagnetic fluctuations while enforcing MHD equilibrium implies that the plasma kinetic pressure is not constant while instability occurs. Such an instability does not fit in the conventional entropy mode category, but might be better referred to as an electromagnetic universal instability. The dispersion relation derived in our previous work assumed singly charged ions and neglected electron finite Larmor radius (FLR) effects -- a common practice for analyzing ion-scale instabilities. In this paper, we extend our derivation to include arbitrary charge state, and full electron FLR and Debye shielding effects, and hence attain a more accurate generalized dispersion relation applicable to a wider range of plasmas. In particular, we examine solutions in an electron-positron (pair) plasma environment, where electron FLR effects strongly influence the overall plasma behavior. Notably, we find that the instability persists in this simple plasma system.\n\nElectron-positron plasma research is an active area of inquiry in plasma physics and astrophysics. Though it is conceptually simple, producing a sufficient number of electron-positron pairs to form a plasma and studying its collective behavior before annihilation in the laboratory is challenging. Even though the original idea of a pair plasma experiment and the first theoretical study of its properties (e.g., the absence of Faraday rotation, ion acoustic and drift waves because of the exact mass asymmetry) dates back to the late 1970s,~\\cite{tsytovich1978laboratory} only very recently have active experiments on pair plasma been proposed~\\cite{chen2011towards,pedersen2012plans} and carried out.~\\cite{chen2015scaling,saitoh2015efficient,warwick2017experimental} Two main ongoing pair plasma experiments include the A Positron-Electron Experiment (APEX) in Germany and the experiment in Lawrence Livermore National Laboratory (LLNL) in the United States. APEX produces non-relativistic positrons by the pair production process from absorption of MeV $\\gamma$-radiation in platinum, and plans to trap (and neutralize) them in a magnetic dipole~\\cite{hergenhahn2018progress}. The LLNL experiment generates a relativistic pair plasma via energetic short-pulse laser irradiation and plans to confine it in a magnetic mirror.~\\cite{jens2019}\nIn part motivated by such experiments, theoretical and numerical investigations of instability and transport in pair plasmas have also surged in the past few years.\nFor example, gyrofluid~\\cite{kendl2017interchange} and linear gyrokinetic~\\cite{kennedy2018linear,kennedy2020linear} simulations were performed to study pair plasma's stability in dipole and tokamak\/stellarator configurations.\nMeanwhile, recent electrostatic~\\cite{helander2014microstability} and electromagnetic (with incompressible $B_\\parallel$)~\\cite{helander2016gyrokinetic} work concludes that all microinstabilities are absent in pair plasmas with a homogeneous magnetic field.\nFurther studies suggest that under certain circumstances microinstabilities do exist in a slab pair plasma, e.g., current driven instabilities are supported in a sheared slab,~\\cite{zocco2017slab} drift instabilities can reappear when ion impurities are present and the ion fraction exceeds some threshold,~\\cite{mishchenko2018gyrokinetic}, alternatively density or temperature gradient driven instabilities could be excited in non-neutral electron-positron plasmas.~\\cite{kennedy2019local} The analysis we performed in this paper allows $\\delta B_\\parallel\\neq 0$, therefore, our finding that the $\\delta B_\\parallel$-universal instability can be unstable in pair plasmas does not contradict previous studies, but rather is an extension with a relaxed assumption on the guide field.\n\nThis paper is organized as follows. The generalized dispersion relation of the slab universal instability is derived in section~\\ref{sec:gdr}.\nDiscussions concerning the impact of electron finite Larmor radius (FLR) and Debye shielding effects are presented in section~\\ref{sec:flr}. \nIn section~\\ref{sec:ep} we show that this instability persists in a simple electron-positron pair plasma slab. Section~\\ref{sec:conc} summarizes our key findings.\n\n\\section{Generalized dispersion relation} \\label{sec:gdr}\n\nConsider a quasi-neutral plasma consisting of electrons and one other species with positive charge (e.g., positrons or cations) in a two-dimensional $(xy)$ slab with a guiding magnetic field $\\boldsymbol{B}(x)$ aligned with the $z$ direction. For simplicity we still refer to this positively charged species as ions with charge $q_i{=}Ze$ and mass ratio $\\mu{=}m_i\/m_e$. The equilibrium plasma pressure is then $p_0{=}p_{0i}{+}p_{0e}{=}n_{0i}T_{0i}(1{+}Z\\tau_e)$ where the quasi-neutral condition $n_{0e}{=}Zn_{0i}$ has been applied and $\\tau_e{=}T_{e0}\/T_{i0}$ is the ratio between electron and ion temperatures. Note that the quasi-neutral condition also ensures $L_{n_i}{=}L_{n_e}{=}L_n$ where the characteristic gradient scale length for a quantity $f$ is $L_f=f\/f'$, with $f'=\\partial f\/\\partial x$. The equilibrium pressure balance condition $p_0{+}B_0^2\/(8\\pi){=}\\text{constant}$, therefore, implies that $L_B^{-1}{=}-\\beta L_p^{-1}\/2$, or\n\\begin{equation}\\label{eq:mhd_eq}\n\\frac{L_{n_i}}{L_B}=-\\frac{\\beta_i\\alpha_0}{2},\n\\end{equation}\nwith\n\\begin{equation}\n\\alpha_0=1+\\eta_i+Z\\tau_e+Z\\tau_e\\eta_e,\\\n\\end{equation}\n$\\beta_{\\alpha}=8\\pi p_{\\alpha 0}\/B_0^2$, and the characteristic length ratios are $\\eta_\\alpha=(n_0T_{\\alpha 0}')\/(n_0'T_{\\alpha 0})$ for $\\alpha=e,i$.\nEnforcement of the equilibrium pressure balance condition within the slab geometry (i.e., when the magnetic tension force associated with magnetic curvature is negligible) is proven to be crucial in order to avoid a specious instability driven by the pressure gradients, even in low $\\beta$ cases.~\\cite{rogers2018gyrokinetic}\n\nDefining the thermal speed $v_{t \\alpha}=\\sqrt{{2T_{\\alpha 0}}\/{m_\\alpha}}$, the gyrofrequency $\\omega_{c\\alpha}=q_\\alpha B_0\/(m_\\alpha c)$, and the gyroradius $\\rho_\\alpha=v_{t\\alpha}\/\\omega_{c\\alpha}$, assuming $k_\\parallel=0$ and the perturbed magnetic field $\\tilde{\\boldsymbol{B}} \\propto e^{-i\\omega t+ik_\\perp y} \\hat{\\boldsymbol{z}}$, expanding the linearized Vlasov equation and distribution functions according to the gyrokinetic ordering $\\epsilon\\sim\\rho_\\alpha\/L$, and then substituting the perturbed electron and ion distribution function in Amp\\`ere's law and Gauss's law, yields the generalized dispersion relation of the slab $\\delta B_\\parallel$-universal instability\n\\begin{equation}\\label{eq:gdr}\nI_{\\phi Q} I_{BA}=I_{BQ}I_{\\phi A}=2\\left(I_{\\phi A}\\right)^2\/(Z\\beta_i)\n\\end{equation}\nwith\n\\begin{eqnarray}\nI_{\\phi Q}&=&2\\int_0^\\infty dv v e^{-v^2}\\left[\nZ\\left(J_0^2\\frac{\\bar{\\omega}_i}{\\omega_{bi}}-1\\right)+\\left(J_0^2\\frac{\\bar{\\omega}_e}{\\omega_{be}}-1\\right)\\tau_i\\right] - k^2\\frac{\\lambda_{De}^2}{\\rho_i^2}\\tau_i \\ , \\label{eq:IphiQt}\\\\\nI_{BA}&=&1+4\\int_0^\\infty dv v^3 e^{-v^2}\\left( J_1^2\\frac{\\beta_i}{k^2}\\frac{\\bar{\\omega}_i}{\\omega_{bi}}+\\frac{\\mu}{Z^2\\tau_e}J_1^2\\frac{\\beta_e}{k^2}\\frac{\\bar{\\omega}_e}{\\omega_{be}}\\right)\\ ,\\label{eq:IBAt}  \\\\\nI_{\\phi A}&=&-2Z\\beta_i \\int_0^\\infty dv v^2 e^{-v^2}\\left( J_0J_1\\frac{1}{k}\\frac{ \\bar{\\omega}_i}{\\omega_{bi}} +\\frac{\\mu^{1\/2}}{Z\\tau_e^{1\/2}}J_0J_1\\frac{1}{k}\\frac{ \\bar{\\omega}_e}{\\omega_{be}}\\right). \\label{eq:IphiAt}\n\\end{eqnarray}\nHere $\\tau_i=T_{i0}\/T_{e0}=\\tau_e^{-1}$, $v=v_{\\perp}\/v_{t}$,\n\\begin{eqnarray}\n\\bar{\\omega}_i=\\omega-\\frac{k}{2}\\left[ 1+\\eta_i \\left(v^2-1\\right)\\right] \\ , \\quad \\omega_{bi}= \\omega+\\frac{\\beta_i\\alpha_0}{4}kv^2 , \\\\\n\\bar{\\omega}_e=\\omega+\\frac{Z\\tau_e k}{2}\\left[ 1+\\eta_e \\left(v^2-1\\right)\\right] \\ , \\quad \\omega_{be}=\\omega-\\frac{\\beta_e\\alpha_0}{4}kv^2,\n\\end{eqnarray}\n$J_0$ and $J_1$ are Bessel functions of the first kind, the arguments of $J_{0,1}$ for ions and electrons are $kv$ and $-Z\\sqrt{\\tau_e\/\\mu}\\,\nkv$ respectively, and $\\omega,k$ are normalized according to\n\\begin{equation}\n    \\omega=\\omega_\\text{phys}L_n\/v_{ti}, \\quad k=k_\\text{phys}\\rho_i.\n\\end{equation}\n\nA complete description of the derivation of the generalized dispersion relation is outlined in the supplementary material. In such derivation no assumption was made on charge state, full electron FLR effects were retained and the perturbed electric field was calculated through Gauss's law. \nAs a result, the new dispersion relation is not only a function of $k_\\perp$, plasma $\\beta$, temperature ratios $\\tau_{i,e}$ and characteristic length ratios $\\eta_{i,e}=L_n\/L_{T_{i,e}}$, but also depends on the charge state $Z$, the ion-electron mass ratio $\\mu$ and the Debye length $\\lambda_{D_e}$.\n\nAs expected, if one takes small argument expansions of Bessel functions \n\\begin{equation}\n    J_0(b){=}1{-}b^2\/4{+}O(b^4),\\quad J_1(b){=}b\/2{-}b^3\/16{+}O(b^5)\n\\end{equation}\nand considers dense plasmas ($\\lambda_{De}\/\\rho_i{\\ll} 1$) with singly charged ions ($Z{=}1, \\mu{=}m_i\/m_e{\\gg} 1$ so that only leading order FLR effects of electrons are kept), then the above equations recover the dispersion relation without higher order electron FLR and Debye shielding correction (i.e., equations (17)-(19) of reference~\\cite{rogers2018gyrokinetic}).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f1_kbeta_scan.jpg}\n\t\\caption{Slab universal mode (a) frequency $\\omega$ and (b) linear growth rate $\\gamma$ verses $k_\\perp,\\beta$ in hydrogen plasma ($\\mu=1836,Z=1$) with $\\tau_e=1,\\eta_i=-1,\\eta_e=2$ and $\\lambda_{D_e}\\approx0$.}\n\t\\label{fig:lgr}\n\\end{figure}\n\nBecause the slab $\\delta B_\\parallel$-universal mode is also driven by density and temperature gradients, like conventional entropy modes, it shares characteristics of the entropy modes.\nIn general, the linear growth rate of the slab $\\delta B_\\parallel$-universal mode vanishes in the long wavelength $k\\rho_i\\ll 1$ limit and peaks near $k\\rho_i\\sim 1$. This mode tends to propagate in the electron diamagnetic direction at low $k$ while it reverses to the ion diamagnetic direction at high $k$ (the real frequency $\\omega$ changes from negative to positive) as elucidated in figure~\\ref{fig:lgr}. In addition, this mode is somewhat ``universal\" -- for any slab plasma with fixed $\\beta$ and varying guide magnetic field (i.e., $L_B\\neq \\infty$ or equivalently the plasma pressure $p$ is inhomogeneous), there is always a constrained parameter region (negative $\\eta_i$ and\/or $\\eta_e$) in which this mode is unstable.\n\nWithout further simplification, the generalized dispersion relation of the slab universal instability (equation~\\ref{eq:gdr}) is too complicated to be solved analytically. We therefore solve it using a numerical root-finding algorithm. In order to validate the generalized linear dispersion relation and verify that it is solved correctly, the numerical solutions of equation~\\ref{eq:gdr} are benchmarked with results of GENE's linear eigenvalue solver.~\\cite{jenko2000electron} As shown in figure~\\ref{fig:gene}, overall good agreement has been achieved over a wide range of parameters; a mild discrepancy only appears at moderate $k$ for low $\\beta$ and extremely hot ions (red line and diamonds in figure~\\ref{fig:gene}).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f2_gene.pdf}\n\t\\caption{Linear growth rate of hydrogen plasma from general dispersion relation (solid lines) along with result from GENE eigenvalue solver (diamond markers).}\n\t\\label{fig:gene}\n\\end{figure}\n\nSome analytical progress can be made, however, in simpler limiting cases. In the low $\\beta$ and long wavelength limit, an analytical dispersion relation is derived by taking the small argument approximations of Bessel functions $J_0$ and $J_1$.\nDefining the normalized phase velocity $u=2\\omega\/k$ and the normalized Debye length $\\bar{\\lambda}_{D_e}^2=2\\lambda_{D_e}^2\\tau_i\/(Z\\rho_i^2)$, to $O(\\beta)$, $O(Z)$ and $O(k^2)$, the dispersion relation becomes the quadratic equation\n\\begin{equation}\\label{eq:lowkdr}\n    a_2u^2+a_1u+a_0=0\n\\end{equation}\nwith\n\\begin{eqnarray}\na_2&=&\\left(1+\\frac{1}{\\mu}+\\bar{\\lambda}_{D_e}^2\\right)\\left(1+\\beta\\right)=\\left(1+\\frac{1}{\\mu}+\\bar{\\lambda}_{D_e}^2\\right)\\left[1+\\beta_i\\left(1+Z\\tau_e \\right) \\right]\\ , \\\\\na_1&=&-\\alpha_{1i}\\left(1+\\beta\\right)+\\beta_i\\left( \\frac{\\alpha_0}{2}-\\alpha_4-\\bar{\\lambda}_{D_e}^2\\alpha_4 \\right) +\\frac{1}{\\mu}\\left[ Z\\tau_e\\alpha_{1e}\\left(1+\\beta\\right)-\\beta_i\\left( \\frac{Z\\tau_e\\alpha_0}{2}+\\alpha_4 \\right)\\right]\\ , \\\\\na_0&=&\\beta_i\\left[ \\alpha_{1i}\\alpha_4-\\frac{\\alpha_0\\alpha_{2i}}{2}-\\frac{Z\\tau_e}{\\mu}\\left( \\alpha_{1e}\\alpha_4+\\frac{Z\\tau_e\\alpha_0\\alpha_{2e}}{2}\\right) \\right]\\ .\n\\end{eqnarray}\nand\n\\begin{equation}\n\\alpha_{1i,e}=1+\\eta_{i,e}\\ , \\quad \\alpha_{2i,e}=1+2\\eta_{i,e}\\ , \\quad  \\alpha_4=1-Z^2\\tau_e^2+2\\left(\\eta_i-Z^2\\tau_e^2\\eta_e\\right),\n\\end{equation}\nThe instability condition thus becomes\n\\begin{equation}\n    \\Delta=a_1^2-4a_0a_2<0\n\\end{equation}\nand the linear growth rate is\n\\begin{equation}\\label{eq:lowkgamma}\n    \\gamma_\\text{est}=\\text{Im}(\\omega)=\\frac{|\\Delta|^{1\/2}k}{4a_2}.\n\\end{equation}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f3_lowkcomp.pdf}\n\t\\caption{Numerical (red) and analytical (blue) growth rates in singly charged neon plasma for $k_\\perp\\rho_i=0.1,\\tau_e=2,\\eta_i=-1,\\eta_e=2$ and $\\lambda_{D_e}=0$.}\n\t\\label{fig:lowbeta}\n\\end{figure}\n\nDetails of the derivation are outlined in appendix~\\ref{app:lowbeta}. As plotted in figure~\\ref{fig:lowbeta}, compared to the numerical solution of full dispersion relation in equation~\\ref{eq:gdr}, the low $\\beta$ long wavelength estimate in equation~\\ref{eq:lowkgamma} typically yields an accurate linear growth rate in the low $\\beta$ regime ($\\beta \\leq 0.1$) but fails to predict instability as $\\beta$ approaches unity. \nNevertheless, it is still a very useful formula as it retains electron FLR and Debye shielding effects and is applicable to arbitrary $\\tau_{i,e}$. Therefore, we will use it to explore the stability conditions, quantify the impact of electron FLR effects and help us understand instability properties in the low $\\beta$ long wavelength regime.\n\n\\section{Electron FLR and Debye shielding effects}\\label{sec:flr}\nIn the gyrokinetic study of ion-scale instabilities, a common means of simplifying calculations is to drop high order electron FLR effects because they are normally two orders of magnitude smaller than ion FLR effects. Although recent multi-scale nonlinear simulations~\\cite{maeyama2015cross,howard2015multi} show that electron scale instability can influence ion scale transport due to cross-scale coupling, neglecting electron FLR effects in the linear analysis of ion-scale modes is still considered justified, at least for electron-ion plasmas.\nIn this section, we explore the impact of electron FLR effects on the slab $\\delta B_\\parallel$-universal instability within the linear gyrokinetic analysis, especially when the electron gyro-radius is no longer too small to be neglected.\nWe define the ratio of the electron and ion gyro-radius $\\delta=|\\rho_e\/\\rho_i|=Z\\sqrt{\\tau_e\/\\mu}$ as a (rough) indicator of the relative influence of electron and ion FLR effects.\nTypically, $\\delta$ is much smaller than unity. For example, $\\delta \\approx0.02$ for hydrogen plasma with $T_i\\sim T_e$, and $\\delta \\approx0.05$ for fully ionized neon plasma with $T_i\\sim T_e$. \n\nWe first quantify the corrections to our previous analysis of slab $\\delta B_\\parallel$-universal instability (without electron FLR effects)~\\cite{rogers2018gyrokinetic} due to electron FLR effects.\nFigure~\\ref{fig:eflr_tau1} compares the instability conditions and linear growth rates of the slab $\\delta B_\\parallel$-universal instability based on the low $\\beta$ long wavelength linear growth rate expression~\\ref{eq:lowkdr} with and without electron FLR effects for a hydrogen plasma at $\\beta=0.02,k_\\perp\\rho_i=0.1$ and $\\lambda_{D_e}\\approx0$.\nAs $\\delta\\propto\\sqrt{\\tau_e}$, equal temperature ($\\tau_e=1$), hot electron ($\\tau_e=5$) and extremely hot electron $\\tau_e=20$ cases are chosen.\nIn the equal temperature ($\\tau_e=1,\\delta=0.02$) and hot electron ($\\tau_e=5,\\delta=0.05$) plasmas, the electron FLR correction is too small to be distinguished, indicating that neglecting the electron FLR effects is indeed justified for these cases.\nFor the extremely hot electron case ($\\tau_e=20,\\delta=0.1$), including electron FLR effects has a stabilizing effect on the instability. For this very hot electron system the unstable region in $\\eta_{e,i}$ space shrinks slightly and the maximum linear growth rate decreases by $25\\%$, yet the overall shape remains qualitatively the same.\nTherefore, we conclude that the additional electron FLR effects do not introduce any qualitative change in the properties of the slab $\\delta B_\\parallel$-universal instability, at least for the electron-ion plasma parameters; and the findings based on our previous analysis~\\cite{rogers2018gyrokinetic} are still valid.\nIt is important to point out that this instability occurs in a parameter regime with $\\eta_i<0$ and\/or $\\eta_e<0$, i.e., density and temperature gradients are in opposite directions. Although less common, negative $\\eta$ could exist, for example, in a disrupted tokamak plasma. It also can be deliberately attained by locally heating electrons and\/or ions and fueling in a laboratory setting.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f4_eflr_tau_eta_scan_beta002.pdf}\n\t\\caption{Slab universal instability linear growth rate as a function of $\\eta_e,\\eta_i,\\tau$ for hydrogen plasma ($\\mu=1836,Z=1$) at $k_\\perp\\rho_i=0.1,\\beta=0.02,\\lambda_{D_e}=0$ with and without electron FLR correction.}\n\t\\label{fig:eflr_tau1}\n\\end{figure}\n\nIn order to further investigate the impact of electron FLR effects, stability analyses are performed for six different types of plasmas with different ion species. We now consider singly charged argon plasmas ($\\mu=73,351$), hydrogen plasmas ($\\mu=1836$) and plasmas with artificially reduced ion-electron mass ratios $\\mu\\in\\{100,40,20,10\\}$, with fixed total plasma $\\beta=0.1,\\tau_e=1,k_\\perp\\rho_i=0.1$ and $\\lambda_{D_e}\\approx 0$. These plasmas hence correspond to $\\delta\\in\\{0.004,0.023,0.1,0.158,0.224,0.316\\}$ respectively.\nFigure~\\ref{fig:eflr_mu1} shows how the instability condition and linear growth rate evolve as the ion mass decreases, or $\\delta$ increases. The stabilizing effect due the electron FLR contribution is more profound in this mass scan. For $\\delta \\leq 0.1$ (singly charged argon and hydrogen plasmas), the unstable region in $\\eta_{e,i}$ space barely changes and the peak linear growth rates remains the same. As $\\delta$ continues increasing to sub-unity ($\\sim 0.3$), electron FLR effects start to have substantial influence on the instability: the unstable region becomes more constrained in $\\eta_{e,i}$ space and the maximum linear growth rate reduces to roughly $1\/3$ of its original value -- both indicate that the slab $\\delta B_\\parallel$-universal instability is stabilized by the electron FLR effects. This conclusion is further verified by solving the generalized dispersion relation in equation~\\ref{eq:gdr} for plasmas of different ion mass $\\mu$ at $Z=1,\\beta=0.1,\\tau_e=1,\\eta_e=2,\\eta_i=-1$ and $\\lambda_{D_e}\\approx 0$. Figure~\\ref{fig:eflr_mu2} demonstrates that as the electron FLR effects increase, not only do the linear growth rates diminish, but also the mode  favors travel in the electron direction (negative in our notation) at low $k$, and the propagation transition point (i.e., $\\omega=0$) is pushed towards higher $k$.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f5_eflr_mu_eta_condition.pdf}\n\t\\caption{Slab universal instability linear growth rate as a function of $\\eta_e,\\eta_i,\\mu$ for $k_\\perp\\rho_i=0.1,\\beta=0.1,\\tau_e=1,\\lambda_{D_e}=0$.}\n\t\\label{fig:eflr_mu1}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f6_eflr_mu_gdr.pdf}\n\t\\caption{(a) Real frequencies and (b) linear growth rates of the slab universal instability for singly charged plasma ($Z=1$) with various ion mass and $\\beta=0.1,\\tau_e=1,\\eta_e=2,\\eta_i=-1,\\lambda_{D_e}=0$.}\n\t\\label{fig:eflr_mu2}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f7_klambda_scan.jpg}\n\t\\caption{Stabilization of the slab universal instability for hydrogen plasma with $\\beta=1,\\tau_e=1,\\eta_i=-1$ and $\\eta_e=2$ as normalized Debye length $\\bar{\\lambda}_{D_e}=\\lambda_{D_e}\/\\rho_i$ increases.}\n\t\\label{fig:klambda_scan}\n\\end{figure}\n\nThe impact of Debye shielding on the slab universal instability is also studied. Not surprisingly, it is always stabilizing and starts to play an important role once $\\lambda_{D_e}$ approaches $\\rho_i$ as shown in figure~\\ref{fig:klambda_scan}. \nPhysically, the slab $\\delta B_\\parallel$-universal mode discussed in this paper is an electromagnetic instability -- perturbed electron and ion distribution functions produce a perturbed electrostatic potential $\\tilde{\\phi}$ as well as a perturbed current $\\tilde{j}$; also, the characteristic wavelength of this mode $\\lambda\\sim 1\/k_\\perp\\sim \\rho_i$.\nOn the other hand, by definition, the Debye length $\\lambda_{D}$ measures how far the electrostatic effects can extend in a plasma -- with each distance of $\\lambda_D$, the effective electric potential decreases by a factor of $1\/e$. \nIf $\\lambda_D \\ll \\lambda$, Debye shielding has little influence on $\\tilde{\\phi}$; however, as the Debye length increases towards $\\lambda$, the electrostatic potential (and its perturbation) will be attenuated more and more strongly; eventually, when $\\lambda_D \\gg \\lambda$, $\\tilde{\\phi}$ will be completely screened.\n\nThe Debye shielding effect can also be understood algebraically with the aid of the low $\\beta$ long wavelength analytical formula in equation~\\ref{eq:lowkdr}. If one neglects electron FLR effects ($\\mu\\to\\infty$), then $a_2\\approx (1+\\bar{\\lambda}_{D_e}^2)(1+\\beta)$, $a_1 \\approx -\\alpha_{1i}(1+\\beta)+\\beta_i(\\alpha_0\/2-\\alpha_4-\\bar{\\lambda}_{D_e}^2\\alpha_4)$, and the instability condition with $\\bar{\\lambda}_{D_e}$ ordering yields\n\\begin{equation}\n    \\Delta\\sim  \\beta_i^2\\alpha_4^2\\bar{\\lambda}_{D_e}^4 -O(\\bar{\\lambda}_{D_e}^2)<0.\n\\end{equation}\nClearly, given a large enough $\\bar{\\lambda}_{D_e}$, the $O(\\bar{\\lambda}_{D_e}^4)$ term will dominate and $\\Delta>0$. In other words, the slab universal instability will be completely stabilized if the plasma density is sufficiently low such that $\\bar{\\lambda}_{D_e}\\gg 1$ or $\\lambda_{D_e}\\gg\\rho_i$. \nHowever, in many space and laboratory magnetized plasmas, $\\rho_i\\gg\\lambda_{D_e}$; e.g., for solar wind plasma $\\rho_i\\sim 5\\times10^4 \\text{m}\\gg \\lambda_D\\sim 10\\text{m}$, for LAPD plasma $\\rho_i\\sim 2\\times10^{-3} \\text{m}\\gg \\lambda_D\\sim 1\\times10^{-5}\\text{m}$, and for tokamak plasma $\\rho_i\\sim 2\\times10^{-3} \\text{m}\\gg \\lambda_D\\sim 2\\times10^{-5}\\text{m}$. Therefore Debye shielding stabilization is expected to be weak in these plasma systems.\n\n\\section{Slab $\\delta B_\\parallel$-universal instability in pair plasma}\\label{sec:ep}\nIn an electron-ion plasma, $\\mu=m_i\/m_e\\gg 1$ and $T_e\\sim T_i$ so that $\\delta\\ll 1$ and neglecting higher order electron FLR effects is justified as discussed in section~\\ref{sec:flr}. However, this is not the case for electron-positron pair plasmas. \nDue to its unique mass symmetry feature ($m_i=m_p=m_e$ where subscript $p$ refers to positron), this simple plasma system has some fascinating properties. For example, large temperature separation is not expected in pair plasma as the ratios $\\tau_{ee}:\\tau_{pp}:\\tau_{ep}=1:(m_p\/m_e)^1\/2:m_p\/m_e=1:1:1$ where $\\tau_{rs}$ is the collision time between species $r$ and species $s$.~\\cite{iwamoto1993collective} Therefore, in a pair plasma, $\\delta=\\sqrt{\\tau_e}\\sim 1$; in other words, electron FLR effects are almost always equally as important as the ion (or, positron) FLR effects in a pair plasma.\nAnother interesting consequence of mass symmetry is that many instabilities (e.g., the electrostatic drift instability) will be absent in such a system.~\\cite{tsytovich1978laboratory}\n\nAs shown in figure~\\ref{fig:eflr_mu1}, the slab $\\delta B_\\parallel$-universal instability is stabilized when $\\mu$ decreases (i.e., electron FLR effects become more important) -- the unstable region in $\\eta$ space shrinks and the peak linear growth rate drops. One therefore might expect that this instability could be completely suppressed as $\\mu$ is further decreased. However, we find that though suppressed substantially, the slab universal instability still exists when $\\mu=1$ (figure~\\ref{fig:ep_tau1}). In fact, the electron FLR effects alone cannot completely stabilize the slab $\\delta B_\\parallel$-universal instability in a pair plasma: when $\\delta=|\\rho_e\/\\rho_p|=\\sqrt{\\tau_e}<1$, the electron FLR effects tend to stabilize the instability; while for $\\delta>1$ the electron FLR has a destabilizing effect on the instability.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f8_ep_tau_eta.pdf}\n\t\\caption{Slab universal instability linear growth rate as a function of $\\eta_e,\\eta_i,\\tau$ for electron-positron plasma ($\\mu=1,Z=1$) at $k_\\perp\\rho_i=0.1,\\beta=0.1$. Red line denotes $L_p^{-1}=0$, i.e., plasma pressure is uniform.}\n\t\\label{fig:ep_tau1}\n\\end{figure}\n\nWe therefore conclude that the slab $\\delta B_\\parallel$-universal instability due to plasma inhomogeneity persists in the pair plasma. Our result, however, does not contradict the previous finding that ``there are no linear gyrokinetic instabilities in a pair plasma embedded in a constant magnetic field, regardless of the size of the density and temperature gradients.\"~\\cite{helander2016gyrokinetic} This is because (1) this instability only exists when the magnetic field is not uniform (i.e., no intersection with the red lines in figure~\\ref{fig:ep_tau1}); (2) we assume here that the guiding magnetic field is compressible ($\\delta B_\\parallel \\neq 0$) instead of incompressible ($\\delta B_\\parallel=0$).\n\nFor electron positron pair plasma ($\\mu=Z=1$), the general dispersion relation in equation~\\ref{eq:gdr} becomes\n\\begin{equation}\\label{eq:ep_gdr}\nI_{\\phi Q} I_{BA}=(2\/\\beta_p)\\left(I_{\\phi A}\\right)^2\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\begin{aligned}\nI_{BA}&=1+4\\int_0^\\infty d\\bar{v}_{\\perp} \\bar{v}_{\\perp}^3 e^{-\\bar{v}_{\\perp}^2}\\left( J_1^2\\frac{\\beta_p \\bar{\\omega}_p}{k^2\\rho_p^2\\omega_{bp}}+J_1^2\\frac{\\beta_e \\bar{\\omega}_e}{k^2\\rho_e^2\\omega_{be}}\\right)\\ , \\\\\nI_{\\phi Q}&=2\\int_0^\\infty d\\bar{v}_{\\perp} \\bar{v}_{\\perp} e^{-\\bar{v}_{\\perp}^2}\\left[\nJ_0^2\\frac{\\bar{\\omega}_p}{\\omega_{bp}}-1-\\left(J_0^2\\frac{\\bar{\\omega}_e}{\\omega_{be}}-1\\right)\\frac{T_{p0}}{T_{e0}}\n\\right]\\ ,\\\\\nI_{\\phi A}&=-2\\beta_p \\int_0^\\infty d\\bar{v}_{\\perp} \\bar{v}_{\\perp}^2 e^{-\\bar{v}_{\\perp}^2}\\left( J_0J_1\\frac{ \\bar{\\omega}_p}{k\\rho_p \\omega_{bp}} +J_0J_1\\frac{ \\bar{\\omega}_e}{k\\rho_e \\omega_{be}}\\right)\\ .\n\\end{aligned}\n\\end{eqnarray}\nHere\n\\begin{equation}\n\\bar{\\omega}_p=\\omega-(k\/2)\\left[1+\\eta_p\\left(v^2-1\\right)\\right] \\ ,\\quad \\bar{\\omega}_e=\\omega+(\\tau_e k\/2)\\left[1+\\eta_e\\left(v^2-1\\right)\\right]\\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\omega_{bp}=\\omega-(k\/2)(L_n\/L_B)v^2 \\ ,\\quad \\omega_{be}=\\omega+(\\tau_e k\/2)(L_n\/L_B)v^2\\ .\n\\end{equation}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f9_ep_tau_gdr.pdf}\n\t\\caption{(a) Real frequencies and (b) linear growth rates of the slab universal instability in a pair plasma with $\\beta=0.1$.}\n\t\\label{fig:ep_tau2}\n\\end{figure}\n\nTo highlight the characteristics of this slab universal mode in a pair plasma, we solve the general dispersion relation of equation~\\ref{eq:ep_gdr} and show, in figure~\\ref{fig:ep_tau2}, the frequencies and growth rates as a function of $k_\\perp\\rho_p$ for three different (yet somewhat symmetric) sets of parameters. The self-symmetric setting (i.e., electrons and positrons have the same temperature profile as $\\tau_e=T_e\/T_p=1,\\eta_i=\\eta_e=-0.75$) has balanced electron and positron FLR effects ($\\delta=1$), and results in a \\textit{purely growing} instability, i.e., the real mode frequency $\\omega$ is zero. This can be understood via the low-$\\beta$, long wavelength analytical formula in section~\\ref{sec:gdr}. With these self-symmetric parameters, the linear coefficient $a_1$ in the quadratic equation vanishes (assume $\\bar{\\lambda}_{D_e}=0$), and the solutions become two purely imaginary numbers corresponding to a decaying mode and an instability. \n\nThe mirror settings (case A: $\\tau_e=0.5,\\eta_p=-0.8,\\eta_e=-0.4$ and case B: $\\tau_e=2.,\\eta_p=-0.4,\\eta_e=-0.8$) reflect temperature profiles between electrons and positrons, and produce two anti-symmetric real frequency $k-$spectra and two identical linear growth rate curves (in physical units).\nIn figure~\\ref{fig:ep_tau2}, the anti-symmetric and identical dispersion relations are less obvious because they are plotted in normalized units with respect to case-specific parameters. Recall that for a fixed $\\beta$, the positron temperature in case A ($\\tau_e=0.5$) is twice as much as in case B ($\\tau_e=2$) as $T_p\\propto\\beta_p=\\beta\/(1+\\tau_e)$, resulting in a $\\sqrt{2}$ larger gyro-radius $\\rho_p$ and thermal velocity $v_{tp}$ and hence the differences in figure~\\ref{fig:ep_tau2} (i.e., curves of case A are stretched along $x$ axis and compressed along $y$ axis by $\\sqrt{2}$ compared to the curves of case B). Take for example the most unstable point of two dispersion relations -- case A at $k_\\perp\\rho_p^A=1.01,\\gamma_\\text{max}\/(v_{ti}^A L_n)=0.0267$ and case B at $k_\\perp\\rho_p^B=0.71,\\gamma_\\text{max}\/(v_{ti}^B L_n)=0.0377$, are overlaid with each other in physical units or in normalized units with a fixed positron\/ion temperature (as shown in figure~\\ref{fig:ep_tau3}).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f10_ep_tau_gdr2.pdf}\n\t\\caption{Real frequencies (blue) and linear growth rates (red) of the slab universal instabilities in a pair plasma with $\\beta=0.1$ for case A (solid lines) and B (dashed lines) normalized with respect to case B positron\/ion temperature.}\n\t\\label{fig:ep_tau3}\n\\end{figure}\n\nIt is interesting to see that the mode propagation direction completely reverses once the electron FLR effects exceeds positron FLR effects (i.e., $\\delta>1$) as indicated in figure~\\ref{fig:ep_tau2}(a).\nThis can be understood by considering a slab pair plasma with species gyrating around a guiding magnetic field unstable to the slab $\\delta B_\\parallel$-universal instability. The excitation of this instability requires a non-uniform magnetic field but does not depend on the magnetic field direction. If electrons and positrons are suddenly swapped and the guiding field direction is simultaneously flipped, the new system has virtually the same assembly of particles gyrating in the same direction; one would thus expect the same instability with the mode propagating in the opposite direction with respect to the guiding field direction (as it is flipped).\n\nThe authors would like to remark on the stabilizing effect of Debye shielding in the pair plasma as well. So far our analysis has assumed that the pair plasma is dense enough so that $\\lambda_D\/\\rho_p\\ll 1$; however, this condition is usually not satisfied for present-day laboratory produced pair plasmas (e.g., $\\rho_p\\sim 100~\\mu$m while $\\lambda_D\\sim 3.6$ m in APEX~\\cite{hergenhahn2018progress}, and $\\rho_p\\sim 0.04$ cm while $\\lambda_D\\sim 1$ cm in the LLNL experiment~\\cite{jens2019}) for which the density is low. In our studies Debye shielding effect are always stabilizing, as illustrated in figure~\\ref{fig:ep_lambda}. In the case of $\\tau_e=1$ (so $\\rho_p=\\rho_e=\\rho$) the Debye shielding effect starts to play a role when $\\lambda_D\/\\rho>0.1$, and has a profound impact on the linear growth rate when $\\lambda_D\/\\rho\\sim 1$. This slab $\\delta B_\\parallel$-universal mode is (almost) completely suppressed when $\\lambda_D\/\\rho\\sim 100$. Hence, observation of slab universal instability requires that $\\lambda_D\/\\rho<1$, or equivalently, the plasma number density exceeds a critical value $n_c=B^2\/(8\\pi m_e c^2)$; $n_c$ is also termed the Brillouin limit.~\\cite{brillouin1945theorem} Because Debye shielding stabilizes nearly all instabilities, similar conclusions were drawn in previous studies on pair plasmas, e.g., gyrokinetic GS2 simulations of interchange instability in electron-positron plasma confined in a tokamak configuration~\\cite{pedersen2003prospects} and in a linear electrostatic gyrokinetic stability analysis.~\\cite{helander2014microstability}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\textwidth]{.\/f11_lambda.pdf}\n\t\\caption{Linear growth rate of the slab universal instability in a pair plasma for $\\tau_e=1,\\beta=k_\\perp \\rho=0.1, \\eta_p=\\eta_e=-0.75$ varies with Debye length $\\lambda_D$.}\n\t\\label{fig:ep_lambda}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec:conc}\nIn this paper, the generalized electromagnetic $\\delta B_\\parallel$-universal instability dispersion relation in a collisionless slab plasma is derived by performing a gyrokineitic linear stability analysis accounting for electron finite Larmor radius (FLR) and Debye shielding effects. This ion-scale microinstability, driven by density and temperature gradients, vanishes in the long wavelength limit ($k_\\perp\\to 0$) and typically has a peak growth rate at $k_\\perp\\rho_i\\sim 1$. It also tends to reverse its propagation direction from the electron to the ion diamagnetic direction as $k$ increases. \n\nWe find that the electron FLR effects, at low $k$, generally stabilize the slab $\\delta B_\\parallel$-universal instability, resulting in a more constrained unstable region in the $\\eta_i-\\eta_e$ parameter space and reduced linear growth rates. Although this stabilization effect is weak for electron-ion plasmas where the electron gyro-radius is normally two orders of magnitude smaller than ion gyro-radius ($\\delta=|\\rho_e\/\\rho_i|=Z\/\\sqrt{m_eT_e\/(m_iT_i)}\\sim O(10^{-2})$), it starts to play a role once $\\delta>0.1$ (e.g., in cold ion plasma or artificially reducing the ion-electron mass ratio). \nHowever, electron FLR effects cannot completely stabilize this instability. As a result, we find that this instability persists in the electron-positron pair plasma where the conventional electrostatic universal instability (i.e., drift-wave) is absent due to the unique mass symmetry.\nSeveral interesting features of this instability are observed in the pair plasma. When electrons and positrons are at thermal equilibrium (i.e., $T_p=T_i,L_{T_p}=L_{T_e}$ so $\\delta=1$), the slab $\\delta B_\\parallel$-universal mode becomes a purely growing instability (i.e., $\\omega=0$; or the mode stops propagating). As the electron FLR effects exceed the positron (ion) FLR effects ($\\delta>1$), it starts to drive, or destabilize, the slab $\\delta B_\\parallel$-universal instability and to reverse mode propagation direction.\nWe have also shown that Debye shielding has a strong stabilization effect once the Debye length $\\lambda_D$ approaches $\\rho_i$, as the electrostatic potential fluctuations $\\tilde{\\phi}$ associated with the instability are effectively screened.\n\nIt is worth pointing out that the gyrokinetic linear analysis of slab universal instability can be further extended, either to the relativistic limit, and to dipole and mirror fields such as those planned for use in confining laboratory-produced pair plasmas.\nThe nonlinear dynamics of the slab universal instability, and the turbulent transport it induces, remain unexplored research areas. The results may differ from the linear prediction presented here due to the nonlinear excitation of zonal flows and\/or electromagnetic fluctuations. We look forward to carrying out local gyrokinetic simulations that shed light on these topics.\n\n\\section*{Supplementary Material}\nSee supplementary material for the complete derivation of the generalized dispersion relation of the slab $\\delta B_\\parallel$-universal instability.  \n\n\\section*{Data Availability Statement}\nThe data shown in this article are available from the corresponding author upon reasonable request.\n\n\\begin{acknowledgments}\nB. Zhu and M. Francisquez thank M. J. Pueschel, P. Terry and A. Friedman for the useful discussions and suggestions.\nB. Zhu and X. Xu are supported by DOE contract DE-AC52-07NA27344 through the Lawrence Livermore National Laboratory. M. Francisquez is supported by DOE contract DE-FC02-08ER54966. B. Rogers is supported by DOE-SC-0010508. This research used resources of the Discovery cluster supported by the Research Computing Group at Dartmouth College.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\indent\n\nThe Large Electron Positron collider (LEP) at CERN and the Stanford\nLinear Collider (SLC) are powerful $e^+e^-$ machines operating\nat the $Z$ peak, which can confront theoretical predictions of\nthe minimal Standard Model (SM) with experimental results to a\nhigh accuracy.\nA full analysis of all the electroweak precision data\nincluding those of the year 1995 will either establish the SM up\nto one-loop electroweak level or signal the onset of new physics.\nIn this context, analyzing electroweak oblique parameters~\\cite{STU}\nhas become a common strategy to test the viability of models beyond\nthe SM. The electroweak oblique parameters are sensitive physical\nquantities, when the new-physics interactions couple predominantly\nto $W$ and $Z$ bosons. However, it is imperative to explore\nadditional observables that could be particularly sensitive to\nother sectors of the SM.\n\nIn this paper, we will study a new set of {\\em complementary }\nleptonic observables and explicitly demonstrate the severe limitations\nthat can impose on model building of three-generation extensions of the SM.\nThe set of observables comprises flavour-changing leptonic decays\nof the $Z$ boson~\\cite{LFCNC,KPS}, universality-breaking parameters\n$U_{br}$ for the diagonal decays $Z\\to l\\bar{l}$~\\cite{BKPS}, and\nuniversality-violating parameters $\\Delta{\\cal A}_{l_1l_2}$ based on\nlepton asymmetries measured at LEP and\/or SLC~\\cite{BP}.\nFor the sake of illustration, we will consider a minimal left-right\nsymmetric model (LRSM)~\\cite{MS} described in Section~2.\nSuch a model can naturally generate vector--axial (V--A) as well\nas V+A flavour-dependent $Zl\\bar{l}$ couplings leading to\nnew physics effects that can be probed via the leptonic observables\nmentioned above.\nThis set of observables will be discussed in some detail in Section 3.\nIn Section 4, we will give numerical estimates of these leptonic\nobservables within the framework of a minimal LRSM and investigate\ntheir potential of effectively constraining this model. Furthermore,\nattention will be paid to possible LRSM contributions to $R_b$.\nSection 5 contains our conclusions.\n\n\n\n\\setcounter{equation}{0}\n\\section{The LRSM}\n\\indent\n\nLeft-right symmetric theories based on the gauge group $SU(2)_R\\times\nSU(2)_L\\times U(1)_{B-L}$~\\cite{PS,MS} were motivated from the fact that\nthe spontaneous breakdown of gauge and discrete symmetries can be\naccomplished on the same footing. Such models can naturally arise\nfrom $SO(10)$ grand unified theories via the breaking pattern~\\cite{PS,FM}\n\\begin{displaymath}\nSO(10) \\to SU(4)_{PS}\\times SU(2)_R\\times SU(2)_L \\to SU(3)_c\\times SU(2)_R\n\\times SU(2)_L \\times U(1)_{B-L} \\to \\mbox{SM}.\n\\end{displaymath}\nWe will, however, focus our analysis on the LRSM described in~\\cite{MS}.\n\nIn the LRSM, right-handed neutrinos together with the\nright-handed charged leptons form 3 additional weak isodoublets in\na three generation model. The classification of the quark sector\nproceeds in an analogous way. To be specific, the assignment of quantum\nnumbers to fermions under the gauge group $SU_R(2)\\times SU(2)_L \\times\nU(1)_{B-L}$ is arranged as follows:\n\\begin{eqnarray}\nL_L' = \\left( \\begin{array}{c} \\nu'_l \\\\ l' \\end{array} \\right)_L\\ :\\ (0,1\/2,-1) &&\\qquad\nL_R' = \\left( \\begin{array}{c} \\nu'_l \\\\ l' \\end{array} \\right)_R\\ :\\ (1\/2,0,-1),\\\\\nQ_L' = \\left( \\begin{array}{c} u' \\\\ d' \\end{array} \\right)_L\\ :\\ (0,1\/2,1\/3) &&\\qquad\nQ_R' = \\left( \\begin{array}{c} u' \\\\ d' \\end{array} \\right)_R\\ :\\ (1\/2,0,1\/3),\n\\end{eqnarray}\nwhere the prime superscript of the fermionic fields simply denotes\nweak eigenstates.\nIn order to break the left-right gauge symmetry down to $U(1)_{em}$~\\cite{MS},\nwe have to introduce one Higgs bidoublet,\n\\begin{equation}\n\\Phi\\ =\\ \\left( \\begin{array}{cc} \\phi^0_1 & \\phi^+_1 \\\\\n                          \\phi^-_2 & \\phi^0_2 \\end{array} \\right),\n\\end{equation}\nthat transforms as $(1\/2^*,1\/2,0)$ and two complex Higgs triplets,\n\\begin{equation}\n\\Delta_L\\ =\\ \\left( \\begin{array}{cc} \\delta_L^+\/\\sqrt{2} & \\delta_L^{++} \\\\\n                               \\delta_L^0 & -\\delta_L^+\/\\sqrt{2}\n\\end{array} \\right)\\qquad \\mbox{and}\\qquad\n\\Delta_R\\ =\\ \\left( \\begin{array}{cc} \\delta_R^+\/\\sqrt{2} & \\delta_R^{++} \\\\\n                               \\delta_R^0 & -\\delta_R^+\/\\sqrt{2}\n\\end{array} \\right),\n\\end{equation}\nwith quantum numbers $(0,1,2)$ and $(1,0,2)$, respectively. For simplicity,\nwe will consider that only $\\<\\phi^0_1\\>=\\kappa_1\/\\sqrt{2}$ and\n$\\<\\delta^0_R\\>=v_R\/\\sqrt{2}$ acquire vacuum expectation values (vev's).\nIn practice, this can be accomplished by imposing invariance of the general\nHiggs potential under judicious discrete symmetries of the bidoublet $\\Phi$\nand the Higgs triplets $\\Delta_{L,R}$~\\cite{IL}.\nIn fact, we will concentrate on case~(d) of Ref.~\\cite{GGMKO},\nto which the reader is referred for more details.\nIn case (d), it is $\\<\\delta^0_L\\>= \\<\\phi^0_2\\>=0$,\nimplying that the charged gauge bosons $W_L$ and $W_R$ represent\nalso physical states with masses $M_L=M_W$ and $M_R$, respectively. The\nmassive neutral gauge bosons $Z_L$ and $Z_R$ mix one another with a small\nmixing angle of order $\\kappa_1^2\/v_R^2\\sim 10^{-2}$.\nTo a good approximation, we will therefore neglect\n$Z_L-Z_R$ mixing effects in our calculations.\n\nSuch a minimal LRSM allows the presence of baryon--lepton ($B-L$)\nviolating operators in the Yukawa sector. In fact, the\n$B-L$-violating interactions are introduced by the triplet fields\n$\\Delta_{L,R}$ and give rise to Majorana mass terms $m_{M_{ij}}$\nin the following\nway:\n\\begin{equation}\n{\\cal L}^{B-L}_{int} = - \\frac{\\sqrt{2} m_{M_{ij}}}{2v_R}\\Big(h_{ij}\n\\bar{L}'^C_{L_i}\\, \\varepsilon_{ij} \\Delta_L L_{L_j}'\\ +\\\n\\bar{L}'^C_{R_i}\\, \\varepsilon_{ij} \\Delta_R L_{R_j}'\\Big)\\quad +\\quad \\mbox{H.c.}\n\\end{equation}\nHere, $\\varepsilon_{ij}$ stands for the usual Levi-Civita tensor and the parameters\n$h_{ij}=1$ if left-right symmetry is forced explicitly. However, a\nphenomenological analysis of muon and $\\tau$ decays shows that\n$h_{ij}\\ll 1$~\\cite{GGMKO}. As a result,\n$\\delta_L^+$ and $\\delta^{++}_L$ loop effects have been found to be\nnegligible~\\cite{MAPS}.\n\nIn case (d), the neutrino mass matrix takes the general seesaw-type\nform\n\\begin{equation}\nM^\\nu\\ =\\ \\left( \\begin{array}{cc} 0 & m_D\\\\ m_D^T & m_M \\end{array} \\right),\\label{Mnu}\n\\end{equation}\nwhere $M^\\nu$ is $6 \\times 6$-dimensional matrix. In Eq.~(\\ref{Mnu}), $m_D$\nis a Dirac mass term connecting the left-handed neutrinos with the\nright-handed ones. Relevant theoretical and phenomenological aspects\nrelated to such neutrino mass models may be found in Ref.~\\cite{ZPC}.\nThe matrix $M^\\nu$ can always be diagonalized by a unitary $6\\times 6$\nmatrix $U^\\nu$ according to the common prescription $U^{\\nu T}M^\\nu U^\\nu\n=\\hat{M}^\\nu$. After diagonalization, one gets\n6 physical Majorana neutrinos $n_i$ through the\ntransformations\n\\begin{equation}\n\\left(\\begin{array}{c} \\nu_L' \\\\ \\nu'^{C}_R \\end{array} \\right)_i\\ =\\\n\\sum_{j=1}^{2\\mbox{{\\footnotesize n}}_G} U^{\\nu\\ast}_{ij}\\ n_{Lj}\\quad \\mbox{and}\\quad\n\\left(\\begin{array}{c} \\nu'^C_L \\\\ \\nu'_R \\end{array} \\right)_i\\ =\\\n\\sum_{j=1}^{2\\mbox{{\\footnotesize n}}_G} U^\\nu_{ij}\\ n_{Rj}.\n\\end{equation}\nThe first $\\mbox{n}_G =3$ neutral states, $\\nu_i$ ($\\equiv n_i$ for $i=1,\\dots,\\mbox{n}_G$),\nare identified with the known $\\mbox{n}_G$ light neutrinos,\nwhile the remaining $\\mbox{n}_G$ mass eigenstates, $N_j$ ($\\equiv n_{j+\\mbox{{\\footnotesize n}}_G}$ for\n$j=1,\\dots,\\mbox{n}_G$), are heavy Majorana neutrinos predicted by the model.\nIn addition to the leptonic sector, the quark sector of such an extension\ncontains non-SM couplings of the fermionic fields to the gauge and Higgs\nbosons. Part of the LRSM couplings has been listed in~Ref.~\\cite{IL,GGMKO,GZ}.\nRelevant Feynman rules and additional discussion is given in Appendix A.\n\nAdopting the conventions of Ref.~\\cite{ZPC},\nthe interactions of the Majorana neutrinos, $n_i$, and charged\nleptons, $l_i$, with the gauge bosons $W^\\pm_L\\ (\\equiv W^\\pm)$ and $Z_L$\nare correspondingly mediated by the mixing matrices\n\\begin{equation}\nB^L_{lj}\\ = \\sum\\limits_{k=1}^{\\mbox{{\\footnotesize n}}_G} V^L_{lk} U^{\\nu\\ast}_{kj}\\quad\n\\mbox{and}\\quad\nC^L_{ij}\\ =\\ \\sum\\limits_{k=1}^{\\mbox{{\\footnotesize n}}_G}\\ U^\\nu_{ki}U^{\\nu\\ast}_{kj}, \\label{BL}\n\\end{equation}\nwith $l=1,\\dots,\\mbox{n}_G$ and $i,j=1,\\dots,2\\mbox{n}_G$. By analogy,\nthere exist mixing matrices $B^R_{li}$ and $C^R_{ij}$ given by\n\\begin{equation}\nB^R_{lj}\\ = \\sum\\limits_{k=\\mbox{{\\footnotesize n}}_G +1 }^{2\\mbox{{\\footnotesize n}}_G} V^R_{lk} U^{\\nu\\ast}_{kj}\\quad\n\\mbox{and}\\quad\nC^R_{ij}\\ =\\ \\sum\\limits_{k=\\mbox{{\\footnotesize n}}_G +1 }^{2\\mbox{{\\footnotesize n}}_G}\\ U^\\nu_{ki}U^{\\nu\\ast}_{kj},\n\\label{BR}\n\\end{equation}\nwhich are responsible for the couplings of $W^\\pm_R$ and $Z_R$ to\ncharged leptons and Majorana neutrinos. In Eqs.~(\\ref{BL}) and (\\ref{BR}),\nthe unitary $\\mbox{n}_G\\times \\mbox{n}_G$-matrices $V^L$ and $V^R$ are responsible\nfor the diagonalization of the charged lepton mass matrix via biunitary\ntransformations.\n\nDue to the specific structure of $M^\\nu$ in Eq.~(\\ref{Mnu}),\nthe flavour-mixing matrices $B^L$ and $C^L$\nsatisfy a number of identities that may be found in~\\cite{APetal}\nThese identities, which result from the requirement of unitarity\nand renormalizability of the theory, turn out to be very useful\nin deriving model-independent relations between the mixings $B^L_{li}$,\n$C^L_{ij}$ and heavy neutrino masses. In a two generation mixing model,\nwe have~\\cite{AP1,IP}\n\\begin{equation}\nB^L_{lN_1}\\ =\\ \\frac{\\rho^{1\/4} s^{\\nu_l}_L}{\\sqrt{1+\\rho^{1\/2}}}\\ , \\qquad\nB^L_{lN_2}\\ =\\ \\frac{i s^{\\nu_l}_L}{\\sqrt{1+\\rho^{1\/2}}}\\ ,\n\\end{equation}\nwhere $\\rho=m^2_{N_2}\/m^2_{N_1}\\ (\\ge 1)$\nis a mass ratio of the two heavy Majorana\nneutrinos $N_1$ and $N_2$ present in such a model,\nand $s^{\\nu_l}_L$ is $L$-violating mixings defined as~\\cite{LL}\n\\begin{equation}\n(s^{\\nu_l}_L)^2 \\ \\equiv\\  \\sum\\limits_{i=\\mbox{{\\footnotesize n}}_G +1}^{2\\mbox{{\\footnotesize n}}_G} |B^L_{li}|^2\n\\ \\simeq\\ \\left( m_D^\\dagger \\frac{1}{m_M^2} m_D \\right)_{ll}.\n\\end{equation}\nFurthermore, the mixings $C^L_{N_iN_j}$ are determined by\n\\begin{eqnarray}\nC^L_{N_1N_1} &=& \\frac{\\rho^{1\/2}}{1+\\rho^{1\/2}}\\ \\sum\\limits_{i=1}^{\\mbox{{\\footnotesize n}}_G}\n(s^{\\nu_i}_L)^2, \\qquad C^L_{N_2N_2}\\ =\\ \\frac{1}{1+\\rho^{1\/2}}\\\n\\sum\\limits_{i=1}^{\\mbox{{\\footnotesize n}}_G} (s^{\\nu_i}_L)^2, \\nonumber\\\\\nC^L_{N_1N_2}&=& -C^L_{N_2N_1}\\ =\\ \\frac{i\\rho^{1\/4}}{1+\\rho^{1\/2}}\\\n\\sum\\limits_{i=1}^{\\mbox{{\\footnotesize n}}_G} (s^{\\nu_i}_L)^2.\n\\end{eqnarray}\n\nAt this stage, it should be noted that $M^\\nu$ of Eq.~(\\ref{Mnu})\ntakes the known seesaw form~\\cite{YAN} in case $m_M\\gg m_D$.\nNevertheless, this mass-scale hierarchy can dramatically be relaxed in a\ntwo-family seesaw-type model, which can radiatively induce light-neutrino\nmasses\nin agreement with experimental upper bounds~\\cite{ZPC}. The light-heavy\nneutrino mixings\n$s^{\\nu_l}_L$ of such scenarios  can, in principle, scale as\n$s^{\\nu_l}_L \\sim m_D\/m_M$ rather than the known seesaw relation\n$s^{\\nu_l}_L \\sim \\sqrt{m_{\\nu_l}\/m_N}$. This implies that\nhigh Dirac components are allowed to be present in $M^\\nu$\nand only the ratio $m_D\/m_M$ ($\\sim s^{\\nu_l}_L$) gets limited by a global\nanalysis of low-energy data. Recently, such an analysis has been\nperformed in Ref.~\\cite{Cliff}, in which the combined effect of\nall possible effective operators of the charged- and neutral-current\ninteractions is considered. Although a careful analysis\ncan provide some model-dependent caveats, we will, however,\nconsider the following conservative upper bounds for the $L$-violating\nmixings:\n\\begin{equation}\n(s^{\\nu_e}_L)^2,\\  (s^{\\nu_\\mu}_L)^2 \\ <\\ 0.010,\\qquad\n(s^{\\nu_\\tau}_L)^2\\ <\\ 0.040,\\qquad \\mbox{and}\n\\qquad (s^{\\nu_e}_L)^2 (s^{\\nu_\\mu}_L)^2\\ < \\ 1.\\ 10^{-8}.\\label{mix}\n\\end{equation}\nFor example, the last constraint in Eq.~(\\ref{mix}) comes from\nthe nonobservation of decays of the type $\\mu\\to e\\gamma,\\ eee$,\nor the absence of $\\mu -e$ conversion events in nuclei.\n\nIn LRSMs, the mixing matrices $B^L$, $C^L$, $B^R$ and $C^R$ obey\nthe useful relations\n\\begin{equation}\n\\sum\\limits_{i=1}^{2\\mbox{{\\footnotesize n}}_G} B^L_{l_1i}B^R_{l_2i} =  0,\\qquad\n\\sum\\limits_{l=1}^{\\mbox{{\\footnotesize n}}_G} B^{R\\ast}_{li}B^R_{lj} =  C^R_{ij},\\qquad\nC^{L\\ast}_{ij}\\ +\\ C^R_{ij}\\ = \\ \\delta_{ij}. \\label{idR}\n\\end{equation}\nIn a two-generation mixing model, Eq.~(\\ref{idR}) together with\nEq.~(\\ref{BR}) can be used to obtain the mixings\n\\begin{eqnarray}\nB^R_{l_1N_1} &=& \\cos\\theta_R\\sqrt{1-C^L_{N_1N_1}},\\qquad\nB^R_{l_2N_1} \\ =\\ -\\sin\\theta_R\\sqrt{1-C^L_{N_1N_1}}\\nonumber\\\\\nB^R_{l_1N_2} &=& -\\cos\\theta_R\\frac{C^L_{N_1N_2}}{\\sqrt{1-C^L_{N_1N_1}}}\n\\, -\\, i\\sin\\theta_R\\left( \\frac{(1-C^L_{N_1N_1})(1-C^L_{N_2N_2})-\n|C^L_{N_1N_2}|^2 }{1-C^L_{N_1N_1}} \\right)^{1\/2},\n\\nonumber\\\\\nB^R_{l_2N_2} &=& \\sin\\theta_R\\frac{C^L_{N_1N_2}}{\\sqrt{1-C^L_{N_1N_1}}}\n\\, -\\, i\\cos\\theta_R\\left(\\frac{(1-C^L_{N_1N_1})(1-C^L_{N_2N_2})-\n|C^L_{N_1N_2}|^2 }{1-C^L_{N_1N_1}} \\right)^{1\/2}. \\ \\ \\\n\\end{eqnarray}\nConsequently, the leptonic sector of this two generation scenario\ndepends only on five free parameters;\nthe masses of the two heavy Majorana neutrinos, $m_{N_1}$ and $m_{N_2}$\n[or equivalently $m_{N_1}$ and $\\rho$], the mixing angles $(s^{\\nu_i}_L)^2$,\nwhich are, however, constrained by low-energy data, and an unconstrained\nCabbibo-type angle $\\theta_R$.\n\n\\setcounter{equation}{0}\n\\section{SLC and LEP observables}\n\\indent\n\nIn this section, we will define more precisely the framework of our\ncalculations. In the limit of vanishing charged lepton masses, the\namplitude responsible for the decay $Z\\to l_1\\bar{l}_2$\ncan generally be parametrized as\n\\begin{equation}\n{\\cal T}_l\\ =\\ \\frac{ ig_w}{ 2c_w}\\,\n\\varepsilon^\\mu_Z\\, \\bar{u}_{l_1}\\gamma_\\mu [g^{l_1l_2}_L\\mbox{P}_L\\\n+\\ g^{l_1l_2}_R\\mbox{P}_R ] v_{l_2}, \\label{Ampl}\n\\end{equation}\nwhere $g_w$ is the usual electroweak coupling constant, $\\varepsilon_Z^\\mu$ is\nthe $Z$-boson polarization vector, $u\\ (v)$ is the Dirac spinor of\nthe charged lepton $l_1\\ (l_2)$,\n$\\mbox{P}_L(\\mbox{P}_R)=(1-(+)\\gamma_5)\/2$, and $c^2_w=1-s^2_w=M^2_W\/M^2_Z$.\nIn Eq.~(\\ref{Ampl}), we have defined\n\\begin{equation}\ng^{l_1l_2}_{L,R}=g^l_{L,R}+\\delta g^{l_1l_2}_{L,R},\\qquad\ng^l_L =\\sqrt{\\rho_l}(1-2\\bar{s}^2_w),\\qquad g^l_R=-2\\sqrt{\\rho_l}\\bar{s}^2_w,\n\\end{equation}\nwhere $\\rho_l$, $\\bar{s}_w$, $\\delta g^l_{L,R}\\ (\\equiv\\delta g_{L,R}^{ll})$\nare obtained beyond the Born\napproximation and are renormalization scheme dependent. It should be noted\nthat $\\rho_l$, $\\bar{s}_w$ introduce universal oblique corrections~\\cite{STU},\nwhereas $\\delta g^{l_1l_2}_{L,R}$ are flavour dependent. Obviously, an\nanalogous expression will be valid for the decay $Z\\to b\\bar{b}$, as soon as\n$b$-quark mass effects can be neglected.\n\nIt is  convenient to reexpress the flavour-dependent electroweak\ncorrections in terms of the loop functions $\\Gamma^L_{l_1l_2}$\nand $\\Gamma^R_{l_1l_2}$ as follows:\n\\begin{displaymath}\n\\delta g^{l_1l_2}_L\\ =\\ \\frac{\\alpha_w}{2\\pi} \\, \\Gamma^L_{l_1l_2}, \\qquad\n\\delta g^{l_1l_2}_R\\ =\\ \\frac{\\alpha_w}{2\\pi} \\, \\Gamma^R_{l_1l_2}.\n\\end{displaymath}\nThe nonoblique loop functions $\\Gamma^L_{l_1l_2}$\nand $\\Gamma^R_{l_1l_2}$ depend on whether the underlying theory\nis of V--A or V+A nature. In Appendix B, we have analytically derived\nthe loop functions  $\\Gamma^L_{l_1l_2}$ and $\\Gamma^R_{l_1l_2}$ in the\ncontext of LRSMs.\nIt is then straightforward to obtain the branching ratio for the possible\ndecay of the $Z$ boson into two different charged leptons\n\\begin{equation}\nB(Z\\to \\bar{l}_1 l_2+l_1\\bar{l}_2)\\ =\\ \\frac{\\alpha_w^3}{48\\pi c^2_w}\\,\n\\frac{M_Z}{\\Gamma_Z} \\Big[ |\\Gamma^L_{l_1l_2}|^2+|\\Gamma^R_{l_1l_2}|^2\\Big],\n\\label{Blfv}\n\\end{equation}\nwith $\\alpha_w=g^2_w\/4\\pi$.\nSuch an observable is constrained by LEP results to be, {\\em e.g.},\n$B(Z\\to e\\tau)\\hspace{-0.1cm}\\stackrel{\\displaystyle <}{\\sim} 10^{-5}$~\\cite{PDG}.\n\nAnother observable that has been introduced in~\\cite{BKPS} is the\nuniversality-breaking parameter $U_{br}^{(l_1l_2)}$.\nTo leading order of perturbation theory,\n$U_{br}^{(l_1l_2)}$ is given by\n\\begin{eqnarray}\nU_{br}^{(l_1l_2)} &=& \\frac{\\Gamma (Z\\to l_1\\bar{l}_1)\\ -\\\n\\Gamma (Z\\to l_2\\bar{l}_2)}{\\Gamma (Z\\to l_1\\bar{l}_1)\\ +\\\n\\Gamma (Z\\to l_2\\bar{l}_2)}\\ - \\ U_{br}^{(l_1l_2)}(\\mbox{PS})\\nonumber\\\\\n&=& \\frac{g^l_L (\\delta g_L^{l_1} - \\delta g^{l_2}_L)\\ +\\\ng^l_R (\\delta g^{l_1}_R - \\delta g_R^{l_2} )}{g^{l2}_L\\ +\\ g^{l2}_R}\\nonumber\\\\\n&=& U_{br}^{(l_1l_2)}(\\mbox{LH})\\ +\\ U_{br}^{(l_1l_2)}(\\mbox{RH}), \\label{Ubr}\n\\end{eqnarray}\nwhere $U_{br}^{(l_1l_2)}(\\mbox{PS})$ characterizes known phase-space\ncorrections coming from the nonzero masses of the charged leptons\n$l_1$ and $l_2$ that can always be subtracted, and\n\\begin{eqnarray}\nU_{br}^{(l_1l_2)}(\\mbox{LH})&=&\n\\frac{g^l_L(\\delta g_L^{l_1}- \\delta g_L^{l_2})}{(g^{l2}_L+g^{l2}_R)}\\ =\\\n\\frac{\\alpha_w}{2\\pi}\\frac{g^l_L}{g^{l2}_L+g^{l2}_R}\\, \\Re e e(\\Gamma^L_{l_1l_1}\n-\\Gamma^L_{l_2l_2}) \\\\\nU_{br}^{(l_1l_2)}(\\mbox{RH})\n&=&\\frac{g^l_R(\\delta g_R^{l_1}- \\delta g_R^{l_2})}{g^{l2}_L+g^{l2}_R}\\ =\\\n\\frac{\\alpha_w}{2\\pi}\\frac{g^l_R}{g^{l2}_L+g^{l2}_R}\\, \\Re e e(\\Gamma^R_{l_1l_1}\n-\\Gamma^R_{l_2l_2})\n\\end{eqnarray}\nTo make contact with the corresponding observable given in~\\cite{PDG},\none can easily derive the relation\n\\begin{displaymath}\n\\frac{\\Gamma (Z\\to l\\bar{l})}{\\Gamma (Z\\to l'\\bar{l}')}\n= 2 U^{(ll')}_{br} + 1.\n\\end{displaymath}\n\nThe results of a combined analysis at LEP\/SLC regarding lepton universality\nat the $Z$ resonance can be summarized~\\cite{LEP,SLC}\nas follows:\n\\begin{eqnarray}\n|U_{br}^{(ll')}| & < & 5.\\ 10^{-3}\\qquad (\\mbox{SM}:0),\\nonumber\\\\\n{\\cal A}_\\tau ({\\cal P}_\\tau) &=& 0.143 \\pm 0.010\\qquad (\\mbox{SM}:0.143),\\nonumber\\\\\n{\\cal A}_e ({\\cal P}_\\tau) &=& 0.135 \\pm 0.011,\\nonumber\\\\\n{\\cal A}_{FB}^{(0,l)} &=& 0.0170\\pm 0.0016\\qquad (\\mbox{SM}:0.0153),\\nonumber\\\\\n{\\cal A}_{LR} (SLC)& =& 0.1637 \\pm 0.0075.\\label{Exp}\n\\end{eqnarray}\nIn parentheses, we quote the theoretical predictions obtained in the SM.\n{}From (\\ref{Exp}), we find that the experimental sensitivity to\n$\\Delta{\\cal A}_{\\tau e}$ is about $7\\%$, ($4\\%$) for LEP (SLC).\nNote that ${\\cal A}_e$ should equal the left-right asymmetry, ${\\cal A}_{LR}$,\nmeasured at SLC.\nFurthermore, it is worth mentioning that ongoing SLC experiments\nare measuring the observable\n\\begin{equation}\nA_{LR}^{FB}(f)\\ =\\ \\frac{\\Delta\\sigma( e^-_Le^+\\to f\\bar{f})_{FB}\n- \\Delta\\sigma( e^-_Re^+\\to f\\bar{f})_{FB}}{\\Delta\\sigma( e^-_Le^+\\to\nf\\bar{f})_{FB} + \\Delta\\sigma( e^-_Re^+\\to f\\bar{f})_{FB}}=\\frac{3}{4}\n{\\cal P}_e {\\cal A}_f, \\label{AFBLR}\n\\end{equation}\nThe forward-backward left-right asymmetry for individual flavours will\nbe an interesting alternative of establishing possible\ndeviations from SM universality in lepton asymmetries.\n\nLepton asymmetries [or equivalently forward-backward asymmetries]\ncan also play a crucial r\\^ole to constrain new physics. Here, we will\nbe interested in experiments at LEP\/SLC that measure the observable\n\\begin{eqnarray}\n{\\cal A}_l \\ &=&\\ \\frac{\\Gamma (Z\\to l_L \\bar{l})\\ -\\ \\Gamma (Z\\to l_R \\bar{l})}{\n\\Gamma (Z\\to l \\bar{l})}\\ =\\ \\frac{g^{ll2}_L\\ -\\ g^{ll2}_R}{g^{ll2}_L\\ +\\\ng^{ll2}_R} \\nonumber\\\\\n &=& \\frac{ g^{l2}_L -g^{l2}_R + 2(g^l_L\\delta g^l_L - g_R\\delta g^l_R) }{\ng^{l2}_L + g^{l2}_R + 2(g^l_L\\delta g^l_L + g^l_R\\delta g^l_R)}.\\label{Al}\n\\end{eqnarray}\nIn view of the recent discrepancy of more than $2\\sigma$\nbetween the experimental results of ${\\cal A}_{LR}$ at SLC and\n${\\cal A}_e$ at LEP, we are motivated to use the nonuniversality\nparameter of lepton asymmetries~\\cite{BP}\n\\begin{equation}\n\\Delta{\\cal A}_{l_1l_2}\\ =\\ \\frac{{\\cal A}_{l_1}\\ -\\ {\\cal A}_{l_2}}{{\\cal A}_{l_1}\\ +\\ {\\cal A}_{l_2}}\n\\ =\\ \\frac{1}{\\bar{{\\cal A}}_l} \\Big( U_{br}^{(l_1l_2)}(\\mbox{LH})\\ -\\\nU_{br}^{(l_1l_2)}(\\mbox{RH}) \\Big)\\ -\\ U_{br}^{(l_1l_2)}, \\label{DA}\n\\end{equation}\nwhere $\\bar{{\\cal A}}_l$ may be given by the mean value of the\ntwo lepton asymmetries ${\\cal A}_{l_1}$ and ${\\cal A}_{l_2}$.\nAt this point, it should be stressed that requiring $U_{br}^{(l_1l_2)}=0$ does\n{\\em not necessarily} imply $\\Delta{\\cal A}_{l_1l_2}=0$. As we will later see,\nin LRSMs one can naturally encounter the possibility, in which\n$U_{br}(\\mbox{LH})\\simeq -U_{br}(\\mbox{RH})$ while $\\Delta{\\cal A}_{l_1l_2}$\nbecomes sizeable. Moreover, the physical quantities $U_{br}^{(l_1l_2)}$\nand $\\Delta{\\cal A}_{l_1l_2}$ do not depend explicitly on universal electroweak\noblique parameters, especially when the latter ones may poorly constrain\nsuch three-generation scenarios~\\cite{BKPS}.\n\nAnother observable which will still be of interest is\n\\begin{equation}\nR_b \\ =\\ 0.2202 \\pm 0.0020\\qquad (\\mbox{SM}:0.2158).\n\\end{equation}\nIf the measurement at LEP is correct, $R_b$ turns out to be about $2\\sigma$\noff from the theoretical prediction of the minimal SM.\nNew physics contributions to $R_b$ can be conveniently calculated\nthrough~\\cite{BDV}\n\\begin{equation}\nR_b\\ =\\ 0.22\\Big[ 1+0.78\\nabla_b^{(SM)}(m_t)-0.06\\Delta\\rho^{(SM)}(m_t)\\Big],\n\\end{equation}\nwhere $\\nabla_b^{(SM)}(m_t)$ and $\\Delta\\rho^{(SM)}(m_t)$ contains the\n$m_t$-dependent parts of the vertex and oblique corrections, respectively.\nPractically, only $\\nabla_b^{(SM)}(m_t)$ gives significant negative\ncontributions to $R_b$, which behave, in the large top-mass limit,\nas~\\cite{BPS}\n\\begin{equation}\n\\nabla_b^{(SM)}(m_t)\\ \\simeq\\ -\\frac{20\\alpha_w s^2_w}{13\\pi}\\,\n\\frac{m^2_t}{M^2_Z}.\n\\end{equation}\nIf there are new physics effects contributing to $\\nabla_b^{(SM)}(m_t)$,\nthese can be estimated by\n\\begin{equation}\n\\nabla_b^{(new)}(m_t)\\ =\\ \\frac{\\alpha_w}{2\\pi}\\,\n\\frac{g^b_L\\Re e e(\\Gamma^L_{bb}(m_t)-\\Gamma^L_{bb}(0))\n+g^b_R\\Re e e(\\Gamma^R_{bb}(m_t)-\\Gamma^R_{bb}(0))}{g^{b2}_L\\ +\\ g^{b2}_R}\\, ,\n\\label{Rbnew}\n\\end{equation}\nwhere $g^b_L=1-2s_w^2\/3$ and $g^b_R=-2s^2_w\/3$.\nIn the next section, we will analyze numerically the size of new physics\neffects expected in LRSM.\n\n\\setcounter{equation}{0}\n\\section{Numerical results and discussion}\n\\indent\n\nSince there is a large number of free parameters that could vary\nindependently in the LRSM, we have fixed to typical values\nall of them except of one each time and investigated the behaviour\nof our observables as a function of the remaining kinematic variable.\nMore explicitly, we have found that $\\delta^{++}_R$ quantum corrections\nto the effective $Zl_R\\bar{l}_R$ coupling shown in Fig.~1 are very small,\nsince $M_{\\delta^{++}_R}> 5$~TeV for phenomenological reasons~\\cite{GGMKO}.\nThe very same lower mass bound should obey the flavour-changing scalars\n$\\phi^{0r}_2\\ (\\equiv\\Re e e(\\phi^0_2)\/\\sqrt{2})$ and $\\phi^{0i}_2\\\n(\\equiv\\Im m m(\\phi^0_2)\/\\sqrt{2})$~\\cite{GGMKO}.\nHowever, the mass difference between the two flavour-changing scalars\nshould not be too large because the latter would lead to large negative\ncontributions to $R_b$ (we will discuss the consequences from a large\nmass-difference realization between the flavour-changing scalars\nat the end of this section). In our estimates, we have\nassumed that $\\phi^{0r}_2$ and $\\phi^{0i}_2$ are nearly degenerate\nand heavier than 5 TeV. In such a case, loop effects involving\nflavour-changing scalars are found to be vanishingly small.\n\nIn order to increase the predictability of our LRSM but still keep our\nanalysis on a general basis, we shall consider a two-generation\nmixing scenario.\nThen, the free parameters of our minimal model are: the lepton-violating\nmixings $(s^{\\nu_l}_L)^2$ [which are, however, constrained to some extend by\na global analysis of low-energy data], the two heavy neutrino masses\n$m_{N_1}$ and $m_{N_2}$ [which have been taken to be at the same\nmass scale $m_N$], the masses of the charged gauge boson $M_R$ and\nits orthogonal associate scalar $M_h$, and a Cabbibo-type\nangle $\\theta_R$ that rotates the right-handed charged leptons to\nthe corresponding mass eigenstates.\n\nIn Figs.~2(a)--(d), we present plots of $B(Z\\to e^-\\tau^++e^+\\tau^-)$\nas a function of $m_N$, $M_R$, $M_h$, and $\\theta_R$ while keeping fixed the\nremaining kinematic parameters. In Fig.~2(a), we see the characteristic\n{\\em quadric}, $m^4_N\/M^4_W$, dependence on the branching\nratio~\\cite{KPS,APetal}.\nThe dashed, dotted and dash-dotted lines represent results coming purely\nfrom the $SU(2)_R$ sector for $(s^{\\nu_\\tau}_L)^2=0.040,\\ 0.030$, and\n$0.020$, respectively. The solid lines~$i$, $ii$, and $iii$ correspond\nto a complete computation for the three different lepton-violating\nmixings mentioned above.\nIf we assume some typical values for\nthe rest of the parameters, {\\em i.e.}~$M_R=0.4$~TeV, $M_h=30$~TeV,\nand $\\theta_R=0$, we find that $B(Z\\to e^-\\tau^+ + e^+\\tau^-)\\hspace{-0.1cm}\\stackrel{\\displaystyle <}{\\sim}\n2.\\ 10^{-6}$ for $m_N=3$~TeV. Although the size of new physics effects\nmay be probed at LEP, the reported value is\n$B(Z\\to e\\tau)<10^{-5}$ and it does not yet impose rather severe constraints\non the parameter space of the theory. This conclusion is also supported\nby Figs.~2(b)--(d). In Fig.~2(c), it is worth observing the logarithmic\ndependence of the mass ratio $M_h\/M_R$ on the branching ratio, which\ncan also render the decay channel $Z\\to e\\tau$ measurable. In Fig.~2(d),\none can further see the strong dependence of $\\theta_R$ on\n$B(Z\\to e^-\\tau^+ + e^+\\tau^-)$. However, a similar, though complementary,\nbehaviour will be found to be present in the observables $U_{br}$ and\n$\\Delta{\\cal A}$.\n\nWe are now proceed by examining numerically the dependence of the\nuniversality-breaking parameter $U_{br}$ as a function of various\nkinematic variables shown in Figs.~3(a)--(d).\nAgain, we observe the nondecoupling behaviour of the heavy neutrino\nmass in the observable $U_{br}$~\\cite{BKPS}.\nThe size of new physics becomes significant for $m_N\\hspace{-0.1cm}\\stackrel{\\displaystyle >}{\\sim} 3$ TeV,\n{\\em i.e.} $U_{br}\\sim 4.-5.\\ 10^{-3}$. In Fig.~3(b), we see that the\nvalue of $U_{br}$ decreases rapidly as $M_R$ increases. In Fig.~3(c),\nwe remark again the logarithmic dependence $M^2_h\/M^2_R$ on $U_{br}$.\nIn our estimates, we have used a Cabbibo-type angle $\\theta_R=45^0$, which\nturns out to be a rather moderate value as is displayed in Fig.~3(d).\nAs has been mentioned above, the electroweak corrections originating\ngenuinely from the $SU(2)_R$ sector depend on the angle $\\theta_R$.\nLooking at Fig.~3(d), one can readily see that the choice $\\theta_R=45^0$\ngives smaller effects of nonuniversality in the leptonic\npartial widths of the $Z$ boson.\nIf we had chosen $\\theta_R=-45^0$, we would have obtained much\nstronger combined bounds on the mass parameters and $L$-violating\nmixings of the LRSM.\n\nOne may get the impression that new-physics effects can be\nminimized by selecting $\\theta_R$ to lie in a specific range.\nThis is, however, not true, since the universality-breaking\nparameter $\\Delta{\\cal A}_{l_1l_2}$ will play a complementary r\\^ole\nas is shown in Figs.~4(a)--(d). In Fig.~4, we list the results\nafter adding both contributions coming from $SU(2)_L$ and $SU(2)_R$\ngauge sectors.\nThus, we may be sensitive up to $m_N\\hspace{-0.1cm}\\stackrel{\\displaystyle <}{\\sim} 1.5$~TeV for\n$(s^{\\nu_\\tau}_L)^2=0.04$ and $(s^{\\nu_e}_L)^2=0.01$ (see Fig.~4(a)).\nIn Fig.~4(b), we display the decoupling effect of a very heavy $W_R$.\nIn Fig.~4(c), we find again the logarithmic enhancement caused by\nthe nondegeneracy between $W_R^\\pm$ and $h^\\pm$.\nIn addition, it should be noted that interesting phenomenology can\nonly arise for relatively light $W_R$ bosons,\n{\\em i.e.}~$M_R\\hspace{-0.1cm}\\stackrel{\\displaystyle <}{\\sim} 1$~TeV. The latter observation can also be verified\nfrom Fig.~4(d), in which $\\Delta{\\cal A}$ is drawn as a function of $\\theta_R$\nfor $M_R=0.4,\\ 0.6,$ and 0.8 TeV. Furthermore, one can easily recognize the\ncomplementary r\\^ole that $B(Z\\to l_1l_2)$, $U_{br}$, and $\\Delta{\\cal A}$\nplay as far as $\\theta_R$ is concerned, when comparing Figs.~2(d), 3(d), and\n4(d). For example, the choice $\\theta_R=-45^0$ would make\n$\\Delta{\\cal A}$ more difficult to observe, whereas $U_{br}$ becomes larger\nfor this value of $\\theta_R$. Of course,\nscenarios where $M_R$ is at the TeV scale may not be\ncompatible with $K_L-K_S$ phenomenology if we assume an exact\nleft-right symmetry in the Yukawa sector of the model. Nevertheless,\nin LRSMs that possess nonmanifest or pseudomanifest left-right\nsymmetry, such a constraint is not valid any longer~\\cite{LSS}.\n\nIn the following, we will try to address the question whether\nthere exist possibilities of inducing positive contributions\nto $R_b$ within our LRSM. As has already been noticed\nin Section 3, only positive contributions to $R_b$ are of\npotential interest, which will help to achieve a better agreement between\ntheoretical prediction and the experimental value of $R_b$. In LRSM,\nwe first consider the Feynman graphs 1(m) and 1(n), where the external\nleptons are replaced by $b$-quarks and virtual down-type quarks are running\nin the place of charged leptons. The interaction\nLagrangians of the flavour-changing scalars $\\phi^{0r}_2$ and $\\phi^{0i}_2$\nwith the $d,\\ s,\\ b$ quarks can be obtained by Eq.~(A.17) after making\nthe obvious replacements. These couplings are enhanced, as they are\nproportional to the top-quark mass. In fact, the flavour-changing scalars\ngenerate effective $Zb\\bar{b}$ couplings of both V--A and V+A nature.\nIn the limit $M_{\\phi^{0r}_2},\\ M_{\\phi^{0i}_2}\\gg M_Z$,\nthe effective $Zb\\bar{b}$ couplings take the simple form\n\\begin{eqnarray}\n\\Re e e(\\Gamma^R_{bb}(m_t)-\\Gamma^R_{bb}(0)) &=&  \\frac{1}{8} |V^R_{tb}|^2\n\\frac{m^2_t}{M^2_W}\\, \\left( \\,\n\\frac{\\lambda_S +\\lambda_I}{2(\\lambda_S-\\lambda_I)}\\,\n\\mbox{ln}\\frac{\\lambda_S}{\\lambda_I}\\, -\\, 1\\, \\right), \\label{RbL}\\\\\n\\Re e e(\\Gamma^L_{bb}(m_t)-\\Gamma^L_{bb}(0)) &=& - \\frac{1}{8} |V^L_{tb}|^2\n\\frac{m^2_t}{M^2_W}\\, \\left( \\, \\frac{\\lambda_S +\\lambda_I}{2(\\lambda_S\n-\\lambda_I)}\\, \\mbox{ln}\\frac{\\lambda_S}{\\lambda_I}\\, -\\, 1\\, \\right),\n\\label{RbR}\n\\end{eqnarray}\nwhere $\\lambda_S$ and $\\lambda_I$ are defined in Appendix B after Eq.~(B.1).\nThe analytic function in the parentheses of the r.h.s.\\ of\nEqs.~(\\ref{RbL}) and (\\ref{RbR}) is always positive and equals zero\nwhen the two scalars $\\phi^{0r}_2,\\ \\phi^{0i}_2$ are degenerate.\nSubstituting Eqs.~(\\ref{RbL}) and (\\ref{RbR}) into Eq.~(\\ref{Rbnew}),\none easily finds that the SM value of $R_b$ is further decreased.\nThis leads automatically to the restriction\n\\begin{equation}\nM_{\\phi^{0r}_2}\\ \\simeq\\  M_{\\phi^{0i}_2}. \\label{phi}\n\\end{equation}\nThe mass relation~(\\ref{phi}) has been used throughout our numerical\nestimates.\n\nAnother place that may lead to positive contributions to $R_b$ are\ndue to diagrams similar to Figs.~1(h) and 1(d). Indeed, an analogous\ncalculation gives\n\\begin{equation}\n\\Re e e(\\Gamma^R_{bb}(m_t)-\\Gamma^R_{bb}(0)) \\ =\\ -\\frac{1}{4} |V^R_{tb}|^2\n\\frac{m^2_t}{M^2_W}\\, s^2_\\beta c^2_\\beta\\,\n\\left(\\, \\frac{\\lambda_h +\\lambda_R}{2(\\lambda_h\n-\\lambda_R)}\\, \\mbox{ln}\\frac{\\lambda_h}{\\lambda_R}\\, -\\, 1\\, \\right).\n\\label{Rbh+}\n\\end{equation}\nHowever, the l.h.s.\\ of Eq.~(\\ref{Rbh+}) is proportional to $s^2_\\beta=\nM^2_W\/M^2_R$ yielding a rather small effect. If we insist in cancelling\nthe negative SM vertex correction $\\nabla_b^{SM}(m_t)$\nthrough the contribution (4.4), we find the highly unnatural mass ratio\n\\begin{displaymath}\n\\frac{M_h}{M_R}\\ \\simeq \\ e^{70}.\n\\end{displaymath}\nThe latter also demonstrates the difficulty of obtaining radiatively\npositive contributions to $R_b$ within the LRSM. In Ref.~\\cite{Rizzo},\nit has been shown that $Z_L-Z_R$ mixing effects could help in producing\ncontributions of either sign to $R_b$. We will not pursue this\ntopic here.\n\n\\section{Conclusions}\n\\indent\n\nWe have shown that lepton-flavour violating $Z$-boson decays,\nlepton universality in the decays $Z\\to l\\bar{l}$, and universality\nof lepton asymmetries at LEP\/SLC represent a set of complementary\nobservables and can hence impose severe limitations\non model-building in the leptonic sector. For our illustrative purposes,\nwe have considered a LRSM with two-generation mixing. We have found that the\nobservables $B(Z\\to l_1l_2)$, $U_{br}^{l_1l_2}$, and $\\Delta{\\cal A}_{l_1l_2}$\nare sensitive to different parameter-space regions of this minimal scenario.\nFor instance, if $(s^{\\nu_\\tau}_L)^2=0.03$, $(s^{\\nu_e}_L)^2=0.01$,\n$M_R=0.4$ TeV, and $M_h=30$ TeV, then heavy neutrinos are found to\nhave masses that do not exceed 2 TeV for any value of the Cabbibo-type\nangle $\\theta_R$. On the other hand, constraints on new physics from $R_b$\nprefer scenarios, in which flavour-changing scalars are degenerate\nin mass.\n\nIt may be worth remarking again the fact that LRSMs\ncan naturally predict $U_{br}\\simeq 0$, for some choice of parameters,\nwhich could naively be interpreted that lepton universality is preserved\nin nature. As has been shown in this paper, universality violation\ncan manifest itself in lepton asymmetries $\\Delta{\\cal A}_{l_1l_2}$ as well.\nThis is, however, not an accidental feature of the LRSM but may have\na general applicability to unified models, such as supersymmetric\nextensions of the SM~\\cite{BP}. In general,\nsuch theories can naturally generate both nonuniversal V--A and V+A\n$Zf\\bar{f}$-couplings yielding effects that may be detected by\ncurrent experiments at LEP and SLC.\n\n\n\n\\vskip2cm\n\\noindent\n{\\bf Acknowledgements.} I wish to thank Richardo Barbieri, Jose Bernab\\'eu,\nRoger Phillips, and John Thompson for discussions and comments.\n\n\\newpage\n\\setcounter{section}{0}\n\\def\\Alph{section}.\\arabic{equation}{\\Alph{section}.\\arabic{equation}}\n\\begin{appendix}\n\\setcounter{equation}{0}\n\\section{Feynman rules in the LRSM}\n\\indent\nAlthough some of the Feynman rules required in our problem were given in\nRef.~\\cite{IL,GGMKO}, we list all the relevant Feynman rules and Lagrangians\ngoverning the interactions of the gauge and Higgs bosons with leptons and\nneutrinos, as well as the trilinear couplings of the bosons. The covariant\nderivative acts on the Higgs multiplets as follows:\n\\begin{eqnarray}\nD_\\mu\\Phi &=& \\partial_\\mu\\Phi + i\\frac{g_L}{2} \\vec{\\sigma}\\vec{W}_{L\\mu}\\Phi\n- i\\frac{g_R}{2} \\Phi \\vec{\\sigma} \\vec{W}_{R\\mu},\\\\\nD_\\mu\\Delta_{R,L} &=& \\partial_\\mu\\Delta_{R,L}  + i\\frac{g_{R,L}}{2}\n[\\vec{\\sigma} \\vec{W}_{R,L\\mu},\\Delta_{R,L}]\n+ig' B_\\mu\\Delta_{R,L},\n\\end{eqnarray}\nwhere $\\sigma_i$ are the known $2\\times 2$ Pauli matrices, and $g_L$ ($g_R$)\nare the $SU(2)_L$ ($SU(2)_R$) weak coupling constants which will be\nset equal to $g_w=g_L=g_R$ ($g_w$ is the usual $SU(2)_L$ weak coupling constant\nin the SM).\nTo facilitate our computational task, we will further assume that the\ncorresponding neutral gauge boson $Z_L$ is the $Z$  of the SM\nto a good approximation.\nAlso, we will list the novel LRSM interactions together with the SM\ncouplings in order to avoid possible ambiguities between relative signs.\n\nThe trilinear couplings of gauge, Higgs, and would-be Goldstone bosons\nmay therefore be obtained by (all momenta flow into the vertex)\n\\begin{eqnarray}\nZ_{L\\nu}(r)W_{L\\lambda}^+(p)W_{L\\mu}^-(q)&:&-ig_wc_w\nf_{\\mu\\nu\\lambda}(r,q,p),\\\\\nZ_{L\\nu}(r)W_{R\\lambda}^+(p)W_{R\\mu}^-(q)&:&\n                             ig_w \\frac{s^2_w}{c_w}f_{\\mu\\nu\\lambda}(r,q,p),\\\\\nZ_{L\\nu}(r)G_L^+(p)W_{L\\mu}^-(q)&:& -ig_w M_W\\frac{s^2_w}{c_w}\\mbox{g}^{\\mu\\nu}, \\\\\nZ_{L\\nu}(r)G_R^+(p)W_{R\\mu}^-(q)&:& ig_w M_W\\frac{s^2_\\beta-s^2_w}{s_\\beta c_w}\n                                                               \\mbox{g}^{\\mu\\nu}, \\\\\nZ_{L\\nu}(r)h^+(p)W_{R\\mu}^-(q)&:& -ig_w M_W\\frac{c_\\beta}{c_w}\\mbox{g}^{\\mu\\nu}, \\\\\nZ_{L\\mu}(r)G^+_L(p)G_L^-(q)&:& -i\\frac{g_w}{2c_w} (1-2s^2_w) (p-q)_\\mu, \\\\\nZ_{L\\mu}(r)G^+_R(p)G_R^-(q)&:& -i\\frac{g_w}{2c_w} (s^2_\\beta-2s^2_w)\n                                                                  (p-q)_\\mu, \\\\\nZ_{L\\mu}(r)h^+(p)h^-(q)&:& -i\\frac{g_w}{2c_w} (c^2_\\beta-2s^2_w)\n                                                                  (p-q)_\\mu, \\\\\nZ_{L\\mu}(r)G^\\pm_R(p)h^\\mp(q)&:& \\mp i\\frac{g_w}{2c_w} s_\\beta c_\\beta\n                                                                 (p-q)_\\mu, \\\\\nZ_{L\\mu}(r)\\phi^{0r}_2(p)\\phi^{0i}_2(q)&:& \\frac{g_w}{2c_w} (p-q)_\\mu, \\\\\nZ_{L\\mu}(r)\\delta^{++}_R(p)\\delta^{--}_R(q)&:& 2i\\frac{g_w}{c_w}s^2_w(p-q)_\\mu.\n\\end{eqnarray}\nHere, we have defined $s_\\beta = \\sqrt{1-c^2_\\beta}=M_W\/M_R$ and\nthe Lorentz tensor $f_{\\mu\\nu\\lambda}(r,q,p)=(r-q)_\\lambda\\mbox{g}_{\\mu\\nu} +\n(q-p)_\\nu \\mbox{g}_{\\lambda\\mu} + (p-r)_\\mu\\mbox{g}_{\\nu\\lambda}$.\n\nThe corresponding couplings of the gauge, Higgs, and would-be Goldstone\nbosons to the charged leptons and neutrinos can be read off from the\nLagrangians\n\\begin{eqnarray}\n{\\cal L}^{W_R}_{int} &=& -\\frac{g_w}{\\sqrt{2}} W_R^{-\\mu}\\, B^R_{li}\\\n\\bar{l}\\gamma_\\mu \\mbox{P}_R n_i\\quad +\\quad \\mbox{H.c.},\\\\\n{\\cal L}^{G_R^-}_{int} &=& -\\frac{g_w}{\\sqrt{2}M_W}\\, s_\\beta \\, G_R^{-}\\, B^R_{li}\\\n\\bar{l}\\Big[ m_l\\mbox{P}_R - m_{n_i}\\mbox{P}_L\\Big] n_i\\quad +\\quad \\mbox{H.c.},\\\\\n{\\cal L}^{h^-}_{int} &=& \\frac{g_w}{\\sqrt{2}M_W}\\, c_\\beta\\, h^{-}\\,\n\\bar{l}\\left[ B^R_{li} m_l\\mbox{P}_R - B^R_{lj}\\left(\\delta_{ji}-\n\\frac{C^{R\\ast}_{ji}}{c^2_\\beta} \\right)m_{n_j}\\mbox{P}_L\\right]n_i\\quad\n+\\quad \\mbox{H.c.},\\ \\\\\n{\\cal L}^{\\phi_2^0}_{int} &=& -\\frac{g_w}{2M_W} \\phi_2^{0r}\\,\n\\bar{l}_1\\Big[ B^L_{l_1j}m_{n_j}B^{R\\ast}_{l_2j}\\mbox{P}_R +\nB^R_{l_1j}m_{n_j}B^{L\\ast}_{l_2j}\\mbox{P}_L\\Big]l_2\\nonumber\\\\\n&&-\\frac{ig_w}{2M_W} \\phi_2^{0i}\\,\n\\bar{l}_1\\Big[ B^L_{l_1j}m_{n_j}B^{R\\ast}_{l_2j}\\mbox{P}_R -\nB^R_{l_1j}m_{n_j}B^{L\\ast}_{l_2j}\\mbox{P}_L\\Big]l_2,\\\\\n{\\cal L}^{\\delta_R^{++}}_{int} &=& \\frac{g_w}{2\\sqrt{2}M_W} \\frac{s_\\beta}{c_\\beta}\n\\delta_R^{++}\\,\n\\bar{l}_1^C\\, B^{R\\ast}_{l_1j}m_{n_j}B^{R\\ast}_{l_2j}\\mbox{P}_R\\, l_2 \\quad\n+\\quad \\mbox{H.c.},\n\\end{eqnarray}\nwhere the mixing matrices $B^R$ and $C^R$ are defined in Section 2.\nThe couplings of $Z_L\\ (\\equiv Z)$ to Majorana neutrinos\nmay be found in Ref.~\\cite{ZPC}.\n\n\n\\setcounter{equation}{0}\n\\section{The nonoblique {\\boldmath $Zl\\bar{l}$} vertex}\n\\indent\n\nWe analytically evaluate the loop amplitudes in the limit of vanishing\nexternal lepton masses. We adopt dimensional regularization in conjunction\nwith the reduction algorithm of Ref.~\\cite{PV}. Unlike the metric notation\nof Ref.~\\cite{PV}, we use the Minskowskian metric,\n$\\mbox{g}^{\\mu\\nu}=\\mbox{diag}(1,1,...,-1)$.\n\nThe nonoblique effective $Zll'$ vertex function is similar to the one\nobtained in~\\cite{KPS}. Its analytic form is given by (summation over\nrepeated indices implied)\n\\begin{eqnarray}\n\\Gamma^L_{ll'} & =& B^{L}_{li}B^{L\\ast}_{l'i} \\Bigg\\{\\delta_{ij}\n\\Bigg[c^2_w\\lambda_Z\\Big(C_{11}(\\lambda_i,1,1)+C_{23}(\\lambda_i,1,1)-C_{22}(\\lambda_i,1,1)\\Big)\n\\nonumber\\\\\n&&+ 6c^2_wC_{24}(\\lambda_i,1,1)-s^2_w\\lambda_i C_0(\\lambda_i,1,1) + \\frac{1}{2}(1-2s^2_w)\n\\Big(\\lambda_i C_{24}(\\lambda_i,1,1) \\nonumber\\\\\n&&+ \\frac{1}{2}\\lambda_i B_1(0,\\lambda_i,1) + B_1(0,\\lambda_i,1)\\Big)\\Bigg]\\nonumber\\\\\n&&+ C^{L}_{ij}\\Bigg[ -C_{24}(1,\\lambda_i,\\lambda_j)-\\frac{1}{2}\\lambda_Z\\Big( C_0(1,\\lambda_i,\\lambda_j)\n+ C_{11}(1,\\lambda_i,\\lambda_j)\\nonumber\\\\\n&&+ C_{23}(1,\\lambda_i,\\lambda_j)-C_{22}(1,\\lambda_i,\\lambda_j)\\Big)\n- \\frac{1}{4}\\lambda_i\\lambda_j C_0(1,\\lambda_i,\\lambda_j)\\Bigg]\\nonumber\\\\\n&&+ \\frac{1}{2}C^{L\\ast}_{ij}\\sqrt{\\lambda_i\\lambda_j}\\Bigg[ C_0(1,\\lambda_i,\\lambda_j)+\n\\frac{1}{2}\\lambda_Z \\Big(C_{23}(1,\\lambda_i,\\lambda_j)-C_{22}(1,\\lambda_i,\\lambda_j)\\Big)+\nC_{24}(1,\\lambda_i,\\lambda_j)\\Bigg]\\nonumber\\\\\n&& +\\frac{1}{4}C^{R}_{ij}\\sqrt{\\lambda_i\\lambda_j}\\Bigg[2C_{24}(0,\\lambda_S,\\lambda_I)-\nC_{24}(\\lambda_S,0,0)-C_{24}(\\lambda_I,0,0)+\\frac{1}{2}\\nonumber\\\\\n&&+ s^2_w\\Big( B_1(0,0,\\lambda_S) + B_1(0,0,\\lambda_I)\\Big) -\\, \\frac{1}{2}\\lambda_Z\\Big(\nC_{23}(\\lambda_S,0,0)-C_{22}(\\lambda_S,0,0) \\nonumber\\\\\n&&+ C_{23}(\\lambda_I,0,0)-C_{22}(\\lambda_I,0,0)\\Big) \\Bigg]\\, \\Bigg\\}\\, ,\n\\end{eqnarray}\nwhere $\\lambda_i=m^2_i\/M^2_W$, $\\lambda_Z=M^2_Z\/M^2_W$, $\\lambda_S=M^2_{\\phi^{0r}_2}\/M^2_W$,\nand $\\lambda_I=M^2_{\\phi^{0i}_2}\/M^2_W$. Note that there is a contribution\nproportional to $C^R_{ij}$ that originates solely from the Higgs sector\nof the LRSM. In the notation of~\\cite{Bernd}, the first three of the\nsix arguments of the $C$ functions are always evaluated at $(0,\\lambda_Z,0)$.\n\nIn the LRSM, virtual neutrinos and Higgs scalars induce a nonuniversal $Z$\nboson coupling to right-handed charged leptons, $\\Gamma^R$, as\nshown in Fig.~1. The contributions of the individual graphs  to $\\Gamma^R$\nare listed below\n\\begin{eqnarray}\n\\Gamma^R_{ll'}(a) &=& B^R_{li}B^{R\\ast}_{l'i} s^2_w\\Bigg[\\lambda_Z\\Big(\nC_{22}(\\lambda_i,\\lambda_R,\\lambda_R) - C_{23}(\\lambda_i,\\lambda_R,\\lambda_R) -C_{11}(\\lambda_i,\\lambda_R,\\lambda_R) \\Big)\\nonumber\\\\\n&&-6C_{24}(\\lambda_i,\\lambda_R,\\lambda_R)\\Bigg],\\\\\n\\Gamma^R_{ll'}(b+c) &=& B^R_{li}B^{R\\ast}_{l'i} (s^2_w-s^2_\\beta)\\lambda_i\nC_0(\\lambda_i,\\lambda_R,\\lambda_R),\\\\\n\\Gamma^R_{ll'}(d) &=& \\frac{1}{2}B^R_{li}B^{R\\ast}_{l'i} s^2_\\beta\n(s^2_\\beta-2s^2_w)\\lambda_i C_{24}(\\lambda_i,\\lambda_R,\\lambda_R),\\\\\n\\Gamma^R_{ll'}(e+f) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\n(\\delta_{ij}s^2_\\beta\\lambda_i -\\sqrt{\\lambda_i\\lambda_j} C^L_{ij})\\Big( C_0(\\lambda_i,\\lambda_R,\\lambda_h)+\nC_0(\\lambda_j,\\lambda_R,\\lambda_h)\\Big),\\\\\n\\Gamma^R_{ll'}(g) &=& \\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\\sqrt{\\lambda_k\\lambda_n}\\left(\n1-\\frac{2s^2_w}{c^2_\\beta}\\right) (s^2_\\beta\\delta_{ki} -\nC^L_{ki})(s^2_\\beta\\delta_{in}-C^L_{in})C_{24}(\\lambda_i,\\lambda_h,\\lambda_h),\\\\\n\\Gamma^R_{ll'}(h+i) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\\sqrt{\\lambda_i\\lambda_j}\ns^2_\\beta (s^2_\\beta\\delta_{ij} -C^L_{ij})\\Big( C_{24}(\\lambda_i,\\lambda_R,\\lambda_h)+\nC_{24}(\\lambda_j,\\lambda_R,\\lambda_h)\\Big),\\\\\n\\Gamma^R_{ll'}(j) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\\Bigg\\{\nC^{L\\ast}_{ij}\\Bigg[ 1 - 2 C_{24}(\\lambda_R,\\lambda_i,\\lambda_j) -\\lambda_Z\\Big(\nC_0(\\lambda_R,\\lambda_i,\\lambda_j) + C_{11}(\\lambda_R,\\lambda_i,\\lambda_j)\\nonumber\\\\\n&& +C_{23}(\\lambda_R,\\lambda_i,\\lambda_j) - C_{22}(\\lambda_R,\\lambda_i,\\lambda_j) \\Big) \\Bigg]\\ +\nC^L_{ij}\\sqrt{\\lambda_i\\lambda_j}C_0(\\lambda_R,\\lambda_i,\\lambda_j) \\Bigg\\},\\\\\n\\Gamma^R_{ll'}(k) &=& -\\frac{1}{4}B^R_{li}B^{R\\ast}_{l'j}s^2_\\beta\n\\sqrt{\\lambda_i\\lambda_j} \\Bigg\\{ C^L_{ij}\\Bigg[ 2C_{24}(\\lambda_R,\\lambda_i,\\lambda_j) -\\frac{1}{2}+\\lambda_Z\\Big(\nC_{23}(\\lambda_R,\\lambda_i,\\lambda_j)\\nonumber\\\\\n&&- C_{22}(\\lambda_R,\\lambda_i,\\lambda_j) \\Big) \\Bigg]\n- C^{L\\ast}_{ij}\\sqrt{\\lambda_i\\lambda_j}C_0(\\lambda_R,\\lambda_i,\\lambda_j) \\Bigg\\},\\\\\n\\Gamma^R_{ll'}(l) &=& -\\frac{1}{4}B^R_{lk}B^{R\\ast}_{l'n}\n\\sqrt{\\lambda_k\\lambda_n} \\frac{1}{c^2_\\beta}(s^2_\\beta\\delta_{ki}-C^L_{ki})\n(s^2_\\beta\\delta_{nj}- C^L_{jn} )\n\\Bigg\\{ C^L_{ij}\\Bigg[ 2C_{24}(\\lambda_h,\\lambda_i,\\lambda_j) -\\frac{1}{2}\\nonumber\\\\\n&&+\\lambda_Z\\Big( C_{23}(\\lambda_h,\\lambda_i,\\lambda_j) - C_{22}(\\lambda_h,\\lambda_i,\\lambda_j) \\Big) \\Bigg]\n- C^{L\\ast}_{ij}\\sqrt{\\lambda_i\\lambda_j}C_0(\\lambda_h,\\lambda_i,\\lambda_j) \\Bigg\\}\\ \\\\\n\\Gamma^R_{ll'}(m+n) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\nC^L_{ij}\\sqrt{\\lambda_i\\lambda_j}C_{24}(0,\\lambda_S,\\lambda_I),\\\\\n\\Gamma^R_{ll'}(o+p) &=& \\frac{1}{8}B^R_{li}B^{R\\ast}_{l'j}\nC^L_{ij}(1-2s^2_w)\\sqrt{\\lambda_i\\lambda_j} \\Bigg[ 2C_{24}(\\lambda_S,0,0)+ 2C_{24}(\\lambda_I,0,0)\n         -1\\nonumber\\\\\n&& + \\lambda_Z\\Big( C_{23}(\\lambda_S,0,0)-C_{22}(\\lambda_S,0,0)+\nC_{23}(\\lambda_I,0,0)-C_{22}(\\lambda_I,0,0) \\Big) \\Bigg], \\\\\n\\Gamma^R_{ll'}(q) &=& \\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\nC^{R\\ast}_{ij}s^2_w\\frac{s^2_\\beta}{c^2_\\beta}\n\\sqrt{\\lambda_i\\lambda_j}C_{24}(0,\\lambda_\\delta,\\lambda_\\delta),\\\\\n\\Gamma^R_{ll'}(r) &=& -\\frac{1}{8}B^R_{li}B^{R\\ast}_{l'j}\nC^{R\\ast}_{ij}s^2_w\\frac{s^2_\\beta}{c^2_\\beta}\n\\sqrt{\\lambda_i\\lambda_j} \\Bigg[ 2C_{24}(\\lambda_\\delta,0,0)- \\frac{1}{2}+\\lambda_Z\\Big(\nC_{23}(\\lambda_\\delta,0,0)\\nonumber\\\\\n&&-C_{22}(\\lambda_\\delta,0,0) \\Big) \\Bigg],\n\\end{eqnarray}\nwhere $\\lambda_h=M^2_{h^+}\/M^2_W$ and $\\lambda_\\delta=M^2_{\\delta^{++}_R}\/M^2_W$.\nIn addition to the irreducible three-point functions, we should take\nwave-function renormalization constants into account (Figs.~1(A)--(F)).\nThese additional nonuniversal corrections generated by the selfenergies\nare calculated to give\n\\begin{eqnarray}\n\\Gamma^R_{ll'}(A) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\ns^2_w \\Big(1+2B_1(0,\\lambda_i,\\lambda_R)\\Big), \\\\\n\\Gamma^R_{ll'}(B) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\ns^2_w s^2_\\beta\\lambda_i B_1(0,\\lambda_i,\\lambda_R), \\\\\n\\Gamma^R_{ll'}(C) &=& -\\frac{1}{2}B^R_{li}B^{R\\ast}_{l'j}\n\\frac{s^2_w}{c^2_\\beta} \\sqrt{\\lambda_k\\lambda_n} (s^2_\\beta\\delta_{ki} -C^L_{ki})\n(s^2_\\beta\\delta_{in} -C^L_{in})B_1(0,\\lambda_i,\\lambda_h),\\\\\n\\Gamma^R_{ll'}(D+E) &=& -\\frac{1}{4}B^R_{li}B^{R\\ast}_{l'j}C^L_{ij}\ns^2_w \\sqrt{\\lambda_i\\lambda_j} \\Big( B_1(0,0,\\lambda_S) + B_1(0,0,\\lambda_I) \\Big),\\\\\n\\Gamma^R_{ll'}(F) &=& \\frac{1}{8}B^R_{li}B^{R\\ast}_{l'j}C^{R\\ast}_{ij}\ns^2_w\\frac{s^2_\\beta}{c^2_\\beta} \\sqrt{\\lambda_i\\lambda_j} B_1(0,0,\\lambda_\\delta).\n\\end{eqnarray}\nThe sum of Eqs.~(B.2)--(B.19) should be free from UV divergences,\nwhen $l\\neq l'$. This can easily be verified by employing the\nidentities that the mixing matrices $B^{L,R}$ and $C^{L,R}$\nobey (see also discussion in Section 2). An ultimate check for\nthe correctness of our analytic results is the vanishing of\nall terms involving $s^2_w$ in the limit $\\lambda_Z\\to 0$, due to\nelectromagnetic gauge invariance.\n\n\n\n\\end{appendix}\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn Ref.~\\cite{2N3N} a new formulation for the 2N and 3N bound states in three dimensions has been proposed.\nIn this technique momentum vectors are taken as variables, avoiding a traditional partial-wave decomposition. In addition\nspin operators occurring as scalar products of spin and momentum vectors - shortly called spin-momentum\noperators - are evaluated analytically by means of trace operations. In this approach a NN force is\nemployed using its most general operator structure, i.e. as  sum of 6 spin-momentum operators multiplied with\nscalar functions of momenta. A spin-\\linebreak momentum operator representation is used as well for\nthe 2N and 3N bound states, as in Refs.~\\cite{deut} and \\cite{oph3}, respectively.\n\nWe extend the technique developed in Ref.~\\cite{2N3N} to NN scattering. This would be an alternative to other\nthree-\\linebreak dimensional approach formulated in a momentum-helicity basis~\\cite{NN3D}. In addition we\nintroduce a new set of spin-\\linebreak momentum operators different from the one used in \\linebreak\nRef.~\\cite{2N3N}. We find one of the spin-momentum operators in Ref.~\\cite{2N3N} violates time reversal \nand, therefore, has to be multiplied with a time-reversal violating scalar function. Here we prefer to work with \noperators, which are also invariant with respect to time reversal.   \nThe idea is to apply the general operator structure not only to the NN force but also to\nthe NN T-matrix. The goal is then to find the scalar functions in the expansion of the T-matrix into the\nspin-momentum operators. First, we insert the spin-momentum operators expansions of the NN interaction and\nT-matrix into the Lippmann-Schwinger equation. Next by analytical evaluation we remove the spin dependence\nyielding finally a set of coupled equations for the scalar functions of the T-matrix. Finally we connect\nthe T-matrix to the anti-symmetrized scattering amplitude parameterized by the \\linebreak Wolfenstein parameters.\n\n\\section{Formulation}\\label{formulation}\n\n\\subsection{The general operator structure of NN potential}\n\nThe general operator structure of NN potential reads\n\\begin{equation}\nV^ { t m_t}({\\bf p'}, {\\bf p}) = \\sum_{j=1}^ 6 v_j^ { t m_t} ({\\bf p'}, {\\bf p}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) , \\label{vinw}\n\\end{equation}\nwith $V^ { t m_t}({\\bf p'}, {\\bf p})$ being the NN potential projected on the NN total isospin states $ \\mid t m_t\\rangle$ as\n\\begin{equation}\nV^ { t m_t}({\\bf p'}, {\\bf p}) = \\langle  t m_t \\mid V({\\bf p'}, {\\bf p}) \\mid t m_t\\rangle .\n\\end{equation}\nThe scalar functions $v_j^ { t m_t} ({\\bf p'}, {\\bf p})$ depend only on the vector momenta.\nthe\n$w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})$ are a set of spin-momentum operators,\n\\begin{eqnarray}\nw_1 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})&  = &  1\\cr\nw_2 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})&  = & {\\mbox{\\boldmath$\\sigma$}}_1 \\cdot {\\mbox{\\boldmath$\\sigma$}}_2\\cr\nw_3 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p)}&  = & i \\; ( {\\mbox{\\boldmath$\\sigma$}}_1\n+ {\\mbox{\\boldmath$\\sigma$}}_2 ) \\cdot ( {\\bf p} \\times {\\bf p'})\\cr\nw_4 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})&  = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot ( {\\bf p} \\times {\\bf p'}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot ( {\\bf p} \\times {\\bf p'})\\cr\nw_5 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})&  = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot  ({\\bf p'} + {\\bf p}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot  ({\\bf p'} + {\\bf p})\\cr\nw_6 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})&  = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot ( {\\bf p'} - {\\bf p}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot  ( {\\bf p'} - {\\bf p}) ,\n\\end{eqnarray}\nwhich is time-reversal invariant. As an example a leading order (LO) chiral NN potential is given as~\\cite{evgeny.report} \\pagebreak\n\\begin{eqnarray}\nV_{LO}({\\bf p'}, {\\bf p}) & = & -\\frac{1}{(2\\pi)^3}  \\frac{g_A^2}{4 F_\\pi^2} \\frac{ w_6 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})}{( {\\bf p'} - {\\bf p})^2 +M_\\pi^2} {\\mbox{\\boldmath$\\tau$}_1} \\cdot {\\mbox{\\boldmath$\\tau$}_2}  \\cr\n&& + \\frac{C_S}{(2\\pi)^3}w_1 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& + \\frac{C_T}{(2\\pi)^3}w_2 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) .\n\\end{eqnarray}\n\n\\subsection{The deuteron}\n\nWe briefly describe the formulation for the deuteron. The deuteron has total spin $1$ and isospin $0$.\nIn spin-momentum operator representation the deuteron state is given as~\\cite{deut}\n\\begin{equation}\n\\Psi_{m_d}({\\bf p}) = \\langle {\\bf p} | \\Psi_{m_d}\\rangle = \\sum_{ k=1}^2 \\phi_k(p) \\;  b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) | 1 m_d \\rangle , \\label{deutwf}\n\\end{equation}\nwhere $| 1 m_d \\rangle$ is the total-spin state with magnetic quantum number $m_d$, $\\phi_k(p)$ scalar functions depending on the magnitude of momenta only, and $b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})$ spin-momentum operators given as\n\\begin{eqnarray}\nb_1( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) & = & 1 \\cr\nb_2( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) & = & {\\mbox{\\boldmath$\\sigma$}}_1 \\cdot {\\bf p} \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot {\\bf p} - \\frac{1}{3} p^2 .\n\\end{eqnarray}\nThe scalar functions $\\phi_k(p)$ are connected to the standard partial-wave projected deuteron\n s-wave $\\psi_0(p)$ and d-wave $\\psi_2(p)$ by~\\cite{deut}\n\\begin{eqnarray}\n\\psi_0 (p) & = & \\phi_1 (p) \\cr\n\\psi_2 (p) & = & \\frac{4 p^2} { 3 \\sqrt{2}} \\, \\phi_2 (p)  .\n\\end{eqnarray}\nInserting $\\Psi_{m_d}({\\bf p})$ of Eq.~(\\ref{deutwf}) and $V^ { t m_t}({\\bf p'}, {\\bf p})$ of Eq.~(\\ref{vinw}) into the Schr\\\"odinger equation  for the deuteron in integral form,\n\\begin{equation}\n\\Psi_{m_d}({\\bf p}) = \\frac{1}{ E_d -\\frac{p^2}{m}} \\int d^3 p' V^ {00}({\\bf p}, {\\bf p'}) \\Psi_{m_d}({\\bf p'}),\n\\end{equation}\nyields\n\\begin{eqnarray}\n\\lefteqn{\\sum_{ k=1}^2 \\phi_k(p) \\;  b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) | 1 m_d \\rangle} && \\cr\n&& \\qquad = \\frac{1}{ E_d -\\frac{p^2}{m}} \\int d^3 p' \\sum_{j=1}^ 6 v_j^ {00} ({\\bf p}, {\\bf p'}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}, {\\bf p'}) \\cr\n&& \\qquad \\quad \\sum_{ k'=1}^2 \\phi_{k'}(p) \\;  b_{k'}( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}) | 1 m_d \\rangle . \\label{dschr}\n\\end{eqnarray}\n\nTo remove the spin dependence from Eq.~(\\ref{dschr}) we project Eq.~(\\ref{dschr}) on $\\langle 1 m_d | b_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})$ from the left and sum up over $m_d$. We obtain\n\\begin{eqnarray}\n\\sum_{ k=1}^2 A^ d_{i k}(p) \\phi_{k} (p) & = & \\frac{1}{ E_d-\\frac{p^2}{m}}\\int d^3 p' \\sum_{j=1}^6 v_j^ {00}( {\\bf p},{\\bf p'}) \\cr\n&& \\sum_{k'=1}^2 B^ d_{ijk'} ( {\\bf p},{\\bf p'})\\phi_{k'}(p') ,\n\\end{eqnarray}\nwhich is a set of two coupled equations for $\\phi_{k} (p)$, with $A^ d_{i k}(p)$ and $B^ d_{ijk'} ( {\\bf p},{\\bf p'})$ being defined as\n\\begin{equation}\nA^ d_{i k} ( p) \\equiv \\sum_{ m_d = -1}^{1 } \\langle 1 m_d|\nb_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})\n b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})| 1 m_d \\rangle \n\\end{equation}\n\\begin{eqnarray}\nB^ d_{ijk'} ( {\\bf p},{\\bf p'}) & \\equiv & \\sum_{ m_d = -1}^{1 } \\langle 1 m_d|\nb_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) w_j( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}, {\\bf p'}) \\cr\n&& \\qquad b_{k'}( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'})| 1 m_d \\rangle .\n\\end{eqnarray}\nThe functions $A^ d_{i k}(p)$ and $B^ d_{ijk'} ({\\bf p},{\\bf p'})$ are scalar functions of the vectors ${\\bf p}$ and ${\\bf p'}$, and need to be calculated only once.\nAs example we have e.g.\n\\begin{eqnarray}\nA^d_{11} (p) & = &  3\\cr\nA^d_{22} (p) & = &  \\frac{8}{3} p^4 \\cr\nB^d_{141} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p}\\times {\\bf p'})^2 \\cr\nB^d_{151} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p'}+ {\\bf p})^2 \\cr\nB^d_{161} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p'}- {\\bf p})^2 .\n\\end{eqnarray}\n\n\\subsection{The NN scattering}\n\nThe operator structure given in Eq.~(\\ref{vinw}) for NN force can also be applied to the NN T-matrix as\n\\begin{equation}\nT^ { t m_t}({\\bf p'}, {\\bf p}) = \\sum_{j=1}^ 6 t_j^ { t m_t} ({\\bf p'}, {\\bf p}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) , \\label{tinw}\n\\end{equation}\nwith $t_j^ { t m_t} ({\\bf p'}, {\\bf p})$ being the scalar functions to be found. Inserting both\nthe expansion in Eqs.~(\\ref{vinw}) and (\\ref{tinw}) into the Lippmann-Schwinger equation,\n\\begin{eqnarray}\nT^ { t m_t}({\\bf p'}, {\\bf p}) & = & V^ { t m_t}({\\bf p'}, {\\bf p}) \\cr\n&& + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\int d{\\bf p''} \\frac{V^ { t m_t}({\\bf p'}, {\\bf p''}) T^ { t m_t}({\\bf p''}, {\\bf p})}{p^2 + i\\epsilon -p''^2} , \\qquad\n\\end{eqnarray}\nwhere $\\mu$ is the reduced mass of the NN system, leads to\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k=1}^ 6 t_k^ { t m_t} ({\\bf p'}, {\\bf p}) \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})} && \\cr\n&& \\qquad = \\sum_{k=1}^ 6 v_k^ { t m_t} ({\\bf p'}, {\\bf p}) \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& \\qquad \\quad + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\int d{\\bf p''} \\frac{1}{p^2 + i\\epsilon -p''^2} \\cr\n&& \\qquad \\quad \\sum_{j=1}^ 6 v_j^ { t m_t} ({\\bf p'}, {\\bf p''}) \\; w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p''}) \\cr\n&& \\qquad \\quad \\sum_{k'=1}^ 6 t_{k'}^ { t m_t} ({\\bf p''}, {\\bf p}) \\; w_{k'}({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p''}, {\\bf p}) . \\label{ls}\n\\end{eqnarray}\nWe remove the spin dependence from Eq.~(\\ref{ls}) by multiplying from the left with $w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})$ and perform the trace. \nThis leads to \\pagebreak\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k = 1}^6 A_{ik}({\\bf p'}, {\\bf p}) t_k^{tm_t}({\\bf p'}, {\\bf p}) } && \\cr\n&& \\qquad = \\sum_{k = 1}^6 A_{ik}({\\bf p'}, {\\bf p}) v_k^{tm_t}({\\bf p'}, {\\bf p}) \\cr\n&& \\qquad \\quad + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\sum_{j,k' = 1}^6 \\int d{\\bf p''}\n   \\frac{1} {p^2 + i\\epsilon - p''^2} \\cr\n&& \\qquad \\quad B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})\n                v_j^{tm_t}({\\bf p'}, {\\bf p''}) t_{k'}^{tm_t}({\\bf p''}, {\\bf p}) , \\label{nnls}\n\\end{eqnarray}\nwhere $A_{ik}({\\bf p'}, {\\bf p})$ and $B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})$ are defined as\n\\begin{eqnarray}\nA_{ik}({\\bf p'}, {\\bf p}) & \\equiv & Tr \\{w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& \\quad \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})\\} \\\\\nB_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p}) & \\equiv & Tr \\{w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p''}) \\cr\n&& \\quad \\; w_{k'}({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p''}, {\\bf p})\\}\n\\end{eqnarray}\nAgain the functions\n $A_{ik}({\\bf p'}, {\\bf p})$ and $B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})$ are scalar functions of the momenta ${\\bf p}$ and ${\\bf p'}$, and need to be evaluated only once.\nAs example we show here,\n\\begin{eqnarray}\nA_{24}({\\bf p}', {\\bf p}) & = & A_{42}({\\bf p}', {\\bf p}) = 4 ({\\bf p} \\times {\\bf p}')^2 \\cr\nA_{56}({\\bf p}', {\\bf p}) & = & A_{65}({\\bf p}', {\\bf p}) = 4 (p'^2 - p^2)^2  \\cr\nB_{122}({\\bf p}', {\\bf p}'', {\\bf p}) & = & B_{212}({\\bf p}', {\\bf p}'', {\\bf p}) = B_{221}({\\bf p}', {\\bf p}'', {\\bf p}) = 12 \\cr\nB_{124}({\\bf p}', {\\bf p}'', {\\bf p}) & = & B_{214}({\\bf p}', {\\bf p}'', {\\bf p}) = 4 ({\\bf p} \\times {\\bf p}'')^2 \\cr\nB_{144}({\\bf p}', {\\bf p}'', {\\bf p}) & = & 4 \\{({\\bf p}'' \\times {\\bf p}') \\cdot ({\\bf p} \\times {\\bf p}'')\\}^2\n\\end{eqnarray}\nEquation (\\ref{nnls}) is a set of six coupled equations for $t_k^{tm_t}({\\bf p'}, {\\bf p})$, which can e.g. be solved\nas a system of linear equations.\n\nThe NN scattering observables can be calculated from the anti-symmetrized scattering amplitude $M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})$, \\linebreak which is defined as\n\\begin{eqnarray}\n\\lefteqn{M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})} && \\cr\n&& \\; \\equiv \\langle t m_t | \\langle m_1' m_2'| \\langle {\\bf p'}| M ( 1 - P_{12}) | {\\bf p}\\rangle | m_1 m_2\\rangle | t m_t\\rangle\n\\end{eqnarray}\nand can be parameterized by the Wolfenstein parameters $a^{t m_t}({\\bf p'},{\\bf p})$, $c^{t m_t}({\\bf p'},{\\bf p})$, $m^{t m_t}({\\bf p'},{\\bf p})$, $g^{t m_t}({\\bf p'},{\\bf p})$, $h^{t m_t}({\\bf p'},{\\bf p})$ \\linebreak as~\\cite{book}\n\\begin{eqnarray}\n\\lefteqn{M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})} && \\cr\n&& \\quad = a^ {tm_t}({\\bf p'},{\\bf p}) \\; \\langle m_1' m_2'| w_1( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad -  i \\frac{ c^ {tm_t}({\\bf p'},{\\bf p})}{ | {\\bf p} \\times {\\bf p'}|} \\;\n \\langle m_1' m_2'|w_3 ({\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})| m_1 m_2\\rangle\\cr\n&& \\qquad +  \\frac{ m^ {tm_t}({\\bf p'},{\\bf p})}{ | {\\bf p} \\times {\\bf p'}|^ 2}\n \\langle m_1' m_2'| w_4( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad +  \\frac{ g^ {tm_t}({\\bf p'},{\\bf p})+h^ {tm_t}({\\bf p'},{\\bf p})}{ ( {\\bf p} + {\\bf p'})^ 2} \\cr\n&& \\qquad \\quad \\langle m_1' m_2'|  w_5 ( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad +  \\frac{ g^ {tm_t}({\\bf p'},{\\bf p})-h^ {tm_t}({\\bf p'},{\\bf p})}{ ( {\\bf p} - {\\bf p'})^ 2} \\cr\n&& \\qquad \\quad \\langle m_1' m_2'| w_6( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle . \\quad \\label{wpar}\n\\end{eqnarray}\nThus, finally we connect the scattering amplitude or similarly the Wolfenstein parameters to the scalar function \\linebreak $t_j^ { t m_t} ({\\bf p'}, {\\bf p})$. This can be accomplished by means of Eq.~(\\ref{wpar}) and the relation\nbetween the M- and T-matrix given as\n\\begin{equation}\nM = -\\mu (2\\pi)^2 T .\n\\end{equation}\nWe obtain\n\\begin{eqnarray}\na^ {tm_t} & = &  t_1 + (-)^ t \\; \\Big[ \\frac{1}{2} \\tilde t_1 + \\frac{3}{2} \\tilde t_2 \\cr\n&& +   \\frac{1}{2} p^ 4 ( 1-x^ 2)  \\tilde t_4 + p^ 2 ( 1-x)  \\tilde t_5 + p^ 2 (\n1+x) \\tilde t_6\\Big] \\cr\nc^ {tm_t} & = &  i p^ 2 \\sqrt{1 - x^ 2}  \\left( t_3 - (-)^ t \\tilde t_3 \\right) \\cr\nm^ {tm_t} & = &  t_2 + p^ 4 ( 1 - x^ 2)  t_4\n +   (-)^ t \\; \\Big[ \\frac{1}{2} \\tilde t_1 - \\frac{1}{2} \\tilde t_2 \\cr\n&& + \\frac{1}{2} p^ 4 ( 1- x^ 2) \\tilde t_4 -   p^ 2 (1-x)\\tilde t_5 - p^ 2 ( 1+x) \\tilde t_6 \\Big]\\cr\ng^ {tm_t} & = &  t_2 + p^ 2 ( 1+x)  t_5  + p^ 2( 1-x)  t_6 \\cr\n&& +  (-)^ t \\; \\Big[  \\frac{1}{2} \\tilde t_1 - \\frac{1}{2} \\tilde t_2\n- \\frac{1}{2} p^ 4 ( 1-x^ 2) \\tilde t_4 \\Big]\\cr\nh^ {tm_t} & = &  p^ 2(1+x)  t_5 - p^ 2(1-x) t_6 \\cr\n&& +  (-)^ t \\; \\Big[- p^ 2(1-x)  \\tilde t_5 + p^ 2(1+x)  \\tilde t_6 \\Big] . \\label{wpart},\n\\end{eqnarray}\nwhere $x = {\\bf \\hat p'} \\cdot {\\bf \\hat p}$\nNote that in Eq.~(\\ref{wpart}) we drop ${\\bf p}$ and ${\\bf p'}$ for simplicity and apply the following notation,\n\\begin{eqnarray}\nt_j & \\equiv & t_j^ {t m_t} ( {\\bf p'}, {\\bf p}) \\cr\n\\tilde t_j &\\equiv & t_j^ {t m_t} ( {\\bf p'}, -{\\bf p}) .\n\\end{eqnarray}\n\n\\section{Summary}\n\nWe propose a new technique to calculate the 2N system as function of momentum vectors, i.e.\n without employing a partial wave decomposition. The technique\nis  useful especially in energy regions of hundreds of MeV or when considering the NN t-matrix as input to a three-body calculation. Based on scalar interactions, the scattering of three-bosons has been successfully carried out up to the GeV regime, formulating the Faddeev equations as functions of vector momenta~\\cite{Liu:2004tv}. The formulation of NN scattering presented here is an important\nstep on the way of performing realistic three-body scattering calculations at higher energies.\n\n Based on the general operator\nstructure of the NN interaction we derive the formulation in a spin-momentum operator representation. Here\nthe NN potential, the T-matrix, and the deuteron state are expanded in a set of scalar products\nof spin operators and momentum vectors. We derive a set of two coupled equations for the deuteron wave function\ncomponents, which are connected to the standard partial wave projected wave function s- and d-wave in a simple\nmanner. In case of the NN scattering we obtain a set of six coupled equations for the scalar functions defining\nthe NN T-matrix, and therefore,  the scattering amplitude in the Wolfenstein representation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec1intro}}\n\nMultiple testing (MT) and control of the false discovery rate (FDR, \\cit\n{Benjamini:1995}) has been conducted in a variety of scientific endeavours\nincluding the microarray study in \\cite{Efron:2001} and the analysis of\nspatial data in \\cite{Benjamini:2007}. It is very challenging to develop MT\nprocedures with a desired FDR when the test statistics are strongly dependent.\nRecently, \\cite{Fan:2012} considered MT in the following setting. Let \n\\mathbf{Z}\\sim \\mathsf{N}_{m}\\left( \\boldsymbol{\\mu },\\boldsymbol{\\Sigma \n\\right) $ be an $m$-dimensional Normal random vector (rv) with mean $\\boldsymbol\n\\mu }=\\left( \\mu _{1},...,\\mu _{m}\\right) ^{T}\\in \\mathbb{R}^{m}$ and a\nknown correlation matrix $\\boldsymbol{\\Sigma }$ $\\geq 0$. Given an\nobservation $\\mathbf{z}=\\left( z_{1},...,z_{m}\\right) ^{T}$ of $\\mathbf{Z}$,\nthe $i$th null $H_{i}:\\mu _{i}=0$ versus its alternative $H_{i}^{\\ast }:\\mu\n_{i}\\neq 0$ is tested using $p_{i}=2\\Phi \\left( -\\left\\vert z_{i}\\right\\vert\n\\right) $ and some $t\\in \\left[ 0,1\\right] $ such that $H_{i}$ is rejected\nif and only if $p_{i}\\leq t$, where $\\Phi \\left( \\cdot \\right) $ is the\ncumulative distribution function (cdf) of the standard Normal rv. To tackle this MT\nproblem, \\cite{Fan:2012} proposed the technique of principal factor approximation (PFA) to decompose the dependence embedded in $\\boldsymbol\n\\Sigma }$ and used Theorem 6 in \\cite{Lyons:1988} together with the PFA to\nobtain the strong law of large numbers (SLLN) for the normalized number of conditional rejections\n\\begin{equation}\n\\tilde{R}\\left(  t\\right)  =m^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{k}\\right)\n=m^{-1}\\sum_{i=1}^{m}1_{\\left\\{  p_{i}\\leq t|\\mathbf{\\tilde{w}}_{k}\\right\\}\n}\\label{eqb1a\n\\end{equation}\nfor arbitrary $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ and $\\boldsymbol{\\Sigma\n\\geq0$, where $1_{A}$ is the indicator of a set $A$ and $\\mathbf{\\tilde{w\n}_{k}\\sim\\mathsf{N}_{k}\\left(  0,\\mathbf{I}\\right)  $ are called by\n\\cite{Fan:2012} the ``principal factors'\n(see \\autoref{SecRevisitPFA} for more details). The SLLN for $\\tilde\n{R}\\left(  \\cdot\\right)  $ (and that for another related quantity) enables\n\\cite{Fan:2012} to derive an almost sure (a.s.) approximation to the false\ndiscovery proportion (FDP, \\cite{Genovese:2002}) and it is considered a\ntheoretical breakthrough in FDR control of MT under dependence.\n\n\nUnfortunately, we found that the arguments given by the authors of \\cite{Fan:2012} do not seem to\nbe fully sufficient to yield the SLLN for $\\tilde{R}\\left(  \\cdot\\right)  $\nfor arbitrary $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ and $\\boldsymbol{\\Sigma\n\\geq0$, mainly due to their delicate handle of the involved functional\nremainders in the Taylor expansions and their use of some implicit assumptions\nto derive certain auxiliary asymptotic assertions. In this article, we will\nprovide a complete justification of this law with clarifications on all needed\nassumptions. In \\autoref{SecRevisitPFA} we will revisit the technique of PFA\nand provide examples on different speeds of the PFA to components of the original rv. In\n\\autoref{Sec:ReviewFan}, we will briefly review the strategy on the proof used\nin \\cite{Fan:2012} and point out the extra arguments needed to justify this law.\nIn \\autoref{SecLLNDisRejProc}, we present our detailed proof. Our\nreformulation provides an integrated view on how the speed of the PFA to\n$\\mathbf{Z}$, the dependence among $z_{i}$'s, and the magnitudes of $\\mu_{i\n$'s should interact with each other in order to validate this law. We end the\narticle with a short discussion in \\autoref{SecDiscussion}.\n\n\\section{PFA with different component speeds\\label{SecRevisitPFA}}\n\nLet the triple $\\left(  \\Omega,\\mathcal{F},\\mathbb{P}\\right)  $ be the probability\nspace on which all involved rv's are defined. We restate the technique of PFA\ndeveloped in \\cite{Fan:2012} as follows. Let $\\mathbf{w}=\\left(\nw_{1},...,w_{m}\\right)  ^{T}\\sim\\mathsf{N}_{m}\\left(  \\mathbf{0\n,\\mathbf{I}\\right)  $. The spectral decomposition of $\\boldsymbol{\\Sigma}$\nwith eigenvalues (counting multiplicity) $\\lambda_{1,m}\\geq\\lambda_{2,m\n\\geq\\cdots\\geq\\lambda_{m,m}\\geq0$ and corresponding eigenvectors\n$\\boldsymbol{\\gamma}_{i}=\\left(  \\gamma_{i1},...,\\gamma_{im}\\right)  ^{T}$ for\n$i=1,...,m$ implie\n\\begin{equation}\n\\mathbf{Z}=\\boldsymbol{\\mu}+\\boldsymbol{\\eta}+\\mathbf{v}\\text{,}\n\\label{eqb103\n\\end{equation}\nwhere $\\boldsymbol{\\eta}=\\left(  \\eta_{1},...,\\eta_{m}\\right)  ^{T}$ with\n$\\eta_{i}=\\sum\\nolimits_{j=1}^{k}\\sqrt{\\lambda_{j,m}}\\gamma_{ij}w_{j}$ and\n$\\mathbf{v}=\\left(  v_{1},...,v_{m}\\right)  ^{T}$ with\n$v_{i}=\\sum\\nolimits_{j=k+1}^{m}\\sqrt{\\lambda_{j,m}}\\gamma_{ij}w_{j}$ for any\n$0\\leq k\\leq m$. In (\\ref{eqb103}), $\\boldsymbol{\\eta}=\\mathbf{0}$ is set when\n$k=0$. From (\\ref{eqb103}), we see that $\\omega_{i,m}=\\mathbb{V}\\left[\n\\eta_{i}\\right]  =\\sum\\nolimits_{j=1}^{k}\\lambda_{j,m}\\gamma_{ij}^{2}$ and\n$\\sigma_{i,m}=\\mathbb{V}\\left[  v_{i}\\right]  =1-\\omega_{i,m}$, where\n$\\mathbb{V}\\left[  \\cdot\\right]  $ denotes the variance operator.  Note that\n$\\left(  w_{1},...,w_{k}\\right)  ^{T}$ is exactly $\\mathbf{\\tilde{w}}_{k}$.\n\nFor (\\ref{eqb103}) with $k\\geq1$, \\cite{Fan:2012} pointed out that for a fixed\n$\\delta>0$ there always exists some $1\\leq k\\leq m$ ($k$ can depend on $m$)\nsuch tha\n\\begin{equation}\n\\vartheta_{m}=m^{-1}\\sqrt{\\lambda_{k+1,m}^{2}+...+\\lambda_{m,m}^{2}}\\leq\nCm^{-\\delta}\\text{ for sufficiently large }m\\label{eqb23\n\\end{equation}\nfor some finite constant $C>0$. It should be noted\nthat (\\ref{eqb23}) never holds for any $\\delta>0$ when $k=0$ and\n$\\boldsymbol{\\Sigma}>0$. Further, \\cite{Fan:2012} defined\n\\[\na_{i,m}=\\left(  1-\\omega_{i,m}\\right)  ^{-1\/2\n\\]\nwith the implicit assumption that $\\omega_{i,m}<1$ for $1\\leq i\\leq m$.\nNote that $a_{i,m}\\geq1$ for any $1\\leq i\\leq m$ and finite $m$.\n\nLet $cov\\left(  \\cdot,\\cdot\\right)  $ be the covariance operator and\n$\\mathbf{A}=cov\\left(  \\mathbf{v},\\mathbf{v}\\right)  =\\left(  q_{ij}\\right)\n_{m\\times m}$. The magnitudes of $\\left\\{  a_{i,m}\\right\\}  _{i=1}^{m}$ play a\ncrucial role in the asymptotic analysis on $\\tilde{R}\\left(  \\cdot\\right)  $\nsince they control the speed of PFA via $\\boldsymbol{\\eta}$ to $\\mathbf{Z}$\nand affect the dependence structure, i.e., entries of $\\mathbf{A}$, among\ncomponents of $\\mathbf{v}$. Most importantly, $\\left\\{  a_{i,m}\\right\\}\n_{i=1}^{m}$ affect the remainders of the Taylor expansion used in\n\\cite{Fan:2012}. Let\n$\\mathbf{T}_{m}=\\left(  \\gamma_{ij}\\right)  _{m\\times m}$, $\\mathbf{D\n_{m}=diag\\left\\{  \\lambda_{1,m},...,\\lambda_{m,m}\\right\\}  $ and\n$\\boldsymbol{\\Sigma}=\\boldsymbol{\\Sigma}_{m}$,\nwhere the subscript $m$ denotes the dimension of matrices.\nWe give an example where $\\omega_{i,m}=1$, i.e.,\n$a_{i,m}=\\infty$ for $1\\leq i\\leq k$ for some $k$.\n\n\\begin{lemma}\n\\label{Lm:BlockOrtho}For all even $m\\geq4$ and any $\\boldsymbol{\\mu\n\\in\\mathbb{R}^{m}$, there exists $\\boldsymbol{\\Sigma}_{m}>0$, a block diagonal\n(not diagonal) matrix, such that $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left(\n\\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right)  $ obeys (\\ref{eqb103}) and\nthat (\\ref{eqb23}) holds with $k=2^{-1}m$, $C=1$ and $\\delta\\in\\left(\n0,2^{-1}\\right)  $ but $a_{i,m}=\\infty$ for $1\\leq i\\leq k$ and $a_{i,m}=1$\nfor $k+1\\leq i\\leq m$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, we construct the needed positive eigenvalues $\\left\\{  \\lambda\n_{i,m}\\right\\}  _{i=1}^{m}$ (with $\\lambda_{i,m}\\geq\\lambda_{i+1,m}$). Pick\n$\\delta\\in\\left(  0,2^{-1}\\right)  $, $k=2^{-1}m$ and $\\left\\{  \\varepsilon\n_{j}\\right\\}  _{j=1}^{k}$ such that $0<\\varepsilon_{j}<\\varepsilon_{j+1}<1$\nfor all $1\\leq j\\leq k-1$. Let $\\lambda_{k+j,m}=1-\\varepsilon_{j}$ and\n$\\lambda_{j,m}=1+\\varepsilon_{j}$ for $1\\leq j\\leq k$. Then $\\sum_{i=1\n^{m}\\lambda_{i,m}=m$. Since $\\min\\left\\{  \\sqrt{2}m^{1\/2-\\delta},1\\right\\}\n=1$ whenever $m\\geq1$ and\n\\[\n\\lambda_{k+1,m}\\leq\\sqrt{2}m^{1\/2-\\delta\n\\]\nimplies\n\\[\nm^{-1}\\sqrt{\\sum\\nolimits_{j=k+1}^{m}\\lambda_{j,m}^{2}}\\leq2^{-1\/2\nm^{-1\/2}\\lambda_{k+1,m}\\leq m^{-\\delta}\\text{,\n\\]\nit follows that the choice of $\\left\\{  \\lambda_{i,m}\\right\\}  _{i=1}^{m}$,\n$C$, $\\delta$ and $k$ validates (\\ref{eqb23}).\n\nNext, we construct the desired $\\boldsymbol{\\Sigma}_{m}$ and $\\mathbf{Z}$.\nKeep $k=2^{-1}m$. Let $\\mathbf{Q}_{1}\\in\\mathcal{O}_{k}$ and $\\mathbf{Q\n_{2}\\in\\mathcal{O}_{m-k}$, where $\\mathcal{O}_{n}$ denotes the set of $n\\times\nn$ orthogonal matrices. Define $\\mathbf{T}_{m}=diag\\left\\{  \\mathbf{Q\n_{1},\\mathbf{Q}_{2}\\right\\}  $. Then, $\\mathbf{T}_{m}=\\left(  \\gamma\n_{ij}\\right)  _{m\\times m}$ is orthogonal such that $\\max_{1\\leq i\\leq k\n\\max_{k+1\\leq j\\leq m}\\gamma_{ij}=0$ but $\\sum_{j=k+1}^{m}\\gamma_{ij}^{2}=1$\nfor all $k+1\\leq i\\leq m$. Now setting $\\mathbf{Z}=\\boldsymbol{\\mu\n+\\mathbf{T}_{m}\\sqrt{\\mathbf{D}_{m}}\\mathbf{w}$ for any $\\boldsymbol{\\mu\n\\in\\mathbb{R}^{m}$ gives $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu\n},\\boldsymbol{\\Sigma}_{m}\\right)  $ with $\\boldsymbol{\\Sigma}_{m\n=\\mathbf{T}_{m}\\mathbf{D}_{m}\\mathbf{T}_{m}^{T}$. Obviously, $\\mathbf{Z}$\nadmits decomposition (\\ref{eqb103}) with this $k=2^{-1}m$ and\n$\\boldsymbol{\\Sigma}_{m}>0$ is a block diagonal matrix. However, $\\omega\n_{i,m}=1$ for $1\\leq i\\leq k$ and $\\omega_{i,m}=0$ for $k+1\\leq i\\leq m$,\nmeaning that $a_{1}=...=a_{k}=\\infty$ and $a_{k+1}=...=a_{m}=1$. This\ncompletes the proof.\n\\end{proof}\n\nIt can be easily seen that $\\boldsymbol{\\Sigma}_{m}>0$ is a block diagonal\nmatrix if and only $\\mathbf{T}_{m}$ is so. The Normal rv $\\mathbf{Z\n\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right)  $\nprovided in \\autoref{Lm:BlockOrtho} has a block diagonal correlation matrix\n$\\boldsymbol{\\Sigma}_{m}$ and has a decomposition $\\mathbf{Z}=\\boldsymbol{\\mu\n}+\\boldsymbol{\\eta}+\\mathbf{v}$ where $\\eta_{i}=0$ a.s. for $1\\leq i\\leq k$\nand $v_{i}=0$ a.s. for $k+1\\leq i\\leq m$. Such Normal rv's $\\mathbf{Z}$\npresent a simpler case for multiple testing which $\\mu_{i}$'s are zero since\n$\\left(  z_{1},...,z_{k}\\right)  ^{T}$ are independent of $\\left(\nz_{k+1},...,z_{m}\\right)  ^{T}$, the latter of which has weakly dependent\ncomponents as defined by \\cite{Fan:2012}.\n\nWe have the following example for which each $a_{i,m}$ for $1\\leq i\\leq m$ is\nfinite for any finite $m$:\n\n\\begin{lemma}\n\\label{Lm:DensestOrtho}For any $m\\geq2$, there exists an orthogonal matrix\n$\\mathbf{T}_{m}=\\left(  \\gamma_{ij}\\right)  _{m\\times m}$ such that\n$\\gamma_{ij}\\neq0$ for all $1\\leq i\\leq j\\leq m$. Thus, for any\n$\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ there exits $\\boldsymbol{\\Sigma}_{m}>0$\nfor which $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu\n,\\boldsymbol{\\Sigma}_{m}\\right)  $ admits (\\ref{eqb103}) for any $1\\leq k\\leq\nm$ such that $1<a_{i,m}<\\infty$ for each $1\\leq i\\leq m$ and finite\n$m$. In particular, for $m\\geq4$ even, $\\boldsymbol{\\Sigma}_{m}>0$ can be\nchosen to induce (\\ref{eqb23}) with $k=  2^{-1}m  $, $C=1$ and\n$\\delta\\in\\left(  0,2^{-1}\\right)  $.\n\\end{lemma}\n\n\\begin{proof}\nFirst, we show the existence of such a $\\mathbf{T}_{m}$. Let $$S^{m-1}=\\left\\{\n\\mathbf{x}\\in\\mathbb{R}^{m}:\\left\\Vert \\mathbf{x}\\right\\Vert =1\\right\\}  .$$\nDenote by $\\left\\langle \\cdot,\\cdot\\right\\rangle $ the inner product in\nEuclidean space and by $^{\\perp}$ the orthogonal complement with respect to\n$\\left\\langle \\cdot,\\cdot\\right\\rangle $. Pick $\\mathbf{u}=\\left(\nu_{1},...,u_{m}\\right)^{T}  \\in S^{m-1}$ such that $0<\\min_{1\\leq i\\leq m}$\n$\\left\\vert u_{i}\\right\\vert <\\max_{1\\leq i\\leq m}$ $\\left\\vert u_{i\n\\right\\vert <1$ and $2u_{i}^{2}\\neq1$ for all $1\\leq i\\leq m$. Define\n$\\Pi=\\left\\{  \\mathbf{x}\\in\\mathbb{R}^{m}:\\left\\langle \\mathbf{x\n,\\mathbf{u}\\right\\rangle =0\\right\\}  $. Then $\\Pi$ is a hyperplane in\n$\\mathbb{R}^{m}$ with normal $\\mathbf{u}$. Let $L=\\left\\{  x\\mathbf{u\n:x\\in\\mathbb{R}\\right\\}  $. Then $\\Pi=L^{\\perp}$. Let $\\tilde{T}_{m}$ be the\nreflection with respect to $\\Pi$ (that keeps $\\Pi$ invariant but flips\n$\\mathbf{u}$). Then $\\tilde{T}_{m}\\mathbf{x}=\\mathbf{x}-2\\left\\langle\n\\mathbf{x},\\mathbf{u}\\right\\rangle \\mathbf{u}$ for all $\\mathbf{x\n\\in\\mathbb{R}^{m}$. In particular, $\\tilde{T}_{m}\\mathbf{e}_{i}=\\mathbf{e\n_{i}-2\\left\\langle \\mathbf{e}_{i},\\mathbf{u}\\right\\rangle \\mathbf{u\n=\\mathbf{e}_{i}-2u_{i}\\mathbf{u}$, where $\\mathbf{e}_{i}\\in\\mathbb{R}^{m}$ has\nthe only non-zero entry, $1$, at its $i$th entry. By the construction of\n$\\mathbf{u}$, for each $1\\leq i\\leq m$ each entry of $\\tilde{T}_{m\n\\mathbf{e}_{i}$ is non-zero. Consequently, the matrix $\\mathbf{T}_{m}$ with\nthe $m$ columns $\\boldsymbol{\\gamma}_{i}=\\tilde{T}_{m}\\mathbf{e}_{i}=\\left(\n\\gamma_{i1},...,\\gamma_{im}\\right)^{T}  $ is orthogonal and none of the\n$\\gamma_{ij}$'s is zero.\n\nNow we construct the needed $\\boldsymbol{\\Sigma}_{m}>0$. Take any $m$ positive\nnumbers $\\left\\{  \\lambda_{i,m}\\right\\}  _{i=1}^{m}$ and set $\\mathbf{D\n_{m}=diag\\left\\{  \\lambda_{1,m},...,\\lambda_{m,m}\\right\\}  $. Then\n$\\mathbf{Z}=\\boldsymbol{\\mu}+\\mathbf{T}_{m}\\sqrt{\\mathbf{D}_{m}}\\mathbf{w}$\nfor any $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ satisfies $\\mathbf{Z\n\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right)  $\nwith $\\boldsymbol{\\Sigma}_{m}=\\mathbf{T}_{m}\\mathbf{D}_{m}\\mathbf{T}_{m}^{T}$.\nFurther, (\\ref{eqb103}) holds and $1<a_{i,m}<\\infty$ for each each $1\\leq\ni\\leq m$ and all finite $m$. Specifically, for $m\\geq4$ even, if $\\left\\{\n\\lambda_{i,m}\\right\\}  _{i=1}^{m}$ is chosen to be those given in the proof of\n\\autoref{Lm:BlockOrtho}, then (\\ref{eqb23}) holds with the desired $k$, $C$\nand $\\delta$. The proof is completed.\n\\end{proof}\n\nLet $\\mathcal{O}_{n}$ denote the set of $n\\times n$ orthogonal matrices and\n$\\mathcal{S}_{n}$ the set of $n\\times n$ positive-definite matrices. We\nprovide the third example where $\\limsup_{m\\rightarrow\\infty}a_{i,m}<\\infty$\nfor some $1\\leq i\\leq m$.\n\n\\begin{corollary}\n\\label{Lm:MidOrtho}For $m\\geq4$ even and $\\boldsymbol{\\mu}_{m}\\in\n\\mathbb{R}^{m}$, there exists $\\left\\{  \\boldsymbol{\\Sigma}_{m}\\right\\}  _{m\\geq4}$\nwith $\\boldsymbol{\\Sigma}_{m}\\in\\mathcal{S}_{n}$ such that the following hold:\n\n\\begin{enumerate}\n\\item $\\liminf_{m\\rightarrow\\infty}\\lambda_{m,m}=\\lambda_{0}$ for some\n$\\lambda_{0}>0$.\n\n\\item For each $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu\n_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $, (\\ref{eqb103}) holds for each $1\\leq\nk\\leq m$ and (\\ref{eqb23}) holds with $k=  2^{-1}m  $, $C=1$ and\n$\\delta\\in\\left(  0,2^{-1}\\right)  $.\n\n\\item $1<a_{i,m}<\\infty$ for each $1\\leq i\\leq m$ and finite $m$ but\n$\\limsup_{m\\rightarrow\\infty}a_{m,m}<\\infty$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nWe will employ the techniques used in the previous proofs. Take the\neigenvalues $\\left\\{  \\lambda_{i,m}\\right\\}  _{i=1}^{m}$ constructed in the\nproof of \\autoref{Lm:BlockOrtho} but restrict $\\varepsilon_{2^{-1}m}$ to be\nsuch that $\\liminf_{m\\rightarrow\\infty}\\varepsilon_{2^{-1}m}=\\varepsilon_{0}$\nfor some $\\varepsilon_{0}>0$. Take the $\\mathbf{u}$ and $\\tilde{T}_{m}$\nconstructed in the proof of \\autoref{Lm:DensestOrtho} but let $u_{m}=u_{0}$\nfor a fixed, small positive constant $u_{0}$ (e.g., $u_{0}=10^{-5}$ can be\nused); take the $\\mathbf{T}_{m}=\\left(  \\gamma_{ij}\\right)  _{m\\times m}$\ninduced by $\\tilde{T}_{m}$ under the canonical orthonormal basis $\\left\\{\n\\mathbf{e}_{i}\\right\\}  _{i=1}^{m}$ such that the $i$th column of\n$\\mathbf{T}_{m}$ is $\\tilde{T}_{m}\\mathbf{e}_{i}$. Then none of the entries $\\gamma_{ij}$ of\n$\\mathbf{T}_{m}$ is zero, $\\gamma_{im}=-2u_{i}u_{0}$ for $1\\leq i\\leq m-1$ but\n$\\gamma_{mm}=1-2u_{0}^{2}$. So,\n\\[\n\\sigma_{m,m}=\\sum\\nolimits_{j=k+1}^{m}\\lambda_{j,m}\\gamma_{mj}^{2}\\geq\n\\lambda_{m,m}\\gamma_{mm}^{2}=\\lambda_{m,m}\\left(  1-2u_{0}^{2}\\right)  ^{2\n\\]\nand\n\\[\n\\liminf_{m\\rightarrow\\infty\n}\\sigma_{m,m}\\geq\\left(  1-\\varepsilon_{0}\\right)  \\left(  1-2u_{0\n^{2}\\right)  ^{2}\\text{.\n\\]\nTherefore,\n\\[\n\\limsup_{m\\rightarrow\\infty}a_{m,m}=\\left(  \\liminf_{m\\rightarrow\\infty\n\\sigma_{m,m}^{1\/2}\\right)  ^{-1}\\leq\\dfrac{1}{\\left(  1-2u_{0}^{2}\\right)\n\\sqrt{1-\\varepsilon_{0}}}<\\infty\\text{.\n\\]\nThe proof is completed.\n\\end{proof}\n\nIn fact, we can further construct more elaborate sequence of $\\left\\{\n\\mathbf{Z}_{m}\\right\\}  _{m}$ with $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left(\n\\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $ such that\namong $\\left\\{  a_{i,m}\\right\\}  _{i=1}^{m}$ all the following three types of\nbehaviour occur for some $1\\leq i,i^{\\prime},i^{\\prime\\prime}\\leq m$:\n\n\\begin{enumerate}\n\\item $1<a_{i,m}<\\infty$ for each $m$ but $\\limsup_{m\\rightarrow\\infty}a_{i,m}<\\infty$.\n\n\\item $1<a_{i^{\\prime},m}<\\infty$ for each $m$ and $\\lim_{m\\rightarrow\\infty}a_{i^{\\prime},m}=\\infty$.\n\n\\item $a_{i^{\\prime\\prime},m}=\\infty$\nfor some finite $m$.\n\\end{enumerate}\n\n\\begin{corollary}\n\\label{Lm:MidOrthoB}For large and even $m\\geq8$ and $\\boldsymbol{\\mu}_{m\n\\in\\mathbb{R}^{m}$, there exists $\\left\\{  \\boldsymbol{\\Sigma}_{m}\\right\\}  _{m\\geq\n8}$ with $\\boldsymbol{\\Sigma}_{m}\\in\\mathcal{S}_{n}$ such that the following hold:\n\n\\begin{enumerate}\n\\item $\\liminf_{m\\rightarrow\\infty}\\lambda_{m,m}=\\lambda_{0}$ for some\n$\\lambda_{0}>0$.\n\n\\item For each $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu\n_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $, (\\ref{eqb103}) holds for each $1\\leq\nk\\leq m$ and (\\ref{eqb23}) holds with $k= 2^{-1}m  $, $C=1$ and\n$\\delta\\in\\left(  0,2^{-1}\\right)  $.\n\n\\item $1<a_{i,m}<\\infty$ for each $1\\leq i\\leq m-1$ and finite $m$ and but $a_{m,m}=\\infty$.\n\n\\item $\\lim_{m\\rightarrow\\infty}a_{1,m}=\\infty$ and $\\limsup_{m\\rightarrow\\infty\n}a_{k+1,m}<\\infty$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nTake the eigenvalues $\\left\\{  \\lambda_{i,m}\\right\\}  _{i=1}^{m}$ constructed\nin the proof of \\autoref{Lm:MidOrtho}. Take $\\mathbf{u}=\\left(\nu_{1},...,u_{m}\\right)  ^{T}\\in S^{m-1}$ such that $u_{k+1}=\\tilde{u}_{0}$ for\nsome fixed, small constant $0<\\tilde{u}_{0}<8^{-1}$, $u_{m}=2^{-1}\\sqrt{2}$,\n$u_{i}=0$ for $i=k+2,...,m-1$, and $u_{i}>0$ for $1\\leq i\\leq k$ but\n$\\lim_{m\\rightarrow\\infty}u_{1}=0$. Define $\\Pi$ and $L$ as in\n\\autoref{Lm:DensestOrtho} and let $\\tilde{T}_{m}$ be the reflection with\nrespect to $\\Pi$. Then $\\tilde{T}_{m}\\mathbf{e}_{i}=\\mathbf{e}_{i\n-2u_{i}\\mathbf{u}$. Let the matrix $\\mathbf{T}_{m}$ have its $i$th column\n$\\boldsymbol{\\gamma}_{i}=\\tilde{T}_{m}\\mathbf{e}_{i}$. We see that\n\\begin{align*}\n\\sigma_{k+1,m}  & =\\lambda_{k+1,m}\\left(  1-2u_{k+1}^{2}\\right)  ^{2\n+\\lambda_{m,m}\\left(  2u_{k+1}u_{m}\\right)  ^{2}\\\\\n& \\geq\\left(  1-\\varepsilon_{0}\\right)  \\left[  \\left(  1-2\\tilde{u}_{0\n^{2}\\right)  ^{2}+2\\tilde{u}_{0}^{2}\\right]\n\\end{align*}\nand $\\sigma_{k+1,m}<1$ for any such $m$. However, $\\sigma_{1,m}=4\\lambda\n_{k+1,m}u_{1}^{2}\\tilde{u}_{0}^{2}+2^{-1}\\lambda_{m,m}u_{1}^{2}$, $\\sigma_{m,m}=0$ and $0<\\sigma_{i,m}<\\infty$ for\n$i\\notin\\left\\{  k+1,m\\right\\}  $ for any such $m$. Therefore,\n\\[\n\\limsup_{m\\rightarrow\\infty}a_{k+1,m}=\\left(  \\liminf_{m\\rightarrow\\infty\n}\\sigma_{k+1,m}^{1\/2}\\right)  ^{-1}<\\infty\n\\]\nand $\\lim_{m\\rightarrow\\infty}a_{1,m}=\\left(  \\liminf_{m\\rightarrow\\infty\n}\\sigma_{1,m}^{1\/2}\\right)  ^{-1}=\\infty$. The proof is completed.\n\\end{proof}\n\nThe examples we constructed demonstrate the different\nbehaviour of $\\left\\{  a_{i,m}\\right\\}  _{i=1}^{m}$ in the PFA to\n$\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu}_{m\n,\\boldsymbol{\\Sigma}_{m}\\right)  $ as $m$ changes and hence the need for a\ncareful analysis of the terms involving $a_{i,m}$'s and ratios between the\n$a_{i,m}$'s. However, such an analysis has not been explicitly and carefully conducted in\n\\cite{Fan:2012}.\n\n\\section{Review of strategy in \\cite{Fan:2012}\\label{Sec:ReviewFan}}\n\nLet $\\mathbb{E}$ be the expectation operator. Before we start reviewing the\nstrategy in \\cite{Fan:2012} to show the SLLN for $\\tilde{R}\\left(\n\\cdot\\right)  $, we quote as \\autoref{LemmaLyons} Theorem 6 of\n\\cite{Lyons:1988} as follows:\n\n\\begin{lemma}\n[\\cite{Lyons:1988}]\\label{LemmaLyons}Let $\\left\\{  X_{n}\\right\\}\n_{n=1}^{\\infty}$ be a sequence of real-valued random variables such that\n$\\mathbb{E}\\left[  \\left\\vert X_{n}\\right\\vert ^{2}\\right]  \\leq1$. If\n\\ $\\left\\vert X_{n}\\right\\vert \\leq1$ a.s. and\n\\[\n\\sum\\nolimits_{N=1}^{\\infty}N^{-1}\\mathbb{E}\\left[  \\left\\vert N^{-1\n\\sum\\nolimits_{n=1}^{N}X_{n}\\right\\vert ^{2}\\right]  <\\infty\\text{,\n\\]\nthen $\\lim_{m\\rightarrow\\infty}N^{-1}\\sum_{n=1}^{N}X_{n}=0$ a.s..\n\\end{lemma}\n\nBy (\\ref{eqb23}) and the Cauchy-Schwarz inequality\n\\begin{equation}\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert\n\\leq\\vartheta_{m}\\leq Cm^{-\\delta}\\text{,}\\label{eqb15\n\\end{equation}\nwhere $q_{ij}=cov\\left(  v_{i},v_{j}\\right)  $ and $\\rho_{ij}= q_{ij\n\\sigma_{i,m}^{-1\/2}\\sigma_{j,m}^{-1\/2}$ if $\\sigma_{i,m}\\sigma_{j,m}\\neq0$ and $\\rho\n_{ij}=0$ otherwise. Namely, $\\left\\{  v_{i}\\right\\}  _{i=1}^{m}$ are\n\\textquotedblleft weakly dependent\\textquotedblright\\ as termed in\n\\cite{Fan:2012}. This implies that $\\mathbf{Z}$ conditional on\n$\\boldsymbol{\\eta}$ (or equivalently $\\mathbf{\\tilde{w}}_{k}$) has weakly\ndependent components. Hence it may be possible to apply \\autoref{LemmaLyons}\nto $\\left\\{  X_{i}\\right\\}  _{i=1}^{m}$ with $X_{i}=1_{\\left\\{  p_{i}\\leq\nt|\\mathbf{\\tilde{w}}_{k}\\right\\}  }-\\mathbb{E}\\left[  1_{\\left\\{  p_{i}\\leq\nt|\\mathbf{\\tilde{w}}_{k}\\right\\}  }\\right]  $  as $m \\rightarrow \\infty$ to yield the SLLN for $\\tilde{R}\\left(\nt\\right)  =m^{-1}\\sum_{i=1}^{m}X_{i}$. Sinc\n\\begin{align}\n\\mathbb{V}\\left[  m^{-1}\\sum\\nolimits_{i=1}^{m}X_{i}\\right]   &  =m^{-2\n\\sum\\nolimits_{i=1}^{m}\\mathbb{V}\\left(  X_{i}\\right)  +2m^{-2}\\sum_{1\\leq\ni<j\\leq m}cov\\left(  X_{i},X_{j}\\right)  \\label{eqb38}\\\\\n&  \\leq4m^{-1}+2m^{-2}\\sum\\nolimits_{1\\leq i<j\\leq m}cov\\left(  X_{i\n,X_{j}\\right)  \\text{,}\\nonumber\n\\end{align}\nwe see that i\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{1\\leq i<j\\leq m}cov\\left(  X_{i},X_{j}\\right)  \\leq\nMm^{-\\delta_{1}}\\label{eqb2\n\\end{equation}\nfor some $\\delta_{1}>0$, then\n\\begin{equation}\n\\sum_{m=1}^{\\infty}m^{-1}\\mathbb{V}\\left[  m^{-1}\\sum\\nolimits_{i=1}^{m\nX_{i}\\right]  \\leq4\\sum_{m=1}^{\\infty}m^{-2}+2\\sum_{m=1}^{\\infty\nm^{-1-\\delta_{1}}<\\infty\\text{,}\\label{eqb4\n\\end{equation}\nwhere $M>0$ is a generic constant that can assume different (and appropriate)\nvalues at different occurrences. Once (\\ref{eqb4}) is established,\n\\autoref{LemmaLyons} immediately implie\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}\\left\\vert \\tilde{R}\\left(  t\\right)  -\\mathbb{E\n\\left[  \\tilde{R}\\left(  t\\right)  \\right]  \\right\\vert =0\\text{\na.s..}\\label{eqb3\n\\end{equation}\n\nTo establish (\\ref{eqb2}), \\cite{Fan:2012} applied Taylor expansion with\nrespect to $q_{ij}^{1\/2}$ (when $q_{ij}>0$) to each ter\n\\[\n\\zeta_{ij}=\\mathbb{P}\\left(  p_{i}\\leq t,p_{j}\\leq t|\\mathbf{\\tilde{w}\n_{k}\\right), i\\neq j\n\\]\nand this produces a total of $2^{-1}m\\left(  m-1\\right)  $ functional\nremainders whose various orders are mainly affected by the magnitudes of\n$\\left\\{  a_{i,m}\\right\\}  _{i=1}^{m}$, the ratios between certain pairs among $\\left\\{\na_{i,m}\\right\\}  _{i=1}^{m}$, and the corresponding functional mean values.\nHowever, it seems that \\cite{Fan:2012} failed to fully (and correctly)\nhandle all these remainders for which $\\left\\vert \\rho_{ij}\\right\\vert $ are\nclose $1$ and $a_{i,m}a_{j,m}^{-1}$ (or $a_{i,m}^{-1}a_{j,m}$) grows\nexponentially fast as $m$ increases.\n\n\\section{SLLN for normalized number of conditional rejections\\labe\n{SecLLNDisRejProc}}\n\nWe are ready to provide the extra arguments for the complete proof of the SLLN for $\\tilde{R}\\left(\n\\cdot\\right)  $. We state the key result of \\cite{Fan:2012} in our notations as follows:\n\n\\begin{proposition}\n[\\cite{Fan:2012}]\\label{Prop2Fan2012}Suppose $\\mathbf{Z}_{m}\\sim\\mathsf{N\n_{m}\\left(  \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $ with\n$\\boldsymbol{\\Sigma}_{m}\\geq0$. Choose an appropriate $k$ such that\n(\\ref{eqb23}) holds for some $\\delta>0$. The\n\\begin{equation}\n\\mathbb{V}\\left[  m^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{^{k}}\\right)  \\right]\n=O_{\\mathbb{P}}\\left(  m^{-\\delta}\\right)  \\label{9\n\\end{equation}\nan\n\\begin{equation}\n\\mathbb{P}\\left\\{  \\lim_{m\\rightarrow\\infty}\\left\\vert m^{-1}R\\left(\nt|\\mathbf{\\tilde{w}}_{^{k}}\\right)  -\\mathbb{E}\\left[  m^{-1}R\\left(  t|\\mathbf{\\tilde\n{w}}_{^{k}}\\right)  \\right]  \\right\\vert =0\\right\\}  =1\\text{.}\\label{eqb5\n\\end{equation}\n\n\\end{proposition}\n\nTo present our proof, we introduce some notations and sets. For any two positive sequences\n$\\left\\{  x_{m},y_{m}:m\\geq1\\right\\}  $, we write $x_{m}\\asymp y_{m}$ if and only if\n$0<\\liminf_{m\\rightarrow\\infty}x_{m}y_{m}^{-1}\\leq\\limsup_{m\\rightarrow\\infty\n}x_{m}y_{m}^{-1}<\\infty$. Define three sets for the magnitudes of $\\left\\{\na_{i,m}\\right\\}  _{i=1}^{m}$ a\n\\[\n\\left\\{\n\\begin{tabular}\n[c]{l\n$E_{1}=\\left\\{  i\\in\\mathbb{N}:a_{i,m}<\\infty\\text{ for any } m\\text{ and\n}\\lim_{m\\rightarrow\\infty}a_{i,m}=\\infty\\right\\}  $,\\\\\n$E_{2}=\\left\\{  i\\in\\mathbb{N}:a_{i,m}=\\infty\\text{ for some } m\\right\\}\n$,\\\\\n$E_{3}=\\left\\{  i\\in\\mathbb{N}:\\limsup_{m\\rightarrow\\infty}a_{i,m\n< \\infty\\right\\}  $,\n\\end{tabular}\n\\ \\ \\ \\right.\n\\]\nwhere it is understood that $a_{i,m}$ is undefined when $i>m$ for each finite\n$m$. Additionally, define the set counting how many pairs of $v_{i}$ and\n$v_{j}$ can be highly dependent but not necessarily linearly dependent a.s. a\n\\[\nS_{\\varepsilon,m}=\\left\\{  \\left(  i,j\\right)  :1\\leq i<j\\leq m,\\left\\vert\n\\rho_{ij}\\right\\vert >1-\\varepsilon\\right\\}\n\\]\nfor $\\varepsilon\\in\\left(  0,1\\right)  $ since $m^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert\n=O\\left(  m^{-\\delta}\\right)  $ does not necessarily impl\n\\[\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert \\rho_{ij}\\right\\vert\n=O\\left(  m^{-\\delta}\\right)  \\text{.\n\\]\n\n\n\\autoref{Prop2Fan2012} is reformulated as follows:\n\n\\begin{theorem}\n\\label{MainTheoremNormalOneThresholdWeak}Let $\\mathbf{Z}_{m}\\sim\n\\mathsf{N}_{m}\\left(  \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $\nwith $\\boldsymbol{\\Sigma}_{m}\\geq0$ and choose $k=k_{m}$ such that (\\ref{eqb23})\nholds for some $\\delta>0$. Suppose the following hold:\n\n\\begin{enumerate}\n\\item When $E_{1}\\neq\\varnothing$, there exists some $q\\in\\mathbb{N}$\nindependent of $m$ such tha\n\\begin{equation}\na_{\\left(  m\\right)  }\\asymp a_{\\left(  1\\right)  }^{q}\\text{,} \\label{eqb22b\n\\end{equation}\nwhere $a_{\\left(  1\\right)  }=\\min_{i\\in E_{1},1\\leq i\\leq m}a_{i,m}$ and\n$a_{\\left(  m\\right)  }=\\max_{i\\in E_{1},1\\leq i\\leq m}a_{i,m}$.\n\n\\item For some $\\tilde{\\varepsilon}\\in\\left(  0,1\\right)  $,\n\\begin{equation}\n\\limsup_{m\\rightarrow\\infty}m^{-2+\\delta}\\left\\vert S_{\\tilde{\\varepsilon\n}\\right\\vert <\\infty\\text{.}\\label{eqb24b\n\\end{equation}\n\n\\end{enumerate}\n\nThen\n\\begin{equation}\n\\mathbb{V}\\left[  \\tilde{R}\\left(  t\\right)  \\right]  =O\\left(  m^{-\\delta\n}\\right)  \\label{eqb20\n\\end{equation}\nholds except on the event\n\\begin{equation}\nG_{t,\\varepsilon}=\\cup_{m\\geq1}\\cup_{i\\in E_{1},1\\leq i\\leq m}\\left\\{\n\\omega\\in\\Omega:\\left\\vert \\pm z_{t\/2}-\\eta_{i}-\\mu_{i}\\right\\vert\n<\\varepsilon\\right\\}  \\label{eqb19\n\\end{equation}\nfor any $\\varepsilon >0$, where $z_{t\/2}=\\Phi^{-1}\\left(  t\/2\\right)  $ and $\\Phi^{-1}$ is the inverse\nof $\\Phi$.\n\\end{theorem}\n\n\\begin{proof}\nLet $Y_{i}=1-X_{i}$. Then $cov\\left(  Y_{i},Y_{j}\\right)  =cov\\left(\nX_{i},X_{j}\\right)  $ and (\\ref{eqb38}) is equivalent to\n\\begin{align*}\n\\mathbb{V}\\left[  m^{-1}\\sum\\nolimits_{i=1}^{m}X_{i}\\right]   &  \\leq\n4m^{-1}+2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in S_{\\tilde{\\varepsilon},m\n}cov\\left(  Y_{i},Y_{j}\\right)  \\\\\n&  +2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in S_{\\tilde{\\varepsilon},m}^{C\n}cov\\left(  Y_{i},Y_{j}\\right)  \\\\\n&  \\leq4m^{-1}+Mm^{-\\delta}+2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in\nS_{\\tilde{\\varepsilon},m}^{C}}cov\\left(  Y_{i},Y_{j}\\right)  \\text{,\n\\end{align*}\nsince (\\ref{eqb24b}) implies $m^{-2+\\delta}\\left\\vert S_{\\tilde{\\varepsilon\n,m}\\right\\vert <M$ for all $m\\in\\mathbb{N}$, where $A^{C}$ denotes the complement of a\nset $A$. This means we only need to show\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in S_{\\tilde{\\varepsilon},m}^{C\n}cov\\left(  Y_{i},Y_{j}\\right)  =O\\left(  m^{-\\delta_{1}}\\right\n\\label{eqb6\n\\end{equation}\nfor some $\\delta_{1}>0$, Since $cov\\left(  X_{i},X_{j}\\right)  =0$ for all $\\,1\\leq j\\leq m$ when $i\\in\nE_{2}$, the inequality in (\\ref{eqb6}) is not affected by such summands\n(asymptotically). Therefore, it suffices to sho\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m\n}cov\\left(  Y_{i},Y_{j}\\right)  =O\\left(  m^{-\\delta_{1}}\\right\n\\label{eqb7}\n\\end{equation}\nwhere\n\\[\nI_{\\tilde{\\varepsilon},m}=\\left\\{ \\left(  i,j\\right): \\left(  i,j\\right)  \\in S_{\\tilde\n{\\varepsilon},m}^{C}\\cap\\left(  \\left(  E_{1}\\otimes E_{1}\\right)  \\cup\\left(\nE_{3}\\otimes E_{3}\\right)  \\right)  \\right\\}\n\\]\nand $\\otimes$ denotes the Cartesian product. We can break $I_{\\tilde\n{\\varepsilon},m}=I_{\\tilde{\\varepsilon},m}^{+}\\cup$ $I_{\\tilde{\\varepsilon\n,m}^{-}$ and sho\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+\n}cov\\left(  Y_{i},Y_{j}\\right)  =O\\left(  m^{-\\delta_{1}}\\right)  \\label{eqb8\n\\end{equation}\nan\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{-\n}cov\\left(  Y_{i},Y_{j}\\right)  =O\\left(  m^{-\\delta_{1}}\\right)  \\text{,\n\\label{eqb9\n\\end{equation}\nwhere $I_{\\tilde{\\varepsilon},m}^{+}=\\left\\{  \\left(  i,j\\right)  \\in\nI_{\\tilde{\\varepsilon},m}:\\rho_{ij}\\geq0\\right\\}  $ and $I_{\\tilde\n{\\varepsilon},m}^{-}=\\left\\{  \\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon\n},m}:\\rho_{ij}<0\\right\\}  $. Since the techniques of proving (\\ref{eqb8}) and\n(\\ref{eqb9}) are the same, in the sequel we will just show (\\ref{eqb8}).\n\nLet $h_{1}\\left(  x\\right)  =\\left(  b^{2}-x^{2}\\right)  ^{-1\/2}$\nfor $\\vert b \\vert \\neq \\vert x \\vert$. Then\n\\begin{equation}\nh_{1}\\left(  x\\right)  =h\\left(  0\\right)  +h_{1}^{\\prime}\\left(\n\\theta\\right)  \\left(  x-0\\right)  =b^{-1}+\\dfrac{\\theta}{\\left(  b^{2\n-\\theta^{2}\\right)  ^{3\/2}}x\\label{eqb75\n\\end{equation}\nfor some $\\theta$ strictly between $0$ and $x$, where $h_{1}^{\\prime}\\left(\n\\theta\\right)  =\\theta\\left(  b^{2}-\\theta^{2}\\right)  ^{-3\/2}$. Set\n$o\\left(  x\\right)  =h_{1}^{\\prime}\\left(  \\theta\\right)  x$, which does not\nnecessarily satisfy $\\lim_{x\\rightarrow0}\\left\\vert x^{-1}o\\left(  x\\right)\n\\right\\vert =0$ if $\\lim_{x\\rightarrow0}b=0$. By Taylor\nexpansion, we have, for $\\left\\vert b\\right\\vert \\neq\\left\\vert x\\right\\vert\n$\n\\begin{align*}\n&  \\Phi\\left(  \\dfrac{xz+bc}{\\left(  b^{2}-x^{2}\\right)  ^{1\/2}}\\right)  \\\\\n&  =\\Phi\\left(  c+r_{1}\\left(  c,x\\right)  \\right)  \\\\\n&  =\\Phi\\left(  c\\right)  +\\phi\\left(  c\\right)  xzb^{-1}+\\phi\\left(\nc\\right)  \\left(  xz+bc\\right)  o\\left(  x\\right)  +\\dfrac{1}{2}\\left[\n-\\theta_{\\ast}\\phi\\left(  \\theta_{\\ast}\\right)  \\right]  r_{1}^{2}\\left(\nc,x\\right)  \\\\\n&  =\\Phi\\left(  c\\right)  +\\phi\\left(  c\\right)  xzb^{-1}+\\tilde{o}\\left(\nc,x\\right)  \\text{,\n\\end{align*}\nfor some $\\theta_{\\ast}$ strictly between $c$ and $c+r_{1}\\left(  c,x\\right)\n$, where $\\phi\\left(  t\\right)  =\\dfrac{d}{dt}\\Phi\\left(  t\\right)  $,\n$\\tilde{o}\\left(  c,x\\right)  =r_{2}\\left(  c,x\\right)  +r_{3}\\left(\nc,x\\right)  $ and\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\nr_{1}\\left(  c,x\\right)  =xzb^{-1}+\\left(  xz+bc\\right)  o\\left(  x\\right)\n\\text{,}\\\\\nr_{2}\\left(  c,x\\right)  =-\\dfrac{1}{2 \\sqrt{2 \\pi}}\\theta_{\\ast}e^{-\\theta_{\\ast}^{2\n\/2}r_{1}^{2}\\left(  c,x\\right)  \\text{,}\\\\\nr_{3}\\left(  c,x\\right)  =\\phi\\left(  c\\right)  \\left(  xz+bc\\right)  o\\left(\nx\\right)  \\text{.\n\\end{array}\n\\right.  \\label{eqb11\n\\end{equation}\n\n\nWe will now omit the subscript $m$ in $a_{i,m}$ and write $a_{i,m}$ as $a_{i\n$. Set $b_{ij}=a_{i}^{-1}a_{j}^{-1}$, $r_{1,i}=-z_{t\/2}-\\eta_{i}-\\mu_{i}$,\n$r_{2,i}=z_{t\/2}-\\eta_{i}-\\mu_{i}$ and $c_{l,i}=a_{i}r_{l,i}$ with $l=1,2$.\nSince $z_{t\/2}\\leq0$ for $t\\in\\left[  0,1\\right]  $, then $p_{i}\\geq t$ if and\nonly if $c_{2,i}\/a_{i}\\leq v_{i}\\leq c_{1,i}\/a_{i}$. Using\nthe formula from \\cite{Plackett:1954} or page 192 of \\cite{Tong:1990} for\n$0\\leq\\rho_{ij}<1$, we hav\n\\begin{align}\n\\mathbb{E}\\left[  Y_{i}Y_{j}\\right]   &  =\\mathbb{P}\\left(  \\left\\{\np_{i}>t,p_{j}>t|\\mathbf{\\tilde{w}}_{k}\\right\\}  \\right)  \\label{eqb27}\\\\\n&  =\\mathbb{P}\\left(  \\left\\{  c_{2,i}\/a_{i}<v_{i}<c_{1,i}\/a_{i},c_{2,j}\/a_{j\n<v_{j}<c_{1,j}\/a_{j}\\right\\}  \\right)  \\nonumber\\\\\n&  =\\int_{-\\infty}^{+\\infty}\\left[  \\varphi\\left(  \\rho_{ij},c_{1,i},z\\right)\n-\\varphi\\left(  \\rho_{ij},c_{2,i},z\\right)  \\right]  \\nonumber\\\\\n&  \\text{ \\ \\ \\ }\\times\\left[  \\varphi\\left(  \\rho_{ij},c_{1,j},z\\right)\n-\\varphi\\left(  \\rho_{ij},c_{2,j},z\\right)  \\right]  \\phi\\left(  z\\right)\ndz\\text{,}\\nonumber\n\\end{align}\nwhere $\\varphi\\left(  x_{1},x_{2},z\\right)  =\\Phi\\left(  \\dfrac{\\sqrt{x_{1\n}z+x_{2}}{\\left(  1-x_{1}\\right)  ^{1\/2}}\\right)  $ for $0\\leq x_{1}<1$. Set\n$b=b_{ij}^{1\/2}$ and $x=q_{ij}^{1\/2}$, for which a summation involving $x$\nand\/or $b$ is summing $q_{ij}^{1\/2}$ and\/or $b_{ij}^{1\/2}$ over certain pairs\nof $\\left(  i,j\\right)  $. Applying Taylor expansion to\n$\\varphi\\left(  \\rho_{ij},c_{l,i},z\\right)  $ but only to the second order\nwith respect to $x$ (in \\cite{Fan:2012} this expansion is to the third order) give\n\\[\n\\varphi\\left(  \\rho_{ij},c_{l,i},z\\right)  =\\Phi\\left(  \\dfrac{xz+bc_{l,i\n}{\\left(  b^{2}-x^{2}\\right)  ^{1\/2}}\\right)  =\\Phi\\left(  c_{l,i}\\right)\n+\\phi\\left(  c_{l,i}\\right)  b^{-1}xz+\\tilde{o}\\left(  c_{l,i},x\\right)\n\\]\nand\n\\begin{align*}\nd_{i} &  =\\varphi\\left(  \\rho_{ij},c_{1,i},z\\right)  -\\varphi\\left(  \\rho\n_{ij},c_{2,i},z\\right)  \\\\\n&  =\\Phi\\left(  c_{1,i}\\right)  +\\phi\\left(  c_{1,i}\\right)  b^{-1}xz+\\left[\n-\\Phi\\left(  c_{2,i}\\right)  -\\phi\\left(  c_{2,i}\\right)  b^{-1}xz\\right]\n+\\Delta\\tilde{o}_{i}\\left(  c,x\\right)  \\\\\n&  =\\left[  \\Phi\\left(  c_{1,i}\\right)  -\\Phi\\left(  c_{2,i}\\right)  \\right]\n+\\left[  \\phi\\left(  c_{1,i}\\right)  -\\phi\\left(  c_{2,i}\\right)  \\right]\nb^{-1}xz+\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\\\\n&  =\\Delta\\Phi\\left(  c_{i}\\right)  +\\Delta\\phi\\left(  c_{i}\\right)\nb^{-1}xz+\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\text{,\n\\end{align*}\nwhere the corresponding functional mean value $\\theta_{\\ast}$ in (\\ref{eqb11})\nis denoted by $\\theta_{l,i}$, the mean value $\\theta$ in (\\ref{eqb75}) by\n$\\theta_{i}$, and\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\Delta\\Phi\\left(  c_{i}\\right)  =\\Phi\\left(  c_{1,i}\\right)  -\\Phi\\left(\nc_{2,i}\\right)  \\text{,}\\\\\n\\Delta\\phi\\left(  c_{i}\\right)  =\\phi\\left(  c_{1,i}\\right)  -\\phi\\left(\nc_{2,i}\\right)  \\text{,}\\\\\n\\Delta\\tilde{o}_{i}\\left(  x\\right)  =\\tilde{o}\\left(  c_{1,i},x\\right)\n-\\tilde{o}\\left(  c_{2,i},x\\right)  \\text{.\n\\end{array}\n\\right.  \\label{eqb121\n\\end{equation}\nTherefor\n\\begin{align*}\nd_{i}d_{j}= &  \\Delta\\Phi\\left(  c_{i}\\right)  \\Delta\\Phi\\left(  c_{j}\\right)\n+\\Delta\\Phi\\left(  c_{i}\\right)  \\Delta\\phi\\left(  c_{j}\\right)\nb^{-1}xz+\\Delta\\Phi\\left(  c_{i}\\right)  \\Delta\\tilde{o}_{j}\\left(  x\\right)\n\\\\\n&  +\\Delta\\phi\\left(  c_{i}\\right)  b^{-1}xz\\Delta\\Phi\\left(  c_{j}\\right)\n+\\Delta\\phi\\left(  c_{i}\\right)  \\Delta\\phi\\left(  c_{j}\\right)  b^{-2\nx^{2}z^{2}+\\Delta\\phi\\left(  c_{i}\\right)  b^{-1}xz\\Delta\\tilde{o}_{j}\\left(x\\right)\\\\\n&  +\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\Phi\\left(  c_{j}\\right)\n+\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\phi\\left(  c_{j}\\right)\nb^{-1}xz+\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\tilde{o}_{j}\\left(\nx\\right)\n\\end{align*}\nan\n\\[\n\\int d_{i}d_{j}\\phi\\left(  z\\right)  dz=\\Delta\\Phi\\left(  c_{i}\\right)\n\\Delta\\Phi\\left(  c_{j}\\right)  +\\Delta\\phi\\left(  c_{i}\\right)  \\Delta\n\\phi\\left(  c_{j}\\right)  b^{-2}x^{2}+\\int g_{ij}\\left(  \\cdot\\right)\n\\phi\\left(  z\\right)  dz\\text{,\n\\]\nwher\n\\begin{equation}\ng_{ij}\\left(  \\cdot\\right)  =\\Delta\\Phi\\left(  c_{i}\\right)  \\Delta\\tilde\n{o}_{j}\\left(  x\\right)  +\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\n\\Phi\\left(  c_{j}\\right)  +\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\n\\tilde{o}_{j}\\left(  x\\right)  \\text{.}\\label{eqb82\n\\end{equation}\nConsequentl\n\\begin{equation}\ncov\\left(  X_{i},X_{j}\\right)  =\\Delta\\phi\\left(  c_{i}\\right)  \\Delta\n\\phi\\left(  c_{j}\\right)  b^{-2}x^{2}+\\int g_{ij}\\left(  \\cdot\\right)\n\\phi\\left(  z\\right)  dz\\text{.}\\label{eqb79\n\\end{equation}\n\n\nIt is easy to see that the set $\\left\\{  \\mathbf{\\tilde{w}}_{k\n:\\boldsymbol{\\eta}=z_{t\/2}\\mathbf{1}-\\boldsymbol{\\mu}_{m}\\text{ or\n}\\boldsymbol{\\eta}=-z_{t\/2}\\mathbf{1}_{m}-\\boldsymbol{\\mu}_{m}\\right\\}  $ is\n$\\mathbb{P}$-null. Therefore\n\\[\n\\mathbb{P}\\left(  D^{\\ast}\\right)  \\leq\\lim_{m\\rightarrow\\infty}\\sum_{i=1\n^{m}\\mathbb{P}\\left(  \\bigcup\\nolimits_{l=1}^{2}\\left\\{  r_{l,i}=0\\right\\}\n\\right)  =0\\text{,\n\\]\nwhere $D^{\\ast}=\\lim_{m\\rightarrow\\infty}\\bigcup\\nolimits_{i=1}^{m\n\\bigcup\\nolimits_{l=1}^{2}\\left\\{  r_{i,l}=0\\right\\}  $. Further, on the event\n$\\tilde{D}=\\left(  D^{\\ast}\\cup G_{t,\\varepsilon}\\right)  ^{C}$ with\n$G_{t,\\varepsilon}$ defined in (\\ref{eqb19}), we have\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}\\min_{i\\in E_{1},1\\leq i\\leq m}\\min\\left\\{\n\\left\\vert r_{1,i}\\right\\vert ,\\left\\vert r_{2,i}\\right\\vert \\right\\}\n\\geq\\varepsilon>0\\text{.}\\label{eqb81\n\\end{equation}\nIn what follows, we will just work on the event $\\tilde{D}$,\nwhere (\\ref{eqb81}) will help us obtain uniform boundedness of the\nfunctional coefficients in (\\ref{eqb82}).\n\nFrom (\\ref{eqb81}), we se\n\\[\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\n\\sup_{\\left(  r_{1,i},r_{2,i},a_{i}\\right)\n}\\left\\{  \\left\\vert \\phi\\left(  c_{1,i}\\right)  -\\phi\\left(  c_{2,i}\\right)\n\\right\\vert a_{i}\\right\\}  \\leq M .\n\\]\nThu\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left\\vert \\Delta\\phi\\left(  c_{i}\\right)\n\\Delta\\phi\\left(  c_{j}\\right)  \\right\\vert b^{-2}x^{2}\\leq Mm^{-2\\delta\n}\\text{.}\\label{eqb56\n\\end{equation}\nClearly\n\\begin{align*}\n\\left\\vert g_{ij}\\left(  \\cdot\\right)  \\right\\vert  &  =\\left\\vert \\Delta\n\\Phi\\left(  c_{i}\\right)  \\Delta\\tilde{o}_{j}\\left(  x\\right)  +\\Delta\n\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\Phi\\left(  c_{j}\\right)  +\\Delta\n\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\tilde{o}_{j}\\left(  x\\right)\n\\right\\vert \\\\\n&  \\leq2\\left\\vert \\Delta\\tilde{o}_{j}\\left(  x\\right)  \\right\\vert\n+2\\left\\vert \\Delta\\tilde{o}_{i}\\left(  x\\right)  \\right\\vert +\\left\\vert\n\\Delta\\tilde{o}_{i}\\left(  x\\right)  \\Delta\\tilde{o}_{j}\\left(  x\\right)\n\\right\\vert \\text{.\n\\end{align*}\nIf $\\Delta\\tilde{o}_{j}\\left(  x\\right)  $ for $j=1,...,m$ are dominated by a polynomial\n$g^{\\ast}\\left(  \\left\\vert x\\right\\vert ,\\left\\vert z\\right\\vert \\right)  $\nwith uniformly bounded functional coefficients without the constant term, the\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\leq m^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}\n}\\tilde{g}\\left(  \\left\\vert x\\right\\vert \\right)  \\leq Mm^{-\\beta_{0}\\delta\n}\\text{,}\\label{eqb49\n\\end{equation}\nfor $0<\\beta_{0}\\leq2$, where $\\tilde{g}\\left(  \\cdot\\right)  $ is a\npolynomial in $\\left\\vert x\\right\\vert $ and $\\beta_{0}>0$ is the smallest\ndegree of $\\left\\vert x\\right\\vert $ in $g^{\\ast}\\left(  \\left\\vert\nx\\right\\vert ,\\left\\vert z\\right\\vert \\right)  $ and we have used\n(\\ref{eqb15}) and\n\\begin{equation}\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert ^{\\beta\n}=O\\left(  m^{-\\delta\\min\\left\\{  \\beta,2\\right\\}  }\\right)  \\label{eqb17\n\\end{equation}\nfor all $\\beta\\in(0,2]$ derived from (\\ref{eqb23}) by H\\\"{o}lder's inequality.\nObviously, (\\ref{eqb49}) implie\n\\begin{align}\n&  \\text{ \\ }m^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert \\Delta\\tilde{o\n_{i}\\left(  x\\right)  \\right\\vert \\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\label{eqb46}\\\\\n&  \\leq m^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left(  \\int\\left\\vert \\Delta\\tilde{o\n_{i}\\left(  x\\right)  \\right\\vert ^{2}\\phi\\left(  z\\right)  dz\\right)\n^{1\/2}\\nonumber\\\\\n&  \\text{ \\ \\ \\ }\\times\\left(  \\int\\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right)  \\right\\vert ^{2}\\phi\\left(  z\\right)  dz\\right)  ^{1\/2}\\nonumber\\\\\n&  \\leq Mm^{-2\\beta_{0}\\delta}\\text{,}\\nonumber\n\\end{align}\nfrom which (\\ref{eqb8}) holds with $\\delta_{1}=2\\beta_{0}\\delta$.\nTherefore, it suffices to justify (\\ref{eqb49}).\n\nBy their definitions\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\n\\tilde{o}\\left(  c_{l,i},x\\right)  =\\phi\\left(  c_{l,i}\\right)  x^{2\nzh_{1}^{\\prime}\\left(  \\theta_{i}\\right)  +\\phi\\left(  c_{l,i}\\right)\nbc_{l,i}h_{1}^{\\prime}\\left(  \\theta_{i}\\right)  x-\\dfrac{1}{2}\\theta\n_{l,i} \\phi \\left(\\theta_{l,i}\\right) r_{1}^{2}\\left(  c_{l,i},x\\right)  \\text{,}\\\\\n\\Delta\\tilde{o}_{i}\\left(  x\\right)  =r_{3,1}\\left(  c_{i},x\\right)\nx^{2}z+r_{3,2}\\left(  c_{i},x\\right)  x+r_{4}\\left(  c_{i},x\\right)  \\text{,\n\\end{array}\n\\right.\n\\]\nwher\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\nr_{3,1}\\left(  c_{i},x\\right)  =\\phi\\left(  c_{1,i}\\right)  h_{1}^{\\prime\n}\\left(  \\theta_{i}\\right)  -\\phi\\left(  c_{2,i}\\right)  h_{1}^{\\prime}\\left(\n\\theta_{i}\\right)  \\text{,}\\\\\nr_{3,2}\\left(  c_{i},x\\right)  =\\left[  c_{1,i}\\phi\\left(  c_{1,i}\\right)\nh_{1}^{\\prime}\\left(  \\theta_{i}\\right)  -c_{2,i}\\phi\\left(  c_{2,i}\\right)\nh_{1}^{\\prime}\\left(  \\theta_{i}\\right)  \\right]  b\\text{,}\\\\\nr_{4}\\left(  c_{i},x\\right)  =-\\dfrac{1}{2}\\theta_{1,i}\\phi\\left(\n\\theta_{1,i}\\right)  r_{1}^{2}\\left(  c_{1,i},x\\right)  +\\dfrac{1}{2\n\\theta_{2,i}\\phi\\left(  \\theta_{2,i}\\right)  r_{1}^{2}\\left(  c_{2,i\n,x\\right).\n\\end{array}\n\\right.\n\\]\nFrom\n\\[\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in S_{\\tilde{\\varepsilon},m}^{C\n}\\left\\vert \\rho_{ij}\\right\\vert \\leq1-\\tilde{\\varepsilon}\\text{,\n\\]\nit follows that\n\\begin{equation}\nb^{-2}\\tilde{\\rho}\\geq b^{-2}\\left\\vert \\theta\/b\\right\\vert \\left[  1-\\left(\nx\/b\\right)  ^{2}\\right]  ^{-3\/2}\\geq\\left\\vert h_{1}^{\\prime}\\left(\n\\theta\\right)  \\right\\vert \\geq\\dfrac{\\left\\vert \\theta\\right\\vert }{b^{3\n}\\text{,}\\label{eqb37\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{\\rho}=\\left(  1-\\tilde{\\varepsilon}\\right)  \\left[  1-\\left(\n1-\\tilde{\\varepsilon}\\right)  ^{2}\\right]  ^{-3\/2}\\text{.}\\label{eqb57\n\\end{equation}\nUsing (\\ref{eqb81}), (\\ref{eqb22b}) and the property of $\\phi\\left(\n\\cdot\\right)  $, we hav\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left(  r_{l,i},a_{i}\\right)\n}\\left\\vert \\phi\\left(  c_{l,i}\\right)  h_{1}^{\\prime}\\left(  \\theta\n_{i}\\right)  \\right\\vert \\leq M\\label{eqb51\n\\end{equation}\nan\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left(  r_{l,i},a_{i},a_{j}\\right)\n}\\left\\vert c_{l,i}\\phi\\left(  c_{l,i}\\right)  h_{1}^{\\prime}\\left(\n\\theta_{i}\\right)  a_{i}^{-1\/2}a_{j}^{-1\/2}\\right\\vert \\leq M\\text{.\n\\label{eqb52\n\\end{equation}\nSo\n\\[\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{c_{1,i},c_{2,i}}\\left\\vert\n\\phi\\left(  c_{1,i}\\right)  h_{1}^{\\prime}\\left(  \\theta_{i}\\right)\n-\\phi\\left(  c_{2,i}\\right)  h_{1}^{\\prime}\\left(  \\theta_{i}\\right)\n\\right\\vert \\leq M\n\\]\nan\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left(  r_{1,i},r_{2,i},a_{i\n,a_{j}\\right)  }\\left\\vert \\tau_{ij}\\left(  c_{i},\\theta_{i}\\right)\n\\right\\vert \\leq M\\text{.}\\label{eqb18\n\\end{equation}\nwhere\n\\[\n\\tau_{ij}\\left(  c_{i},\\theta_{i}\\right)  =\\left[  c_{1,i}\\phi\\left(\nc_{1,i}\\right)  h_{1}^{\\prime}\\left(  \\theta_{i}\\right)  -c_{2,i}\\phi\\left(\nc_{2,i}\\right)  h_{1}^{\\prime}\\left(  \\theta_{i}\\right)  \\right]  a_{i\n^{-1\/2}a_{j}^{-1\/2}\\text{.\n\\]\nTherefore\n\\begin{align}\n& m^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert r_{3,1}\\left(  c_{i},x\\right)\n\\right\\vert x^{2}\\left\\vert z\\right\\vert \\phi\\left(  z\\right)  dz\\label{eqb53\n\\\\\n& \\leq Mm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}x^{2}=Mm^{-2\\delta}\\nonumber\n\\end{align}\nan\n\\begin{align}\n&  m^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert r_{3,2}\\left(  c_{i},x\\right)\n\\right\\vert \\left\\vert x\\right\\vert \\phi\\left(  z\\right)  dz\\label{eqb54}\\\\\n&  \\leq Mm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left\\vert x\\right\\vert \\leq Mm^{-\\delta\n}\\text{,}\\nonumber\n\\end{align}\nwhere we have used the inequality (\\ref{eqb15}) and (\\ref{eqb17}).\n\nThe last term $\\varkappa_{m}=m^{-2}\\sum_{1\\leq i<j\\leq m}\\int\\left\\vert\nr_{4}\\left(  c_{i},x\\right)  \\right\\vert \\phi\\left(  z\\right)  dz$ needs\nintricate treatment. The summands in $\\varkappa_{m}$ that involve $a_{i}$ with\n$i\\in E_{3}$ do not inflate the order of $\\varkappa_{m}$ to be greater than\n$m^{-\\delta}$ since such $a_{i}$'s are uniformly bounded from both above and\nbelow, $\\phi\\left(  \\cdot\\right)  $ is rapidly decreasing and smooth, and\n(\\ref{eqb15}) holds. Therefore, to assess the order of $\\varkappa_{m}$, it\nsuffices to deal with its summands that involve $a_{i}$ and $a_{j}$ with\n$\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}=I_{\\varepsilon,m}^{+}\\cap\\left(  E_{1}\\otimes\nE_{1}\\right)  $ and work with the set $B\\left(  \\sqrt{\\left\\vert c\\right\\vert\n}\\right)  =\\left\\{  z:\\left\\vert z\\right\\vert \\leq\\sqrt{\\left\\vert\nc\\right\\vert }\\right\\}  $, where $\\left\\vert c \\right\\vert$ can be either $\\left\\vert c_{1,i} \\right\\vert$ or\n$\\left\\vert c_{2,i} \\right\\vert$ (which is eventually sufficiently large). Since\n\\begin{align*}\n\\liminf_{\\left\\vert c\\right\\vert \\rightarrow\\infty}\\dfrac{r_{1}\\left(\nc,x\\right)  }{c} &  =\\liminf_{\\left\\vert c\\right\\vert \\rightarrow\\infty\n}\\left(  \\dfrac{xzb^{-1}}{c}+\\dfrac{bh_{1}^{\\prime}\\left(  \\theta\\right)\nxc}{c}+\\dfrac{xzh_{1}^{\\prime}\\left(  \\theta\\right)  x}{c}\\right)  \\\\\n&  \\geq\\liminf_{\\left\\vert c\\right\\vert \\rightarrow\\infty}\\left(\nbh_{1}^{\\prime}\\left(  \\theta\\right)  x+\\dfrac{z}{c}h_{1}^{\\prime}\\left(\n\\theta\\right)  x^{2}\\right)  \\\\\n&  \\geq\\liminf_{\\left\\vert c\\right\\vert \\rightarrow\\infty}bh_{1}^{\\prime\n}\\left(  \\theta\\right)  x\\geq0\n\\end{align*}\nregardless of the sign of $x$, we see that $c$ and $r_{1}\\left(  c,x\\right)  $\nhave the same sign, i.e., $cr_{1}\\left(  c,x\\right)  \\geq0$. This means either\n$c+r_{1}\\left(  c,x\\right)  <\\theta_{\\ast}<c<0$ or $0<c<\\theta_{\\ast\n<c+r_{1}\\left(  c,x\\right)  $,$\\ $an\n\\[\n2\\max\\left\\{  \\left\\vert r_{1}\\left(  c,x\\right)  \\right\\vert ,\\left\\vert\nc\\right\\vert \\right\\}  \\geq\\left\\vert \\theta_{\\ast}\\right\\vert >\\max\\left\\{\n\\left\\vert r_{1}\\left(  c,x\\right)  \\right\\vert ,\\left\\vert c\\right\\vert\n\\right\\}  \\text{.\n\\]\nNoticing furthe\n\\begin{align*}\n\\limsup_{m\\rightarrow\\infty}\\dfrac{r_{1}\\left(  c,x\\right)  }{c} &\n\\leq\\limsup_{m\\rightarrow\\infty}\\left(  1+\\left\\vert \\dfrac{z}{c}\\right\\vert\n\\right)  \\left\\vert b^{2}h_{1}^{\\prime}\\left(  \\theta\\right)  \\right\\vert \\\\\n&  \\leq\\limsup_{m\\rightarrow\\infty}\\left(  1+\\left\\vert \\dfrac{z\n{c}\\right\\vert \\right)  \\left\\vert \\left[  1-\\left(  1-\\tilde{\\varepsilon\n\\right)^{2}  \\right]  ^{-3\/2}\\right\\vert \\\\\n&  \\leq\\left(  1-\\tilde{\\varepsilon}\\right)  ^{-1}\\tilde{\\rho}<\\infty\\text{,\n\\end{align*}\nwe se\n\\[\n\\left\\{\n\\begin{array}\n[c]{c\n\\left\\vert r_{1}\\left(  c,x\\right)  \\right\\vert \\leq M\\left(  1-\\tilde{\\varepsilon}\n\\right)  ^{-1}\\tilde{\\rho}\\left\\vert c\\right\\vert \\text{,}\\\\\nM\\left(  1-\\tilde{\\varepsilon}\\right)  ^{-1}\\tilde{\\rho}\\left\\vert c\\right\\vert\n\\geq\\left\\vert \\theta_{\\ast}\\right\\vert >\\left\\vert c\\right\\vert \\text{,\n\\end{array}\n\\right.\n\\]\nan\n\\[\n\\left\\vert r_{2}\\left(  c,x\\right)  \\right\\vert \\leq\\left\\vert \\theta_{\\ast\n}e^{-\\theta_{\\ast}^{2}\/2}r_{1}^{2}\\left(  c,x\\right)  \\right\\vert \\leq\nM\\left\\vert c\\right\\vert ^{3}\\exp\\left(  -\\left\\vert c\\right\\vert\n^{2}\/2\\right)  \\text{,\n\\]\nwhere we used the fact that $t\\phi\\left(  t\\right)  $ is decreasing in $t>0$.\nFro\n\\begin{align*}\n&  r_{1}^{2}\\left(  c,x\\right)  \\\\\n&  =x^{2}b^{-2}z^{2}+2x^{3}b^{-1}h_{1}^{\\prime}\\left(  \\theta\\right)\nz^{2}+2x^{2}ch_{1}^{\\prime}\\left(  \\theta\\right)  z+x^{4}h_{1}^{\\prime\n2}\\left(  \\theta\\right)  z^{2}\\\\\n&  +2x^{3}bch_{1}^{\\prime2}\\left(  \\theta\\right)  z+b^{2}c^{2}h_{1}^{\\prime\n2}\\left(  \\theta\\right)  x^{2}\\\\\n&  =r_{1,1}\\left(  c,z\\right)  x^{2}+2r_{1,2}\\left(  c,z\\right)  x^{3\n+h_{1}^{\\prime2}\\left(  \\theta\\right)  x^{4}z^{2}\\\\\n&  \\leq r_{1,1}^{\\ast}\\left(  c,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)\nx^{2}+r_{1,2}^{\\ast}\\left(  c,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)\n\\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(  \\theta\\right)  \\left\\vert\nc\\right\\vert x^{4}\\text{,\n\\end{align*}\nwe see\n\\[\n\\int_{B\\left(  \\sqrt{c}\\right)  }\\left\\vert r_{2}\\left(  c,x\\right)\n\\right\\vert \\phi\\left(  z\\right)  dz\\leq M\\left\\vert c\\right\\vert \\exp\\left(\n-\\left\\vert c\\right\\vert ^{2}\/2\\right)  g^{\\ast\\ast}\\left(  \\left\\vert\nx\\right\\vert \\right)  \\text{,\n\\]\nwher\n\\[\ng^{\\ast\\ast}\\left(  \\left\\vert x\\right\\vert \\right)  =r_{1,1}^{\\ast}\\left(\nc,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)  x^{2}+r_{1,2}^{\\ast}\\left(\nc,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)  \\left\\vert x\\right\\vert\n^{3}+h_{1}^{\\prime2}\\left(  \\theta\\right)  \\left\\vert c\\right\\vert x^{4\n\\]\nan\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\nr_{1,1}\\left(  c,z\\right)  =b^{-2}z^{2}+2ch_{1}^{\\prime}\\left(  \\theta\\right)\nz+b^{2}c^{2}h_{1}^{\\prime2}\\left(  \\theta\\right)  \\text{,}\\\\\nr_{1,2}\\left(  c,z\\right)  =b^{-1}h_{1}^{\\prime}\\left(  \\theta\\right)\nz^{2}+bch_{1}^{\\prime2}\\left(  \\theta\\right)  z\\text{,}\\\\\nr_{1,1}^{\\ast}\\left(  c,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)\n=b^{-2}\\left\\vert c\\right\\vert +2\\left\\vert c\\right\\vert ^{3\/2}\\left\\vert\nh_{1}^{\\prime}\\left(  \\theta\\right)  \\right\\vert +b^{2}c^{2}h_{1}^{\\prime\n2}\\left(  \\theta\\right)  \\text{,}\\\\\nr_{1,2}^{\\ast}\\left(  c,h_{1}^{\\prime}\\left(  \\theta\\right)  \\right)\n=b^{-1}h_{1}^{\\prime}\\left(  \\theta\\right)  \\left\\vert c\\right\\vert\n+\\left\\vert bh_{1}^{\\prime2}\\left(  \\theta\\right)  \\right\\vert \\left\\vert\nc\\right\\vert ^{3\/2}\\text{.\n\\end{array}\n\\right.\n\\]\nSince $\\varepsilon a_{i}\\leq\\left\\vert c_{l,i}\\right\\vert $ for $l=1,2$ and $1\\leq\ni\\leq m$ by (\\ref{eqb81}) and $\\lim_{m\\rightarrow\\infty}a_{i}=\\infty$, we ge\n\\[\n\\sup_{m\\geq1}\\max_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\sup_{c,b}\\left\\{  \\left\\vert c\\right\\vert\n^{3}\\exp\\left(  -\\left\\vert c\\right\\vert ^{2}\/2\\right)  b^{-2}h_{1}^{\\prime\n2}\\left(  \\theta\\right)  \\right\\}  \\leq M\n\\]\nand $M\\left\\vert c\\right\\vert \\exp\\left(  -\\left\\vert c\\right\\vert\n^{2}\/2\\right)  g^{\\ast\\ast}\\left(  \\left\\vert x\\right\\vert \\right)  $ is thus\ndominated by a polynomial with uniformly bounded coefficients whose lowest\ndegree in $\\left\\vert x\\right\\vert $ is $2$. Hence\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\int_{B\\left(  \\sqrt{c}\\right)  }\\left\\vert\nr_{2}\\left(  c,x\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\leq\nMm^{-2\\delta}\\text{.}\\label{eqb55\n\\end{equation}\n\n\nOn the other hand, on $\\mathbb{R}\\backslash B\\left(  \\sqrt{c}\\right)  $ we\nhav\n\\begin{align*}\n&  \\left\\vert r_{1}^{2}\\left(  c,x\\right)  \\right\\vert \\\\\n&  \\leq\\left\\vert r_{1,1}\\left(  c,z\\right)  \\right\\vert x^{2}+2\\left\\vert\nr_{1,2}\\left(  c,z\\right)  \\right\\vert x^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right)  x^{4}z^{2}\\\\\n&  \\leq\\left[  b^{-2}z^{2}+2\\left\\vert ch_{1}^{\\prime}\\left(  \\theta\\right)\nz\\right\\vert +b^{2}c^{2}h_{1}^{\\prime2}\\left(  \\theta\\right)  \\right]  x^{2}\\\\\n&  +\\left[  \\left\\vert 2b^{-1}h_{1}^{\\prime}\\left(  \\theta\\right)\nz^{2}+2bch_{1}^{\\prime2}\\left(  \\theta\\right)  z\\right\\vert \\right]\n\\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(  \\theta\\right)  x^{4}\\\\\n&  \\leq\\left[  b^{-2}\\left\\vert c\\right\\vert +2\\left\\vert ch_{1}^{\\prime\n}\\left(  \\theta\\right)  \\right\\vert \\left\\vert c\\right\\vert ^{1\/2}+b^{2\nc^{2}h_{1}^{\\prime2}\\left(  \\theta\\right)  \\right]  x^{2}\\\\\n&  +2\\left(  b^{-1}h_{1}^{\\prime}\\left(  \\theta\\right)  \\left\\vert\nc\\right\\vert +2bch_{1}^{\\prime2}\\left(  \\theta\\right)  \\left\\vert c\\right\\vert\n^{1\/2}\\right)  \\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right)  x^{4}\\\\\n&  \\leq M\\left(  \\left\\vert c\\right\\vert ^{3}+\\left\\vert h_{1}^{\\prime}\\left(\n\\theta\\right)  \\right\\vert \\left\\vert c\\right\\vert ^{3\/2}+c^{2}h_{1}^{\\prime\n2}\\left(  \\theta\\right)  \\right)  x^{2}\\\\\n&  +M\\left(  h_{1}^{\\prime}\\left(  \\theta\\right)  \\left\\vert c\\right\\vert\n^{2}+h_{1}^{\\prime2}\\left(  \\theta\\right)  \\left\\vert c\\right\\vert\n^{3\/2}\\right)  \\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right)  x^{4}\\\\\n&  \\leq M\\left\\vert c\\right\\vert ^{3}\\left(  x^{2}+\\left\\vert x\\right\\vert\n^{3}+x^{4}\\right)  \\leq M\\left\\vert c\\right\\vert ^{3}x^{2\n\\end{align*}\nan\n\\begin{align*}\n\\left(  \\int_{\\mathbb{R}\\backslash B\\left(  \\sqrt{c}\\right)  }\\left\\vert\nr_{1}\\left(  c,x\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\right)  ^{2} &\n\\leq\\int_{\\mathbb{R}\\backslash B\\left(  \\sqrt{c}\\right)  }\\left\\vert r_{1\n^{2}\\left(  c,x\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\\\\n&  \\leq x^{2}\\int_{\\mathbb{R}\\backslash B\\left(  \\sqrt{c}\\right)  }M\\left\\vert\nc\\right\\vert ^{3}\\phi\\left(  z\\right)  dz\\leq Mx^{2\n\\end{align*}\nsince $\\lim_{\\left\\vert c\\right\\vert \\rightarrow\\infty}\\left\\vert c\\right\\vert\n^{s^{\\prime}}\\phi\\left(  \\left\\vert c\\right\\vert \\right)  =0$ for any\n$s^{\\prime}>0$. This mean\n\\[\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\int_{\\mathbb{R}\\backslash B\\left(  \\sqrt\n{c}\\right)  }\\left\\vert r_{2}\\left(  c,x\\right)  \\right\\vert \\phi\\left(\nz\\right)  dz\\leq Mm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\left\\vert x\\right\\vert\n=Mm^{-\\delta}\\text{,\n\\]\nwhich, with (\\ref{eqb55}), implie\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\int\\left\\vert r_{2}\\left(  c,x\\right)\n\\right\\vert \\phi\\left(  z\\right)  dz\\leq Mm^{-\\delta}\\text{.}\\label{eqb66\n\\end{equation}\nTherefor\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{\\ast}}\\int\\left\\vert r_{4}\\left(  c_{i},x\\right)\n\\right\\vert \\phi\\left(  z\\right)  dz\\leq Mm^{-\\delta}\\text{.}\\label{eqb67\n\\end{equation}\nCombining (\\ref{eqb53}), (\\ref{eqb54}) and (\\ref{eqb67}) give\n\\begin{equation}\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{+}}\\int\\left\\vert \\Delta\\tilde{o}_{i}\\left(\nx\\right)  \\right\\vert \\phi\\left(  z\\right)  dz\\leq Mm^{-\\delta\n\\text{.}\\label{eqb76\n\\end{equation}\nThus (\\ref{eqb49}) and (\\ref{eqb46}) hold with $\\delta_{1}=2\\delta$. Consequently\n\\[\nm^{-2}\\sum_{\\left(  i,j\\right)  \\in I_{\\varepsilon,m}^{+}}\\int\\left\\vert\ng_{ij}\\left(  \\cdot\\right)  \\right\\vert \\phi\\left(  z\\right)  dz=\n\\left(  m^{-2\\delta}\\right)  \\text{,\n\\]\nwhich, with (\\ref{eqb56}), implies $\\mathbb{V}\\left[  m^{-1}\\sum\n\\nolimits_{i=1}^{m}X_{i}\\right]  =O\\left(  m^{-2\\delta}\\right)  $\nexcept on the event $G_{t,\\varepsilon}$. This completes the proof.\n\\end{proof}\n\nUsing \\autoref{LemmaLyons}, we have the following corollary from\n\\autoref{MainTheoremNormalOneThresholdWeak}.\n\n\\begin{corollary}\n\\label{ASconvNormal}The same conditions of\n\\autoref{MainTheoremNormalOneThresholdWeak} impl\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}\\left\\vert m^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{^{k\n}\\right)  -\\mathbb{E}\\left[  m^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{^{k\n}\\right)  \\right]  \\right\\vert =0\\label{10\n\\end{equation}\noutside the event $G_{t,\\varepsilon}$.\n\\end{corollary}\n\nFrom the proof of \\autoref{MainTheoremNormalOneThresholdWeak}, we see that\namong the functional remainders there are mainly two types that can diverge\nto infinity as $m\\rightarrow\\infty$:\n\n\\begin{enumerate}\n\\item $\\phi\\left(  a_{i,m}r_{i}\\right)  h^{\\prime}\\left(  \\theta_{i}\\right)\na_{i,m}$ for which $\\left\\vert a_{i,m}r_{i}\\right\\vert =O\\left(  1\\right)  $,\n$a_{i,m}\\rightarrow\\infty$ and $\\left\\vert h^{\\prime}\\left(  \\theta\n_{i}\\right)  \\right\\vert \\rightarrow\\infty$, where $r_{i}=\\pm z_{t\/2}-\\eta\n_{i}-u_{i}$.\n\n\\item $\\phi\\left(  a_{i,m}r_{i}\\right)  h^{\\prime}\\left(  \\theta_{i}\\right)\na_{i,m}^{-1}a_{j,m}$ for which $\\left\\vert a_{i,m}r_{i}\\right\\vert\n\\rightarrow\\infty$, $r_{i}=O\\left(  1\\right)  $, $a_{i,m},a_{j,m\n\\rightarrow\\infty$, $h^{\\prime}\\left(  \\theta_{i}\\right)  \\rightarrow\\infty$\nbut $a_{j,m}\\geq e^{Ma_{i,m}^{2}}$ for some $M>0$ large when $m$ is large enough.\n\\end{enumerate}\nSuch terms will very likely inflate $\\mathbb{V}\\left[  \\tilde{R}\\left(  t\\right)  \\right]\n$ out of the desired order $m^{-\\delta}$ for the SLLN for $\\tilde{R}\\left(\nt\\right)  $ to hold. In fact, the event $G_{t,\\varepsilon}$ contains all\n$\\omega\\in\\Omega$ for which the above two cases can happen but it does not\nnecessarily have diminishing probability.\n\nWe briefly describe the roles the two additional conditions (\\ref{eqb22b}) and\n(\\ref{eqb24b}). When $a_{i,m}\\rightarrow\\infty$ but $a_{i.m}<\\infty$ for\nfinite $m$, $a_{i,m}$ represents the rate\nat which $\\tilde{\\eta}_{i}=\\eta_{i}+\\mu_{i}$ stochastically approximates\n$Z_{i}$, and condition (\\ref{eqb22b}) requires that the relative rate at which\n$a_{i,m}$ $\\rightarrow\\infty$ can not be exponential. Condition (\\ref{eqb24b})\ncontrols how many pairs of $\\left(  v_{i},v_{j}\\right)  $, $i\\neq j$ can be\nhighly correlated and restricts the contribution of such pairs in the variance\nof $m^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{k}\\right)  $. It also controls the\n\\textquotedblleft speed\\textquotedblright\\ at which $\\mathbf{\\Sigma}_{m}>0$\ncan approach to singularity. In addition, it ensures the validity and accuracy of the expansion\n(\\ref{eqb27}) (since (\\ref{eqb27}) is undefined when $\\left\\vert \\rho\n_{ij}\\right\\vert =1$), and prevents $h_{1}^{\\prime}\\left(  \\theta\\right)  $ in\n(\\ref{eqb37}) from diverging to infinity (unexpectedly fast). Further, it induces (\\ref{eqb57}),\nand together with condition (\\ref{eqb22b}) it validates (\\ref{eqb53}),\n(\\ref{eqb54}) and (\\ref{eqb67}).\nThese extra sufficient conditions ensure the uniform boundedness of the\ninvolved functional remainders in such Taylor expansions and induce (\\ref{eqb20})\noutside the event $G_{t,\\varepsilon}$.\n\n\\section{Discussion\\label{SecDiscussion}}\n\nUnder two more conditions rather than none in \\cite{Fan:2012}, we have\nreformulated the SLLN for the normalized number of conditional rejections when\ntesting which components of a Normal rv $\\mathbf{Z}_{m}\\sim\\mathsf{N\n_{m}\\left(  \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right)  $ have zero means when its\ncorrelation matrix $\\mathbf{\\Sigma}_{m}$ is known. Our proof shows that the speed of the\nPFA to the original Normal rv, the degree of dependency between components of\nthe Normal rv, and the magnitudes of the conditional component means should be compatible\nwith each other in order to yield such a law. Relaxation of (\\ref{eqb22b}) and\n(\\ref{eqb24b}) is possible. But such a\nlaw may not hold for arbitrary $\\boldsymbol{\\mu}_{m}$ and arbitrary\n$\\mathbf{\\Sigma}_{m}\\geq0$, since, on the even\n\\[\nH_{t,\\varepsilon,\\tilde{\\varepsilon}}=\\left\\{  \\omega\\in\\Omega:1-\\tilde\n{\\varepsilon}\\leq\\rho_{ij}<1,a_{i,m}^{-2}\\log a_{j,m}\\geq1,i,j\\in E_{1},i\\neq\nj\\right\\}  \\cap G_{t,\\varepsilon\n\\]\nwhen $0<\\max\\left\\{  \\varepsilon,\\tilde{\\varepsilon}\\right\\}  \\ll1$\n\\[\n\\mathbb{P}\\left\\{  \\lim_{m\\rightarrow\\infty}m^{-2+\\delta}\\sum\\nolimits_{1\\leq\ni<j\\leq m}\\int\\left\\vert \\Delta\\tilde{o}_{j}\\left(  x\\right)  \\right\\vert\n\\phi\\left(  z\\right)  dz=\\infty\\right\\}  >0\n\\]\n(see (\\ref{eqb121}) for the definition of $\\Delta\\tilde{o}_{j}\\left(\nx\\right)  $) and\n\\begin{equation}\n\\mathbb{P}\\left\\{  \\limsup_{m\\rightarrow\\infty}m^{\\delta}\\mathbb{V}\\left[\nm^{-1}R\\left(  t|\\mathbf{\\tilde{w}}_{k}\\right)  \\right]  =\\infty\\right\\}\n>0\\label{eqb104\n\\end{equation}\ncan happen for any $\\delta >0$, which then invalidates (\\ref{9}) and (\\ref{eqb5}).\nSince estimating the FDP and control of the FDR under other strong but\nunknown types of dependency among the test statistics still remains a highly\nchallenging problem, such an area of research requires\ndevelopment of new methods and clarification of the boundaries beyond which\ncertain methods become fragile or invalid.\n\n\\bibliographystyle{imsart-nameyear}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOne important parameter for the determination of the unitarity triangle\nof the CKM matrix is $V_{ub}$. While its phase can only be determined \nfrom non--leptonic decays, its magnitude will be determined from \nsemileptonic processes. Various methods have been proposed to extract \n$V_{ub}$ from both inclusive as well as exclusive semileptonic decays, \nand the final numbers after the era of the $B$ factories will probably \nbe obtained from a mixture of both exclusive and inclusive methods. \n\nThe disadvantage of exclusive decays is their dependence on form factors\nwhich are nonperturbative quantities. Modelling these form factors \nnecessarily introduces some uncontrollable systematic uncertainty into \nthe determination of $V_{ub}$ and hence a model independent method is\ndesirable. At present the values of $V_{ub}$ extracted from exclusive \nsemileptonic $B\\to\\pi$ decays using lattice QCD, QCD sum rules and \nquark models range from $|V_{ub}| = (2.5 - 4.5)\\times10^{-3}$. Clearly\nthis is not a satisfactory situation, in particular in view of the \ndata to be expected from the $B$ factories.  Although there have been\nrecent efforts to update existing models (see\ne.g. \\cite{hphx9711268}--\\nocite{hphx9712399,hphx9801421}\\cite{hphx9801443}), \nthere is still room for some improvement.\n\nThere are constraints on form factors originating from their analyticity \nproperties and the unitarity of the underlying theory. The ideas \nhow to implement these constraints are in fact quite old \n\\cite{prvxd4x3519}--\\nocite{prvxd3x2807,prvxd4x725,prvxd4x2020}\\cite{npxb189x157}\nand have recently attracted renewed attention. In particular, combining these\nunitarity bounds with lattice data \\cite{hphx9509358} has lead to relatively\ntight and model independent bounds on the form factors of $B \\to \\pi$\ntransitions.\n\nIn the present paper we improve the application of unitarity bounds \nin such a way that not only points at which the form factor is known\n(e.g. from lattice data or heavy quark relations) can be included, \nbut also the slope and higher derivatives at some point, \nwhich could be known for instance from \nsum rule considerations. At the point of maximal \nmomentum transfer the derivatives are correlated with the value of the \nform factor due to analyticity and we shall discuss the restrictions\nobtained from this in some detail.   \n\nA similar improvement of the unitarity bounds has been proposed by\nBoyd et al.~\\cite{hphx9702300,hphx9705252}, where the form factors for \n$B \\to \\pi \\ell \\nu$ are expanded in a cleverly chosen conformal \nvariable $z$ and some of the relations we use in the present paper appear \nin a similar form in these references. \n\nIn the next section we shall summarize the formalism as described in \n\\cite{npxb189x157}.\nThe improved bounds are considered in section \\ref{secimprov} where we\ngeneralize the method to include known slopes and higher \nderivatives. We make use of the correlation between the derivatives \nand the value of the form factor taken at $q^2_{max}$ to tighten \nthe bounds even without using any additional physics input. In order \nto get useful bounds some physics input is necessary and we chose to \nuse the chiral limit which is valid close to the end point. This has \na strong effect since any knowledge of the form factors in the \nend point region tightens the bounds significantly.  \nWith this input we study numerical examples. \n\\section{Bounds on Form Factors}\n\\label{secbounds}\nFor later use we summarize in this section the formalism how to derive \nbounds on the form factors using only perturbative QCD, unitarity and \nthe ana\\-ly\\-ti\\-ci\\-ty of the form factors in the complex plane. \n\nWe choose to describe the hadronic matrix element of the semileptonic \n$\\bar B^0 \\to \\pi^+ \\ell^- \\bar\\nu_\\ell$ decays with the following two \nform factors:\n\\begin{eqnarray}\n\\label{anfang}\n\\langle \\pi^+(p')|V^\\mu|\\bar B^0(p)\\rangle\n&=&\\left(p^\\mu+{p'}^\\mu-\\frac{M^2-m^2}{q^2}q^\\mu\\right)f^+(q^2)\\\\\n&&+\\frac{M^2-m^2}{q^2}q^\\mu f^0(q^2)\\nonumber\n\\end{eqnarray}\nwhere $M^2=m_B^2$, $m^2=m_\\pi^2$, $V^\\mu=\\bar u\\gamma^\\mu b$ and $q=p-p'$.\nIn this notation $q^2$ runs from $q^2_{min}=m_\\ell^2$ to\n$q^2_{max}=(M-m)^2$. Throughout this paper we will neglect the lepton masses\nand therefore set $q^2_{min}=0$. Furthermore, this special choice of\ndecomposition into Lorentz vectors leads to form factors which have to satisfy\nthe kinematical constraint\n\\begin{equation}\n\\label{kinconst}\nf^+(0) = f^0(0).\n\\end{equation}\nThe form factors are real functions of a real variable but it is\nconvenient to think of them as analytic functions in the complex\n$q^2$--plane. \n\nTo derive bounds on $f^+(q^2)$ and $f^0(q^2)$ we consider the\ntwo--point function \n\\begin{eqnarray}\n\\label{2pfunction}\n\\Pi^{\\mu\\nu} &\\equiv& i\\int\\intf{x}\\ex{iq\\cdot\nx}\\langle0|T\\{V^\\mu(x)V^{\\nu\\dagger}(0)\\}|0\\rangle \\nonumber\\\\\n&\\equiv& -(g^{\\mu\\nu}q^2 - q^\\mu q^\\nu)\\Pi_T(q^2) + q^\\mu q^\\nu \\Pi_L(q^2),\n\\end{eqnarray}\nwhere $V^\\mu=\\bar u\\gamma^\\mu b$ as above and with\n$\\Pi_{T\/L}(q^2)$ corresponding to the propagation of a $J^P=1^-\/0^+$\nparticle. This two--point function can be (and has been) reliable evaluated\nin the deep euclidean region where $-q^2 \\equiv Q^2 \\gg \\Lambda^2\\unter{QCD}$,\ni.e. at an energy scale where perturbative QCD is appliccable.\n\nIncluding a sum over all possible intermediate states\n$\\Gamma$ we get the following result for the imaginary part of\n (\\ref{2pfunction}): \n\\begin{eqnarray}\n\\label{states}\n-(g^{\\mu\\nu} q^2 - q^\\mu q^\\nu)\\im{\\Pi_T(q^2)}+q^\\mu q^\\nu\\im{\\Pi_L(q^2)}\n\\nonumber \\\\\n= \\frac 12\\sum_\\Gamma(2\\pi)^4 \\delta^{(4)}(q-p_\\gamma)\\langle0|V^\\mu(0)|\\Gamma\n\\rangle\\langle\\Gamma|V^{\\nu\\dagger}(0)|0\\rangle.\n\\end{eqnarray}\nWe therefore get an equation for the spectral functions $\\im{\\Pi_{T\/L}(q^2)}$\n(i.e. for the absorptive parts of $\\Pi_{T\/L}(q^2)$) which we can relate to the\nreal parts using the substracted dispersion relations\n\\begin{eqnarray}\n\\label{displ}\n\\chi_L(Q^2)\\equiv\\left(-\\pabl{}{Q^2}\\right)(-Q^2 \\Pi_L(Q^2)) =\\frac\n1\\pi\\int_0^\\infty\\into{t} \\frac{t\\im{\\Pi_L(t)}}{(t+Q^2)^2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{dispt}\n\\chi_T(Q^2)\\equiv\\frac12\\left(-\\pabl{}{Q^2}\\right)^2(-Q^2 \\Pi_T(Q^2)) =\\frac\n1\\pi\\int_0^\\infty\\into{t} \\frac{t\\im{\\Pi_T(t)}}{(t+Q^2)^3}.\n\\end{eqnarray}\n\nWe now restrict ourselves to include only contributions of the $B^*$ and\nthe $B\\pi$ states in the sum over all intermediate states in\n (\\ref{states}).\nIt is possible to discard all other intermediate states because \n (\\ref{states}) is a sum of positive terms if the indices are treated\nsymmetrically. Using isospin and crossing symmetry we can use\n(\\ref{anfang}) to express (\\ref{states}) by the two form factors.\nProjecting out now the transversal\/longitudinal parts, one gets the\ninequalities\n\\begin{eqnarray}\n\\label{ineql}\n\\im{\\Pi_L(t)}\\ge\\frac 32\\frac{t_+ t_-}{16\\pi}\\sqrt{(t-t_+)(t-t_-)}\\frac\n{|f^0(t)|^2}{t^3}\\Theta(t-t_+)\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{ineqt}\n\\im{\\Pi_T(t)} &\\ge&\\pi\\left(\\frac{m_{B^*}}{f_{B^*}}\\right)^2\n\\delta(t-m_{B^*}^2) \\nonumber\\\\\n&+& \\frac32 \\frac\n1{48\\pi}\\frac{[(t-t_+)(t-t_-)]^{3\/2}}{t^3}|f^+(t)|^2\\Theta(t-t_+),\n\\end{eqnarray}\nwhere $t=q^2$ and $t_\\pm = (M\\pm m)^2$.\n \nIt is now possible to get bounds on the form factors if one inserts\n(\\ref{ineql},\\ref{ineqt}) in (\\ref{displ},\\ref{dispt}). Since the l.h.s. of \n(\\ref{displ},\\ref{dispt}) can be calculated for $Q^2 \\gg \\Lambda^2\\unter{QCD}$\nusing perturbative QCD one gets inequalities which restrict the form factors. \nThese inequalities take the form (in shorthand notation) \n\\begin{eqnarray}\n\\label{shorthand}\nJ(Q^2) \\ge \\frac 1\\pi \\int_{t_+}^\\infty\\into{t} k(t,Q^2)|f(t)|^2\n\\end{eqnarray}\nwhere $J(Q^2)$ denotes the QCD input (i.e. the perturbative calculation of\n$\\chi_{T\/L}$) including the $B^*$--resonance in case of $f^+$\nand $k(t,Q^2)$ is a known kinematical function. The exact value\/structure of \n$J(Q^2)$ and $k(t,Q^2)$ does of course depend on the form factor under\nexamination.\n\nTo translate the inequality (\\ref{shorthand}) in constraints on the form factor\nfor values of $t$ in the range $[0,t_-]$ (which is the kinematical region of \nphysical interest), we map the complex $t$--plane into the unit disc with \nthe conformal transformation\n\\begin{eqnarray}\n\\frac{1+z}{1-z} = \\sqrt{\\frac{t_+-t}{t_+-t_-}}\n\\end{eqnarray}\nso that (\\ref{shorthand}) becomes\n\\begin{eqnarray}\n\\label{main}\nJ(Q^2)\\ge\\int_{|z|=1} \\frac{\\mbox{d}z}{2\\pi i z}|\\phi(z,Q^2) f(z)|^2\n\\end{eqnarray}\nwith $f(z)\\hat = f(t(z))$.\nHere we have used the fact, that $k(t,Q^2)$ is a positive definite\nquantity so that $\\phi(z,Q^2)\\equiv\\sqrt{k(t(z),Q^2)}$ times the squareroot\nof the Jacobian of the transformation. \n\nThe value of the form factor $f(z)$ for any point $z(t)$ is accessible \nby defining an inner product\n\\begin{eqnarray}\n\\langle g|h \\rangle = \\int_{|z|=1} \\frac{\\mbox{d}z}{2\\pi i z} \\bar g(z)h(z)\n\\end{eqnarray}\nand by considering the product $\\langle g_t|\\phi f\\rangle$ where\n\\begin{eqnarray}\ng_t = \\frac 1{1-\\bar z(t)z},\n\\end{eqnarray}\nso that $f(z(t))$ has no poles in the range $[0,t_-]$. Cauchy's \ntheorem now yields\n\\begin{eqnarray}\n\\label{einf}\n\\langle g_t|\\phi f\\rangle = \\phi(z(t),Q^2) f(z(t)).\n\\end{eqnarray}\nOn the other hand, if there is a pole at $t=t_p$ away from the cut in the\ncomplex $t$--plane (i.e. for $t>t_-$), one would obtain\n\\begin{eqnarray}\n\\langle g_t|\\phi f\\rangle = \\phi(z(t),Q^2) f(z(t)) + \\frac{\\mbox{Res}\n\\{\\phi f,z(t_p)\\}}{z(t_p)-z(t)}.\n\\end{eqnarray}\nThe residue can either be approximated or eliminated completely by the \ntransformation \\cite{plxb301x257}\n\\begin{eqnarray}\n\\label{phitrafo}\n\\phi(z,Q^2)\\quad \\to \\quad \\phi_p(z,Q^2)\\equiv \\phi(z,Q^2)\\frac {z-z(t_p)}\n{1-\\bar z(t_p)z}\n\\end{eqnarray}\nwhere $t_p$ is assumed to lie in the range $[t_-,t_+]$ so that $\\phi_p$ is \npositive for $z=z(t)$ with $t$ in $[0,t_-]$. This replacement cancels the\npole of $f(t)$ so that (\\ref{einf}) holds. The crucial property of this\ntransformation is the fact that, since $|(z-z(t_p))\/(1-\\bar z(t_p)z)| =1$\nfor $z$ on the unit circle, $\\langle \\phi f|\\phi f\\rangle=\\langle \\phi_p\nf|\\phi_p f\\rangle$ and the QCD constraints are left unchanged.\n\n\\begin{figure}[ht]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{without.ps} \\\\\n\\caption{Bounds on the form factors $f^0(t)$ (on the left side, $t$ increasing \nto the left) and $f^+(t)$ (on the right side), derived\nwithout any additional constraints and plotted over the whole kinematical\nrange of $[0,t_-]$ ($t$ in $\\mbox{GeV}^2$).\n\\label{figwithout}} \n\\end{figure}\n\nBecause of the positivity of the inner product we have\n\\begin{eqnarray}\n\\Det{\n\\begin{array}{cc}\n\\langle \\phi f |\\phi f \\rangle & \\langle \\phi f |g_t \\rangle \\\\\n\\langle g_t |\\phi f \\rangle & \\langle g_t |g_t \\rangle\n\\end{array}\n} \\ge 0,\n\\end{eqnarray}\nwhich, by eliminating $\\langle\\phi f|\\phi f\\rangle$ with (\\ref{main}), \ngives us the following bounds on the form factors\n\\begin{eqnarray}\n\\label{withoutpoints}\n|f(t)|^2 \\le J(Q^2)\\frac1{1-z^2(t)}\\frac1{|\\phi(z(t),Q^2)|^2}\n\\end{eqnarray}\nwhere we have made use of the fact that $z(t)$ is real for $t$ in $[0,t_-]$.\n\nThese Bounds, which are derived using only perturbative QCD and which do not\ndepend on any model assumptions or input parameters (they are therefore quite\nloose), are plotted in figure \\ref{figwithout}. They were calculated using\n$Q^2=0$ in $J(Q^2)$ at order ${\\cal O}(\\alpha_s)$ \n\nIf we know the value of the form factor $f_{i}\\equiv f(t_{i})$\nat $n$ points $i=1,\\dots, n$, we can define a matrix $\\cal{M}$\n\\begin{eqnarray}\n\\label{mat}\n\\cal{M} = \\left(\n\\begin{array}{ccccc}\n\\langle \\phi f |\\phi f \\rangle & \n\\langle \\phi f |g_t \\rangle &\n\\langle \\phi f |g_{t_1} \\rangle &\n\\cdots &\n\\langle \\phi f |g_{t_n} \\rangle \\\\\n\n\\langle g_t |\\phi f \\rangle & \n\\langle g_t |g_t \\rangle &\n\\langle g_t |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_t |g_{t_n} \\rangle \\\\\n\n\\langle g_{t_1} |\\phi f \\rangle & \n\\langle g_{t_1} |g_t \\rangle &\n\\langle g_{t_1} |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_{t_1} |g_{t_n} \\rangle \\\\\n\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\n\\langle g_{t_n} |\\phi f \\rangle & \n\\langle g_{t_n} |g_t \\rangle &\n\\langle g_{t_n} |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_{t_n} |g_{t_n} \\rangle\n\\end{array}\n\\right)\n\\end{eqnarray}\nand obtain, using again the positivity of the inner product,\n\\begin{eqnarray}\n\\label{ineq}\n\\det{\\cal M} \\ge 0.\n\\end{eqnarray}\nSince $\\langle \\phi f |\\phi f \\rangle$ is eliminated using (\\ref{main})\nand since all components except $f(t)$ \nof this matrix are known --- they are either constants or functions of $t$ ---\nthe inequality (\\ref{ineq}) leads to bounds on the\nform factors $f(t)$. The inclusion of some known values  (which i.e. are \npredicted by a model or which one can get by lattice calculations, see\n\\cite{hphx9509358}) therefore will give us more stringent bounds \n\\begin{eqnarray}\nF_{lo}(t,t_i, f_i) \\le f(t) \\le F_{up}(t,t_i, f_i)\n\\end{eqnarray}\nwith $F_{lo}$ and $F_{up}$ calculable functions of $t$ with the parameters \n$t_i,f_i$.\n\nThe upper and lower bounds can be written as\n\\begin{eqnarray}\n\\label{bounds}\nF_{up,lo}(t,Q^2) =\n\\frac{-\\beta(t)\\pm\\sqrt{c(Q^2)\\cdot\\Delta(t)}}{\\alpha\\cdot\\phi(t,Q^2)}.\n\\end{eqnarray} \nThe shape of $F_{up,lo}(t,Q^2)$ as functions of $t$ depend on the values \nfor $t_{i}$ and $f_{i}$. In (\\ref{bounds}), $\\alpha$ is a scaling constant,\n$\\beta(t)$ gives roughly the shape of the form factor while the squareroot in\n(\\ref{bounds}) distinguishes the upper from the lower bound. In the squareroot,\n$c(Q^2)$ is a constant in $t$ which depends only on the energy scale $Q^2$,\nand $\\Delta(t)$\nis a function with zeros in every $t_{i}$. The reason why the function\n$\\Delta(t)$ should behave like this is quite clear: \nsince we fixed the value of the form factor at certain points $t_i$, \nthe upper and lower bound should coincide in these points\n(for an exact definition of the functions appearing in (\\ref{bounds}) \nsee \\cite{hphx9509358}). \n\n\\section{Improving the Bounds}\n\\label{secimprov}\nThe method of including `fixed points' (i.e. to fix the value of the\nform factor at certain kinematical points) which we summarized in \n\\ref{secbounds} is well known and used throughout\nthe literature. However, this formalism can be applied in a slightly different\nway: to include the slope or even higher derivatives of the form factor. \nThis is desirable because it is possible to obtain the slope \ne.g.\\ for low $t$ from QCD sum rules or for $t$ near the kinematical end point\nfrom the chiral limit. \n\nConsider two fixed points $t_1$ and $t_2=t_1 + \\epsilon$ with $\\epsilon$\narbitrary but small. We get the coresponding values of $f(t_i)$ via a\nTaylor expansion: \n\\begin{eqnarray}\nf(t_1) &\\equiv& f_1 \\\\\nf(t_2) &=& f(t_1) + \n           \\epsilon \\left. \\frac{d}{dt}f(t) \\right|_{t=t_1} \\, . \n\\end{eqnarray}\nWe can now calculate \nthe bounds depending on these values for $t_1,t_2,f_1$ and $f_2$ \nand take the limit $\\epsilon \\to 0$. It is useful to define a function \n$\\psi (z,Q^2) \\equiv f(z)\\phi(z,Q^2)$, in terms of which we obtain \n\\begin{eqnarray}\n\\label{alptraum}\n\\psi_{up,lo} &=&\n  \t\\frac{(1-z_1^2)(\\psi_1(1-2zz_1+z_1^2)+\\psi_1'(z-z_1)(1-z_1^2))}\n\t{(1 - z z_1)^2} \\nonumber\\\\\n&\\pm& \\frac{(1-z_1^2)(z-z_1)^2}{(1-zz_1)^2\\sqrt{1-z^2}} \\times \\\\\n&&\n\t\\sqrt{J-\\psi_1^2(1+z_1^2)+2\\psi_1\\psi_1'z_1(1-z_1^2)\n\t-[\\psi_1']^2(1-z_1^2)^2} \\nonumber\n\\end{eqnarray}\nwhere $z_1\\equiv z(t_1)$, $\\psi_1 \\equiv \\psi (z_1,Q^2)$ \nand \n$$ \n\\psi_1' = \\left. \\frac{\\partial}{\\partial z} \\psi (z,Q^2) \\right|_{z=z_1} \\, .\n$$\n\nIt is obvious how to extend this to higher derivatives (including e.g.\\ \nthe curvature of the form factor), but the corresponding equations become\ntedious. In parts of the numerical analysis presented below the curvature\nhas been included.  \n\nIt has been observed (see \\cite{hphx9702300,hphx9705252} and\n\\cite{hphx9603414,hphx9712417}) that the inequality \n(\\ref{main}) also yields restrictions for the derivatives of the form \nfactor in the end point $t_-$. We shall use this input to restrict the \npossible values of the form factor at $t_-$ in terms of the slope and \nthe curvature. In order to do this we rewrite  (\\ref{main}) into the \nform\n\\begin{eqnarray} \\label{main1}\n\\frac 1{2\\pi}\\int_0^{2\\pi}\\into{\\Theta} |\\psi(\\ex{i\\Theta})|^2 \\le J(Q^2).\n\\end{eqnarray}\nIn the case of $f^0$ it is now possible to\nperform a Taylor expansion around $z_0 = 0$ ($t=t_-$) which is convergent \nin the full unit disc, since $f^0$ does not have poles or cuts in this \nregion. This is due to the fact that no physical intermediate states can \ncontribute. We therefore get the inequality\n\\begin{eqnarray}\n\\label{ineq1}\nJ(Q^2) \\ge\\psi^2(0)+\\sum_{n=1}^\\infty \\left(\\frac1{n!}\\right)^2\\psi^{(n)^2}(0).\n\\end{eqnarray}\nwhere $\\psi^{(n)}(0)$ denotes the $n$th derivative of $\\psi$ at the point\n$z=0$. Note that the form factor, $z(t)$ and thus also $\\psi$ are real\nin the region $0 \\le t \\le t_-$.\n\nThe form factor $f^+$ is not analytic inside the unit disc, since there \nis a contribution of the $B^*$ in this channel. Hence one expects that the \nTaylor expansion for this form factor converges only within the circle\n$0 \\le |z| \\le |z(m_{B^*}^2)|$. The integral in (\\ref{main1}) runs over \nthe unit circle and thus the integration contour lies outside the radius \nof convergence of the Taylor series. In order to take into account the \n$B^*$--pole one has to expand in a Laurent series or to  subtract the pole. \nHowever, closer inspection reveal that these two methods are equivalent.  \nWe choose to subtract the pole and thus define a function\n\\begin{eqnarray}\n\\tilde f(z) = f(z) - \\frac{\\mbox{Res}\\{f,z_p\\}}{z-z_p}\\frac{\\phi(z_p)}{\\phi(z)}\n\\end{eqnarray}\nand obtain\n\\begin{eqnarray}\n\\label{ftilde1}\n\\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\psi(\\ex{i\\Theta})|^2 \n&=& \\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\tilde \\psi(\\ex{i\\Theta})\n+\\frac{\\mbox{Res}\\{\\phi f,z_p\\}}{\\ex{i\\Theta} - \\ex{i\\Theta_p}}|^2 \\\\\n\\label{ftilde2}\n&=& \\frac{\\mbox{Res}^2\\{\\phi f,z_p\\}}{1-z_p^2} +\n\\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\tilde \\psi(\\ex{i\\Theta})|^2 \n\\end{eqnarray}\nwhere we define $\\tilde \\psi = \\tilde f \\phi$. Note that the terms \nlinear in $\\tilde \\psi$ and $1\/({\\ex{i\\Theta} - \\ex{i\\Theta_p}})$ vanish, \nsince only terms without $\\Theta$ dependence contribute to the integral.  \n\nThus we obtain instead of (\\ref{ineq1}) \n\\begin{eqnarray} \n\\label{ineq2}\nJ(Q^2)-\\frac{\\mbox{Res}^2\\{\\phi f,z_p\\}}{1-z_p^2} \\ge\\tilde\\psi^2(0)+\n\\sum_{n=1}^\\infty \\left(\\frac1{n!}\\right)^2\\tilde\\psi^{(n)^2}(0).\n\\end{eqnarray}\n\n\\begin{figure}[htp]\n\\epsfxsize=9cm\n\\leavevmode\n\\centering\n\\epsffile[70 130 540 650]{ellipse.ps} \\\\\n\\caption{Possible combinations of point, slope and curvature at the \nend point $f^+(t_-)$, allowed parameter triplets lie inside the ellipsoid. \nTo clarify the po\\-si\\-tion of the ellipsoid, it's contours were drawn on \nthree faces of the cube. \n\\label{figellipse}}\n\\end{figure}\n\nIf one redefines $J(Q^2)$ to include the residue and if one \nuses the analytic funtion $\\tilde f$, (\\ref{ineq2}) is the same \nas (\\ref{ineq1}).\n\nSince (\\ref{ineq1}) is a sum of positive terms, we can \ncut off the sum at some value of $n$ and get (e.g. for $n\\le2$)\n\\begin{eqnarray}\n\\label{kurz}\n[(\\phi f)(0)]^2 + [(\\phi f)'(0)]^2 + \\frac14 [(\\phi f)''(0)]^2 < J(Q^2)\n\\end{eqnarray}\nwhere we re--substituted $\\psi$.\n\nSuch equations give us ellipsoids in the parameter space of value, slope\nand higher derivatives of the form factor at the kinematical end point, where\nall allowed parameter combinations lie inside the ellipsoid. If we take, for\nexample, $n\\le2$ as in (\\ref{kurz}) we would get a three--dimensional\nellipsoid in point--slope--curvature--space (see figure \\ref{figellipse}).\n\nThe resulting constraints are not as good as the ones that can be obtained \nfor $B \\to D$ transitions \\cite{hphx9603414,hphx9712417}; this is due to \nheavy quark symmetries, which are not as useful  in $B \\to \\pi$ decays.  \n\nHowever, one may exploit the chiral symmetry for the light degrees of \nfreedom as an additional physics input. These symmetries hold at \nsmall momenta of the pions and are therefore applicable in the end \npoint region where the momentum transfer to the leptons becomes maximal. \nOne may combine heavy quark and chiral symmetry as in \n\\cite{prvxd45x2188}--\\nocite{hphx9602353}\\cite{hphx9707410} and compute \nthe form factors in this limit. One finds \\cite{prvxd45x2188}    \n\\begin{eqnarray}\n\\label{chiralform}\nf^+(t) &=& \\frac{f_B}{2f_\\pi}\\left[1+\\frac{g(M^2-m^2+t)}{M^2+m^2+2\\Delta M\n-t}\\right] \\nonumber\\\\\nf^0(t) &=& \\frac{f_B}{2f_\\pi}\\left[1+\\frac{g(M^2-m^2+t)}{M^2+m^2+2\\Delta M-t}\n\\right. \\\\\n&&\\hspace{1.8cm}\n+\\left.\\frac{t}{M^2-m^2}\\left\\{1-\\frac{g(3M^2-m^2+t)}{M^2+m^2+2\\Delta M-t}\n\\right\\}\\right] \\nonumber\n\\end{eqnarray}\nwhich results in the residue\n\\begin{eqnarray}\n\\label{chiralres}\n\\mbox{Res}\\{f,t_p\\} = -\\frac {f_B}{f_\\pi}gM(M+\\Delta)\n\\end{eqnarray}\nat\n\\begin{eqnarray}\nt_p = M^2+m^2+2\\Delta M\n\\end{eqnarray}\nwhere $\\Delta = m_{B^*} - m_B$. \nThe additional parameters which appear in these relations are\nthe chiral coupling constant $g$ which \ndescribes the $BB^*\\pi$--coupling and the ratio of the decay constants \n$f_B \/ f_\\pi$. \n\nThe form factors (\\ref{chiralform}) are valid in the chiral limit, \nwhich holds only in a small piece of the kinmatically allowed region. \nIn order to extend this to the full range, a variety of \nans\\\"atze have been invented to extend this pole--behaviour to small\n$t$ (see e.g. \\cite{zpxc29x637}--\\nocite{hphx9401303} \\nocite{zpxc48x663}\n\\cite{prvlx68x2887}); however, this makes the extraction ov $V_{ub}$ \nmodel dependent. \n\nWe will now use the improved unitarity bounds and \nthe chiral limit to derive bounds on the form factors. These bounds \nhave the advantage to be model independent, however, as we shall see, \nthey are not very tight over the whole range of $t$. This is mainly\ndue to the fact that the parameters $g$ and $f_B\/f_\\pi$ suffer from large \ntheoretical uncertainties. \n\n\\begin{figure}[htp]\n\\epsfxsize=9cm\n\\leavevmode\n\\centering\n\\epsffile[70 130 540 650]{chiralcone.ps} \\\\\n\\caption{Possible combinations of point, slope and curvature at the end point\n$f^+(t_-)$ using the chiral limit.\n\\label{figchiralcone}}\n\\end{figure}\n\nIn the following we shall consider $f^+$ only; here we can avoid \nthe problem of the large uncertainties in $g$ and $f_B\/f_\\pi$\nby considering fractions like\n$f^{+\\prime}(t_-)\/f^{+} (t_-)$ and $f^{+\\prime\\prime}(t_-)\/f^{+} (t_-)$  \nin which the $f_B\/f_\\pi$--dependence drops out and \nthe remaining $g$--dependence is small, since terms involving $g$ \nare suppressed:\n\\begin{eqnarray}\n\\frac{{f^+}'(t_-)}{f^+(t_-)} &=& \\frac {M+\\Delta}{2 M (M-m) (\\Delta+m)}\n\\frac{1}{1+\\frac{\\Delta+m}{g(M-m)}} \\, , \\\\\n\\frac{{f^+}''(t_-)}{f^+(t_-)} &=& \\frac {1}{2 M^2 (M-m) (\\Delta+m)}\n\\frac{1}{1+\\frac{\\Delta+m}{g(M-m)}} \\, .\n\\end{eqnarray}\n\nThus in the chiral limit the ratios \n$f^{+\\prime}(t_-)\/f^{+} (t_-)$ and $f^{+\\prime\\prime}(t_-)\/f^{+} (t_-)$  \nare practically fixed and we can \nexpress the slope $s$ and the curvature $c$ in terms of the value $p$ \nof $f^+$ at $t_-$. This would yield a straight line in a $p,s,c$ plot; \nhowever, varying the coupling $g$ in a generous range between $0.25$ and \n$0.5$ \\cite{hphx9605342} and allowing for derivations of the chiral limit\nof about 15\\%, we obtain the rectangular cone shown in \nfigure~\\ref{figchiralcone}.       \n\nIn case of the residue it is not possible to get rid of the $g$-- and\n$f_B\/f_\\pi$--parameter, so one has to take the full \nuncertainties into account\nwhen calculating the bounds. One could also choose to drop the residue \ncompletely and work with the redefined $\\phi_p(z,Q^2)$ of (\\ref{phitrafo}) \nbut we decided to use (\\ref{chiralres}) because  \nthe resulting bounds are more stringent.\n\nUsing (\\ref{chiralform}) and their first derivatives at the end point as\ninput parameters in (\\ref{alptraum}) (the extension to higher derivatives\nis trivial) we get upper and lower bounds for $0\\le t\\le t_-$ which depend\nstrongly on the chiral parameters $g$ and $f_B\/f_\\pi$. In addition to\nthat, a combination of figure \\ref{figellipse} and \\ref{figchiralcone} can be\nused to constrain the allowed range for the end point value of $f(t_-)$ and\nits derivatives.\n\nThe combination of unitarity bounds, extended to include derivatives, and \nthe chiral limit now allows us to calculate bounds for the form factors \n$f^+(t)$ and $f^0(t)$.\n\n\\section{Numerical Results and Discussion}\n\\label{seccalc}\nInserting $t_1=t_-$ ($z_1=0$) in (\\ref{alptraum}) one gets\n\\begin{eqnarray}\n\\psi_{up,lo} &=&\n  \t(\\psi_1+\\psi_1'z) \\pm \\frac{z^2}{\\sqrt{1-z^2}}\n\t\\sqrt{J-\\psi_1^2-[\\psi_1']^2}\n\\end{eqnarray}\nwhich leads to the bounds\n\\begin{eqnarray}\n\\label{endbounds}\nF_{up,lo} = \\frac{f_1 \\phi_1 + (f_1 \\phi_1)'z}{\\phi(z,Q^2)} \\pm\n\t\t\\frac{z^2}{\\phi(z,Q^2)}\\sqrt{\\frac{J-(f_1 \\phi_1)^2 - \n\t\t[(f_1 \\phi_1)']^2}{1-z^2}}.\n\\end{eqnarray}\nWe used $Q^2=0$ for the QCD calculations and\ninserted all possible combinations of end point value and slope into\n(\\ref{endbounds}). We also allowed \nvariation of $g$ and $f_B\/f_\\pi$ from $0.25\\le g\\le 0.5$ and \n$1\\le f_B\/f_\\pi\\le 1.7$ (\\cite{hphx9605342, hlax9710057}). \nWithin this variation we determined the bounds \nfor fixed parameter values and finally took the loosest ones as our result.\n\nWe scanned the parameter space of possible $ps$--pairs by dividing\nthe allowed $p$--range into small, equally spaced intervals.\nFor each of these possible $p$--values, there exists an allowed $s$--range\nwhich is also divided into small intervalls. A similar procedure was applied \nfor the chiral input parameters.\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile{pieces.ps} \\\\\n\\caption{Bounds on the form factor $f^+(t)$, derived by using a fixed\nend point and varying the slope according to the chiral limit\n\\label{figpieces}} \n\\end{figure}\n\nFigure~\\ref{figpieces} shows the resulting bounds for the form factor $f^+$. \nThe values of the form factor at $t_-$ vary between 3.2 and 13.5 due to the \nuncertainties in $f_B\/f_\\pi$ and $g$. The two solid lines represent the \nresult for the maximal value of the form factor at $t_-$, where the upper \n(lower) line corresponds to the minimal (maximal) possible value for the \nslope at $t_-$. Similarly, we have plotted the corresponding lines for \nintermediate values of the form factor at $t_-$ down to the minimal value, \nrepresented by the long--dashed line. \n\nThe upper and lower bounds coincide at $t=0$ due to the kinematical\nconstraint (\\ref{kinconst}). We have computed $f^0$ in the same way\nas we did for $f^+$, scanning the parameter space for $p$ and $s$ at $t=t_-$. \nThe resulting bounds for \n$f^0$ at $t=0$ are much tighter that those for $f^+ (0)$ and hence the \nkinematical constraint has a significant effect on $f^+$.  \nIn order to be conservative we have varied the parameters for both form\nfactors independently; that is \nwe did not use the correlation between the form factors implied by the \nchiral limit, namely that at $t=t_-$ they are both given by the same two \nparameters $f_B\/f_\\pi$ and $g$.  \n\nIn figure \\ref{figwith} we have combined the results for $f^+$ and $f^0$, \nplotting the upper bound for the maximal values of $f^+ (t_-)$ and \n$f^0 (t_-)$ with the minmal slopes and the lower bounds for the minmal \nvalues of $f^+ (t_-)$ and $f^0 (t_-)$ with the maximal slopes.  \nCompared to the standard method (see figure \\ref{figwithout}) \nthe inclusion of the slope and the chiral limit has significantly \nimproved the bounds. We have to point out, however, that not any \ncurve within these two bounds is allowed as a form factor, since QCD \nimplies relations between slopes and the form factor values. This can \nbe seen in figure~\\ref{figpieces}.\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{wholeslope.ps} \\\\\n\\caption{Bounds on the form factors $f^0(t)$ (on the left side, $t$ increasing \nleftward) and $f^+(t)$ (on the right side, $t$ increasing rightward) derived \nwith restriction on possible $ps$--pairs at $f(t_-)$ and including the\nkinematical constraint, plotted over the whole\nkinematical range of $[0,t_-]$ ($t$ in $\\mbox{GeV}^2$). \n\\label{figwith}} \n\\end{figure}\n\nFinally one can also use model input into the machinery of unitarity \nbounds. In particular, a model can be used to obtain points and curvatures\naway from $t_-$.   \n\nIn figure \\ref{figmodel2p} and \\ref{figmodel2a} we have used the \nISGW II model \\cite{hphx9503486} as an input. Fig.\\ref{figmodel2p}\nshows the bounds obtained by the standard formalism using two \npoints ($t=10$ GeV${}^2$ and $t_-$) from this model, while in \nfig.\\ref{figmodel2a} we use these two points but also the derivative\nat $t_-$, obtained again from ISGW II, as input parameters. The dashed line \nis the model prediction itself. \n \nAnother commonly used model is the BSW model \\cite{zpxc29x637}. \nIn fig.\\ref{figpoint} we use this model to determine the value of \nthe form factor $f^+$ at $t_-$, while in fig.\\ref{figslope} also \nthe slope at $t_-$ was taken from the model. It is interesting to \nnote that if one also includes the curvature of $f^+$ at $t_-$ \nin the BSW model --- as in fig.\\ref{figcurve} --- \nthe lower bound shifts significantly upwards, such \nthat the model becomes inconsistent with the bounds. This indicates that \nthe curvature of the BSW model is too large in the end point, since \nthe lower bound comes out to be quite high; its value at $t=0$\nis about one, which is in contradiction not only with the BSW model, \nbut also with sum rule calculations \\cite{hphx9305348,hphx9802394}.  \n \n\\begin{figure}[hbp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{model2pkc.ps} \\\\\n\\caption{Bounds on $f^0$ (left) and $f^+$ (right) derived by using the ISGW II\nmodel and using the values at $t=10\\mbox{ GeV}^2$ and $t=t_-$ as input\nparameters. The bounds are drawn solid, the dashed line corresponds to the\nmodel. \n\\label{figmodel2p}} \n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{model2akc.ps} \\\\\n\\caption{Same as in figure \\ref{figmodel2p} but including the slope at $t=t_-$\nas well.\n\\label{figmodel2a}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modpoint.ps} \\\\\n\\caption{Bounds on $f^+$ derived by using the BSW model and using the value \nat $t=t_-$ as input parameter. The bounds are drawn solid, the dashed line \ncorresponds to the model. \n\\label{figpoint}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modslope.ps} \\\\\n\\caption{Same as in figure \\ref{figpoint} but including the slope at $t=t_-$\nas well.\n\\label{figslope}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modcurve.ps} \\\\\n\\caption{Same as in figure \\ref{figslope} but including the curvature at \n$t=t_-$ as well.\n\\label{figcurve}} \n\\end{figure}\n\nThe pole form of the BSW model is motivated by the form factor in the \nchiral limit and thus it is no surprise that they have similar curvatures \nof $f^+$ at $t_-$. We take this as an indication \nthat one should not exploit the chiral limit to obtain derivatives\nof the form factors beyond the slope; going beyond the slope requires\nto take into account higher orders in chiral perturabtion theory. \n\n\\section{Conclusions} \n\\label{secconc}\nUsing unitarity and analyticity to derive bounds on form factors \nusing input from perturbative QCD has become a widely used tool, \nsince this method allows to constrain form factors over the whole\nrange of $q^2$ in a model independent way. \n\nThis method has attracted some attention in $B$ decays and has been \ncombined with heavy quark symmetry; for the $b \\to c$ transitions \nthis has lead to stringent constraints on the form factors, i.e.\nthe Isgur--Wise function. \n\nIn $B \\to \\pi$ transitions heavy quark symmetries are not as useful \nand it is necessary to obtain model independent information from \nother sources. The general bounds in $B \\to \\pi$ decays \nare not very tight and hence also not very useful. \nIncluding points where the form factor is known from other sources\nimproves the bounds; this method has been combined with lattice data\nto obtain model independent bounds for the form factors for \n$B \\to \\pi$ transitions. \n\nIn the present paper we have focussed also at the $B \\to \\pi$ decays\nand improved the bounds on the relevant form factors one more step by \nincluding slopes and higher derivatives of form factors, which in \nsome cases may be known from other sources. Again from unitarity \nthe value, the slope, and even higher derivatives of the form factor at \n$t_-$ are constrained to lie inside an ellipsoid which can be used \nas an input into the machinery we have proposed here. \n\nStill this does not tighten the bounds very much and some more physics\ninput beyond perturbative QCD is needed. Close to the kinematical end point \n$t_-$ chiral \nperturbation theory is valid and we used the chiral limit of the \nform factors as an input. With this we arrived at bounds which are\nmuch tighter than the general bounds obtained without any knowledge \non the form factors. The model independent results of our paper are \nshown in fig.\\ref{figpieces} and fig.\\ref{figwith}.   \n \nFinally one may also combine models with the unitarity bounds from QCD. \nTaking points, slopes and curvatures from these models one may test \nthe consistency of a given model with QCD. An extensive analysis of \nthe various models is beyond the scope of the present paper; we   \nonly considered the ISGW II and the BSW model as examples. Although \nboth models lie within the bounds given in fig.\\ref{figwith} (the \nISGW II touching the lower bound at $t_-$), this still does not \nmean that they are consisten with QCD; the curvature of the BSW model \nat $t_-$ turns out to be incompatibel with the QCD constraints. \n\n\\section*{Acknowledgements}\nWe thank Changhao Jin, who visited Karlsruhe in the early stage of this\nwork, for useful discussions on this subject. This work was supported by \nthe ``Graduiertenkolleg: Elementarteilchenphysik an Be\\-schleu\\-ni\\-gern''\nand the ``Forschergruppe: Quantenfeldtheorie, Computeralgebra und \nMonte Carlo Simulationen'' of the Deut\\-sche For\\-schungs\\-ge\\-mein\\-schaft.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nThe task of program synthesis is to automatically generate a program that is\nconsistent with a specification such as a set of input-output examples, and has\nbeen studied since the early days of Artificial\nIntelligence~\\citep{waldinger1969prow}. There has been a lot of recent progress\nmade on \\emph{neural program induction}, where novel neural architectures\ninspired from computation modules such as RAM, stack, CPU, turing machines, and\nGPU~\\citep{graves2014neural,joulin2015inferring,nram,graves2016hybrid,npi,kaiser2015neural}\nhave been proposed to train these architectures in an end-to-end fashion to\nmimic the behavior of the desired program. While these approaches have achieved\nimpressive results, they do not return explicit interpretable programs, tend not\nto generalize well on inputs of arbitrary length, and require a lot of examples\nand computation for learning each program. To mitigate some of these\nlimitations, \\emph{neural program synthesis}\napproaches~\\citep{JohnsonHMHLZG17,NeurosymbProgsynth,RobustFill} have been\nrecently proposed that learn explicit programs in a Domain-specific language\n(DSL) from as few as five input-output examples. These approaches, instead of\nusing a large number of input-output examples to learn a single program, learn a\nlarge number of different programs, each from just a few input-output examples.\nDuring training, the correct program is provided as reference, but at test time,\nthe learnt model generates the program from only the input-output examples.\n\n\n\n\n\\input{motiv_figure.tex}\n\n\nWhile neural program synthesis techniques improve over program induction\ntechniques in certain domains, they suffer from two key limitations. First,\nthese approaches use supervised learning with reference programs and suffer from\nthe problem of \\textit{Program Aliasing}: For a small number of input-output\nexamples, there can be many programs that correctly transform inputs to outputs.\nThe problem is the discrepancy between the single supervised reference program\nand the multitude of correct programs. Figure \\ref{fig:progsynth} shows an\nexample of this: if maximizing the probability of ground truth program,\npredicting Program B would be assigned a high loss even though the two programs\nare semantically equivalent for the input-output example.\nMaximum likelihood training forces the model to learn to predict ground truth\nprograms, which is different from the true objective of program synthesis:\npredicting \\emph{any} consistent program. To address this problem, we alter the\noptimization objective: instead of maximum likelihood, we use policy gradient\nreinforcement learning to directly encourage generation of \\emph{any} program\nthat is consistent with the given examples.\n\n\nThe second limitation of neural program synthesis techniques based on sequence\ngeneration paradigm~\\citep{RobustFill} is that they often overlook the fact that\nprograms have a strict syntax, which can be checked efficiently. Similarly to\nthe work of~\\citet{NeurosymbProgsynth}, we explore a method for leveraging the\nsyntax of the programming language in order to aggressively prune the\nexponentially large search space of possible programs. In particular, not all\nsequences of tokens are valid programs and syntactically incorrect programs can\nbe efficiently ignored both during training and at test time. A syntax checker\nis an additional form of supervision that may not always be present. To address\nthis limitation, we introduce a neural architecture that retains the benefits of\naggressive syntax pruning, even without assuming access to the definition of the\ngrammar made in previous work~\\citep{NeurosymbProgsynth}. This model\nis jointly conditioned on syntactic and program correctness, and can implicitly\nlearn the syntax of the language while training.\n\n\n\nWe demonstrate the efficacy of our approach by developing a neural program\nsynthesis system for the Karel programming language~\\citep{pattis1981karel}, an\neducational programming language, consiting of control flow constructs such as\nloops and conditionals, making it more complex than the domains tackled by\nprevious neural program synthesis works.\n\nThis paper makes the following key contributions:\n\\begin{itemize}\n\\item We show that Reinforcement Learning can directly optimize for generating\n  any consistent program and improves performance compared to pure supervised\n  learning.\n\\item We introduce a method for pruning the space of possible programs using a\n  syntax checker and show that explicit syntax checking helps generate better\n  programs.\n\\item In the absence of a syntax checker, we introduce a model that jointly\n  learns syntax and the production of correct programs. We demonstrate this\n  model improves performance in instances with limited training data.\n\\end{itemize}\n\n\\section{Related work}\n\nProgram synthesis is one of the fundamental problems in Artificial Intelligence.\nTo the best of our knowledge, it can be traced back to the work\nof~\\citet{waldinger1969prow} where a theorem prover was used to construct LISP\nprograms based on a formal specification of the input-output relation. As formal\nspecification is often as complex as writing the original program, many\ntechniques were developed to achieve the same goal with simpler partial specifications in\nthe form of input-output (IO)\nexamples~\\citep{amarel1970representations,summers1977methodology}. Rule-based\nsynthesis approaches have recently been successful in delivering on the promise\nof Programming By Example~\\citep{lieberman2001your}, the most widely known\nexample being the FlashFill system~\\citep{FlashFill} in Excel. However, such\nsystems are extremely complicated to extend and need significant development\ntime from domain experts to provide the pruning rules for efficient search.\n\nAs a result, the use of Machine Learning methods have been proposed, based on\nBayesian probabilistic models~\\citep{liang2010learning} or Inductive Logic\nprogramming~\\citep{muggleton1991inductive,muggleton2014meta} to automatically\ngenerate programs based on examples. Recently, inspired by the success of Neural\nNetworks in other applications such as vision~\\citep{krizhevsky2012nips} or\nspeech recognition~\\citep{graves2013speech} differentiable controllers were made\nto learn the behaviour of programs by using gradient descent over differentiable\nversion of traditional programming concepts such as memory addressing\n\\citep{graves2014neural}, manipulating stacks\n\\citep{joulin2015inferring,grefenstette2015learning}, register machines\n\\citep{nram}, and data manipulation~\\citep{neelakantan2015neural}. These\napproaches to program induction however tend to struggle with generalization,\nespecially when presented with inputs of a different dimension than the one they\nwere trained with and require a very large amount of training data. Some\nexceptions to this include Neural Programmer Interpreters~\\citep{npi} and its\nextensions~\\citep{nplattice,cai2017making} that learn from program traces rather\nthan only examples. However, they still learn a different model for each program\nand are computationally expensive, unlike our system that uses a single model\nfor learning a large number of programs.\n\nA series of recent works aim to infer explicit program source code with the\nassumption that code structure provides an inductive bias for better\ngeneralization. In particular, explicitly modeling control flow statements such\nas conditionals and loops can often lead to programs capable of generalizing,\nregardless of input size~\\citep{terpret,anc,diffForth}. One remaining\ndrawback of these approaches is the need to restart learning from scratch for\neach new program. Thus they are largely unsuited for the situation of\nsynthesizing a new program on-the-fly from very few examples.\nThe latest developments use large datasets of artificially generated programs\nand learn to map embeddings of IO examples to information about the programs to\ngenerate. \\citet{balog2016deepcoder} produce scores over attributes, to be used\nas heuristics to speed up search-based techniques. \\citet{NeurosymbProgsynth}\nuse their dataset to learn probability over the expansion rules of a predefined\ngrammar, while \\citep{RobustFill} directly predict the source code of the\nprograms. These last two methods use supervised training to maximize the\nlikelihood of a single reference program, while we directly optimize for the\ngeneration of any consistent program.\n\nOur approach to optimize program\ncorrectness is similar in spirit to advances in Neural Machine\nTranslation~\\citep{googleNMT,ranzato2015sequence} that leverage\nreinforcement learning to optimize directly for evaluation\nmetrics. Taking advantage of the fact that programs can be\nsyntactically checked and unit tested against the specification\nexamples, we show how to improve on those\nREINFORCE~\\citep{williams1992simple} based methods. Recently,\n\\citet{guu2017language} proposed a method similar to ours based on\nMaximum Marginal Likelihood to generate programs based on a\ndescription in natural language. From an application point of view,\nour target domain is more complex as our DSL includes control flow\noperations such as conditionals and loops.  Moreover, natural language\nutterances fully describe the steps that the program needs to take,\nwhile learning from IO examples requires planning over potentially\nlong executions. Their approach is more akin to inferring a formal\nspecification based on a natural language description, as opposed to\nour generation of imperative programs.\n\nIncorporating knowledge of the grammar of the target domain to enforce\nsyntactical correctness has already proven useful to model arithmetic\nexpressions, molecules~\\citep{kusner2017grammar}, and\nprograms~\\citep{NeurosymbProgsynth, yin17acl}. These approaches define the model over the\nproduction rules of the grammar; we instead operate directly over the terminals\nof the grammar. This allows us to learn the grammar jointly with the model in\nthe case where no formal grammar specification is available. Our approach is\nextremely general and can be applied to the very recently proposed methods for\ninferring and executing programs for visual reasoning~\\citep{JohnsonHMHLZG17} that\nto the best of our knowledge does not directly explictly encourage  grammar\nconsistency of the resulting program.\n\n\n\n\n\n\\section{Problem overview}\nBefore describing our proposed methods, we establish the necessary notation,\npresent the problem setting and describe the general paradigm of our approach.\n\n\\subsection{Program Synthesis formulation}\nTo avoid confusion with probabilities, we will use the letter $\\lambda$ to\ndenote programs. $I$ and $O$ will be used to denote respectively input states\nand output states and we will use the shortcut $IO$ to denote a pair of\ncorresponding input\/output examples. A state constitutes what the programs are\ngoing to be operating on, depending on the application domain. In FlashFill-type\napplications~\\citep{NeurosymbProgsynth,RobustFill}, Input and Output states would\nbe strings of characters, while in our Karel environment, states are grids\ndescribing the presence of objects. If we were to apply our method to actual\nprogramming languages, states would represent the content of the machine's\nregisters and the memory.\n\nAt training time, we assume to have access to $N$ training samples, each\ntraining sample consisting of a set of $K$ Input\/Output states and a program\nimplementing the mapping correctly:\n\\begin{equation}\n    \\mathcal{D} = \\Bigg\\{\\ \\Big( \\ioset{k}{i}{1..K}, \\ \\lambda_i \\Big)\\ \\Bigg\\}_{i=1..N}  \\text{such that: } \\quad  \\lambda_i(I^k_i) = O^k_i \\quad \\forall i \\in 1 .. N, \\quad \\forall k \\in 1 .. K\n\\end{equation}\nwhere $\\lambda_i(I^k_i)$ denotes the resulting state of applying the program\n$\\lambda_i$ to the input state $I^k_i$. Our goal is to learn a synthesizer $\\sigma$\nthat, given a set of input\/output examples produces a program:\n\\begin{equation}\n    \\sigma : \\quad  \\left\\{ IO^k \\right\\}_{k=1..K} \\quad \\longrightarrow \\quad \\hat{\\lambda}\n\\end{equation}\nWe evaluate the programs on a set of test cases for which we have both\nspecification examples and held-out examples:\n\\begin{equation}\n    \\mathcal{D}_{\\text{test}}= \\Bigg\\{\\ \\Big( \\ioset{k_\\text{spec}}{j}{1..K} , \\quad  \\ioset{k_\\text{test}}{j}{K+1..K^\\prime}  \\Big) \\ \\Bigg\\}_{j=1..N_\\text{test}}\n\\end{equation}\nAt test time, we evaluate the performance of our learned synthesizer by\ngenerating, for each sample in the test set, a program $\\hat{\\lambda_j}$. The\nmetric we care about is \\emph{Generalization}:\n{\\small\n\\begin{equation}\n  \\label{eq:generalization}\n  \\Big\\{\\ j \\in \\{1..N_\\text{test}\\}\\ \\text{such that } \\  \\hat{\\lambda_j}(I^k_j) == O^k_j \\quad \\forall k \\in \\left\\{1..K^\\prime\\right\\} \\Big\\} \\text{ where } \\hat{\\lambda_j} = \\sigma\\left( \\ioset{k_\\text{spec}}{j}{1..K} \\right).\n\\end{equation}\n}\n\n\\subsection{Neural Program Synthesis Architecture}\nSimilar to \\citet{RobustFill} we use a sequential LSTM-based~\\citep{lstm} language model, conditioned on an\nembedding of the input-output pairs. Each pair is encoded independently by a convolutional neural network (CNN)\nto generate a joint embedding. A succinct description of the architecture can be\nfound in section \\ref{subsec:exp_domain} and the exact dimensions are available in\nthe supplementary materials.\n\n\\begin{figure}[t]\n  {  \\noindent\\resizebox{\\textwidth}{!}{\n      \\input{figures\/model\/model.tex}\n    }}\n  \\caption{Architecture of our model. Each pair of Input-Output is embedded\n    jointly by a CNN. One decoder LSTM is run for each example, getting fed in a\n    concatenation of the previous token and the IO pair embedding (constant\n    across timestep). Results of all the decoders are maxpooled and the\n    prediction is modulated by the mask generated by the syntax model. The\n    probability over the next token is then obtained by a Softmax transformation.}\n  \\label{fig:model_arch}\n\\end{figure}\n\n\nEach program is represented by a sequence of tokens $\\lambda = [s_1, s_2, ...,\ns_L]$ where each token comes from an alphabet $\\Sigma$. We model the program one\ntoken at a time using an LSTM. At each timestep, the input consists of the\nconcatenation of the embedding of the IO pair and of the last predicted token.\nOne such decoder LSTM is run for each of the IO pairs, all using the same\nweights. The probability of the next token is defined as the Softmax of a linear\nlayer over the max-pooled hidden state of all the decoder LSTMs. A schema\nrepresenting this architecture can be seen in Figure~\\ref{fig:model_arch}.\n\nThe form of the model that we are learning is:\n{\\small\n\\begin{equation}\n  p_{\\theta}(\\lambda_i \\ | \\ \\ioset{k}{i}{1..K}) = \\prod_{t=1}^{L_i}\\ p_\\theta(s_t \\ | \\ s_1, ...,s_{t-1}, \\ioset{k}{i}{1..K})\n\\end{equation}\n}\nAt test time, the most likely programs are obtained by running a beam search. One\nof the advantages of program synthesis is the ability to execute hypothesized programs. Through execution, we remove syntactically incorrect programs\nand programs that are not consistent with the observed examples. Among the remaining programs, we\nreturn the most likely according to the model.\n\n\\section{Objective Functions}\n\n\\subsection{Maximum Likelihood optimization}\nTo estimate the parameters $\\theta$ of our model, the default solution is to\nperform supervised training, framing the problem as Maximum Likelihood\nestimation. \\citet{RobustFill} follow this approach and use stochastic gradient\ndescent to solve:\n\\begin{equation}\n  \\label{eq:mle-obj}\n  \\theta^{\\star} = \\argmax_{\\theta} \\prod_{i=1..N} p_{\\theta}(\\lambda_i\\ | \\ \\ioset{i}{i}{1..K}) = \\argmax_{\\theta} \\sum_{i=1..N} \\log \\left( p_{\\theta}( \\lambda_i \\ | \\  \\ioset{k}{i}{1..K})  \\right)\n\\end{equation}\n\nHowever, this training objective exhibits several drawbacks. First, at\ntraining time, the model is only exposed to the training data distribution,\nwhile at test time, it is fed back the token from its own previous predictions.\nThis discrepancy in distribution of the inputs is well known in Natural Language\nProcessing under the name of \\textit{exposure bias}~\\citep{ranzato2015sequence}.\n\nMoreover, this loss does not represent the true objective of program synthesis.\nIn practice, any equivalent program should be as valid a prediction as the\nreference one. This property, that we call \\textit{program aliasing}, is not\ntaken into account by the MLE training.\nIdeally, we would like the model to learn to reason about the necessary steps\nneeded to map the input to the output. As a result, the loss shouldn't penalize\ncorrect programs, even if they do not correspond to the ground truth.\n\n\n\\subsection{Optimizing Expected Correctness}\n\\label{subsec:expected-correctness}\nThe first modification that we propose is to change the target objective to\nbring it more in line with the goal of program synthesis. We replace the\noptimization problem of (\\ref{eq:mle-obj}) by\n\\begin{equation}\n  \\label{eq:RL}\n  \\theta^{\\star} = \\argmax_{\\theta} \\mathcal{L}_{R}(\\theta), \\qquad \\text{ where } \\mathcal{L}_{R}(\\theta) = \\sum_{i..N} \\left( \\sum_{\\lambda} p_{\\theta}(\\lambda \\ | \\ioset{k}{i}{1..K})\\ R_i(\\lambda)\\right),\n\\end{equation}\nwhere $R_i(\\lambda)$ is a reward function designed to encode the quality of\nthe sampled programs. Note that this formulation is extremely generic and would\nallow to represent a wide range of different objective functions. If we assume\nthat we have access to a simulator to run our programs, we can design $R_i$ so\nas to optimize for generalization on held-out examples, preventing the model to\noverfit on its inputs. Additional property such as program conciseness, or\nruntime efficiency could also be encoded into the reward.\n\nHowever, this expressiveness comes at a cost: the inner sum in (\\ref{eq:RL}) is\nover all possible programs and therefore is not tractable to compute. The\nstandard method consists of approximating the objective by defining a Monte\nCarlo estimate of the expected reward, using $S$ samples from the model. To\nperform optimization, an estimator of the gradient of the expected reward is\nbuilt based on the REINFORCE trick~\\citep{williams1992simple}.\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:Rl-sample}\n  \\mathcal{L}_{R}(\\theta) &\\approx \\sum_{i=1..N}\\sum_{r=1}^{S} \\frac{1}{S}\\ R_i(\\lambda_r), \\qquad \\text{ where } \\lambda_r \\sim p_\\theta(\\  \\cdot \\  |  \\ioset{k}{i}{1..K})\\\\\n  \\nabla_{\\theta} \\mathcal{L}_R(\\theta) &\\approx \\sum_{i=1..N} \\sum_{r=1}^{S} \\frac{1}{S} \\ R_i(\\lambda_r) \\nabla_{\\theta}\\log\\left(  p_\\theta(\\  \\lambda_r \\  |  \\ioset{k}{i}{1..K}) \\right)\n   \\end{split}\n\\end{equation}\n\nHowever, given that we sample from a unique model, there is a high chance that\nwe will sample the same programs repeatedly when estimating the gradient. This\nis especially true when the model has been pre-trained in a supervised manner. A\ndifferent approach is to approximate the distribution of the learned\ndistribution by another one with a smaller support.\n\nTo obtain this smaller distribution, one possible solution is to employ the $S$\nmost likely samples as returned by a Beam Search. We generate the embedding of\nthe IO grids and perform decoding, keeping at each step the $S$ most likely\ncandidates prefixes based on the probability $p_{\\theta}$ given by the model. At\nstep $t$, we evaluate $p_{\\theta}(s_1 \\dots s_t, \\ioset{k}{i}{1..K})$ for all\nthe possible next token $s_t$ and all the candidates $(s_1 \\dots s_{t-1})$\npreviously obtained. The $S$ most likely sequences will be the candidates at the\n$(t+1)$ step. Figure \\ref{fig:beamsearch} represents this process.\n\n\\begin{figure}[t]\n\t\\noindent\\resizebox{\\textwidth}{!}{\n    \\input{figures\/beamsearch\/bs.tex}\n\t\t}\n    \\caption{\\label{fig:beamsearch}Approximation using a beamsearch. All\n      possibles next tokens are tried for each candidates, the $S$ (here 3) most\n      likely according to $p_\\theta$ are kept. When an End-Of-Sequence token\n      (green) is reached, the candidate is held out. At the end, the most likely\n      complete sequences are used to construct an approximate distribution,\n      through rescaling.}\n\\end{figure}\n\n\nBased on the final samples obtained, we define a probability distribution to use\nas an approximation of $p_{\\theta}$ in \\eqref{eq:RL}. As opposed to\n\\eqref{eq:Rl-sample}, this approximation introduces a bias. It however has the\nadvantage of aligning the training procedure more closely with the testing\nprocedure where only likely samples are going to be decoded. Formally, this\ncorresponds to performing the following approximation of the objective function,\n($\\text{BS}(p_\\theta, S)$ being the $S$ samples returned by a beam search with\nbeam size $S$):\n\\begin{equation}\n  \\label{eq:rl-bs}\n  \\begin{split}\n    & \\mathcal{L}_R(\\theta) \\approx \\sum_{i..N} \\left( \\sum_{\\lambda} q_{\\theta}(\\lambda \\ | \\ioset{k}{i}{1..K})\\ R_i(\\lambda)\\right)\\\\\n    & \\text{where } \\qquad q_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K}) = \\begin{cases}\n      \\frac{p_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K})}{\\sum_{\\lambda_r \\in \\text{BS}(p_{\\theta}, S)} p_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K})} &\\qquad \\text{ if } \\lambda_r \\in \\text{BS}(p_{\\theta}, S)\\\\\n      0 &\\qquad \\text{otherwise}.\n      \\end{cases}\n  \\end{split}\n\\end{equation}\nWith this approximation, the support of the distribution $q_{\\theta}$ is much\nsmaller as it contains only the $S$ elements returned by the beam search. As a\nresult, the sum become tractable and no estimator are needed. We can simply\ndifferentiate this new objective function to obtain the gradients of the loss\nwith regards to $p_\\theta$ and use the chain-rule to subsequently obtain\ngradient with regards to $\\theta$ necessary for the optimization.\nNote that if $S$ is large enough to cover all possible programs, we recover the\nobjective of \\eqref{eq:RL}.\n\n\nBased on this more tractable distribution, we can define more complex objective\nfunctions. In program synthesis, we have the possibility to prune out several\npredictions by using the specification. Therefore, we can choose to go beyond\noptimizing the expected reward when sampling a single program and optimize the\nexpected reward when sampling a bag of $C$ programs and keeping the best one.\nThis results in a new objective function: {\\small\n  \\begin{equation}\n    \\label{eq:rl-div-gen}\n    \\theta^{\\star} = \\argmax_{\\theta} \\sum_{i=1..N} \\left( \\sum_{ \\{\\lambda_{1},..,\\lambda_{C}\\} \\in {\\text{BS}(p_{\\theta, s})^C}} \\left[ \\max_{j \\in {1..C}} R_i(\\lambda_j)\\right] \\left( \\prod_{r \\in {1..C}} q_{\\theta}\\left(\\lambda_r \\ | \\ioset{k}{i}{1..K}\\right)  \\right)  \\right),\n  \\end{equation}}\nwhere $q_{\\theta}$ is defined as previously. We argue that by optimizing this\nobjective function, the model gets the capability of ``hedging its bets'' and\nassigning probability mass to several candidates programs, resulting in a higher\ndiversity of outputs.\n\nIn the special case where the reward function only takes values in $\\{0, 1\\}$,\nas it is when we are using correctness as a reward, this can be more easily computed as:\n{\\small\\begin{equation}\n  \\label{eq:rl-div-zeroone}\n    \\theta^{\\star} = \\argmax_{\\theta} \\sum_{i=1..N} \\left(1  - \\left(\\sum_{\\lambda_r \\in \\text{BS}(p_{\\theta}, S)} [R_i(\\lambda_r) == 0]\\  q_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K}) \\right)^C \\right)\n  \\end{equation}}\nThe derivation leading to this formulation as well as a description on how to\nefficiently compute the more general loss\\eqref{eq:rl-div-gen} can be found in\nappendix \\ref{app:div_details}.\nNote that although this formulation brings the training objective function\ncloser to the testing procedure, it is still not ideal. It indeed makes the\nassumption that if we have a correct program in our bag of samples, we can\nidentify it, ignoring the fact that it is possible to have some incorrect program\nconsistent with the IO pairs partial specification (and therefore not prunable).\nIn addition, this represents a probability where the $C$ programs are sampled\nindependently, which in practice we wouldn't do.\n\n\\section{Model}\n\\subsection{Conditioning on the syntax}\n\\label{subsec:handwritten}\nOne aspect of the program synthesis task is that syntactically incorrect\nprograms can be trivially identified and pruned before making a prediction.\nAs a result, if we use \\texttt{stx} to denote the event that the sampled program\nis syntactically correct, what we care about modeling correctly is\n$p( \\lambda \\ | \\ioset{k}{i}{1..K}, \\mathtt{stx} )$.\nUsing Bayes rule, we can rewrite this:\n\\begin{equation}\n  \\label{eq:bayes}\n  \\begin{split}\n  p\\left( \\ \\lambda \\ |\\ \\ioset{k}{i}{1..K}, \\mathtt{stx}\\right) &\\propto p\\left(\\ \\mathtt{stx}\\ |\\ \\ioset{k}{i}{1..K}, \\lambda\\ \\right) \\times p\\left(\\ \\lambda \\ |\\ \\ioset{k}{i}{1..K}\\ \\right)\\\\\n  &\\propto p\\left(\\ \\mathtt{stx}\\ |\\ \\lambda\\ \\right) \\times p\\left(\\ \\lambda \\ |\\  \\ioset{k}{i}{1..K}\\right)\n  \\end{split}\n\\end{equation}\nWe drop the conditional dependency on the IO pairs in the second line\nof~(\\ref{eq:bayes}) as the syntactical correctness of the program is independent\nfrom the specification when conditioned on the program. We can do the same operation at the token level, denoting by\n$\\mathtt{stx}_{1 \\dotsc t}$ the event that the sequence of the first $t$ tokens $s_1 \\dotsb s_t$ doesn't\ncontain any syntax error and may therefore be a prefix to a valid program.\n\\begin{equation}\n  \\label{eq:tokenbayes}\n  \\begin{split}\n    p\\left( \\ s_t \\ |\\ s_1 \\dotsb s_{t-1}, \\ioset{k}{i}{1..K}, \\mathtt{stx}_{1 \\dotsc t}\\right) \\propto \\quad& p\\left( \\ \\mathtt{stx}_{1 \\dotsc t}\\ | s_1\\dotsb s_t\\right) \\times \\\\\n   & \\quad \\ p\\left(\\ s_t\\ |\\ s_1 \\dotsb s_{t-1}, \\ioset{k}{i}{1..K}\\right)\n  \\end{split}\n\\end{equation}\nGiven a grammar, it is possible to construct a checker to determine\nvalid prefixes of programs. Example applications include\ncompilers to signal syntax errors to the user and autocomplete features of\nIntegrated Development Environments (IDEs) to restrict the list of suggested\ncompletions.\n\nThe quantity $ p\\left( \\ \\mathtt{stx}_{1 \\dotsc t}\\ |\\ s_1\\dotsb s_t\\right)$ is\ntherefore not a probability but can be implemented as a deterministic process\nfor a given target programming language. In practice, this is implemented by\ngetting at each timestep a mask \\mbox{$M = \\{-\\inf, 0\\}^{ | \\ \\Sigma \\ |}$}\nwhere $M_j = -\\inf$ if the $j$-th token in the alphabet is not a valid token in\nthe current context, and 0 otherwise. This mask is added to the output of the\nnetwork, just before the Softmax operation that normalizes the output to a\nprobability over the tokens.\n\nConditioning on the syntactical correctness of the programs provides several\nadvantages: First, sampled programs become syntactically correct by\nconstruction. At test time, it allows the beam search to only explore useful\ncandidates. It also ensures that when we are optimizing for correctness, the\nsamples used to approximate the distribution are all going to be valid\ncandidates. Restricting the dimension of the space on which our model is defined\nalso makes the problem simpler to learn.\n\n\n\n\n\\subsection{Jointly learned syntax}\n\\label{subsec:jointGrammar}\nIt may not always be feasible to assume access to a syntax checker. In general,\nwe wish to retain the syntax checker's ability to aggressively prune the search\nin program space without requiring access to the syntax itself. To this end, we\npropose to represent the syntax checker as a neural network module and learn it\njointly. Similar to the base model, we implement learned syntax checking using\nan LSTM $g_\\phi$. Comparing to the decoder LSTM, there are two major\ndifferences:\n\\begin{itemize}\n\\item The syntaxLSTM is conditioned only on the program tokens, not on the IO pairs. This ensures that the learned checker models only the syntax of the language.\n\\item The output of the syntaxLSTM is passed through an elementwise $x \\mapsto -\n  \\exp(x)$ activation function and added to the decoder LSTM's output. Similar\n  to the mask in Section~\\ref{subsec:handwritten}, the exponential activation\n  function allows the syntaxLSTM to output high penalties to any tokens deemed\n  syntactically incorrect.\n  %\n\\end{itemize}\n\nThe addition of the syntaxLSTM doesn't necessitate any change to the training\nprocedure as it is simply equivalent to a change of architecture. However, in\nthe supervised setting, when we have access to syntactically correct programs,\nwe have the possibility of adding an additional term to the loss\n~(\\ref{eq:mle-obj}) to prevent the model from masking valid programs:\n\\begin{equation}\n  \\label{eq:stx_loss}\n  \\mathcal{L}_{\\mathtt{syntax}} = - \\sum_{i=1 \\dotsb N} \\sum_{t= 1 \\dotsb L} g_{\\phi}\\left(\\ s^i_t\\  | s^i_1 \\dotsb s^i_{t-1}\\right), \\qquad \\text{ where } \\lambda_i = [s^i_1 , s^i_2 \\dotsb s^i_L]\n\\end{equation}\nThis loss penalizes the syntaxLSTM for giving negative scores to each token\nbelonging to a known valid program. We use the reference programs as example of\nvalid programs when we perform supervised training.\n\n\\section{Experiments}\n\\subsection{The domain: Karel}\n\\label{subsec:exp_domain}\nThe Karel programming language is an educational programming\nlanguage~\\citep{pattis1981karel}, used for example in Stanford CS introductory\nclasses~\\citep{cs106} or in the Hour of Code initiative~\\citep{hoc}. It features\nan agent inside a gridworld (See Figure \\ref{fig:progsynth}), capable of moving\n(\\texttt{move}, \\texttt{turn\\{Left,Right\\}}), modifying world state\n(\\texttt{\\{pick,put\\}Marker}), and querying the state of the nearby environment\nfor its own markers (\\texttt{markerPresent}, \\texttt{noMarkerPresent}) or for\nnatural obstacles (\\texttt{frontIsClear}, \\texttt{leftIsClear},\n\\texttt{rightIsClear}).\n  Our goal is to learn to generate a program in the Karel\nDSL given a small set of input and output grids. The language supports for\nloops, while loops, and conditionals, but no variable assignment. Compared to\nthe original Karel language, we only removed the possibility of defining\nsubroutines. The specification for the DSL can be found in\nappendix~\\ref{sec:karel-spec}.\n\nTo evaluate our method, we follow the standard practice~\\citep{RobustFill,\n  NeurosymbProgsynth,neelakantan2015neural, balog2016deepcoder} and use a\nsynthetic dataset generated by randomly sampling programs from the DSL. We\nperform a few simple heuristic checks to ensure generated programs have\nobservable effect on the world and prune out programs performing spurious\nactions (e.g. executing a \\texttt{turnLeft} just after a \\texttt{turnRight} for\nexample). For each program, we a set of IO pairs are generated by sampling\nrandom input grids and executing the program on them to obtain the corresponding\noutput grids. A large number of them are sampled and 6 are kept for each\nprogram, ensuring that all conditionals in a program are hit by at least one of\nthe examples. The first 5 samples serve as the specification, and the sixth one\nis kept as held-out test pair. 5000 programs are not used for training, and get\nsplit out between a validation set and a test set.\n\n\nWe represent the input and output elements as grids where each cell in the grid\nis a vector with 16 channels, indicating the presence of elements\n(\\texttt{AgentFacingNorth}, \\texttt{AgentFacingSouth}, $\\cdots$,\n\\texttt{Obstacle}, \\texttt{OneMarkerPresent}, \\texttt{TwoMarkersPresent},\n$\\cdots$). The input and output grids are initially passed through independent\nconvolution layers, before being concatenated and passed through two\nconvolutional residual blocks and a fully connected layer, mapping them to a\nfinal 512-dimensional representation. We run one decoder per IO pair and perform\na maxpooling operation over the output of all the decoders, out of which we\nperform the prediction of the next token. Our models are implemented using the\nPytorch framework~\\citep{pytorch}. Code and data will be made available.\n\n\n\\subsection{Results}\nWe trained a variety of models on the full Karel dataset containing 1-million\nexamples as well as a reduced dataset containing only $10,000$ examples. In\ngeneral, the small dataset serves to help understand the data efficiency of the\nprogram synthesis methods and is motivated by the expected difficulty of\nobtaining many labeled examples in real-world program synthesis domains.\n\nThe Karel DSL was previously used by ~\\citet{metainduction} to study the\nrelative perfomances of a range of methods depending on the available amount of\ndata. The task considered was however different as they attempted to perform\nprogram induction as opposed to program synthesis. Rather than predicting a\nprogram implementing the desired transformation, they simply output a\nspecification of the changes that would result from applying the program so a\ndirect number comparison wouldn't be meaningful.\n\nModels are grouped according to training objectives. As a baseline, we use\n\\textbf{MLE}, which corresponds to the maximum likelihood objective\n(Eq.\\ref{eq:mle-obj}), similar to the method proposed by ~\\citet{RobustFill}.\nUnless otherwise specified, the reward considered for our other methods is\ngeneralization: +1 if the program matches all samples, including the held out one\nand 0 otherwise. \\textbf{RL} uses the expected reward objective\n(Eq.\\ref{eq:RL}), using REINFORCE to obtain a gradient\nestimate  (Eq.\\ref{eq:Rl-sample}). \\textbf{RL\\_beam} attempts to solve the proxy\nproblem described by Equation~\\eqref{eq:rl-bs} and \\textbf{RL\\_beam\\_div} the\nricher loss function of Equation~\\eqref{eq:rl-div-gen}.\n\\textbf{RL\\_beam\\_div\\_opt} also optimizes the loss of\nequation~\\eqref{eq:rl-div-gen} but the reward additionally includes a term\ninversly proportional to the number of timesteps it takes for the program to\nfinish executing. All RL models are initialized from pretrained supervised\nmodels.\n\n\\setlength{\\floatsep}{5pt}\n\\begin{table}\n  \\small\n  \\renewcommand{\\arraystretch}{1}\n  \\setlength{\\tabcolsep}{0.7ex}\n  \\begin{tabular}{ l@{\\hspace*{3ex}}cc@{\\hspace*{3ex}}cc}\n    \\toprule\n    & \\multicolumn{2}{c}{\\textbf{Full Dataset}} & \\multicolumn{2}{c}{\\textbf{Small Dataset}} \\\\\n    \\cmidrule(lr{3ex}){2-3}\\cmidrule(lr{3ex}){4-5}\n    \\textbf{Top-1} & \\multicolumn{1}{l}{\\textbf{Generalization}} & \\multicolumn{1}{l}{\\textbf{Exact Match}} & \\multicolumn{1}{l}{\\textbf{Generalization}} & \\multicolumn{1}{l}{\\textbf{Exact Match}} \\\\\n    \\midrule\n    \\textbf{MLE} & 71.91 & \\textbf{39.94} & 12.58 & 8.93 \\\\\n    \\textbf{RL} & 68.39 & 34.74 & 0 & 0 \\\\\n    \\textbf{RL\\_beam} & 75.72 & 8.21 & \\textbf{25.28} & \\textbf{17.63} \\\\\n    \\textbf{RL\\_beam\\_div} & 76.20 & 31.25 & 23.72 & 16.31 \\\\\n    \\textbf{RL\\_beam\\_div\\_opt} & \\textbf{77.12} & 32.17 & 24.24 & 16.63 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{\\textbf{RL\\_beam} optimization of program correctness results in\n    consistent improvements in top-1 generalization accuracy over supervised\n    learning \\textbf{MLE}, even though the exact match of recovering the\n    reference program drops. The improved objective function results in further\n    improvements.}\n  \\label{tab:rl_results}\n\\end{table}\n\n\\paragraph{Optimizing for correctness (RL):\\ }\\mbox{}\nResults in Table \\ref{tab:rl_results} show that optimizing for the expected\nprogram correctness consistently provides improvements in top-1 generalization\naccuracy. \\textbf{Top-1 Generalization Accuracy} (Eq. \\ref{eq:generalization})\ndenotes the accuracy of the most likely program synthesized by beam search\ndecoding having the correct behaviour across all input-output examples. We\ndidn't perform any pruning of programs that were incorrect on the 5\nspecification examples. The improved performance of RL methods confirms our\nhypothesis that better loss functions can effectively combat the program aliasing problem.\n\nOn the full dataset, when optimizing for correctness, \\textbf{Exact Match\n  Accuracy} decreases, indicating that the RL models no longer prioritize\ngenerating programs that exactly match the references. On the small dataset,\n\\textbf{RL\\_beam} methods improves both exact match accuracy and generalization.\n\nComparing RL\\_beam to standard RL, we note improvements across all levels of\ngeneralization. By better aligning the RL objective with the sampling that\nhappens during beam search decoding, consistent improvements can be made in\naccuracy. Further improvements are made by encouraging diversity in the beam of\nsolutions (RL\\_beam\\_div) and penalizing long running programs\n(RL\\_beam\\_div\\_opt).\n\nIn the settings where little training data is available, RL methods\nshow more dramatic improvements over MLE, indicating that data\nefficiency of program synthesis methods can be greatly improved by\nusing a small number of samples first for supervised training and\nagain for Reinforcement Learning.\n\nAs a side note, we were unable to achieve high performance when\ntraining the RL methods from scratch. The necessity of extensive\nsupervised pretraining to get benefits from Reinforcement Learning\nfine-tuning is well-known in the Neural Machine Translation\nliterature~\\citep{ranzato2015sequence,googleNMT,\n  wiseman2016sequence,bahdanau2016actor}.\n\nTable \\ref{tab:beam_results} examines the top-1, top-5, and top-50\ngeneralization accuracy of supervised and RL models. RL\\_beam methods performs\nbest for top-1 but their advantage drops for higher-rank accuracy. Inspection of\ngenerated programs shows that the top predictions all become diverse\nvariations of the same program, up to addition\/removal of no-operations\n(\\texttt{turnLeft} followed by \\texttt{turnRight}, full circle obtained by a\nseries of four \\texttt{turnLeft}). The RL\\_beam\\_div objective helps alleviate\nthis effect as does RL\\_beam\\_div\\_opt, which penalizes redundant programs.\nThis is important as in the task of Program Synthesis, we may not necessarily\nneed to return the most likely output if it can be pruned by our specification.\n\n\\begin{table}\n  \\begin{floatrow}\n    \\capbtabbox[.4\\textwidth]{%\n      \\small\n      \\renewcommand{\\arraystretch}{1}\n      \\setlength{\\tabcolsep}{0.7ex}\n      \\begin{tabular}{ l@{\\hspace*{1ex}}c@{\\hspace*{1ex}}c@{\\hspace*{1ex}}c}\n        \\toprule\n        \\textbf{Generalization} & Top-1 & Top-5 & Top-50 \\\\\n        \\midrule\n        \\textbf{MLE} & 71.91 & 79.56 & \\textbf{86.37} \\\\\n        \\textbf{RL\\_beam} & 75.72 & 79.29 & 83.49 \\\\\n        \\textbf{RL\\_beam\\_div} & 76.20 & 82.09 & 85.86 \\\\\n        \\textbf{RL\\_beam\\_div\\_opt} & \\textbf{77.12} & \\textbf{82.17} & 85.38 \\\\\n        \\bottomrule\n      \\end{tabular}\n    }{%\n      \\caption{\\label{tab:beam_results}\\textbf{Top-k accuracies: } MLE shows\n        greater relative accuracy increases as k increases than RL. Methods\n        employing beam search and diversity objectives reduce this accuracy\n        gap by encouraging diversity in the beam of partial programs.}\n    }\n    \\hfill\n    \\capbtabbox[.55\\textwidth]{%\n      \\small\n      \\renewcommand{\\arraystretch}{1}\n      \\setlength{\\tabcolsep}{0.7ex}\n      \\begin{tabular}{ l@{\\hspace*{3ex}}ccc@{\\hspace*{3ex}}ccc  @{\\hspace*{6ex}}ccc @{\\hspace*{3ex}}ccc}\n        \\toprule\n        \\textbf{Top-1 Generalization} & \\textbf{Full Dataset} & \\textbf{Small Dataset} \\\\\n        \\midrule\n        \\textbf{MLE} & 71.91 & 12.58 \\\\\n        \\textbf{MLE\\_learned} & 69.37 & \\textbf{17.02} \\\\\n        \\textbf{MLE\\_handwritten} & 72.07 & 9.81 \\\\\n        \\textbf{MLE\\_large} & \\textbf{73.67} & 13.14 \\\\\n        \\bottomrule\n      \\end{tabular}\n    }{%\n      \\caption{\\label{tab:syntax_results}\\textbf{Grammar prunes the space of\n          possible programs}: On the full dataset, handwritten syntax checking\n        MLE\\_handwritten improves accuracy over no grammar MLE, although\n        MLE\\_large shows that simply adding more parameters results in even\n        greater gains. On the small dataset, learning the syntax MLE\\_learned\n        outperforms handwritten grammar and larger models.}\n    }\n  \\end{floatrow}\n\\end{table}\n\n\\paragraph{Impact of syntax:}\\mbox{}\nWe also compare models according to the use of syntax:\n\\textbf{MLE\\_handwritten} denotes the use of a handwritten syntax\nchecker (Sec \\ref{subsec:handwritten}), \\textbf{MLE\\_learned} denotes\na learned syntax (Sec \\ref{subsec:jointGrammar}), while no suffix\ndenotes no syntax usage. Table \\ref{tab:syntax_results} compares\nsyntax models.\n\nOn the full dataset, leveraging the handwritten syntax leads to\nmarginally better accuracies than learning the syntax or using no\nsyntax. Given access to enough data, the network seems to be capable\nof learning to model the syntax using the sheer volume of training\nexamples.\n\nOn the other hand, when the amount of training data is limited,\nlearning the syntax produces significantly better performance. By\nincorporating syntactic structure in the model architecture and\nobjective, more leverage is gained from small training\ndata. Interestingly, the learned syntax model even outperforms the\nhandwritten syntax model. We posit the syntaxLSTM is free to learn a\nricher syntax. For example, the syntaxLSTM could learn to model the\ndistribution of programs and discourage the prediction of not only\nsyntactically incorrect programs, but also the unlikely ones.\n\nTo control for the extra parameters introduced by the syntaxLSTM, we\ncompare against MLE\\_large, which uses no syntax but features\na larger decoder LSTM, resulting in the same number of parameters as\nMLE\\_learned. Results show that the larger number of parameters is not\nenough to explain the difference in performance, which again indicates\nthe utility of jointly learning syntax.\n\n\\paragraph{Analysis of learned syntax:}\\mbox{}\nSection~\\ref{subsec:jointGrammar} claimed that by decomposing our\nmodels into two separate decoders, we could decompose the learning so\nthat one decoder would specialize in picking the likely tokens given\nthe IO pairs, while the other would enforce the grammar of the\nlanguage. We now provide experimental evidence that this decomposition\nhappens in practice.\n\\setcounter{figure}{1}\n\\begin{wrapfigure}[25]{R}{.4\\textwidth}\n  \\floatbox[\\nocapbeside]{table}[\\FBwidth]\n  {\\caption{Importance of Syntax}\\label{tab:syntaxDecomp}}\n  {\n    \\noindent\\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{@{}c@{\\hspace*{5ex}}cc@{}}\n      \\toprule\n      \\twoLines{\\% Synctactically}{Correct}& \\twoLines{Joint}{Model} & \\twoLines{Without}{Learned Syntax}\\\\\n      \\midrule\n      Amongst Top1 & 100 \\% & 0 \\% \\\\\n      Amongst Top5 & 100 \\% & 0 \\% \\\\\n      Amongst Top50 & 100 \\% & 0 \\% \\\\\n      Amongst Top100 & 99.79 \\% & .04 \\% \\\\\n      \\bottomrule\n    \\end{tabular}}\n    }\n    \\vspace{5pt}\n    \\ffigbox{\\caption{Syntax Comparison}\\label{fig:grammarComp}}\n    {\\begin{subfloatrow}\n        \\ffigbox[\\FBwidth][][]{\\caption{Manual}\\label{subfig:manGrammar}}\n        {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/manual_syntax.png}}\n        \\ffigbox[\\FBwidth][][]{\\caption{Learned}\\label{subfig:learnedGrammar}}\n        {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/learned_syntax.png}}\n        \\ffigbox[\\FBwidth][][]{\\caption{Diff}\\label{subfig:diffGrammar}}\n        {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/diff.png}}\n      \\end{subfloatrow}\n      \\caption{\\label{fig:grammarComp}Syntax Comparison}\n    }\n  \\end{wrapfigure}\n\nTable~\\ref{tab:syntaxDecomp} shows the percentage of syntactically\ncorrect programs among the most likely predictions of the \\textbf{MLE\n  + learned} model trained on the full dataset. Both columns\ncorrespond to the same set of parameters but the second column doesn't\napply the syntaxLSTM's mask to constrain the decoding process. The\nprecipitous drop in syntax accuracy indicates the extent to which the\nprogram decoder has learned to rely on the syntaxLSTM to produce\nsyntactically correct programs.\n\nFigure~\\ref{fig:grammarComp} compares the syntax masks generated by\nthe learned and handwritten syntax models while decoding Program A in\nFigure~\\ref{fig:progsynth}. (\\ref{subfig:manGrammar}) shows output of\nthe handwritten syntax checker; (\\ref{subfig:learnedGrammar}) shows\nsyntaxLSTM output. White cells indicates tokens that are labeled\nsyntactically correct at each decoding step.\n\nFigure \\ref{subfig:diffGrammar} analyzes the difference between the\nhandwritten and learned syntax masks. White indicates similar output,\nwhich occupies the majority of the visualization. Blue cells\ncorrespond to instances where the syntaxLSTM labeled a token correct\nwhen it actually was syntactically incorrect. This type of error can\nbe recovered if the program decoder predicts those tokens as unlikely.\n\nOn the other hand, red cells indicate the syntaxLSTM predicted a valid\ntoken is syntactically incorrect. This type of error is more dangerous\nbecause the program decoder cannot recover the valid token once it is\ndeclared incorrect. The majority of red errors correspond to tokens\nwhich are rarely observed in the training dataset, indicating that the\nsyntaxLSTM learns to model more than just syntax - it also captures\nthe distribution over programs. Given a large enough dataset of real\nprograms, the syntaxLSTM learns to consider non-sensical and unlikely\nprograms as syntactically incorrect, ensuring that generated programs\nare both syntactically correct and likely.\n\n\n\n\\section{Conclusion}\nWe presented two novel contributions to improve state-of-the-art neural program\nsynthesis techniques. Our first contribution uses Reinforcement Learning to\noptimize for generating any consistent program, which we show helps in improving\ngeneralization accuracy of the learned programs for large training datasets. Our\nsecond contribution incorporates syntax checking as an additional conditioning\nmechanism for pruning the space of programs during decoding. We show that\nincorporating syntax leads to significant improvements with limited training\ndatasets.\n\n\\clearpage\n{\\small\n\\bibliographystyle{plainnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nBesov spaces $B^s_{p,q}(\\mathbb{R}^n)$  were  introduced  by Besov \\cite{besov}.  This scale of  spaces has been a favorite through the years, with thousands  of  references available.  Perhaps two  of its most interesting features is that many earlier classes of function spaces appears in this scale, as Sobolev spaces, and also that there are many equivalent ways to define $B^s_{p,q}(\\mathbb{R}^n)$, in such way you can pick the more suitable one for your purpose. The reader may see Stein \\cite{stein}, Peetre \\cite{peetre},  Triebel \\cite{tbook}  for an introduction of Besov spaces on $\\mathbb{R}^n$. For a historical account on Besov spaces and related topics, see  Triebel \\cite{tbook} and the more short but useful Jaffard \\cite{jaffard}, Yuan,  Sickel and  Yang  \\cite{ysy}  and Besov and Kalyabin \\cite{besov2}.\\\\ \n\nIn the last decades there were a huge amount of activity on the generalisation  of  harmonic analysis (see Deng and Han \\cite{deng}), and Besov spaces in particular,   to less regular {\\it phase spaces}, replacing $\\mathbb{R}^n$ by something with a poorer structure. It turns out that for {\\it small $s > 0$} and $p,q\\geq 1$, a proper definition demands  strikingly weak assumptions.  There is a large body of literature that provides a definition and properties of Besov spaces on {\\it homogeneous spaces}, as defined by Coifman and Weiss \\cite{cw}. Those are quasi-metric spaces with a doubling measure, which includes in particular Ahlfors regular metric spaces. We refer to the  pionner work of Han and  Sawyer \\cite{hs} and   Han,  Lu and Yang  \\cite{han2} for Besov spaces  on homogeneous spaces and more recently  Alvarado and   Mitrea \\cite{sharp} and Koskela, Yang, and Zhou \\cite{koskela2} (in this case for metric measure spaces) and Triebel \\cite{fractal}\\cite{fractal2} for Besov spaces on fractals. There is a long list of examples of homogeneous spaces in Coifman  and Weiss \\cite{examples}. The d-sets as  defined in Triebel \\cite{fractal} are also examples of homogeneous spaces. \n\\vspace{5mm}\n\n\\noindent {\\it We give a very elementary (and yet pratical) construction of   Besov  spaces $\\mathcal{B}^s_{p,q}$, with $p,q\\geq 1$ and $0 < s < 1\/p$, for  measure spaces endowed with   a grid, that is, a sequence of finite partitions of measurable sets satisfying certain mild properties. }\n\\vspace{5mm}\n\nThis construction is close to the  martingale Besov spaces  as defined by Gu and Taibleson \\cite{martingale}, however we  deal with \"nonisotropic  splitting\" in our grid without   applying the Gu-Taibleson recalibration procedure to our grid, which  simplifies the definition and it allows a wider class of examples. \\\\\n\nThe main tool in this work is the concept of {\\it atomic decomposition}.  An atomic decomposition is the representation of each function in a space of functions  as a (infinite) linear combination of fractions of the so-called {\\bf atoms}. The advantage of atomic decompositions in that the  atoms are functions that are  often far more regular than a typical element of  the space.  But a distinctive feature (compared with  Fourier series with either Hilbert basis  or unconditional basis)  is that  such atomic decomposition is in general {\\it not} unique.  However, in a successful atomic decomposition of a normed space  of functions we can attribute  a \"cost\" to each possible representation, and the norm of the space is equivalent to the minimal cost (or infimum) among all representations.  A function represented by a single atom  has norm at most one, so the term \"atom\" seems to be appropriated. \\\\\n\n Coifman \\cite{coifman} introduced the atomic decomposition of the  real Hardy space $H^p(\\mathbb{R})$ and Latter \\cite{latter} found an atomic decomposition for $H^p(\\mathbb{R}^n).$ The influential work of   Frazier and Jawerth \\cite{fj} gave us an atomic decomposition for the  Besov spaces $B^s_{p,q}(\\mathbb{R}^n)$. In the context of homogeneous spaces, we have results by Han,  Lu and Yang  \\cite{han2} on atomic decomposition of Besov spaces by H\\\"older atoms.  \\\\\n \n Closer to the spirit  of  this work we have the atomic decomposition of Besov space $B^s_{1,1}([0,1])$, with $s\\in (0,1)$,  by de Souza \\cite{souzao1}  using special atoms, that we call  {\\bf Souza's atoms} (see also De Souza \\cite{souzao2} and de Souza, O'Neil and Sampson  \\cite{sn}). A Souza's atom $a_J$ on an   interval $J$ is quite simple, consisting of a  function whose support is $J$ and $a_J$ is {\\it constant} on $J$. \\\\\n \n We also refer to the results on the B-spline atomic decomposition of the Besov space  of the unit cube of $\\mathbb{R}^n$ in DeVore and  Popov \\cite{spline2} (with  Ciesielski \\cite{spline1} as a precursor), that in the case $0< s < 1\/p$ reduces to an atomic decomposition by Souza's atoms, and the work of  Oswald \\cite{oswald0} \\cite{oswald} on finite element approximation in bounded polyhedral domains on $\\mathbb{R}^n$.\\\\\n \nOn the  study  of Besov spaces on $\\mathbb{R}^n$ and smooth manifolds,  Souza's atoms may seem to have  setbacks that restrict  its usefulness. They are not smooth, so it is fair to doubt  the effectiveness  of  atomic decomposition by Souza's atoms to obtain  a better understanding  of a  partial differential equation or  to represent  data  faithfully\/without artifacts, a constant concern in applications of  smooth wavelets (see Jaffard, Meyer and Ryan \\cite{jaffard2}). \\\\\n\nOn the other hand, in the study of   ergodic properties  of piecewise smooth dynamical systems the {\\it transfer operators} plays a huge role. Those operators acts on  Lebesgue spaces $L^1(m)$,  but they are more useful if one can show it has a (good) action on more regular function spaces. Unfortunately  in many cases the transfer operator  does {\\it not}  preserve neither smoothness nor even continuity of a function.  Discontinuities   are a fact of life in this setting, and they do not go away as in certain dissipative PDEs, since the positive $1$-eigenvectors of this operator, of utmost importance importance in its  study, often have discontinuities themselves. The works of  Lasota and Yorke \\cite{ly}  and Hofbauer and  Keller  \\cite{hk3} are  landmark results in this direction. See  also Baladi \\cite{bb} and Broise \\cite{broise} for more details.   Atomic decomposition with Souza's atoms, being the simplest possible kind of atom with discontinuities, might come in handy  in these cases. That was  a major motivation for this work. \\\\\n\nBesides this fact, in  an {\\it abstract} homogeneous space higher order smoothness does not seem to be a very useful concept, since  we can define $B^s_{p,q}$ just for small values of $s$, so atomic decompositions by  Souza's atoms sounds far more attractive.  \\\\\n\n\nIndeed, the simplicity of Souza's atoms allow us to  throw away the necessity of a metric\/topological structure on the phase space.  We define  {\\bf Besov spaces} on every  measure space with a non atomic finite measure, provided we endowed it  with a {\\it good grid}. A {\\bf good grid}  is just a sequence of finite partitions of measurable sets satisfying certain mild properties.   We give the  definition of Besov space on measure space with a (good) grid in Part II. \\\\\n\n\n \nOn the literature we usually see a known space of functions be described using atomic decomposition. This typically comes after a careful study of such space, and it is often a challenging  task. More rare is the path we follow here. We {\\it start} by {\\it defining }  the Besov spaces {\\it through}  atomic decomposition by  Souza's atoms. This construction of the spaces and the study of its basic properties, as its completeness and its compact inclusion in Lebesgue spaces, uses fairly general and simple arguments, and it does not depend on the particular nature of the atoms used, except for very mild conditions on its regularity. In Part I we describe this construction in full generality. \n\\vspace{5mm}\n\n\\noindent {\\it We construct Besov-ish   spaces. There are far more general than Besov spaces.  In particular the nature of the atoms may change with its location and scale in the phase space and the grid itself can be  very irregular.} \n\\vspace{5mm}\n\n\nThe main result in Part I is Proposition \\ref{trans}. Due the generality of the setting, its  statement  is hopeless technical in nature, however this very powerful result has a simple meaning. Suppose we  have an atomic decomposition of a function. If we replace each of those atoms by a combination of atoms (possibly of a different class) in such way we do not change the location of the support and also  the \"mass\" of the representation is concentrate pretty much on the same original scale, then we obtain a new atomic decomposition, and the cost of this atomic decomposition can be estimate by the cost of the original atomic decomposition. We will use this result many times along this work. This is obviously  connected with  {\\it almost diagonal operators}  as in   Frazier and Jawerth \\cite{fj}\\cite{fj2}.\\\\\n\nIn Part II we also offer a detailed analysis of the Besov spaces defined there. Since we defined it using combinations of Souza's atoms, it is not clear {\\it a priori} how rich are those spaces. So \n\\vspace{5mm}\n\n\\noindent {\\it  We give a bunch of alternative characterisations of these Besov spaces.  We  show that using more  flexible classes of atoms (piecewise H\\\"older atoms, $p$-bounded variation atoms and even Besov atoms with higher regularity), we obtain the same Besov space. This often allows us to easily verify if  a given function belongs to $B^s_{p,q}$. We also have  a mean oscillation characterisation in the spirit of Dorronsoro \\cite{mo} and Gu and Taibleson  \\cite{martingale}, and we also obtained  another one using  Haar wavelets.}\n\\vspace{5mm} \n\n\n  Haar wavelets were introduced by Haar  \\cite{haar}  in the  real line.  The elegant construction of unbalanced Haar  wavelets in general measure spaces with a grid by  Girardi and Sweldens \\cite{gw}  will play an essential role here. If in general homogeneous spaces the Calder\\'on reproducing formula appears to be the bit of  harmonic analysis that survives in it and it allows  to carry out the theory, in finite measure spaces with a good grid (and in particular {\\it compact } homogenous spaces) full-blown Haar systems are  alive and well. Recently  a Haar system was used by Kairema, Li, Pereyra and Ward \\cite{ka} to study dyadic versions of the Hardy and BMO spaces in homogeneous spaces.  \\\\\n \n We also provided a few applications in part III. In particular we study the boundedness of pointwise multipliers acting in the Besov space. Since it is very easy to multiply Souza's atoms, the proofs of these results are very short and easy to understand, generalising some of the results for Besov spaces in $\\mathbb{R}^n$ by  Triebel\\cite{multi} and Sickel \\cite{sickel}. We also study the boundedness of left composition in  Besov spaces of measure spaces with a grid, similar to some results for $B^s_{p,q}(\\mathbb{R}^n)$ in  Bourdaud and Kateb \\cite{k1} (see also Bourdaud and Kateb \\cite{k0}\\cite{kateb1}).\\\\\n\n\nIt may come as a surprise to the reader  that Besov spaces on  compact homogeneous spaces as defined by  Han,  Lu and Yang  \\cite{han2} (and in particular  Gu-Taibleson recalibrated martingale Besov spaces \\cite{martingale})  are indeed {\\it particular cases} of Besov spaces defined here, provided $0< s< 1\/p$ and $s$  is small.  We show this in a forthcoming work \\cite{smania-homo}.\n\\vspace{5mm} \n\n\n\n\n\n\n\\section{Notation} We will use $C_1, C_2, \\dots...$ for positive constants and  $\\lambda_1, \\lambda_2, \\dots$ for positive constants smaller than one.  Whenever a (non-generic) constant is defined, it appears in blue. Every other appearance of the same constant appears in red, and if you click on it  the link will send you to its definition. \n\n\n\\begin{table}[h]\n  \\centering\n  \\caption{Most important notation\/symbols\/constants}\n  \\label{tab:table1}\n  \\begin{tabular}{cc}\n    \\toprule\n    Symbol & Description\\\\\n    \\midrule\n    $I$ & phase space\\\\\n     $m$ & finite measure in  the phase space $I$\\\\\n     $a_P, b_P$ & an atom supported on P\\\\\n     $\\mathcal{A}$ & a class of atoms \\\\\n     $\\mathcal{A}(Q)$ & set of atoms of class $\\mathcal{A}$ supported on $Q$ \\\\\n     $\\mathcal{A}^{sz}_{s,p}$& class of $(s,p)$-Souza's atoms \\\\\n     $\\mathcal{A}^{h}_{s,\\beta,p}$& class of $(s,\\beta,p)$-H\\\"older atoms \\\\\n     $\\mathcal{A}^{bv}_{s, \\beta,p}$& class of $(s,\\beta,p)$-bounded variation atoms \\\\\n     $\\mathcal{A}^{bs}_{s,\\beta,p,q} $ & class of $(s,\\beta,p,q)$-Besov's atoms \\\\\n     $\\mathcal{P}$ & grid of  $I$\\\\\n    $\\mathcal{P}^k$ & family  subsets of $I$ at the $k$-th level of $\\mathcal{P}$\\\\\n    $\\Crr{menor} \\leq \\Crr{maior}$ & describes geometry of the grid $\\mathcal{P}$\\\\\n    $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ & (s,p,q)-Banach space defined by the class of atoms $\\mathcal{A}$\\\\\n    $\\mathcal{B}^s_{p,q}$ or  $\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})$ & (s,p,q)-Banach space defined by Souza's atoms\\\\\n    $P, Q, W$ & elements of the grid $\\mathcal{P}$\\\\\n    $L^p$ or $L^p(m)$ & Lesbesgue spaces of  $(I,\\mathbb{A}, m)$ \\\\\n    $|\\cdot|_p$ & norm in $L^p$, $p \\in (0,\\infty].$ \\\\\n    $p'$ & $1\/p+ 1\/p'=1$, with $p\\in [1,\\infty]$.  \\\\\n    $\\rho$ & $\\min\\{1,p,q\\}$.\\\\\n    $\\hat{t}$  & $\\max\\{t,1\\}.$\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\vspace{1cm}\n\n\\newpage \n\n\\centerline{ \\bf I. DIVIDE AND RULE.}\n\\addcontentsline{toc}{chapter}{\\bf I. DIVIDE AND RULE.}\n\\vspace{1cm}\n\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent   In Part I. we are going to assume  $s > 0$, $p \\in (0,\\infty)$ and $q\\in (0,\\infty].$ \\end{minipage} \n}\n\\ \\\\ \\\\\n\n\\section{Measure spaces and grids}\\label{partition}   Let $I$ be a measure space with a $\\sigma$-algebra $\\mathbb{A}$ and $m$ be a  measure on $(I,\\mathbb{A})$, $m(I)<  \\infty$. Given a measurable set $J\\subset  I$ denote $|J|=m(J)$.  We denote the Lebesgue spaes of $(I,\\mathbb{A},m)$ by $L^p$. A {\\bf grid}  is a sequence of finite  families of measurable sets with positive measure  $\\mathcal{P}= (\\mathcal{P}^k)_{k\\in \\mathbb{N}}$, so that at least one of these families is non empty and\n\\begin{itemize}\n\\item[${\\Cll[G]{fin}}$.] Given $Q\\in \\mathcal{P}^k$, let\n$$\\Omega_Q^k=\\{ P \\in \\mathcal{P}^k\\colon \\ P\\cap Q\\neq \\emptyset \\}.$$\nThen\n$$\\Cll{mult1}=\\sup_k \\sup_{Q \\in \\mathcal{P}^k} \\# \\Omega_Q^k< \\infty.$$\n\\end{itemize}\n\n  Define  $|\\mathcal{P}^k|=\\sup \\{|Q|\\colon Q \\in \\mathcal{P}^k\\}$.  To simplify the notation we also assume that $P\\neq Q$ for every $P\\in \\mathcal{P}^i$ and $Q\\in \\mathcal{P}^j$ satisfying $i\\neq j$. We often abuse notation using $\\mathcal{P}$ for both   $(\\mathcal{P}^k)_{k\\in \\mathbb{N}}$ and $\\cup_k \\mathcal{P}^k$. \n \\begin{remark} \n  There are plenty of examples of spaces with  grids. Perhaps the simplest one is obtained considering  $[0,1)$ with the Lebesgue measure and the {\\bf dyadic grid} $\\mathcal{D}=(\\mathcal{D}^k)_k$ given by\n  $$\\mathcal{D}^k=\\{[i\/2^k,(i+1)\/2^k), \\ 0\\leq i < 2^k   \\}.$$\n  We can also consider the dyadic grid $\\mathcal{D}_n=(\\mathcal{D}^k_n)_k$ of $[0,1)^n$, endowed with the Lebesgue measure,  given by\n   $$\\mathcal{D}^k_n=\\{J_1\\times\\cdots \\times J_n, \\ with \\ J_i\\in \\mathcal{D}^k \\}.$$\n  and also the corresponding $d$-adic grids replacing $2^k$ by $d^k$ in the above definitions. The above grids are somehow special since they are nested sequence of partitions of the phase space $I$ and all elements on the same level has exactly the same measure. \n  \\end{remark}\n  \n    \\begin{remark} Indeed, any  measure space with a finite non-atomic measure can be endowed with a grid  made of a nested sequence of partitions and such that all elements on the same level has exactly the same measure, since any such measure space is measure-theoretically the same that a finite interval with the Lebesgue measure.\n  \\end{remark}\n  \n   \\begin{remark} If we  consider a (quasi)-metric space $I$ with a finite measure $m$, we would like to construct \"nice\" grids on $(I,m)$. It turns out that if $(I,m)$ is a homogeneous space one can construct a nested sequence of partitions of the phase space $I$ and all elements on the same level are open subsets and have \"essentially\" the same measure. This is an easy consequence of a famous result by Christ \\cite{christ}. See \\cite{smania-homo}.\n   \\end{remark}\n  \n  \\begin{remark} One can constructs grids for smooth compact manifolds and bounded polyhedral domains in $\\mathbb{R}^n$ using successive subdivisions of an initial triangulation of the domain (see for instance Oswald \\cite{oswald0}\\cite{oswald} ).\n  \\end{remark}\n  \n\n  \n \n\n\n\\section{A bag  of tricks.}\n\n Following closely the notation of Triebel \\cite{fractal}, consider the set $\\ell_q(\\ell_p^{\\mathcal{P}})$ of all indexed sequences $$x=(x_{P})_{P\\in \\mathcal{P}},$$ with $x_{P}\\in \\mathbb{C}$, satisfying  \n$$|x|_{\\ell_q(\\ell_p^{\\mathcal{P}})}= \\Big( \\sum_k \\big(\\sum_{P\\in \\mathcal{P}^k} |x_{P}|^p \\big)^{q\/p} \\Big)^{1\/q} < \\infty,$$\nwith the usual modification when  $q=\\infty$. Then  $(\\ell_q(\\ell_p^{\\mathcal{P}}),|\\cdot|_{\\ell_q(\\ell_p^{\\mathcal{P}})})$ is a complex $\\rho$-Banach space  with $\\rho= \\min \\{1,p,q\\}$, that is,  $d(x,y)=|x-y|_{\\ell_q(\\ell_p^{\\mathcal{P}})}^\\rho$ is a complete metric in $\\ell_q(\\ell_p^{\\mathcal{P}}).$\n\n\nThe following is  a pair of arguments we will use along this paper to estimate norms in $\\ell_p$ and $\\ell_q(\\ell_p^{\\mathcal{P}})$. Those are very elementary, and we do not claim any originality. We collect them here to simplify the exposition. The reader can skip this for the cases  $p, q \\geq 1$, when the  results bellow reduce  to the familiar H\\\"older's and Young's inequalities. Their proofs  were mostly based  on  \\cite[Proof of Theorem 3.1]{fj}. Recall that for $t \\in (0,\\infty]$ we defined $\\hat{t}=\\max \\{ 1, t\\}$.\n\n\\begin{proposition}[H\\\"older-like trick] \\label{holder}Let $t\\in (0, \\infty)$ and $q\\in (0,\\infty]$. Let $a=(a_k)_k, b=(b_k)_k,c=(c_k)_k$ nonnegative   sequences  such that for every $k$\n$$a_k^{1\/\\hat{t}} \\leq C^{1\/\\hat{t}} b_k^{1\/\\hat{t}}c_k^{1\/\\hat{t}}.$$\nThen if $q< \\infty$ we have\n$$(\\sum_k a_k^{1\/\\hat{t}})^{\\hat{t}\/t} \\leq  C^{1\/t} \\Cll{co}(t,q,b)\\Big(  \\sum_k c_k^{q\/t}\\Big)^{1\/q},$$\nand if  $q=\\infty$ \n$$(\\sum_k a_k^{1\/\\hat{t}})^{\\hat{t}\/t} \\leq  C^{1\/t} \\Crr{co}(t,q,b)\\sup_k c_k^{1\/t}.$$\nwhere  \n\\begin{itemize} \n\\item[A.] If $t\\geq 1$ and $q\\geq 1$ then $\\Crr{co}(t,q,b)=(\\sum_k b_k^{q'\/t})^{1\/q'} $ if  $q > 1$, \\\\ and  $\\Crr{co}(t,1,b)=\\sup_k b_k^{1\/t}$  if  $q=1$.\n\\item[B.] If $t\\geq 1$ and $q\\leq  1$ then $\\Crr{co}(t,q,b)=\\sup_k b_k^{1\/t}$.\n\\item[C.] If $t< 1$ and $q\/t\\geq 1$ then $\\Crr{co}(t,q,b)=(\\sum_k b_k^{(q\/t)'})^{1\/(t(q\/t)')} $ if $q< t$, \\\\ and \n$\\Crr{co}(t,q,b)=\\sup_k b_k^{1\/t}$ if $q=t$. \n\\item[D.] If $t< 1$ and $q\/t<   1$ then $\\Crr{co}(t,q,b)= \\sup_k b_k^{1\/t} $.\n\\end{itemize}\n\n\\end{proposition}  \n\\begin{proof}   We have \n\n\\noindent {\\it Case A.}  If $t\\geq 1$ and $q\\geq 1$, by the H\\\"older inequality for the pair $(q,q')$\n$$\\sum_{k} a_k^{1\/\\hat{t}}=\\sum_k a_k^{1\/t} \\leq C^{1\/t} (\\sum_k b_k^{q'\/t})^{1\/q'}    \\big(  \\sum_k c_k^{q\/t}\\big)^{1\/q}.$$\n\n\n\\noindent {\\it Case B.} If $t\\geq 1$ and $q\\leq 1$ then   the triangular inequality for $|\\cdot|^q$ implies\n\\begin{eqnarray*}\n(\\sum_{k} a_k^{1\/\\hat{t}})^q&=&(\\sum_k a_k^{1\/t})^q \\leq C^{q\/t} \\big(   \\sum_k b_k^{1\/t}c_k^{1\/t}\\big)^q \\\\\n&\\leq& C^{q\/t}   \\sum_k \\big(  b_k^{1\/t}c_k^{1\/t}\\big)^q  \\\\\n&\\leq& C^{q\/t}   (\\sup_k b_k^{q\/t})\\Big( \\sum_kc_k^{q\/t}\\Big)\n\\end{eqnarray*}\n\n\\noindent {\\it Case C.} If $t< 1$ and $q\/t \\geq 1$ then $\\hat{t}\/t=1\/t$ and by the  H\\\"older inequality for the pair $(q\/t,(q\/t)')$\n$$\\sum_{k} a_k^{1\/\\hat{t}}\\leq C (\\sum_k b_k^{(q\/t)'})^{1\/(q\/t)'}    \\big(  \\sum_k c_k^{q\/t}\\big)^{t\/q},$$\n\n\\noindent {\\it Case D.} if $t<  1$ and $q\/t<  1$ then  using the triangular inequality for $|\\cdot|^{q\/t}$\n\\begin{align*} (\\sum_k a_k^{1\/\\hat{t}})^{q\/t} &\\leq C^{q\/t} \\big(\\sum_k  b_k c_k\\big)^{q\/t}\\\\\n&\\leq C^{q\/t} \\sum_k  \\big(b_k c_k\\big)^{q\/t}\\\\\n&\\leq C^{q\/t} (\\sup_k b_k^{q\/t}) \\Big(  \\sum_k c_k^{q\/t}\\Big). \\end{align*} \n\\end{proof}\n\n\\begin{proposition}[Convolution  trick]\\label{young} Let $p,q\\in (0, \\infty)$. Let $a=(a_k)_{k\\in \\mathbb{Z}}, b=(b_k)_{k\\in \\mathbb{Z}},c=(c_k)_{k\\in \\mathbb{Z}}\\geq 0$ be such that for every $k$\n$$a_k^{1\/\\hat{p}} \\leq C^{1\/\\hat{p}} \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{1\/\\hat{p}}c_i^{1\/\\hat{p}}.$$\nThen\n$$(\\sum_k a_k^{q\/p})^{1\/q} \\leq  C^{1\/p}    \\Crr{co2}(p,q,b) \\Big(  \\sum_k c_k^{q\/p}\\Big)^{1\/q},$$\nwhere $\\Cll{co2}\\geq 1$ satisfies \n\\begin{itemize} \n\\item[A.] If $p\\geq 1$ and $q\\geq 1$ then $\\Crr{co2}(p,q,b)= \\sum_{k\\in \\mathbb{Z}} b_k^{1\/p}$. \n\\item[B.] If $p\\geq 1$ and $q\\leq  1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k^{q\/p})^{1\/q}$.\n\\item[C.] If $p< 1$ and $q\/p\\geq 1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k)^{1\/p}$.\n\\item[D.] If $p< 1$ and $q\/p<   1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k^{q\/p})^{1\/q}$\n\\end{itemize}\n\n\\end{proposition}  \n\\begin{proof} We have\n\n\n\\noindent {\\it Case A.}  If $p\\geq 1$ and $q\\geq 1$, by the Young's inequality for the pair $(1,q)$ \n$$(\\sum_k a_k^{q\/p})^{1\/q} \\leq  C^{1\/p}    \\big( \\sum_k b_k^{1\/p}\\big) \\Big(  \\sum_k c_k^{q\/p}\\Big)^{1\/q}.$$\n\n\\noindent {\\it Case B.} If $p\\geq 1$ and $q\\leq 1$ then   the triangular inequality for $|\\cdot|^q$ and the Young's inequality for the pair $(1,1)$ imply \n\\begin{eqnarray*}\n\\sum_{k} a_k^{q\/p}&=& C^{q\/p} \\sum_k \\big(  \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{1\/p}c_i^{1\/p} \\big)^q \\\\\n&\\leq&C^{q\/p}   \\sum_k \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{q\/p}c_i^{q\/\\hat{p}}  \\\\\n&\\leq& C^{q\/p}   (\\sum_k b_k^{q\/p})\\Big( \\sum_k c_k^{q\/p}\\Big)\n\\end{eqnarray*}\n\n\\noindent {\\it Case C.} If $p< 1$ and $q\/p \\geq 1$ then by the  Young's  inequality for the pair $(1, q\/p)$\n$$(\\sum_{k} a_k^{q\/p})^{p\/q} \\leq C (\\sum_k b_k)    \\big(  \\sum_k c_k^{q\/p}\\big)^{p\/q}.$$\n\n\\noindent {\\it Case D.} If $p<  1$ and $q\/p<  1$ then  using the triangular inequality for $|\\cdot|^{q\/p}$ and the Young's inequality for the pair $(1,1)$ \n\\begin{eqnarray*} \\sum_k a_k^{q\/p}&\\leq& C^{q\/p} \\sum_k \\big(\\sum_{i\\in \\mathbb{Z}}  b_{k-i} c_i\\big)^{q\/p}\\\\\n&\\leq& C^{q\/p} \\sum_k  \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{q\/p}  c_i^{q\/p}\\\\\n&\\leq& C^{q\/p} (\\sum_k b_k^{q\/p}) \\Big(  \\sum_k c_k^{q\/p}\\Big).\n\\end{eqnarray*} \n\\end{proof}\n\n\\section{Atoms}\n\n Let $\\mathcal{P}$ be a grid.  Let $p \\in [1,\\infty), u \\in [1,\\infty]$,  and $s> 0$. A family of  {\\bf atoms } associated with $\\mathcal{P}$ of type $(s,p,u)$ is an indexed family $\\mathcal{A}$ of pairs   $(\\mathcal{B}(Q),\\mathcal{A}(Q))_{Q\\in \\cup_k \\mathcal{P}^k}$ where  \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{banach}}$.] $\\mathcal{B}(Q)$ is a complex Banach space contained in $L^{pu}$.\n\\item[${\\Cll[A]{suporte}}$.] If $\\phi \\in \\mathcal{B}(Q)$ then $\\phi(x)=0$ for every $x \\not\\in Q$.\n\\item[${\\Cll[A]{convex}}$.]  $\\mathcal{A}(Q)$ is a convex subset of $\\mathcal{B}(Q)$ such that $\\phi \\in \\mathcal{A}(Q)$ if and only if $\\sigma~\\phi~\\in~\\mathcal{A}(Q)$ for every $\\sigma \\in \\mathbb{C}$ satisfying $|\\sigma|=1$.\n\\item[${\\Cll[A]{atom}}$.] We have $$|\\phi|_{pu} \\leq |Q|^{s-\\frac{1}{u'p}}$$ for every $\\phi \\in \\mathcal{A}(Q)$. \n\\end{itemize}\n\n We will say that $\\phi \\in \\mathcal{A}(Q)$ is an $\\mathcal{A}$-atom of type $(s,p,u)$ supported on $Q$ and that $\\mathcal{B}(Q)$ is  the local Banach space on $Q$. Sometimes we also assume \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{compact}}$.]  For every $Q\\in \\mathcal{P}$ we have that $\\mathcal{A}(Q)$ is a compact subset in the strong topology of    $L^p$.\\\\\n\\end{itemize}\nor\n\\begin{itemize}\n\\item[${\\Cll[A]{compactw}}$.] We have $p\\in [1,\\infty)$ and  every $Q\\in \\mathcal{P}$ the set  $\\mathcal{A}(Q)$ is a sequentially compact subset in the {\\it weak } topology   of $L^p$.\\\\\n\\end{itemize}\nor even \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{finite}}$.]  For every $Q\\in \\mathcal{P}$ we have that $\\mathcal{B}(Q)$ is finite dimensional and  $\\mathcal{A}(Q)$ contains a neighborhood of $0$ in $\\mathcal{B}(Q)$.\\\\\n\\end{itemize}\n \n \n We provide examples of classes of atoms in Section \\ref{secatom}.\n\n\\section{Besov-ish spaces} \n\n Let $p \\in (0,\\infty)$, $u \\in [1,\\infty]$, $q \\in (0,\\infty]$, $s> 0$,  $\\mathcal{P}=(\\mathcal{P}^k)_{k\\geq 0}$ be a grid  and let $\\mathcal{A}$ be a family of atoms of type $(s,p,u)$.  We will also assume that  \n\\begin{itemize}\n\\item[${\\Cll[G]{sum}}.$] We have\n$$\\Cll{decaypart} =    \\Crr{co}(p,q, (|\\mathcal{P}^k|^s)_k) < \\infty.$$\nand\n$$\\lim_k |\\mathcal{P}^k|=0.$$\n\\end{itemize}\nThe {\\bf Besov-ish space} $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$  is  the set of all complex valued functions $g \\in L^p$ that  can  be represented by an absolutely convergent series on $L^p$\n\\begin{equation} \\label{rep} g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k}    s_Q a_Q\\end{equation}\nwhere $a_Q$ is in $\\mathcal{A}(Q)$, $s_Q \\in \\mathbb{C}$  and with finite {\\bf cost }\n\\begin{equation} \\label{rep2} \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}    |s_Q|^p)^{q\/p} \\big)^{1\/q} < \\infty.\\end{equation}\nNote that the inner sum in (\\ref{rep}) is finite. By absolutely convergence in $L^p$ we mean that \n$$\\sum_{k=0}^{\\infty} \\big| \\sum_{Q \\in \\mathcal{P}^k}    s_Q a_Q  \\big|_p^{p\/\\hat{p}} < \\infty.$$ \nThe series in (\\ref{rep}) is called a {\\bf $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$-representation} of the function $g$. \nDefine\n$$|g|_{\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})} = \\inf  \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}     |s_Q|^p )^{q\/p} \\big)^{1\/q} ,$$\nwhere the infimum runs over all possible representations of $g$ as in (\\ref{rep}).  \n\nQuite often along this work, when it is obvious the choice of the measure space $I$ and\/or the grid $\\mathcal{P}$  we will write either $\\mathcal{B}^s_{p,q}(\\mathcal{P}, \\mathcal{A})$ or even $\\mathcal{B}^s_{p,q}( \\mathcal{A})$ instead of $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$.  Whenever we write just $\\mathcal{B}^s_{p,q}$ it means that we choose the Souza's atoms $\\mathcal{A}^{sz}_{s,p}$, with parameters $s$ and $p$,  a class of atoms  we  properly define in Section \\ref{souzaa}. \n\n\n\n\n\n\n\\begin{proposition}\\label{lp}   Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Let $t \\in (0,\\infty)$ be such that \n\\begin{equation}\\label{c11} s-\\frac{1}{p}+\\frac{1}{t} \\geq 0, \\ p \\leq t \\leq pu\\end{equation} \nand suppose\n\\begin{equation}\\label{decay}  \\Cll{kt}=\\Crr{mult1}^{1+1\/t}  \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) < \\infty,\\end{equation} \nThen for every coefficients $s_Q$ satisfying (\\ref{rep2}) and every $\\mathcal{A}$-atoms $a_Q$  on $Q$ the series (\\ref{rep}) converges absolutely on $L^t$. Indeed \n\\begin{align}\\label{flp}\n|\\sum_{Q \\in \\mathcal{P}^k}   s_Q a_Q|_t&\\leq  \\Crr{mult1}^{1+1\/t}   |\\mathcal{P}^k|^{s-\\frac{1}{p} +\\frac{1}{t}} \\Big(  \\sum_{Q \\in \\mathcal{P}^k} |s_Q|^p   \\Big)^{1\/p}\n\\end{align} \nand\n\\begin{equation}\\label{inclulp}  |g|_{t} \\leq  \\Crr{kt} | \\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}.\\end{equation}\n\\end{proposition}\n\\begin{proof}  Firstly note that if $p\\leq t \\leq  pu$\n\\begin{align*}  |a_P|^t_t &= \\int |a_P(x)|^t  1_P\\ dm(x)\\\\\n&\\leq  ||a_P|^t|_{\\frac{pu}{t}}  |1_P|_{\\frac{pu}{pu-t}} \\\\\n&\\leq  |a_P|_{pu}^t |P|^{\\frac{pu-t}{pu}}\\\\\n&\\leq |P|^{ts-\\frac{t}{u'p}} |P|^{\\frac{pu-t}{pu}} =|P|^{t(s-\\frac{1}{p} +\\frac{1}{t})}.\\end{align*} \nConsequently\n\\begin{align}\n|\\sum_{Q \\in \\mathcal{P}^k}   s_Q a_Q|_t^{t}&\\leq   \\int |\\sum_{Q \\in \\mathcal{P}^k}   s_Q a_Q|^t \\ dm\\nonumber \\\\\n&\\leq  \\sum_{Q \\in \\mathcal{P}^k}  \\int_Q |\\sum_{P \\in \\mathcal{P}^k}   s_P a_P|^t  \\ dm \\nonumber \\\\\n&\\leq   \\sum_{Q \\in \\mathcal{P}^k}  \\int_Q |\\sum_{P \\in \\Omega_Q^k}   s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq  \\Crr{mult1}^{t} \\sum_{Q \\in \\mathcal{P}^k}  \\int_Q  \\sum_{P \\in \\Omega_Q^k}  |s_P a_P|^t \\ dm  \\nonumber \\\\\n&\\leq  \\Crr{mult1}^{t}  \\sum_{Q \\in \\mathcal{P}^k}  \\sum_{P \\in \\Omega_Q^k}  \\int   |s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t}\\sum_{Q \\in \\mathcal{P}^k}  \\sum_{P \\in \\Omega_Q^k} \\int  |s_P|^t |a_P|^t_t \\ dm \\ \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t}   \\sum_{Q \\in \\mathcal{P}^k} \\sum_{P \\in \\Omega_Q^k}   |P|^{t(s-\\frac{1}{p} +\\frac{1}{t})} |s_P|^t  \\nonumber \\\\\n&\\leq  \\Crr{mult1}^{t+1}  \\sum_{P \\in \\mathcal{P}^k}  |P|^{t(s-\\frac{1}{p} +\\frac{1}{t})} |s_P|^t   \\ \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t+1}  |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} \\sum_{P \\in \\mathcal{P}^k} |s_P|^t  \\ \\nonumber \n\\end{align} \nBy Proposition \\ref{holder} (H\\\"older-like trick) and $p\\leq t$ we have \n\\begin{align}\n|\\sum_k \\sum_{Q \\in \\mathcal{P}^k}   s_Q a_Q|_t &\\leq   \\Big( \\sum_k\\big( |\\sum_{Q \\in \\mathcal{P}^k}   s_Q a_Q|_t^{t}\\big)^{1\/\\hat{t}}   \\Big)^{\\hat{t}\/t} \\nonumber \\\\ \n&\\leq  \\Crr{mult1}^{1+\\frac{1}{t}}  \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) \\Big(  \\sum_k \\big( \\sum_{P \\in \\mathcal{P}^k} |s_P|^t  \\big)^{q\/t}\\Big)^{1\/q}. \\nonumber  \\\\\n&\\leq  \\Crr{mult1}^{1+\\frac{1}{t}}  \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) \\Big(  \\sum_k \\big( \\sum_{P \\in \\mathcal{P}^k} |s_P|^p \\big)^{q\/p}\\Big)^{1\/q}. \\nonumber \n\\end{align} \n\\end{proof}\n\\begin{remark}\\label{sharp} Note that due ${\\Crr{sum}}$ if  $t=p$ then  $\\Crr{kt} <\\infty$.  Sometimes it is convenient to use sharper estimates than (\\ref{flp}) and (\\ref{inclulp}) replacing  $|\\mathcal{P}^k|$ by the sequence \n$$C^k =\\max \\{ |Q|\\colon \\ Q\\in \\mathcal{P}^k, \\ s_Q\\neq 0\\}.$$\nFor instance, if $s_Q=0$ for every $Q\\in \\mathcal{P}^k$ with $k\\leq N$, then we can replace $ \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) $ by  $\\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})}1_{(N,\\infty)}(k) )_k ) $ in (\\ref{decay}). Here $1_{(N,\\infty)}$ denotes  the indicator function of the set $(N,\\infty)$.\n\\end{remark}\n\n\n\n\n\\begin{proposition}  Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Then  $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ is a complex  linear space and $|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is a $\\rho$-norm, with $\\rho=\\min\\{1,p,q\\}$. Moreover the linear inclusion\n$$\\imath\\colon  (\\mathcal{B}^s_{p,q}(\\mathcal{A}),|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})\\rightarrow (L^p,|\\cdot|_p)$$\nis continuous.  \\end{proposition} \n\\begin{proof} Le $f , g  \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$. Then there are $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations \n$$ f = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k}    s_Q' a_Q'   \\  and \\ g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k}    s_Q a_Q.$$\nLet $sgn \\ 0 =0$ and $sgn \\ z =z\/|z|$ otherwise. Of course\n\\begin{equation} \\label{quo}  \\sum_{Q \\in \\mathcal{P}^k}    c_Q b_Q = \\sum_{Q \\in \\mathcal{P}^k}    s_Q' a_Q'  + \\sum_{Q \\in \\mathcal{P}^k}    s_Q a_Q,\\end{equation} \nwhere\\footnote{We don't need to worry so much if  $|s_Q| + |s_Q'|=0$, since in this case  $c_Q=0$ and we can choose $b_Q$ to be an arbitrary atom (for instance $a_Q$) in such way that (\\ref{quo}) holds.  For this reason we are not going to explicitly deal with similar situations ( that is going to appear quite often) along the paper.} \n$$b_Q =   \\frac{|s_Q'|}{|s_Q| + |s_Q'|} sgn(s_Q') a_Q'     +  \\frac{|s_Q|}{|s_Q| + |s_Q'|}  sgn(s_Q') a_Q,$$\nand\n$$c_Q = |s_Q'| + |s_Q|.$$\nNote that $sgn(s_Q') a_Q', sgn(s_Q) a_Q$ are atoms due ${\\Crr{convex}}$. So  by ${\\Crr{convex}}$  we have that $b_Q$ is also an atom, since it  is a convex combination of atoms. In particular \n$$\\sum_k  \\sum_{Q \\in \\mathcal{P}^k}    c_Q b_Q $$\nconverges absolutely in $L^p$ to $f+g$. It remains to prove that this is a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation of $f+g$. Indeed\n$$\\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}    |c_Q|^p)^{q\/p} \\big)^{\\rho\/q} \\leq \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}    |s_Q'|^p)^{q\/p} \\big)^{\\rho\/q} + \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}    |s_Q|^p)^{q\/p} \\big)^{\\rho\/q}.$$\nTaking the infimum over all possible $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations of $f$ and $g$ we obtain\n$$|f+g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho \\leq  |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho  + |g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho.$$\nThe identity $ |\\gamma f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}=  |\\gamma| |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is obvious.  By Proposition \\ref{lp} we have that  if $|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}=0$ then $|f|_p=0$, so $f=0$, so $|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is a $\\rho$-norm, moreover  (\\ref{inclulp}) tell us that $\\imath$ is continuous. \n\\end{proof}\n\\begin{proposition}\\label{compa2} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Suppose that $g_n$ are functions in $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ with  $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations \n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a $\\mathcal{A}$-atom supported on $Q$, satisfying \n\\begin{itemize}\n\\item[i.] There is $C$ such that for every $n$ \n\\begin{equation}\\label{estt}  (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C.\\end{equation} \n\\item[ii.] For every $Q\\in \\mathcal{P}$ we have that $s_Q=\\lim_n s_Q^n$ exists. \n\\item[iii.] For every $Q\\in \\mathcal{P}$ there is  $a_Q\\in \\mathcal{A}(Q)$ such that \n\\begin{enumerate}\n\\item  either the sequence $a_Q^n$ converges to $a_Q$  in the strong topology  of $L^p$, or\n\\item we have  $p\\in [1,\\infty)$ and $a_Q^n$ weakly converges to  $a_Q$.\n\\end{enumerate}\n\\end{itemize}\nthen $g_n$ either strongly or weakly  converges in $L^p$, respectively,  to $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$,  where  $g$ has  the   $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation\n\\begin{equation}\\label{repp} g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q a_Q\\end{equation} \nthat satisfies \n\\begin{equation}\\label{estt2} (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C\\end{equation} \n\\end{proposition}\n\\begin{proof}By (\\ref{estt}) it follows that (\\ref{estt2}) holds and that (\\ref{repp}) is indeed a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation of a function $g$. It remains to prove that $g_n$ converges to $g$ in $L^p$ in the topology under consideration. Given $\\epsilon > 0$,  fix  $N$ large enough such that \n$$\\Crr{mult1}^{1+1\/p}  \\Crr{co}(p,q, ( |\\mathcal{P}^k|^{ps}1_{[N,\\infty)}(k) )_k )   (2C^\\rho+1)^{1\/\\rho}< (\\epsilon\/2)^{\\hat{p}\/p}.$$\nWe can write\n$$g_n-g= \\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q) + \\sum_{k> N} \\sum_{Q\\in {\\mathcal P}^k}  c_Q^n b_Q^n,$$\nwhere\n$$b_Q^n =   \\frac{|s_Q^n|}{|s_Q^n| + |s_Q|} sgn(s_Q^n) a_Q^n     +  \\frac{|s_Q|}{|s_Q^n| + |s_Q|}  sgn(-s_Q) a_Q,$$\nis an atom in $\\mathcal{A}(Q)$, and \n$$c_Q^n = |s_Q^n| + |s_Q|.$$\nNote that  the series in the r.h.s. converges absolutely  in $L^p$. Of course\n$$ (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|c_Q^n|^p)^{q\/p})^{1\/q}\\leq (2C^\\rho+1)^{1\/\\rho},$$\nSo by (\\ref{inclulp})  in Proposition \\ref{lp} (see also Remark \\ref{sharp}) we have \n$$\n|\\sum_{k> N} \\sum_{Q \\in \\mathcal{P}^k}   c_Q^n b_Q^n|_p\\leq \\Crr{mult1}^{1+1\/p}  \\Crr{co}(p,q, ( |\\mathcal{P}^k|^{ps}1_{[N,\\infty)}(k) )_k )   (2C^\\rho+1)^{1\/\\rho}  < (\\epsilon\/2)^{\\hat{p}\/p}. \n$$\nIn the case $ii.1$, note that if $n$  is large enough  then \n$$| \\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q)|_p^{p\/\\hat{p}}< \\epsilon\/2,$$\nand consequently $|g-g_n|_p^{p\/\\hat{p}} < \\epsilon$.  So $g_n$ strongly converges to $g$. \n\n\\noindent In the case $ii.2$, given $\\phi\\in (L^p)^\\star$, with $p\\geq 1$ we have that for $n$  large enough\n$$|\\phi(\\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q))|\\leq |\\phi|_{(L^p)^\\star} \\epsilon\/2,$$\nand of course\n$$\n|\\phi(\\sum_{k> N} \\sum_{Q \\in \\mathcal{P}^k}   c_Q^n b_Q^n|_p)| \\leq  |\\phi|_{(L^p)^\\star}  \\epsilon\/2,\n$$\nso $g_n$ weakly converges to $g$ in $L^p$. \n\\end{proof}\n\n\\begin{corollary}\\label{compa1}  Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$, and \n\\begin{enumerate}\n\\item either ${\\Crr{compact}}$ or\n\\item we have $p\\geq 1$ and  ${\\Crr{compactw}}$.\n\\end{enumerate}\nThen\n\\begin{itemize}\n\\item[i.] Let $g_n \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ be such that $|g_n|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq C$ for every $n$. Then there is a subsequence that converges either strongly or weakly in $L^p$, respectively,  to some $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ with $|g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq~C$.\n\\item[ii.] In both cases $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a complex $\\rho$-Banach space, with $\\rho=\\min\\{1,p,q\\}$, \n\\item[iii] If ${\\Crr{compact}}$ holds then  the inclusion \n$$\\imath \\colon (\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}) \\rightarrow (L^p, |\\cdot|_p)$$\nis a compact linear inclusion. \n\\end{itemize} \n\\end{corollary} \n\\begin{proof}[ Proof of i.] There are $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations\n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a $\\mathcal{A}$-atom supported on $Q$ and\n\\begin{equation} \\label{sed0}  (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C+\\varepsilon_n,\\end{equation}\nand  $1\\geq \\varepsilon_n\\to 0$.\nIn particular, $|s_Q^n|\\leq C+1$. Since the set $\\cup_k \\mathcal{P}^k$ is countable,  by the Cantor  diagonal argument, taking a subsequence we can assume  that $s_Q^n\\to_n s_Q$ for some $s_Q \\in \\mathbb{C}$.  Due ${\\Crr{compact}}$ (${\\Crr{compactw}}$) and the Cantor  diagonal argument, we can suppose that  $a_Q^n$ strongly (weakly) converges  in $L^p$ to  some $a_Q \\in \\mathcal{A}(Q)$. We set \n$$g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Qa_Q.$$\nBy Proposition \\ref{compa2}  we conclude that  $g\\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ with  $|g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq C$, and that $g_n$ converges to $g$ in $L^p$.\n\\end{proof}\n\\begin{proof}[Proof of ii.] Let $g_n$ be a Cauchy sequence on $\\mathcal{B}^s_{p,q}(\\mathcal{A})$. By Proposition \\ref{lp} we have that $g_n$ is also a Cauchy sequence in $L^p$. Let $g$ be its limit in $L^p$. By Corollary   \\ref{compa1}.i  have that $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$. Note that for large $m$ and $n$  \n$$|g_n-g_m|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq \\epsilon,$$\nand $g_n - g_m$ converges to $g_n-g$ in $L^p$, so again by Corollary  \\ref{compa1}.i  we have that \n$$|g_n-g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq \\epsilon,$$\nso $g_n$ converges to $g$ in $\\mathcal{B}^s_{p,q}(\\mathcal{A})$.\n\\end{proof}\n\n\\begin{proof}[Proof of iii.]  It follows from i. \\end{proof}\n\nThe proof of the following resukt is quite similar. \n\\begin{corollary}\\label{compa12}  Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$, and \n\\begin{enumerate}\n\\item either ${\\Crr{compact}}$ or\n\\item we have $p\\geq 1$ and  ${\\Crr{compactw}}$.\n\\end{enumerate}\nThen for every $f\\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ there is a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation\n$$f=\\sum_k \\sum_{P\\in \\mathcal{P}^k}   c_P a_P$$\nsuch that \n$$|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}=\\Big( \\sum_k \\big(\\sum_{P\\in \\mathcal{P}^k}  |c_P|^p\\big)^{q\/p} \\Big)^{1\/q}.$$\n\\end{corollary}\n\n\nWe refer to Edmunds and  Triebel \\cite{et} for more information on compact linear transformations between quasi-Banach spaces.\n\n\\begin{corollary} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. If  for every $Q\\in \\mathcal{P}$ we have that $\\mathcal{B}(Q)$ is finite-dimensional and $\\mathcal{A}(Q)$ is a closed subset of $\\mathcal{B}(Q)$    then \n $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a $\\rho$-Banach space, with $\\rho=\\min\\{1,p,q\\}$.\n\\end{corollary}\n\\begin{proof} Since all norms are equivalent in $\\mathcal{B}(Q)$ we have that ${\\Crr{atom}}$ implies that $\\mathcal{A}(Q)$ is a closed and bounded subset of $\\mathcal{B}(Q)$, so it is compact. By Corollary \\ref{compa1}.ii it follows that $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a $\\rho$-Banach space.\n\\end{proof}\n\n\\section{Scales of spaces} \n\n\\label{escala} Note that a family of atoms $\\mathcal{A}$ of type $(s,p,u)$ induces an one-parameter scale \n$$\\tilde{s}\\rightarrow \\mathcal{A}_{\\tilde{s},p},$$\nwhere $\\mathcal{A}_{\\tilde{s},p}$ is the family of atoms of type $(\\tilde{s},p,u)$ defined by \n$$\\mathcal{A}_{\\tilde{s},p}(Q)= \\{ |Q|^{\\tilde{s}-s}a_Q\\colon \\ a_Q\\in \\mathcal{A}\\}.$$\nMoreover  a family of atoms $\\mathcal{A}$ of type $(s,p,\\infty)$ induces a two-parameter scale\n$$(\\tilde{s},\\tilde{p})\\rightarrow \\mathcal{A}_{\\tilde{s},\\tilde{p}},$$\nwhere $\\mathcal{A}_{\\tilde{s},\\tilde{p}}$ is the family of atoms of type $(\\tilde{s},\\tilde{p},\\infty)$ defined by \n$$\\mathcal{A}_{\\tilde{p},\\tilde{s}}(Q)= \\{ |Q|^{\\tilde{s}-s+1\/p-1\/\\tilde{p} }a_Q\\colon \\ a_Q\\in \\mathcal{A}\\}.$$\n\n\n\n\\begin{proposition} \\label{compa} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$. Suppose that the $(s,p,\\infty)$-atoms $\\mathcal{A}$ satisfy ${\\Crr{banach}}$-${\\Crr{atom}}$. Let $0\\leq s < \\tilde{s} $ and $q,\\tilde{q} \\in [1,\\infty]$.  Suppose\n$$\\Big(  \\sum_{k}  |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q}  < \\infty.$$\nThen \n\\begin{itemize}\n\\item[A.] We have $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\subset {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ and the inclusion is a continuous linear map.\n\\item[B.] Suppose that also satisfies ${\\Crr{compact}}$.  Let $g_n \\in \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})$ be such that $|g_n|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$ for every $n$. Then there is a subsequence that converges in ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ to some $g \\in \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{s,p})$ with $|g|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$. \n\\item[C.] Suppose that also satisfies ${\\Crr{finite}}$. The inclusion $\\imath\\colon \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\mapsto {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ is a compact linear map. \n\\end{itemize}\n\\end{proposition}\n\\begin{proof} Consider a $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})$-representation \n$$f=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Qa_Q,$$\nSince $a_Q$ is an $\\mathcal{A}_{\\tilde{s},p}$-atom, we have that $b_Q=a_Q|Q|^{s-\\tilde{s}}$ is an $\\mathcal{A}_{s,p}$-atom. In particular, we can write\n$$f=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q|Q|^{\\tilde{s}-s}b_Q.$$\nIf  $k\\geq k_0$ then \n\\begin{eqnarray} \n(\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{1\/p} &\\leq&  |\\mathcal{P}^k|^{\\tilde{s}-s} (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{1\/p} \\nonumber \\\\ \n&\\leq&  |\\mathcal{P}^k|^{\\tilde{s}-s}   \\Big(\\sum_{k\\geq k_0 } (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{\\tilde{q}\/p}\\Big)^{1\/\\tilde{q}},\\nonumber\n\\end{eqnarray}\nso \n\\begin{eqnarray} \n&& \\Big( \\sum_{k\\geq k_0} (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\  &\\leq& \\Big(  \\sum_{k\\geq k_0}  |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q}   \\Big(\\sum_{k\\geq k_0 } (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{\\tilde{q}\/p}\\Big)^{1\/\\tilde{q}}.\\label{uyuy}\n\\end{eqnarray}\n\n\\noindent {\\it Proof of A.} In particular taking $k_0=0$ we conclude that $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\subset {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ and\n$$|f|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{p,s})}\\leq   \\Big(  \\sum_{k}  |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q}  | f|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{p,\\tilde{s}})}.$$\n\n\\noindent {\\it Proof of B.}  By definition, there exist  $s^n_Q\\in \\mathbb{C}$, such that\n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a  $\\mathcal{A}_{\\tilde{s},p}$-atom supported on $Q$ and\n\\begin{equation} \\label{sed}  (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{\\tilde{q}\/p})^{1\/\\tilde{q}}\\leq C+\\varepsilon_n,\\end{equation}\nwhere $\\varepsilon_n\\to 0$.\nIn particular, $|s_Q^n|\\leq C+\\varepsilon_n$. Since the set $\\cup_k \\mathcal{P}^k$ is countable,\n  by the Cantor  diagonal argument, taking a subsequence we can assume that \n  $s_Q^n\\to s_Q$ and   (due ${\\Crr{compact}}$) that $a_Q^n$ \nconverges in $\\mathcal{B}(Q)$ and $L^p$ to some $a_Q\\in \\mathcal{A}_{\\tilde{s},p}$.  By Lemma \\ref{compa2} the sequence $g_n$ converge in $L^p$ to a function $g$ such that $|g|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$ and with $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{s,p})$-representation\n$$g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q a_Q.$$\nIt remains to show that the convergence indeed occurs in the topology of  ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$. For  every $k_0\\geq 0$  and $\\delta > 0$ we can write \n$$g_n-g=  \\sum_{k< k_0} \\sum_{Q\\in {\\mathcal P}^k} \\delta d_Q^{n} + \\sum_{k\\geq k_0} \\sum_{Q\\in {\\mathcal P}^k}  |Q|^{\\tilde{s}-s} c_Q^{n} b_Q^{n},$$\nwhere \n$$d_Q^{n}= \\frac{1}{\\delta}(s_Q^na_Q^n - s_Q a_Q),$$\nand with $b_Q^{n}\\in \\mathcal{A}_{s,p}$ given by \n$$b_Q^{n} =   \\frac{ |Q|^{s-\\tilde{s}} |s_Q^n|}{|s_Q^n| + |s_Q|} sgn(s_Q^{n}) a_Q^{n}     +  \\frac{ |Q|^{s-\\tilde{s}} |s_Q|}{|s_Q^n| + |s_Q|}  sgn(-s_Q) a_Q,$$\nand\n$$c_Q^{n} = |s_Q^n| + |s_Q|.$$\nNote that $b_Q^{n}  \\in \\mathcal{A}_{s,p}$.  Given $\\epsilon > 0$, choose $k_0$ such that \n $b_Q^{n}  \\in \\mathcal{A}_{s,p}$.  Given $\\epsilon > 0$, choose $k_0$ such that \n$$ \\Big(  \\sum_{k\\geq k_0}  |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q}   (2C+1) \\leq (\\epsilon\/2)^{1\/\\rho}.$$\nBy  (\\ref{uyuy}) and (\\ref{sed})  for each  $n$ large enough we have\n$$\n \\Big( \\sum_{k\\geq k_0} (\\sum_{Q\\in {\\mathcal P}^k}|c_Q^{n}|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{q\/p} \\Big)^{1\/q} \\leq \\Big(  \\sum_{k\\geq k_0}  |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q}   (2C+1) < (\\epsilon\/2)^{1\/\\rho}.\n$$\nIn particular\n$$\\big|  \\sum_{k\\geq k_0} \\sum_{Q\\in {\\mathcal P}^k}  |Q|^{\\tilde{s}-s} c_Q^{n} b_Q^{n}\\big|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho < \\epsilon\/2.$$\nChoose $\\delta > $ such that \n$$\n \\Big( \\sum_{k<  k_0} (\\sum_{Q\\in {\\mathcal P}^k}\\delta^p\\Big)^{q\/p} \\Big)^{\\rho\/q}<   \\epsilon\/2.\n$$\n\nDue ${\\Crr{finite}}$  there is $\\eta > 0$ such that for every $Q\\in \\mathcal{P}^k$, with $k< k_0$, if $h \\in \\mathcal{B}(Q)$ satisfies  $|h|_{\\mathcal{B}(Q)}\\leq \\eta$ then $h \\in  \\mathcal{A}_{p,s}(Q)$. Since $\\lim_n s_Q^na_Q^n=s_Q a_Q$ in $\\mathcal{B}(Q)$ we conclude that for $n$ large enough we have \n$$    d_Q^{n}=\\frac{1}{\\delta}(s_Q^na_Q^n - s_Q a_Q) \\in \\mathcal{A}_{s,p}(Q)$$\nfor every  $Q\\in \\mathcal{P}^k$, with $k< k_0$. In particular\n$$\\big|\\sum_{k< k_0} \\sum_{Q\\in {\\mathcal P}^k} \\delta d_Q^{n}\\big|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho<  \\epsilon\/2.$$\nWe conclude that\n$$| g_n-g|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho<  \\epsilon,$$\nfor $n$ large enough, so the sequence $g_n$  converges (due  Corollary \\ref{compa1})  to $g$ in the topology of ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Transmutation of atoms} \nIt turns out that sometimes a Besov-ish space can be obtained using different classes of atoms. The key result in Part I is the following \n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{transm.pdf}\n\\caption{ In the transmutation of atoms we replace  atoms whose supports are represented in orange (supports with distinct scales are drawn in distinct lines for a better understanding) by linear combination of atoms  (whose supports are represented  in green), in such way we do not change too much  the location of the support and also  most of the  \"mass\" of the representation is concentrated  on the same original scale.   }\n\\end{figure}\n\n\n\n\\begin{proposition}[Transmutation of atoms] \\label{trans} Assume  \\\\ \n\n\\begin{itemize}\n\\item[{\\bf I.}] Let  $\\mathcal{A}_2$    be  a class of $(s,p,u_2)$-atoms  for  a grid  $\\mathcal{W}$, satisfying  ${\\Crr{banach}}$-${\\Crr{compact}}$ and ${\\Crr{fin}}$-${\\Crr{sum}}$. Let $\\mathcal{G}$ be also a  grid satisfying ${\\Crr{fin}}$-${\\Crr{sum}}$.\n\\item[{\\bf II.}] Let $k_i\\in \\mathbb{N}$ for $i\\geq 0$ be a sequence such that there is  $\\alpha > 0 $ and $A, B\\in \\mathbb{R}$ satisfying\n$$ \\alpha i +A    \\leq   k_i\\leq \\alpha i +B $$\nfor every $i$.\n\\item[{\\bf III.}] There is $\\lambda \\in(0,1)$ such that the following holds. For every $Q\\in \\mathcal{G}$ and $P\\in \\mathcal{W}$ satisfying $P\\subset Q$ there  are atoms $b_{P,Q} \\in \\mathcal{A}_2(P)$ and  corresponding $s_{P,Q}\\in \\mathbb{C}$ such that \n$$h_Q=\\sum_k  \\sum_{P\\in \\mathcal{W}^k, P\\subset Q} s_{P,Q} b_{P,Q}.$$\nis a  $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation of a function $h_Q$, with $s_{P,Q}=0$ for every $Q\\in \\mathcal{G}^i$,  $P \\in \\mathcal{W}^k$ with $k < k_i$  and moreover\n\\begin{equation}\\label{ad}  \\sum_{P\\in \\mathcal{W}^k, P \\subset Q} |s_{P,Q}|^p   \\leq \\Cll{rf}  \\lambda^{k-k_i}.\\end{equation}\nfor every $k\\geq k_i$.  \\\\ \\\\\n\\end{itemize}\nLet\n$$\\mathcal{H}^k= \\bigcup_{Q\\in \\mathcal{G}} \\{ P \\subset Q\\colon \\ P\\in \\mathcal{W}^k \\ and  \\ s_{P,Q}\\neq 0      \\}. $$\nThen  \\\\ \\\\\n\\begin{itemize}\n\\item[{\\bf A.}] For every coefficients $(c_Q)_{Q\\in \\mathcal{G}}$ such that \n$$\\big( \\sum_i  \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{q\/p}\\big)^{1\/q}<\\infty$$\nwe have that the sequence\n\\begin{equation}\\label{seq33} N\\mapsto \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i }c_Q h_Q\\end{equation}\nconverges in $L^p$ to a function in $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$ that has a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation \n\\begin{equation}\\label{cs} \\sum_k \\sum_{P\\in \\mathcal{H}^k}     m_P d_P\\end{equation}\nwhere $m_P \\geq 0$ for every $P$ and \n\\begin{eqnarray} \\label{igual1} &&\\big(\\sum_k \\big( \\sum_{\\substack{ P \\in \\mathcal{W}^k\\\\ P\\in \\mathcal{H}}} |m_{P}|^p \\big)^{q\/p}\\big)^{1\/q}  \\nonumber \\\\ \n&\\leq& \\Crr{mult1}  \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q} \\big( \\sum_i  \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{q\/p}\\big)^{1\/q}\n\\end{eqnarray}\nHere $\\Cll{m1}=\\max \\{\\ell \\in \\mathbb{N}, \\ell < \\alpha\\}+1$ and $b=(b_n)_{n\\in \\mathbb{Z}}$ is defined  by \n$$b_n = \\begin{cases}    \\lambda^{ \\alpha n} &   \\text{ if } n>   \\frac{A}{\\alpha} - 1,\\\\\n0  &   \\text{ if } n\\leq    \\frac{A}{\\alpha} - 1,\n \\end{cases}\n $$\n\n\\item[{\\bf B.}] Suppose that the   assumptions of A. hold and that  $s_{P,Q}$ are non negative real numbers and $b_{P,Q} > 0$  on $P$ for every   $P, Q$.   Then $m_P\\neq 0$ and $d_P \\neq  0$ on $P$ imply that  $P\\subset  supp \\ h_Q$ for some $Q\\in \\mathcal{W}^k$ satisfying $c_Q \\neq 0$ and $s_{P,Q}  >0$.  If we additionally  assume that $c_Q \\geq 0$ for every $Q$ then $m_P\\neq 0$ also   implies $d_P >  0$ on $P$.\n\\item[{\\bf C.}] Let $\\mathcal{A}_1$     be a   class of $(s,p,u_1)$-atoms for  the  grid  $\\mathcal{G}$ satisfying  ${\\Crr{banach}}$-${\\Crr{atom}}$. Suppose that there is $\\lambda < 1$ such that   for every atom $a_Q\\in \\mathcal{A}_1(Q)$ we can find $s_{P,Q}$ and $b_{P,Q}$ in III. such that  $h_Q=a_Q$. Then  $$\\mathcal{B}^s_{p,q}(\\mathcal{A}_1)\\subset \\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$$ and this inclusion is continuous. Indeed\n$$|\\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)} \\leq  \\Crr{mult1}  \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q} |\\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}_1)}$$\nfor every  $\\phi \\in \\mathcal{B}^s_{p,q}(\\mathcal{A}_1).$ \n\\end{itemize}\n\\end{proposition}\n\\begin{proof} \nFor every $P\\in \\mathcal{H}^k$, with $k\\in \\mathbb{N}$,  and $N\\in \\mathbb{N}\\cup \\{\\infty\\}$ define\n$$m_{P,N}=\\sum_{i\\leq N} \\sum_{\\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|=\\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|$$\nDue $\\Crr{sum}$ this sum has a finite number of terms. If this sum has  zero terms define $m_{P,N}=0$ and let $d_{P,N}$ be  the zero function. Otherwise define \n\\begin{equation}\\label{from} d_{P,N}= \\frac{1}{m_{P,N}}\\sum_{i\\leq N} \\sum_{\\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} c_Q s_{P,Q} b_{P,Q}.\\end{equation} \nWe have that $d_{P,N}$  is an $\\mathcal{A}_2(P)$-atom.  \n\\vspace{5mm} \n\n\\noindent {\\bf Claim I.} {\\it We claim that for $N\\in \\mathbb{N}$}\n $$\\sum_{k}   \\sum_{P\\in \\mathcal{H}^k} m_{P,N}  d_{P,N} = \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i}c_Q h_Q.   $$\nNote that if $Q\\in \\mathcal{G}^i$ then due (\\ref{flp}), with $t=p$  and (\\ref{ad})  we have\n$$\\sum_k  \\big| \\sum_{\\substack{P\\in \\mathcal{H}^k\\\\ P\\subset Q}} s_{P,Q} b_{P,Q}\\big|_p^{p\/\\hat{p}} < \\infty.$$\nConsequently we can do the following manipulation in $L^p$\n\\begin{eqnarray*}\n \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i} c_Q h_Q&=& \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i}  \\sum_k  \\sum_{\\substack{ P\\in \\mathcal{H}^k\\\\  P\\subset Q}} s_{P,Q} b_{P,Q}\\\\\n &=& \\sum_k \\sum_{ P\\in \\mathcal{H}^k}    \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i}   \\sum_{P\\subset Q} s_{P,Q} b_{P,Q}\\\\\n &=& \\sum_k \\sum_{ P\\in \\mathcal{H}^k}   m_{P,N}  d_{P,N}.\n \\end{eqnarray*}      \nThis concludes the proof of Claim I. \n\\vspace{5mm}\n\n\\noindent {\\bf Claim II.} {\\it For every $N\\in \\mathbb{N}\\cup \\{\\infty\\}$ we claim that \n\\begin{equation}\\label{a222}  \\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,N}  d_{P,N}\\end{equation}\nis a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation and }\n\\begin{align}&   \\big(\\sum_k \\big( \\sum_{ P\\in \\mathcal{H}^k} |m_{P,N}|^p \\big)^{q\/p}\\big)^{1\/q} \\nonumber \\\\\n\\label{ess} &\\leq \\Crr{mult1}  \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q}\\big( \\sum_i  \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{q\/p}\\big)^{1\/q}.\n\\end{align}\nIndeed \n\\begin{align*} \n\\Big( \\sum_{P \\in \\mathcal{H}^k} |m_{P,N}|^p \\Big)^{1\/\\hat{p}}&= \\Big(  \\sum_{P \\in \\mathcal{W}^k} \\big( \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|\\big)^p \\Big)^{1\/\\hat{p}} \\\\\n&\\leq  \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}}  \\big( \\sum_{P \\in \\mathcal{W}^k} \\big( \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|\\big)^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}}  \\big( \\sum_{P \\in \\mathcal{W}^k} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q|^p |s_{P,Q}|^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}}  \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}}  \\big(\\sum_{Q\\in \\mathcal{G}^i}   |c_Q|^p  \\sum_{\\substack{P \\in \\mathcal{W}^k\\\\ P \\subset Q}}  |s_{P,Q}|^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq  \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}}  \\sum_{\\substack{k_i\\leq k \\\\ i\\leq N}}  \\lambda^{(k-k_i)\/\\hat{p}}   \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{1\/\\hat{p}} \\\\\n&\\leq  \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}}  \\sum_{\\substack{\\alpha i + A\\leq k}}  \\lambda^{(k-\\alpha i -B)\/\\hat{p}}   \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{1\/\\hat{p}} . \\numberthis \\label{for}\n\\end{align*}\nIf $\\alpha=1$ and $A=B=0$ then this is a convolution, so we can use Proposition \\ref{young} (the convolution trick)  and it easily follows (\\ref{igual1}). In the general case, consider\n$$u_k=\\sum_{P \\in \\mathcal{H}^k} |m_{P,N}|^p  \\text{  and }  c_i=\\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p$$\nEvery $k\\in \\mathbb{N}$ can be  written in an {\\it unique  way} as $k=\\alpha j_k+ \\ell_k+r_k$, with $j_k \\in \\mathbb{N}$, $\\ell_k \\in \\mathbb{N}$, $\\ell_k+r_k < \\alpha$ and $r_k \\in [0,1)$.   Fix  $\\ell \\in [0,\\alpha)\\cap\\mathbb{N}$ and $j\\in \\mathbb{N}$. Then there is at most one  $k'\\in \\mathbb{N}$ such that $\\ell_k=\\ell$ and $j_k=j$. Indeed, if $k'= \\alpha j + \\ell+r'$ and $k''= \\alpha j + \\ell+r''$, with $r',r''\\in [0,1)$ and $\\ell+r'$ and $\\ell+r''$ smaller than $\\alpha$, then $k'-k''=r'-r''\\in (-1,1)$, so $k'=k''$ and $r'=r''$. If such $k'$ exists, denote $k(\\ell,j)=k'$ and $r(\\ell,j)= k(\\ell,j)-\\alpha j -\\ell$ and $a_{\\ell,j}=u_{k(\\ell,j)}$. Otherwise let $a_{\\ell,j}=0$. Then (\\ref{for}) implies \n\\begin{align*}\na_{\\ell,j}^{1\/\\hat{p}}&\\leq \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}}  \\sum_{\\substack{\\alpha i + A\\leq  \\alpha j+ \\ell+r(\\ell,j)}}  \\lambda^{( \\alpha j+ \\ell+r(\\ell,j)-\\alpha i -B)\/\\hat{p}} c_i^{1\/\\hat{p}}  \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}} \\lambda^{-B\/\\hat{p}}  \\sum_{\\substack{ i \\leq   j+ \\frac{-A+\\ell+r(\\ell,j)}{\\alpha} }  }  \\lambda^{ \\alpha (j- i)\/\\hat{p}} c_i^{1\/\\hat{p}}\\\\\n&\\leq  \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}}   \\lambda^{-B\/\\hat{p}}   \\sum_{\\substack{ i <    j+ \\frac{-A}{\\alpha} +1}}  \\lambda^{ \\alpha (j- i)\/\\hat{p}} c_i^{1\/\\hat{p}}\\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}}   \\Crr{rf}^{1\/\\hat{p}}  \\lambda^{-B\/\\hat{p}}   \\sum_{i \\in \\mathbb{Z} }  b_{j-i}^{1\/\\hat{p}} c_i^{1\/\\hat{p}}\n\\end{align*}\nHere $b_n= \\lambda^{ \\alpha n} $ if  $n> A\/\\alpha - 1$, and $b_n=0$ otherwise. Fixing $\\ell \\in \\mathbb{N}, \\ \\ell< \\alpha$, Proposition \\ref{young} (the convolution trick) gives us \n\\begin{eqnarray*} K_\\ell= \\big(\\sum_{\\substack{k\\in \\mathbb{N} \\\\ \\ell_k=\\ell}}  u_{k}^{q\/p}\\big)^{1\/q} &=&\\big(\\sum_{j} a_{\\ell,j}^{q\/p}\\big)^{1\/q}\\\\\n&\\leq&  \\Crr{mult1}  \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\big( \\sum_i  \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{q\/p}\\big)^{1\/q}.\n\\end{eqnarray*} \nand\n\\begin{align} \\label{bbbb} \\big(\\sum_k \\big( \\sum_{ P\\in \\mathcal{H}^k} |m_{P,N}|^p \\big)^{q\/p}\\big)^{1\/q}&=  \\big(\\sum_k u_k^{q\/p}\\big)^{1\/q}  \\nonumber \\\\\n&=  \\big(\\sum_{0\\leq \\ell < \\alpha} \\sum_{\\substack{k\\in \\mathbb{N} \\\\ \\ell_k=\\ell}}  u_{k}^{q\/p}\\big)^{1\/q}=  \\big(\\sum_{0\\leq \\ell < \\alpha}K_\\ell^q \\big)^{1\/q}\\nonumber \\\\\n &\\leq \\Crr{mult1}  \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q}  \\big( \\sum_i  \\big( \\sum_{Q\\in \\mathcal{G}^i}  |c_{Q}|^p  \\big)^{q\/p}\\big)^{1\/q}.\n\\end{align}\nThis implies in particular that the sum  in (\\ref{a222}) is a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation. This proves Claim II.  \n \\vspace{5mm}\n  \n  \\noindent {\\bf Claim III.} {\\it We have that in the strong topology of  $L^p$\n$$\\lim_{N\\rightarrow \\infty}  \\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,N}  d_{P,N} =\\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,\\infty}  d_{P,\\infty}.   $$}\nFor each $P\\in \\mathcal{H}$ the sequence\n\\begin{equation}\\label{seqdf} N\\mapsto m_{P,N}\\end{equation} \nis eventually constant, therefore convergent. The same happens with \n\\begin{equation}\\label{seqdf2} N\\mapsto d_{P,N}.\\end{equation}\n\n\n\\noindent Estimate (\\ref{ess}) and Proposition \\ref{compa2} imply that  that (\\ref{seq33}) converges in $L^p$ to a function    with $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation  (\\ref{a222}) with $N=\\infty$. This concludes the proof of Claim III. \n\nThen Claim I, II, and III imply A. taking $m_p=m_{P,\\infty}$ and $d_P=d_{P,\\infty}.$ We have that  C. is an immediate consequence of A.  Note that (\\ref{from}) and A. give B.\n\\end{proof}\n\\vspace{1cm}\n\n\\section{Good grids} \\label{goodgrids} A $(\\Cll[c]{menor},\\Cll[c]{maior})$-{\\bf good  grid} , with $0< \\Crr{menor}<  \\Crr{maior} <  1$, is a grid  $\\mathcal{P}= (\\mathcal{P}^k)_{k\\in \\mathbb{N}}$  with the following properties:\n\\begin{itemize}\n\\item[${\\Cll[G]{g2}}.$] We have $\\mathcal{P}^0=\\{I\\}$.\n\\item[${\\Cll[G]{g3}}.$] We have $I=\\cup_{Q \\in \\mathcal{P}^k} Q$  (up to a set of zero $m$-measure).\n\\item[${\\Cll[G]{g4}}.$] The elements of the  family $\\{  Q \\}_{Q\\in \\mathcal{P}^k} $ are pairwise disjoint. \n\\item[${\\Cll[G]{g5}}.$] For every $Q\\in \\mathcal{P}^k$ and $k > 0$ there exists $P \\in \\mathcal{P}^{k-1}$ such that $Q\\subset P$.  \n\\item[${\\Cll[G]{g6}}.$] We have\n$$\\Crr{menor} \\leq \\frac{|Q|}{|P|} \\leq \\Crr{maior}$$\nfor every $Q\\subset P$ satisfying  $Q\\in \\mathcal{P}^{k+1}$ and $P\\in \\mathcal{P}^{k}$ for some $k\\geq 0$. \n\\item[${\\Cll[G]{g7}}.$]  The family $\\cup_k \\mathcal{P}^k$ generate the  $\\sigma$-algebra $\\mathbb{A}$. \n\\end{itemize}\n\n\\section{Induced  spaces} \nConsider a Besov-ish space $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$, where $\\mathcal{P}$ is a  good grid. Given $Q\\in \\mathcal{P}^{k_0}$, we can  consider the sequence of finite families of subsets $\\mathcal{P}_Q=(\\mathcal{P}_Q^i)_{i\\geq 0}$ of $Q$  given by \n$$\\mathcal{P}^i_Q=\\{ P\\in \\mathcal{P}^{k_0+i}, P\\subset Q\\}.$$\nLet $\\mathcal{A}_Q$ be the restriction of the indexed family $\\mathcal{A}$ of pairs   $(\\mathcal{B}(P),\\mathcal{A}(P))_{P\\in \\mathcal{P}}$ to indices  belonging to $\\mathcal{P}_Q$. Then we can consider the {\\bf induced} Besov-ish space $\\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q)$. Of course the inclusion\n$$i\\colon \\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q) \\rightarrow \\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$$\nis well defined and  it is a weak contraction, that is\n$$|f|_{\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})}\\leq |f|_{\\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q)}.$$\nUnder the degree of generality we are considering here, the {\\bf restriction } transformation\n$$r\\colon \\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A}) \\rightarrow L^p$$\ngiven by $r(f)= f\\cdot 1_Q$, is a bounded linear transformation, however it is easy to find examples of Besov-ish spaces where $f 1_Q \\not\\in \\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q).$\n\n\n\n\n\n\n\n\n\n\\section{Examples of classes of atoms.}\\label{secatom}\n\nThere are many classes of atoms one may consider. We list here just  a few of them. \n\n\n\\subsection{Souza's  atoms}\\label{souzaa}\n\nLet $Q\\in \\mathcal{P}$. A {\\bf $(s,p)$-Souza's atom}  supported  on $Q$ is a function $a\\colon I \\rightarrow \\mathbb{C}$ such that $a(x)=0$ for every $x \\not\\in Q$ and $a$ is constant on $Q$, with \n$$|a|_\\infty\\leq |Q|^{s-1\/p}.$$\n\nThe set of Souza's atoms supported  on $Q$ will be denoted by $\\mathcal{A}^{sz}_{s,p}(Q).$ A {\\bf canonical Souza's atom} on $Q$ is the Souza's atom such that  $a(x)= |Q|^{s-1\/p}$ for every $x \\in Q$.  Souza's atoms are $(s,p,\\infty)$-type atoms.\n\n\n\n\n\n\n\n\n\n\n\n \\subsection{H\\\"older atoms} \\label{hatoms} Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\\cdot,\\cdot)$, such that every $Q\\in \\mathcal{P}$ is a bounded set and there is $\\Cll[c]{hh},\\Cll[c]{hhh} \\in (0,1)$ such that\n$$\\Crr{hh} \\leq \\frac{diam \\ P}{diam \\ Q} \\leq \\Crr{hhh}$$\nfor every $P\\subset Q$ with $P \\in \\mathcal{P}^{k+1}$ and $Q \\in \\mathcal{P}^{k}$.  Additionally assume  that there is $\\Cll{kki}\\geq 0$ and $D\\geq 0$ such that \n$$\\frac{1}{\\Crr{kki}^{D}}|Q| \\leq (diam \\ Q)^D \\leq \\Crr{kki}^{D} |Q|.$$\n\nLet \n$$0< s <  \\frac{1}{p}, \\ s < \\beta.$$   For every $Q\\in \\mathcal{P}$, Let  $C^\\alpha(Q)$ be the Banach space of all functions  $\\phi$ such that $\\phi(x)=0$ for $x \\not\\in Q$, and   \n$$|\\phi|_{C^\\alpha(Q)}= |\\phi|_\\infty + \\sup_{\\substack{x,y \\in Q\\\\ x\\neq y}} \\frac{|\\phi(x)-\\phi(y)|}{d(x,y)^\\alpha} < \\infty.$$\nLet $\\mathcal{A}^{h}_{s,\\beta,p}(Q) \\subset C^{\\beta D}(Q)$ be the convex subset of all functions $\\phi$ satisfying \n$$\\sup_{\\substack{x,y \\in Q\\\\ x\\neq y}} \\frac{|\\phi(x)-\\phi(y)|}{d(x,y)^{D \\beta}} \\leq  |Q|^{s-1\/p-\\beta} \\   and \\ |\\phi|_{\\infty} \\leq |Q|^{s-1\/p}.$$\nWe say that $\\mathcal{A}^{h}_{s,\\beta,p}(Q)$ is the   set of {\\bf $(s,\\beta,p)$-H\\\"older atoms } supported on $Q$. Of course $\\mathcal{A}^{h}_{s,\\beta,p}$-atoms are  $(s,p,\\infty)$-type atoms and $\\mathcal{A}^{sz}_{s,p}(Q)\\subset \\mathcal{A}^{h}_{s,\\beta,p}(Q)$.\n\n\n\\subsection{Bounded variation atoms}  Now suppose that  $I$ is an interval of $\\mathbb{R}$ with  length $1$, $m$ is the Lebesgue measure on it and the partitions in the  grid $\\mathcal{P}$ are partitions by intervals.  \nLet $Q$ be an interval and $s \\leq   \\beta$, $p\\in [1,\\infty)$.  A $(s,\\beta,p)$-bounded variation  atom on $Q$ is a function $a\\colon \\mathbb{R} \\rightarrow \\mathbb{C}$ such that $a(x)=0$ for every $x \\not\\in Q$,\n$$|a|_\\infty\\leq |Q|^{s-1\/p}.$$ \nand\n$$var_{1\/\\beta}(a,Q) \\leq |Q|^{s-1\/p}.$$ \nHere $var_{1\/\\beta}(\\cdot,Q)$ is the pseudo-norm \n$$var_{1\/\\beta}(a,Q) = \\sup (\\sum_i |a(x_{i+1})-a(x_i)|^{1\/\\beta})^{\\beta},$$\nwhere the sup runs over all possible sequences $x_1 < x_1 < \\dots < x_n$, with $x_i$ in the  interior of $Q$.\nWe will denote the set of  bounded variation  atoms on  $Q$ as $\\mathcal{A}^{bv}_{s, p,\\beta}(Q)$. Bounded variation atoms are also $(s,p,\\infty)$-type atoms.\n\\vspace{1cm}\n\n\n\n\\newpage\n\n\\centerline{ \\bf II. SPACES DEFINED BY  SOUZA'S ATOMS.}\n\\addcontentsline{toc}{chapter}{\\bf II. SPACES DEFINED BY  SOUZA'S ATOMS.}\n\\vspace{1cm}\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent   In Part II. we  suppose that $s> 0$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. \\end{minipage} \n}\n\\ \\\\ \\\\\n\n\n\\section{Besov spaces in a measure space with a good grid}\n\n\n We will study the Besov-ish spaces  $\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$ associated with the measure space with a good grid $(I, \\mathcal{P},m)$. We denote  $\\mathcal{B}^s_{p,q}=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$. Note that $\\mathcal{A}^{sz}_{s,p}$ satisfies ${\\Crr{banach}}$-${\\Crr{finite}}$. Note that by Proposition \\ref{lp} there is $\\beta > 1$ such that $\\mathcal{B}^s_{p,q} \\subset L^\\beta$.\n \nIf  $p\\in [1,\\infty)$, $q\\in [1,\\infty]$ and  $0< s< \\frac{1}{p}$ we wil say that  $\\mathcal{B}^s_{p,q}$ is a   {\\bf Besov space}. \n\n\n\\section{Positive cone}\n\nWe say that $f$ is $\\mathcal{B}^s_{p,q}$-positive if there is a $\\mathcal{B}^s_{p,q}$-representation\n$$f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} c_Pa_P$$\nwhere $c_P\\geq 0$ and $a_P$ is the standard $(s,p)$-Souza's atom supported on $P$. The set of all $\\mathcal{B}^s_{p,q}$-positive functions is a convex cone in $\\mathcal{B}^s_{p,q}$, denoted $\\mathcal{B}^{s+}_{p,q}$. We can define a ``norm\" on $\\mathcal{B}^{s+}_{p,q}$ as\n$$|f|_{\\mathcal{B}^{s+}_{p,q}}=\\inf   \\Big( \\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k}  c_P^p \\big)^{q\/p} \\Big)^{1\/q},$$\nwhere the infimum runs over all possible $\\mathcal{B}^s_{p,q}$-positive representations of $f$. Of course for every $f,g \\in \\mathcal{B}^{s+}_{p,q}$ and $\\alpha \\geq 0$ we have \n$$|\\alpha f|_{\\mathcal{B}^{s+}_{p,q}}=\\alpha  | f|_{\\mathcal{B}^{s+}_{p,q}}, \\ |f+g|_{\\mathcal{B}^{s+}_{p,q}}\\leq  |f|_{\\mathcal{B}^{s+}_{p,q}}+  |g|_{\\mathcal{B}^{s+}_{p,q}}, \\   | f|_{\\mathcal{B}^{s}_{p,q}} \\leq | f|_{\\mathcal{B}^{s+}_{p,q}}.$$\nMoreover if $f\\in \\mathcal{B}^s_{p,q}$   is a real-valued function then one can find $f_+,f_-\\in \\mathcal{B}^{s+}_{p,q}$ such that $f=f_+-f_-$ and $$|f_+|_{\\mathcal{B}^{s+}_{p,q}} \\leq |f|_{\\mathcal{B}^{s}_{p,q}} \\ and \\  |f_-|_{\\mathcal{B}^{s+}_{p,q}} \\leq |f|_{\\mathcal{B}^{s}_{p,q}}.$$\n\nAn obvious but important observation is\n\n\\begin{proposition} if $f \\in \\mathcal{B}^{s+}_{p,q}$ then its support    \n$$supp \\ f = \\{ x\\in I\\colon \\ f(x)\\neq 0\\}.$$\nis (up to a set of zero measure) is a countable union of elements of $\\mathcal{P}$. \n\\end{proposition} \n\n\n\n\\section{Unbalanced Haar wavelets} \\label{haar} Let $\\mathcal{P}=(\\mathcal{P}^k)_k$ be a good grid. For every $Q \\in \\mathcal{P}^k$ let $\\Omega_Q=\\{ P_1^Q,\\dots,P_{n_Q}^Q\\}$, $n_Q\\geq 2$,  be  the family of  elements  $\\mathcal{P}^{k+1}$ such that $P_i^Q \\subset Q$ for every $i$, and ordered in some arbitrary way.  The elements of  $\\Omega_Q$ will be called   {\\bf children} of $Q$.  Note that every $Q\\in \\mathcal{P}$ has at least two children. We will use one of the method described (Type I tree with the logarithmic subtrees construction) in  Girardi and Sweldens \\cite{gw} to construct an unconditional basis of $L^{\\beta}$, for every  $1< \\beta <\\infty$. \n\nLet   $\\mathcal{H}_Q$ be the family of pairs $(S_1,S_2)$, with $S_i \\subset \\Omega_Q$ and $S_1\\cap S_2=\\emptyset$, defined  as\n$$\\mathcal{H}_Q=\\cup_{j\\in \\mathbb{N}} \\mathcal{H}_{Q,j},$$\nwhere $\\mathcal{H}_{Q,j}$ are constructed recursively  in the  following way. Let $\\mathcal{H}_{Q,0}=\\{ (A,B)\\}$, where $A=\\{ P_1^Q,\\dots,P_{[n_Q\/2]}^Q   \\}$ and $B=\\{ P_{[n_Q\/2]+1}^Q,\\dots,P_{n_Q}^Q   \\}$. Here $[x]$ denotes the integer part of $x\\geq 0$.   Suppose that we have defined  $\\mathcal{H}_{Q,j}$. For each element   $(S_1,S_2)\\in \\mathcal{H}_{Q,j}$, {\\it fix}  an ordering $S_1=\\{ R_1^1, \\dots, R_{n_1}^1  \\}$ and $S_2=\\{ R_1^2, \\dots, R_{n_2}^2  \\}$. For each  $i=1,2$ such that  $n_i\\geq 2$, define $T_i^1=\\{ R_1^i,\\dots,R_{[n_i\/2]}^i  \\}$ and $T_i^2=\\{ R_{[n_i\/2]+1}^i,\\dots,R_{n_i}^i   \\}$ and  add  $(T_i^1,T_i^2)$ to  $\\mathcal{H}_{Q,j+1}$. This defines $\\mathcal{H}_{Q,j+1}.$\n\nNote that since $\\mathcal{P}$ is a good grid we have $\\mathcal{H}_{Q,j}=\\emptyset$ for large $j$ and indeed \n$$\\sup_{Q\\in \\mathcal{P}} \\#\\mathcal{H}_Q < \\infty.$$\n\nDefine $\\mathcal{H}=\\cup_{Q\\in \\mathcal{P}} \\mathcal{H}_Q$.  For every $S=(S_1,S_2) \\in\\mathcal{H}_Q$ define\n\n$$\\phi_{S} =     \\frac{1}{m_{(S_1,S_2)}}\\Big( \\frac{\\sum_{P \\in S_1} 1_{P}}{\\sum_{P \\in S_1} |P| }   -  \\frac{\\sum_{R \\in S_2} 1_{R}}{\\sum_{R \\in S_2} |R|} \\Big)$$\nwhere \n$$m_{(S_1,S_2)}=  \\big(  \\frac{1}{\\sum_{P \\in S_1} |P|}  + \\frac{1}{\\sum_{R \\in S_2} |R|}   \\big)^{1\/2} .$$\nNote that \n$$\\int_Q \\phi_S  \\ dm=0.$$\nSince $1\\leq \\#S_i\\leq 1\/\\Crr{menor}$ we have\n$$\\Crr{menor}|Q|\\leq \\sum_{P \\in S_i} |P|\\leq  \\frac{\\Crr{maior}}{\\Crr{menor}} |Q|,$$\nso\n$$\\Big(\\frac{2\\Crr{menor}}{\\Crr{maior}}\\Big)^{1\/2}\\frac{1}{|Q|^{1\/2}} \\leq m_{(S_1,S_2)}\\leq \\Big(\\frac{2}{\\Crr{menor}}\\Big)^{1\/2}\\frac{1}{|Q|^{1\/2}}$$\nConsequently \n\\begin{align}\\label{estphi} \\frac{\\Crr{c1}}{|Q|^{1\/2}}  &\\leq \\frac{1}{m_{(S_1,S_2)}}    \\min \\{  \\frac{1}{\\sum_{P \\in S_1} |P| },   \\frac{1}{\\sum_{R \\in S_2} |R| }    \\} \\nonumber \\\\\n&\\leq |\\phi_{S}(x)|\\leq    \\frac{1}{m_{(S_1,S_2)}}    \\max \\{  \\frac{1}{\\sum_{P \\in S_1} |P| },   \\frac{1}{\\sum_{R \\in S_2} |R| }    \\}\\nonumber \\\\\n&\\leq \\frac{\\Crr{c2}}{|Q|^{1\/2}} .\\end{align}\nfor every $x \\in \\cup_{P\\in S_1\\cup S_2} P.$  Here \n$$\\Cll{c1}= \\frac{\\Crr{menor}^{3\/2}}{\\sqrt{2}\\Crr{maior}}  \\ and \\ \\Cll{c2}=  \\frac{\\Crr{maior}^{1\/2}}{\\sqrt{2}\\Crr{menor}^{3\/2}}+1. $$\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{haartudo.pdf}\n\\end{figure}\nLet \n$$\\hat{\\mathcal{H}}= \\{ I\\}\\cup \\mathcal{H}$$ and define\n$$\\phi_I= \\frac{1_I}{|I|^{1\/2}}.$$\nThen  by  Girardi and Sweldens \\cite{gw}  we have that \n$$\\{ \\phi_S\\}_{S\\in \\hat{\\mathcal{H}}}$$\nis an unconditional basis of $L^\\beta$ for every $\\beta > 1$.   \n \n  \\section{Alternative characterizations I: Messing with norms.}\\label{alt1}\n We are going to describe three norms that are equivalent to $|\\cdot|_{\\mathcal{B}^s_{p,q}}$. Their advantage is that they are far more concrete, in the sense that we do not need to consider arbitrary atomic decompositions to define them. \n\n \\subsection{Haar  representation}  \nFor every $f\\in L^\\beta$, $\\beta > 1$,  the series \n\\begin{equation}\\label{fs22}  f =    \\sum_{S\\in \\hat{\\mathcal{H}}} d_S^f \\phi_S \\end{equation} \nis converges unconditionally  in $L^\\beta$, where\n $d_S^f = \\int f \\phi_S \\ dm.$ \n We will call the r.h.s. of (\\ref{fs22}) the {\\bf Haar representation} of $f$. \n Define\n $$ N_{haar}(f)=|I|^{1\/p-s-1\/2} |d_I^f|+ \\Big( \\sum_{k}  \\big(  \\sum_{\\substack{ _{Q\\in \\mathcal{P}^{k}}}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\sum_{_{S\\in \\mathcal{H}_Q}}   |d_S^f|^p\\big)^{q\/p} \\Big)^{1\/q},$$\n\n\n\n\\subsection{Standard atomic representation} Note that \n$$k_I^fa_I= d_I^f \\phi_I,$$\nwhere  $k_I= |I|^{1\/p-s-1\/2}d_I^f$ and $a_I$ is the canonical Souza's atom on $I$.  Let  $S\\in \\mathcal{H}$. Then $S\\in \\mathcal{H}_Q$, with $S=(S_1,S_2)$ and some $Q\\in \\mathcal{P}^k$, with $k\\geq 0$.  It is easy to see that for every $P\\in S_1\\cup S_2$ the function \n $$a_{S,P}= \\frac{ |Q|^{1\/2}}{\\Crr{c2}} |P|^{s-1\/p}  \\phi_S 1_{P}$$\nis a Souza's atom on $P$. Choose \n $$c_{S,P}^f=   \\Crr{c2} |Q|^{-1\/2}  |P|^{1\/p-s}  d_S^f. $$\n Note that\n\\begin{equation} \\label{estcsp} |c_{S,P}^f|\\leq   \\Crr{c2} \\max\\{ \\Crr{maior}^{1\/p-s}, \\Crr{menor}^{1\/p-s}\\}   |Q|^{1\/p-s-1\/2}  |d_S^f|.\\end{equation} \nFor every child $P$ of $Q \\in \\mathcal{P}^k$, $k\\geq 0$,  define \n $$\\tilde{a}_P^f=   \\frac{1}{\\tilde{k}_P^f}  \\sum_{S=(S_1,S_2)\\in \\mathcal{H}_Q}   \\sum_{P\\in S_1\\cup S_2} c_{S,P}^f a_{S,P},$$\n where  \n\\begin{equation}\\label{hh} \\tilde{k}_P^f =   \\sum_{S=(S_1,S_2)\\in \\mathcal{H}_Q}   \\sum_{P\\in S_1\\cup S_2} |c_{S,P}^f|.\\end{equation}\nThe (finite) number of terms on this sum depends only on the geometry of $\\mathcal{P}$. \nThen $\\tilde{a}_P^f$ is a Souza's  atom on $P$ and\n$$ \\sum_{S\\in \\mathcal{H}_Q}   d_S^f \\phi_S =    \\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\_{P\\subset Q}}}  \\tilde{k}_P^f \\tilde{a}_P^f.$$\n Let $a_P$ be  the canonical $(s,p)$-Souza's atom on $P$ and choose $x_P\\in P$.  Denote\n$$k_P^f= \\frac{ \\tilde{a}_P^f(x_P)}{|P|^{s-1\/p}} \\tilde{k}_P^f= \\frac{1}{|P|^{s-1\/p}}\\sum_{S\\in \\mathcal{H}_Q}   d_S^f \\phi_S(x_P)=\\frac{1}{|P|^{s-1\/p}} \\int   f \\sum_{S\\in \\mathcal{H}_Q} \\phi_S(x_P)\\phi_S \\ dm.$$\nIn particular for  every $P$ \n $$f\\mapsto k^f_P$$\n extends to a bounded linear functional in $L^1$. We have $|k_P^f|\\leq \\tilde{k}_P^f$ and \n\\begin{align} f &= k_I^f a_I  + \\sum_k  \\sum_{Q\\in \\mathcal{P}^{k}}\\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\_{P\\subset Q}}} k_P^f a_P \\nonumber \\\\\n \\label{sumf}   &= \\sum_i  \\sum_{Q\\in \\mathcal{P}^{i}} k_Q^f a_Q.\n\\end{align}\n where this series converges unconditionally in $L^\\beta$. Here $a_P$ is the canonical Souza's atom. We will call the r.h.s. of (\\ref{sumf}) the {\\bf standard atomic  representation} of $f$. Let\n $$  N_{st}(f)=|k_I^f|+ \\Big(  \\sum_{k\\geq 1} \\big( \\sum_{Q\\in \\mathcal{P}^{k}}  |k_Q^f|^p    \\big)^{q\/p}\\Big)^{1\/q}.$$\n \n \\subsection{Mean oscillation}\n Define for $p\\in [1,\\infty)$\n$$osc_p(f,Q)= \\inf_{c \\in \\mathbb{C}} (\\int_Q |f(x)-c|^p \\ dm(x) )^{1\/p},$$\nand \n$$osc_\\infty(f,Q)=\\inf_{c \\in \\mathbb{C}} |f-c|_{L^\\infty(Q)}.$$\nDenote for every $p\\in [1,\\infty)$ and $q \\in [1,\\infty]$\n\\begin{equation} \\label{oc} osc^s_{p,q}(f)= \\Big(  \\sum_k \\big(  \\sum_{Q\\in \\mathcal{P}^k} |Q|^{-sp} osc_p(f,Q)^p  \\big)^{q\/p} \\Big)^{1\/q},\\end{equation} \nwith the obvious adaptation for $q=\\infty$.  Let $$ N_{osc}(f)=|I|^{-s}|f|_p+osc^s_{p,q}(f).$$\n \n \n\n \\subsection{These norms are equivalent} We have \n \n \\begin{theorem} \\label{alte} Suppose $s> 0$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. Each one of the norms  $|f|_{\\mathcal{B}^s_{p,q}}$, $N_{st}(f)$, $N_{haar}(f)$, $N_{osc}(f)$ is finite if and only if  $f\\in \\mathcal{B}^s_{p,q}$. Furthermore these norms are equivalent on $\\mathcal{B}^s_{p,q}$. Indeed\n\\begin{equation}\\label{in1}  |f|_{\\mathcal{B}^s_{p,q}}\\leq N_{st}(f),\\end{equation}\n\\begin{equation}\\label{in2}N_{st}(f)\\leq \\Cll{e} N_{haar}(f),\\end{equation} \n\\begin{equation}\\label{in3}N_{haar}(f)\\leq \\Crr{c2} N_{osc}(f),\\end{equation}\n\\begin{equation}\\label{in4}N_{osc}(f)\\leq \\Cll{no}  |f|_{\\mathcal{B}^s_{p,q}}.\\end{equation}\nwhere\n$$\\Crr{e}=1+\\Crr{c2} \\max\\{ \\Crr{maior}^{1\/p-s}, \\Crr{menor}^{1\/p-s}\\}  \\Crr{menor}^{-2-1\/p},$$\n $$\\Crr{no}=\\Crr{mult1}^{1+1\/p}  \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{sp} )_k )|I|^{-s}+\\frac{1}{1-\\Crr{maior}^s}.$$\n \\end{theorem}\n \\begin{proof} The inequality (\\ref{in1}) is obvious. To simplify the notation we write $d_S, k_P$ instead of $d_S^f, k_P^f$.\\ \\\\ \\\\\n \\noindent {\\bf Proof of (\\ref{in2}).} The  number of terms in the r.h.s. of (\\ref{hh}) depends only on the geometry of $\\mathcal{P}$. Indeed \n \\begin{equation}\\label{numero}  \\sup_{Q \\in \\mathcal{P}} \\sum_{S=(S_1,S_2) \\in \\mathcal{H}_Q} \\#(S_1 \\cup S_2) \\leq \\frac{1}{\\Crr{menor}^2}.\\end{equation} \n  Consider the standard atomic representation of $f$ given by  (\\ref{sumf}). Note that  by (\\ref{estcsp})\n\\begin{align*} \n\\sum_{Q\\in \\mathcal{P}^{k}} \\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\ _{P\\subset Q}}} |k_P|^p &\\leq   \\sum_{Q\\in \\mathcal{P}^{k}} \\Big(\\sum_{_{S\\in \\mathcal{H}_Q}}   \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}} |c_{S,P}| \\Big)^p  \\\\\n&\\leq  \\Crr{c2}^p \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\}  \\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\Big( \\sum_{_{S\\in \\mathcal{H}_Q}}   \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}}   |d_S|\\Big)^p   \\\\\n&\\leq   \\Crr{c2}^p  \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\}\\Crr{menor}^{-2p} \\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\sum_{_{S\\in \\mathcal{H}_Q}}   \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}}   |d_S|^p   \\\\\n&\\leq   \\Crr{c2}^p  \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\} \\Crr{menor}^{-2p-1} \\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\sum_{_{S\\in \\mathcal{H}_Q}}   |d_S|^p.\n\\end{align*}\nfor every $k$. Consequently\n\\begin{align*} \n& |k_I|+ \\Big( \\sum_{k\\geq 1} \\big( \\sum_{P \\in \\mathcal{P}^{k}} |k_P|^p\\big)^{q\/p}  \\Big)^{1\/q} \\leq \\\\\n&\\leq   |I|^{_{\\frac{1}{p}-s-1\/2}} |d_I|  + \\Crr{c2} \\max\\{ \\Crr{maior}^{_{\\frac{1}{p}-s}}, \\Crr{menor}^{_{\\frac{1}{p}-s}}\\}\\Crr{menor}^{_{-2-\\frac{1}{p}}}  \\Big( \\sum_k  \\big(  \\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\sum_{_{S\\in \\mathcal{H}_Q}}   |d_S|^p\\big)^{q\/p} \\Big)^{1\/q}.\n\\end{align*}\nThis completes the proof of (\\ref{in2}). \\\\ \\\\\n\\noindent {\\bf Proof of (\\ref{in3}).} Note that\n$$|d_I|\\leq  \\int_I |f| |\\phi_I|\\ dm\\leq |f|_p |I|^{1\/2-1\/p}.$$Given $\\epsilon > 0$ and  $Q \\in \\mathcal{P}$, choose $c_Q\\in Q$ such that \n$$\\Big( \\int_Q |f - c_Q|^p \\ dm \\Big)^{1\/p} \\leq (1+\\epsilon)  osc_p(f,Q).$$\n Since $\\phi_S$ has zero mean on $Q$ for every $S\\in \\mathcal{H}_Q$ we have \n\\begin{align*} \n&\\big(\\sum_{Q\\in \\mathcal{P}^{k}}|Q|^{1-sp -\\frac{p}{2}}  \\sum_{S\\in \\mathcal{H}_Q}  |d_S|^p\\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}  \\sum_{S\\in \\mathcal{H}_Q}  |\\int f \\phi_S \\ dm |^p \\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}}  \\sum_{S\\in \\mathcal{H}_Q}  |\\int_Q f \\phi_S -c_Q \\phi_S\\ dm |^p \\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}}  |Q|^{_{1-sp -\\frac{p}{2}}}  \\sum_{S\\in \\mathcal{H}_Q} (\\int_Q |f - c_Q| |\\phi_S| \\ dm )^p \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2}\\big(\\sum_{\\substack{_{Q\\in \\mathcal{P}^{k}}\\\\ _{Q\\subset J}}}  |Q|^{_{1-sp -\\frac{p}{2}}}   \\sum_{S\\in \\mathcal{H}_Q} ((1+\\epsilon) osc_p(f,Q) |Q|^{1\/p'-1\/2} )^p \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2}(1+\\epsilon) \\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp +p\/p'-p}}   \\sum_{S\\in \\mathcal{H}_Q} osc_p(f,Q)^p  \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2} (1+\\epsilon)\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{-sp}}  \\sum_{S\\in \\mathcal{H}_Q}  osc_p(f,Q)^p  \\big)^{1\/p}.\n\\end{align*}\nSince $\\epsilon$ is arbitrary, this concludes the proof of (\\ref{in3}). \\\\ \\\\\n\\noindent {\\bf Proof of (\\ref{in4}).} Finally note that  if $f \\in \\mathcal{B}^s_{p,q}$ and $\\epsilon > 0$ then there is a $\\mathcal{B}^s_{p,q}$-representation of $f$ \n$$f= \\sum_{P \\in \\mathcal{P}} k_P a_P.$$\n such that \n $$ \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k}    |k_Q|^p)^{q\/p} \\big)^{1\/q} \\leq (1+\\epsilon)  |f|_{\\mathcal{B}^s_{p,q}}.$$\nFor each $J \\in \\mathcal{P}^{k_0}$, choose  $x_J \\in J$. Then \n\\begin{align}  & \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp }  osc_p(f,J)^p  \\Big)^{1\/p} \\nonumber \\\\\n&\\leq   \\Big(  \\sum_{J\\in \\mathcal{P}^{k_0} }  |J|^{-sp} \\int_J |f(x) -    \\sum_{\\substack{_{Q \\in \\mathcal{P}}\\\\ _{J \\subset Q}}} k_Q a_Q(x_J)|^p \\ dm   \\Big)^{1\/p} \\nonumber \\\\\n&\\leq  \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} }  |J|^{-sp} \\int |\\sum_{k> k_0}  \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} k_R a_R |^p \\ dm  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq \\Big( \\int |  \\sum_{k>  k_0}\\sum_{J\\in \\mathcal{P}^{k_0} }    |J|^{-s}  \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} k_R a_R |^p \\ dm  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq  \\sum_{k>  k_0} \\Big( \\int | \\sum_{J\\in \\mathcal{P}^{k_0} }    |J|^{-s}  \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} k_R a_R |^p \\ dm  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq  \\sum_{k> k_0} \\Big(   \\sum_{J\\in \\mathcal{P}^{k_0} }  |J|^{-sp}\\int   |    \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} k_R a_R |^p \\ dm  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq  \\sum_{k> k_0} \\Big(   \\sum_{J\\in \\mathcal{P}^{k_0} } \\Big( \\frac{\\sup \\{ |R|\\colon R \\in \\mathcal{P}^k, R  \\subset J \\}}{ |J|}\\Big)^{sp}  \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} |k_R|^p  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq  \\sum_{k> k_0} \\Big(  \\Crr{maior}^{sp(k-k_0)}   \\sum_{J\\in \\mathcal{P}^{k_0} }  \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R  \\subset J}}} |k_R|^p  \\Big)^{1\/p}  \\nonumber \\\\\n&\\leq  \\sum_{k> k_0}  \\Crr{maior}^{s(k-k_0)}  \\big( \\sum_{R \\in \\mathcal{P}^k} |k_R|^p  \\big)^{1\/p}.  \\nonumber \n\\end{align}\nThis is  a convolution, so \n\\begin{align*} \\Big( \\sum_{k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp }  osc_p(f,J)^p \\Big)^{q\/p} \\Big)^{1\/q} &\\leq \\frac{1+\\epsilon}{1-\\Crr{maior}^s} |f|_{\\mathcal{B}^s_{p,q}}.\\end{align*}\nand since $\\epsilon > 0$ is arbitrary,  by Proposition \\ref{lp} we obtain \n\\begin{align*} &|I|^{-s}|f|_p + \\Big( \\sum_{k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp }  osc_p(f,J)^p  \\Big)^{q\/p} \\Big)^{1\/q} \\\\\n&\\leq \\Big(\\Crr{mult1}^{1+1\/p}  \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{sp} )_k )|I|^{-s}+\\frac{1}{1-\\Crr{maior}^s}\\Big) |f|_{\\mathcal{B}^s_{p,q}}.\\end{align*}\nThis proves (\\ref{in4}).\n \\end{proof}\n \n The following is an important consequence of this section. \n \\begin{corollary} \\label{fou} For each $P\\in \\mathcal{P}$ there exists a linear functional in $L^1$ \n $$f  \\mapsto k_P^f$$\n with the following property. The so-called  standard $\\mathcal{B}^s_{p,q}$-representation of $f\\in \\mathcal{B}^s_{p,q}$ given by \n$$ f = \\sum_k \\sum_{P \\in \\mathcal{P}^{k}} k_P^f a_P,$$\nsatisfies \n$$\\Big(\\sum_k  \\big( \\sum_{P\\in \\mathcal{P}^k}  |k_P^f|^p\\big)^{q\/p}\\Big)^{1\/p}\\leq \\Crr{c2}  \\Crr{e}\\Crr{no} |f|_{\\mathcal{B}^s_{p,q}}.$$\n\n\n \n\n \\end{corollary}\n \n \n  \\section{Alternative characterizations II: Messing with atoms.}\n Here we move to  alternative descriptions of $\\mathcal{B}^s_{p,q}$ which are quite different of those in  Section \\ref{alt1}. Instead of choosing  a definitive representation of elements of $\\mathcal{B}^s_{p,q}$ we indeed give atomic decompositions of $\\mathcal{B}^s_{p,q}$ using far more general classes of atoms.\n \n \\subsection{Using Besov's atoms}\n\nThe advantage of Besov's atoms is that it is a wide and general class of atoms, that includes even unbounded functions. They can  be considered in {\\it every} measure space   endowed with a good grid as in Section \\ref{partition}. Moreover  in appropriate settings it contains  H\\\"older and bounded variations atoms, which will be quite useful in the get other characterizations of $\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$. The atomic decompositions of  $B^s_{p,q}(\\mathbb{R}^n)$ by Besov's atoms  were considered  by Triebel \\cite{ns} in the case $s>0$, $p=q\\in [1,\\infty]$ and \nby Schneider and Vyb\\'\\i ral \\cite{corjan} in the case $s>0$, $p,q\\in [1,\\infty]$. They are refered there as \"non-smooth atomic decompositions\". \n\n\nLet $s < \\beta$ and $\\tilde{q}\\in [1,\\infty]$. A  {\\bf $(s,\\beta,p,\\tilde{q})$-Besov atom}  on the interval $Q$ is a function $a \\in \\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})$ such that  $a(x)=0$ for $x \\not\\in Q$,\nand\n$$|a|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})}\\leq \\frac{1}{\\Crr{ba}} |Q|^{s-\\beta}.$$\nwhere \n$$\\Cll{ba}=\\Crr{mult1}^{1+1\/p}  \\Big( \\sum_k \\Crr{maior}^{k\\beta\\tilde{q}}    \\Big)^{1\/\\tilde{q}}\\geq 1.$$\nThe family of $(s,\\beta,p,\\tilde{q})$-Besov atoms supported on $Q$ will be denoted by  $\\mathcal{A}^{bv}_{s,\\beta,p,\\tilde{q}}(Q)$.  Naturally $\\mathcal{A}^{sz}_{s,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q)$. By Proposition \\ref{lp}  we have\n\\begin{eqnarray}\n|a|_{p}&\\leq&  \\frac{\\Crr{mult1}^{1+1\/p}}{\\Crr{ba}}  \\Big( \\sum_k |\\mathcal{P}^k_Q|^{\\beta\\tilde{q}}    \\Big)^{1\/\\tilde{q}} |a|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})}\\\\\n&\\leq&  |Q|^{s-\\beta} |Q|^{\\beta} =  |Q|^s=  |Q|^{s-\\frac{1}{\\infty}}. \n\\end{eqnarray} \nso a $(s,\\beta,p,\\tilde{q})$-Besov atom is an atom of  type $(s,p,1)$. \\\\ \\\\\n\nThe following result says there are many ways to define $\\mathcal{B}^s_{p,q}$ using various classes of atoms. \n\n\\begin{proposition}[Souza's atoms and  Besov's  atoms]\\label{besova}Let $\\mathcal{P}$ be a good grid. Let $\\mathcal{A}$ be a class of $(s,p,u)$-atoms , with $u\\geq 1$, such that  for some $s< \\beta$, $\\tilde{q}\\in [1,\\infty]$, and $\\Crr{56}$, $\\Crr{566} \\geq 0$ we have that  for every $Q\\in \\mathcal{P}$ \n$$\\frac{1}{\\Cll{56}} \\mathcal{A}^{sz}_{s,p}(Q) \\subset \\mathcal{A}(Q)\\subset  \\Cll{566} \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q).$$\nThen \n $$\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A})=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}).$$ \n Moreover\n$$|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})} \\leq \\Crr{56} |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})} \\  and \\  |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})}  \\leq \\frac{\\Crr{566}}{1-\\Crr{maior}^{\\beta-s}}|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}.$$\n\\end{proposition} \n\\begin{proof} The  first inequality is obvious. To prove the second inequality, recall that due Proposition \\ref{trans} it is enough to show  the following claim\n\\vspace{5mm}\n\n\\noindent {\\it Claim. Let  $b_J$ be a  $(s,\\beta,p,\\tilde{q})$-Besov atom on $J\\in \\mathcal{P}^j$.  Then for every $P\\subset J$ with $P\\in \\mathcal{P}$ there is  $m_P \\in \\mathbb{C}$ such that \n$$b_J= \\sum_{P\\subset J} m_P a_P,$$\nwhere $a_P$ is the canonical $(s,p)$-Souza's atom on $P$ and \n$$  \\Big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |m_P|^p\\Big)^{1\/p} \\leq   \\frac{2}{\\Crr{ba}}  \\Crr{maior}^{(k-j)(\\beta-s)}.$$}\n\\vspace{5mm} \n\n\n\\noindent Indeed, since\n$$|b_J|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(J,\\mathcal{P},\\mathcal{A}^s_{\\beta,p})}\\leq \\frac{1}{\\Crr{ba}} |J|^{s-\\beta},$$\nthere exists a $\\mathcal{B}^\\beta_{p,\\tilde{q}}(J,\\mathcal{P},\\mathcal{A}^s_{\\beta,p})$-representation\n$$b_J=\\sum_{P\\in \\mathcal{P}, P\\subset J} c_P d_P,$$\nwhere $d_P$ is the canonical $(\\beta,p)$-Souza's atom on $P$ and \n$$\\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |c_P|^p\\big)^{1\/p} \\leq \\Big(\\sum_i \\big( \\sum_{P\\in \\mathcal{P}^i, P\\subset J} |c_P|^p  \\big)^{\\tilde{q}\/p}    \\Big)^{1\/\\tilde{q}}\\leq \\frac{2}{\\Crr{ba}}   |J|^{s-\\beta}.$$\nThen \n$$a_P = |P|^{s-\\beta}d_P$$\nis a $(s,p)$-Souza's atom and \n$$b_J=\\sum_{P\\in \\mathcal{P}, P\\subset J} m_P a_P,$$\nwith $m_P = c_P |P|^{\\beta-s}$ and \n\\begin{align} \\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |m_P|^p\\big)^{1\/p} &=  \\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} \\Big(\\frac{|P|}{|J|}\\Big)^{p(\\beta-s)} |J|^{p(\\beta-s)}|c_P|^p\\big)^{1\/p}  \\nonumber \\\\\n&\\leq \\Crr{maior}^{(k-j)(\\beta-s)}  |J|^{\\beta-s}  \\big(  \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |c_P|^p\\big)^{1\/p}  \\nonumber    \\\\\n&\\leq  \\frac{2}{\\Crr{ba}}  \\Crr{maior}^{(k-j)(\\beta-s)}. \\nonumber\n\\end{align}\n\\end{proof} \n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \\subsection{Using H\\\"older atoms} \\label{hatoms2} Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\\cdot,\\cdot)$ and a good grid satisfying the assumptions in Section \\ref{hatoms}. \n\n\\begin{proposition} \\label{hold} Suppose\n$$0< s  < \\beta < \\tilde{\\beta},$$\n$p\\in [1,\\infty)$ and  $\\tilde{q}\\in [1,\\infty].$  For every $Q\\in \\mathcal{P}$ we have \n\\begin{equation}\\label{final1} \\Cll{at3} \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q),\\end{equation}  for some $\\Crr{at3} > 0$. Moreover\n\\begin{equation}\\label{final2} \\Cll{at4} \\mathcal{A}^{h}_{s,\\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q),\\end{equation}  for some $\\Crr{at4} > 0.$ In particular\n\\begin{equation}\\label{inclusion1}\\mathcal{B}^s_{p,q}=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{h}_{s,\\tilde{\\beta},p})=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{h}_{s,\\beta,p}(Q))\\end{equation}\nand the corresponding norms are equivalent.\n\\end{proposition} \n\\begin{proof} Let $\\phi \\in \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)$.  Then $\\phi$ has a continuous extension to $\\overline{Q}$.  So firstly we  assume that    $\\phi\\geq 0$ has a continuous extension to $\\overline{Q}$. Define \n$$c_Q= \\min \\phi(Q) |Q|^{1\/p-\\beta}.$$\nand for every  $P\\subset W\\subset Q \\in \\mathcal{P}^j$ with $P \\in \\mathcal{P}^{k+1}$ and $W \\in \\mathcal{P}^{k}$ define\n\\begin{equation} \\label{cv} c_P= (\\inf \\phi (P) -\\inf \\phi (W))|P|^{1\/p-\\beta}\\end{equation}\nOf course in this case $c_P\\geq 0$ and\n\\begin{align*} |c_P| &\\leq (\\inf \\phi (P) -\\inf \\phi (W))|P|^{1\/p-\\beta} \\\\\n&\\leq  |Q|^{s-1\/p-\\tilde{\\beta}}  (diam \\ W )^{\\tilde{\\beta}D} |P|^{1\/p-\\beta}\\\\\n& \\leq \\frac{\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^{\\tilde{\\beta}}}\\Big( \\frac{|P|}{|Q|}\\Big)^{1\/p} |Q|^{s-\\tilde{\\beta}} |P|^{\\tilde{\\beta}-\\beta}\\\\\n& \\leq \\frac{\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^{\\tilde{\\beta}}} \\Cll{super}  \\Big( \\frac{|P|}{|Q|}\\Big)^{1\/p} |Q|^{s-\\beta} \\Big(\\frac{|P|}{|Q|}\\Big)^{\\tilde{\\beta}-\\beta}.\n\\end{align*}\nHere $\\Crr{super}= \\sup_{Q\\in \\mathcal{P}}|Q|^{\\tilde{\\beta}-\\beta}.$\nConsequently \n\\begin{align}\\label{adapt} \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} |c_P|^p   & \\leq  |Q|^{p(s-\\beta)}   \\Crr{maior}^{(k+1-j)p(\\tilde{\\beta}-\\beta)} \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} \\frac{ \\Crr{super}^p\\Crr{kki}^{D\\tilde{\\beta}p}}{\\Crr{hh}^{p \\beta}} \\frac{|P|}{|Q|} \\nonumber \\\\\n&\\leq |Q|^{p(s-\\beta)} \\frac{ \\Crr{super}^p \\Crr{kki}^{D\\tilde{\\beta}p}}{\\Crr{hh}^{\\beta p}} \\Crr{maior}^{(k+1-j)p(\\tilde{\\beta}-\\beta)}.\\end{align}\nso\n\\begin{align*}\\Big(  \\sum_k \\big( \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} |c_P|^p\\big)^{\\tilde{q}\/p} \\Big)^{1\/{\\tilde{q}}}  &\\leq  \\frac{ \\Crr{super}\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^\\beta} \\Big(\\frac{1}{1-\\Crr{maior}^{\\tilde{q} (\\tilde{\\beta}-\\beta) }}\\Big)^{1\/\\tilde{q}}  |Q|^{s-\\beta}.\\end{align*}\nThis implies that \n$$\\tilde{\\phi}=\\sum_{\\substack{ P\\in \\mathcal{P}\\\\ P\\subset Q}} c_P a_P, $$\nwhere $a_P$ is the canonical $(\\beta,p)$-atom on $P$,  is a $\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q,\\mathcal{P}_Q\\mathcal{A}^{sz}_{\\beta,p})$-representation of  a function $\\tilde{\\phi}$.  From (\\ref{cv}) it follows that\n$$\\sum_{k\\leq N} \\sum_{P\\in \\mathcal{P}^k} c_P a_P(x)= \\min \\phi(W)$$\nfor every $x \\in W\\in \\mathcal{P}^{N}.$ In particular\n$$\\lim_{N\\rightarrow \\infty} \\sum_{k\\leq N} \\sum_{P\\in \\mathcal{P}^k} c_P a_P(x) =\\phi(x).$$\nfor almost every $x$, so $\\tilde{\\phi}=\\phi$. So\n\\begin{equation}\\label{inbv} |\\Crr{at3}\\phi|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q,\\mathcal{P}_Q,\\mathcal{A}^{sz}_{\\beta,p})}\\leq  \\frac{1}{2} |Q|^{s-\\beta}\\end{equation} \nwhere\n$$\\Crr{at3}= \\Big( 2\\frac{\\Crr{super} \\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^\\beta} \\Big(\\frac{1}{1-\\Crr{maior}^{\\tilde{q}(\\tilde{\\beta}-\\beta)}}\\Big)^{1\/\\tilde{q}}\\Big)^{-1}.$$\n\nIn the general case, note that $\\phi_+(x)=\\max\\{ \\phi,0\\}, \\phi_-(x)=\\min\\{-\\phi,0\\} \\in \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)$ and $\\phi=\\phi_+-\\phi_-$. Applying (\\ref{inbv}) to $\\phi_+$ and $\\phi_-$ we obtain (\\ref{final1}).\n\nThe second inclusion (\\ref{final2}) in the proposition can be obtained taking $\\tilde{\\beta}=\\beta$ in (\\ref{adapt}) and a few modifications in the above argument.  By Proposition \\ref{besova} we have (\\ref{inclusion1}).\n\\end{proof}\n\n\\subsection{Using bounded variation atoms}  Now suppose that  $I$ is an interval of $\\mathbb{R}$ of length $1$, $m$ is the Lebesgue measure on it and the partitions in $\\mathcal{P}$ are partitions by intervals. \n\n\\begin{proposition}  If \n$$0< s\\leq \\beta \\leq \\frac{1}{p}$$\nthen  $$\\Crr{at1} \\mathcal{A}^{bv}_{s, \\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q).$$\nfor every $Q\\in \\mathcal{P}.$\nIf $$0< s\\leq \\beta < \\tilde{\\beta} \\leq   \\frac{1}{p}$$           \nthen  $$\\Crr{at2} \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,q}(Q)$$\nfor every $q \\in [1,\\infty]$. In particular\n$$\\mathcal{B}^s_{p,q}= \\mathcal{B}^s_{p,q}(\\mathcal{A}^{bv}_{s, \\beta,p})=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}).$$\nand the corresponding norms are equivalents.\n\\end{proposition} \n\\begin{proof} Suppose\n$$0< s \\leq  \\beta  \\leq \\tilde{\\beta}\\leq 1$$\nLet  $Q\\in \\mathcal{P}^j$ and $a_Q\\in \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)$. We have\n$$|a_Q|_p\\leq (   |Q||Q|^{sp-1}   )^{1\/p}=|Q|^s\\leq \\Cll{supe} |Q|^{s-\\beta},$$\nwhere $\\Crr{supe}= \\sup_{Q\\in \\mathcal{P}} |Q|^\\beta.$\nNote that  for every $k\\geq j$\n\\begin{align*}\n&\\sum_{W\\in \\mathcal{P}^k} |W|^{-\\beta p} osc_p(a_Q,W)^p  \\\\\n &\\leq  \n\\sum_{W \\subset Q, W\\in \\mathcal{P}^k} |W|^{1-\\beta p} osc_\\infty(a_Q,W)^p  \\\\\n&\\leq \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k}  |W|^{\\frac{1-\\beta p}{1-\\tilde{\\beta} p}} \\big)^{1-\\tilde{\\beta} p}  \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k}   osc_\\infty(a_Q,W)^{1\/\\tilde{\\beta}}   \\big)^{\\tilde{\\beta} p} \\\\ \n&\\leq   \\Big( \\max_{W \\subset Q, W\\in \\mathcal{P}^k} |W|^{\\frac{(\\tilde{\\beta}-\\beta) p}{1-\\tilde{\\beta} p} } \\Big)^{1-\\tilde{\\beta} p}  \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k}  |W| \\big)^{1-\\tilde{\\beta} p}   (var_{1\/\\tilde{\\beta}}(a_Q,Q))^{p}  \\\\\n&\\leq \\Crr{maior}^{(k-j)(\\tilde{\\beta}-\\beta)p}|Q|^{(\\tilde{\\beta}-\\beta)p}   |Q|^{1-\\tilde{\\beta} p} |Q|^{sp-1}\\\\\n&\\leq \\Crr{maior}^{(k-j)(\\tilde{\\beta}-\\beta)p}  |Q|^{(s-\\beta) p}.\n\\end{align*}\nNote that in the case  $\\tilde{\\beta}p=1$ the argument above needs a simple modification. For $k < j$ let $W_Q\\in \\mathcal{P}^k$ be such that $Q\\subset W_Q$. Then \n\\begin{align*}\n&\\sum_{W\\in \\mathcal{P}^k} |W|^{-\\beta p} osc_p(a_Q,W)^p \\leq  |W_Q|^{-\\beta p} |Q|  |Q|^{sp -1}   \\\\\n&\\leq  |W_Q|^{-\\beta p} |Q|^{sp}  =  \\Big( \\frac{|Q|}{|W_Q|}\\Big)^{\\beta p}  |Q|^{(s-\\beta)p}\\leq \\Crr{maior}^{(j-k)\\beta p}  |Q|^{(s-\\beta)p}.\n\\end{align*}\nBy Theorem \\ref{alte} we have that if $\\beta=\\tilde{\\beta}$ then  $\\Cll{at1} \\mathcal{A}^{bv}_{s, \\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q)$ for some $\\Crr{at1} > 0$, and if $\\beta < \\tilde{\\beta}$ we have   that for every $q \\in [1,\\infty)$ \n\\begin{align*}\n&|I|^{-s}|a_Q|_p+\\Big( \\sum_{k} \\big( \\sum_{W\\in \\mathcal{P}^k} |W|^{-sp} osc_p(a_Q,W)^p \\big)^{q\/p}\\Big)^{1\/q} \\\\\n&\\leq \\Big(  \\Crr{supe} |I|^{-s}+\\big( \\frac{1}{1-\\lambda^{q(\\tilde{\\beta}-\\beta)}}    +  \\frac{1}{1-\\Crr{maior}^{q\\beta}}  \\big)^{1\/q} \\Big)   |Q|^{s-\\beta} .\\\\\n\\end{align*}\nso $\\Cll{at2} \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,q}(Q)$ for some $\\Crr{at2} > 0$.\n\\end{proof}\n\n \n \n\n\n\\section{Dirac's approximations} \nWe will use the Haar basis and notation defined by Section \\ref{haar}. For every $x_0 \\in I$ and $k_0 \\in \\mathbb{N}$  define the finite family \n$$\\mathcal{S}_{x_0}^{k_0} =   \\{ S \\colon \\ S=(S_1,S_2) \\in \\mathcal{H}(Q),\\ with \\ Q \\in \\mathcal{P}^k, \\ k < k_0, \\ x_0 \\in \\cup_{a=1,2} \\cup_{P \\in S_{a}} P   \\}$$\nLet $N(x_0,k_0)=\\# \\mathcal{S}_{x_0}^{k_0}$. Then we can enumerate the elements $$S^1, S^2, \\dots, S^{N(x_0,k_0)}$$ of $\\mathcal{S}_{x_0}^{k_0}$ \nsuch that $S^i=(S^i_1,S^i_2)$ satisfies\n$$x_0 \\in \\cup_{P \\in S^i_{a_i}} P $$\nfor some $a_i \\in \\{1,2\\}$ and\n$$\\cup_{P \\in S^{i+1}_1\\cup S^{i+1}_2} P     \\subset \\cup_{Q \\in S^{i}_1\\cup S^{i}_2} Q$$ \nfor every $i$. \nLet \n$$\\psi_0    =   \\frac{1_{I}}{|I|}$$\nand define for $i >0$ \n\\begin{eqnarray*}\n\\psi_{i}&=& (-1)^{a_{i}+1}\\frac{\\sum_{R \\in S^{i}_{3-a_{i}}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} \\big( \\frac{ \\sum_{P \\in S^{i}_1}1_{P}}{\\sum_{P \\in S^{i}_1}|P|} -  \\frac{ \\sum_{R \\in S^{i}_2}1_{R}}{\\sum_{R \\in S^{i}_2}|R|} \\big)  \\\\\n&=&   (-1)^{a_{i}+1} \\frac{\\sum_{R \\in S^{i}_{3-a_{i}}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} m_{S^i} \\phi_i.\n\\end{eqnarray*}\nOne can prove by induction on $j$ that for $j > 0$\n$$\\sum_{i=0}^j  \\psi_i = \\frac{\\sum_{P \\in S^{j}_{a_j}}1_{P} }{\\sum_{P \\in S^{j}_{a_j}}|P|}.$$\nand in particular \n$$\\sum_{i=0}^{N(x_0,k_0)}  \\psi_i (x_0)= \\frac{1_{Q_{k_0}} }{|Q_{k_0}|},$$\nwhere $x_0 \\in Q_{k_0} \\in \\mathcal{P}^{k_0}.$ In other words \n\\begin{equation}\\label{part}  \\frac{1}{ |I|^{1\/2} }\\phi_{I} + \\sum_{i=1}^{N(x_0,k_0)}  (-1)^{a_i+1}  \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|}  \\Big)  m_{S^i}    \\phi_{S^i}=\\frac{1_{Q_{k_0}} }{|Q_{k_0}|},\\end{equation}  \nNote that \n\\begin{align}\n\\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|}  \\Big) m_{S^i}&= \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|}  \\Big) \\Big(    \\frac{\\sum_{R \\in S_2^i} |R| + \\sum_{P \\in S_1^i} |P|  }{(\\sum_{R \\in S_2^i} |R| ) (\\sum_{P \\in S_1^i} |P|)}   \\Big)^{1\/2} \\nonumber \\\\\n\\label{igual} &= \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|}  \\Big)^{1\/2}  \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}}  \n\\end{align}\nMultiplying (\\ref{part}) by $f$ and integrating it  term by term, and using (\\ref{igual}) we obtain \n$$\\frac{d_{I}}{|I|^{1\/2}} + \\sum_{i=1}^{N(x_0,k_0)}  (-1)^{a_i+1}  \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|}  \\Big)^{1\/2}  \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}}    d _{S^i} =\\int f \\cdot \\frac{1_{Q_{k_0}}}{|Q_{k_0}|} \\ dm.$$\n\n\\noindent If  $f \\in L^\\beta$, with $\\beta > 1$,   it  can be written as \n$$f = \\sum_{S\\in \\hat{\\mathcal{H}}(\\mathcal{P})} d_S\\phi_S$$\nwith $d_S= \\int f \\phi_S \\ dm$, where this series converges  unconditionally  on $L^\\beta$. Let\n\\begin{equation}\\label{k0} f_{k_0} = d_{I} \\phi_{I}+ \\sum_{k < k_0} \\sum_{Q\\in \\mathcal{P}^k}\\sum_{S\\in \\mathcal{H}(Q)} d_S\\phi_S.\\end{equation}\nThen\n\\begin{align} f_{k_0}(x_0)  &= d_{I} \\phi_{I} +\\sum_{i=1}^{N(x_0,k_0)} d_{S^i} \\phi_{S^i}(x_0).\\nonumber \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}}+  \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1}d_{S^i}  \\frac{1}{m_{S^i}}  \\frac{1}{\\sum_{P \\in S_{a_i}^i} |P| }  \\nonumber \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}}+\\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} d_{S^i}  \\Big(    \\frac{(\\sum_{R \\in S_2^i} |R| ) (\\sum_{P \\in S_1^i} |P|)}{\\sum_{R \\in S_2^i} |R| + \\sum_{P \\in S_1^i} |P|  }   \\Big)^{1\/2} \\frac{1}{\\sum_{P \\in S_{a_i}^i} |P| }  \\nonumber  \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}} +   \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} d_{S^i}   \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|}  \\Big)^{1\/2}  \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}}   \\nonumber \\\\\n&= \\int f \\cdot \\frac{1_{Q_{k_0}}}{|Q_{k_0}|} \\ dm. \\nonumber \n\\end{align}\nLet \n\\begin{equation}\\label{reep} f =  \\sum_k  \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P,\\end{equation}\nbe the series given by (\\ref{sumf}). Note that\n\\begin{equation} f_{k_0} = \\sum_{k\\leq k_0}  \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P,\\end{equation}\n\nConsequently \n\n\\begin{proposition}[Dirac's Approximations]\\label{boup} Let $f \\in L^\\beta$, with $\\beta > 1$. Let\n$$ f = \\sum_k  \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P.$$\n\\begin{itemize}\n\\item[A.] If  this representation is either as in (\\ref{sumf}) or $k_P\\geq 0$ for every $P$ then we have For every $Q \\in \\mathcal{P}$ \n$$\\Big| \\sum_{J\\in \\mathcal{P}, Q \\subset J} k_J  |J|^{s-1\/p} \\Big| \\leq |f1_Q|_\\infty.$$\n\\item[B.]  In the case of the representation  (\\ref{sumf}) we also have \n$$\\sum_{J\\in \\mathcal{P}, Q \\subset J} k_J  |J|^{s-1\/p} =\\int f \\cdot \\frac{1_{Q}}{|Q|} \\ dm.$$\n\\end{itemize}\n\\end{proposition} \n\\begin{proof} We have that $A.$ is obvious if $k_P\\geq 0$ for every $P$. In the other case note that \nfor  every $x_0 \\in Q\\in \\mathcal{P}^{k_0}$ we have\n $$f_{k_0}(x_0)=\\sum_{J\\in \\mathcal{P}, Q\\subset J} k_J  |J|^{s-1\/p}=  \\int f \\cdot \\frac{1_{Q}}{|Q|} \\ dm.$$\nso A. and  B. follows.\n\\end{proof}\n\n\n\\newpage\n\n\\centerline{ \\bf III. APPLICATIONS.}\n\\addcontentsline{toc}{chapter}{\\bf  III. APPLICATIONS.}\n\\vspace{1cm}\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent   In Part III. we  suppose that $0< s<1\/p$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. \\end{minipage} \n}\n\\ \\\\ \\\\\n\n \n\\section{Pointwise multipliers acting on $\\mathcal{B}^s_{p,q}$}\n\nHere we will apply the previous sections to study pointwise multipliers of $\\mathcal{B}^s_{p,q}$. To be more precise, Let $g\\colon I \\rightarrow \\mathbb{C}$ be a measurable function. We say that $g$ is  a pointwise multiplier acting on $\\mathcal{B}^s_{p,q}$ if  the transformation\n$$G(f)=gf$$\ndefines  a bounded operator in $\\mathcal{B}^s_{p,q}$. We denote the set of pointwise multipliers by $M(\\mathcal{B}^s_{p,q})$. We can consider the norm on $M(\\mathcal{B}^s_{p,q})$ given by\n$$|g|_{M(\\mathcal{B}^s_{p,q})}=\\sup\\{|gf|_{\\mathcal{B}^s_{p,q}}\\ s.t. \\ |f|_{\\mathcal{B}^s_{p,q}}\\leq 1    \\}.$$\nOf course a necessary condition for  a function to be a multiplier is that \n\n$$\\mathcal{B}^s_{p,q,selfs}=\\{ g\\in \\mathcal{B}^s_{p,q}\\colon \\sup_{a_Q \\in \\mathcal{A}^{sz}_{s,p}} |ga_Q|_{\\mathcal{B}^s_{p,q}} < \\infty\\}$$ \nDenote\n$$|g|_{\\mathcal{B}^s_{p,q,selfs}}=\\sup_{a_Q \\in \\mathcal{A}^{sz}_{s,p}} |ga_Q|_{\\mathcal{B}^s_{p,q}}.$$      \nThe linear space $\\mathcal{B}^s_{p,q,selfs}$ endowed with $|\\cdot|_{\\mathcal{B}^s_{p,q,selfs}}$is a normed space introduced by  Triebel \\cite{ns}. We have \n$$|g|_{\\mathcal{B}^s_{p,q,selfs}}\\leq |g|_{M(\\mathcal{B}^s_{p,q})}$$ \nIn the following three propositions we see that  many results of  Triebel \\cite{ns}   and  Schneider and Vyb\\'\\i{ral}  \\cite{corjan} for  Besov spaces in $\\mathbb{R}^n$ can be easily moved to our setting. The simplest  case  occurs when $p=q=1$.\n\n\n\n\\begin{proposition} We have that $M(\\mathcal{B}^s_{1,1})=\\mathcal{B}^s_{1,1,selfs}$.\n\\end{proposition} \n\\begin{proof} Let $g \\in \\mathcal{B}^s_{1,1,selfs}.$ Given $f\\in  \\mathcal{B}^s_{1,1}$ and  $\\epsilon > 0$  one can find a $\\mathcal{B}^s_{1,1}$-representation \n$$f=\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qa_Q,$$\nwhere $a_Q$ is a $(s,1)$-Souza's atom and \n$$\\sum_k \\sum_{Q\\in \\mathcal{P}^k} |c_Q| < (1+\\epsilon) |f|_{\\mathcal{B}^s_{1,1}}$$\nso \n$$\\sum_k \\sum_{Q\\in \\mathcal{P}^k} |c_Q ga_Q|_{\\mathcal{B}^s_{1,1}} < (1+ \\epsilon) |g|_{\\mathcal{B}^s_{1,1,selfs}}|f|_{\\mathcal{B}^s_{1,1}}.$$ \nand consequently \n$$|gf|_{\\mathcal{B}^s_{1,1}}=|\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qga_Q|_{\\mathcal{B}^s_{1,1}}\\leq (1+ \\epsilon)  |g|_{\\mathcal{B}^s_{1,1,selfs}}|f|_{\\mathcal{B}^s_{1,1}}.$$\nSince $\\epsilon$ is arbitrary we get\n$$|gf|_{\\mathcal{B}^s_{1,1}}\\leq  |g|_{\\mathcal{B}^s_{1,1,selfs}} |f|_{\\mathcal{B}^s_{1,1}}.$$\n\\end{proof} \n\n\n\\begin{lemma}\\label{incint3} Let $W\\in \\mathcal{P}$. The restriction application\n$$r\\colon \\mathcal{B}^s_{p,q}(I, \\mathcal{P}, \\mathcal{A}^{sz}_{s,p})\\rightarrow  \\mathcal{B}^s_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})$$\ngiven by $r(f)=1_Wf$ is continuous. Indeed there is $\\Cll{incint} \\geq 1$, that does not depend on $W$, such that \n\\begin{enumerate}\n\\item[A.] For every $f \\in \\mathcal{B}^s_{p,q}$ we have \n\\begin{equation}\\label{incint2} |1_Wf|_{\\mathcal{B}^s_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})}\\leq \\Crr{incint} |f|_{\\mathcal{B}^s_{p,q}}. \\end{equation}\nIn particular $1_W\\in M(\\mathcal{B}^s_{p,q})$.\n\\item[B.] For every  $\\mathcal{B}^{s+}_{p,q}$-representation \n$$f=\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qa_Q$$\n one can find a $\\mathcal{B}^{s+}_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})$-representation \n$$1_Wf= \\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} d_Qa_Q$$\nsuch that \n\\begin{equation}\\label{incint2b}\n\\Big( \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k}  (d_Q)^p\\big)^{q\/p} \\Big)^{1\/q} \\leq \\Crr{incint} \\Big( \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k}  (c_Q)^p\\big)^{q\/p} \\Big)^{1\/q} . \n\\end{equation} \nMoreover  $d_Q\\neq 0$ implies $Q\\subset \\ supp \\ f$.\n\\end{enumerate}\n\\end{lemma} \n\\begin{proof} Let $Q\\in \\mathcal{P}.$ Denote by $a_Q$ the canonical $(s,p)$-Souza's atom supported on $Q$.  If $W\\cap Q=\\emptyset$ we can write $1_Wa_Q= 0a_Q.$ If $Q\\subset W$ then $1_Wa_Q= 1a_Q.$ If $W\\subset Q$ then\n$$1_Wa_Q= \\Big(\\frac{|W|}{|Q|}\\Big)^{1\/p-s}a_W,$$\nwhere\n$$ \\Big(\\frac{|W|}{|Q|}\\Big)^{1\/p-s}\\leq \\Crr{maior}^{(k_0(W)-k_0(Q))(1\/p-s)}$$\nIn every case we can write\n$$h_Q=1_Wa_Q = \\sum_k \\sum_{\\substack{P\\in \\mathcal{P}^k\\\\ P\\subset Q}}    s_{P,Q} a_P,$$\nwith \n$$ \\sum_{\\substack{P\\in \\mathcal{P}^k\\\\ P\\subset Q}}    |s_{P,Q}| ^p\\leq \\Crr{maior}^{(k-k_0(Q))(1-sp)}$$\nBy Proposition \\ref{trans}.A and \\ref{trans}.B  there is $\\Crr{incint}$ such that  A. and  B.  hold. \n\\end{proof}\n\n\\begin{proposition} We have that $ \\mathcal{B}^s_{p,q,selfs} \\subset L^\\infty$ and this inclusion is continuous.\n\\end{proposition}  \n\\begin{proof} Let $g\\in \\mathcal{B}^s_{p,q,selfs}$. Then    $ga_Q\\in \\mathcal{B}^s_{p,q}$ and by Lemma \\ref{incint3} we have\n$$|g1_Q|_{ \\mathcal{B}^s_{p,q}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{s,p})}\\leq \\Crr{incint}|g1_Q|_{\\mathcal{B}^s_{p,q}} \\leq \\Crr{incint} |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\n\nBy Proposition \\ref{lp}  (taking t = $p$) we have  $(s,p)$-Souza's atom $a_Q$ we have \n$$|ga_Q|_p\\leq   \\Crr{mult1}^{1+1\/p} \\Crr{co}(p,q, ( |\\mathcal{P}^k_Q|^{sp})_k ) \\Crr{incint}  |g|_{\\mathcal{B}^s_{p,q,selfs}} < \\Cll{nova} |Q|^s  |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\nfor some constant $\\Crr{nova}$. In other words\n$$\\Big( |Q|^{sp-1} \\int_Q |g|^p \\ dm \\Big)^{1\/p}  \\leq \\Crr{nova} |Q|^s  |g|_{\\mathcal{B}^s_{p,q,selfs}},$$\nso\n$$\\frac{1}{|Q|}  \\int_Q |g|^p \\ dm \\leq   \\Crr{nova}^p   |g|_{\\mathcal{B}^s_{p,q,selfs}}^p, $$\nfor every $Q\\in \\mathcal{P}$.  Due to the fact that $\\cup_k\\mathcal{P}^k$ generates the $\\sigma$-algebra $\\mathbb{A}$, by  L\\'evy's Upward Theorem (see Williams \\cite{martin})  for almost every $x\\in I$ the following holds.  If $x\\in Q_k\\in \\mathcal{P}^k$ then \n$$\\lim_k \\frac{1}{|Q_k|}  \\int_{Q_k} |g|^p \\ dm = |g(x)|^p.$$\nSo \n$$|g|_\\infty\\leq  \\Crr{nova}   |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\n\\end{proof} \n\n\\begin{figure}\n\\includegraphics[scale=0.5]{narch.pdf}\n\\caption{ Non-Archimedean  behaviour. Illustration for Proposition \\ref{sepa}. The filled regions are the supports of the functions $g_i$. the squares are the elements of the grid on which there are atoms contributing for  the representation of $f$. Every square intercepts at most one support, so each atom ``sees'' only the function whose support  is nearby.}\n\\end{figure}\n\n\\subsection{Non-Archimedean  behaviour in $\\mathcal{B}^\\beta_{p,\\tilde{q},selfs}$} If we have a sequence $g_i \\in M(\\mathcal{B}^s_{p,q})$ we can get the naive  estimate \n$$|(\\sum_i g_i)f|_{\\mathcal{B}^s_{p,q}}\\leq  \\big( \\sum_i |g_i|_{M(\\mathcal{B}^s_{p,q})}\\big)|f|_{\\mathcal{B}^s_{p,q}}.$$\n\nBut it is remarkable that sometimes one can get a far better estimate.  To state the result we need to define\n\n$$|g|_{\\mathcal{B}^{s,t}_{p,q,selfs}}=\\sup_{\\substack{ a_Q \\in \\mathcal{A}^{sz}_{s,p}\\\\ Q\\in \\mathcal{P}^j \\\\ j\\geq t}} |ga_Q|_{\\mathcal{B}^s_{p,q}}.$$\nIt is easy to see that this norm is equivalent to $|\\cdot|_{\\mathcal{B}^{s}_{p,q,selfs}}$.\n\n\\begin{proposition} \\label{sepa} Let $\\beta > s$. There is $\\Cll{gen}$ with the following property. Let  $g_i \\in  \\mathcal{B}^{\\beta,t_i}_{p,\\tilde{q},selfs}$, with $i\\in \\Lambda \\subset \\mathbb{N}$,  and $t_i \\in \\mathbb{N}$.\n\nConsider a function $f$ with  a $\\mathcal{B}^s_{p,q}$-representation \n$$ f=\\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nsatisfying \n\\begin{itemize}\n\\item[A.] We have \n$$\\sup_{\\substack{ Q\\in \\mathcal{P} \\\\ c_Q\\neq 0}}  \\  \\sum_{_{Q\\cap supp \\ g_i \\neq \\emptyset}} |g_i|_{ \\mathcal{B}^{\\beta,t_i}_{p,\\tilde{q},selfs}} \\leq N.$$\n\\item[B.]  If $Q\\in \\mathcal{P}^k$ satisfies $c_Q\\neq 0$ and $Q\\cap supp \\ g_i \\neq \\emptyset$ then $k\\geq t_i$. \n\\end{itemize}\nThen we can find a $\\mathcal{B}^s_{p,q}$-representation\n\\begin{equation} \\label{rre} (\\sum_i g_i)f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} d_Q a_Q\\end{equation} \nsuch that \n\\begin{align}\\label{reep} &\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |d_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\\n&\\leq \\Crr{gen} N \\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |c_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \n\\end{align} \n\\end{proposition} \n\\begin{proof} It is enough to prove the result for the case when  $\\Lambda$ is finite. Let $Q\\in \\mathcal{P}^{k_0}$ with $c_Q\\neq 0$.  There is   $\\{i_1,\n\\dots, i_j\\}\\subset \\Lambda$,  such that   \n\\begin{equation}\\label{aa1} (\\sum_i g_i ) a_Q=  \\sum_{\\ell\\leq j} g_{i_\\ell} a_Q.\\end{equation} \nand $Q\\cap supp \\ g_{i_\\ell} \\neq \\emptyset$ for every $\\ell$. In particular $k_0\\geq \\max_\\ell t_{i_\\ell}.$\nBy Lemma \\ref{incint3} for each $\\ell\\leq j$ we can find a $\\mathcal{B}^\\beta_{p,q} $-representation \n\\begin{equation}\\label{hjk} g_{i_\\ell}a_Q= \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} \\tilde{s}_{P,Q}^\\ell b_{P}\\end{equation}\nsuch that $b_P$ is the canonical $(\\beta,p)$-Souza's atom supported on $P$ and \n$$\\Big( \\sum_k \\big(  \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}}  |\\tilde{s}_{P,Q}^\\ell|^p   \\big)^{\\tilde{q}\/p}  \\Big)^{1\/\\tilde{q}}\\leq 2\\Crr{incint}  |g_{i_\\ell}|_{\\mathcal{B}^{\\beta,t_{i_\\ell}}_{p,\\tilde{q},selfs}} .$$\nSince $b_P= |P|^{\\beta -s}  a_P$, where $a_P$ is the canonical $(s,p)$-Souza's atoms supported on $P$, we can write\n$$g_{i_\\ell}a_Q= \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} s_{P,Q}^\\ell a_{P},$$\nwith $s_{P,Q}^\\ell=\\tilde{s}_{P,Q}^\\ell|P|^{\\beta -s}$ satisfying \n\\begin{align*}\n &\\big( \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}}  |s_{P,Q}^\\ell|^p \\big)^{1\/p} \\\\\n&\\leq  2\\Crr{incint}  |g_{i_\\ell}|_{\\mathcal{B}^{\\beta,t_{i_\\ell}}_{p,\\tilde{q},selfs}}  \\Crr{maior}^{(\\beta-s)(k-k_0)} \\sup_{Q\\in \\mathcal{P}}|Q|^{\\beta-s},\n\\end{align*} \nso we can write\n$$(\\sum_i g_i ) a_Q=  \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} s_{P,Q} a_{P},$$\nwith \n$$s_{P,Q}=\\sum_\\ell s_{P,Q}^\\ell$$\n satisfying \n$$\\big( \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}}  |s_{P,Q}|^p \\big)^{1\/p} \\leq  2 N \\Crr{incint} \\Crr{maior}^{(\\beta-s)(k-k_0)} \\sup_{Q\\in \\mathcal{P}}|Q|^{\\beta-s}.$$\nBy Proposition \\ref{trans}.A we can find a $\\mathcal{B}^s_{p,q}$-representation (\\ref{rre}) satisfying (\\ref{reep}). \n\\end{proof}\n\n\\begin{remark}\\label{posrem} If $g$ is    $\\mathcal{B}^\\beta_{p,q}$-positive we can define\n$$|g|_{\\mathcal{B}^{s+,t}_{p,q,selfs}}=\\sup_{\\substack{ a_Q \\in \\mathcal{A}^{sz}_{s,p}\\\\ Q\\in \\mathcal{P}^j \\\\ j\\geq t}} |ga_Q|_{\\mathcal{B}^{s+}_{p,q}}.$$\n If we assume additionally that $g_i$ are $\\mathcal{B}^{\\beta,t_i}_{p,q}$-positive, Proposition \\ref{sepa} remains true if we replace all the instances  of $|\\cdot|_{\\mathcal{B}^{s,t}_{p,q,selfs}}$ by $|\\cdot|_{\\mathcal{B}^{s+,t}_{p,q,selfs}}$  in its statement. Moreover  by Proposition \\ref{trans}.B and Lemma \\ref{incint3}. B we can conclude that\n \\begin{itemize}\n \\item[i.]  if $c_Q\\geq 0$ for every $Q$ then $d_Q\\geq 0$ for every $Q$,\n \\item[ii.] If $Q$ is such that $d_Q\\neq 0$ then $Q\\subset supp \\ g_i$, for some $i\\in \\Lambda$. \n\\end{itemize}\n\\end{remark} \n\n\\begin{corollary}\\label{23er} For every $\\beta > s$ and $\\tilde{q}\\in [1,\\infty]$  we have  $\\mathcal{B}^\\beta_{p,\\tilde{q},selfs} \\subset M(\\mathcal{B}^s_{p,q})$. Moreover this inclusion is continuous. \n\\end{corollary} \n\n\n\n\n\\subsection{Strongly regular domains }\\label{srd} We may wonder on which conditions the characteristic function of a set  $\\Omega$ is a pointwise multiplier in $\\mathcal{B}^{s}_{p,q}$.  \n\\begin{definition} A measurable set $\\Omega\\subset I$ is  a {\\bf $(\\alpha,\\Cll{rp2},\\Cll{k1} )$-strongly regular domain}  if  for every $Q\\in \\mathcal{P}^j$, with $j\\geq \\Crr{k1}$,  there is  family $\\mathcal{F}^k(Q\\cap \\Omega) \\subset \\mathcal{P}^k$  such that \n\\begin{itemize}\n\\item[i.] We have $Q\\cap \\Omega = \\cup_{k} \\cup_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} P$.\n\\item[ii.] If $P,W \\in \\cup_{k} \\mathcal{F}^k(Q\\cap \\Omega)$ and $P\\neq W$ then $P\\cap W=\\emptyset$. \n\\item[iii.] We have\n\\begin{equation} \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} |P|^{\\alpha}\\leq \\Crr{rp2}  |Q|^{\\alpha}.\\end{equation} \n\\end{itemize}\n\\end{definition} \n\n\nThe following result can be associated  with results in Triebel \\cite{ns} for $B^s_{p,p}(\\mathbb{R}^n)$, specially when we consider the setting of Besov spaces in compact homogenous spaces. See Section \\ref{srd} for details.See also Schneider and Vyb\\'\\i ral \\cite{corjan}.\n\n\\begin{proposition} \\label{pos2} If $\\Omega$ is a $(1-\\beta p,\\Crr{rp2},\\Crr{k1} )$-strongly regular domain then \n\\begin{equation}\\label{fimm} |1_\\Omega|_{\\mathcal{B}^{\\beta+,{\\Crr{k1}}}_{p,\\infty,selfs}}\\leq   \\Crr{rp2}^{1\/p}.\\end{equation}\n\\end{proposition} \n\\begin{proof} Given $Q\\in \\mathcal{P}^j$, with $j\\geq \\Crr{k1}$  we can write\n\\begin{equation} \\label{poi} 1_\\Omega a_Q =    \\sum_{k}   \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} \\Big(\\frac{|P|}{|Q|} \\Big)^{1\/p-\\beta}   a_P.\\end{equation} \nwhere $a_P$ is a $(\\beta,p)$-atom.  Note that \n$$  \\big(   \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} \\Big(\\frac{|P|}{|Q|} \\Big)^{1-\\beta p} \\big)^{1\/p} \\leq \\Crr{rp2}^{1\/p},$$\nso (\\ref{fimm}) holds.\n\\end{proof}\n\n\\begin{proposition}[Pointwise Multipliers I] \\label{pm1} There is $\\Cll{gen2}$ with the following property. Suppose that  $\\Omega_i$ are  $(1-\\beta p,K_i,t_i)$-strongly regular domains, $i\\in \\Lambda \\subset \\mathbb{N}$, and $\\Theta_i>  0$ for every $i\\in \\Lambda_i$.\nConsider a function $f$ with  a $\\mathcal{B}^s_{p,q}$-representation \n$$ f=\\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nsatisfying  \n\\begin{itemize}\n\\item[A.] We have \n$$\\sup_{\\substack{ Q\\in \\mathcal{P} \\\\ c_Q\\neq 0}}  \\  \\sum_{_{Q\\cap \\Omega_i \\neq \\emptyset}} \\Theta_i K_i^{1\/p} \\leq N.$$\n\\item[B.]  If $Q\\in \\mathcal{P}^k$ satisfies $c_Q\\neq 0$ and $Q\\cap \\Omega_i \\neq \\emptyset$ then $k\\geq t_i$. \n\\end{itemize}\n  Then we can find a $\\mathcal{B}^s_{p,q}$-representation\n\\begin{equation} \\label{rre2} (\\sum_i \\Theta_i 1_{\\Omega_i})f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} d_Q a_Q\\end{equation} \nsuch that \n\\begin{align}\\label{reep2} &\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |d_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\\n&\\leq \\Crr{gen2} N\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |c_Q|^p \\big)^{q\/p} \\Big)^{1\/q}.\n\\end{align} \nMoreover \n\\begin{itemize}\n\\item[i.] If $Q$ satisfies  $d_Q\\neq 0$ then $Q\\subset \\Omega_i$ for some $i \\in \\Lambda$. \n\\item[ii.] If $c_Q\\geq 0$ for every $Q$ then $d_Q\\geq 0$ for every $Q$. \n\\end{itemize}\n\\end{proposition} \n\\begin{proof} It follows from Proposition \\ref{sepa}, Proposition \\ref{pos2} and Remark \\ref{posrem}. \n\\end{proof} \n\n\n\n\n\\subsection{Functions on $\\mathcal{B}^{1\/p}_{p,\\infty}\\cap L^\\infty$  } \n\nWe would like to give explicit examples of multipliers in $\\mathcal{B}^s_{p,q}$. One should compare the following result with the study by Triebel\\cite{multi}  of the regularity of  the multiplication on Besov spaces. See also Maz'ya  and  Shaposhnikova \\cite{sob} for more information   on multipliers in classical Besov spaces.\n\n\n\\begin{proposition}[Pointwise multipliers II] \\label{mult} Let $g\\in  \\mathcal{B}^{1\/p}_{p,\\infty}\\cap L^\\infty$. Then the multiplier operator\n$$G\\colon \\mathcal{B}^s_{p,q} \\rightarrow \\mathcal{B}^s_{p,q}$$\ndefined by $G(f)=gf$ is a well defined and bounded operator acting on $(\\mathcal{B}^s_{p,q},|~\\cdot~|_{\\mathcal{B}^s_{p,q}})$.\nIndeed\n$$|G|_{\\mathcal{B}^s_{p,q}}\\leq \\Crr{e}\\Crr{no}  \\frac{|g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} }{1-\\Crr{maior}^{1\/p-s}} +|g|_\\infty,$$\nwhere $\\Crr{e}=\\Crr{e}(1\/p,p,\\infty)$ and $\\Crr{no}=\\Crr{no}(1\/p,p,\\infty)$ are as in Corollary \\ref{fou}.\n\\end{proposition} \n\\begin{remark} \\label{pos3} We can get a similar result replacing $\\mathcal{B}^{a}_{p, b}$ by $\\mathcal{B}^{a+}_{p, b}$ everywhere. \n\\end{remark} \n\\begin{proof} Let  $a_Q=|Q|^{s-1\/p}1_Q$ be  the canonical $(s,p)$-Souza's atom on $Q$ and $b_J=1_J$ be  the canonical $(1\/p,p)$-Souza's atom on $J$. Given $\\epsilon > 0$, let \n$$f = \\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nbe  a $\\mathcal{B}^s_{p,q}$-representation of $f$ such that \n$$ \\Big(  \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k} |c_Q|^p\\big)^{q\/p} \\Big)^{1\/q} \\leq (1+\\epsilon)|f|_{\\mathcal{B}^s_{p,q}}$$\nand\n$$g =\\sum_k  \\sum_{J\\in \\mathcal{P}^k} e_J b_J$$\nbe  a $\\mathcal{B}^{1\/p}_{p,\\infty}$-representation of $g$ given by Corollary \\ref{fou} (in the case of Remark \\ref{pos3} we can consider an optimal $\\mathcal{B}^{1\/p}_{p,\\infty}$-positive representation of $g$). We claim that \n$$u_1 =\\sum_j  \\sum_{J \\in \\mathcal{P}^j} \\big(\\sum_{J\\subset Q, Q\\neq J,Q\\in \\mathcal{P}} \\big( \\frac{|J|}{|Q|}\\big)^{1\/p-s}  c_Q e_J  \\big)  a_J, \\ and $$\n$$u_2 = \\sum_k \\sum_{Q \\in \\mathcal{P}^k} \\big( \\sum_{Q\\subset J,  J\\in \\mathcal{P} } c_Q e_J \\big)  a_Q,$$\nare $\\mathcal{B}^s_{p,q}$-representations of functions $u_i \\in \\mathcal{B}^s_{p,q}$. Firstly note that the inner sums are finite. Moreover  if $J\\in \\mathcal{P}^j$ we denote by \n$Q_k(J)$ the unique element  of $\\mathcal{P}^k$, with $k\\leq j$ that satisfies  $J \\subset Q_k(J)$ then \n\\begin{align}& \\ \\big( \\sum_{J \\in \\mathcal{P}^j } \\big|\\sum_{k \\leq j}  \\Big( \\frac{|J|}{|Q_k(J)|}\\Big)^{1\/p-s}  c_{Q_k(J)} e_J\\big|^p \\big)^{1\/p} \\nonumber  \\\\\n&\\leq  \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)}  \\big( \\sum_{J \\in \\mathcal{P}^j }| c_{Q_k(J)} e_J|^p \\big)^{1\/p} \\nonumber  \\\\\n&\\leq \\big( \\sum_{J \\in \\mathcal{P}^j }|e_J|^p \\big)^{1\/p}  \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)}  \\max_{Q\\in \\mathcal{P}^k}  |c_Q|\\nonumber  \\\\\n&\\leq \\Big( \\max_j \\big( \\sum_{J \\in \\mathcal{P}^j }|e_J|^p \\big)^{1\/p} \\Big) \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)}  (\\sum_{Q\\in \\mathcal{P}^k}  |c_Q|^p)^{1\/p} \\nonumber  \\\\\n&\\leq \\Crr{e}\\Crr{no}  |g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)}  (\\sum_{Q\\in \\mathcal{P}^k}  |c_Q|^p)^{1\/p} \\nonumber  \n\\end{align} \nThe right hand side is a convolution, so we can easily get \n$$ |u_1|_{\\mathcal{B}^s_{p,q}} \\leq (1+\\epsilon) \\Crr{e}\\Crr{no} \\frac{|g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} }{1-\\Crr{maior}^{1\/p-s}}     |f|_{\\mathcal{B}^s_{p,q}}.$$\nMoreover by Proposition \\ref{boup}.B, with $s=1\/p$, we obtain\n\\begin{align}& \\Big( \\sum_{Q \\in \\mathcal{P}^k} \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J} c_Q e_J \\big|^p  \\Big)^{1\/p} \\nonumber  \\\\\n&\\leq  \\Big( \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J} e_J \\big|^p  \\Big)^{1\/p} \\nonumber  \\\\\n&\\leq  \\Big(  \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\Big)^{1\/p} |g|_{\\infty}. \\nonumber  \n\\end{align} \nSo\n$$ |u_1|_{\\mathcal{B}^s_{p,q}} \\leq  (1+\\epsilon) |f|_{\\mathcal{B}^s_{p,q}} |g|_\\infty .$$\nWe claim  that $gf= u_1+u_2$. Indeed  let\n$$f_{k_0} = \\sum_{k<k_0} \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nand\n$$g_{k_0} = \\sum_{k<k_0} \\sum_{J\\in \\mathcal{P}^k} e_J b_J$$\nBy Proposition \\ref{lp} we have \n$$\\lim_{k_0} |g_{k_0}-g|_{p'}=0$$\nand\n $$\\lim_{k_0} |f_{k_0}-f|_{p}=0.$$\nSo\n$$\\lim_{k_0}|f_{k_0}g_{k_0}-fg|_1=0.$$\nNote that \n\\begin{itemize}\n\\item[A.] If $Q\\subset J$ then  $a_Q b_J= a_Q$,\n\\item[B.] If $J \\subset Q$ then \n$$a_Q b_J =  \\big( \\frac{|J|}{|Q|}\\big)^{1\/p-s} a_J.$$\n\\end{itemize}\nSo\n\\begin{eqnarray*}\nf_{k_0}g_{k_0}&=& \\sum_{k<k_0} \\sum_{Q\\in \\mathcal{P}^k}  \\sum_{i<k_0} \\sum_{J\\in \\mathcal{P}^i} e_J c_Q a_Qb_J\\\\\n&=&  \\sum_{k<k_0} \\sum_{Q\\in \\mathcal{P}^k}  \\big( \\sum_{\\substack{J\\in \\mathcal{P}\\\\ Q\\subset J}} e_Jc_Q\\big)a_Q +  \\sum_{k<k_0} \\sum_{Q\\in \\mathcal{P}^k}  \\sum_{i<k_0} \\sum_{\\substack{J\\in \\mathcal{P}^i\\\\J\\subset Q\\\\J\\neq Q}}  e_Jc_Q \\big( \\frac{|J|}{|Q|}\\big)^{1\/p-s} a_J \\\\\n&=&  \\sum_{k<k_0} \\sum_{Q\\in \\mathcal{P}^k}  \\big( \\sum_{\\substack{J\\in \\mathcal{P}\\\\ Q\\subset J}} e_Jc_Q\\big)a_Q +   \\sum_{i<k_0} \\sum_{J \\in \\mathcal{P}^i}  \\Big(  \\sum_{\\substack{Q\\in \\mathcal{P}\\\\J\\subset Q\\\\J\\neq Q}}  e_Jc_Q \\big( \\frac{|J|}{|Q|}\\big)^{1\/p-s}\\Big) a_J\\\\\n&=&u_{1,k_0}+u_{2,k_0}.\n\\end{eqnarray*}\nNote that  \n$$\\lim_{k_0} |u_{r,k_0}-u_r|_1=0 \\ and \\   |u_{r,k_0}|_{B^s_{p,q}}\\leq |u_{r}|_{B^s_{p,q}} \\ for \\ r=1,2.$$\nNow we can use Corollary \\ref{compa1}.i to conclude the proof.\n\\end{proof}\n\n\n\n\n\\section{ $\\mathcal{B}^s_{p,q}\\cap L^\\infty$ is a quasi-algebra}\n\nMultipliers in $\\mathcal{B}^s_{p,q}\\cap L^\\infty$ are indeed much easier to come by.\n\n\\begin{proposition}[Pointwise multipliers III] \\label{mult33} Let $g,f  \\in \\mathcal{B}^{s}_{p,q}\\cap L^\\infty$. Then $g\\cdot f  \\in \\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ and \n$$|f\\cdot g|_{\\mathcal{B}^s_{p,q}}+|f\\cdot g|_{\\infty}\\leq\\Crr{e}\\Crr{no} (|f|_{\\mathcal{B}^{s}_{p,q}}  +|f|_\\infty)( |g|_{\\mathcal{B}^{s}_{p,q}} +|g|_\\infty).$$\nSo $\\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ is a quasi Banach algebra. Here  $\\Crr{e}=\\Crr{e}(s,p,q)$ and $\\Crr{no}=\\Crr{no}(s,p,q)$ are as in Corollary \\ref{fou}.\n\\end{proposition} \n\\begin{proof} Of course $|f\\cdot g|_\\infty\\leq |f|_\\infty|g|_\\infty$. Let  $a_Q=|Q|^{s-1\/p}1_Q$ be  the canonical $(s,p)$-Souza's atom on $Q$. Let \n$$f = \\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nand\n$$g =\\sum_k  \\sum_{J\\in \\mathcal{P}^k} e_J a_J$$\nbe  $\\mathcal{B}^s_{p,q}$-representations of $f$ and $g$  given by Corollary \\ref{fou}. We claim that \n$$u_1 =\\sum_k  \\sum_{Q \\in \\mathcal{P}^k} \\big( \\sum_{Q\\subset J,  J\\in \\mathcal{P} } |J|^{s-1\/p} c_Q e_J \\big)  a_Q,$$\n$$u_2 =\\sum_k  \\sum_{J \\in \\mathcal{P}^k} \\big(\\sum_{J\\subset Q, Q\\neq J,Q\\in \\mathcal{P}} |Q|^{s-1\/p}  c_Q e_J  \\big)  a_J.$$\nare $\\mathcal{B}^s_{p,q}$-representations of functions $u_i \\in \\mathcal{B}^s_{p,q}$. Moreover by Proposition \\ref{boup}.A  we have \n\\begin{align}& \\Big( \\sum_{Q \\in \\mathcal{P}^k} \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J}  |J|^{s-1\/p}  c_Q e_J \\big|^p  \\Big)^{1\/p} \\nonumber  \\\\\n&\\leq  \\Big( \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J}  |J|^{s-1\/p}  e_J \\big|^p  \\Big)^{1\/p} \\nonumber  \\\\\n&\\leq  \\Big(  \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\Big)^{1\/p} |g|_{\\infty}. \\nonumber  \n\\end{align} \nSo \n$$ |u_1|_{\\mathcal{B}^s_{p,q}} \\leq \\Crr{e}\\Crr{no}  |g|_\\infty |f|_{\\mathcal{B}^s_{p,q}},$$\nand by an analogous argument\n$$ |u_2|_{\\mathcal{B}^s_{p,q}} \\leq \\Crr{e}\\Crr{no} |f|_\\infty |g|_{\\mathcal{B}^s_{p,q}}.$$\nDefine $f_{k_0}$ and $g_{k_0}$ as in the proof of Proposition \\ref{mult}. By Proposition \\ref{boup} we have  $|f_{k_0}|\\leq |f|_\\infty$ and $|g_{k_0}|\\leq |g|_\\infty$. Since $\\lim_{k_0} f_{k_0}=f$ and $\\lim_{k_0} g_{k_0}=g$ in $L^p$, we can assume, taking a subsequence if necessary, that $f_{k_0}g_{k_0}$ converges pointwise to $fg$. So by the Theorem of Dominated Convergence we have $\\lim_{k_0} f_{k_0}g_{k_0}= fg$ in $L^1$. \nFinally note that if $J \\subset Q$ then \n$$a_Q a_J =  |Q|^{s-1\/p} a_J,$$\nNow we can use the same argument as in the proof  of Proposition \\ref{mult}  to conclude that $gf= u_1+u_2$.  This concludes the proof. \n\\end{proof}\n\n\\subsection{Regular domains} \\label{cf} Here we will give sufficient conditions for the characteristic function of a set to define a bounded pointwise multiplier either on $\\mathcal{B}^{s}_{p,q}\\cap L^\\infty$. For every set  $\\Omega$, let \n$$k_0(\\Omega)=\\min \\{k\\geq 0 \\colon \\ \\exists P \\in \\mathcal{P}^k \\ s.t. \\ P \\subset \\Omega     \\} $$\n\n\\begin{definition} We say that a countable family of pairwise disjoint measurable sets $\\{\\Omega_r\\}_{r\\in \\Lambda}$  is  {\\bf  $(\\alpha, \\Cll{domain},\\Cll[c]{domainc})$-regular family} if  one can  find  families $\\mathcal{F}^k(\\Omega_r)\\subset \\mathcal{P}^k$, $k\\geq  k_0(\\Omega_r)$,  such that \n\\begin{itemize}\n\\item[A.] We have $\\Omega_r = \\cup_{k\\geq k_0(\\Omega_r)} \\cup_{Q\\in \\mathcal{F}^k(\\Omega_r)} Q$.\n\\item[B.] If $P,Q \\in \\cup_{k\\geq k_0(\\Omega_r)} \\mathcal{F}^k(\\Omega_r)$ and $P\\neq Q$ then $P\\cap Q=\\emptyset$. \n\\item[C.] We have\n\\begin{equation}\\label{dom} \\sum_{r\\in \\Lambda} \\sum_{Q\\in \\mathcal{F}^k(\\Omega_r)} |Q|^{\\alpha}\\leq \\Crr{domain} \\Crr{domainc}^{k-k_0(\\cup_r \\Omega_r)} |\\cup_r \\Omega_r|^{\\alpha}.\\end{equation} \n\\end{itemize}\nWe say that a measurable set $\\Omega$ is a $(\\alpha, \\Crr{domain},\\Crr{domainc})$-regular domain if $\\{\\Omega\\}$  is a $(\\alpha, \\Crr{domain},\\Crr{domainc})$-regular family.\n\\end{definition}\n\n\\begin{proposition} Let $\\beta > s$. Every $(1-\\beta p, C,0)$-strongly regular domain is a  $(1-sp,C',\\Crr{maior}^{(\\beta-s)p})$-regular domain,\nfor some $C'$. \n\\end{proposition}\n\n\\begin{proof} Consider a  $(1-\\beta p,\\Crr{rp2},0)$-strongly regular domain $\\Omega$. There are at most $\\Crr{menor}^{-k_0(\\Omega)}$ elements in $\\mathcal{P}^{k_0(\\Omega)}$ and\n$$\\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{-k_0(\\Omega)}  \\leq \\frac{|Q|}{|W|}\\leq \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)}$$\nfor every $Q,W\\in \\mathcal{P}^{k_0(\\Omega)}$. Consequently\n$$\\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{-k_0(\\Omega)}  \\leq \\frac{|\\Omega|}{|Q|}\\leq \\Crr{menor}^{-k_0(\\Omega)} \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)}$$\nfor each $Q \\in \\mathcal{P}^{k_0(\\Omega)}$.\nFor every $Q\\in \\mathcal{P}^{k_0(\\Omega)}$  there is a family $\\mathcal{F}^k(Q\\cap \\Omega)$ such that \n$$\\sum_k \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} P= Q\\cap \\Omega$$\nand\n$$ \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\leq C  |Q|^{1-\\beta p}.$$\nLet \n$$\\mathcal{F}^k(\\Omega)=\\cup_{Q\\in \\mathcal{P}^{k_0(\\Omega)} } \\mathcal{F}^k(Q\\cap \\Omega).$$\nWe have\n\\begin{align*}  \\sum_{Q\\in  \\mathcal{P}^{k_0(\\Omega)}}\\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-s p}&=\\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p} |P|^{(\\beta-s)p}\\\\\n&\\leq  \\sum_{Q\\in  \\mathcal{P}^{k_0(\\Omega)}} \\big(\\max_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{(\\beta-s)p} \\big)  \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\\\\n&\\leq  \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p}  \\sum_{Q\\in  \\mathcal{P}^{k_0(\\Omega)}} |Q|^{(\\beta-s)p}  \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\\\\n&\\leq  C \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p}  \\sum_{Q\\in  \\mathcal{P}^{k_0(\\Omega)}} |Q|^{(\\beta-s)p} |Q|^{1-\\beta p} \\\\\n& \\leq C \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} \\sum_{Q\\in  \\mathcal{P}^{k_0(\\Omega)}} |Q|^{1-s p}\\\\\n&\\leq   C  \\Crr{menor}^{-k_0(\\Omega)}   \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)(1-s p)}   \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} |\\Omega|^{1-s p}   \n\\end{align*} \nThis concludes the proof. \\end{proof}\n\\begin{remark} Suppose that there is $\\Cll{compara}$ such that for every $k$ and every $Q, W\\in \\mathcal{P}^k$ we have\n$$\\frac{1}{\\Crr{compara}}\\leq    \\frac{|Q|}{|W|}\\leq \\Crr{compara},$$\nand \n$$\\#\\{  P\\in \\mathcal{P}^{k_0(\\Omega)}\\colon \\ P\\cap \\Omega\\neq \\emptyset \\}\\leq \\Cll{numero},$$\nThen it is easy to see that one can choose $C'= \\Crr{numero}\\Crr{compara}C$.\n\n\n\\end{remark} \n\n\nThe following result is similar to results for Sobolev spaces  by Faraco and Rogers \\cite{faraco}. See also Sickel \\cite{sickel}.\n\n\\begin{corollary} If $\\{ \\Omega_r\\}_{r\\in \\Lambda}$ is a $(1-ps, \\Crr{domain},\\Crr{domainc})$-regular family  then there is $\\Cll{rel}$ such that  for every  $g\\in \\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ and $r\\in\\Lambda$ we can find a  $\\mathcal{B}^{s}_{p,q}$-representation\n\\begin{equation}\\label{pdd} g\\cdot 1_{\\Omega_r}= \\sum_k \\sum_{Q\\in \\mathcal{P}^k, Q \\subset \\Omega_r }  d_Q^r   a_Q,\\end{equation}\nsuch that \n\\begin{equation}\\label{hiip1}  \\Big( \\sum_j \\big(  \\sum_r    \\sum_{\\substack{ Q\\in \\mathcal{P}^j \\\\  Q \\subset \\Omega_r }}  |d_Q^r|^p )^{q\/p}  \\Big)^{1\/q} \\leq \\Crr{rel}  |g|_{\\mathcal{B}^{s}_{p,q}}.\\end{equation}\nNote that \n$$\\Omega=\\cup_r\\Omega_r$$\nis  a $(1-ps, \\Crr{domain},\\Crr{domainc})$-regular domain and \n $F(g)=g1_{\\Omega}$ is a bounded operator in $\\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ satisfying \n\\begin{equation}\\label{G} |F|_{\\mathcal{B}^{s}_{p,q}\\cap L^\\infty}\\leq  \\Crr{e}\\Crr{no} \\Big( 1+  \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}\\Big).\\end{equation}\nMoreover\n\\begin{equation}\\label{estG} |1_\\Omega|_{\\mathcal{B}^s_{p,q}}\\leq  \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}.\\end{equation}\n\\end{corollary} \n\\begin{proof} Notice that\n$$f=1_{\\cup_r \\Omega_r}= \\sum_k  \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q,$$\nwhere $c_Q=|Q|^{1\/p-s}$ for every $Q\\in \\cup_k \\cup_r \\mathcal{F}^k(\\Omega_r)$ and $c_Q=0$ otherwise.\nLet \n$$g =\\sum_k  \\sum_{J\\in \\mathcal{P}^k} e_J a_J$$\nbe  $\\mathcal{B}^s_{p,q}$-representations  $g$  given by Corollary \\ref{fou}. Consider $u_1, u_2$ as in the proof of Proposition \\ref{mult33}. By Proposition \\ref{boup} we can get exactly the same estimate as in the proof of Proposition \\ref{mult33}.\n\nNote that those  $Q\\in \\mathcal{P}^k$ for which the corresponding atom $a_Q$  has  a no vanishing coefficient in the definition of $u_1$ belongs to $\\cup_r \\cup_j \\mathcal{F}^j(\\Omega_r)$, and moreover every $J\\in \\mathcal{P}^k$ for which the corresponding atoms $a_J$  has  no vanishing coefficients in the definition of $u_2$ is contained in some  $Q\\in  \\mathcal{F}^j(\\Omega_r)$, for some $j$ and $r$. In particular $J\\subset \\Omega_r$.  So (\\ref{pdd}) holds, with\n$$d_Q^r= \\big( \\sum_{Q\\subset J,  J\\in \\mathcal{P} } |J|^{s-1\/p} c_Q e_J \\big)  + \\big(\\sum_{Q\\subset J, J\\neq Q,J\\in \\mathcal{P}} |J|^{s-1\/p}  c_J e_Q  \\big) $$\nfor every $Q\\subset \\Omega_r$.  \n\nNote also that\n\\begin{eqnarray*}\n&&\\Big( \\sum_k \\big(\\sum_r \\sum_{Q\\in \\mathcal{F}^k(\\Omega_r)} |Q|^{1-sp}\\big)^{q\/p} \\Big)^{1\/q}\\\\\n&\\leq&  \\Crr{domain}^{1\/p} \\Big( \\sum_{k\\geq k_0(\\cup_r \\Omega_r)}   \\Crr{domainc}^{(k-k_0(\\Omega))q\/p} \\Big)^{1\/q} |\\Omega|^{1\/p-s}\\\\\n&\\leq&  \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}.\n\\end{eqnarray*}\nso (\\ref{estG})\nand consequently (\\ref{G}) hold. \n\\end{proof}\n\n\\begin{remark} Using the methods in Faraco and Rogers \\cite{faraco} one can show that  quasiballs in $[0,1]^n$ (and in particular  quasidisks in $[0,1]^2$, that is, domains delimited by quasicircles) give  examples of regular domains in $[0,1]^n$ endowed with the  good grid of dyadic $n$-cubes and the Lebesgue measure $m$.\n\\end{remark}\n\n\n\\section{A remarkable description of $\\mathcal{B}^s_{1,1}$.}\n\nWhen $p=q=1$ (and $s >0$  small), something curious happens. We can skip the good grid and characterise the Besov space  $\\mathcal{B}^s_{1,1}$ of a homogeneous space using regular domains.   Fix $\\Crr{domain}\\geq 1$ and $\\Crr{domainc} \\in (0,1)$. Let $\\mathcal{W}$ be the family of all $(1-s, \\Crr{domain},\\Crr{domainc})$-regular domains. Of course $\\mathcal{P}\\subset \\mathcal{W}$. Let $\\hat{\\mathcal{W}}$ be a family of sets satisfying \n$$\\mathcal{P}\\subset \\hat{\\mathcal{W}} \\subset \\mathcal{W}$$\nDefine $B^{1-s}$  as the set of all functions $f \\in L^{1\/(1-s)}$ that can be written as\n\\begin{equation}\\label{ree} f=\\sum_{i=0}^{\\infty}   c_i \\frac{1_{A_i}}{|A_i|^{1-s}},\\end{equation}\nwhere $A_i\\in  \\hat{\\mathcal{W}}$ for every $i\\in \\mathbb{N}$  and \n$$\\sum_i |c_i| < \\infty.$$\nIt is easy to see that \n$$|f|_{1\/(1-s)}\\leq \\sum_i |c_i|.$$\nDefine\n$$|f|_{B^{1-s}}=\\inf \\sum_i |c_i|,$$\nwhere the infimum runs over all possible representations (\\ref{ree}). One can see that $(B^{1-s},|\\cdot|_{B^{1-s}})$ is a normed vector space.\n\n\\begin{proposition} \\label{rema}We have that $B^{1-s}=\\mathcal{B}^{s}_{1,1}(\\mathcal{P})$ and the corresponding norms are equivalent.\\end{proposition}\n\\begin{proof} Note that (\\ref{estG})  says that there is $C$ such that if  $A\\in  \\mathcal{W}$ then   $1_{A} \\in \\mathcal{B}^s_{1,1}(\\mathcal{P})$  and $$|1_A|_{\\mathcal{B}^s_{1,1}(\\mathcal{P})}\\leq C |A|^{1-s}.$$\nIn particular if $f$ has a representation (\\ref{ree}) we conclude that\n$$|f|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}\\leq  C  |f|_{B^{1-s}}.$$\n In particular $B^{1-s}\\subset \\mathcal{B}^{s}_{1,1}(\\mathcal{P}).$ On the other hand if $g\\in \\mathcal{B}^{s}_{1,1}(\\mathcal{P}).$ then we can  write\n$$g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k}    s_Q \\frac{1_Q}{|Q|^{1-s}}$$\n$$\\sum_{P\\in \\mathcal{P}} |s_Q|= \\sum_{k=0}^{\\infty}  \\sum_{Q \\in \\mathcal{P}^k}    |s_Q|  < \\infty.$$\nand $|g|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}$ is the infimum of $\\sum_{P\\in \\mathcal{P}} |s_Q|$ over all possible representations. In particular  $g\\in B^{1-s}$\nand\n$$|g|_{B^{1-s}}\\leq |g|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}.$$\n\\end{proof}\n\n\n\\begin{remark} Let $I=[0,1]$ with the dyadic grid $\\mathcal{D}$ and the Lebesgue measure $m$. We prove in Part IV that $\\mathcal{B}^s_{1,1}(\\mathcal{D})$, with $0< s<1$, is the  Besov space $B^s_{1,1}([0,1])$, and its norms are equivalent. Note that every interval $[a,b]\\subset [0,1]$ is a $(1-s, 2,2^{s-1})$-regular domain. So we can apply Proposition \\ref{rema} with  $\\hat{\\mathcal{W}}=\\{[a,b], \\ 0\\leq a < b\\leq 1\\}.$  That is, $f$ belongs to $B^s_{1,1}([0,1])$ if and only if it can be written as in (\\ref{ree}), where every $A_i$ is an interval and $\\sum_i |c_i| < \\infty$, and the norm in $B^s_{1,1}([0,1])$ is equivalent to the infimum  of $\\sum_i |c_i|$ over all possible such representations. This characterisation of the Besov space $B^s_{1,1}([0,1])$  was first obtained   by Souza \\cite{souzao1}.\n\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Left compositions.} The folllowing  result generalizes a well-known result  on left composition operators acting  on Besov spaces of $\\mathbb{R}^n$.  See  Bourdaud and Kateb \\cite{k1}  \\cite{k0}\\cite{kateb1} for recents developments on the study of left compositions on  Besov spaces of $\\mathbb{R}^n$. \n\n\\begin{proposition} \\label{expo} Let \n$$g\\colon I \\rightarrow \\mathbb{C}$$ be a Lipchitz function such that $g(0)=0$. Then the left composition\n$$L_g\\colon \\mathcal{B}^s_{p,q} \\rightarrow \\mathcal{B}^s_{p,q}$$\ndefined by $L_g(f)=g\\circ f$ is well defined and\n$$|g\\circ f|_p+ osc^s_{p,q}(g\\circ f)\\leq K (|f|_p +osc^s_{p,q}(f)),$$\nwhere $K$ is the  Lipchitz constant of $g$. Consequently there exists $C$ such that \n$$|L_g(f)|_{\\mathcal{B}^s_{p,q}}\\leq C|f|_{\\mathcal{B}^s_{p,q}}$$\nfor every $f\\in \\mathcal{B}^s_{p,q}$.\n\\end{proposition}\n\\begin{proof} Note that \n\\begin{align}\nosc_p(g\\circ f, Q)&=\\inf_{a\\in \\mathbb{C}}\\Big(  \\int _Q |g(f(x))-a|^p \\ dm(x) \\Big)^{1\/p} \\nonumber \\\\\n& \\leq \\inf_{a\\in \\mathbb{C}}\\Big(  \\int _Q |g(f(x))-g(a)|^p \\ dm(x) \\Big)^{1\/p}  \\nonumber \\\\ \n& \\leq K  \\inf_{a\\in \\mathbb{C}}\\Big(  \\int _Q |f(x)-a|^p \\ dm(x) \\Big)^{1\/p} = K osc_p(f, Q).\n\\end{align}\nSo it easily follows that $osc^s_{p,q}(g\\circ f)\\leq K osc^s_{p,q}(f)$. Of course $|g\\circ f|_p\\leq K|f|_p$.  In particular $g\\circ f \\in \\mathcal{B}^s_{p,q}.$\n\\end{proof}\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\input{intro}\n\n\\section{Background}\n\\label{sec:bg}\n\\input{background}\n\n\\subsection{Motivation}\n\\label{sec:motiv}\n\\input{motivation}\n\n\\section{Gossip Service}\n\\label{sec:protocol}\n\\input{protocol}\n\n\\section{Performance Evaluation}\n\\label{sec:impl}\n\\input{implementation}\n\n\\section{Related Work}\n\\label{sec:related}\n\\input{related}\n\n\\section{Conclusion}\n\\label{sec:concl}\n\\input{conclusion}\n\n\n\\section{Availability of Code}\n\nThe source code developed and used for the performance evaluation comprised in this paper is available as open source, allowing the experiments to be reproduced.\nIn detail, our implementation of WS-Gossip on the Web Services for Devices (WS4D) Java Multi Edition DPWS Stack (JMEDS) is available at \\url{https:\/\/github.com\/filipecampos\/ws_gossip}.\nThe code used in the WS-Eventing scenarios, as well as for setting up and controlling all the experiments, is available at \\url{https:\/\/github.com\/filipecampos\/ws_gossip_tests}.\n\n\n\\section*{Acknowledgments}\n\\small{\nThis work has been partially supported by the Portuguese National Science Foundation FCT - Funda\u00e7\u00e3o da Ci\u00eancia e Tecnologia, through grant SFRH\/BD\/66242\/2009.\n}\n\n\\bibliographystyle{abbrv}\n\n\\subsection{Service Oriented Standards}\n\n\\subsubsection{Eventing and messaging}\n\nThe WS-Eventing specification supports simple\npublisher-subscriber interaction, by defining how Web Services can subscribe to\nor accept subscriptions for event notification messages\\,\\cite{ws_eventing}.\n\nThe specification defines the four following roles:\n\\begin{description}\n \\item [Event Source]Sends notifications on triggered events,\nand accepts requests for creating subscriptions.\n \\item [Subscription Manager]Manages event subscriptions. \nIt notifies \\textbf{Subscribers} when their subscriptions are terminated unexpectedly,\n and replies to their subscription management enquiries, such as subscription's\nstatus retrieval, renewal or deletion.\n \\item [Event Sink]Receives event notification messages.\n \\item [Subscriber]Contacts an \\textbf{Event Source} to create a subscription to\nmanifest the interest of its associated \\textbf{Event Sink} to be notified on the\noccurrence of some event.\nnotifications. It is also responsible for issuing subscription management\nrequests to the \\textbf{Subscription Manager}.\n\\end{description}\n\nIn the simplest scenario, with only two intervening Web Services, a \\textbf{Publisher}\nwill comprise both the \\textbf{Event Source} and the \\textbf{Subscription \nManager} roles, whereas the entity that accumulates both the\n\\textbf{Subscriber} and \\textbf{Event Sink} roles will be referred as a \\textbf{Subscriber}.\n\n\\begin{figure}[htbp]\n \\centering\n\\includegraphics[width=.8\\textwidth]{ws_eventing}\n\\caption{WS-Eventing components.}\n\\label{fig:ws_eventing}\n\\end{figure}\n\nWS-Eventing defines the concept of delivery mode, in order to better adapt the\ngeneral publish\/subscribe pattern to scenarios with different event delivery\nrequirements. The default delivery mode of this specification is the single\ndelivery asynchronous \\textit{push} mode. But, for instance, situations with slow event\nconsumers where they poll for event messages may be preferred, in order to\ncontrol the rate of message arrival and avoid overwhelming them if the rate of\ngeneration and transmission of event messages is far superior to their\nprocessing rate.\n The subscription request message defines the specific delivery mode to be used in\nnotifying the identified \\textbf{Event Sink}, and new delivery modes can be freely used,\nif both event sources and consumers support them. In the event that a \\textbf{Subscriber}\nrequests a delivery mode that is not supported by the \\textbf{Event Source}, it will\nrespond signaling this situation, and it may convey a list of the supported\ndelivery modes.\nThis specification proposes, as an example, that notifications can be wrapped in\na standard message instead of the default unwrapped mode where each notification\nis transmitted as a message typed according to the event's action.\n\nAlthough it lacks explicit support for brokered dissemination, it embodies a\nflexible filtering mechanism in the base specification, favoring lightweight\nimplementations and many-to-one dissemination scenarios. And since it was backed\nby major vendors, such as IBM, Microsoft or TIBCO, it has therefore been the\npreferred choice for connected devices, namely, within WS-Management\\,\\cite{ws_man} and DPWS.\n\nThe alternative family of standards OASIS WS-Notification\\,\\cite{ws_bn,ws_brn,ws_t} also provides, besides simple notification and subscription mechanisms, extensible topic definition and brokered dissemination.\n\n\nSince the early 1990s, Reliable Messaging has been seen as a solution for such\nscenarios by the IT community, and so, several message queueing technologies\nhave been used, such as IBM's WebSphereMQ and Microsoft's MSMQ, in addition to\nreliable publish\/subscribe technologies, such as Tibco Rendezvous. In an effort\nto bridge all these different technologies, the Java Message Service (JMS) API\nwas developed by the Java Community Process. Some of these technologies were\nadapted to Web Services. However, due to the exploitation of proprietary\nprotocols, interoperability can only be achieved recurring to gateways that\ntranslate between specific pairs of environments.\n\nWith the emergence of Web Services as the preferred integration solution for\ndistributed systems, WS-Reliable\\-Mes\\-sag\\-ing (WS-RM)\\,\\cite{wsrm} is the\ncurrently adopted standard for achieving reliable message exchange between\ndistributed applications in the presence of software component, system and\nnetwork failures. Due to the interoperable nature of Web Services, WS-RM allows\nto bridge two different infrastructures, such as different operating or\nmiddleware systems, into an end-to-end model where messages are exchanged\nreliably\\,\\cite{ibm_secure}. So, this standard ensures the interoperability of\nservices in what comes to\nReliable Messaging, which also simplifies the development of services,\nsince they must implement the protocols, minimizing the number of errors in\nbusiness logic\\,\\cite{ibm_secure}.\n\nWS-RM distinguishes all the entities\ninvolved in an interaction, as well as the various meanings of the terms\n\\textit{send}, \\textit{transmit}, \\textit{receive} and \\textit{deliver}, as they\nrelate to different components. In that sense, the basic model of\nWS-RM is described in Figure\\,\\ref{rm_fig} and it\nincludes four distinct entities:\n\\begin{description}\n\\item [Application Source]Service or application logic that \\textit{sends}\nthe message to the \\textbf{RM Source};\n\\item [RM Source]Physical processor or node that performs the actual wire\ntransmission;\n\\item [RM Destination]Target processor or node that \\textit{receives} the\nmessage and then \\textit{delivers} it to the \\textit{application destination};\n\\item [Application Destination]Target service of the message.\n\\end{description}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.6\\textwidth]{ws_rm}\n\\caption{WS-ReliableMessaging basic interaction.}\n\\label{rm_fig}\n\\end{figure}\n\nThese nodes are endpoints, which according to the WS-RM\nstandard, represent addressable entities that send and receive Web Services\nmessages.\n\nThe basic mechanism of the standard works, in a simplified way, as follows: the source\nnode sends a Web Service message containing a WS-RM\nheader, which is received by the destination node that then replies by sending\nan acknowledgment message to the source node.\n\nThere are several types of assurances defined in WS-RM, in terms of message\ndelivery:\n\\begin{description}\n\\item [AtMostOnce]A message is delivered at most once, but it might not be\ndelivered at all.\n\\item [AtLeastOnce]A message is delivered at least once, but it could be\ndelivered more times.\n\\item [ExactlyOnce]This type is a combination of the previous two. A message is\ndelivered only once.\n\\item [InOrder]When there are several ordered messages, they are delivered in\nthe same order as they were sent.\n\\end{description}\nEach reliable message exchanging sequence will enforce the message delivery guarantees according to the type defined in the sequence's creation.\n\n\n\\subsubsection{Transactional Coordination}\n\nThe term coordination sometimes refers to a type of orchestration that is\ndefined in the WS-Coordination specification\\,\\cite{wscoor}. It\nspecifies an extensible framework for \\textit{context} management, that provides\ncoordination for the actions of distributed applications. This coordination is\nachieved through provided protocols that support distributed applications, for\ninstance, those that need to reach consistent agreement on the outcome of\ndistributed transactions.\n\nAn application service can create a context needed to propagate coordination\ninformation to other services involved in some activity. These services will\nthen need to register as participants of that activity. For this purpose, the\napplication must include the created \\textit{coordination context} in the\nmessages that it sends to the referred services.\n\nA \\textit{coordination context} can be transmitted using application-specific\nmechanisms, such as the header element of a SOAP application message. This kind\nof conveyance is commonly referred to as flowing the \\textit{context}.\n\nThe structure of a \\textit{context} and the requirements to propagate it between\ncooperating services are also defined in WS-Coordination, and can depend on the\ntype of coordination that is used. A \\textit{coordination context} contains\ninformation on:\n\\begin{itemize}\n\\item how to access a coordination registration service;\n\\item the coordination type;\n\\item relevant extensions.\n\\end{itemize}\n\nThis framework also enables existing transaction processing, workflow, and other\nsystems for coordination to hide their proprietary protocols and to operate in\nan heterogeneous environment.\n\nThis specification is not sufficient by itself to coordinate Web Services, since\nit provides only a coordination framework, leaving undefined the concrete\nprotocol and targeted coordination type. The standards WS-Atomic{\\-}Transaction (WS-AT)\\,\\cite{wsat} and\nWS-BusinessActivity (WS-BA)\\,\\cite{wsba} implement the WS-Coordination framework and also extend it \nby defining their own coordination type: short-term atomic transactions, and\nlong-running business activities, respectively.\n\nThe WS-AT specification defines a\nprotocol that can be plugged into WS-Coordination to provide an adaptation for\nWeb Services of the classic \\textit{2PC}\\,\\cite{understanding_soa} mechanism, making the changes,\nresulting from the activity of some service, persistent. It\nis often said that this protocol does not adapt well to Web Services.\nNonetheless, it is adequate for interoperability across short-lived, co-located\nservices that need to ensure consistent, all-or-nothing results for a\ntransaction.\n\nA \\textit{2PC} process\nconsists on a poll conducted by the \\textit{coordinator} that will lead it to\nsend two alternative directives to all the participants in the transaction:\n\t\\begin{description}\n\t \\item [commit]If all of the registered services have responded\nindicating that the changes were successful.\n\t \\item [rollback]If at least one of the registered services\nfails to respond or responds indicating a failure. \n\t\\end{description}\n\nA service that participates in an atomic transaction can register for more than\none of the different types of coordination protocols, as defined in the WS-AT\nspecification:\n\n\t\\begin{description}\n\t\\item [Two-Phase Commit]Coordinates registered participants to\nreach a commit or abort decision, and ensures that all participants are informed\nof the final result. It has two variants:\n\t\t\\begin{description}\n\t\t\\item [Volatile 2PC]Participants manage volatile\nresources such as a cache register or a window manager.\n\t\t \\item [Durable 2PC]Participants manage durable\nresources such as a da{\\-}ta{\\-}base register or a file.\n\t\t\\end{description}\n\t \\item [Completion]Initiates commit processing when an\napplication tells the \\textit{coordinator} to either try to commit or abort an\natomic transaction. Based on the registered participants for each protocol, the\n\\textit{coordinator} begins with \\textit{Volatile 2PC} and then proceeds through\n\\textit{Durable 2PC}. After the transaction has completed, a status is returned\nto the application and the final result to the service that initiates the\ntransaction (\\textit{initiator}), if it has registered for this protocol.\n\t\\end{description}\n\n\\subsubsection{Combining services}\n\nOne of the main benefits of using standards lies in the ability to combine them due\nto their well known interfaces and behavior, in order to extract the most suitable features for each scenario.\n\nFor instance, both of the aforementioned event-driven specifications, WS-Eventing and WS-Notification, can\nbe combined with a coordination protocol in\norder to guarantee the atomicity of an event\\,\\cite{4279671}. Although this\ncomposition ensures that notifications reach all the relevant targets, it proves\nto be a heavy process for resource constrained devices, specially in scenarios\nwith a large number of targets, or with a large amount of communication errors,\nwhere WS-RM could help mitigating them, but also at the cost of increasing\nresource consumption.\n\nWS-RM can be combined with several WS-* standards. On the one hand, it can\nimprove its features by leveraging:\n\\begin{itemize}\n \\item WS-Addressing, which enables the identification of messages and addresses\nof\nendpoints. This specification was modified to accommodate some needs of the WS-RM\nspecification, like the reuse of a message ID when retransmitting identical\nmessages to counter communication errors\\,\\cite{soa_concepts_tech_design};\n \\item WS-Security, to protect the integrity and confidentiality of the\nexchanged messages;\n \\item WS-Policy, to specify the delivery assurance, among other requirements,\nfor a\nsequence\\,\\cite{soa_concepts_tech_design,soa_field_guide,soa_approach}.\n\\end{itemize}\nOn the other hand, due to its ability to ensure reliable communication between\ntwo endpoints, WS-RM can be leveraged by other standards, such as\nWS-Eventing, WS-Notification, WS-AT, WS-BA and WS-Coordination to\nachieve reliable communication among the intervening parties.\n\nAlthough the WS-RM specification allows to condition\nservice activities, it is different from WS-AT or\nWS-BA, in the sense that a coordinating entity is not\nneeded to inspect the progress of the activities, being the reliability rules\nconveyed as SOAP headers in the exchanged\nmessages\\,\\cite{soa_concepts_tech_design}.\n\nRegarding the questions posed in the beginning of this section, WS-RM would be a \nsuitable standard to ensure point-to-point reliable message delivery. However, it\nwould be very inefficient and poise a heavy weight on the message sender in terms \nof processing power, if there are lots of message recipients or if lots of errors \noccur. In order for WS-RM to guarantee atomic delivery to all targets, it would have to \nrely on WS-AT, or a similar protocol based in WS-Coordination, which would, once again, increase the\nconsumption of the sender's processing and communication resources, due to the additional message traffic.\n\n\n\\subsection{Gossip}\n\nIn computer networking, gossiping describes the process where a participant that\nintends to disseminate some information chooses a small random subset of other\nparticipants and forwards the information to them. Each of these destinations,\nupon receiving the information, repeats the same procedure, hence, the gossip\nmoniker. This procedure mimics also how epidemics spread in populations and,\ntherefore, are also known as epidemic protocols\\,\\cite{Eugster:2004p10747}.\n\n\\subsubsection{Reliability and Scale}\n\n\\begin{figure}[t]\n \\centering\n \\begin{tikzpicture}\n  \\begin{axis}[xlabel=Fanout (f),ylabel=(\\%),ymax=130]\n   \\addplot table[header=false,x index=0,y index=1]{gossip.dat};\n   \\addplot table[header=false,x index=0,y index=2]{gossip.dat};\n   \\legend{Average Receivers,Atomic Runs}\n  \\end{axis}\n \\end{tikzpicture}\n\\caption{Reliability of gossip (250 participants, 10 dissemination runs,\nvariable fanout).}\n\\label{fig:gosreliability}\n\\end{figure}\n\nMost interestingly, gossip protocols don't need a reactive mechanism to deal\nwith failures, namely, buffering, acknowledgement, retransmission, and garbage\ncollection, which account for most of the complexity in common communication\nprotocols. Instead, reliability is proactively achieved by the protocol's\ninherent redundancy and randomization, that cope with both process and network\nlink failures.\n\nThe expected probability for a message being delivered to each destination and\nto all destinations as a whole can be derived directly from protocol parameters\n$f$, the number of targets that are locally selected by each process for\ngossiping, and $r$, maximum number of times a message is relayed before being\nignored. Figure~\\ref{fig:gosreliability} illustrates the impact of these parameters\nby showing simulation results of disseminating 10 messages to 250 receivers,\nwith $r=5$ and a variable $f$. Notice that with $f>4$ each destination gets each\nmessage with a very high probability. With $f>7$, each message is atomically\nreceived by all destinations also with a very high probability.\n\nBy adjusting $r$ and $f$ parameters according to system size and expected\nfaults, gossip can be configured such that any desired average number of\nreceivers successfully get the message. Better yet, parameters can be set such\nthat the message is atomically delivered to all the receivers with high\nprobability leading to guaranteed atomic delivery\\,\\cite{1297243}. The key to\nscalability is that the required fanout configuration is at worst\nlogarithmically proportional to system size.\n\n\\subsubsection{Variants}\n\nThere are two main variants of gossip protocols\\,\\cite{892324}, which provide\ndifferent message exchange patterns and performance trade-offs. In \\emph{push\ngossip}, a node that becomes aware of new information, conveys it immediately to\ntarget nodes. This variant is adequate for one-to-many dissemination of small\nmessages and events. With \\emph{pull gossip}, instead of gossiping upon arrival\nof new information, a node periodically selects a number of peers and asks them\nfor new information. It has been shown that combining \\emph{push} and \\emph{pull\ngossip} results in dissemination being achieved in a lower number of\nsteps\\,\\cite{892324} and provides a generic framework for gossiping that can be\ntailored for multiple purposes by parameterizing it with different aggregation\nfunctions\\,\\cite{newscast03}.\n\nIn addition, lazily deferring the transmission of payload improves performance\nin heterogeneous networks, allowing gossip protocols to approximate ideal\nresource usage efficiency\\,\\cite{Pereira06}. Such \\emph{lazy} variants are most\nuseful when the data payload is very large, but also when it is very likely that\nthe data is already known throughout the network.\n\nFinally, there are two options regarding relaying duplicate messages. In the\n\\textit{infect-and-die} model, a participant that receives the message (i.e. is\ninfected), sends the received message to other nodes, and then never sends it\nagain, becoming dead in the analogy with epidemics. In the\n\\textit{infect{\\--}forever} model, also known as\n\\textit{balls-and-bins}\\,\\cite{koldehofe02simple}, a participant might relay\nreceived message multiple times, possibly until $r$ rounds are reached. This\nlast alternative has the advantage of requiring no state at participants to\nrecall recently relayed messages. On the other hand, it usually requires more\nnetwork resources as the relay limit has to be set conservatively.\n\n\\subsubsection{Membership Management}\n\nA key component of a gossip protocol is the ability to obtain random subsets of\nparticipants to direct messages at in each gossip operation. This component has\nto provide an uniform random sample and, as much as possible, drawn from a\ncurrent view of operational participants\\,\\cite{1045666}. The first option is to\nshare the full list of participants, allowing each of them to locally draw\nsubsets as desired\\,\\cite{312207}. This is adequate when the list does not\nchange frequently, to avoid taxing the network with constant updates, and is\nsmall enough to fit each participants memory.\n\nIf these conditions are not met, it has also been shown that sufficiently good\nrandom samples can be obtained by having each participant keep a small partial\nview of the system, which is itself maintained using a gossip\nprotocol\\,\\cite{945507,Eugster:2004p10747}. A particularly simple but effective\napproach\\,\\cite{Voulgaris:2005p6235} is allowing a node to exchange some\nelements in its local list with the same number of elements from some other\nnode. This progressively shuffles the list of each participant and leads to the\ndesired uniform random sample. By adding a time-based lease and renewal\nmechanism, it also deals with participants entering and leaving the system.\n\n\n\\subsection{Experimental setting}\n\nExperimental evaluation is done using the Minha middleware test platform\\,\\cite{Carvalho:2011:EED:2093185.2093188,minha_url}, which virtualizes multiple devices within a single JVM while simulating the performance characteristics of a real system.\nIt also allowed us to inject network faults to better assess the reliability of WS-Gossip.\n\n\nEach test corresponds to the simulation of the runtime of a given number of devices collocated in the same LAN, in a single host with the following configuration: 64-bit Ubuntu Server 10.04.4 Linux, two 12-core AMD Opteron\\textsuperscript{TM} Processor 6172, 2.1GHz, 128 GB RAM, 64-bit Sun Microsystems Java SE 1.6.0\\_26.\n\nThe evaluation consists in executing a periodic event dissemination, for the mentioned scenarios, where a new value is propagated from a single producer device to a given number of consumer devices.\nA centralized managing device was used to control peer management\\footnote{Discovery proxy is not yet implemented in JMEDS 2.0 beta 3a. Instead we used a custom registry service.} and the execution of the test.\n\nThe following scenarios were analyzed:\n\\begin{description}\n\\item [WS-Eventing]A publish\/subscribe communication protocol was selected as it is one of the most used event dissemination patterns. Hence, the WS-Eventing standard, as provided by JMEDS, was evaluated using HTTP\/TCP communication.\n\\item [WS-Gossip]The \\textit{push} variant of WS-Gossip was selected to be evaluated in conjunction with SOAP-over-UDP.\nTwo different scenarios were evaluated for WS-Gossip in terms of communication errors to compare the achieved reliability and latency degradation.\nEach of these scenarios designation is then suffixed with (0\\% Loss), when there are no message losses, and with (10\\% Loss), when 10\\% of communication losses are introduced by the Minha simulator.\n\\end{description}\n\nThe execution procedure of each test comprised the following steps:\n\\begin{enumerate}\n \\item The manager and the producer devices are started.\n \\item The consumer devices are then started. In WS-Eventing,\nthey subscribe with the producer as soon as they are started. In WS-Gossip, the manager, informs each consumer of its neighbors according with the configured fanout value, so they can convey new messages to them.\nFor  both scenarios, the manager verifies if all the devices started correctly before signaling the producer to start the dissemination.\n \\item The producer begins disseminating events periodically, which are propagated across the network.\n \\item The producer terminates and notifies the manager.\n \\item The manager informs, sequentially, all the devices about the file they should write their run statistics to.\n\\end{enumerate}\nThe tests for each scenario consisted in 5 runs for each given number of devices, where 120 events were periodically emitted with an interval of 5 seconds.\n\nThe interval between the initial emission of a message and its reception by a consumer was measured in nanoseconds since Minha enables the execution of all the intervening devices inside a single JVM on a single host.\nThe sampling of the instant of emission was performed right before the producer sends a message, and the reception time measurement was done in the first operation of the method invoked to deal with a new message at a consumer.\n\nIn WS-Gossip, the used values for the fanout parameter were computed according to\\,\\cite{amk-from-epidemics-to-dc}, taking into account the number of devices, as well as an expected error rate (e) of 5\\% and a delivery assurance (p) of 99\\%, ranging from a value of 8 for 10 devices to 11 for 250 devices.\nIn these very same scenarios, the publisher is randomly selected from all the nodes, contrarily to the WS-Eventing scenario where the publisher is the first device.\n\n\\subsection{Results and discussion}\n\n\\begin{figure}[htbp]\n \\centering\n \\begin{tikzpicture}\n  \\begin{axis}[xlabel=devices,ylabel=ms,ymax=130, legend columns=1, legend style={\n \tat={(1.03,0.48)},\n\tanchor=west}]\n   \\addplot table[header=false,x index=0,y index=1]{pushudp-0.dat};\n   \\addplot table[header=false,x index=0,y index=1]{pushudp-10.dat};\n   \\addplot table[header=false,x index=0,y index=1]{notif.dat};\n   \\legend{WS-Gossip(0\\% Loss),WS-Gossip(10\\% Loss),WS-Eventing}\n  \\end{axis}\n \\end{tikzpicture}\n \\caption{WS-Eventing vs. WS-Gossip (latency).}\n \\label{fig:wse_wsg}\n \\end{figure}\n \n\\begin{figure}[htbp]\n \\centering\n \\begin{tikzpicture}\n  \\begin{axis}[xlabel=devices,ylabel=hops,ymin=0, legend columns=1, legend style={\n \tat={(1.03,0.48)},\n\tanchor=west}]\n   \\addplot table[header=false,x index=0,y index=3]{pushudp-0.dat};\n   \\addplot table[header=false,x index=0,y index=3]{pushudp-10.dat};\n   \\legend{WS-Gossip(0\\% Loss),WS-Gossip(10\\% Loss)}\n  \\end{axis}\n \\end{tikzpicture}\n \\caption{Average hops to delivery in WS-Gossip.}\n \\label{fig:hops}\n\\end{figure}\n\nResults presented in Figures~\\ref{fig:wse_wsg} and \\ref{fig:hops} are the average of all 5 runs for each scenario.\nFor latency measurements, the first and the last 10 iterations were discarded in order to minimize the effect of Java JIT compilation, although it also masks the delay of TCP connection establishment in WS-Eventing.\n \nIn Figure\\,\\ref{fig:wse_wsg}, the message delivery latency of the WS-Eventing grows linearly with the number of targets, from 10 to 126 milliseconds, whereas that of WS-Gossip(0\\% Loss) is very small and grows very slowly, between 2.8 to 10.5 milliseconds.\nThis can be justified by scattering the load of propagating a message throughout an entire network, by the devices on that network, instead of overloading a single device, such as the publisher in WS-Eventing.\nThe message delivery latency of WS-Gossip(10\\% Loss) is very close to that of WS-Gossip(0\\% Loss), suffering a small increase of around 0.1 milliseconds.\n \nFigure\\,\\ref{fig:hops} presents the logarithmic growth of the average number of hops a message goes through from emission to reception in WS-Gossip, between 1.2 to 2.64 hops, confirming that the gossip protocol scales logarithmically with system size. This figure also shows that the introduction of communication losses has little effect on the number of hops a message goes through, with an average increase of around 0.6 hops in WS-Gossip(10\\% Loss) compared to the baseline scenario, where no messages are lost in the network.\n \nMessage delivery rate is not presented graphically since it is 100\\% both in WS-Eventing and in WS-Gossip(0\\% Loss) and it is always greater than 99.9\\% in WS-Gossip(10\\% Loss), and most frequently 100\\%, even with a rate of communication losses that corresponds to the double of the expected 5\\%.\n\nTo conclude the analysis of the results, considering an environment with \\textit{n} devices, where the WS-Eventing producer will always have to send \\textit{n} messages for each event, whereas gossip peers will send a number of messages equal to its fanout \\textit{f}, thus spreading the load throughout the network which results in savings in the consumption of resources by the producer for cases where \\textit{n} $ > $ \\textit{f}.\n\n\n\n\n\\subsection{Header information}\n\nAs previously stated, the unit of information being gossiped is the SOAP envelope. Messages in a gossip interaction contain an entry in the SOAP header section of the SOAP envelope describing how to relay such messages. These are initialized by the initiator device, either within a shadow service or by a gossip-aware client. Moreover, there is also the assumption of WS-Addressing\\,\\cite{ws_a} providing a unique identifier for each message and support for asynchronous replies. Briefly, it contains the following information:\n\\begin{description}\n\\item [Scope\/Type]As defined by WS-Discovery, this field implicitly describes the set of targets. Devices can be configured to relay messages only within a specific scope and type.\n\\item [Fanout]The number of peers to target in each interaction.\n\\item [Hops]The remaining number of hops. This must be decremented by each device that relays the message. The message is discarded when it reaches zero.\n\\item [IdTTL]The time that each device should buffer the message identifier for duplicate detection. If this is set to zero, the protocol degenerates to the \\textit{balls-and-bins} variant\\,\\cite{ballsandbins}.\n\\item [DataTTL]The time that each device should buffer the message itself for retransmission in lazy gossip variants. If this is set to zero, the protocol will never issue advertisements and will always use an \\textit{eager} variant.\n\\item [Filter]An optional item, specifying a rule to filter replies, which must be specified using XSLT. Valid rules are configured by the deployer and advertised as policies by the shadow service.\n\\end{description}\n\n\\subsection{Operation styles}\n\nSOAP and WSDL support several operation styles\\,\\cite{soap,wsdl_11,wsdl_12}. Besides a typical client-server interaction (i.e. \\textit{request-response}), it is also possible to have input-only operations (i.e. \\textit{one-way}), output-only operations (i.e. \\textit{notification}), and call-back operations (i.e. \\textit{solicit-response}). It is also possible that a \\textit{two-way} operation leads to multiple replies. These different operation styles allow WS-Gossip to support different gossip variants in addition to the previously described eager \\textit{push-style}, such as the \\textit{lazy} and the \\textit{pull} variants.\n\nGossiping in \\textit{one-way} and \\textit{notification} operations is handled as described previously: Upon reception of a message, it is propagated and no reply is expected. In \\textit{request-reply} and \\textit{solicit-response} operation, the message is propagated and then all replies received are propagated back to the initiator. This requires the initiator's address to be stored alongside with the message identifier used for duplicate detection during the specified \\textbf{IdTTL}. Consider the following example: A \\textit{request-response} to query available disk space of servers in a data center. A client invokes the operation on the shadow service, which eventually reaches all targets. All responses then travel back along the same tree implicitly created by the request message and will eventually reach the initiator.\n\nAn alternative is to make use of a filter. This can omit or aggregate replies according to a rule specified when gossip is initiated. \nConsider the following example: The same \\textit{request-response} operation is used to determine which server has the most available disk space in a data center. This requires that upon deployment, devices are configured to support the maximum filter on the disk space query operation. A client invokes the operation on the shadow service, which eventually reaches all targets. Responses then travel back along the same tree implicitly created by request message, but they are buffered and filtered such that only the maximum discovered downstream is returned by each peer. Each peer's reply is sent as soon as all its targets have replied, with a value or with a fault, or when a timeout expires.\n\n\\subsection{Gossip styles}\n\nIn addition to eager \\textit{push-style} gossip described so far, \\textit{lazy} and \\textit{pull} variants are supported as follows. Besides offering the same port type as the hosted service, the shadow gossip service provides a gossip port with the following operations:\n\\begin{description}\n\\item [Push]Alternative to directly using the interface. This allows a set of messages to be submitted in a single interaction.\n\\item [PushIds]Informs the target that a number of messages are locally available. These should then be requested using \\textbf{Fetch}.\n\\item [Pull]Returns currently buffered messages during a time interval specified as a parameter.\n\\item [PullIds]Variant of the previous operation, which requests identifiers instead of the actual messages. These can then be requested using \\textbf{Fetch}.\n\\item [Fetch]Returns currently buffered messages, as specified by a list of identifiers provided as a parameter.\n\\end{description}\nGossip variants can be achieved through the composition of the previous operations. Namely, \\textit{lazy push} is obtained by using \\textbf{PushIds} instead of \\textbf{Push} and then waiting for \\textbf{Fetch} to be used later on selected identifiers. \\textit{Eager pull} is obtained by periodically invoking \\textbf{Pull}. Finally, \\textit{lazy pull} is obtained by periodically invoking \\textbf{PullIds} and then using \\textbf{Fetch} on the resulting identifiers that are unknown.\n\nThe gossip variant chosen for each operation depends on configuration by the service deployer. In particular, the optimum configuration for \\textit{push} gossip is to use the \\textit{eager} variant for early rounds and then \\textit{lazy}. For \\textit{pull} gossip, the \\textit{lazy} variant is interesting for very large payloads. The combination of both \\textit{push} and \\textit{pull} is known to ensure rapid and robust dissemination of information\\,\\cite{892324,312207}.\n\n\\subsection{Peer service}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.75\\textwidth]{peers-crop}\n\\caption{Overview of peer management.}\n\\label{fig:memb}\n\\end{figure}\n\nBy default, WS-Gossip does not need an explicit peer management service. Instead, each gossip interaction can be configured with a scope or a service type that is then used to discover the full set of reachable peers through WS-Discovery. This is most useful in scenarios where a discovery proxy device exists, since a set of peers can be obtained efficiently by querying the proxy. This leads to a configuration with centralized peer information while information dissemination is distributed, which is adequate for scenarios with low churn and relatively high messaging rate.\n\nIf a proxy is not available, the usage of the Ad-Hoc mode of WS-Discovery would lead to a large number of multicast messages that would most likely defeat the purpose of gossip. Instead, our proposal allows that peers discovered to be cached locally and exchanged with other peers to implicitly create an overlay network using the Newscast protocol\\,\\cite{newscast03}.\nThe structure of the stored peer information comprises a list where each service instance is represented by an entry that contains the following elements:\n\\begin{description}\n\\item [Address]Corresponds to the service endpoint address.\n\\item [Type]Identifies the type of the service.\n\\item [DeviceId]Identifies the device where the service is hosted. This field is not applicable to services that are not associated with any device.\n\\item [Heartbeat]Counter that is incremented as messages, such as the invocation of the \\textbf{Exchange} operation, are issued by other peer.\n\\end{description}\nThis information is exchanged among different devices and also updated through the examination of WS-Dis\\-cov\\-er\\-y multicast messages issued by target services entering or leaving the network.\nPeriodically, if the instance has not received a request for exchanging its membership information during a certain time frame, it selects another instance of the peer service to which it sends such a request containing the list of the known endpoints. Upon reception of such a message, the contacted instance returns to the requester its own list of known endpoints, and merges it with the received one.\n\nThe heartbeat counter of a service instance that never sends a new message, or eventually sends but without reaching a peer service instance, stays unchanged, implying that it will move towards the end of the membership list as the counter of other services is being updated and new services are discovered. That service instance will eventually be discarded when the cache of the Peer Service reaches the configured maximum size.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:introduction}\n\nIndependence in graph products has been studied by many authors but almost always in the context of the independence number, commonly denoted by $\\alpha$.\nWe mention just samples  of papers concerning the independence number of a Cartesian product $\\alpha(G \\,\\square\\, H)$ (see~\\cite{f-2011, hk-1996, js-1994, k-2005, nr-1996})\nand  of a direct product $\\alpha(G \\times H)$ (see~\\cite{jk-1998, kr-2022, nr-1996}).  In addition, for both of these two products some investigation has also been done\non the so-called ultimate independence ratios,  $\\lim_{m \\to \\infty}\\frac{\\alpha(\\,\\square\\,_{i=1}^m G)}{n(G)^m}$ and $\\lim_{m \\to \\infty}\\frac{\\alpha(\\times_{i=1}^m G)}{n(G)^m}$.\nSee for example~\\cite{al-2007, bnr-1996, hyz-1994, t-2014}.\n\n\nNowakowski and Rall~\\cite{nr-1996} studied the behavior of a number of domination, independence and coloring type invariants on nine associative graph products whose edge sets depend on the edge sets of both factors.  In particular, they proved some lower and upper bounds for the\ncardinality of a smallest maximal independent set, the independent domination number, of these products.  For an excellent survey of independent domination\nsee the paper~\\cite{gh-2013} by Goddard and Henning.\nIn this work we will focus on the independent domination number of the direct product of two graphs.  In particular, we are interested\nin how the independent domination number of a direct product relates to the independent domination numbers of the two factors.\nIn the process we give a counterexample to the following conjecture of Nowakowski and Rall.\n\\begin{conj} {\\rm \\cite[Section 2.4]{nr-1996}}  \\label{conj:lowerbound}\nFor all graphs $G$ and $H$, $i(G \\times H) \\ge i(G)i(H)$.\n\\end{conj}\nIn fact, we prove a stronger result; namely\n\\begin{thm} \\label{thm:extremecounterexample}\nFor any positive integer $n$ such that $n>10$, there exists a pair of graphs $G$ and $H$ such that\n$\\min\\{i(G),i(H)\\}=n+2$ and $i(G \\times H) \\leq 12$.\n\\end{thm}\n\nThe organization of the paper is as follows.  In the next section we provide necessary definitions and several previous results.\nIn Section~\\ref{sec:productwithcomplete} we restrict our attention to direct products in which one of the factors is a complete\ngraph, and introduce a method for calculating the independent domination number of $G \\times K_n$ in terms of minimizing\na certain kind of labelling of $V(G)$.  Using this scheme we find the values of $i(P_m \\times K_n)$ and $i(C_m \\times K_n)$.\nLower bounds for $i(G \\times H)$, in terms of other domination-type invariants of $G$ and $H$, are given in Section~\\ref{sec:lowerbounds}.\nThe main result of the paper is in Section~\\ref{sec:counterexamples} where we give an infinite collection of counterexamples to\nConjecture~\\ref{conj:lowerbound} and prove Theorem~\\ref{thm:extremecounterexample}.\n\n\n\\section{Definitions and preliminary results} \\label{sec:defns}\n\nWe denote the order of a finite graph $G=(V(G),E(G))$ by $n(G)$.  For a positive integer $n$ we let $[n]=\\{1,\\ldots,n\\}$; the vertex set of the complete graph $K_n$\nwill be $[n]$ throughout.  A subset $D\\subseteq V(G)$ \\emph{dominates} a subset $S \\subseteq V(G)$ if $S \\subseteq N[D]$.  If $D$ dominates~$V(G)$, then we will also say that $D$ dominates the graph $G$ and that $D$ is a \\emph{dominating set} of $G$.  If $D$, in addition to being a dominating set\nof $G$, has the property that every vertex in $D$ is adjacent to at least one other vertex of $D$, then $D$ is a \\emph{total dominating set} of $G$. The\n\\emph{total domination number} of $G$ is the minimum cardinality among all total dominating sets of $G$; it is denoted $\\gamma_t(G)$.  The \\emph{$2$-packing number} of $G$, denoted $\\rho(G)$, is the largest cardinality of a vertex subset $A$ such that the distance in $G$ between $a_1$ and $a_2$ is at least $3$ for every pair $a_1,a_2$\nof distinct vertices in $A$.\nA set $I \\subseteq V(G)$ is an \\emph{independent dominating} set if $I$ is simultaneously independent and dominating.  This is equivalent to $I$ being a maximal independent set with respect to set inclusion.  The \\emph{independence number} of $G$ is the cardinality, $\\alpha(G)$, of a largest independent set in $G$.  We denote by $i(G)$ the smallest cardinality of a maximal independent set in $G$; this invariant is called the \\emph{independent domination number} of $G$.\n\nThe \\emph{direct product}, $G\\times H$, of graphs $G$ and $H$ is defined as follows:\n\\begin{itemize}\n\\item $V(G \\times H)=V(G) \\times V(H)$;\n\\item $E(G \\times H)= \\{ (g_1,h_1)(g_2,h_2) \\, \\colon\\, g_1g_2 \\in E(G) \\,\\,\\text{and}\\,\\,h_1h_2 \\in E(H) \\}$\n\\end{itemize}\nThe direct product is both commutative and associative.   For a vertex $g$ of $G$, the \\emph{$H$-layer over $g$} of $G\\times H$ is the set $\\{ \\, (g,h) \\mid h\\in V(H) \\,\\}$, and it is denoted by $\\LSs g H$.  Similarly, for $h \\in V(H)$, the \\emph{$G$-layer over $h$}, $G^h$, is the set $\\{ \\, (g,h) \\mid g\\in V(G) \\,\\}$.  Note that each $G$-layer and each $H$-layer is an independent set in $G\\times H$.  The \\emph{projection to $G$} is the map $p_G: V(G\\times H) \\to V(G)$ defined by $p_G(g,h)=g$.  Similarly, the \\emph{projection to $H$} is the map $p_H: V(G\\times H) \\to V(H)$ defined by $p_H(g,h)=h$.  If $A \\subseteq V(G\\times H)$ and $g \\in V(G)$, then we employ\n$\\LSs g A$ to denote $A \\cap\\, \\LSs g H$.  Similarly, $A^h=A \\cap G^h$ for a vertex $h$ of $H$.\n\nThe following result of Topp and Volkmann will be useful in establishing our main results.\n\\begin{lem} {\\rm \\cite[Proposition 11]{tv-1992}} \\label{lem:inverseimage}\nLet $H$ be a graph with no isolates.  If $I$ is a maximal independent set of any graph $G$, then $I \\times V(H)$ is a maximal independent set of $G \\times H$.\n\\end{lem}\n\n\\medskip\nAs an immediate consequence of Lemma~\\ref{lem:inverseimage} we get a lower bound for $\\alpha(G \\times H)$, which is well-known, and an upper bound for $i(G \\times H)$.\nBoth were established earlier by Nowakowski and Rall~\\cite{nr-1996}.\n\\begin{cor} {\\rm \\cite[Table 3]{nr-1996}}            \\label{cor:triviallower}\nIf both $G$ and $H$ have no isolated vertices, then\n\\begin{itemize}\n\\item $\\alpha(G \\times H) \\ge \\max \\{\\alpha(G) n(H), \\alpha(H) n(G)\\}$;\n\\item $i(G \\times H) \\le \\min \\{i(G) n(H), i(H) n(G)\\}$.\n\\end{itemize}\n\\end{cor}\n\n\\section{Independent domination in $G \\times K_n$} \\label{sec:productwithcomplete}\n\nIn this section we focus on direct products in which one of the factors is a complete graph, and we will use notation introduced in our paper~\\cite{kr-2022}.\n\nLet $I$ be a maximal independent set of $G \\times H$.  Suppose $g$ is a vertex of $G$ such that $\\LSs{g}{I} \\not=\\emptyset$ but  $\\LSs{g}{I} \\not= {\\LSs{g}{H}}$.  Let $(g,h)\\in {\\LSs{g}{H}- \\,\\LSs{g}{I}}$.  Since $I$ is a dominating set of $G \\times H$, it follows that there exists $g' \\in N_G(g)$ and $h'\\in N_H(h)$ such that $(g',h') \\in I$.  Note that  such a vertex $h'$ does not belong to $N_H(p_H(\\LSs{g}{I}))$.  For if $h'x \\in E(H)$ for some $(g,x) \\in I$, then $(g',h')$ and $(g,x)$ are adjacent vertices of $I$, which\nis a contradiction.  However, it is possible that $h' \\in p_H(\\LSs{g}{I})$.\n\nConsider now the special case $G \\times K_n$ for $n \\ge 2$.  The following lemma is from~\\cite{kr-2022}.  For the sake of completeness we give its short proof.\n\n\\begin{lem} {\\rm \\cite[Lemma 9]{kr-2022}}\\label{lem:sizeoflayers}\nLet $n \\ge 2$ and let $G$ be any graph.  If $I$ is any maximal independent set of $G \\times K_n$, then $\\left| I \\cap {\\LSs{g}{K_n}} \\right | \\in \\{0,1,n\\}$, for any $g \\in V(G)$.\n\\end{lem}\n\\begin{proof} If $n=2$, then the conclusion is obvious.  Assume $n\\ge 3$ and suppose for the sake of contradiction that $\\left| I \\cap {\\LSs{g}{K_n}} \\right |=m$ for some $2 \\le m <n$.  Assume without loss of generality that $\\{(g,1),(g,2)\\} \\subset I$.  Let $i \\in [n]$ such that $(g,i) \\not\\in I$. As above, there exists $g' \\in N_G(g)$ and $j \\in N_{K_n}(i)$ such that $(g',j)\\in I$.  Since $n \\ge 3$, we infer that $j \\not=1$ or $j \\not =2$.  This implies that $(g',j) \\in N(\\{(g,1),(g,2)\\})$, which contradicts the independence of $I$.  Therefore,\n$\\left| I \\cap {\\LSs{g}{K_n}} \\right | \\in \\{0,1,n\\}$.\n\\end{proof}\n\nThe following result gives tight upper and lower bounds for $i(G \\times K_2)$.\n\\begin{thm} \\label{thm:lowerandupper2}\nIf $G$ is any graph with no isolated vertices, then\n\\[\\gamma_t(G) \\le i(G \\times K_2) \\le \\min\\{2i(G),n(G)\\}\\,.\\]\n\\end{thm}\n\\begin{proof}  The upper bound follows from Corollary~\\ref{cor:triviallower}.  Let $M$ be an independent dominating set of $G \\times K_2$ such that $i(G \\times K_2)=|M|$.  If $M^1$ or $M^2$ is empty, say $M^2=\\emptyset$, then $(g,1) \\in M$ for every $g\\in V(G)$.  Hence, $|M|=n(G)\\ge \\gamma_t(G)$.  Thus, assume that $M^1\\not= \\emptyset$ and $M^2\\not=\\emptyset$.  For each $(g,2) \\in M$, choose $g' \\in N_G(g)$.  Let $\\widehat{M}= M^1 \\cup \\{(g',1)\\,\\colon\\,(g,2) \\in M\\}$.  It is clear that $\\widehat{M}$ dominates $G^2$.  For if $(g,2) \\in M$, then $(g',1) \\in \\widehat{M}$ and $(g',1)$ is a neighbor of $(g,2)$.  If $(g,2) \\not\\in M$, then $M^1$ contains a neighbor of $(g,2)$ since $M$ is a dominating set of $G \\times K_2$.  Consequently, $p_G(\\widehat{M})$ is a total dominating set of $G$, and we get\n\\[\\gamma_t(G) \\le |p_G(\\widehat{M})| \\leq |\\widehat{M}| \\le |M|=i(G \\times K_2)\\,.\\]\n\\end{proof}\n\nAny graph $G$ that has a vertex of degree $n(G)-1$ shows that the lower bound in Theorem~\\ref{thm:lowerandupper2} is tight.  For the upper bound let $G=K_{n,n}$.  Since $G \\times K_2 =2G$, we see that the upper bound is also tight.\n\n\nLet $I$ be any maximal independent set of $G \\times K_n$ and let $n\\ge 2$ be a positive integer.  As in~\\cite{kr-2022} we use Lemma~\\ref{lem:sizeoflayers} to define a weak partition of $V(G)$.  We will say this weak partition is \\emph{generated by} or \\emph{corresponds to} $I$.\n(A weak partition of a set $X$ is a collection of pairwise disjoint subsets of $X$, in which some may be empty, whose union is $X$.)  In particular, $V_0,V_1,\\ldots,V_n,V_{[n]}$ defined by\n\\begin{enumerate}\n\\item[{\\bf(a)}] $V_0=\\{ g \\in V(G)\\,\\colon\\, I \\cap {\\LSs{g}{K_n}} = \\emptyset\\}$;\n\\item[{\\bf(b)}] For each $k \\in [n]$, $V_k=\\{ g \\in V(G)\\,\\colon\\, I \\cap {\\LSs{g}{K_n}} =\\{(g,k)\\} \\}$;\n\\item[{\\bf(c)}] $V_{[n]}=\\{ g \\in V(G)\\,\\colon\\, I \\cap {\\LSs{g}{K_n}}={\\LSs{g}{K_n}} \\}$.\n\\end{enumerate}\nis a weak partition.  Furthermore, the following four conditions hold.\n\\begin{enumerate}\n\\item For $k \\in [n]$, if $u \\in V_k$ and $v \\in V(G)-(V_0 \\cup V_k)$, then $uv \\not\\in E(G)$.\n\\item For $k \\in [n]$, if $V_k$ is not empty, then no vertex of $V_k$ is isolated in $G[V_k]$.\n\\item The set $V_{[n]}$ is independent in $G$.\n\\item For each $g \\in V_0$, either $N_G(g) \\cap V_{[n]} \\not=\\emptyset$ or $g$ has a neighbor in at least two of the sets $V_1,\\ldots,V_n$.\n\\end{enumerate}\n\nConversely, for a given weak partition of $V(G)$ that satisfies these four conditions, it is clear how to construct a maximal independent set $D$ of $G \\times K_n$.\nWe then say this weak partition \\emph{constructs} $D$.  The independent domination number of $G \\times K_n$ can be computed in the following way.\n\\begin{equation} \\label{eqn:indepdomlabel}\ni(G \\times K_n)= \\min\\{n\\cdot |V_{[n]}|+ \\sum_{k=1}^n|V_k|\\}\\,,\n\\end{equation}\nwhere the minimum is computed over all weak partitions $V_0,V_1,\\ldots,V_n,V_{[n]}$ that satisfy conditions $1{-}4$ above.\n\nWe used a computer program to compute the independent domination numbers of the direct product of small paths and small cycles with complete graphs.  In Table~\\ref{Tab:Special} we accumulate some of these values.  Because of the smallest independent dominating sets produced by our software, it is clear that these same values hold if $K_3$ is replaced by $K_n$ in the direct product for any $n \\ge 4$.\n\n\\begin{table}[ht!]\n\\begin{center}\n\\begin{tabular} { |c | c c c c c c c c c c | }\n\\hline\n$m$  &3&4&5&6&7&8&9&10&11&12 \\\\ \\hline\n$i(P_m \\times K_3)$ &3&4&4&5&6&6&7&8&8&9 \\\\ \\hline\n$i(C_m \\times K_3)$ &3&4&5&4&5&6&6&7&8&8 \\\\ \\hline\n\\end{tabular}\n\\caption{Some independent domination numbers}\\label{Tab:Special}\n\\end{center}\n\\end{table}\n\n\nInstead of specifically listing the sets $V_0,V_1,\\ldots,V_n,V_{[n]}$ we can instead represent this weak partition by a labelling of the vertices of $G$.\nA vertex $x$ of $G$ is labelled with the symbol $\\ast \\in \\{0,1,\\ldots,n,[n]\\}$ if and only if $x \\in V_{\\ast}$.  (Note that we are allowing labels\nto  be used more than once.) For any weak partition\n$V_0,V_1,\\ldots,V_n,V_{[n]}$ (equivalently any labelling) of $V(G)$ that satisfies the four conditions, we say it is  \\emph{legal} and has\n\\emph{weight} $n\\cdot |V_{[n]}|+ \\sum_{k=1}^n|V_k|$.  For the purposes of this paper, we\ncall any weak partition (equivalently, any labelling) of $V(G)$ \\emph{optimum} if it attains the minimum weight in~\\eqref{eqn:indepdomlabel} above.\nWe illustrate this labelling in Figure~\\ref{fig:labeling}\non a cycle of order $16$ that defines a maximal independent set of the direct product $C_{16} \\times K_3$.  Note that $16=6r+p$ for $r=2$ and $p=4$.\nThis labelling is part of the pattern denoted by $(1,1,0,2,2,0)^2(3,3,3,0)$, which means label the vertices of $C_{16}$ in consecutive order by\nrepeating the sequence of six  labels $1,1,0,2,2,0$ two ($r=2$) times followed by the sequence of four ($p=4$) labels $3,3,3,0$ one time.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.45,style=thick]\n\\def\\vr{5pt}\n\\path (-16,0) coordinate (0); \\path (-14,0) coordinate (1);\n\\path (-12,0) coordinate (2);  \\path (-10,0) coordinate (3);\n\\path (-8,0) coordinate (4);  \\path (-6,0) coordinate (5);\n\\path (-4,0) coordinate (6);  \\path (-2,0) coordinate (7);\n\\path (0,0) coordinate (8);  \\path (2,0) coordinate (9);\n\\path (4,0) coordinate (10);  \\path (6,0) coordinate (11);\n\\path (8,0) coordinate (12);  \\path (10,0) coordinate (13);\n\\path (12,0) coordinate (14);  \\path (14,0) coordinate (15);\n\\draw (0) .. controls (-1,4) .. (15);\n\\draw (0) -- (1); \\draw (1) -- (2);  \\draw (2) -- (3);  \\draw (3) -- (4);  \\draw (4) -- (5);  \\draw (5) -- (6);\n \\draw (6) -- (7);  \\draw (7) -- (8);  \\draw (8) -- (9);  \\draw (9) -- (10);  \\draw (10) -- (11);  \\draw (11) -- (12);\n \\draw (12) -- (13);  \\draw (13) -- (14);  \\draw (14) -- (15);\n\n\n\\foreach \\i in {0,...,15}\n{  \\draw (\\i)  [fill=white] circle (\\vr); }\n\\foreach \\i in {2,5,8,11,15}\n\\draw[anchor = north] (\\i) node {$0$};\n\\foreach \\i in {0,1,6,7}\n\\draw[anchor = north] (\\i) node {$1$};\n\\foreach \\i in {3,4,9,10}\n\\draw[anchor = north] (\\i) node {$2$};\n\\foreach \\i in {12,13,14}\n\\draw[anchor = north] (\\i) node {$3$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Labeling of $C_{16}$}\n\\label{fig:labeling}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\begin{prop} \\label{prop:pathscycles}\nLet $m$ and $n$ be positive integers with $m \\ge 3$ and $n \\ge 3$.  Then,\n\\begin{itemize}\n\\item[{\\rm(a)}] $i(P_m \\times K_2)=2\\lceil \\frac{m}{3} \\rceil$.\n\\item [{\\rm(b)}] $i(C_m \\times K_2)= \\lceil \\frac{2m}{3} \\rceil$ if $m$ is odd, and $i(C_m \\times K_2)= 2\\lceil \\frac{m}{3} \\rceil$ if $m$ is odd.\n\\item [{\\rm(c)}] $i(C_m \\times K_n)=m$ for $3 \\le m \\le 5$, and $i(C_m \\times K_n)=\\left\\lceil \\frac{2m}{3} \\right \\rceil$, for every $m \\ge 6$.\n\\item [{\\rm(d)}] $i(P_m \\times K_n)=\\left\\lceil \\frac{2m+2}{3} \\right \\rceil$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof} Note that $C_m \\times K_2=C_{2m}$ if $m$ is odd, and $C_m \\times K_2=2C_{m}$ if $m$ is even.  Also, for any $m$,\n$P_m \\times K_2=2P_m$.   Statements (a) and (b) now follow from $i(C_k)=i(P_k)=\\lceil k\/3 \\rceil$.\n\nNow consider $C_m \\times K_n$ for some $n \\ge 3$.  It is easy to check that the vertex labellings $(1,1,1)$, $(1,1,1,1)$ and\n$(1,1,1,1,1)$  of $C_3$, $C_4$ and $C_5$ respectively are optimum for the direct products $C_m \\times K_n$ for $3 \\le m \\le 5$.\nNow, let $m \\ge 6$.  Table~\\ref{Tab:Largerm} presents labelling patterns of $V(C_m)$ and $V(P_m)$, based on the congruence of $m$ modulo $6$,\nthat establish the upper bound of $\\lceil 2m\/3 \\rceil$ for $i(C_m \\times K_n)$ and of $\\left\\lceil \\frac{2m+2}{3} \\right \\rceil$ for $i(P_m \\times K_n)$.\n\nWe may assume that in any optimum labelling of $V(C_m)$ no vertex receives the label $[n]$.  For, suppose some vertex $x$ is labelled $[n]$. By conditions $1$ and $3$\nboth of the neighbors of $x$ are labelled $0$.  For the vertices of $C_m$ within distance $2$ of $x$, the labelling sequence\n$(i,0,[n],0,i)$ can be replaced with $(i,i,0,i,i)$  and the sequence  $(i,0,[n],0,j)$, for $i \\neq j$, can be replaced with $(i,i,0,j,j)$.\nThis in turn implies that at most one vertex of any  three consecutive\nvertices of $C_m$ can be labelled $0$.  That is, at most $\\lfloor m\/3 \\rfloor$ vertices can be labelled $0$.  Since\n$m=\\lfloor m\/3\\rfloor +\\lceil 2m\/3 \\rceil$, we get $i(C_m \\times K_n) \\ge \\lceil 2m\/3 \\rceil$.  This establishes statement (c).\n\nWe use essentially the same reasoning to prove the lower bound for $i(P_m \\times K_n)$ with one small difference, this being that the path\nhas two vertices of degree $1$.  This forces several additional cases.  Suppose the path $P_m$ is $v_1,v_2,\\ldots,v_{m-1},v_m$.\nWe claim that without loss of generality we may assume that any\noptimum labelling of $V(P_m)$ does not use the label $[n]$.  We modify any labelling that has such a vertex $x$ with label $[n]$ in such a way\nthat the weight is not increased.  Note that every neighbor of $x$ is labelled $0$.  Suppose first that $x$ and the vertices within distance $2$\nof $x$ all have degree $2$. As above,  the labelling sequence $(i,0,[n],0,i)$ can be replaced with $(i,i,0,i,i)$  and the sequence\n$(i,0,[n],0,j)$, for $i \\neq j$, can be replaced with $(i,i,0,j,j)$.  The labelling sequence $([n],0,[n],0,i)$ can be replaced with $([n],0,i,i,i)$;\nsimilarly $(i,0,[n],0,[n])$ can be replaced with $(i,i,i,0,[n])$.  Now suppose that $x=v_2$.\nThe sequence $(0,[n],0,i,i)$ can be replaced with $(i,i,i,i,i)$, the sequence $(0,[n],0,[n],0)$ can be replaced with\n$(i,i,0,[n],0)$, and $(0,[n],0,0,[n])$ can be replaced with $(i,i,i,0,[n])$.  The case $x=v_{m-1}$ is handled in a similar fashion.\nFinally, if $0$ and $[n]$ are the only labels used, then the only case to consider is when one or both of the end vertices is labelled $[n]$. Suppose\n$v_1$ is labelled $[n]$.  The labelling sequence $([n],0,[n],0,[n])$ can be replaced with $(i,i,i,0,[n])$,  the sequence $([n],0,[n],0,0)$\ncan be replaced with $(i,i,i,i,0)$, and $([n],0,0,[n],0)$ can be replaced with $(i,i,0,[n],0)$.  The case where $v_m$ is labelled $[n]$ is handled\nsymmetrically.  Thus, we may assume that no optimum labelling of $V(P_m)$ uses the label $[n]$.\n\nTherefore, any optimum labelling of $V(P_m)$ has weight $\\sum_{k=1}^n|V_k|$ since $V_{[n]}=\\emptyset$.  By considering the three cases of $m$ modulo\n$3$ and using conditions $1,2$ and $4$, it is now easy to see that at most $\\left \\lfloor \\frac{m-2}{3} \\right \\rfloor$ vertices of $P_m$ can be labelled $0$.\nSince $m=  \\left \\lfloor \\frac{m-2}{3} \\right \\rfloor + \\left \\lceil \\frac{2m+2}{3} \\right \\rceil$, it follows that any legal\nlabelling of $V(P_m)$ has weight at least $\\left \\lceil \\frac{2m+2}{3} \\right \\rceil$.  This lower bound coincides with the values\ngiven in Table~\\ref{Tab:Largerm}, which finishes the proof.\n\\end{proof}\n\\begin{table}\n\\begin{center}\n\\begin{tabular} { |c | c | c |}\n\\hline\n$m=$ & labelling pattern of $C_m$ & labelling pattern of $P_m$  \\\\ \\hline\n$6r$ & $(1,1,0,2,2,0)^r$ & $(1,1,0,2,2,0)^{r-1}(1,1,0,2,2,2)$\\\\ \\hline\n$6r+1$ & $(1,1,0,2,2,0)^{r-1}(1,1,0,2,2,2,0)$ & $(1,1,0,2,2,0)^{r-1}(1,1,1,0,2,2,2)$\\\\ \\hline\n$6r+2$ & $(1,1,0,2,2,0)^{r-1}(1,1,0,2,2,2,2,0)$ & $(1,1,0,2,2,0)^r(1,1)$\\\\ \\hline\n$6r+3$ & $(1,1,0,2,2,0)^r (3,3,0)$ & $(1,1,0,2,2,0)^r(1,1,1)$\\\\ \\hline\n$6r+4$ & $(1,1,0,2,2,0)^r (3,3,3,0)$ & $(1,1,0,2,2,0)^r(1,1,1,1)$\\\\ \\hline\n$6r+5$ & $(1,1,0,2,2,0)^r (3,3,3,3,0)$ & $(1,1,0,2,2,0)^r(1,1,0,2,2)$\\\\ \\hline\n\\end{tabular}\n\\caption{Construction of minimum independent dominating sets}\\label{Tab:Largerm}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\section{Lower bounds} \\label{sec:lowerbounds}\n\nIn the section we prove some lower bounds for $i(G \\times H)$ in terms of other domination-type graphical invariants.\nUsing an argument similar to that in the proof of Theorem~\\ref{thm:lowerandupper2}, we can establish a lower bound for the independent\ndomination number of two graphs, neither of which has an isolated vertex.\n\n\\begin{prop} \\label{prop:generallower}\nIf $G$ and $H$ are any two graphs that both have minimum degree at least 1, then\n\\[i(G\\times H) \\ge \\max\\{\\rho(G)\\gamma_t(H),\\rho(H)\\gamma_t(G)\\}\\,.\\]\n\\end{prop}\n\\begin{proof}   Let $I$ be an independent dominating set of $G \\times H$ of smallest cardinality. For a vertex $g$ of $G$, let $J_g=I \\cap (N_G(g) \\times V(H))$.  Since $I$ is a dominating set, each vertex of $\\{g\\} \\times V(H)$ is either in $I$ or is adjacent to a vertex in $J_g$.  Furthermore, since $I$ is independent, exactly one of these holds for each vertex\nin $\\{g\\} \\times V(H)$.  For each $h\\in V(H)$ such that $(g,h) \\in I$, fix a single neighbor $(g',h')$ of $(g,h)$.  Finally, let\n$\\widehat{J}_g = J_g \\cup \\{(g',h')\\,\\colon\\,(g,h) \\in I\\}$.  Note that $|\\{(g',h')\\,\\colon\\,(g,h) \\in I\\}|\\le |\\LSs{g}{I}|$ and that\nthe projection $p_H(\\widehat{J}_g)$ is a total dominating set of $H$. Let $A$ be a maximum 2-packing of $G$.  It now follows that\n\\[|I|=\\sum_{x \\in V(G)}|\\LSs{x}{I}| \\ge \\sum_{g\\in A}|I \\cap(N_G[g]\\times V(H))|\\ge \\sum_{g\\in A}|\\widehat{J}_g|\\ge \\sum_{g\\in A}|p_H(\\widehat{J}_g)|\\ge \\rho(G)\\gamma_t(H)\\,.\\]\nBy reversing the roles of $G$ and $H$ in the above argument we get the desired conclusion.\n\\end{proof}\n\nIn Section~\\ref{sec:counterexamples} we demonstrate the existence of pairs of graphs $G$ and $H$ such that\n$i(G \\times H) < \\min\\{i(G),i(H)\\}$.  The following corollary to Proposition~\\ref{prop:generallower} shows that when\nboth factors are claw-free this is not possible.\n\n\n\\begin{cor}\nIf $G$ and $H$ are both claw-free with no isolated vertices, then\n\\[i(G \\times H)\\geq \\max\\{i(G),i(H)\\}\\,.\\]\n\\end{cor}\n\\begin{proof}\nFrom Proposition~\\ref{prop:generallower} it follows directly that $i(G \\times H)\\geq \\max\\{\\gamma(G),\\gamma(H)\\}$.\nThe result now follows  since $G$ and $H$ are claw-free, which implies that  $\\gamma(G)=i(G)$ and $\\gamma(H)=i(H)$.\n\\end{proof}\n\n\n\\begin{prop} For any connected graphs $G$ and $H$,\n\\[i(G\\times H) \\ge \\max\\left\\{\\frac{{n(H)}{\\gamma(G)}}{\\Delta(H)+1}, \\frac{{n(G)}{\\gamma(H)}}{\\Delta(G)+1} \\right\\}\\,.\\]\n\\end{prop}\n\n\\begin{proof} Let $D$ be an independent dominating set of $G\\times H$. For each $v \\in V(H)$, let $X_v = p_G(D \\cap(V(G) \\times N_H[v]))$.\nNote that $X_v$ is a dominating set of $G$. Moreover, $\\sum_{v \\in V(H)} |X_v|$ counts each vertex of $D$ at most $\\Delta(H)+1$ times. Thus,\n\\[|D| \\ge \\frac{\\sum_{v\\in V(H)} |X_v|}{\\Delta(H) + 1} \\ge \\frac{n(H)\\gamma(G)}{\\Delta(H)+1}.\\]\nThe result now follows by interchanging the roles of $G$ and $H$.\n\\end{proof}\n\n\n\n\n\nIf both factors of a direct product are connected and bipartite, then we get a lower bound just in terms of the domination numbers of the factors.\n\\begin{prop} If $G$ and $H$ are two connected bipartite graphs, then\n\\[i(G\\times H) \\ge \\max\\{2\\gamma(G), 2\\gamma(H)\\}\\,.\\]\n\\end{prop}\n\n\\begin{proof} Let $D$ be an independent dominating set of $G\\times H$. Let $A_G,B_G$ be the bipartition of $V(G)$ and $A_H,B_H$ be the\nbipartition of $V(H)$.  Note that $G\\times H$ is disconnected since no vertex in $(A_G\\times B_H) \\cup (B_G \\times A_H)$ is adjacent to a vertex in\n$(A_G \\times A_H) \\cup (B_G \\times B_H)$.  Thus, it suffices to show that\n\\[|D \\cap ( (A_G\\times B_H)\\cup (B_G\\times A_H))| \\ge \\gamma(G).\\]\nLet $A_2=p_G(D \\cap (A_G \\times B_H))$ and let $A_1=A_G-A_2$.  Similarly, we let $B_2=p_G(D \\cap (B_G \\times A_H))$ and let $B_1=B_G-B_2$.\nWe claim that $A_2 \\cup B_2$ is a dominating set of $G$. To see this, fix $x \\in A_1$.  Since $A_G\\times B_H$ is an independent set,\nevery vertex in $\\{x\\} \\times B_H$ is dominated by some vertex in $B_2 \\times A_H$.  This implies that $B_2$ dominates $A_1$, and\na similar argument shows $A_2$ dominates $B_1$.  Thus, $A_2 \\cup B_2$ is a dominating set of $G$, and we infer that\n$|D \\cap ( (A_G\\times B_H)\\cup (B_G\\times A_H))|\\geq \\gamma(G)$.  Similarly, $|D \\cap ( (A_G\\times A_H)\\cup (B_G\\times B_H))| \\geq \\gamma(G)$,\nand it follows that $|D| \\geq 2\\gamma(G)$.\nInterchanging the roles of $G$ and $H$ in the above argument shows that $|D| \\geq 2\\gamma(H)$, which finishes the proof.\n\\end{proof}\n\n\\medskip\n\n\\section{Counterexamples to Conjecture~\\ref{conj:lowerbound}} \\label{sec:counterexamples}\n\n\n\nWe now present counterexamples to Conjecture~\\ref{conj:lowerbound}.  Let $m$ and $r$ be positive integers larger than $2$.  Let $A,B,C$ and $D$ be pairwise disjoint\nindependent sets of cardinality $m$.  The graph $X_m$ has vertex set and edge set defined as follows.\n\\begin{itemize}\n\\item $V(X_m)=A\\cup B\\cup C \\cup D \\cup \\{x_1,x_2,x_3,x_4\\}$.\n\\item $E(X_m)=\\{x_1w\\,:\\,w \\in A\\cup C\\}\\cup \\{x_2w\\,:\\,w \\in B\\cup D\\} \\cup \\{x_3w\\,:\\,w \\in A\\cup B\\} \\cup \\{x_4w\\,:\\,w \\in C\\cup D\\} \\cup \\{x_1x_2,x_3x_4\\}$.\n\\end{itemize}\nFor example, the graph $X_3$ in shown in Figure~\\ref{fig:X3}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.6,style=thick]\n\\def\\vr{3pt}\n\\path (-2,0) coordinate (0); \\path (2,0) coordinate (1);\n\\path (-2,6) coordinate (2);  \\path (2,6) coordinate (3);\n\\path (-6,3) coordinate (4);  \\path (-5,3) coordinate (5);\n\\path (-4,3) coordinate (6);  \\path (-3,3) coordinate (7);\n\\path (-2,3) coordinate (8);  \\path (-1,3) coordinate (9);\n\\path (1,3) coordinate (10);  \\path (2,3) coordinate (11);\n\\path (3,3) coordinate (12);  \\path (4,3) coordinate (13);\n\\path (5,3) coordinate (14);  \\path (6,3) coordinate (15);\n\\draw (0) -- (1); \\draw (2) -- (3);\n\\foreach \\i in {4,...,6}\n{  \\draw (0) -- (\\i); \\draw (2) -- (\\i); }\n\\foreach \\i in {10,...,12}\n{  \\draw (0) -- (\\i); \\draw (3) -- (\\i); }\n\\foreach \\i in {7,...,9}\n{  \\draw (1) -- (\\i); \\draw (2) -- (\\i); }\n\\foreach \\i in {13,...,15}\n{  \\draw (1) -- (\\i); \\draw (3) -- (\\i); }\n\\draw (0)  [fill=white] circle (\\vr); \\draw (1)  [fill=white] circle (\\vr);\n\\draw (2)  [fill=white] circle (\\vr); \\draw (3)  [fill=white] circle (\\vr);\n\\foreach \\i in {4,...,15}\n{  \\draw (\\i)  [fill=white] circle (\\vr); }\n\\draw[anchor = south] (2) node {$\\mathbf x_3$}; \\draw[anchor = south] (3) node {$\\mathbf x_4$};\n\\draw[anchor = north] (0) node {$\\mathbf x_1$}; \\draw[anchor = north] (1) node {$\\mathbf x_2$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The graph $X_3$}\n\\label{fig:X3}\n\\end{figure}\n\n\n\nWe claim that $i(X_m)=m+2$. The set $D \\cup \\{x_1,x_3\\}$ is an independent dominating set of $X_m$, and hence $i(X_m) \\le m+2$.  Now let $J$ be any independent dominating set of $X_m$.  If $J$ has a nonempty intersection with one of $A,B,C$ or $D$, then $J$ contains all the vertices of that set.  On the other hand, no independent subset of $\\{x_1,x_2,x_3,x_4\\}$ dominates all of $A\\cup B\\cup C \\cup D$.  This establishes the claim.\n\nLet $H_r$ be the complete multipartite graph of order $2r$ in which each of the\n$r$ partite sets has cardinality $2$.  More specifically, let the partite sets be $\\{u_i,v_i\\}$ for $i\\in [r]$.  It is straightforward to check that the set $I$ defined by\n$I=(\\{x_1,x_2\\} \\times \\{u_1,v_1\\}) \\cup (\\{x_3,x_4\\} \\times \\{u_2,v_2\\})$ is an independent dominating set of $X_m \\times H_r$.  Therefore,\n\\[i(X_m \\times H_r) \\le 8 < (m+2)2=i(X_m)i(H_r)\\,.\\]\nThis shows that Conjecture~\\ref{conj:lowerbound} is false.  Moreover, it shows that the difference $i(G)i(H)-i(G \\times H)$ can be arbitrarily large.  In fact,\nfor $m\\ge 7$ we see that $i(X_m \\times H_r) < i(X_m)$.\n\n\nSince the above shows there exist pairs of graphs $G$ and $H$ such that $i(G \\times H)$ is not only strictly smaller than $i(G)i(H)$\nbut can be smaller than $\\max\\{i(G),i(H)\\}$, we are led to the following obvious question.\n\n\\begin{ques} \\label{ques:1}\nIs $i(G\\times H) \\ge \\min\\{i(G),i(H)\\}$ for every pair of graphs $G$ and $H$?\n\\end{ques}\n\\medskip\n\nWe now prove Theorem~\\ref{thm:extremecounterexample}, which answers the above question in the negative and in fact shows\nthat the difference $\\min\\{i(G),i(H)\\}- i(G\\times H)$ can be arbitrary large.\n\n\\medskip\n\\noindent \\textbf{Theorem~\\ref{thm:extremecounterexample}} \\emph{\nFor any positive integer $n$ such that $n>10$, there exists a pair of graphs $G$ and $H$ such that\n$\\min\\{i(G),i(H)\\}=n+2$ and $i(G \\times H) \\leq 12$.\n}\n\\begin{proof}\nFor each positive integer $n$ such\nthat $n>10$, we now define a pair of graphs $G_n$ and $H_n$.\n\nLet ${\\cal A}$ be the collection of subsets of $[6]$ defined by\n\\[{\\cal A}= \\{\\{3, 4, 5, 6\\}, \\{2, 5, 6\\}, \\{1, 2, 3, 4\\}, \\{1, 3, 4, 6\\}, \\{1, 2, 5\\}\\}\\,,\\]\nand let $A_s=\\{u_s,v_s\\}$, for each $s \\in [6]$.  For each $J \\in {\\cal A}$, we let $A_J$ be an independent set of $n$ vertices.\nThe graph $G_n$ has vertex set\n\\[V(G_n) = \\left(\\bigcup_{s=1}^6 A_s\\right) \\cup \\left(\\bigcup_{J\\in{\\cal A}} A_J\\right )\\,.\\]\nThe only edges of $G_n$ are given by the following three conditions.\n\\begin{itemize}\n\\item For each $s \\in [6]$, the vertex $u_s$ is adjacent to $v_s$.\n\\item For each $J \\in {\\cal A}$ and for every $s \\in J$, each of the $n$ vertices of $A_J$ is adjacent to both vertices of $A_s$.\n\\item Each of the sets $A_1 \\cup A_5$, $A_1 \\cup A_6$, $A_2 \\cup A_3$, $A_2 \\cup A_4$,  $A_2 \\cup A_6$,  $A_3 \\cup A_5$,  and $A_4 \\cup A_5$ induces a clique in $G_n$.\n\\end{itemize}\n\nWe claim that $i(G_n) = n+2$. To see this, observe first that $\\{u_1, u_2\\} \\cup A_{\\{3, 4, 5, 6\\}}$  is an independent dominating set of $G_n$.\nTo see that $i(G_n) \\ge n+2$, let $X = \\cup_{i=1}^6 A_i$.  It is easy to see that the only maximal independent sets in $G_n[X]$ are the following:\n\\begin{enumerate}\n\\item[(a)] $\\{x, y\\}$ where $x \\in A_1$ and $y \\in A_2$,\n\\item[(b)] $\\{x, y, z\\}$ where $x \\in A_1$, $y\\in A_3$, and $z \\in A_4$\n\\item[(c)] $\\{x, y\\}$ where $x \\in A_2$ and $y \\in A_5$\n\\item[(d)] $\\{x, y, z\\}$ where $x \\in A_3$, $y\\in A_4$, and $z \\in A_6$\n\\item[(e)] $\\{x, y\\}$ where $x \\in A_5$ and $y \\in A_6$\n\\end{enumerate}\n\nMoreover, for each maximal independent set $I$ of $G_n[X]$ listed above, there exists a $J \\in {\\cal A}$ such that\nno vertex of $A_J$ is adjacent to a vertex of $I$. Thus, $i(G_n) \\ge n+2$.\n\\vskip5mm\nThe graph $H_n$ is defined in a similar way.\nLet ${\\cal B}$ be the collection of subsets of $[6]$ defined by\n\\[ {\\cal B}= \\{\\{2, 3, 4, 6\\}, \\{2, 3, 4, 5\\}, \\{1, 3, 5, 6\\}, \\{1, 2, 4, 6\\}, \\{1, 3, 4, 5\\}, \\{1, 2, 3, 6\\}, \\{1, 4, 5, 6\\}\\}\\,.\\]\nFor each $K \\in {\\cal B}$ we let $B_K$ be an independent set of $n$ vertices.\nThe vertex set of $H_n$ is given by\n\\[V(H_n) = \\{y_1, y_2, y_3, y_4, y_5, y_6\\} \\cup \\left(\\bigcup_{K\\in{\\cal B}} B_K\\right )\\,,\\]\nand the edge set of $H_n$ is given by the following two conditions.\n\n\\begin{itemize}\n\\item $\\{y_1y_2, y_1y_3, y_1y_4, y_2y_5, y_3y_6, y_3y_4, y_4y_6, y_5y_6\\} \\subset E(H_n)$\n\\item For every $K \\in {\\cal B}$, the vertex $y_k$ is adjacent to each vertex of $B_K$ if and only if $k \\in K$.\n\\end{itemize}\n\nWe claim that $i(H_n) = n+2$. One can easily verify that the only maximal independent sets in the induced subgraph\n$H_n[\\{y_1, y_2, y_3, y_4, y_5, y_6\\}]$ are the following: $\\{y_1, y_5\\}$, $\\{y_1, y_6\\}$, $\\{y_2, y_4\\}$, $\\{y_3, y_5\\}$,  $\\{y_2, y_6\\}$,\n$\\{y_2, y_3\\}$, and $\\{y_4, y_5\\}$.\n\n\nMoreover, for each maximal independent set $I$ of $H_n[\\{y_1, y_2, y_3, y_4, y_5, y_6\\}]$ listed above, there exists a set $B_K$ such that no vertex of $B_K$\nis adjacent either vertex of $I$. Thus, $i(H_n) \\ge n+2$. On the other hand, $\\{y_1, y_5\\} \\cup B_{\\{2, 3, 4, 6\\}}$ is an independent dominating set of $H_n$.\n\nTherefore, we have shown that $i(G_n)=i(H_n)=n+2$.  We claim that the set $D$ defined by $D = \\cup_{s=1}^6 \\{(u_s, y_s), (v_s,y_s)\\}$\nis an independent dominating set of $G_n \\times H_n$.\nIt is clear that $\\{(u_s, y_s), (v_s,y_s)\\}$ is independent for each $s \\in [6]$.  Now suppose $(a, y_j)$ and $(b, y_k)$ are adjacent where $a \\in A_j$ and\n$b \\in A_k$  for $1 \\le j\\le k \\le 6$.  It follows that $ab \\in E(G_n)$ and $y_jy_k \\in E(H_n)$. However, by construction, each vertex of $A_j$ is adjacent to each vertex of $A_k$ only if\n$\\{y_j, y_k\\}$ is an independent set in $H_n$, which is a contradiction.  Hence, $D$ is independent in $G_n \\times H_n$.\n\n Now we verify that $D$ dominates $G_n\\times H_n$. First, we show that all vertices of $X\\times \\{y_1, y_2, y_3, y_4, y_5, y_6\\}$ are dominated.\n\\begin{itemize}\n\\item $A_1 \\times \\{y_1\\}$ dominates $A_1 \\times \\{y_1, y_2, y_3, y_4\\}$, $A_5 \\times \\{y_2, y_3, y_4\\}$ and $A_6 \\times \\{y_2, y_3, y_4\\}$.\n\\item $A_2 \\times \\{y_2\\}$ dominates $A_2 \\times \\{y_1, y_2, y_5\\}$, $A_3 \\times \\{y_1, y_5\\}$, $A_4\\times \\{y_1, y_5\\}$, and $A_6\\times \\{y_1, y_5\\}$.\n\\item $A_3 \\times \\{y_3\\}$ dominates $A_3 \\times \\{y_1, y_3, y_4, y_6\\}$, $A_2\\times \\{y_1, y_4, y_6\\}$, and $A_5\\times\\{y_1, y_4, y_6\\}$.\n\\item $A_4 \\times \\{y_4\\}$ dominates $A_4\\times \\{y_1, y_3, y_4, y_6\\}$ and $A_2 \\times \\{y_3\\}$.\n\\item $A_5 \\times \\{y_5\\}$  dominates $A_1 \\times \\{y_6\\}$, $A_3 \\times \\{y_2\\}$, and $A_4\\times \\{y_2\\}$.\n\\item $A_6 \\times \\{y_6\\}$  dominates $A_1\\times \\{y_5\\}$.\n\\end{itemize}\n\nNext, let $J \\in {\\cal A}$ and let $g \\in A_J$.  It is easy to see that $\\{y_j: j \\in J\\}$ is a total dominating set of $H_n$.  This implies that\n$\\cup_{j \\in J} \\{(u_j,y_j),(v_j,y_j)\\}$ dominates $\\LSs {g}{H_n}$.\nFinally, let $K \\in {\\cal B}$ and let $h \\in B_K$.  Again it is straightforward to verify that $\\cup_{k \\in K}A_k$ totally dominates $G_n$.\nIt follows that $\\cup_{k \\in K} \\{(u_k,y_k),(v_k,y_k)\\}$ dominates $G_n^h$.\n\nTherefore, $D$ is an independent dominating set of $G_n \\times H_n$ and\n\\[i(G_n \\times H_n) \\le |D|=12 < n+2=\\min\\{i(G_n),i(H_n)\\}\\,.\\]\n\\end{proof}\n\n\\section{Conclusion}\nNowakowski and Rall posited the following list of conjectures involving a direct or Cartesian product in \\cite{nr-1996}.\n\\begin{conj} {\\rm \\cite[Section 2.4]{nr-1996}} For all graphs $G$ and $H$\n\\begin{enumerate}\n\\item[1.] $ir(G\\Box H) \\ge ir(G)ir(H)$\n\\item[2.] $i(G\\times H) \\ge i(G)i(H)$\n\\item[3.] $\\gamma(G\\Box H) \\ge \\gamma(G)\\gamma(H)$ (Vizing's conjecture)\n\\item[4.] $\\Gamma(G\\times H) \\ge \\Gamma(G)\\Gamma(H)$; $\\Gamma(G\\Box H) \\ge \\Gamma(G)\\Gamma(H)$\n\\end{enumerate}\n\\end{conj}\n\nBre\\v{s}ar proved that $\\Gamma(G\\Box H) \\ge \\Gamma(G)\\Gamma(H)$ in \\cite{b-2005} and Bre\\v{s}ar, Klav\\v{z}ar, and Rall proved that $\\Gamma(G\\times H) \\ge \\Gamma(G)\\Gamma(H)$ in \\cite{bkr-2007}. It is still unknown whether $ir(G\\Box H) \\ge ir(G)ir(H)$ ($ir$ denotes the lower irredundance number), and Vizing's conjecture remains unsettled. In this paper, we proved that there exist pairs of graphs for which $i(G\\times H) < \\min\\{i(G), i(H)\\}$. We also studied the behavior of $i(G\\times K_n)$ for a general graph $G$ and were able to provide the exact values for $i(G\\times K_n)$ when $G \\in \\{P_m, C_m\\}$.\n\nConsider the following computational problem.\n\\begin{center}\n\\fbox{\\parbox{0.85\\linewidth}{\\noindent\n{\\sc Independent Domination of Direct Products}\\\\[.8ex]\n\\begin{tabular*}{.93\\textwidth}{rl}\n{\\em Input:} & A graph $G$, a positive integer $n \\geq 3$ and an integer $k$.\\\\\n{\\em Question:} & Is $i(G \\times K_n) \\leq k$?\n\\end{tabular*}\n}}\n\\end{center}\n\nAs presented in Section~\\ref{sec:productwithcomplete}, showing that $i(G \\times K_n) \\leq k$ is equivalent to finding a weak partition $V_0,V_1,\\ldots,V_n,V_{[n]}$  of $V(G)$\nthat satisfies the four conditions necessary to construct an independent dominating set such that the weight is at most $k$.  We pose the following problem.\n\n\\begin{prob}\nDetermine the complexity of {\\sc Independent Domination of Direct Products}\n\\end{prob}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Intro}Introduction}\nThere has been a renewed interest in the iron-based superconductors since the discovery of superconductivity (SC) at about 30\\,K in the $A_{0.8+\\delta}$Fe$_{1.6+\\beta}$Se$_{2}$ ($A$ = K, Rb, Cs or Tl\/K) \\cite{KFeSe-prb,RbFeSe32K,CsFeSe-MSC,TlFeSe-Mott} compounds due to their unprecedented physical properties, such as the coexistence of high temperature SC with strong antiferromagnetic (AFM) order \\cite{CsFeSe-MSC,KMSC-neutron,CsMSC-neutron,Magn-MSC,Magnon-MSC}. However, whether SC and AFM order coexist microscopically or SC only occurs in the non-magnetic phase is still highly debated since some reports support the coexistence picture \\cite{CsFeSe-MSC,KMSC-neutron,CsMSC-neutron,Magn-MSC,Magnon-MSC} and others favor the phase separation scenario \\cite{PS-XRD,Ricci-XRD,PS-Moss,PS-ARPES}. Local probe techniques such as muon-spin relaxation\/rotation ($\\mu$SR) \\cite{CsFeSe-MSC} and M\\\"ossbauer spectroscopy (MS) \\cite{PS-Moss,Moss-Ryan,Moss-Nowik,Moss-Li} have shown that a two-component picture is inescapable to describe the system correctly, namely, all samples are phase-separated into a major AFM phase and a minor paramagnetic (PM) phase. Nuclear magnetic resonance (NMR) \\cite{NMR-prl} and $\\mu$SR \\cite{uSR-PM} experiments reveal that the PM phase becomes superconducting below $T_c$. However, the scenario that only the PM phase becomes superconducting alone can not explain all the above mentioned experiments. Thus further studies on these compounds are desired to settle the debate. In this case, an investigation on the local magnetic property is helpful in understanding the correlation between SC and magnetic ordering of these systems.\n\nMS has been proved to be a very useful tool to probe local specific information of the iron-based superconductors \\cite{PS-Moss,Moss-Ryan,Moss-Nowik}. Especially, when possible coexistence of magnetic order and SC presents in the same sample, a MS study might reveal rich information. So far, only a few work using MS to study these materials \\cite{PS-Moss,Moss-Ryan,Moss-Nowik} have been reported. A detailed study focusing on the temperature dependence of the local magnetic field at the iron site near the superconducting transition temperature is still missing, which might hold the key to understand the interesting interplay between AFM order and SC. Therefore, in the present work, MS was used to study the magnetic structure and temperature dependence of the hyperfine magnetic field (HMF) at the iron nucleus of K$_{0.84}$Fe$_{1.99}$Se$_2$ single crystals. The results provide evidence that a spin excitation gap opens up before entering the SC state. Using a simple spin model, we show that the ferromagnetically coupled (FMC) four spins can be viewed as a net spin, which couples antiferromagnetically with each other to form the checkerboard-like AFM structure.\n\n\n\\section{\\label{sec:Experiment}Experiments}\nSingle crystals of potassium intercalated iron-selenides of nominal composition K$_{0.8}$Fe$_2$Se$_2$ were grown by the self melting method similar to previous reports \\cite{KFeSe-prb,CsFeSe-MSC}. Stoichiometry of high purity K pieces, Fe and Se powders were mixed and put in a sealed quartz tube. The samples were heated to 1273\\,K slowly, kept for 2\\,h, cooled down to 973\\,K at the rate of 5\\,K\/h and then furnace cooled to room temperature by shutting down the furnace. The resulting plate-like crystals with a shiny surface are of a size up to $6\\times4\\times2$\\,mm$^3$. The actual composition is determined to be K$_{0.84}$Fe$_{1.99}$Se$_2$ by energy dispersive X-ray spectrum (EDXS).\n\nSingle crystal x-ray diffraction (XRD) measurements were performed on a Philips X'pert diffractometer with Cu K$_{\\alpha}$ radiation. AC susceptibility measurements were carried out through a commercial (Quantum Design) superconducting quantum interference device (SQUID) magnetometer. Transmission M\\\"ossbauer spectra were recorded using a conventional constant acceleration spectrometer with a $\\gamma$-ray source of 25\\,mCi $^{57}$Co in palladium matrix moving at room temperature. The absorber was kept static in a temperature-controllable cryostat filled with helium gas. The velocity of the spectrometer was calibrated with $\\alpha$-Fe at room temperature and all the isomer shift quoted in this work are relative to that of the $\\alpha$-Fe.\n\n\n\\section{\\label{sec:Results}Results and Discussion}\nSingle crystal x-ray diffraction pattern of K$_{0.84}$Fe$_{1.99}$Se$_2$ is shown in the inset of Fig. \\ref{Fig1}. As can be seen, only ($00l$) diffraction peaks are observed, indicating the crystallographic $c$-axis is perpendicular to the plane of the plate-like single crystal. Interestingly, two sets of ($00l$) reflections corresponding $c_1$=14.098\\,{\\AA} and $c_2$=14.272\\,{\\AA} are observed, which are attributed to the inhomogeneous distribution of the intercalated K atoms \\cite{AFeSe-TwoSet}. The temperature dependence of the AC susceptibility of K$_{0.84}$Fe$_{1.99}$Se$_2$ single crystal measured along the $ab$-plane with $H_{ac}$=1\\,Oe and $f$=300\\,Hz is shown in Fig. \\ref{Fig1}. The onset superconducting transition temperature, $T_c$, is determined to be 28\\,K from the real part of the susceptibility. The superconducting volume fraction is estimated to be $\\sim$80\\% at 2\\,K.\n\n\\begin{figure}[htp]\n\\includegraphics[width=8 cm]{Fig1-ACXRD\n\\caption{\\label{Fig1} Temperature dependence of AC susceptibility of the K$_{0.84}$Fe$_{1.99}$Se$_2$ crystal measured along the $ab$-plane with $H_{ac}$=1\\,Oe and $f$=300\\,Hz. Inset is the single crystal X-ray diffraction pattern of K$_{0.84}$Fe$_{1.99}$Se$_2$ crystal.}\n\\end{figure}\n\n\nM\\\"ossbauer spectra of a mosaic of single crystal flakes, oriented on a thin paper underlayer so that the $c$-axis is perpendicular to the plane of the M\\\"ossbauer absorber, recorded below room temperatures are shown in Fig. \\ref{Moss}. All spectra share similar spectral shapes and are fitted with two components: a dominant magnetic sextet and a nonmagnetic quadrupole doublet, with \\textsc{MossWinn 4.0} \\cite{MossWinn} programe.\n\n\\begin{figure}[htp]\n\\includegraphics[width=8 cm]{Fig2-Moss\n\\caption{\\label{Moss} M\\\"ossbauer spectra taken at indicated temperatures of the K$_{0.84}$Fe$_{1.99}$Se$_2$ single crystal. The M\\\"ossbauer absorber was prepared with well-cleaved flake-like single crystals, which were put together with the $c$-axis aligned perpendicular to the plane of the M\\\"ossbauer absorber.}\n\\end{figure}\n\nMake an intense study of the doublet, one finds that the two peaks are strongly polarized with an intensity ratio close to 3:1. This means that the orientation of the main axis of the electric field gradient (EFG) of the PM phase is parallel to the $c$ axis of the crystal, which agrees well with the fitted angle of $\\theta_{pm}$=8(3)$^{\\circ}$ by V. Ksenofontov \\textit{et al} \\cite{PS-Moss}. Due to the low statistical of the data and to simplify the fitting procedure, we fitted the doublet assuming that $\\theta_{pm}$=0$^{\\circ}$. The derived isomer shift and quadrupole splitting at 15\\,K is found to be $\\delta$=0.631\\,mm\/s and $eQV_{zz}\/2$=-0.272\\,mm\/s, respectively. These hyperfine parameters are close to the reported values of $\\beta$-FeSe \\cite{FeSe-PseudoPhase,MossFeSe}, while the quadrupole splitting is a little bit smaller than the corresponding doublet in the Rb$_{0.8}$Fe$_{1.6}$Se$_2$ compound \\cite{PS-Moss}, which might be due to different stoichiometries of these samples. Therefore, we may attribute the doublet to the FeSe phase (pseudo-FeSe phase), which corresponds to the FeSe$_4$ tetrahedrons that have K vacancy neighbors in the crystal structure. The coexistence of nonmagnetic pseudo-FeSe phase with the main AFM phase can be naturally understood in the phase separation scenario, which is supported by scanning nanofocused x-ray diffraction \\cite{PS-XRD,Ricci-XRD}, NMR \\cite{NMR-prl}, and previous M\\\"ossbauer \\cite{PS-Moss} studies.\n\nIn order to get a better fit of the spectra, special care should be taken in adjusting the sextet. In our previous manuscript \\cite{Moss-Li}, we fitted the spectra with a relatively small EFG value, assuming that the axis of the main component of the EFG coincide with the crystallographic $c$-axis and the direction of the magnetic moments of the Fe atoms. A close inspection of the spectra reveals that this procedure can not account for the slightly asymmetries of the line pairs (1,6) and (3,4) and the positions of the line pairs (2,5) and (3,4) as pointed out by V. Ksenofontov \\textit{et al} \\cite{PS-Moss}. Therefore, in the present paper we refitted the spectra according to the procedures given by V. Ksenofontov \\textit{et al}, which solves the static Hamiltonian for mixed magnetic and quadrupole interactions with arbitrary relative orientation. The asymmetry parameter $\\eta=(V_{xx}-V_{yy})\/V_{zz}$ is assumed to be zero to further simplify the problem since the fitted value for Rb$_{0.8}$Fe$_{1.6}$Se$_2$ is rather small $\\sim0.1$ \\cite{PS-Moss}. Fitting the spectra yields an averaged relative intensity of 75\\% for the AFM phase, which is significantly smaller than the reported value of 88\\% for Rb$_{0.8}$Fe$_{1.6}$Se$_2$ \\cite{PS-Moss} and K$_{0.8}$Fe$_{1.76}$Se$_2$ \\cite{Moss-Ryan} compounds. This may be caused by the high amounts of iron in our K$_{0.84}$Fe$_{1.99}$Se$_2$ crystal, which may favor the pseudo-FeSe phase. The derived hyperfine parameters for the AFM phase are $\\delta$=0.654\\,mm\/s and $eQV_{zz}\/2$=1.172\\,mm\/s with an angle $\\theta_{afm}$=44(1)$^{\\circ}$ between the axis of $V_{zz}$ and the HMF $B_{hf}$=28.32\\,T at 15\\,K, compares well with previously reported values of some similar compounds \\cite{PS-Moss,Moss-Nowik}.\n\n\nIn order to get a better understanding of the magnetic properties in these materials, we investigated the temperature dependence of the AFM order parameter. The temperature dependence of the HMF, $B_{hf}(T)$, at the Fe site in K$_{0.84}$Fe$_{1.99}$Se$_2$ is depicted in Fig. \\ref{HF}, together with different fitting results. Similar to the behavior of neutron powder diffraction (NPD) (101) magnetic Bragg peak intensity profile \\cite{KMSC-neutron}, the HMF shows an plateau below $\\sim50\\,K$ and then decreases gradually with increasing temperature. A simple Brillouin function was used to fit the HMF data in a previous work \\cite{Moss-Ryan} and a rough agreement was found in the temperature range 10-530\\,K. However, as can be seen from Fig. \\ref{HF}, the Brillouin function (dot line) together with the power law (dashdotted line) fails to describe the low temperature behavior of the HMF. As is well known, in the temperature range of $T\\ll T_N$, the decrease in HMF with increasing temperature can be well explained by spin excitations \\cite{SpinWave-book} within the spin wave theory. For a three dimensional antiferromagnet, the temperature dependence of HMF at low temperatures follows \\cite{SpinWave-book2},\n\\begin{eqnarray}\nB_{hf}(T) = B_{hf}(0)(1 - C T^{2}e^{-\\Delta E\/k_BT})\n\\label{MSEG}\n\\end{eqnarray}\nwhere $C$ is a constant that contains the spin wave stiffness. $\\Delta E$ is the spin excitation gap (SEG), which is necessary to reproduce the plateau of the HMF at low temperatures. Applying equation (\\ref{MSEG}) to the data yields the following results, $B_{hf}(0)$=28.47\\,T, $\\Delta E$=63\\,K ($\\sim$5.5\\,meV). Interestingly, the fitted $\\Delta E\\sim$5.5\\,meV has a nonzero value, suggesting that a substantial SEG due to spin anisotropy opens up above the the superconducting transition temperature. Actually, the SEG has been predicted theoretically \\cite{SpinWave-Theory} and was recently observed by neutron scattering studies in Rb$_{0.89}$Fe$_{1.58}$Se$_2$ \\cite{SEG-RbFeSe} compound.  SEG was also observed in YBa$_2$Cu$_3$O$_{7-\\delta}$ (YBCO) and La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) \\cite{SEG-Cuprates,Kofu-SEG,Millis-SEG1993,Millis-SEG1994,Bourges-SEG,Anderson-SEG} cuprate superconductors and whether the SEG is related to superconductivity is still an open question. There is experimental evidence that well-defined SEG ($\\sim$6\\,meV) in the incommensurate spin fluctuations is observed in the superconducting state only for samples close to the optimal doping \\cite{optimaldoping} for LSCO systems. And a rough proportionality between T$_C$ and SEG was also observed for YBCO superconductors: in the weakly doped region, $E_G\\approx k_BT_C$, while in the heavily doped region, $E_G\\approx 3.8k_BT_C$ \\cite{SEGPropTC,Dai2001,Regnault1994}. Thus, SEG and SC might closely related with each other and a thorough investigation of the evolution of SEG and $T_C$ with different carrier-doping levels is highly desired. In this aspect, in-depth neutron scattering studies of the AFM spin excitation spectrum may yield fruitful information on this issue.\n\n\\begin{figure}[htp]\n\\includegraphics[width=8 cm]{Fig3-Bhf\n\\caption{\\label{HF} Temperature dependence of the hyperfine field, $M(T)$, extracted from least-squares fits of the M\\\"ossbauer spectra. Fitting to the $M(T)$ data with different theories are also compared, the power law (dashdotted line), Brillouin function (dot line) and gaped spin wave theory (solid line). As can be seen, the 3D spin wave theory with an energy gap of $\\Delta E \\sim$5.5\\,meV can better reproduce the low-temperature plateau of the hyperfine field (see text).}\n\\end{figure}\n\nTo understand the temperature dependence of the HMF and estimate the magnetic exchange interactions of our sample, we go to the novel magnetic ordering structure of this compound below $T_N$. The magnetic moments of the four irons in each $\\sqrt{5}\\times\\sqrt{5}$ unit cell align ferromagnetically along the crystalline $c$-axis \\cite{KMSC-neutron,PS-Moss}. And the ferromagnetic blocks interact with each other antiferromagnetically to form a block-checkerboard AFM pattern. Though accredited values of exchange interactions in this system are not reached, strong interactions within the FMC blocks have been predicted theoretically and observed experimentally \\cite{SpinWave-Theory,SEG-RbFeSe}. A recent neutron scattering experiment showed that the acoustic spin waves between $\\sim$9\\,meV to $\\sim$70\\,meV arise mostly from AFM interactions between the FMC blocks, while the optical spin waves associated with the exchange interactions of iron spins within the FM blocks are above $\\sim$80\\,meV \\cite{SEG-RbFeSe}.\n\nConsidering the energy scales of the acoustic and optical spin waves and the large energy separation between them, it is reasonable to assume that, at low temperatures, the decrease in HMF with increasing temperature below room temperature is controlled by the AFM interactions of the FMC blocks. Thus, by fitting the temperature dependence of the HMF data we can deduce the effective interaction, $J_{eff}$, between two nearest FMC blocks, as illustrated in Fig. \\ref{Heisenberg} (a). To simplify the calculation, we make a premise that the iron spins in the FMC block fluctuate coherently. This is reasonable at least at temperatures below room temperature due to the strong ferromagnetic interactions within the FMC block. In this case, an FMC block can be regarded as a super-spin with $S_{eff}=8$.\n\n\\begin{figure}[htp]\n\\includegraphics[width=8 cm]{Fig4-Heisenberg\n\\caption{\\label{Heisenberg} (a) schematic representation of the effective interaction model used in the text. $J_0$ represents the effective interaction constant within each FMC block and is strong enough with respect to the inter-block interaction constant $J_{eff}$ to force the four spins fluctuate coherently at finite temperatures. (b) temperature dependence of the HMF together with fitting results using equation (\\ref{BhfT}) (see text).}\n\\end{figure}\n\nIn the simplest case of two interacting spins, the energy levels are $E(S)=J_{eff}S(S+1)$, where $\\vec{S}=\\vec{S_1}+\\vec{S_2}$ and $\\vec{S_1}$ and $\\vec{S_2}$ are the angular momenta of the two coupled spins. The HMF probed by M\\\"ossbauer measurements will be $B_{hf}=C\\langle S_Z\\rangle$, where $\\langle S_Z\\rangle$ is the expectation value of the $z$ component of $\\vec{S_i}$, and reaches its maximum value at ground state (zero temperature). While at elevated temperatures, states with higher $S$ are accessible, which results in the decrease of HMF as observed above. If we assume the relaxation between the electronic states is fast with respect to the Larmor precession time, we can express the finite temperature HMF as $B_{hf}(T)=B_{hf}(0)(1-\\sum_Sh_Sn_S)$, and\n\\begin{eqnarray}\n\\small\n\\sum_Sh_Sn_S= \\frac{\\sum\\limits_{S=0}^{16}\\sum\\limits_{S_z=0}^S h(S,S_Z)e^{-J_{eff}S(S+1)\/k_BT}}{\\sum\\limits_{S=0}^{16}\\sum\\limits_{S_z=-S}^S e^{-J_{eff}S(S+1)\/k_BT}},\n\\label{BhfT}\n\\end{eqnarray}\nwhere $n_S$ is the populations and $h(S,S_Z)$ is the decrease in HMF corresponding to state $|S,S_Z\\rangle$. If we assume that $h(S,S_Z)$ is proportional to $S_Z$ with the same proportionate constant for all $S$ states, $h(S,S_Z)=h_0S_Z$, then we can fit the experimental data with equation (\\ref{BhfT}). As can be seen from Fig. \\ref{Heisenberg} (b), a good agreement between the theoretical curve and the experimental data can be obtained and the fitted parameters are $J_{eff}$=22.8\\,meV, $B_{hf}(0)$=28.44\\,T and $h_0$=0.697\\,T.\n\nTo see the efficiency of our simple model in describing the low-energy spin excitations, we compare our results with that deduced from the effective spin Hamiltonian model, which has been widely used to describe the ground state and spin excitations for this type of compounds. Usually, the Hamiltonian involves intra-block nearest and second nearest neighbor interactions $J_1$, $J_2$ and the inter-block nearest and second nearest neighbor interactions $J_1'$, $J_2'$. Even the third nearest neighbor interactions $J_3$, $J_3'$ have been adopted to fit the spin wave spectra by Miaoyin Wang \\textit{et al} \\cite{SEG-RbFeSe}. In terms of the $J_1$-$J_1'$-$J_2$-$J_2'$-$J_3$-$J_3'$ model, the low-energy spin waves can be approximately described by ($J_1'+2J_2'+2J_3)S\/4\\sim$17\\,meV. Obviously, our results of $J_{eff}$ agrees reasonably well with the neutrons scattering results, which proves the validity of our above assumption.\n\n\n\n\n\\section{\\label{sec:Conclusion}Concluding remarks}\nHigh quality single crystals of K$_{0.84}$Fe$_{1.99}$Se$_2$ have been prepared and studied by M\\\"ossbauer spectroscopy. Temperature dependence of the hyperfine magnetic field is well explained within the gaped spin wave theory. Fitting the experimental data yields a spin excitation gap of about 5.5\\,meV\/63\\,K. Supposing the blocked spins fluctuate coherently, the effective exchange interaction between these coupled spin blocks is estimated to be $J_{eff}$=22.8\\,meV, which agrees reasonably well with previous NPD estimated value ($J_{eff}\\sim$17\\,meV).\n\n\n\n\\begin{acknowledgments}\nWe thank T. Xiang, P. C. Dai and Y. Z. You for useful discussions.\nThis work was supported by the National Natural Science Foundation of China under Grants No. 10975066.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSelf-excited dynamo action refers to the instability of a plasma in\na non-magnetic equilibrium state to amplify magnetic fields within the\nframework of resistive magnetohydrodynamics (MHD).\nCharge separation effects are assumed absent, i.e.\\ no battery-type\neffects are explicitly involved, although they do play a role in producing\na weak initial seed magnetic field that is needed to provide a perturbation\nto the otherwise field-free initial state.\nA dynamo instability may occur when the magnetic Reynolds number is large\nenough, i.e.\\ the fluid motions and the scale of the domain are large enough.\nThis instability is normally a linear one, but some dynamos\nare subcritical and require then a finite-amplitude initial field.\nIn the linear case one speaks about slow and fast dynamos depending on\nwhether or not the growth rate of the dynamo scales with resistivity.\nFor fast dynamos the growth rate scales with the rms velocity of the\nflow, which is turbulent in most cases.\nDynamos saturate when the magnetic energy becomes comparable with the\nkinetic energy.\nSome of the kinetic energy is then channelled through the magnetic\nenergy reservoir and is eventually dissipated via Joule heating.\n\nThere are several applications where one considers non self-excited\ndynamo action.\nExamples can be found in magnetospheric physics and in plasma physics\nwhere one is interested in the electromotive force induced by a flow\npassing through a given magnetic field.\nLater in this paper we will discuss the reversed field pinch (RFP)\nbecause of its connection with the $\\alpha$ effect that plays\nan important role in astrophysical dynamos.\nThe physics of the RFP has been reviewed by Ortolani \\& Schnack (1993)\nand, in a broader context with reference to astrophysical large-scale\ndynamos, by Ji \\& Prager (2002) and Blackman \\& Ji (2006).\n\nThe $\\alpha$ effect is one of a few known mechanisms able to generate\nlarge-scale magnetic fields, i.e.\\ fields whose typical length scale\nis larger than the scale of the energy carrying motions.\nIt was also one of the first discussed mechanisms able to produce\nself-excited dynamo action at all.\nIndeed, Parker (1955) showed that the swirl of a convecting flow under\nthe influence of the Coriolis force can be responsible for producing\na systematically oriented poloidal magnetic field from a toroidal field.\nThe toroidal field in turn is produced by the shear from the differential\nrotation acting on the poloidal field.\nParker's paper was before Herzenberg (1958) produced the first existence\nproof of dynamos.\nUntil that time there was a serious worry that Cowling's (1933) anti-dynamo\ntheorem might carry over from two-dimensional fields to three-dimensional\nfields, as is evident from sentimental remarks made by Larmor (1934).\n\nLarmor (1919) proposed the idea of dynamo action in the\nastrophysical context nearly 100 years ago.\nNowadays, with the help of computers, it is quite easy to solve the\ninduction equation in three dimensions in simple geometries and obtain\nself-excited dynamo solutions with as little as $16^3$ mesh points\nusing, for example, the sample ``\\texttt{kin-dynamo}'' that comes with\nthe {\\sc Pencil Code} (\\url{http:\/\/pencil-code.googlecode.com}).\n\nIn addition to dynamos in helical flows, which can generate large-scale\nfields, there are also dynamos in non-helical flows that produce only\nsmall-scale fields.\nThis possibility was first addressed by Batchelor (1950) based on\nthe analogy between the induction equation and the vorticity equation.\nAgain, this was not yet very convincing at the time.\nThe now accepted theory for small-scale dynamos was first proposed by\nKazantsev (1968) and the first simulations were produced by\nMeneguzzi et al.\\ (1981).\nSuch simulations are computationally somewhat more demanding and require\nat least $64^3$ mesh points (or collocation points in spectral schemes).\nIn the past few years this work has intensified (Cho et al.\\ 2002,\nSchekochihin et al.\\ 2002, Haugen 2003).\nWe will not discuss these dynamos in the rest of this paper.\nInstead, we will focus on large-scale dynamos.\nMore specifically, we focus here on a special class of large-scale\ndynamos, namely those where kinetic helicity plays a decisive role\n(the so-called $\\alpha$ effect dynamos).\nNevertheless, we mention at this point two other mechanisms that\ncould produce large-scale magnetic field without net helicity.\nOne is the incoherent $\\alpha$--shear effect that was originally proposed\nby Vishniac \\& Brandenburg (1997) to explain the occurrence of large-scale\nmagnetic fields in accretion discs, and later also for other astrophysical\napplications (e.g., Proctor 2007).\nIt requires the presence of shear, because otherwise only small-scale\nmagnetic fields would be generated (Kraichnan 1976, Moffatt 1978).\nThe other mechanism is the shear--current effect of\nRogachevskii \\& Kleeorin (2003), which can operate if the turbulent\nmagnetic diffusion tensor is anisotropic, so the mean electromotive\nforce from the turbulence is given by $-\\eta_{ij}\\overline{J}_j$ such that the\nsign of $\\eta_{ij}\\overline{U}_{i,j}$ is positive, and that this quantity is\nbig enough to overcome resistive effects.\nHere, a comma denotes partial differentiation, $\\overline{\\bm{U}}$ is the mean flow,\n$\\overline{\\mbox{\\boldmath $J$}}{}}{$ is the mean current density,\nand summation over repeated indices is assumed.\nThis effect too requires shear, because otherwise $\\eta_{ij}\\overline{U}_{i,j}$\nwould be zero.\nSimulations show large-scale dynamo action in the presence\nof just turbulence and shear, and without net helicity, but there\nare indications that this process may also be just the result of incoherent\n$\\alpha$--shear dynamo action (Brandenburg et al.\\ 2008a).\n\nThroughout this paper, overbars denote suitable spatial averages\nover one or two coordinate directions.\nFurthermore, we always assume that the scale of the energy-carrying eddies\nis at least three times smaller than the scale of the domain.\nWe refer to this property as ``scale separation''.\nIn this sense, scale separation is a natural requirement, because we\nwant to explain the occurrence of fields on scales large compared with\nthe scale of the energy-carrying motions.\nScale separation has therefore nothing to do with a gap in the kinetic\nenergy spectrum, as is sometimes suggested.\n\nMuch of the work on $\\alpha$ effect dynamos has been done in the\nframework of analytic approximations.\nHowever, this is only a technical aspect that is unimportant for the\nactual occurrence of large-scale fields under suitable conditions.\nThis has been demonstrated by numerical simulations, as will be discussed\nbelow.\n\n\\section{Helical large-scale dynamos}\n\nA possible way of motivating the physics behind helical dynamo action\nis the relation to the concept of an inverse turbulent cascade.\nThis particular idea was first proposed by Frisch et al.\\ (1975) and is\nbased on the conservation of magnetic helicity,\n\\begin{equation}\nH_{\\rm M}=\\int_V\\mbox{\\boldmath $A$} {}\\cdot\\mbox{\\boldmath $B$} {}\\,{\\rm d} {} V,\n\\end{equation}\nwhere $\\mbox{\\boldmath $A$} {}$ is the magnetic vector potential and $\\mbox{\\boldmath $B$} {}=\\mbox{\\boldmath $\\nabla$} {}\\times\\mbox{\\boldmath $A$} {}$\nis the magnetic field in a volume $V$.\n\nIt is convenient to define spectra of magnetic energy and magnetic\nhelicity, $E_{\\rm M}(k)$ and $H_{\\rm M}(k)$, respectively.\nAs usual, these spectra are obtained by calculating the three-dimensional\nFourier transforms of magnetic vector potential and magnetic field,\n$\\hat{\\bm{A}}_{\\bm{k}}$ and $\\hat{\\bm{B}}_{\\bm{k}}$, respectively,\nand integrating $|\\hat{\\bm{B}}_{\\bm{k}}|^2$ and\nthe real part of $\\hat{\\bm{A}}_{\\bm{k}}\\cdot\\hat{\\bm{B}}_{\\bm{k}}^*$ over shells of constant\n$k=|\\bm{k}|$ to obtained $E_{\\rm M}(k)$ and $H_{\\rm M}(k)$, respectively.\n(Here, an asterisk denotes complex conjugation.)\nThese spectra are normalized such that\n$\\int E_{\\rm M}(k)\\,{\\rm d} {} k=\\bra{\\mbox{\\boldmath $B$} {}^2}\/2\\mu_0$ and\n$\\int H_{\\rm M}(k)\\,{\\rm d} {} k=\\bra{\\mbox{\\boldmath $A$} {}\\cdot\\mbox{\\boldmath $B$} {}}$ for $k$ from 0 to $\\infty$,\nwhere angular brackets denote volume averages over a periodic domain\nand $\\mu_0$ is the vacuum permeability.\nUsing the Schwartz inequality one can then derive the so-called realizability\ncondition,\n\\begin{equation}\nk|H_{\\rm M}(k)|\/2\\mu_0\\leq E_{\\rm M}(k).\n\\end{equation}\nFor fully helical magnetic fields with (say) positive helicity,\ni.e.\\ $H_{\\rm M}=2\\mu_0 E_{\\rm M}(k)\/k$, one can show that energy\nand magnetic helicity cannot cascade directly, i.e.\\ the interaction\nof modes with wavenumbers $\\bm{p}$ and $\\bm{q}$ can only produce fields whose\nwavevector $\\bm{k}=\\bm{p}+\\bm{q}$ has a length that is equal or smaller than the maximum\nof either $|\\bm{p}|$ or $|\\bm{q}|$ (Frisch et al.\\ 1975), i.e.\\\n\\begin{equation}\n|\\bm{k}|\\leq\\max(|\\bm{p}|,|\\bm{q}|).\n\\label{kpq}\n\\end{equation}\nThis means that magnetic helicity and magnetic energy are transformed to\nprogressively larger length scales.\nA clear illustration of this can be seen in decaying helical turbulence.\n\\FFig{InvCasc} shows magnetic energy spectra from a simulation of\nChristensson et al.\\ (2001) at different times for a\ncase where the initial magnetic field was fully helical and had a spectrum\nproportional to $k^4$ with a resolution cutoff near the largest possible\nwavenumber.\nNote that the entire spectrum appears to shift to the left, i.e.\\ toward\nlarger length scales, in an approximately self-similar fashion.\nThe details of the argument that led to \\Eq{kpq} are due to\nFrisch et al.\\ (1975), and can also be found in the reviews by\nBrandenburg et al.\\ (2002) and Brandenburg \\& Subramanian (2005a).\n\n\\begin{figure}[t!]\\begin{center}\n\\includegraphics[width=.6\\columnwidth]{InvCasc}\n\\end{center}\\caption[]{\nMagnetic energy spectra at different times\n(increasing roughly by a factor of 2).\nThe curve with the right-most location of the peak corresponds to\nthe initial time, while the other lines refer to later times (increasing\nfrom right to left). Note the\npropagation of spectral energy to successively smaller wavenumbers $k$,\ni.e.\\ to successively larger scales.\nAdapted from Christensson et al.\\ (2001).\n}\\label{InvCasc}\\end{figure}\n\nIn the non-decaying case, when the flow is driven by energy input at some\nforcing wavenumber $k_{\\rm f}$, the inverse cascade is clearly seen if there is\nsufficient scale separation, i.e.\\ if $k_{\\rm f}$ is large compared with the\nsmallest wavenumber $k_1$ that fits into a domain of size $L=2\\pi\/k_1$.\nAn example is shown in \\Fig{FpMkt2}, where kinetic energy is injected\nat the wavenumber $k_{\\rm f}=30k_1$.\nIt is evident that there are two local maxima of spectral magnetic\nenergy, one at the forcing wavenumber $k_{\\rm f}$, and another one at a smaller\nwavenumber that we call $k_{\\rm m}$, which is near $7k_1$ in \\Fig{FpMkt2}.\nDuring the kinematic stage the entire spectrum moves upward, with the\nspectral energy increasing at the same rate at all wavenumbers.\nEventually, when the field has reached a certain level, the spectrum\nbegins to change its shape and the second local maximum at $k_{\\rm m}$\nmoves toward smaller values, suggestive of an inverse cascade.\nHowever, a more detailed analysis (Brandenburg 2001) shows that the\nenergy transfer is nonlocal, i.e.\\ most of the energy is transferred\ndirectly from the forcing wavenumber to the wavenumber where most of\nthe mean field resides.\nThis suggests that we have merely a nonlocal inverse {\\it transfer} rather than\na proper inverse {\\it cascade}, where the energy transfer would be local in\nspectral space.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[width=.5\\textwidth]{pMkt2}\\caption{\nMagnetic energy spectra for a run with forcing at $k=30$. The times,\nin units of $(c_{\\rm s} k_1)^{-1}$,\nrange from 0 (dotted line) to 10, 30, ..., 290 (solid lines). The thick\nsolid line gives the final state at $c_{\\rm s} k_1 t=1000$,\ncorresponding to $\\urmsk_{\\rm f} t\\approx2000$ turnover times.\nHere, $c_{\\rm s}$ is the sound speed and $u_{\\rm rms}$ is the turbulent rms velocity.\nNote that at early times\nthe spectra peak at $k_{\\max}\\approx7k_1$. The $k^{-1}$ and $k^{+3\/2}$\nslopes are given for orientation as dash-dotted lines.\nAdapted from Brandenburg (2001).\n}\\label{FpMkt2}\\end{figure}\n\nThe position of the local maximum can readily be explained by mean-field\ndynamo theory with an $\\alpha$ effect.\nThe evolution equation of such a dynamo is\n\\begin{equation}\n{\\partial\\overline{\\mbox{\\boldmath $B$}}{}}{\\over\\partial t}=\\mbox{\\boldmath $\\nabla$} {}\\times\\alpha\\overline{\\mbox{\\boldmath $B$}}{}}{+\\eta_{\\rm T}\\nabla^2\\overline{\\mbox{\\boldmath $B$}}{}}{,\n\\label{MFD}\n\\end{equation}\nwhere $\\alpha$ is a pseudo-scalar, $\\eta_{\\rm T}=\\eta+\\eta_{\\rm t}$ is the sum of\nmicroscopic Spitzer resistivity and turbulent resistivity\\footnote{Note\nthat resistivity and magnetic diffusivity differ by a $\\mu_0$ factor.\nHere, we always mean the magnetic diffusivity, although we use the two\nnames sometimes interchangeably.}, and an overbar denotes a suitably\ndefined spatial average (e.g.\\ planar average).\nAssuming $\\overline{\\mbox{\\boldmath $B$}}{}}{=\\hat{\\bm{B}}_{\\bm{k}}\\exp(\\lambda t+{\\rm i}\\bm{k}\\cdot\\bm{x})$ with\neigenfunction $\\hat{\\bm{B}}_{\\bm{k}}$, one finds the dispersion relation to be\n(Moffatt 1978)\n\\begin{equation}\n\\lambda(k)=|\\alpha|k-\\eta_{\\rm T} k^2,\n\\end{equation}\nwhere $k=|\\bm{k}|$.\nThe maximum growth rate is attained for a value of $k$ where\n${\\rm d} {}\\lambda\/{\\rm d} {} k=0$, i.e.\\ for $k=k_{\\rm m}=\\alpha\/2\\eta_{\\rm T}$.\nThe migration of the spectral maximum to smaller $k$ can then be\nexplained as the result of a suppression of $\\alpha$.\nIn other words, as the dynamo saturates, $\\alpha$ decreases, and so does\n$k_{\\rm m}$, which corresponds to the spectral maximum moving to the left.\n\nIt should be noted that there is also the possibility of a suppression\nof $\\eta_{\\rm t}$, which would work the other way, so that this interpretation\nmight not work.\nIndeed, there are arguments for a suppression of $\\eta_{\\rm t}$ that would be as\nstrong as that of $\\alpha$, but this applies only to the two-dimensional\ncase and has to do with the conservation of the mean-squared vector\npotential in that case (Gruzinov \\& Diamond 1994).\nSimulations also find evidence that in three dimensions the suppression\nof $\\alpha$ is stronger than that of $\\eta_{\\rm t}$ (Brandenburg et al.\\ 2008b).\n\nIn the following we discuss in detail the role played by magnetic helicity.\nThis has been reviewed extensively in the last few years\n(Ji \\& Prager 2002, Brandenburg \\& Subramanian 2005, Blackman \\& Ji 2006).\n\n\\section{Slow saturation}\n\nIn a closed or periodic domain, the saturation of a helical large-scale\ndynamo is found to be\nresistively slow and the final field strength is reached with a time\nbehavior of the form (Brandenburg 2001)\n\\begin{equation}\n\\bra{\\overline{\\mbox{\\boldmath $B$}}{}}{^2}(t)=B_{\\rm eq}^2{k_{\\rm m}\\overk_{\\rm f}}\n\\left[1-e^{-2\\eta k_{\\rm m}^2(t-t_{\\rm s})}\\right]\n\\quad\\mbox{for $t>t_{\\rm s}$},\n\\label{SlowSat}\n\\end{equation}\nwhere $B_{\\rm eq}$ is the equipartition field strength and $t_{\\rm s}$ is\nthe time when the slow saturation phase begins.\nWe emphasize that it is the microscopic $\\eta$ that enters \\Eq{SlowSat},\nand that the relevant length scale, $2\\pi\/k_{\\rm m}$, is that of the\nlarge-scale field, so the saturation behavior is truly very slow.\n\nThe reason for this slow saturation behavior is related to the conservation\nof magnetic helicity, which obeys the evolution equation\n(e.g., Ji et al.\\ 1995, Ji 1999)\n\\begin{equation}\n{{\\rm d} {}\\over{\\rm d} {} t}\\bra{\\mbox{\\boldmath $A$} {}\\cdot\\mbox{\\boldmath $B$} {}}=-2\\eta\\mu_0\\bra{\\mbox{\\boldmath $J$} {}\\cdot\\mbox{\\boldmath $B$} {}}\n-\\bra{\\mbox{\\boldmath $\\nabla$} {}\\cdot\\mbox{\\boldmath $F$} {}_{\\rm H}}.\n\\label{dABdt}\n\\end{equation}\nwhere $\\mbox{\\boldmath $F$} {}_{\\rm H}$ is the magnetic helicity flux, but for the periodic\ndomain under consideration we have $\\mbox{\\boldmath $\\nabla$} {}\\cdot\\mbox{\\boldmath $F$} {}_{\\rm H}=0$.\nClearly, in the final state we have then $\\bra{\\mbox{\\boldmath $J$} {}\\cdot\\mbox{\\boldmath $B$} {}}=0$.\nThis can only be satisfied for nontrivial helical fields if\nsmall-scale and large-scale fields have values of opposite sign,\nbut equal magnitude, i.e.\\ $\\bra{\\mbox{\\boldmath $j$} {}\\cdot\\mbox{\\boldmath $b$} {}}=-\\bra{\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{}$,\nwhere $\\mbox{\\boldmath $B$} {}=\\overline{\\mbox{\\boldmath $B$}}{}}{+\\mbox{\\boldmath $b$} {}$ and $\\mbox{\\boldmath $J$} {}=\\overline{\\mbox{\\boldmath $J$}}{}}{+\\mbox{\\boldmath $j$} {}$ are the decompositions of\nmagnetic field and current density into mean and fluctuating parts.\nHere we choose to define mean fields as one- or two-dimensional coordinate\naverages.\nExamples include planar averages such as $xy$, $yz$, or $xz$ averages\nin a periodic Cartesian domain, as well as one-dimensional averages\nsuch as $y$ or $\\phi$ averages in Cartesian or spherical domains,\n$(r,\\theta,\\phi)$, where the other two directions are non-periodic.\n\n\\EEq{SlowSat} can be derived under the assumption that large-scale and\nsmall-scale fields are fully helical with\n$\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{=k_{\\rm m}^2\\overline{\\mbox{\\boldmath $A$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{=\\mp k_{\\rm m}\\overline{\\mbox{\\boldmath $B$}}{}}{^2$\nand $\\bra{\\mbox{\\boldmath $j$} {}\\cdot\\mbox{\\boldmath $b$} {}}=\\pmk_{\\rm f}\\bra{\\mbox{\\boldmath $b$} {}^2}\\approx\\pmk_{\\rm f}B_{\\rm eq}^2$,\nwhere upper and lower signs refer to positive and negative\nhelicity of the small-scale turbulence and we have assumed that\nthe small-scale field has already reached saturation, i.e.\\\n$\\bra{\\mbox{\\boldmath $b$} {}^2}\\approxB_{\\rm eq}^2\\equiv\\mu_0\\bra{\\rho\\mbox{\\boldmath $u$} {}^2}$, where $\\rho$ is\nthe density and $\\mbox{\\boldmath $u$} {}=\\mbox{\\boldmath $U$} {}-\\overline{\\bm{U}}$ is the fluctuating velocity.\n\nA mean-field theory that obeys magnetic helicity conservation was\noriginally developed by Kleeorin \\& Ruzmaikin (1982) and has recently\nbeen applied to explaining slow saturation (Field \\& Blackman 2002,\nBlackman \\& Brandenburg 2002, Subramanian 2002).\nThe main idea is that the $\\alpha$ effect has two contributions\n(Pouquet et al.\\ 1976),\n\\begin{equation}\n\\alpha=\\alpha_{\\it K}+\\alpha_{\\it M},\n\\label{alphaKM}\n\\end{equation}\nwhere $\\alpha_{\\it K}=-{\\textstyle{1\\over3}}\\tau\\overline{\\mbox{\\boldmath $\\omega$} {}\\cdot\\mbox{\\boldmath $u$} {}}$ is the usual kinetic\n$\\alpha$ effect related to the kinetic helicity,\nwith $\\mbox{\\boldmath $\\omega$} {}=\\mbox{\\boldmath $\\nabla$} {}\\times\\mbox{\\boldmath $u$} {}$ being the vorticity, and\n$\\alpha_{\\it M}={\\textstyle{1\\over3}}\\tau\\overline{\\mbox{\\boldmath $j$} {}\\cdot\\mbox{\\boldmath $b$} {}}\/\\rho_0$ is a magnetic\n$\\alpha$ effect that can, for example, be produced by the growing\nmagnetic field in an attempt to conserve magnetic helicity.\n(Here, $\\rho_0$ is an average density, but we note that there is at present\nno adequate theory for compressible systems with nonuniform density.)\n\nNote that $\\alpha_{\\it M}$ is related to the small-scale current helicity\nand hence to the small-scale magnetic helicity which, in turn, obeys\nan evolution equation similar to \\Eq{dABdt}, but with an additional\nproduction term, $2\\overline{\\mbox{\\boldmath ${\\cal E}$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{$, that arises from mean-field\ntheory via $\\overline{\\mbox{\\boldmath ${\\cal E}$}}{}}{=\\alpha\\overline{\\mbox{\\boldmath $B$}}{}}{-\\eta_{\\rm t}\\mu_0\\overline{\\mbox{\\boldmath $J$}}{}}{$, and thus from\nthe $\\alpha$ effect itself.\nThe equation for $\\alpha_{\\it M}$ has then the form\n\\begin{equation}\n{\\partial\\alpha_{\\it M}\\over\\partial t}=-2\\etatk_{\\rm f}^2\n\\left({\\overline{\\mbox{\\boldmath ${\\cal E}$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{\\overB_{\\rm eq}^2}+{\\alpha_{\\it M}\\overR_{\\rm m}}\\right)\n-\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_\\alpha,\n\\label{dalphaMdt}\n\\end{equation}\nwhere $R_{\\rm m}=\\eta_{\\rm t}\/\\eta$ is a measure of the ratio of turbulent to\nmicroscopic magnetic diffusivity.\nWith $\\eta_{\\rm t}=u_{\\rm rms}\/3k_{\\rm f}$ (Sur et al.\\ 2008) we can relate this to\nthe more usual definition for the magnetic Reynolds number,\n$\\tildeR_{\\rm m}=u_{\\rm rms}\/\\etak_{\\rm f}$, via $\\tildeR_{\\rm m}=3R_{\\rm m}$.\nFurthermore, we have allowed for the possibility of fluxes of magnetic and\ncurrent helicities that also lead to a flux of $\\alpha_{\\it M}$.\nSuch fluxes are primarily important in inhomogeneous domains and especially\nin open domains where one can have an outward helicity flux (Ji 1999).\n\nThe use of \\Eq{alphaKM} is sometimes criticized because it is based on\na closure assumption.\nIndeed, there are questions regarding the meaning of the term\n$\\overline{\\mbox{\\boldmath $j$} {}\\cdot\\mbox{\\boldmath $b$} {}}$ and whether it really applies to the actual\nfield, or the field in the unquenched case.\nThis has been discussed in detail in a critical paper by\nR\\\"adler \\& Rheinhardt (2007).\nPart of this ambiguity can already be clarified in the low conductivity limit.\nSur et al.\\ (2007) have shown that one can express $\\alpha$ either\ncompletely in terms of the helical properties of the velocity field or,\nalternatively, as the sum of two terms, a so-called kinetic $\\alpha$\neffect and an oppositely signed term proportional to the helical part\nof the small scale magnetic field.\nHowever, it is fair to say that the problem is not yet completely understood.\nThe strongest argument in favor of \\Eqs{alphaKM}{dalphaMdt} is that\nthey reproduce catastrophic (i.e.\\ $R_{\\rm m}$-independent) quenching of $\\alpha$\nand that this approach has led to the prediction that such quenching can be\nalleviated by magnetic helicity fluxes.\nThis prediction has subsequently been tested successfully on various\noccasions (Brandenburg 2005, K\\\"apyl\\\"a et al.\\ 2008).\nFinally, it should be noted that \\Eq{alphaKM} has also been confirmed\ndirectly using turbulence simulations (Brandenburg et al.\\ 2005c, 2007).\n\nThe idea to model dynamo saturation and suppression of $\\alpha$ by solving\na dynamical equation for $\\alpha_{\\it M}$ is called dynamical quenching.\nIn addition to the resistively slow saturation behavior described by\n\\Eq{SlowSat}, this approach has also been applied to decaying turbulence\nwith helicity (Yousef et al.\\ 2003; Blackman \\& Field 2004), where the\nconservation of magnetic helicity results in a slow-down of the decay.\nThis can be modeled by an $\\alpha$ effect that offsets the turbulent\ndecay proportional to $\\eta_{\\rm t} k^2$ such that the decay rate becomes nearly\nequal to the resistive value, $\\eta k^2$.\nThis is explained in detail in the following section.\n\n\\section{Decay in a Cartesian domain}\n\\label{Sdynquench}\n\nIn the context of driven turbulence, the properties of solutions of a\ndecaying helical magnetic field were studied earlier by\nYousef et al.\\ (2003), who found that for fields with\n$\\overline{\\mbox{\\boldmath $B$}}{}}{^2\/B_{\\rm eq}^2\\ga R_{\\rm m}^{-1}$ the decay of $\\overline{\\mbox{\\boldmath $B$}}{}}{$ is slowed\ndown and can quantitatively be described by the dynamical quenching model.\nThis model applies even to the case where the turbulence is nonhelical and\nwhere there is initially no $\\alpha$ effect in the usual sense.\nHowever, the magnetic contribution to $\\alpha$ is still non-vanishing,\nbecause the $\\alpha_{\\it M}$ term is driven by the helicity of the\nlarge-scale field.\n\nTo demonstrate this quantitatively, Yousef et al.\\ (2003) have adopted\na one-mode approximation with $\\overline{\\mbox{\\boldmath $B$}}{}}{=\\hat{\\bm{B}}(t)\\exp({\\rm i} k_1z)$,\nand used the mean-field induction equation together with the dynamical\n$\\alpha$-quenching formula \\eq{dalphaMdt},\n\\begin{equation}\n{{\\rm d} {}\\hat{\\bm{B}}\\over{\\rm d} {} t}={\\rm i}\\bm{k}_1\\times\\hat{\\mbox{\\boldmath ${\\cal E}$} {}}\n-\\eta k_1^2\\hat{\\bm{B}},\n\\label{ODE1}\n\\end{equation}\n\\begin{equation}\n{{\\rm d} {}\\alpha\\over{\\rm d} {} t}=-2\\eta_{\\rm t} k_{\\rm f}^2\n\\left({{\\rm Re}(\\hat{\\mbox{\\boldmath ${\\cal E}$} {}}^*\\cdot\\hat{\\bm{B}}) \\over B_{\\rm eq}^2}\n+{\\alpha\\overR_{\\rm m}}\\right),\n\\label{ODE2}\n\\end{equation}\nwhere the flux term is neglected,\n$\\hat{\\mbox{\\boldmath ${\\cal E}$} {}}=\\alpha\\hat{\\bm{B}}-\\eta_{\\rm t}{\\rm i}\\bm{k}_1\\times\\hat{\\bm{B}}$\nis the electromotive force, and $\\bm{k}_1=(0,0,k_1)$.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[width=0.5\\textwidth]{pselected_decay}\\caption{\nDynamical quenching model with helical and nonhelical initial fields,\nand an additional $\\eta_{\\rm t}$ quenching function,\n$\\eta_{\\rm t}=\\eta_{\\rm t0}\/(1+\\tilde{g}|\\overline{\\mbox{\\boldmath $B$}}{}}{|\/B_{\\rm eq})$.\nThe quenching parameters are $\\tilde{g}=0$ (solid line) and 3 (dotted line).\nThe graph for the nonhelical cases has been shifted in $t$ so that one\nsees that the decay rates are asymptotically equal at late times.\nThe value of $\\eta_{\\rm T}$ used to normalize the abscissa is based\non the unquenched value.\nAdapted from Yousef et al.\\ (2003).\n}\\label{Fpselected_decay}\\end{figure}\n\n\\FFig{Fpselected_decay} compares the evolution of $\\overline{\\mbox{\\boldmath $B$}}{}}{\/B_{\\rm eq}$\nfor helical and nonhelical initial conditions, $\\hat{\\bm{B}}\\propto(1,{\\rm i},0)$\nand $\\hat{\\bm{B}}\\propto(1,0,0)$, respectively.\nIn the case of a nonhelical field, the decay rate is not quenched at all,\nbut in the helical case quenching sets in for\n$\\overline{\\mbox{\\boldmath $B$}}{}}{^2\/B_{\\rm eq}^2\\ga R_{\\rm m}^{-1}$.\nThe onset of quenching at $\\overline{\\mbox{\\boldmath $B$}}{}}{^2\/B_{\\rm eq}^2\\approx R_{\\rm m}^{-1}$\nis well reproduced by the simulation.\nIn the nonhelical case, however, some weaker form of quenching sets in\nwhen $\\overline{\\mbox{\\boldmath $B$}}{}}{^2\/B_{\\rm eq}^2\\approx1$.\nWe refer to this as standard quenching (e.g.\\ Kitchatinov et al.\\ 1994)\nwhich is known to be always present; see \\Eq{SlowSat}.\nBlackman \\& Brandenburg (2002) found that, for a range of different\nvalues of $R_{\\rm m}$, $\\tilde{g}=3$ results in a good description of\nthe simulations of cyclic $\\alpha\\Omega$-type dynamos that were reported\nby Brandenburg et al.\\ (2002.).\n\n\\section{Relevance to the reversed field pinch}\n\nThe dynamical quenching approach has been applied to modeling\nthe dynamics of the reversed field pinch, where one has an initially\nhelical magnetic field of the form\n\\begin{equation}\n\\overline{\\mbox{\\boldmath $B$}}{}}{=\\hat{\\bm{B}}\\pmatrix{0\\cr J_1(kr)\\cr J_0(kr)},\n\\end{equation}\nwhere we have adopted cylindrical coordinates $(r,\\theta,z)$.\nIn a cylinder of radius $R$ such an initial field becomes kink unstable\nwhen $kR\\ga\\pi$.\nBoth laboratory measurements (e.g.\\ Caramana \\& Baker 1984) and numerical\nsimulations (Ho et al.\\ 1989) confirm the idea that the field-aligned\ncurrent leads to kink instability, and hence to small-scale turbulence\nand thereby to the emergence of $\\eta_{\\rm t}$ and $\\alpha$.\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=.6\\columnwidth]{ZT40a}\n\\end{center}\\caption[]{\nTime evolution of toroidal flux in the RFP experiment of\nCaramana \\& Baker (1984) compared with a calculation with no dynamo effect.\nThe decay around $t=18$ms is due to termination of the applied electric field.\nCourtesy of E. J. Caramana and D. A. Baker.\n}\\label{toroidal_flux}\\end{figure}\n\nJust as in the case discussed in \\Sec{Sdynquench}, the emergence of $\\alpha$\nslows down the decay in such a way that the toroidal field remains nearly\nconstant and is maintained against resistive decay; see \\Fig{toroidal_flux}.\nThe details of this mechanism have been discussed by Ji \\& Prager (2002).\nIn particular, they show that the parallel electric field cannot be\nbalanced by the resistive term alone, and that there must be an additional\ncomponent resulting from small-scale correlations of velocity and magnetic\nfield, $\\overline{\\mbox{\\boldmath $u$} {}\\times\\mbox{\\boldmath $b$} {}}$, that explain the observed profiles of\nmean electric field and mean current density.\nFurthermore, the parallel electric field reverses sign near the edge\nof the device, while the parallel current density does not.\nAgain, this can only be explained by additional contributions from\nsmall-scale correlations of velocity and magnetic field.\nThe RFP experiment also shows that magnetic helicity evolves on time scales\nfaster than the resistive scale, which is only compatible with the presence\nof a finite magnetic helicity flux divergence (Ji et al.\\ 1995, Ji 1999).\n\n\\section{Magnetic helicity in realistic dynamos}\n\nAstrophysical dynamos saturate and evolve on dynamical time scales and\nare thus not resistively slow.\nCurrent research shows that this can be achieved by expelling\nmagnetic helicity from the domain through helicity fluxes.\nThis is why we have allowed for the $\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_\\alpha$ term\nin \\Eq{dalphaMdt}.\nSince the magnetic $\\alpha$ effect is proportional to the current\nhelicity of the fluctuating field, the $\\overline{\\mbox{\\boldmath $F$}}{}}{_\\alpha$ flux should\nbe proportional to the current helicity flux of the\nfluctuating field.\n\nThe presence of the flux term generally lowers the value of\n$|\\alpha_{\\it M}|$, and since the $\\alpha_{\\it M}$ term quenches the total\nvalue of $\\alpha$ ($=\\alpha_{\\it K}+\\alpha_{\\it M}$), the effect of this helicity flux\nis to alleviate an otherwise catastrophic quenching.\nIndeed, in an open domain and without a flux divergence, the $\\alpha_{\\it M}\/R_{\\rm m}$\nterm in \\Eq{dalphaMdt} can result in ``catastrophically low'' saturation field\nstrengths that are by a factor $R_{\\rm m}^{1\/2}$ smaller than the equipartition\nfield strength (Gruzinov \\& Diamond 1994; Brandenburg \\& Subramanian 2005b).\n\nMagnetic helicity obeys a conservation law and is therefore\nconceptually easier to tackle than current helicity.\nHowever, there is the difficulty of gauge dependence of magnetic\nhelicity density and its flux.\nIt is therefore safer to work with the current helicity, which is\nalso the quantity that enters in \\Eqs{alphaKM}{dalphaMdt}.\nHowever, more work needs to be done to establish the connection between\nthe two approaches.\n\nOver the past 10 years there has been mounting evidence that the Sun\nsheds magnetic helicity (and hence current helicity) through coronal\nmass ejections and other events.\nUnderstanding the functional form of such fluxes is very much a matter\nof ongoing research (Subramanian \\& Brandenburg 2004, 2006,\nBrandenburg et al.\\ 2009).\nIn the following section we present a simple calculation that allows\nus to estimate the amount of magnetic helicity losses required for the\nsolar dynamo to work.\n\n\\section{Estimating the required magnetic helicity losses}\n\nIn order to estimate the magnetic helicity losses required to alleviate\ncatastrophic quenching we make use of the relation between the $\\alpha$ effect\nand the divergence of the current helicity flux\n(Brandenburg \\& Subramanian 2005a),\n\\begin{equation}\n\\alpha={\\alpha_{\\it K}+R_{\\rm m}\\left[\n\\eta_{\\rm t}\\mu_0\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{\/B_{\\rm eq}^2\n-\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}\/(2 k_{\\rm f}^2B_{\\rm eq}^2)\n-\\dot\\alpha\/(2\\eta_{\\rm t} k_{\\rm f}^2)\\right]\n\\over1+R_{\\rm m}\\overline{\\mbox{\\boldmath $B$}}{}}{^2\/B_{\\rm eq}^2},\n\\label{QuenchExtra2}\n\\end{equation}\nwhere $\\dot\\alpha=\\partial\\alpha\/\\partial t$ and $\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}$\nis the mean flux of current helicity from the small-scale field,\n$\\overline{(\\mbox{\\boldmath $\\nabla$} {}\\times\\mbox{\\boldmath $e$} {})\\times(\\mbox{\\boldmath $\\nabla$} {}\\times\\mbox{\\boldmath $b$} {})}$, where $\\mbox{\\boldmath $e$} {}$ is\nthe fluctuating component of the electric field; see also\nSubramanian \\& Brandenburg (2004).\nIn the steady-state limit and at large $R_{\\rm m}$ we have\n\\begin{equation}\n\\alpha\\approx\n\\eta_{\\rm t}{\\mu_0\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{\\over\\overline{\\mbox{\\boldmath $B$}}{}}{^2}\n-{\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}\\over2 k_{\\rm f}^2\\overline{\\mbox{\\boldmath $B$}}{}}{^2}.\n\\label{QuenchExtra3}\n\\end{equation}\nWe neglect the $\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{$ term, because the catastrophic\nquenching in dynamos with boundaries has never been seen to be alleviated\nby this term (Brandenburg \\& Subramanian 2005b).\nThus, we use \\Eq{QuenchExtra3} to estimate $\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}$ as\n\\begin{equation}\n\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}=2\\alpha k_{\\rm f}^2\\overline{\\mbox{\\boldmath $B$}}{}}{^2.\n\\end{equation}\nNext, we take the volume integral over one hemisphere, i.e.\\\n\\begin{equation}\n{\\cal L}_{\\rm C}\\equiv\\oint_{2\\pi}\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}\\cdot{\\rm d} {}\\mbox{\\boldmath $S$} {}\n=\\int\\mbox{\\boldmath $\\nabla$} {}\\cdot\\overline{\\mbox{\\boldmath $F$}}{}}{_{\\rm C}\\,{\\rm d} {} V\n={2\\pi\\over3}R^3\\bra{2\\alpha k_{\\rm f}^2\\overline{\\mbox{\\boldmath $B$}}{}}{^2},\n\\end{equation}\nwhere ${\\cal L}_{\\rm C}$ is the ``luminosity'' or ``power'' of current helicity.\nWe estimate $\\bra{\\alpha}$ using $\\alpha\\Omega$ dynamo theory\nwhich predicts that (e.g., Robinson \\& Durney 1982, see also\nBrandenburg \\& Subramanian 2005a)\n\\begin{equation}\n\\alpha k_1\\Delta\\Omega\\approx\\omega_{\\rm cyc}^2,\n\\end{equation}\nwhere $\\omega_{\\rm cyc}=2\\pi\/T_{\\rm cyc}$ is the cycle frequency\nof the dynamo, $T_{\\rm cyc}$ is the 22 year cycle period of the Sun,\n$\\alpha$ is assumed constant over each hemisphere, and $\\Delta\\Omega$\nis the total latitudinal shear, i.e.\\ about $0.3\\Omega$ for the Sun.\nThere is obviously an uncertainty in relating local values of\n$\\alpha$ to volume averages.\nHowever, if we do set the two equal, we obtain at least an upper limit\nfor ${\\cal L}_{\\rm C}$.\nWe may then relate this to the luminosity of {\\it magnetic} helicity,\n${\\cal L}_{\\rm H}$, that we assume to be proportional to ${\\cal L}_{\\rm C}$\nvia\n\\begin{equation}\n{\\cal L}_{\\rm C}=k_{\\rm f}^2{\\cal L}_{\\rm H}.\n\\label{LCLH}\n\\end{equation}\nWith this we find for the total magnetic helicity loss over half a\ncycle (one 11 year cycle)\n\\begin{equation}\n{\\textstyle{1\\over2}}{\\cal L}_{\\rm H}T_{\\rm cyc}\\leq{4\\pi\\over3}R^3\n{\\omega_{\\rm cyc}^2\\over k_1\\Delta\\Omega}T_{\\rm cyc}\\bra{\\overline{\\mbox{\\boldmath $B$}}{}}{^2}\n={4\\pi\\over3}LR^3{\\omega_{\\rm cyc}\\over\\Delta\\Omega}\\bra{\\overline{\\mbox{\\boldmath $B$}}{}}{^2},\n\\end{equation}\nwhere we have estimated $k_1=2\\pi\/L$ for the relevant wavenumber of the\ndynamo in terms of the thickness of the convection zone $L$.\nInserting now values relevant for the Sun, $L=200\\,{\\rm Mm}$, $R=700\\,{\\rm Mm}$,\n$\\omega_{\\rm cyc}\/\\Delta\\Omega=10^{-2}$, and $\\overline{\\mbox{\\boldmath $B$}}{}}{\\sim300\\,{\\rm G}$,\nwe obtain\n\\begin{equation}\n{\\textstyle{1\\over2}}{\\cal L}_{\\rm H}T_{\\rm cyc}\\leq10^{46}\\,{\\rm Mx}^2\/\\mbox{cycle}.\n\\label{estimate}\n\\end{equation}\nThis is comparable to earlier estimates based partly on observations\n(Berger \\& Ruzmaikin 2000; DeVore 2000) and partly on turbulence simulations\n(Brandenburg \\& Sandin 2004), but we recall that \\Eq{estimate} is only\nan upper limit.\nFurthermore, the connection between current helicity and magnetic helicity\nassumed in relation \\eq{LCLH} is quite rough and has only been seriously\nconfirmed under isotropic conditions.\nIn addition, it is not clear that the gauge-invariant magnetic helicity\nflux defined by Berger \\& Field (1984) and Finn \\& Antonsen (1985)\nis actually the quantity of interest.\nFollowing earlier work of Subramanian \\& Brandenburg (2006), the\ngauge-invariant magnetic helicity is not in any obvious way related\nto the current helicity which, in turn, is related to the density\nof the flux linking number and is at least approximately equal to\nthe magnetic helicity in the Coulomb gauge.\n\nIn summary, although magnetic helicity is conceptually advantageous\nin that it obeys a conservation equation, the difficulty in dealing\nwith a gauge-dependent quantity can be quite serious.\nMoreover, as emphasized before, it is really the current helicity that\nis primarily of interest, so it would be useful to shift attention from\nmagnetic helicity fluxes to current helicity fluxes.\n\n\\section{Conclusions}\n\nIn this review we have attempted to highlight the importance of\nmagnetic helicity in modern nonlinear dynamo theory.\nMuch of the early work on the reversed field pinch since the\nmid 1980s now proves to be extremely relevant in view of the\npossibility of resistively slow saturation by a self-inflicted\nbuild-up of small-scale current helicity, $\\overline{\\mbox{\\boldmath $j$} {}\\cdot\\mbox{\\boldmath $b$} {}}$,\nas the dynamo produces large-scale current helicity, $\\overline{\\mbox{\\boldmath $J$}}{}}{\\cdot\\overline{\\mbox{\\boldmath $B$}}{}}{$.\nSince this concept is not yet universally accepted, the additional evidence\nfrom the reversed field pinch experiment can be quite useful.\n\nThe interpretation of resistively slow saturation has led to the\nproposed solution that magnetic helicity fluxes (or current helicity\nfluxes) are responsible for removing excess small-scale magnetic\nhelicity from the system.\nThis allows the dynamo to reach saturation levels that can otherwise\nbe $R_{\\rm m}^{1\/2}$ times smaller than the equipartition value given\nby the kinetic energy density of the turbulent motions.\nThe consequences of this prediction have been tested in direct simulations\n(Fig.~3 of Brandenburg 2005 and Fig.~17 of K\\\"apyl\\\"a et al.\\ 2008),\nconfirming thereby ultimately basic aspects of incorporating the\nmagnetic helicity equation into mean-field models.\nHowever, more progress is needed in addressing questions regarding the\nrelative importance of magnetic and current helicity fluxes through the\nsurface compared to diffusive fluxes across the equator\n(Brandenburg et al.\\ 2009).\n\nWe reiterate that the reversed field pinch experiment is not\n{\\it directly} relevant to the self-excited dynamo, but rather\nits nonlinear saturation mechanism.\nSo far, successful self-excited dynamo experiments have only been\nperformed with liquid sodium (Gailitis et al.\\ 2000;\nStieglitz \\& M\\\"uller 2001; Monchaux et al.\\ 2007).\nThis may change in future given that one can usually achieve much\nhigher magnetic Reynolds numbers in plasmas than in liquid metals\n(Spence et al.\\ 2009).\nIt might then, for the first time, be possible to address experimentally\nquestions regarding the relative importance of magnetic and current\nhelicity fluxes for dynamos.\n\n\\subsection*{Acknowledgments}\n\nI thank the referees for detailed and constructive comments\nthat have helped to improve the presentation.\nThis work was supported in part by the Swedish Research Council,\ngrant 621-2007-4064, and the European Research Council under the\nAstroDyn Research Project 227952.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFinancial options are securities that\nconvey rights to conduct specific trades in the future.\nFor example, an S\\&P\\,500\\xspace \\emph{call option} with \\emph{strike price} 4500 and \\emph{expiration date} December 31, 2021, provides the right to buy one share of S\\&P\\,500\\xspace at $\\$4500$ on that day.%\n\\footnote{For simplicity, examples are given without the typical 100x contract multiplier.}\nA standard financial option is a derivative instrument of a \\emph{single} underlying asset or index, and its payoff is a function of the underlying variable. \nThe example option contract pays $\\max\\{S-4500,0\\}$, where $S$ is the value of the S\\&P\\,500\\xspace index on the expiration date.%\n\\footnote{\n\tThroughout this paper, we restrict to \\emph{European options} which can be exercised only at expiration.\n\tThe settlement value is calculated as the opening value of the index on the expiration date or the last business day (usually a Friday) before the expiration date.\n\tIn cash-settled markets, instead of actual physical delivery of the underlying asset, the option holder gets a cash payment that is equivalent to the value of the asset.\n} \nInvestors trade options to hedge risks and achieve certain return patterns, or to speculate about the movement of the underlying asset price, buying an option when its price falls below their estimate of its expected value. \nThus, option prices reveal the collective risk-neutral belief distribution of the underlying asset's future value.\n\nDespite the significant volume of trade and interest in option contracts, financial exchanges that support trading of standard options have two limitations that compromise its expressiveness and efficiency.\n\\textit{First}, the exchange offers only a selective set of markets for options written on a specific underlying asset (i.e., option chain), in which each market features a predetermined strike price and expiration date.\nFor example, as of this writing, the Chicago Board Options Exchange (CBOE) offers 120 distinct strike prices, ranging from 1000 to 6000 at intervals of 25, 50, or 100, for S\\&P\\,500\\xspace options expiring by the end of 2021. \nWhile traders can engineer custom contracts (e.g., a S\\&P\\,500\\xspace call option with a strike price of \\$4520) by simultaneously purchasing multiple available options in appropriate proportions, they must monitor several markets to ensure that a bundle can be constructed at a desired price. \nOften, execution risk and transaction costs prevent individual traders from carrying out such strategies. \nAs the prescriptive markets prevent traders from conveniently expressing custom strike values, the exchange may fail to aggregate supply and demand requests of greater detail, leading to a loss of economic efficiency.\n\n\\textit{Second}, each market within an option chain independently aggregates and matches orders in regard to a single contract with a specified option type, strike price, and expiration date, despite its interconnectedness to other option markets and their common dependency on the underlying asset.\nSuch independent market design fails to notify traders of strictly better, cheaper options\nand generically allows arbitrage opportunities.\nMoreover, investments get diluted across independent markets even when participants are interested in the same underlying asset.\nThis can lead to the problem of thin markets, where few trades happen and bid-ask spreads become wide.\nEven for some of the most actively traded option families, empirical evidence has shown that liquidity can vary much across option types and strikes.\n\\citeauthor{Cao2010} studied eight years of options trading data and find consistently lower liquidity in puts and deep in-the-money options~\\cite{Cao2010}.\n\n\\subsection{An Exchange for Standard Financial Options}\nThe paper first focuses on the design of an exchange for standard financial options to address limitations discussed above.\nWe propose a mechanism to aggregate and match orders on options across different strike prices that are logically related to the same underlying asset.\nAs a result, the exchange operates a consolidated market for each underlying security and expiration, where traders can specify custom options of arbitrary strike value.\nThe mechanism is an exchange that generalizes the double auction; it is not a market maker or any individual arbitrageur. \nIt works by finding a match that maximizes net profit subject to zero loss even in the worst case, and thus poses no risk to the exchange regardless of the value of the underlying security at expiration.\nWe show that the mechanism is computationally efficient, and key market operations (i.e., match and price quotes) can be computed in time polynomial in the number of orders.\nWe conduct experiments on real-market options data, using our mechanism to consolidate outstanding orders from independently-traded options markets.\nEmpirical results demonstrate that the proposed mechanism can match options that the current exchange cannot and provide more competitive bid and ask prices.\nThe improved efficiency and expressiveness may help to aggregate more fine-grained information and recover a complete and fully general probability distribution of the underlying asset's future value at expiration.\n\n\\subsection{An Exchange for Combinatorial Financial Options}\nIn the second part of the work, we generalize standard financial options to define \\textit{combinatorial financial options}---derivatives that give contract holders the right to buy or sell \\emph{any linear combination} of underlying assets at any strike price.\nFor example, a call option written on ``$1$\\texttt{AAPL}\\xspace$+1$\\texttt{MSFT}\\xspace'' with strike price 300 specifies the right to buy one share of Apple \\emph{and} one share of Microsoft at $\\$300$ total price on the expiration date, whereas a call on ``$2$\\texttt{AAPL}\\xspace$-1$\\texttt{MSFT}\\xspace'' with strike price 50 confers the ability to buy two shares of Apple and sell one share of Microsoft at expiration for a net cost of \\$50.\nCombinatorial options allow traders to conveniently and precisely hedge their exact portfolio, replicating any payoff functions that standard options can achieve and exponentially more. \nSome of the standard options on mutual funds and stock indices, like the S\\&P\\,500\\xspace and Dow Jones Industrial Average (DJI), are indeed combinatorial options written on the respectively predefined portfolios. \nThe CBOE recently launched options on eleven Select Sector Indices,%\n\\footnote{\\url{https:\/\/markets.cboe.com\/tradable_products\/sp_500\/cboe_select_sectors_index_options\/}}\neach of which can be considered as a combinatorial option of a pre-specified linear combination of stocks representing one economic sector.\nThe goal of this work is to allow traders to create options on their own custom indices, representing any linear combination of stocks and sectors that they want.\n\nWe design an exchange for combinatorial financial options.\nSuch a market enables the elicitation and recovery of future correlations (e.g., complimentary or substitution effects) among assets.\nHowever, as traders are offered the expressiveness to specify weights (or shares) for multiple underlying assets, new challenges arise and the thin market problem exacerbates.\nOpening a new, separate exchange for each new combination of stocks and weights would rapidly grow intractable.\nNaively matching buy and sell orders for only the exact same portfolio may yield few or no trades, despite plenty of acceptable trades among orders.\n\nExtending the mechanism that consolidates standard options on a single underlying security, we propose an optimization formulation that can match combinatorial options written on different linear combinations of underlying assets.\nThe mechanism maximizes net profit subject to no risk to the exchange, regardless of (any combination of) values of all assets at expiration. \nWe show that the proposed matching mechanism with increased expressiveness, however, comes at the cost of higher computational complexity: the optimal clearing of a combinatorial options market is coNP-hard. \n\nWe demonstrate that the proposed mechanism can be solved relatively efficiently by exploiting constraint generation techniques.\nWe propose an algorithm that finds the exact optimal-matching solution by satisfying an increasing set of constraints iteratively generated from different future values of the underlying assets.\nIn experiments on synthetic combinatorial orders generated from real-market standard options prices, we show that the matching algorithm terminates quickly, with its running time growing linearly with the number of orders and the size of underlying assets. \n\n\\subsection{Roadmap}\nSection~\\ref{sec:related} discusses related work on option pricing and combinatorial markets.\nSection~\\ref{sec:notations} introduces notations and background on the current options market.\nWe first focus on the market design for standard financial options. \nSection~\\ref{sec:standard_option} introduces the formal setting of a consolidated options market with limit orders, and proposes a matching mechanism for options across types and strikes (\\Mech{single_match}).\nSection~\\ref{sec:combo_options}, our main contribution, defines a combinatorial financial options exchange (Mechanism \\ref{mech:combo_match}), analyzes the computational complexity of optimally clearing such a market (Theorem~\\ref{thm:NP-complete} and Theorem~\\ref{thm:coNP-hard}), and proposes a constraint generation algorithm to find the exact optimal match (Algorithm~\\ref{algo:comb_match}). \nSection~\\ref{sec:exp_standard_option} and Section~\\ref{sec:exp_combo_option} evaluate the two proposed mechanisms on real-market standard options data and synthetically generated combinatorial options data, respectively.\nSection~\\ref{sec:conclusion} concludes and discusses possible future directions.\n\n\\section{Related Work}\n\\label{sec:related}\n\\subsection{Rational Option Pricing}\nOur proposed market designs relate to arbitrage conditions that have been studied extensively in financial economics ~\\cite{Modigliani1958,Varian1987}.\nIn short, an arbitrage describes the scenario of ``free lunches''---configurations of prices such that one can get something for nothing.\nThe matching operation of an exchange (i.e., the auctioneer function) can be considered as arbitrage elimination: matching orders in effect works by identifying combinations of orders that reflect a risk-free surplus (i.e., gains from trade). \nIn the classic double auction, a match accepting the highest buy order at, say, \\$12 and the lowest sell order at \\$10 in effect yields an arbitrage surplus of \\$2 that can go to the buyer, the seller, the exchange, or split among them in any way. \n\n\\citeauthor{Merton1973} first investigated the no-arbitrage pricing for options, stating the necessity of convexity in option prices~\\cite{Merton1973}.\nOther relevant works examine no-arbitrage conditions for options under different scenarios, such as modeling the stochastic behavior of the underlying asset (e.g., the Black-Scholes model)~\\cite{bs_model,pricing_discrete_time} and considering the presence of other types of securities (e.g., bonds and futures)~\\cite{Garman1976, Ritchken1985}.\nThe most relevant work to our proposed approach is by \\citeauthor{Herzel2005}, who makes little assumption on the underlying process of the asset price and other financial instruments~\\cite{Herzel2005}.\nThe paper proposes a linear program to check the convexity between every strike-price pair; it finds arbitrage opportunities that yield positive cash flows now and no liabilities in the future on European options written on the same underlying security.\n\nOur first contribution on a market design that consolidates standard options generalizes Herzel's by adding an extra degree of freedom to incorporate potential future payoff gain or loss into the matching.\nTo our knowledge, no prior work has defined general combinatorial financial options or investigated the matching mechanism design and complexity for such a market.\n\\subsection{Combinatorial Market Design}\nMuch prior work examines the design of combinatorial markets, both exchanges and market makers \\cite{Chen2008b,Hanson03}, for different applications including prediction markets with Boolean combinations \\cite{Fortnow2005}, permutations \\cite{Chen2007}, hierarchical structures \\cite{Guo2009}, tournaments \\cite{Chen2008a}, and electronic sourcing \\cite{Sandholm2007}. \n\\citeauthor{DudikEtAl13} show how to employ constraint generation in linear programming to keep complex related prices consistent~\\cite{DudikEtAl13}. \n\\citeauthor{KroerDudik16} generalize this approach using integer programming~\\cite{KroerDudik16}.\n\\citeauthor{Rothschild14} investigate misaligned prices for logically related binary-payoff contracts in prediction market, and uncover persistent arbitrage opportunities for risk-neutral investors across different exchanges~\\cite{Rothschild14}.\n\nDesigning combinatorial markets faces the tradeoff between expressiveness and computational complexity: \ngiving participants\ngreater flexibility to express preferences can help to elicit better information and increase economic efficiency, but in the meantime leads to a more intricate mechanism that is computationally harder.\nSeveral works have formally described and quantified such tradeoff \\cite{Benisch2008,Golovin2007}, and studied how to balance it by exploiting the outcome space structure and limiting expressivity \\cite{Chen2008b,XiaPe11,LaskeyEtAl18,Dudik2020}.\nOur work here contributes to this rich literature by designing a vastly more expressive version of the popular financial options market. \nThe state space and payoff function for combinatorial financial options differ from the previously studied combinatorial structures in both meaning and computational complexity.\nThe associated matching problem is novel (with its new form of payoff function), and is not a specialization of previously studied combinatorial prediction market.\n\n\\section{Background and Notations}\n\\label{sec:notations}\nThere are two types of options, referred to as \\emph{call} and \\emph{put} options.\nWe denote a call option as $C(S, K, T)$ and a put option as $P(S, K, T)$, which respectively gives the option buyer the right to \\emph{buy} and \\emph{sell} an underlying asset $S$ at a specified \\emph{strike price} $K$ on the \\emph{expiration date} $T$.\nIn the rest of this paper, we omit $T$ from the tuples for simplicity, as the mechanism aggregates options within the same expiration.\n\nThe option buyer decides whether to exercise an option.\nSuppose that a buyer spends \\$8 and purchases a call option, $C(S\\&P\\,500\\xspace, 4500, 20211231)$.\nIf the S\\&P\\,500\\xspace index is \\$4700 at expiration, the buyer will pay the agreed strike \\$4500, receive the index, and get a \\textit{payoff} of \\$200 and a \\textit{net profit} of \\$192 (assuming no time value). \nIf the S\\&P\\,500\\xspace price is \\$4200, the buyer will walk away without exercising the option.\nTherefore, the payoff of a purchased option is \n\\[\\Psi := \\max\\{\\chi(S-K), 0\\},\\]\nwhere $S$ is the value of underlying asset at expiration and $\\chi \\in \\{-1, 1\\}$ equals 1 for calls and $-1$ for puts.\nAs the payoff for a buyer is always non-negative, the seller receives a \\emph{premium} now (e.g., \\$8) to compensate for future obligations.\n\nOption contracts written on the same underlying asset, type, strike, and expiration are referred to as an \\emph{option series}.\nConsider options of a single security offering both calls and puts, ten expiration dates and fifty strike prices.\nAll option series render a total of a thousand markets, with each maintaining a separate \\emph{limit order book}.\nIn such a market, deciding the existence of a transaction takes $\\mathcal{O}(1)$ time by comparing the best bid and ask prices, and matching an incoming order can take up to $\\mathcal{O}(n_\\text{o})$ time depending on its quantity, where $n_\\text{o}$ is the number of orders on the opposite side of the order book.\n\n\\section{Consolidating Standard Financial Options}\n\\label{sec:standard_option}\nThis section proposes a mechanism to consolidate buy and sell orders on standard financial options written on the same underlying asset across different types and strike prices.\nThe model does not make any assumption on the option's pricing model or the stochastic behavior of the underlying security.  \n\\subsection{Matching Orders on Standard Financial Options}\nConsider an option market in regard to a single underlying asset $S$ with an expiration date $T$\\@.\nTraders can specify orders on such options with any positive strike value.\nThe market has a set of buy orders, indexed by $m \\in \\{1, 2, ..., M\\}$, and a set of sell orders, indexed by $n \\in \\{1, 2, ..., N\\}$. \nBuy orders are represented by a type vector $\\boldsymbol{\\phi} \\in \\{-1, 1\\}^M$ with each entry specifying a put or a call option, a strike price vector $\\boldsymbol{p} \\in \\mathbb{R}_{+}^M$, and bid prices $\\boldsymbol{b} \\in \\mathbb{R}_{+}^M$.\nSell orders are denoted by a separate type vector $\\boldsymbol{\\psi} \\in \\{-1, 1\\}^N$, a strike vector ${\\boldsymbol{q}} \\in \\mathbb{R}_{+}^N$, and ask prices $\\boldsymbol{a} \\in \\mathbb{R}_{+}^N$.\n\nThe exchange aims to match buy and sell orders submitted by traders.\nSpecifically, it decides the fraction $\\boldsymbol{\\gamma} \\in [0, 1]^M$ to sell to buy orders and the fraction $\\boldsymbol{\\delta} \\in [0, 1]^{N}$ to buy from sell orders.\nWe start with a formulation that matches orders by finding a common form of arbitrage in which the exchange maximizes net profit at the time of order transaction, subject to \\textit{zero} payoff loss in the future for all possible states of $\\S$:\n\\begin{align}\n\t\\max \\limits_{\\boldsymbol{\\gamma}, \\boldsymbol{\\delta}} & \\displaystyle \\quad \\boldsymbol{b}^\\top \\boldsymbol{\\gamma} - \\boldsymbol{a}^\\top \\boldsymbol{\\delta} \\notag\\\\\n\t\\text{s.t.} & \\displaystyle \\quad \\,\\,\\underbrace{\\!\\!\\sum_{m} \\gamma_m \\max\\{\\phi_m(S - p_m), 0\\}\\!\\!}_{\\Psi_{\\Gamma}}\\,\\, - \\,\\,\\underbrace{\\!\\!\\sum_{n} \\delta_n \\max\\{\\psi_n(S - q_n), 0\\}\\!\\!}_{\\Psi_{\\Delta}}\\,\\,\\leq 0, \\quad \\forall S \\in [0, \\infty) \\notag\n\\end{align}\nHere, the first term $\\Psi_{\\Gamma}:=\\sum_{m} \\gamma_m \\max\\{\\phi_m(S - p_m), 0\\}$ in the constraint calculates the total payoff of sold options as a function of $S$, which is the obligation or liability of the exchange at the time of option expiration.\nThe second term $\\Psi_{\\Delta} := \\sum_{n} \\delta_n \\max\\{\\psi_n(S - q_n), 0\\}$ computes the total payoff of bought options, by which the exchange has the right to exercise.\nThe constraint guarantees that the liability of the exchange does not exceed its payoff, regardless of the value of $S$ on the expiration date.\nWe denote options bought by the exchange as Portfolio $\\Delta$ and options sold as Portfolio $\\Gamma$, and the constraint enforces that Portfolio $\\Delta$ \\emph{(weakly) dominates} Portfolio $\\Gamma$.\n\nWe further extend the above formulation.\nInstead of restricting the exchange to zero loss at expiration, we allow it to take into account the potential (worst-case) deficit or gain in the future.\nWe denote this extra degree of freedom as a decision variable $L \\in \\mathbb{R}$, and have the following matching mechanism:\n\\begin{align}\\tag{M.1}\n\\label{mech:single_match}\n\\max \\limits_{\\boldsymbol{\\gamma}, \\boldsymbol{\\delta}, L} & \\displaystyle \\quad \\boldsymbol{b}^\\top \\boldsymbol{\\gamma} - \\boldsymbol{a}^\\top \\boldsymbol{\\delta} - L\\\\\n\\text{s.t.} & \\displaystyle \\quad \\,\\,\\underbrace{\\!\\!\\sum_{m} \\gamma_m \\max\\{\\phi_m(S - p_m), 0\\}\\!\\!}_{\\Psi_{\\Gamma}}\\,\\, - \\,\\,\\underbrace{\\!\\!\\sum_{n} \\delta_n \\max\\{\\psi_n(S - q_n), 0\\}\\!\\!}_{\\Psi_{\\Delta}}\\,\\,\\leq L, \\quad \\forall S \\in [0, \\infty) \\label{eq:single_constraint}\n\\end{align}\n\nThe constraint now guarantees that the difference between the liability and the payoff of the exchange will be bounded by $L$, regardless of the value of $S$ on the expiration date. \nFollowing the definition below, we say that the constraint enforces Portfolio $\\Delta$ to \\emph{(weakly) dominate} Portfolio $\\Gamma$ with some constant offset $L$.\n\\begin{definition} [Payoff dominance with an offset. Adapted from~\\citeauthor{Merton1973}~\\cite{Merton1973}]\n\t\\label{def:opt_dominate}\n\n\tPortfolio $\\Delta$ \\emph{(weakly) dominates} Portfolio $\\Gamma$ with an offset $L$, if the payoff of Portfolio $\\Delta$ plus a constant $L$ is\n\tgreater than or equal to that of Portfolio $\\Gamma$ for all possible states of the underlying variable at expiration. \n\tPortfolio $\\Gamma$ is said to be \\emph{(weakly) dominated} by Portfolio $\\Delta$ with an offset $L$.\n\\end{definition}\nSince this potential gain or loss is further incorporated in the objective at the time of matching, the mechanism \\ref{mech:single_match}  guarantees no overall loss for each match, and enjoys an extra degree of freedom to trade.\nWe give two motivating examples based on real-market options data to illustrate the economic meaning and the usefulness of a flexible $L$.\nEach of the matches shown below would not have been found if we restrict a zero payoff loss for the exchange.\n\\Ex{nonneg_L} showcases the scenario where $L$ allows the exchange to take a (worst-case) deficit at the time of option expiration, if it is preemptively covered by a surplus (i.e., revenue) at the time of order transaction.\n\\begin{example} [A match with a positive $L$]\n\t\\label{ex:nonneg_L}\n\n\tWe use \\ref{mech:single_match} to consolidate options of Walt Disney Co. (\\texttt{DIS}\\xspace) that are priced on January 23, 2019 and expire on June 21, 2019.\n\tWe find the following match, where each order would not transact in its corresponding independent market. \n\tThe exchange can\n\t\\begin{itemize}\n\t\t\\setlength{\\itemsep}{2pt}\n\t\t\\item sell to the buy order on $C(\\texttt{DIS}\\xspace, 110)$ at bid \\$7.2,\n\t\t\\item sell to the buy order on $P(\\texttt{DIS}\\xspace, 150)$ at bid \\$38.75,\n\t\t\\item buy from the sell order on $C(\\texttt{DIS}\\xspace, 150)$ at ask \\$0.05,\n\t\t\\item buy from the sell order on $P(\\texttt{DIS}\\xspace, 110)$ at ask \\$5.1.\n\t\\end{itemize}\n\tTherefore, the exchange gets an immediate gain of \\$40.8 ($7.2+38.75-5.1-0.05$).\n\n\tFigure~\\ref{subfig:lgtzero} plots the payoffs of bought and sold options as a function of DIS, showing that the exchange will have a net liability of $\\$40$ (i.e., $L=40$), regardless of the DIS value at expiration.\n\tThe exchange makes a net profit of \\$0.80 from the match at no risk.\n\t\\qed\n\\end{example}\n\\Ex{neg_L} below demonstrates a different scenario where the decision variable $L$ allows the exchange to have a temporary deficit (i.e., expense) at the time of order transaction, if it is guaranteed to earn it back later at the time of option expiration. \n\\begin{example} [A match with a negative $L$]\n\t\\label{ex:neg_L}\n\tWe consolidate options of Apple Inc. (\\texttt{AAPL}\\xspace) that are priced on January 23, 2019 and expire on January 17, 2020.\n\tWe find the following match, where the exchange can\n\t\\begin{itemize}\n\t\t\\setlength{\\itemsep}{2pt}\n\t\t\\item sell to the buy order on $C(\\texttt{AAPL}\\xspace, 160)$ at bid \\$14.1,\n\t\t\\item sell to the buy order on $P(\\texttt{AAPL}\\xspace, 80)$ at bid \\$0.62,\n\t\t\\item buy from the sell order on $C(\\texttt{AAPL}\\xspace, 80)$ at ask \\$74.2,\n\t\t\\item buy from the sell order on $P(\\texttt{AAPL}\\xspace, 160)$ at ask \\$19.1.\n\t\\end{itemize}\n\tThe match incurs an expense of \\$78.58 now ($14.1+0.62-74.2-19.1$), and yields a guaranteed payoff of \\$80 (i.e., $L=-80$) at expiration. \n\tFigure~\\ref{subfig:lstzero} depicts the respective payoffs of bought and sold options.\n\tFrom the match, we can infer an interest rate of 1.82\\%, calculated by $78.58 e^{r \\Delta t} = 80$.\n\t\\qed\n\\end{example}\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{subfigure}{0.47\\columnwidth}\t\n\t\n\t\t\\includegraphics[width=0.9\\columnwidth]{opt_img\/DIS_payoff.pdf}\n\t\t\\caption{Payoffs of standard options bought and sold in \\Ex{nonneg_L} as a function of DIS value.}\n\t\t\\label{subfig:lgtzero}\n\t\\end{subfigure}\n\t\\hspace{0.04\\columnwidth}\n\t\\begin{subfigure}{0.47\\columnwidth}\t\n\t\n\t\t\\includegraphics[width=0.9\\columnwidth]{opt_img\/AAPL_payoff.pdf}\n\t\t\\caption{Payoffs of standard options bought and sold in \\Ex{neg_L} as a function of AAPL value.}\n\t\t\\label{subfig:lstzero}\n\t\\end{subfigure}\n\n\n\n\n\n\t\\caption[Payoffs of the matched options as a function of the value of the underlying asset at expiration.]{Payoffs of the matched options as a function of the value of the underlying asset at expiration. Fig.~\\ref{subfig:lgtzero} shows the case of $L>0$, and Fig.~\\ref{subfig:lstzero} the case of $L<0$.}\n\t\\label{fig:arb_example}\n\\end{figure}\n\nWe now analyze the complexity of running \\Mech{single_match}.\nThe left-hand side of constraint \\eqref{eq:single_constraint} is a linear combination of max functions, and thus is a piecewise linear function of $S$.\nTherefore, it suffices to solve \\ref{mech:single_match} by satisfying constraints defined by $S$ at each breakpoint.\nIn our case, breakpoints of the constraint \\eqref{eq:single_constraint}\nare the defined strike values in the market, plus two endpoints: $\\boldsymbol{p} \\cup {\\boldsymbol{q}} \\cup \\{0, \\infty\\}$.\nLet $n_K$ denotes the number of distinct strike values in the market, which is bounded above by\n$n_{\\text{orders}} = M+N$, the total number of orders in the market.\nTherefore, \\ref{mech:single_match} is a linear program that has $n_K+2$ payoff constraints, and requires time polynomial in the size of the problem instance to solve.\nWe defer the complete proof to Appendix A.1.\n\\begin{theorem}\n\t\\label{thm:consolidate_standard_options}\n\t\\Mech{single_match} matches financial options written on the same underlying asset and expiration date across all types and strike prices in time polynomial in the number of orders.\n\\end{theorem}\n\n\n\\paragraph{Remarks.}\nSeveral extensions can be directly applied to the mechanism:\n\\begin{enumerate}\n    \\setlength{\\itemsep}{0pt}\n    \\item We can incorporate the time value of investments by multiplying $L$ by a (discount) rate in the objective of \\ref{mech:single_match}.\n    \\item We note that matching solutions returned by \\ref{mech:single_match} may involve fractional shares of stocks. Ideally, an exchange (e.g., cash-settled markets) allows it, and if an exchange (e.g., physical-settled markets) does not, we would need to normalize or round.\n    \\item Mechanism~\\ref{mech:single_match} identifies a maximal bundle of orders that can be accepted without risk to the exchange, but does not specify what to do with the surplus if any. The surplus can be split arbitrarily among the involved traders and the exchange. We note that while the surplus is guaranteed to be nonnegative, it may have an uncontingent (cash) component and a state-contingent component.\n\\end{enumerate}\n\n\\subsection{Quoting Prices for Standard Financial Options}\nA standard exchange maintains the best quotes (i.e., the highest bid and lowest ask) for each independent options market.\nThis section extends \\Mech{single_match} to quote the most competitive prices for any \\emph{custom} option of a specified type and strike by considering all other options related to the same underlying security.\nWe describe the price quote procedure in an arbitrage-free options market\\footnote{That is, no match will be returned by \\Mech{single_match}.} in regard to a single underlying asset $S$ with an expiration date $T$, represented by $(\\boldsymbol{\\phi}, \\boldsymbol{p}, \\boldsymbol{b}, \\boldsymbol{\\psi}, {\\boldsymbol{q}}, \\boldsymbol{a})$. \nWe defer the detailed proof of correctness to Appendix A.2.\n\n\\begin{enumerate}\n\t\\setlength{\\itemsep}{8pt}\n\t\\item The best bid $b^*$ for an option $(\\chi, S, K)$ is the maximum gain of selling a portfolio of options that is \\emph{weakly dominated} by $(\\chi, S, K)$ with some constant offset $L$.\n\t\n\tWe derive $b^*$ by adding the option $(\\chi, S, K)$ to the sell side of the market indexed $N+1$ (as the exchange buys from sell orders), initializing its price $a_{N+1}$ to 0, and solving for \\ref{mech:single_match}. \n\tThe best bid $b^*$ is then the returned objective.\n\t\n\t\\item The best ask $a^*$ for an option $(\\chi, S, K)$ is the minimum cost of buying a portfolio of options that \\emph{weakly dominates} $(\\chi, S, K)$ with some constant offset $L$.\n\t\n\tWe derive $a^*$ by adding the option $(\\chi, S, K)$ to the buy side of the market indexed $M+1$, initializing its price $b_{M+1}$ to a large number (e.g., $10^6$), and solving for \\ref{mech:single_match}. \n\tThe best ask $a^*$ is then $b_{M+1}$ minus the returned objective.\n\\end{enumerate}\nIn the case of matching orders with multiple units, it is necessary to consider all orders in the market.\nFor deciding the existence of a match or quoting (instantaneous) prices, however, we only need to consider a set of orders that have the most competitive prices.\nWe define a set with such orders as a frontier set $\\mathcal{F}$.\n\\begin{definition} [A Frontier Set of Options Orders]\n\t\\label{def:opt_frontier}\n\tAn option order is in the \\emph{frontier set} $\\mathcal{F}$ if its bid price is greater than or equal to the maximum gain of selling a weakly dominated portfolio of options with some offset $L$, or if its ask price is less than or equal to the minimum cost of buying a weakly dominant portfolio of options for some offset $L$.\n\\end{definition}\n\n\\begin{corollary}\n\t\\label{coro:frontier_set}\n\t\\Mech{single_match} determines the existence of a match and returns the instantaneous price quote in time polynomial in $\\card{\\mathcal{F}}$.\n\\end{corollary}\n\nThe proof of Corollary~\\ref{coro:frontier_set} is deferred to Appendix A.3,\nwhich shows that in order to determine the existence of a match or quote the most competitive prices (i.e., the highest bid and the lowest ask) for any target option $(\\chi, S, K)$, it suffices to consider options orders in $\\mathcal{F}$ and run \\Mech{single_match}.\nThe runtime complexity follows immediately from \\Thm{consolidate_standard_options}.\n\n\\section{Combinatorial Financial Options}\n\\label{sec:combo_options}\nThis section introduces \\emph{combinatorial financial options} and designs a market to trade such options. \nCombinatorial options extend standard options to more general derivative contracts that can be written on any linear combination of $U$ underlying assets.\nWe formally define a combinatorial financial option and its specifications.\n\\vspace{1ex}\n\\begin{definition} [Combinatorial Financial Options]\n\tConsider a set of $~U$ underlying assets.\n\tCombinatorial financial options are derivative contracts that specify the right to buy or sell a linear combination of the $U$ underlying assets at a strike price on an expiration date.\n\tEach contract specifies a call or put type $\\chi \\in \\{1, -1\\}$, a weight vector $\\boldsymbol{\\omega} \\in \\mathbb{R}^{U}$, a strike price $K \\geq 0$, and an expiration date $T$.\n\tIt has a payoff of $\\max\\{\\chi(\\boldsymbol{\\omega}^\\top \\S-K), 0\\}$, where $\\S \\in \\mathbb{R}^U_{\\geq 0}$ is a vector of the underlying assets' values at $T$.\n\n\n\n\\end{definition}\n\\vspace{2ex}\nConsider a combinatorial option $C(\\texttt{MSFT}\\xspace-\\texttt{AAPL}\\xspace, 0)$ that has weight $1$ for \\texttt{MSFT}\\xspace, weight $-1$ for \\texttt{AAPL}\\xspace, and a strike price of zero.\nAn investor who buys the option bets on the event that Microsoft outperforms Apple Inc., and will exercise it if $S_\\texttt{MSFT}\\xspace > S_\\texttt{AAPL}\\xspace$.\nThus, unlike standard options that will pay off due to price changes of a single security, combinatorial options bet on relative movements between assets or groups of assets, thus enabling the expression of future correlations among different underlying assets.\n\nWe note several distinctions and interpretations in regard to the definition of combinatorial financial options:\n\\begin{enumerate}\n    \\setlength{\\itemsep}{6pt}\n    \n    \\item Distinction between a combinatorial option and a combination of standard options. \n    \n    Consider the above combinatorial option $C(\\texttt{MSFT}\\xspace-\\texttt{AAPL}\\xspace, 0)$ and a combination of two standard options $C(\\texttt{MSFT}\\xspace, K)$ and $P(\\texttt{AAPL}\\xspace, K)$. One would exercise the combinatorial option and possess the portfolio, i.e., long a share of \\texttt{MSFT}\\xspace and short sell a share of \\texttt{AAPL}\\xspace, as long as $S_\\texttt{MSFT}\\xspace \\geq S_\\texttt{AAPL}\\xspace$, whereas in the combination of standard options, one would exercise both and possess the same portfolio, if $S_\\texttt{MSFT}\\xspace\\geq K$ \\emph{and} $S_\\texttt{AAPL}\\xspace \\leq K$.%\n    \\footnote{In terms of possessing the portfolio \\texttt{MSFT}\\xspace-\\texttt{AAPL}\\xspace, $C(\\texttt{MSFT}\\xspace-\\texttt{AAPL}\\xspace, 0)$ captures all combination of two standard options of the form, $C(\\texttt{MSFT}\\xspace, K)$ and $P(\\texttt{AAPL}\\xspace, K)$ for all positive $K$.}\n    Thus, the exercise of a combinatorial option does not imply a simultaneous exercise of all standard options in the combination, and vice versa, the simultaneous buy and sell of several standard financial options with different underlying securities and strike prices (e.g., multi-leg options orders) cannot replicate the payoff of a combinatorial option.\n\n    \\item Distinction between a call and a put combinatorial option.\n    \n    We note that the distinction between a call and a put for combinatorial options depends on the strike price and coefficients that one specifies in a contract.\n    For instance, in the above example, $C(\\texttt{MSFT}\\xspace-\\texttt{AAPL}\\xspace, 0)$ is identical to $P(\\texttt{AAPL}\\xspace-\\texttt{MSFT}\\xspace, 0)$, as they have the same payoff function $\\max\\{S_\\texttt{MSFT}\\xspace-S_\\texttt{AAPL}\\xspace, 0\\}$.\n    Despite the different interpretations and expressions, we follow the convention of standard financial options, and have the strike price always be non-negative.\n\\end{enumerate}\n\n\n\\subsection{Matching Orders on Combinatorial Financial Options}\nThe increased expressiveness in combinatorial options brings new challenges in market design: \nonly matching buy and sell orders on options related to the same assets and weights may yield few or no trades, despite plenty of profitable trades among options written on different portfolios.\nWe start by giving the following motivating examples to illustrate such scenarios.\n\\begin{example}[Matching combinatorial option orders]\n\t\\label{ex:combo_options}\n\tConsider a combinatorial options market with four orders\n\t\\begin{itemize}\n\t\t\\setlength{\\itemsep}{2pt}\n\t\t\\item $o_1$: buy one $C(1\\texttt{AAPL}\\xspace+2\\texttt{MSFT}\\xspace, 300)$ at bid \\$110;\n\t\t\\item $o_2$: buy one $C(1\\texttt{AAPL}\\xspace+1\\texttt{MSFT}\\xspace, 300)$ at bid \\$70;\n\t\t\\item $o_3$: sell one $C(1\\texttt{AAPL}\\xspace+3\\texttt{MSFT}\\xspace, 300)$ at ask \\$160;\n\t\t\\item $o_4$: sell one $C(1\\texttt{AAPL}\\xspace, 250)$ at ask \\$5.\n\t\\end{itemize}\n\tThe exchange returns no match if it only considers options related to the same combination of assets.\n\tHowever, a profitable match does exist.\n\tThe exchange can sell to $o_1$ and $o_2$ and simultaneously buy from $o_3$ and $o_4$ to get an immediate gain of \\$15 (110+70-160-5).\n\tFigure~\\ref{fig:combo_match_example} plots the overall payoff (which is always non-negative), as a function of $S_\\texttt{AAPL}\\xspace$ and $S_\\texttt{MSFT}\\xspace$ on the expiration date:\n\t%\n\t{\n\t\t\\begin{align}\n\t\t\\Psi := &\\max\\{S_\\texttt{AAPL}\\xspace+3S_\\texttt{MSFT}\\xspace-300, 0\\} + \\max\\{S_\\texttt{AAPL}\\xspace-250, 0\\} - \\notag\\\\\n\t\t&\\max\\{S_\\texttt{AAPL}\\xspace+2S_\\texttt{MSFT}\\xspace-300, 0\\} - \\max\\{S_\\texttt{AAPL}\\xspace+S_\\texttt{MSFT}\\xspace-300, 0\\},\\notag\n\t\t\\end{align}\n\t}\n\tTherefore, the exercises of options cannot subtract from the \\$15 immediate gain, but could add to it, depending on the future prices of the two stocks.\n\n\n\t\\qed\n\\end{example}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.45\\columnwidth]{opt_img\/combo_example}\n\t\\caption[Payoff of combinatorial options matched in \\Ex{combo_options} as a function of $S_\\texttt{AAPL}\\xspace$ and $S_\\texttt{MSFT}\\xspace$.]{Payoff of combinatorial options matched in \\Ex{combo_options} as a function of $S_\\texttt{AAPL}\\xspace$ and $S_\\texttt{MSFT}\\xspace$. The example demonstrates the case of $L=0$.}\n\t\\label{fig:combo_match_example}\n\\end{figure}\n\nIn the above example, the exchange can consider matching each individual buy order sequentially to some combination of sell orders.\nFor example, the exchange can first sell to buy order $o_1$ and buy from sell orders $\\frac{2}{3}o_3$ and $\\frac{1}{3}o_4$, and then sell to buy order $o_2$ and buy from sell orders $\\frac{1}{3}o_3$ and $\\frac{2}{3}o_4$.\nBoth are profitable trades subject to no loss, leading to the same ultimate match as in Example~\\ref{ex:combo_options}.\nThe next example shows that such sequential matching of each individual buy order to a combination of sell orders may fail to find valid trades.\n\\begin{example}[Matching combinatorial option orders in batch]\n\t\\label{ex:combo_options_2}\n\tConsider the following four combinatorial options orders\n\t\\begin{itemize}\n\t\t\\setlength{\\itemsep}{2pt}\n\t\t\\item $o_1$: buy one $C(\\text{A}+\\text{B}, 10)$ at bid \\$6;\n\t\t\\item $o_2$: buy one $C(\\text{B}+\\text{C}, 7)$ at bid \\$6;\n\t\t\\item $o_3$: sell one $C(\\text{A}+\\text{B}+\\text{C}, 7)$ at ask \\$10;\n\t\t\\item $o_4$: sell one $C(\\text{B}, 3)$ at ask \\$2.\n\t\\end{itemize}\n\tNo match will be found if we consider each buy order individually: covering a sold combinatorial option in $o_1$ or $o_2$ requires buying the same fraction of $o_3$, which is at a higher price and will incur a net loss.\n\tHowever, a valid match does exist by selling to $o_1$ and $o_2$ and buying from $o_3$ and $o_4$.\n\tIt costs the exchange \\$0, and can yield a positive payoff in the future. \n\tThe exchange has\n\t%\n\t{\n\t\t\\begin{align*}\n\t\t\\max\\{S_A+S_B-10, 0\\} + \\max\\{S_B+S_C-7, 0\\}\n\t\t\\leq \\max\\{S_A+S_B+S_C-7, 0\\} + \\max\\{S_B-3, 0\\},\n\t\t\\end{align*}\n\t}\n\t%\n\tmeaning the liability will always be no larger than the payoff of bought options, for all non-negative $S_A, S_B, S_C$.%\n\t\\footnote{To see this, we can add $\\max\\{S_A-7, 0\\}$ on both sides of the inequality, and have $\\max\\{S_A-7, 0\\} + \\max\\{S_B+S_C-7, 0\\} \\leq \\max\\{S_A+S_B+S_C-7, 0\\}$ and $\\max\\{S_A+S_B-10, 0\\} \\leq \\max\\{S_A-7, 0\\} + \\max\\{S_B-3, 0\\}$ by Jensen's inequality.}\n\t\\qed\n\\end{example}\n\nWe now introduce the formal setting of a combinatorial financial options market.\nA combinatorial options market is a two-sided market with a set of buy limit orders, indexed by $m \\in \\{1, 2, ..., M\\}$, and a set of sell limit orders, indexed by $n \\in \\{1, 2, ..., N\\}$. \nBuy orders are represented by a type vector $\\boldsymbol{\\phi} \\in \\{1, -1\\}^M$ with each entry specifying a put or a call combinatorial option, a weight matrix $\\boldsymbol{\\alpha} \\in \\mathbb{R}^{U \\times M}$ specifying the linear combinations, a strike vector $\\boldsymbol{p} \\in \\mathbb{R}_{\\geq 0}^M$, and a bid-price vector $\\boldsymbol{b} \\in \\mathbb{R}_{+}^M$.\nSell orders are defined by a separate type vector $\\boldsymbol{\\psi} \\in \\{1, -1\\}^N$, a weight matrix $\\boldsymbol{\\beta} \\in \\mathbb{R}^{U \\times N}$, a strike vector ${\\boldsymbol{q}} \\in \\mathbb{R}_{\\geq 0}^N$, and an ask-price vector $\\boldsymbol{a} \\in \\mathbb{R}_{+}^N$.\nSimilar to a standard options market, the exchange decides the fraction $\\boldsymbol{\\gamma} \\in [0, 1]^M$ to sell to buy orders and the fraction $\\boldsymbol{\\delta} \\in [0, 1]^{N}$ to buy from sell orders to maximize net profit.\nWe generalize the proposed mechanism \\ref{mech:single_match} for standard options to facilitate trading combinatorial options:\n\\begin{align}\\tag{M.2}\n\\label{mech:combo_match}\n\\max \\limits_{\\boldsymbol{\\gamma}, \\boldsymbol{\\delta}, L} & \\displaystyle \\quad \\boldsymbol{b}^\\top \\boldsymbol{\\gamma} - \\boldsymbol{a}^\\top \\boldsymbol{\\delta} - L\\\\\n\\text{s.t.} & \\displaystyle \\quad \\displaystyle \\sum_{m} \\gamma_m \\max\\{\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m), 0\\} -   \\sum_{n} \\delta_n \\max\\{\\psi_n(\\boldsymbol{\\beta}_n^\\top \\S - q_n), 0\\} \\leq L \\quad \\forall \\S \\in \\mathbb{R}_{\\geq 0}^U \\label{eq:constraint}\n\\end{align}\n\nHowever, unlike \\ref{mech:single_match}, it is no longer feasible to solve the optimization problem \\ref{mech:combo_match} by enumerating the constraint at each breakpoint, which requires iterating over every combination of underlying asset values.\nThe number of constraints can grow exponentially as $\\mathcal{O}(2^{M+N})$ or $\\mathcal{O}((M+N)^U)$ by Sauer's Lemma.\\footnote{In fact, we can write \\ref{mech:combo_match} as an exponential-sized linear program.}\n\nWe analyze the complexity of finding the optimal match in a combinatorial options market, i.e., solving for \\ref{mech:combo_match}.\nWe first show in the following theorem that given a market instance, it is NP-complete to decide if a certain matching assignment, $\\boldsymbol{\\gamma}$ and $\\boldsymbol{\\delta}$, violates the constraint \\eqref{eq:constraint} for a fixed $L$. \n\\begin{theorem}\n\t\\label{thm:NP-complete}\n\tConsider all combinatorial options in the market $(\\boldsymbol{\\phi}, \\boldsymbol{\\alpha}, \\boldsymbol{p}, \\boldsymbol{\\psi}, \\boldsymbol{\\beta}, {\\boldsymbol{q}})$. \n\tFor any fixed $L$, it is NP-complete to decide \n\t\\begin{itemize}\n\t\t\\setlength{\\itemsep}{5pt}\n\t\t\\item Yes: $\\boldsymbol{\\gamma} = \\boldsymbol{\\delta} = \\mathbf{1}$ violates the constraint in \\ref{mech:combo_match} for some $\\S$,\n\t\t\\item No: $\\boldsymbol{\\gamma} = \\boldsymbol{\\delta} = \\mathbf{1}$ satisfies the constraint in \\ref{mech:combo_match} for all $\\S$,\n\t\\end{itemize}\n\teven assuming that each combinatorial option is written on at most two underlying assets.\n\\end{theorem}\n\\begin{proof}\n\tThe decision problem is in NP.\n\tGiven a certificate which is a value vector $\\S \\in \\mathbb{R}^U_{+}$, we plug $\\S$ into the constraint of \\ref{mech:combo_match} to compute the payoff and check whether it is less than $L$.\n\tThis takes time $\\mathcal{O}(U(M+N))$.\n\t\n\tFor the NP-hardness, we prove by reducing from the Vertex Cover problem --- given an undirected graph $G=(V,E)$ and an integer $k$, decide if there is a subset of vertices $V' \\subseteq V$ of size $k$ such that each edge has at least one vertex in $V'$.\t\n\t\n\t\tGiven a Vertex Cover instance $(G, k)$, we construct an instance of the combinatorial options matching problem.\n\t\tLet the set of underlying assets correspond to vertices in $G$, i.e., $U=|V|$.\n\t\tFor each vertex indexed $i$, we associate four options with it, one on the buy side and three on the sell side. They have payoff functions as follows:\n\t\t\\begin{align*}\n\t\tf_i &=\\max\\{2K_1S_i-K_1, 0\\}, &g^{(1)}_i &=\\max\\{K_1S_i, 0\\},\\\\\n\t\tg^{(2)}_i &=\\max\\{K_2S_i-K_2, 0\\}, &g^{(3)}_i &=\\max\\{S_i, 0\\},\n\t\t\\end{align*}\n\t\twhere we choose $K_1$ and $K_2$ for some large numbers with $K_2 \\gg K_1$.\n\t\tFor example, we choose $K_1=10|E|$ and $K_2=100|E|$.\n\t\tFor each edge $e = (i, j)$, we define two options that involve its two end-points $i$ and $j$, one on the buy side and one on the sell side.\n\t\tThey have payoff functions:\n\t\t\\begin{align*}\n\t\tf_e &= \\max\\{S_i+S_j, 0\\}, &g_e = \\max\\{S_i+S_j-1, 0\\}.\n\t\t\\end{align*}\n\t\tFinally, we include one sell order on an option with payoff $g^\\star = \\max\\{|E|-k-L-0.5,0\\}$. %\n\t\n\t\tSince $L$ is fixed in advance, we assume $|E|-k-L-0.5>0$ without loss of generality.\n\t\tThus, we have $M=|V|+|E|$ buy orders and $N=3|V|+|E|+1$ sell orders.\n\t\tThe construction takes time polynomial in the size of the Vertex Cover instance.\n\t\t\n\t\t\\emph{Suppose the Vertex Cover instance is a Yes instance.}\n\t\tWe assume that $\\{v_1, v_2, ..., v_k\\}$ is a vertex cover.\n\t\tWe show that assigning $S_1,S_2,...,S_k$ to 1 for the selected underlying assets (i.e., vertices) and 0 for the rest unselected ones gives an $\\S$ that violates the constraint \\eqref{eq:constraint}.\n\t\tThe left-hand side of the constraint \\eqref{eq:constraint} is\n\t\t\\begin{align}\n\t\t\\label{eq:LHS}\n\t\tz := & \\,\\,\\underbrace{\\!\\!\\sum_{i \\in V}\\left(f_i-g^{(1)}_i-g^{(2)}_i-g^{(3)}_i\\right)\\!\\!}_{z_v}\\,\\,+\\,\\,\\underbrace{\\!\\!\\sum_{e \\in E} (f_e-g_e)\\!\\!}_{z_e}\\,\\,-g^\\star.\n\t\t\\end{align}\n\t\tFor $S_i\\in\\{0,1\\}$, it is easy to see that $f_i-g_i^{(1)}-g_i^{(2)}=0$.\n\t\tThus, we have $z_v = -\\sum_{i \\in V}g^{(3)} = -k$ by our assignment.\n\t\tSince at least one of $S_i,S_j$ is $1$ for any edge $(i,j)\\in E$, we have $f_e - g_e = 1$ and $z_e = \\card{E}$.\n\t\tTherefore, we have \n\t\t\\begin{align*}\n\t\tz = & -k + |E| - (|E| - k - L - 0.5) = L + 0.5 > L.\n\t\t\\end{align*}\n\t\t\n\t\t\\emph{Suppose the Vertex Cover instance is a No instance.}\n\t\tWe aim to show that for the given $\\boldsymbol{\\gamma}$ and $\\boldsymbol{\\delta}$, there does not exist an $\\S$ that violates the constraint.\n\t\tWe prove by maximizing $z$ and demonstrating $z \\leq L$.\t\n\t\tWe start by proving the following claim.\n\t\t\\begin{claim}\n\t\t\t\\label{thm:claim1}\n\t\t\tFor an optimal $z$, we have $S_i \\in \\{0,1\\}$.\n\t\t\\end{claim}\n\t\tWe prove by contradiction. First assume $S_j > 1$ for some $j$.\n\t\tWe rewrite Eq.~\\eqref{eq:LHS} and have\n\t\t\\begin{align*}\n\t\tz := & \\,\\,\\underbrace{\\!\\!f_j-g^{(1)}_j-g^{(2)}_j-g^{(3)}_j\\!\\!}_{z_j}\\,\\, + \\,\\,\\underbrace{\\!\\!\\sum_{i\\in V\\setminus j}\\left(f_i-g^{(1)}_i-g^{(2)}_i-g^{(3)}_i\\right)\\!\\!}_{z_i}\\,\\, + \\,\\,\\underbrace{\\!\\!\\sum_{e \\in E} (f_e-g_e)\\!\\!}_{z_e}\\,\\,-g^\\star.\n\t\t\\end{align*}\n\t\tWe first analyze $z_j$ and have\n\t\t\\begin{align*}\n\t\tz_j = & \\max\\{2K_1 S_j-K_1, 0\\} - \\max\\{K_1S_j, 0\\} - \\max\\{K_2 S_j-K_2, 0\\} - \\max\\{S_j, 0\\}\\\\\n\t\t= & 2K_1 S_j-K_1 - K_1S_j - (K_2S_j-K_2) - S_j \\tag{by assumption of $S_j > 1$}\\\\\n\t\t= & K_2 - K_1 - (K_2-K_1+1)S_j.\n\t\t\\end{align*}\n\t\tRecall that we choose $K_2 \\gg K_1$, e.g., $K_1=10|E|$ and $K_2=100|E|$.\n\t\tThus, we have $z_j$ increase with rate $K_2-K_1+1$, as $S_j$ decreases uniformly.\n\t\tSince $z_e$ decreases at most $|E|$ and the rest two terms, $z_i$ and $g^\\star$, do not depend on $S_j$, decreasing $S_j$ increases $z$.\n\t\tIt is sub-optimal to have $S_j > 1$ for some $j$.\n\t\t\n\t\tNext, we assume $0 \\leq S_j \\leq 1$ for some $j$ and have\n\t\t\\begin{align*}\n\t\tz_j\n\t\n\t\t&= \\begin{cases}\\displaystyle\n\t\t-K_1S_j-S_j &\\quad 0 \\leq S_j \\leq 0.5,\\\\\n\t\t\\displaystyle\n\t\tK_1(S_j-1)-S_j &\\quad 0.5 < S_j \\leq 1.\\\\\n\t\t\\end{cases}\n\t\t\\end{align*}\n\t\tAs $K_1$ is large, by a similar argument analyzing the growth rate of each term, we show that $z_j$ (and also $z$) increases by assigning $S_j$ to $0$ if $0 \\leq S_j \\leq 0.5$ and by assigning $S_j$ to $1$ if $0.5 < S_j \\leq 1$.\n\t\t\n\t\tWe finish proving Claim~\\ref{thm:claim1}, and now have $S_i \\in \\{0,1\\}$.\n\t\tFollowing Eq.~\\eqref{eq:LHS}, we have\n\t\t\\[z = -\\sum_{i \\in V}S_i + \\sum_{e \\in E} (f_e-g_e) - (|E|-k)+L+0.5.\\]\n\t\tThe goal is to maximize $z$ and show $z\\leq L$.\n\t\tAs $S_i$ is an integer, to show $z\\leq L$, it suffices to show that $\\sum_{e \\in E} (f_e-g_e)-\\sum_{i \\in V}S_i<|E|-k$.\n\t\tWe prove by contradiction, assuming $\\sum_{e \\in E} (f_e-g_e)-\\sum_{i \\in V}S_i \\geq |E|-k$.\n\t\tRecall that to have $f_e-g_e = 1$ for $e=(i,j)$, we need at least one of $S_i,S_j$ to be $1$.\n\t\tWe consider the following two possible cases, and aim to refute them:\n\t\t\\begin{enumerate}[(a)]\n\t\t\t\\setlength{\\itemsep}{5pt}\n\t\t\t\\item $\\sum_{e \\in E} (f_e-g_e)=|E|$ and $\\sum_{i \\in V}S_i \\leq k$.\n\t\t\t\n\t\t\tThis means we cover all edges with at most $k$ vertices assigned to 1.\n\t\t\t\n\t\t\t\\item $\\sum_{e \\in E} (f_e-g_e)<|E|$ and $\\sum_{i \\in V}S_i < k$.\n\t\t\t\n\t\t\tFor any $e$ with $f_e-g_e=0$, we can assign 1 to one of its end-points, and have $\\sum_{e \\in E} (f_e-g_e)=|E|$ without changing $\\sum_{e \\in E} (f_e-g_e)-\\sum_{i \\in V}S_i$. This leads back to (a).\n\t\t\\end{enumerate}\n\t\tBoth cases contradict to the fact that the Vertex Cover instance is a No instance.\n\t\tWe have $\\sum_{e \\in E} (f_e-g_e)-\\sum_{i \\in V}S_i < |E|-k$ as desired, and thus $z \\leq L$ for all $\\S$.\n\\end{proof}\n\\vspace{1ex}\n\nUsing a slightly stronger version of \\Thm{NP-complete} (see Theorem 5 in Appendix B.1),\nwe show that optimal clearing of a combinatorial options market is coNP-hard.\nWe defer the proof of \\Thm{coNP-hard} to Appendix B.1\n\\begin{theorem}\n\t\\label{thm:coNP-hard}\n\tOptimal clearing of a combinatorial options market $(\\boldsymbol{\\phi}, \\boldsymbol{\\alpha}, \\boldsymbol{p}, \\boldsymbol{b}, \\boldsymbol{\\psi}, \\boldsymbol{\\beta}, {\\boldsymbol{q}}, \\boldsymbol{a})$ is coNP-hard, even assuming that each combinatorial option is written on at most two underlying assets.\n\\end{theorem}\n\\subsection{A Constraint Generation Algorithm to Match Combinatorial Financial Options}\nSince it is no longer practical to solve \\ref{mech:combo_match} by identifying all constraints defined by different combinations of underlying asset values, we propose an algorithm (\\Algo{comb_match}) that finds the exact optimal match through iterative constraint generation.\nWe first explain how \\Algo{comb_match} works, and then prove that it is an equivalent formulation of \\ref{mech:combo_match}.\n\nAt the core of \\Algo{comb_match} is a constraint generation process, where a new optimization problem is defined per iteration to find a violated constraint.\nThe constraint set $\\mathcal{C}$ includes the $\\S \\in \\mathbb{R}^U_{\\geq 0}$ value vectors that define different constraints. We can plug each $\\S$ back into the constraint to define $\\boldsymbol{f}$ and $\\boldsymbol{g}$, which are the realized payoffs of bought and sold options with respect to the $\\S$.\nWe start with the constraint set $\\mathcal{C}$ of a zero vector, meaning that all underlying assets have price zero at expiration.\nIn each iteration, the upper-level optimization problem (\\ref{eq:combo_upper}) computes the optimal match (i.e., $\\boldsymbol{\\gamma}^*$, $\\boldsymbol{\\delta}^*$, and $L^*$) that satisfies this restrictive set of generated constraints.\nThus, it is a linear program with $\\card{\\mathcal{C}}$ constraints.\nWe then use the lower-level optimization problem (\\ref{eq:comb_milp}) to find the $\\S^*$ value vector that violates the returned matching solution the most.\nThat is, given $\\boldsymbol{\\gamma}^*$, $\\boldsymbol{\\delta}^*$, and $L^*$, it generates the most adversarial realization of $\\S$ that yields the worst-case payoff loss for the exchange at expiration.\nWe show that this lower-level optimization can be formulated as a mixed-integer linear program.\nWe then include this generated $\\S^*$ vector into the constraint set.\n\nThe exact optimal match is returned when the lower-level \\ref{eq:comb_milp} gives an objective value of zero, meaning there exists no such $\\S$ that violates the upper-level matching assignment.\nTherefore, the final number of constraints in $\\mathcal{C}$ is the number of iterations that takes \\Algo{comb_match} to terminate.\nThe algorithm trivially terminates finitely, but similar to the simplex method, it has no guarantee on the rate of convergence.\nWe later evaluate its performance on synthetic combinatorial options markets of different scales, and demonstrate that \\Algo{comb_match} converges relatively fast, with the number of iterations increasing linearly in the size of a market instance.\n\nWe next prove \\Algo{comb_match} returns the same optimal clearing solution as \\ref{mech:combo_match}.\nWe first show in the Lemma that \\ref{eq:comb_milp} finds the value vector $\\S$ that violates the constraint \\eqref{eq:constraint} in \\ref{mech:combo_match} the most.\n\\begin{algorithm}[t]\n\t\\caption{Match orders in a combinatorial options market.}\n\t\\label{algo:comb_match}\n\t\\begin{algorithmic}[1]\n\t\t\\Statex \\textbf{Input:}~%\n\t\tA combinatorial options market defined by\n\t\n\t\t$(\\boldsymbol{\\phi}, \\boldsymbol{\\alpha}, \\boldsymbol{p}, \\boldsymbol{b}, \\boldsymbol{\\psi}, \\boldsymbol{\\beta}, {\\boldsymbol{q}}, \\boldsymbol{a})$.\n\t\t\\Statex \\textbf{Output:}~%\n\t\tAn optimal clearing that matches $\\boldsymbol{\\gamma}^*$ buy orders to $\\boldsymbol{\\delta}^*$ \n\t\n\t\tsell orders.\n\t\t\\medskip\n\t\t\\State Initialize $z \\gets \\infty$, $\\S \\gets \\boldsymbol{0}$, \n\t\t\\Statex ~\\hphantom{Initializ} $\\boldsymbol{f} \\gets \\max\\{\\boldsymbol{\\phi}(\\boldsymbol{\\alpha}^\\top \\S-\\boldsymbol{p}), \\boldsymbol{0}\\}$, $\\boldsymbol{g} \\gets \\max\\{\\boldsymbol{\\psi}(\\boldsymbol{\\beta}^\\top \\S-{\\boldsymbol{q}}), \\boldsymbol{0}\\}$, $\\mathcal{C} \\gets \\{(\\boldsymbol{f}, \\boldsymbol{g})\\}$.\n\t\n\t\n\t\t\\While {$z > 0$}\n\t\t\\State Solve the following upper level LP and get the optimal $(\\boldsymbol{\\gamma}^*, \\boldsymbol{\\delta}^*, L^*)$\n\t\n\t\t\\begin{align*}\n\t\t\\label{eq:combo_upper}\n\t\t&\\max \\limits_{\\boldsymbol{\\gamma}, \\boldsymbol{\\delta}, L}  \\quad \\boldsymbol{b}^\\top \\boldsymbol{\\gamma} - \\boldsymbol{a}^\\top \\boldsymbol{\\delta} - L \\tag{M.3U}\\\\\n\t\t&\\quad\\text{s.t.} \\quad \\boldsymbol{\\gamma}^\\top \\boldsymbol{f} - \\boldsymbol{\\delta}^\\top \\boldsymbol{g} \\leq L \\quad &\\forall (\\boldsymbol{f}, \\boldsymbol{g}) \\in \\mathcal{C}\n\t\t\\end{align*}\n\t\t\\State Given $(\\boldsymbol{\\gamma}^*, \\boldsymbol{\\delta}^*, L^*)$, solve the following lower level MILP  and get the optimal\n\t\t\\Statex ~\\hphantom{w } $(\\S^*, \\boldsymbol{f}^*, \\boldsymbol{g}^*, \\boldsymbol{I}^*, z^*)$\n\t\t\\begin{align}\n\t\t\t\\label{eq:comb_milp}\n\t\t\n\t\t\n\t\t\n\t\t\t& \\max \\limits_{\\S, \\boldsymbol{f}, \\boldsymbol{g}, \\boldsymbol{I}} \\quad z := \\boldsymbol{\\gamma}^\\top \\boldsymbol{f} - \\boldsymbol{\\delta}^\\top \\boldsymbol{g} - L \\tag{M.3L}\\\\\n\t\t\t& \\text{  s.t.} \\quad \\quad \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\geq \\mathcal{M}(\\mathcal{I}_m-1) \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\leq \\mathcal{M} \\mathcal{I}_m \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad f_m \\leq \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) - \\mathcal{M}(\\mathcal{I}_m-1) \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad f_m \\leq \\mathcal{M} \\mathcal{I}_m \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad \\mathcal{I}_m \\in \\{0, 1\\} \\qquad \\qquad \\forall m \\in \\{1,...,M\\} \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad g_n \\geq \\psi_n(\\boldsymbol{\\beta}_n^\\top \\S - q_n) \\notag\\\\\n\t\t\t& \\hphantom{\\text{  s.t.}} \\quad \\quad g_n \\geq 0 \\qquad \\qquad \\qquad \\forall n  \\in \\{1,...,N\\} \\notag\n \t\t\\end{align}\n\t\t\\State $\\mathcal{C} \\gets \\mathcal{C} \\cup (\\boldsymbol{f}^*, \\boldsymbol{g}^*)$, $z\\gets z^*$\n\t\t\\EndWhile\n\t\t\\State \\Return $\\boldsymbol{\\gamma}^*$ and $\\boldsymbol{\\delta}^*$\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\\begin{lemma}\n\tGiven fixed $\\boldsymbol{\\gamma}$, $\\boldsymbol{\\delta}$, and $L$ for a combinatorial options market $(\\boldsymbol{\\phi}, \\boldsymbol{\\alpha}, \\boldsymbol{p}, \\boldsymbol{b}, \\boldsymbol{\\psi}, \\boldsymbol{\\beta}, {\\boldsymbol{q}}, \\boldsymbol{a})$, \\ref{eq:comb_milp} returns the value of underlying assets $\\S$ that violates the constraint of \\ref{mech:combo_match} the most.\n\\end{lemma}\n\\begin{proof}\n\tFirst, it is easy to see that the formulation below returns the $\\S$ that violates constraint \\eqref{eq:constraint} the most, since we will have the largest feasible $\\boldsymbol{f}$ and the smallest feasible $\\boldsymbol{g}$ at the optimum.\n\tThat is, $ f_m = \\max\\{\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m), 0\\}$ and $g_n = \\max\\{\\psi_n(\\boldsymbol{\\beta}_n^\\top \\S - q_n), 0\\}$.\n\t\t\\begin{align}\n\t\t& \\max \\limits_{\\S, \\boldsymbol{f}, \\boldsymbol{g}} \\quad \\boldsymbol{\\gamma}^\\top \\boldsymbol{f} - \\boldsymbol{\\delta}^\\top \\boldsymbol{g} - L \\notag\\\\\n\t\t& \\text{  s.t.} \\quad \\quad f_m \\leq \\max\\{\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m), 0\\} &\\forall m \\in \\{1,...,M\\} \\notag\\\\\n\t\t& \\phantom{\\text{  s.t.}} \\quad \\quad g_n \\geq \\psi_n(\\boldsymbol{\\beta}_n^\\top \\S - q_n) \\notag\\\\\n\t\t& \\phantom{\\text{  s.t.}} \\quad \\quad g_n \\geq 0 &\\forall n  \\in \\{1,...,N\\} \\notag\t\n\t\t\\end{align}\n\t\n\tIt remains to show the set of constraints related to any buy order $m$ in \\ref{eq:comb_milp} is equivalent to $f_m \\leq \\max\\{\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m), 0\\}$.\n\tIn short, the set of constraints in \\ref{eq:comb_milp} linearize the max functions by using the big-$\\mathcal{M}$ trick on each binary decision variable $\\mathcal{I}_m$, where $\\mathcal{M}$ is a large constant, say $10^6$.\n\t\n\tConsider each case of $\\mathcal{I}_m \\in \\{0, 1\\}$.\n\tThe first two constraints, $\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\geq \\mathcal{M}(\\mathcal{I}_m-1)$ and $\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\leq \\mathcal{M} \\mathcal{I}_m$, guarantee that $\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\geq 0 \\iff \\mathcal{I}_m=1$.\n\tThen, following the third and fourth constraints, i.e., $f_m \\leq \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) - \\mathcal{M}(\\mathcal{I}_m-1)$ and $f_m \\leq \\mathcal{M} \\mathcal{I}_m$, we have \n\t\\[f_m \\leq 0 \\quad \\text{if $\\mathcal{I}_m = 0$}; \\quad f_m \\leq \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\quad \\text{if $\\mathcal{I}_m = 1$.}\\]\n\tTherefore, at the optimum, we have \n\t\\[f_m = 0 \\quad \\text{if $\\mathcal{I}_m = 0$}; \\quad f_m = \\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m) \\quad \\text{if $\\mathcal{I}_m = 1$,}\\]\n\twhich is equivalent to $f_m = \\max\\{\\phi_m(\\boldsymbol{\\alpha}_m^\\top \\S - p_m), 0\\}$.\n\n\\end{proof}\nTherefore, when \\ref{eq:comb_milp} returns an objective value of zero, the constraint \\eqref{eq:constraint} in \\ref{mech:combo_match} is satisfied for all $\\S$, and \\Algo{comb_match} returns a valid match that optimizes for overall profit.\n\\begin{theorem}\n\t\\label{thm:equivalent_form}\n\tGiven a combinatorial options market instance $(\\boldsymbol{\\phi}, \\boldsymbol{\\alpha}, \\boldsymbol{p}, \\boldsymbol{b}, \\boldsymbol{\\psi}, \\boldsymbol{\\beta}, {\\boldsymbol{q}}, \\boldsymbol{a})$, \\Algo{comb_match} returns the optimal clearing defined in \\ref{mech:combo_match}.\n\\end{theorem}\n\n\\section{Experiments: Real-Market Standard Financial Options}\n\\label{sec:exp_standard_option}\nWe first evaluate the proposed mechanism~\\ref{mech:single_match} that matches orders on standard options across types and strike values.\nWe conduct empirical analysis on the OptionMetrics dataset provided by the Wharton Research Data Services (WRDS), which contains real-market option prices, i.e., the best bid and ask prices, for each market defined by an underlying asset, an option type, a strike price, and an expiration date.%\n\\footnote{Our data includes American options that allow exercise before expiration. \n\tIn practice, American options are\n\talmost always more profitable to sell than to exercise early \\cite{Singh2019}.\n\tIn experiments, we ignore early exercise and treat them as European options.}\nWe choose options data on 30 stocks that compose the DJI, as these stocks have actively traded options that cover a wide range of moneyness and maturity levels.\nThere are a total of 25,502 distinct options markets for the 30 stocks in DJI on January 23, 2019,\ncovering around 12 expiration dates for each stock.\n\nWe use \\ref{mech:single_match} to consolidate the outstanding buy orders and sell orders from independently-traded options markets that associated with the same security and expiration date.\nThis reduces the original 25,502 distinct markets to a total of 366 markets, which have standard options across different types and strikes.\nWe run \\ref{mech:single_match} on these consolidated markets to\n\t\\begin{enumerate}[(1)]\n\t\t\\setlength{\\itemsep}{2pt}\n\t\t\\item find trades that the current independent-market design cannot match,\n\t\t\\item compute new price quotes implied by this consolidated market design, and\n\t\t\\item compare matches and price quotes to those of the case when we restrict $L$ to 0.\n\t\\end{enumerate}\n\nOut of the 366 consolidated options markets, we find profitable matches in 150 markets, which failed to transact under the independent market design.\nAmong those trades, 94 have non-negative $L$ (i.e., the case of \\Ex{nonneg_L}), making an average net profit of \\$1.03, with a maximum of \\$9.64; \nthe remaining 56 have negative $L$ (i.e., the case of \\Ex{neg_L}),  implying an average interest rate of 0.7\\%, with a maximum at 2.02\\%.\nWhen we restrict $L$ to 0 (i.e., $L$ is no longer a decision variable), we are able to find matches in 74 markets.\nDetailed statistics for options of each stock are available in the Appendix C.\n\nFor the arbitrage-free consolidated markets (i.e., markets with no match returned by \\ref{mech:single_match}), we find that approximately 49\\% of the orders belong to the frontier set.\nUsing these orders to derive the most competitive bids and asks, we find that the bid-ask spreads can be reduced by 73\\%, from an average of 80 cents for each option series in the independent markets to 21 cents in consolidated options markets.\nFor the case of $L$ set to 0, the bid-ask spreads can be reduced by 52\\%.\n\nThese results show that the exchange, by consolidating options markets across types and strike prices, can potentially achieve a higher economic efficiency, matching orders that the current independent design cannot and providing more competitive bid and ask prices.\n\n\\section{Experiments: Synthetic Combinatorial Options Market}\n\\label{sec:exp_combo_option}\nSince there is no combinatorial option traded in financial markets, we evaluate the proposed Algorithm~\\ref{algo:comb_match} on synthetic combinatorial options, with prices calibrated using real-market standard options written on each related underlying security.%\n\\footnote{We adopt the same dataset as Section~\\ref{sec:exp_standard_option}, and use standard options that expire on Febuary 1, 2019, to calibrate order prices for generated combinatorial options.}\nWe implement Algorithm~\\ref{algo:comb_match} using Gurobi 9.0~\\cite{gurobi}.\nWe are interested in quantifying the performance of \\Algo{comb_match} in parametrically different markets that vary in the likelihood of matching, the number of orders, and the number of underlying assets.\nWe first describe our synthetic dataset.\n\n\\subsection{Generate Synthetic Orders}\nWe generate combinatorial options markets of $U$ underlying assets.\nEach combinatorial option is written on a combination of two stocks, $S_i$ and $S_j$, randomly selected from the $U$ underlying assets.\nThis gives a total number of $U \\choose 2$ asset pairs.\nWeights for the selected assets, $w_i$ and $w_j$, are picked uniformly randomly from $\\{\\pm1, \\pm2, \\dotsc, \\pm9\\}$ and are preprocessed to be relatively prime.\n\nWe generate strikes and bid\/ask prices using real-market standard options data related to each individual asset to realistically capture the value of the synthetic portfolio.\nLet $\\mathcal{K}_i$ and $\\mathcal{K}_j$ respectively denote the set of strike prices offered by standard options on each selected asset.\nWe generate the strike $K$ by first sampling two strike values, $k_i \\sim \\mathcal{K}_i$ and $k_j \\sim \\mathcal{K}_j$, and scaling them by the associated weights to get $K = w_ik_i+w_jk_j$.\nIf $K$ is positive, we have a call option.\nInstead, if $K$ is negative, we generate a put option, and update the strike to $-K$ and weights to $-w_i$ and $-w_j$ to comply with the representation and facilitate payoff computations.\n\nWe randomly assign each option to the buy or sell side of the market with equal probability, and generate a bid or ask price accordingly.\nSimilar to the price quote procedure in Section~\\ref{sec:standard_option}, we derive the bid $b$ by calculating the maximum gain of selling a set of standard options whose payoff is dominated by the combinatorial option of interest, and compute the ask $a$ by calculating the minimum cost of buying a set of standard options whose payoff dominates the generated conbinatorial option.\nWe add noises around the derived prices to control the likelihood of matching in a market, and set final prices to $b(1+\\zeta)$ or $a(1-\\zeta)$, where $\\zeta \\sim [0, \\eta]$ and $\\eta$ is a noise parameter.\n\n\\subsection{Evaluation}\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{subfigure}{0.48\\columnwidth}\t\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\columnwidth]{opt_img\/combo_vary_noise_level.pdf}\n\t\t\\caption{Vary noise $\\eta$ added to order prices in markets with $U=4$ and $n_\\text{orders}=150$.}\n\t\t\\label{subfig:combo_plots_a}\n\t\t\\vspace{2ex}\n\t\\end{subfigure}\n\t\\hspace{0.02\\columnwidth}\n\t\\begin{subfigure}{0.48\\columnwidth}\t\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\columnwidth]{opt_img\/combo_vary_book_size.pdf}\n\t\t\\caption{Vary number of orders $n_\\text{orders}$ in markets with $U=4$ and $\\eta = 2^{-4}$.}\n\t\t\\label{subfig:combo_plots_b}\n\t\t\\vspace{2ex}\n\t\\end{subfigure}\n\t\\hspace{0.02\\columnwidth}\n\t\\begin{subfigure}{0.48\\columnwidth}\t\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\columnwidth]{opt_img\/combo_vary_stock_size.pdf}\n\t\t\\caption{Vary size of underlying assets $U$ in markets with $n_\\text{orders} = 150$ and $\\eta = 2^{-4}$.}\n\t\t\\label{subfig:combo_plots_c}\n\t\\end{subfigure}\n\t\\caption[Results of using \\Mech{combo_match} to match orders in synthetic combinatorial options markets.]{Results of using \\Mech{combo_match} to match orders in synthetic combinatorial options markets. The number of generated constraints (solid lines) and the net profits (dashed line), as the markets vary in price noise, the number of orders, and the size of underlying assets. Red lines represent markets that offer a restrictive set of asset pairs, which covers all $U$ underlying assets.}\n\t\\label{fig:combo_plots}\n\\end{figure}\nWe explore a wide range of markets that vary in price noise $\\eta$, the number of orders $n_\\text{orders}$, and the number of underlying assets $U$.\nFor each market, we measure (1)~the number of iterations (i.e., the total number of constraints generated) that \\Algo{comb_match} takes to find an exact optimal clearing and (2)~the net profit made from the trade.\nFor all experiments, we show results averaged over 40 simulated markets, with the error bars denoted one standard error around the means.\n\nWe first validate that as larger noise is added to the derived bids and asks, the likelihood of matching in our simulated combinatorial options market becomes higher.\nWe generate markets with four underlying assets (arbitrarily selected from the 30 stocks in DJI) and 150 synthetic combinatorial options orders, and vary the noise level $\\eta \\in \\{2^{-7}, 2^{-6}, 2^{-5}, 2^{-4}, 2^{-3}\\}$. \nFigure~\\ref{subfig:combo_plots_a} plots the averaged results.\nAs expected, the net profit made from the optimal match increases, as $\\eta$ increases.\nMoreover, we find that as the matching probability increases, the number of iterations that takes to find the optimal solution consistently decreases.\nThis makes sense as intuitively, in thin markets where few trades are likely to occur, the lower-level optimization program will keep coming up with candidate $\\S$ values to refute a large number of matching proposals until convergence.\n\nFigure~\\ref{subfig:combo_plots_b} further quantifies the change in iteration numbers and net profits, as we vary the number of combinatorial options orders.\nWe fix these markets to have four underlying assets with a price noise of $2^{-4}$. \nAs we see from Figure~\\ref{subfig:combo_plots_b}, as a market aggregates more orders, transactions are more likely to happen, leading to larger net profits.\nWe also find that the number of generated constraints grows (sub)linear in the number of orders.\nSince different $\\S$ values are generated to define payoffs of \\emph{distinct} options and the number of distinct options increases sublinear in the number of total orders, the rate of increase in the number of iterations tends to decrease as a market aggregates more orders.\n\nFinally, we evaluate how \\Algo{comb_match} scales to markets with increasingly larger numbers of underlying assets.\nIn this case, as the dimension of $\\S$ becomes large, the number of asset-value combinations grows exponentially.\nFigure~\\ref{subfig:combo_plots_c} (black lines) demonstrates a much faster increase in the number of iterations and a steady decrease in the net profit.%\n\\footnote{We report average runtimes to quantify the impact of increasing constraints.\n\tThe average times (in seconds) that \\Algo{comb_match} computes the optimal match are 4, 31, 47, 59, 68, and 72 for the respective markets with $U \\in \\{2, 4, 8, 12, 16, 20\\}$.\n}\nIt suggests that as the market provides a large set of underlying assets (e.g., all 30 stocks in DJI), the thin market problem may still arise even when the mechanism facilitates matching options written on different combinations of underlying assets.\nHere, in this set of experiments, we make the assumption that every asset pair in the $U \\choose 2$ is equally likely to be traded.\nIn real markets, investors may be more interested in certain asset pairs, trading them more frequently than the others.\nBased on such observations, a market can specify a prescriptive set of asset pairs, $\\mathcal{P}$, to alleviate the thin market problem.\nFor the experiments, we choose $\\card{\\mathcal{P}} = U$ and have underlying securities in those asset pairs cover the $U$ assets.\nTraders can choose from any asset pairs within this prescriptive set and specify custom weights for each security.\nFigure~\\ref{subfig:combo_plots_c} (red lines) shows that such prescriptive design may indeed attenuate the thin market problem, leading to a faster convergence and higher profits.\n\\section{Discussion}\n\\label{sec:conclusion}\nThe paper proposes market designs to facilitate trading standard options and the more general \\emph{combinatorial financial options}.\nWe start by examining standard financial exchanges that operate separate options markets and have each independently aggregate and match orders on options of a designated type, strike price, and expiration.\nWhen logically related financial markets run independently, traders may remove arbitrage and close bid-ask spreads themselves. \nOur OptionMetrics experiments show that they do so suboptimally.\nExecution risk and transaction cost often prevent arbitrage opportunities from being fully removed.\nMoreover, profits\ncan flow to agents with better computational power and little information. \nOur consolidated market design, by putting computational power into the exchange, supports trading related options across different strikes and reduces bid-ask spreads, thus reducing the arms race among agents and rewarding informed traders only.\n\nWe extend standard options to define combinatorial financial options.\nWe are interested in designing a fully expressive combinatorial options market that allows options to be specified on all linear combinations of assets. \nIn such a market, traders are granted the convenience and expressiveness to speculate on correlated movements among stocks and more precisely hedge their risks (e.g., in one transaction, purchase a put option on their exact investment portfolio).\nIncreasing expressiveness may also benefit the exchange: the market is able to incorporate information of greater detail to optimize outcome, improving economic efficiency, and obtain a surplus from trade to split between the exchange and traders.\nLike many other combinatorial markets, the combinatorial options market comes at the cost of a higher computational complexity: optimal clearing of such a market is coNP-hard.\nWe propose a constraint generation algorithm to solve for the exact optimal match, and demonstrate its viability on synthetically generated combinatorial options markets of different scales. \n\nTwo immediate questions arise from our work.\n\\emph{First}, can we design computationally efficient matching algorithms by limiting expressivity within a combinatorial options market or imposing certain assumptions on the underlying assets?\nOne next step is to explore naturally structured markets where combinations are limited to components in a graph of underlying assets. \nOne special case is a hierarchical graph \\cite{Guo2009}, for example, the S\\&P\\,500\\xspace, sectors like travel and technology, subsectors like airlines and internet within those, etc.\nAnother direction is to relax our current constraint that guarantees no loss for all possible states, assuming or learning a probability distribution over the security's future price (e.g., recovering probability from option prices~\\cite{recover_prob_options}).\n\n\\emph{Second}, when operating such a combinatorial options market, how should an exchange set the market clearing rule?\nWhile the proposed mechanism (\\ref{mech:combo_match}) and matching algorithm (\\Algo{comb_match}) allow flexibility in the timing of market clear operations, and so can work in either continuous or periodic (batch) form, adopting an appropriate clearing rule is important and can affect both computational and economic efficiency.\nOur experiments on synthetic combinatorial options market (Section~\\ref{sec:exp_combo_option}) provide some preliminary understanding.\nResults on markets with low noise suggest potential limitations of continuous clearing: as there is no match among existing orders and the matching probability is low with only one incoming order, the lower-level program may take a long time to refute a large number of matching proposals until convergence. \nSurplus can also be low in a continuous clearing market that looks for immediate trades.%\n\\footnote{We conduct preliminary experiments on synthetic data and find that the average net profit achieved in a market with continuous clearning is about 30\\% lower than that of a market with batch clearing of the same 100 orders.}\nBatch clearing might be suitable for this combinatorial market, achieving a more practical convergence speed and higher surplus.\nThe next question becomes how frequently should the market clear? \nLike discussed in prior works studying the proposal of having financial markets move from continuous to batch clearing \\cite{budish2014,budish2015,brinkman2017empirical}, choosing the appropriate clearing interval is an art.\nIn our case, it depends on the market scale (e.g., the number of underlying assets and orders) and liquidity (e.g., matching probability), and a designer needs to balance price discovery and matching surplus (longer batches improve trading surplus, but can significantly slow down price discovery).\nAnalysis based on empirical mechanism design~\\cite{emd,brinkman2017empirical} may be useful to help a market designer set the optimal clearing interval and maximize efficiency.\n\nWe leave these questions open for future research.\n\n\\begin{acks}\nWe thank Miroslav Dud\\'ik and Haoming Shen for helpful discussions. \nWe are also grateful to the anonymous reviewers for constructive feedback. \n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe study on the adaptive control of robot manipulators with dynamic parameter uncertainty has a long and rich history (see, e.g., the early results in \\cite{Craig1987_IJRR,Slotine1987_IJRR,Middleton1988_SCL}), and the employment of adaptive control provides robot manipulators with the ability of performing tasks in the unknown environment. The recent advances in adaptive robot control occur in \\cite{Cheah2003_TRA,Cheah2006_IJRR,Cheah2006_TAC,Braganza2005_CDC,Dixon2007_TAC} aiming at handling the kinematic parameter uncertainty. Kinematic uncertainty is frequently encountered as the robots perform various work in the task space (e.g., Cartesian space or image space) (see, e.g., \\cite{Cheah2003_TRA,Liu2006_TRO}), among which is the now actively studied visual servoing problem (see, e.g., \\cite{Liu2006_TRO,Wang2010_TCST,Wang2015_Aut}). These control schemes (e.g., \\cite{Cheah2003_TRA,Cheah2006_IJRR,Cheah2006_TAC,Braganza2005_CDC,Dixon2007_TAC,Wang2010_TCST,Wang2015_Aut}) are characterized by the use of an approximate Jacobian matrix (due to the kinematics uncertainty), and the prominent part of the control scheme may be the {approximate transpose Jacobian control} with\/without a {kinematic parameter adaptation law}.\n\n At the present stage, one may say that the stability properties of the adaptive Jacobian control system under both the uncertain kinematics and dynamics are fully addressed, as can be seen in the above mentioned results, yet it remains unclear about the performance of the system in the sense that some performance issues regarding, e.g., tracking accuracy and transient response, are not adequately studied. In fact, the performance of the now commonly adopted transpose Jacobian feedback (e.g., \\cite{Cheah2006_IJRR,Cheah2006_TAC,Braganza2005_CDC,Wang2010_TCST,Wang2015_Aut}), as stated in \\cite{Craig2005_Book}, is not desirable especially when the manipulator moves in a large range although the transpose Jacobian feedback for robot task-space control problem shows excellent stability property (refer to the pioneering work in \\cite{Takegaki1981_ASME} on the regulation problem and to \\cite{Cheah2006_IJRR,Cheah2006_TAC} on the tracking problem). Another commonly adopted task-space control approach is inverse Jacobian feedback (see, e.g., \\cite{Craig2005_Book}), and the stability analysis of the inverse Jacobian feedback for regulation problem is given in \\cite{Cheah2005_TRO}, which seems much more involved than that of its counterpart (i.e., transpose Jacobian feedback).\n\n It is well known that the performance of a linear time-invariant system is ensured by appropriately designating the poles of the closed-loop system. For the nonlinear robotic system, this is almost not achievable except for the known parameter case (e.g., the standard computed torque control can result in a linear error dynamics with guaranteed performance---see \\cite{Spong2006_Book,Slotine1991_Book}). {Let us now contemplate the standard control problem for a frictionless mass that is governed by $m\\ddot{y}=u$, where $y\\in R$ denotes the position of the mass, $m\\in R$ the mass and $u\\in R$ the control input. From the standard linear system theory, if the control $u$ takes a PD action, the design of the gains must take into account the mass property of the system and one advisable design is to choose mass-dependent gains. This standard idea has already appeared in the robot control problem with or without dynamic uncertainties, e.g., the computed torque control actually takes inertia-matrix-dependent PD control plus certain feedforward terms (see, e.g., \\cite{Spong2006_Book,Slotine1991_Book}), and the adaptive control in \\cite{Slotine1989_Aut} chooses the feedback gain based on the estimated inertia matrix (see \\cite[Sec. 3.2]{Slotine1989_Aut}).} However, it is unclear {how to ensure the performance of the robot system under both the kinematic and dynamic uncertainties}. There is also some work addressing the performance in the robust control framework (e.g., \\cite{Yao1996_ASME}), yet the gain selection is conservative.\n\n\n\n In this paper, we propose a separation approach to the adaptive control problem for robots with both the uncertain kinematics and dynamics, and two adaptive controllers are proposed. The proposed first controller can also ensure, in the sense of certainty equivalence, the performance of the closed-loop system with essentially the same modification as in \\cite{Slotine1989_Aut}. The superior\/desirable properties of the proposed approach are summarized as follows.\n  \\begin{enumerate}\n\\item It realizes the separation of the kinematic and dynamic loops (i.e., the separation is realized in the case that the joint velocity tracking error is guaranteed to be square-integrable and bounded) thanks to the employment of the kinematic parameter adaptation law (or the new definition of the joint reference velocity) and that of the control law (with the same structure as the Slotine and Li adaptive scheme in the task space \\cite[Sec.~3]{Slotine1987_IJRR}) that does not use the approximate transpose Jacobian matrix, while the two loops are coupled and mixed in most existing results (e.g., \\cite{Cheah2006_IJRR,Cheah2006_TAC,Braganza2005_CDC,Wang2013_TAC});\n\n   \n\n    \\item the proposed first controller with appropriate modifications that follow the result in \\cite{Slotine1989_Aut} improves the performance of the closed-loop system, extending the scheme in \\cite{Slotine1989_Aut} to be capable of handling both the kinematic and dynamic uncertainties, {without the expense of conservative gain selection (e.g., \\cite{Yao1996_ASME})}.  It is also shown that even under constant-gain feedback, the proposed controller tends to give better performance than the approximate transpose Jacobian feedback.\n\n \\end{enumerate}\nWe would like to emphasize that the separation property stated in 1) becomes more prominent in industrial robotic applications in that the joint velocity control mode is very common in most industrial manipulators. {Under the joint velocity control mode, we cannot modify the joint servoing module and what we can design is the joint velocity command}. {The separation property of the proposed controllers makes one reduced case of our main result serve well for this application scenario [i.e., taking the joint reference velocity as the joint velocity command of the joint servoing module (see Remark 3)], while the adaptive transpose Jacobian control does not fit this circumstance due to the coupling nature of the adaptive transpose Jacobian feedback in the torque input}.\n\n\n\n\n\n\\section{Kinematics and Dynamics}\n\nLet $x\\in R^n$ be the position of the end-effector in the task space (e.g., Cartesian space or image space), and it is relevant to the joint position via the nonlinear mapping \\cite{Craig2005_Book,Spong2006_Book}\n\\be\n\\label{eq1}\nx=f(q)\n\\ee\nwhere $q\\in R^n$ denotes the joint position, and $f: R^n\\to R^n$ is the mapping from joint space to task space.\n\nDifferentiating (\\ref{eq1}) with respect to time gives the relation between the task-space velocity and joint-space velocity \\cite{Craig2005_Book,Spong2006_Book}\n\\be\n\\label{eq2}\n\\dot x=J(q)\\dot q\n\\ee\nwhere $J(q)\\in R^{n\\times n}$ is the Jacobian matrix. In the case that the kinematic parameters are unknown, we cannot obtain the task-space position\/velocity by the direct kinematics given above. Instead, we assume that certain task-space sensors (e.g., a camera) are employed to give the task-space position\/velocity information. The kinematics (\\ref{eq2}) has the following linearity-in-parameters property \\cite{Cheah2006_IJRR}.\n\n\\emph{Property 1:} The kinematics (\\ref{eq2}) depends linearly on a constant parameter vector $a_k$, which gives rise to\n\\be\n\\label{eq:a1}\nJ(q)\\xi=Y_k(q,\\xi)a_k\n\\ee\nwhere $\\xi\\in R^n$ is a vector and $Y_k(q,\\xi)$ is the kinematic regressor matrix.\n\n\nThe equations of motion of the manipulator can be written as \\cite{Slotine1991_Book,Spong2006_Book}\n\\begin{equation}\n\\label{eq3}\nM(q)\\ddot q+C(q,\\dot q)\\dot q+g(q)=\\tau\n\\end{equation}\nwhere $M \\left( {q }\n\\right) \\in R^{n\\times n}$ is the inertia matrix, $C \\left( {q ,\\dot\n{q}} \\right) \\in R^{n\\times n}$ is the Coriolis and centrifugal matrix,\n$g \\left( {q } \\right) \\in R^n$ is the gravitational torque, and\n$\\tau  \\in R^n$ is the joint control torque. For the convenience of later reference, three well-understood properties associated with the dynamics (\\ref{eq3}) are listed as follows (see, e.g., \\cite{Slotine1991_Book,Spong2006_Book}).\n\n\\textit{Property 2:} The inertia matrix $M (q )$ is symmetric and uniformly positive\ndefinite.\n\n\\textit{Property 3: }The Coriolis and centrifugal matrix $C (q ,\\dot q )$ can be\nappropriately determined such that $\\dot {M}(q) - 2C(q,\\dot q)$ is skew-symmetric.\n\n\\textit{Property 4: }The dynamics (\\ref{eq3}) depends linearly on a\nconstant parameter vector $a_d $, which leads to\n\\begin{equation}\n\\label{eq4}\nM \\left( q  \\right)\\dot \\zeta  + C \\left( q ,\\dot {q} \\right)\\zeta + g \\left( {q } \\right) = Y_d ( q ,\\dot {q}\n,\\zeta ,\\dot \\zeta )a_d\n\\end{equation}\nwhere $\\zeta \\in R^n$ is a differentiable vector, $\\dot {\\zeta }$ is the time derivative of $\\zeta $, and $Y_d( {q ,\\dot {q} ,\\zeta ,\\dot {\\zeta }} )$ is the dynamic\nregressor matrix.\n\n\n\\section{Adaptive Control}\n\nIn this section, we investigate the adaptive controller design for the robot manipulator given by (\\ref{eq2}) and (\\ref{eq3}), and the control objective is to drive the robot end-effector to asymptotically track a desired trajectory in the task space, i.e., to ensure that $x-x_d\\to 0$ as $t\\to\\infty$, where $x_d$ denotes the desired task-space trajectory and it is assumed that $x_d$, $\\dot x_d$ and $\\ddot x_d$ are all bounded.\n\n\\subsection{Adaptive Controller I}\n\nFollowing \\cite{Cheah2006_IJRR,Cheah2006_TAC}, we define a joint reference velocity using the estimated Jacobian matrix as\n\\be\n\\label{eq6}\n\\dot q_r =\\hat{J}^{-1}(q)\\dot x_r\n\\ee\nwhere $\\dot x_r=\\dot x_d- \\alpha \\Delta x$, $\\Delta x=x-x_d$ denotes the task-space position tracking error, $\\alpha$ is a positive design constant, and $\\hat{J}(q)$ is the estimated Jacobian matrix which is obtained by replacing $a_k$ in $J(q)$ with its estimate $\\hat{a}_k$.\nDifferentiating equation (\\ref{eq6}) with respect to time gives the joint reference acceleration\n\\be\n\\label{eq7}\n\\ddot q_r=\\hat J^{-1}(q)\\Big[\\ddot x_r-\\dot {\\hat J}(q)\\dot q_r\\Big].\n\\ee\nLet us now define a sliding vector\n\\be\n\\label{eq8}\ns=\\dot q-\\dot q_r\n\\ee\nand using (\\ref{eq2}), (\\ref{eq:a1}), and (\\ref{eq6}), we can rewrite equation (\\ref{eq8}) as\n\\begin{align}\n\\label{eq9}\ns=&J^{-1}(q)\\left[\\dot x-J(q)\\dot q_r\\right]=J^{-1}(q)\\left[\\dot x-\\hat{J}(q)\\dot q_r+Y_k(q,\\dot q_r)\\Delta a_k\\right]\\nn\\\\\n=& J^{-1}(q)\\left[\\Delta \\dot x+\\alpha \\Delta x+Y_k(q,\\dot{q}_r)\\Delta a_k\\right]\n\\end{align}\nwhich can further be written as \\cite{Ma1995_ICRA,Cheng2009_Aut}\n\\be\n\\label{eq10}\n\\Delta \\dot x=-\\alpha\\Delta x-Y_k(q,\\dot q_r)\\Delta a_k+J(q)s\n\\ee\nwhere $\\Delta a_k=\\hat{a}_k-a_k$ is the kinematic parameter estimation error.\n\n\n\nThe control law is given as\n\\be\n\\label{eq11}\n\\tau= -K s+ Y_d(q,\\dot q,\\dot q_r,\\ddot q_r)\\hat{a}_d\n\\ee\nwhere $K\\in R^{n\\times n}$ is a symmetric positive definite matrix. The estimated dynamic parameter $\\hat{a}_d$ (i.e., the estimate of $a_d$) is updated by\n\\begin{align}\n\\label{eq12}\n\\dot{\\hat{a}}_d=&-\\Gamma_d Y_d^T(q,\\dot q,\\dot q_r,\\ddot q_r)s\n\\end{align}\nwhere $\\Gamma_d$ is a symmetric positive definite matrix. The estimated kinematic parameter $\\hat{a}_k$ is updated by the direct adaptation law\n\\be\n\\label{eq13}\n\\dot{\\hat{a}}_k=\\Gamma_k Y_k^T(q,\\dot q_r)[(\\beta\/\\alpha)\\Delta \\dot x+\\Delta x]\n\\ee\n where $\\Gamma_k$ is a symmetric positive definite matrix and $\\beta\\in[0,1]$ is a design constant.\n\n\\emph{Remark 1:} The differences between the adaptive controller here and the one in \\cite{Cheah2006_IJRR,Cheah2006_TAC} are that 1) the feedback part in (\\ref{eq11}) can be rewritten as $-K J^{-1}(q)\\left[\\Delta \\dot x +\\alpha \\Delta x+Y_k(q,\\dot q_r)\\Delta a_k\\right]$, which can thus be intuitively interpreted as {inverse Jacobian feedback of both the task-space tracking error and the kinematic parameter estimation error} rather than the approximate transpose Jacobian feedback and 2) the kinematic parameter adaptation law (\\ref{eq13}) is based on an adaptive regressor that depends on the joint reference velocity $\\dot q_r$ rather than $\\dot q$.  The dynamic parameter adaptation law (\\ref{eq12}) is actually the same as the one in \\cite{Cheah2006_TAC}. The control law (\\ref{eq11}) expands the inverse Jacobian based task-space adaptive scheme in \\cite[Sec.~3]{Slotine1987_IJRR} to additionally include the inverse Jacobian feedback of the kinematic parameter estimation error, which supplies our controller with the ability of handling the kinematic uncertainties.\n\n\nSubstituting the control law (\\ref{eq11}) into the dynamics (\\ref{eq3}) yields\n\\be\n\\label{eq14}\nM(q)\\dot s+C(q,\\dot q)s=-K s +Y_d(q,\\dot q,\\dot q_r,\\ddot q_r)\\Delta a_d\n\\ee\nwhere $\\Delta a_d=\\hat{a}_d-a_d$. The closed-loop robotic system can be described by\n\\be\n\\label{eq15}\n\\begin{cases}\n\\Delta \\dot x=-\\alpha\\Delta x-Y_k(q,\\dot q_r)\\Delta a_k+J(q)s,\n\\\\\nM(q)\\dot s+C(q,\\dot q)s=-K s +Y_d(q,\\dot q,\\dot q_r,\\ddot q_r)\\Delta a_d,\n\\end{cases}\n\\ee\nand the adaptation laws (\\ref{eq12}) and (\\ref{eq13}).\n\nWe are presently ready to formulate the following theorem.\n\n\\emph{Theorem 1:} {Suppose that the estimated Jacobian matrix $\\hat J(q)$ is nonsingular and that all joints of the manipulator are revolute. Then, the control law (\\ref{eq11}), the dynamic parameter adaptation law (\\ref{eq12}), and the kinematic parameter adaptation law (\\ref{eq13}) ensure that the task-space tracking errors converge to zero, i.e., $\\Delta x\\to 0$ and $\\Delta \\dot x\\to 0$ as $t\\to\\infty$.}\n\n\\emph{Proof: } Following \\cite{Slotine1987_IJRR,Ortega1989_Aut}, we take into account the Lyapunov-like function candidate $V_1=(1\/2)s^T M(q)s+(1\/2)\\Delta a_d^T \\Gamma_d^{-1}\\Delta a_d$, and differentiating $V_1$ with respect to time along the trajectories of the second subsystem in (\\ref{eq15}) and of the adaptation law (\\ref{eq12}) and using {Property 3}, we obtain\n$\n\\dot V_1=-s^T K s\\le 0\n$, which implies that $s\\in {\\cal L}_2\\cap {\\cal L}_\\infty$ and  $\\hat a_d\\in {\\cal L}_\\infty$\n\nSince $J(q)$ is bounded (by the assumption that all joints of the manipulator are revolute), we have that $J(q)s\\in {\\cal L}_2$, and thus, there exists a constant $l_M>0$ such that $\\int_0^t s^T J^T(q)J(q)sdr\\le l_M$ for all $t\\ge 0$. Then, consider the following quasi-Lyapunov function candidate\n\\begin{align}\n\\label{eq17}\nV_2=&\\frac{1-\\beta}{2}\\Delta x^T \\Delta x+\\frac{1}{2\\alpha}\\bigg[{l_M-\\int_0^t s^T J^T(q)J(q)sdr}\\bigg]\\nn\\\\\n&+\\frac{1}{2}\\Delta a_k^T \\Gamma_k^{-1}\\Delta a_k\n\\end{align}\nwhere the second term of $V_2$ follows the result in \\cite[p.~118]{Lozano2000_Book}, and taking the derivative of $V_2$ along the first subsystem in (\\ref{eq15}) gives\n\\begin{align}\n\\label{eq18}\n\\dot{V}_2=&-\\alpha(1-\\beta)\\Delta x^T\\Delta x- (1-\\beta)\\Delta a_k^T Y_k^T(q,\\dot q_r)\\Delta x+\\Delta a_k^T \\Gamma_k^{-1}\\dot{\\hat a}_k\\nn\\\\\n&+(1-\\beta)\\Delta x^T J(q)s-\\frac{1-\\beta}{2\\alpha}s^T J^T(q)J(q)s\\nn\\\\\n&-\\frac{\\lambda}{2\\alpha} s^T J^T(q)J(q)s.\n\\end{align}\nUsing $J(q)s=\\Delta \\dot x+\\alpha\\Delta x+Y_k(q,\\dot q_r)\\Delta a_k$ [from (\\ref{eq10})], we can write (\\ref{eq18}) as\n\\begin{align}\n\\label{eq:a5}\n\\dot{V}_2=&-\\alpha(1-\\beta)\\Delta x^T\\Delta x- (1-\\beta)\\Delta a_k^T Y_k^T(q,\\dot q_r)\\Delta x+\\Delta a_k^T \\Gamma_k^{-1}\\dot{\\hat a}_k\\nn\\\\\n&+(1-\\beta)\\Delta x^T J(q)s-\\frac{1-\\beta}{2\\alpha}s^T J^T(q)J(q)s\\nn\\\\\n&-\\frac{\\beta}{2\\alpha} \\big(\\Delta \\dot x+\\alpha\\Delta x\\big)^T\\big(\\Delta \\dot x+\\alpha\\Delta x\\big)\\nn\\\\\n&-\\Delta a_k^T Y_k(q,\\dot q_r)[(\\beta\/\\alpha)\\Delta \\dot x+\\beta\\Delta x]\\nn\\\\\n&-\\frac{\\beta}{2\\alpha}\\Delta a_k^T Y_k^T(q,\\dot q_r)Y_k(q,\\dot q_r)\\Delta a_k.\n\\end{align}\nUsing the inequality\n$\n\\Delta x^T J(q)s\\le (\\alpha\/2)\\Delta x^T \\Delta x+\\left[1\/(2\\alpha)\\right]s^T J^T(q)J(q)s\n$ from the standard result concerning the basic inequalities\nand substituting the adaptation law (\\ref{eq13}) into\nequation (\\ref{eq:a5}) yields\n\\be\n\\label{eq:a6}\n\\dot{V}_2\\le-\\frac{\\alpha(1-\\beta)}{2}\\Delta x^T\\Delta x-\\frac{\\beta}{2\\alpha} \\big(\\Delta \\dot x+\\alpha\\Delta x\\big)^T\\big(\\Delta \\dot x+\\alpha\\Delta x\\big)\\le 0\n\\ee\nwhich directly gives the result that $\\hat{a}_k\\in {\\cal L}_\\infty$ and that $\\Delta x\\in {\\cal L}_2\\cap {\\cal L}_\\infty$ in the case $\\beta<1$. In the case $\\beta=1$, we obtain from (\\ref{eq:a6}) that $\\Delta\\dot x+\\alpha\\Delta x\\in {\\mathcal L}_2$, and further that $\\Delta x\\in {\\mathcal L}_2\\cap{\\mathcal L}_\\infty$ according to the input-output properties of exponentially stable and strictly proper linear systems \\cite[p.~59]{Desoer1975_Book}.\n\n\nFrom equation (\\ref{eq6}), {if the estimated Jacobian matrix $\\hat{J}(q)$ is nonsingular}, we have that $\\dot{q}_r\\in {\\cal L}_\\infty$ since $\\dot{x}_r\\in {\\cal L}_\\infty$. Then, we obtain that  $\\dot q\\in {\\cal L}_\\infty$ since $s\\in {\\cal L}_\\infty$, and that $\\dot x\\in{\\cal L}_\\infty$ based on (\\ref{eq2}). Therefore, $\\Delta x$ must be uniformly continuous, and from the properties of square-integrable and uniformly continuous functions \\cite[p.~117]{Lozano2000_Book}, we obtain $\\Delta x\\to 0$ as $t\\to\\infty$. From (\\ref{eq13}), we have that $\\dot{\\hat{a}}_k\\in \\cal L_\\infty$ since $Y_k(q,\\dot q_r)$ and $\\Delta x$ are both bounded, which then implies the boundedness of $\\dot{\\hat{J}}(q)$. Thus, from (\\ref{eq7}), we obtain that $\\ddot q_r\\in {\\cal L}_\\infty$. From (\\ref{eq14}), we obtain that $\\dot s\\in {\\cal L}_\\infty$ by using {Property 2}. This leads us to obtain that $\\ddot q=\\ddot q_r+\\dot s\\in {\\cal L}_\\infty$, and that $\\ddot x\\in {\\cal L}_\\infty$ from the differentiation of equation (\\ref{eq2}), i.e., $\\ddot x=J(q)\\ddot q+\\dot J (q)\\dot q$. Therefore, $\\Delta \\ddot x\\in {\\cal L}_\\infty$, and then $\\Delta \\dot x$ is uniformly continuous. Due to the result that $\\Delta x\\to 0$ as $t\\to\\infty$, we obtain from Barbalat's Lemma \\cite{Slotine1991_Book} that $\\Delta \\dot x\\to 0$ as $t\\to\\infty$. \\hfill {\\small $\\blacksquare$}\n\n\\subsection{Adaptive Controller II}\n\nWe now present an adaptive controller that also has the separation property but uses different joint reference velocity and kinematic parameter adaptation law. This controller relies on the joint reference velocity defined as\n\\be\n\\label{eq:a8}\n\\dot q_r=\\hat J^{-1}(q)\\dot x_d-\\alpha \\hat J^T(q)\\Delta x.\n\\ee\nWe then have that\n\\be\n\\label{eq:a9}\n\\Delta \\dot x=-\\alpha\\hat J(q)\\hat J^T (q)\\Delta x-Y_k(q,\\dot q)\\Delta a_k+\\hat J(q)s.\n\\ee\n\nThe control law and the dynamic parameter adaptation law are still (\\ref{eq11}) and (\\ref{eq12}) yet with $\\dot q_r$ given by (\\ref{eq:a8}). The kinematic parameter adaptation law is given as\n\\be\n\\label{eq:a10}\n\\dot{\\hat a}_k=\\Gamma_k Y_k^T(q,\\dot q)\\Delta x.\n\\ee\n\n\\emph{Theorem 2:} The control law (\\ref{eq11}) and dynamic parameter adaptation law (\\ref{eq12}) with $\\dot q_r$ given by (\\ref{eq:a8}), and kinematic parameter adaptation law (\\ref{eq:a10}) ensure that  $\\Delta x\\to 0$ and $\\Delta \\dot x\\to0$ as $t\\to\\infty$ provided that the estimated Jacobian matrix $\\hat J(q)$ is nonsingular.\n\n\\emph{Proof:} For the dynamic loop, we can directly obtain that $s\\in{\\mathcal L}_2\\cap {\\mathcal L}_\\infty$ and $\\hat a_d\\in{\\mathcal L}_\\infty$, by following similar procedures as in the proof of Theorem 1. Then, there exists a positive constant $l_M^\\ast$ such that $\\int_0^t s^T(r)s(r)dr\\le l_M^\\ast$, $\\forall t\\ge 0$. Consider the following quasi-Lyapunov function candidate\n\\be\n\\label{eq:a11}\nV_2^\\ast=\\frac{1}{2}\\Delta x^T \\Delta x+\\frac{1}{2\\alpha}\\left[l_M^\\ast-\\int_0^t s^T(r)s(r)dr\\right]+\\frac{1}{2}\\Delta a_k^T\\Gamma_k^{-1}\\Delta a_k\n\\ee\nand we obtain\n\\be\n\\dot V_2^\\ast\\le -(\\alpha\/2)\\Delta x^T \\hat J(q)\\hat J^T(q)\\Delta x\\le 0\n\\ee\nwhere we have used the following result from the standard basic inequalities\n\\be\n\\Delta x^T\\hat J(q)s\\le (\\alpha\/2)\\Delta x^T \\hat J(q)\\hat J^T(q)\\Delta x+1\/(2\\alpha)s^T s.\n\\ee\nThen, we can show the convergence of the task-space tracking errors, using similar procedures as in the proof of Theorem 1. \\hfill {\\small $\\blacksquare$}\n\n\\emph{Remark 2:} The second adaptive controller achieves separation by using a strong feedback of the tracking error $\\Delta x$ in the definition of the joint reference velocity (this form of reference velocity appears in the context of global task-space control with known kinematic parameters \\cite{Li2013_Aut}). This can be noticed more clearly from (\\ref{eq:a9}) and the final equivalent feedback gain is $\\alpha\\hat J(q)\\hat J^T(q)$ at the task-space velocity level. Due to this, the second adaptive controller is applicable to robots with prismatic joints, in contrast with the proposed first adaptive controller. The kinematic parameter adaptation law (\\ref{eq:a10}) is the same as the direct version of that in \\cite{Cheah2006_IJRR} (i.e., with the prediction error being removed), and the separation of the kinematic and dynamic loops is mainly attributed to the new definition of the joint reference velocity (\\ref{eq:a8}) and the control law (\\ref{eq11}).\n\n\n\\emph{Remark 3:} From the separation analysis in the proofs of Theorem 1 and Theorem 2, we can obtain two adaptive kinematic schemes. One is given by the joint reference velocity (\\ref{eq6}) and kinematic parameter adaptation law (\\ref{eq13}), and the other is given by  (\\ref{eq:a8}) and (\\ref{eq:a10}). Both of them are expected to serve well for industrial robotic applications. In fact, the joint velocity servoing module can generally ensure that the joint velocity tends sufficiently fast to the joint reference velocity in the sense that the joint velocity tracking error $s=\\dot q-\\dot q_r$ is square-integrable and bounded. Hence, $J(q)s\\in {\\mathcal L}_2\\cap {\\mathcal L}_\\infty$ and $s\\in{\\mathcal L}_2\\cap {\\mathcal L}_\\infty$. Consider the same quasi-Lyapunov function as (\\ref{eq17}) or (\\ref{eq:a11}), and then the tracking error convergence can be obtained by following similar analysis as in the proof of Theorem 1 or Theorem 2.\n\n\n\\emph{Remark 4:} {The assumption that the manipulator is away from the singular configuration and the use of the parameter projection algorithms ensure that the estimated Jacobian matrix $\\hat J(q)$ is nonsingular in the parameter adaptation process (see, e.g., \\cite{Cheah2006_IJRR,Cheah2006_TAC,Dixon2007_TAC})}.\n\n\\section{Performance of the System}\n\n In this section, we show how the first adaptive controller given in Sec. III improves the performance of the robotic system under both the uncertain kinematics and dynamics by a suitable nonconservative modification. This modification follows \\cite{Slotine1989_Aut}, yet in the context of task-space robot control with kinematic uncertainties. The extension turns out to be direct thanks to the formulation in the previous section, yet, here, our emphasis is on demonstrating {why this modification implies potentially good performance}. Moreover, it will be shown that even under constant-gain feedback, inverse Jacobian feedback yields potentially better performance than transpose Jacobian feedback.\n\nFollowing \\cite{Slotine1989_Aut}, we specify the feedback gain $K$ in (\\ref{eq11}) as\n\\be\n\\label{eq21}\nK=\\lambda_c \\hat M(q)\n\\ee\nwith $\\lambda_c$ being a positive design constant and meanwhile modify the adaptation law (\\ref{eq12}) as\n\\be\n\\label{eq22}\n\\dot{\\hat a}_d=-\\Gamma_d Y_d^T(q,\\dot q,\\dot q_r,\\ddot q_r^\\ast)s\n\\ee\n  where $\\hat M(q)$ is the estimated inertia matrix obtained by replacing $a_d$ in $M(q)$ with $\\hat a_d$, and $\\ddot q_r^\\ast=\\ddot q_r-\\lambda_c s$. The selection (\\ref{eq21}) and the modification (\\ref{eq22}) yields (the same as the case in \\cite{Slotine1989_Aut})\n$\n\\dot V_1=-\\lambda_c s^T M(q)s\\le 0.\n$\nThen, the same result as in {Theorem 1} follows.\n\n{Let us now focus on interpreting the performance issues from two perspectives. First, the derivative of $V_1$ can further be written as\n\\begin{align}\\dot V_1\n=-\\lambda_C\\underbrace{\\psi^T J^{-T}(q)M(q) J^{-1}(q)\\psi}_{V^\\ast}\n\\end{align}\nwith $\\psi=\\Delta \\dot x +\\alpha \\Delta x+Y_k(q,\\dot q_r)\\Delta a_k$, which implies the exponential convergence of $\\Delta \\dot x+\\alpha\\Delta x+Y_k(q,\\dot q_r)\\Delta a_k$ with the rate $\\lambda_c$ in the case that the dynamic parameter is known [in this case, $V_1=(1\/2)V^\\ast$] and further the exponential convergence of $\\Delta x$ with the rate $\\min\\left\\{\\lambda_c,\\alpha\\right\\}$ in the case that the kinematic parameter is known (i.e., the performance is quantified in the sense of certainty equivalence).}\nOn the other hand, based on (\\ref{eq9}), we can rewrite the definitions of $\\dot q_r,\\ddot q_r$ in (\\ref{eq6}) and (\\ref{eq7}) as\n\\begin{align}\n\\dot q_r=&J^{-1}(q)\\left[\\dot x_r-Y_k(q,\\dot q_r)\\Delta a_k\\right]\n\\\\\n\\ddot q_r=&J^{-1}(q)\\left[\\ddot x_r-\\dot Y_k(q,\\dot q_r)\\Delta a_k-Y_k(q,\\dot{q}_r)\\dot{\\hat{a}}_k\\right]-J^{-1}(q)\\dot J(q)\\dot q_r.\n\\end{align}\nWith the above new formulation of $\\dot q_r$ and $\\ddot q_r$, the control law can now be written as\n\\begin{align}\n\\label{eq26}\n\\tau=& \\hat M(q) J^{-1}(q)\\Big[\\ddot x_d-(\\alpha+\\lambda_c)\\Delta \\dot x-\\alpha\\lambda_C\\Delta x\\nn\\\\\n&-\\left(\\dot{Y}_k(q,\\dot q_r)+\\lambda_c Y_k(q,\\dot q_r)\\right)\\Delta a_k-Y_k(q,\\dot q_r)\\dot{\\hat{a}}_k\\Big]\\nn\\\\\n&+\\left[\\hat C(q,\\dot q)J^{-1}(q)-\\hat M(q)J^{-1}(q)\\dot J(q)J^{-1}(q)\\right]\\nn\\\\\n&\\times\\left[\\dot x_r-Y_k(q,\\dot q_r)\\Delta a_k\\right]+\\hat g(q).\n\\end{align}\n{which is quite similar to the certainty-equivalence form of the task-space inverse dynamics (see, e.g., \\cite{Luh1980_TAC,Craig2005_Book,Spong2006_Book}) and thus ensures the performance in the sense of certainty equivalence, where $\\alpha$ and $\\lambda_c$ act as the quantification of the performance (e.g., speed of response and robustness)}. We now present the following theorem to summarize the above result.\n\n{\\emph{Theorem 3:} The adaptive controller (\\ref{eq26}), (\\ref{eq22}), and (\\ref{eq13}) ensures the exponential convergence of the task-space position tracking error with the rate $\\min\\{\\lambda_c,\\alpha\\}$ in the sense of certainty equivalence.}\n\n{\nTo further clarify the potential benefit of the choice of the feedback gain (\\ref{eq21}), let us consider a scenario that a convergence rate of the task-space tracking error $\\gamma^\\ast$ (in the sense of certainty equivalence) is required by certain task. To serve this requirement, we only need to specify $\\lambda_c$ and $\\alpha$ as $\\lambda_c=\\alpha=\\gamma^\\ast$ as using the control scheme here. In contrast, if still using the constant gain feedback (similar to the case in \\cite{Yao1996_ASME}), the convergence rate of $s$ becomes $\\lambda_{\\min}\\{K\\}\/\\lambda_{\\max}\\{M(q)\\}$ and thus the best choice is perhaps to specify $K$ and $\\alpha$ such that $\\alpha=\\gamma^\\ast$ and $\\lambda_{\\min}\\{K\\}\\ge\\gamma^\\ast \\lambda_{\\max}\\{M(q)\\}$. Due to the uncertainty of $M(q)$, conservativeness is generally inevitable.}\n\n\\emph{Remark 5:} The adaptive controller given by (\\ref{eq26}), (\\ref{eq22}), and (\\ref{eq13}) yields\n\\be\n\\label{eq28}\n\\begin{cases}\n\\Delta \\dot x=-\\alpha\\Delta x-Y_k(q,\\dot q_r)\\Delta a_k+J(q)s,\\\\\nM(q)\\dot s+C(q,\\dot q)s\\\\\n=-\\lambda_c M(q) s+Y_d(q,\\dot q,\\dot q_r,\\ddot q_r^\\ast)\\Delta a_d.\n\\end{cases}\n\\ee\nThe feedback gains in both the systems of (\\ref{eq28}) are inertia-dependent (the apparent inertia of the first subsystem in (\\ref{eq28}) can be considered as $I_n$). This gives the additional demonstration on why the adaptive scheme given by (\\ref{eq26}), (\\ref{eq22}), and (\\ref{eq13}) implies good task-space tracking performance.\nThe approximate transpose Jacobian feedback adopted in \\cite{Cheah2006_IJRR} [i.e., $-\\hat J^T(q) K\\hat J(q)s$] with the gain selection (\\ref{eq21}) and appropriate modification of the dynamic parameter adaptation law would render the feedback gain (with respect to $s$) in the closed-loop dynamics as $-\\lambda_c\\hat J^T(q)M(q) \\hat J(q)$, which, in most cases, cannot match $M(q)$. This also suggests that for the task-space tracking problem, approximate transpose Jacobian control (e.g., \\cite{Cheah2006_IJRR,Cheah2006_TAC}) may not be preferred.\n\n\\emph{Remark 6:} The assertion in Remark 5 holds even for the case of constant-gain feedback (i.e., $K$ is chosen to be constant). It is well known that the task-space inertia $J^{-T}(q)M(q)J^{-1}(q)$ involves the inversion of the Jacobian matrix. In the case of using approximate transpose Jacobian feedback as is the case in \\cite{Cheah2006_IJRR,Cheah2006_TAC}, the inversion of the transpose of the approximate Jacobian matrix would cancel the transpose of the approximate Jacobian matrix and render the feedback gain to be $K$, which implies that we have to rely on the constant gain $K$ to compensate for the task-space inertia $J^{-T}(q)M(q)J^{-1}(q)$. In the case of using inverse Jacobian feedback without involving the transpose of the approximate Jacobian matrix as in our result, the task-space formulation renders the inverse Jacobian feedback premultiplied by $J^{-T}(q)$, i.e., using the Jacobian-dependent varying gain $J^{-T}(q)K J^{-1}(q)$ to compensate for the varying task-space inertia $J^{-T}(q)M(q)J^{-1}(q)$, which tends to be much easier. The performance superiority of inverse Jacobian feedback control is thus obvious.\n\n\\emph{Remark 7:}  In the visual tracking problem for robots with uncertainties in the camera model and\/or manipulator kinematics, most results, e.g., \\cite{Cheah2007_ICRA,Wang2010_TCST,Wang2012b_Mech,Li2012_ACC,Wang2015_Aut}, are fully\/partly based on the approximate transpose Jacobian feedback. The use of constant-gain feedback in the joint space (i.e., in the form $-K s$) occurs \\cite{Lizarralde2013_Aut} [which employs the indirect kinematic parameter adaptation law, and additionally require the persistent excitation of the kinematic regressor $Y_k(q,\\dot q)$ so that the convergence of the tracking error is ensured], and also appears in \\cite{Braganza2005_CDC,Wang2010_TCST,Wang2012b_Mech,Li2012_ACC} as part of the overall feedback action (the use of $-K s$ alone in this case, yet, cannot ensure stability), which is the same as the one proposed in our result and may also be interpreted as inverse Jacobian feedback, yet the rationality\/interpretation of doing so and the performance issues associated with the closed-loop system are not adequately addressed. The adaptive kinematic regressor used in the first adaptive controller is the same as the work in \\cite{Ma1995_ICRA,Cheng2009_Aut} (where the work in \\cite{Ma1995_ICRA} handles the control of attitude-controlled space manipulators using adaptive Jacobian technique with a coupled stability analysis for the kinematic and dynamic loops), and furthermore if we removed the velocity tracking error $\\Delta \\dot x$ in (\\ref{eq13}), the kinematic parameter adaptation law (\\ref{eq13}) would be the same as the one in \\cite{Ma1995_ICRA,Cheng2009_Aut}. {It is worth remarking that the adaptive controller given by (\\ref{eq11}), (\\ref{eq12}), and (\\ref{eq13}) with $\\lambda=0$ is quite similar to the one in \\cite{Ma1995_ICRA} (with its journal version in \\cite{Ma1996_CTA}, which is mainly pursued in the Chinese control literature), and due to the reason of language, it passed out of the knowledge of the international community. The main novel points of our first result given in Sec. III-A, in comparison with \\cite{Ma1995_ICRA}, lies in the proposed more general direct kinematic parameter adaptation law, the separation analysis (by using a quasi-Lyapunov analysis), and the clarification of the separation property (rationality) of using inverse Jacobian feedback as well as the adaptive kinematic regressor matrix (which then implies its potential applications to robots having an unmodifiable joint servoing controller yet admitting the design of the joint velocity command).} In addition, the first adaptive controller is demonstrated to be convenient for accommodating the performance issues.\n\n\n\n\n\n\n\n\n\n\\section{Simulation Results}\n\nLet us consider a standard 2-DOF (degree-of-freedom) planar manipulator that grasps an unknown tool. The physical parameters of the 2-DOF manipulator are not listed for saving space.\nThe sampling period is chosen as 5 ms. The desired trajectory of the manipulator end-effector is chosen as\n$\nx_d=[1.6754+0.3\\cos \\pi t,\n3.9950+0.3\\sin\\pi t]^T\n$.\n\nFor the first adaptive controller given in Sec. III-A, the controller parameters $K$, $\\alpha$, $\\Gamma_d$, and $\\Gamma_k$ are chosen as $K=30 I_2$, $\\alpha=10$, $\\beta=0.5$, $\\Gamma_d=200 I_4$, and $\\Gamma_k=300 I_3$, respectively. The initial parameter estimates are chosen as $\\hat{a}_d(0)=\\left[0,0,0,0\\right]^T$ and $\\hat a_k(0)=\\left[4.0,5.0,2.0\\right]^T$, while their actual values are $a_d=[7.9628, -0.9600, 19.2828,$\n$10.1495]^T$ and $a_k=\\left[2.0000,3.3856,0.8000\\right]^T$. Simulation results are shown in Fig. 1 and Fig. 2. For the second adaptive controller, the controller parameters are chosen to be the same as the first except that the design parameter $\\alpha$ is decreased to $\\alpha=1.5$ (since in this case, the equivalent feedback gain contains the transpose of the estimated Jacobian matrix). Simulation results are plotted in Fig. 3 and Fig. 4.\n\n\nUnder the same context, we also conduct the simulation when the controller given in \\cite{Cheah2006_IJRR,Cheah2006_TAC} is adopted. The control law in this case employs the approximate transpose Jacobian feedback $-\\hat{J}^T(q)K \\hat{J}(q)s$ and the kinematic parameter adaptation law takes the form $\\dot{\\hat{a}}_k=\\Gamma_k Y_k^T(q,\\dot q)[(\\beta\/\\alpha)\\Delta \\dot x+\\Delta x]$. The controller parameters are chosen to be the same as in the first adaptive controller. Simulation results in this context are shown in Fig. 5 and Fig. 6.\n\nOne obvious difference between the simulation results under the first controller and those under the one in \\cite{Cheah2006_IJRR,Cheah2006_TAC} is that the first controller results in better tracking accuracy [approximately $0.0015$ m (Fig. 1) versus $0.006$ m (Fig. 5) after $t=6$ s] and more adequate utilization of the joint torques (see Fig. 2 as compared with Fig. 6). The tracking accuracy under the second adaptive controller (after $t=6$ s) is comparable to that under the one in \\cite{Cheah2006_IJRR,Cheah2006_TAC}.\n\nThe tracking error with the controller under the estimated-inertia-based feedback action $-\\lambda_c\\hat{M}(q)s$ is shown in Fig. 7, where we choose $\\lambda_c=\\alpha=10$ so that the closed-loop dynamics is approximate to a critically damped linear dynamics, and the other parameters are chosen to be the same as those in the first adaptive controller. The main superiority may lie in the fact that responses of the tracking errors become more uniform and the tracking errors converge faster as compared with the first adaptive controller using constant-gain feedback (see Fig. 1).\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\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig1}\n\\caption{End-effector position tracking errors (first adaptive controller).}\\label{fig:side:a}\n\\end{minipage\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig2}\n\\caption{Joint control torques (first adaptive controller).} \\label{fig:side:b}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig3}\n\\caption{End-effector position tracking errors (second adaptive controller).}\\label{fig:side:a}\n\\end{minipage\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig4}\n\\caption{Joint control torques (second adaptive controller).} \\label{fig:side:b}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig5}\n\\caption{End-effector position tracking errors (adaptive transpose Jacobian feedback).}\\label{fig:side:a}\n\\end{minipage\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig6}\n\\caption{Joint control torques (adaptive transpose Jacobian feedback).} \\label{fig:side:b}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=3.3in]{Fig7}\n\\caption{End-effector position tracking errors (estimated-inertia-based feedback).} \\label{fig:side:b}\n\\end{minipage}\n\\end{figure}\n\n\n\n\\section{Conclusion and Discussion}\n\nIn this paper, we consider the adaptive tracking problem for robot manipulators subjected to both the kinematic and dynamic uncertainties. We propose two adaptive controllers that enjoy the separation property. The performance is then shown to be conveniently ensured under the first adaptive controller in the sense of certainty equivalence, with essentially the same modification of the control law and of the dynamic parameter adaptation law as in \\cite{Slotine1989_Aut}. Our study also suggests that to obtain a potentially good task-space tracking performance, adaptive inverse Jacobian feedback seems preferable than the commonly adopted adaptive transpose Jacobian feedback (e.g., \\cite{Cheah2006_IJRR,Cheah2006_TAC}).\n\nOne desirable feature of the proposed control schemes is that the separation of the kinematic and dynamic loops makes one reduced version of our control scheme rather suitable for industrial robotic applications. {This originates from the fact that the kinematic control law (represented by the joint reference velocity) plus the kinematic parameter adaptation law will ensure the convergence of the task-space tracking errors so long as the joint servoing loop (commonly embedded in most industrial robots) can ensure that the joint velocity tends sufficiently fast to the joint reference velocity in the sense that the joint velocity tracking error is square-integrable and bounded}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe {\\em cycle-cocycle reversal system} of a graph, introduced in \\cite{gioan2007enumerating}, consists of equivalence classes of orientations of the graph with respect to the {\\em cycle-cocycle reversal relation}, that is, two orientations are equivalent if they differ by successive reversals of directed (co)cycles.\nAs proven in the same paper, the number of cycle-cocycle reversal classes of $G$ equals the evaluation $t(G;1,1)$ of its Tutte polynomial, which is also the number of spanning trees of $G$.\nThis result can be thought as a ``linear algebra free'' formulation of Kirchhoff's Matrix-Tree Theorem.\nIn particular, the cycle-cocycle reversal system is related to various combinatorial objects associated to the graph Laplacian, notably the {\\em sandpile group} (also known as the {\\em critical group} or {\\em Jacobian group} in the literature).\nThe notion of {\\em chip-firing} (related to the {\\em abelian sandpile model}) can also be partially interpreted by {\\em cycle-cocycle reversals}.\nWe refer the reader to \\cite[Section 5]{gioan2007enumerating} and \\cite{backman2014riemann} for details.\n\n\\smallskip\n\nThe above setup, and further results involving Tutte polynomial evaluations for related reversal classes, were generalized to {\\em regular matroids} in \\cite{gioan2008circuit} (in terms of a {\\em circuit-cocircuit reversal system}).\nThis provides an approach to generalize the theory of sandpile groups and chip-firing to regular matroids (different from the approach in \\cite{MerinoDissertation}).\nSuch topic has been investigated since then in \\cite{bby2017geometric} in a unifying way.\n\n\\smallskip\n\nThe circuit-cocircuit reversal system is actually defined for general oriented matroids, and it was shown in \\cite{gioan2008circuit} that the aforementioned Tutte polynomial evaluations are not available for $U_{2,k}$, $k\\geq 4$.\nSince $U_{2,4}$ is the excluded minor for the class of regular matroids within oriented matroids, it was expected that these enumerative results are not available when the oriented matroid is not regular; we will prove this rigorously in this note.\nIn particular, we extend the results in \\cite{gioan2008circuit} to an equivalence of (oriented) matroid properties.\nWe use a noteworthy general relation between circuit-cocircuit reversal classes and activity classes of (re)orientations, which are known to be enumerated using the same Tutte polynomial evaluations, and have been introduced in the context of {\\em active bijections} \\cite{gioan2002thesis,gioan2005activity, gioanlasvergans2018AB, gioanlasvergans2018AB2}.\nWe also use this to give a short proof of results in~\\cite{gioan2008circuit}.\n\n\\section{Preliminaries}\n\nWe assume that the reader is familiar with the basic theory of oriented matroids \\cite{bjorner1999oriented}.\nGiven an oriented matroid $M$ on $E$, we identify the set of its reorientations with $2^E$ via the bijection associating $A\\subseteq E$ with $_{-A}M$.\n\nLet $M$ be an oriented matroid on $E$.\nFollowing \\cite{gioan2008circuit}, let us write $M \\sim -_CM$ if $C$ is a positive circuit or cocircuit of $M$ (we say that $-_CM$ is obtained from $M$ by a {\\em circuit or cocircuit reversal}, respectively).\nApplying the same rule to reorientations $-_{A}M$ for $A\\subseteq E$ (i.e., writing $-_{A}M\\sim-_{C\\triangle A}M$ when $C$ is a positive circuit or cocircuit of $-_{A}M$) and taking the transitive closure of the relation, we obtain an equivalence relation, whose equivalence classes are called {\\em circuit-cocircuit reversal classes} of reorientations of~$M$.\nAllowing only the use of positive circuits, resp. positive cocircuits, yields by the same way the \\emph{circuit reversal classes}, resp. the \\emph{cocircuit reversal classes}.\nAs observed in \\cite{gioan2008circuit}, circuit reversals act on the totally cyclic part of $M$ (the union of positive circuits of $M$) and cocircuit reversals act on the acyclic part of $M$ (the union of positive cocircuits of $M$).\n\nIt was shown and geometrically illustrated in \\cite[Proposition 2 and Figure 1]{gioan2008circuit} that, for any integer $k$, the number of acyclic cocircuit reversal classes of a uniform oriented matroid $U_{2,k}$ equals $1$, or $2$, if $k$ is even, or odd, respectively.\nOn the other hand, it is well-known that an oriented matroid $M$ is regular if and only if $U_{2,4}$ is not a minor of $M$. Indeed, regular matroids are precisely the orientable binary matroids \\cite[Theorem 7.9.3]{bjorner1999oriented}, so the claim follows from \\cite[Theorem 6.5.4]{oxley2006matroid}.\n\n\\medskip\n\nNow, let $M$ be an oriented matroid on a linearly ordered set $E$.\nConsider a reorientation $-_AM$ of $M$ such that $A$ does not contain the minimum element of a positive circuit or cocircuit of $-_AM$, we call such a reorientation \\emph{circuit-cocircuit minimal} (with respect to $M$); the terminology here is from \\cite{backman2014riemann}, and it is called \\emph{active fixed and dual-active fixed} in \\cite{gioanlasvergans2018AB, gioanlasvergans2018AB2}.\nSimilarly, we can define a \\emph{circuit minimal} (or \\emph{active-fixed}), resp. a \\emph{cocircuit minimal}  (or \\emph{dual-active-fixed}),  reorientation $-_AM$ of $M$ when $A$ does not contain the minimum element of a positive circuit, resp. cocircuit.\nLet us denote by $t(M;x,y)$ the Tutte polynomial of $M$.\nFrom the works on {\\em active bijections} \\cite{gioan2002thesis,gioan2005activity, gioanlasvergans2018AB, gioanlasvergans2018AB2}, we have:\n\n\\begin{theorem} \\label{CCMO_main}\nLet $M$ be an oriented matroid on a linearly ordered set.\nThen\n \\begin{enumerate}[leftmargin=10mm]\n \\item $t(M;1,1)=\\#$ circuit-cocircuit minimal reorientations of $M$,\n \\item $t(M;1,2)=\\#$ cocircuit minimal reorientations of $M$,\n \\item $t(M;2,1)=\\#$ circuit minimal reorientations of $M$,\n \\item $t(M;1,0)=\\#$ (circuit-)cocircuit minimal acyclic reorientations of $M$,\n \\item $t(M;0,1)=\\#$ circuit(-cocircuit) minimal totally cyclic reorientations of~$M$.\n \\end{enumerate}\n\\end{theorem}\n\nLet us explain this briefly; details can be found in \\cite{gioan2002thesis,gioan2005activity, gioanlasvergans2018AB, gioanlasvergans2018AB2}.\nThe {\\em active partition} of $M$ is a partition of its ground set induced by taking differences of unions of positive circuits\/cocircuits whose the minimum element is greater than a given element.\nTherefore, the minimum elements of the parts are precisely the minimum elements of some positive circuits\/cocircuits (called {\\em active\/dual-active elements}).\nActivity classes of reorientations are the sets of reorientations obtained from a given reorientation by arbitrarily reorienting parts of its active partition.\nIt turns out that all reorientations obtained by this way share the same active partition. Hence activity classes partition the set of reorientations.\nBy choosing a suitable reorientation for each part, each activity class contains a unique circuit-cocircuit minimal reorientation, which can be thought of as a representative of the class.\nFinally, using a classical formula of the Tutte polynomial in terms of orientation activities \\cite{lasvergnas1984activity}, one enumerates activity classes and gets the above evaluations.\n\n\\section{Results}\n\nWe first give a noteworthy property relating reversal classes and activity classes.\n\n\\begin{proposition} \\label{prop:CCMO_rep}\nLet $M$ be an oriented matroid on a linearly ordered ground set.\nEvery circuit-cocircuit reversal class of $M$ contains at least one circuit-cocircuit minimal  reorientation.\n\\end{proposition}\n\nThe following proof is essentially given in \\cite{backman2018partial} for graphs, but for the sake of interest and completeness, we include it here.\nA corollary of the proof is that a minimal reorientation can be obtained greedily. Moreover, we note that there is an interpretation using combinatorial commutative algebra \\cite[Section 4]{backman2018partial}.\n\n\\begin{proof}\nStart with an arbitrary reorientation of $M$, and greedily reorient any positive (co)circuit whose minimal element is in the set of reoriented elements with respect to $M$.\nOnce the procedure stops, we will have a circuit-cocircuit minimal reorientation equivalent to the starting reorientation, so it suffices to show the procedure always terminates.\nIf this is not the case, then, since the number of reorientations is finite, without loss of generality, we must return to the starting reorientation.\nLet $e$ be the minimal element that was reoriented (which must occur at least twice) in the process.\nWhen $e$ was reoriented for the first time, we must have reoriented it to remove it from the set of reoriented elements with respect to $M$, so the second reorientation is not valid, a contradiction.\n\\end{proof}\n\nBy combining Theorem \\ref{CCMO_main} and Proposition \\ref{prop:CCMO_rep}, we can use the set of circuit-cocircuit minimal reorientations (and variations thereof) as an intermediate object, and get the following corollary concerning the enumeration of reversal classes in terms of the Tutte polynomial.\n\n\\begin{corollary} \\label{coro:CCMO_red}\nLet $M$ be an oriented matroid on a linearly ordered ground set.\nThe number of circuit-cocircuit reversal classes is at most $t(M;1,1)$, with equality if and only if no two circuit-cocircuit minimal reorientations are contained in the same class.\nThe number of acyclic cocircuit reversal classes is at most $t(M ; 1, 0)$, with equality if and only if no two acyclic cocircuit minimal reorientations are contained in the same class.\nAnalogous statements hold for each of the other settings in Theorem \\ref{CCMO_main}.\n\\end{corollary}\n\n\\begin{proof}\nThe first statement follows from comparing Equation~(1) of Theorem \\ref{CCMO_main} and Proposition \\ref{prop:CCMO_rep}.\nThe variations follow from comparing the other equations of Theorem \\ref{CCMO_main} and the corresponding counterparts of Proposition \\ref{prop:CCMO_rep}, since circuit and cocircuit reversals preserve the acyclic and totally cyclic parts of reorientations.\n\\end{proof}\n\n\\medskip\n\nNow we prove the main theorem of this note, which includes the original results of \\cite{gioan2008circuit} (the enumerations when $M$ is regular, with a new proof) and their converses.\n\n\\begin{theorem} \\label{thm}\nLet $M$ be an oriented matroid. Consider the following six statements:\n\\begin{enumerate}\n\\item $M$ is regular,\n\\item $t(M;1,1)=\\#$ circuit-cocircuit reversal classes of $M$,\n\\item $t(M;1,2)=\\#$ cocircuit reversal classes of $M$,\n\\item $t(M;2,1)=\\#$ circuit reversal classes of $M$,\n\\item $t(M;1,0)=\\#$ acyclic (circuit-)cocircuit reversal classes of $M$,\n\\item $t(M;0,1)=\\#$ totally cyclic circuit(-cocircuit) reversal classes of $M$.\n\\end{enumerate}\n\nThen we have the following implications: (1) implies all other statements; (2), (3), (4) each implies (1); (5) implies (1) if $M$ has no loops; (6) implies (1) if $M$ has no coloops.\nIn particular, if $M$ has no loops nor coloops, then all statements are equivalent. Moreover, if any of the equalities fail, then the left hand side is larger.\n\\end{theorem}\n\n\\begin{proof} Let us separate implications.\n\n\\smallskip\n\n$\\bullet$ $(1)\\Rightarrow (2)$. We give an alternative proof to that of \\cite{gioan2008circuit}.\nBy Corollary \\ref{coro:CCMO_red}, it suffices to show that every circuit-cocircuit reversal class contains a unique circuit-cocircuit minimal reorientation.\nWe claim that any two reorientations within a reversal class differ by a disjoint union of positive circuits and cocircuits, which will imply that at most one of them can be minimal, thus proving the implication.\n\nBy induction and restricting to the totally circuit part (the acyclic part follows from duality), it suffices to show that if $C$ is a positive circuit of $M$ and $D$ is a positive circuit of $-_{C}M$, then $C\\triangle D$ is a disjoint union of positive circuits of $M$.\nBy \\cite[Corollary 7.9.4]{bjorner1999oriented}, we may assume that some totally unimodular matrix $Q$ realizes $M$.\nBy total unimodularity, if $C$ is a positive circuit of $M$, then the characteristic vector $\\rchi_C\\in\\{0,1\\}^E$ of $C$ is in the kernel $\\ker(Q)$ of $Q$; similarly,  since $D$ is a positive circuit of $-_CM$, $\\rchi_{D\\setminus C}-\\rchi_{C\\cap D}\\in\\ker(Q)$.\nSince their sum $\\rchi_{C\\triangle D}$ is in $\\ker(Q)$, $C\\triangle D$ is a positive vector and contains some positive circuit $C_1$ of $M$, thus $\\rchi_{(C\\triangle D)\\setminus C_1}=\\rchi_{C\\triangle D}-\\rchi_{C_1}\\in\\ker(Q)$.\nProceeding by induction, we can write $C\\triangle D$ as a disjoint union of positive circuits.\n\n\\smallskip\n\n$\\bullet$ $(1)\\Rightarrow (3),(4),(5),(6)$. Alternatively to \\cite{gioan2008circuit}, the proofs are similar to the above one by restricting to circuit reversals only, then restricting to totally cyclic reorientations only, and then taking duals.\n\n\\smallskip\n\n$\\bullet$ $(5)\\Rightarrow (1)$ assuming $M$ has no loops. \nSuppose $M$ is not regular. Then it has a minor $M\/A\\setminus B$ that is isomorphic to $U_{2,4}$.\nBy Corollary \\ref{coro:CCMO_red}, it suffices to show that there are two cocircuit minimal acyclic orientations in the same (circuit-)cocircuit reversal acyclic class.\nUp to enlarging $A$, we can assume that $M\/A$ is loopless and that the rank of $M\/A$ equals $2$. \nSince $M$ is loopless, up to reorientation, we can assume that $M$ and $M\/A$ are acyclic.\nLet $C$ and $D$ be the two positive cocircuits of $M\/A$ (which are also cocircuits of $M$).\nThus, $-_CM$ and $-_DM$ are in the same acyclic reversal class.\nDenote $S=C\\triangle D$.\nSince the rank of $M\/A$ is $2$, any cocircuit of $M\/A$ is the union of all parallel classes but one.\nSince $S$ is the union of  two parallel classes ($C\\setminus D$ and $D\\setminus C$) of $M\/A$, no cocircuit of $M$ is contained in $S$ (otherwise, the complement of $S$ is a parallel class in $M\/A$, and reducing parallel classes of $M\/A$ yields $U_{2,3}$, a contradiction).\nChoose a linear ordering of $E$ such that elements of $S$ are greater than elements of $E\\setminus S$, then $S$ does not contain the minimum element of a cocircuit (otherwise it would contain a cocircuit), thus $-_CM$ and $-_DM$ are both cocircuit minimal.\n\n\\smallskip\n\n$\\bullet$  $(2)\\Rightarrow (1)$. Suppose $M$ is not regular.\nThen $M'$, the oriented matroid obtained from removing all loops of $M$, is also not regular.\nBy the implication $(5)\\Rightarrow (1)$ for $M'$, there exist distinct acyclic reorientations $-_{A}M'$ and $-_{B}M'$ that are (circuit-)cocircuit reversal equivalent and both (circuit-)cocircuit minimal.\nNow $-_{A}M$ and $-_{B}M$ are circuit-cocircuit reversal equivalent reorientations of $M$ that are both circuit-cocircuit minimal.\nThe implication follows from Corollary \\ref{coro:CCMO_red}.\n\n\\smallskip\n\n $\\bullet$ $(3)\\Rightarrow (1)$. The proof is the same as for $(2)\\Rightarrow (1)$ except that, at the end, $-_{A}M$ and $-_{B}M$ are cocircuit reversal equivalent reorientations of $M$ that are both cocircuit minimal.\nThe implication again follows from Corollary \\ref{coro:CCMO_red}.\n\n\\smallskip\n\n $\\bullet$ $(4)\\Rightarrow (1)$, and $(6)\\Rightarrow (1)$ assuming $M$ has no coloops.  The two implications are the dual statements of $(3)\\Rightarrow (1)$ and $(5)\\Rightarrow (1)$, respectively.\n\n\\smallskip\n\nFinally, let us mention that the relations between the implication $(5)\\Rightarrow (1)$ and the other ones could also be handled from the decomposition into acyclic and totally cyclic parts, along with the convolution formula for the Tutte polynomial, similarly as in \\cite{gioan2008circuit}.\n\\end{proof}\n\nWe end with an open question.\nBy direct computation, the number of circuit-cocircuit reversal classes and the number of bases differ by a rather large margin for small non-regular oriented matroids.\nSo, does there exist an absolute constant $K>1$ such that the number of bases of a non-regular oriented matroid is at least $K$ times the number of circuit-cocircuit reversal classes?\n\n\\section*{Acknowledgements}\n\nThe second author, Chi Ho Yuen, would like to thank Matthew Baker for suggesting the problem, and Spencer Backman for introducing him the work of the first author, Emeric Gioan.\nThe role of the first author of this paper has been to simplify and extend\na preprint written at the initiative of the second author.\n\n\\vspace{-2mm}\n\n\\bibliographystyle{plain}\n\n\\input{Oriented-matroid-reversal-classes.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n  For a Josephson junction created by a superconductor-insulator-superconductor interface, the Josephson relation $I_s = I_c \\sin \\phi$ relates the supercurrent $I_s$ to the phase difference $\\phi$ of the order parameter  across the junction \\cite{Josephson}.  $I_c$ is the critical current of the junction, which is the maximal supercurrent across the junction.\nThis connection is the defining functionality of quantum mechanical devices, such as superconducting quantum interference devices (SQUIDs).\n\n\n\n\n The on-going study of Josephson junctions (JJs) was broadened  in scope with the design of Josephson junctions in ultracold atom systems. \nThis led to atomic JJs \\cite{Inguscio, Oberthaler, Steinhauer, Thywissen, Betz, Inguscio2, Inguscio3, Schmiedmayer2018}, supercurrent dynamics in ring condensates \\cite{Campbell2011, Campbell2014, Campbell2014N, Amy2014}, dc-SQUIDs  \\cite{Ryu2013, CampbellSQ}, and quantum transport \\cite{Ventra2015, Esslinger2017}.\nJosephson tunneling between two condensates was studied in Ref.  \\cite{Smerzi1997} and their theoretical investigation reported in Refs. \\cite{Fantoni2000, Strinati2007, Dalfovo2007, Salasnich2009, Leggett1998, Ippei2005, Band2011, Leggett2001, Modugno2017, Zwerger1, Zaccanti, Marc}.   \nCurrent phase relations of atomic JJs were measured in Refs. \\cite{Niclas, Roati2019}. \nThe decay of a supercurrent due to phase slip dynamics was discussed in Refs. \\cite{Roati2018, RoatiAIP, Proukakis}. \nTemperature dependence of the phase coherence was measured in Ref. \\cite{Oberthaler2006}.\n\n\n\nJosephson junctions are utilized in phenomenological models of high-temperature superconductors, which describe these materials as stacks of two-dimensional (2D) systems coupled by JJs.  \nJosephson junction arrays  also serve as a  model to describe transport phenomena in optically driven high-temperature superconductors \\cite{Junichi}.\n2D systems such as thin film superconductors \\cite{ThinSC} or Josephson junction arrays \\cite{Cuccoli}\nundergo a Berezinskii-Kosterlitz-Thouless (BKT) transition within the XY universality class \\cite{Berezinskii, Kosterlitz, Kosterlitz2}. \nThe superfluid phase has quasi-long-range order and is characterized by a scale-invariant exponent $\\tau$ of the single-particle correlation function $g_1$ that decays algebraically at large distances, $g_1(r) \\sim |r|^{-\\tau\/4}$. \nAt the transition the exponent assumes the critical value $\\tau_c=1$ which is accompanied by a universal jump of the superfluid density.   \n\n\n   Recently, Ref. \\cite{Niclas} reported on a study on Josephson junction dynamics of an ultracold 2D gas of $^{6}$Li atoms, which is realized by separating two uniform 2D clouds with a tunneling barrier. \nUsing a strong barrier higher than the mean-field energy, the experiments measure the current-phase relation of an ideal junction, and the critical currents in the crossover from tightly bound molecules to weakly bound Cooper pairs. \n\n\n\n\n\n \n  \n  \n  In this paper,  we establish a connection between the quasi-order scaling of 2D superfluids and the critical current of the Josephson junction.  \nSpecifically, we demonstrate the BKT scaling of the critical current of a Josephson junction coupling two 2D Bose gases using classical field simulations. \nThe Josephson junction is created by a tunneling barrier between two 2D clouds of $^{6}\\mathrm{Li}_2$ molecules, motivated by the experiments of Ref. \\cite{Niclas}.\nWe find an interplay of the bulk and junction dynamics that is influenced by the barrier height and the temperature. \nDepending on these parameters, we find multimode (MM) and second-harmonic (SH) contributions to the current-phase relation.  \nFor large barrier heights we find that the junction dynamics displays ideal Josephson junction (IJJ) behavior, i.e. it  obeys the nonlinear current-phase relation $I(\\phi)=I_c \\sin(\\phi)$.\nWe map out the MM, the SH, the IJJ, and an overdamped regime as a function of barrier height and temperature.\nWe determine the critical current numerically based on the current and phase dynamics at the barrier. \nThe numerically obtained critical current, and an analytical estimate that we derive, show excellent agreement with the experimental values of Ref. \\cite{Niclas}. \nWe confirm the BKT scaling of the critical current by performing simulations for varying system sizes. \nThe exponent of the critical current across the transition demonstrates agreement with the exponent of the corresponding equilibrium system, if the system is in the IJJ regime. \nFinally, we address the damping of the current and identify the damping mechanism which is due to phonon excitations in the bulk, and the nucleation of vortex-antivortex pairs in the junction. \n\n\n\n\n\n\nThis paper is organized as follows.\nIn Sec. \\ref{sec:method} we describe our simulation method. \nIn Sec. \\ref{sec:bjj} we show the condensate dynamics and its dependence on the barrier height and the temperature. \nIn Sec. \\ref{sec:jc} we determine the critical current for an ideal junction and compare it to the simulation and the experiment. \nIn Sec. \\ref{sec:exp} we show the power-law scaling of the critical current  for varying system sizes. \nIn Sec. \\ref{sec:damping} we discuss the dissipation mechanism of the current, and in Sec. \\ref{sec:con} we conclude.\n\n\n\n\n\n\n\n\n\n \n\n\\begin{figure*}\n\\includegraphics[width=1.00\\linewidth]{density}\n\\caption{\\textbf{Dynamical regimes.} (a) Time evolution of the density $\\delta n(x, t) = n(x,t) - n(x)$, which is averaged over the $y$ direction and the ensemble, for $\\tilde{V}_0=0.0$, $0.8$, and $2.0$. $n(x)$ is the equilibrium density. \nPanel (b) shows the corresponding phase evolution $\\delta \\phi(x, t) = \\phi(x,t) -  \\phi_\\mathrm{m} (t)$ of a single trajectory of the ensemble.  \n$ \\phi_\\mathrm{m}$ is the mean phase.\nWe imprint a phase of $\\phi_0=\\pi\/4$ on the left reservoir at $t=0$ for $n= 2.25\\, \\mu \\mathrm{m}^{-2}$ and $T\/T_0=0.3$. \nThe barrier width $w$ is denoted by the two vertical dotted lines. \nPanels (c, d) show the current $I_0$ as a function of $\\phi_0$ for various $\\tilde{V}_0$ at $T\/T_0=0.1$ and $0.3$, respectively.  \nThe continuous lines are fits with the fitting function $I(\\phi_0) = I_1 \\sin(\\phi_0) + I_2 \\sin(2\\phi_0)$. \n(e) The dynamical regimes are multimode (MM), second-harmonic (SH), ideal Josephson junction (IJJ), and overdamped (OD), see text.   }\n\\label{Fig:dynamics}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Simulation method}\\label{sec:method}\n\n Motivated by the experiments \\cite{Niclas} we study 2D clouds of  $^{6}\\mathrm{Li}_2$ molecules confined in a box of dimensions $L_x \\times L_y$. \nWe simulate the dynamics using the c-field method of Ref. \\cite{Singh2017}. \nThe system is described by the Hamiltonian\n\\begin{equation} \\label{eq_hamil}\n\\hat{H}_{0} = \\int \\mathrm{d}{\\bf r} \\Big[  \\frac{\\hbar^2}{2m} \\nabla \\hat{\\psi}^\\dagger({\\bf r}) \\cdot \\nabla \\hat{\\psi}({\\bf r})  + \\frac{g}{2} \\hat{\\psi}^\\dagger({\\bf r})\\hat{\\psi}^\\dagger({\\bf r})\\hat{\\psi}({\\bf r})\\hat{\\psi}({\\bf r})\\Big].\n\\end{equation}\n$\\hat{\\psi}$ ($\\hat{\\psi}^\\dagger$) is the bosonic annihilation (creation) operator. \nThe interaction $g$ is given by $g=\\tilde{g} \\hbar^2\/m$, where $\\tilde{g}$ is the dimensionless interaction, and $m$ the molecular mass.\n$\\tilde{g}$ is determined by $\\tilde{g}=\\tilde{g}_0\/\\bigl(1- \\frac{\\tilde{g}_0}{2\\pi} \\ln(2.09 k_\\mathrm{F} \\ell_z) \\bigr)$, with  $\\tilde{g}_0= \\sqrt{8 \\pi} a_s\/\\ell_z$ \\cite{Turlapov2017}. $a_s$ is the molecular s-wave scattering length, $\\ell_z= \\sqrt{\\hbar\/(m \\omega_z)}$ the harmonic oscillator length in the transverse direction, and $k_\\mathrm{F}$ the Fermi wavevector.\nWe discretize space on a lattice of size $N_x \\times N_y$ and a discretization length $l=0.5\\, \\mu \\mathrm{m}$. \nWithin the c-field representation we replace the operators $\\hat{\\psi}$ in Eq. \\ref{eq_hamil} and the equations of motion by complex numbers $\\psi$. We sample the initial states in a grand canonical ensemble having chemical potential $\\mu$ and temperature $T$ via a classical Metropolis algorithm. \nWe choose the system parameters, such as the density $n$, $\\tilde{g}$, and $T$ to be close to the experiments.  \nIn particular, we choose $n \\approx 1.24\\, \\mu \\mathrm{m}^{-2}$, $T\/T_0=0.3$, and $L_x \\times L_y = 20 \\times 40\\, \\mu \\mathrm{m}^2$, which are the same as in Ref. \\cite{Niclas}.  The critical temperature $T_0$ is estimated by $T_0 = 2\\pi n \\hbar^2\/(m k_\\mathrm{B} {\\cal{D}}_c)$, where ${\\cal{D}}_c= \\ln(380\/\\tilde{g})$ is the critical phase-space density \\cite{Prokofev2001}. \nWe vary $\\tilde{g}$ in the range $1-4$ to cover the BEC regime of the experiments. \nFor additional simulations, we  use $n \\approx 2.25\\, \\mu \\mathrm{m}^{-2}$ and $\\tilde{g}=1.8$, while we vary $T\/T_0$ and the box size.\n\n \n\n\n\n   To create the Josephson junction we add a barrier term $\\mathcal{H}_{\\mathrm{b}}(t) = \\int \\mathrm{d}{\\bf r}\\, V({\\bf r},t) n({\\bf r})$, where $n({\\bf r})$ is the density at the location  ${\\bf r}=(x,y)$. The barrier potential $V({\\bf r},t)$ is given by \n\\begin{equation}\\label{eq_pot} \nV({\\bf r},t)  = V_0 (t) \\exp \\bigl(- 2 (x-x_0)^2\/w^2 \\bigr),\n\\end{equation}\nwhere $V_0(t)$ is the time-dependent strength and $w$ the width. The potential is centered at $x_0= L_x\/2$. \nWe choose $w$ in the range $(0.85 - 2)\\, \\mu \\mathrm{m}$ and $V_0$ in the range $ V_0\/\\mu \\equiv \\tilde{V}_0 = 0-6$, \nwhere $\\mu = gn$ is the mean-field energy. \nWe ramp up $V_0$ linearly over $150\\, \\mathrm{ms}$ and then wait for $50\\, \\mathrm{ms}$. \nThis splits the system in $x$-direction into two uniform 2D clouds, which we refer to as the left and right reservoir. \nTo create a phase difference across the junction, we imprint a phase  $\\phi_0$ on the left reservoir, \nresulting in the phase difference $\\phi_0= \\phi_L - \\phi_R$, where $\\phi_L$ ($\\phi_R$) is the mean phase of the left (right) reservoir.\nThe sudden imprint of the phase and the barrier lead to the dynamics displayed in Fig. \\ref{Fig:dynamics}.   \nWe calculate the $x$ component of the  current density defined as: \n\\begin{align}\nj(x)= \\frac{\\hbar}{2imN_y} \\sum_{y} ( \\psi^\\ast_{{\\bf r}} \\psi_{{\\bf r} + l \\hat{e}_x }  - \\psi_{{\\bf r}} \\psi^\\ast_{{\\bf r} + l \\hat{e}_x } ).\n\\end{align}\nWe calculate $j_1$ with ${\\bf r}= (x_0  - l \\hat{e}_x, y )$,  and $j_2$ with ${\\bf r}= (x_0 ,  y )$.\n$j= \\langle j_1 + j_2 \\rangle\/2$ gives the averaged current density at the barrier center. \nThis current fulfills the continuity equation of the density imbalance between the two reservoirs. \nThe time evolution of the current is determined by $I(t) = j(t) L_y$, see Fig. \\ref{Fig:damping}(a).  \nWe fit $I(t)$ with the function $f(t)=  I_0 e^{-\\Gamma t} \\sin(\\omega t + \\theta)$ to determine\nthe magnitude $I_0 \\equiv |I_0| $, the damping rate $\\Gamma$, the frequency $\\omega$, and the phase shift $\\theta$.\n\n  \n \n\n\n\n  \n  \n  \n  \n  \n  \n\n\\section{Bulk versus junction dynamics}\\label{sec:bjj}\n\n    To characterize the junction dynamics we analyze the time evolution of the density and the phase of the system. \nAs an illustration, we choose $n= 2.25\\, \\mu \\mathrm{m}^{-2}$, $T\/T_0=0.3$, and $\\tilde{g}= 1.8$. \nWe imprint a phase of $\\pi\/4$ on the left reservoir at $t=0$, for $\\tilde{V}_0=0$, $0.8$, and $2$. \nWe use $w=0.85\\, \\mu \\mathrm{m}$, which in terms of the healing length $\\xi$ is $w\/\\xi=2.4$, where $\\xi =\\hbar\/\\sqrt{2\\mu m}=0.35\\, \\mu \\mathrm{m}$.\nIn Fig. \\ref{Fig:dynamics}(a) we show the time evolution of the density $\\delta n (x, t)= n(x,t)- n(x)$, which is averaged over the y direction and the thermal ensemble. $n(x)$ is the averaged equilibrium density profile. \nFor no barrier, $\\tilde{V}_0=0$, the phase imprint creates two density pulses that are visible as density increase and decrease in the left and right reservoir, respectively. \nThe density pulses propagate in opposite direction with the sound velocity and are reflected by the box edges.\nFor $\\tilde{V}_0=0.8$, the barrier confines the density wave partially in each of the reservoirs.\nFor $\\tilde{V}_0=2$, the density waves are well confined within the reservoirs as flow between the reservoirs is obstructed by the barrier. \nInstead, the density waves tunnel across the barrier, resulting in coherent Josephson oscillations between the reservoirs. \n\n   \n   \n  \n    In Fig. \\ref{Fig:dynamics}(b) we show the time evolution of the phase $\\delta \\phi (x,t) = \\phi(x,t) -  \\phi_\\mathrm{m} (t)$ of a single trajectory, for the same $\\tilde{V}_0$ as in Fig. \\ref{Fig:dynamics}(a). \n$ \\phi_\\mathrm{m}$ is the mean global phase.   \nThe sudden imprint of phase adds the mean phase difference $\\phi_0 = \\phi_L - \\phi_R $ between the left and right reservoir at $t=0$. \nDuring the time evolution a phase gradient develops within the reservoirs, which results in $\\phi_{L\/R}$ being different from the phases close to the junction. \nFor $\\tilde{V}_0=0$, the phase evolves linearly with the distance. \nAs the barrier height is increased, the phase gradient within the reservoirs decreases.\nFor $\\tilde{V}_0=2$, the phase gradient within the reservoir almost vanishes, resulting in \n$\\phi_{L\/R}$ being the same as the phases in direct vicinity of the junction.   \nThis corresponds to IJJ dynamics. \n \n\n\n\n  We now examine the current-phase relation (CPR) of the junction. \nWe determine the current $I_0$ by fitting the time evolution of the current $I(t)$ to a damped sinusoidal function as described in Sec. \\ref{sec:method}.\nTo obtain the CPR we calculate $I_0$ as a function of $\\phi_0$. \nIn Fig. \\ref{Fig:dynamics}(c) we show  $I_0(\\phi_0)$ for various values of $\\tilde{V}_0$ at  $T\/T_0 =0.1$,  and in Fig. \\ref{Fig:dynamics}(d)  at $T\/T_0=0.3$. \nWe analyze these CPR curves by fitting them with a multi-harmonic fitting function  $I(\\phi_0) = \\sum_{n=1}^{n_\\mathrm{max}} I_n \\sin(n \\phi_0) $, where we choose $n_\\mathrm{max} = 5$.\nWe find that the result of these fits can be grouped into three regimes, as a function of the barrier height $\\tilde{V}_0$ and the temperature. \nIf the coefficient $I_1$ is dominant, the CPR reduces to the form of an IJJ, $I(\\phi_0) =  I_1 \\sin( \\phi_0)  $.\nIf both $I_1$ and $I_2$ are non-negligible, we refer to the regime as second harmonic (SH) regime. \nIn Fig. \\ref{Fig:dynamics}(e) we indicate the crossover from IJJ to SH by depicting the ratio $I_2\/I_1$. \nFor lower temperatures and barriers, higher harmonical contributions become important. \nWe indicate the multimode (MM) regime if $\\sum_{n>2} (I_n\/I_1)^2  > 0.02$, see also Appendix \\ref{ap:sec:cpr}.\nWe note that CPR deviations were pointed out  for $\\tilde{V}_0 \\lesssim 1$ by Refs. \\cite{Strinati2007, Piazza2010, Watanabe2009} .\nFurthermore, we indicate the overdamped (OD) regime based on the analysis shown in Sec.  \\ref{sec:damping}.\n\n\n\n\n  The results depicted in  Fig. \\ref{Fig:dynamics}(e) demonstrate that the IJJ regime is strongly sensitive to the temperature and the barrier height. \n This derives from the properties of the dynamical evolution shown in Figs.  \\ref{Fig:dynamics}(a) and (b).\n The initial phase imprint creates phonon pulses in the two reservoirs. \n For low temperatures and small barrier heights these pulses are weakly damped, which leads to the multimode regime. \n For increasing temperature and barrier height, fewer and fewer of the phonon modes of the system contribute. \n The increasing barrier height leads to a long tunneling time, which exceeds the damping time of more and more phonon modes, until the phase dynamics reduces to the dynamics of two global phases for each reservoir, as visible in Figs. \\ref{Fig:dynamics}(a) and (b). \n However, if the barrier is increased further, eventually the dynamics become overdamped. Here, the two reservoirs dephase on the timescale of the tunneling rate.   \nWe note that  the temperature dependence of the current-phase relation was measured in Nb\/InAs\/Nb junctions by Ref. \\cite{Ebel2002} and in a weak link of $^4$He superfluids by Ref. \\cite{Packard2006}. \n\n\n\n\n\n\n\n\n\n\\section{Critical current}\\label{sec:jc}\n\n\\begin{figure}\n\\includegraphics[width=1.00\\linewidth]{Ic}\n\\caption{\\textbf{Critical current.} (a) Time evolution of the junction current $I(t)$ versus phase $\\phi_j(t)$ for $\\tilde{V}_0=1$, $2$, and $3$, depicted in the $I - \\phi_j$ plane. \nThe critical current $I_c$ is determined by the linear slope (dashed lines) of $I\/\\sin \\phi_j$. \nPanels (b-d) show $I_c (\\tilde{V}_0)$ for $T\/T_0= 0.1$, $0.2$, and $0.3$, respectively, for $w\/\\xi=2.9$ (squares) and $4.3$ (circles).\nThe continuous lines are the estimates of Eq. \\ref{eq:jc}.  \n}\n\\label{Fig:Ic}\n\\end{figure}\n\n\n\n\n  To determine the Josephson critical current, we calculate the junction phase  $\\phi_j  \\equiv  \\langle  \\phi_j  \\rangle = \n \\langle \\phi_{2, \\mathrm{m}} - \\phi_{1, \\mathrm{m}}  \\rangle$, where  $ \\phi_{1\/2, \\mathrm{m}} = \\phi_{x_0 \\mp 2 l \\hat{e}_x } $ are the mean phases calculated by taking an average of the fields over the $y$ direction. \nWe use the same density $n$ as above, and $T\/T_0=0.2$. \nWe calculate the time evolution of the current $I(t)$ and the phase $\\phi_j(t)$.\nIn Fig. \\ref{Fig:Ic}(a) we show the time evolution of $I(\\phi_j)$ for $\\tilde{V}_0=1$, $2$, and $3$.\nIn this small $\\phi_j$ regime a linear behavior is observed. \nThe width of the distribution increases with increasing $\\tilde{V}_0$ due to increased phase fluctuations across the barrier.\nWe determine the critical current $I_c$ by the slope $I\/\\sin \\phi_j$.  \nWithin the IJJ regime, this coincides with $I_1$.\n$I_c$ decreases with increasing $\\tilde{V}_0$, with $I_c = 83.8$, $15$, and $2.9 \\, \\mathrm{ms}^{-1}$ for $\\tilde{V}_0=1$, $2$, and $3$, respectively.       \nIn Figs. \\ref{Fig:Ic}(b)-(d) we show $I_c(\\tilde{V}_0)$ for $T\/T_0 = 0.1$, $0.2$, and $0.3$, and the barrier widths $w\/\\xi=2.9$ and $4.3$. \nAs expected, $I_c$ is smaller for wider barriers. \n$I_c$ also decreases with the temperature as we show below.\n\n\n\n\n   We derive an analytical estimate of the critical current by solving the mean-field equation and by considering single-particle tunneling across a rectangular barrier of height $V > \\mu$ and width $d$, see Appendix \\ref{ap:sec:Ic}. \nWe obtain the current density \n\\begin{align}\\label{eq:jc:phij}\nj = j_c \\sin \\phi_j,\n\\end{align}\nwith \n\\begin{align}\\label{eq:jc}\nj_c = 2 n_0 \\frac{\\mu}{V + \\sqrt{V^2 - \\mu^2\/2}}  \\frac{ \\hbar \\kappa }{m} \\exp(-\\kappa d)\n\\end{align}\nand\n\\begin{align}\\label{eq:ka}\n\\kappa^2 = \\frac{m}{\\hbar^2} \\biggl( 3V - 2 \\mu - \\sqrt{V^2 - \\mu^2\/2} \\biggr). \n\\end{align}\n$n_0$ is the condensate density. $\\kappa$ is the damping parameter of the exponential wavefunction inside the barrier, \nwhich is determined variationally, and includes the mean-field repulsion under the barrier. \nWe interpret this result for $j_c$ as the product of the density $n_0 \\mu\/\\bigl( V+ \\sqrt{V^2 - \\mu^2\/2} \\bigr)$ at the barrier boundary and the velocity $\\hbar \\kappa\/m$ at the barrier center.  \nAlternatively, it is instructive to rewrite Eq. \\ref{eq:jc} in terms of the bulk current density $c n_0$, and the tunneling amplitude $t_0(\\tilde{V}, d)$ across the barrier as\n\\begin{align}\\label{eq:jc2}\nj_c = c n_0 t_0(\\tilde{V}, d),\n\\end{align}\nwith\n\\begin{align}\\label{eq:t0}\nt_0(\\tilde{V} ,d) &=  2\\sqrt{2} \\frac{\\sqrt{ 6\\tilde{V} -4 - \\sqrt{4 \\tilde{V}^2 -2} } }{2 \\tilde{V} + \\sqrt{4\\tilde{V}^2-2}}  \\nonumber \\\\\n&\\quad \\times \\exp \\Bigl(-\\frac{d}{2\\xi} \\sqrt{ 6\\tilde{V} -4 - \\sqrt{4 \\tilde{V}^2 -2} }   \\Bigr).\n\\end{align}\n$c= \\sqrt{\\mu\/m}$ is the sound velocity, $\\xi$ the healing length, and $\\tilde{V} = V\/\\mu$ the scaled barrier height. \nWe note that $j_c$ is described in terms of the bulk current and the tunneling amplitude for a 3D condensate in Refs.  \\cite{Zwerger1, Zaccanti}.\nWe determine the estimate $I_c= j_c Ly$ by using  $V=V_0$ and $d \\approx  1.2 w$,  and by determining $c(T)$ and $n_0$ numerically. \n$V_0$ and $w$ are of the Gaussian barrier used in the simulation.\nThis value for $d$ is set by fitting the simulated critical currents in the IJJ regime, which is close to our assumption $d \\approx w$ used in Ref. \\cite{Niclas}.\nIn Figs. \\ref{Fig:Ic}(b)-(d) we show the estimates  as a function of $\\tilde{V}_0$ for $T\/T_0=0.1$, $0.2$, and $0.3$.\nThe estimates agree with simulated critical currents at all $\\tilde{V}_0 > 1$, for all $T\/T_0$. \nThe agreement is particularly good for the IJJ regime. \nThis suggests that the barrier reduction of $I_c$ is due to the tunneling amplitude $t_0(\\tilde{V}_0, w)$ that decreases with increasing $\\tilde{V}_0$ and $w$. We note that the barrier width reduction follows an exponential behavior given by Eq. \\ref{eq:jc}.\n\n\n\n\n\n\n\n\\begin{figure}[]\n\\includegraphics[width=1.00\\linewidth]{Ic_Temp}\n\\caption{\\textbf{Temperature dependence.} \n(a) $I_c (\\tilde{V}_0)$ for $w\/\\xi=2.9$ and various $T\/T_0$.  \n(b) Mean bulk current $I_\\mathrm{B}$ (circles) determined from the normalized critical currents shown in the inset, see text.\nThe error bar denotes the standard deviation. \nThe bulk values of the current determined using the condensate density and phonon velocity are shown by the diamonds. \n}\n\\label{Fig:ic_temp}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=1.00\\linewidth]{comparison}\n\\caption{\\textbf{Comparison to the experiments.} The measurements of the critical current (crosses) are compared to the simulations (squares) and the analytical estimate (continuous line) for various $\\ln(k_\\mathrm{F} a_\\mathrm{2D}  )$ in the BEC regime. \n$k_\\mathrm{F}$ is the Fermi wavevector and $a_\\mathrm{2D} $ is the 2D scattering length.\nWe use $n \\approx 1.24\\, \\mu \\mathrm{m}^{-2}$, $T\/T_0 \\approx 0.3$, $w=0.81\\, \\mu \\mathrm{m}$, and $\\tilde{V}_0=1.4$, which are the same as the experiments.  \nThe shaded areas include a $15\\%$ uncertainty of $\\tilde{V}_0$ as in experiment.  \nThe measurement data is from Ref. \\cite{Niclas}.\n}\n\\label{Fig:comp}\n\\end{figure}\n\n\n\n \n \n   In Fig. \\ref{Fig:ic_temp}(a) we show $I_c$ as a function of $\\tilde{V}_0$ for $w\/\\xi =2.9$ and various $T\/T_0$.\nAs $T\/T_0$ increases, $I_c(\\tilde{V}_0)$ decreases with a rather sudden jump to very low values for $T\/T_0 > 0.6$. \n$I_c$ and the bulk current are connected according to Eq. \\ref{eq:jc2} via $ I_\\mathrm{B} = I_c(\\tilde{V}_0) \/ t_0(\\tilde{V}_0, w)$. \nWe thus divide $I_c(\\tilde{V}_0)$ by $t_0(\\tilde{V}_0, w)$, see inset of Fig. \\ref{Fig:ic_temp}(b). \nThe results are almost independent of $\\tilde{V}_0$ as expected. \nBy taking an average over the range $\\tilde{V}_0 =1-2$, we obtain the mean value of $I_\\mathrm{B}$ which is shown in Fig. \\ref{Fig:ic_temp}(b). \nThe mean $I_\\mathrm{B}$ decreases with increasing $T\/T_0$ and becomes small due to increased thermal fluctuations for $T\/T_0 \\geq 0.6$.\nWe compare this result to the actual bulk value $I_\\mathrm{B} = c(T) n_0 L_y$ by determining $n_0$ and $c(T)$ numerically for various $T\/T_0$. \nThis bulk result agrees at intermediate temperatures, where the system is near the IJJ regime. \nAt low temperatures, the system is in the MM regime, where the above estimate is not valid. \nThe bulk current is linked to the BKT scaling exponent that we determine in Sec. \\ref{sec:exp}.    \n\n\n\n   \n    Finally, we compare the simulations to the measurements that are performed at several interaction strengths in the BEC regime \\cite{Niclas}.  \nWe use the range $\\tilde{g}=1 - 4$, and determine the critical current $I_c$ as described above.  \nIn Fig. \\ref{Fig:comp} we show the simulated $I_c$ as a function of the interaction parameter $\\ln(k_\\mathrm{F} a_\\mathrm{2D} )$. \n$k_\\mathrm{F}$ is the Fermi wavevector and $a_\\mathrm{2D}  =2.96 \\ell_z \\exp \\bigl( -\\ell_z\\sqrt{\\pi}\/a_{\\mathrm{3D}} \\bigr)$ is the 2D scattering length, where $a_\\mathrm{3D} = a_s$ is the 3D scattering length.  \nThe simulation results are consistent with the experimental results within the error bars of the measurement.\nFor $\\ln(k_\\mathrm{F} a_\\mathrm{2D} )= -1$, the simulation results start to deviate from the experimental result, because the system enters the strongly interacting regime, where the c-field approach starts to deviate systematically.   \nThe shaded areas reflect the error bars of theory when assuming a $15\\%$ uncertainty of $\\tilde{V}_0$ as in experiment.  \nWe calculate the estimate of Eq. \\ref{eq:jc} by determining the condensate density numerically for all interactions. \nThe condensate fraction $n_0\/n$ has weak interaction dependence and is about $62\\%$. \nWe show the analytical estimates in Fig. \\ref{Fig:comp}.\nThe estimates, including the $15\\%$ uncertainty of $\\tilde{V}_0$, are consistent with both simulation results and the experimental results.\n\n\n\n\n\n\n\\section{Berezinskii-Kosterlitz-Thouless scaling}\\label{sec:exp}\n\\begin{figure}\n\\includegraphics[width=1.00\\linewidth]{tau}\n\\caption{\\textbf{Determining the scaling exponent.} \n(a) Condensate fraction $n_0\/n$ of the equilibrium system as a function of system length $L$ on a log-log scale for various $T\/T_0$.\n(b) Averaged $n_0\/n$ determined from the critical current density $j_c$ via Eq. \\ref{eq:jc2}.  \nThe continuous lines are the algebraic fits.\nThe exponential fit (dashed line) is a good fit for $T\/T_0=0.9$ in panel (a).  \n(c) Extracted exponents of the equilibrium system (squares) and $j_c$ (circles). \nThe black cross corresponds to the measurements of $j_c$ in Ref. \\cite{Niclas}.\n  }\n\\label{Fig:tau}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=1.00\\linewidth]{damping}\n\\caption{\\textbf{Current damping.} \n(a) Oscillations of the current $I(t)$ for $T\/T_0=0.1$, $0.3$, and $0.5$. We use $\\tilde{V}_0 = 2$ and $w=1 \\, \\mu \\mathrm{m}$.\n(b) Normalized oscillation frequency $\\omega\/\\omega_0$ and (c) scaled damping rate $\\Gamma\/\\omega$, as a function of $\\tilde{V}_0$, for the temperatures as in panel (a). \n$\\omega_0$ is the sound frequency. \nThe red shaded area denotes the overdamped regime.  \n }\n\\label{Fig:damping}\n\\end{figure}\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=1.00\\linewidth]{vortex}\n\\caption{\\textbf{Damping mechanism.} \n(a) Phase distribution $\\phi(x,y) = \\phi(x,y) -  \\phi_\\mathrm{m}$ of a single trajectory at $t=3.6\\, \\mathrm{ms}$, after the phase imprint, for $T\/T_0=0.1$, $0.3$, and $0.5$. $ \\phi_\\mathrm{m}$ is the mean global phase.\nThe corresponding bulk condensate fractions are $n_0\/n=0.88$, $0.64$, and $0.40$, respectively.\nThe barrier parameters are the same as in Fig. \\ref{Fig:damping}(a). The barrier width is indicated by the two vertical dotted lines.\nThe circles and the triangles denote vortices and antivortices, respectively. \nThe box dimensions are $20 \\times 40\\,\\mu \\mathrm{m}^2$.   \nFor the time evolution dynamics see Ref. \\cite{note_1}. \n(b) Average vortex number $N_v$ and (c) differential vortex number $N_v - N_{v,0}$, depicted as a function of $\\tilde{V}_0$, \nwhere $N_{v,0}$ is $N_v(\\tilde{V}_0 =0)$. \nThe vertical dashed lines in panel (c) mark the onset of barrier-induced vortices for $T\/T_0=0.1$, $0.3$, and $0.5$ at $\\tilde{V}_c =1.47$, $0.58$, and $0.29$, respectively.   }\n\\label{Fig:vortex}\n\\end{figure*}\n\n\n\n\n\n\n\n  We have shown above that the critical current depends on the condensate density. We use that  dependence to  extract the scaling exponent of the quasi-condensate.\nThe condensate density scales algebraically with the system size as \n\\begin{align}\\label{eq:n0}\nn_0 \\approx n \\Bigl( \\frac{L}{r_0}  \\Bigr)^{-\\tau\/4}, \n\\end{align}\nwhere $\\tau (T)$ is the temperature-dependent exponent, $L$ the system length, and $r_0$ the short-range cutoff of the order of $\\xi$. \nEmploying this scaling, we first determine $\\tau (T)$ for a square-shaped system in equilibrium. \nWe choose $n \\approx 2.25\\, \\mu \\mathrm{m}^{-2}$, $\\tilde{g}=1.8$, and $L$ in the range $(2-128)\\, \\mu \\mathrm{m}$. \nWe calculate $n_0$ as a function of $L$ for various $T\/T_0$ for a system with periodic boundary conditions.\nIn Fig. \\ref{Fig:tau}(a) we show the condensate fraction $n_0\/n$ as a function of $L$ for various $T\/T_0$. \nAt low and intermediate $T\/T_0$, $n_0$ shows a power-law behavior as it decreases linearly with $L$ on a log-log scale. \nAt high $T\/T_0$, $n_0$ deviates from this power-law scaling and instead shows an exponential  behavior \nwhich is characteristic of the thermal phase \\cite{Singh2014}. \nTo confirm the power-law scaling, we fit $n_0\/n$ to Eq. \\ref{eq:n0}, with $\\tau$ and $r_0$ as fitting parameters.\nWe show the fits in Fig. \\ref{Fig:tau}(a). \nThe algebraic fits describe the behavior very well for $T\/T_0 \\leq 0.6$, whereas they fail to capture the dependence for $T\/T_0 > 0.6$. \nFor $T\/T_0=0.9$ the dependence is captured by the exponential fit, while for $T\/T_0=0.7$ the algebraic fit is better than the exponential fit. \n\n\n\n   Next, we determine the scaling exponent of the critical current density.\nWe choose $n$ and $\\tilde{g}$ as above, and $L$ in the range $(40-128)\\, \\mu \\mathrm{m}$.\nFor the barrier, we use $w\/\\xi= 2.9$ and $\\tilde{V}_0$ in the range $\\tilde{V}_0=1.2-2$. \nWe calculate the critical current density $j_c$ as a function of $L$ for various $T\/T_0$, \nwhere $j_c$ is determined by the critical current described in Sec. \\ref{sec:jc}. \nWe obtain the condensate density $n_0$ by normalizing $j_c$ with the sound velocity $c(T)$ and the tunneling amplitude $t_0(\\tilde{V}_0, w)$, see Eq. \\ref{eq:jc2}.\nWe then average $n_0$ over the barrier heights employed. \nWe expand on this normalization in Appendix \\ref{ap:sec:scale}.\nIn Fig. \\ref{Fig:tau}(b) we show the condensate fraction $n_0\/n$ that is determined from the critical current density as a function of $L$ for various $T\/T_0$. \n$n_0\/n$ shows a power-law behavior for $T\/T_0 \\leq 0.6$, \nwhich is confirmed by the algebraic fits shown in Fig. \\ref{Fig:tau}(b). \n   In Fig. \\ref{Fig:tau}(c) we show the temperature dependence of  $\\tau$ for the equilibrium system and $\\tau$ determined from the critical current density.  \n The equilibrium value of  $\\tau$ increases linearly at low and intermediate $T\/T_0$, \nwhile it deviates from linear behavior at high $T\/T_0$ in the crossover regime. \nWe estimate the transition temperature with the BKT critical value $\\tau_c=1$, which gives $T\/T_0 \\approx 0.6$. \nWe note that this value of the critical temperature is renormalized to a lower value in thermodynamic limit \\cite{Giorgini2008}.  \nWe also note that this estimate is below $T_0$ of a weakly interacting 2D Bose gas \\cite{Prokofev2001}.\nAbove the transition, $\\tau$ increases rapidly with the temperature. \nThis temperature dependence of $\\tau$ across the transition is described by the RG equations \\cite{LM2017}.\nStudies of BKT scaling in ultracold gases were reported in Refs. \\cite{Hadzibabic2006, Murthy2015, Igor2016}.\nIn addition to the equilibrium value of $\\tau$ we show the value of $\\tau$ based on the critical current scaling. \nThe results show excellent agreement with the exponents of the equilibrium system for the temperatures $0.2 < T\/T_0 < 0.55$, \nand follow the qualitative behavior outside of this temperature range. \nThe deviations below $T\/T_0=0.2$ are due to multimode dynamics that influence the results of the critical current density, while the deviations for $T\/T_0 \\geq 0.55$ are due to thermal excitations at the barrier, and the onset of overdamped dynamics. \n\n\n     Furthermore, we determine the exponent $\\tau$ from the measurements of the critical currents shown in Fig. \\ref{Fig:comp}. \nMaking use of  Eq.  \\ref{eq:jc2}, these measurements yield the condensate fraction $n_0\/n = 0.72 (8)$.  \nFor the system size $L$ in the experiment and  $r_0= \\xi$, the scaling of Eq. \\ref{eq:n0} results in an exponent of $\\tau= 0.32 (12)$. \nWe show this value of $\\tau$ for $T\/T_0=0.3$ in Fig.  \\ref{Fig:tau}(c), which agrees with the exponents of the simulated critical current density and the equilibrium system. \n  \n\n\n\n\n\n\\section{Current damping and dissipation mechanism}\\label{sec:damping}\n\n\n  Here we analyze the damping of the supercurrent and identify the associated dissipation mechanism.\nAs an illustration, we choose $w\/\\xi=2.9$ and $\\tilde{V}_0=2.0$, and calculate the time evolution of the current $I(t)$ as described above. \nIn Fig. \\ref{Fig:damping}(a) we show $I(t)$ for $T\/T_0=0.1$, $0.3$, and $0.5$.\nThe current oscillations are underdamped at $T\/T_0=0.1$ and $0.3$.\nThe damping increases with increasing $T\/T_0$.\nFor $T\/T_0=0.5$,  the current undergoes an overdamped motion. \nTo quantify this observation we determine the oscillation frequency $\\omega$ and the damping rate $\\Gamma$ as described in Sec. \\ref{sec:method}. \nIn Fig. \\ref{Fig:damping}(b) we show $\\omega\/\\omega_0$ determined as a function of $\\tilde{V}_0$ for $T\/T_0=0.1$, $0.3$, and $0.5$.\n$\\omega_0$ is the oscillation frequency for  $\\tilde{V}_0=0$, which we refer to as the sound frequency. \nAs $\\tilde{V}_0$ increases, $\\omega\/\\omega_0$ decreases.  This decrease is more pronounced for higher temperatures.\nWe note that the dependence of $\\omega\/\\omega_0$ on the barrier height differs qualitatively between the low $(\\tilde{V}_0 < 1)$ and high $(\\tilde{V}_0 > 1)$ barrier regimes. \nIn Fig. \\ref{Fig:damping}(c) we show the results of $\\Gamma\/\\omega$ as a function of $\\tilde{V}_0$.\nFor $T\/T_0=0.1$, $\\Gamma\/\\omega$ is small at all $\\tilde{V}_0$, confirming underdamped motion.  \nFor $T\/T_0=0.3$, $\\Gamma\/\\omega$ increases at high $\\tilde{V}_0$ and is generally below $0.5$ that we use as the definition of the temperature-induced overdamped limit.  \nFor $T\/T_0=0.5$, $\\Gamma\/\\omega$ increases rapidly with $\\tilde{V}_0$ and reaches the overdamped limit at $\\tilde{V}_0 \\geq 1.5$.\n\n\n\n\n    To identify the origin of the damping we examine the phase dynamics of a single trajectory of the ensemble. \nWe calculate the phase $\\phi(x,y)= \\phi(x,y)-  \\phi_\\mathrm{m}$ for the same parameters as in Fig. \\ref{Fig:damping}(a), \nwhere $\\phi_\\mathrm{m}$ is the mean global phase. \nIn Fig. \\ref{Fig:vortex}(a) we show $\\phi(x,y)$ at $t =3.6 \\, \\mathrm{ms}$, after the phase imprint, for $T\/T_0=0.1$, $0.3$, and $0.5$.\nThe phase imprint develops a phase difference between the two reservoirs and a corresponding phase gradient across the barrier. \nAt low temperature the reservoir phase is weakly fluctuating as demonstrated by $\\phi(x,y)$ at $T\/T_0=0.1$.\nThe fluctuations of the phase increase with increasing $T\/T_0$. \nThe reservoir phase is moderately and strongly fluctuating for $T\/T_0=0.3$ and $0.5$, respectively. \nThis results in the creation of vortices, which is confirmed by the calculation of the phase winding around the plaquettes of the numerically introduced lattice. \nWe calculate the phase winding around the lattice plaquette of size $l\\times l$ using $\\sum_{\\Box} \\delta \\phi(x,y) = \\delta_x\\phi(x,y) + \\delta_y\\phi(x+l,y)+\\delta_x\\phi(x+l,y+l)+\\delta_y\\phi(x,y+l)$, \nwhere the phase differences between sites are taken to be $\\delta_{x\/y} \\phi(x,y)  \\in (-\\pi, \\pi]$. \nWe show the calculated phase windings in Fig. \\ref{Fig:vortex}(a). \nWe identify a vortex and an antivortex by a phase winding of $2\\pi$ and $-2\\pi$, respectively. \nFor $T\/T_0=0.1$, we observe only one vortex-antivortex pair inside the barrier and no vortices in the bulk.\nThis scenario changes due to increased thermal fluctuations at high temperatures. \nFor $T\/T_0=0.3$, there is nucleation of multiple vortex pairs inside the barrier, and a few vortex pairs near the box edges. \nFor $T\/T_0=0.5$, we observe proliferating vortices in the regions of low densities around the barrier, and in the bulk.\n  \n\n\n\n  To understand the role of vortex fluctuations at the barrier,  we calculate the total number of vortices $N_v$ and average it over the thermal ensemble.    \nIn Fig. \\ref{Fig:vortex}(b) we plot $N_v$ as a function of $\\tilde{V}_0$ for the same values of $T\/T_0$ as in Fig. \\ref{Fig:vortex}(a).\nAt $T\/T_0 =0.1$, $N_v$ remains close to zero for barrier heights below a threshold value and beyond this $N_v$ increases with increasing $\\tilde{V}_0$.\nAt high temperatures the system features thermal vortices even in the absence of the barrier and the onset of barrier-induced vortices occurs at a $\\tilde{V}_0$ lower than that at low temperature.  \nTo determine this threshold $\\tilde{V}_c$ we calculate the differential vortex number $N_v (\\tilde{V}_0) = N_v - N_{v,0}$, where $N_{v,0} = N_v (\\tilde{V}_0=0)$. \nWe show $N_v (\\tilde{V}_0)$ in Fig. \\ref{Fig:vortex}(c).\nWe define $\\tilde{V}_c$ for which $N_v (\\tilde{V}_0)$ approaches $1$. \nThis gives $\\tilde{V}_c =1.47$, $0.58$, and $0.29$ for $T\/T_0=0.1$, $0.3$, and $0.5$, respectively. \nThis onset of vortices is associated with the damping of the oscillation shown above.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:con}\n\n   In this paper, we have established a direct connection between the Josephson critical current and BKT scaling in an ultracold 2D Bose gas using classical field simulations.  \nFor this, we have examined the dynamics across a Josephson junction created by a tunnel barrier between two uniform 2D clouds of $^{6}\\mathrm{Li}_2$ molecules, which is motivated by the experiments of Ref. \\cite{Niclas}.\nBased on the current-phase relation, we have mapped out the multimode, the second-harmonic (SH), the ideal junction (IJJ), and the overdamped regime as a function of the barrier height and the temperature. \nFor the IJJ regime, we have derived an analytical estimate of the critical current, which is in good agreement with the simulations and the experiments \\cite{Niclas}. \nWe have demonstrated the BKT scaling of the critical current numerically by varying the system size. \nThe scaling exponents of the critical current are in agreement with the exponents of the corresponding equilibrium system. \nFinally, we have addressed the damping of the current, which is due to phononic excitations in the bulk, and the nucleation of vortex pairs in the junction. \n \n \n  In conclusion, we have discussed the  dynamics of atomic clouds in 2D, coupled via a Josephson junction, which  results in the hybridization of the bulk and tunneling dynamics. As such, it combines and relates two foundational effects of quantum physics, in particular condensation and Josephson oscillations. Our results demonstrate a method to measure a static property of  many-body order, in particular the condensate density, via a dynamical  oscillatory process, in particular Josephson oscillations. Both the principle of this method, as well as the presented discussion of dynamical regimes of this system, can be applied to a wide range of quantum gas systems, to gain insight into their dynamical and static properties.\n \n\n   \n\n   \n   \n  \n\n  \n\n\n \n \n\n  \n\\section*{acknowledgements}\n\nWe thank Markus Bohlen for his contributions on experimental work,  Thomas Lompe and Henning Moritz for their contributions during this joint work and careful reading of the manuscript, and Francesco Scazza and Alessio Recati for stimulating discussions. \nThis work was supported by the European Union's Seventh Framework Programme (FP7\/2007-2013) under grant agreement No. 335431 and by the DFG in the framework of SFB 925 and the excellence clusters `The Hamburg Centre for Ultrafast Imaging'- EXC 1074 - project ID 194651731 and `Advanced Imaging of Matter' - EXC 2056 - project ID 390715994.\n\n\n\n\n  \n  \n   \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe rapid process (r-process) is known to be responsible for about\nhalf of the abundances of elements heavier than Fe in solar-system\nmaterial. Since the reaction occurs through very neutron-rich,\nunstable nuclei, the understanding of the process is still a\nchallenging issue in nuclear physics. The astrophysical sites of this\nprocess are still unclear, though numerical simulations have been made\nfor many possibilities (e.g. Wanajo \\& Ishimaru 2006).\n\nAn important progress in this field in the past decade is the\ndiscoveries of very metal-poor stars having large over-abundances of\nr-process elements. Such objects are expected to record the yields of\nnucleosynthetic events in the early Galaxy including the r-process.\nSeveral objects having very large excesses of r-process elements\n(e.g. Eu) with respect to other metals ([Eu\/Fe]$>1$) are known in the\nGalactic halo \\citep{sneden03, hill02, christlieb04, barklem05}, and\nare called as 'r-II' stars \\citep{beers05}. A remarkable result for\nthese stars is that the abundance patterns of neutron-capture elements\nare very similar. Objects having such large excesses of r-process\nelements are only found at the metallicity around [Fe\/H]$=-3.0$, and\nno object having [Eu\/Fe]$\\gtrsim+1.0$ is known in the range of\n[Fe\/H]$>-2.5$ in the Galactic halo, though the reason for such\na metallicity distribution is still unknown.\n\nHowever, the situation is quite different in some dwarf spheroidal\ngalaxies (dSph), in which high abundance ratios of r-process elements\nare found even in stars with relatively high metallicity. The chemical\nabundance studies for individual stars in the local group dSph made\nlarge progresses in the past several years, thanks to high resolution\nspectrographs mounted at 8--10~m telescopes (e.g. Shetrone et\nal. 2001, 2003; Venn et al. 2004). One surprising result derived by\nthese abundance studies is the existence of stars showing extremely\nlarge enhancements of heavy neutron-capture elements with relatively\nhigh metallicity. The most remarkable star is the red giant COS~82 in\nthe Ursa Minor dSph. High resolution spectroscopy of this object was\nfirst obtained by Shetrone et al. (2001; in their\npaper, this star is referred to as '199'), who reported that this star\nis moderately metal-poor ([Fe\/H]$\\sim -1.5$) but shows large\nenhancement of heavy neutron-capture elements ([Eu\/Fe]=1.49). Our\nprevious observing obtained a red spectrum of this\nstar with higher quality, and has successfully detected some other\nheavy elements (e.g. Dy, Er), confirming that the abundance pattern of\nelements heavier than Ba almost completely agrees with the r-process\nabundance pattern estimated from solar-system material (Sadakane et\nal. 2004).  This result is surprising, because no moderately\nmetal-poor star ([Fe\/H]$\\gtrsim-2$) in our Galaxy is known to date to\nhave such large excesses of r-process elements. Tsujimoto \\& Shigeyama\n(2002) applied their supernova-driven star formation model and\nproposed that this phenomenon can be explained by assuming a large\nvelocity dispersion of the gas from which low mass stars like COS~82\nformed.\n\nIn this Letter, we report the detection of the Th absorption line at\n5989~{\\AA} for this object by the re-analysis of our high resolution\nspectrum obtained by \\citet{sadakane04}. The actinide Th has an\nisotope whose half life is 14.05 Gyr, and is produced only by the\nr-process. The production of actinides is very sensitive to the\nenvironment of the sites (e.g. entropy, timescale of the explosion)\naccording to model calculations for the r-process. Therefore, the Th\nabundance can be a very strong constraint on the study of the origin\nof the neutron-capture elements in this object. While Th abundance has\nbeen determined for more than ten field metal-poor stars and globular\ncluster stars, our measurement for COS~82 is the first determination\nof an actinide abundance for an extra Galacitic object. We also\ndiscuss the possibility to apply the Th abundance ratio to estimate\nthe age of this object.\n\n\\section{Observation, Analysis and Results}\n\nA high resolution spectrum of COS~82 was obtained by\n\\citet{sadakane04} with the Subaru Telescope High Dispersion\nSpectrograph (HDS, Noguchi et al. 2002). The spectrum covers\n4400--7100~{\\AA} with a resolving power of 40,000.  The\nsignal-to-noise ratio per 1.8~km$^{-1}$ pixel is 60 at 5900~{\\AA}.\nStandard data reduction procedures are carried out with the IRAF\nechelle package\\footnote{IRAF is distributed by the National Optical\n  Astronomy Observatories, which is operated by the Association of\n  Universities for Research in Astronomy, Inc. under cooperative\n  agreement with the National Science Foundation.} as described by\n\\citet{aoki05}. Equivalent widths for isolated absorption lines are\nmeasured by fitting gaussian profiles.\n\nA standard abundance analysis was made for the measured equivalent\nwidths using the model atmospheres of \\citet{kurucz93}. The analysis\nwas made primarily based on the line list of \\citet{sadakane04}. We\nadded some weak Fe {\\small I} lines, which enable one to determine the\nmicro-turbulence more accurately.  The effective temperature ($T_{\\rm\neff}$) of 4300~K, determined by \\citet{sadakane04}, was adopted in the\nanalysis. The micro-turbulence ($v_{\\rm turb}$) and gravity ($g$) are\ndetermined so that the derived Fe abundance is not dependent on the\nstrengths of Fe {\\small I} lines nor on the ionization stage,\nrespectively. The results ($\\log g=0.6$~dex and $v_{\\rm\nturb}=1.7$~km~s$^{-1}$) agree with those of \\citet{sadakane04} within\nthe estimated errors. The metallicity  is assumed to be the\nFe abundance ([Fe\/H]). The Fe abundance derived by the present\nanalysis is [Fe\/H]$=-1.42$, adopting the solar abundance of\n$\\log\\epsilon$(Fe) = 7.45 \\citep{asplund05}. The results of the abundance\nanalysis are given in table \\ref{tab:res}.\n\nThe Th abundance is determined from the Th {\\small II} 5989~{\\AA}\nline. The oscillator strengths of Th lines, including the 5989~{\\AA}\nline, are obtained by \\citet{nilsson02}.  This line was investigated by\n\\citet{yushchenko05} for the field metal-poor star HD~221170. The line\nwas measured for the same object, using a higher quality spectrum, by\n\\citet{ivans06}, who reported that the abundance determined from this\nline agrees with those from other Th {\\small II} lines. We performed\nan analysis for HD~221170 using the spectrum obtained with the Subaru\nTelescope and the model atmosphere \\citep{kurucz93} with the \nparameters adopted by \\citet{ivans06}, and confirmed that the\nderived Th abundance agrees very well with their result. We note that\nthe partition function of the Th {\\small II} was calculated using the\nenergy levels listed by \\citet{bw92}.\n\nThough telluric absorption lines, probably due to water vapor, exist\nin this wavelength range, we found that the Th {\\small II} 5989~{\\AA}\nline of COS~82 is not distinctively affected by them.\nFigure~\\ref{fig:sp} shows the observed spectrum along with the\nsynthetic ones for three Th abundances. The data for spectral lines\nother than the Th {\\small II} line were adopted from the list of\n\\citet{kurucz95}. The wavelength of the observed spectrum is\ndetermined from 137 Fe {\\small I} lines in the spectrum. We identified\nthe absorption feature at 5989.3~{\\AA} as a Nd {\\small II} line.\nAlthough a Nd {\\small II} line at 5989.378~{\\AA} with the lower\nexcitation potential of 6005.270 cm$^{-1}$ is listed by\n\\citet{kurucz95}, J. E. Lawler (private communication) suggests that\nthe line should be the transition from 19758.540 to 3066.755\ncm$^{-1}$ at $\\lambda_{\\rm air} =$ 5989.312~{\\AA}. Since the transition\nprobability of this line is unknown, we assume it to be $\\log\ngf=-2.05$. This $gf$-value and the Nd abundance determined from other\nspectral regions well reproduce the strength of the line for COS~82 as\nwell as for HD~221170.\n\nThe Th abundance is determined by $\\chi^{2}$-fitting of synthetic\nspectra to the observed one from 5988.9 to 5989.2~{\\AA}, which is not\nseverely affected by the 5989.3~{\\AA} feature. The derived abundance\nis $\\log \\epsilon$(Th)$=-0.25$. The 2$\\sigma$ level of the fitting\nerror is 0.07~dex. The continuum level is estimated for the spectrum\naround the absorption features. The uncertainty of the continuum\nplacement ($\\sim 0.5$\\%) results in an possible error of 0.03~dex in\nthe Th abundance. We estimate the line broadening for relatively clean\nlines in the spectrum, and found that the line widths are primarily\ndetermined by the instrumental broadening, and no significant\nbroadening by macro-turbulence nor rotation is detected. The possible\nerrors in the Th abundance due to the adopted broadening parameter\n($\\pm 0.5$~km~s$^{-1}$) is 0.03~dex.\n\nThe abundances of other elements are determined using the line list of\n\\citet{sadakane04} with some modifications. The transition\nprobabilities of Nd {\\small II}, Sm {\\small II} and Gd {\\small II} are\nupdated, adding several new lines to the list, using the data of\n\\citet{hartog03}, \\citet{lawler06} and \\citet{hartog06}, respectively.\nThe effects of hyperfine splitting for Ba, La, Eu, and Pr are included\nusing the line data obtained by \\citet{mcwilliam98},\n\\citet{lawler01a}, \\citet{lawler01b}, and \\citet{ginibre89},\nrespectively. We also measure the O abundance from the [O {\\small I}]\n6300.3~{\\AA} line. The results are listed in table~\\ref{tab:res}. An\nupper limit of the Os abundance is derived from the Os {\\small I}\n5584~{\\AA} line. Estimates of upper limits for other neutron-capture\nelements are also attempted, but no meaningful result is derived from\nour spectrum. The abundances of Fe and most neutron-capture\nelements derived by the present analysis agrees with the results of\n\\citet{sadakane04} within the errors (see below). Exceptions are Eu\nand Gd abundances, for which the differences between the two\nmeasurements are about 0.25~dex. This would be due to the difference\nof the spectral features and line data adopted in the analysis. The\nabundances of Fe and some neutron-capture elements determined by\n\\citet{shetrone01} agree well with our results, while their Ce and Sm\nabundances are more than 0.3~dex higher than ours. We suspect that\ntheir results are uncertain because of the low S\/N ratio in the blue\nregion of their spectrum where Ce and Sm lines used in their analysis\nexist.\n\nThe abundance uncertainties for elements for which more than 10 lines\nare measured are estimated by $\\sigma N^{-1\/2}$, where $\\sigma$ is the\nstandard deviation of the abundances from individual lines and $N$ is\nthe number of lines used. A typical uncertainty of the equivalent\nwidth measurement from our spectrum is 6~m{\\AA}.  We performed\nabundance analyses for Fe {\\small I} lines with equivalent widths from\n30 to 200~m{\\AA}, changing the equivalent widths by 6~m{\\AA}. The\neffects of the change of equivalent widths on the derived abundances\nare 0.15--0.20~dex, which agree with the $\\sigma$ (0.18~dex) for Fe\n{\\small I} ($\\sigma_{\\rm Fe I}$). For elements whose abundances are\ndetermined from less than 10 lines, the random error is estimated by\n$\\sigma_{\\rm Fe I}N^{-1\/2}$. Errors due to the uncertainty of\natmospheric parameters are estimated by calculating abundances\nchanging atmospheric parameters by $\\Delta(T_{\\rm eff})$=150~K,\n$\\Delta(\\log g)$=0.3~dex, $\\Delta$([Fe\/H])=0.2~dex, and $\\Delta(v_{\\rm\nturb})=0.3$~km~s$^{-1}$. These errors are added, in quadrature, to the\nrandom errors estimated above, and are given in table~\\ref{tab:res}\n($\\sigma_{\\rm total}$). We found, however, that the abundances of the\nneutron-capture elements determined from the ionized species\nsystematically scales by changes of the adopted atmospheric\nparameters. For the discussion based on the abundance pattern, we\ncalculate the average of the abundance changes for heavy\nneutron-capture elements (La--Er) by changing the atmospheric\nparameters, and adopt the deviation from this systematic abundance\nchange for individual elements as the errors due to the uncertainties\nof atmospheric parameters. This results in quite small errors ($\\leq\n0.05$~dex). These errors and the random errors are added in\nquadrature, and are also given in table~\\ref{tab:res} as $\\sigma_{\\rm\nn-cap}$.\n\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(80mm,150mm){fig1.ps}\n  \\end{center} \n\\caption{\nComparisons of synthetic spectra for the region including the Th II\n5989~{\\AA} line with the observed spectrum. The Th abundances assumed\nin the calculations are $\\log\\epsilon$(Th)$= -0.15, -0.25, -0.35$\n(solid lines), and $-\\infty$ (dotted line). }\\label{fig:sp}\n\\end{figure}\n\n(see text).\n\n\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(80mm,150mm){fig2.ps}\n  \\end{center} \n\\caption{\nUpper panel: Scaled abundances of neutron-capture elements for COS~82\n(filled circles), CS~22892--052 (open circles), CS~31082--001\n(crosses), and the solar-system r-process component (solid line). The\nabundances are normalized to the average of the abundances of the 10 elements\nfrom Ba to Er in the logarithmic scale. Lower panel: Abundance\ndifferences in the logarithmic scale between COS~82 and CS~22892--052\n(open circles) and that between COS~82 and the solar-system r-process\ncomponent (filled circles). Solid lines indicate the averages of the\nabundance differences for Ba--Er.}\\label{fig:pattern}\n\\end{figure}\n\n\\begin{longtable}{lrrrrrr}\n\\caption{Elemental Abundances Results}\\label{tab:res}\n\\hline\n\\hline\nspecies & $\\log\\epsilon_{\\rm sun}$ & $\\log\\epsilon$& [X\/Fe] & $N$ & $\\sigma_{\\rm totol}$ & $\\sigma_{\\rm n-cap}$ \\\\\n\\endhead\n\\hline\nFe I  & 7.45 & 6.03 &  ...  & 137 & 0.27 & ... \\\\\nFe II & 7.45 & 6.02 &  ...  &  13 & 0.18 & ... \\\\\nO I   & 8.70 & 7.45 &  0.17 &   1 & 0.2  & ... \\\\\nY II  & 2.21 & 1.22 &  0.42 &   6 & 0.23 & 0.14 \\\\\nZr II & 2.59 & 1.82 &  0.65 &   1 & 0.24 & 0.19 \\\\\nBa II & 2.17 & 1.36 &  0.60 &   3 & 0.30 & 0.21 \\\\\nLa II & 1.13 & 0.52 &  0.81 &   8 & 0.16 & 0.06 \\\\\nCe II & 1.63 & 0.76 &  0.55 &   4 & 0.20 & 0.14 \\\\\nPr II & 0.80 & 0.20 &  0.82 &   7 & 0.19 & 0.11 \\\\\nNd II & 1.45 & 1.00 &  0.97 &  41 & 0.17 & 0.05 \\\\\nSm II & 0.98 & 0.90 &  1.34 &   2 & 0.21 & 0.13 \\\\\nEu II & 0.52 & 0.34 &  1.24 &   3 & 0.17 & 0.11 \\\\\nGd II & 1.09 & 1.10 &  1.43 &   3 & 0.16 & 0.12 \\\\\nDy II & 1.17 & 1.13 &  1.38 &   3 & 0.17 & 0.10 \\\\\nEr II & 0.97 & 0.73 &  1.18 &   3 & 0.17 & 0.11 \\\\\nTh II & 0.09 &-0.25 &  1.08 &   1 & 0.15 & 0.08 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{longtable}\n\n\\section{Discussion and concluding remarks}\n\nThe upper panel of figure~\\ref{fig:pattern} shows the abundance patterns\nof neutron-capture elements for COS~82, the solar-system r-process\ncomponent \\citep{simmerer04}, and the two r-process-enhanced field\nstars CS~22892--052 \\citep{sneden03} and CS~31082--001\n\\citep{hill02}. The abundances are normalized to the average of Ba--Er\nones. As reported by \\citet{sadakane04}, the overall abundance pattern\nof heavy neutron-capture elements (Ba--Er) in COS~82 agrees well with\nthe r-process abundance pattern. In the lower panel, the abundance\ndifferences between COS~82 and the solar-system r-process component, as\nwell as that between COS~82 and CS~22892--052, are shown. The Th\nabundance of COS~82 is lower than the normalized value of the\nsolar-system r-process component, while that agrees very well with the\nvalue of CS~22892--052. This point is discussed below in more\ndetail. The abundance of the light neutron-capture element Y in COS~82\nis also lower than the normalized value in the solar-system r-process\ncomponent as found in field r-process-enhanced stars.\n\n\n\nAlthough such small deviation exists, an important result derived from\nthe comparison is that the abundance pattern of neutron-capture\nelements, including Th, in COS~82 agrees with the pure r-process\nabundance pattern. According to model calculations for the r-process,\nthe actinide abundances with respect to other neutron-capture elements\nare very sensitive to the model parameters (e.g. electron fraction,\nentropy; Wanajo et al. 2002). The agreement of the Th abundance ratio\nto other neutron-capture elements in COS~82 with that of CS~22892--052\nindicates that the mechanism and astrophysical sites that are\nresponsible for the heavy elements in COS~82 is common with those for\nthe r-process-enhanced field stars. Recent abundance studies for\nr-process-enhanced stars suggest that the r-process occurs under a\nquite limited condition (e.g. supernova explosions whose progenitors\nhave mass in a very narrow range). Our result shows that this is also\napplied to stars even in dSph. \n\nThe metallicity of COS~82 is more than one order of magnitude higher\nthan those in the field r-II stars ([Fe\/H]$\\sim -3$). The chemical\ncomposition of field stars with such metallicity is believed to be\naffected by a number of nucleosynthesis events.  However, the\nmetallicity, as well as the amount of neutron-capture elements, should\nbe related to how the yields from a nucleosynthesis event are captured\nby the clouds from which next generation, low mass stars formed, and\nthe neutron-capture elements of COS~82 might be provided by a single\nevent. For instance, \\citet{tsujimoto02} proposed that the large\nexcesses of neutron-capture elements in a few objects in dSph,\nincluding COS~82, are the result of a larger velocity dispersion of\nthe gas from which low mass stars like COS~82 formed in dSph than in\nthe Galactic halo.  That is, the yields of one supernova are recorded\nin the next generations of stars with smaller mixing with\nmetal-deficient (or metal-free) gas in dSph.\n  \nFinally, we discuss an estimate of the age of COS~82 from the abundance\nratio between Th and other stable elements, which could be an\nindependent calibration for the age measurements for this galaxy. If\nthe age of this star is shown to be quite long ($>10$~Gyr), the star\nformation of this dSph should be completed with a short time scale.\nTh has a half life of 14.05 Gyr, and the age of the object (more\nstrictly, the duration from the r-process event) is estimated from the\nTh abundance ratio to other stable elements, if its initial abundance\nratio is given. Here the average of abundances of the nine elements\nfrom La to Er is used as the reference \\footnote{Ba is excluded from\nthe reference elements, because its abundance error is relatively\nlarge due to the strengths of the lines used in the analysis. The\nconclusion in this paragraph is, however, unchanged if this element is\nincluded in the calculation.} (E$_{\\rm stable}$).  The Th\/E$_{\\rm\nstable}$ ratio of COS~82 agrees very well with that of CS~22892--052,\nwhile that is 0.3~dex lower than the value in the r-process component in\nsolar-system material (figure~\\ref{fig:pattern}). \n\nUnfortunately, the abundance ratio of Th\/E$_{\\rm stable}$ produced by\nthe r-process is not constant. Indeed, the actinides including Th are\nsignificantly enhanced, with respect to other neutron-capture\nelements, in the r-II star CS~31082--001, compared to that of the\nsolar-system r-process component \\citep{hill02}. However, studies of\nTh abundances for a number of r-process enhanced field stars show that\nthe abundance ratios distribute in a rather narrow range ($-0.6\\leq\n$Th\/E$_{\\rm stable} \\leq -0.1$), and CS~22892--052 is one of the\nobjects that have the lowest ratio. Assuming that the initial\nabundance ratio of Th\/E$_{\\rm stable}$ for COS~82 is within this\nrange, and that the field halo stars including CS~22892--052 are quite\nold (their ages are around 12--13 Gyr), COS~82 is also an old object.\n(If a higher value of the initial Th\/E$_{\\rm stable}$ ratio is\nassumed, the derived age becomes longer).  \n\nThe error ($\\sigma_{\\rm n-cap}$) in the determination of the Th\nabundance ratio is 0.08~dex, while the scatter of the abundances of\nthe above nine stable elements around those of the solar-system\nr-process component, as well as of CS~22892--052, is 0.11~dex. We\nestimate the error of the Th\/E$_{\\rm stable}$ to be 0.13~dex, which\nresults in the uncertainty of the age estimate to be 6 Gyr. Hence,\nthough the object is suggested to be as old as field halo stars, the\nage estimate using Th chronometer gives only a weak constraint. A\nmeasurement of U\/Th ratio, which is a more sensitive chronometer, for\nthis object will enable one to estimate the age with a much smaller\nerror ($\\lesssim$ 2 Gyr). \\\\\n\n\nWe are grateful to Dr. J. E. Lawler for providing useful Nd {\\small\n  II} line data. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION \\label{intro}}\n\n PSR B0628$-$28\\, is a bright radio pulsar (Large, Vaughan \\& Wielebinski,\n1969). Its characteristic spin-down age is $\\sim 2.75 \\times 10^6$\nyears. The pulsar's period and period derivative are $P=1.24$ s and\n$\\dot{P}=7.1\\times 10^{-15}$, implying a spin-down luminosity of $\\log\n\\dot{E} = 32.16\\,\\,\\mbox{erg s}^{-1}$ and a magnetic field at the\nneutron star poles of\\, $\\log B_\\perp=12.48$ G. The distance to the\npulsar as listed in the Princeton Pulsar Catalog (Taylor, Manchester\n\\& Lyne 1993) and the ATNF online pulsar database is 2.14 kpc (Taylor\n\\& Cordes 1993; Manchester et al.~2005). Using the NE2001 Galactic\nfree electron density model of Cordes \\& Lazio (2002) which\nbuilds upon and supersedes the Taylor \\& Cordes (1993) model by\nexploiting new observations and methods, the radio dispersion measure\nof $\\mbox{DM}=34.36\\, \\mbox{pc cm}^{-3}$ implies that the pulsar might\nbe closer at a distance of only 1.45 kpc.\n The pulsar parameters and the emission properties observed in other\n X-ray detected rotation-powered pulsars (cf.~Becker \\& Tr\\\"umper\n 1997; Becker \\& Aschenbach 2002 and references therein), seem to\n indicate that PSR B0628$-$28\\ is an ordinary, old\\footnote{in standards of\n X-ray detected non-recycled pulsars} radio pulsar which is\n detectable just at the threshold of sensitivity of the current\n generation of X-ray observatories.  The results from ROSAT and\n Chandra observations, however, suggest that there might be more to\n expect from this pulsar.\n\n PSR B0628$-$28\\, was first noticed to have a faint potential X-ray counterpart\n in the ROSAT all-sky survey, RX J0630.8$-$2834\\, (Becker et al.~1993). The pulsar\n was therefore re-observed in October 1992 and April 1993 in two short\n pointed ROSAT observations of $\\sim 2.5$ ksec and $\\sim 6.3$ ksec\n exposures. Both observations confirmed the detection of the X-ray\n source RX J0630.8$-$2834\\, and collected 27 source counts from it in the ROSAT\n PSPC (Position Sensitive Proportional Counter). The low number of \n counts did not allow to obtain more than a rough hardness \n ratio\\footnote{Ratio of the number of counts detected in the soft \n spectral bands vs.~that detected in the hard bands.} which suggested \n that RX J0630.8$-$2834\\, has a medium soft X-ray spectrum. A timing analysis was \n not possible either, thus leaving the association between RX J0630.8$-$2834\\, and \n PSR B0628$-$28\\, an open issue and based on a rough positional agreement only.\n\n Such an association, however, was regarded to be unlikely based on\n the estimated energy budget of the putative X-ray counterpart. The\n X-ray luminosity of RX J0630.8$-$2834\\, extrapolated to the pulsar's distance was\n found to be $\\sim 37\\%$ of the pulsar spin-down energy. Its X-ray\n conversion factor thus would be few hundred times higher than the\n average X-ray conversion factor $\\sim 10^{-3}\\,\\dot{E}$ which is\n observed in the $0.1-2.4$ keV band from other X-ray detected\n rotation-powered pulsars (Becker \\& Tr\\\"umper 1997).\n\n Follow-up optical imaging and spectroscopic observations with the ESO\n New Technology Telescope (NTT) in La Silla (Chile) were performed to \n assess the nature of the unidentified X-ray source RX J0630.8$-$2834. Various optical \n sources were identified in the $\\sim 25\"$ wide ROSAT PSPC error box, \n which could not be conclusively ruled out as potential optical counterparts \n of RX J0630.8$-$2834.\n\n Recently, \\\"Ogelman and Tepedelenlioglu (2002) have observed RX J0630.8$-$2834\\, as\n part of the Chandra-AO3 guest observer program. The source, CXC J063049.4$-$283443, was\n clearly detected with the ACIS$-$S\\footnote{See \n \\protect{$\\mbox{http:}\/\/\\mbox{cxc.harvard.edu}\/\\mbox{cdo}\/\\mbox{user\\_guide.html}$} \n for a detailed description of Chandra, its instrumentation and various\n observing modes.} detector in an 17 ksec exposure in\n which 189 source counts were recorded from it.  Although representing\n an improvement over the ROSAT observations, the low number of source\n counts did not support a detailed spectral analysis which could\n discriminate between various possible spectral models. Fixing the\n column absorption to a value corresponding to a neutral hydrogen\n density of $1.07\\times 10^{21}\\,\\mbox{cm}^{-2}$ (as deduced from the\n pulsar's radio dispersion measure by assuming 10 H-Atoms per free\n electron), both a power law model with photon-index $\\alpha=2.45\\pm\n 0.15$ and a thermal blackbody model with $T=(4.53\\pm 0.13) \\times\n 10^6$ K and $R_{bb}\\sim 1\\,\\mbox{km}$ describe the energy spectrum of\n CXC J063049.4$-$283443\\, equally well.  As the frame time of the ACIS-S in the selected\n mode was close to the pulsars rotation period, a timing analysis did\n not yield a significant result and prevented the derivation of\n reliable upper limits for the fraction of pulsed photons. However,\n the high spatial resolution of Chandra along with its superior\n pointing accuracy allowed the determination of a much more accurate\n position of the putative X-ray counterpart RX J0630.8$-$2834\\ than before.  The\n Chandra position of RX J0630.8-2834 was found to match the position\n of PSR B0628$-$28\\, by 1.5 arcsec! Any offset from the pulsar radio position\n due to its proper motion (Brisken et al.~2003) is well within the\n uncertainties of the Chandra position. Hence, the positional\n coincidence between RX J0630.8$-$2834\\, (CXC J063049.4$-$283443\\, resp.) strongly suggests that this\n X-ray source is indeed the counterpart of PSR B0628$-$28.\n\n In this paper we report on X-ray, optical and radio observations of\n PSR B0628$-$28\\, which were made with XMM-Newton, the ESO NTT in La Silla and\n the Lovell telescope at the Jodrell Bank Observatory in order to\n explore the spectral and timing emission properties of this\n interesting pulsar. The paper is organized in the following manner:\n in \\S2 we describe the optical, radio and XMM-Newton observations of\n PSR B0628$-$28\\, and provide the details of the data processing and data\n filtering. The results of the spectral and timing analysis are given\n in \\S3. A summary and concluding discussion is presented in \\S4.\n\n\n\\section{OBSERVATIONS AND DATA REDUCTION\\label{obs}}\n\n\\subsection{OPTICAL OBSERVATIONS OF PSR B0628$-$28\\label{optical_obs}}\n\n Two images of 600s exposure\n time in the B and V filters, pointed on the position of PSR B0628$-$28, were\n made on Dec.~15, 1999 with the ESO NTT Multi-Mode Instrument (EMMI). \n The $B$ (422.3 nm) image was taken with the blue arm,\n equipped with a Tektronix $1024 \\times 1024$ chip having 24 $\\mu$\n pixels size (0.37 arcsec), while the $V$ (542.6 nm) image was\n acquired with the red arm mounting a Tektronix $2086 \\times 2048$\n chip with 24 $\\mu$ pixel size (0.27 arcsec).  Two images of the\n Landolt field Ru149 were observed soon after for the purpose of\n calibrating the two images.\n\n Bias and flat-field were applied as usual, using calibration\n sets acquired at evening twilight, and the reduction of the images\n was performed with both DAOPHOT (Stetson 1987) and SExtractor\n (Bertin and Arnouts, 1996) to take into account the presence of\n extended sources and to check the output photometry\n results. Photometric calibration was performed by using the\n Ru149 field and the average La Silla extinction terms (-0.22 for B\n and -0.12 for V, as found in the instrument web-page at ESO La Silla\n site). The night was photometric and with a good seeing, 0.75 arcsec\n in V.\n\n No source is detected at the position of PSR B0628$-$28. Figure\n \\ref{optical_images} shows a close-up view of the pulsar field as\n seen in the visible (V) and blue (B) bands. 3$\\sigma$ upper limits\n assuming a point-source and correcting for the interstellar\n extinction are $V=26.1\\,\\mbox{mag}$ and $B=26.2\\,\\mbox{mag}$,\n respectively. Values for E(B-V) were derived from dust maps\n (Schlegel et al.~1998). Converting these upper\n limits\\footnote{Information on the conversion from the magnitude\n scale to the photon flux in Jy can be found at\n http:\/\/www.astro.utoronto.ca\/$\\sim$patton\/astro\/mags.htm} to a photon\n flux upper limit yields $1.322 \\times 10^{-7} Jy$ for the V-band and\n $1.423 \\times 10^{-7} Jy$ for the B-band, respectively.\n\n\\subsection{RADIO OBSERVATIONS OF PSR B0628$-$28\\label{radio_obs}}\n\n The ephemerides for the analysis of the X-ray data were obtained from\n radio observations and the measurement of pulse times--of--arrival\n (TOAs) using the 76-m Lovell radio telescope at Jodrell Bank\n Observatory. Table \\ref{t:radio} summarizes the radio ephemerides of\n PSR B0628$-$28. A dual-channel cryogenic receiver system sensitive to two\n orthogonal polarizations was used predominantly at frequencies close\n to 1400 MHz. The signals of each polarization were mixed to an\n intermediate frequency, fed through a multi-channel filter-bank and\n digitized.  The data were de-dispersed in hardware and folded\n on--line according to the pulsar's dispersion measure and topocentric\n period.  The folded pulse profiles were stored for subsequent\n analysis. In a later off--line processing step, every sub-integration\n spoiled by RFI was removed, the polarizations combined and the\n remaining sub-integrations averaged to produce a single\n total--intensity profile for the observation.  TOAs were subsequently\n determined by convolving, in the time domain, the averaged profile\n with a template corresponding to the observing frequency. The\n uncertainty on the TOA was found using the method described by Downs\n \\& Reichley (1983), which incorporates the off--pulse rms noise and\n the `sharpness' of the template.  These TOAs were corrected to the\n solar system barycenter using the Jet Propulsion Laboratory DE200\n solar system ephemeris (Standish 1982). More details can be found in\n Hobbs et al.~(2004). Spectral data from PSR B0628$-$28\\, were obtained from\n the compilation of Maron et al.~(2000).\n\n The emission beam geometry of PSR B0628$-$28\\, is not known but can be\n estimated from fitting the canonical rotating vector model to the\n polarization position angle from radio data (e.g.~Lorimer \\& Kramer\n 2005). We used the polarization profiles observed at 408 MHz and 1400\n MHz by Gould \\& Lyne (1998, see reference for details about observing\n set-up and calibration procedure) to determine the inclination of the\n magnetic axis to the rotation axis to be $\\alpha=70^o$. The impact\n angle of the line of sight with the magnetic axis is fitted to be\n $\\beta=-12^o$. As usual, the small duty cycle of \n polarized emission available for modeling produces a distinct \n correlation between the fit results for $\\alpha$ and $\\beta$. We \n demonstrate this by showing the $\\chi^2$-sphere for the 1400-MHz \n least-squares fit in $\\alpha$ and $\\beta$ in Figure \\ref{polarisation}. \n Acceptable fits lie in the inside of the sickle shaped goodness of \n fit contours. At the higher frequency of 1400~MHz shown here, the \n formal global minimum of the least-squares fit resides at the edge \n of the contours main body, in a small isolated point located at \n $\\alpha=11^o$ and $\\beta=-3^o$ (cf.~Figure \\ref{polarisation}).  In\n order to decide, which solution in this rather large solution space\n is physically more acceptable, we can make use of the pulse shape\n information: the measured 50\\% width of the radio pulse amounts to\n $\\sim 17^o$.  From studies of a large sample of radio pulsars, one\n finds a relationship between the beam radius and the spin period (see\n Lorimer \\& Kramer 2005 and references therein). Based on such\n results, for a spin period of 1.244~s, one expects a beam width of\n $\\sim 11^o$ for a line of sight cutting through the center of the\n emission cone (i.e.~$\\beta \\sim 0^o$).  An observed pulse width\n larger than this value may be a good indication of the possibility\n that the magnetic inclination angle is small, producing a path of the\n line-of-sight through the beam that is curved rather than\n straight. The fairly shallow gradient of the position angle curve may\n also add weight to this small-$\\alpha$ solution, but a combination of\n the $\\chi^2$-contours obtained for 408~MHz and 1400~MHz makes a\n nearly orthogonal solution $\\alpha\\sim 70^o, \\beta\\sim-12$ the most\n likely one. A sketch which illustrates possible beam geometries is \n given in Figure \\ref{PSRB_geometry}.\n\n \\subsection{XMM-NEWTON OBSERVATIONS OF PSR B0628$-$28 \\label{xray_obs}}\n\n PSR B0628$-$28\\, was observed with XMM-Newton\\footnote{See http$:\/\/$xmm.vilspa.esa.es$\/$\n for a description of XMM-Newton, its instrumentation and the various detector\n modes available for observations.} on April 20, 2004 (XMM rev.~773)\n for a total on-source time of $\\sim 48\\,500$s. We used the EPIC-PN\n camera as the prime instrument and operated both MOS1\/2 cameras in\n PrimeFullWindow mode to obtain imaging and spectral data. The EPIC-PN\n camera was setup to operate in PrimeLargeWindow readout mode which\n provides imaging, spectral and timing information with a temporal\n resolution of 48 ms. The higher temporal resolution of the EPIC-PN\n small-window mode would have been attractive but an overkill in view\n of its $\\sim 30\\%$ higher dead-time (see e.g.~Becker \\& Aschenbach\n 2002 for a summary of XMM-Newton instrument modes suitable for pulsar\n studies). The medium filter was used for the MOS1\/2 cameras and the\n thin filter for the EPIC-PN. The expected lower efficiency of the RGS\n means that it is of limited use for the given exposure time. The\n optical field is well known from our NTT\/EMMI observations so that we\n do not report on the analysis of XMM's optical monitor (OM)\n data. This data, which were taken with the OM in standard\n configuration are superseded in sensitivity by our ground based\n ESO\/NTT observation (see \\S \\ref{optical_obs}).  A summary of\n exposure times, instrument modes and filters used for the X-ray\n observation is given in Table ~\\ref{t:xray_obs}.\n\n XMM-Newton data have been seen to show timing discontinuities in the\n photon arrival times with positive and negative jumps of the order of\n one to several seconds (Becker \\& Aschenbach 2002; Kirsch et\n al.~2003). Inspecting the log-files from our processing of raw data\n we found that the EPIC-PN data of PSR B0628$-$28\\, exhibit a clock\n discontinuity showing one negative jump of 15s. We therefore used \n a release track version of the XMM-Newton SAS (Standard Analysis Software) \n for the analysis of the EPIC-PN data. This software detects and corrects \n most of the timing discontinuities during data processing. In addition, known \n timing offsets due to ground station and space craft clock propagation \n delays are corrected by this software in using new reconstructed time \n correlation (TCX) data. Barycenter correction of the EPIC-PN data and \n all other analysis steps were performed by using SAS Version 6.1.\n\n Data screening for times of high background was done by inspecting\n the light-curves of the MOS1\/2 and PN data at energies above 10\n keV. Apart from having a rather high sky background contribution,\n strong X-ray emission from soft proton flares are seen at various\n times during the observation. Creating the light-curves with bins of\n 100s, we rejected those bins where the MOS1\/2 light-curves had more\n than 130 cts\/bin. For the EPIC-PN data we rejected times with more\n than 90 cts\/bin (for comparison, data with a typical low sky\n background require the rejection of events at a level of $\\sim10$\n cts\/bin).  The data screening reduced the effective exposure time for\n the MOS1\/2 to 35.6 ksec and 34.6 ksec, respectively. For the EPIC-PN\n data the effective exposure time was reduced to 31.7 ksec. The net\n exposure time after rejecting times of high sky background thus is\n only about 74\\% of the requested observing time.\n\n For the spectral analysis based on the MOS1\/2 data we used only those\n events with a detection {\\em pattern} between $0-12$ (i.e. single,\n double and triple events) and the {\\em flag} parameter set to less\n than or equal to 1. The latter criterion excludes events which are\n located near to a hot pixel, or to a bright CCD column, or which are\n near to the edge of the CCD. For the EPIC-PN timing and spectral\n analyzes, we used only single and double events, i.e.~those which\n have a pattern parameter of less than, or equal to, 4 and a flag\n parameter equal to zero. The energy range of the MOS1\/2 and EPIC-PN\n CCDs was restricted to $0.2-10$ keV for the spectral and timing\n analysis.\n\n \\section{ANALYSIS OF THE XMM-DATA OF PSR B0628$-$28\\label{PSRB}}\n\n The putative X-ray counterpart of PSR B0628$-$28\\, is detected with high\n significance in both the MOS1\/2 and EPIC-PN data. The XMM-Newton net\n counting rates within the $0.2-10$ keV band are $0.0047\\pm 0.0008$\n cts\/s (MOS1), $0.0042 \\pm 0.0008$ cts\/s (MOS2), and $0.021 \\pm 0.003$\n cts\/s (EPIC-PN), respectively. A maximum-likelihood source-detection\n did not yield any evidence for a spatial extent of RX J0630.8$-$2834\\, of larger\n than 15 arcsec, corresponding to the HEW (Half Energy Width) of the \n instruments' point spread function.\n\n\n \\subsection{Timing Analysis\\label{PSRB_timing}}\n\n We used the EPIC-PN large-window mode data for the timing\n analysis. Given the pulsar period of $\\sim 1.244$s the temporal\n resolution of 48ms allows us to perform a detailed timing analysis in\n searching for X-ray pulsations and to construct a pulse profile with\n up to $\\sim 26$ independent phase bins, if supported by the photon\n number statistics.\n\n Events were selected from a circle of 25 arcsec radius centered on\n RX J0630.8$-$2834. This extraction region contains 80\\% of the point source\n flux. For the barycenter correction we applied the standard\n procedures for XMM-Newton data using {\\em barycen-1.17.3}\\\/ and the\n JPL DE200 Earth ephemeris (Standish 1982) to convert photon arrival\n times from the spacecraft to the solar system barycenter (SSB) and\n the barycentric dynamical time (TDB). The pulsar radio timing\n position (cf.~Table \\ref{t:radio}) was used for the barycenter\n correction. The spin-parameters $f$ and $\\dot{f}$ of PSR B0628$-$28\\, are\n known with high precision from our contemporaneous radio\n observations, covering the mean epoch MJD=53063.4104692111723\n (TDB@SSB) of the XMM-Newton observation. PSR B0628$-$28\\, is not known to show\n timing irregularities (glitches) so that we can fold the photon\n arrival times using the pulsar's radio ephemeris (see Table\n \\ref{t:radio}).\n The statistical significance for the presence of a periodic signal\n was obtained from a $Z^2_n$-test with $1-10$ harmonics in combination\n with the H-Test to determine the optimal number of harmonics (De\n Jager 1987; Buccheri \\& De Jager 1989). The optimal number of phase\n bins for the representation of the pulse profile was determined by\n taking into account the signal's Fourier power and the optimal number\n of harmonics deduced from the H-Test (see Becker \\& Tr\\\"umper 1999\n and references therein).\n\n Within the $0.2-10$ keV energy band, 1290 events were available for\n the timing analysis of which $\\sim 35\\%$ are estimated to be\n background. The $Z^2_n$-test gave 66.1 for $n=4$ harmonics\n ($Z^2_1=48.41$). According to the H-Test, the probability of\n measuring $Z^2_4=66.1$ by chance is $\\sim 3 \\times 10^{-11}$ thus\n confirming RX J0630.8$-$2834\\, (CXC J063049.4$-$283443, resp.) to be the counterpart of PSR B0628$-$28\\, and\n establishing it firmly to be an X-ray pulsar!\n\n Figure \\ref{PSRB_pulseprofiles} depicts the X-ray pulse profile of\n PSR B0628$-$28\\, for the $0.2-10$ keV energy band.  As indicated by the higher\n harmonic content and as can be seen easily in the figure, the pulse\n shape is not sinusoidal but double peaked, with a main broad peak of \n width $\\sim 180^\\circ$ and a narrow pulse component of $\\sim 45^\\circ$\n width longitude. By taking the center of mass of both pulse components as \n a reference point, the separation in phase between both peaks is $\\sim 0.4$ \n ($144^\\circ$ longitude). The fraction of pulsed events in the \n $0.2-10$ keV energy range is determined to be of $39 \\pm 6 \\%$ by \n using a bootstrap method (Swanepoel, de Beer \\& Loots 1996; Becker \n \\& Tr\\\"umper 1999). Restricting the timing analysis to the $0.2-1.0$ \n keV and  $1.0-2.1$ keV energy bands yields the pulsed fractions \n $41 \\pm 7\\%$ and $42 \\pm 10\\%$, respectively, which do not indicate \n a significant energy dependence over the $0.2-10$ keV bandpass, \n although the decreasing signal to noise ratio beyond $\\sim 2.1$ keV  \n makes any conclusion for this energy band merely tentative. \n\n Comparing the X-ray pulse profile with the one taken with the Jodrell\n Bank radio telescope at 1.4 GHz shows that both profiles are markedly \n different. Measuring the TOAs of the radio and X-ray pulses shows that \n the narrow X-ray pulse leads the radio pulse by $\\sim 0.2$ ($72^\\circ$  \n longitude) in phase. The radio pulse itself leads the broader X-ray \n pulse by about the same amount of $72^\\circ$ longitude. The arrival of \n the radio pulse thus appears to be in the middle between the arrival \n of the two X-ray pulse components.\n\n We note that uncertainties of the XMM-Newton clock against UTCs are not \n relevant as those are on a scale of $\\sim 100\\,\\mu s$ (Becker et al.~2005, \n in prep.) and thus are a factor of $\\sim 1000$ smaller than the bin width  \n of the X-ray pulse profile shown in Figure \\ref{PSRB_x_radio_profiles}.\n\n\n \\subsection{Spectral Analysis\\label{PSRB_spectral}}\n\n The energy spectrum of PSR B0628$-$28\\, was extracted from the MOS1\/2 data by\n selecting all events detected in a circle of radius 50 arcsec\n centered on the pulsar position. Using the XMM-Newton\/EPIC-MOS model\n point spread function, 90\\% of all events of a point source are\n within this region. The background spectrum was extracted from an\n annulus of outer radius 85 arcsec surrounding PSR B0628$-$28. For the EPIC-PN\n data we used an extraction radius of 31 arcsec centered on PSR B0628$-$28.\n This selection region includes $\\sim 85\\%$ of the point source\n flux. As the instrument focus is relatively close to the edge of the\n PN-CCD, we extracted the background spectrum from a source free\n region about one arc-minute east of the pulsar. Out-of-time events\n prevent us from extracting the background spectrum from a region\n located below the source and along the CCD read-out direction.\n\n In total, the extracted spectra include 814 source counts from the\n EPIC-PN camera and 430 source counts from the EPIC-MOS1\/2\n detector. The spectral data were dynamically binned so as to have at\n least 30 counts per bin. Model spectra were then simultaneously fit\n to both the PN and MOS1\/2 data.\n\n Amongst the single component spectral models, a power law model was\n found to give the statistically best representation ($\\chi^2$ =34.6 for\n 37 dof) of the observed energy spectrum. A single blackbody model\n did not give acceptable fits ($\\chi^2$ =65.1 for 37 dof) and can be \n rejected.  The power law model yields a column absorption of\n $N_H=6.0_{-1.8}^{+3.1}\\times 10^{20}\\,\\mbox{cm}^{-2}$, a photon-index\n $\\alpha = 2.63_{-0.15}^{+0.23}$ and a normalization of\n $1.0_{-0.1}^{+0.13}\\times 10^{-5}$ photons cm$^{-2}$ s$^{-1}$\n keV$^{-1}$ at $E=1$ keV. The errors represent the $1\\sigma$\n confidence range for one single parameter of interest. \n For the unabsorbed energy flux we measured $f_x=\n 3.36_{-0.17}^{+0.22}\\times 10^{-14}\\, \\,{\\rm ergs\\,\\, s}^{-1}\\,{\\rm\n cm}^{-2}$ in the $0.5-10$ keV band, yielding an X-ray luminosity of\n $L_x=8.42_{-0.54}^{+0.42} \\times 10^{30}\\,{\\rm ergs\\,\\, s}^{-1}$ for\n a distance of 1.45 kpc.  For the ROSAT energy band, $0.1-2.4$ keV, we \n measured the flux to be $f_x=9.4_{-2.3}^{+5.0}\\times 10^{-14}\\,\\,\n {\\rm ergs\\,\\, s}^{-1}\\,{\\rm cm}^{-2}$, yielding an X-ray luminosity \n of $L_x=2.4_{-0.6}^{+1.25}\\times 10^{31} \\, {\\rm ergs\\,\\, s}^{-1}$.  \n These luminosities imply the huge rotational energy to X-ray energy\n conversion factors $L_x\/\\dot{E}= 58.3\\times 10^{-3}$ within $0.5-10$\n keV and $163.4\\times 10^{-3}$ if transformed to the ROSAT band (see\n \\S\\ref{discussion} for a discussion).\n\n The best-fit power law spectrum and residuals are shown in\n Figure~\\ref{PSRB_pl_spectrum}.  Contour plots showing the relationship \n between the photon index and the column absorption for various confidence \n levels are shown in Figure \\ref{PSRB_pl_contour}. \n\n Testing composite spectral models consisting of two Planckian components \n or of a Planckian plus power law component resulted in fits which had a \n chi-square of 28.6 (for 35 dof), i.e.~a goodness comparable to what we found \n in the single component power law fit. The F-test statistic for adding the \n extra blackbody spectral component to the power law model yields a \n probability of only 97.2\\%, i.e.~slightly more than $2\\sigma$. The \n justification to include a thermal component to the power law spectrum \n thus is not particularly strong by statistical means. \n\n The parameters fitted for the composite Planckian plus power law model are a \n column absorption of $N_H \\le 2.5 \\times 10^{20}\\,\\mbox{cm}^{-2}$,\n a photon-index $\\alpha = 2.27^{+0.23}_{-0.13}$ and a normalization of the power\n law component of $4.8^{+1.6}_{-0.9} \\times10^{-6}$ photons cm$^{-2}$ s$^{-1}$\n keV$^{-1}$ at $E=1$ keV.  The blackbody temperature and the size of the projected\n emitting area are $kT = 0.25_{-0.04}^{+0.05}$ keV and $R_{bb}= 69_{-25}^{+30}$\\,m, \n assuming a pulsar distance of 1.45 kpc. For the unabsorbed energy flux and \n luminosity we compute $f_x = 2.8^{+3}_{-1} \\times 10^{-14} \\,\\mbox{erg s}^{-1}\\mbox{cm}^{-2}$\n and $L_x = 7.0^{+8}_{-2}  \\times 10^{30}\\,\\mbox{erg s}^{-1}$ for the $0.5-10$ keV band. For \n the $0.1-2.4$ keV energy band we compute the model flux to be \n $f_x = 4.0^{+6.0}_{-1.7} \\times 10^{-14} \\,\\mbox{erg s}^{-1}\\mbox{cm}^{-2}$, yielding an \n X-ray luminosity of   $L_x = 1.0^{+1.5}_{-0.5}  \\times 10^{31}\\,\\mbox{erg s}^{-1}$.\n The rotational energy to X-ray energy conversion factors obtained from this combined \n model are $L_x\/\\dot{E}= 48.4\\times 10^{-3}$ within $0.5-10$ keV and $69.8 \\times 10^{-3}$ \n if transformed to the ROSAT band.\n\n In computing the relative contributions of the thermal and non-thermal spectral components\n we find that for the best fitting parameters $\\sim 20\\%$ of the X-ray flux within the \n $0.1-2.4$ keV band could be of thermal origin and emitted from heated polar caps. In\n stretching the errors to the limits, however, a maximum of up to $\\sim 100\\%$ thermal \n flux can not be excluded for this energy band. At higher and lower energies non-thermal \n emission will dominate though. Figure \\ref{PSRB_bb_pl_model} illustrates the relative \n contributions of the thermal and non-thermal spectral components as indicated by the best \n fitting model parameters. \n\n Defining the size of a presumed polar cap as the foot points of the neutron star's dipolar \n magnetic field, the radius of the polar cap area is given by $\\rho=\\sqrt{2\\pi R^3\/c P}$ with \n $R$ being the neutron star radius, $c$ the velocity of light and P the pulsar rotation period \n (see e.g.~Michel 1991).  For PSR B0628$-$28\\, with a rotation period of 1.244 s this yields a polar cap \n radius of $\\rho\\sim 130$ m which is not too different from the size of the blackbody emitting \n area fitted in the composite blackbody plus power law model for an assumed distance of 1.45 kpc. \n\n The two component blackbody model would yield an alternative description of the spectrum as well, \n based on thermal emission mechanisms only. The parameters fitted by this model are a column \n absorption of $N_H \\le 0.9 \\times 10^{20}\\,\\mbox{cm}^{-2}$, blackbody temperatures of \n $\\mbox{kT}_1=0.142_{-0.01}^{+0.22}$ keV, $\\mbox{kT}_2=0.402_{-0.04}^{+0.09}$ keV, and radii of\n the projected emitting areas of $R_1=278.8_{-59.2}^{ +116.4}$\\,m, $R_2=34.8_{-11.33}^{+9.75}$\\,m.\n\n With a spin-down age of $\\sim 2.7\\times 10^6$ years PSR B0628$-$28\\, should\n still have some residual heat content from its birth event. Depending\n on the equation of state the surface temperature could be in the\n range $\\sim 1-3 \\times 10^5$ K (cf.~Becker \\& Pavlov 2002 and\n references therein) and then would contribute on a low level to the\n detected soft X-ray emission. Clearly, the power law spectral model\n fit does not require this extra thermal component by statistical\n means but to estimate the upper limit for the surface temperature of\n PSR B0628$-$28\\, we have added a blackbody component to the best fitting power\n law model.  We then calculated the confidence ranges of the blackbody\n normalization and temperature by leaving all other parameters\n free. The resulting contours, computed for two parameters of\n interest, are shown in Figure \\ref{PSRB_cooling_pl_contour}. For the\n thermal emission to be emitted from the whole surface of a neutron\n star of radius 10 km we find a $2\\sigma$ surface temperature upper\n limit of $T_s^\\infty < 5.3 \\times 10^5$ K which is somewhat above the\n temperatures predicted by cooling models (e.g.~Page \\& Applegate\n 1992; Yakovlev et al.~1999).\n\n The spectral parameters of the various models fitted to the energy spectrum \n of PSR B0628$-$28\\, are summarized in Table \\ref{spectral_fits_J0628}.\n\n\n\n\\subsection{Multi-wavelength Spectrum\\label{PSRB_multi_spec}}\n\n In order to construct a broadband spectrum combining all spectral\n information available from PSR B0628$-$28\\, we adopted the radio spectrum from\n Maron et al.~(2000) and plotted it in Figure\n \\ref{PSRB_broadband_spectrum} together with the optical V- and B-band\n upper limits from our ESO\/NTT observations and the XMM-Newton\n observed pulsar spectrum.  The radio spectrum has been measured up to\n 10.6~GHz and can be modeled by a simple power-law with an average\n photon index of $-2.9 \\pm 0.1$ (Maron et al. 2000), even though\n in contrast to measurements reported by Reyes et al.~(1995), the\n spectrum does show a turn-over at low frequencies (see Maron et\n al.~2000 and references therein). The flux density in the radio part of\n the spectrum, which is supposed to be due to coherent radiation, is\n several orders of magnitude greater than the extrapolated optical or\n X-ray flux densities.\n\n Extrapolating the power law spectrum which describes the XMM-Newton\n data to the optical V- and B-bands yields a photon flux which exceeds\n the measured upper limits by more than an order of magnitude. This\n suggest that the broadband spectrum, if entirely non-thermal, has to\n break somewhere before or in the soft channels of the X-ray\n spectrum. To test this hypothesis we have fitted a broken power law\n model to the XMM-Newton data and found it providing a better\n description ($\\chi^2$ = 27.576 for 37 dof) of the observed X-ray\n spectrum than the single power law model does. A broken power law\n model which is in agreement with the optical V- and B-band upper\n limits and the XMM-Newton data has its break point fitted at\n $E_{break}= 0.85^{+0.13}_{-0.16}\\,\\mbox{keV}$. The photon-index for\n $E < E_{break}$ and $E > E_{break}$ is found to be $\\alpha_1=\n 1.45^{+0.04}_{-0.003}$ and $\\alpha_2 =2.71^{+0.34}_{-0.26}$,\n respectively, with a normalization of $1.6 \\times10^{-5}$ photons\n cm$^{-2}$ s$^{-1}$ keV$^{-1}$ at 1 keV. The column absorption is\n compatible with an upper limit of $N_H=1.5\\times\n 10^{20}\\,\\mbox{cm}^{-2}$.  The errors represent the $1-\\sigma$\n confidence range calculated for two parameters of interest.\n\n\n \\section{DISCUSSION \\& SUMMARY\\label{discussion}}\n\n We have investigated the optical and X-ray emission properties of \n RX J0630.8$-$2834\\, and identified it to be the X-ray counterpart of the old \n rotation-driven pulsar PSR B0628$-$28\\, by detecting its X-ray pulses at the \n radio pulsar's spin frequency. Its X-ray pulse profile is characterized by \n a single broad peak and a second narrow pulse component which leads the \n main pulse by $\\sim 144^\\circ$.  A comparison of the pulse profiles\n observed in the $0.2-10$ keV band and at 1.4 GHz shows that the  \n X-ray pulse profile is markedly different from the single narrow\n peaked profile observed in the radio band. The XMM-Newton observed \n pulsed fraction is  $(39\\pm 6)\\%$ with no strong energy dependence  \n in the $0.2-10$ keV bandpass. The pulsar's X-ray spectrum is very well \n described by a power law spectrum, indicating that non-thermal radiation  \n processes dominate the pulsar's X-ray emission in the $0.2-10$ keV band. \n A $\\sim 20\\%$ thermal flux contribution from heated polar caps is inferred \n from the best fitting parameters of a combined power law plus Planckian \n spectrum.  \n\n The emission beam geometry of PSR B0628$-$28\\, is estimated from radio polarization \n data taken at 408 MHz and 1400 MHz. A formal best fit of the 1400 MHz data \n yields $\\alpha\\sim 11^\\circ$ for the inclination of the magnetic axis\n to the rotation axis and $\\beta\\sim -3^\\circ$ for the impact angle of the\n line of sight. A combination of results obtained from 408 MHz and 1400 MHz\n data, however, makes a nearly orthogonal solution with $\\alpha\\sim 70^\\circ$ \n and $\\beta\\sim -12^\\circ$ the most likely one. We would like to point out,\n however, that in the case the beam geometry would be $\\alpha\\sim 11^\\circ$,\n $\\beta\\sim -3^\\circ$ (which currently can not be ruled out) it would disclose \n us a full view of the pulsar polar cap.\n\n Comparing the pulsar's X-ray luminosity, $L_x(\\mbox{0.1-2.4 keV}) =\n 2.36_{-0.57}^{+1.25}\\times 10^{31} \\,{\\rm ergs\\,\\, s}^{-1}$, as inferred\n by the power law spectral model and estimated under the assumption of a \n pulsar distance of 1.45 kpc, with the pulsar's spin-down energy, $\\dot{E}= \n 1.445\\times 10^{32}\\,\\mbox{erg\/s}$, we find that PSR B0628$-$28\\, emits the huge \n amount of $\\sim 16\\%$ of its spin-down energy into the soft X-ray band. If\n confirmed, PSR B0628$-$28\\, would be the first X-ray over-luminous rotation\n powered pulsar identified among all $\\sim 1400$ radio pulsars known\n today. It is not known what could cause such a high X-ray luminosity. Among \n the most possible scenarios are extreme re-heating from vortex creep, \n accretion from the ISM, decay of the pulsar magnetic field, or extreme\n polar cap heating (cf.~Harding \\& Muslimov 2001,2002), though all this \n is not observed so far in any of the other X-ray detected rotation \n powered pulsars. \n\n Well known pulsars which, according to their timing parameters, fall\n into the same category as PSR B0628$-$28\\, are PSRs B1929+10, B0950+08,\n B0823+26 and J2043+2740. All four have been detected by ROSAT and\n ASCA at the limits of their sensitivity and have been investigated\n using XMM-Newton recently (Becker et al.~2004; 2005), but none of \n them shows evidence for the X-ray over-luminosity observed in PSR B0628$-$28. \n Thus, is PSR B0628$-$28\\, unique among all known rotation powered pulsars? At \n least its observed spectral and temporal emission\n properties do not support this view.  The results found for PSRs\n B1929+10, B0950+08, B0823+26 and J2043+2740 are all in line with the\n emission properties observed in PSR B0628$-$28. As for PSR B0628$-$28, the X-ray\n emission from these old pulsars seems to be  dominated by \n non-thermal radiation processes. None of the observed spectra \n {\\em required} adding a thermal component consisting of either a hot \n polar cap or surface cooling emission to model the data. The X-ray \n spectrum of PSR B0950+08 is best described by a single power law of \n photon-index $\\alpha=1.93^{+0.14}_{-0.12}$. Its X-ray emission is \n pulsed with two peaks per period and a phase separation of $\\sim 144^\\circ$ \n between the two pulse components. Its pulsed fraction is $(28 \\pm 6)\\%$ in\n the $0.2-10$ keV band. The spectral and temporal emission properties\n observed from PSR B1929+10 and PSR B0823+26 are very\n similar to those seen in PSR B0950+08 and PSR B0628$-$28 (Becker et al.~2004). \n Their energy spectra can be adequately described by a single power law with \n photon-index\n $\\alpha = 2.72^{+0.11}_{-0.09}$ and $\\alpha=2.5^{+0.9}_{-0.45}$,\n respectively. Fitting combined Planckian plus power law models\n to the observed spectral data from these pulsars, however, is possible\n as well and indicates thermal contributions from heated polar caps\n to the soft X-ray bands of the order of $\\sim 10-40\\%$ according to\n the best fitting model spectra. The fraction of pulsed photons for \n all these pulsars are in the range of $\\sim 30-50\\%$. \n \n The similarity in the emission properties to other X-ray detected\n pulsars from the same class does not support the assumption that\n there is anything special in the emission properties of PSR B0628$-$28\\, which\n justifies the high spin-down to X-ray energy conversion. The\n spin-down to X-ray energy conversion efficiency for the sample of\n ROSAT detected rotation-powered pulsars was investigated by Becker \\&\n Tr\\\"umper (1997) who described the available data with the relation\n $L_x\\sim 10^{-3}\\,\\dot{E}$. This relation is corrected for possible\n thermal contributions e.g.~in cooling neutron stars and was found to\n fit the data within the $0.1-2.4$ keV band assuming isotropy. Arguing\n that the $2-10$ keV energy band should be better suited for this\n study because any thermal contribution and spectral fitting\n uncertainties from interstellar absorption is minimized, Possenti et\n al.~(2002) performed a similar analysis using ROSAT, ASCA and\n BeppoSAX data of a somewhat larger sample than was available\n before. For faint sources like the old and nearby radio pulsars and\n most millisecond pulsars, Possenti et al.~(2002) obtained the flux in\n the $2-10$ keV band by simply upscaling the ROSAT results from Becker \\& \n Tr\\\"umper(1997). Those\n results, however, are not very meaningful, especially if there is no\n spectral information available and a simple counts-to-energy\n conversion requires the assumption of a spectral shape, as it was the\n case for the sample of old non-recycled pulsars.  Here, PSR B1929+10\n was taken as a prototype and X-ray fluxes were obtained by assuming\n that the X-ray emission from old pulsars originates entirely from \n heated polar caps.  Our new spectral results obtained for PSR B0628$-$28\\, \n and for the other old rotation-powered pulsars in recently XMM-Newton\n observations (cf.~Becker et al.~2004; 2005), however, shows that this\n assumption is no longer valid.\n\n Consequently, we do not follow their conclusions e.g.~of maximum \n X-ray emissivity but assume that the conversion efficiency $L_x\/ \\dot{E}\\,$ \n for PSR B0628$-$28\\, within $0.1-2.4$ keV is in the same range of \n $x=(0.5 - 5)\\times 10^{-3}$ than observed by Becker et al.~(2004; 2005) \n in PSR B1929+10 ($3.44\\times 10^{-3}$), B0950+08 ($2.85\\times 10^{-4}$), \n B0823+26 ($5.1\\times 10^{-4}$), and J2043+2740 ($5\\times 10^{-4}$) using \n XMM-Newton. We are therefore in favor of the interpretation that in\n $L_x=4 \\pi d^2_{DM} f_x$ the \n distance of 1.45 kpc, based on the radio dispersion measure and the Galactic \n free electron density model NE2001 (Cordes \\& Lazio 2002), is overestimated. \n\n If we take the errors in the X-ray flux measured from PSR B0628$-$28\\, into\n account, we obtain a pulsar distance $d(x)=\\sqrt{\\dot{E} x \/ 4 \\pi\n f_x}$ which is about 5 to 22 times closer than the radio dispersion\n measure inferred distance.  Table \\ref{distance_scales} summarizes\n the scaling parameters which were computed for the measured soft\n X-ray flux but with assumed spin-down conversion efficiencies.\n\n Distances based on the NE2001 Galactic free electron density model\n are estimated to be, on average, correct within a factor of $\\sim\n 20\\%$. However, as the Galactic free electron distribution is seen to\n vary significantly on small and large scales by more than two orders\n of magnitude it might not be excluded that for single pulsars the\n model can over- or underestimate the free electron density and thus\n yields a wrong distance estimate.  Although it is difficult to\n imagine that the model predicted distance is off by a factor 10 to\n 20, a factor of $\\sim 3-5$ might not be excluded in rare cases.\n\n For example, for a pulsar distance of 300pc the NE2001 model predicts\n a dispersion measure $DM=\\int_{0}^{300pc} n_e\\, dl$ of\n $4.25\\,\\mbox{pc cm}^{-3}$. If compared with the measured value of\n $34.36\\,\\mbox{pc cm}^{-3}$ this means that any extra unmodeled\n material along the line of sight to PSR B0628$-$28\\, would need to contribute\n a radio dispersion measure of $\\sim 30\\,\\mbox{pc cm}^{-3}$. This\n correction would not necessarily be due to an unmodeled H-II region\n (typically $DM\\sim 1000 \\,\\mbox{pc cm}^{-3}$) but would be only\n slightly higher than, but still comparable to the typical \"clump\"\n added in the NE2001 model for discrepant lines of sight, e.g.~towards\n PSR J1031-6117, PSR J1119-6127, and PSR J1128-6219 (Cordes \\& Lazio\n 2003).  Although these \"clumps\" in the NE2001 model do not\n necessarily represent physical objects, we did inspect the Digitized\n Sky Survey images and the images from our ESO\/NTT observations to see\n whether they show any evidence of enhanced diffuse emission near to\n the region of PSR B0628$-$28. No noticeable source was recognized that could\n account for this extra correction. Inspecting the Southern H-Alpha\n Sky Survey Atlas (Gaustad et al.~2001) revealed some indications on a\n larger scale of $\\sim 10^\\circ$ for a H-II region to the north of\n PSR B0628$-$28, but it is unlikely that this could affect the distance\n estimate by a factor of 5 to 10 or even more (J.Lazio priv.~com.).\n\n One way to evaluate a possible range of distances is by comparing the\n transverse velocity, derived from the measured proper motion and an\n assumed distance, to typical pulsar velocities. For instance, for the\n 1.45 kpc inferred from NE2001, the proper motion measurement of 48.7\n mas yr$^{-1}$ (Hobbs et al.~2004) translates into a transverse speed\n of 335 km s$^{-1}$ which compares to a mean 2-D speed as found for a\n large sample of pulsars of $246\\pm22$ km s$^{-1~}$ (Hobbs et\n al.~2005). For the larger distance of 2.15 kpc derived using the\n Taylor \\& Cordes (1993) model, the velocity increases to a rather\n large but still plausible value of 497 km s$^{-1}$. If the pulsar\n were a factor of 5 or 22 closer, than the transverse speed drops to\n 67 km s$^{-1}$ and 15 km s$^{-1}$, respectively. Unless the\n non-measurable radial speed happens to be large, the latter \n velocity is extremely small and quite unlikely.\n\n Although the distance of the pulsar might be the most obvious\n parameter which may lead to a too high X-ray conversion efficiency,\n there might be others. The conversion efficiency here and in Becker\n \\& Tr\\\"umper (1997) is for isotropy.  Would it be possible that the\n pulsar has an atypically small beaming angle?  or that it could be a\n coaction of several effects like beaming, distance, peculiarities of\n its magnetosphere which cause the apparent higher X-ray efficiency?\n\n So far PSR B0628$-$28\\, was not included in any radio parallax measurement\n program.  The pulsar's radio emission is strong enough so that a\n radio parallax could be measured with the VLBA. This will be done on\n a time scale of about two years and will provide us with the missing\n information on whether it is \"simply\" the pulsar distance which\n causes the inferred huge X-ray conversion or whether PSR B0628$-$28\\, bears\n more interesting things to discover.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nHuman intelligence exhibits \\emph{compositional generalization}, the ability to understand and produce unseen combinations of seen components \\cite{chomsky1965aspects,fodor1988connectionism}.\nFor example, if a person knows the meaning of ``run'' and ``run twice'', once she learns the meaning of a new word ``jump'', she can immediately understand the meaning of ``jump twice''.\nThis compositional generalization is essential in human cognition, enabling humans to learn semantics from very limited data and extend to unseen data \\cite{montague1974formal,partee1984compositionality,lake2017building}.\nTherefore, it is widely believed that compositional generalization is a key property towards human-like AI \\cite{mikolov2016roadmap,bengio2019towards,mollica2020composition}.\nHowever, in recent years, accumulating evidences have shown that neural sequence-to-sequence (seq2seq) models exhibit limited compositional generalization ability~\\cite{lake2018generalization,hupkes2020compositionality,keysers2020measuring,furrer2020compositional}.\n\n\n\nOn the other hand, semi-supervised learning \\cite{chapelle2009semi} is a machine learning paradigm that alleviates the limited-data problem through exploiting a large amount of unlabelled data, which has been successfully used in many different areas such as machine translation \\cite{sennrich2016improving,he2016dual,skorokhodov2018semi} and semantic parsing \\cite{yin2018structvae,cao2019semantic}.\nIn sequence-to-sequence (seq2seq) tasks, unlabelled data (i.e., monolingual data) are usually cheap and abundant, containing a large amount of combinations that are unseen in limited labelled data (i.e., parallel data).\nTherefore, we propose a hypothesis that \\emph{semi-supervised learning can enable seq2seq models understand and produce much more combinations beyond labelled data, thus tackling the bottleneck of lacking compositional generalization}.\nIf this hypothesis holds true, the lack of compositional generalization would no longer be a bottleneck of seq2seq models, as we can simply tackle it through exploiting large-scale monolingual data with diverse combinations.\n\nIn this work, we focus on \\emph{Iterative Back-Translation (IBT)} \\cite{hoang2018iterative}, a simple yet effective semi-supervised method that has been successfully applied in machine translation.\nThe key idea behind it is to iteratively augment original parallel training data with pseudo-parallel data generated from monolingual data.\nTo our best knowledge, iterative back-translation has not been studied extensively from the perspective of compositional generalization.\nThis is partially because a concern about \\emph{the quality of pseudo-parallel data}:\ndue to the problem of lacking compositional generalization, for non-parallel data with unseen combinations beyond the parallel training data, pseudo-parallel data generated from them will be error-prone.\nIt is natural to speculate that errors in pseudo-parallel data are going to be reinforced and then even harm the performance.\n\nThis paper broadens the understanding of iterative back-translation from the perspective of compositional generalization, through answering three research questions:\n\\textbf{RQ1. } How does iterative back-translation affect neural seq2seq models' ability to generalize to more combinations beyond parallel data?\n\\textbf{RQ2. } If iterative back-translation is useful from the perspective of compositional generalization, what is the key that contributes to its success?\n\\textbf{RQ3. } Is there a way to further improve the quality of pseudo-parallel data, thereby further improving the performance?\n\n\\textbf{Main Contributions.}\n(1) We empirically show that iterative back-translation substantially improves the performance on compositional generalization benchmarks (CFQ and SCAN) (Section \\ref{section:c1}).\n(2) To understand what contributes to its success, we carefully examine the performance gains and observe that iterative back-translation is effective to correct errors in pseudo-parallel data  (Section \\ref{section:c2}).\n(3) Motivated by this analysis, we propose \\emph{curriculum iterative back-translation} to further improve the quality of pseudo-parallel data, thereby improving the performance of iterative back-translation (Section \\ref{section:c3}).\n\n\\section{Background}\n\n\\subsection{Compositional Generalization Benchmarks}\n\\label{section:tasks}\n\nIn recent years, the compositional generalization ability of DNN models has become a hot research problem in artificial intelligence, especially in Natural Language Understanding (NLU), because compositional generalization ability has been recognized as a basic but essential capability of human intelligence~\\cite{lake2017building}.\nTwo benchmarks, SCAN \\cite{lake2018generalization} and CFQ \\cite{keysers2020measuring}, have been proposed for measuring the compositional generalization ability of different machine learning-based NLU systems.\nOur experiments are also conducted on these two benchmarks.\n\n\\subsubsection{SCAN}\nThe SCAN dataset consists of input natural language commands (e.g., ``\\emph{jump and look left twice}'') paired with output action sequences (e.g., ``\\emph{JUMP LTURN LOOK LTURN LOOK}'').\nDifficult tasks are proposed based on different data split settings, e.g.,\n(1)~\\textbf{ADD\\_JUMP}: the pairs of train and test are split in terms of the primitive \\emph{JUMP}.\nThe train set consists of \\emph{(jump, JUMP)} and all pairs without the primitive \\emph{JUMP}.\nThe rest forms the test set.\n(2)~\\textbf{LENGTH}: Pairs are split by action sequence length into train set ($\\leq 22$ tokens) and test set ($\\geq 24$ tokens). (3)~\\textbf{AROUND\\_RIGHT}: The phrase ``\\emph{around right}\" is held out from the train set, while ``around\" and ``right\" appear separately.\n(4)~\\textbf{OPPOSITE\\_RIGHT}: This task is similar to AROUND\\_RIGHT, while the held-out phrase is ``\\emph{opposite right}''.\n\n\\subsubsection{CFQ}\nThe \\emph{Complex Freebase Questions (CFQ)} benchmark \\cite{keysers2020measuring} contains input natural language questions paired with their output meaning representations (SPARQL queries against the Freebase knowledge graph).\nTo comprehensively measure a learner's compositional generalization ability, CFQ dataset is splitted into train and test sets based on two principles:\n(1) \\emph{Minimizing primitive divergence}: all primitives present in the test set are also present in the training set, and the distribution of primitives in the training set is as similar as possible to their distribution in the test set.\n(2) \\emph{Maximizing compound divergence}: the distribution of compounds (i.e., logical substructures in SPARQL queries) in the training set is as different as possible from the distribution in the test set.\nThe second principle guarantees that the task is compositionally challenging.\nFollowing these two principles, three different \\emph{maximum compound divergence (MCD)} dataset splits are constructed.\nThe mean accuracy of standard neural seq2seq models on the MCD splits is below 20\\%.\nMoreover, this benchmark also constructs 3 MCD data splits for SCAN dataset, and the mean accuracy of standard neural seq2seq models is about 4\\%.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth,clip=true]{pictures\/IBT\/bt.png}\n\\caption{Iterative back-translation: exploiting monolingual data to augment parallel training data.\nThe solid lines mean that the src2trg\/trg2src model generates pseudo-parallel data from monolingual data;\nthe dashed lines mean that parallel and pseudo-parallel data are used to train the models.}\n\\label{fig:ibt}\n\\end{figure}\n\n\\subsection{Iterative Back-Translation~(IBT)}\n\nIterative back-translation~\\cite{hoang2018iterative} has been proved as an effective method utilizing bi-directional monolingual data to improve the performance of machine translation models.\nAlgorithm~\\ref{alg:ibt} describes the training process of iterative back-translation.\nFirstly, the source-to-target (src2trg) model and target-to-source (trg2src) model are initially trained on the original parallel data ($D_p$) for $K$ steps (Line 1).\nAfter this initial stage, each training step samples a batch of monolingual data from one direction and back-translate them to the other direction.\nThen it would perform two mini-batches to train source-to-target model using the given parallel data and pseudo backtranslated parallel data separately, and also another two mini-batches for training target-to-source model.\nFigure~\\ref{fig:ibt} visualizes the iterative training process.\n\n\\begin{algorithm}[h]\n\\small\n\t\\caption{Iterative Back-Translation}\n\t\\textbf{Input:} parallel data $D_p$, monolingual data $D_{src}$, $D_{trg}$ \\\\\n\t\\textbf{Output:} source-to-target model $M_{\\rightarrow}$ and target-to-source model $M_{\\leftarrow}$\n\t\\begin{algorithmic}[1]\n\t\t\\State \\textbf{Initial Stage}: Separately train source-to-target model $M_{\\rightarrow}$ and target-to-source model $M_{\\leftarrow}$ on $D_{p}$ for K steps\n\t\t\\While{$M_{\\rightarrow}$ and $M_{\\leftarrow}$ have not converged  \\textbf{or} step $\\leq$ N}\n\t\t    \\State Sample a batch $B_{t}$ from $D_{trg}$ and use  $M_{\\leftarrow}$ to create $B_p=\\{(\\hat{s}, t) | t\\in B_t\\}$ \n\t\t    \\State Sample a batch $B_{s}$ from $D_{src}$ and use  $M_{\\rightarrow}$ to create $B_p'=\\{(s, \\hat{t}) | s\\in B_s\\}$\n\t\t \n\t\t \n\t\t \n\t\t    \\State Train source-to-target model $M_{\\rightarrow}$ on $D_{p}$\n\t\t    \\State Train source-to-target model $M_{\\rightarrow}$ on $B_p$\n\t\t    \\State Train target-to-source model $M_{\\leftarrow}$ on $D_{p}$\n\t\t    \\State Train target-to-source model $M_{\\leftarrow}$ on $B_{p}'$\n\t    \\EndWhile\n\t\\end{algorithmic}\n\\label{alg:ibt}\n\\end{algorithm}\n\n\\begin{table*}[htbp]\n\\small\n\\caption{Performance (accuracy) on SCAN Tasks. Cells with a white background are results obtained in previous papers; cells with a grey background are results obtained in this paper.}\n\\centering\n\\begin{tabular}{lccccccccc}\n\\toprule[1pt]\n Models  & ADD\\_JUMP & LENGTH  &AROUND\\_RIGHT& OPPOSITE\\_RIGHT & MCD1 & MCD2 & MCD3\\\\\n\\midrule\nLSTM+Att &$0.0\\pm0.0$&$14.1$ & $0.0\\pm0.0$ & $16.5\\pm6.4$ &$6.5\\pm3.0$ & $4.2\\pm1.4$& $1.4\\pm0.2$\\\\\nTransformer &$1.0\\pm0.6$& $0.0$& $53.3\\pm10.9$ & $3.0\\pm6.8$ & $0.4\\pm0.2$ & $1.6\\pm0.3$&$0.88\\pm0.4$\\\\\nUni-Transformer& $0.3\\pm0.3$ & $0.0$ & $47.0 \\pm10.0$ &15.2$\\pm$13.0& $0.5\\pm0.1$ & $1.5\\pm0.2$&$1.1\\pm0.4$\\\\\n\\midrule\nSyn-att&$91.0\\pm27.4$&$15.2\\pm0.7$&$28.9\\pm34.8$&$10.5\\pm8.8$&-&-&-\\\\\nCGPS&$98.8\\pm1.4$& $20.3\\pm1.1$ &$83.2\\pm13.2$ &$89.3\\pm15.5$ &$1.2\\pm1.0$ & $1.7\\pm2.0$& $0.6\\pm0.3$ \\\\\nEquivariant&$99.1\\pm0.0$&$15.9\\pm3.2$&$\\mathbf{92.0\\pm0.2}$&-&-&-&-\\\\\nGECA & $87.0$ &- & $82.0$&- &-&-&-\\\\\nMeta-seq2seq&$99.9$&$99.9$&$16.6$&-&-&-&-\\\\\nT5-11B&$98.3$&$3.3$&$49.2$&$\\mathbf{99.1}$ &$7.9$&$2.4$ &$16.8$ \\\\\n\\midrule[1pt]\n GRU+Att~(Ours) & \\cellcolor{lightgray}$0.6$ &\\cellcolor{lightgray}$14.3$&\\cellcolor{lightgray} $23.8$ & \\cellcolor{lightgray}$0.27$  &\\cellcolor{lightgray} $18.7$ &\\cellcolor{lightgray}$32.0$& \\cellcolor{lightgray}$42.4$ \\\\\n\\quad +mono30&\\cellcolor{lightgray}$\\mathbf{99.6}$&\\cellcolor{lightgray}$\\mathbf{77.7}$&\\cellcolor{lightgray}$37.8$&\\cellcolor{lightgray}$95.8$ &\\cellcolor{lightgray} $\\mathbf{64.3}$&\\cellcolor{lightgray}$\\mathbf{80.8}$&\\cellcolor{lightgray}$\\mathbf{52.2}$ \\\\\n\\midrule\n\\quad +mono100&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$99.9$&\\cellcolor{lightgray}$42.9$&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$87.1$&\\cellcolor{lightgray}$99.0$&\\cellcolor{lightgray}$64.7$\\\\\n\\quad +transductive&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$99.3$&\\cellcolor{lightgray}$34.5$&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$74.8$ &\\cellcolor{lightgray}$99.7$ &\\cellcolor{lightgray}$77.4$  \\\\\n\\bottomrule[1pt]\n\\end{tabular}\n\\label{tab:scan}\n\\end{table*}\n\n\\section{IBT for Compositional Generalization}\n\\label{section:c1}\n\nTo examine the effectiveness of IBT for compositional generalization, we conduct a series of experiments on two compositional generalization benchmarks (SCAN and CFQ, introduced in Section~\\ref{section:tasks}).\n\n\\subsection{Setup}\n\\label{sec:ibt_setup}\n\\subsubsection{Monolingual Data Settings}\nIn our experiments, we conduct different monolingual data settings as follows:\n\\begin{itemize}\n    \\item \\textbf{+transductive}:\n    \\emph{use all test data as monolingual data.}\n    We use this setting to explore how iterative back-translation performs with the most ideal monolingual data.\n    \\item \\textbf{+mono100}:\n    \\emph{use all dev data as monolingual data.}\n    In this setting, test data do not appear in the monolingual data, while the monolingual data contains much more unseen combinations beyond the parallel training data.\n    \\item \\textbf{+mono30}:\n    \\emph{randomly sample 30\\% source sequences and 30\\% target sequences from dev data.}\n    This setting is more realistic than ``+tranductive'' and ``+mono100'', because there is no implicit correspondence between source-side and target-side monolingual data (i.e., for a sequence in source-side monolingual data, the target-side monolingual data do not necessarily contain its corresponding target sequence, and vice versa).\n    Therefore, in our analysis of experimental results, we regard results of \\textbf{``+mono30\"} as the actual performance of iterative back-translation.\n\\end{itemize}\n\nNote that because there are no dev data in SCAN tasks, we randomly hold out half of the original test data as the dev data (the held-out dev data and the remaining test data are not overlapped).\n\n\\subsubsection{Implementation Details} We implement iterative back-translation based on the code of UNdreaMT\\footnote{https:\/\/github.com\/artetxem\/undreamt}~\\cite{artetxe2018iclr}. For CFQ, we use a 2-layer GRU encoder-decoder model equipped with attention. We set the size of both word embedding and hidden states to 300. We use a dropout layer with the rate of 0.5 and the training process lasts 30000 iterations with batch size 128. For SCAN, we also use a 2-layer GRU encoder-decoder model with attention. Both of the embedding size and hidden size are set to 200. We use a dropout layer with the rate of 0.5 and the training process lasts 35000 iterations with batch size 64. We set $K=5000$ steps in Algorithm~\\ref{alg:ibt} for both CFQ and SCAN benchmarks.\n\\subsubsection{Baselines}\n(i) \\textbf{General seq2seq models}:~LSTM, Transformer and Uni-Transformer~\\cite{keysers2020measuring}. (ii)~\\textbf{SOTA models on SCAN\/CFQ benchmarks}:  Syn-att~\\cite{russin2019compositional}, CGPS~\\cite{li2019compositional},  Equivariant~\\cite{gordon2019permutation}, GECA~\\cite{andreas-2020-good}, Meta-seq2seq~\\cite{lake2019compositional} and T5-11B~\\cite{furrer2020compositional}. \n\n\\begin{table}[tbp]\n\\small\n\\caption{Performance (accuracy) on CFQ tasks.}\n\\centering\n\\begin{tabular}{lcccc}\n\\toprule[1pt]\n Models & MCD1 & MCD2 & MCD3 \\\\\n\\midrule\nLSTM+Attn & $28.9\\pm1.8$ & $5.0\\pm0.8$ & $10.8\\pm0.6$  \\\\\nTransformer & $34.9\\pm1.1$ & $8.2\\pm0.3$ & $10.6\\pm1.1$ \\\\\nUni-Transformer & $37.4\\pm2.2$ & $8.1\\pm1.6$ & $11.3\\pm0.3$\\\\\nCGPS & $13.2\\pm3.9$ & $1.6\\pm0.8$ & $6.6\\pm0.6$\\\\\nT5-11B & $61.4\\pm4.8$&$30.1\\pm2.2$&$31.2\\pm5.7$\\\\\n\\midrule[1pt]\nGRU+Attn~(Ours) &\\cellcolor{lightgray}$32.6\\pm0.22$ &\\cellcolor{lightgray}$6.0\\pm0.25$ & \\cellcolor{lightgray}$9.5\\pm0.25$\\\\\n\\quad +mono30 & \\cellcolor{lightgray}$\\mathbf{64.8\\pm4.4}$&\\cellcolor{lightgray}$\\mathbf{57.8\\pm4.9}$&\\cellcolor{lightgray}$\\mathbf{64.6\\pm4.9}$\\\\\n\\midrule\n\\quad +mono100&\\cellcolor{lightgray}$83.2\\pm3.1$&\\cellcolor{lightgray}$71.5\\pm6.9$&\\cellcolor{lightgray}$81.3\\pm1.6$\\\\\n\\quad +transductive &\\cellcolor{lightgray}$88.4\\pm0.7$ &\\cellcolor{lightgray}$81.6\\pm6.5$ &\\cellcolor{lightgray}$88.2\\pm2.2$\\\\\n\\bottomrule[1pt]\n\\end{tabular}\n\\label{tab:cfq}\n\\end{table}\n\n\\subsection{{Observations}}\n\\begin{figure*}[t]\n\\small\n\\centering\n\\includegraphics[width=0.9\\textwidth]{pictures\/IBT\/accuracy.jpg}\n\\caption{Quality~(accuracy scores) of generated target data during the training procedure.}\n\\label{fig:accuracy_scores}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{pictures\/IBT\/bleu.jpg}\n\\caption{Quality~(BLEU scores) of generated source data during the training procedure.}\n\\label{fig:bleu_scores}\n\\end{figure*}\nWe report the experimental results in Table~\\ref{tab:scan} (SCAN) and~\\ref{tab:cfq} (CFQ).\nNotice that we pay more attention to MCD splits (on both SCAN and CFQ), since it has been proven that they are more comprehensive for evaluating compositional generalization than other splits \\cite{keysers2020measuring,furrer2020compositional}.\nOur observations are as follows:\n\n\\textit{1.Iterative Back-translation substantially improves the performance on compositional generalization benchmarks.}\nFirstly, IBT can always bring large performance gains to our baseline (GRU+Att).\nThen, compared to the previous state-of-the-art models, IBT can consistently achieve remarkable performance on all tasks, while those SOTA models only show impressive performances on ADD\\_JUMP \/ LENGTH \/ AROUND\\_RIGHT \/ OPPOSITE\\_RIGHT tasks of SCAN, but perform poorly on MCD tasks of SCAN\/CFQ.\nMoreover, IBT also achieves higher performance than T5-11B, which is a strong baseline that incorporates rich external knowledge during pre-training stage.\n\n\n\n\n\\textit{2. Better monolingual data, better results.} \nAs described in section~\\ref{sec:ibt_setup}, the quality of monolingual data are gradually improved from  ``+mono30\" to ``+mono100\" then to ``+transductive\". In most cases, it's as expected that ``+transductive\" performs better than ``+mono100\", and ``+mono100\" performs better than ``+mono30\".\n\n\n\n\\section{Secrets behind Iterative Back-Translation}\n\\label{section:c2}\n\nSection \\ref{section:c1} shows that iterative back-translation substantially improves seq2seq models' ability to generalize to more combinations beyond parallel data, but it is still unclear how and why iterative back-translation succeeds (RQ 2).\n\nTo answer this research question, we first empirically analyze pseudo-parallel data quality during the training process of iterative back-translation, and find that errors are increasingly corrected (Section \\ref{section:quality}).\nThen, we conduct ablation experiments to further illustrate this observation.\nWe find that:\n\\emph{even the error-prone pseudo-parallel data are beneficial.}\nWe speculate the reason is that knowledge of unseen combinations are implicitly injected into the model, thereby the src2trg model and the trg2src model can boost each other rather than harm each other (Section \\ref{section:static}).\nMoreover, pseudo-parallel data in iterative back-translation contain perturbations, helping models correct errors more effectively (Section \\ref{section:otf}).\nIn this section, all experiments are conducted on the CFQ benchmark, as it is much more complex and comprehensive than SCAN.\n\n\\begin{figure*}[t]\n\\small\n\\centering\t\n    \\begin{subfigure}{.35\\textwidth}\n    \t\\centering\n    \t\\includegraphics[width=\\textwidth]{pictures\/IBT\/acc_hist.pdf}\n    \t\\caption{Accuracy of Src2trg models}\n    \t\\label{fig:acc}\n    \\end{subfigure}\n    \\begin{subfigure}{.35\\textwidth}\n    \t\\centering\n    \t\\includegraphics[width=\\textwidth]{pictures\/IBT\/bleu_hist.pdf}\n    \t\\caption{BLEU of Trg2src models}\n    \t\\label{fig:bleu}\n    \\end{subfigure}\n\\caption{Ablation experiments for understanding the key factors contribute to the performance gain.\nThe comparison of baseline and BT indicates that even error-prone pseudo-parallel data are beneficial due to the injected implicit knowledge of unseen combinations (Section \\ref{section:static}).\nThe comparison of BT and BT+OTF indicates that perturbations brought by the on-the-fly mechanism can prevent models learning specific incorrect bias, thus improving the performance (Section \\ref{section:otf}).\nIBT performs best because it uses both source-side and target-side monolingual data while others use only target-side monolingual data.}\n\\label{fig:stanford_res}\n\\end{figure*}\n\n\\subsection{Quality of Pseudo-Parallel Data}\n\\label{section:quality}\n\nWe use Figure \\ref{fig:accuracy_scores} and \\ref{fig:bleu_scores} to get a better sense of the quality of pseudo-parallel data during the training process.\nThe x-axis represents the number of training steps: in the first 5000 steps, we train the src2trg model and the trg2src model with only parallel data;\nthe iterative back-translation process starts from step 5001 (dashed vertical lines).\nFor each step, we use the model to generate pseudo-parallel data from all monolingual data, and then evaluate the generated data quality.\nSpecifically:\nthe src2trg model generates a SPARQL query for each natural language utterance in the source-side monolingual data, and we define the quality of these generated SPARQL queries (i.e., target-side sequences) as the accuracy of them;\nthe trg2src model generates a natural language utterance for each SPARQL query in the target-side monolingual data, and we define the quality of these generated natural language utterances (i.e., source-side sequences) as the BLEU score~\\cite{papineni2002bleu} of them.\n\nWe observe that \\emph{iterative back-translation can increasingly correct errors in pseudo-parallel data.}\nEven though the pseudo-parallel data are error-prone at the end of initial stage~(dashed horizontal lines), the accuracy\/BLEU score has increased dramatically since the iterative back-translation process starts.\n\n\n\n\n\\subsection{Impact of Error-Prone Pseudo-Parallel Data}\n\\label{section:static}\n\nTo explore the reasons why iterative back-translation can increasingly correct error pseudo-parallel data, we investigate whether models can benefit from the initially generated pseudo-parallel data which are error-prone.\nIf yes, errors are going to be reduced during the iterative training process;\notherwise, errors are going to be reinforced.\n\nToward this end, we conduct two ablation experiments with ``+mono30\" monolingual data setting:\n\n\\begin{itemize}\n\\item \\textbf{BT-Src2trg}: the standard back-translation method for training the src2trg model (i.e., BT in Figure \\ref{fig:acc}).\nIt consists of three phases:\nfirst, train a src2trg model and a trg2src model using parallel data;\nthen, generate pseudo-parallel data from target-side monolingual data using the trg2src model;\nfinally, tune the src2trg model using the union of parallel data and pseudo-parallel data.\n\\item \\textbf{BT-Trg2src}: the standard back-translation method for training the trg2src model (i.e., BT in Figure \\ref{fig:bleu}).\nIt also consists of three phrases dual to those in BT-Src2trg.\n\\end{itemize}\n\n\n\n\n\n\n\nAccording to the performance of BT and baseline (i.e., src2trg\/trg2src models trained only on parallel data) in Figure \\ref{fig:stanford_res}, we observe that \\emph{even error-prone pseudo-parallel data can be beneficial.}\nSpecifically, though the initial pseudo-parallel data are error-prone (on average, 21.9\\% BLEU for Trg2src and 16.0\\% accuracy for Src2trg), they can still improve each other (5.4\\% BLEU and 4.0\\% accuracy gains for Trg2src and Src2trg models, respectively).\n\nA reasonable illustration for the observation is that: even though pseudo-parallel data are error-prone, it can still implicitly inject the knowledge of unseen combinations into the src2trg\/trg2src models, thereby boosting these models rather than harming them.\n\n\\subsection{Impact of Perturbations}\n\\label{section:otf}\n\n\nPrevious studies show that \\emph{perturbations (or noises)} in pseudo data play an important role in semi-supervised learning \\cite{zhang2016exploiting,edunov2018understanding,he2019revisiting,xie2020self}.\n\nIn iterative back-translation, the on-the-fly mechanism (i.e., models will be tuned each mini-batch on-the-fly with a batch of pseudo-parallel data) brings perturbations to pseudo-parallel data.\nDifferent batches of pseudo-parallel data are produced by different models at different steps, rather than the same model.\nThis makes errors more diverse, thus preventing the src2trg\/trg2src model learning specific incorrect bias from specific dual model.\nTherefore, it is reasonable to speculate that some performance gains are brought by such perturbations.\n\nWe conduct two ablation experiments (with ``+mono30\" monolingual data setting) to study the impact of such perturbations:\n\n\\begin{itemize}\n\\item \\textbf{BT-Src2trg-with-OTF}:\nthis method can be seen as iterative back-translation without source-side monolingual data (i.e., BT+OTF in Figure \\ref{fig:acc}).\nSpecifically, during the iterative training process, the trg2src model will only be tuned with parallel data.\nAs Figure \\ref{fig:bleu_scores} shows, the quality of pseudo-parallel data produced by the trg2src model would keep error-prone if it is only tuned on parallel data (the baseline curves in Figure \\ref{fig:bleu_scores}).\nWe want to see whether the src2trg model could be improved if the trg2src model keeps providing error-prone pseudo-parallel data.\n\\item \\textbf{BT-Trg2src-with-OTF}:\nthis method can be seen as iterative back-translation without target-side monolingual data (i.e., BT+OTF in Figure \\ref{fig:bleu}).\nWe want to see whether the trg2src model could be improved if the src2trg model keeps providing error-prone pseudo-parallel data.\n\\end{itemize}\n\n\n\n\n\nAccording to the performance of BT and BT+OTF in Figure \\ref{fig:stanford_res}, we find that perturbations brought by the on-the-fly mechanism do help models benefit from error-prone pseudo-parallel data more effectively.\nFor example, on average, the accuracy of BT-Src2trg-with-OTF is 33.3\\%, while the accuracy of BT-Src2trg is only 20.1\\%.\nAccording to our statistics, each sequence in target-side monolingual data has 49.3 different back-translations during the iterative training process.\nThe perturbations bring 13.3\\% performance gain, even if the quality of pseudo-parallel data has almost no improvement (see the baseline curves in Figure \\ref{fig:bleu_scores}).\n\nThis experimental results support our hypothesis that \\emph{perturbations brought by the on-the-fly mechanism contributes a lot to the success of iterative back-translation in compositional generalization benchmarks}.\n\n\\section{Curriculum Iterative Back-Translation}\n\\label{section:c3}\n\nAs discussed in Section \\ref{section:c2}, the major reason for why iterative back-translation succeeds is that it can increasingly correct errors in pseudo-parallel data.\nBased on this observation, it is reasonable to speculate that iterative back-translation can be further improved through injecting inductive bias that can help errors be reduced more efficiently.\nTowards this end, we consider curriculum learning~\\cite{bengio2009curriculum} as a potential solution: start out iterative back-translation with easy monolingual data (of which the generated pseudo-parallel data are less error-prone), then gradually increase the difficulty of monolingual data during the training process.\nBased on this intuition, we propose \\emph{Curriculum Iterative Back-Translation (CIBT)}.\n\n\\begin{table*}[t]\n\\small\n    \\centering\n    \\caption{Performance~(accuracy) of curriculum iterative back-translation.}\n\t\\begin{tabular}{lcccccc}\n\t    \\hline\n    \t\\multirow{2}{*}{} & \\multirow{2}{*}{IBT} & \\multicolumn{5}{c}{CIBT with hyperparameter $c$ (steps in each stage)} \\\\ \\cline{3-7}\n    \t &  & 2000 & 2500 & 3000 & 3500 & 4000 \\\\ \\cline{1-1} \\cline{2-2} \\cline{3-7}\n         MCD1 & $  64.8 \\pm 4.4 $ & $  66.1 \\pm 5.0 $ & $  66.0 \\pm 4.8 $ & $  \\textbf{66.6} \\pm 5.4 $ & $  65.9 \\pm 3.7 $ & $  65.4 \\pm 3.8 $ \\\\ \n        MCD2 & $  57.8 \\pm 4.9 $ & $  68.6 \\pm 2.6 $ & $  \\textbf{69.1} \\pm 3.1 $ & $  68.0 \\pm 1.9 $ & $  66.8 \\pm 2.4 $ & $  65.4 \\pm 3.1 $ \\\\\n        MCD3 & $  64.6 \\pm 4.9 $ & $  70.2 \\pm 4.9 $ & $  68.4 \\pm 7.0 $ & $  \\textbf{70.4} \\pm 4.8 $ & $  69.2 \\pm 4.1 $ & $  67.0 \\pm 6.3 $ \\\\ \n        Mean & $  62.4 \\pm 6.1 $ & $  \\textbf{68.3} \\pm 4.1 $ & $  67.8 \\pm 4.7 $ & $  \\textbf{68.3} \\pm 4.1 $ & $  67.3 \\pm 3.4 $ & $  65.9 \\pm 4.1 $ \\\\ \\hline\n    \\end{tabular}\n\t\\label{cl_results_summary}\n\\end{table*}\n\n\\subsection{Method}\n\n\\subsubsection{Problem Formulation}\n\nWe denote source-side and target-side monolingual data in iterative back-translation as $D_{src}$ and $D_{trg}$, respectively.\nSuppose that we have a simple heuristic algorithm $\\mathcal{C}$, which can divide $D_{src}$ into $n$ parts (denoted as $D_{src}^{(1)}$, $D_{src}^{(2)}$, ..., $D_{src}^{(n)}$), and $D_{trg}$ can also be divided into $n$ parts (denoted as $D_{trg}^{(1)}$, $D_{trg}^{(2)}$, ..., $D_{trg}^{(n)}$).\nThey are sorted by difficulty in ascending order:\nthe initial src2trg model performs roughly better on $D_{src}^{(i)}$ than on $D_{src}^{(j)}$ if $i<j$;\nand the initial trg2src model performs roughly better on $D_{trg}^{(i)}$ than on $D_{trg}^{(j)}$ if $i<j$.\nWe define that $(D_{src}^{(1)}, D_{trg}^{(1)}), (D_{src}^{(2)}, D_{trg}^{(2)}), ..., (D_{src}^{(n)}, D_{trg}^{(n)})$ are $n$ curriculums from simple to difficult.\nOur goal is to improve the performance of iterative back-translation with this additional curriculum setting.\n\n\\subsubsection{Method Details}\n\nAlgorithm \\ref{cl alg} details the workflow of curriculum iterative back-translation.\n\n\\begin{algorithm}[!h]\n\\small\n\t\\caption{Curriculum Iterative Back-Translation}\n\t\\label{cl alg}\n\t\\textbf{Input:} $M_{\\rightarrow}$ and $M_{\\leftarrow}$, the src2trg and trg2src models that have been initially trained on parallel data; $D_{src}$ and $D_{trg}$, source-side and target-side monolingual data; $\\mathcal{C}$, curriculum segmentation algorithm; $c$, the number of steps each stage should be trained; $n$, number of curriculums.\n\t\\begin{algorithmic}[1]\n\t\t\\State Use $\\mathcal{C}$ to split $D_{src}$ into $D^{(1)}_{src}, D^{(2)}_{src}, ..., D^{(n)}_{src}$\n\t\t\\State Use $\\mathcal{C}$ to split $D_{trg}$ into $D^{(1)}_{trg}, D^{(2)}_{trg}, ..., D^{(n)}_{trg}$\n\t\t\\State $D'_{src}=\\emptyset, D'_{trg}=\\emptyset$\n\t\t\\For{$t= 1, 2, ..., n$} \\algorithmiccomment{$n$ stages}\n\t\t\\State $D'_{src} = D'_{src} \\cup D^{(t)}_{src}$\n\t\t\\State $D'_{trg} = D'_{trg} \\cup D^{(t)}_{trg}$\n\t\t\\State Iterative back-translation ($c$ steps) to update $M_{\\rightarrow}$ and $M_{\\leftarrow}$, with $D'_{src}$ and $D'_{trg}$ as monolingual data\n\t\t\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Experimental Setup}\nWe evaluate the effectiveness of curriculum iterative back-translation on the CFQ ``+mono30\" setting.\n\n\\subsubsection{Curriculum Segmentation.}\nDefining the curriculum segmentation algorithm $\\mathcal{C}$ is critical in our method.\nIn previous research work on curriculum learning, many metrics for curriculum segmentation have been proposed, such as length, frequency, and perplexity.\n\nIn curriculum iterative back-translation, $\\mathcal{C}$ is used to split both $D_{src}$ and $D_{trg}$.\nWe denote that $\\mathcal{C}_i(s)$ is an indicator that takes the value 1 if and only if $\\mathcal{C}$ put the sample $s$ into the $i$-th curriculum.\nIt is a straightforward assumption that: for each $i = 1, 2, ..., n$, the performance of curriculum iterative back-translation would be better if $\\mathcal{C}_i(x)=\\mathcal{C}_i(y)$ for any source-side sequence $x$ paired with its ground truth target-side sequence $y$.\nIntuitively, we want to ensure that $x$ and $y$ are in the same curriculum.\n\nBased on this principle, we define $\\mathcal{C}$ based on the number of entities occurring in each source\/target sample.\nFor example, consider the source sequence \\textit{``Were M1 and M3 distributed by M0's employer and distributed by M2?\"}.\nIt contains 4 entities (M0, M1, M2, and M3), and its corresponding target sequence (a SPARQL query) also contains these 4 entities.\nTherefore, we put them into the same curriculum.\nSpecifically, we define 6 curriculums, which contain monolingual data with $(\\leq 1) \/ 2 \/ 3 \/ 4 \/ 5 \/ (\\geq 6)$ entities.\n\n\nFigure \\ref{baseline_acc_in_each_curriculum} shows that: in this entity-based curriculum setting, curriculum 1-6 are indeed arranged in order from simple to difficult.\nSpecifically, we evaluate the performance of the baseline model (i.e., the seq2seq model trained on parallel data) on $D_{src}^{(1)}, D_{src}^{(2)}, ..., D_{src}^{(6)}$, respectively.\nWe observe that the baseline model performs better on curriculum 1-2 than on curriculum 3-6, which indicates that this entity-based curriculum setting is reasonable.\n\n\\subsubsection{Hyperparameters}\nWe evaluate the performance of curriculum iterative back-translation with varying $c$ (the number of steps each stage should be trained): 2000\/2500\/3000\/3500\/4000.\nTo be fair, after $n\\times c$ steps, we continue to train on complete monolingual data (i.e., $D_{src}$ and $D_{trg}$) for $(25,000-n\\times c)$ steps, thus keeping the number of overall training steps the same.\n\n\\subsection{Results and Analysis}\n\\subsubsection{Curriculum learning benefits iterative back-translation.}\n\n\\begin{figure}[t]\n\\centering\n\\small\n\\includegraphics[width=0.9\\columnwidth,clip=true]{pictures\/CL\/baseline_accuracy_in_each_curriculum.pdf}\n\\caption{Performance of baseline model in each curriculum.}\n\\label{baseline_acc_in_each_curriculum}\n\\end{figure}\n\n\\begin{figure}[t]\n\\small\n\\centering\n\\includegraphics[width=0.7\\columnwidth,clip=true]{pictures\/CL\/performance_on_first_k_curriculums.pdf}\n\\caption{Performance on different subsets. This figure indicates that curriculum learning is more beneficial to difficult data (larger $k$) than simple data (smaller $k$).}\n\\label{influence_of_each_curriculum}\n\\end{figure}\n\nTable \\ref{cl_results_summary} shows the main results. CIBT consistently outperforms IBT in all three data splits (MCD1\/MCD2\/MCD3) and all hyperparameter settings ($c$ =$2000 \/ 2500 \/ 3000 \/ 3500 \/ 4000$). For MCD1\/MCD2\/MCD3 tasks of CFQ, CIBT outperforms IBT by 1.8\\%\/11.3\\%\/5.8\\% accuracy scores.\nAn interesting observation is that incorporating curriculum learning into iterative back-translation brings more stable performance:\nIBT performs much worse on MCD2 (57.8\\%) than on MCD1\/MCD3 (64.8\\%\/64.6\\%), while CIBT's improvement on MCD2 (11.3\\%) is much better than on MCD1\/MCD3 (1.8\\%\/5.8\\%), making the performances on MCD1\/MCD2\/MCD3 comparable (66.6\\%\/69.1\\%\/70.4\\%).\n\n\\subsubsection{Curriculum learning is more beneficial to difficult data than simple data.}\nFigure \\ref{influence_of_each_curriculum} further illustrates how curriculum learning improves the performance.\nWe use $k$ to denote test cases that would be put into the first $k$ curriculums, then obtain 6 subsets of test data: $T_1$, $T_2$, ..., $T_6$.\nWe have $T_1\\subset T_2\\subset ... \\subset T_6$.\nAs we can observe in Figure \\ref{influence_of_each_curriculum}, on the simplest subset ($T_1$), the performances of IBT and CIBT are very close.\nIf we consider $T_2$, a subset which is a bit more difficult than $T_1$, CIBT performs slightly better than IBT.\nSimilarly, we observe that: more difficult a subset of test data is, more gains CIBT brings.\nTherefore, we can conclude that curriculum learning is more beneficial to difficult data than simple data.\n\n\n\n\\section{Related Work}\n\n\\subsection{Compositional Generalization}\n\\label{sec:related_cg}\nRecently, exploring compositional generalization of DNN models has attracted a large attention on different topics, e.g., visual-question answering \\cite{hudson2018compositional}, algebraic compositionality \\cite{veldhoen2016diagnostic,saxton2018analysing}, and logic inference \\cite{bowman2015tree,mul2019siamese}.\nIn this work, we focus on compositional generalization in language, which is benchmarked by SCAN \\cite{lake2018generalization} and CFQ \\cite{keysers2020measuring} tasks.\n\nWhile mainstream DNN models are proved to exhibit limited compositional generalization ability in language \\cite{lake2018generalization,hupkes2020compositionality,keysers2020measuring,furrer2020compositional}, there have been many efforts to address this limitation, including\ndesign specialized architectures~\\cite{russin2019compositional,li2019compositional,gordon2019permutation,guo2020hierarchical,liu2020compositional}, data augmentation~\\cite{jia-liang-2016-data,andreas-2020-good}, meta-learning~\\cite{lake2019compositional,nye2020learning} and pre-training models~\\cite{furrer2020compositional}.\nComparing to these previous work, this paper incorporates semi-supervised learning into compositional generalization, which is task-agnostic and model-agnostic, thus improving the performance on both SCAN and CFQ benchmarks.\n\n\\subsection{Iterative Back-translation}\nThe idea of iterative back-translation dates back at least to \\emph{back-translation}, which simply augments parallel training data using pseudo-parallel data generated from only target-side monolingual data.\nBack-translation has been proven effective in statistical machine translation \\cite{goutte2009learning,bojar2011improving} and neural machine translation \\cite{sennrich2016improving,edunov2018understanding,imamura2018enhancement,xia2019generalized,wang2019improving}.\nIterative back-translation \\cite{hoang2018iterative,cotterell2018explaining,lample2018unsupervised,conneau2019cross} is an extension of back-translation, in which forward and backward direction models generate pseudo-parallel data for each other and improve each other, bringing stronger empirical performance.\n\nTo our best knowledge, this paper is the first to study iterative back-translation from the perspective of compositional generalization.\nWe demonstrate that iterative back-translation can help seq2seq models make better generalizations to much more combinations, thus broadening the understanding of iterative back-translation.\n\n\\section{Conclusion}\nIn this paper, we revisit iterative back-translation from the perspective of compositional generalization.\nWe find that iterative back-translation is an effective method that exploits a large amount of monolingual data to enable seq2seq models better generalize to more combinations beyond limited parallel data.\nWe also find that iterative back-translation has the ability to increasingly correct errors in pseudo-parallel data, thus contributing to its success.\nTo encourage this behaviour, we propose curriculum iterative back-translation, which explicitly improves the quality of pseudo-parallel data, thus further improving the performance. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{Sec:Intro}\n\\par Wireless communication has become an increasingly important part of human existence, a reality re-emphasized by the recent pandemic \\cite{saeed2020wireless}. The ubiquitous presence of wireless signals brings about significant advantages in terms of connectivity for communication and environmental awareness. However, this comes at the potential cost of users' data privacy and security \\cite{furqan2021wireless}. In the current work, we focus on this problem in the context of cooperative communication in the presence of untrusted relays.\n\n\\par \\Ac{MIMO} is considered to be an essential enabler for \\ac{5G} (and even Wi-Fi) and beyond wireless networks \\cite{IEEE_specturm}. However, services such as \\ac{mMTC} and \\ac{IoT} cannot always ensure the availability of hardware capabilities that can support \\ac{MIMO} operation. This led to the emergence of \\textit{cooperative communication} paradigm around the turn of the century, where the aim is to improve the \\ac{QoS} of the users through cooperation. Cooperative communication, while generally applicable to different network realizations, is more suited to \\textit{ad hoc} and \\acp{WSN} compared to the cellular systems \\cite{nosratinia2004cooperative}. Unlike the latter, the former two systems lack a sophisticated authentication mechanism. This may lead to violation of a user's data confidentiality by the very nodes that are expected to improve the communication via spatial diversity. Accordingly, there exists a plethora of approaches for \\ac{PLS} in cooperative communication, which can be broadly categorized into \\textit{cooperative relaying}, \\textit{cooperative jamming} and their hybrid combinations. Both these approaches usually comprise of two phases, where the first one is broadcast of information from the source. In the second step, the appropriate relay forwards the message to the destination. In the case of relaying, the selection of the helper (relay) is done to degrade the eavesdropper's interception capability. In the case of jamming, the source itself, the destination, or any other helper node transmits noise-like signals such that the eavesdropper cannot hear the message \\cite{pahuja2019cooperative}.\n\n\\par Despite the clear advantage in terms of degraded eavesdropping capability \\cite{jameel2018comprehensive}, the cooperative jamming techniques have various limitations. For instance, in the case of friendly jamming the presence of a trustable node is necessary \\cite{zhang2012physical}. Furthermore, the jamming signal from this helper can also degrade the destination's performance. Destination-based jamming techniques resolve this issue by exploiting their prior knowledge of the jamming signal. However, they require full-duplexing capabilities to achieve spatial diversity \\cite{zhao2018secrecy}. In both cases, jamming signals are to be transmitted throughout the first phase (where the source broadcasts the signal) of cooperative communication, which is unsuitable for power-limited devices. To this end, we propose the design of a jamming signal which is only transmitted for part of the broadcast phase, yet its effect is spread throughout the signal duration by exploiting the properties of \\ac{OFDM} signals and \\ac{FFT} operation. Since the jamming signal is only transmitted during the \\ac{CP} part at the destination, it also eliminates the need for full-duplexing capability. \n\n\\par The rest of this article is organized as follows. Section \\ref{Sec:SysModel} describes the general system model for \\ac{AAF} cooperative relaying. The proposed approach for \\ac{CP} jamming is discussed in Section \\ref{sec:Proposed}. Simulation results and related discussions are covered in \\ref{sec:Results} while Section \\ref{sec:Conclusion} concludes the work and provides some future directions. \n\n\\section{System Model}\n\\label{Sec:SysModel}\n\\par Without loss of generality, we assume a dual-hop half-duplex relay-aided system with \\ac{AAF} protocol \\cite{liu2009cooperative}. As shown in Fig. \\ref{fig:SysModel}, it consists of a source that wants to communicate with a destination in the presence of an untrusted relay, where each node consists of a single antenna. Here, the goal is to use the advantages provided by the relay without letting it decode any information. The system is based on \\ac{OFDM} modulation with $N$ subcarriers. The channels corresponding to source-destination ($H_{sd}$), source-relay ($H_{sr}$), and relay-destination ($H_{rd}$) links are assumed to be slowly varying Rayleigh fading with $L$ exponentially decaying taps. A \\ac{CP} of length $L$ is introduced to convert the multipath frequency channel into flat fading subchannels. \n\n\\par At the transmitter, the frequency domain \\ac{OFDM} symbols $\\mathbf{{S}}=\\begin{bmatrix} S_{0} & S_{1} &...& S_{N-1}\\end{bmatrix}^{\\textrm T} \\in \\mathbb{C}^{[N \\times 1]}$ are passed through the \\ac{IFFT} process as $s[n]=\\sum_{k=0}^{N-1} S_{k} \\exp^{\\frac{j 2 \\pi nk}{N}}$ and a \\ac{CP} of length $L$ is added. The transmitted signal with \\ac{CP} can be written as \n$\\mathbf{{s}}=[s[N-L+1], \\dots, s[N-1], s[0], s[1], \\dots, s[N-L], s[N-L+1], \\dots,  s[N-1] ]$. Finally, the source node broadcasts the information to relay and destination. The received signal at the destination in the frequency domain on $k$-th subcarrier can be represented by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_sd}\nY_{sd}(k) &= \\sqrt{P_1} H_{sd}(k) S(k) + V_{sd}(k) ,\\\\k &= 1,\\dots, K;~n = 1, \\dots, N,\n\\end{aligned}\n\\end{equation}\nwhere $H_{sd}(k)$ is the \\ac{CFR} of the $k$-th subcarrier between source and destination, the transmitted power by the source is represented by $P_{1}$, $S(k)$ is the symbol transmitted by source node on $k$-th subcarrier, and $V_{sd}(k)$ represents \\ac{AWGN} on the $k$-th subcarrier with variance $N_{0}\/2$. Similarly, the received signal at relay is given by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_sr}\nY_{sr}(k) &= \\sqrt{P_1} H_{sr}(k) S(k) + V_{sr}(k),\n\\end{aligned}\n\\end{equation}\nwhere $H_{sr}(k)$ is the \\ac{CFR} of the $k_{th}$ subcarrier for source-relay and $V_{sr}(k)$ represents the \\ac{AWGN}. The relay node will normalize the received signal before retransmission by the factor  \n\\begin{equation}\n\\beta (k)=\\sqrt \\frac{1}{P_1 H_{sr}(k)+ N_{0}}.\n\\end{equation}\n\nFinally, the received signal at destination from the relay can be given as \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_rd}\nY_{rd}(k) &= \\sqrt{P_2} H_{rd}(k) \\Big(\\beta (k) Y_{sr}(k))\\Big) + V_{rd}(k), \n\\end{aligned}\n\\end{equation}\nwhere $H_{rd}(k)$ is the frequency domain channel attenuation on the $k$-th subcarrier between relay and destination, the transmitted power is represented by $P_{2}$, and  $V_{rd}(k)$ shows  \\ac{AWGN} on the $k$-th subcarrier. The receiver will finally combine $Y_{rd}(k)$ signal with $Y_{sd}(k)$ given in (\\ref{eq:Y_sd}) by applying \\ac{MRC} and decode the information \\cite{seo2006maximum}.\n\n\\section{Proposed Approach}\n\\label{sec:Proposed}\n\\begin{figure}[t!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.9\\columnwidth]{Figures\/SysModelv3.pdf}\\label{fig:SysModel}} \n    \\subfigure[]{\\includegraphics[width=0.9\\columnwidth]{Figures\/Proposedv3.pdf}\\label{fig:Proposed}} \n    \\caption{Illustration of the proposed approach in a relaying scenario. (a) A typical dual-hop relay aided system comprising of a single source, relay and destination each. (b) The user signal is transmitted from source ($S$) at time $t_0$, reaching the relay ($R$) and destination ($D$) at times $t_1$ and $t_2$, respectively, while the jamming signal is transmitted by $D$ at $t_2$ and reaches $R$ at $t_3$.}\n    \\label{fig:eqqAPP}\n\\end{figure}\n\n\\par This section presents the details of the proposed algorithm. The basic idea here is to devise a method that enables us to utilize the advantages of the untrusted relay while keeping the information secure from being eavesdropped on by the same node. Moreover, we want to achieve this goal without using an external helper or full-duplex receiver. It should be noted that in the case of half-duplex destination, diversity gain is nullified since the destination transmits jamming signal throughout the first phase rendering it incapable of receiving the broadcast copy of the signal. The proposed approach, on the other hand, provides the required spatial diversity obtained from the signals received in both phases while ensuring that the relay is unable to intercept the data signal. This is achieved by the transmission of jamming signal over the \\ac{CP} duration of the received \\ac{OFDM} signal in the first phase.\n\nThe proposed algorithm has two phases.\n\\begin{itemize}\n    \\item \\textbf{Phase-1}: Source transmits a communication signal while destination sends a jamming signal during a portion of signal duration.\n    \\item \\textbf{Phase-2}: The relay forwards a signal that it receives during phase-1 to the destination for final combining and decoding at the destination. \n\\end{itemize}\n\\noindent The following text provides details of the aforementioned phases of the proposed approach.\n\\subsection{Phase-1}\n\n\\par In phase-1, the source node broadcasts an \\ac{OFDM} signal to destination and untrusted relay over multi-path Rayleigh fading channel. The received signals at destination and relay are given in (\\ref{eq:Y_sd}) and (\\ref{eq:Y_sr}), respectively. The relay forwards the received signals using \\ac{AAF} protocol to the destination in phase-2 to enhance the reliability of communication. However, there is a possibility that the relay may try to intercept and decode the data signal for malicious purposes. To ensure reliable communication while ensuring security from the untrusted relay, we propose that the destination transmits the jamming signal over the \\ac{CP} duration of the received signal right before receiving the \\ac{OFDM} signal. Even though the jamming signal overlaps with only a portion of the time-domain transmitted signal, its effect spreads over all the data-containing subcarriers once the \\ac{FFT} process is applied at the relay. This received signal at the relay, containing the jamming signal arriving from the destination is given by\n\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:y_sr_hat}\n\\hat{Y}_{sr}(k) &= \\sqrt{P_1} H_{sr}(k) S(k) + V_{sr}(k) \\\\&+\\sqrt{P_j} H_{rd}(k) J(k)+ V_{rd}(k),\n\\end{aligned}\n\\end{equation}\nwhere $J(k)$ is the jamming signal and the power of the jamming signal transmitted by the destination is represented by $P_{j}$. It should be noted that (\\ref{eq:y_sr_hat}) contains the additional jamming term compared to (\\ref{eq:Y_sr}), hence the term $\\hat{Y}_{sr}(k)$ is used instead of ${Y}_{sr}(k)$.\n\n\\par Figure \\ref {fig:Proposed} provides a temporal illustration of phase-1 and how the signals arrive at both destination and relay. The transmission from the source initiates at time $t_0$. The signal arrives at relay and destination nodes at times times $t_1$ and $t_2$, respectively. The destination then transmits the jamming signal, equal in length to the \\ac{CP} itself and represented by a checkered block in the figure, immediately upon receiving its signal. The jamming signal reaches the relay at $t_3$. The starting point of the jammed portion of the relay's signal in time can be calculated as\n\\begin{equation}\nt_d = t_3 - t_1 = \\frac{d_{sd}+d_{rd}-d_{sr}}{c},\n\\end{equation}\nwhere $d_{sd}$ indicates the distance between source and destination, $d_{sr}$ is the distance between source and relay, and $d_{rd}$ is the distance between relay and destination. As mentioned earlier, as long as part of the jamming signal overlaps with the data portion at the relay, its decoding capability is degraded. The overlap itself is ensured by the propagation delay between the transceivers. \n\n\\subsection{Phase-2}\n\\par In phase-2, the relay transmits the signal to the destination after employing \\ac{AAF} protocol. The received signal at the destination from the relay without jamming is given by (\\ref{eq:Y_rd}). Replacing ${Y}_{sr}(k)$ with $\\hat{Y}_{sr}(k)$ in (\\ref{eq:Y_rd}) and expanding it gives us\n\\begin{equation}\n\\begin{aligned}\n\\hat{Y}_{rd}(k) &= \\sqrt{P_2} H_{rd}(k)\\big(\\beta(k) \\sqrt{P_1} H_{sr}(k) S(k)\\\\& +  V_{sr}(k)+\\sqrt{P_j} H_{rd}(k) J(k)+ V_{rd}(k)\\big).\\\\ \n\\end{aligned}\n\\end{equation}\n\n\\par The receiver will first remove the jamming signal from $\\hat{Y}_{rd}(k)$. Since the jamming signal is known at the destination and \\ac{CP} is discarded in the receiver anyway, the jamming signal does not affect the signal of the legitimate receiver. However, some reflected components of the jamming signal may leak into the actual data and cause a self-interference. In this study, we assume that the receiver is able to cancel that interference by estimating the channel around itself. Afterwards, it employs \\ac{MRC} \\cite{seo2006maximum} to combine the resultant signal with the $Y_{sd}(k)$ that it received during phase-1.  \n\n\\par Here it is important to highlight that unlike conventional \\ac{PLS} techniques \\cite{hamamreh2018classifications}, the proposed method does not require channel state information at the transmitter \\cite{ankarali2016channel}. This renders the proposed approach suitable for scenarios where the transmitter\/source node has limited capabilities in terms of computation, channel estimation, etc. \n\n\n\\begin{table}[t!]\n\\centering \\renewcommand{\\arraystretch}{1.25}\n\\caption{Simulation parameters and assumptions} \\label{tab:sim_parameters}\n\\resizebox{0.9\\columnwidth}{!}{%\n\\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Parameter}      & \\textbf{Value}                \\\\ \\hline\nSimulation environment             & Urban macro                   \\\\ \\hline\nCarrier frequency $(f_c)$     & 2 GHz                         \\\\ \\hline\nShadow fading standard deviation $(\\sigma)$        & 4 dB               \\\\ \\hline\nSource\/relay transmit power ($P_{Tx}$)  & 23 dBm                        \\\\ \\hline\nNoise power density $(N_0)$     & -174 dBm\/Hz                    \\\\ \\hline\nSource-destination distance     & 1000 m \n\\\\ \\hline\nSource-relay distance     & (100, 200, ... , 900) m \n\\\\ \\hline\nCIR length ($L$)      & 32 \n\\\\ \\hline\nModulation      & QPSK\n\\\\ \\hline\nFFT Size ($N$)      & 256\n\\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\section{Simulation Results and Discussion}\n\\label{sec:Results}\n\\par As mentioned earlier, this work considers the presence of a single antenna source, destination, and relay nodes in an urban macro environment. As the ratio between received data and jamming signals depends heavily on the position of the relay, the following pathloss model is used \\cite{3GPP_38_901}:\n\\begin{equation}\n\\label{pathloss}\n    PL = 22\\log_{10}(d)+20\\log_{10}(f_c)+28+\\sigma,\n\\end{equation}\nwhere $d$ is the distance between the transceivers in meters, $f_c$ is the carrier frequency in GHz, and $\\sigma$ represents the shadow fading modeled as a zero-mean log-normal distribution. The source and destination nodes are assumed to be $1$km apart, with the relay occupying possible positions at multiples of $100$m from them. The transmission power of both source and relay is selected as $23$dBm, which is typical of a cellular \\ac{UE}. Each link experiences a slow Rayleigh fading channel, with an exponentially decaying power delay profile containing a \\ac{CIR} of $L = 32$ taps, which is also the length of the \\ac{CP} used (unless mentioned otherwise). Perfect synchronization and channel estimation is assumed for all links. Noise \\ac{PSD}, $N_0$, is taken to be $-174$dBm\/Hz. The simulation results are averaged over $5000$ \\ac{OFDM} blocks, where each block contains $256$ \\ac{QPSK} symbols. A summary of these simulation parameters is provided in Table \\ref{tab:sim_parameters}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{Figures\/Relay_Dest.pdf}\n    \\caption{BER performance comparison of relay and destination. Relay's performance is seen to be degraded severely with the proposed CP jamming scheme.}\n    \\label{fig:rel_dest_comparison}\n\\end{figure}\n\n\\par Figure \\ref{fig:rel_dest_comparison} shows the performance of relay and destination nodes in terms of \\ac{BER}. This is used to represent the \\textit{security gap}, which is quantified by the gap in error rates (bit, symbol, packet, etc.) of the legitimate and illegitimate receivers \\cite{hamamreh2018classifications}. The horizontal axis represents the position of the relay, where zero is the location of the source itself, while $1$ represents the location of the destination. For this simulation, both data and jamming signals are assumed to have the same power levels ($23$ dBm). The two curves for relay represent the two cases of normal relay operation (without jamming) and with the proposed jamming approach, respectively, while the curves for destination represent its performance in phase-1 (where it only receives the broadcast signal from the source) and when it applies \\ac{MRC} to the two copies received via source and relay. It can be seen that the relay's performance is significantly better than the destination when neither jamming nor combining is performed. This is simply due to the physical closeness of the relay to the source, and the consequent decrease in pathloss. It is shown in \\cite{chen2009multi} that distance-based cooperative protocol selects the relay closest to the destination as the optimal one. Our results also re-iterate this, as the performance of destination after \\ac{MRC} is significantly improved when the relay moves closer to it. The proposed algorithm also increases the \\ac{BER} of the untrusted relay at this point, leading to increased security gap.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{Figures\/CP_Pjam.pdf}\n    \\caption{Effect of jamming power and \\ac{CP} duration.}\n    \\label{fig:CP_Pjam}\n\\end{figure}\n\n\\par The interception capability of the eavesdropping relay depends on the jamming signal received. Accordingly, we look at the effect of different power levels of the jamming signals ($P_J$) as well as its different durations, as shown in Fig. \\ref{fig:CP_Pjam}. For the former, we have considered three cases, i.e., $P_J = [0.25, 0.5, 1]*P_{Tx}$. It can be seen that variation in $P_J$ has a clearly distinguishable effect on the relay's \\ac{BER}. For instance, when the relay is at the middle of the source and destination nodes, its \\ac{BER} goes from $0.019$ to $0.05$ as $P_J$ increases from $0.25*P_{Tx}$ to $P_{Tx}$. To see the effect of the length of the jamming signal, we looked at three cases where both the length of \\ac{CP} and jamming were varied, i.e., \\ac{CP} ratios of $1\/16, 1\/8, 1\/4$ were used (which were kept equal to \\ac{CIR} itself). It is observed that the length of the jamming signal itself has no visible effect on the \\ac{BER} performance. This observation also encourages us to use the minimum possible \\ac{CP} length in the system without compromising on the security performance of the proposed algorithm.\n\n\\section{Conclusion and Future Directions}\n\\label{sec:Conclusion}\nWireless communication has become exceedingly important in our daily lives. However, one of the major concerns in future networks is the privacy of user data. To this end, this work tries to address the eavesdropping problem caused by untrusted relays. The proposed approach involves the transmission of a jamming signal from the destination in the interval while it receives the \\ac{CP} part of the signal. Unlike the conventional solutions which require jamming signals to be transmitted throughout the broadcast phase, the proposed method only requires the signal to be transmitted for a fraction of it.\n\n\\par It should be noted that even though the current work focuses on the cooperative communication scenario, specifically untrusted relays, the concept presented in this work is much more general and can be applied to the eavesdropping problem in any \\ac{OFDM} system. Moreover, in such systems the jamming signal may also corrupt the (blind) synchronization attempted by any illegitimate receiver. Further work can also be carried out to optimize the relay placement and jamming power allocation to maximize the security gap in cooperative systems. Moreover, the proposed technique can be used for secure key exchange between the legitimate nodes in the presence of an untrusted relay. Furthermore, the proposed algorithm can also be applied in cases when there is no direct path between source and destination.\n\n\\section*{Acknowledgment}\nThis work was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant No. 120C142. The work of Haji M. Furqan was supported by HISAR Lab at TUBITAK BILGEM, Gebze, Turkey.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Lorenz 1984 Model}\nThe study of atmospheric dynamics is challenging due to its complex and chaotic nature. Lorenz \\cite{Lorenz:1984atmosphere} proposed in 1984 a simplified model for representing Hadley circulation of air in the atmosphere:\n\\begin{align}\n\\frac{dX}{dt}&=-Y^2-Z^2-aX+aF\\\\\n\\frac{dY}{dt}&=XY-bXZ-Y+G\\\\\n\\frac{dZ}{dt}&=bXY+XZ-Z\n\\end{align}\n\nThe equations are, in his words, \"the simplest model capable of representing an unmodified or modified Hadley circulation, determining its stability, and, if it is unstable, representing a stationary or migratory disturbance\" \\cite{Lorenz:1984atmosphere}. Lorenz uses the variable $X$ to represent the intensity of the symmetric globe-encircling westerly wind current, and also the poleward temperature gradient. The variables $Y$ and $Z$ represent the cosine and sine phase of a chain of superposed large-scale eddies. The parameters $a$, $b$, $F$ and $G$ may be chosen within certain bounds.\n\nThe model demonstrates chaotic behaviour for certain sets of parameters. In want of analytical solutions, initial-value problems are traditionally solved purely numerically by the use of finite steps in the temporal domain. The time steps of explicit time advance methods for general, space-dependent problems are restricted to small values through constraints such as the Courant-Friedrichs-Lewy (CFL) condition, and implicit schemes require time-consuming matrix operations at each time step. Semi-implicit methods allow large time steps and more efficient matrix inversions than those of implicit methods, but may feature limited accuracy. In this work we suggest an alternative, time-spectral approach for solution of equations (1)-(3). \n\n\\subsection{Generalized Weighted Residual Method}\nThe Generalized Weighted Residual Method (GWRM) differs from traditional spectral methods for initial-value problems \\cite{Gottlieb:1} in that also the time domain is treated spectrally \\cite{Scheffel:GWRM1}. As a result the GWRM eliminates grid causality conditions such as the CFL condition, being associated with time stepping algorithms. Although the problems to be solved are assumed causal, the method is acausal in the sense that the time dependence is calculated by a global minimization procedure (the weighted residual formalism) acting on the time integrated problem. Recall that in standard WRM \\cite{Fletcher:1}, initial value problems are transformed into a set of of coupled ordinary, linear or non-linear, differential equations for the time-dependent expansion coefficients. These are solved using finite differencing techniques. \n\nThe GWRM enables semi-analytical solutions;  finite approximate solutions are obtained as analytical Chebyshev expansions. Not only temporal and spatial, but also a physical parameter domain may be treated spectrally, being of interest for carrying out scaling dependence in a single computation. Chebyshev polynomials are used for the spectral representation. These have several desirable qualities. They converge rapidly to the approximated function, they are real and can straightforwardly be converted to ordinary polynomials and vice versa, their minimax property guarantees that they are the most economical polynomial representation, they can be used for non-periodic boundary conditions (being problematic for Fourier representations) and they are particularly apt for representing boundary layers where their extrema are locally dense \\cite{Scheffel:GWRM1}.\n\nIn standard text books on spectral methods for differential equations, time-spectral methods are usually touched upon only briefly and dismissed on the grounds that they are expensive \\cite{Canuto:1,Boyd:1}. Historically, a number of authors have investigated various suggestions for and aspects of spectral methods in time. \nIn 1979 a pseudo-spectral method, based on iterative calculation and an approximate factorization of the given equations, was suggested in \\cite{Morchoisne:1}. Also, some early ideas were not developed further by Peyret and Taylor in \\cite{Peyret:1}.\n\nIn 1986 and 1989, Tal-Ezer \\cite{Tal-Ezer:1,Tal-Ezer:2},  proposed time-spectral methods for linear, periodic hyperbolic and parabolic equations, respectively, using a polynomial approximation of the evolution operator in a Chebyshev least square sense. Periodicity was assumed for the spatial domain through use of the Fourier spectral approximation.\nThe method extends the traditional ${\\Delta}t=O(1\/N{^2})$ stability criterion for explicit algorithms, where the space resolution parameter $N=O(1\/{\\Delta}x)$, to higher efficiency resulting in the stability condition ${\\Delta}t=O(1\/N)$). This approach to extend the time step in explicit methods was further studied in \\cite{Kosloff:1}. The method is not widely used; a reason for this may be its complexity and its restriction to certain classes of problems. Later, Luo extended the method to more general boundary conditions and multiple spatial dimensions \\cite{Luo:1}.\n\nIerley et al. \\cite{Ierley:1} solved a a class of nonlinear parabolic partial differential equations with periodic boundary conditions using a Fourier representation in space and a Chebyshev representation in time. Similarly as for the GWRM, the Burger equation and other problems were solved with high resolution. Tang and Ma \\cite{Tang:1} also used a spatial Fourier representation for solution of parabolic equations, but introduced Legendre Petrov-Galerkin methods for the temporal domain.\n\nIn 1994, Bar-Yoseph et al \\cite{Zrahia:1,Bar-Yoseph:1} used space-time spectral element methods for solving one-dimensional nonlinear advection-diffusion problems and second order hyperbolic equations. Chebyshev polynomials were later employed in space-time least-squares spectral element methods \\cite{Maerschalck:1}.\n\nA theoretical analysis of Chebyshev solution expansion in time and one-dimensional space, for equal spectral orders, was given in \\cite{Dutt:1}. The minimized residuals employed were however different from those of the GWRM.\n\nMore recently Dehghan and Taleei \\cite{Dehghan:1} found solutions to the non-linear Schr\\\"{o}dinger equation, using a time-space pseudo-spectral method where the basis functions in time and space were constructed as a set of Lagrange interpolants.\n\nTime-spectral methods feature high order accuracy in time. For implicit finite difference methods, deferred correction may provide high order temporal accuracy \\cite{Gustafsson:1,Gustafsson:2}. A relatively recent approach to increase the temporal efficiency of finite difference methods is time-parallelization via the parareal algorithm \\cite{Lions:1}. This method, however, features rather low parallel efficiency and improvements have been suggested, for example the use of spectral deferred corrections \\cite{Emmett:1}.\n\nAn interesting Jacobian-free Newton-Krylov method for implicit time-spectral solution of the compressible Navier-Stokes equations has recently been put forth by Attar \\cite{Attar:1}. \n\nA time-spectral method for periodic unsteady computations, using a Fourier representation in time, was suggested in \\cite{Gopinath:1} and further developed in \\cite{Sicot:1} and \\cite{Luder:1}. A generalization to quasi-periodic problems was developed in \\cite{Mavriplis:1}. \n\nIn summary, although time-spectral methods have been explored in various forms by several authors during the last few decades, and were found to be highly accurate, the GWRM as described in \\cite{Scheffel:GWRM1} has not been pursued. The present work contributes to the evaluation of this method.\n\n\nThe structure of the paper is as follows. In section 2 we briefly review the general GWRM formalism for solving a set of pde's but subsequently restrict us to a discussion of an optimized solution of the ode's (1)-(3). A major goal of this study is to evaluate possible advantages of the GWRM in relation to finite difference methods (FDM). Thus a comparative numerical study of convergence, accuracy, and efficiency for short time interval solutions are presented in section 3. We have selected the explicit fourth order Runge-Kutta method (RK4) to represent efficient FDM. Comparisons will also be made with the implicit second order Lobatto IIIA (trapezoidal rule) and fourth order Lobatto IIIC methods, the latter being particularly apt for stiff problems \\cite{Brenan:Lobatto}. The chaotic character of the Lorenz equations is predominant in longer time calculations. In section 4 we determine long time accuracy and efficiency as well as the Liapunov exponent for the scenario studied here, employing both the RK4 and the GWRM. In section 5 the predictability of the GWRM is studied. We find that the efficiency of the GWRM compares well with and may exceed that of the RK4 method. A discussion follows in section 6 and conclusions are given in section 7.\n\n\\section{GWRM formalism}\n\nThe GWRM \\cite{Scheffel:GWRM1} solves a system of initial-value partial differential equations\n\\begin{equation}\n\\frac{\\partial\\mathbf{u}}{\\partial t}=D\\mathbf{u}+f\n\\end{equation}\nHere $D$ denotes a linear or nonlinear matrix differential operator that may depend on physical variables ($t$, $\\mathbf{x}$, and $\\mathbf{u}$) as well as on physical parameters (denoted $\\mathbf{p}$), and $f$=${f}(t,\\mathbf{x};\\mathbf{p})$ is a known source or forcing term. An approximate solution ansatz is assumed as a truncated multivariate series of first kind Chebyshev polynomials $T$, which becomes\n\\begin{equation}\n\\mathbf{u}(t,x;p)=\\sum_{q=0}^{Q}\\sum_{r=0}^{R}\\sum_{s=0}^{S}\\mathbf{a}_{qrs}T_q(\\tau)T_r(\\xi)T_s(P)\n\\end{equation}\n\nfor the case of one spatial dimension $x$ and one parameter $p$. The coefficients $\\mathbf{a}_{qrs}$ of this analytical expression are obtained from the Galerkin weighted residual method. For a single differential equation in $u$ it involves integration\n\\begin{equation}\n\\int_{t_0}^{t_1}\\int_{{x_0}}^{{x_1}}\\int_{{p_0}}^{{p_1}}RT_q(\\tau)T_r(\\xi)T_s(P)w_tw_xw_pdtd{x}d{p}=0\n\\end{equation}\nover the entire computational domain and solution for all indices $q,r,s$. The following weight factors are used \\cite{Mason:1}:\n\\begin{equation}\nw_t=(1-\\tau^2)^{\\frac{1}{2}},\nw_x=(1-\\xi^2)^{\\frac{1}{2}},\nw_p=(1-P^2)^{\\frac{1}{2}}\n\\end{equation}\n$R$ is the residual of the approximation \\cite{Fletcher:1}, defined as\n\\begin{equation}\nR=u(t,{x};{p})-\\left(u(t_0,{x};{p})+\\int_{t_0}^{t}(Du+f)dt'\\right)\n\\end{equation}\nand $\\tau$, $\\xi$ and $P$ are given by ($z$ can be any of $t$, $x$ or $p$ and $z_0$ and $z_1$ represent interval boundaries.)\n\\begin{equation}\n\\tau=\\frac{t-A_t}{B_t},\n\\xi=\\frac{x-A_x}{B_x},\nP=\\frac{p-A_p}{B_p},\n\\end{equation}\n\\begin{equation}\nA_z=\\frac{z_1+z_0}{2},\nB_z=\\frac{z_1-z_0}{2}\n\\end{equation}\n\nCarrying out the integrations in equation (6), the following system of algebraic equations results for the Chebyshev coefficients: \n\\begin{equation}\n\ta_{qrs}=2\\delta_{q0}b_{rs}+A_{qrs}+F_{qrs}\n\\end{equation}\nfor which $b_{rs}$ are the Chebyshev coefficients of the initial condition, and $A_{qrs}$ are obtained from the expansion\n\\begin{equation}\n\\int_{t_0}^{t}Dudt'=\\sum_{q=0}^{Q}\\sum_{r=0}^{R}\\sum_{s=0}^{S}A_{qrs}T_q(\\tau)T_r(\\xi)T_s(P)\n\\end{equation}\nshowing that equation (11) is a linear\/nonlinear equation in the parameters $a_{qrs}$ depending on the form of equation (4). The coefficients $F_{qrs}$ are obtained by a similar procedure to that in (12). Boundary conditions enter by replacing high-end $r$ indexed coefficients of (11) with corresponding spectral boundary equations. For the system of ode's (1)-(3), the forcing term and boundary conditions are not applicable and equations (11)-(12) are replaced by\n\\begin{equation}\n\t\\mathbf{a}_{q}=2\\delta_{q0}\\mathbf{b}+\\mathbf{A}_{q}\n\\end{equation}\n\\begin{equation}\n\\mathbf{b}=(X(0),Y(0),Z(0)),  ~~~ \\int_{t_0}^{t}D\\mathbf{u}dt'=\\sum_{q=0}^{Q}\\mathbf{A}_{q}T_q(\\tau)\n\\end{equation}\nwhere the array $\\mathbf{a}_q$ contains the Chebyshev coefficients of all $X$, $Y$ and $Z$ variables.\n\nTo iteratively solve the system of non-linear algebraic equations for the Chebyshev coefficients in the GWRM, a semi-implicit root-solver (SIR) has been developed \\cite{Scheffel:SIR}. This solver has better global convergence characteristics than Newton's method and avoids landing on local minima, as may occur when employing Newton's method with line-search. In GWRM applications, the handling of matrix equations in SIR for solution of equations (11) or (13) is the bottleneck both in terms of memory use and efficiency. In unoptimized cases, memory scales with the total number of Chebyshev coefficients to the second power, here as $(3(Q+1))^2$, and the number of operations scales to the third power, here $(3(Q+1))^3$. There exist, however, optimizing metods not touched upon here to reduce these scalings substantially. \n\nFor the present ode's to be solved, the maximum Chebyshev order $Q$ will be 10 or less, so that the main factor governing efficiency is rather the rate of convergence, determined by the number of SIR iterations which, in turn, depend on the length of the time interval used. A single GWRM solution for temporal domains exceeding about two Lorenz time units can generally not be computed due to the complexity of the present equations and the spurious roots that are generated by the nonlinear algebraic terms obtained by transforming equations (1)-(3) to equation (13). The time domain is thus divided into a number of partially overlapping subintervals of typical length 0.5-2 Lorenz time units. End conditions at each time interval are used as initial conditions for the subsequent interval. The solutions are finally joined to constitute a piece-wise analytical solution for the entire time domain. It should be noted that the lengths of the time intervals exceed the length of finite difference time steps with orders of magnitude. In this work, GWRM time intervals are typically a factor 100 longer than the RK4 time steps for comparable accuracy. \n\nFurthermore, the GWRM code employed here uses an adaptive algorithm to control the length of each time interval in order to guarantee convergence. If the accuracy requirements are not fulfilled, a shorter time interval will be chosen and the coefficients will be recalculated. The amount of overlap between subsequent time domains can be arbitrarily chosen in order to provide optimized two-point contact and enhanced convergence. \n\nGenerally, the convergence of iterative solvers like SIR are strongly dependent on the choice of starting conditions. An important property of the GWRM is that convergence is guaranteed as the time interval is decreased. This is because the solution then approaches the initial state, the coefficients of which are used as initial iterate for SIR. Moreover we will see that for simulations that are performed as perturbations of a \"base run\", use of the latter solution as initial state significantly reduces the number of required SIR iterations. The GWRM efficiency then exceeds that of RK4 and implicit methods by a wide margin. \n\n\n\\section{Convergence, accuracy, and efficiency}\n\nWe are interested in solving equations (1)-(3) accurately for long times up to some 100 time units. Consequently it is of interest to optimize the GWRM solution in individual time intervals. We may thus ask: is it preferable to use higher order approximations in long time intervals rather than to use lower order approximations in shorter intervals? To find out,  we solve equations (1)-(3) for $a = 0.25$, $b = 4.0$, $F = 8.0$ and $G = 1.0$, which parameters correspond to chaotic behaviour. The initial conditions used are well inside the attractor domain; $(X(0),Y(0),Z(0))=(0.96,-1.1,0.5)$. A corresponding GWRM solution for the time interval $[0,30]$ is shown in FIG. 1.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width=5.0in]{Lorenz_1984_XYZ.eps}\n\t\\caption{GWRM solution of Lorenz equations (1)-(3) for initial values $(X(0),Y(0),Z(0))=(0.96,-1.1,0.5)$ and parameters $a = 0.25$, $b = 4.0$, $F = 8.0$ and $G = 1.0$. From top to bottom; $X(t)+8, Y(t)+4, Z(t)$. The time domain was divided into $N=71$ subintervals during the time-adaptive computation.}\n\\end{figure}\n\nIn order to control GWRM accuracy we primarily need to consider the parameters used in SIR, the accuracy of which is essentially determined by three parameters. The first is the round-off error which, for the Maple computations of this paper, is globally set to $10^{-14}$. Second, a parameter $tol$ governs the mean absolute difference between successively iterated Chebyshev coefficients. Third, a parameter $\\epsilon$ controls the numerical ratio between the highest and lowest two mode number Chebyshev coefficients (absolute values); this ratio is small for well resolved solutions. \n\nIn TABLE~\\ref{Table_maxinterval} approximate maximum time interval lengths, corresponding to convergent GWRM solutions, are shown for different Chebyshev orders $Q$. CPU times for each case are also given. We have here set $tol=10^{-5}$. SIR is run in standard mode, without sub-iterations; see \\cite{Scheffel:SIR}. The convergence of Newton's method would be insufficient for some of these cases. When solving for longer time intervals than those of TABLE~\\ref{Table_maxinterval}, the GWRM may still converge but to spurious solutions that are not causally related to the initial conditions assumed. For the same reason, values of $Q$ beyond 8 do not result in convergent solutions in longer time intervals. We can now answer the question posed above; it is usually more efficient to employ lower values of $Q$ in time subintervals of some optimized length than to use a higher value in a single interval with the same total length. \n\n\\begin{table}[h!]\n\t\\caption{Maximum time interval lengths allowing GWRM convergence, and corresponding CPU times in seconds.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t$Q$&Time interval&CPU time\\\\\\hline\n\t\t$4$&$1.0$&0.031\\\\\\hline\n\t\t$5$&$1.3$&0.046\\\\\\hline\n\t\t$6$&$1.6$&0.047\\\\\\hline\n\t\t$7$&$1.9$&0.047\\\\\\hline\n\t\t$8$&$2.0$&0.062\\\\\\hline\n\t\t$9$&$1.7$&0.062\\\\\\hline\n\t\t$10$&$1.8$&0.078\\\\\\hline\n\t\\end{tabular}\\label{Table_maxinterval}\n\\end{center}\n\\end{table}\n\nWe need also to consider accuracy. In order to determine the values required of the parameters $Q$, $tol$ and $\\epsilon$ for a certain accuracy, comparisons with highly accurate solutions of (1)-(3) should be carried out. The solutions are obtained from Maple's built-in ode solver dsolve, for which we have set the relative error to $10^{-10}$. A series of computations have been made for the time interval $[0,2]$. The automatic time-adaptive algorithm, here using a time interval overlap of length $10^{-8}$ time units, checks for convergence. If convergence is not obtained, the time interval is halved. Every $M$th interval, where $M$ can be set (it is here 10), the code attempts to extend the time interval by a given factor, here 1.5. Obtained accuracy, in terms of maximum error of either of the $X$, $Y$, or $Z$ variables, are displayed in TABLE~\\ref{Table_max_error_at_2}. Again we have set $tol=10^{-5}$. The number of time subintervals used is denoted by $N$.\n\n\\begin{table}[h!]\n\t\\caption{Maximum GWRM absolute error of $X,Y,Z$ at $t=2$ for $tol=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$\\epsilon$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&4&$7$&0.078&$7.9\\times10^{-2}$\\\\\\hline\n\t\t$10^{-1}$&6&$3$&0.062&$1.6\\times10^{-2}$\\\\\\hline\n\t\t$10^{-1}$&8&$3$&0.078&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-1}$&10&$3$&0.109&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-2}$&4&$12$&0.109&$1.9\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&6&$3$&0.078&$1.6\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&8&$3$&0.094&$1.1\\times10^{-3}$\\\\\\hline\n\t\t$10^{-2}$&10&$3$&0.109&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-3}$&4&$28$&0.218&$1.7\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&6&$7$&0.125&$3.1\\times10^{-4}$\\\\\\hline\n\t\t$10^{-3}$&8&$3$&0.078&$1.1\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&10&$3$&0.110&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&4&$176$&0.998&$1.2\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&6&$18$&0.250&$1.5\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&8&$7$&0.156&$1.2\\times10^{-6}$\\\\\\hline\n\t\t$10^{-5}$&10&$7$&0.250&$6.8\\times10^{-7}$\\\\\\hline\n\t\t$10^{-5}$&12&$3$&0.140&$2.9\\times10^{-6}$\\\\\\hline\n\t\\end{tabular}\\label{Table_max_error_at_2}\n\\end{center}\n\\end{table}\n\nIt is seen that, as far as accuracy is concerned, low $Q$ values are undesirable. In order to satisfy the $\\epsilon$ criterion, the corresponding solutions need a large number of time subintervals and are thus costly. The solution error at the end of the computation (\"Error\" in TABLE~\\ref{Table_max_error_at_2}) need be of order $10^{-3}$ for the computations of sections IV-V of this study. Thus $Q$-values in the range of 8-10 are found to be optimal with respect to both accuracy and efficiency. The root solver SIR is here best run in Newton mode to reduce the number of iterations required. The initial time interval is set to $[0,2\/N_t]$ with $N_t=3$.\n\nNext, we compare with the RK4 method. The single parameter that controls accuracy is the step length parameter $h$. In TABLE~\\ref{Table_RK4_at_20} the maximum absolute error of $X,Y,Z$ at $t=20$ is computed for various step lengths. Corresponding results for the GWRM are displayed for different values of $\\epsilon$ and $tol$ (with $N_t=50$) in TABLE~\\ref{Table_GWRM_fixed_tol_at_20} and TABLE~\\ref{Table_GWRM_fixed_epsilon_at_20} respectively. It is seen that for low accuracy RK4 is somewhat more efficient than GWRM. For higher accuracy (better than $10^{-3}$) the GWRM is more efficient than RK4. This is particularly pronounced when the time overlap (\"handshaking\") procedure in GWRM is omitted. It should be noted that similar results as in TABLE~\\ref{Table_GWRM_fixed_tol_at_20} and TABLE~\\ref{Table_GWRM_fixed_epsilon_at_20} are obtained when the initial conditions $(X(0),Y(0),Z(0))$ are varied. The results are thus representative. \n\nRK4 is an explicit method. Are implicit algorithms like the second-order Trapezoid or the fourth-order Lobatto IIIC methods possibly more efficient? In TABLES \\ref{Table_Trapezoid_at_20} and \\ref{Table_LobattoIIIC_at_20} we show results for these methods for the same step lengths that were used for RK4 in TABLE III. It is seen that both methods are more costly than RK4 and GWRM for the same accuracy. The Lobatto IIIC method is more than an order of magnitude slower.\n\n\\begin{table}[h!]\n\t\\caption{RK4 method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$0.21$&$1.3\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$0.41$&$1.5\\times10^{-1}$\\\\\\hline\n\t\t$0.02$&1000&$1.01$&$4.8\\times10^{-3}$\\\\\\hline\n\t\t$0.01$&2000&$1.96$&$3.2\\times10^{-4}$\\\\\\hline\n\t\t$0.005$&4000&$3.91$&$5.7\\times10^{-5}$\\\\\\hline\t\n\t\\end{tabular}\\label{Table_RK4_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[h!]\n\t\\caption{GWRM maximum absolute error of $X,Y,Z$ at $t=20$ for $tol=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$\\epsilon$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&6&$38$&0.48&$9.1\\times10^{-1}$\\\\\\hline\n\t\t$10^{-1}$&8&$37$&0.80&$1.8\\times10^{-1}$\\\\\\hline\n\t\t$10^{-1}$&10&$37$&1.12&$5.5\\times10^{-3}$\\\\\\hline\n\t\t$10^{-2}$&6&$51$&0.64&$1.3\\times10^{-1}$\\\\\\hline\n\t\t$10^{-2}$&8&$38$&0.73&$2.8\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&10&$37$&1.14&$5.5\\times10^{-3}$\\\\\\hline\t\t\n\t\t$10^{-3}$&6&$72$&0.84&$7.2\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&8&$50$&0.91&$2.3\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&10&$38$&1.05&$2.4\\times10^{-3}$\\\\\\hline\t\t\n\t\t$10^{-5}$&6&$185$&1.84&$1.2\\times10^{-3}$\\\\\\hline\n\t\t$10^{-5}$&8&$83$&1.42&$3.8\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&10&$55$&1.44&$5.6\\times10^{-5}$\\\\\\hline\n\t\\end{tabular}\\label{Table_GWRM_fixed_tol_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[H]\n\t\\caption{GWRM maximum absolute error of $X,Y,Z$ at $t=20$ for $\\epsilon=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$tol$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&6&$173$&1.20&$1.1\\times10^{0}$\\\\\\hline\n\t\t$10^{-1}$&8&$95$&1.12&$1.5\\times10^{0}$\\\\\\hline\n\t\t$10^{-1}$&10&$58$&1.08&$1.9\\times10^{0}$\\\\\\hline\n\t\t$10^{-2}$&6&$200$&1.65&$4.8\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&8&$84$&1.22&$2.0\\times10^{-1}$\\\\\\hline\n\t\t$10^{-2}$&10&$55$&1.17&$2.5\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&6&$185$&1.73&$1.8\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&8&$83$&1.28&$4.6\\times10^{-4}$\\\\\\hline\n\t\t$10^{-3}$&10&$55$&1.30&$4.2\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&6&$185$&1.84&$1.2\\times10^{-3}$\\\\\\hline\n\t\t$10^{-5}$&8&$83$&1.42&$3.8\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&10&$55$&1.44&$5.6\\times10^{-5}$\\\\\\hline\n\t\\end{tabular}\\label{Table_GWRM_fixed_epsilon_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[H]\n\t\\caption{Trapezoid method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$0.84$&$1.1\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$1.44$&$6.3\\times10^{-1}$\\\\\\hline\n\t\t$0.02$&1000&$3.33$&$8.0\\times10^{-1}$\\\\\\hline\n\t\t$0.01$&2000&$6.89$&$2.1\\times10^{-1}$\\\\\\hline\n\t\t$0.005$&4000&$12.6$&$8.0\\times10^{-2}$\\\\\\hline\n\t\t\\end{tabular}\\label{Table_Trapezoid_at_20}\n\\end{center}\n\\end{table}\n\n\\begin{table}[H]\n\t\\caption{Lobatto IIIC method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$4.99$&$5.8\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$9.32$&$4.8\\times10^{-2}$\\\\\\hline\n\t\t$0.02$&1000&$20.9$&$1.3\\times10^{-3}$\\\\\\hline\n\t\t$0.01$&2000&$40.4$&$4.6\\times10^{-5}$\\\\\\hline\n\t\t$0.005$&4000&$78.0$&$3.5\\times10^{-5}$\\\\\\hline\t\n\t\\end{tabular}\\label{Table_LobattoIIIC_at_20}\n\\end{center}\n\\end{table}\n\n\\section{Long time behaviour}\nTo examine the accuracy of numerical solutions in absence of analytical solutions, a common practice is to study the convergence of different solutions, controlled by some parameter, towards a high accuracy solution. This approach will now be used in a comparative study of the accuracy of RK4, Lobatto IIIC and the GWRM for the Lorenz 1984 problem in the time interval $[0,30]$. Chaotic systems like the Lorenz 1984 equations are characterized by the fact that two initially adjacent states (with separation $E_{initial}$) will, even in absence of numerical errors, deviate at a rate \n\\begin{equation}\nE(t)=E_{initial}e^{\\lambda t}\n\\end{equation}\nwhere $\\lambda$ is the Lyapunov exponent \\cite{Lyapunov:1892thesis}. \n\nOur numerical results show that $\\ln E$ typically grows rapidly when $t$ is small and then linearly when $t$ is large. This suggests an approximate dependence of the form $E(t)=F(t)+Ce^{At}$ where $F(t)$ is some non-exponential function, playing a dominant role when $t$ is small. For large $t$, the exponential factor takes over and the logarithmic graph is linear. \n\nIn light of the above, we have found the following numerical procedure to be useful. For each of the numerical methods, a set of 100 solutions have been produced employing a set of selected accuracies. Each solution is compared with a high accuracy solution that uses exactly the same initial conditions, and the deviation as a function of time has been computed as described below. To optimize statistics, the attractor space is scanned by letting the end state of each high accuracy solution constitute the initial state for the new pair of solutions to be computed. The graphs $\\ln(E)$ vs $t$ are subsequently generated by taking the arithmetic mean of the calculations.\n\nThe expression\n\\begin{equation}\n\\ln(E)=E_0+At\n\\end{equation}\nis fitted to the graphs over the entire temporal domain. We find that the slope $A$ is quite independent of the choice of method or accuracy, which suggests that it is determined by the Lorenz system (1)-(3) itself indicating that $A=\\lambda$. The accuracy of the method can thus be solely estimated from $E_0$. Since the $\\ln E(t)$ graph is similar for each variable $X,Y,Z$, as can be seen from FIG. 2, we will mainly focus our discussion on the $X$ dimension.  \n\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ex.eps}\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ey.eps}\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ez.eps}\n\t\\caption{GWRM: $\\ln E_{X,\\epsilon,tol}$, $\\ln E_{Y,\\epsilon,tol}$ and $\\ln E_{Z,\\epsilon,tol}$ vs $t$ for $tol=10^{-5}$ and different $\\epsilon$.}\n\\end{figure}\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{GWRM_wohs_tol_Ex.eps}\n\n\n\t\\caption{GWRM: $\\ln E_{X,\\epsilon,tol}$ vs $t$ for $\\epsilon=10^{-5}$ and different $tol$.}\n\t\\end{figure}\nTwo factors control the accuracy of the GWRM method: the SIR parameters $tol$ and $\\epsilon$.\nThe deviation for each pair of GWRM computations is calculated by\n\\begin{equation}\nE_{k,\\epsilon,tol}(t)=|{u_{k}(t)}_{\\epsilon,tol}-{u_{k}(t)}_{10^{-5},10^{-5}}|\n\\end{equation}\nwhere $u_{k}$ denote the components of the vector $(X,Y,Z)$. Plots of $ln(E_{k,\\epsilon,tol}(t))$ for various $\\epsilon$ are given in FIG. 2 and a plot showing the deviations due to different $tol$ is shown in FIG. 3. Least square fitted parameters of (16) are listed in TABLE~\\ref{Table_errorfit_GWRM}.\n\\begin{table}[h!]\n\t\\caption{GWRM: Fitted parameters of Equation(17).}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$\\varepsilon$&tol&$A$&$E_0$\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-4}$&0.27&-17.9\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-3}$&0.27&-12.9\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-2}$&0.27&-7.4\\\\\\hline\n\t\t$1\\times 10^{-4}$&$1\\times 10^{-5}$&0.27&-14.5\\\\\\hline\n\t\t$1\\times 10^{-3}$&$1\\times 10^{-5}$&0.27&-11.4\\\\\\hline\n\t\t$1\\times 10^{-2}$&$1\\times 10^{-5}$&0.27&-9.1\\\\\\hline\t\n\t\\end{tabular}\\label{Table_errorfit_GWRM}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{RK4_Ex.eps}\n\n\n\t\\caption{RK4: $\\ln E_{X,h}$ vs $t$ with a variation of $h$.}\n\\end{figure}\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{Lobatto3C_Ex.eps}\n\n\n\t\\caption{Lobatto IIIC: $\\ln E_{X,h}$ vs $t$ with a variation of $h$.}\n\\end{figure}\nThe explicit classical RK4 and implicit Lobatto IIIC finite difference methods are chosen for comparisons.\nFor FDM, the controlling factor of accuracy is the time step $h$. The deviation is thus calculated as\n\\begin{equation}\nE_{k,h}(t)=|{u_{k}(t)}_{h}-{u_{k}(t)}_{5\\times 10^{-3}}|\n\\end{equation}\nDeviations $E(t)$ for different time step lengths are plotted in FIG:s 4 and 5 respectively and the fitted parameters are listed in TABLE~\\ref{Table_errorfit_RK4} and TABLE~\\ref{Table_errorfit_LobattoIIIC}.\n\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\caption{RK4: fitted parameters of Equation(17).}\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t$h$&$A$&$E_0$\\\\\\hline\n\t\t\t$1\\times 10^{-1}$&0.27&-4.1\\\\\\hline\n\t\t\t$5\\times 10^{-2}$&0.27&-6.9\\\\\\hline\n\t\t\t$2\\times 10^{-2}$&0.27&-10.8\\\\\\hline\n\t\t\t$1\\times 10^{-2}$&0.27&-13.6\\\\\\hline\n\t\\end{tabular}\\label{Table_errorfit_RK4}\n\t\\end{center}\n\\end{table}\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\caption{Lobatto IIIC: fitted Value of Equation(17).}\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t$h$&$A$&$E_0$\\\\\\hline\n\t\t\t$1\\times 10^{-1}$&0.27&-6.8\\\\\\hline\n\t\t\t$5\\times 10^{-2}$&0.27&-9.1\\\\\\hline\n\t\t\t$2\\times 10^{-2}$&0.27&-12.8\\\\\\hline\n\t\t\t$1\\times 10^{-2}$&0.27&-15.4\\\\\\hline\n\t\\end{tabular}\\label{Table_errorfit_LobattoIIIC}\n\t\\end{center}\n\\end{table}\nThe accuracy in terms of $E(0)$ of the fourth order FDM, the RK4 and Lobatto IIIC methods, is found to depend strongly on the step size; approximately $E(0)\\propto h^{3.8}$. The accuracy scalings of the GWRM are $E_0\\propto \\text{$\\epsilon$}^{1.2}$ for $tol=10^{-5}$ and $E_0\\propto \\text{$tol$}^{2.3}$ for $\\epsilon=10^{-5}$; see TABLE XI. Comparing TABLES \\ref{Table_errorfit_GWRM} and \\ref{Table_errorfit_RK4} with run times in TABLES \\ref{Table_RK4_at_20}, \\ref{Table_GWRM_fixed_tol_at_20} and \\ref{Table_GWRM_fixed_epsilon_at_20} it is again seen that the GWRM is more efficient at higher accuracies than the finite difference methods. \n\n\\begin{table}[h!]\n\\center\n\\caption{Accuracy comparison.}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tMethod&$\\epsilon$&$tol$&$h$&$E_0$\\\\\\hline\n\t\tRK4&&&$1\\times 10^{-1}$&-4.1\\\\\\hline\n\t\t&&&$5\\times 10^{-2}$&-6.9\\\\\\hline\n\t\t&&&$2\\times 10^{-2}$&-10.8\\\\\\hline\n\t\t&&&$1\\times 10^{-2}$&-13.6\\\\\\hline\n\t\tLobatto IIIC&&&$1\\times 10^{-1}$&-6.8\\\\\\hline\n\t\t&&&$5\\times 10^{-2}$&-9.1\\\\\\hline\n\t\t&&&$2\\times 10^{-2}$&-12.8\\\\\\hline\n\t\t&&&$1\\times 10^{-2}$&-15.4\\\\\\hline\n\t\tGWRM&$1\\times 10^{-2}$&$1\\times 10^{-5}$&&-9.1\\\\\\hline\n\t\t&$1\\times 10^{-3}$&$1\\times 10^{-5}$&&-11.4\\\\\\hline\n\t\t&$1\\times 10^{-4}$&$1\\times 10^{-5}$&&-14.5\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-2}$&&-7.4\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-3}$&&-12.9\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-4}$&&-17.9\\\\\\hline\n\t\\end{tabular}\\label{Table_efficiency}\n\\end{table}\n\nIn weather forecasting, running a number of simulations with slightly perturbed initial conditions is a common practice to enhance predictability. The GWRM is particularly well suited for these tasks, since the Chebyshev coefficients for the \"base\" run solution may be used as good initial iterates in SIR when computing the perturbed scenarios. This substantially reduces the number of iterations and the CPU time. In the next section predictability is studied in a comparison between the GWRM and the RK4 methods.  \n\n\\section{Predictability}\n\n\\noindent The Lorenz equations, and many weather related phenomena, are inherently chaotic. In other words, it is improbable that a long-term solution is representative if the initial condition varies slightly from the \"exact\" value. The error will eventually saturate at some level which implies that a random initial condition can be chosen without affecting the error growth, thus predictability is lost. The characteristic time $T$ when a perturbed solution diverges significantly from the true solution is thus of importance. It is in the time interval $[0,T]$ that predictions are meaningful.\n\nIt is of interest to compare the GWRM and RK4 method with regards to predictability. Our analysis goes as follows: a solution $\\mathbf{u_{1}}$ is obtained for $t=50$, where $\\mathbf{u}$ again denotes the vector $(X(t),Y(t),Z(t))$, and $\"1\"$ specifies a base run of the Lorenz equations with initial conditions  $\\mathbf{V}=(X(0),Y(0),Z(0))$. Next we perturb the initial conditions slightly; $\\mathbf{V}'=\\mathbf{V}+\\mathbf{\\delta}$, where the three components of the vector $\\mathbf{\\delta}$ are randomly distributed so that $0<\\delta<0.01$. The simulation is subsequently run again to obtain a perturbed solution $\\mathbf{u}'_{n}$. The deviations $e_{k,n}=(u'_{k,n}-u_{k,1})^2$ between the two simulations are then calculated for $k=1,2,3$. The same procedure is used for all perturbed scenarios, and then summed over the scenarios in order to obtain an error growth $E_{k}(t)=\\frac{1}{N}{\\sum}(u'_{k,n}-u_{k,1})^2, ~n=2..N$, where $N$ is the number of scenarios. Both RK4 and GWRM undergo the same analysis with comparable accuracies, which is chosen as the average accuracy of the three variables X, Y and Z, whereby the CPU times used are collected in TABLE XII. \n\nThe GWRM is found to be more than four times as efficient as RK4. As was shown in section III, this is partly due to the high efficiency of the GWRM for high accuracy, but mainly due to that for these perturbative runs, the initial iterates used by SIR are closer to the solution thus reducing the number of iterations needed.\n\n\\begin{table}[H]\n\\center\n\\caption{Error growth comparison between GWRM and RK4.}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tNumerical Method & Av. acc. & Steps & $\\epsilon$ & \\# Runs & CPU time [min] \\\\\\hline\n\t\tGWRM & 5.6e-6 & & $10^{-5}$ & 250 & 17.9 \\\\\\hline\n\t\tRK4 & 5.9e-6 & 15000 & & 250 & 83.5 \\\\\\hline\n\t\\end{tabular}\\label{Table_errorgrowth}\n\\end{table}\n\nThe most evident advantage that GWRM has over RK4 is that the time sub-intervals are typically two orders of magnitude larger. The time-adaptive scheme also allows for longer time intervals in smooth regions and shorter intervals in high gradient regions. This reduces the computation time for the method. The error growth in these studies show that the Lorenz equations are highly sensitive to perturbed initial conditions, and this will have a dominating effect where weather predictions are concerned, as is well established. This shows the need for a numerical model that is efficient at higher accuracies. As can be seen from FIG. \\ref{fig:1}, RK4 and GWRM give very similar error growths for the same accuracies. It is also seen that the characteristic time $T$ before which predictions are valid is about $T=10$ time units, whereafter the error growths tend to saturate.\n\n\\begin{figure}[H]\n\\center\n\t\\includegraphics[width=2.3in]{ETx_t50_P3.eps}\n\t\\includegraphics[width=2.3in]{ETy_t50_P3.eps}\n\t\\includegraphics[width=2.3in]{ETz_t50_P3.eps}\n\t\\caption{Error growth analysis using 250 perturbed scenarios, with comparable accuracy for the GWRM and RK4. GWRM: \\textit{brown line}, $m=8$, $tol=10^{-5}$ and $\\epsilon=10^{-5}$. RK4: \\textit{black dash line}, steps=15000.}\n\t\\label{fig:1}\n\\end{figure}\n\n\n\\section{Discussion}\nThe GWRM is a time-spectral method and differs substantially from traditional time-stepping methods. In this study a first assessment is made of its potential for modelling advanced NWP problems by addressing the Lorenz 1984 chaotic equations. The results are only tentative with respect to NWP, but motivate further studies where space dependence is taken into account. Of particular interest here is that recent development of the GWRM for problems in magnetohydrodynamics has lead to algorithms where high efficiency in handling the spatial dimensions have been obtained, using subdomains. The physical equations of each spatial subdomain may here be solved locally whereas the internal boundary condition equations are solved globally. This enables parallelization of the temporal domain.\n\nAn interesting feature of the GWRM is the possibility to average over the fast time scale for problems with several time scales \\cite{Scheffel:GWRM1}. A consequence is enhanced efficiency, since only lower order Chebyshev expansions are needed. This approach would be of considerable interest when performing multiple perturbation analysis of a base scenario because fewer scenarios would have to be computed, enhancing efficiency. For the present strongly nonlinear problem, however, high accuracy was found necessary for convergence to the correct solution. The multiple roots of the nonlinear algebraic system of equations (13) are closely distributed. Thus SIR easily departs from the correct solution, corresponding to the initial conditions given, for low temporal resolution. An effort to circumvent this tendency was made by linearising equations (1)-(3) before solution using SIR and gradually adding the nonlinear terms during the iterations. This approach is again not successful because of the tendency of root solvers to lock at the positions of the roots corresponding to the linear equations when the nonlinear terms are included. \n\nAnother question of interest is whether very long time intervals ($>>1$ time unit) could be employed, using the GWRM. The lesson learned from various problems solved so far is that the manageable interval length is indeed problem dependent. For non-smooth problems like the present, the time interval length is limited even if the Chebyshev order is increased because, again, of the spurious solutions found.\n\nAlthough a causal problem is solved, the GWRM need not necessarily be used as a causal method since both initial and end conditions may be applied. This is an interesting possibility for time-spectral methods which, however, requires further theoretical understanding. Various combination of initial and end conditions have indeed been tried, all within the attractor domains of the solutions, but no improvement of convergence was found.  \n\nUsually overlapping time intervals, employing two point contact, improves convergence. It is notable, and worthy of further study, that the best results (maximum efficiency) were here obtained using one point contact.\n\nSummarizing, the fact that the GWRM competes well with standard finite difference methods in solving the demanding Lorenz 1984 equations and that very efficient GWRM techniques have been developed for pde's where spatial dimensions are included, suggests that usage of time-spectral methods for NWP is well worth further exploration.  \n\n\n\\section{Conclusion}\nIn this work, a preliminary evaluation of a recently developed time-spectral method GWRM is carried out with respect to potential use for numerical weather prediction. The Lorenz (1984) chaotic equations are solved. The efficiency and the behaviour of the error growth with time is compared to traditional explicit and implicit finite difference methods. In particular, the optimal length of the solution time intervals have been determined, the accuracy of the method has been studied in detail and predictability has been investigated. \n\nIt is found that GWRM efficiency is in parity of, or better than, the finite difference methods. Efficiency is further enhanced for cases where several perturbed scenarios need be computed. This is mainly due to that GWRM time intervals are two orders of magnitude larger than those of finite difference methods. Furthermore, the GWRM solutions are analytical Chebyshev series expansions. These findings, and the existence of efficient algorithms for time parallelisation of spatially dependent pde's, are encouraging for future studies of time-spectral methods for NWP. \n\n\\section{Acknowledgement}\n\\noindent The authors would like to thank Dr {\\AA}ke Johansson at SMHI (Swedish Meteorological and Hydrological Institute) for inspiration and for pointing at the relevance of solving the Lorenz equations for a primary evaluation of the GWRM as a method for NWP.\n\n\\label{}\n\n\n\n\n\\section*{References}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Categories of causal r\\textsc{pes}\\ and \\textsc{rcn}}\\label{sec:category}\nOccurrence nets and \\textsc{pes} es are equipped with morphisms and turned  into categories that are\nrelated by suitable functors. \nIn this section, we extends such constructions to \\textsc{rcn}\\ and causal-r\\textsc{pes}. \nWe start by recalling the notions of morphisms for occurrence nets and prime event structures.\n\\begin{definition}\\label{de:occ-net-morph}\n Let $C_0 = \\langle B_0, E_0, F_0, \\mathsf{c}_0\\rangle$ and $C_1 = \\langle B_1, E_1, F_1, \\mathsf{c}_1\\rangle$\n be two occurrence nets. Then, an occurrence net \\emph{morphism} from $C_0$ to $C_1$ is a pair $(\\beta,\\eta)$ where\n $\\beta$ is a relation between $B_0$ and $B_1$, $\\eta : E_0\\to E_1$ is a partial mapping such that\n \\begin{itemize}\n   \\item $\\beta(\\mathsf{c}_0) = \\mathsf{c}_1$ and for each $b_1\\in \\mathsf{c}_1$ there exists a unique\n         $b_0\\in \\mathsf{c}_0$ such that $b_0 \\beta b_1$, and\n   \\item for all $e_0\\in E_0$, when $\\eta(e_0)$ is defined and equal to $e_1$ then \n         $\\beta(\\pre{e_0}) = \\pre{\\eta(e_0)} = \\pre{e_1}$\n         and $\\beta(\\post{e_0}) = \\post{\\eta(e_0)} = \\post{e_1}$.\n \\end{itemize}\n\\end{definition}\nAs we consider just occurrence nets, we have also that if $b_0 \\beta b_1$ and $(e_1,b_1)\\in F_1$, then there\nexists a unique event $e_0\\in E_0$ such that $\\eta(e_0) = e_1$ and $(e_0,b_0)\\in F_0$. \nConsequently, if a condition $b_1$ in $C_1$ is related to a condition $b_0$ in $C_0$ \nby $\\beta$, then the events producing these conditions are related \nby $\\eta$; moreover, there is a unique event $e_0$ in $C_0$ mapped to the event $e_1$. \nMorphisms on occurrence nets compose and the identity mapping is defined as\nthe identity relation on conditions and the \nidentity mapping on events. \nWe have the category having as objects the occurrence nets and as morphisms the \noccurrence nets morphisms, which will be denoted with $\\mathbf{Occ}$.\n\n\\begin{definition}\\label{de:pes-morph}\n Let $P_0 = (E_0, <_0, \\#_0)$ and $P_1 = (E_1, <_1, \\#_1)$ be two \\textsc{pes} es. Then,  a\n \\emph{\\textsc{pes}\\ morphism} from $P_0$ to $P_1$ is a partial mapping $f : E_0 \\to E_1$ such that\n \\begin{enumerate}\n   \\item for all $e_0\\in E_0$ if $f(e_0)$ is defined then $\\hist{f(e_0)} \\subseteq f(\\hist{e_0})$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and\n         $f(e_0) \\#_1 f(e_0')$ then $e_0 \\#_0 e_0'$, and \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and equal, then\n         $e_0 \\neq e_0'$ implies $e_0 \\#_0 e_0'$.    \n \\end{enumerate}\n\\end{definition}\n\nAgain \\textsc{pes}\\ morphisms compose, and we have the category $\\mathbf{PES}$ \n whose objects are \\textsc{pes} es\nand whose arrows are the \\textsc{pes}\\ morphisms.\n\nThe  categories $\\mathbf{Occ}$ and $\\mathbf{PES}$ are related as follows:  each occurrence net is associated with a prime event structure \nas stated\nin Proposition~\\ref{pr:on-to-pes}; each occurrence net morphism\n $(\\beta,\\eta)$ is associated with  $\\eta$, which turns out to be a \\textsc{pes}\\ morphism. We denote such functor as\n$\\mathcal{P}$  (as in Proposition~\\ref{pr:on-to-pes}).\nConversely, the mapping $\\mathcal{E}$ introduced in Proposition~\\ref{pr:pes-to-on} can be turned into a functor, \nas shown in \\cite{Win:ES}.  \nA \\textsc{pes}\\ morphism $f$ is turned into an occurrence net morphism as follows:\nthe relation $\\beta$ on conditions is defined such that (i) $(a,A) \\beta (a',A')$ when $a = \\bot = a'$, \n$A' = f(A) \\neq \\emptyset$, and $|A| = 1$, and (ii) $(e,A) \\beta (f(e),A')$ when $f(e)$ is defined and\n$A' = f(A) \\neq \\emptyset$;  the partial mapping on events is just $f$. It is then easy to\nsee that such definition indeed conforms an occurrence net morphism. \nThe nice result is that these functors form a coreflection, where $\\mathcal{C}$ is the right\nadjoint and $\\mathcal{E}$ the left adjoint.\n\nWe now recall the notion of r\\textsc{pes}\\ morphisms introduced in \\cite{GPY:CatRES}.\n\n\\begin{definition}\\label{de:rpes-morph}\n  Let $P_0 = (E_0, U_0, <_0, \\#_0, \\prec_0, \\triangleright_0)$ and $P_1 = (E_1, U_1, <_1, \\#_1, \\prec_0, \\triangleright_0)$ be two r\\textsc{pes}. Then, an\n \\emph{r\\textsc{pes}\\ morphism} from $P_0$ to $P_1$ is a partial mapping $f : E_0 \\to E_1$ such that\n \\begin{enumerate}\n   \\item for all $e_0\\in E_0$ if $f(e_0)$ is defined then \n         $\\setcomp{e_1\\in E_1}{e_1 <_1 f(e_0)} \\subseteq \\setcomp{f(e)}{e <_0 e_0}$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and\n         $f(e_0) \\#_1 f(e_0')$ then $e_0 \\#_0 e_0'$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and equal, then\n         $e_0 \\neq e_0'$ implies $e_0 \\#_0 e_0'$, \n   \\item for all $e_0\\in E_0$ and $e\\inU_0$ such that\n         $f(e_0)$ and $f(e)$ are both defined  and $f(e_0) \\triangleright_1 f(e)$ then\n         $e_0 \\triangleright_0 e$, and\n   \\item for all $e_0\\inU_0$ if $f(e_0)$ is defined then \n         $\\setcomp{e_1\\in E_1}{e_1 \\prec_1 \\underline{f(e_0)}} \\subseteq \\setcomp{f(e)}{e <_0 \\underline{e_0}}$.          \n \\end{enumerate}\n\\end{definition}\nThe notion of morphism above generalise the one on \\textsc{pes}\\ by requiring that prevention is preserved as\nwell as the reverse causality relation. \nIn \\cite{GPY:CatRES} it is also shown that \nr\\textsc{pes}\\ and r\\textsc{pes}\\ morphisms form a category, which is called $\\mathbf{RPES}$. \nWe restrict our attention to the subcategory $\\mathbf{cRPES}$, which \n has causal-r\\textsc{pes} as objects and r\\textsc{pes}\\ morphisms as arrows.\n\n\\begin{proposition}\\label{pr:full-subcat}\n $\\mathbf{cRPES}$ is a full subcategory of $\\mathbf{RPES}$.\n\\end{proposition}\n\nWe now turn our attention to reversible causal nets and introduce the notion of morphisms for reversible causal nets.\n\n\\begin{definition}\\label{de:rcn-morph}\n Let $R_0 = \\langle B_0, E_0, U_0, F_0, \\mathsf{c}_0\\rangle$ and \n $R_1 = \\langle B_1, E_1, U_1, F_1, \\mathsf{c}_1\\rangle$ be two \\textsc{rcn}{s}.\n Then an \\textsc{rcn}\\ \\emph{morphism} from $R_0$ to $R_1$ is the pair $(\\beta,\\eta)$ such that\n \\begin{itemize}\n   \\item $(\\beta,\\eta)$ restricted to the occurrence nets  \n         $\\langle B_0, E_0\\setminus U_0, F'_0, \\mathsf{c}_0\\rangle$ to\n         $\\langle B_1, E_1\\setminus U_1, F'_1, \\mathsf{c}_1\\rangle$ is an occurrence net morphism, \n         and\n   \\item $\\eta(U_0) \\subseteq U_1$ and if $e\\in U_0$ and $\\eta(e)$ is defined then\n         also $f(e')$ is defined, where $e' = h(e)$.      \n \\end{itemize}\n\\end{definition}\nIt is straightforward to check that \\textsc{rcn}\\ morphisms compose and that the identity relation and the identity mapping on events\nconforms a \\textsc{rcn}\\  morphism. Hence,  \\textsc{rcn}\\ together with \\textsc{rcn}\\ morphism form a category, which we call\n$\\mathbf{rOcc}$. \n\nThen, the definition of $\\mathcal{C}$ in Proposition~\\ref{pr:rce-to-rpes} can be \\changed{easily}{} extended to a functor.\n\\begin{proposition}\\label{pr:C-is-functor} \n $\\mathcal{C} : \\mathbf{rOcc} \\to \\mathbf{cRPES}$ acting on objects as in Proposition~\\ref{pr:rce-to-rpes}\n and on morphisms $(\\beta,\\eta) : R_0 \\to R_1$ as $\\eta$ restricted to the events that are not\n reversing ones, is a functor.\n\\end{proposition}\n\nAlso the construction in Definition~\\ref{de:rpestorcn} can be extended to a functor. \n\n\\begin{proposition}\\label{pr:E-is-functor} \n $\\mathcal{E}_{r} : \\mathbf{cRPES} \\to \\mathbf{rOcc}$ acting on objects as in Definition~\\ref{de:rpestorcn}\n and on morphisms as stipulated in for occurrence net, requiring that reversing events are\n preserved, is a functor.\n\\end{proposition}\n\nAlong the lines of \\cite{Win:ES} we can establish a relation between the categories\n$\\mathbf{cRPES}$ and  $\\mathbf{rOcc}$.\n\n\\begin{theorem}\\label{th:coreflection} \n $\\mathcal{E}_{r}$ and $\\mathcal{C}_{r}$ form a coreflection, where $\\mathcal{C}_{r}$ is the right\nadjoint and $\\mathcal{E}_{r}$ the left adjoint.\n\\end{theorem}\n\n\n\n\\subsection{From r\\textsc{pes}\\ to \\textsc{rcn}}\nCorrespondingly to what is usually done when relating nets to event structures, we show that \nif we focus on causal r\\textsc{pes}{es} then we can relate them to reversible occurrence nets.\nThe construction is indeed quite standard (see \\cite{Win:ES,BCP:LenNets} among many others),\nbut we do need a further observation on causal r\\textsc{pes}.\n\n\\begin{proposition}\\label{pr:pes-associated-to-crpes}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}\\ and\n let $<^{+}$ be the transitive closure of $<$. Then, $\\#$ is inherited along  $<^{+}$,\n \\emph{i.e.} $e\\ \\#\\ e' <^{+} e''\\ \\Rightarrow\\ e\\ \\#\\ e''$.\n\\end{proposition}\nA consequence of this proposition is that the conflict relation is fully characterized by the\ncausality relation, and the same intuition for introducing reversible causal net can be used in\nassociating a net to a causal r\\textsc{pes}\\ like the one used to associate an occurrence net to a \\textsc{pes}. \n\n\\begin{definition}\\label{de:rpestorcn}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}, and \n  $\\bot\\not\\in E$ be a new symbol. Define \n $\\mathcal{E}_{r}(\\mathsf{P})$ as the Petri net $\\langle B, \\hat{E}, F, \\mathsf{c}\\rangle$ where\n \\begin{itemize}\n    \\item $B = \\setcomp{(a,A)}{a\\in E\\cup\\setenum{\\bot} \\land A \\subseteq E \\land \\CF{A} \\land \n    a \\neq \\bot\\ \\Rightarrow\\ \\forall e\\in A.\\ a \\ll e}$,\\smallskip\n   \\item $\\hat{E} = E\\times\\setenum{\\mathtt{f}}\\  \\cup\\  U\\times\\setenum{\\mathtt{r}}$,\n   \\item $F = \\setcomp{(b,(e,\\mathtt{f}))}{b = (a,A)\\ \\land\\ e\\in A} \\quad \\cup\\quad\n              \\setcomp{((e,\\mathtt{f}),b)}{b = (e,A)}\\quad\\cup\\quad$ \\phantom{asdfa}\n              $\\setcomp{(b,(e,\\mathtt{r}))}{b = (e,A)}\\quad\\cup\\quad\n              \\setcomp{((e,\\mathtt{r}),b)}{b = (a,A)\\ \\land\\ e\\in A}$,\n                          and\n   \\item $\\mathsf{c} = \\setcomp{(\\bot,A)}{A \\subseteq E\\ \\land\\ \\CF{A}}$,                                               \n \\end{itemize}\n\\end{definition} \nIn essence the construction above takes the \\textsc{pes}\\ associated to an r\\textsc{pes}\\ and\nconstructs the associated occurrence net, which is then \\emph{enriched} with \nthe reversing events (transitions). The result is a reversible occurrence net.\n\n\\begin{proposition}\\label{prp:rpes_to_cnet}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}. Then  \n $\\mathcal{E}_{r}(\\mathsf{P}) = \\langle B, \\hat{E}, U\\times\\setenum{\\mathtt{r}}, F, \\mathsf{c}\\rangle$ as defined\n in Definition \\ref{de:rpestorcn} is a reversible occurrence net with respect to $U\\times\\setenum{\\mathtt{r}}$.\n\\end{proposition} \n\\begin{theorem}\\label{th:cccrpestorcn}\n Let $\\mathsf{P}$ be a causal \n r\\textsc{pes}. Then  \n $X'$ is a configuration of $\\mathcal{E}_{r}(\\mathsf{P})$ iff $X$ \n is a configuration of $\\mathsf{P}$, where $X' = \\setcomp{(e,\\mathtt{f})}{e\\in X}$.\n\\end {theorem}\nClearly, if we start from a reversible causal net, we get a r\\textsc{pes}\\ from which a reversible causal\nnet can be obtained having the same states (up to renaming of events).\n\\begin{corollary}\n Let $R$ be a \\textsc{rcn}. Then $\\states{\\mathcal{E}_{r}(\\mathcal{C}_{r}(R))} = \\states{R}$.\n\\end{corollary}\n\n\\begin{example}\n Consider the r\\textsc{pes}\\ with four events $\\setenum{e_1,e_2,e_3,e_4}$ such that $e_1 < e_3$ and $e_2 < e_4$,\n $e_1$ is in conflict with $e_2$ and this conflict is inherited along $<$. Furthermore, let \n $e_1$ and $e_3$ be reversible, and $e_3 \\triangleright \\underline{e}_1$. The construction \nin Definition~\\ref{de:rpestorcn} gives the net below.\n \\begin{center}\n  \\input{rpesrcnfig.tex}\n  \\end{center}\n\\end{example}\n\n\n\n\\section{Conclusions and future works}\\label{sec:conc}\nThe constructions we have proposed to associate a reversible causal net to a causal \nreversible prime event structure, and vice versa, are certainly driven by the classical ones \n(see \\cite{Win:ES}) for relating occurrence nets and prime event structures. \nThe consequence of this approach is that the causality relation, either the one given in a r\\textsc{pes}\\ or\nthe one induced by the flow relation in the occurrence net obtained ignoring the reversing\nevents, is the one driving the construction. One of the other two relations of an r\\textsc{pes}\\ is substantially\nignored (and we obtain from a \\textsc{rcn}\\ a causal r\\textsc{pes}\\ where the reverse causality relation just \nsays that an event can be reversed only after it has occurred) whereas the second (prevention)\nis tightly related to the causality relation: $b$ is caused by $a$ precisely when $b$ prevents \nof undoing of $a$.\nThe notion of reversible causal net we have proposed suggests this construction, so the problem \nof finding which kind of net would correspond to, for example, a cause-respecting or even\nan arbitrary r\\textsc{pes}\\ remains open and\ncertainly deserves to be investigated.\n\nIt is however interesting to observe that the construction in Definition \\ref{de:rpestorcn}\ngives a reversible causal net even when the r\\textsc{pes}\\ one started with is not a causal r\\textsc{pes}. \n Consider the r\\textsc{pes}\\ with two events $\\setenum{e_1, e_2}$ such that\n $e_1 < e_2$ and where the conflict and the prevention relations are empty.\n The only reversible event is $e_1$ and $e_1 \\prec \\underline{e}_1$.\n The set $\\{e_2\\}$ is a reachable configuration: we can remove $e_1$ from a reachable configuration \n$\\setenum{e_1, e_2}$ by performing the event $\\underline{e}_1$. \nThis is an example of out-of-causal order computation. \nGiven this r\\textsc{pes}, our construction produces the following \\textsc{rcn}, which does not have $\\{e_2\\}$ among its configurations.\n %\n \\begin{center}\n   \\input{counterex.tex}\n \\end{center}\n\n\\noindent\nThe constructions we have proposed are somehow the more adherent to what is usually done, based on the \ninterpretation that \\emph{causality} implies that the event causing some other event somehow\nproduces something that it is used by the latter. \nThis is not the only interpretation of what causality could mean. \nIn fact, causality is often confused with the observation that two causally related events appear\nordered in the same way in each possible execution, and when we talk about ordered execution, \nit should be stressed that this can be achieved in several ways, for instance using \\emph{inhibitor}\narcs. \nConsider the net $C$\n\\begin{center}\n\\input{inhiscausal.tex}\n\\end{center}\nthe event $e_2$ can be executed only after the event $e_1$ has been executed. However, $e_1$ \ndoes not produce a token (resource) that must be used by $e_2$. If we simply make\nthe event $e_1$ reversible but do nothing to prevent reversing of $e_1$ before $e_2$ is reversed,\nthen we would obtain the net $C'$. We could do better in $C''$ where we model the prevention using the\nso-called \\emph{read arcs} \\cite{MR:CN}. Hence, using the inhibitor or read arcs seem feasible way\nforwards to capture more precise the new relations of  r\\textsc{pes} es, including prevention. \nA similar approach has been already pursued in\n\\cite{CP:soap17} to model the so called modifiers that are able to change the causality pattern of\nan event.\nThis suggests that, for arbitrary r\\textsc{pes} es, we need to find relations different from the flow \nrelation to capture faithfully (forward and reverse) causal and prevention dependencies.\nThis will be the subject of future research.\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nEvent structures and nets are closely related. Since the seminal papers\nby Nielsen, Plotkin and Winskel \\cite{NPW:PNES} and Winskel \\cite{Win:ES}, the relationship among\nnets and event structures has been considered as a pivotal characteristic of\nconcurrent systems, as testified by numerous papers in the literature addressing\n event structures and nets. \nThe ingredients of an event structure are, beside a set of events, a number of relations that are\nused to express which events can be part of a configuration (the snapshot of a concurrent \nsystem), modelling a consistency predicate, and how events can be added to reach another \nconfiguration, modelling the dependencies among the (sets of) events.\nOn the net side, the ingredients boil down to constraints on how transitions may be executed,\nand usually have a structural flavour.\n\nSince the introduction of event structures there has been a flourish of\ninvestigations into the possible relations among events, giving rise to a number\nof different definitions of event structures.\nWe recall some of them, without the claim of completeness. \nFirst to mention are the classical \\emph{prime} event structures \n\\cite{Win:ES} where\nthe dependency between events, called \\emph{causality}, is given by a partial order and the consistency\nis determined by a \\emph{conflict} relation.\n\\emph{Flow} event structures \\cite{Bou:FESFN} drop the requirement that the dependency should\nbe a partial order, and \\emph{bundle} event structures \\cite{Langerak:1992:BES} are able to represent \nOR-causality by allowing each event to be caused by a member of a bundle of events.\n\\emph{Asymmetric} event structures \\cite{BCM:CNAED} introduce the notion of weak causality that can \nmodel asymmetric conflicts. \\emph{Inhibitor} event structures \\cite{BBCP:rivista} are able\nto faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. \nIn \\cite{BCP:LenNets} event structures where the causality relation may be circular are\ninvestigated, and in \\cite{AKPN:lmcs18} the notion of dynamic causality is considered.\nFinally, we mention the quite general approach presented in \\cite{GP:CSESPN}, where there is\na unique relation, akin to a \\emph{deduction relation}. \nTo each of the aforementioned event structures \na particular class of nets corresponds. To prime event structures we have \n\\emph{occurrence nets}, to flow event structures we have flow nets, to bundle event structures\nwe have \\emph{unravel nets} \\cite{CaPi:PN14}, to asymmetric and inhibitor event structures we\nhave \\emph{contextual nets} \\cite{BCM:CNAED,BBCP:rivista}, to event structures with circular\ncausality we have \\emph{lending nets} \\cite{BCP:LenNets}, to those with dynamic causality we have\n\\emph{inhibitor unravel nets} \\cite{CP:soap17} and finally to the ones presented in \n\\cite{GP:CSESPN}  \\emph{1-occurrence nets} are associated.\n\nRecently a new type of event structure tailored to model \\emph{reversible} computation has been\nintroduced and studied \\cite{PU:jlamp15,UPY:RES-NGC18}. In particular, in \\cite{PU:jlamp15},\n\\emph{reversible prime event structures} have been introduced. In this kind of event structure \ntwo relations are added: \nthe \\emph{reverse causality} relation and the \\emph{prevention} relation. The first \none is a standard dependency relation: in order to reverse an event some other events must be\npresent. The second relation, on the contrary, identifies those events whose presence \\emph{prevents}\nthe event being reversed. \nThis kind of event structure is able to model different\nflavours of reversible computation\nsuch as causal-consistent reversibility~\\cite{rccs,ccsk,rhotcs} and out-of-causal-order \nreversibility~\\cite{PhiUliYuen12,UK16}. \nCausally consistent reversibility relates reversibility with causality: an event can be undone provided that all of its effects have been undone. This allows the system to get back to a past state, which was possible to reach by just the normal (forward) computation.\nThis notion of reversibility is natural in reliable distributed systems since when an error occurs the system tries to go back to a past consistent state. Examples of how causal consistent reversibility is used to model reliable systems are  \ntransactions~\\cite{DanosK05,LaneseLMSS13} and rollback protocols~\\cite{VassorS18}.\nAlso, causally consistent reversibility can be used for program analysis and  debugging \\cite{GiachinoLM14,LanesePV19}. \n On the other hand, out-of-causal-order reversibility  does not preserve causes, and it is suitable to model biochemical reaction where, for example, a bond can be undone leading to a different state which was not present before.\n\nReversibility in Petri nets has been studied in \\cite{PhilippouP18,MMU:coordination19} with two different approaches. \nIn \\cite{PhilippouP18} reversibility for acyclic Petri net is solved by\nrelying on a new kind of tokens, called bonds, that\nkeep track of the execution history. Bonds are rich enough for allowing other approaches to\nreversibility such as out-of-causal order and causal consistent reversibility. In\n \\cite{MMU:coordination19}\na notion of \\emph{unfolding} of a P\/T (place\/transition) net, where all the transitions can be reversed, has been\nproposed. In particular,  by resorting to standard notions of the Petri net  theory \\cite{MMU:coordination19} provides a causally-consistent reversible semantics \nfor P\/T nets. This exploits the well-known unfolding of P\/T nets into occurrence nets \n\\cite{Win:ES}, and is done by adding for each transition its reversible counterpart. \n \n\nIn this paper we start\nour research quest towards studying what kind of nets can be\nassociated to reversible prime event structures. To this aim we introduce the notion of\na \\emph{reversible causal net} which is an occurrence net enriched with \ntransitions which operationally\n\\emph{reverse} the effects of executing some others\n\n\nWe associate to each reversing transition (event in the occurrence net dialect) a unique transition, \nwhich is in charge of producing\n the effects that the reversing transition\nhas to undo.\nTo execute a reversing event in a reversible causal net \nthe events caused by the event to be reversed (the one associated to the reversing one)\nhave to be reversed as well, and if this is not possible then the reversing event cannot be\nexecuted. This corresponds, in the reversible event structure, to the fact that such \nevents \\emph{prevent} the reversing event from happening. A reversible causal \nnet where the reversing events \nhave been removed is just an occurrence net.\nThis discussion suggests the easiest way of relating\nreversible causal nets and reversible prime event structure: the causal relation is\nthe one induced by the occurrence net, and the prevention relation is the one induced\nby the events that are in the future of the one to be reversed. The reversible causality relation\nis the basic one: in order to reverse an event the event itself must be present. \nWhat is obtained from a reversible causal net is a \\emph{causal} reversible prime event structure,\nwhich is a subclass of reversible prime event structures.  \n\nWhen we start from a causal reversible prime event structure, it is possible to obtain\na reversible causal net. The ingredients that are used are just the causal relation and\nthe set of reversible events. \nThus, if we start from a causal\nreversible prime event structure, the obtained reversible causal net  \nhas the same set of configurations.\nHence, the precise correspondence is between causal reversible prime event structures and \nreversible causal nets. \nThis relation is made clear also by turning causal reversible prime event structures and\nreversible causal nets into categories. Then the constructions associating reversible causal nets to \ncausal reversible prime event structures can be turned into functors and these functors form a coreflection.\nThis implies that the notion of reversible causal net is the appropriate one when dealing with causal\nprime event structures.\n\n\\paragraph{Structure of the paper.}\nSection \\ref{sec:preliminaries} reviews some preliminary notions for nets and\nevent structures, including reversible prime event structures. Section \\ref{sec:cn-and-pes} recalls\nthe well-known relationship between prime event structures and occurrence nets. The core of the\npaper is Section \\ref{sec:rcn-and-rpes} where we first introduce reversible causal nets and then\nwe show how to obtain a reversible causal net from an occurrence net. We then \nshow how to associate a causal reversible prime event structure to a reversible causal net, \nand vice versa. \nWe sum up our findings in Section \\ref{sec:category} where we introduce a notion of morphism for \nreversible causal nets and show that this gives a category which is related to the subcategory of causal\nreversible prime event structures.\nSection \\ref{sec:conc} concludes the paper\n\n\n\\section{Omitted Proofs}\\label{sec:proof}\n\\input{proofs}\n\\end{document}\n\n\\section{Occurrence nets and prime event structures}\\label{sec:cn-and-pes}\nWe now review the notion of occurrence nets~\\cite{NPW:PNES,Win:ES}. \nGiven a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$,\nwe \nwrite $<_N$ for transitive closure of $F$, and $\\leq_N$ for the reflexive closure of $<_N$.\n We say $N$ is \\emph{acyclic} if  $\\leq_N$ is a partial order.\nFor occurrence nets, we adopt the usual convention and refer to places and transitions respectively \n as {\\em conditions} and {\\em events}, and correspondingly use $B$  and \n$E$ for the sets of conditions and events.\n We will often confuse conditions with places and \nevents with transitions.\n\\begin{definition}\n An \\emph{occurrence net} (\\textsc{on}) $C = \\langle B, E, F, \\mathsf{c}\\rangle$ is an acyclic, safe net \n satisfying the\n  following restrictions:\n  \\begin{itemize}\n    \\item $\\forall b\\in \\mathsf{c}$. $\\pre{b} = \\emptyset$,\n    %\n    \\item $\\forall b\\in B$. $\\exists b'\\in \\mathsf{c}$ such that $b' \\leq_C b$,\n    %\n    \\item $\\forall b\\in B$. $\\pre{b}$\n          is either empty or a singleton,\n   \n    \\item for all $e\\in E$ the set $\\hist{e} = \\setcomp{e' \\in E}{e'\\leq_C e}$ is finite,\n          and\n   \n    \\item  $\\#$ is an irreflexive and symmetric relation defined as follows:\n          \n           \\begin{itemize}\n             \\item $e\\ \\#_0\\ e'$ iff $e, e' \\in E$, $e\\neq e'$ and \n                   $\\pre{e}\\cap\\pre{e'}\\neq \\emptyset$,\n             %\n             \\item $x\\ \\#\\ x'$ iff $\\exists y, y'\\in E$ such that $y\\ \\#_0\\ y'$ and $y \\leq_C x$ and \n                   $y' \\leq_C x'$.\n             \\end{itemize}\n     \\end{itemize}\n\\end{definition}\nThe intuition behind occurrence nets is the following: each condition $b$ represents \nthe occurrence of a token,\nwhich is produced by the \\emph{unique} event in $\\pre{b}$, \nunless $b$ belongs to the initial marking,\nand it is used by only one transition (hence if $e, e'\\in\\post{b}$, then $e\\ \\#\\ e'$).\nOn an occurrence net $C$ it is natural to define a notion of \\emph{causality} among elements of the \nnet: we say that $x$ is \\emph{causally dependent} on $y$ iff $y \\leq_C x$.\n \nOccurrence nets are often the result of the \\emph{unfolding} of a (safe) net. \nIn this perspective an occurrence net is meant to describe precisely the non-sequential semantics\nof a net, and each reachable marking of the occurrence net corresponds to a reachable marking\nin the net to be unfolded. Here we focus purely on occurrence nets and not on the nets\nthey are unfoldings of.\n\n\\begin{definition}\\label{de:cn-conf}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be a \\textsc{on}\\ and \n $X\\subseteq E$ be a subset of events. Then $X$ is a \\emph{configuration} of\n $C$ whenever $\\CF{X}$ and $\\forall e\\in X$. $\\hist{e}\\subseteq X$.\n The set of configurations of the occurrence net $C$ is denoted by $\\Conf{C}{\\textsc{on}}$.\n\\end{definition}\n\nGiven an occurrence net $C = \\langle B, E, F, \\mathsf{c}\\rangle$ and a state\n$X \\in \\states{C}$, it is easy to see that it is \\emph{conflict free}, \\emph{i.e.}\n$\\forall e, e'\\in X$. $e\\neq e'\\ \\Rightarrow\\ \\neg (e\\ \\#\\ e')$, and \\emph{left closed},\n\\emph{i.e.} $\\forall e \\in X$. $\\setcomp{e'\\in E}{e'\\leq_C e}\\subseteq X$.\n\n\\begin{proposition}\\label{pr:states-are-conf}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net and $X\\in \\states{C}$. Then\n $X\\in \\Conf{C}{\\textsc{on}}$.\n\\end{proposition}\nWe now recall the connection among occurrence nets and prime event structures~\\cite{Win:ES}.\n\\begin{proposition}\\label{pr:on-to-pes}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net. Then\n $\\mathcal{P}(C) = (E, \\leq_C, \\#)$\n  is a \\textsc{pes}, and \n $\\Conf{C}{\\textsc{on}} = \\Conf{\\mathcal{P}(C)}{\\textsc{pes}}$.\n\\end{proposition}\n\n\\begin{example}\\label{ex:cn}\n\nFigure~\\ref{fig:occ-nets} illustrates some  (finite) occurrence nets. We can associate \\textsc{pes} es to them as follows.\nThe net $C_1$ has two concurrent events, which hence are neither causally ordered nor in conflict,\nconsequently both $<$ and $\\#$ are  empty. The two events $e_1$ and $e_2$ in  $C_2$ are in conflict,\nnamely $e_1\\ \\#\\ e_2$, while they are causally ordered in $C_3$, namely $e_1 < e_2$, and not in conflict.\nFinally, in $C_4$ we have $e_1< e_3$ and $e_2<e_4$ and $e_1\\ \\#\\ e_2$. Additionally, conflict inheritance \ngive us $e_1\\ \\#\\ e_4$, $e_2\\ \\#\\ e_3$ and $e_3\\ \\#\\ e_4$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\input{cn-example.tex}\n\\caption{Some occurrence nets}\n\\label{fig:occ-nets}\n\\end{center}\n\\end{figure}\n\n\\end{example}\n\nConversely, every \\textsc{pes}\\ can be associated with an occurrence net. \n\n\\begin{proposition}\\label{pr:pes-to-on}\n Let $P = (E, <, \\#)$ be a \\textsc{pes}\\ and let $\\bot\\not\\in E$ be a new symbol. \n Then $\\mathcal{E}(P) = \\langle B, E, F, \\mathsf{c}\\rangle$ \n\n defined as follows \n  \\begin{itemize}\n    \\item $B = \\setcomp{(a,A)}{a\\in E\\cup\\setenum{\\bot} \\land A \\subseteq E \\land \\CF{A} \\land \n    a \\neq \\bot\\ \\Rightarrow\\ \\forall e\\in A.\\ a < e}$,\n   \\item $F = \\setcomp{(b,e)}{b = (a,A)\\ \\land\\ e\\in A}\\cup \\setcomp{(e,b)}{b = (e,A)}$, and\n   \\item $\\mathsf{c} = \\setcomp{(\\bot,A)}{A \\subseteq E\\ \\land\\ \\CF{A}}$,                                               \n \\end{itemize}\n is an occurrence net, and $\\Conf{P}{\\textsc{pes}} = \\Conf{\\mathcal{E}(P)}{\\textsc{on}}$.\n\\end{proposition}\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n We denote with  $\\ensuremath{\\mathbb{N}}$ the set of natural numbers.\n %\n Let $A$ be a set, a {\\em multiset\\\/} of $A$ is a function $m : A\n \\rightarrow \\ensuremath{\\mathbb{N}}$.\n %\n The set of multisets of $A$ is denoted by $\\mu A$.  \n %\nWe assume the usual operations on multisets such as union $+$ and difference $-$.\n %\n \n We write $m \\subseteq m'$ if $m(a) \\leq m'(a)$ for all $a \\in A$.  \n %\n For $m\\in \\mu A$, we denote with $\\flt{m}$ the multiset defined as $\\flt{m}(a)\n = 1$ if $m(a) > 0$ and $\\flt{m}(a) = 0$ otherwise. \n %\n When a multiset $m$ of $A$ is a set, \n \\emph{i.e.} $m = \\flt{m}$, we write\n $a\\in m$ to denote that $m(a) \\neq 0$, and often confuse the\n multiset $m$ with the set $\\setcomp{a\\in A}{m(a) \\neq 0}$. \n %\n Furthermore we use the standard set operations like $\\cap$, $\\cup$ or\n $\\setminus$.\n \n Given a set $A$ and a relation $<\\ \\subseteq A\\times A$, we say that $<$ is \n an \\emph{irreflexive} partial order whenever\nit is irreflexive and transitive. \nWe shall write $\\leq$ for the reflexive closure of a  partial order $<$.\n \n\\subsection{Petri nets} \nWe review the notion of Petri net along with some auxiliary notions.\n\\begin{definition}\n   A \\emph{Petri net} is a 4-tuple \n   $N = \\langle S, T, F, \\mathsf{m}\\rangle$, where\n   $S$ is a set of {\\em places} and $T$ is a set of {\\em transitions} \n   (with $S \\cap T = \\emptyset$), \n  \n   {$F \\subseteq (S\\times T)\\cup (T\\times S)$}\n    is the \n    \\emph{flow} relation, and\n   $\\mathsf{m}\\in \\mu S$ is called the {\\em initial marking}.\n \\end{definition}\n Petri nets are depicted as usual. \n %\n Given a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$ and  $x\\in S\\cup T$, we define the following\n multisets:  \n\n {$\\pre{x} = \\{y\\ |\\ (y,x)\\in F\\}$}\n  and \n\n {$\\post{x} = \\{y\\ |\\ (x,y)\\in F\\}$}.\n If $x\\in S$ then \n\n\n $\\pre{x} \\in \\mu T$ and  $\\post{x} \\in \\mu T$;\n analogously, \n if $x\\in T$ then $\\pre{x}\\in\\mu S$ and $\\post{x} \\in \\mu S$.\n %\n A multiset of transitions $A\\in \\mu T$, called \\emph{step}, \n is enabled at a marking $m\\in \\mu S$, denoted by $m\\trans{A}$,\n whenever $\\pre{A} \\subseteq m$, where $\\pre{A} = \\sum_{x\\in\\flt{A}}\\ A(x)\\cdot\\pre{x}$. \n %\n A step $A$ enabled at a marking $m$ can \\emph{fire} and its firing produces \n the marking $m' = m - \\pre{A} + \\post{A}$, where \n $\\post{A} = \\sum_{x\\in\\flt{A}}\\ A(x)\\cdot\\post{x}$.\n The firing of $A$ at a marking $m$ is denoted by $m\\trans{A}m'$.\n %\n We assume that each transition $t$ of a net $N$ is such that $\\pre{t}\\neq\\emptyset$,\n meaning that no transition may fire \\emph{spontaneously}.\n %\n Given a generic marking $m$ (not necessarily  the initial one), \n the (step) \\emph{firing sequence} ({shortened as} \\textsf{fs}) of  \n $N = \\langle S, T, F, \\mathsf{m}\\rangle$  starting at $m$ is\n defined as: \n   ($i$)  $m$ is a firing sequence (of length 0), and \n %\n  ($ii$) if $m\\trans{A_1}m_1$ $\\cdots$ $m_{n-1}\\trans{A_n}m_n$ is a firing sequence \n        and $m_n\\trans{A}m'$,  \n        then also $m\\trans{A_1}m_1$ $\\cdots$ $m_{n-1}\\trans{A_n}m_n\\trans{A}m'$\n        is a firing sequence.\n %\n Let us note that each step $A$ such that $|A| = n$ \n can be written as $A_1 + \\cdots + A_n$ where for each $1 \\leq i\\leq n$ it holds that \n $A_i = \\flt{A_i}$ and  $|A_i| = 1$, and $m\\trans{A}m'$ iff for each decomposition\n of $A$ in $A_1 + \\cdots + A_n$, we have that $m\\trans{A_1}m_1\\dots m_{n-1}\\trans{A_n}m_n = m'$.\n %\n When $A$ is a singleton, \\emph{i.e.}\n $A = \\setenum{t}$, we write \n $m\\trans{t}m'$.   \n\n The set of firing sequences of a net $N$ \n starting at a marking $m$ is denoted by $\\firseq{N}{m}$ and it is ranged over by $\\sigma$.\n Given a \\textsf{fs}\\ $\\sigma = m\\trans{A_1}\\sigma'\\trans{A_n}m_n$, we denote with\n $\\start{\\sigma}$  the marking $m$, with $\\lead{\\sigma}$ the marking $m_n$ \n and with $\\remains{\\sigma}$ the \n \\textsf{fs}\\ $\\sigma'\\trans{A_n}m_n$. \n %\n Given a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$, a marking $m$ is \\emph{reachable} \n iff there exists a \n \\textsf{fs}\\ $\\sigma \\in \\firseq{N}{\\mathsf{m}}$ \n such that $\\lead{\\sigma}$ is $m$. \n %\n The set of reachable markings of $N$ is\n $\\reachMark{N} = \\bigcup_{\\sigma\\in\\firseq{N}{\\mathsf{m}}} \\lead{\\sigma}$.\n %\n Given a \\textsf{fs}\\ $\\sigma = m\\trans{A_1}m_1\\cdots m_{n-1}\\trans{A_n}m'$, \n we write  \n $X_{\\sigma} = \\sum_{i=1}^{n} A_i$\n for the multiset of transitions associated to  \\textsf{fs}. \nWe call $X_{\\sigma}$ a \\emph{state} of the net and write\n \\(\n   \\states{N} = \\setcomp{X_{\\sigma}\\in \\mu T}{\\sigma\\in\\firseq{N}{\\mathsf{m}}}\n \\)\n for the set of states of $N$. \n \n \n\n\\begin{definition}  \n A net $N = \\langle S, T, F, \\mathsf{m}\\rangle$ is said to be \\emph{safe} if\n each marking $m\\in \\reachMark{N}$ is such that $m = \\flt{m}$. \n\\end{definition}\n\nIn this paper we consider safe nets $N =  \\langle S, T, F, \\mathsf{m}\\rangle$ \nwhere each transition can be fired, \\emph{i.e.}\n$\\forall t\\in T\\ \\exists m\\in \\reachMark{N}.\\ m\\trans{t}$, and each place is marked in a \ncomputation, \\emph{i.e.} $\\forall s\\in S\\ \\exists m\\in \\reachMark{N}.\\ m(s) = 1$.\n\n\\subsection{Prime event structures}\n\nWe now recall the notion of prime event structure~\\cite{Win:ES}.\n\n\\begin{definition}\\label{de:pes-winskel}\n A \\emph{prime event structure ({\\textsc{pes}})} is a triple $P = (E, <, \\#)$, where \n \\begin{itemize}\n  \\item $E$ is a  countable set of \\emph{events},\n  \\item $<\\ \\subseteq E\\times E$ is\nan irreflexive partial order called the \\emph{causality relation}, such that\n        $\\forall e\\in E$. $\\setcomp{e'\\in E}{e' < e}$ is finite, and \n  \\item $\\#\\ \\subseteq E\\times E$ is a \\emph{conflict relation}, which is irreflexive,\n    symmetric and \\emph{hereditary} relation with respect to $<$: if  $e\\ \\#\\ e' < e''$, then\n    $e\\ \\#\\ e''$ for all $e,e',e'' \\in E$.\n \\end{itemize}       \n\\end{definition}\n\nGiven an event $e\\in E$,  $\\hist{e}$  denotes the set $\\setcomp{e'\\in E}{e'\\leq e}$. \n{A subset of events $X \\subseteq E$ is left-closed if $\\forall e\\in X. \n\\hist{e}\\subseteq X$.\n} \nGiven a\n subset $X\\subseteq E$ of events, we say that $X$ is \\emph{conflict free} iff \nfor all $e, e'\\in X$ it holds that \n$e\\neq e'\\ \\Rightarrow\\ \\neg(e\\ \\#\\ e')$, and we denote it with $\\CF{X}$. \nGiven $X\\subseteq E$ such that $\\CF{X}$ and $Y\\subseteq X$, then also $\\CF{Y}$.\n\nWhen adding reversibility to \\textsc{pes} es, conflict heredity may not hold.\nTherefore, we rely on a weaker form of \\textsc{pes}\\ by following  the approach in~\\cite{PU:jlamp15}.\n\\begin{definition}\\label{de:pre-pes}\n   A \\emph{pre-\\textsc{pes}} (p\\textsc{pes}) is a triple $P = (E, <, \\#)$, where\n   \\begin{itemize}\n    \\item $E$ is a set of \\emph{events},\n    \\item $\\#\\ \\subseteq E\\times E$ is an irreflexive and symmetric relation,\n    \\item $<\\ \\subseteq E\\times E$ is an irreflexive partial order such that\nfor every $e\\in E$. $\\setcomp{e'\\in E}{e' < e}$ is finite and\n                  conflict free, and\n    \\item $\\forall e, e'\\in E$. if $e < e'$ then not $e\\ \\#\\ e'$.\n   \\end{itemize}\n  \\end{definition}\n{A p\\textsc{pes}\\ is  a prime event structure in which conflict heredity does not hold, \nand since every \\textsc{pes}\\ is also a p\\textsc{pes}\\ the notions and results stated below \nfor p\\textsc{pes} es also apply to \\textsc{pes} es.\n}\n\n\\begin{definition}\\label{de:ppes-conf-enab}\n   Let $P = (E, <, \\#)$ be a p\\textsc{pes}\\ and\n  \n   $X\\subseteq E$ such that $\\CF{X}$.\n   For $A\\subseteq E$, we say that $A$ is \\emph{enabled}\n   at $X$ if\n   \\begin{itemize}\n    \\item $A\\cap X = \\emptyset$ and $\\CF{X\\cup A}$, and\n    \\item $\\forall e\\in A$. if $e' < e$ then $e'\\in X$.\n   \\end{itemize}\n   If $A$ is enabled at $X$, then $X \\stackrel{A}{\\longrightarrow} Y$ where $Y = X\\cup A$.\n  \\end{definition}\n\\begin{definition}\\label{de:rpes-forwconf}\n   Let $P = (E, <, \\#)$ be a p\\textsc{pes}\\ and\n  \n   {$X\\subseteq E$ s.t. $\\CF{X}$}.\n   $X$ is a \\emph{forwards reachable configuration} if there exists a sequence\n  \n   {$A_1,\\ldots,A_n$}, such that\n   \\[ X_i\\stackrel{A_i}{\\longrightarrow} X_{i+1} \\ \\mathit{ for all}\\ i, \\ \\mathit{and}\\\n       X_1 = \\emptyset \\ \\mathit{and}\\ X_{n+1}=X.\n       \\]\nWe write $\\Conf{P}{p\\textsc{pes}}$ for the set of all  (forwards reachable) configurations of $P$.\n  \\end{definition}\nWhen a p\\textsc{pes}\\ is a \\textsc{pes}\\ we shall write $\\Conf{P}{\\textsc{pes}}$ instead of $\\Conf{P}{p\\textsc{pes}}$, with \n$\\Conf{P}{\\textsc{pes}}=\\Conf{P}{p\\textsc{pes}}$ holding.\nFrom a p\\textsc{pes}\\ a \\textsc{pes}\\ can be obtained.\n\n\n\\begin{definition}\\label{de:hereditary-closure}\n Let $P = (E, <, \\#)$ be a p\\textsc{pes}. Then $\\mathsf{hc}(P) = (E, <, \\sharp)$ is the \n \\emph{hereditary closure} of $P$, where $\\sharp$ is\n derived by using the following rules \n \\[ \n\\begin{array}{ccccc}\n\\infer{e\\ \\sharp\\ e'}{e\\ \\#\\ e'} & \\qquad & \\infer{e\\ \\sharp\\ e''}{e\\ \\sharp\\ e'\\ \\ \\ \\ \\ e' < e''}\n & \\qquad  & \\infer{e\\ \\sharp\\ e'}{e'\\ \\sharp\\ e}\\\\\n\\end{array}\n\\]\n\\end{definition}\nThe following proposition relates p\\textsc{pes}\\ to \\textsc{pes}\\ \\cite{PU:jlamp15}.\n\\begin{proposition}\\label{pr:ppes-prop}\n Let $P = (E, <, \\#)$ be a p\\textsc{pes}, then\n \\begin{itemize}\n   \\item $\\mathsf{hc}(P) = (E, \\leq, \\sharp)$ is a \\textsc{pes},\n   \\item if $P$ is a \\textsc{pes}, then $\\mathsf{hc}(P) = P$, and\n   \\item $\\Conf{P}{p\\textsc{pes}} = \\Conf{\\mathsf{hc}(P)}{\\textsc{pes}}$.\n \\end{itemize}\n\\end{proposition}\n\n\\subsection{Reversible prime event structures} \nWe now focus on the notion of \n\\emph{reversible prime event structure}. The definitions and the results in this subsection \nare drawn from \\cite{PU:jlamp15}. In reversible event structures some \nevents are categorised as \\emph{reversible}. The added relations\nare among events and those representing the \\emph{actual} undoing of the reversible events.\nThe undoing of events is represented by \\emph{removing} them (from a configuration), and is achieved\nby \\emph{executing} the appropriate \\emph{reversing} events.\n\n\\begin{definition}\\label{de:rpes}\n A \\emph{reversible prime event structure} (r\\textsc{pes}) is the tuple \n $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ where \n $(E, <, \\#)$ is a p\\textsc{pes}, $U\\subseteq E$ are the\n \\emph{reversible\/undoable}\n\n  events\n (with reverse events being denoted by \n {$\\underline{U} = \\setcomp{\\underline{u}}{u\\inU}$ and disjoint from $E$, i.e., $\\un U \\cap E = \\emptyset$})\nand\n { \\begin{itemize}\n   \\item $\\triangleright\\ \\subseteq E\\times \\underline{U}$ is the \\emph{prevention} relation,\n   %\n   \\item $\\prec\\ \\subseteq E\\times \\underline{U}$ is the \\emph{reverse causality} relation and\n         it is such that $u\\prec \\underline{u}$ for each $u\\inU$ and \n         $\\setcomp{e\\in E}{e\\prec\\underline{u}}$ is finite and conflict-free for every $u\\inU$, \n   %\n   \\item if $e \\prec \\underline{u}$ then not $e \\triangleright \\underline{u}$,\n   %\n   \\item the \n         \\emph{sustained causation}  $\\ll$  is a transitive relation defined such that $e \\ll e'$ if $e < e'$ and if $e\\inU$, then \n         $e' \\triangleright \\underline{e}$, and \n   %\n   \\item $\\#$ is hereditary with respect to $\\ll$: if $e\\ \\#\\ e' \\ll e''$, then $e\\ \\#\\ e''$.\n \\end{itemize}\n}\n\\end{definition}\nThe ingredients of a r\\textsc{pes}\\ partly overlap with those of a \\textsc{pes}: there is a causality\nrelation ($<$) and a conflict one ($\\#$) and the two are related by the \\emph{sustained causation} relation $\\ll$. The new ingredients are the \n\\emph{prevention} relation and the \\emph{reverse causality} relation.\nThe prevention relation states that certain events should be absent when trying to reverse an event, e.g., \n{$e\\triangleright \\underline{u}$}\nstates that  {$e$} should be absent when reversing {$u$}.  The reverse causality relation\n{$e \\prec \\underline{u}$} says that \n{$\\underline{u}$} can be executed \nonly when $e$ is present.\n\n\n \n\\begin{example}\\label{ex:cr not causal}\nLet $\\mathsf{P} = (E,U,<,\\#,\\prec,\\mathrel{\\triangleright})$ where\n$E = U = \\{a,b,c\\}$, $a<b$ and $a \\prec \\un a$, $b \\prec \\un b$, $c \\prec \\un c$, $c \\prec \\un a$ \nwith $b \\mathrel{\\triangleright} \\un a$ and no conflict.\nThen $a \\ll b$ because $a < b$ and $b \\mathrel{\\triangleright} \\un a$.\n$\\mathsf{P}$ states that $b$ causally depends on $a$ and that $c$ is concurrent w.r.t. both $a$ and $b$. \nNote that  every event is reversible in $\\mathsf{P}$ because $U = E$.  As expected, the reverse causality relation $\\prec$\nis defined such that every reverse event requires the presence of the corresponding reversible event, i.e., $e\\prec \\un e$ for all $e\\in E$. \nAdditionally, it also requires $c \\prec \\un a$, i.e.,  $a$ can be reversed only when $c$ is present. The prevention \nrelation states that $a$ cannot be reversed when $b$ is present, i.e., $b \\mathrel{\\triangleright} \\un a$.\n\\end{example}\n\\begin{definition}\\label{de:rpes-conf-enab}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be an r\\textsc{pes}\\ and\n  $X\\subseteq E$ be a set of events such that $\\CF{X}$. For\n $A\\subseteq E$ and $B\\subseteq U$, we say that $A\\cup \\underline{B}$ is \\emph{enabled}\n at $X$ if\n \\begin{itemize}\n  \\item $A\\cap X = \\emptyset$, $B \\subseteq X$ and $\\CF{X\\cup A}$, \n  \\item $\\forall e\\in A, e'\\in E$. if $e' < e$ then $e'\\in X\\setminus B$,\n  \\item $\\forall e\\in B, e'\\in E$. if $e' \\prec \\underline{e}$ then \n        $e' \\in X\\setminus (B\\setminus\\setenum{e})$,\n  \\item $\\forall e\\in B, e'\\in E$. if $e' \\triangleright \\underline{e}$ then $e'\\not\\in X\\cup A$.      \n \\end{itemize}\n If $A\\cup \\underline{B}$ is enabled at $X$ then \n $X \\stackrel{A\\cup\\underline{B}}{\\longrightarrow} Y$ where $Y = (X\\setminus B)\\cup A$.\n\\end{definition}\n\\begin{example}\\label{ex:rpes-red}\nConsider  the r\\textsc{pes}\\ in Example~\\ref{ex:cr not causal}.\nWe have, e.g.,  $\\emptyset \\stackrel{\\{a,c\\}}{\\longrightarrow} \\{a,c\\} \n \\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{c\\}$ and $\\emptyset \\stackrel{\\{a\\}}{\\longrightarrow} \\{a\\}\n \\stackrel{\\{b\\}}{\\longrightarrow} \\{a, b\\} \n \\stackrel{\\{c, \\un b\\}}{\\longrightarrow} \\{a,c\\} \\stackrel{\\{b\\}}{\\longrightarrow} \\{a,b,c\\}$. \n \\end{example}\nReachable configurations are subsets of events which can be reached from the empty set by performing\nevents or undoing previously performed events.\n\\begin{definition}\\label{de:rpes-conf}\nLet $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}\\ and\n let $X\\subseteq E$ be a subset of events such that $\\CF{X}$.\n We say that $X$ is a \\emph{(reachable) configuration} if there exist two sequences\n of\n sets $A_i$ and $B_i$, for $i=1,\\ldots,n$,  such that\n \\begin{itemize}\n  \\item $A_i\\subseteq E$ and $B_i\\subseteq U$ for all $i$, and\n  \\item $X_i\\stackrel{A_i\\cup\\underline{B_i}}{\\longrightarrow} X_{i+1}$ for all $i$ with \n   $X_1 = \\emptyset$ and $X_{n+1}=X$.\n \\end{itemize}\nThe set of configurations of $\\mathsf{P}$ is denoted by $\\Conf{\\mathsf{P}}{r\\textsc{pes}}$.\n\\end{definition}\n\\begin{example} \nThe  set of configurations of $\\mathsf{P}$ defined in Example~\\ref{ex:cr not causal} is \n$\\Conf{\\mathsf{P}}{r\\textsc{pes}} = \\{\\emptyset, \\{a\\}, \\{c\\}, \\{a,b\\}, \\{a,c\\}, \\{a,b,c\\}\\}$\nas illustrated by the sequences shown in Example~\\ref{ex:rpes-red}.\n \\end{example}\n\n\nAs discussed in Section~\\ref{sec:intro}, r\\textsc{pes} es accommodate different flavours of \nreversibility. Hereafter, we  focus on \\emph{causal} reversibility \\cite{Lanese14}, \nwhich is one of the most common models of \nreversibility in distributed systems, in which an event can reversed \nonly when all the events it has caused have already been  reversed.\n  \n  \\begin{definition}\\label{de:cr-and-causal-rpes}\n   Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be an r\\textsc{pes}. Then $\\mathsf{P}$ is \n   \\emph{cause-respecting} if for any $e, e'\\in E$, if $e < e'$ then $e \\ll e'$. \n   $\\mathsf{P}$ is \\emph{causal} if for any $e\\in E$ and $u\\inU$ the following holds:\n\t $e \\prec \\underline{u}$ iff  $e = u$, and  \n    $e \\triangleright \\underline{u}$ iff $u < e$.\n  \\end{definition}\n  %\n \n\\begin{example}\\label{ex:cr not causal2}\nThe r\\textsc{pes}\\ $\\mathsf{P}$ in Example~\\ref{ex:cr not causal} is a cause-respecting r\\textsc{pes}.\nHowever $\\mathsf{P}$ is not causal because of $c \\prec \\un a$, which means that $c$ has to be present \nfor $a$ to be reversed even if $c$ does not causally depend on $a$.  If we remove $c \\prec \\un a$ then we obtain a causal r\\textsc{pes}.\n\\end{example}\n  \n\n\\begin{example}\\label{out-of-causal-order}\nAn example of out-of-causal order reversibility can be obtained from the definition of \nthe r\\textsc{pes}\\ $\\mathsf{P}$ in \nExample~\\ref{ex:cr not causal} by replacing $b \\mathrel{\\triangleright} \\un a$ by \n$a \\mathrel{\\triangleright} \\un b$. Then, we have $\\emptyset \\stackrel{\\{a\\}}{\\longrightarrow} \\{a\\}\n\\stackrel{\\{b,c\\}}{\\longrightarrow} \\{a,b,c\\}\\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{b,c\\}$.\n Note that $a$ can be reversed even in the presence of the event $b$, which causally \ndepends on $a$.\n\\end{example}\n\nCause-respecting and causal r\\textsc{pes} es enjoy the following useful properties \\cite{PU:jlamp15}.\n \n\\begin{proposition}\\label{pr:rpes-causal-cause-respecting}\nLet $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}.\nLet $X$ be a left-closed and conflict-free subset of events in $E$ and\nlet $A, B\\subseteq U$. Then\n     \\begin{itemize}\n\\item if $\\mathsf{P}$ is cause-respecting and $X \\stackrel{A\\cup \\underline{B}}{\\longrightarrow} X'$, \n      then $X'$ is also left-closed,\n\\item if $\\mathsf{P}$ is cause-respecting and\n            $X \\stackrel{\\underline{B}}{\\longrightarrow} X'$, then\n            $X' \\stackrel{B}{\\longrightarrow} X$,\n\\item if $\\mathsf{P}$ is causal and\n            $X \\stackrel{A\\cup \\underline{B}}{\\longrightarrow} X'$, then\n            $X' \\stackrel{B\\cup\\underline{A}}{\\longrightarrow} X$.\n     \\end{itemize}\n\\end{proposition}\n\n\\begin{example}\\label{not-left-closed}\nThe above properties do not hold when a r\\textsc{pes}\\ is not cause-respecting \/ causal. Consider the \nr\\textsc{pes}\\ in Example~\\ref{out-of-causal-order}. We have that $\\{a,b,c\\}\\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{b,c\\}$ but \n$\\{b,c\\}$ is not left-closed.\n\\end{example}\n\n\nA particular r\\^ole will be played by the configurations that can be reached without\nexecuting any reversible events.\n\\begin{definition}\\label{de:rpes-forwconf}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}\\ and\n  $X\\in\\Conf{\\mathsf{P}}{r\\textsc{pes}}$ be a configuration.\n $X$ is \\emph{forwards reachable} if there exists a sequence\n of sets $A_i\\subseteq E$, for $i=1,\\ldots,n$, such that \n {$\n X_i\\stackrel{A_i}{\\longrightarrow} X_{i+1}$ for all  $i$, with  \n     $X_1 = \\emptyset$ and $X_{n+1}=X$.\n}\n\\end{definition}\nAlthough $\\{b,c\\}$ in Example~\\ref{not-left-closed} is a reachable configuration it is \nnot forwards reachable. However, the configurations of a cause-respecting r\\textsc{pes}\\ are forwards reachable\n\\cite{PU:jlamp15}.\n\\begin{proposition}\n     Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a cause-respecting r\\textsc{pes}, and\n     let $X$ be configuration of $\\mathsf{P}$. Then $X$ is forwards reachable.\n\\end{proposition}\n\n\n\n\\section{Achieving non causal consistent reversibility}\nIn the previous section we have shown how reversible causal nets and reversible prime event\nstructures are related, and we have seen that ....\n\nto do:\n\\begin{enumerate}\n \\item define a notion of net where causality is present but each event can be undone easily,\n       using inhibitor arcs\n \\item how to enforce $\\triangleright$ to be cause-respecting also on these net? \n \\item do we have always have \\emph{single} events? \n\\end{enumerate}\n\\section{Reversible causal nets and reversible prime event structures}\\label{sec:rcn-and-rpes}\nWe now introduce\nthe notion of \\emph{reversible causal nets}.\nA similar notion has been proposed in~\\cite{MMU:coordination19} for adding\n causally-consistent reversibility to Petri nets by making  reversible every event in\nthe unfolding of the net. \nIn this work we deal with a generalised version of \nreversible causal nets in which\n transitions may be irreversible, i.e.,  \nwe do not require every transition of a net to be undoable.\n\nThe intuition behind reversible causal nets is the following: we add special\ntransitions (events in the classical occurrence net terminology) to an occurrence net which, \nwhen executed, \\emph{undo} \nthe execution of other (standard) transitions. When we remove these special transitions from a\nreversible causal net we obtain a standard occurrence net.\n\n\n\\begin{definition}\\label{def:rcn}\n A \\emph{reversible causal net} (\\textsc{rcn})  is a tuple $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ \nwhere $\\langle B, E, F, \\mathsf{c}\\rangle$ is a safe net such that \n \\begin{itemize}\n   \\item $U\\subseteq E$ and $\\forall u\\in U$. $\\exists!\\ e\\in E\\setminusU$ such that\n         $\\pre{u} = \\post{e}$ and $\\post{u} = \\pre{e}$, \n   \\item $\\forall e, e'\\in E$. $\\pre{e} = \\pre{e'}\\ \\land\\ \\post{e} = \\post{e'}\\ \n         \\Rightarrow\\ e = e'$, \n        \n  \n   \\item $\\bigcup_{e\\in E} (\\pre{e}\\cup\\post{e}) = B$, and        \n   \\item $C_{E\\setminus U} = \\langle B, E\\setminus U, F', \\mathsf{c}\\rangle$ is an occurrence net, \n         where $F'$ is the restriction of $F$ to the transitions in $E\\setminus U$.\n \\end{itemize}        \n\\end{definition}\nThe events in $U$ are the reversing ones and\nwe often say that a reversible causal net $R$ is reversible \\emph{with respect to} $U$. \nWe  write $\\overline{E}$ \n for the set of events $E\\setminus U$ and\n$C_{\\overline{E}}$ instead of $C_{E\\setminus U}$.\nThe first condition in Definition~\\ref{def:rcn} implies that \n each reversing event $u\\inU$ is associated with  a unique event $e$ that \ncauses the effects that  $u$ is intended to \\emph{undo}; hence $e$ here is a\n\\emph{reversible} event. Moreover, \nthe second condition ensures that there is an injective\nmapping $h : U \\to E$ that associates  each event  ${u\\inU}$ with a different \nevent $e\\in E$ such that $\\pre{e} = \\post{u}$ and $\\post{e} = \\pre{u}$, in other words, each reversible event has exactly one reversing event.\nThe third requirement guarantees that all conditions (places) of the net appear at least in the pre or the postset of some event  (transitions), i.e., \nthere are no isolated conditions. \nThe last condition ensures that the net obtained by deleting all reversing events is an\noccurrence net.\n\n\\begin{example}\\label{ex:rcn}\n We present some reversible causal nets in Figure~\\ref{fig:rcn}. The reversing events are drawn in red, and\n their names are underlined.\n  The events $e_1$ and $e_2$ in $R_1$ are both reversible, while \n  $e_1$ is the only  reversible event in $R_2$. In $R_3$ the events $e_1$, $e_3$ and $e_4$ are the reversible ones.\n\n \\begin{figure}[t]\n \\begin{center}\n  \\input{rcn-example.tex}\n \\end{center}\n \\caption{Some reversible causal nets}\n \\label{fig:rcn}\n \\end{figure}\n\\end{example}\n\nWe prove that the set of reachable markings of a reversible causal net is not influenced \nby performing reversing events.\n\\begin{proposition}\\label{pr:reachable-markings-of-rcn}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}.\n Then $\\reachMark{R} = \\reachMark{C_{\\overline{E}}}$.\n\\end{proposition}\n\nA consequence of the above proposition is the following corollary, which \nestablishes\nthat each marking can be reached by using just \\emph{forward events}.\n\\begin{corollary}\\label{cor:a}\n Let $C = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}\\\n and $\\sigma$ be a \\textsf{fs}. Then, there exists a \\textsf{fs}\\ $\\sigma'$ such that\n $X_{\\sigma'}\\subseteq \\overline{E}$ and $\\lead{\\sigma} = \\lead{\\sigma'}$.\n\n\\end{corollary}\n\n\\begin{definition}\\label{de:conf-rcn}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}, and\n  $X \\subseteq \\overline{E}$ be a subset of forward events. \n Then,  $X$ is a configuration of $R$ iff $X$ is a configuration of $C_{\\overline{E}}$.\n The set of configurations of $R$ is as usually denoted with $\\Conf{R}{\\textsc{rcn}}$.\n\\end{definition}\nA configuration of a reversible causal net $R$ with respect to $U$ is a subset of  \n$E\\setminus U$;\nconsequently, the reversing events (i.e., the ones in $U$) that may have been executed to reach a particular \nmarking are not considered as part of the configuration.\nObserve that, differently from \noccurrence net, $\\states{R} \\neq \\Conf{R}{\\textsc{rcn}}$ because the former may contain also reversing events.\nHowever, as a consequence of Corollary \\ref{cor:a}, there is no loss of information.\n\nWe show how to construct a reversible causal net from an occurrence net, once we have\nidentified the events to be \\emph{reversed}.\n\n\\begin{definition}\\label{de:constructing-a-reversible-causal-net}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net and $U \\subseteq E$ be a set \n of reversible events. Define $\\mathsf{R}(C) = \\langle B, \\hat{E}, U\\times\\setenum{r}, \\hat{F}, \\mathsf{c}\\rangle$ \n be the net where $\\hat{E}$ and $\\hat{F}$ are defined as follows:\n  \\begin{itemize}\n   \\item $\\hat{E} = E\\times\\setenum{\\mathtt{f}} \\ \\cup\\  U\\times\\setenum{\\mathtt{r}}$, and\n   %\n   \\item \n$\\hat{F} = \n\t\t\\setcomp{(b, (e,\\mathtt{f}))}{(b,e)\\in F} \\quad\\cup\\quad \\setcomp{ ((e,\\mathtt{f}),b)}{(e,b)\\in F}\\quad\\cup$ \\phantom{asdfasdfas} \\phantom{asdfa}\n\t\t$\\setcomp{(b, (e,\\mathtt{r}))}{(e,b)\\in F}\\quad\\cup\\quad \\setcomp{ ((e,\\mathtt{r}),b)}{(b,e)\\in F}$\n  \\end{itemize}\n  The mapping $h : U\\times\\setenum{\\mathtt{r}} \\to E\\times\\setenum{\\mathtt{f}}$ is defined as\n  $h(e,\\mathtt{r}) = (e,\\mathtt{f})$. \n\\end{definition}\nThe construction above simply adds as many events (transitions) as those to\nbe reversed. The preset of each added event is the postset of the\ncorresponding event to be reversed, and its postset is defined as the preset of the event to \nbe reversed.\nThe events in $U\\times\\setenum{\\mathtt{r}}$ are the reversing events.\n\n\\begin{proposition}\\label{prop:rev_net}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net, $U\\subseteq E$ be the subset of\n reversible events, and \n $\\mathsf{R}(C) = \\langle B, \\hat{E}, U\\times\\setenum{\\mathtt{r}}, \\hat{F}, \\mathsf{c}\\rangle$ be the net \n  in Definition \\ref{de:constructing-a-reversible-causal-net}. Then, $\\mathsf{R}(C)$ is a reversible causal net with respect to\n $U\\times\\setenum{\\mathtt{r}}$.\n\\end{proposition}\n\\begin{example}\\label{ex:reversing}\n Consider the occurrence net $C_1$ in Figure~\\ref{fig:occ-nets}, and assume that both events are reversible. The \n net $R_1$ in Figure~\\ref{fig:rcn} is $\\mathsf{R}(C_1)$ (after renaming events with\n the convention that $(e,\\mathtt{f})$ is named as $e$ and $(e,\\mathtt{r})$ as $\\underline{e}$). The \\textsc{rcn}\\ $R_3$ in Figure~\\ref{fig:rcn} is \n $\\mathsf{R}(C_4)$, with $C_4$ in Figure~\\ref{fig:occ-nets} and the set of reversible events  $U = \\{e_1,e_2,e_4\\}$.\n\\end{example}\n\n\\subsection{From \\textsc{rcn}\\ to r\\textsc{pes}}\nAs it is usually done for causal nets,\nwe now associate each reversible causal net with a reversible prime event structure.\nGiven an \\textsc{rcn}\\ $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$, we denote the\nset of events $\\setcomp{e'}{e  <_R e'}$ by $\\future{e}$. Observe that this set is not necessarily conflict-free.\n\n\\begin{proposition}\\label{pr:rce-to-rpes}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$, \n then $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ is its associated r\\textsc{pes}, where\n \\begin{itemize}\n  \\item $E' = \\overline{E}$ and $U'= h(U)$,\n  \\item $<$ is\n       \n         $<_{C_{\\overline{E}}}$.\n  \\item $\\#$ is the conflict relation defined on the occurrence net $C_{\\overline{E}}$,\n  \\item $e\\ \\triangleright\\ \\underline{e'}$ whenever $e\\in \\future{e'}$, \n  \\item $e\\ \\prec\\ \\underline{e'}$ whenever $e = e'$, and\n  \\item $\\ll = <$.         \n \n \\end{itemize}\n\\end{proposition}\n\n\n\\begin{example}\n Consider the reversible causal net $R_3$ in Figure~\\ref{fig:rcn}. The associated r\\textsc{pes}\\ has\n the events $\\setenum{e_1,e_2,e_3,e_4}$ and the reversible events $\\setenum{e_1,e_3,e_4}$. \n The causality relation of the associated p\\textsc{pes}\\ \n is $e_1 < e_3$, $e_2 < e_4$, the \n conflict relation is \\emph{generated} by $e_1 \\# e_2$, and it is inherited along $\\ll$, which \n coincides with $<$. The reverse causality stipulates that $e_1 \\prec \\underline{e}_1$, \n $e_3 \\prec \\underline{e}_3$ and $e_4 \\prec \\underline{e}_4$ and finally\n $e_3 \\triangleright \\underline{e}_1$, as to be allowed to undo $e_1$ it is necessary to\n undo $e_3$ first.\n\\end{example}\n\nThe following result states that the r\\textsc{pes}\\ associated to a reversible causal net is \ncausal, hence cause-respecting.\n\\begin{proposition}\\label{prp:rpes_respect}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$ and \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be the associated r\\textsc{pes}. \n Then $\\mathcal{C}_{r}(R)$ is a\n causal r\\textsc{pes}.\n\\end{proposition}\n\nWe show that each configuration of a \\textsc{rcn}\\ is a configuration of the corresponding\nr\\textsc{pes},  and vice versa.\n\\begin{theorem}\\label{th:rnctorpes-conf-correspond}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$ and \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be the associated r\\textsc{pes}. \n Then $X \\subseteq E'$ is a configuration of $R$ iff $X$ is a configuration of $\\mathcal{C}_{r}(R)$.\n\\end{theorem}\n\n\nWe stress that a reversing event in a reversible causal net is enabled at a marking when\nthe conditions in the postset of the event to be reversed are marked. \nThis may happen only\nwhen all the events that causally depend on the event to be reversed have either been\nexecuted and reversed or not been executed at all.\nThus every \\textsc{rcn}\\ enjoys \\emph{causally consistent} reversibility~\\cite{rccs,rhotcs}, \nand consequently cannot implement the\nso called \\textit{out-of-causal order} reversibility~\\cite{PhiUliYuen12,UK16}.\nContrastingly,  r\\textsc{pes} es are able to model \\textit{out-of-causal order} reversibility (as illustrated in  Example~\\ref{out-of-causal-order}).\n\n\n\n\nThe proposition below establishes a  correspondence between the steps in a reversible causal net\nand the sequences of reachable configurations of the r\\textsc{pes}\\  associated to that net. \nProposition~\\ref{prop:mixed_step} below formalises what is called \\emph{mixed-reverse} transitions in~\\cite{EKM:petri_net}. \nWe now introduce some auxiliary notation. Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}, and $X\\subseteq E$ be a \nconfiguration of $R$, we write  $\\mathsf{mark}(X)$ to denote the marking reached \nafter executing the events\nin $X$; this marking can be expressed as $(\\mathsf{c}\\cup\\post{X})\\setminus \\pre{X}$.\n\n\\begin{proposition}\\label{prop:mixed_step}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net and  \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be its associated r\\textsc{pes}. \n Let $X\\in\\Conf{R}{\\textsc{rcn}}$ and  $A\\subseteq E$ be a set of events such that \n $\\mathsf{mark}(X)\\trans{A}$. Then $\\hat{A}\\cup \\underline{B}$ is enabled at $X$ in\n $\\mathcal{C}_{r}(R)$, where $\\hat{A} = \\setcomp{e\\in A}{e\\not\\in U}$ and\n $\\underline{B} = \\setcomp{e\\in A}{e\\in U}$.\n\\end{proposition}\n\n\n\\endinput \n\\michelenote{to be explained better that with \\textsc{rcn}\\ the only possible associated event structure\nis a cause-respecting and causal r\\textsc{pes}, as we have to take into account that a reverse event to be\nexecuted in the net needs that its preset is marked...}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMultiple photoionization of atoms and molecules belongs to the most sensitive probes of electron correlations \\cite{SPDI-Review}. This is distinctly exemplified by the process of single-photon double ionization (SPDI) which would not exist in the absence of electron correlations. Since photoabsorption is induced by a one-body operator, the ejection of the second electron relies exclusively on the interaction among the electrons in the target.\\footnote{We note that SPDI may also proceed through a shake-off mechanism \\cite{SPDI-Review} which relies on the sudden termination of electron-electron interaction present in the initial bound state.} SPDI has most comprehensively been studied in helium, which represents the prototype atom for this process.\n\nApart from the direct channel of SPDI, double electron emission can also occur after resonant photoexcitation of an autoionizing state which lies in the double continuum \\cite{resonant-SPDI}. This channel requires a more complex atomic structure than in helium. It has been studied thoroughly in alkaline-earth metal atoms (such as Be, Mg, Ca) which exhibit a heliumlike $ns^2$ electron configuration in the outer shell \\cite{SPDI-alkali, SPDI-Mg-I, SPDI-Mg-II, SPDI-Ca-I}. Since the latter is well separated from the inner shells both spatially and energetically, these atoms may be considered as quasi-two-electron systems. Resonant SPDI has been measured, for example, around the $3p\\to 3d$ resonance \nin Ca \\cite{SPDI-Ca-exp} and the $2p\\to 3d$ resonance in Mg \\cite{SPDI-Mg-exp}.\n\nAutoionization processes can also occur when an atom is not isolated in space but in close vicinity to another atom, so that the electronic structures at both centers are coupled by long-range interactions. \nFor example, two-center electron correlations can drive radiationless decay of an inner-valence vacancy in one of the atoms, in case when a single-center Auger decay of this vacancy is energetically forbidden. In such a situation, the vacancy may decay by transferring excitation energy to the neighboring atom. This process of interatomic Coulombic decay (ICD) \\cite{ICD, ICDres, ICDrev} can be much faster than a single-center radiative decay. It has been observed in a  variety of systems, comprising noble gas dimers \\cite{dimers} and clusters \\cite{clusters}. The closely related process of two-center photoionization (2CPI) was also studied, both theoretically \\cite{2CPI, Perina} and experimentally in He-Ne dimers \\cite{2CPIexp} and Ne-Ar clusters \\cite{Hergenhahn}. A similar energy transfer process was identified in the photoionization of helium droplets doped with rare gas atoms \\cite{droplets}.\n\nInteratomic electron correlations may also lead to double ionization after single-photon absorption. Double ionization of helium dimers was observed after irratiation with $\\approx 64$\\,eV synchrotron photons \\cite{SPDI-He2}. The process was shown to rely on a knock-off reaction: the first electron, which is photoejected from one of the helium atoms, kicks out the second electron from the other helium atom in an ($e$, $2e$) collision. Double ionization of magnesium in Mg-He clusters was demonstrated to be largely enhanced due to electron-transfer-mediated decay for photon energies above the helium ionization threshold \\cite{SPDI-ETMD-theo, SPDI-ETMD-exp}. Also here, the photoabsorption first produces a He$^+$ ion which, subsequently, is neutralized via electron transfer from a Mg atom. Simultaneously, a second electron is ejected from Mg to keep the energy balance. Finally, double ICD has been predicted to occur in endohedral fullerenes \\cite{DICD-theo}. For example, an excited Mg$^+$ ($2p^53s^2$) ion embedded in C$_{60}$ can decay into its ground state, forming a Mg$^+$($2p^63s)$@C$_{60}^{2+}$ complex. A related experiment has been conducted very recently on alkali dimers such as K-Rb attached to helium droplets \\cite{DICD-exp}. Excitation energy from the helium 1$s$2$s$ state was transfered to the dimer, leading to double ionization with subsequent dissociation into K$^+$ and Rb$^+$.\n\n\\begin{figure}[t]  \n\\vspace{-0.25cm}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{2C_SPDI.eps}\n\\end{center}\n\\vspace{-0.5cm} \n\\caption{Scheme of two-center single-photon double ionization. First, atom $B$ is resonantly photoexcited. Afterwards, upon radiationless energy transfer to atom $A$ via a two-center Auger decay, the latter is doubly ionized. The process involves three active electrons and relies on both interatomic and intraatomic electron correlations, as indicated by the dashed arrows.}\n\\label{figure1}\n\\end{figure}\n\nIn the present paper, we study another interatomic mechanism of SPDI, which may proceed in two-center heteroatomic systems. The process leads to double ionization of an atom $A$ in the presence of a neighboring atom $B$. It is schematically illustrated in Fig.~\\ref{figure1}. The energy of the incident photon is assumed to be resonant with a dipole-allowed bound-bound transition in atom $B$ and to exceed the double ionization threshold of atom $A$. In a first step, atom $B$ is resonantly excited by photoabsorption, creating an autoionizing resonance state in the double continuum of the two-atomic system. Afterwards, the excitation energy is transfered via interatomic electron correlations to atom $A$ where it leads to double ionization due to intraatomic correlations between the two active electrons at center $A$.\\footnote{In the context of ICD, the second step in two-center SPDI has been termed participator double resonant interatomic Coulombic decay (pDRICD); see Fig.~2\\,a) in Ref.~\\cite{ICDres}.} We show that the resonant two-center SPDI can dominate over the direct SPDI of atom $A$ by several orders of magnitude. It can even be substantially stronger than the direct single ionization of atom $A$. \n\nOur paper is organized as follows. In Sec.~II we present our theoretical considerations of two-center SPDI. After formulating the general framework, an analytical expression for the double-to-single ionization ratio will be obtained. In Sec.~IV we illustrate our findings by some numerical examples and discuss their physical implications.  Concluding remarks are given in Sec.~IV. Atomic units (a.u.) will be used throughout unless otherwise stated.\n\n\n\\section{Theory of two-center single-photon double ionization}\n\nLet us consider a system consisting of two atoms, $A$ and $B$, separated \nby a sufficiently large distance $R$ such that their individuality \nis basically preserved. The atoms, which are initially in their ground states,  \nare exposed to a resonant electromagnetic field. The latter \nwill be treated as a classical electromagnetic wave of linear polarization, \nwhose electric field component reads \n\\begin{eqnarray}\n{\\bf F}({\\bf r},t)= {\\bf F}_0 \\cos\\left(\\omega t - {\\bf k} \\cdot {\\bf r}\\right).\n\\label{field}\n\\end{eqnarray}\nHere $\\omega = c k $ and ${\\bf k}$ \nare the angular frequency and wave vector,  \nand ${\\bf F}_0=F_0\\,{\\bf e}_z$ denotes the field strength vector\nwhich is chosen to define the $z$ direction.\n\nAssuming the atoms to be at rest,  \nwe take the position of the nucleus of atom $A$  \nas the origin and denote the coordinates \nof the nucleus of atom $B$, the two (active) electrons of atom $A$ \nand that of atom $B$ by ${\\bf R}$, ${\\bf r}_j$ ($j\\in\\{1,2\\}$) \nand ${\\bf r}_3 ={\\bf R} + \\boldsymbol{\\xi}$, \nrespectively, where $\\boldsymbol{\\xi}$ \nis the position of the electron of atom $B$ \nwith respect to its nucleus. \nLet atom $B$ have an excited state $\\chi_e$\nreachable from the ground state $\\chi_g$ by a dipole-allowed transition. \n\nThe total Hamiltonian describing the\ntwo atoms in the external electromagnetic field reads \n\\begin{eqnarray} \nH =  \\hat{H}_0 + \\hat{V}_{AB} + \\hat{W},  \n\\label{hamiltonian}  \n\\end{eqnarray} \nwhere $ \\hat{H}_0 $ is the sum of the Hamiltonians \nfor the noninteracting atoms $A$ and $B$,  \n$\\hat{V}_{AB}$ the interaction between the atoms \nand $\\hat{W} = \\hat{W}_A + \\hat{W}_B$ the interaction \nof the atoms with the electromagnetic field.\nWithin the dipole approximation and length gauge, the interaction $\\hat{W}$ reads   \n\\begin{eqnarray} \n\\hat{W} = \\sum_{j=1,2,3} {\\bf F}({\\bf r}_j={\\bf 0},t) \\cdot {\\bf r}_j\\ .\n\\label{W} \n\\end{eqnarray} \nThe two terms with $j\\in\\{1,2\\}$ compose the interaction $\\hat{W}_A$ with the electrons in atom $A$,\nwhereas the term with $j=3$ the interaction $\\hat{W}_B$ with the electron in atom $B$.\nFor electrons undergoing electric dipole transitions, the interatomic \ninteraction reads \n\\begin{eqnarray} \n\\hat{V}_{AB} &=& \\sum_{j=1,2}\\left( \\frac{{\\bf r}_j\\cdot \\boldsymbol{\\xi}}{R^3} \n- \\frac{ 3 ({\\bf r}_j\\cdot{\\bf R})(\\boldsymbol{\\xi}\\cdot{\\bf R})}{R^5} \\right)\\ .\n\\label{V_AB} \n\\end{eqnarray} \nIt is assumed that $\\omega_{ge} R \/c  \\ll 1$, where $\\omega_{ge}$ \nis the atomic transition frequency and $c$ the speed of light, such that \nretardation effects can be neglected\n\\footnote{Retardation effects in interatomic energy-transfer processes have \nrecently been studied in Refs.~\\cite{Buhmann,Salam}; see also Ref.~\\cite{2CPI}.}\n\nIn the process of two-center SPDI one has essentially three different basic two-electron configurations, which are schematically illustrated in Fig.~\\ref{figure1}:\n(I) $\\Psi_{g,g} = \\Phi_{g}({\\bf r}_1, {\\bf r}_2) \\chi_g(\\boldsymbol{\\xi})$ with total energy $E_{g,g} = \\varepsilon_0 + \\epsilon_1$, where both atoms are in the corresponding ground states $\\Phi_g$ and $\\chi_g$;\n(II) $\\Psi_{g,e} = \\Phi_{g}({\\bf r}_1, {\\bf r}_2) \\chi_e(\\boldsymbol{\\xi})$ with total energy $E_{g,e} = \\varepsilon_0 + \\epsilon_1$, in which atom $A$ is in the ground state while atom $B$ is in the excited state $\\chi_e$;\n(III) $\\Psi_{{\\bf p}_1,{\\bf p}_2,g} = \\Phi_{{\\bf p}_1,{\\bf p}_2}({\\bf r}_1,{\\bf r}_2) \\chi_g(\\boldsymbol{\\xi})$ with total energy $E_{{\\bf p}_1, {\\bf p}_2, g} = \\varepsilon_{p_1} + \\varepsilon_{p_2} + \\epsilon_0$, where both electrons of atom $A$ have been emitted into the continuum with asymptotic momenta ${\\bf p}_1$ and ${\\bf p}_2$, while the electron of atom $B$ has returned to the ground state. \n\nWithin the second order of time-dependent perturbation theory, the probability amplitude for two-center SPDI can be written as\n\\begin{eqnarray}\nS^{(2)}_{{\\bf p}_1,{\\bf p}_2}\\! &=&\\! -\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Psi_{{\\bf p}_1,{\\bf p}_2,g}|\\hat{V}_{AB}| \\Psi_{g,e} \\rangle\\, e^{-i(E_{g,e}-E_{{\\bf p}_1,{\\bf p}_2,g})t}\\nonumber\\\\\n& & \\times \\int_{-\\infty}^{t} dt'\\, \\langle \\Psi_{g,e}|\\hat{W}_{B}| \\Psi_{g,g} \\rangle\\, e^{-i(E_{g,g}-E_{g,e})t'}\n\\label{S1}\n\\end{eqnarray}\n\nBy performing the inner time integral, we obtain\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Psi_{{\\bf p}_1,{\\bf p}_2,g}|\\hat{V}_{AB}| \\Psi_{g,e} \\rangle\\,\\frac{F_0}{2} \\nonumber\\\\\n& & \\times\\,\\frac{\\langle \\chi_e|\\xi_z| \\chi_g \\rangle}{\\epsilon_0+\\omega-\\epsilon_1+\\frac{i}{2}\\Gamma}\\,e^{-i(E_{g,g}+\\omega-E_{{\\bf p}_1,{\\bf p}_2,g})t}\\ .\\nonumber\\\\\n\\label{S2}\n\\end{eqnarray}\nHere we have applied the rotating wave approximation and inserted the total width $\\Gamma=\\Gamma_r+\\Gamma_a$ of the excited state $\\chi_e$ in atom $B$. It accounts for the finite lifetime of this state and consists of the radiative width $\\Gamma_r$ and the two-center Auger (ICD) width $\\Gamma_a$. Taking also the outer time integral, we arrive at\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\pi\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)| \\Phi_g \\rangle\\cdot\\left( {\\bf e}_z - \\frac{3R_z}{R^2}\\,{\\bf R}\\right) \\nonumber\\\\\n& & \\times\\ \\frac{F_0}{R^3}\\, \\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2}{\\Delta+\\frac{i}{2}\\Gamma}\\ \\delta(\\varepsilon_{p_1} + \\varepsilon_{p_2} - \\varepsilon_0 - \\omega)\\ .\\nonumber\\\\\n\\label{S3}\n\\end{eqnarray}\nwhere the detuning from the resonance $\\Delta = \\epsilon_0+\\omega-\\epsilon_1$ has been introduced.\nThe delta function in Eq.~\\eqref{S3} displays the law of energy conservation in the process.\n\nFrom the transition amplitude we can obtain the fully differential ionization cross section in the usual way by taking the absolute square and dividing it by the interaction time $\\tau$ and the incident flux $j=\\frac{cF_0^2}{8\\pi \\omega}$, that is\n\\begin{eqnarray} \n\\frac{{\\rm d}^6\\sigma^{(2)}}{{\\rm d}^3p_1\\,{\\rm d}^3p_2} = \\frac{1}{(2\\pi)^6 j\\tau}\\,\\left|S^{(2)}_{{\\bf p}_1,{\\bf p}_2}\\right|^2\\ .\n\\label{CS}\n\\end{eqnarray}\nThe factor $(2\\pi)^{-6}$ arises from the fact that the continuum states in our calculations are normalized to a quantization volume of unity. Performing the integration over $\\varepsilon_{p_2}$ with the help of the $\\delta$-function in Eq.~\\eqref{S3}, we obtain the five-fold differential cross section\n\\begin{eqnarray} \n\\frac{{\\rm d}^5\\sigma^{(2)}}{{\\rm d}\\varepsilon_{p_1}{\\rm d}^2\\Omega_{p_1}{\\rm d}^2\\Omega_{p_2}} &=& \\frac{\\omega\\,p_1\\,p_2}{(2\\pi)^4 c R^6}\\ \\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^4}{\\Delta^2+\\frac{1}{4}\\Gamma^2}\\nonumber\\\\\n& & \\hspace{-2cm} \\times\\, \\big| \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)| \\Phi_g \\rangle \\cdot\\left( {\\bf e}_z - 3\\cos\\theta_R\\,{\\bf e}_R\\right) \\big|^2\\ ,\\nonumber\\\\\n\\label{5CS}\n\\end{eqnarray}\nwhere we have introduced the unit vector ${\\bf e}_R={\\bf R}\/R$ along the internuclear separation and the angle $\\theta_R$ between ${\\bf R}$ and the field direction.\n\nWe can draw a comparison with the direct SPDI of atom $A$ by the electromagnetic field. The corresponding probability amplitude in the first order of perturbation theory is given by \n\\begin{eqnarray}\nS^{(1)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|\\hat{W}_A| \\Phi_g \\rangle\\, e^{-i(\\varepsilon_g-\\varepsilon_{{\\bf p}_1,{\\bf p}_2})t}\\nonumber\\\\\n&=& -i\\,\\frac{F_0}{2}\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)\\cdot{\\bf e}_z| \\Phi_g \\rangle \\nonumber\\\\\n& & \\times\\ 2\\pi\\delta(\\varepsilon_{p_1} + \\varepsilon_{p_2} - \\varepsilon_0 - \\omega)\\ .\n\\label{S-1C}\n\\end{eqnarray}\nFor the special cases, when the separation vector ${\\bf R}$ between the atoms $A$ and $B$ is oriented either along the field direction or perpendicular to it, we can cast Eq.\\,\\eqref{S3} into the form\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& \\frac{\\alpha}{R^3}\\,\\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2}{\\Delta+\\frac{i}{2}\\Gamma}\\,S^{(1)}_{{\\bf p}_1,{\\bf p}_2}\n\\label{S4}\n\\end{eqnarray}\nwhere $\\alpha=-2$ for ${\\bf R}\\parallel {\\bf F}_0$ and $\\alpha=1$ for ${\\bf R}\\perp {\\bf F}_0$. When the field is exactly resonant with the transition $\\chi_g\\to\\chi_e$ in atom $B$ and the interatomic distance is sufficiently large, so that $\\Gamma_a\\ll\\Gamma_r$, this expression becomes \n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& \\frac{3\\alpha}{2i} \\left(\\frac{c}{\\omega R}\\right)^3\\,S^{(1)}_{{\\bf p}_1,{\\bf p}_2}\\ .\n\\label{S5}\n\\end{eqnarray}\nHere we have used that the radiative width is given by $\\Gamma_r=\\frac{4\\omega_{ge}^3}{3c^3}\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2$ with $\\omega_{ge}=\\epsilon_1-\\epsilon_0=\\omega$. \n\nFrom Eq.\\,\\eqref{S5} we can infer that the cross section for two-center SPDI can be largely enhanced by a factor $[c\/(\\omega R)]^6\\gg 1$ as compared with the usual one-center process of SPDI, where the neighboring atom $B$ is not involved. For example, assuming $\\omega\\approx 21.2$\\,eV (corresponding to the first excitation energy in helium) and $R=10$\\,\\AA, an enormous enhancement by 6 orders of magnitude results. This implies further that two-center SPDI can even exceed the direct single ionization of atom $A$ considerably. Because the double-to-single ionization ratio is typically of the order $10^{-2}$, an enhancement by four orders remains. We point out that, nevertheless, two-center SPDI is not the dominant ionization channel because {\\it single} ionization of atom $A$ via 2CPI is considerably stronger, being enhanced over direct single photoionization by a factor $[c\/(\\omega R)]^6$ as well \\cite{2CPI}. The ratio of two-center SPDI--to--2CPI is thus of the order $10^{-2}$, just like the ratio between the corresponding one-center processes.\n\nThe processes of two-center SPDI and direct SPDI lead to the same final state, since atom $B$ eventually returns to its ground state and thus serves as a catalyzer. Therefore, the corresponding probability amplitudes  \\eqref{S3} and \\eqref{S-1C} are generally subject to quantum interference. However, for parameters where the two-center channel strongly dominates, the interference is of minor importance and may be neglected.\n\n\n\\section{Numerical Results and Discussion}\n\nNext we illustrate some characteristic properties of two-center SPDI by numerical examples. Our focus lies on the angular distribution of the two electrons emitted from atom $A$, which strongly depends on the position ${\\bf R}$ of the neighboring atom $B$, as we will show. \n\n\\begin{figure*}[htb]\n\t\\begin{center}\n\t\\includegraphics[width=15cm]{Bild1.eps}\n\t\t\\caption{Five-fold differential cross section \\eqref{5CS} of two-center SPDI in a He-Li$^{2+}$ system (in units of 10$^{-3}$\\,a.u.). The photon energy, $\\omega=91.8$\\,eV, is resonant with the $1s\\to2p$ transition in Li$^{2+}$ and the field strength is $F_0=10^{-5}$\\,a.u. ($\\approx 5\\times 10^4$\\,V\/cm). The atoms are separated along the field direction by $R=10$\\,a.u. From (a)\\,-\\,(c) in the upper row the polar emission angle $\\theta_1$ of the first electron is varied from $0^{\\circ}$ to $45^{\\circ}$ in steps of $15^{\\circ}$, as indicated by the red arrows. Similarly, from (d)\\,-\\,(f) in the lower row it is varied further from $60^{\\circ}$ to $90^{\\circ}$. The blue line shows a polar plot of the angular distribution of the second electron. Both electrons have the same energies $\\varepsilon_{p_1}=\\varepsilon_{p_2}\\approx 6.8$\\,eV and azimuthal angles $\\varphi_1 = \\varphi_2 = 0^{\\circ}$.}\n\t\t\\label{Fig2}\n\t\\end{center}\n\\end{figure*}\n\nA simple diatomic system is used as generic model to illustrate the effects. Since most of the studies on SPDI have considered helium atoms, we also use helium as atom $A$ in our model system. For the helium ground state, we apply a radially correlated wave function of the form\n\\begin{eqnarray}\n\\Phi_g({\\bf r}_1,{\\bf r}_2) = N\\left( e^{-\\alpha r_1}e^{-\\beta r_2} + e^{-\\alpha r_2}e^{-\\beta r_1}\\right)\\ ,\n\\end{eqnarray}\nwith $\\alpha=1.18853$, $\\beta=2.18317$ determined from a variational calculation \\cite{Keller} and $N$ denoting the normalization constant. The corresponding ground state energy amounts to $\\varepsilon_0\\approx-2.876$\\,a.u. which differs from the exact value by 1\\,\\%. To describe the double continuum of helium we employ 2C wave functions, consisting of a symmetrized product of two Coulomb waves $\\psi^{(-)}_{{\\bf p}_j}({\\bf r}_j)$ \\cite{LandauQM} with nuclear charge $Z_A=2$. Final-state correlations are accounted for by inclusion of a Gamov factor $G({\\bf p}_1-{\\bf p}_2)$ \\cite{LandauQM} which depends on the relative momentum between the electrons:\n\\begin{eqnarray}\n\\Psi_{{\\bf p}_1,{\\bf p}_2}({\\bf r}_1,{\\bf r}_2)\\!\\! &=&\\!\\! \\frac{1}{\\sqrt{2}}\\Big[ \\psi^{(-)}_{{\\bf p}_1}({\\bf r}_1)\\psi^{(-)}_{{\\bf p}_2}({\\bf r}_2)+\\psi^{(-)}_{{\\bf p}_1}({\\bf r}_2)\\psi^{(-)}_{{\\bf p}_2}({\\bf r}_1) \\Big]\\nonumber\\\\\n& & \\times\\ G({\\bf p}_1-{\\bf p}_2)\\ .\n\\end{eqnarray}\nInto the fully differential ionization cross section, the absolute square $|G({\\bf p})|^2=\\frac{2\\pi}{p}\\left( {\\rm e}^{2\\pi\/p}-1\\right)^{-1}$ of the Gamov factor enters.\nWe note that calculations of one-center double ionization of helium using these wave functions show good agreement with more sophisticated theories in terms of the shape of the photoelectron angular distributions (see, e.g., \\cite{Keller, Maulbetsch1993, Macri}; the {\\it absolute} value of the differential cross section is not well reproduced, though).\nApplication of these wave functions offers the advantage that the matrix elements in Eq.~\\eqref{S3} can be evaluated analytically by standard means.\n\n\\begin{figure*}[htb]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.95]{Bild2.eps}\n\t\t\\caption{Same as Fig.~\\ref{Fig2} but this time the polar emission angle of the first electron is kept fixed ($\\theta_1=0^\\circ$) while the orientation of the Li$^{2+}$ ion relative to the He atom varies: (a) $\\theta_R=22.5^\\circ$, (b) $\\theta_R=45^\\circ$, (c) $\\theta_R=67.5^\\circ$, (d) $\\theta_R=90^\\circ$. The straight grey lines show the resulting effective polarization axis [see Eq.~\\eqref{eff-pol}]. }\n\t\t\\label{Fig3}\n\t\\end{center}\n\\end{figure*}\n\nIn order to provide sufficient energy to overcome the double ionization threshold of helium, a Li$^{2+}$ ion is assumed to constitute the neighboring atomic center $B$ in our generic model system. The latter is hydrogenlike with nuclear charge $Z_B=3$. The photon energy $\\omega=\\epsilon_1-\\epsilon_0=3.375$\\,a.u. is assumed to be resonant with the $1s\\to2p$ transition in Li$^{2+}$. For the internuclear distance the value $R=10$\\,a.u. is taken throughout. At this distance, the radiative width largely exceeds the two-center Auger width, so that $\\Gamma\\approx \\Gamma_r$. The relative enhancement between two-center SPDI in He-Li$^{2+}$ versus ordinary one-center SPDI of helium amounts to $[c\/(\\omega R)]^6\\sim 4.4\\times 10^3$.\n \nWe have calculated photoelectron angular distributions in the case of coplanar emission, with the azimuthal angles of both ejected electrons set to $\\varphi_1 = \\varphi_2 = 0$. Their polar angles $\\theta_1$ and $\\theta_2$ are measured with respect to the polarization direction of the external field \\eqref{field}, which was chosen to define the $z$ direction. We assume equal energy sharing between the electrons. \n\nFigure 2 shows polar plots of the five-fold differential cross section \\eqref{5CS} when the interatomic separation vector lies parallel to the field axis (${\\bf R}\\parallel {\\bf F}_0$). In panels (a)\\,-\\,(f) the polar emission angle of the first electron is varied stepwise from $0^{\\circ}$ to $90^{\\circ}$. We observe that the angular distributions closely agree with those known from one-center SPDI of helium, as a comparison with results from the literature reveals (see, e.g., Fig.~1 in \\cite{Brauning} for $\\theta_1\\in\\{0^{\\circ}, 30^{\\circ}, 60^{\\circ}, 90^{\\circ}\\}$ and Fig.~4\\,(c) in \\cite{Thumm} for $\\theta_1\\approx \\pm75^{\\circ}$). With the rotation of the emission angle $\\theta_1$ of the first electron, the angular distribution of the second electron changes accordingly. The repulsion of the two electrons becomes apparent this way, but it does not lead to back-to-back emission. Instead a preferred relative angle between $\\theta_1$ and $\\theta_2$ of about $100^\\circ$--$140^\\circ$ appears. \nFor symmetry reasons, during ionization along one of the coordinate axes ($\\theta_1=0^\\circ$ or 90$^\\circ$), two emission directions are equally probable for the second electron. For emission angles $\\theta_1$ in between, one direction is preferred. \n\nThe angular emission patterns are modified when the relative orientation between the atoms is changed. This is displayed in Fig.~\\ref{Fig3} where the first electron is always ejected under $\\theta_1=0^\\circ$ and the internuclear angle $\\theta_R$ is varied. A very strong sensitivity on the position of the Li ion is found.\n\n\\begin{figure*}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.97]{Bild3.eps}\n\t\t\\caption{Density plots of the five-fold differential cross section \\eqref{5CS} for two-center SPDI in a He-Li$^{2+}$ system, depending on the polar emission angles of both electrons. From (a)\\,-\\,(e) the relative interatomic orientation is varied from $\\theta_R=0^\\circ$ to $90^\\circ$ in steps of $22.5^\\circ$. Panel (f) shows the joint angular distribution when an average over the interatomic orientation is taken. The other parameters are as in Fig.~\\ref{Fig2}.}\n\t\t\\label{Fig4}\n\t\\end{center}\n\\end{figure*}\n\nThe shapes of the photoelectron angular distributions can be understood by inspection of Eqs.~\\eqref{S3} and \\eqref{S-1C}. In case of one-center SPDI, the interaction operator $({\\bf r}_1 + {\\bf r}_2)\\cdot{\\bf e}_z$ appears, with the field polarization vector ${\\bf e}_z$, which leads to the emission patterns known for this process. In case of two-center SPDI, a similar interaction operator arises, but the field polarization is replaced by an effective polarization vector\n\\begin{eqnarray}\n\t{\\bf n}_{\\text{eff}}={\\bf e}_z - 3\\cos\\theta_R\\,{\\bf e}_R\n\t\\label{eff-pol}\n\\end{eqnarray}\nwhich represents a linear combination of the field polarization and the interatomic orientation vector.\nNote that ${\\bf n}_{\\text{eff}}$ is generally not normalized but has length $|{\\bf n}_{\\text{eff}}|=(1+3\\cos^2\\theta_R)^{1\/2}$. This effective polarization vector encodes the impact of the relative position of atom $B$ on the angular distribution of two-center SPDI. \n\nThe angular spectra in Fig.~\\ref{Fig2} refer to $\\theta_R=0^\\circ$, where ${\\bf n}_{\\text{eff}}=-2\\,{\\bf e}_z$. Thus, the effective polarization is (anti)parallel to the field polarization which explains, why the photoelectron emission patterns for two-center and one-center SPDI have identical shapes. \n\nConversely, in Fig.~\\ref{Fig3}\\,(a)\\,-\\,(c) the effective polarization direction differs from the field polarization and, thus, from the emission direction of the first electron, as indicated by the straight grey lines. The angle between ${\\bf e}_z$ and the effective polarization {\\it axis} amounts to about 30$^\\circ$ in panel (a), 75$^\\circ$ in panel (b), and 60$^\\circ$ in panel (c). Accordingly, Fig.~\\ref{Fig3}\\,(a) resembles Fig.~\\ref{Fig2}\\,(c), Fig.~\\ref{Fig3}\\,(b) resembles Fig.~\\ref{Fig2}\\,(e), and Fig.~\\ref{Fig3}\\,(c) resembles Fig.~\\ref{Fig2}\\,(d), provided a proper rotating reflection is applied. [In fact, if $\\theta_R$ is changed and at the same time $\\theta_1$ is laid along the corresponding effective polarization direction, an angular distribution equivalent to Fig.~2\\,(a) results, which is only rotated by the angle $\\theta_R$ (not shown).]\nFor $\\theta_R=90^\\circ$ [see Fig.~\\ref{Fig3}\\,d)], one has ${\\bf n}_{\\text{eff}}={\\bf e}_z$ and the emission pattern coincides with the one in Fig.~\\ref{Fig2}\\,(a). Only the absolute values of the five-fold differential cross sections differ by a factor of 4, because the effective polarization vector in Fig.~\\ref{Fig2}\\,(a) has length $|{\\bf n}_{\\text{eff}}|=2$.\n\nOur findings are confirmed by Fig.~\\ref{Fig4} which shows two-dimensional density plots of the joint angular distribution of both electrons. In panels (a)\\,-\\,(e) the interatomic orientation is stepwise changed from $\\theta_R=0^\\circ$ to $90^\\circ$. The emission pattern in Fig.~\\ref{Fig4}\\,(a), where ${\\bf R}\\parallel {\\bf F}_0$ and thus ${\\bf n}_{\\text{eff}}=-2\\,{\\bf e}_z$, agrees with the corresponding result for one-center SPDI (see Fig.~4\\,(a) in \\cite{Thumm}). \nFour pronounced emission peaks appear which lie symmetrically to the vertical line defined by $\\theta_1 + \\theta_2 = 360^\\circ$. These four peaks shift gradually as $\\theta_R$ increases in panels (b)\\,-(d) and eventually reach their initial positions in Fig.~\\ref{Fig4}\\,(e), where ${\\bf R}\\perp {\\bf F}_0$ and thus ${\\bf n}_{\\text{eff}}={\\bf e}_z$ holds. Consequently, the distributions in panels (a) and (e) are identical in shape and only differ in absolute value by a factor of 4 because of the different values of $|{\\bf n}_{\\text{eff}}|$.\n\nIn an experimental situation it can be difficult to fix the interatomic orientation between atoms $A$ and $B$. If, for example, a gas of van-der-Waals dimers $A$-$B$ was used, the internuclear axis would be randomly oriented. Therefore, we have averaged our results over the interatomic orientation, which results in the angular distribution in Fig.~\\ref{Fig4}\\,(f). The four emission peaks are smeared out into four parallel emission stripes which have the same slope and now extend over the complete plot range. \n\nOur calculations in this section were performed considering a He-Li$^{2+}$ model system in order to demonstrate some general features of two-center SPDI angular distributions. One should note, however, that it is very difficult to prepare such a system for conducting a dedicated experiment. Besides, the presence of the doubly ionized lithium ion will cause energy shifts in the neighbouring helium atom, which were not taken into account here.\n\nIn view of a potential experimental observation of two-center SPDI one could employ an earth-alkaline metal, such as calcium or magnesium, as atom $A$ in conjunction with helium as atom $B$. Using photon energies $\\omega=21.2$\\,eV (23.1\\,eV), which are resonant to the $1s\\to 2p$ ($1s\\to 3p$) transition in helium, double ionization of Ca (Mg) could be probed. The corresponding thresholds are 18.0\\,eV and 22.7\\,eV, respectively \\cite{NIST}. These target systems could be realized, for example, by Ca (or Mg) atoms attached to helium droplets (see, e.g., \\cite{Ca-He} for a related experiment). Due to the smaller energy transfers and interatomic distances involved, the relative enhancement between two-center and one-center SPDI would be much larger here than in the He-Li$^{2+}$ model system considered above.\n\n\n\\section{Conclusion}\nSingle-photon double ionization in two-center systems of atoms $A$ and $B$ was studied, where resonant photoexcitation of $B$ leads to double ionization of $A$ via the combined action of interatomic and intraatomic electron-electron correlations. It was shown that, due to its resonant character, two-center SPDI can largely dominate over the ordinary one-center SPDI of atom $A$ for interatomic distances up to few nanometers. \n\nThe influence of atom $B$ was also revealed in the angular distributions of emitted photoelectrons. They depend sensitively on the direction of an effective polarization vector, which arises as linear combination of the external field polarization and the interatomic separation vector. This effective polarization direction determines the shape of angular distributions from two-center SPDI in the same way as the field polarization vector does in case of one-center SPDI. The corresponding effects were illustrated by considering a simple two-center model system.\n\nOur predictions could be tested, for example, in Ca-He or Mg-He systems, which possess low double-ionization thresholds. When the interatomic orientation is not fixed but randomly distributed in the target, the correspondingly averaged photoelectron angular distribution of two-center SPDI looks qualitatively different from the one-center case. This characteristic feature could help to experimentally identify the two-center process, in addition to its strong resonant enhancement over one-center SPDI in terms of the total cross section.\n\n\\section*{Acknowledgement}\nThis work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No. 349581371 (MU 3149\/4-1 and VO 1278\/4-1). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe occurrence of spontaneous symmetry breaking in atomic nuclei leads to \nvarious nuclear shapes\nwhich can usually be described by the parametrization of the nuclear surface or the\nnucleon density distribution~\\cite{Bohr1998_Nucl_Structure_1,Ring1980}.\nIn mean-field calculations, the following parametrization\n\\begin{equation}\n \\beta_{\\lambda\\mu} = {4\\pi \\over 3AR^\\lambda} \\langle Q_{\\lambda\\mu} \\rangle,\n \\label{eq:01}\n\\end{equation}\nis usually used, where $Q_{\\lambda\\mu}$ are the mass multipole operators.\nIn Fig.~\\ref{Pic:deformations}, a schematic show is given for some typical nuclear shapes.\nThe majority of observed nuclear shapes is of spheroidal form which\nis usually described by $\\beta_{20}$.\nHigher-order deformations with $\\lambda>2$ such as $\\beta_{30}$\nalso appear in certain atomic mass regions~\\cite{Butler1996_RMP68-349}.\nIn addition, non-axial shapes in atomic nuclei, in particular,\nthe nonaxial-quadrupole (triaxial) deformation $\\beta_{22}$ has been\nstudied both experimentally and\ntheoretically~\\cite{Starosta2001_PRL86-971,Odegard2001_PRL86-5866,Meng2010_JPG37-064025}.\nThe influence of the nonaxial octupole $\\beta_{32}$ deformation\non the low-lying spectra has been also investigated~\\cite{Hamamoto1991_ZPD21-163,Skalski1991_PRC43-140,%\nLi1994_PRC49-R1250,Takami1998_PLB431-242,Yamagami2001_NPA693-579,%\nDudek2002_PRL88-252502,Dudek2006_PRL97-072501,Olbratowski2006_IJMPE15-333,%\nZberecki2006_PRC74-051302R,Dudek2010_JPG37-064032}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{1.0\\columnwidth}{!}{%\n \\includegraphics{deformations} }\n\\end{center}\n\\caption{\\label{Pic:deformations}(Color online)\nA schematic show of some typical nuclear shapes. \nFrom left to right, the 1st row: (a) Sphere, (b) Prolate spheroid,\n(c) Oblate spheroid, (d) Hexadecapole shape, \nand the second row: (e) Triaxial ellipsoid,\n(f) Reflection symmetric octupole shape, (g) Tetrahedron,\n(h) Reflection asymmetric octupole shape with very large quadrupole deformation\n    and large hexadecapole deformation.\nTaken from Ref.~\\cite{Lu2012_PhD}.\n}\n\\end{figure}\n\nIn nuclear fission study,\nvarious shape degrees of freedom play important and different roles\nin the occurrence and in determining the heights of the inner and outer barriers\nin actinide nuclei (in these nuclei double-humped fission barriers usually appear).\nFor example, the inner fission barrier is usually lowered when the triaxial\ndeformation is allowed, while for the outer barrier the reflection asymmetric\n(RA) shape is favored~\\cite{Pashkevich1969_NPA133-400,Moeller1970_PLB31-283,%\nGirod1983_PRC27-2317,Rutz1995_NPA590-680,Abusara2010_PRC82-044303,%\nPrassa2012_PRC86-024317,Prassa2013_PRC88-044324}.\n\nIn order to give a microscopic and self-consistent description of\nthe potential energy surface (PES) with more shape degrees of freedom included,\nmulti-dimensional constrained covariant density functional theories were developed\nrecently~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}.\nIn these theories, all shape degrees of freedom $\\beta_{\\lambda\\mu}$ \nwith even $\\mu$ are allowed.\nIn this contribution, we present two recent applications of these theories:\nthe PES's of actinide nuclei and\nthe non-axial reflection-asymmetric $\\beta_{32}$ shape in some transfermium\nnuclei.\nIn Section~\\ref{sec:formalism}, the formalism of our multi-dimensional constrained\ncovariant density functional theories will be given briefly.\nThe results and discussions are presented in Section~\\ref{sec:results}.\nFinally we give a summary in Section~\\ref{sec:summary}.\n\n\n\\section{Formalism}\n\\label{sec:formalism}\n\nThe details of the formalism for covariant density functional theories\ncan be found in Refs.~\\cite{Serot1986_ANP16-1,Reinhard1989_RPP52-439,%\nRing1996_PPNP37-193,Vretenar2005_PR409-101,Meng2006_PPNP57-470,Niksic2011_PPNP66-519}.\nThe CDFT functional in our multi-dimensional constrained calculations\ncan be one of the following four forms:\nthe meson exchange or point-coupling nucleon interactions combined with\nthe non-linear or density-dependent couplings~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,%\nLu2012_EPJWoC38-05003,Lu2013_arXiv1304.6830}.\nHere we show briefly the one corresponding to the non-linear point coupling\n(NL-PC) interactions.\nThe starting point of the relativistic\nNL-PC density functional is the following Lagrangian:\n\\begin{equation}\n \\mathcal{L} = \\bar{\\psi}(i\\gamma_{\\mu}\\partial^{\\mu}-M)\\psi\n              -\\mathcal{L}_{{\\rm lin}}\n              -\\mathcal{L}_{{\\rm nl}}\n              -\\mathcal{L}_{{\\rm der}}\n              -\\mathcal{L}_{{\\rm Cou}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n \\mathcal{L}_{{\\rm lin}} & = & \\frac{1}{2} \\alpha_{S} \\rho_{S}^{2}\n                              +\\frac{1}{2} \\alpha_{V} \\rho_{V}^{2}\n                              +\\frac{1}{2} \\alpha_{TS} \\vec{\\rho}_{TS}^{2}\n                              +\\frac{1}{2} \\alpha_{TV} \\vec{\\rho}_{TV}^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm nl}}  & = & \\frac{1}{3} \\beta_{S} \\rho_{S}^{3}\n                              +\\frac{1}{4} \\gamma_{S}\\rho_{S}^{4}\n                              +\\frac{1}{4} \\gamma_{V}[\\rho_{V}^{2}]^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm der}} & = & \\frac{1}{2} \\delta_{S}[\\partial_{\\nu}\\rho_{S}]^{2}\n                              +\\frac{1}{2} \\delta_{V}[\\partial_{\\nu}\\rho_{V}]^{2}\n                              +\\frac{1}{2} \\delta_{TS}[\\partial_{\\nu}\\vec{\\rho}_{TS}]^{2}\n \\nonumber \\\\\n &  & \\mbox{}                 +\\frac{1}{2} \\delta_{TV}[\\partial_{\\nu}\\vec{\\rho}_{TV}]^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm Cou}} & = & \\frac{1}{4} F^{\\mu\\nu} F_{\\mu\\nu}\n                             +e\\frac{1-\\tau_{3}}{2} A_{0} \\rho_{V} ,\n\\label{eq:lagrangian}\n\\end{eqnarray}\nare the linear coupling, nonlinear coupling, derivative coupling,\nand the Coulomb part, respectively.\n$M$ is the nucleon mass, $\\alpha_{S}$, $\\alpha_{V}$, $\\alpha_{TS}$,\n$\\alpha_{TV}$, $\\beta_{S}$, $\\gamma_{S}$, $\\gamma_{V}$, $\\delta_{S}$,\n$\\delta_{V}$, $\\delta_{TS}$, and $\\delta_{TV}$ are coupling constants\nfor different channels and $e$ is the electric charge.\n$\\rho_{S}$, $\\vec{\\rho}_{TS}$, $\\rho_{V}$, and $\\vec{\\rho}_{TV}$ are the isoscalar density,\nisovector density, time-like components of isoscalar current, and time-like components of\nisovector current, respectively.\nThe densities and currents are defined as\n\\begin{eqnarray}\n      \\rho  _{S} = \\bar{\\psi}\\psi , & \\qquad &\n \\vec{\\rho}_{TS} = \\bar{\\psi}\\vec{\\tau}\\psi ,\n \\nonumber \\\\\n      \\rho_{V} = \\bar{\\psi} \\gamma^{0} \\psi , & \\qquad &\n \\vec{\\rho}_{TV} = \\bar{\\psi} \\vec{\\tau} \\gamma^{0} \\psi.\n \\label{eq:densities}\n\\end{eqnarray}\nStarting from the above Lagrangian, using the Slater determinants as\ntrial wave functions and neglecting the Fock term as well as the contributions\nto the densities and currents from the negative energy levels, one\ncan derive the equations of motion for the nucleons,\n\\begin{equation}\n \\hat{h}\\psi_{i} = \\left(\\bm{\\alpha}\\cdot\\vec{p}+\\beta(M+S(\\vec{r}))+V(\\vec{r})\\right)\\psi_{i}\n                 = \\epsilon_{i}\\psi_{i}\n ,\n\\end{equation}\nwhere the potentials $V(\\bm{r})$ and $S(\\bm{r})$ are calculated as\n\\begin{eqnarray}\nS & = & \\alpha_{S}\\rho_{S}+\\beta_{S}\\rho_{S}^{2}+\\gamma_{S}\\rho_{S}^{3}+\\delta_{S}\\triangle\\rho_{S}\\nonumber \\\\\n &  & +\\left(\\alpha_{TS}\\rho_{TS}+\\delta_{TS}\\triangle\\rho_{TS}\\right)\\tau_{3}  ,\\\\\nV & = & \\alpha_{V}\\rho_{V}+\\gamma_{V}\\rho_{V}^{3}+\\delta_{V}\\triangle\\rho_{V}W\\nonumber \\\\\n &  & +\\left(\\alpha_{TV}\\rho_{TV}+\\delta_{TV}\\triangle\\rho_{TV}\\right)\\tau_{3}  .\n\\end{eqnarray}\n\nAn axially deformed harmonic oscillator (ADHO) basis is adopted for solving the\nDirac equation~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,Lu2011_PRC84-014328}.\nThe ADHO basis is defined as the eigen solutions of the Schrodinger\nequation with an ADHO potential~\\cite{Gambhir1990_APNY198-132,Ring1997_CPC105-77},\n\\begin{eqnarray}\n \\left[ -\\frac{\\hbar^{2}}{2M} \\nabla^{2} + V_{B}(z,\\rho) \\right] \\Phi_{\\alpha}(\\bm{r}\\sigma)\n & = &\n E_{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma)\n ,\n \\label{eq:BasSchrodinger-1}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n V_{B}(z,\\rho) = \\frac{1}{2} M ( \\omega_{\\rho}^{2}\\rho^{2} + \\omega_{z}^{2}z^{2} )\n ,\n\\end{equation}\nis the axially deformed HO potential and $\\omega_{z}$ and $\\omega_{\\rho}$\nare the oscillator frequencies along and perpendicular to $z$ axis, respectively.\nThese basis states are also eigen functions of the $z$ component of the\nangular momentum $j_{z}$ with eigen values $K=m_{l}+m_{s}$.\nFor any basis state $\\Phi_{\\alpha}(\\bm{r}\\sigma)$, the time reversal state\nis defined as $\\Phi_{\\bar{\\alpha}}(\\bm{r}\\sigma)=\\mathcal{T}\\Phi_{\\alpha}(\\bm{r}\\sigma)$,\nwhere $\\mathcal{T}=i\\sigma_{y}K$ is the time reversal operator and\n$K$ is the complex conjugation.\nApparently we have $K_{\\bar{\\alpha}}=-K_{\\alpha}$\nand $\\pi_{\\bar{\\alpha}}=\\pi_{\\alpha}$.\nThese basis states form a complete set for expanding any two-component spinors.\nFor a Dirac spinor with four components,\n\\begin{equation}\n \\psi_{i}(\\bm{r}\\sigma) =\n \\left( \\begin{array}{c}\n        \\sum_{\\alpha}f_{i}^{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma) \\\\\n        \\sum_{\\alpha}g_{i}^{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma)\n        \\end{array}\n \\right),\n\\end{equation}\nwhere the sum runs over all the possible combination of the quantum\nnumbers $\\alpha=\\{n_{z},n_{r},m_{l},m_{s}\\}$, and $f_{i}^{\\alpha}$ and\n$g_{i}^{\\alpha}$ are the expansion coefficients.\nIn practical calculations, one should truncate the basis\nin a proper way~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,Lu2011_PRC84-014328}.\n\nThe nucleus is assumed to be symmetric under the $V_4$ group, that is, all the\npotentials and densities can be expanded as \n\\begin{equation}\n f(\\rho,\\varphi,z) =  f_{0}(\\rho,z) \\frac{1}{\\sqrt{2\\pi}}\n+ \\sum_{n=1}^{\\infty} f_{n}(\\rho,z) \\frac{1}{\\sqrt{\\pi}}\\cos(2n\\varphi),\n\\end{equation}\n\nThe PES is obtained by the constrained self-consistent calculation,\n\\begin{equation}\n E^{\\prime} = E_{{\\rm RMF}} +\n              \\sum_{\\lambda\\mu} \\frac{1}{2} C_{\\lambda\\mu}Q_{\\lambda\\mu} ,\n\\end{equation}\nwhere the variables $C_{\\lambda\\mu}$'s change their values during the iteration.\n\nBoth the BCS approach and the Bogoliubov transformation are implemented\nin our model to take into account the pairing effects.\nFor convenience, we name the MDC-CDFT with the BCS approach for the pairing \nas the MDC-RMF model and that with the Bogoliubov transformation as the MDC-RHB model.\nMore details of the multi-dimensional constraint covariant density\nfunctional theories can be found in Refs.~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}.\n\n\n\\section{Results and discussions}\n\\label{sec:results}\n\n\\subsection{PES's of actinides} \n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.9\\columnwidth}{!}{%\n \\includegraphics{Pu240PES1d.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240-1d}(Color online)\nPotential energy curves of $^{240}$Pu with various self-consistent symmetries imposed.\nThe solid black curve represents the calculated fission path with $V_4$ symmetry imposed:\nthe red dashed curve is that with axial symmetry (AS) imposed,\nthe green dotted curve that with reflection symmetry (RS) imposed,\nthe violet dot-dashed line that with both symmetries (AS \\& RS) imposed.\nThe empirical inner (outer) barrier height $B_\\mathrm{emp}$ is denoted by the grey square (circle).\nThe energy is normalized with respect to the binding energy of the ground state.\nThe parameter set used is PC-PK1.\nTaken from Ref.~\\cite{Lu2012_PRC85-011301R}.\n}\n\\end{figure}\n\nIn Refs.~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}, one- (1-d),\ntwo- (2-d), and three-dimensional (3-d) constrained calculations were \nperformed for the actinide nucleus $^{240}$Pu.\nThe MDC-RMF model with the parameter set PC-PK1~\\cite{Zhao2010_PRC82-054319} was used.\nIn Fig.~\\ref{Pic:PU240-1d} we show the 1-d potential energy curves \nfrom an oblate shape with $\\beta_{20}$ about $-0.2$ to the fission\nconfiguration with $\\beta_{20}$ beyond 2.0\nwhich are obtained from calculations with different self-consistent symmetries\nimposed: the axial (AS) or triaxial (TS) symmetries combined with\nreflection symmetric (RS) or asymmetric cases.\nThe importance of the triaxial deformation on the inner barrier and\nthat of the octupole deformation on the outer barrier are clearly seen:\nThe triaxial deformation reduces the inner barrier height by more than 2 MeV\nand results in a better agreement with the empirical value~\\cite{Abusara2010_PRC82-044303};\nthe RA shape is favored beyond the fission isomer and lowers very much\nthe outer fission barrier~\\cite{Rutz1995_NPA590-680}.\nBesides these features, it was found for the first time that the outer\nbarrier is also considerably lowered by about 1 MeV when the triaxial\ndeformation is allowed.\nIn addition, a better reproduction of the empirical barrier height can be seen for\nthe outer barrier.\nIt has been stressed that this feature can only be found when the axial and\nreflection symmetries are simultaneously broken~\\cite{Lu2012_PRC85-011301R}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.95\\columnwidth}{!}{%\n \\includegraphics{Pu240_Inner.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240_2dI}(Color online)\nPotential energy surfaces of $^{240}$Pu in the $(\\beta_{20},\\beta_{22})$ plane\naround the inner barrier.\nThe energy is normalized with respect to the binding energy of the ground state.\nThe least-energy fission path is represented by a dash-dotted line.\nThe saddle point is denoted by the full star.\nThe contour interval is 0.5 MeV.\n}\n\\end{figure}\n\nTwo-dimensional PES's in the $(\\beta_{20},\\beta_{22})$ plane near the inner and \nouter barriers are shown in Figs.~\\ref{Pic:PU240_2dI} and~\\ref{Pic:PU240_2dO}, respectively.\nStarting from the axially symmetric ground state, the nucleus goes through the \ntriaxial valley to the isometric state. \nThe inner barrier is located at $\\beta_{20} \\approx 0.65$ and $\\beta_{22} \\approx 0.06$. \nThe isomeric state keeps an axially symmetric shape.\nAs $\\beta_{20}$ further increases, the nucleus goes through a triaxial valley\nagain, and then goes fission. \nThe outer barrier is located at $\\beta_{20} \\approx 1.21$, $\\beta_{22} \\approx 0.02$, \nand $\\beta_{30} \\approx 0.37$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.95\\columnwidth}{!}{%\n \\includegraphics{Pu240_Outer.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240_2dO}(Color online)\nPotential energy surfaces of $^{240}$Pu in the $(\\beta_{20},\\beta_{22})$ plane\naround the outer barrier.\nThe energy is normalized with respect to the binding energy of the ground state.\nThe least-energy fission path is represented by a dash-dotted line.\nThe saddle point is denoted by the full star.\nThe contour interval is 0.25 MeV.\n}\n\\end{figure}\n\nA systematic study of even-even actinide nuclei has been carried out\nand the results were presented in Ref.~\\cite{Lu2013_arXiv1304.2513}\nwhere we have shown that the triaxial deformation lowers\nthe outer barriers of these actinide nuclei by about $0.5 \\sim 1$ MeV \n(about $10 \\sim 20 \\%$ of the barrier height).\n\n\\subsection{$Y_{32}$-correlations in $N=150$ isotones}\n\nIt has been anticipated that the tetrahedral shape \n($\\beta_{\\lambda\\mu} = 0$, if $\\lambda\\ne3$ and $\\mu\\ne2$) may appear in the ground states of\nsome nuclei with special combinations of the neutron and\nproton numbers~\\cite{Li1994_PRC49-R1250,Dudek2010_JPG37-064032,Dudek2002_PRL88-252502}.\nThe tetrahedral symmetry-driven quantum effects may also lead to a large increase \nof binding energy in superheavy nuclei~\\cite{Chen2013_NPR30-278}. \nHowever, no solid experimental evidence has been found for nuclei with\ntetrahedral shapes.\nOn the other hand, $\\beta_{32}$ deformation may appear together with other shape\ndegrees of freedome, say, $\\beta_2$.\nFor example, it has been proposed that the non-axial\noctupole $Y_{32}$-correlation results in the experimentally observed low-energy $2^-$\nbands in the $N = 150$ isotones~\\cite{Robinson2008_PRC78-034308} and the RASM calculations\nreproduces well the experimental observables of these $2^-$ bands~\\cite{Chen2008_PRC77-061305R}.\n\nIn Ref.~\\cite{Zhao2012_PRC86-057304} the non-axial reflection-asymmetric $\\beta_{32}$\ndeformation in $N=150$ isotones, namely $^{246}$Cm, $^{248}$Cf,\n$^{250}$Fm, and $^{252}$No was investigated using the MDC-RMF model\nwith the parameter set DD-PC1~\\cite{Niksic2008_PRC78-034318}.\nIt was found that \ndue to the interaction between a pair of neutron orbitals,\n$[734]9\/2$ originating from $\\nu j_{15\/2}$ and\n$[622]5\/2$ originating from $\\nu g_{9\/2}$, and\nthat of a pair of proton orbitals,\n$[521]3\/2$ originating from $\\pi f_{7\/2}$ and\n$[633]7\/2$ originating from $\\pi i_{13\/2}$,\nrather strong non-axial octupole $Y_{32}$ effects appear in $^{248}$Cf and\n$^{250}$Fm which are both well deformed with large axial-quadrupole deformations,\n$\\beta_{20} \\approx 0.3$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.90\\columnwidth}{!}{%\n \\includegraphics{Cf248_b32_Pair_bin} }\n\\end{center}\n\\caption{\\label{fig:b32} (Color online)\nThe binding energy $E$ (relative to the ground state) for $^{248}$Cf\nas a function of the non-axial octupole deformation parameter $\\beta_{32}$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:b32}, the potential energy curve,\ni.e., the total binding energy as a function of $\\beta_{32}$ was shown\nfor $^{248}$Cf.\nAt each point of the potential energy curve, the energy is minimized\nautomatically with respect to other shape degrees\nof freedom such as $\\beta_{20}$, $\\beta_{22}$, $\\beta_{30}$, and $\\beta_{40}$, etc.\nOne finds in this curve a clear pocket with the depth more than 0.3 MeV.\nSimilar potential energy curve was also obtained for $^{250}$Fm.\nFor $^{246}$Cm and $^{252}$No, only a shallow minimum develops along\nthe $\\beta_{32}$ shape degree of freedom.\nIt was also shown that the evolution of the non-axial octupole $\\beta_{32}$ effect along\nthe $N=150$ isotonic chain is not very sensitive to the form of\nthe energy density functional and the parameter set we used~\\cite{Zhao2012_PRC86-057304}.\n\nBoth the non-axial octupole parameter $\\beta_{32}$ and the energy gain due to\nthe $\\beta_{32}$-distortion reach maximal values at $^{248}$Cf \nin the four nuclei along the $N=150$ isotonic chain~\\cite{Zhao2012_PRC86-057304}.\nThis is consistent with the analysis given in\nRefs.~\\cite{Chen2008_PRC77-061305R,Jolos2011_JPG38-115103} and the experimental observation\nthat in $^{248}$Cf, the $2^-$ state is the lowest among these\nnuclei~\\cite{Robinson2008_PRC78-034308}.\nThese results indicate a strong $Y_{32}$-correlation in these nuclei.\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this contribution we present the formalism and some applications of the multi-dimensional\nconstrained relativistic mean field (MDC-RMF) models in which all shape degrees of freedom\n$\\beta_{\\lambda\\mu}$ with even $\\mu$ are allowed.\nThe potential energy surfaces (curves) of actinide nuclei \nand the effect of the triaxiality on the first and second fission barriers \nwere investigated.\nIt is found that besides the octupole deformation, the triaxiality also\nplays an important role upon the second fission barriers.\nThe non-axial reflection-asymmetric $\\beta_{32}$ shape in $N=150$ isotones were studied\nand rather strong non-axial octupole $Y_{32}$ effects have been found in $^{248}$Cf and\n$^{250}$Fm which are both well deformed with large axial-quadrupole deformations,\n$\\beta_{20} \\approx 0.3$.\n\n\n\\begin{ack}\nThis work has been supported by Major State Basic Research Development \nProgram of China (Grant No. 2013CB834400), \nNational Natural Science Foundation of China (Grant Nos. 11121403, 11175252, \n11120101005, 11211120152, and 11275248), \nthe Knowledge Innovation Project of Chinese Academy of Sciences (Grant No. KJCX2-EW-N01). \nThe results described in this paper are obtained on the ScGrid of Supercomputing\nCenter, Computer Network Information Center of Chinese Academy of Sciences.\n\\end{ack}\n\n\n\n\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nIn the past years, our group has extensively used the {\\it Hubble Space Telescope} ({\\it HST}\\,) archive to study star clusters in both Magellanic Clouds \\citep[e.g.][and references therein]{milone2009a, milone2020a}. The exquisite stellar photometry and astrometry provided by {\\it HST}, together with the most advanced techniques for the analysis of astronomical \nimages \\citep[e.g.][]{anderson2008a, sabbi2016a, bellini2017a}, has provided significant advances in understanding Magellanic-Cloud star clusters and their stellar populations.\n\nInspired by the discovery that the color-magnitude diagram (CMD) of the Large Magellanic Cloud (LMC) cluster NGC\\,1806  is not consistent with a single isochrone \\citep{mackey2007a}, we started a series of papers to investigate the so-called extended-main sequence turn off phenomenon (eMSTO), in clusters with ages from about one to 2.3\\,Gyr.    \n The main results include the discovery that the eMSTO is a common feature of Magellanic Cloud clusters \\citep{milone2009a}, the early discoveries of split main sequence (MS) in young Magellanic Cloud clusters \\citep{milone2013a, milone2015a, milone2016a, milone2017a}, and the characterization of the multiple populations in young and intermediate-age LMC and Small Magellanic Cloud (SMC) clusters \\citep{milone2018a}. We provided the first direct evidence, based on high-resolution spectra, that the blue and red MS are made up of stellar populations with different rotation rates \\citep{marino2018a} and the color and magnitude of an eMSTO star depend on stellar rotation  \\citep{dupree2017a, marino2018a}.    \n  The high-precision photometry resulting from this project has been instrumental to shed light on the physical mechanisms that are responsible for generating multiple populations in young clusters and has been used both by our team and by other groups to constrain the effect of rotation and stellar mergers on the eMSTO and the split MS \\citep[e.g.][]{bastian2009a, dantona2015a, dantona2017a, wang2022a, cordoni2022a} and the contribution of variable stars on the eMSTO \\citep{salinas2018a}.\n \n Although our main purpose consisted in investigating the eMSTO phenomenon, the resulting photometric and astrometric catalogs have  been used for various investigations of stellar astrophysics, including multiple stellar populations in Magellanic Cloud globular clusters \\citep[GCs,][]{lagioia2019a, lagioia2019b, milone2020a, dondoglio2021a}, photometric binaries \\citep[][]{milone2009a, milone2013a}, Be stars \\citep[][]{milone2018a, hastings2021a}, and extinction \\citep{demarchi2020a}.\n \n Driven by these results, we decided to homogeneously analyze all archival images collected with the Ultraviolet and Visual Channel (UVIS) and the Near Infrared Channel (NIR) of Wide Field Camera 3 (WFC3) and with the Wide Field Channel of the Advanced Camera for Surveys (WFC\/ACS) on board {\\it HST}.    \n  In this work, we present high-precision stellar positions and magnitudes for stars in 101 fields of Magellanic Clouds that include 113 star clusters.\n \n \n The paper is organized as follows. Section \\ref{sec:data} describes the dataset and the methods used for homogeneously reducing the data and presents the CMDs. The methods for correcting the photometry for differential reddening and the differential-reddening maps are discussed in Section \\ref{sec:reddening}, while Section \\ref{sec:centri} is dedicated to the determination of the cluster centers. \n Absolute stellar proper motions are derived in Section \\ref{sec:pms}. Section \\ref{sec:secrets}\n provides some scientific cases that arise from early inspection of our catalogs. Finally, we report in the appendix the serendipitous discovery of one gravitational lens and two stellar clusters.\n \n\\section{Data and data analysis}\\label{sec:data}\nThe dataset used in this paper comprises images collected through the UVIS\/WFC3 and NIR\/WFC3, and WFC\/ACS on board {\\it HST}.\nThe images include 84 known star clusters in the  Large Magellanic Cloud (LMC) and 29 clusters in the Small Magellanic Cloud (SMC). These clusters span wide intervals of age and stellar density, from sparse star-forming regions to old and dense GCs. \n The main properties of the available exposures are listed in Table\\,\\ref{tab:data}.\n\nPhotometry and astrometry are obtained from calibrated, flat-fielded WFC3\/NIR ({\\rm \\_flt}) exposures, while in the case of UVIS\/WFC3 and WFC\/ACS data we used the calibrated, flat-fielded exposures corrected for the effects of the poor charge-transfer efficiency (CTE) of the detectors \\citep[{\\rm \\_flc,}][]{anderson2010a}.\nStars are measured by means of distinct approaches\n that work best in different brightness regimes, \n as discussed in the following subsections. \n\n\\subsection{First-pass photometry}\n\nWe accounted for spatial variations of the Point-Spread Function (PSF) by using the grids of library PSFs provided by Jay Anderson for each filter and camera. The PSFs can change from one exposure to another due to focus variations produced by the breathing of {\\it HST}, small guiding inaccuracies, and residual CTE. To derive the optimal PSF we perturbed the library PSFs by using a version of the \\citet{anderson2006a} computer program adapted to UVIS\/WFC3 and WFC\/ACS \\citep[see also][]{bellini2013a}. In a nutshell, we divided each image into a grid containing $n \\times n$ cells, with $n$ ranging from 1 to 5. Bright, isolated, and unsaturated stars within each cell are fitted by the library PSF model, and the residuals of the fit are iteratively used to improve the PSF model itself.\nWe calculated the appropriate PSF model of each star  based on its location in the detector by linearly interpolating the four nearest PSFs of the grid \\citep{anderson2000a}.\nThe number of cells in the grid has been fixed with the aim of obtaining the best quality-fit parameters for bright stars and depends on the number of available reference stars used to constrain the PSF perturbation in the cell. \n\n\nThese PSFs are then used to measure the magnitudes and positions of unsaturated stars in each image. Saturated stars in the UVIS\/WFC3 and WFC\/ACS images are measured using the methods by \\citet{gilliland2004a} and \\citet{gilliland2010a}. These authors noted that the total number of electrons of saturated stars in the UVIS\/WFC3 and WFC\/ACS detectors is conserved and this information is preserved in the ${\\rm \\_flt}$ images with gain=2.\nHence, we measured each saturated star in an aperture of 5-pixel radius and added the contiguous saturated pixels that had bled outside this radius \\citep[see][for details]{anderson2008a}. \n\nAll catalogs derived from each filter and camera have been tied to the same photometric zero point, corresponding to the zero point of the deepest exposure in the filter that we used as a reference frame to construct the photometric  master frame. To do this, we used the bright, unsaturated stars that are well-fitted by the PSF to calculate the  difference between the magnitudes in the master frame and in each exposure. We used the mean of these magnitude differences to transform stars measured in each exposure into this reference frame.\n\nStellar positions are corrected for geometric distortion by using the solutions provided by \\citet{anderson2006b} for WFC\/ACS and \\citet{bellini2009a} and \\citet{bellini2011a} for UVIS\/WFC3.\nThe coordinates of stars in all images of each cluster are transformed  into a common reference system based on Gaia Early Data Release 3 (eDR3) catalogs  \\citep[][]{gaia2020a}, in such a way that the abscissa and the ordinate are aligned with the West and North direction, respectively.  \nWe first de-projected the right ascension and declination into the plane tangential to the center of the main cluster in the field. \nWe assumed  for these coordinates a scale factor of 0.04 arcsec per pixel. \nWe first used bright, unsaturated stars that are well-fitted by the PSF to derive the six-parameter linear transformations used to convert the coordinates of all stars in each exposure into this reference frame.   \nThen, we derived the $3\\sigma$-clipped average stellar positions to derive a new astrometric catalog, that we used as a master frame to improve the transformations.  \n\n\\subsection{Multi-pass photometry}\nThe main outcomes from first-pass photometry, including  PSF models, coordinate transformations, photometric zero points, stellar magnitudes, and positions, are used to simultaneously identify and measure all point-like sources in all exposures.   \nTo do this, we used the FORTRAN computer program KS2 developed by Jay Anderson \\citep[e.g.\\,][]{sabbi2016a, bellini2017a, nardiello2018a}, which is the evolution of {\\it kitchen sink}, originally written to reduce WFC\/ACS images  \\citep{anderson2008a}. \nKS2 exploits various iterations to find and measure stars. It first identifies the brightest and most isolated stars, calculates their fluxes and positions, and subtracts them from the image. In the subsequent iterations, it finds, measures, and subtracts  stars that are gradually fainter and closer to neighbor stars. \n We used the stellar positions and magnitudes derived from first-pass photometry to generate appropriate masks for bright stars, including saturated ones. These masks optimize the detection and measurement of faint sources that are close to bright stars. They also minimize the detection of spurious sources that are typically associated with diffraction spikes and other structures of the stellar profile.  \n\nThis program adopts three distinct methods to measure stars, each providing optimal photometry for different ranges of stellar luminosity and density.  \n\\begin{itemize}\n    \\item Method I is optimal for relatively bright stars. It provides accurate measurements of all stars that generate distinct peaks within their local 5$\\times$5-pixel raster after neighbor stars are subtracted. \n    Each star is measured by using the PSF model corresponding to its position, while the sky level is estimated from the annulus between 4 and 8 pixels from the center of the star.\n    \\item Method II provides the best photometry for faint stars, which do have not enough flux to provide robust fits with the PSF. After subtracting neighbor stars, KS2 performs the aperture photometry of the star in the $5\\times5$ pixel raster. Each pixel is properly weighted to ensure low weight to those pixels contaminated by nearby stars. The sky is calculated as in Method I.\n    \\item Method III provides the best photometry in very crowded regions and for faint stars when a large number of exposures are available. It works as Method II, but aperture photometry is calculated over a circle with a radius of 0.75 pixels and the local sky in the annulus between 2 and 4 pixels from the position measured during the finding stage. \n\\end{itemize}\n    Stellar fluxes and positions are measured in each exposure separately and then are properly averaged together to derive our best determinations of magnitudes and positions. \n\n    Figure\\,\\ref{fig:methods} compares the CMDs of stars in the field of view of Lindsay\\,1 obtained from the three methods. We have chosen this GC as an example because of the deep F275W and F814W photometry available, which comprises 16 and 7 exposures in F275W and F814W, with total integration times of 27,341s and  2,206s, respectively. \n    A visual comparison of the top panels reveals that methods\\,II and III are optimal for faint stars as they provide well-defined MSs. The latter method provides slightly better photometry for stars at the bottom of the MS alone, whereas Method\\,II provides the best photometry for the remaining faint MS stars as highlighted in the middle panels, where we show the zoomed CMDs for MS stars with instrumental $-6<$F275W$<-4$ mag\\footnote{Instrumental magnitudes are defined as the $-$2.5 log$_{10}$ of the detected photo-electrons.}. Clearly, the MS plotted in the central panel is much narrower and better defined than that shown in the left and right panels.\n    On the contrary, Method I provides the best photometry for stars with bright instrumental magnitudes as demonstrated by the narrow red-giant and sub-giant branch (RGB and SGB) sequences in the bottom-left F555W vs.\\,F555W$-$F814W CMDs.\n For each field, we derived three distinct catalogs from Methods\\,I, II, and III. The science results shown in this paper, which are all focused on bright stars, are based on the photometry derived from Method\\,I.\n\n\\subsection{Photometry calibration}\nPhotometry of each filter and camera has been calibrated to the Vega mag system  by computing the aperture correction to the PSF-fit-derived magnitudes and applying  to the corrected instrumental magnitude a photometric zero-point.\n To calculate the aperture corrections, we used  unsaturated and isolated stars only.\n \nWe measured aperture magnitudes within circular regions of $\\sim$0.4 and 0.5 arcsec radius for UVIS\/WFC3 and WFC\/ACS, respectively. To do this, we used the drizzled  and CTE-corrected (${\\rm \\_drc,}$) images, which are normalized to 1s exposure time. Aperture photometry has been calibrated by adding to these instrumental magnitudes the corresponding aperture corrections and the zero points \\citep{bohlin2016a, deustua2017a}.  \nFinally, we calculated the 3 $\\sigma$-clipped average of the difference between instrumental PSF magnitudes and calibrated aperture magnitudes for the stars in common. The resulting average values are then added to all the stars to derive calibrated magnitudes.       \n\n\\subsection{Quality parameters}\\label{subsec:quality}\nThe computer program KS2 computes for each star various parameters that can be used as diagnostics of the photometric and astrometric quality. For each filter it provides three main quantities:\n\\begin{itemize}\n\n   \\item the $RADXS$ parameter is a shape parameter that indicates the amount of flux that exceeds the predictions from the best-fitting PSF \\citep{bedin2008a}. It is defined as $RADXS=(\\sum_{\\rm i,j} pix_{\\rm i,j}-PSF_{\\rm i,j})\/10^{-{\\rm mag}\/2.5}$ where the sum is calculated within  an annulus between 1.0 and 2.5 pixels from the center of the star and is normalized to the star's total flux. This quantity is negative when the object is sharper than the PSF (e.g.\\,cosmic rays and PSF artifacts) and it is positive when the object is broader than the PSF (e.g.\\,galaxies). The perfect PSF fit corresponds to $RADXS$=0. \n\n    \\item the quality-fit parameter, $qfit$, which is indicative of the goodness of the PSF fit. It is defined as $qfit=\\frac{\\sum_{i,j} pix_{\\rm i,j} PSF_{\\rm i,j}}{\\sqrt{\\sum_{\\rm i,j} pix^{2}_{\\rm i,j} PSF^{2}_{\\rm i,j}}}$, and it is calculated in a 5$\\times$5 pixel area centered on the star, and  $pix_{\\rm i,j}$ and $PSF_{\\rm i,j}$ are the values of the pixel and the best-fitting PSF model, respectively, estimated in the pixel (i,j).    It ranges from unit, in the case of a perfect fit, to zero.\n     \n        \\item the root mean scatter of the magnitude determinations, $rms$.\n\\end{itemize}\n\nAs an example, in the top-left panels of Figure \\ref{fig:selezioni} we plot the $RADXS$ and $qfit$ parameters derived from F336W photometry of the star cluster Reticulum as a function of the F336W instrumental magnitude. Top-right panels show the analogous figures but for the F814W filter. The azure lines are drawn by hand with the criteria of separating the bulk of well-measured point-like sources from sources that are poorly fitted by the PSF model.\nThe bottom panels compare the CMD of stars that pass the selection criteria in both filters and the CMD of stars that have been rejected in at least one filter.\nAlthough the magnitude $rms$ is another diagnostic of photometric quality, we prefer not to use it to select the sample of stars with high-quality measurements to avoid excluding variable stars. \n\\begin{figure} \n\\centering\n\\includegraphics[width=8.5cm,trim={9.5cm 0.0cm 9.5cm 0.0cm},clip]{methodsLR.pdf}\n \\caption{Comparison of the instrumental F275W vs.\\,F275W$-$F814W CMDs of stars in the field of view of the star cluster Lindsay\\,1 as derived from Method I (top-left), Method II (top-middle) and Method III (top-right). Well-measured stars are colored black, while stars with poor photometry are plotted with light gray dots. Azure crosses mark stars where the F814W magnitude is derived from saturated images. Middle panels are zoomed-in views of the top-panel CMD around the MS. Bottom panels compare the region of the instrumental F555W vs.\\,F555W$-$F814W CMDs populated by bright stars with F555W$<-9.75$ mag. Calibrated magnitudes and colors are indicated by the top and right axes.  } \n \\label{fig:methods} \n\\end{figure} \n\n\n\n\\begin{centering} \n\\begin{figure} \n  \\includegraphics[width=9.5cm,trim={0.5cm 5.5cm 1cm 4.0cm},clip]{selezioni.pdf}\n \\caption{\\textit{Top panels}. $RADXS$ and $qfit$ parameters derived from F336W  (left) and F814W (right) photometries of stars in the field of view of Reticulum. The azure lines separate well-measured stars (black dots) from poorly measured sources (gray crosses).\n  The bottom panels show the instrumental F336W vs.\\,F336W$-$F814W CMD for stars that pass the selection criteria in both filters (left) and for the remaining stars (right).}\n \\label{fig:selezioni} \n\\end{figure} \n\\end{centering} \n\n\n\n\n\\subsection{The color-magnitude diagrams}\\label{subsec:CMDs}\nIn the four panels of Figure\\,\\ref{fig:HESS} we show the $m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ (left) and the $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ (right) Hess diagrams of all the observed fields in the LMC (top) and SMC (bottom).\nClearly, these diagrams reveal the complexity of stellar populations in the Magellanic Clouds, from bright and blue MSs composed of young and metal-rich stars to old and metal-poor stellar populations characterized by blue and faint MSs and faint RGBs.\n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=13.5cm,trim={0.0cm 0.0cm 10.0cm 1.0cm},clip]{ALLcmd.pdf}\n\n \\caption{$m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ (left panels) and $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ (right panels) Hess diagrams for all LMC (top) and SMC stars (bottom).}\n \\label{fig:HESS} \n\\end{figure*} \n\nTo further illustrate the variety of stellar populations and environments contained in the dataset of this paper, we show in Figure\\,\\ref{fig:CMDs} the stacked images and the CMDs of stars in three distinct fields that host stellar populations with different stellar densities, ages, and metallicities. The F475W image and the CMD of stars in the field around the open cluster NGC\\,1966 are plotted in the top panels.  This region, which has never been studied with {\\it HST}, hosts a conspicuous population of very young stars\n\n  that populate the\n   upper MS and the pre-MS.\n   The CMD also reveals old-RGB and red-clump stars that likely belong to foreground and background LMC old stellar populations. \n   Notably, the region hosts various nebula like NGC\\,1965 and the gas nebula around the Wolf-Rayet star HD-269546 \\citep[ the brightest star visible in the stacked image, ][]{westerlund1964a}, which are visible here in unprecedented detail.\nThe figures in the middle and bottom panels refer to the regions around the intermediate-age cluster NGC\\,2121 (age $\\sim 3$ Gyr) and the dense and old GC NGC\\,2210 (age $\\sim 12$ Gyr), respectively. \n\n\\begin{figure*} \n\\centering\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc1966.pdf}\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc2121.pdf}\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc2210.pdf}\n \\caption{Stacked images and CMDs of stellar fields  with different ages and stellar densities.  North is up and East to the right.} The top panels show the F475W image and the $m_{\\rm F475W}$ vs.\\,$m_{\\rm F475W}-m_{\\rm F814W}$ CMD of stars in the star-forming region around the very young cluster NGC\\,1966. The middle and bottom panels illustrate the F814W stacked images and the CMDs of the intermediate-age cluster NGC\\,2121 (age $\\sim 3$ Gyr) and the old GC NGC\\,2210 (age $\\sim 12$ Gyr), respectively. \n \\label{fig:CMDs} \n\\end{figure*} \n\nThe CMDs are used to estimate age, distance modulus, (m$-$M$)_{0}$, metallicity, [M\/H], and reddening, E(B$-$V), by using isochrones from the Padova database \\citep{marigo2017a}.  \n To minimize the contamination of field stars, we excluded from the analysis the stars at a large distance from the cluster center. Moreover, we statistically subtracted the field stars from the CMD of cluster members by using the method of \\citet{gallart2003a}, in close analogy with what is done in previous papers from our group \\citep[e.g.\\,][]{marino2014a, milone2018a}. In a nutshell, we defined by eye a region that is centered on the cluster and includes the bulk of cluster stars (hereafter cluster field) and a reference field with the same area, and at a large distance from the cluster center, which is mostly composed of field stars. \n We associated with each star in the reference field, the star in the cluster field at the smallest distance in the CMD, where the distance is defined as \\\\\n $\n {\\rm distance}=\\sqrt{(k \\times {\\Delta \\rm color})^{2} + {(\\Delta \\rm magnitude)^{2}} }\n $\\\\\n where $\\Delta$color and $\\Delta$magnitude are the color and magnitude differences, respectively, and $k$ is a factor enhancing the difference in color with respect to the magnitude difference, which is derived as in \\citet[][see their section 3.1]{marino2014a}. These stars are excluded from the comparison with the isochrones.\nThe cluster parameters and the best-fitting isochrones are used in Section\\,\\ref{sec:reddening} to estimate differential reddening maps and to investigate the eMSTO phenomenon (Section\\,\\ref{sec:secrets}). \nTo find the best-fitting isochrone, we used the CMD from the  UVIS\/WFC3 and\/or WFC\/ACS photometry providing the widest color baseline, thus maximizing the sensitivity to metallicity. \nHowever, when photometry in optical filters is available, we excluded the UV filters F225W, F275W, F336W, and F343N from the analysis to minimize the effect of multiple populations \\footnote{These UV filters encompass various molecular bands that include carbon, nitrogen, and oxygen.\nTheir fluxes are sensitive to the abundances of these elements, which are not constant within clusters with multiple populations.  Hence, the CMDs made with UV filters  may lead to less  accurate determinations of the cluster parameters than optical CMDs.}. Indeed, these UV filters are sensitive to the potential effects due to stellar populations with different nitrogen and oxygen abundances, \\citep[e.g.][]{marino2008a, milone2020a, dondoglio2021a}, which are typical features of GCs older than $\\sim$2.3 Gyr, and to stellar populations with different rotation rates \\citep[e.g.][]{dantona2015a, milone2016a, li2017a, marino2018a}, which are present in all clusters younger than $\\sim$2 Gyr.\n\nThe observed CMDs are compared with grids of isochrones with different reddening values, distances, metallicities, and ages. The resulting best-fitting parameters are provided in  Table\\,\\ref{tab:info} and are estimated as follows.\n\n We first determined the isochrone and the values of reddening and distance modulus that, based on the visual comparison with the CMD,  provide the best match with the CMD. Then, we improved the determination of the best-fitting parameters using the following iterative approach. We fixed the values of age, distance, and reddening and better constrained the cluster metallicity by comparing the slopes of the fiducial lines of the observed RGB and the MS of the CMDs, and the slope of the corresponding magnitude intervals of the isochrones. \n\n  %\n  Then, we assumed the metallicity value corresponding to the minimum difference between the slopes of the observed CMDs and the isochrones to improve the estimates of reddening, age, and distance modulus. To do this, we \n   adopted the criteria of obtaining the best match between the isochrone and the observed CMD, {which may change for clusters with different ages.}\n\nThe best-fitting parameters of clusters older than $\\sim 2.3$\\,Gyr were estimated by determining the isochrone that best fits the CMD from the MSTO through the SGB \\citep[e.g.][]{dotter2010a}.  Specifically, we calculated the $\\chi^{2}$ values of the distances in the CMD between the fiducial lines of the MSTO and SGB stars and the isochrone. The best-fitting values of age, reddening and distance modulus are derived by means of $\\chi^{2}$ minimization.\nA visual inspection at the CMDs reveals that all clusters between $\\sim 10$ Myr and $\\sim$2.5 Gyr exhibit eMSTOs. Since the eMSTOs challenge their age determinations we provide two age values.\nWe list in column 11 of Table\\,\\ref{tab:info} the age of the isochrone that best fits the lower part of the eMSTO. Clearly, this age value would represent the oldest cluster stars, if the eMSTO is entirely due to age variation. \nAlternatively, if the eMSTO is entirely due to rotation, our age estimate would provide an upper limit to cluster age, as the fast-rotating stars populate the lower part of the eMSTO \\citep[e.g.][]{dupree2017a, marino2018a, marino2018b, kamann2020a}. Hence, we provide in column 12 of Table\\,\\ref{tab:info} the age of the isochrone that best fits the upper part of the eMSTO.\n In the clusters younger than $\\sim 10$\\,Myr, where it is challenging to identify the MSTO, our age determination is largely based on evolved stars.   In these young clusters and in the clusters with the eMSTO, the age, distance modulus, and reddening were derived by eye.\n\n To quantify the typical precision of the values of metallicity, age, reddening, and distance modulus inferred by the isochrones we applied the following procedure to four couples of clusters with different ages. The photometry of the clusters of each pair comes from datasets with large differences in the number of images and in the total exposure times. Hence, the range of uncertainties on the fitting parameter inferred from each couple of clusters would comprise the parameters' uncertainties of all studied clusters with similar ages.\n\n We first linearly added to the slopes of the fiducial lines of the RGB and MS stars that were used to constrain the metallicity, the corresponding errors. Hence, we derived the best value of [Fe\/H] that corresponds to the isochrone that provides the best match with the used slope. We repeated the same  procedure but by using the slopes of the MS and RGB fiducials after subtracting the errors. \n  We consider the semi-difference between the maximum and the minimum [M\/H] value, $\\Delta$[M\/H]  as a quantity indicative of the precision of our metallicity estimate.\n\nSimilarly, we shifted each point of the fiducial line of  the MS and the SGB to the bright and blue side of the CMD,  perpendicular to the isochrone. We indicate the resulting line as blue-shifted fiducial. The shift is applied in such a way that 68.27\\% of the stars on the blue side of the original fiducial line are located on the red side of the blue-shifted fiducial. We applied a similar procedure to derive a red-shifted fiducial line.\nHence, we repeated four times the procedure described above to estimate the values of age, reddening, and distance modulus but by assuming the various combinations of the largest and minimum values of [M\/H] and the blue-shifted and red-shifted fiducials.\nWe consider the semi-differences between the maximum values of age ($\\Delta$age), distance modulus, ($\\Delta$(m$-$M)$_{0}$), and reddening ($\\Delta$E(B$-$V)), as a proxy of the precision of the estimates of the corresponding quantities.\n\nThe results are listed in Table\\,\\ref{tab:errori} for the pairs of clusters of old GCs  NGC\\,2005 and NGC\\,1939 (ages of $\\sim 13$\\,Gyr), intermediate age clusters Kron\\,3 and Kron\\,1 (ages of $\\sim$6--7 Gyr). We also investigated the $\\sim$2 Gyr-old clusters NGC\\,1846 and Hodge\\,7 and the young clusters NGC\\,1866 and BSDL\\,1650 (ages of $\\sim$300 Myr).\n\n\n\n\\section{Differential reddening}\\label{sec:reddening}\nTo derive high-resolution reddening maps, we applied to our dataset the method originally developed by \\citet{milone2012a} to correct the ACS\/WFC F606W and F814W magnitudes of Galactic GCs for differential reddening \\citep[see also][]{bellini2017a, jang2022a}. \nThe main difference of the adopted procedure is that the catalogs of several GCs comprise photometry in more than two bands. \nThe main steps of our iterative method, which is illustrated in Figure \\ref{fig:red} for NGC\\,416, can be summarized as follows:\n\\begin{itemize}\n    \\item We built the $m_{\\rm F814W}$ vs.\\,$m_{\\rm X}-m_{\\rm F814W}$ diagrams, where X=F275W, F336W, F343N, F438W, F555W, and F814W. Each diagram has been used to gather information on differential reddening from a sample of reference stars.\n    Reference stars are selected in the CMD region where the reddening direction defines a wide angle with the cluster fiducial line in such a way that we can easily disentangle the effect on stellar colors and magnitudes due to differential reddening from the shift due to photometric uncertainties.\n    As an example, panel a of Figure \\ref{fig:red} highlights in black the selected reference stars of NGC\\,416 in the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD.\n    \n    \\item \n    We first derived the reddening direction corresponding to each star as $\\theta={\\rm arctan} \\frac{A_{\\rm X}}{A_{\\rm X}-A_{\\rm F814W}}$, where $A_{\\rm X}$ and $A_{\\rm F814W}$ are the absorption coefficients in the X and F814W bands, respectively. To derive them, we identified the point on the best-fitting isochrone with the same $m_{\\rm X}$ magnitude as the reference star and calculate the $m_{\\rm X}$ and $m_{\\rm F814W}$ magnitude differences with the corresponding point of the isochrone with E(B$-$V)=0 mag. This procedure allows us to account for the dependence of reddening direction from the total amount of reddening and  from its spectral type.   \n    As an example, panel a of Figure\\,\\ref{fig:red} shows the reddening direction associated with the reference star indicated by the red cross.\n    \n    \\item We translated the CMD into a new reference frame where the origin corresponds to the reference stars as illustrated in panels a and b of Figure \\ref{fig:red}. This CMD is rotated  counterclockwise by an angle $\\theta$ so that the abscissa and the ordinate of the new reference frame are parallel and orthogonal, respectively, to the reddening direction. \n    \n    \\item We generated the fiducial line of MS, SGB, and RGB stars, which we plot as a continuous red line in panel b. To do this, we divided the sample of MS stars into 'ordinate' intervals. For each bin, we calculated the median abscissa associated with the median 'ordinate' of the stars in the bin. The fiducial line has been derived by linearly interpolating these median points.\n     \n    \\item We calculated the distance of the reference star from the fiducial line along the reddening direction, $\\Delta x'$ as shown in panel b for a reference star of NGC\\,416 that we marked with a large red cross \\footnote{ The differential reddening is responsible for shifting the stars along the reddening line. The amount of such shift, which is proportional to the amount of reddening along the line of sight, depends on the star's position in the field of view.  As a consequence, the stars in the different regions of the field are systematically shifted towards larger or lower values of x' with respect to the cluster fiducial line depending on whether they are affected by a larger or smaller amount of reddening with respect to the median cluster reddening (see Figure 5 for an example). On the contrary, photometric errors are responsible for a random scatter along the fiducial line, but such a scatter is essentially not dependent on the reddening direction and the position of the star in the field of view.}.\n    \n    \\item We calculated the projection of $\\Delta x'$ along the $m_{\\rm X}-m_{\\rm F814W}$ color direction, $\\Delta$ ($m_{\\rm X}-m_{\\rm F814W}$) and plotted this quantity for the available X filters as shown in panel c  of Figure \\ref{fig:red}.  The observed values of $\\Delta$ ($m_{\\rm X}-m_{\\rm F814W}$) are compared with corresponding quantities derived from the isochrones and corresponding to reddening variations ranging from $\\Delta$ E(B$-$V)=$-$0.3 to 0.3 mag in steps of 0.001 mag. The value of $\\Delta$ E(B$-$V) that provides the minimum $\\chi^{2}$ is assumed as the best differential-reddening estimate associated with the reference star marked with the red cross.\n    \n   \n    %\n   \n    %\n\\end{itemize}\n\nTo derive the amount of differential reddening associated with each star in the catalog, we selected   a sample of N spatially nearby reference stars, (light-blue crosses in Figure\\,\\ref{fig:red}) as shown in panel d. The best determination of differential reddening is provided by the median of the $\\Delta$ E(B$-$V) values of these  $N$ neighbors. We excluded the target star from its own differential reddening determination.\n  We derived various determinations of differential reddening  by assuming different values of $N$, from 35 to 95 in steps of 5 and from 100 to 150 in steps of 10. For each determination, we   calculated the pseudo-color distances between the value of $x'$ of the reference stars, corrected for differential reddening, and the fiducial line of Figure\\,\\ref{fig:red}. \n  We assumed that our best determination of differential reddening is given by the value of $N$ that provides the minimum value of the r.m.s of these distances. In particular, we used N=75 for NGC\\,416.\n\nAs an example, Figure\\,\\ref{fig:redmap} shows the reddening map in the direction of NGC\\,416 and compares the original CMD to the CMD corrected for differential reddening. A collection of reddening maps for six clusters is provided in Figure \\ref{fig:redmaps}. \n\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=21.0cm,trim={0.5cm 1.2cm 0.0cm 0.0cm},clip]{red12.pdf}\n\n \\caption{This figure illustrates the procedure to estimate the amount of differential reddening associated with the target star represented with the large blue dot. \n Panel a shows the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD of all the stars. Reference stars, are located between the two dotted gray lines and are colored black, whereas the neighboring reference stars are marked with light-blue crosses.  The gray continuous lines are the abscissa and the ordinate of the rotated reference frame centered on the reference star marked with the large red cross, while the red arrow indicates the reddening direction. Panel b shows the same stars as panel a but in the rotated reference frame. The red continuous line is the fiducial of reference stars and the inset highlights the relative position between one reference star and the fiducial.   \n  Panel c represents the values of $\\Delta x'$ inferred from different filters (red dots). Gray crosses are the corresponding values derived for $\\Delta$ E(B$-$V) ranging from 0.01 to 0.10 mag in steps of 0.01 mag, while the black crosses provide the best fit to the observations and correspond to $\\Delta$ E(B$-$V)=0.058 mag.  Finally, the finding chart zoomed in around the target is illustrated in panel d. See the text for details.\n }\n \\label{fig:red} \n\\end{figure*} \n\n\n\\section{Cluster centers}\\label{sec:centri}\n\nTo determine the coordinates of the center of each star cluster we followed the procedure described in \\citet{cordoni2020b}. In a nutshell, we first selected by eye a sample of probable cluster members based on their location in the CMD and smoothed their stellar spatial distribution with a Gaussian kernel of fixed size. The kernel size has been chosen with the criteria of favoring the overall shape of the cluster, instead of the small-scale structures. \nWe derived five contour lines within 50 arcsec from the cluster center and interpolated each of them with an ellipse by using the algorithm by \\citet{halir1998a}.\nOur best cluster-center determination corresponds to the median value of the centers of the ellipses, while the corresponding uncertainty has been estimated as the dispersion of the center determinations inferred from each ellipse. \n Due to the low number of stars, it was not possible to apply the method above in 13 poorly-populated star clusters, namely BRHT\\,5b, BSDL\\,1650, KMK\\,8827, KMHK\\,1073, KMHK\\,8849, OGLE-CL-LMC390, NGC\\,1749, NGC\\,290, NGC\\,1850A, NGC\\,1858, NGC\\,1938 and NGC\\,1966. For these clusters, we provide raw center determination based on the peak of the histogram distributions of the coordinates of the probable cluster members.\nResults are provided in Table \\ref{tab:info}.\n\n\n\\begin{figure*} \n\\centering\n  \\includegraphics[width=16.0cm,trim={3cm 0.0cm 3.0cm 0.0cm},clip]{redmapLR.pdf}\n\n \\caption{\\textit{Left.} Comparison between the original $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD of NGC\\,416 (top) and the CMD corrected for differential reddening (bottom). \\textit{Right.} Differential-reddening map in the direction of NGC\\,416. The levels of gray are proportional to the reddening variation as indicated on the top-right. The panels on the right show $\\Delta$ E(B$-$V) against the abscissa for stars in eight ordinate intervals. Similarly, the panels on the top represent the reddening variation as a function of the ordinate for stars in eight intervals of X. The field is centered around the center of NGC\\,416 (X,Y=6019,5507) and the X and Y axis are parallel to the right ascension and declination direction, respectively. We adopted a scale of 0.04 arcsec per pixel.}\n \\label{fig:redmap} \n\\end{figure*} \n\n\n\n\\begin{figure*} \n\\centering\n  \\includegraphics[width=16.0cm,trim={3cm 0.0cm 3.0cm 0.0cm},clip]{redmaps.pdf}\n\n \\caption{Differential-reddening maps of the regions in front of HW\\,57, NGC\\,416, NGC\\,1751, NGC\\,1850, NGC\\,1852 and NGC\\,1856.}\n \\label{fig:redmaps} \n\\end{figure*} \n\n\\begin{figure} \n\\centering\n\\includegraphics[height=12cm,trim={1.0cm 5.8cm 5.2cm 4.5cm},clip]{PMGAIA.pdf}\n\\caption{This figure illustrates the procedure to identify stars that, based on the position in the CMDs from {\\it HST} photometry and in the proper-motion diagram from GAIA eDR3, are probable members of NGC\\,1806. The probable members are represented with red circles in the CMDs plotted in the top panels and    the bottom-left panel and in the proper motion diagram shown in the bottom-right panel. Stars that are not located on the main evolutionary sequences in at least one CMD are represented with blue-starred symbols. The arrow plotted in the proper-motion diagram indicates the mean cluster motion, while the circle is used to select the stars that are not included in the sample of probable cluster members, due to their large proper motions (aqua-starred symbols). See the text for details.  }\n \\label{fig:PMGAIA} \n\\end{figure} \n\\begin{figure*} \n\\centering\n\\includegraphics[height=9cm,trim={0.0cm 1cm 0.0cm 1cm},clip]{pmsLR.pdf}\n \\caption{ \\textit{Left.} Coordinates, in degrees, relative to the SMC center of the studied SMC and LMC clusters. The arrows in the top panel are indicative of the absolute proper motions of each cluster, while the  bottom panel represents  the proper motions of LMC and SMC clusters after subtracting the average motion of the corresponding galaxy.  \\textit{Right.} Proper motions of stars brighter than g$_{\\rm BP}=16.0$ mag in the region around the LMC and the SMC (black points). The studied LMC and SMC clusters are plotted in all panels with red and aqua dots, respectively.  }\n \\label{fig:PMs} \n\\end{figure*} \n\\section{Proper motions}\\label{sec:pms}\n\nTo estimate the absolute proper motions of the studied clusters, we combined information from {\\it HST} photometry and Gaia eDR3 proper motions. Specifically, for each cluster, we selected by eye stars that,  based on their positions in all available CMDs, are probable cluster members.  Then, we used the Gaia eDR3 catalog to select stars with magnitude $g_{\\rm BP}<19.0$ mag, which according to the criteria by \\citet{cordoni2018a} have high-quality proper motions. The average proper motion of each cluster has been calculated as the 3-$\\sigma$ clipped average of the proper motions of selected cluster members for which are available both {\\it HST} photometry and Gaia eDR3 high-quality proper motions.  \nWe estimated the corresponding uncertainty by following the method of \\citet{vasiliev2019a}, which accounts for systematic errors.\n\n The main steps of the procedure used to derive the absolute proper motion are illustrated in Figure\\,\\ref{fig:PMGAIA} for NGC\\,1806. For this cluster, we have photometry in five photometric bands of UVIS\/WFC3 and WFC\/ACS. We constructed ten CMDs of stars in the FoV of NGC\\,1806 including four $m_{\\rm F814W}$ vs.\\,$m_{\\rm X}-m_{\\rm F814W}$ CMDs, where X=F336W, F343N, F435W, and F555W, three $m_{\\rm F555W}$ vs.\\,$m_{\\rm X}-m_{\\rm F555W}$ CMDs, where X=F336W, F343N, and F435W, two $m_{\\rm F435W}$ vs.\\,$m_{\\rm X}-m_{\\rm F435W}$ CMDs, where X=F336W and F343N, and the    $m_{\\rm F343N}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F343N}$ CMD. For each CMD, we selected by eye the stars that, based on their colors and magnitudes, are located on the main cluster evolutionary sequences. As an example, the  stars that, based on their positions in all CMDs,  likely belong to the RGB, AGB, and red clump of NGC\\,1806 are colored black in the three CMDs of Figure\\,\\ref{fig:PMGAIA}. The colored symbols mark  stars with available Gaia eDR3 proper motions in both the CMDs and in the proper-motion diagram. The stars that do not belong to the RGB, AGB, and red clump of NGC\\,1806 in at least one CMD are represented with blue-starred symbols and are not included in the determination of the cluster proper motion. \n We also excluded the selected stars with proper motions that differ from the average cluster motion by more than three times the proper motion dispersion (i.e.\\, the stars outside the black circle shown in the bottom-right panel of Figure\\,\\ref{fig:PMGAIA} represented with aqua-starred symbols). The remaining stars are marked with red open dots.\n\n\n\n\n\n\nResults are provided in Table\\,\\ref{tab:data}. The left panels of Figure\\,\\ref{fig:PMs} show the positions of the studied LMC and SMC clusters relative to the SMC center. In the top-left panel, we associate to each cluster the corresponding proper-motion vector, while in the bottom-left panel we show the proper-motion residuals after subtracting to LMC and SMC clusters the average motion of the corresponding Magellanic Cloud from  \\citet{gaia2018a}\\footnote{Although the investigation of the Magellanic Clouds' rotation is beyond our scope, we note that no clear rotation pattern is evident from the bottom-left panel of Figure\\,\\ref{fig:PMs}. This statement, which is based on a visual inspection of this figure, seems to contrast with the evidence of the LMC rotation pattern shown by \\citet{vandermarel2014a, helmi2018a}. We also note that the right panel of Figure\\,\\ref{fig:PMs} highlights the relative motions within the SMC following the pattern of the SMC tidal expansion along the bridge and counter-bridge as detected in previous works \\citep[e.g.][]{zivick2018a, piatti2021a, dias2021a, schmidt2022a}.}. \nThe proper motion diagram is plotted in the right panel of Figure\\,\\ref{fig:PMs} and reveals that, based on proper motions, all clusters are consistent with being either LMC or SMC members. \n\n\\subsection{Proper motions from Gaia eDR3 and {\\it HST} }\nFor  thirteen GCs, we take advantage of having more than one epoch observations with appropriate signal-to-noise ratio and temporal baselines to disentangle the internal kinematics of Magellanic Cloud stars and separate cluster members and field stars by using {\\it HST} data alone. Detailed information on the {\\it HST} images available for these clusters are provided in Table\\,\\ref{tab:dataPM}. Relative {\\it HST} motions are then transformed into absolute motions based on Gaia eDR3 proper motions.\n\nTo derive relative proper motions we applied to our dataset the procedure described by \\citet[][]{piotto2012a} and described in the following for NGC\\,1978. \nIn a nutshell, we first identified the distinct groups of images collected at the same epoch through the same filter and camera. We reduced each group of images, separately, as described in Section\\,\\ref{sec:data}, and obtained the corresponding astrometric and photometric catalogs. \n\n The reference frame defined by the first-epoch images collected through the reddest filter is adopted as a master frame. The coordinates of stars in each catalog are transformed into the master frame by means of six-parameter linear transformations \\citep{anderson2006a}. \n To minimize the effect of possible small residual distortions we applied local transformations based on the nearest 70 reference stars. Target stars are never  included in the calculation of their own transformations.   \n \n The abscissa and the ordinate of each star, expressed in milliarcsec, are plotted against the epoch, expressed in years, as shown in panels a1--a4 of Figure\\,\\ref{fig:MetPM} for two stars in the field of view of NGC\\,1978. For simplicity, in this figure, we show the displacements DX and DY, and the time relative to the stellar position and time at the first epoch. \n  These points are finally fitted with a weighted least-squares straight line, whose slope corresponds to the best proper motion estimate.   \n \nThe selection of the stars used to derive the transformation is a critical step for accurate proper-motion determination. Hence, we selected bright and unsaturated stars that pass the criteria of selection  discussed in Section\\,\\ref{subsec:quality}. We derived proper motions relative to a sample of cluster members that have been selected iteratively. As a consequence, the average relative motion of the cluster is set to zero.\nWe first identified probable cluster stars that, based on all available CMDs, lie on the main evolutionary sequences and used them to derive initial proper motion estimates.\nThen, we iteratively excluded those stars that do not share the same motion as the bulk of cluster members (i.e.\\,stars with proper motions greater than three times the proper-motion dispersion of cluster stars).\nPanels b and c of Figure\\,\\ref{fig:MetPM} mark with black points the probable cluster members that we selected for deriving stellar proper motions in the $m_{F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD and in the $m_{\\rm F814W}$ vs.\\,$DR=\\sqrt{DX^{2}+DY^{2}}$ plane,  whereas aqua crosses are probable cluster members. Gray points mark the remaining stars with cluster-like proper motions, i.e.\\, the saturated stars, the faint stars, and the stars that do not lie on the main evolutionary sequences in the CMDs.\n\nTo transform relative proper motions into absolute ones we derived the difference between the relative proper motions derived from {\\it HST} images and the absolute proper motions from Gaia eDR3 for an appropriate sample of stars.\n  Specifically, this selected sample includes stars with high-quality relative proper motions (i.e.\\,bright, unsaturated stars that pass the criteria of selection of Section\\,\\ref{subsec:quality}).  In addition, the selected sample includes stars that, based on the proper motion uncertainties and on the values of the Renormalized Unit Weight Error (\\texttt{RUWE}), the astrometric\\_gof\\_al\n(\\texttt{As\\_gof\\_al}) parameters of the Gaia eDR3 catalog   have accurate Gaia eDR3 absolute proper motions. We refer to papers by \\citet{cordoni2018a, cordoni2020a}, for details on the procedure.\n  The sample includes both cluster and field stars, with the exception of a few stars with parallaxes significantly larger than zero.\n\nThe $3\\sigma$-clipped mean differences of the proper motion along each direction  ($\\mu_{\\alpha} {\\rm cos}{\\delta}$ and $\\mu_{\\delta}$)  are considered as the best estimate of the zero points of the motions and are used to convert relative proper motions into absolute ones.\nAs an example, panels d1 and d2 of Figure\\,\\ref{fig:MetPM} show the histogram distributions of the quantities $\\Delta \\mu_{\\alpha} {\\rm cos} {\\delta}$=$\\mu_{\\alpha} {\\rm cos} \\delta$-DX and $\\Delta \\mu_{\\delta}$=$\\mu_{\\delta}$-DY for stars in the field of view of NGC\\,1978.\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=9.5cm,trim={2cm 0cm 1.0cm 0cm},clip]{MetPMLR.pdf}\n\\caption{Procedure to estimate absolute proper motions. Panels a1 and a2 show the displacements along the X and Y directions in four epochs of a probable field star (blue triangles) relative to the mean motion of NGC\\,1978. Similarly, panels a3 and a4 show the displacements of a candidate cluster member (red dots). The $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of stars in the field of view of NGC\\,1978 is plotted in panel b, while panel c shows relative stellar proper motions against $m_{\\rm F814W}$. Aqua crosses are probable field stars selected on the basis of their proper motions. The stars used as references to calculate relative proper motions are colored black, while the remaining stars with cluster-like proper motions are gray. Panels d1 and d2 show the histogram of the difference between our relative proper motions and the absolute proper motions from Gaia eDR3. }\n \\label{fig:MetPM} \n\\end{figure*} \n\nThe proper motion diagrams for NGC\\,1978 stars are plotted in the left panels of Figure\\,\\ref{fig:NGC1978vi} in four distinct magnitude bins. These diagrams can be  used to separate the bulk of cluster members (black dots) from  probable field stars (red crosses). Here, the red circles that enclose the NGC\\,1978 stars have radii equal to 2.5$\\sigma$, where $\\sigma$ is the average between the $\\sigma-$clipped dispersion values of $\\mu_{\\alpha} {\\rm cos} \\delta$ and $\\mu_{\\delta}$.  For illustration purposes, we only mark in red the most-evident field stars with $\\mu_{\\rm \\alpha} \\cos{\\delta} >1.6$ mas\/yr and a distance of more than 0.2 mas\/yr from the average motion of NGC\\,1978, while the remaining stars are colored gray. \n The  $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of probable cluster members and field stars is shown in the right panel of  Figure\\,\\ref{fig:NGC1978vi}.\n\\begin{figure} \n\\centering\n \\includegraphics[height=9.0cm,trim={0.1cm 5.cm 0.0cm 3.5cm},clip]{NGC1978pmS.pdf}\n\n \\caption{Proper motion diagrams of stars in the field of view of NGC\\,1978 in four F814W magnitude intervals (left). The $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of stars in the left panels is plotted on the right. Stars within the red circles plotted in the left panels are considered probable cluster members and are colored black, whereas  the most-evident field stars are represented with red crosses. The remaining stars are colored gray. See the text for details.}\n \\label{fig:NGC1978vi} \n\\end{figure} \n\n\n\\section{A saucerful of secrets}\\label{sec:secrets}\nThe photometry and astrometry of this work are  exquisite tools  to investigate various astrophysical topics. In this section, we provide further examples of science outcomes that arise from visual inspections of the photometric diagrams and of the proper-motion diagrams.  \nSpecifically, in Section\\,\\ref{sec:NEWeMSTO} we report the discovery of eMSTOs in the clusters KMHK\\,361 and NGC\\,265. Section\\,\\ref{sec:AGEemsto} compares the CMDs of LMC clusters younger than $\\sim$2.3 Gyr and investigates the color and magnitude distribution of eMSTO in clusters with different ages. \nGaps and color discontinuities along the MS of NGC\\,1783 are investigated in Section\\,\\ref{sub:zigzag} while Section\\,\\ref{sec:NGC1783} provides evidence of new features along the eMSTO and the upper MS of NGC\\,1783. \nFinally, Section\\,\\ref{sec:pm} is focused on the proper motions of the star clusters and of Magellanic Cloud stellar populations in eleven fields.\n\n\\subsection{Clusters without previous evidence of eMSTO}\\label{sec:NEWeMSTO}\nFigure\\,\\ref{fig:NEWeMSTO} provides evidence that the CMDs of the star clusters KMHK\\,361 (age of 1.35 Gyr) and NGC\\,265 (age of 450 Myr) are not consistent with a single isochrone.\n In this figure, we compare the CMDs of stars in circular fields centered on the cluster (hereafter cluster fields) and in reference fields of the same area. We adopted radii of 20 and 24 arcsec for KMHK\\,361 and NGC\\,265, respectively, enclosing the bulk of cluster stars.  To minimize the contamination from cluster stars, the reference fields are as\n  far away from the cluster centers as possible, while still being within the FoV.\n\n By assuming a uniform distribution of field stars in the small {\\it HST} field of view, the distribution of stars in the reference-field CMD is indicative of the  contamination due to field stars. \n \n Clearly, KMHK\\,361 exhibits an eMSTO, which cannot  be explained by field-star contamination alone. \n Similarly, NGC\\,265 shows an intrinsic eMSTO. The upper MS is split in the F435W magnitude interval between $\\sim$21 and 22 mag, with the red MS hosting about two-thirds of MS stars. The two MSs merge around $m_{\\rm F435W} \\sim 22.5$ mag. The comparison between the CMDs of stars in the field and reference fields reveals that the split MS and the eMSTO are not due to field-star contamination.\n \n The visual inspection of the CMDs from our survey suggests that all clusters with ages between $\\sim 0.1$ and $\\sim$2.3 Gyr exhibit the eMSTO (Cordoni et al.\\,in preparation).\n These findings corroborate the evidence that eMSTOs are common features of clusters younger than $\\sim$2.3 Gyr, while split MSs are widespread phenomena among clusters younger than $\\sim$0.8 Gyr \\citep[e.g.][]{milone2009a, milone2022a, niederhofer2015a, goudfrooij2011a, li2017a, correnti2017a}. \n\n\n\n\n\\begin{figure*} \n\\centering\n  \\includegraphics[height=6.5cm,trim={0.5cm 5cm 0.2cm 4.5cm},clip]{KMHK361.pdf}\n    \\includegraphics[height=6.5cm,trim={0.5cm 5cm 0.2cm 4.5cm},clip]{NGC0265.pdf}\n\n\\caption{CMDs of the clusters KMHK\\,361 and NGC\\,265 without previous evidence of eMSTOs. Stars in the cluster field and reference field of each cluster are represented with black points and aqua crosses, respectively. See text for details.}\n \\label{fig:NEWeMSTO} \n\\end{figure*} \n \n\n\n\n\\subsection{The eMSTO in clusters of different age}\\label{sec:AGEemsto}\n\\begin{centering} \n\\begin{figure*} \n \\includegraphics[height=17cm,trim={0.1cm 5cm 0.0cm 4.5cm},clip]{emstoLMC.pdf}\n\n \\caption{Collection of $M_{\\rm F336W}$ vs.\\,$M_{\\rm F336W}-M_{\\rm F814W}$ CMDs of LMC clusters younger than 2.5 Gyr. All panels have the same scale and are zoomed around the MS, while the insets highlight the MSTO. Clusters are sorted by age.}\n\n\n\n \\label{fig:emstoLMC} \n\\end{figure*} \n\\end{centering} \n\nOur dataset provides a unique opportunity for comparing CMDs of clusters with different ages derived with homogeneous methods. As an example, we take advantage of the collection of $M_{\\rm F336W} vs.\\, M_{\\rm F336W}-M_{\\rm F814W}$ CMDs shown in Figure \\ref{fig:emstoLMC} to investigate how the eMSTO phenomenon changes as a function of cluster age. In this figure, star clusters\n are sorted by age, from $\\sim$10 Myr (NGC\\,1818) to $\\sim$2.5 Gyr (NGC\\,1978). \n The observed magnitudes have been converted into absolute ones by  adopting the values of distance modulus and reddening listed in Table\\,\\ref{tab:info}. \n\nA visual inspection of this figure corroborates the previous conclusion that the split MS is visible in all LMC clusters younger than $\\sim$800 Myr (from NGC\\,1818 to NGC\\,1953) and seems to disappear at older ages \\citep{milone2018a}.\nThe color separation between the blue and red MSs approaches its maximum value around the MSTO  and decreases towards faint luminosities \\citep{milone2016a}.\nAs pointed out by \\citet{wang2022a}, the gap between the blue and red MSs significantly narrows down around $M_{\\rm F336W}=1.0$ mag, \n which is the luminosity level where the fraction of blue-MS stars approaches its minimum value \\citep{milone2018a}.\nNoticeably, this magnitude value  corresponds to an MS mass of $\\sim$2.5 $\\mathcal{M}_{\\odot}$, where the slowly rotating component of MS field stars disappears \\citep{zorec2012a}.\n\nWe also confirm that the eMSTO is a ubiquitous feature of LMC clusters younger than $\\sim$2.3 Gyr. It is visible in all clusters where the turn-off is brighter than the MS bending around $M_{\\rm F336W}=3.0$ mag and disappears in NGC\\,1978 \\citep[e.g.][]{milone2009a, goudfrooij2014a}. \nSince the MS bending is due to a change in the stellar structure, the eMSTO is  associated with stars with radiative envelopes alone. In addition, the split MS is visible among stars brighter than the MS bend.\n\nFigure\\,\\ref{fig:emstoLMC} reveals that the color and magnitude distributions of stars across the eMSTO significantly change from one cluster to another. As an example, the Hess diagrams plotted in the top panels (a1, a2, and a3) of Figure \\ref{fig:cumulative} suggest that most TO stars of NGC\\,1868 populate the bright and blue region of the eMSTO, whereas NGC\\,2173 shows higher stellar density on the bottom-red side of its eMSTO. NGC\\,1852 seems to show an intermediate distribution. \n \n  To parametrize the stellar distribution of eMSTO stars in the CMDs,\n   we adopted the procedure illustrated in Figure \\ref{fig:cumulative}b for NGC\\,1852.\n  We defined a new reference frame where the origin, {\\it O}, is set by hand on the bright and blue side of the eMSTO, and the abscissa, {\\it X'}, envelopes the bright part of the eMSTO and points towards the red.\n  We derived the red and blue fiducials of the eMSTO in the new  reference frame and represented them as red and blue lines in the CMD of Figure \\ref{fig:cumulative}b. To derive the fiducials, we follow the recipe by \\citet{milone2017a}, which is based on the naive estimator \\citep{silverman1986a}. We first divided the eMSTO into a series of bins with fixed  pseudo-magnitude,  {\\it $\\delta$Y'}. The bins are defined over a grid of points separated by intervals of  fixed pseudo-magnitude (s=$\\delta$Y\/3). For each interval, we calculate the 4$^{\\rm th}$ and the 96$^{\\rm th}$ percentile of the {\\it X'} distribution and associated these values with the mean pseudo-magnitude {\\it Y'} of stars in the bin. These values are then linearly interpolated to derive the red and blue boundaries of the eMSTO. These lines are used to calculate the quantity\n  \\begin{equation}\n\\Delta_{\\rm X'}=\\frac {X'-X'_{\\rm blue\\,fiducial}} {X'_{\\rm red\\,fiducial}-X'_{\\rm blue\\,fiducial}}  \n  \\end{equation}\n   that is defined in such a way that the stars on the blue and red fiducials have $\\Delta_{X'}$=0 and 1, respectively.\n  \n   Figure\\,\\ref{fig:cumulative} compares the kernel density  (panel c) and the cumulative distributions of $\\Delta_{X'}$ (panel d) for NGC\\,1852 (black), NGC\\,1868 (aqua), and NGC\\,2173 (orange).\n   We confirm the visual impression of a predominance of blue eMSTO stars in NGC\\,1868, whereas the $\\Delta_{ X'}$ distribution of NGC\\,2173 is peaked towards the red. NGC\\,1852 has an intermediate distribution.\n\n\n\\begin{centering} \n\\begin{figure} \n \\includegraphics[height=8.0cm,trim={1.0cm 5.5cm 0.0cm 4.5cm},clip]{cumulative.pdf}\n\n \\caption{$m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ Hess diagrams of NGC\\,1868 (a1), NGC\\,1852 (a2), and NGC\\,2173 (a3) zoomed around the eMSTO. \n The blue and red lines are the boundaries of the eMSTOs.  Panel b illustrates the scheme to derive the $\\Delta_{X'}$ quantity for eMSTO stars, while the corresponding kernel-density distributions and cumulative distributions are plotted in panels c and d, respectively, for NGC\\,1868 (aqua), NGC\\,1852 (black) and NGC\\,2173 (orange). See the text for details.}\n \\label{fig:cumulative} \n\\end{figure} \n\\end{centering} \n\nTo quantify the $\\Delta_{X'}$ differences among the various clusters, we define two quantities: i) the area, A, below the cumulative curve shown in Figure\\,\\ref{fig:cumulative} and ii) the median value of $\\Delta_{\\rm X'}$, $<\\Delta_{\\rm X'}>$. If the distribution is dominated by blue and bright MSTO stars we would expect large values of A and small values of $<\\Delta_{\\rm X'}>$, while a predominance of faint and red eMSTO stars corresponds to small A and large $<\\Delta_{\\rm X'}>$. Results are shown in Figure\\,\\ref{fig:RelEMSTO}, where we plot both quantities against cluster age. \n  LMC clusters\n  (red dots in Figure\\,\\ref{fig:RelEMSTO}) exhibit a strong anti-correlation between A and age and a correlation between  $<\\Delta_{\\rm X'}>$ and age, as also indicated by the values of the Spearman's rank correlation coefficients of $-$0.91 and 0.92, respectively.  \nIntriguingly, the $\\sim$2 Gyr old SMC clusters\n\n  NGC\\,411 and NGC\\,416 exhibit larger values of A and smaller values of  $<\\Delta_{\\rm X'}>$ than LMC clusters with similar ages, with NGC\\,411 having the largest differences. The small statistical sample of clusters prevents us from reaching a firm conclusion on whether NGC\\,411 is an outlier or SMC and LMC clusters exhibit different trends.\n\nThe multiple populations of young and  intermediate-age star clusters share common features that have been instrumental to shed light on the origin of split MSs and eMSTOs \\citep[see][for a recent review]{milone2022a}. \nAs an example, the eMSTO width depends on cluster age. Specifically, if the eMSTO is interpreted as an age spread, the resulting age range is proportional to cluster age \\citep[e.g.][]{niederhofer2015a, cordoni2018a}.\nMoreover, the fractions of stars along the blue and the red MS correlate with stellar mass. The fraction of blue-MS stars varies from $\\sim 40\\%$ among stars with masses of $\\sim 1.5 \\mathcal{M_{\\odot}}$ to $\\sim 15\\%$  among $\\sim 2.5-3.0 \\mathcal{M_{\\odot}}$ stars. It  arises again in more massive stars, up to $\\sim 40\\%$ in $\\sim 5.0 \\mathcal{M_{\\odot}}$-stars.\nThe fractions of blue- and red-MS stars do not depend on other properties of the host cluster like the global cluster's mass \\citep[][]{milone2018a}. These results have been instrumental to demonstrate that rotation plays a major role in shaping the eMSTOs and the split MSs of Magellanic-Cloud clusters.\n\nThe evidence that the  $\\Delta_{X'}$ distribution of stars along the eMSTO depends on cluster age provides a potential further constraint to the eMSTO phenomenon.\nTo start investigating the physical reasons responsible for the relations shown in Figure \\ref{fig:RelEMSTO}, we used stellar models from the Padova database  \\citep{marigo2017a} to simulate a group of CMDs of non-rotating stellar populations with ages of 100, 200, 500, 1,000, 1,250, 1,500 and 2,000 Myrs and internal age spreads. We assumed a flat distribution and maximum width corresponding to the average age variations inferred by \\citet{cordoni2018a} for Magellanic Cloud clusters with the same age.  We derived the A and  $<\\Delta_{\\rm X'}>$ quantities for each simulated CMD by using the same procedure adopted for real stars and plotted the resulting values against the oldest age of the simulated stellar population (open triangles of Figure\\,\\ref{fig:RelEMSTO}). \n\n Similarly, we simulated another group of CMDs for coeval stellar populations where 33\\% of stars have no rotation, whereas the remaining  67\\% of stars have rotation equal to 0.9 times the breakout value. The simulated diagrams have ages of  100, 150, 500, 800, and 1,250 Myr and are derived by means of Geneva models \\citep{ekstrom2012a, ekstrom2013a, mowlavi2012a, wu2016a}. \n We assumed random viewing-angle distributions and adopted the gravity-darkening model by \\citet{espinosa2011a} and the limb-darkening effect \\citep{claret2000a}. Stellar magnitudes for the available {\\it HST} filters have been derived using the model atmospheres by Castelli \\& Kurucz (2003). \n The resulting A and  $<\\Delta_{\\rm X'}>$ quantities are represented with  filled diamonds in Figure\\,\\ref{fig:RelEMSTO}. \n  For completeness, we used the Geneva models to simulate non-rotating stellar populations with internal age spreads, in close analogy with what we did with the Padova models. Results are represented with filled triangles.\n \n \n Clearly, the  A and  $<\\Delta_{\\rm X'}>$ quantities inferred from both groups of simulated diagrams provide poor fits to the observations. This fact indicates that internal age variation alone is not responsible for the eMSTO when we assume a flat age distribution for all clusters.\n Similarly, rotation alone is not responsible for the eMSTO when we assume two populations for all clusters: one of non-rotating stars and one of fast rotators with $\\omega=0.9\\omega_{c}$.\n \n  It is now widely accepted that the luminosity of eMSTO stars depends on gravity darkening and that its  effect is strong for large values of the ratio between the rotational velocity and the critical velocity. Our results could indicate that this ratio increases when stars age on the MS as suggested by \\citet{hastings2020a}.\n \n To properly constrain the contribution of rotation and age variation on the eMSTO, it is mandatory to extend the analysis to simulated diagrams that account for different internal age distributions, different rotation-rate distributions \\citep[e.g.][]{huang2006a, huang2010a, goudfrooij2018a}, and that account for both age variations and stellar populations with different rotation rates. \n \n  The interpretation of the eMSTO phenomenon should also account for binary evolution effects \\citep[e.g.][]{wang2022a}. As an example, the stellar models by \\citet{wang2020a} show that the fraction of evolutionary-driven mergers rises for smaller stellar masses, at the expense of the binaries that survive the mass transfer and produce spun-up accretors. An appropriate comparison between the observations illustrated in  Figures\\,\\ref{fig:cumulative} and \\ref{fig:RelEMSTO} and the predictions of stellar models that account for binary evolution is mandatory to shed light on the effect of binary evolution on the eMSTO.  \n \n\n\n\\begin{figure} \n\\centering \n \\includegraphics[height=5.0cm,trim={0.0cm 5.5cm 0.0cm 12.0cm},clip]{RelEMSTO.pdf}\n\n %\n\n \\caption{Area below the $\\Delta_{\\rm X'}$ cumulative curve, A, (left) and median $\\Delta_{\\rm X'}$ value as a function of cluster age for LMC (red dots) and SMC (aqua dots) clusters with the eMSTO.  Open and filled triangles are inferred from simulated CMDs of non-rotating stellar populations with different ages  derived from the Padova and Geneva database, respectively. The diamonds correspond to coeval stellar populations with different rotation rates from the Geneva database. \nSee text for details.}\n \\label{fig:RelEMSTO} \n\\end{figure} \n\n\\subsection{A population of UV-dim stars along the eMSTO of NGC\\,1783}\\label{sub:1783emsto}\n \n\\begin{centering} \n\\begin{figure*} \n \\includegraphics[height=9.0cm,trim={0.1cm 4.cm 0.0cm 3.5cm},clip]{NGC1783pm.pdf}\n  \\includegraphics[height=9.0cm,trim={0.1cm 4.cm 0.0cm 3.5cm},clip]{NGC1783nuv.pdf}\n\n \\caption{Proper motion diagrams of stars in the field of view of NGC\\,1783 in five F438W magnitude intervals (left). The $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD for stars in the left panels is plotted on the middle. Stars within the black circles plotted in the proper motion diagrams are considered probable cluster members and are colored black, whereas field stars are represented with aqua crosses. The right panel shows the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD for probable cluster members, while the inset represents the Hess diagram of the CMD region around the upper MS.} \n \\label{fig:NGC1783pm} \n\\end{figure*} \n\\end{centering} \nThe stellar proper motions derived from our dataset allow the partial separation of bright field stars from NGC\\,1783 cluster members, thus providing new insights on its stellar populations. \nThe left panels of Figure\\,\\ref{fig:NGC1783pm} show the proper-motion diagram for stars in the field of view of NGC\\,1783 in five magnitude bins. The black circles are centered on the absolute proper motion of NGC\\,1783, and are used to separate probable cluster members (black points) from field stars (aqua crosses). \n\nThe corresponding $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD (middle panel) highlights several characteristics of NGC\\,1783 in unprecedented detail. These include the eMSTO \\citep{mackey2008a, milone2009a, goudfrooij2014a} together with a well-populated sequence of MS-MS binaries with large mass ratio \\citep{milone2009a}. The SGB also exhibits intrinsic broadening in color and magnitude, with the majority of stars populating the upper SGB.  \nMoreover, the CMD reveals a broad, possibly dual, sequence of stars  brighter and bluer than the turn-off.\n This blue sequence, which will be investigated in detail in Section\\,\\ref{sec:NGC1783} was first identified by \\citet{li2016a} who associated it with the young stellar populations within NGC\\,1783. Their result has been challenged by \\citet{cabreraziri2016a} who suggested that the blue sequence is composed of field stars. \n\nHere, we focus on the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD of NGC\\,1783, which is illustrated in the right panel of Figure\\,\\ref{fig:NGC1783pm}. \nAn unexpected feature of this CMD is the sparse cloud of stars on the red side of the eMSTO. These stars, which we dub UV-dim, are marked with red triangles in the left panel of Figure\\,\\ref{fig:NGC1783emsto} where we reproduce the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD  zoomed around the eMSTO.\nUV-dim stars comprise a small fraction of $\\sim$7\\% of the total number of eMSTO stars with $20.4<m_{\\rm F555W}<21.5$ mag.\nWe used the same colors to represent these stars in the other panels of Figure\\,\\ref{fig:NGC1783emsto} and showed that they define distinct sequences in both $m_{\\rm F435W}$ vs.\\,$m_{\\rm F343N}-m_{\\rm F435W}$ and $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMDs. If the extreme position in the left-panel CMD is due to observational errors alone, the selected stars would have the same probability of having redder or bluer $m_{\\rm F343N}-m_{\\rm F435W}$ and $m_{\\rm F555W}-m_{\\rm F814W}$ colors than the bulk of MSTO stars. On the contrary, the presence of distinct sequences demonstrates that the extreme red $m_{\\rm F275W}-m_{\\rm F438W}$ colors of the selected stars are intrinsic.\n We note that the Be stars, which are commonly observed in Magellanic Cloud clusters younger than $\\sim 300$ Myr \\citep{keller2000a, bastian2017a, correnti2017a, milone2018a},  also exhibit redder colors  than the bulk of eMSTO stars in CMDs composed of F275W and F336W filters.\n\nIn the following, we explore the possibility that the extreme F275W$-$F438W colors of UV-dim stars are an effect connected to the stellar rotation.\n  It is well known that stellar rotation diminishes the effective temperature and the luminosity of a star, with fast-rotating MSTO stars being redder and dimmer than slow rotators. The position of a star along the eMSTO depends on the effects of limb and gravity darkening and on the viewing angle of the stellar rotation axes with respect to the line of sight. In this context, UV-dim stars \n  \n   would comprise of fast rotators that are seen equator-on, as these stars appear colder and fainter than pole-on fast rotators. \n  \n \n  To qualitatively explore this suggestion, we produced simulations based on the isochrones  from the Geneva database \\citep[][]{mowlavi2012a, ekstrom2012a, ekstrom2013a,  georgy2014a} in close analogy with what we did in Section\\,\\ref{sec:AGEemsto}. \n\n   In the top panels of Figure \\ref{fig:CLOUDsim} we simulated a population of non-rotating stars (aqua points), which includes the 33\\% of the total number of stars, and a population of fast rotators, where the stellar rotation corresponds to 0.9 times the breakout value ($\\omega\/\\omega_{\\rm c}=0.9$, black points). \n   The simulated fraction of binaries is 0.3 and is similar to the observed binary fractions of intermediate-age LMC star clusters \\citep{milone2009a}.\n\n   We note that non-rotating stars are located on the red and faint side of the eMSTO in the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ and $m_{\\rm F435W}$ vs.\\,$m_{\\rm F343N}-m_{\\rm F435W}$ CMDs, similarly to the clouds of UV-dim stars observed in the corresponding diagrams of NGC\\,1783. However, these stars define a narrow sequence that overlaps with the red portion of the eMSTO, in contrast with what is observed in the cloud of stars of NGC\\,1783 where we observe a broad color distribution that extends towards the red of the bulk of eMSTO stars.  In addition, simulated stars exhibit fainter $m_{\\rm F555W}$ magnitudes and redder $m_{\\rm F555W}-m_{\\rm F814W}$ colors than the bulk of eMSTO, in disagreement with what is observed in the $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD. Hence, we conclude that the  cloud of stars in NGC\\,1783 is not consistent with a population of non-rotating stars. \n   \n   In the bottom panels of Figure\\,\\ref{fig:CLOUDsim} we limit the analysis to fast-rotating stars only, where the position of a star in the eMSTO strongly depends on the limb and gravity darkening and on the viewing angle.\n    We select by hand a sample of eMSTO stars with red $m_{\\rm F275W}-m_{\\rm F438W}$ colors and faint $m_{\\rm F438W}$ magnitudes (red points in Figure\\,\\ref{fig:CLOUDsim}).\n  Clearly, the hypothesis that stars with a certain range of viewing angle correspond to the cloud of NGC\\,1783 stars is challenged by the position of the selected stars in the optical CMD, where they define a narrow sequence in the middle of the eMSTO.\n  Further, it seems unlikely that the spread of the turnoff can be entirely attributed to the viewing angle of stars rotating close to breakout value $\\sim$2 Gyrs after their formation.\n\n  \n   As an alternative, circumstellar disks could be responsible for absorbing the UV radiation and the consequent cloud of stars on the red side of the eMSTO. Debris disks, possibly associated with planet formation, are frequently observed around A-type stars \\citep[e.g.][and references therein]{eiroa2013a}.\n   A challenge is the location of UV-dim stars in the $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD, which would imply that their disks poorly affect the emergent optical radiation. On the contrary, dust absorption is strongly dependent on the wavelength and is much more significant in the UV than in the optical. Hence, circumstellar dust in a disk could explain the location of these stars in the CMD (D'Antona et al., in preparation).\n   \n   In this scenario, the disks are associated with stars that are currently non-rotating so that they can distribute along a narrow sequence in the optical CMD. \n  \n  \n   If the disk formation is associated with rotation, these stars should have experienced fast rotation in their lifetime.\n\n\n\n\n\n\n\\begin{figure} \n\\centering\n \\includegraphics[height=5.0cm,trim={1.2cm 6.0cm 0.0cm 12.0cm},clip]{NGC1783emsto.pdf}\n\n \\caption{$m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ (left), $m_{\\rm F435W}$ vs.\\,$m_{\\rm F343N}-m_{\\rm F435W}$ (middle), and $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ (right) CMDs of proper-motion selected NGC\\,1783 stars. Stars in the red tail of the eMSTO, selected from the left-panel CMD, are colored red.}\n \\label{fig:NGC1783emsto} \n\\end{figure} \n\n\n\n\n\\begin{figure} \n\\centering\n \\includegraphics[height=8.0cm,trim={0.5cm 6.0cm 0.0cm 5.0cm},clip]{CLOUDsimLR.pdf}\n\n \\caption{Simulated CMDs of two stellar populations of fast-rotating stars ($\\omega\/\\omega_{\\rm c}=0.9$, black points) and non-rotating stars (aqua points). Red points in the bottom-panel diagrams mark the sample of fast rotating stars selected by hand and located on the red side of the eMSTO in the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD. Simulations are derived from Geneva isochrones.}\n \\label{fig:CLOUDsim} \n\\end{figure} \n\n\\subsection{A zigzag across the MS of NGC\\,1783}\\label{sub:zigzag}\nA visual inspection of the CMD of Figure\\,\\ref{fig:NGC1783pm} shows another intriguing detail of the CMD of NGC\\,1783. As highlighted by the Hess diagram in the inset, the upper MS runs in a zigzag, with two main discontinuities ($m_{\\rm F438W} \\sim 22.0$ and 22.4) and various sudden changes of slope around $m_{\\rm F438W}=21.9, 22.1, 22.3$ and 22.5.\n\u00a0\n\nWe compare in Figure\\,\\ref{fig:NGC1783iso} the observed upper MS of NGC\\,1783 with the isochrones from Padova \\citep[left,][]{marigo2017a}. The faint MS discontinuity corresponds to effective temperature $T_{\\rm eff}=6,900$K and mass \u00a0$\\mathcal{M}$=1.26 $M_{\\odot}$, whereas stars on the bright MS discontinuity have $T_{\\rm eff}=7,250$K and $\\mathcal{M}$=1.19 $M_{\\odot}$. \n\nWe tentatively associate the hotter gap of the NGC\\,1783 MS with the original B{\\\"o}hm-Vitense gap.\nA gap along the MS at $T_{\\rm eff} \\sim 7,500$K has been first predicted by \\citet[][]{bohmvitense1970a} and observed in the nearest Galactic open clusters \\citep[e.g.][]{bohmvitense1974a, debruijne2000a, debruijne2001a}. The B{\\\"o}hm-Vitense gap is associated with sudden changes in the structure of convective atmospheres. It has been interpreted as a color effect, due to the fact that the temperature gradient in deep atmospheric layers becomes smaller than the radiative gradient. As an alternative, it is the effect of temperature inhomogeneities produced by photospheric granulation \\citep[e.g.][]{bohmvitense1982a}.\n\nThe colder discontinuity of the NGC\\,1783 MSs could correspond to a distinct MS gap, which was earlier investigated by \\citet{dantona2002a}.\n Indeed, fainter MS gaps have been observed around $T_{\\rm eff} \\sim 7,000$K \\citep[e.g.][]{rachford2000a}.\nAt this temperature, convection begins in stellar envelopes, and an increasing amount of the stellar exterior \n becomes convective as the mass and effective temperature decrease. Also, the eMSTO disappears below $\\sim 7,000$K, confirming that fast-rotating MS stars are only  present at hotter temperatures. Indeed, the external turbulence brakes the envelope rotation. \nClearly, the change of stellar structure results in a variation of the MS slope. \nRecent works provide evidence of an MS kink at similar temperatures in several Galactic and Magellanic Cloud clusters, where the split MS, which is associated with stellar populations with different rotation rates, merges into a single MS \\citep[e.g.][]{dantona2017a, milone2018a, marino2018a, goudfrooij2018a}. \u00a0\n\nThe fainter MS gap has been investigated in the Hyades by \\citet{dantona2002a} based on the full spectrum of turbulence model by \\citet{canuto1996a}, which \u00a0predicts that\u00a0the depth of the convective envelope suddenly changes within a narrow range of stellar mass and around \n $T_{\\rm eff} \\sim 6,800$K. They concluded that the gap is an effective-temperature effect associated with the sharp effective-temperature difference between stars that are only convective in the surface layers  and stars with well-developed convective interiors. \n\nHowever, these results are based on poorly-populated CMDs of open clusters, which often make it challenging to assess the statistical significance of the gaps.\nThe high-precision {\\it HST} photometry of populous star clusters may overcome this limitation and provides new insights on the MS region in the temperature range between $\\sim$6,500K and 7,500K.\n\nAs shown in Figure\\,\\ref{fig:NGC1783iso}, neither the isochrones from the Padova group nor those from the BaSTI \\citep[middle,][]{pietrinferni2004a}, and MESA \\citep[][]{choi2016a, dotter2016a, paxton2011a} databases reproduce the observed MS discontinuities.\nThis fact corroborates the conclusion that  these stellar models, poorly reproduce the CMD region where the stellar atmosphere changes from radiative to convective.\n\n \n\\begin{centering} \n\\begin{figure} \n \\includegraphics[height=4.5cm,trim={0cm 0.0cm 0.0cm 0.0cm},clip]{NGC1783_isoc_2LR.pdf}\n\n \\caption{Reproductions of the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD of Figure\\,\\ref{fig:NGC1783pm} zoomed around the upper MS. The red, blue, and green lines superimposed on each CMD are the best-fitting non-rotating isochrones from the Padova (left), BaSTI (middle), and MESA (right) databases. A rotating MESA isochrone with $\\omega=0.4 \\omega_{\\rm c}$, with $\\omega_{\\rm c}$ being the breakout velocity, is plotted in the right panel in cyan.}\n \\label{fig:NGC1783iso} \n\\end{figure} \n\\end{centering} \n\n\n\n\\subsection{Search of multiple generations in NGC\\,1783} \\label{sec:NGC1783}\n\nIn the past years, astronomers have dedicated huge efforts to searching for young star clusters that are analogous to old GCs. Indeed, they may provide a snapshot of multiple populations shortly after their formation.\nThe two sequences of stars in the field of view of the $\\sim$1.5 Gyr-old cluster NGC\\,1783 are a hotly debated case because they are consistent with younger stellar populations of $\\sim$440 and 520 Myr.\nAfter statistically subtracting the contribution of field stars from the CMD of stars around NGC\\,1783, \\citet{li2016a}  concluded that the blue MSs are cluster members. They  suggested that these young stars are the signature of burst-like star formation and that NGC\\,1783 has experienced multiple bursts of star formation with an age difference of a few hundred million years. In this scenario, NGC\\,1783 is eventually the young counterpart of GCs with multiple populations. \n \n This result has been challenged by \\citet{cabreraziri2016a}, who suggested that the background subtraction method adopted by Li and collaborators may not remove contaminating field stars. Hence, they concluded that there is no evidence for multiple generations within NGC\\,1783 and that the young populations are field LMC stars along the same line of sight of NGC\\,1783.\n\n As anticipated in Section\\,\\ref{sub:1783emsto}, the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD plotted in Figure\\,\\ref{fig:NGC1783pm} clearly reveals the stellar sequence first investigated by \\citet{li2016a}, whereas stellar proper motions allow disentangling the bulk of cluster members and field stars.\n To investigate whether NGC\\,1783 hosts a population of bright and hot MS stars, we combined information from photometry and stellar proper motions as illustrated in Figure\\,\\ref{fig:NGC1783y}.\n  We first used the dashed rectangle plotted in the top-left panel of Figure \\ref{fig:NGC1783y} to select a sample of stars on the blue side of the cluster MSTO in the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD. \n \n In the top-right panels, we compare the CMDs for stars in the dashed-line rectangle located within and outside a radius equal to 45 arcsec from the cluster's center. The external region has a $\\sim$four-time wider area than the internal one.\n \n The proper motion diagram plotted in the bottom-left panel of Figure\\,\\ref{fig:NGC1783y} shows that NGC\\,1783 stars are partially separated from LMC stars.\n  We draw the red circle to separate the bulk of cluster members from field stars and represent these stars with black circles and aqua-starred symbols, respectively, in the bottom-left and in the top panels of Figure\\,\\ref{fig:NGC1783y}.  \n  \n  We find that the sample of selected blue stars in the CMD comprises 18 field stars in the internal region, while 64 field stars belong to the external region. Their ratio of about four is comparable with the ratio of the corresponding field-of-view areas as expected if field stars have uniform spatial distribution. On the contrary, the number of stars with cluster-like proper motion in the internal and external field are 53 and 64 and are comparable with each other thus indicating that they unlikely belong to the LMC field population.\n  \n Noticeably, the distribution of stars in the proper motion diagram (i.e.\\,the stellar abscissas and ordinates and their density) is well reproduced by a function composed of the sum of two 3D Gaussian  functions, that we derived by means of least squares minimization. For illustration purposes, we show in Figure\\,\\ref{fig:NGC1783y} the histogram distributions of $\\mu_{\\rm \\alpha}$ cos $\\delta$ and $\\mu_{\\delta}$ together with the corresponding two-dimensional Gaussian functions.\n \n    Clearly, the stars with cluster-like proper motions   selected in the bottom-left panel of Figure\\,\\ref{fig:NGC1783y}  may also include field stars. To estimate the fraction of field stars that contaminate the sample of probable cluster members we used \n the best-fitting 3D Gaussian functions to simulate the proper motions plotted on the bottom-right panel of  Figure\\,\\ref{fig:NGC1783y}. Here, we show a subsample of 198 simulated stars, which is the same number of observed stars. Simulated field stars are represented with starred symbols and cluster stars with small dots. Clearly, a fraction of field stars (red starred symbols) have cluster-like proper motions while some cluster members (blue dots) lie outside the red circle.  In particular, the fraction of field stars within the red circle, with respect to the number of cluster members is $\\sim$5\\%. These facts demonstrate that the majority  ($\\sim$ 95\\%) of stars with cluster-like proper motions selected in Figure\\,\\ref{fig:NGC1783y}  are cluster members.\n \n In summary, our proper-motion-based results confirm the conclusion of Li and collaborators that the blue sequences are composed of genuine members of NGC\\,1783.\nMore sophisticated analysis is mandatory to understand whether the blue sequence is associated with young stellar populations as suggested by Li and collaborators or whether it is composed of blue stragglers. \n  The cluster members on the bright and blue side of the red clump are consistent both with young red clump stars and with binary systems composed of red-clump stars.\n \n\n\\begin{figure*} \n\\centering\n\n\n\n\n \\includegraphics[height=11.0cm,trim={3.0cm 0cm 3.0cm 0cm},clip]{NGC1783yLR.pdf}\n\n \\caption{ \\textit{Top.} Reproductions of the CMDs of NGC\\,1783 of Figure\\,\\ref{fig:NGC1783pm}. Right panels are zoomed-in views of the CMD region on the bright-blue side of the MSTO (dashed rectangle in the top-left panel) for stars with radial distances from the cluster center smaller and larger than 45 arcsec. \n \\textit{Bottom.} Proper motion diagram of stars plotted in the top-left panel (left). The red circle separates stars with a cluster-like motion from the bulk of field stars. Black points and aqua-starred symbols mark the selected bright-blue stars. The corresponding histogram distributions for $\\mu_{\\rm \\alpha}$ cos $\\delta$ and $\\mu_{\\delta}$ are also represented on the top and right side of the panel. The best fit bi-Gaussian functions are represented with gray lines, and the two Gaussian components are colored aqua and red. The bottom-right panel shows the simulated proper motions for cluster members and field stars.  Black and blue dots indicate NGC\\,1783 stars with cluster-like and field-like proper motions, respectively, while red color is used to distinguish field stars with cluster-like proper motions from the remaining field stars (aqua starred symbols). \n }\n \\label{fig:NGC1783y} \n\\end{figure*} \n\n\n\n\n\n\n\n\\subsection{Relative proper motions of stellar populations in the LMC and the SMC} \\label{sec:pm}\nFigures\\,\\ref{fig:PMsSMC} and \\ref{fig:PMsLMC} show the CMDs and the proper motion diagrams  of stars in the fields of view of five SMC clusters, namely Kron\\,34, NGC\\,294, NGC\\,339, NGC\\,416, and NGC\\,419, and three LMC clusters, namely NGC\\,1755, NGC\\,1801 and NGC\\,1953.\nSince we focus on the internal kinematics of LMC and SMC stars, we restrict the analysis to the magnitude interval that provides the most precise proper-motion determinations. \nThe stellar concentrations around the center of each diagram are composed of cluster members and their broadening is mostly due to observational uncertainties. Indeed, the star-to-star scatter associated with the internal motions of cluster members is negligible with respect to proper motion errors at the distance of the Magellanic Clouds. \nOn the contrary, field stars exhibit broad proper-motion distributions, which are significantly wider than what is expected from observational uncertainties alone. \n\nWe identified in each CMD a group of stars with blue $m_{\\rm F336W}-m_{\\rm F814W}$ colors, which mostly comprise the young stellar populations of the host galaxy, and a group of old stars with red colors.   Furthermore, we selected a sample of very-young LMC stars in NGC\\,1801 and NGC\\,1953 that define the bluest and brightest MS in their CMD (aqua triangles). The selected groups of old and young stars, identified in the CMDs, are highlighted with red and blue symbols, respectively, in the proper motion diagrams plotted in the third and fourth columns of panels. \n  A visual inspection of these figures suggests that the proper motions of young and old SMC stars typically show different ellipticities, whereas the differences are less pronounced for LMC stellar populations. \n\nResults are illustrated in Figure\\,\\ref{fig:ellissi} and  summarized in Table\\,\\ref{tab:ellissi}, where we provide for each population the median proper motions relative to the main cluster in the field, the ellipticity of the best-fitting ellipse that encloses 90\\% of stars ($\\epsilon=1-b\/a$, where $a$ and $b$ are the minor and major axes of the ellipse), and the position angle $\\theta$.\nAs shown in Figure\\,\\ref{fig:PMsSMC}, young SMC stars exhibit flat proper motion distributions, where the eccentricity of the best-fitting ellipses ranges from $\\epsilon \\sim 0.3$ in NGC\\,416 to $\\epsilon \\sim 0.6$ in NGC\\,339. The proper motion distributions of the old stellar populations have smaller eccentricity values, between $\\epsilon \\sim 0.1$ in NGC\\,416 and NGC\\,339 and $\\epsilon \\sim 0.3$ in NGC\\,419.  In all cases, the major axis of the best-fitting ellipses roughly follows the direction North West - South East, thus pointing towards the LMC.   \nSimilar conclusions are derived by \\citet{massari2021a} based on high-precision proper motions and stellar photometry from {\\it HST} of NGC\\,419. These authors demonstrated that it is possible to separate cluster members from SMC field stars  by using stellar kinematics. Moreover, they identified a kinematic stellar component that they associated with the Magellanic Bridge. Although our results do not provide evidence for populations of field stars with distinct kinematics, the flattened proper motion distributions would reflect the flow motion of stars from the SMC to the LMC.\n Further evidence of SMC star clusters showing a relative motion pointing towards the LMC is provided by \\citet{zivick2018a, piatti2021a, dias2021a, schmidt2022a}.\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=13.0cm,trim={7cm 1cm 9cm 0cm},clip]{PMsSMCLR.pdf}\n\n \\caption{First-column panels show the CMDs of stars in the field of view of five SMC clusters with available proper motions.  The  corresponding proper motion diagrams  are plotted in the second column, while the third and fourth columns represent the proper motions of candidate old and young field stars selected in the CMDs and colored red and blue, respectively. \n  Red and blue ellipses provide the best-fitting of the distributions of candidate old and young field stars in the proper motion diagram. \n  The Hess diagrams of the proper-motion distributions are shown in all proper-motion diagrams.  }\n \\label{fig:PMsSMC} \n\\end{figure*} \n\n Young LMC stars exhibit flatter proper-motion distributions than old LMC stars, in close analogy with what is observed for the SMC. However, the ellipses that best fit the proper motions of LMC stars in the fields of NGC\\,1755, and NGC\\,1953 have different orientations than the corresponding ellipses inferred for stars in the direction of NGC\\,1801.  Very young stars in the field of view of NGC\\,1953 have more clustered proper motions with respect to the remaining young stars. Interestingly,   the selected young stars show some hints of split MS in the CMD. A spectroscopic investigation is mandatory to understand whether the split is due to stellar populations with different rotation rates, similar to what is observed in the star clusters with similar ages, or to differences in distance, age, and\/or chemical composition.\n The proper motion distributions of stars in the direction of the three analyzed LMC clusters are nearly circular, in contrast with what is observed for young stars in both Magellanic Cloud clusters and old SMC stars. \n \n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=15.0cm,trim={3cm 0cm 3cm 0cm},clip]{PMsLMCLR.pdf}\n \\caption{Similar to Figure\\,\\ref{fig:PMsSMC} but for the LMC clusters NGC\\,1755, NGC\\,1801, and NGC\\,1953. Very young stars in the fields of view of NGC\\,1801 and NGC\\,1953 are represented with aqua triangles.}\n \\label{fig:PMsLMC} \n\\end{figure*} \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=8.5cm,trim={6.0cm 0cm 4cm 0cm},clip]{ellissiLR.pdf}\n \\caption{Positions of candidate young LMC and SMC stars relative to the center of the SMC (top). The bottom panel shows the density distributions of candidate old LMC and SMC stars. \n The ellipses that provide the best fit of proper-motion distributions in young and old stars in the fields of eight clusters are colored blue and red, respectively.   }\n \\label{fig:ellissi} \n\\end{figure} \n\n\n         \n\\subsubsection{The massive star-forming region NGC\\,346}\\label{sub:NGC346}\n\nNGC\\,346 is a very young SMC star cluster \\citep[age $\\sim$ 3 Myr,][]{bauret2003a, sabbi2007a} that is responsible for the excitation of the surrounding H${\\rm II}$ region N\\,66. \nA stacked F814W image of NGC\\,346 and its neighborhoods is shown in the left panel of Figure\\,\\ref{fig:NGC346} where we mark with an azure circle the central part of the NGC\\,346 star-forming region.    \n\nThe intermediate-age star cluster BS\\,90 is also visible to the north of NGC\\,346 and is highlighted by the red circle in the left panel of Figure\\,\\ref{fig:NGC346}. The proper motion diagram of all stars with $m_{\\rm F814W}$ between 18.4 and 21.4 mag is plotted in the middle panel of Figure\\,\\ref{fig:NGC346} and comprises stars with the best proper-motion quality \\footnote{ Specifically, the selected stars pass the criteria of selection based on the $RADXS$ and $qfit$ parameters discussed in Section\\,\\ref{subsec:quality} and that are not saturated in the long-exposure F555W and F814W images (see Table\\,\\ref{tab:dataPM} for details on the dataset). Moreover, we only included stars with small proper-motion errors, when compared to the bulk of stars with similar magnitudes. To select them, we first plotted the proper-motion uncertainty against the F814W magnitude. Then, we divided the magnitude interval into various 0.25 mag bins and calculated the median proper-motion uncertainty for each bin. We computed the absolute values of the difference between the uncertainty of each star and the median value and estimated the 68.27$^{\\rm th}$ percentile of the corresponding distribution ($\\sigma$). We added three times $\\sigma$ to the median uncertainty of each bin  and associated this value with the median magnitude of the stars in the bin. Finally, these points are linearly interpolated and the stars that are located below this line in the proper-motion uncertainty vs.\\,magnitude plane are considered as well measured. }. Stars within the regions centered on NGC\\,346 and BS\\,90 (azure and red points, respectively) define two distinct clumps in the proper motion diagram, which are mostly composed of cluster members.    \nWe selected probable cluster members with proper motions smaller than four times the r.m.s of the proper motion distributions (stars within the circles) and calculated the median values of $\\mu_{\\rm \\alpha} cos{\\delta}$ and $\\mu_{\\rm \\delta}$. Results are listed in Table\\,\\ref{tab:n346}. The fact that NGC\\,346 and BS\\,90 exhibit different proper motions demonstrates that they are distinct clusters projected onto the same field of view.\n\nThe probable members of NGC\\,346 and BS\\,90, selected from both stellar proper motions and positions, are marked with azure and red points in the CMD in the right panel of Figure\\,\\ref{fig:NGC346}. The sample of selected NGC\\,346 stars defines a well-populated MS and upper pre-MS, whose large broadening is indicative of a significant amount of differential reddening.  On the contrary, BS\\,90 exhibits narrow SGB and RGB sequences, and a well-defined red clump, thus confirming that this cluster is poorly affected by differential reddening \\citep[][]{sabbi2007a}. Specifically, the average reddening variation in the field of view within 36 arcsec from the center of BS\\,90 never exceeds $\\Delta$E(B$-$V)$=0.013$ mag, with  $\\sim 68 \\%$ of the stars  having $\\Delta$E(B$-$V) values within 0.004 mag from the average reddening.  This fact demonstrates that this cluster is in the foreground with respect to the region of NGC\\,346 and N\\,66.\n\\begin{centering} \n\\begin{figure*} \n \\includegraphics[height=7.5cm,trim={0.1cm 4cm 0.0cm 1.0cm},clip]{NGC346ima.pdf}\n\n \\caption{Stacked F814W WFC\/ACS image of the SMC field that includes the clusters NGC\\,346 and BS\\,90, which are highlighted by the azure and red circle, respectively (left). The middle panel shows the proper motion diagram, while the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD is represented in the right panel. \n  Only stars in the F814W magnitude interval between the two horizontal lines in the CMD are plotted in the proper-motion diagram, where the stars in the NGC\\,346 and BS\\,90 regions defined in the left panel are colored blue and red, respectively. Blue and red symbols in the right-panel CMD mark proper-motion selected cluster members in the NGC\\,346 and BS\\,90 regions. See text for details.}\n \\label{fig:NGC346} \n\\end{figure*} \n\\end{centering} \n\n The CMD of stars in the field of view of NGC\\,346 comprises stars in different evolutionary stages for which high-precision proper motions are available, including pre-MS stars, MS stars, and evolved stars in the SGB, RGB, and red-clump phases. \n  As widely discussed in literature works \\citep[e.g.][]{sabbi2007a, cignoni2010a, cignoni2011a}, this field hosts a conspicuous population of pre-MS stars that are highlighted in the top-left panel of Figure \\ref{fig:NGC346pop}. In particular, we used orange and yellow colors to represent two samples of pre-MS\\,I and pre-MS\\,II stars with high-precision proper motions, which lie respectively inside and outside the central region of NGC\\,346. The remaining pre-MS stars are colored black.\n   As indicated in Table\\,\\ref{tab:n346}, the proper motion distributions of the two groups of pre-MS stars share the same mean motion as NGC\\,346 indicating that stars in both the central region and in the outskirts share the same mean motions (top-middle panel of Figure\\,\\ref{fig:NGC346pop}), although the latter  exhibits a wider proper motion dispersion. \n   The distribution of pre-MS stars across the field of view highlights the distinctive structure of the NGC\\,346 region described by \\citet{contursi2000a}, including the  low-density filament oriented to the north-east direction (spur), and the fan-shaped structure (bar), that hosts the majority of pre-MS stars.\n  \n  \n  MS stars are investigated in the middle panels of Figure\\,\\ref{fig:NGC346pop}, and comprise stars of the young population of the SMC. The proper motion diagram reveals that the bulk of MS stars (hereafter MS\\,I stars, aqua points) exhibit a proper-motion distribution similar to the NGC\\,346 pre-MS stars. In addition, we note a tail of stars in the proper motion diagram that points towards the LMC and that we colored blue and name MS\\,II. \n  Both groups of selected MS stars seem diffused over the whole field, but the MS\\,I stars define some stellar overdensities that trace the bar and possibly, some clumps of the Spur. \nWe suggest that the MS\\,II is mostly composed on SMC field stars that follow an elliptical proper motion distribution, in close analogy with what is observed for the other analyzed SMC young stars. On the contrary, MS\\,I stars comprise both field SMC stars, and stars of the NGC\\,346 region.\n \n Finally, the old SMC field stars are investigated in the bottom panels of Figure\\,\\ref{fig:NGC346pop}. Although this stellar population hosts some stars up to ages of more than 10 Gyr, it is mostly associated to a major star-formation episode that occurred between $\\sim 3$ and 5 Gyr ago. The RGB stars of the old populations with radial distance smaller than 36 arcsec from the center of BS\\,90 are marked with red points. These stars exhibit similar proper motion distribution as young stars and pre-MS stars in the star-forming region of BS\\,90 (bottom-middle panel) and are uniformly distributed across the entire field of view (bottom-right panel of Figure \\ref{fig:NGC346pop}). Note, the motion of the bulk of old SMC field stars is significantly different from that of BS\\,90, despite this cluster having an age  similar to most old field stars. \n\n\\begin{figure*} \n\\centering\n %\n %\n \\includegraphics[height=14.0cm,trim={4.5cm 0cm 3.0cm 0cm},clip]{NGC346popLR.pdf}\n\n \\caption{The three left panels are reproductions of the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMDs for stars in the field of view of NGC\\,346 and are used to select various groups of stars that we represented with colored symbols. \n The middle panels show the proper motions for stars in the magnitude interval delimited by the horizontal lines in the corresponding CMDs, whereas the right panels show the coordinates of the selected stars.\n The top panels are focused on candidate pre-MS stars. Specifically, bright pre-MS within 28 arcsec from the center of NGC\\,346 (pre-MS\\,I sample) are represented with orange symbols, the remaining bright pre-MS stars (pre-MS\\,II sample) are colored yellow, while the black symbols in the top panels mark faint pre-MS stars.  Bright MS stars are investigated in the middle panels. Blue and aqua colors mark MS\\,I and MS\\,II stars, which have radial distances larger than 28 arcsec from the center of NGC\\,346, but different proper motion distributions. The remaining bright-MS stars are colored black. The bottom panels highlight the selected RGB stars with radial distances from the center of BS\\,90 larger (red) and smaller than 60 arcsec (black). }\n \\label{fig:NGC346pop} \n\\end{figure*} \n\n\n\n\\section{Summary and conclusions}\\label{sec:summary}\nWe have used the {\\it HST} archive to retrieve ACS\/WFC, UVIS\/WFC3, and NIR\/WFC3 images of 101 fields in the direction of the LMC and the SMC. These images include  29 SMC clusters and 84 LMC clusters. We derived high-precision photometry and astrometry by using the methods and the computer programs developed by Jay Anderson and his collaborators and obtained high-resolution reddening maps in the direction of each  cluster.\nWe provide accurate determinations of cluster centers and estimate distance modulus, reddening, metallicity, and age by comparing the CMDs with Padova isochrones \\citep{marigo2017a}.\nMoreover, we calculated proper motions for cluster and field stars in twelve stellar fields that have been multi-epoch observations. \n\nThe exquisite photometry, astrometry, and proper motions presented in this paper have the potential to shed light on a variety of astrophysical phenomena. As an example, we present here some results that are evident from visual inspection of the CMDs and the proper motion diagrams.\n\n\\begin{itemize}\n\n    \\item {\\bf New insights into the eMSTO phenomenon.}\n    \n    The photometric catalogs, derived from homogeneous  data reduction, allow accurate comparison of clusters with different ages.  \n    The analysis of the $m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMDs of 19 LMC clusters in a wide range of ages between $\\sim$20 Myr and 2 Gyr, reveals that the distribution of stars along the eMSTO significantly changes from one cluster to another and depends on GC age. While the eMSTOs of young clusters are dominated by blue and bright eMSTO stars, the fraction of stars in the red and faint eMSTO increases in older clusters. \n    This property of eMSTO stars provides a new observational constraint to understand the physical mechanism that is responsible for the eMSTO. \n    \n    We also provide the first evidence of eMSTO in the LMC intermediate-age cluster KMHK\\,361 and in the SMC young star cluster NGC\\,265, where we also detect a split MS, with the blue MS hosting about one-third of MS stars.  This finding corroborates the conclusion that the eMSTO is a universal feature of the CMD of clusters younger than $\\sim$2 Gyr and the split MS is a common phenomenon that occurs in clusters younger than $\\sim 1$ Gyr. \n    \n\\item {\\bf A new feature along the eMSTO.} \nWe find that about 7\\% of eMSTO stars in NGC\\,1783 exhibit redder $m_{\\rm F275W}-m_{\\rm F438W}$ and  $m_{\\rm F343N}-m_{\\rm F438}W$  colors than the remaining eMSTO stars. They show a wide color broadening up to $\\sim$0.2 mag in the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD, but the color spread decreases to less than 0.1 mag in the  $m_{\\rm F435W}$ vs.\\,$m_{\\rm F343N}-m_{\\rm F435W}$ diagram. On the contrary, when observed in optical CMDs, these stars define a narrow sequence and have intermediate colors relative to the remaining eMSTO stars.\n    \n    \\item     {\\bf Hunting for multiple star-formation bursts in intermediate-age star clusters.}\nIt has been suggested that the bright and blue stars of the CMD of NGC\\,1783 are cluster members and correspond to young stellar generations \\citep{li2016a}. This result, which would be a major step towards the understanding of multiple populations in GCs, has been challenged by \\citet{cabreraziri2016a}, who suggested that the blue sequences are composed of field stars. \n\nThe catalogs of the present work include proper motions of stars in the field of view of NGC\\,1783 thus providing additional information on the origin of the blue sequences.  \n In particular, we support the conclusion that most bright blue MS stars are consistent with being cluster members, thus excluding that the blue MS are artifacts produced by poor statistical subtraction of the field.\n  Our results, based on proper motion analysis, do not allow us to infer whether the blue sequence is composed of MS stars of a young stellar generation or are the blue straggler sequences of NGC\\,1783. \n\n\\end{itemize}\n\nIn addition to disentangling cluster members and field stars, proper motions allow the investigation of the internal kinematics of stellar populations in the Magellanic Clouds.\n We have identified two groups of young and old Magellanic-Cloud field stars in the FoVs of five SMC clusters, namely KRON\\,34, NGC\\,294, NGC\\,339, NGC\\,416, and NGC\\,419, and of three LMC clusters NGC\\,1755, NGC\\,1801 and NGC\\,1953.\n\nThe proper motions of young SMC stars exhibit elliptical  distributions with high ellipticity values and major axes that point toward the LMC. \nThe flattened proper-motion distributions would be associated with the Magellanic bridge and represent the dynamic signature of the flow motion of stars from the SMC to the LMC. Our results corroborate the evidence that SMC stars are affected by the LMC \\citep[e.g.][]{piatti2015a}.\nOld and young SMC stars exhibit different kinematics.\nThe proper motions of the old SMC stars are also oriented towards the LMC and have elliptical distributions  but with lower values of ellipticity and, in most cases, different centers.  The different proper-motion distributions of old and young stars could reflect, in part, the presence of different young and old bridges.\n \n The young and the old LMC stars also exhibit different motions on the plane of the sky. While the proper motions of the old populations show nearly circular distributions, young LMC stars have more flattened proper-motion distributions, with different orientations of the best-fitting ellipses. \n\n\\section*{Acknowledgments} \n\\small\nWe thank the anonymous referee for various suggestions that improved the quality of the manuscript.\nThis work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research innovation programme (Grant Agreement ERC-StG 2016, No 716082 'GALFOR', PI: Milone, http:\/\/progetti.dfa.unipd.it\/GALFOR).\nAPM, MT, and ED acknowledge support from MIUR through the FARE project R164RM93XW SEMPLICE (PI: Milone). APM and MT  have been supported by MIUR under PRIN program 2017Z2HSMF (PI: Bedin). This research was supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) through project number CE170100013.\nThis work is based on observations made with the NASA\/ESA {\\it Hubble Space Telescope}, obtained from data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.\n\\section*{Data availability}\nThe data underlying this article will be shared on reasonable request to the corresponding author. \\\\\n\n\\section*{Appendix. Serendipitous discoveries}\nWe report the serendipitous findings of two new star clusters, hereafter clusters 1 and 2. Two zoom-ins of the region around cluster 1 are provided in the top-left panels of Figure\\,\\ref{fig:galfor2}, where we provide the monochromatic images in the F475W and F850LP bands. The cluster, which is centered around RA=00:45:21.28, DEC=$-$73:12:18.1, J2000 is clearly visible in the F850LP image, while the most prominent feature of the F475W image is a nebula that envelopes it. The top-right panel shows the $m_{\\rm F775W}$ vs.\\,$m_{\\rm F775W}-m_{\\rm F850LP}$ CMD for all stars in the WFC\/ACS field that includes cluster 1. \nThe bright star close to the cluster center is classified as the H$\\alpha$ emission-line star MA93-99 by \\citet{meyssonnier1993a}.\n We defined a circle centered on the cluster with a 4 arcsec radius (hereafter cluster region) and represented with black circles all stars within this region. We also plot the stars in a randomly selected reference region with the same area as the cluster region with aqua crosses.\n Clearly, we note an overdensity of stars in the cluster region with red $m_{\\rm F775W}-m_{\\rm F850LP}$ colors and luminosity fainter than $m_{\\rm F775W} \\sim 23$. These stars are qualitatively consistent with a population of pre-MS stars.\n\nCluster\\,2 is located around RA=05:03:55.87 DEC=$-$66:24:36.9, J2000 and is shown in the F160W image plotted on the bottom-left of Figure\\,\\ref{fig:galfor2}. The $m_{\\rm F160W}$ vs.\\,$m_{\\rm F110W}-m_{\\rm F160W}$ CMD plotted in the right panel reveals an overdensity of red stars in the cluster field, which suggests that cluster\\,2 hosts a conspicuous population of pre-MS stars. \n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=11cm,trim={2cm 0cm 2cm 0.0cm},clip]{galfors.pdf}\n\n \\caption{{\\textit{Top panels}}. Stacked images in F475W and F850LP centred on the cluster\\,1 discovered in this paper (left). The $m_{\\rm F775W}$ vs.\\,$m_{\\rm F775W}-m_{\\rm F850LP}$ diagram is plotted on the right for all stars in the ACS\/WFC field (gray points), stars in the cluster\\,1 field (black circles) and in the reference field (aqua crosses). {\\textit{Bottom panels}}. Stacked image in F160W for the discovered cluster\\,2 (left) and $m_{\\rm F160W}$ vs.\\,$m_{\\rm F110W}-m_{\\rm F160W}$ CMD (right). Gray, black, and aqua symbols indicate stars in the WFC3\/NIR field, stars in the cluster field, and in the reference field, respectively. }\n \\label{fig:galfor2} \n\\end{figure*} \n\nAs a further outcome of the survey, we report the serendipitous discovery of a gravitational lens in the field of view of the SMC  star cluster Lindsay\\,38.\n The F814W stacked image shown in Figure \\ref{fig:lente} reveals an elongated arc-like structure around what appears to be a single early-type galaxy centered on the coordinates RA=00 48 57.01, DEC=$-$69 51 30.4, J2000. Its spatial extension is such that fine structures of the source are detectable in the arc. This object may correspond to a strong lensing configuration that implies remarkable alignment between lens and source along the line of sight.\n Due to their intrinsic symmetry, such a peculiar configuration would permit to constrain with great accuracy the total enclosed mass within the projected Einstein radius \\citep[e.g.][and references therein]{bettinelli2016a}. \n\\begin{figure} \n\\centering\n \\includegraphics[height=7.0cm,trim={8cm 4cm 6cm 4.0cm},clip]{CandidateLgL38.pdf}\n\n \\caption{Stacked F814W WFC\/ACS image of the field around the   early-type galaxy centered at RA=00 48 57.01, DEC=$-$69 51 30.4, J2000. Note the elongated arc-like structure due to gravitational lensing. }\n \\label{fig:lente} \n\\end{figure} \n\n\n\n\n\\input{tabDATA.tex}\n\\input{TabINFO.tex}\n\n\\begin{table}\n  \\caption{ The quantities listed in this table are indicative of the precision of values of distance modulus, reddening, age, and metallicity inferred from isochrone fitting. The horizontal lines separate the couples of clusters with similar ages and different photometric qualities. See text for details. }\n\\begin{tabular}{l c c c c}\n\\hline \\hline\nID         & $\\Delta$(m$-$M)$_{0}$ & $\\Delta$E(B$-$V) & $\\Delta$age & $\\Delta$[M\/H] \\\\\n           & [mag] & [mag] & [Myr] & [dex] \\\\\n\\hline\nNGC\\,2005  & 0.10 & 0.010 & 500 & 0.10 \\\\\nNGC\\,1939  & 0.15 & 0.015 & 1000 & 0.15 \\\\\n\\hline\nKron\\,3    & 0.10 & 0.010 & 200  & 0.10 \\\\\nKron\\,1    & 0.10 & 0.020 & 350  & 0.15  \\\\\n\\hline\nNGC\\,1846  & 0.10 & 0.010 & 75  & 0.10  \\\\\nHodge\\,7   & 0.15 & 0.020 & 150 &  0.15 \\\\\n\\hline\nNGC\\,1866  & 0.10 & 0.010 & 30  & 0.10 \\\\\nBSDL\\,1650 & 0.20 & 0.035 & 100 & 0.20  \\\\\n\\hline\n\\end{tabular}\n\\label{tab:errori}\n\\end{table}\n\n\n\\input{tab.tex}\n\n\\begin{table*}\n  \\caption{Proper motions, relative to the main cluster in the FoVs, of field stellar populations represented with red, blue, and aqua colors in Figures\\,\\ref{fig:PMsSMC} and \\ref{fig:PMsLMC}. For each population we provide the ID of the reference cluster, the number of stars, $N$, the median proper motions ($\\delta \\mu_{\\rm \\alpha} \\cos{\\delta}$ and $\\delta \\mu_{\\rm \\delta}$, the ellipticity, $\\epsilon$, and the position angle, $\\theta$, of the best-fitting ellipse.}\n\\begin{tabular}{l l l c c c c}\n\\hline \\hline\nID & Population & $N$  & $\\delta \\mu_{\\rm \\alpha} \\cos{\\delta} $ &   $\\delta \\mu_{\\rm \\delta}$  & $\\epsilon$ & $\\theta$ \\\\\n   &  &   & [mas yr$^{-1}$] &   [mas yr$^{-1}$]  &  & [deg] \\\\\n\\hline\nKRON\\,34 & Red   &539 &    0.001$\\pm$0.004 &    0.003$\\pm$0.004  & 0.23$\\pm$0.05 & 43$\\pm$9 \\\\ \n         & Blue  &193 &    0.031$\\pm$0.006 & $-$0.013$\\pm$0.004  & 0.34$\\pm$0.06 & 28$\\pm$5 \\\\ \n\\hline\nNGC\\,294 & Red   &657 &    0.041$\\pm$0.004 & $-$0.026$\\pm$0.003  & 0.19$\\pm$0.06 & 37$\\pm$6 \\\\ \n         & Blue  &256 &    0.045$\\pm$0.004 & $-$0.024$\\pm$0.004  & 0.42$\\pm$0.07 & 28$\\pm$6 \\\\ \n\\hline\nNGC\\,339 & Red   & 54 &    0.014$\\pm$0.014 &    0.013$\\pm$0.012  & 0.10$\\pm$0.08 & 33$\\pm$11 \\\\ \n         & Blue  &192 &    0.110$\\pm$0.011 & $-$0.059$\\pm$0.009  & 0.62$\\pm$0.03 & 38$\\pm$3 \\\\ \n\\hline\nNGC\\,416 & Red   &330 &    0.017$\\pm$0.005 &    0.047$\\pm$0.004  & 0.08$\\pm$0.07 & 19$\\pm$13 \\\\ \n         & Blue  &688 &    0.109$\\pm$0.006 &    0.017$\\pm$0.004  & 0.30$\\pm$0.05 & 26$\\pm$6 \\\\ \n\\hline\nNGC\\,419 & Red   &211 & $-$0.070$\\pm$0.008 &    0.050$\\pm$0.008  & 0.31$\\pm$0.07 & 33$\\pm$6 \\\\ \n         & Blue  &267 & $-$0.077$\\pm$0.009 &    0.054$\\pm$0.006  & 0.42$\\pm$0.07 & 29$\\pm$5 \\\\ \n\\hline\nNGC\\,1755& Red   &296 &    0.077$\\pm$0.006 &    0.169$\\pm$0.007  & 0.02$\\pm$0.08 & 86$\\pm$16 \\\\ \n         & Blue  &183 & $-$0.019$\\pm$0.005 &    0.139$\\pm$0.005  & 0.17$\\pm$0.07 &314$\\pm$14 \\\\ \n\\hline\nNGC\\,1801& Red   &321 &    0.058$\\pm$0.007 & $-$0.036$\\pm$0.007  & 0.05$\\pm$0.06 & 12$\\pm$12 \\\\ \n         & Blue  &209 &    0.001$\\pm$0.005 &    0.009$\\pm$0.005  & 0.20$\\pm$0.13 & 48$\\pm$16 \\\\ \n         & Aqua  & 44 &    0.023$\\pm$0.012 &    0.003$\\pm$0.017  & 0.09$\\pm$0.12 & 46$\\pm$18 \\\\ \n\\hline\nNGC\\,1953& Red   &441 &    0.095$\\pm$0.007 &    0.039$\\pm$0.006  & 0.00$\\pm$0.05 &360$\\pm$13 \\\\ \n         & Blue  &186 & $-$0.029$\\pm$0.006 & $-$0.007$\\pm$0.006  & 0.28$\\pm$0.06 &317$\\pm$16 \\\\ \n         & Aqua  &53  & $-$0.092$\\pm$0.009 & $-$0.022$\\pm$0.008  & 0.17$\\pm$0.11 &323$\\pm$17 \\\\\n\\hline\n\\end{tabular} \n \\label{tab:ellissi}\n \\end{table*}\n\n\\begin{table*}\n  \\caption{Proper motions relative to NGC\\,346 and proper-motion dispersions  for the clusters NGC\\,346 and BS\\,90 and for the selected populations of pre-MS, MS, and RGB field stars.}\n\\begin{tabular}{l c c c c c c}\n\\hline \\hline\nID & $\\delta \\mu_{\\rm \\alpha} \\cos{\\delta}$ & $\\delta \\mu_{\\rm \\delta} $ &   $\\sigma \\mu_{\\rm \\alpha} \\cos{\\delta}$  & $\\sigma \\mu_{\\rm \\delta} $ & $N$ \\\\\n     & [mas yr$^{-1}$] & [mas yr$^{-1}$]    & [mas yr$^{-1}$] & [mas yr$^{-1}$] & \\\\\n\n\\hline\nNGC\\,346  &   0.000$\\pm$0.001  &    0.000$\\pm$0.001 & 0.028  & 0.025  &  945\\\\ \nBS\\,90    &$-$0.016$\\pm$0.001  &    0.153$\\pm$0.001 & 0.036  & 0.036  &  2220\\\\ \npre-MS\\,I &$-$0.032$\\pm$0.005  &    0.019$\\pm$0.004 & 0.083  & 0.075  &  345\\\\ \npre-MS\\,II &   0.004$\\pm$0.004  &    0.007$\\pm$0.004 & 0.045  & 0.044  & 162 \\\\\nMS\\,I    &$-$0.024$\\pm$0.002  &    0.005$\\pm$0.002 & 0.059  & 0.062  &  2136\\\\ \nMS\\,II   &   0.188$\\pm$0.003  & $-$0.034$\\pm$0.004 & 0.079  & 0.088  &  582  \\\\ \nRGB      &$-$0.031$\\pm$0.003  &    0.024$\\pm$0.008 & 0.084  & 0.089  &  713  \\\\ \n\\hline\n         \\end{tabular} \n \\label{tab:n346}\n \\end{table*}\n\n\\bibliographystyle{mnras}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Outline}\n\\label{s1}\nIn the last decades, the interest in fractional Brownian motion, first introduced in \\cite{kolmogorov} and referred to as {\\sl fBm} in the sequel, has increased enormously \nas one important ingredient of fractal models in the sciences. The paper \\cite{mandel} has been one of the\nkeystones that has attracted the attention of part of  the probabilistic community to this challenging object.\nSome of the research on {\\sl fBm} has significantly influenced the present state of the art of Gaussian processes\n(see for instance  \\cite{bn}, \\cite{np}, \\cite{pipitaqu}, just to mention \na few). An important aspect of the study of {\\sl fBm} lies in the domain of stochastic analysis. Since this process is neither a semimartingale \nnor a Markov process, It\\^o's theory does not apply. For values of the Hurst parameter $H$ greater than $\\frac{1}{2}$ -the regular case- integrals of Young's\ntype and fractional calculus techniques have been considered (\\cite{zaehle1}, \\cite{zaehle2}).\nHowever, for $H$ less than $\\frac{1}{2}$ this approach fails. The integral representation of {\\sl fBm}\nas a Volterra integral with respect to the standard Brownian motion has been successfully exploited in setting up a stochastic calculus where classical tools of \nGaussian processes along with fractional and Malliavin calculus are combined.\nA pioneering work in this context is \\cite{decreustu}, then \\cite{alomanu2}, \\cite{cacomon},\n\\cite{d} and also \\cite{feypradel}.\nSince then, there have been many contributions to the subject. Let us refer to \\cite{n} for enlightening contents\nand a pretty complete list of references. Rough path analysis (see  \\cite{lq}) provides a new approach somehow related to Young's\napproach. \n\nOne of the main reasons for developing a stochastic calculus based on {\\sl fBm} is mathematical modeling. The theory of ordinary and partial differential equations\ndriven by a fractional noise is nowadays a very active field of research. Some of the motivations come from a number of applications in engineering, biophysics and \nmathematical finance; to refer only to a few, let us mention \\cite{dms}, \\cite{ks}, \\cite {roger}. There are also purely mathematical motivations. Problems studied so far \nrange from the existence, the uniqueness, the regularity and the long-time behaviour of solutions to large deviations, support theorems and the analysis of the law of the solutions using Malliavin calculus. \nWithout aiming to be exhaustive, let us refer to \\cite{b-h}, \\cite{f},  \\cite{guletin}, \\cite{huokszang}, \n \\cite{klinzaehle}, \\cite{lq},\n\\cite{maslonua}, \\cite{m-ss}, \\cite{nuarasca}, \\cite{nuavu}, \\cite{q-t} and \\cite{tintuvien}) \nfor a reduced sample of published work. \n\nThis paper aims to pursue the investigations of \\cite{nuavu}, where \nthe authors develop an existence and uniqueness theory of variational\nsolutions for a class of non-autonomous semilinear partial differential\nequations driven by an infinite-dimensional multiplicative fractional noise\nthrough the construction and the convergence of a suitable Faedo-Galerkin scheme.\n\nAs is the case for deterministic partial differential equations, a recurrent\ndifficulty  is the necessity to decide \\textit{ab\ninitio} what solution concept is relevant, since there are several \\textit{a\npriori} non-equivalent possibilities to choose from. Thus, while in\n \\cite{nuavu}  two notions of \\textit{variational solution} that\nare subsequently proved to be indistinguishable are introduced, the focus in \\cite{guletin}\nor \\cite{maslonua} is rather on the idea of\\textit{\\ mild solution}, \nnamely, a solution which can be expressed as a nonlinear integral\nequation that involves the linear propagator of the theory without any\nreference to specific classes of test functions. Consequently, this leaves\nentirely open the question of knowing whether the variational and mild notions\nare in some sense equivalent, and indeed we are not aware of any connections\nbetween them thus far in this context. For equations of the type considered in this\narticle but driven by standard Wiener processes, this issue was addressed\nin \\cite{sansovu1}. In \\cite{denistoica} a similar question was analyzed for a class of very  general \nSPDEs driven by a finite-dimensional Brownian motion.\n\nIn this article we consider the same class of equations as in \\cite{nuavu}. \nWe develop an existence and uniqueness  theory of mild solutions and  prove the indistinguishability of variational and mild solutions.\nWe also prove the H\\\"older continuity of their sample paths.\n\nBefore defining the class of problems we shall investigate, let us fix the notation.\nAll the functional spaces we introduce are real and we use the standard\nnotations for the usual Banach spaces of differentiable functions, of\nH\\\"{o}lder continuous functions, of Lebesgue integrable functions and for the\nrelated scales of Sobolev spaces defined on regions of Euclidean space used for instance in  \\cite{adams}. \nFor $d\\in\\mathbb{N}^+$ let $D\\subset\\mathbb{R}^{d}$ be an open and  bounded set whose boundary $\\partial D$ is\nof class $\\mathcal{C}^{2+\\beta}$ for some $\\beta\\in(0,1)$ (see, for instance,\n\\cite{eidelzhita} and \\cite{ladyuralsolo} for a definition of\nthis and related concepts). We will denote by $(.,.)_2$ the standard inner product in \n$L^2({\\mathbb{R}}^d)$, by $(.,.)_{\\mathbb{R}^d}$ the Euclidean inner product in $\\mathbb{R}^d$\nand by $|.|$ the associated Euclidean norm.\n\n Let $(\\lambda_{i})_{i\\in\\mathbb{N}^{+}}$ be any sequence of positive real numbers such that\n$\\sum_{i=1}^{+\\infty}\\lambda_{i}<+\\infty$ and $(e_{i})_{i\\in\\mathbb{N}^{+}}$  an orthonormal basis of $L^{2}(D)$ such that \n$\\sup_{i\\in\\mathbb{N}^{+}}\\left\\|  e_{i}\\right\\|  _{\\infty}<+\\infty$ (the existence of such a basis follows from \n\\cite{ovespelc}). We then define the linear, self-adjoint, positive,\nnon-degenerate trace-class operator $C$ in $L^{2}(D)$ by $Ce_{i}=\\lambda\n_{i}e_{i}$ for each $i$. In the sequel we write $\\left(  \\left(  B_{i}\n^{H}(t)\\right)  _{t\\in\\mathbb{R}^{+}}\\right)  _{i\\in\n\\mathbb{N}^{+}}$ for a sequence of one-dimensional, independent, identically\ndistributed fractional Brownian motions with Hurst parameter \n$H\\in(0,1)$, defined on the complete probability space\n$\\left(  \\Omega,\\mathcal{F},\\mathbb{P}\\right)  $ and starting at the origin.\nWe introduce the $L^{2}(D)$-valued fractional Wiener process $\\left(\nW^{H}(.,t)\\right)  _{t\\in\\mathbb{R}^{+}}$ by setting\n\\begin{equation}\nW^{H}(.,t):=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\ne_{i}(.)B_{i}^{H}(t), \\label{1}\n\\end{equation}\nwhere the series converges a.s. in the strong topology of $L^{2}(D)$, by virtue\nof the basic properties of the $B_{i}^{H}(t)$'s and the fact\nthat $C$ is trace-class.\n\nLet $T>0$, $\\alpha\\in(1-H,\\frac{1}{2})$ and $(F(t), t\\in [0,T] )$ be a stochastic process taking values in the space of \nlinear bounded operators on $L^2(D)$. Assume that\n\\begin{equation*}\n\\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}<+\\infty,\n\\end{equation*}\nwhere for a function $f:[0,T]\\to L^2(D)$,\n\\begin{equation*}\n\\Vert f\\Vert_{\\alpha,1}= \\int_0^T\\left(\\frac{\\Vert f(s)\\Vert_2}{s^\\alpha}+\\int_0^s \\frac{\\Vert f(s)-f(r)\\Vert_2}{(s-r)^{\\alpha+1}}dr \\right) ds.\n\\end{equation*}\nFollowing \\cite{maslonua} we can define a pathwise generalized Stieltjes integral (see also \\cite{zaehle1})\n\\begin{equation*}\n\\int_0^T F(s) W^H(ds):=\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}} \\int_0^T F(s)e_i B_i^H(ds),\n\\end{equation*}\nwhich satisfies the property\n\\begin{align*}\n\\left\\Vert\\int_0^T F(s) W^H(ds)\\right\\Vert_2&\\le \\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}\n\\left(\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\Lambda_\\alpha(B_i^H)\\right).\n\\end{align*}\nHere $\\Lambda_\\alpha(B_i^H)$ is a positive random variable defined in terms of a Weyl derivative (see \\cite{maslonua}, Equation (2.4)),\nsatisfying $\\sup_{i\\in\\mathbb{N}^+}E\\left(\\Lambda_\\alpha(B_i^H)\\right)<+\\infty$, as is proved in \\cite{nuarasca}, Lemma 7.5.\nConsequently, if $\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}<+\\infty$, the random variable \n\\begin{equation}\nr_{\\alpha}^{H}:=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\Lambda_{\\alpha}(B_{i}^{H}). \\label{29}\n\\end{equation}\nis finite, a.s., and then\n\\begin{equation}\n\\label{i}\n\\left\\Vert\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}} \\int_0^T F(s)e_i B_i^H(ds)\\right\\Vert_2\\le \nr_{\\alpha}^H \\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}.\n\\end{equation}\n\\smallskip\n\nNext, we introduce the class of real, parabolic,\ninitial-boundary value problems formally given by\n\\begin{align}\ndu(x,t)  &  = \\left(\\operatorname*{div}(k(x,t)\\nabla\nu(x,t))+g(u(x,t))\\right)  dt+h(u(x,t))W^{H}(x,dt),\\nonumber\\\\\n(x,t)  &  \\in D\\times \\left(  0,T\\right]  ,\\nonumber\\\\\nu(x,0)  &  =\\varphi(x),\\text{ \\ \\ }x\\in\\overline{D},\\nonumber\\\\\n\\frac{\\partial u(x,t)}{\\partial n(k)}  &  =0,\\text{ \\ \\ }(x,t)\\in\\partial\nD\\times\\left(  0,T\\right]  , \\label{2}\n\\end{align}\nwhere the last relation stands for the conormal derivative of $u$ relative to the matrix-valued field $k$.\n\nIn the next section we shall give a rigorous meaning to such a  formal expression and for this, we shall use the \npathwise integral described before.\n\nIn the sequel we write $n(x)$ for the unit outer normal vector at $x\\in$ $\\partial D$ and introduce \nthe following set of assumptions:\n\\begin{description}\n\\item{(C)} The square root $C^{\\frac{1}{2}}$ of the covariance operator is\ntrace-class, that is, we have $\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}<+\\infty$.\n\\item{(${\\rm K}_{\\beta,\\beta^{\\prime}}$)} The entries of $k$ satisfy $k_{i,j}%\n(.)=k_{j,i}(.)$ for all $i,j\\in\\left\\{  1,...,d\\right\\}  $ and there exists a\nconstant $\\beta^{\\prime}\\in\\left(  \\frac{1}{2},1\\right]  $ such that\n$k_{i,j}\\in\\mathcal{C}^{\\beta,\\beta^{\\prime}}(\\overline{D}\\times\\left[0,T\\right])$\nfor each $i,j$. In addition, we have\n$k_{i,j,x_{l}}:=\\frac{\\partial k_{i,j}}{\\partial x_{l}}\\in\\mathcal{C}\n^{\\beta,\\frac{\\beta}{2}}(\\overline{D}\\times\\left[  0,T\\right])$\nfor each $i,j,l$ and there exists a constant $\\underline{k}\\in\\mathbb{R}%\n_{\\ast}^{+}$ such that the inequality\n$\n\\left(  k(x,t)q,q\\right)  _{\\mathbb{R}^{d}}\\geq\\underline{k}\\left|  q\\right|\n^{2}\n$\nholds for all $q\\in\\mathbb{R}^{d}$ and all $(x,t)\\in\\overline{D}\\times\\left[\n0,T\\right]$.\\\\\nFinally, we\nhave\n$$\n(x,t)\\mapsto\\sum_{i=1}^{d}k_{i,j}(x,t)n_{i}(x)\\in\\mathcal{C}^{1+\\beta\n,\\frac{1+\\beta}{2}}(\\partial D\\times\\left[  0,T\\right])\n$$\nfor each $j$ and the conormal vector-field $(x,t)\\mapsto\nn(k)(x,t):=k(x,t)n(x)$ is outward pointing, nowhere tangent to $\\partial D$\nfor every $t$.\n\\item{(L)} The functions $g,h:\\mathbb{R\\mapsto R}$ are Lipschitz continuous.\n\\item{(I)} The initial condition satisfies $\\varphi\\in\\mathcal{C}^{2+\\beta}%\n(\\overline{D})$ and the conormal boundary condition relative to $k$.\n\\end{description}\n\n\n\nFinally, we consider the following assumption which also appears in \\cite{nuavu}: \n\\smallskip\n\n\\noindent (${\\rm H}_\\gamma$)   The derivative $h^{\\prime}$ is H\\\"{o}lder continuous with exponent $\\gamma\\in\\left(0,1\\right]$\nand bounded; moreover, the Hurst parameter satisfies $H\\in\\left(\\frac{1}{\\gamma+1},1\\right)$.\n\nNotice that if the derivative $h^{\\prime}$ is Lipschitz continuous this amounts to assuming $H\\in\\left(\\frac{1}{2},1\\right)$.\n\n\nProblem (\\ref{2}) is identical to the\ninitial-boundary value problem investigated in \\cite{nuavu}, up to Hypotheses\n(K$_{\\beta,\\beta^{\\prime}}$) which imply Hypotheses (K)\nof that article. This\nimmediately entails the existence of what is called there a variational\nsolution of type II for (\\ref{2}), henceforth simply coined variational solution. \nWith (K$_{\\beta,\\beta^{\\prime}}$) we have the existence and regularity properties of the Green function\nassociated with the differential operator governing (\\ref{2}). We shall give more details on this in the \nnext section.\n\\medskip\n\nWe organize this article in the following way. In Section \\ref{s2} we first recall\nthe notion of {\\it variational solution} and introduce a\nnotion of {\\it mild solution} for (\\ref{2}) by means of a family of evolution\noperators in $L^{2}(D)$ generated by the corresponding deterministic \nGreen's function. We then proceed by stating our main\nresults concerning the existence, uniqueness, and H\\\"{o}lder regularity of the mild\nsolution along with its indistinguishability from the\nvariational solution when $h$ is an affine function. \nThe section ends with a discussion about the results and methods of their proofs. \nThese are gathered in Section \\ref{s3}.\n\\bigskip\n\n\n\n\\section{Statement and Discussion of the Results}\n\\label{s2}\n\nIn the remaining part of this article we write $H^{1}(D\\times(0,T))$ for the\nisotropic Sobolev space on the cylinder $D\\times(0,T)$, which consists of all\nfunctions $v\\in L^{2}(D\\times(0,T))$ that possess distributional derivatives\n$v_{x_{i}}$, $v_{\\tau}\\in L^{2}(D\\times(0,T))$.\nThe set of all $v\\in H^{1}(D\\times(0,T))$ which do not depend\non the time variable identifies with $H^{1}(D)$, the usual Sobolev\nspace on $D$ whose\nnorm we denote by\n$\\left\\|  .\\right\\|  _{1,2}$. \n\nFor $0<\\alpha<1$\nwe introduce the Banach space $\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$\nof all Lebesgue-measurable mappings $u:\\left[  0,T\\right]  \\mapsto L^{2}(D)$\nendowed with the norm \n\\begin{equation}\n\\left\\|  u\\right\\|  _{\\alpha,2,T}^{2}:= \\left(\\sup_{t\\in\\left[0,T\\right]}\\left\\|u(t)\\right\\|_{2}\\right)^2 + \\int_{0}^{T}dt\\left(  \\int_{0}\n^{t}d\\tau\\frac{\\left\\|  u(t)-u(\\tau)\\right\\|  _{2}^{{}}}{(t-\\tau)^{\\alpha+1}\n}\\right)  ^{2}<+\\infty. \\label{3}\n\\end{equation}\nNotice that $\\Vert \\cdot\\Vert_{\\alpha,1}\\le c\\left\\| \\cdot \\right\\|  _{\\alpha,2,T}$,  \nand also that the spaces $\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$ decrease when $\\alpha$ increases. \n\nWe recall the following notion introduced in \\cite{nuavu}, in which the function $x\\mapsto v(x,t)\\in L^{2}(D)$\nis interpreted as the Sobolev trace of $v\\in H^{1}(D\\times(0,T))$ on the\ncorresponding hyperplane. \n\n\\begin{definition}\n\\label{d1}\nFix $H\\in\\left(\\frac{1}{2},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{1}{2}\\right)$. We\nassume that conditions (C), (L) are satisfied and that the initial condition $\\varphi$ belongs to $L^2(D)$. In addition we suppose that  the symmetric matrix valued function $k$ satisfies\n\\begin{equation*}\n\\underline k |q|^2 \\le (k(x,t)q,q)_{\\mathbb{R}^d}\\le \\overline k |q|^2,\n\\end{equation*}\nfor any $q\\in\\mathbb{R}^d$ and some positive constants $\\underline k$, $\\overline k$\nindependent of $x$ and $t$.\n\nUnder these conditions, the $L^{2}(D)$-valued random field $\\left( \nu_{V}(.,t)\\right)  _{t\\in\\left[  0,T\\right]  }$ defined and measurable on\n$\\left(  \\Omega,\\mathcal{F},\\mathbb{P}\\right)  $ is a \\textit{variational\nsolution }to Problem (\\ref{2}) if\n\n(1)  $u_{V}\\in L^{2}(0,T;H^{1}(D))\\cap\\mathcal{B}^{\\alpha,2}%\n(0,T;L^{2}(D))$ a.s., which means that \n\\[\n\\int_{0}^{T}dt\\left\\|  u_{V}(.,t)\\right\\|  _{1,2}^{2}=\\int_{0}^{T}dt\\left(\n\\left\\|  u_{V}(.,t)\\right\\|  _{2}^{2}+\\left\\|  \\nabla u_{V}(.,t)\\right\\|\n_{2}^{2}\\right)  <+\\infty\n\\]\nand $\\left\\|  u_{V}\\right\\|  _{\\alpha,2,T}<+\\infty$\nhold a.s.\n\n(2) The integral relation\n\\begin{align}\n\\int_{D}dx\\ v(x,t)u_{V}(x,t)  &  =\\int_{D}dx\\ v(x,0)\\varphi(x)+\\int_{0}^{t}\nd\\tau\\int_{D}dx\\ v_{\\tau}(x,\\tau)u_{V}(x,\\tau)\\nonumber\\\\\n&  -\\int_{0}^{t}d\\tau\\int_{D}dx\\left(  \\nabla v(x,\\tau),k(x,\\tau)\\nabla\nu_{V}(x,\\tau)\\right)  _{\\mathbb{R}^{d}}\\nonumber\\\\\n&  +\\int_{0}^{t}d\\tau\\int_{D}dxv(x,\\tau)g(u_{V}(x,\\tau))\\nonumber\\\\\n&+\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left(v(.,\\tau),h(u_{V}(.,\\tau))e_{i}\\right)  _{2}B_{i}^{H}(d\\tau).\n \\label{4}\n\\end{align}\nholds a.s. for every $v\\in$ $H^{1}(D\\times(0,T))$ and every $t\\in\\left[\n0,T\\right] $.\n\\end{definition}\n\n\nWith the standing hypotheses we easily infer that each term in\n(\\ref{4}) is finite a.s. In particular, Hypothesis (C) and the fact that $h$ is Lipschitz continuous,\nalong with (\\ref{i})  imply the\nabsolute convergence, a.s., of the series of the last term in (\\ref{4}).\n\n\n\n\\bigskip\n\nLet\n$G:\\overline{D}\\times\\left[  0,T\\right]  \\times\\overline{D}\\times\\left[  0,T\\right]\n\\diagdown\\left\\{  s,t\\in\\left[  0,T\\right]  :s\\geq t\\right\\} \\to\\mathbb{R}$\nbe the parabolic Green's function associated with\nthe principal part of (\\ref{2}). Assume that (K$_{\\beta,\\beta^{\\prime}}$) holds; it is well-known that \n$G$ is a continuous function, twice continuously differentiable in $x$, once continuously\ndifferentiable in $t$. For every $(y,s)\\in D\\times (0,T]$, it is also a classical solution to the linear initial-boundary\nvalue problem\n\\begin{align}\n\\partial_{t}G(x,t;y,s)  &  =\\operatorname*{div}(k(x,t)\\nabla_{x}\nG(x,t;y,s)),\\text{ \\ \\ }(x,t)\\in D\\times\\left(0,T\\right],\\nonumber\\\\\n\\frac{\\partial G(x,t;y,s)}{\\partial n(k)}  &  =0,\\text{ \\ \\ \\ }(x,t)\\in\n\\partial D\\times\\left(  0,T\\right], \\label{6}\n\\end{align}\nwith\n\\[\n\\int_{D}dyG(.,s;y,s)\\varphi(y):=\\lim_{t\\searrow s}\\int_{D}dyG(.,t;y,s)\\varphi\n(y)=\\varphi(.),\n\\]\nand satisfies the heat kernel estimates\n\\begin{equation}\n\\left|  \\partial_{x}^{\\mu}\\partial_{t}^{\\nu}G(x,t;y,s)\\right|  \\leq\nc(t-s)^{-\\frac{d+\\left|  \\mu\\right|  +2\\nu}{2}}\\exp\\left[  -c\\frac{\\left|\nx-y\\right|  _{{}}^{2}}{t-s}\\right]  \\label{7}\n\\end{equation}\nfor $\\mu=(\\mu_{1},...,\\mu_{d})\\in\\mathbb{N}^{d}$, $\\nu\\in\\mathbb{N}$ and\n$\\left|\\mu\\right| +2\\nu\\leq 2$, with $\\left|  \\mu\\right|  =\\sum_{j=1}^{d}\n\\mu_{j}$ (see, for instance, \n\\cite{eidelzhita} or\n\\cite{ladyuralsolo}). In particular, for $\\left|  \\mu\\right|  =\\nu=0$ we have\n\\begin{equation}\n\\left|  G(x,t;y,s)\\right|  \\leq c(t-s)^{-\\frac{d}{2}}\\exp\\left[\n-c\\frac{\\left|  x-y\\right|  _{{}}^{2}}{t-s}\\right]. \\label{8}\n\\end{equation}\nWe shall refer to (\\ref{8}) as the {\\it Gaussian\nproperty} of $G$. \n\nWe can now define the notion of\nmild solution for (\\ref{2}).\n\n\\begin{definition}\n\\label{d2}\nFix $H\\in\\left(\\frac{1}{2},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{1}{2}\\right)$. Assume that the hypotheses (C), (K$_{\\beta,\\beta^{\\prime}}$), (L)\nhold and that the initial condition $\\varphi$ is bounded.\n\nUnder these assumptions, the $L^{2}(D)$-valued random field $\\left(\nu_{M}(.,t)\\right)  _{t\\in\\left[  0,T\\right]  }$ defined and measurable on\n$\\left(  \\Omega,\\mathcal{F},\\mathbb{P}\\right)  $ is a \\textit{mild}\n\\textit{solution }to Problem (\\ref{2}) if the following two conditions are satisfied:\n\n(1) $u_{M}\\in L^{2}(0,T;H^{1}(D))\\cap\\mathcal{B}^{\\alpha,2}\n(0,T;L^{2}(D))$ a.s.\n\n(2) The relation\n\\begin{align}\nu_{M}(.,t)  &  =\\int_{D}dy\\ G(.,t;y,0)\\varphi(y)+\\int_{0}^{t}d\\tau\\int\n_{D}dy\\ G(.,t;y,\\tau)g\\left(  u_{M}(y,\\tau)\\right) \\nonumber\\\\\n&+\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left(  \\int\n_{D}dy\\ G(.,t;y,\\tau)h\\left(u_{M}(y,\\tau)\\right)  e_{i}(y)\\right)\nB_{i}^{H}(d\\tau) \\label{9}\n\\end{align}\nholds a.s. for every $t\\in\\left[  0,T\\right]  $ as an equality in $L^{2}(D)$.\n\\end{definition}\n\nWe shall prove in Lemma \\ref{l2} that with the standing assumptions, each term in (\\ref{9}) indeed\ndefines a $L^{2}(D)$-valued stochastic process.\n\\medskip\n\nThe main results of this article are gathered in the next theorem.\n\\bigskip\n\n\\begin{theorem}\n\\label{t1}\nAssume that Hypotheses ($C$), ($K_{\\beta,\\beta^{\\prime}}$), ($L$), ($I$) and ($H_{\\gamma}$) hold; then\nthe following statements are valid:\n\n\\begin{description}\n\\item{(a)} Fix $H\\in\\left(\\frac{1}{\\gamma+1},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{\\gamma}{\\gamma+1}\\right)$. Then\nProblem (\\ref{2}) possesses a variational solution\n$u_V$; moreover, every such variational solution is a mild solution $u_M$ to (\\ref{2}).\nMore presicely, for every $t\\in\\left[  0,T\\right]$, $u_V(.,t)=u_M(.,t)$  a.s. in $L^{2}(D)$.\n\n\n\\item{(b)} Fix $H\\in\\left(\\frac{1}{\\gamma+1}\\vee\\frac{d+1}{d+2},1\\right)$ and then $\\alpha\\in\\left(1-H,\\frac{\\gamma}{\\gamma+1}\\wedge\\frac{1}{d+2}\\right)$. Assume in addition that $h$ is an affine function. Then $u_V$  is the unique variational solution to\n (\\ref{2}), while $u_M$ is its unique mild solution. \n \n\\item{(c)} Let $H$ and $\\alpha$ be as in part (b).\nThen every mild solution $u_{M}$ to Problem (\\ref{2}) \nis H\\\"{o}lder continuous with respect to the time variable. More precisely,\nthere exists a positive, a.s. finite  random variable $R_{\\alpha}^{H}$\nsuch that the estimate\n\\begin{equation}\n\\left\\|  u_{M}(.,t)-u_{M}(.,s)\\right\\|  _{2}\\leq R_{\\alpha}^{H}\n\\left|  t-s\\right|  ^{\\theta}\\left(  1+\\left\\|u_{M}\\right\\|  _{\\alpha,2,T}\\right)\n\\label{h1}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[  0,T\\right] $ and every\n$\\theta\\in\\left( 0,\\left(\\frac{1}{2}-\\alpha\\right)\\wedge\\frac{\\beta}{2}\\right)$.\n \\end{description}\n\\end{theorem}\n\n\n\n\n{\\bf Remarks}\n\\begin{enumerate}\n\\item The existence of a mild solution will be proved by reference to\nthe existence of a variational solution. \nThis is in contrast with the method of \\cite{maslonua}, in which the\nauthors prove the existence of mild solutions for a class of\n\\textit{autonomous}, parabolic, fractional stochastic initial-boundary value\nproblems by means of Schauder's fixed point theorem. Their method thus\nrequires the construction of a continuous map operating in a compact and\nconvex set of a suitable functional space.  \nIf $h$ is an affine function, the arguments of the proof of Statement (b)\n(see (\\ref{322})) show that a similar approach might be possible for\nour equation.\nTo the best of our knowledge, there exists as yet no such direct way to prove the\nexistence of mild solutions to (\\ref{2}) for a non affine $h$. \n\n\\item If $h$ is not affine, the question of uniqueness remains unsettled. \nIn fact, uniqueness could be proved if we were able to extend the  inequality (\\ref{322})\nto any Lipschitz function $h$. This does not seem to be a trivial point, due to the form of the second term in the right-hand side of (\\ref{3}).\n\n\\item If $h$ is an affine function, Theorem \\ref{t1} establishes the complete\nindistinguishability of mild and variational solutions, although we do not\nknow whether this property still holds for a general $h$.\n\n\\item If $h$ is a constant function the setting of the problems and their proofs become much \nsimpler. Indeed, in Definitions \\ref{d1} and \\ref{d2} the space $\\mathcal{B}^{\\alpha,2}(0,T; L^2(D))$\ncan be replaced by the larger one $L^\\infty(0,T; L^2(D))$, consisting of Lebesgue-measurable\nmappings $u:[0,T] \\to L^2(D)$ such that $\\sup_{t\\in[0,T]}\\Vert u(t)\\Vert_2 <\\infty$. This can be checked\nby going through the proofs of  \\cite{nuavu} and Lemma \\ref{l2} of Section \\ref{s3}.\nMoreover, the range of values of $\\theta$ in statement (c) of Theorem \\ref{t1} can be extended to the interval $\\left(0,\\frac{\\beta}{2}\\right)$.\nThis can be easily checked by going through the proof of Proposition \\ref{p4}, by checking first that the right-hand side\nof the inequalities (\\ref{40}), (\\ref{41}) can be replaced by $c(t-s)^{\\delta} (s-\\tau)^{-\\delta}$ and \n$c(t-s)^{\\frac{\\delta}{2}} (s-\\tau)^{-\\frac{1}{2}}(\\tau-\\sigma)^{\\frac{1}{2}(1-\\delta)}$, respectively.\n\n\\item \nBy using the \\textit{factorization method}, and under a different set of assumptions on the range of admissible values of $H$ and $\\alpha$,\nwe can obtain a different range of values for the H\\\"older exponent which in general do not provide as good an estimate as (\\ref{h1}) does.\nWe deal with this question in Proposition \\ref{pfact}. \nThe \\textit{factorization method}\nhas been introduced in \\cite{DP-K-Z} and since then\nextensively used for the analysis of the sample paths of solutions to\nparabolic stochastic partial differential equations (see, for instance,\n\\cite{sansovu1}). \n\\end{enumerate}\n\n\\section{Proofs of the Results}\n\\label{s3}\n\nIn what follows we write $c$ for all the irrelevant deterministic constants that occur in\nthe various estimates. \nWe begin by\nrecalling that the uniformly elliptic partial differential operator with\nconormal boundary conditions in the principal part of (\\ref{2}) admits a\nself-adjoint, positive realization $A(t):=-\\operatorname*{div}(k(.,t)\\nabla)$\nin $L^{2}(D)$ on the domain\n\\begin{equation}\n\\mathcal{D}(A(t))=\\left\\{  v\\in H^{2}(D):\\left(  \\nabla\nv(x),k(x,t)n(x)\\right)  _{\\mathbb{R}^{d}}=0,\\text{ \\ }(x,t)\\in\\partial\nD\\times\\left[  0,T\\right]  \\right\\}  \\label{11}\n\\end{equation}\n(see, for instance, \\cite{lions}). An important consequence\nof this property is that the parabolic Green's function $G$ is also, for every\n$(x,t)\\in D\\times\\left(  0,T\\right]  $ with $t>s$, a classical solution to the\nlinear boundary value problem\n\\begin{align}\n\\partial_{s}G(x,t;y,s)  &  =-\\operatorname*{div}(k(y,s)\\nabla_{y}\nG(x,t;y,s)),\\text{ \\ \\ }(y,s)\\in D\\times\\left(  0,T\\right]  ,\\nonumber\\\\\n\\frac{\\partial G(x,t;y,s)}{\\partial n(k)}  &  =0,\\text{ \\ \\ \\ }(y,s)\\in\n\\partial D\\times\\left(  0,T\\right]  , \\label{12}\n\\end{align}\ndual to (\\ref{6}) (see, for instance, \\cite{eidelzhita} or \\cite{friedman});\nthis means that along with (\\ref{7}) we also have \n\\begin{equation}\n\\left|  \\partial_{y}^{\\mu}\\partial_{s}^{\\nu}G(x,t;y,s)\\right|  \\leq\nc(t-s)^{-\\frac{d+\\left|  \\mu\\right|  +2\\nu}{2}}\\exp\\left[  -c\\frac{\\left|\nx-y\\right|  _{{}}^{2}}{t-s}\\right]  \\label{13}\n\\end{equation} \nfor $\\left|  \\mu\\right|  +2\\nu\\leq2$. We now use these facts to prove in the next lemma\nestimates for $G$, which we shall invoke repeatedly in\nthe sequel to analyze various singular integrals. For the sake of clarity we\nlist those inequalities by their chronological order of appearance in the\nproofs below.\n\n\\begin{lemma}\n\\label{l1}\nAssume that Hypothesis ($K_{\\beta,\\beta^{\\prime}}$) holds.\n Then, for all \n$x,y\\in D$ and for every $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$ we have the\nfollowing inequalities.\n\\begin{description}\n\\item{(i)} For all $t, \\tau, \\sigma \\in[0,T]$ with $t>\\tau>\\sigma$ and some $t^*\\in(\\sigma,\\tau)$,\n\\begin{align}\n&  \\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| \\nonumber\\\\\n&  \\leq c\\left(  t-\\tau\\right)  ^{-\\delta}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast\n})^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|  x-y\\right|  _{{}}^{2}}{t-t^{\\ast\n}}\\right].  \\label{14}\n\\end{align}\n\\item{(ii)} For all $t,s,\\tau\\in[0,T]$ with $t>s>\\tau$ and some $\\tau^*\\in(s,t)$,\n\\begin{align}\n&  \\left|  G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|  ^{{}}\\nonumber\\\\\n&  \\leq c\\left(  t-s\\right)  ^{\\delta}(s-\\tau)^{-\\delta}(\\tau^{\\ast}\n-\\tau)^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|  x-y\\right|  _{{}}^{2}}\n{\\tau^{\\ast}-\\tau}\\right]  \\label{15}\n\\end{align}\nand\n\\begin{align}\n&  \\left|  G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|  ^{\\delta}\\nonumber\\\\\n&  \\leq c\\left(  t-s\\right)  ^{\\delta}(s-\\tau)^{-\\frac{d+2}{2}\\delta+\\frac\n{d}{2}}(\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|\nx-y\\right|  _{{}}^{2}}{\\tau^{\\ast}-\\tau}\\right].  \\label{16}\n\\end{align}\n\\item{(iii)} For all $t, s, \\tau, \\sigma\\in[0,T]$ with $t>s>\\tau>\\sigma$, \n\\begin{equation}\n\\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|  ^{1-\\delta}\n\\leq c\\left(  \\tau-\\sigma\\right)  ^{1-\\delta}(s-\\tau)^{-\\frac{d+2}\n{2}(1-\\delta)} \\label{17}\n\\end{equation}\nuniformly in $t$.\n\\end{description}\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }By applying successively (\\ref{8}), the\nmean-value theorem for $G$ and (\\ref{13}) with $\\left|\n\\mu\\right|  =0$ and $\\nu=1$ we may write\n\\begin{align*}\n&  \\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| \\\\\n&  \\leq\\left(  \\left|  G(x,t;y,\\tau)\\right|  +\\left|  G(x,t;y,\\sigma)\\right|\n\\right)  ^{1-\\delta}\\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|  ^{\\delta}\\\\\n&  \\leq c\\left(  (t-\\tau)^{-\\frac{d}{2}}+(t-\\sigma)^{-\\frac{d}{2}}\\right)\n^{1-\\delta}(\\tau-\\sigma)^{\\delta}\\left|  G_{t^{\\ast}}(x,t;y,t^{\\ast})\\right|\n^{\\delta}\\\\\n&  \\leq c(t-\\tau)^{-\\frac{d}{2}(1-\\delta)}(t-t^{\\ast})^{-\\frac{d+2}{2}\n\\delta+\\frac{d}{2}}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast})^{-\\frac{d}{2}}\n\\exp\\left[  -c\\frac{\\left|  x-y\\right|  _{{}}^{2}}{t-t^{\\ast}}\\right] \\\\\n&  \\leq c\\left(  t-\\tau\\right)  ^{-\\delta}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast\n})^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|  x-y\\right|  _{{}}^{2}}{t-t^{\\ast\n}}\\right]\n\\end{align*}\nfor some $t^{\\ast}\\in(\\sigma,\\tau)$, since $-\\frac{d+2}{2}\\delta+\\frac{d}{2}<0$\nand $-\\frac{d}{2}(1-\\delta)-\\frac{d+2}{2}\\delta+\\frac{d}{2}=-\\delta$. This\nproves (\\ref{14}). Up to some minor but important changes, the\nremaining inequalities can all be proved in a similar way. \\hfill $\\blacksquare$\n\n\\bigskip\n\nEstimate (\\ref{14}) now allows us to prove that our notion of\nmild solution in Definition \\ref{l2} is indeed well-defined; to this end for arbitrary mappings $\\varphi$ and $u$\ndefined on $D$ and $D\\times [0,T]$, respectively, we introduce\nthe functions $A(\\varphi)$, $B(u)$, $C(u): D\\times\\left[0,T\\right]  \\mapsto\\mathbb{R}$ by\n\\begin{align}\nA(\\varphi)(x,t)   &  :=\\mathit{\\ }\\int_{D}dy\\ G(x,t;y,0)\\varphi\n(y),\\label{19}\\\\\nB(u)(x,t)  &  :=\\mathit{\\ }\\int_{0}^{t}d\\tau\\int_{D}dy\\ G(x,t;y,\\tau\n)g\\left(  u(y,\\tau)\\right)  ,\\label{20}\\\\\nC (u)(x,t)  &  :=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}\n}\\int_{0}^{t}\\left(  \\int_{D}dy\\ G(x,t;y,\\tau)h\\left(  u(y,\\tau)\\right)\ne_{i}(y)\\right)  B_{i}^{H}(d\\tau), \\label{21}\n\\end{align}\nand prove the following result.\n\n\\begin{lemma}\n\\label{l2}\nThe hypotheses are the same as in Definition \\ref{d2}.\n Then, for every\n $u\\in\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$ we have $A(\\varphi)(.,t) , B(u)(.,t)\\in L^{2}(D)$, and also\n$C(u)(.,t)\\in L^{2}(D)$ a.s., for every $t\\in\\left[0,T\\right]$.\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\\textbf{Proof. }The assertion is evident for  $A(\\varphi)(.,t)$, \nsince $\\varphi$ is bounded and (\\ref{8}) holds.\nAs for $B(u)(.,t)$, we infer from the Gaussian property of $G$ that\nthe measure $d\\tau dy\\left|  G(x,t;y,\\tau)\\right|  $ is finite on $\\left[\n0,T\\right]  \\times D$ uniformly in $(x,t)\\in D\\times\\left[  0,T\\right]  $, so\nthat by using successively Schwarz inequality with respect to this measure\nalong with Hypothesis (L) for $g$ we obtain\n\\begin{align*}\n&  \\left|B(u)(x,t)\\right|  \\leq\\int_{0}^{t}d\\tau\\int_{D}dy\\left|\nG(x,t;y,\\tau)g\\left(  u(y,\\tau)\\right)  \\right| \\\\\n&  \\leq c\\left(  \\int_{0}^{t}d\\tau\\int_{D}dy\\left|  G(x,t;y,\\tau)\\right|\n\\left(  1+\\left|  u(y,\\tau)\\right|  ^{2}\\right)  \\right)  ^{\\frac{1}{2}}\n\\end{align*}\nfor every $x\\in D$. We then get the inequalities\n\\begin{align*}\n&  \\left\\|B(u)(.,t)\\right\\|  _{2}^{2}=\\int_{D}dx\\left|  \\int\n_{0}^{t}d\\tau\\int_{D}dy\\ G(x,t;y,\\tau)g\\left(  u(y,\\tau)\\right)  \\right|  ^{2}\\\\\n&  \\leq c\\int_{0}^{t}d\\tau\\int_{D}dy\\left(  1+\\left|  u(y,\\tau)\\right|\n^{2}\\right) \n\\leq c\\left(  1+\\int_{0}^{t}d\\tau\\left\\|  u(.,\\tau)\\right\\|  _{2}\n^{2}\\right) < +\\infty. \n\\end{align*}\n\n It remains to show that $\\left\\|C(u)(.,t)\\right\\|  _{2}^{2} <+\\infty$ a.s. for every $t\\in\\left[  0,T\\right] $. \n\nDefine the functions $f_{i,t}(u):\\left[0,t\\right)  \\mapsto L^{2}(D)$ by\n\\begin{equation}\nf_{i,t}(u)(.,\\tau):=\\int_{D}dy\\ G(.,t;y,\\tau)h\\left(  u(y,\\tau)\\right)\ne_{i}(y). \\label{22}\n\\end{equation}\nWe shall prove that \n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{t}\nf_{i,t}(u)(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|  _{2}\n \\leq c\\ r_{\\alpha}^{H}\\left(  1+\\left\\|  u\\right\\|\n_{\\alpha,2,T}\\right),  \\label{23}\n\\end{equation}\na.s., where $r_\\alpha^H$ is the a.s.  finite and positive random variable\ndefined in (\\ref{29}).\n\nIndeed, by using an argument similar to the one above, since $h$ is Lipschitz\ncontinuous and $\\sup_{i\\in\\mathbb{N}^{+}}\\left\\|  e_{i}\\right\\| _{\\infty\n}<+\\infty$, we first obtain\n\\begin{equation}\n\\sup_{i\\in\\mathbb{N}^{+}}\\left\\|  f_{i,t}(u)(.,\\tau)\\right\\|  _{2}\\leq c\\left(  1+\\left\\|\nu(.,\\tau)\\right\\|  _{2}\\right)  \\label{24}\n\\end{equation}\nfor every $\\tau\\in\\left[0,t\\right)$. Furthermore,\nfor every $x\\in D$ and all $\\sigma,\\tau\\in\\left[  0,t\\right)  $ with\n$\\tau>\\sigma$ we have\n\\begin{align*}\n&\\left|  f_{i,t}(u)(x,\\tau)-f_{i,t}(u)(x,\\sigma)\\right| \n \\leq c\\Big(\\int_{D}dy\\left|  G(x,t;y,\\tau)\\right|  \\left|  u(y,\\tau\n)-u(y,\\sigma)\\right| \\\\\n&\\quad \\quad  +\\int_{D}dy\\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|  (1+\\left|\nu(y,\\sigma)\\right|)\\Big),\n\\end{align*}\nso that we get successively\n\\begin{align*}\n&  \\left|  f_{i,t}(u)(x,\\tau)-f_{i,t}(u)(x,\\sigma)\\right|  ^{2}\\\\\n&  \\leq c\\int_{D}dy\\left|  G(x,t;y,\\tau)\\right|  \\left|  u(y,\\tau\n)-u(y,\\sigma)\\right|  ^{2}\\\\\n&  +c\\int_{D}dy\\left|  G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|  \\left(  1+\\left|\nu(y,\\sigma)\\right|  ^{2}\\right) \\\\\n&  \\leq c\\int_{D}dy\\left|  G(x,t;y,\\tau)\\right|  \\left|  u(y,\\tau\n)-u(y,\\sigma)\\right|  ^{2}\\\\\n&  +c\\left(  t-\\tau\\right)  ^{-\\delta}(\\tau-\\sigma)^{\\delta}\\int\n_{D}dy(t-t^{\\ast})^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|  x-y\\right|  _{{}\n}^{2}}{t-t^{\\ast}}\\right]  \\left(  1+\\left|  u(y,\\sigma)\\right|  ^{2}\\right)\n\\end{align*}\nfor some $t^{\\ast}\\in(\\sigma,\\tau)$ and for every $\\delta\\in\\left(  \\frac\n{d}{d+2},1\\right)$. This is achieved by using Schwarz inequality with respect to the finite\nmeasures $dy\\left|  G(x,t;y,\\tau)\\right|  $ and $dy\\left|  G(x,t;y,\\tau\n)-G(x,t;y,\\sigma)\\right|  $ on $D$, respectively, along with\n(\\ref{14}).  We then integrate the preceding estimate with\nrespect to $x\\in D$ and apply the Gaussian property of $G$ to eventually\nobtain\n\\begin{align}\n&\\sup_{i\\in\\mathbb{N}^{+}}\\left\\|  f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\right\\|_{2}\\nonumber\\\\\n&\\leq c\\left(\\left\\|  u(.,\\tau)-u(.,\\sigma)\\right\\|  _{2}\n+\\left(  t-\\tau\\right)  ^{-\\frac{\\delta}{2}}(\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(  1+\\left\\|  u(.,\\sigma)\\right\\|  _{2}\\right)\\right).\\label{25}\n\\end{align}\nTherefore, by applying (\\ref{i}) we have\n\\begin{align}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{t}\nf_{i,t}(u)(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|  _{2}\\nonumber\\\\\n&\\le r_\\alpha^H\\sup_{i\\in\\mathbb{N}^+} \\int_{0}^{t}d\\tau\\left(  \\frac{\\left\\|\nf_{i,t}(u)(.,\\tau)\\right\\|  _{2}}{\\tau^{\\alpha}}+\\int_{0}^{\\tau}\nd\\sigma\\ \\frac{\\left\\|  f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\right\\|  _{2}\n}{\\left(  \\tau-\\sigma\\right)  ^{\\alpha+1}}\\right) \\nonumber\\\\\n&\\le c r_\\alpha^H\\left(  1+\\int_{0}^{t}d\\tau\\ \\frac{\\left\\|\nu(.,\\tau)\\right\\|  _{2}^{{}}}{\\tau^{\\alpha}}+\\int_{0}^{t}d\\tau\\int_{0}^{\\tau\n}d\\sigma\\ \\frac{\\left\\|  u(.,\\tau)-u(.,\\sigma)\\right\\|  _{2}^{{}}}{\\left(\n\\tau-\\sigma\\right)  ^{\\alpha+1}}\\right. \\nonumber\\\\\n&  +\\left.  \\int_{0}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{0}^{\\tau\n}d\\sigma(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(  1+\\left\\|\nu(.,\\sigma)\\right\\|  _{2}\\right)  \\right)  \\label{26}\n\\end{align}\na.s. \n \nLet us now examine more closely the singular integrals in the\nabove terms. On the one hand, we may write\n\\begin{equation}\n\\int_{0}^{t}d\\tau\\ \\frac{\\left\\|u(.,\\tau)\\right\\|_2}{\\tau^{\\alpha}}\n+\\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\ \\frac{\\left\\|u(.,\\tau)-u(.,\\sigma)\\right\\|_2}{\\left(\\tau-\\sigma\\right)^{\\alpha+1}}\n\\leq c\\,\\left\\|  u\\right\\|_{\\alpha,2,T},  \\label{27}\n\\end{equation}\nby using Schwarz inequality relative to the measure $d\\tau$ on $(0,t)$ in\nthe last two integrals along with (\\ref{3}). On the other hand, \nin (\\ref{26}) the exponent $\\delta$ can be taken arbitrarly close to $1$; consequently \nour range of values of $\\alpha$ allows the condition $2\\alpha<\\delta$ to be satisfied.\nThus we can\nintegrate the singularities of the time increments in the last line of\n(\\ref{26}) and get the bound\n\\begin{equation}\n\\int_{0}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{0}^{\\tau}d\\sigma\n(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(  1+\\left\\|  u(.,\\sigma)\\right\\|  _{2}\\right)\n\\leq c\\left(1+\\sup_{t\\in[0,T]}\\left\\|u(.,t)\\right\\|_2\\right).\n\\label{28}\n\\end{equation}\nTherefore, we can\nsubstitute (\\ref{27}), (\\ref{28}) into (\\ref{26}) to\nobtain (\\ref{23}). \n\n\\hfill $\\blacksquare$\n\n\\bigskip\n\nIn order to relate the notions of variational and mild solution, we recall\nthat the self-adjoint operator $A(t)=-\\operatorname*{div}(k(.,t)\\nabla)$\ndefined on (\\ref{11}) generates the family of evolution operators\n$U(t,s)_{0\\leq s\\leq t\\leq T}$ in $L^{2}(D)$ given by\n\\begin{equation}\nU(t,s)v=\\begin{cases}\nv,&\\text{if}\\ \\ s=t,\\\\\n\\int_{D}dy\\ G(.,t;y,s)v(y), &\\text{if}\\ \\ t>s,\n\\label{30}\n\\end{cases}\n\\end{equation}\nand that each such $U(t,s)$ is itself self-adjoint (see, for instance,\n\\cite{tanabe}), which means that the symmetry property \n\\begin{equation}\nG(x,t;y,s)=G(y,t;x,s) \\label{symmetry}\n\\end{equation}\nholds for every $(x,t;y,s)\\in\\overline{D}\\times\\left[  0,T\\right]\n\\times\\overline{D}\\times\\left[  0,T\\right]  \\diagdown\\left\\{  s,t\\in\\left[0,T\\right]  :s\\geq t\\right\\}$. \n\n\\bigskip\n\n\\noindent{\\bf Proof of Statement (a) of Theorem \\ref{t1}}\n\\medskip\n\nThe existence of a variational solution $u_V$ was proved in Theorem of \\cite{nuavu}.\n\nIn order to prove that every variational solution is mild, we follow the same approach as in Theorem 2 of \\cite{sansovu1}.\nFor the sake of completeness, we sketch the main ideas. \n\nWe shall check that the $L^{2}(D)$-valued stochastic process\n\\begin{align*}\n&  u_{V}(.,t)-\\int_{D}dy\\ G(.,t;y,0)\\varphi(y)-\\int_{0}^{t}d\\tau\\int\n_{D}dy\\ G(.,t;y,\\tau)g\\left(  u_{V}(y,\\tau)\\right) \\\\\n&  -\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\int_{0}^{t}\\left(\\int_D dy\\ G(.,t;y,\\tau)h\\left(u_{V}(y,\\tau)e_i(y)\\right)\\right)\nB_i^{H}(d\\tau)\n\\end{align*}\nis a.s. orthogonal for every $t\\in\\left[  0,T\\right]  $ to the dense subspace\n$\\mathcal{C}_{0}^{2}(D)$ consisting of all twice continuously differentiable\nfunctions with compact support in $D$. To this end, for every $v\\in\\mathcal{C}_{0}^{2}(D)$\nand all $s,t\\in\\left[  0,T\\right]$ with $t\\geq s$\nwe define $v^{t}(.,s):=U(t,s)v$, that is, \n\\begin{equation}\nv^{t}(x,s)=\n\\begin{cases}\nv(x),& \\text{if}\\ \\ s=t,\\\\\n\\int_{D}dyG(y,t;x,s)v(y),& \\text{if}\\ \\ t>s,\n\\end{cases}\n\\label{31}\n\\end{equation}\nfor every $x\\in D$ by taking (\\ref{30}) and (\\ref{symmetry}) into\naccount. It then follows from (\\ref{12}), (\\ref{symmetry}) and Gauss'\ndivergence theorem that $v^{t}\\in H^{1}(D\\times(0,T))$, and that for every $t\\in\\left[  0,T\\right]$, the relation\n\\begin{equation}\n\\int_{0}^{t}d\\tau\\int_{D}dx\\ v_{\\tau}^{t}(x,\\tau)u_{V}(x,\\tau)=\\int_{0}^{t}\nd\\tau\\int_{D}dx\\left(  \\nabla v^{t}(x,\\tau),k(x,\\tau)\\nabla u_{V}\n(x,\\tau)\\right)  _{\\mathbb{R}^{d}} \\label{32}\n\\end{equation}\nholds a.s.\nTherefore, we may take (\\ref{31}) as a test function in\n(\\ref{4}), which, as a consequence of (\\ref{32}),\nleads to the relation\n\\begin{align*}\n(v,u_{V}(.,t))_{2}  &  =(v^{t}(.,0),\\varphi)_{2}+\\int_{0}^{t}d\\tau\n(v^{t}(.,\\tau),g(u_{V}(.,\\tau)))_{2}\\\\\n&  +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left(\nv^{t}(.,\\tau),h(u_{V}(.,\\tau))e_{i}\\right)  _{2}B_{i}^{\\text{\\textsc{H}}%\n}(d\\tau),\n\\end{align*}\nvalid a.s. for every $t\\in\\left[0,T\\right]$. After some rearrangements, the substitution of\n(\\ref{31}) into the right-hand side of the preceding expression then\nleads to the equality\n\\begin{align*}\n(v,u_{V}(.,t))_{2}  &  =\\left(  v,\\int_{D}dyG(.,t;y,0)\\varphi(y)\\right)_{2}\\\\\n&+\\left(  v,\\int_{0}^{t}d\\tau\\int_{D}dyG(.,t;y,\\tau)g\\left(  u_{V}\n(y,\\tau)\\right)  \\right)_{2}\\\\\n&  +\\left(  v,\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\int_{0}^{t}\\left(\\int_D dy\\  G(.,t;y,\\tau)h\\left(  u_{V}(y,\\tau)e_i(y)\\right)\\right)\nB_i^{H}(d\\tau)\\right)_{2},\n\\end{align*}\nwhich holds for every $t\\in\\left[  0,T\\right]$ a.s.  and every\n$v\\in\\mathcal{C}_{0}^{2}(D)$, thereby leading to the desired orthogonality\nproperty. \\hfill $\\blacksquare$\n\\bigskip\n\\newpage\n\n\\noindent{\\bf Proof of Statement $(b)$ of Theorem \\ref{t1}}\n\\bigskip\n\n\\noindent Under the standing assumptions, we already know from \\cite{nuavu}\nthat the variational solution is unique. Moreover, we have just proved that every variational solution is also\na mild solution. Hence, it suffices to prove that uniqueness holds\nwithin the class of mild solutions.\nTo this\nend, let us write $u_{M}$ and $\\tilde{u}_{M}$ for any two such solutions\ncorresponding to the same initial condition $\\varphi$; from (\\ref{9}) and\n(\\ref{19})-(\\ref{21}) we have\n\\begin{align}\n&  \\left\\Vert u_{M}(.,t)-\\tilde{u}_{M}(.,t)\\right\\Vert _{2}\\nonumber\\\\\n&  \\leq\\left\\Vert B(u_{M})(.,t)-B(\\tilde{u}_{M})(.,t)\\right\\Vert\n_{2}+\\left\\Vert C(u_{M})(.,t)-C(\\tilde{u}_{M})(.,t)\\right\\Vert _{2}\n\\label{302}\n\\end{align}\na.s. for every $t\\in\\left[  0,T\\right]  $.\n\nWe proceed by estimating both terms\non the right-hand side of (\\ref{302}). \nSince $g$ is Lipschitz, we have\n\\begin{equation}\n\\left\\| B\\left(  u_M\\right)  (.,t)-B(\\tilde{u}_M)(.,t)\\right\\|  _{2}^{2}\n\\leq c\\int_{0}^{t}d\\tau\\left\\|  u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|  _{2}^{2} \\label{303}\n\\end{equation}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\nIn order to analyze the second term or the right-hand side of (\\ref{302}) we will need the following preliminary result.\n\n\\begin{lemma}\n\\label{l5}\nThe hypotheses are the same as in part (a) of Theorem \\ref{t1} and let\nthe $f_{i,t}(u)$'s be the functions given by (\\ref{22}). Then,\nthe estimate\n\\begin{equation}\n\\sup_{(i,t)\\in\\mathbb{N}^+\\times[0,T]} \\left\\| f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\\right\\|_2\n\\leq c\\left\\|  u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|_{2}\n\\label{304}\n\\end{equation}\nholds a.s. for every $\\tau\\in\\left[0,t\\right)$.\n\nMoreover, if $h$ is an affine function we\nhave\n\\begin{align}\n&  \\sup_{i\\in\\mathbb{N}^+}\\left\\|  \\ f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\n -f_{i,t}(u_M)(.,\\sigma)+f_{i,t}(\\tilde{u}_M)(.,\\sigma)\\right\\|_{2}\\nonumber\\\\\n&\\quad  \\leq c(t-\\tau)^{-\\delta}(\\tau-\\sigma)^{\\delta}\\left\\|\nu_M(.,\\sigma)-\\tilde{u}_M(.,\\sigma)\\right\\|  _{2}\\nonumber\\\\\n& \\quad +c\\left\\|  u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)\n+\\tilde{u}_M(.,\\sigma)\\right\\|  _{2} \\label{305}\n\\end{align}\na.s. for all $t,\\tau,\\sigma\\in\\left[0,T\\right]$ with $t>\\tau>\\sigma$ and every $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }Up to minor modifications, we can prove (\\ref{304}) as we\nargued in the proof of (\\ref{24}).\n\nFor the proof of (\\ref{305}) we first write\n\\begin{align*}\n&\\left\\|  \\ f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\n -f_{i,t}(u_M)(.,\\sigma)+f_{i,t}(\\tilde{u}_M)(.,\\sigma)\\right\\|_{2}^2\\\\\n &\\quad \\le 2\\left(F^1(i,t,\\tau,\\sigma)+F^2(i,t,\\tau,\\sigma)\\right),\n \\end{align*}\n with\n \\begin{align*}\n& F^1(i,t,\\tau,\\sigma)\\\\\n&= \\int_D dx\\left\\vert\\int_D dy\\ e_i(y)\\left(G(x,t;y,\\tau)-(G(x,t;y,\\sigma)\\right)\\left(u_M(y,\\tau)-\\tilde u_M(y,\\tau)\\right)\\right\\vert^2,\\\\\n&F^2(i,t,\\tau,\\sigma)\\\\\n&=\\int_D dx\\left\\vert\\int_D dy\\ e_i(y) G(x,t;y,\\sigma)\\left(u_M(y,\\tau)-\\tilde u_M(y,\\tau)-u_M(y,\\sigma)+\\tilde u(y,\\sigma)\\right)\\right\\vert^2.\n\\end{align*}\nFrom the Gaussian property of $G$ we clearly see that  $F^2(i,t,\\tau,\\sigma)$ is  bounded above by the square of the last term of (\\ref{305}). \nMoreover, by applying first (\\ref{14}) and then Schwarz inequality we obtain\n\\begin{equation*}\nF^1(i,t,\\tau,\\sigma)\\le c(t-\\tau)^{-2\\delta}(\\tau-\\sigma)^{2\\delta}\\left\\|u_M(.,\\sigma)-\\tilde{u}_M(.,\\sigma)\\right\\|_{2}^2.\n\\end{equation*}\nHence (\\ref{305}) is proved.\n\n\\hfill $\\blacksquare$\n\\medskip\n\nThe preceding result now leads to the following estimate for the second term\non the right-hand side of (\\ref{302}).\n\n\\begin{lemma}\n\\label{l6}\nThe hypotheses are those of Theorem \\ref{t1} part (b).\nThen we have\n\\begin{align}\n&  \\left\\|C(u_M)(.,t)-C(\\tilde{u}_M\n)(.,t)\\right\\|  _{2}\\nonumber\\\\\n&\\quad  \\leq cr_{\\alpha}^{H}\\left(  \\int_{0}^{t}d\\tau\\left(\n\\frac{1}{\\tau^{\\alpha}}+\\frac{1}{\\left(  t-\\tau\\right)  ^{\\alpha}}\\right)\n\\left\\|  u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|  _{2}\\right. \\nonumber\\\\\n&\\quad  +\\left.  \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\frac{\\left\\|  u_M\n(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)+\\tilde{u}_M(.,\\sigma\n)\\right\\|  _{2}}{\\left(  \\tau-\\sigma\\right)  ^{\\alpha+1}}\\right)\n\\label{306}\n\\end{align}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }From (\\ref{21}), (\\ref{22}),\nand by using (\\ref{i}), (\\ref{304}), (\\ref{305}), we have\n\\begin{align}\n&  \\left\\| C(u_M)(.,t)-C(\\tilde{u}_M\n)(.,t)\\right\\|  _{2}\\nonumber\\\\\n&  \\leq cr_{\\alpha}^{H}\\left(  \\int_{0}^{t}d\\tau\\frac{\\left\\|\nu_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|  _{2}}{\\tau^{\\alpha}}\\right.\n\\nonumber\\\\\n&  +\\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma(t-\\tau)^{-\\delta}\n(\\tau-\\sigma)^{\\delta-\\alpha-1}\\left\\|  u_M(.,\\sigma\n)-\\tilde{u}_M(.,\\sigma)\\right\\|  _{2}\\nonumber\\\\\n&  +\\left.  \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\frac{\\left\\|  u_M\n(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)+\\tilde{u}_M(.,\\sigma\n)\\right\\|  _{2}}{\\left(  \\tau-\\sigma\\right)  ^{\\alpha+1}}\\right)\n\\label{307}\n\\end{align}\na.s. for every $t\\in\\left[  0,T\\right]$.\n\nFurthermore, by swapping each\nintegration variable for the other in the second term on the right-hand side\nand by using Fubini's theorem we may write\n\\begin{align*}\n&  \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma(t-\\tau)^{-\\delta}\n(\\tau-\\sigma)^{\\delta-\\alpha-1}\\left\\|  u_M(.,\\sigma\n)-\\tilde{u}_M(.,\\sigma)\\right\\|  _{2}\\\\\n&  =\\int_{0}^{t}d\\tau\\left\\|  u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|\n_{2}\\int_{\\tau}^{t}d\\sigma(t-\\sigma)^{-\\delta}(\\sigma-\\tau\n)^{\\delta-\\alpha-1}\\\\\n&  =c\\int_{0}^{t}d\\tau\\frac{\\left\\|  u_M(.,\\tau)-\\tilde{u}_M\n(.,\\tau)\\right\\|  _{2}}{\\left(  t-\\tau\\right)  ^{\\alpha}},\n\\end{align*}\nafter having evaluated the singular integral explicitly in terms of Euler's\nBeta function; this is possible since we can choose $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$ such that $\\alpha<\\delta$. The\nsubstitution of the preceding expression into (\\ref{307}) then proves\n(\\ref{306}). \\hfill $\\blacksquare$\n\\bigskip\n\nIn what follows, we write $R$ for all the  irrelevant a.s. finite and positive\nrandom variables that appear in the different estimates, unless we specify these\nvariables otherwise.\nThe preceding inequalities then lead to a\ncrucial estimate for $z_{M}:=u_{M}-\\tilde{u}_{M}$ with respect to the norm in\n$B^{\\alpha,2}(0,t;L^{2}(D))$.\n\\begin{lemma}\n\\label{l30}\nWe assume the same hypotheses as in part (b) of Theorem \\ref{t1}.\nThen we have\n\\begin{align*}\n\\left\\Vert z_{M}\\right\\Vert _2^{2}  &  \\leq R\\left(  \\int_{0}^{t}d\\tau\n\\sup_{\\sigma\\in\\lbrack0,\\tau]}\\left\\Vert z_{M}(.,\\sigma)\\right\\Vert_{2}^{2}\\right. \\nonumber\\\\\n&  +\\left.  \\int_{0}^{t}d\\tau\\left(  \\int_{0}^{\\tau}d\\sigma\\frac{\\left\\Vert\nz_{M}(.,\\tau)-z_{M}(.,\\sigma)\\right\\Vert _{2}}{\\left(  \\tau-\\sigma\\right)^{\\alpha+1}}\\right)^{2}\\right) \n\\end{align*}\na.s. for every $t\\in\\left[  0,T\\right]$.\n\\end{lemma}\n\n\\noindent{\\bf Proof.}\nWe apply Schwarz inequality relative to the measure $d\\tau$ on\n$(0,t)$ to both integrals on the right-hand side of (\\ref{306}). This leads\nto\n\\begin{align}\n&  \\Vert C(u_{M})(.,t)-C(\\tilde{u}_{M})(.,t)\\Vert_{2}^{2}\\nonumber\\\\\n&  \\leq R\\left(  \\int_{0}^{t}d\\tau\\Vert z_{M}(.,\\tau)\\Vert_{2}^{2}+\\int\n_{0}^{t}d\\tau\\left(  \\int_{0}^{\\tau}d\\sigma\\frac{\\Vert z_{M}(.,\\tau\n)-z_{M}(.,\\sigma)\\Vert_{2}}{(\\tau-\\sigma)^{\\alpha+1}}\\right)  ^{2}\\right)\n\\label{3701}\n\\end{align}\na.s. for every $t\\in\\left[  0,T\\right]  $. This estimate along with (\\ref{302}),\n(\\ref{303}) yields the result.\n\n \\hfill $\\blacksquare$\n\nAs a consequence of the preceding Lemma we obtain\n\\begin{align}\n\\left\\Vert z_{M}\\right\\Vert _{\\alpha,2,t}^{2}  &  \\leq R\\left(  \\int_{0}^{t}d\\tau\n\\sup_{\\sigma\\in\\lbrack0,\\tau]}\\left\\Vert z_{M}(.,\\sigma)\\right\\Vert_{2}^{2}\\right. \\nonumber\\\\\n&  +\\left.  \\int_{0}^{t}d\\tau\\left(  \\int_{0}^{\\tau}d\\sigma\\frac{\\left\\Vert\nz_{M}(.,\\tau)-z_{M}(.,\\sigma)\\right\\Vert _{2}}{\\left(  \\tau-\\sigma\\right)^{\\alpha+1}}\\right)^{2}\\right)  \\label{308},\n\\end{align}\nby the very definition of the norm $\\left\\Vert \\cdot\\right\\Vert _{\\alpha,2,t}^{2}$.  \n\n\\bigskip\n\nWe proceed by analyzing further the second\nterm on the right-hand side of (\\ref{308}), so as to eventually obtain an inequality of Gronwall type\nfor $\\Vert z_M\\Vert_{\\alpha,2,t}^2$.  \n\nFirst we introduce some notation.\nFor $0\\le \\tau<s\\le t\\le T$, we set\n\\begin{equation}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau):=f_{i,t}(u_{M})(.,\\tau)-f_{i,s}(u_{M})(.,\\tau),\n\\label{39}\n\\end{equation}\nwhere the $f_{i,t}(u_{M})$'s are given by (\\ref{22}).\n\nBy reference to (\\ref{9}),\nwe may write\n\\begin{align}\n&  z_M(.,\\tau)-z_M(.,\\sigma)\\nonumber\\\\\n&  =\\int_{\\sigma}^{\\tau}d\\rho\\int_{D}dyG(.,\\tau;y,\\rho)\\left(  g(u_M\n(y,\\rho))-g(\\tilde{u}_M(y,\\rho))\\right) \\nonumber\\\\\n&  +\\int_{0}^{\\sigma}d\\rho\\int_{D}dy\\left(  G(.,\\tau;y,\\rho)-G(.,\\sigma\n;y,\\rho)\\right)  \\left(  g(u_M(y,\\rho))-g(\\tilde{u}_M\n(y,\\rho))\\right) \\nonumber\\\\\n&  +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{\\sigma}^{\\tau}\\left(\nf_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M\n)(.,\\rho)\\right)  B_{i}^{H}(d\\rho)\\nonumber\\\\\n&  +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{\\sigma}\\left(\nf_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\rho)-f_{i,\\tau,\\sigma}^{\\ast\n}(\\tilde{u}_M)(.,\\rho)\\right)  B_{i}^{H}(d\\rho)\n\\label{309}\n\\end{align}\nfor all $\\sigma,\\tau\\in\\left[  0,t\\right]  $ with $\\tau>\\sigma$\n\nOur next goal is to estimate the $L^{2}(D)$-norm of each contribution on\nthe right-hand side of (\\ref{309}). Regarding the first two terms\nwe have the following result.\n\n\\begin{lemma}\n\\label{l7}\nThe hypotheses are the same as in part (a) of Theorem \\ref{t1}; then we\nhave\n\\begin{align}\n&  \\left\\|  \\int_{\\sigma}^{\\tau}d\\rho\\int_{D}dy\\ G(.,\\tau;y,\\rho)\\left(\ng(u_M(y,\\rho))-g(\\tilde{u}_M(y,\\rho))\\right)  \\right\\|\n_{2}\\nonumber\\\\\n&  \\leq c\\left(  \\tau-\\sigma\\right)  ^{\\frac{1}{2}}\\left(  \\int_{\\sigma}\n^{\\tau}d\\rho\\left\\|  z_M(.,\\rho)\\right\\|  _{2}^{2}\\right)\n^{\\frac{1}{2}}\n\\label{310}\n\\end{align}\nand\n\\begin{align}\n&  \\left\\|  \\int_{0}^{\\sigma}d\\rho\\int_{D}dy\\left(  G(.,\\tau;y,\\rho\n)-G(.,\\sigma;y,\\rho)\\right)  \\left(  g(u_M(y,\\rho\n))-g(\\tilde{u}_M(y,\\rho))\\right)  \\right\\|  _{2}\\nonumber\\\\\n&  \\leq c\\left(  \\tau-\\sigma\\right)  ^{\\frac{\\delta}{2}}\\left(  \\int\n_{0}^{\\sigma}d\\rho\\left(  \\sigma-\\rho\\right)  ^{-\\delta}\\left\\|  z_M\n(.,\\rho)\\right\\|  _{2}^{2}\\right)  ^{\\frac{1}{2}}\n\\label{311}\n\\end{align}\na.s. for all $\\sigma,\\tau\\in\\left[  0,t\\right] $ with \n$\\tau>\\sigma$ and every $\\delta\\in\\left(  \\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n{\\bf Proof:} The inequality (\\ref{310}) follows by applying Schwarz inequality and using the Gaussian property along with\nassumption (L). As for (\\ref{311}), we first apply Schwarz inequality  with respect to the measure on $[0,\\sigma]\\times D$ given by\n$|G(x,\\tau;y,\\rho)-G(x,\\sigma;y,\\rho)|d\\rho\\ dy$ and then (\\ref{15}).\n\n\\hfill $\\blacksquare$\n\n\n\n\nNext, we turn to the analysis of the third term on the right-hand side of\n(\\ref{309}).\n\n\\begin{lemma}\n\\label{l8}\nWith the same hypotheses as in part (b) of Theorem \\ref{t1}, we have\n\\begin{align*}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{\\sigma}^{\\tau\n}\\left(  f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M\n)(.,\\rho)\\right)  B_{i}^{H}(d\\rho)\\right\\|  _{2}\\\\\n&  \\leq R\\left(  \\int_{\\sigma}^{\\tau}\nd\\rho\\left(  \\frac{1}{\\left(  \\rho-\\sigma\\right)  ^{\\alpha}}+\\frac{1}{\\left(\n\\tau-\\rho\\right)  ^{\\alpha}}\\right)  \\left\\|  z_M(.,\\rho)\\right\\|\n_{2}\\right.  \\\\\n&  +\\left.  \\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}d\\varsigma\n\\frac{\\left\\|  z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\|  _{2}\n}{\\left(  \\rho-\\varsigma\\right)  ^{\\alpha+1}}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[0,t\\right]$ with \n$\\tau>\\sigma$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} \nIn terms of the variables $\\tau,\\rho$ and $\\varsigma$,\ninequalities (\\ref{304}), (\\ref{305}) of Lemma \\ref{l5} now read\n\\begin{equation}\n\\sup_{(i,\\tau)\\in\\mathbb{N}^+\\times [0,T]}\\left\\|  \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}\n(\\tilde{u}_M)(.,\\rho)\\right\\|  _{2}\n\\leq c\\left\\|  z_M(.,\\rho)\\right\\|_{2} \\label{312}\n\\end{equation}\nand\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+} \\left\\|  \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}\n(\\tilde{u}_M)(.,\\rho)\\ -f_{i,\\tau}(u_M)(.,\\varsigma\n)+f_{i,\\tau}(\\tilde{u}_M)(.,\\varsigma)\\right\\|  _{2}^{{}}\\nonumber\\\\\n&  \\leq c(\\tau-\\rho)^{-\\delta}(\\rho-\\varsigma)^{\\delta}\n\\left\\|  z_M(.,\\varsigma)\\right\\|  _{2}\n+c\\left\\|  z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\|  _{2},\n\\label{313}\n\\end{align}\nrespectively. Thus, by an extended version of (\\ref{i}) for indefinite generalized Stieltjes integrals\n(see Proposition 4.1 in \\cite{nuarasca}),\n\\begin{align*}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{\\sigma}^{\\tau}\n\\left(  f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M)(.,\\rho)\\right)  B_{i}^{H}(d\\rho)\\right\\|  _{2}\\\\\n&  \\leq r_\\alpha^H \\sup_{i\\in\\mathbb{N}^+}\\Big\n\\Big(\\int_{\\sigma}^{\\tau}d\\rho\\frac{\\left\\|\\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M%\n)(.,\\rho)\\right\\|_{2}}{\\left(  \\rho-\\sigma\\right)  ^{\\alpha}}\\\\\n&  +\\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}\\frac{d\\varsigma}{\\left(\\rho-\\varsigma\\right)^{\\alpha+1}}\\\\\n&\\times\\left\\|  \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M)(.,\\rho)\\ -f_{i,\\tau}(u_M)(.,\\varsigma)+f_{i,\\tau}(\\tilde{u}_M)(.,\\varsigma)\\right\\|_2\\Big)\\Big) \\\\\n&  \\leq R\\left(  \\int_{\\sigma}^{\\tau}d\\rho\n\\frac{\\left\\|  z_M(.,\\rho)\\right\\|  _{2}}{\\left(  \\rho\n-\\sigma\\right)  ^{\\alpha}}+\\int_{\\sigma}^{\\tau}d\\rho\\left(  \\tau-\\rho\\right)\n^{-\\delta}\\int_{\\sigma}^{\\rho}d\\varsigma\\left(  \\rho-\\varsigma\n\\right) ^{\\delta-\\alpha-1}\\left\\|  z_M(.,\\varsigma\n)\\right\\|  _{2}\\right. \\\\\n&  +\\left.  \\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}d\\varsigma\n\\frac{\\left\\|  z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\|_{2}\n}{\\left(  \\rho-\\varsigma\\right)  ^{\\alpha+1}}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[  0,t\\right]  $ with $\\tau>\\sigma$ and every\n$\\delta\\in\\left(  \\frac{d}{d+2},1\\right)$. But the second term on the\nright-hand side is equal to\n\\begin{equation*}\nc\\int_{\\sigma}^{\\tau}d\\rho\\left(  \\tau-\\rho\\right)  ^{-\\alpha}\\left\\|\nz_M(.,\\rho)\\right\\|  _{2},\n\\end{equation*}\nas can be easily checked by applying Fubini's theorem and by evaluating the\nresulting inner integral in terms of Euler's Beta function. \\hfill $\\blacksquare$\n\\bigskip\n\nAs for the analysis of the fourth term on the right-hand side of\n(\\ref{309}) we need some preparatory results. In particular we shall use the estimate for \ntime increments of the Green function, valid for any  $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$: \n\\begin{align}\n& \\left|G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma)+G(x,s;y,\\sigma)\\right|\\nonumber\\\\\n& \\leq \\left( \\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|^{\\delta}\n+\\left|G(x,t;y,\\sigma)-G(x,s;y,\\sigma)\\right|^{\\delta}\\right) \\nonumber\\\\\n&  \\times\\left( \\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|^{1-\\delta}\n+\\left|G(x,s;y,\\tau)-G(x,s;y,\\sigma)\\right|^{1-\\delta}\\right) \\nonumber\\\\\n&  \\leq \\left( t-s\\right)^\\delta(s-\\tau)^{-\\frac{d+2}{2}\\delta+\\frac{d}{2}}\n\\left( \\tau-\\sigma\\right) ^{1-\\delta}(s-\\tau)^{-\\frac{d+2}{2}(1-\\delta)}\\nonumber\\\\\n& \\times\\left( (\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\n\\exp\\left[ -c\\frac{\\left|x-y\\right|^{2}}{\\tau^{\\ast}-\\tau}\\right]  \n+(\\sigma^{\\ast}-\\sigma)^{-\\frac{d}{2}}\\exp\\left[-c\\frac{\\left| x-y\\right|^{2}}{\\sigma^{\\ast}-\\sigma}\\right]  \\right) \\nonumber\\\\\n&  =c\\left(  t-s\\right)  ^{\\delta}\\left(  s-\\tau\\right)  ^{-1}\\left(\n\\tau-\\sigma\\right)  ^{1-\\delta}\\nonumber\\\\\n&  \\times\\left(  (\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|\nx-y\\right|  _{{}}^{2}}{\\tau^{\\ast}-\\tau}\\right]  +(\\sigma^{\\ast}\n-\\sigma)^{-\\frac{d}{2}}\\exp\\left[  -c\\frac{\\left|  x-y\\right|^{2}}{\\sigma^{\\ast}-\\sigma}\\right]  \\right),  \\label{44}\n\\end{align}\nwith $\\tau^*, \\sigma^* \\in(s,t)$, which follows from (\\ref{16})--(\\ref{17}).\n\n\n\\begin{lemma}\n\\label{l9}\nThe hypotheses are the same as in part (b) of Theorem \\ref{t1} and the\n$f_{i,\\tau,\\sigma}^{\\ast}(u)$'s are the functions given by\n(\\ref{39}). Then, the estimates\n\\begin{equation}\n\\sup_{i\\in\\mathbb{N}^+}\\left\\|\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M)\n(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M\n)(.,\\rho)\\right\\|  _{2}\n\\leq c(\\tau-\\sigma)^{\\frac{\\delta}{2}}(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left\\|z_M(.,\\rho)\\right\\|  _{2} \\label{314}\n\\end{equation}\nand\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+}\\left\\|\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M\n)(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M\n)(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\varsigma\n)+\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M)(.,\\varsigma\n)\\right\\|  _{2}\\nonumber\\\\\n&  \\leq c(\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(  (\\sigma-\\rho)^{-\\frac{1}{2}\n}(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)}\\left\\|  z_M(.,\\varsigma\n)\\right\\|  _{2}\\right. \\nonumber\\\\\n&  +\\left.  (\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left\\|  z_M\n(.,\\rho)-z_M(.,\\varsigma)\\right\\|  _{2}\\right)  \\label{315}\n\\end{align}\nhold a.s. for all $\\tau,\\sigma,\\rho,\\varsigma\\in\\left[ 0,T\\right]$ with \n$\\tau>\\sigma>\\rho>\\varsigma$ and every $\\delta\\in\\left(  \\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} It follows from the same type of \narguments as those outlined in the\nproof of Lemma \\ref{l5}. For the proof of (\\ref{314}) the key estimate is (\\ref{15}). For (\\ref{315}), we also apply (\\ref{15}) along with\n(\\ref{44}). \n\n \\hfill $\\blacksquare$\n\\bigskip\n\nThe last relevant $L^{2}(D)$-estimate regarding (\\ref{309}) is then\nthe following.\n\\begin{lemma}\n\\label{l10}\nThe hypotheses are the same as in part (b) of Theorem \\ref{t1}. Then we\nhave\n\\begin{align*}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{\\sigma\n}\\left(  f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\rho)-f_{i,\\tau,\\sigma\n}^{\\ast}(\\tilde{u}_M)(.,\\rho)\\right)  B_{i}^{\\text{\\textsc{H}}\n}(d\\rho)\\right\\|  _{2}\\\\\n&  \\leq R (\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left(  \\frac{1}\n{\\rho^{\\alpha}}+\\frac{1}{(\\sigma-\\rho)^{\\alpha}}\\right)  \\left\\|\nz_M(.,\\rho)\\right\\|  _{2}\\right. \\\\\n&  +\\left.  \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\int\n_{0}^{\\rho}d\\varsigma\\frac{\\left\\|  z_M(.,\\rho)-z_M\n(.,\\varsigma)\\right\\|  _{2}}{\\left(  \\rho-\\varsigma\\right)  ^{\\alpha+1}\n}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[  0,t\\right]$ with \n$\\tau>\\sigma$ and every $\\delta\\in\\left(  \\frac{d}{d+2},1-2\\alpha\\right)$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} By applying (\\ref{i}), \ntogether with (\\ref{314}), (\\ref{315}), we get\n\\begin{align*}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{\\sigma\n}\\left(  f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\rho)-f_{i,\\tau,\\sigma\n}^{\\ast}(\\tilde{u}_M)(.,\\rho)\\right)  B_{i}^{\\text{\\textsc{H}}\n}(d\\rho)\\right\\|  _{2}\\\\\n&  \\leq R (\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\frac{\\left\\|\nz_M(.,\\rho)\\right\\|  _{2}}{\\rho^{\\alpha}}\\right. \\\\\n&  +\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{1}{2}}\\int_{0}^{\\rho\n}d\\varsigma(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)-\\alpha-1}\\left\\|\nz_M(.,\\varsigma)\\right\\|  _{2}\\\\\n&  +\\left.  \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\int\n_{0}^{\\rho}d\\varsigma\\frac{\\left\\|  z_M(.,\\rho)-z_M(.,\\varsigma)\n\\right\\|  _{2}}{\\left(  \\rho-\\varsigma\\right)  ^{\\alpha+1}\n}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[  0,t\\right]  $ with $\\tau>\\sigma$. But for every \n$\\delta\\in\\left(  \\frac{d}{d+2},1-2\\alpha\\right)  $, we have \n\\begin{align*}\n& \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{1}{2}}\\int_{0}^{\\rho}\nd \\varsigma(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)-\\alpha-1}\\left\\|\nz_M(.,\\varsigma)\\right\\|  _{2}\\\\\n&  =c\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\alpha-\\frac{\\delta}{2}}\\left\\|\nz_M(.,\\rho)\\right\\| _{2}.\n\\end{align*}\nThis yields the result.\n\n\\hfill$\\blacksquare$\n\\bigskip\n\nLet us go back to the inequality (\\ref{308}). Owing to (\\ref{309}) and by using the estimates\n(\\ref{310}), (\\ref{311}) together with Lemmas \\ref{l8} and \\ref{l10}, we have\n\\begin{equation}\n\\label{3160}\n\\Vert z_M\\Vert_{\\alpha,2,t}^2 \\le R \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]}\\Vert z_M(.,\\rho)\\Vert_2^2\n+\\sum_{k=1}^6 \\left[I_k(\\tau)\\right]^2\\right)\n\\end{equation}\na.s., where\n\\begin{align*}\nI_1(\\tau)&= \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{\\frac{1}{2}+\\alpha}}\n\\left(\\int_\\sigma^\\tau d\\rho \\Vert z_M(.,\\rho)\\Vert_2^2\\right)^{\\frac{1}{2}},\\\\\nI_2(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\left(\\int_0^\\sigma d\\rho (\\sigma-\\rho)^{-\\delta}\\Vert z_M(.,\\rho)\\Vert_2^2\\right)^{\\frac{1}{2}},\\\\\nI_3(\\tau)&= \\int_0^\\tau \\frac{d\\sigma }{(\\tau-\\sigma)^{\\alpha+1}}\n\\int_\\sigma^\\tau d\\rho\\left(\\frac{1}{(\\rho-\\sigma)^\\alpha}+\\frac{1}{(\\tau-\\rho)^\\alpha}\\right )\\Vert z_M(.,\\rho)\\Vert_2,\\\\\nI_4(\\tau)&=\\int_0^\\tau \\frac{d\\sigma }{(\\tau-\\sigma)^{\\alpha+1}}\n\\left(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right),\\\\\nI_5(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\int_0^\\sigma \\frac{d\\rho}{(\\sigma-\\rho)^{\\frac{\\delta}{2}}}\\left(\\frac{1}{\\rho^\\alpha}+\\frac{1}{(\\sigma-\\rho)^\\alpha}\\right) \\Vert z_M(.,\\rho)\\Vert_2,\\\\\nI_6(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\int_0^\\sigma \\frac{d\\rho}{(\\sigma-\\rho)^{\\frac{\\delta}{2}}} \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}.\\\\\n\\end{align*}\nSet $T_k(t)= \\int_0^t d\\tau\\ \\left[I_k(\\tau)\\right]^2$, $k=1, \\ldots, 6$.\n\nThe function $\\sigma\\mapsto (\\tau-\\sigma)^{-\\frac{1}{2}-\\alpha}$ is integrable on $(0,\\tau)$ for $\\alpha\\in(0,\\frac{1}{2})$. Thus\nwe have \n\\begin{equation} \n\\label{316}\nT_1(t)\\le c \\int_0^t d\\tau  \\Vert z_M(.,\\tau)\\Vert_2^2.\n\\end{equation}\n Since we can choose $\\delta>2\\alpha$, we have that $\\sigma\\mapsto (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}$ is integrable on $(0,\\tau)$. Then, applying Schwarz inequality with respect to the measure \n given by $d\\sigma(\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}$,  we obtain\n \\begin{align}\n \\label{317}\nT_2(t)&\\le c \\int_0^t d\\tau\\ \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\left(\\int_0^\\sigma d\\rho\\ (\\sigma-\\rho)^{-\\delta}\n\\Vert  z_M(.,\\rho)\\Vert_2^2\\right)\\nonumber\\\\\n&\\le c\\int_0^t d\\tau \\left(\\sup_{0\\le \\rho\\le \\tau}\\Vert z_M(\\cdot,\\rho\\Vert_2)^2\\right)\\left(\\int_0^\\tau d\\sigma (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\sigma^{1-\\delta}\\right)\\nonumber\\\\\n&\\le c\\int_0^t d\\tau \\left( \\sup_{0\\le \\rho\\le \\tau}\\Vert z_M(\\cdot,\\rho\\Vert_2)^2\\right),\n\\end{align}\nwhere in the last inequality we have used that $\\alpha+\\frac{\\delta}{2}<1$ along with the definition of Euler's Beta function.\n\nBy integrating one obtains\n\\begin{equation*}\n\\int_\\sigma^\\tau d\\rho\\left(\\frac{1}{(\\rho-\\sigma)^\\alpha}+\\frac{1}{(\\tau-\\rho)^\\alpha}\\right )= \\frac{2(\\tau-\\sigma)^{1-\\alpha}}{1-\\alpha}.\n\\end{equation*}\nMoreover, the function $\\sigma\\mapsto (\\tau-\\sigma)^{-2\\alpha}$ is integrable on $(0,\\tau)$. Consequently,\n\\begin{align}\n\\label{318}\nT_3(t)&\\le c \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert  z_M(.,\\rho)\\Vert_2^2\\right) \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{-2\\alpha}\\nonumber\\\\\n &\\le c  \\int_0^t d\\tau \\left( \\sup_{\\rho\\in[0,\\tau]} \\Vert  z_M(.,\\rho)\\Vert_2^2\\right).\n\\end{align}\nFor any $\\tau\\in (0,t)$, set\n\\begin{equation*}\nI_\\tau= \\int_0^\\tau d\\sigma \\ (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\left(\\int_0^\\sigma d\\rho\\ (\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left(\\frac{1}{\\rho^\\alpha}\n+\\frac{1}{(\\sigma-\\rho)^\\alpha}\\right)\\right).\n\\end{equation*}\nIt is a simple exercise to check that for $\\alpha+\\frac{\\delta}{2}<1$,\n$\\sup_{\\tau\\in[0,t]} I_\\tau < +\\infty$.\n\nSince\n\\begin{equation*}\nT_5(t) \\le \\int_0^t d\\tau\\ I_\\tau^2 \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert  z_M(.,\\rho)\\Vert_2^2\\right),\n\\end{equation*}\nwe conclude that\n\\begin{equation}\n\\label{319}\nT_5(t) \\le c \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert  z_M(.,\\rho)\\Vert_2^2\\right).\n\\end{equation}\n\n\nFix $\\eta\\in(0,1)$ so that $\\sigma\\mapsto (\\tau-\\sigma)^{-\\eta}$ is integrable on $(0,\\tau)$. Applying Schwarz inequality first with respect to the\nmeasure $d\\sigma (\\tau-\\sigma)^{-\\eta}$, and then with respect to the Lebesgue measure on the interval $(\\sigma,\\tau)$ yields\n\\begin{align*}\nT_4(t)&= \\int_0^t d\\tau \\Big( \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^\\eta} (\\tau-\\sigma)^{-\\alpha-1+\\eta}\\\\\n&\\quad \\times \\Big(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\Big)\\Big)^2\\\\\n&\\le c \\int_0^t d\\tau\\  \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^\\eta} (\\tau-\\sigma)^{-2\\alpha-2+2\\eta}\\\\\n&\\quad \\times \\left(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\\\\n&\\le c \\int_0^t d\\tau\\  \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{\\eta -2\\alpha-1}\\\\\n&\\quad \\times  \\int_\\sigma^\\tau d\\rho \\left( \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2.\n\\end{align*}\nBy choosing $\\eta>2\\alpha$, the function $\\sigma\\mapsto (\\tau-\\sigma)^{\\eta -2\\alpha-1}$ is integrable on $(0,\\tau)$. Thus, from the preceding inequalities\nwe obtain\n\\begin{align}\n\\label{320}\nT_4(t)&\\le c \\int_0^t d\\tau\\  \\int_0^\\tau d\\rho \\left( \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\  \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2\\ .\n\\end{align}\nBy Fubini's theorem and evaluations based upon Euler's Beta function, we have\n\\begin{align}\n\\label{321}\nT_6(t)&= \\int_0^t d\\tau \\Big(\\int_0^\\tau d\\rho \\Big( \\int_\\rho^\\tau d\\sigma (\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\Big)\\nonumber\\\\\n&\\quad\\times \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\Big)^2\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\left(\\int_0^\\tau d\\rho \\left(\\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\right)\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\  \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2.\n\\end{align}\n\n\n\n\nFinally, inequalities (\\ref{3160}) to (\\ref{321}) imply\n\\begin{equation}\n\\label{322}\n\\Vert z_M\\Vert_{\\alpha,2,t}^2 \\le R \\int_0^t d\\tau\\ \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2\\ \n\\end{equation}\na.s.\nBy Gronwall's lemma, this clearly implies the uniqueness of the mild solution.\nNow the proof of  part (b) of Theorem \\ref{t1} is complete. \\hfill $\\blacksquare$\n\\bigskip\n\n\n\\noindent{\\bf Proof of Statement $(c)$ of Theorem \\ref{t1}}\n\\bigskip\n\nWe investigate\neach of the functions (\\ref{19})--(\\ref{21}) separately.\n\n\\begin{proposition}\n\\label{p2}\nAssume that Hypotheses ($K_{\\beta,\\beta^{\\prime}}$) and ($I$) hold. Then, there exists $c\\in(0,+\\infty)$ \nsuch that the estimate\n\\begin{equation}\n\\left\\|A(\\varphi)(.,t) -A(\\varphi)(.,s)\\right\\|  _{2}\\leq\nc\\left|  t-s\\right|  ^{\\theta^{\\prime}} \\label{33}\n\\end{equation}\n\\textit{holds for all }$s,t\\in\\left[  0,T\\right]  $ and every $\\theta^{\\prime\n}\\in\\left(  0,\\frac{\\beta}{2}\\right]  $.\n\\end{proposition}\n\n\\noindent\\textbf{Proof. }Relation (\\ref{19}) defines a classical solution to\n(\\ref{2}) when $g=h=0$, so that the standard regularity theory for\nlinear parabolic equations gives $(x,t)\\mapsto A(\\varphi)\n(x,t)\\in\\mathcal{C}^{\\beta,\\frac{\\beta}{2}}(\\overline{D}\\times\\left[\n0,T\\right]  \\mathbb{)}$ (see, for instance, \\cite{eidelzhita}), from which\n(\\ref{33}) follows immediately. \\hfill $\\blacksquare$\n\n\\bigskip\n\nRegarding (\\ref{20}) we have the following result.\n\\begin{proposition}\n\\label{p3}\nAssume that the same hypotheses as in \nTheorem \\ref{t1} (a) hold and let $u_{M}$ be any mild solution to (\\ref{2}).\nThen, there exists $c\\in(0,+\\infty)$ such that the\nestimate\n\\begin{equation}\n\\left\\|B(u_{M})(.,t) - B(u_{M})(.,s)\\right\\|  _{2}\\leq\nc\\left|  t-s\\right|  ^{\\theta^{\\prime\\prime}}\\left(1+\\sup_{t\\in[0,T]}\\left\\|  u_{M}(.,t)\\right\\|_2\\right)  \\label{34}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[  0,T\\right]$ and every\n$\\theta^{\\prime\\prime}\\in\\left(  0,\\frac{1}{2}\\right)$.\n\\end{proposition}\n\n\\noindent\\textbf{Proof. } Without restricting the generality, we may assume that $t>s$.\nWe have\n\\begin{align}\n&B(u_{M})(.,t) - B(u_{M})(.,s)\n=\\int_{s}^{t}d\\tau\\int_{D}dy\\ G(.,t;y,\\tau)g\\left(  u_{M}(y,\\tau)\\right)\n\\nonumber\\\\\n&\\quad \\quad  +\\int_{0}^{s}d\\tau\\int_{D}dy\\left(  G(.,t;y,\\tau)-G(.,s;y,\\tau)\\right)\ng\\left(  u_{M}(y,\\tau)\\right),  \\label{35}\n\\end{align}\nand remark that in order to keep track of the increment $t-s$ we can estimate\nthe first term on the right-hand side of (\\ref{35}) by using the same\nkind of arguments as we did in the first part of the proof of Lemma \\ref{l2}. For\nevery $x\\in D$ this gives\n\\begin{align*}\n&  \\int_{s}^{t}d\\tau\\int_{D}dy\\ \\left|  G(x,t;y,\\tau)g\\left(  u_{M}\n(y,\\tau)\\right)  \\right| \\\\\n&  \\leq c(t-s)^{\\frac{1}{2}}\\left(  \\int_{s}^{t}d\\tau\\int_{D}dy\\left|\nG(x,t;y,\\tau)\\right|  \\left(  1+\\left|  u_{M}(y,\\tau)\\right|  ^{2}\\right)\n\\right)  ^{\\frac{1}{2}},\n\\end{align*}\nso that we eventually obtain\n\\begin{equation}\n\\left\\|  \\int_{s}^{t}d\\tau\\int_{D}dy\\ G(.,t;y,\\tau)g\\left(  u_{M}(y,\\tau)\\right)  \\right\\|  _{2}\n\\leq c(t-s)^{\\frac{1}{2}}\\left(  1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right)\n\\label{36}\n\\end{equation}\na.s. for all $s,t\\in\\left[  0,T\\right]  $ with $t>s$. In \\ a similar manner,\nwe can keep track of the increment $t-s$ in the second term on the right-hand\nside of (\\ref{35}) by using (\\ref{15}). We thus have\n\\begin{align}\n&  \\left\\|  \\int_{0}^{s}d\\tau\\int_{D}dy\\left(  G(.,t;y,\\tau)-G(.,s;y,\\tau\n)\\right)  g\\left(  u_{M}(y,\\tau)\\right)  \\right\\|  _{2}^{2}\\nonumber\\\\\n&  \\leq c\\int_{0}^{s}d\\tau\\int_{D}dy\\int_{D}dx\\left|  G(x,t;y,\\tau\n)-G(x,s;y,\\tau)\\right|  \\left(  1+\\left|  u_{M}(y,\\tau)\\right|  ^{2}\\right)\\nonumber \\\\\n&  \\leq c(t-s)^{\\delta}\\int_{0}^{s}d\\tau(s-\\tau)^{-\\delta}\\left(  1+\\left\\|\nu_{M}(.,\\tau)\\right\\|  _{2}^{2}\\right)\\nonumber \\\\\n&  \\leq c(t-s)^{\\delta}\\left(  1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2^{2}\\right)\\label{37}\n\\end{align}\nfor every $\\delta\\in\\left(  \\frac{d}{d+2},1\\right)  $, \na.s. for all $s,t\\in\\left[  0,T\\right]  $ with $t>s$.  This last relation holds\n\\textit{a fortiori} for each $\\delta\\in\\left(  0,1\\right)$, so that\n(\\ref{36}) and (\\ref{37}) indeed prove (\\ref{34}).\n\n \\hfill $\\blacksquare$\n\n\\bigskip\n\nAs for the stochastic term (\\ref{21}), we have the following.\n\n\\begin{proposition}\n\\label{p4}\nAssume the same hypotheses as in \nTheorem \\ref{t1} (c), and let $u_{M}$ be any mild solution to (\\ref{2}).\nThen, there exists $c\\in(0,+\\infty)$ such that the\nestimate\n\\begin{equation}\n\\left\\|C(u_{M})(.,t) - C(u_{M})(.,s)\\right\\|  _{2}\\leq\ncr_{\\alpha}^{H}\\left|  t-s\\right|  ^{\\theta^{\\prime\n\\prime\\prime}}\\left(  1+\\left\\|  u_{M}\\right\\|_{\\alpha,2,T}\\right)\n\\label{38}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[  0,T\\right]$ and every\n$\\theta^{\\prime\\prime\\prime}\\in\\left(  0,\\frac{1}{2}-\\alpha\\right)$.\n\\end{proposition}\n\nThe proof of Proposition \\ref{p4} is more complicated than that of Proposition \\ref{p3}.\nWe begin with a\npreparatory result whose proof is based on inequalities\n(\\ref{15})--(\\ref{17}).\n\n\n\\begin{lemma}\n\\label{l3}\nWith the same hypotheses as in part (a) of Theorem \\ref{t1}, \n the estimates\n\\begin{equation}\n\\sup_{i\\in \\mathbb{N}^+}\\left\\|  f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)\\right\\|  _{2}\\leq c\\left(\nt-s\\right)  ^{\\frac{\\delta}{2}}\\left(  s-\\tau\\right)  ^{-\\frac{\\delta}{2}\n}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right)  \\label{40}\n\\end{equation}\n\\textit{and }\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+}\\left\\|  f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}^{\\ast}(u_{M}\n)(.,\\sigma)\\right\\|  _{2}\\nonumber\\\\\n&  \\leq c\\left(  t-s\\right)  ^{\\frac{\\delta}{2}}\\left(  s-\\tau\\right)\n^{-\\frac{\\delta}{2}}\\left\\|  u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|\n_{2}\\nonumber\\\\\n&  +c\\left(  t-s\\right)  ^{\\frac{\\delta}{2}}\\left(  s-\\tau\\right)  ^{-\\frac\n{1}{2}}\\left(  \\tau-\\sigma\\right)  ^{\\frac{1}{2}(1-\\delta)}\\left(1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2\\right)  \\label{41}\n\\end{align}\nhold a.s. for every $\\delta\\in\\left(  \\frac{d}{d+2},1\\right)$ and for all $\\sigma,\\tau\\in\\left[  0,s\\right)$ with $\\tau>\\sigma$ in\n(\\ref{41}).\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }The proof of (\\ref{40}) is analogous to that of\n(\\ref{25}) and is thereby omitted. As for (\\ref{41}), by using\n Schwarz inequality relative to the measures $dy\\left|\nG(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|$ and\n\\[\ndy\\left|  G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma)+G(x,s;y,\\sigma)\\right|\n\\]\non $D$ along with Hypothesis (L) for $h$, we get\n\\begin{align}\n&  \\left\\|  f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}^{\\ast}(u_{M}\n)(.,\\sigma)\\right\\|  _{2}^{2}\\nonumber\\\\\n&  \\leq c\\int_{D}dx\\int_{D}dy\\left|  G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|\n\\left|  u_{M}(y,\\tau)-u_{M}(y,\\sigma)\\right|  ^{2}\\nonumber\\\\\n&  +c\\int_{D}dx\\int_{D}dy\\left|  G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma\n)+G(x,s;y,\\sigma)\\right|\\nonumber\\\\\n&\\quad\\quad \\times \\left(  1+\\left|  u_{M}(y,\\sigma)\\right|\n^{2}\\right) \\nonumber\\\\\n&  \\leq c\\left(  t-s\\right)  ^{\\delta}\\left(  s-\\tau\\right)  ^{-\\delta\n}\\left\\|  u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|  _{2}^{2}\\nonumber\\\\\n&\\leq c\\left(  t-s\\right)  ^{\\delta}\\left(  s-\\tau\\right)  ^{-1}\\left(\n\\tau-\\sigma\\right)  ^{1-\\delta}\\left(1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2^2\\right),  \\label{42}\n\\end{align}\na.s. for all $s,t,\\sigma,\\tau\\in\\left[  0,T\\right]  $ with $t\\geq\ns>\\tau>\\sigma$ and every $\\delta\\in\\left(  \\frac{d}{d+2},1\\right)  $, as a\nconsequence of (\\ref{15}), (\\ref{44}) and the Gaussian property.\n\n\n\\hfill $\\blacksquare$\n\\bigskip\n\n\\noindent\\textbf{Proof of Proposition \\ref{p4}.} For $t>s$ we write\n\\begin{align}\nC(u_{M})(.,t) - C(u_{M})(.,s) &= \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{s}^{t}f_{i,t}\n(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\nonumber\\\\\n& +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{s}f_{i,t,s}^{\\ast\n}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau). \\label{45}\n\\end{align}\nIn order to estimate the first\nterm on the right-hand side of\n(\\ref{45}), we can start by using inequalities (\\ref{24}) and\n(\\ref{25}) to obtain\n\\begin{align}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{s}^{t}\nf_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|  _{2}\n\\nonumber\\\\\n&  \\leq cr_{\\alpha}^{H}\\left(  \\int_{s}^{t}\\frac{d\\tau}\n{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\frac{\\left\\|  u_{M}(.,\\tau)\\right\\|\n_{2}^{{}}}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\int_{s}^{\\tau}d\\sigma\n\\frac{\\left\\|  u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|  _{2}^{{}}}{\\left(\n\\tau-\\sigma\\right)  ^{\\alpha+1}}\\right. \\nonumber\\\\\n&  +\\left.  \\int_{s}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{s}^{\\tau\n}d\\sigma(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(  1+\\left\\|\nu_{M}(.,\\sigma)\\right\\|  _{2}\\right)  \\right)  \\label{46}\n\\end{align}\na.s. for every $s,t$ $\\in\\left[  0,T\\right]  $ with $t>s$ and each $\\delta\n\\in\\left(  \\frac{d}{d+2},1\\right)$.\n\nFurthermore, we have\n\\begin{align}\n&  \\int_{s}^{t}\\frac{d\\tau}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\frac{\\left\\|\nu_{M}(.,\\tau)\\right\\|  _{2}^{{}}}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\int\n_{s}^{\\tau}d\\sigma\\frac{\\left\\|  u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|\n_{2}^{{}}}{\\left(  \\tau-\\sigma\\right)  ^{\\alpha+1}}\\nonumber\\\\\n&  \\leq c\\left(  \\left(  t-s\\right)  ^{1-\\alpha}\\left(  1+\\left\\|\nu_{M}\\right\\|  _{\\alpha,2,T}\\right)  +(t-s)^{\\frac{1}{2}}\\left\\|\nu_{M}\\right\\|  _{\\alpha,2,T}\\right) \\nonumber\\\\\n&  \\leq c(t-s)^{\\frac{1}{2}}\\left(  1+\\left\\|  u_{M}\\right\\|_{\\alpha,2,T}\\right)  \\label{47}\n\\end{align}\nsince $\\alpha<\\frac{1}{2}$. Moreover,\n\\begin{align}\n&  \\int_{s}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{s}^{\\tau}d\\sigma\n(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(  1+\\left\\|  u_{M}\n(.,\\sigma)\\right\\|  _{2}\\right) \\nonumber\\\\\n&  \\le c\\ \\left(1+\\sup_{t\\in[0,T]}\\left\\|  u_{M}(.,t)\\right\\|_2\\right)\\int_{s}^{t}d\\tau\\  (t-\\tau)^{-\\frac{\\delta}{2}}\\int_s^\\tau d\\sigma\\ \\left(\\tau-\\sigma\\right)\n^{\\frac{\\delta}{2}-\\alpha-1} \\nonumber\\\\\n&  \\leq c(t-s)^{1-\\alpha}\\left(1+\\sup_{t\\in[0,T]}\\left\\|  u_{M}(.,t)\\right\\|_2\\right),  \\label{48}\n\\end{align}\nby virtue of the convergence of the\nintegral, which can be expressed in terms of Euler's Beta function \nsince $\\alpha<\\frac{\\delta}{2}$. The substitution of (\\ref{47}) and (\\ref{48}) into\n(\\ref{46}) then leads to the inequality\n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{s}^{t}\nf_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|  _{2}\\leq\ncr_{\\alpha}^{H}(t-s)^{\\frac{1}{2}}\\left(  1+\\left\\|\nu_{M}\\right\\|_{\\alpha,2,T}\\right)  \\label{hoelder8}\n\\end{equation}\na.s. for every $s,t$ $\\in\\left[  0,T\\right]  $ with $t>s$. \n\nIt remains to\nestimate the second term on the right-hand side of (\\ref{45}). \nFrom (\\ref{i}) with $T$ replaced by $s$, we have\n\\begin{align*}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\n\\leq r_\\alpha^H\\\\\n&\\times\\sup_{i\\in\\mathbb{N}^+}\\int_{0}^{s}d\\tau\\left(  \\frac{\\left\\|\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)\\right\\|  _{2}}{\\tau^{\\alpha}}+\\int\n_{0}^{\\tau}d\\sigma\\frac{\\left\\|  f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}\n^{\\ast}(u_{M})(.,\\sigma)\\right\\|  _{2}}{\\left(  \\tau-\\sigma\\right)\n^{\\alpha+1}}\\right).\n\\end{align*}\n\nBy substituting (\\ref{40}) and (\\ref{41}) we  obtain\n\\begin{align}\n&  \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\\nonumber\\\\\n&  \\leq cr_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{s}d\\tau\\left(  s-\\tau\\right)  ^{-\\frac{\\delta}{2}}\\tau^{-\\alpha\n}\\left(1+\\sup_{t\\in[0,T]}\\left\\|  u_{M}(.,t)\\right\\|_2\\right)  \\right.\n\\nonumber\\\\\n&  +\\int_{0}^{s}d\\tau\\left(  s-\\tau\\right)  ^{-\\frac{\\delta}{2}}\\int_{0}\n^{\\tau}d\\sigma\\frac{\\left\\|  u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|_{2}\n}{\\left(  \\tau-\\sigma\\right)  ^{\\alpha+1}}\\nonumber\\\\\n&  +\\left.  \\int_{0}^{s}d\\tau\\left(  s-\\tau\\right)  ^{-\\frac{1}{2}}\\int\n_{0}^{\\tau}d\\sigma\\left(  \\tau-\\sigma\\right)  ^{\\frac{1}{2}\\left(\n1-\\delta\\right)  -\\alpha-1}\\left(1+\\sup_{t\\in[0,T]}\\left\\|  u_{M}(.,t)\\right\\|_2\\right)  \\right) \\nonumber\\\\\n&  \\leq c\\ r_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(1+\\left\\|  u_{M}\\right\\|_{\\alpha,2,T}\\right)\\nonumber\\\\\n&\\quad\\times \\left(1+\\int_{0}^{s}d\\tau\\left(  s-\\tau\\right)  ^{-\\frac{1}{2}}\\int_{0}^{\\tau}d\\sigma\n\\left(  \\tau-\\sigma\\right)  ^{\\frac{1}{2}\\left(  1-\\delta\\right)-\\alpha-1}\\right),\\label{49}\n\\end{align}\nwhere we have got the last estimate using Schwarz inequality with\nrespect to the measure $d\\tau$ on $\\left(  0,s\\right)$ along with\n(\\ref{3}) in the first two integrals on the right-hand side.\n\nBy imposing the additional restriction $\\delta<1-2\\alpha$, we have\n\\begin{equation*}\n\\int_{0}^{s}d\\tau\\left(  s-\\tau\\right)  ^{-\\frac{1}{2}}\\int_{0}^{\\tau}d\\sigma\n\\left(  \\tau-\\sigma\\right)  ^{\\frac{1}{2}\\left(  1-\\delta\\right)-\\alpha-1}\n<+\\infty.\n\\end{equation*}\nThus, we have proved that\n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\|  \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\\leq cr_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(\n1+\\left\\|  u_{M}\\right\\|_{\\alpha,2,T}\\right)  \\label{50}\n\\end{equation}\na.s. for all $s,t$ $\\in\\left[  0,T\\right]  $ with $t>s$ and every $\\delta\n\\in\\left( \\frac{d}{d+2},1-2\\alpha\\right)$. The existence of this restricted\ninterval of values of $\\delta$ is possible by our choice of $\\alpha$. Relations\n(\\ref{45}), (\\ref{hoelder8}) and (\\ref{50}) clearly \nyield (\\ref{38}) with \n$\\theta^{\\prime\\prime\\prime}\n=\\frac{\\delta}{2}\\in\\left(  0,\\frac{1}{2}-\\alpha\\right)  $.\n \\hfill $\\blacksquare$\n\n\\bigskip\n\nIt is immediate that Propositions \\ref{p2} to \\ref{p4} imply statement $(c)$ of  Theorem \\ref{t1}.\nNotice that $R_\\alpha^H = c(1+r_\\alpha^H)$, with $r_\\alpha^H$ defined in (\\ref{29}).\n\\bigskip\n\nFinally, we give an alternate to the result proved before, as mentioned in Section \\ref{s2}, Remark 5.\n\n\\begin{proposition}\n\\label{pfact}\nThe assumptions are as in Theorem \\ref{t1} part (a). Then,\n\\begin{equation}\n\\Vert C(u_{M})(.,t)-C(u_{M})(.,s)\\Vert_{2}\\leq R |t-s|^{\\theta^{\\prime\\prime\\prime\\prime}}\n\\left(1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right)\n\\label{323}\n\\end{equation}\nholds a.s. for all $s,t\\in\\lbrack0,T]$ and every \n$\\theta^{\\prime\\prime\\prime\\prime}\\in\\left(  0,\\frac{2}{d+2}\\wedge\\frac{1}{2}\\right)$. Consequently, \n\\begin{equation}\n\\label{3231}\n\\Vert u_{M})(.,t)-u_{M})(.,s)\\Vert_2\\le R |t-s|^\\theta \\left(1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right),\n\\end{equation}\na.s. for all $s,t\\in\\lbrack0,T]$ and each $\\theta\\in\\left(0,\\frac{2}{d+2}\\wedge\\frac{\\beta}{2}\\right)$.\n\\end{proposition}\n{\\bf Proof.} \nWe use the factorization method we alluded to in Section \\ref{s2}. For this we express \n$C(u_{M})(.,t)$ in terms of the auxiliary $L^{2}(D)$-valued process\n\\begin{equation*}\nY_{\\varepsilon}(u_{M})(.,t):=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}%\n\\int_{0}^{t}(t-\\tau)^{-\\varepsilon}f_{i,t}(u_{M})(.,\\tau)B_{i}^H(d\\tau)\n\\end{equation*}\ndefined for every $\\varepsilon\\in\\left(  0,\\frac{1}{2}\\right)  $. In fact, by\nrepeated applications of Fubini's theorem and by using the fundamental\nproperty $U(t,\\tau)U(\\tau,\\sigma)=U(t,\\sigma)$ for the evolution operators\ndefined in (\\ref{30}) we obtain\n\\begin{align}\nC(u_{M})(.,t)  &  =\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}%\n^{t}f_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\nonumber\\\\\n&  =\\frac{\\sin(\\varepsilon\\pi)}{\\pi}\\int_{0}^{t}d\\tau(t-\\tau)^{\\varepsilon\n-1}\\int_{D}dyG(.,t;y,\\tau)Y_{\\varepsilon}(u_{M})(y,\\tau) \\label{factorization}%\n\\end{align}\nfor every $t\\in\\left[  0,T\\right]$ a.s. \n\nNext we prove that a.s., \n\\begin{equation}\n\\sup_{t\\in\\lbrack0,T]}\\Vert Y_{\\varepsilon}(u_{M})(.,t)\\Vert_{2}\\leq R\\left(\n1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right).  \\label{324}\n\\end{equation}\nIndeed, according with (\\ref{i}),\n\\begin{align*}\n&\\Vert Y_{\\varepsilon}(u_{M})(.,t)\\Vert_{2}\\le r_\\alpha^H \\sup_{i\\in\\mathbb{N}^+} \\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\\\\n&\\quad\\times\\left\\{\\frac{\\Vert f_{i,t}(u)(.,\\tau)\\Vert_2}{\\tau^\\alpha}+\\int_0^\\tau d\\sigma \\frac{\\Vert f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\Vert_2 }{(\\tau-\\sigma)^{\\alpha+1}}\\right\\}.\n\\end{align*}\nFrom the estimate (\\ref{24}) and the definition of the Beta function, we have\n\\begin{equation*}\n\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\frac{\\Vert f_{i,t}(u)(.,\\tau)\\Vert_2}{\\tau^\\alpha}\\le c\\left(1+\\sup_{\\tau\\in[0,t]} \\Vert u(.,\\tau)\\Vert_2\\right).\n\\end{equation*}\nSince $\\varepsilon\\in(0,\\frac{1}{2})$, applying Schwarz's inequality yields\n\\begin{align*}\n&\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\int_0^\\tau \\frac{\\Vert u(.,\\tau)-u(.,\\sigma)\\Vert_2}{(\\tau-\\sigma)^{\\alpha+1}}\\\\\n&\\quad\\le \\left[\\int_0^t  d\\tau\\ (t-\\tau)^{-2\\varepsilon}\\right]^{\\frac{1}{2}} \\left[\\int_0^t d\\tau\\ \\left(\\int_0^\\tau d\\sigma \\frac{\\Vert u(.,\\tau)-u(.,\\sigma)\\Vert_2}{(\\tau-\\sigma)^{\\alpha+1}}\\right)^2\\right]^{\\frac{1}{2}}\\\\\n&\\quad\\le c \\Vert u\\Vert_{\\alpha,2,t}.\n\\end{align*}\nMoreover, for any $\\delta\\in(0,\\frac{1}{2})$ such that $\\frac{\\delta}{2}-\\alpha>0$, we have\n\\begin{equation*}\n\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon-\\frac{\\delta}{2}} \\int_0^\\tau d\\sigma (\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(1+\\Vert u(.,\\sigma)\\Vert_2\\right)\n\\le c\\left(1+\\sup_{\\sigma\\in[0,t]}\\Vert u(.,\\sigma)\\Vert_2\\right).\n\\end{equation*}\nBy virtue of (\\ref{25}) the two above estimates imply\n\\begin{align*}\n\\int_0^t d\\tau\\int_0^\\tau d\\sigma (t-\\tau)^{-\\varepsilon}\\frac{\\Vert f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\Vert_2 }{(\\tau-\\sigma)^{\\alpha+1}}\n\\le c \\left(1+\\Vert u\\Vert_{\\alpha,2,t}\\right).\n\\end{align*}\nThis ends the  proof of (\\ref{324}).\n\\smallskip\n\n\n We can now proceed by estimating the\ntime increments of $C(u_{M})$ using (\\ref{factorization}) and (\\ref{324}) rather than\nwith the expressions of Proposition \\ref{p4}. For this\nwe  follow the arguments of the proof of (66) in Proposition 6 of \\cite{sansovu1}\nto see that,  by choosing\n $\\theta^{\\prime\\prime\\prime\\prime}\\in\\left(0,\\frac{2}{d+2}\\wedge\\frac{1}{2}\\right)$ with the additional restriction\n$\\varepsilon\\in\\left(  \\theta^{\\prime\\prime\\prime\\prime},\\frac{2}{d+2}%\n\\wedge\\frac{1}{2}\\right)$, we obtain\n\\begin{align*}\n&  \\left\\Vert C(u_{M})(.,t)-C(u_{M})(.,s)\\right\\Vert _{2}\\\\\n&  \\leq c\\left(\\left\\Vert \\int_{s}^{t}d\\tau(t-\\tau)^{\\varepsilon-1}\\int\n_{D}dyG(.,t;y,\\tau)Y_{\\varepsilon}(u_{M})(y,\\tau)\\right\\Vert _{2}\\right. \\\\\n&  +\\left.  \\left\\Vert \\int_{0}^{s}d\\tau\\int_{D}dy\\left(  (t-\\tau\n)^{\\varepsilon-1}G(.,t;y,\\tau)-(s-\\tau)^{\\varepsilon-1}G(.,s;y,\\tau)\\right)\nY_{\\varepsilon}(u_{M})(y,\\tau)\\right\\Vert _{2}\\right) \\\\\n&  \\leq R\\left(  \\left\\vert t-s\\right\\vert ^{\\varepsilon}+\\left\\vert\nt-s\\right\\vert ^{\\theta^{\\prime\\prime\\prime\\prime}}\\right)  (1+\\Vert\nu_{M}\\Vert_{\\alpha,2,T})\n\\leq R |t-s|^{\\theta^{\\prime\\prime\\prime\\prime}}\\left(\n1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right),\n\\end{align*}\nproving (\\ref{323}).\n\nFinally, this estimate along with those established in Propositions \\ref{p2} and \\ref{p3} provide (\\ref{3231}) and finish the proof of\nthe proposition.\n\n\\hfill $\\blacksquare$\n\n\n\n\n\\bigskip\n\n\\noindent\\textbf{Acknowledgements.}\nThe research of the first author concerning this paper was completed at the Institute Mittag-Leffler\nin Djursholm. \nThe research of the second author was supported in part by the \nInstitute of Mathematics of the University of Barcelona where this work was begun, and in\npart by the ETH-Institute of Theoretical Physics in Zurich. They would like to\nthank the three institutions for their very kind hospitality.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:Intro}\n\nThe global star formation rate (SFR) of large galaxy samples at intermediate redshift has been an area of active study for the past decade \\citep{Steidel99, Hopkins06, Zhu09, Gilbank10}. Each SFR study has chosen a particular star-formation rate indicator (or combination of indicators) from a variety of possible measures. The indicators associated with star formation cover a broad range of wavelengths of the electromagnetic spectrum from x-ray to radio. The most studied measures include UV luminosity, which indicates the amount of massive star formation within the composite stellar population of a galaxy \\citep{Heckman98}, and the luminosity of nebular emission lines such as H$\\alpha$ and [\\ion{O}{2}], which measure the amount of ionizing radiation from massive stars \\citep[hearafter M06]{Kennicutt98, Kewley04, Moustakas06}. Additionally, deep observations from \\emph{Spitzer} have tied the mid-IR continuum emission at $24\\micron$ to compact star formation regions in local galaxies \\citep{Calzetti07, Kennicutt09}.\n\nAll SFR diagnostics essentially measure the energy output of young stars either through direct observation of the UV flux or indirect observation of the ionizing luminosity through nebular emission lines or reprocessed emission from dust. Regardless of the diagnostic used, the SFR indicators have systematic effects which must be removed in order to give an accurate measure of the SFR. The primary physical effect that dominates systematic uncertainty in restframe UV\/optical observations is the amount of dust extinction in the host galaxy. Galaxy dust extinction attenuates the UV continuum or emission line luminosity and biases the indicators to lower SFRs \\citep{Kennicutt98, Kewley02, Kewley04}. Further, the amount of dust extinction within the star-forming galaxy population also correlates with galaxy luminosity \\citep{WH96}, stellar mass \\citep{Brinchmann04, Noeske07b, Elbaz07, Gilbank10}, and star-formation history \\citep{Noeske07a}, presumably due to increased metallicity and dust production from massive stars and mass-loss return to their ISMs. Producing accurate dust-corrected SFRs from UV continuum or nebular emission lines requires at minimum a measure of the dust extinction through diagnostics such as the UV\/optical slope \\citep{Calzetti94, Kong04} or the Balmer decrement (H$\\alpha$\/H$\\beta$) assuming case B recombination for typical galaxy environments \\citep{Calzetti01}. \n\nAdditional systematic uncertainties can also limit the SFR accuracy after dust extinction has been removed. For example, emission line luminosities from [\\ion{O}{2}] and [\\ion{O}{3}] depend on other nebular characteristics such as the chemical abundance and excitation states of the gas \\citep{KD02}. M06 estimates that even after an ideal Balmer-decrement dust correction is applied, [\\ion{O}{2}]-based SFRs have a lower error limit of 32\\% for galaxies with typical metallicities and can have considerably larger error for more extreme abundances. Further, the gas can be excited from sources that are not associated with actual star formation, such as the hard-ionizing background radiation created by AGN, which can be a dominant effect in quiescent galaxies \\citep{Yan06}. \n\nBeyond the UV and optical regime, IR measurements can be used to trace galaxy SFR \\citep{Kennicutt98, Daddi07}; UV photons from massive stars are absorbed by dust within the host galaxy ISM and re-emmited at IR wavelengths. IR-based SFRs are most accurate for heavily attenuated environments such as young starbursts where dust-corrections and measurements of the UV luminosity are difficult. However, IR observations tend to underestimate the dust luminosity in normal spirals where much of the UV radiation is unabsorbed.  IR luminosities can also have systematic variations due to differing dust geometries in the host galaxy or contamination from old stellar populations, particularly in red-sequence galaxies \\citep[hereafter S09]{Salim09}. Nevertheless, recent work has calibrated the SFR to IR luminosity in various bands for these dusty galaxies \\citep{Zhu08, Calzetti09, Rieke09} and calibrated the $24\\micron$ luminosity to H$\\alpha$ luminosity to provide a measure of both the SFR and account for typical dust attenuation encountered across a broad range of blue galaxies \\citep{Kennicutt09}. \n\nOne way to mitigate these various systematic effects is to obtain integrated spectral and photometric measurements over a wide wavelength range, providing constraints on the dust extinction and independent cross-checks in SFR. In S09, galaxy template SEDs were fit to measurements spanning FUV to $K$-band wavelengths available from the All-Wavelength Extended Groth Strip International Survey (AEGIS) \\citep{Davis07}. The S09 SFRs were based primarily on measurements of the UV luminosity and are combined with a Baysian analysis of stellar population synthesis models to fit the template SEDs. By simultaneously fitting across the panchromatic data and correcting for various astrophysical effects such as dust extinction, UV upturn in red galaxies, and SFR timescales, the dust-corrected SFRs are robust against large systematic errors that would typically limit SFR measurements at intermediate redshifts across galaxy types. We consider the SED SFRs fit to UV\/optical measurements to be the least biased measure of SFR in red galaxies, particularly when the UV upturn is taken into account \\citep{Han07}. These SED-based SFRs provide a reference for the SFR calibration presented here and will be discussed further in Section~\\ref{sec:Datasets}.\n\nThe goal of this work is to take advantage of these robust SED-fit SFRs and provide a SFR calibration for optically-selected galaxy samples where wide multi-wavelength coverage or emission line data may not be available. Specifically, we wish to develop a suitable calibration using SFR data in the AEGIS field and extend that calibration to $z\\sim1$ galaxies in the DEEP2 Redshift Survey (Davis, 2003) outside the AEGIS field. Because the SFR calibration is based on a large galaxy sample, the final means-tested SFR calibration will not be highly accurate for individual galaxies. However, it will reproduce the global SFR trends seen in the $z\\sim1$ AEGIS sample and allow us to confidently study the galaxy environment at these redshifts with respect to SFR. In \\S\\ref{sec:Datasets} we describe the volume-limited galaxy sample selected from the DEEP2 survey that is matched to the S09 AEGIS SFRs. In \\S\\ref{sec:SFRcorr}, we investigate which combination of DEEP2 measurements, observed [\\ion{O}{2}] luminosity, restframe $B$-band luminosity, or restframe color, that are most correlated with SFR and will therefore provide the best leverage in calibration. Our SFR model for these tests uses a weighted linear combination of observed parameters (not corrected for dust extinction), and we investigate which parameter combination delivers the most accurate SFR calibration for galaxies with both red and blue optical restframe colors (\\S\\ref{sec:MBUBcal} and \\S\\ref{sec:RedBlueCal}). In \\S\\ref{sec:OIISFRs}, we describe two empirical SFR estimators which use local [\\ion{O}{2}] luminosity measurements and correct the SFRs for systematic effects via M$_B$ or stellar mass. We then compare these diagnostics to our best-fit SFR calibration at intermediate redshifts, and we present our conclusions in \\S\\ref{sec:Conclusions}. Throughout this work, we adjust all masses and SFRs to a \\citet{Salpeter55} IMF and assume a $\\Lambda$CDM cosmology ($\\Omega_{M}=0.3$, $\\Omega_{\\Lambda}=0.7$, $h=0.7$). All magnitudes used in this study are in the AB system.\n\n\\section{Datasets}\n\\label{sec:Datasets}\n\nThe main sample for this work is drawn from the DEEP2 survey \\citep{Davis03} where the [\\ion{O}{2}] doublet emission lines have been measured with the DEIMOS spectrograph on Keck \\citep{Faber03}. The DEEP2 survey contains four fields that are separated across the Northern sky for year-round observation, and one of the fields (Field 1) overlaps the AEGIS survey in the Extended Groth Strip (EGS). Within the EGS, DEEP2 targets all galaxies to R$_{AB}<24.1$ and is limited to $z<1.4$, beyond which the [\\ion{O}{2}]  emission line falls outside the observed wavelength range (6500-9200\\AA). In the non-EGS fields, targets selected by DEEP2 are limited by design at a lower redshift of $z>0.7$ due to an optical $BRI$ color cut. The total DEEP2 sample with good redshifts ($>95\\%$ confidence, $z$ quality flag $>3$) is 31,656 galaxies. When available, the emission line equivalent widths in DEEP2 are measured using a nonlinear least-squares fit of a Gaussian to the emission line profiles. A line flux is then computed by measuring the continuum in a 20-60\\AA\\ window around the line and calibrating the flux to $K$-corrected $RI$ photometry. This method takes into account both the throughput and slit losses for DEEP2 galaxies of which nearly all have an effective radius less than the 1\" DEIMOS slit width (Weiner et. al, 2012, in preparation; description of method in \\citealp{Weiner07, Zhu09}).\n\nTo build a sample of $z\\sim1$ SFRs, we match the DEEP2 galaxy sample to available panchromatic data from the AEGIS survey. The matched AEGIS sample contains a total of 5345 objects and is a subset of the total available DEEP2 spectra in the EGS field. From this sample, we obtained SFR estimates from template SEDs which are fit to UV, optical, and IR $K$-band photometry in S09. The composite templates are constructed from stellar population synthesis models of \\citet{BC03} augmented with a two-component dust attenuation model of \\citet{CF00} to account for dust associated with various star-formation histories and the intergalactic medium. The templates used in S09 also take into account extreme blue horizontal branch stars which are thought to be responsible for the ``UV upturn\" seen in some early-type galaxies with old stellar populations \\citep[see][for details of the SED models]{Salim07}. For 45\\% of the S09 sample, the SED-fit SFRs have been compared with $24\\micron$ measurements for both red sequence and blue cloud galaxies. The total IR luminosities are inferred from the $24\\micron$ flux densities and galaxy redshifts by fitting to IR SED templates \\citep{DH02}. S09 found that SED SFRs of the blue galaxy population (NUV-R$<3$) were consistent with the measured IR luminosities to 0.3 dex RMS. The SFRs for red galaxies (NUV-R$>5$) were less well matched in IR luminosity due to the increasing contribution of light from old stellar populations in the IR measurements. Because S09 accounts for the latter systematic effect though template fitting,  the S09 SFRs based on UV SEDs are a better estimate of the true SFR than the $24\\micron$ measurements in red galaxies. \n\nThe stellar masses for our galaxy sample are computed from the \\citet{Bell03} color-M\/L relations, modified to DEEP2 redshifts through the \\citet{Weiner09} calibration.  The color-M\/L calibration is a linear set of equations that require several measured quantities, include M$_{B}$,  restframe (U-B) and (B-V) colors, and galaxy redshift.  Independently, we confirm that our calculated stellar masses from these relations agree with stellar masses derived from $K$-band measurements \\citep{Bundy06} to 0.3 dex RMS with no systematic offset. As an additional consistency check, we find that the color-M\/L stellar masses agree with those generated by the more sophisticated S09 SED-fitting techniques to within 0.3 dex. We use stellar masses from the M\/L-color calibration rather than stellar masses obtained from the SED fitting because we want the input parameters in our SFR calibration to be easily reproducible and independent from the SED fitting that yields our fiducial SFRs. \n\n\\subsection{Sample selection}\nTo produce a statistical sample volume-limited in M$_{B}$ for both red and blue galaxies, we restrict the matched DEEP2 \/ AEGIS galaxy sample to M$_{B}<-20$ for $0.74<z<1.0$, resulting in 1039 total galaxies. The selected redshift range and galaxy luminosity limit is similar to color-independent, volume-limited DEEP2 samples used in previous environmental studies \\citep{Cooper06, Gerke07, Coil08}. Approximately 17\\% of the selected red galaxies and 9\\% of the blue galaxies in our sample have no measured amount of L[\\ion{O}{2}].  The L[\\ion{O}{2}] measurement failures in blue galaxies are likely due to interactions of the emission line with bright sky emission lines, which happens at an nearly equal rate for both blue and red galaxies over this redshift range. The incompleteness due to sky line interaction is generally independent of the galaxy restframe $B$-band luminosity, color, and redshift. Removing the galaxies without measured [\\ion{O}{2}] reduces the sample size to 885 DEEP2 galaxies matched with AEGIS and restricts our SFR calibration analysis to galaxies with a SED-fit SFR limit of log($\\psi$)$>-1.0$  and [\\ion{O}{2}] line luminosity limit of log(L[\\ion{O}{2}])$>39.7$ at $z\\sim1$.\\footnote{The SFR $\\psi$ is in units of M$_{\\sun}$ yr$^{-1}$, and L[\\ion{O}{2}] is in ergs s$^{-1}$ throughout this study.}\n\nWhile blue galaxies in our sample generally have well detected L[\\ion{O}{2}] measurements ($>3\\sigma$), a significant fraction of the red galaxies with L[\\ion{O}{2}] measurements have large error due to inherently lower line luminosities. Restricting the L[\\ion{O}{2}] measurements to better than 3$\\sigma$ in DEEP2\nreduces the sample to 80\\% of the volume-limited blue galaxies and 27\\% of red galaxies. For comparison, \\citet{Yan06} found that 35\\% of all volume-limited red galaxies in SDSS had [\\ion{O}{2}] detected at a $3\\sigma$ level in EW. Because we are interested in testing SFR calibrations for a uniform sample with both L[\\ion{O}{2}]  and M$_B$, we wish not to limit the completeness in M$_B$, which is well measured for all galaxies, due to the higher L[\\ion{O}{2}] errors.  Therefore, we opt to keep all galaxies with measured L[\\ion{O}{2}] in our sample and weight the SFR fits by the measurement errors.  \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/OII-SEDsfr-Selection.pdf}\n\\caption{Comparison of the S09 SFRs for blue galaxies from $0.74<z<1.4$ with observed L[\\ion{O}{2}] in DEEP2. The plotted points are for both uncorrected (black triangles) and dust corrected (red diamonds) SFR values. The green stars denote the rejected S09 SFRs for heavily dust-extincted blue galaxies that require a factor $>20$ in dust correction. Note that the uncorrected SFRs are tightly correlated with observed L[\\ion{O}{2}], indicating that the dust correction introduces systematic uncertainty into the corrected SFR values.}\n\\label{fig:SFRreject}\n\\end{figure}\n\nWe place one final restriction on the sample to remove galaxies that have a large dust-corrected SFRs in the S09 data. Figure~\\ref{fig:SFRreject} shows the dust-corrected and uncorrected S09 SFRs in the matched DEEP2 \/ AEGIS sample as a function of observed [\\ion{O}{2}] luminosity from DEEP2.  We find that the uncorrected UV\/optical SED SFRs and observed L[\\ion{O}{2}] are tightly correlated due to similar dust extinction properties in the UV. After the dust correction is applied, a small population (13\\%) of galaxies have SFRs that are $>3\\sigma$ outliers from the mean dust-corrected distribution. These ``dusty\" star-forming galaxies have SFR corrections of more than a factor of 20 and are noted in S09 as primarily intermediate color (green valley) galaxies. Since we are attempting to calibrate SFRs at $z\\sim1$ with limited optical observations, we assume that the dust extinction for these obscured galaxies cannot be accurately measured on an individual galaxy basis (usually done through IR measurements) and remove them from the following analysis. While removing the outliers allows our calibrations to more closely track the ``typical\" galaxy in our sample, this underlying assumption restricts the SFR calibrations in this study to galaxy samples with moderate dust extinction and we acknowledge that our results will not be accurate for galaxies that require extraordinarily large dust corrections. The sample used in our SFR calibration is reduced to a total of 771 galaxies after we remove these galaxies.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SelectionHists.pdf}\n\\caption{Histograms of the selected and matched AEGIS \/ DEEP2 sample from $0.74<z<1.0$ (red, dashed) and the entire selected DEEP2 galaxy sample over the same redshift range in the EGS (black, solid).  The DEEP2 EGS sample histograms have been normalized to the total number in the AEGIS matched sample to show the relative shape of the distributions. The dotted vertical lines mark the median values for each sample. }\n\\label{fig:SampleHists}\n\\end{figure}\n\nTo confirm that our selected and matched AEGIS \/ DEEP2 sample is a representative subset of optically selected $z\\sim1$ galaxies, we compare the measured parameter distributions for the matched AEGIS \/ DEEP2 sample to the full EGS DEEP2 galaxy sample using the same selection cuts. This test must be restricted to the EGS field because the original DEEP2 target selection within the EGS was color-independent, whereas the three other DEEP2 fields were color selected.  Figure~\\ref{fig:SampleHists} shows the selected sample histograms for [\\ion{O}{2}] line luminosity, restframe B-band luminosity, stellar mass, and redshift. Because the matched and selected galaxy sample is roughly 10\\% of the entire DEEP2 EGS sample, the DEEP2 histograms have been normalized to the total matched AEGIS sample for easy comparison of the distributions. We find that the AEGIS \/ DEEP2 matched sample is an accurate subset of volume-limited, optically-selected galaxies in this redshift range and produces equivalent median values as the full EGS DEEP2 sample with log(L[\\ion{O}{2}])=41.5, M$_{B}=-20.8$, log(M$_{*}$\/M$_{\\sun}$)=10.3 and redshift $z=0.88$.\n\n\n\\section{SFR correlations}\n\\label{sec:SFRcorr}\n\n\\subsection{Trends in blue and red galaxies}\n\\label{sec:Galtrends}\nIn this study, we explore the systematic correlation between the dust-corrected SFR and three measured physical parameters: 1) observed [\\ion{O}{2}] line luminosity, 2) restframe $B$ magnitude, and 3) restframe (U-B) color. These parameters are based $only$ on optical measurements (i.e. no UV or IR data iss required) and therefore may be readily produced over wide fields where spectroscopic or photometric redshifts exist. The correlated trends of observed L[\\ion{O}{2}] and M$_B$ with SFR are shown in Figure~\\ref{fig:SFRtrends}, where we have separated the least-squares fits of both red and blue galaxies using the restframe color criteria of \\citet{Willmer06}. The linear fits are weighted by the measurement error in both the independent and dependent variables plus an additional ``intrinsic scatter\" term added in quadrature (see Section~\\ref{sec:FitMethod} for more detail). In the case of L[\\ion{O}{2}], we plot the measurements with $<2\\sigma$ significance using an open circle to indicate that they are less important in the fit.\n\nSeveral trends are apparent in the figure: \\emph{i}) (U-B) color clearly separates the red galaxies with lower SFRs from the blue galaxies with higher SFRs, \\emph{ii}) the red galaxies clearly have a greater scatter about the mean regression line in both observed L[\\ion{O}{2}] and M$_B$ relative to the blue galaxies, and \\emph{iii}) the red and blue galaxies have similar SFR slopes in M$_B$ but have different slopes in L[\\ion{O}{2}]. The similar SFR slope between red and blue galaxies and restframe M$_B$ might indicate that the galaxy $B$-band luminosity is a more accurate indicator of the corrected SFR regardless of galaxy color (albeit with a small offset between SFR zeropoints), but the SFR variation among the red galaxy population is too large for a definitive statement. Figure~\\ref{fig:SFRtrends} also shows that there is a large variation in SFR for blue galaxies of a fixed L[\\ion{O}{2}], demonstrating that observed [\\ion{O}{2}] line luminosities with small measurement errors may still have large systematic variations in corrected SFR due to the extinction effects. \n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[height=3.0in]{.\/SFRcorr-2parm.pdf}\n\\caption{Correlated trends in SFR with respect to observed L[\\ion{O}{2}] and M$_B$ for the restframe red (red circles) and blue (black triangle) galaxies in our matched sample. The SFRs used in this study are computed from UV-optical data fit to SED templates in S09 and the linear trends are weighted by the measurement errors. The points with error bars in the upper left of each plot show the median error for the color-separated galaxy measurements. In the case of L[\\ion{O}{2}], open symbols mark the galaxies where the [\\ion{O}{2}] luminosity has less than a $2\\sigma$ measurement in DEEP2. }\n\\label{fig:SFRtrends}\n\\end{figure*}\n\nThe difference in the SFR scatter of red and blue galaxies relative to their mean trends in M$_B$ and L[\\ion{O}{2}] occurs for several reasons. Red galaxies are typically quiescent in star formation and have inherently lower UV flux than that of the blue galaxies, leading to higher measurement uncertainty in red galaxy SED SFRs. Similarly, the measurements in L[\\ion{O}{2}] have higher measurement error in red galaxies simply because the emission line fluxes are fainter than blue galaxies. For red galaxies, we also expect the measurements are biased by [\\ion{O}{2}] emission not associated with star formation. \\citet{Yan06} showed that L[\\ion{O}{2}] is dominated by AGN emission for a significant fraction of the local red galaxy population and that this emission is not representative of true star formation. We will study the extent to which the observed L[\\ion{O}{2}] can be used as a direct proxy for quiescent galaxy SFR in the following section.\n\n\\subsection{Fitting Methodology}\n\\label{sec:FitMethod}\nTo obtain an empirical SFR calibration for our matched DEEP2 \/ AEGIS sample, we perform a weighted $\\chi^2$ fit using observed L[\\ion{O}{2}] and M$_B$, where the galaxies have been separated into red and blue subsamples by restframe (U-B) color. The galaxy SFR fits follow a simple linear combination model according to \n\\begin{equation}\n\\psi_{model}= C_0 + \\sum_{i=1}^n C_iP_i,\n\\label{eq:fitmodel}\n\\end{equation}\nwhere $n$ is the number of observed parameters for the linear model and $C_i$ is the scalar fit coefficient for the measured parameter $P_i$. The linear SFR model is then fit to the corrected S09 SFRs, $\\psi_{S09}$, by minimizing\n\\begin{equation}\n\\chi^2=\\sum_{k=1}^m \\frac{(\\psi_{model} - \\psi_{S09})_k^2}{(\\sigma_{o}^2+\\sigma_{i}^2)_k},\n\\label{eq:chisquare}\n\\end{equation}\nwhere $k$ is the index of $m$ sample galaxies. The $\\chi^2$ is weighted by the quadrature sum of the observed parameter variance, $\\sigma_{o}^2$, and the intrinsic scatter in the distribution, $\\sigma_{i}^2$. Typically, the error in $\\psi_{S09}$ dominates the measured variance when compared to the observed independent variables of M$_B$ and L[\\ion{O}{2}]. However, in the case of the low [\\ion{O}{2}] luminosity in red galaxies, it is important to include the luminosity error to properly weight the fit. The intrinsic scatter is computed through a bootstrap method by varying the $\\sigma_{i}$ term until the reduced $\\chi^2\/NDF=1$, ensuring that the computed fit coefficient errors are properly scaled and that the fit is unbiased given differing errors in the dependent and independent variables. We refer the reader to the appendix of \\citet{Weiner06} for details of weighting fits with intrinsic scatter.\n\nOnce the $\\chi^2$ fit is minimized, we subtract the best-fit SFR from the matched S09 SFR and measure the RMS of the residual errors. We also judge the performance of the calibration by measuring the correlation between residual SFR errors and the measured parameters of M$_B$, L[\\ion{O}{2}], and stellar mass. A large residual correlation is an indication that the calibration process has not fully characterized the SFR distribution and that a better calibration may be sought. Because stellar mass and restframe colors are correlated by color-M\/L relation, we expect that any residual systematic correlated error in color from our SFR calibration will also produce a residual error correlated with the calculated stellar masses.\n\n\\subsection{Color-separated Empirical SFR Calibration}\n\\label{sec:RedBlueCal}\nAs discussed in Section~\\ref{sec:Galtrends} and shown in Figure~\\ref{fig:SFRtrends}, we initially separate the matched sample by restframe blue and red color and fit the linear model of Equation~\\ref{eq:fitmodel} with  L[\\ion{O}{2}] and M$_B$ as the independent variables. We report the best-fit coefficients and parameter errors of this calibration in Table~\\ref{tab:SFRresults}. In Figure~\\ref{fig:OIIbluered}, we show that by fitting only L[\\ion{O}{2}] as an independent parameter in each galaxy type, we can achieve a SFR fit that produces residual errors with an RMS of 0.27 dex scatter for blue galaxies, 0.46 dex for red galaxies, and 0.31 dex for all galaxies in the sample. This result suggests that if restframe colors and [\\ion{O}{2}] measurements are available, L[\\ion{O}{2}] can be a decent tracer of the average SFR in large galaxy samples even though AGN activity may be contributing to the [\\ion{O}{2}] flux in individual red galaxies.  However, there remains a residual systematic correlation along M$_B$ after calibration, as indicated by the Pearson correlation coefficient of $r^2=0.18$. The residual trend indicates that the full behavior of SFR has not been calibrated with L[\\ion{O}{2}] alone and that some improvement can still be made to reduce the scatter in the calibrated SFRs. This residual correlation is likely due to the degenerate effects of dust reddening on the observed [\\ion{O}{2}] luminosity, $B$-band galaxy luminosity, and restframe color.\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=6.5in]{.\/resid-2color-o2wt-O2.pdf}\n\\caption{SFR fit residuals using L[\\ion{O}{2}] as the independent parameter in the SFR calibration. Fits for red (circles) and blue (triangles) galaxies are derived separately. The points with error bars in the upper left of each plot shows the median errors for each galaxy color, and the Pearson correlation coefficient values for the combined sample are shown in the upper right. In the case of stellar mass, the error on log(M$_*$\/M$_{\\sun}$) is calculated from the RMS of the DEEP2 masses to the fit S09 masses and therefore accounts for the systematic error in estimating stellar mass from the color-M\/L relation.}\n\\label{fig:OIIbluered}\n\\end{figure*}\n\nSimilarly, we obtain a SFR calibration by fitting the linear trends in M$_B$ separately for red and blue galaxies (top plot of Fig~\\ref{fig:MBOIIbluered}) and the fit coefficients are tabulated in Table~\\ref{tab:SFRresults}. The blue galaxy SFRs are highly correlated with $B$-band luminosity (Fig.~\\ref{fig:SFRtrends}), which provides greater leverage in the calibration over L[\\ion{O}{2}]. The fit SFR residual errors from the M$_B$ fit show a reduced scatter for blue galaxies (0.21 dex), an increased scatter for red galaxies (0.54 dex), and no change in the scatter for the combined sample (0.30 dex). There is less correlation in M$_B$ (by construction) and stellar mass, but we observe that the SFR residual errors are now correlated with L[\\ion{O}{2}], particularly for the red galaxies.  Again, the residual correlations in L[\\ion{O}{2}] after calibrating with M$_B$ likely results from the differing amount of dust extinction inherent in these two observed, uncorrected measures.\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=6.5in]{.\/resid-2color-o2wt-MB.pdf}\\\\\n\\vspace{0.1in}\n\\includegraphics[width=6.5in]{.\/resid-2color-o2wt-O2-MB.pdf}\\\\\n\\caption{SFR fit residuals using only M$_B$ (top) and both L[\\ion{O}{2}] and M$_B$ (bottom) as the independent parameters in the SFR calibration. Fits for red and blue galaxies are derived separately.}\n\\label{fig:MBOIIbluered}\n\\end{figure*}\n\n\n\\begin{table*}[tb]\n\\caption{Summary of best fit coefficients for SFR calibration separated by galaxy color}\n\\begin{center}\n\\begin{tabularx}{0.89\\textwidth}{cccccccc}\n\\hline\n\\hline\n\\multicolumn{2}{c}{Fit Parameters\\T} & \\multicolumn{3}{c}{Blue Galaxies} & \\multicolumn{3}{c}{Red Galaxies} \\\\\n$P{_1}$ & $P_{2}$ & $C_{0,b}$ & $C_{1,b}$  &  $C_{2,b}$ &  $C_{0,r}$ & $C_{1,r}$  & $C_{2,r}$  \\\\\n\\hline\nL[\\ion{O}{2}]$^a$\\T & --~~&~~0.722 $\\pm$ 0.028 & 0.470 $\\pm$ 0.038 & --~~&~~0.351 $\\pm$ 0.040 & 0.763 $\\pm$ 0.077 & --\\\\[3pt]\nM$_{B}$$^b$ & --~~&~~1.099 $\\pm$ 0.008 & -0.356 $\\pm$ 0.013 & --~~&~~0.065 $\\pm$ 0.064 & -0.304 $\\pm$ 0.083 & --\\\\[3pt]\nL[\\ion{O}{2}]~~\\B& M$_B$~~&~~1.003 $\\pm$ 0.024 & 0.143 $\\pm$ 0.031 & -0.346 $\\pm$ 0.014~~&~~0.302 $\\pm$ 0.055 & 0.738 $\\pm$ 0.081 & -0.088 $\\pm$ 0.069 \\\\[3pt]\n\\hline\n\\hline\n\\end{tabularx}\n\\end{center}\n\\hspace{0.8in}$^a$ The [\\ion{O}{2}] luminosity has a zeropoint of L[\\ion{O}{2}]=$10^{41}$ ergs s$^{-1}$ cm$^{-2}$. \n\\vspace{-0.00in}\n\n\\hspace{0.8in}$^b$ The $B$-band luminosity has a zeropoint of M$_B$=-21. \\\\\n\\label{tab:SFRresults}\n\\end{table*}\n\\begin{table}[thb]\n\\caption{Intrinsic scatter and residual errors for SFR calibrations separated by galaxy color$^a$}\n\\begin{center}\n\\begin{tabularx}{0.98\\columnwidth}{cccccccc}\n\\hline\n\\hline\n\\multicolumn{2}{c}{Parameters\\T}  & \\multicolumn{2}{c}{Intrinsic Scatter} & $\\Delta_{mean}$ & \\multicolumn{3}{c}{Residual RMS} \\\\\n$P{_1}$ \\B& $P_{2}$ &~~blue & red & all & blue & red & all \\\\\n\\hline\nL[\\ion{O}{2}]\\T & -- &~~0.22 & 0.29 & 0.00 & 0.27 & 0.46 & 0.31 \\\\[3pt]\nM$_{B}$ & -- &~~0.15 & 0.49 & -0.01 & 0.21 & 0.54 & 0.30 \\\\[3pt]\nL[\\ion{O}{2}] & M$_B$ &~~0.13 &  0.29 & -0.01 & 0.21 & 0.45 & 0.27 \\\\[3pt]\n\\hline\n\\hline\n\\end{tabularx}\n\\end{center}\n\\vspace{0.04in}$^a$ SFR errors are in units of dex.\n\\label{tab:SFRresids}\n\\end{table}\nWe can also combine both M$_B$ and L[\\ion{O}{2}] measurements to reduce the residual SFR correlations below what can be obtained by fitting each independent parameter alone. The combined calibration fit achieves an RMS scatter of 0.21, 0.45, and 0.27 dex for the blue, red, and combined galaxy sample, respectively (c.f. bottom of Fig.~\\ref{fig:MBOIIbluered}).  The inclusion of L[\\ion{O}{2}] with M$_B$ actually provides constraint on the \\emph{red} galaxies, reducing the scatter from 0.54 to 0.45 dex. This result may seem counterintuitive since the measurement error of L[\\ion{O}{2}] is larger for red galaxies than blue galaxies and AGN are known to contribute significantly the [\\ion{O}{2}] luminosity.  However, it is worth noting that galaxies with red restframe colors will span larger range of dust extinction values and SFR, from elliptical galaxies with low dust content and low dust-corrected SFR values to late-type galaxies with heavy dust extinction and high dust-corrected SFR. Even with significant measurement error, we find that the additional information from L[\\ion{O}{2}] to M$_B$ helps provide a valuable constraint on the dust-corrected SFR in red galaxies.  The best fit coefficients for all three color-separated SFR calibrations presented in this section are summarized in Table~\\ref{tab:SFRresults} and the residual RMS scatter and mean residual offsets are reported in Table~\\ref{tab:SFRresids}.\n\n\\subsection{Constructing a single SFR calibration}\n\\label{sec:MBUBcal}\nWhile separating the red and blue galaxy populations enables a SFR calibration across galaxy types with minimal residual correlation and scatter, it inherently requires all three measurement parameters considered in this study (L[\\ion{O}{2}] , M$_B$, and restframe color) to utilize the relation. Ideally, one would like a single, simple relation to calibrate all galaxies to the SED SFRs while minimizing the number of required measurement parameters.  As shown in Section~\\ref{sec:Galtrends}, the red and blue galaxy SFRs clearly have differing trends in the observed parameters and the intrinsic color dependence cannot be ignored. For example, a SFR calibration using L[\\ion{O}{2}] for all galaxies \\emph{without} color information produces a 35\\% increase in the SFR fit residual scatter with large residual correlations in L[\\ion{O}{2}], M$_B$, and M$_{*}$. As an alternative to separating galaxies by restframe color, we can add restframe color information directly into the SFR calibration and fit the entire galaxy sample simultaneously. Including the restframe color as an independent parameter in the fit produces a continuous SFR calibration between galaxy types and helps break the degeneracies between the galaxy color, SFR, and dust reddening. \n\n\\begin{figure*}[thb]\n\\centering\n\\includegraphics[width=6.5in]{.\/resid-o2wt-MB-UB.pdf}\\\\\n\\vspace{0.1in}\n\\includegraphics[width=6.5in]{.\/resid-o2wt-MB-UB-UB2.pdf}\\\\\n\\caption{SFR fit residuals using M$_{B}$ and (U-B) as the dependent parameter in the SFR calibration for all galaxies. The top figure shows the fit residuals from a linear fit in both independent variables, while the bottom figure shows the result of a second-order polynomial in (U-B).}\n\\label{fig:MBUB}\n\\end{figure*}\n\nFigure~\\ref{fig:SFRtrends} demonstrated that the dust-corrected SED SFRs are highly correlated with galaxy $B$-band luminosity, which provides excellent calibration leverage for blue galaxy SFRs and marginal leverage for red galaxy SFRs. We also observed that the correlation between SFR and M$_B$ between red and blue galaxies may have similar slopes with only a difference in the zeropoint of the relation. Motivated by these observed trends, we use the form of Eq.~\\ref{eq:fitmodel} and perform a fit with both restframe M$_B$ and (U-B) color for all galaxies; the resulting residual errors are shown in Figure~\\ref{fig:MBUB} (top plot).  We find that the SFR behavior of the full galaxy sample is not well described by a simple linear fit in M$_B$ and (U-B) alone.  The SFR calibration produces residual errors with $\\sigma_{all}=0.33$ dex scatter for the full sample and large residual error correlated in M$_B$ and stellar mass. The residual correlations are apparently non-linear, particularly in the case of stellar mass, and therefore are not fully characterized by the Pearson correlation coefficient alone.  \n\nBecause the fit residuals in the M$_{B}$ and (U-B) color calibration are seemingly non-linear, we modify our linear SFR calibration model (Eq.~\\ref{eq:fitmodel}) to include a second-order term in (U-B) (bottom plot of Figure~\\ref{fig:MBUB}).  The inclusion of this non-linear color term produces an better fit to the dust-corrected SED SFRs for the full galaxy sample. The final RMS scatter in the SFR fit residuals are  $\\sigma_{blue}=0.19$ dex, $\\sigma_{red}=0.47$ dex, and $\\sigma_{all}=0.26$ dex. The fit residuals from this calibration are \\emph{statistically} better than all other methods considered in this study,  and the calibration does not require L[\\ion{O}{2}] to be included in the fit as was necessary in \\S\\ref{sec:RedBlueCal}. Further, the SFR calibration leaves no significant correlations in M$_B$, M$_{*}$, \\emph{and} L[\\ion{O}{2}] (r$^{2}<0.032$). \n\nThe reason for the poor fit using only linear terms in M$_B$ and (U-B) alone can be seen by studying the behavior of the galaxy SFRs in the M$_B$ - (U-B) color plane.  Figure~\\ref{fig:S09SFRcontours} shows the S09 SFRs matched to sources within our DEEP2 sample from $0.74<z<1.4$; the overplotted contours correspond to lines of average SFR with log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$.  The SED-fit SFR behavior are decidedly non-linear in M$_B$ and (U-B) color. For the blue galaxies, the SFRs primarily increase with M$_{B}$ with only a weak dependence on (U-B) color. This SFR behavior in the blue cloud is in agreement with measured SFRs at $z=1.4$ \\citep{Weiner09} and is consistent with the assumption that the UV slope and (U-B) colors are correlated, therefore associating M$_B$ with UV flux. However, as star formation trends from active to quiescent around (U-B)=1, the SFR contours rapidly transition to much lower SFRs and have a weaker dependence on M$_{B}$. \n\\begin{figure}[tb]\n\\centering\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-S09-contour.pdf}}\n\\caption{The M$_{B}$ - (U-B) plane with SFRs measured in the AEGIS survey (S09) and matched to the DEEP2 catalog from $0.74<z<1.4$.  The contour lines correspond to average SFRs of log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$.}\n\\label{fig:S09SFRcontours}\n\\end{figure}\n\nIn addition to the differing SFR trends in M$_{B}$ between red and blue galaxies, the differential extinction between independent fit parameters, such as the uncorrected $B$-band luminosity and observed [\\ion{O}{2}] luminosity, can couple the fit parameters such that independent linear coefficients are not an accurate description of the global SFR trends.  Using an additional second-order (U-B) color term in the SFR calibration allows an extra degree of freedom within the fit to simultaneously correct for both intrinsic galaxy color between red and blue galaxies and dust reddening effects.  By separating the degenerate color effects with an additional color fit parameter, we can effectively reduce the systematic coupling in the SFR fit residuals. \n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=6.5in]{.\/resid-o2wt-MB-UB-BV.pdf}\\\\\n\\caption{SFR fit residuals using MB, (U-B), and (B-V) as the independent parameters in the SFR calibration for all galaxies. The residual errors are tabulated in Table~\\ref{tab:SFRtotresults}. }\n\\label{fig:MBUBBV}\n\\end{figure*}\nKnowing that additional color information can produce an accurate SFR calibration, we also generate a fit to the SED SFRs using M$_B$, (U-B), and (B-V) colors (see Figure~\\ref{fig:MBUBBV}). The fit residuals in this ``multicolor\" calibration are very similar to the second-order (U-B) calibration, with $\\sigma_{all}=0.27$ dex scatter and small residual correlations in all three measurement parameters ($r^{2}<0.045$).  Again, we see that an additional color parameter allows the model more freedom to decouple the intrinsic galaxy color and the effects of dust in the SFR. The combined sample SFR calibration coefficients and fit residual errors for each of these fit models is tabulated in Table~\\ref{tab:SFRtotresults}.\n\nWe note that it is not entirely surprising that a useful empirical SFR calibration can be achieved using observed $B$-band luminosity and multiple restframe colors; these measures are similar to the quantities that construct the SED-fit SFRs. Either the multicolor or second-order (U-B) calibration can be used to produce statistically similar SFR calibrations within our volume-limited ($0.74<z<1$) DEEP2 \/ AEGIS sample. However, we will see in the following section the multicolor calibration is better behaved in the $M_{B}$ - (U-B) plane when extended to the full $0.74<z<1.4$ DEEP2 sample.\n\n\\subsection{Trends in M$_B$ - (U-B) plane}\n\\label{sec:contours}\nAn additional comparison diagnostic of our SFR calibration models can be observed in the M$_{B}$ - (U-B) plane. Figure~\\ref{fig:2colorSFRcont} shows the SFRs calibrated from M$_{B}$ and L[\\ion{O}{2}] using the bimodal restframe color-selected calibration detailed in Section~\\ref{sec:RedBlueCal}. The galaxy data points are taken for the full DEEP2 sample from $0.74<z<1.4$ and are color-coded with $\\Delta$log($\\psi$)=0.3 dex SFR bins. The overplotted solid lines correspond to contours of constant SFR with log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$ from the S09 AEGIS sample (c.f. Figure~\\ref{fig:S09SFRcontours}). In general, the color-separated SFR calibration tracks the S09 contours well, particularly in the blue galaxies where the SFR dependence is strongly correlated with M$_B$ and independent of restframe color. However, the drastic split between the galaxy colors near the green valley is clearly not representative of the true SFR transition at (U-B)$\\sim$1. Additionally, bright galaxies just below the (U-B)=1 line have overpredicted SFRs relative to the S09 contours while bright red galaxies just redward of the line have underpredicted SFRs.  Figure~\\ref{fig:S09SFRcontours} shows that there is some ``mixing\" of SFRs for the brightest galaxies with optically red restframe colors, indicating that galaxies with (U-B)$>1$ at these redshifts can either be old early-type galaxies with low SFR or heavily-reddened late-type galaxies with a young stellar population and high SFR.\n \n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-2color-MB-O2cal-contour-th.pdf}\n\\caption{The diagnostic M$_{B}$ - (U-B) plane with SFRs calibrated using the M$_{B}$ - L[\\ion{O}{2}]  calibration split between red and blue galaxies (dotted line). The solid lines correspond to constant contours of SFR generated from the matched S09 AEGIS galaxy sample with log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$ and are plotted here for easy reference to the original calibration data. See Section~\\ref{sec:RedBlueCal} for details of the color-split SFR calibration.}\n\\label{fig:2colorSFRcont}\n\\end{figure}\n\nAs developed in Section~\\ref{sec:MBUBcal}, the SFR transition from red to blue galaxies can be \\emph{approximated} by a second-order polynomial fit in M$_{B}$ and (U-B) color that is valid within the range of SFRs and errors considered in our matched DEEP2 \/ AEGIS sample. The left hand plot of Figure~\\ref{fig:SFRcontours} shows the calibrated SFRs for the full DEEP2 sample from $0.74<z<1.4$ using the [M$_{B}$, (U-B), (U-B)$^2$] calibration parameter model (see Table~\\ref{tab:SFRtotresults}). Compared to the S09 SFR contours, the global SFR trends are reproduced by this calibration and SFRs smoothly transition between galaxies of differing restframe color. However, there is some discrepancy for faint blue galaxies; the SFRs become increasingly dependent on (U-B) color for (U-B)$<0.5$ and M$_{B}>-20$. This unphysical behavior is a consequence of the second-order term in the fit where there are few matched S09 data points to constrain the SFR calibration. \n\\begin{figure*}[tb]\n\\centering\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-MB-UB-UB2cal-contour-th.pdf}}\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-MB-UB-BVcal-contour-th.pdf}}\n\\caption{The diagnostic M$_{B}$ - (U-B) plane with calibrated $0.74<z<1.4$ DEEP2 SFRs using M$_{B}$ and a second-order polynomial in (U-B) restframe colors (left) and the SFR calibration using M$_{B}$, (U-B), and (B-V) restframe colors (right).}\n\\label{fig:SFRcontours}\n\\end{figure*}\n\nThe right hand plot of Figure~\\ref{fig:SFRcontours} shows the linear SFR calibration using the multicolor calibration with M$_B$, (U-B), and (B-V) color used as independent fit parameters. Similar to the second-order M$_{B}$ - (U-B) SFR calibration, the multicolor calibration breaks some of the degeneracy in intrinsic galaxy color and dust reddening and therefore produces a smooth transition in (U-B) color between red and blue galaxies. However, the multicolor calibration remains insensitive to restframe colors for (U-B)$<1$ and therefore does not produce an unphysical turnover in the SFR dependence for faint blue galaxies.  We also note that multicolor SFR calibration produces increased SFRs for the brightest red galaxies (M$_{B}<-22.5$ and (U-B)$>1$). The S09 data shows that bright star-forming galaxies with heavily-reddened restframe colors do exist in this region. However, because the transition from high to low SFR is a strong function of (U-B) color, the SFRs in this region will not be highly accurate on an individual galaxy basis, but the SFR trend for the entire galaxy population will on average be correct. \n\nWhile the two calibrations used in Figure~\\ref{fig:SFRcontours} are statistically equivalent in comparison to the S09 matched sample, we opt for the linear multicolor calibration to produce better qualitative agreement for faint blue galaxies. Figure~\\ref{fig:SFRfit} shows the SFR produced by the multicolor calibration matches the ``true\" SED SFRs very well, with a residual scatter of $\\sigma_{all}=0.28$ dex for all galaxies with no offset in the mean of the distribution. In fact, the scatter is slightly reduced relative to the residual scatter seen in the original volume-limited calibration generated between $0.74<z<1$. The additional high redshift galaxies from $1<z<1.4$ are predominantly blue galaxies which have a more accurate SFR calibration than red galaxies through $M_{B}$.  The additional blue galaxies reduce the average residual scatter for the combined galaxy sample. \n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-SFRfit.pdf}\n\\caption{SFR fit residuals using MB, (U-B), and (B-V) as the independent parameters in the SFR calibration for all DEEP2 galaxies from $0.74<z<1.4$.}\n\\label{fig:SFRfit}\n\\end{figure}\n\n\\section{Comparison to local [\\ion{O}{2}] SFR calibrations}\n\\label{sec:OIISFRs}\nThe [\\ion{O}{2}] emission line has long been studied as a diagnostic for star formation in active galaxies \\citep{Gallagher89, Hogg98, Kennicutt98, Teplitz03}. To remove the effects of dust extinction and oxygen abundance \/ metallicity, [\\ion{O}{2}]  is typically calibrated to H$\\alpha$ emission in the local universe, where H$\\alpha$ luminosity is directly proportional to the number of ionizing photons available from massive stars.  The H$\\alpha$ luminosity is also more robust against systematic error than other emission lines as it is generally insensitive to the ISM metallicity, experiences less dust attenuation, and absorption due to the underlying stellar continuum can be easily modeled \\citep{MK06}. A calibration of the [\\ion{O}{2}]\/H$\\alpha$ ratio can be readily performed to good accuracy at low redshifts where both lines can be simultaneously observed \\citep[M06]{Kewley04}. At higher redshifts, the H$\\alpha$ line typically becomes difficult to measure as it moves to NIR wavelengths, leaving estimates of the galaxy SFR dependent on assumptions of the mean [\\ion{O}{2}]\/H$\\alpha$ ratio, extinction, and metallicity of lower-redshift samples. \n\nIn this section, we study two empirical [\\ion{O}{2}]-based SFR calibrations measured in the local universe and compare the results to the multicolor $z\\sim1$ SFR calibration developed in \\S\\ref{sec:MBUBcal}. We reiterate that these empirical [\\ion{O}{2}] SFR diagnostics are measured from large galaxy samples and are intended to represent the mean trends of the galaxy population. Applying the SFR calibrations on a individual galaxy-by-galaxy basis may lead to large variations and systematic error when compared to individual galaxy SFR diagnostics. Additional measures of dust extinction and metallicity should be obtained to accurately calculate an individual galaxy SFR.  \n\n\\subsection{[\\ion{O}{2}] SFR calibration through M$_B$}\n\\label{sec:M06cal}\nTo mitigate the systematic effects that limit [\\ion{O}{2}] as a SFR tracer, M06 uses an empirical calibration of the [\\ion{O}{2}] SFR against the restframe $B$-band magnitude, which is correlated with both the galaxy extinction and metallicity. M$_B$ therefore serves as a proxy for the variation in the [\\ion{O}{2}]\/H$\\alpha$ ratio from the local average value.  The M06 calibration produces good agreement (0.28 dex RMS scatter) with H$\\alpha$-based SFRs at low redshift. M06 performed a comparison with galaxies at higher redshift ($z\\sim0.7-1.5$) and found that the dust-corrected [\\ion{O}{2}]\/H$\\alpha$ ratio remains roughly constant, although brighter galaxies tend to have lower ratios (see Fig. 22 of M06). The trend of lower observed [\\ion{O}{2}]\/H$\\alpha$ ratios for brighter and more massive star-forming galaxies has been further confirmed with NIR measurements and tied to increased reddening through independent measurements of the extinction \\citep{Weiner07}. To obtain SFRs for our sample, we follow the recommendation of M06 and interpolate between the median SFR values (their Table 2) using L[\\ion{O}{2}] as a tracer of the SFR and M$_B$ to correct the SFR for dust extinction. The absolute $B$-band magnitudes assumed in M06 are in the Vega system, but we convert them to AB magnitudes for consistency within this study.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/OIIvsMB2.pdf}\n\\caption{The observed [\\ion{O}{2}] emission line luminosity as a function of restframe $B$-band luminosity for the local sample measured in M06 (black triangles) and for DEEP2 galaxies from $0.74<z<1.4$ (gray contours). The vertical and horizontal lines mark the volume limits for blue DEEP2 galaxies at $z=1$ (dot-dashed) and $z=1.4$ (dotted). The solid (black) line is a weighted linear fit to the M06 data in the plotted range, and the dashed line is the same fit with a L[\\ion{O}{2}] zeropoint shifted by +0.35 dex.}\n\\label{fig:MBvsOII}\n\\end{figure}\nWe stress that one should not $directly$ apply the locally-derived M06 SFR calibration using both observed L[\\ion{O}{2}] and restframe M$_{B}$ measured for galaxies at intermediate redshift. Applying the local SFR relations and dust corrections to galaxies at intermediate redshifts can introduce considerable systematic uncertainty into the SFR calculation. It is well known that the $B$-band luminosity function has evolved since $z=1$ \\citep{Willmer06, Faber07}, where the mean M$_B$ for integrated galaxy samples has dimmed by $Q\\approx1.3$ magnitudes between $z=1$ and $z=0$. To show the discrepancy between the M06 local SFR calibration and that of a similar L[\\ion{O}{2}] and M$_{B}$ relation constructed at $z\\sim1$, we plot in Figure~\\ref{fig:MBvsOII} the raw M06 data for star-forming local galaxies (M$_B<-18$) and $0.74<z<1.4$ blue galaxies from DEEP2. The blue lines in the figure correspond to the approximate volume limits of the DEEP2 measurements at $z<1$ (dot-dashed line) and $z<1.4$ (dotted line). For reference, we plot the weighted linear fit to the M06 data set with a black solid line, showing the trend of the galaxy continuum luminosity to be correlated with the observed [\\ion{O}{2}] luminosity and therefore SFR for blue galaxies. Clearly, the DEEP2 data extends to higher luminosities in both L[\\ion{O}{2}] and M$_{B}$ than M06, making the application of a SFR calibration at intermediate redshift heavily dependent on an extrapolation from the local values. It is also clear that the DEEP2 blue galaxy samples lie well above the mean local trend, however it is not immediately apparent whether the DEEP2 luminosities are brighter due to an evolution in M$_{B}$, L[\\ion{O}{2}], or both.\n\nAssuming \\emph{no} evolution in M$_{B}$, the dashed line in Fig.~\\ref{fig:MBvsOII} shows that the aggregate sample of DEEP2 blue galaxies has a $\\sim0.35$ dex increase in observed [\\ion{O}{2}] luminosity, and one would incur a similar bias in the mean SFR of $z\\sim1$ galaxies. A more accurate calibration of the SFR at $z\\sim1$ using the M06 data should at minimum account for the evolution in galaxy luminosity per fixed stellar mass. We therefore correct the measured values M$_B$ from DEEP2 galaxies to include a dimming factor of $Q=1.3$ magnitudes per unit redshift before applying the local M06 calibration. This simple luminosity correction factor implicitly assumes that both the [\\ion{O}{2}]\/H$\\alpha$ ratio and stellar mass (or the mass-metallicity relation) has evolved less than the galaxy luminosity, an assumption in agreement with recent studies (\\citealp{KK04, Zahid11}; J. Moustakas et. al, 2011, in preparation).  \n\\\\\n\\subsection{[\\ion{O}{2}] SFR calibration through Stellar Mass}\n\\label{sec:G10cal}\nRecent efforts to use L[\\ion{O}{2}] as a SFR diagnostic have focused on incorporating stellar mass into the calibration. Stellar mass has been shown to be correlated with many galaxy properties, including luminosity, metallicity, dust extinction, and the SFR in SDSS galaxies \\citep{Brinchmann04, Blanton09, Garn10}. The stellar mass estimate serves as a proxy for these many complex factors and therefore accounts for the systematic differences in the observed [\\ion{O}{2}]\/H$\\alpha$ ratio.  Following the formalism of \\citet[hereafter G10]{Gilbank10}, we construct an empirical calibration of the [\\ion{O}{2}] SFR from \\citet{Kennicutt98} using\n\\begin{equation}\n{\\rm SFR_{L[O~\\Rmnum{2}]} }= \\frac{10^{0.4A_{H\\alpha}}}{{[O~\\Rmnum{2}]_{obs}}\/[H{\\alpha}]_{obs}}\\frac{L[O~\\Rmnum{2}]}{1.4\\times10^{-41} {\\rm erg~s}^{-1}}\n\\label{eq:Gilbank}\n\\end{equation}\nin log(M$_{\\sun}$ yr$^{-1})$, where {[O~\\Rmnum{2}]$_{obs}$\/[H${\\alpha}$]$_{obs}$ is the average reddened line flux ratio between [\\ion{O}{2}] and H$\\alpha$ for the total sample of galaxies and $A_{H\\alpha}$ is the extinction of H$\\alpha$ line emission due to dust.  In the case of DEEP2 galaxies, we use a ratio of observed [\\ion{O}{2}]\/H$\\alpha$=0.48 measured indirectly through [\\ion{O}{2}]\/H$\\beta$ from $0.73<z<0.87$ and H$\\beta$\/H$\\alpha$ from $0.33<z<0.39$ at M$_H$=-21 \\citep{Weiner07}.  The value of  [\\ion{O}{2}]\/H$\\alpha$ is in agreement with local measurements found in \\citet{Kennicutt92}, but we caution that this value is not singular and can vary greatly depending on the galaxy sample selection. For the dust attenuation $A_{H\\alpha}$, we use the mass-dependent dust extinction model of \\citet{GB10}, who performed a PCA analysis of SDSS DR7 galaxies using SFR, stellar mass, and metallicity to produce a best fit model of the H$\\alpha$ absorption. \\citeauthor{GB10} found that a simple polynomial model in stellar mass best described the dust extinction using  \n\\begin{equation}\nA_{H\\alpha} = 0.91+ 0.77 X + 0.11X^2 - 0.09 X^3, \n\\end{equation}\nwhere $X$ = log(M$_{*}$\/$10^{10}$ M$_{_{\\sun}}$). Implicitly, this model accounts for deviations from the observed [\\ion{O}{2}]\/H$\\alpha$ ratio and the observed differences in metallicity and luminosity that are correlated with the dust extinction. The data used to construct the model cover a large dynamic range of stellar mass for ``normal\" galaxies, $8.5<{\\rm log}(\\rm M_{*}\/\\rm M_{sun})<11.5$, which spans the majority of the $z\\sim1$ DEEP2 sample. \n\n\\subsection{SFR Comparisons}\n\\label{sec:SFRcomp}\n\n\\subsubsection{SED-fit SFRs compared to L[\\ion{O}{2}] SFRs}\nTo understand how the M06 and G10 calibrations compare to actual SFRs measured at $z\\sim1$, we again use the S09 SFR data from our volume-limited DEEP2 \/ AEGIS sample. Because these calibrations were derived from local samples of star-forming galaxies, we restrict the sample to only galaxies with blue restframe colors and convert all SFRs to a Salpeter IMF for these comparisons.  \n\n\\label{sec:S09vsOII}\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/OIISFRdiffs.pdf}\n\\caption{Comparison of the S09 SFRs to calculated SFRs using L[\\ion{O}{2}]  measurements and empirical dust corrections from the current epoch. The dashed (red) lines show the weighted linear least squares fit of the data points and the solid (black) line is the 1:1 correspondence. The top plot shows the S09 SFRs compared to the M06 empirical SFR relation using L[\\ion{O}{2}] and M$_{B}$. The M06 calibration has been adjusted by $Q=1.3$ magnitudes per unit redshift in M$_B$ to account for the luminosity evolution at $z\\sim1$. The dotted (red) line shows the M06 calibration before the luminosity evolution is taken into account.  The bottom plot shows the S09 SFRs and the G10 SFRs which uses L[\\ion{O}{2}] and stellar mass for calibration. }\n\\label{fig:OIISFRdiff}\n\\end{figure}\nFigure~\\ref{fig:OIISFRdiff} shows the comparison between the L[\\ion{O}{2}]-based SFR calibrations (M06 and G10) and the S09 SFRs from $0.74<z<1.0$. The dashed lines in the figure show the linear least squares fit of the sample weighted by the estimated errors in both measures of the SFR.  The upper panel of the plot shows the M06 calibration produces a 0.29 dex RMS residual scatter relative to the S09 data with a mean offset of -0.02 dex. The RMS scatter measured from the M06 calibration extrapolated to our sample is 0.08 dex larger than the residual error generated from the blue galaxy SFR calibration using L[\\ion{O}{2}] and M$_{B}$ (see \\S\\ref{sec:RedBlueCal}). These results indicate that extrapolating the SFR behavior through L[\\ion{O}{2}] and M$_{B}$ from the local relations to $z\\sim1$ can contribute to the residual scatter, which is not unexpected given the systematic uncertainty in the galaxy dust corrections and average $B$-band luminosity evolution. We also plot a dotted line to represent the best fit trend of the M06 calibration without accounting for the luminosity evolution between $z=0.88$ and the current epoch. The difference between the SFR distribution means with and without luminosity evolution results in a bias of 0.25 dex. \n\nThe bottom panel of Figure~\\ref{fig:OIISFRdiff} shows the comparison of the G10 calibrated SFRs to the S09 SFRs.  We find that the SFR distributions agree with 0.27 dex RMS scatter and have a mean offset of -0.06 dex. While the mean offset is larger than the M06 calibration, the scatter is slightly smaller in the G10 case. Remarkably, the relation between stellar mass and $A_{H\\alpha}$ in Equation~\\ref{eq:Gilbank} produces a relatively accurate calibration at $z\\sim1$ without having to account for any evolution. This result indicates that the dust extinction correction required for [\\ion{O}{2}]-based SFRs is better correlated and more constant with stellar mass than restframe M$_{B}$. \n\nWe note that the slopes of the weighted linear fits shown in Figure~\\ref{fig:OIISFRdiff} are not well matched to a 1:1 correspondence. This is due in part to the large amount of scatter ($\\sim0.28$ dex) and SFR measurement error relative to the overall SFR range probed ($\\sim1.5$ dex) in the volume-limited blue galaxy sample. In order to better test the overall accuracy of the [\\ion{O}{2}] SFR calibration slope, we must increase the sample size and probe a wider range of SFRs.\n\n\\subsubsection{Multicolor SFRs compared to L[\\ion{O}{2}] SFRs}\n\\label{sec:MYSFRvsOII}\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SFRdiffs.pdf}\n\\caption{Comparison of the M06 and G10 L[\\ion{O}{2}]-based SFR calibrations to the multicolor SFR calibration for $0.74<z<1.4$ DEEP2 galaxies. The dashed (red) lines show the weighted linear regression of the data points and the solid (black) line is the 1:1 correspondence. Similar to Figure~\\ref{fig:OIISFRdiff}, the M06 calibration has been adjusted by $Q=1.3$ magnitudes per unit redshift in M$_B$ to account for the luminosity evolution at $z\\sim1$. }\n\\label{fig:SFRdiff}\n\\end{figure}\nWe now compare the multicolor SFR calibration presented in \\S\\ref{sec:MBUBcal} to the M06 and G10 SFR calibrations applied at intermediate redshifts, where we have extended the calibrations to include both red and blue galaxies from $0.74<z<1.4$ with measured L[\\ion{O}{2}].  As mentioned in \\S\\ref{sec:Galtrends}, determining a SFR for a red galaxy based on the observed [\\ion{O}{2}] luminosity can have large systematic uncertainty. However, there are some applications for which determining a rough SFR for large galaxy samples can still produce interesting global measures, e.g in measuring how the SFR correlates with environment \\citep{Cooper06}. It is also interesting to check for systematic biases in [\\ion{O}{2}] SFR calibrations when they are applied to red galaxies at $z\\sim1$. \n\nAs shown in the top panel of figure~\\ref{fig:SFRdiff}, we find that the M06 empirical calibration from L[\\ion{O}{2}] and M$_B$ is a reasonable match to our multicolor SED-fit SFR calibration \\emph{after} a correction is applied for luminosity evolution in M$_B$ (see \\S\\ref{sec:M06cal}). The residual median offset between the SFR calibrations is 0.05 dex with a RMS residual error or 0.26 dex after the offset is removed.  The residual RMS scatter has been reduced slightly over the previous volume-limited blue galaxy sample because we have added in many star-forming blue galaxies at $z>1$. The SFR calibration slope differs by 11\\% per decade, perhaps due to additional systematic bias such as an evolution in the [\\ion{O}{2}]\/H$\\alpha$ ratio between $z=1$ and the current epoch. Alternatively, the disagreement could be due to differential luminosity evolution as our assumption of a constant $Q=1.3$ mag\/$z$ corresponds primarily to $L_{*}$ galaxies.  \n\nThe bottom panel of Figure~\\ref{fig:SFRdiff} shows that the G10 calibration, modified to correct dust extinction in L[\\ion{O}{2}]  through the stellar mass, is an excellent match to our multicolor SFR calibration. The mean of the SFR distributions have a very small offset of 0.01 dex and the slope of the SFRs are in agreement to better than 2\\% per decade.  Overall, both the M06 and G10 SFR calibrations seem to be generally consistent with our multicolor SFR calibration, an encouraging result considering the multicolor calibration is based only on broadband optical colors and restframe magnitude. We do not observe a significant systematic bias in the red galaxy SFRs estimated through either L[\\ion{O}{2}]-based SFR calibrations relative to our multicolor SFR calibration.\n\nFinally, we compare the SED-fit SFR contours in the restframe color-magnitude plane to those generated from the M06 and G10 calibrations. As shown in Figure~\\ref{fig:OIISFRcontours}, both L[\\ion{O}{2}]-based calibrations produce similar behavior as the S09 SFR contours in the color-magnitude plane, but the G10 calibration is slightly better at reproducing the color-independent SFR behavior seen in the blue cloud and therefore is in better agreement with our multicolor calibration for blue galaxies (c.f. Figure~\\ref{fig:SFRdiff}). Combined with the previous comparison to the SED-fit SFRs, these results show that our SFR calibration using only restframe optical colors and $B$-band luminosity are at least as accurate as using a local L[\\ion{O}{2}]-based SFR relation and extrapolating it to $z\\sim1$.  \n\n\\begin{figure*}[thb]\n\\centering\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-M06cal-contour-th.pdf}}\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-G10cal-contour-th.pdf}}\n\\caption{The M$_{B}$ - (U-B) plane with calibrated $0.74<z<1.4$ DEEP2 SFRs using the M06 empirical SFR calibration (left) and the G10 empirical SFR calibration (right) based on local SFR measurements.  The contour lines correspond to the S09 SFRs of log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$.}\n\\label{fig:OIISFRcontours}\n\\end{figure*}\n \n\\section{Conclusions}\n\\label{sec:Conclusions}\nWe present a calibration of the star formation rate for DEEP2 galaxies using SED-fit SFRs from AEGIS and observed correlations in [\\ion{O}{2}], M$_{B}$, and restframe colors. Within a volume-limited sample of galaxies between $0.74<z<1.0$, we formulate and minimize a weighted $\\chi^2$ fit with respect to these parameters  to determine a simple, robust empirical SFR calibration. This calibration is useful for estimating the global SFR behavior for a large sample of optically-measured galaxies at $z\\sim1$, but it is not recommended for use on an individual galaxy basis. We summarize our main results as follows. \n\n\\begin{enumerate}\n\\item{Red sequence and blue cloud galaxies at $z\\sim1$ have large differences in their SFRs. As such, an accurate SFR calibration must minimally separate the galaxies into two populations based on restframe color {\\it or} include the restframe color into the calibration. }\n\\item{We find that a single linear relation in M$_{B}$ and restframe (U-B) and (B-V) colors breaks the observed degeneracies in the intrinsic galaxy color and dust reddening and results in a SFR calibration within the accuracy of the available data. The calibration produces fit residual errors with 0.27 dex RMS scatter for the total sample ($\\sigma_{blue}=0.20$ dex, $\\sigma_{red}=0.47$ dex) and no significant systematic residual correlations with L[\\ion{O}{2}], M$_{B}$, or the color-M\/L stellar mass. Additional constraint from observed [\\ion{O}{2}] luminosity does not improve the calibration accuracy, likely due to the fact that the SED-fit SFRs are highly correlated with the galaxy UV\/optical continuum for blue galaxies and L[\\ion{O}{2}] is less correlated with star formation in red galaxies.} \n\\item{The multicolor SFR calibration developed from the matched DEEP2 \/ AEGIS sample can be extended to all DEEP2 galaxies from $0.74<z<1.4$ without a significant increase in SFR residual scatter. The fit SFR behavior in the M$_B$ - (U-B) color plane accurately reflects the average galaxy population in the S09 data without introducing significant discontinuities between red and blue galaxies.} \n\\item{We compare the SED-fit SFRs to two calibrations (M06 and G10) which use local [\\ion{O}{2}] luminosity measurements as a primary indicator of SFR and apply empirical dust corrections through observed M$_B$ or stellar mass. We find that the RMS scatter between the calculated SFRs are 0.29 dex (M06) and 0.27 dex (G10), which is $\\sim0.08$ dex larger than the residual scatter generated from the best SFR calibrations in this study for the same volume-limited blue galaxy sample.}\n\\item{The empirical L[\\ion{O}{2}] - M$_B$ SFR calibration found in M06 should be corrected for the luminosity evolution in star-forming galaxies when applied at $z\\sim1$.  Applying the locally-measured M06 relation at intermediate redshift and not correcting for the mean galaxy luminosity evolution will overestimate the $z=1$ SFRs by 0.3 dex or more.  An L[\\ion{O}{2}]-based SFR calibration that uses stellar mass to correct for the dust extinction (G10) does not require a large correction for evolution, indicating that stellar mass is better correlated with extinction than $M_{B}$ and has less evolution from $z\\sim1$ to the current epoch.}\n\\item{Finally, we compare the multicolor SFR calibration recommended in this study to the L[\\ion{O}{2}]-based SFR calibrations from M06 and G10. We find that our empirical calibration is in good agreement with both of the L[\\ion{O}{2}] calibrations for all galaxies within the SFR calibration accuracy. We conclude that the multicolor SFR calibration fit from the AEGIS SED-fit SFRs is equivalent in accuracy to the local L[\\ion{O}{2}] SFR calibrations extrapolated to $z\\sim1$.}\n\\end{enumerate}\n\nWe gratefully thank the DEEP2 and AEGIS team for making their data public. Funding for the DEEP2 survey has been provided by NSF grants AST95-09298, AST-0071048, AST-0071198, AST-0507428, and AST-0507483 as well as NASA LTSA grant NNG04GC89G. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The DEEP2 team and Keck Observatory acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community and appreciate the opportunity to conduct observations from this mountain. This study makes use of data from AEGIS, a multiwavelength sky survey conducted with the $Chandra$, $GALEX$, $Hubble$, Keck, CFHT, MMT, $Subaru$, Palomar, $Spitzer$, VLA, and other telescopes and supported in part by the NSF, NASA, and the STFC. This work was also sponsored in part by the United States Department of Energy under contract No. DE-AC02-05CH11231. \n\n\\include{SFRcal.bbl}\n\n\\clearpage\n\\begin{sidewaystable*}[p]\n\\caption{Summary of best fit coefficients, coefficient error, and fit residual errors for the full sample SFR calibration}\n\\begin{center}\n\\begin{tabularx}{0.98\\textwidth}{llllccccccccccccccc}\n\\hline\n\\hline\n\\multicolumn{4}{c}{Fit Parameters\\T} & \\multicolumn{5}{c}{Coefficients} & \\multicolumn{5}{c}{Coefficient Error} & Intrinsic Scatter$^a$ & \\multicolumn{4}{c}{Fit Residuals$^a$ } \\\\\n$P_{1}$\\B & $P_{2}$ & $P_{3}$ & $P_{4}$~~&~~$C{_0}$ & $C{_1}$  & $C{_2}$ & $C_{3}$ & $C_{4}$~~&~~$\\sigma_{0}$ & $\\sigma_{1}$ & $\\sigma_{2}$ & $\\sigma_{3}$ & $\\sigma_{4}$~~&~~$\\sigma_{i}$~~&~~$\\sigma_{red}$ & $\\sigma_{blue}$ & $\\sigma_{all}$ & $\\Delta_{mean}$ \\\\\n\\hline\nM$_{B}$\\T & (U-B) & -- & -- & 2.039 & -0.408 & -1.436 & -- & --  & 0.043 & 0.020 & 0.054 & -- & -- & 0.27 & 0.48 & 0.25 & 0.33 & -0.01\\\\[3pt]\nM$_B$ & (U-B) & (U-B)$^2$ & -- & 0.381 & -0.424  & 2.925 & -2.603 & -- & 0.082 &  0.015 & 0.206 & 0.122 & -- & 0.19 &  0.47 & 0.19 & 0.26 & -0.02\\\\[3pt]\nM$_B$ \\B& (U-B) & (B-V) & -- & 1.199 & -0.431 & -11.251 & 15.446  & -- & 0.055 & 0.016 & 0.526 & 0.821 & -- & 0.20 & 0.47 & 0.20 & 0.27 & -0.01\\\\[3pt]\n\\hline\nL[\\ion{O}{2}]\\T & (U-B) & -- & -- & 0.536 & 0.705 & 0.005  & -- & -- & 0.064 & 0.035 & 0.062 &  -- & -- & 0.24 & 0.45 & 0.28 & 0.33 & -0.02\\\\[3pt]\nL[\\ion{O}{2}] & (U-B) & (U-B)$^2$ & -- & -0.290 & 0.599 & 2.50 & -1.601 & --  & 0.097 & 0.034 & 0.245 & 0.154 & -- & 0.24 &  0.46 & 0.28 & 0.33 & -0.01\\\\[3pt]\nL[\\ion{O}{2}] \\B& (U-B) & (B-V) & -- & 0.187 & 0.629 & -5.818 & 8.954 & -- & 0.068 & 0.034 & 0.641 & 0.977 & -- & 0.22 & 0.44 & 0.27 & 0.31 & -0.01\\\\[3pt]\n\\hline\nL[\\ion{O}{2}]\\T & M$_B$ & (U-B) & -- & 1.203  & 0.420 & -0.284 & -0.637  & -- & 0.073 & 0.037 & 0.0202 & 0.073  & -- & 0.20 & 0.44 & 0.23 & 0.30 & -0.02\\\\[3pt]\nL[\\ion{O}{2}] & M$_B$ & (U-B) & (U-B)$^2$ & 0.270 & 0.157 & -0.381  & 2.721 & -2.325 & 0.079 & 0.034 & 0.017 & 0.193 & 0.128 & 0.14  & 0.44 & 0.20 & 0.26 & 0.00\\\\[3pt]\nL[\\ion{O}{2}] \\B& M$_B$ & (U-B) & (B-V) & 0.921 & 0.217 & -0.367 & -9.309 & 12.990 & 0.066 & 0.0.035 & 0.018 & 0.562 & 0.833 & 0.16 & 0.43 & 0.20 & 0.26 & -0.01\\\\[3pt]\n\\hline\n\\hline\n\\end{tabularx}\n\\end{center}\n\\hspace{0.8in}$^a$ SFR error is in units of dex.\n\\label{tab:SFRtotresults}\n\\end{sidewaystable*}\n\\clearpage\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\begin{tabular}{ccc}\n    \\includegraphics[width = 0.27\\linewidth]{img\/ill1}  &\n    \\includegraphics[width = 0.27\\linewidth]{img\/ill2} & \n    \\includegraphics[width = 0.27\\linewidth]{img\/ill3}  \\\\         \n    \n    {\\small (a)} &  {\\small (b)} & {\\small (c)} \n  \\end{tabular}\n\\caption{Illustration of the guidance process. (a) Calibration Wizard proposes the next best pose based on the previous calibration results. (b) The camera should be moved towards the proposed target pose. (c) When the camera is close enough to the suggested pose, the system acquires an image and then proposes a next pose. Demo code:~\\url{https:\/\/github.com\/pengsongyou\/CalibrationWizard}.}\n\\label{fig:wizard-process}\n\\end{figure*}\n\n\nCamera calibration is a prerequisite to many methods and applications in computer vision and photogrammetry, in particular for most problems where 3D reconstruction or motion estimation is required.\nIn this paper we adopt the popular usage of planar calibration objects, as introduced\\footnote{Planar targets were used before, but essentially in combination with motion stages, in order to effectively generate 3D targets \\cite{tsai1987versatile,lenz1988techniques,wei1993complete,li1994camera}} by \\cite{sturm1999plane,zhang2000flexible} and made available through OpenCV~\\cite{opencv} and Matlab~\\cite{bouguet2000matlab}, even though our approach can be directly applied to 3D calibration objects too.\n\nIt is well known that when using planar calibration targets, the accuracy of the resulting calibration depends strongly on the poses used to acquire images.\nFrom the theoretical study of degenerate sets of camera poses in \\cite{sturm1999plane,zhang2000flexible}, it follows for example intuitively that it is important to vary the orientation (rotation) of the camera as much as possible during the acquisition process.\nIt is also widely known to practitioners that a satisfactory calibration requires images such that the target successively covers the entire image area, otherwise the estimation of radial distortion and other parameters usually remains suboptimal.\nWe have observed that inexperienced users usually do not take calibration images that lead to a sufficiently accurate calibration.\n\nSeveral efforts have been done in the past to guide users in placing the camera.\nIn photogrammetry for instance, the so-called network design problem was addressed, through an off-line process: how to place a given number of cameras such as to obtain an as accurate as possible 3D reconstruction of an object of assumed proportions \\cite{Mason1995a,Olague2002a}.\nOptimal camera poses for camera calibration have been computed in~\\cite{ricolfe2011camera}, however only for constrained camera motions and especially, only for the linear approach of \\cite{zhang2000flexible}, whereas we consider the non-linear optimization for calibration.\nAlso, these poses are difficult to realize, even for expert users.\nThe ROS~\\cite{quigley2009ros} monocular camera calibration toolbox provides text instructions so that users can move the target accordingly.\nMore recently, some cameras, such as the ZED stereo system from StereoLabs, come with software that interactively guides the user to good poses during the calibration process; these poses are however pre-computed for the particular stereo system and this software cannot be used to calibrate other systems, especially monocular cameras.\n\nIn this paper we propose a system that guides the user through a simple graphical user interface (GUI) in order to move the camera to poses that are optimal for calibrating a camera. Optimality is considered for the bundle adjustment type non-linear optimization formulation for calibration. For each new image to be acquired, the system computes the optimal pose, i.e.\\ which adds most new information on intrinsic parameters, in addition to that provided by the already acquired images.\nThe most closely related works we are aware of are \\cite{richardson2013aprilcal,rojtberg2018efficient}. They also suggest next best poses to the user. However, unlike ours, they are both strategy-based methods, where suggestions are selected from a fixed dataset of pre-defined poses, which may not be enough for various camera models or calibration targets.\nIn our approach, each new suggested pose results from a global optimization step.\nFurthermore, we propose a novel method for incorporating the uncertainty of corner point positions rigorously throughout the entire pipeline, for calibration but also next best pose computation.\nOur approach is not specific to any camera model; in principle, any monocular camera model can be plugged into it, although tests with very wide field of view cameras need to be done.\n\nThe paper is organized as follows. Section~\\ref{sec:method} describes the theory and mathematical details of Calibration Wizard. Section~\\ref{sec:autocorrelation} shows how to incorporate corner point uncertainty in the process.\nExperiments are reported in Section~\\ref{sec:results}, followed by conclusions in Section~\\ref{sec:conclusions}.\n\n\\section{Methodology}\n\\label{sec:method}\n\nOur goal is to provide an interactive guidance for the acquisition of good images for camera calibration.\nAn initial calibration is carried out from a few (typically 3) freely taken images of a calibration target.\nThe system then guides the user to successive next best poses, through a simple GUI, cf.\\ Fig~\\ref{fig:wizard-process}.\nThe underlying computations are explained in the following subsections.\n\n\\subsection{Calibration formulation}\n\\label{method-formulation}\n\nLet the camera be modeled by two \\textbf{local projection functions} $x=q_x(\\Theta, S), y=q_y(\\Theta, S)$ that map a 3D point $S$ given in the local camera coordinate system, to the image coordinates $x$ and $y$.\nThese functions depend on intrinsic parameters $\\Theta$ (assumed constant across all images).\nFor example, the standard 3-parameter pinhole model consisting of a focal length $f$ and principal point $(u,v)$, i.e.\\ with $\\Theta=(f, u, v)^\\top$, has the following local projection functions (a full model with radial distortion is handled in supplementary material, section 3):\n\\begin{equation}\nq_x(\\Theta,S) = u + f \\frac{S_1}{S_3}\n\\enspace\\enspace\\enspace\nq_y(\\Theta,S) = v + f \\frac{S_2}{S_3}\n\\label{eq:pinhole}\n\\end{equation}\n\nLet camera pose be given by a 6-vector $\\Pi$ of extrinsic parameters.\nWe use the representation $\\Pi=(t_1,t_2,t_3,\\alpha,\\beta,\\gamma)^\\top$, where $t=(t_1,t_2,t_3)^\\top$ is a translation vector and the 3 angles define a rotation matrix as the product of rotations about the 3 coordinate axes: $R = R_z(\\gamma) R_y(\\beta) R_x(\\alpha)$.\n$R$ and $t$ map 3D points $Q$ from the world coordinate system to the local camera coordinate system according to:\n\\begin{equation}\n\tS = RQ + t\n\\end{equation}\nOther parameterization may be used too, e.g.\\ quaternions.\n\nA camera with pose $\\Pi$ is thus described by two \\textbf{global projection functions} $p_x$ and $p_y$:\n\\begin{align}\n\tp_x(\\Theta, \\Pi, Q) &= q_x(\\Theta, RQ+t) = q_x(\\Theta, S)\\\\\n\tp_y(\\Theta, \\Pi, Q) &= q_y(\\Theta, RQ+t) = q_y(\\Theta, S)\n\\end{align}\nSince a planar calibration target is used, the 3D points $Q$ are pre-defined and their corresponding $Z$ coordinates $Q_3$ are set to $0$.\nWe now consider $m$ images of a target consisting of $n$ calibration points.\nInputs to the calibration are thus the image points $(x_{ij}, y_{ij})$ for $i=1\\cdots m$ and $j=1\\cdots n$, which are detected by any corner detector, e.g.\\ the OpenCV \\verb+findChessboardCorners+ function~\\cite{opencv}.\nFor ease of explanation, we assume here that all points are visible in all images, although this is not required in the implementation.\n\nOptimal calibration requires a non-linear simultaneous optimization of all intrinsic and extrinsic parameters (bundle adjustment).\nThis comes down to the minimization of the geometric reprojection error \\cite{hartley2003multiple}:\n\\begin{equation}\n   \\min_{\\Theta, \\{\\Pi_i\\}} \\sum_{i,j} \\left( x_{ij} - p_x(\\Theta, \\Pi_i, Q_j) \\right)^2 + \\left( y_{ij} - p_y(\\Theta, \\Pi_i, Q_j) \\right)^2\n   \\label{eq:cost}\n\\end{equation}\nUsually, local non-linear least square optimizers are used, such as Levenberg-Marquardt.\nOur system is independent of the optimizer used; all it requires is the computation, at the found solution, of the partial derivatives of \\eqref{eq:cost}, see next.\n\n\\subsection{Computation of next best pose}\n\\label{ss:next_pose}\n\nWe suppose that we have already acquired $m$ images and estimated intrinsic parameters and poses from these, by solving \\eqref{eq:cost}.\nThe goal now is to compute the next best pose; the objective is to reduce, as much as possible, the expected uncertainty on the estimated intrinsic parameters.\n\nLet us consider the Jacobian matrix $J$ of the cost function~\\eqref{eq:cost}, evaluated at the estimated parameters.\n$J$ contains the partial derivatives of the cost function's residuals, i.e.\\ of terms\n$\\hat{x}_{ij} = x_{ij} - p_x(\\Theta, \\Pi_i, Q_j)$ and $\\hat{y}_{ij} = y_{ij} - p_y(\\Theta, \\Pi_i, Q_j)$.\n$J$ contains one row per residual.\nIts columns are usually arranged in groups, such that the first group contains the partial derivatives\nwith respect to the intrinsic parameters $\\Theta$, and subsequent groups of columns, the derivatives\nrelative to extrinsic parameters of the successive images.\nThe highly sparse form of $J$ is thus as follows (we assume here that there are $k$ intrinsic parameters):\n\\begin{equation}\n\\hspace{-0.7em}\nJ = \\begin{pmatrix}\nA_1 & B_1 & 0 & \\cdots & 0 \\\\\nA_2 & 0 & B_2 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_m & 0 & 0 & \\cdots & B_m\n\\end{pmatrix}\n\\label{eq:jacob2}\n\\end{equation}\n\\begin{equation*}\n    A_i = \\begin{pmatrix}\n\\frac{\\partial \\hat{x}_{i1}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{x}_{i1}}{\\Theta_k} \\\\\n\\frac{\\partial \\hat{y}_{i1}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{y}_{i1}}{\\Theta_k} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial \\hat{x}_{in}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{x}_{in}}{\\Theta_k} \\\\\n\\frac{\\partial \\hat{y}_{in}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{y}_{in}}{\\Theta_k}\n\\end{pmatrix}\n\\enspace\nB_i = \\begin{pmatrix}\n\\frac{\\partial \\hat{x}_{i1}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{x}_{i1}}{\\Pi_{i,6}} \\\\\n\\frac{\\partial \\hat{y}_{i1}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{y}_{i1}}{\\Pi_{i,6}} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial \\hat{x}_{in}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{x}_{in}}{\\Pi_{i,6}} \\\\\n\\frac{\\partial \\hat{y}_{in}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{y}_{in}}{\\Pi_{i,6}}\n\\end{pmatrix}\n\\end{equation*}\nwhere $A_i$ are matrices of size $2n \\times k$, containing the partial derivatives of residuals with respect to $k$ intrinsic parameters,\nwhereas $B_i$ are matrices of size $2n \\times 6$, containing the partial derivatives with respect to extrinsic parameters.\nNow, the so-called information matrix is computed as $J^\\top J$:\n\\begin{equation}\nJ^\\top J = \\begin{pmatrix}\n\\sum\\limits_i A_i^\\top A_i & A_1^\\top B_1 & A_2^\\top B_2 & \\cdots & A_m^\\top B_m \\\\\nB_1^\\top A_1 & B_1^\\top B_1 & 0 & \\cdots & 0 \\\\\nB_2^\\top A_2 & 0 & B_2^\\top B_2 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nB_m^\\top A_m & 0 & 0 & \\cdots & B_m^\\top B_m\n\\end{pmatrix}\n\\label{eq:infoMat}\n\\end{equation}\nLike $J$, $J^\\top J $ is highly sparse.\nImportantly, its inverse $(J^\\top J)^{-1}$ provides an estimation of the covariance matrix of the estimated intrinsic and extrinsic parameters.\n\nFor camera calibration, we are only interested in the covariance matrix of the intrinsic parameters, i.e.\\ the upper-left $k\\times k$ sub-matrix of $(J^\\top J)^{-1}$.\nDue to the special structure of $(J^\\top J)^{-1}$, it can be computed efficiently.\nLet\n\\begin{align*}\n    U &= \\sum_i A_i^\\top A_i \\\\\n    V &= \\text{diag}\\left(B_1^\\top B_1, \\cdots, B_m^\\top B_m\\right) \\\\\n    W &= \\begin{pmatrix} A_1^\\top B_1 & \\cdots & A_m^\\top B_m \\end{pmatrix}\n\\end{align*}\nThen,\n\\begin{equation}\n    J^\\top J = \n        \\begin{pmatrix}\n            U & W\\\\ W^\\top & V\n        \\end{pmatrix}\n\\end{equation}\nand as described in~\\cite{hartley2003multiple}, the upper-left sub-matrix of $(J^\\top J)^{-1}$ is given by $\\Sigma = (U-W V^{-1} W^\\top)^{-1}$, i.e.\\ the inverse of a $k\\times k$ symmetric matrix.\n\nLet us now return to our goal, determining the next best pose $\\Pi_{m+1}$.\nThe outline of how to achieve this is as follows.\nWe extend the Jacobian matrix in Eq.~\\eqref{eq:jacob2} with a part corresponding to an additional image, whose pose is\nparameterized by $\\Pi_{m+1}$.\nThe coefficients in $A_{m+1}$ and $B_{m+1}$, are thus functions of $\\Pi_{m+1}$.\nNaturally, $\\Pi_{m+1}$ is also implicitly embedded in the intrinsic parameter's covariance matrix associated with this extended system.\nTo reduce the uncertainty of the calibration, we wish to determine $\\Pi_{m+1}$ that makes $\\Sigma$ as ``small'' as possible.\nInspired by~\\cite{haner2015phd}, we choose to minimize the trace of this $k \\times k$ matrix.\nSince we wish to compute the next best pose within the entire 3D working space, we use a global optimization method.\nOur experiments suggest that simulated annealing~\\cite{van1987simulated} or ISRES~\\cite{runarsson2005search} work well for this small optimization problem~\\footnote{We did not consider the stopping criterion in the current version, but one could simply stop our method when the relative residual of the trace of the covariance matrix mentioned above is smaller than a threshold.}.\nEspecially the latter works in interactive time. \n\n\nNote that the computation of the partial derivatives used to build the $A_i$ and $B_i$ matrices can be done very efficiently using the chain rule.\nFurther, computation of $\\Sigma$ for different trials of $\\Pi_{m+1}$ can also be done highly efficiently by appropriate pre-computations of the parts of matrices $U$ and $W V^{-1} W^\\top$ that do not depend on $\\Pi_{m+1}$. See more details in the supplementary material, section 2. \n\n\\section{Taking into Account the Uncertainty of Corner Points}\n\\label{sec:autocorrelation}\n\n\\begin{figure*}[t]\\centering\n{\\renewcommand{\\arraystretch}{0.4}\n\\begin{tabular}{cccccc}\n\\multirow{3}{*}[1.2cm]{\\includegraphics[width = 0.33\\linewidth]{img\/graph_blur_255.png}} &\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/30_0.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/50_0.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/70_0.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/90_0.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/110_0.png} \\\\\n& \\includegraphics[width = 0.07\\linewidth, frame]{img\/30_1.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/50_1.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/70_1.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/90_1.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/110_1.png} \\\\\n& \\includegraphics[width = 0.07\\linewidth, frame]{img\/30_2.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/50_2.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/70_2.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/90_2.png}&\n\\includegraphics[width = 0.07\\linewidth, frame]{img\/110_2.png}\\\\\n& $30^\\circ$& $50^\\circ$& $70^\\circ$& $90^\\circ$& $110^\\circ$\n\\end{tabular}}\n\\caption{Uncertainty of corner position as a function of opening angle and blur.\nLeft: plot of the first eigenvalue of the autocorrelation matrix, over opening angle and for different blur levels (Gaussian blur for $\\sigma=0,1,2,3$).\nRight: corners for different opening angles and blur levels ($\\sigma=0,1,2$) and computed $95\\%$ confidence level uncertainty ellipses (enlarged 10$\\times$ for display).}\n\\label{fig:corner_uncertainty}\n\\end{figure*}\n\n\n\n\nSo far, we have not used information on the uncertainty of corner point positions:\nin Eq.~\\eqref{eq:cost}, all residuals have the same weight (equal to $1$).\nIdeally, in any geometric computer vision formulation, one should incorporate estimates of\nuncertainty when available.\nIn the following, we explain this for our problem, from two aspects: first for the actual\ncalibration, i.e.\\ the parameter estimation in Eq.~\\eqref{eq:cost}.\nSecond, more originally, for computing the next best pose.\n\n\\subsection{Corner Uncertainty in Calibration}\n\nConsider a corner point extracted in an image; the uncertainty of its position can be\nestimated by computing the autocorrelation matrix $C$ for a window of a given size around\nthe point (see for instance \\cite{harris1988a}).\nConcretely, $C$ is an estimate of the inverse of the covariance matrix of the corner position.\nNow, let $C_{ij}$ be the autocorrelation matrix for the $j$th corner in the $i$th image.\nThe $C_{ij}$ can be incorporated in the calibration process by inserting the block-diagonal\nmatrix composed by them, in the computation of the information matrix of Eq.~\\eqref{eq:infoMat}:\n\\begin{equation}\nH = J^\\top \\text{diag}(C_{11}, C_{12}, \\cdots, C_{1n}, C_{21}, \\cdots, C_{mn}) J\n\\label{eq.corrected_information_matrix}\n\\end{equation}\n\nThis uncertainty-corrected information matrix can then be used by Gauss-Newton or\nLevenberg-Marquardt type optimizers for estimating the calibration (for other optimizers,\nthe autocorrelation matrix may have to be used differently) as well as for the quantification\nof the uncertainty of the computed intrinsic parameters.\n\n\\subsection{Next Best Pose Computation}\n\nThe second usage of corner uncertainty concerns the computation of the next best pose.\nIn particular, for each hypothetical next pose that we examine, we can project the points of the calibration target to the image plane, using this hypothetical pose and the current estimates of intrinsic parameters.\nThis gives the expected positions of corner points, if an actual image were to be taken from that pose.\nImportantly, what we would like to compute in addition, is the expected uncertainty of corner extraction, i.e.\\ the uncertainty of the corner positions extracted in the expected image.\nOr, equivalently, the autocorrelation matrices computed from the expected pixel neighborhoods around these corner points.\nIf we are able to do so, we can plug these into the estimation of the next best pose, by inserting the expected autocorrelation matrices in Eq.~\\eqref{eq:infoMat} the same way as done in Eq.~\\eqref{eq.corrected_information_matrix}.\n\nBefore explaining how to estimate expected autocorrelation matrices, we describe the benefits of this approach.\nIndeed, we have found that without doing so, the next best pose is sometimes rather extreme,\nwith a strong grazing viewing angle relative to the calibration target.\nThis makes sense in terms of pure geometric information contributed by such a pose,\nbut is not appropriate in practice, since with extreme viewing angles image corners are\nhighly elongated: they may be difficult to extract in the actual image and also, their uncertainty is very large in one direction.\nWhile this may be compensated by using images acquired from poses with approximately\nperpendicular viewing directions one from another, it is desirable and indeed more principled to fully integrate\ncorner uncertainty right from the start in the computation of the next best pose.\n\nLet us now explain how to compute expected autocorrelation matrices of corner points, for a hypothetical next pose.\nThis is based on a simple reasoning.\nThe overall shape of an image corner (in our case, a crossing in a checkerboard pattern), is entirely represented by the ``opening angle'' of the corner.\nWhat we do is to precompute autocorrelation matrices for synthetic corners for the entire range of opening angles, cf.\\ Fig.~\\ref{fig:corner_uncertainty}: the top row on the right shows ideal corners generated by discretizing continuous black-and-white corners, for a few different opening angles.\nFor each of them, the autocorrelation matrix (cf.\\ \\cite{harris1988a}) was computed; as mentioned, its inverse is an estimate of the corner position's covariance matrix.\nThe figure shows the plots of $95\\%$ uncertainty ellipses derived from these covariance matrices (enlarged 10 times, for better visibility).\nSuch ideal corners are of course not realistic, so we repeat the same process for images blurred by Gaussian kernels of different $\\sigma$ (2nd and 3rd rows of the figure).\nNaturally, blurrier corner images lead to smaller autocorrelation matrices and larger uncertainty ellipses.\n\nOne may note several things.\nFirst, between $30^\\circ$ and $90^\\circ$ opening angles, the largest uncertainty differs by a factor of about $2$, see the uncertainty ellipses in Fig.~\\ref{fig:corner_uncertainty}.\nSecond, intuitively, the uncertainty ellipse of a corner with opening angle $\\alpha$ is the same as with opening angle $180^\\circ - \\alpha$, but turned by $90^\\circ$ (cf.\\ the 3rd and 5th columns in Fig.~\\ref{fig:corner_uncertainty}, for $70^\\circ$ and $110^\\circ$).\nHence, the eigenvalues of the autocorrelation matrix of a corner with opening angle $\\alpha$, are the same as that for $180^\\circ - \\alpha$, but they are ``swapped'' (associated with the respective opposite eigenvector).\n\nThe left part of Fig.~\\ref{fig:corner_uncertainty} shows plots of the first eigenvalue (associated with eigenvector $(0,1)$) of the autocorrelation matrix $C$ as a function of opening angle.\nDue to the above observation, the second eigenvalue (associated with eigenvector $(1,0)$) associated with opening angle $\\alpha$ is simply given by the first eigenvalue associated with $180^\\circ - \\alpha$.\nThe graphs on the left of the figure confirm that increasing blur decreases the autocorrelation matrices eigenvalues.\nLet us note that we also simulated Gaussian pixel noise on the corner images; even for larger than realistic noise levels, the impact on the results shown in Fig.~\\ref{fig:corner_uncertainty}, was negligible.\n\nLet us finally explain how to use these results.\nFirst, we determine the average blur level in the already acquired images, from the strength of image gradients across edges, and then select the graph in Fig.~\\ref{fig:corner_uncertainty} associated with the closest simulated blur (for more precision, one could also compute the graph for the actual detected blur level).\nLet the function represented by the graph be $f(\\alpha)$ -- one can represent it as a lookup table or fit a polynomial to the data of the graph (we did the latter).\nThis allows to compute the diagonal coefficients of the autocorrelation matrix from the opening angle,\nas $f(\\alpha)$ and $f(180^\\circ - \\alpha)$.\nSecond, so far we have only considered axis-aligned corners.\nIf we now consider a corner with opening angle $\\alpha$, but that is rotated by an angle $\\beta$, then\nits autocorrelation matrix is nothing else than:\n\\begin{equation}\n\\footnotesize\nC = \\begin{pmatrix} \\cos\\beta & -\\sin\\beta \\\\ \\sin\\beta & \\cos\\beta \\end{pmatrix}\n\\begin{pmatrix} f(\\alpha) & 0 \\\\ 0 & f(180^\\circ - \\alpha) \\end{pmatrix}\n\\begin{pmatrix} \\cos\\beta & \\sin\\beta \\\\ -\\sin\\beta & \\cos\\beta \\end{pmatrix}\n\\end{equation}\n\nWe now have all that is needed to incorporate corner uncertainty in next best pose computation.\nFor each hypothetical pose we project, as shown above, all calibration points.\nFor each point (corner), using its neighbors, we can compute the opening angle $\\alpha$\nand rotation angle $\\beta$ and thus, the expected autocorrelation matrix $C$.\nIt can then be inserted in the computation of the information matrix, like in Eq.~\\eqref{eq.corrected_information_matrix}.\n\nThe effect of this approach on proposing the next best pose is to strike a balance between maximizing pure geometric ``strength'' of a pose (often achieved by strong tilting of the calibration pattern) and maximizing corner extraction accuracy (in fronto-parallel poses, corners are overall closest to exhibiting right angles, i.e.\\ where their auto-correlation matrices have maximal trace).\n\n\\subsection{Possible Extensions}\n\\label{sec:autocorrelation.next}\n\nSo far we have described the basic idea for incorporating corner uncertainty.\nThe following extensions may be applied; we plan this for future work.\nThe values plotted in Fig.~\\ref{fig:corner_uncertainty} are obtained for corners exhibiting the full range of 256 greylevels (black to white).\nIn real images, the range is of course smaller.\nIf the difference between largest and smallest greylevels is $x$, then the plotted values (left of Fig.~\\ref{fig:corner_uncertainty}) are divided by $255^2\/x^2$ (easy to prove but not shown due to lack of space).\nIn turn, the uncertainty ellipses are scaled up by a factor of $255\/x$.\nThis should be taken into account when predicting auto-correlation matrices for the next pose.\nThe range of greylevels depends on various factors, such as distance to the camera and lighting conditions.\nOne can learn the relationship between pose and greylevel range for a given calibration setup as part of the calibration process and use it to predict the next best pose.\n\nSimilarly, one might predict expected blur per corner, based on a learned relationship between blur and distance to camera, e.g.\\ by inferring the camera's depth of field during calibration. We observed that within the depth of field, image blur is linearly related to the distance to the camera.\n\nUsing these observations should allow to further improve the next best pose proposal, by achieving an even better compromise between geometric information of the pose and accuracy of image processing (here, corner extraction), both of which influence calibration accuracy.\n\n\n\\section{Results and Evaluation}\n\\label{sec:results}\n\nSynthetic and real-world experiments are performed here to evaluate the effectiveness of our Calibration Wizard.\nNote that in the optimization process, we ensure that all corner points should be within the image plane, otherwise the optimization loss is set to an extremely large value. \n\n\\subsection{Synthetic evaluations}\nTo assess the proposed system, we simulate the process of camera calibration with pre-defined intrinsic parameters, with Matlab. Here we first briefly introduce the procedure of producing random checkerboard poses.\n\n\\textbf{Data preparation.}\nFirst, $9\\times 6$ target points are defined, with $Z$ components set to $0$.\nNext, the 3D position of the virtual camera is created, with $X$ and $Y$ coordinates proportional to $Z$ within a plausible range.\nThen the camera is first oriented such that its optical axis goes through the center of the target and finally, rotations about the three local coordinate axes by random angles between $-15^{\\circ}$ and $15^{\\circ}$ are applied.\nNow, from the position, rotation matrix and the given intrinsic parameters, we can project the 3D target points to the image, and finally add zero-mean Gaussian noise with the same noise level to them.\nMoreover, we ensure that all 54 points are located within the field of view of a $640\\times 480$ image plane.\n\n\n\\textbf{Evaluation of accuracy and precision.}\nWe primarily compare the calibration accuracy obtained from random images, with that from images acquired as proposed by our system, with and without taking into account the autocorrelation matrix explained in the previous section.\nTo this end, the experimental process is as follows: create 3 initial random images, based on which we have 3 paths to acquire the calibration results:\n\\begin{itemize}\n    \\setlength\\itemsep{0.1em}\n\t\\item Produce many other random images \n\t\\item Obtain 17 images proposed by the wizard, so $3 + 17 = 20$ images in total \n\t\\item Obtain 17 wizard images taking the autocorrelation matrix into account \n\\end{itemize}\n$100$ trials are performed for each experiment, hence we acquire $100$ samples of intrinsic parameters for each.\nIn the first test, we set $f = 800, (u,v) = (320,240)$ and radial distortion coefficients $ k_1 = 0.01, k_2 = 0.1$. \nFig.~\\ref{fig:syn-test1} illustrates the statistical results of the estimated focal length from $100$ trials. \nIt can be easily noticed that the focal length acquired using our wizard is not only much closer to the ground truth, but also more concentrated (precise) than the estimation from pure random images.\nFor example, the estimated focal length acquired from only $3$ random + $4$ wizard images has outperformed the one from 20 random images.\nMoreover, not shown in the graph: $3$ random + $17$ wizard images still give higher accuracy than $60$ random images, which directly demonstrates the usefulness of our approach. \n\\begin{figure}[t]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.48\\linewidth]{img\/synFequal800_mean_smallK}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synFequal800_std_smallK}\n  \\end{tabular}\n\\caption{Comparisons of the focal length estimated from three schemes on synthetic data: randomly taken images, calibration wizard and wizard using autocorrelation matrix. $f = 800, (u,v) = (320,240), k_1 = 0.01, k_2 = 0.1$. Initial calibration was done with $3$ random images. Left: Mean values of the estimated focal length, where the {\\color{red} red} dashed line represents the ground truth $f=800$. Right: Standard deviations of the estimated focal length. Wizard images provide significantly more accurate and precise calibration results than random ones.}\n\\label{fig:syn-test1}\n\\end{figure}\n\n\\begin{figure}[!t]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.48\\linewidth]{img\/synFequal800_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synFequal800_std}  \\\\         \n    \\includegraphics[width = 0.48\\linewidth]{img\/synK1equal05_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synK1equal05_std}  \\\\             \n    \\includegraphics[width = 0.48\\linewidth]{img\/synK2equal1_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synK2equal1_std}  \\\\                 \n  \\end{tabular}\n\\caption{Comparisons of the intrinsic parameters estimated from three schemes on synthetic data: randomly taken images, calibration wizard without and with autocorrelation matrix. $f = 800, (u,v) = (320,240), k_1 = \\mathbf{0.5}, k_2 = \\mathbf{1}$. Wizard images achieve superior performance over random images on all intrinsic parameters. Considering the autocorrelation matrices can further provide the most accurate and precise estimation outcomes.}\n\\label{fig:syn-test2}\n\\end{figure}\n\nHowever, in this experiment, we notice that our system does not show much advantage over the randomly-taken images of the estimated distortion coefficients $k_1$ and $k_2$.  \nThus, a second experiment is performed with larger radial distortion coefficients $k_1 = 0.5$ and $k_2 = 1$, while the focal length and principal point stay the same. \nFig.~\\ref{fig:syn-test2} shows the effectiveness of the proposed system, especially with the consideration of autocorrelation matrix for target points.\nWhen the radial distortion is large, we notice that not only both distortion coefficients, but also the focal length and principle points (not shown here) estimated from purely random images deviate much from the ground truth, as was also reported in~\\cite{weng1992camera}. \nIn contrast, our system still manifests the ability of centering around the ground truth with incomparably low standard deviation.  \nFurthermore, compared to the simple case of the proposed system, both first-order statistics features appear to be most desirable when considering the autocorrelation matrices for the feature points.\n\n\n\\textbf{Robustness to noise.}\nWe are also interested in the performance of our approach with respect to the level of noise added to 2D corner points.\nIn this experiment, we compare 4 different configurations: $20$ random images, $40$ random images, $3$ random + 17 wizard images and $3$ random + 17 wizard-Auto images.\nZero-mean Gaussian noise with standard deviation of $0.1, 0.2, 0.5, 1$ with respect to $2$ pixels has been added to the image points respectively, and the comparisons are shown in Fig.~\\ref{fig:syn-test3}.\nSpecifically, it can be distinctly seen from the figure that, even when unrealistically strong noise is added ($\\sigma = 2$), both versions of our approach (3 random + 17 wizard images) still provide better accuracy than even 40 random images.\nMore synthetic experiments can be found in the supplementary material.\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.48\\linewidth]{img\/synNoise-mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synNoise-std}\n  \\end{tabular}\n\\caption{Comparisons among various calibration schemes of the robustness to noise. Zero-mean Gaussian noise with standard deviation of $0.1, 0.2, 0.5, 1$ respectively $2$ pixels is added to 2D target points. The focal length estimated from both of our methods with only 20 images, is more accurate (left) and precise (right) than that from 40 random images, especially when the noise level is high.}\n\\label{fig:syn-test3}\n\\end{figure}\n\n\n\n\n\\subsection{Real-world evaluations}\n\\label{sec:result-real}\n\nAlthough the performance of the Calibration Wizard has been demonstrated for the synthetic data, we ultimately want to evaluate the effectiveness of its proposed next best pose on real-world examples.\nWe designed two experiments for this purpose where we also compare with calibrations obtained with freely taken images. Evaluating calibration results is difficult since ground truth is not readily available; we thus devised two experiments where calibration quality is assessed through evaluating results of applications -- pose estimation and SfM.\nWe used the commonly-used Logitech C270H HD Webcam in our experiments. It has an image size of $640\\times 480$ and around $60^\\circ$ field of view.\nFig.~\\ref{fig:img-example} provides some sample calibration images.\nOne may notice that wizard-suggested images indeed correspond to poses often chosen by experts for stable calibration: large inclination of the target along the viewing direction, targets covering well the field of view and\/or reaching the image border.\n\nIn the following, we denote ``$x$-free'' the calibration results from $x$ images acquired freely by an experienced user using OpenCV, compared to ``$x$-wizard'' where guidance was used. \n\n\n\n\\noindent\\textbf{Pose estimation.}\nSimilar to the experiment performed in~\\cite{ha2017deltille}, in order to quantitatively evaluate the quality of camera calibration, we design the first real-world experiment where,\napart from the images used for calibration, we then also acquire a number of extra checkerboard images which are only used for evaluation,\ncf.\\ Fig.~\\ref{fig:pnp}. \n\nFirst, $4$ corner points are utilized to calculate the pose with EP$n$P~\\cite{lepetit2009epnp}, given the intrinsic parameters provided by the calibration.\nThen, since we have assumed the $Z$ components of the target points to be $0$ in the world coordinate system, it is straightforward to back-project the remaining $50$ points to 3D, onto the target plane, using the calibrated intrinsic parameters and the computed pose (cf.\\ Fig.~\\ref{fig:pnp} right).\nThe smaller Euclidean distance between the back-projected and theoretical 3D points, the better the calibration.\n\nThere are 80 images in total for testing so we have $50\\times 80 = 4,000$ points for assessment.\nThe mean and standard deviation of the 4,000 distance errors are applied as metric.\nTable~\\ref{tab:3d-test} demonstrates that our system, when using only 15 images for calibration, still exceeds the performance of using 50 freely acquired images and exceeds that of using 20 such images by about 5\\%.\nThis seemingly small improvement may be considered as significant since it may be expected that differences are not large in this experiment. Even with a moderately incorrect calibration, pose estimation from 4 outermost target points will somewhat balance the reconstruction errors for the inner corner points.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.44\\linewidth]{img\/pnp-min} &\n    \\includegraphics[width = 0.47\\linewidth,frame]{img\/pnp_comp_new}\n  \\end{tabular}\n\\caption{Pose estimation test. Left: checkerboard image where $4$ {\\color{green}green} corner points are used for pose estimation, and the remaining $50$ {\\color{red}red} points for reconstruction. Right: $50$ ground-truth points in black and residuals between them and the reconstructed corner points in {\\color{red}red} (enlarged 50 times for visualization).}\n\\label{fig:pnp}\n\\end{figure}\n\n\\begin{table}[!htb]\\centering\n\\caption{Pose estimation test with Logitech C270H HD Webcam.}\n\\footnotesize\n\\begin{tabular}{c|c|c||c|c|c}\n\\hline\n        & mean  & std   &                           & mean           & std \\\\ \\hline \\hline\n\\; 3-free  & 0.856 & 1.130 & 3-free + 4-wizard           & 0.862          & 1.155          \\\\\n10-free & 0.815 & 1.115 & 3-free + 6-wizard           & 0.783          & 1.092          \\\\\n20-free & 0.802 & 1.115 & 3-free + 9-wizard           & 0.788          & 1.104          \\\\\n50-free & 0.789 & 1.108 & \\textbf{3-free + 12-wizard} & \\textbf{0.763} & \\textbf{1.082} \\\\ \\hline\n\\end{tabular}\n\\label{tab:3d-test}\n\\end{table} \n\nWe also tested our approach on the FaceTime HD camera of a MacBook Pro. This camera has higher resolution and different field of view compared to other commonly use webcams, so it is a suitable alternative to show the robustness of our method. As shown in Table~\\ref{tab:3d-test-mac}, adding only one or two wizard images can largely reduce the Euclidean distance and outperforms the results from freely taking many more images.\n\n\\begin{table}[!htb]\\centering\n\\caption{Pose estimation test with FaceTime HD camera.}\n\\footnotesize\n\\begin{tabular}{c|c|c||c|c|c}\n\\hline\n        & mean  & std   &                           & mean           & std \\\\ \\hline \\hline\n\\; 3-free  & 2.503 & 2.557 & 3-free + 1-wizard      & 1.455          & 1.630          \\\\\n10-free & 1.664 & 1.839 & \\textbf{3-free + 2-wizard}           & \\textbf{1.165}          & \\textbf{1.491}          \\\\\n20-free & 1.255 & 1.606 & & &          \\\\\\hline\n\\end{tabular}\n\\label{tab:3d-test-mac}\n\\end{table} \n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width = 0.18\\linewidth]{img\/rand1.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/rand2.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/rand3.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/rand5.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/rand4.jpg}\\\\\n\\includegraphics[width = 0.18\\linewidth]{img\/wizard1.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/wizard2.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/wizard3.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/wizard5.jpg}\n\\includegraphics[width = 0.18\\linewidth]{img\/wizard4.jpg}\n\\caption{Sample images used for the calibration in real-world tests. Top row: freely-taken images. Bottom row: wizard guided images.}\n\\label{fig:img-example}\n\\end{figure*}\n\n\n\\noindent\\textbf{Structure from Motion test.}\nIn this last experiment, we assess our Calibration Wizard by investigating the quality of 3D reconstruction in a structure-from-motion (SfM) setting. \nThe object to be reconstructed is the backrest of a carved wooden bed, as shown in Fig.~\\ref{fig:sfm}.\nWe devised a simple but meaningful experiment to evaluate the quality of the calibration, as follows.\nWe captured images from the far left of the object and gradually move to the right side, and then proceed backwards and return to the left, approximately to the starting point.\nThe acquired images are then provided as input to VisualSfM~\\cite{wu2013towards}; we added an identical copy of the first image of the sequence, to the end of the sequence, but without ``telling'' this to the SfM tool and without using a loop detection method during SfM.\nThe purpose of doing so is: if calibration is accurate, the incremental SfM should return poses for the first image and the added identical last image, which are close to one another.\nMeasuring the difference in pose is not sufficient since the global scale of the reconstruction can be arbitrarily chosen by the SfM tool for each trial.\nSo instead, we project all 3D points that were reconstructed on the basis of interest points extracted in the first image, using the pose computed by SfM for the identical last image, and measure the distance between the two sets of 2D points such constructed.\nThis distance is independent of the scene scale and is thus a good indicator of the quality of the SfM result which in turn is a good indicator of the quality of the calibration used for SfM.\n\n\nNote that we only match two consecutive frames instead of full-pairwise matching within the given sequence.\nIn this case, 2D errors are accumulated so the reconstruction results highlight the calibration accuracy more strongly.\n\nThe experiment is described as follows.\nWe first obtain the 5-parameter calibration result (including two radial distortion coefficients), from 3 freely acquired images (``3-free'').\nThen, on the one hand, another 17 images are taken, from which the intrinsic parameters of ``7-free'' and ``20-free'' are obtained.\nOn the other hand, we take another 4 sequential images proposed by the Calibration Wizard, where we get the intrinsic parameters of ``3-free + 2-wizard'' and ``3-free + 4-wizard''.\nAnd now, we load VisualSfM with intrinsic parameters of these five configurations respectively, along with the backrest sequence taken by the same camera.\nIt is worth mentioning that we conduct five trials of VisualSfM for each configuration in order to lessen the influence of the stochastic nature of the SfM algorithm.\n\\begin{figure}[!htb]\n\\centering\n    \\includegraphics[width = 0.9\\linewidth]{img\/pano} \n    \\includegraphics[width = 0.9\\linewidth]{img\/capture1_edit} \n\\caption{Structure from motion test.\nTop: Panorama stitched by Hugin (\\url{http:\/\/hugin.sourceforge.net}), showing the test scene.\nBottom: Result of applying VisualSfM~\\cite{wu2013towards} to build a 3D model.\nWe started capturing images from the left and moved clockwise, finally came back approximately to the starting point.}\n\\label{fig:sfm}\n\\end{figure}\n\n\n\\begin{table}[t]\n\\centering\n\\caption{2D errors of SfM tests under various calibration schemes.}\n\\begin{tabular}{c||r@{}l|r@{}l|r@{}l}\n\\hline\nCalibration scheme & \\multicolumn{2}{c|}{mean} & \\multicolumn{2}{c|}{std} & \\multicolumn{2}{c}{median} \\\\\n\\hline \\hline\n3-free                      & 43 & .6 & 11 & .5 & 44 & .3 \\\\  \n7-free                      & 30 & .5 & 11 & .7 & 31 & .7 \\\\\n20-free                     & 15 & .7 & 10 & .5 & 16 & .1      \\\\\n3-free $+$ 2-wizard \\quad & 17 & .4 & 10 & .8 & 13 & .2    \\\\\n3-free $+$ 4-wizard \\quad & \\textbf{14} & \\textbf{.4} & \\textbf{9} & \\textbf{.1} & \\textbf{10} & \\textbf{.6}  \\\\\n\\hline      \n\\end{tabular}\n\\label{tab:sfm}\n\\end{table}\nResults are listed in Table~\\ref{tab:sfm}, where we evaluate the 2D errors across all 5 trials.\nSome observations can be made:\nwith only 5 images in total, ``3-free + 2-wizard'' has already provided an accuracy competitive to 20 freely-taken images.\nBoth ``7-free'' and ``3-free + 4-wizard'' use 7 images for calibration, but it can be clearly noticed that the latter one has far lower errors in all aspects.\nIt is reasonable to conclude that our method notably improves the quality of calibration and 3D reconstruction with a considerably small number of calibration images.\n\nFinally, it is worth mentioning that all real-world experiments were performed with a 2.7 GHz Intel i5 CPU (no GPU used). To compute the next best pose with a $9\\times6$ target, our un-optimized C++ code took about $0.4s$ for 3 target images and $1.5s$ for 15 images (increasing roughly linearly per image), but we found that 10 images are usually sufficient for a good calibration. \n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nCalibration Wizard is a novel approach which can guide any user through the calibration process.\nWe have shown that accurate intrinsic parameters can be obtained from only a small number of images suggested by this out-of-the-box system.\nSome ideas for future work were already mentioned in section \\ref{sec:autocorrelation.next}.\nWe also plan to apply the approach to very wide field of view cameras such as fisheyes.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Another Synthetic Experiment}\n\\label{s.experiments}\nWhen users try to calibrate a camera, they tend to randomly move the camera around and get as many images as possible for calibration. With our calibration wizard, we have already shown the superiority over such randomly-captured case in the main paper.\nNow, considering that calibration is an interactive procedure, i.e., a user can move the camera around, we could actually keep all the intermediate frames when moving from one position to the next, and then all these frames can be used for calibration. \nHere we perform another synthetic test to validate that the calibration can still be improved by moving to optimal poses under this scenario.\nProvided the frame rate is $25$ fps and it takes 1 second to move from one pose to the next, 25 images can then be acquired within \\textit{one path}.\nTo this end, the experimental process is as follows.\n\nFirst, we create 4 random poses using exactly the same way described in the main paper, from which we could acquire 3 paths by pose interpolation. Thus, there are $3\\times 25 = 75$ frames in total for the case ``Random''. This process mimics moving the camera from one pose to another continuously for three times.\n\nThen, we randomly take 5 images within the first path to get the initial calibration for our wizard. The wizard then proposes a next best pose and while the simulated camera moves there, 25 additional images for calibration are acquired. This is done 3 times, for 75 additional calibration images in total.\nThis input leads to the results labeled as ``Wizard'' in the following.\nThe same process is also applied for taking autocorrelation matrix into account in next best pose computation (``Wizard-Auto'')\n\nFinally, we simply compare the calibration results acquired from random-generated paths and wizard-generated ones. \nAs illustrated in Fig.~\\ref{fig:syn-test21_supp}, we can clearly notice that the calibration results from our system have higher precision and accuracy than the results from random paths.\n\nWe also perform the comparison with respect to the level of noise added to 2D corner points.\nIn this experiment, zero-mean Gaussian noise with standard deviation of $0.1, 0.2, 0.5, 1$ respectively $2$ pixels has been added to corner points, and the comparisons are shown in Fig.~\\ref{fig:syn-test3_supp}.\nAgain, even when unrealistically strong noise is added ($\\sigma = 2$), our system provides much better estimation of the focal length. Moreover, when considering the autocorrelation matrices, our system seems more robust to the noise.\n\nNote that we only select the results for some intrinsic parameters for these two experiments, but similar results can be expected for other parameters like principal points.\n\n\n\n\n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.48\\linewidth]{img\/synF_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synF_std}  \\\\         \n    \\includegraphics[width = 0.48\\linewidth]{img\/synK1_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synK1_std}  \\\\             \n    \\includegraphics[width = 0.48\\linewidth]{img\/synK2_mean}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/synK2_std}  \\\\                 \n  \\end{tabular}\n\\caption{Comparisons of the intrinsic parameters estimated from three schemes on synthetic data: the paths generated from random poses, generated from the poses proposed by calibration wizard without and with autocorrelation matrix. $f = 800, (u,v) = (320,240), k_1 = \\mathbf{0.5}, k_2 = \\mathbf{1}$. {\\color{red} Red} dashed lines represent the ground truth values. Wizard images achieve superior performance over random images on all intrinsic parameters.}\n\\label{fig:syn-test21_supp}\n\\end{figure}\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{tabular}{cc}\n    \\includegraphics[width = 0.48\\linewidth]{img\/noise_75_mean.eps}  &\n    \\includegraphics[width = 0.48\\linewidth]{img\/noise_75_std.eps}\n  \\end{tabular}\n\\caption{Comparisons of the calibration schemes concerning the robustness to added noise. Zero-mean Gaussian noise with standard deviation of $0.1, 0.2, 0.5, 1$ respectively $2$ pixels is added to the 2D target points. The intrinsic parameters estimated from the wizard-generated paths (shown are results for the focal length), are more accurate (left) and precise (right) than random-generated paths, especially when the noise level is high. The total number of image is 75.}\n\\label{fig:syn-test3_supp}\n\\end{figure}\n\n\n\n\\section{An efficient way of computing the covariance matrix of intrinsic parameters}\n\\label{s.computations}\n\nAs already explained in the paper, the Jacobian matrix $J$ can be denoted as:\n\\begin{equation}\nJ = \\begin{pmatrix}\nA_1 & B_1 & 0 & \\cdots & 0 \\\\\nA_2 & 0 & B_2 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_m & 0 & 0 & \\cdots & B_m\n\\end{pmatrix}\n\\end{equation}\n\nLet us examine in detail the incorporation of the autocorrelation matrices of image corner points in the computation of the information matrix, as shown in Eq.~(9) in the paper:\n\\begin{equation}\nH = J^\\top \\text{diag}(C_{11}, C_{12}, \\cdots, C_{1n}, C_{21}, \\cdots, C_{mn}) J\n\\label{eq.information_matrix}\n\\end{equation}\n\nLet us decompose the matrices $A_i$ and $B_i$ appearing in the definition of the $J$, into matrices $A_{ij}$ and $B_{ij}$ containing the residual associated with a single image point each (point $j$ in image $i$):\n\\[\nA_{ij} = \\begin{pmatrix}\n\\frac{\\partial \\hat{x}_{ij}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{x}_{ij}}{\\Theta_k} \\\\\n\\frac{\\partial \\hat{y}_{ij}}{\\Theta_1} & \\cdots & \\frac{\\partial \\hat{y}_{ij}}{\\Theta_k}\n\\end{pmatrix}\n\\enspace\nB_{ij} = \\begin{pmatrix}\n\\frac{\\partial \\hat{x}_{ij}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{x}_{ij}}{\\Pi_{i,6}} \\\\\n\\frac{\\partial \\hat{y}_{ij}}{\\Pi_{i,1}} & \\cdots & \\frac{\\partial \\hat{y}_{ij}}{\\Pi_{i,6}}\n\\end{pmatrix}\n\\]\n\nThe $A_{ij}$ are of size $2\\times k$ ($k$ is the number of intrinsic parameters) and the $B_{ij}$ of size $2\\times 6$.\nWe then have:\n\\[\nA_i = \\begin{pmatrix} A_{i1} \\\\ \\vdots \\\\ A_{in} \\end{pmatrix}\n\\enspace\nB_i = \\begin{pmatrix}  B_{i1} \\\\ \\vdots \\\\ B_{in}\n\\end{pmatrix}\n\\]\n\nNow, the information matrix $H$ of Eq.~\\eqref{eq.information_matrix} can be structured as:\n\\begin{equation}\nH = \\begin{pmatrix}\n    U & W\\\\ W^\\top & V\n\\end{pmatrix}\n\\label{eq:infoMat_new}\n\\end{equation}\n\nwhere \n\\begin{eqnarray}\nU & = & \\sum_i U_i \\\\\n\\text{with}\\ U_i & = & \\sum_j A_{ij}^\\top C_{ij} A_{ij} \\\\\nW & = & \\begin{pmatrix} W_1 & \\cdots & W_m \\end{pmatrix} \\\\\n\\text{with}\\ W_i & = & \\sum_j A_{ij}^\\top C_{ij} B_{ij} \\\\\nV & = & \\begin{pmatrix} V_1 & & & \\\\ & V_2 & & \\\\ & & \\ddots & \\\\ & & & V_m \\end{pmatrix} \\\\\n\\text{with}\\ V_i & = & \\sum_j B_{ij}^\\top C_{ij} B_{ij}\n\\end{eqnarray}\n\n$U$ is a symmetric matrix of size $k\\times k$, $W$ is of size $k \\times (6m)$ (remember that $m$ is the number of images) and $V$ is a block-diagonal matrix consisting of $m$ symmetric $6\\times6$ matrices $V_i$.\n\nAs described in~\\cite{hartley2003multiple}, the upper-left sub-matrix of $H^{-1}$ is given by \n\\begin{equation}\n\\Sigma = (U-W V^{-1} W^\\top)^{+}\n\\label{eq:sigma_big}\n\\end{equation}\nwhich is a pseudo-inverse of a $k\\times k$ symmetric matrix.\nLet us expand this using the above definitions of $U, V$ and $W$.\n\nUsing the above definitions of $U, V$ and $W$, we can rewrite $\\Sigma$ as:\n\\begin{equation}\n\\Sigma = \\left\\{ \\sum_i \\left( U_i - W_i V_i^{-1} W_i^\\top \\right) \\right\\}^{+}\n\\label{eq.covariance}\n\\end{equation}\n\nThe computation is efficient.\nThe $V_i$ are symmetric and of size $6\\times6$, hence their inversion is a small problem.\nOther than that, the computation only involves products and sums of small matrices and one final inversion of a $k \\times k$ symmetric matrix.\n\nLet us now consider the computation of the next best pose.\nAs explained in the paper, we use global optimization algorithms for this purpose.\nThey will need to compute $\\Sigma$ and its trace multiple times, for multiple hypothetical next poses.\nIt is thus interesting to study how to cut computation times by performing adequate pre-computations.\nThis is very simple in our case.\nLet image $m+1$ be the next image, for which to compute the optimal pose.\nImages $1$ till $m$ are thus already acquired and we have computed intrinsic and pose parameters for them.\nAll we need to do is to compute once the following part of equation (\\ref{eq.covariance}), for images $1$ to $m$:\n\\begin{equation}\nA = \\sum_{i=1}^m \\left( U_i - W_i V_i^{-1} W_i^\\top \\right)\n\\end{equation}\n\nThen, for each hypothetical pose for image $m+1$, we only need to compute\n\\begin{equation}\n\\Sigma = \\left\\{ A + U_{m+1} - W_{m+1} V_{m+1}^{-1} W_{m+1}^\\top \\right\\}^{+}\n\\end{equation}\nThis computation is efficient: besides the computation of partial derivatives of the projection function for the hypothetical new pose (see section \\ref{s.derivatives} for details) and the sum or multiplication of small matrices (of size at most $(max(k,6) \\times max(k,6)$), it only involves the inversion of a symmetric matrix of size $6\\times6$ and of a symmetric matrix of size $k\\times k$.\n\nNote that in this section we handled the incorporation of autocorrelation matrices.\nIf one wishes to work without them, it suffices to replace the $C_{ij}$ by identity matrices.\n\n\\section{Details on the computation of partial derivatives in $J$}\n\\label{s.derivatives}\n\nIn the following, we use the following shorthand notations for partial derivatives, which makes for easier reading.\nIf $A$ is a vector of size $a$ and $B$ a vector of size $b$ (with possibly, $a=1$ or $b=1$), then we write:\n\\begin{equation}\n\\frac{\\partial A}{\\partial B}\n=\n\\begin{pmatrix}\n\\frac{\\partial A_1}{\\partial B_1} & \\cdots & \\frac{\\partial A_1}{\\partial B_b} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial A_a}{\\partial B_1} & \\cdots & \\frac{\\partial A_a}{\\partial B_b}\n\\end{pmatrix}\n\\end{equation}\n\nAlso, if $A$ is a matrix and $B$ a scalar, then $\\frac{\\partial A}{\\partial B}$ is the matrix gathering the partial derivatives of the coefficients of $A$ relative to the scalar $B$, in the same order as they appear in $A$.\n\nAs we have seen in the paper, $3D$ to $2D$ projections can be written as:\n\\begin{align}\n\tp_x(\\Theta, \\Pi, Q) &= q_x(\\Theta, RQ+t) = q_x(\\Theta, S)\\\\\n\tp_y(\\Theta, \\Pi, Q) &= q_y(\\Theta, RQ+t) = q_y(\\Theta, S)\n\\end{align}\n\nThanks to the chain rule, the derivatives of residuals $\\hat{x}$ (for their definition, see section 2 of the main paper) with respect to extrinsic parameters can be decomposed as:\n\\begin{eqnarray}\n\\frac{\\partial \\hat{x}}{\\partial R} & = &\n\\frac{\\partial p_x}{\\partial R} =\n\\frac{\\partial q_x}{\\partial S}\\frac{\\partial S}{\\partial R} \\label{eq.pd1} \\\\\n\\frac{\\partial \\hat{x}}{\\partial t} & = & \\frac{\\partial p_x}{\\partial t} = \\frac{\\partial q_x}{\\partial S}\\frac{\\partial S}{\\partial t} \\label{eq.pd2}\n\\end{eqnarray}\nand likewise for partial derivatives of $\\hat{y}$.\nHere, we use the informal notation\n\\begin{equation}\n\\partial R = \\partial \\begin{pmatrix} \\alpha \\\\ \\beta \\\\ \\gamma \\end{pmatrix}\n\\end{equation}\nwhere the three angles $\\alpha, \\beta, \\gamma$ parameterize the rotation matrix $R$ as shown on page 2 of the main paper and as reproduced here:\n\\begin{equation}\nR = R_z(\\gamma) R_y(\\beta) R_x(\\alpha)\n\\end{equation}\n\n$\\partial q_x\/\\partial S$ and $\\partial q_y\/\\partial S$ depend on the projection functions for the camera model used for calibration (see below for an example, the pinhole model with one radial distortion coefficient), while $\\partial S\/\\partial R$ and $\\partial S\/\\partial t$ are generic for all camera models.\nRemember that\n\\begin{equation}\nS = RQ+t\n\\label{eq.S}\n\\end{equation}\nThus:\n\\makeatletter\n    \\def\\tagform@#1{\\maketag@@@{\\normalsize(#1)\\@@italiccorr}}\n\\makeatother\n{\\small\n\\begin{align}\n\\label{eq:dSdR}\n\\frac{\\partial S}{\\partial R} &=\n  \\begin{pmatrix}\n  \\frac{\\partial S_1}{\\partial \\alpha} & \\frac{\\partial S_1}{\\partial \\beta} & \\frac{\\partial S_1}{\\partial \\gamma} \\\\\n  \\frac{\\partial S_2}{\\partial \\alpha} & \\frac{\\partial S_2}{\\partial \\beta} & \\frac{\\partial S_2}{\\partial \\gamma} \\\\\n  \\frac{\\partial S_3}{\\partial \\alpha} & \\frac{\\partial S_3}{\\partial \\beta} & \\frac{\\partial S_3}{\\partial \\gamma}\n  \\end{pmatrix}\n= R  \\arraycolsep=1.4pt\n  \\begin{pmatrix}\n  Q_1 & Q_3 & -Q_2 \\\\\n  -Q_3 & Q_2 & Q_1 \\\\\n  Q_2 & -Q_1 & Q_3\n  \\end{pmatrix}\\\\\n\\label{eq:dSdt}  \n\\frac{\\partial S}{\\partial t} &=\n  \\begin{pmatrix}\n  \\frac{\\partial S_1}{\\partial t_1} & \\frac{\\partial S_1}{\\partial t_2} & \\frac{\\partial S_1}{\\partial t_3} \\\\\n  \\frac{\\partial S_2}{\\partial t_1} & \\frac{\\partial S_2}{\\partial t_2} & \\frac{\\partial S_2}{\\partial t_3} \\\\\n  \\frac{\\partial S_3}{\\partial t_1} & \\frac{\\partial S_3}{\\partial t_2} & \\frac{\\partial S_3}{\\partial t_3}\n  \\end{pmatrix}\n = I_{3\\times3}   \n\\end{align}}\n\nThe derivation is straightforward. \n\n\n\n\n\n\n\\subsection{Example: pinhole model with one radial distortion coefficient}\n\nHere we use the pinhole model with one radial distortion coefficient as an example. \nIt can be easily generalized to other more complicated models.\nThe model can be represented as:\n\\begin{eqnarray}\nq_x(S) & = & u + (1 + k_1 r^2) f \\frac{S_1}{S_3} \\\\\nq_y(S) & = & v + (1 + k_1 r^2) f \\frac{S_2}{S_3}\n\\end{eqnarray}\nwhere $r = \\sqrt{(\\frac{S_1}{S_3})^2 + (\\frac{S_2}{S_3})^2} = \\frac{1}{S_3}\\sqrt{S_1^2 + S_2^2}$. \nWe define $\\Theta = (f,u,v, k_1)$.\n\nFirst, the partial derivatives of the local projection functions with respect to $S$ can be written as:\n\\begin{equation}\n\\begin{pmatrix}\n\\frac{\\partial q_x}{\\partial S} \\\\\n\\frac{\\partial q_y}{\\partial S}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\frac{\\partial q_x}{\\partial S_1} & \\frac{\\partial q_x}{\\partial S_2} & \\frac{\\partial q_x}{\\partial S_3}\\\\\n\\frac{\\partial q_y}{\\partial S_1} & \\frac{\\partial q_y}{\\partial S_2} & \\frac{\\partial q_y}{\\partial S_3}\\\\\n\\end{pmatrix}\n\\end{equation}\nwhere\n\\begin{align*}\n    \\frac{\\partial q_x}{\\partial S_1} &= \\frac{f}{S_3} + \\frac{f k_1}{S_3^3}( 3S_1^2 +S_2^2 ) \\\\\n    \\frac{\\partial q_x}{\\partial S_2} &= \\frac{2 f k_1 S_1 S_2}{S_3^3} \\\\\n    \\frac{\\partial q_x}{\\partial S_2} &= -\\frac{f S_1}{S_3^2} - \\frac{3 f k_1 S_1 (S_1^2 + S_2^2)}{S_3^4}\\\\\n    \\frac{\\partial q_y}{\\partial S_1} &= \\frac{2 f k_1 S_1 S_2}{S_3^3} \\\\\n    \\frac{\\partial q_y}{\\partial S_2} &= \\frac{f}{S_3} +  \\frac{f k_1}{S_3^3}(S_1^2 + 3S_2^2 ) \\\\\n    \\frac{\\partial q_y}{\\partial S_2} &= -\\frac{f S_2}{S_3^2} - \\frac{3 f k_1 S_2 (S_1^2 + S_2^2)}{S_3^4}\\\\\n\\end{align*}\nThis, together with Eq.~\\eqref{eq:dSdR} and \\eqref{eq:dSdt},\nallows to compute the partial derivatives of residuals relative to extrinsic parameters in the Jacobian matrix $J$, by inserting into Eq.~\\eqref{eq.pd1} and \\eqref{eq.pd2} (and likewise for partial derivatives of $\\hat{y}$).\n\nAs for the partial derivatives relative to intrinsic parameters, they can be computed as:\n\\begin{align}\n\\frac{\\partial \\hat{x}}{\\partial \\Theta} & =\n\\frac{\\partial q_x}{\\partial \\Theta} = \\begin{pmatrix} (1 + k_1 r^2)\\frac{S_1}{S_3}\n& 1 \n& 0 \n& r^2 f \\frac{S_1}{S_3} \n\\end{pmatrix} \\\\\n\\frac{\\partial \\hat{y}}{\\partial \\Theta} & =\n\\frac{\\partial q_y}{\\partial \\Theta} = \n\\begin{pmatrix} (1 + k_1r^2)\\frac{S_2}{S_3}\n& 0 \n& 1 \n& r^2 f\\frac{S_2}{S_3} \n\\end{pmatrix} \n\\end{align} \nNow, we have everything needed for building $J$.\n\n\\section{Uncertainty map}\n\\label{sec:uncertainty-map}\nHere we introduce the concept of uncertainty map which could be an effective tool to visualize the quality of the current calibration. Also, the uncertainty map can be used to evaluate the evolution of the quality along the calibration process when adding more and more images. \n\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\begin{tabular}{cccc}\n    \\includegraphics[width = 0.28\\linewidth, trim = 10em 10em 10em 2.5em, clip]{img\/umap_3.png}  &\n    \\includegraphics[width = 0.28\\linewidth, trim = 10em 10em 10em 2.5em, clip]{img\/umap_3+2.png} & \n    \\includegraphics[width = 0.28\\linewidth, trim = 10em 10em 10em 2.5em, clip]{img\/umap_3+2w.png} &\n    \\includegraphics[width = 0.041\\linewidth]{img\/colorbar.png} \n    \\\\     \n    {\\small (a)} &  {\\small (b)} & {\\small (c)} &\n  \\end{tabular}\n\\caption{Illustrations for the uncertainty maps and the effectiveness of our camera wizard. The size of the map is the same as acquired image size $640\\times 480$. (a) Initial calibration results with 3 freely taken images. On top of the 3 freely-taken images, (b) add two freely taken images. (c) add two images proposed by wizard. We can clearly visualize: (1) the more calibration images, the lower the uncertainty (2) the uncertainty obtained from using wizard images is much lower than with freely taken ones (cf. the scale on the right hand side).}\n\\label{fig:umap-illu}\n\\end{figure*}\n\nGiven the $k \\times k$ covariance matrix $\\Sigma$ of intrinsic parameters, an uncertainty map is defined as the expected uncertainties of the local projection model across the image plane. \nThat is to say, we propagate the uncertainties of intrinsic parameters to each pixel on the image area.\nThe quality of calibration at any stage of the calibration process can be visualized as shown in Fig.~\\ref{fig:umap-illu}.\n\nMathematically speaking, for each pixel $(x,y)$ on the image, the uncertainty propagation process can be formulated as:\n\\begin{equation}\n\\begin{split}\n    \\Gamma&(x,y) =\\\\& \\begin{pmatrix}[1.5] \\frac{\\partial q_x(\\Theta,S)}{\\partial \\Theta} |_{S=S(x,y,\\Theta)} \\\\ \\frac{\\partial q_y(\\Theta,S)}{\\partial \\Theta} |_{S=S(x,y,\\Theta)} \\end{pmatrix}\t\n    \\Sigma\n    \\begin{pmatrix}[1.5] \\frac{\\partial q_x(\\Theta,S)}{\\partial \\Theta}  |_{S=S(x,y,\\Theta)} \\\\ \\frac{\\partial q_y(\\Theta,S)}{\\partial \\Theta}  |_{S=S(x,y,\\Theta)} \\end{pmatrix}^\\top\t\n    \\label{eq:uncertainty-propergate}\n\\end{split}    \n\\end{equation}\nwhere $\\Gamma$ is a $2\\times 2$ covariance matrix expressing the uncertainty per image point $(x,y)$.\nTo compute this, we first need to back-project the image point to a 3D point $S$ (any 3D point along the line of sight associated with the image point will do).\nWe denote this as $S(x,y,\\Theta)$ in the above equation: the back-projection depends on the image point coordinates $x$ and $y$ and of course on the intrinsic parameters $\\Theta$.\nThen, we propagate the uncertainty on the intrinsic parameters (covariance matrix $\\Sigma$) in a standard way through the forward projection, as shown in the above equation, to obtain the covariance matrix $\\Gamma$.\n\nNote that, we are aware that minimizing over this pixel reprojection uncertainty is better than over the trace of the covariance matrix $\\Sigma$. The main problem with minimizing reprojection uncertainty is computation time, incompatible with real-time. In this paper we stuck to the trace, a commonly used simple cost function. In the future, we will consider the reprojection uncertainty for a subset of pixels. \n\nIn the following, we provide an example for the process of acquiring the uncertainty map and some analysis.\n\n\\subsection{Example: 3-parameter pinhole model}\n\nFor this simple camera model, the uncertainty map can be analyzed theoretically, as shown in the following.\nWe remind the definition of the 3-parameter pinhole model, where the vector of intrinsic parameters is $\\Theta=(f,u,v)^\\top$.\n\n\\begin{eqnarray*}\nq_x(\\Theta,S) & = & u + f \\frac{S_1}{S_3} \\\\\nq_y(\\Theta,S) & = & v + f \\frac{S_2}{S_3}\n\\end{eqnarray*}\n\nThese are the ``forward'' projection equations.\nAs explained above, to construct the uncertainty map, we should back-project each pixel to 3D, i.e.\\ to some 3D point $S$ which, if forward-projected, gives rise to the original pixel.\nOne possibility for the back-projection is as follows:\n\\[\nS(x,y,\\Theta) = \\begin{pmatrix} x-u \\\\ y-v \\\\ f \\end{pmatrix}\n\\]\n\nWe now compute the partial derivatives appearing in Eq.~\\eqref{eq:uncertainty-propergate}:\n\\begin{eqnarray*}\n\\begin{pmatrix}\n\\frac{\\partial q_x(\\Theta,S)}{\\partial \\Theta} |_{S=S(x,y,\\Theta)} \\\\\n\\frac{\\partial q_y(\\Theta,S)}{\\partial \\Theta} |_{S=S(x,y,\\Theta)}\n\\end{pmatrix} & = &\n\\begin{pmatrix}\n\\frac{S_1}{S_3} & 1 & 0 \\\\\n\\frac{S_2}{S_3} & 0 & 1\n\\end{pmatrix} |_{S=S(x,y,\\Theta)} \\\\\n& = &\n\\begin{pmatrix}\n\\frac{x-u}{f} & 1 & 0 \\\\\n\\frac{y-v}{f} & 0 & 1\n\\end{pmatrix}\n\\end{eqnarray*}\n\nInserting this in Eq.~\\eqref{eq:uncertainty-propergate}, we get the desired covariance matrices per image point:\n\\begin{equation*} \n\\Gamma(x,y) =\n\\begin{pmatrix}\n\\frac{x-u}{f} & 1 & 0 \\\\\n\\frac{y-v}{f} & 0 & 1\n\\end{pmatrix}\n\\Sigma\n\\begin{pmatrix}\n\\frac{x-u}{f} & \\frac{y-v}{f} \\\\\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix}\n\\end{equation*}\n\n\nTo analyze the nature of the uncertainty map, we can imagine it as the coefficients of an ``uncertainty ellipse'' centered at every image pixel.\nOnce we acquire $\\Gamma$ for each pixel, the trace, larger eigenvalue or determinant of $\\Gamma$ can be used to measure the impact of the uncertainty of intrinsic parameters, on this pixel.\nIn Fig.~\\ref{fig:umap-illu}, we generate the map by creating an image where the value of every pixel is given by the trace of $\\Gamma$.\n\nWe may further analyze the nature of these uncertainty maps as follows.\nWhen computing the trace of $\\Gamma$ in detail, we get:\n\\begin{align*}\n    tr(\\Gamma(x,y)) = \\Sigma_{22} + &\\Sigma_{33} + \\frac{\\Sigma_{11}}{f^2} \\left[ (x-u)^2 + (y-v)^2 \\right]\\\\\n&+ \\frac{2}{f} \\left[ \\Sigma_{12} (x-u) + \\Sigma_{13}(y-v) \\right]\n\\end{align*}\nThis is quadratic with respect to $x$ and $y$. The uncertainty map, if visualized in 3D (as height map above the $x-y$-plane), can be shown to be a circular paraboloid.\nThe global minimum of the trace is attained at\n\\[\nx^* = u - f \\frac{\\Sigma_{12}}{\\Sigma_{11}}\n\\enspace\\enspace\\enspace\ny^* = v - f \\frac{\\Sigma_{13}}{\\Sigma_{11}}\n\\]\nand has the following value:\n\\begin{equation*}\n\\Sigma_{22} + \\Sigma_{33} - \\frac{\\Sigma_{12}^2 + \\Sigma_{13}^2}{\\Sigma_{11}}\n\\end{equation*}\nA few observations are as follows. \nIf the covariance between the focal length and the principal\npoint coordinates ($\\Sigma_{12}$ and $\\Sigma_{13}$) approaches zero, then the minimum of the uncertainty map coincides with the\nprincipal point and its value there is the sum of the variances of the principal point coordinates.\nAnd if the variance of the focal length tends to zero, the uncertainty map tends to being uniform.\nTherefore, when the calibration result gets more accurate (low uncertainty), the range of the uncertainty map will become smaller, and its minimum will be approaching the principal point.\nFinally, an empirical observation is that the covariance matrices $\\Gamma$, when visualized as uncertainty ellipses, tend to be oriented towards the minimum of the uncertainty map (i.e.\\ at each image point, the associated uncertainty ellipse has an axis that ``points'' towards the minimum of the uncertainty map).\n\nMore analysis, also for more complex camera models, is possible.\n\nIn summary, we believe that this concept is a principled way of displaying and interpreting the uncertainty of a calibration estimate.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \\label{intro}\n\n On a smooth connected   compact   $4$-manifold $M$, the {\\em Weyl functional} \n\\begin{equation}\n\\label{weylfun}\n\\mathcal{W}(g) := \\int_M\\| W\\|_{g}^2~d\\mu_g~,\n\\end{equation}\n quantifies  the deviation  of  a Riemannian metric\n$g$ from  local conformal flatness. \nHere $W$ denotes the {\\em Weyl tensor} of $g$, which is the piece of the Riemann curvature of $g$ complementary to the Ricci tensor, while  \n the norm $\\|\\cdot\\|_g$  and the volume form $d\\mu_g$ in the integrand\nare    those associated with the given metric $g$. \n The Weyl functional \\eqref{weylfun} is invariant   not only under the  action of \nthe diffeomorphism group (via pull-backs), but also under the action  of the smooth positive functions ${\\zap f}: M\\to {\\mathbb R}^+$\n by conformal rescaling $g \\rightsquigarrow {\\zap f}g$. \n\n\nIt is both natural and  useful to study   metrics that are critical points of Weyl functional. The   Euler-Lagrange equations for  such  a metric can be expressed \n\\cite{Bach,bes}  as \n$B=0$,\nwhere the {\\em Bach tensor} $B$ is defined by\n$$\nB_{ab}:=(\\nabla^c\\nabla^d+\\frac{1}{2}r^{cd})\nW_{acbd}, \n$$\nso  these critical metrics  are  said to be \n{\\em Bach-flat}.  For reasons reviewed  in \\S \\ref{pdq},  every $4$-dimensional \n {conformally  Einstein} metric  is Bach-flat, as is every  {anti-self-dual} metric.\n Conversely, if  the Bach-flat manifold $(M^4,g)$ also happens to be {\\em K\\\"ahler}, Derdzi\\'{n}ski \\cite[Prop. 4]{derd} discovered that \n the  geometry of $g$ must locally be of one of these two types near a generic point. Our purpose here is to \n sharpen this observation into a global classification of solutions. Our main result is the following:\n \n\n \n \n\n\n\n\\begin{main} \\label{summary}\nLet $(M^4, g, J)$ be a  compact connected  Bach-flat  K\\\"ahler surface. \nThen $g$ is either anti-self-dual, or else is  conformally Einstein on an open dense subset\nof $M$. Moreover, the geometric behavior of $(M^4, g, J)$ fits into  exactly one slot of the following \nclassification scheme: \n\\begin{enumerate}[{\\rm I.}]\n\\item The scalar curvature satisfies $s > 0$ everywhere. In this case,  there are  just two possibilities:\n\\begin{enumerate}[{\\rm (a)}]\n\\item $(M,g, J)$ is K\\\"ahler-Einstein, with $\\lambda >  0$; or else \n\\item $(M, s^{-2}g)$ is Einstein,  with $\\lambda >  0$, but has holonomy $\\mathbf{SO}(4)$. \n\\end{enumerate} \n\\item The scalar curvature satisfies $s \\equiv 0$. There are again two possibilities: \n\\begin{enumerate}[{\\rm (a)}]\n\\item $(M, g, J)$ is K\\\"ahler-Einstein, with $\\lambda =  0$; or else \n\\item  $(M,J)$ is a (possibly blown-up) ruled surface, and $g$ is anti-self-dual, but $M$ is not even homeomorphic to an Einstein manifold. \n\\end{enumerate} \n\\item The scalar curvature satisfies $s < 0$ somewhere. Then there are again exactly two possibilities: \n\\begin{enumerate}[{\\rm (a)}]\n\\item $(M,g, J)$ is K\\\"ahler-Einstein, with $\\lambda < 0$; or else \n\\item $(M,J)$ is a (possibly blown-up) ruled surface, and  $s$  vanishes exactly along a smooth connected totally umbilic  hypersurface $\\mathcal{Z}^3\\subset M^4$. \nMoreover, $M- \\mathcal{Z}$ has precisely \n two connected components,  and on both  of these $h:=s^{-2}g$  is a complete \nEinstein metric with $\\lambda < 0$. \n\\end{enumerate}\n\\end{enumerate}\n\\end{main} \n\n\nA great deal is already known about most  cases in this classification:\n\n\\begin{itemize}\n\\item \nA complex surface admits \\cite{chenlebweb,sunspot,tian} a metric of class {I} iff it has $c_1> 0$. Moreover, this metric is always unique \\cite{bandomab,lebuniq} up to \ncomplex automorphisms and homotheties. \n\\begin{itemize}\n\\item The relevant metric is of type {I(a)} iff the complex automorphism group is reductive. \n\\item  Up to isometry and rescaling, there are \\cite{lebuniq} exactly two solutions of type {I(b)}. \n\\end{itemize}\n\\item Metrics of class {II} are  generally called  scalar-flat K\\\"ahler metrics. These are exactly the K\\\"ahler metrics that are anti-self-dual, in the sense that\nthe self-dual Weyl tensor $W_+$  vanishes identically. \n\\begin{itemize}\n\\item The metrics of type {II(a)}   are often called Calabi-Yau metrics.  A K\\\"ahler-type complex surface admits such  metrics  \\cite{yauma} iff it has  $c_1=0$ mod torsion. \n This happens \n  exactly for  the minimal complex surfaces of  Kodaira dimension $0$ for which  $b_1$ is even.   When a complex surface admits such a metric,  there is  then  exactly one \nsuch metric in every K\\\"ahler class. \n\\item  Any complex surface that  admits a metric of type {II(b)} must be projective-algebraic and have  Kodaira dimension $-\\infty$; \nthey all violate the Hitchin-Thorpe inequality, and most of them are non-minimal. \nSuch solutions exist in \ngreat abundance \\cite{klp,leblown,rolsing}; in particular,   any    complex surface with $b_1$ even and \nKodaira dimension $-\\infty$   has blow-ups  that admit metrics of this type. \n\\end{itemize}\n\\item The two cases  that make up class {III} are wildly different. \n\\begin{itemize}\n\\item A complex surface admits a metric of  type {III(a)} iff \\cite{aubin,yau} it has $c_1 < 0$. This happens \\cite[Prop. VII.7.1]{bpv} iff the complex surface \nis minimal,  has Kodaira dimension $2$,  and contains no rational curve of  self-intersection $-2$.\n\n\n\\item  By contrast, any complex surface admitting a metric of type {III(b)} must have Kodaira dimension $-\\infty$. Infinitely many solutions  of this type are currently known\n\\cite{hwasim,tofribach}. However,  the known solutions all display  peculiar features that  seem most likely to  just be \nartifacts of  the method of construction. \n\\end{itemize}\n\\end{itemize}\n\nOur exposition begins, in \\S \\ref{pdq}, with a review of some key background facts that will help set the stage  for our main results. We then develop \n the basic trichotomy of solutions  in \\S \\ref{tri}.  The proof of Theorem \\ref{summary} is then completed in \\S \\ref{rapt} by carefully proving various auxiliary \nassertions regarding  specific cases  in our classification scheme. The article then concludes by proving some additional  results\nabout specific types of solutions, with a particular focus on  interesting open problems.\n\n\n \n\\section{The Weyl Functional}\n\\label{pdq}\n\n\n     \nIf $(M,g)$ is any smooth compact oriented Riemannian $4$-manifold, \nthe Thom-Hirzebruch signature theorem $\\mathbf{p_1} = 3\\tau$ implies\na Gauss-Bonnet-like integral formula \n\\begin{equation}\n\\label{th} \n\\tau (M) = \\frac{1}{12\\pi^2}   \\int_{M}\\left( |W_+|^{2}_g- |W_+|^{2}_g\\right) d\\mu_{g}\n\\end{equation}\nfor the signature $\\tau = b_+ - b_-$ of $M$. In this formula,  $W_\\pm = (W\\pm \\star W)\/2$  denotes\nthe  self-dual (respectively, anti-self-dual) \npart of the Weyl curvature \n$${W^{ab}}_{cd} ={\\mathcal{R}^{ab}}_{cd} - 2 {r}^{[a}_{[c}\\delta^{b]}_{d]}+\\frac{s}{3} \\delta^a_{[c}\\delta^b_{d]}$$\nof the given metric $g$, here expressed in terms of the Riemann curvature tensor $\\mathcal{R}$, Ricci tensor $r$, and scalar curvature $s$.\nWe remind the reader of the fundamental fact that ${W^a}_{bcd}$ is conformally invariant;  this  in particular explains\nthe conformal invariance of the Weyl functional   \\eqref{weylfun}. \nBut now notice  that \\eqref{th}  allows one to re-express the Weyl functional \\eqref{weylfun} \nas\n\\begin{equation}\n\\label{funky}\n{\\mathcal W}(g)= \n -12\\pi^2 \\tau (M) + 2\\int_M | W_+|^2 d\\mu ~.\n\\end{equation}\n\nNow, for  any  smooth 1-parameter family of metrics \n$$g_t:= g +t\\dot{g}+ O(t^2)$$ \nthe first variation of the Weyl functional is   given by \n\t$$\\left. \\frac{d}{dt}{\\mathcal W}(g_{t})\\right|_{t=0} =\n\t-\\int {\\dot{g}}^{ab} B_{ab} ~d\\mu $$\nwhere \\cite{Bach,bes} the {\\em Bach tensor} $B$ is given by\n\\begin{equation}\n\\label{bach1}\nB_{ab}=(\\nabla^c\\nabla^d+\\frac{1}{2}r^{cd})\nW_{acbd}\n\\end{equation}\nNotice that the contracted Bianchi identity\n$$\\nabla^aW_{abcd} = \\nabla_{[c} r_{d]b} + {\\textstyle \\frac{1}{6}} g_{b[c} \\nabla_{d]}s$$\n implies that any Einstein metric \nsatisfies the {\\em Bach-flat} condition $B=0$. However, since \n the conformal invariance of the Weyl functional  also makes it\nclear  that  the  Bach-flat condition is \nconformally invariant, it follows that any conformally Einstein $4$-dimensional metric \n is automatically Bach-flat.  \n \n \n On the other hand,  any oriented Riemannian 4-manifold satisfies the remarkable identity\n$$(\\nabla^a\\nabla^b + \\frac{1}{2}r^{ab}){(\\star W)}_{cabd}  =0, $$\nwhich encodes the fact that the first variation of \\eqref{th} is zero. \nThis allows one to  rewrite \\eqref{bach1} as \n \\begin{equation}\n\\label{bach2}\n B_{ab}= (2\\nabla^c\\nabla^d+r^{cd})(W_+)_{acbd} . \n\\end{equation}\nIn particular, any metric with $W_+\\equiv 0$ is automatically Bach-flat. Indeed, \nequation \\eqref{funky} shows that such {\\em anti-self-dual} metrics are  actually\nminimizers of the Weyl functional, and so must  satisfy the associated Euler-Lagrange equation \n$B=0$. \n  \nThe Bach tensor is automatically symmetric, trace-free, and divergence-free. This reflects the\nfact  that  $-B$ is the  gradient of $\\int|W|^2d\\mu$,\nwhich  is invariant under diffeomorphisms and rescalings. \nSince  $B$ must therefore be $L^2$-orthogonal to any tensor\nfield of the form $ug_{ab}$ or $\\nabla_{(a} v_{b)}$, we  have\n\\begin{equation}\n\\label{harmonica}\nB_{ab}= B_{ba}, ~~ {B_a}^a=0, ~~\\nabla^aB_{ab}=0\n\\end{equation}\nfor any $4$-dimensional Riemannian metric.  \n\nWe now narrow our discussion to the case of K\\\"ahler metrics. \nFor any K\\\"ahler metric $g$ on a complex surface $(M,J)$,   with the orientation \ninduced by $J$, the self-dual Weyl tensor\nis given by \n\\begin{equation}\n\\label{kahlerweyl}\n{(W_+)_{ab}}^{cd}= \\frac{s}{12}\\left[\n\\omega_{ab}\\omega^{cd} - \\delta_{a}^{[c} \\delta_{b}^{d]}+ {J_a}^{[c}{J_b}^{d]}\n\\right]\n\\end{equation}\nand so \nis completely determined by the \nscalar curvature and the K\\\"ahler form $\\omega = g(J \\cdot , \\cdot)$. \nIn particular,  we therefore have \n\\begin{equation}\n\\label{sebastian}\n|W_+|^2 = \\frac{s^2}{24}, \n\\end{equation}\na key fact whose lack of conformal invariance ceases to seem  paradoxical as soon as one   recalls \nthat the K\\\"ahler condition isn't conformally invariant either. \nIn conjunction, equations \\eqref{funky} and \\eqref{sebastian} now tell us that any Bach-flat K\\\"ahler metric is a critical point of the Calabi functional \n\\begin{equation}\n\\label{calfun}\n{\\mathcal C} (\\omega ) =  \\int s^2 d\\mu ~,\n\\end{equation}\nconsidered either as a functional on a fixed K\\\"ahler class $\\Omega =[\\omega ]$, or\non the entire space of K\\\"ahler metrics, with $\\Omega$ allowed to vary. \nIn particular, a conformally  Einstein, K\\\"ahler metric $g$ must\nbe an extremal K\\\"ahler metric in the sense of Calabi \\cite{calabix}.\nOne of  several equivalent characterizations of an extremal metric is the requirement   that \n $\\xi :=J\\nabla s$  be a Killing field of $g$. \n\n\n\nPlugging \\eqref{kahlerweyl}  into \\eqref{bach2}, we now obtain a concrete  formula \n$$B_{ab}= \\frac{s}{6}\\mathring{r}_{ab} + \\frac{1}{4}{J_a}^c{J_b}^d\\nabla_c\\nabla_d s\n+\\frac{1}{12} \\nabla_a\\nabla_b s + \\frac{1}{12}g_{ab}\\Delta s$$\nfor the Bach tensor of any K\\\"ahler metric, where \n$\\mathring{r}$ denotes  the trace-free part \n$$\\mathring{r}_{ab} = r_{ab} - \\frac{s}{4}g_{ab}$$\nof the Ricci curvature. \nSetting $J^*(B) = B(J\\cdot , J \\cdot )$, we next decompose the Bach tensor \n$$B=B^\\boxplus +B^\\boxminus$$\ninto its $J$-invariant and $J$-anti-invariant parts, and observe that \n\\begin{eqnarray}\n\\label{johann} \nB^\\boxplus &:=& \\frac{1}{2}\\left[ B + J^*(B)\\right]=\n\\frac{1}{6} \\Big[ s \\mathring{r} + 2 \\Hess^\\boxplus_0(s)\\Big] \\\\\nB^\\boxminus&:=& \\frac{1}{2}\\left[ B - J^*(B)\\right]=\\frac{1}{12} \\Big[  \\Hess (s) -\nJ^*\\Hess (s) \\Big] ~.\\nonumber \n\\end{eqnarray}\nHere $\\Hess = \\nabla\\nabla$ denotes  the Hessian of a function, and \n $\\Hess^\\boxplus_0$ is its trace-free, $J$-invariant  part. \nNow notice that,  since $s$ is real-valued, $\\Hess (s) =\nJ^*\\Hess (s)$ if and only if  \n$$\\nabla_{\\bar{\\mu}}\\nabla^{\\nu} s = g^{{\\nu}\\bar{\\lambda}}\\nabla_{\\bar{\\mu}}\\nabla_{\\bar{\\lambda}}s= 0,$$\nand this is exactly Calabi's equation $\\overline{\\partial}\\nabla^{1,0}s =0$ for an extremal K\\\"ahler metric. Consequently, \na K\\\"ahler metric $g$ is extremal iff its Bach tensor $B$ is $J$-invariant. \nWhen this happens,  conditions  \\eqref{harmonica} then tells us that $\\psi = B(J \\cdot , \\cdot )$ is a harmonic \nanti-self-dual $2$-form, and  the symmetric  tensors\n$g+ tB$ are therefore   $J$-compatible \nK\\\"ahler metrics for all small $t$. Since $\\dot{g}=B$ for this variation of the metric, and since $-B$ is the gradient of $\\mathcal{W}$, \nit therefore follows  \\cite{chenlebweb}    that   any critical point of the Calabi functional $\\mathcal{C}$ on the space of {all} \n$J$-compatible K\\\"ahler metrics must   actually be Bach-flat. \n\n\n\n\n\n\nRevisiting   \\eqref{johann}  now reveals  that a  K\\\"ahler metric is Bach-flat iff \nit is extremal  and satisfies\n\\begin{equation}\n\\label{rico}\n0= s\\mathring{r} + 2  \\Hess_0(s).\n\\end{equation}\nHowever, \n the trace-free Ricci tensor $\\mathring{r}$ always transforms  \\cite{bes} under conformal changes $g\\rightsquigarrow u^2g$ \nby \n\\begin{equation}\n\\label{riforma}\n\\mathring{r}\\rightsquigarrow \n\\hat{\\mathring{r}}= \\mathring{r} +2 u \\Hess_0 (u^{-1})~.\n\\end{equation}\nThus, as was first pointed out by Derdzi\\'{n}ski \\cite{derd}, \nthe peculiar  conformal rescaling $h=s^{-2}g$ of a \nBach-flat K\\\"ahler metric satisfies $\\mathring{r}=0$, and so is locally Einstein, \non the (possibly empty) open set where  $s\\neq 0$. We  will  now begin to systematically explore the  global ramifications of this observation. \n\n\n\\section{The Basic Trichotomy} \n\\label{tri}\n\nIn this section, we will study the global behavior of the scalar curvature $s$ on a   compact Bach-flat  K\\\"ahler surface. \nOur approach hinges on a  general  property of   strictly extremal K\\\"ahler manifolds: \n\n\n\\begin{lem} \n\\label{bott} \nLet  $(M^4, g,J)$  be a  compact connected extremal K\\\"ahler surface whose scalar curvature $s$ is non-constant.  Then \n $s : M \\to {\\mathbb R}$ is  a generalized Morse function in the sense of Bott \\cite{bottcrit}. In other words, \nthe  locus where  $\\nabla s =0$ is a disjoint union $\\bigsqcup C_j \\subset M$ of compact submanifolds, and the Hessian $\\Hess(s):= \\nabla \\nabla s$ \nis non-degenerate  on the normal bundle  $(TC_j)^\\perp$ of each  $C_j$. Moreover,  each  submanifold $C_j$ is  either a single point or \n a smooth compact connected complex curve.  \n\\end{lem}\n\n\\begin{proof} Since $s$ is non-constant, $(M,g,J)$ is a {\\em strictly} extremal K\\\"ahler manifold, and $\\xi= J\\nabla s$ is a {\\em non-trivial} Killing field. The critical points\nof $s$ are exactly the fixed points of the flow $\\{ \\Phi_t : M\\to M~|~ t\\in {\\mathbb R}\\}$ generated by $\\xi$, and since the diffeomorphisms $\\Phi_t$\nare all isometries of $(M,g)$, every connected component $C_j$ of the fixed point set is \\cite{kobfixed} a totally geodesic submanifold. \n\nAt  $p\\in C_j$, let $v\\in (T_pC_j)^\\perp$ be a unit vector normal to $C_j$, and let \n$\\gamma : {\\mathbb R} \\to M$ be the unit speed geodesic through $p= \\gamma (0)\\in C_j$ with initial tangent vector $v$. Since $\\xi=0$ only at \n$\\bigsqcup C_k$,  we then have \n$\\xi|_{\\gamma (t)}\\neq 0$ for all sufficiently \nsmall $t> 0$. However, since $\\xi$ is Killing, $\\xi\\circ \\gamma$ is a Jacobi field along $\\gamma$. Because \n$\\xi|_{\\gamma (0)}=0$, we must therefore have $\\nabla_{\\gamma^\\prime(0)}\\xi\\neq 0$, as \nJacobi's equation is a linear second order ODE, and $\\xi$ would therefore vanish identically along \n$\\gamma$ if the initial value $(\\xi|_p , (\\nabla_v\\xi)|_p)$ of this solution vanished. \nThis shows that  that  $v$ cannot belong to \n  the kernel of $v\\mapsto (\\Hess  s)(v, \\cdot) = \\omega ( \\cdot , \\nabla_v\\xi)$;  and since $v$ is an arbitrary unit normal vector, it follows that the restriction of the Hessian \n$\\nabla \\nabla s$ to the normal bundle $(TC_j)^\\perp$  must be non-degenerate. This shows  that  $s: M\\to {\\mathbb R}$ is  a  Morse-Bott function, as claimed. \n\nIn particular, the tangent space $TC_j$ of any component of the critical locus must  exactly coincide with the kernel of the Hessian $\\nabla\\nabla s$\nat any point. But since $\\nabla^{1,0}s$ is a holomorphic vector field, this Hessian must be $J$-invariant. Hence $TC_j= \\ker \\Hess (s)$ is also $J$-invariant, and $C_j\\subset M$\nis therefore a complex submanifold. Since $s$ is non-constant by assumption, and since $M$ is assumed to have complex dimension $2$, the components $C_j$\ncan only have complex dimension $0$ or $1$. Each component $C_j$ of the critical locus is therefore either a single point or a  totally geodesic compact complex curve. \n\\end{proof}\n\nThis immediately tells us something useful about  the zero set\n$$\\mathcal{Z}:=  \\{ p\\in M~|~ s(p) = 0\\}$$ \nof the scalar curvature. \n\n\\begin{lem} \\label{lemon} Let $(M,g,J)$ be a compact extremal K\\\"ahler manifold. \n If $s\\not\\equiv 0$, then the open subset $M-\\mathcal{Z}$ is dense in $M$. \n\\end{lem}\n\\begin{proof} If $s$ were a non-zero constant, $\\mathcal{Z}$ would be empty, and there would be nothing to prove. We may thus assume from now on that \n$s$ is non-constant. Lemma \\ref{bott} then tells us that $\\nabla s$ and $\\nabla \\nabla s$ can never vanish at the same point.\nIn particular,  if $p$ is any point where $s(p)=0$,\n Taylor's theorem with remainder  allows us to construct a  short embedded curve $\\gamma : (-\\varepsilon , \\varepsilon) \\to M$ through $p= \\gamma (0)$ \n on which $s\\circ \\gamma$ vanishes only at the origin. Hence  every point of $\\mathcal{Z}$ belongs\nto the closure of $M-\\mathcal{Z}$. This shows that $M-\\mathcal{Z}$ is dense, as claimed. \n\\end{proof}\n\nWe now specialize to the case of {\\em Bach-flat} K\\\"ahler surfaces. Since  equation \\eqref{johann} tells us  that  these K\\\"ahler manifolds\n are in particular extremal,  the  \nabove Lemmata therefore   automatically apply. However, we have already observed  that the equation  $B=0$ can be expressed as \n$$0= \\mathring{r} + 2 s^{-1} \\Hess_0 s$$\non the open set  $M-\\mathcal{Z}$ defined by  $s\\neq 0$, and, by equation \\eqref{riforma},  this is exactly equivalent to saying that \nthe metric $h:= s^{-2}g$ on $M-\\mathcal{Z}$  satisfies $\\mathring{r}=0$. \nSince the doubly-contracted  Bianchi identity $2\\nabla\\cdot r =  \\nabla s$  implies that a $4$-manifold \nwith $\\mathring{r}=0$ \nmust have locally constant scalar curvature, \nthis means that the function $\\kappa$ defined by\n\\begin{equation}\n\\label{capital}\n\\kappa =  -6s \\Delta s - 12 |\\nabla s|^2 + s^3\n\\end{equation}\nis locally constant on $M-\\mathcal{Z}$; indeed,  on this open set \n$$\\kappa=s^3 ( 6\\Delta + s) s^{-1}$$\nexactly represents the scalar curvature of the local Einstein metric $h=s^{-2}g$. \nOn the other hand, \nsince elliptic regularity  implies  \\cite{ls2}\n that any extremal K\\\"ahler metric \nis smooth with respect to the complex atlas, \nour definition \\eqref{capital} of  $\\kappa$ certainly guarantees that it is a smooth function on all of $M$. \nThese facts now allow us to   deduce the following: \n\n\n\n\n\n\\begin{lem} On any compact connected Bach-flat K\\\"ahler surface $(M,g,J)$, \nthe smooth function $\\kappa : M\\to {\\mathbb R}$ defined by \\eqref{capital} is necessarily constant. \n\\end{lem}\n\\begin{proof} If $s\\equiv 0$, equation \\eqref{capital} immediately tells us that \n  $\\kappa\\equiv 0$, and we are done. We may therefore assume henceforth that  $s\\not\\equiv 0$. Now  notice that \nthe smooth $1$-form $d\\kappa$  vanishes on the  set $M-\\mathcal{Z}$,  since on this set $\\kappa$ is locally the scalar curvature of the Einstein metric $h=s^{-2}g$, \nand is therefore locally constant. \nBut since $(M-\\mathcal{Z} ) \\subset M$ is dense  by Lemma \\ref{lemon}, it therefore follows that $d\\kappa \\equiv 0$ by continuity. \nIntegration on paths thus  shows  that $\\kappa$ is constant, as claimed. \n\\end{proof}\n\n\nThe sign of $\\kappa$ thus provides a basic trichotomy that will form the basis of our classification of these manifolds. However, the\nsign of $\\kappa$ also has a direct interpretation in terms of the behavior of the scalar curvature of the given K\\\"ahler metric: \n\n\n\\begin{lem} \n\\label{sign}\nOn any compact connected Bach-flat K\\\"ahler surface $(M,g,J)$, \nthe minimum value $\\min s$ of the scalar curvature  of $g$  has exactly the same sign (positive, negative, or zero) as the constant $\\kappa$. \n\\end{lem}\n\\begin{proof}\nWhen the scalar curvature $s$ is constant,  \\eqref{capital} says $\\kappa = s^3 = (\\min s)^3$, so the claim obviously holds. \nOtherwise, $s$ is non-constant, and  Lemma \\ref{bott} tells us that $\\Hess (s)\\neq 0$ at any critical point of $s$. In particular, if\n$p\\in M$ is a point where $s$ achieves its minimum, $\\Delta s :=   - \\nabla^a\\nabla_a s <  0$ at $p$, since we now know that \n$\\Hess s= \\nabla\\nabla s$ must be positive semi-definite and non-zero at\na minimum.    However, evaluation of  equation \\eqref{capital} at $p$ tells us that   $$ \\kappa= s(p)~\\left[s^2 - 6\\Delta s\\right](p),$$\nsince $|\\nabla s|^2(p)=0$. Since $[s^2 - 6 \\Delta s ](p) > 0$, this shows that \n $\\kappa$ and $s(p) = \\min s$ must have the same sign,  and the result therefore follows.  \n\\end{proof}\n\n\nOur next result leads to  a complete understanding of   the   $\\kappa=0$ case. \n\n\\begin{prop} A  compact connected K\\\"ahler surface $(M,g,J)$ is Bach-flat  and has $\\kappa=0$ if and only if its scalar curvature $s$ vanishes identically. \n\\label{sfk}\n\\end{prop}\n\\begin{proof} \nIf  our K\\\"ahler surface $(M,g,J)$ \nhas $s\\equiv 0$, it is anti-self-dual by \\eqref{sebastian},\nand therefore Bach-flat; the fact that such a manifold has $B=0$ \nis also directly confirmed by  \\eqref{johann}. \nInspection of equation \\eqref{capital} now reveals that it also has $\\kappa=0$.\n\n\nConversely, \nif $(M,g,J)$ is a Bach-flat K\\\"ahler surface with \n$\\kappa =0$, then Lemma \\ref{sign}  tells us that  $\\min s =0$. Thus,   $\\mathcal{Z}= s^{-1}(0)$ is non-empty, and every $p\\in \\mathcal{Z}$\nis a minimum of $s$. \nWe will now argue by contradiction, and assume  that \n  $s\\not\\equiv 0$. This implies    that $s$ is   non-constant, so \nLemma \\ref{bott} now tells us  that $\\Hess(s):= \\nabla \\nabla s$ must be non-zero at any point where $\\nabla s=0$. \nBut since any point $p\\in \\mathcal{Z}$ is  a minimum of $s$,  this means  that  $\\Hess(s)\\neq 0$ at every point of $\\mathcal{Z}$. \nHowever, equation \\eqref{rico} tells us  that the trace-free part $\\Hess_0(s)$ of the Hessian {\\em does} vanish \nalong the locus $\\mathcal{Z}$ defined by  $s=0$.  Thus, for every point $p\\in \\mathcal{Z}$, there is\na constant $a=a(p)\\neq 0$ such that \n\\begin{equation}\n\\label{ompholos}\n\\nabla \\nabla s = 2a g\n\\end{equation}\nat $p$. Since $p$ is a minimum of $s$, we  must moreover have $a > 0$. Hence every  $p\\in\\mathcal{Z}$ is a non-degenerate local minimum of $s$,\nand it therefore \n follows that $\\mathcal{Z}$ is discrete. Since $M$ is compact,  this then implies that  $\\mathcal{Z}$ is  finite. \n\nNow let $p\\in \\mathcal{Z}$ be any point where $s$ vanishes, and let $\\varrho$ be the Riemannian distance from $p$ in $(M, g)$. Since  \\eqref{ompholos}\nguarantees that $\\nabla \\xi = 2a J$ at $p$, \n the isometry $\\Phi_{\\pi \/2a}$, gotten by flowing along $\\xi$ for time $t={\\pi}\/{2a}$, therefore fixes $p$,  but  reverses the direction of each geodesic through $p$.  \nThe Taylor expansion of $s$ in geodesic normal coordinates $x^j$ centered at $p$ therefore contains only terms of even order, and \\eqref{ompholos} therefore tells us that \n$$s = a  \\varrho^2 + O (\\varrho^4 ).$$\nOn the other hand,  we also have\n\\begin{eqnarray*}\ng_{jk} &=& \\delta_{jk}+ O(\\varrho^2)\\\\\ng_{jk,\\ell}&=&O(\\varrho)\n\\end{eqnarray*}\nin geodesic normal  coordinates. \nIf we now pass to inverted coordinates $\\tilde{x}^j= x^j\/(a \\varrho^2)$ and set $\\text{\\cyr   ya} :=\\sqrt{\\sum_j (\\tilde{x}^j)^2}= 1\/(a \\varrho)$, the metric  $h=s^{-2}g$ thus \nsatisfies \n\\begin{eqnarray*}\nh_{jk} &=& \\delta_{jk}+ O(\\text{\\cyr   ya}^{-2})\\\\\nh_{jk,\\ell}&=&O(\\text{\\cyr   ya}^{-3})\n\\end{eqnarray*}\n so that $(M-\\mathcal{Z} , h)$ is {\\em asymptotically flat}. However, since $\\kappa=0$, the Einstein metric $h$ is actually Ricci-flat.\n This in particular implies   \\cite{bartnik} that each end of $(M-\\mathcal{Z}, h)$ has mass zero. The positive mass theorem \\cite{syaction}\n therefore asserts that \n$(M-\\mathcal{Z} , h)$ is isometric to Euclidean  ${\\mathbb R}^4$. In particular, $g$ is conformally flat on $M-\\mathcal{Z}$, and so has \n$W_+\\equiv 0$ on this open dense set. But  by \\eqref{sebastian}, this means that the K\\\"ahler metric $g$ satisfies $s\\equiv 0$ on $M-\\mathcal{Z}$.\nBut $M-\\mathcal{Z}$ is by definition precisely the set where $s\\neq 0$, so this is a contradiction!  In other words,  $M-\\mathcal{Z}$ must actually be empty, \nand  any compact Bach-flat K\\\"ahler surface with $\\kappa =0$ must therefore  have \n$s\\equiv 0$, as claimed. \n \\end{proof}\n\n\n\n\n\n\\begin{rmk} The above proof can be recast in a way that avoids using the positive mass theorem. Indeed,  the  volume growth \nof an asymptotically Euclidean Ricci-flat manifold with one end must be exactly Euclidean in the large-radius limit, and would be even larger if there were several ends. \nThe Bishop-Gromov inequality \\cite{bishop,gromhaus} thus forces the exponential map of any Ricci-flat asymptotically flat manifold to actually be an isometry. This shows \nthat the only such manifold is Euclidean space. \n\nThe  final contradiction could also have been rephrased  so as to  emphasize   topology instead of  geometry. \nFor example, if $M-\\mathcal{Z}$ were diffeomorphic to  ${\\mathbb R}^4$, it  would  only have\none end, so  $\\mathcal{Z}$ would necessarily  consist of a single point $p$, and  $M$  would have  to be homeomorphic to $S^4= {\\mathbb R}^4\\cup \\{ p\\}$. \nBut this is absurd, because, for example, the K\\\"ahler class $[\\omega]$  of $(M,g,J)$ is a non-zero element of $H^2(M,  {\\mathbb R})$, while  $H^2(S^4, {\\mathbb R})=0$. \nAlternatively, one could obtain a contradiction at this same juncture \n by  emphasizing that $(M,J)$ is by hypothesis a complex surface, whereas  $S^4$ does not  \\cite{wusphere} even admit  an almost-complex structure.\n\\end{rmk}\n\nIt now  remains for us  to analyze the two cases $\\kappa > 0$ and $\\kappa < 0$. The first of these is simpler, and  is quite thoroughly understood. \n \n\\begin{prop} \n\\label{pop}\nIf the constant $\\kappa$ is positive, the scalar curvature $s$ of the extremal K\\\"ahler manifold $(M,g,J)$ is  everywhere positive. Consequently,   \n$\\mathcal{Z}$ is empty, \nand $(M,h)$ is a compact   Einstein $4$-manifold  with positive Einstein constant. \n\\end{prop}\n\\begin{proof} If $\\kappa > 0$,  Lemma \\ref{sign} tells us that $\\min s > 0$, too. Thus $s> 0$ everywhere on $M$. Hence \n$\\mathcal{Z} =s^{-1}(0) = \\varnothing$, and  $M- \\mathcal{Z} = M$. Thus $h=s^{-2}g$ is a globally defined \nEinstein metric on $M$, with positive Einstein constant $\\lambda = \\kappa \/4$. \n\\end{proof}\n\nPrevious  results \\cite{lebuniq} therefore provide a complete classification of $\\kappa > 0$  solutions.  We will say more about this classification in \\S \\ref{conclusion}  below.\n\n\\bigskip \n\nBy contrast,  the  $\\kappa < 0$ case is distinctly  more complicated: \n\n\n\\begin{prop}\n\\label{understand}\nIf  the constant $\\kappa$ is negative, and if $(M,g, J)$ is not K\\\"ahler-Einstein, then $\\mathcal{Z}$ is a smooth connected real hypersurface $\\mathcal{Z}^3 \\subset M^4$, \nand  the  complement $M-\\mathcal{Z}$ of this hypersurface consists of exactly two connected components $M_+$ and $M_-$.  The Einstein manifolds $(M_\\pm , h:=s^{-2}g)$ are \nboth complete, and have\nnegative Einstein constant. Moreover, these two manifolds are both Poincar\\'e-Einstein, with the same conformal infinity $(\\mathcal{Z} , [g|_\\mathcal{Z} ])$. \n\\end{prop}\n\n\\begin{proof}\nWhen $\\kappa < 0$,   Lemma \\ref{sign}  tells us that $\\min s < 0$. If $s$ is constant, the constant is thus negative, and $s$ is  nowhere zero. Equation \n \\eqref{rico} thus implies  that the K\\\"ahler metric $g$ is actually Einstein. \n \nWe may thus henceforth assume  that $s$ is non-constant. Lemma \\ref{bott} therefore tells us  that \n$s: M \\to {\\mathbb R}$ is  a Morse-Bott function. \nIn particular, $\\Hess s\\neq 0$ at any critical point of $s$. \nIf $p$ is a point where $s$ attains its minimum, we therefore have $\\Delta s = - \\nabla^a\\nabla_a s< 0$ at $p$. However, since $\\kappa < 0$, we also have $s (p ) = \\min s < 0$\nby Lemma \\ref{sign}. Thus $s\\Delta s > 0$ at $p$. On the other hand, since  $\\nabla s=0$ at any\n minimum, \\eqref{capital} tells us that \n$6 s\\Delta s = s^3 - \\kappa$\nat $p$.  Hence  $\\min (s^3 -\\kappa) = [s^3-\\kappa] (p) = [6s\\Delta s](p) > 0$, and we therefore have \n $s^3 - \\kappa > 0$ on all of $M$. Equation \\eqref{capital} now  tells us that \n $$s\\Delta s + 2|\\nabla s|^2 = \\frac{s^3- \\kappa}{6} > 0$$\neverywhere. In particular, we have $s\\Delta s > 0$ at every critical point of $s$.\n\nIf $q$ is now a point where $s$ attains its maximum, our Morse-Bott argument  predicts that $\\Delta s = -\\nabla^a\\nabla_a s> 0$ at $q$, since $\\Hess s$ is negative semi-definite and \nnon-zero at $q$. Since we have shown that $s\\Delta s > 0$ at every critical point, and so  in particular at  $q$,   it therefore follows that $\\max s = s (q) > 0$. \n\n\nSince $M$ is connected, and since $s$ takes on both positive and negative values, \nthe locus $\\mathcal{Z}$ defined by $s=0$ must therefore be  non-empty. \nMoreover,  since  $s\\Delta s > 0$ at every critical point of $s$, it follows that the locus $\\mathcal{Z}$ defined by $s=0$  cannot contain any critical points. \nIn other words,  $0$ is a regular value of $s$, and $\\mathcal{Z}=s^{-1}(0)$ is a therefore a smooth compact non-empty   real hypersurface\nin $M$.  \n\n\n\n\nNow since $s: M\\to {\\mathbb R}$ is a  Morse-Bott function, \nand since $\\Hess s$ is $J$-invariant, any complex-codimension-one component $C_j$ of the critical set \nmust be a local maximum or a local minimum of $s$. The  other critical points of $s$ are isolated, and  $\\Hess s$ is non-degenerate at these  critical points,\nwith  index $0$, $2$, or $4$; those of index $0$ are local minima, those of index $4$ are local maxima, and those of\nindex $2$ are saddle points where the  Hessian is of type $({+}{+}{-}{-})$. \nThis  dictates   the manner in which the sub-level sets $M_t := s^{-1} \\left( (-\\infty , t]\\right)$ can change as we increase $t$. \nIndeed, for regular values $t_1 < t_2$,  the sub-level set \n$M_{t_2}$ is obtained  from  from the lower sub-level set $M_{t_1} $ in a manner determined by the critical points  with  $t_1< s < t_2$\nand, up to homotopy,  is gotten by adding\n\\begin{itemize}\n\\item   a disjoint,  unattached point  for each isolated local minimum; \n\\item a disjoint, unattached connected Riemann surface for each non-isolated  local minimum; \n\\item a $2$-disk, attached along its $S^1$ boundary, for each saddle point; \n\\item a $4$-disk, attached along its $S^3$ boundary, for each isolated local maximum; and \n\\item a $2$-disk bundle over a connected Riemann surface, attached along its circle-bundle boundary, for each non-isolated\nlocal maximum. \n\\end{itemize}\nSince these  operations always \nentail adding a path-connected space along a path-connected boundary,   different path \ncomponents of $M_{t_1}$ always survive as separate path components of $M_{t_2}$. \nIt therefore follows that only one \n of the $C_j$ can be  a  local minimum $s$, \nsince two different local minima \nwould necessarily end up in different connected components of the connected $4$-manifold $M$.   \nIn particular,   the set of all local minima of $s$ is actually the set of all {\\em global} minima, and must   \n either  be a connected compact\ncomplex curve or just a single point.   Looking at the same picture upside down, \n we similarly  see that the set of  local maxima of $s$ coincides with the set of global maxima,\n and must  either  be a single point or a  connected compact complex curve. \n \n The open set $M_-$ defined by $s < 0$ \n is  the interior of the compact manifold-with-boundary $M_0:= s^{-1}((-\\infty , 0])$,\n and is therefore homotopy equivalent to it. On the other hand, since  $M= M_t$ for any \n $t > \\max s$, we see that $M_0$ and $M$ have the same number of connected components.\n Hence $M_-$ must be connected. However, since $-s: M\\to {\\mathbb R}$ is also a Morse-Bott function, \n  the same reasoning also applies when we ``turn the picture upside down.'' The set \n $M_+$ defined by \n $s > 0$ is  therefore connected, too. \n Thus $M-\\mathcal{Z}= M_-\\sqcup M_+$ consists of exactly two connected components, as claimed. \n \nSimilar reasoning  allows us to understand how  $\\partial M_t$ changes as we vary $t$. In the region $\\min s < s < \\max s$, \n$s$ is a Morse function in the standard  sense, and only has   critical points  of index $2$. If $t_1$ and $t_2$ are regular values of $s$ with \n$\\min s < t_1 < t_2 < \\max s$, \nit therefore follows that $\\partial M_{t_2}$ is obtained from $\\partial M_{t_1}$ by performing surgeries in dimension $2-1=1$.\nIn other words, every time one passes a critical point, one  just modifies the $3$-manifold $\\partial M_{t_1}$   by performing a {\\em Dehn surgery}; that is, \none just removes   a solid torus $S^1 \\times D^2$,  and then glues \nit  back  in via  a self-diffeomorphism of its $S^1 \\times S^1$ boundary.\n Since $M_t$ is  connected when $t = \\min s+ \\epsilon$ for sufficiently small\n$\\epsilon> 0$, and because    Dehn surgery on a connected $3$-manifold always produces another connected $3$-manifold, \n it follows that $\\partial M_t$ is connected for any regular value $t \\in (\\min s , \\max s )$. In particular, since $0$ is a regular value\n of $s: M\\to {\\mathbb R}$, \nit follows  that $\\mathcal{Z}= \\partial M_0$ is connected, as claimed. \n\n\nSince the  metric $h=s^{-2}g$ on the interior $M_-$ of the compact manifold-with-boundary $M_0$ is obtained by rescaling a smooth metric\n$g$ by the inverse-square of a non-negative  function $-s: M_0\\to {\\mathbb R}$ which vanishes only at  $\\partial M_0= \\mathcal{Z}$ and has non-zero \nnormal derivative along the boundary, the Einstein manifold \n$(M_-,h)$ is  conformally compact \\cite{grahlee,lebhspace,mazco} and hence, in particular,  is complete. By the same reasoning, $(M_+, h)$ is \nalso  conformally compact, and therefore  complete. Since, by construction, both of  these manifolds have conformal infinity $(\\mathcal{Z},  [g|_\\mathcal{Z}])$ and \nEinstein constant $\\kappa\/4 < 0$, we have thus established all of our claims. \n\\end{proof}\n\n\n\\begin{rmk}\nA key point in the above result is  that $\\max s > 0$ when $\\kappa < 0$ and $s$ is non-constant. This can also be proved  in  the following interesting way:\n\n\n Since the flow of $\\xi=J\\nabla s$ preserves both $g$ and \nthe scalar curvature $s$ of $g$,  it therefore also preserves the conformally rescaled metric $h=s^{-2}g$  on the open set\n$M-\\mathcal{Z}$ where $h$ is defined. If  we  had  $\\max s < 0$ for a solution with $s$ non-constant,  $\\mathcal{Z}$ would be empty, \nand $(M, h)$ would be a compact Einstein manifold with Einstein constant $\\kappa \/4 < 0$ which supported a   Killing field\n$\\xi\\not\\equiv 0$. But any Killing field $\\xi$ satisfies the Bochner formula \\cite{bochner} \n\\begin{equation}\n\\label{bow}\n0 = \\frac{1}{2}\\Delta |\\xi |^2 + |\\nabla \\xi |^2 - r(\\xi , \\xi)\n\\end{equation}\nand since $(M,h)$ would have negative Ricci curvature $r=\\frac{\\kappa}{4}h$, this  immediately leads to a contradiction, as the \nright-hand-side would be strictly positive at a maximum of $|\\xi|^2$.  Hence $(M,g)$  must    have $\\max s \\geq 0$. \nHowever,  equation \\eqref{capital} tells us that $|\\nabla s|^2 = - \\kappa\/12 > 0$ along the locus $\\mathcal{Z}$ where $s=0$.\nThe maximum of $s$  therefore cannot be achieved  at a point where $s=0$, and \nit thus   follows that we must  actually have     $\\max s > 0$, as claimed.  \n\\end{rmk}\n\n\n\\section{The Proof of Theorem \\ref{summary}}\n\\label{rapt}\n\nThe trichotomy laid out in the previous section proves most of Theorem \\ref{summary}. It remains only to prove the auxiliary claims made  regarding the \nsecond case of each of our three classes. In doing so, we will make  repeated use of the following observation:\n\n\n\n\n\\begin{lem}\n\\label{lulu}\n If  the compact connected complex surface $(M^4,J)$ admits a strictly extremal K\\\"ahler metric $g$,\nthen  $(M,J)$ is ruled, and    $\\tau (M) \\leq 0$. \n\\end{lem}\n\\begin{proof} \nBy hypothesis,  the scalar curvature $s$ of $g$ is non-constant, and\n$\\xi = J \\nabla s$ is  a non-trivial Killing field. If $p\\in M$ is a minimum of the scalar curvature $s$, it is   fixed\nby the flow of $\\xi$, and \nthe  action of $\\xi$ is therefore completely determined, via the exponential map of $g$, \nby the induced isometric action on \n$T_pM$ generated   by $\\nabla \\xi|_p$. \nHowever, the same observation  allows us to identify the  isotropy subgroup of $p$ in the identity component   $\\Iso_0 (M,g)\\subset \\Aut (M,J)$\nof the isometry group \nwith a subgroup of $\\mathbf{U}(2)$. Since closure of the  $1$-parameter group \nof isometries generated by $\\xi$ is a closed connected Abelian subgroup of the istropy group of $p$,\nand since $\\mathbf{U}(2)$ has rank $2$, this implies  that either $\\xi$ is periodic, or that the closure of the  group\nit generates is a $2$-torus in $\\Iso_0 (M,g)\\subset \\Aut_0(M,J)$.  \n\nLet us first consider what happens if the isotropy group of $p$ contains a $2$-torus. \nSince the action of this torus on $M$ is modelled, near $p$, on the action of $\\mathbf{U}(1) \\times \\mathbf{U}(1)\\subset \\mathbf{U}(2)$\non $\\mathbb C^2$, the generators give rise to \ntwo global \nholomorphic vector fields $\\Xi_1$ and $\\Xi_2$  which both vanish at $p$, but which are linearly independent at generic  nearby points. \nThus $\\Theta =\\Xi_1\\wedge \\Xi_2$ is a holomorphic section  of the anti-canonical bundle $K^{-1}$ \nwhich vanishes at $p$, but which does not vanish identically. If $\\phi$ is a holomorphic section of $K^{\\ell}$, \n$\\ell > 0$,  then the contraction $\\langle  \\phi , \\Theta^{\\otimes \\ell} \\rangle$ \nis  a global holomorphic function \non $M$, and so must be  constant. However,  since  $\\langle  \\phi , \\Theta^{\\otimes \\ell} \\rangle$  certainly vanishes at $p$, this constant function consequently    vanishes\nidentically. Since $\\Theta^{\\otimes \\ell}$  spans  the fiber of $K^{-\\ell}$ at a generic point, this shows\nthat the holomorphic differential $\\phi$ vanishes on an open  set, and therefore vanishes identically. Thus, the plurigenera $p_\\ell(M) = h^0(M, \\mathcal{O} (K^\\ell))$, $\\ell > 0$,  must  all vanish,\nand  the  Kodaira dimension of $(M,J)$ must be $-\\infty$. \n\nOn the other hand,  if $\\xi=J\\nabla s$ is instead periodic,  then $(M,J)$ contains a family of  rational curves. Indeed, if $\\xi$ has period $\\lambda$, then \n$2\\nabla^{1,0}s = \\nabla s - i\\xi$  generates a holomorphic action of $\\mathbb C\/\\langle i \\lambda \\rangle$, which we now identify\nwith the punctured complex plane $\\mathbb C^\\times$ via $\\zeta \\mapsto \\exp ( 2\\pi \\zeta\/\\lambda)$. We will next show that the generic orbit of the resulting \n$\\mathbb C^\\times$-action on $(M,J)$ can be compactified  by adding two fixed points, corresponding to  $0$ and $\\infty$, to produce a rational curve in $M$.  \nIndeed,  we already saw in the proof of Proposition \\ref{understand}\nthat the only critical points of $s$ that are not global maxima or minima are  saddle points where $\\Hess s$ is of type \n$({+}{+}{-}{-})$. Since  only a $1$-real-parameter family of trajectories of $\\nabla s$ ascends to such a saddle point, the \ngeneric trajectory of $\\nabla s$  misses every saddle point, and therefore flows from the minimum value to the maximum value of $s$. \nNow recall that $\\min s$ is either attained at a unique point $p_-$, or is attained along a single connected totally geodesic Riemann surface, which we will \ncall $\\Sigma_-$; similarly, $\\max s$ is either attained at a unique point $p_+$, or is attained along a single connected totally geodesic Riemann surface\n$\\Sigma_+$. The behavior of the flow lines of $\\nabla s$ near $\\Sigma_\\pm$ is particularly simple; asymptotically, they simply approach $\\Sigma_\\pm$ orthogonally,\nat a unique point, since in exponential coordinates $\\xi$ just generates rotation in $\\mathbb C^2= \\{ (z,w)\\}$ about the $z$-axis, and since \n$\\nabla J=0$, this implies that $\\nabla^{1,0}s$  is  therefore given in these coordinates by a constant times $w\\frac{\\partial}{\\partial w} + O(|(z,w)|^3)$. \nOn the other hand, near $p_\\pm$, the periodicity of $\\xi$ analogously allows one to express $\\nabla^{1,0}s$ as a constant times\n$k z\\frac{\\partial}{\\partial z} + \\ell w\\frac{\\partial}{\\partial w} + O(|(z,w)|^3)$ for suitable non-zero integers $k$ and $\\ell$. In either case, \nwe  not only see  that a  generic $\\mathbb C^\\times$-orbit may be completed as a holomorphic map ${\\mathbb C \\mathbb P}_1\\to M$, but also \nthat, as we vary the orbit by moving our initial data though a sufficiently small   holomorphic disk $D_\\varepsilon$  transverse to a given generic $\\mathbb C^\\times$-orbit,\nthe map $D_\\varepsilon\\times \\mathbb C^\\times\\to M$  induced by the action extends continuously as a map $\\Phi:  D_\\varepsilon\\times{\\mathbb C \\mathbb P}_1\\to M$; and \na variant of  the proof of the Riemann removable singularities theorem then shows that this continuous extension $\\Phi$ is  necessarily holomorphic.  \nIn particular, any $\\phi\\in \\Gamma (M, \\mathcal{O}(K^{\\ell}))$ pulls back as a holomorphic section \n$\\Phi^*\\phi$ of $\\mathcal{O}(K^{\\ell})$ on  $D_\\varepsilon\\times{\\mathbb C \\mathbb P}_1$.\nBut   the restriction of $K^{\\ell}_{D_\\varepsilon\\times{\\mathbb C \\mathbb P}_1}$ to any \n$\\{ \\mbox{point}\\} \\times {\\mathbb C \\mathbb P}_1$\nhas negative degree $-2\\ell$, so it follows that  $\\Phi^*\\phi\\equiv 0$. However, since $\\Phi$  is a local biholomorphism  on  $D_\\varepsilon\\times \\mathbb C^\\times$, \nit follows that $\\phi$ must itself vanish on an open set. Hence $\\phi \\equiv 0$ \nby uniqueness of analytic continuation. \nThus, the plurigenera $p_\\ell(M) = h^0(M, \\mathcal{O} (K^\\ell))$, $\\ell > 0$,  must  all vanish,\nand we thus conclude that   the  Kodaira dimension of $(M,J)$ must be $-\\infty$, just as in the previous  case. \n\nSince $(M,J)$ is of K\\\"ahler type, \nthe Eniques-Kodaira classification \\cite{bpv,GH} therefore implies that it is rational or ruled. \nHowever, since $g$ is a strictly extremal K\\\"ahler metric on $(M,J)$, the Futaki invariant of $(M,J, [\\omega ])$ must also be non-zero \\cite{calabix2}.\nSince the Futaki invariant $\\mathfrak{aut}(M) \\to \\mathbb C$ kills the derived Lie algebra $[\\mathfrak{aut}(M) , \\mathfrak{aut}(M)]$,\nthis  means that  $\\Aut_0 (M, J)$  cannot be semi-simple. Consequently,   $(M,J)$ \ncannot be  ${\\mathbb C \\mathbb P}_2$. Since $(M,J)$  is rational or ruled, it   is therefore  either a geometrically ruled surface, or a blow-up \nof a geometrically ruled surface. In particular,  $\\tau (M) \\leq 0$, as claimed. \n\\end{proof}\n\n\n\nThis now allows us to prove a useful fact  regarding  case  {I(b)}: \n\n\\begin{prop} \n\\label{holistic}\nLet $(M,g,J)$  be a compact connected  Bach-flat K\\\"ahler surface with $\\kappa> 0$. If $g$ is not K\\\"ahler-Einstein,   then the  conformally related\nEinstein metric $h= s^{-2}g$ \nhas holonomy $\\mathbf{SO}(4)$. \n\\end{prop}\n\\begin{proof} By Proposition  \\ref{pop},  $(M^4,g,J)$ must be  a strictly extremal K\\\"ahler surface, and Lemma \\ref{lulu} therefore tells us that \n$(M,J)$ is a ruled surface with $\\tau (M) \\leq 0$. \nOn the other hand, $M$ also \nadmits a globally defined Einstein metric  $h=s^{-2}g$  with $\\lambda > 0$, so Bochner's theorem \\cite{bochner}\nimplies that $b_1(M)=0$.  \nSurface classification \\cite{bpv} therefore tells us that  $(M,J)$ is rational, and so in particular is   simply connected. \n\n\n Since $M^4$ is simply connected, the holonomy of any metric on $M$ thus coincides with its restricted holonomy, and so \n is a compact connected Lie subgroup of $\\mathbf{SO}(4)$. However, the action of $\\mathbf{SO}(4)$ on $\\Lambda^2 = \\Lambda^+ \\oplus \\Lambda^-$ gives rise to an isomorphism \n$\\mathbf{SO}(4)\/{\\mathbb Z}_2 = \\mathbf{SO}(3)\\times \\mathbf{SO}(3)$, where the two copies of $\\mathbf{SO}(3)$ act  on $\\Lambda^+$ and $\\Lambda^-$, respectively,\nvia the tautological $3$-dimensional representation. Thus, if the holonomy group were smaller than $\\mathbf{SO}(4)$, its image in at least one  factor\n$\\mathbf{SO}(3)$ would have to be contained in $\\mathbf{SO}(2)$, and \nsome non-zero self-dual or anti-self-dual\n$2$-form  would therefore  be fixed by the holonomy group. Rescaling this $2$-form $\\alpha$ to have norm $\\sqrt{2}$ with respect to $h$ and then extending it\nas a parallel form to all of $M$,  the almost-complex structure $\\mathscr{J}$ defined by $\\alpha= h(\\mathscr{J}\\cdot , \\cdot )$  would then be \nintegrable, and  $(M,h, \\mathscr{J})$ would then be a K\\\"ahler-Einstein manifold. More specifically,  a parallel self-dual $\\alpha$ would give rise to a $\\mathscr{J}$  compatible\nwith the same orientation as the original complex structure $J$, while  a parallel  anti-self-dual  $\\alpha$ would instead \ngive rise to a $\\mathscr{J}$  compatible\nwith the opposite orientation. \n\n\n\nNow  the Bach-flat K\\\"ahler metric $g$ is non-Einstein \nby assumption. \nSince $h=s^{-2}g$ is  Einstein  and is conformally related to $g$, it therefore follows that \n$$\\int_M s^2_g d\\mu_g > \\int_M s^2_h d\\mu_h  ,$$\nas can either be  read off  from the conformal invariance of \n$$\\int_M \\left( \\frac{s^2}{24} - \\frac{|\\mathring{r}|^2}{2}\\right) d\\mu = 8\\pi^2 \\chi (M) - \\int_M |W|^2 d\\mu $$\nor deduced from the more general fact  \\cite{obata} that Einstein metrics are always Yamabe minimizers. \nSince the K\\\"ahler metric $g$ satisfies \\eqref{sebastian},  we thus have \n$$\\int_M |W_+|_h^2 d\\mu_h = \\int_M |W_+|_g^2 d\\mu_g =  \\int_M \\frac{s^2_g}{24} d\\mu_g > \\int_M \\frac{s^2_h}{24} d\\mu_h. $$\nIt therefore follows that $h$ cannot satisfy \\eqref{sebastian}, and so cannot  be K\\\"ahler in a manner compatible with the given orientation of $M$. On the other hand, \nequation \\eqref{th}  tells us that \n$$\\int_M |W_-|_h^2 d\\mu_h = -12\\pi^2 \\tau (M) + \\int_M |W_+|_h^2 d\\mu_h$$\nand since $\\tau(M) \\leq 0$, we therefore have  \n$$\\int_M |W_-|_h^2 d\\mu_h \\geq \\int_M |W_+|_h^2 d\\mu_h > \\int_M \\frac{s^2_h}{24} d\\mu_h .$$\nThis shows that $h$ cannot satisfy the reverse-oriented analog of  \\eqref{sebastian}, \nand hence  cannot be K\\\"ahler with respect to a reverse-oriented complex structure, either. \nThis proves that, when $s_g$ is non-constant and positive,  the holonomy group of $(M,h)$ is exactly $\\mathbf{SO}(4)$, as claimed. \n\\end{proof}\n\nConcerning case {II(b)}, we have the following:\n\n\\begin{prop}\n\\label{scalar-flat}\n If $\\kappa=0$ and $(M,g,J)$ is not K\\\"ahler-Einstein, \nthen  $(M,J)$ is a (possibly blown-up) ruled complex surface with $c_1^2 < 0$. \nMoreover,  no $4$-manifold homeomorphic to $M$ ever admits an Einstein metric. \n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{sfk}, a Bach-flat K\\\"ahler surface $(M,g,J)$ with $\\kappa =0$ must have $s\\equiv 0$. \nSince one can decompose the the Ricci form $\\rho$ of any K\\\"ahler surface  as \n$$\\rho = \\frac{s}{4} \\omega + \\mathring{\\rho}$$\nwhere $\\mathring{\\rho}\\in \\Lambda^-= \\Lambda^{1,1}_{\\mathbb R} \\cap (\\omega)^\\perp$ is the primitive part of $\\rho\\in \\Lambda^{1,1}$, we see  that $\\rho$ is anti-self-dual \nwhenever $g$ is scalar-flat K\\\"ahler. It follows \\cite{lsd} that any scalar-flat K\\\"ahler surface satisfies\n$$4\\pi^2 c_1^2 (M, J) =  \\int_M \\mathring{\\rho}\\wedge \\mathring{\\rho} = -\\int_M \\mathring{\\rho}\\wedge \\star \\mathring{\\rho} =-\\int_M |\\mathring{\\rho}|^2 d\\mu,$$\nso that $c_1^2\\leq 0$, with equality if and only if $g$ is Ricci-flat; cf. \\cite{laf}.  In particular, if $(M,g,J)$ is not K\\\"ahler-Einstein, \nwe have\n$$(2\\chi + 3\\tau )(M) = c_1^2 (M,J) < 0.$$\nHowever, the Hitchin-Thorpe inequality \\cite{bes,gray-ht,hit,tho} tells us that the homotopy invariant $(2\\chi + 3\\tau )(M)$\nmust be non-negative for any compact oriented $4$-dimensional Einstein manifold. Thus, if $\\kappa =0$ and \n$(M,g,J)$ is not K\\\"ahler-Einstein, $M$ cannot even be homeomorphic to an Einstein manifold. \n\nOn the other hand, the plurigenera $p_\\ell(M) = h^0(M, \\mathcal{O} (K^\\ell))$, $\\ell > 0$, \n of a non-Ricci-flat scalar-flat K\\\"ahler surface must all vanish \\cite{lsd,yauruled}. Indeed, if\n$\\phi$  is a holomorphic section of $K^{\\ell}$, the Ricci form  satisfies \n\\begin{equation}\n\\label{plurigenus}\n\\ell \\rho = i\\partial\\bar{\\partial}\\log |\\phi|^2\n\\end{equation}\naway from the zero locus of $\\phi$. \nTaking the inner product of both sides with $\\omega$  thus yields\n$$ 0 = -\\ell s = \\Delta \\log |\\phi|^2$$\nwherever $\\phi\\neq 0$. Since $|\\phi|^2$ must have a maximum on $M$, the strong maximum principle \\\n\\cite{giltrud} therefore says that $\\log |\\phi|^2$ is constant away from the zero set of $\\phi$.\nIf $\\phi$ does not vanish identically, it therefore  has constant non-zero norm, and \nequation \\eqref{plurigenus} then says that $(M,g)$ is  Ricci-flat. \nIf $(M,g,J)$ is  not K\\\"ahler-Einstein, its  plurigenera must therefore all vanish, as claimed. The Kodaira dimension of $(M,J)$ is therefore\n$-\\infty$, and the Enriques-Kodaira classification \\cite{bpv} therefore \ntells us that $(M,J)$ is rational or ruled. Morever, since $c_1^2 < 0$, it certainly can't be ${\\mathbb C \\mathbb P}_2$. We thus   conclude that  $(M,J)$  can actually be \nobtained from some ruled surface by blowing up $\\geq 0$ points. \n\\end{proof}\n\nPutting these facts together, we now immediately have:\n\n\\begin{prop} \n\\label{planb}\nSuppose that $(M,g,J)$ is  a compact connected Bach-flat K\\\"ahler surface which is not K\\\"ahler-Einstein. Then $(M,J)$ is ruled.\n\\end{prop}\n\\begin{proof} If \n$g$ has $\\mathring{r}\\not\\equiv 0$, \n equation \\eqref{rico}  tells us that the Bach-flat K\\\"ahler manifold $(M,g,J)$  has either  $s$ non-constant   or  $s\\equiv 0$. \nIf  $g$ has $s\\equiv 0$, then  Proposition \\ref{scalar-flat} then tells us that $(M,J)$ is ruled. Otherwise, \n$g$ must be a strictly extremal K\\\"ahler metric, and Lemma \\ref{lulu} then again  guarantees that  $(M,J)$ is ruled, as claimed. \n\\end{proof}\n\nFinally, regarding solutions of type {III(b)}, we have the following:\n\n\\begin{prop} \n\\label{omphalos} \nLet  $(M,g,J)$ be  a compact connected Bach-flat K\\\"ahler surface with $\\kappa < 0$  that  is not K\\\"ahler-Einstein. \nThen the connected real hypersurface $\\mathcal{Z}\\subset M$ is totally umbilic with respect to $g$. Moreover, the Weyl curvature\n$W$ of $(M,g)$ vanishes identically along $\\mathcal{Z}$. \n\\end{prop}\n\\begin{proof} While both of these features are general consequences \\cite{fefgra,lebhspace} of the fact that $h=s^{-2}g$ is Poincar\\'e-Einstein, \nwe will give quick, self-contained proofs that  supply further information in the present special context. \n\nLet $\\check{g}= g|_{\\mathcal{Z}}$ denote the induced Riemannian metric (or {\\em first fundamental form}) of our hypersurface,  \nand let $\\gemini$ denote the second fundamental form (or {\\em shape tensor}) of $\\mathcal{Z}$. To say that $\\mathcal{Z}$ is totally \numbilic just means that $\\gemini = {\\zap f} \\check{g}$ for some function ${\\zap f} : \\mathcal{Z} \\to {\\mathbb R}$, which is  then  just the mean curvature of $\\mathcal{Z}$. \nHowever,   the second fundamental form  can be expressed as    $\\gemini = (\\nabla \\nu)|_{\\mathcal{Z}}$\nin terms  of any  unit   $1$-form   $\\nu$  on $M$ which is  normal to $\\mathcal{Z}$ along this hypersurface, and which is \ncompatible with the given orientations of $M$ and $\\mathcal{Z}$.  \nHowever,  since $\\mathcal{Z}$ is defined by $s=0$, and since equation \\eqref{capital} tells us that \n$|\\nabla s|^2 = - \\frac{\\kappa}{12}$\nalong $\\mathcal{Z}$, it follows  that $\\nu = 2\\sqrt{3\/|\\kappa|} \\, ds$ is a valid choice for this  unit  co-normal  field, provided\n we orient $\\mathcal{Z}$  in the corresponding manner.  Thus \n$$\\gemini = \\left. \\left( 2\\sqrt{\\frac{3}{|\\kappa|}}  \\Hess s \\right)\\right|_{\\mathcal{Z}}$$\ngives us a  convenient formula  for the second fundamental form of $\\mathcal{Z}$. However,  equation \\eqref{johann} \ntells us that \n$\n\\Hess_0 s =0\n$\nalong the locus  where $s=0$, so \n$\\Hess s = - (\\Delta s\/4)g$\nalong $\\mathcal{Z}$, and hence \n\\begin{equation}\n\\label{omphaloi}\n \\gemini  = - \\left( \\frac{\\sqrt{3}}{2\\sqrt{|\\kappa|}}\\Delta s\\right) \\check{g}.\n\\end{equation}\nThis shows that $\\mathcal{Z}$ is totally umbilic, as claimed. \n\nOn the other hand, since the K\\\"ahler metric $g$ has $s=0$ along $\\mathcal{Z}$, equation \\eqref{sebastian} tells us that $W_+=0$ there, too. \nThus $W= W_-$ at $\\mathcal{Z}$. However, if we let  $\\nu$ denote the  unit $1$-form normal to  $\\mathcal{Z}$,\nthere is a natural bundle isomorphism \n\\begin{eqnarray*} \nT^*\\mathcal{Z}&\\longrightarrow& \\Lambda^-|_\\mathcal{Z}\\\\\n\\theta &\\longmapsto&(\\nu \\wedge \\theta) - \\star (\\nu \\wedge \\theta).\n\\end{eqnarray*}\nApplying the identity \n$$W_- (\\varphi - \\star \\varphi, \\varphi - \\star \\varphi) -  W_+ (\\varphi + \\star \\varphi, \\varphi + \\star \\varphi) = - 4 \\mathcal{R} (\\varphi , \\star \\varphi ),$$\nto $\\varphi = \\nu \\wedge \\theta$, and remembering that $W_+=0$ along $\\mathcal{Z}$, \nwe therefore see that $W=W_-$ is completely characterized along $\\mathcal{Z}$ by knowing    ${{\\mathcal{R}^1}_{234}=W^1}_{234}$ for those  \northonormal co-frame $\\{ e^1, \\ldots , e^4\\}$ in  which $e^1=\\nu$. \nBut we can easily understand these components, using the fact that $W$ is conformally invariant. Indeed, since a \nconformal change $g\\rightsquigarrow u^2 g$ changes the second fundamental form by $\\gemini\\rightsquigarrow u\\gemini + \\langle du, \\nu\\rangle \\check{g}$, a hypersurface\nis totally umbilic  iff it can be made totally geodesic by  a conformal change. But the unit normal $1$-form  \n$\\nu$ is parallel along any  totally geodesic hypersurface. In such a conformal gauge, we therefore  have  ${\\mathcal{R}^1}_{234} ={W^1}_{234}=0$. Conformal invariance thus  \n allows us to deduce that \n$W$ vanishes identically along $\\mathcal{Z}$, as claimed. \n\\end{proof}\n\nTheorem \\ref{summary} is now an immediate consequence. Indeed, the basic trichotomy into solution classes {I}, {II}, and {III} just reflects the sign of \n$\\kappa$, or equivalently, by Lemma \\ref{sign}, the sign of $\\min s$. \n Propositions \\ref{sfk}, \\ref{pop}, and \\ref{understand} then explain the basic features of each class. If the solution is K\\\"ahler-Einstein, \nit is then of type {I(a)}, {II(a)}, or {III(a)}, depending on the sign of $\\kappa$. Otherwise, the underlying complex surface is ruled by Proposition \\ref{planb},\nand the remaining  claims made about solutions of type {I(b)}, {II(b)}, and {III(b)} are then proved by Propositions  \\ref{holistic},  \\ref{scalar-flat}, \nand \\ref{omphalos}, respectively. \n\n\n\\section{Problems and Perspectives} \n\\label{conclusion}\n\n\n\\bigskip\n\nAs mentioned in \\S \\ref{intro},  solutions of class {I} have been completely classified \\cite{lebuniq}. \nOne key fact that enabled this classification was the observation \\cite{lebhem}  that  in this case \n$(M,J)$ has  $c_1> 0$,    and  hence is  a Del Pezzo surface \\cite{delpezzo}. \nIndeed, since $s\\neq 0$,  we can represent $2\\pi c_1$ by  $\\tilde{\\rho}:=\\rho + 2i\\partial\\bar{\\partial} \\log |s|$, \nwhere $\\rho$ is the Ricci form of $(M,g,J)$, \nand equation \\eqref{rico} then tells us that  \n$\\tilde{\\rho}=q(J\\cdot, \\cdot)$, where the symmetric tensor field  $q$ is given by \n\\begin{equation}\n\\label{rep}\nq = \\frac{2s+ \\kappa s^{-2}}{12} g + s^{-2} |\\nabla s|^2 g^\\perp .\n\\end{equation}\nHere, $g^\\perp$ is defined to be zero on the span of $\\nabla s$ and $J\\nabla s$, but to coincide with $g$ on the orthogonal complement of this subspace; \nwhile this of course means that $g^\\perp$ is only defined away from the critical points of $s$,  the tensor field  $|\\nabla s|^2g^\\perp$  has a unique smooth extension\nacross the critical points, which is explicitly given by declaring it to be zero at this exceptional set. Since $\\kappa > 0$ implies that $s> 0$ everywhere, \nit follows that $q > 0$ for a solution of type {I}, and that $(M,J)$ is therefore \na Del Pezzo surface when $\\kappa$ is positive. Conversely, one can  show \\cite{chenlebweb,tian,sunspot}\nthat every Del Pezzo $(M,J)$ admits a $J$-compatible class-{I} solution, and that this solution is moreover unique \\cite{bandomab,lebuniq} up to complex automorphism and rescaling. In fact, the solution is K\\\"ahler-Einstein except in exactly two cases, namely  the blow-up  of ${\\mathbb C \\mathbb P}_2$ at one or two distinct points. \nThus, solutions of type {I(a)} and {I(b)} are distinguished by  whether or not the Lie algebra of holomorphic vector fields on $(M,J)$ is reductive. \n\nBy further elaboration on   this idea, one is led to the following  result: \n\n\\begin{prop}\n\\label{beth}\nLet $g$ and $\\tilde{g}$ be be two  $J$-compatible Bach-flat K\\\"ahler metrics on the same compact complex surface $(M,J)$.\nIf these two solutions have different types, according to the classification scheme of Theorem \\ref{summary}, then one is of type {\\rm II(b)}, and the other\nis of type {\\rm III(b)}. \n\\end{prop}\n\\begin{proof}\nSolutions of type {II(a)} and {III(a)} are distinguished from the others by the Kodaira dimension of $(M,J)$. On the other hand, we have just seen that solution of \nclass {I} exist precisely on complex surfaces with $c_1 > 0$, and   solutions  of type {I(a)} are then  distinguished from those of type {I(b)} by  whether \nthe  Lie algebra of holomorphic vector fields is reductive. \nIt thus  only remains  to show that the existence of solution of class {I} precludes the existence of \na solution of type {II(b)} or {III(b)}. However, any extremal K\\\"ahler metric on a Del Pezzo surface has positive scalar curvature \\cite[Lemmata A.2 and B.2]{lebhem10}.\nConsequently, the presence of  a Bach-flat K\\\"ahler metric of class {I} automatically precludes the existence of a solution of any other class.\\end{proof}\n\n\nThis makes the following piece of\nspeculation seem irresistible: \n\n\n\n\n\\begin{conj} \n\\label{wild}  \nOn a fixed compact complex surface $(M,J)$, any \n pair of  \\linebreak $J$-compatible Bach-flat K\\\"ahler metrics are necessarily of  the same type, \nin the sense of    Theorem \\ref{summary}. \n\\end{conj}\n\n\nAlthough the currently known solutions of type {III(b)} will almost certainly \n turn out to be atypical in many respects,   these   known examples  all live on geometrically ruled surfaces,\nwhich necessarily have   $\\tau =0$. \nThe following result thus explains why Conjecture \\ref{wild} is supported by all  known examples: \n\n\\begin{prop}\nNo compact complex surface $(M,J)$ of signature $\\tau=0$ can  admit a pair of $J$-compatible \nBach-flat K\\\"ahler metrics that have different types,  in the sense of    Theorem \\ref{summary}. \n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{beth}, it suffices to show that there cannot simultaneously be a solution $g$ of type {II(b)} and a solution $\\tilde{g}$ of type {III(b)}. \nHowever, any solution of class {II} is scalar-flat K\\\"ahler, and therefore has $W_+=0$ by equation \\eqref{sebastian}; and such a solution is\nof type {II(b)} iff it is not Ricci-flat.  If we now assume \nthat $\\tau (M) =0$, equation \\eqref{th} then tells us that $W_-=0$, making  $(M,g)$ is conformally flat, as well as scalar-flat. The Weitzenb\\\"ock formula\nfor $2$-forms thus simplifies to say that the Hodge Laplacian on $2$-forms coincides with the Bochner Laplacian $\\nabla^*\\nabla$, and \n any harmonic $2$-form must therefore be parallel. Since the assumption that $\\tau =0$ also  forces $b_-=b_+\\neq 0$, \n our metric    $g$ is K\\\"ahler with respect to both orientations, and, since $g$ is not flat, it follows  \\cite{burbar,lsd} that  $(M,J)$ is therefore a geometrically \nruled surface of the form $\\Sigma\\times_\\gamma {\\mathbb C \\mathbb P}_1$\n for some representation $\\gamma: \\pi_1(\\Sigma)\\to \\mathbf{PSU}(2)$. The given scalar-flat metric $g$ is then a twisted product metric, \n obtained by equipping $\\Sigma$ and ${\\mathbb C \\mathbb P}_1$ with metrics of constant (and opposite) Gauss curvature, and thus   belongs to a $2$-parameter family \n of constant-scalar-curvature metrics gotten by rescaling $\\Sigma$ and ${\\mathbb C \\mathbb P}_1$ by arbitrary positive constants. \n Since $b_2 (M)=2$,  this shows that the Futaki invariant is identically zero on an \n open set of the K\\\"ahler cone. However, \nthe Futaki invariant is  quite generally a real-analytic function of the K\\\"ahler class, as can be seen   by locally sweeping out\n the K\\\"aher cone by real-analytic families of real-analytic K\\\"ahler metrics. Hence the Futaki invariant of $(M,J)$ vanishes for every K\\\"ahler class, \n and it  therefore follows \\cite{calabix2} that  any \n extremal K\\\"ahler metric on $(M,J)$ must  have constant scalar curvature. Consequently,  $(M,J)$ cannot admit a strictly extremal K\\\"ahler metric, \n and no $J$-compatible  solution $\\tilde{g}$ of type {III(b)} can  therefore exist on $(M,J)$. \n\\end{proof}\n\nThere is a more fundamental reason to hope that  Conjecture \\ref{wild} might be true. Recall that Bach-flat metrics are critical points of\nthe Weyl functional, and that Bach-flat K\\\"ahler metrics are therefore, in particular, critical points of the Calabi energy \\eqref{calfun} on the space\nof K\\\"ahler metrics compatible with a given complex structure. However, the known examples are always \\cite{simtofri} actually {\\em absolute minima} for\nthe latter problem. \n\n\\begin{conj} \n\\label{thing} \nLet $g$ be a Bach-flat K\\\"ahler metric on a compact complex surface $(M,J)$. Then $g$ is an absolute minimizer of \nthe Calabi energy $\\mathcal{C}$ on the space of all $J$-compatible K\\\"ahler metrics on $M$. \n\\end{conj}\n\nThis conjecture is an easy exercise for solutions of type {I(a)}, {II(a)}, {II(b)}, and {II(a)}. It is moreover also true for solutions of type {I(b)}, \nalthough the proof \\cite{lebuniq} is much more subtle   in this case. By contrast,\nwe do not currently know whether  Conjecture \\ref{thing} holds \n for {\\em general} solutions of this type  {III(b)}, although \nit does in fact hold \\cite{simtofri} \n for all {\\em known} solutions.  Notice that Conjecture \\ref{thing} would certainly \nimply Conjecture \\ref{wild}, since any solution of type {III(b)} necessarily  has $\\mathcal{C}> 0$, whereas any solution of type {II(b)} obviously  has\n$\\mathcal{C}=0$. \n\n\n\\bigskip\n\nThese remarks make it obvious that solutions of type {III(b)} represent the area where our understanding of the subject \nremains most deficient. Still, it is not hard to prove a bit more about them. For example: \n\n\n\n\\begin{prop} Let $(M,g, J)$ be a solution of type {\\rm III(b)}.  Then the complete Einstein Hermitan manifold $(M_-, h,J)$ of Proposition \\ref{understand} \nhas numerically positive canonical line bundle $K_{M_-}$. In other words, every compact holomorphic curve $C\\subset M_-\\subset M$ satisfies $c_1\\cdot C < 0$. \n\\end{prop}\n\\begin{proof} The flow of $-\\nabla s= J \\xi$ for positive time is holomorphic, and preserves the region $M_-\\subset M$ where $s< 0$; moreover,  \n $s$ is  non-increasing under the flow.   However, since \nthe holomorphic vector field $\\nabla s - i \\xi$ has zeroes, its contraction with any holomorphic $1$-form vanishes identically, and\nthe induced action on the Albanese torus is therefore zero. By duality, the induced action on the Picard torus is also trivial, \nso the action sends any holomorphic curve to a curve to which it is linearly equivalent. In particular, any compact holomorphic curve $C\\subset M_-\\subset M$ \ngives rise to the same divisor line bundle $L\\to M$ as any of its images under the downward flow of the gradient vector field of $s$. Taking the \nlimit in $\\mathbb{P}[\\Gamma (M,\\mathcal{O}( L))]$, we can thus represent the limit of the the images of $C$ under the downward flow\nby a (typically singular) curve in $M_-$ which is sent to itself by the action of $\\mathbb C^\\times$. Such a curve is a sum, with non-negative  integer coefficients, \nof the curve $\\Sigma_-$ at which the minimum of $s$ is achieved (assuming  the minimum does not occur at an isolated  point) and \nof  ``vertical'' rational curves arising from flow lines descending from  saddle points of $s$ in $M_-\\subset M$. It therefore suffices to check\nthat $c_1$ is negative on $\\Sigma_-$ and on any curve tangent to $\\nabla s$ and $\\xi = J \\nabla s$. However, we can \nrepresent $2\\pi c_1$ on $M_-$ by $\\tilde{\\rho}:=\\rho + 2i\\partial\\bar{\\partial} \\log |s|$, \nwhere $\\rho$ is the Ricci form of $(M,g,J)$, and the corresponding symmetric tensor field is once again given by \\eqref{rep}. \nAt critical points of $s$, or in directions tangent to the space of $\\nabla s$ and $\\xi$, this expression  simplifies \nto just become $(2 s+ \\kappa s^{-2}) g \/12$, which is negative-definite on the region $M_-$ given by $s < 0$, since \nwe also have $\\kappa < 0$ for a solution of type {III(b)}. This shows that  the given curve $C$ is homologous to a curve\n$\\tilde{C}$ on which $c_1\\cdot \\tilde{C} < 0$, and we therefore have $c_1 \\cdot C < 0$, too, as claimed. \n\\end{proof}\n\nOn the other hand, this says nothing at all about\n $(M_+, J)$, and this is definitely  not a mere  matter of accident. \nFor example, when $(M,J)$ is a Hirzebruch surface, \n$M_-$ is a tubular neighborhood of a rational curve on which $c_1$ is negative, while  \n$M_+$ is a tubular neighborhood of a curve on which  $c_1$ is positive. Curiously enough, \n$(M_-, h)$ and $(M_+, h)$ are \n in fact\nactually isometric\\footnote{These  complete Einstein manifolds   seem to have  been first  discovered  \nby B\\'erard-Bergery \\cite{beber}, who made a systematic study of  cohomogeneity-one Einstein metrics. \nThey have subsequently been rediscovered several times by various groups of physicists \\cite{chamblin,papo}.} in these examples; however, \nthis is not  a  paradox,  because the relevant  isometry is orientation-reversing, and \nso does not intertwine the given complex structures  on $M_\\pm$ in any  direct manner.  \n\n\\bigskip \n\nWe can also prove some things  about the real hypersurface $\\mathcal{Z}\\subset M$: \n\n\\begin{prop} Let $(M,g, J)$ be a Bach-flat  K\\\"ahler surface of type {\\rm III(b)}, and let $\\mathcal{Z} \\subset M$ be the smooth \n real hypersurface given by $s=0$. Then the compact connected $3$-manifold $\\mathcal{Z}$ is Seifert-fibered. Moreover, \n the restriction of $\\xi$ to ${\\mathcal{Z}}$ is a non-trivial Killing field of constant length with respect to the induced metric \n  $\\check{g}= g|_{\\mathcal{Z}}$, and its orbits are therefore geodesics of $\\check{g}$. \n The flow of $\\xi$ moreover preserves the  CR structure induced on $\\mathcal{Z}$ by $(M,J)$, and, at any   $p\\in \\mathcal{Z}$, the following are equivalent:\n \\begin{itemize}\n  \\item  the Levi form of the induced CR structure is non-degenerate; \n \\item the Ricci curvature of $\\check{g}$ is positive in the direction of $\\xi$; \n \\item  the second fundamental form $\\gemini$ of $\\mathcal{Z}\\subset M$ is non-zero; and \n \\item the extrinsic Laplacian $\\Delta_g s$ of the scalar curvature is non-zero. \n \\end{itemize}\n\\end{prop}\n\\begin{proof}\nEquation \\eqref{capital} tells us that $|\\xi|^2 = |\\nabla s|^2 = - \\kappa \/12 >  0$ along $\\mathcal{Z}$, so the restriction of the Killing field $\\xi$ to $\\mathcal{Z}$\ndoes indeed have constant, non-zero  length. Since  the closure of the group of isometries generated by $\\xi$ is a compact connected Abelian \nLie group, and hence a torus, we can approximate $\\xi$ uniformly by non-zero periodic Killing fields, and the choice of such an approximation\n then endows  $\\mathcal{Z}$ with \na circle action for which all isotropy groups are finite, thereby giving it a Seifert-fibered structure. Since \n$\\xi$ is a Killing field of constant length with respect to \n$\\check{g}$, we also have \n$$\\xi^a \\nabla_a \\xi_b = - \\xi^a \\nabla_b \\xi_a = - \\frac{1}{2}  \\nabla_b |\\xi|^2 =0,$$\non $(\\mathcal{Z},\\check{g})$, and the trajectories of $\\xi$ are therefore geodesic. Finally, since the flow of $\\xi$ on $M$ preserves both $J$ and $s$, \nthe  flow acts on the hypersurface $s=0$  by CR automorphisms. \n\n\nSince $s$ is a non-degenerate defining function for $\\mathcal{Z}$, the restriction of $i\\partial \\bar{\\partial} s$ to the CR tangent space of $\\mathcal{Z}$\nexactly represents the Levi form. On the other hand, equation \\eqref{rico} tells us that $i\\partial \\bar{\\partial} s$ is a multiple of the K\\\"ahler form \n$\\omega$ along the locus $s=0$, so we therefore conclude that the Levi-form is non-degenerate exactly at there points of $\\mathcal{Z}$ where $\\Delta s \\neq 0$. \nOn the other hand,  equation \\eqref{omphaloi} tells us  that the  second fundamental form of  $\\mathcal{Z}$ is also non-zero exactly at points where \n$\\Delta s \\neq 0$. Finally, since the unit normal vector field is a constant multiple of $J\\xi$, \nthe restriction of \n$\\gemini ( J\\cdot  , \\cdot )$ to  $\\xi^\\perp \\subset T\\mathcal{Z}$ is a non-zero constant times the intrinsic covariant derivative $\\nabla \\xi$, and since \n$\\mathcal{Z}$ is umbilic, it therefore follows that $\\gemini \\neq 0$ exactly when $|\\nabla \\xi|^2$;  but \nthe Bochner Weitzenb\\\"ock formula \\eqref{bow} for a Killing field tells us that $|\\nabla \\xi|^2 \\equiv r(\\xi , \\xi)$ on $(\\mathcal{Z},\\check{g})$, so the positivity \nof the Ricci-curvature of $\\check{g}$ in the direction of $\\xi$ is also equivalent to all the other conditions under discussion. \n\\end{proof}\n\nIn the known examples, $\\mathcal{Z}$ is actually  strictly pseudo-convex. Is this a general feature of all solutions, or\nis it a mere artifact, reflecting fact that the known solutions have universal covers of cohomogeneity one? \n\n\n\\bigskip\n\n\nAs long as we  only  consider Bach-flat K\\\"ahler metrics that are compatible with some  fixed complex structure $J$ on $M$, \nConjecture \\ref{wild} claims that \n the solution type, as per Theorem \\ref{summary},  should be completely determined by $(M,J)$. \n While  there is a preponderance of evidence in favor of such a conjecture, it is also important to notice \nthat the type of the solution is certainly not just determined by the diffeotype of $M$ alone. \nFor example, while the smooth manifolds $S^2 \\times S^2$ and ${\\mathbb C \\mathbb P}_2\\# \\overline{{\\mathbb C \\mathbb P}}_2$ each  support\na unique complex structure with $c_1 > 0$,  each also carries an infinite number of other complex structures  realized by the various Hirzebruch surfaces\n$\\mathbb{P}( \\mathcal{O} \\oplus \\mathcal{O} (\\ell))\\to {\\mathbb C \\mathbb P}_1$, $\\ell \\geq 0$. Now,  every Hirzebruch surface with $\\ell > 2$  carries \\cite{hwasim}\na Bach-flat\nK\\\"ahler metric of type {III(b)}, in contrast to the solutions of type {I(b)} that instead exist when $\\ell = 0$ and $1$. Similarly, \nthe $4$-manifolds arising as $S^2$-bundles over curves of genus $\\geq 2$   carry solutions of both of type {II(b)} and {III(b)}; but \nthis form of peaceful co-existence  is once again only made made possible by allowing the complex structure to vary.  \n\nNonetheless, Theorem \\ref{summary} does have  consequences that do primarily  reflect the   differential topology of the underlying $4$-manifold: \n\n\n\\begin{prop} \n\\label{ez} \nLet $M$ be the underlying smooth $4$-manifold of a  {non-minimal} compact  complex surface of Kodaira dimension $\\geq 0$. \nThen there is  no complex structure $J$ on $M$ for which $(M,J)$ admits a  Bach-flat K\\\"ahler   metric. Moreover, for\ncomplex surfaces of Kodaira dimension $1$, the existence of Bach-flat K\\\"ahler metrics is similarly  obstructed even when the surface is minimal. \n\\end{prop}\n\\begin{proof} On a compact complex surface  $(M,J)$ of Kodaira dimension $\\geq 0$, Theorem \\ref{summary} tells us that any Bach-flat K\\\"ahler metric\nmust be K\\\"ahler-Einstein, with Einstein constant $\\lambda \\leq 0$. This in particular either means that  $c_1^{\\mathbb R}=0$ or  $c_1 < 0$.\nHence  $(M,g)$ must be minimal and have Kodaira dimension $0$ or $2$. However, for complex surfaces of K\\\"ahler type,  \nSeiberg-Witten theory implies  \\cite{lky,morgan} that Kodaira dimension is a diffeomorphism invariant,\nand  that non-minimality is a moreover a diffeomorphism  invariant whenever the Kodaira dimension is $\\geq 0$. \nThus the operative obstruction really just reflects the differential topology of $M$, in a manner that is  insensitive to the detailed complex geometry \nof the given  $J$. \n\\end{proof}\n\nFinally, it should perhaps also be emphasized that \nProposition \\ref{ez}  certainly does not obstruct the existence of  more \ngeneral Bach-flat  metrics. Indeed, a result of  Taubes \\cite{tasd} implies that any complex surface has blow-ups that  admit\nanti-self-dual metrics. Thus,  there are certainly many non-minimal complex surfaces of each possible Kodaira dimension $\\geq 0$\nthat   do indeed  admit Bach-flat  metrics; it's just  that these metrics don't happen to be conformally  K\\\"ahler!  \n\n\n\n\\vfill\n\n\\noindent \n{\\bf Acknowledgments:} This paper is  dedicated to the fond memory of my  friend  Gennadi Henkin,\nin the belief that he would have \nbeen intrigued  by the role  played   by CR manifolds, and would have formulated   perceptive, helpful  questions that could  have\npushed this  investigation even  further. \nI would also like to thank another old friend, Robin Graham, for some useful pointers  regarding  Poincar\\'e-Einstein\nmetrics. Finally, I would like to thank my former student Christina T{\\o}nnesen-Friedman for clarifying some important details  of  her beautiful construction of   explicit examples. \n\\pagebreak \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\input{tex\/intro2}\n\n\n\\section{Related work}\n\\input{tex\/related-work}\n\\label{sec:related-work}\n\n\n\\section{The proposed framework}\n\\input{tex\/method2}\n\n\n\\section{Experiment, results and evaluation}\n\\input{tex\/reseval}\n\n\\section{Floorplan generation}\n\\label{sec:floorplan_generation}\n\\input{tex\/generative}\n\n\\section{User study}\n\\input{tex\/user-study}\n\n\n\n\\section{Conclusion}\n\\input{tex\/conclusion}\n\n\n\n\n\\section*{Declarations}\n\n\\subsection{Funding}\nNot applicable\n\\subsection{Conflicts of interest} \nThe authors declare that they have no conflict of interest.\n\n\\subsection{Availability of data and material (data transparency)}\nThe dataset is public and the reference to data is available inside paper.\n\n\\subsection{Code availability (software application or custom code)}\nNot applicable\n\n\\subsection{Authors' contributions}\nNot applicable\n\n\\bibliographystyle{spmpsci}     \n\n\n\\subsection{TEMPORARY RESULTS}\n\n\\begin{table*}[]\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n                                        & \\textbf{Walk Length}        & \\textbf{Proxy Seen In Training or Not} & \\textbf{Rank of Graph by itself{[}1, 2, 3, 4, 5{]}(\\%)} & \\textbf{Rank of Proxy{[}1, 2, 3, 4, 5{]}(\\%)} \\\\ \\hline\n\\multirow{6}{*}{\\textbf{Random Walk 1}} & \\multirow{2}{*}{\\textbf{3}} & \\textbf{Y}                                                &  {[}99.9, 0.1, 0, 0, 0{]} & {[}0.1, 49, 2, 1, 1{]}                                                \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                & {[}100, 0, 0, 0, 0{]}                                                        & {[}0, 40, 1, 1, 0.7{]}                                                \\\\ \\cline{2-5} \n                                        & \\multirow{2}{*}{\\textbf{5}} & \\textbf{Y}                                                & {[}100, 0, 0, 0, 0{]}                                                        & {[}0, 75, 4, 2, 1.6{]}                                                 \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                &{[}100, 0, 0, 0, 0{]}                                                         &  {[}0, 41, 1, 1, 0.9{]}                                         \\\\ \\cline{2-5} \n                                        & \\multirow{2}{*}{\\textbf{7}} & \\textbf{Y}                                                & {[}100, 0, 0, 0, 0{]}                                   & {[}0, 57, 2, 1, 1{]}                       \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                &{[}47, 18, 11, 9, 5{]}                                                          &{[}52, 74, 1, 0.5, 0.4{]}                                                  \\\\ \\hline\n\\multirow{6}{*}{\\textbf{Random Walk 2}} & \\multirow{2}{*}{\\textbf{3}} & \\textbf{Y}                                                &   {[}100, 0, 0, 0, 0{]}                                                      &  {[}0.        , 83.5, 3.2, 1.9, 1.32{]}                                                \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                & {[}99.2, 0.7, 0.        , 0.        , 0.        {]}                                                        &   {[}0.7, 54, 2.8, 1.8, 1.2{]}                                               \\\\ \\cline{2-5} \n                                        & \\multirow{2}{*}{\\textbf{5}} & \\textbf{Y}                                                & {[}100, 0, 0, 0, 0{]}                                                         &  {[}1, 44, 2 , 1, 1.1{]}                                                \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                &   {[}100, 0, 0, 0, 0{]}                                                      &  {[}2, 47, 2.54, 1.69, 1.17 {]}                                                \\\\ \\cline{2-5} \n                                        & \\multirow{2}{*}{\\textbf{7}} & \\textbf{Y}                                                &   {[}83, 16, 0, 0, 0{]}                                                      &  {[}16, 16, 0, 0, 0{]}                                                \\\\ \\cline{3-5} \n                                        &                             & \\textbf{N}                                                & {[}100, 0, 0, 0, 0{]}                                                        &   {[}0.03, 27, 1, 1, 1{]}                                               \\\\ \\hline\n\\end{tabular}\n\\caption{Two different version of random walks with different walk lengths are used for converting graphs to sequences. For each walk length the random walks run 10 times to generate sequences and 1 time for creating a proxy graphs. Then  We tested the models in two states whether proxy graphs are seen in training or not(Proxy Seen In Training or Not). The next column shows the rank percentage of query graph by itself within 5 nearest neighbours. The last column shows rank percentage of proxy graphs within five nearest neighbours. In a good model we should have as more as possible the graphs by theirself in first rank and more proxy graphs in nearest ranks. {\\color{red} Progress May 15, model rank: Random Walk 1 walk length 5, Random Walk 2 walk length 5}}\n\\end{table*}\n\n\\subsection{Pairwise Similarity between Floorplans}\n{\\color{red} TODO: Major changes, it is similar to SIMAUD}\n\nIn this use case we demonstrate a pairwise comparison between input floorplan and its second nearest neighbour from the embedding space. Underline graphs of both floorplans are compared using a popular graph similarity distance metric, Graph Edit Distance (GED)~\\cite{bunke1983distance}. The edit distance between $G1$ and $G2$, GED(G1, G2), is the count of edit operations that transform $G1$ into $G2$, where the edit operations on a graph $G$ can be an insertion or deletion of an edge or node. All edit operations have the same constant cost which is $1$. If two graphs are identical, their GED is $0$.\n\nFirst, we retrieved a semantically similar design (second nearest neighbour) from the embedding space for the given input floorplan such that they have similar structural attributes but significantly different behavioral attributes, Figure~\\ref{fig:application_similarity_static}. The GDP value is reported as $0$, showcasing that the graphs are similar in their design semantics. Average exit flow values and color-coded density heatmaps are also shown for both floorplans. Input floorplan yielded comparatively lower exit flow than its nearest neighbour found from the embedding.\n\nIn next scenario, we retrieved a behaviorally similar floorplan (second nearest neighbour) from the embedding space for the given floorplan such that they have similar behavioral attributes but significantly different design semantics. A GDP value of $10$ is reported from the graphs comparison, showcasing the two graphs are design semantically different. However, their corresponding floorplans yielded similar exit flow values. A GDP graph transformation as well as density heatmaps are shown, Figure~\\ref{fig:application_similarity_dynamic}. \n\n\n\\begin{figure}[H]\n    \\centering\n    \n    \\begin{subfigure}[t]{0.9\\linewidth}\n    \\begin{tabular}{c}\n        \\includegraphics[width=1.0\\linewidth]{images\/applications\/case1_1_v2.pdf}\n    \\end{tabular}\n    \\caption{Two floorplans which are structurally similar (Graph Edit Distance (GED) = $0$) but have different crowd behavioral attributes.}\n    \\label{fig:application_similarity_static}\n    \\end{subfigure}\n    \n    \\par\\bigskip\n    \n    \\begin{subfigure}[t]{0.9\\linewidth}\n    \\begin{tabular}{c}\n        \\includegraphics[width=1.0\\linewidth]{images\/applications\/case1_2_v4.pdf}\n    \\end{tabular}\n    \\caption{Two floorplans which are behaviorally similar (having similar average exit flows) but design semantically different (Graph Edit Distance (GED) = $10$).}\n    \\label{fig:application_similarity_dynamic}\n    \\end{subfigure}\n    \n    \\caption{Pairwise similarity comparison between input floorplan and queried nearest neighbour from the embedding space.}\n    \\label{fig:application_similarity}\n\\end{figure}\n\n\n\n\\subsection{Behavioral and Geometrically-powered Floorplans Retrieval}\n\n{\\color{red} TODO: Rewriting}\n\nIn this use case we demonstrate the potential of using our embedding approach to retrieve similar and related floorplans. We trained three models with the architecture described. The difference between these models is the considered features. First model is trained only with design semantic features, second only with behavioral features and the third with all of the features.\n\nFirst, we trained the models over houseExpo dataset (Fig.~\\ref{fig:case11}). The top row is the baseline and the only considered feature is node degree. All the neighbours have same structure with same node degree but different design semantic features. The middle row showcases an example for the model which considered only design semantic features. This embedding setting contains a $12$ dimensional design semantic vector, $11$ for room types and $1$ for room dimensions. Given an input floorplan, a set of $5$ nearest neighbors from the embedding space are retrieved. For an embedding space to be meaningful and valid, a queried graph (input floorplan) should have itself as the first nearest neighbor during a retrieval, and therefore, in the figure, the first neighbor is same as the queried floorplan itself. The first 4 nearest neighbours have almost same structure (same node types with different sizes). The last nearest neighbour also has same structure with one extra node. But still the majority of its sequences are similar to queried graph. The bottom row showcases an example of floorplans retrieval from a collectively trained model with both design semantic and behavioral features. In this case, the features dimension for each node is 21. The top 5 nearest neighbours for the same input floorplan are retrieved from the embedding. Since in this case all of the features are considered, the neighbours must follow similar structure, similar design semantic and also similar behavioral pattern. The behavioral performance of the floorplans are visualized as color-coded crowd trajectories in the images by room type color.\n\nTo show performance of our model over more complicated datasets, the model is trained one time with design semantic and another time with all features over our synthesized indoor dataset. For both cases the top 5 nearest neighbours are retrieved and depicted in Fig. \\ref{fig:case22}.  Fig. \\ref{fig:case33} depicts the top 5 nearest neighbours from the model trained over our urban dataset only with design semantic features.\n\n\n\n\n\\subsection{Generation of a Composite Floorplan}\n\n{\\color{red} It is not complete}\n\nEmbedding spaces have commonly been used to make predictions and retrieving similar objects given some input criterion or an object. In this use case we demonstrate that our embedding methodology can also be used to generate a compound floorplan with collective features given multiple floorplans with a set of completely or partially different features. Figure~\\ref{fig:application_generation} showcases one such example. Two floorplans with $2$ and $3$ different room types respectively, are given as input to our autoencoder, and the first nearest neighbour is a composite floorplan consist of $5$ different room types. It not just contains all the features from input floorplans but also maintains a significant proportion of their geometric symmetries.\n\n\\begin{figure*}[tbp]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/applications\/application_generation_v3.pdf}\n    \\caption{Generation of a single floorplan with combined features from an embedding space given two floorplans with different set of features as input. Nodes of underline graph for each floorplan are color-coded based on room types.}\n    \\label{fig:application_generation}\n\\end{figure*}\n\n\n\\begin{comment}\n\\begin{table}[htp]\n\\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n\\textit{\\textbf{Ranks\/Models}}          & \\textbf{Model#1} & \\textbf{Model#2} & \\textbf{Model#3} \\\\ \\hline\n{First rank(query graph itself)} & 100\\%                          & 100\\%                   & 100\\%                 \\\\ \\hline\n{Second rank(proxy graph)}       & 86\\%                           & 83\\%                    & 85\\%                  \\\\ \\hline\n{Third rank(proxy graph)}        & 13\\%                           & 13\\%                    & 14\\%                  \\\\ \\hline\n{Fourth rank(proxy graph)}       & 1\\%                            & 3\\%                     & 1\\%                   \\\\ \\hline\n{Fifth rank(proxy graph)}        & 0                              & 2\\%                     & 0                     \\\\ \\hline\n\\end{tabular}\n\\caption{This table shows the percentage of the floorplans that have them-self as first nearest neighbours and the rank percentage of their proxy graphs.}\n\\label{tbl:percentt}\n\\end{table}\n\\end{comment}\n\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{cc}\n        \\includegraphics[width=0.99\\textwidth]{images\/experiments\/jjk_with_crowd_traces_v2.pdf}\n    \\end{tabular}\n    \\caption{The top 5 nearest neighbours from the embedding space for three different models trained on HouseExpo dataset. The top row shows baseline graphs with their respective degrees. In the second row, neighbors are retrieved from the model trained with design semantic features. Third row shows neighbors which are retrieved from a model jointly trained with design semantic and human behavioral features.}\n    \\label{fig:case11}\n\\end{figure*}\n\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{c}\n        \\includegraphics[width=0.99\\textwidth]{images\/experiments\/jjk2.pdf}\n    \\end{tabular}\n    \\caption{The top 5 nearest neighbours from the embedding space for two different models trained on synthesized indoor dataset. Top: neighbors which are retrieved from the model trained with design semantic features. Bottom: neighbors which are retrieved from a model jointly trained with design semantic and human behavioral features.}\n    \\label{fig:case22}\n\\end{figure*}\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth, height=5.5cm]{images\/expert_study\/experts-preferences_v3.pdf} \\\\\t\n\t\n\t\\caption{\\label{figure:experts-pref} The green portion of the bars show the average rating of experts' identification of similarity between input floorplan and the retrieved neighbors from the embedding space. }\n\\end{figure}\n\\subsection{Embedding With Design Semantic Features}\n\n\n\n\n\n\n\n\n\n\n\n    \n    \n    \n    \n    \n    \n\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{Sampling from posterior distribution}\nVAE learns the data distribution instead of deterministic mapping. Therefore, we can sample from these posterior distributions for generating new data. As mentioned in Sec.~\\ref{sec:training}, for each graph, we run random walk 11 times to generate 11 sets of node and edge sequences. To generate a new floorplan, we select a floorplan and a set from its 11 sets of node and edge sequences.\nBy decoding the samples from posterior distribution of these sequences, we get new sequences. There could be different strategies to produce a new floorplan with these newly generated sequences. We select the node with the highest degree that is repeated in all sequences as the main node. Therefore, the arrangement of other rooms can be fixed relatively. These new sequences give us information about room types and square footage. Our method does not encode the geometry shape of rooms, therefore, we can assume the room shapes are similar to the originally selected floorplan or any arbitrary shapes that satisfy the generated square footages (Fig. \\ref{fig:generr_posterior}). Generating new floorplans with this approach is limited and gives us similar floorplans in terms of the number of rooms and room types, the only changes in new floorplans are the square footage and geometry shape of rooms. The square footage generated in the new sequences do not have the same value for each room, however, with considering the original sequences as references, the average of generated square footage can be used for a new floorplan.\n\n\n\n\\subsection{Homotopies}\n\nVAE makes a continues embedding space, and it allows interpolation in this space. We used the concept of homotopy that means the set of points on the line between two embedding vectors. Instead of a set of random points, we limit our experiment to the point in the middle of the line. We can select two random sequences from two random floorplans, and after encoding them, we can calculate the difference of their embedding vectors. Adding half of their difference to the base vector and decoding it gives us a new sequence. Fig. \\ref{fig:generr} shows an example.\nBy replacing the newly generated sequence with the old random sequence from the original floorplan,\nwe can generate new floorplans. It can happen between any other random sequences as well, and in this way, we can generate more derivative samples. Different strategies for interpolation and generating new floorplans could be used here. With homotopy we do not have the mentioned limitations in sampling from posterior distribution. Floorplans with varying room types can be generated as a result of interpolation, and the only limitation is the geometry shape of rooms, which is not encoded. To address this limitation, we can assume the geometry shapes are the same as reference floorplans or any arbitrary shapes that satisfy the generated square footages.\n\n\\subsection{Contributions}\n\nOur contributions can be summarized as follow: (i) a workflow to represent floorplans as attributed graphs, augmented with design semantic and crowd behavioral features generated by running crowd simulations, (ii) a novel unsupervised generative deep-learning model to learn a meaningful vector representation of floorplans using  LSTM Variational Autoencoder, (iii) generation of new floorplans using the proposed model, and (iv) a user study to evaluate the qualitative performance of our approach, and (v) provision of a publicly released dataset of floorplans of indoor environments which are augmented with semantic and behavioral features. The design semantic features are extracted by our automated tool and the human behavioral features are generated by hours of running simulation.\n\n\\subsection{Floorplan dataset}\n\\label{subsec:dataset}\n\nWe used the houseExpo dataset \\cite{li2019houseexpo} that includes $35,357$ 2D floorplan layouts in JavaScript Object Notation format. There are 25 room types in this dataset where some of them share similar semantic labels (e.g. toilet and bathroom or terrace and balcony). We reduced the types to 10, including unknown types. This reduction is done by removing less common types based on reported statistical metrics in the dataset (e.g. freight elevator) and considering a unique type for similar propose components. The final room types are Bedroom, Bathroom, Office, Garage, Dining Room, Living Room, Kitchen, Hall, Hallway and Unknown. Unknown type is considered for room segments with a noisy label.\n \nAdditionally, we remove from the set the floorplans with inaccurate or missing labels. At the end of this process, we obtain $8,729$ floorplan layouts. This preprocessing is done to make the dataset solid for training. Corrupted data will decrease the training accuracy and, consequently, the test accuracy with undesirable outcomes of misleading the model.\n\n\\subsubsection{houseExpo++ dataset}\n\nThe original houseExpo dataset includes $35,126$ 2D floorplans. For each floorplan, the number of rooms, bounding box of the whole floorplan, a list of vertices and a dictionary of room categories, as well as their bounding boxes are provided~\\cite{li2019houseexpo}. While we can use the provided bounding boxes of the rooms for segmentation, these bounding boxes are not accurate, so we use them only for labeling. We compile these floorplans (in JSON format) to images. Then, we segment the images to find the rooms, their connections if there are any, the direction of connections and their square footage. The provided bounding boxes in the original dataset are used for assigning labels to room segments by the criterion of maximum overlapping. The described processes are done with our automated tool by image processing techniques. Moreover, we convert these JSON-formated floorplans to files in a readable format with our 3D crowd simulator (SteerSuite), and by running simulations, we record the human behavior features (features are provided in Table~\\ref{tab:featt}). We call this augmented dataset as \\textit{houseExpo++} that is publicly available at: \\url{https:\/\/github.com\/VahidAz\/Floorplan_dataset}. Since not all floorplans have accurate labels, we prune them and use $8,729$ floorplans for training and experiments (please refer to section~\\ref{subsec:dataset} for more details).\n\n\n\n\n\n\n\n\n\n\\subsection{Floorplans to attributed graphs}\n\\label{sec:feature_encode}\nAfter pruning the dataset, we represent each floorplan with a graph.\nThe rooms compose nodes, and the edges are their connectivity if there is an immediate door between the room pairs. For this conversion, we compiled houseExpo samples as images. Then we utilized a series of image processing techniques for room segmentation and finding their connectivity. \nGraph structure resembles the structure of floorplans like the number of rooms and their connectivity. However, considering floorplan structure is necessary but not enough. In order to have a better and more meaningful representation, we need to integrate high-level design semantic features. Moreover, humans are the inhabitant of these buildings, and their interaction with the environment provides valuable implicit information. Integration of how they interact with environments is necessary in term of safety or other type of design metrics like visibility and accessibility. Therefore, we augmented the graphs with both high-level design semantic features and human behavioral features. The nodes are augmented with both design semantic and human behavioral features and the edges only with design semantic features (Table \\ref{tab:featt}).\n\n\n\n\n\\begin{table}[]\n\\setlength{\\tabcolsep}{3pt}\n\\caption{\\label{tab:featt} Features on nodes and edges.}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|l|}{}                                                     & \\begin{tabular}[c]{@{}c@{}}Feature\\\\ Classes\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Feature\\\\ Types\\end{tabular}                                                                                                                                                                                    & Dimension                                                                 \\\\ \\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Node\\\\ Features\\end{tabular}} & \\begin{tabular}[c]{@{}c@{}}Design\\\\ Semantic\\\\\\hline\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Square Footage\\\\ Room types \\\\\\hline\\end{tabular}                                                                                                                                                                        & \\begin{tabular}[c]{@{}c@{}}1\\\\ 10\\\\\\hline\\end{tabular}                            \\\\ \n                                                                         & Behavioral                                                & \\begin{tabular}[c]{@{}c@{}}Not completed agents\\\\ Max evacuation time\\\\ Min evacuation time\\\\ Exit flow rate\\\\ Completed agents\\\\ Max traveled distance\\\\ Avg evacuation time\\\\ Avg traveled distance\\\\ Min traveled distance\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}c@{}}Edge\\\\ Feature\\end{tabular}                   & \\begin{tabular}[c]{@{}c@{}}Design\\\\ Semantic\\end{tabular} & Direction                                                                                                                                                                                                                                  & 4  \\\\ \\hline                                                                      \n\\end{tabular}\n\\end{table}\n\nThe design semantic features include room type, square footage and the connection direction. The room types represented with a 10 dimensional one-hot vector where $roomType_i(i) = 1$ if the type is $i^{th}$ type and other entities are zero. The square footage represented with a scalar value and direction of connection with a 4 dimensional one-hot vector. We considered four main directions: North, East, South and West. Thus, $direction_i(i) = 1$ if direction belongs to $i^{th}$ direction and other entities are zero. The room types are provided as a label in dataset and both square footage and direction of connection are extracted by image processing techniques. Note that since we are not given the cardinal directions, we considered the top left corner of floorplan images as origin. Hence, the $+y$ axis points to north, and other directions are considered relatively. In addition, the direction between rooms are bidirectional and for simplicity we considered the direction from node (i.e., room) with the highest degree (room with more connections) to node with low degree. It is because usually the node (i.e., room) with the highest degree is the main room in the floorplan like the living room or hallway. The room type and square footage is specific to each room and we consider them as node features. Since the direction of the connection is a shared property between room pairs, we add it to the edge features (edge between room pairs).\n\nThe human behavioral features are generated by simulation (Fig.~\\ref{fig:sim}). They include metrics regarding evacuation time, traveled distance, flow rate and the number of successful\/unsuccessful agents to exit from the corresponding building (Table \\ref{tab:featt}). To generate these behavioral features, we converted 2D floorplans to 3D models loadable in a crowd simulator, SteerSuite \\cite{singh2009open}.  The simulator automatically populates virtual agents in each room with the target to exit the floorplan and features are calculated. All features in this class are presented with one scalar value with total dimension 9. These features are generated for each room, hence we added them to the nodes feature vector. At the end of this step, each floorplan is represented with an attributed graph $G=(V, E)$ in which $V$ denotes its vertex set (room segments) and $E \\subseteq V \\times V$ denotes its edges (connectivity between room pairs). Each node $v$ has a $20$ dimensional feature vector $F_{v}$ and each edge $e$ has a $4$ dimensional feature vector $F_{e}$.\n\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{images\/bim_model_new.PNG} \\\\\n    \\caption{ \\label{fig:sim} Crowd simulations are used to compute behavioral features for the floorplans.}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Floorplan embedding}\n\nWe convert the floorplans to attributed graphs with features on both nodes and edges (Sec.~\\ref{sec:feature_encode}). These graphs represent floorplan geometry as well as their design semantics and behavioral features. They can be used directly for floorplan representation. However, graph analysis is expensive in term of computation and space cost. This challenge is addressed by proposing efficient methods for graph analysis like \\cite{kumar2016g,low2012distributed,gonzalez2014graphx} but yet they are not enough efficient and these methods do not represent graphs with a compact numerical vector. Another solution for addressing the complexity of graph analysis is graph embedding. Graph embedding maps the graphs to a low dimensional space while their properties and information are maximally preserved. In this low dimensional space the graphs with similar properties are close. We have different type of graphs like heterogeneous graph, homogeneous graph, attributed graph and etc. It means the input for graph embedding methods are varying and a single method can not handle all types. Moreover, graphs can be labeled or unlabeled. Graph embedding can mainly be divided to node embedding, edge embedding, hybrid embedding and whole-graph embedding   \\cite{DBLP:journals\/corr\/abs-1709-07604}.\n\nIn this paper, we are dealing with whole attributed graph embedding. It is because we want to represent each attributed graph (attributed floorplan) with a vector, and attributes present both on nodes and edges. These vectors encode graph (floorplan) structure as well as their design semantics and behavioral features. In addition, these graphs are unlabeled, i.e., we do not have label for each graph to perform supervised classification or regression. Moreover, the size of graph (in terms of the number of nodes) varies and is relatively small. Note that we can use other type of embedding like node embedding and then use the average (or other type of aggregation) of the node embedding vectors as the whole-graph representation. However, with this strategy the whole graph structure is not captured properly and does not lead to accurate vector representation~\\cite{bai2019unsupervised,taheri2018learning}. \n\n\nThere are quite a few works for the whole graph embedding.\nSome whole-graph embedding methods rely on the efficient calculation of graph similarities in graph classification tasks \\cite{shervashidze2011weisfeiler,niepert2016learning,mousavi2017hierarchical,bai2019unsupervised}. These methods are supervised and need a labeled dataset. In addition, they are designed for unattributed graphs. On the contrary, our graphs are attributed and unlabeled. Therefore, these types of methods are not applicable for our problem, and we need an unsupervised method. Graph2Vec~\\cite{narayanan2017graph2vec} is an unsupervised method that by maximizing the likelihood of graph subtrees given graph embedding, generates vector representations. However, since this model uses subgraphs, the graph global information is not captured properly \\cite{taheri2018learning}. In addition, this method is not applicable for graphs with attributes both on nodes and edges. In \\cite{taheri2018learning} for capturing the whole-graph structures, they take advantage of random walk for converting graphs to a set of sequences. It is shown using random walk leads to better representation in comparison to the adjacency matrix, because the random walks capture more than the immediate neighbourhood. \nThen a LSTM autoencoder is presented to learn graph representations. However, this method suits unattributed graphs.\n\nSentences are presented with a sequence of words. In \\cite{DBLP:journals\/corr\/BowmanVVDJB15,wang2019topic} LSTM Variational Autoenocder is used for text and sentence embedding and generation. The performance of converting graphs to sequences in \\cite{taheri2018learning} and the methods proposed in \\cite{DBLP:journals\/corr\/BowmanVVDJB15,wang2019topic,taheri2018learning} motivate us to convert our graphs to sequences and propose a novel LSTM Variational Autoencoder model that suits our unlabeled attributed graphs (both on nodes and edges). In particular, we convert each floorplan graph into a set of sequences (which will be described in Sec.~\\ref{sec:graph2seq}), and we propose a generative model that maps our graphs (in sequences) to a d-dimensional space $\\theta : G \\rightarrow R^d$.\nThe proposed model is detailed in the next section.\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Model}\nWe present a novel LSTM Variational Autoencoder architecture illustrated in Fig.~\\ref{fig:framework}. LSTM is a special kind of Recurrent Neural Network (RNN). It is designed for learning long-term dependencies by introducing state cell \\cite{hochreiter1997long} to address the RNN problem with long term sequences \\cite{bengio1994learning}. \nAutoencoders are a type of unsupervised neural network with two connected networks. The first network is an encoder that converts the inputs to latent vectors in a low dimensional space. The second network is the decoder, which reconstructs the original input vector from latent vectors \\cite{kramer1991nonlinear}. However, the vanilla autoencoders map each input to a constant vector. In other words, the encoded vectors are not continuous, and interpolation is not allowed. Variational Autoencoder (VAE) is designed to address the limitation of vanilla autoencoders as a generative model. VAE is a generative model that learns the Probability Density Function (PDF) of the training data \\cite{kingma2013auto}. \n\n\nAs mentioned we have attributes both on nodes and edges with different dimensions. To address this difference in dimension, we consider two parallel LSTM VAE, one for node sequences and one for corresponding edge sequences. Let $S_n = {s_{n1}, s_{n2}, ..., s_{ni}}$ be a node sequence and $S_e = {s_{e1}, s_{e2}, ..., s_{ei-1}}$ be the corresponding edge sequence. We aim to learn an encoder and decoder to map between the space of these two sequences and their continuous embedding $z \\in R^d$. In each branch, the encoder is defined by a variational posterior $q_\\phi(z|S_n)$ and $q_\\phi(z|S_e)$ and the decoder by a generative distribution\n$p_\\theta(S_n|z)$ and $p_\\theta(S_e|z)$, where $\\theta$ and $\\phi$ are learned parameters. For each branch, loss function has two terms (Eq. \\ref{eq:eq1}). The first term is reconstruction error that we used Mean Square Error (MSE). This term encourages the decoder to learn to reconstruct the data $\\bar S_n,  \\bar S_e$. The second term which is a regularizer is Kullback-Leibler divergence to penalize loss if the encoder outputs representations that are different than a standard normal distribution $N(0, 1)$ \\cite{simonovsky2018graphvae}.  In training, two branch loss functions are summed for back-propagation (Eq. \\ref{eq:eq1}).\n\n\n\n\n\\begin{multline} \\label{eq:eq1}\nLoss_{total} = (||S_n-\\bar S_n||^2 + KL[q_\\phi(z|S_n), N(0, 1)]) + \\\\\n(||S_e-\\bar S_e||^2 + KL[q_\\phi(z|S_e), N(0, 1)])\n\\end{multline}\n\n\n\nIn both branches for the encoder we have a LSTM layer with 256 units. In the following we have two fully connected layer with dimension 16 for generating $\\mu$ and $\\sigma$. Then we have the sampling and finally we have the decoder with one LSTM layer with 256 units for reconstructing the input sequences. The Adam \\cite{kingma2014adam} is used as optimizer. Before training, each graph is converted to a set of sequences (Sec.~\\ref{sec:graph2seq}) and these sequences are used as input for training. After training, each graph is represented by averaging its sequences' embedding vectors. \n\n\n\n\n\\subsection{Graphs to sequences}\n\\label{sec:graph2seq}\nGraphs can converted to sequences by methods including but not limited to random walk or Breadth First Search (BFS). In \\cite{taheri2018learning} the random walk, BFS and shortest path between all pairs of nodes are utilized. The experiments show sequences generated by random walk lead to a better vector representation. The reason is random walk captures more than immediate neighbours of nodes. Random walk is introduced in \\cite{Perozzi_2014} for converting graphs to sequences. In this version, we pick a node, and then we pick one of its edges randomly to move to the next node. We repeat this procedure until we get a walk of some predefined length (length of a walk is defined by the number of nodes on the walk, and walk that is shorter than the predefined length will be padded into the predefined length). Later, two other versions are proposed.\n\n\\textit{Random Walk 1}.  In \\cite{grover2016node2vec} the random walk is modified to have two parameters $Q$ and $P$. Parameter $Q$ is the probability of discovering the undiscovered parts of the graph and parameter $P$ is the probability for returning to the previous node.  We call the random walk proposed in \\cite{grover2016node2vec} `Random Walk 1'.\n\n\\textit{Random Walk 2}. In \\cite{taheri2018learning} the random walk is modified by adding probability $1\/D(N)$, where $D(N)$ is the degree of node $N$. In this walk we start from a node and the next node will be selected by its probability $1\/D(N)$. We call this random walk presented in \\cite{taheri2018learning} as `Random Walk 2'.\n\nWe used both walks with different walk lengths to find the best performer walk and walk length.\nBesides node sequences, the edge sequences are captured at the same time. Both of nodes and edges sequences are used for training the model.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Dataset}\n\\label{subsec:dataset}\n\n{\\color{blue}\nWe used the houseExpo dataset \\cite{li2019houseexpo} that includes $35,000$ 2D floorplan layouts in JavaScript Object Notation (JSON) format. There are 25 component types in this dataset where some of them share similar semantic labels (e.g. toilet and bathroom or terrace and balcony). We reduce the types to 10 including unknown label \\ref{tbl:roomtypes} by removing less common semantics based on reported statistical metrics in dataset (e.g. freight elevator) and considering an unique type for similar proposes components.\nAdditionally, we remove from the set the floorplans with inaccurate or missing labels. At the end of the process, we obtain 8775 floorplan layouts. Figure~\\ref{fig:rawf} shows an image of a raw floorplan compiled as image.\n}\n\n\\begin{table}[hbt]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\n\\textbf{Room types} & \\begin{tabular}[c]{@{}c@{}}Bathroom\\\\ Office\\\\ Garage\\\\ Dining\\_Room\\\\ Bedroom\\\\ Living\\_Room\\\\ Kitchen\\\\ Hall\\\\ Hallway\\\\ Unknown\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\label{tbl:roomtypes}\n\\caption{Available room types in datasets.}\n\\end{table}\n\n\n\n\\subsection{Floorplans to Graphs and Feature Augmentation}\n\n{\\color{blue}\n\nIn this step, raw floorplans are represented as an attributed graph in which rooms compose nodes and edges are their connectivity if there is an immediate door between room pairs. For this conversion houseExpo samples are compiled as image and by image processing techniques room segments and their connectivity are recognized. The connection direction is recognized as a part of this process. We consider four directions East, West, North and South. Finding connection between two room segments is from the room with high degree to node with low degree since our graphs is not directed.\n\nFloorplans structure like number of rooms and their connectivity is encoded in graph structure. However, capturing structure is necessary but not enough. So, we are augmenting graphs nodes with design semantic and human behavioral features and edges are with direction features shown in Table \\ref{table:feat}.\nDesign semantic features include room type with 10 dimensional one-hot vector and square footage with one scalar value. Human behavioral features are generated by simulation and include metrics regarding evacuation time, traveled distance, flow rate and the number of successful\/unsuccessful agents to exit from corresponding building. All features in this class are presented with a scalar value and total dimension is 9.\nTo generate these behavioral features the 2D floorplans are converted to 3D models loadable in a crowd simulator, SteerSuits \\cite{singh2009open}.  The simulator automatically populates virtual agents in each room with the target to exit the floorplan and features are calculated.\n\nAt the end of this step, the floorplans are presented as an attributed graph $G=(V, E)$ in which $V$ denotes its vertex set (room segments) and $E \\subseteq VxV$ denotes its edges (connectivity between room pairs). Each node $v$ has a constant dimensional feature vectors $F_{v}$. Moreover, each edge $e$ has a constant feature vectors $F-{e}$. The nodes feature vector dimension is 11 and edge feature vector dimension is 4.\n}\n\n\n\\begin{table}[htb]\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multirow{3}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Node\\\\ Features\\end{tabular}}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Feature \\\\ Classes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Feature \\\\ Types\\end{tabular}}                                                                                                                                                                                   & \\textbf{\\begin{tabular}[c]{@{}c@{}}Feature\\\\ Dimension\\end{tabular}}               \\\\ \\cline{2-4} \n                                                                                  & \\textbf{\\begin{tabular}[c]{@{}c@{}}Design \\\\ Semantic\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Square Footage\\\\ Room Type\\end{tabular}}                                                                                                                                                                         & \\textbf{\\begin{tabular}[c]{@{}c@{}}1\\\\ 11\\end{tabular}}                            \\\\ \\cline{2-4} \n                                                                                  & \\textbf{Behavioral}                                                 & \\textbf{\\begin{tabular}[c]{@{}c@{}}Not completed agents\\\\ Max evacuation time\\\\ Min evacuation time\\\\ Exit flow rate\\\\ Completed agents\\\\ Max traveled distance\\\\ Ave evacuation time\\\\ Avg traveled distance\\\\ Min traveled distance\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\\\ 1\\end{tabular}} \\\\ \\hline\n\\textbf{\\begin{tabular}[c]{@{}c@{}}Edge\\\\ Features\\end{tabular}}                  & \\textbf{\\begin{tabular}[c]{@{}c@{}}Design\\\\ Semantic\\end{tabular}}  & \\textbf{Direction}                                                                                                                                                                                                                                  & \\textbf{4}                                                                         \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Floorplan Embedding}\n{\\color{blue} \nWe convert floorplans to attributed graphs which both nodes and edges have feature. These graph are representing floorplan geometry as well as it design semantics and behavioral features. These graphs can be used directly for comparing floorplans. However, Graph analysis are expensive in term of computation and space cost. This challenge addressed by proposing efficient methods for graph analysis like \\cite{kumar2016g} but still they are not efficient and these methods do not represent graphs with a vector. Another solution for addressing complexity of graph analysis is graph embedding. Graph embedding maps the graphs to a low dimensional space while their properties are preserved. Particularly we are looking for a mapping function to map graph G to a n-dimensional vector which graph properties are reserved and graphs with similar properties are close in this n-dimensional embedding space.Graph embedding can mainly divided to node embedding, edge embedding, hybrid embedding and whole-graph embedding   \\cite{DBLP:journals\/corr\/abs-1709-07604}.\nIn this study, the aim is to embed whole floorplan(graph) and other types are not in our focus. There is possibility to use partial embedding like node embedding and use the average of the node embedding vectors as whole-graph representation. But since in partial embedding the whole graph structure is not captured, they do not lead to good performance \\cite{taheri2018learning}. \n\nSome whole-graph embedding methods relying on efficient calculation of graph similarities in graph classification tasks \\cite{shervashidze2011weisfeiler}. These methods are supervised and need labeled dataset. Since, none of our datasets are labeled we need an unsupervised method. Graph2Vec\\cite{narayanan2017graph2vec} is an unsupervised method that by maximizing the likelihood of graph subtrees given graph embedding generates representations. Since this model uses subgraphs for finding representation, the graph global information is not captured and causes low performance. In \\cite{taheri2018learning} for capturing the whole-graph structures, they converted a graph to a set of sequences and proposed a LSTM autoencoder to learn graph representations. This method suits unattributed graphs and nodes are embedded for generating nodes feature vectors prior training. In \\cite{DBLP:journals\/corr\/BowmanVVDJB15} and \\cite{wang2019topic} Variational autoenocder is used for text and sentence embedding and generation which in both inputs are sequences. It motivates us to adapt their model both for embedding and generating floorplans \\ref{fig:method}.\n\nLong Short Term Memory, \n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Graphs to Sequences}\n{\\color{blue}\nWe can convert graphs to sequences by methods including not limited to Random walk or Breadth First Search(BFS). In \\cite{taheri2018learning} they used Random walk, BFS and shortest path between all pairs of nodes. The experiments show sequences generated by random walk lead to a better embedding performance.\n\nRandom walk is used in \\cite{Perozzi_2014} for converting graphs to sequences. In this version we pick a node and then randomly we pick an edge to move to its neighbours. We repeat this procedure until we get a walk of some predefined length. In \\cite{grover2016node2vec} the random walk is modified to have two parameters $Q$ and $P$. Parameter $Q$ is the probability of discovering the undiscovered parts of the graph and parameter $Q$ is the probability for returning to the previous node. In \\cite{taheri2018learning} they modified the random walk with adding probability $1\/D(N)$, where $D(N)$ is the degree of node $N$. In this walk we start from a node and the next node will be selected by its probability $1\/D(N)$.\n\nWe used both walks with different walk length to figure out which walk and which walk length leads to better result for us. We call the random walk proposed in \\cite{grover2016node2vec} 'Random Walk 1' and the one presented in \\cite{taheri2018learning} as 'Random Walk 2'. Beside node sequences, the edge sequences captured and then both are used for training the model.\n\n\n}\n\n\n\n\n\n\n\n\\subsection{Model}\n{\\color{blue}\nIn this paper we present a LSTM variational Auto-encoder model for embedding our attributed graphs Fig. \\ref{fig:vae_model}. As explained we have attributes both in nodes and edges with different dimension. To handle this difference in dimensionality, we consider two parallel LSTM VAE, one for node sequences and one for corresponding sequences. In training phase both branch loss function is added up. \n}\n\n{\\color{red} TODO: Completing this part, maybe it will integrated somewhere else.}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=15cm, height=5cm]{images\/vae_imgs\/model.png} \\\\\n\t\\caption{\\label{fig:vae_model} {\\color{blue} The proposed model with two parallel LSTM VAE, one for node sequences and one for corresponding edge sequences. Each branch loss function is added up at the end in training.} {\\color{green} Does this reflect the most recent developments ? We need to greatly improve this image. Perhaps Honglu can help}\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\\begin{comment}\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{c}\n        \\includegraphics[width=14cm,height=6cm]{images\/Model_2.png}\n    \\end{tabular}\n    \\caption{\\textcolor{blue}{MK: I would consider integrating this into Figure 1 and making Figure 1 more complex} Proposed model, LSTM Autoiencoder. Right part shows the encoder that has 2 layer LSTM, first one has 84 and second one has 64 LSTM units. Left part shows decoder that has two LSTM layer, first one has 64 and the second one has 84 LSTM units(opposite to encoder).}\n    \\label{fig:modell}\n\\end{figure*}\n\\end{comment}\n\n\\subsection{Floorplan representation}\nFloorplan representation aims for representing floorplans with numerical vectors that their structure, as well as their features, are encoded in these vectors. \nTo the best of our knowledge, representing floorplans by numerical vectors are not done to date. There are some prior works for retrieving similar floorplans with representing floorplans as images or graphs. They can be mainly divided into three categories: image-based, graph-based and symbol-spotting methods.\n\n\n\n\n\n\\subsubsection{Image-based methods} \nSeveral approaches based on conventional image processing techniques for comparing floorplans are proposed.\nIn these approaches, floorplans are represented as images and Histogram of Oriented Gradients(HOG)~\\cite{dalal2005histograms}, Bag of Features(BOF)~\\cite{lazebnik2006beyond}, Local Binary Pattern(LBP)~\\cite{ahonen2006face} and Run-Length Histogram~\\cite{de2013runlength} have been utilized for extracting features from these images. Then, these extracted features are used for comparison and retrieving floorplans. By emerging Convolutional Neural Networks (CNN), In~\\cite{sharma2017daniel} a deep CNN is presented for feature extraction to address the limitation of conventional image processing techniques for extracting features. These method suits object-centric floorplans datasets in which floorplans are annotated with furniture or specific visual symbols. However, these features are not semantics and do not correctly capture the high-level design structures. Moreover, human behavioral features are not considered in these methods, whereas occupants' behaviours (i.e., trajectories) are often highly correlated with environments~\\cite{sohn2020laying}.\n\n\n\n\\subsubsection{Graph-based methods}\nIn this category, floorplans are represented with graphs and graph matching methods are utilized for measuring their similarity. Different strategies are used for representing floorplans as a graph. In \\cite{sharma2016unified} rooms are nodes, and edges capture the adjacency between the rooms. In addition, nodes are augmented with furniture types annotated in floorplans. \nIn~\\cite{sharma2018high}, the graphs are augmented with more attributes like room area and furniture style in three different representation layers. Since in these works, floorplans are represented with graphs, all of them capture the floorplans structure, and some of them add attributes to nodes. However, mostly they do not include semantic and high-level features as well as human behavioral features. Moreover, all of these methods use graphs as a final representation and do not provide numerical vectors. These methods use graph matching methods for finding similarity, which directly applied over graphs.\n\n\n\n\n\n\n\\subsubsection{Symbol-Spotting methods}\nSymbol spotting is a special case of Content-based Image Retrieval (CBIR) \\cite{heylighen2001case,Richter2007}  which is used for document analysis. By giving a query, system retrieves zones from the documents which are likely to contain the query. Queries could be a cropped or hand-sketched image. Pattern recognition techniques are used in symbol spotting methods like moment invariants such as Zernike moments in ~\\cite{lambert1995line}. Reducing search space in symbols spotting methods is proposed based on hashing of shape descriptors of graph paths (Hamiltonian paths) in ~\\cite{dutta2011symbol}.\nSIFT\/SURF~\\cite{weber2010scatch} features being efficient and scale-invariant are commonly used for spotting symbols in graphical documents. Symbol spotting methods are applied to small datasets which do not have complex images, and they are only applicable for retrieval purpose.\n\n\n\n\n\\subsection{Floorplan generation}\nFloorplan generation aims for generating floorplan designs automatically by satisfying some constraints like room sizes and adjacency between rooms. We can divide them into two groups: Procedural\\slash Optimization-based methods, and recently deep learning methods.   \n\n\\subsubsection{Procedural\\slash Optimization-based methods}\nIn these methods, the constraints are manually defined, and optimization methods are used for constraint satisfaction to generate new floorplans. In \\cite{merrell2010computer}, they used bayesian network to learn synthesizing floorplans with given high-level requirements. In \\cite{rodrigues2013evolutionary,rodrigues2013evolutionary1} they proposed an enhanced Evolutionary Strategy (ES) with a Stochastic Hill Climbing (SHC) technique for floorplan generation.\n\n\n\\subsubsection{Deep Learning methods}\nIn \\cite{wu2019data} a deep network was proposed for converting a given floorplan layout as input to a floorplan with predicting rooms and walls location. In \\cite{chaillou2019ai+} they proposed a method comprising of three deep network models to generate floorplans. In the first step, the model generates the layout, then room locations and finally, furniture locations. In this model, users are in loop, and they can modify the input for the next steps. In \\cite{hu2020graph2plan} they proposed a framework based on deep generative network. \nUsers specify some properties like room count, and their model converts a layout graph, along with a building boundary, into a floorplan.  A graph-constrained generative adversarial network is proposed in \\cite{nauata2020housegan}. They took an architectural constraint as a graph (i.e., the number and types) and produced a set of axis-aligned bounding boxes of rooms.\n\n\n\\subsection{Comparison to prior work}\n\nPrevious works do not represent floorplans with numerical vectors. They usually overlooked the design semantic and human behavioral features. In this paper, we are presenting floorplans with numerical vectors, encoded with design semantic and human behavioral features, and room directions. In addition, we also utilize generative models for addressing floorplan generation at the same time.\n\nThe proposed method is an extension of a recent approach~\\cite{azizifloorplan} with the following notable extensions: \n\n\\begin{itemize}\n    \\item A novel LSTM Variational Autoencoder with two branches for representing attributed graphs with features on both nodes and edges with numerical vectors. In addition, we also include rooms' directions as edge attributes to maintain layout symmetry.\n    \n    \\item We study and showcase the generative power of our model for generating new floorplans.\n    \n    \\item We conducted a user study to evaluate the qualitative performance of our embedding approach.\n\\end{itemize}\n\n\\noindent We believe these extensions significantly expand upon the previous conference paper and merit consideration for the journal. \\newline\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Training}\n\\label{sec:training}\nAs described in Sec.~\\ref{sec:graph2seq}, we converted graphs to nodes and edge sequences. We utilized both mentioned walks with walk length 3, 5 and 7. For random walk 1, we set both $Q$ and $P$ to 0.5.\nFor each graph, we run random walk $11$ times.\nTherefore, we have 11 sets of node and edge sequences for each graph. Out of 10, one of them is considered as proxy set (proxy graph). These proxy sequences are used later for evaluation.\nIn total we have 306440 nodes sequences and 306440 corresponding edge sequences. The sequences with length less than target length are padded with zero. We trained three models with considering different set of features.\n\n\\textit{Model 1}. In this model we only considered the design semantic features on nodes. We removed the edge branch and the model is trained only with nodes sequences. The dimension of node features is 11.\n\n\\textit{Model 2}. In this model we considered design semantic features both on nodes and edges. The model is trained with both branches. \nThe features dimension on nodes is 11 and on edges is 4.\n\n\\textit{Model 3}. In this model we considered all design semantic features and human behavioral features. The model is trained with two branches. The features dimension on nodes is 20 and on edges is 4.\n\nAll three models have the same described architecture and loss function. Note that in Model 1, the edge branch is removed and we have only node branch's loss function. \nBut, in other two models, the loss is summation of two branches loss. The learning rate was set empirically as 0.001. All three models are trained on a machine with 32 GB RAM, 12 * 3.50 GHz cores CPU and Quadro K620 GPU with 2 GB Memory. In average each model takes about 4 hours for training with 50 epochs.\n\n\n\n\n\n\n\n\n\n\\subsection{Quantitative evaluation}\nThis section presents quantitative results from the experiments.\n\n\\subsubsection{Nearest neighbours ranks}\n\\label{sec:nn_rank}\nAs mentioned, by embedding, the graphs are mapped to a continues embedding space. The graphs with similar structure and properties should be close to each other in the embedding space. The closeness of two floorplans can be measured by the euclidean distance of the two corresponding embedding vectors (smaller distance denotes higher similarity).\nIf this embedding space is well constructed, similar floorplans in terms of structure, semantic and behavioral properties should be close to each other.\nTherefore, for each graph (called as query graph), we compute the euclidean distance from this graph to other graphs (including itself) and rank the other graphs for this graph according to the distance.\nWe hypothesize that each graph should find itself as the first nearest neighbour and its proxy graph (a different set of sequences for query graph) in close ranks. For this study, we use the model 3 to obtain the top 5 nearest neighbours for each floorplan in the learned embedding space. We calculate the percentage of graphs that have themselves in the first rank and the percentage of proxy graphs in the other four ranks. The table \\ref{tbl:nnr} shows these percentages with different walk lengths for both random walks.\n\n\n\n\n\n\\begin{table}[]\n\\caption{\\label{tbl:nnr} Nearest neighbour ranks (Sec.~\\ref{sec:nn_rank}) with two types of random walks and walk length 3, 5 or 7. For each graph,  we  compute  the  euclidean  distance  from  this graph  to  other  graphs  (including  itself and a proxy graph which is a different set of sequences of the query graph)  and  rank  the other graphs for this graph according to the euclidean distance. Each  graph  should  find  itself  as the first nearest neighbour and its proxy graph in close ranks. We calculate the percentage of graphs that have themselves in the first rank and the percentage of proxy graphs in the other top four ranks (e.g., `[0, 75, 4, 2, 2]' in the table denotes that 75$\\%$ graphs have their proxy as the top 2 nearest neighbor, 4$\\%$ as  top 3, 2$\\%$ as  top 4, and 2$\\%$ as top 5). This analysis showcases that walk length 5 can lead to better performance, and the random walk  2 is superior. }\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|l|}{}                                  & \\begin{tabular}[c]{@{}c@{}}Walk\\\\ Length\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Rank of\\\\ query floorplan\\\\ within 5 NN\\end{tabular}                               & \\begin{tabular}[c]{@{}c@{}}Rank of\\\\ proxy graph\\\\ within 5 NN\\end{tabular}                                  \\\\ \\hline\n\\begin{tabular}[c]{@{}c@{}}Random\\\\ Walk 1\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}3\\\\ 5\\\\ 7\\end{tabular}     & \\begin{tabular}[c]{@{}c@{}}{[}100, 0, 0, 0, 0{]}\\\\ {[}100, 0, 0, 0, 0{]}\\\\ {[}100, 0, 0, 0, 0{]}\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}{[}0, 49, 2, 1, 1{]}\\\\ {[}0, 75, 4, 2, 2{]}\\\\ {[}0, 57, 2, 1, 1{]}\\end{tabular}   \\\\ \\hline\n\\begin{tabular}[c]{@{}c@{}}Random\\\\ Walk 2\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}3\\\\ 5\\\\ 7\\end{tabular}     & \\begin{tabular}[c]{@{}c@{}}{[}100, 0, 0, 0, 0{]}\\\\ {[}100, 0, 0, 0, 0{]}\\\\ {[}100, 0, 0, 0, 0{]}\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}{[}0 , 83, 3, 2, 1{]}\\\\ {[}0, 94, 2 , 1, 1{]}\\\\ {[}0, 89, 3, 1, 1{]}\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\nAs table \\ref{tbl:nnr} shows, in both random walks, walk length 5 leads to better performance. In addition, the random walk 2 is superior. By random walk 2 and walk length 5, each graph by itself is in the first rank and $94\\%$ of proxy graphs in second rank. Since proxy graphs are a different set of sequences on the graphs, if the model performs properly, a good percent of the proxy graphs should be present in top ranks. Random walk 2 performs better since it captures our graphs structure better because in graphs (i.e., floorplans) we have always a main node (i.e., room) with high degree. Then moving toward this node gives the sequences that capture our graph structure better. The walk length has dependency to size of the available graphs in dataset. For us walk length 5 is the suitable length since in both random walks, the embedding performance is better in compare to walk length 3 and 5.\n\n\n\n\n\n\n\n\\subsubsection{Clustering}\nAs mentioned in previous section floorplans with similar properties are close in the embedding space. This similarity is in term of floorplans structure, design semantics and human behavioral features. There are many parametric method for clustering like KMeans \\cite{wagstaff2001constrained} that we need to give the number of clusters as input parameter. Since we do not want to limit ourself to a specified number of clusters, we used Density Based Spatial Clustering of Applications with Noise (DBSCAN). It is a non parametric clustering method based on density. Each dense region (close packed points) represents a cluster and the points in low density regions are marked as outliers. It has two parameters, the minimum number of points in each cluster and the maximum distance between two samples in each cluster \\cite{ester1996density}.\n\nWe run DBSCAN over our embedding space to cluster it and then we used TSNE \\cite{maaten2008visualizing} to reduce vectors dimension to two. The Fig. \\ref{fig:clus} shows the resulting clusters over 1000 samples. We evaluated some clusters to see whether the samples in clusters follow the same pattern or not. We observed almost in all considered clusters the number of nodes, node degrees, node types and features are close. Except features closeness we can calculate the standard deviation for number of nodes, average of node degrees and node types for each cluster. We calculated the mentioned metrics for all clusters out of 1000 samples and their averages are provided in Table \\ref{tbl:stdd}. In Fig. \\ref{fig:clus}, two sample floorplans from one of the clusters depicted which shows the graph properties are encoded accurately with our model.\n\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{images\/clustring.pdf} \\\\\n    \\caption{ \\label{fig:clus}The clusters after running DBSCAN over 1000 random samples. \n   \n    Two samples from one of the cluster are shown. They have the same number of nodes, same node degrees and similar room types. This shows the embedding space indeed captures the design semantics of floorplans.}\n\\end{figure}\n\n\n\\begin{table}[]\n\\centering\n\\caption{Average of standard deviation for number of nodes, node degrees average and node types in clusters out of 1000 samples.} \\label{tbl:stdd}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n             & \\textbf{\\begin{tabular}[c]{@{}c@{}}Number\\\\ of nodes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Average\\\\ of node degrees\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Node\\\\ types\\end{tabular}} \\\\ \\hline\n\\textbf{STD} & 0.16                                                               & 0.09                                                                       & 1.07                                                          \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Qualitative evaluation}\nThis section presents qualitative results from the experiments.\n\n\\subsubsection{Nearest neighbours (NNs) }\n\\label{sec:quali_nn}\nAs described we trained three models with different set of features. Each model makes an embedding space.  We selected three random floorplans and found their top 5 nearest neighbours in the corresponding embedding space of each model. The Fig. \\ref{fig:nnnn1} shows the query floorplans and their top 5 nearest neighbours. For each sample, first row shows the NNs in the first model's embedding space, second line shows the NNs in the second model's embedding space and third row shows the NNs in the third model's embedding space. As shown in the image, with the first model, the floorplans have the same structure in term of the rooms numbers, room (node) degrees and room types. But the room arrangements are not similar. In the second model, since the edge features are added, the high rank NNs follow the same arrangement and with moving toward low rank NNs the arrangement similarity is dcreased. However, they have similar structure yet. In the third model, the human behavior features are added as well and now floorplans with similar behavioral features get close to query floorplans. The last row for each sample shows the visualization of crowd flow rate. The numbers inside floorplans in first and second row shows the square footage of each room. In third rows the numbers depict flow rates. Please note, as mentioned in section \\ref{sec:feature_encode}, the north is at the top and other directions are recognized correspondingly.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth,height=\\textheight]{images\/Results_v2.png} \\\\\n\t\\caption{\\label{fig:nnnn1}  The top 5 NNs for three floorplans found with three models. The first row shows the NNs with first model, second row shows the NNs with second model and third row shows the NNs with third model. The color of the room denotes the room type, and the numerical value inside each node denotes the square footage of the room. In the third row of each sample, the crowd flow is visualized and the numbers depicts flow rates. See Sec.~\\ref{sec:quali_nn} for details.}\n\\label{fig:model}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth]{images\/genn2_v2.pdf} \\\\\n\n\\caption{\\label{fig:generr} Floorplan generation with interpolating in embedding space. We select two random sequences from two random floorplans and after encoding them, we calculate the difference of their embedding vectors. Adding half of their difference to the base vector and decoding it gives us a new sequence.  These new sequence can be used for generating new floorplans.\nSee Sec.~\\ref{sec:floorplan_generation} for details.}\n\\end{figure*}\n\n\n\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth]{images\/gen1.pdf} \\\\\n\n\\caption{\\label{fig:generr_posterior} Floorplan generation with sampling from posterior distributions. To generate a new floorplan, we select a floorplan from our dataset and a set from this floorplan's 11 sets of node and edge sequences. By decoding the samples from posterior distribution of these sequences, we obtain new sequences and then map them into new floorplans. }\n\\end{figure*}\n\n\\section{Introduction}\n\\label{intro}\nYour text comes here. Separate text sections with\n\\section{Section title}\n\\label{sec:1}\nText with citations \\cite{RefB} and \\cite{RefJ}.\n\\subsection{Subsection title}\n\\label{sec:2}\nas required. Don't forget to give each section\nand subsection a unique label (see Sect.~\\ref{sec:1}).\n\\paragraph{Paragraph headings} Use paragraph headings as needed.\n\\begin{equation}\na^2+b^2=c^2\n\\end{equation}\n\n\\begin{figure}\n  \\includegraphics{example.eps}\n\\caption{Please write your figure caption here}\n\\label{fig:1}      \n\\end{figure}\n\\begin{figure*}\n  \\includegraphics[width=0.75\\textwidth]{example.eps}\n\\caption{Please write your figure caption here}\n\\label{fig:2}      \n\\end{figure*}\n\\begin{table}\n\\caption{Please write your table caption here}\n\\label{tab:1}      \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\nfirst & second & third  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nnumber & number & number \\\\\nnumber & number & number \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Apparatus} Floorplans are presented as 2D blueprints (e.g. a top-down skeletal view of an environment layout). The users (e.g. study participants) viewed these blueprints as high-resolution images on their own computer screens via an online survey. For model (1), each room in a floorplan is annotated with room dimension (e.g., square footage area) and color-coded with respect to its room type. The annotation for model (2) is similar to model (1) with an addition of edges between rooms and their respective directions (e.g., North, East, West, South). For model (3), we showed color-coded trajectories of virtual occupants from the rooms they spawned-in to the exit, along with square footage area of each. \n\n\\subsection{Participants} Ten ($10$) domain experts from the architecture community ($4$ female and $6$ male) voluntarily participated in the user study. Table~\\ref{table:experts-info} shows the demographic information and domain knowledge of the experts. On average, all the participants had above-average experience and expertise in interpreting architecture designs and were Knowledgeable of computational tools for design space exploration (self-reported).\n\n\\subsection{Procedure and Task} The user study is conducted as an online survey and delivered in four parts. In part (a), users are asked to provide their demographic information and report the domain knowledge and expertise in architecture and urban design. In part (b), users are presented with five different input floorplans. For each input floorplan, a sequence of 5 nearest neighbours are presented in a randomized order, which are retrieved using model (3), and presented to the users ``without'' any visual annotations. Users are asked to interactively reorder the given sequence of floorplans (e.g., via drag and drop), based on their perceived \"similarity\" of these floorplans with respect to input floorplan. The ordering sequence is arranged such that, more a floorplan is towards left in the order, the nearest it gets to the input floorplan. In parts (c), (d) and (e), the nearest neighbours are retrieved using models (1), (2) and (3) respectively, for the same five input floorplans which are used in part (b). In parts (c), (d), and (e), the floorplans are presented to the users ``with'' visual annotations for their respective features. We estimated that the user study will take up to $15$ minutes at maximum to complete. \n\n\\subsection{Independent and Dependent Variables} Input floorplans and the retrieved nearest neighbours from the models are the primary independent variables. The rearranged sequences of floorplans by the users are the only dependent variables. \n\n\\subsection{Results} Figure~\\ref{figure:user-sequences} shows the user-ordered sequences of the nearest neighbours for the three models from user study.  The colored bars for each neighbour of an input floorplan represent the number of users who correctly perceived the order of the neighbour in the given sequence. Overall, about $28.68$\\% of the neighbours are accurately ordered in their sequences based on their perceived similarity with respect to input flooplans for model (1), $59.28$\\% for model (2) and about $77.6$\\% for model (3), collectively by all the users. These results highlight that users least performed when they had to perceive the similarity between floorplans by considering the design semantic features alone, whereas they performed comparatively better when presented with the neighbours annotated with edge and\/or behavioural attributes. The users performed the best for the model (3) when presented with the floorplans visually annotated with design semantics (e.g., room types), edge (e.g., movement direction of the agents), and behavioural (e.g., movement flow of the agents) features. The findings from the user study suggest that both of our hypothesis stand valid.\n\n\nWe also wanted to analyze the users' performance in perceiving the ordering sequence of the neighbours when floorplans are not visually annotated with their respective features. To test this, we used the input floorplans and their neighbours from model (3) and presented them as model (0) in the user study. These floorplans were presented to the users without any visual annotations. This was so we could analyze how important is the visual annotation of the features, and its significance to assist users in perceiving the neighbours in their correct order. Interestingly, about $ 45.68$\\% of the neighbours were accurately ordered in their sequences based on their perceived similarity with respect to input flooplans for model (0). This result revealed that the annotations for design semantic features, alone, are not a good representative to convey the spatial feature information of the floorplans. As well, that the users better perceive the floorplans retrieved from the embedding space that is trained not only with the design semantic feature alone but also with the additional edge or\/and dynamic behavioural features.\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\n\t\\begin{tabular}{cc}\n\t\n\t    {{Model (0): Design Semantic + Edge + Behaviour}} & {{Model (1): Design Semantic}} \\\\\n\t    (\\textit{Without Annotation}) & (\\textit{With Annotation}) \\\\\n\t    \n\t    \\includegraphics[width=0.45\\linewidth]{images\/user_study\/new\/model_0_user_sequences.pdf} & \\includegraphics[width=0.45\\linewidth]{images\/user_study\/new\/model_1_user_sequences.pdf} \\\\\n\t    \n\t    {} & {} \\\\\n\t    \n\t    {{Model (2): Design Semantic + Edge}} & {{Model (3): Design Semantic + Edge + Behaviour}} \\\\\n\t    (\\textit{With Annotation}) & (\\textit{With Annotation}) \\\\\n\t    \n\t    \\includegraphics[width=0.45\\linewidth]{images\/user_study\/new\/model_2_user_sequences.pdf} & \\includegraphics[width=0.45\\linewidth]{images\/user_study\/new\/model_3_user_sequences.pdf} \\\\\n\t    \n\t    {\\hspace{1.3cm} Participants} & {\\hspace{1.3cm} Participants} \\\\\n\t    \n    \\end{tabular}\n   \n\t\\caption{\\label{figure:user-sequences} The accuracy of user-ordered sequences of the nearest neighbours. Colored bars for each neighbour of an input floorplan represent the number of users who correctly perceived the order of the neighbour in the given sequence. Gray bars are for the nearest neighbours which are queried using model (3) but were presented to users without any annotations, Green bars are for nearest neighbours which are queried using model (1) and were presented with annotations, blue bars using model (2) --  with annotations, and orange bars using model (3) --  with annotations. }\n\\end{figure*}\n\n\\begin{comment}\n\\begin{figure*}[t]\n\t\\centering\n\t\n\t\\begin{tabular}{c}\n\t\n\t    {\\textit{Model (1): Design Semantic}} \\\\\n\t    \n\t    \\includegraphics[width=0.98\\linewidth]{images\/user_study\/pilot\/model_1_user_sequences_trend_v2.pdf} \\\\\n\t    \n\t    {\\textit{Model (2): Design Semantic + Edge}} \\\\\n\t    \n\t    \\includegraphics[width=0.98\\linewidth]{images\/user_study\/pilot\/model_2_user_sequences_trend_v2.pdf} \\\\\n\t    \n\t    {\\textit{Model (3): Design Semantic + Edge + Behaviour}} \\\\\n\t    \n\t    \\includegraphics[width=0.98\\linewidth]{images\/user_study\/pilot\/model_3_user_sequences_trend_v2.pdf} \\\\\n\t    \n    \\end{tabular}\n    \n\t\\caption{\\label{figure:user-sequences-trend} The proximity of user-ordered sequences of nearest neighbours for the input floorplans with respect to their similarity ranks retrieved from the embedding models. The default ordering of nearest neighbours from the models are $1--5$ (vertical axis). User-ordered sequences for each user are shown in same color for all the inputs and across all three models.}\n\\end{figure*}\n\\end{comment}\n\n\t\n\t    \n\n\t    \n\t    \n\t    \n\t    \n\t    \n\t    \n\t    \n\t    \n    \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nSolving systems of linear equations given by symmetric diagonally matrices (SDD) is of interest to researchers in a variety of fields. Such constructs, for example, are used to determine solutions to partial differential equations~\\cite{c18} and computations of maximum flows in graphs~\\cite{c19,c20}. Other application domains include machine learning~\\cite{c21,c22}, and computer vision~\\cite{c34}\\footnote{This research is supported in parts by by ONR grant Number N00014-12-1-0997 and AFOSR grant FA9550-13-1-0097.}. \n\nMuch interest has been devoted to determining fast algorithms for solving SDD systems. Recently, Spielman and Teng~\\cite{c24}, utilized the multi-level framework of~\\cite{c25}, pre-conditioners~\\cite{c26}, and spectral graph sparsifiers~\\cite{c27}, to propose a nearly linear-time algorithm for solving SDD systems. Further exploiting these ingredients, Koutis \\textit{et. al}~\\cite{c28, c29} developed an even faster algorithm for acquiring $\\epsilon$-close solutions to SDD linear systems. Further improvements have been discovered by Kelner \\textit{et. al}~\\cite{c30}, where their algorithm relied on only spanning-trees and eliminated the need for graph sparsifiers and the multi-level framework.  \n\nMotivated by applications, much progress has been made in developing  parallel versions of these algorithms. Koutis and Miller~\\cite{c31} proposed an algorithm requiring nearly-linear work (i.e., total number of operations executed by a computation) and $m^{\\sfrac{1}{6}}$ depth (i.e., longest chain of sequential dependencies in the computation) for planar graphs. This was then extended to general graphs in~\\cite{c32} leading to depth close to $m^{\\sfrac{1}{3}}$. Since then, Peng and Spielman~\\cite{c11} have proposed an efficient parallel solver requiring nearly-linear work and poly-logarithmic depth without the need for low-stretch spanning trees. Their algorithm, which we provide a distribute construction for, requires sparse approximate inverse chains~\\cite{c11} which facilitates the solution of the SDD system. \n\nLess progress, on the other hand, has been made on the distributed version of these solvers. Contrary to the parallel setting, memory is not shared and is rather distributed in the sense that each unit abides by its own memory restrictions. Furthermore, communication in a distributed setting fundamentally relies on message passing through communication links. Current methods, e.g., Jacobi iteration~\\cite{c16,c17}, can be used for such distributed solutions but require substantial complexity. In~\\cite{c33}, the authors propose a gossiping framework for acquiring a solution to SDDM systems in a distributed fashion. \nRecent work~\\cite{c34} considers a local and asynchronous solution for solving systems of linear equations, where they acquire a bound on the number of needed multiplication proportional to the degree and condition number of the graph for one component of the solution vector. \n\n\n\\textbf{Contributions:} In this paper, we propose a fast distributed solver for linear equations given by symmetric diagonally dominant M-Matrices. Our approach distributes the parallel solver in~\\cite{c11} by considering a specific approximated inverse chain which can be computed efficiently in a distributed fashion. We develop two versions of the solver. The first, requires full communication in the network, while the second is restricted to R-Hop neighborhood of nodes for some $R \\geq 1$. Similar to the work in~\\cite{c11}, our algorithms operate in two phases. In the first, a ``crude'' solution to the system of equations is retuned, while in the second a distributed R-Hop restricted pre-conditioner is proposed to drive the ``crude'' solution to an $\\epsilon$-approximate one for any $\\epsilon > 0$. Due to the distributed nature of the setting considered, the direct application of the sparsfier and pre-conditioner of Peng and Spielman~\\cite{c11} is difficult due to the need of global information. Consequently, we propose a new sparse inverse chain which can be computed in a decentralized fashion for determining the solution to the SDDM system. \n\nInterestingly, due to the involvement of powers of matrices with eigenvalues less than one, our inverse chain is substantially shorter compared to that in~\\cite{c11}. This leads us to a distributed SDDM solver with lower computational complexity compared to state-of-the-art methods.  Specifically, our algorithm's complexity is given by  \\[\\mathcal{O}\\left(n^{3}\\frac{\\bm{\\alpha}}{R}\\frac{\\bm{W}_{\\text{max}}}{\\bm{W}_{\\text{min}}}\\log\\left(\\frac{1}{\\epsilon}\\right)\\right),\\] with $n$ being the number of nodes in graph $\\mathcal{G}$, $\\bm{W}_{\\text{max}}$ and $\\bm{W}_{\\text{min}}$ denoting the largest and smaller weights of the edges in $\\mathcal{G}$, respectively, $\\bm{\\alpha}=\\min\\left\\{n,\\frac{d_{\\text{max}}^{R+1}-1}{d_{\\text{max}}-1}\\right\\}$ representing the upper bound on the size of the R-Hop neighborhood $\\forall \\bm{v} \\in \\mathcal{V}$, and $\\epsilon \\in (0,\\frac{1}{2}]$ being the precision parameter. Furthermore, our approach improves current linear methods by a factor of $\\log n$ and by a factor of the degree  compared to~ \\cite{c34} for each component of the solution vector.\n\nHaving developed such distributed solvers, we next contribute by proposing an accurate distributed Newton method for network flow optimization. Exploiting the sparsity pattern of the dual Hessian, we propose a Newton method for network optimization that is both faster and more accurate than state-of-the-art techniques. Our method utilizes the proposed SDDM distributed solvers to approximate the Newton direction up to any arbitrary $\\epsilon >0$. The resulting algorithm is an efficient and accurate distributed second-order method which performs almost identically to exact Newton. We analyze the properties of the proposed algorithm and show that, similar to conventional Newton methods, superlinear convergence within a neighborhood of the optimal value is attained.  We finally demonstrate the effectiveness of the approach in a set of experiments on randomly generated and Barbell networks. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The parallel SDDM Solver}\\label{Sec:ParallelSolver}\nWe now review the parallel solver for symmetric diagonally dominant (SDD) linear systems~\\cite{c11}.\n\n\n\n\n\\subsection{Problem Setting}\\label{Sec:ProbSetting}\nAs detailed in~\\cite{c11}, SDDM solvers consider the following system of linear equations:\n\\begin{equation}\\label{lin_sys}\n\\bm{M}_{0}\\bm{x} = \\bm{b}_{0}\n\\end{equation}\nwhere $\\bm{M}_{0}$ is a Symmetric Diagonally Dominant M-Matrix (SDDM). Namely, $\\bm{M}_{0}$ is symmetric positive definite with non-positive off diagonal elements, such that for all $i=1,2,\\ldots, n$:\n\\begin{equation*}\n\\left[\\bm{M}_{0}\\right]_{ii} \\ge -\\sum_{j=1, j\\ne i}^{n}\\left[\\bm{M}_{0}\\right]_{ij}.\n\\end{equation*}\nThe system of Equations in~\\ref{lin_sys} can be interpreted as representing an undirected weighted graph, $\\mathcal{G}$, with $\\bm{M}_{0}$ being its Laplacian. Namely, $\\mathcal{G} = \\left(\\mathcal{V},\\mathcal{E},\\bm{W}\\right)$, with $\\mathcal{V}$ representing the set of nodes, $\\mathcal{E}$ denoting the edges, and $\\bm{W}$ representing the weighted graph adjacency. Nodes $\\bm{v}_i$ and $\\bm{v}_j$ are connected with an edge $\\bm{e}=\\left(i,j\\right)$ iff $\\bm{W}_{ij}> 0$, where: \n\\begin{equation*}\n\\bm{W}_{ij} = -\\left[\\bm{M}_{0}\\right]_{ij}.\n\\end{equation*}\nFollowing~\\cite{c11}, we seek $\\epsilon$-approximate solutions to $\\bm{x}^{\\star}$, being the exact solution of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$, defined as:\n\\begin{definition}[$\\epsilon-$ Approximate Solution]\nLet $\\bm{x}^{\\star}\\in \\mathbb{R}^{n}$ be the solution of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$. A vector $\\tilde{\\bm{x}}\\in \\mathbb{R}^{n}$ is called an $\\epsilon-$ approximate solution, if:\n\\begin{equation}\n\\left|\\left|\\bm{x}^{\\star} - \\tilde{\\bm{x}}\\right|\\right|_{\\bm{M}_{0}} \\le \\epsilon\\left|\\left|\\bm{x}^{\\star}\\right|\\right|_{\\bm{M}_{0}}, \\ \\ \\ \\text{where $\\left|\\left|\\bm{u}\\right|\\right|^{2}_{\\bm{M}_0} = \\bm{u}^{\\mathsf{T}}\\bm{M}_{0}\\bm{u}$.\n}\n\\end{equation}\n\\end{definition}\n\nThe R-hop neighbourhood of node $\\bm{v}_{k}$ is defined as $\\mathbb{N}_{R}\\left(\\bm{v}_{k}\\right) = \\{\\bm{v}\\in \\mathcal{V}: \\text{dist}\\left(\\bm{v}_{k}, \\bm{v}\\right)\\le R\\}$. We also make use of the diameter of a graph, $\\mathcal{G}$, defined as $\\text{diam}\\left(\\mathcal{G}\\right) = \\max_{\\bm{v}_{i},\\bm{v}_{j}\\in \\mathcal{V}}\\text{dist}\\left(\\bm{v}_i,\\bm{v}_j\\right)$. \n\n\\begin{definition}[Sparsity Pattern]\nWe say that a matrix $\\bm{A} \\in \\mathbb{R}^{n\\times n}$ has a sparsity pattern corresponding to the R-hop neighborhood if $\\bm{A}_{ij} = 0$ for all $i = 1,\\ldots, n$ and for all $j$ such that $\\bm{v}_j\\notin \\mathbb{N}_{R}\\left(\\bm{v}_i\\right)$. \n\\end{definition}\n\nWe will denote the spectral radius of a matrix $\\bm{A}$ by $\\rho\\left(\\bm{A}\\right) = \\max{\\left|\\bm{\\lambda}_i\\right|}$, where $\\bm{\\lambda}_i$ represents an eigenvalue of the matrix $\\bm{A}$. Furthermore, we will make use of the condition number\\footnote{Please note that in the case of the graph Laplacian,  the condition number is defined as the ratio of the largest to the smallest nonzero eigenvalues.}, $\\kappa\\left(\\bm{A}\\right)$ of a matrix $\\bm{A}$ defined as $\\kappa=\\left|\\frac{\\bm{\\lambda}_{\\text{max}}\\left(\\bm{A}\\right)}{\\bm{\\lambda}_{\\text{min}}\\left(\\bm{A}\\right)}\\right|$.  In~\\cite{c11} it is shown that the condition number of the graph Laplacian is at most \\[\\mathcal{O}\\left(n^{3}\\frac{\\bm{W}_{\\text{max}}}{\\bm{W}_{\\text{min}}}\\right),\\] where $\\bm{W}_{\\text{max}}$ and $\\bm{W}_{\\text{min}}$ represent the largest and the smallest edge weights in $\\mathcal{G}$. Finally, the condition number of a sub-matrix of the Laplacian is at most $\\mathcal{O}\\left(n^4\\frac{\\bm{W}_{\\text{max}}}{\\bm{W}_{\\text{min}}}\\right)$, see~\\cite{c11}.  \n\n\n\\subsection{Standard Splittings \\& Approximations}\\label{Sec:Standard}\nOur first contribution is a distributed version of the parallel solver for SDDM systems of equations previously proposed in~\\cite{c11}. Before detailing our solver, however, we next introduce basic mathematical machinery needed for developing the parallel solver of~\\cite{c11}. The parallel solver commences by considering the standard splitting of the symmetric matrix $\\bm{M}_{0}$: \n\\begin{definition}\nThe standard splitting of a symmetric matrix $\\bm{M}_{0}$ is: \n\\begin{equation}\n\\bm{M}_{0} = \\bm{D}_{0} - \\bm{A}_{0}.\n\\end{equation}\n\\end{definition}\nHere, $\\bm{D}_0$ is a diagonal matrix consisting of the diagonal elements in $\\bm{M}_{0}$ such that: \n\\begin{equation*}\n\\left[\\bm{D}_{0}\\right]_{ii} = \\left[\\bm{M}_{0}\\right]_{ii} \\ \\ \\forall i=1,2,\\dots, n.\n\\end{equation*}\nFurthermore, $\\bm{A}_0$ is a non-negative symmetric matrix such that:\n\\begin{align*}\n\\left[\\bm{A}_0\\right]_{ij}=\n\\begin{cases}\n-\\left[\\bm{M}_{0}\\right]_{ij} &: \\text{if $i \\ne j$,} \\\\\n0 &: \\text{otherwise.}\n\\end{cases}\n\\end{align*}\n\nTo quantify the quality of the acquired solutions, we define two additional mathematical constructs. First, the Loewner ordering is defined as: \n\\begin{definition}\nLet $\\mathcal{\\bm{S}}_{(n)}$ be the space of $n \\times n$-symmetric matrices. The Loewner ordering $\\preceq$ is a partial order on $\\mathcal{\\bm{S}}_{(n)}$ such that $\\bm{Y}\\preceq \\bm{X}$ if and only if $\\bm{X} - \\bm{Y}$ is positive semidefinite.\n\\end{definition}\n\nHaving defined the Loewner order, we next define the notion of approximation for matrices ``$\\approx_{\\alpha}$'': \n\\begin{definition}\\label{Def:Ordering}\nLet $\\bm{X}$ and $\\bm{Y}$ be positive semidefinite symmetric matrices. Then $\\bm{X}\\approx_{\\alpha} \\bm{Y}$ if and only iff\n\\begin{equation}\ne^{-\\alpha}\\bm{X} \\preceq \\bm{Y} \\preceq e^{\\alpha}\\bm{X}\n\\end{equation}\nwith $\\bm{A}\\preceq \\bm{B}$ meaning $\\bm{B} - \\bm{A}$ is positive semidefinite.\n\\end{definition}\n\nBased on the above definitions, the following lemma represents the basic characteristics of the $\\approx_{\\alpha}$ operator:\n\\begin{lemma}~\\cite{c11}\\label{approx_lemma_facts}\nLet $\\bm{X},\\bm{Y},\\bm{Z}$ and, $\\bm{Q}$ be symmetric positive semi definite matrices. Then\n\\begin{enumerate}\n\\item[] (1) If $\\bm{X}\\approx_{\\alpha} \\bm{Y}$, then $\\bm{X} + \\bm{Z} \\approx_{\\alpha} \\bm{Y} + \\bm{Z}$, \n\\item[] (2) If $\\bm{X}\\approx_{\\alpha} \\bm{Y}$ and $\\bm{Z}\\approx_{\\alpha} \\bm{Q}$, then $\\bm{X} + \\bm{Z} \\approx_{\\alpha} \\bm{Y} + \\bm{Q}$, \n\\item[] (3) If $\\bm{X}\\approx_{\\alpha_1} \\bm{Y}$ and $\\bm{Y} \\approx_{\\alpha_2} \\bm{Z}$, then $\\bm{X} \\approx_{\\alpha_1 + \\alpha_2} \\bm{Z}$\n\\item[] (4) If $\\bm{X}$, and $\\bm{Y}$ are non singular and $\\bm{X}\\approx_{\\alpha} \\bm{Y}$, then $\\bm{X}^{-1}\\approx_{\\alpha} \\bm{Y}^{-1}$, \n\\item[] (5) If $\\bm{X}\\approx_{\\alpha} \\bm{Y}$ and $\\bm{V}$ is a matrix, then $\\bm{V}^{\\mathsf{T}}\\bm{X}\\bm{V}\\approx_{\\alpha}\\bm{V}^{\\mathsf{T}}\\bm{Y}\\bm{V}$.\n\\end{enumerate}\n\\end{lemma}\n\nSince the parallel solver returns an approximation, $\\bm{Z}_{0}$, to $\\bm{M}_{0}^{-1}$ (see Section~\\ref{Sec:ParallelSolver}), the following lemma shows that ``good'' approximations to $\\bm{M}^{-1}_0$ guarantee ``good'' approximate solutions to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$.\n\\begin{lemma}\\label{lemma_approx_matrix_inverse}\nLet $\\bm{Z}_0\\approx_{\\epsilon}\\bm{M}^{-1}_0$, and $\\tilde{\\bm{x}} = \\bm{Z}_0\\bm{b}_0$, then $\\tilde{\\bm{x}}$ is $\\sqrt{2^{\\epsilon}(e^{\\epsilon} - 1)}$ approximate solution to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$.\n\\end{lemma}\n\\begin{proof}\nThe proof can be found in the appendix. \n\\end{proof}\n\n\\subsection{The Solver}\\label{Parallel:SDDM}\\label{Sec:ParrallelSolver}\nThe parallel SDDM solver proposed in~\\cite{c11} is a parallelized technique for solving the problem of Section~\\ref{Sec:ProbSetting}. It makes use of inverse approximated chains (see Definition~\\ref{Def:InvChain}) to determine $\\tilde{\\bm{x}}$ and can be split in two steps. In the first, Algorithm~\\ref{Algo:Inv}, a ``crude'' approximation, $\\bm{x}_{0}$, of $\\bm{\\tilde{x}}$ is returned. $\\bm{x}_{0}$ is driven to the $\\epsilon$-close solution, $\\tilde{\\bm{x}}$, using Richardson Preconditioning in Algorithm~\\ref{Algo:Inv2}. Before we proceed, we start with the following two Lemmas which enable the definition of inverse chain approximation. \n\n\\begin{lemma}~\\cite{c11}\\label{SDDM_splitting_lemma}\nIf $\\bm{M} = \\bm{D} - \\bm{A}$ is an SDDM matrix, with $\\bm{D}$ being positive diagonal, and $\\bm{A}$ denoting a non-negative symmetric matrix, then $\\bm{D} - \\bm{A}\\bm{D}^{-1}\\bm{A}$ is also SDDM.\n\\end{lemma}\n\n\\begin{lemma}~\\cite{c11}\\label{approx_inverse_formulae_lemma}\nLet $\\bm{M} =\\bm{D} - \\bm{A}$ be an SDDM matrix, where $\\bm{D}$ is positive diagonal and $\\bm{A}$ a symmetric matrix. Then \n\\begin{align}\\label{Inv_of_SDDM}\n\\left(\\bm{D}-\\bm{A}\\right)^{-1} &= \\frac{1}{2}\\Big[\\bm{D}^{-1} + \\left(\\bm{I} + \\bm{D}^{-1}\\bm{A}\\right)\\left(\\bm{D} - \\bm{A}\\bm{D}^{-1}\\bm{A}\\right)^{-1} \\left(\\bm{I} + \\bm{A}\\bm{D}^{-1}\\right)\\Big].\n\\end{align}\n\\end{lemma}\n\n\nGiven the results in Lemmas \\ref{SDDM_splitting_lemma} and \\ref{approx_inverse_formulae_lemma}, we now can consider inverse approximated chains of $\\bm{M}_0$:\n\\begin{definition}\\label{Def:InvChain}\nLet $\\mathcal{C} = \\{\\bm{M}_0, \\bm{M}_1, \\ldots, \\bm{M}_d\\}$  be a collection of SDDM matrices such that $\\bm{M}_i = \\bm{D}_i - \\bm{A}_i$, with $\\bm{D}_i$ a positive diagonal matrix, and $\\bm{A}_i$ denoting a non-negative symmetric matrix. Then $\\mathcal{C}$ is an inverse approximated chain if there exists  positive real numbers $\\epsilon_0, \\epsilon_1, \\ldots, \\epsilon_d$ such that: \n\\begin{enumerate}\n\\item[](1) $\\bm{D}_i - \\bm{A}_i \\approx_{\\epsilon_{i-1}} \\bm{D}_{i-1} - \\bm{A}_{i-1}\\bm{D}^{-1}_{i-1}\\bm{A}_{i-1} \\ \\ \\forall i = 1,\\ldots, d$, \n\\item[] (2) $\\bm{D}_i \\approx_{\\epsilon_{i-1}}\\bm{D}_{i-1}$, and \n\\item[] (3) $\\bm{D}_d\\approx_{\\epsilon_d} \\bm{D}_d - \\bm{A}_d$. \n\\end{enumerate}\n\\end{definition}\n \nIt is shown in~\\cite{c11} that an approximate inverse chain allows for ``crude'' solutions to the system of linear equations in $\\bm{D}_{0}-\\bm{A}_{0}$ in time proportional to the number of non-zeros entries in the matrices in the inverse chain. Such a procedure is summarized in the following algorithm: \n\n\\begin{algorithm}\\label{Algo:CrudeParallel}\n  \\caption{$\\text{ParallelRSolve}\\left(\\bm{M}_0,\\bm{M}_1,\\ldots, \\bm{M}_d, \\bm{b}_0\\right)$}\n  \\label{Algo:Inv}\n  \\begin{algorithmic}[1]\n\t\\STATE \\textbf{Input}: Inverse approximated chain, $\\{\\bm{M}_0,\\bm{M}_1,\\ldots, \\bm{M}_d\\}$, and $\\bm{b}_0$ being \n\t\\STATE \\textbf{Output}: The ``crude'' approximation, $\\bm{x}_0$, of $\\bm{x}^{\\star}$    \n    \\FOR  {$i=1$ to $d$} \n    \t\\STATE $\\bm{b}_i = \\left(\\bm{I}+\\bm{A}_{i-1}\\bm{D}^{-1}_{i-1}\\right)\\bm{b}_{i-1}$\n    \\ENDFOR \n    \\STATE $\\bm{x}_d = \\bm{D}^{-1}_d\\bm{b}_d$\n    \\FOR  {$i=d-1$ to $0$} \n    \t\\STATE $\\bm{x}_{i} = \\frac{1}{2}\\left[\\bm{D}^{-1}_i\\bm{b}_{i} + \\left(\\bm{I} + \\bm{D}^{-1}_i\\bm{A}_i\\right)\\bm{x}_{i+1}\\right]$\n    \\ENDFOR \n    \\STATE \\textbf{return} $x_0$ \n  \\end{algorithmic}\n\\end{algorithm}\nOn a high level, Algorithm~\\ref{Algo:CrudeParallel} operates in two phases. In the first (i.e., lines~3-5) a forward loop (up-to the length of the inverse chain $d$) computes intermediate vectors $\\bm{b}_{i}$ as:\n\\begin{equation}\n\\label{Eq:bUpdate}\n\\bm{b}_{i}=\\left(\\bm{I}+\\bm{A}_{i-1}\\bm{D}_{i-1}^{-1}\\right)\\bm{b}_{i-1},\n\\end{equation}\nfor $i=\\{1,\\dots,d\\}$. These can then be used to compute the ``crude'' solution $\\bm{x}_{0}$ using a ``backward'' loop (i.e., lines~7-9). Consequently, the crude solution is computed iteratively backwards as: \n\n\\begin{equation*}\n\\bm{x}_{i} = \\frac{1}{2}\\left[\\bm{D}^{-1}_i\\bm{b}_{i} + \\left(\\bm{I} + \\bm{D}^{-1}_i\\bm{A}_i\\right)\\bm{x}_{i+1}\\right], \n\\end{equation*}\nwith $\\bm{x}_{d}=\\bm{D}_{d}^{-1}\\bm{b}_{d}$ and $\\bm{b}_{i}$ as defined in Equation~\\ref{Eq:bUpdate}. The quality of the ``crude'' solution returned by the Algorithm is quantified in the following lemma:\n\\begin{lemma}~\\cite{c11}\\label{Rude_Alg_guarantee_Lemma}\nLet $\\{\\bm{M}_0, \\bm{M}_1,\\ldots, \\bm{M}_d\\}$ be the inverse approximated chain and denote $\\bm{Z}_0$ be the operator defined by $\\text{ParallelRSolve}\\left(\\bm{M}_0, \\bm{M}_1,\\ldots, \\bm{M}_d, \\bm{b}_0\\right)$, namely, $\\bm{x}_0 = \\bm{Z}_0\\bm{b}_0$. Then\n\\begin{equation}\\label{appr_inv_express}\n\\bm{Z}_0\\approx_{\\sum_{i=0}^{d}\\epsilon_i}\\bm{M}^{-1}_0\n\\end{equation}\n\\end{lemma}\nHaving returned a ``crude'' solution to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$, the authors in~\\cite{c11} obtain arbitrary close solutions using the \\emph{preconditioned Richardson iterative scheme}. The first step in the exact solver is the usage of Algorithm~\\ref{Algo:CrudeParallel} to obtain the ``crude'' solution $\\chi$. This is then updated through the loop in lines~4-8 to obtain an $\\epsilon$-close solution to $\\bm{x}^{\\star}$, see Algorithm~\\ref{Algo:Inv2}.\n\\begin{algorithm}[h!]\n  \\caption{$\\text{ParallelESolve}\\left(\\bm{M}_0, \\bm{M}_1,\\ldots, \\bm{M}_d,  \\bm{b}_0, \\epsilon\\right)$}\n  \\label{Algo:Inv2}\n  \\begin{algorithmic}[1]\n\t\\STATE \\textbf{Input}: Inverse approximated chain $\\{\\bm{M}_0,\\bm{M}_1,\\ldots, \\bm{M}_d\\}$, $\\bm{b}_0$, and $\\epsilon$. \n\t\\STATE \\textbf{Output}: $\\epsilon$ close approximation, $\\tilde{\\bm{x}}$, of $\\bm{x}^*$\n\t\\STATE \\textbf{Initialize}: $\\bm{y}_0 = 0$; \\\\\n\t$\\chi = \\text{ParallelRSolve}\\left(\\bm{M}_0,\\bm{M}_1,\\ldots, \\bm{M}_d, \\bm{b}_0\\right)$ (i.e., Algorithm~\\ref{Algo:Inv})\n    \\FOR  {$k=1$ to $q$}\n    \t\\STATE $\\bm{u}_{k}^{(1)} = \\bm{M}_0\\bm{y}_{k-1}$\n    \t\\STATE $\\bm{u}_{k}^{(2)} = \\text{ParallelRSolve}\\left(\\bm{M}_0, \\bm{M}_1,\\ldots, \\bm{M}_d, \\bm{u}_{k}^{(1)}\\right)$\n    \t\\STATE $\\bm{y}_{k} = \\bm{y}_{k-1} - \\bm{u}_{k}^{(2)} + \\chi$ \n    \\ENDFOR \n    \\STATE $\\tilde{\\bm{x}} = \\bm{y}_q$\n    \\STATE \\textbf{return} $\\tilde{\\bm{x}}$ \n  \\end{algorithmic}\n\\end{algorithm}\nFollowing the analysis in~\\cite{c11}, Lemma~\\ref{Exact_Alg_guarantee_lemma} provides the iteration count needed by Algorithm~\\ref{Algo:Inv2} to arrive at $\\tilde{\\bm{x}}$:\n\\begin{lemma}~\\cite{c11}\\label{Exact_Alg_guarantee_lemma}\nLet $\\{\\bm{M}_0, \\bm{M}_1\\ldots \\bm{M}_d\\}$ be an inverse approximated chain such that $\\sum_{i=1}^{d}\\epsilon_i < \\frac{1}{3}\\ln2$. Then $\\text{ParallelESolve}\\left(\\bm{M}_0, \\bm{M}_1,\\ldots, \\bm{M}_d, \\bm{b}_0, \\epsilon\\right)$ requires $q$ iterations to arrive at an $\\epsilon$ close solution of $\\bm{x}^{\\star}$ with:  \n$q = \\mathcal{O}\\left(\\log\\frac{1}{\\epsilon}\\right)$.\n\\end{lemma}\n\n\\section{Distributed SDDM Solvers}\nHaving introduced the parallel solver, next we detail our first contribution by proposing a distributed solver for SDDM linear systems. In particular, we develop two versions. The first, requires full communication in the network, while the second restricts communication to the R-Hop neighborhood increasing its applicability. Not only our solver improves the computational complexity of distributed methods for system of equations represented by an SDDM matrix, but can also be applied to a variety of fields including distributed Newton methods, computer vision, among others.\n\n \nTo compute the solution of SDDM systems in a distributed fashion, we follow a similar strategy to that of~\\cite{c11} with major differences. Our distributed solver requires two steps to arrive at an $\\epsilon$-close approximation to $\\bm{x}^{\\star}$. Similar to~\\cite{c11}, the first step adopts an inverse approximated chain to determine a ``crude'' solution to $\\bm{x}^{\\star}$. The inverse chain proposed in~\\cite{c11} can not be computed in a distributed fashion rendering its immediate application to our setting difficult. Hence, we par-ways with~\\cite{c11} by proposing an inverse chain which can be computed in a distributed fashion. This chain, defined in Section~\\ref{Sec:Chain}, enables the distributed computation of both a crude and exact solution to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$. Interestingly, due to the involvement of matrices with eigenvalues less than 1, the length, $d$, of our inverse chain is substantially shorter compared to that of~\\cite{c11}, allowing for fast and efficient distributed solvers. Given the crude solution, the second step computes an $\\epsilon$-close approximation to $\\bm{x}^{\\star}$.  This is achieved by proposing a distributed version of the Richardson pre-conditioning scheme. Definitely, this step is also similar in spirit to that in~\\cite{c11}, but generalizes the aforementioned authors' work into a distributed setting and allows for $\\epsilon$-close approximation to $\\bm{x}^{\\star}$ for \\emph{any arbitrary} $\\epsilon > 0$. Main results on the full communication version of the solver are summarized in the following theorem: \n\n\\begin{theorem}\\label{Main_Theorem}\nThere exists a distributed algorithm,\n\\[\\mathcal{A}\\left(\\{[\\bm{M}_0]_{k1},\\ldots [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, \\epsilon\\right),\\] that computes $\\epsilon$-close approximations to the solution of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$ in \n$\\mathcal{O}\\left(n^2\\log\\kappa \\log\\left(\\frac{1}{\\epsilon}\\right)\\right)$ time steps, with $n$ the number of nodes in $\\mathcal{G}$, $\\kappa$ the condition number of  $\\bm{M}_0$, and $[\\bm{M}_{0}]_{k\\cdot}$ the $k^{th}$ row of $\\bm{M}_{0}$, as well as $\\epsilon\\in \\left(0, \\frac{1}{2}\\right]$ representing the precision parameter. \n\\end{theorem}\n\n\nThe above distributed algorithms require no knowledge of the graph's topology, but do require the information from all other nodes (i.e., full communication) for computing solutions to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$. In a variety of real-world applications (e.g., smart-grids, transportation) load, capacity, money and resource restrictions pose problems for such a requirement. Consequently, we extend the previous solvers to an R-Hop version in which communication is restricted to the R-Hop neighborhood between nodes for some $R\\geq 1$. Again we follow a two-step strategy, where in the first we compute the crude solution and in the second an $\\epsilon$-close approximation to $\\bm{x}_{0}$ is determined using an ``R-Hop restricted'' Richardson pre-conditioner.  These results are captured in the following theorem: \n\n\\begin{theorem}\\label{Main_TheoremTwo}\nThere is a decentralized algorithm, \n\\[\\mathcal{A}(\\{[\\bm{M}_0]_{k1},\\ldots [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, R, \\epsilon),\\] that uses only $R$-Hop communication between the nodes and computes $\\epsilon$-close solutions to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$ in \\[\\mathcal{O}\\left(\\left(\\frac{\\alpha\\kappa}{R} + \\alpha Rd_{max}\\right) \\log\\left(\\frac{1}{\\epsilon}\\right)\\right)\\] time steps, with $n$ being the number of nodes in $\\mathcal{G}$, $d_{max}$ denoting the maximal degree, $\\kappa$ the condition number of $\\bm{M}_0$, and $\\alpha = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$ representing the upper bound on the size of the R-hop neighborhood $\\forall \\bm{v}\\in \\mathcal{V}$, and $\\epsilon\\in (0, \\frac{1}{2}]$ being the precision parameter. \n\\end{theorem}\n\nThe remainder of the section details the above distributed solvers and provides rigorous theoretical guarantees on the convergence and convergence rates of each of the algorithms. We start by describing solvers requiring full network communication and then detail the R-Hop restricted versions. \n\n\\subsection{Full Communication Distributed Solvers}\nAs mentioned previously, our strategy for a distributed implementation of the parallel solver in~\\cite{c11} requires two steps. In the first a ``crude'' solution is returned, while in the second an $\\epsilon$-close approximation (for any arbitrary $\\epsilon >0$) to $\\bm{x}_{0}$ is computed. \n\n\\subsubsection{``Crude'' Distributed SDDM Solvers}\\label{Sec:Chain}\nThe distributed crude solver, represented in Algorithm~\\ref{Algo:DisRudeApprox}, resembles similarities to the parallel one of~\\cite{c11} with major differences. On a high level, Algorithm~\\ref{Algo:DisRudeApprox} operates in two distributed phases. In the first, a forward loop computes intermediate $\\bm{b}$ vectors which are then used to update the crude solution of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$. The crucial difference to~\\cite{c11}, however, is the distributed nature of these computations. Precisely, the algorithm is responsible for determining the crude solution for each node $\\bm{v}_{k} \\in \\mathcal{V}$. Due to such distributed nature, the inverse approximated chain used in~\\cite{c11} is inapplicable to our setting. Therefore, the second crucial difference to the parallel SDDM solver is the introduction of a new chain which can be computed in a distributed fashion. Starting from $\\bm{M}_{0} = \\bm{D}_{0} - \\bm{A}_{0}$, our ``crude'' distributed solver makes use of the following collection as the inverse approximated chain: \n\\begin{equation}\\label{Eq:InverseChain}\n\\mathcal{C} = \\{\\bm{A}_0,\\bm{D}_0,\\bm{A}_1, \\bm{D}_1,\\ldots, \\bm{A}_d, \\bm{D}_d\\},\n\\end{equation}\nwhere \n$\\bm{D}_{k}=\\bm{D}_{0}$, \\text{and} $\\bm{A}_{k}=\\bm{D}_{0}\\left(\\bm{D}_{0}^{-1}\\bm{A}_{0}\\right)^{2^{k}}$, \nfor $k=\\{1,\\dots, d\\}$ with $d$ being the length of the inverse chain. Note that since the magnitude of the eigenvalues of $\\bm{D}^{-1}_0\\bm{A}_0$ is strictly less than 1, $\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^{2^{k}}$ tends to zero as $k$ increases which reduces the length of the chain needed. This length is explicitly computed in Section~\\ref{Sec:Length} for attaining $\\epsilon$-close approximations to $\\bm{x}^{\\star}$. \n\nIt is relatively easy to verify that $\\mathcal{C}$ is an inverse approximated chain, since: \n\\begin{enumerate}\n\\item[] (1) $\\bm{D}_i - \\bm{A}_i \\approx_{\\epsilon_{i-1}} \\bm{D}_{i-1} - \\bm{A}_{i-1}\\bm{D}^{-1}_{i-1}\\bm{A}_{i-1}$ with $\\epsilon_i = 0$ for $i = 1,\\ldots, d$,\n\\item[] (2)  $\\bm{D}_{i}\\approx_{\\epsilon_{i-1}}\\bm{D}_{i-1}$ with $\\epsilon_i = 0$ for $i = 1,\\ldots, d$, and \n\\item[] (3) $\\bm{D}_d\\approx_{\\epsilon_d}\\bm{D}_d - \\bm{A}_d$.\n\\end{enumerate}\n\n \n\n\nAlgorithm~\\ref{Algo:DisRudeApprox} returns the $k^{th}$ component of the approximate solution vector, $[\\bm{x}_{0}]_{k}$. As inputs it requires the inverse chain of Equation~\\ref{Eq:InverseChain}, the $k^{th}$ component of $\\bm{b}_{0}$, and the length of the inverse chain. Namely, each node, $\\bm{v}_k\\in \\mathcal{V}$, receives the $k^{th}$ row of $\\bm{M}_0$, the $k^{th}$ value of $\\bm{b}_{0}$ (i.e., $[\\bm{b}_0]_k$), and the length of the inverse approximated chain $d$ and then operates in two parts. In the first (i.e., lines~1-8) a forward loop computes the $k^{th}$ component of $\\bm{b}$ exploiting the distributed inverse chain, while in the second a backward loop (lines~9-17) is responsible for computing the $k^{th}$ component of the``crude'' solution $[\\bm{x}_{0}]_{k}$ which is then returned. Essentially, in both the forward and backward loops each of the $\\bm{b}$ and $\\bm{x}$ vectors are computed in a distributed fashion based on the relevant components of the matrices, explaining the usage of $\\mathbb{N}_{\\cdot}$ loops in Algorithm~\\ref{Algo:DisRudeApprox}. \n\n\\begin{algorithm}[t!]\n  \\caption{\n \t\t \\hspace{0em} $\\text{DistrRSolve}\\Big(\\left\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\right\\}, \t [\\bm{b}_0]_k, d\\Big)$\n\t   }\n\t   \\label{Algo:DisRudeApprox}\n  \\begin{algorithmic}[1]\n\t\\STATE \\textbf{Part One: Computing $[\\bm{b}_{i}]_{k}$}\n\t\\STATE $[\\bm{b}_1]_k = [\\bm{b}_0]_k + \\sum_{j: \\bm{v}_j\\in \\mathbb{N}_{1}\\left(\\bm{v}_k\\right)}[\\bm{A}_0\\bm{D}^{-1}_0]_{kj}[\\bm{b}_{0}]_j$\t\n\t\\FOR {$i = 2$ to $d$}\n\t\t\\FOR {$j: \\bm{v}_j \\in \\mathbb{N}_{2^{i-1}}\\left(\\bm{v}_k\\right)$}\n\t\t\t\\STATE$\n\t\t\t\\left[(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-1}}\\right]_{kj} =  \\sum_{r=1}^{n}\\frac{[\\bm{D}_0]_{rr}}{[\\bm{D}_0]_{jj}}\\left[(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-2}}\\right]_{kr} \\left[(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-2}}\\right]_{jr}$\n\t\t\\ENDFOR \n\t\t\\STATE $[\\bm{b}_i]_k = [\\bm{b}_{i-1}]_k + \\sum_{j: \\bm{v}_j \\in \\mathbb{N}_{2^{i-1}}\\left(\\bm{v}_k\\right)}\\left[(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-1}}\\right]_{kj}[\\bm{b}_{i-1}]_{j}$\n\t\\ENDFOR\n\t\\\\\\hrulefill\n\t\\STATE \\textbf{Part Two: Computing $[\\bm{x}_{0}]_{k}$}\n\t\\STATE $[\\bm{x}_d]_k = \\sfrac{[\\bm{b}_d]_k}{[\\bm{D}_0]_{kk}}$\n\t\\FOR {$i = d - 1$ to $1$}\n\t\t\\FOR {$j: \\bm{v}_j \\in \\mathbb{N}_{2^i}(\\bm{v}_k)$}\n\t\t\t\\STATE$\n\t\t\t\\left[(\\bm{D}^{-1}_0\\bm{A}_0)^{2^i}\\right]_{kj} = \n\t\t\t\\sum_{r=1}^{n}\\frac{[\\bm{D}_0]_{jj}}{[\\bm{D}_0]_{rr}}\\left[(\\bm{D}^{-1}_0\\bm{A}_0)^{2^{i-1}}\\right]_{kr} \\left[(\\bm{D}^{-1}_0\\bm{A}_0)^{2^{i-1}}\\right]_{jr}$\n\t\t\\ENDFOR \n\t\t\\STATE$\n\t\t[\\bm{x}_i]_k =  \\frac{[\\bm{b}_i]_k}{2[\\bm{D}_0]_{kk}} + \\frac{[\\bm{x}_{i+1}]_{k+1}}{2} +\\frac{1}{2}\\sum_{j: \\bm{v}_j \\in \\mathbb{N}_{2^{i}}(\\bm{v}_k)}\\left[(\\bm{D}^{-1}_0\\bm{A}_0)^{2^i}\\right]_{kj}[\\bm{x}_{i+1}]_j$\n\t\\ENDFOR \n\t\\STATE $[\\bm{x}_0]_k = \\frac{[\\bm{b}_0]_k}{2[\\bm{D}_{0}]_{kk}} + \\frac{[\\bm{x}_{1}]_k}{2} + \\frac{1}{2}\\sum_{j:\\bm{v}_j\\in \\mathbb{N}_{1}(\\bm{v}_k)}[\\bm{D}^{-1}_0\\bm{A}_0]_{kj}[\\bm{x}_1]_j$\n    \\STATE \\textbf{return:} $[\\bm{x}_0]_k$ \n  \\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{Theoretical Guarantees of Algorithm~\\ref{Algo:DisRudeApprox}:} Due to the modifications made to the original parallel solver, new theoretical analysis quantifying convergence and accuracy of the returned ``crude'' solution is needed. We show that $\\text{DistrRSolve}$ computes the $k^{th}$ component of the ``crude'' approximation of $\\bm{x}^{\\star}$ and provide time complexity analysis. These results are summarized in the following lemma\\footnote{For ease of presentation, we leave the proof of the lemma to the appendix.}: \n\n\\begin{lemma}\\label{Rude_Dec_Alg_guarantee_Lemma}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting of $\\bm{M}_{0}$. Let $\\bm{Z}^{\\prime}_0$ be the operator defined by $\\text{DistrRSolve}([\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, d)$ (i.e., $\\bm{x}_0 = \\bm{Z}^{\\prime}_0\\bm{b}_0$). Then\n\\begin{equation*}\n\\bm{Z}^{\\prime}_0\\approx_{\\epsilon_d} \\bm{M}^{-1}_0.\n\\end{equation*}\nMoreover, Algorithm~\\ref{Algo:DisRudeApprox} requires  $\\mathcal{O}\\left(dn^2\\right)$ time steps. \n\\end{lemma}\n\nIn words, Lemma~\\ref{Rude_Dec_Alg_guarantee_Lemma} states that Algorithm~\\ref{Algo:DisRudeApprox} requires $\\mathcal{O}\\left(dn^2\\right)$ to arrive at an $\\epsilon_{d}$ approximation to the real inverse $\\bm{M}_{0}^{-1}$, where this approximation is quantified using Definition~\\ref{Def:Ordering}: \n\\begin{equation*}\ne^{-\\epsilon_{d}} \\bm{Z}_{0}^{\\prime}\\preceq \\bm{M}_{0}^{-1}\\preceq e^{\\epsilon_{d}}\\bm{Z}^{\\prime}_{0}.\n\\end{equation*}\n\n$\\bm{Z}_{0}^{\\prime}$ can then be used to compute the crude solution as $\\bm{x}_{0}=\\bm{Z}_{0}^{\\prime}\\bm{b}$. Note that the accuracy of approximating $\\bm{M}_{0}$ is limited to $\\epsilon_{d}$ motivating the need for an ``exact'' distributed solver reducing the error to any $\\epsilon>0$.  \n\n\n\n\n\\subsubsection{``Exact'' Distributed SDDM Solvers}\nHaving introduced $\\text{DistrRSolve}$, we are now ready to present a distributed version of Algorithm~\\ref{Algo:Inv2} which enables the computation of $\\epsilon$ close solutions for $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$. Contrary to the work of~\\cite{c11}, our algorithm is capable of acquiring solutions up to any arbitrary $\\epsilon > 0$. Similar to $\\text{DistrRSolve}$, each node $\\bm{v}_k\\in \\mathcal{V}$ receives the $k^{th}$ row of $\\bm{M}_0$, $[\\bm{b}_0]_k$, $d$ and a precision parameter $\\epsilon$ as inputs. Node $\\bm{v}_k$ then computes the $k^{th}$ component of the $\\epsilon$  close approximation of $\\bm{x}^{\\star}$ by using $\\text{DistrRSolve}$ as a sub-routine and updates the solution iteratively as shown in lines~2-6 in Algorithm~\\ref{Alg_ExactBla}.\n\n\n\\begin{algorithm}[h!]\n  \\caption{$\\text{DistrESolve}\\left(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, d, \\epsilon\\right)$}\n  \\begin{algorithmic}[1]\n\\STATE \\textbf{Initialize}: $[\\bm{y}_0]_k = 0$; \n$[\\chi]_k = \\text{DistrRSolve}\\left(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, d\\right)$ (i.e., Algorithm~\\ref{Algo:DisRudeApprox})\t\n\t\\FOR {$t=1$ to $q$}\n\t\t\\STATE $\\left[\\bm{u}^{(1)}_{t}\\right]_k = [\\bm{D}_0]_{kk}[\\bm{y}_{t-1}]_k - \\sum_{j: \\bm{v}_j\\in \\mathbb{N}_{1}(\\bm{v}_k)}[\\bm{A}_{0}]_{kj}[\\bm{y}_{t-1}]_j$\n\t\t\\STATE $\\left[\\bm{u}^{(2)}_{t}\\right]_k = \\text{DistrRSolve}(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, \\left[\\bm{u}^{(1)}_{t}\\right]_k, d,)$\n\t\t\\STATE $[\\bm{y}_t]_k = [\\bm{y}_{t-1}]_k - \\left[\\bm{u}^{(2)}_{t}\\right]_k + [\\chi]_k$ \n\t\\ENDFOR \n    \\STATE $[\\tilde{\\bm{x}}]_k = [\\bm{y}_q]_k$\n    \\STATE \\textbf{return} $[\\tilde{\\bm{x}}]_k$ \n  \\end{algorithmic}\n  \\label{Alg_ExactBla}  \n\\end{algorithm}\n\n\n\\textbf{Analysis of Algorithm~\\ref{Alg_ExactBla}:} Here, we again provide the theoretical analysis needed for quantifying the convergence and computational time of the exact algorithm for returning $\\epsilon$-close approximation to $\\bm{x}^{\\star}$. The following lemma shows that $\\text{DistrESolve}$ computes the $k^{th}$ component of the $\\epsilon$-close approximation of $x^{\\star}$:\n\\begin{lemma}\\label{Dist_Exact_algorithm_guarantee_lemma}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting. Further, let $\\epsilon_d < \\frac{1}{3}\\ln2$ in the nverse approximated chain $\\mathcal{C} = \\{\\bm{A}_0,\\bm{D}_0,\\bm{A}_1, \\bm{D}_1,\\ldots, \\bm{A}_d, \\bm{D}_d\\}$. Then $\\text{DistrESolve}(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [b_0]_k, d, \\epsilon)$ requires $\\mathcal{O}\\left(\\log\\frac{1}{\\epsilon}\\right)$ iterations to return the $k^{th}$ component of the $\\epsilon$ close approximation for $\\bm{x}^{\\star}$.\n\\end{lemma}\n\nThe above lemma proofs that the algorithm requires $\\mathcal{O}(\\log \\frac{1}{\\epsilon})$ iterations for attaining for returning the $k^{th}$ of the $\\epsilon$-close approximation to $\\bm{x}^{\\star}$. Consequently, the overall complexity can be summarized as: \n\\begin{lemma}\\label{time_complexity_of_distresolve}\nLet $\\bm{M}_0 =\\bm{D}_0 - \\bm{A}_0$ be the standard splitting. Further, let $\\epsilon_d < \\frac{1}{3}\\ln2$ in the inverse approximated chain $\\mathcal{C} = \\{\\bm{A}_0,\\bm{D}_0,\\bm{A}_1, \\bm{D}_1,\\ldots, \\bm{A}_d, \\bm{D}_d\\}$. Then, $\\text{DistrESolve}(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, d, \\epsilon)$ requires $\\mathcal{O}\\left(dn^2\\log(\\frac{1}{\\epsilon})\\right)$ time steps.\n\\end{lemma}\n\n\\subsubsection{Length of the Inverse Chain}\\label{Sec:Length}\nBoth introduced algorithms depend on the length of the inverse approximated chain, $d$. Here, we provide an analysis to determine the value of $d$ which guarantees $\\epsilon_d < \\frac{1}{3}\\ln2$ in $\\mathcal{C} = \\{\\bm{A}_0,\\bm{D}_0,\\bm{A}_1, \\bm{D}_1,\\ldots, \\bm{A}_d, \\bm{D}_d\\}$: \n\\begin{lemma}\\label{eps_d_lemma}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting and $\\kappa$ denote the condition number of $\\bm{M}_0$. Consider the inverse approximated chain \n\\begin{equation*}\n\\mathcal{C} = \\{\\bm{A}_0,\\bm{D}_0,\\bm{A}_1, \\bm{D}_1,\\ldots, \\bm{A}_d, \\bm{D}_d\\},\n\\end{equation*}\nwith $d =  \\lceil \\log \\left(2\\ln\\left(\\frac{\\sqrt[3]{2}}{\\sqrt[3]{2} - 1}\\right)\\kappa\\right)\\rceil$, then $\n\\bm{D}_0\\approx_{\\epsilon_d} \\bm{D}_0 - \\bm{D}_0\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^{2^d},$ \nwith $\\epsilon_d < \\frac{1}{3}\\ln2$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof will be given as a collection of claims:\n\n\\begin{claim}\nLet $\\kappa$ be the condition number of $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$, and $\\{\\lambda_i\\}^{n}_{i=1}$ denote the eigenvalues of $\\bm{D}^{-1}_0\\bm{A}_0$. Then,\n$|\\lambda_i| \\le 1 - \\frac{1}{\\kappa}$,  \nfor all $i=1,\\ldots, n$\n\\end{claim}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\n\n\nNotice that if $\\lambda_i$ represented an eigenvalue of $\\bm{D}^{-1}_0\\bm{A}_0$, then $\\lambda^{r}_i$ is an eigenvalue of $\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^r$ for all $r\\in \\mathbb{N}$. Therefore, we have \n\\begin{equation}\\label{spect_radius}\n\\rho\\left(\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^{2^d}\\right)\\le \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d}\n\\end{equation}\n\n\\begin{claim}\nLet $\\bm{M}$ be an SDDM matrix and consider the splitting $\\bm{M} =\\bm{D} -\\bm{A}$, with $\\bm{D}$ being non negative diagonal and $\\bm{A}$ being symmetric non negative. Further, assume that the eigenvalues of $\\bm{D}^{-1}\\bm{A}$ lie between $-\\alpha$ and $\\beta$. Then, \n\\[\n(1 - \\beta)\\bm{D} \\preceq \\bm{D} -\\bm{A} \\preceq (1 + \\alpha)\\bm{D}.\\] \n\\end{claim}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\n\nCombining the above results, gives\n\\[\\left[1 - \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d}\\right]\\bm{D}_d\\preceq \\bm{D}_d - \\bm{A}_d\\preceq\\left[1 + \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d}\\right]\\bm{D}_d. \\]\nHence, to guarantee that $\\bm{D}_d\\approx_{\\epsilon_d}\\bm{D}_d - \\bm{A}_d$, the following system must be satisfied: \n\\begin{align*}\ne^{-\\epsilon_d} &\\le 1 - \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d},  \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ e^{\\epsilon_d} \\ge 1 + \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d}.\n\\end{align*}\nIntroducing $\\gamma$ for $\\left(1 - \\frac{1}{\\kappa}\\right)^{2^d}$, we arrive at: \n\\begin{align*}\n\\epsilon_d &\\ge \\ln\\left(\\frac{1}{1 - \\gamma}\\right),  \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\epsilon_d \\ge \\ln(1 + \\gamma).\n\\end{align*}\nHence, $\\epsilon_d \\ge \\max\\left\\lbrace \\ln\\left(\\frac{1}{1 - \\gamma}\\right), \\ln(1 + \\gamma)\\right\\rbrace = \\ln\\left(\\frac{1}{1 - \\gamma}\\right)$. \nNow, notice that if $d = \\lceil \\log c \\kappa\\rceil$ then,\n$\\gamma = \\left(1 - \\frac{1}{\\kappa}\\right)^{2^d} =  \\left(1 - \\frac{1}{\\kappa}\\right)^{{c}\\kappa} \\le \\frac{1}{e^c}$. \nHence, $\\ln\\left(\\frac{1}{1 - \\gamma}\\right) \\le \\ln\\left(\\frac{e^c}{e^c - 1}\\right)$. This gives $c = \\lceil 2\\ln\\left(\\frac{\\sqrt[3]{2}}{\\sqrt[3]{2} - 1}\\right)\\rceil$, implying $\\epsilon_d = \\ln\\left(\\frac{e^c}{e^c - 1}\\right) < \\frac{1}{3}\\ln2$.\n\\end{proof}\n\nUsing the above results the time complexity of $\\text{DistrESolve}$ with $d =  \\lceil \\log \\left(2\\ln\\left(\\frac{\\sqrt[3]{2}}{\\sqrt[3]{2} - 1}\\right)\\kappa\\right)\\rceil$ is $\\mathcal{O}\\left(n^2\\log\\kappa \\log(\\frac{1}{\\epsilon})\\right)$ times steps, which concludes the proof of Theorem \\ref{Main_Theorem}.\n\n\n\\subsection{R-Hop Distributed Solvers}\nThe above version of the distributed solver requires no knowledge of the graph's topology, but does require the information from all other nodes. Next, we will outline an R-Hop version of the algorithm in which communication is restricted to the R-Hop neighborhood between nodes. Due to such communication constraints, the R-Hop solver is general enough to be applied in a variety of fields including but not limited to, network flow problems (see Section~\\ref{Sec:Newton}). Along with Theorem~\\ref{Main_TheoremTwo}, the following corollary summarizes the results of the R-Hop distributed solver: \n\n\\begin{corollary}\\label{CorollaryTwo}\nLet $\\bm{M}_0$ be the weighted Laplacian of $\\mathcal{G} = \\left(\\mathcal{V},\\mathcal{E},\\bm{W}\\right)$. There exists a decentralized algorithm that uses only $R$-hop communication between nodes and computes $\\epsilon$ close solutions of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$ in\n $\\mathcal{O}\\left(\\frac{n^3\\alpha}{R}\\frac{\\bm{W}_{\\text{max}}}{\\bm{W}_{\\text{min}}}\\log(\\frac{1}{\\epsilon})\\right)$ time steps, with $n$ being the number of nodes in $\\mathcal{G}$, $\\bm{W}_{\\text{max}}, \\bm{W}_{\\text{min}}$ denoting the largest and the smallest weights of edges in $\\mathcal{G}$, respectively, $\\alpha = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$ representing the upper bound on the size of the R-hop neighborhood $\\forall \\bm{v}\\in \\mathcal{V}$, and $\\epsilon\\in (0, \\frac{1}{2}]$ being the precision parameter. \n\\end{corollary}\n\nSimilar to development of the full communication solver, the R-Hop version also requires two steps to attain the $\\epsilon$-close approximation to $\\bm{x}^{\\star}$, i.e., the ``crude R-Hop'' and the ``exact R-Hop'' solutions. \n\n\\begin{algorithm}[t!]\n  \\caption{$\\text{RDistRSolve}\\left(\\mathcal{C}, [\\bm{b}_0]_k, d, R\\right)$}\n  \\label{Algo:DistRHop}\n  \\begin{algorithmic}[1]\n\t\\STATE \\textbf{Part One:}\t\n\t\\STATE $\\{[\\bm{C}_0]_{k1},\\ldots,[\\bm{C}_0]_{kn}\\} = \\text{f}_0\\left([\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}, R\\right)$\n\t\\STATE $\\{[\\bm{C}_1]_{k1},\\ldots,[\\bm{C}_1]_{kn}\\} = \\text{f}_1\\left([\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}, R\\right)$\n\\\\\\hrulefill\t\n\t\\STATE \\textbf{Part Two:}\n\t\\FOR {$i=1$ to $d$}\n\t\\STATE $l_{i-1} = \\sfrac{2^{i-1}}{R}$\n\t\\STATE $[\\bm{u}^{(i-1)}_{1}]_k = [\\bm{A}_0\\bm{D}^{-1}_0\\bm{b}_{i-1}]_k$\n\t\\STATE $[\\bm{u}^{(i-1)}_{l_{i-1}}]_k = \\text{f}_2([\\bm{u}^{(i-1)}_{1}]_k)$\n\t\t\t\\STATE $[\\bm{b}_i]_k = [\\bm{b}_{i-1}]_k + [\\bm{u}^{(i-1)}_{l_{i-1}}]_k$\\\\\n\t\n\t\t\n\t\\ENDFOR \n\t\t\\\\\\hrulefill\n\t\\STATE \\textbf{Part Three:}\n\t\\STATE $[\\bm{x}_d]_k = \\sfrac{[\\bm{b}_d]_k}{[\\bm{D}_0]_{kk}}$\n\t\\FOR {$i=d-1$ to $1$}\n\t\\STATE $l_i = \\sfrac{2^i}{R}$\n\t\\STATE $[\\bm{\\eta}^{(i+1)}_{1}]_k = [\\bm{D}^{-1}_0\\bm{A}_0\\bm{x}_{i+1}]_k$\n\t\\STATE $\\left[\\bm{\\eta}^{i+1}_{l_i}\\right]_k  = \\text{f}_3([\\bm{\\eta}^{(i+1)}_{1}]_k)$\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t \n\t\t\t\\STATE $[\\bm{x}_i]_k = \\frac{1}{2}\\left[\\frac{[\\bm{b}_i]_k}{[\\bm{D}_0]_{kk}}  + [\\bm{x}_{i+1}]_k + [\\bm{\\eta}^{i+1}_{l_i}]_k \\right]$\\\\\n\t\n\t\\ENDFOR \n\t\\STATE $[\\bm{x}_0]_k = \\frac{1}{2}\\left[\\frac{[\\bm{b}_0]_k}{[\\bm{D}_0]_{kk}} + [\\bm{x}_1]_k + [\\bm{D}^{-1}_0\\bm{A}_0\\bm{x}_1]_k \\right]$\n    \\STATE \\textbf{return} $[\\bm{x}_0]_k$ \n  \\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{``Crude'' R-Hop Distributed Solver}\nThe ``crude R-Hop'' solver uses the same inverse approximated chain as that of the full communication version (see Equation~\\ref{Eq:InverseChain}) to acquire a ``crude'' approximation for the $k^{th}$ component $\\bm{x}_{0}$ while only requiring R-Hop communication between the nodes. Algorithm~\\ref{Algo:DistRHop} represents the ``crude'' R-Hop solver requiring the inverse chain, $k^{th}$ component of $\\bm{b}_{0}$, length of the inverse chain $d$, and the communication bound $R$ as inputs. Namely, each node $\\bm{v}_{k} \\in \\mathcal{V}$ receives the $k^{th}$ row of $\\bm{M}_{0}$ , $k^{th}$ component, $[\\bm{b}_{0}]_{k}$, of $\\bm{b}_{0}$, the length of the inverse chain, $d$, and the local communication bound\\footnote{For simplicity, $R$ is assumed to be in the order of powers of 2, i.e., $R=2^{\\rho}$.}  $R$ as inputs to output the $k^{th}$ component of the ``crude'' approximation to $\\bm{x}^{\\star}$.  Algorithm~\\ref{Algo:DistRHop} operates in three major parts. Due to the need of the R-powers of $\\bm{A}_{0}\\bm{D}_{0}^{-1}$ and $\\bm{D}_{0}\\bm{A}_{0}^{-1}$, the first step is to compute such matrices in a distributed manner. \n\n\n\\begin{algorithm}[t!]\n  \\caption{$\\text{f}_0\\left([\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}, R\\right)$}\n  \\label{Alg_4}\n  \\begin{algorithmic}[1]\n\t\\FOR {$l = 1$ to $R - 1$}\n\t\t\\FOR {$j$ s.t.$\\bm{v}_j\\in \\mathbb{N}_{l+1}(\\bm{v}_k)$}\n\t\t\t\\STATE\n\t\t\t$\\left[(\\bm{A}_0\\bm{D}^{-1}_0)^{l+1}\\right]_{kj} = \\sum\\limits_{r:\\bm{v}_r\\in \\mathbb{N}_1(v_j)}\\frac{[\\bm{D}_0]_{rr}}{[\\bm{D}_0]_{jj}}[(\\bm{A}_0\\bm{D}^{-1}_0)^l]_{kr}[\\bm{A}_0\\bm{D}^{-1}_0]_{jr}$\n\t\t\\ENDFOR\n\t\\ENDFOR\n\t\\STATE \\textbf{return} $\\bm{c}_0 = \\{[(\\bm{A}_0\\bm{D}^{-1}_0)^{R}]_{k1},\\ldots,[(\\bm{A}_0\\bm{D}^{-1}_0)^{R}]_{kn} \\}$\n  \\end{algorithmic}\n\\end{algorithm}\n\nGiven the inverse chain and the communication bound, R, $\\text{f}_{0}(\\cdot)$ and $\\text{f}_{1}(\\cdot)$ serve this cause as detailed in Algorithms~\\ref{Alg_4} and~\\ref{Alg_5}, respectively. Essentially, these algorithms execute  multiplications needed for determining $\\left(\\bm{A}\\bm{D}_{0}^{-1}\\right)^{R}$ and $\\left(\\bm{D}\\bm{A}_{0}^{-1}\\right)^{R}$ in a distributed fashion looping over the relevant hops of the network. For a node, $\\bm{v}_{k} \\in \\mathcal{V}$, the $k^{th}$ component of these powers are returned to Algorithm~\\ref{Algo:DistRHop} as $[\\bm{C}_{0}]_{ki}=\\left[\\left(\\bm{A}_{0}\\bm{D}_{0}^{-1}\\right)^{R}\\right]_{ki}$ and $[\\bm{C}_{1}]_{ki}=\\left[\\left(\\bm{D}_{0}\\bm{A}_{0}^{-1}\\right)^{R}\\right]_{ki}$ for $i=\\{1,\\dots, n\\}$; see Part One in Algorithm~\\ref{Algo:DistRHop}. \n\n\nSimilar to the full communication version, the second two parts of the ``crude R-Hop'' solver run two loops. In the first, the $k^{th}$ component of $\\bm{b}_{i}$ is computed by looping forward through the inverse chain, while in the second the $k^{th}$ component of the crude solution is determined by looping backwards. \n\\begin{algorithm}[t!]\n  \\caption{$\\text{f}_1([\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}, R)$}\n  \\label{Alg_5}\n  \\begin{algorithmic}[1]\n\t\\FOR {$l = 1$ to $R - 1$}\n\t\t\\FOR {$j$ s.t.$\\bm{v}_j\\in \\mathbb{N}_{l+1}(\\bm{v}_k)$}\n\t\t\t\\STATE$\n\t\t\t\\left[(\\bm{D}^{-1}_0\\bm{A}_0)^{l+1}\\right]_{kj} = \\sum\\limits_{r:\\bm{v}_r\\in \\mathbb{N}_1(\\bm{v}_j)}\\frac{[\\bm{D}_0]_{jj}}{[\\bm{D}_0]_{rr}}[(\\bm{D}^{-1}_0\\bm{A}_0)^l]_{kr}[\\bm{D}^{-1}_0\\bm{A}_0]_{jr}$\n\t\t\\ENDFOR\n\t\\ENDFOR\n\t\\STATE \\textbf{return} $\\bm{c}_1 = \\{[(\\bm{D}^{-1}_0\\bm{A}_0)^{R}]_{k1},\\ldots,[(\\bm{D}^{-1}_0\\bm{A}_0)^{R}]_{kn} \\}$\n  \\end{algorithmic}\n\\end{algorithm}\n\n\nThe second part of the solver is better depicted in the flow diagrams of Figures~\\ref{Fig:FlowDiagramOne} and~\\ref{Fig:FlowDiagramTwo}. Within the first loop running through the length of the inverse chain, the condition $i-1 < \\rho$ is checked. In case this condition is true, $\\bm{A}_{0}$, $\\bm{D}_{0}$, and the previous iteration vector $\\bm{b}_{i-1}$ are used to update $\\bm{b}_{i}$ as shown in Figure~\\ref{Fig:FlowDiagramOne}. \n\n\\begin{equation*}\n\\left[\\bm{b}_{i}\\right]_{k}=\\left[\\bm{b}_{i-1}\\right]_{k}+\\left[\\bm{u}_{2^{i-1}}^{(i-1)}\\right]_{k}\n\\end{equation*}\n\n\nThis is performed using another loop constructing a series of $[\\bm{u}_{j}^{(i-1)}]_{k}$\nvectors for $j=1,\\dots, 2^{i-1}$, used to update the $k^{th}$ component of $\\bm{b}_{i}$ at the $i^{th}$ iteration:\nAt the next $i^{th}$ iteration, the condition $i-1 < \\rho$ is checked again. In case this condition is met, the previous computations are executed again. Otherwise, the commands depicted in Figure~\\ref{Fig:FlowDiagramTwo} run. Here, a temporary variable denoting the fraction to the communication bound $R$, $l_{i-1}=\\frac{2^{i-1}}{R}$, is used to determine the upper iterate bound in $j$. Again throughout this loop, a series of $\\bm{u}$ vectors used to update $[\\bm{b}_{i}]_{k}$ are constructed. \n\nHaving terminated the forward loop, the $k^{th}$ component of the ``crude'' solution is computed by looping backward through the inverse approximated chain (see part three in Algorithm~\\ref{Algo:DistRHop}). For $i$ running backwards to $1$, an R-Hop condition, $\\rho$, is checked. In case $i < \\rho$, the following update is performed:\n\\begin{equation*}\n[\\bm{\\eta}_{j}^{(i+1)}]_{k}=\\left[\\bm{D}^{-1}_{0}\\bm{A}_{0}\\bm{\\eta}_{j-1}^{(i+1)}\\right]_{k},\n\\end{equation*}\nfor $j=2,\\dots, 2^{i}$, $i=d-1,\\dots, 1$, and \n\\begin{equation*}\n\\left[\\bm{\\eta}_{1}^{(i+1)}\\right]_{k}=\\left[\\bm{D}^{-1}_{0}\\bm{A}_{0}\\bm{x}_{i+1}\\right]_{k}. \n\\end{equation*}\n\\begin{figure*}[t!]\n\\centering\n\\vspace{-.5em}\n\\subfigure[]{\n\t\\label{Fig:FlowDiagramOne}\n\\includegraphics[width=0.43\\textwidth]{PartOne}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{Fig:FlowDiagramTwo}\n\\includegraphics[width=0.5\\textwidth]{PartTwo}\n}\n\\caption{The left figure depicts the first condition of part two in Algorithm~\\ref{Algo:DistRHop} which has to be checked. In case $i-1 < \\rho$, the computations shown in the figure are executed. The right figure handles the case when $i-1 \\geq \\rho$. Here, the computations shown in the figure are executed. Note, that the chains are constructed using the computations returned by $\\text{f}_{0}(\\cdot)$ and $\\text{f}_{1}(\\cdot)$.}\n\\end{figure*}\n\n\n \n\n\n\n\nThe role of $\\bm{\\eta}$ are backward intermediate solutions needed for updating the ``crude'' solution to $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$: \n\\begin{equation*}\n[\\bm{x}_{i}]_{k}=\\frac{1}{2}\\left[\\frac{[\\bm{b}_{i}]_{k}}{[\\bm{D}_{0}]_{kk}}+[\\bm{x}_{1}]_{k}+[\\bm{\\eta}^{(i+1)}_{2^{i}}]\\right]_{k},\n\\end{equation*}\nfor a node $\\bm{v}_{k} \\in \\mathcal{V}$. In case $i \\geq \\rho$ a similar set of computations are executed for updating the crude solution using: \n\\begin{equation*}\n[\\bm{x}_{i}]_{k}=\\frac{1}{2}\\left[\\frac{[\\bm{b}_{i}]_{k}}{[\\bm{D}_{0}]_{kk}}+[\\bm{x}_{1}]_{k}+[\\bm{D}_{0}^{-1}\\bm{A}\\bm{x}_{1}]\\right]_{k}.\n\\end{equation*}\n\n\\textbf{Analysis of Algorithm~\\ref{Algo:DistRHop}} Similar to the previous section, we next provide the theoretical analysis needed for quantifying the performance of the crude R-Hop solver. The following Lemma shows that $\\text{RDistRSolve}$ computes the $k^{th}$ component of the ``crude'' approximation of $\\bm{x}^{\\star}$ and provides the algorithm's time complexity:\n \n\\begin{lemma}\\label{r_hop_rude_lemma}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting and let $\\bm{Z}^{\\prime}_0$ be the operator defined by $\\text{RDistRSolve}$, namely, $\\bm{x}_0 = \\bm{Z}^{\\prime}_0\\bm{b}_0$, then\n$\\bm{Z}^{\\prime}_0\\approx_{\\epsilon_d} \\bm{M}^{-1}_0$.  \nFurthermore, $\\text{RDistRSolve}$ requires \n\\[\\mathcal{O}\\left(\\frac{2^d}{R}\\alpha + \\alpha Rd_{max}\\right),\\] where $\\alpha = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$ to arrive at $\\bm{x}_{0}$. \n\\end{lemma}\n\n\\begin{proof}\nThe proof of the above Lemma can be obtained by proving a collection of claims:\n\\begin{claim}\\label{claim_1}\nMatrices $\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^r$ and $\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{r}$ have sparsity patterns corresponding to the $r$-Hop neighborhood for any $r\\in \\mathbb{N}$.\n\\end{claim}\n\nThe above claim is proved by induction on $R$. We start with the base case: for $R = 1$,\n\\begin{align*}\n[\\bm{A}_0\\bm{D}^{-1}_0]_{ij} =\n\\begin{cases}\n &\\frac{[\\bm{A}_0]_{ij}}{[\\bm{D}_0]_{ii}} \\ \\ \\  \\text{if } j: \\bm{v}_j\\in \\mathbb{N}_{1}(\\bm{v}_i) \\\\\n &0 \\ \\ \\ \\text{otherwise.}\n\\end{cases}\n\\end{align*}\nTherefore, $\\bm{A}_0\\bm{D}^{-1}_0$ has a sparsity pattern corresponding to the $1$-Hop neighborhood. \nAssume that for all $1\\le p\\le R-1$, $\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^p$ has a sparsity pattern corresponding to the $p-hop$ neighborhood. Consider, $\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{r}$\n\\begin{equation}\\label{sparisty_lemma_2}\n[(\\bm{A}_0\\bm{D}^{-1}_0)^{R}]_{ij} = \\sum_{k=1}^{n}[(\\bm{A}_0\\bm{D}^{-1}_0)^{R-1}]_{ik}[\\bm{A}_0\\bm{D}^{-1}_0]_{kj}\n\\end{equation}\nSince $\\bm{A}_0\\bm{D}^{-1}_0$ is non negative, then $[(\\bm{A}_0\\bm{D}^{-1}_0)^{R}]_{ij} \\ne 0$ iff there exists $k$ such that $\\bm{v}_k\\in \\mathbb{N}_{R-1}(\\bm{v}_i)$ and $\\bm{v}_k\\in \\mathbb{N}_1(\\bm{v}_j)$, namely, $\\bm{v}_j\\in \\mathbb{N}_{R}(\\bm{v}_i)$. The proof can be done in a similar fashion for $\\bm{D}^{-1}_0\\bm{A}_0$.\n\\end{proof}\n\nThe next claim provides complexity guarantees for $\\text{f}_{0}(\\cdot)$ and $\\text{f}_{1}(\\cdot)$ described in Algorithms~\\ref{Alg_4} and~\\ref{Alg_5}, respectively. \n\\begin{claim}\\label{claim_2}\nAlgorithms~\\ref{Alg_4} and~\\ref{Alg_5} use only the R-hop information to compute the $k^{th}$ row of $\\left(\\bm{D}^{-1}_0\\bm{A}_0\\right)^{R}$ and $\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{R}$, respectively, in $\\mathcal{O}\\left(\\alpha Rd_{max}\\right)$ time steps, where $\\alpha = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$.  \n\\end{claim}\n\\begin{proof}\nThe proof will be given for $\\text{f}_{0}(\\cdot)$ described in Algorithm~\\ref{Alg_4} as that for $\\text{f}_{1}(\\cdot)$ can be performed similarly. Due to Claim~\\ref{claim_1}, we have\n\\begin{align}\\label{eq_1}\n\\left[\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{l+1}\\right]_{kj} &= \\sum\\limits_{r=1}^{n}\\left[\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{l}\\right]_{kr}\\left[\\bm{A}_0\\bm{D}^{-1}_0\\right]_{rj} = \\sum\\limits_{r: \\bm{v}_r\\in \\mathbb{N}_1(\\bm{v}_j)}\\left[\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{l}\\right]_{kr}\\left[\\bm{A}_0\\bm{D}^{-1}_0\\right]_{rj}\n\\end{align} \nTherefore at iteration $l+1$, $\\bm{v}_k$ computes the $k^{th}$ row of $\\left(\\bm{A}_0\\bm{D}^{-1}_0\\right)^{l+1}$ using: \n\\begin{enumerate}\n\\item[] (1) the $k^{th}$ row of $(\\bm{A}_0\\bm{D}^{-1}_0)^{l}$, and\n\\item[] (2) the $r^{th}$ column of $\\bm{A}_0\\bm{D}^{-1}_0$.\n\\end{enumerate}\nNode $\\bm{v}_{r}$, however, can only send the $r^{th}$ row of $\\bm{A}_{0}\\bm{D}^{-1}_{0}$ making $\\bm{A}_{0}\\bm{D}^{-1}_{0}$ non-symmetric. Noting that\n$\\sfrac{[\\bm{A}_0\\bm{D}^{-1}_0]_{rj}}{[\\bm{D}_0]_{rr}} = \\sfrac{[\\bm{A}_0\\bm{D}^{-1}_0]_{jr}}{[\\bm{D}_0]_{jj}}$, since $\\bm{D}^{-1}_0\\bm{A}_0\\bm{D}^{-1}$ is symmetric, leads to \n$[(\\bm{A}_0\\bm{D}^{-1}_0)^{l+1}]_{kj} = \n\\sum\\limits_{r: \\bm{v}_r\\in \\mathbb{N}_1(\\bm{v}_j)}\\frac{[\\bm{D}_0]_{rr}}{[\\bm{D}_0]_{jj}}[(\\bm{A}_0\\bm{D}^{-1}_0)^{l}]_{kr}[\\bm{A}_0\\bm{D}^{-1}_0]_{jr}$. \nTo prove the time complexity guarantee, at each iteration $\\bm{v}_k$ computes at most $\\alpha$ values, where $\\alpha  = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$ is the upper bound on the size of the R-hop neighborhood $\\forall \\bm{v}\\in \\mathcal{V}$.  Each such computation requires at most $\\mathcal{O}(d_{max})$ operations. Thus, the overall time complexity is given by $\\mathcal{O}(\\alpha Rd_{max})$.\n\\end{proof}\n\nWe are now ready to provide the proof of Lemma \\ref{r_hop_rude_lemma}. \n\\begin{proof}\nFrom \\textbf{Parts Two} and \\textbf{Three} of Algorithm~\\ref{Algo:DistRHop}, it is clear that node $\\bm{v}_k$ computes $[\\bm{b}_1]_k,[\\bm{b}_2]_k,\\ldots, [\\bm{b}_d]_k$ and $[\\bm{x}_d]_k, [\\bm{x}_{d-1}]_k,\\ldots, [\\bm{x}_0]_k$, respectively. These are determined using the inverse approximated chain as follows\n\\begin{align}\\label{eq_4}\n\\bm{b}_i &= (\\bm{I} + (\\bm{A}_{i-1}\\bm{D}^{-1}_{i-1})\\bm{b}_{i-1}  \n=\\bm{b}_{i-1} + (\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-1}}\\bm{b}_{i-1} \\\\ \\nonumber\n\\bm{x}_{i} &= \\frac{1}{2}[\\bm{D}^{-1}_i\\bm{b}_i + (\\bm{I} +\\bm{D}^{-1}_i\\bm{A}_i)x_{i+1}] =\\frac{1}{2}[\\bm{D}^{-1}_0\\bm{b}_i + \\bm{x}_{i+1} + (\\bm{D}^{-1}_0\\bm{A}_0)^{2^i}\\bm{x}_{i+1}]\n\\end{align}\n\nConsidering the computation of $[\\bm{b}_1]_k,\\ldots, [\\bm{b}_d]_k$ for $\\rho> i - 1$, we have\n\\begin{align*}\n[\\bm{b}_{i}]_k &= [\\bm{b}_{i-1}]_k + [(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-1}}\\bm{b}_{i-1}]_k = [\\bm{b}_{i-1}]_k + [\\underbrace{\\bm{A}_0\\bm{D}^{-1}_0 \\ldots \\bm{A}_0\\bm{D}^{-1}_0}_{2^{i-1}}\\bm{b}_{i-1}]_k \\\\\n&= [\\bm{b}_{i-1}]_k + [\\underbrace{\\bm{A}_0\\bm{D}^{-1}_0 \\ldots \\bm{A}_0\\bm{D}^{-1}_0}_{2^{i-1}-1}\\bm{u}^{(i-1)}_1]_k \\dots = [\\bm{b}_{i-1}]_k + \\left[\\bm{u}^{(i-1)}_{2^{i-1}}\\right]_k\n\\end{align*}\n\\text{with $\\bm{u}^{(i-1)}_{j+1} = \\bm{A}_0\\bm{D}^{-1}_0\\bm{u}^{(i-1)}_{j}$ for $j = 1,\\ldots 2^{i-1}-1$.} Since $\\bm{A}_0\\bm{D}^{-1}_0$ has a sparsity pattern corresponding to 1-hop neighborhood (see Claim~\\ref{claim_1}),  node $\\bm{v}_k$ computes $\\left[\\bm{u}^{(i-1)}_{j+1}\\right]_k$, based on $\\bm{u}^{(i-1)}_{j}$, acquired from its 1-Hop neighbors. It is easy to see that $ \\forall i \\ \\text{such that} \\ i - 1 < \\rho$ the computation of $[\\bm{b}_i]_k$ requires $\\mathcal{O}\\left(2^{i-1}d_{max}\\right)$ time steps. Thus, the computation of $[\\bm{b}_1]_k, \\ldots, [\\bm{b}_{\\rho}]_k$ requires $\\mathcal{O}(2^{\\rho}d_{\\text{max}}) = \\mathcal{O}(Rd_{\\text{max}})$. Now, consider the computation of $[\\bm{b}_i]_k$ but for $i - 1 \\geq \\rho$ \n\\begin{align*}\n[\\bm{b}_{i}]_k &= [\\bm{b}_{i-1}]_k + [(\\bm{A}_0\\bm{D}^{-1}_0)^{2^{i-1}}\\bm{b}_{i-1}]_k  =[\\bm{b}_{i-1}]_k + [\\underbrace{\\bm{C}_0\\ldots \\bm{C}_0}_{l_{i-1}}\\bm{b}_{i-1}]_k \\\\\n&=[\\bm{b}_{i-1}]_k + [\\underbrace{\\bm{C}_0 \\ldots \\bm{C}_0}_{l_{i-1}-1}\\bm{u}^{(i-1)}_1]_k  =[\\bm{b}_{i-1}]_k + \\left[\\bm{u}^{(i-1)}_{l_{i-1}}\\right]_k\n\\end{align*}\nwith $\\bm{C}_0 = (\\bm{A}_0\\bm{D}^{-1}_0)^R$,  $l_{i-1} = \\frac{2^{i-1}}{R}$, and $\\bm{u}^{(i-1)}_{j+1} = \\bm{C}_0\\bm{u}^{(i-1)}_{j}$ for $j=1,\\ldots, l_{i-1} - 1$.\nSince $\\bm{C}_0$ has a sparsity pattern corresponding to R-hop neighborhood (see Claim~\\ref{claim_1}), node $\\bm{v}_k$ computes $[\\bm{u}^{(i-1)}_{j+1}]_k$ based on the components of $\\bm{u}^{(i-1)}_{j}$ attained from its R-hop neighbors. For each $i$ such that $i - 1 \\geq \\rho $ the computing $[\\bm{b}_i]_k$ requires $\\mathcal{O}\\left(\\frac{2^{i-1}}{R}\\alpha\\right)$ time steps, where $\\alpha = \\min\\left\\{n, \\frac{\\left(d^{R+1}_{\\text{max}} - 1\\right)}{\\left(d_{\\text{max}} - 1\\right)}\\right\\}$ being the upper bound on the number of nodes in the $R-$ hop neighborhood $\\forall \\ \\bm{v}\\in \\mathcal{V}$. Therefore, the  overall computation of $[\\bm{b}_{\\rho + 1}]_k, [\\bm{b}_{\\rho + 2}]_k, \\ldots, [\\bm{b}_d]_k$ is achieved in $\\mathcal{O}\\left(\\frac{2^d}{R}\\alpha\\right)$ time steps. Finally, the time complexity for the computation of all of the values $[\\bm{b}_1]_k,[\\bm{b}_2]_k,\\ldots, [\\bm{b}_d]_k$ is $\\mathcal{O}\\left(\\frac{2^d}{R}\\alpha + Rd_{max}\\right)$. Similar analysis can be applied to determine the computational complexity of $[\\bm{x}_d]_k,[\\bm{x}_{d-1}]_k,\\ldots, [\\bm{x}_1]_k$, i.e., \\textbf{Part Three} of Algorithm~\\ref{Algo:DistRHop}. We arrive at\n$\\bm{Z}^{\\prime}_0\\approx_{\\epsilon_d} \\bm{M}^{-1}_0$. \nFinally, using Claim \\ref{claim_2}, the time complexity of $\\text{RDistRSolve}$ (Algorithm~\\ref{Algo:DistRHop}) is $\\mathcal{O}\\left(\\frac{2^d}{R}\\alpha + \\alpha Rd_{max}\\right)$.\n\\end{proof} \n\\subsubsection{``Exact'' R-Hop Distributed Solver}\nHaving developed an R-hop version which computes a ``crude'' approximation to the solution of $\\bm{M}_{0}\\bm{x}=\\bm{b}_{0}$, now we provide an exact R-hop solver presented in Algorithm~\\ref{Alg_ExactRHop}. Similar to $\\text{RDistRSolve}$, each node $\\bm{v}_k$ receives the $k^{th}$ row $\\bm{M}_0$, $[\\bm{b}_0]_k$, $d$, $R$, and a precision parameter $\\epsilon$ as inputs, and outputs the $k^{th}$ component of the $\\epsilon$ close approximation of vector $\\bm{x}^{\\star}$. \n\n\\begin{algorithm}\\label{Algo:EDistR}\n  \\caption{ \\hspace{-.5em} $\\text{EDistRSolve}\\left(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{b}_0]_k, d, R,\\epsilon\\right)$}\n  \\label{Alg_ExactRHop}\n  \\begin{algorithmic}\n\\STATE \\textbf{Initialize}: $[\\bm{y}_0]_k = 0$, and $[\\bm{\\chi}]_k = \\text{RDistRSolve}(\\{[M_0]_{k1},\\ldots, [M_0]_{kn}\\}, [b_0]_k, d, R)$\t\n\t\\FOR {$t=1$ to $q$}\n\t\t\\STATE  \\hspace{-2em} $[\\bm{u}^{(1)}_{t}]_k = [\\bm{D}_0]_{kk}[\\bm{y}_{t-1}]_k - \\sum_{j: \\bm{v}_j\\in \\mathbb{N}_{1}(\\bm{v}_k)}[\\bm{A}_{0}]_{kj}[\\bm{y}_{t-1}]_j$\n\t\t \\STATE \\hspace{-2em} $[\\bm{u}^{(2)}_{t}]_k = \\text{RDistRSolve}(\\{[\\bm{M}_0]_{k1},\\ldots, [\\bm{M}_0]_{kn}\\}, [\\bm{u}^{(1)}_{t}]_k, d, R)$\n\t\t\\STATE  \\hspace{-2em} $[\\bm{y}_t]_k = [\\bm{y}_{t-1}]_k - [\\bm{u}^{(2)}_{t}]_k + [\\bm{\\chi}]_k$ \n\t\\ENDFOR \n    \\STATE \\textbf{return} $[\\tilde{\\bm{x}}]_k = [\\bm{y}_q]_k$\n  \\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{Analysis of Algorithm~\\ref{Alg_ExactRHop}:} The following Lemma shows that $\\text{EDistRSolve}$ computes the $k^{th}$ component of the $\\epsilon$ close approximation to $\\bm{x}^{\\star}$ and provides the time complexity analysis. \n\n\\begin{lemma}\\label{Dist_Exact_algorithm_guarantee_lemma_Bla}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting. Further, let $\\epsilon_d < \\sfrac{1}{3}\\ln2$. Then Algorithm~\\ref{Alg_ExactRHop} requires $\\mathcal{O}\\left(\\log\\frac{1}{\\epsilon}\\right)$ iterations to return the $k^{th}$ component of the $\\epsilon$ close approximation to $\\bm{x}^{\\star}$.  \n\\end{lemma}\n\nThe following Lemma provides the time complexity analysis of $\\text{EDistRSolve}$:\n\n\\begin{lemma}\\label{time_complexity_of_distresolve_Bla}\nLet $\\bm{M}_0 = \\bm{D}_0 - \\bm{A}_0$ be the standard splitting and let $\\epsilon_d < \\sfrac{1}{3}\\ln 2 $, then $\\text{EDistRSolve}$ requires \n$\\mathcal{O}\\left(\\left(\\sfrac{2^d}{R}\\alpha + \\alpha Rd_{max}\\right)\\log\\left(\\sfrac{1}{\\epsilon}\\right)\\right)$ time steps. Moreover, for each node $\\bm{v}_k$, $\\text{EDistRSolve}$ only uses information from the R-hop neighbors.\n\\end{lemma}\n\n\n\\textbf{Length of the Inverse Chain:} Again these introduced algorithms depend on the length of the inverse approximated chain, $d$. The analysis in Section~\\ref{Sec:Length} can be applied again to determine the $d =  \\lceil \\log \\left(2\\ln\\left(\\frac{\\sqrt[3]{2}}{\\sqrt[3]{2} - 1}\\right)\\kappa\\right)\\rceil$ as the length of the inverse chain.\n\n\n\n\n\\subsection{Comparison to Existing Literature}\nAs mentioned before, the proposed solver is a distributed version of the parallel SDDM solver of~\\cite{c11}. Our approach is capable of acquiring $\\epsilon$-close solutions for arbitrary $\\epsilon>0$ in $\\mathcal{O}\\left(n^{3}\\frac{\\bm{\\alpha}}{R}\\frac{\\bm{W}_{\\text{max}}}{\\bm{W}_{\\text{min}}}\\log\\left(\\frac{1}{\\epsilon}\\right)\\right)$, with $n$ the number of nodes in graph $\\mathcal{G}$, $\\bm{W}_{\\text{max}}$ and $\\bm{W}_{\\text{min}}$ denoting the largest and smaller weights of the edges in $\\mathcal{G}$, respectively, $\\bm{\\alpha}=\\min\\left\\{n,\\frac{d_{\\text{max}}^{R+1}-1}{d_{\\text{max}}-1}\\right\\}$ representing the upper bound on the size of the R-Hop neighborhood $\\forall \\bm{v} \\in \\mathcal{V}$, and $\\epsilon \\in (0,\\frac{1}{2}]$ as the precision parameter. After developing the full communication version, we proposed a generalization to the R-Hop case where communication is restricted. \n\nOur method is faster than state-of-the-art methods for iteratively solving linear systems. Typical linear methods, such as Jacobi iteration~\\cite{c16}, are guaranteed to converge if the matrix is \\emph{strictly} diagonally dominant. We proposed a distributed algorithm that generalizes this setting, where it is guaranteed to converge in the SDD\/SDDM scenario. Furthermore, the time complexity of linear techniques is $\\mathcal{O}(n^{1+\\beta} \\log n)$, hence, a case of strictly diagonally dominant matrix $\\bm{M}_{0}$ can be easily constructed to lead to a complexity of $\\mathcal{O}(n^{4}\\log n)$. Consequently, our approach not only generalizes the assumptions made by linear methods, but is also faster by a factor of $\\log n$. Furthermore, such algorithms require average consensus to decentralize vector norm computations. Contrary to these methods which lead to additional approximation errors to the real solution, our approach resolves these issues by eliminating the need for such a consensus framework. \n\nIn centralized solvers, nonlinear methods (e.g., conjugate gradient descent~\\cite{c37,c36}, etc.) typically offer computational advantages over linear methods (e.g., Jacobi Iteration) for iteratively solving linear systems. These techniques, however, can not be easily decentralized. For instance, the stopping criteria for nonlinear methods require the computation of weighted norms of residuals (e.g., $||\\bm{p}_{k}||_{\\bm{M}_{0}}$ with $\\bm{p}_{k}$ being the search direction at iteration $k$). To the best of our knowledge, the distributed computation of weighted norms is difficult. Namely using the approach in~\\cite{c38}, this requires the calculation of the top singular value of $\\bm{M}_{0}$ which amounts to a power iteration on $\\bm{M}_{0}^{\\mathsf{T}}\\bm{M}_{0}$ leading to the loss of sparsity. Furthermore, conjugate gradient methods require global computations of inner products.\n\nAnother existing method which we compare our results to is the recent work of the authors~\\cite{c34} where a local and asynchronous solution for solving systems of linear equations is considered. In their work, the authors derive a complexity bound, for one component of the solution vector, of $\\mathcal{O}\\left(\\min\\left(d\\epsilon^{\\frac{\\ln d}{\\ln ||\\bm{G}||_{2}}}, \\frac{d n \\ln \\epsilon}{\\ln ||\\bm{G}||_{2}}\\right)\\right)$, with $\\epsilon$ being the precision parameter, $d$ a constant bound on the maximal degree of $\\mathcal{G}$, and $\\bm{G}$ is defined as $\\bm{x}=\\bm{G}\\bm{x}+\\bm{z}$ which can be directly mapped to $\\bm{A}\\bm{x} = \\bm{b}$. The relevant scenario to our work is when $\\bm{A}$ is PSD and $\\bm{G}$ is symmetric. Here, the bound on the number of multiplications is given by $\\mathcal{O}\\left(\\min\\left(d^{\\frac{\\kappa(\\bm{A})+1}{2}\\ln\\frac{1}{\\epsilon}}, \\frac{\\kappa(\\bm{A})+1}{2}n d \\ln\\frac{1}{\\epsilon}\\right)\\right)$, with $\\kappa(\\bm{A})$ being the condition number of $\\bm{A}$. In the general case, when the degree depends on the number of nodes (i.e., $d = d(n)$), the minimum in the above bound will be the result of the second term ( $\\frac{\\kappa(\\bm{A})+1}{2}n d \\ln\\frac{1}{\\epsilon}$) leading to $\\mathcal{O}\\left(d(n) n \\kappa(\\bm{A})\\ln \\frac{1}{\\epsilon}\\right)$. Consequently, in such a general setting, our approach outperforms~\\cite{c34} by a factor of $d(n)$.\n\n\\textbf{Special Cases:} To better understand the complexity of the proposed SDDM solvers, next we detail the complexity for three specific graph structures. Before deriving these special cases, however, we first note the following simple yet useful connection between weighted and unweighted Laplacians of a graph $\\mathcal{G}$. Denoting by $\\bm{B}$ the incidence matrix of $\\mathcal{G}$ and $\\bm{W}$ a diagonal matrix with edge weights as diagonal elements, we can write: \n\\begin{align*}\n\\mathcal{L}_{\\mathcal{G}}&=\\bm{B}^{\\mathsf{T}}\\bm{B} \\ \\ \\ \\text{and} \\ \\ \\ \\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})} = \\bm{B}^{\\mathsf{T}}\\bm{W}\\bm{B}.\n\\end{align*}\nHence, we can easily establish:\n\\begin{align*}\n\\mu_{n}\\left(\\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})}\\right)\\leq \\bm{w}_{\\text{max}}\\mu_{n}\\left(\\mathcal{L}_{\\mathcal{G}}\\right) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mu_{2}\\left(\\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})}\\right) \\geq \\bm{w}_{\\text{min}}\\left(\\mathcal{L}_{\\mathcal{G}}\\right).\n\\end{align*}\n\nThis implies that the condition number of the weighted Laplacian satisfies: \n\\begin{align*}\n\\kappa\\left(\\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})}\\right)=\\frac{\\mu_{n}\\left(\\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})}\\right)}{\\mu_{2}\\left(\\mathcal{L}_{\\mathcal{G}}^{(\\text{weighted})}\\right)} \\leq \\frac{\\bm{w}_{\\text{max}}}{\\bm{w}_{\\text{min}}}\\kappa\\left(\\mathcal{L}_{\\mathcal{G}}\\right),\n\\end{align*}\nwith $\\bm{w}_{\\text{min}}$ and $\\bm{w}_{\\text{max}}$ are the minimal and maximal edge weights of $\\mathcal{G}$. Using the above, we now consider four different graph topologies: \n\\subsubsection{Path Graph}\nSimilar to the hitting time of a Markov chain on a path graph which is given by $\\mathcal{O}(n^{2})$, the time complexity of the R-Hop SDDM solver is given by\\footnote{Due to space constraints, the proofs can be found in the appendix.}: \n\\begin{corollary}\nGiven a path graph $\\mathcal{P}_{n}$ with $n$ nodes, the time complexity of the R-Hop SDDM solver is given by:\n\\begin{equation*}\n\\text{T}_{\\text{SDDM}}(\\mathcal{P}_{n})=\\mathcal{O}\\left(n^{2} \\log \\frac{1}{\\epsilon}\\right),\n\\end{equation*}\nfor any $\\epsilon > 0$ and for $k=m=\\mathcal{O}(\\sqrt{n})$.\n\\end{corollary}\n\n\n\\subsubsection{Grid Graph} Recognizing that a grid graph $\\mathcal{G}_{k \\times m}$ can be represented as a product of two path graphs, $\\mathcal{G}_{k \\times m}=\\mathcal{P}_{k} \\times \\mathcal{P}_{m}$, our solver's computational time can be summarized by: \n\\begin{corollary}\nGiven a grid graph, $\\mathcal{G}_{k \\times m}$, the time complexity of the distributed SDDM-solver can be bounded by: \n\\begin{equation*}\n\\text{T}_{\\text{SDDM}}\\left(\\mathcal{G}_{k \\times m}\\right) = \\mathcal{O}\\left(n\\log\\frac{1}{\\epsilon}\\right),\n\\end{equation*}\nfor any $\\epsilon>0$. \n\\end{corollary}\n\n\n\\subsubsection{Scale-Free Networks (Polya-Urn Graphs)} Using the results developed in~\\cite{c39,c40} the total time complexity of the distributed R-Hop solver is bounded by: \n\\begin{corollary}\nGiven a scale-free network, $\\mathcal{G}_{\\text{SN}(n)}$, the time complexity of the R-Hop SDD solver for R=1 is given by: \n\\begin{equation*}\n\\text{T}_{\\text{SDD}}\\left(\\mathcal{L}_{\\mathcal{G}_{\\text{SN}(n)}}\\right)=\\mathcal{O}\\left(n^{2} \\log n \\log \\frac{1}{\\epsilon}\\right).\n\\end{equation*}\n\\end{corollary}\n\n\\subsubsection{$d$- Regular Ramanujan Expanders} For $d$ regular Ramanujan expanders in which $d$ does not depend on $n$, we have $\\mu_{n}\\left(\\mathcal{L}_{\\text{RExp}(d)}\\right) \\leq 2d$ and $\\mu_{2}\\left(\\mathcal{L}_{\\text{RExp}(d)}\\right) \\geq d-2\\sqrt{d-1}$. Hence, the time complexity of the SDD-solver is given by constant time. \n\n\nThe developed R-Hop distributed SDDM solver is a fundamental contribution with wide ranging applicability. Next, we develop one such application from. We apply our solver for proposing an efficient and accurate distributed Newton method for network flow optimization. Namely, the distributed SDDM solver is used for computing the Newton direction in a distributed fashion up-to any arbitrary $\\epsilon >0$. This results in a novel distributed Newton method outperforming state-of-the-art techniques in both computational complexity and accuracy.\n\n\n\n\n\n\\section{Distributed Newton Method for Network Flow Optimization}\\label{Sec:Newton}\n\n\nConventional methods for distributed network optimization are based on sub-gradient descent in either the primal\nor dual domains, see. For a large class of problems, these techniques yield iterations that can be\nimplemented in a distributed fashion using only local information. Their applicability, however, is limited by increasingly slow convergence rates. Second order Newton methods~\\cite{c4} are known to overcome this limitation leading to improved convergence rates.\n\n\nUnfortunately, computing exact Newton directions based only on local information is challenging. Specifically, to determine the Newton direction, the inverse of the dual Hessian is needed. Determining this inverse, however, requires global information. Consequently, authors in~\\cite{c5,c6,c7} proposed approximate algorithms for determining these Newton iterates in a distributed fashion. Accelerated Dual Descent (ADD)~\\cite{c5}, for instance, exploits the fact that the dual Hessian is the weighted Laplacian of the network and performs a truncated Neumann expansion of the inverse to determine a local approximate to the exact direction. ADD allows for a tradeoff between accurate Hessian approximations and communication costs through the N-Hop design, where increased N allows for more accurate inverse approximations arriving at increased cost, and lower values of N reduce accuracy but improve computational times. Though successful, the effectiveness of these approaches highly depend on the accuracy of the truncated Hessian inverse which is used to approximate the Newton direction. As shown later, the approximated iterate can resemble high variation to the\nreal Newton direction, decreasing the applicability of these techniques.\n\n\\textbf{Contributions:} Exploiting the sparsity pattern of the dual Hessian, here we tackle the above problem and propose a Newton method for network optimization that is both faster and more accurate. Using the above developed solvers for SDDM linear equations, we approximate the Newton direction up-to any arbitrary precision $\\epsilon> 0$. This leads to a distributed second-order method which performs almost identically the exact Newton method.  Contrary to current distributed Newton methods, our algorithm is the first which is capable of attaining an $\\epsilon$-close approximation to the Newton direction up to any arbitrary $\\epsilon >0$. We analyze the properties of the proposed algorithm and show that, similar to conventional Newton methods, superlinear\nconvergence within a neighborhood of the optimal value\nis attained. \n\nWe finally demonstrate the effectiveness of the approach in a set of experiments on randomly generated and Barbell networks. Namely, we show that our method is capable of significantly outperforming state-of-the-art methods in both the convergence speeds and in the accuracy of approximating the Newton direction.\n\n\n\\subsection{Network Flow Optimization}\n\nWe consider a network represented by a directed graph $\\mathcal{G}=\\left(\\mathcal{N},\\mathcal{E}\\right)$ with node set $\\mathcal{N}=\\{1,\\dots, N\\}$ and edge set $\\mathcal{E}=\\{1,\\dots, E\\}$. The flow vector is denoted by $\\bm{x}=\\left[x^{(e)}\\right]_{e\\in\\mathcal{E}}$, with $x^{(e)}$ representing the flow on edge $e$. The flow conservation conditions at nodes can be compactly represented as \n\\begin{equation*}\n\\bm{A}\\bm{x}=\\bm{b},\n\\end{equation*}\nwhere $\\bm{A}$ is the $N \\times E$ node-edge incidence matrix of $\\mathcal{G}$ defined as \n\n\\begin{displaymath}\n   \\bm{A}_{i,j} = \\left\\{\n     \\begin{array}{lr}\n       1 & \\text{if edge $j$ leaves node $i$} \\\\\n      - 1 & \\text{if edge $j$ enters node $i$} \\\\\n      0 & \\text{otherwise,}\n     \\end{array}\n   \\right.\n\\end{displaymath} \nand the vector $\\bm{b} \\in \\bm{1}^{\\perp}$ denotes the external source, i.e., $b^{(i)} > 0$ (or $b^{(i) } < 0$) indicates $b^{(i)}$ units of external flow enters (or leaves) node $i$. A cost function $\\bm{\\Phi}_{e}: \\mathbb{R} \\rightarrow \\mathbb{R}$ is associated with each edge $e$. Namely, $\\bm{\\Phi}_{e}(x^{(e)})$ denotes the cost on edge $e$ as a function of the edge flow $x^{(e)}$. We assume that the cost functions $\\bm{\\Phi}_{e}$ are strictly convex and twice differentiable. Consequently, the minimum cost network optimization problem can be written as \n\\begin{align}\n\\label{Eq:OptimAll}\n&\\min_{\\bm{x}} \\sum_{e=1}^{E} \\bm{\\Phi}_{e}(\\bm{x}^{(e)}) \\\\ \\nonumber\n&\\text{s.t. $\\bm{A}\\bm{x}=\\bm{b}$}\n\\end{align}\n\nOur goal is to investigate Newton type methods for solving the problem in~\\ref{Eq:OptimAll} in a distributed fashion. Before diving into these details, however, we next present basic ingredients needed for the remainder of the paper. \n\n\\subsection{Dual Subgradient Method} \nThe dual subgradient method optimizes the problem in Equation~\\ref{Eq:OptimAll} by descending in the dual domain. The Lagrangian, $l(\\cdot): \\mathbb{R}^{E} \\times \\mathbb{R}^{N} \\rightarrow \\mathbb{R}$, is given by \n\\begin{equation*}\nl (\\bm{x},\\bm{\\lambda}) = -\\sum_{e=1}^{E} \\bm{\\Phi}_{e}({x}^{(e)}) + \\bm{\\lambda}^{\\mathsf{T}} (\\bm{A}\\bm{x}-\\bm{b}). \n\\end{equation*}\n\nThe dual function $q(\\bm{\\lambda})$ is then derived as \n\\begin{align*}\nq(\\bm{\\lambda}) &= \\inf_{\\bm{x} \\in \\mathbb{R}^{E}} l(\\bm{x},\\bm{\\lambda}) = \\inf_{\\bm{x} \\in \\mathbb{R}^{E}} \\left(-\\sum_{e=1}^{E} \\bm{\\Phi}_{e} (x^{(e)}) + \\bm{\\lambda}^{\\mathsf{T}}\\bm{A}\\bm{x}\\right) - \\bm{\\lambda}^{\\mathsf{T}}\\bm{b} \\\\ \n& = \\sum_{e=1}^{E} \\inf_{x^{(e)} \\in \\mathbb{R}} \\left(-\\bm{\\Phi}_{e}(x^{(e)}) + \\left(\\bm{\\lambda}^{\\mathsf{T}}\\bm{A}\\right)^{(e)}x^{(e)}\\right) - \\bm{\\lambda}^{\\mathsf{T}}\\bm{b}.\n\\end{align*}\nHence, it can be clearly seen that the evaluation of the dual function $q(\\bm{\\lambda})$ decomposes into E one-dimensional optimization problems. We assume that each of these optimization problems have an optimal solution, which is unique by the strict convexity of the functions $\\bm{\\Phi}_{e}$. Denoting the solutions by $x^{(e)} (\\bm{\\lambda})$ and using the first order optimality conditions, it can be seen that for each edge, e, $x^{(e)}(\\bm{\\lambda})$ is given by\\footnote{Note that if the dual is not continuously differentiable, the a generalized Hessian can be used.}\n\\begin{equation}\n\\label{Eq:MapBack}\nx^{(e)}(\\bm{\\lambda})=[\\dot{\\bm{\\Phi}}_{e}]^{-1}\\left(\\lambda^{(i)}-\\lambda^{(j)}\\right),\n\\end{equation}\nwhere $i \\in  \\mathcal{N}$ and $j \\in \\mathcal{N}$ denote the source and destining nodes of edge $e=(i,j)$, respectively (see~\\cite{c6} for details). Therefore, for an edge $e$, the evaluation of $x^{(e)}(\\bm{\\lambda})$ can be performed based on local information about the edge's cost function and the dual variables of the incident nodes, $i$ and $j$. \n\nThe dual problem is defined as $\\min_{\\bm{\\lambda}\\in \\mathbb{R}^{N}} q(\\bm{\\lambda})$. Since the dual function is convex, the optimization problem can be solved using gradient descent according to\n\\begin{equation}\n\\label{Eq:GD}\n\\bm{\\lambda}_{k+1} = \\bm{\\lambda}_{k} - \\alpha_{k}\\bm{g}_{k} \\hfill \\ \\ \\text{for all $k \\geq 0$,}\n\\end{equation}\nwith $k$ being the iteration index, and $\\bm{g}_{k}=\\bm{g}\\left(\\bm{\\lambda}_{k}\\right)=\\nabla q(\\bm{\\lambda}_{k})$ denoting the gradient of the dual function evaluated at $\\bm{\\lambda}=\\bm{\\lambda}_{k}$. Importantly, the computation of the gradient can be performed as $\\bm{g}_{k}=\\bm{A}\\bm{x}\\left(\\bm{\\lambda}_{k}\\right)-\\bm{b}$, with $\\bm{x}(\\lambda_{k})$ being a vector composed of $x^{(e)}(\\bm{\\lambda}_{k})$ as determined by Equation~\\ref{Eq:MapBack}. Further, due to the sparsity pattern of the incidence matrix $\\bm{A}$, the $i^{th}$  element, $g_{k}^{(i)}$, of the gradient $\\bm{g}_{k}$ can be computed as \n\\begin{equation}\n\\label{Eq:GDDist}\ng_{k}^{(i)}=\\sum_{e=(i,j)} x^{(e)}(\\bm{\\lambda}_{k}) - \\sum_{e=(j,i)} x^{(e)}(\\bm{\\lambda}_{k}) - b^{(i)}.\n\\end{equation}\n\nClearly, the algorithm in Equation~\\ref{Eq:GD} can be implemented in a distributed fashion, where each node, $i$, maintains information about its dual, $\\bm{\\lambda}_{k}^{(i)}$, and primal, $x^{(e)}(\\bm{\\lambda}_{k})$, iterates of the outgoing edges $e=(i,j)$. Gradient components can then be evaluated as per~\\ref{Eq:GDDist} using only local information. Dual variables can then be updated using~\\ref{Eq:GD}. Given the updated dual variables, the primal variables can be computed using~\\ref{Eq:MapBack}.\n\nAlthough the distributed implementation avoids the cost and fragility of collecting all information at centralized location, practical applicability of gradient descent is hindered by slow convergence rates. This motivates the consideration of Newton methods discussed next. \n\n\\subsection{Newton's Method for Dual Descent}\nNewton's method is a descent algorithm along a scaled version of the gradient. Its iterates are typically given by \n\\begin{equation}\n\\bm{\\lambda}_{k+1} = \\bm{\\lambda}_{k}+\\alpha_{k}\\bm{d}_{k} \\hfill \\ \\ \\  \\text{for all $k \\geq 0$,}\n\\end{equation}\nwith $\\bm{d}_{k}$ being the Newton direction at iteration $k$, and $\\alpha_{k}$ denoting the step size. The Newton direction satisfies\n\\begin{equation}\n\\label{Eq:Hessian}\n\\bm{H}_{k}\\bm{d}_{k}=-\\bm{g}_{k},\n\\end{equation} \nwith $\\bm{H}_{k}=\\bm{H}(\\bm{\\lambda}_{k})=\\nabla^{2}q(\\bm{\\lambda}_{k})$ being the Hessian of the dual function at the current iteration $k$. \n\n\n\n\n\n\\subsubsection{Properties of the Dual and Assumptions}\nHere, we detail some assumptions needed by our approach. We also derive essential Lemmas quantifying properties of the dual Hessian. \n\\begin{assumption}\nThe graph, $\\mathcal{G}$, is connected, non-bipartite and has algebraic connectivity lower bound by a constant $\\bm{\\omega}$. \n\\end{assumption}\n\n\\begin{assumption}\\label{Ass:Two}\nThe cost functions, $\\bm{\\Phi}_{e}(\\cdot)$, in Equation~\\ref{Eq:OptimAll} are \n\\begin{enumerate}\n\\item twice continuously differentiable satisfying \n\\begin{equation*}\n\\gamma \\leq \\ddot{\\bm{\\Phi}}_{e}(\\cdot) \\leq \\Gamma,\n\\end{equation*}\nwith $\\gamma$ and $\\Gamma$ are constants; and\n\\item Lipschitz Hessian invertible for all edges $e\\in \\mathcal{E}$\n\\begin{equation*}\n\\left|\\frac{1}{\\ddot{\\bm{\\Phi}}_{e}(\\bm{x})} - \\frac{1}{\\ddot{\\bm{\\Phi}}_{e}(\\hat{\\bm{x}})}\\right| \\leq {\\delta}\\left|\\bm{x}-\\hat{\\bm{x}}\\right|.\n\\end{equation*}\n\\end{enumerate}\n\\end{assumption}\n\nThe following two lemmas~\\cite{c5,c6} quantify essential properties of the dual Hessian which we exploit through our algorithm to determine the approximate Newton direction. \n\n\\begin{lemma}\\label{lemma:Crap}\nThe dual objective $q(\\bm{\\lambda})=\\bm{\\lambda}^{\\mathsf{T}}(\\bm{A}\\bm{x}(\\bm{\\lambda})-b)-\\sum_{e}\\bm{\\Phi}_{e}(\\bm{x}(\\lambda))$ abides by the following two properties~\\cite{c5,c6}:\n\\begin{enumerate}\n\\item The dual Hessian, $\\bm{H}(\\bm{\\lambda})$, is a weighted Laplacian of $\\mathcal{G}$:\n\\begin{equation*}\n\\bm{H}(\\bm{\\lambda})=\\nabla^{2}q(\\bm{\\lambda})=\\bm{A}\\left[\\nabla^{2}f(\\bm{x}(\\bm{\\lambda}))\\right]^{-1}\\bm{A}^{\\mathsf{T}}. \n\\end{equation*}\n\\item The dual Hessian $\\bm{H}(\\bm{\\lambda})$ is Lispshitz  continuous with respect to the Laplacian norm (i.e., $||\\cdot||_{\\mathcal{L}}$) where $\\mathcal{L}$ is the unweighted laplacian satisfying $\\mathcal{L}=\\bm{A}\\bm{A}^{\\mathsf{T}}$ with $\\bm{A}$ being the incidence matrix of $\\mathcal{G}$. Namely, $\\forall \\bm{\\lambda}, \\bar{\\bm{\\lambda}}$: \n\\begin{equation*}\n||\\bm{H}(\\bar{\\bm{\\lambda}})-\\bm{H}(\\bm{\\lambda})||_{\\mathcal{L}} \\leq B ||\\bar{\\bm{\\lambda}}-\\bm{\\lambda}||_{\\mathcal{L}},\n\\end{equation*}\nwith $B=\\frac{\\mu_{n}(\\mathcal{L})\\bm{\\delta}}{\\gamma \\sqrt{\\mu_{2}(\\mathcal{L})}}$ where $\\mu_{n}(\\mathcal{L})$ and $\\mu_{2}(\\mathcal{L})$ denote the largest and second smallest eigenvalues of the Laplacian $\\mathcal{L}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\nThe following lemma follows from the above and is needed in the analysis later: \n\\begin{lemma}\\label{lemma:B}\nIf the dual Hessian $\\bm{H}(\\bm{\\lambda})$ is Lipschitz continuous with respect to the Laplacian norm $||\\cdot||_{\\mathcal{L}}$ (i.e., Lemma~\\ref{lemma:Crap}), then for any $\\bm{\\lambda}$ and $\\hat{\\bm{\\lambda}}$ we have \n\\begin{equation*}\n||\\nabla q(\\hat{\\bm{\\lambda}})-\\nabla q({\\bm{\\lambda}}) - \\bm{H}(\\bm{\\lambda})(\\hat{\\bm{\\lambda}}-\\bm{\\lambda})||_{\\mathcal{L}} \\leq \\frac{B}{2}||\\hat{\\bm{\\lambda}} - \\bm{\\lambda}||_{\\mathcal{L}}^{2}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\n\n\nAs detailed in~\\cite{c6}, the exact computation of the inverse of the Hessian needed for determining the Newton direction can not be attained exactly in a distributed fashion. Authors in~\\cite{c5,c6} proposed approximation techniques for computing this direction. The effectiveness of these algorithms, however, highly depends on the accuracy of such an approximation. In this work, we propose a distributed approximator for the Newton direction capable of acquiring $\\epsilon$-close solutions for any arbitrary $\\epsilon$. Our results show that this new algorithm is capable of significantly surpassing others in literature where its performance accurately traces that of the standard centralized Newton approach. \n\\subsection{Accurate Distributed Newton Methods}\nUsing the results of the distributed R-Hop solver, we propose a novel technique requiring only R-Hop communication for the distributed approximation of the Newton direction. Given the results of Lemma~\\ref{lemma:Crap}, we can determine the approximate Newton direction by solving a system of linear equations represented by an SDD matrix\\footnote{For ease of presentation, we refrain some of the proofs to the appendix.} with $\\bm{M}_{0}=\\bm{H}_{k}=\\bm{H}(\\bm{\\lambda}_{k})$.  \n\nFormally, we consider the following iteration scheme:\n\\begin{equation}\n\\bm{\\lambda}_{k+1}=\\bm{\\lambda}_{k} + \\alpha_{k}\\tilde{\\bm{d}}_{k},\n\\end{equation}\nwith $k$ representing the iteration number, $\\alpha_{k}$ the step-size, and $\\tilde{\\bm{d}}_{k}$ denoting the approximate Newton direction. We determine $\\tilde{\\bm{d}}_{k}$ by solving $\\bm{H}_{k}\\bm{d}_{k}=-\\bm{g}_{k}$ using Algorithm~\\ref{Algo:EDistR}. It is easy to see that our approximation of the Newton direction, $\\tilde{\\bm{d}}_{k}$, satisfies\n\\begin{align*}\n||\\tilde{\\bm{d}}_{k} - \\bm{d}_{k}||_{\\bm{H}_{k}} &\\leq \\epsilon ||\\bm{d}_{k}||_{\\bm{H}_{k}} \\  \\ \\ \\text{with} \\ \\ \\ \\tilde{\\bm{d}}_{k} =-\\bm{Z}_{k}\\bm{g}_{k},\n\\end{align*}\nwhere $\\bm{Z}_{k}$ approximates $\\bm{H}^{\\dagger}_{k}$ according to the routine of Algorithm~\\ref{Algo:EDistR}. The accuracy of this approximation is quantified in the following Lemma \n\n\\begin{lemma}\\label{Lemma:Bla}\nLet $\\bm{H}_{k}=\\bm{H}(\\bm{\\lambda}_{k})$ be the Hessian of the dual function, then for any arbitrary $\\epsilon > 0$ we have  \n\\begin{equation*}\ne^{-\\epsilon^{2}} \\bm{v}^{\\mathsf{T}} \\bm{H}_{k}^{\\dagger} \\bm{v} \\leq \\bm{v}^{\\mathsf{T}}\\bm{Z}_{k}\\bm{v}\\leq e^{\\epsilon^{2}}\\bm{v}^{\\mathsf{T}}\\bm{H}_{k}^{\\dagger}\\bm{v}, \\ \\ \\ \\ \\ \\ \\forall \\bm{v} \\in \\bm{1}^{\\perp}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\n\\subsubsection{Convergence Guarantees}\nGiven such an accurate approximation, next we analyze the iteration scheme of our proposed method showing that similar to standard Newton methods, we achieve superlinear convergence within a neighborhood of the optimal value. We start by analyzing the change in the Laplacian norm of the gradient between two successive iterations \n\\begin{lemma}\nConsider the following iteration scheme $\\bm{\\lambda}_{k+1}=\\bm{\\lambda}_{k} + \\alpha_{k}\\tilde{\\bm{d}}_{k}$ with $\\alpha_{k} \\in (0,1]$, then, for any arbitrary $\\epsilon>0$, the Laplacian norm of the gradient, $||\\bm{g}_{k+1}||_{\\mathcal{L}}$, follows:\n\\begin{align}\\label{Eq:NormG}\n||\\bm{g}_{k+1}||_{\\mathcal{L}} &\\leq \\left[1-\\alpha_{k} +\\alpha_{k}\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]||\\bm{g}_{k}||_{\\mathcal{L}}  + \\frac{\\alpha_{k}^{2}B\\Gamma^{2}(1+\\epsilon)^{2}}{2\\mu^{2}_{2}(\\mathcal{L})}||\\bm{g}_{k}||_{\\mathcal{L}}^{2},\n\\end{align}\nwith $\\mu_{n}(\\mathcal{L})$ and $\\mu_{2}(\\mathcal{L})$ being the largest and second smallest eigenvalues of $\\mathcal{L}$, $\\Gamma$ and $\\gamma$ denoting the upper and lower bounds on the dual's Hessian, and $B \\in \\mathbb{R}$ is defined in Lemma~\\ref{lemma:B}. \n\\end{lemma}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\n\nAt this stage, we are ready to present the main results quantifying the convergence phases exhibited by our approach:\n\\begin{theorem}\nLet $\\gamma$, $\\Gamma$, $B$ be the constants defined in Assumption~\\ref{Ass:Two} and Lemma~\\ref{lemma:Crap}, $\\mu_{n}(\\mathcal{L})$ and $\\mu_{2}(\\mathcal{L})$ representing the largest and second smallest eigenvalues of the normalized laplacian $\\mathcal{L}$, $\\epsilon \\in \\left(0, \\frac{\\mu_{2}(\\mathcal{L}}{\\mu_{n}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right)$ the precision parameter for the SDDM solver, and letting the optimal step-size parameter $\\alpha^{*}=\\frac{e^{-\\epsilon^{2}}}{(1+\\epsilon)^{2}}\\left(\\frac{\\gamma}{\\Gamma}\\frac{\\mu_{2}(\\mathcal{L})}{\\mu_{n}(\\mathcal{L})}\\right)^{2}$. Then the proposed algorithm given by the $\\bm{\\lambda}_{k+1}=\\bm{\\lambda}_{k}+\\alpha^{*}\\tilde{\\bm{d}}_{k}$ exhibits the following three phases of convergence:\n\\begin{enumerate}\n\\item \\textbf{Strict Decreases Phase:} While $||\\bm{g}_{k}||_{\\mathcal{L}} \\geq \\eta_{1}$:\n\\begin{equation*}\nq(\\bm{\\lambda}_{k+1})-q(\\bm{\\lambda}_{k}) \\leq -\\frac{1}{2} \\frac{e^{-2\\epsilon^{2}}}{(1+\\epsilon)^{2}}\\frac{\\gamma^{3}}{\\Gamma^{2}}\\frac{\\mu_{2}^{2}(\\mathcal{L})}{\\mu_{n}^{4}(\\mathcal{L})}\\eta_{1}^{2}.\n\\end{equation*} \n\\item \\textbf{Quadratic Decrease Phase:} While $\\eta_{0} \\leq ||\\bm{g}_{k}||_{\\mathcal{L}}\\leq \\eta_{1}$:\n\\begin{equation*}\n||\\bm{g}_{k+1}||_{\\mathcal{L}} \\leq \\frac{1}{\\eta_{1}}||\\bm{g}_{k}||_{\\mathcal{L}}^{2}.\n\\end{equation*}\n\\item \\textbf{Terminal Phase:} When $||\\bm{g}_{k}||_{\\mathcal{L}} \\leq \\eta_{0}$: \n\\begin{equation*}\n||\\bm{g}_{k+1}||_{\\mathcal{L}}\\leq \\sqrt{\\left[1-\\alpha^{*}+\\alpha^{*}\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]}||\\bm{g}_{k}||_{\\mathcal{L}},\n\\end{equation*}\n\\end{enumerate}\nwhere $\\eta_{0}=\\frac{\\bm{\\xi}(1-\\bm{\\xi})}{\\bm{\\zeta}}$ and $\\eta_{1}=\\frac{1-\\bm{\\xi}}{\\bm{\\zeta}}$, with \n\\begin{align}\\label{Eq:Aux}\n\\bm{\\xi}&=\\sqrt{\\left[1-\\alpha^*+\\alpha^*\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]} \\ \\ \\ \\text{with} \\ \\ \\ \\bm{\\zeta}=\\frac{B(\\alpha^{*}\\Gamma(1+\\epsilon))^{2}}{2\\mu_{2}^{2}(\\mathcal{L})}\n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nWe will proof the above theorem by handling each of the cases separately. We start by considering the case when $||\\bm{g}_{k}||_{\\mathcal{L}} > \\eta_{1}$ (i.e., \\textbf{Strict Decrease Phase}). We have:\n\\begin{align*}\nq(\\lambda_{k+1}) & = q(\\bm{\\lambda}_{k}) + \\bm{g}_{k}^{\\mathsf{T}}(\\bm{\\lambda}_{k+1}-\\bm{\\lambda}_{k}) +\\frac{1}{2}(\\bm{\\lambda}_{k+1}-\\bm{\\lambda}_{k})^{\\mathsf{T}}\\bm{H}(\\bm{z})(\\bm{\\lambda}_{k+1}-\\bm{\\lambda}_{k}) \\\\ \n& = q(\\bm{\\lambda}_{k}) + \\alpha_{k} \\bm{g}_{k}^{\\mathsf{T}}\\tilde{\\bm{d}}_{k} +\\frac{\\bm{\\alpha}_{k}^{2}}{2} \\tilde{\\bm{d}}_{k}^{\\mathsf{T}}\\bm{H}(\\bm{z})\\tilde{\\bm{d}}_{k} \\leq q(\\bm{\\lambda}_{k}) + \\alpha_{k}\\bm{g}_{k}^{\\mathsf{T}}\\tilde{\\bm{d}}_{k}+\\frac{\\alpha_{k}^{2}}{2\\gamma}\\tilde{\\bm{d}}^{\\mathsf{T}}_{k} \\mathcal{L}\\tilde{\\bm{d}}_{k},\n\\end{align*} \nwhere the last steps holds since $\\bm{H}(\\cdot) \\preceq \\frac{1}{\\gamma}\\mathcal{L}$. Noticing that $||\\tilde{\\bm{d}}_{k}||_{\\mathcal{L}}^{2} \\leq \\frac{\\Gamma^{2}(1+\\epsilon)^{2}}{\\mu_{2}^{2}(\\mathcal{L})}||\\bm{g}_{k}||_{\\mathcal{L}}^{2}$ (see Appendix), the only remaining step needed is to evaluate $\\bm{g}_{k}^{\\mathsf{T}}\\tilde{\\bm{d}}_{k}$. Knowing that $\\tilde{\\bm{d}}_{k} = -\\bm{Z}_{k}\\bm{g}_{k}$, we recognize\n\\begin{align*}\n\\bm{g}_{k}^{\\mathsf{T}}\\tilde{\\bm{d}}_{k} &= - \\bm{g}_{k}^{\\mathsf{T}}\\bm{Z}_{k}\\bm{g}_{k} \\leq e^{-\\epsilon^{2}}\\bm{g}_{k}^{\\mathsf{T}}\\bm{H}_{k}^{\\dagger}\\bm{g}_{k} \\ (\\text{Lemma~\\ref{Lemma:Bla}}) \\\\\n&\\leq -\\frac{e^{-\\epsilon^{2}}}{\\mu_{n}(\\bm{H}_{k})}\\bm{g}_{k}^{\\mathsf{T}}\\bm{g}_{k} \\leq -\\frac{e^{-\\epsilon}}{\\mu_{n}(\\mathcal{L})}\\bm{g}_{k}^{\\mathsf{T}}\\bm{g}_{k} \\leq -\\frac{e^{-\\epsilon^{2}}\\gamma}{\\mu_{n}(\\mathcal{L})}\\frac{\\bm{g}_{k}^{\\mathsf{T}}\\mathcal{L}\\bm{g}_{k}}{\\mu_{n}(\\mathcal{L})} = \\frac{e^{-\\epsilon^{2}}\\gamma}{\\mu_{n}^{2}(\\mathcal{L})}||\\bm{g}_{k}||_{\\mathcal{L}}^{2},\n\\end{align*}\nwhere the last step follows from the fact that $\\forall \\bm{v}\\in \\mathbb{R}^{n}: \\bm{v}^{\\mathsf{T}}\\bm{v} \\geq \\frac{\\bm{v}^{\\mathsf{T}}\\mathcal{L}\\bm{v}}{\\mu_{n}(\\mathcal{L})}$. Therefore, we can write\n\\begin{equation*}\nq(\\bm{\\lambda}_{k+1})-q(\\bm{\\lambda}_{k}) \\leq -\\left[\\alpha_{k}\\frac{e^{-\\epsilon^{2}}\\gamma}{\\mu_{n}^{2}(\\mathcal{L})}-\\alpha_{k}^{2}\\frac{\\Gamma^{2}(1+\\epsilon)^{2}}{2\\gamma\\mu_{2}^{2}(\\mathcal{L})}\\right]||\\bm{g}_{k}||_{\\mathcal{L}}^{2}.\n\\end{equation*}\nIt is easy to see that $\\alpha_{k}=\\alpha^{*}=\\frac{e^{-\\epsilon^{2}}}{(1+\\epsilon)^{2}}\\left(\\frac{\\gamma}{\\Gamma}\\frac{\\mu_{2}(\\mathcal{L})}{\\mu_{n}(\\mathcal{L})}\\right)^{2}$ minimizes the right-hand-side of the above equation. Using $||\\bm{g}_{k}||_{\\mathcal{L}}$ gives the constant decrement in the dual function between two successive iterations as \n\\begin{equation*}\nq(\\bm{\\lambda}_{k+1})-q(\\bm{\\lambda}_{k}) \\leq -\\frac{1}{2} \\frac{e^{-2\\epsilon^{2}}}{(1+\\epsilon)^{2}}\\frac{\\gamma^{3}}{\\Gamma^{2}}\\frac{\\mu_{2}^{2}(\\mathcal{L})}{\\mu_{n}^{4}(\\mathcal{L})}\\eta_{1}^{2}.\n\\end{equation*}\nConsidering the case when $\\eta_{0} \\leq ||\\bm{g}_{k}||_{\\mathcal{L}}^{2}\\eta_{1}$ (i.e., \\textbf{Quadratic Decrease Phase}), Equation~\\ref{Eq:NormG} can be rewritten as\n\\begin{equation*}\n||\\bm{g}_{k+1}||_{\\mathcal{L}}\\leq \\bm{\\xi}^{2}||\\bm{g}_{k}||_{\\mathcal{L}} + \\bm{\\zeta}||\\bm{g}_{k}||_{\\mathcal{L}}^{2},\n\\end{equation*}\nwith $\\bm{\\xi}$ and $\\bm{\\zeta}$ defined as in Equation~\\ref{Eq:Aux}. Further, noticing that since $||\\bm{g}_{k}||_{\\mathcal{L}} \\geq \\eta_{0}$ then $||\\bm{g}_{k}||_{\\mathcal{L}}\\leq \\frac{1}{\\eta_{0}}||\\bm{g}_{k}||_{\\mathcal{L}}^{2}=\\frac{\\bm{\\zeta}}{\\bm{\\xi}(1-\\bm{\\xi})}||\\bm{g}_{k}||_{\\mathcal{L}}^{2}$. Consequently the quadratic decrease phase is finalized by\n\\begin{align*}\n||\\bm{g}_{k+1}||_{\\mathcal{L}} \\leq \\bm{\\zeta}\\left(\\frac{\\bm{\\xi}}{1-\\bm{\\xi}}+1\\right)||\\bm{g}_{k}||_{\\mathcal{L}}^{2}&=\\frac{\\bm{\\zeta}}{1-\\bm{\\xi}}||\\bm{g}_{k}||_{\\mathcal{L}}^{2} =\\frac{1}{\\eta_{1}}||\\bm{g}_{k}||_{\\mathcal{L}}^{2}.\n\\end{align*}\nFinally, we handle the case where $||\\bm{g}_{k}||_{\\mathcal{L}}\\leq \\eta_0$ (i.e., \\textbf{Terminal Phase}). Since $||\\bm{g}_{k}||_{\\mathcal{L}}^{2} \\leq \\eta_{0}||\\bm{g}_{k}||_{\\mathcal{L}}$, it is easy to see that \n\\begin{align*}\n||\\bm{g}_{k+1}||_{\\mathcal{L}} &\\leq (\\bm{\\xi}^{2}+\\bm{\\zeta}\\eta_{0})||\\bm{g}_{k}||_{\\mathcal{L}}=(\\bm{\\xi}^{2}+\\bm{\\xi}(1-\\bm{\\xi})) ||\\bm{g}_{k}||_{\\mathcal{L}} \\\\\n& =\\bm{\\xi}||\\bm{g}_{k}||_{\\mathcal{L}} = \\sqrt{\\left[1-\\alpha^*+\\alpha^*\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]}||\\bm{g}_{k}||_{\\mathcal{L}}.\n\\end{align*}\n\\end{proof}\n\n\\subsubsection{Iteration Count and Message Complexity}\nHaving proved the three convergence phases of our algorithm, we next analyze the number of iterations needed by each phase. These results are summarized in the following lemma: \n\n\\begin{lemma}\\label{Lemma:Iter}\nConsider the algorithm given by the following iteration protocol: $\\bm{\\lambda}_{k+1}=\\bm{\\lambda}_{k}+\\alpha^{*}\\tilde{\\bm{d}}_{k}$. Let $\\bm{\\lambda}_{0}$ be the initial value of the dual variable, and $q^{*}$ be the optimal value of the dual function. Then, the number of iterations needed by each of the three phases satisfy: \n\\begin{enumerate}\n\\item The {\\textbf{strict decrease phase}} requires the following number of iterations to achieve the quadratic phase: \n\\begin{equation*}\nN_{1} \\leq C_{1} \\frac{\\mu_{n}(\\mathcal{L})^{2}}{\\mu_{2}^{3}(\\mathcal{L})} \\left[1-\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]^{-2},\n\\end{equation*}\nwhere $C_{1}=C_{1}\\left(\\epsilon, \\gamma, \\Gamma, \\bm{\\delta}, q(\\bm{\\lambda}_{0}), q^{\\star} \\right)=2\\bm{\\delta}^{2}(1+\\epsilon)^{2}\\left[q(\\bm{\\lambda}_{0})-q^{\\star}\\right]\\frac{\\Gamma^{2}}{\\gamma}$.\n\\item The \\textbf{quadratic decrease phase} requires the following number of iterations to terminate: \n\\begin{equation*}\nN_{2} = \\log_{2}\\left[\\frac{\\frac{1}{2}\\log_{2}\\left(\\left[1-\\alpha^{*}\\left(1-\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right)\\right]\\right)}{\\log_{2}(r)}\\right],\n\\end{equation*}\nwhere $r=\\frac{1}{\\eta_{1}}||\\bm{g}_{k^{\\prime}}||_{\\mathcal{L}}$, with $k^{\\prime}$ being the first iteration of the quadratic decrease phase. \n\\item The radius of the {\\textbf{terminal phase}} is characterized by:\n\\begin{equation*}\n\\rho_{\\text{terminal}} \\leq \\frac{2\\left[1-\\epsilon\\frac{\\mu_{n}(\\mathcal{L})}{\\mu_{2}(\\mathcal{L})}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]}{e^{-\\epsilon^{2}\\gamma\\bm{\\delta}}}\\mu_{n}(\\mathcal{L})\\sqrt{\\mu_{2}(\\mathcal{L})}.\n\\end{equation*}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nSee Appendix. \n\\end{proof}\n\nGiven the above result, the total message complexity can then be derived as:\n\\begin{equation*}\n\\mathcal{O}\\left(\\left(N_{1}+N_{2}\\right)n\\beta\\left(\\kappa(\\bm{H}_{k})\\frac{1}{R}+R\\text{d}_{\\max}\\right)\\log\\left(\\frac{1}{\\epsilon}\\right)\\right).\n\\end{equation*} \n\n\\subsubsection{Comparison to Existing Literature}\nRecent progress towards distributed second-order methods applied to network flow optimization adopt an approximation scheme based on the properties of the Hessian. Most of these methods (e.g.,~\\cite{c7}) handle the simpler consensus setting and thus are not directly applicable to our more general setting. Closest to this work is that proposed in~\\cite{c5}, where the authors approximate the Newton direction by truncating the Neumann expansion of the pseudo-inverse of the Hessian. This truncation, however, introduces additional error to the computation of the Newton direction leading to inaccurate results especially on large networks. The primary contrast to this work is that our method is capable of acquiring $\\epsilon$-close (for any arbitrary $\\epsilon$) approximation to the Newton direction. In fact, the proposed method is almost identical to the exact Newton direction computed in a centralized manner (see Section~\\ref{Sec:ExpNewton}). \n\nNext, we formally derive the iteration counts for our method on four-special cases as benchmark comparisons. Since the main contribution (due to the presence of $\\log \\log (\\cdot)$ term in $N_{2}$) in the iteration count is given by the strict decrease phase (see Lemma~\\ref{Lemma:Iter}), we develop these results in terms of $N_{1}$. Also note that the corollaries provided below are substantially harder to acquire when compared to the standard Newton analysis due to the dependence of some constants, e.g, $m$ and $M$ on the graph structure. Opposed to the analysis performed by other methods, our analysis explicitly handles such dependencies leading to more realistic and accurate mathematical insights. This explains the relatively high dependency on $n$ in some cases. Note, that in such case where these relations (i.e., parameter dependency on the graph structure) were not taken into account, substantial decrease in the complexity can be achieved at the compensate of accurate mathematical description.  \n\n\\textbf{Path Graph:} \n\\begin{corollary}\nGiven a path graph $\\mathcal{P}_{n}$ with $n$ nodes, the strict-decrease phase of the distributed Newton method is given by:\n\\begin{equation*}\nN_{1}\\left(\\mathcal{P}_{n}\\right)=\\mathcal{O}\\left(n^{6}\\left[1-\\epsilon\\frac{4n^{2}}{\\pi^{2}}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]^{-2}\\right)\n\\end{equation*}\n\\end{corollary}\n\n\\textbf{Grid Graph:} \n\\begin{corollary}\nFor a grid graph $\\mathcal{G}_{k \\times m}$ the strict-decrease phase of the distributed Newton method is given by:\n\\begin{equation*}\nN_{1}\\left(\\mathcal{G}_{k \\times m}\\right)=\\mathcal{O}\\left(n^{3}\\left[1-\\epsilon\\frac{8n^{2}}{\\pi^{2}\\left(k^{2}+\\frac{n^{2}}{k^{2}}\\right)}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]^{-2}\\right)\n\\end{equation*}\n\\end{corollary}\n\n\\textbf{Scale-Free Graph:} \n\\begin{corollary}\nFor a scale free grid graph $\\mathcal{G}_{\\text{SN}(n)}$ with $n$, the strict-decrease phase is given by:\n\\begin{equation*}\nN_{1}\\left(\\mathcal{G}_{\\text{SN}(n)}\\right)=\\mathcal{O}\\left(n^{4}\\log n\\left[1-\\epsilon\\frac{n^{\\frac{3}{2}}\\log n}{4}\\sqrt{\\frac{\\Gamma}{\\gamma}}\\right]^{-2}\\right)\n\\end{equation*}\n\\end{corollary}\n\n\\textbf{d-Regular Ramanujan Expanders:} For such expanders the iteration count for the strict-decrease phase can be bounded by a constant. \n\n\\subsection{Experiments and Results}\\label{Sec:ExpNewton}\nWe evaluated the proposed distributed second-order method in three sets of experiments on randomly generated and Barbell networks. The goal was to assess the performance on networks exhibiting good and bad mixing times. We compared our algorithm's performance to: 1) exact-newton computed in a centralized fashion, 2) Accelerated Dual Descent (ADD) with two different splittings~\\cite{c5,c6}, 3) dual sub-gradients, and 4) the fully distributed algorithms for convex optimization~\\cite{c41} (FDA). An $\\epsilon$ of $\\frac{1}{10,000}$, a feasibility threshold of $10^{-5}$, and an R-Hop of 1 were provided to our SDDM solver for determining the approximate Newton direction. For all other methods free parameters were chosen as specified by the relevant papers. Feasibility and objective values were used as performance measures. \n\n\n\\subsubsection{Experiments \\& Results on Random Graphs} \nTwo experiments on small (20 nodes, 60 edges) and large (50 nodes, 150 edges) random graphs were conducted. The random graphs were constructed in such a way that  edges were drawn uniformly at random. The flow vectors, $\\bm{b}$, were chosen to place source and sink nodes $\\text{diam}(\\mathcal{G})$ apart.  \n\\begin{figure*}[t!]\n\\centering\n\\vspace{-.5em}\n\\subfigure[]{\n\t\\label{fig:PerfSM}\n\\includegraphics[width=0.49\\textwidth]{feas2060}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{fig:PerfCP}\n\\includegraphics[width=0.5\\textwidth]{obj2060}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{fig:TrajSM}\n\\includegraphics[width=0.49\\textwidth]{feas_50_150}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{fig:TrajCP}\n\\includegraphics[width=0.5\\textwidth]{Obj50_150}\n}\n\n\\caption{Performance metrics on two randomly generated networks, showing the primal objective, $f\\left(\\bm{x}_{k}\\right)$, and feasibility $||\\bm{A}\\bm{x}_{k}-\\bm{b}||$ as a function of the number of iterations $k$ on a loglog scale. On a relatively small network (i.e., $20$ nodes and $60$ edges- Figures (a) and (b)) we outperform all method by an order of magnitude and trace the trajectory of exact Newton computed in a centralized fashion. On larger networks (i.e., 50 nodes and 150 edges-Figures (c) and (d)), SDDM-Newton is superior to all other algorithms, where it converges 5 orders of magnitude faster.}\n\\label{Fig:ResBenchmark}\n\\end{figure*}\n\n\n\nResults summarizing the primal objective value, $f([\\bm{x}]^{(e)})=\\exp([\\bm{x}]^{(e)})+\\exp(-[\\bm{x}]^{(e)})$, and feasibility, $||\\bm{A}\\bm{x}_{k} - \\bm{b}||$, on both networks are shown in Figure~3. On small networks (i.e., 20 nodes and 60 edges), all algorithms perform relatively well. Clearly, our proposed approach (titled SDDM-Newton in the figures) outperforms, ADD, Sub-gradients, and the approach in~\\cite{c41} with about an order of magnitude. Another interesting realization is that SDDM-Newton accurately tracks the exact Newton method with its direction computed in a centralized fashion. The reason for such positive results, is that our algorithm is capable of approximating the Newton direction up-to-any arbitrary $\\epsilon >0$ while abiding by the R-Hop constraint. For larger networks, Figures~\\ref{fig:TrajSM} and~\\ref{fig:TrajCP}, SDDM-Newton is highly superior compared to other approaches. Here, our algorithm is capable of converging in about 3-5 orders of magnitude faster, i.e., we converge in about 3000 iterations as opposed to 6000 for ADD which showed the best performance among the other techniques. It is worth noting that we are again capable of tracing exact Newton due to the accuracy of our approximation. \n\n\n\\subsubsection{Experiments \\& Results on Bar Bell Graphs} \nIn the second set of experiments we assessed the performance of SDDM-Newton on a barbell graph, Figure~\\ref{Fig:BarBellConstructions}, of 60 nodes. The graph consisted of two 20 node cliques connected by a 20 nodes line graph is depicted in. We assign directed edges on the graph arbitrary and solve the network flow optimization problem with a cost of $f([\\bm{x}]^{(e)})=\\exp([\\bm{x}]^{(e)})+\\exp(-[\\bm{x}]^{(e)})$. \n\n\\begin{wrapfigure}{l}{0.5\\textwidth}\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{BarBell}\n  \\end{center}\n  \\caption{A high-level depiction of a barbell graph showing two cliques connected by a line graph.}\n  \\label{Fig:BarBellConstructions}\n\\end{wrapfigure}\n\nResults in Figure~\\ref{fig:PerfSMTw} and~\\ref{fig:PerfCPTw} demonstrate the superiority of our algorithm to state-of-the-art methods. Here, again we are capable of converting faster than other techniques in about 2 magnitudes faster. It is worth noting that the second-best performing algorithm is ADD which is capable of converging in almost 3000 iterations. \n\n\\begin{figure*}[t!]\n\\centering\n\\vspace{-.5em}\n\\subfigure[]{\n\t\\label{fig:PerfSMTw}\n\\includegraphics[width=0.5\\textwidth]{feasBarBell60}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{fig:PerfCPTw}\n\\includegraphics[width=0.49\\textwidth]{objBarbell60}\n}\n\\caption{Feasibility and objective results on a 60 node bar graph. Again in these sets of experiments, our method is capable of outperforming all other methods. Furthermore, SDDM-Newton is again capable of tracing the exact Newton method.}\n\\end{figure*}\nFinally, we repeated the same experiments on a larger bar bell network formed of 120 nodes (two cliques with 40 nodes connected by a 40 node line graph). These results, demonstrated in Figures~\\ref{fig:PerfSMTwo} and~\\ref{fig:PerfCPTwo}, again validate the previous performance measures showing even better performance in both the objective value and feasibility. \n\\begin{figure*}[t!]\n\\centering\n\\vspace{-.5em}\n\\subfigure[]{\n\t\\label{fig:PerfSMTwo}\n\\includegraphics[width=0.5\\textwidth]{BarBellFeasBig}\n}\n\\hfill\\hspace{-.3in}\\hfill\n\\subfigure[]{\n\t\\label{fig:PerfCPTwo}\n\\includegraphics[width=0.49\\textwidth]{BarBellLargeObjective}\n}\n\\caption{Feasibility and objective results on a 120 node bar graph. Again in these sets of experiments, our method is capable of outperforming all other methods. Furthermore, SDDM-Newton is again capable of tracing the exact Newton method.}\n\\end{figure*}\n\n\\subsubsection{Measuring Message Complexity}\nThough successful, the experiments performed in the previous section show accuracy improvements without demonstrating per-iteration message complexities needed. To have a fair comparison to state-of-the-art methods, in this section we report such results on four different network topologies. Here, we show that our method requires a relatively slight increase in message complexity to trace the exact Newton direction. \n\n\nThese per-iteration values were determined by deriving bounds to each of the benchmark algorithms. For all methods except SDDM-Newton and that in~\\cite{c5}, such complexity can be bounded by $\\mathcal{O}(d_{\\text{max}})$ with $d_{\\text{max}}$ being the maximal degree. As for SDDM-Newton, the per-iteration complexity is upper-bounded by $\\mathcal{O}(\\bm{\\kappa}(\\mathcal{L}_{\\mathcal{G}}) d_{\\text{max}})$, with $\\bm{\\kappa}(\\mathcal{L}_{\\mathcal{G}})$ being the condition number of the graph Laplacian. For the algorithm in~\\cite{c5} the message complexity satisfies $\\mathcal{O}(\\bm{\\kappa}(\\mathcal{L}_{\\mathcal{G}}) d_{\\text{max}} \\log n)$ with $n$ being the total number of nodes in $\\mathcal{G}$. Immediately, we recognize that our algorithm is faster by a factor of $\\log n$ compared to that in~\\cite{c5}. Compared to other techniques, however, our method is slower by a factor of $\\bm{\\kappa}(\\mathcal{L}_{\\mathcal{G}})$. \n\\begin{wrapfigure}{l}{0.6\\textwidth}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth, height=2.1in]{MessageComplexityBla}\n\\end{center}\n\\caption{Message complexity results comparing our method to state-of-the-art techniques on four different network topologies. Graphs 1 and 2 correspond to random networks with 20 nodes, 60 edge and 100 nodes and 500 edge, respectively. Graphs 3 and 4 report message complexities needed for two barbell graphs. }\n\\label{Fig:barbell}\n\\end{wrapfigure}\n\nTo better quantify such a difference, we perform four sets of experiments on two randomly generated and two barbell networks with varying size. The random networks consisted of 20 nodes, 60 edges, and 120 nodes and 500 edges, respectively. Moreover, the barbell graphs followed the same construction of the previous section with sizes varying from 60 to 120 nodes. Comparison results showing the logarithm of the message complexity are reported in the bar graph of Figure~\\ref{Fig:barbell}. These demonstrate that SDDM-Newton requires a slight increase in the message complexity to trace the exact Newton direction. \n\n\\section{Conclusion}\n\nIn this paper we proposed a distributed solver for linear systems described by SDDM matrices. Our approach distributes that in~\\cite{c11} by proposing the usage of an inverse approximated chain which can be computed in a distributed fashion. Precisely, two solvers were proposed. The first required full communication in the network, while the second restricts communication to the R-Hop neighborhood between the nodes. \n\nWe applied our solver to network flow optimization. This resulted in an efficient and accurate distributed second-order method capable of tracing exact Newton computed in a centralized fashion. We showed that similar to standard Newton, our methods are capable of achieving superlinear convergence in a neighborhood of the optimal solution. We extensively evaluated the proposed method on both randomly generated and barbell graphs. Results demonstrate that our method outperforms state-of-the-art techniques. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA Voronoi summation formula of an automorphic form is a \nPoisson summation formula weighted by Fourier coefficients of the automorphic form with twists by some additive arithmetic functions. \nThe Voronoi formula of an automorphic form on GL(2) is well known and has a lot of application.\n\n\n\nA Voronoi formula of an automorphic form on GL(3) was firstly proved  by Miller and Schmid in \\cite{millerschmid1}. Later, a Voronoi formula was established for GL($n$) for $n\\geq 4$ in \\cite{millerschmid2}, \\cite{goldfeldli1}, and \\cite{goldfeldli2}. \nAn adelic version was established in \\cite{ichinotemplier}.\nThe Voronoi formula has important application, such as a nontrivial bound on the exponential sum in  \\cite{miller} and subconvexity bounds in  \\cite{lisubconvexity}\n\n\nMore recently, Li and Miller established a different type of Voronoi formula on GL(4)\\footnote{Their result was announced at\n\\textit{Representation Theory, Automorphic Forms, and Complex Geometry, a conference in honor of the 70th birthday of Wilfried Schmid\n} in May 2013. The slides of Li's talk are available in the link: \\url{http:\/\/www.math.harvard.edu\/conferences\/schmid_2013\/docs\/li.pdf}}. Their formula is an equality between a weighted sum of Fourier coefficients of an automorphic form on GL(4) \nand a dual sum, both twisted by classical Kloosterman sums. This formula is different from those in \\cite{millerschmid2}, \\cite{goldfeldli1}, \\cite{goldfeldli2}, and \\cite{ichinotemplier}\nand inspires us to develop this paper.\n\n\nIn this paper, we discover more Voronoi formulae on GL($n$) when $n\\geq 4$. One of these new formulae overlaps the formula of Li and Miller. Our formula on GL($n$) is an equality between a sum of Fourier coefficients of an automorphic form twisted by hyper-Kloosterman sums of dimension $(k-1)$ and a dual sum twisted by hyper-Kloosterman sums of dimension $(n-k-1)$. The Voronoi formula of \\cite{millerschmid1}, \\cite{millerschmid2}, \\cite{goldfeldli1}, and \\cite{goldfeldli2} are the case of $k=1$. Li and Miller's Voronoi formula on GL(4) is the case of $n=4$ and $k=2$.\nFor an automorphic form on GL($n$), there are $[n\/2]$ different Voronoi formulae, which correspond to $k=1,\\dots,[n\/2]$ respectively. \n\n\nFor an integer $k\\geq 1$ define the $(k-1)$-dimensional hyper-Kloosterman sum by\n$$Kl_k(m,q)=\\sum_{\\underset{\\scriptstyle (x_1,q)=\\dots=(x_{k-1},q)=1}{x_1, \\dots, x_{k-1}\\mod q}} e\\bra{\\frac{x_1+\\dots+x_{k-1}+m\\overline{x_1\\dots x_{k-1}}}{q}}$$ for a prime number $q$.\nWe state our formulae in the following.\n\n\n\n\n\n\n\n\\begin{theorem}\\label{main}\nLet $\\pi$ be an even Maass cusp form for $\\ssl{n}{Z}$ \nof the spectral parameter $(\\lambda_1,\\dots,\\lambda_n)\\in\\mathbb{C}^n$ and $n\\geq 2$. Let $A(m_1,\\dots,m_{n-1})$ be the Fourier coefficient of $\\pi$. Let \n$\\omega\\in C_c^\\infty(0,\\infty)$ be a test function with $\\tilde{\\omega}$ its Mellin transform\nand let $q$ be a prime number, $a$ and $\\overline{a}$ two integers with $a\\overline{a}\\equiv 1\\mod q$. \nFor $1\\leq k \\leq n-1,$ we have \n\\begin{align*}\n&\\sum_{m=1}^\\infty A(1,\\dots,1,m)Kl_k(am,q)\\omega(m)\\\\\n&\\quad+\\sum_{2\\leq l\\leq k}\\sum_{m=1}^\\infty(-1)^{l+k}q^{l-1} A(1,\\dots,1,\\overset{ \\text{position }l}{\\overbrace{q,1,\\dots,1,m}})\\omega\\bra{{m}{q^l}}\\\\\n\\quad&=\\frac{q^k}{2} \\sum_{m=1}^\\infty \\frac{A(m,1,\\dots,1)}{m}\\\\\n\\quad&\\quad\n\\times\\bra{Kl_{n-k}(\\bar am,q)\\bra{\\Omega_++\\Omega_-}\\bra{\\frac{m}{q^n}}\n+Kl_{n-k}(-\\bar am,q)\\bra{\\Omega_+-\\Omega_-}\\bra{\\frac{m}{q^n}}}\\\\\n\\quad&\\quad+\\sum_{2\\leq l\\leq n-k}\\sum_{m=1}^\\infty (-1)^{n-k+l}q^{k-1} \\frac{A(\\overset{\\text{position } l}{\\overbrace{m,1,\\dots,1,q}},1,\\dots,1)}{m}\\Omega_+\\bra{\\frac{m}{q^{n-l}}},\n\\end{align*}\nwhere $\\Omega_{\\pm}$ is defined as $$\\Omega_+(x)=\\frac{1}{2\\pi i}\\int_{\\Re {s}=-\\sigma}\\tilde{\\omega}(s)\\pi^{-n(1\/2-s)}\\prod_{j=1}^n \\Gamma\\bra{\\frac{1-s-\\overline{\\lambda_j}}{2}}\\Gamma\\bra{\\frac{s-{\\lambda_j}}{2}}^{-1}x^s\\;{\\rm{d}} s$$\nand$$\n\\Omega_-(x)=\\frac{1}{2\\pi i}\\int_{\\Re {s}=-\\sigma}\\tilde{\\omega}(s)i^{-n}\\pi^{-n(1\/2-s)}\\prod_{j=1}^n \\Gamma\\bra{\\frac{2-s-\\overline{\\lambda_j}}{2}}\\Gamma\\bra{\\frac{s+1-{\\lambda_j}}{2}}^{-1}x^s\\;{\\rm{d}} s.$$\n\\end{theorem}\n\\begin{proof}\nThis is literally the sum of Theorem \\ref{maineven} and Theorem \\ref{mainodd}.\n\\end{proof}\n\n\n\n\n\n\n\nOur proof is based on the functional equations of the twisted $L$- functions of $\\pi$ by (multiplicative) Dirichlet characters. \nThe relationship between the Voronoi formulae and the functional equations of $L$-functions is known to experts.\nThe same method was used in Section 4 of \\cite{goldfeldli1} and in          proving Lemma 1.3 in \\cite{buttcane} for a special case on GL(3). \n\n\nOur proof does not depend on automorphicity but depends only on the family of the functional equations of GL(1)-twisted $L$-functions. \nFunctional equations of GL(1) twists are consequences of automorphicity but they do not imply automorphicity on GL($n$) when $n\\geq 3$.\nIn view of the converse theorem on GL($n$) of Cogdell and Piatetski-Shapiro \\cite{converse}, a proof based on GL(1)-twisted functional equations is stronger than a proof based on automorphicity. \n\n\nOur theorem and proof can be easily applied to many conjectural functorial lifts. \nFor example, although the Rankin-Selberg convolutions on GL($m$)$\\times$GL($n$) are not generally known to be automorphic on GL($m\\times n$), except the few cases of GL(2)$\\times$GL(2) and GL(2)$\\times$GL(3), the functional equations of GL($m$)$\\times$GL($n$)$\\times$GL(1) are well known. \nThus, we have the Voronoi formulae for the Rankin-Selberg convolutions on GL($m$)$\\times$GL($n$). Considering the Langlands-Shahidi method and the general Rankin-Selberg method, there are numerous functorial cases to have Voronoi formulae without being known to be automorphic.\n\n\n\nThe obvious weakness of our result is that the denominator $q$ in the hyper-Kloosterman sums has to be a prime number. More general formulae with arbitrary integer $q$ awaits exploration.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgment}\nThe author wants to thank Wenzhi Luo, Roman Holowinsky, and Eyal Kaplan for helpful discussion. The author wants to thank Dorian Goldfeld for introducing this topic to him. \n\n\n\\section{Background}\nLet $\\pi$ be an even Maass cusp form for $\\ssl{n}{Z}$ with the spectral parameter $(\\lambda_1,\\dots,\\lambda_n)\\in \\mathbb{C}^n$. Let $A(*,\\dots,*)$ be the Fourier-Whittaker coefficient of $\\pi$. We have $$A(\\pm m_1,\\dots,\\pm m_{n-1})=A(m_1,\\dots,m_{n-1}),$$\n$$\\overline{A(m_1,\\dots,m_{n-1})}=A(m_{n-1},\\dots,m_1),$$\nand $A(1,\\dots,1)=1.$ We refer to \\cite{goldfeld} for the definitions and the basic results of Maass forms for $\\ssl{n}{Z}$.\nThe dual (contragredient) Maass form of $\\pi$ is denoted by $\\tilde{\\pi}$. Let $B(*,\\dots,*)$ be the Fourier-Whittaker coefficient of $\\tilde\\pi$ and we have \n$$\\overline{A(m_1,\\dots,m_{n-1})}=B(m_{n-1},\\dots,m_1).$$\n\n\n\n\nThe standard $L$-function of $\\pi$ is $$L(s,\\pi)=\\sum_{m=1}^\\infty \\frac{A(1,\\dots,1,m)}{m^s}.$$\nIt satisfies the functional equation \n\\begin{equation}\\label{FEstandard}\nL(s,\\pi) = G_+(s)L(1-s,\\tilde \\pi),\\end{equation}\nwhere $$G_+(s)= \\pi^{-n(1\/2-s)}\\prod_{j=1}^n \\Gamma\\bra{\\frac{1-s-\\overline{\\lambda_j}}{2}}\\Gamma\\bra{\\frac{s-{\\lambda_j}}{2}}^{-1}.$$\n\nLet $q$ be a prime number. Let $\\psi$ be a nontrivial Dirichlet character mod $q$ and $\\tau(\\psi)$ its Gauss sum.\nThe $\\psi$-twisted $L$-function is $$L(s,\\pi\\times\\psi)=\\sum_{m=1}^\\infty \\frac{\\psi(m)A(1,\\dots,1,m)}{m^s}.$$\n\n\nLet $k$ be an integer with $1\\leq k \\leq n-1$. In the case of $\\psi(-1)=1$, the twisted $L$- function satisfies the functional equation\n$$L(s,\\pi\\times \\psi) =\\tau(\\psi)^{n} q^{-ns}G_+(s)L(1-s,\\tilde{\\pi}\\times \\bar \\psi),$$ \nwhich can be rewritten as \n\\begin{equation}\\label{FEeven}\n\\tau(\\bar\\psi)^kL(s,\\pi\\times \\psi) =\\tau(\\psi)^{n-k} q^{k-ns}G_+(s)L(1-s,\\tilde{\\pi}\\times \\bar \\psi).\n\\end{equation}\nIn the case of $\\psi(-1)=-1$, the twisted $L$- function satisfies the functional equation\n$$L(s,\\pi\\times \\psi) =\\tau(\\psi)^{n} q^{-ns}G_-(s) L(1-s,\\tilde{\\pi}\\times \\bar \\psi),$$\nwhich can be rewritten as \n\\begin{equation}\\label{FEodd}\n\\tau(\\bar\\psi)^kL(s,\\pi\\times \\psi) =\\tau(\\psi)^{n-k} q^{k-ns}G_-(s) L(1-s,\\tilde{\\pi}\\times \\bar \\psi),\n\\end{equation}\nwhere \n$$G_-(s)=i^{-n}\\pi^{-n(1\/2-s)}\\prod_{j=1}^n \\Gamma\\bra{\\frac{2-s-\\overline{\\lambda_j}}{2}}\\Gamma\\bra{\\frac{s+1-{\\lambda_j}}{2}}^{-1}.$$\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Hyper-Kloosterman Sums}\nLet $q$ be a prime number. For abbreviation, we define $e(x)=\\exp(2\\pi i x)$.\nLet $k$ be a positive integer.\nThe $(k-1)$-dimensional hyper-Kloosterman sum is defined as\n$$Kl_k(m,q)=\\sum_{\\underset{\\scriptstyle (x_1,q)=\\dots=(x_{k-1},q)=1}{x_1, \\dots, x_{k-1}\\mod q}} e\\bra{\\frac{x_1+\\dots+x_{k-1}+m\\overline{x_1\\dots x_{k-1}}}{q}}$$\nand we shall note that if $q|m$,\n\\begin{equation}\\label{kloostermandegenerate0}\nKl_k(m,q)=(-1)^{k-1}.\n\\end{equation}\nIn the degenerate case of $k=1$, we have the additive character $$Kl_1(m,q) = e\\bra{\\frac{m}{q}}.$$ When $k=2$, \nthe 1-dimensional hyper-Kloosterman sum is identical to the classical Kloosterman sum.\n\n\nWe define $\\underset{\\psi\\mod q}{\\sum {}^*}$ as the summation over nontrivial Dirichlet characters $\\psi$ of mod $q$.\nIt is known \n\\begin{equation}\\label{kloostermaneven}\n\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\psi)^k \\bar{\\psi}(m)=\n\\begin{cases}\n\\frac{q-1}{2}\\bra{Kl_{k}(m,q)+Kl_{k}(-m,q)}-(-1)^k,&\\text{ if }(m,q)=1\\\\\n0, &\\text{ otherwise}.\n\\end{cases}\n\\end{equation}\nIt is also known\n\\begin{equation}\\label{kloostermanodd}\n\\underset{\\underset{\\scriptstyle{ \\psi(-1)=-1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\psi)^k \\bar{\\psi}(m)=\\frac{q-1}{2}\\bra{Kl_{k}(m,q)-Kl_{k}(-m,q)}.\n\\end{equation}\nWe will use the family of (multiplicative) Dirichlet characters to recover the additive characters and the hyper-Kloosterman sums by the two formulae above.\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\section{Even Part}\nLet $q$ be a prime number, $a$ and $\\overline{a}$ two integers with $a\\overline{a}\\equiv 1\\mod q$. \nLet $k$ be an integer and $1\\leq k \\leq n-1$.  \n\nDefine \ntwo Dirichlet series\n$$L_q(s,\\pi)=\\sum_{q|m,m>0}\\frac{A(1,\\dots,1,m)}{m^s}$$ \nand $$L_q(s,\\tilde\\pi)=\\sum_{q|m,m>0}\\frac{A(m,1,\\dots,1)}{m^s}.$$ \nDefine \ntwo Dirichlet series\n$$\\mathsf{Z}(s,q)=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\bar\\psi)^k \\psi(a) L(s,\\pi\\times \\psi) + (-1)^k\\bra{L(s,\\pi)-qL_q(s,\\pi)}$$\nand $$\\tilde{\\mathsf{Z}}(s,q)=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\psi)^{n-k} \\psi(a)L(s,\\tilde\\pi \\times\\psibar) + (-1)^{n-k}\\bra{L(s,\\tilde\\pi)-qL_q(s,\\tilde\\pi)}.$$\n\n\n\n\\begin{lemma}\\label{zdirichlet}\nWe have the identities \n$$\\mathsf{Z}(s,q)=\\frac{1}{2}\\sum_{m=1}^\\infty \\bra{Kl_k(am,q)+Kl_k(-am,q)} \\frac{A(1,\\dots,1,m)}{m^s}$$\nand \n$$\\tilde{\\mathsf{Z}}(s,q)=\\frac{1}{2}\\sum_{m=1}^\\infty\n\\bra{Kl_{n-k}(\\bar am,q)+Kl_{n-k}(-\\bar am,q)}\n \\frac{A(m,1,\\dots,1)}{m^{s}}.$$\n\\end{lemma}\n\n\n\n\\begin{proof}\nThis follows from  \\reff{kloostermandegenerate0} and \\reff{kloostermaneven},\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\nFor $1\\leq l \\leq n-1$, define a Dirichlet polynomial \n$$H_l(s,q)=\\sum_{i=l}^{n-1} (-1)^{i}\\frac{A(1,\\dots,1,\\overset{ \\text{position }i}{\\overbrace{q,1,\\dots,1}})}{q^{is}}+\\frac{(-1)^n}{q^{ns}},$$\nwhere $q$ appears on the $i^{\\text{th}}$ position from the right. \nWe define another Dirichlet polynomial \n$$\\tilde H_l(s,q)=\\sum_{i=l}^{n-1} (-1)^{i}\\frac{A(\\overset{\\text{position } i}{\\overbrace{1,\\dots,1,q}},1,\\dots,1)}{q^{is}}+\\frac{(-1)^n}{q^{ns}},$$\nwhere $q$ appears on the $i^{\\text{th}}$ position from the left.\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{termsleft}\nWe have the identities \n\\begin{equation}\\label{x1}L(s,\\pi) H_l(s,q) = (-1)^l\\sum_{m=1}^\\infty \\frac{A(1,\\dots,1,\\overset{\\text{position }l}{\\overbrace{q,1,\\dots,1,m}})}{q^{ls}m^s}\\end{equation} \nfor $2\\leq l\\leq n-1$\nand \n\\begin{equation}\\label{x2}\nL(s,\\pi) H_1(s,q)=-L_q(s,\\pi).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy the Hecke relation \n\\begin{multline*}\n\\aleft{i}A(1,\\dots,1,m)\\\\=\\begin{cases}\nA(1,\\dots,1,\\overset{\\text{position }i}{\\overbrace{q,1,\\dots,1,m}})+\nA(1,\\dots,1,\\overset{\\text{position }i+1}{\\overbrace{q,1,\\dots,1,m\/q}}),&\\text{ if }q|m\\\\\nA(1,\\dots,1,\\overset{\\text{position }i}{\\overbrace{q,1,\\dots,1,m}}),&\\text{ otherwise,}\n\\end{cases}\n\\end{multline*}\nwe have \\reff{x1}\nfor $2\\leq l\\leq n-2$\n after multiplying the Dirichlet series $L(s,\\pi) H_l(s)$ term by term.\nSimilarly, by the Hecke relation\n$$A(1,\\dots,1,q)A(1,\\dots,1,m)=\n\\begin{cases}A(1,\\dots,1,mq)+A(1,\\dots,1,q,m\/q),&\\text{ if }q|m\\\\\nA(1,\\dots,1,mq),&\\text{ otherwise,}\n\\end{cases}\n$$\nand \n$$A(q,1,\\dots,1)A(1,\\dots,1,m)=\n\\begin{cases}A(q,1,\\dots,1,m)+A(1,\\dots,1,m\/q),&\\text{ if }q|m\\\\\nA(q,1,\\dots,1,m),&\\text{ otherwise,}\n\\end{cases}\n$$\nwe have \\reff{x2}, and \\reff{x1} for $l=n-1$, respectively.\n\\end{proof}\n\n\n\nBy a similar proof, we have the following lemma for $\\tilde{\\pi}$.\n\n\\begin{lemma}\\label{termsright}\nWe have the identities\n$$L(s,\\tilde\\pi) \\tilde{H}_l(s,q) = (-1)^l\\sum_{m=1}^\\infty \\frac{A(\\overset{\\text{position } l}{\\overbrace{m,1,\\dots,1,q}},1,\\dots,1)}{q^{ls}m^s}$$\nfor $2\\leq l\\leq n-1$\nand\n$$L(s,\\tilde\\pi) \\tilde H_1(s,q)=-  L_q(s,\\tilde\\pi).$$\n\\end{lemma}\n\n\n\n\n\\begin{lemma}\\label{cruciallemma}\nWe have the following identity between Dirichlet polynomials\n\\begin{multline*}\n\\frac{1}{q}+H_1(s,q)+\\sum_{2\\leq l\\leq k}(q^{l-1}-q^{l-2})H_l(s,q)\\\\\n=(-1)^n q^{k-ns}\\bra{\\frac{1}{q}+\\tilde H_1(1-s,q)+\\sum_{2\\leq l\\leq n- k}(q^{l-1}-q^{l-2})\\tilde H_l(1-s,q)}.\\end{multline*}\n\\end{lemma}\n\\begin{proof}\nThe left side equals\n$$\\frac{1}{q}+\\sum_{i=1}^{n-1} (-1)^{i}q^{\\min\\{i,k\\}-1}\\frac{A(1,\\dots,1,\\overset{ \\text{position }i}{\\overbrace{q,1,\\dots,1}})}{q^{is}}+\\frac{(-1)^nq^{k-1}}{q^{ns}}.$$\nThe right side equals\n\\begin{align*}\n\\frac{1}{q}+\\sum_{i=1}^{n-1} &(-1)^{i}q^{\\min\\{i,n-k\\}-1}\\frac{A(\\overset{\\text{position } i}{\\overbrace{1,\\dots,1,q}},1,\\dots,1)}{q^{i-is}}+\\frac{(-1)^nq^{n-k-1}}{q^{n-ns}}\\\\\n&=\\frac{1}{q}+\\sum_{i=1}^{n-1} (-1)^{n-i}q^{\\min\\{n-i,n-k\\}-1}\\frac{A(1,\\dots,1,\\overset{ \\text{position }i}{\\overbrace{q,1,\\dots,1}})}{q^{(n-i)-(n-i)s}}+\\frac{(-1)^nq^{n-k-1}}{q^{n-ns}}\n\\\\\n&=\n\\frac{(-1)^nq^{-k-1}}{q^{-ns}}+\\sum_{i=1}^{n-1} (-1)^{n-i}q^{\\min\\{0,i-k\\}-1}\\frac{A(1,\\dots,1,\\overset{ \\text{position }i}{\\overbrace{q,1,\\dots,1}})}{q^{-(n-i)s}}\n+\\frac{1}{q}\n\\end{align*}\nAfter multiplying $(-1)^nq^{k-ns}$, we match term by term with the left side.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Theorem \\ref{maineven}}\nLet $\\omega(x)\\in C_c^\\infty(0,\\infty)$ be a test function. Let\n$$\\tilde{\\omega}(s)=\\int_0^\\infty\\omega(x)x^{s-1} \\;{\\rm{d}} x$$ be the Mellin transform of $\\omega$. \nDefine \n\\begin{equation}\\label{omegaplus}\n\\Omega_+(x)=\\frac{1}{2\\pi i}\\int_{\\Re {s}=-\\sigma}\\tilde{\\omega}(s) G_+(s)x^s\\;{\\rm{d}} s.\n\\end{equation}\nWe have the Mellin inversion theorem\n\\begin{equation}\\label{mellininversion}\n\\omega(x)=\\frac{1}{2\\pi i}\\int_{\\Re {s}=\\sigma} x^{-s}\\tilde{\\omega}(s)\\;{\\rm{d}} s.\n\\end{equation}\n\n\n\n\n\n\n\n\\begin{theorem}\\label{maineven}\nLet $\\pi$ be an even Maass cusp form for $\\ssl{n}{Z}$ with $n\\geq 2$. Let $A(m_1,\\dots,m_{n-1})$ be the Fourier coefficient of $\\pi$.  Let \n$\\omega\\in C_c^\\infty(0,\\infty)$ be a test function\nand let $q$ be a prime number, $a$ and $\\overline{a}$ two integers with $a\\overline{a}\\equiv 1\\mod q$. \nFor $1\\leq k \\leq n-1,$ we have \n\n\\begin{align*}\n&\\frac{1}{2}\\sum_{m=1}^\\infty A(1,\\dots,1,m)\\bra{Kl_k(am,q)+Kl_k(-am,q)}\\omega(m)\\\\\n&+\\sum_{2\\leq l\\leq k}\\sum_{m=1}^\\infty(-1)^{l+k}q^{l-1} A(1,\\dots,1,\\overset{ \\text{position }l}{\\overbrace{q,1,\\dots,1,m}})\\omega\\bra{{m}{q^l}}\\\\\n&\\quad\\quad=\\frac{q^k}{2}\\sum_{m=1}^\\infty \\frac{A(m,1,\\dots,1)}{m}\\bra{Kl_{n-k}(\\bar am,q)+Kl_{n-k}(-\\bar am,q)}\\Omega_+\\bra{\\frac{m}{q^n}}\\\\\n&\\quad\\quad\\quad+\\sum_{2\\leq l\\leq n-k}\\sum_{m=1}^\\infty (-1)^{n-k+l}q^{k-1} \\frac{A(\\overset{\\text{position } l}{\\overbrace{m,1,\\dots,1,q}},1,\\dots,1)}{m}\\Omega_+\\bra{\\frac{m}{q^{n-l}}},\n\\end{align*}\nwhere $\\Omega_+$ is defined in \\reff{omegaplus}.\n\\end{theorem}\n\n\n\n\n\\begin{proof}\nBy the functional equations \\reff{FEstandard} and \\reff{FEeven} and Lemma \\ref{cruciallemma}, we have\n\\begin{align*}\n\\mathsf{Z}(&s,q)+(-1)^kq\\sum_{2\\leq l\\leq k}(q^{l-1}-q^{l-2})H_l(s,q)L(s,\\pi)\\\\\n&=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\bar\\psi)^k\\psi(a)L(s,\\pi\\times \\psi) \\\\\n&\\quad+(-1)^k\\bra{L(s,\\pi)-qL_q(s,\\pi)+q\\sum_{2\\leq l\\leq k}(q^{l-1}-q^{l-2})H_l(s,q)L(s,\\pi)}\n\\\\\n&=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\bar\\psi)^k\\psi(a)L(s,\\pi\\times \\psi) \\\\&\n\\quad +(-1)^k{q\\bra{\\frac{1}{q}+H_1(s,q)+\\sum_{2\\leq l\\leq k}(q^{l-1}-q^{l-2})H_l(s,q)}L(s,\\pi)}\n\\\\\n&=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\psi)^{n-k}\\psi(a)q^{k-ns}G_+(s)L(1-s,\\tilde{\\pi}\\times\\bar{\\psi})\n\\\\\n&\\quad+(-1)^{n-k}q\\bra{\\frac{1}{q}+\\tilde H_1(1-s,q)+\\sum_{2\\leq l\\leq n-k}(q^{l-1}-q^{l-2})\\tilde H_l(1-s,q)}q^{k-ns}G_+(s)L(1-s,\\tilde{\\pi})\\\\\n&= q^{k-ns}G_+(s)\\bra{\\tilde{\\mathsf{Z}}(1-s,q)+(-1)^{n-k}q\\sum_{2\\leq l\\leq n-k}(q^{l-1}-q^{l-2})\\tilde H_l(1-s,q)L(1-s,\\tilde{\\pi})}.\\end{align*}\n\n\nIntegrating on both ends with $\\tilde{\\omega}$ and shifting the integral on the left from $\\Re {s}=-\\sigma$ to $\\Re {s}=\\sigma$, we have \n\\begin{align*}\n&\\int_{\\Re {s}=\\sigma} \\bra{\\mathsf{Z}(s,q)+(-1)^kq\\sum_{2\\leq l\\leq k}(q^{l-1}-q^{l-2})H_l(s,q)L(s,\\pi)} \\tilde{\\omega}(s) \\;{\\rm{d}} s\\\\\n&=\\int_{\\Re {s}=-\\sigma} q^{k-ns}G_+(s)\\times\\\\\n&\\quad\\;\\quad\\;\\bra{\\tilde{\\mathsf{Z}}(1-s,q)+(-1)^{n-k}q\\sum_{2\\leq l\\leq n-k}(q^{l-1}-q^{l-2})\\tilde H_l(1-s,q)L(1-s,\\tilde{\\pi})} \\tilde{\\omega}(s) \\;{\\rm{d}} s.\n\\end{align*}\nRecalling Lemma \\ref{zdirichlet}, Lemma \\ref{termsleft}, and Lemma  \\ref{termsright}, we complete the proof with the Mellin inversion \\reff{mellininversion}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Odd Part}\nDefine $$\\mathsf{Y}(s,q)=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=-1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\bar\\psi)^k \\psi(a) L(s,\\pi\\times \\psi) $$\nand $$\\tilde{\\mathsf{Y}}(s,q)=\\underset{\\underset{\\scriptstyle{ \\psi(-1)=-1}}{\\psi\\mod q}}{\\sum {}^*} \\tau(\\bar\\psi)^{n-k} \\psi(a) L(s,\\tilde\\pi \\times\\psibar).$$\nBy the function equation \\reff{FEodd}, we have \n\\begin{equation}\\label{odd}\n\\mathsf{Y}(s,q)\n=q^{k-ns}G_-(s)\\tilde{\\mathsf{Y}}(1-s,q).\n\\end{equation}\n\n\n\\begin{lemma}\\label{zzdirichlet}\nWe have the identities \n$$\\mathsf{Y}(s,q)=\\frac{1}{2}\\sum_{m=1}^\\infty \\bra{Kl_k(am,q)-Kl_k(-am,q)} \\frac{A(1,\\dots,1,m)}{m^s}$$\nand \n$$\\tilde{\\mathsf{Y}}(s,q)=\\frac{1}{2}\\sum_{m=1}^\\infty\n\\bra{Kl_{n-k}(\\bar am,q)-Kl_{n-k}(-\\bar am,q)}\n \\frac{A(m,1,\\dots,1)}{m^{s}}.$$\n\\end{lemma}\n\n\\begin{proof}\nThis follows from \\reff{FEodd}.\n\\end{proof}\n\n\n\nLet $\\omega(x)\\in C_c^\\infty(0,\\infty)$ be a test function. Let\n$\\tilde{\\omega}(s)=\\int_0^\\infty\\omega(x)x^s \\;{\\rm{d}} x\/x$ be its \nMellin transform. \nDefine \n\\begin{equation}\\label{omegaminus}\\Omega_{-}(x)=\\frac{1}{2\\pi i}\\int_{\\Re {s}=-\\sigma}\\tilde{\\omega}(s) G_{-}(s)x^s\\;{\\rm{d}} s.\\end{equation}\nIntegrating with $\\tilde{\\omega}$ on both sides of \\reff{odd} and shifting the integral on the left to $\\Re {s}=\\sigma$, we get $$\\frac{1}{2\\pi i}\\int_{\\Re {s}=\\sigma}\\mathsf{Y}(s,q)\\tilde{\\omega}(s)\\;{\\rm{d}} s=\\frac{1}{2\\pi i}\\int_{\\Re {s}=-\\sigma} q^{k-ns}G_-(s)\\tilde{\\mathsf{Y}}(1-s,q)  \\tilde{\\omega}(s) \\;{\\rm{d}} s,$$\nwhich, along with Lemma \\ref{zzdirichlet} and \\reff{mellininversion}, gives us the following theorem.\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{mainodd}\nLet $\\pi$ be an even Maass cusp form for $\\ssl{n}{Z}$ with $n\\geq 2$. Let $A(m_1,\\dots,m_{n-1})$ be the Fourier coefficient of $\\pi$.  Let \n$\\omega\\in C_c^\\infty(0,\\infty)$ be a test function \nand let $q$ be a prime number, $a$ and $\\overline{a}$ two integers with $a\\overline{a}\\equiv 1\\mod q$. \nFor $1\\leq k \\leq n-1,$ we have \n\\begin{multline*}\n\\frac{1}{2}\\sum_{m=1}^\\infty A(1,\\dots,1,m)\\bra{Kl_k(am,q)-Kl_k(-am,q)}\\omega(m)\\\\\n=\\frac{q^k}{2} \\sum_{m=1}^\\infty \\frac{A(m,1,\\dots,1)}{m}\\bra{Kl_{n-k}(\\bar am,q)-Kl_{n-k}(-\\bar am,q)}\\Omega_-\\bra{\\frac{m}{q^n}},\n\\end{multline*}\nwhere $\\Omega_{-}$ is defined in \\reff{omegaminus}.\n\\end{theorem}\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}\nThe Cosmic Microwave Background Polarization (CMBP) is a powerful tool\nfor investigating the early Universe. \nFor instance,\ninflationary models lead to a well defined peak--pattern of the\n$E$--mode power spectrum and its precise measurement\nprovides a check of the inflationary paradigm\nitself \\citep{ko99}.\n\nThe second and fainter CMBP component -- the $B$--mode -- \nis predicted to directly probe Inflation. \nIn fact, the emission on a few degrees\nscale is related to the amount of gravitational waves (GW) generated by the Inflation,\nand allows measurement of its main parameters, such as the\nenergy density of the Universe when this event\noccurred \\citep{ka98}. \nOn subdegree scales, instead, the $B$-mode is\ncontaminated by the $E$-mode via galaxy gravitational \nlensing, allowing though a way \nto determine the matter fluctuation power spectrum \n\\citep{zaldarriaga98}.\nThe first steps into $E$-mode measurements has begun with \nthe detections of the DASI, CAPMAP and CBI experiments \n\\citep{leitch04,barkats04,readhead04}. However,\nwe are far from a complete characterization\nand the $B$-mode is still elusive.\n\nSeveral astrophysical sources of polarized emission in the foreground must\nbe considered as they could contaminate the tiny CMBP signal.\nAmong them, the diffuse synchrotron emission\nof the Galaxy dominates at low frequencies and it is expected to be\nthe major contaminant up to 100~GHz. \nRecently, evidence for a significant anomalous dust emission,\ncompetitive with the synchrotron up to $50$~GHz, has been found\n(e.g. \\citealt*{deoliveira04,fink04}).\nAlthough potentially relevant  \nin total intensity, the small polarization fraction \npredicted \\citep{lazarian00} seems to leave synchrotron as the\nleading polarized component.\n\nThe study of the synchrotron emission\nis thus crucial for CMBP experiments. Besides the estimate of the\ncontamination level, it allows \nthe tuning of foreground separation procedures to extract \nthe cosmic signal \n(e.g. \\citealt{tegmark00}) and \nthe selection of optimal sky areas for observation.\n\nPolarization measurements with large sky coverage has becoming \navailable at frequencies up to 1.4~GHz \n(e.g. see \\citealt{carretti05} and references therein \nfor a review of the available data), \nalthough their sensitivities and angular resolutions do not fully cover \nthe CMBP needs.\nAn ideal field for CMBP studies is the high Galactic latitude \nfield ($b \\sim -38^\\circ$) in the region observed by the BOOMERanG \nexperiment.  This field, positioned at \n$\\alpha = 5^{\\rm h}$, $\\delta = -49^{\\circ}$, \nhas low levels of synchrotron emission and has been\nselected as the target area for \nCMBP experiments (e.g. BaR-SPOrt, \\citealt{cortiglioni03}, and\nBOOMERanG-B2K, \\citealt{masi02}).\nThe first measurement of the polarized synchrotron\nemission in low-emission areas was reported by \\citet{be03}\nin this field at 1.4 GHz with the Australia Telescope\nCompact Array (ATCA).  \nThe analysis of \nthese observations has allowed the\nfirst estimate of the synchrotron contamination on the CMBP spectra\nat high latitude \\citep{carretti05}.\nHowever, these authors found that at 1.4 GHz the area \nis likely to be affected by Faraday Rotation (FR) \neffects, which could enhance the detected emission \nand modify the spectra by\nsteepening their slopes.  In particular, their analysis shows that, \nat 1.4~GHz this area is in an intermediate state between negligible \nand significant FR effects. \nSince FR has a square \ndependence on the wavelength ($\\delta\\varphi\\propto\\lambda^2$),\nhigher frequencies should \nallow observations of the region without \nsignificant effects. Finally, the 1.4~GHz data have an \nangular scale sensitivity range limited by the interferometric \nnature of the observations, and cover only the angular scales up to \n15~arcmin.\n\nTo overcome these constraints, we have conducted \nsingle dish observations of this area \nat 2.3~GHz with the Parkes Radio telescope.\nThese observations also allow us to extend \nthe angular scale sensitivity toward degree scales, \nleading to a firmer assessment of the contamination\nof the CMBP $B$--mode generated by GW, which peaks on \n$\\sim 2^\\circ$ scales.\n\nThe paper is organized as follows: in Section~\\ref{obsSec} we present\ndetails of the observations along with the discussion of the obtained\nmaps.  In Section~\\ref{specSec} we show the analysis of the angular\npower spectra, while in Section~\\ref{discSec} we discuss the\nimplications for CMBP measurements.\n\n\\section{Observations}\\label{obsSec}\n\n\\begin{figure*}\n  \\includegraphics[angle=0, width=0.4\\hsize]{fig1a.ps}\n  \\includegraphics[angle=0, width=0.4\\hsize]{fig1b.ps}\n  \\includegraphics[angle=0, width=0.4\\hsize]{fig1c.ps}\n  \\includegraphics[angle=0, width=0.4\\hsize]{fig1d.ps}\n\\caption{Maps of the Stokes parameters $I$ (top-left), \n         $Q$ (bottom-left) and $U$ (bottom-right) of the observed area\n         at 2.3~GHz. The polarized intensity $L=\\sqrt{Q^2+U^2}$ is shown as well\n         (top-right). Units are mK.\\label{mapFig}\n}\n\\end{figure*}\n\nWe have made single dish observations of the \n$2^\\circ \\times 2^\\circ$ area centred in \n$\\alpha = 5^{\\rm h}00^{\\rm m}00^{\\rm s}$,~$\\delta = -49^{\\circ} 00^\\prime 00^{\\prime\\prime}$ \n(J2000) ($l\\sim255^\\circ$,~$b\\sim-38^\\circ$) \nwith the Parkes Radio telescope at 2.3~GHz on 17--23 August 2004.  \nThe field was surveyed with\n$2\\degr$ scans in both RA and Dec.  Scans were separated by 3~arcmin, \nproviding an adequate sampling of the $8\\farcm8$ telescope beam.  \nWe used the dual circular polarization Galileo receiver\nin combination with a wide band 13~cm feed. \nThe two total intensity channels and the two linear Stokes parameters,\n$Q$ and $U$, were formed from the auto and cross correlations of the two\nreceiver outputs.\nThe digital correlator was configured to provide \n1024 250~kHz-channels, giving a total bandwidth of 256 MHz centered on\n2300 MHz.\nIn the subsequent analysis \nthe channels have been grouped in 8 $\\times$ 32~MHz.\nThe system showed an optimal sensitivity in a 200~MHz wide band. After the\nselection for RF interferences, a total \neffective bandwidth of $\\Delta\\nu = 128$~MHz has remained with a\ncentral frequency of 2332~MHz.\n\nThe sources 1934-638 and 3C138 have been used for total intensity and\npolarization calibration, respectively.  \nThe absolute polarization state of 3C138 was determined using\nATCA one week prior to the Parkes\nobservations.  Although 3C138 is variable, the timescale is such that\nthe one week lag between observations is insignificant \\citep{padrielli87}.\n\nImages of the Stokes parameters $I$, $Q$ and $U$ were constructed\nthrough an iterative map--making procedure, \nbased on the estimation and removal of a linear \nbaseline from each scan. The\nimage produced at each iteration is used in the \nnext to subtract the sky signal from the data and\nimproves the baseline evaluation (see \\citealt{sbarra03} \nfor the basic equations and Carretti \\& Poppi, in preparation, \nfor an implementation at the Medicina Radio telescope).\nDiffering from \\citet{sbarra03}, a linear behaviour for the \nbaseline is allowed. \nAs usual, the removal of a linear baseline leads to the loss of the\nabsolute emission levels in the images.  \n\nThe Stokes $I$, $Q$, $U$ and linearly polarized intensity, \n$L=\\sqrt{Q^2 + U^2}$, images\nare shown in Figure~\\ref{mapFig}.  The images cover an area of \n$2\\degr \\times 2\\degr$, have a pixel size of 3~arcmin, \nand have an rms pixel sensitivity of 1.0~mJy~beam$^{-1}$ or 800~$\\mu$K\nin brightness temperature units (the gain is 0.77~Jy\/K). \nFinally, they have been smoothed to a FWHM of $10\\farcm 6$.\n\nThe total intensity emission is dominated by point sources, \ncorresponding to the strongest sources in the 1.4~GHz image\n\\citep{carretti05}.\nThe polarization images are dominated by diffuse emission.\nStructures are present on all angular scales up to the field size,\nwith the largest feature extending from north to south.\nExcluding the one evident point source, the polarized intensity image\npeaks at about 5~mK, while the peak-to-peak variation\nof $U$ is of about 10~mK ($Q$ shows less power).\n\nThe small range of overlapping angular scales in the 1.4 and 2.3 GHz\nobservations makes\na direct comparison difficult.\nHowever, it is worth noting the filamentary structure about 1~degree long\npresent at 1.4~GHz has no counterpart at 2.3~GHz \nbut approximately\ncorresponds with the maximum gradient of the 2.3~GHz $U$ map.\nIt is possible that this structure is generated by a Faraday\nscreen at 1.4~GHz, as supposed also by \\citet{be03}, but the structure\ndisappears at 2.3~GHz, where FR is weaker.\n\nAn image of the polarization angle is shown in Figure~\\ref{paFig}.\nThe pattern is\nuniform inside the large structures \nbut sudden changes of about $90\\degr$ are observed when crossing\na zero in polarized intensity.\nThese features can have physical origins, such as the effects of a\nFaraday screen or a sudden change in the magnetic field direction.\nHowever, here they correspond to a change of sign\nof $U$ (the dominant component) \nand are characterized by a\n$\\sim 90^\\circ$ change of polarization angle.  \nThis could be simply generated by a mean value removal. \nThe addition of a constant value to $U$ would eliminate this\nchange of sign, keeping more uniform the polarization angle pattern.\nMap-making procedures can generate such a situation, \ntherefore, besides the presence of either a\nFaraday screen or a rapid change of the magnetic field, the sudden\nchanges in polarization angle of our map can be related to this\nnon-physical cause.  \nOnly observations of a larger area can provide the missing mean \nemission and distinguish the two cases.\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=0.85\\hsize]{fig2.ps}\n\\caption{Polarization angle map. The vector length is proportional to the \n         polarized intensity.\\label{paFig}\n}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=0.85\\hsize]{fig3.ps}\n\\caption{Angular power spectra $C^E_\\ell$\nand $C^B_\\ell$ in the observed area together with their power law fits.\n$C^B_\\ell$ is slightly shifted to the right to avoid confusion.\nThe fit to the 1.4~GHz $E$--mode power spectrum scaled up to 2.332~GHz \n($C_{1.4}^E$) is also displayed \n(a frequency spectral slope of $\\alpha = -2.8$ is assumed).\n\\label{specFig}}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=0.85\\hsize]{fig4.ps}\n\\caption{Fits to power spectra $C^X_{\\ell}$ computed at $\\ell = 1000$\nfor both 1.4~GHz and 2.3~GHz. $E$-- and $B$--mode are almost overlapped at 1.4~GHz, \nso that the latter has been omitted to avoid confusion. \nSimilarly, $C^B_{2.3}$ is slightly shifted left to avoid confusion.\nThe dotted line represents the 1.4~GHz value extrapolated \nusing a brightness temperature slope of $\\alpha = -2.8$.\n\\label{spec2Fig}}\n\\end{figure}\n\n\n\n\n\\section{Power Spectrum Analysis}\n\\label{specSec}\nWe compute the power spectrum of the $E$-- and $B$--modes of the\npolarized component through the Fourier\ntechnique of \\citet{seljak97}.  The results are shown in\nFigure~\\ref{specFig} together with power law fits\nto the equation:\n\\begin{equation}\n    C_{\\ell}^X = C_{500}^{X} \\left({\\ell \\over 500}\\right)^{\\beta_X},\n    {\\rm with\\;\\;} X=E,\\,B,\n\\end{equation}\nwhere $\\ell$ is the multipole, corresponding to\nthe angular scale $\\theta \\approx 180^\\circ \/ \\ell$.\nThe spectra of $E$-- and $B$--mode have similar power, except at\nsmaller $\\ell$, where the poor statistics allows larger deviations\nbetween the two components.\nThe results of the fits are given in Table~\\ref{specTab}.\n\\begin{table}\n \\centering\n  \\caption{Fit parameters for $E$ and $B$ spectra.\\label{specTab}}\n  \\begin{tabular}{@{}lcc@{}}\n  \\hline\n   Spectrum     &   $C_{500}^X$ [$10^{-12}$~K$^2$]     &   $\\beta_X$        \\\\\n  \\hline\n  $C^E_\\ell$ &  $21 \\pm 2 $  &  $-1.46 \\pm 0.14$ \\\\\n  $C^B_\\ell$ &  $37 \\pm 6 $  &  $-1.87 \\pm 0.22$ \\\\\n  \\hline\n  \\end{tabular}\n \\label{powFitTab}\n\\end{table}\n\nThe comparison with the spectra obtained in the same area at 1.4~GHz\n\\citep{carretti05}\nallows interesting considerations.\nFigure~\\ref{specFig} plots the 1.4~GHz spectrum scaled to \n2.3~GHz with a brightness temperature frequency spectral\nslope $\\alpha = -2.8$, \nwhile Figure~\\ref{spec2Fig} shows the $E$--mode \npowers for $\\ell=1000$ at 1.4 and 2.3~GHz.\nThe $E$--mode emission level of our 2.3~GHz observations is \nsignificantly weaker than expected for a synchrotron spectrum\nwith slope of $\\alpha = -2.8$.\nIn fact, a frequency spectral index of\n$\\tilde{\\alpha}_E = -3.6 \\pm 0.15$ is\nrequired to fit the amplitudes. A similar view is given\nby the $B$--mode, although the needed slope is flatter\n($\\tilde{\\alpha}_B = -3.3 \\pm 0.2$).\nSuch steep slopes are unlikely at these frequencies for synchrotron\nemission \\citep{platania98}.  We suggest that the amplitudes at 1.4~GHz are\naffected by FR.\nThis suggestion is supported by \\citet{carretti05}, who find that\nrandomization of polarization angles induced by FR\ncan transfer power from large to small angular scales, enhancing\nthe power on subdegree scales.  At 2.3~GHz, where Faraday effects are\nless than at 1.4~GHz, the observed polarization is more closely related\nto the intrinsic emission, as discussed in \\citet{carretti05}.\nThus it seems clear that observations at higher frequencies\nare more robust for CMB extrapolation.\n\n\nThe comparison between the slopes of the angular power spectra\nprovides a similar view. At 2.3~GHz $\\beta_E$ is flatter than\nat 1.4~GHz, where a value of $\\beta_E^{1.4}\\sim -1.97\\pm 0.08$ has been\nmeasured. The values measured at 2.3~GHz are closer to the mean value\n$\\beta_X^{GP} \\sim -1.6$ ($X=E,B$) obtained on the Galactic plane\nat the same frequency \\citep{bruscoli02}.  \n\\citet{carretti05} discussed the\neffects of the FR on the power spectrum slope,\nfinding that steepening can occur if FR action is\nsignificant. In this context, the flatter slope\nmeasured at 2.3~GHz can be interpreted as FR effects being\nweak relative to those at 1.4~GHz.\nWe note that the $B$--mode cannot be fit well with a power law\n(see Figure~\\ref{specFig}), so that\nthe resulting larger error makes $\\beta_B$ compatible with both\n$\\beta_B^{1.4} = -1.98 \\pm 0.07$ and $\\beta_X^{GP}$. Therefore, \nno firm statements can be made about the comparison of \n$\\beta_B^{1.4}$ and $\\beta_B$.\n\n\n\n\\section{Discussion}\n\\label{discSec}\n\nThe steep spectral index needed to match the 1.4~GHz emission level,\nthe regularity in the polarization angle pattern,  and the flatter\nslope of the 2.3~GHz $E$--mode angular power spectrum, indicate that our\nmaps are weakly affected by FR effects. Consequently, \nthey provide the most reliable estimates of the contamination of the CMBP\nby synchrotron emission at high Galactic latitudes. Moreover, these data\nexplore larger angular scales than possible with the 1.4~GHz\nobservations (limited to scales smaller than 15$\\arcmin$), allowing\nestimates closer to the peak of the $B$--mode emission at about\n$2^\\circ$ ($\\ell \\sim 100$).\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=0.85\\hsize]{fig5.ps}\n\\caption{Angular power spectra $C^E_\\ell$ (top)\nand $C^B_\\ell$ (bottom)\nextrapolated to 32~GHz and 90~GHz, respectively, \nof both the 2.3 and 1.4~GHz data.\nA frequency spectral slope of $\\alpha = -3.1$ is assumed.\nSpectra expected for the CMBP are shown for comparison (solid) by using\ncosmological parameters determined with the WMAP data\n\\citep{spergel03}. For the $B$--mode, a tensor-to-scalar power ratio\n$T\/S = 0.01$ is  assumed.\\label{specCMBFig}}\n\\end{figure}\n\nThe extrapolations up to 32~GHz ($E$--mode) and 90~GHz ($B$--mode)  are\nshown in Figure~\\ref{specCMBFig}, where the typical spectral index\n$\\alpha = -3.1$ of the 1.4--23~GHz range has been assumed\n\\citep{bernardi04}. \nThe contamination of the $E$--mode is significantly\nlower than that estimated with the 1.4~GHz data: a factor two better is\nseen at $\\ell\\sim 1000$, while the flatter slope of the 2.3~GHz angular\nspectrum increases that factor at larger scales to about~4 at \n$\\ell = 400$. \nNote that the  $\\ell$-range now covered allows a direct estimate\non angular scales usually probed by 30~GHz experiments (e.g. the\n32~GHz channel of BaR-SPOrt with a FWHM~=~$0.4^\\circ$).\nThis improves the expected detectability of the CMBP $E$--mode signal,\nby avoiding the uncertainties of an angular extrapolation.  \nAt 32~GHz, the first CMBP peak  ($\\ell\\sim 400$) is about 3 orders \nof magnitude above the\nsynchrotron emission (i.e. a factor 30 in signal). This makes us\nconfident that in this area the cosmic signal is detectable by\nexperiments in this band. \nEven considering an uncertainty \nof $\\Delta\\alpha=0.2$ in the\nspectral index \\citep{platania98,bennett03}, the extrapolation would\nchange by a factor $\\sim 3$ in spectrum, not significantly affecting \nthe previous conclusion. \nIn addition, our estimate strengthens\nthe results obtained by the CMBP experiments, which\nfind indications that at 30 GHz the emission of this \nforeground is not dominant \\citep{leitch04,readhead04}.\nAt 90 GHz the frame is even better: the\nsynchrotron $E$-mode is 5~orders of magnitude weaker than the CMB,\ngiving a very negligible contamination in the CMB frequency window.\n\n\n\nOur measurements allow even larger improvements in the expected\ndetectability of the $B$--mode component.\nThe improved coverage of the\nangular spectrum gives a more reliable estimate of the $B$--mode\ncontamination near the  $\\ell\\sim 100$ CMBP peak.  \nThe power of this peak is expected to vary with the \ntensor-to-scalar perturbation ratio\n$T\/S$ so measuring the level of the GW\nbackground in the early Universe. \nOur data suggest that at 90~GHz the synchrotron contamination \ndominates the CMBP signal for models with  $T\/S < 0.01$,\nallowing the detection of this still unknown parameter well below\nits present upper limit ($T\/S < 0.90$ -- 95\\% C.L.:\n\\citealt{spergel03}).\n\n\nThe resulting picture is encouraging:  \nin this area of the sky the CMBP $E$--mode is expected a factor 30\nlarger than the synchrotron emission at $\\sim 30$~GHz, \nso appearing free of this contaminat, \neven allowing for possible uncertainties in the frequency \nextrapolation.  \nAt 90~GHz the contamination on\nthe $E$--mode is negligible, while that on\nthe $B$--mode appears to dominate the CMB signal only for\nmodels with $T\/S < 0.01$.\nThis low level is beyond the capability of the the PLANCK mission \n($\\Delta\\,(T\/S)\\sim 0.06$ -- 95\\%~C.L.: see \\citealt{tegmark00}) \nand only future experiments devoted to the search of the $B$-mode will\nbe limited by the foreground emission in this sky area.\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=0.90\\hsize]{fig6.ps}\n\\caption{Frequency behaviour of the synchrotron\nand dust $B$--mode in our patch. The quantity\n$\\sqrt{\\ell (\\ell + 1) C^B_l \/ (2 \\pi)}$ in $\\ell = 100$ is shown\nbecause a fair estimate of the signal on the scale where the CMB\n$B$--mode peaks ($\\theta \\sim 2^\\circ$).\nThe CMB level expected in the case of $T\/S = 0.01$ is  plotted\nfor comparison.\nSynchrotron emission is estimated by extrapolating to $\\ell = 100$\nthe fit of Table~\\ref{specTab}.\nWe assume a slope of $\\alpha = -3.1$ for the frequency behaviour.\nDust emission is evaluated from the total intensity\nupper limit measured with the BOOMERanG experiment at $b=-38^\\circ$\n\\citep{masi01}. A slope of $\\alpha = 2.2$\nis assumed for the frequency extrapolation \\citep{bennett03}.\nTwo values of polarization fraction (5\\% and 20\\%)\nare used to bracket the 10\\% deduced by \\citet{benoit04}\nfor high Galactic latitudes.}\n\\label{foregFig}\n\\end{figure}\n\nAlthough an accurate estimate of the dust\ncontribution in the area cannot be done due to the many uncertainties,\nwe indicate some plausible upper limits in Figure~\\ref{foregFig}.\nThese, and our new lower estimate of the synchrotron component, \nsuggest that the thermal dust could become the most important source of\nforeground  noise at 90~GHz. This would move the optimal window for\n$B$--mode investigations from 90--100~GHz toward 70~GHz,\nas for the CMB anisotropy \\citep{bennett03}.\nBetter estimates of the optimum $B$--mode observing frequency require a\ndirect measure of the polarized dust emission in this sky area, \npossibly at higher frequency where the dust\nemission is stronger.\n\n\n\n\\section*{Acknowledgments}\n\nThis work has been carried out in the frame of the SPOrt experiment, \na program funded by ASI (Italian Space Agency).\nG.B. and S.P. acknowledge ASI grants.\nWe wish to thank Bob Sault for \nthe ATCA observations of 3C138, \nWarwick Wilson for his support in the\nWide Band Correlator set-up, and\nJohn Reynolds for his support in the receiver set-up.\nPart of this work is based on observations taken with\nthe Parkes Radio telescope, which is part of the Australia Telescope,\nfunded by the Commonwealth of Australia for operation\nas a National Facility managed by CSIRO.\nWe acknowledge the use of the CMBFAST package.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction-1}\n\nA \\emph{conformal symplectic structure} on a manifold $M$ is a generalization of a symplectic structure. Locally, a conformal symplectic manifold is equivalent to a symplectic manifold, but the local symplectic structure is only well-defined up to scaling by a constant, and the monodromy of the local symplectic structure around curves may induce these rescalings. To our knowledge, the notion was first introduced by Vaisman in \\cite{Vaisman_lck} and \\cite{Vaisman}. They also appeared in work of Guedira and Lichnerowicz in \\cite{Guedira_lichnerow}. It was later studied by Banyaga \\cite{Banyaga_propertieslcs}, among many others.\\footnote{In previous literature, this structure is often called a \\emph{locally} conformal symplectic structure, which is a more accurate term. But also more cumbersome.} More recently a first general existence result was given in \\cite{Apo_Dlou} for complex surfaces with odd first Betti number, it was later proved in \\cite{E_M_Symp_Cob} that an almost symplectic manifold with non zero first Betti number admits a conformal symplectic structure, providing a large class of examples of such structures.\n \nMore formally, we can take a number of equivalent definitions. We can say that a conformal symplectic structure on $M$ is an atlas of charts to $\\mathbb{R}^{2n}$, so that the transition maps $\\psi_{ij}$ satisfy $\\psi_{ij}^*\\omega_\\text{std} = c_{ij}\\omega_\\text{std}$, for constants $c_{ij} > 0$. Equivalently, let $E \\to M$ be a flat, orientable, real line bundle. Then a conformal symplectic structure is a $2$-form on $M$ taking values in $E$, which is closed and non-degenerate.\n\nTaking a connection on $E$ leads to the most tractable definition. A \\emph{conformal symplectic structure} on $M$ is a pair $(\\eta, \\omega) \\in \\Omega^1M \\times \\Omega^2M$, so that $d\\eta = 0$, $d\\omega = \\eta \\wedge \\omega$, and $\\omega^{\\wedge n} \\neq 0$. Because the choice of connection is non-canonical, $(\\eta, \\omega)$ defines the same conformal symplectic structure as $(\\eta + df, e^f\\omega)$, for any $f \\in C^\\infty M$. (The choice of a specific $\\eta$ representing $[\\eta]$ is roughly analogous to a choice of contact form for a given contact structure, $\\eta$ is called a \\textit{Lee form} of the conformal symplectic structure since such a form appeared in the work of Lee \\cite{Lee_evendim}.)\n\nFor an example of a conformal symplectic manifold, let $\\beta$ be a closed $1$-form on a manifold $Q$, let $\\lambda = p\\cdot dq$ be the tautological $1$-form on $T^*Q$, let $\\eta \\in \\Omega^1(T^*Q)$ be the pullback of $\\beta$ under the projection, and let $\\omega = d\\lambda - \\eta \\wedge \\lambda$.\n\nConformal symplectic manifolds enjoy many of the properties that make symplectic manifolds interesting. The definitions of Lagrangian, isotropic, etc. are exactly the same, and there is a natural notion of exact conformal symplectic structures, and exact Lagrangians inside them. They have a natural Hamiltonian dynamics (a smooth function defines a flow preserving the structure). They satisfy a Moser-type theorem, which implies that Darboux's theorem and the Weinstein tubular neighborhood theorems hold in this context (a small neighborhood of any Lagrangian is equivalent to the example above). When restricted to a coisotropic, the kernel of $\\omega$ is a foliation, and in the case the leaf space is a manifold it inherits a conformal symplectic structure. The Poisson bracket on Hamiltonians intertwines the Lie bracket.\n\nHowever, many of the more modern methods in symplectic geometry cannot be easily generalized, and often the theorems fail to hold true. For example, suppose that $\\beta \\in \\Omega^1 Q$ is a $1$-form which never vanishes. Then in the conformal symplectic manifold $(T^*Q, \\eta, \\omega)$ defined above, the zero section $Z = \\{p=0\\}$ is displaceable by Hamiltonian isotopy. In fact, if $\\varphi_t$ is the Hamiltonian isotopy generated by the Hamiltonian $H = 1$, then $\\varphi_t(Z) \\cap Z = \\varnothing$ for any $t>0$.\n\nFrom the point of view of Floer theory, the problem with conformal symplectic structures is that we cannot even get started. The set of almost complex structures $J$ compatible with $\\omega$ is still a non-empty contractible space, and $\\overline\\partial_J$ is still an elliptic operator, but because $\\omega$ is not closed we have no bounds on energy, and therefore we do not expect Gromov compactness to hold, even for conformal symplectic manifolds which are both closed and exact. We discuss explicit examples suggesting failure of Gromov compactness in Section \\ref{sec:failure-compactness}. Whether Gromov compactness can be generalized to this context by defining a more sophisticated compactification remains to be seen. \n\nThe main theorem of this paper proves the following an analogous of Laudenbach-Sikorav's Theorem \\cite{Lau_Sik} in the context of conformal symplectic manifolds. We let $H^\\text{Nov}_*(L, [\\eta]; \\mathbb{F})$ be the Novikov homology of $L$ in the homology class $[\\eta] \\in H^1(L; \\mathbb{R})$.\n\n\\begin{thm}\\label{thm:rigidprinc}\nLet $\\eta$ be a closed $1$-form on an orientable manifold $Q$, and let $\\mathbb{F}$ be a field. Let $\\phi$ be an Hamiltonian diffeomorphism of the conformal symplectic manifold $(T^*Q,\\eta,d\\lambda-\\eta\\wedge\\lambda)$ (where $\\lambda$ is the canonical form) such that $\\phi(Q_0)$ intersect $Q_0$ transvesally, then $\\#\\{\\phi(Q_0) \\cap Q_0\\}\\geq \\operatorname{rk} H^\\text{Nov}_*(Q, [\\eta]; \\mathbb{F})$. If $Q$ is non-orientable then $\\#\\{\\phi(Q_0) \\cap Q_0\\}\\geq \\operatorname{rk} H^\\text{Nov}_*(Q, [\\eta]; \\mathbb{Z}_2)$\n\\end{thm}\n\n\\begin{coro}\\label{cor:C0 close}\nLet $(M, \\eta, \\omega)$ be a conformal symplectic manifold, and let $L \\subseteq M$ be a Lagrangian. If $\\varphi_t$ is any $C^0$ small Hamiltonian isotopy, then $\\#\\{\\varphi_1(L) \\cap L\\} \\geq \\operatorname{rk} H^\\text{Nov}_*(L, [\\eta]|_L; \\mathbb{Z}_2)$. If $L$ is oriented we may replace $\\mathbb{Z}_2$ with any field $\\mathbb{F}$.\n\\end{coro}\n\nIn order to prove Theorem \\ref{thm:rigidprinc} we prove an analogous to Sikorav's Theorem \\cite{Sikorav_famgen} about persistence of generating families which allows to provide bounds for intersection of Lagrangian submanifolds in conformal contangent in terms of stable $\\eta$-critical points of function. \n\nThe layout of the paper follows. Section \\ref{sec:introduction} introduces the basic definitions and theorems in conformal symplectic geometry. We include a number of propositions which are not necessary for the proof of Theorem \\ref{thm:rigidprinc}, with the hope of giving a large-scale introduction to the theory, particularly for symplectic and contact geometers. \n\nSection \\ref{sec:overv-morse-novik} gives a brief overview of Morse-Novikov homology needed for the proof of Theorem \\ref{thm:rigidprinc}. Finally, in Section \\ref{sec:rigid-lagr-inters} we complete the proof.\n\n\\section*{Acknowledgements}\n\\label{sec:aknowledgments}\nThe authors thank Vestislav Apostolov and Fran\u00e7ois Laudenbach for inspiring discussions. The first author benefited from the hospitality of several institutions and wishes to thank the institute Mittag-Leffler in Stockholm, CIRGET in Montr\u00e9al and MIT in Cambridge for the nice work environment they provided. The second author would like to thank Universit\\'e de Nantes and the Radcliffe Institute for Advanced Study for their pleasant work environments. B. Chantraine is partially supported by the ANR project COSPIN (ANR-13-JS01-0008-01) and the ERC starting grant G\\'eodycon. E. Murphy is partially supported by NSF grant DMS-1510305 and a Sloan Research Fellowship.\n\n\\section{Main definitions}\n\\label{sec:introduction}\n\n\\subsection{Conformal symplectic manifolds.}\n\\label{sec:conf-sympl-manif}\n\n\\begin{prop}\\label{prop:equiv def}\nLet $M$ be a $2n$-manifold. The following are equivalent:\n\n\\begin{itemize}\n\\item An atlas of charts $M = \\bigcup U_i$, $\\varphi_i:U_i \\to \\mathbb{R}^{2n}$, so that the transition maps $\\psi_{ij} = \\varphi_i \\circ \\varphi_j^{-1}$ preserve the standard symplectic form up to scaling by a positive local constant: $\\psi_{ij}^*(\\omega_\\text{std}) = c_{ij} \\omega_\\text{std}$. Two atlases are considered equivalent if they admit a common refinement.\n\n\\item A flat, real, orientable line bundle $E \\to M$, and a $2$-form $\\sigma \\in \\Omega^2(M, E)$, so that $\\sigma$ is non-degenerate (as a map $TM \\to T^*M \\otimes E$) and closed (as a form with values in a flat line bundle). Two such structures $(E_1, \\sigma_1)$, $(E_2, \\sigma_2)$ are considered equivalent if there is an isomorphism $\\varphi: E_1 \\to E_2$ covering the identity map, so that $\\varphi^*\\sigma_2 = \\sigma_1$.\n\n\\item A pair $(\\eta, \\omega) \\in \\Omega^1M \\times \\Omega^2M$, so that $d\\eta = 0$, $d\\omega = \\eta \\wedge \\omega$, and $\\omega^{\\wedge n} \\neq 0$. $(\\eta, \\omega)$ is equivalent to $(\\eta + df, e^f\\omega)$ for any $f \\in C^\\infty M$.\n\\end{itemize}\n\nAny of these structures is called a \\emph{conformal symplectic structure} on $M$.\n\\end{prop}\n\n\\begin{proof}\nGiven an atlas, the association of the positive number $c_{ij}$ to every intersection $U_i \\cap U_j$ can be thought of as the clutching function for a principal $GL^+(1, \\mathbb{R})$-bundle on $M$, with the discrete topology. That is, the numbers $c_{ij}$ define an orientable flat line bundle, $E$, and the form $\\sigma =\\varphi_i^*\\omega_\\text{std}$ is well defined as a $2$-form taking values in $E$. It is closed and non-degenerate, since these are local conditions. \n\nGiven a pair $(E, \\sigma)$, we can identify $E \\cong M \\times \\mathbb{R}$ as smooth vector bundles globally, since $E$ is a real, orientable line bundle. A choice of flat connection on $E$ then is simply a closed $1$-form $\\eta \\in \\Omega^1(M)$. In this way we can identify $\\sigma$ as an ordinary $2$-form $\\omega \\in \\Omega^2(M)$. Then to say that $\\omega$ is closed as a $2$-form with values in $E$ is equivalent to the statement $d\\omega - \\eta \\wedge \\omega = 0$. The choice of connection $\\eta$ is not canonical; gauge symmetries of $E$ act by $\\eta \\mapsto \\eta + df$ for any $f \\in C^\\infty(M)$, and this gauge symmetry acts on $\\omega$ by $\\omega \\mapsto e^f\\omega$.\n\nGiven a pair $(\\eta, \\omega)$ as above, consider the covering space $\\pi: \\widetilde M \\to M$ associated to $[\\eta]$. Then $\\pi^*\\eta = d\\theta$ for some $\\theta \\in C^\\infty \\widetilde M$, and for any covering transformation $g \\in \\pi_1M \/ \\ker [\\eta] \\subseteq \\operatorname{Diff}\\widetilde M$, we have $\\theta \\circ g = \\theta + \\langle[\\eta], g \\rangle$. Let $\\widetilde\\omega = e^{-\\theta}\\pi^*\\omega \\in \\Omega^2\\widetilde M$. Then $\\widetilde\\omega$ is a symplectic form, and $g^*\\widetilde\\omega = e^{-\\langle[\\eta], g \\rangle}\\widetilde\\omega$. Therefore by taking a Darboux atlas on $(\\widetilde M, \\widetilde\\omega)$, we get a conformal symplectic atlas on $M$.\n\\end{proof}\n\n\\begin{definition}\\label{def:deta}\nLet $\\eta$ be a closed $1$-form on a manifold $M$. The \\emph{Lichnerowicz-De Rham differential} on $\\beta \\in \\Omega^*(M)$ is defined as $d_\\eta \\beta=d\\beta-\\eta\\wedge\\beta$. \n\\end{definition}\n\nFrom the fact that $\\eta$ is closed and of odd degree we get that $d_\\eta^2=0$ and from the fact that $\\eta$ has degree $1$ we see that $d_\\eta$ is degree $1$ as well. Perhaps the most essential difference between $d_\\eta$ and $d$ is that $d_\\eta$ does not satisfy a Stokes' theorem. We note two important formulas:\n\n$$d_{\\eta + df}\\beta = e^f d_\\eta(e^{-f}\\beta)$$\n$$\\mathcal{L}_X\\beta = X \\, \\lrcorner \\, d_\\eta\\beta + d_\\eta(X \\, \\lrcorner \\, \\beta) + \\eta(X)\\beta$$\n\nwhere $\\mathcal{L}_X$ is the Lie derivative along the vector field $X$. By passing to the covering space of $M$ defined by $\\ker [\\eta]$, it follows that the homology $H^*(\\Omega^*M, d_\\eta)$ is isomorphic to $H^*(M; [\\eta], \\mathbb{R})$, the homology of $M$ with local coefficients defined by the homomorphism $[\\eta]: \\pi_1M \\to \\mathbb{R}$. \n\n\\begin{note}\nFor concreteness, we will state definitions and prove propositions in the setup where a conformal symplectic structure is a pair $(\\eta, \\omega)$, and then when relevant show that the definitions are invariant under gauge equivalence $\\eta \\rightsquigarrow \\eta + df$. By working directly with the complex $\\Omega^*(M, E)$ for the flat line bundle $E$, the arguments are more elegant, but also they are likely more opaque.\n\\end{note}\n\n\\begin{definition}\\label{def:basics}\nLet $(M, \\eta, \\omega)$ be a conformal symplectic structure. We say that a submanifold $L \\subseteq M$ is \\emph{isotropic} if $\\omega|_L = 0$, \\emph{coisotropic} if $TL^{\\perp \\omega} \\subseteq TL$, and \\emph{Lagrangian} if it is isotropic and coisotropic. \n\nWe say that $(\\eta, \\omega)$ is \\emph{exact} if $\\omega = d_\\eta\\lambda$ for some $\\lambda \\in \\Omega^1M$. In this case we say that $\\lambda$ is a \\emph{Liouville form} for $(\\eta, \\omega)$, and the vector field $Z_\\lambda$ defined by $Z_\\lambda \\, \\lrcorner \\, \\omega = \\lambda$ is called the \\emph{Liouville vector field}.\\footnote{This usage is somewhat different from the standard symplectic case, where a Liouville form is required to be a primitive of $\\omega$ and also satisfy a convexity condition near the boundary\/infinite portion of $M$. The same conditions on $Z_\\lambda$ make sense in this context, so they could easily be imposed.} If $L \\subseteq M$ is a Lagrangian in $(M, \\eta, d_\\eta\\lambda)$, we say that $L$ is \\emph{exact} if $\\lambda|_L = d_\\eta h$ for some $h \\in C^\\infty L$.\n\\end{definition}\n\nNotice that all of the above definitions are well defined up to gauge equivalence, because if we replace $(\\eta, \\omega)$ with $(\\eta + df, e^f\\omega)$, we can also replace $\\lambda$ and $h$ with $e^f\\lambda$ and $e^fh$. In particular, the Liouville vector field $Z_\\lambda$ is well defined independent of gauge, it is \\emph{not} rescaled by $e^f$. Note also that there is nothing precluding the existence of a closed, exact conformal symplectic manifold.\n\n\\begin{ex}\\label{ex:confsymplectisation}\nLet $(Y,\\lambda)$ be a manifold with a $1$-form $\\lambda$. We denote by $S^1_T$ the quotient $\\mathbb{R}\/T\\mathbb{Z}$ and parametrise it with the coordinate $\\theta$. On $S_T^1\\times Y$ let $\\eta = -d\\theta$. Then $\\omega =d_\\eta\\lambda =d\\lambda + d\\theta\\wedge\\lambda$, so $\\omega$ is non-degenerate if and only if $\\lambda$ is a contact form. Furthermore, given another contact form defining the same cooriented contact structure, $e^f\\lambda$, we have that the conformal Liouville manifold $(S_T^1 \\times Y, \\eta, \\lambda)$ is gauge equivalent to $(S_T^1 \\times Y, \\eta+df, e^f\\lambda)$, which is conformal symplectomorphic to $(S_T^1\\times Y, \\eta, e^f\\lambda)$ under the coordinate change $(\\theta, y) \\mapsto (\\theta - f(y), y)$.\n\nThe minimal cover which makes $\\eta$ exact is $\\mathbb{R}\\times Y\\rightarrow S^1\\times Y$, and the conformal Liouville structure $(\\eta, \\lambda)$ pulls back to $(-dt, \\lambda)$, which is gauge equivalent to the exact symplectic structure $e^t\\lambda$, known as the symplectization of $(Y,\\ker\\lambda)$. Hence, the conformal symplectic manifold will be called the \\textit{conformal symplectization} of $(Y,\\xi)$. We denote this manifold by $S^\\text{conf}_T(Y, \\ker\\lambda)$. (Notice that $\\langle[\\eta], S^1 \\times \\{\\text{point}\\}\\rangle = -T$, so the choice of $T$ affect the conformal symplectomorphism type.) If the choice of $T$ is irrelevant, we will use the notation $S^\\text{conf}(Y, \\ker\\lambda)$.\n\nNow, let $\\Lambda \\subseteq Y$ be a Legendrian submanifold. Then the cylinder $\\mathbb{R}\\times \\Lambda$ is an exact Lagrangian in the symplectization of $(Y, \\ker \\lambda)$, and furthermore it is invariant under the covering transformations of $\\mathbb{R}\\times Y\\rightarrow S^1\\times Y$. Therefore it descends to an exact Lagrangian submanifold $S^1 \\times \\Lambda$ of $S^\\text{conf}(Y, \\ker\\lambda)$. More generally, if $\\Sigma$ is a Lagrangian cobordisms from $\\Lambda$ to itself which is cylindrical outside of $(0,T)\\times Y$ then $\\Sigma$ descends to a Lagrangian submanifold of $S^\\text{conf}_T(Y, \\ker\\lambda)$.\n\\end{ex}\n\nThis paper will focus primarily on the following example:\n\n\\begin{ex}\\label{ex:cotangent}\nLet $\\beta$ be a closed $1$-form on a smooth manifold $Q$, let $\\pi$ be the projection $T^*Q\\rightarrow Q$, and let $\\lambda_\\text{std}$ be the tautological form on $T^*Q$. Define $\\eta:=\\pi^*\\beta$. Then $(T^*Q,\\eta, \\lambda_\\text{std})$ is a conformal Liouville manifold (since $\\eta\\wedge \\lambda_\\text{std} \\wedge d\\lambda_\\text{std}^{n-1}=0$). We will denote this conformal Liouville manifold by $T^*_\\beta Q$.\n\nLet $\\alpha:Q \\rightarrow T^*Q$ be a $1$-form. Then $\\alpha^*\\lambda_\\text{std}=\\alpha$ and thus $\\alpha(Q)$ is Lagrangian in $T_\\beta^*Q$ if and only if $d_\\beta \\alpha = 0$. Furthermore this Lagrangian is exact if and only if $\\alpha$ is $d_\\beta$-exact.\n\\end{ex}\n\n\\subsection{Conformal symplectic transformations.}\n\\label{ssec:conf-sympl-transf}\n\nA diffeomorphism between conformal symplectic manifolds $\\varphi: (M_1, \\eta_1, \\omega_1) \\to (M_2, \\eta_2, \\omega_2)$ is called a \\emph{conformal symplectomorphism} if $\\varphi^*\\eta_2 = \\eta_1 + df$ and $\\varphi^*\\omega_2 = e^f \\omega_1$ for some $f \\in C^\\infty M_1$. We denote by $\\operatorname{Symp}(M,\\eta,\\omega)$ the group of conformal symplectomorphisms of $(M,\\eta,\\omega)$. The Lie algebra $\\mathfrak{S}\\operatorname{ymp}(M,\\eta,\\omega)$ of this group is given by vector fields $X$ satisfying $\\mathcal{L}_X\\omega=f\\omega$ and $\\mathcal{L}_X\\eta=df$ for some $f$, and using Cartan's formula we see this is equivalent to\n\\begin{align*}\nd_\\eta(X \\, \\lrcorner \\, \\omega) + \\eta(X)\\omega &= \\mathcal{L}_X \\omega = f \\omega\\\\\nd(\\eta(X)) &= \\mathcal{L}_X\\eta =df\n\\end{align*}\n\nTherefore $X$ is a conformal symplectic vector field if and only if $d_\\eta(X\\, \\lrcorner \\, \\omega) = c\\omega$ for a constant $c \\in \\mathbb{R}$. \n\nThe subalgebra of $\\mathfrak{S}\\operatorname{ymp}(M,\\eta,\\omega)$ which are $\\omega$-dual to $\\eta$-closed $1$-forms (i.e. vector fields $X$ with $d_\\eta(X\\, \\lrcorner \\, \\omega) = 0$) will be called \\textit{divergence free conformal symplectic vector fields}. Notice the following fact: a conformal symplectic manifold is exact if and only if it admits a conformal symplectic vector field which is not divergence-free. In this case, after choosing a Liouville form $\\lambda$, we see that every conformal symplectic vector field is of the form $X + cZ_\\lambda$, where $X$ is divergence-free.\n\n\\begin{definition}\nThe conformal symplectic vector fields which are dual to $\\eta$-exact forms is called \\emph{Hamiltonian vector fields}. For any function $H \\in C^\\infty M$, the Hamiltonian vector field $X_H$ associated to $H$ is defined by $X_H \\, \\lrcorner \\, \\omega = - d_\\eta H$. A conformal symplectomorphism of $M$ which is the time-$1$ flow of a path of Hamiltonian vector fields will be called a \\textit{Hamiltonian diffeomorphism}.\n\\end{definition}\n\nNotice that the association $H \\rightsquigarrow X_H$ depends on $\\eta$, but the algebra of Hamiltonian vector fields (and therefore the group of Hamiltonian diffeomorphisms) does not. Using $X_H^\\eta$ to denote this dependence, we immediately have $X_H^\\eta = X_{e^f H}^{\\eta + df}$. To phrase the dependence in a gauge free way, let $E$ be the flat line bundle determined by $[\\eta]$, then Hamiltonians are taken to be $H \\in \\Omega^0(M, E)$. Since $\\omega \\in \\Omega^2(M, E)$, the equation $X_H \\, \\lrcorner \\, \\omega = - d_E H$ defines $X_H$ unambiguously.\\footnote{The reader experienced with contact geometry will be familiar with this situation: the contact vector field $X_H$ associated to a Hamiltonian $H \\in C^\\infty M$ depends on a choice of contact form, but the correct way to define the relationship without reference to a contact form is to take contact Hamiltonians $H \\in \\Omega^0(M, TM\/\\xi)$.}\n\n\\begin{definition}\nGiven a conformal symplectic structure with a chosen connection $\\eta$, we define the \\emph{Lee vector field} to be the Hamiltonian vector field generated by $H = 1$. We sometimes denote this vector field as $R_\\eta = X_1$.\n\\end{definition}\n\n\\begin{ex}\nGiven a contact manifold $(Y, \\ker \\alpha)$, we defined the conformal symplectization in Example \\ref{ex:confsymplectisation} by $(S^1 \\times Y, \\eta = -d\\theta, \\omega = d_\\eta\\alpha)$. In this case, the Lee vector field of $\\eta$ is equal to the Reeb vector field of $\\alpha$.\n\\end{ex}\n\nNote that this exhibit a first difference with the symplectic case:  this flow has no fixed point for small time. This is an example of a more general phenomenon: given a conformal symplectic manifold $(M, \\eta, \\omega)$, if $\\eta$ can be chosen to have no zeros, then $R_\\eta$ is a Hamiltonian vector field with no zeros, and therefore we can construct many Hamiltonian diffeomorphisms with no fixed points. Another place where this condition is relevant is about Lagrangian displaceabality:\n\n  \\begin{ex}\n    Given a cotangent bundle $T_\\beta^*Q$ with Liouville structure given by a closed one form $\\beta$ on $Q$ as in Example \\ref{ex:cotangent}, the Liouville vector field $Z_\\lambda$ is given by  the standard Liouville vector field on $T^*Q$. The Lee vector field $R_\\eta$ corresponds to the symplectic vector field dual to $\\eta$ (in the standard symplectic sense), i.e. its flow acts fiberwise, and translates the vertical fiber $T_q^*Q$ in the direction $\\beta_q$. In particular, if $[\\beta] \\in H^1Q$ can be represented by a non-vanishing closed $1$-form, then there exists an autonomous Hamiltonian flow on $T^*_\\beta Q$ so that $\\varphi_H^t(Q) \\cap Q = \\varnothing$ for \\emph{any} $t>0$.\n  \\end{ex}\n\nGiven an exact conformal symplectic manifold $(M, \\eta, d_\\eta \\lambda)$, there is some interplay between the vector fields $Z_\\lambda$ and $R_\\eta$. Always, we have $$\\lambda(R_\\eta) = \\omega(Z_\\lambda, R_\\eta) = - \\eta(Z_\\lambda).$$ Furthermore, suppose that $X \\in \\ker d\\lambda$. Since $\\omega = d_\\eta\\lambda = d\\lambda - \\eta \\wedge \\lambda$, we get that $X \\, \\lrcorner \\, \\omega = \\lambda(X)\\eta - \\eta(X)\\lambda$, and therefore $X = \\lambda(X)R_\\eta - \\eta(X)Z_\\lambda$ since $\\omega$ is non-degenerate. It follows that, if $d\\lambda$ has non-zero kernel, it must be equal to the span of $R_\\eta$ and $Z_\\lambda$. Plugging either $R_\\eta$ or $Z_\\lambda$ into the original equation, we see that $d\\lambda$ has non-zero kernel if and only if $\\eta(Z_\\lambda) = -1$, if and only if $\\lambda(R_\\eta) = 1$. \n\nIn the literature, exact conformal symplectic manifolds satisfying $\\eta(Z_\\lambda) = -1$ everywhere are often referred to as conformal symplectic manifolds \\emph{of the first kind}. Notice that this condition is not gauge invariant, and therefore it can be difficult to tell when a given conformal symplectic manifold is gauge-equivalent to one satisfying this property. However, this condition has strong implications, as shown in \\cite[Theorem A]{Bazzoni_conf}: if $M$ is compact and $\\eta(Z_\\lambda) = -1$ everywhere, and if $\\{\\eta = 0\\}$ has at least one compact leaf, then $M$ is conformal symplectomorphic to the suspension of a strict contactomorphism of a contact manifold $Y$. \n\n\\subsection{Moser's theorem and tubular neighborhoods}\n\\label{ssec:moser}\nWe have the following version of Moser's Theorem (first proved in \\cite{Banyaga_propertieslcs}).\n\n\n\\begin{thm}\\label{thm:Moser}\nLet $(M, \\eta)$ be a closed smooth manifold equipped with a closed $1$-form, and let $\\{\\omega_t\\}_{t \\in [0,1]}$, be a path of conformal symplectic structures in the same homology class, i.e. $\\omega_t = \\omega_0 + d_\\eta\\lambda_t$. Then there is an isotopy $\\varphi_t: M \\to M$ and a path of smooth functions $f_t \\in C^\\infty M$ so that $\\varphi_t^*\\eta = \\eta + df_t$ and $\\varphi_t^*\\omega_t = e^{f_t}\\omega_0$. That is, an exact homotopy of conformal symplectic structures is always induced by an isotopy (and gauge transformations)\n\\end{thm}\n\n\\begin{proof}\nWe find a vector field $X_t$ suitable for $X_t = \\dot{\\varphi_t} \\circ \\varphi_t^{-1}$, and then using compactness of $M$ integrate $X_t$. Differentiating the desired equations gives\n$$\\varphi_t^*\\left(\\mathcal{L}_{X_t}\\eta\\right) = d\\dot f_t$$\n$$\\varphi_t^*\\left(\\dot\\omega_t + \\mathcal{L}_{X_t}\\omega_t\\right) = \\dot f_t e^{f_t}\\omega_0 = \\dot f_t \\varphi_t^*\\omega_t.$$\n\nLetting $\\mu_t = \\dot f_t \\circ \\varphi_t^{-1}$, we have\n$$d(\\eta(X_t)) = \\mathcal{L}_{X_t}\\eta = d\\mu_t$$\n$$\\dot\\omega_t + d_\\eta(X_t \\, \\lrcorner \\, \\omega_t) + \\eta(X_t)\\omega_t = \\mu_t\\omega_t.$$\n\nDefining $X_t$ by $X_t \\, \\lrcorner \\, \\omega = \\dot\\lambda_t$, and taking $\\mu_t = \\eta(X_t)$, we see that this system has a solution.\n\\end{proof}\n\nJust as in the symplectic case, the Moser theorem for conformal symplectic structures is used to prove a Weinstein neighborhood theorem for Lagrangians in conformal symplectic manifolds, showing that Example \\ref{ex:cotangent} is a universal model. More precisely we have the following result, which appears in the recent paper \\cite{Otiman_Stanciu_darboux}. It also follows from a more general version for coisotropic manifolds, appearing in \\cite[Theorem 4.2]{VanLe_Oh_cositrop}.\n\n\\begin{thm}\\label{thm:lag nbhd}\nLet $(M,\\eta,\\omega)$ be a conformal symplectic manifold and $L\\subset M$ a compact Lagrangian. Let $\\beta=\\eta|_L$. Then there exists a neighborhood $\\mathcal{N} \\subseteq M$ of $L$ so that $\\mathcal{N}$ is conformally symplectomorphic to a neighborhood $U$ inside $(T^*_\\beta L, \\pi^*\\beta, d_{\\pi^*\\beta}\\lambda_\\text{std})$, sending $L \\subseteq \\mathcal{N}$ to the zero section $\\{p=0\\} \\subseteq U$.\n\\end{thm}\n\n\\begin{proof}\nLetting $\\mathcal{N}$ be a tubular neighborhood of $L$, we have a diffeomorphism $\\psi: T^*L \\to \\mathcal{N}$ sending the zero section $Z \\subseteq T^*L$ to $L\\subseteq \\mathcal{N}$, so that $\\psi^*\\omega|_{T_ZT^*L} = d_{\\pi^*\\beta}\\lambda_\\text{std}|_{T_ZT^*L}$. $\\psi^*\\eta$ and $\\pi^*\\beta$ are homologous, so by taking a gauge transformation on $\\mathcal{N}$ we can assume that they are equal. \n\nSince $\\psi^*\\omega_1|_Z = 0$, therefore $\\psi^*\\omega = d_{\\pi^*\\beta}\\lambda_1$ is exact (the Lichnerowicz-De Rham complex is homotopy invariant). We can furthermore assume that $\\lambda_1|_{T_ZT^*L} = 0$ by addition of a $d_{\\pi^*\\beta}$-closed $1$-form. Let $\\lambda_t = t\\lambda_1 + (1-t)\\lambda_\\text{std}$, and $\\omega_t = d_\\eta\\lambda_t$. By taking a smaller neighborhood if necessary, we may assume that $\\omega_t$ is non-degenerate for all $t \\in [0,1]$.\n\nWe then run the Moser method, as above. The vector field $X_t$ obtained is zero along $Z$, and therefore by shrinking $\\mathcal{N}$ further if necessary, we get a conformal symplectomorphism from a neighborhood of $Z \\subseteq T^*_\\beta L$ to $\\mathcal{N}$. \n\\end{proof}\n\n\\subsection{Flexibility and rigidity}\\label{ssec:formal}\n\nGiven a closed manifold $M^{2n}$, we would like to know when $M$ admits a conformal symplectic structure. If so, how many distinct structures are there up to isotopy. There are some basic obstructions\/invariants. For ease of notation we focus on the exact case.\n\n\\begin{definition}\nLet $M^{2n}$ be a closed manifold. An \\emph{exact almost conformal symplectic structure} (or EACS structure) is a pair $(a, w)$, where $a \\in H^1(M; \\mathbb{R})$, and $w \\in \\Omega^2M$ is a non-degenerate $2$-form.\n\\end{definition}\n\nAny exact conformal symplectic structure $(\\eta, d_\\eta\\lambda)$ defines an almost conformal symplectic structure with $a = [\\eta]$, $w = d_\\eta\\lambda$. Therefore, for a manifold to admit an exact conformal symplectic structure, it must also admit an EACS structure. Similarly, for two conformal symplectic structures to be isotopic, they necessarily must be homotopic through EACS structures (where $w$ is homotoped but $a$ is fixed). The benefit of passing to EACS structures is that they are classified purely by algebraic topology.\n\nWe show that that the classification of exact conformal symplectic structures is strictly more subtle than their almost conformal counterpart.\n\n\\begin{prop}\\label{prop:nonflexible}\nThere exists a manifold $M$ of any dimension and and $ [\\eta] \\neq 0 \\in H^1M$, which admits two exact conformal symplectic structures $(\\eta, d_\\eta\\lambda_1)$ and $(\\eta, d_\\eta\\lambda_2)$ which are homotopic through non-degenerate $2$-forms, but they are not conformally symplectomorphic.\n\\end{prop}\n\n\\begin{proof}\nWe take $M = S^1 \\times S^{2n-1}$. Let $\\xi_\\text{ot} = \\ker \\alpha_\\text{ot}$ be an overtwisted contact structure (see \\cite{BEM}) on $S^{2n-1}$, which is homotopic through almost contact structures to the standard contact structure $\\xi_\\text{std}$. \n\nAs the symplectization of $(S^{2n-1},\\xi_\\text{ot})$ is different form the one of $(S^{2n-1},\\xi_\\text{std})$ (actually the second cannot embed in the first as $(S^{2n-1},\\xi_\\text{std})$  is fillable) we know that those two structure are not equivalent.\n \nMoser's theorem therefore implies that they are not homotopic through exact conformal symplectic structures. But they are homotopic through non-degenerate $2$-forms, since the original contact structures are homotopic through almost contact structures.\n\\end{proof}\n\n\\begin{remark}\n  As shown in \\cite{Murphy_closedlagsymplectisation}, when $n>2$ the symplectization $(\\mathbb{R} \\times S^{2n-1}, e^r\\alpha_\\text{ot})$ contains an exact Lagrangian, $L \\subseteq \\mathbb{R} \\times S^{2n-1}$. Let $R > 0$ be a constant so that $L \\subseteq (0, R) \\times S^{2n-1}$. Then by identifying $S^1 = \\mathbb{R} \/ R\\mathbb{Z}$ $L$ embeds as an exact Lagrangian into the conformal symplectization $(M, \\eta = -d\\theta, \\lambda_1 = \\alpha_\\text{ot})$.\n\nHowever, there can be no nulhomologous exact Lagrangian in the conformal symplectization of the standard contact structure, $(S^1 \\times S^{2n-1}, \\eta, \\lambda_2 = \\alpha_\\text{std})$. For if there was, it would lift to a compact exact Lagrangian in the universal cover, which is conformally symplectomorphic to $\\mathbb{C}^n \\setminus \\{0\\}$, which then contradicts Gromov's theorem. This shows that those two conformal symplectic structure can be distinguished by their Lagrangian submanifolds. \n\\end{remark}\nFor $2n = 2$ we know that conformal symplectic structures are equivalent to almost conformal symplectic structures, since all $2$-forms are exact when $[\\eta] \\neq 0$. \n\nOn the question of existence, a nearly complete answer was established in \\cite{E_M_Symp_Cob}:\n\n\\begin{thm}\nLet $M$ be a closed manifold with an EACS structure $(a, w)$, where $a \\neq 0$ is in the image $H^1(M; \\mathbb{Z}) \\to H^1(M, \\mathbb{R})$. Then for all sufficiently large $C \\in [1, \\infty)$, there is an exact conformal symplectic structure $(\\eta, d_\\eta\\lambda)$, with $[\\eta] = Ca$ and $d_\\eta\\lambda$ being homotopic to $w$ through non-degenerate $2$-forms.\n\nIn particular, given any closed manifold $M$ satisfying $H^1(M; \\mathbb{R}) \\neq 0$ and having a non-degenerate $2$-form, $M$ admits an exact conformal symplectic structure.\n\\end{thm}\n\nThis follows indeed from the existence result of symplectic structure on almost symplectic cobordisms by cutting along an hypersurface Poincar\u00e9 dual to $[\\eta]$ which is non-separating by hypothesis. Making then the cobordism symplectic between the same contact structure on the ends leads to a symplectic cobordisms whose ends can be identified to give a conformal symplectic manifold.\n\nIt is an open question whether we can in general construct conformal symplectic structures where $[\\eta]$ is a prescribed $1$-form, either in the case where $[\\eta]$ is small, or when $[\\eta]: \\pi_1M \\to \\mathbb{R}$ has non-discrete image. Note that in \\cite{Apo_Dlou}, there is one example that shows that after fixing a complex structure J, the set of $[\\eta]$ for which there exists a conformal symplectic structure compatible with $J$ consists of a single point. \n\n\\begin{remark}\nFrom this theorem, it is easy to construct non-exact conformal symplectic structures: given an exact symplectic structure $(\\eta, d_\\eta\\lambda)$ and a class $b \\in H^2(M; [\\eta], \\mathbb{R})$, choose a $d_\\eta$-closed $2$-form $B \\in \\Omega^2M$ representing $b$. Then for sufficiently large $C$, $\\omega = Cd_\\eta\\lambda + B$ is a conformal symplectic structure.\n\\end{remark}\n\n\\begin{remark}\nThe same strategy also works for constructing conformal symplectic manifolds with convex contact boundary: any contact manifold which admits an almost complex filling admits a conformal symplectic filling. Indeed from such a formal filling the main theorem in \\cite{E_M_Symp_Cob} allows us to construct a symplectic cobordisms from $(S^{2n-1},\\xi_\\text{ot})$ to $(S^{2n-1},\\xi_\\text{ot})\\sqcup (Y,\\xi)$, and gluing the two $(S^{2n-1},\\xi_\\text{ot})$ leads to a conformal symplectic filling of $(Y, \\xi)$.\n\nAs stated there, the theorem only applies to cobordisms with tight convex boundary when $n > 2$. However, after an examination of the proof there it is clear that the result applies when $n=2$ as well, as long as a single component of the convex boundary is overtwisted.\n\\end{remark}\n\n\\subsection{Pseudo-holomorphic curves and a ``failure'' of Gromov compactness.}\n\\label{sec:failure-compactness}\n\nInspired from successes in symplectic and contact geometry, we might try to develop a theory of pseudo-holomorphic curves for conformal symplectic manifolds. Similar to the symplectic case, in a conformal symplectic manifold the space of compatible almost complex structures $J$ is contractible, and the equation $\\overline{\\partial}_J u = 0$ is elliptic. But when trying to prove a compactness theorem, we run into an immediate problem: the Lichnerowicz-De Rham complex does not satisfy Stokes' theorem, and without this we have no way to give a $C^0$ bound on holomorphic energy. We give here two examples of holomorphic curves arising naturally which suggest bad compactness properties.\n\nConsider the conformal sympectisation of an overtwisted sphere $Y = S^{2n-1}_\\text{ot}$ with a generic contact form. When equipped with a cylindrical almost complex structure $J$, there is a holomorphic plane which is asymptotic to one of the Reeb orbits, as shown in \\cite{Albers_Hofer}. In the conformal symplectization $S^\\text{conf}(Y)$ this gives a holomorphic plane, which looks something like a leaf of a Reeb type foliation. Though such curves are handled in the symplectization setting by SFT compactness \\cite{Bourgeois_&_Compactness} and \\cite{Abbas_book}, this sort of compactness strongly relies on having contact asymptotics and a cylindrical complex structure, which is not a natural notion in general conformal symplectic manifolds.\n\nA second phenomena is given by the conformal symplectic filling of $S^3_{\\text{ot}}$ described in the previous section. The bishop family near the elliptic point of an overtwisted disk leads to a family of holomorphic curves which cannot be compactified in a standard way. We suspect that such a family breaks into curves involving half-plane such as those described before.\n\n\\subsection{Contactisation, reduction, and generating functions}\n\\label{ssec:cont-red-gen}\n\n\\begin{definition}\\label{def:contactization}\nIf $(M,\\eta,\\lambda)$ is a conformal Liouville manifold, we define the \\emph{contactisation} of $(M, \\eta, \\lambda)$ to be the manifold $M\\times \\mathbb{R}$ equipped with the contact structure given by the kernel of $\\alpha=dz-z\\eta-\\lambda=d_\\eta z-\\lambda$ (note that $\\alpha$ is a contact form if and only if $d_\\eta \\lambda$ is non-degenerate).\n\\end{definition}\n\nNotice that an exact Lagrangian submanifold $L$ lifts to a Legendrian submanifold $\\widetilde{L}= \\{(q,f(q)); q \\in L\\}$ where $f$ is such that $d_\\eta f=\\lambda|_L$.\n\nIn the cotangent case, the space $\\mathcal{J}^1_\\beta(Q):=T^*_\\beta Q\\times \\mathbb{R}$ with the contact form $d_\\beta z-\\lambda$ is called the \\emph{$\\beta$-jet bundle}, natural Legendrian submanifolds arise as $\\beta$-jets of functions $f$, i.e. $j^1_\\beta f(q) = (q,d_\\beta f_q,f(q))$. Note that $\\mathcal{J}^1_\\beta(Q)$ is actually contactomorphic to $\\mathcal{J}^1(Q)$ by the map $\\varphi(q,p,z) = (q,p-z\\beta_q,z)$. However, this contactomorphism is not compatible with the projection to $T^*Q$ and therefore does not identify the symplectic properties of $T^*Q$ with the conformal symplectic properties of $T^*_\\beta Q$. Rather, it maps $\\beta$-jets of functions to (regular) jets of functions.\n\nThe Reeb vector field of $\\alpha$ is given by $R_\\alpha=(1+\\lambda(R_\\eta))\\partial_z-R_\\eta$.\n\n\\begin{ex}\n  Let $(M,\\alpha)$ be a contact manifold. Then the contact form on $M\\times S^1\\times\\mathbb{R}$ is $dz-zd\\lambda-\\alpha$ the Reeb vector field is just the original Reeb vector field seen as vector fields invariant by $S^1\\times\\mathbb{R}$.\n  The contactization of a conformal symplectization is a contact manifold associated to $(M,\\ker \\alpha)$, this manifold is the product of the original contact manifold and $T^*S^1$ with the Liouville structure given by $pd\\theta-dp$.\n\\end{ex}\n\nLet $(M,\\eta,\\omega)$ be a conformal symplectic manifold, and let $N\\subset M$ be a coisotropic submanifold (i.e. such that $TN^{\\perp \\omega}\\subset TN$). Since $d\\omega=\\eta\\wedge \\omega$ it follows from the Fr\u00f6benius integrability theorem that the distribution $\\ker(\\omega|_N)$ is integrable. Note that for a vector field $X$ in $\\ker \\omega|_N$ we have that\n\\begin{align}\\label{eq:3}\n \\mathcal{L}_X(\\omega|_N)&=\\eta(X)\\omega \\\\\n\\mathcal{L}_X\\eta&=d(\\eta(X)).\\nonumber\n\\end{align}\n\n\nLet $N^\\omega$ be the leaf space of the associated foliation and assume that $N^\\omega$ is a manifold. It follows from equation \\eqref{eq:3} that on charts of $N^\\omega$ there is a well define conformal symplectic structure and thus that $N^\\omega$ is a conformal symplectic manifold. This conformal symplectic manifold is called the \\textit{conformal symplectic reduction} of $N$.\n\n\\begin{ex}\n  Let $Q,Q'$ be two manifolds and $\\beta,\\beta'$ closed one forms on $Q,Q'$. And let $T^*_{\\beta\\oplus\\beta'}(Q\\times Q')\\simeq T^*_\\beta Q\\times T_{\\beta'}Q'$ be the conformal symplectic manifold associated to $\\pi^*\\beta\\oplus(\\pi')^*\\beta'$. Then $T^*_\\beta Q\\times Q'_0$ is cositropic and its reduction gives back $T^*_\\beta Q$.\n\\end{ex}\n\nConsider a Lagrangian submanifold $L$ of $(M, \\eta, \\omega)$ and assume the coisotropic $N$ intersects $L$ transversely. Let $\\varphi$ be the projection $N\\rightarrow N^\\omega$. Then the manifold $L_N=\\varphi(L\\cap N)$ is an immersed Lagrangian submanifold of $N^\\omega$. (This is linear algebra and thus equivalent to the symplectic case and follows from\\cite[Section 5.1]{Bates_Wein} for instance).\n\nNow given a map $F:Q\\times Q'\\rightarrow \\mathbb{R}$, we can take the graph to get a Lagrangian section $d_{\\beta\\oplus\\beta'}F$ of $T^*_{\\beta \\oplus \\beta'}(Q \\times Q')$, and apply symplectic reduction as above to the coisotropic $T^*_\\beta Q \\times Q'_0$ to obtain an immersed exact Lagrangian submanifold of $T_{\\beta}Q$. When $Q'=\\mathbb{R}^m$ (and $\\beta'=0$) and $F$ is quadratic at infinity we denote the corresponding Lagrangian $L_F$. $F$ is called a \\textit{$\\beta$-generating family} for $L_F$.\n\nNote that $L_F$ lifts to a Legendrian submanifold $\\Lambda_F$ of $\\mathcal{J}_\\eta^1(Q)$ and through the contactomorphism $\\phi$ defined above we get a Legendrian submanifold of $\\mathcal{J}^1(Q)$, and $F$ is a generating family (in the standard sense) for $\\phi(\\Lambda_F)$. Chekanov persistence's Theorem \\cite{Chekanov_Quasifunctions_Generating} therefore implies that if $L$ in $T_\\beta^*Q$ admits a generating family and $L_t$ is an isotopy of $L$ through exact Lagrangians then $L_1$ admits a $\\beta$-generating family as well.\n\nWe remark that if $L$ is given by a $\\beta$-generating family $F:Q\\times \\mathbb{R}^{m}$ then zeros of $d_\\eta F$ (called $\\eta$-critical points of $F$) corresponds to intersection points between $L$ and the zero section. All together, this implies\n\n\\begin{thm}\\label{stablecrit}\nLet $\\beta$ be a closed $1$-form on a closed manifold $Q$, and define $\\operatorname{Stab}_{\\beta}(Q) = \\min\\limits_F \\#\\{q \\in Q | d_\\beta F(q)=0\\}$, where $F$ is taken among all functions $F:Q\\times \\mathbb{R}^m\\rightarrow \\mathbb{R}$ which are quadratic at infinity, for all $m \\in \\mathbb{N}$. That is, $\\operatorname{Stab}_\\beta(Q)$ is the stable $\\beta$-critical number of $Q$. Let $L \\subseteq T^*_\\beta Q$ be a Lagrangian which is Hamiltonian isotopic to the zero section $Z$. Then the number of intersections between $L$ and $Z$ is at least $\\operatorname{Stab}_{\\beta}(Q)$.\n\\end{thm}\n\nForm the previous theorem we find that in order to have a lower bounds on the number of intersection points of a deformation a the $0$-section in a cotangent bundle we need to estimate the number of $\\beta$-critical points of a generating family $F$. To get such estimates we study the Morse-Novikov homology of a deformation of the function $\\ln |F|$.\n\n\\section{Overview of Morse-Novikov homology}\n\\label{sec:overv-morse-novik}\n\nIn this section we present the basics of the construction of the Morse-Novikov complex associated to a closed $1$-forms on $Q$. We also recall its essential properties we need to prove Theorem \\ref{thm:rigidprinc}. We refer the reader to more comprehensive references like \\cite{Farber_oneforms} and \\cite{Pajitnov_circledvalued} for more details.\n\nWe assume that $Q$ is connected. A closed $1$-forms $\\eta$ is \\textit{Morse} if near each of its zeros it is the derivative of a non-degenerate quadratic form $h$. It follows from Morse's Lemma that any $1$-forms whose graph intersect the $0$-section transversely is Morse, therefore closed $1$-forms are generically Morse. We refer to a zero of a closed $1$-form $\\eta$ as a \\textit{critical point} of $\\eta$. Given a critical point $q$ of $\\eta$ the number of negative eigenvalues of the quadratic form $h$ such that near $q$, $\\eta=dh$ is independent of the coordinate system, this number is called the \\textit{index} of $q$ written $I_\\eta(q)$. \n\nGiven a metric $g$ on $Q$ we define the vector field $\\nabla\\eta$ to be the dual of $\\eta$ with respect to $g$. We denote by $x_t^\\eta$ the flow of $\\nabla\\eta$. We will always assume that there is a compact subset $K$ such that \n\\begin{itemize}\n\\item The frontier of $K$ is a manifold with corners.\n\\item All critical points of $\\eta$ are in $K$ (in particular there are finitely many critical points),\n\\item $\\nabla\\eta$ is complete, so $x_t^\\eta$ is defined for all $t \\in \\mathbb{R}$,\n\\item For every component $V$ of $Q\\setminus K$ either for all $q\\in V$ and $t\\geq 0$ $x_t^\\eta(q)\\not\\in K$ or for all $q\\in V$ and $t\\leq 0$ $x_t^\\eta(q)\\not\\in K$. \n\\end{itemize}\nAny critical points $q$ have stable and unstable manifolds $W^s(q)$ and $W^u(q)$ defined by\n$W^s(q)=\\{q'|\\lim_{t\\rightarrow \\infty}x_t^\\eta(q')=q\\}$ and $W^u(q)=\\{q'|\\lim_{t\\rightarrow -\\infty}x_t^\\eta(q')=q\\}$.\n\nNote that $W^u(q)$ is a disk of dimension $I_\\eta(q)$ and $W^s(q)$ is a disk of dimension $n-I_\\eta(q)$. We say that the pair $(\\eta,g)$ is \\textit{Morse-Smale} if for any critical points $q,q'$ of $\\eta$ the disks $W^u(q)$ and $W^s(q')$ intersects transversely. In this situation $W^s(q)\\cap W^u(q')$ are open manifolds of dimension $I_\\eta(q)-I_\\eta(q')$ with an action of $\\mathbb{R}$ on them which is free (unless $q=q'$).\n\nFor a ring $R$ we denote by $\\Lambda_\\eta(Q,R)$ the completion of the group ring $R[\\pi_1(Q)]$ given by series of the form $\\sum_ia_ig_i$ where $\\lim_{i\\rightarrow\\infty}\\int_{g_i}\\eta=-\\infty$.\n \nNow for any critical points $q$ of $\\eta$ we choose \n\\begin{enumerate}\n\\item A path $g_q$ from $q$ to the base point.\n\\item An orientation of the tangent space of $q$ (we do not assume $Q$ is orientable).\n\\end{enumerate}\n\nGiven two critical points $q,q'$ of $\\eta$ such that $I_\\eta(q)-I_\\eta(q')=1$ then any component $\\gamma$ of $W^s(q)\\cup W^u(q')$ is copy of $\\mathbb{R}$ which compactifies to a path from $q$ to $q'$ which, together with the capping paths $g_q$ and $g_{q'}$, gives an element $g_\\gamma$ of $\\pi_1(Q)$.\n\nThis gives the series $u_{q,q'}=\\sum_\\gamma\\pm g_\\gamma$, which is in $\\Lambda_\\eta(Q,R)$ since we follow negative trajectories of $\\nabla\\eta$. The sign $\\pm$ is determined by whether or not the chosen orientation of $T_qQ$ transports to the one of $T_{q'}Q$ along the path $\\gamma$.\n\nThe Morse-Novikov complex of $(\\eta,g)$ is given by $C_k(\\eta,R)=\\oplus_{I_\\eta(q)=k}\\Lambda(Q,R)\\langle q\\rangle$ with differential $d(q)=\\sum_{I_\\eta(q')=k-1}u_{q,q'}q'$. We have that $d^2=0$ and the homology of $(C_*,d)$ is called the Morse-Novikov homology of $\\eta$, denoted $H^{\\text{Nov}}_*(\\eta)$.\n\nWhen $Q$ is compact it follows from a Theorem of Sikorav in \\cite{Sik_Thesis} (see also \\cite{Poz_Thesis}) that this homology is the homology of $Q$ with local coefficients in $\\Lambda_\\eta(Q, R)$. Notice that if $Q$ is compact with boundary and if $\\nabla \\eta$ points inward the boundary then the same results holds.\n\nWhen $R$ is a principal ring, $\\Lambda_\\eta (Q,R)$ is a principal ring as well and we call the $k-th$ \\textit{Novikov-Betti number} $b^k_\\eta$ of $\\eta$ the rank of the free part of $H^{\\text{Nov}}_k(\\eta)$. For compact oriented closed manifold they satisfy $b^k_\\eta=b^{n-k}_\\eta$ (see \\cite[Corollary 2.9]{Pajitnov_circledvalued} and \\cite[Section 1.5.3]{Farber_oneforms}), for non-oriented on the same is true if $R=\\mathbb{Z}_2$. \n\nNote that $\\Lambda_\\epsilon(Q,R)$ is a module over $R[\\pi_1(Q)]$, the Morse-Novikov homology of $\\eta$ is isomorphic to the homology with local coefficient in $H_*(K,\\partial_- K;\\Lambda_\\epsilon(K,R))$ (see \\cite{Latour}) where $\\partial_- K$ (resp $\\partial_+ K$) denotes the points where $\\nabla \\eta$ points outward (resp. inward) $K$.\n\n\n \\section{Rigidity of Lagrangian intersection}\n\\label{sec:rigid-lagr-inters}\nWe are now ready to prove Theorem \\ref{thm:rigidprinc}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:rigidprinc}]\nFrom Theorem \\ref{stablecrit} it remains to find a lower bound of the number of zeroes of $dF-F\\beta$ where $F$ is a function on $Q\\times\\mathbb{R}^m$, so that there is a compact ball $B \\subseteq \\mathbb{R}^m$ such that outside of $Q\\times B$, $F(x,\\xi)=q(\\xi)$ where $q$ is a non-degenerate quadratic form of index $k$. Because we can stabilize $F$ we can assume that $k>0$ without losing generality. Up to a small perturbation not affecting the zeroes we can also assume that $\\beta$-critical values of $F$ are not $0$. Fix a metric $g$ on $Q$ inducing the product metric $g\\oplus g_0$ on $Q\\times \\mathbb{R}^m$, we denote by $\\nabla: T^*(M\\times \\mathbb{R}^m)\\rightarrow T(M\\times \\mathbb{R}^m)$ the map induced by this metric.\n\nChoose $\\varepsilon>0$ such that for all $(x,\\xi)$ such that $|F(x,\\xi)|<2\\varepsilon$ we have $d_\\beta F(x,\\xi)\\not=0$ and $dF(x,\\xi)\\not=0$. Let $X_+:=\\{(x,\\xi)|F(x,\\xi)\\geq\\varepsilon\\}$, $X_-:=\\{(x,\\xi)|F(x,\\xi)\\leq-\\varepsilon\\}$ and $X_0:=\\{(x,\\xi)|-\\varepsilon\\leq F(x,\\xi)\\leq \\varepsilon\\}$.\n\nLet $G$ be a function on $Q\\times \\mathbb{R}^N$ so that $G(x,\\xi)=|F(x,\\xi)|$ outside of $X_0$, depends only on $\\xi$ outside of $Q\\times B$ and $G(x, \\xi) > 0$ everywhere. Note that outside of $Q\\times B$ the gradient vector field of $d_\\beta|F|$ is transverse to $\\partial X_\\pm$ so we can ensure that $\\nabla(dG - G\\beta) \\pitchfork \\partial X_0$. \n\nWe then let $H := \\log G$, and notice that the zeros of $\\gamma = dH - \\beta$ are the same as the zeros of $G(dH - \\beta) = dG - G\\beta$. Therefore, the zeros of $dF - F\\beta$ are the same as the zeros of $\\gamma$ on $X_+ \\cup X_-$, though $\\gamma$ will also have additional zeros in $X_0$. Since $\\gamma$ is closed and $[\\gamma] = [\\beta]$ we can now relate this to $H^\\text{Nov}_*(Q, [\\beta])$.\n\nAt first, we work with $H^\\text{Nov}_*$ as homology with local coefficients in $\\Lambda_\\beta(Q, \\mathbb{F})$, rather than Morse-Novikov homology. Throughout the next section, $H_*^{\\text{Nov}}$ will always be taken in the homology class $[\\beta]$. Let $r = \\operatorname{rk}(H_*^\\text{Nov}(Q, [\\beta]))$, and consider the exact sequence of the pair $(X_0, X_0 \\setminus Q \\times B)$.\n\n$$\n\\xymatrix{H_*^\\text{Nov}(X_0, X_0 \\setminus Q \\times B) \\ar[r]^{[-1]} &H_*^\\text{Nov}(X_0 \\setminus Q \\times B) \\ar[d]\\\\\n&H_*^\\text{Nov}(X_0) \\ar[ul]&}  \n$$\n\nOutside of $B$, we have that $F(x, \\xi) = q(\\xi)$, and therefore $X_0 \\setminus Q \\times B \\cong Q \\times S^{k-1} \\times S^{m-k-1} \\times (-\\varepsilon, \\varepsilon) \\times (0, \\infty)$. Thus \n$$\\operatorname{rk}(H_*^\\text{Nov}(X_0 \\setminus Q \\times B)) = \\operatorname{rk}(H^{\\text{Nov}}_*(Q) \\otimes H_*(S^{k-1}\\times S^{m-k-1})) = 4r,$$\n and it follows that either $H_*^\\text{Nov}(X_0, X_0 \\setminus Q \\times B)$ or $H_*^\\text{Nov}(X_0)$ must have total rank at least $2r$. We consider each case separately, with the goal of proving the inequality \n \\begin{equation}\\label{eq:1}\n \\operatorname{rk}(H_*^\\text{Nov}(X_+\\partial X_+)) + \\operatorname{rk}(H_*^\\text{Nov}(X_-, \\partial X_-)) \\geq r.\n \\end{equation}\n\n\\emph{Case (1) \\quad $\\operatorname{rk}(H_*^\\text{Nov}(X_0)) \\geq 2r$:}\n\nConsider the Mayer-Vietoris sequence for $(X_+ \\cup X_0) \\cup (X_- \\cup X_0)$, which is\n\n$$\n\\xymatrix{H_*^\\text{Nov}(Q \\times \\mathbb{R}^m) \\ar[r]^{[-1]} & H_*^\\text{Nov}(X_0) \\ar[d]\\\\\n&H_*^\\text{Nov}(X_+ \\cup X_0) \\oplus H_*^\\text{Nov}(X_- \\cup X_0)\\ar[ul]&}  \n$$\n\nThis implies that $$\\operatorname{rk}(H_*^\\text{Nov}(X_+ \\cup X_0)) + \\operatorname{rk}(H_*^\\text{Nov}(X_- \\cup X_0)) \\leq \\operatorname{rk}(H_*^\\text{Nov}(X_0)) + r.$$ \n\nBy looking at the exact sequence of the pair $(X_+ \\cup X_0, X_0)$, we get\n\n$$\\operatorname{rk}(H_*^\\text{Nov}(X_+ \\cup X_0, X_0)) \\geq \\operatorname{rk}(H_*^\\text{Nov}(X_0)) - \\operatorname{rk}(H_*^\\text{Nov}(X_+ \\cup X_0)),$$\n\nand we get a similar inequality for the pair $(X_- \\cup X_0, X_0)$. Rearranging these three inequalities gives\n\n\\begin{align*}\n\\operatorname{rk}(H_*^\\text{Nov}&(X_+ \\cup X_0, X_0)) + \\operatorname{rk}(H_*^\\text{Nov}(X_+ \\cup X_0, X_0)) \\geq \\\\\n& 2 \\operatorname{rk}(H_*^\\text{Nov}(X_0)) - \\operatorname{rk}(H_*^\\text{Nov}(X_+ \\cup X_0)) - \\operatorname{rk}(H_*^\\text{Nov}(X_- \\cup X_0)) \\geq \\\\\n& \\operatorname{rk}(H_*^\\text{Nov}(X_0)) - r \\geq r,\n\\end{align*}\n\nwhich proves the inequality \\ref{eq:1} after applying excision.\n\n\\emph{Case (2) \\quad $H_*^\\text{Nov}(X_0, X_0 \\setminus Q \\times B) \\geq 2r$:}\n\nConsider the Mayer-Vietoris sequence:\n\n\\begin{equation}\\label{eq:2}\n\\xymatrix{H_*^\\text{Nov}(Q \\times \\mathbb{R}^m, Q \\times \\mathbb{R}^m \\setminus Q \\times B) \\ar[r]^{[-1]} & H_*^\\text{Nov}(X_0, X_0 \\setminus Q \\times B) \\ar[d]\\\\\n& \\hspace{-2.2 in} H_*^\\text{Nov}(X_+ \\cup X_0, (X_+ \\cup X_0)\\setminus Q \\times B) \\oplus H_*^\\text{Nov}(X_- \\cup X_0, (X_- \\cup X_0)\\setminus Q \\times B)\\ar[ul]&}  \n\\end{equation}\n\nNotice that the pair $(X_+ \\cup X_0, (X_+ \\cup X_0) \\setminus Q \\times B)$ is homeomorphic to $(X_+, X_+ \\setminus Q \\times B)$, since $\\nabla F$ presents $X_0$ as a collar neighborhood of $\\partial X_+ \\cup X_0$. Using excision we have that $$H_*^\\text{Nov}(X_+, X_+\\setminus Q \\times B) \\cong H_*^\\text{Nov}(X_+ \\cap Q \\times B, X_+ \\cap \\partial (Q \\times B)).$$ Furthermore since we are using field coefficients, and using the hypothesis of either orientability of $Q$ or $\\mathbb{F} = \\mathbb{Z}_2$ we can apply Poincar\\'e duality to obtain $$H_*^\\text{Nov}(X_+\\cap Q \\times B, X_+ \\cap \\partial (Q \\times B)) \\cong H_{m+n-*}^\\text{Nov}(X_+ \\cap Q \\times B, (\\partial X_+)\\cap Q \\times B).$$\n\nWe also note that the pair $(X_+ \\cap Q\\times B, (\\partial X_+) \\cap Q\\times B)$ is homeomorphic to $(X_+, \\partial X_+)$, since $F$ is standard outside $Q \\times B$. In summation we obtain an isomorphism\n\n$$H_*^\\text{Nov}(X_+ \\cup X_0, (X_+ \\cup X_0)\\setminus Q \\times B) \\cong H_{m+n-*}^\\text{Nov}(X_+, \\partial X_+),$$\n\nand also a similar isomorphism for $X_-$. Using excision and K\\\"unneth we also see that $$H_*^\\text{Nov}(Q \\times \\mathbb{R}^m, Q \\times \\mathbb{R}^m \\setminus Q \\times B) \\cong H^\\text{Nov}_*(Q) \\otimes H_*( \\mathbb{R}^m, \\mathbb{R}^m \\setminus B) \\cong H^\\text{Nov}_{*+m}(Q).$$ \n\nPutting these ingredients together, the exact sequence \\ref{eq:2} becomes\n\n$$\n\\xymatrix{H_{*+m}^\\text{Nov}(Q) \\ar[r]^{[-1]} & H_*^\\text{Nov}(X_0, X_0 \\setminus Q \\times B) \\ar[d]\\\\\n&  H_{m+n-*}^\\text{Nov}(X_+, \\partial X_+) \\oplus H_{m+n-*}^\\text{Nov}(X_-, \\partial X_-)\\ar[ul]&}\n$$\n\nThis immediately implies the inequality \\ref{eq:1}.\n\nWe now return to Morse-Novikov homology. By our construction of $\\gamma = dH - \\beta$, we have that $-\\nabla \\gamma$ is transverse to $\\partial X_+$, and pointing out of $X_+$. Furthermore $-\\nabla \\gamma$ is transverse to $X_+ \\cap \\partial (Q \\times B)$, pointing into $X_+ \\cap \\partial (Q \\times B)$. Therefore the Morse-Novikov homology of $\\gamma$ on the manifold $X_+ \\cap Q \\times B$ is computing the homology $H^\\text{Nov}(X_+ \\cap Q \\times B, (\\partial X_+) \\cap Q \\times B, [\\beta])$, which is isomorphic to $H^\\text{Nov}_*(X_+, \\partial X_+, [\\beta])$. Similarly the Morse-Novikov homology of $\\gamma$ on the manifold $X_- \\cap Q\\times B$ computes $H^\\text{Nov}_*(X_-, \\partial X_-, [\\beta])$. Inequality \\ref{eq:1} therefore implies that $\\gamma$ must have at least $r$ critical points on $X_+ \\cup X_-$, which completes the proof.\n\\end{proof}\n\n\\bibliographystyle{plain}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Large deviation analysis of the weights of energy eigenstates after quantum quench}\n\tWe here consider the origin of the absence of thermalization after quantum quench in an integrable system [7]. In the absence of thermalization, an athermal eigenstate must have an exponentially larger weight than a typical weight of thermal eigenstates, because the number of athermal eigenstates is exponentially smaller than that of thermal eigenstates (i.e., $D_{\\mathrm{out}}\/D$ decays exponentially). We can directly see this by large deviation analysis. Specifically, we consider the XX model (6) of the main text, and evaluate the sum of the weights of athermal eigenstates: \n\t\\begin{equation}\n\tP_{\\mathrm{out}}:=\\sum_{\\alpha}\\left|\\braket{E_{\\alpha}|\\psi}\\right|^2\\theta \\br{\\left|\\braket{\\hat{O}}_{\\alpha}-\\braket{\\hat{O}}_{\\mathrm{MC}}\\right|-\\varepsilon},\n\t\\end{equation}\n\twhere $\\ket{\\psi}$ is the initial state before quench, and the summation is taken over all the eigenstates. As the initial state, we adopt the ground state of the Hamiltonian before the quench:\n\t\\begin{equation}\n\t\\hat{H}_{0}:=-\\sum_{i=1}^{L}\\sbr{\\cre{b}{i}\\ani{b}{i+1}+h.c.}+U\\sum_{i=1}^{3L\/4}\\cre{b}{i}\\ani{b}{i},\n\t\\end{equation}\n\twhere we take the limit $U\\rightarrow+\\infty$. The inset of Fig. \\ref{pout} shows a schematic of the quench.\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=0.5\\linewidth]{.\/pout.eps}\n\t\t\\caption{The system size-dependence of $P_{\\mathrm{out}}$ defined in Eq. (S1). (Inset) Schematic of the quantum quench: An infinitely high potential wall is removed.}\n\t\t\\label{pout}\n\t\\end{figure}\n\t\n\tFigure~\\ref{pout} shows our numerical result of $P_{\\mathrm{out}}$ for observables $\\hat{h}$ and $\\hat{f}_0$ defined in the main text, where $\\varepsilon$ is taken to be $\\varepsilon=\\left|\\mathrm{Tr}\\sbr{\\hat{O}\\rho_{\\mathrm{DE}}}-\\braket{\\hat{{O}}}_{\\mathrm{MC}}\\right|$. We do not show the result for $\\hat{n}_1\\hat{n}_2$ because the difference between the diagonal average and the MC average is too small to see the large deviation property of $P_{\\mathrm{out}}$. As seen from Fig.~\\ref{pout}, $P_{\\mathrm{out}}$ approximately equals $1\/2$ regardless of the system size for $L \\geq 32$, which implies the absence of thermalization in the thermodynamic limit. We can thus conclude that the sampling of energy eigenstates by the quench is exponentially biased to athermal eigenstates.\n\t\n\t\\newpage\n\t\\section{The parameter-dependence of $D_{\\mathrm{out}}\/D$}\n\tIn this section, we show the parameter-dependence of the ratio of athermal eigenstates $D_{\\mathrm{out}}\/D$ in order to check the validity of our conclusions in the main text. When we change one parameter to investigate the dependence of $D_{\\mathrm{out}}\/D$ on it, we fix the other parameters to the same value as used in the main text. In the results shown below, the lack of the data point means that $D_{\\mathrm{out}}\/D$ is exactly zero.\n\t\n\t\\subsection{Integrable system}\n\tWe first show the parameter-dependence of $D_{\\mathrm{out}}\/D$ for the integrable system (6) of the main text. The parameters used in the main text are given by $E=0$, $\\Delta E=0.1L$, $\\delta=0.1$, and $\\varepsilon=0.01, 0.05, 0.2$ for $\\hat{n}_1\\hat{n}_2$, $\\hat{h}$, and $\\hat{f}_0$, respectively.\n\t\n\tFigure \\ref{int_epsilon} shows the $\\varepsilon$-dependence of $D_{\\mathrm{out}}\/D$, where $D_{\\mathrm{out}}\/D$ decays exponentially for all $\\varepsilon$. In addition, we found that $D_{\\mathrm{out}}\/D$ with large $\\varepsilon$ decays faster than with small $\\varepsilon$, but never goes to zero. This implies that the weak ETH holds but the strong ETH does not hold.\n\t\n\tFigure \\ref{int_energy} shows the $E$-dependence of $D_{\\mathrm{out}}\/D$, where $D_{\\mathrm{out}}\/D$ decays exponentially except for the case of $\\hat{f}_0$ with $E=0.2L$. A possible origin of this exceptional behavior is simply the large finite size effect; For a larger system, $D_{\\mathrm{out}}\/D$ is expected to decay exponentially in any energy range. \n\t\n\tFigure \\ref{int_delta} shows the $\\delta$-dependence of $D_{\\mathrm{out}}\/D$. In the case that $\\delta$ is too small, like $\\delta=0.01$, the analysis does not work well for small $L$. This is because the number of energy eigenstates in the MC average is very small when $\\delta$ and $L$ are too small. Even with such a small $\\delta$, $D_{\\mathrm{out}}\/D$ decays exponentially independent of $\\delta$, when $L$ becomes large.\n\t\n\tFigure \\ref{int_DeltaE} shows the $\\Delta E$-dependence of $D_{\\mathrm{out}}\/D$. $D_{\\mathrm{out}}\/D$ decays exponentially except for the case that $\\Delta E$ is too small. When $\\Delta E$ is too small, we cannot obtain the enough number of eigenstates for statistical analysis.\n\t\n\tIn summary, the exponential decay of $D_{\\mathrm{out}}\/D$ is universal and independent of the parameters $\\varepsilon$, $E$, $\\delta$ and $\\Delta E$, except for some extreme cases.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/int_epsilon}\n\t\t\\caption{The $\\varepsilon$-dependence of $D_{\\mathrm{out}}\/D$ for the integrable system.}\n\t\t\\label{int_epsilon}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/int_energy}\n\t\t\\caption{The $E$-dependence of $D_{\\mathrm{out}}\/D$ for the integrable system.}\n\t\t\\label{int_energy}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/int_delta}\n\t\t\\caption{The $\\delta$-dependence of $D_{\\mathrm{out}}\/D$ for the integrable system.}\n\t\t\\label{int_delta}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/int_DeltaE}\n\t\t\\caption{The $\\Delta E$-dependence of $D_{\\mathrm{out}}\/D$ for the integrable system.}\n\t\t\\label{int_DeltaE}\n\t\\end{figure}\n\t\n\t\n\t\\newpage\n\t\\subsection{Non-integrable system}\n\tWe next show the parameter-dependence of $D_{\\mathrm{out}}\/D$ for the non-integrable system (7) of the main text with $\\lambda = 1$. In this model, the parameters used in the main text are given by $E=0$, $\\Delta E=0.1L$, $\\delta=0.1$, and $\\varepsilon=0.003, 0.015, 0.05$ for $\\hat{n}_1\\hat{n}_2$, $\\hat{h}$ and $\\hat{f}_0$, respectively.\n\t\n\tFigure \\ref{non_epsilon} shows the $\\varepsilon$-dependence of $D_{\\mathrm{out}}\/D$. We see that $D_{\\mathrm{out}}\/D$ decays double exponentially in $L$ regardless of $\\varepsilon$. In contrast to the integrable system, $D_{\\mathrm{out}}\/D$ with large $\\varepsilon$ and large $L$ becomes exactly zero for all the observables. This is a clear evidence of the validity of the strong ETH. \n\t\n\tFigure \\ref{non_energy} shows the $E$-dependence of $D_{\\mathrm{out}}\/D$. The double-exponential decay is seen in all the energy range.  \n\t\n\tFigure \\ref{non_delta} shows the $\\delta$-dependence of $D_{\\mathrm{out}}\/D$. We found a similar trend to the results for the integrable system. If $\\delta$ is too small, like $\\delta=0.01$, $D_{\\mathrm{out}}\/D$ does not show the double exponential decay for small $L$, because the number of eigenstates in the energy shell is too small. However, even with such a small $\\delta$, $D_{\\mathrm{out}}\/D$ decays double exponentially for large $L$.\n\t\n\tFigure \\ref{non_DeltaE} shows the $\\Delta E$-dependence of $D_{\\mathrm{out}}\/D$. We see that $D_{\\mathrm{out}}\/D$ decays double exponentially regardless of the value of $\\Delta E$.\n\t\n\tIn the above results, the double exponential decay is universally seen for all the parameters except for a few extreme cases, which supports our conclusion that the strong ETH holds for non-integrable systems independently of the choices of the parameters.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/non_epsilon}\n\t\t\\caption{The $\\varepsilon$-dependence of $D_{\\mathrm{out}}\/D$ for the non-integrable system.}\n\t\t\\label{non_epsilon}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/non_energy}\n\t\t\\caption{The $E$-dependence of $D_{\\mathrm{out}}\/D$ for the non-integrable system.}\n\t\t\\label{non_energy}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/non_delta}\n\t\t\\caption{The $\\delta$-dependence of $D_{\\mathrm{out}}\/D$ for the non-integrable system.}\n\t\t\\label{non_delta}\n\t\\end{figure}\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{.\/non_DeltaE}\n\t\t\\caption{The $\\Delta E$-dependence of $D_{\\mathrm{out}}\/D$ for the non-integrable system.}\n\t\t\\label{non_DeltaE}\n\t\\end{figure}\n\t\n\t\\section{The operator norm of the commutator}\n\tIn this section, we show the operator norm of the commutator $\\Delta_{\\hat O} := \\|i[\\hat{H}_{XXX}, \\hat O]\\|\/\\|\\hat O\\|$ for $\\hat O = \\hat n_1\\hat n_2$, $\\hat h$, and $\\hat f_0$. The norm $\\Delta_{\\hat O}$ quantifies a ``distance'' between the observable $\\hat O$ and the Hamiltonian $\\hat{H}_{XXX}$, and therefore a smaller value of $\\Delta_{\\hat O}$ implies smaller effective randomness of $\\hat O$ with respect to $\\hat{H}_{XXX}$. Figure~\\ref{norm} shows the $\\lambda$-dependence of $\\Delta_{\\hat O}$. We find that for $\\lambda \\geq 0.5$, the largest value of $\\Delta_{\\hat O}$ is given by $\\hat h$, followed by $\\hat n_1\\hat n_2$ and $\\hat f_0$ in descending order. This order is the same as that of $c$ and $sz$ in Fig~4~(b) and (c) in the main text. This result supports our argument that a smaller distance between $\\hat O$ and $\\hat{H}_{XXX}$ implies smaller values of $c$ and $sz$. Furthermore, for $\\lambda \\geq 0.5$, $\\Delta_{\\hat O}$ is almost independent of $\\lambda$. This is consistent with Fig.~4~(b) in the main text, in which $c$ is also almost independent of $\\lambda$ in the same range.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.5\\linewidth]{.\/norm}\n\t\t\\caption{The $\\lambda$-dependence of $\\Delta_{\\hat O}$, which is numerically obtaind for $L = 21$.}\n\t\t\\label{norm}\n\t\\end{figure}\n\t\n\t\\section{The $\\lambda$-dependence of $s$}\n\tWe here show the values of entropy density $s$ in detail. We numerically evaluated $s$ through the fitting of $D:= |\\mathcal{M}(0, 0.1L)|$ by $f(L) = ae^{sL}$. Table~\\ref{entropy_density} shows the results of the fitting, where $s$ approximately equals $0.6$ and is almost independent of $\\lambda$. \n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\begin{tabular}{|c||c|c|}\n\t\t\t\\hline\n\t\t\t$\\lambda$& $s$ & Asymptotic Standard Error\\\\\n\t\t\t\\hline\\hline\n\t\t\t1 &\t0.598202  &\t0.0008833\\\\\n\t\t\t0.8 & 0.597839 &\t0.0002985\\\\\n\t\t\t0.5 &\t0.601356 &\t0.0005085\\\\\n\t\t\t0.2&0.608291&0.0006695 \\\\\t0.1&0.609596&0.0009544 \\\\\t0.05&0.616175&0.002512\\\\\n\t\t\t0&0.615605&0.001714\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{The $\\lambda$-dependence of $s$.}\n\t\t\\label{entropy_density}\n\t\\end{table}\n\t\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ion temperature measurements in the scrape-off layer (SOL) are of high interest because of incomplete understanding of transport processes and plasma-wall interaction. Although models predicting such transport exist \\cite{esel}, the ion temperature and its fluctuations are neglected in most of the models partially due to the lack of systematic experimental data with temporal resolution on the turbulent and blob filament time scale. Such cold ion approximation is in contradiction with measured profiles of $T_i$ in SOL, which typically exceed those of $T_e$ \\cite{kocan-ti-profile}. Recent modelling \\cite{esel-recent}\\cite{esel-recent2} with finite $T_i$ showed that ion temperature influences drift wave growth rate and filament propagation speed. However, the role of $T_i$ fluctuations is not yet completely understood.\n\nThe measurements of ion temperature cannot be obtained using the standard sweeping Langmuir probe because of the excessive electron current, which masks the changes of ion current for probe potential above the plasma potential. Several techniques for for measuring the ion temperature in the tokamak scrape-off layer have been developed, the most commonly used being the Katsumata probe \\cite{katsumata} and the retarding field analyzer (RFA) \\cite{rfa}\\cite{brunner}\\cite{pitts}. Although it is possible to achieve ion temperature on the filament timescale ($\\mu$s) using such diagnostics \\cite{kocan-ion-fluct}, it is technologically challenging and requires condition averaging.\n\nIn order to facilitate fast $T_i$ measurements, an E$\\times$B analyzer has been used earlier on DITE \\cite{matthews} and Asdex\\cite{staib} tokamak. We have refined the design of the ExB analyzer with aid of numerical simulations. \n\n\\section{Principle of the E$\\times$B analyzer}\n\\label{intro}\n\nThe E$\\times$B analyzer consists of an array of collectors and a pair of planar electrodes located in a cavity behind a slit plate perpendicular to the tokamak magnetic field B (see Fig. \\ref{schema}). The slit plate protects the internal components from the plasma heat fluxes and reflects the electrons back into the plasma. The dispersion of the ion guiding centers from the slit axis parallel to B (i.e. along the collector array) $\\Delta_x$ can be expressed as:\n\\begin{equation}\n              \\Delta_x = \\frac{E}{B}L \\frac{1}{v_{||}},\n\\label{def}\n\\end{equation}\nwhere $L$ is the length of the electrodes inside cavity and $v_{||}$ is the ion parallel velocity. \\cite{matthews}.  The electric field $E$ is created by a potential difference between the electrodes (dubbed top and bottom), which are separated by distance $d$.\n\\begin{equation}\n             E = (V_{top}  - V_{bottom})\/d\n\\label{def-E}\n\\end{equation}\n\nThe parallel velocity consists of the original velocity at which ion enters the sheath in front of slit plate and an additional component due to acceleration caused by potential inside the cavity $V_{mid}$\n\\begin{equation}\n            v_{||} = v_{i} + \\sqrt{\\frac{2e (-V_{mid} + V_{plasma})}{m_i}}.\n\\label{eq-acc}\n\\end{equation}\n, where  $V_{mid}$ is the potential along ion beam path inside the cavity, which is typically located half way between the planar electrodes .\n\\begin{equation}\n            V_{mid} = (V_{top}  + V_{bottom})\/2\n\\label{eq-vmid}\n\\end{equation}\n Since in the experiment the magnitude of potential $V_{mid}$ is typically much larger than the plasma potential, we neglect the later and assume it equal to zero. The acceleration inside cavity to $V_{mid}$ introduces maximum deflection $ \\Delta_{x,max}$ for particles with zero initial velocity.\n\\begin{equation}\n           \\Delta_{x,max} =\\frac{(V_{top} - V_{bottom}) L}{B d \\sqrt{(e -(V_{top}  + V_{bottom})\/m_i))}}\n\\label{eq-max-dx}\n\\end{equation}\nIn the experiment one of the electrodes (in case of AUG $V_{top}$) is typically fixed at zero voltage in order not to exceed, so $\\Delta_{x,max}$ depends only on the applied voltage on $V_{bottom}$.\n\\begin{equation}\n           \\Delta_{x,max} =\\frac{L \\sqrt{-V_{bottom}}}{B d \\sqrt{e\/m_i}}\n\\label{eq-max-dx-vbot}\n\\end{equation}\nThe choice of $V_{bottom}$ hence determines the span over which the ions will be spread at the collectors.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.66]{exb_schema_v2.eps}\n\\caption{Schematic top and side view of the E$\\times$B analyzer.}\n\\label{schema}\n\\end{center}\n\\end{figure}\n\n\nMeasurements by RFAs in the tokamak SOL showed that ions in the high-energy range of the parallel ion distribution are characterized by the Maxwellian distribution of the parallel speeds \\cite{kocan-jnm} and therefore suitable for fitting by using the formula:\n\\begin{equation}\n            I_c \\sim exp \\left ( - {v^2_{i}} \\over {T_i} \\right) \n\\end{equation}\nor by replacing the parallel velocity by displacement $ \\Delta_x$ \n\\begin{equation}\nI_c(\\Delta_x)=I_0 exp \\left (  {E^2 L^2} \\over {B^2 \\Delta_x^2 T_i} \\right). \n\\end{equation}\nIn practice we measure currents from collectors at different $\\Delta_x$ and by fitting $I_c(\\Delta_x)$ with an exponential, we can obtain $T_i$.\n\n\n\n\n\\section{Design considerations}\n\\subsection{Probe head dimensions}\nThe probe head is designed for use on the horizontal reciprocating manipulators on AUG and COMPASS, which have compatible interfaces. The reciprocation allows to acquire measurement inside the SOL without damaging the diagnostics by enormous heat fluxes. On AUG,  multiple strokes during a discharge with typical stroking frequency 1 Hz are possible, COMPASS manipulator allows single reciprocation due to limited duration of the discharge.  The manipulator imposes upper limit on the diameter of the probe head  - maximum 60 mm is allowed. Due to high heat fluxes in the SOL especially during ELMs, the analyzer has to be protected by a carbon cover, which leaves effectively some 50 mm of inner space for the analyzer itself. This limit determines the maximum size of the inner cavity and as such is crucial for the design of the diagnostics. The remaining parts of this section describe parts of the analyzer which are of special attention.\n\n\n\\subsection{Entrance slit - geometry}\n\\label{section-slit}\nThe entrance slit is the plasma-facing part of the analyzer and serves to stop electrons from entering the cavity. Unlike the RFA, where parasitic electron current itself degrades the measurement, in case of the analyzer, it has to be avoided mainly due to unwanted side effects. The electrons can ionize neutrals within the analyzer cavity, which would then contribute to the collector currents. The main goal of the design thus to optimize  the slit geometry in order to allow measurable ion current inside the cavity (while acting as a barrier for electrons). At the same time the slit has to be capable of withstanding the high heat loads when operating in the vicinity of last closed flux surface or during the ELMs.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{spice-slit.eps}\n\\caption{Potential distribution around the entrance slit (left) and profile along the slit axis (right). Slit width (100 $\\mu$A) equals to $13 \\times \\lambda_D$.}\n\\label{spice}\n\\end{center}\n\\end{figure}\n\n\nIn previous experiments with RFAs (which also required biased slit)[REF - Pitts thesis] \\cite{kocan-jnm}, the slit width was chosen to be comparable to $\\lambda_D$ to prevent potential leaking inside the cavity (which would admit electrons to surpass the slit). However, dedicated particle-in-cell (PIC) simulations using a SPICE2 code\\cite{komm-textor}\\cite{renaud-spice} revealed that this criterion can be somehow relaxed and slits as wide as $10\\times\\lambda_D$ still do not admit electrons inside the cavity, as shown in Fig. \\ref{spice}. The simulation was performed for the worst case scenario, which can be foreseen on AUG ($n_e$ = 1$\\times$10$^{19}$ m$^{-3}$, $T_e$=10 eV, $T_i$ = 20 eV, B= 1.9 T) with very short Debye length. The figure shows the distribution of self-consistently calculated potential (A) as well as the potential profile along the slit axis (B). The profile illustrates two important features of the simulated slit. The potential barrier for the electrons is as deep as -9 k$T_e$ (the slit is biased to -13 k$T_e$, meaning 260 V below the plasma potential). At the same time the space charge due to ion beam does not result in potential structures surpassing the plasma potential, so the space charge does not block the ions from entering the cavity.  A thinner slit ($~\\sim \\lambda_D$) would ensure full potential barrier at -13 $kT_e$ but it would also admit much smaller (and thus more difficult to measure) ion current inside the cavity.\n\nIncreasing the slit length is a simple mean to increase the collector currents. Since all the particles inside the cavity should be affected by the same potential, the slit has to oriented so that the longer side is parallel to the collector array (ie. perpendicular to the individual collectors).  For this reason, the slit should not be larger than the collector width ($\\sim$1 mm). \n\nAnother mean of increasing the admitted current without enlarging the slit is to increase the ion transmission by using a knife-edge shaped slit \\cite{pitts} (see Fig. \\ref{schema-slit}) instead of a rectangular one. The transmission was first simulated using a Monte-Carlo code for varying angle $\\alpha$ of the opening between 0 and 25 degrees. Deuterons with mono-energetic parallel speeds and Maxwellian perpendicular velocities (characterized by $T_i$=40eV) were launched at random position along the slit entrance. Simulated slit was  100 $\\mu m$ wide and $1000 \\mu m$ long, cut into 2 mm thick slit plate. The slit transmission factor (i.e. the number of transmitted ions $N_{trans}$ to the total number of injected ions $N_ {inj}$) calculated for different $\\alpha$ and $v_{||}$ is shown in  Fig. \\ref{trans}. The ion transmission increases with $\\alpha$ and is almost insensitive to $v_{||}$ for large $v_{||}$ from which $T_i$ is derived in experiment. This is a substantial improvement over the rectangular slit (shown as red line in the figure). It should be noted that all ions are accelerated in the parallel direction by 2-10 $T_e$ due to the potential drop in the Debye sheath in front of the slit plate and thus arrive at the slit plate with relatively large $v_{||}$. The angle $\\alpha=20^\\circ$ was chosen for the ExB analyzer. \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{slit-v2.eps}\n\\caption{Schematic view of the entrance slit.}\n\\label{schema-slit}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.1]{trans.eps}\n\\caption{Transmission factor $N_{trans}\/N_{inj}$ as a function of parallel velocity $v_x$ for varying angle $\\alpha$ of the slit (2 mm thick). The red line shows the transmission factor of a rectangular slit. }\n\\label{trans}\n\\end{center}\n\\end{figure}\n\n\n\nIt has to be noted that such high transmission was not achieved in experiment. When comparing the input current (calculated by the slit plate current density integrated over the slit cross-section) with total ion current registered by all collectors, the effective transmission was typically close to $\\sim 15\\%$, which is similar to RFA experiments \\cite{brunner}\\cite{kocan-jnm}. The strong attenuation is caused by electric fields inside the slit volume, which deform the ion trajectories and in many cases prevent them from passing through the slit. The problem was simulated using a 3D particle-in-cell code SPICE3\\cite{komm-katsumata} and the effective slit transmission is shown in Fig. \\ref{slit-transmission}. Despite the absence of siginificant space charge effects inside the slit volume, the slit transmission for self-consisted potential profile was only $20\\%$, while for vacuum potential it has reached $50\\%$ (which is however still lower than MC code prediction). \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.4]{transmission.eps}\n\\caption{Current flowing through the analyzer cavity for vacuum potential (red) and self-consistent potential calculated by SPICE3 code (blue). Theoretical current based on current density arriving at the slit plate and slit cross-section marked in magenta.}\n\\label{slit-transmission}\n\\end{center}\n\\end{figure}\n\n\n\n\nIn order to study the power handling of the slit plate under the plasma exposure, the slit plate was modelled by the two-dimensional finite difference heat transfer code which accounts for the variation of tungsten properties with the temperature. In the simulation the plasma facing part of the slit was irradiated by the steady-state heat flux of 10 MW\/m$^2$, expected during the ELM in the AUG far SOL, lasting 2 seconds. Such long heat pulse duration corresponds to rather extreme conditions. The typical ELM duration and frequency in AUG are, respectively, $\\sim$4ms and 50Hz, meaning that the slit plate will on average experience the heat loads of about 2MW\/m$^2$ during the ELMy H-mode. Moreover, the probe is typically maintained in the plasma only for several tens of miliseconds.\nThe heat transfer from the slit to the support structure was neglected. Fig \\ref{FEM}A shows the time evolution of the temperature of the slit leading edge (at which the highest temperature was observed), with $T_{max}$=2900K, well below the tungsten melting temperature (3695K). Fig \\ref{FEM}B shows the temperature distribution within the slit (for t=1.7 s), which shows very small difference of temperatures. Despite the exaggerated pulse duration, the melting of the slit leading edges is therefore highly unlikely.      \n\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.20]{heat_duo_cut.eps}\n\\caption{Temperature of the hottest point at the slit during a 2 s pulse with 10 MW\/m$^2$ load.}\n\\label{FEM}\n\\end{center}\n\\end{figure}\n\n\\subsection{Space charge effects inside the cavity}\nIn the previous section, the slit was optimized to admit maximum ion current inside the cavity. Since the cavity is substantially larger (33x6x15 mm$^3$) than in case of modern RFAs , concerns about the possible build-up of space charge inside the cavity had to be addressed. Such space charge would be deprimental for the measurement for a number of reasons: It could prevent the ions from reaching the collectors, deform their velocity distribution function and suppress the E field created by the electrodes. Analytical calculations following simple analytical approach \\cite{brunner} indeed indicated that for the expected ion fluxes the space charge would be well developed.  The problem was first targeted by 2D particle-in-cell simulations, which confirmed potential structures surpassing the plasma potential. However, dedicated 3D particle-in-cell simulations using SPICE3 code \\cite{komm-katsumata} showed that the space charge is negligible, as shown in Fig.\\ref{potential-3d}   (0.1x1.0 mm slit was simulated in COMPASS SOL plasma $T_e = 20 eV$, $n_e$ = $1 \\times 10^{18}$ m$^{-3}$, B=1 T ). This is due to the fact, that simulating slit of finite width (which is not possible in 2D approach) brings additional attenuation of the ion flow on slit sides (the slit length is comparable to $r_{Li}$). At the same time it allows to capture how the ion beam expands inside the cavity volume, which reduces local charge density.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.3]{potential-3d.eps}\n\\caption{Potential calculated by SPICE3 code for a slit 0.1x1.0 mm. Vacuum potential (left) and self-consistent potential (right).}\n\\label{potential-3d}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\section{Signal analysis}\nUsing a guiding center approach, one could expect that the current density distribution on the collectors could be directly associated with the shape of the parallel ion velocity distirbution function. However, since the cavity size is not significantly larger than the ion Larmor radius, finite Larmor effects have to be taken into account. Ions are admitted inside the analyzer cavity through a thin slit, which forces them to follow a cycloid with node separation \\cite{matthews}\n\n\\begin{equation}\n\\Delta_n = \\frac{2 \\pi m_i E_y }{e B^2},\n\\end{equation}\n\nDepending on the length of the cavity, the current density profile can be significantly modified, as shown in Fig. \\ref{nodes}, where current distribution along the $x$ coordinate (direction of the E$\\times$B drift) are shown as a function of cavity length $a$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{nodes.eps}\n\\caption{Current density (in arbitrary units) in the direction of E$\\times$B drift (x) as a function of cavity length $a$. Particles are injected at $x$=0 and $a$ = 0.  }\n\\label{nodes}\n\\end{center}\n\\end{figure}\n\n\nThese profiles were obtained using a simple test particle Monte Carlo (MC) code, which calculates ion trajectories inside the cavity. The input parallel distribution velocity function was taken from a 1D quasineutral kinetic model of the scrape-off-layer \\cite{qpic}, which satisfies Bohm criterion. The perpendicular velocity distribution function was assumed Maxwellian.  The particles were generated at random positions inside the slit $z(t_0)$ and their motion was then resolved analytically assuming static homogeneous B field. The time needed to  pass through the cavity is $t_{fin}$.\n\\begin{equation}\nt_{fin} = v_x\/L,\n\\end{equation}\nwhere x direction is parallel to the magnetic field.  The particle reaches its final position $z(t_{fin})$ at x=L\n\\begin{equation}\nz(t_{fin}) = \\rho_L (sin \\psi - sin(\\psi + \\Omega_L t_{fin})) + \\frac{E_y}{B} t_{fin} + z(t_0),\n\\end{equation}\nHere $\\psi$ is the phase between $y$ and $z$ velocity component (generated from uniform distribution), $\\rho_L$ is the Larmor radius and $\\Omega_L$ the Larmor frequency.\n\nThe MC simulation assumes constant drift velocity (and so $E_y$) and potential equal to $V_{mid}$ along the ion beam path. Particle-in-cell simulations in the previous section shown, that self-consistent E field can have significant impact on slit transmission, so the MC code was benchmarked with SPICE3 simulation. Since a full 3D3V simulation of the entire cavity is computationally extremely demanding (especially in terms of required RAM), only a single selected case could be compared with the much easier MC calculations. This simulation was carried out for the same plasma parameters as in section \\ref{section-slit} and required a grid of 192$\\times$256$\\times$1280 cells, the largest case calculated by SPICE3 to date, with complete runtime of approximately 2 months. One electrode was left at 0, the other biased to -100V. This is lower voltage difference than used in experiment but allowed to capture the current density distribution within the available volume (the cavity had to be shortened in the direction of E$\\times$B drift).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{PIC_vs_MC_pot.eps}\n\\caption{Profiles of $y$ component of E field and potential along the B field inside analyzer's cavity. PIC results are plotted in solid lines, MC assumptions dashed. Position of electrode in PIC simulation indicated in orange.}\n\\label{PIC_vs_MC_pot}\n\\end{center}\n\\end{figure}\n\nThe profiles of $y$ component of E field and potential along the magnetic field line which intercepts the slit are shown in Fig. \\ref{PIC_vs_MC_pot}. It can be seen that MC assumptions (plotted in dashed lines) are simplifying the real profiles from PIC simulation but the difference is not dramatic. In order to get better match in terms of applied electric field, one can shorten the cavity length in the MC simulation, so that the effect on ions will be comparable to PIC run. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{current_profile_PIC_vs_MC.eps}\n\\caption{Profiles of current density along $x$ direction for PIC (black) and MC (colored) simulations. The PIC profile was shifted by 0.34 mm to facilitate comparison with MC results, original PIC profile shown in grey.}\n\\label{PIC_vs_MC_cur}\n\\end{center}\n\\end{figure}\nCurrent density profiles obtained by both codes are shown in Fig. \\ref{PIC_vs_MC_cur} (particles were injected at $x$ = 0.0). The position of the peak in PIC results was shifted by 0.34 mm with respect to MC profiles, so in order to facilitate comparison of the two methods, it was shifted respectively. Since the fit of the $T_i$ is done with respect to the peak maximum, this does not influence further analysis. It can be seen, that PIC profiles best match MC profile at $z$ = $-22.0$ (green line) but the MC profile is not strongly dependent on $z$ (note that z=0 represents slit position and in PIC simulation the electrodes are located between $z$ = -4.0 and -30.0 mm). Although the profiles are not completely identical, one has to keep in mind that for the analysis only the rising (left) slope is being used, and that the current is collected by collectors typically $\\sim$0.9 mm wide, for which the agreement is sufficient. Since the computational demands of the PIC simulations does not allow for large number of runs, which are needed for calibration of the results,  In the further analysis we will rely on the MC code with shortened cavity length $a$ by 3.0 mm. The MC code simulates an infinitely thin slit plate with rectangular slit. Originally, the slit cross-section 1.0$\\times$0.1 mm$^2$ was used, however this simplification was later refined to account for knife-edge profile and Larmor losses at slit edges. An effective slit length $L_{eff}$ was calculated using slit length in the middle of slit plate, from which ion Larmor radius $r_{Li}$ was substracted on each side. For COMPASS parameters, this resulted in reduction from 1.0 to 0.5 mm.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{effective_slit.eps}\n\\caption{Calculation of the effective slit length $L_{eff}$.}\n\\label{eff_slit}\n\\end{center}\n\\end{figure}\n\nUsing the MC code, the current density profile was mapped onto an array of synthetic collectors. Depending on the collector current distribution, two techniques can be used to fit the $T_i$. If the rising edge of the current profile is distributed on more than 2 collectors, $T_i$ is obtained using an exponential fit to the collector currents, where collector positions are translated to parallel velocities using equations \\ref{def} and \\ref{eq-acc}. This technique allows also to obtain an estimation of error of the fit. If the rising edge covers only two collectors, a simple 2-point interpolation is used:\n\\begin{equation}\nT_i = \\frac{m_i E L}{2 e B_{loc}}  \\left (\\frac{1}{\\Delta^2_x(2)} - \\frac{1}{\\Delta^2_x(1)}  \\right ) \/ log \\left (\\frac{I_{col}(1)}{I_{col}(2)} \\right),\n\\label{fit}\n\\end{equation}  \nwhere indexes $1$ and $2$ denote the two collectors respectively.\n\n The obtained ion temperature was compared with the temperature of injected ions. The systematic difference between these two values was characterized by coefficient $k$\n\\begin{equation}\n              T_{i_{real}} = k T_{i_{simulated}},\n\\end{equation}\nwhich in principle can depend on a large number of parameters -  analyzer's dimensions, electric field applied between the electrodes $E_y$, tokamak magnetic field B, potential between the electrodes along the ion beam $V_{middle}$ and ion temperature itself. The quality of the fit also depends on the size and spacing of the collectors and amount of parasitic noise in acquired signals. In the design consideration some of the parameters (such as the $B$ field and cavity size) are determined by constraints specific to each machine, some (such as the  $E_y$ field) can be adjusted to improved the quality of acquired signals. The aim of this study is not to determine an universal scaling of the $k$ parameter, which could be used for any tokamak but rather than that present scaling for two machines, where the specific probe head was utilized - COMPASS and AUG (see Fig. \\ref{k-lmode}). Since the probe head is not capable of measuring the local plasma potential, it remains a free parameter in the scaling. However, for L-mode studies, it has been observed \\cite{adamek-lmode} in AUG that the fluctuations of plasma potential are relatively small, typically around +30V. The obtained scaling of $k$ parameter does not show a substantial variation of the $k$ parameter with plasma potential, which is due to condition $|V_{mid}| > |V_{plasma}|$. The scaling of $k$ parameter for AUG was obtained using the 2-point interpolation technique using voltage difference 200 V between the electrodes ($E_y$ = 33 kV\/m), for COMPASS the exponential fitting has been used with voltage difference 300 V ($E_y$ = 50 kV\/m).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{duo-k-scaling_L-mode.eps}\n\\caption{Scaling of the $k$ parameter:  for AUG L-mode conditions (left) obtained by 2-point interpolation method ($E_y$ = 33 kV\/m) and for COMPASS L-mode conditions (right) obtained by exponential fit ($E_y$ = 50 kV\/m) for different values of plasma potential ($V_{sheath}$). }\n\\label{k-lmode}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Electron collector current}\nThe first experiment revealed parasitic electron current on all collectors, which was interfering with the measurements.  The origin of the electrons inside the cavity was not immediately evident, however an effective mean on reducing the current was found out in experiment. The voltages on electrodes were arranged so that  $V_{mid} < V_{slit}$. This setup however enhanced acceleration of ions and consequently reduced the dispersion due to applied electric field. An alternative solution was to introduce a protective grid in front of the collector array. When this grid was biased so that $V_{grid} <  V_{slit}$, the electron current was successfully suppressed. This indicated that the electrons are secondaries originating from the back side of the slit plate and thus repelled by a potential lower than $V_{slit}$. Such secondary electrons would however have to be created by impact of either ions or electrons on the back side of the slit, which was in contradiction with a simple model of the analyzer's behavior. The particle-in-cell simulations, which were originally used to assess the space charge effects inside the cavity, revealed that due to strong E field inside the slit volume, the ion trajectories are deformed so that some ions are reflected towards the back side of the slit plate (example trajectories are shown in Fig. \\ref{trajectories} right).\n\n\n\n\\section{Design of the analyzer}\n\n\n The analyzer designed for AUG and COMPASS tokamak (as shown in Fig. \\ref{rozstrel}) is contained within a 5 mm thick carbon cylindrical housing (60 mm in diameter, 150 mm length), which is attached to the internal structure by four TZM fixation bolts. The cavity entrance is located 15 mm bellow the probe head ending, where the housing thickness is reduced to 3 mm around an orifice of approx. 12 mm$^2$.  Since the dimensions of the orifice (approx. 5$\\times$3 mm) is larger than ion Larmor radius ($r_L <$ 1 mm), we assume that the ion flux arriving into the slit is not attenuated by the housing. We also use the orifice area as an effective collecting area for ions falling onto the slit plate, neglecting possible finite Larmor effects.\n\nThe entrance slit is machined into a 2 mm thick  tungsten plate following the design described in section \\ref{section-slit} using the spark erosion technique. First measurements were performed with a 1 mm long and 100 $\\mu$m wide slit with edge opening at  20$^\\circ$.  The slit is oriented with its longer side along the collector array. \n\nThe internal cavity is made of Boron Nitride parts with two copper planar electrodes. The electrodes covered 30 out of the 36 mm of total distance from slit to the collector array. The electrode separation  is 6 mm, which is more than the expecred ion Larmor radii and allows to create sufficient electric fields ($\\sim$ 50 kV\/m) using available power sources without an elevated risk of arcing (up to 400V).\n\nThe collector array is printed on a FR4 PCB (made by Pragoboard company), which has significant advantages over other approaches, such as the sandwich design\\cite{renaud-sandwich}. Risk of short-circuits between the segments is minimal, it occupies less space and in principle segments with varying size are easy to produce. For the first experiment we used an array of 12 collectors, each 0.9 mm wide with 0.3 separation.  The number of segments is limited by the number of pins in the manipulator interface (22 in AUG, 18 in COMPASS) and number of available channels in fast data acquisition system (8 during the first experiment). The upper part of the analyzer is connected to the manipulator interface by two stainless steel rods (see Fig. \\ref{photo}). \n\nThe signals from collectors are carried by standard vacuum-compatible Huber-Suhner cables to the manipulator interface, which consists of a matrix of SNB connectors. The cables run through the manipulator towards the current pre-amplifier (total length of cables from collectors to the amplifier is $\\sim$4 meters).\n\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.33]{rozstrel.eps}\n\\caption{Schematic view of the ExB analyzer probe head.}\n\\label{rozstrel}\n\\end{center}\n\\end{figure}\n\n\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{photo.eps}\n\\caption{Front view of the analyzer with the tungsten slit plate in front.}\n\\label{photo}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Voltage setup}\nThe following arrangement of voltages was used in the first experiments, which were focused on measurements of turbulence in SOL. The slit plate was biased to -160 V and the grid in front of the collector array to -180 V. The top electrode was left at 0 V, while the bottom electrode was set to -200V. Assuming that the plasma potential is positive in SOL [REF Adamek], the potential inside the cavity is $< V_{plasma}$ and so ions can freely propagate. In order to determine $\\Delta_x$, the knowledge of precise alignment of local B field with the analyzer is needed in order to determine the position at which field lines passing through the slit intersect the collector array. Such precise alignment ($<1^\\circ$) cannot be measured a priori, since the orientation of field lines depend on local value of safety factor $q$ but  has to be determined in the experiment for a given location of the probe. In order to do so, a short synchronouse voltage pulse was introduced on all components, which dropped the voltage to 0V. During the pulse, electrons can propagate inside the cavity along the field line and mark the collector, which is aligned with the slit.\n\n\\section{Summary}\nAn instrumental study of the E$\\times$B analyzer for COMPASS and AUG tokamaks has been performed by means of MC and PIC simulations. The geometry of the slit with knife-edge opening at 20 degrees has been found convenient in terms of its transmission and power handling. A series of MC simulations was used to identify combinations of applied $E$ field and bias voltage, which are suitable for  $T_i$ measurements in the conditions expected in the SOL. PIC simulations we performed in order to investigate the possibility of a space charge build up in the entrance slit. The analyzer was successfully constructed and tested on AUG to obtain temperature measurements at the timescale of blobs. \n\\medskip \n\\section{Acknowledgements}\nThis work was partly supported by MSMT Project $\\#$LM2011021 and by EFDA Fellowship programme.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTwo parties, Alice and Bob, wish to discuss some rather sensitive information. They want to have a private conversation. A third party, Eve, wishes to listen in on this private conversation. Since Eve's propensity for eavesdropping is well-known, Alice and Bob try to design a system that provides secure communication. Since Eve is the determined sort, she will search for and exploit any weaknesses in the system created by Alice and Bob. Alice and Bob will eventually discover their system leaks (Eve is a notorious gossip) and they will either try to patch it or create a new system. Escalation ensues, and being three clever people, increasingly clever methods are devised for secure communication and for cracking those systems, or, cryptography and cryptanalysis, respectively.\n\nSo far, the only clever cryptographic method that cryptanalysts have not cracked is the one-time pad. The one-time pad creates a ciphertext using a key at least as long as the plaintext input and the key is used only once. If Alice and Bob both possess the same secret key, Alice can encipher the plaintext, send the result to Bob, and he can decipher with his copy of the key. The one-time pad has been proven perfectly secure\\cite{shannon:49}, but it suffers from some non-trivial requirements. Namely, perfectly secure system needs a true random number generator and a secure way to distribute the key. Current cryptographic random number generators remain unproven. Distributing the key appears to provide the exact same challenge cryptographers were trying to address in the first place: sending a secure message. It turns out that both of these requirements may be satisfied by leveraging quantum mechanics.\n\\section{Quantum Cryptography}\nApproaching the two requirements for a one-time pad with quantum mechanics returns promising results. First, using quantum mechanics to generate random numbers seems ideal\\cite{bell:64}. In fact, commercial quantum random number generators are available\\cite{idQ}. Second, one can imagine using quantum mechanics to set up a secure short-term communication channel for distributing the key. One quantum key distribution (QKD) protocol serves as a foundation for nearly all of the rest. The Bennett-Brassard 1984 (BB84) relies on two principles of quantum mechanics:\n\\begin{enumerate}\n\\item One cannot take a measurement without disturbing the system.\n\\item One cannot clone an unknown quantum state.\n\\end{enumerate}\nUnder this protocol, Alice will send quantum bits (qubits) to Bob in the form of photons transmit the key. Should Eve intercept those photons, she will not be able to ``read'' the key without alerting Alice and Bob to her presence. A closer look at the basics of the protocol will illustrate how this is true.\n\nThis protocol uses polarization encoding.\nPhotons are prepared in four types of polarizations: horizontal, vertical, and two diagonals. Figure~\\ref{pe} illustrates these four encodings. Actually, there is another polarization, circular, but we will come back to that. Say vertical and $-45\\,^{\\circ}$ diagonal polarization represent 0, and horizontal and $+45\\,^{\\circ}$ diagonal polarization represent $1$. To measure a qubit, Alice and Bob will use one of two bases. One base, X, measures diagonally polarized photons. The other, Z, measures horizontally and vertically polarized photons. Figure~\\ref{bennetbrassard} shows how the BB84 protocol works. Now, that is not to say one cannot measure horizontally polarized photons in the X base. In fact, measuring with the \"wrong\" base reveals the beauty of this protocol.\n\\begin{table}[hbp]\n\\begin{tabular}{|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Polarization Encoding}\\\\\n\\hline\n\\includegraphics{polarEncod0} & $0$ \\\\\n\\hline\n\\includegraphics{polarEncod1} & $1$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Typical configuration for polarization encoding.\\label{pe}}\n\\end{table}\n\n\\begin{figure*}[tbp] \\centering\n\\includegraphics[width=350px]{bb84}\n\\caption[BB84 Protocol]{\n\\begin{inparaenum}[\\itshape a\\upshape)]\n\\item Basic setup between Alice and Bob. The classical channel is insecure and it is a non-issue for Eve to eavesdrop on communication over this line. The quantum channel is where the key is transmitted.\n\\item Alice sends photons polarized in such a way that if they are measured in its complementing base, it will produce a $1$ or (exlusive or) $0$ reliably.\n\\item Bob measures the photons with his random basis, and sends the basis over the classical line to Alice.\n\\item Alice tells Bob which elements of his basis were compatible with her original signal. The qubits that Bob measured ``correctly'' make up the \\emph{sifted-key}.\n\\end{inparaenum}\n\\label{bennetbrassard}\n}\n\\end{figure*}\n\nThe first principle of quantum mechanics, mentioned above, says that measuring disturbs the system. This means that the quantum state collapses to a classical state upon measurement. For example, if one measures a horizontally polarized photon in the Z base, the measurement will return a 1. If one measures a horizontally polarized photon in the X base, the measurement will return a 0 or a 1 with equal probability. The quantum state collapses to either $-45\\,^{\\circ}$ or $+45\\,^{\\circ}$ diagonal polarization. When measuring in the Z base, the quantum state still collapses, but it collapses to horizontally polarized every time. The results are the same for all four of the polarizations. Therefore, measuring in the ``right\" base produces one of the two results (1 or 0) every time, while measuring in the ``wrong\" base produces one of the two results with 50\\% probability. Alice will send a series of qubits so Bob can measure with a series of bases, called a basis, and get a string of 1s and 0s.\n\nHow is this useful? If Alice wants to send a key via photons to Bob, she will have to tell him, over classical communication lines, which basis to measure with. Otherwise, 50\\% of the 1s and 0s Bob possesses at the end will not match Alice's bitstring. If Alice communicates the basis to Bob over classical lines, it will be a non-issue for Eve to eavesdrop on the classical line, intercept the qubits, measure in Alice's basis, and generate and send qubits to Bob with the same polarization as Alice's original signal. Eve will have the same key as Alice and Bob and can eavesdrop on their private conversation.\n\nAlice does not send her basis, so Bob measures in a randomly chosen basis. This results in 50\\% error in Bob's key. After Bob does his measurement, he sends \\emph{his} basis to Alice. Alice compares her basis his and tells Bob which elements of his basis match hers. Alice and Bob both discard the mismatches in their results, producing a common \\emph{sifted key}.\n\nEve cannot perform the same attack she used when Alice sent the basis measurement to Bob. Alice's qubits have come and gone, so there is no opportunity to measure in Bob's basis unless Eve can discover it in advance. This leaves Eve with no choice but to measure the qubits in her own random basis. However, 50\\% of Eve's basis does not match Alice's basis. Eve does not know which elements of her basis are ``wrong\", but she must send something on to Bob or he and Alice will simply discard the missing qubit bringing Eve no closer to her goal. The best Eve can do is send a guess.\n\nIf Eve guesses the base to send to Bob, half of those will be incongruent with Alice's original qubits. Alice and Bob will only use about half of the qubits. Now, half of those sifted qubits will produce unmatching measurements for Alice and Bob (ie. Eve has introduced a 25\\% error rate in the sifted key between Alice and Bob). The error rate in the sifted key will alert Alice and Bob to Eve's presence, and they can discard the key ensuring Eve gains no useful information.\n\nWhy can Eve not simply copy the qubit to send to Bob? Then, she could wait until Bob announces his basis and perform the same measurement. Going back to the second featured principle of quantum mechanics this protocol relies on, one sees that cloning unknown quantum states is impossible\\cite{wootters:84}. Since Eve cannot clone the qubit, her only option that remains is sending a guess.\n\nTheoretically, this protocol is provably secure.\nHowever, the real-world implementation proves less than ideal.\nFurthermore, quantum operations suffer from errors like classical operations.\\cite{kak:96,kak:99}\n\\section{Faked-state Attack}\nThe BB84 protocol relies on the system's ability to send and detect single photons. Sending single photons at an exact time is a problem\\cite{bell:64}, so current systems send a very weak light signal instead. These signals are not quite single photons. Likewise, the detectors emulate detecting single photons by picking up very weak signals. It is this reliance on weak signals that eavesdroppers will attack. In the faked-state attack, Eve manipulates this weakness in Bob's detectors to force him to measure in the same basis as Eve\\cite{makarov:05}.\n\nThe faked-state attack exploits a weakness in the apparatus of QKD systems, namely, the diode used to detect photons\\cite{makarov:06,makarov:09}. The avalanche photodiode (APD) detector is supposed to detect single photons, or at least approximate the detection of single photons. However, an APD requires a recharge time of roughly $1\\mu$s before it is ready to detect another photon. This normally would not be a problem because under expected use, weak laser signals would be sent infrequently enough for the detector to recharge. Makarov discovered that if you shine continuous light into the APD, the detector does not recharge and is reduced to a classical photodiode. Furthermore, by manipulating his continuous beam of light, he can make the detector click when he wants or he can blind the detector from seeing valid input.\n\nNow, the apparatus has four detectors, one for each polarization. Eve will intercept the photon from Alice and prepare a faked-state to send to Bob. They say faked because Eve is not sending Bob a quantum state, but she is making Bob's apparatus think it is detecting a quantum state. Anyway, Eve will intercept Alice's photon, and she will prepare her faked state in the opposite base with the opposite bit she detects. For example, if Eve detects a 0 in the X base, she will prepare a 1 in the Z base for Bob. Here, she exploits the detectors. At the same time she sends her prepared fake state, Eve will blind Bob's 1-bit detectors. If he measures in the same base as Eve, X, he has a 50\\% chance of detecting a 0 or nothing at all. If he measures in a base different than Eve's, Z, he is guaranteed to detect nothing at all. Figure~\\ref{fsa} illustrates how this works given Alice's choice of sending 0 in the X base. Doing this, Eve guarantees Bob's apparatus only detects the same bit, in the same base, she detected.\n\\begin{table*}[tbp] \\centering\n\\begin{tabular}{|c c c c c c|}\n\\hline\nAlice & Eve's Base & Eve's Measurement & Eve Sends & Bob's Base & Bob's Measurement\\\\\nX0 & X & 0 & Z1 & X & 0 or no detection\\\\\nX0 & X & 0 & Z1 & Z & no detection\\\\\nX0 & Z & 0 & X1 & X & no detection\\\\\nX0 & Z & 0 & X1 & Z & 0 or no detection\\\\\nX0 & Z & 1 & X0 & X & no detection\\\\\nX0 & Z & 1 & X0 & Z & 1 or no detection\\\\\n\\hline\n\\end{tabular}\n\\caption[Eve's faked-stage attack]{Eve's faked-stage attack. Note Eve's measurement in the bottom for has a 50\\% chance of detecting a 0 or a 1 since she uses an incompatible base to Alice's base.\\label{fsa}}\n\\end{table*}\n\nOne might think Bob not detecting 50\\% of Alice's signals would raise concerns, but under normal conditions with today's technology, Bob's apparatus will only detect a small portion of the photons sent by Alice. The blame for this lies with the system's reliance on psuedo-qubits, and this reliance is a result of the protocol's requirement that the system send and register single photons. However, not all QKD protocols have this requirement.\n\n\\section{Three-Stage Protocol}\nImagine Alice wants to send a secret item to Bob. She puts the item in a box and locks the box with her padlock. Then, she sends the box to Bob. When it arrives, Bob puts his own padlock on the box. Now, the box has two padlocks on it. Then, he sends it back to Alice. Alice unlocks her padlock and sends it back. Upon arrival this time, the box only has Bob's padlock on it. He unlocks it and withdraws the secret item. Assuming these padlocks are indestructible and can only be unlocked with their respective owners' key, Eve cannot obtain the secret item.\n\nIn the three-stage protocol\\cite{kak:06}, the secret item is the symmetric key Alice and Bob want to use for a one-time pad.\nThe padlocks are transformations\\cite{kak:00} applied to the qubits.\nIt is important that these transformations are commutative (ie. for transformations U and V, UV = VU).\nAlice will perform her transformation (U\\textsubscript{A}) on the qubit and sends it to Bob.\nBob applies his transformation (U\\textsubscript{B}) and sends it back to Alice.\nAlice performs her inverse transformation (U\\textsubscript{A}\\textsuperscript{$-$}) and sends it back to Bob.\nBob applies his inverse transformation (U\\textsubscript{B}\\textsuperscript{$-$}) and measures the qubit in the predetermined base to get the bit.\nFigure~\\ref{3stage} illustrates this transaction.\n\n\\begin{figure*}[tbp] \\centering\n\\includegraphics[width=400px]{kak09}\n\\caption[Kak's Three-stage Protocol]{\n\\begin{inparaenum}[\\itshape a\\upshape)]\n\\item A qubit is emitted from the source, and Alice's transformation is applied. The qubit travels to Bob, where Bob's transformation is applied.\n\\item The qubit sent back to Alice, now the result of two transformations. Alice's inverse transformation is applied canceling out her initial transformation.\n\\item The qubit is sent to Bob one last time, now with only his transformation applied. Bob's inverse transformation is applied, and Bob's apparatus measures the qubit.\n\\end{inparaenum}\n\\label{3stage}\n}\n\\end{figure*}\n\nUnlike BB84, the three-stage protocol does not rely on Alice and Bob sending single photons.\nThe three-stage protocol provides another important advantage.\nWhen she attacks communication under the BB84, Eve knows that the qubits will arrive in one of two polarization bases.\nThen, she must only discern one bit of information.\nIn the three-stage protcol, qubits can be sent with \\emph{any} polarization.\nThey are limited only by the precision of the equipment.\n\nFor example, let Alice and Bob use rotation for their transformations.\nTheir systems perform one of 1024 possible rotations and the appropriate inverse rotation, respectively.\nEve can easily cut into the line\\cite{makarov:05} and siphon off some of the photons in any stage and allow the rest to continue to their intended destination.\nThen, she can send these siphoned photons through a series of filters to discover the angle.\nIf Eve can siphon off and discover the angle at all three stages of the protocol, she can determine Alice's and Bob's transformations.\nHowever, note that to determine the angle with 100\\% certainty, Eve would have to siphon off at least 1024 photons (assuming she has the technology to split and direct them to 1024 different filters).\nIf Alice and Bob send less photons than this threshold, Eve's attempt will ``use up'' the photons, and she will be detected.\n\nSurely, Bob has to use a detector similar to the APD to perform the final measurement.\nCan Eve manipulate a beam of light to create a faked state within Bob's apparatus such that she can learn the key?\nAssume Eve can create faked states within Bob's equipment.\nShe make his equipment register a 1, a 0, or nothing at all.\nEve cannot leverage this in any way to gain secret information.\nNotice that intercepting the signal at any stage, Eve will possess a transformed qubit.\nMeasuring the transformed qubit has a 50\\% chance of coinciding with Alice's original input.\nShe can force Bob's apparatus to register the same result, but the 50\\% error rate will betray her eavesdropping.\n\nEve cannot leverage the faked states attack on the three-stage protocol, but she can employ a simpler intercept-resend attack.\nSince it is safe to assume that Eve can obtain the same kind of equipment as Alice and Bob, she could cut into the line, pose as Bob and complete a transaction with Alice, and then pose as Alice and complete a transaction with Bob.\nNow, she is privy to any secret communication over the quantum channel.\n\nOne possible solution to this attack is to apply classical cryptography\\cite{chen:09} to ensure the message's authenticity.\nAnother solution uses trusted certificates created using quantum mechanics\\cite{perkins:06}.\n\n\\section{Conclusion}\nReal-world implementations of quantum cryptography protocols are not as completely secure as one may hope when looking at the theory.\nSimply put, the fault lies with the apparatus.\nIn a more abstract sense, the fault lies with trying to emulate quantum states.\nThese emulations are imperfect and attackers can exploit the imperfections.\nThe Kak protocol offers advantages, but it is yet to be seen if it can be implemented for real-world applications and how it holds up against other attacks.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction }\nDe Broglie-Bohm Mechanics also known as pilot-wave formulation was formulated by D. Bohm in \\cite{Bohm:1951xw,Bohm:1951xx} based on previous\nwork of L. de Broglie\n\\cite{Broglie1,Broglie2} who suggested that particle trajectories are guided by waves. In this interpretation, individual quantum system is formed by a point particle \\emph{and} a guiding wave. This is in sharp contrast with orthodox or Copenhagen interpretation of quantum mechanics, where an individual quantum system is simultaneously both a wave and particle. This system exhibits its particle or wave origin in dependence on the experiments. Say differently, in orthodox formulation of quantum mechanics does not make sense to speak about  particle's position  until its position is measured while in de Broglie-Bohm mechanics particle still has its own trajectory even if its form can differ from the classical one. It turns out that de Broglie-Bohm mechanics is,\nin words of John Bell \\cite{Bell:1987hh} \"Bohmian\nmechanics is a correct interpretation of quantum phenomena that exactly reproduce the predictions of the orthodox interpretation! \"\n\\footnote{For review and extensive list of references, see\n\\cite{Berndl:1995ze,Tumulka:2018vot,Holland,Bohm}.}.\nThis theory  claims, that no rule of the quantum theory is broken when we include a single trajectory to specify the state of a system more finely than does the wave function. There is however an important point that whenever we want to calculate the trajectory we \\emph{firstly} have to know the wave function \\emph{whose determination is independent} on the path. This is main\nobjection to the de Broglie-Bohm theory since the concept of trajectory of particle can be interpreted as an redundant addendum and as an interpretative element rather than an intrinsic ingredient of quantum theory.\n\nHowever there could be deeper reason why we should consider trajectory of particle as well defined physical object. The clue to this problem is in the similarity between the Schr\\r{o}dinger equation in the position representation suitable expressed as two real equations. In more details, we can always write\nthe wave function in the form $\\psi=\\sqrt{\\rho}e^{iS}$, where $\\rho$ and $S$ are two real functions. Then it can be shown that the equation of motion for $S$ has the form of the quantum version of Hamilton-Jacobi (HJ) equation while the equation of motion for $\\rho$ corresponds to the equation of continuity.\nThis is very nice picture and one can ask the question whether similar analysis can be extended to another situations. In fact, the goal of this paper is to perform similar analysis in case of relativistic massive particle in general background metric and gauge field and also to the case of particle in Newton-Cartan background\n\\footnote{For related work, see \\cite{Nikolic:2012wj}.}. We show, following very nice analysis reviewed in\n\\cite{Oriols}, that even the classical relativistic particle and particle in Newton-Cartan background can be described with the help of wave function. We also show that fundamental difference between classical and quantum description is in the\nthe fact that the equations of motion that follow from the equations of motion for complex field in general background allow superposition of two solutions while in case of the classical description this is not possible. The same situation occurs in case\nof particle in Newton-Cartan background. More precisely, we firstly determine an action for particle in Newton-Cartan background implementing null dimensional reduction\n\\cite{Festuccia:2016caf,Julia:1994bs}. Then we use the relation between conjugate momenta and partial derivative of the action in order to find\nHJ equation for particle in Newton-Cartan background that has not been determined before. As the next step we determine an action for Schr\\r{o}dinger field in Newton-Cartan background again using null dimensional reduction. Then we write this field using polar form and derive equations of motion where the first one correspond to the quantum HJ equation for particle in Newton-Cartan background while the second one corresponds to the equation of continuity. We also find an action for the complex scalar field that describes classical Newton-Cartan particle and we again show that the crucial difference between quantum and classical description is in the fact that linear combinations of two solutions of the classical equations of motion is not their solutions.\n\nThis paper is organized as follows. In the next section (\\ref{second}) we review basic facts\nabout Hamilton-Jacobi theory which is central for formulation de Broglie-Bohm theory and relation between classical and quantum physics. In section (\\ref{third}) we analyze relativistic massive particle in general background and find classical and quantum form of HJ equation for it. We also find Lagrangian for quantum relativistic particle. In section (\\ref{fourth}) we\nextend this analysis to the case of particle in Newton-Cartan background. Finally in conclusion (\\ref{fifth}) we outline our results and suggest possible extension of this work.\n\\section{ Review of Hamilton-Jacobi Theory}\\label{second}\nIn this section we briefly review basic facts about Hamilton-Jacobi theory, following\n\\cite{Oriols}, for more detailed treatment, see for example \\cite{Goldstein}.\nLet us consider an action defined as $S=\\int d\\lambda L(X,\\dot{X})$ where\n$X$ label position of particle in target space time and\n where $\\dot{X}=\\frac{dX}{d\\lambda}$. All $X's$ depend on $\\lambda$ which parameterizes world-line of particle. Let $\\lambda_0$ and $\\lambda_f$ are initial and final values of world-line parameters and let $X_0=X[\\lambda_0], X_f=X[\\lambda_f]$ are initial and final\npositions of the particle. Classical trajectory, which we denote as $X_p[\\lambda]$, is a trajectory that solves the equations of motion corresponds to the stationary value of the action functional\n\\begin{equation}\nS[X,X_0,\\lambda_0,X_f,\\lambda_f]=\n\\int_{\\lambda_0}^{\\lambda_f}d\\lambda L(X,\\dot{X}) \\ .\n\\end{equation}\nMore precisely, let $X_{np}(\\lambda)=X_p(\\lambda)+\\delta X(\\lambda)$ is a trajectory that differs from the physical one by small displacement parameterized by $\\delta X$. We only demand that initial and final positions of these two trajectories are the same so that $\\delta X(\\lambda_0)=\\delta X(\\lambda_f)=0$. Then the action evaluated on the second trajectory is equal to\n\\begin{eqnarray}\n& &S[X_{np},X_0,X_f]=\\int_{\\lambda_0}^{\\lambda_f}\nd\\lambda L(X_p+\\delta X,\\dot{X}_p+\\delta \\dot{X}_p)=\n\\int_{\\lambda_0}^{\\lambda_f}d\\lambda L(X_p,\\dot{X}_p)+\n\\nonumber \\\\\n& &+\\int_{\\lambda_0}^{\\lambda_f}\nd\\lambda \\left[\\frac{\\partial L}{\\partial X}-\\frac{d}{d\\lambda}\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\right]\\right]_{X=X_p}\\delta X(\\lambda)+\n\\left.\\frac{\\partial L}{\\partial \\dot{X}}\\delta X\\right|_{\\lambda_0}^{\\lambda_f}=\n\\nonumber \\\\\n& &+\\int_{\\lambda_0}^{\\lambda_f}\nd\\lambda \\left[\\frac{\\partial L}{\\partial X}-\\frac{d}{d\\lambda}\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\right]\\right]_{X=X_p}\\delta X(\\lambda) \\ ,\n\\nonumber \\\\\n\\end{eqnarray}\nwhere we used the fact that\n $\\delta X(\\lambda_0)=\\delta X(\\lambda_f)=0$. Now the fact that\nphysical trajectory corresponds to the stationary point of the action means that\n$S[X_{np}]-S[X_p]=0$ for any small variation $\\delta X$ which can be also written as $\\delta S=0$. As a result we derive Lagrangian equations of motion\n\\begin{equation}\n\\left[\\frac{\\partial L}{\\partial X}-\\frac{d}{d\\lambda}\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\right]\\right]_{X=X_p}=0 \\ .\n\\end{equation}\nOf course, this result  is well known. On the other hand it is less known\nthat the action formalism is more  powerful and allows more general analysis. For example,\nwe would like to analyze an ensemble of physical trajectories that differ in initial or final conditions. For that reason we define\n\\begin{equation}\nS_{p-\\delta \\lambda}=S(X_{p-\\delta \\lambda})=\n\\int_{\\lambda_0}^{\\lambda_f+\\delta \\lambda}d \\lambda\nL(X_{p-\\delta \\lambda}(\\lambda),\\dot{X}_{p-\\delta \\lambda}[\\lambda]) \\ ,\n\\end{equation}\nwhere $X_{p-\\delta \\lambda}[\n\\lambda]$ is physical trajectory that has identical initial and final positions as $X_p[\\lambda]$ but taking a longer time $\\lambda_f+\\delta \\lambda$.\nLet us write this new physical trajectory as\n$X_{p-\\delta \\lambda}(\\lambda)=\nX_p(\\lambda)+\\delta X(\\lambda)$ for $\\lambda\\in [\\lambda_0,\\lambda_f]$. For $\\lambda$ that is outsides from this interval we have\n\\begin{equation}\\label{Xpexpansion}\nX_{p-\\delta \\lambda}[\\lambda_f+\\delta \\lambda]=\nX_p[\\lambda_f]+\\dot{X}_p[\n\\lambda_f]\\delta \\lambda+\\delta X[\\lambda_f+\\delta \\lambda] \\ .\n\\end{equation}\nSince by definition we have\n\\begin{equation}\\label{X-p}\nX_{p-\\delta \\lambda}[\\lambda_f+\\delta \\lambda]=X_p[\\lambda_f]\n\\end{equation}\nthe equation (\\ref{Xpexpansion}) implies\n\\begin{equation}\n\\delta X[\\lambda_f+\\delta \\lambda]=-\\dot{X}_p[\\lambda_f]\\delta \\lambda \\ .\n\\end{equation}\nThen we can again write\n\\begin{eqnarray}\\label{Sdeltap}\n& &S_{p-\\delta \\lambda}=\\int_{\\lambda_0}^{\\lambda_f+\\delta \\lambda}\nd\\lambda \\left[L(X,\\dot{X})_{X=X_p}+\\left(\\frac{\\partial L}{\\partial X}\\delta X-\n\\frac{d}{d\\lambda}\\left[\n\\frac{\\partial L}{\\partial \\dot{X}}\\right]\\right)_{X=X_p}\\delta X\\right]\\nonumber \\\\\n& &+\\left[\\frac{\\partial L }{\\partial \\dot{X}}\\delta X\\right]_{\n\\lambda_0}^{\\lambda_f+\\delta \\lambda} \\ .  \\nonumber \\\\\n\\end{eqnarray}\nTo proceed further we note that we have\n\\begin{eqnarray}\n\\int_{\\lambda_0}^{\\lambda_f+\\delta \\lambda}L(X)_{X=X_p}=\n\\int_{\\lambda_0}^{\\lambda_f}L(X)_{X=X_p}+\\int_{\\lambda_f}^{\\lambda_f+\\delta \\lambda}L(X)_{X=X_p}\n=S_p+L[X_p]\\delta t \\ . \\nonumber \\\\\n\\end{eqnarray}\nFinally note that the last term in (\\ref{Sdeltap}) can be written as\n\\begin{eqnarray}\n\\left[\\frac{\\partial L}{\\partial\\dot{X}}\\delta X\\right]_{\\lambda_0}^{\\lambda_f+\\delta \\lambda}=\n\\frac{\\partial L}{\\partial \\dot{X}}(\\lambda_f+\\delta \\lambda)\n\\delta X(\\lambda+\\delta \\lambda)=\n-p(\\lambda_f+\\delta \\lambda)\\dot{X}(\\lambda_f)\n\\nonumber \\\\\n\\end{eqnarray}\nusing (\\ref{X-p}).\nFinally, since $X_p$ is physical trajectory we find that the second term on the first line in (\\ref{Sdeltap}) is zero. Putting these terms together we find\n\\begin{equation}\nS_{p-\\delta \\lambda}=S_\\lambda+L\\delta \\lambda-p\\dot{X}\\delta \\lambda=S_p-H\\delta \\lambda\n\\end{equation}\nthat can be written as\n\\begin{equation}\\label{partialSlambdaf}\n\\frac{\\partial S(X_p(\\lambda_f))}{\\partial \\lambda_f}=\n-H(X_p(\\lambda_f)) \\ .\n\\end{equation}\nAs the next step we analyse the value of the action $S_{p-\\delta X}$ when\nwe modify the final position $X_f+\\delta X$ without modification of the initial\nand final times.  The final position on the new physical trajectory $X_{p-\\delta X}(\\lambda)=X[\\lambda]+\\delta X[\\lambda]$ means\n\\begin{equation}\n\\delta X[\\lambda_0]=0 \\ , \\quad \\delta X[\\lambda_f]=\\delta X_f \\ .\n\\end{equation}\nLet us now evaluate the action on this new trajectory\n\\begin{eqnarray}\\label{SdeptaX}\n& &S_{p-\\delta X}=\\int_{\\lambda_0}^{\\lambda_f}d\\lambda L(X)|_{X=X_p+\\delta X}=\\nonumber \\\\\n& &=\\int_{\\lambda_0}^{\\lambda_f}d\\lambda L[X_p]+\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\delta X\\right]_{\\lambda_0}^{\\lambda_f}\n+\\int_{\\lambda_0}^{\\lambda_f}\nd\\lambda \\left[\\frac{\\partial L}{\\partial X}-\\frac{d}{d\\lambda}\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\right]\\right]_{X=X_p}\\delta X \\ .\n\\end{eqnarray}\nThe first term on the second line in (\\ref{SdeptaX})\nis $S_p$. Further, the last expression on the second line in (\\ref{SdeptaX})\nvanishes  due to the fact that the physical trajectory solves the\nequation of motion.\nFinally the second term is equal to\n\\begin{eqnarray}\n\\left[\\frac{\\partial L}{\\partial \\dot{X}}\\delta X\\right]_{\\lambda_0}^{\\lambda_f}=\n\\left. \\frac{\\partial L}{\\partial \\dot{X}}\\right|_{X=X_p}\\delta X_f=p[\\lambda_f]\\delta X_f \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nThen (\\ref{SdeptaX}) can be written as\n\\begin{equation}\\label{partSx}\n\\frac{\\partial S}{\\partial X_f}=\\lim \\frac{S_{p-\\delta X}-S_p}{\\delta X_f}=\np[\\lambda_f] \\ .\n\\end{equation}\nIn summary, the variation of the action with respect to final point is equal to momentum\nof the trajectory at the final time. Then collecting all terms together we obtain\nHamilton-Jacobi equation\n\\begin{equation}\n\\frac{\\partial S}{\\partial \\lambda}+H\\left(X,\\frac{\\partial S}{\\partial X},\\lambda\\right)=0 \\ .\n\\end{equation}\n\\section{Relativistic  Classical and Quantum Particle}\\label{third}\n\\subsection{Derivation of Relativistic Hamilton-Jacobi equation}\nLet us consider a relativistic particle with the action\n\\begin{equation}\\label{relpartaction}\nS=-m\\int d\\lambda \\sqrt{-g_{MN}\\dot{X}^M\\dot{X}^N}+q\\int d\\lambda A_M \\dot{X}^M \\ ,\n\\end{equation}\nwhere $X^M,M=0,\\dots,D-1$ label position of particle in target space-time\nwith general metric $g_{MN}(X)$ and gauge field $A_M$. Further, $m$ is the mass of the particle while $q$ is its charge. Now from the action (\\ref{relpartaction}) we derive following conjugate momenta\n\\begin{equation}\\label{relp}\np_M=m\n\\frac{g_{MN}\\dot{X}^N}{\\sqrt{-g_{MN}\\dot{X}^M\\dot{X}^N}}+qA_M \\ .\n\\end{equation}\nIt is easy to see that the\n Hamiltonian is zero\n\\begin{equation}\nH=p_M\\dot{X}^M-L=0\n\\end{equation}\nand hence Hamilton Jacobi equation implies $\\frac{\\partial S}{\\partial \\lambda}=0$. In other words in case of the relativistic theory there is no\nexplicit dependence of the action on parameter $\\lambda$. In order to find\nrelativistic HJ equation we firstly use the definition of $p_M$ given in\n(\\ref{relp}) and the relation between partial derivative of action and conjugate momenta as was given in (\\ref{partSx}). Explicitly, using (\\ref{relp}) we derive\nclassical HJ equation for relativistic particle\n\\begin{equation}\\label{relpartHJ}\n(\\partial_M S-qA_M)g^{MN}(\\partial_MS-qA_M)+m^2=0 \\ .\n\\nonumber \\\\\n\\end{equation}\n\\subsection{Ensemble of Trajectories}\nThe solution of the HJ equation is function $S$ that depends on space-time coordinates $x$. Then it is more natural to use it for the description of ensemble of trajectories where each trajectory differs by initial position and velocity. In order to describe\nthis fact we introduce $\\rho$ as a function of space-time points that describe\ndistribution of the initial positions at time $\\lambda_0$. These trajectories then evolve according to HJ equations. Further, let us define following particle four-velocity\n\\begin{equation}\nu^M(\\lambda)=\\frac{\\dot{X}^M}{\\sqrt{-g_{MN}\\dot{X}^M\\dot{X}^N}} \\ .\n\\end{equation}\nTo see that this is a correct form of the particle four-velocity let us presume that there is well defined relation between arbitrary parameter $\\lambda$ and proper-time $\\tau$ in the form $\\lambda=\\lambda(\\tau)$. Then we can rewrite $u^M$ as\n\\begin{equation}\nu^M=\\frac{\\frac{dX^M}{d\\tau}\\frac{d\\tau}{d\\lambda}}{\n\t\\sqrt{-g_{MN}\\frac{dX^M}{d\\tau}\\frac{dX^N}{d\\tau}}\\frac{d\\tau}{d\\lambda}}=\n\\frac{dX^M}{d\\tau}\n\\end{equation}\nsince $\\sqrt{-g_{MN}\\frac{dX^M}{d\\tau}\\frac{dX^N}{d\\tau}}=1$.  On the other hand we know that $u^M$ can be expressed with the help of the partial derivative of $S$ and external gauge field $A_M$ as\n\\begin{equation}\nu^M=g^{MN}(\\partial_N S-qA_N) \\ .\n\\end{equation}\nHence it is natural to define current density in the form\n\\begin{equation}\nJ^M=\\rho \\sqrt{-g}g^{MN}(\\partial_NS-qA_N) \\\n\\end{equation}\nwhich is current defined in each point of  space-time. This current is\nconserved\n\\begin{equation}\n\\partial_M J^M=\\partial_M (\\sqrt{-g}\\rho g^{MN}(\\partial_NS-qA_N))=0 \\\n\\end{equation}\nwhich expresses the fact that   trajectory cannot be created or destroyed.\nIn summary, in case of the ensemble of trajectories of  classical relativistic particle we have following two equations\n\\begin{eqnarray}\\label{classsreleqm}\n& &\\partial_M[\\rho \\sqrt{-g}g^{MN}(\\partial_NS-qA_N)]=0\n\\ , \\nonumber \\\\\n& &(\\partial_M S-qA_M)g^{MN}(\\partial_NS-qA_N)+m^2=0 \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nWe see that even classical particle has description with the help of two fields\n$S(x)$ and $\\rho(x)$, where $S(x)$ is action function that solves classical relativistic Hamilton Jacobi equation while $\\rho(x)$ describes a distribution of\ntrajectories from ensemble over space-time. It is also nice that\n these equations of motion can be derived from the action\n\\footnote{We work in units where $\\hbar=1$.}\n\\begin{equation}\\label{Iclassrel}\nI_{class}=-\\int d^Dx \\sqrt{-g}(\\rho g^{MN}(\\partial_MS-qA_N)(\\partial_MS-qA_M)+m^2\\rho)\n \\ .\n\\end{equation}\nIn fact, performing variation of the action (\\ref{Iclassrel}) with respect to $\\rho$ we obtain the second equation in (\\ref{classsreleqm}) while performing\nvariation with respect to $S$ we obtain  first equation in (\\ref{classsreleqm}).\nAs the next step we introduce complex field $\\psi_{clas}=\\sqrt{\\rho}e^{iS}$ so that we can write\n\\begin{equation}\n\\rho=\\psi_{clas} \\psi^*_{clas} \\ , \\quad  S=\\frac{1}{2i}\\ln \\frac{\\psi_{clas}}{\\psi^*_{clas}} \\ .\n\\end{equation}\nThen it is easy to see that the action (\\ref{Iclassrel}) can\nbe written in the form\n\\begin{eqnarray}\nI_{class}\n&=&\\int d^Dx \\sqrt{-g}\\left(-\\frac{1}{2}D_M\\psi (D_N\\psi)^*g^{MN}+\\right.\n\\nonumber \\\\\n& & \\left.+\n\\frac{\\psi^*_{clas}}{4\\psi_{clas}}\nD_M\\psi_{clas} D_N\\psi_{clas} g^{MN}+\n\\frac{\\psi_{clas}}{4\\psi_{clas}^*}(D_M\\psi_{clas})^*(D_N\\psi^{clas})^*g^{MN}+m^2\n\\psi_{clas} \\psi^*_{clas}\\right) \\ ,\n\\nonumber \\\\\n\\end{eqnarray}\nwhere\n\\begin{equation}\nD_M\\psi_{clas}=\\partial_M\\psi_{clas}-iqA_M\\psi_{clas} \\ .\n\\end{equation}\nNote that the equation of motion that follow from the action above takes the form\n\\begin{eqnarray}\\label{eqkkclas}\n& &\t\\frac{1}{2}D_M[\\sqrt{-g}g^{MN}\n\tD_N\\psi_{clas}]\n+\\frac{1}{4\\psi_{clas}}D_M\\psi_{clas}D_N\\psi_{clas}g^{MN}\\sqrt{-g}+m^2\\psi_{clas}\\sqrt{-g}-\n\\nonumber \\\\\n&-&\\frac{\\psi_{clas}}{4(\\psi_{clas})^*}\n(D_M\\psi_{clas})^*(D_N\\psi_{clas})^*g^{MN}\\sqrt{-g}\n-\\frac{1}{2}D_M\\left[\\frac{\\psi_{clas}}{\\psi_{clas}^*}\n\\sqrt{-g}g^{MN}(D_N\\psi_{clas})^*\\right]=0 \\ .\n\\nonumber \\\\\n\\end{eqnarray}\t\nThe crucial point of this equation is the lack of linearity of solutions. In fact, if $\\psi^1_{clas}$ and $\\psi^2_{clas}$ are  two solutions of the equation (\\ref{eqkkclas})\nthen their linear combinations $a\\psi_{clas}^1+b\\psi_{clas}^2$, where $a,b$ are two complex numbers cannot be solutions of (\\ref{eqkkclas}). This is fundamental difference between classical and quantum relativistic particle.\n\\subsection{Quantum Relativistic Particle}\nNow we consider complex field action\n\\begin{equation}\nI=-\\int d^Dx \\sqrt{-g}[g^{MN}D_M \\Psi (D_N \\Psi)^*+m^2\\Psi \\Psi^*] \\ ,\n\\end{equation}\nwhere $D_M\\Psi=\\partial_M\\Psi-iqA_M\\Psi$. The equation of motion derived\nfrom this action has the form\n\\begin{equation}\nD_M[\\sqrt{-g}g^{MN}D_N\\Psi]-m^2\\Psi=0\n\\end{equation}\nthat are clearly linear which is fundamental difference between classical\nand quantum description.\n\nAs the next step we introduce polar decomposition of  $\\Psi$ as\n\\begin{equation}\n\\Psi=\\sqrt{\\rho}e^{iS} \\ ,\n\\end{equation}\nso that we have\n\\begin{equation}\nD_M\\Psi=\\frac{1}{2\\sqrt{\\rho}}\\partial_M\\rho e^{iS}+i\\sqrt{\\rho}(\\partial_M S-qA_M)e^{iS} \\ .\n\\end{equation}\nThen it is easy to see that the action has the form\n\\begin{eqnarray}\nI=-\\int d^Dx \\sqrt{-g}\\left(g^{MN}\\frac{1}{4\\rho}\\partial_M\\rho\n\\partial_N\\rho+\\rho  g^{MN}(\\partial_M S-qA_M)(\\partial_NS-qA_N)+m^2\\rho \\right)\n\\nonumber \\\\\n\\end{eqnarray}\nwith following equations of motion\n\\begin{eqnarray}\\label{KGeqm}\n& &-\\sqrt{-g}\\frac{1}{4\\rho^2}g^{MN}\\partial_M\\rho\\partial_N\\rho\n-\\partial_M[\\sqrt{-g}g^{MN}\\frac{1}{4\\rho}\\partial_N\\rho]+\n\\nonumber \\\\\n& &+\\sqrt{-g}g^{MN}(\\partial_MS-qA_M)(\\partial_N S-qA_N)+m^2\\sqrt{-g}=0 \\ ,\n\\nonumber \\\\\n& &\\partial_M[\\sqrt{-g}g^{MN}\\rho(\\partial_NS-qA_N)]=0 \\ .\n\\end{eqnarray}\nNote that the first equation can be interpreted as the quantum relativistic\nHamilton-Jacobi equation when we write it in the form\n\\begin{eqnarray}\n& &g^{MN}(\\partial_M S-qA_M)(\\partial_NS-qA_N)+m^2+Q=0 \\ , \\nonumber \\\\\n& & Q=-\\frac{1}{4\\rho^2}g^{MN}\\partial_M \\rho \\partial_N\\rho-\\frac{1}{\\sqrt{-g}}\n\\partial_M[\\sqrt{-g}g^{MN}\\frac{1}{4\\rho}\\partial_N\\rho]  \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nNow we would like to ask a question what is a consequence of the quantum\npotential $Q$ on the dynamics of quantum particle. Since $p_M=\\partial_MS$ we can\nrewrite the HJ equation into the form\n\\begin{equation}\n(p_M-qA_M)g^{MN}(p_N-qA_N)+m^2+Q=0 \\\n\\end{equation}\nwhich is nothing else than the quantum version of Hamiltonian constraint. As\na consequence the Hamiltonian has the form\n\\begin{equation}\nH_Q=\\Gamma\\left( (p_M-qA_M)g^{MN}(p_N-qA_N)+m^2+Q\\right)\n\\ ,\n\\end{equation}\nwhere $\\Gamma$ is Lagrange multiplier corresponding to the Hamiltonian constraint.\nLet us now derive corresponding Lagrangian. Using the Hamiltonian given above we obtain\n\\begin{equation}\n\\dot{X}^M=\\pb{X^M,H_Q}=2\\Gamma g^{MN}(p_N-qA_N)\n\\end{equation}\nand it is easy to find\n\\begin{equation}\nL_Q=p_M\\dot{X}^M-H_Q=\n\\frac{1}{4\\Gamma}\\dot{X}^Mg_{MN}\\dot{X}^N-\\Gamma (m^2+Q) \\ .\n\\end{equation}\nFinally we can eliminate $\\Gamma$ using corresponding equations of motion\n\\begin{equation}\n-\\frac{1}{4\\Gamma^2}\n\\dot{X}^Mg_{MN}\\dot{X}^N-(m^2+Q)\n=0\n\\end{equation}\nthat has solution\n\\begin{equation}\n\\Gamma=\\frac{\\sqrt{-g_{MN}\\dot{X}^M\\dot{X}^N}}{2\\sqrt{m^2+Q}} \\ .\n\\end{equation}\nInserting back to the Lagrangian we derive the final form of the\nLagrangian for quantum relativistic particle in Bohmian mechanics\n\\begin{equation}\nL_Q=-\\frac{1}{2}\\sqrt{m^2+Q}\\sqrt{-g_{MN}\\dot{X}^M\\dot{X}^N}+qA_N\\dot{X}^N \\ .\n\\end{equation}\nWe mean that this is very remarkable result. In fact, solving the equations of motion that follow from this Lagrangian we derive quantum trajectories for relativistic particle where these trajectories are strongly affected by quantum potential $Q$ which depends on function $\\rho$. It is further very interesting that the quantum potential appears in the relativistic Lagrangian under square root together with the rest mass of the particle. Of course, in order to determine quantum trajectories we have to know the form of $\\rho$ that is determined by second equation in (\\ref{KGeqm}) that, since it contains $S$ has to be solved together with the first equation in (\\ref{KGeqm}).\n\\section{Particle in Newton-Cartan Background and Its Hamilton-Jacobi Equation}\n\\label{fourth}\nIn this section we find Hamilton-Jacobi equation for particle in Newton-Cartan  background. To do this we find an action for massive particle in this background with the help of null dimensional reduction, following\n\\cite{Festuccia:2016caf,Julia:1994bs}.\nExplicitly, let us consider an action for massive particle in $D+1$ dimensions with the background metric $g_{AB}$ and the gauge field $A_A$ where $A=0,\\dots,D$. The particle action has standard form\n\\begin{equation}\\label{massparticleaction}\nS=-T\\int d\\lambda \\sqrt{-\\gamma_{AB}\\dot{X}^A\\dot{X}^B}+q\\int d\\lambda A_M\n\\dot{x}^M \\ ,\n \\end{equation}\n where $T$ is the mass of particle and where $q$ is its charge. As the next\n step we determine Hamiltonian for this particle. From (\\ref{massparticleaction})\nwe find following conjugate momenta\n\\begin{equation}\np_A=T\\frac{\\gamma_{AB}\\dot{X}^B}{\\sqrt{-\\gamma_{AB}\\dot{X}^A\\dot{X}^B}}+q A_A\n\\end{equation}\nthat implies following Hamiltonian constraint\n\\begin{equation}\n\\mH_\\tau=(p_A-qA_A) \\gamma^{AB}(p_B-qA_B)+T^2\\approx 0 \\ .\n\\end{equation}\nLet us now consider background metric which possesses a null\nisometry that is generated by coordinates $\\partial_u$\n\\begin{equation}\nds^2=\\gamma_{AB}dx^A dx^B=2\\tau_\\mu dx^\\mu(du-M_\\nu dx^\\nu)+h_{\\mu\\nu} dx^\\mu dx^\\nu \\ , \\quad \\mu,\\nu=0,1,\\dots,D-1\n\\end{equation}\nso that\n\\begin{equation}\\label{gammametric}\n\\gamma_{\\mu u}=\\gamma_{u\\mu}=\\tau_\\mu \\ , \\quad \\gamma_{\\mu\\nu}=\nh_{\\mu\\nu}-\\tau_\\mu M_\\nu -\\tau_\\nu M_\\mu\\equiv \\bh_{\\mu\\nu}\n\\\n\\end{equation}\nand also\n\\begin{equation}\n\\sqrt{-\\gamma}=e \\ , \\quad\ne=\\det \\left(\\tau_\\mu, e_\\mu^{ \\ a}\\right) \\ .\n\\end{equation}\nNote that the metric inverse to (\\ref{gammametric}) has the form\n\\begin{equation}\\label{gammametricinv}\n\\gamma^{uu}=2\\Phi \\ , \\quad  \\gamma^{u\\mu}=-\\hv^\\mu \\ , \\quad  \\gamma^{\\mu\\nu}=h^{\\mu\\nu} \\ ,\n\\end{equation}\nwhere we defined spatial vierbein $e^\\mu_{ \\ a}$ and $e_\\mu^{ \\ a} \\ , a=1,\\dots,D-1$ as\n\\footnote{We follow convention used in \\cite{Festuccia:2016caf}.}\n\\begin{equation}\nh_{\\mu\\nu}=e_\\mu^{ \\ a}e_\\nu^{ \\ b}\\delta_{ab} \\ , \\quad\nh^{\\mu\\nu}=e^\\mu_{ \\ a}e^\\nu_{ \\ b}\\delta^{ab} \\  , \\quad\ne^\\mu_{ \\ a}e_\\mu^{ \\ b}=\\delta_a^b \\ .\n\\end{equation}\nWe also introduce temporal vector $v^\\mu$ that obeys the relation\n\\begin{equation}\nv^\\mu\\tau_\\mu=-1 \\ , \\quad v^\\mu e_\\mu^{ \\ a}=0 \\ , \\quad\n\\tau_\\mu e^\\mu_{ \\ a}=0 \\ .\n\\end{equation}\nFinally we introduced Galilean invariant objects $\\hv^\\mu, \\he_\\mu^{ \\ a}$ and\n$\\Phi$ defined as\n\\begin{eqnarray}\n\\hv^\\mu=v^\\mu-h^{\\mu\\nu}M_\\nu \\ , \\quad\n\\he_\\mu^{ \\ a}=e_\\mu^{ \\ a}-M_\\nu e^\\nu_{ \\ b}\\delta^{ba}\n\\tau_\\mu \\ , \\quad\n\\Phi=-v^\\mu M_\\mu+\\frac{1}{2}h^{\\mu\\nu}M_\\mu M_\\nu \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nWe further presume that the gauge field has the form $A_A=(A_u,A_\\mu)=\n(\\varphi,\\bar{A}_\\mu-\\varphi M_\\mu)$.\nNow with the help of this metric we perform dimensional reduction.\nTo do this we have to presume that all fields do not depend on $u$.\nThen using the previous form of the Hamiltonian constraint and the form\nof the background field we find that the Hamiltonian constraint has the form\n\\begin{eqnarray}\n& &\\mH_\\tau=2(p_u-q\\varphi)\\Phi (p_u-q\\varphi)\n-2(p_u-q\\varphi) \\hv^\\mu (p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu))+\\nonumber \\\\\n& &+h^{\\mu\\nu}(p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu))( p_\\nu-q(\\bar{A}_\\nu-\\varphi M_\\nu))+T^2\\approx 0 \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nLet us now assume that the particle has constant momentum along $u-$direction\nthat we denote as $m$. This is certainly reasonable assumption due to the fact that  background fields do not depend on $u-$coordinate.\n We also see that it is possible to take the limit\n$T\\rightarrow 0$ at the Hamiltonian constraint. In the end we obtain\nHamiltonian constraint in the form\n\\begin{eqnarray}\n& &\\mH_\\tau=2(m-q\\varphi)\\Phi (m-q\\varphi)\n-2(m-q\\varphi) \\hv^\\mu (p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu))+\\nonumber \\\\\n& &+h^{\\mu\\nu}(p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu))( p_\\nu-q(\\bar{A}_\\nu-\\varphi M_\\nu))\\approx 0 \\ , \\quad\nH=\\Gamma \\mH_\\tau  \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nIn order to find corresponding Lagrangian we determine\nequations of motion for $X^\\mu$\n\\begin{equation}\n\\dot{X}^\\mu=\\pb{X^u,H}=\\Gamma 2h^{\\mu\\nu}(p_\\nu-q(\\bar{A}_\\nu-\\varphi M_\\nu))-2(K-q\\varphi)\\Gamma\\hv^\\mu \\ ,\n\\end{equation}\nwhere $\\Gamma$ is Lagrange multiplier corresponding to the constraint $\\mH_\\tau\\approx 0$. Then we obtain Lagrangian for particle in Newton-Cartan background in the form\n\\begin{eqnarray}\\label{LNCparticle}\n& &\\mL=p_\\mu \\dot{X}^\\mu-H=\\nonumber \\\\\n&=&\\Gamma (p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu)) h^{\\mu\\nu}(p_\\nu-q(\\bar{A}_\\nu-\n\\varphi M_\\nu))-\n2(m-q\\varphi)^2\\Phi\\Gamma+qA_\\mu\\partial_\\lambda x^\\mu \\ . \\nonumber \\\\\n\\end{eqnarray}\nTo proceed further we use the fact that\n\\begin{eqnarray}\n& &\\he_\\mu^{ \\ a}\\dot{X}^\\mu=2\\Gamma e^\\nu_{ \\ c}\\delta^{ca}(p_\\nu-q(\\bar{A}_\\nu-\n\\varphi M_\\nu))\n\\ ,  \\nonumber \\\\\n& &\\dot{X}^\\nu \\he_\\nu^{ \\ a}\\delta_{ab}\\he_\\mu^{ \\ b}\\dot{X}^\\mu=\n4\\Gamma^2 (p_\\mu-q(\\bar{A}_\\mu-\\varphi M_\\mu)) h^{\\mu\\nu}(p_\\nu-q(\\bar{A}_\\nu-\\varphi M_\\nu))\n\\nonumber \\\\\n\\end{eqnarray}\nso that the Lagrangian (\\ref{LNCparticle})  has the form\n\\begin{equation}\n\\mL=\\frac{1}{4\\Gamma}\\dot{X}^\\mu \\he_\\mu^{ \\ a}\\delta_{ab}\n\\he_\\nu^{ \\ b}\\dot{X}^\\nu-2(m-q\\varphi)^2\\Phi\\Gamma+q A_\\mu\\dot{X}^\\mu \\ .\n\\end{equation}\nFinally we solve the equations of motion for $\\Gamma$ and we get\n\\begin{equation}\n\\tau_\\mu \\dot{X}^\\mu=2\\Gamma(m-q\\varphi)\n\\end{equation}\nthat implies $\\Gamma=\\frac{1}{2(K-q\\varphi)}\\dot{X}^\\mu \\tau_\\mu\n$ and hence the Lagrangian density has the form\n\\begin{eqnarray}\\label{mLfinalNC}\n\\mL&=&\\frac{(m-q\\varphi)}{2\\tau_\\mu\\dot{X}^\\mu}\\dot{X}^\\mu\n\\he_\\mu^{ \\ a}\\delta_{ab}\\he_\\nu^{ \\ b}\\dot{X}^\\nu\n-(m-q\\varphi)\\Phi\\tau^\\mu\\dot{X}^\\mu+aA_\\mu\\dot{X}^\\mu=\\nonumber \\\\\n&=&\\frac{(m-q\\varphi)}{2\\tau_\\mu \\dot{X}^\\mu}\\dot{X}^\\mu\n\\bh_{\\mu\\nu}\\dot{X}^\\nu+q A_\\mu\\dot{X}^\\mu \\ ,\n\\nonumber \\\\\n\\end{eqnarray}\nwhere in the last step we used the fact that\n\\begin{equation}\n\\he_\\mu^{ \\ a}\\delta_{ab}\\he_\\nu^{ \\ b}=\\bh_{\\mu\\nu}+2\\Phi\\tau_\\mu\n\\tau_\\nu \\ .\n\\end{equation}\nThis is an action for the massive particle in Newton-Cartan background and in\nthe background gauge field that coincides with the action found in\n\\cite{Festuccia:2016caf}.\n\nNow we derive HJ equation for particle in NC background. We start with the\n conjugate momenta that follow from (\\ref{mLfinalNC})\n\\begin{equation}\\label{pmupartialS}\np_\\mu=-\\frac{(m-q\\varphi)}{2(\\tau_\\mu\\dot{X}^\\mu)^2}\\tau_\\mu\\dot{X}^\\rho\\bh_{\\rho\\sigma}\n\\dot{X}^\\sigma+qA_\\mu+\\frac{(m-q\\varphi)\\bh_{\\mu\\nu}\\dot{X}^\\nu}{\\tau_\\rho\n\t\\dot{X}^\\rho}=\\partial_\\mu S \\ .\n\\end{equation}\nAs the next step we\nmultiply this expression with $v^\\mu$ and  we obtain\n\\begin{equation}\nv^\\mu(\\partial_\\mu S-q A_\\mu+(m-q\\varphi)M_\\mu)=\\frac{(m-q\\varphi)}{2(\\tau_\\mu\\dot{X}^\\mu)^2}\n\\dot{X}^\\rho h_{\\rho\\sigma}\\dot{X}^\\sigma\n\\end{equation}\nOn the other hand when we multiply (\\ref{pmupartialS})  with $h^{\\nu\\mu}$ we get\n\\begin{equation}\nh^{\\nu\\mu}(\\partial_\\mu S-qA_\\mu)=\n\\frac{(m-q\\varphi)}{\\tau_\\rho\\dot{X}^\\rho}h^{\\nu\\mu}h_{\\mu\\rho}\\dot{X}^\\rho\n-(m-q\\varphi)h^{\\nu\\mu}M_\\mu \\ .\n\\end{equation}\nCollecting there results together we obtain following classical HJ equation for massive and charged particle in Newton-Cartan background\n\\begin{eqnarray}\\label{HJNCparticle}\n& &(\\partial_\\mu S-qA_\\mu+(m-q\\varphi)M_\\mu)h^{\\mu\\nu}\n(\\partial_\\nu S-qA_\\nu+(m-q\\varphi)M_\\nu)-\\nonumber \\\\\n& & -2(m-q\\varphi)v^\\mu\n(\\partial_\\mu S-qA_\\mu+(m-q\\varphi)M_\\mu)=0 \\ .\n\\nonumber \\\\\n\\end{eqnarray}\n  For further purposes we rewrite this equation into the form\n\\begin{eqnarray}\n& &2(m-q\\varphi)\\hv^\\mu (\\partial_\\mu S-qA_\\mu)-(\\partial_\\mu S-qA_\\mu)h^{\\mu\\nu}(\\partial_\\nu S-qA_\\nu)-2(m-q\\varphi)^2\\Phi=0\n\\ .\n\\nonumber \\\\\n\\end{eqnarray}\nIn order to find its quantum version we start with the action for complex scalar\nfield in NC background. We again derive this action using   null dimensional reduction.\nMore explicitly, let us consider  an action  for complex scalar field in\n$D+1$ dimensions\n\\begin{equation}\nI=\\int d^{D}x\\sqrt{-\\gamma}\n\\left(-\\gamma^{AB}D_A \\Psi D_B\\Psi^*\\right) \\ ,\n\\end{equation}\nwhere $D_A=\\partial_A \\Psi-iq A_A\\Psi$.\nNow  we presume that the\nbackground metric is given in\n(\\ref{gammametric}) and (\\ref{gammametricinv}). We further presume\nthat all fields do not depend on $u$ and impose following ansatz for the  field $\\Psi$\n\\begin{equation}\n\\Psi=e^{imu}\\psi \\ , \\quad  \\partial_u \\Psi=\nim \\Psi \\ .\n\\end{equation}\nThen  we obtain an action for Schr\\r{o}dinger\nfield in NC background\n\\begin{eqnarray}\\label{SchNCb}\n& &I=\\int d^{D}xe( i(m-q\\varphi)\\hv^\\mu(D_\\mu\\psi)^*\\psi-i(m-q\\varphi)\\hv^\\mu D_\\mu\\psi\n\\psi^*-\\nonumber \\\\\n& & -2\\Phi (m-q\\varphi)^2\\psi \\psi^*-h^{\\mu\\nu}D_\\mu\\psi (D_\\nu\\psi)^*) \\ ,\n\\nonumber \\\\\n\\end{eqnarray}\nwhere\n\\begin{equation}\nD_\\mu\\psi=\\partial_\\mu\\psi-iq A_\\mu \\psi \\ .\n\\end{equation}\nFrom (\\ref{SchNCb})  we could find equations of motion for $\\psi$ which are clearly linear.\n\nAs the next step we use polar parametrization of $\\psi$ defined as\n\\begin{equation}\n\\psi=\\sqrt{\\rho}e^{iS} \\ ,\n\\end{equation}\nwhere $\\rho$ and $S$ are real. Inserting this form of $\\psi$ into\n(\\ref{SchNCb}) we obtain\n\\begin{eqnarray}\\label{SchactSrho}\n& &I^{sch}\n=\\int d^{D}x e\\left(2(m-q\\varphi)\\rho \\hv^\\mu (\\partial_\\mu S-qA_\\mu)\n-2\\Phi(m-q\\varphi)^2\\rho-\\right.\\nonumber \\\\\n& &\\left.-\\frac{1}{4\\rho}h^{\\mu\\nu}\\partial_\\mu\\rho \\partial_\\nu\\rho-\n\\rho h^{\\mu\\nu}(\\partial_\\mu S-qA_\\mu)(\\partial_\\nu S-qA_\\nu)\\right) \\ .\n\\nonumber \\\\\n\\end{eqnarray}\nFrom (\\ref{SchactSrho}) we obtain  following equations of motion\n\\begin{eqnarray}\n& &\\partial_\\mu[e(m-q\\varphi)\\rho \\hv^\\mu]-\\partial_\\mu[e\\rho h^{\\mu\\nu}\n(\\partial_\\nu S-qA_\\nu)]=0 \\ , \\nonumber \\\\\n& &2e(m-q\\varphi)\\hv^\\mu(\\partial_\\mu S-qA_\\mu)-2e\\Phi(m-q\\varphi)^2+\n\\nonumber \\\\\n& &+\\frac{1}{4\\rho^2}eh^{\\mu\\nu}\\partial_\\mu\\rho\\partial_\\nu\\rho+\n\\partial_\\mu[\\frac{1}{2\\rho}eh^{\\mu\\nu}\\partial_\\nu\\rho]\n-eh^{\\mu\\nu}(\\partial_\\mu S-qA_\\mu)(\\partial_\\nu S-qA_\\nu)=0 \\ .  \\nonumber \\\\\n\\end{eqnarray}\nThe first equation can be interpreted as continuity equation  that expresses\nconservation of the trajectories in the ensemble of  all possible trajectories of massive\nparticle in NC background while the second one can be written into the form\n\\begin{eqnarray}\n& & 2(m-q\\varphi)\\hv^\\mu(\\partial_\\mu S-qA_\\mu)-h^{\\mu\\nu}(\\partial_\\mu S-qA_\\mu)\n(\\partial_\\nu S-qA_\\nu)-\\nonumber \\\\\n& & -2\\Phi (m-q\\varphi)^2+Q=0 \\ , \\nonumber \\\\\n\\end{eqnarray}\nwhere $Q$ is defined as\n\\begin{equation}\nQ=\\frac{1}{4\\rho^2}h^{\\mu\\nu}\\partial_\\mu\\rho \\partial_\\nu\\rho+\\frac{1}{2e}\n\\partial_\\mu\\left[\\frac{1}{\\rho}eh^{\\mu\\nu}\\partial_\\nu\\rho \\right] \\ .\n\\end{equation}\nIt is clear that classical action corresponds to the situation when $Q=0$. Then the equations of motion for $S$ and $\\rho$ can be derived from the action\n\\begin{equation}\nI^{sch}_{clas}=\\int d^{D}x\ne(2(m-q\\varphi)\\rho \\hv^\\mu (\\partial_\\mu S-qA_\\mu)-2\\Phi (m-q\\varphi)^2\\rho\n-\\rho h^{\\mu\\nu}(\\partial_\\mu S-qA_\\mu)(\\partial_\\nu S-qA_\\nu)) \\\n\\end{equation}\nor equivalently using complex field $\\psi_{clas}=\\sqrt{\\rho}e^{iS}$\n\\begin{eqnarray}\n& &I^{sch}_{clas}=\\int d^{D}x e\\left(i(m-q\\varphi)\\hv^\\mu (\\psi_{clas} (D_\\mu\\psi_{clas})^*-\\psi^*_{clas}\nD_\\mu\\psi_{clas})-2\\Phi (m-q\\varphi)^2\\psi_{clas} \\psi^*_{clas} \\right.\\nonumber \\\\\n& & \\left.-\\frac{1}{2}h^{\\mu\\nu}\nD_\\mu\\psi_{clas} (D_\\nu\\psi_{clas})^*+\n +\\frac{1}{4}h^{\\mu\\nu}\\frac{D_\\mu\\psi_{clas} D_\\nu\\psi_{clas}}{\\psi_{clas}\\psi_{clas}}+\n\\frac{1}{4}h^{\\mu\\nu}\\frac{(D_\\mu\\psi_{clas})^*(D_\\nu\\psi_{clas})^*}{\\psi^*_{clas}\n\\psi^*_{clas}}\\right) \\ .  \\nonumber \\\\\n\\end{eqnarray}\nWe again see that resulting equations of motions will be non-linear due to the\nlast two terms on the second line and hence superposition of two solutions of the equations of motion of classical particle in NC background is not solutions of these equations.\n\nFinally we will try to find quantum version of Lagrangian for massive particle\nin NC background starting with the quantum HJ equation. Replacing $\\partial_\\mu S$ with $p_\\mu$ we find that it corresponds to the Hamiltonian constraint in the form\n\\begin{eqnarray}\n& & \\mH_\\tau^{q}=\n2(m-q\\varphi)\\hv^\\mu(p_\\mu-qA_\\mu)-h^{\\mu\\nu}(p_\\mu-qA_\\mu)\n(p_\\nu-qA_\\nu)-\\nonumber \\\\\n& &-2\\Phi (m-q\\varphi)^2+Q=0 \\  \\nonumber \\\\\n\\end{eqnarray}\nso that the Hamiltonian is $H^q=\\Gamma \\mH_\\tau^q$,  where $\\Gamma$ is corresponding Lagrange multiplier. With the help of this Hamiltonian\nwe find\n\\begin{equation}\n\\dot{X}^\\mu=\\pb{X^\\mu,H^q}=2\\Gamma ((m-q\\varphi)\\hv^\\mu-h^{\\mu\\nu}(p_\\nu-qA_\\nu))\n\\end{equation}\nand hence\n\\begin{eqnarray}\nL^q=p_\\mu\\dot{X}^\\mu-H^q\n=\\frac{(m-q\\varphi)}{2\\dot{X}^\\mu\\tau_\\mu}\\dot{X}^\\mu\\bh_{\\mu\\nu}\\dot{X}^\\nu+qA_\\mu+\n\\frac{1}{m-q\\varphi}Q\\tau_\\mu\\dot{X}^\\mu \\\n\\end{eqnarray}\nusing the fact that from the equation of motion for $X^\\mu$ we obtain\n\\begin{equation}\n\\Gamma=-\\frac{\\dot{X}^\\mu\\tau_\\mu}{2(m-q\\varphi)} \\ .\n\\end{equation}\nWe see fundamental difference between relativistic quantum particle and quantum particle in NC background which is in the presence of the quantum potential in both Lagrangians where in case of Newton-Cartan quantum particle the quantum potential is simply added to the Lagrangian of classical particle while in case of quantum relativistic particle it appears under square root together with $m^2$.\n\\section{Conclusion}\\label{fifth}\nIn this short note we performed analysis of classical and quantum particles in relativistic and Newton-Cartan background in the framework of de Broglie-Bohm quantum mechanics. We determined classical and quantum version of HJ equation for relativistic particle and discuss properties of the wave function description of both theories. We performed the same analysis in case of particle in Newton-Cartan background where we determined classical and quantum HJ equation. We mean that this is really interesting result since we found covariant version of non-relativistic de-Broglie-Bohm quantum mechanics.\n\nDe Broglie-Bohm formulation of quantum mechanics is very fascinating and certainly deserves further study and analysis. In particular, it is very interesting that Lagrangian for quantum particle contains quantum potential that depends on derivatives of density of trajectories at space-time point. Further,\nit can be shown that the Schr\u00f6dinger equation\nmay be recast as a self-contained second-order Newtonian law for a\ncongruence of spacetime trajectories, see for example\n\\cite{Holland2}. It would be certainly very interesting to perform similar analysis in case of massive particle in Newton-Cartan background. We hope to return to this problem in near future.\n\n\n\n {\\bf Acknowledgement:}\n\n This  work  was\nsupported by the Grant Agency of the Czech Republic under the grant\nP201\/12\/G028.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\\label{introduction}\n\nSince 1958, translators and linguists have published work on translation processes~\\cite{VinayDarbelnet1958,Newmark1981,ChuquetPaillard1989,Molina_HurtadoAlbir_2002}.\nThey distinguish literal translations from other translation processes at subsentential level. Consider these two human non-literal translation examples: \nthe first translation preserves exactly the meaning,\nwhere the fixed expression \\textit{\u00e0 la hauteur de}~`to the height of' has a figurative sense which means \\textit{capable of solving}; while the second one is more complicated, there exists a textual inference between the source text and the translation. \n\n\\vspace*{0.5em}\n\n\\noindent \\textit{(1.EN) a solution \\textbf{that's big enough to solve} our problems} \\\\\n\\textit{(1.FR) une solution \\textbf{\u00e0 la hauteur de} nos probl\u00e8mes}  \\\\\n\\textit{(2.EN) and \\textbf{that scar has stayed with him} for his entire life} \\\\\n\\textit{(2.FR) et que, toute sa vie, \\textbf{il a souffert de ce traumatisme}} \\\\\n(`he has suffered from this traumatism') \n\n\\vspace*{0.5em}\n\nNon-literal translations can bring difficulties for automatic word alignment~\\cite{Dorr_DUSTer_2002,Deng2017}, or cause meaning changes in certain cases. However, to the best of our knowledge, there has not been effort to automatically classify these fine-grained translation processes to benefit downstream natural language processing tasks. For example, \\gls{mt} techniques have been leveraged for paraphrase extraction from bilingual parallel corpora~\\cite{BannardCallison_Burch2005,Mallinson2017}. The assumption is that two monolingual segments are potential paraphrases if they share common translations in another language. Currently the largest paraphrase resource, PPDB (ParaPhrase DataBase)~\\cite{Ganitkevitch2013}, has been built following this method. Nonetheless, Pavlick \\textit{et al.}~\\cite{PPDB_relation2015} revealed that there exist other relations (\\textit{i.e. Entailment (in two directions), Exclusion, Other related and Independent})\\footnote{Exclusion: X is the contrary of Y; X is mutually exclusive with Y. Other related: X is related in some other way to Y. (\\textit{e.g. country \/ patriotic}). Independent: X is not related to Y.} than strict equivalence (paraphrase) in PPDB. Non-literal pivot translations inside the parallel corpora could break the strict equivalence between the candidate paraphrases extracted, whereas they have not received enough attention during this corpora exploitation.\n\nFrom a linguistic point of view, apart from the word-for-word literal translation, different versions of human translations reflect the richness of human language expressions, where various translation processes could be employed. Furthermore, because of the existing differences between languages and cultures, non-literal translation processes are sometimes inevitable to produce correct and natural translations. The fine-grained phrase-level translation processes could help foreign language learners to better compare the language being learned with\nanother language already mastered. \n \nBased on the theories developed in translation studies and through manually annotating and analyzing an English-French parallel corpus, Zhai \\textit{et al.}~\\cite{zhai2018b} have proposed a typology of translation processes adapted to their corpus. In this work our main contribution is proposing an automatic classification of translation processes at subsentential level, based on these annotated examples. From the aspect of granularity and our goal of better controlling paraphrasing process or helping foreign language learners, it is different from the task of filtering semantically divergent parallel sentence pairs to improve the performance of \\gls{mt} systems~\\cite{Carpuat2017,Vyas2018,Pham2018}. Experimental results show that we can distinguish non-literal translation processes from literal translation with an accuracy of 87.09\\%, and 55.20\\% for classifying among non-literal multi-classes. \n\nIn the present paper, after reviewing related work, we describe the manual annotation and the data set. Exploited features and different neural network architectures will be presented, followed by experimental results and error analysis. Finally we conclude and present the perspectives of this work.\n\n\n\\section{Related Work}\n\\label{relatedWork}\n\nTranslators and linguists have proposed several typologies to characterize different translation processes. Vinay and Darbelnet~\\cite{VinayDarbelnet1958} identified direct and oblique translation processes, the latter being employed when a literal translation is unacceptable, or when structural or conceptual asymmetries arising between the source language and the target language are non-negligible. Following studies include, among others, the work of Newmark~\\cite{Newmark1981,newmark1988}, Chuquet and Paillard \\cite{ChuquetPaillard1989}. More recently, Molina and Hurtado Albir~\\cite{Molina_HurtadoAlbir_2002} proposed their own categorization based on studying the translation of cultural elements in the novel \\textit{A Hundred Years of Solitude} from Spanish to Arabic. \n\nNon-literal translations or cross-language divergences have been studied to improve \\gls{mt} related techniques. In order to enable more accurate word-level alignment, Dorr \\textit{et al.}~\\cite{Dorr_DUSTer_2002} proposed to transform English sentence structure to more closely resemble another language. A translation literalness measure was proposed to select appropriate sentences or phrases for automatically constructing \\gls{mt} knowledge~\\cite{imamura2003automatic}. Using a hierarchically aligned parallel treebank, Deng and Xue~\\cite{Deng2017} semi-automatically identify, categorize and quantify seven types of translation divergences between Chinese and English.\\footnote{Lexical encoding; difference in transitivity; absence of language-specific function words; difference in phrase types; difference in word order; dropped elements; structural paraphrases.} Based on the syntactic and semantic similarity between bilingual sentences, Carl and Schaeffer~\\cite{Carl_Schaeffer_2017} developed a metric of translation literality. We have drawn inspiration from these preceding work for our feature engineering. \n\nRecently, different models have been proposed to automatically detect translation divergence in parallel corpora, with the goal of automatically filtering out divergent sentence pairs to improve \\gls{mt} systems' performance. An SVM-based cross-lingual divergence detector was introduced \\cite{Carpuat2017}, using word alignments and sentence length features. Their following work~\\cite{Vyas2018} proposed a Deep Neural Network-based approach. This system could be trained for any parallel corpus without any manual annotation. They confirmed that these divergences are a source of performance degradation in neural machine translation. Pham \\textit{et al.}~\\cite{Pham2018} built cross-lingual sentence embeddings according to the word similarity with a neural architecture in an unsupervised way. They measure the semantic equivalence of a sentence pair to decide whether to filter it out. \n\nAnother task studying human translations concerns automatic post-editing~\\cite{APE_2018}. The aim is evaluating systems for automatically correcting translation errors of an unknown ``black box'' \\gls{mt} engine, by learning from human revisions of translations produced by the same engine. Evaluation metrics include TER~\\cite{snover2006study}, BLEU~\\cite{papineni2002bleu} and manual evaluation. The task that we propose here is different from these attempts, which either filter semantically divergent sentence pairs to improve the performance of \\gls{mt} systems; or automatically correct machine translation errors to improve the translation quality. Our task of classifying translation processes (in two classes or in multi-classes) at subsentential level is a stand-alone task. One of our long term objectives is leveraging this automatic classification to better control phrase-level paraphrase extraction from bilingual parallel corpora. \n\n\n\\section{Manual Annotation and Data Description}\n\\label{annotation}\n\nIn order to model translation choices made by human translators at subsentential level, Zhai \\textit{et al.}~\\cite{zhai2018b} have annotated a trilingual parallel (English-French, English-Chinese) corpus of TED Talks\\footnote{https:\/\/www.ted.com\/} with translation processes. The corpus is composed of transcriptions and human translations of oral presentations. The inter-annotator agreement (Cohen's Kappa)~\\cite{Cohen1960} for annotating the English-French and English-Chinese control corpus is 0.67 and 0.61, both around the substantial agreement threshold. This indicates that the task of manual annotation is already complicated. Readers can find more details of corpus construction in the article~\\cite{zhai2018b}. \n\nThe automatic classification is conducted on the English-French pair in this work. We present in the table~\\ref{tab:definition} a brief definition, a typical example and the number of instances for each category to be automatically classified.\\footnote{Note that there are other detailed annotation rules in the annotation guidelines.} We combine \\textit{Transposition} and \\textit{Mod+Trans} in a category \\textit{Contain\\_Transposition}, where \\textit{Modulation} is considered as a neutral part. We will work on the classification of the pair English-Chinese once the annotation phase is finished. In this work, we conduct experiments in a simplified scenario, where we already know the boundaries of bilingual pairs, and we only predict the translation process. For example, given the pair \\textit{deceptive $\\rightarrow$ une illusion} in a pair of bilingual sentences, the goal is to predict its label \\textit{Contain\\_Transposition}. \n\n\\vspace*{-1em}\n\n\\begin{table}[!ht]\n  \\centering \n   \n    \\caption{Definition, typical example and number of instances for each translation process to be automatically classified. The instances were manually annotated in an English-French parallel corpus of TED Talks. We combine \\textit{Transposition} and \\textit{Mod+Trans} in a category \\textit{Contain\\_Transposition} for the automatic classification.}\n    \\begin{tabularx}{\\textwidth}{l|L{9.5cm}}\n    \\toprule\n    Translation Process \n    & Definition and typical example \\\\ \\midrule\n    Literal & Word-for-word translation, also concerns lexical units in multiword form.  \\\\ \n    (3771) & \\textit{certain \\textbf{kinds} of $\\rightarrow$ certains \\textbf{types} de} \\\\ \\hline \n    \n    Equivalence & Non-literal translation of proverbs or fixed expressions; a word-for-word translation makes sense but the translator expresses differently, without changing the meaning and the grammatical classes. \\\\\n    (289) & \\textit{\\textbf{back then} $\\rightarrow$ \\textbf{\u00e0 l'\u00e9poque} (`at that time')} \\\\ \\hline\n    \n    Generalization & Several source words or expressions could be translated into a more general target word or expression, the translator uses the latter to translate.  \\\\ \n    (86) & \\textit{as we \\textbf{sit here} in ... $\\rightarrow$ alors que nous \\textbf{sommes} \u00e0 ... (`as we are at ...')} \\\\  \\hline\n    \n    Particularization & The source word or expression could be translated into several target words or expressions with a more specific meaning, and the translator chooses one of them according to the context. \\\\ \n    (215) & \\textit{the idea I want to \\textbf{put out} is ... $\\rightarrow$ l'id\u00e9e que je veux \\textbf{diffuser} c'est ... (`the idea I want to spread is ...')} \\\\ \\hline\n    \n    Modulation & Metonymical and grammatical modulation~\\cite{ChuquetPaillard1989}; change the point of view; the meaning could be changed. \\\\\n    (195) & \\textit{\\textbf{that scar has stayed with him} $\\rightarrow$ \\textbf{il a souffert de ce traumatisme} (`he has suffered from this traumatism')} \\\\ \\hline\n    \n    Transposition & Change grammatical classes without changing the meaning. \\\\\n    (289) & \\textit{unless \\textbf{something changes} $\\rightarrow$ \u00e0 moins qu'\\textbf{un changement ait lieu} (`unless a change occurs')} \\\\ \\hline\n    \n    Mod+Trans & Combine the transformations of \\textit{Modulation} and of \\textit{Transposition}, which could make the alignment difficult. \\\\ \n    (53) & \\textit{this is a completely \\textbf{unsustainable} pattern $\\rightarrow$ il est absolument \\textbf{impossible de continuer sur} cette tendance (`it is completely impossible to continue on this trend')} \\\\ \n    \\bottomrule \n    \\end{tabularx}\n    \\label{tab:definition}\n\\end{table}\n\n\n\\section{Automatic Classification}\n\\label{classification}\n\nWe have tried two approaches for the automatic classification. Since the size of the cross validation data set is quite small, we first compare different statistical machine learning techniques with feature engineering. We also build different neural network architectures which we explain below. \n\n\\subsection{Feature Engineering with Statistical Machine Learning Techniques}\n\nWe describe below the features exploited in this work. The tag sets of English and French for \\gls{pos} tagging, constituency parsing and dependency parsing have been converted into three compact and unified tag sets~\\cite{UniversalPosTag2012}.  \n\n1) The \\gls{pos} tagging is done by \\textit{Stanford CoreNLP}~\\cite{stanford-coreNLP2014} for the two languages. \nOn source and target side, for each \\gls{pos} tag, the number of its occurrence is counted in a vector. \nWe also calculate the cosine similarity between these two vectors (on all words and only on content words).\\footnote{The tags of content words include: ADJ, ADV, NOUN, PROPN, VERB. If a segment does not contain any content word, the original segment is used.}\n\n2) We verify the pattern of \\gls{pos} tag sequence changing according to a manual list, for example the pair \\textit{methodologically $\\rightarrow$ de fa\u00e7on m\u00e9thodologique} `methodologically' corresponds to the pattern \\textit{ADV $\\rightarrow$ ADP NOUN ADJ}.  \n\n3) The number of tokens in the two segments ($l_e$, $l_f$), the ratio of these numbers ($l_e\/l_f$, $l_f\/l_e$), the distance Levenshtein~\\cite{Levenshtein1966} between the segments. \n\n4) The constituency parsing is done by \\textit{Bonsai}~\\cite{Bonsai2010} for French, by \\textit{Stanford CoreNLP} for English. We compare the \\gls{pos} tags for a pair of words, the non-terminal node tags for a pair of segments, the tag category (\\textit{e.g.} verb $\\rightarrow$ verb phrase) for a word translated by a segment or vice versa. \n\n5) The dependency parsing is done by \\textit{Stanford CoreNLP} for the two languages. Inside the segments, the number of occurrence of each dependency relation is counted. Outside the segments, among the words linked at source and target side, we filter those which are aligned in the sentence context. Then the number of occurrence of each dependency relation between the words in segments and these context words is counted.  \n\n6) The cosine similarity is calculated between the embeddings from \\textit{ConceptNet Numberbatch}~\\cite{conceptNet5_5_2017}. \nThis resource is multilingual and the system based on \\textit{ConceptNet} took the first place in the task ``Multilingual and Cross-lingual Semantic Word Similarity'' of SemEval2017~\\cite{Semeval2017_task2,ConceptNet_semeval2017}. \nCertain multi-word expressions have their own embeddings in this resource. Otherwise, we calculate the average of embeddings only on content words. The same features are calculated for lemmatized segments.\\footnote{The lemmatization is done by \\textit{Stanford CoreNLP} and \\textit{Tree Tagger}~\\cite{Treetagger1995} for English and French.} \n\n7) The resource \\textit{ConceptNet}~\\cite{conceptNet5_5_2017} also provides assertions in triplet: a pair of words or expressions linked by a relation.\nIn this multilingual resource, we verify if an English-French pair is directly linked; indirectly linked by another French segment or simply not linked.\\footnote{The EN-FR and FR-FR assertions are used in this work.} Three forms are tested: original form, lemmatized form and lemmatized filtered form.\\footnote{We filter the words in a manual list, for example the light verbs, determinants, pronouns, \\textit{etc.}} \n\n8) On the lemmatized filtered form, we calculate the percentage of tokens which are linked with a relation of derivation, based on the resource \\textit{ConceptNet}. For example \\textit{deceptive} and \\textit{illusion} `illusion' are not directly linked in the resource, but they are both linked to \\textit{illusoire} `illusory'. Hence we consider that there exists a link of derivation between them.\n\\\\\n\nFor the three following features, we have exploited the lexical translation probability table generated by the statistical word alignment tool \\textit{Berkeley Word Aligner}~\\cite{BerkeleyAligner2006}, trained on an English-French parallel corpus composed of TED Talks and a part of Paracrawl corpus (in total 1.8M parallel sentence pairs and 41M English tokens).\\footnote{https:\/\/wit3.fbk.eu\/, https:\/\/paracrawl.eu\/index.html}\n\n9) The entropy of the distributions of lexical translation probabilities~\\cite{Entropy1990,Carl_Schaeffer_2017}, calculated according to this equation: $H(X) = \\sum_{i}P(x_i)I(x_i) = -\\sum_{i}P(x_i)log_eP(x_i)$. We calculate the average entropy on content words. A bigger entropy indicates that the words have more general meanings or they are polysemous. The same feature is calculated on the lemmatized content words.  \n\n10) The bidirectional lexical weighting on content words, by supposing a \\textit{n-m} alignment $a$ between the segments ($\\bar{e}$ and $\\bar{f}$). In the scheme proposed by Koehn \\textit{et al.}~\\cite{lexicalWeighting2003} (equation~\\ref{eq:lexWeighting}), to calculate the direct lexical weighting, each of the English words $e_i$ is generated by aligned foreign words $f_j$ with the word translation probability $w(e_i|f_j)$. And similarly for the reverse lexical weighting $lex(\\bar{f}|\\bar{e},a)$. The same feature is calculated for lemmatized content words. This feature could reflect the alignment confidence between a pair of segments. \n\n\\begin{equation}\n    \\label{eq:lexWeighting}\n    lex(\\bar{e}|\\bar{f}, a) = \\prod_{i=1}^{length(\\bar{e})} \\frac{1}{|\\{j|(i,j) \\in a\\}|} \\sum_{\\forall(i,j)\\in a} w(e_i|f_j)   \n\\end{equation}\n\n11) The sum of lexical translation probability differences between the human translation and the most probable translation according to the probability table. \nFor each source word, we take the target word in human translation with the biggest probability. \nAccording to this method, we also count the unaligned words \nto calculate a ratio on the total number of tokens on each side. These features are calculated in the two directions of translation. \n\nWe use the toolkit \\textit{Scikit-Learn}~\\cite{sklearn2011} to train different statistical machine learning classifiers.\\footnote{The code and data set is publicly available at https:\/\/github.com\/YumingZHAI\/ctp.} \n\n\n\\subsection{End-to-end Neural Network Architectures}\n\nThe source and target phrases are encoded using a bidirectional encoder with \\gls{gru} (size $10$). The outputs of forward and backward recurrent networks are concatenated to form the source and target phrase representations (size $20$). After the encoder layer we have tried two different architectures. The first one is to build an alignment matrix for the source-target phrases, using the dot product of the two representations, inspired by these two work~\\cite{Legrand2016,Pham2018}. Then a \\textit{Convolutional Neural Network} (CNN) classifier is applied to this alignment matrix, which is composed of one convolution layer followed by pooling. Since the shape of the alignment matrix varies from one source-target pair to another, an adaptive pooling is used~\\cite{pyramidpooling2015}. The output of the pooling layer is fed into a fully-connected layer followed by a linear layer as the output. In the second architecture, the source and target outputs of the encoder layer are averaged over time steps to produce two fixed-dimensional vectors, which are further concatenated (size $40$) and fed into a \\gls{mlp} classifier. The hidden layer of \\gls{mlp} includes $10$ hidden units with tanh non-linearity.  \n\nThe length of our phrases is usually short, especially for word-for-word \\textit{Literal} instances. In order to build a more robust alignment matrix and to avoid the out-of-vocabulary problem, we finally choose to use character embeddings. As shown in table~\\ref{tab:neuralclf-results-binary}, for the embedding layer, we have tried respectively randomly initialized character embeddings (size $10$), and training our own word embeddings using skipgram model of \\textit{FastText}~\\cite{fasttext_2016} on a TED Talks corpus (around 3M tokens for both English and French), with a word-embedding size of $100$, minimum n-gram of~$3$, and maximum n-gram of~$6$.\nAll the models have been trained in $200$ epochs, with a learning rate of $0.0001$ using Adam optimizer and the minibatch size of $20$. Dropout has been applied to all layers except the output and embedding layers. \n\n\\vspace{-1em}\n\n\\section{Experimental Results and Analysis}\n\\label{experiment}\n\nThe table~\\ref{tab:neuralclf-results-binary} and~\\ref{tab:neuralclf-results-multi} show the results of our classifiers using end-to-end neural network architectures, for binary classification (balanced distribution) and multi-class classification. For the binary classification, \\textit{Non\\_literal} (NL) class has in total 1127 instances, and 1127 \\textit{Literal} (L) instances are randomly chosen. Besides the preprocessing steps of lowercasing and correcting minor spelling errors, for the neural classifiers, we also normalized the clitic forms to complete words (\\textit{e.g. 're $\\rightarrow$ are}), and normalized digits to letter form (\\textit{e.g. 42 $\\rightarrow$ four two}). The architecture using word embeddings and \\gls{mlp} obtain better results and is faster than the other two architectures. However, the current data set is too small for neural architectures to produce satisfactory results.  \n\n\\vspace{-2em}\n\n\\begin{table}[!ht]\n\\parbox{.5\\linewidth}{\n\\centering\n\\caption{Binary classification \\\\(balanced distribution)}\n\\begin{tabular}{l|l|c|c|c} \n    \\hline\n    \\multicolumn{2}{c|}{Architecture}  & Accuracy & F1 (L) & F1 (NL) \\\\ \\hline\n    \\multicolumn{5}{c}{Randomly initialized character embedding} \\\\ \\hline\n    \\multicolumn{2}{c|}{CNN} & 59.99\\%  & 0.60   & 0.60   \\\\  \n    \\multicolumn{2}{c|}{MLP} & 71.16\\%  & 0.71   & 0.71  \\\\ \\hline   \n    \\multicolumn{5}{c}{Pre-trained fasttext word embedding} \\\\ \\hline\n    \\multicolumn{2}{c|}{MLP} & \\textbf{71.25\\%}  & 0.71   & 0.71  \\\\ \n    \\hline \n\\end{tabular} \n\\label{tab:neuralclf-results-binary}\n}\n\\hfill\n\\parbox{.5\\linewidth}{\n\\centering\n\\caption{Multi-class classification \\\\(five non-literal classes)}\n\\begin{tabular}{l|l|c|c|c} \n    \\hline\n    \\multicolumn{2}{c|}{Architecture}  & Accuracy & Micro-F1 & Macro-F1 \\\\ \\hline\n    \\multicolumn{5}{c}{Randomly initialized character embedding} \\\\ \\hline\n    \\multicolumn{2}{c|}{CNN} & 34.08\\%  & 0.34 & 0.20     \\\\  \n    \\multicolumn{2}{c|}{MLP} & 40.74\\%  & 0.41 & 0.34      \\\\ \\hline   \n    \\multicolumn{5}{c}{Pre-trained fasttext word embedding} \\\\ \\hline\n    \\multicolumn{2}{c|}{MLP} & \\textbf{43.22\\%}  & 0.43 & 0.34  \\\\  \n    \\hline \n\\end{tabular}  \n\\label{tab:neuralclf-results-multi}\n}\n\\end{table}\n\nThe number of all non-literal instances (1127) is only one third of \\textit{Literal} instances (3771). Considering this important difference, for the statistical machine learning classifiers, we first evaluated them under these configurations:  \n\n-~six classes (\\textit{Literal, Equivalence, Generalization, Particularization, Modulation, Contain\\_Transposition}). We first put all \\textit{Literal} instances. Then to have an approximately balanced class distribution, we randomly take 200 instances for \\textit{Literal}. \n\n-~two classes (\\textit{Literal} and \\textit{Non\\_literal}), with three distributions (3:1, 2:1, 1:1). The distribution 3:1 is the natural distribution in the data set. The instances of \\textit{Literal} have been extracted randomly for the last two distributions. \n\n-~five classes (only non-literal categories). \n\nFor each configuration, we have tuned the hyperparameters of different classifiers. We evaluate them by five-fold cross-validation,\\footnote{\nStratifiedKFold is used for cross-validation, where the folds are made by preserving the percentage of samples for each class.} using the metrics such as the average accuracy of five folds, the micro average and macro average F1-score~\\cite{Tsoumakas2011}. The \\textit{DummyClassifier} is used as a baseline, which generates random predictions by respecting the distribution of training classes. \n\nFirst, we attempted a direct classification into six classes (see table~\\ref{tab:recap-configs}). The best results by \\textit{RandomForest} reflect the difficulty of the task in multi-classes. On the other hand, we observe the potential of our features on classifying the category \\textit{Literal} when the number of instances increases. As a result, we decide to divide the problem: conduct first a binary classification, and secondly a multi-class classification among the non-literal categories. \n\n\\vspace{-0.5cm}\n\n\\begin{table}[!ht]\n    \\centering\n    \\caption{Classification results under different configurations, using all features}\n    \\begin{tabular}{l|c|c|c|c}\n        \\hline\n        Distribution of classes  & Classifier                           &  Accuracy     & Micro-F1    & Macro-F1 \\\\ \\hline\n        \\multicolumn{5}{l}{\\textbf{Six classes}}  \\\\ \\hline\n         \\multirow{2}{*}{six classes, with 3771 \\textit{Literal}} & Dummy &  60.76\\%     & 0.61        & 0.15    \\\\\n                                                          & RandomForest &  \\textbf{83.10\\%}     & 0.83        & 0.44    \\\\ \\hline\n                                                          \n        \\multirow{2}{*}{six classes, with 200 \\textit{Literal}} & Dummy &   18.92\\%     & 0.19        & 0.16    \\\\       \n                                                          & RandomForest &  57.04\\%     & 0.57        & 0.52    \\\\ \\hline \n        \\multicolumn{5}{l}{\\textbf{Two classes}}  \\\\ \\hline\n        \\multirow{2}{*}{\\textit{Literal} (3) : \\textit{Non\\_literal} (1)} & Dummy        &  65.84\\%     & 0.66  & 0.52    \\\\                                                                                               & RandomForest &  \\textbf{90.16\\%}     & 0.90  & 0.86 \\\\ \\hline \n        \\multirow{2}{*}{\\textit{Literal} (2) : \\textit{Non\\_literal} (1)} & Dummy        &  56.43\\%     & 0.56  & 0.51    \\\\\n                                                                          & RandomForest &  88.85\\%     & 0.89  & 0.88    \\\\ \\hline\n        \\multirow{2}{*}{\\textit{Literal} (1) : \\textit{Non\\_literal} (1)} & Dummy        &  53.19\\%     & 0.53  & 0.53    \\\\       \n                                                                          & RandomForest &  \\textbf{87.09\\%}     & 0.87  & 0.87    \\\\ \n         \\hline\n        \\multicolumn{5}{l}{\\textbf{Five classes}}  \\\\ \\hline\n        \\multirow{2}{*}{Five non-literal classes}                         & Dummy        & 20.32\\%      & 0.20  & 0.18 \\\\\n                                                                          & RandomForest & \\textbf{55.10\\%}      & 0.55  & 0.47  \\\\ \n        \\hline \n    \\end{tabular}\n    \\label{tab:recap-configs}\n\\end{table}\n\nFor the binary classification, the two best classifiers are \\textit{RandomForest} and \\gls{mlp}. Furthermore, \\textit{RandomForest} has better performance than the two combined by the method \\textit{hard voting} or \\textit{soft voting}. The table~\\ref{tab:recap-configs} presents the results under three different class distributions. From the natural distribution (3:1) to our artificial balanced distribution by randomly choosing \\textit{Literal} instances (thus both class have 1127 instances), the average F1-score for the class \\textit{Non\\_literal} increases from 0.78 to 0.88. We will continue to test this tendency when a larger data set is available. Table~\\ref{tab:recap-configs} also shows the results for the classification into five non-literal classes using all features, and the average F1-score for each non-literal category are shown in table~\\ref{tab:5classes-eachClass}. The category \\textit{Generalization} has many fewer instances than the other categories, which need to be augmented; there exist many confusions between \\textit{Modulation} and the other categories, which suggests rather a review of annotation guidelines. \n\n\\vspace*{-0.8cm}\n\n\\begin{table}[!ht]\n    \\centering\n    \\caption{Average F1-score for each non-literal class, using all features}\n    \\begin{tabular}{l|c|c|c|c|c}\n    \\hline\n    Category & Equivalence & Generalization & Particularization & Modulation & Contain\\_Transposition  \\\\ \\hline \n    Nb. instances & 289       & 86           & 215             & 195      & 342               \\\\ \\hline\n    Average F1 & 0.51 & 0.25 & 0.56 & 0.36 & 0.68  \\\\ \\hline \n    \\end{tabular}\n    \\label{tab:5classes-eachClass}\n\\end{table}\n\n\\vspace*{-0.3cm}\n\nThe table~\\ref{tab:feature-ablation} recapitulates the best performance on binary classification (balanced distribution) and on the classification of five non-literal classes, using the most helpful set of features. \nWith the best performing classifier \\textit{RandomForest}, we have investigated the performance of features one by one and also grouped them: \\textit{PoS\\_tagging} (feature 1, 2), \\textit{surface} (feature 3), \\textit{syntactic\\_analysis} (feature 4, 5), \\textit{external\\_resource} (feature 6, 7, 8) and \\textit{word\\_alignment} (feature 9, 10, 11). For binary classification, feature 10 (bidirectional lexical weighting) is most helpful, which generates average F1-score of 0.78 for \\textit{Literal} and 0.80 for \\textit{Non\\_literal} by itself. The group of features \\textit{word\\_alignment} contributes the most for the binary classification. The combination of all features generates the best results, which remain the same if we remove the feature 4 (constituency parsing), 7 (how the pair is linked in the resource \\textit{ConceptNet}) and the features on \\gls{pos} tagging apart from the vector counting the occurrence of each tag. The features in float form generally perform better than those in discrete form (\\textit{e.g.} 0, 1, \\textit{etc.}). Concerning the classification into five non-literal classes, the combination of all features except the group \\textit{external\\_resource} leads to the best results, where the group  \\textit{PoS\\_tagging} and \\textit{syntactic\\_analysis} contribute more than the group \\textit{word\\_alignment} and \\textit{surface}. The accuracy changes from 55.10\\% to 55.20\\% after feature ablation (see table~\\ref{tab:recap-configs}).\n\nOur error analysis shows that in binary classification, it is difficult to distinguish \\textit{Literal} and \\textit{Equivalence}; in multi-class classification, the biggest confusion is between \\textit{Equivalence} and \\textit{Contain\\_Transposition}. Consequently, we conducted another three binary classification experiments (see table~\\ref{tab:groupClass}), where in all configurations each class has 549 instances to make the results comparable: i) \\textit{Literal} vs \\textit{Non\\_literal} ii) \\textit{Literal} combined with \\textit{Equivalence} (E), vs the other classes iii) \\textit{Literal} combined with \\textit{Equivalence} and \\textit{Transposition} (T), vs the other classes. The third configuration is more interesting, because the group of translation processes \\textit{LET} do not bring meaning changes, while the processes \\textit{non-LET} could. The results show that by including \\textit{Transposition} (change grammatical classes without changing the meaning), the performance gets better than only grouping \\textit{Literal} and \\textit{Equivalence}, since we avoid the confusion between \\textit{Equivalence} and \\textit{Transposition}. The better results of binary classification (L vs NL, LET vs non-LET) indicate that in future work we can develop cascading classifiers, namely first separating word-for-word literal translations, or those which do not cause meaning changes, then conducting a finer-grained classification among the other categories. \n\n\n\\begin{table}[!ht]\n    \\centering\n    \\caption{Classification results after feature ablation study}\n    \\begin{tabular}{L{3cm}|c|c|c}\n    \\hline\n     & average accuracy & \\multicolumn{2}{c}{average F1-scores} \\\\  \\hline\n    binary classification (balanced distribution) & \\textbf{87.09\\%} & 0.87 (Literal) & 0.88 (Non\\_literal)  \\\\ \\hline \n    five non-literal classes & \\textbf{55.20\\%} & 0.55 (micro average) & 0.48 (macro average) \\\\ \\hline\n    \\end{tabular}\n    \\label{tab:feature-ablation}\n\\end{table}\n\n\\begin{table}[!h]\n    \\centering\n    \\caption{Classification results after grouping classes, every class has 549 instances}\n    \\begin{tabular}{c|c|c|c}\n     \\hline\n     Configuration & average accuracy & average F1 (class1) & average F1 (class2)  \\\\ \\hline\n     Dummy   & 48.63\\% & 0.49 & 0.49 \\\\ \\hline\n     L vs NL & 85.24\\% & 0.84 & 0.86  \\\\ \n     LE vs non-LE & 75.32\\% & 0.74 & 0.77 \\\\ \n     LET vs non-LET & 79.42\\% & 0.78 & 0.81 \\\\ \\hline  \n    \\end{tabular}\n    \\label{tab:groupClass}\n\\end{table}\n\n\n\\section{Conclusion and Perspectives}\n\nWe have proposed a new Natural Language Processing task of automatically classifying translation processes at subsentential level, based on manually annotated examples from a parallel English-French TED Talks corpus. \nTo the best of our knowledge, these translation processes have not been explicitly exploited during paraphrase extraction from bilingual parallel corpora.\nWith the best performing classifier \\textit{RandomForest} and feature engineering, our empirical results show a best accuracy of 87.09\\% for binary classification (\\textit{Literal} vs \\textit{Non\\_literal}) and 55.20\\% for multi-class classification (\\textit{Equivalence, Generalization, Particularization, Modulation, Contain\\_Transposition}), which are much better than the baseline random classifier. \n\nThis task is complicated, and our exploratory work is restrained by the limited amount of annotated examples. However, our work demonstrates that automatically classifying translation processes seem possible, and the experiments show the directions that we can follow in future work. \nThere is much room to constitute an augmented and balanced data set, on which we will evaluate our classifier to observe the performance. The finer error analysis of the classification results is useful to help the research on corpus annotation and linguistic analysis. \nWe will continue to improve the classifier on English-French, by implementing other features for multi-class classification, and explore more neural architectures. We will also extend our work to English-Chinese translation pairs. \nOne of our long term objectives is leveraging this automatic classification to better control paraphrase extraction from bilingual parallel corpora. \n\n\n\\vspace{-1em}\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDuring the last decade the complexity paradigm \\cite{anderson1972more,Jaguar, pietronero2008complexity} has been successfully applied not only to the area of physical systems but also to many other phenomena, including socio-economic systems \\cite{castellano2009statistical, tacchella2018dynamical}. \\newline \nOne of the most ubiquitous features in the complexity science is the emergence of extreme events, orders of magnitude larger than the typical ones. Severe financial crisis, devastating earthquakes, or deadly wars, all can be interpreted in terms of complex inherent structures which give rise to power law distributed phenomena \\cite{Newman}. Mandelbrot and Taleb have been among the firsts \\cite{mandelbrot2001scaling, taleb2007black} to investigate these phenomena and to stress the limits of standard statistical techniques in such framework, the latter with the introduction of the celebrated Black Swan metaphor. According to Taleb \\cite{taleb2007black} a Black Swan event:\n\t\\begin{itemize}\n\t\t\\item is unexpected and unpredictable;\n\t\t\\item has a great impact, both positive or negative;\n\t\t\\item makes people try to explain its occurrence once it has been observed.\n\t\\end{itemize}\nIf extreme event are found to follow a power law their effects are in some way mitigated because they stop to be totally unexpected. In this case Taleb speaks of Gray Swans \\cite{taleb2007black}, which are unpredictable but not unexpected. However, as we are going to show, even if we know that the underlying distribution is power law-like, Black Swans can still occur due to the possible dynamics of the upper cutoff. \n\n\n\t\nThe First World War and 9\/11 terrorist attacks are often used as examples of Black Swans \\cite{taleb2007black, nafday2009strategies, hajikazemi2016black, ferguson2020black}; however, up to now a scientific and quantitative assessment of Black Swans has been lacking and this sometimes allows policy makers to improperly use this terminology. For instance, during the last months many governments, financial institutions and journals identified the COVID-19 pandemic as a Black Swan, even though the possibility of the spreading of a newborn virus had been already pointed out by several studies and even by Taleb himself \\cite{taleb2007black}. This arbitrariness is favored by the fact that after the efforts of Taleb and Mandelbrot, most of the discussion about Black Swans is still conducted on a qualitative philosophical\/psychological level. \n\t\nIn the following we show, on a quantitative basis, that Black Swans are related to a non-stationarity in the inherent power law distribution, generated by a jump dynamics of the upper cutoff. This implies that when performing risk analysis, the assumption of a time independent probability distribution may result in severe underestimations of extreme events. A central property of Black Swans is that they can not be predicted starting from data about the system under consideration, however, an analysis including also the environment the system is coupled to permits, in some cases, to spot the jumps of the upper cutoff, and so to foresee Black Swans. This new perspective allow us to introduce a quantitative measure of the surprise associated to large events, that we call Blackness, which can be used to scientifically classify them into three distinct sets: White Swans, Grey Swans, and Black Swans. Our quantitative criterion partly confirms what previously stated by Taleb using qualitative arguments, namely that the First World War and 9\/11 terrorist attacks have been Black Swans with respect to the number of casualties, while 1987 Black Monday has been only a Grey Swan. New examples of Black Swans we find are the amount of goals scored by Lionel Messi in LaLiga and the number of casualties of Turkish Airline Flight 981 disaster, since both events drastically overcame the previous estimate of the upper cutoff. \n\t\n\t\n\t\\vspace{1.2em}\n\t\\begin{figure*}[t]\n\t   \t\\centering\n        \\includegraphics[width=0.9\\linewidth]{fig_1_main.jpg}     \n\t    \\caption{\\textbf{Black Swans arise from jumps in the upper cutoff.} \\textbf{a)} Rank-size plot of airline disasters casualties occurred before 1974-03-03 (blue dots) with the corresponding fit to Zipf-Mandelbrot law (red line). The presence of strong deviations from Zipf's law would have suggested that this set completely sampled the inherent distribution, meaning that the maximal possible number of casualties (i.e., the upper cutoff) was approximately two hundred. The stylized black swan corresponds to Turkish Airline Flight 981 disaster. $346$ people died in that circumstance, subverting the estimate of the upper cutoff. This accident, which has been the first to involve a large-body aircraft, can be then considered a Black Swan. \\textbf{b)} Probability distribution of airline disaster casualties. The red dots represent the distribution of accidents occurred between 1949 and 1969, while blue ones show the distribution of those occurred between 1969 and 1989. Approximately the same number of accidents occurred in these two periods ($798$ and $781$, respectively) and so if the distribution would not have changed, then red and blue dots would have spanned the same range of casualties. However, the upper cutoff approximately doubles, making the Black Swan possible.}     \n        \\label{fig:fig_1_main}\n\t\\end{figure*}\n\t\\section{Results}\n\t\\subsection{Zipf-Mandelbrot law}\n\tOur scientific framework to analyze Black Swans is given by Zipf's law and, more in general, power laws. Zipf's law \\cite{Zipf} is an ubiquitous scaling law found in many natural and socio-economical systems \\cite{Newman, blasius2009, furusawa2003, maillart2008, refId0}. Given a system composed of $N$ objects and denoting by $S(k)$ the size of the $k$th largest one, Zipf's law reads\n\t\\begin{equation}\t\n\t\tS(k)=\\frac{S(1)}{k^{\\gamma}},\n\t\t\\label{eq:Zipf}\n\t\\end{equation}\n\twhere $k$ is the rank, $\\gamma$ is the Zipf's exponent, and $S(1)$ is the empirical maximum, that is the largest element or event in the system under consideration. \n\n\tZipf's law is generally visualized by the so called rank-size plot, obtained plotting the ordered sequence of the sizes as function of their position in the sequence; a straight line in loglog scale is thus expected. However, it is common to observe deviations from Zipf's law at low ranks \\cite{de2021dynamical, cristelli2012there, burroughs2001}, which can be quantified by introducing a parameter $Q$ in the so called Zipf-Mandelbrot law \\cite{Mandelbrot}:\n\t\\begin{equation}\t\n\t\tS(k)=\\frac{\\bar{S}}{(k+Q)^{\\gamma}}.\n\t\t\\label{eq:Zipf_Mandelbrot}\n\t\\end{equation}\n\tThe parameter $Q>0$ will play a crucial role in the analysis of Black Swans. As shown in \\cite{de2021dynamical}, the Zipf-Mandelbrot law is observed whenever the inherent distribution is a power law and its parameters are related to those of the probability distribution by the following relations \n\t\\begin{equation}\n\t\t\\begin{cases}\n\t\t\t\\gamma=\\frac{1}{\\alpha-1}\\\\\n\t\t\t\\bar{S}=N^{\\gamma}s_m\\\\\n\t\t\tQ=N\\ton*{\\frac{s_m}{s_M}}^{1\/\\gamma}\n\t\t\\end{cases}\n\t\t\\label{eq:gamma_barS_Q}\n\t\\end{equation}\n\twhere \n\t\\begin{itemize}\n\t\t\\item $\\alpha$ is the exponent of the inherent power law distribution, that is $P(S)\\sim S^{-\\alpha}$;\n\t\t\\item $s_m$ and $s_M$ are the lower and upper cutoffs of the distribution, possibly corresponding to physical limits, that is $P(S)=0$ for $S<s_m$ and $S>s_M$;\n\t\t\\item $N$ is the number of elements in the system or records in the catalog.\n\t\\end{itemize}\n\tThe deviation parameter $Q$ is a measure of the level of sampling \\cite{de2021dynamical} (see Methods for details): for $Q\\approx0$ the underlying distribution is under-sampled and the upper cutoff can not be inferred, while for $Q\\gg1$ it is completely sampled and $s_M$ coincides with the empirical maximum. Indeed by combining Eqs.~\\eqref{eq:Zipf_Mandelbrot} and \\eqref{eq:gamma_barS_Q} we obtain an expression relating the upper cutoff of the distribution to the measurable parameters $Q$, $S(1)$ and $\\gamma$\n\t\\begin{equation}\n\t\ts_M=S(1)\\ton*{\\frac{Q+1}{Q}}^{\\gamma}.\n\t\t\\label{eq:s_M_main}\n\t\\end{equation}\nIn the limit $Q\\to0$ the upper cutoff diverges, meaning that data do not provide a sufficient level of sampling for inferring it. This last expression thus allows, when $Q$ is sufficiently large, to compute the upper cutoff of a power law distribution and will be used in the following to study real systems.\n\t\\vspace{1.5em}\n\n\t\t\\begin{figure*}[t]\n    \t\\centering\n        \\includegraphics[width=\\linewidth]{blackness_new.jpg}\n        \\caption{\\textbf{Blackness.} Blackness of ten events with $90\\%$ confidence bounds. Here the threshold for an event to be considered a Black Swan is $\\beta>1$; only four events out of the ten we considered are Black Swans (Turkish Airlines disaster, Lionel Messi, 9\/11 terrorist attacks and World War One), while the remaining six are Grey Swans. These results have been obtained by mean of Eq.~\\eqref{eq:blackness2}; see Methods for details.}     \n        \\label{fig:fig_2_main}\n\t\\end{figure*}\t\n    \n    \\subsection{The Blackness}\n\tLet us consider a random number generator extracting values from a power law distribution with unknown parameters $\\alpha$, $s_m$ and $s_M$. If we look at the first $N\\ll\\ton*{s_M\/s_m}^{1\/\\gamma}$ numbers, being $Q\\ll1$, the corresponding rank-size plot will be straight \\cite{de2021dynamical}. Moreover, our set under-samples the inherent distribution, and so there would be no surprise if the next draw returns a value much larger than those previously observed, because it is not possible to infer the upper cutoff $s_M$. \t\n\t\n\tIf we keep drawing numbers, sooner or later we will completely sample the inherent distribution, this occurring for $N\\gg\\ton*{s_M\/s_m}^{1\/\\gamma}$. In this way, the empirical maximum becomes very close to the upper cutoff, that can thus be inferred using Eq.~\\eqref{eq:s_M_main}, and nothing unexpected should occur. In this situation Black Swans are absent, but what happens if the upper cutoff increases to $s_M'\\gg s_M$? (Note that this actually happens in various social and biological systems, for instance as a consequence of the coupling with an external system or a technological change). The answer is simple, our apparently perfect knowledge of the underlying distribution becomes problematic as soon as a number close to $s_M'$ is extracted. Without knowing if and when the upper cutoff jumps, there is no way not only to predict, but also to expect such an event. This kind of event is what we will classify as a Black Swan. \n\n\n\t\n\tThis simple example we sketched captures some points that are crucial for understanding the phenomenology of Black Swans (and possibly mitigate their utmost effects):\n\t\\begin{itemize}\n\t\t\\item an arbitrary large event can be or be not a Black Swan depending on the level of sampling of the distribution, so depending on the value of $Q$. A system showing pure Zipf's law ($Q\\approx0$) never gives rise to Black Swans. The idea is that when the level of sampling is low there is no way to characterize the upper limit and so to realize that $s_M \\rightarrow s_M'$.\n\t\t\\item an event can be classified as a Black Swan if and only if it is (much) larger than any event previously observed \\textit{and} the deviation parameter $Q$ of the system is large. Indeed, this implies that it is beyond the estimate of the upper limit;\n\t\t\\item a stationary power law only gives rise to Grey Swans, since when the level of sampling is low there is no surprise if an event much larger than those previously observed occurs, while when the level of sampling is high no event much larger than the observed maximum can occur. Note that for a power law without upper cutoff, the level of sampling is always low, since the support of the distribution is not finite. As a consequence, also in this case, no Black Swan can be observed.\n\t\\end{itemize}\n\n\n   \n   \n   \n   \n\n\tIn order to better clarify these points, we consider a real system showing this kind of dynamics: aircraft accidents. Here the size of the event is given by the number of casualties. In Fig.~\\ref{fig:fig_1_main}a we show the corresponding rank size plot computed by considering all events until the 02-03-1974, the day before the crash of the Turkish Airline Flight 981, which is represented by the Black Swan. Strong deviations from Zipf's law are present, and $Q$ is relatively large. On the basis of the previous considerations, one could have concluded that the upper cutoff of the distribution had already been reached, and so no surprise should have been expected. However, the day after, the Flight 981 crashed, provoking the death of 346 people, a number approximately twice as large as the previous most severe accident. This increase was possible because of the introduction in the late sixties of wide-body aircrafts, that can carry approximately twice as many passengers as the older narrow-body models. This implied a sudden increase of the upper cutoff of the casualties distribution, that we depict in Fig.~\\ref{fig:fig_1_main}b. In this sense we can conclude that the Flight 981 was indeed a Black Swan. Note that, before and after the jump, the scaling exponent remained the same. \\newline\n\nIn summary, usually an empirical power law is characterized by three parameters, the exponent and the two cutoffs, and it must be non stationary in order to give rise to Black Swans. Now, the lower cutoff clearly does not play any role regarding large events, while a variation of the exponent influences the frequency of extreme events, but not their size. As a consequence, the only form of non-stationarity that may produce Black Swans is a jump of the upper cutoff.\n\t\\vspace{1.5em}\t\n\t\t\n\t\\begin{figure*}[t]\n    \t\\centering\n        \\includegraphics[width=\\linewidth]{working_on_16_11_21.jpg}\n        \\caption{\\textbf{Swans Plane.} Visualization in the Swans plane of the outcomes of a truncated stationary power law with $\\alpha=1.5$, $s_m=1$ and $s_M=10^6$ (color map) and of the four Black Swans we identified (white dots). Concerning the stationary power law, the figure shows the frequency of events with $\\mathcal{R}>1$, obtained exploiting a binning procedure in the Swans plane; lighter colors correspond to higher frequencies. The white curve corresponds to the threshold $\\beta=1$ and divides Grey Swans and Black Swans. As expected, the frequency goes to zero above the curve, proving that a stationary power law only produces Grey Swans. Note that as expected all the four Black Swans reside in a region of the Swans Plane where the frequency of power law events is vanishing, so they can not be explained by a stationary power law.}     \n        \\label{fig:Q_ratio_powerlaw}\n\t\\end{figure*}\n\tOnce the phenomenology of extreme events has been discussed, it is simple to derive a quantitative criterion to classify them in Black, Gray, and White Swans: i) an event smaller than the maximum of those previously occurred is a White Swan; ii) a Gray Swan is an event whose size is larger than those previously observed, but that is not unexpected in the sense that it is lower than the estimated upper cutoff; iii) a Black Swan is an event whose size is larger than those previously observed and that is also unexpected, being larger than the expected upper cutoff.\\newline\n\tWe can then define the \\textit{Blackness} $\\beta$ of a new event of size $S_{new}$ as\n\t\\begin{equation}\n\t\t\\beta=\\frac{S_{new}-S(1)}{s_M-S(1)},\n\t\t\\label{eq:blackness1}\n\t\\end{equation}\n\twhere $S(1)$ is the empirical maximum and $s_M$ is the best estimate of the upper cutoff we can infer from the data, that is given by Eq.~\\eqref{eq:s_M_main}. Formalizing the three definitions just mentioned, $\\beta$ can be used to determine the ``color'' of an extreme event:\n\t\\begin{itemize}\n\t\t\\item \\textbf{White Swan:} $S_{new}<S(1) \\rightarrow \\beta<0$;\n\t\t\\item \\textbf{Grey Swan:} $S(1)<S_{new}<s_M\\rightarrow 0<\\beta<1$;\n\t\t\\item \\textbf{Black Swan:} $S_{new}>s_M\\rightarrow \\beta>1$.\n \t\\end{itemize}\n \tAs shown in the methods section, $\\beta$ can be expressed directly in terms of only empirical quantities\n \t\\begin{equation}\n\t\t\\beta=\\frac{\\mathcal{R}-1}{\\ton*{\\frac{Q+1}{Q}}^{\\gamma}-1},\n\t\t\\label{eq:blackness2}\n\t\\end{equation}\n\twhere $\\mathcal{R}=S_{new}\/S(1)$ is the relative size of the new event with respect to the empirical maximum. This last expression shows that $\\beta$ well summarizes the two crucial points we stressed above. Namely $\\mathcal{R}>1$ is a necessary but not sufficient condition for an event to be a Black Swan, since also the deviation parameter $Q$ (or, equivalently, the level of sampling) must be considered. Indeed if $Q=0$, that is for a perfect Zipf's law, the Blackness goes to zero independently of the relative size. This because in the case of a pure Zipf's law the upper cutoff can not be estimated. \\newline\n\t\\vspace{1.5em}\t\n\t\n\t\\subsection{Applications to real systems}\n\t\\subsubsection{Classification of past events}\n\tThe Blackness concept allows to search for Black or Gray swans in any system showing an inherent power law distribution, obtaining an objective and scientific based measure of the ``surprise'' associated to an event. By performing fits to the rank-size plots (see the Method section for details), and by using Eq.~\\eqref{eq:blackness2}, we computed the Blackness of a number of extreme events that previous works have found to be power law distributed \\cite{clauset2009power, malacarne2000regularities, cirillo2020tail, mandelbrot2001scaling, golyk2012self, chatterjee2017fat} and that range from sport to natural disasters and finance; the results are displayed in Fig.~\\ref{fig:fig_2_main}. The ``Blackest'' Black Swan among the events we considered is Turkish Airline disaster, followed by WWI; the size of the latter has been defined as the number of casualties normalized by the world population. For what concerns the WWI, the growing globalization of the world and the extensive usage of new deadly weapons may be the responsible of the jump of the upper cutoff. Surprisingly WWII is only a Grey Swan. Intuitively, the occurrence of the Great War, a conflict much more severe than any previous interstate-war, already proved that the upper cutoff of the distribution had increased, making the second world conflict relatively less surprising in terms of casualties. Another Swan that presents a Blackness lower than one (considering uncertainty) is the 1987 Black Monday. Here the size is given by the module of Dow Jones index daily return; during the Black Monday this index lost about $23\\%$ of its value, the worst fall in its history.  Also other ``officially unexpected'' events in finance and economics are found to be Grey Swans: the oil market volatility due to the spreading of Covid-19 (the event we analyzed is the largest monthly fluctuation of West Texas Intermediate (WTI) index (+75\\%, occurred between March and April 2020). We considered pre-Covid19 data (1986-2019) and the growth of China (the size is defined as the percentage variation of GDPppp and we considered $35$ years returns of countries between the periods 1865-1900, 1900-1935, and 1950-1985. Countries that grew thanks to oil or natural resources have been excluded, but analogous results are found also considering all the countries). It is interesting to note that according to the common wisdom the growth of China is an incredible outlier \\cite{pritchett2014asiaphoria}, while in our framework its growth is a remarkable but not unexpected event, confirming recent analysis based on the methods of analogues \\cite{tacchella2018dynamical, cristelli2015heterogeneous}. \\newline\n\tConversely, the soccer top star Lionel Messi is a Black Swan if measured by the number of goals he scored in LaLiga (455). We can argue that the growth of the upper cutoff is due to the increase of the number of teams taking part in the championships, which, in turn, increased the number of games played in a season and the number of potential goals. Indeed, the previous record holder was Telmo Zarra, who scored 268 goals playing between 1949 and 1957; even if, as Lionel Messi, he scored an average of 0.9 goals per match and played for the same number of seasons (16), the number of teams involved in LaLiga was between 12 and 16, against the 20 of today. Using the same criterion Cristiano Ronaldo, with his $311$ goals, is only a Grey Swan. The September 11 attacks, often used as an archetype of Black Swan events, is characterized by a Blackness much larger than one, confirming the blackness of this terrorist attack in terms of casualties, even if in this case a clear and objective motivation for the jump of the upper cutoff is hard to find. Finally, the Burj Khalifa, that is the tallest building in the world, is only a Grey Swan, despite an height approximately $70\\%$ larger than Taipei 101, the previous record holder.\\newline\t\t\t\t\nSetting $\\beta=1$ in Eq.~\\eqref{eq:blackness2} we can obtain a threshold value $\\mathcal{R}_{th}$ for the relative size dividing Black from Grey Swans:\n\t\\begin{equation}\n\t\t\\mathcal{R}_{th}=\\ton*{\\frac{Q+1}{Q}}^{\\gamma}=\\frac{s_M}{S(1)} \n\t\t\\label{eq:threshold_R}\n\t\\end{equation}\nwhich once again can expressed in terms of only empirical quantities and coincides with the ratio between the estimate of the upper cutoff and the empirical maximum. \nIn this way we can now define the \\textit{Swans plane} $Q-\\mathcal{R}^{1\/\\gamma}$ (Fig.~\\ref{fig:Q_ratio_powerlaw}), since only these two quantities are needed to determine the nature of an extreme event. Three areas can be identified: the White Swans region ($\\mathcal{R}^{1\/\\gamma}<1$, if the event has a size less than the already seen maximum $S(1)$), the Grey Swans region ($1<\\mathcal{R}<\\frac{Q+1}{Q}$, a size higher than the maximum, but not surprising given the upper cutoff estimation) and the Black Swans region ($\\mathcal{R}>\\frac{Q+1}{Q}$, higher than both the maximum and the upper cutoff and so, in this sense, surprising). Note that the trivial White Swan region is not represented in the figure.  \\newline\nAs discussed above, stationary power laws are not expected to produce Black Swans, being such events related to an abrupt non stationarity of the upper cutoff. This is confirmed by Fig.~\\ref{fig:Q_ratio_powerlaw}, where we exploited the Swans plane to visualize the outcomes of a stationary truncated power law.\tMore precisely we generated $N_i=100$ numbers from a stationary truncated power law with parameters $\\alpha=1.5$, $s_m=1$ and $s_M=10^6$, so that the initial deviation parameter is $Q_i=0.1$, and then we extracted other $N_f=10^5$ numbers, so to arrive to a final deviations parameter $Q_f=100$. At each draw we checked if the newcomer has been larger than the previously observed numbers. If this was the case, we fitted the sample available before the new extraction occurred with Zipf-Mandelbrot law, determining the value of $Q$; the full procedure has been repeated $10^3$ times. Brighter colors correspond to a larger frequency and, as expected, frequencies go to zero out of the Grey Swan region delimited by the white curve corresponding to $\\beta=1$. In the same figure we also plotted the four Black Swans we identified, as it is possible to see they all reside in a region of the Swans Plane where the stationary power law produces no events and so they are not compatible with a stationary power law. \\newline\n\\subsubsection{Future Black Swans}\nBy using our estimation of the upper cutoff, given by Eq.~\\eqref{eq:s_M_main}, we can estimate the size of a newcomer event to be a Black Swan. We computed the upper cutoff of five social and natural systems and, in turn, the minimum size for new events to be Black Swans. We show our results in Table \\ref{tab:fig_1_main}. It results that a pandemic should kill more than $5$ billion people - i.e. approximately $40$ times the Black Death, the worst pandemic ever with its $200$ millions deaths - for being considered a Black Swan. This stems from the fact that only few pandemics have been recorded during human history and so the inherent distribution is not much sampled. Conversely, the distributions of wildfires occurred in Alberta (CA) and Italian earthquakes are characterized by an high level of sampling. Indeed a wildfire should be just $1.54$ times larger than the observed maximum\\footnote{August 1981 wildfire, $~10^6$ha burned} for being classified as a Black Swan, while an earthquake should release only $1.22$ times the energy of worst Italian earthquake ever\\footnote{1693 Val di Noto earthquake, $7.32$Mw}. Finally, the distribution of Interstate Wars and Dow Jones index daily returns are completely undersampled and this reflects in the fact that in these systems any event, no matter how large, can be at most a Grey Swan. More details about these database are reported in the Methods section.\n\t\t\n\t\t\\begin{table}[t]\n\t\t\\begin{tabular}{ ||c|c|c|| } \n\t\t\t\\hline\n\t\t\tSystem & $\\mathcal{R}_{th}$ & $s_M$ \\\\\n\t\t\t\\hline\n\t\t\t\\cellcolor{LightRed} Pandemics & $42.29$ & $5.81\\cdot 10^9$ deaths\\\\\n\t\t\t\\cellcolor{LightGreen}Alberta Wildfires &$1.54$& $1.55\\cdot10^6$ha\\\\\n\t\t\t\\cellcolor{LightRed} Interstate Wars & $\\infty$ & $\\infty$\\\\\n\t\t\t\\cellcolor{LightRed} Dow Jones index return & $\\infty$ & $\\infty$\\\\\n\t\t\t\\cellcolor{LightGreen} Italian Earthquake & $1.23$ & $7.46$M\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\textbf{Future Black Swans.} Upper cutoff $s_M$ and relative size $\\mathcal{R}_{th}$ of five events that could be considered Black Swans with a confidence of $90\\%$. Systems highlighted in red are characterized by a low level of sampling, while those highlighted in green present an high level of sampling.}     \n        \\label{tab:fig_1_main}\n\t\\end{table}\t\n\t\n\\section{Discussion}\nIn this work we exploit the connections among the Zipf-Mandelbrot law, the inherent power law, and the level of sampling, discussed in \\cite{de2021dynamical}, to obtain a relation to derive an empirical estimation of the upper cutoff of a power law distribution. We define Black Swans as events whose size is both extreme (i.e., larger than the observed maximum) and unexpected (i.e., larger than the estimated cutoff). We show that three ingredients are needed to produce Black Swans: an inherent truncated power law distribution, an high level of sampling, and a jump dynamics of the upper cutoff. Turkish Airline Flight 981 is a clear-cut example where such a dynamics is particularly evident, since the jump of the upper cutoff has been provoked by a well defined technological advance, the introduction of large body aircrafts. As a consequence, looking only at data regarding a certain system, may result in ignoring the jump of the upper cutoff, not allowing to foresee the possibility of a Black Swan. When performing risk assessment it is instead crucial to analyze also the context the systems lives in, since jumps can be provoked by eternal factors rather than by the laws governing the system itself, and not to rely on the assumption of a stationary probability distribution.\\newline \nWe also introduced a quantitative and scientific measure of very large events, the Blackness, which allows to divide them in White, Grey, and Black Swans. Using this parameter, which depends on empirical quantities only, we checked that stationary power laws only produce Grey Swans and we analyzed several empirical systems in search of Black Swans. We found out that our criterion is in line with most of the qualitative findings of Taleb, since it correctly classifies the World War I and 9\/11 terrorist attacks as Black Swans, while the 1987 Black Monday is classified only as a Grey Swan. We also spotted new examples of Black Swans, such as Turkish Airline disaster and number of goals scored by Lionel Messi. The latter probably connected to an increase of the number of teams involved in LaLiga championship.\nFinally, we determined how large should various events be in order to be classified as Black Swan by estimating the upper cutoff of their inherent power law distributions. For instance, in the case of Italian earthquakes the distribution is highly sampled, and so an earthquake releasing just $1.23$ times the energy of the largest earthquake ever recorded would be a Black Swan. Conversely, the distribution of Dow Jones index daily returns is undersampled and so any fluctuation, no matter how much large, would be only an unsurprising Grey Swan. \\\\\nWe believe that the introduction of a quantitative criterion to classify extreme events in Black Swans (or not) can be extremely useful not only from a scientific point of view, but also to scientifically ground the discussion among public opinion, academia, and policy makers. \n\n\n\t\t\\begin{acknowledgments}\n           Giordano De Marzo is grateful to Federico Attili for useful discussions about the fitting procedure and the computation of confidence bounds.\n        \\end{acknowledgments}\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBragg mirrors, consisting of multiple thin layers of dielectric material, can \nby carefull design reflect for specific wavelenghts of electromagnetic \nradiation.\nWhere single layer mirrors are rather limited in that the incoming grazing \nangles should be below or close to the critical angle of that material, \nmulti-layer mirrors allow for larger angle windows.\nMultilayered thin film mirrors efficiently reflecting X-rays, nowadays find \ntheir application towards a variety of applications such as EUV \nphotolitography, X-ray microscopy, synchrotron radiation, plasma physics and \nastrophysics. For EUV photolitography working around a 13~nm wavelength, \ntypically Mo\/Si multi-layers having a reflectivity of about 60\\% find \ngood use. At lower wavelenghts, however, their reflectivity becomes \nvery low, a region where Ni\/C multi-layers have higher reflectivity. \nSpiga et al. \\citep{Spiga2004}\ne.g. measured a reflectivity of 95\\%.\nFor current hard X-ray focusing telescopes, the use of Pt and the corresponding \nK adsorption edge leads to an upper energy limit of $\\approx$79.4~keV. The \nusage \nof a different heavy element in the multi-layers, such as Ni, can result in an \nextended energy bandwidth into the soft $\\gamma$-ray range as Ni does not have \nadsorption edges in the hard X-ray\/soft $\\gamma$-ray range.\nEspecially depth-graded Ni\/C multi-layers find their use in focusing optics in \nastronomical telescopes that aim for hard X-ray and $\\gamma$-ray observatories \nwith typical energies of 158~keV and 511~keV, important to study respectively \nthe Ni decay and positron-electron annihilation in supernovae \\citep{Buis2006}.\nIn Fig.~\\ref{fig:0} we plotted the calculated X-ray reflectivity curves at \n8.05~keV versus \ngrazing angle and versus wavelength using Fresnel equations and Henke's optical \ndata \\citep{Henke1988}. Calculations are shown for a single Ni\/C multi-layer \nand \na 40 \npairs Ni\/C multi-layer, both with a Ni film thickness of 1.2~nm and a C film \nthickness of 6.7~nm, as suggested for an optic design for use in hard X-ray and \nsoft $\\gamma$-ray astronomy applications using silicon pore optics \n\\citep{Collon2016}. \nFor comparison \nwe plotted the calculated X-ray reflectivity curves for a 40 pairs Mo\/Si \nmulti-layer of 2.76~nm Mo and 4.14~nm Si. Notice the superior \nreflectivity for Ni\/C multi-layers at grazing angles in comparison to the Mo\/Si \nmulti-layers, see Fig.~\\ref{fig:0}(a), and the vanishing reflectivity at lower \nwavelenghts of the Mo\/Si multi-layers in comparison to the Ni\/C multi-layer, \nsee \nFig.~\\ref{fig:0}(b).\n\n\\begin{figure}\n\t\n\\includegraphics[width=8cm]{Fig1a.pdf}\n\\includegraphics[width=8cm]{Fig1b.pdf}\n\\caption{\n(a) Calculated X-ray reflectivity curves at 8.05~keV (Cu K-$\\alpha$) of a Ni\/C \nmulti-layer with 1 and 40 pair(s) of 1.2~nm Ni and 6.7~nm C layers, and of \na Mo\/Si \nmulti-layer with 40 pairs of 2.76~nm Mo and 4.14~nm Si layers.\n(b) Calculated X-ray curves of the Ni\/C multi-layer and Mo\/Si multi-layer \nboth with 40 pairs at wavelengths of 3 to 14.5~nm.}\n\\label{fig:0}\n\\end{figure}\n\nDesigning towards shorter periodic wavelenghts, however, reduces the \nexperimentally \nobserved reflectivity significantly in comparison to the anticipated \ntheoretical reflectivity. This is anticipated to result from the substantial \ninterface roughness, caused by crystallization of Ni and is found to occur \nbelow a thickness of 2~nm followed by interdiffusion between adjacent Ni and C \nlayers worsening the observed interlayer roughness \n\\citep{Krawietz1995, Takenaka1996, Dietsch1998, Georgescu1999, Peng2016}. In \norder to \nreduce the roughness between adjacent layers, Pulsed Laser Depositing (PLD) \nthese films is found to result in smoother films as the deposited particles \nhaving higher kinetic energy of several 100~eV can smoothen the Ni\/C interface. \n\nHere, we investigate the feasibility of producing such thin \nmulti-layers by the use of Pulsed Laser Deposition on a silicon-oxide (SiO) \nsubstrate containing a native oxide. Pulsed Laser \nDeposition is a well studied technique to aid in the synthesis of \nmulticomponent and complex thin films. In comparison to conventional vapor \ndeposition techniques, PLD provides additional parameters thereby in principle \ngiving more flexibility to the (control of the) growth process of thin films. \nBesides this, PLD is known for its high supersaturation in growth, resulting in \na high nucleation density resulting in (ultra)smooth thin film growth \n\\citep{Blank2014}, a necessity for the application described before. Although a \nlot of work is done on the PLD growth of (complex) oxides \\citep{Blank2014}, \nonly \nlittle studies are done on the initial stages of the growth of metallic thin \nfilms \\citep{Krebs1993, Krebs1995, Lunney1995, Faehler1997, Krebs1997, \nDoeswijk2004}. In this work, we study the evolution of metallic thin films \ngrown \nby PLD with emphasis on the process variables. \nBy studying the growth at varying conditions, we seek the optimal PLD growth \nconditions for Ni\/C thin film multi-layers as well as studying the influence on \nthe resulting reflectivity, as these multi-layers could serve as a next \ngeneration coating for application in future hard X-ray and soft $\\gamma$-ray \noptics.\n\nIn this paper, we therefore start by the growth and characterization of \nindividual Ni and C films in order to optimize the individual growth conditions. \nWe then stack these films at elevated temperature to \ntest their thermal stability. As analysis reveals the intermixing of Ni and C \nat elevated temperatures, we summarize by analysis of pairs of Ni\/C \nmulti-layers grown at RT conditions, discussing the experimentally \nobserved reflectivity in view of the theoretically predicted one. This brings us \nto conclusions and outlook towards future applications.\n\n\\section{Experimental details}\nFor comparison to commercially available silicon-oxide pore optics \n\\citep{Collon2016}, we diced a native silicon-oxide wafer in samples of \n5$\\times$5~mm$^2$ which were then ultrasonically degreased in \nacetone and ethanol prior to insertion into the vacuum setup. The SiO \nsample was glued by \nsilver-paste to a sample-holder that could be heated in-situ by laser heating. \nThe sample temperature was measured by a pyrometer at the backside of the \nsample-holder, estimated to result in an error of $\\pm$50$^{\\circ}$C.\n\nA PLD system from Twente Solid State Technology (TSST) {B.V.} was used in \ncombination with \na 248~nm excimer laser\nwith a pulse duration of 25 ns to ablate material from the targets: \nsingle crystal Ni(111) and Highly Oriented Pyrolytic Graphite (HOPG) of ZYA \ngrade. A rectangular mask was \nused to create a well-defined and homogeneous laser profile on the target.\nThe target was positioned 50~mm from the substrate. The laser fluence was \ncontrolled using a variable beam attenuator. By applying a magnification of \n11.5, we achieved a laser energy density of 6~J\/cm$^2$ for the ablation \nof C and 7.5~J\/cm$^2$ for the ablation of Ni, using a spot-size of \n0.6~mm$^2$. Both were scanned during deposition over a width of 5~mm at a speed \nof 0.5~mm\/second. The background gas pressure was in \nthe range of 10$^{-8}$ mbar. \n\nThe growth rate per pulse was obtained by determination of the Kiessig \nfringes in the resulting X-ray reflectivity curve for a thick film grown. For \nNi \nwe determined a growth rate of 1.9$\\times$10$^{-3}$~nm\/pulse where growth was \nperformed at a repetition rate of 8~Hz. The growth rate for C was found to be \n1.8$\\times$10$^{-2}$~nm\/pulse at a repetition rate of 10~Hz. \n\nDuring deposition the surface structure was \nmonitored using Reflective High Energy Electron Diffraction (RHEED) operated at \n30~keV, at pressures up to 0.3~mbar, enabled \ndue to differential pumping. Subsequently, X-ray reflectivity (XRR) and \ndiffraction (XRD) at a Cu K-$\\alpha$ wavelength \nwas used for reflectivity and structural characterization of the multi-layers. \nBy use of ex-situ tapping mode atomic force microscopy (TM-AFM) the resulting \nsurface roughness ($\\sigma_{RMS}$) was determined. The surface roughness (RMS) \nof the \npristine SiO samples was found to be $\\approx$0.58~nm.\n\n\n\\section{Results and discussion}\nTo study and optimize the PLD growth conditions for individual Ni and C films, \ntypical parameters in the PLD method should be considered; the laser fluence, \ngiving the target \nparticles their initial kinetic energy; the laser repetition rate, dictating \nthe growth rate; the process pressure, reducing the kinetic energy of the \nparticles arriving on the substrate as well as the plasma shape \n\\citep{Groenen2015}; and the substrate temperature, enhancing diffusivity \nduring \nthe growth. Here, we focus on the role of process pressure and substrate \ntemperature upon growth, as they mainly govern the roughness of the resulting \nfilm as for \nits reflective properties of the multi-layer we are not interested in obtaining \nhighly crystalline films. \n\n\nAs is described in literature for the growth of thin graphite films, an \nincrease \nin laser energy density results in higher quality films \\citep{SarathKumar2013, \nSarathKumar2013b}. Besides this, a thermal treatment is known to increase the \nthin film quality \\citep{SarathKumar2013b, Tite2014}. As a low surface \nroughness \nis of utmost importance for future applications, we deposited thin graphite \nfilms at the described conditions, an ablation threshold of 6~J\/cm$^2$, for \nincreasing process pressures of respectively 0.027, 0.05 and 0.1~mbar of Ar, as \nan increased process pressure typically leads to more 2D growth for PLD growth \n\\citep{Sturm2000,Scharf2002,Scharf2004,Groenen2015}. For PLD films grown at \nprocess pressures beyond 0.027~mbar, we find however a sharp increase in \nroughness.\nTherefore, the process \npressure throughout this paper was 0.027~mbar of Ar by influx of a 40 \nstandard cubic centimeter (SCCM) Ar flow and subsequent control on the pump \nrate by positioning the gate valve between pump and vacuum \nchamber. \n\nIn comparison to the PLD of oxides, the ablation of metal targets requires \nablation thresholds typically an order of magnitude higher \n\\citep{Doeswijk2004}. \nHowever, too large ablation thresholds in combination with a too low process \npressure results in droplet formation at growth \\citep{Krebs1993, Krebs1995, \nLunney1995, Krebs1997, Faehler1997}. A study on the microstructural evolution \nof Ni films upon increasing ablation threshold \\citep{Kumar2008} revealed an \nincreased clustering and denser packing of nanoparticles. In order to obtain \nsufficiently high deposition rates in high vacuum, laser fluences of more than \n5~J\/cm$^2$ are a necessity. Besides this, it should be noted that in \norder to reduce the droplet density, a sweet spot of ~7-9~J\/cm$^2$ \nexists, as discussed in Fig.~5 of Ref. \\citep{Faehler1997}. Making use of an \nablation threshold of 7.5~J\/cm$^2$ and a \nprocess pressure of 0.042~mbar to reduce the kinetics of the arriving \nparticles, \nwe obtain low roughness Ni surfaces which show no preferential crystallographic \norientation. \n\nIn order to study the quality of the mirror, which depends on the quality of \nevery single layer deposited, we started by analyzing single layers of Ni and C \ndeposited by PLD.\nBefore deposition, the RHEED pattern reveals a clean SiO substrate pattern as \nshown in Fig.~\\ref{fig:3}(a). Upon deposition of C at \nRT, a continuous decay in the intensity of the substrate spots appears. For \nthicker films, clear transmission RHEED patterns of multiwalled carbon \nnanotubes appear, see e.g. Fig.~\\ref{fig:3}(b). The \ntransmission pattern was checked by tilting the sample as well as varying the \nangle of incidence with respect to the surface, see Fig.~\\ref{fig:3}(c-h). By \nthis, substrate features should reveal position changes as they are very \nsensitive to tilt, in contrast to transmission features which keep their \nposition.\n\nIn order to calculate the preferred interplanar distances within \nthe C film, the preferred Debye-Scherrer ring radius is calculated \nby integrating the radial intensity distribution along the periphery of circles \naround the RHEED central spots for increasing radius.\nFig.~\\ref{fig:3}(i) shows the integral of the radial intensity distribution of \nFig.~\\ref{fig:3}(c). \nSimilar to RHEED images described for randomly oriented multiwalled carbon \nnanotube samples, the radial profile of the diffraction intensity is in \ncorrespondence with graphites lattice parameters $a$ and $2b$ as described in \ndetail in Ref.~\\citep{Drotar2001}.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig2.pdf}\n\\caption{(color online) (a) RHEED pattern of the SiO substrate. (b) \nTransmission RHEED pattern of as-deposited C film at an incident angle of \n1$^\\circ$ (b), 1.5$^\\circ$ (c), 2$^\\circ$ (d), 2.5$^\\circ$ (e), 3$^\\circ$ (f), \n3.5$^\\circ$ (g), 4$^\\circ$ (h). All RHEED patterns where taken at a primary \nelectron energy of 30~keV. (i) The integral of the radial intensity \ndistribution of the as grown C film. The labeled arrows \nindicate the expected positions of the Debye-Scherrer rings \ncalculated from powder diffraction intensities.}\n\\label{fig:3}\n\\end{figure}\n\nFor Ni films grown, we only observe a continuous decay in the RHEED intensity \nof the \nsubstrate spots, however, no formation of a transmission RHEED pattern. This is \nalso the case for thicker Ni films, indicative of the amorphous \nmicrostructure of the layer, a necessity for efficient mirrors.\n\nIn Fig.~\\ref{fig:1}(a-c), we show topographic TM-AFM images of the grown 1.2~nm \nthick Ni and 6.7~nm thick C films. The different AFM \nimage sizes reveal roughnesses (RMS) $<$0.1-1.4~nm. In comparison to \nmagnetron-sputtered Ni films \\citep{Peng2016}, where the roughness increases \nfor \ndecreasing film thickness, the roughness at the interface seems rather ill \ndefined. We should however take into account that during the PLD processing of \nmetals, the formation of spherical droplets with sizes in the range of \n0.1-0.2~$\\mu$m are often reported \\citep{Faehler1997, Largeanu2011}.\nFrom a grain analysis on the AFM images shown in Fig.~\\ref{fig:1} we find \nspherical droplets having similar dimensions. \nAlthough larger sized \ndroplets can easily be avoided, these \nsmaller droplets result from the fast heating and cooling occuring \nduring melting of the targets surface and can not be completely avoided. \nLiterature suggests the use of some kind of velocity filter \nin order to obtain droplet free films \\citep{Faehler1997}. By marking these \nsmall \ndroplets and excluding them from the analysis, we find roughnesses \n(RMS) in the range of $<$0.1-0.2~nm. If one would be able to avoid the droplets \nresulting from the heating and cooling at the target, one would obtain PLD \nfilms superior to magnetron-sputtered films.\n\nFor the roughness (RMS) of the C films, see Fig.~\\ref{fig:1}(d-e), we find \nvalues between 0.2-0.8~nm. For these films, we observe an increased number of \nsmall droplets in line with the observations for the Ni films. Excluding the \ndroplets from the image analysis, reveals a roughness (RMS) of 0.2~nm, in \nagreement with the observations for the thickness dependent roughness (RMS) of \nmagnetron-sputtered C films \\citep{Peng2016}. \n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig3.pdf}\n\\caption{(color online) Topographic TM-AFM images of a 1.2~nm deposited Ni film \non the SiO substrate of 10$\\times$10$\\mu$m$^2$ (a), 5$\\times$5$\\mu$m$^2$ (b) \nand 1$\\times$1$\\mu$m$^2$ (c), and of a 6.7~nm thick deposited C film on the SiO \nsubstrate of 10$\\times$10$\\mu$m$^2$ (d), 5$\\times$5$\\mu$m$^2$ (e) and  \n1$\\times$1$\\mu$m$^2$ (f). }\n\\label{fig:1}\n\\end{figure}\n\n\\begin{center}\n\\begin{table}\n\\begin{tabular}{l|l|l}\nAFM image size & $\\sigma_{RMS}$ Ni film & $\\sigma_{RMS}$ C film\\\\\n\\hline\n10$\\times$10$\\mu$m$^2$ & 0.23nm (1.36nm) & 0.24nm (0.78nm)\\\\\n\\hline\n5$\\times$5$\\mu$m$^2$ & 0.14nm (0.91nm) & 0.22nm (0.51nm)\\\\\n\\hline\n1$\\times$1$\\mu$m$^2$ & 0.09nm & 0.11nm\n\\end{tabular}\n\\label{tab:1}\n\\caption{Roughness (RMS) measured for different AFM image sizes, excluding \nmasked nm-sized particles. Between brackets, the roughness (RMS) including the \nentire image.}\n\\end{table}\n\\end{center}\n\nHaving found PLD conditions to grow Ni and C films, both, with surface \nroughnesses $\\leq$0.2~nm for the required thicknesses of respectively 1.2 and \n6.7~nm, we stack both films, thereby changing the growth conditions in \nsequence. For the Ni film we use an ablation threshold of 7.5~J\/cm$^2$ and a \nsubstrate temperature at RT, where for the following C film we use an ablation \nthreshold of 6~J\/cm$^2$ and vary the growth temperature \nas a thermal treatment is known to increase the thin film quality \n\\citep{SarathKumar2013b, Tite2014} and more importantly, to test the thermal \nstability of the multi-layer as it is required in astronomy applications \n\\cite{Collon2016}.\n\nTo test the thermal stability of the grown multi-layer, we followed a different \napproach compared to the straightforward heating of the multi-layer as \ndescribed in Ref.~\\citep{Krawietz1995}. There it is found that the reflectivity \nof the multi-layer is destroyed upon heating beyond 300$^{\\circ}$C, explained \nby the formation of carbides which decompose upon heating at higher \ntemperatures. \nHere, we want to investigate the role of substrate temperature during \ndeposition instead of post-processed as a thermal treatment is known to \nincrease \nthe thin film quality \\citep{SarathKumar2013b, Tite2014}. We therefore use a \ngrowth temperature for the C layer of RT, 300$^{\\circ}$C, 500$^{\\circ}$C and \n700$^{\\circ}$C. \nAs film thicknesses of 1.2~nm (Ni) and 6.7~nm (C) are hard to measure by Cu \nK-$\\alpha$ X-rays as the Kiessig fringes are very wide in \n2$\\theta$, and one typically hits the noise floor \nof the detector for angles beyond 5$^{\\circ}$, we study thicker C \n(75~nm) and Ni (20~nm) films here, in order to make it easier to determine the \nKiessig fringes and to enable for more pronounced diffusivity through \nthe multi-layer. \nCharacterization by X-ray diffraction and reflectivity is plotted \nin Fig.~\\ref{fig:2}(a) and (b).\nIn Fig.~\\ref{fig:2}(a) we plot the X-ray diffraction curve for a \nmulti-layer grown at RT. The pattern reveals no specific \ncrystallographic preferential orientation for Ni(111) at 44.6$^{\\circ}$ \n\\citep{Miller2012}, Ni(200) at 51.8$^{\\circ}$ or Ni(220) at 76.4$^{\\circ}$ \nin agreement with JCPDS No:04-0850.\nSimilar, the X-ray diffraction pattern shows no preferential \ncrystallographic orientation for a crystalline C film, known to result in a \nweak and broad peak around 20$^{\\circ}$ \\citep{SarathKumar2013, \nRuammaitree2013}. \nThe peak at 33$^{\\circ}$ and 69$^{\\circ}$ \ncorresponds to the silicon wafer, used as a substrate. In contrast to \nRef.~\\citep{Krawietz1995}, we do not observe any indication for \ncrystalline Ni(111) nor Ni(200) formation in our curves, although the layers of \nthe multi-layer described here are thicker and the effect is therefore expected \nto be \nmore pronounced. \n\nThe diffraction patterns, see \nFig.~\\ref{fig:2}(a), do not show any formation of carbides (Ni$_3$C) nor highly \ncrystalline Ni structures as compared to Ref.~\\citep{Krawietz1995}. From \nthe bulk phase diagram \\citep{Singleton1989}, these nickel carbide structures \nare anticipated to appear in a narrow temperature window \\citep{Krawietz1995}, \ndepending on the Ni\/C ratio at the interface. As the temperature reading used \nin our experiments might be underestimated, we might have been beyond this \ntemperature window, resulting in the lack of nickel carbide structures.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig4a.pdf}\n\\includegraphics[width=8cm]{Fig4b.pdf}\n\\caption{(a) X-ray diffraction curves of a Ni\/C multi-layer, \n20~nm Ni and 75~nm C, grown at increasing temperature. Spectra are vertically \noffset for presentation. (b) The \ncorresponding X-ray reflectivity curves. Beyond a growth \ntemperature of 300$^{\\circ}$C, the Kiessig fringes are not present.}\n\\label{fig:2}\n\\end{figure}\n\nIn Fig.~\\ref{fig:2}(b) we plot the corresponding X-ray \nreflectivity curves, revealing clear Kiessig fringes, arising from the \nconstructive interference between the X-rays  reflected from the film-vacuum, \nfilm-film and substrate-film interface. \nWe find the reflectivity to be \nfully destroyed upon growth temperatures beyond 300$^{\\circ}$C as can be seen \nby \nthe absence of Kiessig fringes in Fig.~\\ref{fig:2}(b), indicative of ill \ndefined interfaces at the film-vacuum, film-film and substrate-film interface \nin agreement with literature \\citep{Krawietz1995}. Obviously this is undesired \nfor the application in mind.\nNote that the critical angle decreases at increased growth \ntemperature for the C layer, indicative of a decrease of the density of the \nmulti-layer by about 25\\%. Besides this, the periodicity of the Kiessig \nfringes is also shifting at 300$^{\\circ}$C, suggesting C segregation into \nNi described in literature. Partial segregation of the Ni would \ndecrease the thickness of the reflective Ni layer, resulting in an increase in \nthe width of the Kiessig fringes as observed here. This is hinting towards \nshifting of the interface between the C and Ni film, towards deeper depth.\nTo investigate the several interfaces in more detail and  study the cause for \nthe destroyed reflectivity at growth temperatures beyond 300$^\\circ$C, we make \nuse of additional experimental techniques.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig5.pdf}\n\\caption{X-ray Photoelectron Spectroscopy scan at several angles of incidence \n$\\varphi$ around the C 1s peak for a 1.2~nm thick Ni, 6.7~nm thick C \nmulti-layer grown at respectively RT and 700$^\\circ$, revealing hardly any \nnickel carbide formation, corresponding to an energy of $\\sim$282.8~eV.}\n\\label{fig:xps}\n\\end{figure}\n\nTo confirm the lack of nickel carbide formation at elevated temperature, we \nprobed the surface chemical properties by depth-profiling X-ray photoelectron \nspectroscopy (XPS), see Fig.~\\ref{fig:xps}. By variation of the angle of \nincidence, the estimated penetration depth is about 10~nm at $\\varphi=0^\\circ$ \n(for perpendicular incidence), 7~nm at $\\varphi=-35^\\circ$ and 3~nm at \n$\\varphi=70^\\circ$ incidence with respect to the substrates normal. The C 1s \npeak shows omnipresent contributions of the carbon \n$sp^2$ and $sp^3$ peaks at respectively 284.5~eV and 285.4~eV for all angles \nof incidence where the high ratio of $sp^2$:$sp^3$ is indicative for amorphous \ncarbon \\cite{Lascovich1991}. Nickel carbides, found around 282.8~eV \n\\citep{Seo2017}, show only \na faint signature when probing more surface sensitive. The C$-$O$-$C and \nO$-$C$=$O functional groups at respectively 286.5~eV and 288.7~eV seem absent \nas \nthis film has been prepared and transferred to XPS in (ultra)high vacuum \nconditions.\n\nAs the loss of reflectivity can result from an ill defined film-vacuum, \nfilm-film and\/or substrate-film interface, we investigate the film-vacuum \nsurface morphology by AFM, see Fig.~\\ref{fig:5}(a-d).\nThe measured surface \nroughness (RMS) by AFM for C films grown at RT is 0.89~nm, see \nFig.~\\ref{fig:5}. For \nincreasing growth \ntemperature, the roughness (RMS) increases to 2.1~nm at 300$^{\\circ}$C, 37.6~nm \nat 500$^{\\circ}$C and 45~nm at 700$^{\\circ}$C. \nKrawietz et al. \\citep{Krawietz1995} attribute the change in reflectivity \nbeyond \n300$^{\\circ}$C solely to the fragmentation of the Ni layers supported by the \noutdiffusion of C from the Ni layers.\n\n\\begin{figure}[h]\n\\includegraphics[width=8cm]{Fig6.pdf}\n\\caption{(color online) TM-AFM images of 20~nm thick Ni film, grown at RT, \nfollowed by subsequent C film growth at RT (a), 300$^{\\circ}$C (b), \n500$^{\\circ}$C (c) and 700$^{\\circ}$C (d).\n}\n\\label{fig:5}\n\\end{figure}\n\n\n\nIn order to investigate the different interfaces, we grow a multi-layer of 5 \nNi\/C stacks containing a 1.2~nm thick Ni layer followed by a 6.7~nm thick C \nlayer, where the Ni layer was grown at RT and the C layer was grown at \n700$^{\\circ}$C. This sample was then perpendicular cut and grinded carefully to \na thickness where XTEM could \nbe performed. The resulting XTEM image is shown in Fig.~\\ref{fig:4}. At the \nbottom of the image, the highly crystalline Si substrate is imaged along the \n[022]-axis. Above the Si substrate, a thin amorphous oxide layer can be \ndiscriminated. Above this, we find a thick layer of 32-34~nm in thickness, in \ngood agreement with the deposited amount of Ni and C material. In this thick \nlayer, we can not discriminate individual Ni and C layers. Instead, a layer \nof \nrandomly oriented carbon nanotubes can be seen, with intermixed in it, \ncrystalline clumps of Ni, similar to as described in Ref.~\\citep{Peng2016}. \nThe crystalline Ni clumps, having sizes up to 4-5~nm in diameter, suggest \noutdiffusion of C \\citep{Krawietz1995}.\n\n\\begin{figure}\n\t\\includegraphics[width=8cm]{Fig7.pdf}\n\t\\caption{XTEM image of a multi-layer of 5 Ni\/C stacks of 1.2~nm Ni and \n6.7~nm C. The highly crystalline substrate can be determined at the bottom, \nhaving an amorphous oxide layer on top. The following 32-34~nm contains the \nintermixed Ni\/C layers, where intermixing occurred during growth at \n700$^{\\circ}$C.}\n\t\\label{fig:4}\n\\end{figure}\n\nTo study the feasibility of creating high reflectivity multi-layers by PLD \ngrowth, we investigate the resulting reflectivity of several pairs of \nmulti-layers grown.\nAs the growth of multi-layers at elevated temperatures results in intermixing \nof Ni and C destroying its reflective properties, we summarize this paper by \ngrowing a 5 and 10 pairs Ni\/C multi-layer at RT, both with a Ni film thickness \nof 1.2~nm and a C film thickness of 6.7~nm at the same laser energy densities \nand process pressures described before. \nIn Fig.~\\ref{fig:8} we plot the X-ray reflectivity curve for 5 pairs \nof Ni\/C multi-layers. For 10 pairs of Ni\/C multi-layers, see inset of \nFig.~\\ref{fig:8}. \n\nFrom calculated X-ray reflectivity curves at 8.05~keV the resulting \nreflectivity at a grazing angle of $\\theta$=0.6$^\\circ$ is 13.9\\% for 5 pairs \nincreasing to 42.6\\% for 10 pairs of multi-layers. An increase of the \nnumber of pairs of multi-layers results into an increased reflectivity at this \ngrazing angle, see also Fig.~\\ref{fig:0}(a). Analyzing the experimentally grown \npairs of multi-layers in Fig.~\\ref{fig:8}(a) and (b), we find a reflectivity for \nthe 5 and 10 pairs of respectively 13.0\\% and 11.8\\%. Note that the reflectivity \nfor 5 pairs of multi-layers is close to the theoretically predicted value, \nwhereas for an increasing number of pairs the reflectivity does not increase.\nIn order to understand this, we quantify the roughness for the film-vacuum and \nsubstrate-film interface, by modeling the system as film pairs of uniform \n(electronic) density on top of a uniform (electronic) dense substrate. Fitting \nwas done by using the fitting parameters film thickness ($d_{Ni}$ and \n$d_{C}$), assuming constant thicknesses for all layers, interface \nroughness between Ni ($R_{rms}^{Ni}$) and C films \n($R_{rms}^{C}$) and Ni-substrate interface roughness \n($R_{rms}^{substrate}$), a background resulting from scattering and a \nnormalization factor \\cite{GenX}. Note that we fit up to \nlimited 2$\\theta$ angle (see the dashed line in Fig.~\\ref{fig:8}) to ensure the \ndynamical scattering theory is applicable and stay far from the kinetical \nscattering regime \\cite{Jankowski2017,Vlieg2012}.\nFrom this we find a steep roughness increase going from 5 pairs \n($R_{rms}^{C}$=0.27~nm and $R_{rms}^{Ni}$=0.41~nm) to 10 pairs of Ni\/C \nmulti-layers ($R_{rms}^{C}$=1.3~nm and $R_{rms}^{Ni}$=2.1~nm) grown by PLD at \nRT. Note that for 5 pairs, the fitted roughnesses agree well with the studied \nAFM images, see also Fig.~\\ref{fig:1}. For 10 pairs of multi-layers, the \nroughness of samples grown at RT appears to be dictated by the droplets as \ndiscussed in Table~\\ref{tab:1}.\n\n\\begin{figure}\n\t\\includegraphics[width=8cm]{Fig8.pdf}\n\t\\caption{X-ray reflectivy curves for 5 pairs of Ni\/C multi-layers  \nand 10 pairs of Ni\/C multi-layers (inset) with a 1.2~nm Ni and 6.7~nm C film \nthickness. The dashed curves in both graphs have been obtained by fitting as \ndescribed in the text.}\n\t\\label{fig:8}\n\\end{figure}\n\n\\section{Conclusions}\nHere, we investigated the PLD grown low roughness Ni\/C multi-layers by use of \nRHEED, XRD, XRR, TM-AFM, XPS and XTEM. Thin Ni and C layers with \nroughnesses below $\\leq$0.2~nm can be created by PLD at RT. However, due to the \nfast heating and cooling occuring during melting of the targets surface, small \ndroplets during deposition can not be avoided limiting the reflectivity at \ngrazing angles for increased number of pairs of multi-layers.\nGrowth at temperatures of about 300$^{\\circ}$C results in \nintermixing of Ni and C, destroying the reflective properties of the \nmulti-layer. XTEM image analysis of multi-layers grown at elevated \ntemperatures reveals (crystalline) Ni clumps intermixed with C.\nFor future applications in hard X-ray and soft $\\gamma$-ray focusing \ntelescopes, the stability of multi-layers under extreme (temperature) \nconditions \nis required and needs further investigation in order to find technological \nsolutions.\n\n\\section{Acknowledgments}\nThis work is part of the research programme of NanoNext NL project 7B \nMultilayered and artificial materials, project 12 and the research programme of \nthe Foundation for Fundamental Research on Matter (FOM), which is part of the \nNetherlands Organisation for Scientific Research (NWO). \nCross-section TEM imaging has been performed by Dr. Rico Keim. \nThe author is gratefull to Prof.~Dr.~Ing.~G.~Rijnders for his permanent support.\n\n\\section{References}\n\\bibliographystyle{elsarticle-num}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{1. Introduction}\n\nDegenerate energy levels (gapless) can form conducting states even if they are confined to the surface (or to the edge). If their degeneracy is kept intact against external disturbances (temperature ($T$), pressure, electric ($\\textbf{E}$) and magnetic ($\\textbf{B}$) fields), such that the bulk always has a well-defined band gap, then the materials satisfying the above conditions can be grouped into a new class of solids. These new solids are known as the topological insulators (TI). At least, that is the proper and generalized definition one can think of to identify TI based on the crossed energy-level notion in two-dimensional quantum Hall metals, which was first calculated by Hatsugai~\\cite{hat}. This definition also applicable to three-dimensional TI~\\cite{mele,balent}, and it properly rules out free-electron and Fermi-liquid metals, even in the presence of skin effect (frequency-dependent surface conductivity)~\\cite{ash}. In TI, we actually need the bulk to be gapped so that the surface states form a distinct two-dimensional system that can exhibit quantum Hall metallic properties~\\cite{rui,chang}. This means that, one may need to perform quantum Hall effect measurements~\\cite{rui,chang,konig} to confirm whether an insulator is indeed a topological insulator because electric transport measurements~\\cite{barua,vonk}, and other surface analysis techniques, such as the angle-resolved photoemission spectroscopy (ARPES)~\\cite{ylchen}, de Haas-van Alphen~\\cite{kittel} and Shubnikov-de Haas~\\cite{shub} effects are insufficient to claim the bulk is gapped, while the surface is metallic, independent of the bulk.   \n\nOn the other hand, TI are also related to topology, a mathematical notion that deals with `smooth' deformations of any entity~\\cite{naka}, and in our case, this entity strictly refers to two-dimensional surface metallic states or degenerate energy levels. One can think of these metallic states or energy levels as the elements (or open subsets) within an open set that comprises all degenerate energy levels. This means that the energy levels or the subsets are the topology on an open set where this open set and its subsets satisfy some precise conditions, which allow smooth deformations~\\cite{geroch}. Now, any deformation to these energy levels requires one to invoke an arbitrary right-hand side (rhs) action operator~\\cite{naka} that tacks a transition function (namely, a phase factor) on the rhs of an electronic wavefunction~\\cite{and1}. However, it has been rediscovered recently that a wavefunction picks up or drops a phase factor as a result of a physical notion known as the Pancharatnam phase retardation (due to a phase acceleration or deceleration). In fact, when the phase and(or) group momenta(um) of a particular wavefunction changes, then one has to tack the Pancharatnam-Berry (PB) phase factor on the rhs of a wavefunction, which gives rise to the Pancharatnam wavefunction transformation~\\cite{and2} that was originally discovered (formulated and observed) by Pancharatnam using the polarized pencil beams and Poincar${\\rm \\acute{e}}$ sphere~\\cite{pan1,pan2}. This means that, the original Berry's phase~\\cite{berry} is a rediscovered special case (the group momentum of a wavefunction is invariant) within the generalized Pancharatnam phase~\\cite{and1,and2,pan1,pan2}. Compared to a conducting surface, the bulk requires the wavefunction to be transformed beyond the phase factor due to a nonzero energy gap between the valence and conduction bands. Therefore, the above rhs action operator has a precise physical origin within the context of quantum mechanics due to the Pancharatnam wavefunction transformation~\\cite{and1}. \n\nHaving said that, we can now set the next course of action---to understand how the definition for a topological insulator (stated in the first paragraph) can be achieved physically, or, how to identify the relevant physical parameter(s) responsible to obtain such an insulator. In this respect, Kane and Mele~\\cite{kane}, and independently by Bernevig, Hughes and Zhang~\\cite{bern2} have confirmed the Hatsugai's surface energy-level crossings such that one needs a combination of physical properties to activate the required energy-level crossings at the edge or on the surface, while the bulk is of course, remains gapped. The physical phenomena that are at play to produce both gapped bulk and gapless surface states in a single system come from different magnitudes of certain crucial parameters (the bulk has different magnitudes compared to the edge or surface)~\\cite{kane}. For example, the interplay among these critical parameters, namely, the spin-orbit coupling, intersite hopping matrix elements, Rashba coupling and the staggered sublattice potential gives rise to a topologically insulating phase~\\cite{kane}. \n\nNote this, the above critical parameters that eventually give rise to changing electronic structure in the bulk can only be different from its surface if their respective chemical composition and\/or the coordination numbers are themselves different. For example, HgTe\/Hg$_{0.3}$Cd$_{0.7}$Te quantum well for different thickness ($d$) have given rise to a quantum phase transition (QPT) from the usual gapped surface (similar to bulk) to a gapless surface states (while the bulk is still gapped) where the critical thickness here is $d_{\\rm c}$ = 6.3 nm that refers to the width of their devise. However, the existence of odd numbers of spin-polarized edge channels is actually an assumption~\\cite{konig}. But never mind, the said QPT has been correctly associated to changing electronic structure due to spin-orbit coupling (SOC)~\\cite{konig} following the Bernevig-Hughes-Zhang (BHZ) model~\\cite{bern2}. Here, it is alright to not to know exactly what other critical parameter(s) (besides SOC) are responsible for the changing electronic structure (or changing energy(Landau)-level crossing). Anyway, what we wanted to say here is that the relation between changing Landau-level crossing and $d$ due to SOC originate from the changing defect types and composition or more precisely, from changing chemical composition and\/or the coordination numbers with changing $d$. These changes due to defect types and composition will give rise to changing SOC~\\cite{arul9}, which shall be made explicit when we discuss the doping-dependent resistivity in TI.\n   \nThe point that we do not agree with K${\\rm \\ddot{o}}$nig \\textit{et al}.~\\cite{konig} and BHZ~\\cite{bern2} is the assumption that time reversal symmetry (TRS) is not violated for the above devise for different applied gate voltages on the basis of BHZ model~\\cite{bern2}. This invariance of TRS in the presence of electric current (due to spin or charge or both) in any condensed matter system should not be assumed to be true haphazardly due to a well thought-out hypothesis put forth by Messiah~\\cite{albert}. See the additional notes (prior to conclusions) for proper arguments. In particular, the TRS defined in Ref.~\\cite{albert} leading to Kramers degeneracy~\\cite{kramers} is not necessarily required to obtain crossed energy levels due to interplay among the above stated critical physical parameters. Of course, one can construct a particular Hamiltonian to satisfy TRS if the energy levels are crossed (not necessarily Dirac points due to Kramers degeneracy) between the bulk valence and conduction bands~\\cite{konig,ylchen,hasan} such that $E_1(\\textbf{k}) = E_2(-\\textbf{k})$ where $\\textbf{k}$ denotes the wavevector~\\cite{rroy3}. But this construction does not imply that TRS is always preserved by default in the presence of internal electric current---for example, all standard ($\\textbf{B} = 0$) and Hall resistance ($\\textbf{B} \\neq 0$) measurements~\\cite{hall1,klit,tsui} necessarily violate TRS as a result of nonzero internal electric current (static or time-dependent)~\\cite{albert}. For Hall current, one needs to consider Zeeman splitting~\\cite{zeeman1} due to nonzero applied magnetic field, while static current produces static `external' magnetic ($\\textbf{B}'$) field. In our formalism within IET, we will properly consider TRS to address electron conduction in TI.\n\nTherefore, our objective here is to develop the basic mechanism required to consistently explain the electronic transport phenomena and carrier-type transition in TI down to atomic energy levels. We start our analyses by invoking the properties of ionization energies and energy level spacings within the ionization energy theory (IET). Subsequently, we derive the relevant equations required to properly understand the time reversal operation in TI, and then justify the conditions that may give rise to TRS violation in TI. With these knowledge as background, we will first evaluate the influence of the energy-level spacing ($\\xi$, which is also known as the ionization energy) on resistivity and carrier-type transition with respect to different chemical compositions in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se TI. The band structure properties of (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se TI relevant to understand transport phenomena have been investigated by Jinsong Zhang \\textit{et al.}~\\cite{jins} and Dziawa \\textit{et al.}~\\cite{dzia}, respectively. \n\nWe will exploit the relation between $\\xi$ and the electron excitation probability to answer the microscopic origins for (i) the changes in doping-dependent resistivity data (including the carrier-density and electron-ion scattering magnitudes) in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se TI, and (ii) the carrier-type transition ($\\texttt{p}$- to $\\texttt{n}$-type or vice versa) for different chemical compositions and defects in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$. Our research on the above-stated points ((i) and (ii)) can lead us to understand how TI fit properly and consistently within the current knowledge of atoms, metallic Fermi liquid, semiconductors, ferromagnets, Mott-Hubbard insulators, cuprates strange metals and quantum Hall metals. In addition, these points were not addressed nor generalized in earlier reports with respect to different types of atoms, and with respect to non-Kramers degeneracy (crossed energy levels with $\\xi \\neq 0$). \n\n\\section*{2. Ionization energy theory}\n\nIonization energy is defined as the minimum energy required to remove a bound electron (bounded to a nucleus of an atom) to infinity ($\\textbf{r} \\rightarrow \\infty$). In contrast, IET is based on the theorem that states---the minimum energy required to excite or to polarize a bound electron to a finite distance ($\\textbf{r} \\rightarrow \\textbf{r}_{\\rm finite}$) within a quantum system (atomic, molecular or solid) is proportional to the ionization energy defined above. This proportionality is precisely known as the ionization energy approximation~\\cite{arul1,arul2}, which can be written as,\n\\begin {eqnarray}\n\\xi^{\\rm quantum}_{\\rm system} \\propto \\xi^{\\rm constituent}_{\\rm atom}. \\label{eq:1}\n\\end {eqnarray}  \nWithin a quantum system, the energy is quantized and therefore, the ionization energy is also known as the energy level spacing. This energy level spacing can be used to rewrite the standard Schr${\\rm \\ddot{o}}$dinger equation to read~\\cite{arul3,arul4},\n\\begin {eqnarray}\n&&{\\rm i}\\hbar\\frac{\\partial \\Psi(\\textbf{r},t)}{\\partial t} = \\bigg[-\\frac{\\hbar^2}{2m}\\nabla^2 + V_{\\rm IET}\\bigg]\\Psi(\\textbf{r},t) \\nonumber \\\\&& = H_{\\rm IET}\\Psi(\\textbf{r},t) = (E_0 \\pm \\xi)\\Psi(\\textbf{r},t), \\label{eq:2}\n\\end {eqnarray}  \nwhich is known as the IET-Schr${\\rm \\ddot{o}}$dinger equation where $H_{\\rm IET}$ and $V_{\\rm IET}$ are the exact Hamiltonian and potential term, respectively. The properties of electrons are represented by the true and real (not a guessed) wavefunction, $\\Psi(\\textbf{r},t)$, $\\hbar$ is the Planck constant divided by 2$\\pi$ and $m$ denotes electron mass. Here $E_0$ is the ground state energy for temperature ($T$) equals zero Kelvin, in the absence of external disturbances. The sign, $\\pm$ refers to electrons and holes, respectively. The above generalized definitions for $H_{\\rm IET}$ and $V_{\\rm IET}$ imply Eq.~(\\ref{eq:2}) is too general to capture a specific quantum matter. One way to solve Eq.~(\\ref{eq:2}) is to rewrite it in the usual form, \n\\begin {eqnarray}\n\\bigg[-\\frac{\\hbar^2}{2m}\\nabla^2 + V\\bigg]\\Psi(\\textbf{r},t) = H\\Psi(\\textbf{r},t) = E\\Psi(\\textbf{r},t), \\label{eq:3}\n\\end {eqnarray}  \nafter which, one can incorporate the relevant potential terms ($V$) with appropriate approximations, and make do with the guessed wavefunction, satisfying the variational principle to find the energy eigenvalues. Here, we will not follow this approach. Instead, we make use of Eqs.~(\\ref{eq:1}) and~(\\ref{eq:2}) to tackle a specific quantum matter. For example, $\\xi$ will furnish one with the details on the electron excitation probabilities~\\cite{arul3} for a given material with different chemical compositions~\\cite{arul1,arul2}. These probabilities can be properly incorporated into the relevant physical parameters (namely, carrier density, electron-ion interaction term and carrier-type transition) by means of the energy-level spacing renormalization group method~\\cite{arul4}. Our renormalization procedure will renormalize the above physical parameters such that they can be used to determine the microscopic mechanisms responsible for doping-dependent electrical resistivity and carrier-type transition in TI. A specific potential term in Eq.~(\\ref{eq:2}) has been exploited earlier in Refs.~\\cite{qpt,ion}. Note this, one can also reprocess the energy-level spacing cut-off parameter exactly within the Shankar's wavenumber-dependent renormalization technique~\\cite{shank,shank2,shank3}. For an obvious reason (see below), we prefer to work with the energy-level spacing. \n\n\\subsubsection*{2.1. Ionization energies for isolated atoms}\n\nWe now list the chemical elements needed to make TI, calculate their average ionization energies and explain what these averaged values mean. For an hypothetical atom, $\\texttt{X}^{i+}$ (with $i$ number of valence electrons) the averaged $\\xi$ can be calculated from,\n\\begin {eqnarray}\n\\xi^{\\rm constituent}_{\\rm atoms} = \\sum_j\\sum_i^z \\frac{1}{z}\\xi_{j,i}(\\texttt{X}^{i+}_{j}). \\label{eq:4}   \n\\end {eqnarray}\nHere, the subscript $j$ identifies the different types of chemical elements ($\\texttt{X}_j$) in a given TI, while the subscript, $i = 1, 2, \\cdots, z$, counts the number of outer electrons required to interact with its nearest and next nearest neighbors by forming bonds. We list all the essential chemical elements required to form TI in Table~1. Before averaging, the experimental ionization energy values for each outer electron (1$^{\\rm st}$, 2$^{\\rm nd}$, and so on) were obtained from Refs.~\\cite{web,web2}. This is the only external information one needs to proceed. Table~1 also lists the averaged ionization energy values calculated from Eq.~(\\ref{eq:4}) with respect to the stated valence states.    \nWe now briefly state the importance of approximating $\\xi^{\\rm quantum}_{\\rm matter}$ from $\\xi^{\\rm constituent}_{\\rm atom}$ ($\\xi$ for short) via Eqs.~(\\ref{eq:1}) and~(\\ref{eq:4}). A chemical element, $\\texttt{X}_j$ with large $\\xi(\\texttt{X}_j)$ means its valence electrons are not easily excited or polarized, and conversely, the electron-excitation probability and the atomic polarizability of a chemical element ($\\texttt{X}_{j+1}$) are large if its $\\xi(\\texttt{X}_{j+1})$ is small. Furthermore, one can readily use Eq.~(\\ref{eq:1}) to claim that the above probability and polarizability for a given TI get smaller if we systematically substitute one of its constituent chemical elements ($\\texttt{X}_j$, $\\xi(\\texttt{X}_j)$) with another atom ($\\texttt{X}_{j+1}$) that has a larger $\\xi(\\texttt{X}_{j+1})$. The above probability can be derived from the Fermi-Dirac statistics (FDS) with $\\xi$ as an additional restrictive condition~\\cite{arul1,arul2}, and is given by,\n\\begin{eqnarray}\nf(E_0,\\xi) = \\frac{1}{e^{\\lambda[(E_0 + \\xi) - E_{\\rm F}^{0}]}+1}, \\label{eq:5}\n\\end{eqnarray} \nwhere $\\lambda = (12\\pi\\epsilon_0\/e^2)a_{\\rm B}$, $\\epsilon_0$ is the permittivity of free space, $a_{\\rm B}$ is the Bohr radius and $E_{\\rm F}^{0}$ denotes the Fermi level for $T = 0$K and without any external disturbances. Equation~(\\ref{eq:5}) is also known as the ionization energy based FDS ($i$FDS). The atomic polarizability (due to outer electron displacement) can be determined from~\\cite{arul5,arul6},\n\\begin {eqnarray}\n\\alpha(\\xi) = \\sum_j\\sum_i\\frac{Z_je^2}{m}\\exp\\big[\\lambda(E_{\\rm F}^0 - \\xi_i)\\big]\\frac{f_i}{(\\omega_{0i}^2 - \\omega^2)}, \\label{eq:6}\n\\end {eqnarray}  \n$\\omega_{0i}$ is the $i^{\\rm th}$ electron frequency, $Z_j$ is the atomic number of $j^{\\rm th}$ atom, while $f_i$ denotes the strength factor of an $i^{\\rm th}$ polarizable electron in a given atom. Equation~(\\ref{eq:6}) is obtained by neglecting the spontaneous emission (or classically known as the damping factor)~\\cite{arul6}. Apparently, both $f(E_0,\\xi)$ and $\\alpha(\\xi)$ are inversely proportional to $\\xi$. However, IET is inapplicable for free-electron (due to $\\xi = 0$) and Fermi liquid metals (because $\\xi$ is an irrelevant constant), which imply IET is only useful for quantum systems with `relevant' $\\xi$ such that their Hamiltonians can be written in the form of Eq.~(\\ref{eq:2}).     \n\n\\section*{3. Time reversal symmetry}\n\nWe have stated earlier in the introduction that a time-independent internal electric current induces static `external' magnetic field, which actually violates TRS. Here, we will prove that TRS is indeed broken in the presence of `internal' electric current, and in the absence of applied magnetic field as correctly hypothesized by Messiah~\\cite{albert}. Here, the `applied' magnetic field is defined to originate from a source external to the quantum system under investigation, whereas, the `external' magnetic field originates from a constant internal electric current within the observed quantum system. In order to construct the logical proof for this violation, we first need to expose the properties of time reversal operation properly and correctly, going beyond the expositions presented in Ref.~\\cite{albert}. Of course, the mathematical arguments and definitions invoked herein must always be compatible with the physical observations. In the subsequent paragraphs, we will derive the necessary equations and definitions required for a proper time-reversal operation, which then can be used to tackle TRS head-on, and to develop our logical proof on TRS violation due to static internal electric current.\n\n\\subsubsection*{3.1. Time reversal operation for electrons}\n\nTime reversal symmetry in quantum mechanical systems deal with the dynamical state represented by the time-dependent wavefunction. The energy eigenvalue, $E$ obtained from the Schr${\\rm \\ddot{o}}$dinger equation (see Eq.~(\\ref{eq:2}) or~(\\ref{eq:3})) is unique because the wavefunction, $\\Psi(\\textbf{r},t)$ is also unique such that two different wavefunctions, $\\Psi(\\textbf{r},t)$ and $\\Psi'(\\textbf{r},t)$ are not allowed to be the solutions to the same Schr${\\rm \\ddot{o}}$dinger equation simultaneously with the same eigenvalue where $\\Psi(\\textbf{r},t) \\neq \\Psi'(\\textbf{r},t)$. This is a physical requirement. However, TRS requires the time-reversed wavefunction, $\\Psi(\\textbf{r},t)^{\\rm rev}$ to be another valid solution to the same Schr${\\rm \\ddot{o}}$dinger equation, with the same eigenvalue where $\\Psi(\\textbf{r},t)^{\\rm rev} \\neq \\Psi(\\textbf{r},t)$. This inequality ($\\Psi(\\textbf{r},t)^{\\rm rev} \\neq \\Psi(\\textbf{r},t)$) is in violation of the above physical requirement (a unique wavefunction corresponds (one-to-one) to a unique eigenvalue, or precisely $\\Psi(\\textbf{r},t) \\neq \\Psi'(\\textbf{r},t)$ is not allowed). Therefore, the only option we have, which allows both $\\Psi(\\textbf{r},t)^{\\rm rev}$ and $\\Psi(\\textbf{r},t)$ as distinct solutions to the same Schr${\\rm \\ddot{o}}$dinger equation (that gives the same eigenvalue), and satisfies the requirement, $\\Psi(\\textbf{r},t)^{\\rm rev} \\neq \\Psi(\\textbf{r},t)$ is to define $\\Psi(\\textbf{r},t)^{\\rm rev} = \\Psi^*(\\textbf{r},-t)$. We cannot define $\\Psi(\\textbf{r},t)^{\\rm rev} = \\Psi(\\textbf{r},-t)$ because by definition, $\\Psi(\\textbf{r},-t) = \\Psi(\\textbf{r},t)$. Having said that, we can now easily show that $\\Psi^*(\\textbf{r},-t)$ is also a solution to the same Schr${\\rm \\ddot{o}}$dinger equation by taking the complex conjugate (both sides) of Eq.~(\\ref{eq:3}) (after replacing $t$ with $-t$) to arrive at, \n\\begin {eqnarray}\n{\\rm i}\\hbar\\frac{\\partial}{\\partial t}\\Psi^*(\\textbf{r},-t) = H\\Psi^*(\\textbf{r},-t) = E\\Psi^*(\\textbf{r},-t). \\label{eq:7}\n\\end {eqnarray}  \nAs a consequence of the above complex conjugation, we can define the time reversal operator, $\\texttt{K}$ properly by writing $\\Psi(\\textbf{r},t)^{\\rm rev}$ in the form, \n\\begin {eqnarray}\n\\Psi(\\textbf{r},t)^{\\rm rev} = \\texttt{K}\\Psi(\\textbf{r},-t)\\texttt{K}^* = \\Psi^*(\\textbf{r},-t),\\label{eq:8}\n\\end {eqnarray}  \nwhich means that the time reversal operation, in this case, is a form of complex conjugation~\\cite{albert} because the phase factor is time-dependent, and it is not a constant. This operation will leave the real numbers (position, $\\textbf{r}$) and real operators ($Q$) alone ($\\texttt{K}\\textbf{r}Q\\texttt{K}^{*} = \\textbf{r}Q$). In contrast, if $\\texttt{K}$ operates on complex numbers or operators, then $\\texttt{K}$ changes the sign in the following way, $\\texttt{K} {\\rm i}\\texttt{K}^{*} = -{\\rm i}$. Since $\\texttt{K}$ is a complex conjugation operator, one can readily surmise that $\\texttt{K} = \\texttt{K}^{*}$. \n\nFor an electron however, $\\texttt{K}$ is insufficient because an electron is a spin-$\\frac{1}{2}$ particle such that we also need to invert the spin (or change its sign) where each spin component ($\\textbf{S}_x$, $\\textbf{S}_y$ and $\\textbf{S}_z$) should be inverted. This means that, we need a new time reversal operator, $\\texttt{K}_{\\textbf{S}}$, to have the following essential property, \n\\begin {eqnarray}  \n\\frac{\\hbar}{2}\\texttt{K}_{\\textbf{S}}\\sigma_{x,y,z}\\texttt{K}_{\\textbf{S}}^{\\dagger} = -\\frac{\\hbar}{2}\\sigma_{x,y,z},\\label{eq:9}\n\\end {eqnarray}  \nwhere $\\textbf{S}_{x,y,z} = \\frac{\\hbar}{2}\\sigma_{x,y,z}$, $\\sigma_{x,y,z} = \\sigma_{x}, \\sigma_{y}, \\sigma_{z}$, denote the Pauli matrices and $\\texttt{K}_{\\textbf{S}}^{\\dagger}$ is an antiunitary Hermitian conjugate. This inversion (sign change) due to time reversal operation is mandatory because the spin is intrinsically undefined (the spin is not static). From Eq.~(\\ref{eq:8}), $\\texttt{K}$ is a complex conjugation operator, and therefore, it cannot invert the real spin components ($\\sigma_{x}$ and $\\sigma_{z}$), which explains why we need a new time-reversal operator for electrons that has to be defined in a different context. This new operator, $\\texttt{K}_{\\textbf{S}}$ has to be a 2$\\times$2 matrix such that $\\texttt{K}_{\\textbf{S}}$ also needs to invert the real Pauli matrices, namely, $\\sigma_x$ and $\\sigma_z$. This new time-reversal operator for electrons has been defined by Messiah~\\cite{albert},\n\\begin {eqnarray}  \n\\texttt{K}_{\\textbf{S}} = -{\\rm i}\\sigma_y\\texttt{K}, \\label{eq:10} \n\\end {eqnarray}  \nusing a phase-factor type rotation operator. However, we still need to define $\\texttt{K}_{\\textbf{S}}^2$ properly, such that i in Eq.~(\\ref{eq:10}) is left intact because it has a precise physical meaning with respect to spin via the rotation operator. We also need to know $\\texttt{K}_{\\textbf{S}}^2$ so that we can use it to generalize and to understand the time reversal operation for even and odd number of electrons. In particular, we can readily define $\\texttt{K}_{\\textbf{S}}^2 = -{\\rm i}\\sigma_y\\texttt{K}(\\sigma_{x,y,z}\\psi(\\textbf{r},t)){\\rm i}\\sigma_y\\texttt{K}^*$ where we also have defined an arbitrary spin($\\sigma$)-wavefunction($\\Psi$), $\\Psi(\\sigma,\\textbf{r},t) = \\sigma_{x,y,z}\\psi(\\textbf{r},t)$, which is a complex number. One can readily recall the following useful identities, $\\sigma_y\\sigma_x\\sigma_y = -\\sigma_x$, $\\sigma_y\\sigma_y\\sigma_y = -\\sigma_y$, $\\sigma_y\\sigma_z\\sigma_y = -\\sigma_z$, $\\sigma_y\\sigma_y = \\sigma_z\\sigma_z = \\sigma_x\\sigma_x = \\textbf{\\textsl{I}}_2$, $\\det$($\\textbf{\\textsl{I}}_2$) = 1 and $\\texttt{K}\\texttt{K}^* = \\texttt{K}^2 = 1$. For a single electron, $\\texttt{K}_{\\textbf{S}}^2 = -{\\rm i}\\sigma_y(-\\sigma_{x,y,z})\\psi^*(\\textbf{r},-t)(-{\\rm i})(-\\sigma_y)\\texttt{K}\\texttt{K}^* = -1\\sigma_{x,y,z}\\psi^*(\\textbf{r},-t)$. For two electrons, $\\texttt{K}_{\\textbf{S}}^2 = [-1\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)][-1\\sigma_{2;x,y,z}\\psi^*_2(\\textbf{r},-t)] = [\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)][\\sigma_{2;x,y,z}\\psi^*_2(\\textbf{r},-t)]$. Therefore, for $n$ number of electrons, \n\\begin {eqnarray}  \n&&\\texttt{K}_{\\textbf{S}}^2 = (-1)^n\\big[\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)\\big]\\big[\\sigma_{2;x,y,z}\\psi^*_2(\\textbf{r},-t)\\big]\\cdots\\big[\\sigma_{n;x,y,z}\\psi^*_n(\\textbf{r},-t)\\big]. \\nonumber \\\\&& \\label{eq:11} \n\\end {eqnarray}  \nFor even $n$, we can rewrite Eq.~(\\ref{eq:11}) (recall the conditions used to derive Eq.~(\\ref{eq:7})), \n\\begin {eqnarray}\n[\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)][\\sigma_{2;x,y,z}\\psi^*_2(\\textbf{r},-t)] \\cdots &\\longrightarrow& [\\psi^{\\rm real}_1(\\textbf{r},-t)][\\psi^{\\rm real}_2(\\textbf{r},-t)] \\cdots \\nonumber \\\\&& = [\\psi^{\\rm real}_1(\\textbf{r},t)][\\psi^{\\rm real}_2(\\textbf{r},t)] \\cdots, \\nonumber \\\\&& \\label{eq:12} \n\\end {eqnarray}\nwhere we have dropped the spin components (or the Pauli matrices) because real wavefunctions cannot be defined as $\\Psi(\\sigma,\\textbf{r},t) = \\sigma_{x,y,z}\\psi(\\textbf{r},t)$. This means that, a real wavefunction needs to transform in this way, $\\texttt{K}_{\\textbf{S}}\\psi^{\\rm real}_1(\\textbf{r},t)\\texttt{K}^{\\dagger}_{\\textbf{S}} = \\psi^{\\rm real}_1(\\textbf{r},t)$. Moreover, each electron in a system with even number of electrons (see Eq.~(\\ref{eq:12})) can be represented by a real wavefunction, $\\psi^{\\rm real}_1(\\textbf{r},t), \\cdots, \\psi^{\\rm real}_n(\\textbf{r},t)$. Therefore, to satisfy TRS for $n = n_{\\rm even}$, we can either use real $\\psi^{\\rm real}_n(\\textbf{r},t)$ or complex wavefunctions, $\\sigma_{n;x,y,z}\\psi_n(\\textbf{r},t)$. If we were to use complex wavefunctions, then their eigenvalues must be at least doubly degenerate (see the following paragraph). \n\nOn the other hand, for odd number of electrons, $\\texttt{K}_{\\textbf{S}}\\sigma_{1;x,y,z}\\psi_1(\\textbf{r},t)\\texttt{K}^{\\dagger}_{\\textbf{S}} = -\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)$ where $-\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)$ is orthogonal to $\\sigma_{1;x,y,z}\\psi_1(\\textbf{r},t)$ and consequently, the eigenvalue for $-\\sigma_{1;x,y,z}\\psi^*_1(\\textbf{r},-t)$ and $\\sigma_{1;x,y,z}\\psi_1(\\textbf{r},t)$ is (doubly) degenerate. Contrary to even number of electrons, odd $n$ strictly require complex wavefunctions to satisfy TRS, and for $n_{\\rm odd} > 1$, the eigenvalue is at least two-fold degenerate (with even order) due to Kramers degeneracy. In summary, by definition TRS requires gapless (degenerate) energy levels. Add to that, a gapless electronic system with even number of electrons can be represented by a set of real or complex wavefunctions to satisfy TRS. On the other hand, the same system with odd number of electrons needs complex wavefunctions so as not to violate TRS.\n\n\\subsubsection*{3.2. Time reversal violation due to electric current}\n\nFrom the preceding subsections, we have formally and properly shown that any form of electron excitation, from the valence to conduction band violates TRS such that degenerate energy levels are mandatory for TRS---because we also need to independently time-reverse the photon that is responsible for the said excitation. Here, we claim that TRS is violated if there is a static internal electric current within a gapless quantum system. This is not a trivial claim, which exists only as a hypothesis due to Messiah~\\cite{albert}, and therefore, we need some logical arguments to understand it properly. We consider a one-dimensional system for convenience, and using an arbitrary wavefunction, $\\Psi(x,-t)$ where $\\Psi_{\\rm rev}(x,t) = \\Psi^*(x,-t)$ (from Eq.~(\\ref{eq:8})), one can write the flow of probability in this form,\n\\begin {eqnarray}\n\\frac{\\partial}{\\partial t}|\\Psi(x,-t)|^2 = \\Psi^*(x,-t)\\frac{\\partial\\Psi(x,-t)}{\\partial t} + \\frac{\\partial\\Psi^*(x,-t)}{\\partial t}\\Psi(x,-t) \\neq 0,  \\label{eq:13} \n\\end {eqnarray}\nwhere $\\Psi^*(x,-t)$ is a complex conjugate of $\\Psi(x,-t)$, and $-t$ is kept explicit for we are attempting to derive the time reversed probability current, $\\textbf{J}_{\\rm rev}$. We can now rewrite Eq.~(\\ref{eq:3}) to obtain,\n\\begin {eqnarray}\n\\frac{\\partial \\Psi^*(x,-t)}{{\\rm d}t} = \\frac{{\\rm i}\\hbar}{2m}\\bigg[\\frac{\\partial^2\\Psi^*(x,-t)}{\\partial x^2} - \\frac{{\\rm i}}{\\hbar}V\\Psi^*(x,-t)\\bigg].  \\label{eq:14} \n\\end {eqnarray}\nTaking the complex conjugate of Eq.~(\\ref{eq:14}), and using Eq.~(\\ref{eq:13}), we arrive at the time reversed probability current,\n\\begin {eqnarray}\n\\textbf{J}_{\\rm rev} = -\\frac{{\\rm i}\\hbar}{2m}\\bigg[\\frac{\\partial \\Psi(x,-t)}{\\partial x}\\Psi^*(x,-t) - \\Psi(x,-t)\\frac{\\partial \\Psi^*(x,-t)}{\\partial x}\\bigg]. \\label{eq:15} \n\\end {eqnarray}\nNow, using Eq.~(\\ref{eq:3}) directly, we can derive the forward probability current,\n\\begin {eqnarray}\n\\textbf{J}_{\\rm fwd} = \\frac{{\\rm i}\\hbar}{2m}\\bigg[\\Psi(x,t)\\frac{\\partial \\Psi^*(x,t)}{\\partial x} - \\frac{\\partial \\Psi(x,t)}{\\partial x}\\Psi^*(x,t)\\bigg], \\label{eq:16} \n\\end {eqnarray}\nor alternatively, Eq.~(\\ref{eq:16}) can also be obtained from $\\texttt{K}_{\\textbf{S}}\\textbf{J}_{\\rm rev}\\texttt{K}^{\\dagger}_{\\textbf{S}}$ after replacing $-t$ with $t$ and $\\texttt{K}_{\\textbf{S}}(\\partial \\Psi^*(x,t)\/\\partial x)\\texttt{K}^{\\dagger}_{\\textbf{S}} = \\partial \\texttt{K}_{\\textbf{S}}\\Psi^*(x,t)\\texttt{K}^{\\dagger}_{\\textbf{S}}\/\\partial x$. Next, we invoke one of the Maxwell equations within matter in the presence of macroscopic static internal current density (current per unit length, $\\textbf{i}$),\n\\begin {eqnarray}\n\\nabla \\times \\textbf{B}' = \\mu_{\\rm perm}\\textbf{i}_{\\rm fwd} \\propto \\mu_{\\rm perm}\\textbf{J}_{\\rm fwd}, \\label{eq:17} \n\\end {eqnarray}\nwhere $\\mu_{\\rm perm}$ is the permeability constant and the term, `static current' here of course implies $\\partial \\textbf{D}\/\\partial t = 0$ in which, $\\textbf{D}$ is the usual time-dependent electric charge displacement. Equation~(\\ref{eq:17}) for $\\textbf{i}_{\\rm rev}$ is given by,\n\\begin {eqnarray}\n\\nabla \\times \\textbf{B}'' \\propto \\mu_{\\rm perm}\\textbf{J}_{\\rm rev}, \\label{eq:18} \n\\end {eqnarray}\nwhere $\\textbf{B}'$ is the induced external magnetic field due to $\\textbf{i}_{\\rm fwd}$, while $\\textbf{B}''$ is induced by $\\textbf{i}_{\\rm rev}$ such that $\\textbf{B}'$ is or is not equal to $\\textbf{B}''$, and it does not matter. The point is, both $\\textbf{B}'$ and $\\textbf{B}''$ are independent to each other (they have independent sources, $\\textbf{i}_{\\rm fwd}$ and $\\textbf{i}_{\\rm rev}$), they are not zero ($\\textbf{i}_{\\rm fwd}$ and $\\textbf{i}_{\\rm rev}$ are not zero) and they are not `applied' magnetic fields ($\\textbf{i}_{\\rm fwd}$ and $\\textbf{i}_{\\rm rev}$ are the currents within the system). Therefore, when the current is time-reversed, we also need to independently reverse $\\textbf{B}'$. The requirement to time-reverse $\\textbf{B}'$ independently when the current ($\\textbf{i}_{\\rm fwd}$) is reversed strictly means that TRS will be violated if we reverse only the current. In summary, TRS is guaranteed to be violated even in the presence of static (constant) electric current, which also induces the static `external' magnetic field. Here, we do not require any applied magnetic field to violate TRS, as correctly hypothesized by Messiah~\\cite{albert}.\n\n\\textit{Warning}: One should be careful here to understand the points stated above on TRS based on Eqs.~(\\ref{eq:17}) and~(\\ref{eq:18}). For example, we cannot time-reverse $\\textbf{B}'$ in order to reverse the current, $\\textbf{i}_{\\rm fwd}$ because the time reversed $\\textbf{B}'$ will not induce its own current, in agreement with our analyses above. If it induces its own current, $\\textbf{i}_{\\rm rev}$, then $\\textbf{B}' = \\textbf{B}''$, which means $\\textbf{B}''$ exactly cancels $\\textbf{B}'$, and this cancellation symmetry has got nothing to do with our time reversal operation. This is why physical time reversal operation is entirely different from the philosophical one where the latter also implies going back in time by deleting the future. In particular, if we reverse $\\textbf{B}'$ by deleting (or canceling) $\\textbf{B}'$, then $\\textbf{i}_{\\rm fwd}$ is also deleted simultaneously, without the need to delete $\\textbf{i}_{\\rm fwd}$ independently. But the time reversal operation by erasing the future is never allowed in our universe (due to the second law of thermodynamics), and this difference (between the physical and philosophical time reversal operations) is the root cause for the confusion in TRS.    \n\n\\subsubsection*{3.3. Relevant and irrelevant energy-level spacing}\n\n`Relevant' energy level spacing means a particular physical property is directly influenced by $\\xi$, while an `irrelevant' $\\xi$ and $\\xi = 0$ have no influence at all. Examples of trivially relevant energy level spacings~\\cite{arul7} are the atomic energy level spacings~\\cite{web,web2}, band gaps in band insulators and semiconductors~\\cite{ash}, Mott-Hubbard gaps in Mott insulators~\\cite{mott}, and the molecular gaps in molecules~\\cite{ira} (between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO)). All Fermi liquid metals~\\cite{ash} (including quantum Hall metals~\\cite{klit,tsui}) require irrelevant $\\xi$ ($\\xi_{\\rm irr} \\neq 0$)~\\cite{and1}, while $\\xi = 0$ is for free electrons (Fermi gas)~\\cite{arul7}. However, there is a special situation that can give rise to a nontrivially relevant $\\xi$, which is applicable to strange metals in cuprates~\\cite{arul7}, and other strongly correlated metals where $\\xi$ is `special' because it exists within a degenerate electronic system, and its origin is due to different wavefunctions~\\cite{arul8}. In particular, the special energy level spacing theorem reads~\\cite{arul7,arul8}---an electron needs energy supply to occupy another degenerate energy level because the wavefunction representing this particular electron needs to be transformed (beyond the phase factor) to occupy the new energy level. \n\nIn summary, from our previous analyses on TRS, it should be clear now that TRS will be violated in the presence of static electric current regardless of whether $\\xi = 0$, or $\\xi$ is an irrelevant constant, or $\\xi$ is a finite relevant constant. This also means that any violation of TRS does not necessarily imply a given system is gapped because internal electric current has been shown to violate TRS even in gapless systems. Apart from that, even though TRS is known to be responsible for Kramers degeneracy, TI can form non-Kramers degenerate surface states due to many other complicated interactions as pointed out in the introduction, and in Refs.~\\cite{arul7,arul8}.       \n\n\\section*{4. Results and discussion} \n\nWe show here that the metallic property in gapless TI, can be associated to the above special (nontrivially relevant) energy level spacing ($\\xi^{\\rm non}_{\\rm triv}$). Usually, a given compound is defined to be metallic if its resistivity ($\\rho(T)$) gets smaller with lowering the temperature $T$. Both $\\rho(T, x)$ and the carrier-type transition ($\\texttt{n}$- to $\\texttt{p}$-type) have been measured by Jinsong Zhang \\textit{et al}.~\\cite{jins} in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ TI. Apart from metallic resistivity, we will also explain why the bandgap of another TI, namely, Pb$_{1-x}$Sn$_x$Se changes with Sn substitution, which were first measured by Strauss~\\cite{strau} and later confirmed by Dziawa \\textit{et al.}~\\cite{dzia} using ARPES measurements. The latter bandgap analyses are based on the trivially relevant $\\xi_{\\rm triv}$. Since the energy level spacings for both gapless and gapped systems are nonzero and relevant, we are therefore forced to identify them with $\\xi^{\\rm non}_{\\rm triv}$ and $\\xi_{\\rm triv}$ for gapless and gapped systems, respectively.\n\n\\subsubsection*{4.1. Carrier density and electron-ion interaction} \n\nChanges in the magnitude of resistivity in TI or any other strongly correlated solid state systems are usually due to doping $x$, $T$ and $\\textbf{B}$. In the absence of $\\textbf{B}$, and for constant $T$, the microscopic physical parameters that will significantly influence the resistivity with respect to $x$ are the carrier density ($n(\\xi)$) and the scattering amplitude, $|f(\\theta)|$. Here~\\cite{arul2,arul5},  \n\\begin {eqnarray}\nn(\\xi) = \\texttt{C}(T)\\exp{\\big[\\lambda(\\xi - E_{\\rm F}^0)\\big]}, \\label{eq:19} \n\\end {eqnarray}\nwhere $\\texttt{C}(T)$ denotes a collection of fundamental constants, which actually depends on the dimensionality of the system and $\\texttt{C}(T)$ also contains a $T$-dependent parameter. The resistivity,\n\\begin {eqnarray}\n\\rho(x) = \\frac{m}{ne^2\\tau} = \\texttt{C}'\\tau(x)^{-1}\\exp{\\big[\\lambda(\\xi - E_{\\rm F}^0)\\big]}, \\label{eq:20} \n\\end {eqnarray}\nwhere we have suppressed the $T$ dependence because $\\rho(x)$ is for constant $T$. Next, we need to find how the scattering rate, $\\tau(x)^{-1}$ vary with doping. For free-electron and Fermi liquid metals, $\\tau(x)^{-1} \\longrightarrow \\tau^{-1}$ by definition~\\cite{arul7} because the electrons are completely independent of the types of ions such that they interact with neutral phonons, not with the positively charged ions. This means that, $\\tau^{-1}$ does not systematically change with the type of atoms in Fermi liquid~\\cite{arul7}. In this case, the scattering rates are mostly contributed by the well-known electron-electron (e:e) and electron-phonon (e:ph) type collisions. On the other hand, in strongly correlated metals, including metallic TI, we have the usual contribution from $\\tau_{\\rm e:e}^{-1}$ and another dominant contribution from the electron-ion (e:ion) scattering rate, $\\tau(x)_{\\rm e:ion}^{-1}$. We show here that $\\tau(x)_{\\rm e:ion}^{-1}$ is proportional to the scattering amplitude, $|f(\\theta)|$.\n\nTo understand why $\\tau(x)_{\\rm e:ion}^{-1} \\propto |f(\\theta)|$, we use the Born approximation~\\cite{davi} and the renormalized screened Coulomb potential~\\cite{arul5},\n\\begin {eqnarray}\nV_{\\rm e:ion} = \\frac{e^2}{4\\pi\\epsilon_0r}\\exp{\\big[-\\mu_{\\rm IET}r e^{-(1\/2)\\lambda\\xi}\\big]}, \\label{eq:21} \n\\end {eqnarray}\nto obtain the scattering amplitude,\n\\begin {eqnarray}\n|f(\\theta)| \\cong \\frac{2me^2}{4\\pi\\epsilon_0\\hbar^2(\\mu^2_{\\rm IET}e^{-\\lambda\\xi} + \\kappa^2)} , \\label{eq:22} \n\\end {eqnarray}\nwhere $\\mu_{\\rm IET}$ is the constant of proportionality, and as required, Eq.~(\\ref{eq:22}) reduces to the Rutherford scattering amplitude (for two-point charges) when $\\xi \\rightarrow \\infty$~\\cite{arul4}, and it also reduces to the Yukawa scattering amplitude if $\\xi \\rightarrow 0$~\\cite{arul4}. Moreover, $\\kappa = |\\textbf{k}' - \\textbf{k}|$ depicts the changes in the wavevector such that $\\textbf{k}'$ and $\\textbf{k}$ point in the incident and scattered directions, respectively, and during the process, the momentum transfer is given by $\\hbar(\\textbf{k} - \\textbf{k}')$. It is obvious from Eq.~(\\ref{eq:22}) that $\\theta$ is the angle due to scattering between the directions, $\\textbf{k}$ and $\\textbf{k}'$. In addition, Eq.~(\\ref{eq:22}) leads us straight to $|f(\\theta)| \\propto \\xi$ that implies large scattering amplitude is obtained for an ion (scattering center) with large $\\xi$, which in turn justifies the correctness of this relation, $\\tau(x)_{\\rm e:ion}^{-1} \\propto |f(\\theta)|$. As a consequence, the approximate resistivity can be written in the form (using Eq.~(\\ref{eq:20})),\n\\begin {eqnarray}\n\\rho(x) \\propto \\frac{\\texttt{C}''\\exp{\\big[\\lambda(\\xi - E_{\\rm F}^0)\\big]}}{\\mu^2_{\\rm IET}e^{-\\lambda\\xi} + \\kappa^2}, \\label{eq:23} \n\\end {eqnarray}\nwhere $\\texttt{C}''$ is another collection of fundamental constants, which includes $\\texttt{C}$ and $\\texttt{C}'$. Firstly, even though Eq.~(\\ref{eq:23}) is an approximate one, but it exactly captures the sought-after effect of changing carrier density and e:ion scattering strength on $\\rho(x)$, and secondly, the equation also shows that $n(\\xi)$ and $\\tau^{-1}(x)$ do not compete with each other with opposite effects as a result of doping or changing chemical composition. This second point is important such that $\\rho(x)$ (for constant $T$) is guaranteed to decrease if one substitutes a chemical element with another one that has a smaller $\\xi$ because small $\\xi$ leads to large $n(\\xi)$ and $\\tau(x)$ (see Eq.~(\\ref{eq:20})). Thus far, we have ignored the $T$-dependence even though we know both $\\xi(T)$ and $\\tau_{\\rm e:ion}(T)$ do exist simply because their explicit forms are unknown due to their complex $T$-dependences. Logically, $\\tau_{\\rm e:ion}$ has to be $T$-dependent, while $\\xi$ has been experimentally proven to be $T$-dependent by Dionicio~\\cite{dio} using the results of Fukuda \\textit{et al.}~\\cite{fuku}. In particular, Dionicio showed that the valence state of a given multivalent chemical element is $T$-dependent. But anyway, Eq.~(\\ref{eq:23}) is exact with respect to $\\xi$-dependence, and therefore, it is sufficient for our analyses, and to show that indeed non-Kramers degeneracy (due to relevant $\\xi^{\\rm non}_{\\rm triv} \\neq 0$) is responsible for the metallic surface states in TI.      \n\n\\subsubsection*{4.2. Doping-dependent resistivity and carrier-type transition}\n\nWe are basically done deriving all the equations needed to explain the measured resistivity data and the carrier-type transition in Pb$_{1-x}$Sn$_x$Se and (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ TI. Dziawa \\textit{et al.}~\\cite{dzia} have studied the narrow bandgap semiconductor, Pb$_{1-x}$Sn$_x$Se where a topological phase transition is observed for $x = 0.23$ when $T$ is reduced from 300K to 77K. In particular, Pb$_{0.77}$Sn$_{0.23}$Se is a gapped semiconductor for $T = 300$K, and it becomes a gapless TI for $T = 77$K due to band inversion observed indirectly via ARPES. Moreover, for a given $T$ (300K or 195K or 77K), Sn substitutional doping (increasing $x$ from 0 to about 0.3) systematically reduces the bandgap of Pb$_{1-x}$Sn$_x$Se semiconductor from about 0.3 to approximately 0.05 eV (see Fig.~1 in Ref.~\\cite{dzia}). This observation is easily understood within IET by noting that $\\xi$ now represents the bandgap, $\\xi_{\\rm Sn^{2+}} < \\xi_{\\rm Pb^{2+}}$ (see Table~1), and from Eqs.~(\\ref{eq:19}) and~(\\ref{eq:23}), $\\rho(x)$ is predicted to decrease with increasing Sn content, provided that the valence states of the chemical elements (Pb and Sn) do not change significantly due to defects. If the valence states do change, then the analyses become tedious, which have been addressed elsewhere for other strongly correlated materials and oxides~\\cite{arul7,arul9,arul10} where topological insulators are no exception (see below).\n\nIn the absence of defects, the changes on $\\rho(x)$ for Pb$_{1-x}$Sn$_x$Se have been exposed in a straightforward manner within IET. However, the (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ material is reported to have a complicated doping-dependent resistivity by Jinsong Zhang \\textit{et al.}~\\cite{jins}. For example, the measured $\\rho(x)$ first increases with $x$, for up to $x = 0.94$, and then it reduces until $x = 1$ (see Fig.~4 in Ref.~\\cite{jins}). With increasing Sb content ($0 \\leq x \\leq 0.94$), the system also become relatively more insulating. Further Sb substitutional doping ($0.94 < x \\leq 1$) leads to a weaker insulating behavior such that Sb$_2$Te$_3$ is a semimetal with holes as the dominant charge carriers. In contrast, Bi$_2$Te$_3$ is an electron-dominant metal. The above carrier-type transition from an $\\texttt{n}$-type insulator to a $\\texttt{p}$-type semimetal are of course due to defects where the valence states for Sb$^{3+}$ and Te$^{2+}$ are not constants with doping (for $x > 0.94$). \n\nSimilar to Si = [Si$^{4+}$][Si$^{4+}$] semiconductor, (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ is also predominantly covalent bonded and therefore, we can also write it in the form, [(Bi$_{1-x}$Sb$_x$)$_2$]$^{6+}$[Te$_3$]$^{6+}$ = [(Bi$^{3+}_{1-x}$Sb$^{3+}_x$)$_2$][Te$^{2+}_3$]. Using Eqs.~(\\ref{eq:19}) and~(\\ref{eq:23}), and the fact that $\\xi_{\\rm Bi^{3+}} < \\xi_{\\rm Sb^{3+}}$, one can deduce that $\\rho(x)$ should increase with increasing Sb content, which has been observed for $0 \\leq x \\leq 0.94$. Next, to understand the carrier-type transition in this class of material, it is sufficient for us to focus on these two extreme cases, namely, for $x = 0$ ($\\texttt{n}$-type; Bi$^{3+}_{2}$Te$^{2+}_3$) and for $x = 1$ ($\\texttt{p}$-type; Sb$^{3+}_2$Te$^{2+}_3$). We will exploit the carrier-type transition theorem developed in Ref.~\\cite{arul10}, which will also lead us to understand why $\\rho(x)$ have decreased for $x > 0.94$.              \n\nThe carrier-type transition theorem states that $\\texttt{p}$-type materials with relevant $\\xi$ should satisfy this condition~\\cite{arul10}, \n\\begin {eqnarray} \n\\xi_{\\rm acceptor}^{a+} < \\xi_{\\rm host}^{h+},~~~ a < h ~~~{\\rm and} ~~~x_{\\rm acceptor} < y_{\\rm host}, \\label{eq:24} \n\\end {eqnarray} \nwhere $a$ and $h$ denote the acceptor and host valence states, respectively, while $x_{\\rm acceptor}$ and $y_{\\rm host}$ are the respective concentrations of the acceptor and host chemical elements. The acceptor accepts holes from the host such that the electrons from the acceptor are more easily polarizable than that of the host following Eq.~(\\ref{eq:6}). For example, for [(Bi$^{3+}_{1-x}$Sb$^{3+}_x$)$_2$][Te$^{2+}_3$], the subscripts, $2(1-x)$ and 3 are the concentrations for the host chemical elements, Bi and Te for $x < 0.5$. Now, Bi$^{3+}_{2}$Te$^{2+}_3$ is clearly an $\\texttt{n}$-type system because it does not fulfill the above condition where Te$_3$ is both the host and acceptor, which is not allowed from Eq.~(\\ref{eq:24}), and if this is the case, then Eq.~(\\ref{eq:6}) does not allow the creation of holes~\\cite{arul10}. In other words, the inequality, $\\xi_{\\rm acceptor}^{a+} > \\xi_{\\rm host}^{h+}$ does not exist because $\\xi_{\\rm acceptor; Te}^{2+} = \\xi_{\\rm host; Te}^{2+}$ where the latter equality violates the condition stated in Eq.~(\\ref{eq:24}), and therefore, Eq.~(\\ref{eq:6}) cannot be used to create holes. Given this background, Sb$^{3+}_2$Te$^{2+}_3$ is also predicted to be an $\\texttt{n}$-type material. \n\nHowever, Sb$^{3+}_2$Te$^{2+}_3$ can be made to be a $\\texttt{p}$-type material if we introduce some small amount of defects (with $x_{\\rm new} \\ll 2$ and $y \\ll 3$) to give rise to [Sb$^{3+}_{2-x_{\\rm new}}$Sb$^{a'+}_{x_{\\rm new}}$][Te$^{2+}_{3-y}$Te$^{a''+}_{y}$]. Here, if $x_{\\rm new}$ exists, then $y$ should also exist to balance the defects introduced by $x_{\\rm new}$ so as to maintain a proper coordination number, regardless of the crystal growth conditions because the distribution of these defects need not be homogeneous at all. We now evaluate [Sb$^{3+}_{2-x_{\\rm new}}$Sb$^{a'+}_{x_{\\rm new}}$] and [Te$^{2+}_{3-y}$Te$^{a''+}_{y}$] separately. Apparently, Sb$^{3+}_{2-x_{\\rm new}}$ is the host, and Sb$^{a'+}_{x_{\\rm new}}$ is the acceptor where $\\xi_{\\rm Sb}^{3+} > \\xi_{\\rm Sb}^{a'+}$ if $a' < 3+$. Similarly, for [Te$^{2+}_{3-y}$Te$^{a''+}_{y}$], Te$^{2+}_{3-y}$ is the host, while Te$^{a''+}_{y}$ is the acceptor such that $\\xi_{\\rm Te}^{2+} > \\xi_{\\rm Te}^{a''+}$ if $a'' < 2+$. These inequalities satisfy Eq.~(\\ref{eq:24}), and consequently, Sb$^{3+}_{2-x_{\\rm new}}$Sb$^{a'+}_{x_{\\rm new}}$Te$^{2+}_{3-y}$Te$^{a''+}_{y}$ is a $\\texttt{p}$-type TI in the presence of defects. \n\nMoreover, $\\xi_{\\rm Sb}^{3+} > \\xi_{\\rm Sb}^{a'+}$ and $\\xi_{\\rm Te}^{2+} > \\xi_{\\rm Te}^{a''+}$ give rise to decreasing $\\rho(x)$ for $x > 0.94$, which is in agreement with the observed data (see Fig.~4 in Ref.~\\cite{jins}). The resistivity should decrease for $x > 0.94$ because the carrier-type transition to $\\texttt{p}$-type is caused by the increasing defect density ($x_{\\rm new}$ and $y$), namely, Sb$^{a'+}$ and Te$^{a''+}$ for $x > 0.94$, which give rise to increasing number of holes due to these inequalities, $\\xi_{\\rm Sb}^{3+} > \\xi_{\\rm Sb}^{a'+}$ and $\\xi_{\\rm Te}^{2+} > \\xi_{\\rm Te}^{a''+}$. Using the above analyses, one can also theoretically obtain a $\\texttt{p}$-type Bi$^{3+}_{2}$Te$^{2+}_3$ with appropriate defects, Bi$^{a'+}$ and Te$^{a''+}$ where $a' < 3+$ and $a'' < 2+$. The above defect densities can be determined experimentally by using the chemical method to measure the valence states of various chemical elements that was first carried out by Mahendiran \\textit{et al.}~\\cite{mahen}. \n\nIn summary, both Sb$^{3+}_{2}$Te$^{2+}_3$ and Bi$^{3+}_{2}$Te$^{2+}_3$ can be synthesized as $\\texttt{p}$-type materials for $x_{\\rm new} \\neq 0$ and $y = 0$ or $x_{\\rm new} = 0$ and $y \\neq 0$, without requiring both $x_{\\rm new} \\neq 0$ and $y \\neq 0$ as discussed above. We also have used the so-called relevant $\\xi$ to explain the transport properties (in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se TI) and the $\\texttt{n}$- to $\\texttt{p}$-type carrier-type transition in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$, which unequivocally proves that the Kramers degeneracy is not responsible for the metallic surface states, at least in the above stated TI.           \n\n\\section*{5. Additional notes}\n\nUsing the proven TRS violation result, and the notion of energy-level spacing within IET, we have justified that these topological insulators, namely, (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se, in the presence of surface electric current necessarily violate TRS. Even in the absence of IET, TRS is guaranteed to be violated for any solid state system (gapless or gapped) in the presence of electric current. Furthermore, within IET, we have shown that the degeneracy in topological insulators is not of the Kramers-type. To prove this, we derived the temperature-independent resistivity and scattering-rate equations as functions of doping ($x$). The experimental results obtained from the above materials ((Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se) follow exactly as predicted by these equations (Eqs.~(\\ref{eq:23}) and~(\\ref{eq:24})). This means that, the energy levels are crossed (in gapless systems) in such a way that there are nonzero wavefunction-induced energy gaps at the crossing points. This `special' gap exists due to different orthogonalized wavefunctions.\n\nTo reinforce the above non-Kramers degeneracy in topological insulators, we also provide precise explanation on the physics of $\\texttt{n}$- to $\\texttt{p}$-type carrier-type transition in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$. As a matter of fact, the existence of non-Kramers degeneracy in any system, correlated or not, is not surprising because complicated interactions and\/or their interplay indeed have given rise to crossed energy levels. For example, in Fermi metals, metallic heavy fermions, conventional superconductors, strange metals (cuprates and other strongly correlated metals), metallic ferromagnets (manganites), quantum Hall metals (2-dimensional Fermi gas) and semi-metals.   \n\nOur proof of TRS violation in any condensed matter system in the presence of electric current, directly challenges the analysis carried out by K${\\rm \\ddot{o}}$nig \\textit{et al}.~\\cite{konig} and Bernevig, Hughes and Zhang~\\cite{bern2}. They~\\cite{konig,bern2} claimed that the spin-down and spin-up (spin polarized) edge currents do not violate TRS due to Kramers degeneracy. We have proven that, with or without Kramers degeneracy, TRS will be violated in the presence of spinful electric current. Secondly, the degeneracy in topological insulators is not of the Kramers-type, which are unambiguously supported by the doping-dependent experimental results.   \n\n\\section*{6. Conclusions}\n\nWe have theoretically shown that the time reversal symmetry can be violated for all materials with non-zero energy level spacing, even in the presence of degenerate energy levels. In this case, the degeneracy is not of the Kramers-type due to the existence of nontrivially relevant energy level spacing, $\\xi$. To prove the existence of non-Kramers degeneracy in topological insulators, namely, in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and Pb$_{1-x}$Sn$_x$Se, we have exploited the ionization energy theory to explain their transport properties and the $\\texttt{n}$- to $\\texttt{p}$-type carrier-type transition. In doing so, we have unambiguously shown that the nontrivially relevant $\\xi$ played a pivotal role in the metallic states of topological insulators where Kramers degeneracy is not responsible for gapless system with nonzero and relevant $\\xi$. In particular, we have explained the systematic changes to the band gap and the resistivity in Pb$_{1-x}$Sn$_x$Se TI as a result of $\\xi$, which changes with different chemical compositions (due to changing $x$). In addition, $\\xi$ is also found to be responsible for the $\\texttt{n}$- to $\\texttt{p}$-type carrier-type transition and doping-dependent (temperature-independent) resistivity in (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ TI. These physical phenomena were captured entirely by the nontrivially relevant energy level spacing even in the gapless metallic states of topological insulators. \n\nOne can readily extend the analyses performed herein to other doped topological insulators where compatible dopants can be selected from the periodic table of chemical elements, and after estimating their most probable valence states, one can then calculate their average ionization energies to predict and explain the transport phenomena, even in the presence of defects.         \n\n\\section*{Acknowledgments}\n\nI am grateful to the financiers, Sebastiammal Savarimuthu, Arulsamy Innasimuthu, Amelia Das Anthony, Malcolm Anandraj and Kingston Kisshenraj for their kind support and hospitality between Aug 2011 and Aug 2013. I am also grateful to Marco Fronzi (Osaka City University) for his unconditional help in providing most of the references.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\PARstart{B}{enefiting} from the improvement of computing power and big data, deep learning has achieved unprecedented development in many applications, i.e., speech and audio processing \\cite{wang2018supervised}, natural language processing \\cite{7243232}, object detection \\cite{li2016deepsaliency}, and so on. In recent years, it also achieves dramatic development in the field of wireless communications, e.g., modulation classification \\cite{ali2017automatic}, symbol detection \\cite{Ye2017Power}, end-to-end communication \\cite{anderson2018deepmod}, and mobile edge computing \\cite{huang2019deep}, \\cite{huang2018distributed}, \\cite{huang2019multi}.\n\nDeep learning-based modulation classification automatically and efficiently classify received signals without prior knowledge. Modulation classification is a fundamental step for many applications in wireless communication systems, such as spectrum management in cognitive communication systems \\cite{ali2016advances} and unauthorized signal detection in secure communications \\cite{dobre2007survey}, \\cite{jang2018performance}. Traditional modulation classification method either requires high computational complexity or greatly depends on manual operations \\cite{dobre2007survey}. Recently, deep learning is successfully introduced to classify signals \\cite{triantaris2019automatic}, \\cite{o2018over}, \\cite{peng2018modulation}, \\cite{tang2018digital}, \\cite{ramjee2019fast}, \\cite{rajendran2018deep}, which feeds raw signal data or its transforms into a deep neural network and instantly obtains the modulation category at the network output. It achieves higher classification accuracy than traditional methods for automatic modulation classification based on expert features such as higher order cumulants based features \\cite{o2016convolutional}, while requiring a little extra computational overhead computation time.\n\nAlthough deep learning-based approaches can greatly improve the performance of the modulation classifier, it requires a large volume of training radio samples. However, in practice, collecting a large amount of high quality and reliable training radio samples sometimes is costly and difficult. Data augmentation has been widely used to deal with lack of training data by artificially expanding the training dataset with label preserving transformation. Different data augmentation methods have been proposed in the literature, i.e., random cropping, rotation and mirroring in image classification \\cite{krizhevsky2012imagenet}, \\cite{he2016deep} and pitch shifting, time stretching and random frequency filtering in speech recognition\\cite{salamon2017deep}. For deep learning-based radio modulation classification, data augmentation can improve its invariant, especially for small radio signal dataset.\n\nAugmenting modulated radio signal is similar to augment images as shown in Fig.~\\ref{fig1}. Specifically, we consider three basic augmentation methods, i.e. rotation, flip, and Gaussian noise, for both an image and a quadrature phase-shift keying (QPSK) modulated radio signal sample illustrated in constellation diagram. For the image, after rotation or flip augmentation, the same cat is displayed but from different viewpoints. In the constellation diagram of the QPSK modulated radio signal, the black circles indicate four ideal reference points, and the red crosses are the received symbols which are shifted due to the imperfection of transmitter\/receiver hardware and wireless channel \\cite{sankhe2019oracle}. In Fig.~\\ref{fig1}, we consider two received symbols with positive phase shift (1, 1) and (-1, 1), which are counter-clockwise shifted from their reference points. In wireless communication, each received symbol will be demodulated and mapped to one of the reference points based on the transmitted content. After rotation augmentation, two new symbols (-1, 1) and (-1, -1) are generated as shown Fig.~\\ref{fig1}(b), which are also positively phase shifted. Therefore, for the radio modulation classification task considered in this paper, rotating the modulated radio signal is similar to rotating an image, without losing features for classification. However, flipping the radio signal generates two new QPSK modulated symbols whose phases are negatively shifted in the clockwise direction, as shown in Fig.~\\ref{fig1}(c). Although both rotation and flip augmentation methods achieve similar accuracy improvements for image classification \\cite{taylor2017improving}, \\cite{shijie2017research}, it is an open question about which one is preferred for radio modulation classification. After the Gaussian noise augmentation, the image is full of 'snow' and the received radio symbols are deviated as shown in Fig.~\\ref{fig1}(d). Can all these three augmentation methods improve the classification accuracy for deep learning-based radio modulation classification? To the best of our knowledge, the effect of different data augmentation methods on radio modulation classification has not been evaluated yet.\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig1.pdf}\n\t\\caption{Different data augmentation methods for both image and modulated signal: (a) raw data, (b) rotation, (c) flip, (d) Gaussian noise.} \\label{fig1}\n\\end{figure}\n\nIn this paper, we study data augmentation methods for deep learning-based radio modulation classification. Specifically, a state-of-the-art deep learning-based modulation classifier, is used to automatically classify the modulation category of each radio signal sample. Based on the characteristics of the modulated signal, we study three augmentation methods, i.e., rotation, flip, and Gaussian noise. After extensive numerical evaluations on an open radio signal dataset, we obtain the following contributions:\n\n(1)\tWe propose algorithms to augment radio signals at both training phase and inference phase of the deep learning algorithm, which achieves around 2.5\\% improvement on the baseline in terms of classification accuracy.\n\n(2)\tWe discover that the rotation augmentation method outperforms the flip method, both of which achieve higher classification accuracy than the Gaussian noise method.\n\n(3)\tWe propose a joint augmentation policy with both rotation and flip methods for insufficient training dataset. Given only 12.5\\% of training dataset, the joint augmentation method expands the dataset to be a size of 75\\% of the initial dataset and achieves an even higher classification accuracy than the baseline with 100\\% training dataset without augmentation.\n\n(4) With data augmentation, we successfully classify radio samples by using only one half of the sampling points. Therefore, the deep learning model can be simplified with a significantly reduced inference complexity. Furthermore, in the future field deployment, the modulation category can be successfully classified upon receiving only half number of radio sampling points, which greatly reduces the classification response time.\n\nThe remainder of this paper is organized as follows. Section II presents related work. Section III provides an overview of the studied radio signal dataset and the deep learning-based modulation classifier. We introduce three data augmentation methods in Section IV and propose an algorithm to augment signals at both deep learning phases in Section V. In Section VI, we present the simulation setup and the final experimental results. We finally conclude this paper in Section VII. \n\n\\section{Related Work}\n\\subsection{Deep Learning in Radio Modulation Classification}\nDeep learning has been applied to automatically classify radio modulation categories in recent literature. By converting radio signals into images, two convolutional neural network (CNN)-based deep learning models, GoogleNet \\cite{szegedy2015going} and AlexNet \\cite{krizhevsky2012imagenet}, originally developed for image classification, are used for modulation classification \\cite{peng2018modulation}, \\cite{tang2018digital}. The modulation classification accuracy is further improved by a modified deep residual network (ResNet) \\cite{o2018over}, which is fed with the modulated in-phase (I) and quadrature phase (Q) signals. Considering channel interference, the CNN structure also achieves a considerable classification accuracy \\cite{triantaris2019automatic}. In addition to the CNN-based models, the Long Short-Term Memory (LSTM) architecture with time-dependent amplitude and phase information can achieve the state-of-the-art classification accuracy \\cite{rajendran2018deep}. To reduce the training time of deep learning models, different subsampling techniques are investigated in \\cite{ramjee2019fast} which reduce the dimensions of the input signals.\n\n\\subsection{Data Augmentation in Deep Learning}\nData augmentation is widely used in deep learning algorithms to increase the diversity of training dataset, prevent model overfitting, and improve the robustness of the model. For image classification tasks, generic data augmentation methods include flip, rotation, cropping, color jittering, edge enhancement, and Fancy PCA \\cite{taylor2017improving}. Other complex data augmentation methods synthesize a new image from two training images \\cite{inoue2018data} or from Generative Adversarial Nets (GAN) \\cite{antoniou2017data}. Although there are many augmentation methods for images, AutoAugment \\cite{cubuk2018autoaugment} is proposed to automatically search for augmentation policies based on the dataset. In addition to images, augmentation methods such as synonym replacement, random insertion, random swap, and random deletion are used for text classification \\cite{wei2019eda}, where the same accuracy as normal in all training data is achieved when only half of the training data is available. For speech recognition tasks, training audio is augmented by changing the audio speed \\cite{ko2015audio}, warping features, masking blocks of frequency channels, and masking blocks of time steps \\cite{park2019specaugment}.\n\nThere are few related works on data augmentation for radio modulation classification in the literature. The most related work is a GAN based data augmentation method proposed in \\cite{tang2018digital}. The authors first converted the signal samples into Contour Stellar Images which were further used to train the GAN network so as to generate new signal training samples. With GAN-based augmentation, the modulation classification accuracy is improved by no more than 6\\%. However, training GAN network still requires sufficient signal samples to guarantee the convergence. Moreover, as reported in \\cite{tang2018digital}, the classification accuracy based on augmented dataset is lower than on real dataset with the same amount of signal samples. Therefore, an efficient augmentation method for insufficient radio signal dataset is still absent. \n\n\\section{Preliminaries}\n\n\\begin{figure*}[t!] \n\t\\centering\n\t\\includegraphics[width=7.16 in]{Fig2.pdf}\n\t\\caption{Constellation diagrams of 11 modulated signals \\cite{o2016radio} under different SNRs.}\\label{fig2}\n\\end{figure*}\n\nIn this section, we introduce the radio signal dataset and the architecture of the state-of-the-art LSTM model \\cite{hochreiter1997long}, which will be used to evaluate different data augmentation methods presented in Sec. IV.\n\n\\subsection{Radio Signal Dataset}\nWe evaluate the radio signal modulation classification based on an open radio signal dataset, RadioML2016.10a \\cite{o2016radio}. The radio signals in the dataset consider sample rate offset, center frequency offset, multi-path fading and additive white Gaussian noise. Specifically, there are 220,000 modulated radio signal segments belonging to 11 different modulation categories, i.e., binary phase-shift keying (BPSK), QPSK, eight phase-shift keying (8PSK), continuous phase frequency-shift keying (CPFSK), Gauss frequency-shift keying (GFSK), pulse-amplitude modulation four (PAM4), quadrature amplitude modulation 16 (QAM16), quadrature amplitude modulation 64 (QAM64), double-sideband AM (AM-DSB), single-sideband AM (AM-SSB) and wideband FM (WB-FM). Each radio signal sample is composed of 128 consecutive modulated in-phase (I) signal and quadrature phase (Q) signal. The labels of each signal sample include its value of signal-to-noise ratio (SNR) and its corresponding modulation category. There are total 20 different SNRs ranging from -20dB to 18dB with a step size of 2dB. In the dataset, these 220,000 signal samples are uniformly distributed among 11 modulation categories and 20 SNRs. In other words, there are 1,000 signal samples for each modulation category at each SNR. In Fig.~\\ref{fig2}, we plot examples of 11 modulation categories in forms of constellation diagrams under different SNRs. In the following subsection, we introduce a deep learning algorithm which automatically predicts the radio's modulation category based on its raw I\/Q signals.\n\n\\subsection{LSTM Network Architecture}\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig3.pdf}\n\t\\caption{The architecture of LSTM network.} \\label{fig3}\n\\end{figure}\n\nLSTM is a special category of Recurrent neural network (RNN), which is widely used to process time series data. Benefited from a specific LSTM memory cell mechanism, LSTM effectively solves the exploding and vanishing gradient problem of traditional RNN during training process and learns long-term dependencies in sequential data. The LSTM memory cell mainly consists of a forget gate, an input gate and a update gate \\cite{gers1999learning}, which implement selective retention and discard of input information.\n\nThe LSTM network takes each data sample with consecutive modulated in-phase (I) and quadrature phase (Q) signals as input and maps them to a specific modulation category, whose architecture is shown in Fig.~\\ref{fig3}. Specifically, the modulated I\/Q signals are first converted into amplitudes and phases \\cite{rajendran2018deep}, as:\n\\[\\left\\{ \\begin{array}{l}{\\rm A}{\\rm{ = }}\\sqrt {{I^2} + {Q^2}} \\\\\\phi {\\rm{ = arc}}\\tan ({\\rm{Q}}\/I)\\end{array} \\right.,\\]\nwhere $A$ and $\\phi $ represent the amplitude and phase of the modulated signal, respectively. The obtained signals are then fed into a two-layer LSTM network to extract characteristic features, where each layer has 128 LSTM cells. Finally, a fully connected layer with Softmax function is used to map the radio signal sample to one of these 11 modulation categories. Adam optimizer \\cite{kingma2014adam} with dynamic learning rate is used to minimize the cross-entropy loss as follows:\n\\[\\ell  =  - \\sum\\limits_{k = 1}^K {{y_k}\\log ({{\\hat y}_k})},\\]\nwhere $K$ is the number of classes, ${y_k}$ represents the ground truth label, and ${\\hat y_k}$  denotes the probability that the input sample will be predicted as $k{\\rm{ - th}}$ class.\n\n\\section{Data Augmentation Methods}\n\\label{sec:guidelines}\nData augmentation is a method widely used in deep learning because it improves the generalization ability of the model and alleviates overfitting. In this section, we describe in detail three data augmentation methods for modulation signal recognition, including rotation, flip, and Gaussian noise. The dataset is expanded by a scale factor $N$.\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig4.pdf}\n\t\\caption{Constellation diagram of an QPSK radio signal sample with different data augmentation methods.} \\label{fig4}\n\\end{figure}\n\n\\subsection{Rotation}\nBy rotating a modulated radio signal $(I,Q)$ around its origin, we obtain augmented signal sample $(I',Q')$ as follows:\n\\[\\left[ \\begin{array}{l}\nI'\\\\\nQ'\n\\end{array} \\right] = \\left[ \\begin{array}{l}\n\\cos \\theta \\ \\ {\\rm{  - sin}}\\theta \\\\\n\\sin \\theta \\ \\ \\ \\ \\ {\\rm{  cos}}\\theta \n\\end{array} \\right]\\left[ \\begin{array}{l}\nI\\\\\nQ\n\\end{array} \\right],\\]\nwhere $\\theta$ is the angle of rotation. In this paper, the radio signal is rotated in the counter-clockwise direction by $0$, $\\pi {\\rm{\/2}}$, $\\pi$, and ${\\rm{3}}\\pi {\\rm{\/2}}$. In Fig.~\\ref{fig4}(a), we plot the constellation diagram of an QPSK sample where one set of raw data is augmented into four radio signal samples.\n\n\\subsection{Flip}\nFor a given modulated radio signal $(I,Q)$ , we define the horizontal flip by switching the $I$ value to its opposite, as:\n\\[\\left[ \\begin{array}{l}\nI'\\\\\nQ'\n\\end{array} \\right] = \\left[ \\begin{array}{l}\n- I\\\\\n\\ \\ Q\n\\end{array} \\right],\\]\nand define the vertical flip by switching the $Q$ value to its opposite, as:\n\\[\\left[ \\begin{array}{l}\nI'\\\\\nQ'\n\\end{array} \\right] = \\left[ \\begin{array}{l}\n\\ \\ \\ I\\\\\n- Q\n\\end{array} \\right],\\]\nto augment the radio signals. We can perform horizontal flip, vertical flip, or both flips at the same time such that the signal dataset is expanded by a scale factor $N = 4$, as shown in Fig.~\\ref{fig4}(b).\n\n\\subsection{Gaussian Noise}\nBy adding a Gaussian noise ${\\cal N}(0,{\\sigma ^2})$ to the modulated radio signal $(I,Q)$, we obtain the augmented signal sample $(I',Q')$ as:\n\\[\n\\left[ \\begin{array}{l}\nI'\\\\\nQ'\n\\end{array} \\right] = \\left[ \\begin{array}{l}\nI\\\\\nQ\n\\end{array} \\right] + {\\cal N}(0,{\\sigma ^2}),\n\\]\nwhere ${\\sigma ^2}$ is the variance of noise. In Fig.~\\ref{fig4}(c), we show the augmented signal samples by adding Gaussian noise with different standard deviations $\\sigma {\\rm{ = 0}}$, $\\sigma {\\rm{ = 0.0005}}$ $\\sigma {\\rm{ = 0.001}}$ and $\\sigma {\\rm{ = 0.002}}.$ For each data augmentation method, the original radio signal dataset is expanded by a default scale factor $ N = 4$, as illustrated in Fig.~\\ref{fig4}. Note that the Gaussian noise data augmentation is supposed to significantly expand the dataset by choosing enough different values of $\\sigma$. However, in the next section, we show that the Gaussian noise data augmentation is not preferred for radio data augmentation.\n\n\\section{Data Augmentation Time}\nThe execution of a deep learning algorithm includes training phase and inference phase. Data augmentation can be performed in both phases, resulting in three possible combinations of augmentations, i.e., test-time augmentation, train-time augmentation, and train-test-time augmentation.\n\n\\subsection{Train-time augmentation}\nTrain-time augmentation performs data augmentation during the training stage of the model. That is the training dataset is augmented and expanded by a scale factor $N$ while the test dataset remains the same. Taking the rotation data augmentation as an example, the training dataset is expanded from 110,000 radio signal samples to 440,000 samples after train-time augmentation. In general, a larger size of training dataset leads to a higher modulation classification accuracy.\n\n\\subsection{Test-time augmentation}\nTest-time augmentation fuses features of all augmented radio signal samples in inference phase. In the inference phase, one radio signal sample $(I,Q)$ in the test dataset is augmented into $N$ samples $\\{ {(I',Q')^n}|n \\in N\\}$. Then each augmented sample ${(I',Q')^n}$ is fed into the LSTM network, and we obtain a vector of corresponding predicted probabilities $\\hat y_k^n$. The predicted modulation category is decided through summing the predicted probabilities $\\hat{y}_k^n$ over all $N$ augmented samples and choosing the one with maximum conference \\cite{zheng2019fusion}, as:\n\\[\\mathop {\\arg \\max }\\limits_{1 \\le k \\le K} \\sum\\limits_{n = 1}^N {\\hat{y}_k^n}.\\]\n\n\\subsection{Train-test-time augmentation}\nTrain-test-time augmentation conducts both train-time augmentation and test-time augmentation, where both training and test datasets are augmented and expanded by a factor $N$.\n\nIn Fig.~\\ref{fig5}, we numerically study the performance of the data augmentation at different phases, where the rotation augmentation with a scale factor $N=4$ is considered. Comparing with the baseline without augmentation, augmentations at different phases all improve the classification accuracy when the SNR is greater than -10 dB. The train-time augmentation achieves better performance than test-time augmentation, and the train-test-time augmentation generates the highest accuracy. Specifically, comparing with the baseline, the train-test-time augmentation improves the modulation classification accuracy by 8.87\\% when SNR is -6dB and by about 2.2\\% when SNR is greater than 4 dB. In the following numerical studies, we use the train-test-time augmentation by default.\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig5.pdf}\n\t\\caption{Classification accuracy under different augmentation times.} \\label{fig5}\n\\end{figure}\n\n\\section{Augmentation Performance}\nIn this section, we numerically study the performance of different radio data augmentation methods in terms of modulation classification accuracy. The open dataset, RadioML2016.10a, is divided equally into a training dataset and a test dataset, each containing 110,000 radio signal samples. In order to avoid overfitting, we set dropout rate to be 0.5 at both two LSTM layers. The number of training epoch is 80 and the mini-batch size is 128. The value of the learning rate is initially set as 0.001 and is halved when the training accuracy is not improved during three consecutive epochs. The model is implemented based on PyTorch \\cite{paszke2017automatic}.\n\n\\subsection{Augmentations On Full Dataset}\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig6.pdf}\n\t\\caption{Classification accuracy under different augmentation methods.} \\label{fig6}\n\\end{figure}\n\n\\begin{figure*}[t!] \n\t\\centering\n\t\\includegraphics[width=7. in]{Fig7.pdf}\n\t\\caption{Confusion matrices under different data augmentation methods with 100\\% training dataset when SNR is -2dB.}\\label{fig7}\n\\end{figure*}\n\n\\begin{figure*}[t!] \n\t\\centering\n\t\\includegraphics[width=7. in]{Fig8.pdf}\n\t\\caption{Confusion matrices under different data augmentation methods with 100\\% training dataset when SNR is 18dB.}\\label{fig8}\n\\end{figure*}\n\nIn Fig.~\\ref{fig6}, we study the modulation classification accuracies of the LSTM model after deploying all three data augmentation methods presented in Sec. VI. Comparing with the baseline without augmentation, all augmentation methods improve the classification accuracy when the SNR is greater than -10dB, especially for the rotation data augmentation and flip data augmentation. In particular, the rotation data augmentation method achieves the greatest improvement by 8\\% when SNR is between -6dB and -2dB and by about 2\\% at higher SNR ($\\ge$4dB). Meanwhile, the Gaussian noise data augmentation performs better at lower SNR when it is between -16dB and -10dB. Intuitively, adding Gaussian noise reduces the SNR of the original data sample which in turn generates more signal samples with low SNR. However, the improvement is trivial since the resulting classification accuracy is too small, less than 2\\% when SNR is smaller than 10 dB. Therefore, rotation data augmentation and flip data augmentation are more preferred for radio signals in modulation classification.\n\nTo further evaluate the improvements of different augmentation methods on classification accuracy, we present the corresponding confusion matrices these at low SNR (-2dB) and high SNR (18dB) in Fig.~\\ref{fig7} and Fig.~\\ref{fig8}, respectively. Most values at diagonal entries of these matrices are increased after argumentation, which means the modulation classification accuracy are improved. Specifically, the proposed augmentation methods successfully reduce the confusion between QAM16 and QAM64 and solve the short-time observation problem presented in \\cite{o2017introduction}. At low SNR, the LSTM model is difficult to classify 8PSK and QPSK, whose classification accuracy is greatly improved after rotation augmentation as shown in Fig.~\\ref{fig7}. At high SNR, the accuracy of the LSTM model is mainly limited by the confusion between AM-DSB and WBFM, which dues to frequent radio samples without information in the dataset \\cite{o2017introduction}. In general, rotation and flip achieve better classification accuracy than Gaussian noise for all modulation categories.\n\n\\subsection{Augmentations On Partial Dataset}\nIn Fig.~\\ref{fig9}, we further study the performance of different data augmentation methods with insufficient training dataset. To form new training sub-dataset, we randomly sample partial radio signal samples from the initial 110,000 radio signal training samples, i.e., 12.5\\% of the initial training dataset. Then, the LSTM network is trained by feeding the obtained training sub-dataset and is tested with the initial 110,000 radio signal testing samples. Note that 12.5\\% of the training dataset is insufficient to train the LSTM network, resulting a low modulation classification accuracy around 45\\% under high SNR, as shown in Fig.~\\ref{fig9}. After deploying different radio data augmentation methods, the classification accuracy is improved. As expected, both the rotation augmentation and the flip augmentation outperform the Gaussian noise data augmentation. Interestingly, while the training sub-dataset is expanded by a scale factor $N = 4$ after augmentation, in the same size of 50\\% of the initial dataset, the rotation\/flip augmentation achieves a higher classification accuracy, around 0.04\\%-4.03\\%, than the baseline by training the LSTM with 50\\% of the initial training dataset without augmentation. \n\nWe further consider a joint augmentation policy with both rotation and flip methods, which expands the dataset by a scale factor $N = 6$ (with 2 redundant augmented radio signal samples) as shown in Fig.~\\ref{fig4}(a-b). After this joint augmentation, the size of the training dataset is expanded from 12.5\\% to be 75\\% of the initial training dataset. Interestingly, we obtain similar classification accuracies at different SNRs as the baseline with 100\\% training dataset without augmentation, as plotted in Fig.~\\ref{fig9}. Note that such a classification accuracy is achieved by using 25\\% less training data.\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig9v2.pdf}\n\t\\caption{Classification accuracy versus different data augmentation methods under different SNRs with 12.5\\% training dataset.} \\label{fig9}\n\\end{figure}\n\n\\begin{figure*}[t!] \n\t\\centering\n\t\\includegraphics[width=7. in]{Fig10.pdf}\n\t\\caption{Confusion matrices under different data augmentation methods with 12.5\\% training dataset when SNR is 18dB.}\\label{fig10}\n\\end{figure*}\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=3.3 in]{Fig11.pdf}\n\t\\caption{Classification accuracy versus different data augmentation methods under different SNRs with 64-length signal.} \\label{fig11}\n\\end{figure}\n\nTo further evaluate the advantages of joint rotation and flip augmentation, we present confusion matrices in different augmentation methods with 12.5\\% training dataset at 18dB in Fig.~\\ref{fig10}. When training dataset is insufficient, it is difficult to classify BPSK, WBFM, QAM16 and QAM64, whose classification accuracies are significantly improved after joint augmentation. Specifically, in reducing the confusion between QAM16 and QAM64, the joint augmentation performs better than both the rotation augmentation and the flip augmentation.\n\nWe have also evaluated another joint augmentation with all three augmentation methods. However, adding Gaussian noise method to the joint rotation and flip augmentation slightly reduces the classification accuracy. Therefore, we conclude that both rotation and flip methods are preferred for radio data augmentation and they can be jointly applied to further improve the augmentation performance.\n\n\\subsection{Augmentations On short Sample}\nWe further evaluate data augmentation methods for modulated radio signals with fewer sampling points. We halve each original 128-point radio signal sample into two new samples and obtained a new dataset consisting of 440,000 entries of 64-point radio signal samples. Similar to previous evaluations, we randomly choose half of them to the LSTM network, which is further tested with the remaining half dataset. With a shorter radio signal sample, the number of LSTM cells in each LSTM layer in Fig.~\\ref{fig3} is reduced from 128 to 64, resulting a simpler inference model. Specifically, the number of parameters of the LSTM network is reduced from 201.1K to 54.1K and the inference complexity in FLOPs (floating-point operations) is reduced from 2.8K to 1.4K.\n\nIn Fig.~\\ref{fig11}, we evaluate modulation classifications with 64-point radio samples. Without augmentation, 64-point modulated radio samples always lead to lower classification accuracy than the baseline with 128-point, around an 8\\% reduction when SNR is greater than 0 dB. The classification accuracy is improved after deploying either rotation or flip augmentation. Especially, the joint rotation and flip augmentation can achieve 1\\% higher classification accuracy than the baseline under high SNR. Therefore, with  data augmentation, the radio signal modulations can be successfully classified upon receiving only half number of sampling points, which significantly reduces the classification response time.\n\n\\section{Conclusion}\nIn this paper, we studied radio data augmentation methods for deep learning-based modulation classification. Specifically, three typical augmentation methods, i.e., rotation, flip, and Gaussian noise, were studied based on a well-known LSTM model. We first studied radio data augmentations at training and inference phases and revealed that train-test-time augmentation achieves the highest accuracy. Then, we numerically evaluated all three augmentation methods based on the full and partial training dataset. All numerical results show that both the rotation and the flip methods achieve higher classification accuracy than the Gaussian noise method and the rotation method achieves the highest accuracy. Meanwhile, a joint augmentation policy with both rotation and flip methods can further improve the classification accuracy, especially with insufficient training samples. Given only 12.5\\% of initial training dataset, the joint augmentation method expands the dataset to be a size of 75\\% of the initial dataset and obtains even higher than the baseline with 100\\% training datasets without augmentation. Furthermore, after deploying data augmentation, a radio sample can be classified based on only one half of the radio sampling points, resulting in a simplified deep learning model and shorter the classification response time.  \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{i}\n\nThe halo mass function (HMF hereafter) is a unique prediction of\ncosmological models of structure formation. \nThe evolution of the HMF traced by galaxy clusters has been recognized\nsince a long time as a powerful tool to trace the growth of cosmic\nstructures and, therefore, to constrain cosmological parameters\n\\cite[see][for reviews, and references\ntherein]{Rosati2002,Allen2011}. In particular, cosmological\napplications of the HMF require to know its shape and evolution to a\nhigh precision, in order to fully exploit its potential as a\ncosmological tool to be applied to ongoing and future large surveys of\ngalaxy clusters \\citep[e.g.][]{Wu2010,Murray2013}.\n\nN--body simulations covering wide dynamic ranges are nowadays\nproviding rather accurate calibration of the mass function of Dark\nMatter (DM) halos \\citep[e.g.][]{Jenkins2001, Reed2003, Reed2007,\n  Reed2013,Lukic2007,Tinker2008,Crocce2010,Courtin2011,\n  Bhattacharya2011,Angulo2012,Watson2013}. Various extensions of the\nstandard $\\Lambda$CDM cosmology model, such as coupled dark energy\nmodels \\citep[e.g.][]{Cui2012b, Baldi2012}, modified gravity models\n\\citep[e.g.][]{Schmidt2009,Zhang2013,Puchwein2013}, non-Gaussian\ninitial conditions \\citep[e.g.][]{Grossi2009,Pillepich2010}, massive\nneutrinos \\citep[e.g.][]{Brandbyge2010,Ichiki2012,Costanzi2013}, Warm\nDark Matter \\citep[e.g.][]{Schneider2013,Angulo2013}, have been\nstudied using numerical simulations, and their effect on the HMF has\nbeen investigated.\n\nA crucial aspect in the \ncalibration of the HMF is related to the algorithm used to identify\nhalos, the two most widely used being the Friend of Friend (FoF) one\nand the Spherical Overdensity (SO) one. The choice of a specific\nalgorithm clearly impacts both the number of identified halos and\ntheir mass \\citep[e.g.][; see also \\citealt{Knebe2011} for a\ndetailed comparison between different halo finders]{White2001,\n  White2002, Lukic2009, More2011, Watson2013}.\n\nAll the above mentioned HMF calibrations are based on N--body\nsimulations that follow the evolution of a collisionless DM fluid. On\nthe other hand, the presence of baryons is known to add subtle but\nsizeable effects on halo formation and internal structure, whose\ndetails also depend on the physical processes included in the\nnumerical treatment of the baryonic component, such gas\ncooling, star formation and energy feedback from supernovae (SN) and\nActive Galactic Nuclei (AGN) \\citep[e.g.][and references\ntherein]{KravtsovBorgani2012}.\n\nA number of studies based on cosmological hydrodynamical simulations\nhave been recently carried out to analyse in detail the effect of\nbaryonic processes on different properties of the total mass\ndistribution, such as the power spectrum of matter density\nfluctuations \\citep[e.g.]{Rudd2008, Daalen2011, Casarini2012}, the\nhalo correlation functions \\citep[e.g.][]{Zhu2012, Daalen2013}, the\nhalo density profiles \\citep[e.g.][]{Duffy2010, Lin2006} and\nconcentration \\citep[e.g.][]{Rasia2013, Bhattacharya2013}, and the HMF\n\\citep[e.g.][]{Stanek2009, Cui2012a, Sawala2013, Martizzi2013,\nCusworth2013, Balaguera2013, Wu2013}.\n\nAs for the effect of non--radiative hydrodynamics, the presence of\nbaryons has been shown to induce a slight increase of the HMF\n\\citep[][hereafter Paper I]{Cui2012a}. When extra--heating is included,\n\\cite{Stanek2009} found instead a decrease in the HMF. As for the\neffect of radiative cooling, star formation and SN feedback, different\ngroups consistently found an increase of the HMF, an effect that is\nmore evident in the high--mass end\n\\citep{Stanek2009,Cui2012a,Martizzi2013}. On the other hand,\n\\citet{Sawala2013} found that efficient SN feedback produces an\nopposite effect in low--mass halos.\n\nOn the other hand, a number of analyses have shown in the last years\nthat including AGN feedback in cosmological simulations provides\npopulations of galaxy clusters in better keeping with observational\nresults \\citep[e.g.][]{Puchwein2008, ShortThomas2010, Fabjan2010,\n  McCarthy2011, Planelles2013, LeBrun2014}. \nWhen the AGN feedback is included, different results were found by\n\\citet{Martizzi2013} and \\citet{Cusworth2013}. The former showed that\nthe HMF with AGN feedback is higher than the fitting function from\n\\citet{Tinker2008}, while the latter predicted a lower HMF compared to\nthe same fitting function. However, their implementation of the AGN\nfeedback differ.  \\cite{Martizzi2013} described AGN feedback by\ncomputing explicitly gas accretion rates onto super-massive black\nholes (SMBHs) included as sink particles in simulations that also\ninclude radiative cooling, star formation and SN feedback\n\\citep[e.g.][]{Springel2005,BoothSchaye2009}.  \\cite{Cusworth2013}\nincluded AGN feedback by computing the associated feedback energy from\nthe semi--analytic model of galaxy formation by \\cite{Guo2011},\nwithout including radiative cooling and assuming zero mass for gas\nparticles, so that no back-reaction of baryons on the DM distribution\nis allowed.\n\nIn this paper we extend our previous analysis of baryonic effects on\nthe HMF, presented by \\citetalias{Cui2012a}, by also including in our\nsimulations the effect of AGN feedback. We directly compare the HMF\nobtained from DM--only simulations to those produced by radiative\nhydrodynamical simulations both with and without AGN feedback, using\nexactly the same initial conditions, mass and force resolutions. The\nplan of the paper is as follows. In section \\ref{simulation}, we\npresent the simulations analysed in this paper. Section \\ref{halo} is\ndevoted to the description of the halo identification methods. In\nsection \\ref{results} we present the results of our analysis and\ndescribe in detail the differences in the HMF induced by different\nfeedback models. Our results are discussed and summarised in Section\n\\ref{concl}.\n\n\\section{The Simulations}\n\\label{simulation}\n\nThree large--volume simulations are analysed in this paper, namely two\nhydrodynamical simulations which include different description of\nfeedback processes affecting the evolution of baryons and one N--body\nsimulation including only DM particles. Initial conditions for these\nsimulations are the same as described in \\citetalias{Cui2012a} and we refer\nto that paper for further details. The hydrodynamical simulations have\nthe same number dark matter particles ($1024^3$) and gas particles\n($1024^3$). A first hydrodynamical simulation includes radiative\ncooling, star formation and kinetic SN feedback (CSF hereafter), while\nthe second one also includes the effect of AGN feedback (AGN\nhereafter). As for the DM simulation, it starts for the same initial\nconditions as the hydrodynamical simulations, with the gas particles\nreplaced by collisionless particles, so as to have the same\ndescription of the initial density and velocity fields as in the\nhydrodynamical simulations.\n \nThe three simulations have been carried out using the TreePM-SPH code\n{\\small GADGET-3}, an improved version of the {\\small GADGET-2} code\n\\citep{Gadget2}. \nGravitational forces have been computed using a Plummer--equivalent\nsoftening which is fixed to $\\epsilon_{Pl}=7.5h^{-1}$ physical kpc\nfrom $z=0$ to $z=2$, and fixed in comoving units at higher\nredshift. The simulations assume flat $\\Lambda$CDM cosmology with\n$\\Omega_{\\rm m} = 0.24$ for the matter density parameter, $\\Omega_{\\rm\n  b} = 0.0413$ for the baryon contribution, $\\sigma_8=0.8$ for the\npower spectrum normalisation, $n_{\\rm s} = 0.96$ for the primordial\nspectral index, and $h =0.73$ for the Hubble parameter in units of\n100$\\vel {\\rm Mpc}^{-1}$. Initial conditions have been generated at\n$z=49$ using the Zeldovich Approximation for a periodic cosmological\nbox with comoving size $L=410\\Mpc$.\nThe masses of gas and DM particles have a ratio such that to reproduce\nthe cosmic baryon fraction, with $m_g\\simeq 7.36 \\times 10^{8} \\hMsun$\nand $m_{DM}\\simeq 3.54 \\times 10^{9} \\hMsun$, respectively.\n\nIn the hydrodynamical simulations, radiative cooling is computed for\nnon--zero metallicity using the cooling tables by\n\\cite{sutherland_dopita93}, also including heating\/cooling from a\nspatially uniform and evolving UV background. Gas particles above a\ngiven threshold density are treated as multi-phase, so as to provide a\nsub\u2013resolution description of the inter\u2013stellar medium, according to\nthe model described by \\cite{springel_hernquist03}.\nConversion of collisional gas particles into collisionless star\nparticles proceeds in a stochastic way, with gas particles spawning a\nmaximum of two generations of star particles. We also include a\ndescription of metal production from chemical enrichment contributed\nby SN-II, SN-Ia and AGB stars, as described by\n\\cite{tornatore_etal07}. Kinetic feedback is implemented by mimicking\ngalactic ejecta powered by SN explosions, with a wind mass upload\nproportional to the local star-formation rate, $\\dot M_w=\\eta \\dot\nM_*$. In the CSF simulation we use $\\eta=2$ and $v_w = 500\\, {\\rm\n  km}\\, s^{-1}$ for the wind velocity, which corresponds to assuming\nabout unity efficiency for the conversion of energy released by SN-II\ninto kinetic energy for the adopted Salpeter IMF.\n\nAs for the AGN simulation, it includes both the effect of galactic\nwinds, with $v_w = 350\\, {\\rm km}\\, s^{-1}$ and the same mass--load\nparameter $\\eta=2$, along with energy feedback\nresulting from gas accretion onto SMBHs. The model of AGN feedback\nused in this simulation is the same as that adopted by\n\\cite{Fabjan2010} and is largely inspired to the model originally\nintroduced by \\cite{Springel2005b}. SMBHs, seeded with an initial mass\nof $10^6M_\\odot$ in halos resolved with at least 100 DM particles,\nsubsequently grow by merging with other BHs and by gas accretion. The\nlatter proceeds at the Bondi rate and is Eddington--limited. A\nfraction $\\epsilon_r=0.1$ of accreted mass is converted into\nradiation, with a fraction $\\epsilon_f$ of this radiation  thermally\ncoupled to the surrounding gas. We assume $\\epsilon_f=0.1$ which\nincreases by a factor of four whenever accretion takes place at a rate\nof at most one-hundredth of the Eddington limit.  \n\nWe note that the main motivation for efficient SN feedback with $v_w =\n500\\, {\\rm km}\\, s^{-1}$ in the CSF simulations lies in the need of\nreconciling simulation predictions on the cosmic star formation rate\nwith observations, at least at redshift $z>2$, a choice that still\nproduces too efficient star formation at lower redshift\n\\citep[e.g.][]{Tornatore10}. Although AGN feedback is motivated by the\nneed of reducing the star formation rate at lower redshift, still its\neffect is quite significant already at $z\\sim2$. Therefore, in order\nto prevent too strong a reduction of star formation around this\nredshift when SN and AGN feedbacks are both included, we decided to\nreduce by a factor of two the kinetic energy associated to the\nformer. This lowers the resulting wind velocity to $v_w = 350\\, {\\rm\n  km}\\, s^{-1}$.\n\n\\section{Halo identification}\n\\label{halo}\n\nThe two most common methods used for halo identification in\nsimulations are the Friend-of-Friend (FoF) algorithm\n\\citep[e.g.][]{Davis1985} and the spherical overdensity (SO) algorithm\n\\citep{Lacey1994}. The FoF algorithm has only one parameter, $b$,\nwhich defines the linking length as $b l$ where $l=n^{-1\/3}$ is the\nmean inter-particle separation, with $n$ the mean particle number\ndensity. In the SO algorithm, there is also only one free parameter,\nnamely the overdensity $\\Delta_c$. The overdensity determines the\naperture of the sphere, within which the total mean density is\n$\\Delta_c ~ \\rho_{crit}$. Here, $\\rho_{crit}$ is the critical cosmic\ndensity. Each of the two halo finders has its own advantages and\nshortcomings \\citep[see more details in][and references\n  thereon]{Jenkins2001, White2001, \n  Tinker2008}, and the differences between the two methods in terms of\nhalo masses and HMFs have been discussed in several analysis\n\\citep[e.g.][]{White2002, Reed2003, Reed2007, Cohn2008, More2011,\n  Anderhalden2011, Knebe2013, Watson2013}. We adopt both methods to\nidentify halos in this paper.\n\n\\subsection{Friend-of-Friend Halos}\n\nIn our three simulations FoF halos are identified by a on-the-fly FoF\nfinder, with a slight smaller linking length $b=0.16$ compared to\ncommonly used one, $b = 0.2$. Dark matter particles are linked\nfirst. Then, each gas and star particle is linked to the nearest\ndark matter particle, whenever the linking criterion is satisfied.\n\n\\subsection{Spherical Overdensity Halos -- {\\small PIAO}}\n\n\\begin{CJK*}{UTF8}{gkai}\nWe carry out a spherical overdensity (SO) halo search by using an\nefficient memory-controlled parallel {\\bf P}ython spher{\\bf I}c{\\bf\n  A}l {\\bf O}verdensity halo finding code --- {\\small PIAO} (Chinese\ncharacter: \u6f02). This code is based on the standard SO algorithm. Its\naim is not to provide a new halo identification method, but to analyse\nlarge simulations on a small computer server or PC with limited\nmemories. To overcome a memory deficiency problem, we adopt a simple\nstrategy, which is based on splitting the whole simulation box into\nsmall mesh-boxes, and analysing them one-by-one. The details of this\nstrategy and how to incorporate it within the SO method is discussed\nin Appendix \\ref{A:Piao}.  {\\small PIAO} is parallelised with a python\nMPI package (MPI4py) to speed up the calculation by taking advantage\nof multi-core CPUs.\n\\end{CJK*} \n\nWe applied {\\small PIAO} to the three simulations analysed in this\npaper. For all of them, SO halos are identified at three different\noverdensity values\\footnote{In the following, the overdensity value\n  $\\Delta_c$ is expressed in units of the cosmic critical density at a\n  given redshift, $\\rho_c(z)=3H^2(z)\/(8\\pi G)$.}, $\\Delta_C = 2500,\n500, 200$. As detailed in the appendix, local density maxima around\nwhich growing spheres encompassing a given overdensity, are searched\nby assigning density at the positions of particles using 64 SPH neighbours and\nwithout allowing halos to overlap with each other.\n\n\\subsection{Matching halos}\n\n\\label{sec:match}\nSince all three simulations share the same initial\nconditions, dark matter particles have the same progressive\nidentification number (IDs). We exploit this information to match\nhalos identified in different simulations. Using a given halo\nidentified in the\nDM simulation as the reference, a halo in the CSF or AGN simulation is\ndefined to be the counterpart of the DM halo whenever it includes the\nlargest number of DM particles belonging to the latter.  We define the\nmatching rate as the ratio of matched to total number of dark matter\nparticles in the DM halo. Clearly, the larger this rate, the more\naccurate is the matching. In order to avoid multi-matching, i.e. two\nhalos from CSF\/AGN simulation matched to one halo in the DM one, only\nhalos with matching rate larger than $0.5$ are selected. We \nverified that the fractions of matched SO halos for $\\Delta_c = 500$,\nare $97.5\\%$ at $z = 0$, $98.3\\%$ at $z = 0.6$, $98.6\\%$ at $z =\n1.0$ and $99.4\\%$ at $z = 2.2$. Most of the mismatched halos have\nsmaller halo mass, e.g. $85\\%$ of them have halo mass $M_{500} <\n10^{13} \\hMsun$ at $z = 0$. \n\nAt each overdensity $\\Delta_c$, we only consider halos with\n$M_{\\Delta_c} \\geq 10^{12.5} \\hMsun$. With this choice, the smallest\nhalo can still have $\\sim 1000$ particles within the corresponding\n$R_{\\Delta_c}$. However, to allow for a complete matching, we consider\nhalos as small as $M_{\\Delta_c}=10^{11.5} \\hMsun$ in the AGN and CSF\nsimulations to be matched to the halos in the DM simulation. As shown\nby \\cite{Reed2013}, halos resolved with fewer than $N \\sim 1000$\nparticles are unlikely to be used for a high-accuracy HMF measurement.\nFurthermore, \\cite{Watson2013} also pointed out that the correction\nfor low number of particles sampling FoF halos from \\cite{Warren2006}\nis $\\sim 2$ per cent for the FoF halos containing 1000 particles. \nWe used a fixed mass bin $\\Delta \\log M = 0.2$ for the calculation of\nthe HMF, without further correction. As discussed by \\cite{Lukic2007},\nthe uncertainty in the HMF resulting from the choice of the binning is\nnegligible as long as the bin width does not exceed $\\Delta \\log M =\n0.5$.\n\n\\section{Results}\n\\label{results}\n\nBasic information on the number of halos identified by the FoF and SO\nfinders can be obtained from the cumulative HMF. We just mention here\nthat over $10^4$ halos are always found with both methods at $z = 0$\nwith halo mass $M \\geq 10^{12.5} \\hMsun$.  This number can reach $\\sim\n70000$ for FoF halos and for SO halos with $\\Delta_c = 200$. At the\nhighest considered redshift, $z = 2.2$, this number is still $\\sim\n10^4$ for FoF and for SO halos with $M_{200} \\geq 10^{12.5}$. However,\nat the same redshift we only have $\\sim 10^3$ SO halos with $M_{2500}\n\\geq 10^{12.5}\\hMsun$. The CSF simulation has more both SO and FoF\nhalos than the DM one at all redshifts and halo masses, an increase\nthat is less apparent for $\\Delta_c = 200$.  On the contrary, the AGN\nsimulation produces fewer halos of fixed mass than the DM one. Due to\nlimited simulation box size, only a few halos have mass $M \\geq\n10^{15} \\hMsun$ for FoF and SO with $\\Delta_c = 200$. Given the\nlimited dynamical range accessible to our simulations, we attempt in\nthe following to provide fitting expressions to the corrections to the\nHMF induced by baryon effects, while we avoid providing absolute\nfitting functions to the HMF.\n\n\\subsection{The HMF from Friend-of-Friend}\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{HMF_fof.eps}\n\\caption{Baryon effects on the halo mass function (HMF) for halos\n  identified with the Friend-of-Friends (FoF) algorithm. Upper panel:\n  FoF halo mass functions (HMFs). Different line styles are for\n  different redshifts, as indicated in the legend in the bottom left\n  corner, while different colours refer to the different simulations\n  (legend in upper right corner). Lower panel: ratios between each of\n  the HMFs from the hydrodynamical simulations and the HMF of the DM\n  simulation.}\n\\label{fig:hmf_fof}\n\\end{figure}\n\nWe compare in the upper panel of Fig. \\ref{fig:hmf_fof} the HMFs for\n the three different simulations, while the lower panel shows the\nrelative difference between each of the two hydrodynamical simulations\nand the DM one.  As for the effect of the baryon physics described by\nthe CSF model, we note that the difference with respect to the DM case\nhas a clear redshift evolution and halo mass dependence. As redshift\ndecreases from $z = 2.2$ to 0, the HMF ratio drops from $\\sim 1.6$ to\n$\\sim 1.1$, with a weak increasing trend of this ratio with halo mass,\nat all redshifts. Quite remarkably, including AGN feedback has the\neffect of reducing the difference with respect to the DM-only case:\nthe HMF ratio drops to about unity for massive halos with $M_{FoF}\n\\simgt 10^{14} \\hMsun$, while at smaller halo mass it decreases to $\\sim\n0.9$ for $M_{FoF} \\approx 10^{13} \\hMsun$.  Unlike the CSF case,\nthese differences do not show any evidence of redshift evolution\nfrom $z = 1$ to $z = 0.0$. At higher redshift, $z = 2.2$, the HMF\nratio keeps fluctuating around 1, as a consequence of the limited\nstatistics of halos due to the finite box size.\n\n\\subsection{The HMF from Spherical Overdensity}\n\\begin{figure*}\n\\includegraphics[width=1.0\\textwidth]{HMF_SO.eps}\n\\caption{Effect of baryons on the halo mass function (HMF) for\n  spherical overdensity (SO) halos. Each panel is the same as in\n  Fig. \\protect\\ref{fig:hmf_fof}, but for the spherical SO HMFs\n  with halo masses computed for $\\Delta_c = 2500$ (left panel), 500\n  (middle panel) and 200 (right panel). }\n\\label{fig:hmf_so}\n\\end{figure*}\n\nWe compare in the upper panels of Fig. \\ref{fig:hmf_so} the HMFs\nobtained from the SO halo finder at three overdensities, $\\Delta_c =\n2500, 500, 200$ (from left to right), along with the ratios of the\nHMFs from the CSF and AGN simulations with respect to the DM-only\nresult (lower panels).  As expected, baryons have a larger impact at\nthe highest considered overdensity, $\\Delta_c = 2500$. In this case,\nthe ratio between CSF and DM HMFs shows a redshift evolution similar\nto the FoF results but with a higher amplitude, ranging from $\\sim\n1.4$ at $z = 0$ to $\\sim 2.5$ at $z = 2.2$, but with no significant\ndependence on the halo mass. At lower overdensities, $\\Delta_c = 200$\nand 500, the redshift evolution becomes weaker and the differences\nwith respect to the DM case are reduced, with only a $\\simlt 5$ per\ncent difference at $\\Delta_c=200$ \\citepalias[similar results for the\nCSF case were also found by][]{Cui2012a}.\n\nWhen AGN feedback is included in the simulation, the corresponding HMF\ndrops below the HMF from the DM simulation, by an amount that\ndecreases for lower $\\Delta_c$ values, with no evidence for redshift\ndependence of the HMF difference. At $\\Delta_c = 2500$, this ratio has\na weak halo mass dependence, ranging from $\\sim 0.7$ at $10^{12.5}\n\\hMsun$ to $\\sim 0.5$ at $10^{14} \\hMsun$. At $\\Delta_c = 500$ and\n200, the difference between AGN and DM HMFs reduces, with a mild\ndependence on halo mass: $dn\/dn_{DM} \\approx 0.7, \\approx 0.8$ at $M\n\\approx 10^{13} \\hMsun$ to $dn\/dn_{DM} \\approx 0.9, \\approx 1.0$\n($\\Delta_c = 500, 200$, respectively) in the high mass end.\n\nIn general, the effect of including baryons on the HMF goes in the\nsame direction, independent of whether FoF or SO halo finders are\nused. While this holds at a qualitative level, as expected\nquantitative differences between FoF and SO results are found,\nespecially for the AGN case. As we will discuss in the following, the\neffect of including AGN feedback is that of producing halos that are\nless concentrated than in the CSF case. As a result, one expects that\nmatching SO and FoF HMFs requires in the CSF simulation a higher\n$\\Delta_c$ than in the AGN simulation. Many efforts are made to\nrematch the two halo mass functions by tuning $b$ and $\\Delta_c$\n\\citep[for example][]{Lukic2009, Courtin2011, More2011}. However, as\nshown in \\cite{Watson2013}, even in dark-matter-only simulations,\nmatching FoF HMFs to SO HMFs not only depends on the choice of $b$ and\n$\\Delta_c$, but also on the concentration parameter, pseudo mass\nevolution, and the problems inside the two algorithms. These\nquantitative differences between FoF and SO results make this matching\nprogress even more complex if baryon models are taken in account.\n\n\\begin{figure*}\n\\includegraphics[width=1.0\\textwidth]{MD_fit.eps}\n\\caption{Mass dependence of the ratio of masses of matched SO halos\n  computed for $\\Delta_c = 500$ at different redshifts, as reported in\n  the upper--right corner of each panel. Each point represents a halo\n  mass ratio between the matched CSF (red points) or AGN (green\n  points) halos to DM ones, as a function of the mass of the matched\n  halo in the DM simulation. The thick magenta and green lines show\n  the mean values of this ratio within each mass bin for the CSF and\n  AGN simulations, respectively. The best--fitting relation for the\n  mass correction of Eq. \\protect\\ref{eq:1} is shown with the solid\n  black lines.  We note that the same relation provides a good fit at\n  all redshifts, at least up to $z=1$. See Table 1 for the values of\n  the parameters defining this best-fit relation.  }\n\\label{fig:md_so}\n\\end{figure*}\n\nIn order to understand the origin of the baryonic effects on the HMFs\npredicted by our simulations, we further focus on the difference of\nmasses of matched halos at overdensity $\\Delta_c = 500$ (see Section\n\\ref{sec:match} for the description of the matching procedure).  We\nshow in Fig. \\ref{fig:md_so} the ratio between masses of matched halos\nin each one of the two hydrodynamical simulations and in the DM\nsimulation (red and green points for the CSF and AGN case,\nrespectively). In each panel, the thick lines show the mean value of\nthese ratios computed within each mass bin (magenta for CSF and blue\nfor AGN).  As for the CSF case, the effect of baryons is that of\nincreasing halo masses by an amount which is almost independent of\nredshift. At each redshift, the halo mass ratio weakly decreases with\nhalo mass, from $\\sim 1.1$ at $M_{500} = 10^{12.5} \\hMsun$ to $\\sim\n1.05$ at $M_{500} \\simgt 10^{13.5} \\hMsun$, then becoming constant\n\\citepalias[see also][]{Cui2012a}.  As for the AGN simulation, the\neffect of baryons goes in the opposite direction of decreasing halo\nmasses, thereby decreasing the corresponding HMF, as shown in\nFig. \\ref{fig:hmf_so}. Also in this case, there is no evidence for a\nredshift evolution of the halo mass ratio, at least below $z = 1.0$.\nHowever, there is an obvious increase of this ratio with halo mass,\nthat ranges from $\\sim 0.8$ at $M_{500} = 10^{12.5} \\hMsun$ to $\\simeq\n1$ for the most massive halos found in our simulation box. Similar\ntrends are also found for the mass ratio with $\\Delta_c = 2500, 200$,\nboth of which also show no evidence of redshift dependence for both\nhydrodynamical simulations. We verified that using the median value of\nthose data points gives almost identical lines to these mean lines.\nAs discussed in the Appendix C, this effect of reduction of halo\nmasses in the presence of AGN feedback is quite robust against\nnumerical resolution. We refer to this Appendix for a more detailed\ndiscussion of the resolution test that we carried out.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{MD_sr.eps}\n\\caption{The same as the bottom right panel of Fig. \\ref{fig:md_so},\n  but for halo masses in the CSF\/AGN simulations computed within the\n  $R_{500}$ radius of the corresponding halos from the DM\n  simulation. The dashed thick lines are the previous results in\n  Fig. \\ref{fig:md_so} at $z = 0$.}\n\\label{fig:sr}\n\\end{figure}\n\nSince the masses of SO halos are computed by adding up all the\nparticles within a sphere with radius $R_{\\Delta_c}$, it is clear that\na change of the halo density profiles induced by the presence of\nbaryons would also change the corresponding values of\n$R_{\\Delta_c}$. In order to quantify the effect of this variation, we\nalso compute masses for each halo in the CSF\/AGN simulations by using\nthe value of $R_{\\Delta_c}$ of the corresponding halo identified in\nthe DM simulation.  In Fig. \\ref{fig:sr}, we show again the halo mass\ndifference at $\\Delta_c = 500$ after applying this re-tuning of the\nhalo radii. A comparison with the $z = 0$ result from in\nFig. \\ref{fig:md_so} demonstrates that these ratios are only slightly\nshifted towards unity, for both CSF and AGN models. This small change\nimplies that the differences in halo masses are mostly contributed by\nthe baryon effects on the halo density profiles, which can not be\nrecovered by simply changing the halo radius.\n\nIncluding only SN feedback in the form of galactic ejecta is already\nknown not to be able to regulate overcooling at the centre of\nrelatively massive halos, with $M> 10^{12.5} \\hMsun$. Adiabatic\ncontraction \\citep[e.g.][]{gnedin2004}, associated to the condensation\nof an exceedingly large amount of cooled gas, leads then to an\nincrease of density within a fixed halo aperture radius and,\ntherefore, to an increase of the halo mass with respect to the DM\ncase. The opposite effect is instead associated to the inclusion of\nAGN feedback. In this case, the sudden displacement of large amount of\ngas at epochs corresponding to the peak of AGN feedback efficiency,\ntaking place at $z\\sim 2$--3, causes sudden variations of the halo\npotential, which reacts with an expansion, thus decreasing halo masses\n(see discussion in Sect. \\label{gnedin2004} below).\n\n\n\\subsection{Density profiles}\n\\label{sec:profs}\nHaving quantified the variation of halo masses, we now discuss how\nthis variation is associated to changes in the total density profiles\nof halos induced by baryonic processes.\n\n\\begin{figure*}\n\\includegraphics[width=1.0\\textwidth]{DP_tot.eps}\n\\caption{The stacked ratios of cumulative density profiles,\n  $\\rho(<R)$, of matched halos identified in each one of the\n  hydrodynamical simulations and in the DM one. Different panels\n  refer to different halo mass ranges, with the bottom right panel\n  showing the results for all halos with mass $M_{500}\\ge 10^{12.5}\n  \\hMsun$.  In each panel, the red and green solid lines are for the\n  mean density ratios for the CSF and AGN case, respectively. Error\n  bars indicate the $1 \\sigma$ scatter around the mean.  Vertical\n  dashed, dot--dashed and dotted lines indicate the median values of\n  $R_{200}$, $R_{500}$ and $R_{2500}$ computed within each halo mass\n  interval, with magenta and blue colours referring to the CSF and AGN\n  simulations, respectively. We also show with continuous black lines\n  the median values of $R_{2500}$, $R_{500}$ and $R_{200}$ for the DM\n  simulation. The shadow regions show the limits of the gravitational\n  softening.}\n\\label{fig:dpt}\n\\end{figure*}\n\nTo this purpose, we show in Fig. \\ref{fig:dpt} the stacked ratio\nbetween density profiles for halos belonging to different mass\nintervals. We consider in this plot halos that are matched in the\nCSF\/AGN and in the DM simulations. Halos are separated into five mass\nbins, according to the $M_{500}$ halo mass, measured in the DM\nsimulation. For each matched halo, we first compute the ratio of\ncumulative density profiles, $\\rho(<R)$. Then, we stack the density\nprofile ratios for all halos belonging to the same mass bin. The\nstacked density profile ratios for CSF and AGN halos are shown with\nred and green curves, respectively, with error bars indicating the $1\n\\sigma$ intrinsic scatter within each mass interval.\n\nWe note that density profiles for CSF halos are always higher than\nthose for the DM simulation for all mass bins. The effect is stronger\ntowards the cluster centre, as a result of adiabatic contraction\ntriggered by gas condensation from cooling. This result, which is in\nline with those found by several previous analyses\n\\citep[e.g.][]{gnedin2004,puchwein2005}, indicates that the feedback\nincluded in our CSF model is not efficient in counteracting the effect\nof radiative cooling in increasing total density in central halo\nregions. While the effect is quite small at $R_{200}$, it becomes\nsizeable at $R_{2500}$, where density increases by up to about 50 per\ncent for the smallest resolved halos.\n\nIt is quite interesting that a different behaviour is found when AGN\nfeedback in included. In this case, an increase of total density\nassociated to gas condensation is only found at small radii, below\n(20--30) $h^{-1}$kpc, which are however close the smallest scale that\ncan be trusted at the resolution of our simulations. At larger radii\nthe effect of AGN feedback is that of decreasing the halo density\nprofile, an effect that becomes negligible at large radii, approaching\n$R_{200}$, and for the most massive halos found in our\nsimulations. This result is in line with those from previous analyses\nof cluster simulations including AGN feedback\n\\citep[e.g.][]{Duffy2010,Mead2010,Dubois2010,Killedar2012,Martizzi2013,Cui2014}. The\ndecrease of density profiles is caused by the effect of AGN feedback\nthat, at redshifts corresponding to the peak of BH accretion $z\\simeq\n2$--3, causes a sudden expulsion of gas from the potential wells of\nthe cluster progenitor halos \\citep[similar result is also found by\n][]{Dubois2010}. The expulsion of large amount of gas causes potential\nwells to react with some expansion, thereby causing a decrease of the\ndensity in central regions.\n\n\\subsection{Correcting the halo mass function}\n\n\\begin{figure*}\n\\includegraphics[width=1.0\\textwidth]{HMF_SO_fit.eps} \n\\caption{The ratios between the halo mass functions from the CSF (upper\n  panel), AGN (lower panel)\n  simulations and from the DM simulation, after applying the correction\n  to DM halo masses, as described in the text.  From left to right, we\n  show results for overdensities $\\Delta_c = 2500, 500, 200$. Different line\n  styles and colors indicate different redshifts, a shown in the\n  legend in the upper right panel.}\n\\label{fig:cd}\n\\end{figure*}\n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{l||c||c||c|}\n\\hline\n & $\\Delta_c = 2500$ & $\\Delta_c = 500$ & $\\Delta_c = 200$ \\\\\n\\hline\n\\hline\n  AGN & & & \\\\\n\\hline\n$M_0$ & 13.952 & 13.471 & 13.334\\\\\n\\hline\n$\\alpha$ & 0.322 & 0.288 & 0.274\\\\\n\\hline\n\\hline\n  CSF & & & \\\\\n\\hline\n$M_0$ & 13.629 & 14.305 & 14.182\\\\\n\\hline\n$\\alpha$ & 0.295 & 0.085 & 0.045\\\\\n\\hline\n\\end{tabular}\n\\caption{Values of the best--fitting parameters describing the\n  variation of mass of matched halos in the AGN, CSF and in the DM\n  simulation, as described by Eq. \\ref{eq:1}.}\n\\end{centering}\n\\label{Tb:fit}\n\\end{table}\n\nHaving traced the origin of the variation of halo masses induced by\nbaryonic processes, we investigate now whether the application of a\nsuitable correction to halo masses allows one to recover the HMF from\na hydrodynamical simulation, starting from the DM--only HMF. In\n\\citetalias[][]{Cui2012a}, we have already shown that the HMFs from\nnon--radiative and CSF simulations can be corrected to the\ndark-matter-only HMF one, up to a residual scatter of $\\simlt 3$ per\ncent, by adopting a constant halo mass shift. From\nFig. \\ref{fig:md_so}, we note that the mean values of the halo mass\ndifference between AGN and DM simulations (thick blue line) has a\nsignificant mass dependence that needs to be included in the\ncorrection.\n\nAs a convenient relation to describe the mass--dependence of the mean\nmass ratio, we use these sigmoid functions\n\\be\n\\left\\{ \\begin{array}{l l}\nf(x) \\,=\\, \\alpha\/(1+e^{-3 (x-M_0) \/ 2}) + 1 - \\alpha & \\quad \\text{AGN},\\\\\nf(x) \\,=\\, \\alpha\/(1+e^{3 (x-M_0) \/ 2}) + 1 & \\quad \\text{CSF},\n\\end{array} \\right.\n\\label{eq:1}\n\\ee \nwith $M_0$ and $\\alpha$ considered as fitting parameters. Here, $f(x)$ is\nthe halo mass ratio between AGN and DM sets, with $x=\\log\nM_{\\Delta_c,DM}$. Since the halo mass difference shows no evidence for\nredshift evolution, at least for $z \\leq 1.0$, we do not attempt to\nfit a possible redshift dependence of the $M_0$ and $\\alpha$ parameters and\nexclude the results at $z=2.2$ from the analysis.\nThe values of $M_0$ and $\\alpha$ obtained in this way are reported in Table\n1 for the three values of the overdensity $\\Delta_c$ at\nwhich SO halos have been identified.\nThe thin black line shown in Fig. \\ref{fig:md_so} indicates this\nfitting to the correction for $\\Delta_c=500$. We verified that a\nredshift--independent mass correction also holds for the other\n$\\Delta_c$ values up to redshift $z = 1$. Further, we also checked\nthat the fitted parameters show no significant difference, whether we\nchoose to use median mass ratio or the mean mass ratio. \n\nWe show in Fig. \\ref{fig:cd} the ratio between the HMFs from the CSF,\nAGN and from the DM simulations, after correcting halo masses in the\nformer according to Eq. \\ref{eq:1}. In each panel, different\nline-types indicate results at different redshifts with different\npanels referring to different values of $\\Delta_c$.  We note that the\ncorrection to halo masses allows one to recover the DM HMF to good\naccuracy, at least for masses larger than $10^{13} \\hMsun$, the HMFs\nare matched to the DM one with no apparent systematic bias within\nrandom oscillations of $\\simlt 5$ per cent. We note that results\nbecome noisy whenever the sampling effects become important due to the\nlimited halo statistics. This is the case for large masses and,\nespecially, for the highest overdensity $\\Delta_c=2500$. At small\nmasses, below $10^{13} \\hMsun$, we note that the adopted mass\ncorrection systematically produces an overestimate of the corrected\nHMF for AGN set. For the smallest considered mass, $M_{DM} \\ge\n10^{12.5} \\hMsun$, this overestimate is as large as 10--15 per cent,\nthe exact value depending on the overdensity $\\Delta_c$. The reason\nfor this lies in the fact that this fitting of Eq. 1 does not provide\nan accurate description of the mass correction at small halo masses\ndue to the halo mass cut at $10^{12.5} \\hMsun$, as also shown in\nFig. \\ref{fig:md_so}.\n\n\\section{Discussion and Conclusions}\n\\label{concl}\n\nBased on a set of large--scale N-body and hydrodynamical simulations,\nwe presented an analysis of the baryon effects on the halo mass\nfunction, with emphasis on the role played by gas accretion onto\nsuper-massive black holes (SMBHs) and the ensuing AGN feedback. As\nsuch, this analysis extends our previous one, presented in\n\\citetalias{Cui2012a}, to the case in which one accounts for a\nsuitable model for AGN feedback that regulates star formation within\nmassive halos, thereby improving the degree of realism of the\nsimulated galaxy clusters and groups. We compared three simulations: a\nfirst one including only collisionless Dark Matter; a second one\nincluding hydrodynamics with radiative physics and supernova (SN)\nfeedback; a third one which also includes AGN feedback. Based on these\nsimulations, we analyse how the halo mass distribution reacts to\novercooling and, in the presence of AGN feedback, to the sudden\ndisplacements and expulsion of large amount of gas.\n\nWe use both Friends-of-Friends (FoF) and Spherical Overdensity (SO)\nalgorithms to identify halos. FoF halos are identified using a\nstandard (on-the-fly) FoF finder with a linking length $b = 0.16$. As\nfor SO halos, they are identified using an efficient memory-controlled\npython code --- {\\small PIAO}, in which halos are not allowed to\noverlap. We focus on three overdensities $\\Delta_c = 2500, 500, 200$\nfor the SO halos.\n\nThe main results of our analysis are summarised as follows.\n\\begin{description}\n\n\\item[1.] Including AGN feedback in hydrodynamical simulations causes\n  a reduction of the halo mass function (HMF) with respect to the\n  DM--only case, by an amount which depends on the overdensity within\n  which halo mass is measured. This effect amounts to about 20 per\n  cent for overdensity $\\Delta_c=500$, with no evidence of mass and\n  redshift dependence, and increases at higher overdensity. Therefore,\n  our model of AGN feedback reverses the effect of radiative physics with\n  no efficient feedback, which instead leads to an increase of the HMF.\n\\item[2.] The baryonic effects on the HMF can be traced to the effect\n  that different feedback models have on the halo density profiles and\n  total masses. In the absence of AGN feedback, we confirm that halo\n  density profiles steepen as a consequence of adiabatic contraction\n  associated to overcooling, thus causing an increase of halo\n  masses. AGN feedback generates shallower density profiles and a\n  corresponding decrease of halo masses. This effect is caused by the\n  improved regulation of overcooling associated to AGN feedback and to\n  the sudden displacement of large amount of gas, which makes the\n  gravitational potential responding with an expansion. The relative\n  decrease of halo masses is larger in smaller objects, where AGN\n  feedback is relatively more efficient, and at higher overdensities;\n  for $\\Delta_c=500$, it amount to $\\sim 20$ per cent at $\\log\n  M_{500}=12.5$, while becoming negligible for the largest halos found\n  in our simulation box, with $\\log M_{500}\\simeq 15$. Interestingly,\n  this effect is independent of redshift, at least up to $z=1$ where\n  we have a large enough statistics of massive halos.\n\\item[3.] We provide a mass--dependent fit to the halo mass variations\n  induced by baryonic effects. After applying this model for the mass\n  correction to the HMF obtained from the DM--only simulation, we\n  recover the HMF from hydrodynamical simulations, up to a random\n  scatter of $\\simlt 5$ per cent for halos more massive than $10^{13}\n  \\hMsun$, with no significant bias and independently of redshift.\n\\end{description}\n\nOur analysis demonstrates that baryon effects can cause sizeable\nvariations of the HMF and, therefore, affects the measurement of\ncosmological parameters from redshift number counts of galaxy\nclusters. For instance, recent results from the Planck Collaboration\n\\citep[cf. also \\citealt{Spergel2013}]{PlanckXXI} highlights that the\ncosmological model preferred by CMB analysis over-predicts the number\nof clusters expected in the SZ Planck Cluster Survey by about 50 per\ncent. Interestingly, this tension would be relaxed, although by a\nsmall amount, if the HMF calibrated with our AGN simulation is used to\npredict SZ cluster number counts.\n\nIn a recent paper, \\cite{Khandai} also investigated the effect of\nincluding AGN feedback on the HMF. Since they considered simulation\nboxes smaller than our, with size of 100$\\hMsun$, they better probed\nthe low-mass end of the HMF, while having a worse sampling of the high\nmass end. As a result of their analysis, they found that the FoF halo\nmass function can be described with a universal form to a reasonable\naccuracy, even after accounting for baryon effects.\\footnote{After the\n  submission of our paper, a paper by \\cite{Velliscig14} appeared on\n  the arXiv, which also included an analysis of baryon effects on halo\n  masses and HMF, when AGN feedback is also included. Using box size\n  and resolution quite similar to those of our simulations, they\n  confirmed our results on the effect of AGN feedback on the HMF.}\n\nNumerical convergence in the calibration of the HMF from purely\ncollisionless simulations could in principle be reached by ``brute\nforce'' (i.e. increasing the dynamic range accessible and the\ngravitational force integration accuracy). The same may not be true\nwhen the effects of baryons are included. In fact, our analysis shows\nthat baryons can generate variations of the HMF with respect to the DM\ncase which goes in different directions, depending on the nature and,\npossibly, on the numerical implementation of feedback. In this\nrespect, confidence in the calibration of baryon effects in the HMF is\nstrictly related to the level of agreement between observational\nresults and model predictions for a variety of properties of galaxy\nclusters. Current implementations of AGN feedback in cosmological\nsimulations produce cluster populations with an increased degree of\nrealism \\citep[e.g.][ and references therein]{Puchwein2008,Dubois2012,\n  Planelles2013, Planelles2014, LeBrun2014}. As an example, we compare\nin Appendix \\ref{A:sbf} two basic properties involving baryons in\nclusters, namely the stellar mass fraction and the total baryon mass\nfraction, with observational results. The good agreement between the\nAGN simulation and observational results in quite encouraging. Still,\nit is fair to admit that important tensions still exist between a\nnumber of observed and simulated cluster properties, such as the\nthermal structure of cool cores and the properties of the Brightest\nCluster Galaxies \\citep[e.g.][ cf.  \\cite{Kravtsov2014}]{Dubois2011,\n  Ragone2013, Planelles2014}. There is no doubt that providing an HMF,\nthat accounts for the inclusion of the baryon physics at the percent\nlevel of accuracy required by the next generation of large--scale\ncluster surveys, still requires substantial work and, ultimately, a\nprecise numerical description of galaxy formation in a cosmological\nframework.\n\n\\section*{Acknowledgements}\nWe thank the anonymous referee for a careful reading of\nthe manuscript and for useful comments, and\n Francisco Navarro--Villaescusa for his help in using Python\nand MPI4Py. All the figures in this paper are plotted using the python\nmatplotlib package \\citep{Hunter:2007}. Simulations have been carried\nout at the CINECA supercomputing Centre in Bologna, with CPU time\nassigned through ISCRA proposals and through an agreement with the\nUniversity of Trieste.  WC acknowledges a fellowship from the European\nCommission's Framework Programme 7, through the Marie Curie Initial\nTraining Network CosmoComp (PITN-GA-2009-238356), and the supports\nfrom ARC DP130100117, from UWA Research Collaboration Awards and from\nthe Survey Simulation Pipeline (SSimPL;\n{\\texttt{http:\/\/www.ssimpl.org\/}}). We acknowledge financial\nsupport from the PRIN-MIUR 2009AMXM79 Grant, from a PRIN-INAF\/2012\nGrant, from the PD51 INFN Grant and from the ``Consorzio per la Fisica\ndi Trieste''.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\t\\subsection{Motivation}\n\t\n\tWe introduce Riemannian first-passage percolation (FPP) as a model of random geometry in the continuum.  Our work is motivated by standard FPP, where one builds a random distance function on $\\mathbb Z^d$ from independent, identically-distributed ``passage times'' over bonds of the lattice.  We instead construct our distance function on $\\mathbb R^d$ from a random Riemannian metric.  The main result of the paper is a shape theorem for Riemannian FPP:  if $B_t$ is the random Riemannian ball of radius $t$, then $\\tfrac{1}{t}B_t$ tends toward a limiting shape.  From this it follows that a random Riemannian metric is geodesically complete with probability one.\n\t\n\tHammersley and Welsh \\cite{hammersley1965fpp} introduced standard FPP in 1965 in order to model fluid flow through porous media.  Consider the $d$-dimensional lattice $\\mathbb Z^d$ with $d \\ge 2$.  Let $\\{t_b\\}$ be a family of independent, identically distributed, non-negative random variables, indexed by bonds (nearest-neighbor edges) $b$ of the lattice.  For any $z,z' \\in \\mathbb Z^d$, define\n\t\t$$d(z,z') = \\inf_{\\gamma} \\sum_{b \\in \\gamma} t_b,$$\n\twhere the infimum is taken over all lattice paths $\\gamma$ connecting $z$ to $z'$.  This $d$ is a random distance function on $\\mathbb Z^d$.  For a very good introduction to standard FPP, see Howard \\cite{howard2004mfp}.  \n\t\n\tConsider the random distance $d(0, n\\E_1)$ between the origin and the point $n\\E_1 = (n,0,\\dots,0)$.  One wishes to study the asymptotic behavior of this quantity as $n \\to \\infty$.  In \\cite{kingman1968ets}, Kingman formulated his famous subadditive ergodic theorem in order to prove the basic result of FPP:  provided the passage times have finite mean, there exists a non-random constant $\\mu_{\\E_1}$, such that \n\t\t$$\\lim_{n\\to\\infty} \\tfrac{1}{n} d(0,n\\E_1) = \\mu_{\\E_1}$$\n\talmost surely and in $L^1$.  The same is clearly true for all coordinate axes and, more generally, for each direction $v \\in S^{d-1}$, there exists a non-random constant $\\mu_v$, such that\n\t\t\\begin{equation} \\label{kingmandiscrete}\n\t\t\t\\lim_{n\\to \\infty} \\tfrac{1}{n}d(0,\\widetilde{nv}) = \\mu_v \\end{equation}\n\talmost surely and in $L^1$, where $\\widetilde{nv} \\in \\mathbb Z^d$ is the nearest lattice point to $nv$.  The constant $\\mu_v$ is non-zero provided that the probability that $t_b = 0$ is less than the critical percolation probability for $\\mathbb Z^d$ \\cite{howard2004mfp}.\n\n\tThe shape theorem of Cox and Durrett \\cite{cox1981slt} is a stronger result.  Consider\n\t\t$$\\tilde B_t = \\{z \\in \\mathbb Z^d : d(0,z) \\le t\\},$$\n\tthe random ball of radius $t$ in $\\mathbb Z^d$.  This is a lattice object, so we ``inflate'' it to get a continuum one:  for $z \\in \\mathbb Z^d$, let $C_z = [z-1\/2,z+1\/2)^d$ be the unit cube centered at $z$ in $\\mathbb R^d$, and let \n\t\t$$B_t = \\bigcup_{z \\in \\tilde B_t} C_z.$$\n\tWe define the rescaling $\\tfrac{1}{t} B_t$ as the set of all points $x \\in \\mathbb R^d$ such that $tx \\in B_t$.  The shape theorem says that there exists a non-random convex, compact set $A$, depending only on the distribution of $t_b$, such that $\\tfrac{1}{t} B_t \\to A$:  for all $\\epsilon > 0$, with probability one, there exists a time $T$ such that if $t \\ge T$, then\n\t\t$$(1-\\epsilon)A \\subseteq \\tfrac{1}{t} B_t \\subseteq (1+\\epsilon)A.$$\n\t\n\tJust as standard FPP is a model of random geometry on the discrete lattice $\\mathbb Z^d$, Riemannian FPP is a model of random geometry in the continuum.  We consider a random Riemannian metric $g$ on $\\mathbb R^d$ whose distribution is translation-invariant, has finite-range dependence and satisfies certain moment conditions.  By the standard construction in Riemannian geometry, this defines a random distance function $d(x,y)$ in $\\mathbb R^d$.  Kingman's theorem can again be applied to prove that for each unit vector $v$, there exists a non-random constant $\\mu_v \\ge 0$ such that $\\tfrac{1}{n} d(0,nv) \\to \\mu_v$ a.s. and in $L^1$.  From the positive-definiteness of $g$, we prove Theorem \\ref{posconst}:  $\\mu_v > 0$ for all $v \\in S^{d-1}$.  Consider the set\n\t\t$$A = \\{x \\in \\mathbb R^d : |x| \\le \\mu_{x\/|x|}^{-1} \\},$$\n\tand the Riemannian ball of radius $t$ centered at the origin\n\t\t$$B_t = \\{ x \\in \\mathbb R^d : d(0,x) \\le t \\}.$$\n\tTheorem \\ref{shapethm} is the shape theorem for Riemannian FPP, which states that for all $\\epsilon > 0$, with probability one, there exists a random time $T > 0$ such that if $t \\ge T$, then\n\t\t$$(1-\\epsilon)A \\subseteq \\tfrac{1}{t} B_t \\subseteq (1+\\epsilon)A.$$\n\tConsequently, we call $A$ the limiting shape of the model.  \n\t\n\tTheorem \\ref{completeness} follows from the shape theorem:  with probability one, no curve $\\gamma$, parametrized by Riemannian length, reaches infinity in finite time.  When the metric is further assumed to be smooth with probability one, then this is geometrically significant:  by the Hopf-Rinow theorem \\cite{lee1997rmi}, this is equivalent to geodesic completeness of the metric.\n\t\n\n\tTo prove our results, we need some technical estimates on the distance function $d$, which we obtain in Section \\ref{disclemmas}.  To prove these, we discretize the continuum model:  to each point $z \\in \\mathbb Z^d$, we associate a certain value $X_z$ based on the Riemannian metric $g(x)$ over the unit cube $C_z = [z-1\/2,z+1\/2)^d$.  By treating $X_z$ as defining a dependent FPP model on the lattice, we prove some estimates of the ``entropy-energy type'' on $X_z$, and from these derive the desired estimates on $d$.\n\t\n\t\n\t\\subsection{Geometry Background and Notation} \n\t\n\tBefore introducing any probabilistic structure, we introduce some geometric notation.  Consider $\\mathbb R^d$ with $d \\ge 2$ and the standard Euclidean coordinates.  Write\n\t\t$$\\operatorname{SPD} = \\{ \\mbox{symmetric, positive-definite $d \\times d$ real matrices} \\},$$\n\tand let $g \\in C(\\mathbb R^d,\\operatorname{SPD})$ be a continuous matrix-valued function on $\\mathbb R^d$ with values in $\\operatorname{SPD}$.  $g$ defines a Riemannian structure on $\\mathbb R^d$:  for tangent vectors $v, v' \\in T_x \\mathbb R^d$, we consider the inner product $\\langle v, g(x) v' \\rangle$.  For a single vector $v$, we denote by $\\|v\\| = \\sqrt{\\langle v, g(x) v \\rangle}$ and $|v| = \\sqrt{\\langle v, v \\rangle}$ the Riemannian and Euclidean lengths of $v$, respectively.  For a $C^1$-curve $\\gamma : [a,b] \\to \\mathbb R^d$, we define the Riemannian and Euclidean lengths of $\\gamma$ by\n\t\t$$R(\\gamma) = \\int_a^b \\| \\dot \\gamma(t) \\| \\sD t \\qquad \\mathrm{and} \\qquad L(\\gamma) = \\int_a^b | \\dot \\gamma(t) | \\sD t,$$\n\trespectively.  We say that a curve is finite if it has finite Euclidean length; for our model, Theorem \\ref{completeness} will imply that finite curves have finite Riemannian length.  The Riemannian distance between two points $x$ and $y$ is defined by\n\t\t$$d(x,y) = \\inf_\\gamma R(\\gamma),$$\n\twhere the infimum is over all $C^1$-curves $\\gamma$ connecting $x$ to $y$.\n\t\n\tFor a Riemannian metric $g$, we define the real, positive functions\n\t\t$$\\Lambda(x) = \\mbox{maximum eigenvalue of $g(x)$} \\qquad \\mathrm{and} \\qquad \\lambda(x) = \\mbox{minimum eigenvalue of $g(x)$}.$$\n\tFor any $K \\subseteq \\mathbb R^d$, define\n\t\t$$\\Lambda(K) = \\sup_{x\\in K} \\Lambda(x) \\qquad \\mathrm{and} \\qquad \\lambda(K) = \\inf_{x\\in K} \\lambda(x).$$\n\tBy the continuity and positivity of $g$, if $K$ is bounded then\n\t\t$$0 < \\lambda(K) \\le \\Lambda(K) < \\infty.$$\n\tFor $z \\in \\mathbb Z^d$, let $C_z = [z - 1\/2, z + 1\/2)^d$ be the unit cube centered at $z$.  Write \n\t\t$$\\Lambda_z = \\Lambda(C_z) \\qquad \\mathrm{and} \\qquad \\lambda_z = \\lambda(C_z).$$\t\n\n\n\t\n\t\\subsection{Riemannian FPP}\n\t\n\n\t\n\tLet $\\Omega = C(\\mathbb R^d, \\operatorname{SPD})$ and let $\\mathcal F$ be the $\\sigma$-algebra generated by cylinder sets.  Let $\\mathbb P$ be a translation-invariant probability measure on $(\\Omega, \\mathcal F)$ which has finite-range dependence, and consider a random Riemannian metric $g \\in \\Omega$ with distribution $\\mathbb P$.  The finite-range dependence means there exists some $R > 0$ such that if $|x-y|\\ge R$, then $g(x)$ and $g(y)$ are independent.  Furthermore, suppose that $\\Lambda_0$ has a finite moment-generating function.  That is,\n\t\t\\begin{equation} \\label{finmom}\n\t\t\tM(r) = \\mathbb E[ \\E^{r \\Lambda_0} ] < \\infty \\qquad \\mbox{for all $r \\in \\mathbb R$,} \\end{equation}\n\twhere $\\mathbb E$ denotes expectation with respect to $\\mathbb P$.  By translation invariance, the family $\\{\\Lambda_z\\}$ is identically distributed, and we refer to its generic element as $\\Lambda$; similarly for $\\{\\lambda_z\\}$ and $\\lambda$.  By Chebyshev's inequality \\cite{durrett1996probability}, \\eqref{finmom} implies that $\\Lambda$ and $\\lambda$ have exponential tail decay:\n\t\t$$\\mathbb P( \\lambda > u ) \\le \\mathbb P( \\Lambda > u ) \\le M(r) \\E^{-r u},$$\n\tfor all $u > 0$ and $r > 0$.\t\n\t\n\tWe provide a concrete example.\n\t\n\t\\begin{env_exa}\n\t\tLet $c : [0,\\infty) \\to \\mathbb R$ be a compactly-supported covariance function (see \\cite{gneiting2002csc} for examples).  Let $\\xi : \\mathbb R^d \\to \\mathbb R$ be a mean-zero, stationary, istropic Gaussian field with covariance function $c$; that is,\n\t\t\t$$\\mathbb E[\\xi(x)] = 0 \\qquad \\mathrm{and} \\qquad \\mathbb E[\\xi(x)\\xi(y)] = c(|x-y|)$$ \n\t\tfor all $x, y \\in \\mathbb R^d$.  The covariance $c$ must satisfy certain necessary and sufficient conditions \\cite{talagrand1987regularity} for the field $\\xi$ to be everywhere continuous with probability one; suppose this is the case.\n\t\t\n\t\tLet $g : \\mathbb R^d \\to \\mathbb R$ be the diagonal matrix with entries\n\t\t\t$$g_{ii}(x) = \\log(1 + \\E^{\\xi(x)}),$$\n\t\tfor $1 \\le i \\le d$.  This is continuous and positive, so it suffices to show that the assumption \\eqref{finmom} is satisfied.  \n\t\t\n\t\t\\begin{env_pro}\n\t\t\tAssumption \\eqref{finmom} is satisfied for this choice of $g$, and the shape theorem (Theorem \\ref{shapethm}) applies.  Let $A$ be the limiting shape, defined in \\eqref{limshape}.  The measure $\\mathbb P$ is isotropic, so Corollary \\ref{isotropic} implies that $A$ is a Euclidean ball.  Furthermore, if the field $\\xi$ is $C^1$ with probability one, then Corollary \\ref{realized} implies that the metric $g$ is geodesically complete.\n\t\t\\end{env_pro}\n\t\t\\begin{proof}\n\n\t\tAssume for simplicity that $c(0) = 1$.  The Gaussian concentration inequality \\cite{adler07} implies that\n\t\t\t$$\\mathbb P(\\Lambda_0 > u) = \\mathbb P(\\sup \\xi > \\log(e^u - 1) ) \\le \\exp(-\\log(\\E^u-1)^2\/2) \\le 2 \\E^{-u^2\/2}.$$\n\t\tThe fundamental theorem of calculus and Fubini's theorem imply that\n\t\t\t$$\\mathbb E \\E^{r \\Lambda_0} = \\mathbb E \\! \\left( 1 + \\int_0^{\\Lambda_0} r \\E^{ru} \\sD u \\right) = 1 + \\int_0^{\\infty} r \\E^{ru} \\mathbb P( \\Lambda_0 > u ) \\sD u \\le 1 + 2 \\int_0^{\\infty} r \\E^{ru} \\E^{-u^2\/2} \\sD u,$$\n\t\twhich is finite for all $r$.  Thus \\eqref{finmom} is satisfied, and the results of this paper apply to the random Riemannian metric $g$.\n\t\t\\end{proof}\n\t\\end{env_exa}\n\n\tThe random variables $\\lambda_z$ and $\\Lambda_z$ give rise to a dependent FPP model on sites of the lattice $\\mathbb Z^d$.  In Section \\ref{depfpp}, we prove some general estimates for dependent FPP, then in Section \\ref{contapps} we apply these to estimates on our distance function $d$.  Our techniques are based on the energy-entropy methods of mathematical physics, where one shows that an event occurs with extremely low probability over one particular connected set (``high energy''), but sums this over all possible connected sets at the origin (``high entropy'').  One adjusts parameters in the problem so that this sum converges, then applies the Borel-Cantelli lemma.  A large-deviations estimate like \\ref{finmom} is critical: the number of connected sets at the origin grows exponentially in $n$, the size of the sets, so the probabilities must decay exponentially in $n$ for the arguments to hold.\n\n\t\n\t\n\t\n\tIn the standard FPP setting of passage times $t_b$ across bonds $b$, Cox and Durrett \\cite{cox1981slt} prove that a necessary and sufficient condition for a shape theorem is that $\\mathbb E \\min\\{t_1, \\dots, t_{2d}\\}^d < \\infty$, where $t_i$ are $2d$ independent copies of $t_b$.  Thus we believe that our assumption \\eqref{finmom} is not the most general, and can be replaced by a finite moment estimate on $\\Lambda$ instead to prove a more general result.  \n\t\n\tIn probability theory, Kingman's subadditive ergodic theorem \\cite{durrett1996probability} is used to prove that stationary, subadditive sequences obey laws of large numbers.  If $X_{n,m}$ is a non-negative, stationary sequence which satisfies $X_{n,m} \\le X_{n,r} + X_{r,m}$, then $\\tfrac{1}{n} X_{0,n}$ converges almost surely and in $L^1$.  Furthermore, if the sequence is ergodic, this convergence is to a non-random constant.  In our context, the sequence in question is $X_{n,m} = d(nv,mv)$ for a fixed unit vector $v$.  The subadditivity condition is exactly the triangle inequality for $d$, so Kingman's theorem implies that for each $v \\in S^{d-1}$, there exists a non-random constant $\\mu_v \\ge 0$ such that\n\t\t\\begin{equation} \\label{kingman}\n\t\t\t\\lim_{t\\to\\infty} \\tfrac{1}{t} d(0,tv) = \\mu_v \\end{equation}\n\talmost surely and in $L^1$.  The constants $\\mu_v$ may depend on the direction $v$, though if the measure $\\mathbb P$ is isotropic (rotationally-invariant) then $\\mu_v = \\mu$ will be independent of $v$.  We show in Theorem \\ref{posconst} that $\\mu_v > 0$ for all $v$.\n\t\n\t\\begin{env_pro} \\label{contmu}\n\t \t$\\mu_v$ is a continuous function of $v$.\n\t\\end{env_pro}\n\t\\begin{proof}\n\t\tThis remarkably short proof is due to Kesten \\cite{kesten1180arp} (see his Proof of Theorem 1.7 on page 158).  First we show that the function $\\mu_v$ is bounded above as a function of $S^{d-1}$.  Write $v = \\sum_1^d v^i \\E_i$, for the standard basis vectors $\\E_i$ in $\\mathbb R^d$.  We use the triangle inequality to bound $d(0,tv)$ by the sum of the distances between successive points $0$, $t v^1 \\E_1$, $t v^1 \\E_1 + t v^2 \\E_2$ and so on until $tv$.  Translation-invariance implies\n\t \t\t$$\\mathbb E d(0,tv) \\le \\mathbb E d(0,tv^1 \\E_1) + \\dots + \\mathbb E d(0,tv^d \\E_d).$$\n\t \tDividing by $t$ and taking the limit $t \\to \\infty$ gives\n \t\t\t$$\\mu_v = \\lim_{t\\to\\infty} \\tfrac{1}{t} \\mathbb E d(0,tv) \\le \\sum_{i=1}^d \\lim_{t\\to\\infty} \\tfrac{1}{t} \\mathbb E d(0, tv^i \\E_d).$$\n \t\tThe terms which equal zero we may ignore; for the non-zero terms, we make the substitution $t' = |v^i| t,$ so that the right-hand side equals\n \t\t\t$$\\sum_{i=1}^d \\lim_{t'\\to\\infty} \\tfrac{|v^i|}{t'} \\mathbb E d(0, \\pm t' \\E_i) = \\sum_{i=1}^d |v^i| \\mu_{\\E_i} \\le d \\max\\{\\mu_{\\E_i}\\},$$\n \t\tas desired.  \n \t\t \t \t\t\n\t \tNow, consider two different unit vectors $v$ and $v'$, and write $u = \\tfrac{v-v'}{|v-v'|}$.  By the same arguments,\n\t \t\t$$|\\mu_v - \\mu_{v'}| \\le \\lim_{t\\to\\infty} \\tfrac{1}{t} \\mathbb E d(tv,tv') = \\mu_u |v - v'|.$$\n\t \tThis tends to zero as $v' \\to v$ since $\\mu_u$ is bounded above.\n\t\\end{proof}\t\n\t\n\t\n\t\n\t\\section{Discretization Lemmas} \\label{disclemmas}\n\t\n\t\\subsection{Dependent FPP on a Lattice} \\label{depfpp}\n\n\t\tFor a continuous curve $\\gamma$, we would like to introduce a discrete analogue $\\Gamma$ on the lattice.  For example, $z \\in \\Gamma$ if $\\gamma$ meets the cube $C_z$.  However, this set $\\Gamma$ is not connected on $\\mathbb Z^d$; consider the straight line from $0$ to $(1,1)$ in $\\mathbb R^2$.  We get around this by modifying the familiar graph structure of $\\mathbb Z^d$ to introduce a new lattice, which we call the $*$-lattice.  In this section, we prove some estimates for dependent FPP on the $*$-lattice, then in Section \\ref{contapps} we apply these estimates to the continuum model.\n\t\t\n\t\tFor $z \\in \\mathbb Z^d$, we write $z = (z^1, \\dots, z^d)$.  We say that $z, z' \\in \\mathbb Z^d$ are $*$-adjacent if $\\max_{1\\le i\\le d} (z - z')^i \\le 1$.  The $*$-lattice is the graph with vertex set $\\mathbb Z^d$, and edge set given by $*$-adjacency; that is, the usual lattice $\\mathbb Z^d$ along with all the diagonal edges.\n\t\t\n\t\tWe say that a set $\\Gamma \\subseteq \\mathbb Z^d$ is $*$-connected if for all $z, z' \\in \\Gamma$, there is a path from $z$ to $z'$ along the $*$-lattice which remains in the set $\\Gamma$.  Technically, that there is a finite sequence of $*$-adjacent points beginning with $z$ and ending with $z'$, all contained in $\\Gamma$.  \n\t\t\n\t\tLet $S_n$ be the number of $*$-connected sets which contain the origin.\n\t\t\n\t\t\\begin{env_lem} \\label{numsets}\n\t\t\tThere exists $\\sigma$ such that $S_n \\le \\sigma^n$.  Obviously, $\\sigma > 1$.  \n\t\t\\end{env_lem}\n\t\t\\begin{proof}\n\t\t\tClearly, $\\log S_n$ is a non-negative subadditive sequence:\n\t\t\t\t$$\\log S_{n+m} \\le \\log S_n + \\log S_m.$$\n\t\t\tBy Fekete's lemma \\cite{fekete1923verteilung}, there exists a constant $a \\ge 0$ such that $\\log S_n \\le a n$ for all $n$.  Defining $\\sigma = \\E^a$ proves the result.\n\t\t\\end{proof}\n\t\t\n\t\tLet $X_z$ be a stationary, non-negative random field on the $*$-lattice with finite-range dependence, and with a finite moment-generating function\n\t\t\t\\begin{equation} \\label{finmom2}\n\t\t\t\tM(r) = \\mathbb E[\\E^{r X}] < \\infty \\qquad \\mbox{for all $r \\in \\mathbb R$.} \\end{equation}\n\t\tThe finite-range dependence means that there is an integer $R \\ge 1$ such that if $|z - z'| \\ge R$, then $X_z$ and $X_{z'}$ are independent.\n\t\t\n\t\tIf $\\Gamma \\subseteq \\mathbb Z^d$ is a collection of lattice points, we write\n\t\t\t$$X(\\Gamma) = \\sum_{z \\in \\Gamma} X_z,$$\n\t\tand call this the passage time of $\\Gamma$. \n\t\t\n\t\tThe following two lemmas can be thought of as spatial laws of large numbers.  The first says that for sufficiently large $n$, if there is a uniform bound on $X(\\Gamma)\/n$, then there is also a uniform bound on $|\\Gamma|\/n$.  The second lemma reverses the implication, though with different constants.\n\t\t\t\t\n\t\t\\begin{env_lem} \\label{passstep}\n\t\t\tSuppose $X_z$ additionally satisfies\n\t\t\t\t\\begin{equation} \\label{zeroatom}\n\t\t\t\t\t\\mathbb P(X = 0) < \\sigma^{-(2R+1)^d}. \\end{equation}\n\t\t\tFor any $A > 0$ there is a non-random $B > 0$ such that, with probability one, for any sequence $a_n \\in \\mathbb Z^d$, there exists $N > 0$ such that for all $n \\ge N$, if $\\Gamma$ is a $*$-connected set which contains the point $a_n$ and $X(\\Gamma) \\le An$, then $|\\Gamma| \\le Bn.$\n\t\t\\end{env_lem}\n\t\n\t\t\\begin{env_lem} \\label{upbound}\n\t\t\tFor any $B > 0$ there is a non-random $C > 0$ such that, with probability one, for any sequence $a_n \\in \\mathbb Z^d$, there exists $N > 0$ such that for all $n \\ge N$, if $\\Gamma$ is a $*$-connected set which contains the point $a_n$ and $|\\Gamma| \\le Bn$, then $X(\\Gamma) \\le Cn.$\n\t\t\\end{env_lem}\n\t\t\n\t\tSome assumption like \\eqref{zeroatom} is necessary for Lemma \\ref{passstep}.  Let $p = \\mathbb P(X = 0)$, and suppose that $p > p_c$, the critical probability for site percolation on the $*$-lattice.  By percolation theory, with probability one, the set\n\t\t\t$$\\Gamma = \\{ z \\in \\mathbb Z^d : X_z = 0 \\}$$\n\t\tcontains an infinite $*$-connected component.  That is, $|\\Gamma| = \\infty$ but $X(\\Gamma) = 0$.  No such assumption is necessary for Lemma \\ref{upbound}.  \n\t\t\n\t\tIn this paper, our assumptions of non-negativity and \\eqref{zeroatom} are stronger than necessary:  we apply these lemmas only to the positive fields $\\lambda_z$ and $\\Lambda_z$.  However, we anticipate these lemmas to be of independent use in future work on Riemannian FPP, where one may consider fields $X_z$ which take the value $0$ on a cube $C_z$ with small but non-zero probability.  For example, if $E_z$ is the event that $\\Lambda_z \\le h$ for a sufficiently large value of $h$, one may apply these lemmas to $X_z = 1_{E_z}$, the indicator function of $E_z$.  \n\t\t\n\t\tThe generality of the sequence $a_n$ is needed for Lemma \\ref{unifT}.  In the proof of that result, we fix a point $x \\in \\mathbb R^d$, and define the sequence $a_n = \\widetilde{nx} \\in \\mathbb Z^d$ to be the nearest lattice point to $nx$.\n\t\t\n\t\t\n\t\t\\begin{proof}[Proof of Lemma \\ref{passstep}]\n\t\tIn what follows, we assume that $\\Gamma$ is a $*$-connected set.  Fix $A > 0$ and the sequence $a_n$.  Consider the events\n\t\t\t$$E_n = \\{ \\exists ~ \\Gamma \\mathrm{~such~that~} a_n \\in \\Gamma, ~ X(\\Gamma) \\le An,\\mathrm{~and~} |\\Gamma| > Bn \\}.$$\n\t\tWe claim that we can choose an integer $B$ large enough so that $\\mathbb P(E_n)$ decays exponentially in $n$.  From the Borel-Cantelli lemma it will follow that, with probability one, only finitely many of the events $E_n$ occur, which will prove the result.\n\n\t\tLet $z_i$ be a predetermined enumeration of $\\mathbb Z^d$; for example, a spiral path beginning at $a_n$.  For any $*$-connected set $\\Gamma$, we define $\\Gamma' \\subseteq \\Gamma$ by proceeding along the sequence $z_i$ and including each point of $\\Gamma$ at a distance at least $R$ away from the previous points chosen.  In the form of an algorithm:\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item Let $i_1$ be the first index for which $z_{i_1} \\in \\Gamma$.  Let $g_1 = z_{i_1}$.\n\t\t\t\t\\item Given $\\{g_1, \\dots, g_{j-1}\\}$, let $i_j$ be the first index for which $z_{i_j} \\in \\Gamma$ and so that $|z_{i_j} - g_{j'}| > R$ for $1 \\le j' < j$.  Let $g_j = z_{i_j}$.\n\t\t\t\\end{itemize}\n\t\tLet $\\Gamma' = \\{g_1, g_2, \\dots\\}$.  If $\\Gamma$ is finite, then so is $\\Gamma'$.  Let \n\t\t\t$$B(z,R) = \\{z' \\in \\mathbb Z^d : |z-z'| \\le R\\}$$\n\t\tbe the Euclidean ball of radius $R$ centered at $z$ in $Z^d$.  Note that $|B(z,R)| \\le (2R+1)^d$.  To avoid this cumbersome factor $(2R+1)^d$ which appears frequently, for the remainder of this proof we write\n\t\t\t$$K = (2R+1)^d.$$\n\t\tBy construction, the set $\\Gamma$ is covered by taking balls around every point in $\\Gamma'$:\n\t\t\t$$\\Gamma \\subseteq \\bigcup_{z \\in \\Gamma'} B(z,R).$$\n\n\t\tLet $\\Gamma$ be a $*$-connected set described in the event $E_n$, so that\n\t\t\t$$Bn < |\\Gamma| \\le \\sum_{z \\in \\Gamma'} |B(z,R)| \\le |\\Gamma'| K.$$\n\t\tThus, $|\\Gamma'| > Bn\/K$.\n\t\t\n\t\tWe may assume that $\\Gamma$ consists only of the first $Bn$ points it meets of the sequence $z_i$; the non-negativity of $X_z$ implies that the passage time $X(\\Gamma)$ is still at most $An$.  Similarly, we assume $|\\Gamma'| = \\lfloor Bn\/K \\rfloor$, the integer part of $Bn\/K$.  Furthermore, since $\\Gamma' \\subseteq \\Gamma$,\n\t\t\t$$X(\\Gamma') \\le X(\\Gamma) \\le An.$$\n\t\tThus the probability $\\mathbb P(E_n)$ is bounded by\n\t\t\t\\begin{equation} \\label{splitestimate}\n\t\t\t\t\\mathbb P \\left( \\exists ~\\Gamma \\mathrm{~s.t.~} a_n \\in \\Gamma, ~ X(\\Gamma') \\le An, \\mathrm{~and~} |\\Gamma| = Bn \\right) \\le \\sum_{\\Gamma} \\mathbb P \\left( X(\\Gamma') \\le An \\right), \\end{equation}\n\t\twhere the outer sum is taken over all $*$-connected sets $\\Gamma$ for which $a_n \\in \\Gamma$ and $|\\Gamma| = Bn$.  Lemma \\ref{numsets} implies that the number of such sets is bounded by $\\sigma^{Bn}$.\n\t\t\n\t\tThe family of random variables $\\{X_z\\}_{\\Gamma'}$ is independent, since the points $z \\in \\Gamma'$ are separated by distances at least $R$.  Let $\\{X_i\\}$ be $\\lceil Bn\/K \\rceil$ independent copies of $X$.  The exponential Chebyshev inequality \\cite{durrett1988lnp} implies that the right-hand side of \\eqref{splitestimate} is bounded above by\n\t\t\t$$\\sigma^{Bn} \\, \\mathbb P \\left( \\sum_{i=1}^{\\lceil Bn\/K \\rceil} X_i \\le An \\right) \\le \\sigma^{Bn} \\, \\E^{r An} \\, \\mathbb E \\left( \\E^{-r \\sum X_i} \\right)$$\n\t\tfor any $r > 0$.  Again by independence, if we write $M(-r) = \\mathbb E \\E^{-r X}$, this is equal to\n\t\t\t$$\\sigma^{Bn} \\E^{r An} \\left(\\mathbb E \\E^{-r X} \\right)^{\\lceil Bn\/K \\rceil} \\le \\sigma^{Bn} \\E^{r An} M(-r)^{Bn\/K} = \\left( \\sigma \\, \\E^{r A \/ B} \\, M(-r)^{1\/K} \\right)^{Bn}.$$\n\t\n\t\tBy the bounded convergence theorem, as $r$ tends to infinity, $M(-r) \\to \\mathbb P(X = 0)$, which is strictly less than $\\sigma^{-K}$ by assumption \\eqref{zeroatom}.  Let $r$ be large enough so that $M(-r) < \\sigma^{-K}$.  Write $p = \\sigma M(-r)^{1\/K} < 1$, so that\n\t\t\t$$\\mathbb P(E_n) \\le (p \\E^{rA\/B})^{Bn}.$$\n\t\tChoose the number $B$ to satisfy \n\t\t\t\\begin{equation} \\label{defB}\n\t\t\t\trA\/\\log(1\/p) < B < rA\/\\log((1+p)\/2p). \\end{equation}\n\t\tThis implies\n\t\t\t$$\\tfrac{1+p}{2} < p \\E^{r A\/B} < 1.$$\n\t\tThe left inequality will be used later in the proof of Thereom \\ref{posconst}; the right inequality implies that\n\t\t\t$$\\sum \\mathbb P(E_n) \\le \\sum (p \\E^{r A\/B})^{Bn} < \\infty,$$\n\t\tso by the Borel-Cantelli lemma, with probability one, only finitely many of the events $E_n$ hold.\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{proof}[Proof of Lemma \\ref{upbound}]\n\t\tIn what follows, we assume that $\\Gamma$ is a $*$-connected set.  Fix an integer $B > 0$ and the sequence $a_n$.  Consider the events\n\t\t\t$$E_n = \\{ \\exists ~ \\Gamma \\mathrm{~such~that~} a_n \\in \\Gamma,~ X(\\Gamma) > Cn,\\mathrm{~and~} |\\Gamma| \\le Bn \\}.$$\n\t\tWe claim that we can choose $C$ large enough so that $\\mathbb P(E_n)$ decays exponentially in $n$.  From the Borel-Cantelli lemma, with probability one, it will follow that only finitely many of the events $E_n$ occur, which will prove the result.\n\t\n\t\tAs in the proof of Lemma \\ref{passstep}, we consider passage times of subsets of $\\Gamma$ to exploit independence.  Unlike in that proof, it does not suffice to consider just one subset $\\Gamma'$.  Instead, we partition the set into $k$ disjoint subsets $\\Gamma_1, \\dots, \\Gamma_k$, and consider the passage times of each.\n\t\t\n\t\tThere are $k = R^d$ points in the cube $\\{0, \\dots, R-1\\}^d$; order them $z_1, \\dots, z_k$.  We can partition the lattice $\\mathbb Z^d$ into $k$ subsets by considering $R$-translations of these points.  We write $z = (z^1, \\dots, z^d)$ for all $z \\in \\mathbb Z^d$.  For any $*$-connected set $\\Gamma$, we partition it into $k$ subsets by defining\n\t\t\t$$\\Gamma_j = \\{ z \\in \\Gamma : \\mbox{$R$ divides $z^i - z_j^i$ for all $i = 1, \\dots, d$} \\}$$\n\t\tfor $j = 1, \\dots, k$.  Since points in $\\Gamma_j$ are separated by distance at least $R$, for each $j$ the family of random variables $\\{X_z\\}_{\\Gamma_j}$ is independent.\n\t\t\n\t\tLet $\\Gamma$ be a set described in the event $E_n$.  We may assume that $|\\Gamma| = Bn$, since if it is less, the inclusion of additional points will only increase the passage time calculation.  Since\n\t\t\t$$X(\\Gamma) = X(\\Gamma_1) + \\dots + X(\\Gamma_k),$$\n\t\tif $X(\\Gamma) > Cn$, then $X(\\Gamma_j) > Cn\/k$ for some $j$.  Furthermore, $|\\Gamma_j| \\le |\\Gamma| \\le Bn$ for each $j$.\n\t\t\n\t\t As in Lemma \\ref{passstep}, the probability $\\mathbb P(E_n)$ is bounded above by\n\t\t\t\\begin{equation} \\label{splitestimate2}\n\t\t\t\t\\sum_{\\Gamma} \\mathbb P \\left( \\exists ~ j \\in \\{1,\\dots,k\\} \\mathrm{~s.t.~} X(\\Gamma_j) > Cn\/k \\right), \\end{equation}\n\t\twhere again the sum is over $*$-connected sets $\\Gamma$ for which $a_n \\in \\Gamma$ and $|\\Gamma| = Bn$.  For each $j$, the family of random variables $\\{X_i\\}_{\\Gamma_j}$ is independent.  Let $\\{X_i\\}$ be $Bn$ independent copies of $X$.  The exponential Chebyshev inequality \\cite{durrett1988lnp} implies that the right-hand side of \\eqref{splitestimate2} is bounded above by\n\t\t\t$$k \\, \\sigma^{Bn} \\, \\mathbb P \\left( \\sum_{i=1}^{Bn} X_i > Cn\/k \\right) \\le k \\, \\sigma^{Bn} \\, \\E^{-Cn\/k} \\, \\mathbb E \\left( \\E^{\\sum X_i} \\right),$$\n\t\twhere we have added extra independent copies of $X_i$ to make the number of terms exactly $Bn$.  If we write $M = M(1) = \\mathbb E \\E^X$, then this is equal to\n\t\t\t$$k \\, \\sigma^{Bn} \\E^{-Cn\/k} M^{Bn} = k \\left( \\sigma^{B} \\E^{-C\/k} M^{B} \\right)^{n}.$$\n\t\tWe choose $C \\gg 1$ so that $\\sigma^{B} \\E^{-C\/k} M^{B} < 1$.  Thus\n\t\t\t$$\\sum \\mathbb P(E_n) \\le k \\sum \\left( \\sigma^{B} \\E^{-C\/k} M^{B} \\right)^{n} < \\infty,$$\n\t\tso by the Borel-Cantelli lemma, with probability one only finitely many of the events $E_n$ hold.\n\t\\end{proof}\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Applications to Continuum Model} \\label{contapps}\n\t\t\n\tIn this section, we apply the estimates from the previous section to the continuum model.  Lemma \\ref{unifT} says that at large scales, the Riemannian distance function between two points is bounded by a uniform constant $K$ times the Euclidean distance between them.  Theorem \\ref{posconst} says that $\\mu_v$ is positive for all $v$.\n\t\n\tThe cubes $C_z$ form a partition of $\\mathbb R^d$.  For $x \\in \\mathbb R^d$, let $\\tilde x$ be the unique point on the lattice such that $x \\in C_{\\tilde x}$.  The metric $g$ induces two notions of ``passage time'' over a discrete set $\\Gamma$:\n\t\t$$\\Lambda(\\tilde\\gamma) = \\sum_{z \\in \\tilde\\gamma} \\Lambda_z \\qquad \\mathrm{and} \\qquad \\lambda(\\tilde\\gamma) = \\sum_{z \\in \\tilde\\gamma} \\lambda_z.$$\n\n\t\\begin{env_lem} \\label{unifT}\n\t\tThere exists a non-random $K > 0$ such that, with probability one, for all $x \\in \\mathbb R^d$ and $\\rho > 0$, there exists $T(x) > 0$ such that if $t \\ge T$ and $|x-y| \\le \\rho$, then\n\t\t\t$$d(tx,ty) \\le Kt\\rho.$$ \n\t\\end{env_lem}\n\t\\begin{proof}\n\t\tBy rescaling $x$ and $y$, it suffices to prove the lemma with $\\rho = 1$.  \n\t\n\t\tApply Lemma \\ref{upbound} with $B = 2d$ and $X_z = \\Lambda_z$.  Thus there exists a non-random $C > 0$ such that, with probability one, for any sequence $a_n \\in \\mathbb Z^d$, there exists $N > 0$ such that for all $n \\ge N$, if $\\Gamma$ is a finite $*$-connected set which contains the point $a_n$ and $|\\Gamma| \\le 2d n$, then $\\Lambda(\\Gamma) \\le Cn$.\n\t\t\n\t\tLet $K = C\\sqrt{d}$, and suppose that with positive probability, there is some $x \\in \\mathbb R^d$ such that for any $n > 0$, there exists $y$ (depending on $n$) such that $|x - y| \\le 1$ but $d(nx, ny) > Kn$.  We will show that this leads to a contradiction.  Let $N$ be as in Lemma \\ref{upbound} applied with the sequence $a_n = \\widetilde{nx}$.\n\t\t\n\t\tSuppose that $n \\ge N$.  Let $\\gamma$ be the straight-line segment from $nx$ to $ny$.  Let\n\t\t\t$$\\Gamma = \\{ z \\in \\mathbb Z^d : \\gamma \\cap C_z \\ne \\emptyset \\},$$\n\t\tindex the cubes $C_z$ which $\\gamma$ meets.  Note that $\\widetilde{nx} \\in \\Gamma$.  Clearly, \n\t\t\t$$|\\Gamma| \\le 2d |nx-ny| \\le 2d n,$$\n\t\tsince $|x-y| \\le 1$.  By Lemma \\ref{upbound},\n\t\t\t$$\\Lambda(\\Gamma) \\le Cn.$$\n\t\t\n\t\tThe distance between $nx$ and $ny$ is a lower bound for the Riemannian length of $\\gamma$:\n\t\t\t$$d(nx, ny) \\le R(\\gamma).$$\n\t\tFurthermore, since $\\gamma$ is a line segment, the Euclidean length of $\\gamma$ in each cube is at most $\\sqrt{d}$.  Thus we can estimate the Riemannian length of $\\gamma$ by summing $\\Lambda_z$ over $\\Gamma$:\n\t\t\t\\begin{equation} \\label{partitionR}\n\t\t\t\td(nx, ny) \\le R(\\gamma) = \\sum_{z \\in \\Gamma} R(\\gamma \\cap C_z) \\le \\sum_{z \\in \\Gamma} \\Lambda_z \\sqrt{d} = \\Lambda(\\Gamma) \\sqrt{d} \\le C n \\sqrt{d} = Kn, \\end{equation}\n\t\tsince $K = C\\sqrt{d}$.  This contradicts the assumption that $d(nx,ny) > Kn$.\n\t\t\t\\end{proof}\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\t\\begin{env_thm} \\label{posconst}\n\t\tThe constants $\\mu_v$ are all positive.\n\t\\end{env_thm}\n\t\\begin{proof}\n\t\tSuppose $\\mu = \\mu_v = 0$ for some unit vector $v$. \n\t\t\n\t\tLet $\\epsilon > 0$, and apply Lemma \\ref{passstep} with the constant sequence $a_n \\equiv 0$ to $A = 4\\epsilon$ and $X_z = \\lambda_z$.  Thus there exists a non-random $B > 0$ such that, with probability one, there exists $N_1 > 0$ such that for $n \\ge N_1$, if $\\Gamma$ is a $*$-connected set which contains the origin and $\\lambda(\\Gamma) \\le 4\\epsilon n$, then $|\\Gamma| \\le Bn$.  \n\t\t\n\t\tBy Kingman's subadditive ergodic theorem, with probability one there exists $N_2 > 0$ such that if $n \\ge N_2$, then\n\t\t\t\\begin{equation} \\label{kingmanposconst}\n\t\t\t\td(0,nv) \\le \\epsilon n \/ 2, \\end{equation}\n\t\tsince we assumed that $\\mu_v = 0$.\n\t\t\n\t\tLet $N = \\max\\{N_1, N_2\\}$, and suppose $n \\ge N$.  \\emph{A priori}, the distance $d(0,nv)$ need not be realized as the Riemannian length of a curve, so let $\\gamma$ be a $C^1$-curve from $0$ to $nv$ with\n\t\t\t$$R(\\gamma) \\le d(0,nv) + \\epsilon n \/ 2 \\le \\epsilon n,$$\n\t\twhere the second inequality follows from \\eqref{kingmanposconst}.\n\t\n\t\tDefine the discrete set\n\t\t\t\\begin{equation} \\label{Gammadef}\n\t\t\t\t\\Gamma = \\{ z \\in \\mathbb Z^d : L(\\gamma \\cap C_z) \\ge 1\/4 \\}; \\end{equation}\n\t\tthat is, $z \\in \\Gamma$ provided the Euclidean length of $\\gamma$ in the cube $C_z$ is at least $1\/4$.  \n\t\t\n\t\tWe claim that $\\Gamma$ is $*$-connected.  Suppose not, so that the continuum set $W = \\bigcup_{z \\in \\Gamma} C_z$ has at least two components, separated by Euclidean distance at least $1$.  Let $W'$ be the $1\/4$-neighborhood around $W$, so that the components of $W'$ are separated by Euclidean distance at least $1\/2$.  By definition of $\\Gamma$, the curve $\\gamma$ meets each component of $W'$, but not the complement $\\mathbb R^d \\setminus W'$.  Since $\\gamma$ is continuous, this is a contradiction; hence, $\\Gamma$ is $*$-connected. \n\n\t\tIn each cube $C_z$, we can estimate the Riemannian length of $\\gamma$ using $\\lambda_z$:\n\t\t\t$$L(\\gamma \\cap C_z) \\lambda_z \\le R(\\gamma \\cap C_z),$$\n\t\twhere $L$ denotes Euclidean length.  Furthermore, by summing $\\lambda_z$ over the points of $\\Gamma$, we get a lower bound for $R(\\gamma)$:\n\t\t\t$$\\tfrac{1}{4} \\lambda(\\Gamma) \\le \\sum_{z \\in \\Gamma} L(\\gamma \\cap C_z) \\lambda_z \\le \\sum_{z \\in \\Gamma} R(\\gamma \\cap C_z) \\le R(\\gamma) \\le \\epsilon n.$$\n\t\tClearly, $0 \\in \\Gamma$, so Lemma \\ref{passstep} implies that $|\\Gamma| \\le Bn$.\n\n\t\tIn the proof of that lemma, we chose $B$ so that\n\t\t\t$$B \\le \\tfrac{r}{\\log((1+p)\/2p)} A,$$\n\t\tfor positive constants $r$ and $p < 1$ not depending on $A$; see \\eqref{defB}.  Since $A = 4\\epsilon$, if we write $B' = 4r\/\\log((1+p)\/2p)$, then \n\t\t\t\\begin{equation} \\label{GammaB}\n\t\t\t\t|\\Gamma| \\le B' \\epsilon n. \\end{equation}\n\t\t\t\n\t\tLet $z$ be a lattice point in $\\Gamma$ which minimizes the distance $|nv - z|$.  Clearly, $|z| \\ge n\/2$.  Since $\\Gamma$ is $*$-connected and $z \\in \\Gamma$,\n\t\t\t$$|\\Gamma| \\ge n\/2\\sqrt{d}.$$\n\t\tFor small $\\epsilon$, this contradicts \\eqref{GammaB}, so $\\mu$ must be positive.\n\t\\end{proof}\n\t\n\t\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{The Shape Theorem and Consequences} \\label{results}\n\t\n\tDefine the function\n\t\t$$\\mu(x) = \\begin{cases} \\mu_{x\/|x|} |x|, & x \\ne 0 \\\\ 0, & x = 0. \\end{cases}$$\n\tProposition \\ref{contmu} and Theorem \\ref{posconst} imply that $\\mu$ is continuous and, for $x \\ne 0$, strictly positive.  It follows from the triangle inequality for the distance function that $\\mu(x)$ is a norm on $\\mathbb R^d$.  Consider the unit ball in this norm,\n\t\t\\begin{equation} \\label{limshape}\n\t\t\tA = \\{ x : \\mu(x) \\le 1 \\} = \\{ x : |x| \\le \\mu_{x\/|x|}^{-1} \\}, \\end{equation}\n\tas well as the random Riemannian ball of radius $t$ centered at the origin,\n\t\t$$B_t = \\{ x : d(0,x) \\le t \\}.$$\n\t\t\n\t\\begin{env_thm}[Shape Theorem]\t\\label{shapethm}\n\t\t\tFor all $\\epsilon > 0$, with probability one, there exists $T$ such that if $t \\ge T$, then\n\t\t\t\t\\begin{equation} \\label{shapestatement}\n\t\t\t\t\t(1-\\epsilon) A \\subseteq \\tfrac{1}{t} B_t \\subseteq (1+\\epsilon) A.\n\t\t\t\t\\end{equation}  \n\t\\end{env_thm}\n\t\n\tThe set $A$ is called the limiting shape of the random Riemannian metric $g$.  The shape theorem for lattice first-passage percolation was proved by Cox and Durrett \\cite{cox1981slt}; our proof is modeled on the arguments in Durrett \\cite{durrett1988lnp}.\n\t\t\n\t\t\\begin{proof}\n\t\t\tIt suffices to prove the theorem for $\\epsilon \\in (0,1)$.  Let\n\t\t\t\t\\begin{equation} \\label{delta}\n\t\t\t\t\t\\delta < \\min \\left\\{ \\frac{1}{K}, 1 - \\frac{1+\\epsilon^2}{1+\\epsilon} \\right\\}, \\end{equation}\n\t\t\twhere $K$ is as in Lemma \\ref{unifT}.  Let $B^{\\mathrm{E}}(x,r)$ denote the Euclidean ball of radius $r$ centered at $x$.\n\t\t\t\n\t\t\tWe will show that with probability one, there exists $T > 0$ such that if $t \\ge T$, then $(1-\\epsilon)A \\subseteq \\tfrac{1}{t} B_t$.  To do this, we will first prove that for every $x$, there is a random $T(x)$ such that if $t \\ge T$, then the small Euclidean ball $B^{\\mathrm{E}}(x,\\delta\\epsilon^2)$ is contained in $\\tfrac{1}{t} B_t$.  Since the set $(1-\\epsilon)A$ is compact, it can be covered by finitely many balls $B^{\\mathrm{E}}(x_i, \\delta\\epsilon^2)$.  Letting $T = \\max\\{T(x_i)\\}$ will prove the result.\n\t\t\t\n\t\tFix $x \\in (1-\\epsilon)A$.  We claim that with probability one, there exists $T(x)$ such that if $t \\ge T$ and $|x-y| \\le \\delta \\epsilon^2$, then $d(0,ty) \\le t$, hence $B^{\\mathrm{E}}(x,\\delta \\epsilon) \\subseteq \\tfrac{1}{t} B_t$ for $t \\ge T$.  By the triangle inequality,\n\t\t\t$$d(0,ty) \\le d(0,tx) + d(tx,ty).$$\n\t\t\n\t\tThe first term is controlled by Kingman's theorem:  with probability one, there exists $T_1(x)$ such that if $t \\ge T_1$, then\n\t\t\t$$d(0,tx) \\le (1+\\epsilon) t \\mu(x) \\le (1-\\epsilon^2) t,$$\n\t\tsince $\\mu(x) \\le (1-\\epsilon)$.\n\t\n\t\tThe second term is controlled by Lemma \\ref{unifT} applied to this $x$ and $\\rho = \\delta\\epsilon^2$.  With probability one, there exists $T_2(x)$ such that for all $t \\ge T_2$ and $y$ with $|x-y| \\le \\delta \\epsilon^2$, then\n\t\t\t$$d(tx,ty) \\le Kt \\delta \\epsilon^2 < \\epsilon^2 t,$$\n\t\tsince $\\delta < 1\/K$.\n\t\t\n\t\tLet $T(x) = \\max\\{T_1, T_2\\}$.  If $t \\ge T(x)$, then for all $y \\in B^{\\mathrm{E}}(x,\\delta \\epsilon^2)$, $$d(0,ty) \\le (1-\\epsilon^2)t + \\epsilon^2 t = t,$$ implying that $B^{\\mathrm{E}}(x, \\delta \\epsilon^2) \\subseteq \\tfrac{1}{t} B_t$ for all $t \\ge T(x)$.  Let $x_i$ be finitely many points such that $\\bigcup B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2)$ covers $(1-\\epsilon)A$, and let $T = \\max_i T(x_i)$.  Then for all $t \\ge T$,\n\t\t\t$$(1-\\epsilon)A \\subseteq \\bigcup_i B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2) \\subseteq \\tfrac{1}{t} B_t,$$\n\t\tcompleting the lower half of the shape theorem.\n\t\t\n\t\tNow we prove the upper half.  For any $x$, let $T_2(x)$ be as above, so that if $t \\ge T_2$ and $|x-y| < \\delta \\epsilon^2$, then $d(tx,ty) \\le \\epsilon^2 t$.  By Kingman's theorem, with probability one, there exists  $T_3(x)$ such that if $t \\ge T_3$, then\n\t\t\t$$d(0,tx) \\ge  (1-\\delta) t \\mu(x).$$\n\t\tChoose finitely many $x_i \\in 2A \\setminus (1+\\epsilon)A$ such that the closure of $2A \\setminus (1+\\epsilon)A$ is covered by $\\bigcup B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2)$.  Belonging to $2A \\setminus (1+\\epsilon)A$ implies that $\\mu(x_i) > (1+\\epsilon)$.  Let $T = \\max_i \\{T_2(x_i), T_3(x_i)\\}$, and let $t \\ge T$.  Then if $y \\in B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2)$,\n\t\t\t\\begin{eqnarray*}\n\t\t\t \td(0,ty) &\\ge& d(0, tx_i) - d(tx_i, ty) \\\\\n\t\t\t\t\t&\\ge& (1-\\delta)t \\mu(x_i) - \\epsilon^2 t \\\\\n\t\t\t\t\t&\\ge& (1-\\delta)(1+\\epsilon)t - \\epsilon^2 t \\\\\n\t\t\t\t\t&>& (1+\\epsilon^2) t - \\epsilon^2 t = t,\n\t\t\t\\end{eqnarray*}\n\t\twhere the final inequality follows from the assumption \\eqref{delta} on $\\delta$.  Thus $\\bigcup B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2) \\subseteq \\tfrac{1}{t} B_t^c$ so\n\t\t\t$$2A \\setminus (1+\\epsilon)A \\subseteq \\bigcup B^{\\mathrm{E}}(x_i, \\delta \\epsilon^2) \\subseteq \\tfrac{1}{t} B_t^c.$$\n\t\tThe set $\\tfrac{1}{t} B_t$ is connected and contains the origin, hence $\\tfrac{1}{t} B_t \\subseteq (1+\\epsilon)A$ as desired.\n\t\\end{proof}\n\t\n\t\\begin{env_cor} \\label{isotropic}\n\t\tIf the measure $\\mathbb P$ is isotropic (rotationally-invariant), then the limiting shape $A$ is the Euclidean ball of radius $\\mu^{-1}$. \\hfill $\\square$\n\t\\end{env_cor}\n\t\n\tA simple consequence of Lemma \\ref{unifT} is that the convergence \\eqref{kingman} given by Kingman's theorem is uniform:\n\t\n\t\\begin{env_pro} \\label{unifconv}\n\t\tFor all $\\epsilon > 0$, with probability one, there exists $T>0$ such that if $t \\ge T$, then for all $v \\in S^{d-1}$,\n\t\t\t$$\\left| \\tfrac{1}{t} d(0,tv) - \\mu_v \\right| \\le \\epsilon.$$\n\t\\end{env_pro}\n\t\n\t\\begin{proof}\n\t\tSuppose not.  Then with positive probability, there exist $\\epsilon > 0$, $t_n \\to \\infty$ and $v_n \\in S^{d-1}$ such that\n\t\t\t$$\\left| \\tfrac{1}{t_n} d(0,t_n v_n) - \\mu_{v_n} \\right| > \\epsilon,$$\n\t\tfor all $n$.  By compactness of the sphere, a subsequence of $v_n$ converges to some $v \\in S^{d-1}$; assume without loss of generality that $v_n \\to v$.  By the triangle inequality,\n\t\t\t\\begin{equation} \\label{epsunif}\n\t\t\t\t\\epsilon < \\left| \\tfrac{1}{t_n} d(0,t_nv_n) - \\tfrac{1}{t_n} d(0, t_nv) \\right| + \\left| \\tfrac{1}{t_n} d(0,t_nv) - \\mu_v \\right| + \\left| \\mu_{v_n} - \\mu_v \\right|. \\end{equation}\n\t\tThe second and third terms tend to zero a.s. as $n \\to \\infty$ by Kingman's theorem and Proposition \\ref{contmu}, respectively.  By the triangle inequality, the first term is bounded by $\\tfrac{1}{t_n} d(t_nv, t_nv_n)$.  Let $\\rho = \\epsilon \/ 2K$, where $K$ is as in Lemma \\ref{unifT}.  Since for large $n$, $|v - v_n| \\le \\rho$, we can apply that lemma with $x = v$ to get\n\t\t\t$$\\tfrac{1}{t_n} d(t_nv,t_nv_n) \\le K\\rho = \\epsilon \/2,$$\n\t\tfor large $n$ almost surely.  This contradicts \\eqref{epsunif}.\n\t\\end{proof}\n\t\n\t\n\t\\begin{env_thm} \\label{completeness}\n\t \tWith probability one, if $\\gamma$ is a $C^1$-curve parametrized by Riemannian length, then $|\\gamma(t)| < \\infty$ for all $t \\ge 0$. \n\t\\end{env_thm}\n\t\\begin{proof}\n\t\tIt suffices to consider curves starting from the origin.  Suppose that with positive probability, there exists a smooth curve $\\gamma$ parametrized by Riemannian length which starts from the origin and for which\n\t\t\t\\begin{equation} \\label{Tlim}\n\t\t\t\t\\lim_{t \\to T} |\\gamma(t)| = \\infty \\end{equation}\n\t\tfor some finite $T$.  Since $\\gamma$ is parametrized by Riemannian length, for all $t \\ge 0$,\n\t\t\t$$t \\ge d(0,\\gamma(t)).$$\n\t\tProposition \\ref{unifconv} and \\eqref{Tlim} imply that\n\t\t\t$$0 = \\lim_{t\\to T} \\frac{t}{|\\gamma(t)|} \\ge \\lim_{t\\to T} \\frac{d(0,\\gamma(t))}{|\\gamma(t)|} \\ge \\min_{v} \\mu_v > 0,$$\n\t\ta contradiction.\n\t\\end{proof}\n\t\n\tWe can refine this result if we impose an additional smoothness constraint on the metric.  If $g$ is a $C^2$-smooth Riemannian metric, i.e. $g \\in C^2(\\mathbb R^d, \\operatorname{SPD})$, then we can use the calculus of variations to derive the Euler-Lagrange equations for the functional $R$.  These are called the geodesic equations \\cite{lee1997rmi} for the Riemannian metric $g$, and solutions to this system are called geodesics.  The geodesic equations form a second-order system with locally-Lipschitz coefficients, so a geodesic is uniquely determined by its starting point and velocity.  We call a geodesic $\\gamma$ length-minimizing if for all $x, y \\in \\gamma$, the distance $d(x,y)$ is realized as the Riemannian length of the part of the curve $\\gamma$ which connects the two points.  Not all geodesics are length-minimizing; for example, on the sphere, the geodesics are great circles, which do not minimize length past antipodal points.\n\n\tA metric is said to be geodesically complete if for all $x \\in \\mathbb R^d$ and $v \\in T_x \\mathbb R^d$, the unique geodesic $\\gamma$ at $x$ in direction $v$ can be continued for all time.  Part of the Hopf-Rinow theorem \\cite{lee1997rmi} of Riemannian geometry is that geodesic completeness is equivalent to the condition stated in Theorem \\ref{completeness} for our random metric $g$.  A further corollary \\cite{lee1997rmi} is that distances are always realized by geodesics.  Summarizing, we have\n\t\n\t\\begin{env_cor} \\label{realized}\n\t\tSuppose that, in addition to the assumptions of this paper, $g$ is a $C^2$-smooth random Riemannian metric.  Then, with probability one, $g$ is geodesically complete.  Consequently, for all $x, y \\in \\mathbb R^d$, there is a finite length-minimizing geodesic $\\gamma$ connecting $x$ to $y$ such that\n\t\t\t$$d(x,y) = R(\\gamma).$$\n\t\\end{env_cor}\n\n\nWe alert the reader to a different meaning of the word ``geodesic,'' used often in the first-passage percolation literature.  The term is used there to denote a globally length-minimizing path.  This is very different from the standard meaning of the word in differential geometry: as described above, geodesics are the curves which locally minimize length, but not necessarily globally.  We adhere to this meaning in the present paper.\n\n\n\tThe existence of two-sided minimizing paths is an open question for all FPP models.  For two-dimensional standard FPP, Licea, Newman and Piza \\cite{licea1996geodesics, licea1996superdiffusivity, newman1997tds} have a number of results in this direction.  Their work relies on certain curvature assumptions about the limiting shape, which have not been verified for models of independent FPP.  Howard and Newman's \\cite{howard1997euclidean, howard2001special} model of Euclidean FPP, on the other hand, is rotationally invariant.  Consequently, the limiting shape is a Euclidean ball, and they prove many results not available in the lattice setting.  See the excellent survey \\cite{howard2004mfp} for more details.  By Corollary \\ref{isotropic}, the limiting shape for isotropic Riemannian FPP is a Euclidean ball.  Our hope is for this setting to be a fertile ground for adapting the results referenced in this paragraph.\\newline\n\n\t{\\bf Remark added after publication:}  The moment assumptions \\eqref{finmom} and \\eqref{finmom2} can be slightly weakened to:  $$M(r) = \\mathbb E[ \\E^{r \\Lambda_0} ] < \\infty \\qquad \\mbox{for all $r \\le a$},$$ for some $a > 0$.  The only modification to the paper is in the proof of Lemma \\ref{passstep}, where one uses $M = M(a)$ instead of $M = M(1)$.  We leave the details to the reader.\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAqueous solutions under confinement are present and essential in a wide range of chemical and biological processes. Confinement of aqueous electrolytes in carbon nanopores is of particular interest due to its relevance in several technological areas such as elec\\-tro\\-che\\-mi\\-cal double layer capacitors (EDLCs)\\cite{Fic12,Sajjad21} and capacitive deionization.\\cite{Porada13,Luciano20} Confinement alters significantly the properties of the adsorbed species with respect to their bulk counterparts which can in turn affect the performance of these devices. Characterization of quantities such as the hydration structure of the ions in the pores, ion dynamics, ion distribution across pores of different sizes, and ion-carbon wall interactions is thus necessary in order to improve these technologies and has been the subject of numerous studies. \n\nExperimental attempts to understand the interfacial behaviour of aqueous electrolytes under confinement have involved techniques such as X-ray diffraction,\\cite{Ohba12} extended X-ray absorption fine structure,\\cite{Ohkubo02} small-angle X-ray scattering,\\cite{Prehal15,Prehal17} electrochemical quartz crystal microbalance,\\cite{Escobar-Teran22,Wu18} Raman spectroscopy,\\cite{Nunes20} nuclear magnetic resonance spectroscopy,\\cite{Cervini19,Luo15a,Luo15b} and have provided valuable information about the electrolyte confinement in materials such as carbon nanotubes, slit-shaped nanospaces, graphene nanowrinkles, and nanoporous carbons. However, a quantitative microscopic characterization is still out of reach with many investigation techniques, especially for disordered porous carbons, due to their structural complexity and the existence of fast motion in the systems.\n\nThanks to its quantitative and nucleus specific features, nuclear magnetic resonance (NMR) spectroscopy is a remarkable technique to study a wide range of systems. The chemical shifts of NMR active nuclei are sensitive to their local environment making this technique compatible with the investigation of aqueous electrolyte species in the bulk and under confinement. More precisely, the peaks observed for species adsorbed to the carbon surface are shifted in the NMR spectrum relative to the species in the bulk electrolyte allowing for a clear identification and quantification of the confined species.\\cite{Harris96,Dickinson2000,Griffin14,Griffin16} The additional possibility to conduct NMR measurements while applying a potential difference between the electrodes has also been exploited to study charging mechanisms in EDLCs\\cite{Forse16,Griffin14} and even characterize ion diffusion for different states of charge.\\cite{Forse17} \n\nThe chemical shift of the species confined in porous carbonaceous materials is mainly the consequence of the secondary magnetic shielding generated by ring currents arising on the carbon surface when applying a primary magnetic field to the system.\\cite{Lazzeretti2000,Anderson10} This phenomenon can be gauged using a Nucleus Independent Chemical Shift (NICS) approach.\\cite{Chen05,Gershoni-Poranne21} NICS can be evaluated using density functional theory (DFT) calculations and previous studies have shown, amongst other effects, that the shielding effect is larger when the ions are closer to the surface of the carbon.\\cite{Xing14,Forse14} Hence the distribution of the ions at the interface and the porosity of the carbon are expected to have a major influence on the chemical shift of the adsorbed species. The local structure, and especially the size of the aromatic domains, was also shown to have a large effect which could be used to get insights into the structures of a range of porous carbons.\\cite{Forse15b} \n\nWhile the ring currents are predominant in determining the chemical shifts for a range of electrolytes,~\\cite{Forse21,Sasikumar21} other parameters can lead to significant modifications of the chemical shifts. NMR spectra are known to be influenced by the hydration number of the ions under consideration, the dehydration of ions upon entering the micropores can thus lead to large chemical shift changes.\\cite{Luo15b,Gerken02} Due to exchange between the environments inside and outside of the pores and motional averaging, the carbon particle size can also lead to large variations.\\cite{Cervini19} The concentration of ions in organic and aqueous electrolytes adsorbed on microporous carbon can further influence the overall chemical shift.\\cite{Cervini19,Fulik18} The interpretation of NMR spectra is, as a consequence, both rich and complex, and distinguishing the contributions from the above mentioned factors is an essential task for a better understanding of the behaviour of the electrolyte species at a local scale. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.58]{Figures\/Figure1.png}\n\\caption{Depending on the carbonaceous material and electrolyte, several factors can contribute differently to the chemical shifts of electrolyte species and consequently to the chemical shift difference, $\\Delta\\delta$, between bulk (ex-pore) and confined (in-pore) ions.}\n\\label{factors}\n\\end{figure}\n\nWhile DFT calculations are adapted to estimate NMR parameters of small static systems, they are not adequate to represent the variety of environments existing in real systems and the ion motion. Molecular dynamics (MD) simulations are more suited for such studies. Feng~\\textit{et~al.}\\cite{Feng10} conducted si\\-mu\\-la\\-tions of K$^+$ ions in water, in contact with slit pores, and demonstrated that the distribution of these ions in electrified pores of different sizes depend on the ion hydration, the water-water interactions and the pore size. Beckstein~\\textit{et~al.}\\cite{Beckstein04} used molecular simulations of an aqueous electrolyte in contact with different nanopores and channels to evaluate the energy barriers to Na$^+$ ion permeation, related to ion dehydration, through pores with various radii smaller than 1~nm. Another study on MD simulations of an aqueous NaCl electrolyte and carbide derived carbons (of average pore sizes 0.75~nm and 1.0~nm) have revealed that unlike large organic ions dissolved in acetonitrile, the desolvation of Na$^+$ ions is limited in the carbon nanopores.\\cite{ganfoud19} Overall, these works show that different ions are more or less susceptible to dehydration and as a consequence, to fully interpret NMR spectra, one has to consider both the NMR parameters for specific ion configurations and the distribution and exchange of ions between these environments.\n\nIn previous works, we proposed to combine the results from DFT calculations and MD simulations using a mesoscopic model to determine NMR spectra.\\cite{Merlet15,Sasikumar21} This method takes into account information on i)~ion adsorption and organization, ii)~local magnetic shielding, and iii)~pore size distribution. This technique was shown to give results in good agreement with experimental NMR, Raman spectroscopy and pair distribution function analysis for a range of porous carbons.\\cite{Forse15b} The mesoscopic model is also suitable for predicting \\emph{in situ} NMR spectra for supercapacitors based on organic electrolytes.\\cite{Sasikumar21} One of the assets of the model is to be able to modify independently the ion adsorption and local magnetic shielding entries allowing one to assess the relative importance of these two factors. Thanks to this feature, it was possible to demonstrate that, for organic electrolytes, the variations of the magnetic shieldings, as a consequence of the changes in electronic structure with charging (negatively or positively), have a predominant influence on the overall chemical shift compared to ion organization.\\cite{Sasikumar21} This result is consistent with experiments conducted on a range of electrolytes showing similar variations of the chemical shift with applied potential.\\cite{Forse21,Fulik18,Wang11b} \n\nIn this work, we investigate aqueous electrolytes with various alkali metal ions in contact with porous carbon particles corresponding to polyether ether ketone derived carbons (PDCs). The motivation for this study is to explore additional effects observed experimentally with aqueous electrolytes compared to organic electrolytes (see Figure~\\ref{factors}).\\cite{Cervini19} One interesting aspect of the PDCs is that their pore size distributions depend on the activation conditions. Here the PDCs will be referred to using burn-off values which denote the percentage of mass lost after the activation step and is thus indicative of the porosity generated. \n\n\nThe $^7$Li, $^{87}$Rb $^{133}$Cs, and $^1$H NMR spectra for LiCl, RbCl, and CsCl solutions (1M) in various PDCs were simulated using the mesoscopic model previously developed (also referred to as ``lattice model\" in the remainder in the article).\\cite{Merlet15,Sasikumar21} The information on ion adsorption and organization is provided as free energy profiles (extracted from MD simulations and shown in Figure~S2), the information on local magnetic shielding is given by NICS profiles (calculated through DFT) and the pore size distributions are the ones obtained previously from adsorption isotherms (see Figure~S3).\\cite{Cervini19} All data considered correspond to a neutral carbon particle. While the free energy and NICS profiles are calculated in the proximity of a single carbon surface, the lattice model considers slit pores with two carbon surface assuming symmetrical free energies and additive NICS. It is worth noting that any specific interaction or charge transfer between ions \/ water molecules and the carbon surface are neglected at this stage. The chemical shifts of in-pore species relative to the bulk ($\\Delta\\delta_{\\rm NICS}$) were evaluated and compared against experimental chemical shifts obtained by Cervini \\emph{et~al.}.\\cite{Cervini19} Results are shown in Figure~\\ref{compare_shift}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.3]{Figures\/Figure2.png}\n\\caption{Comparison of average chemical shifts of the in-pore alkali metal ions and hydrogen atoms of water relative to the bulk species in various PDCs evaluated by lattice simulations and experiments.\\cite{Cervini19} Different symbols and colors correspond to different nuclei probed. Filled symbols correspond to experimental results while open symbols correspond to lattice model results. The $\\Delta\\delta_{\\rm NICS}$ values for Rb$^+$ and Cs$^+$ are superimposed.}\n\\label{compare_shift}\n\\end{figure}\n\nIn general, the simulated $\\Delta\\delta_{\\rm NICS}$ values decrease with an increase in the burn-off value of the PDC. This is in qualitative agreement with the experiments. The pore size distributions for the PDCs corresponding to the studied range of burn-off values considered show pores in the micropore and meosopore ranges with a large amount of pores around 0.8~nm, 1.1~nm and 2.2~nm.\\cite{Cervini19} With the increase in burn-off value, there is usually an increase of pores in the 2-3~nm range. As DFT calculations show that NICS are larger for small pore sizes\\cite{Xing14,Forse14}, an increase of the average pore size is expected to lead to a smaller $\\Delta\\delta$ values. The data for NICS is included in the lattice model, as well as the pores size distributions, which allows for a reproduction of the $\\Delta\\delta$ variation with burn-off values.\n\nLooking closer at the trend, the $\\Delta\\delta_{\\rm NICS}$ observed for all the species under consideration at a burn-off value of 50\\% are slightly smaller (- 5\\%) than that of 56\\%. In addition to the $\\Delta\\delta_{\\rm NICS}$ values, the lattice model also gives access to the ions or molecules distributions across pores of different sizes. The ion distributions for Li$^+$ and Cs$^+$ in the three PDCs considered are given in Figures~S3 and~S4 (results for Rb$^+$ are very similar to the results for Cs$^+$). For the PDC with 56\\% burn off, the fraction of ions occupying pores sizes in the range 1.9-3.2~nm is larger than for the PDC with 56\\% burn off leading to a smaller $\\Delta\\delta_{\\rm NICS}$ value for this PDC compared to the 50\\% material.\n\nWhile the trend with burn-off values is qualitatively similar for the various PDCs, the variation of the chemical shifts with the porosity, i.e. the slope of the curve, is slightly overestimated by the simulations. This discrepancy can come from the hypotheses made in the lattice model regarding the NICS calculated through DFT (\\emph{e.g.} averages across NICS profiles calculated for different aromatic molecules, positions at which the NICS are calculated close to the carbon surface, ...) which are described in details in previous works.\\cite{Merlet15,Sasikumar21} This could also be due to the difference in the carbon structures used for NMR experiments and adsorption studies to determine the pore size distribution. It is worth mentioning that the determination of pore size distributions is subject to some limitations. Different pore size distributions can be obtained for a given porous carbon depending for example on the probe molecule chosen and the model adopted to interpret the adsorption isotherms.~\\cite{Dombrowski00,Neimark09} Overall, the agreement between the lattice model and experiments for the trend across burn-off values is satisfying. \n\nThe $\\Delta\\delta_{\\rm NICS}$ calculated using the lattice model for all cations ($^7$Li, $^{87}$Rb, $^{133}$Cs) are similar and smaller than the ones of $^1$H for all porous carbons considered. The small difference of 1-2~ppm between $^1$H, $^7$Li and $^{87}$Rb\/$^{133}$Cs can be ascribed to the ion organization at the interface with carbon. Indeed, the free energy profiles extracted from MD simulations show that hydrogen atoms come closer to the carbon surface than the other species, followed by Li$^+$ and by Rb$^+$ and Cs$^+$ (see Figure~S2). As a consequence, the lattice model predicts that the population of Li$^+$ ions (or hydrogen atoms) in small pores is larger than the one of Cs$^+$ (or Rb$^+$) ions leading to a smaller $\\Delta\\delta_{\\rm NICS}$ value for large ions. This is illustrated in Figures~S3 and~S4 giving the distribution of Li$^+$ and Cs$^+$ ions in pores of different sizes. However, one should keep in mind that the lattice model only takes adsorption into account through average free energy profiles obtained at a planar surface and specific energy penalties related to desolvation could change the picture in very small pores. \n\nWhile the lattice model satisfactorily predicts the $\\Delta\\delta$ values of $^1$H and $^7$Li with respect to the experiments\\cite{Cervini19}, the values calculated for $^{87}$Rb and $^{133}$Cs are significantly underestimated, by 4-5 ppm. In addition, while the relative variation in $\\Delta\\delta$ between $^1$H and $^7$Li is reproduced by the lattice model, the relative difference between these two species and $^{87}$Rb\/$^{133}$Cs is completely wrong. This shows that the ion organization at the carbon surface and the resulting distribution in the pores, already taken into account in the lattice model, cannot be responsible for this shift. \n\nIn previous works, it was shown that more than ion organization, notable contributions from ring currents on the total shift are to be expected\\cite{Sasikumar21}. In the case of interest here, variations in ring currents may arise due to specific interactions and charge transfer between the electronic density of the alkali metal ions and the carbon. These effects are neglected in the NICS calculations used so far but they can be explored with DFT. \n\nThe chemical shift profiles for Li$^+$, Rb$^+$, and Cs$^+$ ions at various distances from a circumcoronene molecule were calculated by DFT (see Figure~S5). The shifts calculated, referenced to the bare ion in vacuum, are shown in Fi\\-gu\\-re~\\ref{shift_M+}. For completeness, the chemical shifts for other alkali metal ions in the series (Na$^+$ and K$^+$) are given in Figure~S6. For distances below 0.6~nm, the chemical shifts deviate increasingly from the NICS as the ion size increases, which corresponds to an increasing polarizability. The values for Li$^+$ are extremely close to the NICS values. Overall, more negative chemical shifts compared to NICS are observed for the Li$^+$ and Rb$^+$ with the deviation being much larger for the latter. The deviation in the shift is -0.2~ppm and -3.7~ppm for Li$^+$ and Rb$^+$ respectively at a distance of 0.35~nm. The case of Cs$^+$ is strikingly different from the other ions with an increase of shift by 10.7~ppm compared to the NICS at 0.35~nm. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.3]{Figures\/Figure3A.png}\n\\includegraphics[scale=0.3]{Figures\/Figure3B.png}\n\\caption{a) Chemical shift profiles of various alkali metal ions with respect to vacuum calculated at various distances from the centre of the circumcoronene molecule and NICS values at the same distances. b) $\\Delta\\delta$ values of the in-pore species relative to the bulk species obtained using the lattice model with the NICS profiles (NICS) and the chemical shift profiles in presence of the ions (CSP).}\n\\label{shift_M+}\n\\end{figure}\n\nThe origin of the deviation of the chemical shift of the ion from the NICS is not well understood. As the ring currents are sensitive to the molecule charge,\\cite{Sasikumar21} a charge transfer between the ion and aromatic molecule could be a reason for such a variation. To test this hypothesis, Bader and Mulliken charges on atoms of the cation - circumcoronene systems were calculated.\\cite{g16,bader85,Lu12} The calculations were done for cations 0.35~nm away from the centre of the circumcoronene. The results show evidence of a limited charge transfer of 0.03~e maximum between the cations and the carbon. Interestingly, previous studies have analyzed a small change in the ring currents upon interaction with an ion and the nature of the ion does not affect the ring current changes in the carbon.\\cite{Guell05,Rodriguez-otero08,Mary18}. \n\nOne way to evaluate the variation in ring currents due to the proximity of the ion is the calculation of the NICS(1)\\cite{Chen05}, i.e the chemical shift of a ``ghost\" atom located at 0.1~nm of the ion-circumcoronene system, below the plane of the circumcoronene molecule on the opposite side of the ion. The NICS(1) is evaluated to be -13.6~ppm and -14.0~ppm respectively for Li$^+$ and Cs$^+$. The NICS(1) corresponding to the circumcoronene molecule without any ion nearby is -15.3~ppm. While there is a small change between the systems with and without ion, the variation is probably too small to explain the change in $\\Delta\\delta$ values for large ions. \n\nTo evaluate the influence on the $\\Delta\\delta$ values of the variations in chemical shifts observed for large polarizable ions, with respect to NICS, the lattice simulations were repeated with the chemical shift profiles calculated with the actual ions. The results obtained are shown in Figure~\\ref{shift_M+}. The $\\Delta\\delta_{\\rm CSP}$ and $\\Delta\\delta_{\\rm NICS}$ values are very similar and as such cannot explain the $\\Delta\\delta$ values observed experimentally for Rb$^+$ and Cs$^+$. It is worth noting that the free energy profiles for these large ions (Figure~S2) show a first minimum close to 0.6~nm indicating that very few ions get closer to the carbon than this distance. As the chemical shift profiles and NICS values are not significantly different at these distances even for the large ions, it is not surprising that the variations in chemical shifts due to ion-carbon interactions do not lead to noticeable changes in $\\Delta\\delta$. \n\nOne possibility worth investigating is the fact that MD simulations with planar electrodes, used here to extract free energy profiles, may lead to an underestimation of the amount of ions in smaller pores as shown in other systems.\\cite{Sasikumar21} A more realistic representation of the experiment could be obtained from MD simulations with porous carbon electrodes to check the occupancy of the small pores by the cations. This is usually much more involved computationally and less general so it is out of scope of the current work. Instead, MD simulations of the electrolytes confined in slit pores of width 0.8~nm or 1.1~nm were conducted. These simulations do not show any evidence of cations inside the pore with a width of 0.8~nm. In the slit pore of width 1.1~nm, it was observed that Cs$^+$, Rb$^+$ and Li$^+$ ions occupy preferentially the centre of the pore, approximately 0.55~nm away from the carbon surface (see ion and water densities for the in Figure~S9). Interestingly, the MD simulations suggest that the adsorption of Cs$^+$ ions is slightly more favorable than the one of the other cations, in contrast with the lattice simulation results, but the effect is limited. Following these considerations, no major shift is expected from specific ion-carbon interactions. \n\nOverall, the results from the lattice model, relying on MD simulations and DFT calculations, suggest that ion distributions in pores of different sizes and NICS values are sufficient to explain chemical shifts for Li$^+$ and hydrogen atoms from the water molecules but not for the larger Rb$^+$ and Cs$^+$ ions.  \n\n\nTheoretical calculations have indicated that the hydration number of ions can have a major effect on their chemical shift, this was shown for example for Na$^+$ and F$^-$ ions for various hydration numbers.\\cite{Luo15b,Gerken02} This is not accounted for in the lattice simulations. The adsorption and dehydration of ions in hydrophobic carbon pores are dependent on an energy barrier related to the solvation energy of the ions\\cite{Luo15b,Beckstein04} and on the pore size. The larger ions, Cs$^+$ and Rb$^+$, are more polarizable than the Li$^+$ ion and have a weaker hydration shell that can distort easily as opposed to the case of Li$^+$. They are therefore more prone to dehydration potentially leading to a significant variation of the chemical shift for confined species. Furthermore, in the vicinity of the carbon, the presence of solvent molecules can affect both the interaction of the ion with the carbon\\cite{Rao09} and the carbon itself thus altering the ring currents and the chemical shift. Here, we first investigate the dehydration effect in the bulk before exploring the interface between carbon and ion-solvent complexes. We mainly focus on Cs$^+$ as we expect a similar behaviour for Rb$^+$. \n\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[scale=0.62]{Figures\/Figure4.png}\n\\caption{Scheme depicting the chemical shift variations for different ion-water-circumcoronene systems corresponding to various local environments for the Cs$^+$ and Li$^+$ ions.}\n\\label{dehydration_shifts}\n\\end{figure*}\n\nTo study the effect of the hydration number on the chemical shift, four configurations of a fully hydrated Cs$^+$ ion in the bulk were extracted from the MD simulations and the chemical shift of the hydrated ion relative to the bare ion was assessed for different hydration numbers by removing water molecules from the hydration shell (see Supplementary Information for details). Starting with configurations with 9 water molecules (a frequent arrangement in the MD simulations), water molecules were removed one by one until reaching the bare ion state which serves as a reference. Between the full hydration and the bare ion, the variation in chemical shift is huge, reaching almost 200~ppm. It is worth noting that low solvation numbers (below 4) are not present in solution, as can be seen in the MD simulations (see Figure~S10). For hydration numbers between 9 and 6, the variation of the chemical shift with the removal of one water molecule is relatively small ranging from 2 to 10~ppm. For hydration numbers below 6, the difference in chemical shift becomes larger in the range of 10 to 50~ppm. In addition to the effect of the number of water molecules, the calculations indicate that, for a given hydration number, the orientation of the water molecules and the ion-water distances affect the chemical shift significantly. A variation as large as 75~ppm is observed as local structure changes. \n\nFrom the DFT calculations, it seems that a dehydration of the ions in the pores could lead to a very large shift. To complement this study, the MD simulations were used to determine the distribution of hydration numbers for the electrolyte species in the bulk and under confinement in a 1.1~nm slit pore. The results are shown in Figure~S10. The distributions for the bulk and in-pore species are superimposed indicating that the extent of dehydration is minimal in this case. The ion densities across the pores for Cs$^+$ and water atoms indicate that the ions mostly reside at the centre of the pore in a fully hydrated state while the water molecules line the pore walls (see Figure~S9). The effect of the water reorientation in the pores can also be neglected as Cs$^+$-O, Cs$^+$-H  pair distribution functions are very similar in the bulk and in the pores (see Figure~S11). It is worth noting that the first coordination shell corresponding to chloride ions is also mostly unaffected by the confinement as can be seen from the Cs$^+$-Cl$^-$ pair distribution functions (see Figure~S11). While the situation could be different in slightly smaller pores, the pore size distributions, showing peaks for 0.8~nm and 1.1~nm, and the fact that ions do not enter pores of 0.8~nm in the simulations suggest that no significant difference in $\\Delta\\delta$ is to be expected from a variation in the hydration number. \n\nTo go beyond the effect of dehydration alone and include the influence of the hydration in the vicinity of a carbon surface, ion-water complexes for fully and partially hydrated cations were extracted from the MD simulations and DFT calculations of the chemical shifts of the ions next to a circumcoronene molecule were conducted. The distance of the Cs$^+$ ion from the centre of the closest circumcoronene molecule is set to be 0.35~nm for a partially dehydrated state, which was the closest distance between the Cs$^+$ ion and the carbon surface observed in MD simulations, and 0.45~nm for the fully hydrated configuration, which is a distance commonly observed in the MD simulations. To investigate the chemical shift trend in a slit pore, one more circumcoronene molecule is placed parallel to the first such that the pore width is 1.1~nm. For a detailed understanding of the trends, the chemical shifts of bare ions near a carbon wall and in the slit pore were also evaluated. Similar calculations were conducted for Rb$^+$. For Li$^+$, no desolvation is observed in the MD simulations so a single fully hydrated configuration was considered with a circumcoronene - ion distance of 0.41~nm. The most relevant computed shifts are shown in Figure~\\ref{dehydration_shifts}. The full set of results is given in Supplementary Information (Figures~S12-S16).\n\nThe $\\Delta\\delta$ value calculated between a partially hydrated Cs$^+$ ion in vacuum, i.e. without the influence of any carbon surface, and the same ion-water complex in a slit pore of 1.1~nm, 0.35~nm away from the closest surface, is -13.0~ppm. This is a larger shift than what is observed in experiments but closer to the expected value compared to a simple NICS or bare ion estimation. Indeed, the $\\Delta\\delta$ value calculated between a bare Cs$^+$ ion in vacuum and in the same position in a slit pore is 3.7~ppm. For a fully hydrated Cs$^+$ ion in vacuum and at 0.45~nm away from the closest carbon surface, a case observed much more commonly according to MD simulations, the $\\Delta\\delta$ value calculated is -3.2~ppm. This is this time smaller than the $\\Delta\\delta$ value for the bare ion of -6.3~ppm. \n\nInterestingly, the $\\Delta\\delta$ values calculated for Li$^+$ with or without hydration are very similar (respectively -6.7~ppm and -6.8~ppm). This reinforces the idea that NICS (or chemical shift profiles) are sufficient to estimate the experimental chemical shift difference between bulk and in-pore Li$^+$ ions. On the contrary, for Rb$^+$, large chemical shift differences between hydrated and dehydrated configurations are obtained, in agreement with the fact that NICS (or chemical shift profiles) are, as for Cs$^+$, not sufficient to explain the $\\Delta\\delta$ values obtained. \n\nPrevious DFT calculations conducted in this work suggest that the change in ring currents is not the main origin for the large shift observed. One way to check this is to use the NICS(1) as was done for the bare cations. The NICS(1) is computed to be -14.3~ppm for the partially hydrated Cs$^+$ ion against -13.9~ppm for the bare ion. Clearly the variation in the chemical shift is not due to the alteration of the ring currents which seems insignificant from this calculation. \n\nThe observations made for various hydration numbers and cations underline the challenge of understanding the factors determining the chemical shifts for large polarizable ions such as Cs$^+$ and Rb$^+$ which are largely affected by their hydration shells. Indeed, experimental results also show a variation of the Cs$^+$ chemical shift with ion concentration even in the bulk,~\\cite{Cervini19} an effect we could not explore in this study. To get a better understanding of the relationship between local structure and chemical shift a much broader study including a large number of configurations and, ideally, more realistic nanopores would be needed. Such a study would represent a considerable computational investment and is out of scope of the current work. While the study conducted here on a few configurations is very limited, it proves that $\\Delta\\delta$ values in the range of -9 to -11~ppm, similar to what is measured experimentally, can be reached when partially or fully hydrated ions go from the bulk to a confined state in a nanopore. \n\n\nIn this work, we have used a range of computational methods to investigate different factors affecting the chemical shifts of alkali metal ions adsorbed in nanopores with respect to their counterpart in the bulk aqueous electrolytes. Chemical shift differences calculated between the in-pore and bulk ions, the so-called $\\Delta\\delta$, are systematically compared with experiments conducted with polyether ether ketone derived carbons. A lattice model was first used to explore the importance of the pore size distribution, the ion organization at the carbon surface and the ring currents in the carbon materials in determining the $\\Delta\\delta$ value. The results show that while this approach seems sufficient to evaluate $\\Delta\\delta$ for Li$^+$ and hydrogen atoms from water, it leads to very bad estimations for Cs$^+$ and Rb$^+$. DFT calculations realized on ion-water configurations extracted from MD simulations suggest that the hydration shell has a large effect on the chemical shifts for large polarizable alkali metal ions. While the solvation shell of these ions seems mostly unaffected by the confinement, the inclusion of water molecules in the chemical shifts calculations in the vicinity of a carbon surface is essential to reproduce values in good agreement with experiments. In the future, a better understanding of the relationship between local environment and chemical shift would require additional computational studies on a wide variety of ion-water configurations under confinement.\n\n\\section*{Methods}\n\nA thorough interpretation of the experimental NMR spectra of several aqueous electrolytes in PDCs\\cite{Cervini19} is attempted here through a combination of classical MD simulations, DFT calculations and a previously developed lattice model\\cite{Merlet15,Sasikumar21} which allows one to simulate the NMR spectra of electrolyte species diffusing inside porous carbon materials. In order to model a given system, i.e. a given electrolyte in contact with a PDC, one needs to provide the following information:\n\\begin{itemize}\n\\item a pore size distribution;\n\\item the free energy of ions in pores of different sizes;\n\\item the chemical shift for ions in pores of different sizes.\n\\end{itemize}\nFollowing the lattice model calculations, additional MD simulations and DFT were conducted to test possible origins for the discrepancies between calculations and experiments.\\\\\n\n\n\\emph{Molecular Dynamics simulations}\\\\\n\nThe systems simulated consist in two graphite slabs, placed parallel to each other, either encompassing or surrounded by an aqueous electrolyte in order to represent an unconfined or a nano-confined electrolyte. In the first geometry considered (Figure~S1a), the distance between the graphite slabs is sufficiently large to recover electrolyte bulk properties in the middle of the fluid region. These calculations are used to determine the free energy profiles of ions \/ water molecules adsorbed at planar carbon surfaces. In the second geometry considered (Figure~S1b), a slit pore, with a pore size of 0.8~nm or 1.1~nm, is immersed in the electrolyte. These calculations are used to investigate confinement effects. The choice of the 0.8~nm and 1.1~nm pore widths is based on the pore size distribution obtained for PDCs through adsorption studies\\cite{Cervini19} which show high proportions of pores with these sizes. The electrolytes considered are aqueous solutions of LiCl (1M), RbCl (1M) and CsCl (1M). In all the simulations, 60 ion pairs and 3300 water molecules are used for the electrolyte.\n\nAll-atom MD simulations are conducted using the LAMMPS software.\\cite{LAMMPS} Intermolecular interactions are represented by the sum of an electrostatic and a Lennard-Jones potential. For the water molecules, the commonly used point charge extended (SPC\/E) force field is chosen.\\cite{Berendsen87} The parameters for the alkali metal ions, chloride ions and carbon atoms can be found in published works.~\\cite{Koneshan98,Cole83} The systems are first simulated in the NPT ensemble at 1~atm and 298~K for 1~ns until they reach a constant volume. Following this NPT step, simulations are conducted in the NVT ensemble at 298~K for at least 20~ns (14.5~ns for RbCl) for the first geometry and 10~ns for the second geometry. The results from the last 19~ns of the first geometry (except for RbCl where the last 12~ns are considered) and last 9~ns of the second geometry are considered for the analysis. A timestep of 1~fs is used for all simulations. Additional details on the simulations are given in Supplementary Information. \\\\\n\n\n\\emph{Density Functional Theory calculations}\\\\\n\nAll chemical shift calculations are performed using the Gaussian software\\cite{frisch2013,g16} and the gauge-including atomic orbital GIAO method. A 6-31G(d) basis set and B3LYP exchange-correlation functional are used for NICS calculations. A 3-21G basis set and B3LYP functional are used for chemical shift calculations involving ions considering the basis set availability and computational expense of the calculations involving heavy atoms such as Cs.  \n\nNICS profiles were evaluated using aromatic molecules (coronene, circumcoronene and dicircumcoronene) as model molecules representing the pore surface. Following previous works, the shielding tensors are calculated at various distances of the carbon surface along a line passing through the centre of the molecule, perpendicular to the molecular plane.\\cite{Xing14,Forse14,Kilymis20} The shielding constants are obtained by averaging the diagonal elements of the tensor. The same methodology was used for alkali metal ions at various distances from the centre of the circumcoronene molecule to study the effect of specific ion-carbon interactions (and possible charge transfer) on the chemical shift. The selection of circumcoronene in this case is based on previous simulations which provided a good agreement with the experiments in a range of systems\\cite{Merlet15,Sasikumar21}. The results of the calculations in presence of the alkali metal ions are designated as chemical shift profiles.\n\nChemical shifts of the cations in hydrated clusters, extracted from MD simulations, in the presence and absence of circumcoronene molecules are also determined. To study the effect of the hydration number on the chemical shift, water molecules are removed one at a time starting from a fully hydrated configuration, removing the farthest water molecule first, and the chemical shift is calculated at each step. Several starting configurations were considered to investigate a possible effect of the orientation of neighboring water molecules on the chemical shift of the ions. To evaluate the chemical shifts of fully or partially dehydrated ions in the vicinity of a carbon surface, configurations were extracted from the MD simulations and the chemical shifts were calculated in the presence of circumcoronene at representative distance for the solvation structures considered.\\\\\n \n\n\\emph{Lattice simulations}\\\\\n\nIn the mesoscopic model,\\cite{Merlet15,Sasikumar21} the carbon particles are represented by a cubic lattice, which represents a collection of slit pores. Here, we used 20$\\times$20$\\times$20 lattice sites. PDCs with burn off values of 41\\%, 50\\% and 56\\% are considered for this study through the inclusion of their pore size distributions obtained experimentally from gas adsorption studies.\\cite{Cervini19} Following previous studies, the pore surface distribution is chosen to be a log-normal distribution with a mean of -0.1 and a standard deviation of 0.25 in the case of NICS~\\cite{Merlet15,Anouar19,Sasikumar21} and a pore surface corresponding to circumcoronene in the case of chemical shift profiles.\n\nTo account for the distribution of ions in pores of different sizes in the carbon material, free energy profiles are obtained from MD simulations. The free energy profiles determined in this work for the species of interest in the range of electrolytes studied are shown in Figure~S2. The shielding environments of the ions adsorbed at each site are defined using the NICS profiles or the chemical shift profiles obtained from DFT calculations. \n\nThe ion and water dynamics between the pores of the carbon particle are modelled through kinetic Monte Carlo moves. The adsorbed species explore different shielding environments (and thus different resonance frequencies) throughout their motion. The NMR signal is calculated for each ion or water molecule, and a Fourier transform gives access to the NMR spectrum and effective chemical shift. A more detailed description of lattice simulations is available in published works\\cite{Merlet15,Anouar19}. \n\nThe mesoscopic model allows one to calculate the chemical shift difference, $\\Delta\\delta$, between species present between particles (bulk) and species present in the pores (in-pore) and to analyse the effects of ring currents and ion organization on the total global shift. It is worth noting that the lattice model does not account explicitly for any effects arising from the hydration of the ions on the global shift.\n\n\\begin{acknowledgement} \n\nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement no. 714581). JMG acknowledges funding from EPSRC with grant number EP\/V05001X\/1. This work was granted access to the HPC resources of the CALMIP supercomputing centre under the allocations P17037 and P21014, and of TGCC under the allocations A0070911061 and A0080910463 made by GENCI. The authors acknowledge Luca Cervini for sharing experimental data and for fruitful discussions, and Patrice Simon and Alexander Forse for useful conversations.\n\n\\end{acknowledgement}\n\n\\begin{suppinfo}\n\nSupplementary Information available: details of the molecular dynamics simulations, free energy profiles, pore size distributions and distributions of ions in the different pore sizes, chemical shift profiles for alkali metal ions in the vicinity of circumcoronene, characterization of the ion hydration and its effect on the chemical shift in the bulk, chemical shift for different ion-water-circumcoronene configurations. \n\n\\end{suppinfo}\n\n\\section*{Data availability}\n\nThe program used to do the lattice simulations is available, along with a manual, on Github (https:\/\/github.com\/cmerlet\/LPC3D). 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+{"text":"\\section{Introduction}\nCommodity programmable graphics hardware such as the AMD R600\/R700 and NVIDIA G80\/GT200 architectures have made available on the desktop TFLOP-scale floating-point performance formerly accessible only on dedicated supercomputers, with peak throughput improvements of over an order of magnitude relative to conventional CPUs. This extremely high performance makes GPUs attractive in computational chemistry, as many applications are bound by the limits of available computational power (e.g., the simulation time, simulated system size, and forcefield detail in molecular dynamics are all fundamentally constrained by available computation). Indeed, GPUs have been applied to several important problems in computational chemistry including molecular dynamics \\cite{Friedrichs09,Stone07}, Poisson-Boltzmann electrostatics \\cite{Narumi09}, DFT and MP2 quantum chemistry models \\cite{Yasuda08,Ufimtsev08,Vogt08}, and molecular comparison \\cite{Haque09}. Scientific problems outside chemistry, including biological sequence alignment \\cite{Manavski08} and machine learning \\cite{Catanzaro08} have also shown significant speedups from reimplementation for GPU execution. \n\nWhile GPGPU (general-purpose computation on GPUs) is attractive from the perspective of throughput, its origin in the relatively-error-tolerant area of consumer graphics has raised concerns about reliability. The reliability of GPUs in error-intolerant applications such as scientific simulations is largely unproven. Previous work \\cite{Sheaffer06, Sheaffer07} has questioned the reliability of GPU logic and proposed methods for logic hardening. A particular point of concern, however, is the reliability of memory on GPUs. The lack of ECC (error-checking-and-correcting) functionality in GPGPU systems has been previously noted \\cite{Sheaffer07}. \n\nWhile circumstantial evidence raises reliability questions, no prior work has actually quantified the reliability of GPGPU. Our contribution in this work is a quantification of the reliability of GPU memory subsystems in the context of GPGPU which has important implications for both developers and users of GPU-based scientific codes. We have carried out a large-scale assessment of GPU reliability by using a custom test code to measure error rates on more than 20,000 NVIDIA GPUs on the Folding@home distributed computing platform. In addition to the version of the tester deployed on Folding@home, we have released MemtestG80, a standalone version of our test code, under an open-source LGPL license at \\texttt{https:\/\/simtk.org\/home\/memtest}.\n\nOur experiment comprises over 200 terabyte-hours of memory testing distributed over more than 20,000 individual GPUs worldwide. We also present control experiments carried out on individual nodes and a GPU cluster. Our results demonstrate a detectable, pattern-sensitive rate of memory faults in the installed base of commercial GPU hardware. Specifically, even after controlling for overclocked cards and time of day of test execution (as a proxy for ambient temperature) we find that two-thirds of all tested GPUs exhibit sensitivity to memory faults in a pattern-dependent manner. This error rate depends strongly on board architecture, with devices based on the newer GT200 GPU exhibiting soft error rates nearly an order of magnitude lower than those based on the older G80\/G92 design. \n\nOur paper is organized as follows. We first offer a primer on soft error rate generation and detection mechanisms, and explore prior work (Section \\ref{sec:background}). We then present the design and validation  of MemtestG80, our software-based tester to detect soft errors in logic and memory on NVIDIA GPUs (Section \\ref{sec:memtestG80}), and explain the methodology of our large-scale study of error rates using the Folding@home distributed computing network (Section \\ref{sec:methodology}). In Section \\ref{sec:results}, we present the results of our experiment. We subsequently present detailed analysis of the data in Section \\ref{sec:analysis}, examining of possible error-inducing conditions. We conclude with a discussion of the implications of our findings on the field of reliable GPGPU and steps to be taken by both hardware and software developers.\n\n\\section{Background and Previous Work}\n\\label{sec:background}\nErrors in hardware systems can be classified as ``soft'' or ``hard'': hard errors are triggered by physical hardware defects, whereas soft errors are random, transient faults not due to physical defects. The term ``soft error'' has traditionally referred to radiation-induced upsets in electronic circuits, in which high-energy particles from the environment (e.g., cosmic rays \\cite{Ziegler96} or radiation from chip packaging materials \\cite{Gordon08}) cause erroneous operation by creating localized charge disturbances \\cite{Shivakumar02, Baumann02}. These soft errors are a significant problem in traditional supercomputing, as reflected in the error-checking-and-correcting design of the IBM BlueGene\/L supercomputer \\cite{Adiga02}. \n\nMechanisms other than radiation can also induce transient errors. In memories, reads and writes can occasionally affect the state of adjacent cells; in both memories and logic, timing violations (e.g., from improper specification or overclocking) can cause transient errors \\cite{Cockburn94}. Furthermore, at both the system and chip level, improper signal routing and power design can induce transient errors by various mechanisms such as voltage droop, crosstalk and timing jitter \\cite{Sheaffer06, Metra00}. In this work, we use the term ``soft error'' to refer to the larger class of transient faults, rather than just radiation-induced errors.\n\nWhile memory manufacturers typically do not disclose soft error rate (SER) information \\cite{Cataldo01}, estimates have been drawn for RAM SER from a variety of sources \\cite{Tezzaron04}, ranging from $5 \\times 10^{-14}$ to $4 \\times 10^{-7}$ errors per bit-hour. Data from IBM indicate that in natural environments, even with hundreds of devices under test, more than 1,000 testing hours may be required to accumulate statistically meaningful test results \\cite{Gordon08}. Additionally, possible environmental (e.g., thermal or radiation) effects on SER dictate that a variety of conditions be tested. For example, cosmic ray flux varies by a factor of two depending on latitude \\cite{Ziegler96}, and approximately 13x between sea level and 10,200 ft in altitude \\cite{Gordon08}. Thus, a very-large-scale approach is required in order to efficiently accumulate statistically-significant data regarding SER.\n\nBecause of the significance of SER to the semiconductor and computer industries, much past work has been done on techniques for measuring memory and logic SER. Cockburn \\cite{Cockburn94} presents an introduction to hardware testing methods and fault models used by semiconductor device manufacturers to test memories, and also presents the use of random test patterns as a simple method that achieves good test coverage across a variety of failure modes. Suk and Reddy \\cite{Suk80} establish the importance of using different test patterns to detect pattern-sensitive device faults; however, their presented patterns depend on intimate knowledge of the underlying memory layout. \n\nThese hardware-based testing methods are useful for verifying sensitivity of hardware to design or manufacturing flaws. However, the low radiation flux in most (habitable) environments makes this sort of testing impractical to detect radiation-sensitivity faults in a high-throughput fashion. Consequently, semiconductor manufacturers will sometimes use high-radiation test environments to further characterize devices. Past work includes the use of radioactive isotopes \\cite{Blandford85} and particle accelerators \\cite{Violante07} to bombard chips with high levels of radiation. An alternative approach is the use of high-altitude testing \\cite{Gordon08} (cosmic ray neutron flux rises exponentially with altitude \\cite{Ziegler96}).\n\nHigher-level software and modeling-based techniques have also been proposed for soft error detection and correction. Software hardening techniques can be used to detect logic and memory soft errors \\cite{Nicolescu03, Nicolescu01}. Walcott et al.\\ describe a method for predicting the vulnerability of logic architectures to soft error and the ``architectural vulnerability'' \\cite{Mukherjee05} of different codes running on the same architecure \\cite{Walcott07} and validate their approach through software simulation. Sheaffer et al.\\ also take a simulation-based approach and specifically characterize the architectural vulnerability of graphics algorithms on GPUs \\cite{Sheaffer06}. As their later work points out, however, this characterization is inappropriate for GPGPU, which requires tighter error guarantees than does conventional graphics \\cite{Sheaffer07}.\n\nWhile prior work has laid out testing methods to detect soft errors, no previous studies have applied these techniques to a large installed base to broadly assess the impact of soft errors on the emerging GPGPU platform. Our contributions in this work are twofold: first, a test code using proven memory and logic testing methods to detect soft errors; second, a large body of data using this tester to assess SER on tens of thousands of installed GPUs worldwide.\n\n\\section{Design and Validation of MemtestG80}\n\\label{sec:memtestG80}\nIn this section we describe the design and validation of Mem\\-testG80, our CUDA-based \\cite{Nickolls08} code to test for memory errors on NVIDIA GPUs based on the G80\/Tesla architecture \\cite{Lindholm08}. MemtestG80 is a CUDA implementation of most of the memory tests in Memtest86 \\cite{Memtest86}, a widely-used open-source memory tester for x86-based machines that implements many popular memory-test patterns, including randomized tests and tests for pattern-sensitive errors. Section \\ref{sec:testMI} briefly lists the tests implemented in MemtestG80. For conciseness, we refer the reader to the Memtest86 documentation \\cite{Memtest86} for descriptions of most of the test patterns, and here highlight instead the design decisions made in a parallel GPU implementation of the code. We also detail the implementation of a logic test of our own devising that is unique to MemtestG80. \n\nIn the text that follows, one ``iteration'' of MemtestG80 refers to the execution of each implemented test once. For tests that take place in multiple rounds, every round is executed (with one exception, explained in Section \\ref{sec:procedure}). Such an iteration over 64 MiB of memory typically takes between 1 to 5 seconds to complete, depending on GPU speed; runtime scales linearly with the amount of memory tested.\n\n\\subsection{Offload and Parallelization Scheme}\n\nSeveral design parameters affect the sensitivity and speed of a software-based GPU memory tester. Specifically, the three components of the memory tester --- pattern generation, memory access (writing and reading patterns to\/from memory), and pattern verification --- can be performed either by the CPU or the GPU itself; and if performed on the GPU, can be performed either serially or in parallel across the multiple GPU cores. The decisions made in MemtestG80 are informed both by the assumptions we make about the relative error rates of various system components and by responsiveness requirements dictated by operation on donated distributed-computing resources. \n\nTo improve the speed and responsiveness of the memory tester, all pattern generation, memory access, and verification is done in parallel on the GPU. We assume that the memory error rate is sufficiently low that the on-GPU code (which occupies a much smaller footprint than the tested region) will not be corrupted during the test execution. Performing verification on the GPU leaves the tester vulnerable to GPU logic errors. We therefore implicitly assume that the GPU logic error rate is lower than the GPU memory error rate; however, we verify this assumption by also running a custom logic test that should report errors on architectural paths similar to those used in other parts of the tester. \n\n\\subsection{Logic Testing}\n\nBecause results from the GPU can be passed back to the host CPU only by a copy from the GPU main memory, detection of GPU logic errors under the assumption that memory errors are more frequent than logic errors is nontrivial --- an error in a computed result may be caused by an error in logic or memory. To overcome this problem, a test can be designed which produces the same expected result after varying amounts of logic operations. The same test can be run twice with (for example) four times the number of logic operations in the second execution. Since both tests write the same data to memory, the expected rate of errors due to memory faults will be equal between the two executions; since the latter test runs more logic operations, errors from logic faults should scale with the number of operations.\n\nThe design of our logic test, unique to MemtestG80, is based on the preceding principle. For the core calculation, we use a linear congruential random number generator (LCG) with a short period $k$ starting from zero. Such a generator, when started from zero, will return to zero after a fixed number of iterations $k$. Because the generator only reaches $k$ states, of $2^{32}$ possible (in the 32-bit generator), assuming a uniform probability of error over bits, most logic errors will cause the generator to transition to a state outside the normal operation cycle. Such a state is unlikely to return to zero in the correct number of steps, and therefore whether the generator returns to zero is a good indication of whether a logic error occurred. Our logic test starts the generator from zero and runs it for $k$ or $4k$ cycles, each time writing the results out to memory, reading it back, and verifying that it contains only zeros. Any nonzero values indicate either the presence of a logic or a memory error. Scaling of the number of nonzero values with the number of LCG iterations indicates logic, rather than memory, errors. The use of constant zero as the test pattern further increases the sensitivity of the test to logic rather than memory errors; as we show in Section \\ref{sec:validation}, the constant zero pattern is insensitive to faulty memory.\n\n\n\\subsection{Validation}\n\\label{sec:validation}\nTo validate MemtestG80, we carried out both negative and positive control experiments. Since the purpose of this study is to investigate the latent error rate in GPU hardware, a true negative control (one in which we are guaranteed no errors) is not possible. To minimize the possibility of errors from overheating or power disturbances, the controls were run on machines in controlled-temperature environments with known-good power sources. Shielding against ambient radiation such as cosmic rays requires meters of concrete or rock \\cite{Tezzaron04} and was therefore deemed impractical.\n\nWe ran negative controls on two types of hardware. The first was a GeForce 8800 GTX, a high-end consumer-grade graphics board operating at NVIDIA reference clock frequencies. Secondly, we ran tests on a cluster of 8 NVIDIA Tesla C870 boards. The Tesla C870 is a board designed specifically for GPGPU, is based on the same GPU architecture as the 8800GTX, and should reflect any engineering changes made by NVIDIA for ``pro'' or GPGPU-grade cards. Over 93,000 iterations of MemtestG80 were run over 700MiB of GPU memory on the 8800GTX. An aggregate 1.48 million iterations were run over 320 MiB on each Tesla board. No errors were ever detected on the negative control 8800GTX or Tesla boards, demonstrating that errors detected by MemtestG80 are unlikely to be spurious or due to errors in the code.\n\nTo ensure that MemtestG80 detects errors under conditions known to generate memory errors, we also carried out a positive control experiment. Since overclocking is known to generate memory errors (due to violations of timing constraints on both memory and memory controller circuitry), we used overclocking as our positive control. MemtestG80 was run on a GeForce 9500GT video card (default memory clock rate = 400 MHz) with its memory clock rate set to 400, 420, 430, 440, 450, 475, 500, and 530 MHz. Each test comprised 20 iterations of MemtestG80 (10 at 530MHz because of prompt instability). Between each clock frequency the board was reset to a memory clock of 400MHz and allowed to cool to a constant temperature to avoid thermal effects from affecting test results. We finally ran an additional 20 iterations at 400 MHz to verify that the results were unaffected by the intermediate overclocking. The results of the positive control experiment are presented in Figure \\ref{fig:testSensitivity}.\n\nTwo results deserve special attention. First, both variants of the moving-inversions test (which write the same constant --- zero, all ones, or a random number --- to all words in tested memory, and read it back) are completely insensitive to overclocking-induced errors. This inspires our choice of the all-zero pattern as the logic test pattern, as it seems to be insensitive to memory errors. \n\nSecond, the modulo-20 test is far more sensitive to over\\-clock\\-ing-induced errors than are the other tests, demonstrated by the fact that it started detecting errors at clock rates lower than those required to trigger errors in other tests. The modulo-20 test proceeds in 20 rounds. In round $i$, a 32-bit pattern is written to each memory location whose offset from the start of tested memory is equal to $i$ modulo 20; the bitwise complement of the pattern is then written twice to every other memory location. In the verification kernel, the offsets equal to $i$ modulo 20 are read back and it is verified that they contain the original pattern. This procedure is repeated for $i \\in \\{0,1,2,\\cdots,19\\}$. \n\nThe sensitivity of this test likely stems from the fact that the test's stride of 20 is unlikely to align with any architectural stride in the memory chips (e.g., a row length or chip interleave factor), so test locations are more likely to be physically scattered and to be influenced by neighboring cells. The sensitivity of the modulo-20 test persists in real-world situations and is key to our sampling of error rates. The pattern sensitivity of the memory subsystem is in itself an interesting result, as it demonstrates that the likelihood of encountering a memory error is highly dependent on the access and data patterns.\n\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[height=0.33\\textheight]{images\/testSensitivity_WER.pdf}\n\\caption{Overclocking positive control experiment: word error rate (ratio of incorrect to total words tested) versus clock rate for each memory test type. Both moving inversions tests displayed together as  neither ever reported an error. Dashed lines represent zero errors found at the tested frequency and are arbitrarily set two log units below lowest number of errors. }\n\\label{fig:testSensitivity}\n\\end{figure*}\n\n\n\\section{Experimental Methodology}\n\\label{sec:methodology}\n\\subsection{Testing Procedure}\n\\label{sec:procedure}\nTo carry out the large-scale assessment presented in Section \\ref{sec:results}, MemtestG80 was deployed on the Folding@home (FAH) distributed computing network \\cite{ShirtsPande00,Beberg09}. For each execution of MemtestG80, we collected device information (card name, memory size, and shader-domain clock speed). Because of the widely varying capabilities of GPUs on Folding@home, the size of the tested memory region varied between 32 and 128 MiB; sizes larger than 128MiB were disallowed because of the negative impact on the responsiveness of donors' computers. Because of the high memory bandwidths required on GPUs, it is likely that memory blocks are interleaved across physical memory chips to speed total throughput. We believe that the tested memory region sizes are sufficiently large that they are likely to be spread across all or a substantial fraction of physical chips on the tested devices. The LCG period used in the logic test was 256, 512, or 1024; like the size of the test region, the LCG period varied based on the capability of donor graphics cards. Finally, on Folding@home, only 2 rounds of the modulo-20 test were run per test iteration because of responsiveness concerns. Later rounds were performed in following iterations.\n\nOn these test regions we ran a variable number of test iterations, collecting individual results for every test iteration. Rather than measuring the exact bit-error-rate (which may be unreliable due to GPU logic errors), we check only whether \\emph{any} errors were detected during a test iteration. This ensures the robustness of the test. It is possible that a test which should have returned errors may have its ``error flag'' bits toggled off by a subsequent GPU memory or logic error and will therefore not be detected, creating a false negative. If a test successfully executes, it is possible that a GPU error will toggle an error flag bit on --- but this is in itself a GPU error. Thus, this binary decision approach removes the problem of false positive results. In the worst case, our results will underestimate the error rate; they will never overestimate it.\n\n\\subsection{Test Device Statistics}\n\nThe test was run at least once on over 20,000 distinct GPUs, with an aggregate total of over 4.6 billion test iterations executed. 96.6\\% of our data were collected with a test region size of 64 MiB and an LCG period of 512; 3.2\\% were collected with 32 MiB regions and period-256 LCGs, and the remainder with 128 MiB test regions and period-1024 LCGs. The tested boards are distributed worldwide with excellent coverage of North America and Europe; distributions elsewhere cluster sharply near large population centers (however, besides South Africa, Africa is poorly sampled).\n\nUsing NVIDIA specifications and the shader clock speeds reported by each board, we are largely able to classify boards as overclocked or running at ``stock'' frequencies. In a few cases, identically-named boards have multiple possible stock clock rates (e.g., models that fit into different thermal envelopes); in these cases it is not possible to determine whether or not a board is overclocked. Although memory clock rates will have a larger impact on memory error rate than logic (shader) clock frequencies, memory overclocking can be expected to covary with shader overclocking. The typical reason for Folding@home users to overclock their boards is to increase their throughput on Folding@home molecular dynamics (MD) work units. Empirical testing has shown that shader clock frequencies have more of an impact on MD runtimes than memory clocks, so it is unlikely that a board on Folding@home would have overclocked memory but not overclocked shaders. Some cards are shipped by board vendors in an overclocked state; here too, it would be rare to see overclocked memory without overclocked shaders, as shader clock frequencies can have a major impact on graphics performance. \n\nFigure \\ref{fig:NcardsVcutoff} displays the number of cards that completed a given number of MemtestG80 iterations during the course of this experiment. It further breaks the data into cards running at or below stock frequencies (grouped together as ``stock''), overclocked cards, and cards whose overclocking state is indeterminate. The results show that a slight majority of cards on Folding@home are overclocked; for most iteration count cutoffs, the number of overclocked and stock cards is comparable. \n\nFinally, we achieve good coverage of GPUs across the NVIDIA product line. Table \\ref{tab:cards300k} shows the counts of cards in particular NVIDIA product families that completed at least 300,000 iterations during the course of the experiment. Although the dataset is strongly biased towards NVIDIA consumer graphics cards (GeForce) due to the consumer-oriented nature of Folding@home, we do sample a few professional (Quadro) and GPGPU-dedicated (Tesla) boards.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[trim=1cm 5mm 1cm 8mm,clip,width=\\columnwidth]{images\/Ncards_v_cutoff.pdf}\n\\caption{Number of cards that completed at least a given number of test iterations}\n\\label{fig:NcardsVcutoff}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{| c | c |}\n\\hline\n\\textbf{Card Family} & \\textbf{\\# cards $\\geq$ 300,000 iter.}  \\\\ \\hline \\hline\n\\textit{Consumer graphics cards} & \\textit{4754 total} \\\\ \\hline\nGeForce 8800 & 1779  \\\\ \\hline\nGeForce 9800\/GTS & 1375  \\\\ \\hline\nGeForce GTX  & 1137  \\\\ \\hline\nGeForce 9600  & 368  \\\\ \\hline\nGeForce 8600  & 65  \\\\ \\hline\nGeForce 9500  & 19  \\\\ \\hline\nMobile GeForce & 11 \\\\ \\hline \\hline\n\\textit{Professional graphics cards} & \\textit{36 total} \\\\ \\hline\nQuadro FX & 29  \\\\ \\hline\nQuadroplex 2200 & 6  \\\\ \\hline\nQuadro NVS & 1  \\\\ \\hline \\hline\n\\textit{Dedicated GPGPU cards} & \\textit{22 total} \\\\ \\hline\nTesla T10 & 20  \\\\ \\hline\nTesla C1060 & 2  \\\\ \\hline\n\\end{tabular}\n\\caption{Distribution of cards tested on FAH that ran at least 300,000 test iterations}\n\\label{tab:cards300k}\n\\end{table}\n\n\\section{Results}\n\\label{sec:results}\nWe classified each test iteration returned as having failed or not using the method described in Section \\ref{sec:procedure}. We then inferred an empirical probability of failure (in a given test iteration) for each card tested as the ratio of failed tests to total tests, thereby estimating an empirical probability distribution that any card might have a given probability of failure. To add statistical validity, we applied various cutoffs for the minimum number of test iterations a card must have completed to be used in constructing this empirical card-reliability probability distribution.  Our underlying statistical model is that each card (in combination with its environment) has its own probability of failure $\\mathit{P\\left( fail \\right)}$, and that each card is drawn independently from some underlying distribution $\\mathit{P\\left( P \\left( fail \\right) \\right)}$. In all following plots and analyses, ``$\\mathit{P\\left( fail \\right)}$'' refers to any given card's probability of failing a single MemtestG80 iteration.\n\nFigure \\ref{fig:PfailCDF_few} displays the cumulative distribution functions (integrated probability distributions) derived from this data for 4 values of the iteration threshold. Each trace represents the distribution calculated using a different cutoff for the number of iterations required to have been completed to consider a card for inclusion. A larger number of completed iterations for a card increases the statistical certainty that its probability of failure lies in the given bin of the estimated probability distribution. We present cutoffs only up to 1 million test iterations because the number of cards sampled falls off rapidly past this limit; estimates made using only cards past a cutoff beyond this are not statistically useful.\n\nThe most apparent trend in the data is the strongly bimodal distribution. All the CDFs start with a nonzero value at $\\mathit{P\\left( fail \\right)} = 0$, representing the fraction of cards at each threshold which never failed a test. All CDFs further show a second population with a mean $\\mathit{P\\left( fail \\right)}$ around $2 \\times 10^{-5}$, which represents nearly all the remaining cards. Finally, there is a very small population of cards with failure rates higher than $1 \\times 10^{-4}$, likely representing faulty hardware. This bimodal trend is statistically relevant, as it continues to appear in the data even at the largest cutoff. In particular, the distributions at thresholds of 50,000, 300,000, and 1,000,000 iterations all have similar fractions of cards with zero errors, indicating that this particular population is stable (i.e., measuring with more iterations does not find errors from the zero-error population).\n\nAs these trends are stable with respect to iteration cutoff, it is instructive to examine the distribution of failure probabilities at a single, representative cutoff that maintains statistical validity. Figure \\ref{fig:Pfail_PMFCDF_300k} illustrates both the probability mass function and the cumulative distribution function over failure probabilities at an iteration cutoff of $\\geq$ 300,000 iterations. At this threshold, approximately one-third of cards tested never exhibited a memory error. Nearly all of the remainder had failure probabilities between 0 and $10^{-4}$; only about 2\\% had failure probabilities higher than this.\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[Empirical CDFs of card failure probability at several test-iteration-count thresholds] {\n\\includegraphics[width=0.95\\columnwidth]{images\/CDF_Pfail_few.pdf}\n\\label{fig:PfailCDF_few}\n}\n\\subfigure[Empirical PMF and CDF of card failure probabilities for cards running at least 300,000 MemtestG80 iterations] {\n\\includegraphics[width=0.95\\columnwidth]{images\/PMF_CDF_Pfail_300k.pdf}\n\\label{fig:Pfail_PMFCDF_300k}\n}\n\\caption{Empirical probability mass functions (PMFs) and cumulative distribution functions (CDFs) of card failure probabilities}\n\\end{figure*}\n\n\\section{Analysis}\n\\label{sec:analysis}\nIn this section we explore various hypotheses explaining features of the returned results, breaking the results down by test and by properties of the tested cards. The main statistical methods we apply are an examination of the mutual information between two probability distributions, and the information gain criterion for data partitioning attributes. Colloquially speaking, the mutual information between two distributions is a nonlinear measure of correlation between the two, and the information gain in an attribute measures the amount of variability in an underlying distribution that is explained by the attribute. These techniques are well-known in the statistical literature \\cite{CoverThomas, WittenFrank}.\n\n\\subsection{Hypothesis testing by information gain}\n\\label{sec:hypotest}\n\nTo test our hypotheses we apply the information-theoretic measure known as \\emph{information gain}, which is broadly used in data mining as a heuristic criterion for building decision tree models of data \\cite{WittenFrank}. The hypothesis testing problem is formulated as follows: given a labeled dataset $D$, we partition $D$ according to an indicator variable $V$ into multiple subsets $D_1, D_2, \\cdots, D_{|V|}$. We would like to know how good $V$ is at explaining the variability in $D$. \n\nWe measure the ``variability'' of D and each of its subsets by their respective Shannon entropies, $H(D)$, defined as\n\\begin{displaymath}\n\\nonumber H(D) = -\\sum_{x \\in D} p(x) \\log _2\\left( p(x) \\right)\n\\end{displaymath}\nThe information gain on $D$ from $V$, $I(D;V)$ (also known as the mutual information between $D$ and $V$), is  defined as:\n\\begin{eqnarray}\n\\nonumber I(D;V) &=& H(D) - H(D|V) \\\\\n\\nonumber I(D;V) &=& H(D) - \\sum _{v \\in V} H(D_v)P(V = v)\n\\end{eqnarray}\n\nIf $I(D;V)$ is large compared to $H(D)$, then $V$ explains a significant portion of the distribution of $D$. \n\nTo estimate probability distributions $D$ in our hypothesis testing, we histogrammed failure probabilities on a per-card basis as was done for each distribution in Figure \\ref{fig:PfailCDF_few}, but across the entire range of probabilities from 0 to 1. Although this resolution is too high for the low number of counts at higher probabilities, most of these bins will be zero-valued and will not affect the entropy calculations.\n\n\n\\subsection{Bimodality of P(fail)}\nThe bimodal distribution of card failure probabilities illustrated in Figure \\ref{fig:Pfail_PMFCDF_300k} raises an obvious question: is the existence of cards with nonzero failure probability easily explained through a simple structural or environmental variable, or is it inherent to population of boards? To answer this we tested a set of obvious hypotheses on our data set: that failures are due to overclocking, that they are caused by thermal problems, or that they reflect architectural variability in the hardware.\n\nAs a reference, a perfect indicator variable on this dataset (separating the data into cards which never failed and those which did) has an information gain $I(D;V)$ of 0.8896 bits.\n\n\\subsubsection{Shader Overclocking}\n\nTo test whether shader-overclocked cards are responsible for the second mode in the distribution, around $\\mathit{P\\left( fail \\right)} = 2 \\times 10^{-5}$, we let $D$ be the set of all cards with a known overclocking status, and partitioned it into known stock and known overclocked cards. The entropy of $D$ was calculated to be 6.184 bits, and the mutual information between $D$ and the stock\/overclocked indicator variable was $7.5 \\times 10^{-2}$ bits. $I(D;V)$ for this split is much lower than that for a perfect indicator, indicating that overclocking status explains very little of the information in the distribution. Thus, it is unlikely that overclocking is the cause of bimodality.\n\n\\subsubsection{Time of day}\n\nHigh temperatures are a known cause of transient errors in electronics. Although the APIs used by MemtestG80 do not allow us to monitor board temperatures, the status of Folding@home as a donor project suggests that most tested boards will not be in closely-temperature-controlled environments such as machine rooms. We combine the estimated test runtime, time results were received, and IP geolocation data \\cite{GeoLite} to estimate the local time of day during the. Assuming that the ambient temperature will fluctuate with local time of day, we test the hypothesis that time of day controls the shape of our error distribution. \n\nWe define ``day'' to run from 6am to 6pm, and ``night'' as 6pm to 6am, local to where the test was run. We let $D$ be the set of all tests which both started and ended during the day or during the night, and split it into tests that ran exclusively during the day and tests which ran exclusively at night. This hypothesis has an information gain of 0.0413 bits, indicating that it is exceedingly poor at explaining the shape of the error distribution; local time of day is therefore not an adequate explanation for error rates.\n\n\\subsubsection{Board architecture}\n\nThe GT200 series GPU from NVIDIA has a more advanced memory controller than the original G80\/G92 GPU, supporting (perhaps among other redesigned features) additional memory coalescing modes. To test whether this change in GPU architecture was the cause of bimodality, we let $D$ be all GeForce graphics cards, and partitioned the set into GT200-based and non-GT200-based boards. $I(D;V)$ for this indicator was 0.453 bits, which is a large fraction of the information gained from a perfect indicator. Thus, it is likely that this division significantly explains the error distribution we observe. \n\nFigure \\ref{fig:PfailCDF_few_byarch} shows that GT200-based boards (comprising about one-fifth to one-fourth of our dataset, depending on iteration threshold) were far less likely to fail MemtestG80 iterations --- regardless of iteration threshold, approximately 90\\% of GT200-based cards never reported any errors. The population of GT200-based boards producing errors clusters at a failure probability of $2.2 \\times 10^{-6}$, an order of magnitude lower than the mode failure probability for the overall dataset. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{images\/CDF_Pfail_few_byarch.pdf}\n\\caption{Empirical CDFs of card failure probability at several test-iteration-count thresholds, by architecture (G80\/G92 in red, GT200 in blue)}\n\\label{fig:PfailCDF_few_byarch}\n\\end{figure}\n\n\\subsubsection{Bimodality conclusions}\n\nOur data suggest that the bimodal structure of the failure probability distributions is caused by differing architectures in the boards tested. Specifically, the newer GT200 architecture has an apparent soft error rate nearly tenfold lower than that of G80. The most obvious user-visible enhancement on the GT200 memory controller relative to that on G80 is improved support for coalescing memory operations, or combining multiple memory reads or writes into single transactions. However, this support is insufficient to explain the change in error rates. \n\nUsing the published guidelines for memory coalescing on either architecture \\cite{CudaGuide2.2}, we simulated the memory access patterns for G80 and GT200 on the modulo-20 test (the most sensitive test). As performed in MemtestG80 on Folding@home, the modulo-20 test executes the same number of transactions on both architectures. However, GT200 is able to shrink transactions to be smaller (in terms of bytes) than those performed on G80. As a consequence, G80 generates 16.7\\% more memory traffic (in terms of bytes, not transactions) than GT200 on the test. By itself this does not appear to explain a 10-fold reduction in error probability by GT200. \n\nThe large sample sizes for boards on both architectures make it unlikely that there is a consistent environmental difference between installations of either board type. While it is possible that the error rate is an age-induced effect (G80 is an older architecture than GT200, and it is possible that G80 boards in our sample are physically older than GT200 boards), our data seem to indicate that GT200 is in fact more resistant to soft error generation than is G80.\n\n\\subsection{Impact of errors on molecular dynamics}\n\nTo assess whether memory errors have an impact on scientific computing, we looked for mutual information between the probability that a given card generates memory errors on its Folding@home work units and the probability that the same card triggers an ``early unit end'' (EUE), or simulation failure, on its Folding@home work units. Counting only work units in which at least one MemtestG80 iteration was executed, the mutual information between MemtestG80 errors and Folding@home EUEs was 0.131 bits, compared to overall distribution entropies of 1.965 and 1.018 bits respectively for memory error and EUE distributions. This indicates that MemtestG80 errors likely do not correlate well with EUEs. However, we believe that this measure underreports the true impact on the simulations. EUEs have a variety of causes (including improper simulation setup) which may be unrelated to errors on the board; furthermore, because of the design of the Folding@home client used, certain simulation errors were not reported to the servers and could not be logged. Hence, our results are inconclusive as to the true impact of observed errors on scientific simulations.\n\n\\subsection{Failure modes of tests}\n\\label{sec:testMI}\n\nBy examining the mutual information between the results of each individual test comprising a MemtestG80 iteration, it is possible to better understand the mechanisms triggering failures under various conditions. For each test, we construct a list in which each element corresponds to a single execution of MemtestG80, and the value of each element is the number of failures on that test for that execution. Corresponding elements in each vector map to the same MemtestG80 execution. Each list of failure counts was then histogrammed into 10 bins and normalized to build an empirical probability mass function for the number of failures in that test on a given execution of MemtestG80. Using these probability distributions for tests $X$ and $Y$ we calculated the entropies $H(X)$ and $H(Y)$ according to the formulas in Section \\ref{sec:hypotest}; in this case we use an alternative (equivalent) formulation for the mutual information $I(X;Y)$:\n\\begin{displaymath}\n\\nonumber I(X;Y) = \\sum_{x \\in X} \\sum_{y \\in Y} p(x,y) \\log _2 \\frac{p(x,y)}{p(x)p(y)}\n\\end{displaymath}\n\nThe entropy $H(X)$ can be interpreted as the uncertainty in $X$, as measured by the number of bits required by an optimal code to specify a value from the distribution $p_X(x)$. The mutual information $I(X;Y)$ can be interpreted as the reduction in uncertainty in $X$ caused by knowledge of the value of $Y$, or vice versa (mutual information is symmetric) \\cite{CoverThomas}. Figure \\ref{fig:testMI} shows the ratio of $I(X;Y)$ to $H(X)$ for all tests $X$ and $Y$ used in MemtestG80; this ratio is the fraction of the uncertainty in $X$ explained by knowledge of $Y$. In Figure \\ref{fig:testMI}, the $Y$ (the ``explaining'' distributions) are along the rows; the $X$ (the ``explained'' distributions) are along the columns. The following codes are used to refer to tests within MemtestG80:\n\\begin{description}\n\\item[MI10] Moving inversions, ones and zeros. Writes constant patterns of all-ones and all-zeros to memory.\n\\item[MIR] Moving inversions, random. Writes a constant (host-chosen) pseudorandom number to memory.\n\\item[1WM] Memtest86 variant of walking 1-byte test pattern.\n\\item[1W0\/1] True walking zeros\/ones pattern, 1-byte width.\n\\item[4W0\/1] True walking zeros\/ones pattern, 4-byte width.\n\\item[RB] Random blocks. Writes a different pseudorandom number (generated on the GPU) to each memory block.\n\\item[M20] Modulo-20 test. Described in Section \\ref{sec:validation}.\n\\item[L, L4] Logic test, one or four iterations through LCG cycle.\n\\item[LS(4)] Logic test as L\/L4, but with intermediate LCG state stored in shared memory rather than registers.\n\\end{description}\n\nSeveral interesting trends emerge from this data:\n\\begin{enumerate}\n\\item \\textbf{The Modulo-20 test stands on its own}\\\\\nBoth the M20 column and the M20 row have small values across their lengths, indicating the Modulo-20 test covaried strongly with no other test. This is likely due to the Modulo-20 test's increased sensitivity relative to other tests and reinforces the notion that it probes a different failure mechanism than do other tests.\n\\item \\textbf{The Random Blocks test is a good logic test}\\\\\nAlthough it was not intended as a logic test, the large values in the RB row for the columns corresponding to the LCG-based logic tests indicate that RB does a good job of capturing the errors measured by the LCG tests. Conversely, the small values in the RB column for the LCG tests demonstrate that RB is measuring a superset of errors relative to the LCG tests. This result is reasonable in retrospect: the RB test is very shader-logic intensive. We have designed it around a multithreaded, multi-core Park-Miller Minimal Standard pseudorandom number generator \\cite{Park88}, which in the course of generating a new random number for each memory location performs many more logic operations than any other MemtestG80 test.\n\\item \\textbf{The logic tests measure a distinct failure mode from most memory tests}\\\\\nThe four-iteration variants of the logic test (L4 and LS4) are poorly explained by most memory tests, and in particular, are less-well-explained by the memory tests than are their one-iteration counterparts (L and LS). This is to be expected, as the one-iteration variants are more influenced by memory errors. However, the bright block in the bottom-right of the mutual information plot shows that the logic tests covary strongly among themselves. Furthermore, memory tests have higher mutual information to the L4 test than the LS4 test, indicating that the use of shared memory in the logic test is a significant variable. Together, these results show that the logic tests detect a failure mode distinct from that tested in the memory tests, and that apparent logic errors can be triggered by soft errors in the on-GPU shared memory.\n\\end{enumerate}\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=2.2cm 1.2cm 4.5cm 2cm,clip,width=\\columnwidth]{images\/testMI.pdf}\n\\caption{Mutual information-to-entropy ratios for each test pair. Each entry is the fraction of the entropy of the test in that column explained by the test in that row. Brighter squares indicate that more of the variance of the explained test is explained by the explainer test. Test codes defined in Section \\ref{sec:testMI}.}\n\\label{fig:testMI}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe have presented the first large-scale study of error rates in GPGPU hardware, conducted over more than 20,000 GPUs on the Folding@home distributed computing network. Our control experiments on consumer-grade and dedicated-GPGPU hardware in a controlled environment found no errors. However, our large-scale experimental results show that approximately two-thirds of tested cards exhibited a pattern-sensitive susceptibility to soft errors in GPU memory or logic, confirming concerns about the reliability of the installed base of GPUs for GPGPU computation. We have further demonstrated that this nonzero error rate cannot be adequately explained by overclocking or time of day of execution (a proxy for ambient temperature). However, it appears to correlate strongly with GPU architecture, with boards based on the newer GT200 GPU having much lower error rates than those based on the older G80\/G92 design. While we cannot rule out user error, misconfiguration on the part of Folding@home donors, or environmental effects as the cause behind nonzero error rates, our results strongly suggest that GPGPU is susceptible to soft errors under normal conditions on non-negligible timescales. \n\nOur negative control results suggest (but do not conclusively prove) that with environmental control and the use of dedicated-GPGPU hardware, GPGPU can be reliable. However, our experimental results raise concerns about the reliability of GPGPU on consumer-level hardware as installed in the wild. These data are particularly relevant both to GPU-based distributed computing applications and to vendors of consumer-targeted software that relies on GPU acceleration, such as recent video-encoding applications. We emphasize that although our data were collected only on NVIDIA GPUs, we have no reason to believe that the reliability picture would be significantly different for GPUs from ATI, Intel, Via, or other manufacturers, as the driving forces behind GPU development to date have not emphasized the strict reliability concerns found in GPGPU applications.\n\nWe have furthermore presented the design and validation of MemtestG80, our custom code to test NVIDIA CUDA-enabled GPUs for memory errors. We have released this tool under an open-source LGPL license at \\texttt{https:\/\/simtk.org\/home\/memtest} in the hope that it can be used by others for GPU stress testing or self-checking.\n\n\\subsubsection*{Is dedicated GPGPU hardware the answer?}\n\n\n\nOur work suggests several future avenues of investigation. One question which our sampling is unable to adequately answer is whether professional-level and GPGPU-dedicated boards are significantly more reliable than consumer-grade GPUs. While the underlying architectures are identical to those in consumer-grade cards, this specialist hardware is marketed as being more capable than consumer hardware, and NVIDIA has suggested that Tesla boards are recommended for mission-critical applications. \n\nWhile we were unable to sample a large enough number of Quadro and Tesla cards in our experiment to provide a conclusive answer to this question, the results of our negative control experiment (Section \\ref{sec:validation}) suggest that the Tesla line may truly be more reliable than consumer hardware. In the course of our control experiment, we accumulated approximately 1.48 million MemtestG80 iterations over 8 Tesla boards. Each test operated over 5 times the typical amount of memory tested by a Folding@home MemtestG80 iteration, so the control experiment was equivalent to approximately 7.4 million Folding@home tester iterations, or over 925,000 per card. Neither any memory errors nor any logic errors were ever observed. Because the empirical probability of having a zero-error card in the Folding@home dataset is approximately one-third, the probability that our control Tesla cards were drawn independently from the same distribution is $\\left( \\frac{1}{3} \\right)^8$, or less than 0.01\\% (even less if drawn from the G80-only dataset). While our data do not rule out the possibility that environmental factors are at work (machine room versus uncontrolled environment) or that we tested an unusually good batch of cards, they suggest that the Tesla line are in fact more reliable than consumer-grade hardware.\n\n\\subsubsection*{What can be done?}\n\n\n\nAn obvious first step for software developers is to incorporate active memory test functionality, like that found in MemtestG80, to proactively detect malfunctioning cards. On the hardware side, the addition of parity or ECC functionality to the memory subsystem would guard against memory-induced silent errors. While the addition of partial ECC to the GDDR5 specification is a first step, it is not an end-to-end system, and cannot protect against errors in the memory controller or RAM itself. \n\nWhile memory testing and ECC can guard against errors in GPU memory, our logic tests indicate that transient errors on the GPU itself must also be considered. To combat these sources of faults, it may be necessary to implement measures such as redundant computation, advanced software error-detection \\cite{Nicolescu03, Nicolescu01} or hardware redundancy \\cite{Sheaffer07} mechanisms. \n\nNevertheless, our data demonstrate that it is certainly possible to perform reliable GPGPU computing on consumer-grade hardware, but that doing so requires close attention to the characteristics of the hardware. With the great power of TFLOP-scale computation on a single board comes a great responsibility for the developer to ensure data integrity.\n\n\\section*{Acknowledgments}\nWe foremost thank all the donors on the Folding@home network, without whose donated computer time this study would not have been possible. We further thank Dr. Paul Coteus of IBM for discussions regarding soft error detection and correction mechanisms, Ewen Cheslack-Postava of Stanford for discussions on GPU architectures and the idea of an iterated logic tester, and Adam Beberg, Ewen Cheslack-Postava, Philip Guo, Peter Kasson, and Alex Rasmussen for helpful comments on the manuscript. ISH gratefully acknowledges support from an NSF graduate fellowship. We acknowledge support from NIH (R01-GM062868, U54 GM072970) and NSF (CHE-0535616).\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Summary}\nSynergistic use of controlled sampling from a generative autoencoder and molecular simulation-based screening accelerates design of novel, safe, and potent antimicrobial peptides.\n\n\n\\begin{sciabstract}\n\n\\section*{Abstract}\n\n\nDe novo therapeutic design is challenged by a vast chemical repertoire and multiple constraints, e.g., high broad-spectrum potency and low toxicity. We propose CLaSS (Controlled Latent attribute Space Sampling) \u2014 an efficient computational method for attribute-controlled generation of molecules, which leverages guidance from classifiers trained on an informative latent space of molecules modeled using a deep generative autoencoder.  We screen the generated molecules for additional key attributes by using deep learning classifiers in conjunction with novel features derived from atomistic simulations. The proposed approach is demonstrated for designing non-toxic antimicrobial peptides (AMPs) with strong broad-spectrum potency, which are emerging drug candidates for tackling antibiotic resistance. Synthesis and testing of only twenty designed sequences identified two novel and minimalist AMPs with high potency against diverse Gram-positive and Gram-negative pathogens, including one multidrug-resistant and one antibiotic-resistant K. pneumoniae, via membrane pore formation.  Both antimicrobials exhibit low in vitro and in vivo toxicity and mitigate the onset of drug resistance. The proposed approach thus presents a viable path for faster and efficient discovery of potent and selective broad-spectrum antimicrobials.\n\n\\end{sciabstract}\n\n\\section*{Introduction}\n\n\\textit{De novo} therapeutic molecule design remains a cost and time-intensive process: It typically requires more than ten years and \\$2-3 B USD for a new drug to reach the market, and the success  rate  could be as low as $<1$\\%. \\cite{dimasi2016innovation, desselle2017institutional}. Efficient computational strategies for targeted generation and screening of molecules with desired therapeutic properties are therefore urgently required. As a specific example, we consider here the antimicrobial peptide (AMP) design problem.\nAntimicrobial peptides are emerging drug candidates for tackling antibiotic resistance, \none of the biggest threats in global health, food security, and development. Patients at a higher risk from drug-resistant pathogens are also more vulnerable to illness from viral lung infections like influenza, severe acute respiratory syndrome (SARS), and COVID-19.\nDrug-resistant diseases claim 700,000 lives a year globally~\\cite{UN_2019}, which is expected to rise to 10 million deaths per year by 2050 based on current trends~\\cite{oneill_2016}. Of particular concern are multidrug-resistant Gram-negative bacteria~\\cite{WHO_2019}.\nAntimicrobial peptides that are believed to be the antibiotic of last resort\nare typically 12-50 amino acids long and produced by multiple higher-order organisms to combat invading microorganisms.\nDue to their exceptional structural and functional variety  \\cite{Powers2003}, promising activity, and low tendency to induce (or even reduce) resistance,\nnatural AMPs have been proposed as promising alternatives to traditional antibiotics and as potential next-generation antimicrobial agents \\cite{Mahlapuu2016}.\nMost reported antimicrobials are cationic  and amphiphilic in nature, and possess properties thought to be crucial for insertion into and disruption of bacterial membrane\n~\\cite{Mahlapuu2016}.\n\nRational methods for novel therapeutic peptide design,\nboth in wet lab and \\textit{in silico},  \nheavily rely upon  \nstructure-activity relationship (SAR) studies \\cite{chen2019simulation, torres2018structure, tucker2018discovery, field2013saturation, fjell2012designing, li2017membrane}. Such methods struggle with\nthe prohibitively large molecular space,\ncomplex structure-function relationships, and multiple competing constraints  \nsuch as activity, toxicity, synthesis cost, and stability associated with the design task.\nRecently, artificial intelligence (AI) methods, in particular, statistical learning and optimization-based approaches, have shown promise in designing small- and macro-molecules, including antimicrobial peptides. \nA  comprehensive review of computational methodologies for AMP design can be found in ref \\cite{cardoso2020computer}. \nA conventional approach is to build a predictive model that estimates the properties of a given molecule, which is then used for candidate screening \\cite{jenssen2008qsar, vishnepolsky2019novo, maccari2013antimicrobial, meher2017predicting, thomas2010camp, witten2019deep,xiao2013iamp,veltri2018deep}. Either a manually selected or automatically learned set of compositional, structural, or physicochemical features, or direct sequence is used to build the predictive model. A candidate is typically obtained by combinatorial enumeration of chemically plausible fragments (or sub-sequences) followed by random selection,  from an existing molecular library, or modification thereof. Sequence optimization using genetic algorithms \\cite{porto2018silico, fjell2011optimization}, pattern insertion \\cite{porto2018joker}, or sub-graph matching \\cite{nagarajan2019omega76} has been also used in the development of new drugs, which requires  selection of initial (or template) sequence  and\/or defining patterns.  An alternative  is to develop a generative model \nfor automated \\textit{de novo} design of novel molecules with user-specified properties.  Deep learning-based architectures, such as neural language models as well as deep generative neural networks, have emerged as a popular choice \n\\cite{muller2018recurrent, grisoni2018designing, gupta2018generative,\ngomez1610automatic, jin2018junction, blaschke2018application, chan2019advancing, sanchez2018inverse, nagarajan2018computational}.  Probabilistic autoencoders \\cite{hinton2006reducing, kingma2013auto}, a powerful class of deep generative models, \nhave been used for \nthis design task, which\nlearn a bidirectional mapping of the input molecules (and their attributes) to a continuous latent space.  \n\n\n Earlier deep generative models for targeted generation have often limited the learning to a fixed library of molecules with desired attributes, to restrict the exhaustive search to a defined section of the chemical space. Such an approach can affect the novelty as well as the validity of the generated molecules, as the fixed library represents a small portion of the combinatorial molecular space \\cite{porto2018silico}.\nAlternative methods include Bayesian optimization (BO) on a learned latent space \\cite{gomez1610automatic},  reinforcement learning (RL) \\cite{guimaraes2017objective, popova2018deep}, or semi-supervised learning (SS) \\cite{kang2018conditional}. \nHowever, those approaches require surrogate model fitting (as in  BO), optimal policy learning (as in RL),  or minimizing attribute-specific loss objectives (as in SS),  which suffers from additional computational complexity.  As a result, controlling attribute(s) of designed molecules efficiently continues to remain a non-trivial task. \n\n\n\nTo tackle these challenges, we propose a computational framework for targeted design and screening of molecules, which combines  attribute-controlled deep generative models and physics-driven simulations. \nFor targeted generation, we propose Conditional Latent (attribute) Space Sampling -- CLaSS\nthat leverages guidance from attribute classifier(s) trained on the latent space of the system of interest and uses a rejection sampling scheme for generating molecules with desired attributes. \nCLaSS has several advantages, as it is  efficient and easily repurposable in comparison to existing machine learning algorithms  for targeted generation. \nTo encourage novelty and validity of designed sequences, we performed CLaSS on the latent space of a  deep generative autoencoder that was trained on a larger dataset consisting of all known peptide sequences, instead of a limited number of known antimicrobials. Extensive analyses showed that the resulting latent space is informative of peptide properties. As a result, the antimicrobial peptides generated from this informative space are novel, diverse, valid, and optimized. \n\nAlthough several antimicrobial peptides are in clinical trials \\cite{Mahlapuu2016}, the future design of novel AMP therapeutics requires minimizing the high production cost due to longer sequence length, proteolytic degradation, poor solubility, and off-target toxicity.  A rational path for resolving these problems is to design  short peptides as a minimal physical model\n\\cite{Losasso2019, Cipcigan2018} that captures the high selectivity of natural AMPs. That is, maximizing antimicrobial activity, while minimizing toxicity towards the host.\n\n\n\nTo account for these additional key requirements, such as broad-spectrum potency and low toxicity,  we further provide an efficient \\textit{in silico} screening method that uses deep learning classifiers augmented with high-throughput physics-driven molecular simulations (Fig. \\ref{fig:Overview}). To our knowledge, this is the first computational approach for \\textit{de novo} antimicrobial design that explicitly accounts for broad-spectrum potency and low toxicity, and performs experimental verification of those properties.   Synthesis of  20 candidate sequences (from a pool of $\\sim$ 90,000 generated sequences) that passed the screening enabled discovery of two novel and short length peptides with experimentally validated strong antimicrobial activity against diverse pathogens,  including a hard-to-treat multidrug-resistant Gram-negative \\textit{K. pneumoniae}.  Importantly, both sequences demonstrated  low \\textit{in vitro} hemolysis (HC50) and \\textit{in vivo} lethal (LD50) toxicity.  Circular dichroism experiments  revealed the amphiphilic helical topology of the two novel cationic AMPs.  All-atom simulations of helical YI12 and FK13 show distinct modes of early lipid membrane interaction. Wet lab experiments further confirmed their bactericidal nature; while live imaging with confocal microscopy showed bacterial membrane permeability. Both peptides displayed low propensity to induce resistance onset in \\textit{E. coli} compared to imipenem, an existing antibiotic. No cross-resistance was seen for either YI12 or FK13, when tested using a polymyxin-resistant strain. Taken together, both YI12 and FK13 appear promising therapeutic candidates that deserve further investigation.\nThe present strategy, therefore, provides an efficient \\textit{de novo} approach for discovering novel, broad-spectrum, and low-toxic antimicrobials with therapeutic potential at 10\\% success rate and rapid (48 days) pace.\n\n\n\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*{Results}\n\n\n\n\n\n\\subsection\n{Peptide autoencoder}\n\\label{sec:lvm}\n\nFor modeling the peptide latent space, we used generative models based on a deep autoencoder  \\cite{hinton2006reducing, kingma2013auto} \ncomposed of two neural networks, an encoder and a decoder.   The encoder $q_{\\phi}({\\mathbf{z}}|{\\mathbf{x}})$ parameterized with $\\phi$ learns to map the input ${\\mathbf{x}}$ to a variational distribution, and the decoder $p_{\\theta}({\\mathbf{x}}|{\\mathbf{z}})$ parameterized with $\\theta$ aims to reconstruct the input ${\\mathbf{x}}$ given the latent vector ${\\mathbf{z}}$ from the learned distribution, as illustrated in Fig. \\ref{fig:overview-gen}A. Variational Autoencoder (VAE), the most popular model in this family \\cite{kingma2013auto},  assumes \\emph{latent variable} ${\\mathbf{z}} \\sim p({\\mathbf{z}})$ and follows a simple prior (\\textit{e.g.} Gaussian) distribution.  And the decoder then produces a distribution over sequences given the continuous representation ${\\mathbf{z}}$. \nThus, the generative process is specified as: $p({\\mathbf{x}}) = \\int p({\\mathbf{z}}) p_\\theta({\\mathbf{x}} | {\\mathbf{z}}) d{\\mathbf{z}}$ where we integrate out the latent variable. \nAn alternative (and supposedly improved variant)  of standard VAE is\nWasserstein Autoencoder (WAE) (For details, see SI).\nWithin the VAE\/WAE framework, the peptide generation is formulated as a density modeling problem, \\textit{i.e.} estimating $p({\\mathbf{x}})$ where ${\\mathbf{x}}$ are short variable-length strings of amino acids. The density estimation procedure has to assign a high likelihood to known peptides. Therefore, the model generalization implies that plausible novel peptides can be generated from regions with a high probability density under the model.  Peptide sequences are presented as text strings composed of 20 natural amino acid characters. Only  sequences with length $\\le  25$ were considered for model training and generation, as short AMPs are desired.\n\nHowever, instead of learning a model only over known AMP sequences, one can learn a model over all  peptide sequences reported in the UniProt database \\cite{uniprot} -- an extensive database of protein\/peptide sequences that may or may not have an annotation. For example, the number of annotated AMP sequences is $\\sim$ 9000, and peptide sequences in Uniprot is  $\\sim$ 1.7M, when a sequence length up to 50 is considered.  \nTherefore, we learn a density model of over all known peptides sequences, in this work. The fact also inspires this approach, that unsupervised representation learning by pre-training on a large corpus has recently led to impressive results for downstream tasks in text and speech ~\\cite{ElmoPeters:2018, gpt2:radford2019language, cove:mccann2017learned, devlin2018bert}, as well as in protein biology \\cite{rao2019evaluating,  Madani2020.03.07.982272}. \nAdditionally, in contrast to similar models for protein sequence generation \\cite{riesselman2017deep}, we do not restrict ourselves to learning the density associated with a single protein family or a specific 3D fold.\nInstead, we learn a global model, over all known short peptide sequences expressed in different organisms.\n\nThis global approach should enable meaningful density modeling across multiple families, the interpolation between them, better learning of the ``grammar'' of plausible peptides, and exploration beyond known antimicrobial templates, as shown next.\n\n\n\n\n\nThe advantage of training a WAE (instead of a plain VAE) on peptide sequences \n is evident from the reported evaluation  metrics in Supplementary Information (SI) Table \\ref{tab:model_results} SI. We also observed high reconstruction accuracy and diversity of generated sequences, when the WAE was trained on all peptide sequences, instead of only on AMP sequences ( Table S\\ref{tab:model_results}).  Next, we analyzed the information content of the peptide WAE,  inspired by recent\n investigations in natural language processing. By using the so-called ``probing'' methods,  it has been shown that encoded sentences can retain much linguistic information~\\cite{shi2016does}.\nIn a similar vein, we investigated if the similarity (we use pairwise similarity which is defined by global alignment\/concordance between two sequences) between sequences \\cite{yu2003compositional} is captured by their encoding in the latent ${\\mathbf{z}}$-space, as such information is known to specify the biological function and fold of peptide sequences.\n  Fig. \\ref{fig:wae-latent}A reveals a negative correlation (Pearson correlation coefficient = $-0.63$) between sequence similarities and \nEuclidean distances in the ${\\mathbf{z}}$-space of the  WAE  model, suggesting that WAE intrinsically captures the sequence relationship within the peptide space. The VAE latent space fails to capture such a relation. \n\n\nWith the end-goal of a conditional generation of novel peptide sequences, it is crucial to ensure that the learned encoding in the ${\\mathbf{z}}$-space retains identifiable information about functional attributes of the original sequence. \nSpecifically, we investigate whether the space is \\emph{linearly} separable into different attributes, such that sampling from a specific region of that space yields consistent and controlled generations. For this purpose, we trained linear classifiers for binary  (yes\/no) functional attribute prediction using the ${\\mathbf{z}}$ encodings of sequences (Fig. \\ref{fig:overview-gen}B).\nProbing the ${\\mathbf{z}}$-space modeled by the WAE  uncovers that the space is indeed linearly separable into different functional attributes, as evident from the test accuracy of binary logistic classifiers on test data: The class prediction accuracy of the attribute ``AMP\" using WAE z-classifiers and\nsequence-level classifiers on test data are 87.4 and 88.0, respectively\n(also see Table S\\ref{tab:classifier_results}). Supplementary Table \\ref{tab:AMP-compare} shows the reported accuracy of several existing AMP classification methods, as well as of our sequence-level LSTM classifier. The reported accuracy varies widely from 66.8\\% for iAMP Pred \\cite{meher2017predicting}, to 79\\% for DBAASP-SP \\cite{vishnepolsky2018predictive} that relies on local density-based sampling using physico-chemical features, to 94\\%  for Witten \\textit{et al.} method \\cite{witten2019deep} that uses a convolutional neural net trained directly on a large  (similar to ours) corpus of peptide sequences. Our sequence-level LSTM model shows a comparable 88\\% accuracy. This comparison  reveals a close performance of the ${\\mathbf{z}}$-level classifier, when compared to classifiers reported in literature~\\cite{veltri2018deep,witten2019deep,xiao2013iamp, vishnepolsky2019novo, meher2017predicting, thomas2010camp} or trained in-house that have access to the original sequences.  We emphasize that the goal of this study is not to provide a new AMP prediction method that outperforms  existing machine learning-based AMP classifiers. Rather the goal is to have a predictor trained on latent features resulting in comparable accuracy, which can be used to automatically generate new AMP candidates by conditional sampling directly from the latent space using CLaSS. It should be noted that, comparing different AMP prediction models is non-trivial, as different methods widely vary by training AMP dataset size  (\\textit{e.g.} 712 for AMP Scanner v2 \\cite{veltri2018deep},  3417 for iAMPpred \\cite{meher2017predicting}, 2578 for CAMP \\cite{thomas2010camp}, 140 for DBAASP-SP prediction \\cite{vishnepolsky2019novo}, 1486 for iAMP-2L \\cite{xiao2013iamp}, 4050 for \\cite{witten2019deep}, and 6482 in the present study), sequence length, different definition of AMP and non-AMP, and other data curation criteria. Also, both the z-level and the sequence-level AMP classifiers used in the current study do not require any manually defined set of features, in contrast to many existing prediction tools, e.g.  \\cite{vishnepolsky2019novo} and \\cite{xiao2013iamp}\n\nOn toxicity classification task, a much lower accuracy was found using models trained on latent features, when compared to similar sequence-level deep classifiers~\\cite{gupta2013silico}  (also see Fig.~\\ref{fig:wae-latent} and Table S\\ref{tab:classifier_results}) that report accuracy as high as 90\\%.  \nThese results imply that some attributes, such as toxicity, are more challenging to predict from the learned latent peptide representation; one possible reason can be higher class imbalance in training data (see SI). \n\nWe also investigated the smoothness of the latent space by analyzing the sequences generated  along a linear interpolation vector in the ${\\mathbf{z}}$-space between two distant training sequences (Fig. \\ref{fig:wae-latent}B-C). Sequence similarity, functional attributes (AMP and Toxic class probabilities),  as well as several physicochemical properties including aromaticity, charge, and hydrophobic moment (indicating amphiphilicity of a helix) change smoothly during the interpolation. These results are encouraging, as the WAE latent space trained on the much larger amount of unlabeled data appears to carry significant structure in terms of functional, physicochemical, and sequence similarity.\nFigure \\ref{fig:wae-latent}C also demonstrates that it is possible to identify sequence(s) during linear interpolation that is visibly different from both endpoint sequences, indicating the potential of the learned latent space for novel sequence generation.  \n\n\\subsection*{CLaSS for controlled sequence generation}\n\\label{sec:class}\nFor controlled generation, we aim to control a set of binary (yes\/no) attributes of interest such as antimicrobial function and\/or toxicity. We propose CLaSS - Conditional Latent (attribute) Space Sampling for this purpose. CLaSS leverages attribute classifiers directly trained on the peptide ${\\mathbf{z}}$-space, as those can capture important attribute information (Fig. \\ref{fig:wae-latent}).\nThe goal  is to sample conditionally $p({\\mathbf{x}} | {\\mathbf{a}}_t)$ for a specified target attribute combination ${\\mathbf{a}}_t$.\nThis  task was approached through CLaSS (Fig. \\ref{fig:overview-gen}C), which makes the assumption that attribute conditional density factors as follows:\n$p({\\mathbf{x}}|{\\mathbf{a}}) = \\int \\! \\mathrm{d}z \\, p({\\mathbf{z}}|{\\mathbf{a}}) p({\\mathbf{x}}|{\\mathbf{z}})$\nWe sample  $p({\\mathbf{z}}|{\\mathbf{a}}_t)$\napproximately using rejection sampling from models in the latent ${\\mathbf{z}}$-space appealing to Bayes rule and $p({\\mathbf{a}}_t|{\\mathbf{z}})$ modeled by the attribute classifiers (Fig. \\ref{fig:overview-gen}B-C) (see SI). Since CLaSS only employs simple attribute predictor models and rejection sapling from models of ${\\mathbf{z}}$-space, it is a simple and efficient forward-only screening method. It does not require any complex optimization over latent space, when compared to existing methods for controlled generation, \\textit{e.g.} Bayesian optimization \\cite{gomez2018automatic}, reinforcement learning \\cite{guimaraes2017objective, popova2018deep}, or semi-supervised generative models \\cite{kang2018conditional}. CLaSS is easily repurposable and embarrassingly parallelizable at the same time and does not need defining a starting point in the latent space.  \n\nSince the Toxicity classifier trained on latent features appears weaker (Fig. \\ref{fig:wae-latent}),  antimicrobial function (yes\/no) was used as the sole condition for controlling the sampling from the latent peptide space. Generated antimicrobial candidates were then screened for toxicity using the sequence-level classifier during post-generation filtering. It is  noteworthy, that CLaSS does not perform a heuristic-based search (as in genetic algorithm or node-based sampling, as reported in\\cite{sattarov2019novo}) on  the latent space, rather it relies on a probabilistic rejection sampling-based scheme for attribute-conditioned generation. CLaSS is also different from the local density based sampling approaches (\\textit{e.g.} \\cite{vishnepolsky2019novo}), as those methods rely on clustering the labeled data and then finding the cluster assignment of the test sample by using similarity search, thus is suited for a forward design task. CLaSS, in contrast, is formulated for the inverse design problem, and allows targeted generation by attribute-conditioned sampling from the latent space followed by decoding  (see Methods).\n\n\n\n\n\\subsection*{Features of CLaSS-generated AMPs}\n\\label{sec:seqanalysis} \nTo check the homology of  CLaSS-generated AMP sequences with training data, we performed a BLAST sequence similarity search. We use Expect  value (E-value) for the alignment score of the actual sequences to assess sequence homology, while using other alignment metrics, such as raw alignment scores, percent identity and positive matches, gaps and coverage in alignment, to get an overall sense of sequence similarity. E-value indicates statistical (\\textit{aka.} biological) significance of the match between the query and sequences from a database of a particular size. Larger E-value indicates a higher chance that the similarity between the hit and the query is merely a coincidence, \\textit{i.e.} the query is not homologous or related to the hit.\nWe analyzed the Expect value (E-value) for the matches with the highest alignment score. \n\nTypically E-values $\\le$ 0.001 when querying Uniprot  $nr$ database of size $\\sim$ 220 M are used to infer homology \\cite{pearson2013introduction}.  Since our training database is $\\sim$ 1000 times smaller than Uniprot, an E-value of $\\le$ 10$^{-6}$ can be used for indicating homology.\nAs shown in Table S\\ref{tab:Evalue}, about 14\\% of generated sequences show an E-value of $\\ge$ 10, and another 36\\% have an E-value $>$ 1, when considering  the match with the highest alignment score, indicating insignificant similarity between generated and training sequences. If only the alignments with score $>$ 20  are considered,  the average E-value is found to be  $>$ 2, further implying the non-homologous nature of generated sequences. Similar criteria have also been used for detecting novelty of designed short antimicrobials \\cite{chen2019simulation}. \nCLaSS-generated AMPs are also more diverse, as the unique (\\textit{i.e.} found only once in an ensemble of sequences) $k$-mers ($k$ = 3-6) are more abundant compared to training sequences or their fragments (Figure S\\ref{tab:combined-comp}). \nThese results highlight the ability of the present approach to generate short-length AMP sequences that are, on average,  novel with respect to training data, as well as diverse among themselves.  \n\nDistributions of key molecular features implicated in antimicrobial nature, such as amino acid composition, charge, hydrophobicity (H), and hydrophobic moment ($\\mu$H), were compared between the training and generated AMPs, as illustrated in Fig. \\ref{fig_biometrics}A-D. Additional features are reported in Figure S\\ref{tab:combined-comp}. CLaSS-generated AMP sequences show distinct character:\nSpecifically,  those are richer in R, L, S, Q, and C, whereas A, G, D, H,  N, and W content is reduced, in comparison to training antimicrobial sequences (Fig. \\ref{fig_biometrics}A). We also present the most abundant $k$-mers ($k$=3, 4) in  Figure S\\ref{tab:combined-comp}, suggesting that the most frequent 3 and 4-mers are  K and L-rich in both generated and training AMPs, the frequency being higher in generated sequences.   Generated AMPs are characterized by global net positive charge and aromaticity somewhere in between unlabeled and AMP-labeled training sequences, while the hydrophobic moment is comparable to that of known AMPs (Fig. \\ref{fig_biometrics}B-D and Figure S\\ref{tab:combined-comp}). These trends imply that the generated antimicrobials are still cationic and can form a putative amphiphilic $\\alpha$-helix, similar to the majority of known antimicrobials.  Interestingly, they also exhibit a moderately higher hydrophobic ratio and an aliphatic index compared to training sequences (Figure S\\ref{tab:combined-comp}).  These observations highlight the distinct physicochemical nature of the CLaSS-generated AMP sequences, as a result of the semi-supervised nature of our learning paradigm, that might help in their therapeutic application. For example, lower aromaticity and higher aliphatic index are known to induce better oxidation susceptibility and higher heat stability in short peptides \\cite{li2016molecular}, while lower hydrophobicity is associated with reduced toxicity \\cite{hawrani2008origin}.\n\n\n\n\n\n\n\n\n\n\\subsection*{\\textit{In silico} post-generation screening}\n\\label{sec:res-md} \n\nTo screen the $\\sim$90,000 CLaSS-generated AMP sequences, we first used an independent set of binary (yes\/no) sequence-level deep neural net-based classifiers that screens for  antimicrobial function, broad-spectrum efficacy, presence of secondary structure, as well as  toxicity\n (See Figs. \\ref{fig:Overview}  and  Table S\\ref{tab:classifier_results}). 163 candidates passed this screening, which were then subjected to coarse-grained Molecular Dynamics (CGMD) simulations of peptide-membrane interactions. The computational efficiency of these simulations makes them an attractive choice for high-throughput and physically-inspired filtering of peptide sequences.\n\n\nSince there exists no standardized protocol for screening antimicrobial candidates using molecular simulations, we performed a set of control simulations of known sequences with or without antimicrobial activity. From those control runs,  we found for the first time that the variance of the number of contacts between positive residues and membrane lipids is predictive of antimicrobial activity (Supplementary Fig. \\ref{fig:cgmd}): Specifically, the contact variance differentiates between high potency AMPs and non-antimicrobial sequences with a sensitivity of 88\\% and specificity of 63\\% (see SI). Physically, this feature can be interpreted as measuring the robust binding tendency of a peptide sequence to the model membrane. \nTherefore, we used the contact variance cutoff of $2$ for further filtering of the 163 generated AMPs that passed the classifier screening.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Wet lab characterization}\n\\label{sec:expt}\nA final set of 20 CLaSS-generated AMP sequences that  passed the contact variance-based screening mentioned above, along with their simulated and physico-chemical characteristics, are reported in \n Tables S\\ref{tab:20P-sim} and S\\ref{tab:sicgmd}. Those sequences were tested in the wet lab for antimicrobial activity, as measured using minimum inhibitory concentration (MIC, lower the better) against Gram-positive S. aureus and Gram-negative E. coli (Table S\\ref{tab:20P_MIC}).  \n11  generated non-AMP sequences were also screened for antimicrobial activity (Table S\\ref{tab:11N-MIC}).\nNone of the designed non-AMP sequences showed MIC values that are low enough to be considered as antimicrobials, implying that our approach is not prone to false-negative predictions. We also speculate  that a  domain shift between the AMP test set  and generated sequences from the latent space trained on the labeled and unlabeled peptide sequences likely results in false positive prediction (18 out of 20).\n\n\nAmong the 20 AI-designed AMP candidates, two sequences, \\textit{YLRLIRYMAKMI-CONH2} (YI12, 12 amino acids) and \\textit{FPLTWLKWWKWKK-CONH2} (FK13, 13 amino acids), were identified to be the best with the lowest MIC values (Table     \\ref{tab:tab-mic-compare} and Table S\\ref{tab:20P_MIC}). Both peptides are positively charged and have a nonzero hydrophobic moment (Table S\\ref{tab:sicgmd}), indicating their cationic and amphiphilic nature consistent with known antimicrobials. These peptides were further evaluated against the more difficult-to-treat Gram-negative \\textit{P. aeruginosa},  \\textit{A. baummannii}, as well as a multi-drug resistant Gram-negative \\textit{K. pnuemoniae}. As listed in Fig. \\ref{fig:6_7}, both YI12 and FK13 showed potent broad-spectrum antimicrobial activity with comparable MIC values. We have compared the MIC values of YI12 and FK13 with LLKKLLKKLLKK, an existing alpha helix-forming  antimicrobial peptide with excellent antimicrobial activity and selectivity reported in \\cite{wiradharma2013rationally}. The MIC of LLKKLLKKLLKK against  \\textit{S. aureus}, \\textit{E. coli}, and \\textit{P. aeruginosa} is $>$500, $>$500, and 63, respectively. MIC values of FK13 and FY12 are comparable to that of LLKKLLKKLLKK against \\textit{P. aeruginosa}. However, MIC values of FK13 and YI12 are significantly lower than those of LLKKLLKKLLKK against \\textit{S. aureus} and \\textit{E. coli}, demonstrating greater efficacy of antimicrobial peptides discovered in this study.\nWe also report the results of several existing AMP prediction methods on YI12 and FK13 in Table \\ref{tab:tab-mic-compare}. iAMP Pred \\cite{meher2017predicting} and  CAMP-RF \\cite{thomas2010camp} predict both of them as AMP, where other methods misclassify either of the two. For example, DBAASP-SP \\cite{vishnepolsky2018predictive} that relies on local similarity search misclassifies FK13. Witten \\textit{et al.} \\cite{witten2019deep}  predicts AMP activity against \\textit{S. aureus} for both sequences. The same method does not recognize FK13 as an effective antimicrobial against \\textit{E. coli}.\n\n\n\n\nWe further performed \\textit{in vitro} and \\textit{in vivo} testing for toxicity. \n Based on  activity measure at 50\\% hemolysis ($HC_{50}$) and lethal dose ($LD_{50}$) toxicity values (Table \\ref{tab:tab-mic-compare} and Fig. S\\ref{fig:cd_n_tox}), both peptides appear biocompatible (as the $HC_{50}$ and $LD_{50}$ values are much higher than MIC values), FK13 being more biocompatible than YI12. More importantly, the $LD_{50}$ values of both peptides compare favorably with that of polymyxin B (20.5 mg\/kg) \\cite{rifkind1967prevention}, which is a clinically used antimicrobial drug for treatment of antibiotic-resistant Gram-negative bacterial infection. \n \n\n\n\\subsection*{Sequence similarity analyses}\n\\label{sec:novelty}\nTo investigate the similarity of YI12 and FK13 with respect to training sequences, we analyzed the alignment scores  returned by the BLAST homology search in detail (Fig. S\\ref{fig:blast_result} and  Fig. S\\ref{fig:cd_n_tox}), in line with earlier works \\cite{ronvcevic2018parallel, chen2019simulation}. Scoring metrics include raw alignment score, E-value, percentage of alignment coverage, percentage of identity, percentage of positive matches or similarity,  and percentage of alignment gap (indicating the presence of additional amino acids). BLAST searching with an E-value threshold of 10 against the training database did not reveal any match for YI12, suggesting that there exists no statistically significant match of YI12. Therefore, we further searched for related sequences of  YI12  in the much larger Uniprot database consisting of $\\sim$ 223.5 M non-redundant sequences, only a fraction of which was included in our model training. YI12 shows an E-value of 2.9 to its closest match,\n\nwhich is an 11 residue segment from the bacterial EAL domain-containing protein (Fig. S\\ref{fig:blast_result}). This result suggests that  YI12  shares low similarity, even when all protein sequences in Uniprot are considered. We also performed a BLAST search of YI12 against the PATSEQ database that contains $\\sim$ 65.5 M patented peptides and still received a minimum E-value of 1.66. The sequence nearest to YI12 from PATSEQ is an eight amino acid long segment from a 79 amino acid long human protein, which has with 87.5\\% similarity and only 66.7\\% coverage, further confirming YI12's low similarity to known sequences. \n\n\nFK13 shows less than 75\\% identity, a gap in the alignment, and  85\\% query coverage to its closest match, in the training database, implying  FK13 also shares low similarity  to training sequences (Fig. S\\ref{fig:blast_result}).  YI12 is more novel than FK13, though. The closest match of FK13 in the training database is a synthetic variant of a 13 amino acid long bactericidal domain (PuroA: FPVTWRWWKWWKG) of  Puroindoline-A protein from wheat endosperm.  The antimicrobial and hemolysis activities of FK13 are close to those reported for PuroA \\cite{jing2003conformation, haney2013mechanism}. Nevertheless, FK13 is significantly different from PuroA; FK13  is K-rich and low in W-content, resulting in lower Grand Average of Hydropathy (GRAVY) score ($-0.854$ \\textit{vs.} $-0.962$), higher aliphatic index ($60.0$ \\textit{vs.} $22.3$), and  lower instability index ($15.45$ \\textit{vs.} $58.30$), all together indicative of higher peptide stability. In fact, lower W-content was found beneficial for stabilizing of FK13 during wet-lab experiments, since Tryptophan (W) is susceptible to oxidation in air. Lower W-content has also been implicated in improving  \\textit{in vivo} peptide stability \\cite{mathur2018silico}.\nTaken together, these results illustrate that CLaSS on latent peptide space modeled by the WAE is able to generate novel and optimal antimicrobial sequences by efficiently learning the complicated sequence-function relationship in peptides and exploiting that knowledge for controlled exploration. When combined with subsequent \\textit{in silico} screening,  novel and optimal lead candidates with experimentally confirmed high broad-spectrum efficacy and selectivity are identified at a success rate of 10\\%. The whole cycle (from database curation to wet lab confirmation) took 48 days in total and a single iteration (Fig. \\ref{fig:Overview}). \n  \n\n\n \n \n \\subsection*{Structural and mechanistic analyses}\n \\label{sec:str}\n\n \n We performed all-atom explicit water simulations (see SI) of these two sequences in the presence of a lipid membrane starting from an $\\alpha$-helical structure. Different membrane binding mechanisms were observed for the two sequences, as illustrated in Fig.~\\ref{fig:6_7}A. YI12 embeds into the membrane by using positively charged N-terminal Arginine (R) residues. While FK13 embeds either with N-terminal Phenylalanine (F) or with C-terminal Tryptophan (W) and  Lysine (K). These results provide mechanistic insights into different modes of action adopted by YI12 and FK13 during the early stages of membrane interaction.\n \n Peptides were further experimentally characterized using CD spectroscopy (see SI).\n Both YI12 and FK13 showed random coil-like structure in water, but formed an $\\alpha$-helix in 20\\% SDS buffer (Fig. S\\ref{fig:cd_n_tox}). Structure classifier predictions (see SI) and all-atom simulations   are consistent with the CD resylts. From  CD spectra, $\\alpha$-helicity of YI12 appears stronger than that of FK13, in line with its stronger hydrophobic moment (Table S\\ref{tab:sicgmd}). In summary,  physicochemical analyses and CD spectroscopy together suggest that cationic nature and amphiphilic helical topology are the underlying factors inducing antimicrobial nature in YI12 and FK13.\n \n\n   \n \n To provide insight onto the mechanism of action underlying the antimicrobial activity of YI12 and FK13, we conducted agar plate assay and found that both peptides YI12 and FK13 are bactericidal. There was a 99.9\\% reduction of colonies at 2 x MIC.\n \n Since $\\alpha$-helical peptides like  YI12 and FK13 are known to disrupt membranes by leaky pore formation ~\\cite{kumar2018antimicrobial,guha2019mechanistic}, we performed  live-imaging with confocal fluorescence microscopy for FK13 against \\textit{E. coli} (Figure \\ref{fig:confocal}). The results were compared with that of polymyxin B, that is one of a group of basic polypeptide antibiotics derived from \\textit{B polymyxa}.   After 2 hours of  incubating  E. coli with polymyxin B or with FK13 at 2 x MIC,   confocal imaging was performed.  Emergence of fluorescence, as shown in Figure \\ref{fig:confocal}, confirmed that red propidium iodide (PI) has entered the bacterial cell and interacted with bacterial DNA in the presence of either FK13 or polymyxin B.   This finding implies that both polymyxin and FK13 induce pore formation on the bacteria membrane and allow the PI dye to enter the bacteria. Without the pore formation, the PI dye will not be able to enter the bacteria.\n \n \\subsection*{Resistance analyses}\n Finally, we  performed resistance acquisition studies of E. coli. in the presence of  imipenem, an intravenous $\\beta$-lactam antibiotic,  YI12 or FK13 at concentrations of sub MIC levels. Results shown in Figure \\ref{fig:6_7}B confirm that both YI12 and FK13 do not induce resistance after 25 passages, while E. coli has begun to develop resistance to the antibiotic Imipenem after just 6 passages. We have also investigated the efficacy of these peptides against polymyxin B resistant \\textit{K. pneumoniae}, a strain that is resistant to polymyxin B, a beta-lactum antibiotic, an antibiotic of last resort. Table \\ref{tab:tab-mic-compare} shows the MIC values of   YI12, FK13, and polymyxin B, revealing no  MIC increase for either of the two discovered peptides, when compared to the MIC against the MDR K. pneumoniae stain (from ATCC). In contrast, the MIC of polymyxin B is 2 $\\mu$g\/ml against the same MDR K. pneumoniae stain (from ATCC), but increases to  $>$125 $\\mu$g\/ml once the \\textit{K. pneumoniae} became resistant to polymyxin B. The MIC value of YI12 and FK13 is still lower than that of polymyxin B in the polymyxin B resistant strain, indicating that the resistance to polymyxin B  is not seen towards YI12 and FK13.  Taken together, these results indicate that YI12 and FK13 hold therapeutic potential for treating resistant strains and therefore demand further investigation.\n\n \n\n\n\n\\section*{Discussion and Conclusions}\n\\label{sec:conclusion}\nLearning implicit interaction rule(s) of complex molecular systems is a  major goal of artificial intelligence (AI) research.\nThis direction is critical for designing new molecules\/materials with specific structural and\/or functional requirements, one of the most anticipated and acutely needed applications.\nAntimicrobial peptides considered here represent an archetypal system for molecular discovery problems. They exhibit a near-infinite and mostly unexplored chemical repertoire,\na well-defined chemical palette (natural amino acids), as well as potentially conflicting or opposing design objectives, and is of high importance due to the global increase in antibiotic resistance and a depleted antibiotic discovery pipeline. Recent work has shown that deep learning can be used to help screen libraries of existing chemicals for antibiotic properties \\cite{stokes2020deep}. A number of recent studies have also used AI methods for  design of antimicrobial peptides and provided experimental validation  \\cite{loose2006linguistic,  nagarajan2018computational, maccari2013antimicrobial, porto2018silico, fjell2011optimization, vishnepolsky2018predictive, nagarajan2019omega76}. However, to our knowledge, the present work provides for the first time a fully automated computational framework that combines  controllable generative modeling, deep learning, and physics-driven learning for \\textit{de novo} design of broad-spectrum potent and selective AMP sequences and experimentally validates them for broad-spectrum efficacy and toxicity. Further, the discovered peptides show high efficacy against a strain that is resistant to an antibiotic of last resort, as well as mitigate drug resistance onset.\nWet lab results confirmed the efficiency of the proposed approach for designing novel and optimized sequences with a very modest number of candidate compounds synthesized and tested. The present design approach in this proof-of-concept study yielded a 10\\% success rate and a rapid turnaround of\n48 days, highlighting the importance of combining AI-drive computational strategies with experiments to achieve more effective drug candidates\nThe generative modeling approach presented here can be tuned for not only generating novel candidates, but also for designing novel combination therapies and antibiotic adjuvants, to further advance antibiotic treatments.\n\nSince CLaSS is a generic approach, it is suitable for a variety of controlled generation tasks and can handle multiple controls simultaneously.  The method is simple to implement, fast, efficient, and scalable, as it does not require any optimization over the latent space. CLaSS has additional advantages regarding repurposability, as adding a new constraint requires a simple predictor training. \nTherefore,  future directions of this work will explore the effect of \nadditional relevant constraints, such as the induced resistance, efficacy in animal models of infection, and fine-grained strain-specificity, on the  designed AMPs using the approach presented here. Extending CLaSS application to other controlled molecule design tasks, such as target-specific and selective drug-like small molecule generation is also underway \\cite{chenthamarakshan2020target}.\nFinally,  the AI models will be further optimized in an iterative manner\nby using the feedback from simulations and\/or experiments in an active learning framework. \n\n\n\n \n\n\n\n\\clearpage\n\n\n\n\\subsection*{Online Methods}\n\\subsubsection*{Generative Autoencoders}\nTo learn meaningful continuous latent representations from sequences without supervision, the Variational Autoencoder (VAE) \\cite{kingma2013auto} family has emerged as a principled and successful method.\nThe data distribution $p({\\mathbf{x}})$ over samples ${\\mathbf{x}}$ is represented as the marginal of a joint distribution $p({\\mathbf{x}}, {\\mathbf{z}})$ that factors out as $p({\\mathbf{z}}) p_\\theta({\\mathbf{x}}|{\\mathbf{z}})$.\nThe prior $p({\\mathbf{z}})$ is a simple smooth distribution, while $p_\\theta({\\mathbf{x}} | {\\mathbf{z}})$ is the decoder that maps a point in latent ${\\mathbf{z}}$-space to a distribution in ${\\mathbf{x}}$ data space.\nThe exact inference of the hidden variable ${\\mathbf{z}}$ for a given input ${\\mathbf{x}}$ would require integration over the full latent space: $p({\\mathbf{z}} | {\\mathbf{x}}) = \\frac{p({\\mathbf{z}}) p_\\theta({\\mathbf{x}}|{\\mathbf{z}})}{\\int d{\\mathbf{z}} p({\\mathbf{z}}) p_\\theta({\\mathbf{x}}|{\\mathbf{z}})}$.\nTo avoid this computational burden, the inference is approximated through an inference neural network or encoder $q_\\phi({\\mathbf{z}}|{\\mathbf{x}})$.\nOur implementation follows \\cite{bowman2015large}, where both encoder and decoder are single-layer LSTM recurrent neural networks \\cite{hochreiter1997long},\nand the encoder specifies a diagonal Gaussian distribution, i.e. $q_\\phi(z|x) = N(z; \\mu(x), \\Sigma(x))$ (Fig. \\ref{fig:overview-gen}).\n\nThe basis for auto-encoder training is optimization of an objective consisting of the sum of a reconstruction loss and a regularization constraint loss term: ${\\mathcal{L}}(\\theta, \\phi) = {\\mathcal{L}}_{rec}(\\theta, \\phi) + {\\mathcal{L}}_c(\\phi)$.\nIn the standard VAE objective \\cite{kingma2013auto}, reconstruction loss ${\\mathcal{L}}_{rec}(\\theta, \\phi)$ is based on the negative log likelihood of the training sample, \nand the constraint ${\\mathcal{L}}_c(\\phi)$ uses $D_{KL}$, the Kullback-Leibler divergence:\n\\[\n     \\mathcal{L}_{\\text{VAE}}(\\theta, \\phi) =\n    \\mathbb{E}_{q_\\phi(z|x)}[\\log p_\\theta(x|z)]\n   \n    - D_{KL}(q_\\phi(z|x) || p(z))\n   \n\\]\nfor a single sample.\nThis exact objective is derived from a lower bound on the data likelihood; hence this objective is called the ELBO (Evidence Lower Bound). With the standard VAE, we observed the same posterior collapse as detailed for natural language in the literature \\cite{bowman2015generating},\nmeaning $q(z|x) \\approx p(z)$ such that no meaningful information is encoded in $z$ space.\nFurther extensions include $\\beta$-VAE that adds a multiplier ``weight'' hyperparameter $\\beta$ on the regularization term, and $\\delta$-VAE that encourages the $D_{KL}$ term to be close to a nonzero $\\delta$, \\textit{etc.}, to tackle the issue of posterior collapse. However, finding the right setting that serves as a workaround for the posterior collapse is tricky within these VAE variants. \n\n\nTherefore, many variations within the VAE family have been recently proposed, such as Wasserstein Autoencoder (WAE) \\cite{tolstikhin2017wasserstein, bahuleyan2018probabilistic} and Adversarial Autoencoder (AAE) \\cite{makhzani2015adversarial}.\n\nWAE factors an optimal transport plan through the encoder-decoder pair,\non the constraint that marginal posterior $q_\\phi(z) = \\mathbb{E}_{x \\sim p(x)} q_\\phi(z|x)$\nequals a prior distribution, i.e. $q_\\phi(z)=p(z)$.\nThis is relaxed to an objective similar to $\\mathcal{L}_{\\text{VAE}}$ above. However, \nin the WAE objective \\cite{tolstikhin2017wasserstein}, instead of each individual $q_\\phi({\\mathbf{z}} | {\\mathbf{x}})$, the marginal posterior $q_\\phi({\\mathbf{z}}) = \\mathbb{E}_{\\mathbf{x}} [q_\\phi({\\mathbf{z}}|{\\mathbf{x}})]$ is constrained to be close to the prior $p({\\mathbf{z}})$.\nWe enforce the constraint by penalizing maximum mean discrepancy \\cite{gretton2007kernel} with random features approximation of the radial basis function \\cite{rahimi2007random}:\n${\\mathcal{L}}_c(\\phi) = \\text{MMD}(q_\\phi({\\mathbf{z}}), p({\\mathbf{z}}))$.\nThe total objective for WAE is ${\\mathcal{L}} = {\\mathcal{L}}_{rec} + {\\mathcal{L}}_c$ where we use the reconstruction loss ${\\mathcal{L}}_{rec} = -\\mathbb{E}_{q_\\phi({\\mathbf{z}}|{\\mathbf{x}})}[\\log p_\\theta({\\mathbf{x}}|{\\mathbf{z}})]$. In WAE training with maximum mean discrepancy (MMD) or with a discriminator, we found a benefit of regularizing the encoder variance as in the literature~\\cite{rubenstein2018latent,bahuleyan2018probabilistic}. For MMD, we used a random features approximation of the Gaussian kernel \\cite{rahimi2007random}.\n\nDetails of autoencoder architecture and training, as well as an experimental comparison between different auto-encoder variations tested in this study, can be found in Supplementary Material sections \\ref{si:models}, \\ref{method:training} and \\ref{si:vaewaeaae}. Python codes for training peptide autoencoders are available via github at https:\/\/github.com\/IBM\/controlled-peptide-generation.\n\n\\subsubsection*{CLaSS - Conditional Latent (Attribute) Space Sampling}\n\\label{si:lcs}\nWe propose Conditional Latent (attribute) Space Sampling, CLaSS, a simple but efficient method to sample from the targeted region of the latent space from an auto-encoder, which was trained in an unsupervised manner (Fig. \\ref{fig:overview-gen}). \n\n\n\\textbf{Density Modeling in Latent Space}\nWe assume a latent variable model (e.g., Autoencoder) that has been trained in an unsupervised manner to meet the evaluation criteria outlined in (see SI).\n All training data ${\\mathbf{x}}_j$ are then encoded in latent space: ${\\mathbf{z}}_{j,k} \\sim q_\\phi({\\mathbf{z}} | {\\mathbf{x}}_j)$.\nThese ${\\mathbf{z}}_{j,k}$ are used to fit an explicit density model $Q_\\xi({\\mathbf{z}})$ to approximate marginal posterior $q_\\phi({\\mathbf{z}})$, and a classifier model $q_\\xi(a_i|{\\mathbf{z}})$ for attribute $a_i$ to approximate the probability $p(a_i|{\\mathbf{x}})$.\nThe motivation for fitting a $Q_\\xi({\\mathbf{z}})$ is in order to sample from $Q_\\xi$ rather than $p({\\mathbf{z}})$, since at the end of training the discrepancy between $q_\\phi({\\mathbf{z}})$ and $p({\\mathbf{z}})$ can be significant.\n\nAlthough any explicit density estimator could be used for $Q_\\xi({\\mathbf{z}})$,\nhere we consider Gaussian mixture density models and evaluate negative log-likelihood on a held-out set to determine the optimal complexity.\nWe find 100 components and untied diagonal covariance matrices to be optimal, giving a held-out log likelihood of $105.1$.\nTo fit $Q_\\xi$, we use K=10 random samples from the encoding distribution of the training data,  ${\\mathbf{z}}_{j,k} \\sim q_\\phi({\\mathbf{z}} | {\\mathbf{x}}_j) = \\mathcal{N}(\\mu({\\mathbf{x}}_j), \\sigma({\\mathbf{x}}_j))$, with $k=1 \\dots K$.\n\n Independent simple linear attribute classifiers $q_\\xi(a_i | {\\mathbf{z}})$ are then fitted per attribute.\nFor each attribute $a_i$, the procedure consists of: \n(1) collecting dataset with all labeled samples for this attribute $({\\mathbf{x}}_j, a_i)$, \n(2) encoding the labeled data as before, ${\\mathbf{z}}_{j,k} \\sim q_\\phi({\\mathbf{z}} | {\\mathbf{x}}_j)$, \n(3) fitting $\\xi$, the parameters of logistic regression classifier $q_\\xi(a_i | {\\mathbf{z}})$ with inverse regularization strength $C=1.0$ and 300 lbfgs iterations.\n\n\\textbf{Rejection Sampling for Attribute-Conditioned Generation}\nLet us formalize that there are $n$ different (and possibly independent) binary attributes of interest ${\\mathbf{a}} \\in \\{0,1\\}^n = [a_1, a_2, \\dots, a_n]$, each attribute is only available (labeled) for a small and possibly disjoint subset of the dataset. Since functional annotation of peptide sequences is expensive, current databases typically represent a small  ($\\approx 100-10000$)  subset of the unlabeled corpus.\nWe posit that all plausible datapoints have those attributes, albeit mostly without label annotation. Therefore, the data distribution implicitly is generated as $p({\\mathbf{x}}) = \\mathbb{E}_{{\\mathbf{a}} \\sim p({\\mathbf{a}})} [ p({\\mathbf{x}} | {\\mathbf{a}})]$, where the distribution over the (potentially huge) discrete set of attribute combinations $p({\\mathbf{a}})$ is integrated out, and for each attribute combination the set of possible sequences is specified as $p({\\mathbf{x}} | {\\mathbf{a}})$.\nAs our aim is to sample novel sequences ${\\mathbf{x}} \\sim p({\\mathbf{x}} | {\\mathbf{a}})$, for a desired attribute combination ${\\mathbf{a}} = [a_1, \\dots, a_n]$, \nwe are now able to approach this task through conditional sampling in latent space:\n\\begin{align}\n  p({\\mathbf{x}}|{\\mathbf{a}}) & = \\int \\! \\mathrm{d}z \\, p({\\mathbf{z}}|{\\mathbf{a}}) p({\\mathbf{x}}|{\\mathbf{z}}) \\\\\n  & \\approx \\int \\! \\mathrm{d}z \\, \\hat{p}_\\xi({\\mathbf{z}}|{\\mathbf{a}}) p_\\theta({\\mathbf{x}}|{\\mathbf{z}})\n\\end{align}\nWhere $\\hat{p}_\\xi({\\mathbf{z}}|{\\mathbf{a}})$ will not be approximated explicitly, rather we will use rejection sampling using the models $Q_\\xi({\\mathbf{z}})$ and $q_\\xi(a_i | {\\mathbf{z}})$ to approximate samples from $p({\\mathbf{z}} | {\\mathbf{a}})$.\n\nTo approach this, we first use Bayes' rule and the conditional independence of the attributes $a_i$  conditioned on ${\\mathbf{z}}$, since we assume the latent variable captures all information to model the attributes: $a_i \\perp a_j | {\\mathbf{z}}$ (\\textit{i.e.} two attributes $a_i$ and $a_j$ are independent when conditioned on ${\\mathbf{z}}$)\n\\begin{align}\np({\\mathbf{z}}|{\\mathbf{a}})\n &= \\frac{p({\\mathbf{a}} | {\\mathbf{z}}) q_\\phi({\\mathbf{z}})} {p({\\mathbf{a}})} \\\\\n & = \\frac{ q_\\phi({\\mathbf{z}}) \\prod_i p(a_i | {\\mathbf{z}})} {p({\\mathbf{a}})}\n\\end{align}\n\nThis approximation is introduced to $\\hat{p}_\\xi({\\mathbf{z}}|{\\mathbf{a}})$,\nusing the models $Q_\\xi$ and $q_\\xi$ above:\n\\begin{align}\n\\hat{p}_\\xi({\\mathbf{z}}|{\\mathbf{a}}) &= \\frac{Q_\\xi({\\mathbf{z}}) \\prod_i q_\\xi (a_i | {\\mathbf{z}})}{q_\\xi({\\mathbf{a}})}\n\\label{eq:rejsamplingbase}\n\\end{align}\n\nThe denominator $q_\\xi({\\mathbf{a}})$ in Eq. (\\ref{eq:rejsamplingbase}) could be estimated by approximating the expectation $q_\\xi({\\mathbf{a}}) = \\mathbb{E}_{Q_\\xi({\\mathbf{z}})} q_\\xi({\\mathbf{a}} | {\\mathbf{z}}) \\approx \\frac{1}{N} \\sum_{{\\mathbf{z}}_j \\sim Q_\\xi({\\mathbf{z}})}^N q_\\xi({\\mathbf{a}} | {\\mathbf{z}})$.\nHowever, the denominator is not needed \\textit{a priori} in our rejection sampling scheme,\nin contrast, $q_\\xi({\\mathbf{a}})$ will naturally appear as the rejection rate of samples from the proposal distribution (see below).\n\nFor rejection sampling distribution with pdf $f({\\mathbf{z}})$, we need a proposal distribution $g({\\mathbf{z}})$ and a constant $M$, such that $f({\\mathbf{z}}) \\leq M g({\\mathbf{z}})$ for all ${\\mathbf{z}}$, i.e. $M g({\\mathbf{z}})$ envelopes $f({\\mathbf{z}})$.\nWe draw samples from $g({\\mathbf{z}})$ and accept the sample with probability $\\frac{f({\\mathbf{z}})}{M g({\\mathbf{z}})}  \\leq  1$.\n\nIn the above, to sample from Eq. (\\ref{eq:rejsamplingbase}), we consider ${\\mathbf{a}}$ to be constant.\nWe  perform rejection sampling through the proposal distribution: $g({\\mathbf{z}}) = Q_\\xi({\\mathbf{z}})$ that can be directly sampled.\nNow set $M=1\/q_\\xi({\\mathbf{a}})$ so $M g({\\mathbf{z}}) = Q_\\xi(z) \/ q_\\xi ({\\mathbf{a}})$,\nwhile our pdf to sample from is $f({\\mathbf{z}}) = Q_\\xi({\\mathbf{z}}) \\prod_i q_\\xi (a_i | {\\mathbf{z}}) \/ q_\\xi({\\mathbf{a}})$.\nTherefore, we accept the sample from $Q_\\xi({\\mathbf{z}})$ with probability\n\\[\n\\frac{f({\\mathbf{z}})}{M g({\\mathbf{z}})} = \\prod_i q_\\xi (a_i | {\\mathbf{z}})  \\leq 1\n\\]\nThe inequality trivially follows from the product of normalized probabilities.\nThe acceptance rate is $1\/M = q_\\xi({\\mathbf{a}})$.\nIntuitively, the acceptance probability is equal to the product of the classifier's scores, while sampling from explicit density $Q_\\xi(z)$. \nIn order to accept any samples, we need a region in ${\\mathbf{z}}$ space to exist where $Q_\\xi({\\mathbf{z}}) > 0$ and the classifiers assign a nonzero probability to all desired attributes, \\textit{i.e.} the combination of attributes has to be realizable in ${\\mathbf{z}}$-space.\n\n\n\\clearpage\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/natureBME_fig1_finalV2.pdf}\n\\caption{Overview and timeline of the proposed AI-driven approach for accelerated antimicrobial design.     \n}\n\\label{fig:Overview}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/pipeline_figure-v3.edited.png}\n\\caption{Phases of attribute-controlled peptide sequence generation. (A) Training a generative Autoencoder (AE) model on peptide sequences (AE training in Fig. 1), (B) Mapping sparse peptide attributes to the model's latent ${\\mathbf{z}}$-space and constructing the density model of the ${\\mathbf{z}}$-space (Autoencoder Evaluation in Fig. 1), and (C) Sampling from the ${\\mathbf{z}}$-space using our CLaSS method (Controlled Generation in Fig. 1).  }\n\\label{fig:overview-gen}\n\\end{figure}\n\n\\begin{figure}\n     \\centering\n    \n     \\includegraphics[width=\\textwidth]{figures\/natureBME_fig3_finalV2a.pdf}\n\n    \n\\caption{Characteristics of the generative autoencoder latent space. (A) Relation between sequence similarity and Euclidean distance in latent ${\\mathbf{z}}$-space, when sequences were modeled using WAE (VAE in Inset).   Darker points indicate similarity with itself (\\textit{i.e.} the same exact sequence). \nClassifiers were trained either using WAE ${\\mathbf{z}}$-space encodings (${\\mathbf{z}}$-) or on sequences (sequence-level). Note that the class prediction accuracy of attribute AMP using WAE z-classifiers and sequence classifiers on test data are 87.4 and 88.0, respectively. The same for toxicity attribute are 68.9 and 93.7. \n(B) Decoded sequences and their attributes during a linear interpolation between two distant sequences in the WAE latent space.\nAttributes include (1) physicochemical properties,\n  (2) sequence similarity (\\texttt{evo\\_start}, \\texttt{evo\\_end}) from  endpoint sequences, \nand (3) AMP (\\texttt{z\\_amp}) and Toxic (\\texttt{z\\_tox}) class  probabilities from ${\\mathbf{z}}$-classifiers.\nValues in orange and blue are in the upper and lower quartile, respectively. \nBlack rectangle indicates sequences with low attribute similarity to endpoint sequences.\n(C) As an example of further analysis, we show the relation of AMP class probability and instability index for all candidates in the interpolation and a ball-stick rendering of a selected sequence in the path. An interactive demo of interpolations in peptide latent space is available at https:\/\/peptide-walk.mybluemix.net.\n}\n\\label{fig:wae-latent}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/natureBME_fig4_finalV2.pdf}\n\\caption{Physico-chemical property comparison. (A) Comparison of amino acid composition, (B) global hydrophobicity, (C) hydrophobic moment (C), and (D) charge distribution of CLaSS-generated AMPs with training sequences. Mean and standard deviation  were estimated on three different sets, each consisting 3000 randomly chosen samples. Generated AMP: {\\color{our_orange}orange}; training AMP: {\\color{our_blue}blue}; training unlabeled: {\\color{our_gray}gray}. \n}\n\\label{fig_biometrics}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{figures\/figure_6_7.edited.png}\n\\caption{Atomistic simulation and resistance acquisition studies. (A) Snapshot from all-atom simulation of YI12 (A1)  and FK13 (A2). Selected residues that interact with the membrane are highlighted. (B) Resistance acquisition studies of E. coli. in the presence of sub (1\/2x) MIC levels of imipenem, YI12 and FK13.\n}\n\\label{fig:6_7}\n\\end{figure}\n\n\n\\newcommand{\\specialcell}[2][c]{%\n\\begin{tabular}[#1]{@{}c@{}}#2\\end{tabular}}\n  \n\\begin{table}\n    \\centering\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c}\n        \\toprule\n         Sequence & \\specialcell{SA} & \\specialcell{EC} & \\specialcell{PA } & \\specialcell{AB} & \\specialcell{MDR-KP\n} & \\specialcell{polyR-KP} &  HC$_{50}$ & LD$_{50}$ \\\\ \n        \\hline\n        YI12 & 7.80 & 31.25 & 125.00 & 15.60 & 31.25  & 31 & 125 & 182 \\\\\n        FK13 & 15.60 & 31.25 & 62.50 & 31.25 & 15.60  & 16 & 500 & 158 \\\\\n        \n         \\bottomrule\n    \\end{tabular}\n    \\caption{Antimicrobial activity and toxicity values of YI12 and FK13, two best CLaSS-designed AMPs.\n    MIC values against diverse strains, including SA (S. aureus), EC (E. coli), PA (P. aeruginosa), AB (A. baumannii), one multidrug-resistant (MDR) \\textit{K. pneumonia} (MDR-KP), and one polymyxin B resistant \\textit{K. pneumonia} (polyR-KP). Hemolytic activity measured at 50\\% hemolysis (HC$_{50}$) using rat red blood cells,  and lethal dose toxicity (LD$_{50}$) values are for Balb\/C mice. All reported MIC and HC$_{50}$ values are in $\\mu$g\/mL, while unit for LD$_{50}$ is mg\/kg. As a baseline, MIC value of polymyxin B against polyR-KP is $>$125 $\\mu$g\/mL.}\n    \\label{tab:tab-mic-compare}\n\\end{table}\n\n\n\n\\clearpage\n\n\\section*{Acknowledgments}\nWe acknowledge Youssef Mroueh and Kush Varshney for insightful discussions. We also thank Oscar Chang, Elham Khabiri, and Matt Riemer for help with the initial phase of the work. We would like to acknowledge David Cox, Yuhai Tu, Pablo Meyer Rojas, and Mattia Rigotti for providing valuable feedback on the manuscript. F.C. thanks Patrick Simcock for sharing knowledge. We thank anonymous reviewers for constructive feedback that significantly improved quality of this manuscript.\n\n\n\\section*{Author Information}\nCorrespondence and requests for materials should be addressed to \nPayel Das~(email: daspa@us.ibm.com).\n\n\\textbf{Author Contributions:} P.D., J.C., and A.M. conceived the project; P.D. designed and managed the project; P.D and T.S. designed and implemented the sequence generation and screening framework and algorithm with help from K.W., I.P., and S.G.; Autoencoder experiments were run and analyzed by P.D., T.S., K.W., I.P., P.C., V.C., and C.D.S.; Generated sequences were analyzed in silico by P.D., T.S., H.S., and K.W.; F.C. with help from P.D. and J.C. designed, performed and analyzed the molecular dynamics simulations; Y.Y.Y., J.T., J.H.  performed and analyzed the wet lab experiments; H.S. created final figures; All authors wrote the paper. \n\n\\section{Ethics Statement}\nThe authors declare that they have no competing financial interests. \n\n\\section{Data and materials availability:} A full description of the model and method\nis available in the supplementary materials. Data and code will be available upon request and is also accessible via github at https:\/\/github.com\/IBM\/controlled-peptide-generation\/tree\/master\/models.  \n\n\\section{Supplementary Information}\n\\textbf{Supplementary Information}\nSupplementary tables, figures, descriptions of the dataset and details of models and methods.\n\n\\textbf{Reporting Summary}\n Further information on research design is available in the Nature Research Reporting Summary linked to this article.\n \n \n\n\\clearpage\n\n\n\n\\bibliographystyle{Science}\n\n\n\\section*{Introduction}\n\nIn this file, we present some tips and sample mark-up to assure your\n\\LaTeX\\ file of the smoothest possible journey from review manuscript\nto published {\\it Science\\\/} paper.  We focus here particularly on\nissues related to style files, citation, and math, tables, and\nfigures, as those tend to be the biggest sticking points.  Please use\nthe source file for this document, \\texttt{scifile.tex}, as a template\nfor your manuscript, cutting and pasting your content into the file at\nthe appropriate places.\n\n{\\it Science\\\/}'s publication workflow relies on Microsoft Word.  To\ntranslate \\LaTeX\\ files into Word, we use an intermediate MS-DOS\nroutine \\cite{tth} that converts the \\TeX\\ source into HTML\\@.  The\nroutine is generally robust, but it works best if the source document\nis clean \\LaTeX\\ without a significant freight of local macros or\n\\texttt{.sty} files.  Use of the source file \\texttt{scifile.tex} as a\ntemplate, and calling {\\it only\\\/} the \\texttt{.sty} and \\texttt{.bst}\nfiles specifically mentioned here, will generate a manuscript that\nshould be eminently reviewable, and yet will allow your paper to\nproceed quickly into our production flow upon acceptance \\cite{use2e}.\n\n\\section*{Formatting Citations}\n\nCitations can be handled in one of three ways.  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For the final revision,\nhowever, the \\verb+{figure}+ environment should {\\it not\\\/} be used;\ninstead, the figure captions themselves should be typed in as regular\ntext at the end of the source file (an example is included here), and\nthe figures should be uploaded separately according to the Art\nDepartment's instructions.\n\n\n\n\n\n\n\n\n\\section*{What to Send In}\n\nWhat you should send to {\\it Science\\\/} will depend on the stage your manuscript is in:\n\n\\begin{itemize}\n\\item {\\bf Important:} If you're sending in the initial submission of\n  your manuscript (that is, the copy for evaluation and peer review),\n  please send in {\\it only\\\/} a PDF version of the\n  compiled file (including figures).  Please do not send in the \\TeX\\ \n  source, \\texttt{.sty}, \\texttt{.bbl}, or other associated files with\n  your initial submission.  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The bibliography should not be in a separate file.}  \n  Thus, if the\n  name of your main source document is \\texttt{ltxfile.tex}, you\n  need to include:\n\\begin{itemize}\n\\item \\texttt{ltxfile.tex}.\n\\item \\texttt{ltxfile.aux}, the auxilliary file generated by the\n  compilation.\n\\item A PDF file generated from\n  \\texttt{ltxfile.tex}.\n\n\\end{itemize}\n\\end{itemize}\n\n\n\n\n\\section*{Table of Contents}\n\nSupplementary Text\\\\\nDataset\\\\\nModel and Methods\\\\\nFig. S1 to S4 \\\\\nTables S1 to S8\\\\\nSupplementary References \\textit{(1-67)}\n\\section{Supplementary Text}\n\n\n\n\n\n\n\n\n\\subsection{Conditional Generation with Autoencoders}\nSince the peptide sequences are represented as text strings here, we will be limiting our discussion to literature around text generation with constraints. \nControlled text sequence generation is non-trivial, as the discrete and non-differentiable nature of text samples does not allow the use of a global discriminator, which is commonly used in image generation tasks to guide generation. To tackle this issue of non-differentiability, policy learning has been suggested, which suffers from high variance during training \\cite{yu2017seqgan,guimaraes2017objective}. Therefore, specialized distributions, such as the Gumbel-softmax \\cite{jang2016categorical,kusner2016gans}, a concrete distribution \\cite{maddison2016concrete},  or a soft-argmax function \\cite{zhang2016generating}, have been proposed to approximate the gradient of the model from discrete samples.\n\nAlternatively, in a semi-supervised model setting, the minimization of element-wise reconstruction error has been employed \\cite{kingma2014semi}, which tends to lose the holistic view of a full sentence. Hu et al. \\cite{hu2017toward} proposed a VAE variant that allows both controllable generation and semi-supervised learning. The working principle needs labels to be present during training and encourages latent space to represent them, so the addition of  \n new attributes will require retraining the latent variable model itself.\nThe framework relies on a set of new discrete binary variables in latent space to control the attributes, an ad-hoc wake-sleep procedure. It requires learning the right balance between multiple competing and tightly interacting loss objectives, which is tricky. \n\nEngel et al.~\\cite{engel2017latent} propose a conditional generation framework without retraining the model, similar in concept to ours, by modeling in latent space post-hoc. Their approach does not need an explicit density model in ${\\mathbf{z}}$-space, \nrather relies on adversarial training of generator and attribute discriminator and focuses modifying sample reconstructions rather than generating novel samples. \nRecent Plug and Play Language Model (PPLM) for controllable language generation combines a pre-trained language model (LM) with one or more simple attribute classifiers that guide text generation without any further training of the LM \\cite{dathathri2019plug}. \n\n\\subsection{ Conditional Generation for Molecule Design}\nFollowing the work of G\u00f3mez-Bombarelli et al. \\cite{gomez2018automatic}, Bayesian Optimization (BO) in the learned latent space has been employed for molecular optimization for properties such as drug likeliness (QED) or penalized logP. The standard BO routine consists of two key steps: (i) estimating the black-box function from data through a probabilistic surrogate model; usually a Gaussian process (GP), referred to as the response surface; (ii) maximizing an acquisition function that computes a score that trades off exploration and exploitation according to uncertainty and optimality of the response surface. As the dimensionality of the input latent space increases, these two steps become challenging.  In most cases, such a method is restricted to local optimization using training data points as starting points, as optimizers are likely to follow gradients into regions of the latent space that the model has not been exposed to during training.\nReinforcement learning (RL) based methods provide an alternative approach for molecular optimization \\cite{zhou2019optimization, you2018graph, popova2018deep, Zhavoronkov2019natbio}, in which RL policies are learned by incorporating the desired attribute as part of the reward. However, a large number of evaluations are typically needed for both BO and RL-based optimizations while trading off exploration and exploitation \\cite{korovina2019chembo}. \nSemi-supervised learning has also been used for conditional generation of molecules \\cite{lim2018molecular, kang2018conditional, li2018multi, mendez2020novo}, which needs labels to be available during the generative model training. \n\nCLaSS is fundamentally different from these existing approaches, as it does not need expensive optimization over latent space, policy learning, or minimization of complex loss objectives - and therefore does not suffer from cumbersome computational complexity. Furthermore, CLaSS is not limited to local optimization around an initial starting point. Adding a new constraint in CLaSS is relatively simple, as it only requires a simple predictor training; therefore, CLaSS is easily repurposable. CLaSS is embarrassingly parallelizable as well.  \n\n\n\n\\section{Dataset}\n\n\n\n\\subsection{A Dataset for Semi-Supervised Training of AMP Generative Model}\n\\label{si:data}\n\nWe compiled a new two-part (unlabeled and labeled) dataset for learning a meaningful representation of the peptide space and conditionally generating safe antimicrobial peptides from that space using the proposed CLaSS method. We consider discriminating for several functional attributes as well as for presence of structure in peptides. Only linear and monomeric sequences with no terminal modifications and length up to 50 amino acids were considered in curating this dataset.\nAs a further pre-processing step, the sequences with non-natural amino acids (B, J, O, U, X, and Z) and the ones with lower case letters were eliminated. \n\n\\textbf{Unlabeled Sequences:} The unlabeled data is from Uniprot-SwissProt and Uniprot-Trembl database \\cite{uniprot} and contains just over 1.7 M  sequences, when considering sequences with length up to 50 amino acid.\n\n\\textbf{Labeled Sequences:}\nOur labeled dataset comprises sequences with different attributes curated from a number of publicly available databases~\\cite{singh2015satpdb,pirtskhalava2016dbaasp, khurana2018deepsol, bhadra2018ampep, gupta2013silico}. Below we provide details of the labeled dataset:\n\\begin{itemize}\n    \\item Antimicrobial (8683 AMP, 6536 non-AMP);\n    \\item Toxic (3149 Toxic, 16280 non-Toxic);\n    \\item Broad-spectrum (1302 Positive, 1238 Negative);\n    \\item Structured (1170 Positive, 2136 Negative);\n    \\item Hormone (569 Positive);\n    \\item Antihypertensive (1659 Positive);\n    \\item Anticancer (504 Positive).\n\\end{itemize}\n\n\\subsubsection{Details of Labeled Datasets}\n\\label{si:datadetails}\n\\textbf{Sequences with AMP\/non-AMP Annotation.}\nAMP labeled dataset comprises sequences\nfrom two major AMP databases:  satPDB \\cite{singh2015satpdb} and DBAASP \\cite{pirtskhalava2016dbaasp}, as well as a dataset used in an earlier AMP classification study named as AMPEP \\cite{bhadra2018ampep}. \nSequences with an antimicrobial function annotation in satPDB and AMPEP or a MIC value against any target species less than 25 $\\mu$g$\/$ml in DBAASP \nwere considered as   AMP labeled instances. The duplicates between these three datasets were removed to generate a non-redundant AMP dataset. And, the ones with mean activity against all target species  $>$ 100 $\\mu$g$\/$ml in DBAASP were considered negative instances (non-AMP). Since experimentally verified non-AMP sequences are rare to find, the non-AMP instances in AMPEP were generated from UniProt sequences after discarding sequences that were annotated as AMP, membrane, toxic, secretory, defensive, antibiotic, anticancer, antiviral, and antifungal and were used in this study as well.\n\n\n\\textbf{Sequences with Toxic\/nonToxic Annotation:}\nSequences with toxicity labels are curated from satPDB and DBAASP databases as well as from the ToxinPred dataset \\cite{gupta2013silico}. Sequences with ``Major Function\" or ``Sub Function\" annotated as toxic in satPDB and sequences with hemolytic\/cytotoxic activities against all reported target species less than 200 $\\mu$g$\/$ml in DBAASP were considered as Toxic instances. The toxic-annotated instances from ToxinPred were added to this set after removing duplicates resulting in a total to 3149 Toxic sequences.\nSequences  with hemolytic\/cytotoxic activities $>$ 250 $\\mu$g$\/$ml were considered as nonToxic. The nonToxic instances reported in ToxinPred (sequences from SwissProt or TrEMBL that are not found in search using keyword associated with toxins, \\textit{i.e.} keyword (NOT KW800 NOT KW20) or keyword (NOT KW800 AND KW33090), were added to the nonToxic set, totaling to 16280 non-AMP sequences.  \n\n\\textbf{Sequences with Broad-Spectrum  Annotation}\nAntimicrobial sequences reported in the satPDB or DBAASP database can have both Gram-positive and Gram-negative strains as target groups. We consider such sequences as broad-spectrum. Otherwise, they are treated as narrow-spectrum. Through our filtering, we found  1302 broad-spectrum and 1238 narrow-spectrum sequences.\n\n\\textbf{Sequences with Structure\/No-Structure Annotation:}\nSecondary Structure assignment was performed for structures from satPDB using the STRIDE algorithm~\\cite{Frishman_Argos_1995}. If more than 60\\% of the amino acids are helix or beta-strand, we label it as structured (positive). Otherwise, they are labeled as negative. Through this filtering, we found 1170 positive sequences and 2136 negative sequences.\n\n\\textbf{Peptide Dataset for Baseline Simulations:} For the control simulations, three datasets were prepared. The first two were taken from the satpdb dataset, filtering sequences with length smaller than 20 amino acids. The high potency dataset contains the 51 sequences with the lowest average MIC, excluding sequences with cysteine residues. All these sequences have an average MIC of less than 10 $\\mu$g \/ ml. The low potency dataset contains the 41 sequences with the highest MIC, excluding cysteine residues. All these have an average MIC over 300 $\\mu$g \/ ml.\n\nTo create a dataset of inactive sequences, we queried UniProt using the following keywords: \\emph{NOT keyword:\"Antimicrobial [KW-0929]\" length:[1 TO 20] NOT keyword:\"Toxin [KW-0800]\" NOT keyword:\"Disulfide bond [KW-1015]\" NOT annotation:(type:ptm) NOT keyword:\"Lipid-binding [KW-0446]\" NOT keyword:\"Membrane [KW-0472]\" NOT keyword:\"Cytolysis [KW-0204]\" NOT keyword:\"Cell wall biogenesis\/degradation [KW-0961]\" NOT keyword:\"Amphibian defense peptide [KW-0878]\" NOT keyword:\"Secreted [KW-0964]\" NOT keyword:\"Defensin [KW-0211]\" NOT keyword:\"Antiviral protein [KW-0930]\" AND reviewed:yes}. From this, we picked 54 random sequences to act as the inactive dataset for simulation.\n\n\\section{Model and Methods}\n\\subsection{Autoencoder Details}\n\\label{sec:model}\n\n\n\n\\subsubsection{Autoencoder Architecture}\n\\label{si:models}\nWe investigate two different types of autoencoding approaches: $\\beta$-VAE\\cite{bowman2015generating} and  WAE\\cite{tolstikhin2017wasserstein} in this study.\nFor each of these AEs the default architecture involves bidirectional-GRU encoder and GRU decoder.\nFor the encoder, we used a bi-directional GRU with hidden state size of 80.\nThe latent capacity was set at $D=100$.\n\nFor VAE, we used KL term annealing $\\beta$ from 0 to 0.03 by default.\nWe also present an unmodified VAE with KL term annealing $\\beta$ from 0 to 1.0.\nFor WAE, we found that the random features approximation of the gaussian kernel with kernel bandwidth $\\sigma=7$  to be performing the best. For comparison sake, we have included variations with $\\sigma$ values of 3 and 15 too.\nThe inclusion of $z$-space noise logvar regularization, $R(logVar)$, helped avoiding collapse to a deterministic encoder.\nAmong different regularization weights used,  $1e-3$ had the most desirable behavior on the metrics (see Section \\ref{si:vaewaeaae}).\n\n\\subsubsection{Autoencoder Training}\n\\label{method:training}\n\nWhen training the AE model, we  sub selected sequences with length $\\leq$ the hyperparameter\n$max\\_seq\\_length$.\nFurthermore, both AMP-labeled and unlabeled data were split into train, held out, and test set.\nThis reduces the available sequences for training; \\textit{e.g} for unlabeled set the number of available training sequences are 93k for $max\\_seq\\_length$=25, whereas the number of AMP-labeled sequences was 5000. \nThe sequences with reported activities were considered as confirmed labeled data, and those with confirmed labels were up-sampled at a 1:20 ratio. \nSuch upsampling of peptides with a specific attribute label will allow mitigation of possible domain shift due to unlabeled peptides coming from a different distribution.\n However, the\n benefit of transfer learning from unlabeled to labeled data likely outweighs the effects of domain shift.\n\n\nTo obtain the optimal hyperparameter setting for autoencoder training, we adopted an automated hyperparameter optimization.  Specifically, we performed a grid search in the hyperparameter space and tracked an $L_2$ distance between the reconstructions of held-out data and training sequences, which was estimated using a weighted combination of BLEU, PPL, ${\\mathbf{z}}$-classifier accuracy, and amino acid composition-based heuristics.\nThe best hyperparameter configuration obtained using this process was the following: learning rate = 0.001, number of iterations = 200000, minibatch size = 32, word dropout = 0.3. A beam search decoder was used with a beam size of 5. \n\n\n\n\n\\subsubsection{ Autoencoder Evaluation}\nEvaluation of generative models is notoriously difficult \\cite{theis2015note}.\nIn the variational auto-encoder family, two competing objective terms are minimized: reconstruction of the input and a form of regularization in the latent space, which form a fundamental trade-off \\cite{alemi2017fixing}.\nSince we want a meaningful and consistent latent space,  models that do not compromise the reconstruction quality to achieve lower constraint loss are preferred.\nWe propose an evaluation protocol using four metrics to judge the quality of both heldout reconstructions and prior samples \\cite{sercu2019interactive}. The metrics are\n\\begin{enumerate}[(i)]\n    \\item The objective terms, evaluated on heldout data: reconstruction log likelihood $-\\log p_\\theta({\\mathbf{x}}|{\\mathbf{z}})$ and $D(q^h_\\phi({\\mathbf{z}}) | p({\\mathbf{z}}))$, where $q^h_\\phi({\\mathbf{z}}) = \\frac{1}{N_{hld}}\\sum_{{\\mathbf{x}}^i \\sim \\text{hld}} q_\\phi({\\mathbf{z}}|{\\mathbf{x}}^i)$ is the average over heldout encodings.\n    \\item Encoder variance $\\log(\\sigma_j^2({\\mathbf{x}}^i))$ averaged over heldout samples, in $L^2$ over components $j$. In order to achieve a meaningful latent space, we needed to regularize the encoder variance to not becoming vanishingly small, i.e., for the encoder to become deterministic \\cite{rubenstein2018latent}. Large negative values indicate that the encoder is collapsed to deterministic.\n    \\item Reconstruction BLEU score on held-out samples.\n    \\item Perplexity (PPL) evaluated by an external language model, for samples from prior $p({\\mathbf{z}})$ and heldout encoding $q_\\phi({\\mathbf{z}}|{\\mathbf{x}})$.\n\\end{enumerate}\nNote that (iii) and (iv) involve sampling the decoder (we use beam search with beam 5), which will therefore also take into account any exposure bias \\cite{ranzato2015sequence,bengio2015scheduled}.\nWe propose to evaluate peptide generation using the perplexity under an independently trained language model (iv), which is a reasonable heuristic \\cite{zhao2017adversarially} if we assume the independent language model captures the distribution $p({\\mathbf{x}})$ well. Perplexity of a language model is the inverse probability of the test set, normalised by the number of words.\nA lower perplexity indicates a better model.\n\n\\textbf{External Language Model}\n\nWe trained an external language model on both labeled and unlabeled sequences to determine the perplexity of the generated sequences. Specifically, we used a character-based LSTM language model (LM) with the help of LSTM and QRNN Language Model Toolkit\\cite{meritylm} trained on both AMP-labeled and unlabeled data. We trained our language model with a total of $92624$ sequences, with a maximum sequence length of $25$. Our best model achieves a test perplexity of $13.26$. \n\nTo further validate the performance of our language model, we tested it on sequences with randomly generated synthetic amino acids from the vocabulary of lengths ranging between $10$ to $25$. As expected, we found it to have a high perplexity of $27.29$. Also, when evaluating it for repeated amino acids (sequence consisting of a single token from vocabulary), of length ranging between 10 to 25, we found the perplexity to be very low ($3.48$). Upon further investigation, we observed that the training data consists of amino acids with repeated sub-sequences, which the language model by nature, fails to penalize heavily. Due to this behavior, we can conclude that the perplexity of a collapsed peptide model will be closer to $3.48$ (as seen in the case of $\\beta$-VAE). We have summarized these observation in Table \\ref{tab:model_results}.\n\n\n\\subsubsection{Autoencoder Variants: Comparison}\n\\label{si:vaewaeaae}\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{l|l|rrrr}\n\\toprule\n\\multicolumn{2}{c}{Architecture} & PPL & BLEU & Recon & Encoder variance  \\\\ \\hline\n\\multirow{4}{*}{Reference}   & Repeated Sequences   & 3.48 &  & &    \\\\   \n                      &  Random  & 27.29  &   &  & \\\\   \n                      &  AMP Labeled  & 5.580  &   &  &  \\\\   \n                      &  Labeled + Unlabeled  & 13.26  &  &  & \\\\\n                      &  Peptide LSTM,  \\cite{muller2018recurrent} (Labeled)  & 22.40  &  &  & \\\\\n                      &  Peptide LSTM,  \\cite{muller2018recurrent} (Unlabeled)  & 20.26  &  &  & \\\\\\hline\n \\multicolumn{2}{l}{$\\beta$-VAE (1.0)}                   &\t3.820   &   4e-03   & 2.768 & -2.5e-4   \\\\ \n\\multicolumn{2}{l}{$\\beta$-VAE (0.03)}                  &\t15.34   &   0.475   & 1.075 & -0.620  \\\\\n\\cmidrule{1-6} \\multicolumn{2}{l}{WAE, $\\sigma=3$, $R(logVar)=1e-3$}           &\t13.25 & 0.853  & 0.257 & -3.078 \\\\\n\\multicolumn{2}{l}{WAE, $\\sigma=15$, $R(logVar)=1e-3$}                         &\t12.98 & 0.909  & 0.224 & -4.180 \\\\\n\\multicolumn{2}{l}{WAE, $\\sigma=7$, $R(logVar) =0$}         &\t12.77 & 0.881  & 0.214 & -13.81 \\\\\n\\multicolumn{2}{l}{WAE, $\\sigma=7$, $R(logVar)=1e-2$}      &\t15.16 & 0.665  & 0.685 & -0.3962 \\\\\n \\multicolumn{2}{l}{WAE, $\\sigma=7$, $R(logVar)=1e-3$}          & 12.87 & 0.892  & 0.216 &  -4.141  \\\\ \n \\cmidrule{1-6} \\multicolumn{2}{l}{WAE (trained on AMP-labeled)} &16.12 & 0.510 &  0.354 & -4.316 \\\\ \n \\bottomrule\n\\end{tabular}\n\\caption{Performance of various autoencoder schemes against different baselines. }\n\\label{tab:model_results}\n\\end{table}\n\nThe evaluated metrics on held-out samples for different autoencoder models trained on either labeled or full dataset are also reported in Table \\ref{tab:model_results}. We observed that the reconstruction of WAE is more accurate compared to $\\beta$-VAE: we achieve a  reconstruction error of $0.2163$ and a BLEU score of $0.892$ on a held-out set using WAE with $\\sigma$ of 7 and $R(logVar)$ of 1e-3 (values are $1.079$ and $0.493$ for $\\beta$-VAE with $\\beta$ set to $0.03$). The advantage of using abundant unlabeled data compared to only the labeled ones for representation learning is evident,  as the language model perplexity (PPL, captures sequence diversity) for the WAE model trained on a full dataset is closer to that of the test perplexity ($13.26$), and the BLEU score is also higher when compared to the WAE model trained only on AMP-labeled sequences. For reference, PPL of random peptide sequences is $> 25$, and for repeated sequences is $3.48$. We have also compared the perplexity of the generated sequences using a LSTM model used in \\cite{ muller2018recurrent} and have reported the perplexity in Table \\ref{tab:model_results}.  Results show that the perplexity of the generated samples using the LSTM model is around 22.4, which is close to the random sequence perplexity. In contrast, WAE (as well as VAE with appropriate KL annealing) achieves much lower perplexity that is close to the perplexity of peptide sequences.\nAttribute-controlled sampling using a simple LSTM model is non-trivial.\n\nThe ${\\mathbf{z}}$-classifier trained on the latent space of the best WAE model achieved a test accuracy of 87.4, 68.9, 77.4, 98.3, 76.3\\% for detecting peptides with AMP\/non-AMP, toxic\/non-toxic, anticancer, antihypertensive, and hormone annotation.\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{Post-Generation Screening}\n\n\\subsubsection{Sequence-Level Attribute Classifiers}\n\\label{si:xclf}\nFor post-generation screening, we used four sets of monolithic classifiers that are trained directly on peptide sequences. Each of these binary sequence-level classifiers was aimed at capturing one of the following four properties of peptide sequences, namely, \n\\begin{itemize}\n    \\item AMP\/Non-AMP : Is the sequence an AMP or not?\n    \\item Toxicity\/Non-Toxic : Is the sequence toxic or not?\n    \\item Broad\/Narrow  : Does the sequence show antibacterial activity on both Gram+ and Gram- strains or not?\n    \\item Structure\/No-Structure : Does the sequence have secondary structure or not?  \n\\end{itemize}{}\n\nFor each attribute, we trained a bidirectional LSTM-based classifier on the labeled dataset.\nWe used a hidden layer size of 100 and a dropout of 0.3. Size of dataset used as well as accuracies are  reported in the Table \\ref{tab:classifier_results}.\n\n\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{l|lcc|cc|c}\n\\toprule\n\\multicolumn{1}{c|}{\\textbf{Attribute}} &  \\multicolumn{3}{c|}{\\textbf{Data-Split}} & \\multicolumn{2}{|c|}{\\textbf{Accuracy (\\%)}} & \\textbf{Screening}  \\\\ \n\\cmidrule{2-5} \\cmidrule{5-6}\n{} & \\textit{Train} & \\textit{Valid} & \\textit{Test} & \\textit{Majority Class} & \\textit{Test} &  \\textbf{Threshold} \\\\ \\hline\n\\big\\{AMP , Non-AMP\\big\\} & 6489 & 811 & 812 &  68.9  & 88.0 & 7.944  \\\\ \n\\big\\{Toxic , Non-Toxic\\big\\} & 8153 & 1019 & 1020 &  82.73  & 93.7 & -1.573  \\\\\n\\big\\{Broad, Narrow\\big\\} & 2031 & 254 & 255 & 51.37 & 76.0 & -7.323  \\\\\n\\big\\{Structure , No-Structure\\big\\}  & 2644 & 331 & 331 & 64.65  & 95.1 & -5.382  \\\\\n \\bottomrule\n\\end{tabular}\n\\caption{Performance of classifiers based on different attributes.}\n\\label{tab:classifier_results}\n\\end{table}\n\n\n\\begin{table}[ht]\n    \\centering\n   \n    \\begin{tabular}{p{4cm}|p{2cm}|p{3.5cm}|p{4cm}}\n        \\toprule\n        \\textbf{Classifiers} & Reported Accuracy &  YI12 & FK13  \\\\ \n        \\hline   \n        AMP Classifier (ours) &  88.00 & AMP & AMP \\\\\n        AMP Scanner v2\\cite{veltri2018deep} & 91.00 & Non-AMP & AMP \\\\\n        iAMP Pred  \\cite{meher2017predicting} &  66.80 & AMP & AMP \\\\\n        DBAASP-SP \\cite{vishnepolsky2018predictive} & 79.00 & AMP & Non-AMP \\\\\n        iAMP-2L \\cite{xiao2013iamp} &  92.23 & Non-AMP & AMP \\\\\n        CAMP-RF \\cite{thomas2010camp} & 87.57 & AMP & AMP \\\\\n        Witten E.Coli \\cite{witten2019deep} & 94.30 & AMP & Borderline \\footnotemark  \\\\ \n        Witten S.aureus \\cite{witten2019deep} & 94.30 & AMP & AMP \\\\ \n         \\bottomrule\n    \\end{tabular}\n    \\caption{Reported accuracy and prediction for YI12 and FK13 using different AMP classifiers: AMP Scanner v2 \\cite{veltri2018deep}, iAMP-2L \\cite{meher2017predicting}, DBAASP Prediction \\cite{vishnepolsky2018predictive}, iAMPpred \\cite{xiao2013iamp}, CAMP-RF \\cite{thomas2010camp}, Witten (CNN-70\\%) \\cite{witten2019deep}, and our LSTM-based sequence-based classifier. \n}\n    \\label{tab:AMP-compare}\n\\end{table}{}\n\n\\footnotetext{Witten model predicts logMIC activity values. We followed their criteria for classification: if logMIC is within -1 to 3.5, then the peptide is  active (AMP), if $>3.9$ then inactive (Non-AMP), and between 3.5 to 3.9 is considered borderline. LogMIC values for YI12 (\\textit{E. coli})=1.648,  YI12 (\\textit{S. aureus}=3.607, FK13 (\\textit{E. coli})=1.389,  FK13 (\\textit{S. aureus})=1.634.}\n\nBased on the distribution of the scores (classification probabilities\/logits), we determined the threshold by considering the $50^{th}$ percentile (median) of the scores ( reported in the last column of Table \\ref{tab:classifier_results}). Similarly, we selected a   PPL threshold of $16.04$ that is the $25^{th}$ percentile of the PPL distribution of samples generated from the prior distribution of the best WAE model and also closer to the perplexity of our trained language model on test data.    \n\n\n\n\n\\subsubsection{CGMD Simulations - Contact Variance as a Metric for Classifying Membrane Binding}\n\\label{method:sim}\nGiven a peptide sequence as an input, PeptideBuilder~\\cite{Tien2013} is used to prepare a PDB file of the all-atom representation of the peptide. This is prepared either as an alpha helix (with dihedral angles $\\phi=-57$,  $\\psi=-47$) or as a random coil, with $\\phi$ and $\\psi$ dihedral angles taking random values between $-50^\\circ$ and $50^\\circ$. \n\nThis initial structure is then passed as in input to \\texttt{martinize.py}~\\cite{deJong2012}, which coarse-grains the system. The resulting files are passed into \\texttt{insane.py}~\\cite{Wassenaar2015} to create the peptide membrane system. The solvent is a 90:10 ratio of water to antifreeze particles, with the membrane being a 3:1 mixture of POPC to POPG. The system is 15 nm x 15 nm x 30 nm, with the membrane perpendicular to the longest direction. Ions are added to neutralize the system.\n\nThis is a minimal model of a bacterial membrane that serves as a high-throughput filter. While this model does not replicate the complex physics of the peptide-membrane interactions in exact detail (e.g. membrane composition difference between Gram-positive \\textit{vs.} Gram-negative), it allows us to prioritise the peptides that show the strongest interaction with a membrane for experimental characterisation.\n\nFor the CGMD simulations, we used the Martini forcefield~\\cite{Marrink2007},\nas Martini is optimized for predicting the interactions between proteins and membranes while being computationally efficient,  it is well suited for the task of a quick but physically-inspired filtering peptide sequences.\n\nAfter building, the system is minimized for 50,000 steps using Gromacs 2019.1~\\cite{Berendsen1995, Abraham2015} and the 2.0 version of the Martini forcefield \\cite{Marrink2007}. After minimization, the production run is carried for 1 $\\mu$s at a 20 fs timestep. Temperature is kept constant at 310 K using Stochastic Velocity Rescaling \\cite{Bussi2007} applied independently to the protein, lipid, and the solvent groups. The pressure is kept constant at 1 atmosphere using a Parrinello-Rahman barostat~\\cite{Parrinello1981, Nos1983} applied independently to the same groups as the thermostat. \n\nAfter 1 $\\mu$s of sampling, we estimated the number of peptide-membrane contacts using TCL scripting and in-house Python scripts.\nThe number of contacts between positive residues and the lipid membranes is defined as the number of atoms belonging to a lipid at a distance less than 7.5 \\AA{} from a positive residue.\n\nFor the control simulations, three datasets consisting of reported high-potency AMP, low-potency AMP, and non-AMP sequences were used that are discussed in \\ref{si:datadetails}. We performed a set of 130 control simulations. We found that the variance of the number of contacts (cutoff 7.5 \\AA{}) between positive residues and Martini beads of the membrane lipids is predictive of antimicrobial activity. Specifically, the contact variance distinguishes between high potency and non-antimicrobial sequences with a sensitivity of 88\\% and specificity of 63\\%. \nTo screen, we used a cutoff value of 2 beads for the contact variance. We carried out a set of simulations for the 163 amp-positive and 179 amp-negative generated sequences. \nWe further restricted to sequences that bind in less than $500$ ns during the $1\\,\\mu$s long simulation, so that the contact variance is calculated over at least half of the total simulation time. Only sequences that formed at least $5$ contacts (averaged over the duration of the simulation) were considered. \n\n\n\n\\subsection{All-Atom Simulations}\n\\label{method:aa}\n\nWe used the CHARMM36m \\cite{Huang2016} forcefield to simulate the binding of four copies of YL12 and FK13 to a model membrane. Phi and Psi angles in the initial peptide structure were set to what was predicted using a recent deep learning model \\cite{qin2020artificial}.  A 3:1 DLPC:DLPG bilayer, with shorter tails, was used to speed up the simulation, alongside a smaller water box (98 \\AA{}) than the Martini simulations, to investigate the short-term effect of peptide-membrane interactions.  A 160 ns long trajectory was run for the FK13 system. The length of the YI12 simulation was 200 ns. The number of peptide-membrane and peptide-peptide contacts (using a threshold of 7.5 \\AA{} and ignoring hydrogen atoms) were found to be converged in less time than the maximum simulation length for both systems. \n\nThe bilayer is prepared using CHARMM-GUI~\\cite{Jo2009}, and the peptide sequence is prepared using PeptideBuilder~\\cite{Tien2013}. Solvation and assembly is performed using VMD 1.9.3 \\cite{HUMP96}. The system is simulated using NAMD 2.13 \\cite{Phillips2005}. Temperature is kept constant using a Langevin thermostat and a Nos\\'e-Hoover Langevin piston barostat\nThe Particle-Mesh Ewald method was used for long-range electrostatics. All simulations used a time step of 2 fs.\n\n\\subsection{Peptide Sequence Analysis}\n\\label{method:analysis}\nPhysicochemical properties like aromaticity, Eisenberg hydrophobicity, charge, charge density, aliphatic index, hydrophobic moment, hydrophobic ratio, isoelectric point, and instability index were estimated using the GlobalAnalysis method in modLAMP \\cite{muller2017modlamp}. Protparam tool from Expasy (https:\/\/web.expasy.org\/protparam) was used to estimate the grand average of hydropathicity (GRAVY) score. \n\nPairwise sequence similarity was estimated using a global alignment method, the PAM30 matrix\\cite{yu2003compositional}, a gap open penalty of -9, and a gap extension penalty of -1 using Pairwise2 function of Biopython package \\cite{cock2009biopython}. Higher positive values indicate better similarity. To check the correspondence between sequence similarity and Euclidean distance in ${\\mathbf{z}}$-space, a random set of sequence encodings were first selected, and then sequence similarity and  ${\\mathbf{z}}$-distance with their close latent space neighbors were estimated.\nSequence similarity with respect to a sequence database was estimated using ``blastp-short'' command from NCBI BLAST sequence similarity search tool \\cite{madden2013blast, altschul1990basic}  was used to query generated short sequences by using a word size of 2,  the PAM30 matrix \\cite{yu2003compositional}, a gap open penalty of -9, a gap extension penalty of -1, threshold of 16, $comp\\_based\\_stats$ set to 0 and window size of 15. Alignment score (Bit score), Expect value (E-value), percentage of alignment coverage, percentage of identity, percentage of positive matches or similarity,  percentage of alignment gap were used for analyzing sequence novelty.  \nThe E-value is a measure of the probability of the high similarity score occurring by chance when searching a database of a particular size.  E-values decrease exponentially as the score of the match increases.\nFor the search against patented sequences, we used the PATSEQ database \\cite{Patseq}.\n\n\\subsection{Wet Lab Experiments}\n\\label{sec:exptdetails}\n\n\n\\subsubsection{MIC Measurement}\nAll of the peptides were amidated at their C-terminus to remove the negative charge of the C-terminal carboxyl group. \nAntimicrobial activity of the best AMP hits was evaluated against a broad spectrum of bacteria for minimum inhibitory concentration (MIC), which include Gram-positive \\textit{Staphylococcus aureus} (ATCC 29737), Gram-negative \\textit{Escherichia coli} (ATCC 25922), \\textit{Pseudomonas aeruginosa} (ATCC 9027) and multi-drug resistant \\textit{K. pneumoniae} (ATCC 700603), which produce beta-lactamase SHV 18. The broth microdilution method was used to measure MIC values of the AMPs, and the detailed protocol was reported previously \\cite{chin2018macromolecular,ng2013synergistic}.\n\\subsubsection{Hemolytic Activity}\nThe selectivity of AMPs towards bacteria over mammalian cells was studied using rat red blood cells (rRBCs), which were obtained from the Animal Handling Unit of Biomedical Research Center, Singapore. Negative control: Untreated rRBC suspension in phosphate-buffered saline (PBS); Positive control: rRBC suspension treated with 0.1\\% Triton X. Percentage of hemolysis of rRBCs was obtained using the following formula:\n\\begin{equation}\n    Hemolysis (\\%)=\\frac{O.D._{576 nm} \\textit{ of treated samples } - O.D._{576 nm} \\textit{ of negative control}}{O.D._{576 nm} \\textit{ of positive samples} - O.D._{576 nm} \\textit{ of negative control}}\n\\end{equation}\n\n\\subsubsection{Acute in Vivo Toxicity Analysis}\nThe animal study protocols were approved by the Institutional Animal Care and Use Committee of Biological Resource Center, Agency for Science, technology, and Research (A*STAR), Singapore. LD50 values of the AMPs, the dose required to kill 50\\% mice, were determined using a previously reported protocol \\cite{liu2017highly}. Specifically, female Balb\/c mice (8 weeks old, 18-22 g) were employed. AMPs were dissolved in saline and administered to mice by intraperitoneal (i.p.) injection at various doses. Mortality was monitored for 14 days post-AMP administration, and LD50 was estimated using the maximum likelihood method. \n\\subsubsection{CD Spectroscopy} The peptides were dissolved at 0.5 mg\/mL in either deionized water or deionized water containing 25 mM SDS surfactant. It forms anioic micelles in aqueous solution, which mimic the bacterial membrane. The CD spectra were measured using a CD spectropolarimeter from Jasco Corp. J-810 at room temperature and a quartz cuvette with 1 mm path length. The spectra were acquired by scanning from 190 to 260 nm at 10nm\/min after subtraction with the spectrum of the solvent.\n\n\\subsection{Mechanism study using Confocal microscopy}\nThe mechanism study was performed on an FV3000 Olympus confocal laser scanning microscope. E. coli was grown according to the MIC measurement. Briefly, 500 $\\mu$l of $10^8$ CFU\/ml of bacteria was prepared and placed in a centrifuged tube and 500 $\\mu$l of peptide and polymyxin B at 4xMIC concentration was added. The mixture was incubated at 37 $^{\\circ}$C and shaken at 100 rpm for 2 h. The sample was centrifuged at 4000 rpm for 10 min and supernatant was discarded before fresh 1 ml PBS was added. After washing for 3 times, the pallet was re-suspended with 500 $\\mu$l of glutaraldehyde (2.5\\% v\/v) for 30 min with occasional mixing. The sample was again washed for 3 times and re-suspended in 100 $\\mu$l of PBS and applied onto a poly-L-lysine coated slide. The bacteria was left to attach for an hour before the unattached bacteria was washed away with PBS. The slides was left to air dry before imaging using the FV3000 confocal microscope.\n\n\\subsection{Resistance Study}\nDrug resistance in the E. coli can be induced by treating repeatedly with peptides and imipenem at the sub MIC concentration (1\/2 MIC). MIC experiment was performed as stated earlier using the micro dilution method. The next generation of E. coli can be initiated by growing the bacteria suspension from the sub MIC concentration and performing another MIC assay.  After 18h inoculation, the bacteria grown from sub MIC concentration of the tested peptides\/antibiotics was assayed for MIC. This will continue until we have reached 25 generations. The change in MIC was documented by taking the normalized value of MIC at generation X against the MIC at generation 1.\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n        \\toprule\n        \\textbf{E-value} & $<=0.001$ & $<=0.01$ & $<=0.1$ & $<=1$ & $<=10$ & $>10$ \\\\ \\hline   \n         Labeled & 9.36 & 3.42 & 9.70 & 29.59 & 29.88 & 18.05  \\\\\n         Unlabeled & 5.16 & 4.05 & 9.07 & 30.97 & 36.65 & 14.08  \\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Percentage of CLaSS-generated AMP sequences in different categories of Expect value (E-value). E-value for the match with the highest score was considered, as obtained by performing BLAST similarity search against AMP-labeled training sequences (top row) and unlabeled training sequences (bottom row). \n}\n    \\label{tab:Evalue}\n\\end{table}{}\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\n\\subcaptionbox{Fraction of unique $k$-mer present in different datasets. $k$ = 3-6. The mean and standard errors were estimated on three different sets, each consisting of 3000 randomly chosen samples.}{\\begin{tabular}{c|c|c|c}\n\\toprule\n$k$ & Generated AMP & AMP-labeled Training &  Unlabeled Training \\\\ \\hline\n3 & $0.299 \\pm 0.006$ & $0.223 \\pm 0.006$ &  $0.110 \\pm 0.005$ \\\\\n4 & $0.771 \\pm 0.004$ & $0.607 \\pm 0.002$ &  $0.777 \\pm 0.001$ \\\\\n5 & $0.947 \\pm 0.002$ & $0.706 \\pm 0.002$ &  $0.965 \\pm 0.000$ \\\\\n6 & $0.978 \\pm 0.001$ & $0.742 \\pm 0.002$ &  $0.983 \\pm 0.001$ \\\\\n\\bottomrule\n\\end{tabular}}%\n\\centering\n\\hfill\n\\subcaptionbox{Top 3 $k$-mers ($k$ = 3 and 4) and their corresponding frequency in generated AMPs,  training AMPs, and unlabeled training sequences. The estimated standard deviation values were close to zero.}{\\begin{tabular}{l|ll|ll|ll}\n\\toprule\n\\multirow{2}{*}{Kmers} & \\multicolumn{2}{|c|}{Generated AMP} & \\multicolumn{2}{|c|}{AMP-labeled Training} & \\multicolumn{2}{|c}{Unlabeled Training} \\\\\n\\cmidrule(lr){2-3} \\cmidrule(lr){4-5} \\cmidrule(lr){6-7}\n                                    & top-3         & Frequency        & top-3         & Frequency         & top-3         &  Frequency       \\\\ \\hline\n\\multirow{3}{*}{3}                  & KKK       & $0.011$           &  KKK          & $0.006$           & LLL          &  $0.002$          \\\\\n                                    & LKK       & $0.007$           &  KKL          & $0.004$           & LAA          &  $0.001$         \\\\\n                                    & KLK       & $0.007$           &  GLL          & $0.004$           & RRR          &  $0.001$          \\\\\n\\cmidrule{1-7}                   \n\\multirow{3}{*}{4}                  & KKKK       & $0.005$           &  KKKK          & $0.003$           & PLDL          &  $0.001$          \\\\\n                                    & KLKK       & $0.004$           &  KKLL          & $0.002$           & LDLA          &  $0.001$          \\\\\n                                    & KKLK       & $0.003$           &  KLLK          & $0.002$           & NFPL          &  $0.001$          \\\\\n\\bottomrule\n\\end{tabular}}%\n\n\\hfill\n\\subcaptionbox{Physicochemical properties, such as charge, charge density, aliphatic index, aromaticity, hydrophobicity, hydrophobic moment, hydrophobic ratio, instability index, were estimated on unlabeled training, AMP-labeled training, and generated AMP sequences using CLaSS. Mean, and standard deviation were estimated on three different sets, each consisting of 3000 randomly chosen samples. }{\\begin{tabular}{l|rrr}\n\\toprule\n{} & Generated AMP & AMP-labeled Training & Unlabeled Training \\\\ \\hline\nCharge & $2.695 \\pm 0.039$ & $3.074 \\pm 0.238$ &  $1.172 \\pm 0.218$\\\\\nCharge Density & $0.002 \\pm 0.000$ &  $0.002 \\pm 0.000$ &  $0.001 \\pm 0.000$\\\\\nAliphatic Index & $107.814 \\pm 0.649$ & $101.232 \\pm 1.550$ &  $82.088 \\pm 8.440$ \\\\\nAromaticity & $0.095 \\pm 0.002$ & $0.102 \\pm 0.002$ &  $0.082  \\pm 0.003$ \\\\\nHydrophobicity & $0.048 \\pm 0.007$ & $0.068 \\pm 0.002$ & $0.052 \\pm 0.032$\\\\\nHydrophobic Moment & $0.331 \\pm 0.000$ & $0.335 \\pm 0.019$ & $0.222 \\pm 0.003$ \\\\\nHydrophobic Ratio & $0.446 \\pm 0.001$ & $0.431 \\pm 0.010$ &  $0.425 \\pm 0.013$ \\\\\n \\bottomrule\n\\end{tabular}\n}%\n\\caption{Comparison of CLaSS-generated AMPs with AMP-labeled and unlabeled peptides.}\n\\label{tab:combined-comp}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\n\\multicolumn{1}{|c|}{\\textbf{Sequence}} & \\textbf{Positive Residue} & \\textbf{Binding time (ns)} & \\textbf{Mean}  & \\textbf{Variance } \\\\ \\hline\nYLRLIRYMAKMI & 3 & 210 & 6.45 & 1.27 \\\\ \\hline\nFPLTWLKWWKWKK & 4 & 90 & 5.90 & 1.39 \\\\ \\hline\n\nHILRMRIRQMMT & 3 & 17 & 7.84 & 1.44 \\\\ \\hline\nILLHAILGVRKKL & 3 & 105 & 7.16 & 1.19 \\\\ \\hline\nYRAAMLRRQYMMT & 3 & 19 & 8.79 & 1.25 \\\\ \\hline\nHIRLMRIRQMMT & 3 & 493 & 8.38 & 1.50 \\\\ \\hline\nHIRAMRIRAQMMT & 3 & 39 & 7.20 & 1.39 \\\\ \\hline\nKTLAQLSAGVKRWH & 3 & 177 & 7.62 & 1.46 \\\\ \\hline\n\nHILRMRIRQGMMT & 3 & 62 & 8.37 & 1.53 \\\\ \\hline\nHRAIMLRIRQMMT & 3 & 297 & 7.46 & 1.35 \\\\ \\hline\nEYLIEVRESAKMTQ & 2 & 150 & 6.65 & 1.79 \\\\ \\hline\nGLITMLKVGLAKVQ & 2 & 341 & 8.34 & 1.58 \\\\ \\hline\nYQLLRIMRINIA & 2 & 239 & 6.29 & 1.71 \\\\ \\hline\nVRWIEYWREKWRT & 4 & 125 & 6.41 & 1.28 \\\\ \\hline\nLIQVAPLGRLLKRR & 4 & 37 & 6.52 & 1.24 \\\\ \\hline\nYQLRLIMKYAI & 2 & 192 & 7.75 & 1.86 \\\\ \\hline\nHRALMRIRQCMT & 3 & 80 & 9.15 & 1.27 \\\\ \\hline\nGWLPTEKWRKLC & 3 & 227 & 6.11 & 1.63 \\\\ \\hline\nYQLRLMRIMSRI & 3 & 349 & 8.28 & 1.80 \\\\ \\hline\n\nLRPAFKVSK & 3 & 151 & 7.73 & 1.85 \\\\\n\\hline\n\\end{tabular}\n\\caption{Physics-derived features such as mean and variance of the number of contacts between positive amino acids and membrane beads (that are found to be associated with antimicrobial function in this study), as extracted from CGMD simulations of peptide membrane interactions for the top 20\nAI-designed AMP sequences.}\n\\label{tab:20P-sim}\n\\end{table}\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|l|c|c|r|c|}\n\\toprule\n\\multicolumn{1}{|c|}{\\textbf{Sequence}} & \\textbf{Length} & \\textbf{Charge} & \\textbf{H} & \\textbf{$\\mu$H} \\\\ \\hline\nYLRLIRYMAKMI & 12 & 3.99 & 0.08 & 0.79  \\\\ \\hline\nFPLTWLKWWKWKK & 13 & 5.00 & 0.05 & 0.20 \\\\ \\hline\nHILRMRIRQMMT & 12 & 4.10 & -0.25 & 0.36 \\\\ \\hline\nILLHAILGVRKKL & 13 & 4.09 & 0.27 & 0.33 \\\\ \\hline\nYRAAMLRRQYMMT & 13 & 3.99 & -0.28 & 0.06 \\\\ \\hline\nHIRLMRIRQMMT & 12 & 4.10 & -0.25 & 0.16 \\\\ \\hline\nHIRAMRIRAQMMT & 13 & 4.10 & -0.22 & 0.24  \\\\ \\hline\nKTLAQLSAGVKRWH & 14 & 4.09 & -0.08 & 0.49  \\\\ \\hline\n\nHILRMRIRQGMMT & 13 & 4.10 & -0.19 & 0.27  \\\\ \\hline\nHRAIMLRIRQMMT & 13 & 4.10 & -0.18 & 0.41  \\\\ \\hline\nEYLIEVRESAKMTQ & 14 & 0.00 & -0.16 & 0.26 \\\\ \\hline\nGLITMLKVGLAKVQ & 14 & 3.00 & 0.37 & 0.28 \\\\ \\hline\nYQLLRIMRINIA & 12 & 3.00 & 0.11 & 0.38 \\\\ \\hline\nVRWIEYWREKWRT & 13 & 3.00 & -0.41 & 0.55 \\\\ \\hline\nLIQVAPLGRLLKRR & 14 & 5.00 & -0.12 & 0.38 \\\\ \\hline\nYQLRLIMKYAI & 11 & 2.99 & 0.18 & 0.40 \\\\ \\hline\nHRALMRIRQCMT & 12 & 4.03 & -0.34 & 0.56 \\\\ \\hline\nGWLPTEKWRKLC & 12 & 2.93 & -0.13 & 0.33 \\\\ \\hline\nYQLRLMRIMSRI & 12 & 4.00 & -0.17 & 0.64 \\\\ \\hline\n\nLRPAFKVSK & 9 & 4.00 & -0.17 & 0.70 \\\\\n\\bottomrule\n\\end{tabular}\n\n\\caption{Physico-chemical properties for the top 20 AI-designed AMP sequences.}\n\\label{tab:sicgmd}\n\\end{table}\n\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|l|c|c|}\n\\hline\n\n\\multicolumn{1}{|c|}{\\textbf{Sequence}} & \\textbf{S. aureus ($\\mu$g\/mL)} & \\textbf{E.Coli ($\\mu$g\/mL)} \\\\ \\hline\nYLRLIRYMAKMI-CONH2  & 7.8 & 31.25 \\\\ \\hline\nFPLTWLKWWKWKK-CONH2  & 15.6 & 31.25 \\\\ \\hline\nHILRMRIRQMMT-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nILLHAILGVRKKL-CONH2  & 250 & 250 \\\\ \\hline\nYRAAMLRRQYMMT-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nHIRLMRIRQMMT-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nHIRAMRIRAQMMT-CONH2  & $>$1000 & 1000 \\\\ \\hline\nKTLAQLSAGVKRWH-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\n\nHILRMRIRQGMMT-CONH2  & $>$1000 & $>$1000\\\\ \\hline\nHRAIMLRIRQMMT-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nEYLIEVRESAKMTQ-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nGLITMLKVGLAKVQ-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nYQLLRIMRINIA-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nVRWIEYWREKWRT-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\n\nYQLRLIMKYAI-CONH2  & 125 & 125 \\\\ \\hline\nHRALMRIRQCMT-CONH2  & 1000 & 1000 \\\\ \\hline\nGWLPTEKWRKLC-CONH2  & 1000 & $>$1000 \\\\ \\hline\nYQLRLMRIMSRI-CONH2  & 250 & 500\\\\ \\hline\nFFPLPAISTELKRL-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nLIQVAPLGRLLKRR-CONH2  & $>$1000 & 1000 \\\\ \\hline\n\nLRPAFKVSK-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\n\\end{tabular}\n\\caption{Broad-spectrum MIC values of top 20 AI-designed AMP Sequences}\n\\label{tab:20P_MIC}\n\\end{table}\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|l|c|c|}\n\\hline\n\\multicolumn{1}{|c|}{\\textbf{Sequence}}& \\textbf{S. aureus ($\\mu$g\/mL)} & \\textbf{E.Coli ($\\mu$g\/mL)} \\\\ \\hline\nAMLELARIIGRR-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nIPRPGPFVDPRSR-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nVAKVFRAPKVPICP-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nFPSFTFRLRKWKRG-CONH2 & 62.5 & 62.5 \\\\ \\hline\nRPPFGPPFRR-CONH2  & $>$1000& $>$1000 \\\\ \\hline\nWEEMDSLRKWRIWS-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nRRQAQEVRGPRH-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nKKKKPLTPDFVFF-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nTRGPPPTFRAFR-CONH2  & $>$1000 & $>$1000 \\\\ \\hline\nLALHLEALIAGRR-CONH2  & 250 & $>$1000\\\\ \\hline\n\\end{tabular}\n\\caption{Broad-spectrum MIC values of 11 AI-designed Non-AMP Sequences}\n\\label{tab:11N-MIC}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{figures\/figure-sim_2.pdf}\n\\caption{(A) Snapshot from a coarse-grained molecular dynamics simulation of an AMP (in orange)  binding with a lipid bilayer (gray). \n(B) Confusion matrix of the simulation-based classifier that uses peptide-membrane contact variance as feature for detecting AMP sequences.\n}\n\\label{fig:cgmd}\n\\end{figure}\n\n\\renewcommand{\\thesubfigure}{\\Alph{subfigure}}\n\\begin{figure}\n    \\captionsetup{singlelinecheck = false, justification=justified}\n    \\begin{subfigure}{0.33\\linewidth}\n        \\centering\n            \\caption{}\n            \\includegraphics[scale=0.26]{figures\/AIdesigned_twoAMP_toxicity.png}\n            \\label{fig:toxicity}\n    \\end{subfigure}\n    \\begin{subfigure}{0.32\\linewidth}\n        \\centering\n            \\caption{}\n            \\includegraphics[scale=0.27]{figures\/YI12_CD.png}\n            \\label{fig:YI-CD}\n    \\end{subfigure}\n    \\begin{subfigure}{0.32\\linewidth}\n        \\centering\n            \\caption{}\n            \\includegraphics[scale=0.27]{figures\/FK13-CD.png}\n            \\label{fig:FK-CD}\n        \\end{subfigure}\n        \n       \n        \\center\n       \n\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\toprule\n\\multicolumn{1}{|c|}{\\textbf{Sequence}} & \\textbf{Score} & \\textbf{E-Value}  & \\textbf{\\% Coverage} & \\textbf{\\% Identity} & \\textbf{\\% Positive} & \\textbf{\\% Gap}\\\\ \\hline\n\nYLRLIRYMAKMI   & 21.88 & 1.20 & 66.67 & 75.00 & 75.00 & 75.00 \\\\ \\hline\nFPLTWLKWWKWKK  & 33.30 & 4e-05 & 84.61 & 73.00 & 73.00 & 9.00\\\\ \n\\bottomrule\n\\end{tabular}\n\n\\caption{Percentage of hemolysis of rat red blood cells as a function of peptide concentration. (B) and (C) show CD Spectra of YI12 and FK13 peptide, respectively, at 0.5 mg\/ml concentration in DI water and presence of 20 mM SDS buffer. Both YI12 and FK13 showed a random coil-like structure in the absence of SDS. When SDS was present, both sequences form $\\alpha$-helical structure (evident from the 208 nm and 222 nm peaks). (D) BLAST search results (alignment score, E-value, percentage of alignment coverage, percentage of identity, percentage of positive matches or similarity,  percentage of alignment gap) against full training data for YI12 and FK13.}\n\\label{fig:cd_n_tox}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/Confocal.png}\n\\caption{Confocal images of \\textit{E. coli} (A-B) without any treatment, (C-D) treated with 2xMIC FK13 and (E-F) treated with 2xMIC polymyxin B for 2 hours. Red colour denotes the propidium iodide. Images A, C and E are the dark field image and B, D and F are the bright field image. All scale bars are 10 $\\mu$m.}. \n\\label{fig:confocal}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/figure_6_7b.edited.png}\n\\caption{BLAST search results of YI12 and FK13.}. \n\\label{fig:blast_result}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOne of the many suprising applications of shared entanglement \\cite{mar} is  superdense coding introduced by Bennett and Wiesner \\cite{wei}.\nIn the simplest example of this protocol, two people (Alice and Bob) share a pair of entangled qubits\n(spin half particles or any other two state systems) in a Bell state.\nAlice can then perform any of the four unitary operations given by\nthe identity ${\\bf I}$ or the Pauli matrices ${\\bf {\\sigma_1}} ({\\bf {\\sigma_x}}), {\\bf {\\sigma_2}} ({\\bf {\\sigma_y}})$\nand ${\\bf {\\sigma_3}} ({\\bf {\\sigma_z}})$ on her qubit. Each of these four unitary \noperations map the initial state of the two qubits to a different member\nof the Bell state basis. Clearly, these four orthogonal and therefore fully distinguishable states can be used to encode two bits of information. After encoding her qubit, Alice sends it off to Bob who can extract these two bits of information\nby performing a joint measurement on this qubit and his original qubit. This\napparent doubling of the information conveying capacity of Alice's\nqubit because of prior entanglement with Bob's qubit is referred to as\nsuperdense coding (which we shall henceforth refer to as just\ndense coding). Dense coding has been implemented\nexperimentally with polarization entangled photons \\cite{zei}. Some generalizations of the scheme to pairs of entangled N-level systems in\nnon-maximally entangled states (as opposed\nto qubits in Bell states) \\cite{Bar,haus} and to distributed multiparticle\nentanglement \\cite{bs} have also been studied.\n\n     In the simplest example, the power of the method \nstems from the accessibility of all the four \npossible Bell states through local operations done by Alice alone on her\nqubit \\cite{wei,zei}. Obviously this accessibility is related\nto the fact that the qubits shared by Alice and Bob are\nin an {\\em entangled} state. So an important question to ask is: {\\em In what way does \nthe capacity of conveying information through dense coding depend upon the degree of entanglement\nof the initial shared pair of qubits}? Barenco and Ekert \\cite{Bar} \nand Hausladen {\\em et al.} \\cite{haus} have investigated this\nquestion when the initial shared entangled state is a pure state. Their\nanalysis shows that the amount of information conveyed by the dense coding procedure decreases\nmonotonically from its maximum value (two bits per qubit) with the decrease of the magnitude of the shared entanglement. It becomes one bit per qubit when the\nentanglement  becomes zero. \n\n\n    Recently, there has been a lot of work in quantifying the entanglement\nof systems in mixed states \\cite{ef,vlat1,vlat2,vlat3}. These measures\nof entanglement have their physical interpretation in the process of\nentanglement purification \\cite{pf,Deu,ben2}. A natural question to ask is: {\\em in\nwhat way does the capacity to do dense coding depend on the degree of\nentanglement as specified by these measures?} Answering such a question,\nin effect, would mean {\\em linking up the apparently disconnected concepts of\npurification and channel capacity}. In this paper we take\na step in this direction by obtaining\nbounds on the number of bits of information conveyed per qubit (let us call this {\\bf C} after capacity) during dense coding in terms of the various measures of entanglement. It\nshould be noted that {\\bf C} is a classical capacity as it quantifies\nnumber of bits of information, but the carriers of the information are \nquantum (qubits).\n\n    We also invert our results so that the value of {\\bf C}\ngives us information about the range within which the initial shared\nentanglement lies. In other words, bounds on entanglement measures\n(and hence on purification procedures) can be expressed in terms of\nthe classical capacity {\\bf C}. This might be helpful, because, as\nwe shall show, the classical capacity for dense quantum coding is\nreadily calculable for certain special classes of dense coding protocols.\n\n\n\n\\section{The Holevo function as a measure of C}\n  \n    Suppose, Alice has a set of mixed\nquantum states {$W_i$} at her disposal to convey some classical information (sequences of ones\nand zeros) to Bob. Each state $W_i$ can be regarded as a separate letter.\nAlso suppose that Alice sends the state $W_i$ to Bob with an {\\em a priori}\nprobability $p_i$. The ensemble that Alice uses to communicate with Bob is \ntherefore given by\n\\begin{equation}\n\\label{avg}\nW = \\sum p_i W_i.\n\\end{equation} \nIn the above case, the average number of bits of information that \nAlice can convey to Bob \nper transmission of a letter state is bounded from above by the Holevo function \\cite{khol1}\n\\begin{equation}\n\\label{holv1}\nH = S(W)- \\sum p_i S(W_i),\n\\end{equation}\nwhere $S(\\rho)=-\\mbox{Tr}~\\rho \\log \\rho $ denotes the von Neumann entropy of\nthe state $\\rho$ (Here, and throughout the rest of the paper, $\\log$ stands for $\\log_2$). Using a different notation, the Holevo function can\nbe rewritten as\n\\begin{equation}\n\\label{holv}\nH = \\sum p_i S(W_i||W),  \n\\end{equation}\nwhere\n$S(\\sigma || \\rho )$ stands for $\\mbox{Tr} \\{\\sigma \\log \\sigma - \\sigma \\log \\rho \\}$ \nand is known as the quantum relative entropy between the states $\\sigma$ and\n$\\rho$ \\cite{don}.\nIt was recently shown that this bound can be achieved in the limit of\nan infinite ensemble by appropriate\nBlock coding (grouping together and pruning long strings of letter states to\nrepresent messages) at Alice's end and appropriate measurements at Bob's end\n\\cite{Holevo2}. \n  \n Now consider the case of dense coding. Alice and Bob initially\nshare an entangled pair of qubits in some state $W_0$, which may be mixed. Alice then performs local unitary operations on her qubit to put this shared\npair of qubits in either of the states $W_0, W_1,  W_2$ or $W_3$. In general,\nAlice may use a completely arbitrary set of unitary operations to generate\nthese states:\n\\begin{equation}\n\\label{cg}\nW_i={\\bf U_i}\\otimes {\\bf I}~W_0~{\\bf U_i}\\otimes {\\bf I}.\n\\end{equation}\nIn the above equation, ${\\bf U_i}$  acts on Alice's qubit and\n${\\bf I}$ acts on Bob's qubit.\nBy sending  her encoded qubit to Bob, Alice is essentially communicating with Bob using the states $W_0, W_1,  W_2$ and $W_3$ as separate letters. The number of\nbits she can communicate to Bob using this procedure is thus bounded by\nthe holevo function $H$ given in Eq.(\\ref{holv}). Moreover, if\nsome block coding is done on a large enough collection of qubits in \naddition to the dense coding, then the number of bits of information\ncommunicated is equal to the Holevo function. We will thus take\n\n\\begin{equation}\n\\label{see}\n\\mbox{\\bf C}= H,\n\\end{equation}\nassuming that any additional necessary block coding\nwill automatically be done to supplement the dense coding. Exactly the same\nassumption has been used in Ref.\\cite{haus} to calculate the capacity\nfor dense coding in the case of pure letter states. Eqs.(\\ref{cg}) and \n(\\ref{see}) define the most general version of dense coding\nand we shall refer to this as completely general dense\ncoding (CGCD).\n\n  A simpler example of\ndense coding is the case when the letter states are generated from the\ninitial shared state $W_0$ by\n\n\\begin{eqnarray}\n\\label{w0}\nW_0={\\bf I}\\otimes {\\bf I}~W_0~{\\bf I}\\otimes {\\bf I},\\\\\nW_1={\\bf \\sigma_1}\\otimes {\\bf I}~W_0~{\\bf \\sigma_1}\\otimes {\\bf I}, \\\\\nW_2={\\bf \\sigma_2}\\otimes {\\bf I}~W_0~{\\bf \\sigma_2}\\otimes {\\bf I}, \\\\\nW_3={\\bf \\sigma_3}\\otimes {\\bf I}~W_0~{\\bf \\sigma_3}\\otimes {\\bf I}.\n\\label{w1}\n\\end{eqnarray}\nIn the above set of equations, the first operator of the combination\n${\\bf \\sigma_i} \\otimes {\\bf I}$ acts on Alice's qubit and the second operator\nacts on Bob's qubit. We shall refer to this case (i.e when the letter\nstates are generated by Eqs.(\\ref{w0})-(\\ref{w1})) as simply \ngeneral dense coding (GDC). The generality present in GDC is that Alice\nis allowed to prepare the different letter states with unequal probabilities.\nIn other words, one has to use Eqs.(\\ref{avg})-(\\ref{see}) to estimate\nthe capacity {\\bf C}.\n  \n  In the more special case when Alice not only generates the four letter states\naccording to Eqs.(\\ref{w0})-(\\ref{w1})) but also \nwith equal probability, the ensemble is given by\n\\begin{equation}\n\\label{avg1}\nW = \\frac{1}{4} \\sum_{i=0}^3 W_i.\n\\end{equation}\nand the capacity becomes\n\\begin{equation}\n\\label{c}\n\\mbox{\\bf C} = \\frac{1}{4} \\sum_{i=0}^3 S(W_i||W).  \n\\end{equation} \nWe shall call this simplest case special dense coding (SDC).\nAmong all the possible ways of doing GDC, SDC is the optimal way to \ncommunicate when $W_0$ is a pure state (as we shall show in the next\nsection) or a Bell diagonal state. However, we do not know the optimal way\nto communicate when $W_0$ is a completely general state and CGCD is\nallowed. For most of our paper, we shall\nobtain bounds on the classical capacity {\\bf C} for SDC only. But we shall \npoint out those results which are valid\nfor GDC and CGDC as well. \nThough the main aim of this paper is to establish bounds on \nthe capacity ${\\bf C}$ when the\nletters are mixed states, we shall begin with a calculation of ${\\bf C}$ \nfor pure letter states.  \n\n\n\n\\section{C for SDC with pure letter states}\n\\label{purer}\nConsider the initial shared pure state $W_0$ to be,\n\\begin{equation}\n\\label{p0}\n|\\psi_0\\rangle  =   (a|00\\rangle + b|11\\rangle ).\n\\end{equation}\nThen, according to Eqs.(\\ref{w0})-(\\ref{w1}), the other letter states\nare given by\n \n\\begin{eqnarray}\n|\\psi_1\\rangle  =  (a|10\\rangle + b|01\\rangle ), \\\\\n|\\psi_2\\rangle  =   -i (a|10\\rangle - b|01\\rangle ), \\\\\n|\\psi_3\\rangle  =   (a|00\\rangle - b|11\\rangle ),\n\\label{p3}\n\\end{eqnarray} \nfrom which we obtain $W_i = |\\psi_i\\rangle \\langle \\psi_i|$. As all $W_i$\nare pure states we have\n\\begin{equation}\nS(W_i)=0.\n\\end{equation}\nThus from Eqs.(\\ref{holv1}) and (\\ref{see}) we have\n\\begin{equation}\n\\label{sw}\n\\mbox{\\bf C}=S(W).\n\\end{equation}\nWe will consider only the case of SDC. Thus the ensemble used\nis obtained from Eq.(\\ref{avg1}) to be\n\\begin{eqnarray}\nW &=& \\frac{|a|^2}{2}|00\\rangle\\langle 00| + \\frac{|b|^2}{2}|01\\rangle\\langle 01| \\nonumber \\\\ &+& \\frac{|a|^2}{2}|10\\rangle\\langle 10| + \\frac{|b|^2}{2}|11\\rangle\\langle 11|.\n\\end{eqnarray}\nThus from Eq.(\\ref{sw}) for the capacity {\\bf C}, we get\n\\begin{eqnarray}\n\\mbox{\\bf C} &=& - (|a|^2 \\log \\frac{|a|^2}{2}+ |b|^2 \\log \\frac{|b|^2}{2}) \\\\\n&=& 1 - (|a|^2 \\log |a|^2+ |b|^2 \\log |b|^2).\n\\end{eqnarray}\nNow we should recall that a good measure of entaglement for a pure\nstate of a system composed of two subsystems A and B is given by the\nvon Neumann entropy of the state of either of the subsystems \\cite{ben2,pop}.\nLet us call this measure the von Neumann entropy of entanglement and \nlabel it by $E_v$. Thus\n\\begin{equation}\nE_v(|\\psi\\rangle \\langle \\psi|_{\\mbox \\scriptsize {A+B}})= S {\\large (}\\mbox{Tr}_{\\mbox \\scriptsize {A}}(|\\psi\\rangle \\langle \\psi|_{\\mbox \\scriptsize {A+B}}){\\large )},\n\\end{equation}\nwhere $\\mbox{Tr}_{\\mbox \\scriptsize {A}}$ stands for partial trace over\nstates of system A.\nTherefore, for all the states $W_i$, \n\\begin{equation}\nE_v(W_i) = - (|a|^2 \\log |a|^2+ |b|^2 \\log |b|^2).\n\\end{equation}\nThus,\n\\begin{equation}\n\\label{pure1}\n\\mbox{\\bf C}=1+E_v(W_i).\n\\end{equation}\n  \n  We now prove that for pure\nstates, SDC (using all alphabet states with equal {\\em a priori}\nprobability)\nis the optimal way to communicate among all possible ways of\ndoing GDC (i.e when the letter states are generated \nby Eqs.(\\ref{w0})-(\\ref{w1})). Consider the general case when the states\n$W_i$ are sent with probabilities $p_i$. Then from Eq.(\\ref{avg}) we have\n\\begin{equation}\nW=\\rho_1+\\rho_2,\n\\end{equation}\nwhere\n\n\\begin{eqnarray}\n\\label{ro1}\n\\rho_1 &=& (p_0+p_3)|a|^2 |00\\rangle\\langle 00| + (p_0+p_3)|b|^2|11\\rangle\\langle 11| \\nonumber \\\\ &+& (p_0-p_3)a ^{*}b|11\\rangle\\langle 00| + (p_0-p_3)a b^{*} |00\\rangle\\langle 11|,\n\\end{eqnarray} \nand\n\\begin{eqnarray}\n\\rho_2 &=& (p_1+p_2)|a|^2 |10\\rangle\\langle 10| + (p_1+p_2)|b|^2|01\\rangle\\langle 01| \\nonumber \\\\ &+& (p_1-p_2)a ^{*}b |01\\rangle\\langle 10| + (p_1-p_2)a b^{*}|10\\rangle\\langle 01| .\n\\label{ro2}\n\\end{eqnarray}\nTherefore, $\\rho_1$ and $\\rho_2$ form separate blocks inside the matrix\n$W$ and\n\\begin{equation}\nS(W)=S(\\rho_1)+S(\\rho_2).\n\\end{equation} \nEq.(\\ref{sw}) indicates that we need to choose the probabilities $p_i$ in such\na way that $S(W)$ is maximized. Density matrices with same diagonal elements have the highest von Neumann\nentropy when the nondiagonal elements are zero. Applying this fact to\nEqs.(\\ref{ro1}) and (\\ref{ro2}) we get\n \n\\begin{eqnarray}\n\\label{nond1}\np_0=p_3, \\\\\np_1=p_2.\n\\label{nond2}\n\\end{eqnarray} \nUsing Eqs.(\\ref{nond1}) and (\\ref{nond2}) in the expressions for $\\rho_1$\nand $\\rho_2$ and calculating the entropy $S(W)$ gives\n \n\\begin{eqnarray}\nS(W)&=&-\\sum_{i=0}^1 2p_i(|a|^2 \\log 2p_i|a|^2 + |b|^2 \\log 2p_i|b|^2) \\nonumber  \\\\ &=& -\\sum_{i=0}^1 2p_i \\log 2p_i - (|a|^2 \\log |a|^2 + |b|^2 \\log |b|^2), \n\\end{eqnarray}\nwhere the normalizations of the state amplitudes and the probabilities \n$p_i$ have been used. From analysis of von Neumann entropies it is well\nknown that the expression $-\\sum_{i=0}^1 2p_i \\log 2p_i$ has a maximum\nvalue when both $p_0$ and $p_1$ are equal. Thus $S(W)$, and hence the\nclassical capacity {\\bf C}\nis maximized when\n\\begin{equation}\np_0=p_1=p_2=p_3=\\frac{1}{4}.        \n\\end{equation}\nThus, among all the possible ways of performing GDC, SDC is the optimal way to communicate when pure states are being used\nas letters. From the above result and Eq.(\\ref{pure1}) we can conclude that in the case of GDC\nwith pure letter states we have\n\\begin{equation} \n\\label{pure2}\n\\mbox{\\bf C} \\leq 1+E_v(W_i).\n\\end{equation}\nThis result (Eq.(\\ref{pure2})) had also been obtained in Ref.\\cite{haus} from logical\narguments. Following a procedure analogous to the above proof\nit can be shown that for Bell \ndiagonal letter states SDC is again the optimal way to communicate\namong all the possible ways of doing GDC.\n\n\n\n\\section{Nature of the ensemble used in SDC}\n\n\n In order to derive a lower bound on the classical capacity\n{\\bf C} for SDC we need to prove a \ncrucial lemma concerning the nature of the\nensemble used in SDC: \n  \n   {\\em For states $W_i$ defined\nin accordance to Eqs.(\\ref{w0})-(\\ref{w1}), the ensemble $W$ for SDC as defined\nin Eq.(\\ref{avg1}) is a disentangled state irrespective of the nature\nof $W_0$}.\n\n   To prove this, we have to start by assuming $W_0$ to be\na most general state of two qubits. This is given\nby \\cite{hor1}\n\n\\begin{eqnarray}\n\\label{wo1}\nW_0 &=& \\frac{1}{4}[I\\otimes I + \\sum_m r_m \\sigma_m \\otimes I  \\nonumber \\\\\n&+& I\\otimes \\sum_m s_m \\sigma_m + \\sum_{m,n} t_{mn} \\sigma_m \\otimes \\sigma_n],\n\\end{eqnarray}\nwhere the indeces $m$ and $n$ take on values from $1$ to $3$.\nSubstituting $W_0$ from Eq.(\\ref{wo1}) into Eqs.(\\ref{w0})-(\\ref{w1}) we \nget\n\n\\begin{eqnarray}\n\\label{w11}\nW_1 &=& \\frac{1}{4}[I\\otimes I +r_1 \\sigma_1 \\otimes I- \\sum_{m \\neq 1} r_m \\sigma_m \\otimes I  \n+I\\otimes \\sum_m s_m \\sigma_m \\nonumber \\\\ &+&\\sum_{n} t_{1n} \\sigma_1 \\otimes \\sigma_n - \\sum_{m\\neq 1,n} t_{mn} \\sigma_m \\otimes \\sigma_n], \\\\ \nW_2 &=& \\frac{1}{4}[I\\otimes I +r_2 \\sigma_2 \\otimes I- \\sum_{m \\neq 2} r_m \\sigma_m \\otimes I   \n+I\\otimes \\sum_m s_m \\sigma_m \\nonumber \\\\&+& \\sum_{n} t_{2n} \\sigma_2 \\otimes \\sigma_n -\\sum_{m\\neq 2,n} t_{mn} \\sigma_m \\otimes \\sigma_n], \\\\\nW_3 &=& \\frac{1}{4}[I\\otimes I +r_3 \\sigma_3 \\otimes I- \\sum_{m \\neq 3} r_m \\sigma_m \\otimes I  \n+ I\\otimes \\sum_m s_m \\sigma_m \\nonumber \\\\&+&\\sum_{n} t_{3n} \\sigma_3 \\otimes \\sigma_n - \\sum_{m\\neq 3,n} t_{mn} \\sigma_m \\otimes \\sigma_n]. \n\\label{w12}\n\\end{eqnarray}\nUsing expressions for $W_i$ from Eqs.(\\ref{wo1})-(\\ref{w12}) in Eq.(\\ref{avg1})\nwe get\n\\begin{equation}\nW = I \\otimes \\frac{1}{4}[ I + \\sum_m s_m \\sigma_m].\n\\end{equation}\nThis is very clearly a disentangled state. A plausible physical argument to\nsupport this result can be as follows. We know that an equal mixture of\nthe four Bell states is a disentangled state. So it seems highly likely\nthat an equal mixture of less entangled states\nwill be disentangled as well. This result is, of course,\nvalid only for SDC, where the equal probabilities of the letter states\nresult in the careful cancellation of all the entanglement carrying terms.\nIt allows us to easily derive a lower bound on {\\bf C} for SDC in terms\nof one of the measures of entanglement.\n \n\n\\section{The relative entropy of entanglement as a lower bound on C} \n        To investigate quantitatively the relationship between \nentanglement and {\\bf C} for arbitrary mixed letter states, it is necessary to use some measure \nof entanglement for mixed states. One such measure of entanglement, the relative entropy\nof entanglement, has been\nintroduced in Ref.\\cite{vlat1}. It has been shown to have a statistical\ninterpretation \\cite{vlat2} as well as a physical interpretation \\cite{vlat3,rns}\nin forming bounds on the process of entanglement purification \\cite{ef,pf,Deu,ben2}.    We will use the symbol $E_R$ to represent this measure\nof entanglement. For an\narbitrary  mixed state  $\\sigma$ it is given\nby \n\n\\begin{equation}\nE_R(\\sigma) = \\min_{\\rho \\in {\\cal D}} S(\\sigma||\\rho),\n\\end{equation}\nwhere ${\\cal D}$ is the set of disentangled states. A property which has to\nbe\nsatisfied by any legitimate entanglement measure is its invariance under local unitary operations \\cite{vlat3}.  $E_R(\\sigma)$ is thus\ninvariant under local unitary operations of the state $\\sigma$.\n\n As the state $W$ of the ensemble used for SDC \nhas been shown to be a disentangled state $(i.e.~W \\in {\\cal D})$ in the previous section, \nwe have\n\n\\begin{equation}\nS(W_i||W)  \\geq  \\min_{\\rho \\in {\\cal D}} S(W_i||\\rho) = E_R(W_i).\n\\end{equation}\n\nUsing the above inequality in Eq.(\\ref{c}) for {\\bf C} of SDC we  get\n\\begin{equation}\n\\label{zeq}\n{\\bf \\mbox{C}} = \\frac{1}{4} \\sum_{i=0}^3 S(W_i||W) \\geq \\frac{1}{4} \\sum_{i=0}^3 E_R(W_i).  \n\\end{equation}\nAs each of the states $W_i$ are derived from $W_0$ via local unitary\noperations only,\nwe have\n\\begin{equation}\n\\label{eq}\nE_R(W_i) = E_R(W_0)\n\\end{equation}\nfor all values of $i$. Combining Eqs.(\\ref{eq}) and (\\ref{zeq}) we\nget\n\n\n\\begin{equation}\n{\\bf \\mbox{C}} \\geq  E_R(W_0).  \n\\end{equation}\nThis means that the classical capacity {\\bf C} for SDC is bounded from\nbelow by the relative entropy of entanglement of the initial shared mixed\nstate. This result is however, not generalizable to GDC as it relies\ncrucially on $W$ for SDC being a disentangled state.  \n\n\\section{Average distinguishability as an upper bound on C}\nIn this section, we are going to\ndivert a little bit from the main theme linking {\\bf C} to entanglement measures and point out\nanother interesting bound on {\\bf C} stemming from the mutual distinguishability of the letter states $W_i$.  It is known that \nthe relative entropy $S(W_i || W_j)$ is a kind of statistical measure of the distinguishability  between the quantum states $W_i$ and $W_j$ \\cite{vlat2}.\nThus for GDC one can define an average mutual distinguishability function as\n\n\\begin{equation}\n\\delta(p_i,W_i) = \\sum_i \\sum_j p_i p_j S(W_i || W_j).\n\\end{equation}\nWe show below that this average mutual distinguishability function\nforms an upper bound on {\\bf C}. Putting $W$ from \nEq.(\\ref{avg}) into Eq.(\\ref{see})  we obtain \n\n\n\n\\begin{eqnarray}\n{\\bf \\mbox{C}} &=&  \\sum_{i=0}^3 p_i~S(W_i|| \\sum_{j=0}^3 p_j~W_j) \\nonumber \\\\ &=&  \\sum_{i=0}^3  p_i~S(\\sum_{j=0}^3 p_j~ W_i||\\sum_{j=0}^3 p_j~  W_j),\n\\label{c1}\n\\end{eqnarray}\nwhere the factor $\\sum_{j=0}^3 ~p_j$ has been inserted before $W_i$\nin the second step as its value equals unity. We now use the joint\nconvexity property of the relative entropy \n\n\\begin{equation}\n\tS( \\sum \\lambda_i \\sigma_i ||\\sum \\lambda_i \\rho_i) \\le \\sum \\lambda_i S(\\sigma_i || \\rho_i)\n\t\\label{convex}\n\\end{equation}\nto expand the right hand side of Eq.(\\ref{c1}) and obtain\n \n\n\\begin{equation}\n{\\bf \\mbox{C}} \\leq  \\sum_{i,j=0}^3 p_i p_j~S(W_i || W_j) \n= \\delta(p_i,W_i).  \n\\end{equation}\nNote that the above result is completely general because neither\ndoes it require the special class of local unitary operations given \nby Eqs.(\\ref{w0})-(\\ref{w1}), nor does it require the probabilities\nto be uniform.\nThus the classical capacity for CGDC is bounded from above by\nthe average mutual distinguishability function. Note that we are\npointing  this out just as an interesting bound and it is not\nlinked to the main theme of the paper (relating entanglement measures\nstemming from purification procedures to dense coding capacities).\n\n\n\n\\section{An upper bound on C in terms of the entanglement of formation}\n\nIn this section we find an upper bound on {\\bf C} in terms of yet\nanother measure of entanglement, namely, the entanglement of formation\n$E_F$. Consider a decomposition of an arbitrary mixed state \n$\\sigma$ in terms of pure states $\\sigma_i$ :\n\\begin{equation}\n\\label{dec}\n\\sigma =  \\sum_i \\lambda_i \\sigma_i.\n\\end{equation}   \nThen the entanglement of formation of this state is defined\nas \\cite{ef,wot}\n\\begin{equation}\nE_F(\\sigma) = \\min \\sum_i \\lambda_i E_v (\\sigma_i), \n\\end{equation}\nwhere the minimum is taken over all decompositions of $\\sigma$ of the \ntype given by Eq.(\\ref{dec}). \n     \n  The initial shared state $W_0$ used in dense coding, will, in general, have several decompositions\nin terms of pure states. Let the particular decomposition from which its\nentanglement of formation $E_F(W_0)$ is\ncalculated (referred to as the entanglement minimizing \ndecomposition \\cite{wot}) be \n\\begin{equation}\n\\label{decw}\nW_0 =  \\sum_m q_m W_{0m},\n\\end{equation} \nwhere $W_{0m}$ are pure states and $q_m$ are probabilities.\nAs Eq.(\\ref{decw}) gives the entanglement minimizing decomposition,\nwe have\n\\begin{equation}\nE_F(W_0)=\\sum_m q_m E_v(W_{0m}),\n\\end{equation}\nwhile the normalization of the probabilities imply\n\\begin{equation}\n\\label{norm}\n\\sum_m q_m = 1.\n\\end{equation}\n\n\n As each of the signal states $W_i$ are derived from $W_0$ by local\nunitary operations, they can be decomposed as \n\\begin{equation}\n\\label{decwi}\nW_i = \\sum_m q_m W_{im}, \n\\end{equation}\nwhere each pure state $W_{im}$ is connected to the pure state $W_{0m}$\nby exactly the same local\nunitary operation as that which connects $W_i$ to $W_0$. As any legitimate\nmeasure of entanglement has to remain invariant under local unitary\noperations, we have\n\\begin{equation}\n\\label{im}\nE_v(W_{im})=E_v(W_{0m}),\n\\end{equation}\nand\n\\begin{equation}\n\\label{i}\nE_F(W_i) = E_F(W_0).\n\\end{equation}\n>From Eqs.(\\ref{im}) and (\\ref{i}) we have\n\\begin{eqnarray}\n\\label{equa}\n\\sum_m q_m  E_v(W_{im}) &=& \\sum_m q_m E_v(W_{0m}) \\nonumber \\\\\n&=& E_F(W_0)=E_F(W_i).\n\\end{eqnarray}\n\n\nNow, in the case of SDC, the capacity (from Eq.(\\ref{c})) is\n\\begin{eqnarray}\n{\\bf \\mbox{C}} &=& \\frac{1}{4} \\sum_i  S(W_i||\\frac{1}{4} \\sum_j W_j) \\nonumber \\\\\n &=& \\frac{1}{4} \\sum_i S(\\sum_m q_{m} W_{im}||\\frac{1}{4} \\sum_j \\sum_m q_{m} W_{jm}) \\\\ &=& \\frac{1}{4} \\sum_i S(\\sum_m q_{m} W_{im}|| \\sum_m q_{m} W_m)\n\\label{grhl}\n\\end{eqnarray}\nwhere we have put\n\\begin{equation}\n\\label{ensm}\nW_m = \\frac{1}{4} \\sum_j W_{jm}.\n\\end{equation}\nUsing joint convexity (Eq.(\\ref{convex})) to expand the right hand side\nof Eq.(\\ref{grhl}), we get\n\\begin{eqnarray}\n\\label{less}\n{\\bf \\mbox{C}} & \\leq & \\frac{1}{4} \\sum_i \\sum_m q_{m} S(W_{im}|| W_m) \\nonumber \\\\\n&=& \\sum_m q_{m} ( \\frac{1}{4} \\sum_i  S(W_{im}|| W_m) ). \n\\end{eqnarray}\nNow, for each value of $m$, the expression $\\frac{1}{4} \\sum_i  \nS(W_{im}|| W_m)$ can be regarded as the classical capacity {\\bf C} for SDC with the states\n$W_{im}$ as the four letter states. As each of the\nstates $W_{im}$, are pure, Eq.(\\ref{pure1}) implies\n\\begin{equation}\n\\label{oneE}\n\\frac{1}{4} \\sum_i  \nS(W_{im}|| W_m) = 1+E_v(W_{im}).\n\\end{equation}\nPutting Eq.(\\ref{oneE}) into Eq.(\\ref{less}) we get\n\\begin{equation}\n\\label{qm1}\n{\\bf \\mbox{C}} \\leq \\sum_m q_{m} (1+E_v(W_{im})).\n\\end{equation}\nUsing Eqs.(\\ref{norm}) and (\\ref{equa}) to simplify the right hand side\nof Eq.(\\ref{qm1}) we get\n\\begin{equation}\n\\label{upbnd}\n{\\bf \\mbox{C}} \\leq 1 + E_F(W_i).\n\\end{equation}\n   \n    The above bound is valid for GDC as well. To see this\none just has to repeat the above proof starting with \n\\begin{equation}\n{\\bf \\mbox{C}} = \\sum_i p_i~S(W_i|| \\sum_j p_j W_j)\n\\end{equation}\nand replace Eq.(\\ref{ensm}) with\n\\begin{equation}\n\\label{ensm1}\nW_m =  \\sum_j p_j~W_{jm}.\n\\end{equation}\nEq.(\\ref{less}) then gets replaced by\n\\begin{equation}\n\\label{ii0}\n{\\bf \\mbox{C}} \\leq \\sum_m q_{m} (\\sum_i p_i~S(W_{im}|| W_m)).\n\\end{equation}\nNow note the fact that $\\sum_i p_i~S(W_{im}|| W_m)$ is the\nexpression for the classical capacity {\\bf C} for CGDC with $W_{im}$\nbeing the letter states. When the\nstates $W_{im}$ are generated from $W_{i0}$ according to\nEqs.(\\ref{w0})-(\\ref{w1}) (i.e. when GDC protocol is being followed),\nthen the purity of the states $W_{im}$ guarantees that\n{\\bf C} is less than $1+E_v(W_{im})$ (as shown\nin section \\ref{purer}). Using this fact in Eq.(\\ref{ii0}) we again end up with Eq.(\\ref{upbnd}). Thus even in the case of GDC, the capacity {\\bf C}\nis bounded by $1 + E_F(W_i)$.\n\n\\section{An upper bound on C in terms of the relative entropy of\nentanglement}\nHaving analytically proven that $1 + E_F(W_i)$ is an upper bound\non {\\bf C} for GDC, we now proceed to check whether the even\nsmaller (as proved in Ref.\\cite{vlat3}) quantity $1+E_R(W_i)$ is\nalso an upper bound. However, we do not attempt to prove this analytically\nfor a completely general initial shared state $W_0$. Instead we calculate\nthe capacity {\\bf C} (for SDC only) for those\nspecific classes of the initial shared state $W_0$ whose relative\nentropy of entanglement $E_R(W_0)$ is already known.\nWe then plot\nthis capacity {\\bf C} as a function of the relative entropy of entanglement \n$E_R$ for each of these classes of states and check whether\nthe curve ${\\bf \\mbox{C}}(E_R)$ lies below the plot of $1+E_R$. \n\n   At first we have a look at \nmixed states of the type \\cite{vlat3}\n\n\\begin{equation}\n\\label{lamb}\n\\Lambda_A =  \\lambda |\\Phi^+\\rangle \\langle \\Phi^+| +\n(1-\\lambda) |01\\rangle\\langle 01|,\n\\end{equation}\nwhere $|\\Phi^{+}\\rangle$ is one of the four Bell states which are defined by\n\\begin{eqnarray}\n|\\Phi^{\\pm}\\rangle & = & \\frac{1}{\\sqrt{2}} (|00\\rangle \\pm |11\\rangle ) \\\\\n|\\Psi^{\\pm}\\rangle & = & \\frac{1}{\\sqrt{2}} (|01\\rangle \\pm |10\\rangle )\n\\end{eqnarray}\nWe shall refer to the states described by Eq.(\\ref{lamb}) as \nlambda states of\nthe type A. For them, the relative entropy of entanglement is \\cite{vlat3}\n\\begin{equation} \n E_R(\\Lambda_A)  =  (\\lambda -2)\\log (1-\\frac{\\lambda}{2}) + (1-\\lambda)\n\\log (1-\\lambda),  \n\\end{equation}\nwhile the {\\bf C} for SDC is\n\n\\begin{eqnarray}\n{\\bf \\mbox{C}} &=&  (1-\\lambda)\n\\log (1-\\lambda) + \\frac{1}{2}(\\lambda -2)\\log (1-\\frac{\\lambda}{2}) \\\\\n&+& \\frac{1}{2} \\lambda \\log \\lambda + (1+ \\frac{\\lambda}{2}).\n\\end{eqnarray}\nThe plot of the classical capacity {\\bf C} for these states as a function of $E_R$ of the state\nhas been shown in figure \\ref{lam1} and it is indeed found that\n\n\\begin{equation}\n{\\bf \\mbox{C}} \\leq 1+E_R.\n\\end{equation}\nThe equality holds true only at the two ends of the graph, namely\nat maximal entanglement $E_R=1$ (when $\\sigma_1$ approaches\na Bell state) and zero entanglement $E_R=0$ (when $\\sigma_1$ approaches\n$|01\\rangle\\langle 01|$). \n \n\n\n\n\n\n\nNext we look at states of the type \\cite{vlat3}\n\\begin{equation}\n \\Lambda_B  =  \\lambda |\\Phi^+\\rangle \\langle \\Phi^+| +(1-\\lambda) |00\n\\rangle\\langle 00| \n\\end{equation}\nwhich we call lambda states of the type B. For these \\cite{vlat3}\n\\begin{eqnarray}\n\\label{b1}\nE_R(\\Lambda_B)  &=&  s_+ \\log s_+ + s_- \\log s_-  \\nonumber \\\\\n&-& (1-\\frac{\\lambda}{2}) \\log (1-\\frac{\\lambda}{2}) - \\frac{\\lambda}{2} \\log \\frac{\\lambda}{2} \\;\\; ,\n\\end{eqnarray} \nwhere \n\\begin{equation}\ns_{\\pm} = \\frac{1\\pm \\sqrt{1-2\\lambda (1-\\lambda)}}{2}\n\\end{equation}\nare the eigenvalues of $\\Lambda_B$. The {\\bf C} for SDC in this case\nis given by                        \n\\begin{eqnarray}\n\\label{b2}\n{\\bf \\mbox{C}}&=& s_+ \\log s_+ + s_- \\log s_-\\nonumber \\\\\n&-& (1-\\frac{\\lambda}{2}) \\log \\frac{1}{2}(1-\\frac{\\lambda}{2}) - \n\\frac{\\lambda}{2} \\log \\frac{\\lambda}{4}.\n\\end{eqnarray}\nAs is clear from a simple comparison of\nEqs.(\\ref{b1}) and (\\ref{b2}), for lambda states of the type B,\n\\begin{equation}\n\\label{eq1}\n{\\bf \\mbox{C}} = 1+E_R.\n\\end{equation}\nThus, a curious feature of lambda states of type\nB is that the capacity for SDC is actually always {\\em equal} to\n$1+E_R$. We will discuss a bit more about this curious aspect later in\nthe section.\n  \n    Now we consider another special class of states called\nthe Werner states \\cite{ef,wer} parameterized by a number F called the \nfidelity \\cite{ef} and given by  \n\\begin{eqnarray}\nW_F&=&F|\\Psi^{-}\\rangle \\langle \\Psi^{-}| + \\frac{1-F}{3} (|\\Psi^{+}\\rangle \\langle \\Psi^{+}| \\nonumber \\\\ &+& |\\Phi^{+}\\rangle \\langle \\Phi^{+}|\n+ |\\Phi^{-}\\rangle \\langle \\Phi^{-}|). \n\\end{eqnarray}\nThe relative entropy of entanglement of these states is given by \\cite{vlat3}\n\\begin{equation}\n E_R(W_F) = F\\log F +(1-F)\\log (1-F) + 1, \n\\end{equation}\nand the {\\bf C} for SDC is calculated to be\n\\begin{eqnarray}\n{\\bf \\mbox{C}} &=& 2 + F\\log F +(1-F)\\log \\frac{(1-F)}{3}. \n\\end{eqnarray}\nThe plot {\\bf C} versus $E_R$ has been drawn in Fig.\\ref{wer} and it is \nfound that even in this case, \n\\begin{equation}\n{\\bf \\mbox{C}} \\leq 1+E_R.\n\\end{equation}\n\n\n\n\nNow consider the case of a general Bell diagonal state \n\\begin{eqnarray}\nB_D &=& \\lambda_1 |\\Psi^{-}\\rangle \\langle \\Psi^{-}| + \\lambda_2 |\\Psi^{+}\\rangle \\langle \\Psi^{+}| \\nonumber \\\\\n&+& \\lambda_3 |\\Phi^{+}\\rangle \\langle \\Phi^{+}|\n+\\lambda_4 |\\Phi^{-}\\rangle \\langle \\Phi^{-}|.\n\\end{eqnarray}\nIt has been proved in Ref.\\cite{vlat1} that when all $\\lambda_i \\in [0,\\frac{1}{2}]$ then\n\\begin{equation}\n\\label{zero}\nE_R(B_D)=0,\n\\end{equation}\nwhile when any of the $\\lambda_i$ (say $\\lambda_1$) $\\geq \\frac{1}{2}$,\nthen\n\\begin{equation}\n\\label{fint}\nE_R(B_D)= \\lambda_1\\log\\lambda_1 +(1-\\lambda_1)\\log (1-\\lambda_1) + 1.\n\\end{equation}\nThe {\\bf C} for SDC always turns out to be\n\\begin{equation}\n\\label{ccc}\n{\\bf \\mbox{C}}= 2 + \\sum_i \\lambda_i \\log\\lambda_i.\n\\end{equation}\nFirst consider the case when all $\\lambda_i \\in [0,\\frac{1}{2}]$. From\nEqs.(\\ref{zero}) and (\\ref{ccc}) we find that\n\\begin{eqnarray}\n1+E_R(B_D)-{\\bf \\mbox{C}} &=& - \\sum_i \\lambda_i \\log\\lambda_i -1 \\nonumber\n\\\\ &=&  - \\sum_i \\lambda_i (\\log\\lambda_i+1) \\nonumber \\\\\n&\\geq & 0,\n\\end{eqnarray}\nwhere $\\sum_i \\lambda_i =1$ has been used to proceed from the first to\nthe second step and $ \\log\\lambda_i \\leq -1$ (because $\\lambda_i \\in [0,\\frac{1}{2}]$) has \nbeen used to proceed from the\nsecond to the third step.\n\nNow consider the complementary case ($\\lambda_1 \\geq \\frac{1}{2}$). From\nEqs.(\\ref{fint}) and (\\ref{ccc}) we find that\n\n\\begin{eqnarray}\n1+E_R(B_D)-{\\bf \\mbox{C}} &=& (1-\\lambda_1)\\log (1-\\lambda_1) -\\sum_{i\\neq 1} \\lambda_i \\log\\lambda_i \\nonumber \\\\ &=& (\\sum_{i\\neq 1} \\lambda_i) \\log (\\sum_{j\\neq 1} \\lambda_j)-\\sum_{i\\neq 1} \\lambda_i \\log\\lambda_i \\nonumber \\\\\n&=& \\sum_{i\\neq 1} \\lambda_i (\\log (\\sum_{j\\neq 1} \\lambda_j)- \\log\\lambda_i)\n\\nonumber \\\\ &\\geq& 0,\n\\end{eqnarray}\nwhere in order to proceed from the third to the fourth step we have used\nthe simple fact that $\\log(\\lambda_2+\\lambda_3+\\lambda_4)$ is greater than\neither of the terms $\\log \\lambda_2$,$\\log \\lambda_3 $ or $\\log \\lambda_4$.\nThus for all Bell diagonal states we have\n\\begin{equation}\n{\\bf \\mbox{C}} \\leq 1+E_R.\n\\end{equation}\n \n Now we will point out a curious fact about the situation when any two of the eigenvalues of\na Bell diagonal state (say $\\lambda_3$ and $\\lambda_4$) are zero. When \n$\\lambda_1=\\lambda_2=\\frac{1}{2}$ (which means $E_R=0$), we have {\\bf C}$=1$\n(using Eq.(\\ref{ccc})).\nIn all the other cases (for which we use Eqs.(\\ref{fint}) and (\\ref{ccc})) we have\n\n\\begin{eqnarray}\n1+E_R(B_D)-{\\bf \\mbox{C}} &=& (1-\\lambda_1)\\log (1-\\lambda_1) - \\lambda_2 \\log\n\\lambda_2 \\nonumber \\\\\n&=& 0,\n\\end{eqnarray}\nwhere we have used the simple fact that $\\lambda_1+\\lambda_2=1$. Thus for\nall Bell diagonal states with only two nonzero eigenvalues we have\n\\begin{equation}\n\\label{eq2}\n{\\bf \\mbox{C}} = 1+E_R.\n\\end{equation}\n\n  On the basis of the results obtained in this section let us\n conjecture:\n\n{\\bf Conjecture }: {\\em The {\\bf \\mbox{C}} for SDC with\ncompletely general (possibly mixed) states is bounded from above by $1+E_R$}. \n\n        A great\ndeal of empirical evidence has been presented in this section in support\nof the conjecture. In the next section we proceed to give a heuristic justification in support of our conjecture. In fact, we will try to\njustify an even stronger upper bound on {\\bf C}.\n\n\n \n\n\n\n\\section{SDC and purification}\n\\label{pff}\nTo justify the conjecture of\nthe previous section, we will have to examine the \nfollowing interesting question:\nHow does the capacity {\\bf C}\n change\nif Alice and Bob first locally purify their ensemble and distill Bell states \n(following the optimal purification procedure) and\nfollow this up by SDC?  Various purification procedures\nhave been described in Refs.\\cite{ef,pf,Deu,ben2}. Here we assume that\nAlice and Bob follow the optimal purification process: One which helps\nthem to distill the maximum fraction of Bell states from the initial\nensemble.\nThey will, after optimal purification,\nhave a fraction $E_D$ (where $E_D$ is called the\n{\\em entanglement of distillation})\nshared pairs in Bell states and a fraction $(1-E_D)$ pairs in\na disentangled state. They now complete their 'purification'\nprocess by converting the final subensemble of disentangled pairs to pure\nstates by projective measurements. We refer to such a protocol\nas {\\em complete purification}. After a  complete purification,\nAlice can use the fraction $E_D$ of Bell pairs to send Bob information\nat the rate of 2 bits\/pair and the fraction $(1-E_D)$ of\npure disentangled pairs to send information at\nthe rate of 1 bit\/pair. Thus if Alice and Bob initially shared\n$n$ pairs, the classical capacity after a complete\nand optimal purification procedure is\n\\begin{eqnarray}\n\\label{cppd}\n{\\bf \\mbox{C}}&=& \\frac{1}{n} \\{2 n E_D + n (1-E_D)\\} \\nonumber \\\\\n&=& 1+E_D.\n\\end{eqnarray}\nNote that the above result is only asymptotically true ($n \\rightarrow \\infty$).\nNaively, one might expect this {\\bf C} to be lower than the {\\bf C}\nbefore purification.\nThis is because, as mentioned in Ref.\\cite{ben2}, entanglement\nconcentration is a more destructive process than quantum data\ncompression. Some amount of shared entanglement is destroyed in the process\nof purification. So it might be expected that the final ensemble after a\npurification will be able to convey less classical information\nthan the original ensemble. On the contrary, as we will justify, the\ncapacity for SDC with the purified ensemble is greater than that\nwith the unpurified ensemble when an optimal\nand complete purification protocol is used. The gain comes from the fact that now\nthere are two separate sub-ensembles instead of a single ensemble.\n In this section, we will try to\njustify, albeit heuristically, the following statement (as a more fundamental\nconjecture than the one presented in the previous section):\\\\\n{\\bf Conjecture}: {\\em The capacity for SDC is more when it is preceeded\nby a complete and optimal purification procedure}. \n\n\n We will first consider the special case of Bell diagonal mixed states, for\nwhich the proof of the increase in the SDC capacity on a \ncomplete and optimal purification (our conjecture) can be rigorously proved. For Bell diagonal states\nwith entropy $S(\\rho) \\leq 1$, there is a purification protocol\ncalled hashing (with the distillable fraction being $E_{DH} = 1-S(\\rho)$) \\cite{ef}.\n>From Eq.(\\ref{ccc}) we see that for these Bell diagonal states, the {\\bf C} for SDC\nis equal to $1+E_{DH}$. As hashing may not necessarily be the optimal\nprotocol, we have $E_{DH} \\leq E_D$. This immediately implies ${\\bf \\mbox{C}} \\leq  1+E_D$. For the complementary case of Bell diagonal states with\n$S(\\rho) > 1$, we have (from Eq.(\\ref{ccc}) for SDC), ${\\bf \\mbox{C}} < 1$.\nThis is obviously less than $1+E_D$ for any finite value of the entanglement\nof distillation (i.e for all inseperable states \\cite{hor2}). Thus for all\nBell diagonal states the {\\bf C} for SDC can be improved by a prior optimal\nand complete purification of the ensemble of shared states.\n\n   For the more general case of arbitrary mixed entangled states, the \nproof will, essentially, be heuristic. Our approach will be to split the change in the classical information\ncapacity due to complete and optimal purification into two parts. The first\npart is positive (an increase in capacity) and due to the addition of\nclassical side channels during the purification procedure. These side\nchannels are used by Alice (A) to communicate the results of her measurements\nto Bob (B), or vice versa. As this communication during purification is\nalready conveying information from A to B (or vice versa), the channel\ncapacity of the classical side channels should be directly added to\nthe classical capacity (of course, for this, we have to implicitly assume\nthe additivity of classical capacities). The information conveyed from A to B (or vice versa)\nduring purification is actually used by the two parties to\nprecisely identify their entangled and disentangled subensembles. There is\nalso a negative contribution (decrease in capacity) during the purification\nas a fraction $1-E_D$ of shared pairs loose all their entanglement. Of course\na part of this entanglement is pumped into the entangled subensemble, but the\nremaining is lost. Our job will be to argue that the positive contribution\nin channel capacity due to addition of classical side channels \noutweighs the negative\ncontribution due to loss of entanglement in an optimal and complete\npurification procedure.\n\n\n   \n\n\n\n\n   Consider the\nprocess of purification as a process of information gain \n by A and B. Before the purification,\nboth A and B have an equal amount of knowledge about their\nshared system (they both know the full density matrix). Finally, each of their shared states are pure (because of the completeness of the purification\nprocedure) and both of them have equal knowledge (i.e each know which of the pairs form\nthe disentangled subensemble and which of the pairs form the maximally entangled\nsubensemble). Thus they have both  \ngained an equal amount of information $\\mbox{I}$\/pair about their shared pairs during\nthe process of purification. Now, acquiring a part of this information\nmay not cause any destruction of shared entanglement (lets call it $\\mbox{I}_1$\/pair),\nwhile the remaining part (lets call it $\\mbox{I}_2$\/pair) does cause\na lowering of shared entanglement. We now contend, that this part, \n$\\mbox{I}_2$\/pair has to be equal\nto the number of classical communication channels used in the optimal\npurification procedure. The logic follows from the fact that the optimal\nstrategy would be for A and B to collaborate in such a way that both of\nthem gain\nthe entire information $\\mbox{I}_2$\/pair with the least destruction\nof entanglement. They must then each acquire only a fraction of the\ninformation complementary to the fraction acquired by the other. In this\nway, the total information acquired by A and B from the shared pairs through\ndirect entanglement degrading measurements is $\\mbox{I}_2$\/pair. They can then use a minimum of $\\mbox{I}_2$ classical side channels per pair to\ncommunicate their fraction of the acquired information to the other party.\nOn the other hand, if each wanted to acquire a fraction of the total information\nby direct measurements which was not entirely complementary to the part\nacquired by the other, they would destroy more entanglement than is really\nnecessary. Thus we would expect that the classical side channels contribute to boosting the\ncapacity up by at least an amount $\\mbox{I}_2$\/pair in the optimal purification\nprocedure. \n\n    Now consider how much the capacity decreases due to the degradation of \nshared entanglement during the purification procedure. The classical capacity\nof a fraction $1-E_D$ of the pairs drops from {\\em at most} 2 bits\/pair to\n1 bit\/pair (due to the {\\em completeness} of the purification procedure, the\nfinal capacity cannot be lower than 1 bit\/pair). Thus the drop in\nclassical capacity of each of the finally disentangled pairs \nis not more than 1 bit. This implies that the net decrease in classical\ncapacity of the entire ensemble \ndue to loss of entanglement hasn't been more than $1-E_D$ bits\/pair. \n\n\n    When the information $\\mbox{I}_2$ is greater than 1 bit\/pair, the proof of our\nstatement is straightforward. The increase in capacity due to classical\nside channels (i.e $\\mbox{I}_2$\/pair) is more than 1 bit\/pair, while the\ndecrease in capacity due to loss of entanglement is less than $1-E_D$ bits\/pair.\nAs $1-E_D$ is a fraction (i.e $1-E_D < 1$), the increase in capacity on\npurification overrides the inevitable decrease in capacity due to degradation\nof shared entanglement. Clearly, thus the capacity to do SDC increases\non prior purification when $\\mbox{I}_2 \\geq 1$ bit.\n\n     Now, we have to show that no matter what the value of $\\mbox{I}_2$ is,\nit is {\\em possible} to find a purification protocol for which $\\mbox{I}_2$ is greater than $1-E_D$ bits. In other words, if initially there\nare $n$ shared pairs (with $n$ being large), then the\n number of\nshared pairs which loose all their entanglement on gaining an\ninformation of $\\mbox{I}_2$\/pair can be made less than $n \\mbox{I}_2$ by\nan appropriate choice of the purification protocol. From such a statement it directly\nfollows that the increase in capacity on\npurification overrides the inevitable decrease in capacity due to degradation\nof shared entanglement. A simple way in which one may try to\njustify the above  proposition is from the fact that one\npair generally allows the extraction of upto two bits of information (one bit from each of A's \nand B's qubit). So, it should be possible, in general, to extract $n \\mbox{I}_2$  amount of\ninformation from measurements on less than $n \\mbox{I}_2$ shared pairs (whose\nentanglement is degraded).\nThis intutive understanding leaves\nopen the question as to whether the information $\\mbox{I}_2$\/pair {\\em relevant} to purification\ncan be acquired in this way. For that we resort to explicit exemplification.\nConsider the class of mixed states $\\rho$ (having nonzero entropy  $S(\\rho)$)\nand a diagonal decomposition $\\sum \\lambda_i |\\psi_i\\rangle \\langle \\psi_i|$ in terms of the pure states $|\\psi_i\\rangle$. Also suppose that this\ndiagonal decomposition does not coincide with the entanglement minimizing\ndecomposition \\cite{wot} of $\\rho$. For such states, acquiring the information\nabout the entropy ($S(\\rho)$\/pair) is essential during\npurification, as the final ensemble is pure. We now contend that acquiring this\n$S(\\rho)$ \/pair of information {\\em necessarily} destroys some\namount of shared entanglement. We will justify this by the method \nof contradiction. Suppose, it was really possible to acquire the information\nabout the entropy $S(\\rho)$ without destroying any entanglement. On acquiring an information of the amount equal to $S(\\rho)$\/pair,  A and B\nwill be able to divide their initial ensemble to four seperate pure subensembles,\neach being comprised of one of the states $|\\psi_i\\rangle$. The weight of each of these subensembles will be equal to\nthe eigenvalues $\\lambda_i$. The average entanglement shared by A and B after extracting the information $S(\\rho)$\/pair is thus $\\sum \\lambda_i E_{v}(|\\psi_i\\rangle)$.  However, for the class of states $\\rho$\nthat we are considering, $\\sum \\lambda_i E_{v}(|\\psi_i\\rangle)$ is necessarily greater than the initial\nshared entanglement (as quantified by the entanglement of formation $E_F$).\nThus by purely local actions and classical communications,\nA and B have been able to increase the shared entanglement for these\nclasses of mixed states. This contradicts\nthe definition of entanglement as a quantity that cannot be increased by \nlocal actions and classical communications. Thus, {\\em mixed states with the diagonal\ndecomposition not coincident with the entanglement minimizing decomposition cannot be locally purified\n without necessarily destroying some shared entanglement}. Thus the information\nabout the entropy of these states is of the type $\\mbox{I}_2$ (i.e acquiring the information necessarily causes\ndegradation of entanglement). A corollary which immediately follows (though\nnot directly relevant to the main point of the paper), is: {\\em the optimal purification\nof mixed states with the diagonal\ndecomposition not coincident with the entanglement minimizing decomposition necessarily has a nonzero $\\mbox{I}_2$ and thus necessarily\nrequires the use of classical side channels}. Now consider such states when\ntheir entropy, and thereby $\\mbox{I}_2$, exceeds $1$ bit. Thus the total information $n \\mbox{I}_2$ needed to\nbe acquired is greater than \n$n$ bits. However, purifiability (nonzero $E_D$) implies that the total entanglement loss\ncan be entirely concentrated to the entanglement loss of $n(1-E_D)$ pairs which is less than $n \\mbox{I}_2$ pairs.\nTherefore, it is {\\em possible} to gain the relevant $n \\mbox{I}_2$ bits of information by\ndestroying the entanglement of less than $n \\mbox{I}_2$ pairs. Now we really\nneed an unproven extension of the above statement: {\\em even when} $\\mbox{I}_2\n\\leq 1$ bit and {\\em irrespective of the nature of the decomposition of}\n$\\rho$, it is {\\em possible} to gain the relevant $n \\mbox{I}_2$ amount of information by\ndestroying the entanglement of less than $n \\mbox{I}_2$ pairs. Though this\nmay seem a large extrapolation, it seems highly {\\em plausible} because one\nshared pair allows the extraction of upto two bits of information. Thus we would\nexpect that, no matter what the value of $\\mbox{I}_2$ is, the increase in capacity on\npurification overrides the inevitable decrease in capacity due to degradation\nof shared entanglement.\n\n   \n\n    Thus the capacity to do SDC with the two  pure subenembles (maximally entangled and fully disentangled) after a complete and optimal purification, is\nexpected to be greater\nthan the initial mixed (and uniformly entangled) ensemble. We thus conjecture,\nwith the help of Eq.(\\ref{cppd}), the\nfollowing bound on the capacity of\ndoing SDC with any mixed entangled state\n\n\\begin{equation}\n\\label{bnded}\n{\\bf \\mbox{C}} \\leq  1+E_D.\n\\end{equation}\n\nThis is an even stronger upper bound on the classical capacity than the bound\nconjectured in the previous section as the relative entropy of entanglement\n$E_R$ is necessarily greater than $E_D$ \\cite{vlat3}. From Eq.(\\ref{bnded}),\nit directly follows that ${\\bf \\mbox{C}} \\leq  1+E_R$ (our previous conjecture). \nNote that in this section, this conjecture has been rigourously proved for all Bell diagonal states,\nheuristically proved for all states with entropy $S(\\rho) \\geq 1$ bit (i.e $\\mbox{I}_2 \\geq 1$ bit) and shown to be highly plausible for other types of\nstates.\n\n   \n \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}\nIn this paper we have obtained bounds on the capacity of doing\ndense coding in terms of the different measures of entanglement stemming\nfrom purification procedures. The\n rigorously proved part of the results of this paper is that for SDC one always has\n\\begin{equation}\n E_R \\leq {\\bf \\mbox{C}} \\leq 1+E_F.\n\\end{equation}\nOn the other hand if we allow for a conjecture (well supported by\nexamples and heuristically justified in Section.\\ref{pff}), we have the stronger result\n\\begin{equation}\n E_R \\leq {\\bf \\mbox{C}} \\leq 1+E_R.\n\\end{equation}\nWhat is the significance of bounds on the {\\bf C} for SDC when it can be\nreadily calculated? The importance can be realized when we invert \n the above equation and write\n\\begin{equation}\n {\\bf \\mbox{C}}-1 \\leq E_R  \\leq {\\bf \\mbox{C}}.\n\\end{equation}\nThus by calculating (or measuring) the channel capacity for SDC, we can draw an inference\nabout the range in which the entanglement of the shared states (as quantified\nby $E_R$) lies. Also one of the bounds, namely, ${\\bf \\mbox{C}} \\leq 1+E_F\n$ continues to hold even when we are allowed to vary the a priori probability\nof the various signal states (GDC). Thus, our inequality allows one\nto impose a readily calculable (from the expression for $E_F$ in Ref.\\cite{wot}) limit on the capacity for GDC, without having to optimize GDC over all possible a priori signal probabilities.\n\n    The physical interpretation of the upper bounds becomes clear\nfrom the considerations given in Section.\\ref{pff}. During\noptimal and complete entanglement purification procedures one generally enhances the classical\ncapacity much more due to added classical side channels than the inevitable\ndecrease in capacity brought by the loss of entanglement. The information transferred through these classical side channels \nduring purification helps to identify the entangled and disentangled subensembles. It is through this identification that they directly play the\nrole of boosting the capacity. As after an optimal and complete entanglement purification procedure, the capacity is $1+E_D$, this post-purification\ncapacity is an upper\nbound on the pre-purification capacity. As both the entanglement measures\n$E_F$ and $E_R$ are upper bounds on $E_D$, all our upper bounds follow\nimmediately. It is interesting to note that the upper and the lower bounds\nare trivial in complementary regimes. When ${\\bf \\mbox{C}} \\leq 1$, the upper\nbounds are trivial, but the lower bound ($E_R \\leq {\\bf \\mbox{C}}$) is\nnon trivial. On the other hand when ${\\bf \\mbox{C}} \\geq 1$, the upper\nbounds are non trivial while the lower bound is trivial.\n\n\n     We believe that this paper imparts physical significance to the\nvarious measures of entanglement from the viewpoint of forming bounds on\ncertain kinds of dense coding procedures. This direction of\nphysical interpretation of the entanglement measures is very different\nfrom the standard interpretations which stem from entanglement\ndilution and distillation processes. In a sense, this paper {\\em links up\nthe apparently disconnected notions of entanglement purification\nand dense coding}. The considerations of the previous section implies the\nfollowing lower bound\non the distillable entanglement: $E_D \\geq {\\bf \\mbox{C}}-1$. This is\nimportant because as yet there is no explicit formula for $E_D$ for a\ngeneral mixed entangled state. Thus the readily calculable quantity ${\\bf \\mbox{C}}-1$ (in the case of SDC) may offer a convinient lower bound on\nthe distillable entanglement of a given mixed state.\n  \n   \n \n\n The interesting question to examine is what apart from entanglement\nis involved in determining the dense coding capacity. This may be some measure of distinguishability between\nthe letter states (hinted by the fact that the average mutual distinguishability\nfunction forms an upper bound on the classical\ncapacity {\\bf C}). An aim of further research\nshould be to work towards a complete formula for the capacity of dense\ncoding. Moreover, working with a similar motivation as this paper, one can also try to relate entanglement measures\nstemming from purification procedures to the other uses of shared entanglement\nsuch as teleportation \\cite{ben93} and secret key distribution \\cite{ek91}. \n\n\n\\section{Acknowledgements}\nWe would like to thank Vladimir Buzek, Daniel Jonathan and Peter Knight for valuable \ndiscussions. This work is\nsupported by the Inlaks Foundation, Elsag-Bailey,\nHewlett-Packard, the European TMR networks ERB 4061PL951412 and\nERBFMRXCT96066, the UK Engineering and\nPhysical Sciences Research Council, the European Science Foundation and the\nLeverhume trust.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n1ES\\,1215+303 \\citep[R.A. $= \\rm 12^h17^m52.0819^s$, Dec. $= +30^o07^\\prime 00^{\\prime\\prime}635$, J2000;][]{2011AJ....142...89P}, also known by many other names including Ton\\,605, ON\\,325, B2\\,1215+30 and S3\\,1215+30, is a blazar detected in the very-high-energy (VHE; $\\gtrsim$100~GeV) gamma-ray band. \nBlazars, of which there are, at the time of writing, 72 known to emit VHE radiation\\footnote{\\url{http:\/\/tevcat.uchicago.edu}}, are the most numerous sources detected at these energies comprising approximately one third of the VHE sources. 1ES\\,1215+303 was first discovered at VHE by MAGIC \\citep[the Major Atmospheric Gamma Imaging Cherenkov;][]{Aleksic12} and has been monitored by the Very Energetic Radiation Imaging Telescope Array (VERITAS) at TeV energies since 2008.\n\nThe source exhibited one of the most luminous and large-amplitude flares $E \\gtrsim 90$~GeV ever detected from a VHE blazar measured by VERITAS, when, in 2014, the TeV flux reached 2.4 times the Crab Nebula flux with a variability timescale of $< 3.6$\\,h  \\citep{Abeysekara17B21215}. \nIn the high-energy (HE; $\\approx\\,$100\\,MeV\\,$-$\\,$\\approx\\,$500\\,GeV) gamma-ray band,  1ES\\,1215+303 has been detected by the {\\sl Fermi} Large Area Telescope (LAT), most recently as  4FGL\\,J1217.9+3007  \\citep{2019arXiv190210045T}. \nA high-flux state correlated with that detected in the VHE band was observed at these energies during the luminous and isolated gamma-ray flare of 2014 \\citep{Abeysekara17B21215}. \n\n1ES\\,1215+303 exhibits a double-humped spectral energy distribution (SED) typical of blazars, with the synchrotron peak between radio and X-ray energies and the high-energy peak at GeV\\,$-$\\,TeV energies. \nThe synchrotron peak frequency of 1ES\\,1215+303 has been measured to be $\\nu_{\\text{syn}}>10^{15}$ Hz which led to its classification as either an intermediate-frequency-peaked BL Lac\\footnote{In the classification scheme of \\cite{Padovani95}.} \\citep[IBL; $\\nu_{\\text{syn}}=10^{15.58}$ Hz;][]{Nieppola06} or a high-synchrotron-peaked BL Lac\\footnote{Classification based on the position of the synchrotron peak.} \\citep[HBL; $\\nu_{\\text{syn}}=10^{15.205}$ Hz;][]{3LACAckermann15}. \nThe redshift was measured to be $z = 0.13$ \\citep{Akiyama03}, which was confirmed recently with high signal-to-noise ratio optical spectroscopic data \\citep{Paiano17}, and from Ly$\\alpha$ emission line at $z = 0.1305 \\pm 0.0030$ \\citep{Furniss2019}.\\\\\n\n\\begin{deluxetable*}{l|ccccc}\n\\tablecaption{Overview of the dataset presented in this paper.\\label{tab:dataset}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Instrument} & \\colhead{Waveband} &  \\colhead{Energy} & \\colhead{Date}  & \\colhead{No. of}\\\\\n\\colhead{}           & \\colhead{}         &  \\colhead{range}  & \\colhead{range} & \\colhead{observations\\tablenotemark{a}}\n}\n\\startdata\nVERITAS              & VHE-gamma-ray            & $>$ 200\\,GeV                       & 2009 - 2017 & 87\\\\ \n{\\it{Fermi}}-LAT     & HE-gamma-ray             & 0.1 - 500\\,GeV                 & 2008 - 2017 & 1045\\tablenotemark{b}\\\\\n{\\it{Swift}}-XRT     & X-ray                    & 0.3 - 10\\,keV                  & 2009 - 2017 & 25 \\\\\n{\\it{Swift}}-UVOT    & UV-optical               & 170 - 650\\,nm\\tablenotemark{c} & 2009 - 2017 & 232 \\\\\nTuorla               & Optical                  & $R$-band                             & 2003 - 2017 & 424\\\\\nNOT                  & Optical\\tablenotemark{d} & $R$-band                                 & 2014 - 2017 & 49\\\\\nOVRO                 & Radio                    & 15\\,GHz                            & 2008 - 2017 & 475\\\\\nMets\\\"{a}hovi            & Radio                    & 37\\,GHz                            & 2002 - 2016 & 53\\\\\nVLBA (MOJAVE)        & Radio                    & 15.3\\,GHz                          & 2009 - 2016 & 10 \\\\\nVLBA                 & Radio                    & 22.2 \\& 43.1\\,GHz                    & 2014        & 2 \\\\\n\\enddata\n\\tablenotetext{a}{We list here the number of flux points shown in Figure~\\ref{fig:all-lightcurves} to give an indication of the sampling at each wavelength. For the VLBA observations, we just provide the number of images that were recorded.}\n\\tablenotetext{b}{Number of flux points in the 3-day binned light curve.}\n\\tablenotetext{c}{The UVOT data were taken with six different filters with central wavelengths of 544\\,nm (V filter), 439\\,nm (B filter), 345\\,nm (U filter), 251\\,nm (UVW1 filter), 217\\,nm (UVM2 filter) and 188\\,nm (UVW2 filter) \\citep{2005SSRv..120...95R}.}\n\\tablenotetext{d}{The NOT provided polarization measurements at optical wavelengths.}\n\\end{deluxetable*}\n\nIn this work, we investigate the broadband emission of 1ES\\,1215+303 using multiwavelength (MWL) observations (radio, infrared, optical, ultraviolet, X-ray and gamma-ray) covering the past decade, with a focus on the gamma-ray data. \nGiven that one luminous gamma-ray flare has already been detected, we were interested in exploring the long-term temporal behavior of the source using observations from the {\\sl Fermi}-LAT and VERITAS.\n\n\n\\section{Observations and Data Analysis} \\label{sec:analysis}\n\nAn overview of the observations analyzed for this paper and of the instruments that made them is provided in Table~\\ref{tab:dataset}.\n\n\n\\subsection{VHE Gamma-ray Data: VERITAS}\n\\label{sec:VERITAS}\n\nVERITAS is sensitive to gamma rays in the energy range between $\\approx\\,$85\\,GeV and $>$30\\,TeV \\citep{Park15}. \nIt has a field-of-view (FoV) of $\\approx\\,$3.5$^\\circ$. This makes it possible to observe simultaneously sources with small angular separation such as 1ES\\,1215+303 and 1ES\\,1218+304 (the angular distance between the two is $\\approx\\,$0.76$^\\circ$). They have been monitored regularly since 2008 December. \nThese observations were taken in ``wobble mode''~\\citep{Fomin94} with the source (either 1ES\\,1215+303 or 1ES\\,1218+304) offset by 0.5$^{\\circ}$ from the center of the FoV. \nThe total exposure with 1ES\\,1215+303 in the FoV between 2008 December and 2017 May (after quality selection, before dead-time correction, without accounting for the difference in sensitivity between observations on the two sources) amounts to 175.8\\,h. \nThe VERITAS results on this source between 2008 December and 2012 May were reported in \\cite{Aliu13}, and those between 2013 January and 2014 May, including an extremely luminous flare, in \\cite{Abeysekara17B21215}. \n\nThe VERITAS data were analyzed using two independent packages \\citep{Cogan08, Maier17}, and consistent results were obtained. \nCuts on air shower image parameters optimized for each analysis package for a point source of 2\\% to 10\\% of the Crab Nebula flux with a power-law photon index between 2.5 and 3.0 \\citep{Park15} were used. \n\n\\begin{deluxetable}{cccc}[ht!]\n\\tablecolumns{4}\n\\tabletypesize{\\scriptsize}\n\\setlength\\tabcolsep{1pt}\n\\tablecaption{\\footnotesize VERITAS observations of 1ES\\,1215+303 from 2008 December to 2017 May. The VERITAS observing season runs from the end of the monsoon season ($\\approx\\,$September) until the start of the monsoon season the following year ($\\approx\\,$July) and is divided into periods called ``darkruns'' that are centered on the new moon.\\label{tab:Vobs}}\n\\tablehead{\n\\colhead{Epoch} & \\colhead{Exposure} & \\colhead{Flux $>$200 GeV}& \\colhead{Photon Index} \\\\ \n\\colhead{} & \\colhead{(hr)}  & \\colhead{($\\text{cm}^{-2} \\; \\text{s}^{-1}$)} & \n} \n\\startdata\n2008-2009\t&\t33.8\t&\t$<4.5\\times 10^{-12}$ \t\t&\t-\t\\\\\n2010-2011\t&\t41.9\t&\t$(8.0\\pm0.9)\\times 10^{-12}$ &\t$3.6\\pm0.4$\t\\\\\n2011-2012\t&\t17.5\t&\t$(2.8\\pm1.1)\\times 10^{-12}$ &\t-\t\\\\\n\\hline\n2012-2013 non-flare\t&\t10.8\t&\t$(6.0\\pm1.2)\\times 10^{-12}$ &\t$3.9\\pm0.6$ \\\\\n2013 Feb 07 (2)\t&\t0.5\t\t&\t$(5.1\\pm1.0)\\times 10^{-11}$ &\t$3.7\\pm0.7$ \\\\\n\\hline\n2013-2014 non-flare\t$^\\dagger$ &\t7.4\t&\t$<7.2\\times 10^{-12}$ &\t- \\\\ \n2014 Feb 08 (3)\t&\t0.9\t\t&\t$(5.0\\pm0.1)\\times 10^{-10}$ &\t$3.1\\pm0.1$ \\\\\n\\hline\n2014-2015 non-flare\t&\t14.4\t&\t$(4.2\\pm0.8)\\times 10^{-12}$ &\t$2.8\\pm0.4$ \\\\\n2015 Jan 17 (4)\t&\t0.9\t\t&\t$(5.3\\pm0.5)\\times 10^{-11}$ &\t$3.0\\pm0.2$ \\\\\n\\hline\n2015-2016 non-flare\t&\t22\t&\t$(1.3\\pm0.1)\\times 10^{-11}$ &\t$3.3\\pm0.1$ \\\\\n2016 Apr 09\t(5)&\t0.9\t\t&\t$(3.7\\pm0.5)\\times 10^{-11}$ &\t$3.1\\pm0.3$ \\\\\n\\hline\n2016-2017 non-flare\t&\t24.6\t&\t$(8.0\\pm0.8)\\times 10^{-12}$ &\t$3.9\\pm0.3$ \\\\\n2017 Mar 05 (6)\t&\t0.9\t\t&\t$(5.9\\pm0.9)\\times 10^{-11}$ &\t$2.5\\pm0.4$ \\\\\n2017 Apr 01 (7) &\t2.5\t\t&\t$(9.5\\pm0.6)\\times 10^{-11}$ &\t$3.6\\pm0.1$ \\\\\n\\enddata\n\\tablecomments{The enumeration in parenthesis after the date of a flare corresponds to the flare ID. We refer to Sections \\ref{sec:increasing-flux} and \\ref{sec:seds} for details on the flare ID, the simultaneity of observations with the {\\sl Fermi}-LAT and HE enhanced activity. \\\\\n$^\\dagger$ We reanalyzed the 2013 -- 2014 season non-flare data and report the upper limit of those observations. }\n\\end{deluxetable}\n\nWe found that a power-law model $dN\/dE = N_0 \\left(E\/E_0 \\right)^{-\\Gamma}$ provides a good fit to the VERITAS spectra\n, where $dN\/dE$ is the differential photon flux, $N_0$ is the flux normalization at energy $E_0$, $\\Gamma$ is the photon index, and $E$ is the photon energy. \n\nThe VHE gamma-ray fluxes and best-fit photon indices for 1ES\\,1215+303 for different epochs are shown in Table~\\ref{tab:Vobs}. \nIn most cases, no significant difference was found between the photon index measured during flares and that averaged over the quiescent part of the corresponding season, with the exception of the hard spectrum VERITAS flare on 2017 March 05. \n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=\\textwidth,angle=0]{mwl-LC_v9.png}\n\\caption{The light curves for the various wave\n  bands are shown in descending order of energy from the top to the bottom of the plot. A zoom is provided on the VERITAS data excluding Flare 3. For the XRT panel, the data taken in window-timing (WT) and photon-counting (PC) mode are plotted. For the radio panel, the 37~GHz data with signal to noise ratio S\/N$<4$ are shown in gray. \n  \\label{fig:all-lightcurves}}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{HE Gamma-ray Data: {\\it{Fermi}} Gamma-ray Space Telescope $-$ LAT} \\label{sec:fermi}\n\nThe Large Area Telescope, LAT, on board the {\\it Fermi} Gamma-ray Space Telescope, covers the energy range from $\\approx\\,$20\\,MeV to more than 500\\,GeV \\citep{Atwood09}. The main observation mode of the {\\it Fermi}-LAT is survey mode during which the LAT scans the entire sky every 3 hours.\nWe analyzed the {\\it Fermi}-LAT data from 2008 August 04 (MJD 54682.7), the start of the all-sky survey, up until 2017 September 04 (MJD 58001.0). \nThe data were analyzed using the {\\texttt{Fermi Science Tools}}\\footnote{{\\texttt{Version v10r0p5}; Instrument response functions {\\texttt{P8R2\\_SOURCE\\_V6}}; the \"source\" class events were used.}}. \nWe restricted the photon selection to those with energies between 100~MeV and 500~GeV that had zenith angle of less than 90$^{\\circ}$ in order to reduce contributions from the Earth's limb. \nThey consisted of photons in a circle of radius 10$^{\\circ}$ centered on the position of 1ES\\,1215+303, the region of interest (ROI). \nThe data were modeled using the unbinned maximum likelihood fit method implemented in the {\\it{Fermi}} Science Tools, \\texttt{gtlike}. \nAll of the sources from the third {\\it Fermi}-LAT source catalog, 3FGL \\citep{Acero15}, that lay within a radius of 20$^{\\circ}$ of 1ES\\,1215+303 were included in the background model to ensure that each source that could contribute photons to the ROI was modeled\\footnote{The point spread function (PSF) of the {\\it Fermi}-LAT is approximately 10$^{\\circ}$ at 100\\,MeV at the $95\\%$ containment.}.\n\n\\begin{deluxetable}{ccccc}[ht!]\n\\tablecolumns{4}\n\\setlength\\tabcolsep{1pt}\n\\tablecaption{\\footnotesize {\\it Fermi}-LAT flux and spectral shape of 1ES\\,1215+303 from 2008 August 04, the start of {\\it{Fermi}}-LAT science operations to 2017 September 04, the end of the period covered in this paper. The significance, flux and photon index are provided for the various different epochs listed in the table including each year (360-day bin), the flares (see Section \\ref{sec:increasing-flux} to see how the flaring periods were defined) and for the yearly data excluding the flaring period(s). \\label{tab:LATobs}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Epoch} & \\colhead{State} & Sig. &\\colhead{Flux$_{>\\text{0.1\\,GeV}}$}& \\colhead{$\\Gamma$} \\\\ \n\\colhead{} &  & \\colhead{$\\sigma$} & \\colhead{$10^{-8}\\text{cm}^{-2} \\text{s}^{-1}$} & \n} \n\\startdata\n2008 Nov 17 - 2010 Aug 12 &     &  & & \\\\\n\\&                      & Low & 49.0 & $4.3\\pm 0.3$ & $1.98\\pm 0.03$ \\\\\n2011 Apr 15 - 2012 Apr 09 &  &  &  & \\\\\n2008 Aug 04 - 2009 Jul 30\t& Total & 38.4 &\t$ 5.3\\pm 0.4 $\t&\t$1.94\\pm 0.04$\t\\\\\n2008 Aug 04 - 2009 Jul 30\t& Non-flare & 31.6 &$ 4.3\\pm 0.4 $\t&\t$1.94\\pm 0.04$\t\\\\\n\\hline\n2008 Oct 04 - 2008 Oct 17\t& Flare\\,1 & 26.4 &\t$35.0\\pm 3.5$ &\t$1.92\\pm 0.06$ \\\\\n2009 Jul 30 - 2010 Jul 25\t& Total & 29.3 &\t$4.6\\pm 0.5$ &\t$2.01\\pm 0.05$ \\\\\n2010 Jul 25 - 2011 Jul 20\t& Total & 40.9 &\t$7.2\\pm 0.5$ &\t$1.97\\pm 0.04$ \\\\\n2011 Jul 20 - 2012 Jul 14\t& Total & 32.8 &\t$5.4\\pm 0.5$ &\t$2.00\\pm 0.04$ \\\\\n2012 Jul 14 - 2013 Jul 09\t& Total & 47.0 &\t$7.5\\pm 0.5$ &\t$1.92\\pm 0.03$ \\\\\n2013 Jul 09 - 2014 Jul 04\t& Total & 54.0 &\t$10.1\\pm 0.6$ &  $1.94\\pm 0.03$ \\\\\n2013 Jul 09 - 2014 Jul 04\t& Non-flare & 50.4 &\t$10.0\\pm 0.6$ &  $1.95\\pm 0.03$ \\\\\n2014 Jul 04 - 2015 Jun 29\t& Total & 50.4 &\t$8.7\\pm 0.5$ &\t$1.91\\pm 0.03$ \\\\\n2015 Jun 29 - 2016 Jun 23\t& Total & 54.7 &\t$9.1\\pm 0.5$ &\t$1.90\\pm 0.03$ \\\\\n2016 Jun 23 - 2017 Jun 18\t& Total & 70.1 &\t$12.0\\pm 0.5$ &\t$1.86\\pm 0.02$ \\\\\n2016 Jun 23 - 2017 Jun 18\t& Non-flare & 63.7 &\t$11.2\\pm 0.5$ &\t$1.88\\pm 0.02$ \\\\\n2017 Mar 25 - 2017 Apr 05\t& Flare\\,7 & 25.9 &\t$25.2\\pm 2.8$ &\t$1.74\\pm 0.06$ \\\\\n2017 Apr 09 - 2017 Apr 16\t& Flare\\,8 & 18.9 &\t$28.4\\pm 4.0$ &\t$1.83\\pm 0.08$ \\\\\n2017 Apr 15 - 2017 Apr 23 & Post-flare & 8.6 & $9.5\\pm 2.3$ & $1.89\\pm 0.34$ \\\\\n\\enddata\n\\tablecomments{Sig. stands for significance, while $\\Gamma$ represents the power-law photon  index. \nWe refer to Sections \\ref{sec:increasing-flux} and \\ref{sec:seds} for details on the flare ID, and the simultaneity of observations with VERITAS and GeV enhanced activity.}\n\\end{deluxetable}\n\nTable \\ref{tab:LATobs} shows the best-fit values for the  power-law spectral shape parameter and for the flux obtained for the different epochs, flaring, low state, post-flare, 360-day binned (approximately yearly), and 360-day binned outside flares (non-flare) results. The low state and post-flare states were defined using the Bayesian blocks method as described in Section \\ref{sec:increasing-flux}. \n\nSystematic uncertainties were not included in the reported LAT data. They are estimated to be up to 10\\%, based on the systematic uncertainties on the effective area and on the PSF\\footnote{\\url{https:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/LAT_caveats_p8r2.html}}.\n\n\n\\subsection{X-ray Data: Neil Gehrels {\\it{Swift}} Observatory $-$ XRT} \\label{sec:xrt}\n\nThe X-Ray Telescope \\citep[XRT;][]{2000SPIE.4140...64B} on the \\textit{Neil Gehrels Swift Observatory} \nis sensitive to photons with energies between 0.2 and 10\\,keV \\citep{Gehrels04, Burrows05}. There were 25 pointed \\textit{Swift}-XRT observations within a $10^\\prime$ radius of 1ES\\,1215+303, 20 of which were taken in photon counting mode, and five in windowed timing mode. Only five observations were taken after 2013, one on 2014 February 9 (MJD\\,56697) and four between 2017 April 15 (MJD\\,57858) and 2017 April 23 (MJD\\,57866), all of which were triggered by elevated VHE gamma-ray fluxes detected by VERITAS. \nThe XRT data were initially processed using \\texttt{xrtpipeline}\\footnote{\\texttt{HEASOFT v6.23}, \\texttt{swxrtdas\\_23Jan18\\_v3.4.1} with calibrations from database \\texttt{CALDB 20171113}.}. For subsequent spectral and temporal analysis, we used a circular source region of a radius of 20 pixels ($\\approx\\,47.2''$) and an annular background region with inner and outer radii of 70 and 120 pixels ($\\approx\\,2.75^{\\prime}$--$4.72^{\\prime}$), respectively, both centered on 1ES\\,1215+303. We checked the count rate in the source region for each observation, and confirmed that the pile-up effect is negligible.\n\nThe X-ray spectrum was fit with an absorbed power-law model (\\texttt{wabs*powerlaw}): \n\\begin{equation}\n\\label{eqXspec}\n\\frac{dN}{dE} = e^{-N_H\\sigma(E)} K \\left( \\frac{E}{1\\;\\text{keV}} \\right)^{-\\Gamma}, \n\\end{equation}\nwhere the column density of neutral hydrogen $N_H$ and the photoelectric cross-section $\\sigma(E)$ describe the absorption component, and the normalization $K$ and photon index $\\Gamma$ describe the power-law component. \nWe fixed the column density of neutral hydrogen to $N_H = 1.74  \\times10^{20} \\;\\text{cm}^{-2}$ taken from the Leiden\/Argentine\/Bonn (LAB) survey of Galactic HI \\citep{Kalberla05}. \nThe best-fit photon index, the energy flux between 0.3\\,keV and 10\\,keV, and the goodness of the fit for each observation is shown in Table~\\ref{tab:Xspec} in Appendix~\\ref{app:xrt}. \n\n\n\\subsection{Ultraviolet Data: Neil Gehrels {\\it{Swift}} Observatory $-$ UVOT}\n\\label{sec:UVOT}\n\nThe Ultraviolet\/Optical telescope \\citep[UVOT;][]{2005SSRv..120...95R} on the \\textit{Neil Gehrels Swift Observatory} made many observations of 1ES\\,1215+303 during the time period under study in this paper.\nSpecifically, 232 images containing 1ES\\,1215+303 in the field of view were available (31 with the V filter; 36 with the B filter; 40 with the U filter; 46 with the UVW1 filter; 42 with the UVM2 filter; 37 with the UVW2 filter) and they span the date range from 2009 December 03 (MJD 55168) to 2017 April 23 (MJD 57866). Since UVOT is co-aligned with the XRT, the temporal sampling of the observations from the two instruments is the same.\nThe counts from the source were extracted from a $5.0''$ (radius) aperture around the position of 1ES\\,1215+303.\nThe background counts were estimated using the counts from four neighboring dark-sky regions, each having the same radius as the source region. The magnitude was then computed using the \\texttt{uvotsource}\\footnote{\\texttt{HEASOFT v6.21}, \\texttt{Swift$\\_$Rel4.5(Bld34)$\\_$27Jul2015} with calibrations from \\citet{2011AIPC.1358..373B}.} tool.\nThe counts were first corrected for extinction following the procedure and using the $R_\\textrm{v}[\\equiv A(V)\/E(B-V)]$ value of \\citet{2009ApJ...690..163R}.\nThey were then converted to fluxes using the zero-point values for each of the UVOT filters from \\citet{2008MNRAS.383..627P}. We used the values\\footnote{The wavelength dependent coefficients $a$ and $b$ are defined according to $A_{\\lambda}=E(B-V)[aR_\\textrm{v}+b]$.} of $a$ and $b$ from \\citet{2009ApJ...690..163R}, who computed them following the procedure of \\citet{1989ApJ...345..245C}.\nA value of 0.021 was used for $E(B-V)$ \\citep{2011ApJ...737..103S}; this was accessed through the NASA\/IPAC Extragalactic Database\\footnote{\\url{http:\/\/nedwww.ipac.caltech.edu}}.\n\n\n\\subsection{Optical Data} \\label{sec:opt}\n\n1ES~1215+303 was monitored in the $R$-band at the Tuorla Observatory over the past 15 years as part of the Tuorla blazar monitoring program \\citep{Takalo08, Nilsson2018}. \nWe show the long-term $R$-band flux density in Figure~\\ref{fig:all-lightcurves}. \\\\\n\nThe source was monitored with the Nordic Optical Telescope (NOT). The ALFOSC instrument is used in the standard setup for linear polarization observations ($\\lambda\/2$ retarder followed by a calcite). The observations were performed in the $R$-band from 2014 to 2017 two to four times per month. The data were analyzed as in \\cite{Hovatta2016} with a semi-automatic pipeline using standard aperture photometry and comparison stars procedures.\n\n\\subsection{Radio Data: VLBA} \\label{sec:vlba}\n\n1ES\\,1215+303 was observed with the Long Baseline Observatory's Very Long Baseline Array (VLBA) at 22.2 and 43.1\\,GHz on 2014 November 11\\footnote{The results of these observations are publicly available at \\url{http:\/\/whittierblazars.com\/}} (observation code S7017E3). Approximately two hours of on-source integration time was recorded at each frequency, over a total time span of about seven hours. \nAll observations used a 2\\,Gbps recording rate in a dual-polarization configuration of eight 32\\,MHz channels at matching frequencies in each polarization.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth,angle=0]{vlba22_2.png}\n   \\put(-90,85){2}\n   \\put(-80,102){3}\n   \\put(-66,127){4}\\hspace{20pt}\n\\includegraphics[width=.45\\textwidth,angle=0]{vlba43_2.png}\n   \\put(-120,66){2}\n   \\put(-92,86){3}\n   \\put(-62,139){4}\n\\caption{{\\it Left}: VLBA image at 22.2 GHz. Contours show total intensity, with the lowest contour at 0.129 mJy\\,beam$^{-1}$, and subsequent contours factors of two higher. The peak flux density is 229 mJy\\,beam$^{-1}$. The naturally-weighted beam size is 0.914 by 0.358\\,mas at a position angle of the major axis of $-17.4^{\\circ}$. Sticks show the magnitude of the linearly-polarized flux density (with a scale of 0.1 mas\\,mJy$^{-1}$) and the direction of the EVPA.\nThe color scale indicates fractional polarization. {\\it Right}: VLBA image at 43.1\\,GHz. The lowest contour is 0.308 mJy\\,beam$^{-1}$; the peak flux density is 152 mJy\\,beam$^{-1}$. The naturally-weighted beam size\nis 0.432 by 0.241\\,mas at $1.9^{\\circ}$. The polarized flux density scale of the sticks is 0.05 mas\\,mJy$^{-1}$. The centers of the Gaussian jet components are shown as filled diamonds. The beams are shown in the bottom left-hand corner of each panel as a plus ``$+$''. \\label{fig:vlba22_43}}\n\\end{center}\n\\end{figure*}\n\nWe used the \\texttt{AIPS} software package \\citep{Greisen03} for calibration and fringe-fitting of the correlated visibilities.\nCalibration of the polarization response of the feeds (D-terms) was done through observations of standard calibrator sources.\nCalibration  of  the  electric  vector position angle (EVPA) was done by comparison of  calibrator  sources  to  images  in  the  VLBA Boston monitoring  program, BU-BLAZAR\\footnote{\\url{https:\/\/www.bu.edu\/blazars\/VLBAproject.html}}, \\citep{Jorstad16} or the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments \\citep[MOJAVE;][]{Lister19} databases\\footnote{\\url{http:\/\/physics.purdue.edu\/astro\/MOJAVE\/sourcepages\/1215+303.shtml}}.\nImages were produced using \\textit{clean} and \\textit{self-calibration} in the \\texttt{DIFMAP} software package \\citep{Shepherd97}. All antennas were used for the 43.1~GHz image, and all except Saint Croix were used for the 22.2\\,GHz image.\n\nThe 22.2\\,GHz and 43.1\\,GHz VLBA images are shown in Figure~\\ref{fig:vlba22_43}. \nBoth images exhibit fractional polarization increasing down the jet, relative to the core.\nCircular Gaussian models were fit to the visibilities using the \\texttt{modelfit} routine in \\texttt{DIFMAP}.\nIn addition to the core, three jet components were detected at 22.2\\,GHz, and four jet components were detected at 43.1\\,GHz (with an additional component appearing between the innermost 22.2\\,GHz component and the core). The centers of the Gaussian jet components are shown by filled diamonds on the VLBA images. The parameters of the Gaussian model components are tabulated in Table~\\ref{tab:vlba}.\n\n\n\\begin{deluxetable}{cccccc}[ht!]\n\\tabletypesize{\\scriptsize}\n\\tablecaption{VLBA 43.1, 22.2, and 15.3\\,GHz Gaussian model\ncomponents. \\label{tab:vlba}}\n\\tablehead{\n \\colhead{Flux (Jy)} &  \\colhead{$r$ (mas)} &  \\colhead{P.A. ($^\\circ$)} &  \\colhead{$a^\\dagger$ (mas)}  &  \\colhead{Freq (GHz)} &  I.D. \\\\\n \\colhead{(1)} &  \\colhead{(2)} &  \\colhead{(3)} &  \\colhead{(4)} &  \\colhead{(5)} &  \\colhead{(6)}   \n }\n\\startdata\n     0.127 &        0.03 &        $-16.1$ &       0.04 &   43.1 &   0 \\\\\n     0.044 &         0.13 &          155 &       0.1 &   43.1 &   - \\\\\n     0.014 &         0.47 &          155 &        0.2 &   43.1 &   4 \\\\\n     0.003 &          1.04 &          153 &        0.30 &   43.1 &   3 \\\\\n     0.003 &          1.62 &          147 &        0.39 &   43.1 &   2 \\\\\n\\midrule\n     0.207 &        0.04 &        $-24.4$ &       0.02 &   22.2 &   0 \\\\\n     0.038 &         0.37 &          155 &        0.11 &   22.2 &   4 \\\\\n     0.008 &           1.1 &          151 &        0.30 &   22.2 &   3 \\\\\n     0.004 &          1.72 &          145 &        0.25 &   22.2 &   2 \\\\\n\\midrule\n0.265\t& 0.03 &\t323.1 &\t0.03  & 15.3$^\\ddagger$ & 0 \\\\\n0.033\t& 0.47 &\t152.5 &\t0.12 & 15.3 & 4 \\\\\n0.011\t& 1.06 &\t150.3 &\t0.2 & 15.3 & 3 \\\\\n0.009\t& 1.67 &\t145.6 &\t0.34 & 15.3 & 2 \\\\\n0.013\t& 16.20 &\t143.5 &\t4.41 & 15.3 & 1 \n\\enddata\n\\tablecomments{Columns: (1) flux density of the component, \n\t\t\t(2) and (3) the distance ($r$) and the position angle (P.A.) of the center of the component relative to the origin of the image, \n            (4) the full width at half maximum (FWHM) of the circular Gaussian component, \n            (5) measurement frequency, (6) Identification number of features from (or consistent with) \\citet{Lister19}. \\\\\n$^\\dagger$ The standard deviations of the best-fit Gaussian components are approximately 20\\% of the FWHM beam dimensions.\\\\\n$^\\ddagger$ The 15.3~GHz data correspond to fits using all data from the 10 epochs observed between 2009 and 2016.}\n\\end{deluxetable}\n\n1ES~1215+303 was also observed at 15.3 GHz with the MOJAVE program for 10 epochs between 2009 and 2016. \n\nEmission features derived from a Gaussian model fit to the interferometric visibility data have been identified in the VLBA images at 15.3~GHz. The separations between these emission features and the core at the time of each epoch of observation are shown in the right panel of Figure~\\ref{fig:mojave}, revealing three innermost emission features (components), referenced as 2, 3, and 4. Stationary features are typical in TeV HBLs, being present in the majority of these sources \\citep{Kharb08, Hervet16, Piner18, Lico12}.\nThe mean and standard deviation of the angular separation between the three quasi-stationary components and the core over all epochs are $0.44\\pm0.07$ mas, $1.04\\pm0.09$ mas, and $1.64\\pm0.06$ mas, as shown in Table~\\ref{tab:vlba}. \nThese three stationary components are also resolved in the 22.2 and 43.1-GHz images, and the positions of these three Gaussian components are consistent between the three frequencies. \nThe fourth component observed at 15.3\\,GHz is at a much larger distance from the origin of the images, in a position consistent with a very-long-baseline interferometry  (VLBI) stationary component found at 1.6 and 5\\,GHz \\citep{Giroletti06}.\\\\\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=.4\\textwidth]{1215+303_u_stacked_icc.pdf}\n   \\put(-90,50){1}\n   \\put(-38,145){2}\n   \\put(-31,151){3}\n   \\put(-27,160){4}\n\\includegraphics[width=.555\\textwidth]{1215+303_sepvstime.pdf}\n\\caption{{\\it Left}: The stacked MOJAVE image with the five best-fit Gaussian components from the last epoch on 2016 June 9 overlaid. The standard deviations of the best-fit Gaussian components are approximately 20\\% of the FWHM beam dimensions. The contours show the total intensity, starting at a baseline of 0.2 mJy beam$^{-1}$, and incrementing by factors of $\\sqrt{2}$. Eleven images are stacked here including one from 27 December 1999 which is not shown on the plot on the right. The same circular restoring beam was used for all eleven images. It is shown at the half power level in the bottom left corner as a plus ``$+$''.\n{\\it Right}: The separation between components and the core at the time of each epoch of observation. The innermost three components (designated with number 2, 3, and 4) are quasi-stationary. Robust features which are cross-identified between more than 4 epochs are fitted assuming linear motion.\n\\label{fig:mojave}}\n\\end{center}\n\\end{figure*}\n\nThe components 2, 3, and 4 show subluminal inward apparent speeds respectively of $0.170 \\pm 0.036 \\;c$, $0.246 \\pm 0.055 \\;c$, and $0.194 \\pm 0.040 \\;c$ estimated by MOJAVE. The fact that they have similar inward motions, all consistent with an inward speed of $0.2 \\,c$, suggests that they are due to a downstream shift of the radio core. Indeed, if the three features are stationary shocks, a core shift predicts a similar inward motion for all of them.\nSuch a shift of the radio core can be explained by a slow increase of the jet power over years, which would increase the distance from the supermassive black hole (SMBH) where the jet becomes optically thin in radio. Such a slow power increase is supported by the multi-year increase of the gamma-ray and optical luminosities reported in Section \\ref{sec:temporal}. Similar inward motions have been detected in other BL Lac sources by MOJAVE such as UGC 00773, 3C 66A, and Mrk 421 \\citep{Lister19}.\n\nSince the emission features are quasi-stationary, we show a stacked image of the 15.3 GHz intensity in the left panel of Figure~\\ref{fig:mojave}. The five best-fit Gaussian components from the last epoch on 2016 Jun 9 are shown as red circles. \n\n\n\\subsection{Radio Data: Owens Valley Radio Observatory} \\label{sec:ovro}\n\nWe show the radio flux density measured by the Owens Valley Radio Observatory (OVRO) at 15~GHz over the past decade (2008-2017) in Figure~\\ref{fig:all-lightcurves}, where a total of 475 data points are presented. \nThe procedure of the OVRO data reduction and calibration procedures can be found in \\citet{Richards11}. \n \n \n\\subsection{Radio Data: Mets\\\"{a}hovi} \\label{sec:metsahovi}\n\nWe also show the radio flux density measured by Mets\\\"ahovi Radio Observatory (MRO) at 37\\,GHz in Figure \\ref{fig:all-lightcurves}. The duration of the MRO data are longer than those from OVRO, but the sampling is generally more sparse. \nThe MRO data reduction and analysis procedure can be found in \\citet{Terasranta98}.\nThe radio data were also used in the SED modeling in Section \\ref{sec:modeling}, providing constraints on the less variable jet component. \n\n\n\\section{Temporal studies} \\label{sec:temporal}\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=\\textwidth,angle=0]{Opt-GeV-TeV_LCs_v4.png}\n\\caption{The GeV-TeV full gamma-ray dataset. {\\it Top}: The VERITAS light curve (above 200\\,GeV and excluding Flare 3), with detailed zoom in the VHE flares down to the sub-hour timescales, from year 2015. \\textsf{T}$_{\\rm{0}}$ is in MJD. Upper limits in gray. {\\it Middle}: {\\it Fermi}-LAT 3-day light curve with daily zoom in the Flares 1, 7 and 8 (see text for details). Data points deviating $\\geq 3\\,\\sigma$ from the broken linear function (BLF, dashed line) are shown in black. From these, only the points with two neighbors were used to define the four LAT flares. Bayesian blocks are shown in blue. These were used to define the low state of this source. {\\it Bottom}: Tuorla light curve in gray with seasonal average in black. The last nine years are contemporaneous to the time range in the upper panels. \\label{fig:gev-tev-lcs}}\n\\end{center}\n\\end{figure*}\n\nIn this section we describe various analyses that allow us to exploit the temporal richness of our dataset. \n\n\n\\subsection{The gamma-ray dataset}\\label{sec:gamma}\n\nWe show the nightly VERITAS light curve integrated above 200\\,GeV in the top panel of Figure~\\ref{fig:all-lightcurves}. \nFlux values and their $1\\,\\sigma$ statistical uncertainties are shown only if the data result in a significance value of at least $2\\,\\sigma$, otherwise 95\\% flux upper limits are shown. For the estimation of the integral flux points shown in the LAT light curve, each three-day dataset was subjected to the full likelihood analysis with the spectral parameters of all other sources in the ROI being frozen to those found in the global power-law likelihood analysis. For these short three-day exposures we found no preference for a curved spectral model so the 1ES\\,1215+303 data were modeled as a power law.\n\nAs is discussed in detail in Sections \\ref{sec:increasing-flux} and \\ref{sec:seds}, 1ES 1215+303 flared a number of times at gamma-ray energies during the past decade, labeled Flares from 1 to 8. The names are assigned in chronological order to the gamma-ray flares, independently of whether they occur at HE or VHE. VERITAS gamma-ray flares were observed on six nights, Flares 2 to 7 (Table~\\ref{tab:Vobs}). Two of these were found to have a counterpart at GeV energies, Flares 3 and 7. \nFlares 2 and 3 had a dedicated study reported in \\cite{Abeysekara17B21215} while Flare 1 was analyzed along with 105 sources in \\citet{Abdo10LC} and was not, therefore, subjected to a detailed, individual analysis. \nWe focus on the unpublished observations and, in particular, on Flare 7 that occurred on 2017 April 01 since this is the only unpublished flare with simultaneous LAT-VERITAS data.\n\nNo strong intra-night variability on sub-hour timescales was observed in the light curves with 8-min binning intervals, as can be seen in the insets on the top panel of Figure~\\ref{fig:gev-tev-lcs}.\n\n\n\\subsubsection{Increasing flux trend and definition of flares}\n\\label{sec:increasing-flux}\n\nThe second panel of Figure~\\ref{fig:gev-tev-lcs} shows the {\\it Fermi}-LAT 3-day binned light curve between 2008 August and 2017 September. Flux data and uncertainties are shown when a significance of at least $2\\,\\sigma$ was reached, otherwise 95\\% upper limits are shown. In the following, low flux values were used instead of upper limits for the variability analyses.\n\n\\begin{deluxetable}{l|cccc}[ht!]\n\\tablecaption{ Results of the fit of the {\\it Fermi}-LAT 3-day light curve and Tuorla averaged data.\\label{tab:lat-tuorla}}\n\\tabletypesize{\\tiny}\n\\tablehead{\\noalign{\\vskip4pt}\n\\colhead{Model} & \\colhead{$a$} & \\colhead{$b$} & \\colhead{$t_{\\rm break}$} & \\colhead{$\\chi^2$\/d.o.f.}\\\\\n & \\colhead{($\\text{cm}^{-2} \\, \\text{s}^{-1}$\\,MJD$^{-1}$)} & \\colhead{($\\text{cm}^{-2} \\, \\text{s}^{-1}$)} & \\colhead{(MJD)} & \\colhead{}\n}\n\\startdata\n\\multicolumn{5}{c}{{\\sl Fermi}-LAT}\\\\ \\hline\nConst.  &           NA                    & (5.57$\\pm$0.14)$\\,10^{-8}$ & NA        & 2361\/1043 \\\\\nLinear & (1.92$\\pm$0.14)$\\,10^{-11}$ & $-(1.02\\pm0.08)\\,10^{-6}$ & NA         & 1984.7\/1042 \\\\\nBLF & (2.75 $\\pm$0.27)$\\,10^{-11}$ & (4.00$\\pm$0.20)$\\,10^{-8}$ & 55834$\\pm$134 & 1954.1\/1041 \\\\ \\hline\n\\multicolumn{5}{c}{Tuorla $R$-band}\\\\ \\hline\nConst.  &           NA                   & (2.92$\\pm$0.25)$\\,10^{-3}$ & NA & 102.8\/13 \\\\\nLinear & (5.46$\\pm$1.10)$\\,10^{-7}$ & $-(2.67 \\pm 0.60)\\,10^{-2}$ & NA & 35.4\/12 \\\\\nBLF & (1.73$\\pm$0.44)$\\,10^{-6}$ & (2.58$\\pm$0.15)$\\,10^{-3}$ & 55515$\\pm$297 & 24.0\/11 \n\\enddata\n\\tablecomments{For a linear function $ax+b$, $a$ is the slope and $b$ is the independent term. For a constant function $a$ is not applicable (NA). For the BLF, $a$ is the slope of the linearly increasing section, and $b$ is the value in the constant section.}\n\\end{deluxetable}\n\nThe LAT light curve comprises apparent flaring epochs on top of what looks like a variable baseline flux, which itself is not completely flat.\nIn order to characterize this baseline, we first fit the light curve to a constant flat line ($\\chi^2_{\\rm{red}}=2.26$)\\footnote{The reduced $\\chi^2$ is defined by $\\chi^2_{\\rm{red}}\\equiv \\chi^2$\/d.o.f.}, to a linear function ($\\chi^2_{\\rm{red}}=1.90$) and to a broken linear function (BLF, $\\chi^2_{\\rm{red}}=1.88$).\nA likelihood ratio test shows that the increasing linear function is preferred at the $19.4\\,\\sigma$ level to the constant fit and that the broken linear function (black dashed line in the second panel of Figure \\ref{fig:gev-tev-lcs}) is preferred at the $5.5\\,\\sigma$ level over the increasing linear function. \nThis broken line is composed of first a constant part given by $(4.0\\pm0.2)\\times 10^{-8}\\;\\text{cm}^{-2} \\; \\text{s}^{-1}$, consistent with the Bayesian blocks results (described below); and a linear function of slope $(2.8\\pm0.3)\\times 10^{-11}\\;\\text{cm}^{-2} \\; \\text{s}^{-1}$\\,MJD$^{-1}$ which starts at the break point of MJD $55834\\pm134$ (around September 2011). \\\\\n\nA similar analysis was performed for the Tuorla $R$-band data averaged per season (black squares in the third panel of Figure \\ref{fig:gev-tev-lcs}). It is found that a linear function is preferred at the $8.2\\,\\sigma$ level over a constant function, and that the broken linear function (in the same panel of Figure \\ref{fig:gev-tev-lcs}) ($\\chi^2_{\\rm{red}}=2.2$) is preferred at the $3.4\\,\\sigma$ level over the linear function. \nThe break point found for the Tuorla data is MJD 55515$\\pm$297 (around November 2010), i.e., consistent with the LAT break time. See Table \\ref{tab:lat-tuorla} for details on the results for both datasets. \n\\cite{Lindfors2016} searched for long-term variability trends in Tuorla and 15\\,GHz radio lightcurves from 2008 to 2013. No significant trend was found in radio or optical during this time period. This is not incompatible with our analysis, where the long term flux increase starts around the end of 2010 and become especially visible after 2013. The same study, however, reported to have found a decreasing or increasing trend for a number of other sources in the radio and optical bands.\n\nIn order to identify the LAT flaring epochs we performed a recursive fit on the data that deviated by no more than $3\\,\\sigma$ from the broken linear function (first method). \nThis improved the fit ($\\chi^2_{\\rm{red}}=1.3$) and the results were consistent with those obtained before the $\\geq 3\\,\\sigma$ points were excluded. \nThe points that deviate by $\\geq 3\\,\\sigma$ from this broken line are shown in black in Figure \\ref{fig:gev-tev-lcs}. \nOut of these, only those with at least two neighboring flaring points, also above $3\\,\\sigma$, were used to define a LAT flare \\citep{2015ApJ...814...35C}.\nThis method identified four {\\sl Fermi} flares which we refer to as Flares 1, 3, 7 and 8.\nThe durations of the unpublished flares (1, 7 and 8) are provided in Table \\ref{tab:LATobs} and they are plotted with one-day binning, in order to show their temporal structure in more detail, between the arrow edges in the insets of the second panel of Figure \\ref{fig:gev-tev-lcs}. \nThe peak day of Flare 7 is coincident between {\\sl Fermi} and VERITAS observations, occurring on the night of 2017 April 01 (MJD 57844). This flare is annotated in bold font in the insets of Figure \\ref{fig:gev-tev-lcs}). Details on the searches for simultaneous observations between the LAT and VERITAS are provided in Section \\ref{sec:seds}.\\\\\n\nThe data were also divided into Bayesian blocks \\citep[with a false positive rate, $p_0$, equivalent to $2.84\\,\\sigma$, see eq.\\,(11) of][]{Scargle13}. \nThe prior was chosen so that the flaring periods that we defined using the method described in Section~\\ref{sec:increasing-flux} would be detected by this method, that is, Flares 1, 3, 7 and 8 in the case of the {\\sl Fermi}-LAT.\nThe Bayesian blocks are in general agreement with the increasing trend, i.e., the flux of the blocks shows a mostly increasing trend starting approximately at the break time.\nWe used this method to find the periods during which 1ES\\,1215+303 was in its ``low state'`(see Figure \\ref{fig:gev-tev-lcs}) and also to define the 2017 post-flare state for the SED modeling (Section \\ref{sec:modeling}).\nThe time periods of the different flux states can be found in Table \\ref{tab:LATobs}.\\\\\n\nThe VERITAS light curve is characterized by a baseline at $\\approx 2\\%$ of the Crab Nebula flux. No preference was found for a long-term linear trend. The flares at this wavelength were selected when the photon flux rose above $10\\%$ of the Crab Nebula flux. Between 2013 and 2017, these outbursts were observed at least once per year from 1ES\\,1215+303.\n\n\n\\subsubsection{LAT spectrum and flux}\n\\label{sec:latSpectrumFlux}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\linewidth,angle=0]{glc30_heph_hr60_v7.png}\n\\caption{{\\it Top:} The 30-day binned (black points) and 360-day binned light curves (violet points). {\\it Bottom:} Monthly spectral shape for the 30-day binned (black) and 360-day binned (violet) data. The gray shading in each of the two panels represent the value obtained for the entire 9-year data set.\\label{fig:glc30+heph+hr60}}\n\\end{center}\n\\end{figure}\n\nWe analyzed each year of LAT data leaving both the flux and the photon index free so that the long-term evolution of these values could be investigated.\nThe analysis was also repeated with the flaring epochs excluded  (which are different from the yearly combined datasets only for those three years which included flaring episodes).\nThe results are shown in Figure \\ref{fig:glc30+heph+hr60} and in Table \\ref{tab:LATobs}.\nThe gray shading represents the values for the flux and the photon index obtained for the entire 9-year data set.\n\nThe nominal flux is sufficiently high to allow for binning on short timescales while still being able to extract significant information on the time evolution of the photon indices. Black points in the top and middle panels of Figure \\ref{fig:glc30+heph+hr60} represent the monthly fluxes and photon indices respectively.\nThe data was also analyzed in 60-day bins to calculate the hardness ratio (HR) between two energy ranges, 0.1--1\\,GeV and 1--500\\,GeV. This analysis did not show significant changes in the HR for this source.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth,angle=0]{gammaflux_yearly_v2.png}\n\\caption{Power-law photon index, $\\Gamma$, against flux for the 360-day binned {\\it Fermi}-LAT data. The violet square points show the average value per bin, while black points show the non-flaring state values. Dotted lines show the results of the linear fits. The 360-day light curve and photon indices against time are shown in Figure \\ref{fig:glc30+heph+hr60} for the total data, in violet as well. The dashed lines join the data chronologically, going approximately from left to right, from where the long-term brightening and hardening can be visualised. \\label{fig:index-flux}}\n\\end{center}\n\\end{figure}\n\nThere is strong evidence for a long-term hardening of this source, reaching the $5.0\\,\\sigma$ level with the 30-day binned data, ($4.7\\,\\sigma$ including trials factor by having looked at the 30, 60 and 360-day binned data); $3.6\\,\\sigma$ level for the yearly data bins and $3.2\\,\\sigma$ outside flares with this same binning. We observe a long-term brightening at this binning as well, reaching the $12.8\\,\\sigma$ level for the yearly data bins, and $13.4\\,\\sigma$ outside flares.\nNo photon index\\,-\\,flux or HR\\,-\\,flux correlation was observed for the 30-day or 60-day binned data, respectively. \nFor the 360-day binned data, however, a Pearson correlation parameter of $-0.86$ between the photon index and the flux is obtained for the total data set (violet points in Figure \\ref{fig:index-flux}), and a value of $-0.74$ for the non-flare data (black points in the same figure). \nA likelihood ratio test shows a $3.4\\,\\sigma$ preference, including trials factor (by having looked at the 30, 60 and 360-day binned data), for a linearly decreasing dependence over a constant between the photon index and the flux; which indicates a possible overall ``harder-when-brighter'' trend in this source. \nThe yearly data outside flares also showed a preference at the $2.8\\,\\sigma$ level for a linearly decreasing dependence over a constant.  \nThese data, as well as the linear fits, are shown in Figure \\ref{fig:index-flux} and the details of the fit parameters can be found in Table \\ref{tab:index-flux} in Appendix \\ref{app:yearlyfits}.\nThis ``harder-when-brighter'' trend has been observed in the {\\it Fermi}-LAT data for flat-spectrum radio quasars and low-frequency-peaked BL Lacs \\citep{2010ApJ...710.1271A}. We did not find any photons with $E > 50$\\,GeV associated with any of the flares. The highest energy LAT photon detected had an energy of 466\\,GeV and was detected on 2011 May 01 during a relatively high state of the source that lasted approximately 13 months.\n\n\n\n\\subsection{Multifrequency flux-flux cross-comparison and cross-correlations}\n\\label{sec:fluxflux}\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=.9\\textwidth,angle=0]{mwl-seasoncorr_v2.png}\n\\put(-432,37){{\\tiny S1}}\\put(-433,55){{\\tiny S2}}\\put(-387,68){{\\tiny S3}}\n\\put(-432,77){{\\tiny S4}}\\put(-420,95){{\\tiny S5}}\\put(-352,106){{\\tiny S6}}\n\\put(-348,94){{\\tiny S7}}\\put(-382,119){{\\tiny S8}}\\put(-336,113){{\\tiny S9}}\n\\put(-263,106){{\\tiny S3}}\\put(-264,50){{\\tiny S4}}\\put(-235,94){{\\tiny S5}}\n\\put(-211,68){{\\tiny S7}}\\put(-203,119){{\\tiny S8}}\\put(-188,97){{\\tiny S9}}\n\\put(-80,106){{\\tiny S3}}\\put(-103,49){{\\tiny S4}}\\put(-102,94){{\\tiny S5}}\n\\put(-30,68){{\\tiny S7}}\\put(-47,125){{\\tiny S8}}\\put(-22,96){{\\tiny S9}}\n\n\\caption{Seasonal flux-flux diagrams for VERITAS, the {\\it Fermi}-LAT and Tuorla ($R$-band) energy ranges (in logarithmic scale). The data is labeled from Season~1 (S1), in 2009, to Season~9 (S9), in 2017. The dotted lines join the data chronologically, going approximately from left to right due to the long-term brightening observed in the GeV and optical light curves. The dashed line represents the fit to the expression $\\log_{{10}} (F_{\\rm{LAT}})=a\\,\\log_{{10}} (F_{\\rm{opt}})-b$. The solid line is the fit to the same expression with $a=2$. \\label{fig:ff-year}}\n\\end{center}\n\\end{figure*}\n\nAttempts to search for flux-flux correlations using short time bins failed due to large uncertainties. Furthermore, the cross-correlation function analyses performed showed no evidence for significant inter-band correlation for the data shown in Figure \\ref{fig:all-lightcurves} (see Section \\ref{sec:zdcf} for details). \nWe therefore performed a likelihood analysis of the LAT data using the $R$-band seasonal intervals (when the source was visible to optical telescopes), and analyzed the VHE gamma-ray data from the quiescent state for each year, shown in Table \\ref{tab:Vobs}. \nThe VERITAS data were taken between 2010 and 2017 and thus comprise seven data points, whereas the LAT data start in 2008, and therefore comprise 9 years of data, that is nine data points. \nThe seasonal flux-flux correlations which result from these analyses are shown in Figure~\\ref{fig:ff-year}, in logarithmic scale. \nThe least-squares fits and Pearson correlation coefficients can be found in Table~\\ref{tab:yearlycorr} for the logarithms of the seasonal fluxes for each set of energy bands. \nA strong long-term correlation between the optical and HE gamma-ray bands is found. \n\nWe fitted the (GeV, optical) points with the expression\n$$\\log_{{10}} (F_{\\rm{LAT}})=a\\,\\log_{{10}} (F_{\\rm{opt}})-b$$ \n(dashed line in Figure~\\ref{fig:ff-year}), yielding a slope $a=0.86 \\pm 0.21$ and $b=5.05\\pm 0.49$ with a $\\chi^2\/\\rm{d.o.f.}=41\/6$, and Pearson correlation coefficient of $0.86$. The uncertainties on $a$ and $b$ are obtained after having re-scaled the measurement uncertainties to $\\chi^2\/\\rm{d.o.f.}=1$.\\\\\n\n\\begin{deluxetable}{l|cccc}[ht!]\n\\tablecaption{Seasonal flux logarithm correlations. \\label{tab:yearlycorr}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Energy bands} & \\colhead{Pearson corr.} & \\colhead{Linear fit\\tablenotemark{*}} & $\\chi^2\/d.o.f.$ \\\\ \n\\colhead{}  & \\colhead{coefficient} & \\colhead{slope} &  }\n\\startdata\n  LAT - Optical & 0.86 & 0.86$\\pm$0.21 & 41\/6 \\\\ \n  VERITAS - LAT & 0.59 & 0.63$\\pm$0.62 & 43\/3 \\\\ \n  VERITAS - Optical & 0.44 & 0.06$\\pm$0.80 & 54\/3 \\\\ \n\\enddata\n\\tablenotetext{*}{Uncertainties scaled to $\\chi^2\/d.o.f.$.}\n\\tablecomments{The linear fit slope corresponds to $a$ in a fit to: $\\log(f_1)=a \\,\\log(f_2)+b$, where $f_1$ and $f_2$ are the seasonal fluxes in two different energy bands.}\n\\end{deluxetable}\n \n To our knowledge, this is the first time that such a strong global GeV-optical correlation has been observed over such an extended period of time (more than nine years).\nThe optical emission most likely comes from the synchrotron process and if the gamma-ray photons originate from inverse Compton scattering (ICS), this strong, almost linear ($a=0.86$) correlation is consistent with a long-term variability induced by changes of the Doppler factor or magnetic field of the emitting zone, considering a synchrotron-self-Compton (SSC) scenario. It is also consistent with gamma-ray emission originating from inverse-Compton scattering on an external photon field \\citep[e.g.][]{2011MNRAS.410..368B}. \\\\\n\nIn a SSC scenario, if a change in the number of emitting particles is the cause of the long-term variability, this would induce a quadratic flux-flux correlation ($a = 2$ line in Figure \\ref{fig:ff-year}) between the optical and the gamma-ray data. However, a slope of $a = 2$ is found to be disfavored at the $5.4\\,\\sigma$ level.  \nIf, instead of $\\chi^2\/$d.o.f. re-scaling, we add quadratically a source variability (of $\\approx 30\\%$), obtained from the excess variance analysis per season as in Section \\ref{sec:fluxdist}, we obtain $a=0.83 \\pm 0.33$, which would be preferred over $a=2$ at the $3.6\\,\\sigma$ level.\\\\ \n\nNo evidence for a clear correlation was found between the HE and VHE bands. A weaker correlation is found between the VHE and the optical bands. \nNo long-term correlation was observed between the OVRO data (15\\,GHz) and the optical data or the gamma-ray data.\n\n\n\\subsection{Flux distributions and variability} \\label{sec:fluxdist}\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=.367\\linewidth,angle=0]{variability_v4.png}\\hspace{1.cm}\n\\includegraphics[width=.5\\linewidth,angle=0]{log-normality_v3.png}\n\\caption{{\\sl Left:} The excess variance ($\\overline{\\sigma}_{XS}$) and variability amplitude ($\\overline{F}_{\\rm var}$) for the {\\sl Fermi}-LAT and Tuorla data. {\\sl Right:} LAT, Tuorla and OVRO flux distributions. The (bi)log-normal best fit is shown in solid light blue lines and the (bi)normal in dashed gray lines. The components of the bi-functions are shown in lighter blue for the bi-log-normal and in lighter gray for the normal function.\\label{fig:fdist}}\n\\end{center}\n\\end{figure*}\n\nIn this section we analyze the flux distributions of the {best sampled} light curves from our observing campaign, namely, the OVRO, Tuorla, LAT 3-day binned and VERITAS data. \nThese light curves are probed in order to search for log-normality in the distributions of their fluxes.\n\n\\begin{deluxetable*}{l|cc|ccc|cc|ccc}[ht!]\n\\tablecaption{Widths ($\\sigma$) and goodness of fits ($\\chi^2_{\\rm{red}}$) for normal, bi-normal, log-normal and bi-log-normal fits to the LAT, Tuorla and OVRO flux data. \\label{tab:bi-log-normal}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Dataset} & \\multicolumn{2}{c}{normal} & \\multicolumn{3}{c}{bi-normal} & \\multicolumn{2}{c}{log-normal} & \\multicolumn{3}{c}{bi-log-normal}\\\\\n & \\colhead{$\\sigma$} & \\colhead{$\\chi^2_{\\rm{red}}$} & \\colhead{$\\sigma_1$} & \\colhead{$\\sigma_2$} & \\colhead{$\\chi^2_{\\rm{red}}$} & \\colhead{$\\sigma$} & \\colhead{$\\chi^2_{\\rm{red}}$} & \\colhead{$\\sigma_1$} & \\colhead{$\\sigma_2$} & \\colhead{$\\chi^2_{\\rm{red}}$} }\n\\startdata\n  VERITAS & 0.38$\\pm$0.05 & 0.76   & -                             & -                             &   -   & 0.63$\\pm$0.06 & 0.97 & -             & -             & - \\\\\n  LAT     & 3.9$\\pm$0.3   & 4.12   & -                             & -                             &  -    & 0.43$\\pm$0.02 & 1.42 & -             & -             & - \\\\\n  Tuorla  & -             & -      & 0.5$\\pm$0.1                   & 1.4$\\pm$0.3                   & 1.48  & -             & -    & 0.22$\\pm$0.02 & 0.19$\\pm$0.04 & 1.08 \\\\\n  OVRO    & -             & -      & (3.6$\\pm$0.2)$\\times 10^{-2}$ & (1.5$\\pm$0.4)$\\times 10^{-2}$ & 0.67 & -             & -     & (9.5$\\pm$0.4)$\\times 10^{-2}$ & (2.7$\\pm$0.6)$\\times 10^{-2}$ & 0.82 \\\\\n\\enddata\n\\tablecomments{The log-normal function is given by $f(x)=\\frac{N}{x\\sigma \\sqrt{2\\pi}}\\exp\\left[-\\frac{(\\log{x}-\\mu)^2}{2\\sigma^2}\\right]$. A dash in a given column indicates that the particular function was not fit to that dataset.}\n\\end{deluxetable*}\n\nThis behavior has been studied in other blazars, such as BL\\,Lacertae \\citep{Giebels09}, 1ES\\,1011+496 \\citep{Sinha17} and a population of bright {\\it Fermi} blazars \\citep{Shah18} as well as in other accretion-powered systems \\citep{3LACAckermann15}. \nLog-normal distributions have the property that their means and fluctuations behave linearly on average, and are of interest since they have multiplicative rather than additive properties \\citep{Aitchison1973}. \n\nIn order to estimate the fluctuations in the source flux that are not due to Poisson noise, the excess variance, $\\sigma_{XS}$, was calculated. \nWe binned the flux data points shown in Figure \\ref{fig:all-lightcurves} in segments of equal duration and ensured that each bin contained at least 20 measurements of flux, excluding the flares. \nThe excess variance $\\sigma_{XS}^2 = \\frac{1}{N}\\Sigma^{i=1}_{N}(x-\\overline{x})^2-\\overline{\\sigma^2_i}$ \\citep[][Section B]{Vaughan03}  and the variability amplitude $\\overline{F}_{\\rm var}$ \\citep{Vaughan03} are shown as a function of the flux arithmetic mean in the left hand side of Figure \\ref{fig:fdist}. \nFor the LAT data we obtain $\\sigma_{XS}\\propto (0.25\\pm 0.05)\\,\\overline{\\rm{Flux}}$ ($\\chi^2_{\\rm{red}}=$0.66) and a Pearson correlation coefficient, $\\rho$, of 0.54. \nThe Tuorla data are more sparsely sampled than the LAT data so some of the bins contain fewer than 20 flux measurements (gray points in the bottom left-hand panels of Figure \\ref{fig:fdist}). \nA linear fit to these data outside of the low states yields $\\sigma_{XS}\\propto (0.15\\pm 0.05)\\,\\overline{\\rm{Flux}}$ ($\\chi^2_{\\rm red}=$172.5, $\\rho=0.74$), while a linear fit to the total data set results in $\\sigma_{XS}\\propto (0.16\\pm 0.04)\\,\\overline{\\rm{Flux}}$ ($\\chi^2_{\\rm red}=$134.4, $\\rho=0.80$). \nA similar analysis on the OVRO data did not show significant correlation ($\\rho=-0.20$). \nIt was not possible to perform this analysis on the VERITAS data due to their sparsity.\n\nThe flux distributions of the {\\it Fermi}-LAT, Tuorla and OVRO 15\\,GHz data, and their best fits to the (bi)log-normal (solid light blue) and (bi)normal (gray dashed) functions are shown in the right-hand panels of  Figure~\\ref{fig:fdist}. Both bi-functions consist of two components each, which are shown in lighter colors in the same figure.\nIn the case of the LAT data, flaring states, as they were defined in the previous section, were excluded so as not to favor the log-normal fit (due to a possible bias produced by the elongated tail). \nA Shapiro-Wilk test on the LAT data rejects the normal distribution with a p-value of $4.2\\times 10^{-16}$ and a test statistic of $w=0.87$ \\citep{Shapiro1965}.  \nThe $\\chi^2$ of the fits improve after Poisson noise reduction was applied during faint epochs, reaching the best fit when only data with significance above $3\\,\\sigma$ were included (approximately 60\\% of the data below $3\\,\\sigma$ are located within the low state defined in Section~\\ref{sec:increasing-flux}). \nThe distribution of these data is shown in the top right-hand panel of Figure~\\ref{fig:fdist}. The results of fits to normal ($\\chi^2\/$d.o.f.$=49.4\/12$) and log-normal ($\\chi^2\/$d.o.f.$=17.0\/12$) functions shown in the same figure, are presented in Table~\\ref{tab:bi-log-normal}, where it is observed that the log-normal function provides a much better fit. \nThe middle and bottom panels of the same figure show the Tuorla and OVRO flux distributions, respectively, where, contrary to the LAT data, no periods were excluded on the basis of the flux state of 1ES\\,1215+303. \nThis is because of the relative sparsity in the sampling of these light curves. \nWe observe a double-peaked structure in their flux distributions, possibly due to the fact that both quiescent and flare data are included, or due to the presence of a brighter second quiescent state.\nThe bi-log-normal function does not provide a clear improvement to the fit with respect to the bi-normal function in the case of the Tuorla and OVRO data (see Table~\\ref{tab:bi-log-normal}).\nThe two states of the Tuorla distributions are consistent with the states before and after the break time calculated in Section~\\ref{sec:increasing-flux}. The bi-normal fit results of the OVRO distributions are consistent with the flux density of the states interpreted as quiescent and flaring components by \\cite{Liodakis2017}, which includes data up to February 2016 for this source.\nTwo log-normal states were also previously observed at the IR-optical wavelengths in FSRQ PKS\\,1510-089 \\citep{Kushwaha16}. \nAn analogous analysis performed on the VERITAS data outside flares did not show evidence for a preference for a normal over a log-normal function (see Table~\\ref{tab:bi-log-normal} for the $\\chi^2_{\\rm{red}}$ values). \n\n\n\\subsection{ZDCF} \\label{sec:zdcf}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,angle=0]{1ES1215_zdcf_multipanel_v3.pdf}\n\\caption{The $Z$DCFs between light curves measured at different wavelengths. The pair of wavelengths in each panel is shown in the legend. A positive time lag ($t(X) - t(Y) >0$) between band $X$ and band $Y$ means the emission in band $X$ lags behind that in band $Y$. The vertical dotted lines show the time lag of zero, and the vertical dashed lines show the $1\\,\\sigma$ confidence interval around the maximum-likelihood peak time lag. \\label{fig:ZDCFs}}\n\\end{center}\n\\end{figure}\n\nTo further quantify the inter-band flux-flux correlation from 1ES\\,1215+303, we calculated the $Z$-transformed discrete cross-correlation function \\citep[$Z$DCF;][]{Alexander13} between the light curves from different energy bands, as shown in Figure~\\ref{fig:ZDCFs}. \nThe $Z$DCF method offers a conservative, more efficient estimate of cross-band correlation in light curves, compared to the discrete cross-correlation function \\citep[DCF;][]{Edelson88}. \nTo search for time lags between these energy bands, we used a maximum likelihood function \\citep{Alexander13}. \n\nThe local peak time lag between the 3-day {\\sl Fermi}-LAT and VERITAS light curve data obtained with this method is $t$(VERITAS)$- t$(LAT) $=8^{+11}_{-16}$ days compatible with a zero lag (a positive value indicates that the VERITAS flux is lagging behind the LAT flux).\n\nThere are no significant peaks in the ZDCFs for the optical and gamma or the radio and gamma or the optical and radio fluxes (this last one consistent with \\citet{Lindfors2016}).\n\n\n\\subsection{Power spectral density of the \\textit{Fermi}-LAT light curve} \\label{sec:periodicity-psd}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,angle=0]{1215_PSD_large.pdf}\n\\includegraphics[width=0.45\\textwidth,angle=0]{1215_SuF_Nsim1000_blk_large.pdf}\n\\caption{{\\it Top:} The power spectral density distribution of the 3-day-binned {\\it Fermi}-LAT light curve. The gray points are the periodogram from data \\citep[for details, see][]{TK95}. The red squares are the rebinned periodogram. The dashed line shows a simple power-law fit to the rebinned periodogram. \n{\\it Bottom:} The ``Success Fraction'' of simulated light curves at different power-law index of the power spectral density distribution. \\label{fig:PSD}}\n\\end{center}\n\\end{figure}\n\nThe source exhibits a typical power-law power spectral density (PSD) distribution, commonly observed in AGN. \nThe PSD calculated \\citep{TK95} from LAT data and a simple power-law fit are shown in the top panel of Figure~\\ref{fig:PSD}. \nRed squares represent averages over bins with sizes that follow a geometric series of factor 1.2.\n\nSince power-law PSDs can be distorted by power leakage from longer and shorter timescales, we calculate the ``success fraction'' (SuF) by comparing simulated light curves \\citep{TK95} and the observed one, following the method described in \\citet{Uttley02}. The SuF curve is shown in the bottom panel of Figure~\\ref{fig:PSD}. \n\nThe best-fit power-law index, $0.6\\pm0.1$ is consistent with the relatively wide 90\\% SuF range of 0.38 to 0.68. The SuF curve drops to 0 at indices of 0.3 and 0.9. \nThis suggests that the PSD distribution is relatively flat compared to the typical values between 1 and 2 found in AGNs \\citep[e.g.][]{Uttley02, 2014ApJ...786..143S, 2019ApJ...885...12R}. \n\n\\subsection{ Periodicity analysis of the \\textit{Fermi}-LAT and Tuorla light curves}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,angle=0]\n{LAT_WWZ_LSP_WWZ_d0p00001_t50_marg_300dpi_T_Steve.png}\n\\includegraphics[width=0.5\\textwidth,angle=0]\n{Tuo_WWZ_LSP_WWZ_d0p00005_t25_marg_300dpi_T_noerr.png}\n\\caption{\nThe scalograms from WWZ transform of the {\\it Fermi}-LAT ({\\it top}) and Tuorla ({\\it bottom}) light curves.\nThe Lomb-Scargle periodogram (solid gray line) and the marginal WWZ periodogram (dash-dot blue line) are shown in the right panel of each plot. \n90\\% confidence limits from a purely stochastic model with power-law PSD generated using the method of \\cite{Emmanoulopoulos2013} are also shown, including (dotted gray line) and  excluding (dashed gray line) the effect of the 553 trial frequencies.\n\\label{fig:LSP}}\n\\end{center}\n\\end{figure}\n\nTo test for the presence of periodicity or quasi-periodic optical and gamma-ray oscillation (QPO) of the flux of 1ES\\,1215+303, we calculated the weighted wavelet Z-transform (WWZ; \\citealp{Foster96}) and the Lomb-Scargle periodograms \\citep[LSP;][]{Scargle2} of the {\\it Fermi}-LAT and Tuorla light curves, as shown in Figure~\\ref{fig:LSP}. \nBoth WWZ and LSP are suitable for detecting QPO in unevenly sampled light curves. \nAn excess power at a $\\approx\\,$3-year period appears persistently in the WWZ and LSP of both the {\\it Fermi}-LAT and Tuorla data throughout the observational period. Slightly lower excess power at about a half and a quarter of the $\\approx\\,$3-year period, and the effect of sampling gaps in the optical data are apparent in the WWZ time-frequency plot (scalogram). The {\\it Fermi}-LAT LSP is noisy at shorter periods, while the periodogram (PSD) and the WWZ are much cleaner and are consistent with each other. \n\nThe top right panel of Figure~\\ref{fig:LSP} shows the PSD from the data compared with 90\\% confidence limits (CL) calculated from $4.7\\times10^6$ simulated light curves generated using the method of \\cite{Emmanoulopoulos2013} assuming that the underlying stochastic process has a power-law PSD, and using the flux probability density function (PDF) from the right-hand panels of Figure~\\ref{fig:fdist}. The dashed gray curve shows the CL for an \\textit{a priori} frequency. The dotted gray curve shows the CL that includes the penalty for selecting the frequency with the largest excess \\textit{a posteriori} from the 553 trial frequencies in the PSD. Assuming that the PSD is fully described by this stochastic process, it should be expected that at the 90\\% CL none of the measured PSD powers exceed this dotted gray curve, and indeed none do.\nOur simulations show that the apparent peaks in the LSP power at a $\\approx\\,$3 year period are not significant when the PSD of the underlying stochastic process and the trials factor are taken into account. The fact that the optical data show the same peak at $\\approx\\,$3~years does not lend credence to presence of a true QPO; this should be expected if a single stochastic process is responsible for the optical and gamma-ray light curve. \n\nThe simulated light curves are also used to test whether the trend of linearly increasing flux found in Section~\\ref{sec:increasing-flux} is inconsistent with a stationary stochastic process. We find that a linearly increasing or decreasing trend with a magnitude equal to or greater than that seen in the LAT data is present in approximately 1 in 1,000 simulations ($p=9.6\\times10^{-4}$), equivalent to a significance of $\\approx\\,3.3\\,\\sigma$. The linear trend is therefore only moderately inconsistent with the stochastic modeling.\n\n\n\\subsection{Characterizing the flares} \\label{sec:flares}\n\nWe focus our analysis on the unpublished flares, namely LAT Flare 1, LAT Flare 7 and LAT Flare 8, especially LAT Flare 7, since its peak is coincident with VERITAS Flare 7.\n\nThe decay times of {\\sl Fermi} Flares 1, 7 and 8 were calculated by fitting the 1-day binned light curve to: $F(t)=F_0+F_1\\times 2^{-(t-t_0)\/t_{\\rm{var}}}$.\nThe size, $R$ and Doppler factor, $\\delta$, of the gamma-ray emitting region are related, due to causality, to the variability timescale through: $R\\delta^{-1}\\leq ct_{\\rm{var}}\/(1+z)$. \nThe values found are shown in Table \\ref{tab:tvars}.\n\n\\begin{deluxetable}{l|ccc}[ht!]\n\\tablecaption{The half times for the LAT flares. \\label{tab:tvars}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Flare} & \\colhead{MJD}  & \\colhead{$t_{\\rm{var}}$ UL(90\\%)}\t& \\colhead{$R\\delta^{-1}\\leq$}\\\\\n\\colhead{}           & \\colhead{} & \\colhead{days}\n& \\colhead{$10^{15}$ cm}}\n\\startdata\n  Flare 1 & 54751 & 1.57 & 3.6\\\\\n  {\\bf Flare 7} & {\\bf 57844\\tablenotemark{a}} & {\\bf 0.90} & {\\bf 2.1}  \\\\\n  Flare 8 & 57855 & 1.24 & 2.8\\\\\n\\enddata\n\\tablenotetext{a}{Coincident with a VHE flare.}\n\\end{deluxetable}\n\nA similar fit was performed to the nightly VHE gamma-ray light curve around the time of Flare 7 on 2017 April 01. \nThe exponential decay time was relatively well constrained at $10\\,\\pm \\,2$ days. \nWhile the rise time is less constrained by the fit, we estimate the doubling time to be $<4$ days based on an upper limit measured eight days before the flare. \n\nFrom the SED modeling that we performed (as described in Section~\\ref{sec:modeling}), the Doppler factor for the blob is estimated to be $\\delta=25$. \nFrom fundamental-plane-derived velocity dispersion, \\cite{Woo02} estimated the SMBH mass of the source to be $1.3\\times10^{8} M_\\odot$, which corresponds to a Schwarzschild radius of $R_s \\sim 3.9\\times 10^{11}$\\,m. \nTherefore, the strongest constraint on the size of the emitting region based on the observed fastest gamma-ray variability (shown in Table \\ref{tab:tvars}) is $R\\leq 1350 \\, R_\\text{S}$. \\\\\n\n\n\\section{Spectral Analysis} \\label{sec:spectrum}\n\n\\subsection{LAT long-term SED}\\label{sec:longtermsed}\n\nThree different spectral models were considered to describe the spectrum of 1ES\\,1215+303 as measured by the LAT. These comprised a power-law (described in Section \\ref{sec:VERITAS}), a log-parabola and a power-law sub-exponential cutoff model. \nFor the combined dataset, the curved models were found to be preferred over the power-law model.\n\nFor the individual spectral data points plotted on the SED we used only the power-law model, since for these small data sets we found no preference for curved models. The data were analyzed in energy bands with the spectral parameters of all other sources in the model file being frozen to those values found in the global power-law analysis.\n\nThe power-law (PL) model, $dN\/dE=N_0(E\/E_0)^{-\\Gamma}$, yields an integral flux of $(7.7\\pm 0.2)\\times  10^{-8}$~photons~cm$^{-2}\\,$s$^{-1}$ with a significance of $\\approx 129.1\\,\\sigma$ and a photon index, $\\Gamma$, of $1.92\\,\\pm\\,0.01$ at a decorrelation energy, $E_0$ of 1.36\\,GeV.\nThe log-parabola (LP) model fit, $dN\/dE=N_0(E\/E_b)^{-(\\alpha+\\beta \\log(E\/E_b))}$, where $N_0$ is the normalization and $\\alpha$ and $\\beta$ are the spectral parameters at energy $E_b$, provided an integral flux of $(6.9\\pm 0.2)\\times 10^{-8}$~photons~cm$^{-2}\\,$s$^{-1}$ with a significance of $\\approx 129.3\\,\\sigma$, a spectral slope, $\\alpha$, of 1.86\\,$\\pm$\\,0.01 and a curvature parameter $\\beta$, of 0.039\\,$\\pm$\\,0.006 at the break energy, $E_b$, of 1\\,GeV. \nFrom a power-law sub-exponential cutoff (plSECO) model, $dN\/dE=N_0(E\/E_0)^{-\\gamma_1}e^{-(E\/E_c)^{\\gamma_2}}$, an integral flux of $(7.7\\pm 0.2)\\times 10^{-8}$~photons~cm$^{-2}\\,$s$^{-1}$ was obtained with a significance of $\\approx 129.3\\,\\sigma$, a lower-energy photon index, $\\gamma_1$, of 1.74\\,$\\pm$\\,0.03, a cutoff energy, $E_c$, of 22\\,GeV, an exponent $\\gamma_2$ of 0.40\\,$\\pm$\\,0.06, and decorrelation energy, $E_0$, of 1.36\\,GeV. \nSince the PL and the LP and also the PL and the plSECO are nested models, we use a likelihood ratio test to compare them, $TS_{\\rm{curved}}=2(\\log{L}_{\\rm{curved}}-\\log{L}_{\\rm{PL}})$, where $L$ is the maximum likelihood of the fit. \nLP and plSECO are not nested, therefore, we do not compare them. \nWe find that the LP is preferred over the PL model with a significance of $7.2\\,\\sigma$ while the plSECO is preferred over the PL with a significance of $7.5\\,\\sigma$. \nThese results indicate a preference for curved models \\citep[][]{4lac2019}, which could be an indicator of internal curvature at the source, even before entering the VHE range where the extragalactic background light (EBL) absorption has a considerable impact on the VHE flux.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth,angle=0]{9yrsSED_LAT_v8.png}\n\\caption{SED of the entire {\\it Fermi}-LAT data set (2008-08-04\\,-\\,2017-09-05). The data were analyzed with three different spectral models as described in the text: power-law (dashed), log-parabola (dotted) and power-law sub-exponential cut off (solid line). To visualize the connection with the VHE data, the VERITAS butterfly for the data from 2011 \\citep{Aliu13} was added. The LAT butterflies have been extrapolated to VHE energies and the effects of the EBL included \\citep{Franceschini17}. See details of the {\\sl Fermi}-LAT data in Table \\ref{tab:LATsed9y} in Appendix \\ref{app:LATsed9y}.\\label{fig:9yrsLATsed}}\n\\end{center}\n\\end{figure}\n\nThe three different fit models are shown in Figure \\ref{fig:9yrsLATsed}, where the EBL absorption was taken into account by calculating interpolated values from the model of \\cite{Franceschini17} (and Corrigendum \\cite{Franceschini2018}) at $z=0.131$\\footnote{Only the main paper is cited later in this work.}.  \nThe VERITAS spectrum for the 2011 data \\citep{Aliu13} is shown for visualization.  \nWe found that the HE and VHE data are connected very smoothly. We note that the LAT spectra of the curved models are in better agreement with the VERITAS data (in this case corresponding to an average quiescent state) than the power-law spectral model.\n\n\n\\subsection{GeV-TeV SEDs}\\label{sec:seds}\n\nThe LAT-VERITAS SEDs for the unpublished VHE data are shown in Figure~\\ref{fig:seds}. \nIn 2008, the brightest flare (Flare 1) at GeV energies was detected.\nThere are, however, no corresponding VERITAS data, since 1ES\\,1215+303 observations did not start until 2008 December. \n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=\\textwidth,angle=0]{GeV-TeV_SEDs_v5.png}\n\\caption{SEDs for the LAT and VERITAS data, including flares that have not previously been analyzed. Round points correspond to the {\\it Fermi}-LAT data, while the squares correspond to VERITAS. Data and butterflies for the flaring states are shown in blue and orange. Data in the quiescent state are shown in gray. From 2015 to 2017, the black data points correspond to the total data sets for each season. Power-law and log-parabola butterflies are shown for the black spectra. Only power-law butterflies are shown for the flaring states. Non-coincident GeV-TeV flare SEDs are shown in blue, while the orange SED represents Flare 7.\n\\label{fig:seds}}\n\\end{center}\n\\end{figure*}\n\n\\begin{deluxetable*}{l|cccccc|cccccc}\n\\setlength\\tabcolsep{1.5pt}\n\\tablecaption{Gamma-ray contemporaneous spectral analysis. \\label{tab:spectra}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Year} &\\multicolumn{6}{c}{VERITAS}                                                                                     & \\multicolumn{6}{c}{{\\it{Fermi}}-LAT}                                            \\\\\n\\colhead{}     &\\multicolumn{2}{c}{All}             & \\multicolumn{2}{c}{Flare}           & \\multicolumn{2}{c}{Non-flare}       & \\multicolumn{2}{c}{All}             & \\multicolumn{2}{c}{Flare}           & \\multicolumn{2}{c}{Non-flare}     \\\\\n\\colhead{}     &\\colhead{$\\Gamma$} & \\colhead{Flux} & \\colhead{$\\Gamma$} & \\colhead{Flux} & \\colhead{$\\Gamma$} & \\colhead{Flux} & \\colhead{$\\Gamma$} & \\colhead{Flux} & \\colhead{$\\Gamma$} & \\colhead{Flux} & \\colhead{$\\Gamma$} & \\colhead{Flux}\n}\n\\startdata\n2015          &$3.32\\pm0.18$ & $0.61\\pm0.14$ & $2.96\\pm0.18$ & $4.36\\pm0.99$ & $2.84\\pm0.39$ & $0.56\\pm0.27$ & $1.91\\pm 0.04$ & $5.6\\pm 0.3$ & -- & -- & -- & -- \\\\\n2016           & $3.12\\pm0.13$ & $1.07\\pm0.16$ & $3.06\\pm0.28$ & $2.93\\pm0.89$     & $3.27\\pm0.14$ & $0.78\\pm0.13$ & $1.88\\pm 0.03$ & $7.2\\pm 0.3$ & -- & -- & -- & -- \\\\\n2017           & $3.62\\pm0.10$ & $0.013\\pm0.003 ^\\dagger$        & $3.56\\pm0.13$ &  $0.073\\pm0.023 ^\\dagger$           & $3.94\\pm0.32$           & $0.21 \\pm 0.10$        & $1.85\\pm 0.03$ & $8.6\\pm 0.3$ & $1.61\\pm 0.32$ & $52.3\\pm 25.1$ &  $1.85\\pm 0.04$ & $8.0\\pm 0.5$ \\\\\n\\enddata\n\\tablecomments{The flux value for VERITAS is the normalization (N$_0$) for the differential flux ($dN\/dE$) at energy of 1 TeV in units of 10$^{-12}$TeV$^{-1}$cm$^{-2}$s$^{-1}$. The flux value for {\\it Fermi}-LAT is the normalization (N$_0$) at the decorrelation energy of 1.36\\,GeV in units of 10$^{-12}$MeV$^{-1}$cm$^{-2}$s$^{-1}$. $^\\dagger\\,$Normalization at 3 TeV. }\n\\end{deluxetable*}\n\nThe first panel on the left shows Flare 1 and the low state SED, as defined in Table~\\ref{tab:LATobs}.\nThe 2011 VERITAS butterfly \\citep{Aliu13} is shown since this season belongs to the GeV low state (see Section~\\ref{sec:increasing-flux} for the Bayesian Blocks analysis that was used to define the various emission states of 1ES\\,1215+303).\n\nDuring Flare 4, at VHE, in 2015, there were approximately 40 minutes of simultaneous observations between the LAT and VERITAS; and during Flare 5, also at VHE, in 2016, there were approximately 80 minutes of simultaneous observations between the LAT and VERITAS. No significant HE emission was detected during these simultaneous observations; and no elevated flux was observed in the LAT data for these days. \nVERITAS detected another flare on 2017 March 05, Flare 6,  at a time during which 1ES~1215+303 was not in the LAT FoV. 1ES~1215+303 had been in the FoV of the LAT approximately 2.5 hours before VERITAS started observations, and re-entered the LAT FoV approximately 1 hour after VERITAS finished observing this source during that night.\nNo evidence for an elevated flux was found when the LAT data for this day were analyzed.\nIn 2017, two flares were measured by the LAT with peaks on April 01 and 13 (Flares 7 and 8, respectively; refer to Table \\ref{tab:LATobs} for the duration of these flares). \nLAT Flare 7 had a VHE counterpart (orange), while VERITAS was not observing at the time of Flare 8 at GeV energies (blue). \nThe details of their spectra can be found in Table~\\ref{tab:spectra}.\\\\\n\n\n\\section{Multifrequency radio-to-TeV SED modeling} \\label{sec:modeling}\n\nThe large multiwavelength dataset described in this paper allows us to build broadband SEDs for different periods and states of activity of 1ES~1215+303. \nIn this section, three activity states that have not been examined in previous works are studied: a low, steady state corresponding to the lowest observed \\textit{Fermi}-LAT activity as defined by the Bayesian Block method, the 2017 April 01 GeV-TeV Flare 7, and the subsequent post-flare state from 2017 April 15 to 23.\n\nThese three states are modeled using the ``blob-in-jet\" (Bjet) radiative code from \\cite{Hervet15}. \nGiven the low apparent jet speeds reported in Section \\ref{sec:vlba}, we consider the main emission zone as a continuous high-energy particle flow passing through a stationary shock in the jet. This local plasma flow is identified as a compact spherical blob flow moving at a significant Lorentz factor close to the line of sight.\nWe assume that this blob is filled by an electron (or electron\/positron) population in an isotropic magnetic field.\nWe consider a particle energy distribution which, as a result of injection and cooling, follows a broken power-law function as\n\\begin{equation}\nN_e(\\gamma) = \\left\\lbrace\n     \\begin{array}{ll}\n         N_{e}^{(1)} \\gamma^{-n_1} & \\mathrm{for}~ \\gamma_{\\mathrm{min}} \\leqslant \\gamma \\leqslant \\gamma_{\\mathrm{brk}} \\\\\n         N_{e}^{(2)} \\gamma^{-n_2} & \\mathrm{for}~ \\gamma_{\\mathrm{brk}} \\leqslant \\gamma \\leqslant \\gamma_{\\mathrm{max}}\n     \\end{array}\n     \\right. ,\n\\end{equation}\nwith $N_{e}^{(2)} = N_{e}^{(1)}  \\gamma_{\\rm{brk}}^{(n_2-n_1)}$, and $N_{e}^{(1)}$ the particle density factor set as $N_{e}^{(1)} =N_e(1)$.\n\nThis blob is moving through a conical leptonic plasma jet having a larger radius and a lower flow Lorentz factor. \nThe jet is discretized logarithmically into 50 conical slices along its propagation axis. For the sake of simplicity, each slice has its particle density spectrum considered as a simple power-law function.\nBoth the blob and the jet are radiating in synchrotron and SSC emission. \nWe include the effects of the absorption by the EBL following the model of \\cite{Franceschini17}.\nWe model the data via a ``fit by eye\" process, because the use of a minimization algorithm is very challenging for SSC models due to the strong degeneracies that exist between parameters. Furthermore, it becomes extremely difficult when we have multiple emission zones such as is considered here. Hence the proposed model solutions cannot be considered as the statistically best solutions but are consistent with our assumptions about the underlying emission scenario. The reduced $\\chi^2$ of the fits shown in the following section is for informational purposes only.\n\n\n\\subsection{Low state of 1ES 1215+303}\n\nThe time period corresponding to the low state of the source was defined using the results of the Bayesian block method that was applied to the \\textit{Fermi}-LAT lightcurve (see Fig.~\\ref{fig:gev-tev-lcs}). \nTwo periods between 2008 and 2012 can be considered as the lowest activity state: 2008 November 17 -- 2010 August 12 (MJD 54787--55421) and 2011 April 15 -- 2012 April 10 (MJD 55666--56027). \nThe multiwavelength lightcurves do not show any evidence for an outburst occurring at other wavelengths during these time periods either.\nSuch a long accumulated time of 33 months of low state allows us to have a very well defined {\\it{Fermi}}-LAT spectrum, as well as a well-sampled multiwavelength SED at lower energies. \nIndeed, data from the \\textit{Planck PCCS2} catalog \\citep{Planck16} and the AllWISE Multiepoch Photometry Database\\footnote{\\url{http:\/\/wise2.ipac.caltech.edu\/docs\/release\/allwise\/}} were taken during our defined periods, increasing the broadband coverage.\nThe resulting SED with the favored associated radiative model is presented in Figure~\\ref{fig:SEDmodel_lowstate}, and the model parameters are shown in Table~\\ref{tab::Params_model}. The favored model has a $\\chi^2\/\\rm{d.o.f.} = 364.\/49 = 7.4$ (considering the blob and jet model parameters). The fit quality is strongly impacted by the extremely small uncertainties of the averaged \\textit{WISE} data. Without taking into account \\textit{WISE}, we have $\\chi^2\/\\rm{d.o.f.} = 106.\/45 = 2.4$.\n\n\\begin{figure*}[ht!]\n\t\\begin{minipage}[b]{0.5\\linewidth}\n   \t\t\\centering \\includegraphics[width=9.5cm]{1ES1215_Lowstate_SED2.pdf}\n\t\\end{minipage}\\hfill\n\t\t\\begin{minipage}[b]{0.5\\linewidth}\n      \\centering \\includegraphics[width=9.5cm]{1ES1215_postflare_SED_v2.pdf}\n\t\\end{minipage}\\hfill\n\\caption{Multiwavelength SEDs and models of the source low state (\\textit{Left}), 2017 flare and 2017 post-flare (\\textit{Right}). Plain blue lines are the blob synchrotron and SSC contributions, dot-dashed pink lines are the jet synchrotron and SSC emission, the blue dotted line is the intrinsic SSC emission without EBL absorption, and the thick brown and thick black dot-dashed lines are the sums of all components.\n\\label{fig:SEDmodel_lowstate}}\n\\end{figure*}\n\n\n\\subsubsection{Compact blob}\n\nThe multiwavelength SED from the IR to gamma ray is assumed to be emitted from a compact emission zone, referred to above as the ``blob.\"\nThe SED shows two clear bumps, one peaking in the IR-optical range considered as synchrotron emission and one peaking at high energy considered as being dominated by SSC emission. \nThe apparent contradiction with this observed low frequency synchrotron peak and the HBL classification of the source is further discussed in Section~\\ref{sec:discussion}.\n\nNeither the thermal signature of accretion disk radiation nor a sharp peak at high energy, which would indicate the presence of the external inverse-Compton (EIC) process on the nucleus thermal radiation field, is detected. \nWe therefore consider this process to be negligible, as is often the case for HBL sources.\n\nThe wide gap in energy of about ten orders of magnitude between the synchrotron and SSC peaks implies a very low internal $\\gamma-\\gamma$ opacity to reach the observed energies of $E > 100$\\,GeV. \nA satisfactory solution is found by considering a high Doppler factor value of $\\delta = 25$, associated with the maximum theoretical angle to the line of sight $\\theta \\simeq 2^{\\circ}$.\nAs described in Section~\\ref{sec:flares}, the radius of the emitting region is constrained by taking into account the fastest observed variability of $t_{\\mathrm{var}} = 0.9$ day. \nGiven the Doppler factor considered, this sets an upper limit to the radius of $R \\leq 5.2 \\times 10^{16}$ cm.\n\nThe minimal energy of the radiative electrons is set at the relatively high value of $\\gamma_{\\mathrm{min}} = 4.7\\times 10^{3}$. \nWhile not exceptional in blazar radiative models, such a high $\\gamma_{\\mathrm{min}}$ is often specifically used to describe extreme HBLs \\citep[e.g][]{Aliu14,Archer18}.\nThe blob is matter-dominated with an equipartition ratio between the magnetic field energy density $U_B$ and the particle energy density $U_e$ of $U_B\/U_e = 1.6 \\times 10^{-2}$.\n\n\\subsubsection{Radio jet}\n\\label{sec:Modeling-radio_jet}\n\nThe \\textit{WISE} SED shows a clear luminosity excess in its lowest energy band W4 compared to the other ones, which follow a hard photon index power-law spectrum, as expected for the optically thick blob synchrotron emission.\n\nThis excess can be associated with broader jet emission, dominating the low-energy part of the SED from radio to far infrared. \nAlthough not often modeled, this jet signature is a relatively common HBL feature \\citep[e.g.][]{Katarzynski01, Archer18}.\n\nWith 9 free parameters and only one obvious spectral signature in the radio to far IR, the jet parameters are naturally degenerate. \nIn order to have parameters as physically consistent as possible while keeping a good fit to the data, we constrain several other parameters in addition to the density and Doppler factor that are discussed above. \nWe consider an identical spectral slope for the injected particle spectrum between the blob and the jet and we also assume that the jet is in equipartition.\n\nThe apparent opening angle of the 15.3\\,GHz radio-jet was measured as $\\alpha_\\mathrm{app} = 13.8^{\\circ} \\pm 0.1^{\\circ}$ by \\cite{Pushkarev17} {\\em via} a stacking of the multiple observations of the VLBA referenced in the MOJAVE database. \nThis value confirms the previous measurement of $\\alpha_\\mathrm{app} = 14^{\\circ}$ by \\cite{Hervet16}, which was derived from the same database but based on the evolution of the referenced radio-component sizes.\nThe fact that these two measurements are similar indicates that the jet does not significantly change its direction with the line of sight over time, and that the radio components occupy the full jet cross-section.\n\nFrom the observed jet apparent opening angle and the angle with the line of sight set at $\\theta = 2^{\\circ}$, we can deduce the intrinsic jet opening angle used for the model {\\em via} the relation\n$\\alpha = \\alpha_\\mathrm{app} \\sin(\\theta)$, which leads to $\\alpha\/2 = 0.24^{\\circ}$.\n\n\\begin{deluxetable}{ccc}\n\\tablecaption{Model parameters used for the multiwavelength low state.\\label{tab::Params_model}}\n\\tabletypesize{\\scriptsize}\n    \\tablehead{\n    \\colhead{Parameter} & \\colhead{Value} & \\colhead{Unit}\\\\ \\hline\n    \\colhead{$\\theta$} & \\colhead{$2.0$} & \\colhead{($^{\\circ}$)}\\\\ \\hline\n    \\colhead{Blob} &  &\n    }\n    \\startdata\n    $\\delta$ & $25$ & $-$\\\\ \n    $N_{e}^{(1)}$ & $1.8\\times 10^{6}$ & cm$^{-3}$\\\\ \n    $n_1$ & $2.82$ & $-$\\\\\n    $n_2$ & $3.7$  & $-$\\\\\n    $\\gamma_{\\mathrm{min}}$ & $4.7\\times 10^{3}$ & $-$\\\\\n    $\\gamma_{\\mathrm{max}}$ & $7.0\\times 10^{5}$ & $-$\\\\\n    $\\gamma_{\\mathrm{brk}}$ & $1.5\\times 10^{4}$ & $-$\\\\\n    $B$ & $2.35\\times 10^{-2}$ & G\\\\\n    $R$ & $5.1\\times 10^{16}$ & cm\\\\\n    \\hline\n    Jet\\\\\n    \\hline\n    $\\delta$ & $15$ & $-$\\\\ \n    $N_{e}^{(1)}$ & $1.3\\times 10^{4}$ & cm$^{-3}$\\\\ \n    $n$ & $2.82$ & $-$\\\\\n    $\\gamma_{\\mathrm{min}}$ & $9.0\\times 10^{2}$ & $-$\\\\\n    $\\gamma_{\\mathrm{max}}$ & $3.5\\times 10^{3}$ & $-$\\\\\n    $B_1$ & $3.5\\times 10^{-2}$ & G\\\\\n    $R_1$ & $1.0\\times 10^{17}$ & cm\\\\\n    $L$\\tablenotemark{*} & $1.0\\times 10^{2}$ & pc\\\\\n    $\\alpha\/2$\\tablenotemark{*} & $2.4\\times 10^{-1}$ & $^{\\circ}$\n\\enddata\n\\tablenotetext{*}{\\textit{ Host galaxy frame.}}\n\\end{deluxetable}\n\n\n\\subsection{2017 April flare and post-flare}\n\n\\begin{deluxetable}{ccc}\n\\tablecaption{Model parameters used for the multiwavelength 2017 April 01 flare and post-flare states.\\label{tab::Params_model_flare}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n    \\colhead{Parameter} & \\colhead{Value} & \\colhead{Unit}\\\\ \\hline\n    \\colhead{$\\theta$} & \\colhead{$2.0$} & \\colhead{($^{\\circ}$)}\\\\   \\hline\n    \\colhead{Blob} &  &\n    }\n    \\startdata\n    $\\delta$ & $25$ & $-$\\\\ \n    $N_{e}^{(1)}$ (flare) & $\\boldsymbol{5.5\\times 10^{6}}$ & cm$^{-3}$\\\\ \n    $N_{e}^{(1)}$ (post-flare) & $\\boldsymbol{1.8\\times 10^{6}}$ & cm$^{-3}$\\\\ \n    $n_1$ & $2.9$ & $-$\\\\\n    $n_2$ & $4.5$  & $-$\\\\\n    $\\gamma_{\\mathrm{min}}$ & $4.7\\times 10^{3}$ & $-$\\\\\n    $\\gamma_{\\mathrm{max}}$ & $7.0\\times 10^{5}$ & $-$\\\\\n    $\\gamma_{\\mathrm{brk}}$ & $9.0\\times 10^{4}$ & $-$\\\\\n    $B$ & $5.2\\times 10^{-2}$ & G\\\\\n    $R$ & $5.1\\times 10^{16}$ & cm\n    \\enddata\n\\end{deluxetable}\n\nOn 2017 April 01 (MJD 57844), VERITAS detected its second brightest flare from 1ES\\,1215+303 (referred to as Flare 7). \nThis strong gamma-ray activity was simultaneously detected by \\textit{Fermi}-LAT and was followed by a secondary \\textit{Fermi}-LAT outburst 10 days later which we call {\\sl Fermi} Flare 8 (see Fig.~\\ref{fig:gev-tev-lcs}). \nUnfortunately 1ES\\,1215+303 was not being monitored at any other energies during this time, which prevents us from being able to derive any accurate emission scenario for this April 01 event.\n\nFrom 2017 April 15 to 23 (MJD 57858--57866), the source was monitored at many wavelengths and showed historically high fluxes in the optical, UV, and X-ray bands (see Fig.~\\ref{fig:all-lightcurves}).\nIt is plausible then that the emission zone responsible for the {\\sl Fermi} gamma-ray Flares 7 and 8 was still in its cooling phase during this period.\n\nGiven the many multiwavelength observations available during this post-flare period, we can attempt to derive realistic physical parameters describing the data. \nAs is shown in Figure~\\ref{fig:SEDmodel_lowstate} and Table~\\ref{tab::Params_model_flare}, a particle density decrease of a factor 3 in the emission zone is enough to move from the flare to the post-flare state. \nSuch a decrease matches an interpretation of a flare from a jet overdensity crossing a standing shock.\n\nThe radio jet is assumed to keep a roughly steady flux between all of the states studied. \nThe jet model used for the low state is kept for the 2017 flare\/post-flare, and plays only a very minor role in the total radiative output.\n\nWe considered the same emission zone for all of the SEDs modeled, with a constant plasma flow speed (same Doppler factor and size). \nThe low- and high-state SEDs can be well represented by changing the particle spectrum and the magnetic field parameters. \nThe substantial changes introduced between the low and the high states are an increase of the magnetic field $B$ ($ \\times 2.2$), an increase of the particle spectral break energy $\\gamma_\\mathrm{brk}$ ($\\times 6.0$), and a softening of the particle index after the break $n_2$ ($\\times 1.2$). \nInterpretations of these changes are discussed in Section~\\ref{sec:discussion}. The fit quality of the flare and post-flare states is $\\chi^2_{\\rm flare}\/\\rm{d.o.f.} = 10.5\/0$ and $\\chi^2_{\\rm p-flare}\/\\rm{d.o.f.} = 385\/184 = 2.1$ respectively.\n\n\n\\section{Discussion} \\label{sec:discussion}\n\n\\subsection{Extreme shift of the synchrotron peak frequency}\n\nIn many ways 1ES\\,1215+303 shows typical features of a classical HBL source: it has an FR I radio jet \\citep[at the kpc scale]{Giroletti06}, with multiple stationary radio-components as can be seen from VLBI \\citep{Hervet16,Piner18}, it does not show a thermal accretion disk signature in the blue-UV, nor does it exhibit strong inverse-Compton dominance in the broadband SED.\n\nAn unusual feature, however, is the dramatic change of the synchrotron bump (shape and peak frequency) between low and high activity states. \nThe high state, as observed in the 2017 flare and post-flare SED, presents a synchrotron peak between the UV and soft X-rays, typical of HBLs. Due to the relative flatness of the synchrotron bump it is difficult to determine the precise peak frequency value, but the favored post-flare model shows a synchrotron peak at $\\log_{10}(\\nu_{\\rm{peak}}\/\\rm{Hz}) = 15.75$.\nThe low state is characterized by a much more constrained peak frequency, $\\log_{10}(\\nu_{\\rm{peak}}\/\\rm{Hz})$, of 14.49\\,$^{+0.17}_{-0.54}$ from the model, with boundaries from the IR and optical data (consistent with \\citet{Nilsson2018}'s results, based on the Roma-BZCAT Multi-frequency Catalogue up to 2012). \nThus, if only this low state were considered, this source would be classified as an IBL.\n\nFits to a cubic polynomial function were also performed on the synchrotron bump of the broadband SED; since this is the method followed in the Fourth Catalog of AGNs detected by the {\\sl Fermi}-LAT \\citep[4LAC;][]{4lac2019}. The results were consistent with the blob-in-jet modeling, and are illustrated in Figure \\ref{fig:shift4LAC}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth,angle=0]{shift_4LAC_v1.png}\n\\caption{Photon index versus the logarithm of the frequency of the synchrotron peak. Color markers represent classifications, indicated in the legend, for GeV-detected blazars as published in the 4LAC. 1ES 1215+303 shows a spectral shape characteristic of IBLs during the low state, while exhibiting HBL-like properties during the high state in April 2017. This extreme shift is observed with both the results of the blob-in-jet modeling and the cubic polynomial fit (see text for details).\\label{fig:shift4LAC}}\n\\end{center}\n\\end{figure}\n\nUp to now, the only extreme peak-frequency shift ever observed from mid-IR to X-ray is from the IBL VER\\,J0521+211 (also known as RGB\\,J0521+212) with, however, a lack of optical-UV data during its flare, which prevents any reliable peak shift estimation \\citep{Archambault13}.\nHBLs are also subject to synchrotron peak shifts during flares but to a lesser extent, e.g., between soft\/mid to hard X-rays, making a transition possible from regular to extreme HBL \\citep[e.g][]{Ahnen18}.\nThus, the reported frequency shift in this study is a first for this kind of source, which further increases the diversity of behaviors observed for BL Lacs and raises many questions about the causes of such phenomena.\n\nA critical parameter illustrating this synchrotron peak shift is the Lorentz factor break of the electron spectrum, $\\gamma_{\\mathrm{brk}}$, which increased by a factor of 6 between the low and high states.\nFollowing the common broken power-law description of the particle spectrum, the $\\gamma_{\\mathrm{brk}}$ parameter represents the energy above which the radiative cooling is taking over from the adiabatic (or advective) cooling \\citep[e.g][]{Inoue96}. \nA significant increase of $\\gamma_{\\mathrm{brk}}$, as suggested by the SED modeling, points towards more efficient adiabatic cooling when flaring.\nIn order to have a flare with more efficient non-radiative cooling, the model shown in Figure \\ref{fig:SEDmodel_lowstate} requires a strong increase of the population of injected particles in addition to a local increase of the magnetic field.\nDue to the degeneracy between the magnetic field and the Doppler factor in blazar SSC models, a local increase of the Doppler factor instead of the magnetic field is also a possible explanation.\n\nThe linear flux-flux correlation between the optical and the GeV gamma-ray bands highlighted in Section \\ref{sec:fluxflux}, showing an index ($a=0.86$) of less than 1, is consistent with the fact that a larger variation of the synchrotron peak luminosity than the SSC one was observed in the low state and 2017 post-flare SEDs.\nThe exclusion of a quadratic flux-flux correlation indicates that a change in the number of radiative particles is not the major criterion explaining the common observed variability. However this could be favored for the strongest flares, such as that of 2017 April 01 (see Figure \\ref{fig:SEDmodel_lowstate} and Table \\ref{tab::Params_model_flare}).\n\n\n\\subsection{Multi-year flux increase}\n\nThe broken-line fit of the long term lightcurves is strongly favored over the linear fit for the \\textit{Fermi}-LAT dataset ($5.5 \\,\\sigma$ level), and moderately favored for the optical dataset ($3.4 \\,\\sigma$ level). The times where the break occurs in both datasets are compatible within $1\\,\\sigma$, strengthening the case for a MWL increase of the source activity starting approximately at the time of MJD $55780 \\pm 122$ ($\\sim$ 2011 August).\n\nEven though the LAT linear trend is inconsistent with the stochastic model only at the $3.3\\,\\sigma$ level (see Section \\ref{sec:periodicity-psd}), this long term flux increase of at least 6 years is intriguing and can be caused, in theory, by multiple possible processes such as jet precession, or by an increase in the accretion rate.\n\nThe multiple radio-VLBI observations of the source presented in this work, such as the lack of non-radial motions in the jet, and the straight jet at larger scales discussed in \\cite{Giroletti06} rule out any significant jet precession. \nAlso, jet precession would make the jet width, from stacked radio images, broader than the measured component sizes  (Section \\ref{sec:Modeling-radio_jet}).\nFinally, any precession would likely lead to a long-term rise in the radio emission due to the increase of the Doppler factor. None is observed.\nWe thus consider that the most likely cause of this gamma-ray multi-year flux increase is related to the black hole accretion process. \n\nTidal disruption events (TDEs) are often mentioned when observing multi-year-long flares of supermassive black holes. \nThese should be at a particularly high rate in AGNs due to the interaction of their accretion disk\/torus with nearby stars \\citep{Karas07}.\nIt is, however, very challenging to differentiate a TDE from the natural high-amplitude variability of the accretion disk itself. \nA TDE is usually identified by its strong nuclear ionization and by a specific decreasing flux profile. \nWe do not have access to these observables with the data that we have gathered, which prevents us from any relevant testing of the TDE hypothesis.\n\nThis long-term flux increase can be, however, compared to typical timescales of natural changes that occur in the accretion rate. HBLs are known to be the least powerful blazars and have been associated with a weak accretion mode known as the ``advection dominated accretion flow'' (ADAF).\nIn this case, the typical minimal time for jet loading from a change in the accretion is given by the free-fall timescale $\\tau_{ff}$. From \\cite{Manmoto96} we have\n\\begin{equation}\\small\n\\tau_{ff} = 4.63 \\times 10^{-5} \\left(\\frac{r}{1.0 \\times 10^{3} r_g}\\right)^{3\/2} \\left(\\frac{M_{\\mathrm{BH}}}{10 M_\\odot}\\right) \\mathrm{days}.\n\\end{equation}\n\nBy considering matter loading from the outer part of the ADAF disk, at $r \\sim 3.0 \\times 10^{3} r_g$ \\citep{Narayan96}, and the black hole mass $1.3\\times 10^{8} M_\\odot$ (as discussed in Section~\\ref{sec:flares}), we obtain a typical timescale $\\tau_{ff}$ of 8.7 years.\nThis timescale is similar to the long-term flux increase reported in Section \\ref{sec:increasing-flux} which started around the fall of 2011.\n\nWe found evidence (significance of $4.7\\,\\sigma$) for a long-term spectral hardening trend accompanying the flux increase (see Sections~\\ref{sec:increasing-flux} and \\ref{sec:latSpectrumFlux}). \nSuch a ``harder-when-brighter'' trend (at a $3.6\\,\\sigma$ level in the case of 1ES~1215+303) is typically observed in gamma-ray flat-spectrum radio quasars and intermediate-\/low-frequency peaked BL Lacs \\citep[e.g.,][]{Abdo10}. \nSimilar behavior has been observed in radio galaxies and high-frequency peaked BL Lacs, most commonly in the X-ray band \\citep[e.g.,][]{Brown11, Ahnen16}. \nFrom our SED modeling above, the GeV gamma-ray spectra during higher flux states are indeed harder than the lowest flux state, lending support to the ``harder-when-brighter'' phenomenon. \n\n\n\\subsection{Optical polarization}\n\nThe optical polarization fraction over the 3 years covered by the NOT observations is relatively stable, with values between 5 and 15\\,$\\%$. \nThis relatively low blazar polarization is well within the range of small values typical of HBL sources \\citep{Angelakis16}.\nIn the same paper, it was noted that HBLs tend to concentrate their polarization angle around preferred directions, which is also the case for 1ES\\,1215+303 with small angle variations from 130$^{\\circ}$ to 175$^{\\circ}$.\nThis indicates a stable, nearly toroidal magnetic field structure at the location of the optical emission zone that we described as a compact blob.\n\nThe NOT observations provide good optical polarization coverage around the gamma-ray flare of 2017 April 01. \nDuring this epoch, the polarization angle reached its highest value ($173^{\\circ}$), remaining above $166^{\\circ}$ during the post-flare state. At the same time, the polarization fraction reached its local minimum during the post-flare state. The polarization angle local minimum of the season was $140.6^{\\circ}$, varying a total of $38.4^{\\circ}$ in 2017; while the polarization fraction changed between 5\\% and 10.5\\%.\n\nAlthough this angle shift is much less dramatic than what has been observed in some blazars \\citep[e.g.][]{Abdo10Natur,Marscher10,Kiehlmann16}, it follows a common behavior  associated with gamma-ray flares \\citep{Blinov18,Hovatta2016}. \nThe weak amplitude of the polarization angle shift could find a natural explanation in the observed almost toroidal magnetic structure and a heavily matter-dominated blob, as suggested by the modeling.\n\n\n\\subsection{Log-normal distribution of the optical and HE fluxes}\n\nThe preference for log-normality in the flux distributions of the LAT and Tuorla data {could be} evidence that multiplicative processes \\citep{Aitchison1973} are occurring at these wavelengths, which are, as is discussed in Section \\ref{sec:fluxflux}, strongly correlated over the long term, and which could also be connected due to SSC scattering. \nSeveral hypotheses have been discussed in the literature regarding the nature of the processes behind these observations. \nFor instance, \\cite{Uttley01} attribute them to large, long-time-scale energy releases in the corona, possibly due to magnetic reconnection, initiating avalanche sub-division, which is later superimposed on short-time-scale emissions of energy proportional to the original division. \nThey also mention the natural appearance of these linear relationships in the mechanism proposed by \\cite{lyubarskii97} due to radius-dependent mass-accretion-rate fluctuations producing variations on all time scales in the disk and corona. \nHowever, an interpretation based on additive processes by \\cite{biteau12}, the mini-jets-in-a-jet model, predicts that skewed flux distributions (such as log-normal) could be obtained from the summation of contributions of a large number of mini-jets under specific conditions.\n\n\n\\section{Summary} \\label{sec:summary}\n\nIn this paper we present an analysis of the observations of the HBL 1ES\\,1215+303 between 2008 and 2017 from radio to VHE gamma-ray energies. We summarize our main findings below:\n\n(i) The observations performed by {\\sl Fermi}-LAT in gamma-rays and the Tuorla Observatory in optical show a clear long-term increase of flux over the ten-year period. Both datasets favor a start of this increase around August 2011 ($\\approx\\,$MJD $55780\\pm 122$). No conclusive interpretation is found to explain such a behavior; however, the timescale of this flux increase, while limited by our dataset, is consistent with a process driven by the accretion disk. We can also reject jet precession as the cause of this behavior since precession is not in agreement with the multiple radio-VLBI observations.\n\n(ii) We report the simultaneous coverage of the peak day of Flare 7  between the \\textit{Fermi}-LAT and VERITAS instruments, occurring on the night of 2017 April 01 (MJD 57844).\n\n(iii) An extreme shift of the synchrotron peak frequency from the low state to the 2017 flaring state of the source is observed. \nThis is consistent with a higher break energy of the emitting particles in the flaring state, likely associated with a more efficient adiabatic cooling.\n\n(iv) Three stationary radio features in the innermost jet region are found in the VLBA data at 43.1\\,GHz, 22.2\\,GHz, and 15.3\\,GHz. \nA single-epoch VLBA observation at 43.1\\,GHz produced an image at the highest resolution (at the time of this article) of the jet, revealing a component (unresolved at lower frequencies) very close (0.16\\,mas) to the core. \nStationary components in the vicinity of the radio core are a typical phenomenon in HBLs. \nCombining the SED modeling with this radio behavior, we conclude that this source is a typical HBL even though the synchrotron SED peak lies in the intermediate region when the source is in its lowest state. \n\n(v) We were able to use a two-component (``blob-in-jet\") SSC model to describe multiple flux states of the source. \nThe flaring state is sufficiently described with the same model parameters for the jet component as the low state and with a different particle distribution and magnetic field for the blob component. \n\n(vi) The fluxes measured by the LAT in the HE regime and by Tuorla at optical energies are found to follow a log-normal distribution and to be strongly temporally correlated with one another.  \nThis is consistent with a SSC emission process.\n\n(vii) We searched for evidence of a periodic signal in the Tuorla optical data and in the {\\it{Fermi}}-LAT data, the two datasets for which we have the best-sampled light curves. No evidence for periodicity on any timescale is detected.\\\\\n\nIn the future, studies such as the ones presented should be performed on larger data sets, covering different emission states of the source being studied. Such data are expected to be provided at gamma-ray energies by the Cherenkov Telescope Array \\citep{CTA_book2019}. \n\n\\acknowledgments\n\nThe {\\sl Fermi}-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. \nThese include the National Aeronautics and Space Administration and the Department of Energy in the United States; the Commissariat \\`a l'Energie Atomique and the Centre National de la Recherche Scientifique\/Institut National de Physique Nucl\\'eaire et de Physique des Particules in France; the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy; the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in Japan; and the KA Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden. \nAdditional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'\\'Etudes Spatiales in France. This work performed in part under DOE Contract DE- AC02-76SF00515.\n\nThis research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, and by NSERC in Canada. This research used resources provided by the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy's Office of Science, and resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of VERITAS.\n\nThis research has made use of data from the MOJAVE database that is maintained by the MOJAVE team \\citep{Lister18}. The MOJAVE program is supported under NASA-{\\sl Fermi} grant NNX15AU76G.\n\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\n\nThis 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.\n\nThis research has made use of data from the OVRO 40-m monitoring program \\citep{Richards11} which is supported in part by NASA grants NNX08AW31G, NNX11A043G, and NNX14AQ89G and NSF grants AST-0808050 and AST-1109911.\n\nJ.V. was partially supported by the Alliance program, a partnership between Columbia University, NY, USA, and three major French institutions: \\'Ecole Polytechnique, Paris 1 Panth\\'eon-Sorbonne University and Science Po.\n\nT.S. was supported by the Academy of Finland projects 274477, 284495, and 312496.\n\nW.M. acknowledge support from CONICYT project Basal AFB-170002.\n\nY.Y.K.\\ and A.B.P.\\ were supported by the Russian Foundation for Basic Research (project 17-02-00197), the government of the Russian Federation (agreement 05.Y09.21.0018), and the Alexander von Humboldt Foundation.\n\nS.K. acknowledges support from the European Research Council under the\nEuropean Union's Horizon 2020 research and innovation program, under\ngrant agreement No~771282.\n\n\\software{AIPS (Greisen 2003), DIFMAP (Shepherd 1997), Fermi Science Tools (Fermi Science Support development team 2019)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nQuantum field theory (QFT) techniques provide powerful tools in describing three out of four fundamental interactions. The key problem in applying these methods to quantize gravity is that QFT based on Einstein's general relativity (GR) is perturbatively nonrenormalizable \\cite{'tHooft:1974bx}. However, following Weinberg's asymptotic safety conjecture  \\cite{Weinberg79}, it is possible that in the space of gravitational couplings there exist  non-Gaussian  ultraviolet (UV) fixed point(s) at which nonperturbative QFT techniques  could be used  to define a quantum  theory of gravity valid for any energy scale.\\footnote{There are known examples of perturbatively nonrenormalizable but asymptoticaly safe  QFTs  and  evidence  is growing that it is also the case for gravity  \\cite{Reuter06}.  } Among such techniques lattice methods play an increasingly important  role.\nA good test of any lattice  approach is its ability to \n reproduce  GR in the infrared limit  and also the existence \n of second (or higher) order phase transitions  associated with the perspective UV fixed point(s).\\footnote{Infinite correlation lengths characteristic of such phase transitions make it in principle possible to decrease lattice spacing  to zero, i.e. to investigate the continuum limit.} \nTherefore a study of the phase structure and the order of phase  transitions constitute  first steps in the quest for the continuum theory of quantum gravity. \n\\section{Causal Dynamical Triangulations}\n\nCausal Dynamical Triangulations (CDT) is a lattice model  based on the path integral quantization applied to GR. CDT gives a precise meaning to  a (formal) gravitational path integral\n\\beql{ZCDT}\n{\\cal Z}_{GR}= \\int {\\cal D}_{{\\cal M}}{[g]} e^{iS_{HE[g]}}\\quad \\rightarrow \\quad {\\cal Z}_{CDT} = \\sum_{{\\cal T}}e^{iS_R[{\\cal T}]} \\ ,\n\\end{equation}\napproximating continuous  geometries (described by all physically distinct metric tensors $g$) by  lattices constructed from two types of identical four-dimensional simplicial blocks\nglued together to form triangulations ${\\cal T}$.\nThe key assumption of CDT is an introduction of causal structure by foliating spacetime into Cauchy hyper-surfaces $\\Sigma$ of constant global proper time $T$.  Topologically a triangulation ${\\cal T}$ is $\\Sigma \\times T$, and one  requires that the topology of each spatial slice $\\Sigma$ is fixed. \n Each spatial layer $\\Sigma$ at integer (lattice) time $t$ is by definition constructed from equilateral tetrahedra. The four-simplices interpolate between consecutive spatial  layers of integer $t$  in such a way that also all intermediate Cauchy layers between $t$ and $t+1$ have the requested fixed spatial topology. This can be done by using just two types of four-simplices called the $(4,1)$ simplex and  the $(3,2)$ simplex.\\footnote{The numbers in parentheses  denote vertices lying in $t$ and $t\\pm 1$, respectively.} It is assumed that the interior of each four-simplex is a flat Minkowski  spacetime and local curvature is defined by the way the simplices are glued together. In Eq. \\rf{ZCDT} $S_R$ is the discretized Hilbert-Einstein action $S_{HE}$ obtained  following Regge's method for describing piecewise linear geometries \\cite{Regge:1961px}\n\\beql{SRegge}\nS_{R}=-\\left(\\kappa_{0}+6\\Delta\\right)N_{0}+\\kappa_{4}\\left(N_{(4,1)}+N_{(3,2)}\\right)+\\Delta \\ N_{(4,1)} \\  ,\n\\end{equation} \nwhere $ N_{(4,1)}$,  $ N_{(3,2)}$ and $N_0$ denote the total number of $(4,1)$ simplices, $(3,2)$ simplices and vertices, respectively, while\n$\\kappa_{0}$, $\\Delta$ and $\\kappa_4$ are three  bare coupling constants. They  are functions of the Newton's constant, the cosmological constant and the  asymmetry $\\alpha$ between lengths of time-like and space-like links in the lattice ($a_t^2 = - \\alpha\\  a_s^2$). \n\nIn order to study the regularised path integral one is forced to use  Monte Carlo techniques.\nThis can be done by  applying Wick rotation from positive to negative $\\alpha$ values which changes time-like links\ninto space-like links, i.e. changes real (Lorentzian) time $t^{(L)}$ into imaginary (Euclidean) time $t^{(E)}$ ($t^{(L)} \\to t^{(E)} = i t^{(L)}$), and also changes Lorentzian action into Euclidean action ($S_R^{(L)} \\to S_R^{(E)} = i S_R^{(L)}$).\nAccordingly, the path integral ${\\cal Z}_{CDT}$ \\rf{ZCDT} becomes a partition function which can be studied numerically.\n\n\\begin{figure}[htb]\n\\centerline{\n\\includegraphics[width=7cm]{phasediagramnewCC.pdf}}\n\\caption{Phase structure of 4-dim CDT in the $(\\kappa_0,\\Delta)$ bare couplings plane.  $\\kappa_4$ is fine-tuned to critical value consistent with infinite volume limit. }\n\\label{Fig:F1H}\n\\end{figure}\n\n\\section{Phase structure of 4-dim CDT}\n\nAll  results presented herein  were obtained for a particular choice of  fixed  spatial topology $\\Sigma = S^3$ (3-sphere).\\footnote{Preliminary studies of CDT with toroidal spatial topology confirm the existence of similar phase structure.} Historically, three phases of various spacetime geometry called $A$, $B$ and $C$  were discovered (see Fig. \\ref{Fig:F1H}).\nPhase $A$ (time uncorrelated geometry) and $B$ (time collapsed geometry) do not have clear physical interpretation. The most interesting one is phase $C$, which is separated from phase $A$ by a 1st order transition and from phase $B$ by a 2nd (or higher) order transition  \\cite{Ambjorn:2012ij}. The basic  feature of phase $C$ (now $C_{dS}$) is the emergence of large scale four-dimensional geometry \\cite{Ambjorn05} consistent with semiclassical (Euclidean) de Sitter universe \\cite{Ambjorn:2008wc}. It was also shown that in this phase quantum fluctuation of spatial volume are governed by the minisuperspace reduction of the Hilbert-Einstein action \\cite{Ambjorn:2008wc,Ambjorn:2011ph}. \n\nThe key tool in the analysis of the effective action of CDT was the transfer matrix (TM) parametrised by a spatial volume observable, i.e. the transition amplitude from spatial volume $n$ at (lattice) time $t$ to spatial volume $m$ at time $t+1$. Deep inside phase $C$ ($C_{dS}$ region in Fig. \\ref{Fig:F1H}) the TM  is well parametrised by \\cite{JGSTM}:\n\\begin{equation}\n\\langle n|M_{C_{dS}}|m \\rangle = \\underbrace{\\exp\\Bigg[- \\frac{1}{\\Gamma} \\frac{(n-m)^2}{ (n+m)}\\Bigg] }_{\\text{kinetic part}}  \\underbrace{\\exp\\Bigg[ - \\mu \\left(\\frac{n+m}{2}\\right)^{1\/3} + \\lambda \\left(\\frac{n+m}{2}\\right) \\Bigg] }_{\\text{potential part}} ,\n\\label{TMCdS}\n\\end{equation}\n where $\\Gamma$, $\\mu$ and $\\lambda$ are parameters related to the Newton's constant, the size of the CDT universe and the cosmological constant, respectively. However it was observed that in some region of the parameter space close to phase $A$ ($C_{b}$ region in Fig. \\ref{Fig:F1H}) the TM kinetic part bifurcates, such that \\cite{Ambjorn:2014mra}:\n\\begin{equation}\n\\langle n|M_{C_b}|m \\rangle =\\left[ \\exp \\left( -\\frac{1}{\\Gamma}\\frac{\\Big((n-m) - \\big[c_0 (n+m - s_b) \\big]_+\\Big)^2}{n+m}\\right) + \\right.\n\\label{TMCb}\n\\end{equation}\n$$\n+ \\left. \\exp \\left( -\\frac{1}{\\Gamma}\\frac{\\Big((n-m) + \\big[c_0 (n+m - s_b) \\big]_+\\Big)^2}{n+m}\\right)\\right]\\times  \\text{potential\\_part}[n+m] \\ ,\n$$\nwhich led to a discovery of a new phase transition associated with the parameters $s_b\\to \\infty$ and $c_0\\to 0$, where TM (\\ref{TMCb}) transforms into TM (\\ref{TMCdS}). \n\nThe newly discovered bifurcation phase $C_b$ has many intriguing geometric properties, including very large (potentially infinite)  Hausdorff dimension and also spectral dimension  becoming much larger than 4 for long diffusion times. The $C_{dS}-C_{b}$ phase transition is related to breaking of spatial homogeneity of phase $C_{dS}$ by the appearance  of  compact  spatial volume clusters concentrated around \"singular\" vertices with macroscopically large coordination number present  every  second time layer inside phase $C_b$ \\cite{JHEP1508}.  The volume clusters have (topologically) spherical boundaries and thus can  technically  be called \"black balls\". The \"black balls\" around   singular vertices in time $t$ and $t+2$ are causally connected through those intermediate  4-simplices which also share   one of the \"singular\" vertices. As a result one observes a  marked-out four-dimensional geometric structure related to  evolution  of time-correlated \"black ball\" volume condensations. The geometry of  phase $C_b$ requires further   studies, but a working hypothesis is that what is  observed might be actually a  quantum black hole. \n\nA key question remains about the order of the recently discovered bifurcation transition. In Ref. \\cite{Coumbe:2015oaa} an order parameter related to the appearance of high order vertices   was proposed:\n\\begin{equation}\\label{op}\n\\text{OP}_2= \\frac{1}{2} \\left[ \\Big| O_{max}\\big(t_0\\big) - O_{max}\\big(t_0+1\\big) \\Big|   +     \\Big| O_{max}\\big(t_0\\big) - O_{max}\\big(t_0-1\\big) \\Big|\\right],\n\\end{equation}\nwhere $ O_{max}\\big(t_0\\big)$ is  the highest coordination number among all vertices and $O_{max}\\big(t_0\\pm 1\\big)$ is the highest coordination number of a vertex observed in the neighbouring time slice. The position of the $C_{dS}-C_{b}$ transition is signalled by a peak in susceptibility $\\chi_{\\text{OP}_2} = \\langle \\text{OP}_2 \\rangle - \\langle \\text{OP}_2 \\rangle^2$ and it moves in the CDT bare couplings space $(\\kappa_0,\\Delta)$ when lattice volume $N_{(4,1)}$ is changed. The volume dependence of critical $\\Delta$ (for $\\kappa_0=2.2$ fixed) can be fitted with the following function: \n\\begin{equation}\\label{voldep}\n\\Delta^{crit}\\big(N_{(4,1)}\\big)=\\Delta^{crit}\\big(\\infty\\big) - \\alpha \\ N_{(4,1)}^{\\ -1\/\\gamma} \\ ,\n\\end{equation}\nand  measured value of the critical exponent $\\gamma = 2.71 \\pm 0.34$ strongly supports the conjecture that the bifurcation transition is a 2nd (or higher) order phase transition.\\footnote{For a 1st order transition one should have $\\gamma=1$.}\n\n\n\\section{Summary and conclusions}\nWe have briefly presented the recently updated 4-dim CDT phase diagram, including the semiclassical phase $C_{dS}$ which seems to be  CDT realisation of the correspondence principle, and the newly discovered bifurcation phase $C_b$ with very nontrivial geometric properties. We have provided evidence that the $C_{dS}$-$C_{b}$ phase transition, related to breaking of homogeneity of semiclassical phase, is a 2nd (or higher) order phase transition which may in principle allow one to approach the perspective UV fixed point of quantum gravity \\cite{Ambjorn:2016cpa}. \n\n\n\n\\section{Acknowledgements}\nThe author wishes to  acknowledge the support of the grant DEC- 2012\/ 06\/A\/ST2\/00389 from the National Science Centre Poland. The results described herein were obtained in close collaboration with J. Jurkiewicz, J. Ambj\\o rn,  R. Loll,  A. G\\\"orlich and D.N. Coumbe. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The not-so-simple questions that remain.}\n\nThe recent discoveries in particle physics are built upon the platform created by a solid quantum field theory of the strong interactions: Quantum Chromodynamics (QCD). Without it, we could not \"see\" through the haze of multiple particle production at the LHC and study different decay channels of the Higgs boson. Eventhough QCD is robust enough to encompass perturbative and non-perturbative approaches that allow us to study a variety of phenomena in hadron colliders, we are always tempted to put it to the test in the high temperature and\/or density regimes. For example, if we already know how to study proton-proton collisions ($p+p$ collisions) at LHC energies, starting with parton level cross sections using QCD matrix elements that involve quarks and gluons, that later become hadron level cross sections and can be connected to shape obervables or jet observables, how much of this knowledge can be used to simulate jet formation in nucleus-nucleus collisions ($A+A$ collisions, where $A$ can be any nucleus)?. Do we assume that hadronization and jet formation in the collision of large systems happens just as it does in small systems? It turns out that these questions are just the tip of the iceberg: the physics of heavy-ion collisions at relativistic energies are a portal to new phenomena of the strong interactions. They are the means to explore regimes of QCD under \\textit{extreme} conditions: what happens to strongly interacting matter when it is in a thermal bath and\/or when it is highly compressed? does this mean that we expect ordinary nuclear matter to phase transition into new states of matter? how do we know theoretically and experimentally what are the signals we should look for when these phase transitions occur? Suddenly, heavy-ion collision experiments represent our best tool to learn about the questions that remain: the rich dynamical QCD phenomena that emerges under these conditions, an arena to further extend the vast knowledge we already have on hadron and QCD physics. \n\n\nNow, if you are already an expert on using QCD to study $p+p$ collisions, then you might be surprised to learn that from there, it is an easy road to learn more about $A+A$ collisions. This is the road I plan to take in these couple of lectures, to discuss about the phases of nuclear matter, about QCD now under extreme conditions, about quantum field theory techniques for larger systems and about code used in $p+p$ collisions to be able to simulate $A+A$ collisions. In these two lectures I will aim to give you the basics of heavy-ion collisions at relativistic energies and the physics we can do with them. These lectures will rely heavily on previous enlightening talks and short courses given at different venues (see for example \\cite{Talks2000}) and on various articles and textbooks~\\cite{Busza2018, Wong1994, Csernai1994, Shuryak2004, Vogt2007, Flork2010, Sarkar2010}. \n\n\n\nThe structure of the lectures is as follows: in Section~\\ref{sec:QCDu} we review the essential ideas that helped establish QCD as the theory of strong interactions and use them to have an initial discussion about the nuclear matter phase diagram, then in Section~\\ref{sec:rhics} we review the basic ideas used to study relativistic heavy-ion collisions and the evolution of the system after the collisions, in Section~\\ref{sec:theqgp} we discuss the quark-gluon plasma and its basic properties, in Section~\\ref{sec:phasecritical} we introduce the tools required to study the phase diagram of nuclear matter and the observables with which phase transitions and critical behavior can be probed, finally in Section~\\ref{sec:seasons} we highlight observables that help characterize the QGP formation and evolution. We provide final remarks and future prospects in Section~\\ref{sec:final}.\n\n\n\n\\section{- \\textit{QCD as usual} - said no one, ever.}\n\\label{sec:QCDu}\n\n\n\\subsection{The basics}\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.6]{alfaS.jpg}\n\\caption{Measurements of $\\alpha_s$ as a function of the energy scale $Q$, as reported in the Quantum Chromodynamics review of the Particle Data Group ~\\cite{PDG2018}. From each set of experiments, there is a value of $\\alpha_s$ that has been extracted using different levels of QCD perturbation theory, which are indicated in brackets.}\n\\label{fig:asq2016}\n\\end{figure}\n\n\nIt is now carved in stone (which is to say, it has been established by numerous experiments) that nature favours an $SU(3)_c$ gauge theory as the mathematical framework with huge phenomenological consequences that shape our current understanding of the strong interactions at a fundamental level. As is shown in Fig.~\\ref{fig:asq2016}, QCD, the theory of strong interactions that Nature favours, is everywhere. The emblematic characteristics of this theory make it unique and powerfull: $\\alpha_S$ running with the energy scale of a wide range of experiments and $SU(3)_c$ well established as the underlying symmetry\\footnote{For more interesting discussions about the experimental tests of QCD and about how to extract $\\alpha_s$ beyond perturbative QCD, see for example~\\cite{QCDstatus, stan}.}. This supports the formulation of a microscopic theory in terms of elementary fields (quarks and gluons), whose interactions obey the principles of a relativistic quantum field theory with non-abelian gauge invariance.\n\n\nSince the early development of QCD, people started to amass tools that would allow to study pheonomena at the heart of every experiment conceived, with the highest precision possible. As shown in Fig.~\\ref{fig:qcdish}, nowadays the calculation of observables using perturbative and non-perturbative QCD-inspired tools or QCD-inspired effective theories and their connections, makes it possible to do Standard Model precise calculations and to look for New Physics (see for example~\\cite{higgstools}). QCD in heavy-ion physics has flourished (via many effective theories and extensions to QFT at finite temperature) that provide new tools to describe the initial conditions of ions moving at relativistic velocities, the intermediate stages where the QGP has formed and the later stages where the high temperature and density of gluons and quarks, become hadrons and jets (see for example~\\cite{iancu}).\n\n\n\n\\begin{figure}\n\\begin{center}\n\t\t\\begin{tikzpicture}[scale=0.7,transform shape\n\t\t\t\\Huge\n\t\t\t\\begin{scope}[mindmap, concept color=red, text=white, scale=0.7,transform shape]\n\t\t\t\\tikzstyle{level 1 concept}+=[sibling angle=120]\n\t\t\t\\tikzstyle{level 2 concept}+=[sibling angle=45]\n\t\t\t\n\t\t\t\\tikzstyle{root concept}+=[minimum size=1.5cm, text width=1cm] \n\t\t\t\n\t\t\t\\node[concept] (qcd) {QCD}\n\t\t\t\t[clockwise from = 35]\n\t\t\t\tchild[concept color=blue]{node[concept] (npqcd) {npQCD}\n\t\t\t\t\t[clockwise from = 45]\n\t\t\t\t\tchild[concept color=blue]{node[concept] (lattice) {lattice}}\n\t\t\t\t\tchild[concept color=blue]{node[concept] (sde) {SDEs}}\n\t\t\t\t\tchild[concept color=blue]{node[concept] (sumr) {sum rules}}\n\t\t\t\t}\n\t\t\t\tchild[concept color=green!50!black]{node[concept] (effective) {effective theories}\n\t\t\t\t    [clockwise from = -45]\n\t\t\t\t\tchild[concept color=green!50!black]{node[concept] (hqet) {heavy quark ET}}\n\t\t\t\t\tchild[concept color=green!50!black]{node[concept] (scet) {SCET}}\n\t\t\t\t\tchild[concept color=green!50!black]{node[concept] (nrqcd) {nrQCD}}\n\t\t\t\t}\n\t\t\t\tchild[concept color=black!50!red]{node[concept] (pqcd) {pQCD}\n                    [clockwise from = -140]\n\t\t\t\t\tchild[concept color=black!50!red]{node[concept] (mcs) {loops, legs $\\to$ $\\epsilon$ structure}}\n\t\t\t\t\tchild[concept color=black!50!red]{node[concept] (beyond) {duality}}\n\t\t\t\t\tchild[concept color=black!50!red]{node[concept] (factor) {factorization, jets $\\to$ MCs}}\n\t\t\t\t}\n\t\t\t;\n\t\t\t\\end{scope}\n\t\t\\end{tikzpicture}\n\t\\end{center}\n\\caption{QCD is everywhere. Non-perturbative QCD (npQCD) has a vast reach and we here highlight Schwinger-Dyson equations (SDEs) are solved for the quark and gluon propagator with dressed versions of the QCD vertices to get hadron masses. Lattice QCD can solve numerically the QCD equations of motion and produce incredible results for hadron spectra and thermodynamic properties of nuclear matter. Perturbative QCD (pQCD) was the historical way to establish QCD as a gauge theory of the strong interactions and provided the tools to predict and describe \\textit{jets}, which are now known as the standard signature of hard processes. Throughout the development of hadron and electroweak physics, effective theories have paved the way for modern QCD phenomenology.}\n\\label{fig:qcdish}\n\\end{figure}\n\n\n\n\n\\subsection{The running and confinement}\n\n\nUnder \\textit{ordinary} circumstances, quarks are confined within hadrons. The color potential between quarks inside hadrons at long distances is of order 1 fm and is linear. Separation of quarks requires an infinite amount of energy, so this potential makes hadrons combine into zero net color charge hadrons. From the perturbative regime, confinement can be seen as a direct consequence of the gluon self-interaction. Given that the strength of the strong interaction is characterized by the coupling constant $\\alpha_s= \\frac{g_s^2}{4\\pi}$, we can expose its running (made evident by experiment), by demanding that all relevant observables $\\mathcal{O}(Q^2;\\alpha_s)$ constructed with QCD should be renormalization scale independent. This is encoded in the Renormalization Group Equation as\n\\begin{equation}\n\\left[\\mu^2\\frac{\\partial}{\\partial\\mu^2}+\\beta(\\alpha_s)\\right]~\\mathcal{O}\\left(\\alpha_s,\\frac{Q^2}{\\mu^2}\\right)=0\\nonumber\n\\end{equation}\nwhere the \\textit{beta function} is defined as\n\\begin{eqnarray}\n\\beta(\\alpha_s) \\equiv \\frac{d\\alpha_s}{d\\log\\frac{Q^2}{\\mu^2}}.\n\\label{betabis}\n\\end{eqnarray}\nThe beta function can be obtained perturbatively to study the running of alpha in the high energy regime (small values of  the coupling). At lowest order, \n\\begin{equation}\n\\beta(\\alpha_s)=-b_0\\alpha_s^2 ~~~~~~\\mbox{with}~~~~~~ b_0=\\frac{1}{2\\pi}\\left(\\frac{11}{6}N_c-\\frac{1}{3}N_f\\right),\n\\label{beta}\n\\end{equation}\nwhere $N_c=3$ and $N_f$ are the number of color and flavour degrees of freedom of the theory, respectively. We can now solve Eq.~\\ref{betabis} and ~\\ref{beta}, assuming $b_0>0$ (which is true provided $N_f < 16$) and get the famous \\textit{running} of $\\alpha_s$:\n\\begin{equation}\n\\alpha_s(\\mu^2)=\\frac{1}{b_0\\log(\\mu^2\/\\Lambda^2)}.\\label{e88}\n\\end{equation}\nThe parameter $\\Lambda$ describes the boundary condition of the first order differential equation defining the running of $\\alpha_s$, and corresponds to the scale at which the coupling becomes infinity. Since $\\alpha_s$ is not an observable, it can contain all the terms that are $\\mu$ dependent, in order to achieve a $\\mu$-independent observable $\\mathcal{O}$ that has a power-series representation in terms of $\\alpha_s$. As shown in Fig.~\\ref{fig:screen},  this behaviour shows the difference between the screening and anti-screening efects of the color charge as compared to the electric charge. The observables that can be described where perturbation theory can be applied are usually those where the transferred momentum satisfies $Q^2\\gtrsim 1$ GeV$^2$ which marks the separation between the perturbative and non-perturbative regimes.\n\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.45]{beta-QED-1.jpg} ~\\includegraphics[scale=0.55]{beta-QCD-1.jpg}\n\\caption{On the left we can see the screening effect that quark loops provide for the electric charge, which corresponds to the notion of an effective charge $e(r)$ that becomes smaller with larger distances. This is encoded in the QED beta function $\\beta(r) = -\\frac{de(r)}{d \\log (r)}$, which is positive. On the right we see that the color charge gets screening from the quark loops but antiscreening from the gluon loops, which corresponds to the two contributions with opposite signs to $b_0$ in the QCD beta function. The QCD beta function is negative and so the effective strong coupling becomes small at short distances~\\cite{alan}.}\n\\label{fig:screen}\n\\end{figure}\n\n\n\nWith the aid of Eqs.~\\ref{betabis} and \\ref{beta}, we can define a transferred momentum value $\\Lambda_{QCD}$ small enough such that the coupling blows up\n\\begin{equation}\n1+b_0\\alpha_s(\\mu^2)\\log(\\Lambda_{QCD}^2\/\\mu^2)=0 ~~~~~\\Rightarrow ~~~~~\n   \\Lambda_{QCD}^2=\\mu^2~e^{-\\frac{1}{b_0\\alpha_s(\\mu^2)}}\n\\end{equation}\nwhere $\\Lambda_{QCD}$ is a renormalization scheme dependent quantity. In the $\\overline{\\mbox{MS}}$ scheme and for three active flavors, its value is of order $\\Lambda_{QCD}\\sim 200 - 300$ MeV. This has a huge impact on the phenomenological applications of pQCD, since all dimensionful QCD results with small transferred momentum, scale with $\\Lambda_{\\mbox{QCD}}$. In this context, we are tempted to say that the existence of this scale is a key ingredient for the emergence of baryon masses and thus of the mass of the visible universe, and hastely conclude that this is what explains hadron confinement. Nevertheless, the emergence of confinement is much more than this, so we must pursue an all ecompasing strategy to understand confinement and the breaking of chiral symmetry in QCD, which is also connected to the emergence of hadron masses.\n\n\n\n\n\n\n\\subsection{Chiral symmetry in QCD}\n\\label{sec:chiral}\n\nThe QCD lagrangian\n\\begin{equation}\n   {\\mathcal{L}}_{QCD} = \\sum_{\\alpha=1}^{N_c^2-1} \\sum_{a,b=1}^{N_c} \\sum_{i=1}^{N_f}\\overline{\\psi}_i^a\\left(i\\gamma^\\mu(\\partial_\\mu\\delta^{ab} \n   + i ~g_s ~A_\\mu^{ab}) - m_i\\delta^{ab}\\right)\\psi_i^b\n   -\\frac{1}{4}G^\\alpha_{\\mu\\nu}G_\\alpha^{\\mu\\nu},\n\\label{qcdlag}\n\\end{equation}\nis that of a non-abelian gauge theory with a local symmetry group $SU(N_c)$ with $N_c=3$ where the fundamental fields are matter fields $\\psi^a_i$ (quarks) with masses $m_i$ and massless gauge fields $A^\\alpha_\\mu$ (gluons). In this lagrangian the strong coupling $\\alpha_s=g_s^2\/4\\pi$ enters both in the quark-gluon interaction term $\\overline{\\psi}^a\\gamma^\\mu A^{ab}_\\mu\\psi^b$ (with $A_\\mu^{ab}=A_\\mu^\\sigma(\\tau_\\sigma)^{ab}$) and in the gluon self-interaction term through the field strength tensor, defined as $G^\\alpha_{\\mu\\nu}=\\partial_\\mu A_\\nu^\\alpha - \\partial_\\nu A_\\mu^\\alpha + g_s~ f^{\\alpha\\beta\\sigma}A^\\beta_\\mu A^\\sigma_\\nu$. The $N_f$ quark fields  belong to the fundamental representation of the color group ($N_c$ -dimensional), antiquark fields to the complex conjugate of the fundamental representation (also $N_c$ -dimensional) and gluon fields to the adjoint representation ($N_c^2 - 1$ -dimensional), where $a,b$ run from 1 to $N_c$ and $\\alpha,\\ \\beta,\\ \\sigma$ run from 1 to $N_c^2 - 1$.\n\nIf we imagine a universe in which all the quarks have the same mass $m$, then the quark sector of the QCD lagrangian in Eq.~\\ref{qcdlag} is (we focus on the flavour indices and we omit colour indices for the purpose of the discussion)\n\\begin{equation}\n   {\\mathcal{L}}_{q}=\\sum_{i=1}^{N_f}\\overline{\\psi}_i\\left(i\\gamma^\\mu(\\partial_\\mu\n   + i ~g_s ~A_\\mu) - m\\right)\\psi_i \\nonumber\n\\end{equation}\nThis lagrangian is invariant under continous global \\textit{vector flavour} transformations of $SU(N_f)$ since $\\psi_i \\rightarrow \\psi_i'=e^{-i\\alpha^B(T^B)_i^{\\,\\,\\, j}}\\psi_j$. In fact using Noether's theorem, we find $N_f^2 - 1$ conserved currents associated with this symmetry since $\\partial^\\mu j_\\mu^A=0$ for \n\\begin{equation}\n   j_\\mu^A(x)=\\overline{\\psi}_i(x)\\gamma_\\mu(T^A)^j_{\\,\\, i}\\psi^i_j(x),\n   \\label{jmuA}\n\\end{equation}\nwhere $A,B,C=1,...,N_f^2-1$. The charges (or generators) are obtained from the space integration of the density current $j_0^A$\n\\begin{eqnarray}\n   Q^A=\\int d^3x\\ j_0^A(x), \\nonumber\n\\end{eqnarray}\nwhere $Q^A$ satisfy the $SU(N_f)$ algebra $[Q^A,Q^B]=if^{ABC}Q^C$. Furthermore, current conservation implies that these charges are constant in time, in the sense that they commute with the Hamiltonian $[H,Q^A]=0$ of the theory. The transformation properties of the fields translate into transformation properties of the states (e.g. $Q^A |{\\mathbf{p}},i\\rangle = (T^A)^j_{\\,\\, i}|{\\mathbf{p}},j\\rangle$, supressing spin indices) and if the vacuum state is also invariant under these transformations, all one-particle states of the fundamental representation multiplet, have equal masses $m$. In quantum mechnics, a symmetry of the hamiltonian of the theory reflects in its energy spectrum, most of the time. In this case we have a flavour symmetry transformation that reflects on a degeneracy on the mass spectrum of the theory. This realization of a symmetry is called \\textit{Wigner-Weyl mode}. As we will see in what follows, at energies $1.5 - 3$ GeV there is data showing parity doubling in the hadron spectrum which could be explained through this flavour symmetry in a universe we imagined with quarks that have the same mass. \n\n\nIf we modify the universe slightly to now consider quark flavors with different masses\n\\begin{equation}\n   {\\mathcal{L}}_{q}=\\sum_{i=1}^{N_f}\\overline{\\psi}_i\\left(i\\gamma^\\mu(\\partial_\\mu\n   + i ~g_s ~A_\\mu) - m_i\\right)\\psi_i,\n   \\label{lq}\n\\end{equation}\nthe mass term now spoils $SU(N_f)$ invariance, therefore currents are not conserved ($\\partial^\\mu j_\\mu^A \\neq 0$) and instead of Eq.~\\ref{jmuA}, we find \n\\begin{eqnarray}\n   \\partial^\\mu j_\\mu^A = -i \\sum_{i, j=1}^{N_f}(m_i - m_j)\\overline{\\psi}_i(T^A)^i_{\\,\\, j}.\n   \\nonumber\n\\end{eqnarray}\nWe could still get back the flavour symmetry in an approximate manner: we could divide quarks into two groups \\textit{light quarks} $u,d,s$ and \\textit{heavy quarks} $c,b,t$. The mass difference between each group is large ($\\gtrsim 1$ GeV) but, an approximate flavour symmetry could still be realised for light quarks: $SU(2)$ for $u,d$ or  $SU(3)$ for $u,d,s$, whereby $\\partial^\\mu j_\\mu^A \\sim 0$.\n\n\nQuarks can also be transformed by means of unitary transformations that include the $\\gamma_5$ matrix which are called \\textit{axial flavour} transformations. In infinitesimal form these transformations are $\\delta\\psi_i=-i~\\delta\\alpha^A~(T^A)_i^{\\,\\,\\, j}\\gamma_5\\psi_j$ and the current associated with this symmetry is the axial current\n \\begin{eqnarray}\n   {j_5}_\\mu^A(x)&=&\\overline{\\psi}_i(x)\\gamma_\\mu\\gamma_5(T^A)^j_{\\,\\, i}\\psi^i_j(x).\n\\end{eqnarray}\nAgain, if we consider the universe where the quark masses are equal, then under these transformations the invariance of the lagrangian ${\\mathcal{L}}_q$ in Eq.~\\ref{lq} is spoiled by the mass as $\\delta{\\mathcal{L}}_q=2i~m~\\delta\\alpha^A~\\overline{\\psi}^i(T^A)_i^{\\,\\,\\, j}\\gamma_5\\psi_j$. So, contrary to the invariance of the lagrangian under vector flavour transformations which is attained with quarks of the same mass, invariance under axial flavor transformations is spoiled even in a universe with quarks of the same mass. Now, of course, in a universe where quarks are massless, the quark lagrangian is invariant under both the vector flavour and axial flavour transformations, which together form the \\textit{chiral transformations}. In this case, both the vector and axial currents are conserved: $\\partial^\\mu j_\\mu^A=0, \\partial^\\mu j_{5 \\mu}^A=0$ with the corresponding charges\n\\begin{eqnarray}\n   Q^A=\\int d^3x\\ j_0^A(x),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, Q_5^A=\\int d^3x\\ j_{5 0}^A(x),\n   \\nonumber\n\\end{eqnarray}\nwhere the $Q^A$ and $Q_5^A$ satisfy the commutation relations $\\left[Q^A,Q^B\\right]=if^{ABC}Q^C$, $\\left[Q^A,Q_5^B\\right]=if^{ABC}Q_5^C$ and $\\left[Q_5^A,Q_5^B\\right]=if^{ABC}Q^C$. \n\nIn this context, the axial charges do not form an algebra but, if we define left-handed and right-handed charges as $Q^A_L=\\frac{1}{2}(Q^A - Q^A_5)$ and $Q^A_R=\\frac{1}{2}(Q^A + Q^A_5)$, the newly defined charges decouple and operate separately $\\left[Q^A_L,Q^B_R\\right]=0$, generating each an $SU(N_f)$ group as $\\left[Q^A_L,Q^B_L\\right]=if^{ABC}Q^C_L$ and $\\left[Q^A_R,Q^B_R\\right]=if^{ABC}Q^C_R$. The chiral group is then decomposed into the direct product of two $SU(N_f)$ groups: $SU(N_f)_L\\otimes SU(N_f)_R$. \n\n\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.3]{pr246_15-eps-converted-to.pdf}\\includegraphics[scale=0.3]{pr246_16-eps-converted-to.pdf}\n\\caption{OPAL Collaboration measurement of the $\\rho$ (vector meson, $I^G\/J^P = 1^+\/1^-$) and $a_1$ (axial-vector meson, $I^G\/J^P) = 1^-\/1^+$) mass spectral distributions. The $\\rho$ spectrum peaks at $\\sim 0.6$ GeV$^2$ while the $a_1$ spectrum peaks at $\\sim 1.2$ GeV$^2$~\\cite{parityfig}}.\n\\label{fig:partners}\n\\end{figure}\n\n\nIt turns out that in our universe, chiral symmetry is not exact, quark masses break it explicitly. In practise, in the light quark sector chiral symmetry is approximate, so the breaking can be treated as a perturbation. How can we \\textit{see} that this symmetry is broken? and, what is the signature of this approximate symmetry?. Well, suppose the symmetry is realized in the Wigner-Weyl mode. In the massless quark limit a chiral transformation acting on a massive state changes the parity of the state $Q^A_5 |M,s,{\\mathbf{p}},+,i\\rangle = (T^A)^j_{\\,\\, i}|M,s,{\\mathbf{p}},-,j\\rangle$.\nThis means that we should find parity partners for the massive states. Even in the massive quark limit, when we consider the light quark sector, the degeneracy within parity doublets is lifted, but the masses should remain close to each other. We do not observe parity doublets (copies of, say the neutron and the proton) in the hadron spectrum, so under these conditions, the Wigner-Weyl mode realization of chiral symmetry does not happen. We could then consider another mechanism, the spontaneous breaking of chiral symmetry, also referred to as \\textit{Nambu-Goldstone mode}. In this case the generators do not annihilate the vacuum ($Q^A_5 |0\\rangle \\neq 0$) and in fact new pseudoscalar states are created: $Q^A_5 |0\\rangle = |A,-\\rangle$, with $A=1,...,N_f^2 - 1$.\n\n\n\nIn the massless limit, the charges commute with the hamiltonian therefore these states are $N_f^2 - 1$ pseudoscalar massless particles, Nambu-Goldstone bosons. If $N_f=3$, this would correspond to eight pseudoscalar bosons with small masses. Indeed, the observed pseudoscalar mesons made up of only $u, d, s$ quarks form an octet with $ J^P=0^-$: $\\pi^0, \\eta, \\pi^+, \\pi^-, K^0, K^+, K^-, \\bar{K^0},$ which are the lightest mesons. Beyond the massless limit, an important measurement involves comparing the experimental mass spectral distributions of parity partner mesons. The main ingredients for these mass spectral distributions are the vector and axial correlators $\\mathrm{Im}\\Pi_V(q), \\mathrm{Im}\\Pi_A(q)$. Chiral symmetry implies that these correlators should be identical to all orders in perturbation theory and they were measured at LEP with $\\tau$ decays into even and odd number of pions~\\cite{paritydoubling-exp}. In particular, the seminal measurement of the $\\rho$ (vector meson, $I^G\/J^P = 1^+\/1^-$) and $a_1$ (axial-vector meson, $I^G\/J^P) = 1^-\/1^+$) mass spectral distributions are shown in Fig.~\\ref{fig:partners}. The $\\rho$ spectrum peaks at $\\sim 0.6$ GeV$^2$ while the $a_1$ spectrum peaks at $\\sim 1.2$ GeV$^2$. This large difference cannot be due to finite current quark masses since, as we already discussed, this produces small differences compared with the masses of the mesons themselves. In this case the differences are of the order of the mass of the $\\rho$, so the mechanism that could help explain this, is the spontaneous breaking of chiral symmetry in vaccum\\footnote{For further details see for example Ref.~\\cite{paritydoubling}}.\n\n\n\n\n\n\n\n\\subsection{The many faces\/phases of nuclear matter}\n\\label{sec:faces}\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.25]{phasedg.pdf}~~~\\includegraphics[scale=0.8]{QCD-phaseD-NICA.jpg}\n\\caption{On the left, the phase diagram for QCD matter shows the Hadron Gas phase and the Quark-Gluon Plasma phase in the temperature ($T$) and baryon chemical potential ($\\mu_B$) plane~\\cite{phase-diag-conj}. On the right, the phase diagram for QCD matter shows the Hadron Gas phase, the Quark-Gluon Plasma phase and the Quarkyonic phase in the temperature ($T$) and baryon density ($n$) normalized to the cold nuclei baryon density ($n_o$) plane~\\cite{phase-diag-NICA}.}\n\\label{fig:PhDiag}\n\\end{figure}\n\n\n\nIn heavy-ion collisions at relativistic energies, we reach the conditions under which confinement is no more and we obtain a medium of thermally equilibrated hadronic matter where now the quarks and gluons can move freely. In order to describe these conditions, we identify two important parameters: the temperature $T$ and the baryon number density $n_B$ (or its conjugate variable, the baryon chemical potential $\\mu_B$). Now, as we discussed before, the intrinsic scale in QCD is $\\Lambda_{QCD}\\sim 200$ MeV, so we expect a partons-to-hadrons phase transition around $T\\simeq\\Lambda_{QCD}\\sim 200$ MeV and $n_B\\sim \\Lambda_{QCD}^3\\sim 1$ fm$^{-3}$. Also, the coupling constant runs towards smaller values with increasing energy scale so we can anticipate that both confinement dissolves and this may be somewhat connected to the fact that chiral symmetry breaks. We could argue that QCD matter must undergo phase transitions at high baryon and\/or energy densities. The anticipation and excitement of the posibility to create and probe these \\textit{novel} (back then!) phases of matter was proposed~\\cite{faces} just a decade after the birth of the hadron Quark Model~\\cite{partons} and two years after QCD was established as a non-Abelian quantum field theory with asymptotic freedom~\\cite{octet}\\footnote{For more historical insight into the early developments of the QCD phase diagram and the early discussions on the Quark-Gluon Plasma, see for example Ref.~\\cite{pheno-review-QGP}.}. \n\n\n\n\n\n\n\n\nIn Fig.~\\ref{fig:PhDiag}, the panel on the left is a conjectured phase diagram for QCD matter that shows the Hadron Gas phase and the Quark-Gluon plasma phase separated by a phase transition line that ends in the \\textit{critical end point} (CEP). At $\\mu_B\\sim 0$ the transition is more of a countinous crossover, which as mentioned before, must happen around $\\Lambda_{QCD}$. Recently, lattice calculations and effective model theoretical estimations have provided constraints and have allowed for a better picture of where the CEP might be located and for a better determination of the phase transitions and this has been matched by the identification of observables relevant for criticality studies~\\cite{locationCEP}. This has been accompanied by several new heavy-ion collision experiments that will be able to scan this phase diagram, beyond what current experiments have achieved\\footnote{If you are interested on the thermodynamics and statistical mechanics tools to study QCD and hadrons at high temperature, you can start here~\\cite{QGPbooks} and go from there.}. This is schematically shown in the panel of the right in Fig.~\\ref{fig:PhDiag} where now three phases are separated by a chiral phase transition, a deconfinement transition and a pseudocritical crossover line. The experimental scans using heavy-ion collisions are shown for the CERN-LHC, RHIC-BES, FAIR-SIS and JINR-NICA programs and where the CEP location can be studied using overlapping scan programs. \nThe QGP created at RHIC and LHC at high energies, contains almost equal amounts of matter and antimatter, so the corresponding experimental scan, stays close to the vertical axis: low $\\mu_b$ or $n_B$, high $T$. The planned collisions at FAIR and NICA, will allow to create quark-gluon plasma with an excess of matter over antimatter, and so the corresponding experimental scans will be able to explore the bulk of the phase diagram. \n\n\n\nIn the next sections, we will go deeper into some of these features of the QCD phase diagram. For now, let us turn to heavy-ion physics and collision experiments and discuss the basics of their well-established usefulness to scan this phase diagram. \n\n\n\n\n\n\n\n\n\n\\section{Relativistic heavy ion collisions}\n\\label{sec:rhics}\n\n\\subsection{What happens when two ultra relativistic nuclei collide?}\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.2]{evolution-HICs-NEWa.jpg}~~~~~~~~~~~~~~~~~~~~~~~~~~~\\includegraphics[scale=0.2]{evolution-HICs-NEWb.jpg}\n\\caption{Four stages of a heavy-ion collision and its evolution. (a) Shows the two Lorentz contracted heavy-ions moving towards each other in the CM frame, then (b) shows the overlaping nuclei at the collision stage. (c) Just after the collision the heavy-ions travel accross each other and generate a volume of high temperature and energy density and finally in (d), the system expands and cools down, eventually fragmenting into hadrons that travel to the detector.\\cite{hiccol}}\n\\label{fig:HICcol}\n\\end{figure}\n\n\nIn preparation for a collision, the ions are accelerated to relativistic velocities so much so, that the Lorentz contraction factor $\\gamma$ is of the order of 100 at RHIC and of the order of 2500 at LHC. If we choose the laboratory frame as the center-of-mass (CM) frame in a collider, this means that each incident nucleus is a Lorentz contracted disc of the order of $14\/\\gamma$ fm for Au and Pb nuclei (14 fm being the typical size of these large nuclei). For example, heavy-ion collisions at BNL and CERN have total collision energy ($\\sqrt{s_{NN}}$) per nucleon-nucleon pair in the CM frame of 200 GeV at RHIC and 2.76 TeV at LHC. Fig.~\\ref{fig:HICcol} shows four stages of the heavy-ion collision and its evolution: (a) first the Lorentz contracted heavy-ions come into the collision stage, (b) then they travel accross each other and generate (c) a volume of high temperature and energy density and finally the system expands and cools down, (d) eventually fragmenting into hadrons that travel to the detector. The stage highlighted in panel (c), is just after the collision where the conditions are optimal for creating QGP. \n\nThe incoming ions are very thin discs of interacting transverse color fields and have on average more quarks than antiquarks with color charges, so that they become sources of colored gluons. When the nuclei pass each other, long color fields fill the space between the receding two Lorentz contracted ions, which makes them loose energy and then they gradually decay into $q-\\bar{q}$ pairs and gluons. Now this initial stage is actually dynamic and non-uniform per event, the initial state energy and momentum distributions fluctuate, so modeling the initial state is the first non-trivial task to be approached using high-density\/temperature QCD techniques. Furthermore, we know that most of the incident partons lose energy but only a small - albeit important - fraction of them, will undergo hard interactions. These high-$p_T$ hard-interacting partons will be the seed to produce in-medium jets.\n\n\nOn average the energy density around the midpoint between the two Lorentz contracted ions is far grater than the energy density in a hadron (for example at the LHC with $\\sqrt{s_{NN}} = 2.76$ TeV, $\\langle\\epsilon\\rangle \\sim 12$ GeV\/fm$^3 \\sim 20 ~\\epsilon_{hadron}$). On the other hand, lattice QCD tells us that QCD matter in thermal equilibrium at $T \\sim$ 300 MeV has $\\epsilon \\sim$ $12~ T^4$ = 12.7 GeV\/fm$^3$. This means that the medium formed after heavy-ion collisions cannot be described just as a collection of distinct individual hadrons, rather it has to be made out of a high density of quarks and gluons, given the sheer amount of energy density available in a small volume after the collision. Moreover, before the collision the entropy of the two incident nuclei is practicaly null whereas after the collision the final state can contain as many as $10^4$ particles, so this entropy is produced quickly, in the initial moments after the collision. \n\n\nFurthermore, as can be seen on the left panel in Fig.~\\ref{fig:centrality}, the ions may collide head-on or may only partially overlap at the collision stage, so in the overlap region there are conditions that facilitate QGP formation. This \\textit{centrality} of the collision gives also a characteristic initial shape in the transverse plane to the formed medium, that tends to be more lenticular as the collisions are more head-on or central, as shown in the right hand side of Fig.~\\ref{fig:centrality}. In fact, the deviations from circular symmetry - or lumpiness - in the initial shape of the QGP and fluctuations of the incident nuclei, lead to preassure anisotropies in the hydrodynamic fluid stage of the plasma. This in turn, leads to anisotropies in the expansion velocity of the products of the hadronized plasma, so that these finally produced hadrons display an azimuthal momentum distribution with the footprint of the initial symmetry or lumpiness. These features can be characterized with the extraction of the \\textit{flow coefficients} $v_n$ of the azimuthal particle distribution as\n\\begin{equation}\n\\frac{1}{N} \\frac{dN}{d\\phi} = \\frac{1}{2\\pi}\\left[\n   1 + \\sum_n 2v_n\\cos(n(\\phi - \\Psi)) \\right],\n   \\label{floweq}\n\\end{equation}\nwhere $\\phi$ is the azimuthal angle and $\\Psi$ is the reaction plane, which is determined by the impact parameter and beam direction in a Glauber model, or is determined event-by-event, after the fact, by the weighted projection of the all the $p_T$ tracks.\n\n\n\nThe extreme limit of an off-center collision, is the case where the nuclei miss each other completly. Still these Lorentz-contracted discs contain charges moving at relativistic speeds, so the electromagnetic fields from each ion do interact. These \\textit{ultraperipheral} collisions give rise to $\\gamma + \\gamma$ and $\\gamma + A$ interactions, which dominate the total nucleus-nucleus cross section.\n\n\nThe initial state of the collision can also be characterized using models that allow to correlate the centrality of the collision with the number of \\textit{participant} nucleons that collide with at least another nucleon, the number of \\textit{spectator} nucleons that do not collide and keep on moving along the beam direction and the number of nucleon pairs that collide or \\textit{binary collisions}, assuming transparency of the collision. At early times, simulations show that both spectators and participants create instense but short-lived magnetic fields in the collision zone. The effects of these primordial magnetic fields in the evolution of the collision and the available observables to measure these effects, are a subject of current interest in the community.  \n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.35]{centrality-new.jpg}~~~~~~~~~~~~~~\\includegraphics[scale=0.4]{vN-new.jpg}\n\\caption{On the left, a schematic representation of the collision \\textit{centrality}. The ions may collide head-on or may only partially overlap at the collision stage, so in the overlap region there are conditions that facilitate QGP formation. On the right, the collision centrality promotes a characteristic initial shape in the transverse plane to the formed medium, that tends to be more lenticular as the collisions are more head-on or central \\cite{centralcern, flowcoeff}.}\n\\label{fig:centrality}\n\\end{figure}\n\n\n\n\n\\subsection{How do we access the different stages of a heavy-ion collision?}\n\nEventhough we can use models to simulate the initial condition of a heavy-ion collision as we described before, we only have two quantities under our direct control: which two species of nuclei we accelerate in preparation for the collision and with what amount of energy we make them collide. So with the help of many experiments we have to observe the outcome of the collisions, event by event, and infer the transverse distance between the centers of the colliding nuclei (\\textit{impact parameter} $b$) and the location and motion of partons in nucleons and of nucleons in the colliding nuclei. We can select events with a narrow distribution of impact parmeter or \\textit{centrality bin} only after the experiment or simulation is done, and we can match this result with the impact parameter distributions expected from the initial condition models. Also, we need nuclear and particle physics studies to support these initial conditions models, e.g. the nuclei can be reasonably well approximated by a collection of nucleons, distributed on average according to three dimensional distributions that provide the usual charge distributions for each type of ion.\nMoreover, the average quark and gluon content of the nucleons in nuclei can be given in terms of parton distribution functions (nPDF) which can be sometimes approximated by those of free nucleons (PDF). In fact, the measured hadronic total inelastic cross section can be used to model the nucleons in nuclei as hard spheres with a radius that depends on the collision energy, so that the initial condition of the heavy-ion collision and the subsequent stages can be simulated with well known phenomenological inputs such as $\\sigma^{pp}_{inel}(\\sqrt{s})$ for $p+p$ collisions. \n\n\n\nFor example, we can model colliding nuclei as made of $A$ transparent spheres of radius $\\sqrt{\\sigma^{pp}_{inel}\/4\\pi}$, $A_L$ and $A_R$ being the number of nucleons inside the left- and right-moving nuclei. Then, as shown in Fig.~\\ref{fig:exampleglauber}, in every collision, there are $N_{spec}$ spectator nucleons (dashed circles) that travel down the beam pipe, $N_{part}$ wounded or participant nucleons which collide with at least one other nucleon (solid circles), so that $N_{spec} + N_{part} = A_L + A_R$ and $N_{coll}$ is the number of encounters between nucleons in $A_L$ and in $A_R$. So a toy event with a \\lq\\lq nucleus\\rq\\rq ~of 8 nucleons lined up in a row that collides head-on ($b=0$) with a \\lq\\lq nucleus\\rq\\rq ~of 4 nucleons lined up in a row would have $N_{part} = 12, N_{coll} = 32, N_{spec} = 0$. But in real central heavy-ion collisions, the nucleons at the center of nucleus will hit about 12 nucleons on average, but less if it is located at the edge of the collision. So we expect that in general $N_{coll} \\gg N_{part}$, with a more marked difference for the most central collisions. Indeed, this can be seen in the right panel of Fig.~\\ref{fig:exampleglauber} where the values of $N_{coll}$ and $N_{part}$ are plotted for 1000 events in Au+Au collisions at 200 GeV with respect to the impact parameter. These results were obtained using the so called \\textit{Glauber Model} which is one of the simplest tools that can be used to simulate the initial conditions of heavy-ion collisions, upon which more sophisticated approaches can be constructed where now the Glauber modelling serves as a benchmark. \n\n\n\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.9]{gedanken.jpg}~~~~~\\includegraphics[scale=0.35]{1_Npart_Ncoll_197Au+197Au.png}\n\\caption{On the left, modelling the colliding nuclei as made of transparent spheres where the dashed circles represent spectator nucleons that travel down the beam pipe and the solid circles represent the wounded or participant nucleons which collide with at least one other nucleon. On the right, the number of binary collisions and the number of participants are plotted for 1000 events in Au+Au collisions at 200 GeV with respect to the impact parameter, using MCGlauber\\cite{mcglauber}.}\n\\label{fig:exampleglauber}\n\\end{figure}\n\n\n \n\n\n\n\n\\subsection{How do we study the quark-gluon plasma in heavy-ion collisions?}\n\n\nThe QGP is made of quarks and gluons that are strongly coupled. They form a collective medium that expands and flows as a relativistic hydrodynamic fluid, with low viscosity to entropy density ratio $\\eta\/s \\gtrsim 1\/4\\pi$, during the initial stages after the collision $\\Delta t \\sim 1$ fm\/c\n(in the fluid rest frame). Just after the Lorentz contracted heavy-ions collide, droplets of QGP fluid form and flow hydrodynamically: the initial high pressure generated by the collision, drives fluid motion, expansion, and subsequent cooling, until the energy density of the droplet is of the order or smaller than that of a hadron. This is when the system hadronizes and a mist of hadrons continues to expand, allowing for hadrons to scatter off each other a few times. Finally these hadrons loose sufficient energy to stop scattering and just stream away freely until they are collected by the detector.\n\n\nIn order to study the quark-gluon plasma in heavy-ion collisions we need adequate experimental probes for the different stages of the formation and evolution of the QGP and a model that incorporates the theoretical ideas relevant for different time and energy scales. Some of the most prominent experimental probes that have provided key information about the different stages of the collision are hadronic probes, such as the relative abundances of pions and kaons, electromagnetic probes such as dilepton and photon production, jet quenching and $J\/\\Psi$ resonance production and decay~\\cite{expprobes}. In the last decade, the nuclear physics community has converged towards an initial consensus on a model with marked epochs in the time evolution of heavy-ion collisions. This model not only helps with the construction of useful physical observables, but it also guides the design of physics plans for current and future experiments. The main features of this model of the stages in the evolution of heavy-ion collisions are as follows~\\cite{smallsystem}: \\begin{itemize}\n\\item[I.] The Lorentz-contracted nuclei collide with short transversal time ($\\ll$ 1fm\/c). The energy deposited into the medium mainly through gluon field interactions, creates and inhomogeneous initial condition in the transverse plane.\n\n\\item[II.] At this stage matter is out-of-equilibrium and needs time to equilibrate, so the expansion occurs at almost the speed of light in both the longitudinal direction and radially in the transverse plane.\n\n\\item[III.] Now matter is nearly equilibrated and behaves as a fluid (collective modes). We can use viscous relativistic hydrodynamics with and equation of state from lattice QCD to describe the QGP. The small deviations from equilibrium happen because of shear\/bulk viscous medium.\n\n\\item[IV.] Now, this fluid cools down to about the cross-over temperature of $T \\sim 170$ MeV and breaks up into hadrons. This process of hadronization of fluid modes can be modelled using (e.g.) Cooper-Frye hadronization.\n\n\\item[V.] Finally, the hadrons scatter inelastically until they reach a \\textit{chemical freeze-out}, where no more decays or secondary production is possible and continue to scatter elastically until \\textit{kinetic freeze-out}, where their momentum distribution is set. This then produces hadrons in a final-state with momenta as measured experimentally.\n\\end{itemize}\nThis model of particle production in heavy-ion collisions with an intermediate epoch during which a hydrodynamic fluid forms and expands, is quite different from the ones used for particle production in elementary hadron collisions in which only a few new particles are created. In fact, the question of whether we can create dropplets of QGP in $p+p$ collisions has caused a big excitement and the posibility to have collective effects in relativistic collisions of small systems is now being pursued with both theoretical and experimental approaches\\footnote{For a recent review on small system collectivity effects in relativistic nuclear and hadron collisions see Ref.\\cite{smallsystem}.}.\n\n\n\n\n\\section{The quark-gluon plasma: extreme QCD} \\label{sec:theqgp}\n\nOne way to start exploring the basic properties of the QGP is to observe the hadron mass spectra. At a first glance, we can see that there is a \\lq\\lq QCD mass gap\\rq\\rq: the pion mass is separated from the rest of the hadron masses. So we could imagine a hadronic phase of nuclear matter (pion gas) where the relevant number of degrees of freedom are 3 ($\\pi^-$, $\\pi^0$, $\\pi^+$) and the rest of the hadrons are basically \\lq\\lq excited hadron states\\rq\\rq. Then the corresponding quark-gluon phase (quark-gluon gas) would be one where the relevant degrees of freedom are 37 for two light-quark flavours ($2_{spin}\\times 8_{colour}=16$ for gluons, $(7\/8)\\times 2_{spin} \\times 2_{flavour} \\times 2_{q\\bar{q}} \\times 3_{colour} = 21$ for quarks). So the quark-gluon phase has more than ten times the number of degrees of freedom than the hadron phase. If we use basic thermodynamics, we can do back-of-the-envelope estimates of the preassure of a pion gas ${\\mathrm p}_\\pi$ and compare it with that of a bag of quarks and gluons ${\\mathrm p}_{QGP}$. We will find out that\n$$ {\\mathrm p}_\\pi (T) = 3 ~ \\frac{\\pi^2}{90} T^4 ~~~~~\\ll ~~~~~{\\mathrm p}_{QGP} (T) = 37~ \\frac{\\pi^2}{90} T^4 -B,\n$$\nso even for a bag constant $B^{1\/4} \\sim 200$ MeV, Nature prefers the system with higher preassure for increasing temperatures ($T > 0.68 B^{1\/4}$). So, under these simple arguments a transition from the hadron phase to the deconfined phase is expected~\\cite{QGPbookLR}.\n\n\nLattice QCD has been able to provide the results not only for the preassure ${\\mathrm p}$, but also the energy ${\\mathrm \\varepsilon}$ and entropy ${\\mathrm s}$ densities, for the quark-gluon phase. The left panel on Fig.~\\ref{fig:hotqcd} shows these three thermodynamic quantities using solid curves in a temperature range that starts below the critial temperature and goes higher than the temperatures achieved at the LHC. We can see, for example, that the preassure curve (red) is actually $3 {\\mathrm p}\/T^4$ and the values at both ends of this curve, compare reasonably well with the ones in our back-of-the-envelope estimate, which are for the hadron phase $3\\pi^2\/30$ and $37\\pi^2\/30$ for the quark-gluon phase. Also in this figure, it is clear that the values of these thermodynamic properties increase dramatically for temperatures $T \\gtrsim 140$ MeV and a smooth crossover is apparent at $T_c = (154 ~\\pm ~9)$ MeV. This figure also has the Hadron Resonance Gas (HRG) model curves at low temperatures, which predicts an exponential increase of states. Finally, the non-interacting ideal gas limit is shown at hight temperature which would take an energy density of ${\\mathrm \\varepsilon}\/T^4 = 95\\pi^2\/60$ to have an ideal non-interacting gas of quarks and gluons. Here we could be tempted to study QGP dynamics with Lattice QCD results above the cross over $T_c$, but this conclusion would be based on a \\textit{static} picture, since as we just discussed, thermodynamically this system approaches that of ideal non-interacting gas of quarks and gluons. Now, beyond the static picture, dynamically things look different: the QGP phase is a system more like a viscous hydrodynamic fluid made of strongly interacting quarks and gluons with large cross sections. For example, most of the hadrons produced in heavy-ion collisions at RHIC are best described with a hydrodynamical model of the QGP that expands and generates anisotropies in the transverse momentum distribution, as is shown on the right in Fig.~\\ref{fig:hotqcd} for the case of $v_2$ (second coefficient in r.h.s. of Eq.~\\ref{floweq}), the so called \\textit{elliptic flow}.\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.7]{hotqcd_eos.pdf} ~~~~~~~~~~ \n\\includegraphics[scale=0.4]{ZajcQM18-4.jpg}\n\\caption{On the left, thermodynamic properties calculated with Lattice QCD \\cite{hotqcd}, preassure, energy and entropy densities.  On the right, $v_2$ for most hadrons at RHIC which uses a hydrodynamics description of the QGP \\cite{Zajc, KolbHeinz}.}\n\\label{fig:hotqcd}\n\\end{figure}\n\n\n\nOne important and seminal work for the applicability of viscous hydrodynamics to the description of the QGP, was reported by Danielewicz and Gyulassy back in 1985~\\cite{etaoversbounds}. They came up with a nice way to have bounds for the size of the viscosity in these systems. Since the QGP phase is a system made of strongly interacting quarks and gluons with large cross sections, then we can estimate the viscosity $\\eta$ as the transverse momentum in-medium diffusion coefficient, in terms of the number density $n$, average momentum $\\langle p \\rangle$, mean-free path $\\lambda$ and cross section $\\sigma$ of the medium constituents\n\\begin{equation}\n\\eta \\sim \\frac{1}{3} n \\left\\langle p \\right\\rangle \\lambda ~~~~~~~~~~\\mbox{or}~~~~~~~~~~ \\eta \\sim \\frac{\\left\\langle p \\right\\rangle}{3\\sigma},\n\\label{justeta}\n\\end{equation}\nwhere on the right hand side, we have taken the mean-free path to be $\\lambda = (n\\sigma)^{-1}$, so that we can start to see the immediate implications on the small size of the viscosity for large cross sections and low average momentum. If we consider the QGP regime achieved with large chemical potential and\/or large temperature, plus the fact that quarks and gluons are almost free under these conditions, then naively we expect small particle correlations, so that the mean-free path is small compared with the system size. \n\nIn order to give a better estimate of the viscosity, we can build a quantity that provides a comparison between the viscosity as a \\textit{drag force} and the \\textit{inertial mass} of the volume element of fluid, say\n$$ \\frac{\\mbox{drag force}}{\\mbox{inertial mass}} \\longrightarrow \\frac{\\eta}{\\mathrm{\\varepsilon} + \\mathrm{p}} = \\left(\\frac{\\eta}{s}\\right) \\frac{1}{T}  $$\nwhere the energy and preassure densities can be related thermodynamically to the entropy density $s$ as $\\mathrm{\\varepsilon} + \\mathrm{p} = s~T$ at null chemical potential $\\mu = 0$. Now, we can use the first expression in Eq.~\\ref{justeta} to give a lower bound of $\\eta\/s$: for a system made out of massless non interacting quanta we know that $s = 4n$ and we can estimate $\\left\\langle p \\right\\rangle \\lambda \\gtrsim \\hbar$ so finally the lower bound is\n$ \\frac{\\eta}{s} \\gtrsim \\frac{\\hbar}{4 \\times 3}$. Later, using String Theory, Kovtun and collaborators~\\cite{stringbound} obtained the bound\n$ \\frac{\\eta}{s} \\gtrsim \\frac{\\hbar}{4 \\pi}$.\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.28]{imperfecto.png} \\\\\n\\centering\\includegraphics[scale=0.8]{hydro-now-new.jpg}\n\\caption{In the upper panel, comparison of the ratio $4\\pi ~k~\\eta\/\\hbar s$ in natural units~\\cite{longrangeplan}. On the lower panels, a viscous hydrodynamics calculation for  a collision between heavy ions, as a one-time snapshot at $t= 5$ fm\/c ~\\cite{smallsystem,bigquestions}, where the colour scheme indicates the temperature and the arrows indicate the fluid velocity.}\n\\label{fig:perfecto}\n\\end{figure}\n\n\n\n\nSoon after, the QGP was popularized as the fluid that might have the smallest viscosity or \\textit{fluid imperfection} - a measurement of the departure from ideal hydrodynamics -  with the ratio $4\\pi ~k~\\eta\/\\hbar s$. In natural units then the QGP would have near-fluid perfection in the lower bound for $\\eta\/s$, as is shown in the upper panel of Fig.~\\ref{fig:perfecto} in comparison with ultra-cold atoms, helium and water. So, the extreme conditions that prevail in heavy-ion collisions with extreme expansion rates for the fire ball and  \nthe smallness of $\\eta\/s$ warrants an extended period of applicability of relativistic viscous hydrodynamics (vHydro) between the pre-equilibrium and the final decoupling epochs. This has resulted in an impresive success in the use of vHydro to describe and predict central observables that help characterize the QGP. For example, the lower panel of Fig.~\\ref{fig:perfecto} shows the elliptic flow coefficient $v_2$ for identified hadrons in Pb+Pb collisions at the LHC, compared to hydrodynamic calculations for $\\eta\/s = 0.2$. The strategy used to obtain the curves shown in this figure, is the numerical solution of coupled relativistic hydrodynamic equations, together with a hadronization and flow analysis as was described in the previous section. There are several very successful public and private codes that solve 2+1 and 3+1 vHydro that can be coupled to codes that generate heavy-ion initial conditions and to codes that help in the later stages of the evolution (freeze-out and hadronization). Even a simple linearized version of vHydro equations  \n\\begin{equation}\n\\partial_\\mu \\delta T^{\\mu\\nu} = J^\\nu ; ~~~~~~\\partial_\\mu T_0^{\\mu \\nu} = 0,\n\\label{lhydro}\n\\end{equation}\nwhere the medium total energy-momentum $T^{\\mu\\nu} = T_0^{\\mu\\nu} + \\delta T^{\\mu\\nu}$ with $T_0^{\\mu\\nu}$ the energy-momentum for underlying medium in equilibrium and $\\delta T^{\\mu\\nu}$ a \\textit{small} perturbation ($J^\\nu$ being the source of disturbance), can be solved analytically for certain source models and has provided physical insight into the effects of hydro-modes in experimental observables~\\cite{ushydro}.\n\nFrom the point of view of a microscopic theory, the viscosity is a transport coefficient that quantifies the inefficiency of dissipation in the medium, which would translate into a presistent anisotropy in the energy-momentum tensor $T_{\\mu\\nu}$. Then in the perturbative picture, the QGP would be made up of quasipartice excitations with momenta the size of the dominant scale, the temperature of the system. So the anisotropy in the energy-momentum tensor, arises from the momentum distributions of these quasiparticles. Then, the scattering and spliting of these constituents will drive the system into its equilibrium value. So in this perturbative approach the challenge is to determine the form of the source of anisotropies by the calculation of the collision operator that enables this system relaxation. At fixed order in perturbation theory, this implies solving a Boltzmann equation using finite temperature techniques\n\\begin{equation}\n ( \\partial_t + \\hat{\\mathbf{p}}\\cdot\\nabla_{\\mathbf{x}} ) \nf(\\mathbf{x},\\mathbf{p},t) = - C^{2\\to 2}[f] - C^{1\\to 2}[f],\n\\label{boltzmann}\n\\end{equation}\nwhere $C^{2\\to 2}_s$ is a scattering operator, $C^{1\\to 2}_s$ is a splitting operator and $f(\\mathbf{x},\\mathbf{p},t) = dN\/d^3\\mathbf{x}d^3\\mathbf{p}$ is the phase space distribution for either quarks, antiquarks or gluons. The solution of this equation then, is used to obtain the transport coefficients, via the vHydro equations~\\cite{etaqcd}. So a full hydrodynamical study, will contain the non-linearized version of Eq.~\\ref{lhydro} coupled to a Boltzmann equation simmilar to Eq.~\\ref{boltzmann}\\footnote{For further details on the applicability of relativistic viscous hydrodynamics and for a complete description of the implementation of vHydro beyond the linearized regime, see for example~\\cite{introhydro}.}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{QCD phase diagram and criticality signatures} \n\\label{sec:phasecritical}\n\n\\subsection{Phase transition, tradition vs praxis}\n\nIn general, \\textit{phase transitions} refer to the transformations of a substance -from one matter state to another- as a result of variations of external conditions such as preassure, temperature, etc. Typically, when the substance goes through a phase transition, there are certain quantities that often change in a discontinous matter. Perhaps the most familiar phase transitions of a substance, are those of water: in our daily lives we use water in the liquid, solid and gas phases and under certain conditions of, say, preassure and temperature, we know which phase to expect for water. This information is summarized in phase diagrams in Fig.~\\ref{fig:waterQGP}, where the solid black lines are the values of temperature and preassure under which the phase transitions occur. Notice there is a \\textit{triple point} -three phases coexist- where the phase transition lines intersect and there is a black circle for the \\textit{critical end-point} (CEP) where the phase transition line ends. For water, the CEP is known experimentally but, for the QGP, the CEP is predicted theoretically but not known experimentally. The experimental discovery of this CEP is part of current and future heavy-ion collision experiments and the theoretical connection between regions of criticality on a phase diagram and observables at the experiment, are a subject of much interest.\n\n\n\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.6]{waterQGP.jpg} \n\\caption{On the left, the phase diagram of water which has continous lines representing the phase transitions and a critical end-point. On the right, the phase diagram of nuclear matter also shows the phase transitions with continous lines and a critical end-point, where the cross-over region begins.~\\cite{rajawater}}\n\\label{fig:waterQGP}\n\\end{figure}\n\n\nSince phase transitions are the result of interactions of a large number of particles -so size of the system and number of particles are relevant to the discussion- in technical terms, they occur when the free energy -the energetic balance between changes in internal energy and changes in temperature and entropy- is non-analytic (one of its derivatives diverges) for some values of the relevant thermodynamical variables of the system. So, on the phase transition lines the free energies in both phases coincide. For certain systems such as nuclear matter, it is possible to change the state of the substance, without crossing a phase transition line. In this case, the thermodynamic conditions define a \\textit{cross-over}, rather than a phase transition. On the right panel of Fig.~\\ref{fig:waterQGP}, there is a cross-over region for nuclear matter that transforms between the hadron has phase and the QGP phase, for low baryon chemical potential and high temperature.\n\n\nAgain, in general, phase transitions are characterized as either discontinous or continous. The former are a result of a discontinuous change in entropy at a fixed temperature (the change in entropy corresponds to \\textit{latent heat} $L = T \\Delta S$) and typical examples are solid-liquid and liquid-gas transitions at temperatures below the critical temperature. The latter involve a continuous change in entropy, which means there is no latent heat in the process, such as liquid-gas transitions at temperatures above the critical temperature.\nIn practise, sometimes we do phase transition classification \\`a la Ehrenfest: the order of a transition is the order of the lowest derivative of the free energy, which shows a discontinuity. For example in boiling water, a first order phase transition is that of a discontinuity in the density, the derivative of free energy with respect to chemical potential, and would be labeled a discontinous phase transition under the previous criteria. In a second order phase transition for the magnetization of a ferromagnet, the derivative of the free energy with respect to the external field is continuous, but the \\textit{susceptibility} (the derivative of the magnetization with respect to the external field) is discontinuous and this would be labeled a continous phase transition with the previous criteria. Finally, the modern classification of phase transitions have a more subtle description of them: the first order phase transitions involve latent heat, so the system absorbs or releases heat at a constant temperature, its phases coexist so only some parts of the system have completed the transition; the second order phase transitions are simply continuous transitions and then, the systems' susceptibilities diverge and the characteristic correlation lengths become large.\n\n\n\n\\subsection{Hadron spectra and Hagedorn temperature}\n\n\nIn order to study the impact of the nuclear matter phase transitions in the hadron spectra, we can start with a simple model for the hadron gas phase: an ideal gas of identical neutral scalar particles of mass $m$~\\footnote{For a review on the thermodynamics and hadron spectra of QCD matter is that of Ref.~\\cite{reviewhagedorn}.}. The grand canonical partition function for these neutral scalar particles, contained in a box volume $V$ at temperature $T$, assuming Boltzmann statistics is\n$2\\pi^2\\ln{\\mathcal{Z}}(T,V) = VTm^2~K_2(m\/T)$.\nSo for temperatures much greater than the mass of these particles, the energy density of the system grows as $T^4$, the particle density grows as $T^3$ and the average energy per particle grows as $T$. The more energy available to the system, the higher the temperatures accesible to it and the more energetic its constituents. Now, if we allow these hadrons to interact and we allow for resonance formation, we can use an improved model of an ideal gas like before, but now with a probability to form certain resonances,\n\\begin{equation}\n\\ln{\\mathcal{Z}}(T,V)=\\sum_{i=0}\\frac{VTm_i^2}{2\\pi^2}~\\rho(m_i)~K_2(\\frac{m_i}{T}),\n\\label{resonancemodel}\n\\end{equation}\nwhere we sum over all possible resonances ($i=0$ being the ground state) with weights relative to the ground state, encoded in $\\rho(m)$. In the early days of hadron discoveries, R. Hagedorn ~\\cite{hagedorn} collected the data on the hadron spectra and concluded the hadron spectra could be well described with this model with $\\rho(m)\\propto\\exp(m\/T^H)$ where the \\textit{Hagedorn temperature} is $T^H\\simeq 0.19~\\mbox{GeV}$. This hypothesis of the experimental growth with mass of the number of hadronic resonances is at the core of the phenomenology of particle production as a direct consequence of the phase transition between the QGP and the hadron gas. The amount of data we have accumulated since Hagedorn's time,  means that we can revise this model, use it as a guiding light in the search of possible missing states and explore the effects this would have for hadron flow observables~\\cite{jacki}. Now, if we are in a regime where not only do we have a high temperature $T$ but also a finite baryon chemical potential $\\mu_B$, then this exponetial growth is in direct competition with a Boltzmann factor and $T^H$ is now a fixed temperature limit $T=\\left(1-\\mu_B\/m\\right)T^H$, so in this case, the momentum of the constituents of this hadron gas, does not continue to increase and more species of heavier particles can be produced.\n\n\\subsection{Chiral transition and hadronization}\n\nIn section~\\ref{sec:chiral}, we laid out the ideas about how we should expect a phase transition between a state with light current quarks and state with heavy constituent quarks a \\textit{chiral phase transition}. In other words, we can study the QCD phase diagram at finite $T$ and $\\mu_B$ from the point of view of chiral symmetry restoration, because quarks turn bare in hot and\/or dense energetic nuclear matter. For these studies the typical order parameter to follow in the search of a transition is the \\textit{chiral condensate} $\\langle\\overline{\\psi}\\psi\\rangle$, a quark-antiquark bound state with Bose-Einstein statistics. The easiness of this condensate to form in vacuum is characterized by the value $\\langle\\overline{\\psi}\\psi\\rangle_0=-(0.24$ GeV)$^3$, which sets the scale for the critical temperature of chiral restoration. Chiral perturbation theory provides a strategy to calculate the value of this condensate at finite -but small- temperature and chemical potential and the result shows that the condensate melts, so there is a set of values of temperature and chemical potential that indicate a chiral phase transition in the QCD phase diagram.\n\nIn order to probe this chiral phase transition in experiments, we should keep tabs of the particle abundances and make sure that we have a statistical model that includes variations of these abundances due to a high particle density in the phase transition region\\footnote{For a review of the statistical-thermal model of particle production in heavy-ion collisions is Ref.~\\cite{statmodelrev}.}. For central collisions where $\\mu_B \\sim 1\/\\sqrt{s_{NN}}$ and for current collider based experiments where the highest energies are\nreached -such as the LHC and RHIC- the bayon chemical potential associated to the reaction is the smallest. In these experiments there is then an increase of the \\textit{degree of transparency} of the colliding nuclear matter: the energy injected to the interaction zone, produces roughly an equal number of particles and antiparticles. New experiments -such as NICA and FAIR- have collisions with lower center-of-mass energies, a smaller degree of transparency to create an interaction zone that is rich in baryons.\n\n\n\\subsection{The critical end-point (CEP)}\n\n\n   \n\\begin{figure}\n\\centering\\includegraphics[scale=0.6]{net-proton-1.jpg}\n\\caption{Skewness and kurtosis of net-proton number is reported by the STAR Collaboration~\\cite{sknetproton}.}\n\\label{fig:netproton}\n\\end{figure}\n\nSo far, we have complemented the discussion that we started back in section~\\ref{sec:faces}, on the theoretical ideas and observed phenomena that motivate a conjectured QCD phase diagram for nuclear matter under extreme conditions of temperature and density, such as the ones in Fig.~\\ref{fig:PhDiag}. The main features of the conjectured QCD phase diagram can be summarized as follows: there is a crossover for $\\mu_B=0$, there is a first order phase transition that turns into a second order phase transition somewhere in the middle of the phase diagram and there is a CEP somewhere in the middle of the phase diagram, where the crossover becomes a first order phase transition line\\footnote{For a review on the location of the CEP in the QCD phase diagram see Ref.~\\cite{cep} and references therein.}.\n\nOne way to find experimental evidence of the CEP in relativistic heavy-ion collisions, is to look for event-by-event fluctuations of conserved quantities. The possibility to detect \\textit{non-Gaussian fluctuations} in conserved charges is one of the tools thought to be sensitive to the early thermal properties of the medium created at heavy-ion collision experiments. In principle, the amount of any conserved charge $Q$ in a given volume of phase space $V$ is an integral over the volume of the charge density $n(x)$, but when the measurement of the charge is performed event-by-event for a volume in a thermodynamical system, we expect thermal fluctuations of $Q$. These fluctuations can be quantified with the help of moments of the charge distribution functions e.g. the variance of $Q$ given in terms of a \\textit{correlation function} as \n$$\\langle\\delta Q^2\\rangle_V=\\langle(Q-\\langle Q\\rangle_V)^2\\rangle_V = \\int_Vdx_1dx_2\\langle\\delta n(x_1)\\delta n(x_2)\\rangle.$$\n\nIn general for a conserved quantity $c$, the \\textit{cumulants} $\\langle x^n\\rangle_c$ associated to a probability distribution function ${\\mathcal{P}}(x)$ of a stochastic variable $x$ are\n$$\n\\langle x^n\\rangle_c=\\left.\\frac{d^n}{d\\theta^n}~\\ln G(\\theta)\\right|_{\\theta=0},\n$$\nwhere $\\langle x\\rangle_c=\\langle x\\rangle$, $\\langle x^2\\rangle_c=\\langle x^2\\rangle - \\langle x\\rangle^2 = \\langle \\delta x^2\\rangle$, $\\langle x^3\\rangle_c=\\langle \\delta x^3\\rangle$, $\\langle x^4\\rangle_c=\\langle \\delta x^4\\rangle -3\\langle \\delta x^2\\rangle^2$, etc. For a thermodynamical system the partition function $\\ln{\\mathcal{Z}}$, is the moment generating function - the so called \\textit{cumulant generating function}- $\\ln G$. Since a conserved quantity will be connected to the volume $V$ of the system in a grand canonical ensemble framework, cumulants are extensive quantities. Notice also that non-Gaussian fluctuations will be driven by non-vanishing higher order cumulants. Since the asymmetry and sharpness of a distribution function can also be established through the \\textit{skewness} $S$ and \\textit{kurtosis} $\\kappa$, then there is a direct connection between these two quantities and the cumulants: when the stochastic variable $x$ is normalized to the square root of the variance $\\sigma$, the skewness and the kurtosis are given as the third and fourth-order cummulants: $S=\\langle \\tilde{x}^3\\rangle_c$, $\\kappa=\\langle \\tilde{x}^4\\rangle_c$. If we can describe the fluctuations of conserved charges in heavy-ion experiments using hadron degrees of freedom, then the cumulants should be consistent with those of the Hadron Resonance Gas model (HRG) and so deviations from this model are used as experimental signals of the emergence of non-hadron or non-thermal physics. Near the CEP, higher order cumulants change sign and are sensitive to the increase of correlation lengths. In particular when passing through the CEP in a beam energy scan in heavy-ion collision experiments, the fluctuations of multiplicities or mean transverse momenta of particles are a manifestation of higher non-Gaussian cumulants with a non-monotonic behavior of the correlation length. For example, the top panels in Fig.~\\ref{fig:netproton} show the skewness and kurtosis for net-proton number as reported by the STAR Collaboration for a beam energy scan from 6 GeV to 200 GeV for Au+Au collisions at RHIC. In peripheral and mid-central collisions, the $\\kappa\\sigma^2$ values are close to 1 and the $S\\sigma$ show strong monotonic increase when the energy decreases, giving the first hints of criticality. \n\n\n   \n\n\n\n\\section{Heavy-ion collision physics: season finale and season premiere}\n\\label{sec:seasons}\n\nThe experiements that have established the production of the QGP -RHIC at BNL and LHC at CERN- and are still providing data to characterise it, represent the golden era of heavy-ion collisions. Now, a new era begins with new experiments -NICA at JINR, FAIR at GSI and RHIC at BNL-,  that are going to provide data to scan regions of the QCD phase diagram that we could not access before. Together with heavy-ion physics, astrophysics and condensed matter will continue to enrich the description of the phases of nuclear matter from different perspectives. Eventhough the collisions are at lower energies, this poses new technological challenges both for the preparation of the ion-beam and for the design of detectors that can have de adequate coverage to capture the products of the QGP evolution. Throughout these lectures, we have shown several examples of the theoretical and experimental efforts that provide the most comprehensive picture, up to today, of the QGP and its evolution. In this section we highlight certain aspects of heavy-ion phenomenology that have resolved some issues in the past, but that also present new challenges in this new experimental era.\n\n\n\n\\subsection{$R_{AA}$, jet quenching and correlations}\n\nIn order to quantify the effect of the medium created in heavy-ion collisions in the production of particles, we use the \\textit{nuclear modification factor} $R_{AA}$, which is the ratio of the multiplicity of particles produced in $A+A$ collisions with respect to the multiplicity in $p+p$ collisions, normalized to the average number of binary collisions\n\\begin{equation}\nR_{AA}(p_T)=  \\frac{1}{\\langle N_{\\mbox{\\tiny{coll}}}\\rangle}\\frac{\\frac{dN^{AA}(p_T)}{dp_T}}\n{ \\frac{dN^{pp}(p_T)}{dp_T}}. \\nonumber\n\\end{equation}\nSo, na\\\"ively, we expect this ratio to be closer to one for primordial photons for which the QGP should be transparent, and perhaps smaller than one for hadrons that form at different stages of QGP evolution. In the left panel of Fig.~\\ref{fig:hadquench} we can see the data for $R_{AA}$ for photons and hadrons in the most central $Au+Au$ collisions at $\\sqrt{s_{NN}}=200$ GeV, as reported by the PHENIX collaboration~\\cite{phenixquench}. Indeed hadrons show large suppression -\\textit{quenching}- for a wide range of $p_T$ and photons mostly do not. Looking closely at the plot, we can see that for $p_T <$ 5 GeV there is an excess of photons in Au+Au collisions and there is a slight recovery on the $p_T>$ 5 GeV observed hadron supression. At the LHC this translates into \\textit{jet quenching}, a large -by a factor of 2 for most central collisions- suppression\nof single-inclusive jets produced in Pb+Pb collisions, when compared to p+p collisions, as shown in the right panel of Fig.~\\ref{fig:hadquench}. Jet quenching in heavy-ion collisions is a tool to understanf the formation and evolution of in-medium jets, since there is an enhancement of soft activity at the edges of the jet and far away from the jet. So jet physics in heavy-ion collisions represents one of the most important tools to study both the process of equilibration -through the interactions between the jet and the medium- and gives access to the scale dependence of medium properties. The missing $p_T$ in an event, due to jet quenching, has made it possible to perform in-medium jet-tomography~\\cite{missingpt} and to inspire new developments for in-medium fragmentation and jet-formation in Monte Carlo simulations for heavy-ion experiments. Jet angular and rapidity distributions, have shown that there are \\textit{ridges} or long-range correlations, that support the emergence of collective phenomena. Since these ridges are present at large rapidities, and show enhacement for less central collisions and lower values of $p_T$, then this complements the expected and observed behavior for the values of $v_n$ and further supports the hydrodynamical modelling of the QGP, that we presented back in Sections \\ref{sec:theqgp} and \\ref{sec:phasecritical}. \n\n  \n\\begin{figure}\n\\centering\\includegraphics[scale=0.4]{raa.jpg} ~~~~~~~~~\\includegraphics[scale=0.5]{jetquench.pdf} \n\\caption{Hadron species show \\textit{quenching} at RHIC\\cite{phenixquench} and single-inclusive jet quenching at LHC\\cite{atlasquench}.}\n\\label{fig:hadquench}\n\\end{figure}\n\n\n\n\\subsection{Photons as thermometers and viscometers}\n\nPhotons are a special probe of the QGP that, unlike the hard-probes, can traverse the medium from which they are emitted without further interaction. The photon spectra provides access to the evolution and temperature of the fireball, since they are emitted at all the evolution stages of a heavy-ion collision. Even more, the mechanisms for photo-production are known from many other experiments, so an excess or suppression of photons produced in A+A collisions, would be a signal of new mechanisms possible due to the conditions that prevail in heavy-ion collisions.\n\nPhotons are thermometers of the QGP in the following sense: the effective temperature of the QGP can be extracted from the photon spectrum in the low $p_T$ region, since it represents the inverse logarithmic slope $dN\/dp_T \\sim \\mathrm{Exp}\\left\\{- p_T\/T_{eff}\\right\\}$. These \\textit{thermal photons} provide an estimate of the effective temperature from different experiments, e.g. $T_{eff} = 211 \\pm 19$ MeV in 0-20\\% Au-Au collisions at $\\sqrt{s_{NN}}=200$ GeV at RHIC and $T_{eff} = 304 \\pm 51$ MeV in 0-40\\% Pb-Pb collisions at $\\sqrt{s_{NN}}=2.76$ TeV at LHC. The photon $p_T$ spectrum even shows photons at $T_{eff}<T_c$ with blue-shift to $T_{eff} > Tc$ due to a strong radial hydrodynamic flow for $T_c = 155 - 170$. Recently, there was even a \\textit{direct photon's flow \\lq\\lq puzzle\\rq\\rq} whereby photons from hard scatterings become dominant at high-$p_T$ and are well described by NLO pQCD plus photons from thermal emission of the QGP and the hadronic phase, but there was a photon excess at low-$p_T$ with large values of elliptic flow, that could not be accounted for by the regular mechanisms~\\cite{photonspectra}. This prompted several possibilities for new sources of primordial photons, such as the photo-production via intense magnetic fields present at early stages after the heavy-ion collision~\\cite{photonsB}. Photons are viscometers of the QGP in the following sense: using event-by-event simulations of heavy-ion collisions based on viscous relativistic hydrodynamics, we learn that $\\eta\/s$ suppresses the flow coefficients $v_n$ from the soft photon spectra and the effect is more pronounced for more peripheral collisions~\\cite{photonsviscosity}.\n\n\n\n\n\\section{Summary and open questions}\n\\label{sec:final}\n\nIn these lectures we presented the basic experimental and theoretical tools to study the collisions - and evolution of the colliding system - in heavy-ion experiments, where the QGP is produced, the \\lq\\lq simplest form of complex matter described by the fundamental laws of QCD\\rq\\rq~\\cite{bigquestions}. The QGP is probed with a range of experimental set-ups and we can characterize it with several relevant observables. We have a standard model of heavy-ion collisions, that includes the QGP formation and decay, but that also incorporates the formation of hadrons from the primordial fire ball. The QGP is modeled as a relativistic fluid with very small viscosity and this model can describe most of the flow-related observables across a variety of colliding systems and conditions. Finally, the study of phase transitions in nuclear matter using QCD-inspired models, fuels the experimental plans for the discovery of the critial end-point in this diagram and the ultimate understanding of in-medium hadron formation. As was nicely summarized in Ref.~\\cite{bigquestions}, the field of heavy-ion physics has spread into new theoretical and experimental areas of development that have provided a framework for the exploration of new \\textit{extreme} QCD phenomenology, but that still has many open new and exciting questions to tackle:\n\n\\begin{itemize}\n\\item What new insights can we obtain about well established QCD phenomenology such as hadronization and confinement, from the analysis of heavy ion collision data?\n\\item Do we have an understanding of all the stages of a heavy ion collision? In particular, how well do we understand the initial stages of the collision process, up to the creation of QGP?\n\\item Can we create QGP with collisions of small systems such as proton-proton and proton-light ion collisions?\n\\item What are the properties of QGP?, what have we learned about the dynamics of different probes as they traverse this medium?\n\\item Which aspects of heavy-ion collisions phenomenology involve weakly coupled dynamics or strongly coupled dynamics? \n\\item What are the regions of the nuclear matter phase diagram we can explore using heavy-ion physics?\n\\end{itemize}\n\nI hope that with these lectures, you are motivated to do hot, dense and exciting QCD, and join the efforts of the heavy-ion community in this era where new experiments are helping us explore the phase diagram of nuclear matter.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNeurodegenerative disorders (NDDs), such as Alzheimer's disease (AD), are characterised by the progressive pathological alteration of the brain's biochemical processes and morphology, and ultimately lead to the irreversible impairment of cognitive functions.\nThe correct understanding of the relationship between the different pathological features is of paramount importance for improving the identification of pathological changes in patients, and for better treatment. To this end, the recent availability of collections of imaging and clinical data in NDDs is a unique opportunity to define statistical models for  the {joint modelling} of the temporal changes of {imaging}, {biochemical}, and {clinical} biomarkers.\n\nThe goal of disease progression modeling in NDDs is twofolds: 1) \\emph{quantifying} the dynamics of NDDs along with the related temporal relationship between different biomarkers, and 2) \\emph{staging} patients based on individual observations for diagnostic and interventional purposes. \nThe related challenge is in the definition of optimal methods to integrate and jointly analyze the heterogeneous and highly multi-modal information available to clinicians. Moreover, longitudinal clinical datasets of NDDs generally lack of a well-defined temporal reference, since the onset of the pathology may vary across individuals according to genetic and environmental factors \\cite{yang2011}. Therefore, age or visit date information are biased time references for the individual longitudinal measurements. \n\nTo tackle this problem, in \\cite{donohue2014} the authors proposed to model the temporal biomarker trajectories as a random effect regression problem, building on the well established theory of {self-modeling regression} \\cite{kneip1988}. Practically, the trajectories are modelled by monotonic B-splines, and the estimation is performed by subsequent minimization of the partial residuals sum of squares associated with regression parameters and individual time shift, respectively.\nBased on the assumption of a logistic curve shape for the average biomarker trajectories, \\cite{schiratti2015} frames the random effect regression model in a Riemannian setting, in which the random effects identify individual time-shift and acceleration factors. \nFinally, image-based progression models have been recently proposed \\cite{younes2014,bilgel2015}, based on time-reparameterization of voxel\/mesh-based measures. \nWhile the main focus of current works mainly concerns the modeling of neurodegeneration, the use of disease progression models for predictive purposes is much less investigated. Predictive models of patient staging were proposed within the setting of event based models \\cite{fonteijn2011}, or still through random effect modeling \\cite{guerrero2016}. However, the event based model relies on the coarse binary discretization of the biomarker changes, and does not account for longitudinal observations, while the model proposed in \\cite{guerrero2016} requires cohorts with known disease onset, and therefore does not easily generalise to the study of general clinical populations. \n\nIn this study we propose an unified approach to the consistent disease progression modeling and probabilistic prediction of clinical data, by introducing a Bayesian regression framework \nof imaging and clinical biomarker progressions from time series of individual observations.\nThe model is based on Gaussian process (GP) regression, and is formulated to account for individual random effects and time reparameterization, as well to naturally account for missing observations. The {methodological contribution} of this study consists in reformulating disease progression modeling within a Bayesian context, allowing non-parametric estimates of the biomarker evolution, as well as high flexibility in specifying random effects structure, and time reparameterization models. The uniqueness of the time transformation is enforced by imposing a {monotonicity constraint} on the trajectories, via a prior on the temporal derivatives, while  the model is computationally efficient thanks to the block-wise algebraic structure of the GP covariance function. \nFrom the application point of view, the proposed approach enables novel applications of disease progression modeling such as the \\emph{probabilistic prediction} of disease staging in unseen patients. Moreover, the model naturally accounts for missing data, and allows for uncertainty quantification of both evolutions and parameters.\n\nSection \\ref{section:Model} introduces the proposed GP regression model of joint temporal progression of biomarkers, along with the prior derivatives specification.  The resulting posterior and approximated inference scheme through expectation propagation (EP) is described in Section \\ref{section:Likelihood}. Finally, Section \\ref{section:experiments} illustrates the validation of our model on i) synthetic data and ii) clinical and multivariate imaging measurements from a cohort of 517 amyloid positive individuals of the ADNI database. \nThe experimental results show that our model provides a biologically plausible description of the evolution of AD across the whole disease time-span, as well as statistically significant prediction of disease staging and high classification accuracy on unseen and incomplete testing data.\n\n\n\n\\section{Gaussian process-based random effect modeling of  longitudinal progressions}\\label{section:Model}\nIn what follows, longitudinal measurements of $N_b$ biomarkers $\\{b_1, \\ldots, b_{N_b}\\}$ over time are given for $N$ individuals.\n\nWe represent the longitudinal biomarker's measures associated with each individual $j$ as a multidimensional array $(\\mathbf{y}^j({t_1}), \\mathbf{y}^j({t_2}), \\ldots, \\mathbf{y}^j({t_{k^j}}))^{\\top}$ sampled at $k^j$ multiple time points $ \\mathbf{t} = \\{{t}_{1}, {t}_{2}, \\ldots, {t}_{k^j}\\}$. Although different biomarkers may be in reality sampled at different time-points, for sake of notation simplicity  in what follows we will assume, without loss of generality, that the sampling time is common among them. The observations for individual $j$ at a single time point $t$ are thus a random sample from the following generative model:\n\\begin{align}\\label{GenerativeModel}\n{\\mathbf{y}^j(t)} = \\left({{y}_{b_1}^j(t)}, {{y}_{b_2}^j(t)}, \\ldots, {y}_{b_{N_b}}^j(t)\\right)^{\\top} &= \\mathbf{f}({t}) + {\\boldsymbol{\\nu}^j}({t}) + \\boldsymbol{\\epsilon},\n\\end{align}\nwhere $\\mathbf{f}({t}) = ({{f}_{b_1}}({t}), {{f}_{b_2}}({t})\\ldots, {{f}_{b_{N_b}}}({t}))^{\\top}$ is the fixed effect function modelling the biomarker's longitudinal evolution, $\\boldsymbol{\\nu}^j({t}) = ({\\nu}_{b_1}^j({t}),{\\nu}_{b_2}^j({t}), \\ldots, {\\nu}_{b_{N_b}}^j({t}))^{\\top}$ is the individual random effect, and $\\boldsymbol{\\epsilon} = ({\\epsilon}_{b_1},{\\epsilon}_{b_2}, \\ldots,{\\epsilon}_{b_{N_b}})^{\\top} $ is observational noise independent from time.   \nThe group-wise evolution is modelled as a zero-mean GP, $\\mathbf{f}\\sim \\mathcal{GP}(0,\\Sigma_G)$, the individual random effects are assumed to be Gaussian distributed correlated signal $\\boldsymbol{\\nu}^j\\sim \\mathcal{N}(0,\\Sigma_S)$, while the observational noise is assumed to be a Gaussian heteroskedastic term $\\boldsymbol{\\epsilon} \\sim \\mathcal{N}(0,\\Sigma_{\\epsilon})$, where $\\Sigma_{\\epsilon}$ is a diagonal matrix  $\\mbox{diag}[\\boldsymbol{\\sigma}_{b1}^2,\\boldsymbol{\\sigma}_{b2}^2, \\ldots, \\boldsymbol{\\sigma}_{b_{N_b}}^2 ]$.\\\\\n\\textbf{Fixed Effect Process.} The covariance function $\\Sigma_G$ describes the biomarkers temporal variability, and is represented as a block-diagonal matrix  \n\\begin{eqnarray}\\label{FixCovariance}\n\\Sigma_G(\\mathbf{f},\\mathbf{f}) &= \\mathrm{diag}[ \\Sigma_{b_1}(\\mathbf{f}_{b_1},\\mathbf{f}_{b_1}), \\Sigma_{b_2}(\\mathbf{f}_{b_2},\\mathbf{f}_{b_2}), \\ldots, \\Sigma_{b_{N_b}}(\\mathbf{f}_{b_{N_b}},\\mathbf{f}_{b_{N_b}}) ],\n\\end{eqnarray}\nwhere each block represents the within-biomarker temporal covariance expressed as a negative squared exponential function:\n\\begin{align}\n\\Sigma_{b}(\\mathbf{f}_b(t_1),\\mathbf{f}_b(t_2)) &= \\eta_b \\exp\\left(-\\frac{(t_1-t_2)^2}{2\\,l_b^2}\\right),    \n\\end{align}\nand where the parameters $\\eta_b$ and $l_b$ are the marginal variance and length-scale of the biomarker's temporal evolution, respectively. \\\\\n\\textbf{Individual Random Effects.} \nThe random covariance function $\\Sigma_S$ models the individual deviation from the fixed effect, and is represented as a block-diagonal matrix \n$$\\Sigma_S (\\boldsymbol{\\nu}^j,\\boldsymbol{\\nu}^j)= \\mathrm{diag}[\\,\\Sigma_{b_1}^j(\\boldsymbol{\\nu}_{b_{1}}^j,\\boldsymbol{\\nu}_{b_{1}}^j), \\Sigma_{b_2}^j(\\boldsymbol{\\nu}_{b_{2}}^j,\\boldsymbol{\\nu}_{b_{2}}^j), \\ldots, \\Sigma_{b_{N_b}^j}(\\boldsymbol{\\nu}_{b_{N_b}}^j,\\boldsymbol{\\nu}_{b_{N_b}}^j) ],$$\nwhere each block $\\Sigma_{b}^j$ corresponds to the covariance function associated with the individual process $\\boldsymbol{\\nu}_{b}^j({t})$. Thanks to the flexibility of the proposed generative model, any form of the random effect covariance $\\Sigma_S$ can be easily specified in order to model the subject-specific biomarkers' progression. In what follows we will use a linear covariance form $\\Sigma_{b}^j(\\boldsymbol{\\nu}_{b}^j(t_1), \\boldsymbol{\\nu}_{b}^j(t_2)) = (\\sigma_b^j)^2 \\left((t_1 - \\overline{\\mathbf{t}}) (t_2 - \\overline{\\mathbf{t}}) \\right)$, where $\\overline{\\mathbf{t}}$ is the average observational time for individual $j$, if more than 4 measurements were available, and i.i.d. Gaussian covariance form $\\Sigma_{b}^j(\\boldsymbol{\\nu}_{b}^j(t_1), \\boldsymbol{\\nu}_{b}^j(t_2)) = (\\sigma_b^j)^{2}$ if 2 or 3 measurements were available, while assigning it to 0 otherwise. This choice is motivated by stability concerns, in order to keep the model complexity compatible with the generally limited number of measurements available for each individual. \\\\\n\\textbf{Individual time transformation.}\nThe generative model (\\ref{GenerativeModel}) is based on the key assumption that the longitudinal observations across different individuals are defined with respect to the same temporal reference. This assumption may be invalid when the temporal alignment of the individual observations with respect to the common group-wise model is unknown, for instance in the typical scenario of a clinical trial in AD where the patients' observational time is relative to the common baseline, and where the disease onset is a latent event (past or future) which is not directly measurable.\nWe assume that each individual measurement is made with respect to an absolute time-frame $\\tau$ through a time-warping function $t = \\phi^j(\\tau)$ that models the time-reparameterization with respect to the common group-wise evolution. Model (\\ref{GenerativeModel}) can thus be reparameterized as \n\\begin{align}\\label{GenerativeModelNew}\n\\mathbf{y}^j(\\phi^j(\\tau)) = \\mathbf{f}(\\phi^j(\\tau)) + \\boldsymbol{\\nu}^j(\\phi^j(\\tau)) + \\boldsymbol{\\epsilon}. \n\\end{align}\n\nThe present formulation allows the specification of any kind of time transformation, and in what follows we shall focus on the modelling of a linear reparameterization of the observational time  $\\phi^j(\\tau) = \\tau + d^j$. This modeling assumption is mostly motivated by the choice of working with a reasonably limited number of parameters, compatibly with the generally short follow-up time available per individual (cfr. Table \\ref{Nobs}). Within this setting, the time-shift $d^j$ encodes the disease stage associated with the individual relatively to the group-wise model. \n\nOverall, model (\\ref{GenerativeModelNew}) is  identified by $(N_j + 3) N_b + N_j $ parameters, represented by the fixed effects and noise $\\boldsymbol{\\theta}_G = \\{\\eta_{b_k}, l_{b_k}, \\epsilon_{b_k}\\}_{k=1}^{N_b}$, by the individual random effects parameters $\\boldsymbol{\\theta}_G^j = \\{\\sigma_{b_k}^j\\}_{k=1}^{N_b}$ and by the time-shifts $d^j$. \\\\\n\\textbf{Monotonic constraint in multimodal GP regression.} \nDue to the non-parametric nature of Gaussian process regression, we need an additional constraint on model (\\ref{GenerativeModelNew}) in order to identify a unique solution for the time-warp. By assuming  a steady temporal evolution of biomarkers from normal to pathological values, we shall assume that the biomarker trajectories described by (\\ref{GenerativeModelNew}) follow a (quasi) monotonic behaviour. This requirement can be implemented by imposing a prior positivity constraint on the derivatives of the GP function $f$.  \nInspired by \\cite{riihimaki2010gaussian}, we impose a monotonicity constraint by assuming a probit-likelihood for the  derivative measurements $\\mathbf{m}(t)$ associated with the derivative process $\\mathbf{\\dot{f}}(t) = \\frac{\\mathrm{d} \\mathbf{f}(t)}{\\mathrm{d} t}$ at time $t$:\n\\begin{align}\np(\\mathbf{m}(t)|\\mathbf{\\dot{f}}(t)) &= \\Phi\\left(\\frac{1}{\\lambda}\\mathbf{\\dot{f}}(t)\\right),\n\\end{align}\nwith $\\Phi(z) = \\int_{-\\infty}^z\\mathcal{N}(x|0,1)\\,dx$. The quantity $\\lambda > 0$ is an additional model parameter controlling the degree of positivity enforced on the derivative process, with values approaching to zero for stronger monotonicity constraint. In what follows, the monotonicity of each biomarker is controlled by placing 10 derivative points equally spaced on the observation domain, and by fixing the $N_b$ derivative parameters $\\{\\lambda_{b_k}\\}_{k=1}^{N_b}$ to the value of $e$-6. This choice for the parameter is motivated by the need to enforce a strong monotonicity constraint, necessary for the stability of the initial time-shift estimation. \\\\\n\\textbf{On adding a monotonic constraint to the individual observations.} By following a similar construction, we could equally enforce a monotonic behaviour to the random effects associated with the individual trajectories. This additional constraint would however come with a cumbersome increase of the model complexity, since it would introduce an additional layer of virtual derivative parameters (with associated location) per individual. Moreover, while we are interested in modeling a {globally} monotonic biomarker trajectory on the fixed parameters, we relax this constraint at the individual level, since some subjects may be characterised by non strictly monotonic time-series due to specific clinical conditions.  \n\n\n\n\\section{Joint Model: marginal likelihood and inference}\\label{section:Likelihood}\nGiven the sets of individual biomarker measurements $\\mathbf{y}=\\{(\\mathbf{y}^j(t_i))_{i=1}^{k^j}\\}_{j=1}^{N}$, and of $D$ control derivatives $\\mathbf{m}=\\{m_{b_k}(t_l')\\}_{l=1}^D$ at points $t' = \\{t_l'\\}_{l=1}^D$ for the progression of each biomarker $b_k$, the random effect GP model posterior is:      \n\\begin{align}\np\\left(\\mathbf{f},\\mathbf{\\dot{f}},\\boldsymbol{\\nu}^j|\\mathbf{y},\\mathbf{m} \\right) &= \\frac{1}{Z}p(\\mathbf{f},\\mathbf{\\dot{f}}|t,t')p(\\boldsymbol{\\nu}|t)p(\\mathbf{y}|\\mathbf{f},\\boldsymbol{\\nu})p(\\mathbf{m}|\\mathbf{\\dot{f}})\\nonumber\\\\\n                   &= p(\\mathbf{f},\\mathbf{\\dot{f}}|t,t')p(\\boldsymbol{\\nu}|t)p(\\mathbf{y}|\\mathbf{f},\\boldsymbol{\\nu})\\prod_k\\prod_l\\Phi\\left(\\frac{1}{\\lambda}{\\dot{f}}_{b_k}(t_l')\\right)\\label{posterior},\n\\end{align}\nwhere $\\boldsymbol{\\nu} = \\{\\mathbf{\\nu}^j\\}_{j=1}^N$. Thanks to the linearity of GPs under derivation, we have that $Cov\\left(\\mathbf{f}(t),\\mathbf{\\dot{f}}(t')\\right) = \\frac{\\mathrm{d}Cov(\\mathbf{f}(t),\\mathbf{f}(t'))}{\\mathrm{d}t'}$, and that the joint distribution $p\\left(\\mathbf{f},\\mathbf{\\dot{f}}|t,t'\\right)$ is again a GP:\n\n\\begin{eqnarray}\\label{joint}\np\\left(\\mathbf{f},\\mathbf{\\dot{f}},\\boldsymbol{\\nu}^j|t,t'\\right) &\\sim& \\mathcal{GP}\\left(\\mathbf{f}_{joint}|0,\\Sigma_{joint}\\right) \\nonumber\\\\\n\\mathbf{f}_{joint}  = \\begin{pmatrix}\\mathbf{f}\\\\\n\\mathbf{\\dot{f}}\n\\end{pmatrix} & \\sim & \\mathcal{N}\\left[\\left(\\begin{array}{c}\n0\\\\\n0\n\\end{array}\\right), \\left(\\begin{array}{cc}\n\\Sigma_G(\\mathbf{f}(t),\\mathbf{f}(t)) & \\frac{\\partial\\Sigma_G(\\mathbf{f}(t),\\mathbf{f}(t'))}{\\partial t'}\\\\\n\\frac{\\mathrm{d}\\Sigma_G(\\mathbf{f}(t'),\\mathbf{f}(t))}{\\mathrm{d} t'} & \\frac{\\mathrm{d}^2\\Sigma_G(\\mathbf{f}(t'),\\mathbf{f}(t'))}{\\mathrm{d}{t'}^2}\n\\end{array}\\right)\\right]\\enspace.\\nonumber\n\\end{eqnarray}\n\\subsection{Approximated inference with Expectation Propagation}\nDue to the non-Gaussianity of the derivative likelihood term, the direct inference on the posterior (\\ref{posterior}) is not possible due to its analytically intractable form. For this reason, we employ an approximate inference scheme based on expectation propagation (EP) \\cite{rasmussen2006gaussian,minka2001expectation}. Following \\cite{riihimaki2010gaussian}, we compute an approximated posterior distribution $q\\left(\\mathbf{f},\\mathbf{\\dot{f}},\\boldsymbol{\\nu}^j|\\mathbf{y}^j,\\mathbf{m}\\right)$ by replacing the derivative likelihood terms with local un-normalized Gaussian approximations:\n\\begin{align}\nq\\left(\\mathbf{f},\\mathbf{\\dot{f}},\\boldsymbol{\\nu}^j|\\mathbf{y}^j,\\mathbf{m}\\right) &=\\frac{1}{Z_{EP}} p(\\mathbf{f},\\mathbf{\\dot{f}}|t,t')p(\\boldsymbol{\\nu}|t)p(\\mathbf{y}|\\mathbf{f},\\boldsymbol{\\nu})\\prod_k\\prod_l \\tilde{Z}_{kl}\\mathcal{N}(\\dot{f}_{b_k}(t_l')|\\tilde{\\mu}_{kl},\\tilde{\\sigma}^2_{kl}),\n\\end{align}\nwhere $\\prod_k\\prod_l \\tilde{Z}_{kl}\\mathcal{N}(\\dot{f}_{b_k}(t_l')|\\tilde{\\mu}_{kl},\\tilde{\\sigma}^2_{kl}) = \\mathcal{N}(\\boldsymbol{\\tilde{\\mu}},\\tilde{\\Sigma})\\prod_{k,l}\\tilde{Z}_{kl}$, with $\\boldsymbol{\\tilde{\\mu}} = [\\tilde{\\mu}_{kl}]$, and $\\tilde{\\Sigma}$ is a  diagonal matrix with elements $\\tilde{\\sigma}^2_{kl}$. It follows that the marginal posterior has a Gaussian form, $q\\left(\\mathbf{f},\\mathbf{\\dot{f}},\\boldsymbol{\\nu}^j|\\mathbf{y}^j,\\mathbf{m}\\right)\\sim\\mathcal{N}(\\boldsymbol{\\mu},\\Sigma)$, with $\\boldsymbol{\\mu} = \\Sigma\\tilde{\\Sigma}^{-1}\\boldsymbol{\\tilde{\\mu}}_{joint}$ , and $\\Sigma = (\\Sigma_{joint}^{-1} + \\tilde{\\Sigma}_{joint}^{-1})^{-1}$, where \n\\begin{eqnarray}\n\\boldsymbol{\\tilde{\\mu}}_{joint}  = \\begin{pmatrix}\\mathbf{y}\\\\\n\\boldsymbol{\\tilde{\\mu}}\n\\end{pmatrix} &\\mbox{, and }  & \\tilde{\\Sigma}_{joint} = \\left(\\begin{array}{cc}\n\\Sigma_{\\epsilon} + \\Sigma_{S} & 0\\\\\n0 & \\tilde{\\Sigma}\n\\end{array}\\right)\\enspace.\n\\end{eqnarray}\n\n\\subsubsection{Estimating the EP parameters.} The EP update of the local Gaussian approximation parameters is classically done by iterative moment matching with respect to the product between the cavity distributions $q_{-k'l'}\\left(\\dot{f}_{b_{k'}}(t'_{l'})\\right)$ and the target likelihood term $\\Phi\\left(\\frac{1}{\\lambda}\\dot{f}_{b_{k'}}(t'_{l'})\\right)$.\nIn the GP case the cavity distribution has a straightforward Gaussian form:\n\\begin{eqnarray}\nq_{-k'l'}\\left(\\dot{f}_{b_{k'}}(t'_{l'})\\right) &=& \\int \\prod_{k\\neq k'}\\prod_{l\\neq l'} \\tilde{Z}_{kl}\\mathcal{N}(\\dot{f}_{b_k}(t_l')|\\tilde{\\mu}_{kl},\\tilde{\\sigma}^2_{kl}) d\\dot{f}_{b_k}(t_l')\\nonumber\\\\\n&\\sim&\\mathcal{N}(\\dot{f}_{b_{k'}}(t_{l'}')|\\mu_{-k'l'},\\sigma_{-k'l'}).\\label{cavity}\n\\end{eqnarray}\nAs shown in \\cite{riihimaki2010gaussian} for univariate monotonic regression, moments and updates of the approximation parameters can be computed in an analogous manner as in the classical GP classification problem \\cite{rasmussen2006gaussian}.\n\n\\subsection{Marginal Likelihood and hyper-parameter estimation}\nThe model's log-marginal likelihood under the EP approximation is:\n\\begin{eqnarray}\\label{final_posterior}\n\\log\\mathcal{L} &=& -\\frac{1}{2}\\log|\\Sigma_{joint} + \\tilde{\\Sigma}_{joint}| - \\frac{1}{2}\\boldsymbol{\\tilde{\\mu}}_{joint}^T(\\Sigma_{joint} + \\tilde{\\Sigma}_{joint})^{-1}\\boldsymbol{\\tilde{\\mu}}_{joint} +\\nonumber\\\\ &&\\sum_{k}\\sum_{l}\\frac{(\\mu_{-kl}-\\tilde{\\mu}_{kl})^2}{2(\\sigma_{-kl}^2)+\\tilde{\\sigma}_{kl}^2)} + \\sum_k\\sum_l \\log\\Phi(\\frac{\\mu_{-kl}}{\\sqrt{\\lambda_{k}^2+\\sigma_{-kl}^2)}})+\\nonumber\\\\\n&&\\frac{1}{2}\\sum_k\\sum_l\\log(\\sigma_{-kl}^2+\\tilde{\\sigma}_{kl}^2).\n\\end{eqnarray}\nIn what follows, the optimal parameters are obtained by maximising $\\log\\mathcal{L}$ through conjugate gradient descent, via  alternate optimization between the hyper-parameters $\\boldsymbol{\\theta}_G$ and $\\boldsymbol{\\theta}_G^j$, and the individuals' time-shifts $d^j$. The position of the derivative points was updated at each iteration, according to the changes of the GP domain. Regularisation was also enforced by introducing Gaussian priors for the parameters $\\boldsymbol{\\theta}_G$  and $\\boldsymbol{\\theta}_G^j$. We note that the block structure of the GP covariance allows the computation of the gradients with respect to the biomarkers' and individual parameters by working on matrices of much smaller dimension than the one of the whole GP, thus considerably improving the numerical stability and the computational efficiency of the optimization procedure. \n\n\\subsection{Prediction of observations and individual staging}\nGaussian processes naturally allow probabilistic predictions given the observed data. At any given time point $t^*$, the posterior biomarker distribution has the Gaussian form $p(\\mathbf{f}^*|t^*,\\mathbf{y}, t, \\mathbf{m}, t') \\sim \\mathcal{N}(\\mathbf{f}^*|\\boldsymbol{\\mu}^*, \\Sigma^*)$ with parameters:\n\n\\begin{eqnarray}\n\\boldsymbol{\\mu^*} &=& \\Sigma_G(\\mathbf{f}(t^*),\\mathbf{f}(t))(\\Sigma_{joint} + \\tilde{\\Sigma}_{joint})^{-1}\\boldsymbol{\\tilde{\\mu}}_{joint}\\\\\n\\Sigma^* &=& \\Sigma_G(\\mathbf{f}(t^*),\\mathbf{f}(t^*)) - \\Sigma_G(\\mathbf{f}(t^*),\\mathbf{f}(t))(\\Sigma_{joint} + \\tilde{\\Sigma}_{joint})^{-1}\\Sigma_G(\\mathbf{f}(t),\\mathbf{f}(t^*)).\n\\end{eqnarray}\nWe also derive a probabilistic model for the individual temporal staging  given a set of biomarker observations $\\mathbf{y}^*$, thanks to the Bayes formula:\n\\begin{eqnarray}\np(t^*|\\mathbf{y}^*,\\mathbf{y}, t, \\mathbf{m}, t') = p(\\mathbf{y}^*|t^*,\\mathbf{y}, t, \\mathbf{m}, t')p(t^*)\/p(\\mathbf{y}^*|\\mathbf{y}, t, \\mathbf{m}, t'), \n\\end{eqnarray}\nwhich we compute by assuming an uniform distribution on $t^*$, and by noting that $p(\\mathbf{y}^*|t^*,\\mathbf{y}, t, \\mathbf{m}, t')~\\sim\\mathcal{N}(\\boldsymbol{\\mu}^*,\\Sigma^*+\\Sigma_{\\epsilon})$. \nIn particular, the joint covariance form $\\Sigma_G(\\mathbf{f}(t^*),\\mathbf{f}(t^*))$ can be specified in order to account for incomplete data, and thus generalizes the GP model for predictions in presence of \\emph{missing biomarker observations}. \n\n\\section{Experiments}\\label{section:experiments}\n\\subsection{Synthetic multivariate progressions}\nWe benchmarked the model with respect to synthetic multivariate biomarker progressions.\nWe generated random multivariate sigmoid functions for $N_b$ biomarkers, $\\mathbf{f}(\\tau) = (f_{b_1}(\\tau), f_{b_2}(\\tau), \\ldots, f_{b_{N_b}}(\\tau))^{\\top}$, with $f_{b_k}(\\tau) = 1\/(1+\\exp(-\\alpha_k \\tau))$, $\\tau\\in [0,15]$ and $\\alpha_k \\sim \\mathcal{N}(0,.06) $, and we sampled $N$ individual noisy trajectories at time points $\\tau_k^j$: $\\mathbf{y}_k^j(\\tau_k^j) = f_k(\\tau_k^j) + \\epsilon$, $\\epsilon\\sim\\mathcal{N}(0,\\sigma^2)$.  For each individual we used the same initial sampling time point for every biomarker, while the number of samples per biomarker was allowed to independently vary between 1 and 4. The individual time points were subsequently centered by their mean $\\mu_k^j$ to obtain shifted time-points $t_k^j = \\tau_k^j - \\mu_k^j$ defined in the interval $[-2,2]$. \n\nThe model was applied to estimate biomarker progressions and individual time-shifts with respect to different combinations of trajectory noise $\\sigma$, sample size $N$, and number of biomarkers $N_b$. The accuracy of the model in reconstructing the original time series was quantified by Pearson's correlation between the estimated time-shift $d^j$ and the original individual time reference. The experiments were repeated 10 times for each configuration of parameters $\\sigma \\in \\{0.1, 0.2, 0.3, 0.4\\}$, $N_b\\in\\{4, 8\\}$, and $N\\in\\{20,100\\}$.\\\\\n\\begin{table}[b]\n\\centering\n\\begin{tabular}{c c c c c c || c  c c c c }\n&&\\multicolumn{4}{c}{$N = 20$}&\\multicolumn{4}{c}{$N = 100$}\\\\\\cline{3-10}\n&&\\multicolumn{4}{|c||}{$\\sigma $}&\\multicolumn{4}{c|}{$\\sigma $} \\\\\n & \\multicolumn{1}{c|}{}& 0.1 & 0.2 & 0.3 & 0.4 & 0.1 & 0.2 & 0.3 & \\multicolumn{1}{c|}{.4} \\\\\\cline{2-10}\n\\multirow{2}{*}{$N_b$} & \\multicolumn{1}{c|}{4\\,} &\\multicolumn{1}{c|}{ .95 (.03)}  &\\multicolumn{1}{c|}{ .86 (.08)} &\\multicolumn{1}{c|}{ .71 (.17)} &\\multicolumn{1}{c||}{ .46 (.29) } & \\multicolumn{1}{c|}{ .91 (.04)} & \\multicolumn{1}{c|}{.89(.04)} & \\multicolumn{1}{c|}{.76 (.17)} & \\multicolumn{1}{c|}{.75 (.12)}\\\\\n& \\multicolumn{1}{c|}{8\\,} &\\multicolumn{1}{c|}{ .97 (.01)} & \\multicolumn{1}{c|}{.91 (.06) }& \\multicolumn{1}{c|}{.86 (.06)} & \\multicolumn{1}{c||}{.66 (.3)} & \\multicolumn{1}{c|}{.94 (.04)} & \\multicolumn{1}{c|}{.94 (.02)} & \\multicolumn{1}{c|}{.88 (.06)} & \\multicolumn{1}{c|}{.84 (.07)}\\\\\\cline{2-10}\n\\end{tabular}\\caption{Mean (sd) $R^2$ correlation coefficient across folds between estimated individual time-shifts and ground truth time reference.}\\label{TableSynth}\n\\end{table}\n{\\bf Results.} Table \\ref{TableSynth} reports summary correlations between time-shift estimation and the ground truth individual sampling time. The correlation values are generally high, and increase with lower noise levels. Interestingly, the increase in number of modelled biomarkers is associated with a better performance in recovering the underlying disease staging. We also observe that larger sample sizes are associated with higher correlation values, especially with increasing noise levels. We note however an exception for the case $\\sigma = 0.1$ where, although the overall performance is still high, the correlation slightly decreases with $N=100$.\n\n\n\n\\subsection{Longitudinal modelling of Alzheimer's disease progression}\nWe collected longitudinal measurements for the ADNI individuals with baseline values of CSF A$\\beta$ amyloid lower than the nominal values of 192 pg\/ml. This preliminary selection is aimed to validate the model on a clinical population likely to represent the whole disease time-span.\n\nThe model was trained on a group including normal individuals, mild cognitive impairment subjects converted to AD (MCI conv), and AD patients having at least one measurement for each of the following biomarkers: \\emph{volumetric measures} (hippocampal, ventricular, entorhinal, and whole brain volumes), \\emph{glucose metabolism} (average normalized FDG uptake in prefrontal cortex, anterior cingulate, precuneus and \nparietal cortex),  \\emph{brain amyloidosys} (average normalized AV45 uptake in frontal cortex, anterior cingulate, precuneus and \nparietal cortex), and \\emph{cognitive function} measured by common cognitive questionnaires (ADAS13, RAVLT learning score, FAQ). The testing set was composed by the remaining subjects with at least a missing biomarker, as well as by the subgroup of MCI non converted to AD during the observational time (MCI stable).\nThe volumetric measures were scaled by the individual total intracranial volume, and all the biomarkers measurements were converted into quantile scores (0 to 1 for normal to abnormal values). \nTable \\ref{sociodem} shows baseline clinical and sociodemographic information of the individuals used respectively in training and testing set, while in Table \\ref{Nobs} we report the average follow-up time and the ratio of missing data of the pooled sample.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{ c c c c c c c}\nGroup & N & Age & Sex (\\% females) & ADAS13 & Hippo volume ($mm^3$)& AV45  \\\\\n\\hline\n\\multicolumn{7}{c}{Training data}\\\\\n\\hline\nNL & 76 & 75.8 (6) & 53 & 9.5 (4.4) & 7358 (762) & 1.24 (0.2)\\\\\nMCI conv& 57 & 72.7 (7) & 42 & 20.3 (6.8) & 6464 (861) & 1.44 (0.2)\\\\\nAD & 21 & 72.7 (10) & 43 & 29.3 (8.7) & 5872 (988) & 1.4 (0.2)\\\\\n\\hline\n\\multicolumn{7}{c}{Testing data}\\\\\n\\hline\nNL & 30 & 77.5 (6) & 43 & 11.1 (4.1) & 7137 (800) & 1.03 (0.14)\\\\\nMCI stable& 164 & 73.4 (6.9) & 39 & 15.5 (6.19) & 7028 (1009) & 1.28 (0.2)\\\\\nMCI conv& 71 & 75.4 (6.7) & 40 & 21.3 (5.4) & 5882 (8644) & NA\\\\\nAD & 98 & 74.7 (8) & 40 & 28.6 (8) & 5709 (1105) & 1.59 (0.1)\\\\\n\\end{tabular}\\caption{Baseline clinical and sociodemographic information for the study cohort. NA: no observations available for the considered group. NL: normal individuals, MCI: mild cognitive impairment, AD: Alzheimer's patients. }\\label{sociodem}\n\\end{table}\n\n\\begin{table}[b]\n\\centering\n\\begin{tabular}{ c c c c c c c c c }\nVentr & Hippo & Ent & Whole Brain & ADAS13 & FAQ & RAVLT & AV45 & FDG \\\\\n\\hline\n\\multicolumn{9}{c}{Training data}\\\\\n\\hline\n2.4 (0) & 2.4 (0) & 2.4 (0) & 2.4 (0) &3.2 (0) & 3.2 (0) & 3.2 (0) & 1.3 (0)  & 1.9 (0) \\\\\n\\hline\n\\multicolumn{9}{c}{Testing data}\\\\\n\\hline\n1.8 (.1) & 1.8 (.1) & 1.8 (.1) & 1.8 (.1) & 2.4 (0) & 2.5 (0) & 2.4 (0) & 1.3 (65)  & 1.6 (31) \\\\\n\\end{tabular}\\caption{Average follow-up years and percentage of individuals with missing data for each biomarker (in parenthesis).}\\label{Nobs}\n\\end{table}\n\nThe model was applied in order to estimate the temporal biomarker evolution and the disease stage associated with each individual in training  and testing set. The plausibility of the model was assessed by i) group-wise comparison of the predicted time-shift, ii) prediction of conversion to AD in the MCI testing group, and iii) correlation with respect to the time to AD diagnosis  for the MCI individuals subsequently converted to AD. We finally compared the progression modelled with our approach with respect to the one estimated  with the method proposed in \\cite{donohue2014}. The method was applied to the training data by using the standard parameters defined in the R package \\textsc{grace}\\footnote{https:\/\/mdonohue.bitbucket.io\/grace\/}.\\\\\n{\\bf Results.}\nThe estimated biomarker progression (Figure \\ref{figModelAD}) shows a biologically plausible description of the pathological evolution, compatible with previous findings in longitudinal studies in familial AD \\cite{bateman2012}. The progression is defined on a time scale spanning roughly 20 years, and is characterized at the initial stages by high-levels of AV45, followed by an increase in ventricles volume and abnormality of FDG uptake. These  latter measures are however heterogeneously distributed across clinical groups, and with rather large variability. The evolution is further characterized by increasing abnormality of the volumetric measures, and finally by the steady worsening of neuropsychological scores such as FAQ.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=12cm]{biomarkers.png}\n\\caption{Modelled biomarker progression in A$\\beta$ amyloid positive individuals (solid\/dashed lines: mean $\\pm$ sd).  NL: normal individuals, MCI: mild cognitive impairment converted to AD, AD: Alzheimer's patients.}\n\\label{figModelAD}\n\\end{figure}\nFigure \\ref{figPrediction} shows the posterior predicted distribution of the individual time shift.  Healthy individuals are associated with the early stages of the pathology in both training and testing data, while MCI and AD patients are characterized by respectively intermediate and late predicted progression stages. The group-wise comparison between the expected time-shifts is statistically significant between each group pairs ($p$ $<$1e-$4$). Furthermore, the temporal positioning of the non converting MCI lies between controls and MCI converters: when using the temporal cut-off based on the $10^{th}$ quantile of the time-shift distribution in the training AD population ($t=3.6$ years) we obtained an accuracy of $0.84$ for the discrimination between MCI converters and stable in the testing data. \nWe also measured a negative correlation with respect to the time to AD diagnosis in training, testing, and pooled MCI converter groups, with $R^2$ respectively equal to $-0.20\\, (p=0.1)$, $-0.28\\, (p=0.01)$, and  $-0.28\\, (p=0.001)$. \nFinally, when applying \\cite{donohue2014} to the training data we measured a strong correlation between the resulting individual time-shifts and those obtained with our method ($R^2$ = $0.89$, $p$ $<$1e-$16$). Nevertheless, our estimates provided \\emph{consistently larger effect sizes} for the group-wise separation: (1.96,1.36,0.57) with our methods and (1.74, 1.18, 0.47) with \\textsc{grace} for AD vs NL, MCI vs NL, and AD vs MCI, respectively. \n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=12cm]{predictions.png}\n\\caption{Posterior prediction for the individual time shift in training (top) and testing (bottom) data. Healthy individuals are generally displaced at the early stages of the pathology, while the predictions for MCI and AD patients are associated with respectively intermediate and late progression stages.  NL: normal individuals, MCI: mild cognitive impairment, AD: Alzheimer's patients.  }\n\\label{figPrediction}\n\\end{figure}\n\n\n\\section{Conclusions}\nWe proposed a unified GP-based approach to disease progression modeling from time-series of biomarker measurements enabling novel applications beyond the state-of-art, such as the probabilistic prediction of disease staging in unseen patients.\nFurthermore, the model naturally accounts for missing data, and provides uncertainty quantification of the biomarker evolutions. The model provided remarkable modeling and predictive performance when tested on a large clinical cohort, and thus represents a promising instrument for the analysis of clinical trials data.\n\nSimilarly to \\cite{donohue2014}, in this work we focused on the modeling of disease staging represented by a time translation. However, the proposed framework can naturally account for more complex time transformations, provided that a sufficient number of time points is available for each individual.\nFuture extensions of this model will focus on the quantification of the effect of each biomarkers in the predictive performance, for example by integrating feature selection based on automatic relevance determination. Moreover the present work can be extended to model differential progressions underlying sub-pathologies, by framing the proposed random effect regression within the realm of Gaussian process mixture models. Finally, thanks to the flexibility of our framework, further extension of the model will enable to integrate  within (\\ref{FixCovariance}) a spatio-temporal covariance model (such as the efficient Kronecker form of \\cite{lorenzi2015}), to provide a unified framework for jointly modelling time series of images and scalar biomarkers data in a coherent fully Bayesian setting.\n\n\\section{Acknowledgments}\nEPSRC grants EP\/J020990\/01 and EP\/M020533\/1 support DCA and SO's work on this topic. ML, DCA and SO also received support from the European Union's Horizon 2020 research and innovation programme under grant agreement No 666992 (EuroPOND) for this work. MF gratefully acknowledges support from the AXA Research Fund.\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Shaun Pinder, Haragos P\u00e1l and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive)\n\n\n\n\n\n\n\n[Illustration: \"The token of Hilda!\"]\n\n\n\n\n                            ULRIC THE JARL:\n\n                    A Story of the Penitent Thief.\n\n\n                        BY WILLIAM O. STODDARD.\n\n                            [Illustration]\n\n\n                                LONDON:\n\n         CHARLES H. KELLY, 2, CASTLE STREET, CITY ROAD, E.C.;\n\n                     AND 26, PATERNOSTER ROW, E.C.\n\n                                 1899.\n\n\n\n\n                    [ENTERED AT STATIONERS' HALL.]\n\n\n                   HAYMAN, CHRISTY AND LILLY, LTD.,\n         HATTON WORKS, 113-115, FARRINGDON ROAD, LONDON, E.C.\n\n\n\n\n                               CONTENTS\n\n\n  CHAPTER                                                           PAGE\n\n  I.       Around the Viking House-fire                                7\n\n  II.      The Going Out of the Ice                                   17\n\n  III.     The Launching of \"The Sword\"                               22\n\n  IV.      The Ship \"The Sword\" and the Ice King                      37\n\n  V.       The Unknown Thing                                          45\n\n  VI.      The Fall of the Ice King                                   53\n\n  VII.     The Living Sand                                            60\n\n  VIII.    The Saxon Shore                                            75\n\n  IX.      The Taking of the Trireme                                  86\n\n  X.       The Great Sacrifice of the Druids                          96\n\n  XI.      The Passing of Lars the Old                               108\n\n  XII.     Svein the Cunning Jarl                                    119\n\n  XIII.    Hilda of the Hundred Years                                127\n\n  XIV.     The Jew and the Greek                                     136\n\n  XV.      The Storm in the Middle Sea                               149\n\n  XVI.     The Dead God in Africa                                    165\n\n  XVII.    The Murmuring of the Men                                  181\n\n  XVIII.   The Evil Spirit on \"The Sword\"                            193\n\n  XIX.     In the Night and in the Fire                              202\n\n  XX.      Carmel and Esdraelon                                      214\n\n  XXI.     The Rabbi from Nazareth                                   225\n\n  XXII.    The Tomb Song of Sigurd                                   240\n\n  XXIII.   In a Place Apart at Night                                 255\n\n  XXIV.    The Passing of Oswald                                     266\n\n  XXV.     The Messenger of the Procurator                           276\n\n  XXVI.    The Cunning of Julius                                     284\n\n  XXVII.   The Lion and the Tiger                                    296\n\n  XXVIII.  The Jarl and the Rabbi                                    307\n\n  XXIX.    Beautiful as Aphrodite                                    318\n\n  XXX.     The Javelin of Herod                                      330\n\n  XXXI.    The Places of Sacrifice                                   340\n\n  XXXII.   The Mob of Samaria                                        348\n\n  XXXIII.  The House of Pontius the Spearman                         359\n\n  XXXIV.   The School of Gamaliel                                    371\n\n  XXXV.    In the Court of the Women                                 382\n\n  XXXVI.   The Secret Messenger                                      399\n\n  XXXVII.  The House of Ben Ezra                                     411\n\n  XXXVIII. The Son of Abbas                                          422\n\n  XXXIX.   The Passover Feast                                        438\n\n  XL.     \"A Little While\"                                           448\n\n\n\n\n                             ILLUSTRATIONS\n\n\n                                                                  FACING\n                                                                   PAGE\n\n  \"The token of Hilda!\"                                     Frontispiece\n\n  \"Go forth into the sea, O sword!\"                                   31\n\n  \"Let him win it or perish!\"                                        176\n\n  \"O companion of Hilda!\"                                            263\n\n  \"O thou Jesus, of the sons of the gods!\"                           313\n\n\n\n\n                            ULRIC THE JARL.\n\n\n\n\n                              CHAPTER I.\n\n                     AROUND THE VIKING HOUSE-FIRE.\n\n\nIn the Northland were the roots from which grew the great nations\nwhich now rule the earth. The tribes were many, but the principal\nrepresentative and the absorbent of their thoughts and their traditions\nmay receive from us the general name of Saxons. These were the\nswordsmen of the sea whom the Roman legionaries declared to be the\nhardest fighters they had met, whether on land or water.\n\nIn the Northland were also the germs of political and religious\nliberty, and here were to be found the first forms of our highest faith.\n\nBut the men of the old race sailed southward and then eastward, at the\nfirst, taking their gods with them. Not until centuries later did they\nmarch and conquer this far western world, but we, their children, still\ndevoutly believe that the great God came with them.\n\n                   *       *       *       *       *\n\nThe landward <DW72> of a vast gray granite headland was thickly\ncovered with towering pine trees. Beyond them, inland, lay a snowy\nvalley without woods, and beyond that arose a blue and misty range of\nmountains. There were no trees upon the summit of the headland; only\nbare rocks, storm worn and deeply furrowed, were uplifted to meet the\nbitter wind that swept down over the flinty ice covering of the North\nSea from the yet colder winter which was manufacturing icebergs within\nthe arctic circle. Sheer down, hundreds of feet, the perpendicular\nface of the cliff smote sharply the glittering level that stretched\naway westerly over the sea to the horizon, while an arm of it pushed\nin eastward over the fettered waters of a deep and gloomy fiord,\nrock-bordered.\n\nHere would evidently be a good harbor in summer, when the waters should\nbe free, but now it had a forbidding, dangerous look, and out of the\nfiord poured continually a volume of roaring sound, the solemn organry\nof the wind playing upon the icy and rocky reflectors.\n\nThere was another gigantic sea cliff at a distance of about a mile down\nthe shore, southerly. Between that and the headland the ice line curved\nraggedly inward along the lines of a sheltered cove, which might at\nanother season provide a landing place. Midway, and at the head of the\ncove, there lay, propped up on either side by timbers, the bare hull of\na well-made vessel. It was of goodly size, being over thirty paces in\nlength and of full six paces in width at its middle. At the prow and\nat the stern it was high built, with short decks, under which was room\nfor stowage and for the sheltered sleeping of men. It was lower made\namidships, where were both seats and standing room for rowers, and on\neither side were twenty thole pins. In appearance the hull was somewhat\nflat-bottomed, but it had a keel. At the center arose a stout, high\nmast, but upon it there was yet neither yard nor boom nor sail. Both\nprow and stern were sharply made. Evident was it that she was new and\nhad never yet floated. Her outline was of much beauty, and all her\ntimbers and planks were heavy and strong, that she might battle with\nrough seas and with the ice cakes of the spring breaking. From her prow\nprojected a beak of firmly clamped and tenoned oak, faced and pointed\nwith iron, that she might break not only the waves, but the ribs of\nother ships. All around her and in some parts over her lay the white\nsnow, deeply drifted, but wherever the woodwork was uncovered there\ncould be seen much of skillful carving and smooth polishing.\n\nAt other places along the curve of the cove there were boats and ships,\nlarger and smaller. All were hauled up above high-water mark, and snow\nwas on them. The larger craft seemed to be stanch and seaworthy, but\nnot any of them were equal in size or in strength or in beauty to the\nnew warship.\n\nUpon a straight line inland a hundred fathoms, as if the iron beak were\npointing at it, stood a long, low, irregular building of wood with high\nridged roofs, in which were wide holes at the ridges. From these holes,\nas if they were instead of chimneys, columns of blue smoke were rising\nto be whirled away by the wind. Stonework or brickwork was not to be\nseen. Through the strong timber walls, under the projecting eaves, were\nmany openings, equally cut, window-like, for the entrance of light and\nair on sunny days, but these all were now closed by wooden shutters,\nsome of which were braced from without. The timbers of the house walls\nwere cleanly hewn and skillfully fitted, and they were tightly calked\nwith moss and tempered clay. The roofs were of shingles riven from the\npine trees.\n\nBeyond, landward, there were smaller, ruder structures for the shelter\nof horses, cattle, sheep, and swine, and there were many ricks of hay\nand straw and of yet unthreshed grain. In either direction around the\ncove and scattered irregularly up the valley were a number of less\nextensive buildings for the abode of men. Some of these were mere\nhuts, built ruggedly of timber and unhewn stone. From every roof was\nthere blue smoke rising to testify that there were no empty houses in\nthis seashore village of the vikings. Around the central cluster of\nbuildings there were palisades, but except for these there were no\nsigns of fortifications. It was as if there need be little fear of the\ncoming of any foeman.\n\nBitter and cold and strong was the windstorm that blew across the icy\nsea and smote upon the swaying crowns of the pine forest and howled\namong the bare boughs of the oaks. It came and knocked at the great\ndoor in the front of the house pointed at by the beak which was the\nforefinger of the ship.\n\nThe door swung open for a moment and then it closed, but in that moment\nthere rang out loud voices of rude song and the twanging of sonorous\nharp strings. Also a great blast of fresh, pure air rushed eagerly\ninto the house, where it was much needed. Not but that the vast room,\nlow-walled, high-roofed, was fairly well ventilated in many other ways,\nbut the fire in the middle of its earthen floor was blazing vigorously,\nand not all the smoke might readily escape at the round gap in the roof\nridge over it. Now and then, indeed, the wind blew rudely down through\nthat aperture and sent the smoke clouds eddying murkily among the\nrafters.\n\nBut for the fire blaze and for sundry swinging cressets filled with\nburning pine knots the great hall would have been gloomily dark, but\nthese lights were enough, in spite of the smoke clouds, to show many\nthings which told of what sort this place might be. So also might be\nplainly noted the faces and the forms of the men who sat or stood\naround the fire, or who lay upon the bearskins and the wolfskins that\nwere scattered here and there upon the earth floor and upon the wooden\nsettles along the walls.\n\nA broad table ran across a raised dais at one end of the room, and on\nthis were not only pitchers and mugs of earthenware variously molded,\nwith many drinking horns, but there were also tankards and goblets and\nsalvers of silver, richly designed and graven by the artisans of other\nlands than this. Of the articles of furniture for different uses some\nfew had an appearance of having been brought from far, but the great,\nhigh-backed oaken throne chair behind the long table, at its center,\nwas rich with the grotesquely elaborate carvings of the old North\npeople. On the walls hung shields and arms and armor of many patterns.\nThe steel caps of the vikings hung side by side with visored helmets\nthat told of Greece and Rome and of lands yet further east. There were\nmany men in the room. Some of them were scarred old warriors, but there\nwere youths of all ages above mere boyhood. Likewise were there numbers\nof women.\n\nAs central as was the fire itself were three figures which seemed to\nattract and divide the attention of all the others. On the side of the\nfire toward the door towered one who looked a very embodiment of the\nwarlike young manhood of the race of Odin. His blond beard and mustache\nwere full but not yet heavy. His complexion was fair, notwithstanding\nits weather bronzing, and his steel-blue eyes seemed both to flash and\nto laugh as he stood with folded arms and listened. His dress was\nsimple. His shoes, that arose above his ankles, were well made. Above\nthem were leggings of tanned leather, and he wore a tunic of thick,\nblue woolen cloth. He was unarmed except for the slightly curved,\nbroad-bladed seax in its sheath that hung from his belt. Its blade\nwas not more than a cubit in length. It was sharp on one edge only,\nand it was heavy. The steel hilt and the crosspiece were thick, for\na good grip. It was a weapon terrible to meet if it were in the hand\nof an athlete like this--more than six feet in height, deep-chested,\nlithe and quick of motion--and already the short seax had won for its\nbearers, the Saxons, a dreaded name among all the peoples of the south\ncountries to which their swift keels had carried them.\n\nAt the left of the fire was a large, high-backed chair made of some\nwood which had become almost black with age and smoke. It was not now\noccupied, but in front of it stood the form of a woman, straight as a\npine and taller than any of the men around her. Her face was swarthy,\ndeeply marked, haughty, and her abundant hair fell disheveled down to\nher waist, as white as the drifts upon the mountains. She was clad in\na robe of undyed, grayish wool, falling loosely to her feet. On these\nwere socks and buskins, but her lean, sinewy arms were bare as she\nstretched them out, waving her gnarled old hands in time to the cadence\nof a semimetrical recitation. She spoke in the old Norse tongue, with\na voice upon whose power and mellowness time seemed to have had little\neffect. Every head in the hall bent toward her, as if her words were a\nfascination to her hearers, and none willed to interrupt her.\n\nWeird and wild was the chant of the old saga woman, and the fire in\nher piercing black eyes brightened and dulled or almost went out as she\nsang on, from myth to myth, of the mystical symbolisms of the intensely\npoetic and imaginative North. Gods and demigods and goddesses, heroes\nand heroines, earth forces and spiritual powers, dwarf and giant, gnome\nand goblin, fate maidens, werewolves, serpent lore, the nether frost\nfires, the long night of the utter darkness, the twilight of the gods,\nthe eternal hall of the slain, the city of Asgard--long and wonderful\nwas the saga song of the white-haired woman who had, it was said, seen\nthe ice of more than fivescore winters float out of the North Sea.\n\nShe ceased speaking and sank back into the chair as if all life had\ngone out of her. Rigid and motionless she sat, and there was no light\nin her eyes, but none went near her, nor did any speak. There was\nindeed a momentary outburst of approval, but it hushed itself. Even a\nfierce laugh that came to the lips of the tall young warrior died away\nhalf uttered.\n\nAlmost at the same moment another sound began to fill the hall. It\ncame, at the first, from a large harp that stood a few paces back from\nthe fire. Over the strings of this harp were wandering the long, bony\nfingers of a pair of gigantic hands, while behind it, on a low stool,\nswayed and twisted a form whose breadth of shoulder and length of arm\nwere out of all proportion to its height. The head was bald except for\na fringe of reddish-gray hair above the ears. The face was scarred and\nseamed to distortion, the right eye having been extinguished by a sword\nstroke which, by its furrow, must have half cloven the frontal bone.\nAge was indicated by the tangled gray beard which floated down below\nthe belt, but not in the powerful, rich-toned voice of the harper, for\nthe smoke seemed to eddy and the fire to dance as the harp twanged more\nloudly, and then there came to join it a burst of stormy song--a song\nof battles on the land and on the sea; a song of the mighty deeds done\nby the warriors of old time; a song of fierce and stirring incitement\nto the performance of similar feats by those who listened.\n\nThe harp grew then more softly musical, for he sang of the blue waves\nand the sunny shores of the southern seas; of their islands of beauty;\nof their harbors of peace and their cities of splendor; of temples and\ncastles; of gold and silver and gems; but he seemed to drift beyond all\nthese into a song of something beautiful, which yet was vague and far\naway and indescribable. His thought and word concerning it became like\na refrain, until the minds of all who heard were filled with ideas of\nthe dim and unattainable glory of the land of heroes, the city of the\ngods, the return of the White One, and the rising of the sun that will\nnever set. Like deep answering unto deep were the last utterances of\nharp and harper, and as they suddenly ceased the tall young warrior\nstepped forward two paces and cried loudly:\n\n\"O Hilda! wise woman of the hundred winters, if this is indeed to be\nthy last----\"\n\n\"I shall go out with the spring flood,\" she said, interrupting him,\n\"but thou wilt be upon the sea when they lay me in the cleft between\nthe rocks.\"\n\n\"I will go forth as thou sayest,\" he responded. \"Am I not of the sons\nof the gods? I will sail as my father sailed and as Oswald has sung. I\nwill crush, like him, the galleys of the Romans. I will look upon the\ncities of the east and of the south. I am of Odin's line. I will go\nout in the good ship _The Sword_, and will sail until I see the hero\ngod and the city of the gods and the land of the living sun.\"\n\nLoud now rang the shouts of approval from the bearded vikings as they\nsprang to their feet and began to crowd around their young leader.\n\n\"Go, O Ulric, son of Odin! Sail on into the sunset and the farther\nsea!\" came trumpetlike from the white lips of Hilda.\n\nLow sounds arose, too, from the strings of the harp, but the door swung\nsuddenly open and upon the threshold stood a man garbed in wolfskins.\n\n\"Hael, Ulric the Jarl!\" he shouted, and there were many exclamations\nhere and there around the room.\n\n\"Hael, Wulf the Skater!\" heartily responded Ulric. \"What bringest thou?\"\n\n\"Good tidings!\" replied Wulf, joyously, stepping forward. \"I came down\nthe mountain slide and across the fiord. No other foot will cross\nit this season. During days the ice hath weakened and now the wind\nis changing southerly. There is already a rift in the sky. O son of\nBrander the Brave, be thou ready for the spring outing!\"\n\n\"Odin!\" shouted Ulric. \"Keels for the open sea! Hael to the cruise of\n_The Sword_! Hael to the bright south! And I, Ulric the Jarl, I of the\nsons of the gods, I will go out and I will not return until I have\nlooked into the face of one of the gods. And he will know me, and he\nwill take me by the hand, and he will bid me walk with him into the\ncity of the living sun!\"\n\nGlad were the hearts of all the vikings as they heard, and with one\naccord they shouted loudly:\n\n\"Hael to Ulric the Jarl! Hael to the cruise of _The Sword_! We are his\nmen and with him we will go!\"\n\nLong had been the winter and slow had been the coming of the change for\nwhich men waited. Welcome was Wulf the Skater, but Oswald's fingers\nwere slowly busy among the strings of his harp, and they found strange\nsounds which came out one by one.\n\n\"The message of the harp!\" muttered Hilda. \"It is like the moaning of\nthe sea in the fiord in the long night.\"\n\n\n\n\n                              CHAPTER II.\n\n                       THE GOING OUT OF THE ICE.\n\n\nWulf the Skater brought true tidings to the house of Ulric, the son of\nBrander the Brave, on the day of Saturn. Winter was ending. The word\npassed on from house to house until all in the village came out and\nlooked upward, seeking for the blue rift in the sky. The wind blew not\nnow as in the morning. The north wind had gone elsewhere, and instead\nthere came up from the south a breathing which was fitful and faint at\nfirst. It was cool, also, from having touched the frost faces on its\nway. Only one more hour went by and the sky was almost clear, so that\nthe sun shone down unhindered and his heat was surprisingly strong.\n\nThe south wind grew warmer and more vigorous toward sunset, but with\nhim now came a fog so dense that no man cared to go out into it; for if\nhe did, it was as though darkness touched him. All through the evening\nthe south wind sighed softly among the homes of the vikings, and went\nwandering up the fiords, and felt its way, shivering, across the flinty\nlevels of the frozen sea, but toward the morning of the day of the sun\nthe breeze brought with it, also, to help it, a copious warm rain.\nBefore the noon torrents were leaping down the sides of the mountains\nand the sea was beginning to groan and heave and struggle in its effort\nto take off and put away its winter mail.\n\n\"Harken!\" said Oswald, the harper, as he sat by the now smoldering fire\nin the hall of Ulric's house.\n\n\"I hear,\" said Hilda from her place on the other side of the ash heap.\n\"It is the last time that I shall listen to the song of the outing ice,\nbut I shall feel the wind from the sun land and I shall see the grass\ngreen in the valley before I go. There will be buds on the trees when\nI pass down into the earth to meet my kindred. O what a realm is that!\nThe land of shadows. The under world which has the sod for a roof. But\nthe old runes on the rocks tell of wide places. One may travel far in\nthat land, and where I may go I know not.\"\n\nThe gnarled fingers of Oswald were searching among the strings of his\nharp, but only discords answered his touches.\n\n\"I have heard,\" he said, \"that they hang their shields on the roots of\nthe trees, and they see as we see in a twilight. I think I have heard\nthem harping in the summer nights, when the moon was full and the wind\nwas in the pines. I would that my own harp might be buried with me.\"\n\n\"No need,\" said Hilda. \"They have better harps than thine. They will\ngive thee one. It is well that the weapons of a warrior should be\nplaced beside him in his tomb, but they must be marred in token that he\nuseth them no more. He hath left others for his kinsmen. There are many\ngood swords in the old tombs. One day they will all be opened and the\nblades will be found.\"\n\n\"And also much treasure,\" grumbled Oswald; but his harp twanged angrily\nas he said it, for he had ever been a man to hold fast anything in\nthe shape of coined money or of precious metal. Many were said to be\nthe outland coins in his leather bag in his room at the southerly\nend of the house. He had sometimes shown them to inquiring folk, but\ngrudgingly, and he had always tied them up again tightly, as if he\nfeared that there might be a thief even among the vikings.\n\nHilda arose and walked slowly across the room to the open door. She\nlooked toward the sea, but the mist and the rain were a curtain.\n\n\"Hammers!\" she said. \"I can hear them. Ulric and his men are at work\nupon the ship. She will be ready to launch when the ice goeth out. She\nwill sail to the Middle Sea, but when I look for her I cannot see her\ncome again.\"\n\nOnce more she turned, and this time her slow and stately march carried\nher to the farther end of the hall, on the dais, where many suits of\narmor were hanging. She went straight to one of these and she touched\nit, piece by piece, while Oswald leaned upon his harp and watched her.\n\n\"When the hour was upon me,\" she said, \"I saw the son of Brander in\nbattle, and the men upon whom his ax was falling bore shields like\nthis. There were dark men with them, wearing turbans. It is well. I\nthink that at the end of this cruise he will come to me where I am. It\nwere no shame to his father's son that the valkyrias, when they come to\ncall the hero to Valhalla, should find him circled with slain Romans.\nBrander the Sea King took these arms for his trophies in the great\nfight off the coast of Britain. He drove the Roman galley ashore. He\nburned it with fire. Not one Roman escaped.\"\n\n\"I have seen Britain,\" muttered Oswald.\n\n\"Brander the Brave liked Britain well,\" continued Hilda. \"It is a fair\nland, he said. If he could take more men with him, he would drive out\nof it the Romans and the Britons and keep it. But he said they have no\ngood winters there, and the summers are all too long. It would be no\nland for me. What would I do in an island where the fiords do not shut\nup at the right season? I should perish!\"\n\nVery thoughtful was the face of the tall daughter of the Northland as\nshe passed along, inspecting the armor and talking to herself about its\nvaried history. Some of it had been won in fights with far-away peoples\nbefore she was born, but more of it had been brought into that hall\nbefore her eyes, and she had heard the bringers tell the tales which\nbelonged to its pieces and to the swords and spears. Now, therefore,\nhanging there on the wall, the war treasures of the house of Brander\nwere page-marks for her memory, and she also was a book of the old\nhistory of the Northmen from the days of the gods to this hour of her\nown closing.\n\nSwiftly went by the day of rain and thaw, but their work was tenfold in\nthe night which followed it. The rain fell on the roof in increasing\nabundance, and the wind threw it with force against the sides of the\nhouse. The torrents on the mountains grew into small swift rivers, and\nthey made a continual loud sound of rushing water; but that was not the\ntumult which so filled the air and smote upon the ear. All other sounds\nwere overborne by the booming and groaning of the ice and by the roar\nwith which its loosened edges ground against the granite cliffs in the\nfiords.\n\nThe day of Saturn had been a day of frost and snow and storm until near\nits close. The day of the sun had brought the sun's breath from his\nown land and his smile into the sky, and he had slain the winter at a\nblow. The morrow would be the day of the moon, and before its arrival\ncame now this night of such uproar that Oswald did not care to touch\nhis harp, and the vikings mended their armor and sharpened their swords\nin silence. Hilda also was long silent, nor had Ulric the Jarl spoken\naught that could be heard by all. When at last his voice arose, and men\nput by their work to hear, he gave answer to a question of Tostig the\nRed.\n\n\"Aye!\" he said loudly, \"the ship is ready from stem to stern. We will\nlaunch her behind the ice as it leaveth the shore. We will follow the\nfloes as the tides bear them southward; ever do they melt as they go.\nSo shall no other ship sail before us, and we shall be the first of all\nkeels from the Northland, this year, among the islands of the Middle\nSea.\"\n\nFiercely twanged the harp of Oswald and loud rang the shouts of the men\nwho heard the young jarl speak his purpose, but before the harp could\nsound again Hilda arose in her place.\n\n\"Son of Brander,\" she said, \"thou wilt go. Thou wilt see many things.\nAll day have I been watching thy path, and the clouds are over it. In\nthis thing that I now tell thee, do thou as did thy father: crush the\nkeels of Rome in the seas of Britain and smite the men of Rome on the\nBritish island. And in the end of all thou wilt die, as did thy father,\nat the hand of a spearman of C\u00e6sar.\"\n\n\"So be it,\" shouted Ulric, with a laugh on his lips and a flash of fire\nin his bold, bright eyes; \"I ask no better!\"\n\nHe said no more, but seated himself and began to sharpen his seax on a\nsmooth, hard stone.\n\n\n\n\n                             CHAPTER III.\n\n                     THE LAUNCHING OF \"THE SWORD.\"\n\n\nThe day of the moon, the second day of the week, dawned brightly over\nthe village of the vikings. The faces of the cliffs along the shores of\nthe Northland boomed back continuous echoes of the thunderous reports\nof the splitting ice. The frost had been strong, and the winter mail\nof the sea was thick and hard, but the sun and the lifting tides and\nall the torrents from the mountains made a league, and they were more\npowerful than was the ice. The south wind also helped them.\n\nAll the hours since Wulf the Skater brought the news of the coming\nthaw had been spent by Ulric and his men in getting the good ship\n_The Sword_ ready for the water. No room in her was to be wasted, and\nher hollow, to her very keel, was now closely packed with provisions,\ntaking the place of other ballasting. There were tightly stowed barrels\nof pork and beef, and there were bags and boxes of hard bread, and\ncasks of ale and casks of water. Over the greater part of these were\nplanks fastened down like a deck, for the voyage to be undertaken\npromised to be long, and all except provisions for immediate use must\nbe sealed until a day of need.\n\nThe seats of the rowers were all in, and the short oars, and also the\nlong oars, which a man would stand erect to pull with. The small boats\nwere fastened upon the half decks, fore and aft. The mast was now\nstayed and rigged and the spars and the sail had been swung in their\nplaces. Not of woven stuff was the sail, but of many well-dressed skins\nof leather, that it might toughly withstand any gale.\n\nThere were twenty oars on a side, and the crew who were to do the\nrowing, taking their turns, had been carefully selected during the\nwinter. Their war shields were hung along the bulwarks, and they placed\nthem there with great pride. The chosen men who lived further inland\nwere now arriving, and they were as eager as were the men who dwelt\non the shore. Stalwart and high-hearted were all the vikings who were\nto sail in _The Sword_. Among them were veterans who had fought under\nBrander the Brave, the father of Ulric, and others were youths who were\nnow going out for their first venture in distant seas. Great store\nof weapons went on board, for there had been much making of bows and\narrows and swords and spears and shields all winter. So the gray-headed\nand caretaking warriors declared that the ship was exceedingly well\nprovided.\n\nAt the dawn of the day of the moon Ulric the Jarl stood at high-water\nmark looking seaward.\n\n\"As the tide turneth I shall know,\" he said to those who were with him.\n\"The flood hath lifted the ice, but the ebb must lower it. _The Sword_\nwill be launched at the next high tide if the outing is good.\"\n\nThat might be toward the evening, and word went out so that all might\nbe ready.\n\nThe ship as yet bore no flag, but on the forward half deck stood a\ngreat anvil, carved finely of oak and blackened, and upon the anvil was\nfastened a massive hammer, made in like manner, that Thor the Great,\nthe god of war, the smith god, might go with _The Sword_ into any\nbattle. Now could more fully be seen the carvings and the gildings and\nthe many rich ornamentations which had been lavished upon the ship, and\nmen who now saw her for the first time marveled at her beauty and at\nthe strength of her timbers.\n\n\"Larger ships have been,\" they said, \"but not many, nor was there ever\none that gave better promise of bearing well the shock of another ship\nor the stroke of an ice floe.\"\n\nAll day the sound of harping could be heard in the house, for other\nharpers besides Oswald were now there, and they played and sang in a\nrivalry with each other. Hilda was not to be seen. It was said that\nshe had shut herself up in her own room and would have none speak with\nher. Although the house was thronged, there were none who thought well\nto disturb her. Not many, indeed, were curious enough to pass near\nthe closed door behind which she was believed to be looking into the\ntwilight where the gods live, and out of which come those whose shadows\ndarken the woods at times and whose voices are heard in the night as\nthey talk to one another across the fiords.\n\nThe noon came and at low tide the ice edge was out twenty fathoms\nfrom the shore, leaving clear water behind it. If it should shove in\nagain, there would be no launching, but as the ebb ceased there came an\nunexpected help. A mighty drift of snow and ice had formed, in early\nwinter, hundreds of feet above the level, and yet in a hollow of the\nhigh mountain at the head of the fiord. Hard and strong was the grasp\nof this glacier upon the rocks and trees at its sides, but under it was\na stream which had been covered, though not entirely closed. Above and\nbeyond was now a lake of melted snow, and the water from it was forcing\nits way under the glacier by that rivulet channel, mining, mining,\nmining, until its work was done.\n\nThere was a great sound of breaking, a sound that was sharp, rasping,\nshrieking, as if the mountain uttered a great cry to see the glacier\ntear itself free and spring forward. The screams of a gier-eagle,\nstartled from the withered pine tree on the summit, answered the scream\nof the mountain. Down, down, faster and faster, to the sheer precipice\nat the face of the fiord, and then the glacier itself uttered an awful\nroar as it leaped headlong from the cliff. A thunderous boom responded\nfrom the smitten face of the ice, and through the clefts that were\nmade in all directions the freed salt water bounded high into the\nsunshine, which it had not seen since it was imprisoned in the dark by\nthe winter. The entire mass went over, and with it went the bowlders,\nearth, and trees which it had rent off and brought away. The blow which\nit struck was as a blow from the hammer of Thor, and a vast wave rolled\nout of the fiord, breaking the nearer ice as it went and splitting\nsquare miles of the sea face beyond into floes of a right size for\ndrifting. Out slipped the ice edge at the cove, a hundred fathoms\nfurther. In it came again angrily, but only to retreat once more and\nleave a wider, surer harbor for _The Sword_ to dip her keel into when\nher launching hour should come.\n\nAll things were ready, both at the house and on the shore, when Oswald\nleft his harp to go and speak to one of the maidens, of whom were many\ncome to see the warriors depart.\n\n\"Go thou to Hilda,\" he said. \"Say to her that shortly she will be\nneeded at the ship.\"\n\n\"Come,\" said the maiden to other women who were near her, for she cared\nnot to go alone.\n\nTruly it was not far to go and come, stepped they never so slowly, and\nthey soon brought back word that her door was open, but Hilda they did\nnot find, nor did any know whither she had gone.\n\n\"So?\" said Oswald, thoughtfully. \"Pass thou on, then, and tell this to\nUlric, the son of Brander, for he will understand. Bid Wulf the Skater\nand Tostig the Red that they come now to me.\"\n\nHastily went the maiden, for of this errand she had no fear.\n\nOn the summit of a low hill not more than half a mile from the house\nwas a great heap of stones. Around it, in an oval, standing like\nwatchful sentries, were many great stones, tall and upright. Upon the\nfaces of these uprights were chiseled words in the old runes. A path\nthat led to this hill had been kept open during the winter, and when\nHilda left the house, with none to mark her going, she had walked\nalong this path. The snow in it was soft, taking footprints, and Hilda\nstooped, looking closely at some which were already there. She followed\nthem until they ceased at the heap of stones. She smiled and bowed her\nhead approvingly.\n\n\"Ulric hath been here,\" she said. \"He hath spoken to his father at the\ntomb. The son of the hero will himself be a hero. There is no other\nlike him among the young branches of the tree of Odin.\"\n\nStrong affection sounded in her words concerning the youthful head of\nthe ancient house of Brander the Brave. A flush came for a moment\ninto her withered face, and she stood in silence gazing at the tomb.\nSlowly her arms arose, waving, and her lips opened in a recitative that\nsounded like a song, wherein she was speaking to the father of Ulric\nand to other names than his, calling them her kindred. Louder, more\nweird, mournful, thrilling, grew the tomb song of the old saga woman.\nBut it suddenly ceased, for to her came a response from one that stood\nupon the crest of the central heap of stones.\n\nNot in any human voice of the dead or of the living was her answer,\nbut from the gaunt and grisly shape of a large gray she-wolf,\nfamished-looking, that stood there, snapping fiercely her bloody jaws\nand gazing at Hilda. Then lifted the wolf her head to send forth a\nlong-drawn, wailing howl.\n\nThe long, late winter had been a hard one for all wolves and for other\nwild beasts, for against them the sheepfolds had been well guarded.\nAnd now this hunger-driven monster from the mountains had taken her\nopportunity to venture in almost to the village, finding this day a\nflock without a shepherd. She had ravaged unfought, and now she was\nhere upon the tomb of Brander. Her presence there was as if she had\nbeen a written message to Hilda.\n\n\"Art thou here?\" she exclaimed. \"Aye! Thou art as I saw thee at the\nhouse. Thou art the name of Rome, O bloody mouth! Scourge of the world!\nCurse of all nations! Hungry one! The swords of the Northmen shall yet\nsmite the cubs of the she-wolf in their own den.\"\n\nA sharp, harsh bark, another howl, and a snapping of jaws replied to\nher and then the she-wolf sprang away, disappearing beyond the tomb,\nbut Hilda turned and walked houseward along the path, muttering low as\nshe went.\n\nWhen Tostig the Red and Wulf the Skater came to Oswald, the harper, he\ngave them an errand, for they at once went away together to one of the\nbest made of the stables in the rear of the house. They had not yet\nreturned when Hilda walked past the house and on down to the beach. All\nmen knew that the right hour for the launching of _The Sword_ had come\nwhen Hilda came and stood at the prow of the vessel, laying her hand\nupon it.\n\nShe spoke then but few words, pointing at the heaps of driftwood and\nloose pieces of timber which were there and giving her commands. Those\nwho heard her began to gather all this wood into a great heap. It was\nmore like two heaps, for there was left a bare spot in the middle large\nenough for a yawlboat to have been lodged therein.\n\nUlric, the son of Brander, came and stood by Hilda, and as she looked\nat him the color arose again into her face and a kindly light kindled\nin her eyes. He also smiled at her very lovingly. She spoke a word\nthat none else heard, and he blew three long, powerful blasts upon his\nwar horn. From all directions came in haste the vikings and the other\nshore people and the upland people, both the old and the young, men and\nwomen. From the house came all who were in it. Oswald and the other\nharpers marched to the beach together, bringing their harps.\n\nNow from the stables beyond the house came Tostig the Red and Wulf the\nSkater leading between them, whether he would or not, the snow-white\ncolt which at two years seemed large for a four-year-old, but which\nas yet had neither been bridled nor mounted. That was partly because\nof the spirit that was in him; for none but Ulric or Hilda would he\nwillingly let lay a hand upon him, and his eyes now grew red as if he\nwere fretted overmuch. As he was led along he reared and plunged and\nsnorted furiously, but Tostig and Wulf were strong men and they brought\nhim to the heap of wood and in front of the hollow in its middle.\n\nHilda had brought with her a long polished staff of ash wood, which\nhad something of woven cloth stuff wrapped closely around it. Now she\nmade a sign to Oswald and he struck his harp. So did the other harpers,\nfollowing him, and the sound of their music stirred the blood of all\nwho heard, so that the men shouted and clashed their spears upon their\nshields. Then ceased all the harps but that of Oswald, and he sang a\nsong of war which called upon Odin and all the gods to sail with their\nship, _The Sword_, and give her a successful cruise, with many battles\nand much blood and great plundering and many burnings of the ships and\nof the strongholds of foemen.\n\nThe tide was rising fast, but the ice came no nearer the shore, and it\nwas seen that there would be free searoom for the launching. All things\nelse were ready for this, and the launchers with their hammers and\ntheir handspikes were prepared to go to their places. Oswald ended his\nsong and all looked at Hilda. She did not at once speak, and her face\ngrew ghastly as the face of one from whom life had departed. Taller she\nseemed as she raised her right hand and pointed to the colt.\n\n\"Ulric the Jarl,\" she said, in a hollow voice, but clear, \"son of\nBrander the Brave, heir of the old house of the sea kings, son in the\ntrue line of the hero gods and of Odin, slay now the white horse of\nthe Saxons and launch thy keel into the sea!\"\n\nTostig and Wulf forced back the plunging colt into the hollow between\nthe heaps, and Ulric walked forward, drawing his seax as he went. He\nput his left hand upon the face of the colt and it stood still, looking\nat him and neighing gently, while at every corner of the heaps torches\nof blazing pine were thrust quickly in by old women named for that duty\nby Hilda.\n\nShe had walked away to a little distance from the ship, and she stood\nnow between the sea and the land, upon a spot where the sand was dry\nand smooth. Upon this she drew runes with the point of the staff that\nwas in her hand, all the while chanting a saga which none of those who\nheard her could understand, except that they knew in it the names of\nthe gods.\n\n\"Son of Odin,\" she shouted, \"strike!\"\n\n\"Odin!\" responded Ulric, as he drove his seax to the hilt into the\nbreast and through the heart of the colt.\n\nIt gave one cry that sounded like a human voice in sudden despair. It\nmade one plunging struggle, restrained by Ulric, and then the beautiful\nanimal lay quivering in the hollow. At once a heap of fuel was piled in\nfront of it, concealing the sacrifice to Odin, and the long fingers of\nthe fire seized rapidly upon the dry pine and the cedar and the firwood.\n\nLoudly sounded the harps. Loud was the song in which all voices were\njoining. Out of the fiord came booming a great roar of the sea, for he\nwas smiting his crags and dashing the floes of ice against the granite\nfaces.\n\n[Illustration: \"Go forth into the sea, O sword!\"]\n\nHilda came again to the ship, unfolding as she walked that which\nwas wrapped around her staff, and the south wind that was blowing blew\nit out so that all might see. It was a great banner, for a battlefield\nor for the mast of a warship. It was black, and upon it, fully half\nthe size of the colt which had been slain, was painted the sign of\nthe race of Brander, only to be carried before chiefs of Odin's line,\nthe White Horse of the Saxons. Hilda placed the staff in the hands of\nUlric, and he at once sprang on board the ship. He blew a blast of his\nwar horn, and in a moment all the launchers were at their stations.\nAnother blast, and all the rowers came on board and took their seats,\ntaking hold of the short oars, ready to dip them, while tenscore more\nof vikings, fully mailed and armed, followed and posted themselves\nfore and aft, spear and shield and ax in hand. Ulric the Jarl stood by\nthe hammer of Thor on the fore deck and raised his horn again. At this\nthird blast, as he blew it, the launchers hammered hard and plied their\nhandspikes and their levers.\n\n\"Go forth into the sea, O _Sword_!\" shouted Hilda. \"Thy beak shall\nbreak the ribs of the triremes and thy keel shall plow the seas of the\nsouth!\"\n\nOut sprang the vessel, so deftly shaped, so strongly made, so well\nmanned, and into the sea she glided, while Ulric, the son of Brander,\nlifted high the standard and sounded again his war horn. Every harp\ntwanged its loudest, and every horn on board the ship and on the shore,\nand every voice, joined in the shout of joy that hailed so successful a\nlaunching.\n\n_The Sword_ was now upon the sea, floating at the end of her shore\nhawser, while the crew lowered her anchors from the prow and stern. On\nthe shore the fire flared upward like the streamers in the northern\nsky in winter.\n\nThe pallor on Hilda's face grew ghastlier still, and she walked to the\nhouse, forbidding any to come with her. As she went she muttered:\n\n\"Beautiful is the son of Brander, my boy! my hero! I love him as if I\nwere his mother. Alas, she is not here to love him! O, I am old and it\nmay be that I see not that which I seem to see when my eyes are opened.\nNot so! Him I shall look upon no more, nor upon the ship. I go, for I\nam very old. But I would that the young hero might not go down so soon.\nI would that he might win love and that he might bring home a bride,\nlest the race of Brander the Sea King should die with him. The gods be\nhis guard where he goeth and the valkyrias find him not for a season!\"\n\nSo the lonely old woman went into the house and went to her own room.\nShe had seen the launching of _The Sword_, and the ship was to go out\nwith the outing ice.\n\nRocking at her anchor lay she now, and all along the shore were men and\nwomen who rejoiced to look upon her and to think her the most perfect\nship that had ever been built on the coast of the Northland. The fire\nwas blazing high above the sacrifice to the gods, for many hands were\nready to put on fuel, from time to time, and all knew that it must burn\nuntil _The Sword_ should be out of sight.\n\nIt was when the sun was sinking, and the waves were washing gently and\nmurmuring low along the beach because of the softness of the warm wind\nfrom the south, that Hilda came again, walking hastily. Her head was\ncovered with her hood, and they saw not her face, but she spoke to a\nyouth who stood by a small boat.\n\n\"Take thy boat,\" she said. \"Go thou to the ship. Give Hilda's word\nto Ulric the Jarl. Bid him come to the shore, coming alone, rowing\nhimself. Stay thou there until he returneth. Bid him that not one man\nof those who are now on board shall come again to the shore.\"\n\nThe youth sprang into his boat and went with his message. The men on\nthe ship were greatly busied with stowing of goods and with other care\nfor the fittings of all kinds, but they saw his coming, and Tostig the\nRed hailed him:\n\n\"What doest thou, coming to the ship? Is it not forbidden?\"\n\nThen the youth replied with Hilda's message, and Ulric himself came,\nbut he descended into the boat without speaking while the youth\nclambered on board. It was for him a matter of pride, and a thing to be\nremembered in after days, that his was the last foot of any among the\nshore people to tread the deck of the beautiful ship before she should\nsail for the Middle Sea, and for the fights in which she was to crush\nthe galleys of those far-away nations.\n\nUlric took the oars and rowed to the place where he saw Hilda awaiting\nhim, and she was alone. She had her staff in her hand and she was again\ntracing runes upon the sand. It was the spot where she had stood before\nthe sacrifice was slain, and neither man nor woman would have dared to\ntread upon it until after the next tide. This, when it should come,\nwould wash out the marks which had been made by Hilda. Ulric stepped\nout and drew up his boat and walked near her.\n\n\"I have sent for thee,\" she said, \"to show thee a thing. Thou art\nready, and thy ship. See to it that naught else be sent to her from the\nshore. None of the men must again set foot upon the land. Sail thou\naway this night, and linger not.\"\n\n\"I had so ordered,\" responded Ulric. \"The ice goeth out steadily, and\nwe are to follow it. But I am glad to say this last word with thee, for\nthou art very dear to me.\"\n\n\"More than my son art thou,\" said Hilda, \"because thou art also of the\nsons of the gods.\"\n\n\"There are gods in the south,\" said Ulric, thoughtfully. \"I have it in\nmy mind that I shall see one of them before I return. I would that I\ncould see him in battle, like Thor, or Tiw, or Odin.\"\n\n\"Be thou thyself like one of them,\" said Hilda, and she gazed at him\nlovingly, throwing back her hood.\n\nVery bright were her eyes for a moment and then they grew sad and dim,\nas if a mist from the fiord had floated into them. Ulric looked upon\nher withered face as if also it were beautiful to him, and he said:\n\n\"Thou art a loving woman and true, and I will keep thy bidding on the\nsea and on the land.\"\n\n\"I shall see thee not again,\" she said, \"and I willed to look upon thy\nface this once.\"\n\n\"It may be that thou wilt be here when I return,\" he responded, but she\nshook her head.\n\n\"Son of Odin, not so,\" she said, in a low, soft voice, like that of the\nyoung who love and are parting. \"Me thou wilt not see, and I know not\nif in any manner I am again to see thee. They of that land into which\nI quickly go do sometimes see the people of this land, when the gods\npermit. If so, I will come to thee some evening when there is a silence\naround thee, and I will touch thee on the forehead, thus,\" and she\nleaned forward and kissed him, placing her hands upon his shoulders.\n\n\"I will welcome thee!\" he said, with a great thrill, and she stood\nerect, continuing her last words.\n\n\"I have this much more to tell,\" she said. \"Thou wilt sail far and\ncontend with many. As thou knowest well, thou wilt meet no foemen like\nthe men of Rome, on land or sea. Thou wilt not tarry long in any place,\nfor thou art a viking, and thou hast no home in the south. Thou wilt\ngo on from place to place until thou shalt come to this harbor, or\ncity.\" She pointed at the runes drawn upon the sand at her feet, and he\nreplied:\n\n\"I cannot read them, O Hilda! They are in another tongue. They are\nunlike any that I ever saw.\"\n\n\"Neither can I read them,\" said Hilda. \"But note them with care, for\nwhen thou seest them upon the ground of any land thy voyage is ended.\"\n\nSo Ulric stooped low and studied well the deeply graven furrows which\nthe saga woman, the seeress, had drawn upon the sand. They were in\nshape like this:\n\n[Illustration]\n\n\"Thou seest?\" she said.\n\nBut the runes were close to the water's edge and the tide was coming\nin. At that moment came a great swell out of the fiord, rising and\nsurging along the beach, and it put out a hand of foam, glittering in\nthe light from the setting sun. Hilda stepped back beyond its reach,\nand so did Ulric, for a sound came with it. Back fled the billow,\nbreaking as it went, but it left behind it no trace of those strange\nrunes on the sand.\n\nHilda clasped Ulric in her arms, for a moment, but she did not weep.\n\n\"Go thou to thy ship,\" she said. \"I go to my own place.\"\n\n\"Farewell, my best friend,\" he replied, but she turned and walked away,\nand all who met her made room for her, for a low voice like a wail\ncrept out from under her hood, and she did not walk firmly, as was her\ncustom.\n\n\"Very great was her love for the son of Brander,\" said all of them; and\nthey knew that this was her last season, for she had told them so, even\nat Yule.\n\nUlric rowed to the ship and went on board. The youth returned to the\nshore with his boat. The sailors pulled up the anchors. Then the\nwatchers on the shore saw the long oars go out, the rowers standing\nin their places on either side of the ship, while the young jarl, the\nleader of men, stood alone at the stern, steering with one hand while\nthe other held his war horn. Long and powerful was the blast he blew,\nfor it was a farewell to the Northland and to the people he was to see\nno more. So sailed away the good ship _The Sword_. It had been a grand\nlaunching, but there were those upon the beach who turned and went away\nto their houses mournfully, even weeping.\n\nIn the house of Brander there was silence. Hilda had gone to her own\nroom. All guests had departed. The household folk were for the greater\npart at the beach, by the fire of sacrifice, and Oswald, the harper,\nsat in his place with his harp before him, leaning upon it.\n\n\n\n\n                              CHAPTER IV.\n\n                THE SHIP \"THE SWORD\" AND THE ICE KING.\n\n\nThe morning of the day of Tiw dawned mistily across the cold North Sea.\nEverywhere, as the sun looked in through the floating curtains of fog,\nhe could see steel-blue waves wrestling foamily with the masses of ice.\nWho in this place would imagine that in some other, far away, the same\nsun had found the bright flowers and green leaves of the fully opened\nspring?\n\nThe wind increased with the coming of the sunshine, as if the\nadditional warmth brought to it better strength wherewith to blow away\nthe mists. One mound of white vapor had been thicker and higher than\nits neighbors. It had gathered over something that it was hiding, but\nthe breeze blew now a short, sharp gust and the mound was gone. So was\nuncovered the good ship _The Sword_, and her crew could discern what\nthings might be around them.\n\nUlric the Jarl was standing in full armor on the fore deck. He had been\nwaiting for this clearing, and now he put his horn to his lips. He blew\nit lustily, and all who heard him raised a shout, for they knew that no\nland was in sight and that their voyage had begun.\n\n\"We have gone far in the night,\" said a large man standing near the\njarl. \"But there is much ice. We can do little more than drift, but we\ncan use the oars somewhat.\"\n\n\"We shall go but little faster than doth the ice,\" replied Ulric. \"But,\nO Knud the Bear, thou wilt off with that black shirt of thine when this\nsun is higher.\"\n\nThere was loud laughing at that, for Knud was clad in the warm skins of\nthe bears he had slain. Even upon his head was that which had covered\nthe skull of the largest of them. Good clothing it was for winter time,\nbut it was likely to prove heavy gear for southern wearing.\n\nThe jarl gazed southward, hoping to see open water, but only ice fields\nlay between him and the horizon. The mist was fast disappearing,\nnevertheless, and those who were watching were seeing further; but now\na great cry arose from the stern, where Wulf the Skater was taking his\nturn at the helm.\n\n\"O jarl!\" he shouted. \"Mark! Seest thou how we are pursued? Come\nhither!\"\n\nDown from the fore deck and quickly along the ship to the after deck\nwent Ulric and those who were with him, and there was no need for any\nman to point with his hand as Wulf was pointing.\n\n\"The ice king!\" he said, shivering. \"I told thee how I saw him anchor\noff the North Cape when the leaves fell, and the first freezing put ice\naround him over the calm waters. He came down from his own place that\nfar last summer. He seemeth to me to be as tall as ever, and he hath\nmany strong floes with him.\"\n\nUlric looked, and so did they all, saying nothing at first, for the\nsight was rare. Not often did any mountain of ice float into that\nwater; and here was a mighty one. His peak arose, they could not tell\nhow high, and the sun was glittering gorgeously among his crags.\n\n\"He is moving faster than we are!\" exclaimed Tostig the Red. \"He will\nstrive to overtake us. He could crush us like a nutshell with one of\nhis crags.\"\n\n\"We will keep out of his path if we may,\" responded Ulric. \"But how is\nit that he saileth along so well against the wind without oars? There\nis no tide. If there were any current, it would be with us as much as\nwith him.\"\n\n\"Aye,\" said Tostig the Red. \"But did not Hilda ever tell thee? I have\nheard her speak of these ice kings. The gods that walk on the bottom\nof the sea push him along that he may go south and die, for his time\nhath nearly come. Never, I think, was anything like him seen below the\nfiords until this day.\"\n\nVast, truly, was this ice mountain which was nearing them, propelled by\nsome unseen hand. If there had been a strong undercurrent it would have\nmoved the wonder from the north in precisely this manner. Nevertheless\nall Northmen of the sea knew that any peak of ice above the surface\nmust rest upon a mass of ice seven times greater.\n\nAll the vikings upon _The Sword_ watched earnestly for the next sign of\nwhatever was to come, but Ulric took the helm and sent the rowers to\nthe long oars, two men to each oar. Well and vigorously did they row,\nand the ship was deftly steered into and through one after another of\nthe open channels between the small floes around her. Much distance was\ngained, but at last the ice fields beyond began to close tightly and\nthe rowing ceased.\n\n\"Son of Brander!\" shouted Knud the Bear from the fore deck. \"Mark! The\nfloes are lifting!\"\n\nAll saw that it was true. Under all the nearer ice-pack a hidden\nfield, the forefoot of the iceberg, was slipping steadily on unseen\nuntil those floes rested upon it. And now there came a grating sound\nalong the keel of _The Sword_, and she too was lifted. The ice arose\nwith her, so that she sat firmly in a great cleft of it, remaining\nupright, indeed, but as completely out of water as she had been upon\nthe strand before her launching.\n\nSilent and stern stood Ulric, facing the ice king and asking of\nhimself, \"My voyage hath but begun, and is it ended? Was my ship built\nfor this?\"\n\nNot so was it with the mind of Knud the Bear, for he gazed long and\njoyously upon the untellable beauty and majesty of the ice king, and\nthen with a great laugh he shouted:\n\n\"Sons of the Northland, the gods are with us. They have sent him.\nNothing can stay him. He will carry us fast and far. There will be no\ntoilsome rowing, and we need not care for the direction of the wind.\nThe gods of the frozen sea come with him. They would send us south that\nthey may go and fight the gods of the islands where there is no ice,\nfor they hate them.\"\n\n\"So be it!\" replied Ulric, gloomily, but he looked again and he said to\nKnud, \"I know not the ice gods, but I think there are friends of thine\nyonder. Seest thou?\"\n\nEvery man was gazing, for there was naught else left to do. Around the\npinnacles and the cliffs of the ice king there were sea birds flying\nand screaming. On the snow-packed levels there were brant and geese and\nducks and other fowl that should have been at the south by this time,\nand that would soon, no doubt, be going.\n\n\"Odin the Strong!\" exclaimed Knud, \"I see what thou meanest. I had seen\na white fox, I thought, but yonder are the bears of the night country.\nThey are white, that they may see one another in the dark, and there is\nnothing else that is so fierce as they are.\"\n\n\"Hilda sayeth,\" replied Ulric, \"that all the world north and east of us\nmust forever belong to the sign of the bear. Hast thou ever slain one\nof these white ones?\"\n\n\"Never,\" said Knud. \"I have not hunted to the northward so far as to\nknow much of them. Wulf the Skater hath met them oft enough on the\nnorth coast, but they go back into the night, for they hate the sun. If\nit would not anger the ice king, I would go out and slay one even now.\nBut he brought them with him.\"\n\nSo thought others of the vikings, as if the crew of white monsters now\nclambering nearer over the rugged ridges of the ice were as his own\ncattle to the mighty gnome who had builded this frozen tower for his\ncastle.\n\n\"As many they are,\" said Tostig, \"as the fingers of a hand. I have\nheard that they have no fear of men.\"\n\nIf the bears had no fear, they at least had much curiosity, and they\nwere coming to inquire what this might be that lay upon the ice with so\nmany men walking around within it.\n\nUlric went into the after cabin for a heavier spear than was the light\nweapon he had with him, saying to Knud, \"White bear have I never slain.\nThis chance is mine, but the second fight belongeth to thee. I do not\nrob thee of thy hunt.\"\n\n\"Thine by right, O jarl, is yonder great one,\" replied Knud. \"No man\nmay go before thee unless thou wert hurt or dead. But I warn thee that\nthe long claw, over there, were he to grapple thee, is worse to meet\nthan might be three Romans.\"\n\n\"I would face more than three Romans,\" laughed Ulric. \"But thy pale\nfriend on the floe is a king of bears.\"\n\nHe returned speedily, armed and armored for battle. The spear he\nbrought was long and strong, with a steel crossguard at the heel of\nits broad blade. It was very sharp, but its weight would have been\nunwieldly for a slight man.\n\nTwenty fathoms from the stern of the ship stood the great bear\ngrowling, and the others walked around at a greater distance. He was a\nfathom and a half in length and his paws were tremendous, with claws\nlike reaping hooks. No man ever faced any beast more terrible in aspect\nthan was that angry monster from the darkness which broodeth over the\nforever frozen sea.\n\nDown stepped Ulric, and when he was a few yards from the ship some of\nthe men followed with Knud, but not too near, lest any should seem to\nhelp and so should spoil the honor of the fight.\n\nThe surface of the ice was broken and there were chasms in it, but it\nwas as firm to stand upon as the dry land. Moreover, _The Sword_ was\nnow lying not far away from the mighty perpendicular front of the ice\nking. None knew yet what might be his aspect looking northward, and\nthere were those among the vikings on the ship who shook their heads\ndoubtfully, considering this matter of the bears.\n\nStone still stood this bear, growling at intervals, until the jarl drew\nwithin six paces, holding his spear leveled. Then, with a loud roar\nand a clashing of his teeth, the huge beast made his rush, rising upon\nhis hind feet and spreading his enormous arms to close with Ulric. Had\nhe done so his hug would have been speedy death, but the point of\nthe spear met him firmly, with a thrust which buried the blade to the\ncrossguard midway between his shoulders.\n\n\"That would slay anything else that liveth,\" said Knud to Tostig, \"but\nthe white ones die hard. Mark! the jarl! The son of Brander! It is\ngrand!\"\n\nHis comrades answered with a shout and then they were still, and so\nwere all the vikings, who crowded the decks and bulwarks of the ship,\nlooking on.\n\nHorrible was now the roaring of the bear as he struggled against the\nspear of Ulric, striving to plunge nearer. What tenacity of life must\nhave been his, to fight on with the spear blade in him so deeply!\nAround swung Ulric on the slippery ice and his whole frame was strained\nto its uttermost endurance by the swift changes of that wrestling, but\nthe plunges of the bear forced him backward a fathom at a time. His\nface was now but an arm's length from that of his vast antagonist, and\nthey were looking each other eye to eye. Red and yet full of green fire\nwere the eyes of the bear, and his teeth glistened awfully in their\nranges as his wide jaws opened to gnash them. But that the descendant\nof Odin was many times stronger than other men the combat might here\nhave ended.\n\n\"Slip not now!\" shouted Knud. \"Son of Brander, there is a chasm behind\nthee. Stand fast, if thou canst! Thou art beyond our help!\"\n\nOnly his own length from him was the cleft in the ice floe, and it went\ndown to deep water. If he should fall into it in his heavy armor, none\nmight hope to see him again.\n\nRoar--roar--roar--in dreadful wrath and pain struggled the bear, for\nthis was his death throe; but Ulric's foot found a brace--a break in\nthe ice--and he gathered his last strength, the strength of the sons of\nOdin, the hero might of the old gods.\n\nSnap! The tough ashen shaft of the spear broke at the guard, and both\nbear and hero fell heavily, but Ulric arose with his seax in his hand.\nThe claws of the bear wrenched away his shield as if it had been a\npiece of oaken bark, but the seax was driven in to the hilt, and as it\ncame flashing out the life of the bear came with it. Over he rolled\nwith a loud shriek, that was echoed back from the face of the ice king.\nThen he stretched himself at full length upon the ice and lay still,\nwhile Ulric stepped forward to cut off his forepaws for a token.\n\n\"Hael!\" shouted every voice among the vikings, as the white one rolled\nover. \"Hael to Ulric the Jarl, the son of Brander! The son of Odin!\nHael to the first good death and to the long cruise of _The Sword_!\"\n\n\n\n\n                              CHAPTER V.\n\n                          THE UNKNOWN THING.\n\n\nThe ice king had lost only one of his fierce white flock. It had been\nthe largest of them all, however; and in the latter part of Tiw's day\nthere had been a feast of his flesh. Greatly had the crew of _The\nSword_ enjoyed that feast, and they believed the saying of Knud that\nthere was courage and strength to be gained by such eating after so\nbrave a battle. \"The gods themselves eat mightily,\" he said, \"and they\nhave nothing better than this.\"\n\nDuring that day a number of the vikings went out to explore the ice\nfields somewhat, and they captured many wild fowl easily with bow and\narrow. They reported having seen in the distance other animals, like\ngreat seals or walruses. They also planned to hunt the remaining bears,\nbut the jarl forbade it, being unwilling that they should go far from\nthe ship lest harm should befall them from sudden breaking of the ice.\n\nNevertheless, to all testing, it seemed to be packing even more firmly.\nThe entire visible mass of it drifted steadily southward, as if the ice\nking, or the under gods who were pushing him, knew of the channels by\nwhich they were to steer him into other seas than this.\n\nNight came, and then the day of Odin. But now the worst foe of the ice\nking, deadlier than even the sun, was wearing him away with floods of\nwarm rain. There were rivulets pouring down his sides, and some of\nhis pinnacles and crags came crashing, thundering down from time to\ntime. This was, therefore, not a good day for hunting, and the vikings\npassed it on board the ship, or near it, but not dismally, for there\nwere among them many whose minds and tongues were busy with old voyages\nand old fights, and the land to which they had sailed. Also there were\nsongs to sing, and there was much ale, and no man was hindered from\nfeasting. It was a time, too, for the remembering of sagas, and many\nspoke of Hilda, but Ulric did not utter her name, saying rather that it\nwould be well if Oswald and his harp were on board.\n\nThese two, indeed, the saga woman and the old harper, sat at home in\nthe house of Brander that rainy day, speaking to one another across\nthe ash heap, on which a slow fire smoldered. Their talk was of many\nthings, but from all it would ever come back to some word concerning\nthe ship and her crew and Ulric. To others Hilda had spoken little,\nand they noted that she had not eaten since the launching. Oswald was\nfretful and fitful, and he said that he cared not for harping. In an\nearly hour of the day he had gone out and he had even climbed to the\ncrag on the top of the headland that he might look far to seaward, but\nhe had returned, shaking his head, to say to Hilda:\n\n\"All is ice! She is out of sight, but the floes have closed behind her.\"\n\n\"So they close not before her I care little,\" replied Hilda. \"They\nwill conquer the ice, for the sun will help them, and they are sailing\nnearer the sun.\"\n\nOswald was long silent then, and at last he arose and walked out of\nthe hall while Hilda went to the door and gazed seaward. It was to\nhis own room that the harper made his way, leaving his harp near the\ndais. In a far corner of the house he had been given his place, for he\nwas held in high honor. Nevertheless, it was but small, and bare save\nfor a table and a lamp thereon and a stool. There was, also, a heap of\nskins for warm sleeping, and from under this Oswald drew out something,\nstooping and then looking behind him to be sure the door was closed.\n\"What will the jarl bring me, when he returneth from the southlands?\"\nhe muttered. \"Bright gold, I hope, for there is more to love in the\nyellow, the heavy, than there is in light silver. The touch is not the\nsame, and gold hath a better ring.\"\n\nIt was a bag that he held, untying its mouth, and his hand was now in\nit. He drew out pieces of varied shapes, looking at them and rubbing\nthem with his fingers. \"The faces of kings are on them,\" he said.\n\"Runes of the southlands. I can read some, but all I cannot read. May\nthe gods guide the jarl to places where he will find many like these\nand bring them to me. He careth not for them himself.\"\n\nHilda, standing in the doorway, grew sad and wistful in the face.\n\"Gone,\" she said. \"Gone beyond seeing or hearing. And I love him so! He\nis my hero! My beautiful one! I am old, and I am soon to pass away, and\nI know not clearly whither I go. Sometimes I would that one of the gods\nmight come and tell what things there are in those countries for such\nas I am.\"\n\nThen turned she and went back to her great chair by the fire; but\nUlric also was thinking of her and of Oswald, for he said to Tostig\nand Wulf and those who were with them, under the after deck: \"The\ntongues of the south folk? We do well to talk about them. My father\nknew many. Oswald, the harper, and Hilda could speak with him in all\nof them and they had more that he knew not. She hath learned much in\nher hundred years, and she is not like other women. When I was a child,\nand afterward, in the long winter evenings, when we had naught else to\ndo, I loved to have them teach me, and they said it would be my need\nsome day. I can talk with a Briton or a Roman or a Greek. But Hilda and\nOswald taught me many words of a tongue that belongeth to a people who\nlive on the easterly shore of the Middle Sea. They are a trading folk,\nand our sea kings found them everywhere. They are not like other folk,\nand they have a god of their own, but none of them can tell what he is\nlike. I have thought I would wish to see him, but Hilda sayeth that he\nwill not come out of his own country. And that, too, is much the same\nwith our own gods; but I wish they may go with us now, for some of\nthese southland gods are cunning and strong.\"\n\n\"Not as are the gods of the North,\" said Tostig, sturdily. \"I too have\nheard of these Jews and their god, but I do not care to see either\nhim or any other god. It is more than enough for me when I hear them\nwhispering across the fiords.\"\n\n\"So!\" exclaimed Wulf the Skater. \"I have been out far on the ice, when\nthere was no wind and there was a bright moon, and I have gone landward\nwith speed lest their voices should overtake me. I heard them loudly\nonce, and that night I was chased by many wolves. I slew some, but I\nstopped not for their skins, for the rest were an army.\"\n\n\"Glad am I,\" said Ulric, \"that if I meet one of these gods I can speak\nto him fairly well in his own tongue. How else, for instance, could\nI question this Jew god? We shall sail all around the coasts of the\nMiddle Sea before we come home.\"\n\n\"What couldst thou ask him?\" replied Knud. \"And what thinkest thou he\nmight tell thee?\"\n\n\"One thing that Hilda knew not,\" said Ulric. \"I am curious if the gods\nof those lands know the gods of the North. I would know if this Jew god\nhath ever met with Odin and Thor, and whether or not they are friends.\nIf they have fights, as do our own gods, which of them is the stronger?\nI have thought that if I were a god, I would bring all the others under\nme. It is not managed well.\"\n\n\"I would not have land gods meddling too much with the sea, save in\nbattles,\" said Tostig. \"It is well as it is. But the Middle Sea is\nwide; we may not look upon all of its coasts. There are deep bays and\nmany islands.\"\n\n\"They say,\" responded Ulric, \"that there is an open water leading\nsouthward, and that if one can find it and will sail into it boldly,\nfearing nothing, he may follow its leading until he shall find the city\nof Asgard and the home of the gods. Moreover, there are lands which no\nfoot hath trodden. I would see some of them if they are to be found by\nsailing not too far.\"\n\nSo said they all, and there were other tales to tell concerning seas\nand lands.\n\nThey still were talking of these things when a loud shout from one of\nthe watchers summoned them, and they rushed out to the gunwales and the\ndecks. The rain was no longer falling and the sky was clear, so that\nthey saw well what was doing. The ice king had not at all lost his grip\nupon his own floes, but southward was a vast rift in the ice pack.\nWide and blue was the open water, but it was not very near them, and as\nthey were looking at it from their icy anchorage the watcher shouted\nagain:\n\n\"O Ulric the Jarl, whales! They will come up again from under the\nfloes. I saw them. A great herd!\"\n\nLoud voices replied, inquiring, but they ceased, for the herd quickly\nshowed itself. Many and huge were the whales that emerged, and some of\nthem sprang half their length out of the water.\n\n\"They are pursued!\" exclaimed Knud the Bear. \"I have seen them spring\nin that manner when the swordfish troubled them. But see them flounder\nnow!\"\n\nStrange indeed was the confusion and the tumbling about of this herd of\nthe sea. They were beating the waves into foam, and they were plunging\nhither and thither as if wildly affrighted.\n\n\"I think that it is neither the swordfish nor the thrasher,\" said\nTostig the Red, for he had halfway climbed the mast and he was leaning\nout to see. \"O jarl, it is one of the monsters that Hilda hath told us\nof. She sayeth that only a few are left, for the gods destroyed them\nlest they should eat up all the whales. Look yonder!\"\n\nThey were near enough to see, but could not note clearly until a great\nfragment broke away from the field of ice which carried _The Sword_.\nThrough that chasm at its outer border there came up a shape which was\nnot the head of a whale. It was long, with vast jaws, and in them were\npointed saws of long white teeth, with which it tore terribly the side\nof a tremendous bull whale that was nearest. But the bull whale turned\nand fought him, and there was a vast whirling of foamy water, as the\ntwo sea creatures struggled against each other, beating with heads and\nfins and tails, but the vikings could none the better discern the form\nof the whale's enemy.\n\n\"He is a comrade of the ice king,\" said Wulf the Skater. \"Never before\nwas he seen in these waters. He is somewhat like a snake, but with a\nvast belly. I saw his head once before, long ago. Ten more were with\nme in the ship, and we had been long storm-driven. The old men told me\nmuch about him.\"\n\n\"He could upset a ship,\" said Tostig. \"I am glad we are here on the\nice. But thou mayest have seen another like him.\"\n\n\"Not so said the old men,\" replied Wulf. \"He is alone. There! He\nshoweth again!\"\n\n\"I am glad we have seen him,\" said Ulric. \"But I am more troubled\nconcerning the ice king. See ye not that he is fast melting? I have\nthought that he is beginning to lean this way. We are drifting, truly,\nbut we do not get away from him. We are his prisoners.\"\n\nThey well understood that there might be deadly peril for them in aught\nthat should change the position of the iceberg, but there was naught\nthat they could do, even if sure death were coming. So they preferred\nto gaze after the herd of whales, and every now and then they thought\nthat they caught fresh glimpses of the monster from the under sea, the\nterror of all other monsters. Few of them but had heard and could tell\nold sagas of such creatures, the remnants of the forgotten days, and\nthey agreed that this one was the world-snake that Hilda had sung of as\nthe destroyer.\n\n\"He eateth men joyfully,\" said one, \"when he can get them.\"\n\n\"Hilda said,\" replied Ulric, \"that he cometh among men no more. He\ncannot live in any sea that is plowed by the keels of ships. The gods\nare against him. But now the whales have fled and he hath followed.\"\n\nThen turned they to stare at the ice king, and he seemed as strong as\never. Far away at his right they saw the bears, walking to and fro, and\nthe wind brought from them a sound as if they were moaning.\n\n\n\n\n                              CHAPTER VI.\n\n                       THE FALL OF THE ICE KING.\n\n\nWhen the sun arose upon the fifth day of the week, the day of Thor,\nthe glittering pinnacles of the ice king still towered high above the\nfloes, and these covered the sea as far as the eye could reach. All the\nwhite mass was evidently in motion and the drifting was rapid, but it\nseemed to the vikings as if their danger were striving to push nearer\nto the ship. She was now lying almost within his reach, if he should\nchoose to strike her--and she was but a very small thing. Her crew,\ngoing and coming around her, were but so many specks upon the ice. From\nher masthead still fluttered bravely out her White Horse banner, and\nshe was yet altogether unharmed, but the rowers were at their places\ncontinually.\n\nA prudent captain was the jarl, for, although the men were impatient,\nhe forbade their going far from the ship. He held them back even when\nthe remaining white bears appeared near the feet of the ice king.\n\nKnud was almost angry that he was not permitted to go forth and slay\nthem.\n\n\"One man for each bear, Ulric the Jarl,\" he said. \"It is our right. We\nmay not ever meet them again, and the chance for honor were lost. Thou\nhast won thy pair of claws.\"\n\n\"Thou hast slain bears enough,\" said Ulric. \"Were I to let thee go,\nthou mightest perchance be left behind on the ice, or under it. Small\nhonor in that. I promise thee the next chance to get thyself killed\nfairly.\"\n\n\"I obey,\" growled the grim old hunter, \"for thou art my jarl. But when\nwe return from this cruise I will go with Wulf the Skater into the\nwinter of long night and we will find them there. I will not go to\nValhalla until I have slain one as large as thine.\"\n\n\"Mind not thy bears now,\" responded Ulric. \"Seest thou not? Art thou\nblind?\"\n\nHe blew his horn sharply, and all who were on the ice around the ship\nsprang on board in haste.\n\n\"Mark!\" he shouted. \"Between us and the foot of the ice king there is a\nchasm that widens. We know not when the field may break away. Then he\nwill be upon us. Every man at his place this day!\"\n\nThey who saw could understand, and there was no more talk of hunting.\nEven when a white fox came and looked at them, within bowshot, no arrow\nwent after him.\n\n\"Let him go free,\" said Tostig. \"He hath wild fowl enough for the\ncatching, but he will swim far before he runneth on land again.\"\n\nIt was a time of doubt and of waiting, but the drifting ceased not.\nThere was much discussion at intervals, among even the elder seamen,\nas to precisely in what part of the sea they now might be, for there\nwere no guidings. Toward the sunset, after long hours of idleness that\nbrought weariness, Ulric went and stood by the hammer of Thor on the\nfore deck. Tostig the Red came and stood by him and laid his hand upon\nthe hammer, for Tostig was a smith, as had been his fathers before him.\nNot only could he smelt iron out of the right rock, but he could harden\nit for cutting and for bending and springing. The secret of that art\nwas his inheritance, and Hilda had said that it was a thing that the\nold gods who were dead had brought with them from the east before Asa\nThor's time. It was from a rising-sun land, but a cold one, that Odin\nled his children, said some, and there were runes on the rocks to prove\nit, if they might be read by any now living.\n\n\"We go faster,\" said Tostig. \"We have already gone far this day. If the\ngods were against us, I think they would not so swiftly bear us forward\nwithout wind or work.\"\n\n\"Who knoweth the will of the gods?\" replied Ulric. \"Not thou or I.\nThey puzzle me greatly. I would they might come at times and show\nthemselves. How can one know what to think of a god he hath never seen!\nI mean to look upon one of them, if I may, before I sail back to the\nNorthland. That were a thing worth telling of a winter evening by the\nfire in the hall.\"\n\n\"And have all men answer thee that thou wert lying?\" laughed Knud\ncheerily, from behind Tostig. \"I believe that Hilda seeth them at\nan hour that cometh to her, but I would rather let them alone. I\nwill think well of them if they will but shove us along in the right\ndirection. They work finely now, it seemeth, but the sun goeth down.\nThor hath been friendly to us during all his day, but I doubt if we\nare as safe after he is gone. The morrow will be Freya's day, and she\nmeddleth not overmuch with seafaring matters. \u00c6gir is the god of the\nsea, and of him we know but little, nor of Ran, his wife, nor of his\nnine daughters. They must at this hour be all under the ice doing\nnothing.\"\n\nThe saying of Knud was a thing that it was hard to dispute, but it\nwas in Ulric's mind to wonder whether or not he and his vikings were\ndrifting altogether beyond the help of the old gods of the North.\n\nThe wind began to blow strongly, and the men listened with eager ears,\nfor they thought that they could now and then hear shrill and angry\nvoices from the neighborhood of the ice king. Some of them were like\nshrieks, but these may have been made by the gale itself, blowing among\nthe crags and chasms.\n\n\"We will both eat and drink,\" commanded Ulric. \"Let every man be\nhearty, that he may have his full strength for that which may be before\nhim.\"\n\nAfter he himself had eaten he went to the after deck, putting his hand\nupon the tiller. From that place he might best watch the ice king, and\nthere came others to stand with him, waiting.\n\n\"He is very tall,\" said Ulric, at last. \"I doubt if we shall ever look\nupon his like again. But saw ye ever such moonlight? I have known days\nwhen I could not see so well as I can this night.\"\n\n\"Aye,\" said Wulf. \"I know this moon. It is not such light as ours,\nfor he hath brought it with him. It is the light which the gods make\ninstead of sunlight in his own place, and it will not go south any\nfurther than he goeth. But mark the bears!\"\n\n\"Something troubleth them,\" said Ulric.\n\nAll could see them plainly, and they were like ghosts wandering to and\nfro among the rugged heaps of the ice floes. They were much scattered\nand they moved as if they were hunting for something which they could\nnot find, and they were calling often to each other, moaning as if they\nwere in pain or in great discontent. Sometimes as they did so they\nlifted up their heads toward the moon, but oftener toward the ice king.\n\n\"Look at him now!\" exclaimed Ulric. \"The moon is shining upon him\nwonderfully.\n\n\"It is so,\" said Tostig, \"but I think not of that. Wilt thou note this,\nthat whenever there cometh a boom of the rending ice the bears call out\nto their mates? More than we do they know of such matters. All such\ncreatures have gods of their own, and we may have offended them. I like\nit not.\"\n\n\"The gods of the bears will care for the bears!\" said Knud. \"They have\nnaught to do with men.\"\n\nNevertheless, it was a time for men to speak softly concerning such\nthings when powers whom they saw not and knew not were dragging them\nand their ship along so helplessly. There are times when one feeleth\nthat he can get along well enough without the gods, but this was a\ndifferent matter. All the vikings talked soberly and they were glad\nthat their jarl was a son of Odin.\n\nIt was a strange, solemn, weird night in spite of the moonlight, what\nwith the peril and the moaning bears and the booming ice. After all,\nthey said, Odin himself might not be with them. There had been places,\nas all men knew, where all the gods had abandoned even the bravest of\nthe Northmen. Men like themselves had died without a sword cut or a\nspear thrust. All hope of falling in battle might be lost to them among\nthese treacherous ice floes. It was a short night, if there had been\naught to measure it by, but to the men on _The Sword_ it seemed long\nenough. None cared to go under a deck, but there were some who lay down\nand slept. The moon sank lower and lower and the shadows lengthened\nacross the ice fields, but there was yet a great flood of broken light\nwhen Ulric, the son of Brander, uttered a loud cry and put his war horn\nto his lips. Every man sprang to his feet, for each thought that he\nhad never before heard such a blast as that. A louder sound instantly\nanswered it, but none could tell whether it came from among the ice\npeaks or from down toward the bottom of the sea.\n\n\"The bears are moaning again!\" said Knud. He was ever thinking of his\nbears, but all the rest were hearkening for what might be coming next,\nand they knew not yet the meaning of Ulric's blast.\n\n\"Oars!\" shouted the jarl. \"Every man to his place! There is free water\nsoutherly. The ice king is bowing!\"\n\nLoudly moaned the bears, for a moment, and they seemed to be running\ntoward the ship, as if they would come on board; and Ulric blew his\nhorn again with the notes of battle defiance, but then there burst out\nupon all sides a roaring, splitting, rending sound, such as none of the\nvikings had ever heard before.\n\n\"He hath struck! He is aground!\" shouted Ulric. \"Hark to his breaking!\nHis hour is come!\"\n\nIf that were true, so also it seemed as if the hour of _The Sword_\nhad come, and of all who were on board of her. But the gods were with\nher. If the forefoot of the ice king had indeed caught upon a shoal,\nchecking and breaking him, the shock of that striking had separated the\ngreat floe in front of him so that it might move freely. Still it no\nlonger upheld him, and he suddenly began to pitch forward toward the\nship. Vast was the roll of the sea that swelled away from his pitching,\nand powerfully it uplifted _The Sword_ in her bed of ice.\n\n\"Hold hard, all!\" shouted Ulric. \"Ready with your oars! Odin!\"\n\nUp gazed they then, and the bravest of them shuddered, for the gigantic\nwhite head of the ice king was bowing nearer, as if he would cast\nhimself upon them. On rolled the great wave, steadily, and all along\nthe crest of it the ice it carried was rending into fragments that\nground angrily against each other. The floe that carried _The Sword_\nbecame twain that parted, letting her down and shooting her swiftly\nforward. It was just then that the ice king fell upon his face, his\nuppermost pinnacle almost crashing upon her stern.\n\nThe foaming water dashed across the deck and drenched Ulric at the\ntiller. He was wearing no headpiece now, and the salt spray drops\nglittered brightly among his yellow curls. But they glistened not with\nmoonlight, for while they all had waited and watched the sun had risen\nand his first rays lit the hero face of the son of Odin as he shouted\nto his men to row their best, and as he steered the good ship _The\nSword_ into the open water the White Horse banner of the Saxons floated\ngallantly from the masthead and men sprang to set free the sail.\n\n\"Hael, O Ulric the Jarl!\" shouted Knud the Bear. \"We have a good sea\ncaptain.\"\n\nSo said several of the elder vikings.\n\n\"Hael, all!\" cheerily responded Ulric. \"The ice king hath fallen and we\nshall fear him no more. The gods are with us!\"\n\nLoudly shouted they all, and those who were not rowing clashed their\nswords upon their shields as if they had won a victory.\n\n\"Aye!\" growled Tostig the Red. \"'Tis a stout ship.\"\n\n\n\n\n                             CHAPTER VII.\n\n                           THE LIVING SAND.\n\n\nIt was the time of thaw in the Northland, but the snow and ice go fast\nwhen the winter letteth go its hold. Already great reaches of land\nwere bare, but no man might travel far from his own home because of\nthe floods from the melting. All must wait until days should pass,\nand these were growing longer, but they were full of unrest. Even\nthe cattle in their enclosures lowed impatiently to one another;\nfor the brute creatures know well the signs of the return of green\ngrass to their pastures. In the house of Brander there was no shadow\nbecause of the absence of any who had gone, but these were spoken of\ncheerfully. Moreover, there came boats and larger keels into the cove\nfrom other villages up and down the coast and from out the fiords\nthat were opening. Far and wide had been known the building of _The\nSword_, and many would have been glad to look upon her. All these were\ndisappointed, but there were wise old vikings and jarls of note who\nsaid to Hilda:\n\n\"Thy foster son hath done well. It is like his father. Other keels will\nfollow him speedily, but he will be first to strike.\"\n\nAs if she had been mistress of the house was Hilda, and she entertained\nwell all who came. Reverence was paid her because of her high descent\nand her kinship to Odin the Strong, and because of her hundred winters,\nbut even more because of her learning and her knowledge of the gods.\nMen asked her questions concerning them, and there were those who\nbelieved that she had seen and known more than she would tell.\n\n\"I would not like to anger her,\" said one, \"lest she might afterward\ncome to me in a bad hour, for she hath knowledge of charms and of\nwitchcraft and she can write runes.\"\n\nThere was reason in that, said all, but that she was a kindly woman and\nthat she kept the house of Brander liberally.\n\nMuch time she now spent among the old armor, the trophies on the wall,\nand in the study of such things as had been brought from the lands\naround the Middle Sea. She made Oswald open his bag and she read the\nmany inscriptions upon his coins, and she talked to him of Greece and\nof Rome, where most of them were made. He also knew about his gold and\nsilver pieces, and there were some even of copper for which he had\nnames and values. What good was there in such things in a land like\nthis, where money was not needed?\n\n\"I would that Ulric had them,\" she said. \"He might buy with them\nanother ship, or provisions, or arms.\"\n\n\"Not save of a friend,\" replied Oswald. \"He will need nothing that his\nsword can win for him. It is not the custom of the vikings to be long\nin need.\"\n\nThe household knew by her face that her thoughts were not troubling her\nconcerning Ulric and his men.\n\n\"She hath had no ill token,\" they said. \"It must be that he doeth well.\"\n\nThey knew not of the ice king, nor how narrowly he had missed his last\nangry blow at _The Sword_. But that peril was over and the good ship\nwas flying along in safety, driven by strong rowers, who had also some\nhelp from the sail. They would have had more but that the winds were\nvariable. Therefore the days and the nights went by before they again\nsaw land, and the older seamen knew by that that they had kept in the\nopen sea and were well advanced in their voyage.\n\n\"How fast or how far the ice king bore us I know not,\" said Knud the\nBear, \"but if that headland were not of one of the northern isles, we\nhave seen a cape of North Britain.\"\n\n\"Not so far south as that,\" argued Tostig the Red, \"but all these\ncoasts are bad to land upon. There is naught worth the taking away.\"\n\n\"Our errand is not to them,\" said Ulric. \"We will not waste an arrow\nupon them. I will not let the prow of _The Sword_ touch the sand until\nwe see the mid-coast of the British island----\"\n\n\"We shall see a storm this night,\" interrupted an old viking. \"The wind\nchangeth to the northwest, and Knud may wear his bearskins. It will be\ncold.\"\n\nWhen the night fell all were willing to cover well; but the rowers\nmight rest, for the ship carried her sail all the more safely because\nit was not too large and because she was well laden. There was a spirit\nupon Ulric which kept him at the helm, so that his men needed almost to\ntake him away by force that he might sleep.\n\n\"I would I might see Hilda and have speech with her,\" he said to\nhimself. \"I have strange dreams when I close my eyes. She might tell me\nwhat they mean. Do the gods come to one when he is asleep? I have heard\nso. But they have told me nothing--save that I have dreamed of men who\nwore the armor that hangeth behind the table on the dais. Strong men\nthey were, and dark, and I think they were good swordsmen. Before long\nit may chance that we shall meet a trireme of the Romans if my dreams\nhave that reading. I must burn one of their ships before we pass these\nseas.\"\n\nHeavier blew the gale and higher rose the waves, and _The Sword_ sped\non as if she were a waterfowl, but all on board were willing to be as\nwell covered as was Knud the Bear. The night was dark and the next\nmorning they saw no land. The storm drove them onward steadily all day,\nand now and then they saw ice floating, but no sail of any ship. Again\nthe night came, and the moon was out and the wind lulled, but the waves\nwere still rough.\n\n\"We will not row,\" said Ulric, when they inquired of him. \"There are\ncoasts now not far away. When the dawn cometh we will seek some bay or\nharbor. I have heard that there are villages of North folk hereaway,\nand they would be friendly.\"\n\nSo said they all save Tostig the Red, who laughed somewhat grimly and\nreplied:\n\n\"I think there are villages upon many coasts whereof the folk are\nwilling to be friendly to a crew like this. The seax hath many\nacquaintances who are willing to see him stay quietly in the belt.\"\n\n\"So hath the ax,\" growled old Biorn the Berserker. It was rare for him\nto speak, but he was leaning upon the long handle of his weapon, and\nwhen he lay down on the deck the ax slept beside him.\n\nIt was after the middle watch that night, and Ulric was at the helm.\nHe was steering a straight course southward and the ship was slipping\nquietly over the waves. He was awake, truly, but somehow he seemed to\nhimself to be dreaming almost, and his eyes were downcast. \"The runes\nupon the sand,\" he muttered. \"I can see them now, before the wave\nwashed them away. When and where am I to see them again, and to know\nthat my voyage is ended? Who shall read runes, and how shall I be sure\nthat I am not mistaken? For Hilda will not be there----\"\n\nEven as he spoke there came to his ears a sound, and he looked suddenly\nup, gripping hard the tiller.\n\n\"Faint and far away,\" he exclaimed, \"but it was a trumpet! There are\nthree in the hall at the house and Oswald taught me their soundings.\nUp, all! Rowers to the oars! I will send an answer!\"\n\nLong and powerful was the horn blast that went out across the moonlit\nsea. Clearer and louder than before was the trumpet voice which\ninstantly responded from the right--and that was toward the British\nshore. The men shouted not, for they were listening, and those who\nknew were telling the younger vikings that the jarl had heard from the\nRomans. It was good news to hear, after long waiting, and the rowers\nput out the long oars eagerly.\n\n\"The dawn draweth near,\" shouted Ulric, after blowing his horn again.\n\"We will steer toward yonder trumpet. There will be much music with the\nsun's rising. We will see if the gods of Rome are better than the gods\nof the North in the seas of Britain.\"\n\nLoud voices answered him bidding him lead on; for the blood of the\nvikings was rising hotly, and Biorn the Berserker sharpened the edge of\nhis great ax while he beat the deck with his feet and out through his\nthickly bearded lips there poured, low, but swelling, a song of the\nskalds at the gate of battle.\n\nRed grew the edges of the eastern sky as _The Sword_ pressed her iron\nbeak to the crests of the waves and sprang forward. Joyously rang\nout the war horn of warrior after warrior, for on board were vikings\nof high descent who would not have chosen for their jarl any of less\ndegree than a son of Odin. They were men entitled to go forward into\nthe feast of swords shoulder to shoulder with kings and with chiefs of\nrenown. Said one of them to Ulric:\n\n\"Jarl Ulric, many spears from the stowage. The Romans cast well and\ntheir spears are heavy. I mind not their light javelins nor their\narrows. Close not with any trireme at the first.\"\n\n\"I will be prudent,\" replied Ulric; \"but bring out the spears. There\nare arrow sheaves enough and stones for slinging.\"\n\n\"Let them not ram _The Sword_,\" continued the old fighter. \"Her ribs\nare strong, but so is the beak of a war galley of Rome. Strike her not\nsave amidships.\"\n\nWell was it for older men to counsel so young a leader, but Ulric had\nbeen taught from his infancy not only by Brander the Brave and Oswald,\nbut by all the sea kings and berserkers to whom he had listened while\nthey talked of war around the mid-fire in the old hall. Naught had they\nsaid or sung but he had made its teachings his own against an hour like\nthis.\n\n\"A trireme!\" shouted Knud the Bear as the daylight brightened. \"She is\nof the largest. Helmets and standards and the shields of a cohort of a\nlegion. They are more in number than we are.\"\n\n\"Twice more,\" said the old counselor, \"and her bulk is nearly thrice\nthat of _The Sword_. Beware, O jarl!\"\n\n\"I see her well,\" responded Ulric. \"She is heavy in the water. I think\nshe is overburdened.\"\n\n\"They are swift also,\" said Tostig the Red, \"but that keel cannot turn\nas nimbly as can our own. Let us go nearer!\"\n\n\"Within a spear's cast!\" shouted Ulric, fiercely. \"We will not pass her\nwithout a blow. Wulf, take thou the helm. I will go to the fore deck.\"\n\nThere he stood in the morning light, as the two keels neared each\nother. The Roman trumpets sounded at intervals, and they were answered\nby the war horns of the vikings.\n\n\"She is a splendid war vessel,\" said Ulric to those who were with him.\n\"Never yet have we builded her like. Her bulwarks are higher than ours\nand her sail is many times broader. It is made of woven stuff. Her prow\nis a ram. We must not let her strike us.\"\n\n\"Neither will we strike her,\" said Biorn the Berserker, \"unless we can\nhit her amidships. She is a danger. O jarl, beware! I do not think we\nmay take that trireme, but we can get away from her.\"\n\nSo did not think the trierarch and the centurion on board the trireme.\nHe who was captain of the vessel was of one accord with the officer in\ncharge of the legionaries whom she was conveying. If Ulric could have\nheard them converse as _The Sword_ came toward them, he would have\nlearned somewhat of the estimation in which such as he were held by the\nwolves of Rome.\n\n\"A Saxon pirate, O Lentulus,\" said the trierarch to the man in armor\nat his side. \"It is early in the season for them to be seen in these\nwaters. They are the scourges of the sea.\"\n\n\"And of the shore, friend Comus,\" replied the centurion. \"We will make\nshort work of this one. It is of good size, and it swarmeth with men as\nwith bees.\"\n\n\"Hast thou ever met them in fight?\" asked Comus, \"or is this thy first\nsight of them?\"\n\n\"This is my first service in these waters,\" replied Lentulus, \"but I\nhave heard much of them. I would we had some legions of them to send\nagainst the Parthians, or into Africa. Laurentius had a cohort of them\nwith him in Spain. They make the best of gladiators; C\u00e6sar hath used\nthem in the arena. But it is hard to take them. Let us see if we cannot\nsend him a present of these pirates for the summer games. He is ever in\nneed of good swordsmen.\"\n\n\"Little thou knowest of them,\" laughed Comus. \"We may capture a few\nwounded men. The rest will die fighting.\"\n\nEven while he spoke Tostig the Red was remarking to his friends at the\nstern of _The Sword_, just forward of the deck: \"A fine stone for my\nsling is this. I will strike that high-crested one. There is often much\ntreasure on a trireme, if Thor will let us take her. But the men we\nwant not, nor the keel.\"\n\n\"Burn her,\" they said, \"and throw the soldiers overboard; but the\nRomans die where they stand. We shall take no prisoners but the rowers.\nThe jarl will slay them.\" So without thought of mercy on either side\ndid the two keels draw nearer.\n\nThey were not yet within a spear's cast when they who were with\nTostig stood away from him to give him slinging room. \"He is the best\nslinger,\" they said, \"on all the North coast. Let us see what he can\ndo. He is not a boaster.\"\n\nAs the vessel climbed a wave Tostig poised himself, swinging slowly\nthe leathern thong which upheld the square apron in which his pebble\nrested. Two pounds only in weight it may have been, but it was smooth\nand round from much chafing on the shore of the fiord with other\npebbles as the sea waves had tossed them to and fro in many storms.\nOver the crest of the wave went _The Sword_, and as she did so the\nsling began to whirl swiftly in the hand of Tostig. Hand went to hand\nto give it double force, and then, as the downward plunge of the keel\nwent with him, he gave his might to it and threw.\n\nNone saw the stone, so swiftly did it pass, but the trierarch said to\nthe centurion:\n\n\"O Lentulus, thou art said to be as good a spearman as Pontius of Asia.\nHave thy pilum ready and try thy fortune.\"\n\n\"It is too far,\" said Lentulus, poising his pilum. \"I was in battle\nonce with that same Pontius. Hercules! I am slain!\"\n\nLoud clanged his brazen helmet and prone he fell upon the deck. He did\nnot move again. The stone hurled by Tostig had left him but life enough\nfor that one outcry as it smote him.\n\n\"May all the gods forbid!\" exclaimed Comus. \"What ill fortune is this?\nHe is dead! Toward the pirate! Strike her through and through!\"\n\nEven as he spoke a legionary at his side went down before a second\nstone from the sling of Tostig, and the shouts of the vikings mingled\nwith the clangor of their war horns.\n\nDeft was the steering of Wulf and the swift rush of the trireme was\navoided, _The Sword_ passing her stern so near that every spearman\nmight make a cast. But the legionaries, pilum in hand, had faced the\nfurther bulwark, thinking their foe came that way, and not so many of\nthem were at good stations. Their bowmen also had been deceived, and\ntheir greater number was of no account. Nevertheless, many Roman spears\nflew well, being mostly of the lighter javelins used by them in the\nbeginning of a fight. Easily were these caught upon the broad shields\nof the vikings, as if it were in a mere game at home, and no harm was\ndone by them or by the arrows. Closer were they when they did their\nown throwing, and a hundred heavy spears went hurtling in among the\nlegionaries.\n\n\"Follow!\" shouted Comus. \"Have ready the grapplings! Strike and then\nboard her!\"\n\nA good officer was he, and the rowers as well as the legionaries obeyed\nhim angrily, for they deemed the Northmen insolent in assailing such\nsuperior force.\n\n\"Away!\" shouted Ulric. \"Hael to thee, O Tostig. Get thee to the stern\nand pitch thy pebbles among her rowers.\"\n\nTostig was toiling hard, and so were other good slingers, of whom the\ntrireme seemed to not have any, but _The Sword_ swept on out of range\nwhile her enemy was turning.\n\n\"O jarl,\" said Biorn, \"she is not clumsy, but her steersman went down.\nLet us gain what distance we may. That was a good blow, but we may not\nstrike the next so easily.\"\n\nThe older vikings looked watchfully, as did Biorn, and again they\nsaid: \"Our jarl is young, but this was well done.\"\n\n\"Westward!\" shouted Ulric to Wulf. \"We must lead them toward the land.\nI would I knew this coast.\"\n\n\"That do I,\" said Biorn, \"if we are where I think. There are high\ncliffs, but there is also much marsh land; and off the coast there are\ngreat shallows, worse for a ship than any rocks might be. Watch for\nthem.\"\n\n\"They are our friends,\" said Ulric, \"but they are not friendly to a\ndeep vessel like yonder trireme.\"\n\n\"Aye,\" said Biorn, \"it is our old way of battling such as she is, but\nthere is an evil among these shallows. Hast thou not heard of the sand\nthat is alive? There is much of it hereaway.\"\n\n\"My father warned me of it,\" replied Ulric. \"If horse or man setteth\nfoot upon it, it will seize him and suck him down. But it could not\nswallow a ship.\"\n\n\"Were she a mountain!\" exclaimed Biorn. \"The living sand would be worse\nthan a Roman trireme for _The Sword_ to escape from. Yonder is a land\nline at the sky's edge, and I think I see breakers.\"\n\nThe rowers were rowing well and _The Sword_ had gained a long advantage\nbefore the Roman oarsmen had recovered from their confusion. Now,\nhowever, Ulric upon the foredeck was measuring distances, wave after\nwave, and he spoke out plainly to his men.\n\n\"Swift is _The Sword_,\" he said. \"I had thought that no keel on earth\ncould be swifter, but we are laden heavily; so is the trireme, that she\nturneth not nimbly, but in a straight course she is swifter than are\nwe. She hath many rowers and she is sharp in the prow. She gaineth upon\nus little by little.\"\n\n\"Woe to her,\" responded the vikings. \"She moveth too fast for her good.\"\n\n\"The land riseth fast,\" said Biorn. \"The breakers are not far away.\nUnder them are sand shoals.\"\n\n\"The Roman is but a hundred fathoms behind us,\" replied Ulric. \"Wulf\nthe Skater, steer thou through the breakers. Let us see if she will\ndare to follow.\"\n\nComus, the trierarch, was overeager, or he would have remembered that\nwhich he seemed to have forgotten. They who were with him were stung\nby the death of Lentulus and by the ravages of the Saxon spears and\nstones. None counseled him to prudence, and he dashed on in the foaming\nwake of _The Sword_.\n\n\"Breakers, but no rocks,\" muttered Wulf, as he grasped his tiller\nstrongly. \"Now, if we fill not, we shall dash through. Pull! For the\nNorthland pull!\"\n\nHard strained the rowers. High sprang the curling breakers on either\nhand. Loud rang the shouts and the war horns. But _The Sword_ rose\nbuoyantly over the crown of a great billow and passed on into smoother\nwater.\n\n\"Odin!\" roared Biorn the Berserker. \"The trireme is but fifty paces--\"\n\n\"Struck!\" shouted Ulric. \"On, lest we ourselves may be stranded!\"\n\n\"Deep water here, Jarl Ulric,\" calmly responded an old seaman near him.\n\"We have passed the sand bar. It may be the tide is falling. The gods\nof the sea are against that Roman keel.\"\n\n\"Or they are not with her to-day,\" said Ulric. \"She is held fast. Cease\nrowing and put the sail up again. We will see if there is aught else\nthat we may do. I like not to let her escape me.\"\n\nUp went the sail, and for an hour _The Sword_ did but cruise back and\nforth, only now and then venturing near enough for the hurling of a\nstone or the sending of an arrow. It was then too far for any harm to\nthe Romans, but they could hear the taunting music of the horns.\n\n\"Low tide,\" said Biorn at last, \"and she lieth upon bare sand. We are\nwell away. We can do no more.\"\n\n\"Watch!\" said Ulric. \"They are troubled.\"\n\n\"She lieth too deeply. What is this?\" So asked the Roman seamen of\ntheir captain as they leaned over their bulwarks and studied that bed\nof sand. He answered not, but one, a legionary in full armor, stepped\ndown from the ship to examine more closely--and an unwise man was\nhe. In places the sandy level seemed firm enough, and a horse may\ngallop along a sandy beach after the tide is out and leave but a fair\nhoofprint. That way armies have marched and chariots have driven. There\nwere other patches, however, whereon the sand seemed to glisten and to\nchange in the sunlight, and here there was potent witchcraft working.\nAt these had the sailors been gazing, but the soldier did not reach one\nof them.\n\n\"Back!\" shouted Comus. \"It is the living sand! We are all dead men!\nBack!\"\n\nThe legionary strove to wheel at the word of command, but his feet\nobeyed him not. Even the vikings were near enough to see that the sand\nwas over his ankles.\n\n\"The under gods have seized him,\" muttered Ulric. \"It is from them that\nthe sand liveth. They are angry with him.\n\n\"_Vale! Vale! Vale!_\" shouted the legionary. \"O Comus, I go down! They\nwho dwell below have decreed this. See thou to the ship and follow not\nthe Saxons.\"\n\n\"Follow them?\" exclaimed Comus. \"_Vale_, O comrade! But the trireme\nlieth a handbreadth deeper. She is sinking! O all the gods! Have we\ncome to this ending? Who shall deliver us?\"\n\n\"None, O Comus,\" said a man of dark countenance who leaned over the\nbulwark at his side. \"We have offended the gods and they have left us\nto our fate.\"\n\nLower sank the wooden walls of the great vessel, while her helpless\ncrew and the soldiery stared despairingly at the pitiless sand and at\nthe White Horse flag of the vikings dancing lightly over the sea so\nnear them.\n\n\"Form!\" commanded Comus, and the legionaries fell into ranks all over\nthe vessel. \"Put ye the body of Lentulus upon the deck,\" he said, \"and\nbring me the eagle of the legion. O Lentulus, true comrade, brave\nfriend, we salute thee, for all we who were of thy company go down to\nmeet thee. Behold, we perish!\"\n\nSilent sat the rowers at their oars. The standards fluttered in the\nwind. The trierarch took the eagle and went and stood by the body of\nLentulus.\n\n\"They are brave men, yonder,\" said Biorn the Berserker. \"They will to\ndie in line. So do the Romans conquer all others except the men of the\nNorth.\"\n\n\"They have one trireme the less,\" replied Tostig the Red. \"But they\nhave many more. This is not like burning one. I see no honor to us in\nthis.\"\n\n\"Honor to the gods,\" said Ulric. \"She was too strong for us and Odin\ndestroyed her.\"\n\n\"It is well to have him on our side,\" said Tostig; but Knud the Bear\nlaughed loudly, as was his wont, and said: \"Odin is not a sea god. What\nhath he to do with sand and water? Some other god is hidden under the\nliving sand. We shall leave him behind us when we go away----\"\n\n\"Her bulwarks go under!\" shouted one of the vikings. \"Hark to the\ntrumpets! They go down!\"\n\nThe trumpet blast ceased and there was a great silence, for the like of\nthis had never before been seen.\n\n\"Oars!\" commanded Ulric. \"We will search the coast. Such a warship\nas was this came not hitherward without an errand. She may have had\ncompanions.\"\n\nThe old vikings all agreed with him, and an eager lookout was set, but\nbehind them as they sailed away they saw nothing but a bare bed of\nsand, over which the tide was returning.\n\n\n\n\n                             CHAPTER VIII.\n\n                           THE SAXON SHORE.\n\n\n\"O jarl!\" exclaimed Knud the Bear, in a morning watch, \"we have wasted\ndays in this coasting. The weather hath been rough and the men are\nweary, for we are tightly packed in this ship.\"\n\n\"No longer shouldst thou prevent us from seeking the shore,\" said\nanother. \"I would hunt, and get me some fresh meat.\" There were also\nvoices of impatience and of discontent among the crew.\n\nThe jarl listened, and thoughtfully he responded: \"I have not forgotten\nthat the Romans sail in fleets. We are one keel. If now we have avoided\nany trireme that was company for the one which was swallowed by the\nsand, we have done well. We will steer toward the shore. My father told\nme of such a coast as this.\"\n\n\"As the sun riseth higher,\" said Biorn the Berserker, \"I think I can\nsee a low headland. This is not my first cruising in these seas.\"\n\n\"It is well,\" said the jarl. \"We will go within the headland. If we\nfind a good shore, we will land, for I am of one mind with you.\"\n\nAll the older vikings approved of his prudence, for they knew the\nRomans better than did the younger warriors, full of eagerness. Even\nnow the sailing of _The Sword_ was with caution. The noon drew near and\nthey were close to the headland. It was neither high nor rocky, and on\nit was a forest; but here was a surprise, for the trees growing down to\nthe beach were in full leaf.\n\n\"The winter tarried late in the Northland,\" said the vikings. \"We have\nalso been many days upon our way. The summer is near.\"\n\nThey might also discern patches of green grass, and now Knud shouted\nfrom the fore deck: \"A deep cove, O jarl! It is very deep.\"\n\nUlric was at the helm, and he responded: \"Thou hast good eyes, O Bear.\nWatch thou for rocks and shoals and give me word. Let all eyes watch\nalso for boats or men.\"\n\nThe rowers rowed easily and _The Sword_ slipped on into the cove. Here\nwas dense forest on either side, and there were rocks, but the trees\nwere large and old and there seemed to be little undergrowth, nor was\nthere any sign of the dwellings of men.\n\n\"The Britons,\" said an old viking, \"build not often on the shore. They\nare not seamen. They have no forts but wooden palisades, and they dwell\ninland, where they are more safe. They fight well, but they have little\narmor, and their steel is soft. They are no match for the legions of\nRome.\"\n\nIt was exceedingly still as _The Sword_ went forward. Away at the\nleft a herd of red deer came out under a vast oak and stared at the\nnewcomers. At their head was a stag with branching antlers.\n\n\"Now know we,\" said Biorn the Berserker, \"that no men are near this\nplace, for these creatures are exceedingly timid. But their venison is\nof the best. In Britain are also wild cattle in abundance, and wild\nswine. We will have great hunting before we sail to other places.\"\n\nSwiftly away sped the red deer, for the prow of _The Sword_ touched\nthe strand and Wulf the Skater sprang ashore, followed by a score of\nvikings.\n\n\"On, up the bank!\" shouted the jarl. \"Return and tell what thou seest.\nAll to the shore and stand ready if he findeth an enemy.\"\n\n\"A prudent jarl,\" murmured Biorn the Berserker. \"He will not be\nsurprised.\"\n\nNevertheless, the younger men laughed scornfully, for they liked not\nwell the hard discipline of the jarl, and he brooked no manner of\ndisobedience, as was his right.\n\nBack came one from Wulf the Skater. \"O jarl!\" he shouted. \"A fine\nspring of water. An open glade. Wulf asketh if he shall now cut the\nsaplings.\"\n\n\"I come soon,\" replied the jarl, \"but cut stakes for a palisade leading\ndown to this beach on either hand. Though there be no Romans here,\nthere are Britons not far off.\"\n\nAxes were plying speedily, and while the first fires were kindling many\nsharp stakes were driven, to be woven between with flexible twigs and\nbranches. Such was ever the custom of the Saxons upon a new land, for\nbehind such a wattle-work defense a few warriors may withstand many,\nand light palisades guard well against horsemen. Not all could work in\nthese matters, and twoscore were selected by lot for the first hunting,\ngoing out in four parties, with a command not to venture too far. They\nwere bowmen, but they went in their armor. Before the sun set there was\na good stockade from tree to tree around the spring, with arms that\nreached out on either hand almost to the shore.\n\n\"We will make it stronger,\" said the jarl, \"but behind it we are safe;\nfor we might also retreat to the ship if there were need.\"\n\nNo red deer save one stag and a doe did the hunters bring in, and there\nwould have been a lack of meat but for the slaying by another party of\nfour black cattle, fat and good.\n\n\"O jarl,\" said the men. \"Did we not tell thee? This is better than\nbeing packed so tightly in _The Sword_. This is good venison.\"\n\nWell contented was he also, and he saw that he must humor the men if he\nwere to command them well thereafter. For this reason, therefore, other\nand larger hunting parties went out the next day, and they came home\nheavily laden.\n\n\"O jarl,\" said Tostig the Red, for his party, \"we have also found\npaths, but no men. We saw hills beyond, but a river is between us and\nthem, and a great marsh. I think no Britons come hither across the\nmarsh.\"\n\n\"On the morrow I will go,\" said Ulric. \"I will leave Biorn in command\nof the camp. I have no need for hunting, but I must know the land.\"\n\nBarrels of ale had been brought to the shore, and that night was a\nfeast, with songs and sagas. After the feast the jarl went and lay down\nto sleep under an oak, but his eyes would not close for thinking of the\nNorthland, and of the Middle Sea, and of Asgard.\n\n\"This landing is well,\" he thought, \"and I am glad to be in Britain.\nBut here I may not linger too long. O Hilda of the hundred years, not\nyet hast thou visited me. I wonder if thou or the gods could find me\nthis night under this oak tree. Who should tell thee where to come if\nthou wert seeking me? The gods see everywhere. Biorn sayeth that the\ngods of Britain are gods of the woods, and we are from the sea. I care\nnot much for wood gods.\"\n\nThen he rested, but he arose early and chose the men who were to go\nwith him.\n\n\"Guide me to the river and the marsh,\" he said to Wulf the Skater.\n\n\"I will, O jarl,\" said Wulf; \"but Tostig saw a wild boar yesterday and\nhe hath gone out after him. A vast one, he sayeth, with tusks like a\nwalrus. He will fight well if they can bring him to a fighting.\"\n\n\"Let Tostig win his boar,\" said Ulric. \"We go to the left and we hunt\nnot. I am full of thoughts about this place.\"\n\nA score of vikings were with them, and they marched on in order, two\nand two, as if they had an errand. Grand were the trees, and high, with\nbranches whose foliage made a gloom to walk in.\n\n\"Are we nearly at the marsh?\" asked Ulric at last. \"Here are rocks.\"\n\n\"I know not, O jarl,\" said Wulf. \"We came not so far southerly\nyesterday.\"\n\n\"Hael, Northmen! Hael! But sound no horn! Who are ye?\"\n\nAs if he had suddenly arisen through the ledge of rocks before them,\nupon it stood a tall shape in full armor, spear in hand. From under his\nhelmet tangled white hair fell down to his shoulders, but his right\nhand, holding the spear, was lifted as by one who giveth a command.\n\nAgain he spoke: \"I am Olaf, the son of Hakon, of Droningsfiord. Who are\nye?\"\n\n\"Northmen of thine own land,\" said the jarl. \"I am Ulric, the son of\nBrander. Our ship, _The Sword_, lieth at the shore. How camest thou\nwhere thou art, and who is with thee?\"\n\n\"None are with me,\" said Olaf, sternly. \"We were many, but the Romans\nhave smitten the Saxon shore of Britain and our villages are gone. They\nhave smitten many of the Britons also, and they march to smite them\nagain this day. Tell me, O Jarl Ulric, hast thou seen aught of certain\ntriremes which were to come? I would know if there are more Romans near\nthan I have already counted.\"\n\n\"One hath perished, as I will shortly tell thee,\" said Ulric. \"I have\nseen no other.\"\n\n\"Good!\" said Olaf. \"There floateth one in a harbor not far away, but\nthey who came in her are fewer than when they landed. Twain came, with\na cohort. One hath sailed. Their force was sent to slaughter the Druids\nat their great sacrificing, but first they struck our village at our\nharbor. We fought, but they were too many. I cut my way through the\nranks of their lighter spearmen, and they followed me not far because\nof the nearness of the Britons.\"\n\nOlaf was now descended from the rock and was become as one of them.\nGreat was his wonder at the story of the living sand and the trireme.\n\n\"The gods of the Britons are strong at times,\" he said, \"but they\nare not to be depended on. They have done this because of the great\nsacrifice, that the Romans may not hinder it. Therefore come thou with\nme a little distance and I will show thee a matter. The Romans are\ntangled in a wood. Meddle not thou and thine, however, for thou hast\nanother work to do.\"\n\n\"I meddle not,\" said the jarl, \"but I thank these Druid gods. We were\nclosely pushed and in peril when they ensnared the trireme with their\nsand. I will offend them not, but I would see these great sacrifices\nand I also would offer my token.\"\n\n\"That the Druids will forbid thee,\" said Olaf. \"Follow me quickly to\nthe crown of this ridge, for it is on the bank of the river.\"\n\nEven as he spoke there came to their ears a clangor of trumpets, as if\nmany sounded at once.\n\n\"Romans!\" exclaimed Ulric.\n\n\"Sounding first were they,\" said Olaf, \"but these hoarse ones, very\nloud, are blown by the Druids. Hear, also, the harping. Now look thou,\nfor thou art a captain.\"\n\nThe river before them was but narrow, although it might be deep, and on\nthe other side was a broad open space surrounded by a forest with dense\nundergrowths of bushes, as if it were marshy. In the open was arrayed a\ncohort of Roman soldiers, well ordered, but beyond and in their front\nmight be seen and heard much larger numbers of such as they were, all\ndisarrayed and scattered by the copses. None assailed the cohort in the\nopen, but all the forest swarmed with half-armed Britons, hurling darts\nand plying their light blades. Arrows, also, were flying, and there was\na great tumult of mingled sound.\n\n\"The men in white robes, keeping afar,\" said Olaf, \"are the Druid\npriests. This is as an ambush, and the Romans are falling.\"\n\n\"Their commander hath some wisdom, I think,\" said Ulric. \"His trumpets\ncall back his men for a retreat. He will escape.\"\n\n\"He loseth half his force,\" said Olaf; \"he will lose more as he\nretreateth.\"\n\nFiercer and fiercer arose the sounds of the combat, the shouting, the\nhowling, the twanging of loud harp strings, and the braying of the\ntrumpets. Hard was it for the vikings that they might not have a part\nin such a battle.\n\n\"The Romans are outnumbered,\" said Olaf, \"but they fight well. Their\nretreat will be to the river mouth, where was my village. There have\nthey a camp in our own stockade, and they have also increased it with a\nrampart of earth and palisades. There we must strike them. It is but a\nlittle distance. Come and see.\"\n\n\"But first,\" said Ulric, \"I would see the end of this battle, and I\nwould have speech with a Druid concerning the sacrifices.\"\n\n\"That thou mayest not this day,\" said Olaf, \"and the Romans are cutting\ntheir way through the tumult of half-naked spearmen. Lo, how they\nslay the Britons! But the ranks of their cohort will be thin when the\nremnant reacheth the fort. So hath it often been in their warfare in\nBritain, but each new commander of legionaries cometh here a proud one,\nthinking only of easy victory.\"\n\n\"The darts fly in showers,\" said Ulric, but Wulf the Skater urged him.\n\n\"O jarl!\" he exclaimed. \"The village! The fort! The trireme! Why wait\nwe here? Let us go with Olaf!\"\n\nThe jarl answered not, but walked rapidly, and the rocky ledge grew\nhigher as they went; but there came an end of it.\n\n\"We have walked far,\" said Ulric. \"The way of the Romans was shorter.\nThere come they and their array is not broken. I can see their\ncommander ordering them.\"\n\n\"Thor the Thunderer!\" exclaimed Olaf, \"what havoc the Britons have made\namong them! The gods of the Druids have protected their sacrifices.\"\n\n\"Every Roman left behind hath perished,\" said Ulric. \"Only these are\nalive.\"\n\n\"Not so,\" said Olaf. \"Not a wounded man or one entrapped hath been\nslain. He belongeth to the gods at the place of sacrifice.\"\n\n\"With them as with us,\" said the jarl. \"That is the old North custom. I\nhave seen men slain at the stone of Odin. He who is captured must lose\nhis head. It is well----\"\n\n\"Seest thou?\" loudly demanded Olaf. \"The ruins of our village are yet\nsmoking, although three days have passed. I saw thy ship on the sea\nyesterday, but knew not of thy landing. I meant to watch for thee or\nfor the coming triremes after seeing the battle.\"\n\n\"Yonder trireme at anchor,\" replied the jarl, \"floateth well out from\nthe river mouth. She is large. How shall I take her? For there are yet\nRomans enough to hold her well. I must come to her by night in _The\nSword_.\"\n\nLong and thoughtfully gazed Ulric, studying the position of the trireme\nand the arrival of the beaten Romans at the fort.\n\n\"O jarl,\" said Biorn the Berserker, \"knowest thou not that I am a fish?\nThe trireme is held but by an anchor and a cord of hemp. Go thou and\nbring _The Sword_. When thou art at hand to strike thou mayest have\nthe trireme drifting with the outgoing tide. Strike not when the tide\nrunneth in?\n\n\"Thou canst swim,\" said Ulric, \"and thy seax will sever hemp; but if\nthou waitest here until I come, how wilt thou know in the dark of my\ncoming, or how wilt thou know where to ply the sharp edge?\"\n\n\"When I hear thee whistle thrice,\" said Biorn, \"as if thou wert calling\nthy hawk, I will know of thy coming. If the whistle is from this shore,\nI meet thee here. If it is from seaward, I swim to the trireme. Thou\nwilt know the hemp is severed when thou hearest my own falcon call.\"\n\n\"I go with thee, O jarl!\" shouted Olaf, eagerly, \"that I may be thy\npilot.\"\n\n\"Well for thee, O Biorn the Berserker,\" said Ulric; \"thou art of the\nheroes!\"\n\n\"Here sit I down,\" replied Biorn. \"It is a pleasant place. I think this\ntaking of the trireme will depend upon thee and thy sword more than\nupon a man a fish cutting hemp!\"\n\n\"Haste, now,\" said Ulric to his men. \"_The Sword_ is far from us and\nthis is to be a night of great deeds, and not of ale and feasting.\"\n\nOlaf led, as the guide of their rapid marching, and Biorn sat down upon\na rock to gaze at the doings around the river mouth and at the fort.\n\n\"There come the Britons out of the woods,\" he said to himself. \"If they\nhad been well led they would have pursued more closely--only that few\ncare to press too hard upon even the wreck of a Roman army. Now are all\nthe Romans within the stockade.\"\n\nThe Britons were many, but their prey had escaped them. The camp fort\nwas too strong for them to storm, and their showers of darts flew over\nthe palisades without much harm to any within. The taunting clangor of\ntheir harps and trumpets sounded furiously for a while, and then the\nmultitude swiftly vanished as if it had melted away.\n\n\"If these Britons had a captain,\" said Biorn, \"instead of a herd of\npriests, and if he would arm them well, the Romans would disappear from\nBritain. But I think Ulric the Jarl will find many swords on yonder\ntrireme. Even now they go out in small boats. Biorn the Berserker will\nbe with him when the Saxons are on the Roman deck!\"\n\n\n\n\n                              CHAPTER IX.\n\n                      THE TAKING OF THE TRIREME.\n\n\nThe night was at hand when the jarl and his party arrived at the camp,\nand already all others were around the camp-fires.\n\n\"O jarl!\" shouted Tostig. \"Come thou and see this mighty one! We hauled\nhim hither upon a bundle of branches, and he wearied us with his\nweight.\"\n\n\"Never saw I such a one!\" exclaimed Ulric, gazing at the great boar\nwhich lay at the fire by the spring. \"Was he for thy spear alone?\"\n\n\"For mine!\" said Tostig. \"Now am I even with thee concerning the white\nbear, for this one fought as did the son of the ice king. He nearly\novercame me after he had slain Nef, the son of Ponda, and had rent him\nin pieces. He had no wound from Nef.\"\n\n\"We did watch them,\" said a viking, \"and to Tostig is the honor. If his\nspear had broken, as did thine in the bear, I think Tostig would have\nlost the battle.\"\n\n\"Then had I felt those great tusks,\" laughed Tostig, \"But it will take\nall the night to roast him well.\"\n\n\"He will roast while we fight,\" replied the jarl; \"and some of us will\neat not of him, but in Valhalla. To the ship, all! We go to attack a\nRoman trireme. Let those eat now who have not eaten, taking their meat\nwith them. I leave not a sword here!\"\n\n\"He who would stay behind is nidering!\" shouted Tostig the Red. \"We\nwill follow our jarl to the feast of swords, and they who return\nmay find the boar roasted. Hael to thee, O jarl! Thou bringest good\ntidings.\"\n\nNot until all were in the ship, however, did Ulric explain to his men\nfully and carefully the errand upon which they were going. Wild was\ntheir enthusiasm, and once more the young and the discontented were\nsatisfied with their jarl.\n\n\"He is a son of the gods,\" they said, \"and he will lead us to victory.\"\n\n\"Or to Valhalla,\" growled Knud the Bear. \"Not all of you will eat the\nroasted boar's flesh.\"\n\nThe rowers rowed with power and _The Sword_ went swiftly. Ulric was at\nthe helm, and Olaf was at the prow sending back words of direction. The\ndistance to be traveled was less on the water than on the land, through\nthe forests.\n\n\"I would I knew of the doings of Biorn,\" said one, as the ship rounded\na point and entered the harbor at the river mouth.\n\nThe jarl answered not, but shortly he put his fingers to his lips and\nwhistled thrice.\n\n\"Row slowly, now,\" he said, \"till an answer shall come. I am glad the\nmoon is not yet arisen. We go on behind a curtain.\"\n\nThe jarl's signal had been heard by a man upon whom was only a belt, to\nwhich hung a sheathed seax and a war horn. He stood at the water's edge\nat the harbor side.\n\n\"The jarl cometh!\" he whispered, and he went into the water, making\nno sound. Before that he had crept along the shore, landward, bearing\nhis arms and his armor, and now he had but sixty paces to swim. The\nRoman sentinel on the deck of the trireme heard only the ripple of the\noutgoing tide against her wooden walls.\n\nKnife upon hemp cutteth silently, but soon the sentinel turned with a\nsharp exclamation, for out of the seaward silence there came a long,\nvibrating whistle, another, another, and then from the hollow of a dark\nwave near the trireme there sounded a fourth like unto these three.\nThis last he answered with a shout, and he hurled his pilum at that\ndarkness in the water, but the trireme herself responded with a lurch\nand a yawing as she began to be swept away by the tide. There were\nrowers on board, and they quickly sprang to the oars, but they were\nfew and there was yet no steersman. There were many soldiers also, but\ntheir officer ordered a number of them to the oars, that he might get\nthe ship under control. When, therefore, there came gliding swiftly out\nof the shadows the unlooked-for warship of the Saxons she was alongside\nand her grapplings were made fast with none to hinder.\n\nFrom the opposite side of the Roman vessel, as it were from the water\nitself, now sounded furiously the war horn of Biorn the Berserker.\nFull half of the legionaries rushed in that direction and their hurled\nspears were too hastily lost in the sea. Terribly rang out the war\nhorns and the battle shouts of the Saxons, but the first man of them\non board of the trireme was Ulric the Jarl, and down before his ax\nfell whoever met him. Close behind him were his followers, so that the\nnearer Romans were not only surprised, but outnumbered.\n\nUp the side, near the stern, climbed Biorn the Berserker, and for\na moment he was alone, so quickly had fallen twain who were there.\nTaking in hand the helm, \"Biorn! Biorn the Berserker!\" he shouted. \"O\njarl, I am here! The ship is ours!\" Hard fought the remaining Romans,\nnevertheless, against such odds, but all the rowers were slain at their\noars.\n\n\"It is done!\" said Ulric. \"Silence, all! I have called twice for Biorn.\nWhere is he?\"\n\n\"O jarl, son of Brander the Brave!\" came faintly back from the after\ndeck, \"hast thou fully taken this trireme?\"\n\n\"We have her!\" answered Ulric. \"Thanks to thee, O Biorn! She is thine!\"\n\n\"Odin!\" shouted back the old berserker. \"Then bear thou witness for me,\nat feast and in song, that Biorn, the son of Nar, the sea king, died\nnot by drowning, but by the driven spear of a Roman, in all honor. I go\nto Valhalla as becometh me. Rejoice, therefore, and smite thou these\nRomans once more for me. I die!\"\n\nThere was a silence of a moment on the ship, but then the oldest viking\nof all blew triumphantly his horn and shouted: \"We have heard! Biorn,\nthe hero, hath gone to the hall of the heroes. He died by the spear,\nand not a cow's death. Good is his fortune. Hael to thee, O Biorn! And\nhael to Jarl Ulric, the leader of men.\"\n\nClashed loudly then the shields and spears, but already Saxon hands\nwere upon the oars and Tostig the Red was at the helm, with Olaf by\nhim. Only it might be a dozen warriors had been named by the valkyrias\nto go to Valhalla with Biorn the Berserker, but the Romans whose bodies\nwere cast into the sea were ten times as many.\n\n_The Sword_ and the trireme were now going out with the tide into\nthe open sea and into the darkness, but there had been much sounding\nof trumpets in the camp of the Romans. Few as were the remaining\nlegionaries, they had marched to the shore ready for action. There\nwere small boats at the beach, but it was all too late for any use of\nthese. Those who patrolled and inquired, however, found at the side of\na rock a helmet like a bear's head, a shirt the hide of a bear, two\nheavy spears, an ax--the trophies to them of Biorn the Berserker. These\nwere brought to the centurion in command and he examined them with care.\n\n\"The pirates of the North are here,\" he said. \"Woe is me that ever\nI came to this death coast! Here shall we leave our bones, for the\nBritons will come like locusts, and we have lost our trireme!\"\n\n\"Another ship cometh soon,\" said his friends. \"We may hold the fort\nwell until her arrival. All is not lost.\"\n\n\"Know ye that?\" replied the centurion. \"If the trireme of Lentulus\nwere above the water, she would have arrived long since. He hath never\nfailed an appointment. I think it was his evil demon and not the favor\nof the proconsul that made him the count of the Saxon shore. The fates\nare against us.\"\n\nSo darkly brooded the Romans over their many disasters, while Ulric the\nJarl ordered the steering of his two ships up the coast and into the\ncove where he had first landed.\n\n\"I would have speech with a Druid, if I may,\" he said to Olaf. \"It is\nstrongly upon my mind that I must see this great sacrifice to their\ngods. Manage thou this for me. Thou hast been in league with them.\"\n\n\"What I can do in such a matter I will do,\" said Olaf. \"But, O jarl,\nI have somewhat to say to thee concerning this trireme. Consider her\nwell, for she is a strong warship and there is much room in her.\"\n\n\"Also much plunder,\" said Ulric; \"but that must wait for the day. Each\nman hath his share, and the shares of the slain go to their kindred\nwhen we return.\"\n\n\"So is the North law,\" said Olaf; \"but where shall any man stow that\nwhich may be his prize? _The Sword_ is but a nutshell. Thou wilt think\nof this matter, for thou art jarl.\"\n\nThe night waned toward the dawn and all had need of rest. The ships\nwere anchored, therefore, and the cove was still.\n\nThe trumpets at the Roman camp greeted loudly the sun's rising. The\nsentinels were changed and the patrols came in from the edges of the\nforest to report that no enemy seemed to be coming. The soldiers\nsullenly attended to the customary morning duties of the camp, now and\nthen glancing seaward as if they hoped to see a sail. The centurion in\ncommand walked along the lines of his intrenchments, studying them,\nbut his eyes more often sought the earth. A stalwart man was he, in\nsplendid armor, and his face bore scars of battle. Well had he fought\nthe Britons the day before, but now he loudly exclaimed:\n\n\"O my imprudence! I should have waited for Lentulus and a greater\nforce. Will he never come? But, if he come, the fault of this defeat is\nnot his, but mine. He will be acquitted, and I am left alone to account\nto C\u00e6sar for a lost eagle of a legion!\"\n\nHe smote upon his breast and again he walked onward, downcast and\ngloomy. Once more he spoke, with exceeding bitterness:\n\n\"How shall I answer for the loss of the trireme here in the bay? Will\nnot all men say that I kept no watch?\"\n\nHe stepped upon the rampart and stood still. Near at hand were the\nruins of the Saxon village, but they had ceased smoking and lay black\nand bare as witnesses of the ruthless blow which he had smitten upon\nthe Northmen of the Saxon shore. Beyond were fields which would not\nbe cultivated this season as formerly. There were many corpses yet\nunburied, for the slayers had spared none save boys and girls for the\nslave market. The very young, the very old, even the middle-aged women,\nhad been slain, and the fighting men had fallen with their weapons in\ntheir hands. The prisoners were guarded in a kind of pen at the left,\nand they were many.\n\n\"Petronius,\" shouted the centurion to an officer of rank, \"take with\nthee ten and slay all. We have no conveyance for them. Let not one\nescape.\"\n\nOne order was as another to a Roman soldier, and Petronius answered\nnot, but marched away into the camp, seeking his ten who with him were\nto butcher the prisoners.\n\n\"I am dishonored!\" said the centurion. \"Fate and fortune are against\nme. I can give no reason for the loss of the trireme. I will go down to\nthe shades.\"\n\nSlowly he drew his short-bladed, heavy gladius from its sheath. He\nlooked at it, trying its edge, and he said:\n\n\"Thou hast been with me through many battles, O sword! Thou hast drunk\nthe blood of more lives than I can count. Be thou true to me now, for\nall else is lost.\"\n\nThen he knelt upon the rampart and placed the hilt firmly in the earth,\nthe blade point leaning toward him. He braced himself and cast his\nweight with force. A gasp, a shudder, a struggle of strong limbs, and\nPetronius was in command of the Roman camp, for his superior officer\nwas dead.\n\nThere were many screams at the prison pen, but afterward all was quiet,\nand Petronius returned, to be told of this new misfortune which had\nbefallen.\n\n\"Keep ye good watch,\" he said, \"lest the Britons take us unawares.\nThere is more than one trireme yet to come. But now we will raise the\nfuneral pile of him who lieth here, for he died in all honor.\"\n\nOrders were given and the soldiers brought much wood, but they came and\nwent in silence, for their fates were dark before them.\n\nSo was it with the camp of the Romans; but at the camp of the Saxons,\nat the cove and spring, there was high feasting, for they found the\nwild boar well roasted and the venison was abundant. They needed but\nharps and harpers, for the spirit of song came upon all singers, and it\nwas a day of triumph. Not even the older vikings could say that they\nhad ever heard of the taking of a Roman warship in this wise.\n\n\"Some have the sea kings rammed to sinking,\" they said. \"Some have they\ndriven ashore and some have they burned; but the Romans themselves ever\nburn any keel that they are leaving. Hael to _The Sword_, the victor!\"\n\n\"The smiters of my kindred have themselves been smitten,\" said Olaf,\nthe son of Hakon, but he sat with a fierce fire burning in his eyes and\nhis seax lay bare at his side.\n\n\"We have smitten them upon the sea,\" said Ulric the Jarl, \"but not yet\nupon the land. I may not yet leave Britain. Not until I have kept the\ncounsel of Hilda and my promise to my father at his tomb.\"\n\n\"Do as thou hast said,\" replied Olaf, \"lest evil fortune come to thee.\nBut go thou now and look at the trireme. Is she not thine, to do with\nas thou wilt?\"\n\n\"I will go,\" said Ulric, and with him went only Knud the Bear, by his\nordering.\n\nFirst went they upon _The Sword_, for she was nearer, and she was now\nlashed side by side with the trireme. High above the low bulwarks of\nthe ship from the Northland arose the strong sides of the war vessel\nof C\u00e6sar, and her greater force in fight or in rough seas was evident.\nUlric looked and he thought of the sayings of Olaf, the son of Hakon,\nfor a shrewd suggestion sprouteth in the mind of a wise man like a seed\nsown in a garden.\n\n\"Truly we were overcrowded,\" said Ulric, standing upon the fore deck of\n_The Sword_. \"We are thrice too many souls for so small a ship as this.\nThere was too little room for provisions or for sleeping. There is none\nat all for the storage of spoils. The men will not brook the burning of\nthe shares which may fall to them. They like not my hard ruling even\nthus far.\"\n\n\"O jarl,\" said Knud, \"what sayest thou? Let us not burn good plunder.\nWhat good to win it if we carry it not home with us? I would now go on\nboard the trireme.\"\n\n\"Come,\" said Ulric, and they climbed up over her high bulwark, noting\nhow thick it was and well joined together. Thus they passed from stem\nto stern and in and out of cabins, examining all things--the oars, the\nropes, and the sails.\n\n\"She is provided for a long voyage,\" said the jarl. \"Sawest thou ever\nsuch armor and such store of weapons? We may need them in the southern\nseas.\"\n\n\"That will we,\" replied Knud; \"but I am an old seaman and I was\nthinking of yonder sails. There are twain. They are of strongly woven\nstuff--not skins, like our sail. They will save much rowing. There are\ngood anchors also. Thou sayest well, we are too many in _The Sword_.\"\n\nYet she seemed very beautiful as she lay at the side of the trireme,\nand the jarl remembered how his heart had gone out to her while she was\nbuilding. She had borne him well, also, and she had proved herself.\nWhat might he do with the vessel that he loved? He went on board of her\nagain and he stood by the hammer of Thor on the fore deck.\n\n\"What thinkest thou?\" asked Knud. \"What if I--for I am a smith--put now\nthe anvil and the hammer on the fore deck of the trireme? Will she not\nthen be _The Sword_? Will not Thor and Odin go with her?\"\n\n\"Do even as thou hast said!\" loudly exclaimed Ulric. \"So the gods go\nwith us what matter for a wooden keel?\"\n\nBut his heart smote him sorely.\n\n\"I would,\" he thought, \"that I might have speech with Hilda. I will go\non shore and question Olaf. He is old.\"\n\nOld was he and crafty, for already he had been saying many things to\nthe vikings. He had told them of keels overwhelmed in the storms of\nthe southern seas, or crushed by the rams of Roman warships. He had\nspoken of hungers and thirsts because of lack of room for provisions,\nand of fights lost because there were no more arrows to shoot or spears\nto throw. The young men heard him eagerly, and even the old warriors\nlistened with care. They also called to mind such things and told\nof them, and all who chose to look could see the difference in size\nbetween the two vessels that floated in the cove.\n\n\n\n\n                              CHAPTER X.\n\n                  THE GREAT SACRIFICE OF THE DRUIDS.\n\n\nIn the deep forest stood Olaf, the son of Hakon, and before him stood\na tall, venerable man clad in a robe of white which came down to his\nfeet, whereon were sandals. On his head was naught save abundant gray\nhair and a circlet of beaten gold. On his arms were heavy rings of\ngold, deeply graven, and in his hand was a long white wand, gold tipped.\n\n\"Thou and thy Saxon friends have done well,\" he said in the Latin\ntongue. \"But I like not this message from their jarl.\"\n\n\"He doth but ask of thee, O high priest,\" replied Olaf, \"that he, who\nis not as another man, but is of the sons of the gods of the North, may\nreverence thy gods for the aid they have given him by sea and land, and\nthat he may be present at the great sacrifice, as becometh him. If he\nmay so do, he will give thee a thing the like of which thou hast never\nseen hitherto, and he will smite for thee the Romans.\"\n\n\"Cometh he then from Odin?\" asked the Druid.\n\n\"From Odin,\" said Olaf; \"and of higher rank than he is none among the\nSaxons.\"\n\n\"He is not a king,\" said the Druid, \"but I know of jarls and of their\npedigrees. The Romans at thy village are this day smitten by the\nBritons and we need not his sword. Well is it, however, for him to\ngive a gift. Let him see to it that his offering be right precious. It\nis a day's journey to the sacred place. He may not come down to the\nvalley of the gods, but he may stand upon the hill, among the oaks, and\nafterward I will receive his token.\"\n\n\"So be it, O high priest,\" said Olaf, and he turned away, as did also\nthe Druid.\n\n\"Cunning is he,\" muttered Olaf, as he walked. \"But in us also is there\nprudence and the jarl will be guided in the matter. I think he will not\nfall into this trap of the Britons. They plotted against us before the\nRomans came, and gladly would they see Saxon blood upon the stones of\nsacrifice.\"\n\nSo said he to the jarl at the camp late in the day, and Ulric listened,\npondering.\n\n\"Olaf,\" he said, after a silence, \"Wulf the Skater hath returned from\nlooking at thy place. No other trireme hath arrived, but even while he\nwas watching did the Britons swarm over the palisades. The Romans were\ntoo few to guard their lines, and it was in vain for them to resist a\nmultitude. Thy vengeance is complete.\"\n\n\"The gods have done this,\" said Olaf. \"But what wilt thou do in this\nother matter?\"\n\n\"I will leave a strong guard with the ship,\" said the jarl, \"but with\nthe greater number I will go to look upon the sacrifices. Thou wilt\nguide by a road they know not, and we will defeat their cunning.\"\n\n\"They would not strike thee, I think,\" said Olaf, \"until after the\nsacrifices. This is their reverence to their gods.\"\n\n\"I would I knew,\" said Ulric, \"the name of one of their gods. I will\nnot sacrifice to one to whom I may not speak. He is a breath.\"\n\n\"Thou mayest not enter the sacred valley,\" said Olaf; \"but I have\nsomewhat more to tell thee. Now do I know what is the name of thy\ncaptured trireme.\"\n\n\"The hammer of Thor is on her deck at this hour,\" said the jarl. \"She\nis no longer Roman. But whose is that gilded shape under her beak? It\nseemeth a woman wearing a helmet.\"\n\n\"The Druid told me,\" said Olaf. \"She is Minerva. She is to the Romans\nas are the Nornir. She is both wise and crafty, being a saga woman, and\nthere are runes concerning her.\"\n\n\"She is, then, not of the sea,\" said the jarl. \"I think she will not\ncontend with Thor. It were ill fortune to disturb her, seeing she hath\ndelivered to us the ship; but we must give to it the name of _The\nSword_ or Odin were justly angry, for we gave our keel to him.\n\n\"Thou hast decided well,\" said Olaf; \"but if so, then there must remain\none keel only, not twain. It was commanded thee to burn one ship in\nBritain, and thou mayest not break thy word to the dead and to the\ngods.\"\n\n\"That will I not,\" said Ulric; \"but now we must speedily prepare this\nexpedition.\"\n\nWise had been the work of the tongue of Olaf, for now came the vikings\nto Ulric to speak concerning _The Sword_ and the trireme, so that\nthis which was to be done appeared not as by his ordering, but as the\ncounsel of all.\n\n\"Thou doest well,\" they told him, \"to yield to us in this matter. We\nwill have a larger ship. We will have room for our plunder. We care not\novermuch for thy small keel, and we will burn her at the seaside. Thou\nart our jarl in battle, but thou mayest not rule in all things.\"\n\nNevertheless, they agreed with him all the more readily concerning the\nsacrifices, and those who were to go and those who were to stay by the\nships were chosen by lot lest any should accuse the jarl of unfairness;\nfor it was hoped that here was to be fighting. Not yet had there been\nany division of the spoils because all agreed to wait until a more\nconvenient season, or even until the end of the voyage.\n\n\"They whom the valkyrias do not name,\" said one, \"may apportion\nwhatever may then be found in the ship. There will be fewer weapons,\nperchance, and fewer men.\"\n\nIn the dawn of the next day did the jarl lead out his men, and in the\ndusk did the march end. High and round-topped was the hill in the\nforest to which Olaf guided them, and below was a narrow valley, bare\nof trees. There was yet light to see that in the middle of the valley\nwere many great stones. Some of these stood upright in a wide circle,\nlike the burial stones of the North peoples, but much larger. Other\nstones, long and weighty, lay flat, upheld a little from the ground by\nbowlders under them at either end.\n\n\"They are stones of sacrifice,\" said Olaf. \"On them do they slay both\ncattle and men. But seest thou the cages?\"\n\n\"Penthouses of wood I see,\" said Ulric. \"Very large, but of one story\nand roofed flatly. On the roofs and against the sides are heaps of\nwood. What are these?\"\n\n\"Wait till thou seest,\" said Olaf. \"Their shape on the ground is as the\nbody and the arms and the legs of a man, and there is a meaning in it\nknown to the Druids. They make this wooden man of sacrifice, and they\nfill him full of men and women and children that he may feast. They\nhave made many war captives and they have condemned many for evil-doing\nor for speaking against the Druids.\"\n\n\"Great fires are lighting around the valley and near the stones,\"\nremarked Tostig the Red. \"I have seen many men slain upon stones. It\nis the right place to slay them, where the gods can see all. We shall\nhave a rare treat. But there are hundreds of Britons. They wear little\nclothing.\"\n\n\"They paint themselves blue, instead,\" said Olaf. \"But it keepeth not\nout either the cold or a spear point.\"\n\nMore and more numerous grew the throngs in the valley, coming out from\nunder the trees beyond. Not among them, but walking through them in a\nprocession, came scores at a time of the white-robed Druids, bearing no\narms, but leading with them human beings of both sexes, arm-fettered,\ndefenseless, making no resistance. There was a loud sound of harping\nand chanting as the processions drew near the flat stones.\n\nBehind each of these stood a Druid with a large knife, and before\nhim, stone by stone, was laid a victim. Then fell the knives in quick\nsuccession, with a twanging of harps and a shout, but the Northmen\nsaw no great difference between this offering and such as they had\nwitnessed elsewhere. As the firelight brightened, however, they could\ndiscern that the walls of the wooden man in the middle were open, with\nwide crevices, through which might be seen the naked forms of those who\nwere shut in. They were even crowded, and they uttered loud cries as\nthey saw torches placed against the heaps of wood surrounding the pen.\n\n\"Dry wood,\" said Knud the Bear. \"See how it kindleth! A hot fire! These\nare to be burned for their god? He is a bad one. I like it not. The\nRomans do well to kill these Druids. I would slay them myself.\"\n\nSo said all the vikings, and had there been more of them, they might\nhave vented their anger at this thing. It was not good, even for a god,\nbut the throngs of Britons were well armed, after their fashion, and\nUlric's men were but few in comparison.\n\n\"We would not mind four or five to one,\" he said, \"but we could not\nslay such a multitude. The fires burn terribly! It is not at all like\nkindly slaying with a sword.\"\n\n\"A cut on a man's neck is nothing,\" said Tostig. \"He falleth and that\nis an end. I hope to fall by a sword some day.\"\n\nThe shrieks and cries of agony were dreadful, rising above the twanging\nof the harps and the chanting of the Druids. There was no help for any\nof these who were doomed. Among them, said some of the vikings, must be\nall the Roman prisoners if any had been taken. The burning roofs fell\nin and so did the red blazings of the side walls. Nor did the swarms of\nthe Britons cease to yell with the pleasure of cruelty while they gazed\nupon the frantic struggles of these victims.\n\n\"We have seen enough,\" said Olaf, at last. \"O jarl, we have far to go.\nI hope we may again strike the Romans shortly, but I care not much if\ngood Saxon spears find many marks among the Druids. It would require a\nhost of Saxons to hold this island, killing them all, but I am one who\nwill go back to the North and come again, bringing stout slayers with\nme.\"\n\n\"Some of the white-robed ones come in this direction even now,\"\nresponded the jarl. \"Behind them are spearmen. They must not find us\nupon this hill, but the woods are overdark to march in.\"\n\n\"After we are well covered,\" said Olaf, \"we may kindle torches, but\nthe way by which I lead you is plain and wide, for the war chariots of\nthe British kings have made it in the old days. The Romans now prevent\nthem from having any chariots within their dominions, but there are\nfree tribes beyond their borders. Come!\"\n\n\"On!\" said the jarl. \"This hill was to have been their trap. They seek\nto march around that they may cut off our going. On!\"\n\nSwiftly marched the Saxons for a while, but the darkness of the forest\nwas dense, and now they halted to kindle torches.\n\n\"The Druids and their men carried many and bright ones,\" said Ulric,\n\"so that we saw them enter the woods, but we are too far now for them\nto discern our own.\"\n\nAfter this there were pauses for resting, but the vikings marched on\nuntil the dawn. Then went they forward again, fasting, but at the noon\nthey were greeted by the shouts of the men who held the palisades at\nthe spring.\n\n\"O Tostig the Red,\" responded the jarl, \"hath all been well with thee\nand with the camp?\"\n\n\"Hael, O jarl!\" said Tostig. \"All is well. We have seen Britons at a\ndistance among the trees, but none came near for speech. I think they\nare not overfriendly.\"\n\n\"That are they not, but treacherous,\" said Ulric. \"But now let there be\nroasting and eating and sleeping, and then we shall have new matters\nupon our hands. We have seen things that are worth telling around a\nfire in the winter evenings. I like not these gods of the Britons. They\nare evil-minded.\"\n\nMany were busy at the fires with venison and with fishes which had been\ncaught, but they who had remained at the camp were cooks for the weary\nmen who could tell of this sacrifice of the Druids. As for the jarl,\nhe ate and drank and then he went on board _The Sword_ and lay down to\nsleep upon the after deck, saying little to any man, and Tostig the Red\ncame and sat down by him.\n\nOrders had been given, moreover, and before the setting of the sun both\nkeels were anchored some fathoms out from low-water mark, and only the\nsmall boats were at the beach. It was best, the jarl had said, to trust\ndeep water rather than a stockade after the darkness should come. All\nthe fires in the camp were heaped to burn long, and so were other large\nfires upon the strand. Then came all the vikings on board the ship,\nand there could be no present peril. It was a night of peace, but the\nwatchers saw both dark forms and white ones by the light of the fires,\nand knew that the Britons had come.\n\n\"The white ones are the Druids,\" said Wulf the Skater to his\ncompanions. \"I am not afraid of their gods which have men roasted. I\nhope the jarl will find us a chance to spear priests before we sail\naway from this island.\"\n\nThe rest agreed with him, asking him many questions concerning the\nsacrifices.\n\n\"But for the prudence of the jarl,\" he also told them, \"all we who went\nwould have been taken at a disadvantage in the darkness of the forest.\nThere would have been no fair fighting.\"\n\n\"He is a good battle jarl,\" they said, but it might be seen that among\nthem were some who were not well pleased with his ways.\n\nThere, safe from all assailing, floated the two keels until the dawn.\nThen went some of the men ashore in the small boats, and the fires were\nreplenished for cooking, but none were permitted to wander into the\nwoods. On board the trireme there was much search going on and great\nwas the delight of all over the plunder discovered. Rich indeed was the\nstore of arms, as if it had been intended to refit a cohort or to arm\nnew recruits.\n\n\"It is good, too,\" they said, \"to be able to walk around. There was\nhardly elbow-room on our own keel. But we knew that we must lose some\nand that there would be less crowding when we came home.\"\n\n\"We can give a man to every oar of the trireme,\" said Ulric, \"and yet\nleave threescore to the spears.\"\n\nBut he looked over the bulwark and down into the good ship _The Sword_,\nand his heart smote him sadly, for the very wood she was made of came\nfrom his own trees, and she seemed to look him in the face kindly.\n\nHours went by before there were any newcomers upon the shore, but Olaf\nsaid that there must be patience.\n\n\"Watch also,\" he warned Ulric, \"and let not any Briton come on board.\nWe will meet them in the small boats at the strand.\"\n\nSo it came to be, for at the noon the woods became alive with men.\nForemost came the chief Druid, followed by some of lesser rank and\nby harpers. With them were chiefs of clans of the Britons, each one\ncalling himself a king, but being really less than a Norse jarl in\npower, for he was as a slave to all Druids.\n\n\"These,\" told Olaf, \"make the laws and enforce them. They alone know\nthe sagas of the Britons and what is to be given to the gods. They\nsometimes burn a king if he worketh not their will, and they have magic\narts which make the people fear them. I would slay all such if I were a\nking.\"\n\nHe and Ulric were in the same boat pulling to the strand; and the\nchief Druid was wise, for he came to meet them attended only by two\nother Druids and by seven of his harpers. Behind them under the trees\nclustered the British warriors. They formed no ranks, but they wore\na fierce, warlike appearance. Among them were some in armor that was\nhalf Roman, as if taken in battle. More had Roman swords, but their own\nBritish blades were both short and light. All were armed with javelins,\nbut their shields were of all sorts, only that most of them were made\nof wicker and hide.\n\n\"They are brave enough,\" said Olaf, \"but the Romans seek to prevent\nthem from getting weapons. A Briton might become as good a soldier as\na legionary, with arms and with training. C\u00e6sar is always cunning in\ngovernment.\"\n\n\"Hael, O Druid!\" shouted Ulric. \"I am well pleased to see thee.\"\n\n\"O thou, the jarl of the vikings,\" sternly responded the chief Druid.\n\"Too many came with thee. My permission was but to thee and to Olaf.\nNeither didst thou do reverence to my gods.\"\n\n\"O priest,\" said the jarl, \"I came and I returned as I would. I like\nnot thy gods. What is thy errand with me this day?\"\n\nThe face of Ulric had flushed hotly upon hearing the haughty speech of\nthe Druid, for he was not one to be lightly chidden by any man.\n\n\"O jarl,\" said the Druid yet more sternly, \"I have this also against\nthee, that thou didst promise me a treasure the like of which I never\nsaw before, and thou didst not deliver it. Where is thy great gift?\"\n\n\"O Knud the Bear,\" shouted Ulric, \"row now to the shore and bring to\nthis priest the token of the son of Odin.\"\n\nThe second of the small boats came to the shore and Knud and eight\nother of the tallest vikings, ax in hand, bore out and spread upon the\nearth the tremendous hide of the white bear, the king of bears. From\nthe skull, also, they had reft its whole cover, putting in eyes of\nbright leather. The hide seemed to be longer and broader than in life,\nas if it lay two fathoms from tail to nose.\n\n\"O jarl of the Saxons,\" exclaimed the Druid, \"what is this? I have\nheard of these creatures, but never have I seen one.\"\n\n\"Then have I kept my promise,\" said Ulric. \"Thou mayest hang it in thy\nhouse or in the house of thy gods, as thou wilt, but never was the like\nof it in Britain. He was a son of the ice king. He came from the long\ndarkness, and I slew him with my own hand.\"\n\nAround the jarl stood now a score of vikings; terrible men for a foe\nto look upon, for they were throwers of sudden spears. Still stood the\nchief Druid and his fellows and the harpers, gazing at the great skin,\nand the Britons in the edge of the wood shouted loudly.\n\n\"I agree with thee as to this,\" said the high priest, reluctantly. \"I\naccept thy token, for in it is a meaning that thou knowest not. There\nis an old prophecy concerning the Northern Bear and Britain. Thou hast\ndone well. My quarrel is now with Olaf, who standeth by thee.\"\n\n\"But for him thou wouldst have slain me and mine in thy forest trap on\nthe hill, at the sacrifices,\" answered the jarl, angrily. \"Thy quarrel\nis also with me!\"\n\nThen came the rush of the Britons from the woods, hurling javelins as\nthey came, but the vikings were instantly in their boats, and the high\npriest and all who were with him lay upon the sand, so suddenly were\nthey smitten. From the ships came showers of spears, arrows, stones,\nand the men in the small boats seemed to be unharmed, for their shields\nwere up.\n\n\"Thou sittest very still,\" said Ulric to Olaf. \"What sayest thou? Mine\neyes were upon these blue ones.\"\n\n\"O jarl,\" said Knud the Bear, \"we lifted him in, thinking there might\nstill be life in him, but there is none. The spear of the high priest\nwas strongly driven.\"\n\n\"Hael to thee, O hero!\" shouted the jarl. \"Olaf, the son of Hakon, hath\ngone to Valhalla! He hath died in his armor! Row to the ships. We will\ngo hence and the body of Olaf we will bury in the sea. There shall be\nno lamenting for the son of Hakon.\"\n\nOnly this harm had befallen the Saxons from the treachery of the\nDruids, while the slain lying upon the beach were many. Loudly now\narose the wailing of the Britons, for they had a strange death cry of\ntheir own, long and vibrating, that went far out across the sea.\n\n\"Their gods will be against us,\" said Wulf the Skater. \"We may not now\nlinger long in Britain.\"\n\n\"Very soon,\" said the jarl, \"we will sail for the Middle Sea, but not\nwith two keels. We are too few.\"\n\n_The Sword_ and the trireme, nevertheless, were now going out to sea\nwith all oars, as if to show how many men were needed for this. The\njarl was at the helm of the trireme and his face was clouded.\n\n\"Not yet,\" he said, \"have I smitten the Romans upon the land of\nBritain. That must I do, and I know not how or where. The days go by\nand it will be winter before we reach the Middle Sea. The voyage is\nlong.\"\n\n\n\n\n                              CHAPTER XI.\n\n                     THE PASSING OF LARS THE OLD.\n\n\nSudden is the change from winter to summer in the Northland. The buds\nof the trees get ready under the frost and open to the sunshine as\nsoon as a few days of warmth have told them that they may safely burst\nforth. No full leaves were as yet, but the grass was greening and the\nfisher boats were busy in the fiords.\n\nIn the hall of the house of Brander there were fewer to gather now, in\nthe lengthening evenings, around the central fire, but Oswald's harp\nwas always there. Hilda, from her chair, would often ask him to strike\nup, but there was a lack of spirit in his minstrelsy, and even when she\nspoke to him her voice was weaker and softer than of old. The wrinkles\nupon her face were deepening, and they who looked long at her said to\none another that a light which did not come from the fire played now\nand then across her forehead and around her mouth. At other times she\nwas shut up much in her own room, and it was said that she pored long\nand thoughtfully over polished sheepskins and fragments of gray stone\nwhereon were graven runes that none else might read. Some of these,\nthey said, had been brought by Odin's men when they journeyed from the\nEast into the Northland. Who knew, therefore, but what the runes had\nbeen written in the city of Asgard by the hands of the Asas? It was\nnot well to question over-closely about such things. They said naught\nto her of the matters which were her own, and only once did a little\nmaiden yield to her own curiosity and follow the old saga woman when\nat night she walked out along the path which led to the stones of the\nmighty dead. Afterward she told her mother, and then all the village\nknew, that Hilda did but sit down by the tomb of Brander, weeping\nloudly and talking with him concerning his absent son.\n\n\"It is no wonder,\" said the villagers, \"for she loved Ulric the\nJarl. It is good for all our men that Hilda should speak to the gods\nconcerning their welfare. She knoweth them better than we do, and she\nis to go to them soon. She getteth ready daily.\"\n\nSo fared it in the Northland, but many ships were putting to sea, and\nthere was even jealousy here and there that Ulric and _The Sword_\nshould have gotten away so much in advance of all others. But the ships\nof the vikings would now be so many as to bode ill for the fleets of\nRome and for the merchantmen of the Middle Sea unless C\u00e6sar should send\nforce enough to prevent their coming.\n\n\"Olaf told me,\" said Ulric, talking to Tostig of such matters, \"that\nthe Romans fear the coming of the Saxons. Therefore against our\nvillages as well as against the rebellious Druids came these triremes\nat this time. C\u00e6sar's power in Britain groweth. Around his fortified\ncamps are cities springing up, and he fortifieth also ancient towns.\nWe must come with many keels and a great host when we take this island\naway from C\u00e6sar.\"\n\n\"But I think we will destroy the Britons,\" said Tostig the Red, \"for we\nhave seen that we may not trust them. I like a place where there is so\nmuch good hunting.\"\n\nUlric had been scanning the shore line, for he was steering, and now he\nsaid:\n\n\"We will anchor for the night within yonder rocky point. There is a\nledge there for which I have been seeking.\"\n\nAll day had the two ships been coasting slowly, and the men had\nwondered much what it might be that was in the mind of their jarl, for\nhe was moody. He had also asked many questions of the older vikings.\nThe two ships came to anchor not many fathoms out from the rocky point,\nbut all men were forbidden venturing to the shore.\n\n\"It is not well,\" said Ulric to some who would have landed in the small\nboats. \"If ye but look closely, ye will discern the glimmer of fires\nin the deep forest. Our movement this day hath been followed, and now\na small party might meet too many of their spearmen. They are good\nfighters.\"\n\nThere was much grumbling among the younger men, for they despised this\nprudence of his which ever held them in and thwarted their hot wills,\nbut they had no choice but to obey him concerning the boats.\n\nMore and more plainly through the night darkness might the watchers on\nthe decks discern the fires that were kindled in the woods. The jarl\ngazed at them long, thinking many things concerning the Druids and the\nother Saxon villages of the shore of Britain. He slept after a while to\nthe slow rocking of the ship, and when morn came Wulf the Skater stood\nby him.\n\n\"O jarl,\" he said, \"the Britons build fires along the beach. They swim\nout to us. I have speared four of their swimmers. What do we next?\"\n\nUlric arose and gave orders. Immediately a transfer began from _The\nSword_ to the trireme of all arms and provisions, and the men worked\nrapidly. Only that Wulf worked not, and that an old viking came and\nstood by him at the bulwark.\n\n\"I like it not,\" said Wulf, \"but Ulric is jarl. What sayest thou, Lars\nthe Old, the shipmaker?\"\n\n\"Thou art a seaman,\" said Lars. \"I am of thy mind. I toiled much in the\nshaping and the making of _The Sword_. My heart is heavy.\"\n\n\"So is mine!\" exclaimed Wulf. \"First of all men, after the jarl, did I\ntake her helm. She is Odin's keel. There is bad fortune in leaving her.\"\n\n\"That do I fear,\" said Lars, \"but I leave her not. I was sore smitten\nin the ribs in the fight with the Druids on the beach. I bleed well\nnow. I shall not sail in this trireme.\"\n\n\"Good is thy fate,\" said Wulf. \"Didst thou tell the jarl thou wert\nwounded?\"\n\n\"Not so,\" replied Lars. \"None know but a few of our old vikings. I\nthought not much of it at first, for I have oft been wounded. But now\nthey will soon burn _The Sword_. I command thee that thou lay me upon\nthe fore deck, where was once the hammer of Thor. That is my death\nplace.\"\n\n\"That will I do,\" said Wulf. \"So will say the jarl.\"\n\n\"So do I now say!\" came to them in his own voice, for he also was\nleaning over the rail and he had heard. \"O Lars, I knew not of thy\nhurt, thinking only of Olaf, the son of Hakon. Him have we buried in\nthe sea this day, and thou shalt have thy will. _The Sword_ is nearly\nemptied. We burn her on yonder rocks at the point as the tide falleth.\nWe will lay thee upon her fore deck with thy arms and armor.\"\n\n\"Do thou thy duty by me,\" said Lars, \"that it may be well with thee.\nBut leave not _The Sword_ until every timber shall be burned, lest some\npart of her shall fall into an enemy's hand.\"\n\n\"She is ready!\" exclaimed Ulric. \"We will lift the anchors and move\nboth ships. There will be many to see the burning.\"\n\nTrumpetings and harpings and angry shouts were answering from a throng\nof Britons gathering along the shore. Not any of them could guess as\nyet what would be the next move of the Saxons, but great was their\nwrath that they were able to do no harm.\n\n\"They would we might find reason for landing,\" said Ulric to Wulf, \"but\nI care not to strike them at this place. We would gain nothing.\"\n\n\"O jarl,\" said Wulf, \"Lars, the shipmaker, lieth down. The valkyrias\nare with him.\"\n\n\"He dieth not a cow's death,\" said Ulric, \"but as a true warrior of the\nNorth. It is as he would will, but he still is breathing.\"\n\n\"Yea, but heavily,\" said Wulf. \"I would I were as he is, that I might\nnot leave _The Sword_.\"\n\n\"O Wulf,\" said the jarl, \"thou hast many a feast of swords before thee.\nCheer thee up.\"\n\n\"Jarl Ulric,\" said Wulf, \"do I not know thee? Thou too lovest thy first\nkeel. But I think thou doest wisely. The men have demanded this, and\nthey may not be gainsaid. But I would there had been men enough for\nboth ships, and then I would not have left mine own.\"\n\nOn moved the two keels toward the ledge of rocks, and the tide was\nfalling. They would be bare before long.\n\n\"Row, now!\" shouted the jarl. \"Send _The Sword_ far up upon the ledge.\nShe must be lifted by the rocks till she is out of the water. There\ncome the Britons toward the point. Be ready to strike them! The Druids\nhave gathered an army!\"\n\nNo sail was raised upon either of the ships, but the rowers of the\ntrireme paused while those of _The Sword_ pulled strongly. She was\nlight now, having no stowage or ballast, and quickly her prow was\nthrust high up the ledge between two masses of dark gray stone. Then\nthe trireme was grappled at her stern and many Saxons sprang out upon\nthe ledge. There were several fathoms of water between this and the\nshore.\n\n\"Fast falleth the tide,\" said Ulric. \"Lift ye now Lars the Old, the\nshipmaker, and bear him to the fore deck of _The Sword_. Lay by him his\narms and his armor, breaking the sword and the spear and cleaving the\nshield and mail that no other may ever bear them.\"\n\nThe vikings carried the old warrior quickly, and he uttered no sound.\nThey laid him upon the fore deck and did as Ulric commanded, but the\nhilt of the broken sword, having yet half the length of its bright\nblade, they put into his right hand. In the middle of the ship much\nwood was placed, heaping it, and in this heap a blazing torch was\nthrust. Then all the vikings left _The Sword_, and the greater part of\nher was already out of water.\n\n\"They come in swarms!\" exclaimed Tostig the Red, gazing at the Britons\nwho rushed along the shore toward the point. \"Hael! the fire burneth\nwell! They must not prevent it!\"\n\nUp leaped the long-armed flames, catching the fagots of pine splinters.\n\n\"Burn thou, O _Sword_!\" shouted the jarl. \"I give thee to Odin in the\nfire! Thou art mine own, O good ship from the Northland. I would I\nmight have sailed in thee to the Middle Sea and to the city of the\ngods!\"\n\n\"O jarl,\" said Wulf the Skater, \"even so would I have sailed. I think\nwe shall never see that city. The gods are far away, and I know not if\nthey have any city. I am dark this day, and over me is a cloud.\"\n\nThe jarl spoke not again, but he looked earnestly at _The Sword_ and at\nthat which was threatening along the shore. Still as a stone lay Lars\nthe Old, and some men thought him dead. There were Druids now at the\npoint, and with them were harpers and trumpeters, and the white-robed\nones were chanting to their gods.\n\nThe chanting ceased and a Druid raised his sacred wand, shouting\nfiercely. At that word hundreds of armed Britons began to rush into the\nsea.\n\n\"They are too many,\" said Knud the Bear. \"They do but drown each other.\nThese Druids are not good captains. Therefore are they beaten by the\nRomans in spite of their gods and their sacrifices.\"\n\nThe fire ran everywhere along the bulwarks of _The Sword_ and began to\nclimb over the decks. It climbed the high mast and the wind blew it out\nlike a banner.\n\n\"Odin!\" shouted Ulric. \"The Britons are on the rocks! Smite now!\"\n\nFast flew the arrows and the spears, and almost useless were the\nwicker shields of the Britons. Many of them had none, and their blue\nbodies were plain marks for shaft and stone. They fell in heaps upon\nthe ledge, but a score of them broke through the flames to the very\nfore deck of _The Sword_, and here too the fire was blazing hotly.\nHere before them lay Lars the Old, stretched out as on his funeral\npyre. These were of the best armored of the Britons, and one could\nunderstand that they had thought to take _The Sword_ and push her off,\nthat by her means they might reach the trireme.\n\nNo good captain would so have planned, for such a thing might not be\ndone; but these men were brave, for they stood well and some of them\nhurled their darts vigorously at the vikings, while others strove\nvainly to shove _The Sword_ from the rocks into the sea.\n\nThis thing that came not any man had expected. Just as the strong fire\nin the cabin began to burst up redly through the fore deck, and a\nfiercer flame mounted the after deck, and all the bulwarks were ablaze,\nup to his feet sprang Lars the Old, his gray hair streaming in the\nwind. One blow he struck with his broken sword, burying it in the body\nof a British chief, and then he began to ply his long-handled ax with\nthe strength of one who is dying. Upon him turned the spears and the\nswords of the Britons and he was stricken quickly. He did not shout,\nbut he cleft one more while falling.\n\n\"The hero dieth!\" said the jarl, hurling his spear, and it flew well,\nbut there were not many now upon the fore deck.\n\nMore were swimming from the shore to the ledge, but the fire was\ncompleting its work, and the plan of the Druids was broken altogether.\nWhen once more the wind put aside the black curtain of the smoke it was\nseen that the entire prow had fallen in and that to the very helm the\nflames were fighting joyously.\n\n\"We will stay by until she is burned to her keel,\" said Ulric; \"but now\npull out a little further.\"\n\nSo did they, and the Britons came no more to the ledge, for the prize\nthey had hoped for was a heap of ashes upon the rocks.\n\n\"A good ship was she,\" muttered Knud the Bear. \"She fought well against\nthe ice floes and the storms. May all the gods go with us in this\ntrireme. I would I knew her by some name.\"\n\n\"O Knud,\" loudly responded Ulric, \"I will answer thee. This keel that\nwas Roman hath become Saxon, and her name is now _The Sword_. Else\nwe had not burned the other. The trireme shall be to us as if we had\nbuilded her on the shore of the Northland. She will sail with the\nhammer of Thor and the flag of Odin and not with a Roman god.\"\n\n\"I am better satisfied,\" exclaimed Wulf the Skater. \"But many good\nrowers must take the oars of this trireme in battle. She is heavy.\"\n\n\"I think,\" said Tostig the Red, \"that we are stronger than are the\nhired rowers or the slave rowers of the Romans. Her beak will break the\nribs of another keel and she will do well in storms.\"\n\nThe jarl's eyes were still upon the burning timbers which remained upon\nthe ledge.\n\n\"I will take a boat,\" he said, \"and men with me. We must gather all\nfragments for utter destruction.\"\n\nUpon that duty he went, and it was made complete before the small boat\nreturned to the trireme. All the while many Britons watched them from\nthe shore, but came not against them.\n\n\"Too many of them have been slain,\" said the vikings. \"They like not\nour heavy spears.\"\n\nBefore climbing into the trireme the jarl made them row to her beak,\nthat he might examine well its form and its power for striking a blow,\nand that he might also look more closely at the figurehead.\n\n\"It is much waterworn,\" he said. \"She is the wise woman among the gods\nof the Romans. She will care not much that the hammer of Thor is on the\nfore deck.\"\n\nThe small boat was hoisted to its place and the vikings began to speak\nmore freely of the trireme by her new name of _The Sword_.\n\n\"Up with the sails,\" commanded Ulric. \"The wind is fair. We will go\nsouthward this night, and we will seek the Saxon village that was\ndescribed to me by Olaf, the son of Hakon. But we will not go too fast\nor too far, lest we may pass it in the dark.\"\n\n\"There may be our kinsmen there that need our aid,\" said Knud the Bear.\n\"Seax in hand it would be a pleasure to meet Romans.\"\n\nNow did they begin to discover how much more room there was to walk in\nfrom place to place around the ship, but the younger men praised their\nown prudence for this more than that of Ulric the Jarl. Moreover, to\nplease all, he caused to be brought forth many weapons and much armor.\nThese the men handled curiously, trying on the helmets and the mail and\ntesting the weight of the shields. Garments, also, were given as the\nmen would, and they laughed merrily at each other for the strangeness\nof their changed appearance.\n\nWell out from the land steered the jarl, not knowing the coast, and\nthere was careful watching for breakers which might tell of shoals or\nrocks. He was learning, also, the sailing of this keel and her manner\nof answering the rudder.\n\n\"She is swift,\" he thought, \"and she rideth well the waves. We build\nnot yet such vessels in the Northland, though we have plenty of good\ntimber. She will carry us safely into the Middle Sea, but there is\nroom in her for more men. She requireth too many for her oars. I will\nsail rather than row, lest I breed too much discontent.\"\n\nFar behind him now went out the last burning of the timbers of the good\nkeel he had builded in the Northland, but upon the mast of this which\ncarried him floated still the White Horse flag of the Saxons which had\nbeen given to _The Sword_ by Hilda of the hundred years.\n\n\n\n\n                             CHAPTER XII.\n\n                        SVEIN THE CUNNING JARL.\n\n\nSailing on in the darkness, over an unknown sea, the trireme, which was\nnow the viking ship _The Sword_, moved toward the dawn. None on board\nof her knew the low-lying coast which was in sight when the sun looked\nover the horizon.\n\n\"We are nearer than I deemed,\" said Ulric; but he was at the prow now,\nand an old Danish seaman was at the helm.\n\n\"There are rocks hereaway at the right,\" replied Tostig the Red, \"but\nI can see houses and lines of palisades. The Britons build not such\nhouses. They are like our own.\"\n\n\"There are fields, also, and cattle,\" said Knud the Bear. \"There are\nmen on the beach. Let us sail in. Hark! War horns! We are waited for.\"\n\n\"It is a good harbor,\" said Ulric. \"There are four keels on the strand,\nbut they are small. And there are boats. These are not Romans.\"\n\n\"They will deem that we are,\" said Tostig. \"Thy horn, O jarl.\"\n\n\"Not yet,\" said Ulric. \"We will go nearer. All rowers to the oars! Let\ndown the sail!\"\n\nThen came a surprise to those who were on _The Sword_, so very numerous\nwere the warriors who came down to the shore outside of the lines of\nthe palisades on the harbor side of the village. This, too, was seen\nto be larger as they drew nearer, and some of the houses were as great\nas was the home house of Brander the Brave.\n\n\"It is as Olaf told me,\" thought Ulric. \"The Romans do well to fear the\nSaxons of this coast. We will be friends with these men.\"\n\nThe rowers had brought the ship well in and Ulric stood by the hammer\nof Thor. Three times be blew his horn, standing bareheaded, nor was\nthere any Roman helmet worn by those who were with him. Moreover, the\nbanner on the mast was the White Horse of the Saxons.\n\nHorns answered him, and then there were shouts of greeting, while some\nof the shore men pushed out in a small boat.\n\n\"Come near!\" said Ulric to these. \"I am Ulric the Jarl, the son of\nBrander the Brave. We come in peace. Who are ye?\"\n\nUpon his feet arose a short, squarely made man in the boat. He wore\nfine armor and there was a golden crest upon his steel headpiece.\n\n\"I am Svein Jarl,\" he responded. \"We are Saxons all, and this town on\nthe shore is Rika. Where didst thou win thy keel? I tell thee we are at\npeace with the Romans, as we are with thee.\"\n\n\"So be it,\" said Ulric; but then he told of Olaf and of the Druids and\nof the triremes and of the Roman camp.\n\n\"Strong tryst between me and thee,\" said Svein. \"Thou hast done well.\nOlaf would never make peace because they slew his father, as did they\nthine. They would crucify thee because of thy trireme. But word came\nto me that the Roman consul Licinius is in Britain, and I have sent\nhim bodes, making agreement. We are at war only with the rebellious\nBritons, not with his own. We are too few to contend with Rome. Land\nthou and thine if thou wilt, but see that thou sailest away quickly.\"\n\n\"I understand thee,\" said Ulric. \"I am but one trireme against more\nthan one if the consul sendeth them. But we will not land here. I will\ngo to thy house in greeting, but no more.\"\n\n\"Come,\" said Svein. \"I like thy flag, and I was thy father's sure\ncomrade. The son of Brander is welcome to the house of Svein Jarl.\"\n\nSmall boats from the ship were ready, and in one went Ulric to the\nshore, taking with him many men in the other boats, for he thought: \"I\nknow not Svein well, and Olaf spoke ill of him. He is a friend of the\nRomans.\"\n\nSo said the vikings who remained on the ship, and they kept good watch,\nsaying to one another:\n\n\"We like it not that our jarl should thus venture himself. How know we\nwhat is behind yonder palisades?\"\n\nHearty and kindly were the words spoken to Ulric and his Saxons by the\nwarriors who met them at the beach. Neither did Svein seem to lack in\nany wise, but walked on toward the palisades, bidding the newcomers to\nfollow. At the side of Ulric the Jarl now walked a tall man and large,\nin full armor, but wearing over his shoulder a bearskin.\n\n\"I am Sigurd, the son of Thorolf,\" he said. \"I am a Northman, like\nthyself. The greater part of Svein's men are Danes, as he is. I am not\nwith him, save that my keel was wrecked and I owe him for hospitality.\nBut I am free, having fought for him against the Britons.\"\n\n\"Sail thou with me,\" said Ulric. \"There is room in _The Sword_. Share\nthou fight and prizes by land and sea. Thou art welcome.\"\n\n\"I will put my hands in thine and be thy man,\" said Sigurd. \"Mark thou\nthis, then. When we pass the gate of the palisades many will come and\nrange themselves with thee and me, for they are as I am and would\ndepart from this place. Thou hast thine ax. Be thou ready to smite with\nit, as will I and mine.\"\n\nThen those who looked upon the face of Ulric saw that it became white\nand that his eyes were fiery, flashing blue light, and they thought,\nbut spoke not. \"The jarl is angry! Trouble cometh. We will watch if\nthis is a place of swords.\"\n\nThen again they looked and he seemed taller and his face was red and\nhis eyes were full of glittering, and some trembled, for they said each\nto his mate: \"Seest thou? It is the Odin wrath! Lift thy shield! War\ncometh!\"\n\nOpen swung a wide gate in the palisades and Svein marched in, turning\nto beckon, while many warriors closed in line with the company of Ulric\nand his Saxons; but there were others who remained behind and prevented\nsome from closing the gate. Even as Sigurd had said, when he lifted\nhis hand and made a sign forty and four more who were among Svein's\ngarrison walked along, spear in hand, until they seemed of one band\nwith Ulric's.\n\nBut a sound came loudly, and then another--and another.\n\nSvein stood still and blew upon his war horn, and it was a command to\nhis Danes that they should form as spearmen. From behind a wide house\nrang joyously the note of a Roman trumpet, and a line of legionaries,\nheaded by an officer, began to show itself. The third sound was the\nangry word of Ulric, the son of Brander.\n\n\"Svein Jarl,\" he shouted, \"I know thee. Thou art Svein, son of Hedrig,\nmy father's enemy. Me thou wouldst betray to these wolves of Rome, but\nthou art not able. I will give thee and them to the valkyrias.\"\n\n\"Hold thou, Ulric the Jarl,\" said Svein. \"Thou art caught in a trap.\nThou shalt but give them up their trireme. Thou mayest remain with me.\nLay down thy weapons. Thou and thine are prisoners. We may deal with\nthee as we will.\"\n\nSo said the officer of the legionaries, mockingly, coming forward,\nfollowed by his force. It was but fourscore of men, and they were the\ngarrison of this village, with Svein and his Danes and his Jutlanders.\n\nBut Ulric was a good captain, and he and his Saxons were stepping\nbackward and the gate was still open. Then fell quickly three men who\nstrove to shut it, but they went down by the spears of Sigurd's Saxons.\n\nAt that the Romans charged, and their charge was that of warriors\nexpecting to conquer; but Ulric, the son of Brander, was taller by the\nhead than any among them. He waited not, but stepped out and met them\nin front of the triangle formed by his men, and the flashing of his ax\nwas like the swiftness of the lightning, and his wrath was terrible.\nFast flew the spears on either side, but the Saxons threw first, not\nwaiting, and there were quickly gaps in the Roman line.\n\nNow charged Svein and his followers with shouts of victory, save that a\nnumber of them were Northmen and had no heart to this work. These fell\nback muttering, and one of them said, loudly:\n\n\"Ulric, son of Odin, win thou this fight. The gods of the North be with\nthee. I shed no blood in any such quarrel. I am not a Roman.\"\n\nNevertheless the Saxons from _The Sword_ had been too much outnumbered\nif it had not been for Sigurd and his sailors, for these fought like\nmen who were to die if they did not conquer.\n\nWonderful was the havoc wrought by the ax of Ulric, and the Romans\nfell away from before him. Then picked he up a pilum from the hand\nof a slain legionary and he cast it with his might. Well had it been\nfor Svein the Jarl if his shield had been ready, for the pilum passed\nthrough him at the waist and he would betray no more Saxons. So fell\nthe Roman officer at the hand of Tostig, but the charge had been well\nmade, and only half of Ulric's own men were with him when his triangle\nwas beyond the gate, marching to the shore.\n\n\"Odin!\" he shouted. \"We have slain three for one! Let us burn their\nkeels.\"\n\nBut some of the men who had refused to fight for Svein came around\nby another way and joined the Saxons. Well was it, they said, that\nthe Roman officer had forced Svein to strike at once, for there were\nhundreds of Danish warriors in the upland, and if these had gathered,\nnone of the crew of _The Sword_ could have escaped.\n\nEven now there was preparation for swift following, but Ulric's men\ntook every boat, and the nearest keels on the beach had already fire in\nthem, put there by Sigurd's men and the other Northmen who had deserted\nSvein. These ships were also pushed out into the water that they might\nburn more surely.\n\nWithin the palisades every Saxon who had fallen wounded had already\nbeen slain by the Danes, but these had been sorely smitten and they had\nlost their cunning jarl.\n\nBack now were Ulric and his men on board the trireme, and count was\nmade. \"Thirteen heroes who went to the land with us,\" he said, \"have\ngone to Valhalla. With them went six of Sigurd's company. Therefore,\nwe have ninety more strong men to handle so large a ship and to hold\nspears in battle. The gods are with us, for they have given us a brave\ncombat and a victory.\"\n\nThe keels from the shore were burning hotly, and there might be no\npursuit, but Ulric commanded to lift the sail of _The Sword_, the\ntrireme, and to steer for the open sea.\n\n\"Now do I know,\" said Knud the Bear, \"that Thor came on board with\nhis hammer. We needed more men for the oars, to change hands when one\ncompany is weary. It is good to have the gods with us in such a case.\"\n\nThe wind blew off the land and the ship sailed away gallantly, steering\nsouthward, and Ulric said to those who asked him:\n\n\"We will not again set foot upon the shore of Britain. Our work here is\ndone. We will avoid all keels, friend or foe, that may come near us. We\ngo to the Middle Sea, and our voyage, thus far, is prosperous.\"\n\n                   *       *       *       *       *\n\nThe sun shone brightly in the Northland all that day, but Hilda sat by\nthe fire in the hall of the house of Brander, and she was shivering.\nNear her sat Oswald, the harper.\n\n\"It is cold,\" she said. \"This fire is but red coals and ashes. Let them\nbring wood.\"\n\nSo sat she while they went for wood, and she gazed mournfully into the\ngreat heap of gray and red, dotted with dying embers.\n\n\"I saw not the ship,\" she muttered. \"But I saw Roman helmets. There\nis Ulric, and the Romans go down before him. Where is the ship? I see\nher now, and she is burning. How, then, can Ulric sail away? I read it\nnot, save that he is not slain. O that I could look upon his face again\nbefore I go! How is it that I cannot see the ship? But I knew that she\nwould never come again. It is well that he hath smitten the Romans so\nsoon. I will go to my room, for I am old and the ice is out of the\nfiords and the buds are open and I have seen the grass again. I need\nbut the one token more and then they may lay me away as I have bidden\nthem. Ulric, my beloved! Thou art as my son!\"\n\n\n\n\n                             CHAPTER XIII.\n\n                      HILDA OF THE HUNDRED YEARS.\n\n\n\"Hast thou ever taken a keel into the Middle Sea, O Sigurd, son of\nThorold?\" asked Ulric of his gigantic friend.\n\nThey twain stood together upon the after deck and _The Sword_ was\nsailing but slowly, for the wind was contrary.\n\n\"More than once, O jarl,\" responded Sigurd. \"I have seen the Greek\nislands; I went up the Adriatic Sea with Alfkel the Sea King. We had\nfive keels, and we took great spoil, but only three of our ships ever\nagain touched the shore of the Northland.\"\n\n\"What befell the two that returned not?\" asked the jarl. \"Was it a\nfortune of the sea?\"\n\n\"Not so,\" said Sigurd. \"In that sea the triremes of C\u00e6sar are too many.\nBut thou hast need to consider thy present course. Thou wilt do well to\ncoast along the land easterly after thy last sight of Britain. Between\nthese islands and Spain is a great sea full of storms. Try it not with\na straight passage, but go from point to point, going on shore when\nthou wilt.\"\n\n\"I think it is good counsel,\" said the jarl. \"I have heard of that sea.\nAs to the Adriatic, I would enter it in due season, but first I would\nsee Rome itself, if I might.\"\n\n\"Not if thou go to its port in a keel thou hast won from C\u00e6sar,\" said\nSigurd. \"That were but to offer them thy head. Thou wilt do better\namong the islands and toward the great land that is called Africa.\nThere dwell the black men, and in the inland there are giants wonderful\nto see; and also there are powerful magicians.\"\n\n\"I care not much for them,\" said Ulric, \"although I am curious about\ngiants. Tell me all thou wilt of thy voyages.\"\n\nWillingly did Sigurd tell, and he had seen many wonderful things in the\nsouthlands.\n\n\"I shall gladly see them again,\" he said; and even the next day did\nthis talk go on, for a gale blew and _The Sword_ went before it with\nbut one small sail lifted.\n\nSigurd's men were now as if they had been with Ulric from the first,\nand by them a matter had been told which was now more fully given by\nthe tall viking.\n\n\"Svein concealed it from me,\" he said, \"but an old Dane warned me in\nprivate. The Roman officer of that garrison was but waiting for the\narrival of more legionaries, for Svein's men might not be depended on\nin such an undertaking. It would have included thee and thine. All who\ndid not belong to Svein or who were minded to leave him were to have\nbeen given up as war prisoners to the Romans.\"\n\n\"That they might lose their heads!\" exclaimed Ulric. \"I am glad he is\nslain! It was a dark purpose.\"\n\n\"Thou hast not read it rightly, nevertheless,\" said Sigurd. \"Hast\nthou not heard of the great games and shows of C\u00e6sar and of his chief\nofficers?\"\n\n\"Many a thing have I heard,\" replied Ulric, \"but not from any man who\nhad ever witnessed the things he told of. Hast thou seen?\"\n\n\"No, Jarl Ulric,\" said Sigurd, \"but I have listened to brave men who\nhave looked in upon such things. As to one affair, we learned little\nby little that the proconsul of Britain desired good swordsmen to\ncontend with his trained slaves and with his wild beasts. It was also\nfor his profit to send Saxons as presents to C\u00e6sar to be slain in the\ngreat shows of Rome. For this purpose all we were to be entrapped and\ncaged as soon as their hunting party should become strong enough to\ntake us alive. We were to be set upon unawares. Therefore did we sleep\nby watches, fully armed, for the thing was to be done in the night. So\nwas the idea of Svein, the treacherous, concerning all thy crew.\"\n\n\"He will entrap no more Saxons henceforth,\" said the jarl. \"As for me,\nI would gladly fight a lion or a tiger. It would be great sport. I\nwill try if I may meet these wonderful beasts before I return to the\nNorthland.\n\n\"Thou wilt meet thy lion with full armor,\" responded Sigurd, \"but it\nis not so in these games of the Romans. There is no fair fighting.\nThey arm thee as they see fit. Often thou art not matched with one man\nor with one beast, but with odds, that they may see thee overcome or\ntorn. This is their delight concerning prisoners and malefactors which\ncost them little. They spare their dens of animals and their purchased\ngladiators that they may more cheaply see much blood. But there is\nworse than this among them, for they use the scourge upon us, and a man\nwould rather die ten times than be made to feel the stroke of a whip,\nas if he were a slave.\"\n\n\"If I were indeed lashed,\" growled Ulric, \"it were well for that man,\neven were he C\u00e6sar, not to come near me in after time if there were a\nblade within my reach. There might come a sure cast of a spear, and I\nthrow far.\"\n\n\"This scourging,\" said Sigurd, \"is to break the proud spirit of such\nas thou art. I think thine or mine would not be so destroyed, but\nrather a red fire kindled in ashes that would smolder for a time. But\nthey know us well, these Romans. A captive Saxon is chained as an\nuntameable wild beast until they push him out of his cage into the\narena.\"\n\n\"So slay we all Romans!\" exclaimed the jarl. \"We will count them but\nwolves. But I will see many other cities if I may not go to Rome. The\nwind changeth and I think a storm is upon us.\"\n\nSoon fiercely howled around them an angry north wind, tossing the\nsea in great surges, but the trireme proved herself stanch and well\nbehaved. She held on her way swiftly. Often saw they the land, but\nafter one night more Sigurd called Ulric to a bulwark, at the dawn, and\nhe pointed first westerly.\n\n\"Seest thou,\" he said, \"yonder high white cliffs? We are in the narrow\nsea between Britain and Gaul. We have been driven about too much and\nwe have expended days. Now we may drive southward and we may not meet\nother keels often. The Britons of Gaul are like those of the islands.\nThey are not sea-goers, and they are all under the rule of C\u00e6sar.\"\n\n\"We have no need to strike them,\" said the jarl. \"They are not our\nerrand. We will but sail on as we have planned. Thou hast taught me\nmany things. I thank thee.\"\n\nThe day went by and _The Sword_ drew near the land at times, but it\nwas better to keep well away from an unknown coast. All the crew were\npleased to discover how swiftly they might travel and how readily they\nmight turn so large a vessel.\n\n\"She will do well in battle,\" they said.\n\nAs to the three banks of oars, the jarl angered some by his urgency\nin compelling all to practice their use, that they might become well\nskilled.\n\n\"He is a hard master of a ship,\" said some. \"Do we not know what to do\nwith oars?\" The older men were better satisfied, and they also studied\nthe handling of a trireme.\n\nThe next day _The Sword_ was not far out from the westerly shore of\nGaul and a thing came to Ulric the Jarl as he stood upon the after\ndeck steering and watching the land. He was thinking deeply, also,\nconcerning the gods, and he was remembering those persons whom he had\nleft behind him in the Northland which was now so far away.\n\n\"What is this?\" arose a sudden inquiry in his mind. \"I am not alone! I\nthink that one sitteth by me. I have felt the touch of her hand upon my\nhair, stroking it. There hath been no voice, but the hand is the hand\nof a woman and I know it well of old. I will wait and see if she will\nspeak to me. I have hungered for speech with some whom I may not see. I\nthink that of the unseen ones there must be a great multitude and that\ntheir land must be wide, but no man knoweth what it may be like. In it\nis the city of Asgard. There is Valhalla, and there dwell the heroes\nfrom innumerable battles. I shall not ever be fully contented until I\nhear the valkyrias call my name. But first I would have speech with one\nof these strange gods of the southlands. The Grecians have many, and so\nhave the Romans. I have willed, also, to look upon the face of the god\nof the Jews, for he is said to be a strong one and very beautiful. O\nthou that touchest, I pray thee touch me again.\"\n\nThe wind went softly by him and there came a low whisper in his ear so\nthat he heard it thus, as if it had been a voice:\n\n\"Son of Odin, I have passed. Have passed.\"\n\nMore heard he not, nor did he see any, but at that hour there was a\ngreat silence in the house of Brander in the Northland.\n\nIn her chair sat Hilda, as she was wont, but she was very white, and\nher eyes were shut. Around her stood the household, save that Oswald,\nthe harper, sat with his head bowed upon his harp. Not many men were\nthere, and the women and the maidens did but look at one another and\nat Hilda, for they knew not whether she were living or dead, and they\nfeared to put hands upon her.\n\nThen opened she her eyes and her lips parted.\n\n\"I have seen him,\" she said, \"but the ship is not _The Sword_. I have\nbeen as if I were asleep, but it was no dream. Where my heart is there\nwas I, and I will go to him again. Now, when I sleep again, put ye my\nveil over my face. Let me not fall from my chair, but place me upon my\nbier and make ready to carry me to the cleft of the rocks. If it may\nbe, I will speak once more before I go.\"\n\nSo went she to sleep and they covered her face, but now the women\nwailed loudly and all the men of the household were sent for to come to\nthe hall.\n\n\"Hilda of the hundred winters hath seen the last outing of the ice,\"\nthe women said, \"and now the grass and the leaves have come. She goeth\ndown to her own and she will see the gods.\"\n\nA litter was made and they bore her to her room, for she had given the\nolder women instructions and they knew what to do in such a case. The\nhousehold men came, but they did not stay in the house, for Oswald\nspoke to them and they went out with him to the place of tombs.\n\nThe low hill on which were the standing stones had a face of broken\nrock seaward. In the middle of this face leaned a tall, flat stone, a\nslab of limestone, which had been worked to smoothness on its outer\nside. Upon this surface were many runes graven, in lines and columns,\nand some of them were like small pictures, and more were like letters\nof words that were to be read. The stone was exceedingly heavy, and\nstrong men worked with wooden levers to lay it aside without injuring\nit. When that was done there could be seen a chasm, as if the rock had\nbeen cloven to make an entrance for any who would go in. At this the\nmen looked, but as yet they kept their feet away from it.\n\nAll over the Northland there are such tombs as was this of the house\nof Brander the Brave, the sea king, and in them are the bones of the\nmighty. But in some, as in this, are not buried the heroes after whose\nnames the tombs are called, for they fell upon far-away battlefields or\nin fights at sea, but at their tombs were made sacrifices to the gods,\nnevertheless, and the songs to the dead are to be sung there by their\nkindred. If any man have a hero son, to this place must he come to\nspeak to his father and to the Asas, or he will be accounted nidering\nand unfit to be a jarl and a leader of men.\n\nLow had sunk the sun when a procession walked slowly away from the\nhouse of Brander. The men of best rank and name were proud to be\npermitted to bear the bier of Hilda, as if she had been a princess;\nfor she was of the race of Odin and she had talked with the gods for\na hundred years. Therefore, also, every man wore his full armor; but\nof the women there were some who carried goblets and pitchers which\nhad been Hilda's, of pottery and of bronze and of silver and of beaten\ngold. Others there were who carried her best garments, rending them as\nthey came.\n\n\"She is not to be burned,\" said Oswald. \"She is to be laid in the inner\ncrypt, with her feet toward the east. Her coffin is of wood, and it was\nin her room, but I have brought it. Let her be placed therein.\"\n\nIt was a long box made of planks of the fir tree, and it was large\nenough. In it did they lay the body of Hilda, taking it from the bier.\nThen the strong men bore it into the cleft of the rocks, but not many\nwere permitted to follow and see. Three fathoms deep was the cleft, and\nthen it widened, making a small room, and this could be seen well, for\nsome of the men bore torches. There were other coffins, and there were\nbones and skulls uncovered at the sides and in the corners. There were\nstones also, set up in the form of coffins, and in them were bones and\nmany good weapons, as if to each man had been given shield, ax, sword,\nspear, helmet, and mail, and vessels of pottery and of metal, with good\ngarments. But the arms and armor were for the greater part marred,\nbent, broken, and the garments were rent.\n\n\"Speak not,\" said Oswald, \"but put down the coffin of Hilda here. The\nrunes on the rock beside it were graven by herself for the memory of\nBrander the Brave, for she loved him well.\"\n\nIn the coffin were some things placed. Upon it was laid a plank of fir.\nOn this, then, and on the earth at the head were arranged all other\nmatters brought by the women. Every man walked out then except Oswald,\nand he stood still and spoke to Hilda, but she answered him not. Again\nhe spoke, calling her by name, and those without, in the cleft and\nbeyond it, heard him, and they listened well, but they heard no other\nvoice than his.\n\n\"Hilda!\" he said again. \"Hilda of the hundred winters, daughter of\nOdin, what sayest thou to Oswald, thy friend?\"\n\nThey heard no answer, but Oswald came forth and bade them place the\nstone.\n\n\"Set it well,\" he said, \"for it will not be moved again. The house of\nBrander is ended. There will be no other who will have the right to be\nburied behind the stone.\"\n\nNone answered him, but the women whispered sadly to one another: \"What\nof Ulric the Jarl?\"\n\nThe men followed Oswald to the house, for a feast had been prepared in\nhonor of the daughter of Odin, and the tables were set. Other harpers\nhad come, with chiefs and men of rank, but no other harp might sound\nuntil after that of Oswald.\n\nThe central fire had gone out, but he had bidden them leave the ash\nheap. It was high and gray, and he sat down by it, bringing his harp\nnearer. All who were there had heard him often, but never before, they\nsaid, had they heard him touch his harp as he did now. The music was\nwonderful, and with it arose his voice in marvelous power, for he sang\nof heroes, and of gods, and of the unseen lands where the gods live.\nAlso, before he ceased, he sang of Ulric the Jarl and of the ship _The\nSword_, as if even now he could see her going into battle and hear the\nwarhorn of the son of Brander.\n\nSo was the passing of Hilda of the hundred years.\n\n\n\n\n                             CHAPTER XIV.\n\n                        THE JEW AND THE GREEK.\n\n\n\"O jarl,\" said Sigurd, the son of Thorolf, \"many days have passed since\nwe entered this sea. Thou hast pleased thy crew by landings at harbors.\nThey have also smitten quiet people against thy will, and uselessly.\nThey are hard to govern.\"\n\n\"The thirst of blood cometh upon them,\" said the jarl. \"I would not\nslay any without good need. What knowest thou of this place where we\nare?\"\n\n\"It is the gate of the world, O jarl,\" said Sigurd. \"We have passed\nall Spain and much too long time have we been in our voyaging. This\ngreat cliff upon the Spanish shore is the rock I named to thee at the\nbeginning. Southward, across this narrow strait, is Africa. The Romans\nname this rock the Pillars of Hercules. He is of their old heroes, a\nstrong one, a half god. Not as Thor or Odin. He is of the giants.\"\n\nMany more things said Sigurd, and the vikings thronged around to hear.\nOf the older men, also, were many who knew this place and who had\nwords to speak. The younger men were exultant and their speech was\nboisterous; but the face of the jarl grew harder as he heard them, for\nthey had offended him often by their deeds in Gaul and on the coast\nof Spain, and by their cruelties to peaceable merchant sailors whose\nkeels _The Sword_ had overtaken. \"I am made a pirate against my will,\"\nhe had said of these things. \"The greatest of the sea kings are not\nso, for they have many friends and tributaries among the peoples and\nislands of the Middle Sea.\" Nevertheless, he now spoke loudly to all.\n\n\"Beyond this cape,\" he said, \"is the Middle Sea, which was from the\nfirst the destination of our voyage. Glad am I to have come so far out\ninto the world. From this place onward we are as men who sail into a\nbattle. So will every man bind himself to his obedience, lest his neck\nshall feel a seax.\"\n\nThis was the law of the Northmen upon the sea, and none might complain;\nbut the jarl's hand was upon his sword hilt, and some of the men turned\nto look at each other for a moment.\n\nVery smooth was the water, for there was no wind. The air was soft and\nwarm. Only one bank of the oars propelled _The Sword_. She was now in\nno haste, and all who were on board of her felt their hearts beat with\nrejoicing. To most of them this was their first long war cruise, and\nall things were new, so that they watched eagerly for that which might\nbe next to come.\n\n\"O jarl,\" said Tostig the Red, \"beyond all doubt we shall soon see\ntriremes of the Romans. Will they not at once inquire concerning us?\nWilt thou avoid such a keel or wilt thou hasten into a battle?\"\n\n\"I have considered well,\" said the jarl. \"Of a merchantman we may exact\ntribute, but we need not always destroy. It is not the way of sea\nkings. Prisoners we take not any. A warship of C\u00e6sar we must strike in\nher middle, without warning, that she may go down speedily and that too\nwide a report of our coming may not be given to those who would pursue\nus with a fleet. I know not, after such delays, that we are the first\nof the vikings this year in the Middle Sea.\"\n\n\"O jarl,\" said Knud the Bear, \"care not for that overmuch. We will but\ngo the farther into the sea. I am with thee in thy saying that we must\nsail to the eastern shore of all these waters.\"\n\nThe jarl lifted his war horn and blew long and loudly and his face grew\nbrighter.\n\n\"To Asgard!\" he shouted. \"To the city of the gods! And we will smite\nthe Romans.\"\n\nShield clashed on shield. Horn after horn was blown. The vikings\nshouted joyously and Sigurd the son of Thorolf lifted his great voice\nin a song of war.\n\n\"Easterly!\" commanded Ulric to him who was steering _The Sword_.\n\"The gods of the Northland are with us and our voyage hath been well\nprospered.\"\n\nOn floated the good ship, but she seemed to be sailing over a sea of\npeace, so quiet and so beautiful were both sky and water. An hour went\nby, and now Ulric, sitting on the fore deck, sprang suddenly to his\nfeet, for there came a shout down from Wulf the Skater above the sail\nupon the foremast:\n\n\"O jarl! A sail! Eastward. And no other sail is with her.\"\n\n\"She is our prize!\" shouted the jarl. \"We may not fail of taking the\nfirst keel that we meet, whatever she may be. A man to every oar! But\nlet those who hold the spears put on Roman helmets speedily. Open the\nsheaves of arrows. Bring out spears in abundance.\"\n\nOther commands he gave, and there was no discontented man on board;\nnone who was not willing to do the bidding of his jarl in battle. Then\nwere they glad to be led by a son of Odin; and a hard ruler in a quiet\ntime may be the captain men seek after if an enemy is nearing.\n\nThe jarl bade the rowers to row, but steadily, not wasting their\nstrength, while the Roman helmets were brought out from the stores of\n_The Sword_, and the vikings laughed merrily at each other in this\nstrange disguising.\n\nVery soon they were near enough to learn the kind and the action of\nthis keel which they were to contend with. She was not now attempting\nto either come or go, but she was drifting along over the calm water\nwith her sails flapping lazily against her mast. The vikings might\nsee, however, that her decks were full of armed men, and that she was\na larger vessel than had ever been seen in the seas of the North. Vast\nwas her length and breadth, and she carried five banks of oars instead\nof three, for this was one of the new quinqueremes which C\u00e6sar had\nbuilded for the conveyance of his legions. She was planned, therefore,\nmore for carrying than for speed, although her weight and force might\nbe terrible to crash against another vessel. She was high above the\nwater, like a tower that would be difficult to scale. She had two\nmasts, and on these were bulwarked platforms for archers and slingers.\nShe was as much more than a match for Ulric's keel as had this been for\n_The Sword_, the first, the low-built ship with which he had sailed\nfrom the Northland behind the outing ice, only that the quinquereme was\nless readily to be turned about.\n\nThe officer in command of the Roman warship knew no fear of any foe\nafloat, so sure was he of the superior strength of his vessel, and now\nhe could have no suspicion that an enemy of Rome had come in at this\ntime of the year through the gates of Hercules. He came to the after\ndeck of the quinquereme when his outlooks called him, and his answer to\nthem was haughty.\n\n\"Why did ye disturb me?\" he asked. \"It is but one of the triremes of\nLicinius coming back with tidings for C\u00e6sar. We may hail her, in her\npassing, but we may not hinder her. C\u00e6sar is careful of the bearers of\nhis messages. Men die early who meddle with that which doth not concern\nthem.\"\n\nNo change was made, therefore, in the handling of the quinquereme. The\nrowers sat idly at their places, ready for any orders which might come,\nbut allowing their oars, longer and shorter, to hang in the water, or\nto rest hauled inboard.\n\nNow there came wind enough to fill the sail, and she slipped along\nbetter, while the sailing master came and stood by the haughty\ncenturion.\n\n\"They are in haste,\" he said. \"They row swiftly.\"\n\n\"Well they may,\" replied the officer, \"whether Licinius hath had good\nfortune or whether the fates have been against him. I would not be sent\nto Britain. Too many have gone to ruin on that island.\"\n\n\"It is a bad place,\" said the seaman, \"and all those seas are full of\nSaxons. They are fierce barbarians, but they make good gladiators. I\nwould crucify them all.\"\n\n\"Never spare thou a Saxon,\" said the centurion. \"They are food for the\nsword. Slay every one thou findest on land or sea. Mars be my witness,\nI will spare not one.\"\n\nFor life or for death, therefore, was the swift coming of _The Sword_.\nThe Saxons must overcome the quinquereme, or escape her in some manner,\nor they must die without mercy, and this they knew well.\n\n\"A strong force on board of her,\" said the centurion, as _The Sword_\ndrew nearer. \"But I see no standard save an eagle on the fore deck. She\nhath no officers of rank, and that is strange. I will hail her. Sound\nthou thy trumpet, trumpeter!\"\n\nLoudly rang out the trumpet call, and it was answered by a trumpet\nfrom _The Sword_. But here too was a mystery. The viking who blew was\nbetter used to his war horn, and he knew not that instead of a peaceful\ngreeting he had sounded the notes that bid a Roman legion close with an\nenemy, to win or die.\n\nThe centurion sprang to his feet, for he had been seated.\n\n\"Rowers!\" he shouted, \"to your places! Here is a strange matter! There\nis evil tidings!\"\n\nOther swift orders followed, and every legionary on board the\nquinquereme was at his post, for the Romans are not easily to be taken\nby surprise because of their strict discipline and their rule for\nperpetual readiness by day or night.\n\n\"She is a smaller craft than ours,\" said the sailing master, \"but\nshe is a good one. I know her well, and her sign is Minerva. Who now\ncommandeth her I know not.\"\n\nIn that she was so well known as one of the triremes of the Roman fleet\nin British waters was now a gift of the gods to _The Sword_ and to the\nSaxons. Not the centurion nor his officers nor any seaman or legionary\non board the quinquereme had any thought or suspicion of that which was\nto come.\n\nOnward flashed the swift, strong vessel, the oars of the Northmen\nbiting well the sparkling sea. Fiercely rang the Roman trumpet, warning\nthem to change their course lest there should be a collision. Hoarse\nwere the angry shouts of the astonished centurion, but vain were his\ntoo-long delayed orders to his rowers and his steersman.\n\nOn the fore deck of _The Sword_ stood now a tall shape, wearing indeed\nthe helmet of a Roman, but putting to his lips a war horn of the North.\nBeside him stood what seemed a giant brandishing a spear. The blast was\nsounded and then sped the spear. A hundred more were hurled from _The\nSword_ at the Romans on the decks of the quinquereme. The viking rowers\npulled with their might.\n\nCrash! With a breaking of timbers, a braying of horns, a chorus of\nmocking war cries, the quinquereme was smitten amidships with a force\nwhich threw her legionaries prostrate and sent her rowers from their\noars.\n\nThe centurion was pierced by the spear of Sigurd. The steersman fell\nby a heavy pebble cast by Knud the Bear. The sailing master went down\ntwice smitten.\n\nUp to the masthead of _The Sword_ shot the White Horse flag of the\nSaxons, and the good ship sprang backward with a great rebound, helped\nquickly by the rowers.\n\n\"We have stricken her!\" shouted Ulric. \"The sea poureth into her. Back!\nStrike not again! It is enough!\"\n\nAs the lightning from a clear sky, so was the deathblow given to the\npride and strength of the quinquereme. As a warrior stabbed to the\nheart was she as she leaned over, and as the fatal blue tide poured in\nthrough the deep wound in her side. There was no stanching it. There\nwas no hope. They who had purposed to slay all Saxons were themselves\nto die. On the decks and at the bulwarks, amazed, confounded, the Roman\nsoldiers and sailors stood and gazed in silence at their utterly\nmysterious destroyer. Here was a riddle of the fates and furies which\nnone might read. They knew not even the flag of this strange pirate\nkeel. They only knew that they were going down.\n\nOn the stern of the quinquereme stood three men who were not in armor.\nThey were bearded men and they wore turbans, and they spoke to each\nother in another tongue than the Latin.\n\n\"We may escape,\" one of them said. \"The god of Israel hath heard. We\nare not to be crucified. Let us plunge in and go to yonder ship from\nTarshish. Ben Ezra, what sayest thou?\"\n\n\"Follow me,\" said Ben Ezra, \"ere this accursed quinquereme goeth down\nbearing us with it. On this side, while the Romans are gazing. Take\neach two short oars. We have somewhat to bear with us. Get beyond a\nspear cast as soon as we may.\"\n\nHe was a short man, and old, but his eyes were bright and he seemed a\nbrave one. His two companions were youths. Into the water they slipped\nsilently, as he had said, and they swam well, partly upheld by the\npieces of wood.\n\n_The Sword_ was not receding, but her rowers were pulling easily as\nWulf the Skater steered her around and past the quinquereme. No more\nspears were thrown nor did any arrows fly, but there was a sounding of\nwar horns.\n\nBrave must have been the trumpeter of the legionaries, for he lifted\nhis trumpet and answered defiantly, even while the water rushed in\nthrough the fatal gap in the wooden wall of his sinking vessel.\n\n\"We shall have no prisoners,\" said Knud the Bear. \"I would I knew if\nthey had taken any. What if captured Saxons were on board of her?\"\n\n\"Not at this season of the year,\" said Sigurd. \"But what are those?\nLook yonder! The Romans wear no turbans. O jarl.\"\n\n\"Bid them on board!\" shouted Ulric. \"I would question them. Throw them\na rope!\"\n\nIt was a long thong of twisted hide that was cast out toward Ben Ezra\nand his companions, but it came too late. In a moment their escape\nhad been seen by the legionaries. They were true to their soldier\ndiscipline. They themselves must die, but it was their duty to prevent\nthe departure from the quinquereme of any prisoners. Such as attempted\nit must be slain. So the pila flew fast and even arrows were sent.\n\n\"Ben Ezra, I am smitten!\" gasped one of the younger swimmers.\n\n\"Thou?\" groaned Ben Ezra; but in an instant more, he added: \"O God of\nHosts! My son also! My only son! My Benjamin!\"\n\n\"Father!\" cried out the second youth, in agony, \"the spear of the\nheathen! I die! I die!\"\n\n\"My son!\" again mourned Ben Ezra. \"I care not to live! Let me perish\nwith thee!\"\n\nNevertheless, he had grasped the thong of twisted hide and the instinct\nof self-preservation was strong enough to make him cling to it.\nMoreover, he had taken three of the short oars, instead of only two,\nand on these he was buoying up what seemed a small casket of wood. He\nwas doing so with difficulty, and now he exclaimed:\n\n\"The jewels! The gold! I must not lose them. They are priceless. The\ncenturion knoweth not that I have them. Not only mine are here, but\nthe pr\u00e6tor's also. O Jehovah of Hosts! Thou hast smitten the heathen!\nThat spear fell short. Ha!\"\n\nA pilum struck the oars and Ben Ezra struggled hard for his treasure,\nbut he succeeded in retaining it.\n\nDown sank the two who had been stricken, and in a moment more a strong\nhand of a viking grasped the old man by the shoulder.\n\n\"Courage! Thou art safe!\" he shouted.\n\n\"This first!\" said Ben Ezra, trying to hand him the casket. \"It is\nworth their quinquereme! Ye are Northmen. I am a Jew of Salonica.\nThe Roman robbers plundered my ship unlawfully, and me they meant to\ncrucify, the better to claim my goods. Help me in. I am faint. O my\nson!\"\n\nThey pulled him up over the bulwark with some difficulty, but he spoke\nnot nor did he seem to see anything until he was sure that his casket\nwas in the hands of Ulric the Jarl.\n\n\"Open it not now, O captain of the Saxons,\" said Ben Ezra. \"I have\nmuch to say to thee. When yonder Roman keel goeth down I am no longer\nin peril, for I have kept the law. But the Pr\u00e6tor Sergius of Spain and\nthe commanders of the fleet rob whomsoever they will. Praise God, she\nsinketh fast!\"\n\nIt was even so. The quinquereme was settling in the water and her crew\ncould cast spears no more. They did but stand still and gaze at the sea\nand at their strange enemy, but some of them even now called loudly\nupon their gods, as if there could be any help from them.\n\nShe was a splendid vessel, and her figurehead was a gilded Neptune with\na trident which looked as if it might be of gold. Rich indeed were her\ncarvings and the very handles of her oars were graven and gilded. She\nwas high at prow and stern, a castle of the sea, and the wonder was\nthat she had been cloven at a blow. A lighter vessel with a ram less\nsharp would perhaps have rebounded without doing serious harm, but the\nbeak of _The Sword_ was like a vast spearhead and it had been driven\nhard by the strong arms of the Saxons and by the weight of the trireme.\n\nThe middle parts of the foundering ship could no longer be occupied,\nand the ill-fortuned men who were to perish were now crowded densely\nfore and aft. Even now, however, the legionaries preserved their\ndiscipline, and they slew some of the hired rowers who pressed them\nin too disorderly a manner. These were deaths which were but somewhat\nhastened, yet military order was restored thereby, for the rowers\nfeared the strokes of the pila and the broadswords.\n\n\"They go!\" muttered Ben Ezra. \"So perish all who afflict the chosen\npeople. Rome will yet fall before the sword of Judah and the spear of\nIsrael. Jehovah standeth for his elect. He will have vengeance upon the\nheathen. He will smite through kings in the day of his wrath.\"\n\n\"O Ulric the Jarl,\" said Sigurd, \"thou mayest trust the Jew. He hateth\nRome as we do.\"\n\nThen came Ulric nearer, still watching the quinquereme, but he spoke\nwords to Ben Ezra in a tongue that those who stood by understood not.\n\n\"Father Abraham!\" exclaimed the old man, \"where didst thou learn\nHebrew? I like thee well for this. After yonder quinquereme goeth down\nthou hast cause to consult with me. There are matters thou knowest\nnot.\"\n\n\"Odin!\" exclaimed Ulric. \"Watch! She is pitching forward! They are\nfalling!\"\n\nIn a mass together, as the decks slanted, plunged the overcrowded\nRomans. It was of no avail to struggle or to thrust with sharp weapons.\nAngrily, loudly, in his last desperate valor, blew the trumpeter his\nfinal defiance, but as the blast ended the prow of the quinquereme went\nmadly under, lifting the stern out of water for a moment. Then went\nup a great cry and quickly naught could be seen save a few heads of\nswimmers dotting the blue water.\n\nThe helmets disappeared first because of the weight of armor, but the\nSaxons cast no spears at any who remained, and some who were bareheaded\nseemed to be swimming well.\n\n\"He hath golden hair,\" said Sigurd, pointing at one of these. \"He is no\nRoman. I will call to him in Greek--\"\n\n\"Bid him come,\" said Ulric, \"if so be he is not a Roman. He may live.\"\n\nSigurd sent to the swimmer a few words in a smoothly sounding tongue,\nand the golden-haired youth struck out for the trireme, but he was\nfollowed by twain who were dark and who cursed him in Latin. Well for\nhim that he was the better swimmer, for they strove to grapple him that\nhe might die with them. He might not even then have fully escaped but\nthat Ulric knew their meaning and said to Tostig the Red:\n\n\"I have no spear! Smite those two Romans and save the Greek.\"\n\nNot one spear but twenty sped in answer to that command, and the youth\ncame nearer alone, for there were none to follow him.\n\n\"He was a rower,\" said Ben Ezra. \"He is a slave of the centurion. He is\nfrom Corinth. It is perilous to spare him, lest he might tell of this\nthy doing.\"\n\n\"What harm?\" asked Ulric. \"Can the Romans do more than destroy? I will\nmyself tell them that this is the third of their warships that I have\ntaken from them.\"\n\n\"Thou sailest in one,\" replied Ben Ezra, glancing around him. \"Thou and\nthine are men of valor. But the like of this hath never before been\nseen. A Saxon crew in a Roman trireme fighting the ships of C\u00e6sar!\nMayest thou have a fleet and smite them in the Tiber itself! Now sail\nthou on, for there is another quinquereme and she may not be far away.\nAvoid her, lest thou fall into a snare of presumption.\"\n\n\"Not I,\" laughed Ulric. \"We have done enough this day. Come thou and\ntalk with me, and then I will have speech with the Greek.\"\n\nThe young Corinthian was now aft among the men, and Sigurd was talking\nfreely with him. There were others of the older vikings who had learned\nwords of the Grecian tongue, and they, too, were both speaking and\nhearing.\n\nInto a cabin under the fore deck went Ulric and Ben Ezra, and there\nthey were alone, for none was permitted to follow them.\n\n\n\n\n                              CHAPTER XV.\n\n                     THE STORM IN THE MIDDLE SEA.\n\n\nWide but not high was the space which was inclosed under the fore\ndeck of the trireme _The Sword_. Beneath its floor was much room for\nstowage. The other decks, also, had under them good cabins, suited to\nmany purposes. The decking amidships, whereon tier above tier were\nmade the seats and standing places of the rowers, had openings covered\nby hatches. Down through these, by ladders, might be entered a great\nhollow, and this was for cargo and for sleeping room. Very different\nwas all this from the planning of any vessels which hitherto had been\nbuilded in the Northland. In the cabins under the fore deck were bunks\nfor sailors and soldiers, but all the garnishing was plain. Here, also,\nthere were stores of weapons, with boxes and bales of merchandise. The\ncabins under the after deck were divided and garnished for the uses of\nofficers and men of rank who might at any time be on board.\n\nIt was not long after the sinking of the quinquereme that the jarl and\nthe Jew, Ben Ezra, stood face to face in a small room under the fore\ndeck. Steadily looked Ulric into the face of the Jew.\n\n\"He is old, but he is not aged,\" was his thought concerning this man.\n\"He is tall and broad and strong and heavily bearded. His face proveth\nfor him high intelligence, but it hath deep marks which one may read.\nI think him a subtle man and a keeper of secrets. He is a man of rank\namong his own people, for common men are not as he is. I am glad of\nhim.\"\n\n\"O jarl of the Saxons,\" said Ben Ezra, \"I have blessed thee in my\nsoul, by Jehovah my God, that thou hast utterly smitten to death these\nRomans. Thou didst wisely not to spare any, as they would not have\nspared thee or thine. Thou mayest be sure that if so much as one of\nthem were on board thy ship, he were a danger. I will tell thee of\nmyself.\"\n\n\"Say on,\" said Ulric, \"but speak truly, that it may be well with thee.\"\n\n\"Leader of men,\" said Ben Ezra, \"my life hangeth upon thy life. I am\none with thee. I do but take care for myself in that I am truthful.\nI was informed against in Spain to the pr\u00e6tor because I was rich. I\nwas seized, but I and my son and a Jewish youth, the son of a rabbi,\nescaped from our destroyers. My ship was ready laden and we sailed in\nthe night. The quinquereme was faster and she overtook us. All were\nslain but we three, for they were overfull with rowers and soldiers\nand cared not for more slaves. Even to have escaped the pr\u00e6tor was to\nbe doomed to crucifixion; but they had not yet scourged us, waiting an\nopportunity. O my son! My son! That he might have been spared! For they\nhave slain his mother and his brethren. He was my Benjamin! My youngest\nson! The joy of my heart!\"\n\n\"He was slain by a spear,\" said Ulric, to comfort him. \"He died not on\na bed, that thou shouldst mourn so much for him. Thy god hath done well\nby thee. I saw him swimming bravely till the pilum struck him.\"\n\n\"And the youth, the son of the rabbi Joseph, of Jerusalem!\" groaned Ben\nEzra. \"What shall I say to his father? A fair boy and well favored!\nThey are merciless, for he had done them no wrong.\"\n\n\"Little careth a Roman for that,\" growled Ulric. \"Who is this Greek?\"\n\n\"He was a bondslave of the centurion of the band of legionaries on the\nquinquereme,\" said Ben Ezra. \"His father was a rich man in Corinth,\nbut the proconsul lusted after his goods and he was accused. He was\nbut slain, not crucified, but his older sons went to the arena to feed\nlions, and this Lysias, for his youth and beauty, was kept alive, if\na man be indeed living who is slave to a Roman. Thou mayest trust him\nthat he hated his master.\"\n\n\"For what part didst thou intend to sail,\" asked Ulric; \"seeing the\nRomans could have found thee anywhere on the earth?\"\n\n\"Not so,\" replied Ben Ezra. \"I were safe if once I were in Judea or\nGalilee.\"\n\n\"And where are they?\" asked Ulric.\n\n\"At the eastern shore of this sea, as thou shouldst know,\" said Ben\nEzra, \"is the land of my people. In it are many cities and mountains,\nand its provinces are under different governors. He who is threatened\nby one needs but to flee to another if he can take a gift with him. I\nhave a gift for thee wherewith thou couldst buy a consul, if so be he\nhad no opportunity to rob thee. My goods, all but one casket, went down\nwith the quinquereme. In that casket are gems of my own--\"\n\n\"I want them not,\" said Ulric. \"They are not my prizes. I struck no\nblow for them. Keep thou that which is thine own. I am a Saxon, not a\npirate.\"\n\n\"Thou art a sea king,\" replied Ben Ezra. \"I have had many dealings with\nsuch as thou art. They are not like other men, for they keep faith\nwith strangers. But this, also, I tell thee: as the Roman ship began to\nfill the centurion went mad, it seemed, for he took from his crypt in\nhis cabin his own jewels and some that the pr\u00e6tor would have sent to\nC\u00e6sar to buy a pardon for some of his offenses. These, also, went into\nmy casket, and he placed it on the deck and by it a small bag of gold.\nWith aid of three oars for floats and with strong swimming I rescued\nall, and here they are. Even the centurion knew not their value.\"\n\n\"I know what gold coins are,\" said Ulric, as the bag was opened before\nhim. \"Oswald the Harper taught me concerning money. Are these thine?\"\n\n\"Nay,\" said Ben Ezra, \"they are thine, for they belonged to the\ncenturion. Of the stones I will show thee. That sardonyx is mine. It\nwas graven in Egypt, and on it are words of the wisdom of the priests\nof Isis.\"\n\n\"Runes like Hilda's!\" exclaimed Ulric, gazing earnestly upon the\ncharacters which blended with the varying tints of the beautifully\npolished stone. \"Canst thou read them?\"\n\n\"Not so,\" said Ben Ezra, \"but this sardius, also, is mine. It is a\nstone of the temple of Jehovah at Jerusalem, and on it is his most holy\nname. Touch it not, for thou art of the heathen.\"\n\n\"What have I to do,\" asked Ulric, \"with a matter belonging to your god?\nI have thought that I would like to see him some day. I am sailing to\nfind the city of the gods. He will be there, perhaps, among them.\"\n\n\"He is a great King above all gods,\" said Ben Ezra, reverently, but his\neyes were dwelling upon the glowing, blood-red tint of the inestimable\ngem which bore the holy name.\n\n\"Odin!\" exclaimed Ulric. \"I think your god would be on good terms with\nthe gods of the North, for I have heard well of him. Thou mayest tell\nme more about him some day. But now thou mayest tie up thy gems and\ngive me mine. I have the ship to command, and I care not overmuch for\nstones.\"\n\n\"My sardius alone is worth thy trireme,\" said Ben Ezra, frankly. \"Keep\nthou thy treasure carefully and a day may come when it will be of use\nto thee. Divide not with thy men. Give them other matters.\"\n\nUlric laughed loudly as he responded: \"Good faith is kept among us, O\nJew, but my vikings are welcome to all I possess. The ship itself is\ntheirs, if I am slain, and they will carry to my own house anything\nthat belongs to me. We are not thieves, like the men of the southlands.\"\n\nBen Ezra looked into his face and said: \"Verily thou art my friend! I\nhave not met any like unto thee. I would thou mayest go with me to the\ncity of my God, to Jerusalem. There is his temple and it is the wonder\nof the earth.\"\n\n\"So have I heard,\" said Ulric, joyously. \"I think I will go and see.\nBut whither shall I steer at this hour?\"\n\n\"Toward the coast of Africa,\" said Ben Ezra. \"Thither was I sailing.\nThere are old harbors there for which the Romans take no care. In them\nare pirate peoples, foes of Rome, ancient Carthaginians, Egyptians,\nLibyans; but thou wilt be friends with them.\"\n\n\"_The Sword_ will sail for Africa,\" said Ulric; \"but as for pirates, we\nwill see to that matter.\"\n\n\"Verily there is none like thee!\" exclaimed the Jew. \"Thou art like\nSaul, the son of Kish!\"\n\nInto a small sack of deerskin did Ulric put his jewels, looking at\nthem one by one and admiring their great beauty.\n\n\"Never saw I such before,\" he said. \"They are such as kings wear in the\nsouthlands. I think the gods must have many of them. These white ones\nare pearls, and they are lustrous. The green stones are emeralds.\"\n\n\"They are of great value,\" said Ben Ezra. \"Especially that large,\nflat-faced one. It is engraven with the sign of the sun, and it came\ntherefore from Persia. Thy pearls are from the East, and they are\nwonderful, but some of mine are as large.\"\n\n\"I will keep the gold in my belt pouch,\" said Ulric, \"and thou shalt\nteach me to pay with it.\"\n\n\"Thou shalt not be cheated,\" said Ben Ezra.\n\nThen he took his closed casket with him and walked to the after cabin,\nfor in that was to be his abiding place, and he said that he would\nmourn there for his son.\n\n\"Southward!\" shouted Ulric to the viking at the rudder of the trireme.\n\"We have done well this day, and the night cometh.\"\n\nA wind had arisen and the sails were full, but the men did not seem to\nbe idle. They busied themselves with the tackle and with the stowing\nof the ship, but every now and then each one would step out on deck or\nlean over a bulwark to look long and earnestly across the sparkling sea.\n\n\"This water is very blue,\" said Tostig the Red, \"and so is the sky. O\nKnud, thou hast put away thy bearskins.\"\n\n\"Aye,\" said Knud, \"but how canst thou bear thy mail in such a heat as\nthis? I found this jacket of silk in the after cabin. It is cool and it\nis fine.\"\n\n\"Red as blood it is,\" said Tostig, \"but it would not keep out an arrow.\nThou dost never care much for armor.\"\n\n\"A shield is enough,\" replied Knud, \"and I can catch arrows on my seax.\nI would not be overweighted. I trust the gods will soon send us another\nfight. I would get hand to hand with some good fighter. There is more\npleasure in killing with steel than with the prow of a ship.\"\n\nThe jarl gave orders concerning many things, and then he spoke to\nLysias the Greek. The youth had seated himself in a hollow place\nbetween two oar benches and his face was in his hands, for he was\nweeping.\n\n\"Not often do men weep in the Northland,\" said Ulric, sternly. \"I have\nheard that the Greeks are brave. Why mournest thou? Hast thou not had\ngood vengeance upon the Romans this day? Not one of them escaped. Thou\nshouldst rather be rejoicing.\"\n\n\"Alas! Alas!\" murmured the beautiful youth. \"Corinth! My Sapphira! I\nshall never see her again!\"\n\n\"She was thy love?\" said Ulric, softening somewhat. \"I never had a love\nsave Hilda, the saga woman, and she was a hundred years old. I loved\nher well. Where is thy Sapphira?\"\n\n\"She was more lovely than a dream!\" said Lysias, looking up through his\ntears. \"Her father was Licander, the astrologer, and she was like a\nstar. He knew the heavens and the stars in their courses, and he read\ntheir signs. But he foretold to the Romans their Parthian defeat and\nthey slew him for his bad augury. Of his kindred they left not one, and\nSapphira they sold for a slave to Pontius Pilatus, the procurator of\nJudea. I care not to live, for I have been scourged and I have lost my\nlove.\"\n\nEven as he spoke he threw off a light robe of linen which had covered\nhim, and Ulric saw the half-healed, festering lines of the Roman\nscourge all over the flesh of Lysias.\n\n\"Thou mayest well weep for that!\" exclaimed Ulric, \"if thou art the son\nof a free warrior.\"\n\n\"I did stab three in Spain,\" said Lysias, \"and I had plotted to sink\nthe quinquereme, for she had a bad leak which might be opened.\"\n\n\"Get thou up!\" said Ulric. \"I will gird thee with a sword and give thee\na shield and spear. When thy scourgings are healed thou shalt have\nmail. Thou art strong.\"\n\n\"I have won foot races,\" said Lysias, rising, \"and I can ride any wild\nhorse. I am a bowman and I can cast the javelin far and truly.\"\n\n\"Be more contented, then,\" said Ulric. \"I will give thee chances to\nstrike Romans. There is no need for thee to mourn.\"\n\n\"Thou knowest not love,\" said Lysias; \"but I thank thee, and I would\nhave weapons.\"\n\n\"Come with me,\" said Ulric, and they went together to the after cabin.\n\nThere were doors by which this might be closed, but one of these was\nopen and they went in. Then it could be seen that this cabin space,\nwhich was large, was divided into four apartments by strong wooden\nwalls, each having a door and a window, and in the windows were small\nsheets of glass to let in light and keep out the sea. This first room\nwhere they now were had been the place prepared for some person of high\nrank to occupy, an officer in command of the ship or a high passenger.\nIt was finished in carved wood, with hangings of silk and linen of\nmany colors and of fine needlework. Here, also, were lamps that hung in\ncressets, and there were fixed tables and soft couches and many fair\nweapons and pieces of splendid armor. None of these had the jarl worn\nas yet save a helmet and a rare coat of linked mail richly gilded. Now\nhe selected a good belt, with a sheath and sword, and a long sheathed\ndagger.\n\n\"Throw off thy robe,\" he said to Lysias. \"Put on this tunic and the\nsandals. Belt thee with these.\"\n\nSo the youth did, and it could be seen that his shape was not only\ncomely, but molded for great vigor. The muscles stood out upon his arms\nand shoulders and Ulric himself was but a head the taller.\n\n\"These will soon heal,\" said Ulric, examining the lash cuts. \"Oil them\nwell. I will aid thee. They are now not deep. Thou art a good swimmer.\nI noted thee in the water. Here are thy shield and spear.\"\n\n\"They are Greek, not Roman,\" said Lysias. \"I am glad of that. I want a\nbow and arrows.\"\n\n\"A quiver and a bow are here,\" said Ulric. \"But the arrows are long and\nso is the bow. See if thou canst bend and string it.\"\n\n\"That can I?\" exclaimed Lysias, seizing the bow. \"It is from Sparta,\nfor only the Lacedemonians make them of this length. The Parthian bows\nare shorter, for horsemen, but only a Parthian can bend them--or such\nas I. We are of the ancient Corinthian archers, and there were none\nbetter on earth.\"\n\nHe was bending the strong wood as he talked, and Ulric saw that he did\nso and put on the string of twisted silk with ease. Then took he an\narrow from the quiver and drew it to the head.\n\n\"Thou art the captain,\" he said. \"Thy men call thee the jarl and say\nthou art of the sons of their gods. Canst thou send this arrow farther\nthan I can?\"\n\n\"I will handle thy bow,\" said Ulric.\n\nHe, too, unstrung and strung it and drew the arrow to the head, but he\nsaid, thoughtfully:\n\n\"Thou of the Greeks, I understand thy saying concerning skill. I am\nmany times stronger than thou art, and yet I think thee the better\nbowman. I will call on thee if I would have a sure arrow sent.\"\n\nLysias lifted the spear, which was a fathom long, and light, but he\nlooked around the room and found more of the same pattern and made a\nbundle of them.\n\n\"They are well made,\" he said, \"and their points are of good steel. I\nonce threw one like these through the heart of a man from Athens. He\nwas an enemy of my father. I met him on the seashore and I was quicker\nthan he in casting. He should have worn a thicker breastplate.\n\n\"Hah!\" laughed Ulric. \"I am a spearman, but I prefer the North spear\nand the pilum.\"\n\n\"I like them,\" said Lysias, \"but I know one man that can outthrow thee.\nHe is a Roman knight named Pontius, and they call him the spearman. He\nis the procurator I told thee of. I would I might live until I can kill\nhim. He liveth now in Jerusalem.\"\n\n\"Thither go I!\" exclaimed Ulric. \"I have promised Ben Ezra that I will\ntake him to his own, and I must go to that city and see the temple. I\nhave it in mind that I may see his god. They say he is a good god and a\ngreat fighter like Thor.\"\n\n\"I have heard much of him,\" said Lysias, \"but he is more like Jupiter.\nIf thou wilt land at the island Paphos, I will show thee his statue\nand thou canst see what he is like. We shall hear his voice thunder if\nI read this weather rightly.\"\n\n\"Then he is Thor!\" said Ulric, turning to the door. \"Come! I know not\nthe weather signs of this sea.\"\n\nOut they went and Lysias glanced around the sky. His face was brighter\nnow and he stepped firmly like a warrior.\n\n\"O jarl,\" he shouted, \"I am a seaman also. Take down thy sails quickly!\nPut out a bank of oars. Bid thy steersman keep the head of thy keel\nsouthward, for from thence cometh a tempest. The sky will darken\nrapidly.\"\n\n\"The Greek is right!\" shouted Sigurd. \"I had forgotten the sign of such\na storm, but I call it to mind. It is a strong one.\"\n\nDown came the sails, out went the oars, and the thick haze on the water\nsoutherly, which had been sunlit and fair to look upon, shot up toward\nthe middle heaven, blackening as it went.\n\n\"O jarl,\" said Wulf the Skater, \"thank the gods! We are to see a kind\nof storm that we do not have in our own seas.\"\n\n\"Fine storms come to us in midsummer,\" said Ulric, \"and they roar well\nin the fiords. Will the anger of Thor be louder here? The Greek saith\nthat his Jupiter can thunder, and the Jew told me that his Jehovah is\nalso a thunderer. Are they of kin? They who speak the same tongue are\nof one house.\"\n\nThe Greek was now standing by the anvil and hammer on the fore deck.\n\n\"The sign of this ship was Minerva,\" he muttered, \"but the Saxons\nhave given it to Vulcan. If yonder cloud is indeed of the wind from\nthe African desert we may yet wish that Neptune were our steersman.\nBut what care I for the gods? They were never yet of any use to me. My\nfather made many sacrifices, but the Romans slew him.\"\n\nThere now were sails in sight, but these were fast furling. Most of\nthem were small, but one, at the greater distance, had seemed much\nwider than the rest.\n\n\"I have been watching her,\" said Sigurd to Ulric, speaking of this\ncraft. \"I am not young, but my eyes are the eyes of a falcon. Now that\nher sail is down her oars are out and she steereth toward us. The storm\nwill give her oarsmen enough to do.\"\n\n\"But we must watch her,\" said Ulric. \"Even a merchantman might seek our\ncompany, but she may be a warship.\"\n\n\"So may some of these lesser keels be of the pirates of these coasts,\"\nsaid Sigurd. \"They are many, and we do well if we smite them, for often\nthey are good captures.\"\n\n\"Here cometh the wind!\" shouted Knud the Bear, exultingly. \"The foam\nflyeth!\"\n\nFirst came a sheeted flash of the blinding lightning, and after that\nclosely followed a deep-throated reverberant peal of thunder.\n\n\"The voice of Jah!\" muttered Ben Ezra. \"He hath spoken from his high\nplace.\"\n\n\"Jupiter the Thunderer!\" exclaimed Lysias, still standing by the hammer\nof Thor as if for protection. \"I fear him only at such hours as this;\nbut he is a god of the Romans and I am a Greek. Evil are all gods or I\nshould not have lost my Sapphira. Evil are they and wicked, and they\nhate men, for they destroy us. There is no man but must die, and if the\ngods were good, we might live. But these Saxons are brave seamen!\"\n\nLittle cared they for storms, these sons of the sea kings. They shouted\nand they sang as if they were in a battle, while the waves grew mad\nand boiled frothing around the high wooden walls of _The Sword_. Her\nhead was kept toward the wind and she rode the billows like a vast\nwaterfowl, for the Roman shipbuilders were well skilled.\n\nLess easy must have been the course of a keel that strove to cross the\nsurges with her side to the wind, and it now could be seen that the\nlarge stranger was laboring and that now and then waves broke over her.\n\n\"She bringeth small peril to us,\" said the jarl. \"We will row with but\none bank of oars. Let their rowers weary themselves with three. The\ntrumpeter on her fore deck soundeth a signal.\"\n\n\"Of what good,\" laughed Wulf the Skater, \"is the blowing of a horn in\nsuch a gale as this?\"\n\n\"He sendeth us a signal to heave to and wait for them,\" said Sigurd.\n\"What sayest thou concerning this fellow, O Jew?\"\n\n\"I think her one of the cruisers sent out by the proconsul of Spain,\"\nreplied Ben Ezra. \"They are all weaker vessels than this, but they are\nswift. They protect merchantmen from the African pirates to rob all for\nthe proconsul.\"\n\nThe air grew darker, denser, and the salt spray flew into all faces,\nbut the jarl stood upon the after deck and blew upon his war horn a\nblast louder than that of the Roman trumpet.\n\n\"Thy horn be exalted!\" shouted Ben Ezra. \"It is as the horn of a king!\nMay Jehovah of Hosts be with thee, thou mighty man of valor! Sound\nagain! Let these heathen know that we fear them not.\"\n\n\"But for the storm we might strike them,\" growled Sigurd. \"It is ill to\nlet such a prey go by us.\"\n\nNow was there also a change in the appearance of Ben Ezra. He stood by\nthe jarl as erect as a pine tree. From the stores of _The Sword_ he\nhad provided himself with arms and armor of the best, by permission of\nUlric. The visor of his brazen helmet was open and it might be seen\nthat his dark face glowed like youth as he gazed angrily at the enemy.\n\n\"He is a warrior!\" exclaimed Tostig the Red. \"I like him well. I think\nhe might strike a good blow with that long crooked sword which he hath\nfound. I saw it, but I preferred a straight blade. The shield lifteth\nlightly in his hand and his mail coat fitteth him. He hath put brazen\nguarders upon his arms and legs. A small man should avoid such as he in\nthe press of battle.\"\n\nSo said others of the vikings, but they were watching more closely the\nRoman keel.\n\nThe trumpeter sounded several times and as often did they send back\ndefiances from their war horns.\n\n\"O jarl,\" said Lysias, \"this is the storm which cometh from the African\ndesert. It is not like any other. Not only is there much thunder and\nterrible lightning and strong wind, but I have felt sharp sand upon my\nface. It will blow long and hard, and the waves will not go down, but\nthere will be no more rain. The sky is clearing.\"\n\n\"Thou knowest the storms of thine own sea,\" said Knud the Bear; \"but\nare we far from land?\"\n\n\"No man knoweth that,\" said Lysias; \"but here cometh the Roman, like a\nfool. I would thy jarl might strike him. O jarl, may I use the bow?\"\n\n\"When thou canst,\" said Ulric, \"but the distance is yet too great.\"\n\nLike fierce and angry music rang out the laugh of the Greek, but his\narrow was on the string and he raised the bow.\n\n_The Sword_ sank heavily into the trough of a sea wave and the Roman\nkeel was lifted high upon a surge, just as a long, vivid sheet of\nlightning seemed to bring her nearer by its brightness. Her steersman\nwas a giant, unarmored, straining hard at her tiller and bracing\nhimself. At him was Ulric looking when suddenly he threw up his hands,\nletting the tiller go, and the feathered shaft of the young Greek's\nlong arrow quivered against his naked bosom.\n\n\"Odin! Well shot!\" shouted Ulric, but the bowstring twanged again and\nanother Roman fell upon the deck beside the dead steersman.\n\nLeft to itself and to the will of the wind and the waves, around swung\nthe keel of the Romans, while a great surge poured over her bulwarks\nand her rowers were hurled from their seats. Wild was their shouting\nand another surge poured in.\n\n\"Strike her not!\" said Sigurd. \"Be thou prudent with thine own keel,\nlest thou shouldst be in some manner disabled. Let the Greek send his\narrows, but steer upon thy course.\"\n\nUlric so ordered, but shaft after shaft did Lysias send, not all of\nthem hitting, but not all failing of a mark. Other war horns than that\nof Ulric were sounding and other bows were also quickly plying.\n\n\"I think,\" said Tostig the Red, \"that we have no better bowman than\nthis Greek. He will be a good help in a sea fight. I like well to see\nhis long arrows go so straight to their places. Then the mark goeth\ndown and it is time to laugh.\"\n\nThe Roman rowers were toiling hard to recover control of their vessel,\nbut the Saxons knew little of the astonishment and dismay that reigned\non board of her. Her crew had not thought of an open enemy at the\nfirst. They had deemed _The Sword_ a friend until the sounding of the\njarl's war horn. Even then they had expected no resistance, at least\nno attack, until their steersman fell and a man of rank near him was\npierced by an arrow.\n\nBetter than a sailing vessel can a rowed keel turn her head to the\nwaves, however, and before long the Romans were once more striving to\novertake the Saxons.\n\n\n\n\n                             CHAPTER XVI.\n\n                        THE DEAD GOD IN AFRICA.\n\n\nClouds without rain swept fast across the sky and the waves followed\n_The Sword_ as if they willed to overwhelm her. Well was it that her\nstern was so high and that she was strongly builded. It had seemed,\nalso, that no sea harm had befallen her pursuer, but now the darkness\ndeepened and the watchers on _The Sword_ could no longer discern the\nRoman.\n\n\"O jarl,\" said Sigurd, \"it is a time for prudence. This flying sand\ntelleth of some shore, I think, at no great distance.\"\n\n\"It might be carried far by such a wind,\" said Ulric. \"But Ben Ezra\ntold me of great cities in Africa which have been buried by the sand\nblown from the inner deserts.\"\n\n\"What further counsel hath he?\" asked Sigurd.\n\n\"Answer him, thou,\" said the jarl to Ben Ezra.\n\n\"O warrior of the Saxons,\" said the Jew, \"thou sayest that thou hast\nsailed these seas aforetime. Thou mayest know that the presence of\none Roman trireme portendeth the speedy gathering of a fleet. It were\nwell to destroy this one if she cometh near us again. But we have now\nescaped her pursuing. Let her watchers not see this ship again. I would\nadvise that we now go eastward by the stars, for we may note them at\ntimes through the rifts in the clouds.\"\n\n\"I will so order,\" said the jarl. \"I were not wise to risk harming my\nown keel by a battle among these high waves. It is a peril to a ship to\nbe dashed against even one heavy timber where the aim cannot be made\ncertain. Moreover, we have been long at sea and it were well to seek a\nharbor.\"\n\nBen Ezra said no more, now that his counsel was approved. The head\nof _The Sword_ was turned eastward and all the oars were plying.\nNeither was the wind now so much against her, but the waves were still\ntumultuous. Fast waned the night, growing darker as it passed, and the\njarl himself remained at the helm.\n\n\"I go onward into an unknown sea,\" he thought. \"Who may tell what may\nbe before me? Dawn cometh. There is gray light. O watcher!\"\n\nAnswered him then not a Saxon, but the deep voice of Ben Ezra from the\nforemast.\n\n\"O jarl! A fire! Hold! We near a land!\"\n\n\"Cease rowing, all!\" shouted Ulric. \"O Jew, look again. What seest\nthou?\"\n\n\"Only a dim fire, far in the southward. It is a guide for us, but we\nmay seek it cautiously. The wind goeth down.\"\n\n\"It is so,\" said Knud the Bear. \"It was a hot wind, and this air is\ncooler. I thought we were sailing into a furnace.\"\n\n\"The desert is like a furnace, I have heard,\" said Sigurd. \"Men burn up\nin it and all horses die; but lions live there. How can any beast live\nin a land of fire?\"\n\n\"I know not,\" said Ulric, \"but yonder is a brighter streak of dawn.\nWe shall soon know if the Romans are near us. We may slay them if the\nwater becometh smooth enough for a good fight.\"\n\n\"It would be a grief to all men,\" said Tostig the Red, \"if we lost an\nopportunity. But if this be land, I want some beef.\"\n\n\"Good!\" exclaimed an old viking. \"We had many cattle on the Gaulish\ncoast, but in Spain we got little but sheep. Hereaway may be found\ncattle. We may throw a net, and we may find fishes.\"\n\nThe jarl said nothing, for he watched the sea and the sky and he\nsteered the ship.\n\n\"Nearer!\" shouted Ben Ezra from the mast. \"And the daylight cometh. I\nwatch for the Romans. May the curse of Jehovah be upon them and theirs\nforever!\"\n\nLysias was on the fore deck, and as he heard Ben Ezra he muttered angry\nwords in his own tongue. Then he whispered softly to himself, or to a\nshadow, and his fair face grew white and his teeth ground together as\nif he were in agony. So do they suffer who have lost a love and know\nthat it is forever gone, for Lysias had said:\n\n\"Worse far than if they had slain her! I would that she were dead and I\nwith her. But I may live to slay Romans. Why did this Saxon jarl spare\nany of them? But he is captain, and they say he is a wise one.\"\n\nIn the small wooden fort for slingers and archers, high up the stout\nmast, sat Ben Ezra, and a viking sat with him.\n\n\"O Saxon,\" said the Jew, \"would thy jarl spare them if they came with\nthe day?\"\n\n\"The son of Brander is jarl, not I,\" replied the viking, surlily.\n\"Speak thou not carelessly of the leader of men. Thou art no seaman. He\nwill strike when he is ready. Let that content thee.\"\n\nFor deep and strong was the hold of Ulric upon his older men, by\nreason of his skill as a seaman and as a captain and because of his\ngood fortune; for they saw plainly that Odin and Thor were with him and\nthat the gods of the Middle Sea could do nothing against him. Even the\nice gods had been his friends and the god of the Druids had also helped\nhim, sending him away from Britain unharmed. It was a great thing to\nhave such a jarl, of Odin's line. They all knew, moreover, that Hilda,\nthe saga woman, must by this time have gone down to the gods and that\nshe willed exceedingly well to the crew of _The Sword_ and to her young\nhero.\n\n\"He is truly a leader of men,\" growled the Jew through his thick beard,\n\"but I would once more smite these Philistines of Rome.\"\n\n\"In that I am with thee,\" said the viking, heartily. \"Thou art a good\nsword. I would see thee in battle. It is pleasant to look upon a\nwarrior that slayeth zealously. But our feast of blood will come to us.\nWait.\"\n\nUp sprang the sun above the blue waves of the Middle Sea, and all the\nSaxons shouted joyfully. It was true that there were no enemies in\nsight, nor present hope of any good fighting, but here was a land that\nthey had never seen before. All seamen know the joy there is in finding\na country that is unknown.\n\n\"Hael! O land of the South!\" shouted Tostig the Red. \"Thou hast\nmountains as tall as are those of the North. But this is a bay, a\nharbor, not a fiord.\"\n\n\"What sayest thou, Ben Ezra?\" asked Ulric of the watcher on the mast.\n\n\"Row in!\" replied the Jew. \"There is no other keel in this haven and it\nis a good one. I see no sail nor any boat seaward. This is Africa, and\na city is on the shore, but the fire was at the head of the bay. There\nare rocks ahead. Row around them.\"\n\n\"I see them; a great ledge,\" said Ulric. \"Broken and sharp-toothed are\nthose rocks, and they would wreck any keel that should strike upon\nthem. It is a place of wrecks.\"\n\nThe rowers rowed and _The Sword_ went on through a wide passage at the\nright of the ledge. Then she was in a great basin where many keels\nmight ride at anchor, and before her and on either side of her lay the\nland.\n\nThere seemed but a gentle <DW72> at the seashore. Beyond might be a\nplain for a few miles, and then, lifting their heads so high that they\nentered the dominions of the upper gods to be capped with ice and snow,\nwere the many mountains. Into that upper land no man may enter, for the\nice gods will freeze him and bury him in snow for his insolence.\n\nIt was all exceedingly beautiful, but the rowers now rowed slowly and\nall the other Saxons watched warily as _The Sword_ drew nearer what\nseemed a landing place, a structure of stonework builded far out into\nthe harbor.\n\n\"Bring thy ship to yonder wharf,\" counseled Sigurd. \"No men are to be\nseen, but there are walls and temples and houses. This may be a town of\nthe magicians of Africa. Beware of them, Ulric the Jarl.\"\n\n\"I would I knew who kindled the light,\" said Ulric, thoughtfully. \"If\nwe had sailed toward it in the dark we had perished on that ledge.\"\n\n\"Thereon are fragments of wrecks,\" said Sigurd. \"The breakers there are\nhigh.\"\n\nSo said other of the seamen, but _The Sword_ was now making fast to the\nstone jetty, and Ben Ezra was already out upon it walking shoreward,\nwith his scimiter drawn. He seemed like a younger man. But he was not\nto go alone, for closely behind him hurried Lysias with his bow, and\nKnud the Bear.\n\n\"Here burned the fire,\" said Ulric, a few moments later, pointing at a\nheap of ashes near the head of the jetty. \"There hath been much burning\nof wood at this place.\"\n\nNevertheless he left it behind him and marched rapidly forward. He left\na strong guard with the ship, but he thought it best to enter this\nstrange town with tenscore of armed Saxons arrayed as if they were to\nbe assailed by some enemy.\n\nOn went Ben Ezra, but he met no man, and he came to a wall, in the\nface of which was a ruinous gap where once had been a gate. From\nthis opening it was seen that a broad street led away, bordered by\nruined palaces. At its far end arose one of the temples which had been\ndiscerned from the ship, as it stood upon high ground.\n\n\"Here,\" said Ben Ezra, \"is a city which Jehovah hath smitten for the\nsins of them which dwelt therein.\"\n\nBut he spoke loudly, in the old Hebrew tongue, and at once a voice\nresponded:\n\n\"Who art thou, O Jew, coming hither with a sword? The sword hath\ndeparted from Israel, as it hath from Tyre and Carthage. I am Annibaal,\nthe foe of Rome and of Greece, and I am dying of hunger. Come hither to\nme.\"\n\nAs if without fear Ben Ezra walked toward the sound of that voice not\nmany paces. Then crawled out from behind a fallen column a naked,\nsun-darkened, very hairy shape of a vast man, larger than Sigurd, the\nson of Thorolf, but he lay prone upon the sand gasping. Only one eye\nhad he, for the other was but a hollow socket, and he had but one hand\nand one foot and both of his ears were gone. He was but a mutilated\nremnant of a strong man, and his only weapon was a long straight sword,\nvery bright and seemingly keen, with a golden hilt, whereon were\nglittering gems.\n\n\"O Annibaal,\" said Ben Ezra, in the tongue of Tyre, \"what is this city?\"\n\n\"It is the city of the dead,\" said Annibaal. \"I was a chief of\nCarthage, whereof this was a colony, but some came hither from Tyre,\nand here were already many from Nubia and from Egypt. First the Greeks\nof Alexander harmed us in the old time, but after them, in my day, came\nthe Romans. They smote us hip and thigh, slaying whom they would slay,\nmaking slaves of many, and of me, a prince and captain, they made what\nthou seest, leaving me here alone.\"\n\nAlready Lysias and Knud stood by Ben Ezra, and behind them a few paces\nhalted Ulric the Jarl and his men.\n\n\"I wonder thou didst not die,\" said Ben Ezra, \"or that the lions took\nthee not. I see some of them even now.\"\n\n\"I have slain lions,\" said Annibaal, \"but it is now as if I were\nfriends with them and they harm me not. It is their city and we agree\ntogether. Yet I had at this time no more food and I perish, but I\nlighted my death fire to trap Roman ships to my ledge. I have slain\nmany there, and sometimes I have had joy to hear them when the wind\nbrought their cries to the shore. Their bodies float to the strand and\nthe beasts and the ravens feast upon the wolves of Rome.\"\n\n\"He must die,\" muttered Knud. \"He slayeth sailors. It is not good to\ntrap men, so that they die a cow's death. It is wicked to rob a warrior\nof his right to die in fight or by the righteous breaking up of his\nkeel.\"\n\nSo said other of the vikings, thinking of Valhalla and the gods, for\nthey all were religious men, scorning an evil action. But Ulric had\nsent in haste for food and for water and for ale, commanding that this\nman should be fed.\n\n\"Ye are too late,\" said Annibaal. \"I pray thee, rather, for thou art a\nprince, strike me with thy spear.\"\n\n\"That is a just thing, O jarl,\" said Sigurd. \"He hath been a warrior.\nThou wouldst ask thy kinsman to make the hero spearmark on thee if thou\nwert unluckily perishing in thy bed. Send him marked to his gods, that\nthey take him not for a coward.\"\n\n\"Not yet,\" said Ulric. \"Ben Ezra, talk thou with him as thou wouldst.\"\n\nIn Hebrew and in the tongue of Tyre did the twain converse. When the\nwater came, and the food and ale, Annibaal drank a cup of water, but\nmore he could not do, for he was passing.\n\n\"I have learned much,\" said Ben Ezra; \"but he dieth. Refuse him not\nthy mercy, O jarl. He is a prince, and he is worthy of thy hand. Take\nthou his own sword and smite off his head lest thou fail of a friend in\nthine own hour. Quick ere he fainteth!\"\n\nUlric took the long, beautiful sword, which had slain both men and\nlions, and he struck as became him, for he heard murmurs among the\nSaxons.\n\nAnnibaal had feebly lifted his head to receive the gift he had asked\nfor, and it was severed well, falling upon the sand.\n\n\"Well done, O jarl!\" shouted Knud the Bear. \"I liked him not, but it\nwere shame to let a brave warrior die of thirst. Now do I not fear at\nall to go on into this place, for we have put blood at the gate.\"\n\nThe other Saxons shouted their approval of their jarl's kindness to\nAnnibaal, and they marched forward willingly, blowing their horns and\nclashing their spears upon their shields, for all this great ruin was\nvery wonderful.\n\nThe street was long, and as they went on Sigurd remarked to the jarl:\n\n\"Where there are lions there are no cattle. Where the Romans have been\nthere is left no plunder worth taking. We will but use our eyes till we\ntire and then we will lift our sails and depart.\"\n\nUlric answered not, for a strange look was on his face and his eyes\nwere studying the sword of Annibaal.\n\n\"This hilt hath many runes,\" he said to Ben Ezra. \"Canst thou read\nthem?\"\n\n\"Not so,\" said the Jew, \"but one thing the Carthaginian told thee not.\nI had heard much of this city. It was first builded by the kings of the\nforgotten ages, whereof there are no writings. Our own writings tell\nus somewhat of them. The Egyptian priests know more, but tell it not.\nSo did those of Babylon the elder. Here was a great people, but they\nperished. Even their gods died, being slain by the sword of Jehovah.\"\n\n\"As many gods have been slain by Thor and Odin,\" responded Ulric. \"I\nlike your god, that destroyeth his enemies.\"\n\nMore slowly they walked as they drew near the front of the great temple.\n\n\"The stones of it are large,\" said Ulric to Ben Ezra. \"They are\ngreater than the Druid stones that I saw in Britain.\"\n\n\"I will show thee greater stones than these in the temple of Jehovah at\nJerusalem,\" replied the Jew.\n\n\"I will go there with thee,\" replied Ulric. \"But these are wonderfully\ngraven. Only a good chisel may cut granite rock.\"\n\n\"Their tools were of bronze,\" said the Jew, \"and none but their priests\nknew how to make them. Taller pillars are in Egypt than in Greece\nor Rome, but they are of the old time. No more are set up since the\nEgyptian gods departed. They, too, were overcome by Jehovah.\"\n\n\"He is a great god,\" said Ulric, reverently. \"I would be glad to see\nhim. Let us go up these steps and look in.\"\n\nSome of the vikings paused on the steps and would go no further, for a\nchill was on them in spite of the sunshine. One said to another: \"The\nmagicians may still be here, or some of the old gods of this place.\"\n\n\"The son of Odin need not fear them,\" was answered; \"but we are not as\nhe is. Let us wait until he hath gone in.\"\n\nGreat was their faith in their jarl, but they were disappointed that in\nthis harbor they were to obtain no cattle nor any plunder.\n\nFirst went Lysias, as if he feared not at all; but he had seen many\ntemples, and this was one from which its gods must have gone away,\nleaving it solitary. His bow was in his hand, however, and suddenly he\nstood still, putting a long arrow upon the string in haste.\n\n\"Strike him!\" shouted Ulric. \"He may escape if I try to spear him.\"\n\n\"A splendid lion he is!\" shouted Tostig the Red. \"Thou canst not slay\nhim with thy arrows! Let me go to him!\"\n\nEven at that moment they had passed the portal, for at the top of the\nflight of steps was a level place, stone-floored, surrounded by these\nvast pillars whereof they had been speaking. Across this level was the\nportal, but no doors were in it to hinder. Beyond, as they now saw\nentering, was an open space, a hundred cubits wide and more in length,\nbut it had no roofing. It seemed like a place of assembly, and at its\nfurther end was a high dais, whereon was an altar and behind the altar\nan image. But on the altar couched this lion, tawny and large. His head\nhad been between his paws, but now he arose and sent toward them a roar\nthat was like half-smothered thunder.\n\nThe arrow sped and it smote him in the breast, entering deeply.\n\n\"Odin! What a bound was that!\" exclaimed Ulric, and all the Saxons\nshouted for the pleasure of seeing the stricken beast fly through the\nair toward them.\n\n\"He belongeth to the Greek,\" said Sigurd. \"Spoil not his sport. He\nshooteth well. He is a warrior's son.\"\n\nIt had been a disgrace to any viking to interfere, even if the lion\nshould slay the Greek, but Svip, the son of Leiknar went forward\nwrongfully, lifting his spear. All others did but stand where they were\nand they called out angrily to Svip.\n\n\"He is but a Greek,\" said the son of Leiknar; but the lion sprang again\nand he sprang far, with a short roar which was fierce and guttural,\ntaking Svip for his enemy. Brave was the son of Leiknar, but he knew\nnot aught of lions. Upon him fell the mighty beast, beating down the\nspear with a forepaw. Sharp were the long claws and swift and terrible\nwas the tearing. The shield was no defense and the mail was rent as if\nit had been leather. Torn into fragments was the strong viking ere he\nmight draw his seax, but the bow of Lysias twanged again and his arrow\nsped well.\n\n\"The lion hath no mark but his,\" said Sigurd, the son of Thorolf.\n\"Back! This is his battle. Let him win it or perish!\"\n\nThis was a moment when men look, but do not breathe, for the lion\nturned upon Lysias and the youth faced him boldly, drawing his long\narrow to the head.\n\n\"Well shot!\" shouted Tostig. \"O Greek, thou art a good bowman, but he\nhath thee!\"\n\nThe lion had gathered his strength to spring, but the shaft had gone in\ntoo far. The roar choked in his throat. His limbs refused to cast him.\nHe rolled over, snarling, and pawing at the pavement.\n\n\"I would thou wert a Roman!\" said Lysias. \"But such as thou art have\ntorn my kindred in the arena.\"\n\n\"Slain!\" shouted Sigurd. \"Thou hast done well, O Greek!\"\n\n\"Svip, the son of Leiknar, erred to his death,\" said the jarl. \"The\nfault was his own. But this lion was first smitten upon the stone of\nsacrifice. What sayest thou, O Jew; is there in this any offense to the\ngod of this place?\"\n\n\"There is no god,\" responded Ben Ezra. \"Here are but idols, and upon\ntheir altars couch the beasts of the field. We may go forward. Who\nneedeth to fear gods of stone, which are the work of men's hands and\nwhich neither walk nor speak?\"\n\n\"The lions have no god,\" said Lysias.\n\n[Illustration: \"Let him win it or perish!\"]\n\n\"I would not fear him greatly if they had,\" said an old viking, \"but if\nhe were a man, with a sword in his hand, then I would know what to do\nwith him.\"\n\nSome of the Saxons then declared that they knew what to do with the\nskin of such a lion, and they remained to take it off rather than go\nany nearer to the stone god behind the place of sacrifice. Grand and\nhuge was he, the idol of this broken temple of old time. He was the\nhead of a man upon the body of a beast, carved out of more stones than\none, and he crouched there, looking at them with a stern and terrible\nexpression.\n\n\"I think,\" said Ben Ezra, \"that he is one of the forgotten gods of the\nSidonians. They will not set him up in Egypt, but he was like Jupiter.\"\n\n\"There is no hammer,\" said Sigurd. \"It is not Thor. See the jarl!\"\n\nThey had paused, looking, but the son of Brander the Brave had walked\ncuriously to the side of the god and was studying his marks, for there\nwere many.\n\n\"I would,\" he muttered, \"that Hilda were here, for I think she would\nread. These are like the runes upon the old Odin stone beyond the\nfiord, and they were made when he came from the East. I think this to\nbe one of the Asas; but how came He to make this temple and place it\nhere? The gods do strangely at times.\"\n\nBy him now stood Lysias, and he said: \"O jarl of the Saxons, linger\nnot. The Jew hath found a stone which must be lifted. He waiteth for\nthee.\"\n\nNo message had Ben Ezra sent, but he was stooping over a flat slab in\nthe place of sacrifice. Upon it there were marks of fire and the stone\nwas crumbling.\n\n\"Why lift it?\" asked Ulric, drawing nearer. \"What have we to do with\nthe secrets of the gods? Why should we anger them?\"\n\n\"They are dead,\" said Ben Ezra, \"but I think this to be a door of the\npriests. It is but a broken stone. Give me thy spear.\"\n\n\"Nay,\" said Ulric, \"I can pry with a spear shaft. We will have it up if\nanything may be hidden here for us.\"\n\nThe fire-broken limestone yielded in several pieces to the prying of\nthe tough spear shafts. As its pieces were lifted, or as they fell\naway, behold stone steps, from which all shrank back save the Jew and\nthe son of Odin and the Greek. Even Sigurd held back a moment, saying:\n\n\"I like it not. It is the jarl's place. Let him venture first. He\nknoweth runes that we know not. So doth the Jew, but the Greek is a\nyoung fool.\"\n\nDangerous indeed it was for any man to step into a chamber under the\naltar of a strange god, but when they went down and entered it and\nlooked around there was but little to see.\n\n\"A store of broken weapons and rust-eaten armor,\" said Ulric. \"Some of\nthe hilts and shields are good enough. But there are many skulls and\nbones.\"\n\n\"A crown!\" shouted Ben Ezra, with a round thing in his hand glittering.\n\"Here placed they the ashes of kings from the altar. I know not why\nthey should have buried with one the diadem of his realm. It may be\nthat his dynasty was ended. Many of these stones are rare and precious.\nHere is gold, also, but the silver is of no great value. Let us bear\nall to the ship, for the spoil of this sacred tomb of the kings would\nbuy a Roman province.\"\n\nThe vikings in the outer air were summoned, and now they were not\nunwilling to venture, for the fear of the place had departed when they\nheard again the voice of the jarl. Neither did they care overmuch to\nfind merely the remains of the dead, and they were greatly pleased with\nthe treasures.\n\nBen Ezra bore away one shield which was heavy with gold, and in the\nmiddle of it was a jewel so like a great red eye that the vikings said\nit was looking at them revengefully, and they would not touch it.\n\n\"This place the Romans missed in their search,\" said Ulric. \"Little\nreverence have they for the altars of unknown gods.\"\n\nEven heavy were the burdens carried to the ship, and now all who\nhad been left to guard her were entitled to take their turn in the\nexploration of the city. They went and they came, but they found\nnothing to bring with them and they slew no wild beasts. They reported,\nhowever, that they had seen a leopard and a number of hideous beasts\nwhich Ben Ezra told them were hyenas, which delighted to feast upon the\ndead of battlefields. Successful fishing had been done in the harbor\nwith the small boats, and there was enough for all, but that night\nthere was much murmuring over the lack of fresh meat.\n\n\"Besides,\" said some of the men, \"this strange treasure hath its value,\nbut there hath been no good fighting. When will this jarl of ours\nlead us to a throwing of spears? The months of the summer are already\nwasted.\"\n\nTo these an answer was given by Sigurd, the son of Thorolf, that to\nthem was the fault, for by reason of their unruliness had there been\nneedless landings and delays on the coasts of Gaul and of Spain, and\nidle cruising after fishing boats and empty merchantmen which fought\nnot and paid but little.\n\n\"And the jarl forbade us to slaughter their crews,\" said one. \"I would\nhave slain all.\"\n\nMen who will to find fault may readily prepare a cause. Thus far the\nvoyage of _The Sword_ had been even too prosperous, being guided by\nprudence, and there was lacking the curbing which cometh from wholesome\ndisaster. The weather was all too warm for Northmen, and some few of\nthem had sickened, and of this sickness had four vikings died a cow's\ndeath but for the mark of a spear which was given them by the hands of\nfriends. Now, also, the skin of the lion aroused jealousy against the\nGreek. It was declared that an hour must be found for him to feel an\nedge of a seax, for he was not a Saxon and there should be no outland\nmen like him and Ben Ezra upon a ship from the Northland. The jarl was\ntoo hard in some matters and he was too soft in others. Nevertheless,\ndays went by while all looked at these temples and houses and the\nmighty fortifications. As for the jarl, he explored somewhat, but he\nabode mostly with the ship. He was silent and moody, for there were\nmany things upon his mind.\n\n\"I have come far out into the world,\" he thought. \"I have seen that\nwhich is exceedingly marvelous. I have looked, also, upon the face of\na dead god. Now I will go on until I may have speech with one that is\nliving.\"\n\n\n\n\n                             CHAPTER XVII.\n\n                       THE MURMURING OF THE MEN.\n\n\nOut of the African harbor sailed _The Sword_ with a good wind, and\nthere was no present need for rowing. No longer were the Saxons willing\nto linger in that place and live upon fishes. Small pleasure was to be\nhad there, they said, save to lie at night and listen to the cries of\nmany wild beasts. They had not hunted at night save that one of the\nyouths of Sigurd's party had ventured beyond the jetty foolishly and\nhad not returned. Blood had been found in the morning, but not any of\nhis bones. It had been better if the weather had been rough or if the\nmen had been at the oars, for in their idleness upon this blue and\npeaceful sea was an occasion for discontent.\n\n\"The jarl must do better than this!\" they said to each other, and as\nthey talked of battles the thirst for blood increased among them, for\nit is as a wild fever when it cometh.\n\n\"O jarl,\" said Sigurd, the son of Thorolf, not long after _The Sword_\npassed beyond the ledge whereupon so many had been wrecked by reason\nof the revenge fire of Annibaal, \"I think we do well if we steer now\neastward. We shall find too many Roman triremes in this neighborhood.\"\n\n\"I would seek them,\" said Ulric, \"if not too many of them were\ntogether. Dost thou know of a shore or an island where there are\ncattle?\"\n\n\"Verily I do,\" said Ben Ezra, \"but I know not if we may find it\neasily. We may but sail on. Lysias is with the steersman now, and he is\npointing.\"\n\nVebba, the son of Uric, was at the helm, and he hated the Greek, but he\nlistened, for he could not despise a good bowman.\n\n\"I would carve the blood eagle on thy back,\" he said, laughing, \"but if\nthou wilt guide to where we may slay somebody, thou art better worth\nkilling. I hate thee.\"\n\n\"So do I hate thee,\" said Lysias, boldly, \"but we may not fight on the\nship. I will give thee thy sword play when we get to a good place. But\nI shall strike thy head from thy shoulders.\"\n\n\"Good!\" said Vebba. \"I like thee better. But bring us first to some\ngood fighting.\"\n\nThen went Lysias to Ulric and the Jew, and they conferred somewhat,\nbut Lysias passed from them to the after cabin, and came out bearing\nsomething that he took with him to the after deck.\n\n\"I saw it there,\" said Ulric. \"It is a harp, not half so large as that\nof Oswald's. What can the Greek do with it?\"\n\n\"Wait and see,\" replied Ben Ezra. \"Among the Greeks are those who are\nskilled in music. Hearken!\"\n\nAll ears upon _The Sword_ were suddenly turned to listen, for the harp\nwas a good one.\n\n\"He playeth well!\" said Sigurd. \"No man shall slay him. We needed\nharping.\"\n\n\"Aye,\" replied the discontented men, and then they shouted to Lysias:\n\"Sing!\"\n\nNot at once was he ready to sing, and the harp sounded on as if he\nheard them not.\n\n\"Sing! Sing!\" they shouted again. \"Sing, or we will slay thee!\"\n\n\"Slay on, cowards!\" laughed Lysias, angrily. \"What care I for slaying!\"\n\nFor he had been muttering hoarsely to himself something about Sapphira\nand there were tears in his eyes.\n\n\"Down!\" shouted Sigurd, to a viking who was drawing his seax. \"Harm him\nnot, lest I send thee a spear! I would hear his harp. Down, I say!\"\n\nThe spear of Sigurd was a matter to be avoided, and the seaman left his\nweapon sheathed and sat down. But at that moment arose the voice of\nLysias in a grand Greek song, a song of war and of contending warriors.\n\n\"Right!\" shouted the men to Sigurd. \"Thou shalt slay any that shall rob\nus of our harping. He singeth well.\"\n\nNone would have expected a voice so powerful and so sweet, and they who\nheard it clapped their hands or clashed their spears upon their shields.\n\nThen the war song ended, and the harp began to send out low, sweet\nmusic that made them think of the Northland. They said to one another\nthat now the trees were in leaf, and the grass was green, and the wind\nwas in the pines, and the waves were on the shores, and the voices of\nthe gods could be heard in the fiords. The women and the children, too,\nwere in the houses, or they were caring for the cattle, and the fisher\nboats were out from all the villages. So they grew quiet and looked\nacross the blue waters of the Middle Sea less discontentedly, and the\nthirst for blood waned away for the hour. And yet they knew not that\nnow the Greek was singing in his own tongue of Sapphira the Beautiful,\nand that he did not at all see the ship, or those who were in it, or\nthe sea, but that his eyes, like those of the blind, were seeking far\naway for a face and a form that were out of sight, beyond--he knew not\nwhere.\n\nHis own countenance, with its perfect outlines and its youthful color,\nexhibited his sadness in keeping with the flowing music of his lyre,\nbut he knew not that the eyes of Ulric and of Ben Ezra were reading\nhim. Unlike the rest of the vikings, excepting Sigurd, they understood\nthe words of the song, which was from one of the old poets of the\nbetter days of Greece.\n\n\"I have heard,\" whispered Ulric, \"that even as he saith, the young\nwomen of his people have great beauty.\"\n\n\"Yea,\" returned the Jew, \"I have seen many of them. I have seen this\nSapphira, and she did excel. But no maidens are as those of Israel and\nJudah, the roses of Sharon and the lilies of the valley. Their voices\nare those of birds and their forms are of the heaven. Such was the\nmother of my son in her youth. Such were my daughters. I am glad that\nthey fell by the sword----\"\n\n\"How were they not captured by the Romans?\" asked Ulric.\n\n\"Because of the swords of their husbands and their brethren,\" said the\nJew, calmly. \"All died together, but the fairest of them needed no\nsword save her own. She chose to die by her own hand rather than to\nbecome the sport of the heathen.\"\n\n\"She did well,\" muttered Ulric. \"She was dark and she was beautiful.\nShe was brave and true. I have never loved, but I would I could find\none like her.\"\n\n\"If she were of the race of Abraham,\" replied Ben Ezra, \"she might not\nwed save with one of her own people. That is our law concerning women.\"\n\n\"It is a good law,\" said Ulric. \"Hilda, the saga woman, told me of it.\nShe said that ye have good sagas of your own and that your runes are\nancient. Are there any among you that are descended from the gods?\"\n\n\"We have but one God,\" said Ben Ezra, \"and all we are his children, for\nhe is the creator and father of men.\"\n\n\"He is Odin, the all-father?\" said the young jarl, inquiringly. \"Then,\nwhen I get to Asgard, I shall see him. I have thought much concerning\ngods. That was a strange one in the temple in the city of ruins. He\ngave us much treasure.\"\n\n\"We took it,\" said Ben Ezra.\n\n\"Yea,\" replied Ulric, \"we did so. But the Romans did not find it, nor\nany others that came, until the god who sat there watching permitted it\nto be taken. That was but his stone face that we saw. Thou knowest not\nmuch of gods, to think that he saw us not. Is thy god blind, that thou\ncanst hide away from him?\"\n\n\"Not so,\" said Ben Ezra, thoughtfully. \"Talk no more. The Greek hath\nceased. I think thy men like him better, but there is a spear waiting\nat any hour for either him or me.\"\n\n\"So is mine waiting for him who may cast his own unduly,\" said Ulric,\nangrily, \"and that know they well. But the sun is sinking and a sail is\nin sight. Sigurd seeth afar. He is coming.\"\n\n\"A small trireme,\" said Sigurd, as he drew near. \"I think we must take\nher.\"\n\n\"Take her,\" said the jarl. \"Oars, all! Vebba, son of Uric, steer for\nyonder keel!\"\n\nLoud rang the shouts of the Saxons and the discontented became\ngood-humored, but there was little need for fast rowing. The stranger\nwas nearing them at its best speed, and ere long they could hear the\nsound of a trumpet.\n\n\"The grapplings!\" commanded Ulric. \"If we may not strike her with the\nship, we will board her!\"\n\nSwiftly the two keels approached each other, and rash indeed were the\nRomans, for they were arrogant, not knowing with what they had to deal.\nThey saw the Saxon flag on the mast. They heard the war horns. Many\nmen they saw not at the first, for concealment of his strength was the\nprudence of the jarl, lest his enemy might strive to escape. All the\nmore freely did the fighting men of the small trireme crowd her decks\nand gather at her bulwarks.\n\nEven from afar did the arrows of Lysias and Tostig and other bowmen and\nthe slingstones of Knud begin to go in among them, angering them as\nsome of them fell, hurt or slain. They, too, had bowmen, but neither\ngood nor many, and their arrows were short.\n\nCunningly did Vebba veer away _The Sword_ at the nearing, that a flight\nof spears might hurtle among the Roman soldiers, thinning them. Past\nthem shot the swift keel of the Saxons, only to turn again suddenly,\ncrashing back upon their further banks of oars. They, too, had been\nready for boarding, but their bulwarks were not so high as were those\nof _The Sword_. Her grapnels were well thrown, moreover, and the two\nships were as one when the legionaries made their brave rush to climb\non board their enemy. Well had it been for them if they had been more\nin number. Well if they had not been so rashly self-confident, and if\nthey had not been half beaten by astonishment at the sudden appearing\nof the Northmen at the ship's side.\n\nWith laughter and with mocking did the Saxons hurl their spears and\nthen follow with sword and ax. Over the bulwarks they went, through the\ngaps left by slain Romans, and quickly they went two for one, slaying\njoyously. No Roman thought to surrender, nor was any mercy in the\nhearts of the vikings, but among them all did none smite more eagerly\nthan did Ben Ezra and Lysias.\n\n\"Slay! Slay!\" shouted the Jew. \"O Greek, thou art too slow. Hew down!\nSmite under the fifth rib! Let none escape!\"\n\n\"Good fighters are they!\" shouted Vebba from the after deck of _The\nSword_. \"I will have a fine contest when I slay that Greek. I will\nfight him fairly. But I must get the Jew before me to see how he will\nhandle that crooked blade. He cleft a Roman to the chin. Hah! I am but\nsteersman and I miss the killing.\"\n\nSo did others of the vikings, for there were not enough on the trireme\nto put blood upon every good sword or spear. They were all gone too\nsoon, and there was disappointment. Nevertheless, the legionaries had\ndied hard, and nine of the Northern heroes had gone to Valhalla.\n\n\"To them the gods were kind,\" said Sigurd, \"but this trireme is a fair\nprize. There are ten head of small, fat cattle, besides four fresh\ncarcasses. We must have them on board _The Sword_, with the other\nplunder, before we kindle the fire.\"\n\nThe men were attending to that, for here was their fresh meat without\nthe trouble of landing to find it. All of the slain might be burned\nwith the trireme, with all honor, so there was no more care for that.\nSome Saxons were wounded, but not so that they might die, and there\nwere no prisoners. All provisions and arms were taken over speedily and\nthe good spirits of the men were returning, for none of them waited for\nneedless cooking of the beef that was ready. Roasting might be done\nafterward, but the sharp knife could shred, and a viking cared for\nlittle more at the end of a won battle.\n\n\"Fire, now,\" commanded Ulric, at the last. \"Throw off the grapplings\nand let her drift away. I would see her burn.\"\n\nSo the rowers pulled to a little distance and paused, letting _The\nSword_ rock gently over the soft waves while the fire blazed more and\nmore brightly upon the decks and in the waist of the Roman trireme.\n\n\"She burneth well,\" said Sigurd.\n\n\"So burn every Roman keel!\" exclaimed Ben Ezra. \"Jehovah of Hosts hath\nbeen with me this day, and I have gotten vengeance upon mine enemies.\nMy sword hath been deep in the hearts of the heathen.\"\n\nLysias was silent, but his fair face glistened with pleasure as he\ngazed upon the mounting flames. His lyre was now in his hand again and\nhis fingers wandered over the strings.\n\n\"The harp! the harp!\" shouted some of the vikings. \"If he playeth not,\nwe will slay him.\"\n\n\"An evil spirit is among them,\" muttered the Jew. \"Whence he cometh I\nknow not. Who shall cast him out? for we have neither scribe nor priest\non this accursed vessel. I think that he belongeth to the idol upon the\nfore deck.\"\n\nIn that he spoke of the anvil and hammer of Thor, for to him the Saxons\nascribed the gift of this victory.\n\n\"He is a demon,\" said Ben Ezra, \"and he hath entered into these\nuncircumcised. I would he might lead them to Gehenna.\"\n\n\"The harp! the harp!\" again demanded the vikings, and the voices of the\nrowers were joined to the shouts of those who were feasting.\n\n\"The wind riseth well and so do yonder flames,\" said Lysias; \"but they\nwho are dead feel no pain of burning. Within me is a fire which is a\ncontinual torment. The harp were a relief, and I will sing.\"\n\nIt seemed as if a strange spirit of wild song had come upon him and\nhis lyre. It mattered not greatly that few of the vikings understood\nhis words, so fierce and so triumphant was the music of his singing.\nMoreover, they looked upon his face and it gave them an interpretation,\nfor there was a terrible meaning in its expression.\n\nNow the rowers ceased and the sail was up, but the burning Roman ship\nalso felt the fresh wind, and it was as if it strove to keep them\ncompany while Lysias sang.\n\n\"She will founder shortly,\" said the jarl. \"We are leaving her. I would\nI knew more nearly whitherto we have come. We are far in the Middle Sea\nand we should be near some of its islands.\"\n\n\"Thou knowest,\" remarked Sigurd, the son of Thorolf, \"that opposite\nto the southerly point of Italy there lieth a great island, whereon\nis a volcano, vomiting fire, for under it is the world which burns,\nand there do the gods war with one another. I think we are between\nthat island, which is called Sicily, and a part of Africa. O Jew, what\nsayest thou? Thou hast visited many parts of Africa.\"\n\n\"We have wandered here and there,\" said Ben Ezra. \"The question is\ndifficult. But if yonder haze telleth of the coast of Sicily we may\nmeet another trireme soon. There are many hereabout. They will for the\ngreater part be merchantmen.\"\n\nDown sank the vessel they had burned, with much loud hissing of fire\nmeeting water, and the clouds of smoke and steam went up while the\nSaxons blew their war horns and shouted their exultation. They all had\nfeasted well, however, and those who were not on watch were willing to\nslumber while the increasing gale carried _The Sword_ swiftly toward\nthe east.\n\nAnother was at the helm, and Ulric, the son of Brander, went and sat\ndown upon a silken-covered couch in the after cabin. He was alone, and\nhe brought out his jewels to look at them. They were many and they were\nbeautiful, and he turned them over one by one.\n\n\"Never before,\" he said, \"did I have so good a lamp as this that\nhangeth here. The oil, too, is perfumed and the room is full of a sweet\nodor. These are the ways of the Roman captains and rich men. I may not\nsee Rome, for there are too many quinqueremes in the way, too many\nlegions of warriors on the land. We are few. I do not care much for\ntheir gods, for I have beaten them. I will go on to Asgard, but I will\ngo first to this temple in Jerusalem. Ben Ezra saith that I can buy\nboth priests and governors with these bright stones. But I may have to\nslay my own men if they obey not. If I cut down a few of them the rest\nwill be more peaceable. These Saxons that came with Sigurd hardly call\nme their jarl. If they were dead it would not matter. I will go my own\nway.\"\n\nThe ruby was now in his hand, the great red stone that was graven with\nthe name of the Hebrew god, and among them all there was no other like\nthis. It glowed like fire in the lamplight, and Ulric said: \"It is full\nof blood. It is a stone of stones. But whence came the blood, and how\nis it full of fire? Is he angry with me? I think I will carry his gem\nto him in his temple, and I will tell him I have brought it back. I\nwould not keep from any god that which is his.\"\n\nSo he put it back into the casket and took out an emerald. This, too,\nwas graven with deeply cut runes.\n\n\"One of them,\" said Ulric, \"is like the runes that Hilda showed me in\nthe sand by the sea, but it is alone. I care not until there are three.\nIt is green and wonderful. O Hilda of the hundred years, would that I\ncould show to thee this jewel of the old gods!\"\n\nThe lamp burned low and it was flickering. Without the gale roared\nloudly and the waves beat against the sides of the ship with a groaning\nsound. There was no voice but of the wind and of the surges. The\ncurtains in the cabin swayed to and fro, as did the cresset of the lamp.\n\n\"Hilda! Hilda!\" murmured Ulric, but he saw her not, and even his\nthought of her was confused in his mind. The saga woman was tall and\ndark, but not so tall and fairer was this thought which came before his\neyes as if he were in a dream:\n\n\"So beautiful! So beautiful!\" he said. \"Her eyes are like stars and her\nhair is a cloud of shining curls. Her lips are like the ruby of the\ntemple. I think she is one of the Hebrew maidens that Ben Ezra saith\nexcel all others. I will go to that land and find her, for it must be\nthat she also is of the daughters of the gods. And now I can see Hilda,\nand her hair is white, but her eyes are shut. Therefore I know that\nthey have carried her to the tomb that was made in the rock of Odin. I\nshall see her no more until I get to Asgard. If this is her hand upon\nmy head, she should speak, for I love her well.\"\n\nHe listened, and the lamp went out, but no voice came; and he lay down\nupon his couch, but a fire was kindling in his heart.\n\n\"Lysias loveth Sapphira,\" he thought, \"but thus did I never feel\nbefore. The Hebrew maiden! I would Ben Ezra had not told me of her, for\nnow I can have no other. I had thought that my love would be blue-eyed\nand a daughter of Odin. Shall I not be content if I find that she is\ndark, and that she is a daughter of this Jehovah, the god of the Jews?\nI will go on and I will see what she will say to me.\"\n\nThen he slept, and _The Sword_ swept onward swiftly toward the sunrise.\n\n\n\n\n                            CHAPTER XVIII.\n\n                    THE EVIL SPIRIT ON \"THE SWORD.\"\n\n\nThrough one day more the western gale blew furiously and _The Sword_\nwas driven before it, for none on board cared for any better steering.\nMany vessels were seen from time to time, but all were too busy caring\nfor themselves to pay overmuch attention to a trireme that might be\nfighting the storm as they were. The vikings were at ease concerning\nthe weather, but they grumbled much that the tossing and pitching of\ntheir ship prevented them from making fires wherewith to roast their\nbeef or to broil their fish. On board their Roman keel they had found\ngratings of iron for cooking, better than any of their North making.\nThese gratings were wide, upheld by iron feet, and under them were\nslabs of stone to receive ashes and cinders. Fire would remain upon\nthem well in any ordinary weather, but in such as this the brands and\ncoals might be cast hither and thither. It was not even a time for the\ntelling of sagas nor for the lyre of Lysias, and again the men grew\nmoody and sullen.\n\nThe night returned, and Ulric kept the helm through all its watches,\nfor a heavy weight was on his mind and he had heard from Ben Ezra\nconcerning the evil spirit. \"I would I could slay a demon,\" he had\nanswered, \"but of what good is a spear for an enemy thou canst not see?\nIt were almost as if one fought with a god. I have thought I would like\nto fight with one, but not with Thor or Odin nor with thy Jehovah.\"\n\n\"They who contend with him are broken,\" replied the Jew, \"but I tell\nthee we are far on our way. I think we are not far from Cyprus. We\nmight safely land in one of the havens of that island.\"\n\n\"We might meet a Roman fleet,\" said Ulric.\n\n\"They have none in these waters,\" said Ben Ezra. \"Their merchant ships\nof any consequence go and come in squadrons, well protected, and they\nhave driven out all pirates. They will not be watching, I think, for\nthe coming of such as thou art.\"\n\n\"We now are late in the season,\" replied Ulric. \"I had thought to have\nreached this water before any other keel from the North. We know not\nwhat may have called upon the Romans for watching. I am thinking that\nwhen this wind abateth I must find the men somewhat to occupy them.\"\n\n\"An evil spirit is a busy one,\" said Ben Ezra. \"All thine would find\nenough to do in Cyprus.\"\n\nAfterward many of the men came to the jarl with questionings, and also\nto the Jew and to Lysias. These were looked upon with more favor for\nthe time, for it was said that they might have some worth for piloting.\n\nA night and a day and a night went by and now the waters were again\nquieted. They were even too still, for the rowers had to be sent to the\noars, and the sun looked down upon them with fervent heat, making their\ntoil burdensome. Once more the ship was floating upon an even keel and\nthe men speedily bethought them of the fire gratings. Twain of the fat\ncattle were butchered, and the jarl thought well of it, that the men\nmight be kept in good humor. The fires were lighted, and casks of ale\nwere opened, but the evil spirit was, nevertheless, making himself busy\namong the hearts of the men.\n\nIn the trireme _The Sword_ itself, when she was captured, there had\nbeen a few skins of wine, but it had been red and sour and the vikings\nliked it not. Such as it was it had long since been consumed. In the\nspoil of the burned trireme, however, and hardly noticed at the first,\nthere had been found many wine skins. All had been taken with care, and\nnow one of them was opened to find out what it might be.\n\n\"Dark and sweet and good!\" exclaimed Vebba, the son of Uric. \"I will\nbear a horn of it to the jarl.\"\n\nLarge was the drinking horn, and he filled it to the brim. Sparkles\narose upon the surface of the wine, and it seemed to laugh, as if the\nevil spirit which lived in it were accomplishing his purpose.\n\n\"It is strong,\" said Ben Ezra. \"Drink it not, O jarl, for the demon of\nwine is thine enemy.\"\n\nBut he was too late, for the son of Brander drained the drinking horn\nas if it had held naught but ale. He felt it from his head to his feet,\nas if it had been poured upon a fire that was burning within him, and\nhe stood erect, straightening himself and clinching his hands.\n\n\"Bring me another horn of it,\" he said.\n\n\"That thou shalt not do,\" commanded Ben Ezra, sternly. \"Thou art the\ncaptain. I bid thee drink no more, lest thou lose thy life and thy\nvessel. The demon is upon thee, O jarl! Resist him, or he will bind\nthee hand and foot.\"\n\nThen remembered Ulric a saying of his father and of Hilda, and it was\nas if he had heard her voice saying: \"Son of Odin, beware of the dark\nwine of the south lands, for in it is death.\"\n\n\"Bring me no more,\" he said to Vebba, \"and let the wine skins be cast\ninto the sea.\"\n\nBut the demon had been very busy and from lip to lip had already passed\nthe goblets and the drinking horns. They had been emptied only to be\nquickly filled again, and now the Saxons of Sigurd shouted:\n\n\"Haha! O jarl! Thou wouldst rob us of our feast? We will show thee a\nthing.\"\n\nSigurd himself went among them, but to him, also, they paid no heed,\nand he came back again.\n\n\"I am sleepy,\" said Ulric. \"Wulf the Skater, these three nights I have\nwakened. I will lie down for a while. Take the helm.\"\n\nThen came Tostig the Red and Knud the Bear and four other Saxons of the\nhouse of Brander, and they sat down by Ulric, spear in hand, with their\naxes lying by them. Lysias brought his bow and Ben Ezra closed the\nvisor of his brazen helmet.\n\n\"Trouble cometh,\" he said. \"The heathen are full of wine and of the\nthirst of blood.\"\n\nThere was no quarrel between twain of the vikings that were stepping\nforth upon the fore deck, but they were berserkers, and their seaxes\nwere in their hands, for they were to fight without mail or shields.\n\nSkin after skin of that dark, strong wine was opening, and the men\nloved it, but they would see blood, they said, and the two berserkers\nshouted as they fought.\n\n\"Both of them are down!\" exclaimed Lysias. \"Two more take their places.\nO that the jarl were awake! But I cannot rouse him. Were the Romans to\ncome, we were all dead men.\"\n\nFurious was now the drinking, and a man cast a spear at another without\ncause, laughing to see him struggle and bleed.\n\n\"The evil spirit hath entered them all!\" groaned Ben Ezra. \"This is\nthat which I feared greatly. Every man's sword is against his neighbor.\"\n\nTerrible was that fighting, for warriors who had lost all skill of\nwarding blows or parrying spear casts were still strong to throw or to\nstrike.\n\n\"Where is now this jarl of ours?\" yelled a drunken viking. \"We will see\nif he be a son of Odin. We will slay him and then we may sail at our\npleasure. He hath ruled us with too hard a hand.\"\n\nSteady and stern had indeed been the rule of the son of Brander, and\nhe had brooked no gainsaying, but he had been a prudent captain from\nthe first, and there were a full third of the men now to stand by him\nin his peril. Would there had been more, for on both sides the slain\nwere many. Moreover, when a man went down that was quickly his end, on\nwhichever side he fought, for an enemy came to thrust him.\n\n\"Wake, son of Brander! Wake!\" shouted Tostig in the ear of Ulric. \"Call\nthou upon Odin, thy father, and draw thy sword.\"\n\nWaiting for no orders from any man, Lysias was sending his arrows, sure\nand deep striking, calling out:\n\n\"With me be thou, O Apollo, god of the bow! With me, O Mars, god of\nbattles!\"\n\nBut Ulric opened his eyes slowly and breathed hard. Then he sat up and\nhe saw the men fighting and the blood flowing.\n\n\"Odin!\" he roared, in a voice they had not heard before; but the weapon\nhe lifted was his pole ax, and he rushed forward to the front of his\nfriends.\n\n\"I go with him,\" said Ben Ezra. \"It may be his god hath come to help\nhim. Be with me, O Jehovah of strength!\"\n\n\"We will guard thee at the helm,\" said Knud and Tostig to Wulf the\nSkater. \"This will be ended speedily. Look at the jarl!\"\n\n\"He, too, hath a demon!\" burst from the lips of Ben Ezra, as he saw\nUlric striking. \"They go down before him like corn before the reaper.\"\n\nSigurd had been smitten to the deck by his own Saxons, and Ulric stood\nover him with his ax until the son of Thorolf was hidden by corpses of\nthe slain.\n\nMad with wine and with the fever of the thirst of blood, the rebellious\nvikings fought on, nor would they yield to the command of the son of\nBrander.\n\n\"We will die!\" they said. \"But we will first slay thee. It is a feast\nof swords.\"\n\n\"I would I could spare enough for rowers,\" said Ben Ezra, \"but their\nblood is on their own heads. The evil spirit destroyeth them.\"\n\n\"Thus endeth the cruise of _The Sword_!\" said Ulric, sadly, when at\nlast he might pause for breath. \"Save thee, O Ben Ezra, and Lysias,\nand these few faithful, there are none living save some for whom the\nvalkyrias are calling. What shall we do? for thou art old. What shall\nbe the end of these things?\"\n\nBut Tostig and Knud had watched the falling of Sigurd and they were\nlifting from him the corpses.\n\n\"The sail is up,\" said Ben Ezra. \"Steer eastward, for we may not do\naught else now than land in Syria. Thou and thine shall see Jerusalem.\"\n\n\"So be it!\" said Ulric. \"I think we are none of us wounded. I am not.\"\n\n\"Glad am I of that!\" exclaimed Lysias. \"I feared for thee in that\ncombat. But thou art of the heroes and Jove was thy keeper, with Mars\nand Apollo.\"\n\n\"A feast of blood!\" exclaimed Tostig as he lifted the body of Vebba,\nthe son of Uric, from Sigurd, the son of Thorolf. \"The sea king is not\ndead. He was but stunned.\"\n\nSlowly arose the old warrior until he sat erect and looked around him.\n\n\"I saw them!\" he said, huskily. \"I saw the Nornir in the air above the\nsail. I saw the valkyrias, but they looked sternly at me and passed by.\nWhy, I know not, for I fought well. Odin hath taken many this day. O\njarl, what doest thou?\"\n\n\"Eastward!\" said Ulric. \"Canst thou stand upon thy feet?\"\n\nTostig and Knud aided him, and they brought him a goblet of ale, for\nwine he would not drink.\n\n\"It is well with me now,\" he said. \"My helmet is cloven, but my skull\nis safe. The ax of Vebba was heavy, but he will strike no more. Sad is\nit that he and these are slain. Better had they fallen in a fight with\nthe Romans.\"\n\n\"Not so,\" said Ben Ezra, \"for if some of them were living all we were\ndead. Let us cast them into the sea.\"\n\nWulf the Skater had watched the clouds, and now he said:\n\n\"Ulric the Jarl, if thou wilt, they should be over the sides speedily,\nfor a wind cometh. We shall use no oars henceforth.\"\n\nSad work it was to cast so many forms of dead heroes into the sea, but\nso had it been foredoomed by the Nornir, and there were some of the\nwounded who died while the task went on.\n\nThen Ulric sat down by the hammer of Thor and bowed his head, for his\nheart was heavy.\n\n\"I can sing no song,\" he thought, \"over such a fight as this. I think\nit will now be long before I see the fiords and the hillsides of the\nNorthland. My fate hath changed for me in an hour, and I know not what\ncometh. O Hilda, was this thy dark saying, that I understood not?\"\n\nNo voice responded, nor any motion of the air, but he looked upward and\nhe saw birds that were flying eastward.\n\n\"So will I go,\" he muttered, \"and they who are with me. There is too\nmuch blood upon this keel. I would she were burned with fire, for I\nhate her. The gods of the Romans have had their revenge upon me. I will\nnever again speak lightly of any gods, for they have ways of their own\nand they are cunning. Who shall protect himself against an enemy whom\nhe cannot see?\"\n\nWell blew the wind, and there was little now to be done save to steer\nand to rest. All ate and drank, and Ben Ezra seemed to love that dark,\nstrong wine, but he used it sparingly.\n\n\"It is made in my own land,\" he said, \"but this came from a Greek\nisland, I think. There is good wine in Canaan. I would eat again of\nthe grapes and the pomegranates of Israel and Judah. O my son! That he\nmight have been with me! O my Rachel and my daughters and my firstborn\nand his brethren! The curse of Jehovah be upon Rome forever! Amen!\"\n\nSo the old Hebrew warrior wailed in the bitterness of his soul, and\n_The Sword_ sprang on over the billows, bearing him to his own land,\nbut she was now no longer a warship.\n\n\"We will not count the days,\" said Ulric to Lysias. \"We will speak to\nnone that we may pass.\"\n\n\"Pause not!\" replied Lysias. \"Thou hast thy life yet and I have mine. I\nhave it in my mind that I shall see my Sapphira. I have had a dream in\nthe night and she stood and beckoned me.\"\n\nUlric answered not, but that night he slept upon the deck dreaming, and\nin the morning he thought about his dream also.\n\n\"Hilda was there,\" he said, standing at the helm looking across the\nsea. \"Behind her was the sun rising. Between her and the sun were many\nwarriors, heroes of the gods, armed for battle. There was blood on some\nof them. But at the right hand of Hilda stood that dark and beautiful\none, and there were flowers in her hair, and the flowers were both red\nand white.\"\n\n\n\n\n                             CHAPTER XIX.\n\n                     IN THE NIGHT AND IN THE FIRE.\n\n\nDays come and go and no man may hinder them. The vikings went to and\nfro about _The Sword_ and she seemed lonesome to them, for they were\nfew and she was a great vessel. From time to time many sails were seen\nnear and far, but none gave chase to _The Sword_. Even pirates and all\nmerchantmen avoid what seemeth to be a warship.\n\n\"Winds have been both good and bad for us,\" said the jarl to Ben Ezra\nat the close of a day. \"What thought is in thy mind as to our nearness\nto any land?\"\n\n\"O jarl,\" said Ben Ezra, thoughtfully, \"by the stars that I have\nwatched; by the sun and winds; by the islands which we have passed; by\na dim understanding which cometh to a man in such a case; by all the\nsigns which are given me, we are so near to our destination that we may\nfind a shore this night.\"\n\n\"And if a shore,\" said Ulric, \"what shall it be?\"\n\n\"Even the land that was given to the children of Israel by Jehovah,\ntheir god,\" said the Jew. \"It is ours yet, but the Romans have taken\nthe kingdom from us.\"\n\n\"Their gods are very strong,\" said Ulric, \"and they are exceedingly\ncunning. Else had Thor and Odin saved to us the swords that sailed with\nus from the Northland. Thy god refused to fight with the gods of the\nRomans. I think he was wise in that. But he agreed with them that they\nshould not harm his temple, and I will go and see it. I may meet him.\"\n\n\"Thou wilt not see him,\" said Ben Ezra. \"He was seen by Moses, our\nprophet, but to all others he hath hidden his face.\"\n\n\"I know not that,\" said Ulric. \"They who see the gods are forbidden to\ntell. Hilda, the saga woman, loved me, but she would tell me naught\nconcerning the dead save that they have a country of their own. There\nis much good in that country and when I am slain I shall go to it.\"\n\n\"Thou art to die by the sword?\" asked Ben Ezra. \"How knowest thou that?\"\n\n\"I am of Odin,\" said Ulric, \"and a cow's death is not for me. There\nwill be blood in the hour of my going. If thou seest me on a bed, be\nthou a Saxon unto me, and smite me through with a spear.\"\n\n\"So said Saul, our king, to his armor-bearer at the end of a lost\nbattle,\" said Ben Ezra, marveling somewhat. \"I will do as thou sayest;\nfor verily thou art a jarl and of the princes of the North. Never\nbefore saw I a man like unto thee for battle.\"\n\n\"Save Sigurd, the son of Thorolf, the sea king,\" said Ulric, \"I have\nmet none that might stand before me. He too, is of one line of the hero\nAsas, but not of Odin.\"\n\nBen Ezra was silent, thinking of these things, and _The Sword_ drove\nonward. He and Ulric were at the prow as the darkness deepened. They\ncould see no more save the stars above and the glancing waves around\nthe ship, but they could hear the music of the lyre of Lysias on the\nafter deck. Knud the Bear was at the helm, and all that remained of the\ncrew were gathered there. They cared not to sleep in the cabins or in\nthe bunks, for some of them said that the dead came at night to look\nagain at the keel from which they had departed and that the evil spirit\ncame also.\n\n\"I saw him not,\" said Wulf the Skater, \"but Vebba, the son of Ulric,\nspoke to me, and I think he said the Nornir were at hand. So sayeth\nSigurd, the son of Thorolf.\"\n\nGreatly dispirited were they all, and the lyre was a comfort, but the\nsong of Lysias was low-voiced and sad and they could not understand the\nwords.\n\nNow from the fore deck came back to them one who had heard from the\njarl that they were to look out for a land and be ready to lower the\nsail.\n\n\"Good!\" shouted Tostig the Red. \"O Sigurd, go to the jarl and ask if we\nare steering rightly.\"\n\n\"That are we,\" said Sigurd. \"Seest thou not the north star? Go we not\neastward? What need to trouble the jarl? I would that they who are dead\nhad obeyed him. Then had we all been more joyful.\"\n\n\"Never had crew such adventures as we are having,\" said Knud. \"I think\nwe may gain some good fighting before long. My hand goeth often to the\nhilt of my seax and my blood is unquiet.\"\n\n\"A good sign!\" exclaimed Wulf the Skater. \"I feel better for hearing\nthee. O Greek, sing us a war song!\"\n\nLoudly answered the smitten lyre for a moment, and Lysias obeyed, but\nquickly came back from the fore deck the command of Ulric, the son of\nBrander.\n\n\"Silence, all!\" he shouted. \"There is a trumpet, far away southerly. We\nare too few and we near the land. Hark to the breakers!\"\n\nListening diligently, all ears heard the dashing of that water as if\nupon rocks, and yet again came up from the southward that distant peal\nof the trumpet.\n\n\"Struck!\" suddenly exclaimed Sigurd. \"We go upon a shore. Is this thy\nland, O Jew?\"\n\nNot with a great shock, but glidingly and grating hard, did _The Sword_\ngo on a little while the sail was lowering. Then she stood fast, and\nall on board of her knew that the end of her voyage had come.\n\nNeeding no command, the Saxon sailors made ready two of the small boats\nand prepared to lower them.\n\n\"The trumpet is nearer,\" said Ben Ezra. \"But this ledge of rocks cannot\nbe far from the mainland. Thy men seem to know not of fear and they\nobey thee.\"\n\n\"No Roman arms or armor,\" shouted Ulric. \"We land as Saxons and we will\nleave behind us no token. Kindle a fire amidships.\"\n\nTo his cabin went he and Ben Ezra, and unto them shortly went Lysias,\nbut each prepared bundles of his own to carry to the boats.\n\n\"No man knoweth of thy treasure nor of mine,\" said Ben Ezra to Ulric.\n\"Let the Greek, too, have gold and silver coins, for he will need them.\nHe hath fought well.\"\n\nIn like manner was every man furnished speedily and the burdens were\nnot made uselessly heavy. Nevertheless, Ben Ezra said to Ulric:\n\n\"Never before landed boats of thy people bearing to any shore such\ntreasures as are these. We may buy any Roman governor if in so doing\nwe do not hire him to put us to the sword. We will say that we were\nwrecked, but we must not be seen on the coast.\"\n\nNow the boats were lowered and all entered them, but in every quarter\nof _The Sword_ was a hot fire kindled. The Roman trumpet had not\nsounded again when the Saxons rowed away into the darkness.\n\n\"Row harder!\" commanded Ulric. \"The light of the fire increaseth. We\nknow not how near may be an enemy.\"\n\nWell had he spoken, for the flames were rising furiously and the light\nwind fanned them well.\n\n\"A shore!\" said Sigurd. \"A sandy beach!\" But all others were looking\nback at _The Sword_, to see how fast she was burning, and at that\nmoment there swept past her, outside, as if nearing to grapple her, a\nvast shape of a warship. Then arose suddenly a great volume of shouts\nin the Latin tongue, and the notes of a trumpet sounding commands, but\nUlric said in a low voice to his comrades:\n\n\"A quinquereme! And she also is upon the ledge of rocks. What shall\nsave her from destruction by that fire?\"\n\n\"She cannot escape,\" said Wulf the Skater. \"It is as if we had set a\ngood trap. I think the fire hath already caught her sail. There will\nmany Romans perish this night.\"\n\n\"Pull!\" commanded Ulric. \"The beach! We are here. Haul up the boats.\nOut with all cargo and leave them. Hark to the shouts of them who burn!\"\n\nRashly in swift haste had the Roman warship dashed forward to discover\nwhat might be this unusual thing, of a light that grew and of a crew\nthat replied not to a trumpet of hailing. Not of any rocky ledge had\nher steersman or her sailing master been thinking, and her centurion\nhad deemed it his duty to grapple and to board this strange burning\ntrireme. He would yet have passed her once, only studying her case,\nbut his own ship had smitten a sunken rock, which forced her to swerve\naside heavily, plunging her alongside of her fiery destroyer.\n\nIn vain were then all struggles to release the quinquereme. In vain was\nany effort to extinguish the swiftly devouring flames. Even of small\nboats the Roman ship had but four, and there were sailors who secretly,\nquickly lowered these, dropping into them to row away at once. Of these\nhurrying runaways there were none but hired Ionian rowers, and they\ncared for their lives only.\n\nIll fared it for legionaries in heavy armor, for if they sprang\noverboard, it was to sink. Sad was the fate of many who went into the\nwater, crowding and clinging, for they perished grappling each other in\ntheir astonishment and despair. The Roman warship was on fire from end\nto end, and the side which was not yet burning was toward the sea. What\nwonder that all discipline failed and that all thought of obedience was\ngone? for every bond is loosed by fire.\n\n\"If any follow, they must not find us on the beach,\" said Ben Ezra to\nUlric. \"I can see that the land riseth high and that there are great\nrocks. Let us depart!\"\n\n\"Odin!\" responded the jarl. \"_The Sword_ hath once more smitten the\nRomans. Every man take up his burden. Follow me!\"\n\n\"A good captain,\" muttered Ben Ezra. \"I will cleave unto him. But\nverily our lives are worth but little. I would that we were among the\nmountains, even in Gilboa or in Lebanon, or in the wilderness of Judea.\"\n\n\"Guide thou after daylight cometh,\" said Ulric. \"I would find crags and\ntrees.\"\n\nOn went they, climbing a steep, and ever and anon they looked behind\nto watch the awful splendor of the burning of the two ships upon the\nledge.\n\n\"Here may we halt,\" said Ulric at last. \"We are on a height. It is a\nforest beyond us. The fire burneth lower. There will be no pursuit.\"\n\nThere they sat down, therefore, wearied with their burdens, putting\nthese aside, and ere long they slept, every man, without fear.\n\nAt the ledge of rocks in the sea there was silence, for the two\nships burned to the water's edge and there was little left of them.\nNevertheless, of the swimmers there were a number who reached the\nshore, but all were of sailors unarmored, and no officer or legionary\nwas among them. Here at the beach they found the two small boats\nleft by the Saxons, with oars in them, but the four boats of the\nquinquereme, with the Ionian rowers, had landed further on. There was\nlittle to be done by these exhausted swimmers but to lie down and rest,\nand the Ionians were likewise waiting for day, being full of fear over\nwhat they had been guilty of in taking away the boats of their ship.\nOnly the sword could await them if they were found by a Roman patrol of\nthe coast, for they were to be accounted deserters from their assigned\nposts.\n\nNot long was the remainder of the night. The morn came, and when the\nsun arose Ulric, the son of Brander, sat upon a rock, under an oak\ntree, looking out upon the blue waters of the Middle Sea. Beside him\nsat Sigurd, the son of Thorolf, and scattered around upon the grass\nwere the other Saxons. Lysias stood and leaned against the rock, but\nBen Ezra was nowhere to be seen. In the hand of Ulric was the long,\nstraight sword that had been found with Annibaal at the ruined city on\nthe African shore, but it was sheathed, and the jewels of its golden\nhilt were glittering.\n\n\"There are men upon the shore by our boats,\" said Sigurd. \"They are\nescaped from the burning vessel.\"\n\n\"Look southward!\" replied Ulric. \"A squadron of Roman cavalry. Let us\nsee what they will do, but let us step back behind trees out of their\nsight. They are too many for us.\"\n\n\"Worse than that,\" said Lysias. \"Horsemen might carry an alarm and\nlegionaries on foot might hunt us in these forests.\"\n\nThe cavalry rode fast, and the men at the beach looked mournfully into\neach other's faces, for there was no fleeing from riders. Quickly came\nthese and their officer sprang to the ground, speaking loudly.\n\nThe light of the burning ships had been seen from afar, and even now a\nswift galley had arrived, rowing around the rocks of the ledge, while\nthey who were on board of her studied well the charred fragments.\n\nThe officer questioned with care the rowers, and a small boat from the\ngalley came to the shore with another officer.\n\n\"Were there other boats than these?\" he asked, pointing at the twain\nleft there by the Saxons. \"These are from a warship.\"\n\n\"Yea,\" said the centurion of the cavalry, \"and these deserters took\naway all chance for the escape of our comrades.\"\n\n\"We all swam ashore,\" they said, \"and we found these boats here. Other\nmen than we made off with them, We are innocent.\"\n\nThe two centurions looked at each other and they were of one accord in\nthis matter. At a word of command soldiers dropped from their horses\nsword in hand. At another word the work of punishment began and the\nstern justice of the Roman military law was done in utter injustice,\nfor not one of these who were slain had sinned.\n\n\"They had done somewhat in other days,\" said Ulric, \"and the vengeance\nof their gods found them here, bringing upon them a sword. No man\nescapeth the gods. But I see another man down the beach. He is fleeing\nas if for his life. I think, therefore, that these were not all who\ncame to the shore in some manner.\"\n\nGreat was the wrath and the dismay of all those Romans at this terrible\naffair of wreck and fire, but there was no sign to suggest to them the\npresence of Saxons on the sea or on the land.\n\nUnto the four boatloads of Ionian rowers at their landing place, where\nthey still lingered, came running the one of their number who had gone\nforth as a scout. Pallid with fear and horror he gasped out to them the\nthing that he had seen, and he fell to the sand breathless with running.\n\n\"To the mountains!\" they shouted. \"We are slain if we are found on the\ncoast. They now know not that we are here.\"\n\nThen it could be seen that not only had they taken plenty of weapons\neven in their hasty flight from the burning ship, but that their\napparel was decent. Also their talk indicated that they had many coins\nof money, and that they knew this country whereupon they had landed.\nThey stood still for a moment, and they swore to one another by their\ngods that this should forever be a secret, and then they marched away\nup the steep and were hidden in the forest.\n\nNeither had they failed, in their talk upon the shore, to wonder much\nconcerning the first burning vessel which had been the cause of their\nown disaster. They knew not of the Saxon boats, but they had said of\nthemselves that they would not willingly fall in with any who had\nescaped lest their peril might be increased.\n\n\"It were death,\" they said, \"and we must at once put any such men to\nthe sword.\"\n\nThe Saxon men, whom they did not know, but of whom they had been\nspeaking, were gathered together on the mountain.\n\n\"O jarl,\" said Tostig the Red, \"well that thou didst order us to bring\nprovisions, also, for our first needs. Shall we not now go on into the\nforest and find a place where we may kindle a fire?\"\n\n\"O Tostig,\" said the jarl, \"Ben Ezra is our guide. This is his country.\nWhat sayest thou, O Jew?\"\n\n\"Only this,\" replied Ben Ezra; \"that we are upon Mount Carmel, and that\nthe forests thereof are deep. We are safe if we are prudent. It is a\nwilderness into which not many come at any time, but there are villages\nand cities not far away.\"\n\n\"Lead on thou, then,\" said Ulric. \"Let every man bring all his burden.\nWe will keep up strong hearts, and we will see to what this strange\ncoming on shore will take us.\"\n\nThey had need of cheerful words from their jarl, for upon them all was\na shadow deeper than any of the shadows of the forest. Their faces were\ndark, but among them all was there no face like that of Lysias, the\nGreek. There was no light in it, but rather a bitter sullenness.\n\n\"Sapphira! Sapphira!\" he muttered, walking apart from the rest. \"Am I\nindeed nearing thee? Am I to find thee? Am I, then, to love thee again\nor am I to slay thee? Thou shalt not live to be the bondslave of a\nRoman, even though he be a prince and a ruler!\"\n\nUlric the Jarl heard him. It was as if he had been spoken to concerning\nthe Hebrew maiden whom he had seen with Hilda.\n\n\"I think that she is somewhere in this land to which I have been\nguided,\" he thought. \"I will go on and I may find her. This forest is a\ndense cover of this mountain. I shall be glad to look upon that which\nis beyond it.\"\n\nBen Ezra led onward rapidly, but the way by which he went grew steeper.\nThey came out at last, much heated by their heavy burdens, upon a level\nplace, where were no trees, and here he halted.\n\n\"Here let the fire be made,\" he said to the jarl. \"But if thou and\nSigurd will walk with me a little distance further ye will see\nsomething.\"\n\nGladly did the wearied Saxons pause and make their camp, but their jarl\nand Sigurd followed the Jew. Not far did these go until they came out\nupon a bold, high promontory of rocks.\n\n\"Look!\" said Ben Ezra. \"The Middle Sea.\"\n\nThere were no trees to hinder sight and the air was pure, so that they\nsaw afar. There were many sails and there were also galleys which might\nbe warships.\n\n\"O jarl,\" said Ben Ezra, \"thou art escaped from a Roman fleet. Thou\nwouldst not have done so but for the ledge of rocks and the fire which\ndestroyed thy vessel. Thou art on the front of Carmel. Now turn thee\nto thy left. What seest thou?\"\n\n\"A heap of stones,\" said Ulric. \"They have been shapely, but now they\nare broken down. Was it one of the altars of thy god?\"\n\n\"Not so,\" said Ben Ezra, \"but our fathers made that heap for a sign of\nremembrance. In the ancient days there was on that spot an altar to\nJehovah. Upon it the prophet Elijah sacrificed oxen and the fire of our\ngod came down and consumed both the sacrifice and the altar. Here was\nJehovah's victory over Baal, the god of the heathen, and here were all\nthe priests and prophets of Baal slain with the sword.\"\n\n\"If thy god is here,\" said Ulric, \"I am willing to remain, for I think\nhe hath befriended us. But I have no quarrel with Baal or with any\nother god. I think Odin and Thor to be at peace with thy Jehovah, but I\nlike not at all the cunning gods of the Romans.\"\n\n\"Jehovah destroyeth them in the day of his appointing,\" said Ben Ezra.\n\"They cannot stand against him. He is mighty.\"\n\nThe jarl was silent, gazing out upon the sea, and Sigurd looked around\nhim among the trees.\n\n\"O jarl,\" he said, \"I like not this mountain, full of gods. The men\nhave kindled fires. Let us eat and drink and then let us depart.\"\n\n\n\n\n                              CHAPTER XX.\n\n                         CARMEL AND ESDRAELON.\n\n\n\"Here are boats!\" exclaimed the Roman officer, as he drew rein at the\nplace upon the beach from which the Ionian rowers had fled. \"Then there\nwere more of these cowardly deserters. If all these boats had remained\nwith the ships, how many brave men might have been rescued! We will\nsearch the mountains for these rascals. If C\u00e6sar hath been robbed of\ntwo warships by the fire and the rocks, we will at least avenge the\nshades of our comrades who were left to perish.\"\n\nAn angry man was he, and with good cause so far as these men were\nconcerned, and their crime was well deserving of punishment. He rode\naway with his horsemen, but there would soon be terrible hunters for\nblood among the crags of Carmel. There would, however, be a delay of\nhours before forces could set out from the war garrisons, and meanwhile\nthe Ionians had been pushing their way into the forest.\n\nThey were of one accord that it would not be well for them to continue\nlong in one body, attracting attention, and each man was in dread of\nall his companions, fearing lest their very number should betray him to\nthe sword. They found what seemed a sufficiently hidden camping place\nand they slept. At their breaking of their fast next morning, having\nbut little to break it with, they were apparently almost cheerful,\nchatting lightly among themselves concerning their escape. In that\ncountry, they said, were great numbers of Greeks, who came and went\nunquestioned by the authorities. A few more, if scattered here and\nthere, would go unnoted. Not long time need pass before all of them\nmight be upon the sea again and far away, sailing from the many ports\nof Syria. Not many of them seemed to be warlike men, but it might be\nunderstood, in various forms of speech, that among them was no man who\ngrieved for the destruction of Roman keels and Roman soldiers. Rather\ndid some of them mutter that with their will whole legions had perished\ninstead of half a cohort. They believed themselves to be altogether\nunobserved, but upon them were now gazing eyes of intense hatred from\nthe leafy ambush of some dense thicket at a short distance.\n\n\"O ye who hear me,\" said one of the deserters, loudly, \"know ye this!\nFrom the first ship that struck the reef and began to burn did some\nsurely get to the land. Like us they are now in Carmel. What shall we\ndo with them?\"\n\n\"Slay them!\" sharply responded several voices. \"Lest they prove our\nruin. Slay them without mercy!\"\n\nOne of them was a tall, gray-haired man, with eyes that were set near\ntogether and with a pointed nose. His forehead was high and on it was\nan iron cap. He said:\n\n\"If they be too many, make friends with them at the first, but let none\nescape. I will attend to that.\"\n\nThey listened as if he might be a man of rank and a leader among them,\nbut hidden by the bushes were ears that understood the tongue in which\nhe and they were speaking, and there were other ears which did not\ninterpret.\n\n\"It is of no use to question this Greek of ours, O Knud,\" whispered\none to another of two strong men in the ambush, but his own face and\nhis manner asked a question.\n\n\"Be thou silent, Tostig the Red,\" replied Knud. \"Watch him. Do as he\ndoeth!\"\n\nFor Lysias was muttering low in Greek, \"He betrayed my father in\nCorinth. He would surely destroy me. He is a liar and he must die.\"\n\nTo the head he drew his long arrow, and his companions hindered him\nnot, for his face was burning with wrath, and it pleased them to see\nhim raise his bow.\n\n\"He is a young warrior,\" they thought. \"He knoweth what these have\nspoken.\"\n\nTruly sped the arrow, and the tall old Corinthian traitor was smitten\nthrough the face, so that he spoke no more. Up sprang his companions,\nwild with fear, but another and another of them went down before\nthey could escape among the trees, for the spears of Tostig and Knud\nfollowed the arrows of Lysias and they three followed closely, sword in\nhand.\n\n\"I think,\" said Knud the Bear when he and his friends returned from a\nbrief chasing, \"that too many escaped. I have counted but eleven slain.\nI will ask the Greek his reasons for this when we reach an interpreter.\"\n\n\"Take all coin from these who are slain,\" said Lysias, but he made his\nwords plain by action.\n\n\"They are Greek and Roman coins,\" said Knud. \"We may need them. I am\nlearning much concerning coins. Oswald, the harper, hath many, but I\ncared not for them. A sword is better than money.\"\n\n\"Not in a place of buying,\" laughed Tostig, \"and we are not now an\narmy. We must pay.\"\n\n\"I am not a thief,\" said Knud. \"I will pay, but I shall surely be\ncheated.\"\n\n\"No doubt,\" said Tostig. \"So do we need to take more coins. The Greek\nis right.\"\n\nThen they returned to their camp and Lysias stood before Ulric\nspeaking. The jarl listened with care and he became very thoughtful,\nfor Lysias told him all the words of the Ionians.\n\n\"So, we are to have foot soldiers hunting in these forests,\" he said.\n\"I had thought of that. Thou didst well to slay them. But we who are\nSaxons may not disperse. Go thou and seek thine own safety. Go thou,\nalso, Ben Ezra. Thou art among thine own people.\"\n\n\"Not so,\" said Ben Ezra. \"Let Lysias go, but I remain with thee for\na season. Thou needest a guide. It were well for thee and thy men to\ncross the plains of Esdraelon and get into the mountains of Gilboa. We\nwill go by night, for there is no safety for us in Carmel.\"\n\nTo all the Saxons Ulric interpreted the words of the Jew, and they said\nto him:\n\n\"Thou art the jarl; we will follow thee. But should we not first slay\nthis Lysias?\"\n\n\"Not so,\" said Ulric. \"He hath fought for us this day.\"\n\n\"Not so!\" shouted also Knud the Bear. \"He is a good archer. I will cut\noff the hand that is laid upon him.\"\n\n\"So will I,\" said Tostig, and his seax was in his hand quickly.\n\nThere the matter ended, but Ben Ezra talked with Ulric apart.\n\n\"I send Lysias to Jerusalem,\" he said. \"With him I send a jewel to the\nchief priest and another to the captain of the temple. We will pass\nover to Gilboa. Thence we will go over the Jordan, at the middle ford.\nAfterward we will go down to the wilderness of Judea. In that hiding\nplace no search can find us, as I have often told thee, and it is near\nJerusalem on the east.\"\n\n\"We are a score of men without Lysias,\" said Ulric. \"Shall we march\nnow?\"\n\n\"Come thou first with me,\" said Ben Ezra. \"Not with so much treasure\nmay we cross to Gilboa lest we lose it all on the way. I have found a\ncave in Carmel. Here will we leave the precious stones save a few. I\nswear to thee by my god that I will keep faith with thee.\"\n\n\"I swear not,\" said Ulric, \"for I know not of an oath with a true\ncompanion. Faith of a son of Odin cannot be broken. It is a tryst of\nblood between me and thee.\"\n\n\"Better than any oath,\" said Ben Ezra. \"Knowest thou not, O heathen\njarl, that thou hast covenanted in the name of thy god, whom thou\ncallest thy father?\"\n\n\"Odin!\" exclaimed Ulric. \"So it is. He would be angry with me forever\nif I failed thee in this matter. It is well to beware of provoking the\ngods. See to it that thou anger not thine own.\"\n\nThey walked away together, none following. Not far to go was it before\nthe Jew stood still and looked around him.\n\n\"It is well if we are unseen,\" he said, \"for I have great doubts in my\nmind.\"\n\n\"I see here a great cleft in the face of this crag,\" said Ulric. \"Like\nthis are many entrances of caves in the Northland. I found some among\nthe faces of the fiords. In them are great bones of men and beasts and\nstore of old-time weapons that are made of stone.\"\n\n\"Thou wilt find bones here,\" said the Jew, \"but I think not many\nweapons. The cave is dark, and we will have torches.\"\n\nExceedingly skillful was he in the kindling of a flame among dry\nmosses, and Ulric found withered branches of pine full of resin. A\ntorch for each was lighted, and they went in at the cleft, going\ncautiously.\n\n\"In such places as these dwelt the ancient prophets of Jehovah,\" said\nBen Ezra, \"but now the caves of the land of Israel are the strongholds\nof all robbers. I have heard that there are robbers dwelling in Carmel.\nTurn, now. Let us be sure that no enemy followed us.\"\n\nThe turning was quickly made, for they at that moment heard a sound\nbehind them. Then followed an angry cry and a javelin sped over the\nhead of the stooping Jew to glance from the shield of Ulric. He spoke\nnot, but he threw his spear and drew his seax, for in the cleft passage\nwere armed men. True was the spear-cast and the javelin thrower fell,\nbut over his body sprang Ben Ezra. It was then but a brief struggle\nbetween him in his perfect mail and a robber whose garb was but a tunic.\n\n\"These were but fools,\" said the Jew as his scimiter fell upon a fourth\nof these half-armed men. \"I think they are robbers and that they are\nSamaritans. Accursed are they! I will look to know if there are more of\nthem outside.\"\n\nHe was gone but a moment, and when he returned he exclaimed, hastily:\n\n\"Not any, O jarl! We will leave these bodies here for a token. Now we\nmay enter the cave.\"\n\n\"Touch them not,\" said Ulric. \"Thou art wise. I think that any comrade\nof theirs who may come to see will believe this to be the work of the\nofficers of the law.\"\n\n\"In that were better security for aught that we may will to hide,\" said\nBen Ezra. \"Seest thou now, O jarl? This cave is deep. We will go in\nfurther.\"\n\n\"There are bones to build heaps with,\" replied the jarl. \"Here hath\nbeen a massacre, but these are dry and the slaying was long ago.\"\n\nGloomy and terrible was that deep cave in Carmel, with its dark shadows\nand its whitening skeletons. Among its corners the Jew was searching,\nholding forward his torch.\n\n\"A soft spot in the floor here,\" he said. \"We will dig with our knives.\nWe may come to it again by sure marks, for behind it is the solid rock\nand at its right a fathom and a half is yonder broken altar.\"\n\n\"Knowest thou,\" asked Ulric, \"to what god belongeth this altar? Was it\nthine?\"\n\n\"Nay,\" said the Jew, \"he hath no altars in the caves, but only in the\ntemple at Jerusalem. In the old time was Carmel a stronghold of the\nPhilistines. There have been many gods among these mountains. They were\nall destroyed by Jehovah.\"\n\n\"I would, then, that he might have a care for these treasures of ours,\"\nsaid Ulric, digging rapidly with his broad dagger. \"Go deeper into the\nearth. Make it wider. Now it is enough. O Jew, if thou and I are slain,\nno other hand will ever take out that which we will shortly put in.\"\n\nThe casket and some other matters brought by them were now placed in\nthe cavity which the jarl had dug, and the covering was done with care\nand a removing of surface traces. Then Ulric turned to look upon the\naltar.\n\n\"There are deeply cut runes upon it,\" he said. \"Canst thou read them?\"\n\n\"Nay, but I know that they are Chaldee,\" said the Jew. \"This altar\nis exceedingly old. Who shall say what men and what gods have been\ndwellers in this cave!\"\n\n\"We may now do no more,\" said the jarl. \"We will return to the men. It\nis a good prudence, every way, that we leave a mark of blood at the\nentrance.\"\n\n\"Even so!\" exclaimed Ben Ezra. \"They were robbers, but also are the\nSamaritans the enemies of my people. Now am I sure that Jehovah is with\nthee, and I remember that which is written of such as thou art, that he\nmaketh the heathen his sword.\"\n\nUlric was thinking of another matter.\n\n\"The burdens of the men will still be heavy,\" he said, \"but not now\nwill they carry any weight of provisions. We will obtain pack beasts\nwhen we may. And now we have need for haste lest evil come upon us.\"\n\nThey went out of the cave together and returned to the camp, but Sigurd\nmet them.\n\n\"O jarl,\" he exclaimed. \"Lysias hath disappeared and the men are angry.\nWe had thought he would for a while go with us.\"\n\n\"We will guard our own heads, O Sigurd, the son of Thorolf,\" replied\nUlric. \"We are better without the Greek. He hath gone on an errand. We\nwill but eat and then we will depart, for the Romans come quickly. The\nJew hath a guiding for us.\"\n\nNevertheless, the Saxons all were angry, and they ate in silence. Their\njarl was too soft with strangers, they said to each other. He avoided\ntoo much the shedding of dangerous blood.\n\nHe himself was stern and moody, for he was thinking of his lost ship\nand of the Northland and of Hilda.\n\n\"If she knoweth where I am,\" he thought, \"surely she would give me a\ntoken. I doubt if she can follow me unto this place. How could she find\nme in Carmel?\"\n\nHe stood erect soon, and there was a strong impulse upon him, for he\nlifted his war horn and blew three blasts, toward the sea, and toward\nthe forest, and toward the great crag that standeth on the promontory\nof the mountain. The sea replied not, nor did the forest, but from the\ngreat wall of rock there came back an answer such as will come in the\nwinter time from out the deep throats of the fiords when the gods are\nconversing. Once and again it spoke, and Knud the Bear exclaimed:\n\n\"Odin is here, or Thor, for that is a war horn of the North answering\nthine, O son of Brander. It is a good omen. I like to feel that the old\ngods are with us.\"\n\n\"We will follow thee!\" added Wulf the Skater. \"Go where thou wilt. I\nwill not again forget that thou art of Odin.\"\n\nSo Ulric took up his spear and shield and Ben Ezra led the way; but the\nforest was dense before them and it was a long walk eastward before\nthey came out into an open place.\n\nFrom every lip burst a sudden shout as the Saxons halted to gaze upon\nthat which was before them.\n\n\"The valley of the gods!\" said Ulric.\n\n\"The valley of the slain!\" responded Ben Ezra. \"The plain of Esdraelon.\nThe valley which is before Jezreel. The valley of Decision. O jarl of\nthe Saxons, it is the place of the meeting of the hosts of kings. Since\nthe world was made here hath been the place of battles. Thereon have\nfallen more dead than on any other piece of ground. The chariots and\nthe horsemen have there gone down together.\"\n\n\"Here, then, have Thor and Odin contended with the other gods,\"\nresponded Ulric. \"Thy god hath been here----\"\n\n\"And all the gods of Africa and all the gods of the East!\" shouted the\nJew, enthusiastically. \"Here the hosts of Joshua contended with the\nhosts of Canaan. Here have Judah, and Israel, and Egypt, and Babylon,\nand Nineveh, and Persia, and Greece, and Assyria drawn the sword. In\nthe last days here in Armageddon will perish Gog and Magog, going down\nbefore the spear of Jehovah.\"\n\n\"Glad am I to have seen the place,\" said Ulric, and every viking\nshouted for joy that he had looked upon the greatest battlefield of the\nbroad world.\n\nWell was it worth coming so far to see, and gladly would they have gone\ninto one of those great combats of the kings; but now they were led on\nrapidly, for the day was passing. Not long did it take them to walk\ndown to the level plain, but all the while their eyes were busy.\n\nCities they saw and villages, and many scattered abodes of men. The\nfields were long since reaped, but here had grown much wheat. There\nwere many vineyards, with groves of olive trees and other fruit trees.\nRivers not large but shining. Small hills whereon were towers, as if\nfor watchmen and for garrisons. Names were given to some villages by\nBen Ezra, but the greatest town of all was dimly seen, far away across\nthe plain, and he said it was the ancient city of Jezreel. Beyond all,\ntoward the east, arose mountains in long ridges, and they knew from\nhim that these were the Gilboa to which he was leading them.\n\n\"O Jew,\" said Ulric, \"where halt we this night?\"\n\n\"Not on all the plain,\" said Ben Ezra. \"Even now we near the great\nhighway from the south and in it walks a multitude, but I see no armed\nmen. I think that many eyes are already aware of our coming.\"\n\nThat might well be, for the sunlight flashed upon their armor and their\nhelmets and their spear points, but Tostig answered:\n\n\"O jarl, what care we for armed men? I think the Jew is right. We must\nhasten, even if we have to slay a few Romans.\"\n\n\"None are here,\" said Ben Ezra. \"And the people will trouble us not.\nPontius the Spearman, the procurator of Judea, hath many gladiators and\nhe hath mercenaries whose speech is strange to the nation. None will\nquestion you because ye are not legionaries.\"\n\n\"Well for them that they do not,\" growled Knud the Bear. \"I am no hired\ngladiator. I am a free Saxon. What sayest thou, O jarl?\"\n\n\"Nothing,\" said Ulric, striding forward. \"Let us see what this crowd\nmeaneth.\"\n\n\"We have naught to do with them,\" said Sigurd, \"but I am curious to\nhave a look at the people of the land. None of them can say to himself\nthat we came out of the sea on the other side of Carmel.\"\n\nEvery Saxon was as Sigurd in willing to see the people and to know what\nthis might mean, for there were very many in the highway, men and women\nand children, and there were no horsemen, nor did there seem to be so\nmuch as a spear or a shield among them.\n\n\n\n\n                             CHAPTER XXI.\n\n                       THE RABBI FROM NAZARETH.\n\n\nLysias, the Greek, stood in a copse of thick bushes near the forest\nborder and looked out upon the plain, but not toward Gilboa. He had\nbeen digging in the earth, as Ulric and Ben Ezra had digged in the\ncave, but he had not been hiding treasure. He had but wrapped his\nweapons and his armor in a woolen robe-cloth that he might conceal such\nperilous evidence from inquiring officials of Rome or of any local\nauthority. Earth and flat stones and sods were over them now, and he\nhad made marks upon trees whereby he might find that place again if he\nshould at any future day will to do so. He now walked out beyond the\nbushes with no trace upon him that he had been a warrior.\n\n\"Well was it for me,\" he said, \"that I found such goodly raiment among\nthe spoils of the trireme. Fewer questions are asked of him who is\nhandsomely appareled. Soon I will procure me a beast and I will go with\nall speed to Jerusalem. It is a city to which strangers come from all\nthe world, and he who escapeth into a multitude hideth himself in a\nsolitude.\"\n\nThe tunic which he wore was of silk and his robe was of embroidered\nlinen. Sandals were on his feet and his white turban was of a costly\nsilken fabric. If he had retained any weapon, it was now perfectly\nconcealed. To the eye of one who might chance to meet him he would\nsuggest beauty and riches and peace, and not at all an archer whose bow\nhad sent many messengers of death.\n\n\"Now must I be careful concerning robbers,\" he thought. \"I have both\ngold and jewels with me. But to all who ask my errand I shall be but a\nscholar in the school of Gamaliel at Jerusalem, and therefore I may not\nenter Samaria, but must pass on swiftly. The Romans themselves favor\nall such scholars, and I shall have their protection. Their laws are\ngood and my time for smiting them again hath not come. But never will I\nshow mercy to a Roman.\"\n\nOther things he said concerning the much-vaunted laws and justice of\nthe world's conquerors. Beyond a doubt they not only claimed much in\nthe way of righteousness, and also did many things righteously, but\nbehind this sternly formal justice of theirs, and but little concealed,\nwas a man holding out his hand for bribes, and near him was a place of\nscourging and the sword of a ready executioner.\n\nNevertheless, Lysias walked on joyously. Soon he was in a highway, and\nby it passed through hamlets. He looked inquiringly at all places as\nhe went, but he paused not for conversation with any whom he met or\ngreeted. At last he came to the open gate of a wall, behind which were\na goodly house and some outbuildings of stone. In the gateway stood an\nold man, well appareled, and before him Lysias stood making reverent\nobeisance, as to an elder.\n\n\"I am Simon Ben Assur,\" said the old man. \"Who art thou, O Greek?\"\n\n\"I am Lysias, the scholar, of the school of Gamaliel at Jerusalem,\" he\nreplied. \"I have lost my beast, for he was worthless and he would go\nno further. Hast thou a good ass for sale, that will travel swiftly?\"\n\n\"I see that some one hath sent thee to me,\" replied Ben Assur. \"Thou\nknowest, therefore, that the beast is a swift one.\"\n\n\"Well with thee,\" said Lysias. \"I would buy him but for thy\nextortionate price. Wilt thou now give me an honest bidding, that I may\npay thee and take him away?\"\n\n\"Ha!\" said Ben Assur. \"They told thee my price? There is more which\nthey did not tell thee. The ass is young and there is none swifter than\nhe. He is well trained. The saddle and the bridle are to be purchased\nwith him, as thou needest.\"\n\n\"One needeth them to ride withal,\" said Lysias. \"But every beast hath\nfaults and thine is not worth, upon the market, the half of thy asking.\nI will but look at him and pass on about my business.\"\n\nLoudly laughed Simon then, looking keenly into the handsome face of the\nGreek. He turned and spoke to some one within the inclosure, bidding\nhim bring the ass.\n\n\"O youth,\" he said, \"I mind not that thou hast spoken with that evil\nbeast of a Samaritan. Arcas offereth that he will pay me for the ass\nnext Passover week; and I rejected him not, but told him that the price\nmust now be paid to me in five golden pieces. I will say to thee that\nthe pay days of Arcas never come, and wise men deal not at all with him\nunless he giveth double security.\"\n\n\"I deal not with him,\" said Lysias, \"but I will see thy beast.\"\n\nAnd now a serving man led forth to the gate a large and well-shaped\nanimal, upon which were a fair saddle and bridle.\n\n\"Mount and try him,\" said Simon. \"If thou canst ride at all, thou wilt\nascertain what is under thee; but an unskillful rider may wisely choose\nanother, for he is full of life.\"\n\nLysias sprang to the saddle and rode back and forth along the highway.\n\n\"He must be mine at any price,\" he thought, \"for in his legs is my\nsafety.\"\n\n\"Wilt thou take thy good bargain, O Greek?\" shouted Ben Assur as Lysias\nreturned.\n\n\"He is no good bargain at five pieces,\" said Lysias. \"No ass is worth\nso much. I will give thee one piece--\"\n\n\"Thou art no Samaritan,\" interrupted the old Jew. \"Thou art not Arcas,\nto buy of me and afterward to rob me of my pay with false witnesses\nbefore the magistrate's seat, proving that thou hast already paid me.\nHast thou not two pieces in thy hand? I will give thee a writing of\nsale lest he be taken from thee in Samaria.\"\n\n\"Two I will give,\" said Lysias, after again galloping up and down the\nroad. \"Make out thy bill of sale to Lysias, the scholar. I now return\nspeedily to Jerusalem.\"\n\n\"I think well of thee!\" exclaimed Simon; afterward adding, \"I pray thee\ntake my greeting to the great Rabbi Gamaliel. He knoweth me. I deal\nfairly with thee. I am not ashamed to have thee show unto even him this\nthy purchase.\"\n\nBack into the house he went and he soon returned with a small square\nparchment of a bill of sale. But the coins which he received were heavy\ncoins of Athens and he weighed them thoughtfully in his hand.\n\n\"Good youth,\" he said, \"take thou now the counsel of thy elder. Carry\nnot too many of such as these with thee. Open not thy purse before\nstrangers. Thou art overwell appareled. Get thee as far as the gate\nof a walled town having a garrison before the sun goeth down. Ride\nfast and far that thou mayest be beyond any who might inquire of thee\nconcerning that which is now under thee. Thou hadst better not enter\nSamaria.\"\n\n\"Fare thee well,\" said Lysias, urging the ass promptly. \"I take thy\ncounsel.\"\n\n\"Well for him if he so doeth,\" muttered Ben Assur in the gateway,\n\"since Arcas claimeth the beast as already his own. I will myself now\ndepart for Damascus and the Samaritan devil may seek for his five\npieces where he will. I have beaten him.\"\n\nThe thought then in the mind of Lysias did not err greatly.\n\n\"Something is concealed from me as to this swift one,\" he said to\nhimself. \"I have no business in Samaria that I should risk being robbed\nand then imprisoned as a thief. But if I now meet a Roman patrol, no\nofficer will deem me a pirate coming ashore from a burning trireme with\na band of Saxons.\"\n\nTherefore he blessed his gods for guiding him to the house of Ben\nAssur, and he rode on in safety, but not as yet was there any safety\nfor the others who, like him, had escaped the sea and the fire. Far\nbehind him on Mount Carmel, in a place of few trees, an Ionian sailor\nfell breathless upon the grass while beside him halted a Roman horseman.\n\n\"Get thee up!\" he shouted. \"Answer truly lest I slay thee! Where are\nthy companions?\"\n\n\"Slain by robbers in the armor of Saxons,\" responded the fallen man,\nrising. \"I will tell thee.\"\n\nAnother horseman came galloping to the side of the first and\nlegionaries on foot might be seen not far away. The wisdom of a\ncommander had sent a band of searchers to the side of Carmel toward the\nplain rather than among the crags and forests.\n\nGaining his breath as he could, for he had been running swiftly, the\nIonian told all save that he claimed to have swam to the shore.\n\n\"Thou sawest but three of these Saxons?\" said the officer at last. \"I\nhad no knowledge of any such pirate trireme. The Saxons are to be the\nscourge of the Middle Sea if C\u00e6sar destroyeth them not.\"\n\nMore questions were put to the frightened Ionian, and then he was told:\n\n\"I will not slay thee. Thou wilt come with me to Samaria. Thy testimony\nmust go before the procurator that a fleet may cruise against these\nrovers from the ocean stream. Thy companions that remain must be sought\nout that they may confirm thee.\"\n\nCalm and wise was this man, and he at once sent forward, also, swift\nrunners to ask here and there if anything had been seen of a band, or\nof single men, of the Saxons who had escaped from the trireme.\n\nNow the plain of Esdraelon is wide and the skirts of Carmel are long\nand rugged. There were none who had seen Ulric the Jarl and his\nvikings up to the hour when they walked out into the highway. By his\ndirections, as a prudent captain, they marched orderly, two and two, as\nif they belonged to the auxiliary of some Roman legion and were going\nby due authority.\n\n\"So,\" advised Ben Ezra, \"no man less than a quaternion or a magistrate\nwill run the risk of asking thee a question. No man of the people may\ndemand the errand of a soldier lest harm come to him.\"\n\n\"The multitude hath paused,\" said Sigurd. \"They gather around a man.\nLet us go see.\"\n\nRight and left parted the crowd as the Saxon column marched onward, but\nit halted suddenly, the people closing around and behind it, curiously\nstaring, but not touching nor inquiring whence it came.\n\nThere was an open space on the broad highway, and five paces in front\nof the jarl stood the man of whom Sigurd had spoken. He was of full\nheight and broad, but Ulric said in a low tone to Ben Ezra, in Latin:\n\n\"He looketh not altogether like a Jew. I have seen darker Saxons. I\nthink he is a jarl. Such as he might be a leader of men.\"\n\nProud was the bearing of Sigurd, the son of Thorolf, the sea king; high\nand stern was the aspect of Ulric, the son of Odin; tall and powerful\nmen were all the other vikings; but not among them all was there one\nwith the dignity of this plainly dressed Jew rabbi, who stood there\nunarmed and with only a turban on his head.\n\nHe spoke not now to the Saxons, but before him on the earth rolled and\nwallowed one who seemed in agony. His eyes were starting from their\nsockets and there was foam upon his lips. A shriek burst from him as\nhis convulsed limbs beat the earth.\n\n\"He hath a demon!\" said Ben Ezra to Ulric. \"The evil spirit teareth\nhim. There are many such. Let us see what this rabbi will do. I think\nhim a learned one. Only the learned may deal with demons.\"\n\n\"Come out of him!\" commanded the princely man, stooping to touch the\ndemoniac.\n\nOn his face was a kindly smile, nevertheless, but they saw not his\neyes, for he was looking downward.\n\nWild was the shriek that came back, as if in a fiercer spasm of inward\npain, but a voice followed it, saying:\n\n\"I know thee, who thou art, thou Jesus of Nazareth! Thou holy one of\nGod!\"\n\nAgain he said, \"Come out of him!\" and it was as if some unseen being\ncalled out loudly in an unknown tongue and fled away.\n\nThen arose from the ground the man in whom the evil spirit had been\ndwelling, and he stood erect, unharmed, like other men.\n\n\"A great rabbi!\" whispered Ben Ezra. \"One of the learned, from\nJerusalem. Thou mayest not speak to him while he is healing.\"\n\n\"He that fled called him a son of Odin,\" replied Ulric. \"He looketh\nlike one.\"\n\n\"He may be one of the gods of this land,\" muttered Wulf the Skater. \"I\nlike him not. He commandeth evil spirits and they obey him. I am glad\nthe jarl is also a son of Odin.\"\n\n\"I am glad to have seen a god,\" replied Knud the Bear. \"He is nobler\nthan other men. Let us see what he will do to that crippled one.\"\n\nBent and deformed, as if his arms and legs had little shape left them,\nwas this man whom his friends now half led, half carried, before this\nrabbi of the Jews.\n\n\"Canst thou do anything for him?\" asked one. \"He hath been thus from\nhis birth.\"\n\nNo answer made the man Jesus, but he laid his hand upon the arm of the\ncrippled one.\n\n\"Odin!\" exclaimed Ulric. \"Look! He can stand upon his feet! He lifteth\nhis hands! Thou art right, Ben Ezra. It were evil for me to speak. The\n<DW36> singeth! He is praising his god, and well he may.\"\n\n\"Go thou to the priest at Jerusalem,\" he heard the rabbi from Nazareth\nsay to each in turn of the men who had been cured. \"See thou tell no\nman.\"\n\n\"What meaneth he?\" thought Ulric. \"Have not all we seen with our own\neyes this which hath been done? I would I were healed of something,\nthen would I know what is this secret between them and their god. He\nis a strong one. What will Ben Ezra now say about his Jehovah? I think\nthis may be a stronger god, for Jehovah doth not well guard the Jews\nfrom the Romans.\"\n\nBut there stood the rabbi, Jesus, and he was saying many things to the\nmultitude. Clear was his voice and deep, and they who were not near him\nneeded not to lose a word that he was saying.\n\n\"I understand him not,\" muttered Sigurd. \"I am glad to have seen him,\nbut he is not like our gods of the North. It is time we were marching,\nO jarl.\"\n\n\"Haste then,\" added Ben Ezra. \"This Jesus is a learned rabbi, and he\nhealeth, but the swords of the Romans are not far behind us.\"\n\n\"I would have speech with him before I go,\" said Ulric to Ben Ezra.\n\"What is this that he saith concerning unending life? Do we not all\ndie? Do we not all go to the gods? He is lying. It is not good for a\nson of Odin to lie.\"\n\n\"Speak to him not,\" said Ben Ezra. \"He is touching the sick. Never\nbefore have I seen a rabbi like this.\"\n\n\"He is of the seed of David,\" said a short, dark man who stood near.\n\"He is the Christ that was to come. He is yet to be our King. I am one\nof his disciples. I shall be a prince when he is crowned.\"\n\n\"Thou a prince?\" said Ulric. \"Thou lookest not like a captain of\nwarriors. What couldst thou do in a feast of swords?\"\n\nThe short man shrank away chinking a small bag that was attached to his\nbelt, and his black eyes were glittering with anger.\n\n\"If I were a king,\" said Ulric, \"I would find me better captains than\nhe. I like not his face. He loveth his bag too well. Come on, now!\"\n\nThe order went to his Saxons, but at that moment he heard the rabbi\nsaying: \"Let him sell all that he hath and come and follow me. So shall\nhe have treasure in the heavens.\"\n\n\"Where are they, Ben Ezra?\" asked the jarl.\n\n\"No man knoweth,\" replied Ben Ezra. \"I think they are above the sky. It\nis the place of our people. Thou art a heathen and they have no part\nwith Israel.\"\n\n\"I go to Valhalla and to the city of Asgard,\" said Ulric. \"To the city\nof the gods. I want no treasure in any place of the Jews. Thou mayest\nhave thy heavens to thyself. Lead on!\"\n\nNevertheless, Ulric strode forward and stood for a moment before the\nrabbi looking him in the face.\n\n\"O thou of the sons of the gods,\" he said, \"I also am of the line of\nOdin. I think thou wouldst make a leader of men. I will fight for thee\nif thou wilt.\"\n\n\"Thou art not far from the kingdom,\" said the rabbi, smiling\nwonderfully. \"Go thou thy way, for thou wilt see me again. Thou wilt\ncome unto me in the day in which I shall call thee.\"\n\n\"That will I!\" exclaimed Ulric with an energy that was sudden. \"But I\nthink thou wilt need all the Saxons if thou art to contend with C\u00e6sar.\nIt will be a great battle when his legions meet thee. I have slain many\nRomans already. I am thy man.\"\n\n\"Thou knowest not yet what thou art,\" replied the rabbi, \"but the\nSaxons also are my people. I shall send for them.\"\n\n\"That do thou,\" said Ulric; \"and I, Ulric the Jarl, the son of Brander\nthe Brave, the son of Odin, I will lead them for thee, for I am a jarl\nand a sea king. Fare thee well.\"\n\nNo answer made the rabbi, for he turned to speak to a woman in the\ncrowd, and Ulric turned to walk away with Ben Ezra.\n\n\"The Romans will slay him,\" said Ben Ezra. \"Thou wert imprudent. I\nwonder much. Can this be the Christ that is to come?\"\n\n\"Who, then, is he?\" asked Ulric, and as they went onward the Jew told\nhim many things that were hard to understand.\n\n\"It seemeth to me,\" said the jarl at last, \"that thou speakest a saga\nthat Hilda of the hundred years told me in my childhood. Odin is to\nreturn bringing the gods with him, and some say he hath returned\nalready and that he who saileth far enough to the eastward and\nsouthward may find Asgard. I must see this city, Jerusalem, and its\ntemple, for now I do know that thy Jehovah is a god like Thor or Odin.\"\n\n\"He is the greatest of all gods,\" said Ben Ezra stoutly, \"but this\nrabbi cannot be the Christ. He is but a healer, and there have been\nmany who wrought cures and cast out demons.\"\n\n\"I would he had been with us in _The Sword_,\" replied Ulric, \"in the\nday when the evil spirit took possession of my vikings. But he could\nhave done nothing against the Nornir and the valkyrias. Even Odin could\nnot prevent their calling. It was the time for those men to die.\"\n\n\"I heard this demon that was cast out by the rabbi,\" said Ben Ezra,\n\"but I did not see him. I wonder what he is like?\"\n\n\"I have heard that such are exceedingly wonderful,\" said Ulric. \"They\nare of many shapes, but none are beautiful. Some of them are strong and\nthe gods have to tie them up to trees lest they do mischief.\"\n\n\"So have I heard,\" said the Jew, \"only the demons tied up by thy gods\nare not like our own. We have many, and they seize men by night. They\nserve the magicians.\"\n\n\"I would slay all magicians,\" said Ezra. \"They interfere with the gods\ntoo much. But I see the glint of spears away yonder. I trust there are\nnot too many of them.\"\n\nThey had marched far into Esdraelon and the night was falling. The men\nwere weary and their hearts were heavy.\n\n\"Be thou prudent,\" said Ben Ezra. \"If this be a Roman patrol, smite\nnot, but let me have speech with their officer.\"\n\n\"We may not flee,\" replied the jarl. \"Not only are we overworn, but\nthese are in part mounted men. Silence all! They come!\"\n\nThe Saxons halted, leaning upon their spears, not knowing the purpose\nof their jarl, but trusting him. On toward them rode but three, of whom\none wore a white cloak with a purple border.\n\n\"A Roman of high rank,\" said Ben Ezra. \"Slay him not. The band is\nstrong.\"\n\nNot loudly uttered was the hail of the Roman officer, reining his horse.\n\n\"I am Julius, the centurion of Tiberias,\" he said. \"I know ye, who ye\nare--the gladiators of Caius from Jerusalem for the games at Tiberias.\nYe have taken the wrong road. Who art thou, O Jew?\"\n\n\"I am Ben Ezra, their interpreter,\" replied the Jew. \"Were we not\nforbidden to go by the way of Jezreel?\"\n\nThe centurion laughed freely at that.\n\n\"Caius is careful of his wagers and would not have thy men seen by the\nwrong eyes,\" he said, \"but I have had fortune to beat his cunning by\nthis meeting. I will look well at them. They seem better than any that\nmay be now ready to contend with them.\"\n\n\"Study them well,\" said Ben Ezra, and the centurion rode slowly around\nthe motionless body of Saxons.\n\n\"Would I might slay him!\" muttered Knud the Bear, but none heard.\n\n\"He is a fine mark!\" whispered Wulf the Skater. \"I could spear him off\nhis horse. But the jarl is cunning.\"\n\n\"Cease,\" said Tostig the Red. \"The legionaries are twoscore and we are\nweary.\"\n\n\"O thou,\" said Julius to Ulric, discovering that he was the captain,\n\"thou art a tall one.\"\n\n\"He understandeth Latin,\" said Ben Ezra. \"He is not new, as are the\nothers.\"\n\n\"He looketh a tried swordsman,\" said Julius, for one soldier judgeth\neasily another. \"Saxon, thou wilt win sesterces at Tiberias, but thou\nwilt lose some of thy company.\"\n\n\"Not unless ye have better swords than any we have met,\" replied Ulric.\n\n\"Truly!\" exclaimed Julius, \"this is a deep trick of Caius. He will get\nno foolish wagers from me. But thou, O Saxon, thou wilt have a Numidian\nlion to fight, and he is larger than any Syrian beast. What sayest thou\nto that? Canst thou meet him?\"\n\n\"Judge thou of that when thou seest him before me,\" said Ulric. \"I\nwould gladly meet thy lion if he is a strong one.\"\n\n\"Hard fighters are the Saxons,\" said Julius. \"I will give thy big\nHercules a tiger.\"\n\nHe pointed at Sigurd, and the sea king's face flushed hotly, but he was\nsilent.\n\n\"O Jew,\" said the centurion, \"obey thou Caius lest thou get the\nscourge. Enter not Jezreel. Show not thy gladiators to any. Tell not\nany man that I have seen them and I will give thee ten sesterces. If\nthou tellest, I will reward thee otherwise. Go on a little. Camp in the\nold tower by the highway from Galilee. It hath now no garrison. Thy\nSaxon wolves are guard enough against jackals and robbers.\"\n\n\"I obey thee, O noble Julius,\" said Ben Ezra. \"Thou wilt answer for us\nif we are inquired of concerning this tower?\"\n\n\"I will acquit thee,\" replied the centurion, and he rode away followed\nby his own company.\n\nAll that had been spoken was now interpreted to the Saxons, and it\nseemed to them as if a good jest had been made of this Roman. They\nwere glad, also, of a sure camping place, and they marched on in the\ntwilight; but the Jew purchased for them two fat sheep and a skin of\nwine at a place which they passed in going. Then came they to the empty\ntower at the highway from Galilee, but when they halted Ben Ezra would\nallow none to enter until he had kindled a flame and had made torches.\n\n\"These old towers are abodes of demons,\" he said, \"and the rabbi Jesus\nis not here to cast them out. This Julius may have played a trick upon\nus, sending us to contend with evil spirits which have heretofore\ndriven all garrisons out of this place.\"\n\n\"Have thou thy will,\" said the jarl. \"But a son of Odin careth not much\nfor demons.\"\n\n\n\n\n                             CHAPTER XXII.\n\n                       THE TOMB SONG OF SIGURD.\n\n\nThe broken portal of the old tower in Esdraelon was as the entrance to\na dark cavern, and from it came out a wide-winged owl while Ben Ezra\nwas kindling his flame. Away into the darkness fled the bird of night\nhooting loudly, and the men said to one another:\n\n\"We like not these birds. They are of evil omen. They are friendly with\nbad spirits and the demons have them for their companions.\"\n\nUlric the Jarl stood waiting, and he cared not for the owl, but when a\ntorch was handed to him by the Jew he strode forward, looking warily\naround him as he went, and others followed him closely.\n\nNaught was there to be seen but bare walls of stone and a flight of\nstone steps that were builded spirally, leading upward.\n\n\"O jarl!\" suddenly exclaimed Tostig the Red, going past him, sword in\nhand, \"here, also, are other steps. Look! They go down into the under\nworld. Beneath this tower might be vaults and a prison.\"\n\n\"Such places are ever the abode of the evil spirits,\" said Ben Ezra.\n\"Go not down this at first. It is likely there have been many men slain\nhere, for this tower hath been a place of defense since the old time.\nIt was builded by the Philistines, but the stonework hath been repaired\nby the kings of the nations who came after them.\"\n\nEasy it was to obtain enough of fuel for a bright fire upon the stone\nfloor, and the Saxons loved the light of its blaze, although little\nneed was for warmth. There was a well near by, with a bucket for\nbringing up water and a trough for beasts to drink from. They who\nplanned the tower had provided wisely, but Ben Ezra said of the deep\nwell:\n\n\"Many are the demons which dwell in old wells. They entice men to fall\nin, and they themselves come out to deal evilly with lone wayfarers.\nTherefore some who encamp by the wells are heard of no more. Only the\nvery learned of the rabbis know how to cast them out. Let us hope that\nthis fountain hath been purified.\"\n\n\"The water is good,\" said Knud the Bear, \"and I was thirsty. The gods\nmake wells.\"\n\nThey ate and drank, and then Ulric the Jarl knew that it was his duty\nto further explore the tower. He first climbed the stone stairway to\nthe upper part. Here was no roof, and the walls were notched well for\nbowmen. There was a place, also, for the burning of a beacon light.\n\n\"It is a strong tower,\" said the jarl. \"A few men might keep it against\nmany if the portal had a stout gate with arrow holes. We are garrison\nenough. I will go down.\"\n\nThe stars above were bright, but there was no moon, and nothing could\nhe discern of the plain or of the mountains. He descended the stairway\nand went to the downward steps, taking a larger torch but asking no\ncompany.\n\n\"O jarl,\" said Sigurd, \"have a care for thyself. Thou knowest not who\nmay be the god of this place.\"\n\n\"Odin!\" laughed Ulric. \"Whoever he may be he hath not hindered our\ncoming in. I will see what is below.\"\n\nNone followed him but Tostig the Red, who was ever curious and who had\nno fear of demons, thinking them of no account.\n\n\"O jarl,\" he said, at the bottom of the steps, \"hold up thy torch. This\nwinding stairway hath taken us down two fathoms or more. There is a bad\nsmell. I like it not. I hear something that moveth.\"\n\n\"Help me! For the sake of Jehovah the Blessed!\" gasped a human voice\nnot far away. \"I perish with thirst. They bound me and left me here to\ndie.\"\n\nHe spoke in the old Hebrew tongue, not unlike the tongue which was\ncommonly spoken in that land, and Ulric answered:\n\n\"Who art thou?\"\n\n\"I am Abbas, the merchant, of Jerusalem,\" responded the voice. \"Water!\nWater! They were robbers from Mount Gilboa. I was rash, for I had\nlittle treasure with me. They got but my ass and a bag of denarii, and\nthey were wroth to have so little. This was their hiding place, but\nthey are gone out for prey.\"\n\nOver him stood Ulric, holding the torch, while Tostig with his knife\ncut the hempen fetters and lifted Abbas to his feet. He was naked save\na torn tunic, but he did not seem to be wounded. The Saxons above had\nheard, and a horn of water was brought down by Sigurd, the son of\nThorolf, for Ben Ezra was outside of the tower. Abbas drank, gaining\nstrength, and went up the stairway with little help, while Tostig\nsearched that place in vain for anything worth the taking.\n\n\"They take their spoils elsewhere,\" he muttered, \"but we will care\nlittle for that if Odin hath sent us the slaying of them. I would be\nglad to kill some robbers.\"\n\n\"Men in Galilee owed me money for merchandise,\" explained Abbas as he\nate. \"I came to obtain it, thinking to return in strong company. The\nRomans make the highways safe to all, and I had no fear. But this band\nnumbereth a score. I think they will return before the morning.\"\n\n\"Put out the fire!\" commanded the jarl. \"Every man to his spear and\nshield. We will not let one of them escape. It is evil to leave a man\nto die of thirst instead of giving him the sword.\"\n\n\"The Romans will thank thee well, O chief of the gladiators,\" said\nAbbas. \"They have striven to destroy these robbers of Gilboa, but if\nthese are pressed hardly, they flee across the Jordan. They are from\nthe wilderness.\"\n\nBen Ezra heard standing in the doorway, and he already knew all. To\nUlric he said in the North tongue:\n\n\"Beware whom thou slayest. Thou art but a gladiator in this tower. Thou\nart not here a jarl, to do as thou wilt.\"\n\n\"Ever am I a son of Odin,\" said Ulric. \"I have sold my sword to no man.\nWho shall stay me from slaying? I will spare not one.\"\n\n\"If thou slayest one, slay all,\" said Ben Ezra. \"There is danger in the\nenmity of the men of the wilderness. They forget not, and the next of\nkin may find thee.\"\n\n\"Not if he be wise,\" said Ulric, but he bade his men lie down and rest,\nkeeping watches.\n\nThen spoke to him the Jew Abbas:\n\n\"I will tell thee a thing. Me they may have thought to ransom. I know\nnot. But they will be here at the dawn to lie in wait for a company\nthat cometh from Tiberias with much merchandise. Thou mayest be sure\nthat, if thou slayest them not, then thou and all of thine are to be\nslain.\"\n\n\"That I may well believe,\" said the jarl, \"but they who slay Saxons may\ncount their men and we will count how many remain.\"\n\n\"So be it,\" said Abbas. \"Thou art a tall one. But thou, Ben Ezra, come\nhither and commune with me.\"\n\nSo went they apart and they talked much together in the old Hebrew\ntongue, and it seemed to the jarl that these two Jews might be of kin\nto each other, so many names did they speak of men and of women and of\nplaces.\n\n\"I will trust Ben Ezra,\" he thought, \"but of this Abbas I shall know\nmore at another time. I would see the sun upon his face before I can\nread its meaning.\"\n\nThen came around him and Sigurd all the other Saxons asking curiously\nconcerning all these things which had taken place. They asked about the\ntower and the plain and the mountains until they were satisfied.\n\n\"Thou art a prudent jarl,\" said Tostig the Red, \"but I would rather\nfight lions than to be hidden away among the hills like a wolf. Are\nthere not cities to be seen, and wonderful places? I like not deserts.\"\n\n\"We came out to see the world,\" said Knud the Bear. \"O jarl, there\nmight be excellent fighting if we go in the right direction.\"\n\n\"That would please me also,\" said Wulf the Skater, \"and we may begin\nwith these robbers if they are to come upon us. They may be swordsmen.\"\n\nOther of the vikings spoke strongly, as became warriors, and Ulric\nsaw that they were in earnest. They liked not Gilboa and its caves.\nThey had been shut up on shipboard long and they were in great wonder\nconcerning this country of the Jews.\n\n\"Even so am I,\" Ulric said to them. \"We will go on and see cities, as\nyou desire. We will not be Roman soldiers, but there is no disgrace to\na Northman in slaying a fighting beast or a fighting man. Only I will\nserve no master, even though he be a king. I am of Odin.\"\n\n\"We are as thou art in this matter,\" said the Saxons. \"We will serve\nnone save in thy company, but we pray thee lead us into a better place\nthan this tower or a desert.\"\n\nNow, also, some remembered to speak again of Lysias, the Greek,\nwondering whether or not he had escaped and where he might be. \"Ought\nwe not rather to have slain him?\" they said. \"Who knoweth what report\nhe may send out concerning us?\"\n\n\"He will have good care for his own life in that matter,\" said the\njarl. \"He will be secret for his own sake. Do ye not also remember that\nhe is a good bowman?\"\n\n\"I like him for his archery,\" said Tostig the Red. \"I trust that his\ngods may be with him to help him slay more Romans.\"\n\n\"That seemeth not to be for us,\" said Knud the Bear. \"We are to be\nfriends with them for a season. But I would see a tiger if I may, and\nalso some of these great elephants, which cause me to think of a whale\nwalking upon the land.\"\n\n\"Thou wilt see them at Tiberias if thou goest there,\" said Ben Ezra;\n\"but be careful of thy speech, for thou art now in a Roman land and\nthou art but one man. Thou canst not fight a cohort.\"\n\n\"A warrior may be prudent without dishonor,\" responded Knud. \"I like\nthe Romans better, now I have killed so many of them. They are good\nfighters and they die where they stand, not running away.\"\n\nSo said other of the Saxons, and all slept but the watchers, and the\nnight passed.\n\nIt was in the dull hour before the sun's rising that Abbas, the Jew,\ncame to the jarl and touched him, saying:\n\n\"Arise, O captain of the Saxons. The sentinel at the roadside needeth\nthee.\"\n\n\"Stir up the men,\" spoke Ulric to Sigurd, the son of Thorolf, \"but bid\nthem keep in the tower. Come thou unto me at the road.\"\n\nSo went he out and stood by the sentinel, and with them were Ben Ezra\nand Abbas.\n\n\"O jarl,\" said Wulf the Skater, \"I might not leave my post, but I have\nslain this man that lieth here. What he is I know not, but he crept\nnear me stealthily and I speared him. It was a cast in the dark. He\nweareth a turban.\"\n\n\"A robber from beyond Jordan,\" said Abbas stooping. \"He is a bowman.\nTherefore there are others with him. What sayest thou, captain of the\nSaxons?\"\n\n\"Let no man speak loudly,\" said Ulric. \"Bring no light. I hear horses.\nBe ready. Slay all who come, but give no warning.\"\n\nSo did Sigurd, the son of Thorolf, give direction in the tower, and the\nmen were prudent, waiting for what might come. But Sigurd now stood by\nUlric and seemed like a giant in the gloom. By him stood another Saxon\nquickly, and he was lifting his shield when something smote it, making\nit ring.\n\n\"An arrow,\" he said, \"sent strongly. A dozen men, O jarl!\"\n\n\"Smitten am I!\" shouted Sigurd, but he sprang forward swinging his pole\nax.\n\nUpon him darkly, suddenly, pressed hard a swarm of men, and they were\nas locusts crushed by the foot as his ax fell on them.\n\nUlric stood fast for a moment, but forward with Sigurd went Wulf the\nSkater full of war wrath. More than one arrow rattled on the shield of\nthe jarl, but he had cast his spear and he was now swinging the long,\nstraight sword of Annibaal, the Carthaginian, for men were upon him and\nhe mowed them as rushes.\n\n\"Back to the tower, Ben Ezra!\" exclaimed Abbas.\n\nPast Abbas and Ben Ezra charged four Saxons with Knud the Bear; but the\ntwo Jews went back to the tower, for they were cunning and they willed\nnot to be discovered by these robbers whose vengeance is forever.\n\nMen half armored, moderate in stature, not expecting great resistance,\nwere without hope in such a fray as this. They were there to be\nslaughtered, but at a little distance were others who were on\nhorseback. From among these rode one a little nearer, while the others\nwithheld their archery for fear of hitting their own men.\n\n\"O Abbas, of Jerusalem!\" he shouted. \"Would we had slain thee at once!\nThou hast betrayed us to the Romans. I will yet have revenge upon thee\nand upon thy son. Thou art the father of that Bar Abbas that smote me\nand mine beyond Mach\u00e6rus. May the Romans crucify him!\"\n\nAbbas at the tower heard well, but he replied not, and the Saxons were\nslaying fast the robbers who were on foot. Not one of them escaped, so\nswiftly fell the steel of the strong ones from the Northland.\n\nAgain shouted the man, the robber chief on horseback, shouting to his\nfootmen, but no voice went back to him. Only a spear thrown by Knud the\nBear went through him from breast to back and Ulric blew a blast upon\nhis war horn, for he heard a clash of swords behind him.\n\n\"It is naught!\" shouted Tostig the Red, from the doorway. \"We were\nthree and with us were the two Jews. Some thieves who came here are\ndead, dying easily. Fight on.\"\n\nLoud were the shouts of wrath among the horsemen, and one was\ninterpreted by Abbas to Ulric:\n\n\"He saith 'a Roman garrison is in the tower.' No robber will venture\nnearer.\"\n\n\"Woe to thee, Abbas!\" came fiercely out of the gloom. \"Woe to thee and\nthine! I curse thee by my gods for ever and ever!\"\n\nNo word spoke Abbas, but the horsemen wheeled and rode away swiftly,\nwhile Ulric stooped over one who lay upon the ground.\n\n\"O son of Thorolf!\" he exclaimed, \"I would thou hadst not been smitten.\"\n\n\"That am I,\" said Sigurd. \"The valkyrias have not passed me by. It was\nthe arrow in the dark, and the bowman was near and it pierced my mail.\"\n\n\"Thou didst fight well, being smitten,\" said Ulric, \"for thou art of\nthe heroes.\"\n\n\"Burn me not,\" said Sigurd, \"but bury me by this tower, in my armor,\nlaying my weapons with me. I may need them when I awake among the gods.\nI know not much of these matters, but I have great curiosity.\"\n\n\"Aye,\" said Ulric, \"and if thou seest Hilda of the hundred years, thou\nmayest tell her where I am. Speak thou also to my father, to Brander\nthe Brave, the sea king. Tell him I go on to Asgard, and that I have\nseen one of the gods in this land and that I seek to see him again.\"\n\n\"I also saw him there in the road,\" said Sigurd. \"I think him one of\nthem by his face and by the word of the evil spirits. If thou meetest\nhim again, greet him for me. Give me thy hand, Ulric the Jarl! The\nvalkyrias! Odin!\"\n\nHalf sprang to his feet the mighty son of Thorolf and he uttered a\ngreat cry. Then crashing heavily down he fell prostrate, his shield and\nhis mail clanging. Silently around him stood the Saxons, and one of\nthem said:\n\n\"O jarl, so fall we, one by one. I like it not. We shall never again\nsee the Northland. The gods are against us!\"\n\n\"He died not in his bed,\" said Knud the Bear. \"It is well with him,\nJarl Ulric.\"\n\n\"So die I!\" exclaimed Wulf the Skater. \"Come! Let us dig, for the\nravens must not whet their beaks on the bones of the hero.\"\n\nTherefore, with knives and spearheads and flat stones the Saxons dug a\ndeep hollow in the earth, and into it the sun looked down when he was\nrisen.\n\n\"It will do,\" said Ulric; \"but now we will eat and drink. We have slain\neighteen of these robbers. I would we had slain them all.\"\n\nMany coins had been found upon the dead, especially upon him who had\nbeen mounted, and all these the jarl divided among the men, Ben Ezra\ncounting for him their value.\n\n\"Keep thou some,\" said Knud the Bear.\n\n\"Not so,\" replied Ulric. \"I have enough. I like not too many coins. Ye\nmay need them to buy with. What have I to do with such things?\"\n\n\"Thou art jarl,\" reasoned Knud. \"If thou take them not now we will yet\ncompel thee. Thou canst not do altogether as thou wilt. We think thou\nwilt need many coins. They are the custom of this land.\"\n\n\"So be it,\" said Ulric. \"I am learning much about them. But I would\nrather be rich in cattle and in horses. I have all the lands of\nBrander. I think I will take some coins with me when I go, to keep them\nin a bag like old Oswald, the harper.\"\n\n\"We will pay ours here, I think,\" said Knud. \"But let the Jew make thy\nbargains for thee; for the sons of Odin are not good merchants.\"\n\nBen Ezra spoke then, agreeing well with Knud, but the heart of Ulric\nwas heavy because of Sigurd, for the son of Thorolf had kept good faith\nwith him, and the men who are true to friends are the men who are most\nmissed when the valkyrias come to them.\n\nThere were eating and drinking and there was much curious examination\nof the weapons and clothing and armor of the robbers from beyond\nJordan. Ben Ezra and Abbas answered all questions, but they said, also,\nthat there must be no going away from the tower until a messenger\nshould arrive from Julius or from some other Roman officer.\n\nEven while he was saying this to Ulric there was heard from the\nsouthward along the highway the sound of a trumpet.\n\n\"Whoever cometh,\" said Ben Ezra, \"let me have first speech with him. In\nslaying these who lie here we have been under the orders of Julius, the\ncenturion, and our official responsibility is to him; but he referreth\nus to Caius, of the household of the procurator at Jerusalem. We have\nneed of cunning.\"\n\nThe sun was high now, and Esdraelon was exceedingly beautiful between\nits mountains. It was a plain of brown and green under blue heavens,\na place where the gods might walk; but Ulric, the son of Brander,\nlistened to the trumpet and looked from the bodies of the dead to the\nSaxons, who stood in line on guard at the roadside.\n\n\"This is the valley of battles,\" he said, aloud. \"O Jew, I will heed\nthee. Knowest thou anything of this Julius?\"\n\n\"Not of myself,\" said Ben Ezra, \"but Abbas knoweth of him that he is\nsaid to be a subtle serpent, winning much money on wagers, and that he\nis cruel.\"\n\n\"Mark thou this, then,\" said Ulric. \"I saw in his face a thing that\nI read better now that we have lost a brave swordsman. Deal thou\ncarefully with these who come. I like not this place where too many\nhave fallen, and where thou sayest the multitudes are to perish in the\nlatter days.\"\n\nDark was the brow of the young jarl, and he went and stood by the open\ntomb and by the body of Sigurd, the son of Thorolf.\n\nOut stepped Ben Ezra into the highway, and he stood there making due\nobeisance and uttering a greeting, when a Roman officer wearing a white\ncloak with a purple border drew rein before him.\n\n\"I am Caius, of Thessalonica,\" said the Roman. \"Who art thou and who\nare these?\"\n\n\"If thou art Caius, thou art well arrived,\" said Ben Ezra. \"Thy\nswordsmen rested here at the command of Julius, the centurion, and I\nhave somewhat to tell----\"\n\n\"These, then, were hired for me by that traitor Hyles?\" suddenly\nexclaimed Caius, in wrath. \"And he sent them on to be murdered by\nJulius? Thou knowest not that Hyles was slain in Samaria yesterday?\nTell all!\"\n\nRapidly spoke the Jew, while other horsemen and four chariots halted\nnear in the highway.\n\nCaius dismounted and walked on to where Ulric stood, and the jarl\ngreeted him, pointing down at Sigurd.\n\n\"So! I have lost a good sword by this Julius,\" exclaimed Caius. \"He\nmeant me to lose all that he might win the games. Are any more of thy\nmen hurt?\"\n\n\"None,\" said Ulric in Latin, \"but this was a chief, a hero, a leader of\nmen. Him we must bury before we march.\"\n\n\"I, too, am a soldier!\" shouted Caius. \"He was a brave man! Bury him\naccording to thy custom. Thinkest thou I am a dog? I, too, will stand\nby. Brave men grow scarce. I would that C\u00e6sar had ten legions of such\nas thou art. The new levies are dwarfs!\"\n\nOut went the hand of Ulric freely, for the man's face had scars on it\nand he was of good stature.\n\n\"I will go with thee,\" he said. \"I am Ulric the Jarl, of the sons of\nOdin. It was promised me that I should have a lion to slay and that I\nshould see Jerusalem. Wilt thou keep faith with me?\"\n\n\"No!\" said Caius. \"I will give thee not to a lion; but thou shalt go\nwhere thou wilt, and then thou shalt see Rome and fight before C\u00e6sar.\nWait till thou hast seen this lion prepared for thy destruction. I am\nnot thine enemy to betray thee to ruin.\"\n\n\"I will wait,\" said Ulric, but he turned and beckoned to the Saxons.\n\nAll came and they took up the body of Sigurd, laying it in the deep\ntomb.\n\n\"Put in stones and earth,\" said Ulric; but Caius, of Thessalonica,\nstepped forward and threw in the first handful.\n\n\"Cunning is he,\" whispered Abbas to Ben Ezra. \"He knoweth men. He is\nwinning these Saxons for himself. There are no men more cunning than\nthe Romans.\"\n\nSlowly filled they the tomb, but Ulric stood at the head, looking down,\nand he said aloud: \"Who shall sing the tomb song of Sigurd, the son of\nThorolf?\"\n\n\"Thou, O jarl,\" said Knud the Bear. \"We have no harp nor any saga\nwoman. Sing thou to the hero and to the gods.\"\n\nSong came upon the soul of Ulric and his lips opened--and it was as if\nHilda were with him, for he sang wonderfully. There were women in the\nchariots and they sat listening to the musical voice of the jarl. The\nlegionaries on the horses sat like statues. The Saxons waited, holding\neach his war horn in his hand, as did the jarl, until the tomb was\nfilled, and they laid a broad stone thereon from a ruined part of the\ntower.\n\nUlric lifted his war horn and all the rest did likewise, answering his\nblast and then shouting. He blew again and he cried out:\n\n\"O Sigurd, son of Thorolf, the sea king, I have done as thou didst bid\nme. Bear thou my messages to the dead. Tell them I come. Keep thou a\nplace for me in Valhalla, in the day when the valkyrias come for me.\"\n\n\"Thou hast bidden farewell to thy comrade,\" said Caius, frankly. \"What\ndoest thou with the corpses of these robbers?\"\n\n\"Let the ravens and the wolves care for that matter,\" said Ulric. \"They\nare not ours.\"\n\n\"It is well,\" said the Roman, for there was pride in the manner of\nthe jarl. \"Such work is for slaves, not for thee. An officer will do\nwhatever is needful. Prepare thee to march for Tiberias. Thou wilt have\ngood quarters, near the amphitheater. No man may molest thee, O chief\nof the Saxons. I like thee well, and I would thy tall comrade were\nliving. Subtle indeed is Julius, the gambler, but he hath obtained only\nthe slaying of robbers, and the qu\u00e6stor will but laugh as at a jest.\"\n\nWell pleased were all the Saxons at the respect shown to them and to\ntheir jarl, but they went and looked curiously at the chariots in the\nhighway. They studied well the wheels and the harness, but most of all\ndid they gaze at the charioteers.\n\n\"Now,\" said Knud the Bear, \"I believe that which was told me, for I\nhave seen black men. I must slay one some day that I may know the color\nof his blood and of his flesh. They have strange hair, also, and they\nwear arm rings of silver and rings in their noses and in their ears.\"\n\n\"Those women are like other women,\" thought Ulric. \"Not yet have I seen\nher who stood by Hilda in my dreams. She is tenfold more beautiful than\nany of these.\"\n\nNevertheless, haste was made, and when the trumpet sounded the march\nthe Saxons were ready for the highway; but it was after the middle\nof the day, and Ben Ezra had all directions for the way. On went\nthe chariots and the horsemen, and then Ulric and his men followed,\nsaluting first the tomb of Sigurd.\n\n\n\n\n                            CHAPTER XXIII.\n\n                      IN A PLACE APART AT NIGHT.\n\n\n\"Halt thou! This is the place provided for thy band.\"\n\nSo said to Ulric the Jarl the Roman soldier who stood in the highway\nbefore the inn.\n\nIt was near the setting of the sun and the Saxons were weary with the\nheat. They were thirsty, likewise, and they were glad of a light red\nwine which was brought to them, but Ulric said to the bringers:\n\n\"For me water only. I fear much the evil spirit that hideth in the wine\nof this land. I think he is mine enemy and that my gods are at war with\nhim.\"\n\nSo he drank only water, but they all went in to the supper which had\nbeen ordered for them by Caius. They talked not much with any, for the\npeople of the inn were afraid of them, and men and women and children\nof the neighborhood who came to gaze did so as those who look but in\nreadiness to run away.\n\nThe place was but a hamlet in Esdraelon, and around it were vineyards\nwith many olive trees and fig trees.\n\nThere was a spirit of unrest upon Ulric, the son of Brander, and his\nsoul was troubled within him. He remained not in the inn after supper,\nbut walked out alone fully armed. He conversed in Latin a brief space\nwith the soldier on duty there, asking him questions, and the answers\ndid not please him.\n\n\"Thou wilt feed the beasts of the circus right well,\" said the\nlegionary scornfully. \"They will be hungry when they are let loose\nupon thee and thine. Thou art no Roman. All barbarians are fit to be\ncrucified.\"\n\nDown into his face looked Ulric the Jarl. \"O Roman,\" he said, \"I am a\nmatch for seven such as thou art. I could lift thee above my head and\ncast thee like a stone from a sling. Well said Caius that these new\nlegions were worthless against the strong in battle. Thou hast no part\nin Thor the Hammerer.\"\n\nThe soldier's face was dark with anger, but the jarl laughed and passed\non, and neither of them knew that Knud the Bear in the door of the inn\nhad been balancing his spear.\n\n\"If he lifteth but a hand against the jarl, I will smite him through!\"\nmuttered Knud. \"The jarl is imprudent. I like it not.\"\n\n\"Lower thy spear,\" said Ben Ezra near him. \"There will be no harm to\nthy chief. Thou art overhasty and thou wilt soon die.\"\n\n\"There will be blood at my dying,\" said Knud. \"I will strike for the\njarl if all the legions of C\u00e6sar should come.\"\n\n\"Wait,\" said Ben Ezra. \"Thou wilt find a better hour to use thy spear.\"\n\n\"So be it,\" replied Knud. \"Thou art old and thou art wise and thou\nhatest Romans.\"\n\nOn walked the jarl, but he was thinking, and the thoughts in his mind\nwere heated.\n\n\"Where am I now?\" he said, but not aloud. \"Where is the good ship _The\nSword_? Where are my companions who sailed with me from the Northland?\nWhere is Asgard? I have seen one god, but when shall I look into\nhis face again? When shall I find the maiden who stood by Hilda? My\nheart is on fire when I think of her. None like her was ever seen in\nthe Northland. O Hilda, canst thou tell me does this thy beautiful\ncompanion dwell among the gods? Then will I go to them that I may greet\nher, for she is mine.\"\n\nOther thoughts came to be uttered, but he spoke them not, and he walked\nonward into the deepening gloom. Very dark it was until the moon arose,\nand he knew not that the Saxons at the inn were inquiring angrily\nconcerning him.\n\n\"What are we if we lose our jarl?\" said Wulf the Skater. \"But for him\nwe had been lost long since. We would have no more help of Odin if our\njarl were taken away.\"\n\nBen Ezra and Abbas pacified them, and Tostig the Red said to the others:\n\n\"There are but few Romans near and they are bound under Caius. What\ndanger to the son of Brander were a drove of these Syrian cattle, even\nif they were armed?\"\n\n\"The son of Thorolf was slain by an arrow shot in the dark,\" said a\nviking, surlily. \"The jarl doeth not well to go among arrows. I would\nsee his face again.\"\n\nMurmurs were many, and they all came out and stood before the inn\nexamining their weapons and tightening their mail.\n\nUlric walked on, but not far, in the brightening moonlight.\n\n\"It is like the North country moon in winter,\" he said, for the air was\nclear and many things could be seen as in the day.\n\nBeyond him arose a hill, such as may be in so great a plain, and on it\nthere were ruins, grass-grown and mossy. In the old time there had been\nhere a castle or a pleasure palace, none could tell which, and some of\nthe stones were large, arising as pillars with stones laid across their\nsummits.\n\n\"Not a temple,\" said Ulric, thoughtfully. \"I hope not. I would not go\ntoo near an abiding place of the dead gods. Oft they come back again to\ntrouble men. So saith Ben Ezra. So saith Abbas. They hate men, for men\nworship them no more.\"\n\nHe walked more slowly, thinking of the gods and of Hilda and of the\nstrangeness that he himself was here without a ship or a strong\ncompany, and not knowing what might be before him on the morrow.\n\n\"I am jarl no more,\" he began to say, but at that moment he was\nsuddenly silenced and he stood still to listen.\n\nNot many paces beyond him was an open space on the summit of the hill\nand around it were fallen pillars, many and great, made of white stone.\nFrom this open there arose a voice and the light of the moon trembled\namong the white pillars.\n\n\"He kneeleth!\" said Ulric to himself. \"Ben Ezra called him the rabbi of\nNazareth. If there be dead gods or evil demons here, he feareth them\nnot, for they know him.\"\n\nNot loudly but with exceeding melody of voice the tongue of the\nkneeling man spoke on, and Ulric said:\n\n\"He singeth not to the dead of this place. It is not a saga of heroes.\nHe asketh many things, that they may be given him. I am glad of the old\nHebrew tongue that I understand him somewhat, but much that he speaketh\nI do not understand.\"\n\nSo he listened more, and the voice went on and the moonlight fell\ngloriously upon the face of him who was kneeling.\n\n\"I have been gone long from the inn,\" said Ulric in his thoughts. \"I\nmust return, but I have learned a thing. He is not alone here, as I\nam. The gods are with him, and he talketh with them as one god may\ntalk to another, as friend to friend, right kindly. He is not at war\nwith them, and one of them is his father. I would it were Odin, for I\nlike this god and I like his asking for these things that he needeth.\nI, too, need many things, but Odin is far away and I know him not very\nwell. The face of a god is very beautiful in the moonlight. He is a\ntall, strong man, a good fighter. But the gods have a strength of their\nown, greater than that of men. They can uproot trees and overturn rocks\nand drive the ice out of the fiords. This god could do a great many\nmighty things. I will have a talk with him some day, and I will ask him\nconcerning Asgard.\"\n\nUlric gazed earnestly at the face of Jesus of Nazareth, but the closed\neyes did not open and the wonderful voice continued its many petitions.\n\n\"I would I might see some of the other gods,\" thought the jarl, \"but to\nremain here is not well. He hath come to this place to be alone with\nhis father and his friends, and no brave warrior would be an intruder\nupon the affairs of others. I will go.\"\n\nHe turned and walked away, but his thoughts were dark and heavy within\nhim.\n\n\"This man is of the sons of Odin,\" he said. \"So am I. Therefore he and\nI are of kin, and I would know more of him. I would ask him concerning\nHilda and my father. If he may thus talk with the gods, my right is\nthe same. But he is more than I, for the evil spirits obey him. He is\nno magician, to be friendly with them, but he was not unkind to the\ndemon whom he sent away. If I were a god, I think I would deal well\nwith demons and make them fight for me.\"\n\nSo he communed with himself, walking, until he was loudly greeted at\nthe door of the inn.\n\n\"O jarl,\" shouted Knud, \"thou art safe! I did not know where to search\nfor thee. It is wrong for thee to leave us in this manner.\"\n\n\"O Knud,\" said Ulric, \"I am not a child. The night is quiet. Let us all\nsleep, for the march on the morrow may be long, under a hot sun.\"\n\nThe others reproved him sharply, but they now were glad to rest, and\nthe night waned.\n\nThere was no sound of trumpet at the sun's rising, but a quaternion of\nlegionaries came and the guard was changed. The officer also brought\norders from Caius that the gladiators should move on toward Galilee.\nAlso a chariot came to carry for them their burdens and their heavier\narms and armor, of which there was too much in weight for those who\nwould march rapidly.\n\n\"This is not a country for bearskins,\" said Knud. \"Even Wulf the Skater\nis willing to take off his mail and his helmet. He never would do that\nthing until this day.\"\n\n\"There is no fighting to be done among these vineyards,\" said Wulf,\n\"and I think this red wine maketh one's blood hot. I am thinking that I\nwould gladly see a tiger.\"\n\n\"There will be nothing in this land greater to contend with than\nwas the white bear slain by Jarl Ulric,\" said Tostig the Red. \"The\nchildren of the ice king were strong ones. I would rejoice in ice and\nsnow at this hour.\"\n\n\"It will be long before thou art frozen, O red one,\" laughed another of\nthe Saxons. \"I am melting, like the ice king.\"\n\n\"Thou wilt make less noise when thou fallest,\" said Tostig. \"But cause\nme not to remember too much the sea and the good ship _The Sword_. Such\nthoughts bring me to hate the land, and I listen for the washing of the\nwaves and for the cries of seabirds. It is not good, for the sea is far\naway.\"\n\nSilence came then and the Saxons walked on along the highway, seeing\nall things as they went, but thinking of the blue waters and of the\nplowing keels and of the North. Ulric strode on in advance, and with\nhim were Abbas and Ben Ezra, telling him many things that he might not\nbe ignorant in his dealings with that which was before him.\n\n\"Caius believeth,\" Ben Ezra told him, \"that thou and thy Saxons were\nengaged for him by his bondsman and purveyor Hyles, who was slain at\nSamaria for cheating him. We will have all care concerning that matter,\nbut Julius feareth Caius of Thessalonica because of his near friendship\nwith this Pontius the Spearman, who is master of Judea and Samaria\nunder C\u00e6sar. Win thou the good will of Caius, for he is a man of rank\nand gaineth power. Only trust not any Roman, for they care not for the\nlife of a barbarian more than of a dog.\"\n\nUlric answered little, but he thought, and spoke it not.\n\n\"These twain are Jews, but one is as a free Northman, a warrior, and\nthe other is as a slave in spirit, fearing the Romans even more than\nhe hateth them. I like not Abbas. He would sell me and mine as if we\nwere cattle. Ben Ezra proveth a true friend and I will abide by his\nsayings. Here cometh a party!\"\n\nLooking along the highway at that moment, he saw chariots and horsemen,\nbut no flag nor any armor.\n\n\"Who are these?\" he asked of Ben Ezra.\n\n\"Let them come nearer,\" said the Jew. \"It is likely they are travelers\nof importance. Halt thy men at the roadside and we will see.\"\n\nAt the word the Saxons halted, leaving the road free, and they were all\nwilling to stand and watch this company that came.\n\nFour chariots there were, but the one which came first was gilded\nand carven and was drawn by four white horses. Over it was a silken\ncanopy, and in it sat three veiled women. Of these two were on a front\nseat, behind the charioteers. He who drove was black and exceedingly\nuncomely, and beside him sat a large brown man bearing a spear and\ngirded with a sword. These were turbaned and their apparel was good,\nbut not upon them did the eyes of Ulric linger. On the back seat of the\nchariot, half reclining, was the third woman, and he said to himself:\n\n\"This is the princess and the others are her servants. I would see a\nprincess of this country.\"\n\nForward he strode three paces, and Knud said to Wulf the Skater:\n\n\"How splendid is the youth of our jarl, with his golden hair and his\nface like that of Odin! There is none other like him!\"\n\n[Illustration: \"O companion of Hilda!\"]\n\n\"Beautiful is he!\" exclaimed Wulf. \"But his face is full of pride\nthis day, and I think he is in anger. Speak not to him.\"\n\n\"The woman lifteth her veil,\" said Tostig, \"and she leaneth forward.\nOdin! She is wonderful! Her headdress is of jewels. Mark the jarl!\"\n\nDark yet fair, with the red of the new rose in her cheeks and with\neyes like the lone stars in a winter night, was the young woman who\nso suddenly leaned forth to look at Ulric. Into his eyes, also, came\nflashing a great light and a smile of joy was on his face.\n\n\"O companion of Hilda!\" he shouted in Hebrew. \"How camest thou hither\nfrom thy place among the gods? I am Ulric the Jarl, and I saw thee when\nI was on the sea.\"\n\nSilent was this beautiful princess for a moment, but she grew pale and\nthen red and she seemed to tremble greatly.\n\n\"O maiden,\" said Ben Ezra, \"whoever thou art, drive on. He will not\nharm thee. He is a prince of the Saxons and thou mayest not have\nconversation with him. He is not for such as thou art, O daughter of\nIsrael.\"\n\n\"Hold thou thy peace!\" came from the maiden, as one of high rank may\nspeak. \"Warrior of the Saxons, come thou nearer. Thou didst not see me,\nfor I was never on any ship. What is thy meaning?\"\n\nAlmost at the side of the chariot was the jarl, gazing into her face,\nbut his voice was as the murmur of a harp in the wind when he replied\nto her.\n\n\"O beautiful one!\" he said. \"Princess of the light and of the morning!\nMore beautiful than are the flowers and the stars! Thy face was where\nthe gods live and I saw thee in my dreams. I will give thee this token\nfrom Ulric, of the sons of the gods.\"\n\nHis hand had passed under the mail of his bosom and the bag of gems was\nthere. Now he drew out his hand and he raised before the eyes of the\nJewish maiden the perfect gem of which Ben Ezra had said that it was\npriceless.\n\n\"He must not give her that,\" whispered Abbas.\n\n\"Hinder him not!\" said Ben Ezra. \"Little thou knowest such as he.\nWert thou to interfere now, thy head were at the roadside before thou\ncouldst breathe twice. Leave it upon thy shoulders, madman!\"\n\nAbbas shrank back, clutching his fingers and scowling, but the Jewish\nmaiden's hand was already grasping the jewel and her lips were smiling\nwith a surpassing sweetness.\n\n\"I am Miriam,\" she said, \"and I dwell in Jerusalem. I shall see thee\nno more. But I give thee a ring for my token. Never have I looked upon\nsuch a face as thine.\"\n\nFrom her hand she took a ring, and in it was a large, pure pearl, very\nbrilliant, and the gold of the ring was yellow and heavy.\n\n\"O Miriam,\" said Ulric, in the deep tones of the harp of Oswald, \"I\nwill wear thy ring, but not in battle. I come soon to Jerusalem, and I\nwill meet thee there, or I will meet thee in Asgard, among the gods,\nand I will take thee to the house of my father, Odin. Thou art fit to\nbe a princess in Asgard.\"\n\nHis face was like the sun, but hers grew white again, and she drew her\nveil over it, for Ben Ezra said to Ulric:\n\n\"Let none hear thee, O jarl. I know not this matter. Thy words may\nbring upon her a peril. Harm her not, but be prudent. Thou art a wise\ncaptain. Let her drive on. Forward, charioteer! On thy life linger\nnot!\"\n\n\"Thou art right!\" shouted back the brown man, nodding his head at Ben\nEzra. \"She is my mistress, but she is willful. On!\"\n\nThe black charioteer slackened the reins of his prancing horses and\nthey sprang forward, but a great cry burst from the lips of Miriam, and\nthe word of Hebrew that was in the sound of it was:\n\n\"Farewell, my beloved! I have seen thee!\"\n\n\"Farewell, O princess!\" but in the voice of Ulric, the son of Brander,\nwas a faintness of strong pain, and he turned upon his heel, bowing his\nhead.\n\n\"Speak not to the jarl!\" said Tostig, grasping the arm of Abbas. \"What\nhast thou to do with an affair of a warrior and a woman? Wert thou to\nmeddle with me in such a case, I would cleave thee to the jaws.\"\n\nBut the chariots all moved swiftly away, and so did the horsemen who\nwere with them, but none of these were soldiers, and in the other\nchariots were but servants and much baggage.\n\n\"The jarl hath marched on,\" said Tostig the Red. \"Follow and trouble\nhim not; for that maiden was wonderfully beautiful and she gave him a\nring of remembrance.\"\n\n\n\n\n                             CHAPTER XXIV.\n\n                        THE PASSING OF OSWALD.\n\n\nThe Northland under the autumn sun was as the South, with green fields\nand forests and with glowing blooms upon shrubbery and in the hollows\nof the hills. The fiords were shadowy, with a coolness in the breezes\nwhich breathed among them that was pleasant to the wearied fishermen in\nreturning boats.\n\nUpon the high promontory looking seaward at the north of the cove and\nof the village and of the house of Brander there were no pine trees.\nIts bald granite knob glittered in the waning light so that it might be\nseen from far at sea as if it were a beacon. It was not a place for men\nto seek having no errand to lead them, and not many feet had trodden\nupon it since the world was made.\n\nNevertheless, this place was not at the closing of the day unoccupied,\nand from it there came a sound which went out over the wide water,\nand downward that it might mingle with the voices of the fiords, and\nlandward, also, that it might be joined with the soft sighing and low\nwhispering of the forests. Not loud was this sound at the first, but it\ngrew louder, and then with it went forth a voice.\n\n\"I think my strength faileth me,\" said Oswald, the harper, pausing in\nhis song. \"The harp was overheavy to bring up the mountain. I grow\nold and I am alone. Hilda sleepeth in the tomb of Odin's sons, Ulric\nis afar among unknown seas. Am I to die a cow's death before he\nreturneth? Who is there to make the mark of a spear upon my breast,\nlest I fail of Valhalla? I have fought in many a feast of swords. Why\nam I to perish slowly, without honor? Sad is the fate of Oswald if the\nvalkyrias pass him and leave him to die in his bed.\"\n\nOnce more the song arose, but now his voice was stronger and he sang of\nwar to the rocks and to the trees and to the gods among the fiords. The\nold gier-eagle on the withered pine tree northward listened intently,\nnow and then fanning with his wide wings, until the spirit that was in\nthe harping awakened him well. Loud was the scream that he sent back\nto Oswald, and he dropped suddenly from the branch of the pine tree,\nspreading his pinions and floating over the sea in a wide circle,\nrising as he went.\n\n\"He is free to come and to go,\" mourned Oswald, \"but I am bound at home\nand I shall no more ride the war steeds of the open sea nor hear the\nclang of shields nor see the red blood flow. Where is the good ship\n_The Sword_ this day? Where are Ulric the Jarl and his vikings?\"\n\nLow bowed his head and his hands sought fitfully the strings of his\nharp, bringing out the notes of sorrow.\n\n\"I will arise,\" he said, \"and I will go to Hilda's room. I will play\nto her there and see if she will answer me. She hath not spoken to me\nsince her eyes were closed. But she is with the gods and she hath many\nmatters upon her mind. She hath spoken to Brander the Brave and to\njarls and chiefs and kings that were of old. She hath seen Odin, and\nshe hath heard sagas that we hear not until the return of the gods.\"\n\nHe stood erect upon the rock where he had been sitting, and he was not\nweak, for he shouldered his great harp and bore it with ease as he went\ndown the rugged side of the mountain. Many saw him come, and they who\nwere near enough greeted him, but he paused not to speak. He went not\nthrough the village, among the houses, but along the shore, where the\ntide was coming in and where the waves called out to him as he passed.\nHe turned to listen to them, but across the water came no other voice,\nand he shook his head sadly.\n\n\"Here was _The Sword_ launched,\" he said, halting at the head of the\ncove. \"Here was the White Horse of the Saxons sacrificed to Odin. From\nhence the new keel went out behind the outing ice. Hilda of the hundred\nwinters told me that there would be no return. Is it so? Will the young\njarl never again put his foot upon this beach? Or did she speak only of\nthe vessel? Who may know the counsel of the gods! For they speak unto\nall men in riddles and the meaning thereof is hidden from us.\"\n\nHe turned and walked to the house, passing through the great hall,\nbearing his harp, and he went on to the room of Hilda, looking in.\n\n\"It is empty,\" he said. \"No other hath slept therein since she\ndeparted.\"\n\nBare were the walls, and the floor of cloven pine logs lay black,\nuncovered by rushes. One small table only remained, and upon this was\na Roman lamp of bronze, which Brander, the sea king, had brought back\nfrom one of his voyages to Britain. There was oil in it and a wick, for\nsuch had been a bidding of Hilda to one of the older women and to the\nhousemaidens. They feared much to let that lamp go without filling, if\nthe oil dried away; but it had not been lighted, although a wick was in\nit.\n\n\"I will bring fire,\" said Oswald, and he did so, going out and\nreturning. He set the flame of his small torch to the wick and it\nsurprised him, for it would not burn.\n\n\"O Hilda,\" he exclaimed, \"what is this thing that I cannot light thy\nlamp?\"\n\nThere was no spoken answer, but suddenly the wick took the fire and it\nblazed up a handbreadth, as if for a token.\n\n\"Burn thou, then,\" said Oswald. \"I will sing to her.\"\n\nQuickly they who were in the other rooms of the house and in the hall\nheard the sound of harping and the voice of a wonderful song, for it\nwas as a love song sung to the dead, telling her of the living and\nasking her concerning the gods and of all the places of the gods, where\nshe was dwelling. Men and women listened, looking into one another's\nfaces and whispering low, for the song was very beautiful and the harp\nanswered as if it were alive.\n\nJoyously burned the lamp, with a clear golden flame, as the song went\non, but it at last burned lower and lower and there came a red color\ninto the fire.\n\n\"There hath been much blood!\" exclaimed Oswald. \"I would I had been\nwith the jarl in the feast of swords. The battle is ended!\"\n\nFor the lamp went out and the room was very dark, but he sat in the\ngloom by his harp waiting for what might come.\n\n\"Disturb him not,\" said all the household. \"He ever mourneth for Ulric\nand for Hilda.\"\n\nMuch time went by and now and then there came from that room harp\nnotes, one by one, very faint and low, but Oswald was saying to himself:\n\n\"I have heard and I have not heard. All things are a riddle that I\ncannot read. Surely she touched the harp and her face was in the\nshadows. O Hilda, speak to me, for I am lonely! Tell me that thou hast\nnot forgotten thy kindred!\"\n\nThen fell he down upon his face in a deep swoon, and they who went in\nbecause they heard the sound found, also, the harp lying by him with\nits strings broken. They lifted him and carried him away, taking,\nalso, the harp, but when he again began to breathe and opened his eyes\nthe words that he first uttered were in another tongue than that of\nthe Northland. They heard the name of Hilda, but even when he aroused\nhimself and talked with them he told them naught of what things had\noccurred to him in the room of Hilda, the prophetess. For there are\nsecrets in the lives of men wherein other men have no part, and no\nman openeth his hidden heart unwisely. The thoughts of friends whose\nbodies are far apart are often apt to draw near and to walk the earth\nside by side. Oswald, indeed, was sending his heart out after Hilda\nand after Ulric. If the saga woman had in any manner answered him, no\nman knoweth. Nor can any say that the soul of Ulric was nearer to that\nof Oswald because both were thinking of each other and of her who had\ndeparted from them.\n\nSo may the gods look on from their places and see what men see not, and\nthey may often smile, if they are kindly minded, to see men and women\nmeet and embrace without the touching of the bodily flesh.\n\nThree days went by, and because of a request of Oswald's many messages\nhad gone out from house to house and from village to village, up and\ndown the coast and far inland. To everyone it was told that the hour\nwas at hand and that a token of the gods had come to Oswald, but that\nhe was still living. Upon the fourth day all who were entitled to come,\nby reason of kinship or of their high descent, had arrived. Many men\nand many women had gathered, and among them were those who brought\nharps. These sat apart and they spoke to each other in low voices,\ntuning their harps and listening to the sounds which answered them from\nthe strings.\n\n\"The harp of Oswald is broken,\" said one. \"Who shall take it after him?\"\n\n\"No man,\" replied the oldest of them all. \"It is a harp which came from\nthe East, in the ship of a sea king, and he gave it to the father of\nOswald in the days when Hilda was yet unborn. Upon it are strange runes\nthat none may read.\"\n\n\"It shall rest with him in his grave, then,\" said another, \"but Hilda\nsaid that he would need it not in the place to which he hath gone.\"\n\n\"They have both harps and harpers there,\" said the old man,\nthoughtfully. \"I know not the meaning of Hilda's word. So good a harp\nmust find a player, and I think the gods can mend it. We cannot, for we\nhave no strings like these.\"\n\nBefore them lay the great harp upon the floor of the hall, and one\nlifted it, placing it before a chair as if it might be played upon.\nThere were yet three strings remaining, and the old man sat down in the\nchair and put out his hands, touching, also, the strings which were\nbroken. Not from these, assuredly, came the sound which now fell upon\nthe ears of the gathered vikings, but all were silent, for the spirit\nof song was upon this ancient one whom no man knew. Clear was his\nvoice, but thin, and at times it wavered as if with age and weakness,\nbut he sang the departing song of Oswald and of the old time. Strong\nwere his hands also, for as he ceased he gripped with them and these\nthree strings, also, were snapped asunder with loud twanging.\n\n\"Hilda is right!\" he exclaimed. \"The harp of Oswald is dead. It will\nnever sound again. Build ye a fire, high and hot, and burn upon it this\nframe of wood. I go to Oswald's room.\"\n\nRising from his chair, all saw that he was tall and white-bearded, and\nnone detained him while he went to the room of Oswald.\n\n\"Thou art awakened, O Oswald, the harper?\" he asked, as he entered the\nroom.\n\n\"Waiting for thee, old man,\" came hoarsely from the lips of him who lay\nupon the bed. \"Now lift me up that I may stand erect, and put my sword\nin my hand. I will not die a cow's death, and thou art mine enemy,\nhaving full right in this matter.\"\n\n\"We burn thy harp, as Hilda gave thee directions,\" said the old man.\n\"We bury thee in a coffin at the foot of the great stone, thy arms and\nthy armor with thee.\"\n\n\"Also my bag of coins,\" said Oswald, \"and my cup of silver. I know not\nif I may need them. They have drinking horns in Valhalla. Smite me in\nthe breast. Let the spear mark be a deep one that I may be known as a\nwarrior.\"\n\nIn the doorway and within the room stood now chiefs and heroes and\nthey had heard, and they said to the old one, \"Strike him!\"\n\nDeep and kindly was the thrust of the spear that was given to Oswald,\nand he fell to the ground as if he had fallen in battle, so that all\nthe vikings were satisfied.\n\n\"Art thou to be smitten,\" asked a chief of the old man, \"or goest thou\nhence?\"\n\n\"I am to see the earth put upon him,\" said the old man. \"I came far for\nthis thing, from my place below the great south fiord, toward Denmark.\nAsk me not my name lest there be a blood revenge in the mind of some\nfoolish one. Take Oswald to his tomb and smite me there, for we are to\nbe buried together and my harp goeth with me.\"\n\nAll went out of the room and the bearers brought the body of Oswald,\nthe harpers playing and the women also chanting. The ancient one took\nup his harp, which was not very large, and he seemed joyful as he\nwalked with those who went forth to the place of tombs. The grave of\nOswald was deep and by it lay a coffin of cloven pine pieces. In this\nthey laid him, bending his swords and seax and breaking the shaft\nof his spear. His shield and his mail were broken and all were laid\nupon the body. Then one placed the bag of coins and the goblet at the\nhead and a jarl of rank covered all with a slab of pine, throwing in\nhandfuls of earth and many stones.\n\n\"Art thou ready?\" he asked of the old one.\n\n\"Not thy spear,\" he said. \"Strike with thy sword; and let it be a blow\nthrough the heart. As I cease this song to the gods and to the dead I\nwill lay my harp in the tomb. Strike me then.\"\n\nNow his voice failed him not and he sang well, bringing loud music from\nhis harp.\n\n\"I have fought in fourscore of battles!\" he shouted. \"I have sailed in\nall seas! I have spared none in the feasts of swords! I have seen the\nred blood flow from the hearts of many! I die by the hand of a jarl at\nthe grave of my old foeman. O Oswald, I shall be with thee in Valhalla,\nand there will we cross our swords and fight before the gods. Strike,\nthou of the sword!\"\n\nDown dropped his harp upon the coffin of Oswald and the sword of the\njarl passed through him, flashing and returning. Then the ancient one\nlay upon his harp and earth and stones were thrown in until the tomb\nwas filled and heaped. All the while the other harpers harped and sang,\nso that due reverence was given to the passing of Oswald.\n\n\"Will he see Hilda this night?\" asked one of the women. \"I bade him\ngreet her for me.\"\n\n\"They say that one who dieth must walk alone a little distance,\"\nreplied the other woman, \"and then he cometh to a dog; and he shall\nknow then where to seek a house that he may enter.\"\n\n\"I have heard many things,\" said the first speaker, \"but they do not\nagree. I think we know but little certainly. It would be well if one of\nthe dead were to come back and say what he hath seen.\"\n\n\"I would rather hear a saga,\" said yet another of the women. \"I like\nnot the dead. They are cold and they bring ill fortune. Let them stay\nwith the gods.\"\n\nSo said the greater part, but one woman went away muttering to herself.\n\"The dead! The dead!\" she said. \"They are of no use to us after they\nare buried. They care not for us any more. But I would willingly have\nspeech with one of them if he would not be overchurlish. I will go,\nsome night, and watch at the place of tombs. The witches watch at tombs\nand they see wonders. But it was worth seeing, the slaying of the old\none. He was a brave warrior and he died well.\"\n\nThere was a feast that night in the house of Brander the Brave, for his\nkinsmen and his kinswomen entertained their friends joyfully. There\nwas much singing and harping, and the horns and the cups came and went\noften around the tables. They drank deeply to the success of Ulric, the\nson of Brander, and to the voyage of his good ship _The Sword_, and\nto his return in glory from doing great deeds among the fleets of the\nRomans and among the islands and cities of the Middle Sea.\n\n\"The jarl will come again!\" they shouted. \"And here will he tell us of\nthe feasts of swords and of the crashing of ship against ship. Hael to\nJarl Ulric! Hael!\"\n\n\n\n\n                             CHAPTER XXV.\n\n                   THE MESSENGER OF THE PROCURATOR.\n\n\n\"Not in Samaria this night,\" had Lysias said to himself when he rode\naway upon the swift ass whose ownership might be questioned, \"but there\nare many places by the way wherein a wayfarer may find welcome if he\npayeth.\"\n\nBehind the saddle had been fastened the leathern sack which he had\nbrought with him from _The Sword_. It contained changes of raiment,\nbut little else, for his coins and his jewels, which were not very\nvaluable, were concealed about his person. More than once as he rode on\nhe both thought and spoke concerning Ulric the Jarl and the vikings,\nbut always did he seem well pleased that he was no longer in their\ncompany.\n\n\"The jarl is my sure friend,\" he remarked, \"but some of his tall\ncomrades walk with a hand too near the hilt of a sudden weapon.\"\n\nIt was toward the evening when, after riding through towns and\nvillages, he came to what was evidently a caravansary of good size and\ncleanly keeping at the roadside.\n\n\"Here will I halt,\" he said. \"I am now far escaped from burning wrecks\nand hasty-tempered pirates. I will have this beast of mine well cared\nfor. He showeth no weariness. I think--O ye gods! I know--I am nearer\nmy Sapphira!\"\n\nEre he could dismount, however, before him stood the keeper of the\nhostelry. Such as he are ever ready to greet with smiling faces the\nwell appareled, riding beasts of price. \"He will have money to pay\nwith,\" thought the innkeeper. \"But the land swarmeth with Greeks.\"\n\nLoud and friendly was his greeting, and in a moment more he was made\nto understand that this elegant stranger was Lysias, the student,\nreturning to Jerusalem to the school of Gamaliel from a journey to\nthe Lebanon and to the cities of Galilee. Being a man of Samaria, the\nkeeper was the better pleased that his guest was not a Jew, for of them\nhe spoke with scorn and hatred.\n\n\"O youth,\" he said, as they went into the inn, \"thou art fortunate.\nThou abidest with me this night and on the morrow thy journey will have\ngoodly companionship. There is here a company from Bethsaida and from\nother cities near the sea of Tiberias. They are merchants, and among\nthem are a taxgatherer and one who dealeth in slaves. There is neither\nscribe nor Levite to make thee uncomfortable with his evil speech. May\nthey all perish! It is said that the roads are not entirely safe and\nthe robbers come and go without warning.\"\n\n\"I shall be glad of them,\" said Lysias, \"but I think this village must\nbe safe, for I saw the helmet of a legionary as I rode in.\"\n\n\"Where they are the robbers come not,\" said the keeper, \"but they will\nnot be with thee always on the road.\"\n\nThen walked he away, and Lysias overheard him muttering curses upon all\nRomans and contempt for all Greeks.\n\n\"I think I heard somewhat else,\" thought Lysias, \"and I will look well\nat this company with which I am to journey to Jerusalem. There have\nbeen innkeepers who had no enemies among the robbers and there have\nbeen robbers who paid tribute to all innkeepers. I may not carry a bow,\nbut mine eyes and mine ears may do me good service.\"\n\nVery good was the entertainment given to him and to his comely brown\nbeast, but the departure was early the next morning. Even more in\nnumber than he had expected were these who came out into the road at\nthe door of the inn to go on together.\n\n\"They are of many kinds,\" thought Lysias. \"No twain are alike. I will\nnot have much conversation with them, but I will watch, for I think\nthey know this innkeeper exceedingly well.\"\n\nSo did he, and it was late in the day when he halted upon the summit of\na hill, looking thoughtfully forward and then behind him.\n\n\"O ass,\" he said, \"how fast canst thou gallop if it is to save thy\nmaster's throat from cutting? Thou hast robbers for companions, and\nthey do but await their opportunity, which I have not yet given them by\nthe way.\"\n\nThe ass did but pull at the bit and the rein was loosened that he might\ngo. On the northerly <DW72> of that hill, however, the company of men\nand animals which had seemed but peaceful at the outset had halted for\na rest before ascending the steep. There were now Jews among them, and\nothers of whose race and lineage there might be curious questioning.\nNow, also, there were weapons to be seen, such as privileged merchants\nmight be allowed to carry for their protection, and no doubt they had\nwith them written authority to show to any Roman officer. At the first\nthere had been but a dozen men and the women who were with them, but\nmore had joined at a hamlet upon this side of the city of Samaria, now\nfar behind them. Of these latter was an exceedingly black-browed man,\nhaving but one eye, and he seemed to be a sort of leader and commander\nover the others. To Lysias he had averred that he was a dealer in\ncattle, having a contract with the purveyors of the Roman garrisons.\nThus far he had purchased no beasts, but he had looked covetously at\nthe fine ass which carried the young Greek. At this hour he was saying\nto another of his crew:\n\n\"To-night, then. He hath treasure with him, and the beast will bear\nme swiftly to the wilderness. We will throw his carcass into the pit\nnear the three palms at the crossroad. None will be the wiser and his\nfriends will in vain make inquiry.\"\n\n\"I will stab him as he sleepeth,\" replied the man spoken to. \"The\nRomans care little if there be one Greek the less. We will speak softly\nto him when we catch up with him. I have seen that he hath no manner of\nunquiet mind as to us.\"\n\nOn went the ass, however, at his swiftest pace, even while they were\ntalking, and a long league of the highway did he pass before the\nintending stabber rode over the crest of that hill.\n\n\"Where is the Greek?\" he exclaimed.\n\n\"Ridden on a little,\" replied the evil-faced captain. \"Pursue not, lest\nthou alarm him. He will wait for us. He liketh well to prove the speed\nof his fine beast.\"\n\nHe had not spoken untruly, for Lysias was gladly discovering for the\nfirst time that he had found a treasure with four legs, a swift and\ntireless runner that took pleasure in a race, needing no urging. Only\nin hamlets and villages, of which there were many, was the rein drawn,\nand wayfarers who greeted the rider received but brief responses.\n\n\"Here am I safe!\" he exclaimed at last, \"for yonder on the hill is a\nfort and near it is a camp of Romans. My thieves are no longer a peril.\nGlad am I, too, that I am so far from the Saxon jarl and his pirates.\nTheir short, sharp blades are ever too near even to each other, and a\nspear in the hand of a Saxon is but an eager hunter seeking for a mark.\nI will rest here, and then I will let this beast shorten the road to\nJerusalem.\"\n\nThey who had proposed to take the swift ass from him had also hastened\nsomewhat until they inquired carefully of one whom they met, describing\nLysias and his bearer.\n\n\"Yea,\" said the man, \"ye mean the hasty messenger. He passed me going\nlike the wind. He who sent the message may be sure of its speedy\ndelivery.\"\n\nLoud and fierce were the utterances of the evil-faced one and his\ncompanions, and they cursed their gods for this disappointment. Also\nthey blamed themselves much that they had not sooner taken courage to\nslay the Greek. It was for this, their cowardice and delay, they said,\nthat the gods had mocked them.\n\n\"Never again,\" said the evil-faced one, \"will I throw away a gift that\nthey have placed in my hand. But they might have allowed me the chance\nI had chosen this night. I have but small confidence in the gods. They\nare treacherous.\"\n\nStrong and well made was the Roman camp at the foot of the hill whereon\nwas the castle. There were intrenchments and a mound and lines of\npalisades, and before these there was drawn up a full cohort of\nlegionaries. It was an evening parade, and along the glittering line\nthere rode without companions an eagle-faced man, who wore no armor.\n\n\"Half drilled!\" he muttered, angrily. \"It is well we are at peace. Of\nwhat good were such as these upon the Parthian frontier? Julius C\u00e6sar\nwould never have beaten the Nervii with these dwarfs to face the stout\nbarbarians. He would but have left them to rot in the wilderness, as\nCrassus did his Syrian levies. But I think I can teach these fellows\nthat they cannot trifle with Pontius the Spearman.\"\n\nBackward and all around the cohort rode the wrathful procurator of\nJudea, addressing no man, and then he wheeled and rode out to the\nhighway.\n\n\"O thou upon the swift ass!\" he suddenly shouted. \"Come hither! I\nrequire thee.\"\n\nBowing low, but answering not, Lysias obeyed him, awaiting further\nspeech.\n\n\"Is thy beast as swift as he seemeth?\" said Pontius. \"I know a good\nbeast. Is he tired?\"\n\n\"Never saw I one as swift,\" said Lysias. \"But at the close of a day he\nwere better for a rest.\"\n\n\"He may have a short one,\" said the Roman general, wisely. \"I prepare a\nmessage that will take thy head with it if it be not delivered rightly.\nI have naught here but clumsy beasts that travel a league a day.\"\n\nThen he turned to summon a servant, to whom he gave direction and with\nwhom Lysias rode into the camp, wondering much at his good fortune at\nsuch an hour.\n\n\"This is of Mercurius,\" he thought. \"I have ever thought well of him,\nand my father was once a priest of his temple at Corinth; the god hath\nnow remembered me. To him I owe my prosperity upon this journey, and he\ndid not favor the thieves as is his wont.\"\n\nThe ass had been hard ridden, but seemed not much the worse for it.\nWater and grain were brought to him, and Lysias, also, ate and drank.\nMore time went by than he had expected before a soldier came to summon\nhim to the camp gate.\n\n\"Saddle and mount!\" said the soldier. \"The hand of the procurator is\nheavy.\"\n\nNo answer might be made except to obey, and shortly the young Greek was\nat the gate.\n\n\"Kill not thy beast, lest thou fail of thy errand,\" said the\neagle-faced commander. \"Take thou this letter to the captain of the\nDamascus gate at Jerusalem. This also, and this. He will further\ndeliver them. Abide thou with him until he give thee answers. Bring\nthem to me.\"\n\nFew and brief had been his questionings of the young Greek, the pupil\nof Gamaliel. He was but a tool, an instrument, intelligent, sufficient,\nsure to serve well because of the scourge or the sword, or of reward.\nSo rule the Romans, and they who receive orders from a Roman general\nhesitate not to obey.\n\nSilently sat Lysias until the procurator ceased speaking and motioned\nwith his hand. Then, as if of his own accord, the ass went forward.\nTherefore Lysias had become a royal messenger, whom all men would be\neager to speed upon his way, for the fear of Pontius went with him.\n\n\"Mercurius!\" he shouted, at a goodly distance from the camp. \"Better\nto me art thou than is Jupiter. Now may Venus, also, be my aid if it\nbe true that my Sapphira was sold into the household of this bloody\none. O that I might send an arrow into his heart and a flame into his\ndwelling! But I will not fail of the due delivery of his messages. Who\nknoweth to what the gods may have destined me? Soon will all the Saxons\nperish and no man then will know the manner of my coming into Syria.\nSapphira! Sapphira! O swift beast! O Venus, goddess of love! Let me go\non to Jerusalem that I may once more look into her eyes and hear her\nvoice and touch her hand. She shall not be for another, but for me, for\nthe gods have favored me greatly!\"\n\n\n\n\n                             CHAPTER XXVI.\n\n                        THE CUNNING OF JULIUS.\n\n\n\"O Jew, thou hast brought to Tiberias the gladiators of Caius of\nThessalonica! Woe to thee and to thy accursed race! But I have orders\nconcerning thee and these. They will give us fine sport before long.\"\n\nLow bowed Ben Ezra to the Roman officer of the gate and his reverent\nreply came not fully to the ears of Ulric; but the jarl's face flushed\nhaughtily, for he liked neither the speech nor the manner of the\nRoman. The Saxons, also, were watching their jarl and their faces also\nreddened, the hands of men tightening upon the shafts of their spears.\n\n\"He will be prudent,\" they said, \"but we are not slaves, to be trodden\non. As he doeth so will we.\"\n\nUnto him now the officer turned as if he were looking at some newly\ncaught wild beast.\n\n\"O Saxon,\" he said, \"I have heard of thee. Thou didst well by the\nrobbers, but they cost thy Caius of Thessalonica a tall swordsman.\nNow thou art to be made food for a lion. I shall see thee torn in the\ncircus shortly, please the gods.\"\n\nWith an effort did the jarl steady his temper, but there was a deepness\nin his voice:\n\n\"O Roman, I shall be ready for thy lion. But if thou hast anything\nfurther to say thou mayest say it to Caius of Thessalonica. He is a man\nand he will answer thee.\"\n\n\"What care I for him?\" exclaimed the officer.\n\n\"He will answer thee that thing also,\" said the jarl. \"It is between\nthee and him. I have no words with one who openeth and shutteth a gate.\"\n\n\"I will have thee scourged.\"\n\n\"Silence!\" ordered a stern, hard voice behind him. \"Thou forgettest\nthyself, Demetrius, of the gate. The scourging of gladiators is not\nwith thee. O Saxon, thy answer is good. March on to thy quarters.\"\n\n\"O noble Julius, the centurion,\" replied Ulric, \"thy tower was a fair\nabiding place, and thou wert correct in providing it with a garrison.\"\n\nThe face of Julius flushed somewhat, for the jarl spoke to him as one\ncaptain may to another.\n\n\"I have an account of that affair,\" he said. \"Keep thou thy speech to\nthyself. Thou hast but slain a few robbers.\"\n\n\"I have heard of thee,\" said the jarl, \"that thou art thyself a good\nfighter and entitled to the respect of the brave. Thou hast led a\nlegion to victory in a hard battle. Well with thee!\"\n\nThere is vanity in all men, and the anger passed from Julius while the\nhaughty mannered jarl of the Saxons ascribed to him this fame.\n\n\"I have fought more fights than ever thou hast,\" he said. \"But thou art\na seaman. I would put thee upon a ship if I had one.\"\n\n\"I am of the sea kings,\" said Ulric, \"but yonder water is too small for\na great battle. It is but a fishing pond.\"\n\nThe ground upon which they stood was the high and difficult hill which\nariseth behind the city. This, with its palace gardens, was more than\ntwo leagues in circuit if the wall were measured around it from a\npoint on the south shore to a point on the north shore. But part of\nthis distance was of crags crowned with forts, and much of the city\nwas a suburb, having no wall. Within were temples and great buildings,\nand there was an amphitheater near the shore. The Saxons had wondered\nat the beauty and grandeur of this place as they drew near. They had\nmarched by way of small towns and villages, but up to this hour never\nbefore had one of them seen such a city as Tiberias or such a lake as\nGalilee.\n\n\"Speak no more,\" said Ben Ezra, \"but obey him and march on. Our\nquarters are in the lower town, near the circus. He giveth orders to\nthe guards at the gate.\"\n\nForward strode Ulric, followed by his men, and Julius glanced after\nthem. \"Caius hath beaten me,\" he muttered. \"I have none to contend with\nthese. They must be destroyed by tigers and lions. I will not waste an\nelephant upon them.\"\n\nOnce they were within the wall they could obtain from that height a\nfair view of the city, and they halted as one man.\n\n\"O Jew,\" said Tostig the Red, \"is thy Jerusalem larger and better than\nthis?\"\n\n\"An hundredfold!\" exclaimed Ben Ezra. \"This abomination of the heathen\nis but as a handful compared with Zion, the city of Jehovah, God of\nHosts.\"\n\n\"Then, O jarl,\" said Tostig, \"I will not get myself killed until I have\nseen Jerusalem. Manage thou with care, for I think thou wouldst like to\nsee it thyself.\"\n\n\"So will I,\" replied Ulric. \"But I think we shall suffer no harm in\nthis place. I have not seen any strong men yet except some of these\nJews, who do not carry arms. They would make good fighting men.\"\n\nFor he had looked at all whom he met with the eye of a captain, and the\nrabble of that land did not please him.\n\n\"Thou art right,\" said Ben Ezra. \"Thou hast seen men of the tribe of\nZebulun and of the tribe of Naphtali and some of Ephraim and Manasseh.\nThey are swordsmen if they had a king. Ere long our king cometh. But\nthese heathen of Tiberias are fit only to be crushed under the foot\nlike vipers.\"\n\n\"Speak not so loudly,\" said Abbas at his side. \"Remember that thou art\na Jew, and they hate thee.\"\n\n\"O thou of a weak heart!\" exclaimed Ben Ezra. \"When shall a\nmoney-lender be fit to wear a sword! Knowest thou not that I can lead\nthese Saxons through a host of these dogs of the gentiles? The Romans\nare warriors, but the rabble of Tiberias are scorned even by the\nlepers. Let us go on.\"\n\nFierce was the countenance of the Jew as they went down the long\nstreet, for it was broad and on either side of it were temples and\nshrines.\n\n\"Pollution! Abomination!\" he exclaimed. \"O jarl of the Saxons, these\ngods of Tiberias are but of wood and stone, the work of men's hands.\nThis place is cursed because of them.\"\n\n\"I will inquire shortly of what sort they may be,\" said Ulric. \"I grow\ncurious concerning gods. What need of so many? They would all go down\nbefore the hammer of Thor. Where is thy god that he permitteth them to\nbe here?\"\n\n\"This was never a city of my people,\" said Ben Ezra. \"It is a work of\nthe Greeks and the Romans. In Jerusalem thou wilt see only the temple\nof the living god, and of him thou wilt find no image in stone or wood\nor metal. No man hath ever seen his face.\"\n\n\"I like that,\" said Ulric, striding onward. \"There would be harm if the\ngods were seen too often. I will yet look again into the face of one,\nbut I am of their kindred and Odin is my father. Thy god seemeth a good\none.\"\n\nAll the while the other Saxons gazed as they went, saying not much, but\nwondering, and all who met them stepped aside, for their stature was\ngreat and their arms were splendid. The jarl had bidden them prepare\nfor this on the previous day, and Julius and the gate guards had seen\nNorthmen appareled and armored as if they were now marching to a feast\nof swords.\n\nBehind them now came on their baggage chariot, and shortly it was\njoined by horsemen, servants of Caius, sent by him to care for the\nguidance and welfare of his gladiators.\n\nBefore a palace in the main street of the city, well down toward the\nsea, sat upon their horses two horsemen from whom others reined aside\nrespectfully. These were face to face and they had greeted one another\nwith all ceremony.\n\n\"Thy northern wolves have arrived, O Caius,\" said one. \"But thou art\nshort of a tall fighter.\"\n\n\"So art thou robbed of thy robbers, O Julius,\" replied the friend of\nUlric. \"Thy tower was a subtle trap, but it hath not profited thee\ngreatly.\"\n\n\"Ha!\" responded Julius, mockingly. \"Thou hast lost thy best sword. A\nthousand sesterces that my Numidian lion slayeth thy Saxon chief.\"\n\n\"Wagered!\" exclaimed Caius. \"And a thousand sesterces more that thy\nHyrcanian tiger shall also be slain by the man I will name against him.\"\n\n\"I have thee!\" shouted Julius. \"We will write these wagers with care\nand let thy words be recorded with exactness. If either the lion or the\ntiger shall be slain by a Saxon, I lose that wager, but if a Saxon be\novercome by the tiger or the lion, thou losest.\"\n\n\"Thou hast some cunning of thine own in this,\" laughed Caius, \"but thy\nsesterces and mine will be in the keeping of Sempronius, the judge of\nthe games. I will trust him.\"\n\nEach of these carried a tablet of wax and a pointed stylus of steel\nwherewith they wrote, and the words were compared with care, that they\nmight then be written upon parchments, to be held by the judge of the\ngames.\n\n\"What meaneth he?\" thought Caius as he rode away from the gate. \"I will\nsee the Jew, Ben Ezra, as to this matter. There is a trap. I have not\nyet seen the laws of this circus, and Julius knoweth them well.\"\n\nLike an inn, large and well appointed, was the house to which the\nSaxons were guided, near the circus, and they entered it gladly, for\nthey were as men who were walking on into a new world.\n\n\"O Abbas,\" said Ben Ezra, \"come with me to the amphitheater. I would\ninquire there concerning many things.\"\n\n\"Not so,\" replied Abbas. \"Go thou. I have a friend to commune with and\nI go to meet him.\"\n\n\"O jarl of the Saxons,\" said Ben Ezra, as Abbas departed, \"it is well\nthat he goeth not with us. Come! Trust him not. He is overfond of\nmoney. Thou art a soldier. Thou must see thine enemy. Speak not to any\nman, but hear well. Who here knoweth thy gift of tongues? I am thine\ninterpreter, and be thou as if thou wert deaf.\"\n\nUlric did but bow his head in answer to Ben Ezra, but the other Saxons\nknew the errand of their jarl and approved of his going.\n\nA high arch of marble was the gate of the amphitheater, and on one\nside of it, upon the wall, was a broad tablet of wood. Upon this were\ninscribed many things, and both Ben Ezra and the jarl read them.\n\n\"Speak not,\" said the Jew. \"One cometh to lead us to the dens.\"\n\nThrough the portal they went, guided by a soldier of Julius, and he\nseemed pleased to show them all things. First went they across the\narena, and this was a broad place, egg-shaped, with vast tiers of seats\narising upon all sides. Under these tiers were the keeping places,\nand from some of these came cries and roarings of wild beasts and the\nshouts of men.\n\n\"Here are the prisons of criminals and of captured rebels,\" said the\nsoldier as the guard before a door opened it to let them in, \"but thou\nhast little to do with these. They are to slay each other or to be torn\nby beasts. There are trained swordsmen for thee and thine.\"\n\nNevertheless, he and the jarl and the Jew went into more than one of\nthese prisons, looking well at what they found there.\n\n\"Wretches!\" murmured Ulric. \"Some of them hardly seem like men and\nwomen. It is well for such as they are to be slain quickly. The gods\ncare not for these people, and so they are given to the Romans.\"\n\nNot so thought Ben Ezra, for he beat his breast more than once and he\nwhispered to himself in Hebrew:\n\n\"O God of Israel!\" he gasped. \"Here are of thine own chosen people,\nalso, many scores, taken in the snares of the heathen. Where art thou,\nO Jehovah, that thou hearest not? Canst thou not see this city of\npollution, wherein thy name hath not been written? Unclean! Unclean!\nWoe is me that I am here! It is as Sodom and Gomorrah, and thy fire\nlingereth!\"\n\nWhat he meant Ulric understood only in part, but he saw that many of\nthese who were doomed were Jews.\n\n\"They are not warriors,\" he thought, \"except that some of them are tall\nand strong. They must all die and get out of these prisons, but they go\nnot to Valhalla, and I know not where they go. I care not to slay such\npersons.\"\n\nNow the guard led him and his interpreter to the dens of the animals\nand Ulric was displeased that his men were not with him to see.\n\n\"The wolves,\" he said in Saxon, \"are like those of the North. I think\nnot much of the hyenas, nor of the small leopards. The great leopards\nare fierce beasts and so are the bears, but I could meet one of them.\"\n\nThere were four elephants in one den, and he walked around among them,\nwondering at their size and at their peacefulness, while Ben Ezra told\nhim of their intelligence and of their manner of fighting.\n\nThe jarl did but study them thoughtfully, and now a keeper said to Ben\nEzra:\n\n\"It is known by us that this Saxon is to fight the great lion. Come.\"\n\nThe den was near and in it the lion was pacing to and fro.\n\n\"He is almost as large in body as was the ice bear,\" thought Ulric.\n\"He standeth higher and his head is vast. He is a springing beast. He\nis stronger than the one we saw in Africa. I think he would fail if his\nheart were cloven. Now I will see the tiger.\"\n\nNear was his den also, and he, too, walked to and fro, snarling\nfiercely, for he was hungry.\n\n\"O Abbas,\" said the keeper of the beasts to Ben Ezra, mistaking him,\n\"thou art for Julius in this matter. What thinkest thou of thy Saxon?\nIf he can meet a lion, can he fight, also, the tiger? How will he not\nbe rent quickly when both are let loose upon him!\"\n\n\"Silence, thou unwise one!\" said Ben Ezra. \"Is it for thee to let out\nthis tiger?\"\n\n\"That is my care,\" said the keeper. \"I stand in this small box to throw\nopen the door, and the tiger will be famished on the day of the games.\"\n\n\"Mark thou this thine instruction!\" said Ben Ezra. \"Wait thou not! Send\nout thy tiger when thou hearest the trumpet call for the lion. So shall\nJulius win two thousand sesterces. Hold not thy door till the lion be\nslain, lest thou be smitten with a sword. Thy life for it! The beasts\ngo out together.\"\n\nUlric heard and he understood, for a fire flashed in his eyes, but he\nheld his tongue. \"I am to be torn without hope!\" he thought. \"I am\nbetrayed by Abbas, but I know the thing in the mind of Ben Ezra. He\ndoeth cunningly.\"\n\nSo they walked on across the arena, and as they went Ben Ezra stood\nstill.\n\n\"Here,\" he said in Saxon, \"wilt thou halt if thou art wise. Thou wilt\nhave thy mail on, but only thy sword and thy shield.\"\n\n\"I will wear no armor!\" said Ulric. \"I will bear no weights. What were\nmail and shield against these monsters? I will bring with me the long\nsword of Annibaal. Odin be with me! He who fighteth a lion must spring\nas lightly as doth a lion. He who faceth a tiger must move as the\nlightning or he is lost.\"\n\n\"Thou art wise!\" exclaimed the Jew. \"I have seen no warrior like thee.\nVerily I am true to thee. Sharpen thy sword and let thy hand and thy\nheart be strong. I would that Jehovah of Hosts might fight for thee,\nbut thou art a heathen and thou must look to thine own gods, if so be\nthey can do anything in such a case.\"\n\nDark was the face of the Jew, but he said no more, and they went back\nto the house of the gladiators.\n\nEager were all the Saxons to hear the account of their jarl, and he\ntold them many things, but in the gloom of the evening Caius came and\nhe spoke to Tostig the Red.\n\n\"Thou art to meet a black giant with a net and a trident against thy\nsword and shield,\" he said. \"What thinkest thou, O Saxon? Am I safe to\nwager upon thy success?\"\n\nIt was Abbas who interpreted, but the men had already heard much of\nthese nets and tridents and Tostig stood still for a moment.\n\n\"I have not seen this giant, O Roman captain,\" he said. \"May I be\nguided by my own jarl?\"\n\n\"Verily!\" exclaimed Caius. \"Do thou as he will tell thee, and I know\nnot what it is. O jarl, can he win?\"\n\n\"I saw thy giant,\" said Ulric. \"Tostig the Red will slay him for thee.\nMake thy wagers. I would talk with Abbas.\"\n\n\"So do!\" said Caius, for Ben Ezra had beckoned him and he stepped away\na little.\n\n\"What is it?\" asked Abbas of the jarl.\n\n\"Only this,\" said Ulric. \"I have seen the lion of Julius. He is a great\none. Hath he slain many?\"\n\n\"That I know not,\" said Abbas. \"Why askest thou? What matters it to\nthee?\"\n\n\"Little,\" said Ulric, \"but I was curious,\" and he asked him other\nquestions, keeping him while Ben Ezra talked with Caius, getting full\npermission that the jarl should wear arms of his own choosing and not\nthe armor of a Roman soldier.\n\nCaius rode away and many great ones came and went, as they had been\ndoing; for they who were to make wagers willed to see these pirates of\nCaius, as they called them. Not any, it seemed, went away believing\nthat the jarl could face the Numidian, and they declared that Julius\nwould win his wager.\n\nThen the night passed and in the early dawn Ulric, the son of Brander,\nsat apart by himself sharpening the long, beautiful sword on the stone\nwhich Wulf the Skater had brought to him from the North Cape, at the\nend of the world. To him came then Ben Ezra, looking like one whose\nsoul is burdened within him.\n\n\"O jarl,\" he said, \"the great games are set down for the third day\nhence. Wilt thou then be rested after thy journeying?\"\n\n\"Were I to meet the lion this day,\" replied Ulric, \"I am not weary.\nI care more for the training of Tostig the Red in the matter of this\nblack giant. I pray thee procure for me a net and trident that the\nthing he is to do may not be altogether new to him.\"\n\n\"That will I do,\" said Ben Ezra, \"but thou canst not instruct thyself\nconcerning lions.\"\n\nBefore the close of that day the jarl and Tostig were in a room by\nthemselves, but they told not to any man what they did with these\nstrange weapons whereby so many good swordsmen had been destroyed. That\nday, moreover, and the next day and the next the Saxons wandered much\naround the city of Tiberias, for they were permitted to do so freely,\nand all the people wondered at their stature and their armor.\n\n\"What thinkest thou of all these temples?\" asked Wulf the Skater of\nUlric. \"Would it not be well for thee and Tostig to offer sacrifices to\nsome of these gods?\"\n\n\"What good?\" said the jarl. \"I know them not and they know not me. I\nwould sacrifice to Jehovah if he had an altar here, because he is the\ngod of all this land. I heard Jesus of Nazareth praying to him and\ncalling him his father. If Jesus were here I would ask him that Jehovah\nmight be to me instead of Odin, for I think the North gods are far\naway. Caius may sacrifice to his Roman gods if he will, but thou and I\nhave no business with them.\"\n\n\"Thou art wise, O jarl,\" said Wulf. \"I will waste none of my coins upon\nthese priests and temples.\"\n\n\n\n\n                            CHAPTER XXVII.\n\n                        THE LION AND THE TIGER.\n\n\nSplendid was the appearance of the Saxons on the morning of the great\nday of the games at Tiberias, when they marched around the arena with\nthe jarl at their head, for their arms and armor were bright and their\nbearing was that of warriors accustomed to conquer. They themselves\ngazed, wondering, as they went, at the throngs which crowded the\nrising tiers of seats. Among these were many in gorgeous apparel,\nand the rich women had vied with each other in the colors and shapes\nof their garments and in the gold and jewels of their tiaras and\nother ornaments. There was a place on a lower tier for all the free\ngladiators, and to this the Saxons went after their marching. In it was\na covered stairway going down to the door by which any among them might\nenter a room adjoining the arena to wait for his summons to combat.\nEach company of the trained ones was by itself and they were not too\nnear each other.\n\nJulius and Caius and other great men, with their glittering women, had\na place which was as if it were full of thrones, but in the center of\nthis was one splendid chair in which only a C\u00e6sar or a proconsul might\nat any time presume to sit. It was this day unoccupied, but against\nit leaned the eagle standard of a legion and before it were scattered\nflowers.\n\nThe games began with races, both of footmen and chariots, and in these\nthe multitude were interested greatly, but only they who had wagers\ncared much who might win.\n\nWhen these were over it was time for the shedding of blood, and a band\nof captives were driven in, knowing that their fate had come.\n\n\"I see no swordsmen,\" was in the mind of Ulric. \"Each of these hath a\ndart, but he is naked and so are the women and children.\"\n\nThen uttered he a loud exclamation, for a door under the tiers of seats\nswung open widely and the den behind it vomited wolves famished with\nhunger and thirst.\n\n\"So many!\" said Ulric. \"Where got they so many? This is the cruelty\nof the Romans. I see no sport in this thing. It is but tearing and\nshrieking, for the small darts avail not.\"\n\nNevertheless, many wolves were slain before all the captives were torn\ndown. Men in full armor went out to drive the rest of the beasts back\nto their den, but it was not difficult, for hunger was satiated and a\nwolf might carry with him a torn limb or a fragment of raw flesh.\n\nSwiftly a crowd of bondservants cleansed the arena, and the feast of\nthe wolves had not been long in duration.\n\n\"There cometh now thy giant with the net and trident,\" said Ulric to\nTostig. \"He is very black. He is from Africa. Watch him well, for\nthis thing of his is but a trick of skill. Thou couldst parry that\nthree-pronged spear?\"\n\n\"That can I,\" said Tostig. \"But the net? Let us see what he doeth with\nthat short-legged brown swordsman in mail and helmet.\"\n\nBrave seemed to be the brown warrior, but the net flew over him and\nthe <DW64> stepped backward, dragging. Then it was but as a flash and\nthe trident was driven deeply through mail and breast.\n\nLoud were the plaudits of the multitude, for the pitiless black had\nseemed to show both skill and strength.\n\nThe next comer was a large man, and Ben Ezra, sitting near Ulric,\nground his teeth.\n\n\"A warrior of Israel, from the Lebanon!\" he exclaimed. \"He will but be\nnetted!\"\n\n\"Watch!\" whispered Ulric to Tostig. \"Thy turn cometh next. Mark how he\nfaileth and remember what I taught thee.\"\n\n\"I see his sleight of hand,\" said Tostig. \"I have beaten harder\nfighters than he is. The Jew is snared!\"\n\nLonger this time had been the contest, for the Jew ran, dodging,\nadvancing, retreating, striking, and it was only by his utmost skill\nthat the huge African at last threw over him the fatal net. Even then\nthe trident was parried oft, but it struck and the brave Jew went down.\n\n\"Now!\" said Ulric to Tostig. \"I go with thee. We will show them a\nthing. Let me see thy seax. It is sharp. It will do. Off with thy\narmor! Take this heavy shield and see that thou cast it well.\"\n\nBare, save a cloth around his loins and a helmet on his head, Tostig\nwent out into the arena, and the multitude shouted loudly, but\nJulius bit his lip. \"Here is something more than the Nubian hath yet\nencountered,\" he muttered. \"I would I might change my wagers. Yonder\nSaxon is an athlete for the Olympian games.\"\n\nWell used were the rabble of Tiberias, however, to see their black\nfavorite net his victims. Neither they nor he expected aught but a sure\nand speedy victory.\n\nFacing each other at twenty cubits' distance were now the two\ncombatants, and on the face of Tostig the Red was a smile.\n\n\"Now do I see more plainly the meaning of the jarl!\" he exclaimed. \"Let\nthis black one but cast his net. Thor and Odin! What a simple trick is\nthis to be slain by!\"\n\nThe black uttered a great cry, laughing, as he strode forward, but\nTostig made no retreat. He did but stamp with one foot, balancing\nhimself, and loosened the exceedingly heavy shield upon his left arm,\nto seize it, also, with his right hand.\n\nThrough the air swept the net of peril, whistling as it went, and\nflying, with a wide hollowing, to fall over Tostig as it had fallen\nover many another. Laughed, also, Tostig, throwing with all his\nstrength, and midway in the air the heavy shield struck into the\nhollow of the net, swinging it suddenly downward, but it fell also\nover the points of the lowered trident, tangling it. Around and under\nthe tangle, not touched by it, went the white and muscular shape of\nthe Saxon and the swift seax went twice into the bosom of the African\njuggler with nets.\n\n\"Thy sesterces, O Julius!\" shouted Caius. \"Thy favorite is gone from\nthee. What thinkest thou of my Saxons?\"\n\nTrue gamester was Julius, for his face changed not its proud serenity.\n\n\"I have but learned how a strong swordsman may overcome the weapons of\nNeptune,\" he responded. \"My lion will bring me back my sesterces.\"\n\n\"Well for thee, O jarl!\" muttered Caius. \"My Saxons have a cunning\ncaptain. He is a man to win battles. I must keep him. But great is his\nperil now. Jove guard him lest I lose many sesterces.\"\n\nThe multitude was hoarse with shouting, and now they grew silent, for\nthey knew by the lists that they were next to see a trained swordsman\ntorn asunder by the unconquerable lion from Numidia, the beast which\nhad slain heroes before C\u00e6sar.\n\nThe trumpet had not yet sounded when Ulric, the son of Brander, went\ndown the stairway to the room below where waited for him the master of\nthe games, and upon this man's face was a bitter smile, for he was a\nservant of Julius.\n\n\"O Saxon,\" he said, \"the edict forbiddeth thee to wear mail. Thou hast\nbut a sword and buckler. The lion weareth no armor.\"\n\n\"Ulric the Jarl,\" exclaimed Wulf the Skater, \"this is a trick for thy\ndestruction!\"\n\n\"Wait thou, true friend,\" said the jarl. \"Trust me yet a little. Odin\nis with me this day, and fear not thou these tricksters.\"\n\nThe master of the games understood not the Saxon tongue, but he read\nwell the fierce eyes of Wulf and he fell back a little, for the\nSkater's hand was on his sword-hilt and the Saxons were known to act\nsuddenly.\n\n\"No helmet!\" said the cunning friend of Julius. \"The lion fighteth\nbareheaded.\"\n\nThe sword of Wulf rattled loosely in the sheath as the helmet was put\naside, but he obeyed a sign from Ulric and drew it not.\n\n\"If the jarl be slain,\" he muttered, \"that dog must die. I will see to\nthis matter.\"\n\nKnud the Bear had come down, but he was silent and his face was dark.\nHe and Wulf turned and went up the stairs and so did the master of the\ngames, well satisfied.\n\n\"Now the long sword!\" said Ulric, throwing aside the short falchion\nprovided for him. \"O but its edge is keen!\"\n\nHe heard the trumpet sound and the door before him opened. Then the\ngreat multitude shouted with admiration and the Saxons themselves\nwondered.\n\n\"He is so beautiful!\" exclaimed Tostig the Red. \"O that we must lose\nhim! What shall we do without our jarl?\"\n\n\"Would that I might die with him!\" groaned Wulf the Skater, but Knud\nwas thoughtful.\n\n\"Do we not know him?\" he said. \"Is he not the son of Odin? Are all our\ngods dead? I think the Nornir are not here and that the valkyrias will\nnot come.\"\n\nA tower of white stood the jarl, with but a silken garment from waist\nto knee, and his golden-curled head was a glory. In his hand was the\nAfrican sword, its bright blade and the jewels of its hilt glittering.\n\n\"It is not the sword I sent him,\" muttered Julius. \"That might have\nbroken in his hand, but this will not. He is like Mars! O Caius, what\nthinkest thou of thy barbarian and of thy sesterces?\"\n\n\"Wilt thou double thy wager?\" asked Caius. \"I am pleased with my Saxon\nlion.\"\n\n\"Nay,\" laughed Julius, \"thou wilt have losses enough. Thou wilt see him\ntorn shortly.\"\n\nFor the trumpet spoke again and the lion sprang out of his cage with a\nroar like distant thunder. The sun rays fell upon his face, however,\nand he lifted his head, blinded for a moment. Then he saw the throng\nand he walked along a few paces, as if willing to spring among the\ntiers of seats, but they were high and he looked again around the\narena. Motionless stood Ulric, watching the lion, and between them\nnow was but half the width of the arena. Men breathed not, but leaned\nforward in their places, and now the eyes of the great beast perceived\nthe jarl and he roared with the roar of hunger and wrath.\n\n\"Now for thy Saxon!\" said Julius to Caius. \"I think his hour hath come.\"\n\n\"O jarl!\" murmured Wulf. \"Is it for this thou didst sail to the Middle\nSea? Where is now thy city of Asgard!\"\n\n\"Hark!\" exclaimed Knud the Bear. \"Another cometh! Here is more\ntreachery! A tiger!\"\n\nNot with a roar, but with a snarl that was dreadful did the Hyrcanian\nmonster rush from his den into the arena. He was more terrible to look\nupon than was even the lion, and he paused not in his going. He seemed\nto rush along the ground, crouching stealthily, and he looked longer\nand larger as he went.\n\n\"The jarl is lost!\" said Tostig the Red. \"O to be near with my spear\nfor one cast. This is twain upon one!\"\n\n\"This was thy bargain,\" said Caius to the cunning Julius. \"Thy tiger\nwas to contend with the swordsman of my naming. I have appointed this\nchief.\"\n\n\"So be it!\" said Julius calmly. \"I accept!\"\n\n\"Wait!\" muttered Ben Ezra to the Saxons. \"The beasts have seen each\nother. Mark now the swift movement of the jarl! The lion is about to\nspring! The tiger! O God of Israel, aid thou, even though he be a\nheathen!\"\n\nThe tiger's rush was rapid and Ulric sprang forward as if to meet him;\nbut the lion was in the air with a vast bound, his black mane streaming\nand his teeth showing in the cavern of his jaws.\n\nNot upon Ulric did he alight, however, for at his spring and roar the\ntiger turned in his tracks as toward one who would wrest from him his\nintended prey. Past both of them darted the jarl as the Numidian fell\nheavily upon the Hyrcanian; then his turning was as the light in its\nquickness and he thrust with his might upon the beasts as they grappled\neach other, rolling upon the ground and tearing.\n\n\"He hath cut off one forepaw of the tiger,\" exclaimed Knud. \"That\nthrust was at the lion. Again! Again! Such roaring was never heard.\"\n\nThe wild beasts tore as they roared and the multitude uttered loud\noutcries, but all of the movements in the arena were untellably rapid,\nnor might they who were watching separate Ulric from his two enemies.\nHe was with them at every spring and turn and roll. The long, keen\nsword dripped blood and the white skin of the son of Odin was spattered\nredly, as if he were sorely wounded.\n\n\"If he be slain at all thou losest,\" said Julius.\n\n\"O friend,\" replied Caius, \"be thou contented. Thou must buy thee\nbetter beasts than these.\"\n\n\"Mark!\" exclaimed Tostig the Red. \"That was a thrust behind the\nshoulder. The tiger falleth undermost. O jarl! Beware now of thy lion!\"\n\nOver the dying tiger stood the huge Numidian, panting and roaring, and\nbefore him stood the jarl, looking him in the eyes.\n\n\"Splendid is he!\" exclaimed Ben Ezra. \"Jehovah of Hosts, be with him\nnow! It is the last.\"\n\nForward went the lion, but not with a bound, and he swerved in his\nrush owing to his many wounds. High in the air and over him, in a leap\nfor life, went the son of Odin, and as his feet touched the earth he\nturned, thrusting swiftly, and he sprang again. Wild were the plaudits\nof the multitude, but the lion was staggering and his roar was muffled.\n\n\"One thrust more,\" muttered Ulric. \"I am sorely spent and I bleed.\nHael, Odin! I have cloven his heart! He dieth!\"\n\nThen turned he and walked steadily to the front of the place of the\ngreat ones, while a vast clamor arose in all the tiers of seats.\n\n\"O Saxon,\" said Caius, \"art thou wounded?\"\n\n\"A scratch or two,\" replied the jarl, cunningly. \"Am I to fight another\nlion this day, or wait I until the morrow?\"\n\n\"O Caius, the sesterces are thine,\" said Julius. \"Thy barbarian hath\nwon for thee. Never saw I the like of this.\"\n\n\"To thy place, O jarl,\" shouted Caius. \"I come to thee quickly. Be thou\nsilent!\"\n\nAway strode Ulric, stepping proudly, but the door of the room he sought\nopened as he came.\n\n\"Enter! Enter!\" shouted Knud the Bear. \"O our beloved, art thou slain?\"\n\n\"Water, quickly!\" said Ulric. \"I would drink. Wash me also. Bind up my\nhurts and put on my mail. Let no man see these tearings in my limbs. I\nshall not die!\"\n\n\"Glory to the God of Israel!\" exclaimed Ben Ezra. \"I am the physician\nfor thy hurts. Bring bandages. These are not to death. I feared for\nthee greatly.\"\n\n\"Nevertheless,\" growled Wulf the Skater, \"I will slay that master of\nthe games. O jarl, if we had lost thee!\"\n\nSo said the other Saxons, crowding down to greet him, but the bandages\nwere made firm, the mail and the helmet were put on, and then out\nacross the arena marched they all, the jarl leading them.\n\n\"Truly he is not slain,\" muttered Julius. \"I have lost my beasts and my\nsesterces!\"\n\nAt the great portal, however, Caius waited with a chariot.\n\n\"Not to thy quarters, O Saxon jarl,\" he said. \"I take thee to Capernaum\nfor thy healing. All thy men will follow now, and a ship waiteth at the\nseaside. Julius must not see how thou art wounded. Wilt thou live?\"\n\n\"He will live,\" said Ben Ezra. \"Speak not now. Harm was done by claws,\nbut more by a paw stroke on the head. But for that he had slain them\nsooner, and he was torn only while he was fallen. A hard battle, O\nCaius of Thessalonica.\"\n\n\"He and his have beaten Julius for me,\" said Caius. \"They shall fight\nno more save at Jerusalem and at Rome.\"\n\n\"May we tarry long enough to offer sacrifices to the gods of this\nplace?\" asked Knud. \"I would leave them in good humor. It is well to be\non good terms with the gods.\"\n\n\"What sayest he?\" asked Caius of Ben Ezra, but Ulric himself responded:\n\n\"Peace with thy gods, O Knud. Let alone. I saw when I was under the\ntiger's paw. I thought at first of the valkyrias, but they came not.\nThe gods of this place we will leave here. They are nothing to us.\nCome!\"\n\n\"So be it!\" said Knud. \"I meant only to deal prudently. Thou art our\njarl. We will come.\"\n\nThey lifted Ulric into the chariot, Ben Ezra and Knud and Tostig\nentering with him, and the other Saxons followed, led by Wulf the\nSkater. With them now was Abbas, and he said to Ben Ezra:\n\n\"The keeper of the tiger's cage hath lost his head for letting him out\ntoo soon, and the master of the games lieth slain under the tiers, no\nman knoweth by whose hand.\"\n\n\"They who butcher many,\" said Ben Ezra, \"do well to avoid knives. The\nman with all other men for enemies dieth speedily.\"\n\nBut Wulf the Skater smiled joyously and he said to Lars, the son of\nBeolf, at his side:\n\n\"The Jew is a wise one; but beware thou of Abbas, lest he sell thee.\"\n\nLars looked at the spear in his hand and at Abbas, and he answered not.\n\n\"We have our jarl!\" laughed Wulf as they went forward, and quickly they\nwere at the shore of the Sea of Galilee, and they saw a galley, like\na pleasure boat, rowing rapidly nearer to the place where the chariot\nhalted.\n\n\n\n\n                            CHAPTER XXVIII.\n\n                        THE JARL AND THE RABBI.\n\n\nSoftly and easily may a wounded man be borne along upon cushions over\nsmooth water under a silken canopy. There was no further fatigue for\nthe jarl, the victor, that day, and before its close he lay upon a\ncouch in a room of one of the seaside palaces. All men went out from\nhim save Caius.\n\n\"O jarl, my friend,\" he said, \"I must leave thee. Gain thou thy\nstrength as rapidly as thou mayest. Thy Jew, Ben Ezra, telleth me that\nhe may not tarry here.\"\n\n\"He is not any more needed while I lie thus,\" said the jarl. \"I would\nsee him. If thou art willing, he may go.\"\n\n\"I consent,\" said Caius. \"Thou art interpreter enough for thy men. I\nwill send him to thee, but now I must return to Tiberias, for I have\nmuch upon my hands. May all the gods give thee a speedy recovery, and I\npromise thee that thou shalt yet fight before C\u00e6sar himself. Thou art\nworthy!\"\n\nSo saying, the centurion departed, and in a moment more Ben Ezra came\nand sat down sadly by the side of Ulric.\n\n\"Thou goest from me?\" asked the jarl.\n\n\"Hardly of mine own will,\" replied Ben Ezra, \"but I must go to\nJerusalem, and I will return to thee if thou comest not soon to me. I\ncommit thee to the keeping of Jehovah, my god. Abbas goeth also, and\nthere will be one double tongue the less in Galilee. Fare thee well. I\nhave done for thee what I could.\"\n\n\"O Jew, I thank thee,\" said the jarl. \"Come thou again to me and I will\never welcome thee as if thou wert of my kindred.\"\n\nLittle more did they say, for the jarl was in fever and in pain and the\nhour was late. Ben Ezra departed, but at the door of the room stood\nTostig, spear in hand, although this palace was a place of peace.\n\n\"O Tostig,\" said Ben Ezra. \"I go away for a season. Guard thee well\nyour jarl!\"\n\n\"That will we, O Jew,\" said Tostig. \"There will be swords and spears\naround him by day and night. Whither goest thou?\"\n\n\"To Jerusalem,\" said Ben Ezra, \"and I think I may have somewhat to do\nthere for thy jarl. I love him much. I come again shortly.\"\n\n\"The gods go with thee,\" said Tostig. \"I think thee a brave warrior.\nArt thou sure that the jarl healeth of these hurts?\"\n\n\"No man knoweth surely,\" said Ben Ezra, \"but see ye to it that he hath\nquiet.\"\n\n\"We will care for that,\" said Tostig. \"I have been sore wounded myself,\nand while the cuts were knitting I would fain have cleft the head of\nany who came near me.\"\n\nSo Ben Ezra departed from Tiberias, taking with him Abbas, and the\npalace of the friend of Caius by the Sea of Galilee contained now only\nthe servants of its owner and these who were called the gladiators of\nCaius of Thessalonica. For these there was sufficient occupation of\nmind at the first, for many came to gaze at them, and men of rank,\nalso, were interested, but none might ask undue questions of men whose\nspeech was unknown and whose behavior was silent and haughty. To them,\nalso, not only were all buildings new to be examined, but there were\nfruits and wines and strange ways of living to become accustomed to.\nBoats were there, to be used at any time, and the Saxons talked much\nof the fiords and fishing of their own land while they were amusing\nthemselves upon the Sea of Galilee. Over it did they go from end to end\nthat they might look upon all things upon its shores, and they wondered\nmuch that one small sea should contain such abundance of fishes and\nhave so many towns and cities builded beside it, as if there were no\nother place for the cities of this marvelous land. Few days went by in\nthis manner, but there were other affairs than those of the Saxons.\n\nEver is it true that the cunning, who believe their ways to be\nhidden, are sometimes read as are books in strange tongues read by\nthose who are learned in difficult runes. Julius, the centurion, the\nchief commander of the Roman forces in Galilee, had other hopes and\nambitions than the winning of sesterces in gambling, and he had other\ncunnings besides his tricks of the circus. At this time Herod, tetrarch\nof Galilee and loving to be called a king, was plotting to gain for\nhimself the entire realm which had been ruled by his cruel father,\nHerod the Great. To this end he might require the removal by C\u00e6sar of\nPontius the Spearman from being procurator, and the destruction of\nhis own brother, Herod Antipas, tetrarch of the lands northward of\nGalilee. If, therefore, Herod of Mach\u00e6rus and Julius, the centurion,\nwere working together against the procurator, then the near friend of\nPontius was as a spy and an enemy in their camp. Nevertheless, Caius\nof Thessalonica had been received in Tiberias with all the welcoming\ndue to an exceedingly distinguished visitor, an honored friend.\nNot that Herod was here to meet him at this time, for the tetrarch\npreferred the safekeeping of his Black Castle, Mach\u00e6rus, on the\neasterly side of the Sea of Death, which hath no waves and whereon the\nseabirds die.\n\nCaius, the centurion, walked one evening alone by the shore of the Lake\nof Galilee, and he communed deeply with himself.\n\n\"Thus far Jove hath been with me. I have escaped the treachery of both\nthe wolf Julius and the foxes, the Herods. I do now know that Herod\nAntipas refuseth to join them, to his ruin. Why linger I here, where\nI am not safe for an hour but for the swords of my Saxon gladiators?\nI trust their jarl, for they are his more than mine. He mendeth but\nslowly from the tiger's clawing. I would he were able to ride even in\na chariot, for my errand here is done. Unless he were with me I could\ndo little with his barbarians. Abbas is a traitor, ready for a buyer,\nand I believe him already bought. Ben Ezra--he is a Jew, and every Jew\nhateth every Roman, with good cause. I am glad he hath departed. The\nbarbarians are not so, for they are but gladiators, and this Jarl Ulric\nis not as a common man. I may trust him.\"\n\nSo spoke with himself the grim centurion, the near friend of Pontius\nthe Spearman, considering the affairs of princes and of kingdoms. He\nwalked on, thinking deeply, and ere long he was at the palace by the\nseashore. A legionary stood guard at the portal, but no Saxons were to\nbe seen.\n\nIf one had walked with these at this hour, he would have been at a\nplace from which might be seen the walls of Capernaum. Along the beach\nwere boats and sailing vessels, larger and smaller, and out upon the\nsea were many fishermen. At the water side were some who spread out a\nnet to dry, but above them, on the high ground, had gathered rapidly a\nmingled concourse of people. Said one of the net dryers to another:\n\n\"The rabbi of Nazareth is there. He healeth the people. Only John is\nwith him. We ought not to be here. Let us go to him.\"\n\n\"Did he not bid us go a-fishing?\" replied another. \"We have caught\nmany. It is enough. Let us go.\"\n\nSo left they their net and went up the bank, and as they went they\nheard the voice of the rabbi preaching to the multitude. They listened,\nhastening, and they spoke no more to each other. All utterances were\nstilled save the wonderful voice of the preacher, the music of the\nwaves upon the beach, and the low, painful mutterings of one man who\nhobbled along upon crutches as if to join the gathering.\n\n\"O that I am to be maimed!\" he said. \"I, Ulric, the son of Brander!\nThat I shall no more walk firmly! The tendons and the muscles of my\nlimbs refuse to heal, as if the tiger's claws were poisonous. What\nthinkest thou, Wulf the Skater? Shall we not go on and see this man?\"\n\n\"Thou art faint, O jarl,\" said Wulf. \"It is not well that thou hast\nwalked so far. I fear thou wilt but cure the more slowly. One goeth by\nus! Look at him! Hear him! He is a leper!\"\n\n\"Unclean! Unclean! Unclean!\" a hoarse and croaking sound came to their\nears from the ulcered, shriveled lips of him at whom Wulf pointed.\n\nBehind him were four who carried a sick one in a litter, but they held\nback, not overtaking the leper.\n\n\"Come!\" said Ulric. \"I would look into the face of this god once more.\nWe may hear another of the demons. I have much curiosity concerning\nthem. Put thy arm around me and aid me on.\"\n\n\"Woe is me, son of Brander,\" moaned Wulf, but his strong arm went\naround the waist of his jarl and they walked along.\n\n\"Unclean! Unclean! Unclean!\" the terrible voice repeated, but on the\nbrow of a little knoll the rabbi of Nazareth stood and ceased not his\npreaching.\n\nAll around him were men and women, the old and the young, but these\nstepped suddenly away, as if in fear, while the leper came toward them.\n\n\"He hath no right!\" exclaimed one.\n\n\"Touch him not! Breathe not his breath,\" said another, \"lest thou\nbecome leprous!\"\n\nDown knelt the leper, but the rabbi ceased speaking and looked upon him\nkindly.\n\n\"What wilt thou?\" he asked.\n\n\"That I might be clean,\" gasped the leper.\n\n\"Be thou clean!\" said Jesus.\n\n\"O jarl!\" exclaimed Wulf. \"What is this? He standeth erect! He is\nstrengthening! Would almost that thou wert a Jew, for their god is a\nstrong healer.\"\n\n\"Come!\" said Ulric. \"He hath cured this leper. I will have speech with\nhim. Nearer! I walk more easily. My hurts cease to pain as they did.\nO Wulf, aid me strongly, that I may get to him. Pass me on! I breathe\nmore freely! I strengthen! I fail not! Fear thou not for me that this\nshall do me harm!\"\n\n[Illustration: \"O thou Jesus, of the sons of the gods!\"]\n\n\"O jarl!\" said Wulf. \"This is but a sudden strength that cometh to\nthee. Afterward thou wilt fall!\"\n\n\"On! On!\" exclaimed the jarl. \"I have somewhat to say that I had\nforgotten. I must speak!\"\n\nNear were they now, and the rabbi of Nazareth again ceased speaking as\nhe looked upon the white face of the jarl, but the crutches of Ulric\nhad fallen from his hands and the arm of Wulf seemed still to uphold\nhim.\n\n\"O thou Jesus, of the sons of the gods,\" said the jarl. \"Sigurd, the\nson of Thorolf, hath fallen in battle with robbers, many of whom he\nslew. He bade me that I should see thee again and bring thee his\ngreeting.\"\n\n\"O rabbi of the Jews!\" exclaimed Wulf the Skater, earnestly, \"it is\nUlric, the son of Brander the Brave, of the Northland. His gods are not\nthy gods, for he is a son of Odin, whom thou knowest not. But he is our\njarl and we love him. We pray thee that thou wilt ask of thy god for\nhim that his hurts may be healed and that he may become strong to lead\nus, for we are but as lost children without him.\"\n\nAs yet Jesus answered not, but the jarl stood firmly upon his feet\nand stepped one step nearer, Wulf stepping with him, but of the other\nSaxons was none with them.\n\n\"O rabbi,\" said Ulric, \"I was torn by wild beasts in the arena of\nTiberias. I slew both the lion and the tiger, while they were tearing\neach other. And now I shall be no more a warrior, for my sword\nfalleth from my hand.\" As he spoke he held out the hand which had\nbeen so strong, and which was now so weak, and it was touched by the\noutstretched hand of this rabbi of Nazareth.\n\n\"Go, thou,\" he said. \"Be thou healed. And remember thou that which\nthou hast this day seen and heard. Speak not again now.\"\n\nWulf the Skater took up the crutches, but the jarl put them away,\nsaying:\n\n\"Hath he not bidden us to go our way? Shall we not now do as he hath\nsaid? Come! I walk as if I had not been torn. He is a god!\"\n\n\"O jarl,\" whispered Wulf, trembling, \"what meaneth he? I understand him\nnot. And what is this strange thing which hath come upon thee, as if\nthou wert a Jew? I think his god is a good god and very strong.\"\n\nBut both he and Ulric stepped backward and the rabbi and the man who\nwas leprous stood face to face.\n\n\"Silence, Wulf the Skater!\" whispered Ulric. \"The god hath spoken to me\nas to this one. I have looked into his face. What he hath said I know\nnot, but I go to Caius quickly. Where thou art commanded well do thou\nobey lest evil befall thee.\"\n\n\"Clean! Clean!\" sprang from the lips of the healed leper. \"Hallelujah!\nI glorify the god of Abraham. This man is a great rabbi!\"\n\n\"He is of the sons of the gods, thou stupid one!\" said Ulric. \"I am\nhealed. Who but a god can cure the scratch of a lion or a tiger? He is\nas Odin, and I think they are friends, and that Odin bade him heal me.\nI will fight for him when he gathereth his army. O Wulf the Skater,\ncome! My arm telleth me that I could cast a spear. O thou of Nazareth,\nthank thy father for me, for thou wilt see him before I do. When I am\nslain I shall go to Asgard and I will meet him there, and I hope to\nmeet thee. Also, in thine hour, thou shalt be my captain.\"\n\n\"Go now!\" said Jesus, turning to a sick one.\n\n\"He meaneth he will send for thee,\" said Wulf, walking on at the side\nof Ulric. \"But we need more Saxons for his army if he is to overcome\nthe Roman legionaries. He would do well to gather the sea kings and all\nthe men of the fiords and of the forests. Even from Denmark and the\nislands we might bring to him good fighters. How well could a captain\nkeep his army if he might heal all who were but hurt, losing only the\nheroes for whom the valkyrias had come.\"\n\n\"I walk more strongly!\" said Ulric. \"I would be where I may look at\nmyself, for the marks were deep and they ran as sores. We will go with\nCaius to Jerusalem. I think it well for us that we guard him.\"\n\n\"O jarl,\" said Wulf, \"a friend is a friend, but a Roman valueth a Saxon\nonly for his sword and for his spear. I have thought, indeed, that he\nmight yet give one of us a chance to kill this Julius. I shall not be\nfully contented until I have seen his blood upon a blade of steel.\"\n\nAs a man in a dream walked Ulric, the son of Brander. With him, looking\nat him as they went, walked Wulf the Skater, and now other men drew\nnear.\n\n\"How is it with the jarl?\" asked Knud the Bear. \"He hath no crutches\nthis day.\"\n\n\"He walketh strongly,\" said Tostig the Red. \"His face is ruddy and his\neye is bright. Thou hast been with him, O Wulf; what is this?\"\n\n\"The son of Odin hath had speech with this god of the Jews,\" slowly\nresponded Wulf. \"I myself asked for his healing, but the sons of the\ngods are not like other men. Hold ye your peace, for the jarl was\nbidden to tell no man.\"\n\n\"Let him alone, then,\" said Tostig. \"It is enough that he walketh so\nwell. But yonder is the centurion, Julius, talking with Caius.\"\n\n\"I am to slay him yet,\" said Wulf. \"Watch ye, for we belong to Caius.\"\n\nEnough of Saxon knew their master to gather that saying, and it pleased\nhim well, for he turned and saw blue eyes that flashed a little, and\ndark eyes that seemed to ask his bidding.\n\n\"There is truth in these Saxons!\" he said to himself. \"Were I to\ncommand the death of Julius, he were dead this hour.\"\n\nBut at that moment the voice of Julius rose in a sound of chiding.\n\n\"O Caius,\" he said, \"I did indeed pay my wagers, as became me, but thy\nSaxon died and the payment should be restored to me. If the lion and\nthe tiger slew him, the wager is void.\"\n\n\"Justly spoken, O my friend,\" replied Caius; \"but knowest thou this\nman, or is he dead?\"\n\nThen turned Julius, wondering, for before him stood the son of Brander\nsmiling in a mockery, and saying:\n\n\"Hael to thee, O Julius, the captain! Hast thou any wild beasts with\nthee this day? I am Ulric the Jarl!\"\n\nProud and strong he stood, with the sunlight upon his golden curls and\nthe strength of a hero showing in his movements, but the centurions,\nboth of them, stared at him as if they were in amazement.\n\n\"Thou art not dead?\" said Julius.\n\n\"O jarl, let him take thy hand,\" said Caius. \"Let him be sure of thee\nthat thou art well.\"\n\n\"O Caius,\" said his enemy, \"thy swordsman liveth. I have been\nmisinformed. But how were his wounds that they have healed?\"\n\n\"Scratches!\" said Caius. \"I have care for my gladiators after a fight\nthat they may be ready again. Hast thou any to put against him for a\nthousand sesterces, man for man?\"\n\n\"That have not I!\" exclaimed Julius, looking hard at Ulric. \"He hath\ncost me enough!\"\n\nThen, also, for he was cunning, he understood the looks of the other\nSaxons, closing around the jarl lovingly, and he ground his teeth,\nfor the thought in his mind was: \"They would slay half a cohort of my\ndwarfs. They would slay me, if Caius bade them. I would I had such a\nbodyguard that knew nothing but mine own will.\"\n\nSo thought Caius in his mind, silently, but he said aloud:\n\n\"O Julius, now the games are ended, and my mission to thee from Pontius\nis fulfilled, I will set out on the morrow for Jerusalem. The winter is\nhere. What sayest thou?\"\n\n\"The gods go with thee!\" said Julius. \"Also, if thou art wise, take\nwith thee thy swordsmen. Thou wilt be safe by the way.\"\n\nSo he and Caius walked on by themselves toward the palace and the\nSaxons gathered gladly around their jarl, feeling of his wounds that\nwere healed and wondering greatly at his meeting with this son of the\nunseen god of the Jews.\n\n\n\n\n                             CHAPTER XXIX.\n\n                        BEAUTIFUL AS APHRODITE.\n\n\nAt the Damascus gate of the city of Jerusalem halted a weary-seeming\nass, upon whose back was a dusty and travel-worn rider.\n\n\"Wonderful indeed is the grandeur of this city,\" he had said, as his\njaded beast toiled up the road from the bridge over the Kidron. \"I\nwould willingly have paused longer upon the Mount of Olives, but the\nlash of the procurator is close behind all who ride upon his errands.\nSomewhere in this city of the temple is my Sapphira even now, but how\nshall she be made to know that I am here? Not now, but I will climb\nover all barriers, even these great walls and forts, until I find her.\"\n\nAt the gate was a Roman guard, and to the sentinel on post rode Lysias,\nsaying:\n\n\"O guard! From the procurator to the captain of the gate! In haste! I\nam Lysias, a messenger, with a token in writing. I may not dismount\nuntil he cometh.\"\n\nThe soldier saluted ceremoniously the name and authority of the\nprocurator, but he stirred not from his place. He did but shout loudly,\nand an officer came forth, to whom the Greek repeated his utterances.\n\n\"Sit thou in thy saddle,\" said the officer. \"I may not touch that which\nis in thy keeping. But the centurion cometh shortly--the captain to\nwhom thou art commanded to make thy delivery.\"\n\nNo word spoke Lysias to the important man when he came, but the\nsubofficer made the announcement and the parcel from Pontius the\nSpearman was placed in the right hands.\n\n\"O messenger,\" he said, \"dismount. Thy beast is worn out. So art thou.\nHe will be kept for thee in the stables of the procurator. Thou, too,\nwilt have refreshment. Rest thee and be ready when thy return message\nshall be prepared.\"\n\nHere ended for the present the dangerous responsibilities of Lysias,\nbut in no manner had he yet escaped from the grip which had been put\nupon him. The lodgings to which he was speedily conducted were as a\njail of secure detention and from them he might not think of going\nforth, lest evil should befall him. He might but eat and sleep while\nhis next duties were in course of preparation. Nevertheless rest was\nsweet, and his dreams were free to wander where they would, seeking a\nfair face and welcoming eyes which might not now be far away.\n\nEarly upon the morrow he was summoned to come forth, and he was led to\nthe Damascus gate without having had speech with any save with soldiers\nwho were as his jailers. Here a saddled horse of Arabia awaited him and\nalso a high official, whom he knew not, and the captain of the gate,\nwhom he had already seen.\n\n\"Hear thou with care, O messenger,\" said the latter, sternly, handing\nto him sealed parchments. \"This first to the procurator, from me. These\nfrom the high priest and from the captain of the temple. I give thee,\nalso, a spoken message, which may not be written, for thee to deliver\nand then to forget; for thou art of the household of the procurator,\nand he trusteth thee. Were another to hear these words, lost were his\nhead and thine. Slain is the secret messenger of Herod, and he went\nnot to C\u00e6sar. Caius of Thessalonica is in Galilee watching Julius, the\nsubtle, who plotteth, also, with Herod and with Herod Antipas. Caius\nmay die there, or ere he returneth, but he is trustworthy. Well were it\nthat the procurator should now leave his inspection of the garrisons\nand of Samaria until a better day and that he should now return to\nJerusalem. Go!\"\n\nWords in reply or questioning might not be spoken. Lysias sprang upon\nthe Arabian horse, the letters being hidden in his bosom. Away he rode\ndown into the valley of the Kidron, thinking within himself: \"Great is\nthe peril to him who carrieth the secrets of rulers. Sure is my death\nif I do not this errand well, and yet the very doing of it may bring a\nsword upon me. And now I am indeed of the household of Pontius, wherein\nis hidden my Sapphira. Surely Venus and Juno are with me, and Mercurius\nhimself hath given me this fleet stallion to ride. He goeth like the\nwind.\"\n\nThe remainder of that day went by, and the night also came and went.\nNot any did the messenger have speech with but seemed ready to speed\nhim and glad to see him go from them, as if in having met him might\nbide a future peril. It was only in the forenoon of the next day,\nhowever, that his Arabian steed was halted in the middle of the\nnorthward highway, and before him in a gilded chariot sat Pontius, the\nprocurator, reading slowly and thoughtfully the letters delivered to\nhim by the Greek.\n\n\"Thou hast done well,\" he said. \"Thou art a speedy messenger. Was there\naught else?\"\n\n\"Here are ears near thee, most noble Pontius,\" replied Lysias. \"I pray\nthee bid me be prudent.\"\n\nDown from the chariot sprang the procurator with a fierce flush upon\nhis face.\n\n\"Dismount thee! Come!\" he said. \"Back, all! I would have speech with\nthis man.\"\n\nNot far behind the chariot, but not as if they belonged to the same\ncompany, rode two men upon asses, of whom one said to the other:\n\n\"A messenger, O Ben Ezra. There may be tidings of importance. What\nsayest thou?\"\n\n\"Silence! O Abbas,\" replied the other, \"thus far our god hath\nbefriended us upon our way. Trifle not with the business of the great\nlest the sword seek thee. Thou art overcurious. Let it suffice that we\nare permitted to travel under guard of the procurator's horsemen.\"\n\nAt the roadside now stood he and the Greek and none dared approach\nthem, for the spear of Pontius was in his hand and his brow was dark.\n\"Speak with care!\" he said to Lysias. \"Forget not!\"\n\n\"Thus said the captain of the gate,\" replied Lysias, \"and a centurion\nwho stood by him and who gave me this cornelian for a token, telling me\nnot his name----\"\n\n\"Cornelius of C\u00e6sarea!\" muttered Pontius, but the Greek spoke on,\nuttering exactly the words which had been given him.\n\n\"It is well,\" he said. \"I have word of Caius that he is wise and that\nhis Saxon swordsmen are his bodyguard. More than one secret messenger\nhath been slain, saith Ben Ezra, the bringer of tidings from Galilee.\nTrust him, but not the Jew Abbas who is with him, for he is of Julius.\nI come to Jerusalem quickly. I will give thee a fresh horse in the\nmorning and thou wilt again return, but thou wilt wait for me in mine\nown house. Go, now, and speak to these Jews, questioning them. What\nthey say thou wilt tell me. It is well that thou wilt be in the school\nof Gamaliel and also in the service of the procurator, but let no man\nknow of more than of the school.\"\n\nThe strong man is often in desire of a willing servitor, and it pleased\nPontius that the eyes of the Greek brightened with delight. His lips\nparted also, but the word \"Sapphira\" that was upon them was not uttered\naloud.\n\nThe ruler turned and walked away to his chariot and Lysias remounted\nhis weary horse.\n\n\"I must be cunning with these Jews,\" he thought; \"and in one of them is\nmy deadly peril.\"\n\nThe train passed on and they were riding at his side.\n\n\"Who art thou?\" he asked of Ben Ezra.\n\nThere was no sign of recognition in the face of his former comrade upon\nthe good ship _The Sword_.\n\n\"I am a Jew of Spain,\" he responded, \"and my name is Ben Ezra. I go to\nfulfill a vow in the temple of Jehovah at Jerusalem. Who art thou?\"\n\n\"I am a Greek of Alexandria, named Lysias,\" was replied as cunningly.\n\"I am of the household of the procurator, but I am also a student in\nthe school of the great Gamaliel. Thou doest well to perform thy vows.\nI am now bidden to be with thee. And who is this man?\"\n\n\"I am Abbas of Jerusalem,\" he said for himself, bowing low to one who\nseemed to be trusted by Pontius the Spearman. \"I am a merchant and I\nhave had dealings with the procurator.\"\n\n\"O Abbas,\" said Lysias, \"many have heard of thee. Thou art a lender of\nmoney and thou art hard in thy dealings. Why dost thou pretend that\nthou knowest me not? Hast thou not seen me many times in the markets? I\nthink that thou art never seen in the schools. Tell me, how was it with\nthat trouble of thine that thou didst have before the magistrate? Didst\nthou escape with no more harm than a fine?\"\n\n\"Nay! Nay!\" exclaimed Abbas. \"Speak not too much of that matter. The\njudge compelled that unjust person to pay me my dues and he was cast\ninto prison. I exact no more than my right.\"\n\n\"Thou art, then, a rare money-lender,\" said Lysias; but the cunning of\nthe Greek had succeeded and Abbas was ready thenceforth to say to any\ninquirer that he knew this man well.\n\n\"O youth,\" he said, \"I will talk with thee further concerning certain\nmatters when we may have opportunity. Be not thou too much influenced\nby what thou hearest. Is there any news?\"\n\n\"Tell us what things have occurred,\" added Ben Ezra, \"for we have been\nin Galilee. I journeyed thither as interpreter for the Saxon gladiators\nof Caius, the centurion, of Thessalonica. In his service am I to this\nday.\"\n\n\"A good man and highly honored,\" said Lysias. \"He is a friend of the\nprocurator.\"\n\nSo they rode on conversing, but in Greek. Nor was it difficult as they\nwent for Ben Ezra, even aided by Abbas unwittingly, to inform Lysias\ncompletely concerning the doings of the Saxons.\n\n\"The procurator,\" he said, \"calleth the gladiators of Caius his own.\nThou wilt soon meet them and I will make thee acquainted with them.\"\n\n\"I will gladly have speech with such strange ones,\" said Lysias, \"but\nthe scholars of Gamaliel may not meddle much with the circus.\"\n\nEre long as they rode he and Ben Ezra were able to be out of hearing of\nAbbas and the others, but the speech of the Jew was brief.\n\n\"O Greek,\" he said, \"if thou art imprudent in this matter, for thee is\nnot the scourge, but the sword.\"\n\n\"And for thee crucifixion,\" said Lysias. \"Fear not for me. Thou art as\nI am, and we are one with the jarl and his company.\"\n\nThe place of the procurator's abiding was at hand. It was an ancient\npalace, which was also a fort, and they who occupied it were of high\ndegree. Of them the two Jews and Lysias might see or know but little,\nbut they had quarters assigned them. In the morning orders came to\nLysias only, and he was quickly in the saddle with a message for\nCornelius, the centurion. If he found him not at Jerusalem, he was to\nride on after him, even to C\u00e6sarea.\n\n\"O to be in the procurator's house!\" thought Lysias, \"for she will be\nthere and I shall see her.\"\n\nEven as he rode away from the palace gate, however, bright eyes were\nupon him from a window above and a young girl said in a low, musical\nvoice:\n\n\"O Lysias! Lysias! Do I not know that he is in search of me? Woe to\nhim and woe to me if he should find me! What is this which is come? Am\nI not happy as I am? Surely I do love him. He is very beautiful. He\nloveth me. But what have I, the favorite of the wife of Pontius, to do\nwith him? What have I to do with a love that I lost so long ago and\nthat is gone? It were but a sharp peril now. If I meet him, I can but\ntell him that I am no longer his. He is but a swift messenger of the\nprocurator; a fellow to ride horses and to be scourged if he rideth not\nspeedily. I am one to dwell in palaces, wearing gay apparel and jewels\nand having the favors of the great.\"\n\nFull of pride was her fair face as she spoke, and in it was a scorn\nfor any who were lowly. To her the apparel of her servitude was more\nworth than was the love of a youth who had been robbed of his patrimony\nand whose rank was lost. She sat at the window watching him as he rode\naway, and she sighed deeply.\n\n\"Yes,\" she said, \"I love him, and it is pleasant to love. He is a good\nhorseman. So are all my Roman lovers. What is he compared with a Roman?\nEven the Jews, if they are rich and of power, are better than a poor\nGreek boy, fit only for errands.\"\n\nShe arose and walked away, but a mirror was near and she gazed long at\nher reflection, admiring it greatly.\n\n\"I am as beautiful as Aphrodite, they tell me,\" she said. \"I will\nsacrifice to her this day, and to Juno. There are no gods upon whom\nLysias may call for great gifts. He can bring them no rich offerings,\nwhile I can have oxen slain before the altar. Aye, and I have had\nmen sent to prison and to the arena if they offended me. I sent that\nfoolish Jew girl to the lions at Jerusalem. I taught her better than to\ninterfere with me.\"\n\nHer red lips tightened cruelly, and her eyes were terrible and her\nmovements were lithe as those of a young panther as she walked on along\na corridor. But Lysias galloping northward was alone upon the highway,\nand he shouted aloud:\n\n\"Sapphira! Sapphira! My beautiful one! My beloved! I am drawing nearer\nto thee! Thou art dearer than life and I believe thou art true to thy\nlover. I will find thee yet, and I will look into thine eyes and I will\ntouch thy hand and I will tell thee all that is in my heart.\"\n\nStrong is love and wonderful are its follies and its treacheries, for\neven then his Sapphira sank upon a couch in her own room sighing and\nmurmuring in a low voice:\n\n\"Lysias! Lysias! My beloved! If I have any other lovers I will name\nthem Lysias in my mind, for I do love thee, and love is pleasant.\"\n\nThe procurator made no great haste that morning, although he prepared\nfor journeying. He had many affairs and his messengers came and went,\nand it might be seen that he was a thoughtful governor, attending to\nall who came, only that he sent out some edicts which were full of\nblood and vengeance.\n\nNot long was it before he stood in a private place with Ben Ezra\nquestioning.\n\n\"O Jew,\" he said, \"now thou hast told me how Julius plotted to destroy\nthe Saxon guards of Caius, thou hast told me enough. But for this tall\njarl of thine and his pirates I should never again meet my friend. He\nmay give them to me and I will not waste them in the arena. I know of\na place to which I may send a good sword and where I may not send a\nlegionary.\"\n\nLow bowed the Jew and the unspoken word in his heart was bitter.\n\n\"Do I not know thee?\" he thought. \"Thou treacherous one! Thou wilt send\na Saxon to do a deed, and when it is done for thee thou wilt slay him\nand clear thyself. This is the cunning of the Romans. I will beware of\nthee and thy errands, but I care little for my own neck. O that the\nMessiah, the Prince of Judah, were even now smiting thee and thine from\nthe earth! He cometh soon, I think.\"\n\nSo, bowing as became his station, but guarding well his face and\nletting his eyelids fall over any glitter that might betray him, Ben\nEzra went out of the palace and was joined by Abbas.\n\n\"O my friend,\" said Abbas, \"why linger we?\"\n\n\"We may not linger,\" said Ben Ezra. \"We depart, but thou wilt travel\nalone. I have commands from the procurator. See to it that thou art\nquickly in Jerusalem.\"\n\n\"Whither goest thou?\" asked Abbas.\n\n\"Art thou mad?\" said Ben Ezra. \"Or dost thou know but little of\nPontius? Keep thy questions to thyself and tarry thou not, for I think\nthou hast a spot on thy name. Beware lest it turn into red on thy\ngarments.\"\n\nVery pale was Abbas, but his face was that of a fox with a wolf for his\nfather.\n\n\"O Ben Ezra,\" he said, \"thy counsel is good. But be thou careful of\nthine own head. I can tell much concerning thee.\"\n\n\"In the day that thou chatterest unwisely,\" said Ben Ezra, \"thou wilt\nspread thy arms upon a piece of wood and thou wilt hear the sound of\nhammers. Then thou wilt be set up at a wayside for men to mock thee.\nThe Romans hesitate but little concerning such as thou art.\"\n\nGhastly was now the face of Abbas.\n\n\"O my friend!\" he exclaimed, \"I meant no evil! I will be true to thee!\"\n\n\"Thou wilt remember this thy warning!\" said Ben Ezra, sternly. \"Thou\nwilt not sin against thine own life. If thou shalt at any time err, it\nis no fault of mine. Thy blood is upon thine own head.\"\n\nThey parted one from another, and then came to pass a strange thing,\nfor a servitor led Ben Ezra to the armory of the palace. Here he\nremained but briefly, and when he came out he was armed from head to\nfoot in the panoply allotted to the Jewish servants of the temple under\nits Roman captain. So arrayed he might ride as if he were a Roman\nunder the sure protection of the procurator. A horse was ready for him\nand he mounted, riding to the palace gate. At this place was now the\nprocurator in his chariot.\n\n\"Go thou speedily as thou hast said,\" commanded the procurator. \"Be not\noverhasty, but prudent. If it prove as thou tellest me, well with thee.\"\n\n\"On my head be it,\" said Ben Ezra, and he rode away northward.\n\n\"I have purchased him with a price,\" he said to himself, \"but I will\ndeal truly with the jarl. If some of his treasure and some of mine must\nbe paid as tribute to this Roman governor, all that remaineth--and\nit will be enough for us--will be kept for our own uses. Now for the\ncavern in Carmel, and the journey will be neither long nor unsafe for a\nman traveling with the seal of Pontius.\"\n\nAs for the procurator in his chariot, he, too, had a thought upon his\nmind, and it made his face brighten.\n\n\"The gold is well,\" he thought, \"but the jewels! There is naught else\nfor which C\u00e6sar hath so great a lust. I care little for such things. Of\nwhat value are bright stones except that they will sometimes buy more\nthan will gold or silver? Let the Jew bring his gems and with them I\nwill defeat Herod and Julius.\"\n\nFar on along the southward highway rode Abbas, having a pack beast\nwith him and two fellow-travelers. The Jerusalem road through Judea\nwas accounted safe unless one rode alone or unarmed. Still was his\nface turned backward now and then as that of one who feareth lest he\nmay be followed, for the words of Ben Ezra had been severe, and Abbas\nknew that he who uttered them had been much in conversation with the\nprocurator.\n\n\"He is deep as a well!\" he thought. \"Can he know anything of my\ndealings with Herod? Even now I must go to the ford of the Jordan and\nto Mach\u00e6rus before I go to Jerusalem. Alas! The Black Castle! How many\nhave entered it who never were seen again! Well is it set so near to\nthe Sea of Death! I am a Jew! I hurt not my own people! But it is\nrighteous to profit by the dissensions of the heathen. If Herod and his\nbrother Antipas and this Pontius the Spearman were to slay one another,\nwhat harm to the children of Abraham? Ben Ezra doeth not well to keep\nfaith with a Roman or an Edomite. They have defiled even the temple of\nthe Most High.\"\n\n\n\n\n                             CHAPTER XXX.\n\n                         THE JAVELIN OF HEROD.\n\n\nThe Saxons and their jarl in the palace by the Sea of Galilee were now\nmore impatiently awaiting the orders of Caius of Thessalonica. It was\nat the close of a day that he came to have speech with Ulric, the son\nof Brander, and to wonder again at his swift healing. He examined the\nscars, touching them, and asking many things concerning this learned\nrabbi of Nazareth and of his marvelous cures, for these were things\nwhich no reasonable man might easily believe.\n\n\"Thou hast thy strength again,\" he said at last. \"Never have I\nthought much concerning the gods, but I shall deem it prudent to make\nsacrifices to such as I think may aid me. I have never found them\nprofitable. Take now thy weapons and walk out along the shore with me,\nfor I am restless. I linger here too long on thy account. Come!\"\n\n\"I shall delay thee no longer, O noble Caius,\" said the jarl, \"but well\nam I assured that thou doest well to wear mail and to have thy good\nsword at thy side. Put on thy helmet.\"\n\n\"So do thou,\" said Caius. \"But what said to thee the Jew, thy\ninterpreter? Was it aught more important than thou hast told me?\"\n\n\"Not so,\" said Ulric, \"but the keeper of the tiger's den told much\nunwittingly. The beasts were prepared to win more than sesterces. Had\nI been slain, and Tostig, thou wouldst now have less perfect guarding.\nI will tell thee, O Caius: I like thee well and I am jarl; not another\nwill my men obey. I think thee a good fighter, and such as I am agree\nnot well with cowards or with those who deal in subtleties.\"\n\n\"O jarl,\" said Caius, \"speak not of Julius, the centurion, as if he\nwere a coward, but he is exceedingly deep in his counsels. There is\nmore than thou knowest in this matter. Thou mayest yet have a chance to\nuse thy long, sharp sword again.\"\n\n\"That might please me well,\" said the jarl. \"I like not to leave a\nblade too long in the sheath lest it might rust. But glad am I as we\nwalk to feel no more any hindrance from the work of the tearing claws.\"\n\n\"Well with thee, O jarl!\" exclaimed Caius. \"And now look without\nlooking and mark well without seeming to mark. Seest thou the men in\narmor who have landed from yonder boat at the shore? They walk not\noverrapidly, but they aim to come between us and the palace. Canst thou\nread a riddle?\"\n\n\"I had noted them already,\" said the jarl. \"Men have told me that the\nother shore of this Sea of Galilee belongeth to Herod Antipas, the\nbrother of the Herod who ruleth here under C\u00e6sar. I have heard that men\nwho are hated by the Herods die at distances. But thinkest thou that\neither of them would dare to send a sword against a Roman, and such as\nthou art?\"\n\n\"Consider, O jarl,\" said Caius, calmly. \"Who then would know concerning\nthe sword or him who sent it if thou and I were slain upon this beach\nand our bodies conveyed in yonder boat to be sunken in the sea? Would\nnot the thing be well hidden if the doers of it were shortly also slain\nby Herod Antipas or by his brother, whichever sent them?\"\n\n\"Great would be the inquiry,\" said Ulric.\n\n\"Thou art young!\" said Caius. \"C\u00e6sar might demand my blood of him of\nMach\u00e6rus, in whose land we are, or even of this Julius. What if Antipas\nthus plotted harm to both of them? He could strike them no deeper stab\nthan this! Thy spear, Saxon! O for my shield! I was imprudent!\"\n\n\"Take mine!\" said Ulric, casting his spear. \"I need it not. There are\nnow but four. Ha! A javelin! I caught it! Out with thy sword!\"\n\nEven while talking had they permitted the five men from the boat to\ndraw much nearer and as if unobserved. Sudden and fierce had then begun\nthis assailing. The javelin had been well aimed, but the quick sword of\nthe jarl had parried it. These were men of war who were coming and they\nhad deemed themselves sure of victory, for one had said:\n\n\"On! With him is no one but his tiger-torn gladiator. He hardly may\nstand erect. The centurion is at our mercy. End him!\"\n\n\"Use well the shield,\" said Ulric. \"Thou art thyself a good swordsman.\"\n\nNow he who seemed the leader of these murderers drew back astonished to\nsee how this Saxon, whom he deemed crippled, sprang toward him with a\nwar cry. He was no match for such a one, and his next comrade, turning\naffrighted to see him fall, left his own neck unguarded against the\nsword of Caius. What then were the two who remained against two mighty\nmen of valor?\n\nIll advised had been he who had sent them upon this errand, for the\njarl laughed exultingly to find how well his strength had come back to\nhim.\n\n\"O noble Caius!\" he shouted. \"Thou art a good swordsman. They are all\ndown. But these fellows are Jews. How is this?\"\n\n\"None the less are they from Antipas,\" said Caius. \"I can read his\ncunning. He will say they are but robbers from the rebel bands beyond\nthe Jordan. Therefore I may bring no accusation against him. But I\nthink thou art enough for five such as these. Well is it for me that\nthou art healed. Now will I send word to Julius, and his servants may\nhave the care of this carrion.\"\n\nUlric was silent, looking down upon the slain. \"Jews?\" he said. \"I\nthink now that they are not so, but they are like them. What is thy\nthought, O Caius?\"\n\n\"Samaritans!\" suddenly exclaimed the centurion after a closer\nexamination. \"Not from Antipas. Here is a deeper treachery. These are\nfrom the elder Herod, the fox of Galilee. O jarl, haste! To the palace!\nWe will make ready for our journey. But know thou that our road to\nJerusalem passeth through Samaria, whence these came. Verily I have a\nnew tale to tell the procurator.\"\n\n\"And I have a new thought concerning the keeping of thy life,\" said\nUlric. \"But there will be more than one round shield with thee in\nSamaria. A man needeth to have many eyes in this land.\"\n\nAt that moment, while they still gazed down at the dark yet pallid\nfaces of the dead, they heard near them shouts of angry chiding, but\nthe tongue was not the tongue of that country.\n\n\"O jarl!\" shouted Lars, the son of Beolf, \"we saw thee afar! We came\nin haste! What doest thou here with thy sword in thy hand--thou that\nwert torn by the Roman tiger?\"\n\n\"Woe to thee, O jarl!\" shouted another. \"Thy men should have been with\nthee!\"\n\n\"O Caius,\" exclaimed Tostig the Red, \"thou didst fight for our jarl?\nThen will we fight for thee. Thou hast made good friends this day.\"\n\nSufficiently well did Caius understand Tostig and the others who now\ncame running to see how it might be with the son of Brander, and it\npleased him greatly.\n\n\"I may now depend upon these wolves of the North,\" he thought, \"and\nsore may be my need of such as they who think not but strike, knowing\nonly a friend and a foe and taking no account of numbers against them.\"\n\nThe jarl explained the matter and he seemed to be forgiven, but he and\nthe centurion returned to the palace surrounded by spears ready for the\ncasting.\n\n\"It is well, O jarl,\" said Caius. \"Let all be ready to depart upon the\nmorrow; but I may not go in unseemly haste as in fear.\"\n\n\"Thou wilt go as becometh thee,\" said Ulric. \"He who fleeth unduly from\na sword loseth the regard of brave men. We will be ready.\"\n\nNevertheless, Caius of Thessalonica rode swiftly to the house of Julius\nat Tiberias and was himself the bringer of this tidings.\n\nJulius listened to him in a white wrath. \"O thou, my friend!\" he\nshouted. \"Seest thou not that this thing is aimed at me as much as at\nthee? If thou hadst thus been slain, it had been my utter ruin. Woe to\nthese Herods! They shall both fall by the sword of C\u00e6sar. The gods be\nwith thy Saxons. Thou needest them. Commend me unto Pontius and say to\nhim that thou and I are henceforth one in all these matters. The Herods\nnow seek to stab him also. Let him guard well his head.\"\n\nSo talked they long together in a nearness which they had not known\nbefore, finding themselves in the same peril from the serpents which\nbite in the dark.\n\nFrom the gate of Tiberias on the morrow went out a company worth the\nseeing. Not without armed Roman escort and many bondservants might the\nchariots of so important a man as Caius of Thessalonica set forth.\nWhen to all these were added the vikings, in their best armor and well\nmounted, it was as if a small army had been ordered southward. To the\nplace of parting and of farewell came, also, Julius and many men of\nnote to do all honor to the friend of the procurator.\n\n\"O Caius,\" said Julius, \"I already have a swift messenger from Antipas.\nHe hath sent his horsemen to search the hills beyond the sea and\nTarich\u00e6a. They will ride with all diligence, and beyond doubt they will\nfind some to slay, but thy shield must be nearer to thee than is the\nJordan.\"\n\n\"It will be very near,\" said Caius, smiling, for near him rode Tostig\nthe Red watching all keenly, and his spear was in his hand.\n\nThis, too, saw Julius, and he laughed.\n\n\"O my friend,\" he said, \"it is even so. Fare thee well; but they who\ncome to meet thee should have due warning, for thy protectors are no\nrespecters of persons.\"\n\nAll then rode on, and the Saxons talked much among themselves\nconcerning the things which they had already seen in this land. They\nhad visited all towns and villages around the sea, but none of them\nwere more splendid than Tiberias.\n\n\"I would have visited Capernaum,\" said Ulric.\n\n\"There is no great thing there,\" said Tostig the Red. \"What hadst thou\nin thy mind?\"\n\n\"Only this,\" said the jarl: \"that this son of the old god of the Jews,\nthis rabbi of Nazareth, dwelleth there at times. I owe him thanks and\ngifts for my healing. Also I have it in mind to ask him questions\nconcerning my father, and Hilda, and Valhalla, and Asgard. Hilda I have\nnot seen but in my dream on _The Sword_.\"\n\n\"One was with her, I heard thee say when thou didst meet her. It was\nwell to give her thy ring. I would have done so. But what would this\ngod of the Jews know concerning thy maiden? The gods care not for such\nthings. She was fair to look upon. But, O Ulric the Jarl, I would I\nwere on the sea again!\"\n\nSo said all the vikings many times, but they told the jarl that not in\nany of their goings to Capernaum had they seen Jesus, the rabbi. They\nhad heard of him, that he was away in other places, here and there,\nteaching and preaching and healing many and casting out evil spirits.\n\n\"It is good that he so doeth,\" said Lars, the son of Beolf, \"and that\nhe healed the tiger scratches upon the jarl, but what good is it for\nhim to sing sagas to these people of no account?\"\n\nThere was none to answer him, for even Ulric himself was silent.\nNevertheless, the son of Brander had many thoughts which he did not\nutter and he forgot not any of the words which he had heard spoken by\nthis one who had healed his hurts.\n\n\"I understand them not,\" he said to himself. \"He bade us think of the\ngods, and that I do. Even now I am seeking their city and that I may\nget acquainted with my kindred. How shall I do so completely before\nI am slain? And he who dieth a cow's death, so say the sagas, shall\nnot enter Valhalla, but shall find his place in Hel. I would join the\nheroes of the old time and dwell with Thor and Odin. I think I shall\nknow more after I have seen the city Jerusalem, which Ben Ezra saith is\nlike Asgard. At all events I will sacrifice horses and oxen and sheep\nin the temple of Jehovah as if he were Odin himself, for he is the\nchief god of this wonderful land.\"\n\nMore and more wonderful indeed did it seem to the Saxons as they rode\nonward all that day, for it swarmed with inhabitants and the villages\nand towns were many in number.\n\nIt was at the gate of Jezreel that their company halted, at the setting\nof the sun, and Ulric sat upon his horse looking toward Carmel. Behind\nthe city arose Gilboa, wooded and craggy. Before it stretched Esdraelon.\n\n\"O Wulf the Skater,\" said the jarl, \"do you bear in mind the things\nwhich were said of this city and plain by Ben Ezra and Abbas?\"\n\n\"More was said to thee than to others,\" replied Wulf. \"It is a city of\nsieges and a plain of many battles. I can see the blue ridge of Carmel\nand beyond is the Middle Sea. I would I might see waves this hour and\nsmell the salt air. This is a woeful land, where never is good ice or\ndeep snow. We go on into the winter and we may yet see a snow squall if\nwe are fortunate. But Knud will need no bearskins and Wulf will need no\nskates--and I sicken when I think of such a winter.\"\n\n\"The great battle of the end of the world and the twilight of the\ngods!\" exclaimed Ulric. \"O ye! If Ben Ezra's Jewish sagas lie not, here\nshall we witness the greatest of all the feasts of swords. Here shall\nwe have for our jarl a god, the son of a god, and there will be gods\nand heroes to fight with. I, the son of Odin, will be here! Hael, Odin!\"\n\n\"I will be with thee, then,\" said Knud, \"but if it is soon to come, it\nwere better for some of us to go back to the Northland and return with\nmany keels full of men like ourselves. This god will need Saxons if\nhe is to fight Romans. These Jews will go down like wheat before the\nsickle, for I have been looking at them and at the legionaries.\"\n\n\"Thou art right!\" exclaimed Tostig the Red. \"But there is room on\nthis plain for great armies to meet. They will come from many places,\nAbbas told me, and among them will also be black and yellow men, and\nthere will be great beasts, and the eagles that are wide-winged, and\ncreatures whereof he could not tell me the shape. They may be like the\none we saw come up from under the ice to tear the whales, only that\nsuch as he do not come out upon the land.\"\n\n\"No man knoweth from whence these will come,\" said Knud, \"but some of\nthem are as great serpents with wings. I like not to think of them, for\nthey are full of fire and sulphur, and who can fight well in a smoke\nthat choketh him?\"\n\nAfter this they entered the city of Jezreel, and they wondered greatly\nat the strength of its walls and towers, but they saw not many soldiers.\n\n\"The land is at peace,\" thought Ulric, \"and garrisons may be small. I\nam learning something of war cunning from these Romans. What they take\nthey will hold until a stronger people shall come against them. I know\nof no such people except in the Northlands.\"\n\nYet another thought was in the mind of the jarl, and his eyes wandered\nanxiously wherever he went. In all towns and villages and whenever\ncompanies had been met by the way he had seemed to be searching, and a\nsadness of disappointment was growing upon his face.\n\n\"I heard her say she would see me at Jerusalem,\" he told himself, \"but\nnow the time is long. She may have come hitherward. Of these damsels\nwhom I have seen as I came many are fair to look upon, but none are\nas beautiful as Miriam. Cannot Hilda lead me to her? Shall I indeed\nnot see Miriam until I meet her in Asgard? I would that Caius were in\ngreater haste. We travel slowly.\"\n\nIf he had looked upon fair faces inquiringly with his sad blue eyes,\nalso had all the Saxons laughed to one another quietly to note how many\nwomen put aside their veils a little to turn for another look at the\nface of the jarl.\n\n\"Never before have these seen any like him,\" they said. \"They will not\nsee him again, and he careth not for women save for the one to whom he\ngave a token. He will forever keep his troth with the dark one, the\nbeautiful one, in whose hand he put the ring of the bright red stone as\nwe came through Esdraelon.\"\n\nGood welcome was given to Caius of Thessalonica and his company by the\ngovernor of Jezreel, but the vikings went to their quarters listlessly,\nfor they had all looked across the plain toward Carmel, and the thought\nwithin them was that beyond Carmel was the sea and that upon the sea\nwere ships.\n\n\n\n\n                             CHAPTER XXXI.\n\n                       THE PLACES OF SACRIFICE.\n\n\nQuestions which are asked by the heart of a man may go far. It is as if\nthey were winged and flew on to a chosen place of alighting, as do the\nmessenger doves carrying letters homeward. One of the birds set free\nby the ever-beating heart of Ulric the Jarl found a wonderful resting\nplace.\n\nIt was in a house in a great city, and upon all the earth was nothing\nmore magnificent than this house of houses. Upon the top of a high\nmount in the city was a vast space girded with white walls and towers,\nso that of this whole area was made a fortress of surpassing strength.\nWithin these walls were great buildings not a few and porticos and\nseparated courts for varied uses.\n\nThere was one building which was greater than any of the others,\nand to this as to a center all the many structures related; for\nthe arrangement and the architecture were everywhere exceedingly\nharmonious and convenient. To this greatest building there were several\napproaches, but the main entrance was by an ample ascent of broad stone\nsteps. Beyond the level at the head of this stairway were mighty doors\nwhose surfaces were covered with beaten gold and many designs of golden\nornamentation.\n\nWithin the doors, if one might enter--for here stood ever armed\nguards--they who went on might see yet more splendors of carven\nstonework, whereof some of the stones were rare and precious, and of\ngolden and brazen ornament. Here in high places were altars which\nsmoked with almost unceasing sacrifices. Serving at and about the\naltars were numbers of robed priests with their assistants, and often\nthese were chanting the sagas of their worship, but not in all this\nplace was there any image whereby a stranger might obtain information\nconcerning the shape or person of a god. It was as if he were worshiped\nin ignorance, none having at any time seen him to make a sculpture or a\npainting of his likeness.\n\nIn this inner space or court where were the altars there stood this day\na multitude of men with covered heads, and they now and then uttered\nloud voices in unison, which were responses answering the sagas of the\npriests.\n\nHere were no women, but at the right was a portal and a passage leading\ninto another court, which was also large and splendid. This was the\ncourt of the women, of whom a large number were present, both of the\nyoung and of the old.\n\nThis was the temple of Jehovah, the God of the Jews, in the city of\nJerusalem. To him only were any sacrifices offered upon the altars, and\nthe sagas were chanted that he might hear them if he would, but none\ncould tell whether or not at any time he might be listening. So many\nof the sagas formally besought him not to remain at a distance, but to\ncome to this place and listen and do the things asked for by those who\nbrought to his altars these sacrifices.\n\nSad and sorrowful, yet full of strange music, was the sound of this\nsinging, while the smoke went up from the burnings and while the\ncensers were swung to and fro by the priests to send out upon the air\ntheir clouds of sweet odors. Sad and sorrowful was the pleading, for\nthere cometh a heaviness of soul to him who calleth in vain upon a god\nwho is far away, who is unseen, and who answereth not by voice or sign.\n\nOn the stone pavement, near to a pillar of bright bronze-work and\nsomewhat apart from any of the groups of the other women, knelt one who\nwas veiled and whose voice arose in low murmurings as of a recitation\nand a prayer. The hand which drew her veil more closely was well shaped\nand white and upon one of its fingers was a golden ring among other\nrings less beautiful. So deep was the red light of the ruby in this\nring that its glow seemed hot like fire, and it throbbed as if it had\npulses at the movement of her hand changing the light upon it. Also her\nbosom arose and fell and there were tremors in her voice, and she said,\nwhispering softly in the old Hebrew tongue:\n\n\"O thou who art God over all gods, I have sinned to look upon him,\nfor I am a daughter of Abraham and he is one of the heathen. O that\nhe might also be one of Abraham's children and serve the living God,\neven our God. I have sinned, O Jehovah of Hosts, but I have made my sin\noffering and I have made an offering of atonement also for him.\"\n\nThen the gem flashed a great light, but her hand fell and her veil\nslipped away and the marvel of her face was seen for a moment. Upon it\nwas a smile and a light, and her eyes were closed, but her lips were\nparted.\n\n\"Have I indeed been spoken to?\" she whispered. \"I have been told that\nan angel cometh oft into the court of the women. Never have I seen an\nangel. Who knoweth that one might not come to me? Would he be fairer\nto look upon than was he whom I saw at the wayside? If this be truth,\nthen do I know that my offering hath been accepted and that it is no\nlonger a sin for me to remember him. Woe is me, then, if I am to never\nsee him more! O he was beautiful! Exceedingly! And I have brought into\nthe house of Jehovah this token which he gave me. But what is this\nwhich hath come to me?\"\n\nHer eyes were opened, looking downward, and the red glow of the ruby\nanswered them as if it were speaking to her of love. Then she arose,\ncovering with her long silken veil, and she walked out of the court of\nthe women; but a dove, escaped from the cages of the offerings, flew\nover her head and went out above the great gate and the wall, flying\nswiftly until he disappeared over the Mount of Olives.\n\nOn walked the young woman beyond the temple walls and the sacred mount,\ngoing until she came to a street of palaces, ascending another mount.\nHere shortly she disappeared, but she was more beautiful than any\npalace and in her light stepping there were both gracefulness and a\ngreat pride of manner, as if she were of high degree.\n\nNow at that hour of the evening sacrifice the city was exceedingly\nstill, for men and women everywhere paused in whatever they were\ndoing and turned their faces toward the temple. Horsemen drew rein\nand chariots halted, and there were many who knelt even in the open\nstreets. But of these were none but Jews and Jewish proselytes from\nother nations, and there were those who were worshipers of other gods\nthat were sufficient for them. Roman soldiers who were marching halted\nnot, and of these a body of a hundred spearmen passed out at the\nDamascus gate with an officer at their head.\n\n\"O captain of the gate,\" he shouted, \"yonder cometh a messenger. I will\nawait him.\"\n\n\"Hinder him not!\" replied the keeper of the gate. \"He is known to me.\nIt is the swift messenger of the procurator.\"\n\n\"Am I not captain of the temple?\" shouted the officer so loudly that he\nwho came heard him.\n\n\"If thou art he,\" was uttered, hastily, \"I pray thee come to me!\"\n\nFor the messenger halted, not dismounting.\n\n\"Dog of a Greek!\" exclaimed the captain of the temple, haughtily,\n\"shall I come to thee?\"\n\n\"There are men with thee and in the gate, O captain,\" said Lysias,\nreverently. \"I pray thee permit me to obey the procurator and speak to\nthee only.\"\n\n\"Ho! Thou art right. I come! Hast thou a letter from Pontius?\"\n\n\"This little script only,\" said Lysias, handing him a parchment, \"and\nthese words----\"\n\n\"Utter them quickly!\" said the officer.\n\n\"'Pontius to the captain of the temple: slay the messenger of Herod\nAntipas and let the spy from Mach\u00e6rus not live to sail for Rome. Speed\nthis Lysias to Cornelius, the centurion, and keep him afterward in my\nhouse safely until I come. Let him have speech with no man and let no\nharm come to him.'\"\n\n\"Even so!\" said the captain of the temple. \"Yonder road along the\nvalley of the Kidron bringeth thee to the Joppa gate. From thence is\nthe Joppa highway, and thou wilt find Cornelius at the harbor fort if\nhe hath not departed for C\u00e6sarea. I will give thee a fresh horse.\nTarry not in Joppa or in C\u00e6sarea, but return quickly to me.\"\n\n\"But not to speech with the high priest,\" said Lysias, \"nor to any from\nHerod.\"\n\n\"I will see to that,\" laughed the captain. \"Thou art careful of thy\nhead. Wert thou unmindful of the commands of Pontius, thy shoulders\nwere bare quickly. Thy fresh horse cometh. Mount and ride on.\"\n\nWithout more words Lysias obeyed, but as he rode on along the brook\nKidron he said aloud: \"Well for me that I took rest and food while I\ncould, that I fall not from my horse. I can reach Joppa in due season,\nbut what will yonder captain of the temple do with me when I return? I\nhave heard that the messengers of Roman governors are changed like the\nchanging of guards, and that they who are released go sometimes upon\nerrands from which they do not return. I will sacrifice to Mercury!\"\n\nWhether or not he were weary, Lysias rode well and his fresh horse was\nswift. It was but little to reach the Joppa gate, and the sun was but\nsetting when he turned into the highway leading toward the sea. It was\nbroad and well kept, for chariots and for marching cohorts. Looking\nback, Lysias saw that the gate was closed and none was in the road\nbehind him. Looking forward, he saw no man, but there were houses on\neither side of the way except at one wide, open space which arose at\nthe left in a small hill. Bare was this ascent and he wondered at it,\nsaying to himself:\n\n\"So near the gate and no building thereon? It were a place for one of\nthese outer palaces.\"\n\nHe had paused to fasten the buckle of his bridle and he looked again\nupon the hill, and now shriek after shriek of utter agony came to his\nears from beyond the crest of the ascent. Voice answered unto voice,\nand he shuddered as he heard, but a man in armor came slowly down the\n<DW72>.\n\n\"In the name of the procurator!\" shouted Lysias. \"Is this the Joppa\nroad?\"\n\n\"Art thou of his messengers?\" said the soldier. \"If thou art, thine\nears will tell thee that a score of his enemies are on the wood. This\nplace of skulls will soon smell but badly under this hot sun. Ride on,\nfor this is thy right road.\"\n\n\"This, then, is the hill of crucifixion?\" asked Lysias.\n\n\"Any place will do,\" said the soldier, \"but the procurator humoreth the\nJews and will set up no crosses in the city. The day may come when we\nwill nail them in their temple and set up there an image of Jupiter.\nThey troubled Pontius mightily when we did but carry our eagles to the\ntemple gate, as if one god were not as good as another. What care I for\ngods!\"\n\nLoudly rang again the piercing shrieks while he was speaking, and his\nhard face widened into a grim smile, as if the sounds pleased him. But\nLysias shuddered and his blood ran cold, and he wheeled away to gallop\nout of hearing of those terrible outcries.\n\n\"No Roman may be crucified,\" he exclaimed. \"These are not Romans. To\nthem all other men are less than brutes. I will watch that captain\nof the temple; but whither should I flee from the pursuit of a\nprocurator's executioner?\"\n\nUnder such fear as this dwelt all who were governed by the servants of\nC\u00e6sar, and yet it was said that the common people were more sure of\njustice than from any other rulers if they remained quiet and paid all\ntaxes without murmuring.\n\n\"I will risk all!\" shouted Lysias, \"if I may but once more look into\nthe blue eyes of my Sapphira, for I know she loveth me!\"\n\nThe sun went down as he rode, and the shadows came, and through the\nshadows he galloped on, but now and then it seemed to him as if the\nshrieks from Golgotha were ringing warningly in his ears.\n\n\n\n\n                            CHAPTER XXXII.\n\n                          THE MOB OF SAMARIA.\n\n\nThe city of Jezreel was for Caius of Thessalonica and his train but\na resting place for a night. After leaving behind its towers and the\nvalley of battles, at the side of which it seemed to be posted as a\nsentinel, Ulric the Jarl himself was satisfied with the speed of the\ngoing which brought him to Samaria.\n\nHere, also, as they drew near, the Saxons noted well the fortifications.\n\n\"These walls are old,\" said one. \"Those of Tiberias are newer and\nbetter. I care not for walls. Better is it to fight in the open field,\nwhere swordsmen may come together, shield to shield, in a fair combat.\"\n\nTostig the Red heard, and he shouted loudly:\n\n\"O jarl, not walls! Rather would I have a good keel like _The Sword_\nthan any fort. Towers and walls rest where they are builded, but a ship\nmay sail into new seas. I am hungry for the sea!\"\n\n\"I like not the land at all!\" said Knud the Bear. \"Never again may I be\nfound so far from the rush of waves. I am minded to seek me a keel ere\nlong. I think we shall all die if we may not again see the Northland.\"\n\nHe did but speak for all. While they had been inactive on the shore\nof the Sea of Galilee, and even more after setting out as if to find\nnew adventures, the vikings had returned in their hearts to their old\nmanner of living. They had thought continually of the sea and of ships.\nThey had talked together of the cruise of _The Sword_ and of all the\nstrange things which had befallen them by the way in which they came to\nthis country. They had also told many tales of the great deeds of sea\nkings, but there had been no minstrel or saga woman with them to sing\nthem a saga or to play for them upon a harp. Often, also, did their\nconversings deal with the Northland itself in its summer beauty. They\nlonged for the high mountains and the shadowy coolness of the fiords,\nand for the faces of men and of women and of children on the shores and\nabout the houses. There is ever a kind of sickness which cometh upon\nbrave men in the thinking of such thoughts and in the talking of such\nremembrances afar. So these vikings, who were all that remained of the\nmighty crew of _The Sword_, were not only weary at heart, but almost\nsick in body.\n\n\"A keel?\" said Wulf the Skater to Knud. \"Thou wilt find thee a keel?\nWert thou in thine own seas now thou wouldst find them closed against\nthee. Beautiful would be the ice to look upon. But I think I could make\nme good skates and reach the fiords over the floes.\"\n\nSo said other Saxons, and they did but look listlessly at the walls of\nSamaria.\n\n\"O jarl,\" said Caius from his chariot, \"come thou hither to me. I have\na word for thee.\"\n\nUlric rode to the side of the chariot.\n\n\"What aileth thy men,\" asked the centurion, \"that their faces are so\ncloudy? Are they discontented?\"\n\n\"Not with thee, O noble Caius,\" laughed Ulric, \"but they are ill at\nease on horseback and in peace. They would rather fight for thee than\ntravel like pleasure-seekers. One man is ever afraid that, if this\ncontinueth, he may die in his bed and go to Hel instead of to Valhalla.\"\n\nStern yet pleasant was the countenance of the centurion.\n\n\"I understand thy men,\" he said. \"Let them be posted in the doorway of\nthe house where I abide this night. I have no others here whom I may\ntrust, and this is a city of the enemies of the procurator.\"\n\n\"Thou mayest sleep safely,\" said Ulric. \"I will myself keep that house.\"\n\n\"Thy men could not be bribed,\" said Caius. \"I know that of them.\"\n\n\"They have too many coins already,\" said Ulric. \"But I bade them keep\nall and spend them at Jerusalem. No man need offer them any more. As to\ntreachery, let thine enemy speak of that to Tostig the Red, but first\nlet the seax of Tostig be taken from him.\"\n\n\"I will leave it at his belt,\" said Caius, \"and he may strike with it\nin such a case. But be not thou overhasty with a man of rank, for thou\nwilt be held accountable.\"\n\n\"I will be prudent,\" said Ulric; \"but how is it with thy legionaries?\nIf they are on post, is it not life and death with them?\"\n\n\"Men have died suddenly,\" said Caius, \"with a legionary motionless at\nthe outer door. He stirred not, being as a pillar of wood. Thy men\nwill be free, and will act as if they were hunters of game instead of\nstatues. Thy head is as good as thy hand.\"\n\n\"I will keep thee,\" said Ulric, \"and I would that the men might have a\nchance to draw a sword or throw a spear.\"\n\n\"They will not,\" said Caius. \"There are no men in Samaria who would\ntrifle with such a guard as thy Saxons. Think not but what I will\nremember thee for this matter.\"\n\nThe jarl reined away his horse, thinking deeply.\n\n\"O Caius, do I not know that thou art as other Romans? So soon as thou\nart done with us thou wouldst give us to the lions and look on while we\nwere torn, being amused. Soft words are well enough, however, and thou\nart better than are some of thy people.\"\n\nFor the jarl grew crafty under the burden of leadership, and he seemed\nolder than when he stood with Hilda on the shore of the North sea\nlooking at her runes upon the sand.\n\nA large house like a castle near the eastern wall of the city was\nassigned to so great a man as Caius, but he went the next day to a\nfeast, being entertained by the governor and other notables, among whom\nwere certain lords of Herod's household.\n\n\"It will be late when I return,\" he said to Ulric at his going. \"I will\nsend for thee.\"\n\n\"Not so,\" said the jarl. \"I will come without thy sending. There have\nbeen tumults in Samaria since the sun's rising. There will be good\nspears around thy chariot.\"\n\n\"Do as thou wilt, O jarl,\" said Caius. \"I fear no tumult and I have\ngood attendance.\"\n\n\"Hast thou indeed a guard, and is it not from this man, the governor?\"\nsaid Ulric. \"Leave thou such matters to me, I pray thee, that thou\nmayest at all reach Jerusalem.\"\n\nThe chariot of the centurion rolled away from the palace gate, and\nwith it rode a score of mounted soldiers sent by the governor as a\nguard of honor for his distinguished guest. Hardly were they out of\nsight, however, before the Saxons sprang to their feet at a sudden\nsummons.\n\n\"Spears and shields!\" commanded the jarl. \"Let every man look well to\nhis weapons and to his armor. Be ye all ready to march, but first let\nevery man come to me and report whatever things he hath heard or seen\nthis day.\"\n\nOne had this thing to speak of and another that thing, but for the\ngreater part it all seemed to be of little worth. Their eyes, too,\nhad been better than their ears in a city of an unknown tongue.\nNevertheless, the jarl said to Wulf the Skater:\n\n\"Thou hast scented this danger, then, thou keen old hunter? So is it\nwith me, only that I better understand sayings uttered in my hearing,\nand some who spoke believed that I was as a stone wall, having no ears.\nThey were, therefore, careless. I will say to thee that the soldiers\nwho are now with Caius are all from this new legion wherein Julius was\nfor a while the chief officer. It is for our interest that Caius may\nsuffer no harm. Moreover, we may have some good fighting, and that is\nworth while.\"\n\n\"Thank the gods!\" interrupted Knud the Bear. \"Now may I the more\ncomfortably eat my supper. It is well to have a thoughtful jarl.\"\n\nA city by itself was Samaria, as it had been during long centuries.\nThey who called themselves Samaritans bore deadly hatred to all Jews,\nbut could not prevent them from entering the city and transacting\nbusiness there, although they could have no dealings with the\nSamaritans. All other nationalities came and went freely, and here was\na gathering of the offscourings of the earth. The Jews risk all perils\nfor the sake of traffic, and they had in this matter the protection\nof the Roman laws. Nevertheless, these hatreds were the root of many\ntroubles, and from time to time there had been bloody riots to be\nsuppressed by the legionaries with but small care upon whom their\nswords might fall. It might have been trusted that a Roman of rank like\nCaius would be as safe in Samaria as in Jerusalem or in Rome, and so he\nwould have been but for the intrigues of those who were greater than\nhe. Herod Archelaus, to whom Judea and Samaria had fallen by the will\nof his father, Herod the Great, had forfeited his realm to the Romans\nand it was now ruled by Pontius the Spearman. Both the Herod of the\nBlack Castle, whose legacy had been Galilee and some provinces beyond\nthe Dead Sea, and Herod Antipas, who had inherited large districts at\nthe north and east of Galilee, were plotting to overthrow Pontius and\nalso to defeat each other. The favor of C\u00e6sar was the path to increase\nof power not only for them, but for Roman plotters such as Julius, and\nthere were intrigues against them all at Rome itself. The strifes of\nthose who fought continually for the spoils of Roman conquests were\never records of bloodshed, and no man's life was safe. To be a great\nRoman was to walk on toward destruction.\n\nSplendid was the feast to which Caius went at the palace of the\ngovernor of Samaria, but he was wary and he did not become drunken.\nLong reclined the guests on the couches at the tables, to be served\nwith all the delicacies of the earth. Also there were dancers and mimes\nand musicians. But the end came. Some were to abide in the palace,\nsome were to go to their houses near, in the city. The chariot of Caius\nwaited for him, but as he and his slaves walked out at the main portal\nthey heard a sound of trumpets and great outcries of a multitude.\n\n\"It is nothing,\" said Caius. \"I heard that the rabble had risen against\nthe Jews. Let the legionaries form in the road. Drive on!\"\n\nHe spoke scornfully, but the outcries were near, and now came a great\nrush of men, of whom many were armed. In front of the governor's palace\nwas an open space, into which the multitude was pouring, but from the\nopposite direction came forward another throng of men. In the foreranks\nof these was a small man in armor, with the visor of his helmet closed.\n\n\"Yonder is the chariot of Caius,\" he said. \"Wait only till the Iberians\ncharge. Then slay him and flee. Let the blame fall on the Jews and the\nSamaritans.\"\n\nTwo score were the legionaries, and it was the governor, standing upon\nthe steps of the palace portal, who shouted to them:\n\n\"Charge ye the mob lest they hinder the going of my guest. Slay them! O\nmost noble Caius, I send out also quickly my own guards and servants.\nThou art safe!\"\n\nIf this were indeed the craft of the governor, it was well hidden,\nfor the soldiers went forward smiting all in their way, and armed men\nfrom the palace went also. By this very charge, however, the chariot\nwould have been left alone save for Caius and his charioteers and a few\nmounted bondsmen. Not in the silken robes of a man at a feast was the\ncenturion at this moment, nevertheless. The robes were to be seen in\nthe light of the torches, but they covered good mail and armor, and\nsuddenly upon his head was a helmet and in his hand a pilum.\n\n\"Treachery!\" he shouted. \"The jarl was correct! O for my Saxons!\"\n\n\"Here, O Caius!\" loudly responded a voice from among the shadows of the\npalace front. \"Halt not thy chariot, but drive slowly. We have abundant\njavelins.\"\n\nThe torches held by the bondsmen flared in swinging, more being\nlighted, and past them seemed to go dull red flashes, but these were\nthe bright blades of Syrian darts obtained by Ulric for this business.\nStrong were the arms hurling, and the darts were better than arrows at\nso short a distance.\n\n\"Jupiter Tonans!\" roared Caius. \"I have a sheaf of them here in the\nchariot, for myself and my charioteers. Wise is the Saxon, and he\nprovided them for me!\"\n\nA good thrower was he, and some who had stealthily crept on too nearly\nwere smitten as they sprang forward. Then came the charge which had\nbeen purposed across the open space, but between its front and the\nchariot was a wall of Saxons, in full armor, shouting with the fierce\njoy of battle.\n\nDown went the small leader, cloven to the jaws as if his helmet were of\nwood. Down went his companions rapidly, while the battle laughter of\nthe vikings rang derisively in their ears.\n\nThe other multitude the legionaries were slaughtering pitilessly, but\nthe command of the governor had been to follow, and the soldiers came\nnot back at once.\n\n\"Slay! Slay!\" shouted Caius. \"I come!\"\n\n\"Come not!\" replied Ulric. \"Abide where thou art and press on to thy\nhouse. We will keep these wolves at bay.\"\n\n\"A fight and I not in it?\" said Caius, angrily. \"Commandest thou me?\"\n\n\"In the fight I am jarl!\" said Ulric. \"I am answerable for thy head.\nDrive on!\"\n\n\"Thou art right!\" said Caius, justly. \"On, O charioteer! Obey thou the\nSaxon. I forgot that he is a prince and a captain among his own people.\nI will make him a Roman yet. He should not be a barbarian.\"\n\nHardly might any less than a king, nor even a king except at great cost\nand for policy, obtain Roman citizenship, but this was the meaning of\nthe words of Caius.\n\nThen an arrow flew and struck him upon the left arm, wounding him; but\nhe mentioned it not, for he saw that the charge was broken and that the\nSaxons came to march with the chariot.\n\n\"Not one of them is missing,\" he thought. \"So much for broad shields\nand good mail. The rioters had weapons, but no armor, and they were\nslain as cattle. This arm of mine is but scratched.\"\n\n\"On!\" commanded the jarl, to his men. \"I heard the centurion say he is\nwounded. O Caius, how art thou?\"\n\n\"A sting on my arm,\" replied Caius. \"We shall soon be at the house.\nThis is naught.\"\n\n\"Let me see it speedily,\" said Ulric. \"I have picked up an arrow with a\ngrooved head. Thou knowest what that meaneth.\"\n\n\"Haste! Haste!\" shouted Caius. \"This thing is of Herod, the jackal! I\nam lost.\"\n\nBut the tumult had been stricken to quiet and the ground was strewn\nwith the dead. Now as they went there came swiftly armed horsemen\nof the governor and behind these marched the Iberian legionaries. No\nvisible fault might be charged by Caius upon his host of the feast. Not\nfar was it to his place of abiding, but when the chariot halted there\nhe sprang down and entered in a gloomy silence, followed by the jarl.\n\n\"Home, now,\" commanded the officer of the legionaries. \"Our duty is\ndone.\"\n\nBack with them went all servants of the governor, but Caius was in an\ninner room removing his armor.\n\n\"I wore no armlets,\" he said, \"lest the governor might see them. The\narrow went past my shield while I threw a spear. Thou hast done well, O\nSaxon chief. But for thee I had been murdered. This is a small wound.\"\n\n\"I will suck it for thee before I bind it,\" said Ulric. \"Then watch\nthou if it beginneth to burn, but set thou out hence before dawn.\"\n\n\"That will I this hour,\" said Caius, and orders went forth.\n\nGreat was the declared wrath of the governor of Samaria, for he came\nhimself to inquire concerning the welfare of his guest. Not to him\nwas anything said of a groove in an arrow wherein might be pressed\nsome deadly juice, and he returned to his palace a seeming friend of\nCaius, complaining bitterly of the Jews and Samaritans, more of whom he\nthreatened to slaughter for this night's business.\n\nUlric cared for his men. They had cuts and bruises which they made\nlight of, but among them was no arrow wound. So light a missile would\nhave been stopped by a leathern hauberk, and all their mail was of the\nhighest temper of steel.\n\n\"We will ride soon,\" he told them. \"Be ready to mount and leave this\nplace of thieves.\"\n\n\"I like it well!\" exclaimed Knud the Bear. \"It was not a hard fight, as\nif these fellows had been Danes or Northmen, but I cleft many skulls\nand I think Wulf the Skater killed a score of them. Tostig was unlucky,\nand Ven, the son of Gerta, slew more Samaritans that he did.\"\n\n\"He did not,\" said Wulf. \"Thy counting is not good. And I slew two men\nin armor also.\"\n\n\n\n\n                            CHAPTER XXXIII.\n\n                  THE HOUSE OF PONTIUS THE SPEARMAN.\n\n\nThe road from Samaria to Jerusalem hath many windings and there are\nhills to weary the wayfarer. Climbing one of these slowly was the\nchariot of Caius of Thessalonica. He was lying heavily upon the back\nseat, as one to whom this journey had become an insupportable burden.\n\n\"This long day draweth to its close, O jarl,\" he said to the horseman\nnearest him on the right. \"The roads are worse to pass than were those\nof yesterday. We are now on the level near the ridge of the Mount of\nOlives. Soon we may see the city. My arm burneth and it is swelling.\"\n\n\"I would we were already with thy learned physician,\" replied Ulric.\n\"Be of better cheer. I know little of such matters, but I think thou\ndoest well. I will offer sacrifices for thee in this temple of the\nJews. Hast thou ever done aught against their god? He is revengeful.\"\n\n\"I have not harmed him,\" said Caius. \"I have not slain Jews. Do as thou\nwilt, for at this time there is no other god in Jerusalem. I will pay\nfor thy oxen and Pontius will command the priests to offer them upon\nhis altar. Thinkest thou, O Saxon, that any god hath power to heal the\nwound made by a poisoned arrow?\"\n\n\"That I know not,\" said Ulric. \"I have often wondered much what the\ngods may do. One of them healed me of my hurts from the tiger of\nJulius. Such a god might cast out a poison. He casteth out demons and\nhe healed a leper. He opened the eyes of a blind man. I would that he\nwere now in Jerusalem and that thou mightest look into his face. Also\nI must offer sacrifices of thanks for that matter. It is not right to\nobtain a gift from any god and then not to keep faith with him. A god\nshould be dealt with as if he were a brave warrior.\"\n\n\"Well for thee!\" exclaimed Caius. \"I would indeed that he were here\ninstead of in Galilee. No god may heal aught so far away, and as for\nthis god of the Jews, they will not that a Roman enter his temple.\"\n\n\"Ben Ezra told me of the temple,\" said Ulric, \"that a court is prepared\ninto which all may come. There only will I enter. It is not well to\nanger priests in their temple, for they know the ways of their god and\nwe know them not.\"\n\n\"Thou art young, but thou art cunning,\" said Caius. \"But I have a great\nfear concerning this wound in my arm. It is not like any other, and I\nhave been wounded often. A strange thing is poison. I have considered\nwhy the gods make such a thing and why they put it into the teeth of\nserpents. They are evil!\"\n\n\"A god may need a serpent as thou needest a spear,\" said Ulric. \"It\nis plain to me. If I were a god, I would make what I required for\nmy errands. So do they work with winds and seas and rocks, and with\nthunders and with plagues of many kinds. No man getteth away from\nthem if they have aught against him. Anger not the gods, for they are\npowerful and they are cunning.\"\n\n\"As thou hast said,\" replied Caius, gloomily, \"I have spoken against\nthem at times, and now they have reached me with this Syrian arrow from\nthe quiver of Herod the jackal.\"\n\n\"Odin!\" suddenly exclaimed the jarl; for the overwearied horse under\nhim stood still without a pull of the rein, and before the eyes of the\nSaxons was the City of the Great King, the Holy City, Jerusalem the\nBeautiful.\n\nDeep is the valley of Jehoshaphat, through which runneth the brook\nKidron under its many bridges and between its gardens and palaces.\nBeyond this valley, as the whole company stood still to admire, they\nsaw the mighty walls of the city, high and white, and the castles and\nthe towers, but beyond and above all these, in the bright light of the\ndeclining sun, they saw the glories of the temple which was accounted\none of the seven wonders of the world.\n\n\"It is Asgard!\" said Ulric, thoughtfully, \"and I see the temple of a\ngod that hath power on earth to heal wounds and to give sight, and to\nwhom demons give obedience. I think he is not as are the gods of the\nNorth, and I will ask this son of his more about him.\"\n\nBut the Saxons who were halted with him said one to another:\n\n\"We have come out into the world far enough. We will see this one city\nand we will do somewhat of fighting perhaps. But then we will find a\nkeel, or take one, and we will return to the Northland, whether the\njarl goeth with us or not. The winter of this land is warm, not cold,\nand we may not abide it. We will go into our own fiords as the ice\ncometh out, seeing we may not get there sooner.\"\n\nSo strong is homesickness, and so it will change the hearts and the\nwills of brave men.\n\nAt that hour a youth sat in a vaulted chamber of a great building upon\none of the hills of Jerusalem. Around him the furniture was good,\nbut somewhat plain, and there were weapons and armor of many kinds\nscattered here and there. In a corner was a couch, and there were\nchairs and tables, and on the tables unlighted lamps.\n\n\"I do know,\" he said, \"that Pontius the Spearman is in the city. Why\ndoth he not send for me? I am not in a prison, yet I am not permitted\nto go out into the city since I returned from C\u00e6sarea. The procurator\ncannot think that I know aught more than my messages, nor fear lest\nI should betray him. Why, then, am I shut up in this chamber of the\ncastle?\"\n\nLittle remembered the haughty procurator of so small a matter as a\nyoung Greek messenger for whom he had no present need. Somewhere among\nthe household this Lysias was sure to be awaiting a summons, and there\nwere weighty matters on hand. One was before him pressingly in the\nhall of audience, for he himself stood there angrily reading a written\nscroll which had been brought to him.\n\n\"The high priest and the eagles once more!\" he exclaimed. \"This god\nof the Jews! What is he to me? I anger him not. Little he careth for\nthe standards of the cohorts. Go thou! Tell Caiaphas it shall be as\nhe willeth, and I will send him oxen for his sacrifices. The tribute\ngatherers have brought me even too many horned cattle, and his god may\nhave them.\"\n\nA dignified man, long-robed, gray-bearded, solemn-faced, who stood\nbefore him, bowed low, responding:\n\n\"I hear thee, most noble Pontius. I will bear to the high priest thy\nanswer. It shall be to us as a promise from C\u00e6sar. May the blessing of\nJehovah of Hosts be upon him and thee.\"\n\n\"Go!\" said Pontius, petulantly. \"If he cannot do better for the Romans\nthan he hath done for the Jews, my oxen are but wasted.\"\n\nLowly bowed the Jewish noble, but there was pride in his obeisance, and\nas he went out at the gate he muttered:\n\n\"The gift of Jehovah to these heathen would be the coming of Messiah\nthe Prince and the slaughter of their legions in the valley that is\nbefore Jezreel until the blood should be as a river to wade horses in.\"\n\n\"What thinkest thou, Cornelius,\" said the procurator to a soldier\nof noble presence who stood near him; \"must we yield to these dogs\nforever, with their continual turmoil?\"\n\n\"They have their god,\" said Cornelius. \"I have read much about him. He\nis gone from them for a while, but he hath promised to come back again.\nI think we should make him one of the gods of Rome and set up his image\nin the Pantheon with that of Jupiter.\"\n\n\"That were good policy,\" said Pontius, \"and it would leave these\npriests of his nothing more to complain of. They are a pestilent nest\nof fault-finders and some of them get to the ear of C\u00e6sar, doing us\nmischief; for they are crafty serpents.\"\n\n\"I fear God,\" said Cornelius. \"We are but men and we see but little,\nwhile the eyes of God are everywhere.\"\n\n\"Go thou to Joppa, then,\" said Pontius, \"and let no man pass out of the\nfort without thy knowledge. Thou keepest the gate. Keep it well.\"\n\nSoldierly, friendly, was the parting word of Cornelius to his\ncommander, but he was a free Roman and there was no servility in his\ncourtesy, nor was there any fear.\n\n\"Him, also, I may trust,\" said Pontius, \"but O for the coming of Caius\nof Thessalonica! I will see, also, Lysias, the Greek, and I would that\nBen Ezra were returned from his cave in Carmel with his treasures. I\nwill let him keep a part of them because I have further use for him\nbefore he dieth.\"\n\nIn the strong inner chamber of the procurator's castle Lysias walked\nslowly up and down chafing at his imprisonment, but his eyes glanced\nhither and thither and they were watchful.\n\n\"What!\" he suddenly exclaimed, low-voiced. \"Is the corridor door ajar?\nWould it be my death warrant to look out into the corridor? I am under\nno command not to look, but I may well be prudent where there are so\nmany sharp swords.\"\n\nThe door was but slightly opened, as if he who last passed through had\nshut it carelessly; but there are traps in prison houses, and Lysias\nhesitated, going to listen at the narrow crevice, but not laying a hand\nupon wall or door.\n\n\"No sound,\" he thought. \"I may open and close again. Who knoweth what\nmay be here? I offend no order of any officer.\"\n\nNevertheless, he trembled as he obeyed the strong impulse that was in\nhim. A step forward and he was in the corridor. It was lofty, its floor\nwas of pictured tiling, and it was lighted by windows at each end. Into\nit came another vaulted passage three fathoms away, and he went swiftly\nto that opening.\n\n\"Vast is this palace,\" he was thinking, but at the next beating of his\nheart he went forward with a great bound, for the music of a woman's\nvoice in a gay song fell upon his ear.\n\n\"She is here!\" he exclaimed. \"Now I care not if I die, so I but see\nher.\"\n\nWide open was a door into this second passage and through it poured the\nsong, accompanied by the touching of a small harp. It was a love song,\nand he heard:\n\n    \"Now cometh he, my love,\n      From the land beyond the sea,\n    And the fair wind blowing knoweth,\n      That it bringeth him to me.\"\n\n\"Sapphira! O my beloved! I am here!\"\n\nShe sprang to her feet and the lyre fell from her hand. O she was\nbeautiful, in her sudden astonishment and fear, but he who came toward\nher with open arms seemed even more beautiful than she, for his face\nwas radiant and his eyes were a flame of fire.\n\n\"Sapphira?\"\n\n\"O rash one! Thou art lost! What am I to thee any more? Am I not the\nslave of the procurator of Judea? Thou art not my Lysias; thou art but\na rider of horses.\"\n\nIn her face was a great struggle of pain, nevertheless, and in his was\na whiteness, for he fell upon the floor and lay there moaning.\n\n\"Foolish boy!\" she said, stooping over him. \"I love thee, but I am not\nnow thine, nor can I be. The past is dead, and the gods have bidden us\neternal separation. Destroy me not and destroy not thyself. Go lest the\nsword find thee here! The scourge is close to thee, and sudden death\nboth for thee and me.\"\n\n\"I care not for the scourge or the sword,\" said Lysias, slowly rising\nand gazing at her. \"I care only for thee, O false one! Hast thou\nutterly changed away from me?\"\n\n\"What I was that I am not,\" she said. \"What thou art thou knowest. Art\nthou mad, also, to cast thyself against the power of Pontius? Leave me\nlest I call for help! I will not die on thy account. I love life, and\nlife is full of love for such as I am. What need have I of thee, O lost\nlover?\"\n\nAnger was in her eyes now, and greater fear, for that which she said\nwas true.\n\n\"Kiss me!\" he said, faintly, \"and I will go. The gods have abandoned\nme!\"\n\nThen stepped she forward and kissed him on the lips and a spasm shook\nhim from head to foot, shaking her also.\n\n\"Let thy love die within thee,\" she said, \"and trouble me no more, for\nI live happily in this palace, where all are my friends. Make me not\nthine enemy, for in this thou art a robber.\"\n\n\"That am I,\" he murmured. \"I will go. I came far and risked all to see\nthee. I knew nothing concerning women. Now that I know thee, what thou\nart, I have no need of thee. Love will die, for all else is dead. Sing\nthou thy song, but be sure that all thy roses will wither on thy bosom.\"\n\n\"Cursest thou me?\" she exclaimed. \"Beware what thou sayest! I have\npower!\"\n\n\"As a caged leopard hath power, so hast thou,\" said Lysias. \"I leave\nthee. Be thou a slave, for that is all that is in thee\"--and he was\ngone.\n\nShe stood and looked at the doorway by which he had departed, and her\nlips were without color and her hand was on her bosom.\n\n\"What is this?\" she asked. \"Did I love him better than I knew? Was I\ntoo much in fear that I sent him from me? One cometh who would slay\nhim. It is best that he should go lest he should die. Women must be\nprudent, but this pain is great. I did love him. O that he had not come\nagain, for before he came I was happy. O ye gods, what shall I do? O\nbeautiful Aphrodite, help me, for thou knowest love!\"\n\nIn the corridor lingered Lysias listening, and then he walked on,\nstaggering as he went.\n\n\"O woman! woman!\" he whispered. \"What is woman and what is man? She\nis changed and I change not. I cannot hate her, as I thought I could,\nnow she hath spoken. I will wait cunningly, for I am sure that in this\npalace is one who calleth for my knife or for a spear thrust. I will\nfind him.\"\n\nIn a moment more he was in his own place, still leaving its door ajar,\nas at the first, but he began to search among the weapons and the armor\nin the room, finding a small, sharp blade with an ivory handle, and\nhiding it in his bosom.\n\n\"It will do,\" he said, \"but I would I might wear mail.\"\n\nAt that he was stooping over some fine steelwork and he heard a step\nbehind him. It was a crafty thought which bade him continue his speech.\n\n\"The procurator knoweth me only as a postboy,\" he said. \"I might serve\nhim better in mail. He hath not many who would be true to him as I\nwould. There are those who are false, but I could bring him a good\nsword in a hand he might surely trust.\"\n\n\"O Greek!\" said a deep, stern voice. \"What is this that thou sayest?\nPut on the mail!\"\n\n\"O most noble Pontius!\" exclaimed Lysias, turning, but lifting the\narmor. \"Thou didst not send for me, therefore I came not.\"\n\n\"Speak not,\" said Pontius, \"save to tell me all that thou hast seen and\nheard here and at Joppa and at C\u00e6sarea. I have a work for thee.\"\n\nLysias told all save his meeting with Sapphira, and the procurator\nlistened.\n\n\"Thou hast ears and thou hast eyes,\" he said at last. \"I set thee free\nof all other service but this that I now tell thee. Thou wilt have\nanother abiding place than this, but thou wilt come and go freely among\nmy servants, being known to them as my messenger whom none may hinder.\nNow hath one come from the Damascus gate saying that my friend Caius\nof Thessalonica draweth near, and with him his Saxon gladiators. He\nis wounded, and my physician meeteth him. Go thou. Hear all. See all.\nReport to me of his swordsmen.\n\n\"Now hearken! Among the female slaves of my wife is one in whom is a\nperil, for she is fair. For women I care not, but there are men who\nare fools before bright eyes. In the banquet room and in the balconies\nget thou speech with this Sapphira. She will be spoken to by my wife\nthat she may hide nothing from thee lest she die in the arena. Judea\nand Samaria are worth more to me than is the blood of one fair serpent.\nCome!\"\n\nLysias now stood before the procurator in mail and helmet, girded with\na light sword and bearing a silver-gilded buckler. It was the arms and\narmor of the Syrian mercenaries of Pontius, but as of an officer among\nthem, ordered to duty at the palace.\n\n\"Thou wilt go on foot to the Damascus gate,\" said Pontius. \"The\nphysician waiteth Caius at his house. Deliver this scroll to Caius\nand remain with his company until thou canst bring me exact tidings\nconcerning his wound.\"\n\n\"O most noble Pontius,\" said Lysias, \"I pray thee permission to say\nthis word.\"\n\n\"Say on!\" said the Spearman.\n\n\"Only in being true to thee have I any hope of life, for thy enemies\nare my enemies. I also will at times to attend at the school of\nGamaliel, as I told thee.\"\n\n\"That is thy value to me,\" said Pontius. \"Wert thou any man's\nbondservant, or wert thou other than a youth, a scholar of Gamaliel,\nI would have no use for thee. All they of his manner of teaching are\nhandicraftsmen, even if they are rich. What is thy work?\"\n\n\"I am a shaper of arrows,\" said Lysias, \"and I know the making of a\nbow. Thou mayest yet require to have a sharp arrow sent surely to a\nmark of thy choosing.\"\n\n\"Say thou no more!\" commanded the procurator. \"Thou art wise to\npreserve thy head. Only a fool throweth away his life. Go!\"\n\nFor they had walked out along the passage and before them was a gate\nof the palace. It was not the great gate, but even here were armed\nlegionaries, and their officer and others with him took note of Lysias\nand of the manner of his sending.\n\n\"He is the trusted messenger of the procurator,\" said one. \"I heard of\nhim from the captain of the temple. When he hath borne many messages we\nshall cease to see him.\"\n\nLysias passed on down the steep street in his brilliant armor as one\nhaving a shadow of authority, but his heart was bitter within him.\n\n\"I am to see her again,\" he murmured. \"I would she were dead and I dead\nwith her. I will but live to strike this unknown one, even if I stab\nhim with a blade of Pontius. But I must be cunning with these Saxons.\nDo I not know what manner of pirates they are? Not among any other\ncrew, I think, shall I find men so tall and so strong as are my old\ncomrades from _The Sword_. Their jarl would be a prince of gladiators,\nbut I am not glad that he and his come now to Jerusalem.\"\n\nAway behind him in the palace, in the room where he had met her, sat\nSapphira.\n\n\"What is this?\" she exclaimed. \"Did I not see him walking with the\nprocurator as one walketh with a near friend? Is he, then, more than a\nhorse boy? Is he an officer of the palace, and greater than I? Now am I\nindeed in pain, for I have need of friends. O love! Why was I cruel to\nthee? Come again, O my beloved! My Lysias! I will tell thee that I am\nnot changed! Will he return if I call him? He will, for I am beautiful.\nI am favored by Aphrodite. She will make him bend to me as I will. It\nwas but for a moment, and I was in fear. None must see me this day. I\nwill go at once as if I were summoned by the wife of the procurator.\nWoe to any who shall hinder me.\"\n\nShe caught up and threw over her head a veil and over her body a\nflowing robe of silk embroidered with needlework. Then, as if fear\nhastened her, she flitted away along the main corridor and disappeared.\n\n\n\n\n\n                            CHAPTER XXXIV.\n\n                        THE SCHOOL OF GAMALIEL.\n\n\nWith all honor did the captain of the Damascus gate of Jerusalem\nreceive Caius of Thessalonica, the friend of Pontius the Spearman. The\nchariot halted before the gate and in it sat the stern Roman centurion,\ngiving no external token of a wound or of suffering.\n\n\"O noble Caius,\" said the captain, after his first greeting, \"I have\nthis, also, for thee from the procurator, that his physician, who is\nalso thine, hath gone before thee to thy house. May the gods give him\nboth skill and success.\"\n\n\"I thank the procurator and thee, also,\" said Caius. \"I will now drive\non.\"\n\n\"A moment, O most noble Caius,\" interrupted the officer of the guard.\n\"A messenger even now. He is from the procurator.\"\n\nThere was no stir among the mounted swordsmen who rode before and\nbehind the chariot, but they sent quick glances to each other as their\neyes fell upon this messenger.\n\n\"Silence, O jarl,\" he had said in Greek to Ulric as he drew near him.\n\"I shall go with thee speedily. I thank the gods that I now see thee\nagain. I can do many good things for thee and thine. I pray thee bid\nthem, also, to be as if we were strangers.\"\n\n\"They need no bidding,\" said Ulric. \"Hael to thee.\"\n\nNo further word did either of them speak, but Lysias waited at the side\nof the chariot while Caius read the parchment epistle. It was but\nbrief, and when it was ended the centurion said to Lysias:\n\n\"Go thou and come again. I will answer for thee to Pontius. Say that I\nbid him be with me within the hour lest evil come. Haste! On thy head!\nO charioteer, drive to my house! On, O jarl!\"\n\n\"Behold,\" thought Lysias, \"I am in a sore strait. Pontius will scourge\nme! But I will run.\"\n\nA swift runner was he, even with the mail upon him, and at the gate of\nthe procurator's palace he halted to draw breath.\n\n\"In! In!\" exclaimed the officer of the portal. \"I will announce thee.\nThe procurator giveth a feast, but I may go to him. This must be some\nstrange errand!\"\n\n\"The gods be with thee!\" said Lysias. \"Tell him!\"\n\nIt was but a few terrible moments, full of fear for the young Greek,\nand he stood in an anteroom before the stern Spearman.\n\n\"What did I bid thee?\" he demanded.\n\n\"Slay thou me if thou wilt, most noble procurator,\" bravely responded\nLysias, \"but Caius of Thessalonica sendeth thee greeting and these\nwords: 'Be thou with me within the hour lest evil come.' I beg thee, O\nPontius, let me say this much more: for I heard him whisper, 'Lest he\ngive his power into the hand of him of the Black Castle and his neck to\nthe headsman of C\u00e6sar.' I have not at all disobeyed thee, O Pontius. He\nbade me return to his house for another commandment.\"\n\n\"Be thou there on his arrival and I will count it thy strict\nobedience,\" said Pontius. \"Thou art not a legionary, nor under the law\nof the legion. I think thou servest me well.\"\n\nAway ran Lysias murmuring: \"So narrow is the measure between Roman\nfavor and Roman vengeance! He may die ere I risk his wrath again.\"\n\nNevertheless, it is not easy for one of the great to depart from a\nfeast whereat governors and senators and princes are reclining, and\nPontius went in to pay the duty of host to his many guests, so that\nLysias was in no peril concerning his errand.\n\nThe chariot had reached its halting place and Caius had walked into his\nhouse, upheld somewhat by his pride, but more by the arm of Ulric, the\nson of Brander.\n\nAlready the physician had examined the wound made by the Syrian grooved\narrow.\n\n\"O Saxon,\" he asked, \"thou didst suck this poison well and quickly?\"\n\nUlric did but nod his head.\n\n\"Then know thou, my lord Caius,\" said the man of skill, \"that but for\nthy swordsman thou wert already dead. I will do what I can for thee,\nbut it will be long before thou wilt bear thine armor. This wound must\nbe neither bandaged nor closed, but washed only and kept open. Saxon,\ngive me thy sharpest blade.\"\n\n\"It is my seax,\" said Ulric, drawing it. \"What am I to do?\"\n\n\"Cut into this hard swelling,\" said the physician. \"Cut the depth of\ntwo finger breadths and withdraw thy blade.\"\n\n\"Cut!\" said Caius. \"Am I afraid of an edge?\"\n\n\"So bidden, I will cut,\" said Ulric, and the sword point went into the\nswollen arm.\n\n\"I thought so,\" said the physician. \"With that green corruption\nspurteth out much evil. Widen the cut. Caius is saved. I will put into\nthe gash an ointment that I will bring. It is well for thee, O Caius,\nthat thy strong swordsman is thy trusty friend. I go.\"\n\nBehind them, by express authority, now stood Lysias, listening, and he\nsaid:\n\n\"Most noble Caius, this is my command from the procurator. I must go to\nhim.\"\n\n\"Tell thou him the saying of the physician,\" said Caius. \"Tell him,\nalso, that I change not my greeting. He must come.\"\n\nAgain went Lysias, and again he stood before the procurator telling all\nthat he had heard and seen.\n\n\"Pause thou here a moment,\" said Pontius. \"I would have speech with my\nwife.\"\n\nStill as a statue stood the young Greek, and none who came or went\ndared ask him whence he came, but suddenly an arm was around his neck\nand a kiss was upon his cheek.\n\n\"I am here, beloved, but I may not linger. I will see thee often. I am\nstill thy Sapphira.\"\n\nHe stirred not, spoke not, nor did he turn to see, but there was a\ngrating of teeth.\n\n\"O Lysias! O love!\"\n\n\"Speak not of that which is dead,\" he said to her. \"Go thou thy way.\nThis is no place for the foolishness of unfaithful women. I will indeed\nmeet thee again, but thou art a slave and I am a free warrior. Go!\"\n\nWhite was now the face of Sapphira and her lips were quivering, but she\nwhispered:\n\n\"Scorn me not! I was frightened, and so I was cruel. I do love thee;\nand thou wilt need me in this place, which is as a spider's web. I go.\nFollow me not!\"\n\n\"Follow thee?\" laughed Lysias, scornfully. \"I did follow thee from\nfar, but now I am as a weapon in the hand of the procurator. I shall\nserve not thee, but him.\"\n\n\"Ha!\" muttered one who heard. \"This is, then, the trusted one. Him we\nmust slay.\"\n\nWell for that speaker if his lips had been closed, for in the shadow\nbehind him stood Pontius the Spearman.\n\n\"They who will not betray me must die?\" he thought within himself.\n\"Then do I now know one mark at which my Greek may send his sharpest\narrow and be guiltless. He may slay this Iberian swine with his own\nhand.\"\n\nFor the mutterer was a guest who had risen from a table, and he was one\nwho had been an officer of Herod's household, but was now pretending to\nbe an enemy of the cunning tetrarch, the jackal of the Black Castle.\n\nThe guest returned to his reclining, and Sapphira had vanished as a\nlamp that goeth out, but the procurator came forward.\n\n\"Say to Caius that I come. Abide thou in his house this night and on\nthe morrow until I send for thee, save that thou mayest go in the\nmorning to the school of Gamaliel. Hast thou money for thy uses?\"\n\n\"O most noble Pontius,\" said Lysias, \"the swift ass that was mine own\nis in thy stable. All baggage of mine is in the armory room where thou\ndidst find me. I have gold and silver pieces enough in my pouch for\nthis present. I am not poor, so that what I have be not taken from me.\"\n\n\"I will give orders in these thy matters,\" replied Pontius. \"He who\nserveth me well is rich enough. Thou shalt have thy swift ass and\nsuch other beasts as thou wilt. Go now. I believe thee brave and\nprudent. Thou art young, too, and the girl is fair. Youth is the time\nfor trifling. Provide thee soon a good bow and arrows of thine own\nchoosing.\"\n\n\"Thanks, O noble Pontius! Thanks! I will send sure arrows at thy\nbidding!\" So saying, the young Greek departed.\n\nLong was the conference that night between the Saxons and Lysias.\n\n\"We are little surprised,\" said the jarl, \"for we knew thou wert going\nto this place. Thou art a good fighter and thou hast rightly taken the\nprocurator for thy captain. I have heard that he casteth the pilum even\nbetter than do other Romans. I could follow such a man into battle,\nknowing that he is fitted to lead. Hast thou found thy Sapphira?\"\n\n\"Speak not of her, O jarl,\" said Lysias. \"Ere long thou mayest thyself\nlook upon her, but there is a peril in her name at this hour.\"\n\n\"I read thy face,\" said Ulric. \"Keep thou thine own secret. But thou\nmayest say to Pontius the Spearman that he hath no surer friend than\nCaius of Thessalonica.\"\n\n\"Even now they are together,\" replied Lysias. \"The procurator will know\nall that is known to thy friend, but I fear the careless tongues of thy\nSaxons. They speak to one another concerning triremes and old fights at\nsea. I would they were in their North country.\"\n\n\"So would not I,\" said Ulric, \"unless I were to sail with them. I may\nnot now leave this city of Jerusalem, and to sail to the north were to\nsail into ice fields. We must wait until the spring.\"\n\nNot so thought the homesick vikings in their comfortable lodgings in\nthe house of Caius. Even now they were talking of the sea.\n\n\"It is but a few miles to this seaport called Joppa,\" they said. \"We\nwill learn somewhat concerning the road thither and the shipping. We\nare free men, with the Middle Sea so near at hand.\"\n\nCaius of Thessalonica slept well after his long communing with the\nprocurator, and when he awoke the jarl sat near him.\n\n\"Thou art watchful!\" exclaimed Caius. \"But in Jerusalem I am safe. I\nhave to tell thee, however, that thy gladiators may not abide within\nthe walls. The quarters for such as they are out in the valley of\nJehoshaphat, near the amphitheater. No games are going on at this time,\nbut there will be abundant sport in the days after the Passover feast,\nwhen Herod cometh.\"\n\nThe jarl's brow darkened, but he said only: \"So be it. I will guide\nthem to their place. I myself will inspect the city and the forts and\noffer sacrifices, as I told thee. But this know thou, O noble Caius,\nthat not in this city nor in any other is treachery dead. I fear for\nthee. How is thine arm? I would see it.\"\n\n\"Thou hast knowledge of wounds, but not of poison,\" replied Caius.\n\"Uncover it.\"\n\nThe jarl did so, and he looked thoughtfully at the sore and then at the\nfeverish face of the noble Roman.\n\n\"This man will die slowly,\" he thought, \"but he will die, for this\nwound healeth not. I will not be here when he dieth, lest I be\ndeemed by others only fit food for wild beasts. So will I say to my\ncompanions.\"\n\n\"It changeth little,\" he said aloud to Caius. \"Who shall read a thing\nlike this? I will go and return, but I would my sword might be near\nthee if there is need of it.\"\n\n\"Go, O jarl,\" said Caius. \"I will send for thee if I require thee.\nFulfill thy will concerning the city, for all men may come and go. Only\nthat thou must leave thy weapons from thee or the legionaries will\ndisarm thee. The Jews, also, go unarmed.\"\n\n\"For that I have no care,\" said Ulric, \"but it were a sore thing for\nTostig the Red, for instance, to have no hilt near his grip.\"\n\n\"March them away quickly!\" exclaimed Caius. \"While thou art known to be\nwith me as a guard thou mayest wear thy sword and thy mail. The rules\ngo no further, for there have been many tumults and much bloodshed in\nJerusalem.\"\n\nThe jarl answered not to that, but took his leave, and not at all as a\nservant. Rather did it seem as if the centurion were under his command.\nHe went to his men, and well pleased were they to find their quarters\nwere to be without the walls.\n\n\"O jarl,\" said Wulf the Skater, \"this is much better. I would thou\nwert able now to show us our way to the sea. We have learned much from\nLysias and from others. There is good shipping upon the coast and the\nright keel might be found by brave men.\"\n\n\"Also triremes of C\u00e6sar,\" replied the jarl. \"The coast is well guarded.\nWe will wait a little.\"\n\nOut into the streets they marched, with him at their head, and many\nturned to look upon their array as they went on to the gate. The\ndwellers in Jerusalem were accustomed to seeing various kinds of armed\nmen, but these were unlike any others. Nevertheless, there were devout\nJews who lifted hands to curse them in the name of Jehovah, as heathen\ngladiators whose presence was a pollution of the city of God.\n\nThe amphitheater, when they came to it, was found to be larger than\nthat of Tiberias, with more dens for wild beasts and with a better and\nlonger course for the running of races.\n\n\"I have been told,\" said the circus servitor who guided them, \"that\nHerod the Great delighted much in horses. Also that one value of the\ncircus was as a place of execution for tribes who had rebelled against\nhim. His horsemen on the frontier scouted far and wide for captives and\nhis cages here were ever full.\"\n\n\"I care not for circuses,\" said Wulf the Skater. \"I have seen enough of\nthem.\"\n\n\"And I,\" replied Tostig; \"if I might kill an elephant, it would please\nme. I have a curiosity to know how long it taketh so huge a beast to\ndie.\"\n\n\"Thou wilt see elephants enough,\" said the servitor, \"but they do not\noften spend them upon the games. They are costly, and they come from\nfar. Men and women are plentiful, and they make as good sport in the\nkilling.\"\n\nThe buildings prepared as quarters for trained gladiators, not slaves,\nwere rude but spacious, and here did Ulric leave his friends while he\nreturned to the city, but he remembered the saying of Caius concerning\nhis armor.\n\n\"I may wear a tunic and robe only at most times,\" he said to himself.\n\"But under the tunic may be a coat of fine mail and hidden by the robe\nmay be a seax. I will not be defenseless altogether where there are so\nmany secret daggers as I hear of. I would have speech with Lysias, if I\nmay. I trust him not entirely, and I forget not that he is now of the\nhousehold of the procurator.\"\n\nNot justly altogether was he thinking of the young Greek, for Lysias\nwas a man walking among perils and having a wounded heart under his\nbright mail and his gay apparel. It was but the next day when he made\nhis first entrance at the school of Gamaliel. Celebrated over the\ninhabited earth was this academy, and many came from distant lands\nto hear the teachings of the great and learned rabbi. Among them,\nalso, were those whose real purpose was to obtain for themselves the\nreputation of scholarship through the name of Gamaliel their teacher,\nand they were even as Lysias in that matter. In such a company,\nhowever, small attention was paid to one more young Greek, who seemed\nto be rich, save that none questioned him unwisely after being informed\nthat his protector was Pontius the Spearman. Moreover, if there were\nthose who bowed and made way for him on that account, there were others\nwho bent their brows and drew aside their garments that he should not\ntouch them.\n\n\"Thou art imprudent,\" said an elderly man to one of these. \"Restrain\nthy zeal, I pray thee.\"\n\n\"He is a dog!\" growled the zealot. \"His heathen master slew my father\ncauselessly in the temple, mingling his blood with his sacrifice to\nJehovah. I am of Galilee.\"\n\n\"I will ask thee, then,\" said his adviser, \"sawest thou ever this\nGalilean prophet who cometh from Nazareth? It is said that he worketh\nmany wonders.\"\n\n\"I have seen him,\" said the zealot, \"and wonders he doth work. Hath any\nother rabbi raised the dead? Who else cleanseth a leper or openeth the\neyes of the blind?\"\n\n\"If thou liest not,\" was the surly response, \"he is indeed one of the\nlearned. I will hear his teachings when he cometh to Jerusalem to the\nPassover feast. But he will work no wonders here.\"\n\n\"Knowest thou that?\" sneered the zealot. \"But this thou knowest from\nthe law, that it is not well for thee to speak evil of a rabbi. He who\nrevileth one of the learned goeth to Gehenna.\"\n\n\"I reviled him not!\" exclaimed the adviser, as if in sudden fear. \"I am\na Pharisee of the Pharisees. I am a keeper of the whole law. Verily I\nwill hear thy rabbi when he speaketh. But beware thou of offending the\nprocurator!\"\n\n\"Messiah cometh!\" said the Galilean fiercely. \"He bringeth a sword! He\nwill make his garments red in the blood of the heathen!\"\n\n\"Let not the priests hear thee!\" sharply responded the Pharisee. \"To\nthem only is given the discerning of such matters. Thou wilt yet be\ncast out of the synagogue.\"\n\nThe angry Galilean walked slowly away. \"What know the Pharisees and the\npriests concerning Jesus of Nazareth?\" he muttered. \"I think of him\nthat he is a more learned rabbi than any here in Jerusalem.\"\n\nNow Lysias heard these men, and already had he learned from the Saxons\nin what manner their jarl had been healed of his hurts in Galilee.\n\n\"This prophet!\" he thought. \"I will see him if I may. Alas for me,\nthere is no temple here to Mercurius or to Apollo! I have great need\nto offer sacrifices. No! not to Juno nor to Venus! They have not dealt\nwell with me. I think I shall now hate Sapphira when I see her. How is\nit then that I also love her, seeing that I would slay her if I could?\nThis is that strange thing between a fool and a woman.\"\n\n\n\n\n                             CHAPTER XXXV.\n\n                      IN THE COURT OF THE WOMEN.\n\n\nIt was still the winter time in the Northland, but in Judea the spring\nhad returned. In the lowlands there was already much heat and a swift\ngrowth of all fruits of the earth, but in a high place, like Jerusalem\nupon her hills, the days were cooler and oft the nights were frosty, so\nthat men builded fires in their braziers.\n\n\"This is not according to nature,\" said Lars, the son of Beolf, among\nhis companions. \"We have had no snow save a few flakes, and there hath\nbeen no ice thicker than the blade of my seax. I weary of this land!\"\n\n\"Hael to the Northland!\" exclaimed Tostig the Red. \"Hael to the driving\nstorms and the glittering ice, and to the frost and to the snow!\"\n\n\"I will not stay here,\" said Knud the Bear. \"I will depart from an\naccursed country wherein there is never good winter. But didst thou\nhear the keeper? He saith that ships at Joppa dare not put to sea\nbecause of the rough weather. What seamen are these!\"\n\n\"O men!\" said Wulf the Skater. \"By Odin! If vikings were at the oars\nand if I were at the helm, a keel would seek the open sea.\"\n\n\"We will even go to Joppa when we may,\" said Tostig. \"But our errand\nwill be to the Northland, that we may bring back fleets, and in them\nSaxons, to march with the jarl into the great battle in Esdraelon. We\nare too few.\"\n\n\"I am with thee,\" said another tall viking. \"I have considered this\nmatter, and I think it is also the mind of the jarl. He may not go with\nus, but his secret will is that we go speedily without him. Then will\nhe truthfully say to the Romans that he did not command us to go. I\nwill no longer be shut up in this place as if I were one of the beasts\nin yonder dens waiting for my turn to be made a bloody show of. I am a\nfree warrior, not a caged wild creature. I will go to the sea.\"\n\nOther voices were raised in strong accord with his, and their talk went\non until their minds were on fire and their purposes had become firmly\nfixed, for they were men of experience and of great courage. The jarl\ncame not among them at this time, for he was even then at the temple\ngate inquiring as to the right method of obtaining cattle for his\nsacrifices to Jehovah. A servitor went into one of the inner courts and\nbrought out a dealer who had bullocks at hand, and this man began to\nname prices, counting them in shekels of the temple.\n\n\"What know I of shekels?\" exclaimed Ulric.\n\n\"Thou dost not need to know,\" broke in a voice behind him. \"O jarl, I\nam here. He asketh thee too much. Let me attend to this matter. Well\nfor thee that thou hast it in thy mind to offer sacrifices to the\nliving Jehovah!\"\n\n\"O my friend!\" exclaimed the jarl. \"Glad am I of thy coming! This\ncharge is thine.\"\n\n\"Who art thou that meddlest with another man's affair?\" demanded the\ndealer angrily.\n\n\"Silence, thou!\" was the peremptory reply. \"I am Ben Ezra, the\ninterpreter of Caius of Thessalonica, and this is the captain of his\nguard and of his Saxons. Beware that thou deal not fraudulently with\nany of his people lest I have a hand laid upon thee. I am in my right\nin this matter.\"\n\n\"That do I now admit,\" replied the dealer in a changed manner, \"but I\ncharge him not too much. Come thou and see the cattle.\"\n\nBut the prices he shortly named were less than the half of his former\nasking.\n\n\"Pay him, O jarl,\" said Ben Ezra. \"It is well. Offer thy burnt\noffering, for thou hast great need of the favor of Jehovah in that\nwhich cometh upon thee. I will remain with thee, for I also offer\nsacrifices. O dealer, I will buy of thee. Let the beasts be without\nblemish. I will have, also, a lamb and two doves and wine for the\noblation. Pause not, for I have conferred with the high priest and he\nknoweth my matter, and this is of his direction.\"\n\nBut for the guiding of Ben Ezra the jarl had been dealt with as an\nignorant man, a foreigner having money, but now all things were\naccomplished with order and rectitude. Nevertheless, the jarl was\ndispleased that he was compelled to remain without in the court of the\nheathen, not going near the altar whereon his offerings were burnt.\n\n\"They would prevent such as I am,\" he said, \"from drawing too near\ntheir God and getting acquainted with him. I would both see his face\nand hear his voice. Evil, evil, is this manner of the Jews! Are they of\nhigher degree in the sight of their God than am I, the son of Odin?\"\n\nNevertheless, from the place assigned him he might see all, and there\nhe stood watching the manner of the slaying of his bullocks and the\ngoing up of the great smoke and the swinging of the censers. He\nlistened, also, reverently to the chanting of the priests and the\nLevites and the responses of the Jewish congregations in the other\ncourts.\n\n\"Ben Ezra,\" he remarked, \"might enter the inner court, going where he\nwould, for he is a Jew of high degree. He told me, also, that over\nyonder is the court of the women. I have offered my sacrifice. Why\ndo I linger here?\" For his face grew suddenly pale as if he had been\nstricken through with a spear, and he exclaimed again, \"The court of\nthe women.\"\n\nLoudly swelled the sonorous chorus of the many chanting voices and\nthere came back strange echoes from the inner walls of the temple.\nThe majesty and the splendor of the temple service were unspeakable,\nbut the jarl turned away from it and strode swiftly out of the court\nof the heathen. He walked on until he might stand in a place near the\nbroad passage by which the women worshipers, veiled or unveiled, were\ncontinually coming and going.\n\n\"O Miriam!\" he thought. \"My eyes have sought thee as I have walked\nthe streets of this city. Hilda cometh not any more to counsel me. I\nam dark in all my mind. If thou art not here what do I any longer in\nJerusalem? It is not Asgard, and here are no gods at all. It is but\na city of men like myself, and the women are as other women, and the\nRomans have the rule in spite of this Jehovah.\"\n\nHis thoughts were burning within him and he felt the sickness of\ndisappointment and failure, and his eyes were dull with longing as he\ngazed upon this procession of Hebrew women. Suddenly his heart gave a\ngreat leap, but he stood still, for he heard a voice saying:\n\n\"Miriam! Thy veil! Cover thyself! Yonder Roman stareth at thee!\"\n\n\"I will cleave him to the jaws!\" exclaimed Ulric, turning quickly.\n\nBut before he could move a pace or discern one man from another whom to\nstrike a hand was upon his arm and he heard a whisper which thrilled\nhim from head to foot.\n\n\"I am Miriam! I am now veiled! Harm not thyself nor me. I think he\nheard thee not. Strike not a Roman lest thou be crucified. Follow me,\nO beautiful one. Follow not too nearly, but mark well the house into\nwhich I go. The woman with me is my aunt. The Roman of whom she warned\nme is but a dealer in slaves--but he is a Roman. Come!\"\n\n\"I follow thee, O my beloved,\" whispered the jarl, \"but if he toucheth\nthee, he shall die if he were C\u00e6sar!\"\n\nSunken-eyed, hollow-cheeked, with a forehead low and sloping, was the\ndealer in human cattle who stood shortly at the street side without the\nportal. His lips were moving with an evil expression upon them, and his\neyes had seen too well the exceeding beauty of the Jewish maiden.\n\n\"A thousand sesterces for her at Rome,\" he muttered. \"How shall I\nobtain her? Pontius hath bidden us beware of angering the Jews.\"\n\nThen he came forward a pace and spoke aloud, with small ceremony, to\nUlric.\n\n\"She spoke unto thee, O gladiator. Who is she, and what doest thou\nhere?\"\n\nEven for the sake of Miriam was the jarl somewhat calm in his manner\nand cunning in his speech, but his voice was unpleasant.\n\n\"O Roman,\" he said, \"art thou unwise? Seest thou not that I am a sword?\nOne greater than thou art will answer for my going and coming. I but\ndo his bidding. When thy head passeth suddenly from thy shoulders thou\nwilt ask no more questions concerning a damsel who is guarded by the\nstrong and high one. I will watch for thee henceforth. I am one who\nneedeth not to be commanded a second time concerning a sword cut.\"\n\n\"Aha!\" snarled the dealer. \"I have seen thee heretofore. Thou art\ncaptain of the gladiators of Caius of Thessalonica. I quarrel not with\nhim.\"\n\n\"Nor dost thou need any quarrel with the procurator,\" said Ulric. \"His\narm is longer than thine. Keep back thy foot from unknown ground lest\nthou shalt meet a man coming unto thee in sudden haste.\"\n\nNo word came back, but the man's face darkened venomously, for a Roman\nliketh not a rebuke from a barbarian; but there was fire in the eyes\nof the jarl and his right hand went under his mantle, and the dealer\nunderstood well the meaning of the movement. Nevertheless, a mere\ntrafficker in the flesh of men and women may not wisely stir the wrath\nof a centurion or of a man in authority. A Roman may not be scourged\nor crucified, but he may die suddenly as well as another. So turned he\nsullenly away about his affairs, and the jarl went on his way.\n\nThe streets of Jerusalem are narrow with the exception of the broad\nthoroughfares which lead to the outer gates and the main approaches to\nthe temple. It was a narrow passage between high palace buildings into\nwhich Miriam and her aunt hastily turned their feet not long after,\nescaping from observation by the cruel eyes of the dealer in slaves.\nNo word did they utter, and those whom they met spoke not unto them,\nfor there are laws of privacy and due reserve among the Jews relating\nto the public greeting of women. He who annoyeth them transgresseth and\nis liable to be called to an accounting. They walked onward rapidly,\nand now the way led along the side of a mount beyond which was the\nvalley which divideth the city into, as it were, two cities. Ever\nat a little distance behind them strode a tall shape which did not\nmanifestly appear to pursue them, but for which all other wayfarers\nmade room on approaching.\n\n\"The gladiator seemeth to be in wrath,\" said one who looked upon him.\n\"Beware of the anger of these wild heathen, for they are even as\ntigers, and they know no law.\"\n\nLight was now in his eyes, nevertheless, and his stepping was that of a\nstag upon the hills.\n\n\"I have found her!\" he muttered, joyously. \"I have fulfilled the token\nthat was given me by Hilda in my dream upon _The Sword_. Now shall I\nnot soon see Hilda herself? Hath she not guided me in this, and is she\nnot now with the gods? This may indeed be the city from which I shall\npass on into Asgard. I am glad that I offered sacrifices in the temple\nthis day, for at once have I received this answer from Jehovah that he\nhath shown favor unto me. He is indeed the chief God of this land to\nthis day, for he hath not permitted the Romans to destroy his temple\nnor to slay his priests. I think that if they were to do so, he would\nbe angry and he would surely take his revenge upon them. That would I\ndo if I were a god.\"\n\nThe door of a large house swung open as of itself before Miriam and\nher companion, but Miriam paused upon the threshold. Turning and\nglancing quickly up and down the street, and seeing no peril, she\nraised a hand and beckoned. Ulric came quickly, but Miriam's aunt was\nalready within.\n\n\"Think not to enter with me now,\" said the Jewish maiden, hastily. \"But\ntell me quickly, what art thou in Jerusalem? Why art thou here? What\ndoest thou in Jehovah's temple?\"\n\n\"O Miriam, the beautiful!\" he responded, gazing upon her joyously,\n\"I am even as I said to thee in Esdraelon. I am Ulric, the jarl of\nthe Saxons. I am of Odin's line. Of the sons of the gods. I offered\nsacrifices in the temple of Jehovah asking for thee, and thou seest\nthat he granted my petition.\"\n\nEven as he spoke she stepped back within the doorway, and he also\nentered with her, but as yet the door closed not behind them.\n\n\"I understand thee partly,\" she said, trembling greatly. \"Thou art a\nprince among thine own people. O that thou wert a son of Abraham! O\nthat thou wert not a slayer of men in the circus!\"\n\n\"That I am not!\" exclaimed Ulric. \"Such business is not for me. I am\na free warrior. I go not again into the circus. I am with Caius of\nThessalonica for a season, for I am his friend and his guard. I came\nout from the Northland into the world that I might seek for the city of\nthe gods, that there I may meet my kindred. But I will ask of thee, O\nbeautiful one! O Miriam! how knowest thou Hilda of the hundred years?\"\n\nHer eyes burned earnestly upon him while he was speaking and her face\nwas as the dawn of a new day, for in it there were many changes, the\ncolor coming and departing and the lips quivering.\n\n\"I know her not,\" she said; and now they had drifted on into a small\nanteroom near the door, her veil, also, having been put aside more\nperfectly. \"Who is this Hilda, that thou askest of me such a question?\"\n\n\"Surely thou knowest her?\" he said. \"She is a saga woman of the\nNorthland. She is learned in all the old runes that are written on the\nrocks and on the tombs, and she talketh with the gods in their places.\nI know that it is now many months since she hath been laid in her own\ntomb in the cleft of the rocks, but I saw her with thee, speaking to\nme in a dream, when I was on the sea in my ship. She bade me sail on\nand find thee, and this I have done. Therefore I am glad that I offered\nsacrifices to thy God. Henceforth he shall be to me as Odin, the God\nwho is over all the other gods.\"\n\nShe listened as if his voice were music and as if she willed that he\nmight not cease speaking.\n\n\"Thou hast said!\" she now exclaimed, and a voice behind her, deep and\nsonorous, added:\n\n\"Amen! A great King is he, above all gods. He is the God of gods,\nand beside him there is no other; for Jehovah, our God, is one God,\nand there is none like him. O heathen man, thou hast well spoken.\nThis day hast thou become his servant, for he hath sent unto thee his\ncommandment in a dream, and thou hast obeyed him. Also thou hast done\nwell in offering thy burnt sacrifices.\"\n\n\"That did I according to the directions given me by Ben Ezra from the\npriests,\" said the jarl. \"But who art thou?\"\n\n\"I am Isaac, the aged, the kinsman of this maiden,\" was the response.\n\"O heathen man, I am glad that thou hast powerful friends, for at this\nhour we are among perils, both she and I--and all our house. I will\ntell thee, for one Abbas--accursed be he of Jehovah!--threateneth us\nwith destruction.\"\n\n\"Do I not know him!\" exclaimed Ulric. \"But surely he is nidering! He is\na weak man, and a traitor and a thief. If this be so, his blood be upon\nhis own head, for he must die. I have a matter concerning him that he\nknoweth not. O Miriam, I am a leader of men, and I am not imprudent.\nEvil is he who is careless concerning such as thou art. Tell thou me,\nthat I may have strength to obey thee, do I now remain here longer, or\ndo I depart?\"\n\nAs a man wrestling with himself was the jarl, and her face grew\nwonderfully kind and sweet as she looked upon him; but Isaac now stood\nby her gazing at the jarl, and the wrinkled features of the old man\nwere full of fear and trouble.\n\n\"Depart!\" she said, softly. \"It is enough that I have seen thee again.\nFail not to return, but when thou comest to the door ask only for\nIsaac. O that thou wert of my own people!\"\n\n\"I care not for that matter!\" exclaimed Ulric. \"It will not be long\nbefore I come----\"\n\nBut his eyes were looking down, for upon his own broad, powerful\nhand came, gently alighting as a bird, a whiteness which was lighted\nwonderfully by the red glow of the ruby in the ring. But the hand\nof Miriam lingered not, flitting coyly away as if the bird were\nfrightened, and in the fingers of the jarl, the son of Odin, there was\na strong tremor.\n\n\"Ulric,\" she said, pronouncing his name for the first time, with a\ngreat sigh, \"God hath sent us this promise of deliverance from our\ndestroyers. Thy Hilda was in the Northland?\"\n\n\"An hundred years old was she,\" said the jarl, \"when I bade her\nfarewell. I loved her more than aught else upon earth. She was a\nprincess, and her hair was as the snow, and her smile was exceedingly\ndear to me. Didst thou ever know and love such a one?\"\n\n\"I think she is as Hannah, the prophetess of my people!\" exclaimed\nMiriam. \"But she, too, hath departed. She was a mother in Israel!\"\n\n\"Haste!\" interrupted Isaac. \"Let the young heathen go his way! This is\nunseemly for a maiden of Judah! He may not remain. But, O youth, if\nthou canst do anything, withhold not thy hand.\"\n\n\"Fear not!\" said the jarl. \"I will quickly attend to Abbas and to\nwhoever worketh with him!\"\n\nBut his eyes were gazing deeply into the eyes of Miriam, and it seemed\nas if in this manner they were speaking to one another.\n\n\"Go!\" she whispered. \"Have I not thy ruby? Keep thou, also, my token. I\nam thine!\"\n\n\"O Miriam,\" whispered back Ulric, \"I think thee also a daughter of the\ngods. I go!\"\n\nThe door closed behind him and he strode away, but immediately Isaac\nspoke chidingly.\n\n\"Thou art mad!\" he exclaimed, \"O foolish daughter of Israel! O unwise\ndamsel! What is this stranger unto thee?\"\n\n\"O Isaac, my kinsman,\" she replied, \"this matter concerneth both thy\nlife and mine. Did he not fulfill the law of sacrifices? I will go to\nmy chamber, but I enjoin upon thee that thou greet him kindly when he\nreturneth.\"\n\n\"That much I will do,\" said the old man as one who prudently\nconsidereth a difficult affair. \"Am not I a man of understanding? If\nJehovah hath sent us a sword for our protection, blessed be his name!\nEven this day hath Abbas been with me, and he hath afflicted me sorely.\"\n\n\"What said he?\" she inquired.\n\n\"More than I may wisely tell thee,\" said Isaac. \"Only that he again\nhath demanded thee as the bride of this Tyrsus of Chronea. If thou\nshalt refuse, he will surely bring thee and thy household before a\njudge with whom is a gift and in whose hand is destruction.\"\n\n\"Tell thou that to Ulric the Jarl!\" she said, vehemently. \"Where is now\nthy wisdom? What more, then, hast thou to say? Is not this the spoiling\nof thy goods? If I were given to Tyrsus wouldst thou escape the greed\nof Abbas?\"\n\n\"Father Abraham!\" groaned the old man. \"We are in the power of the\nheathen. Do as thou wilt and I will speak well to thy swordsman.\"\n\nFar down the street, not knowing or caring whither he went, was Ulric\nthe Jarl, but one who stood at the wayside watched his coming and put\nout a hand.\n\n\"Halt, O jarl! Go no further. Such as thou art have need of caution. At\nyonder turn into the valley there are Roman guards and they will arrest\nthee as a gladiator escaped from the circus. Enter not a difficulty.\"\n\n\"O Ben Ezra,\" exclaimed Ulric, \"what sayest thou? Am not I a free\nwarrior?\"\n\n\"Not long wilt thou be free at all,\" said the Jew, \"if thou wanderest\nimprudently. The edicts have been strengthened. The master of the\ngames is a hard man and subtle. Go thou rather to the house of Caius or\nout into the valley of Jehoshaphat.\"\n\n\"Thou art my friend,\" said Ulric, \"and I will ask thee of an important\nmatter. Knowest thou of the doings of Abbas?\"\n\n\"He is in the city,\" said Ben Ezra. \"What is thy need of him? He is\nevil.\"\n\n\"I require of him nothing but his blood upon a blade,\" said Ulric. \"He\nis a plotter against both Caius and the procurator.\"\n\n\"Come thou with me to thy friend's house,\" replied Ben Ezra. \"I know\nthis to be true, but Abbas may not be slain openly.\"\n\n\"If Pontius will command me,\" said Ulric, \"I will bring him this\nserpent's head on the morrow. Otherwise I will guide my own doing. It\nis but a stroke of a sharp sword.\"\n\nLittle said they after that until they were in the house of Caius, but\nwhen they were there it was Ben Ezra and not the jarl who was summoned\nto confer with the centurion. Not long was he absent, but when he\nreturned his face was dark.\n\n\"Trust not a Roman,\" he whispered to Ulric. \"To them all other men are\nbut as cattle. Thou art only a swordsman in the eyes of this Caius.\nSlay not Abbas lest thou anger him. He is thy friend truly, but it is a\nRoman friendship, with a dagger in it. Go thou to thy men. Would thou\nwert on the sea! Thou hast no right to sell them to the circus.\"\n\n\"That will I not!\" said Ulric. \"But I will confer with them speedily.\"\n\nSo went he away, but he went with Ben Ezra.\n\nThere are many cunnings among those who struggle in the net of power,\nand a great subtlety had been born in the mind of Lysias. \"If the\nSaxons remain,\" he had thought, \"I am lost. It is long before they may\nbe slain in the arena. I will go and talk with them again. This galley\nthat, is to bear the messenger of Herod lieth at Joppa.\"\n\nTherefore, even while Ulric had offered his sacrifices, the young Greek\nwas among the Saxons telling them many things.\n\n\"This is no merchant craft,\" he had told them. \"This galley of Herod is\nsmall, but strong for a rough sea. Ye are crew enough.\"\n\n\"That are we,\" said Tostig the Red. \"But the jarl might forbid our\ngoing.\"\n\n\"If ye go not,\" said Lysias, \"ye will be penned as dangerous beasts.\nThe jarl only is secure among the great, his friends. He cannot protect\nyou from the master of the games.\"\n\n\"That dog was here to look at us to-day,\" said Knud the Bear. \"I like\nhim not. I will wear no fetters of his clamping. O ye sons of the free\nvikings, I go to the sea. Who will go with me to take this keel of\nHerod?\"\n\n\"No man will remain behind,\" said Wulf. \"The night shadows come. There\nare horses in the stables. Every man to his armor, and let us take our\ntreasure with us. We will slay as we go and leave behind us a good\nmark.\"\n\nNevertheless, they were prudent, as became warriors who were few in\nnumber, and the guards of the circus had as yet no command concerning\nthem save to let them come and go as they would for a season. The\nstables were near and the horses were many, and with these were only\nslaves who feared to speak to a swordsman. Therefore, if a Saxon came\nto look at beasts or to examine saddles and bridles, no man hindered\nhim. It was but thought that he had curiosity as to trappings which he\nmight use in the games. He did well to take thought concerning his own\nbusiness against the hour when he must slay or be slain. But all the\nwhile a fire burned more hotly in the hearts of the men, for the words\nof Lysias were in full accord with many sayings of the jarl.\n\n\"He hath been troubled in mind concerning us,\" they said. \"He knoweth\nnot what to do. We will take away from him this burden, for we are men\nand we may save ourselves. It is not meet that we should encumber our\njarl unduly. He hath done well with us. He would not have us linger to\nbe slain.\"\n\nNevertheless, the dusky hour was at hand and Ulric came not to them,\nas he at first thought. From the house of Caius he had been silently\nled to the house of Ben Ezra, his friend guiding him as a man who is\nin deep thought. The way seemed one which led toward the valley of\nJehoshaphat, through many streets, but they came to a door before which\nBen Ezra paused and turned to the jarl.\n\n\"I will trust thee,\" he said, \"for it is needful. This is the house of\nmy abiding.\"\n\n\"Not large,\" said Ulric, \"and the front of it is dark and ancient. I\nwill go in with thee.\"\n\n\"In it dwelleth no other beside myself,\" said Ben Ezra, opening the\ndoor with a key. \"But he who knoweth of this place knoweth of death. It\nis a hidden thing in Israel, and I charge thee by thy gods and by the\nwrath of Jehovah, my God, that thou make thyself as one of us to keep\nwell a thing that is shown unto thee in secret.\"\n\n\"I am a keeper of faith,\" said Ulric. \"I will call it a secret of the\ngods, as if it were the tomb of my father. But in this chamber which we\nhave entered I see nothing save plain and simple matters.\"\n\n\"Come further,\" said Ben Ezra, \"for thou hast taken upon thee thy oath.\nDid I not tell thee that I had been to the cave in Carmel and that I\nhad made thy treasure secure?\"\n\n\"It was buried well,\" said Ulric. \"I think no stranger could have found\nit.\"\n\n\"Neither would it have been of any use to thee or me,\" said Ben Ezra.\n\"Couldst thou strike with thy seax if it were buried in a cave in\nCarmel? It is better at thy hand.\"\n\n\"I understand thee,\" exclaimed Ulric. \"At this hour, here in Jerusalem,\nI have need of money. I was never so at any time. It is true that gold\ncoins may be good weapons. I will be glad of them and of the jewels.\"\n\n\"O jarl,\" said Ben Ezra, \"already have I paid much to Pontius the\nSpearman and to the high priest and to the captain of the temple.\nGreedy were they, but I have satisfied them. Of thy share in the matter\nthey know not. Thou hast no need to go to thy men this night, for the\nmorning will do as well, and thou canst plan how they may escape to\ntheir own land.\"\n\n\"So will I do,\" said Ulric. \"I will abide with thee. But this seemeth\nto be as other houses.\"\n\n\"So hath it seemed to any who dwelt here,\" said Ben Ezra, \"unless\nthey were as I am. Is not this back wall strongly made of well-fitted\nmasonwork?\"\n\n\"A well-made wall,\" said Ulric. \"None may break it through. The stones\nare very large.\"\n\nFar back from the street were they now, and the house which had\nappeared small was seen to be of great extent, as if builded down a\nsteep <DW72>. Suddenly the jarl exclaimed:\n\n\"A door of a stone! O Jew, how is it that this great marble turneth at\nthy pushing?\"\n\n\"See thou,\" said Ben Ezra. \"It is set into the wall upon pivotings.\nTherefore it is as firm as the rest of the wall unless it shall be\ntried by one who knoweth the catch-pin at the side. Even then a weak\nhand moveth it not. I will show thee, and then do thou make trial for\nthyself.\"\n\nThe jarl watched and understood.\n\n\"A marvelous trick!\" he exclaimed, opening and shutting the secret\ndoor and finding that much strength was required. \"O Jew, beyond is a\ncorridor of stone, and I see steps which go downward.\"\n\n\"Before thee is a great deep,\" replied Ben Ezra. \"Thou art trusted as\nto this thing in the name of Jehovah. Go in, O jarl of the Saxons. Thou\nwilt go down into the secret chambers of Jerusalem with me.\"\n\n\n\n\n                            CHAPTER XXXVI.\n\n                         THE SECRET MESSENGER.\n\n\nLysias, the Greek, stood reverently before the Roman ruler of\nJerusalem, and the dark, piercing eyes of Pontius were watching his\nface intently.\n\n\"O most noble Pontius,\" said Lysias, \"I have done as thou didst order.\nAll these were the words of Ben Ezra, nor have I failed to tell thee\nevery saying of Abbas and of him who was with him. The messenger from\nMach\u00e6rus goeth swiftly to Joppa and the galley of Herod waiteth for\nhim.\"\n\nThey were standing in the small chamber near the banquet hall, and\nthe voice of Lysias was hushed and tremulous, for the brows of the\nprocurator were knitting and the veins in his temples were swelling.\n\n\"Well for thee, O Greek,\" he muttered, hoarsely. \"But now it is as if\nHerod himself were to be with C\u00e6sar, bringing gifts. The very gods are\nagainst me!\"\n\n\"O most noble Pontius,\" said Lysias, raising his head courageously,\n\"bid me depart and it may be that neither galley nor messenger shall\ncross the sea to Rome.\"\n\n\"I may not hinder a royal messenger,\" said Pontius, gloomily. \"To do so\nwere sure destruction. Thou canst do nothing.\"\n\n\"But if,\" whispered Lysias--\"if Herod, the tetrarch, might know that\nhis galley had departed, and if afterward no man came to tell him of\nher voyage?\"\n\n\"A man may hear good tidings,\" said the procurator, with a dark smile\ndawning in his face. \"But be not thou at any time the bringer of news\nconcerning this galley. Thou hast a letter to bear for me to Cornelius\nat C\u00e6sarea. I bid thee to go by way of Joppa and to return. I now write\nthe parchment. Ride thou thy own swift beast. Whoever may be traveling\nupon their own errands at this time, I meddle not with their affairs.\"\n\n\"Thanks, noble Pontius,\" said Lysias, \"but I will give thee a token. A\nman will come to thee in haste shortly from the keeper of the circus.\nHe will know nothing of the galley of Herod, but he will tell thee of\nher departure from Joppa and of her crew. So shalt thou be sure that I\nknow not aught except my errand to Cornelius.\"\n\nHasty was now the going and the returning of the procurator, but Lysias\nhad now a small tablet and not a parchment to put into his pouch,\nneither looked he upon the writing on the tablet.\n\n\"Go!\" said Pontius. \"I will wait for thy man from the circus. Tell me\nno more!\"\n\nThen passed Lysias out into the corridor and the eyes of Pontius\nfollowed him.\n\n\"Subtle are the Greeks,\" he muttered. \"Already yonder youth knoweth\nenough to kindle a fire that would burn to Tartarus. Let him do this\none thing and I will give him a gift which he hath never yet received.\"\n\nNot far had Lysias gone along the corridor when a hand withheld him and\nthere was a whisper.\n\n\"Lysias! Love! Whither goest thou?\"\n\n\"Sapphira! O beautiful one! I may not linger. I ride swiftly to\nC\u00e6sarea. I will return to thee. Wait thou for me!\"\n\n\"O Lysias! Favored of Aphrodite! Go and return to me. I shall then have\nmany things to tell thee. Then shalt thou know I have loved thee.\"\n\nHer arms were around him, her kiss was upon his lips, and she was gone.\nHe, too, went on in haste, leaving the palace, but she had retreated\ninto an inner chamber, luxuriously furnished, wherein a lamp was\nburning.\n\n\"I will wait here for my mistress,\" she said. \"A strange thing is love,\nfor it may be lighted like this lamp. It may go out and it may burn\nagain if one willeth. I think I must put out this love of mine for\nLysias lest it should burn me. Alas for him or for any who may be made\nthe bearer of secret messages! And I? O Lysias! Well for thee that thou\nknowest not this change which is in store for me. And thou, O beautiful\nAphrodite, be not angry with me that I am to become also a Jewish\nproselyte and offer sacrifices to the God of the Jews. My mistress hath\nbidden me to become free and to wed Ananias. It is better so than to be\na slave, or to throw myself away upon a Greek youth who must shortly\ndisappear. I love not ruin. I am to be rich and I shall be the favorite\nof more gods than one.\"\n\nShe spoke with a triumph upon her face and with exultation in her\nvoice. Then she reclined upon a couch, with the light of the lamp\nshining brilliantly upon her goodly raiment and her beauty, and so she\nawaited the coming of the wife of the procurator.\n\nThrough the Damascus gate passed Lysias, and not long afterward an ass\nhalted near the amphitheater, further down the valley. A slave came out\nto attend to the ass, and was followed by the master of the games.\n\n\"Who art thou?\" he demanded, surlily.\n\n\"See that thou hinder me not,\" said Lysias. \"Look well upon this\nsignet.\"\n\n\"I obey the procurator,\" said the master of the games. \"Do thou his\nbidding. But I will see nothing that thou hast in thy hand by any\ncommandment from him. Hold thou thy peace, O messenger. I meddle not.\"\n\nLysias had dismounted and, without more words, he passed on into the\nquarters of the Saxons. Excepting themselves, no others were present to\nobserve or to hear, but he did not find men who were taking rest. Some\nwere making up packages for carrying, some were examining carefully\ntheir arms and armor as if about to go into battle, but they greeted\nthe Greek heartily. He looked around him for a moment, not without an\nunderstanding of this which they were doing.\n\n\"I am in season, O Tostig the Red,\" he said altogether as if he had\nbeen expected to come. \"But put ye on Roman helmets, every man. Ye are\nto ride fast to Joppa this night. Right glad am I to be your guide, for\nthe roads might prove misleading.\"\n\n\"Hael to thee, O Greek!\" exclaimed Tostig. \"Even now are the horses\nnearly prepared. We will mount at thy bidding. But hast thou at all\nseen the jarl?\"\n\n\"By the will of the gods, and not by his own, he may not now come,\"\nsaid Lysias. \"Were he here, he would say that ye go forth at once and\nthat ye ride well. Mark this saying, however, that there will be one\nat the shore who must by all means enter the galley, but who must not\ntravel far in her.\"\n\n\"It is but a spear thrust,\" said Knud the Bear. \"We will attend to his\ncase.\"\n\nSilently all, but openly and boldly as by men who obeyed a high\ncommand, were the horses led out and mounted. There were also led\nhorses for the packages and for changes, and there was no Roman officer\nof rank at hand to call this doing in question.\n\n\"Ride!\" said Tostig. \"Odin! It will not be well for any who shall cross\nour path!\"\n\nNone was likely to do so. The Romans held Judea by garrisons in forts\nand camps, and not greatly by moving forces. The highway to Joppa would\nbe deserted after nightfall. Who should rashly interfere with mounted\nspearmen, whose very helmets were as a sharp warning to the imprudent?\n\n\"Swiftly! Swiftly!\" exclaimed Lysias, before long. \"We now pass the\nhill of Golgotha. On that mount have many been crucified. Make thee\nsure that ye get well away with this galley of Herod and that no man\nmay find you upon it in after time. I tell you truly that if ye are now\ntaken prisoners ye would but climb yonder Hill of Skulls.\"\n\nSilent were the Saxons at that hard saying, but the horses under them\nappeared to spring forward as if with one accord.\n\nIt was at the foot of a steep declivity that the galloping ceased for a\nbrief resting of the horses, and Tostig exclaimed:\n\n\"O Knud the Bear, this is well. We have gone far. But I like not this\nmanner of departing from our jarl. I think I should have seen his face\nand heard his commandment. Were he to need my sword on the morrow, I\nwould be at his side.\"\n\n\"I also,\" responded Knud. \"We are his own men and he is ours. It is in\nhis heart that we may return to the Middle Sea with a hundred keels.\nWhat, then, would we care for Roman triremes? We could slay all the\nlegionaries in Judea.\"\n\n\"If we might indeed land here again,\" said Wulf the Skater, doubtfully.\n\"At all events we have no more upon our hands this night than to take\nthe keel which is prepared for us and to put to sea.\"\n\nSo said they all, and again they pushed forward, but after a while the\nroad by which they traveled was no longer so rugged and so hilly.\n\n\"We shall kill the horses,\" they said, \"but we may reach the sea before\nthe dawn.\"\n\nSo did it prove, for more than one horse gave out and his rider mounted\nanother from those which were led without any heavy burden. It was yet\ndark, at the last, when Tostig exclaimed:\n\n\"O Greek, I hear the sound of waves upon a beach. Are we now near\nJoppa?\"\n\n\"Too near,\" replied Lysias, \"for into the town itself we may not safely\ngo. We will turn here by this road at the right. If we encounter guards\nor a patrol, let there be no report made of our passing.\"\n\n\"Halt!\" rang out in the road a little ahead of them. \"The password! Who\nare ye?\"\n\nIt was a legionary at his post, a sentry on guard, and to him rode Knud\nthe Bear.\n\n\"I am this,\" he said. \"Take thou my token!\"\n\nDown fell the soldier, speared through the face, so that he spoke not\nagain, and on rode the Saxons toward the sea.\n\n\"We have now only starlight,\" said Lysias, \"but yonder at anchor\nfloateth the galley of Herod, the tetrarch. This is according to the\nsaying of the procurator. All is well, for he who cometh hath not\narrived. There are boats; take them. But here do I leave you, for I\nhave a further errand. Fare thee well.\"\n\n\"Success to thee, O Lysias,\" said Tostig. \"We are thy friends\nhenceforth. Haste thee about thy business. We can care for ourselves\nnow that we see keels and waves.\"\n\nMany voices bade him good speed, and the strong ass appeared but little\nwearied as he sprang away northward along the beach.\n\n\"Glad am I not to be in Joppa this day,\" said Lysias. \"If I am heard\nfrom next at the house of Cornelius at C\u00e6sarea, no man will accuse me\nof having too much acquaintance with the doings of the gladiators of\nCaius. I did but bring to them an order whereof I knew not the meaning.\nI am but a messenger, carrying letters to and fro.\"\n\nNevertheless, his heart was full of great anxiety and he remembered how\ndark had been the hard face of the procurator.\n\nThe fishing boats were many, but only two large ones were taken. Into\nthese the Saxons put their baggage of all kinds, but they drove away\ntheir horses to a good distance down the beach. Then they took the oars\nand in a short rowing they were near the galley.\n\nOver the bulwark leaned an armed man as the boats touched the side.\n\n\"Whence come ye?\" he demanded, but he spoke as to friends, for he was\nat that hour expecting such an arrival and he saw the Roman helmets.\n\nFor a moment no voice replied to him, but the Saxons went quickly over\nthe bulwark.\n\n\"Slay now, but cast none overboard!\" commanded Tostig. \"Here are\nsoldiers sleeping.\"\n\nThese who slept were not many of them Romans, but more were soldiers\nof Herod, Jews and Arabians and Edomites. They died speedily under the\nswift thrusting of the Saxon spears, but the watchman had fallen first.\n\n\"Spare the rowers in their places,\" said Wulf the Skater. \"We will use\nthem.\"\n\nBut these rowers were all slaves, in chains, and they looked upon the\nslaughter in silence, as if it were no affair of their own.\n\n\"Be ready, all!\" suddenly commanded Tostig. \"Utter no sound! A boat\ncometh from the shore, as Lysias gave us to expect. Not from this\nbeach, but from a pier in the harbor. In it are but few men. In the\nprow is some great one, but he weareth no helmet. Let them come on\nboard with all safety, but none in that boat may return to Joppa.\"\n\nFor a cause known to themselves not one of them had any purpose of at\nonce returning. They came swiftly to the galley and all climbed eagerly\non board, casting adrift their boat to float where it would.\n\n\"Away!\" shouted he who seemed their leader, as if speaking to sailors\nwho were under his own direction. \"Row out of the harbor quickly.\nSpeed, or a scourge for every back!\"\n\nSaxon hands were already raising the anchor and the rowers put out\ntheir oars as they were bidden.\n\n\"O all ye gods!\" suddenly cried out the great man, stumbling over a\nfallen soldier. \"What is this? O my destruction! The hand of Pontius\nthe Spearman is here! I perish!\"\n\nThen fell his head upon the deck at the stroke of Knud the Bear, and\nshortly all his companions went down in like manner, for they were\nastonished and they did no fighting.\n\nBeing in fear of death, the rowers rowed with great vigor and Tostig\nwas at the helm.\n\n\"She is swift!\" he exclaimed. \"She is a good keel and she rideth well\nthe waves. We are upon the sea! Hael to the Northland!\"\n\nLoudly shouted all the vikings, clashing their shields, for it was a\njoy to feel the lifting of the long billows and once more to wipe the\nsalt spray from their faces. They rapidly examined the ship from stem\nto stern, but there was much which required more thorough searching.\n\n\"O that the jarl were here with us!\" groaned Tostig. \"The day is here.\nWe, his friends, have escaped from the Romans and from the circus, but\nour jarl is on the land. It is evil! How shall I answer concerning him\nwhen I am inquired of at his own house? Will not some men say that I am\nnidering?\"\n\nLong leagues were rowed and then, for the wind was now right, the sail\nwas lifted.\n\n\"We will cast overboard these who were slain,\" said Knud. \"We will\nweight them, that they may sink. So shall none tell a tale of us to any\nwho may follow. We do as the jarl would have bidden.\"\n\n\"Thou art prudent,\" said Tostig. \"So much for the secret messengers of\nHerod. We have shed blood upon this ship and the gods of the North are\nwith us. Only let us with care avoid all triremes, for we do not need\nto be inquired of by a stronger force.\"\n\n\"This is now the spring,\" said Lars, the son of Beolf. \"If we pause\nnot needlessly, we shall soon reach the fiords, but there will be no\nice in them.\"\n\n\"It is a good cruise,\" said another. \"We may take much plunder by the\nway. Let us now search again the cabins of this galley.\"\n\nMuch that astonished them had been found with those who came on board\nin the boat that they might be slain. More was now discovered in secret\nplaces.\n\n\"Odin!\" exclaimed Knud, examining these matters. \"Here are many coins\nof silver and of gold and a number of bright stones. I think these may\nhave been gifts from Herod, the king, to C\u00e6sar at Rome, but he will not\nsoon see them.\"\n\n\"There are also fine weapons and garments,\" said Wulf. \"It is a very\nrich galley.\"\n\nThe sun was unclouded, the wind blew fairly from the east, the galley\nsped forward gallantly, and the rowers rested upon their seats, but now\nTostig the Red stood upon the after deck with all the Saxons around him.\n\n\"I have heard ye all,\" he said, for they had been speaking many things.\n\"We are of one mind. But if it be your will that I shall now take upon\nme the command of this keel, put ye your hands in mine and give me your\noath to this saying, that we will be satisfied with this great plunder\nwhich we have already taken; that we will keep the open sea, not\nlanding save for food or water; that we will care to take no other keel\nbut this; and that we will sail on until we see the house of Brander\nthe Brave upon the shore of the Northland. After that we will come back\nto the Middle Sea with many swords and we will seek for Ulric, our\njarl, until we find him.\"\n\n\"So will we do!\" shouted Knud the Bear.\n\nOne by one did the Saxons then step forward and put their hands between\nthe hands of Tostig, and the oath was an oath.\n\nNevertheless, they were as men who had sailed away forever, unless the\ngods should see fit to accomplish this their purpose of coming again\nwith a fleet, and with a host to follow Ulric the Jarl and his captain\ninto the great battle in Esdraelon.\n\nWell was it for them that they had thus escaped the sure perils of\nthe circus if they might also escape the many perils of the sea. They\nmight indeed avoid the triremes of Rome, and little cared hardy vikings\nfor rough weather, but the voyage would be long and they were not\nmany spearmen. The slave rowers, however, were sturdy fellows, well\nselected, and these were likely to be better contented with masters who\nflogged them not unduly and who thought it a shame that even a beast,\nbeing their own, should not be well fed and cared for.\n\nAs for Tostig the Red, he had become stern and moody, smiling not\nat all, and he told the rest of the vikings that Ulric, the son of\nBrander, was still his jarl, and that as Ulric, to his thinking, would\nhave directed in any case, so would he order.\n\n\"That will be well for us,\" they said.\n\n\"I have been troubled in my mind,\" he told them. \"I think that I may\nyet slay many Romans at the side of the son of Odin. I myself saw that\nJewish god of his that healed him of his hurts. I heard his words and\nthey were good to hear, although I understood him not very well. If he\nis to be the captain of the host, over the jarl, I am contented. But\nnever yet did I see a better sword than is our jarl.\"\n\n\"Nor did we,\" they answered him. \"We will surely return with thee to\nthe Middle Sea, and our treasure shall go with thine to the making of\nmany great keels and the gathering of the swordsmen of the North.\"\n\nAll things seemed going well with them, but there was, nevertheless, a\nshadow upon the ship, and when the sun was setting Tostig the Red sat\nupon the after deck sharpening his seax upon a stone and now and then\ngazing backward toward the east.\n\n\"Would I were with him this hour,\" he said in a low, sad voice. \"How\nshall the years go by with me henceforth if I am never again to see the\nface of my jarl?\"\n\n\n\n\n                            CHAPTER XXXVII.\n\n                        THE HOUSE OF BEN EZRA.\n\n\nIn the house of Ben Ezra, at the head of the flight of stone steps in\nthe secret passage, Ulric the Jarl stood looking down into a great\ndarkness. But now Ben Ezra came to him, having lighted a large brazen\nlamp which swung like a cresset at the end of a wooden rod a fathom\nlong. The flame of the lamp was very brilliant, but the smoke thereof\nwas unpleasant to the smell because of some strange oil which burned in\nit. Such a lamp might not be lighted at a feast or in the dwellings of\nmen.\n\n\"Follow me, O jarl,\" he said. \"This is the underworld and thou and I\nare alone in this place. But not all the swords of C\u00e6sar could find\nthee if thou wert hidden here. It hath been a refuge for some who fled\nfrom a destroyer.\"\n\n\"O Jew,\" said Ulric, \"I will cover this thy secret. May I fail of\nValhalla, dying as a cow dieth, if I betray thee!\"\n\n\"Come!\" said Ben Ezra, and they went on down the stairway together.\n\nAt the foot of it was a low chamber, the air of which was heavy, but\nBen Ezra turned to the left, and as he lifted his lamp there might be\nseen a narrow cleft in the masonry. A little inside of this cleft there\nwas a barrier of iron-bound woodwork.\n\n\"Lift it away by its hand pieces,\" said Ben Ezra. \"Thou art stronger\nthan I.\"\n\nVery massive was the wooden barrier, but it might be dragged forth and\nlaid upon the floor, and at once a current of cold, damp air poured\nthrough the opening, bringing with it a smell as of earth.\n\n\"I will go in first,\" said Ben Ezra. \"Now will I show thee my crypt.\"\n\nIn a moment more they were stooping over an open coffer, and he said:\n\n\"Here are my treasures and thine, and somewhat which belongeth to thy\nmen. I would they might have it, for we need not any goods but our own.\nThou shalt take away at thy will whatever is thine own.\"\n\n\"I may not remove it now,\" said Ulric, \"save a bag of golden coins. But\nI would ask of thee, if thou wilt tell me, what is this place that we\nare in and how is there such a cavern, with masonwork and corridors and\npillars and cunning doors? Are we to go on into it?\"\n\n\"Thou wilt go no further lest thou lose thyself as in a wilderness,\"\nsaid the Jew, pointing down the passageway.\n\n\"It is like a cave,\" said Ulric. \"I never heard of caves under a city.\"\n\n\"Behold,\" said Ben Ezra, \"the secret of Jerusalem. It is from the\nearliest time. There was a fort here in the days of Adam and here the\ngiants had their dwelling. There are no writings of those ancient\ndays. But on these hills and in these valleys city after city hath\nbeen builded and destroyed. For those walls and buildings much masonry\nwas needed. There were vast halls and hollows made in quarrying stone\nduring ages. Afterward these openings were sealed and made of the\nsecrets of priests and kings. They will not be opened until Messiah\ncometh.\"\n\n\"He is to be thy great king,\" said Ulric. \"What need hath he of caves?\"\n\n\"Not any,\" said Ben Ezra, \"but he will know in what hidden depth he\nshall find the treasures of Adam and of the giants and of the old kings\nand of Solomon, for all are yonder, where none but he may lay a hand\nupon them. Let us go.\"\n\n\"I have seen a wonder,\" said Ulric following his guide. \"But if this\ngod from Nazareth is to be thy king, wilt thou not thyself inform him\nof the way through thy house into his hidden places?\"\n\n\"He will have no need,\" said Ben Ezra; \"but if I saw that he had\nthe right to know, I would tell him. Messiah knoweth all things. As\nfor this rabbi of Galilee, he cometh to Jerusalem even now, for the\nPassover feast draweth near. I would gladly hear him again. During\nyears that are gone there have been many sayings concerning him.\"\n\n\"I know that he hath healed my hurts,\" said Ulric. \"He hath also done\nin like manner by many another. I think that I shall yet be a captain\nof men under him, and the great battle cometh.\"\n\nThey were now in the upper room and the stone door had been closed\nbehind them, swinging upon its pivots.\n\n\"Am I to abide here this night?\" asked Ulric. \"I have an errand of mine\nown in the morning. After that is done I must go to my men. They will\nsurely need counsel and ordering.\"\n\n\"I will now show thee thy chamber wherein thou art to sleep,\" said Ben\nEzra. \"But, I pray thee, do not too many errands within the city walls,\nand neglect not to visit Caius of Thessalonica lest thou lose thy\nstrong friend. It is needful for thee to be seen much at his house.\"\n\n\"I will truly care for him,\" said Ulric. \"It is my duty. But I have a\ngreat concern as to my companions. O that they were even now upon the\nsea and utterly escaped from the circus!\"\n\n\"Else they will surely all be slain,\" said Ben Ezra; but he led the way\nto a place for sleeping and the night closed over all.\n\nWhen the next morning came the watchmen upon the walls of Joppa\ntook note that the swift galley of Herod, the tetrarch, had already\ndeparted. So sent they in their due report, but already had it been\ndiscovered that whoever might now be in her had left behind them\nstrange tokens. In the highway north of the tower came a company of\nlegionaries to change the sentries, and at the turning of the road they\nfound but a dead man, slain by a spear thrust through his head. Who\ncould have done this deed in a day of peace they guessed not at all,\nbut their officer spoke of the Jackal of Mach\u00e6rus. Not long afterward a\nhorseman in bright armor rode along the beach seeing empty boats that\nwere cast up by the waves, and also the empty place where the evening\nbefore the galley had been anchored.\n\n\"I am too late!\" he shouted, angrily. \"The traitor hath escaped to\nRome! What answer shall I give to Herod Antipas? His brother hath again\noutwitted him and I think he is in league with the procurator.\"\n\nFurther up the beach men led along many horses, saddled and bridled,\nwhich they had found astray and ownerless, and this thing also was a\nriddle.\n\nThe governor of Joppa was quickly informed of all, that he might make\nhis report to his commander; but at that hour Pontius the Spearman\nwas sitting in the seat of judgment thinking not of Joppa, and before\nhim came not only his own officers, but Jews, also, and people from\nthe towns and the provinces. Suddenly, however, he turned from aught\nelse to look into the face of one who came in haste, seeming to be\ngreatly disturbed in mind. It was the master of the games who now stood\nopposite the chair of judgment, and at a sign of the procurator's hand\nhe spoke rapidly until he had told his errand, speaking low that none\nelse might hear.\n\n\"O thou,\" said Pontius, calmly, \"go back to thy affairs. I care not\ngreatly what Caius hath done with his gladiators. If indeed they have\nrebelled and if they are in Judea or Samaria, I will retake them for\nhim in season for the games. The fault is not thine.\"\n\n\"O most noble Pontius,\" said the master of the games, \"what sayest thou\nof the Greek? He came unto them, as I testified.\"\n\n\"What is that to thee?\" responded the procurator haughtily. \"Care thou\nfor thy beasts and thy cages. See that thou speak not at all to him of\nthis matter when thou seest him. Go thy way!\"\n\nThe master of the games trembled somewhat as he went forth and Pontius\nfollowed him not with his eyes, but muttered to himself:\n\n\"The Greek hath made good his token of a man from the circus. I will\nnow wait for a word from Joppa, but I will not question him imprudently\nwhen he returneth from C\u00e6sarea.\"\n\nHeavy matters were now coming before him, and among them all was\nnone which seemed to trouble him more than did certain testimonies\nconcerning the evil deeds of robbers from the wilderness of Judea.\n\n\"O ye Jews!\" exclaimed the angry magistrate. \"How shall I execute\njustice when so many of you are in league with these evil-doers? In\nthis city is the refuge of these wicked men. Who will capture for me\nthis Bar Abbas that I may crucify him? He hath kindred among you and ye\nshelter him!\"\n\nLoud were the indignant protests which replied to him from scribes\nand rabbis and rulers, but far back, near the entrance of the hall of\njudgment, were twain who listened eagerly.\n\n\"Father Abraham!\" hoarsely whispered one of these. \"I may not upon any\naccount bring my matter before him. Even now is my son at my house and\nhe brought much profit with him. He is worth more to me than are all\nthe robbers of Gilboa or the tribes beyond Jordan. He must not fall\ninto the hands of these Roman heathen lest I also be destroyed. The God\nof Israel be my protector against the enemy!\"\n\nSearching eyes were upon him, but the thought in the mind of Ulric, the\nson of Brander, was: \"Well for me that I fell in with him as I left the\nhouse of Ben Ezra. Well that I followed him even here. Now will I not\ncease until I know his abiding place.\"\n\nCowering and hiding his face, did Abbas hasten away, hardly daring to\nlook behind him. Many were coming and going, however, and the guilty\ndealer in stolen goods might not take warning of the manner in which he\nwas followed vengefully from street to street.\n\n\"I may not smite him now,\" said the pursuer, \"but he and another will\nshortly be touched by a sharp edge. There entereth he a door and I\nwill leave him for this time. Now must I see Caius and then I go to my\ncompanions. Would that they were on their way to the Northland. Woe to\nme if I bring harm to them!\"\n\nNevertheless, even before going to the house of Caius, the jarl did one\nthing which relieved both his heart and his hands from a heavy weight.\nHe went to the door of a house and he was admitted, but he tarried not\nthere, and he came out again, going on in haste, but with less gloom\nupon his face.\n\n\"She will think well of me!\" he exclaimed. \"I might not have speech\nwith her, but she will look upon my token and she will bless me.\"\n\nIn the house from which he had departed, and in an inner chamber of it,\nstood the Hebrew maiden, and before her was Isaac, the aged, her near\nkinsman. He placed upon a table a heavy bag and he essayed to speak,\nbut his lips trembled and his voice failed him.\n\n\"O Isaac, what is this?\" she exclaimed. \"Where didst thou obtain money,\nseeing the manner in which we are hindered? Hast thou indeed betrayed\nme again by thy weakness?\"\n\n\"Nay! Nay! It is thine!\" shouted Isaac. \"Thy heathen prince came to the\ndoor. I saw him, but he lingered not and none other had speech with\nhim. 'This is for her,' he said. 'Tell her I watch and I return, but\nthat she may not go forth, not even to the temple.' So I brought the\nbag to thee, wondering. Count it, for I have not counted.\"\n\nHe himself untied the bag and poured out the coins upon the table,\ncounting, while Miriam watched as one who seeth dimly in a dream of the\nnight.\n\n\"Ulric, Ulric,\" she muttered, \"thou art more to me than are these. I\nthink of thee that thou art pure gold, but who may weigh thee in the\nbalances? Come to me, for great is my need of thy counsel!\"\n\n\"Of Rome and of Greece and of Judea are these coins,\" said Isaac. \"They\nare thrice our present requirement. Jehovah hath turned to thee the\nheart of this idolater and thou doest well to make him serve thee.\nThou hast the understanding which is given to women. We will pay our\noppressors. We will give a goodly gift to the judge and to the chief\npriests. We will offer a sacrifice of burnt offering of a sweet savor.\nAnd God, even our God, will yet deliver thee also from the hand of this\nheathen gladiator.\"\n\n\"Isaac,\" she exclaimed, \"peace! Speak not of him unduly! Would that a\nfalse judgment concerning money were our only peril.\"\n\n\"O Miriam,\" said the old man, putting the coins in again and tying\nthe bag, \"that also hath been provided for. In this house we may not\nsafely remain, but a sure refuge hath been offered and we shall be for\na season as if we were hidden in a well. One cometh shortly to be our\nguide, and it is needful that thy heathen prince, also, should have\ninformation, for he hath more gold than this and his hand is now open.\"\n\n\"Peace!\" she again exclaimed, but Isaac went out with the bag, saying:\n\n\"Great are the gifts of Jehovah of Hosts! Would that he might now send\nthe sword of this Philistine who loveth her upon the necks of our\nenemies!\"\n\n\"I will wait,\" she was whispering, \"until I see him.\"\n\nLong was the remaining of Ulric the Jarl at the house of Caius of\nThessalonica, but afterward he went out at the Damascus gate purposing\nto visit the amphitheater. He went on down into the valley of\nJehoshaphat, walking slowly, and he came to the bridge over the brook\nKidron, by which he was to pass.\n\n\"O jarl,\" said a youth who waited at the bridge, \"a token from Ben\nEzra!\"\n\n\"None heareth,\" replied Ulric. \"Say on.\"\n\n\"Thy men are not at the circus, he bade me tell thee, and no man\nknoweth whither they are gone. Go thou not thither now, but let the\nhouse of Caius be thy refuge, for there will be an inquiry for thee.\"\n\nStill as a stone stood the jarl while one might breathe three times.\n\n\"I thank thee and him, O youth,\" he said then. \"Go thou to him with his\nword only, that neither he nor I need any to tell us whereunto the sons\nof the Northland have departed. I will do as he hath said.\"\n\nThe youth went from him running, but Ulric did not reenter the city by\nthe Damascus gate.\n\n\"It will be safer to choose another,\" he said. \"I was seen by the\nguards when I came forth. They may by this time have some evil\ncommandment concerning me.\"\n\nSo therefore he made a great circuit of the walls, going far, and even\nafter he selected a gate by which he might prudently go in he seemed\nto have another matter upon his hands. The hours went by, one after\nanother, and it was long after the sunset before he was known to be in\nthe house of Caius. Then speedily he was sent for and he went in to\nwhat was now the sick chamber of the centurion.\n\n\"O jarl,\" said he, \"how is it with thee?\"\n\n\"O most noble Caius,\" replied the jarl, \"I am well, but I am alone in\nJerusalem. All of my companions have returned to their own land.\"\n\n\"Well for thee,\" said Caius. \"Of that I had been informed. A swift\nmessenger from the governor of Joppa brought strange news to the\nprocurator. What sayest thou if thy men have been hired to serve upon a\nship by Herod, the tetrarch? Would they not guard well?\"\n\n\"O Caius,\" said the jarl, \"thou knowest them and I need say no more,\nfor I am ignorant of all this matter save that they are gone.\"\n\n\"I find no fault with thee,\" said Caius. \"The messenger was sent to me\nand I have fully questioned him. Also word came from the procurator\nthat I trouble thee not, for Herod must be allowed to direct his own\naffairs. If he have hired good swordsmen, surely his galley is in safe\nkeeping.\"\n\nUlric looked at him darkly, for the voice of Caius was as of one who\nmocketh bitterly.\n\n\"O Caius,\" said the jarl, \"if thou wilt hear me, I have another affair\nupon my mind. I like not the appearance of thy sore.\"\n\n\"Jarl of the Saxons,\" exclaimed the centurion, \"I seem to myself to be\nrotting away. I am as one who hath the leprosy. But what knowest thou\nof any healing?\"\n\n\"Only this that I have heard this day,\" said Ulric. \"I would have thee\nlive until the arrival of this Jesus of Nazareth. He cometh now to the\nfeast of the Jews. He is of the sons of the gods. Did he not heal me?\nAnd may he not also do something for thee?\"\n\n\"O that he might come quickly!\" said Caius. \"But the gods can do little\nfor such a torment as mine. There are many things which are too much\nfor them. But I will see him when he cometh. I would make him rich with\ngifts if he would heal me.\"\n\n\"I will watch for him,\" said the jarl. \"I may not go again to the\ncircus----\"\n\n\"Go not!\" exclaimed Caius. \"Remain much with me. O Saxon, when this\nfire burneth within me I would gladly fall upon my sword but that it\nwould please my enemies. But if thou goest out now, return quickly. Of\nthis be thou sure in thy mind, that I will not permit thee to enter the\ncircus. Thy sword will have better business. I will speak of thee again\nto the procurator. A messenger from him hath arrived. Leave me with\nhim.\"\n\nMore words might not be spoken and the jarl went out, but it could be\nunderstood that with difficulty did the centurion restrain himself and\nconceal from all the extremity of his suffering from that deadly thing.\n\n\"I will go to the house of Ben Ezra,\" thought Ulric. \"Already have I\nmade sure that there are fewer enemies to bring peril upon Miriam and\nher people. I will see if the Jew hath well attended to his portion of\nher business. But unless help cometh to him speedily, Caius will surely\ndie.\"\n\n\n\n\n                           CHAPTER XXXVIII.\n\n                           THE SON OF ABBAS.\n\n\nLong and thoughtfully and with many questionings did Ben Ezra listen\nto the jarl in an inner chamber of his house. \"Thou hast done well,\"\nhe said, at last, \"but trust thou not too much the favor of the great.\nNeither be thou too sure concerning their power. The leaves fall from\nall the trees in due season. Full of jealousy and of suspicion and of\nmurder are all they who prosper under C\u00e6sar. In the day and in the\nnight is there a weapon not far from any of them. So deal they with\nothers. A Roman friend is ever also a Roman enemy holding a knife, and\nby the hands of their friends do men die continually at Rome.\"\n\n\"That do I believe,\" said Ulric. \"I will be exceedingly prudent. But, O\nmy friend, what hast thou done concerning Miriam?\"\n\n\"I have done all thus far,\" said Ben Ezra. \"I did but need to buy the\ngood-will of the judge and one peril passed away. To him and to another\nI could both pay more and promise more than was in the will of Abbas.\nBut thou, O jarl, hast thou seen the face of this Roman dealer in\nslaves?\"\n\n\"That peril also hath departed,\" said Ulric. \"I am told that a man in\nhaste met him in the valley of Hinnom. The patrol found him there in\ntheir passing, and his head lay at six cubits' length from his body.\"\n\n\"He had many enemies,\" said Ben Ezra, thoughtfully. \"One may even have\nfollowed him from the city. I have now another word for thee concerning\nAbbas, now that thou hast heard the procurator upon the judgment seat\ngive authority to all such as thou art concerning the taking of his\nson. The robbers are a power in Jerusalem and the sword of Bar Abbas is\nagainst thee.\"\n\n\"I have never seen him,\" said Ulric.\n\n\"Then will I tell thee of his face,\" said Ben Ezra, and he minutely did\nso, line by line.\n\n\"Fear not,\" said Ulric. \"I would now know that man if but half his face\nwere shown.\"\n\n\"He is said to be cunning in disguises,\" replied Ben Ezra, \"but his\nbest keeper is the fear of men that by denouncing him they may bring\nupon themselves secret daggers and a vengeance which faileth not.\"\n\n\"No fear have I,\" muttered Ulric, \"but how am I to find him!\"\n\n\"I trust that thou wilt be prospered in that matter,\" said Ben Ezra,\n\"but I have this much more for thee. It behooved me to bring both\nMiriam and Isaac to this house, that I might cover them until this\nperil pass. Roman eyes that thou knowest not may have looked upon her.\nWe must wait. The slaying of one slave dealer may but make needful the\nhiring of another by some great one. The house in which Isaac dwelt was\nbut hired, and another taketh it, that he and his may be thrust out.\nThe net is a wide one.\"\n\n\"Evil! Evil! Evil!\" exclaimed Ulric. \"This city is full of injustice!\nNo such thing could be among the North peoples. I saw no thief ever,\nnor a purchased jarl or judge, until I came southward.\"\n\n\"So doth the whole land groan,\" said Ben Ezra. \"So doth the blood of\nthe innocent cry out unto Jehovah.\"\n\n\"Why, then, answereth he not?\" shouted Ulric, vehemently. \"Surely the\ngod of a people should come to their help in such a distress, else they\nwill surely say of him that he is no god at all.\"\n\n\"So have said many,\" groaned Ben Ezra, \"and I grow weary in heart\nwaiting for a Messiah who doth not come. O that our King were already\nhere. Peace, now, O jarl, concerning him. But I will tell thee of\nMiriam that thou mayest not have speech with her this night. Be thou\nnot also her enemy, to do her harm.\"\n\n\"Where is indeed thy god,\" said Ulric, \"if any hurt may come to such as\nshe is?\"\n\n\"O jarl,\" said Ben Ezra, \"all Jerusalem hath heretofore been heaped\nwith the slain, and the maidens of Zion were led away captive, because\nof the anger of Jehovah. Dost thou not understand? We do suffer for our\nsins and for the sins of our fathers.\"\n\n\"I think the gods do not well in such matters,\" said Ulric. \"They are\nnot just. Surely justice becometh well a brave god. He should not\nstrike down the innocent ones with those who are guilty of evil.\"\n\n\"I know not the counsel of the Most High,\" said Ben Ezra. \"His\njudgments are a great deep, but they are just and righteous altogether.\"\n\n\"No man,\" said the jarl, \"findeth fault with a stroke of a sword\nfairly given, since he who dieth in battle goeth to Valhalla and hath\nattained his inheritance from his brave ancestors. I myself wait for\nthe valkyrias, and I am often weary thinking of the gods and of Asgard.\nWho would avoid a sword if it were in the hand of a brave warrior in\nbattle? Not I, Ulric, the son of Brander.\"\n\n\"Thou art a mighty man of valor,\" said Ben Ezra. \"I have thought of\nthee that thou art almost as a son of Abraham. Go thou to thy sleeping,\nfor this house must be even more thy abiding place than is the house of\nCaius now thy companions are departed.\"\n\nSleep came as to one who is weary both in mind and body, but early upon\nthe morrow the two friends were together again taking counsel.\n\n\"O Jew,\" said the jarl, \"I am ill at ease concerning my men. Would that\nI might see them this day and make sure of their welfare.\"\n\n\"So often doth one think of those who are departed from him,\" said Ben\nEzra; \"but have thou a care that thou inquire not imprudently. All that\nI may learn I will tell thee when thou comest again. It is well for\nthee to go now.\"\n\nOut walked the jarl, going along a corridor which led toward the door\ninto the street.\n\n\"Very wonderful is all this,\" he thought. \"A strange place is this city\nof Jerusalem, with its many rulers and its secrets of the gods and of\nthe old time, and with these things which are done here. Of what good\nis it that it hath so great a temple and so many priests?\"\n\nAt that moment there came to his ears a beautiful, low music murmuring\nthrough the cool air of the corridor.\n\n\"Ulric, art thou here?\"\n\n\"Miriam!\" he exclaimed, turning to listen.\n\n\"O, I thank thee that thou hast come,\" she said. \"I have had such fear\nupon me! Much rather would I die. One moment I must see thee and speak\nto thee! Tarry a moment!\"\n\n\"More I may not do, O Miriam,\" he said, with a great light rising in\nhis eyes. \"But I have given my promise to Ben Ezra and to thy God\nconcerning thee that no harm shall come upon thee. I will but look upon\nthy face.\"\n\n\"Thou art wonderful!\" she said, and then they saw not aught of all the\nworld except each other for that breathing space.\n\n\"O thou,\" she whispered, \"I know not if thou art of the heathen or if I\nam of Abraham's seed. O what but death should part me and thee!\"\n\n\"I think not even death,\" he said, \"seeing that we go to one place\nafter the sword cometh. But if indeed thy Jehovah be a god and if he\nhave given me to thee, I will offer to him many sacrifices. I must go\nforth now, for I have many things to do for thee and for a friend. If\nthis Jesus of Nazareth arriveth, I must have speech with him. I have\ntold thee how he healed me.\"\n\n\"So must I see him,\" she said. \"I listened to him many times in\nGalilee. He is a very learned rabbi and I would hear him again.\"\n\n\"He is more than a rabbi,\" said Ulric. \"He is a god--and he knoweth\nthe other gods. I would ask him concerning Asgard and Valhalla and\nconcerning thee. Thy slave dealer is dead, O Miriam, and soon will I\ndeal justly with this Abbas and those who are with him.\"\n\n\"Thou art my warrior!\" she exclaimed. \"Thou art as one of the heroes of\nIsrael. I trust thee!\"\n\n\"Farewell!\" he said stepping quickly away from her, but no word escaped\nher lips. She did but seem to hold back her hand from its purpose of\ndetaining him, and her breath came and went rapidly as he passed out\nat the door. Then her voice came again and she said, looking upward:\n\n\"O thou Jehovah of Hosts, my God, hast thou not made him also? How am I\nbetter than he that I should be withheld from him? Do I not love him?\"\n\nThe feet of Ulric went but slowly from the house of Ben Ezra, and he\npaid little heed to their guiding, but they brought him to the house of\nCaius of Thessalonica and the warder of the portal stood before him.\n\n\"I have a word for thee, O swordsman,\" he said. \"Thou art well arrived.\"\n\n\"Say on,\" said Ulric.\n\n\"Even now, O Saxon, the procurator himself is with our master Caius\nand with them is the governor of Joppa. Thou mayest not go in. But\npass thou to the Damascus gate, for Caius would know of the arrival of\nthis Galilean healer. It is reported that he is near at hand. Also the\nprocurator would have speech with thee when Caius will send thee to\nhim. These are thy commands and thou wilt do them.\"\n\n\"Say thou, O warder,\" replied Ulric, \"that I go as I am bidden. Well\nfor thee and for us all if the centurion were cured of this evil. Thou\nwilt fall into no better hands than his if he dieth. I am his friend,\nand he suffereth.\"\n\n\"Go! Go!\" said the warder, earnestly. \"All the gods forbid that he\nshould die. If this same Jew rabbi will not heal Caius, it is thy duty\nto slay him with thy sword.\"\n\n\"Speak thou only good concerning him,\" said Ulric, sternly. \"What hast\nthou to do with a sword? I go.\"\n\nThe warder stepped backward a pace, for not many men willed to stand\nbefore the jarl when his hand seemed to be seeking for the hilt of his\nseax.\n\n\"A sudden man!\" he muttered, as he watched his going. \"And they say he\nhath cleft both a lion and a tiger in one combat and that he would wear\nno armor. A man's head might go from his shoulders as if it were but a\nflower from a stem. His eye is a fire!\"\n\n\"I would I better knew the streets,\" said the jarl, as he strode\nswiftly onward. \"I learn them but slowly, for they are very many and\nthey are crooked and the city is great. Whither, now, shall this one\nlead me?\"\n\nAs in an unknown path, therefore, he went on, thinking of many things.\nThe way led him over a hill and through a valley and to a gate in the\nouter wall that he knew well. Here were Roman guards standing at rest,\nhindering none, and Ulric halted near them. \"Many go out,\" he thought,\n\"but a multitude cometh yonder along the road across the Mount of\nOlives. I will wait and see.\"\n\nNearer and nearer along the broad highway poured a vast throng of\npeople, while through the gate passed on a tide which went to mingle\nwith them. Many of those who were coming bore in their hands branches\nof palm trees and they were shouting joyously.\n\n\"What is this which they sing?\" said Ulric. \"What is the meaning of\n'Hosanna in the highest,' and who is David, and what is his son? It is\na saga of the Jews. The whole city is stirred up behind me. This is a\nwonder!\"\n\nAcross the valley and then up toward the gate came on the multitude.\nAmong them were some who took off their outer robes and cast them upon\nthe road before an ass and before his rider, shouting and singing.\n\n\"I have heard them say 'a king,'\" said Ulric. \"But here is no king.\nNone of these men are armed. What saith the procurator to this\nbusiness?\"\n\n\"O gladiator of Caius of Thessalonica,\" suddenly responded a legionary\nof the guard of the gate, \"thou art but a sword. What careth Pontius\nthe Spearman for a mob of women and children? We know thee that thou\nart accounted trustworthy, and thou doest well to inquire concerning\nany tumult of the Jews, but this is no affair of either thine or ours.\"\n\n\"I meddle not,\" said the jarl. \"I am under orders from the centurion\nand from the procurator, but I may watch this matter.\"\n\n\"Watch,\" said the officer. \"Thou art in thy duty. We hinder thee not.\nBut who art thou?\"\n\nThe man whom he now addressed was plainly a Jew, in sordid raiment,\ntall and strong, but who was eyeing the jarl with an evil eye, and his\nmanner was insolent.\n\n\"I am a servant of the high priest and I am here by his command,\" said\nthe Jew. \"There is an order for the arrest of this gladiator.\"\n\n\"Let no accursed Jew take upon him that business,\" laughed the officer.\n\"Thy high priest hath enough to settle with the procurator. But whither\ngoest thou from hence?\"\n\n\"I go to the gate of the valley of Hinnom,\" replied the Jew, \"and thou\nmayest not detain me.\"\n\n\"O officer,\" said Ulric, who had been searching the Jew with keen\ninspection, \"I have an errand to that gate and know not the way\noverwell. I pray thee that thou command him to guide me after I have\nseen this present matter.\"\n\n\"I object not to that,\" said the Jew, with a fierce glitter in his\neyes, \"so that he touch me not to render me unclean against the\nPassover feast.\"\n\n\"Curse thy uncleanness!\" said the Roman, haughtily. \"Thou needest not\nto touch him; but I would he might have a commandment to touch thee.\nO gladiator, I am told that thou art a sure blade, the slayer of the\ngreat Numidian lion. I hope to see thee slay another yet in the circus,\nbut take not the head from this worthless one until thou art duly\nbidden to smite him.\"\n\n\"As thou doest so do I,\" said Ulric. \"Shall a soldier question his\ncaptain?\"\n\n\"Not if that commander be one Pontius the Spearman,\" replied the\nofficer, \"or even Caius of Thessalonica. Thou art right, O gladiator.\nNone will interfere with thee and thy sharp edge.\"\n\n\"Stand by, thou,\" said Ulric to the Jew. \"I will be with thee\npresently.\"\n\nBut now the man became seemingly cringing and friendly in his\ndeportment, bowing low and standing in silence awaiting direction.\n\nNearer and nearer came the multitude along the highway and toward the\ngate. Ulric heard many of them shouting:\n\n\"This is Jesus of Nazareth, the Prophet of Galilee! He is the son of\nDavid! He is the King of Israel. He is the one who was to come! This is\nthe Messiah, the deliverer!\"\n\nOthers there were who loudly gainsaid these acclamations, protesting\nindignantly; but Ulric's thoughts were full of astonishment.\n\n\"I see that the man upon the ass is Jesus. I know that he is of the\nsons of the gods. This is a wonderful affair. But why cometh he now\nwithout an army into a fortified city which hath a Roman garrison?\nOdin! There is no prudence in this coming! They will slay him before he\nhath opportunity to gather men for one good legion.\"\n\nSo pondering in his mind, he watched until the ass and his rider passed\nby him through the gate and into the city.\n\n\"I have again seen his face,\" he said. \"I may not have speech with him\nat this time. But I will take upon me this other errand and see unto\nwhat it will lead me. O thou, my guide, we will depart.\"\n\n\"Come, O gladiator,\" said the Jew. \"It is well for thee to have me with\nthee among so many crooked streets. Touch me not. But what are thy\ncommandments?\"\n\n\"Hold thou thy peace concerning them,\" replied the jarl. \"Lead on!\"\n\nHot wrath burned for a moment in the face of the tall Jew, but he\nobeyed, girding himself and striding forward, but the officer of the\ngate laughed derisively.\n\n\"The dog Jew,\" he said, \"will do well not to stir the temper of a\nSaxon. His head were loosened from his shoulders too quickly. I will\nnot fail to see that fellow in the circus. It mattereth not to me what\nwork there may be for his blade in Hinnom.\"\n\n\"Dost thou not bear in mind,\" said one of the legionaries, \"a certain\nslave dealer and the loosening of his head? This same gladiator was\nseen that day at the Hinnom gate, but the guards were bidden to forget\nhim.\"\n\n\"Thou thyself rememberest too much,\" said the officer, sternly. \"Forget\nthat thou hast seen him here this day. But it is good sport to slay\nJews. I would there might soon be another tumult. I have made the\nfloor of the temple red with the blood of Galileans. The procurator may\nhave a sharp teaching for more of them during this Passover gathering.\"\n\nSo talked the soldiers of Rome, but the jarl was silent and moody as he\nwalked until he and his guide were drawing near to the southerly wall\nof the city. Then he spoke quietly, as a man may speak to his friend,\none whom he hath known well aforetime.\n\n\"O thou,\" he said, \"when hast thou seen thy father, Abbas, and what did\nhe give unto thee concerning me?\"\n\nThe guide turned suddenly, scowling and trembling, but he responded:\n\n\"How knowest thou me? The guards knew me not, nor did any other, for I\nam changed for that reason. What hast thou to do with Abbas, and what\nis thy purpose?\"\n\n\"Thou art but a fool,\" said Ulric. \"I read thy name in thy face. Thou\nart Bar Abbas. I have known thy father many days. Did he not tell thee\nhow I rescued him in the tower in Esdraelon that he died not? But I\nhave thought him a prudent man. How is it that he hath permitted this\nfolly?\"\n\n\"O gladiator,\" said Bar Abbas with a deep, dark smile, \"it is no folly.\nThey who would slay me seek me in the wilderness, not in Jerusalem. A\nman who waiteth within the gate among the legionaries is hidden from\nthe hunters among the hills. I have seen my father and now I go to meet\nhim and his friend the master of the games in the valley of Hinnom, as\nI believe thou, also, art informed beforehand.\"\n\n\"Then thou hast delivered to him thy spoils?\" said Ulric. \"But canst\nthou give me a reason why I should go to meet him in Hinnom?\"\n\n\"That I know not,\" said Bar Abbas. \"But the master of the games is thy\nmaster also. He will give thee thy direction.\"\n\n\"Nevertheless, thou and he are unwise,\" said the jarl. \"I would thou\nwert armed.\"\n\n\"Save my dagger, I am not,\" replied the robber, \"and thou hast no\nweapon.\"\n\n\"A warrior is always armed,\" said Ulric. \"But now we are at the gate\nand here is the officer. Be thou silent.\"\n\n\"Whither goest thou, O gladiator?\" demanded the sentinel. \"This is\nforbidden thee. Thou art too far from the circus.\"\n\n\"Dost thou indeed not know me?\" responded the jarl. \"Or knowest thou\nnot this signet of Caius of Thessalonica?\"\n\n\"I do know thee, who thou art,\" said the officer, \"and I know the\nsignet.\"\n\n\"By commandment of Pontius the Spearman, the procurator,\" said Ulric.\n\n\"I hear,\" said the Roman.\n\n\"Bind thou this Bar Abbas, the robber, and take him to the prison\nand report to the procurator that I have done as he did give me\ninstruction. This thing is upon thy life!\"\n\nForward sprang Bar Abbas dagger in hand, but the strong blow of a\nsoldier smote him to the earth and he was bound with cords.\n\n\"O man,\" shouted the captain of the gate, coming hastily, \"do as he\nbiddeth thee. We also have full commands concerning Bar Abbas. Well do\nI know that this is of the procurator.\"\n\nThen turned he to the jarl.\n\n\"Thou hast more to do, O gladiator?\"\n\n\"I may not answer thee,\" said Ulric, respectfully. \"But now do I go out\ninto the valley to meet one who cometh, and my duty is in my hand. I\nwill return unto thee shortly.\"\n\n\"Thou hast no weapon,\" said the Roman; but upon his face was a look of\nunderstanding, for he was a man of experience and he had been scanning\ncarefully the raiment of Ulric. \"What if an evil person were to meet\nthee?\"\n\n\"O captain,\" said Ulric, \"he who obeyeth a command doeth well. But if I\nreturn not with due speed know thou that I am slain, and inquire into\nthat business.\"\n\n\"That will I do,\" said the officer; \"but they who slay thee may indeed\nneed an inquiry. I think it will not be entirely well with them.\"\n\nThe jarl answered not a word, but he had now upon his mind the things\nthat had been told to him by Bar Abbas on their way, and he went down\ninto the valley, walking rapidly.\n\n\"Before me is a trap,\" he said, \"although it was not set for me, but\nfor some other. I will now fall into it, and glad am I that I am so\nwell prepared. This heavy, sharp-edged gladius is better than my light\nseax.\"\n\nEven then the captain of the gate was replying to a question from the\nquaternion.\n\n\"The gladiator unarmed?\" he exclaimed. \"Do I not know how a sheath will\ncause a wrinkle of a robe to enlarge and stiffen? They who sent him are\nresponsible, not thou or I.\"\n\nThe jarl went on a mile, it might be, and around him was the smoke of\nthe everlasting burning of Hinnom and the smell from the untellable\npollution. Here and there, also, he saw heaps of half-consumed offal in\nwhich many worms were crawling. This place was to all Jews the picture\nand symbol of the punishment of the wicked after death. Not many\nwayfarers were at any time to be encountered here, for all men knew\nthat it was a favorite haunt of evil spirits, of demons, and of robbers.\n\nNevertheless, as the jarl looked forward through the unpleasant\nclouding of smoke he exclaimed, aloud:\n\n\"They come! Yonder is Abbas himself, and with him are four men. They\nride horses. I will wait until they dismount, but woe to me if so much\nas one of them shall escape.\"\n\nHe stood still, making sure of the hilt of his weapon, but the horsemen\ncame near and at once sprang to the ground, coming forward.\n\n\"Knowest thou me?\" said the foremost man. \"Thy fellows have escaped me,\nbut thou shalt not. I will feed thee to the wild beasts!\"\n\n\"O master of the games,\" replied the jarl, \"I am of the household of\nCaius.\"\n\n\"And I am from Pontius the Spearman!\" shouted the master of the games.\n\"O ye of Ethiopia, bind him fast!\"\n\nThe three with him were black slaves, armed with shields and short\nswords and jereeds, but they were naked to their waists.\n\n\"Yield thee, O Saxon,\" cried out Abbas, mockingly. \"I have thy Miriam\nsecurely and she will soon belong to my friend.\"\n\nNow the master of the games was in full armor, but he had turned a\nmoment for the ordering of his slaves. He stooped a little, also, to\nloosen a coil of cordage that was in his hand for the binding of the\njarl, supposing him to be unarmed and helpless against five armed men.\n\nThen swiftly flashed the bright gladius in the hand of Ulric and the\nhead of the master of the games fell to the earth.\n\n\"Thou hast sold Miriam?\" heard Abbas, a hoarse whisper, but he heard no\nmore, for the sword had flashed again.\n\nThe light shaft of a jereed snapped as its blade struck upon the hidden\nmail of the jarl and the black striker fell across the body of Abbas.\nHis next companion was as a defenseless man before the angry might of\nUlric, and hardly was he down before the slave to whom had been given\nthe holding of the horses lay among their hoofs.\n\n\"Sure am I that Abbas is dead,\" said Ulric, stooping over him. \"Not one\nof the others liveth. The horses must now go back at speed. I would not\nhave them seen from the gate.\"\n\nHe pricked them sharply so that they ran in fear. Then he wiped the\ngladius clean, concealed it well, and walked back to the Hinnom gate of\nJerusalem.\n\n\"Hast thou accomplished thy command?\" asked the officer. \"The time hath\nbeen but brief.\"\n\n\"Else were I not here,\" said Ulric. \"There were those who came by\nappointment and one of them was the father of Bar Abbas. The others\nwere but robbers like him.\"\n\n\"O gladiator,\" said the officer, \"so will I report well of thee. I\nthink thou art a sure messenger for an errand of blood.\"\n\nQuestions might not be pressed in such a case, but soldiers were at\nonce sent down the valley to make due inquest.\n\nOnward went Ulric through the streets of the city.\n\n\"O Miriam!\" he groaned. \"Would that I might live for thee! But for this\nday's deed I think that I may soon die. I will now go to the house of\nBen Ezra and I will tell him what hath thus been accomplished for him\nand for her.\"\n\nEven as he went his haste was hindered in a narrow street by a great\nprocession which seemed to be one of rejoicing. Maidens came first,\nwith clashing of cymbals and with singing. Behind them were other\nmusicians not a few and many men and women. Then walked lightly on a\nveiled one in bright robes that were adorned with jewels. Attending her\nand following joyfully was the remainder of the procession.\n\n\"Wilt thou inform me what this may be?\" asked Ulric of one who stepped\napart from the others for a moment.\n\n\"O gladiator,\" replied the Jew, \"this is the wedding of my kinsman,\nAnanias, the son of the money changer of the temple. He marrieth the\nGreek proselyte, Sapphira the beautiful, the freed woman and favorite\nof the wife of the procurator. She hath become a daughter of Abraham.\nShe now goeth to the house of the bridegroom to meet her husband. There\nalso is to be the wedding feast.\"\n\n\"I thank thee,\" said Ulric, but he walked on muttering doubtfully:\n\"Sapphira? Of the household of Pontius the Spearman? I remember well.\nThat was the name of the beloved of Lysias.\"\n\n\n\n\n                            CHAPTER XXXIX.\n\n                          THE PASSOVER FEAST.\n\n\n\"O jarl,\" exclaimed Ben Ezra as they stood together in the house,\n\"would that thou also wert a son of Abraham! But thou hast done a deed\nfor which thou wilt be held to answer. O mighty man of valor, I fear\nthat thy life is forfeited to the law.\"\n\n\"Thinkest thou, O my friend,\" replied Ulric, \"that there is now any\nmore peril to Miriam?\"\n\n\"Not any,\" said Ben Ezra, \"save that all dwellers upon the earth are\never in peril from the evil. Every payment hath been made. Her enemies\nare slain with the sword. She may dwell in peace for a season. But if\nharm cometh to thee----\"\n\n\"Then,\" interrupted the jarl, \"thou knowest that whatever of mine thou\nhast in thy keeping belongeth to Miriam. See thou to that!\"\n\n\"Before Jehovah!\" said Ben Ezra, \"that will I do. The jewels and the\ngold are hers. But what doest thou now, seeing that the officer of the\nHinnom gate will report thee?\"\n\n\"I sleep this night,\" said Ulric. \"After that I purpose going to the\ntemple to hear the words of this son of the gods from Nazareth. I will\nspeak to him concerning Caius. As for this affair of the valley of\nHinnom, it is no secret, and I may not hide myself.\"\n\n\"I also will hear the rabbi from Galilee,\" said Ben Ezra. \"Yesterday\nhe did boldly cleanse the temple of such as were there contrary to the\nlaw.\"\n\nThe jarl listened in silence while the Jew told him many things\nrapidly, but then he said:\n\n\"He is brave. I would I had been with him. I will ask him if he needeth\nnow a good sword. I will do as he shall command me.\"\n\nBut now a servant of Isaac came to summon Ben Ezra, and Ulric was alone.\n\n\"Would that I might see Miriam!\" he murmured, slowly, and a delight\nspoke laughingly in the soft tone of his voice.\n\n\"Ulric, thy Miriam is here! Art thou in any peril? Wilt thou not save\nthyself?\"\n\nShe stood at his side touching him, and his strong arms opened and he\nuttered a great cry, for she glided into them and they were closed\naround her.\n\nWho shall hear or tell the words that are uttered at such a time,\nseeing that they are a thousandfold more than words? He who would\nstrive to repeat them is a foolish one, as if he would echo the\nfar-away music of a song in the night.\n\n\"Thou art safe!\" he said at last. \"That is enough for me. Trouble not\nthy heart overmuch. Only the gods may see that which cometh to us on\nthe morrow. Go thou to thy chamber and thank thy God for me, telling\nhim that I will offer him a great offering and that henceforth he shall\nbe my God also for this thing which he hath done for thee and me.\"\n\nSo she departed as one who must, but who willeth not to go, and the\nnight hours came upon all the city of Jerusalem.\n\nNow at an earlier hour of that day there had been standing in the\nprivate room of Pontius the Spearman a tall and stately matron attired\nin costly garments, and before her stood a youth whose face was full of\ngreat agony.\n\n\"Be thou silent!\" she commanded. \"This was my doing. Questionest thou\nme? What is my freed woman unto such as thou art? Thou hast naught\nto do with Sapphira! Speak not of the matter to the procurator! I do\ncounsel thee well. Thou art but a youth, O Lysias, and in youth there\nis folly!\"\n\nLow bent his head and his bosom heaved with pain, but he was silent.\nThe face of the matron was noble in shape, and not unkindly, but in it\nwas great haughtiness, for the wife of the procurator was as a queen\nand no man might question her will. She looked now at the young Greek,\npitying him for a moment, and then she went from the room, saying no\nmore, for the matter was ended, and he yet stood there alone.\n\n\"All the gods have forgotten me,\" murmured Lysias. \"I will but make my\nreport to the procurator and I will depart--I care not whither.\"\n\nEven as he spoke the ruler of Judea entered the room, striding as if in\nhaste.\n\n\"Thou art here?\" he said, and his face was red, as if in hot anger.\n\"Speak on, O Greek! Tell me of all thy doings, from the first to the\nlast, beginning with Cornelius at C\u00e6sarea.\"\n\n\"O most noble Pontius,\" said Lysias, \"from the centurion, this\nparchment, sealed. He gave me no words to utter.\"\n\n\"I will read,\" said Pontius, but the epistle may have been not only\nbrief but troublesome, for his face darkened yet more angrily.\n\n\"Speak on!\" he commanded, and his messenger told all, to the place\nwhere he had parted from Tostig and the Saxons upon the shore of the\nharbor of Joppa.\n\n\"More than this is already known to me,\" said Pontius. \"Hast thou\nspoken at all of this matter?\"\n\n\"Not to any ear but thine, my lord,\" said Lysias. \"I have been utterly\nprudent. Even the master of the games cannot know concerning thy\ndealing with the secret messenger of Herod.\"\n\n\"Thou knowest?\" almost gasped the procurator. \"Very great is thy\nknowledge. Thou hast done well in this affair. I will give thee now\nanother errand. Call unto me the sentinel in the outer corridor.\"\n\nQuickly Lysias went and returned, bringing with him one of the trusted\nlegionaries of the palace guard who had been on duty.\n\n\"Take thou this youth,\" said Pontius, \"and lead him to the fifth\nchamber of the lower corridor. Summon thou to that room one whom thou\nknowest. Say to him that I will see him again without delay. Then\nreturn thou to thy post.\"\n\n\"Follow!\" said the soldier to Lysias. \"I am bidden to show thee a\ncertain matter.\"\n\nLysias obeyed, but with a faintness coming coldly upon him, but as he\nwent there was a sad thought weighing upon his heart.\n\n\"O that I might but see her! Did she indeed wed him of her own free\nwill? My beloved! O my Sapphira! O my beautiful one! I found thee but\nto lose thee!\"\n\nThere was a stairway, and at the bottom of that there was a long\npassage. It was gloomy and dingy as of a prison, with closed cages\non either side. Here, also, one shortly came and walked with them, a\nshort, broad man in armor, who spoke not.\n\nLysias himself counted the doors.\n\n\"The fifth,\" he said. \"It is open.\"\n\n\"Enter!\" commanded the soldier, but he followed not, and the short,\ndark man went in behind Lysias.\n\nThe door closed clanging, and then there was a silence save for the\nfeet of the departing legionary and a sharp sound of a cry from that\nfifth chamber. A minute passed and then another, and the short, dark\nman came out alone.\n\n\"The Greek,\" he said, \"hath accomplished the errand upon which the\nprocurator sent him. But there is blood upon my hand, and I will wash\nwell before I report to the Spearman lest he inquire of me.\"\n\nAt that hour there was joyous feasting at the house of the father of\nthe Jew Ananias. The bridegroom welcomed his kindred and his friends\nand the red wine was plentiful. In the apartment of the women sat the\nbride arrayed in her jeweled robes. All the women who looked upon her\npraised her, wondering at her great beauty. They said that Ananias had\nwon the pearl, the pearl of pearls, the ruby of rubies, the rose of\nSharon and the lily of the valley. Very joyful, also, was Sapphira, for\nher triumph and her happiness had come to her; but there came a moment\nwhen she suddenly put her hand upon her bosom.\n\n\"Lysias!\" she whispered. \"Did I hear him speak to me? Again! It is\ngone! Thank Aphrodite and thank Juno. It is better to be a wedded\nwoman, a proselyte of the temple, than to be a bondwoman of the\nprocurator.\"\n\nThe days of the wedding feast were to be cut short by the coming of the\nPassover, for only by express permission of the rabbis had the command\nof the wife of Pontius been obeyed at such a time. It was well, they\nadmitted, to change a law to obtain a proselyte from the household of\nthe procurator. The next day, however, would not be altogether sacred,\nand the wedding feast might go on, but it might be extended no further\nlest there should be a grievous sin against the counting of days. When\nthe next day came, therefore, all things belonging to it followed in\ntheir order.\n\nThere was a great gathering in the court of the women in the temple,\nfor here had come the prophet from Galilee, and he was not only\npreaching, but healing also. In front of him where he stood there was\nseated upon the pavement a closely veiled one, whose head was bowed. It\nwas as if she might also be praying silently.\n\nThe sick and the maimed and the blind and many who were in tribulation\ncame and stood by her for a moment to be touched by the rabbi and to\nmake room for others to be healed in like manner. These fell away full\nof joy over that which had come to them, but the veiled one moved not,\nnor did several of the other women who were near. Once only did she\nlift her head, drawing aside her veil, and her voice was low and sweet.\n\n\"O Master, what shall I ask of thee concerning Ulric? Canst thou do\naught for him?\"\n\n\"Be thou contented,\" he said. \"He followeth me.\" He stooped and put\nhis hand upon her head and turned away, for he was departing from that\nplace to the court of the heathen. So she covered her face with her\nveil and left the temple.\n\nIn the court of the heathen was a gathering that was dense for\nmultitude, and here, also, were many who asked for healing. Near to a\npillar by the outer portico stood twain who had just arrived.\n\n\"O Caius,\" said one, \"hast thou strength to stand upon thy feet for a\nlittle?\"\n\n\"Hardly, O jarl,\" said the centurion. \"But I am a Roman. What part have\nI in this Jew rabbi and his god?\"\n\n\"Nay, but stand thou here,\" said Ulric, \"while I go and ask him.\"\n\nOn pressed he through the crowd until he stood before the prophet of\nGalilee.\n\n\"O thou of the sons of the gods,\" he said, \"wilt thou heal a Roman,\nstanding yonder, as thou hast healed me, who am a Saxon? I pray thee\nhave mercy upon him, for he is my friend.\"\n\nNow he had thus interrupted men of dignities and learning who were\nstanding there asking questions of Jesus and gainsaying him, and these\nrebuked the jarl angrily.\n\nThe reply of Jesus was to them in words, but Ulric fell back thinking\nwithin himself: \"His face hath answered me. I know not what this is. I\nwill have speech with him at another time. O that I may be with him in\nthe day of the great battle!\"\n\nSlowly through the throng he went back to Caius at the pillar against\nwhich he had been faintly leaning.\n\n\"O Caius,\" he said, \"I did ask him. Thou wilt yet speak to him for\nthyself.\"\n\n\"Jarl of the Saxons,\" exclaimed Caius, \"I go now to my chariot. Speak\nnot. Seest thou not that I am standing firmly? The pain of the hurt\nhath departed! But here came one with a commandment from the procurator\nbidding thee to his house with speed. Delay not thy going, and deal\nwith him as thou wouldst deal with me. I thank thee and I thank the\nrabbi. Go!\"\n\n\"O gladiator, come thou in haste!\" said one in the raiment of a\nbondservant who stood near. \"The thing is important!\"\n\n\"Tell him I come,\" said Ulric. \"Wait not. I go not in thy company. But\nglad am I, O Caius, my friend, if thou art healed of the poison.\"\n\n\"That I know not,\" said Caius; \"but the burning ceaseth. Return thou\nsoon to me.\"\n\n\"O most noble Caius,\" said Ulric, \"I think this matter of the\nprocurator is already known to me. If I see thee not again, may all be\nwell with thee!\"\n\nHis countenance was bright and his step was firm and he turned away\nfrom Caius, going toward the outer entrance of the court of the women.\nThe distance was but short, and here under the portico waited the\nveiled one.\n\n\"Art thou here?\" she said. \"Hast thou indeed seen him? I spoke to him\nconcerning thee and he told me thou wouldst surely follow him.\"\n\n\"I know not that,\" he said, \"but lift thy veil, O Miriam, that I may\nsee thee--this last seeing. I go hence to death, but O that to thee\nmight come life and joy forever!\"\n\nHer unveiled face before him was white with terror and with agony.\n\n\"O my beloved, what sayest thou?\" she exclaimed. \"To thy death?\"\n\n\"I will wait for thee in Valhalla,\" said the jarl. \"I will have a fair\nhouse for thee in the city of Asgard. There thou shalt live with me\namong the gods. I think this Jesus of Nazareth will also be there, for\nhe is a Son of God and he is my friend and thy friend. Go thou to thy\nhouse. Fare thee well!\"\n\nStrong and brave grew her face and her form was erect when she\nresponded: \"O my beloved, if thou art indeed going now to thy death,\nthen will I also wait and I will come unto thee in thy high place, as\nthou hast said. From the prophet of Galilee have I heard a new thing\nconcerning those who die, that they have a better country than this and\na better city to dwell in. I had not known----\"\n\n\"O Miriam,\" said Ulric, \"it is not new to me. So say all the old sagas\nof the Northland. This have I been taught by Hilda from my childhood.\nShe also will be there, and all my kindred, with thee and me.\"\n\nNone saw, but a swift kiss fell upon her lips and then her veil was\ndrawn, but Ulric went from the portico joyously, exclaiming:\n\n\"I care not now! She may bring me my ruby in the city of the gods, and\nI, the son of Odin, will keep tryst with her whom I love. O Pontius! O\nSpearman! O procurator! I will show thee how little a Saxon jarl careth\nfor the edge of a sword.\"\n\nNevertheless, from that hour onward none saw the jarl, and two days\nwent by. These were days of sorrow and of doubt for Miriam, waiting\nlonely in the house of Ben Ezra. She indeed went forth veiled to listen\nto the preaching of the prophet of Galilee, but ever her eyes were\nsearching among the throngs of hearers for one who came not. \"O that he\nmight have heard these things also,\" she said within her heart. \"Did\nnot Ulric himself say that this is the captain who is also his king?\nHow shall he now follow him into any battle? O that it might be!\"\n\nSo thought Miriam, praying and weeping, and around her were many other\nwomen. \"O weeping one,\" said one of these, \"knowest thou not? The\nMaster himself hath said to us that he is to be crucified!\"\n\n\"Crucified!\" exclaimed Miriam.\n\n\"Yea,\" said the other, \"but that in three days he will arise from the\ndead and that then he will take the kingdom. It is a hard saying.\"\n\n\"That the dead rise we do know,\" replied Miriam, \"but none hath ever\nseen them after their resurrection. I think this saying is like the\nwords of my beloved concerning the city of the gods where I am to\nlive with him. And he--O God of Israel! Where is he now and what hath\nbefallen him?\"\n\nThe evening of that day was set apart for the feast of the Passover.\nMany were gathered to eat of it at the house of Ben Ezra, for the\nkinsfolk of Isaac came also to partake of it. The Scriptures were\nread and hymns and psalms were sung, and they communed sorrowfully\nconcerning the present desolation of their people, the terrors of the\nHerods, the oppression of the Romans, and their fears of the things\nwhich were yet to come upon them. After this some of them slumbered,\nbut not all. There were those who waked and watched, for through\nall the city had gone a saying of Jesus of Nazareth that he was the\nMessiah, and that his kingdom was at hand.\n\nEven the Romans had heard of this saying, but Pontius the Spearman\nhad laughed, for he thought of his forts and his legionaries and\nhe troubled not his mind concerning some unarmed mob of Jewish\nenthusiasts.\n\n\n\n\n                              CHAPTER XL.\n\n                           \"A LITTLE WHILE.\"\n\n\nIt was toward the morning of a new day that one came knocking loudly at\nthe door of the house of Ben Ezra.\n\n\"What wilt thou?\" asked the porter, partly opening the door and looking\nforth.\n\n\"Tell thou to those who are within,\" was responded, \"that the Romans\nand the chief priests have taken the prophet of Galilee by force. He is\nnow at the palace of the procurator and a great multitude gathereth. I\nam a kinsman of Isaac, the aged.\"\n\nSeveral were within hearing and the message passed quickly throughout\nthe house. There was then hurried girding of robes and putting on of\nsandals.\n\n\"We will go forth,\" said Ben Ezra. \"I would see what this thing\nmeaneth. He hath done nothing for which he might be taken, either under\nthe law of the Jews or the law of the Romans.\"\n\nSome said one thing and some another, and so it was over the entire\ncity, for great was the tumult which was arising in Jerusalem. It\nwas said that Jesus had been arrested in the night upon the Mount of\nOlives, beyond the brook Kidron, after he had eaten the Passover in\nthe city with his disciples. Neither he nor they had fought save for a\nblow or two, and no man had been slain. Jesus had been taken before the\nhigh priest and before Herod, the tetrarch, and before the procurator,\nby whom he was now to be judged, the others not having due authority.\nThe tetrarch was in the city at this season by reason of the Passover,\nalthough it was known that he was at enmity with Pontius the Spearman.\n\nThere were many rumors, nor was it easy to determine what report to\nbelieve, but when Ben Ezra and Isaac and their company came to the\npalace of the procurator they saw a strange matter. Outside of the\npalace was a place which was called the Pavement, and to this, and\nnot into the house, the strictest Jews might advance and not become\nunclean, to be afterward unfitted for the Passover worship in the\ntemple. Out of this place had been brought a throne chair of the\nprocurator, and in it he now was seated for judgment, surrounded\nby armed legionaries and men of high degree, as if some matter of\nimportance called for his decision.\n\nBefore him, as one who is accused of some crime and is awaiting\ndecision, stood Jesus of Nazareth, but not as any had ever before seen\nhim. He had been both stripped and scourged, and the soldiers of the\nprocurator, besides beating and mocking him, had derisively arrayed him\nin a purple robe of royalty; but the crown which they had put upon his\nhead was a torture crown, plaited of thorn-tree twigs.\n\nThe procurator himself now spoke, not to the prisoner before him, but\nto the surging mob of Jews upon the Pavement and in the street.\n\n\"Behold the man!\" he said.\n\nThen arose an angry roar of many voices, among which the loudest words\nwere:\n\n\"Crucify! Crucify! Crucify!\"\n\n\"Take him yourselves and crucify him,\" said the procurator, \"for I find\nno crime in him.\"\n\nThen said one to Ben Ezra: \"Already he hath been tried and condemned\nbefore Herod, the king. Also he hath been well examined and scourged\nduly by the procurator. Let him die!\"\n\nThere were many who responded in divers forms of speech to the\nutterance of the procurator, but a ruler among the Jews shouted loudly:\n\n\"We have a law, and by our law he ought to die, because he made himself\nthe Son of God.\"\n\nWhen Pontius heard that he arose and went into the palace for a little\nspace, taking the prisoner with him. What further examination was made\nthus in private the multitude knew not, but when again the procurator\ncame forth, having Jesus brought also, he said to the Jews:\n\n\"Which of these twain shall I release unto you, Bar Abbas, the robber,\nor Jesus who is called the Christ, the King of the Jews?\"\n\nBut they all answered him with shouts of \"Bar Abbas!\" for among the\nrabble were many priests and scribes who were stirring them up to do\nthis thing. Other things were said, both by the procurator and the\naccusers, but it seemed that he would willingly have refrained from\ndoing any further violence to this man.\n\n\"Behold your king!\" he said, at last.\n\n\"Away with him! Crucify! Crucify!\" came back the tumult of fierce\nvoices.\n\n\"Shall I crucify your king?\" he asked.\n\n\"We have no king but C\u00e6sar,\" responded one, and another added: \"If thou\nspare him, thou art no friend of C\u00e6sar.\"\n\nThen a servant of the procurator brought out to him a basin of water,\nand in this he washed his hands, saying to them: \"I am innocent of the\nblood of this righteous person. See ye to it.\"\n\n\"His blood be upon us and upon our children!\" roared the mob.\n\nThen Pontius reentered the palace and the soldiers led away the\nprisoner, for his crucifixion had been commanded, and there is never\nany great delay in the performance of a Roman execution.\n\n\"Let us follow, O Isaac,\" said Ben Ezra.\n\nIn the shadow behind him stood Miriam. \"I also will follow with you,\"\nshe said, \"for Jesus of Nazareth is my King.\"\n\nWithin the palace shortly after this, and in the small chamber near the\nhall of judgment, stood twain who seemed to be having earnest words\nwith one another.\n\n\"O Caius, my friend,\" said the procurator, gloomily, \"am I not in a\nstrait place this day? I have heard thee. Gladly would I grant any\nrequest of thine, as thou knowest. I may not hear thee as to this King\nof the Jews. As to thy gladiator, I would give him back to thee if it\nwere possible, but his evil deeds are too many. Without warrant or\ncommand he slew my slave dealer in the valley of Hinnom. He slew the\nmaster of the games who was over him, and with him also three slaves\nand the Jew merchant Abbas. Moreover, I have word from the proconsul\nof Spain that Saxon pirates under this Ulric the Jarl destroyed two of\nC\u00e6sar's triremes in the British seas. More things than these are justly\ncharged to his account. What say est thou?\"\n\n\"Thou art justified,\" said Caius, reluctantly. \"I may urge thee no\nmore. But I would gladly have saved him. This matter of Jesus of\nNazareth would indeed be brought against thee before C\u00e6sar. It is well\nfor thee that thou art at peace with Herod, the fox.\"\n\n\"I did indeed strive to save the Galilean rabbi,\" said Pontius. \"I\nwill tell thee a thing. My own wife had a dream concerning him and she\nwarned me not to condemn him as of myself. To me, also, he declared\nhimself to be of the gods. I meddle not with them, for little do we\nknow of the gods. But I have this to ask of thee, that thou wilt be my\nwitness of this crucifixion, that I may truly know of whatever shall\nthere occur.\"\n\n\"That will I do!\" exclaimed Caius. \"I also would see how he dieth, for\nI have heard many strange things. It would be a rare thing to see a\ngod upon a cross. Where, now, will be his kingdom and who shall do him\nreverence? I know not, surely, that it was indeed through him that I\nam healed of my hurt. So say a great many others who are cured. Their\nevils have departed from them, they know not how. We do know that no\nman hath such power as this.\"\n\n\"How did he deal with thee?\" asked Pontius.\n\n\"Not at all,\" replied the centurion. \"I stood at a distance when he\nlooked upon me, and I felt the blood changing in my veins. He did not\ntouch me. How, then, was the healing?\"\n\n\"This is wonderful,\" said the procurator. \"I will hear thee again about\nthat matter. Go, now, I pray thee. With him and with thy Saxon there\nwill also be crucified a strong rebel from the Lebanon who was captured\nin Judea. Upon his hands is the blood of many. For this consent of\nthine I thank thee.\"\n\nDuring this time a long procession, accompanied and followed by a\nmixed and growing multitude, was passing slowly through the streets\nof Jerusalem. At its head, although many marched on in advance, were\na quaternion of legionaries and their officer. Close by these were\nfunctionaries of the high priest and rulers of the Jews, with zealous\nscribes and Pharisees and officers from the household of Herod, the\ntetrarch. Next in the procession walked three who bore upon their\nshoulders heavy beams of wood. All three were suffering from the\nlacerations of the Roman scourge, and one was so far weakened that he\nfell under his burden.\n\n\"Bring me hither that huge Jew!\" said the Roman officer in command.\n\n\"I bring him,\" quickly replied a soldier. \"He saith that he is one\nSimon of Cyrene.\"\n\n\"Let him carry the cross for Jesus of Nazareth,\" said the officer. \"We\nmay not be delayed. Scourge him forward!\"\n\nSo again the procession moved on toward the place of execution.\n\nUpon the bosom of each of the condemned ones, to be afterward affixed\nover his head upon his wood of torment, swung a wooden tablet inscribed\nwith his name and with his crime. Of these tablets the first was\nwritten in Latin only, and it told of the rebel of the Lebanon. Upon\nthe second was written:\n\n                     \"ULRIC, THE SAXON MURDERER.\"\n\nUpon the third, a larger tablet, was inscribed, in Latin and in Greek\nand in Hebrew:\n\n          \"THIS IS JESUS OF NAZARETH, THE KING OF THE JEWS.\"\n\nOf this rather than of the others the rabble shouted mockeries as they\nread, for here, they said, was a king upon his way to die as a common\nmalefactor, and for him there was no salvation.\n\nSilent was Ulric the Jarl, even when his eyes met those of Caius of\nThessalonica, but the centurion drew near to him and said:\n\n\"O jarl of the Saxons, I did what I could, but it was beyond my\npower to rescue thee. Thy sword hath fallen upon too many and thy\ncondemnation is just.\"\n\nNo answer made Ulric, and the centurion turned away his horse.\n\nThe gate had been passed and now the low hill of Calvary, or Golgotha,\nwas at hand. The multitude grew as the rising tide of the sea, for all\nJerusalem was stirred by this affair and the prophet of Galilee had\nfriends as well as enemies, and many who came were weeping bitterly.\n\n\"In a strange manner,\" thought the jarl, \"have the valkyrias come for\nme and for him. Where, now, is his father, that he hath thus deserted\nhis son in such a place? Are the Romans more powerful than the gods?\nIt is but little that we must die. Shortly I shall be in Valhalla, and\nI think Hilda will come to meet me at some place that is appointed.\nThere, also, I will wait for Miriam until she shall come. I am glad\nthat I have smitten down her enemies, giving my life for hers, and that\nI have made provision for her welfare.\"\n\nThe summit of the hill was level, and here a space was kept clear that\nthe multitude might not hinder by pressing. Here were three holes in\nthe earth already dug to receive the long timbers after the crosspieces\nand the victims should be spiked upon them.\n\nThe raiment of the condemned was the execution fee of the Roman\nsoldiers, and there was a stripping done, but the tunic of Jesus was\ngambled for by them because it was of one piece, to be spoiled by\ndividing.\n\nThe three crosses now lay upon the sand and Ulric looked earnestly upon\nthem, for a strong and sudden memory came into his mind.\n\n\"The token of Hilda!\" he exclaimed, but in a whisper, hoarse with pain.\n\"These are but as the runes that she showed me upon the sandy beach\nof the North coast before I sailed thence in _The Sword_. Now know I\nthat my voyage is ended, and I die, as she said, by the hands of the\nsoldiers of C\u00e6sar. But I had not thought of such a death as this!\"\n\nFirst of all did the soldiers seize rudely upon Jesus, scoffing at\nhim, and terrible was the swiftly performed work of the driving of the\nspikes, but there was not heard by any a cry of pain.\n\n\"Brave is he!\" thought Ulric. \"I also will hold my peace.\"\n\nFirm, also, was the courage of the rebel Jew from the Lebanon, and the\nmultitude wondered greatly at the fortitude of these who suffered this\nhorror silently.\n\nOne by one did the soldiers and their helpers lift the crosses, fixing\nthem firmly in the earth, and a loud shouting of the rabble arose at\nthe lifting, but there was also weeping and wailing and beating of\nbreasts among the multitude.\n\nAt the foot of the cross of Jesus now knelt women and men to whom he\nspoke, and he also uttered words to some who were not so near.\n\nIn front of the cross whereon the jarl was nailed there came for a\nmoment a veiled one, putting aside her veil and gazing wistfully into\nhis face.\n\n\"O my beloved, thou!\" she exclaimed.\n\n\"Miriam! Loved one!\" he groaned, being in great agony, \"tarry not here!\nLook not upon me! Thine eyes are more than I may bear! Go to thy house!\"\n\nHer lips parted and she strove to speak, but a great tremor shook her,\nand no voice came from her lips except a low, hard cry, having in it\nwhat seemed the name of her god. Then turned she away and she had\nfallen but that the arm of Ben Ezra went quickly around her, and he\ncompelled her to go away a little space that she might kneel and wait.\n\nTime passeth slowly to one who hangeth upon a cross, desiring the\ncoming of the end. The sun beat down hotly. The multitude came and\nwent, and all the open space, to the highway and beyond, was a dense\nthrong.\n\n\"I heard him,\" thought Ulric. \"He hath spoken to his father more than\nonce. If I speak to the gods, are they now near enough to hear me? I\nthink not; but I shall see them shortly.\"\n\nThe man upon the third cross turned now in his writhing and he said to\nJesus:\n\n\"Art not thou the Christ? Save thyself and us!\"\n\nJesus answered not, but the jarl cried out:\n\n\"Dost not thou fear God, seeing thou art in the same condemnation? And\nwe, indeed, justly, for we receive the due reward of our deeds; but\nthis man hath done nothing amiss.\" Then he said to the Christ: \"Jesus,\nremember me when thou comest in thy kingdom!\"\n\nUnto him did Jesus make answer: \"Verily I say unto thee, This day shalt\nthou be with me in the Garden.\"\n\nThen followed a stillness, but the jarl thought of the word which was\ngiven him. \"I knew not of this garden. There it is that I am to be with\nhim and with the gods. There, also, I shall see Hilda, and Miriam will\ndwell with me in the garden. It is enough! I am content!\"\n\nGreat was the cruelty of the Jews and of the rabble, and the hatred of\nsome for Jesus was exhibited in mocking speeches. It was as if they\ntook pleasure in the tokens of his sufferings.\n\nIt was now afternoon, and for some time Jesus had been silent, but\nsuddenly and with a loud voice he cried out:\n\n\"My God! my God! Why hast thou forsaken me?\"\n\nTo that utterance the Jews replied in a manner which Ulric did not\nunderstand, but again Jesus cried out, saying:\n\n\"I thirst!\"\n\nA horrible thirst cometh upon those who are crucified, and a drink of\nvinegar and myrrh with other bitterness is always provided for them by\nsome who are merciful. One ran and took a sponge, soaking it with this\nprovision, and lifted it upon a reed to the lips which were burning.\n\nAt that moment Jesus uttered an exceedingly great voice of pain, and\ninstantly it was seen that his soul had departed from his body. He was\ndead.\n\n\"Would that I were as he!\" thought the jarl, \"that I might be free of\nthis agony and pass on to Valhalla and into this garden to which Jesus\nhath gone before me!\"\n\nThe multitude were not gazing as before upon these who were crucified,\nfor now the light of the sun was withdrawn and a great gloom was over\nall things. The earth quaked under their feet. Great rocks were rended.\nFear fell upon men and women, and with one accord they fled away\ntoward the city, beating their breasts and mourning.\n\nCaius of Thessalonica stood watching these things, and other Romans\nwith him. \"Certainly,\" he exclaimed, \"this was a righteous man. Truly\nthis was the Son of God!\"\n\nBut the Jews had taken thought beforehand for yet another matter. The\nnext day would be their Sabbath, a holy day, and by their law it was\nnot well for one to be left upon a cross over the Sabbath. Therefore\nthey had obtained from the procurator an order that the deaths of\nthese three might be hastened by the breaking of their bones. For this\nbusiness came soldiers with clubs, but they struck not any limb of\nJesus, who was already dead.\n\n\"I have no mark of a spear,\" thought Ulric. \"It is not well. I die\nwithout any wound except of these spikes.\"\n\nNear to him then were these soldiers, but he saw one of them thrust a\npilum blade into the side of Jesus, making a wound from which poured\nboth blood and water. Quickly, now, came merciful relief to the two\nothers, for the soldiers made an end.\n\nAfterward were all the bodies taken down from the crosses, as was\nrequired by the law of the religion of the Jews, and the friends of any\nman were permitted to do their will concerning him.\n\nThe sun had long since set, and the darkness was over the earth, when\na little company of men and women entered the door of the house of Ben\nEzra.\n\n\"O Miriam, my daughter,\" said Isaac, the aged, when they were within,\n\"thou mayest mourn, but be thou comforted. We have buried him in my\nown tomb. And didst thou not hear what was said to him by Jesus of\nNazareth? In him do I now believe. He is God!\"\n\n\"O my beloved!\" wailed Miriam, and she said no more for weeping.\n\n\"Miriam,\" continued Ben Ezra, \"I also believe; trust thou, concerning\nthy husband, that it is well with him!\"\n\n\"Ye are my friends,\" said Miriam. \"I heard the saying, faintly and far.\nThey are at this hour in the garden, do you say? But I am here and I\nam alone, for my love hath been taken from me. Nevertheless, I will be\npatient. It is but for a little while; a little while!\"\n\n\n                               THE END.\n\n\n\n\n                    BOOKS OF TRAVEL AND ADVENTURE.\n\n                         By EGERTON R. YOUNG.\n\n\n=WINTER ADVENTURES OF THREE BOYS IN THE GREAT LONE LAND.=\nCrown 8vo, gilt edges. Eighteen full page Illustrations, 3s. 6d.\n\n\n=SUMMER ADVENTURES OF THREE BOYS IN THE WILD NORTH LAND.=\nThird Thousand. Twenty-eight full page Illustrations. Crown 8vo gilt\nedges, 3s. 6d.\n\n\n=BY CANOE AND DOG TRAIN AMONG THE CREE AND SALTEAUX INDIANS.=\nTwentieth Thousand. With Photographic Portraits of the Rev. E. R.\nYOUNG and Mrs. YOUNG. Map, and Thirty-two Illustrations, 3s. 6d.\n\n  \"Young and old will read this amazing story with delight. His heroic\n  journeys through the snow are described in a way that will secure the\n  attention of all.\"--_Sword and Trowel._\n\n  \"One of the most thrilling narratives of missionary life and adventure\n  ever published.\"--_Birmingham Daily Gazette._\n\n\n=STORIES FROM INDIAN WIGWAMS AND NORTHERN CAMP FIRES.=\nNinth Thousand. Forty-three Illustrations. Imperial 16mo, 3s. 6d.\n\n  \"I regard it as one of the most fascinating, instructive, and\n  stimulating of modern missionary books.\"--Dr. ARTHUR T. PIERSON.\n\n\n=OOWIKAPUN;=\nor, How the Gospel reached the Nelson River Indians. Fourth Thousand.\nIllustrated. Imperial 16mo, 2s. 6d.\n\n  \"Another stirring and delightful volume by the Rev. E. R. Young.\n  It has all the delightful and entertaining features of the best\n  fiction.\"--_Lincolnshire Free Press._\n\n  \"It abounds in fine descriptions of Indian life, with its\n  superstitions, customs, modes of travelling, conflicts with wild\n  beasts, and other thrilling adventures, which will be read with almost\n  breathless excitement.\"--_Leeds Mercury._\n\n\n\n\n               BOOKS OF TRAVEL, ADVENTURE, and HISTORY.\n\n\n=ACROSS THREE OCEANS AND THROUGH MANY LANDS WITH PEN AND CAMERA.= By\nFRED. REYNOLDS. Second Thousand. Imperial 16mo. 96 Illustrations,\nchiefly from Photographs. With Portrait. 3_s._ 6_d._\n\n  \"Mr. Reynolds has produced a bright and chatty volume, and gives an\n  interesting account of each place visited.\"--_Methodist Recorder._\n\n  \"A capital book of travel, well suited as a present for the young.\n  The pictures are a feature of the book; the narrative is bright and\n  chatty.\"--_The Scotsman._\n\n\n=ACROSS SIBERIA ON THE GREAT POST ROAD.= By CHARLES WENYON, M.D. With\nPortrait, Map, and Twenty-seven Illustrations. Third Thousand. Imperial\n16mo, 3_s._ 6_d._\n\n  \"One of the pleasantest books of travel we have read for some time.\n  One lays it down with the feeling of parting from a congenial\n  fellow-traveller on a long and memorable journey.\"--_Sheffield\n  Independent._\n\n\n=TWO MEN OF DEVON IN CEYLON.= A Story of East and West. By S. LANGDON,\nAuthor of \"My Mission Garden,\" etc. Ten Illustrations. Imperial 16mo,\n3_s._ 6_d._\n\n  \"An unusually fine historical romance.\"--_The Christian Endeavour._\n\n  \"The story is exceedingly well told, and is both interesting and\n  instructive.\"--_Glasgow Herald._\n\n  \"An interesting and instructive romance.\"--_Christian Leader._\n\n\n=A SOUTH AUSTRALIAN ROMANCE.= How a Colony was Founded and a Methodist\nChurch Formed. By Rev. JOHN BLACKETT, South Australia. Crown 8vo,\nIllustrated. 2_s._\n\n\n=KENOOSHAO.= A Red Indian Tragedy. By Rev. GEORGE BARNLEY, formerly\nMissionary in the Hudson Bay Territory. Crown 8vo, Illustrated, 1_s._\n\n\n=RAMBLES IN BIBLE LANDS.= By the Rev. RICHARD NEWTON, D.D. Seventy\nIllustrations. Imperial 16mo, 2_s._ 6_d._\n\n  \"An admirable book.\"--_Methodist Recorder._\n\n  \"From the juvenile standpoint we can speak in hearty commendation of\n  it.\"--_Literary World._\n\n\n=OUR INDIAN EMPIRE: ITS RISE AND GROWTH.= By the Rev. J. S. BANKS,\nAuthor of \"Martin Luther, the Prophet of Germany,\" etc., etc.\nThirty-five Illustrations, and a Map. Imperial 16mo, 3_s._ 6_d._\n\n  \"The imagination of the young will be fired by its stirring\n  stories of English victories, and it will do much to make history\n  popular.\"--_Daily Chronicle._\n\n  \"A well-condensed and sensibly-written popular narrative of\n  Anglo-Indian history.\"--_Daily News._\n\n\n=OUR SEA-GIRT ISLE.= English Scenes and Scenery Delineated. By Rev.\nJ. MARRAT. 217 Illustrations, and Map. Revised and Enlarged Edition.\nImperial 16mo, 3_s._ 6_d._\n\n  \"A very pleasant companion.\"--_Daily Telegraph._\n\n  \"Bright and pleasant, full of information and good\n  feeling.\"--_Literary World._\n\n  \"An unusually readable and attractive book.\"--_Christian Age._\n\n\n\n\n                      Pictures of Methodist Life.\n\n                 LANCASHIRE STORIES BY JOHN ACKWORTH.\n\n\n                             =DOXIE DENT.=\n                        A CLOG SHOP CHRONICLE.\nCrown 8vo, Illustrated, Art linen, gilt top, 3_s._ 6_d._ (_New Volume._)\n\n\n                          =BECKSIDE LIGHTS.=\n      FIFTH THOUSAND. Crown 8vo, Art linen, gilt top, 3_s._ 6_d._\n\n  \"His touch is almost as perfect as Mr. Barrie's, and he has to the\n  full the art of presenting his characters in such wise as to leave us\n  with the impression that we have been on intimate terms with living\n  men and women.... We heartily commend this volume to lovers of real\n  life as presented by an artistic temperament.\"--_Daily Chronicle._\n\n\n                        =CLOG SHOP CHRONICLES.=\n      TENTH THOUSAND. Crown 8vo, Art linen, gilt top, 3_s._ 6_d._\n\n  \"Mr. Ackworth has achieved here a distinct success.... The author\n  knows his way to the common human heart. His humour, his pathos,\n  and his at times broad comedy, steeped as they are in the ennobling\n  element of religious faith and love, make us laugh and cry by turns,\n  while they keep us voraciously reading to the end.... There is, in\n  fact, not a story in the book which does not leave us hungering for\n  more.\"--_Christian World._\n\n\n                       =THE SCOWCROFT CRITICS.=\n      FIFTH THOUSAND. Crown 8vo, Art linen, gilt top, 3_s._ 6_d._\n\n\n                           CORNISH SKETCHES.\n\n                     =WHERE THE TAMARISK BLOOMS.=\n                          By REV. JAMES DUNK.\n              Crown 8vo, Art linen, gilt top, 3_s._ 6_d._\n\n \"Drawn with a vividness and subtle charm that must appeal to all who\n love to study the poetry of human nature. Mr. Dunk is a master in the\n art of expression. Each tale is a poem in prose, and his knowledge of\n the heart and mind of the Cornish Methodist is profound, while his\n originality and grace of expression are of a high order.\"--_Birmingham\n Daily Gazette._\n\n\n             SKETCHES OF LINCOLNSHIRE LIFE AND CHARACTER.\n\n                          =KITTIE LONSDALE,=\n                        =AND SOME RUMSBY FOLK.=\n                           By E. M. BRYANT.\n              Crown 8vo, Art linen, gilt top, 3_s._ 6_d._\n\n \"Presented with a vividness and tender sympathy that appeal strongly\n to those who have any knowledge of the reality of the religious life\n of the village Methodists of the past generation. Homely, kindly,\n saturated with a belief in the vitality of religion, these simple\n folk live and move in a lifelike way. Humour and pathos alternate\n with strong religious feeling and simple narrative.\"--_Sheffield and\n Rotherham Independent._\n\n\n                           CHARLES H. KELLY,\n   2, CASTLE STREET, CITY ROAD, E.C.; AND 26, PATERNOSTER ROW, E.C.\n\n\n\n\n                          Transcriber's Note:\n\n\nItalics are indicated by _underscores_.\n\nBolds are indicated by =equal signs=.\n\nSmall capitals have been rendered in full capitals.\n\nA number of minor spelling errors have been corrected without note.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Ulric the Jarl, by William O. Stoddard\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Mark C. Orton, Paul Clark and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive)\n\n\n\n\n\n\n\n[Illustration]\n\n\n\n\n THE SCIENCE OF\n ANIMAL LOCOMOTION\n (ZOOPRAXOGRAPHY)\n\n AN ELECTRO-PHOTOGRAPHIC INVESTIGATION OF\n CONSECUTIVE PHASES OF ANIMAL MOVEMENTS\n\n BY\n\n EADWEARD MUYBRIDGE\n\n EXECUTED AND PUBLISHED UNDER THE AUSPICES OF THE\n UNIVERSITY OF PENNSYLVANIA\n\n DESCRIPTION OF THE APPARATUS\n RESULTS OF THE INVESTIGATION\n DIAGRAMS\n PROSPECTUS\n LIST OF SUBSCRIBERS\n\n EADWEARD MUYBRIDGE\n\n UNIVERSITY OF PENNSYLVANIA\n\n PHILADELPHIA\n\n OR\n 10 HENRIETTA STREET,\n     COVENT GARDEN\n       LONDON\n\n\n\n\nANIMAL LOCOMOTION.\n\n(ZOOPRAXOGRAPHY.)\n\n\n\n\nINTRODUCTORY.\n\n\nIn 1872, the author of the present work at Sacramento, California,\ncommenced an investigation with the object of illustrating by\nphotography some phases of animal movements. In that year his\nexperiments were made with a famous horse--Occident, owned by Senator\nStanford--and photographs were made, which illustrated several phases of\naction while the horse was trotting at full speed, laterally, in front\nof the camera.\n\nThe experiments were desultorily continued; but it was not until 1877\nthat the results of any of them were published.\n\nIn the meanwhile he devised an automatic electro-photographic apparatus,\nfor the purpose of making consecutive photographic exposures at\n_regulated_ intervals of time or of distance. Some of the results of his\nexperiments with this apparatus, which illustrated successive phases of\nthe action of horses while walking, trotting, galloping, &c., were\npublished in 1878, with the title of \"THE HORSE IN MOTION.\" Copies of\nthese photographs were deposited the same year in the Library of\nCongress at Washington, and some of them found their way to Berlin,\nLondon, Paris, Vienna, &c., where they were commented upon by the\njournals of the day.\n\nIn 1882, during a lecture on \"The Science of Animal Locomotion in its\nrelation to Design in Art,\" given at the Royal Institution (see\n_Proceedings_ of the Royal Institution of Great Britain, March 13,\n1882), he exhibited the results of some of his experiments made during a\nfew antecedent years at Palo Alto, California; when he, with the\nzoopraxiscope and an oxy-hydrogen lantern, projected on the wall a\nsynthesis of many of the actions he had analysed.\n\nIt may not be considered irrelevant if he repeats what he on that\noccasion said in his analysis of the quadrupedal walk:--\n\n\"So far as the camera has revealed, these successive foot fallings are\ninvariable, and are probably common to all quadrupeds....\n\n\"It is also highly probable that these photographic\ninvestigations--which were executed with wet collodion plates, with\nexposures not exceeding in some instances the one five-thousandth part\nof a second--will dispel many popular illusions as to the gait of a\nhorse, and that future and more exhaustive experiments, with the\nadvantages of recent chemical discoveries, will completely unveil to the\nartist all the visible muscular action of men and animals during their\nmost rapid movements....\n\n\"The employment of automatic apparatus for the purpose of obtaining a\nregulated succession of photographic exposures is too recent for its\nvalue to be properly understood, or to be generally used for scientific\nexperiment. At some future time the explorer for hidden truths will find\nit indispensable for his investigations.\"\n\nIn 1883, the University of Pennsylvania, with an enlightened exercise of\nits functions as a contributor to human knowledge, instructed the author\nto make, under its auspices, a comprehensive investigation of \"Animal\nLocomotion\" in the broadest significance of the words.\n\n\nA DIAGRAM OF THE STUDIO\n\nand the arrangement of the apparatus used for this purpose is here\ngiven.\n\n[Illustration]\n\nTT represents the track along which the model M was caused to move. B is\nthe background, divided into spaces of 5 centimetres square for the\npurpose of measurement.\n\nL, a horizontal battery of electro-photographic cameras, parallel to the\nline of motion (at a distance of 15 metres or about 48 feet therefrom),\nfor a series of 12 lateral exposures.\n\nR, a vertical battery of electro-photographic cameras, at right angles\nto the lateral battery, for a series of 12 _rear_ foreshortenings.\n\nF, a horizontal battery of electro-photographic cameras, at any suitable\nangle to the lateral battery for a series of _front_ foreshortenings.\n\nO, the position of the electric batteries, a chronograph for recording\nthe time intervals of exposures, and other apparatus used in the\ninvestigation.\n\nA clock-work apparatus, set in motion at the will of the operator,\ndistributed a series of electric currents, and synchronously effected\nconsecutive exposures in each of the three batteries of cameras.\n\nThe intervals of exposures were recorded by the chronograph, and divided\ninto thousandths of a second. These intervals could be varied at will\nfrom seventeen one-thousandth parts of a second to several seconds.\n\nThe task of making the original negatives was completed in 1885; the\nremaining years have been devoted to the preparation of the work for\npublication.\n\n[Illustration:\n\nLATERAL elevation of some consecutive phases of action by representative\nhorses.\n\nEach line illustrates the successive fallings of the feet during a\nsingle stride.\n\nAfter the last phase illustrated, the feet, during continuous motion,\nwill revert practically to their position in the first phase.\n\nThe comparative distances of the feet from each other or from the ground\nare not drawn to scale; and, in any event, would be merely approximate\nfor the succeeding stride.\n\nIn the conjectural stride No. 10, phase 3 is very doubtful, phases 5 and\n7 seem probable in a very long stride.]\n\n\n\n\nDESCRIPTION OF THE PLATES.\n\n\nThe results of this investigation are\n\n=Seven Hundred and Eighty-one Sheets of Illustrations=, containing more\nthan 20,000 figures of men, women, and children, animals and birds,\nactively engaged in walking, galloping, flying, working, jumping,\nfighting, dancing, playing at base-ball, cricket, and other athletic\ngames, or other actions incidental to every-day life, which illustrate\nmotion or the play of muscles.\n\nThese sheets of illustrations are conventionally called \"plates.\"\n\nEach plate illustrates the successive phases of a single action,\nphotographed with automatic electro-photographic apparatus at regulated\nand accurately recorded intervals of time, _consecutively_ from one\npoint of view; or, _consecutively_ AND _synchronously_ from _two_, or\nfrom _three_ points of view.\n\n\n=Each Plate is complete in itself without reference to any other Plate.=\n\nWhen the complete series of twelve consecutive exposures, from each of\nthe three points of view, are included in ONE Plate, the arrangement is\nusually thus:--\n\n +-+-+-+-+-+-+-+-+-+--+--+--+\n | | | | | | | | | |  |  |  | Laterals.\n | | | | | | | | | |  |  |  |\n |1|2|3|4|5|6|7|8|9|10|11|12|\n | | | | | | | | | |  |  |  |\n +-+-+-+-+-+-+-+-+-+--+--+--+\n | | | | | | | | | |  |  |  | Rear Foreshortenings from\n | | | | | | | | | |  |  |  |   points of view on the same\n |1|2|3|4|5|6|7|8|9|10|11|12|   vertical line, at an angle\n | | | | | | | | | |  |  |  |   of 90 deg. from the Laterals.\n +-+-+-+-+-+-+-+-+-+--+--+--+\n | | | | | | | | | |  |  |  | Front Foreshortenings from\n | | | | | | | | | |  |  |  |   points of view on the same\n |1|2|3|4|5|6|7|8|9|10|11|12|   horizontal plane, at suitable\n | | | | | | | | | |  |  |  |   angles from the Laterals.\n +-+-+-+-+-+-+-+-+-+--+--+--+\n\nThe plates are not _photographs_ in the common acceptation of the word,\nbut are printed in PERMANENT INK, from gelatinised copper-plates, by the\nNew York Photo-Gravure Company, on thick linen plate-paper.\n\nThe size of the paper is 45 x 60 centimetres--19 x 24 inches, and the\nprinted surface varies from 15 x 45 to 20 x 30 centimetres--6 x 18 to 9\nx 12 inches.\n\nThe number of figures on each plate varies from 12 to 36.\n\nTo publish so great a number of plates as one undivided work was\nconsidered unnecessary, for each subject tells its own story; and\ninexpedient, for it would defeat the object which the University had in\nview, and limit its acquisition to large Libraries, wealthy individuals,\nor Institutions where it would be beyond the reach of many who might\ndesire to study it.\n\nIt has, therefore, been decided to issue a series of One Hundred Plates,\nwhich number, for the purposes of publication, will be considered as a\n\"COPY\" of the work. These one hundred plates will probably meet the\nrequirements of the greater number of the subscribers.\n\nIn accordance with this view is issued the following\n\n     _PROSPECTUS_\n\n\n\n\n ANIMAL LOCOMOTION,\n\n AN ELECTRO-PHOTOGRAPHIC INVESTIGATION OF CONSECUTIVE PHASES\n OF ANIMAL MOVEMENTS,\n\n BY\n\n EADWEARD MUYBRIDGE.\n\n 1872-1885.\n\n PUBLISHED UNDER THE AUSPICES OF THE\n UNIVERSITY OF PENNSYLVANIA.\n\n _Exclusively by Subscription._\n\n CONSISTING OF A SERIES OF\n\n ONE HUNDRED PLATES,\n\n AT A SUBSCRIPTION PRICE OF\n\n ONE HUNDRED DOLLARS\n     For the United States, or\n\n TWENTY GUINEAS\n     For Great Britain;\n\n Or the equivalent of Twenty Guineas in the gold currency\n of other countries in Europe.\n\n This will be for\n\n Austria,\n     Two Hundred and Ten Florins;\n\n Belgium, France, Italy, and Switzerland,\n     Five Hundred and Twenty-five Francs;\n\n Germany,\n     Four Hundred and Twenty Marks;\n\n Holland,\n     Two Hundred and Fifty Guilders.\n\nThe Plates are enclosed in a strong, canvas-lined, full AMERICAN-RUSSIA\nLEATHER PORTFOLIO.\n\nFor the purpose of placing all of the subscribers upon an equal footing\nin regard to cost, a copy of the work will be sent in the portfolio, and\npacked between boards, to any well-established Institution, or to any\nsubscriber, properly endorsed, to any city in Central or Western Europe,\nor in the United States.\n\n     FREIGHT CHARGES PAID,\n\nif so requested, to the railway station, with the understanding that the\nsubscription price is remitted within one week of the day of the arrival\nof the work at the station.\n\nCustom duties, or any other expenses, if any, at the cost of the\nsubscriber.\n\nAdditional Plates in any required number will be supplied to the\nsubscriber at the same proportionate rate; these, however, must be\nordered at the same time as the subscription Plates.\n\nThe Plates will be supplied\n\n     EXCLUSIVELY TO SUBSCRIBERS.\n\nIt was considered inadvisable to make an _arbitrary_ selection of the\none hundred Plates offered to subscribers, and with the object of\nmeeting, as far as possible, their diverse requirements, they are\ninvited to make their own selection, either from the subjoined list of\nsubjects, or from a detailed catalogue, which will be forwarded free of\nexpense to every subscriber.\n\nThe following are the numbers of Plates published of each class of\nsubjects, from which the subscriber's selection can be made:--\n\n                                                   Plates Published.\n\n Men,      draped                                              6\n  \"        pelvis cloth                                       72\n  \"        nude                                              133\n Women,    draped                                             60\n   \"       transparent drapery and semi-nude                  63\n   \"       nude                                              180\n Children, draped                                              1\n    \"      nude                                               15\n Movements of a man's hand                                     5\n Abnormal movements, men and women, nude and\n     semi-nude                                                27\n Horses walking, trotting, galloping, jumping, &c.            95\n Mules, oxen, dogs, cats, goats, and other domestic\n     animals                                                  40\n Lions, elephants, buffaloes, camels, deer, and other\n     wild animals                                             57\n Pigeons, vultures, ostriches, eagles, cranes, and other\n     birds                                                    27\n                                                            ----\n           Total number of Plates                            781\n Containing more than 20,000 Figures.\n\n=Should the selection be made from the Catalogue, it will be advisable\nto give the Author permission to change any one of the selected Plates\nfor any other illustrating the same action, if, in his judgment, the\nsubstituted Plate illustrates that action with a better model, or in a\nmore perfect manner than the one selected.=\n\n=With regard to the selection of Plates, however, it has been found by\nexperience that unless any special subject or plate is required it will\nbe more satisfactory to the subscriber if he gives the Author GENERAL\nINSTRUCTIONS as to the CLASS of subjects desired and to leave the\nSPECIFIC selection to him.=\n\nMany of the large Libraries and Art or Science Institutions in America\nand in Europe have subscribed for, and have now in their possession, a\ncomplete series of the seven hundred and eighty-one Plates, the\nsubscription price for which is\n\n     FIVE HUNDRED DOLLARS\n\nin the United States,\n\n     ONE HUNDRED GUINEAS\n\nin Great Britain for the complete series, in eight full AMERICAN-RUSSIA\nLEATHER PORTFOLIOS, or if bound in eleven volumes, each plate _hinged_,\nfull American-Russia leather,\n\n     FIVE HUNDRED AND FIFTY DOLLARS\n\nin the United States,\n\n     ONE HUNDRED AND TEN GUINEAS\n\nin Great Britain; or its equivalent for any city in Central or Western\nEurope.\n\nSubscribers who wish to make use of these Plates for the promotion or\ndiffusion of knowledge, or for artistic or scientific purposes, will be\nafforded facilities for acquiring working copies by special arrangement\nwith the Author.\n\n\n\n\nVALEDICTORY.\n\n\nThis is not exactly the place nor the time for the Author to express his\nobligations and thanks to those gentlemen who have assisted him in his\nlabours, but it affords a perhaps not inappropriate opportunity for him\nto pay a tribute of gratitude to his recently deceased friend M.\nMeissonier, without whose enthusiastic encouragement it is probable the\npresent work would never have been undertaken.\n\nIn 1882 he invited his friends to attend an illustrated Lecture given in\nhis studio by the Author, and then referring to a full knowledge of a\nsubject being necessary for it to be truthfully or satisfactorily\ntranslated by the artist, declared how much his own impression of a\nhorse's motion had been changed after having carefully studied its\nconsecutive phases. Attention need not be directed to the modifications\nin the expression of animal movements now progressing in the works of\nthe Painter and the Sculptor.\n\nThe investigations of the Author are so well known, and so generally\nrecognised as affording the only basis of truthful interpretation or\naccurate criticism of Animal Movement, that it is unnecessary to quote\nfrom the many elaborate reviews of \"Animal Locomotion,\" which have been\npublished in the American, English, French, and German Scientific,\nArtistic, and other Journals.\n\nFor the value of the present work to the general student of Nature and\nthe lover of Art, no less than to the Artist and the Archaeologist, the\nPhysiologist and the Anatomist, it is with much pride and gratitude that\nhe refers to the annexed list of some of his European subscribers.\n\n                                                                  E. M.\n\n 10 HENRIETTA STREET,\n     COVENT GARDEN,\n       LONDON,\n         _August 1891_.\n\n\n\n\nSUBSCRIBERS.\n\nThe general or departmental Libraries of the following\n\n\nUNIVERSITIES.\n\n Amsterdam\n Andrews, St.\n Basel\n Berlin\n Bern\n Bologna\n Bonn\n Breslau\n Bruxelles\n Edinburgh\n Erlangen\n Freiburg\n Geneve\n Genova\n Glasgow\n Goettingen\n Griefswald\n Halle\n Heidelberg\n Innsbrueck\n Jena\n Kiel\n Koenigsberg\n Leiden\n Leipzig\n Liege\n Louvain\n Muenchen\n Napoli\n Oxford\n Padova\n Pisa\n Prag\n Roma\n Rostock\n Strassburg\n Torino\n Tuebingen\n Utrecht\n Wien\n Wuerzburg\n Zuerich\n\n\nIMPERIAL, NATIONAL, OR ROYAL ACADEMIES OF FINE ARTS.\n\n Amsterdam\n Antwerpen\n Berlin\n Bern\n Birmingham\n Bologna\n Breslau\n Bruxelles\n Budapest\n Dresden\n Duesseldorf\n Firenzi\n Frankfurt\n Genova\n Gent\n Leipzig\n Liege\n London\n Manchester\n Milano\n Muenchen\n Napoli\n Paris\n Praha\n Roma (_de France_)\n Sheffield\n Torino\n Venezia\n Wien\n Zuerich\n Architectural Institute, Muenchen\n Herkomer School of Art, Bushey\n\n\nART MUSEUMS.\n\n Amsterdam\n Berlin\n Budapest\n\n\nARCHAEOLOGICAL INSTITUTES AND MUSEUMS.\n\n Dresden\n Griefswald\n Heidelberg\n Koenigsberg\n Leipzig\n Prag\n Rostock\n Strassburg\n Wien\n Wuerzburg\n Zuerich\n\n\nINDUSTRIAL ART AND SCIENCE MUSEUMS.\n\n Berlin\n Dublin\n Edinburgh\n Kensington\n Paris\n Wien\n\n\nINDUSTRIAL ART SCHOOLS.\n\n Amsterdam\n Breslau\n Budapest\n Frankfurt\n Nuernberg\n Zuerich\n\n\nLIBRARIES.\n\n The Royal Library, Windsor Castle\n Birmingham, Free Public\n Edinburgh, Advocates'\n Glasgow, Mitchell Free\n Liverpool, Free Public\n London, British Museum\n Manchester, Free Public\n Nottingham, Free Public\n Paris, National Library\n\n\nANATOMICAL INSTITUTES.\n\n Bern\n Breslau\n Freiburg\n Halle\n Innsbrueck\n Kiel\n Koenigsberg\n Leipzig\n Muenchen\n Pisa\n Prag\n Rostock\n Tuebingen\n Wuerzburg\n Zuerich\n\n\nROYAL COLLEGES OF SURGEONS.\n\n Edinburgh\n London\n\n\nPHYSIOLOGICAL INSTITUTES.\n\n Basel\n Berlin\n Bern\n Bologna\n Bonn\n Breslau\n Bruxelles\n Erlangen\n Freiburg\n Genova\n Goettingen\n Griefswald\n Halle\n Heidelberg\n Innsbrueck\n Jena\n Kiel\n Koenigsberg\n Leipzig\n Louvain\n Muenchen\n Napoli\n Prag\n Rostock\n Strassburg\n Torino\n Tuebingen\n Wien\n Wuerzburg\n Zuerich\n\n\nVETERINARY INSTITUTES.\n\n Alfort\n Bern\n Berlin\n Dresden\n\n\nANTHROPOLOGICAL MUSEUMS.\n\n Dresden\n Firenze\n\n\nETHNOLOGICAL, NATURAL HISTORY, AND ZOOLOGICAL INSTITUTES AND MUSEUMS.\n\n Amsterdam\n Bruxelles\n Freiburg\n Kiel\n Leiden\n Liege\n Napoli\n Paris\n Rostock\n\n\nPHYSICAL INSTITUTES.\n\n Basel\n Bologna\n Bruxelles\n Geneve\n Heidelberg\n Padova\n Prag\n Roma\n Rostock\n Utrecht\n\n\nPOLYTECHNIC HIGH SCHOOLS.\n\n Berlin\n Firenze\n Wien\n Zuerich\n\n\nCOLLEGES.\n\n Charterhouse\n Clifton\n Dublin (Trinity)\n Eton\n Owens\n Rossall\n Wellington\n\n\nROYAL PORCELAIN MANUFACTORIES.\n\n Berlin\n Dresden\n\n\nARTISTIC, LITERARY OR SCIENTIFIC CLUBS.\n\n Duesseldorf, _Malkesten_\n Glasgow, _Western_\n London, _Athenaeum_\n Rome, _Internazionale_\n\n       *       *       *       *       *\n\n Agricultural High School of Berlin\n Faculty of Medicine of Paris\n Faculty of Physicians and Surgeons of Glasgow\n Psychological Institute of Leipzig\n Royal College of Physicians, Edinburgh\n Royal Institution, Edinburgh\n Royal Dublin Society\n Royal Society of London\n\nThe names and works of the following subscribers are so well known that\nthe Academical, University, and other honourable distinctions\nappertaining to them are omitted, they being entirely unnecessary:--\n\n\nARTISTS, _Architects, Painters, and Sculptors_.\n\n Albano, Salvatore\n l'Allemand, Sigmund\n Alma-Tadema, L.\n Armitage, E.\n Barabino, Nicolo\n Becker, Carl\n Begas, Reinhold\n Benczur, Gyula\n Berger, Julius\n Behrens, Peter\n Birch, Chas. B.\n Boehm, Sir J. Edgar\n Bonnat, Leon\n Boughton, Geo. H.\n Bouguereau, W. A.\n Braith, Anton\n Brandt, Josef von\n Brausewetter, Otto\n Bridgman, F. A.\n Brock, Thos.\n Canneel\n Carland, Onorato\n Carolus-Durand\n Cavallucci, C. Jacopo\n Cavelier, P. J.\n Charlton, John\n Clay, Sir Arthur\n Coleman, Chas. Caryl\n Coleman, Enrico\n Colin, Paul\n Conti, Tito\n Costa, Giovanni\n Crowe, Eyre\n Dalou, Jules\n Dannat, W. T.\n Davinet, E.\n Davis, H. W. B.\n Defregger, Franz von\n Detaille, Edouard\n Dicksee, Frank\n Diez, Rob.\n Diez, Wm.\n Drion, Prosper\n Dubois, Paul\n Ebner, L.\n Eisenmenger, August\n Ende, Herm\n Ewald, Ernst\n Faed, Thomas\n Falguiere\n Fildes, Luke\n Ford, E. Onslow\n Fremiet, M.\n Frith, W. P.\n Gallegos, Jose\n Garnier, Charles\n Gehrts, Joh.\n Gelli, Edouardo\n Gerome, Jean Leon\n Gilbert, Alfred\n Gilbert, Sir John\n Goodall, Fredk.\n Gordigiani, Michele\n Gow, Andrew C.\n Grosse, Th.\n Gruetzner, Eduard\n Guignard, Gaston\n Gysis, N.\n Haueser, O.\n Hebert, Ernesto\n Herkomer, Hubert\n Hess, Anton\n Higgins, A.\n Huebner, Eduard\n Hunt, Holman\n Janssen, Pet.\n Kampf, Arthur\n Kaulbach, F. A. von\n Kips, A.\n Kirchbach, Fr.\n Klein-Chevalier\n Knaus, Ludwig\n Knight, Ridgway\n Knille, Otto\n Koehler, Robert\n Kopf, Joseph\n Kowalski, A. von\n Kroner, Ch.\n Kruse, Max\n Kuehl, G.\n Kuehn, H.\n Leighton, Sir Frederick\n Lenbach, Franz R. von\n Linton, Sir James D.\n Loefftz, Ludwig R. von\n Long, Edwin\n Lotz, Carl\n Lucas, Seymour\n Luthmer, F.\n MacWhirter, John\n Marks, H. Stacy\n Marshall, W. Calder\n Maurier, George du\n Max, Gabriel\n Meeks, Eugene\n Meissonier\n Menzel\n Meyerheim, Paul\n Millais, Sir John E.\n Miller, Ferdinand R. von\n Molkenbaer, H. B. G.\n Moore, Henry\n Morelli, D.\n Morot, Aime\n Muller, Carl\n Munkacsy, Mich. de\n Murgatroyd, J.\n Muetzel, G.\n Nieper, Ludw.\n Orchardson, W. Q.\n Otto, Heinrich\n Ouless, W. W.\n Papperitz, Georg\n Parsons, Alfred\n Passini, Ludwig\n Piglhein, Bruno\n Portaels\n Powers, Longworth\n Poynter, E. J.\n Prell, H.\n Preyer, Ernest\n Puvis, de Chavennes\n Richmond, W. B.\n Rivalta, Augusto\n Riviere, Briton\n Robert-Fleury, Tony\n Rodin, A.\n Roll\n Roth, Ch.\n Ruemann, Wilh.\n Sant, James\n Sarti, Diego\n Schaper, F.\n Schill, Adolf\n Schilling, Johannes\n Severn, Arthur\n Siemering, R.\n Six, J.\n Sommer\n Stieler, Eugen von\n Story, W. W.\n Sturgess, John\n Sues, Wilh.\n Swan, John M.\n Taylor, Edw. R.\n Teschendorf, E.\n Thiersch, Fredk.\n Thoma, Hans\n Thornycroft, Hamo\n Uhde, F. von\n Vibert, J. G.\n Vinea, Francesco\n Vriendt, de Jules\n Vuillefroy, F. de\n Wagner, Alex.\n Watts, George F.\n Weeks, E. L.\n Weishaupt, Victor\n Wells, Hy. T.\n Werner, A. von\n Whistler, J. McNeil\n Woolner, Thos.\n Zimmermann, Ernst\n Zuegel, H.\n\n\nARCHAEOLOGISTS, MEN OF LETTERS, AUTHORS OF ART WORKS, ETC.\n\n Ball, Valentine\n Berndorf, Otto\n Berlepsch, H. E. von\n Bullen, George\n Coleman, Alexander\n Dickson, Wm. P.\n Donnelly, Genl.\n Duhn, F. von\n Duplessis, Georges\n Eaton, Fredk. A.\n Evans, John\n Falke, J.\n Graf, T. T.\n Hirschfeld, Gustav\n Holmes, Richard R.\n Kekule, Prof.\n Klein, Wilhelm\n Koerte, G.\n Michaelis, Ad.\n Muntz, Eugene\n Obreen, Fr. D. O.\n Overbeck, Johannes\n Pietsch, Ludwig\n Preuner, A.\n Pulszky, Karoli\n Ruskin, John\n Sambuy, Conte Ernesto di\n Schrieber, Th.\n Sittl, K.\n Smith, Genl. Sir R. M.\n Sutton, Chas. W.\n Tedder, Hy. R.\n Thode, H.\n Treu, Georg\n Webster, H. A.\n Wolff, Albert\n\n\nPHYSIOLOGISTS.\n\n Albertoni, Pietro\n Albini\n Aubert, H.\n Bernstein, J.\n Biedermann, W.\n du Bois-Reymond\n Brown-Sequard\n Ewald, R.\n Exner, Sigmund\n Fano, Giulio\n Fick, A.\n Gaule, J.\n Goltz, F.\n Gruetzner, P.\n Heidenhain, R.\n Hensen, V.\n Hering, Ewald\n Hermann, L.\n Kries, J.\n Kronecker, H.\n Kuehne, W.\n Landois, L.\n Luciani, Luigi\n Ludwig, C.\n Marey, E. J.\n Masoin, E.\n Meissner, G.\n Miescher, F.\n Moleschott, Senator J.\n Mosso, A.\n Munk, Hermann\n Pettigrew, J. Bell\n Pflueger, E.\n Rosenthal, I.\n Schiff, M.\n Slosse, A.\n Vintschgau, M. von\n Voit, C. von\n\n\nANATOMISTS.\n\n Braune, Wilh.\n Brunn, A. von\n Cleland, John\n Eisler, P.\n Flemming, W.\n Hasse, C.\n Henke, W. J.\n Humphry, G. M.\n Koelliker\n Marshall, John\n Rabl\n Romiti\n Roux, W.\n Rueckert, J.\n Schwalbe, G.\n Stieda, L.\n Stoehr, Ph.\n Strasser, H.\n Thanhoffer, L. von\n Van Beneden, Edouard\n Virchow, Hans\n Wiedersheim\n\n\nANTHROPOLOGISTS, BIOLOGISTS, PALEONTOLOGISTS, ZOOLOGISTS, ETC.\n\n Acland, Sir H. W.\n Barrier, Gustave\n Blochmann, F.\n Bowman, Sir Wm.\n Brandt, K. E.\n Carpenter, P. Herbert\n Darwin, Francis\n Flower, W. H.\n Galton, Francis\n Guenther, Albert\n Hartog, Marcus\n Haughton, Saml.\n Hollis, W. A.\n Huxley, T. H.\n Jensink, F. A.\n Kerbert, C.\n Lankester, E. Ray\n Lubbock, Sir John\n Mantegazza, Senator\n Meyer, A. B.\n Milne-Edwards\n Mivart, St. George\n Muellenhoff\n Mueller, Max\n Newton, Alfred\n Owen, Sir Richard\n Pasteur, L.\n Romanes, Geo. J.\n Schmidt, Emil\n Schuetz\n Sorby, H. C.\n Swinhoe, Chas.\n Van Wulverhorst\n Virchow, Rudolf\n Weismann, August\n Wundt, W.\n Yseux\n Zittell, C. A. von\n\n\nPHYSICISTS, ETC.\n\n Abney, Capt. W. de W.\n Bellati\n Blazerna, Pietro\n Bramwell, Sir Fredk.\n Bunsen, R.\n Ditscheiner, L.\n Glaisher, James\n Hagenbach-Bischoff\n Helmholtz, H. von\n Huggins, Wm.\n Julius, V. A.\n Mach, E.\n Matthiessen, L.\n Moss, Rich. J.\n Quincke, Georg\n Righi, Augusto\n Rousseau, E.\n Soret, C.\n Tissandier, Gaston\n Thomson, Sir Wm.\n Vogel, H. W.\n Weber, H. F.\n\n       *       *       *       *       *\n\n Moltke, Count von\n Portland, The Duke of\n Wharncliffe, The Earl of\n\n..........\n\n    Transcriber's Note:\n\n    Every effort has been made to replicate this text as faithfully as\n    possible.\n\n    The author spelled Greifswald as Griefswald, Innsbruck as\n    Innsbrueck and Haeuser as Haueser in this text. These spellings have\n    been retained.\n\n    OE ligatures have been expanded.\n    Italic text has been marked with _underscores_.\n    Bold text has been marked with =equals signs=.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Science of Animal Locomotion\n(Zoopraxography), by Eadweard Muybridge\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPressure  \nCooking\n\nby Tom Hirschfeld\n\nA member of Penguin Random House LLC\n\n**Publisher:** Mike Sanders\n\n**Associate Publisher:** Billy Fields\n\n**Managing Editor:** Lori Cates Hand\n\n**Cover and Book Designer:** Becky Batchelor\n\n**Compositor:** Ayanna Lacey\n\n**Proofreader:** Catherine Schwenk\n\n**Indexer:** Brad Herriman\n\n**PRODUCTION, LONDON**\n\n**Digital Producer:** Alex Valizadeh\n\n**Senior Digital Producer:** Miguel Cunha\n\n**DIGITAL OPERATIONS, DELHI**\n\n**Head of Digital Operations:** Manjari Hooda\n\n**Producer:** Rahul Kumar\n\n**Assistant Editor:** Etika Kapil\n\n**DTP Designer:** Manish Bhatt\n\n**Operations Assistant:** Tauhid Nasir\n\nFirst American Edition, 2015  \nPublished in the United States by DK Publishing  \n6081 E. 82nd Street, Indianapolis, Indiana 46250\n\nCopyright \u00a9 2015 Dorling Kindersley Limited\n\nA Penguin Random House Company  \n15 16 17 18 10 9 8 7 6 5 4 3 2 1  \n001\u2013289069\u2013April2016\n\nAll rights reserved.\n\nWithout limiting the rights under the copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of the copyright owner.\n\nPublished in the United States by Dorling Kindersley Limited.\n\neISBN: 978-1-465-45044-9\n\nLibrary of Congress Catalog Card Number: 2015948633\n\n**Note** : This publication contains the opinions and ideas of its author(s). It is intended to provide helpful and informative material on the subject matter covered. It is sold with the understanding that the author(s) and publisher are not engaged in rendering professional services in the book. If the reader requires personal assistance or advice, a competent professional should be consulted. The author(s) and publisher specifically disclaim any responsibility for any liability, loss, or risk, personal or otherwise, which is incurred as a consequence, directly or indirectly, of the use and application of any of the contents of this book.\n\n**Trademarks** : All terms mentioned in this book that are known to be or are suspected of being trademarks or service marks have been appropriately capitalized. Alpha Books, DK, and Penguin Random House LLC cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.\n\nDK books are available at special discounts when purchased in bulk for sales promotions, premiums, fund-raising, or educational use. For details, contact: DK Publishing Special Markets, 345 Hudson Street, New York, New York 10014 or  \nSpecialSales@dk.com.\n\nPrinted in the United States\n\nidiotsguides.com\n\n## **Contents**\n\nIntroduction\n\nPressure Cooker Basics\n\nHow Pressure Cooking Works\n\nWhat Your Pressure Cooker Can Do\n\nTypes of Pressure Cookers\n\nSafety and Maintenance\n\nRelease Methods\n\nUnderstanding PSI\n\nPressure Cooking Vegetables\n\nPressure Cooking Meat\n\nConverting Conventional Recipes\n\nStocking Your Kitchen\n\nPlanning Weeknight Meals\n\nPressure Cooker Recipes\n\nBasics\n\nRich Beef Stock\n\nChicken Stock\n\nVegetable Stock\n\nPressure Cooker Eggs\n\nPressure Cooker Rice\n\nEasy Dried Beans\n\nPressure Cooker Pasta Sauce\n\nBreakfasts\n\nEggs en Cocotte \u00e0 la Cr\u00e8me\n\nEgg Cups\n\nApple Cinnamon Breakfast Oats\n\nSavory Parmesan Steel-Cut Oats\n\nFive-Grain Oatmeal\n\nSausage, Onion, and Gruy\u00e8re Breakfast Casserole\n\nHoney, Raisin, and Quinoa Breakfast Risotto\n\nBacon Jam\n\nAppetizers\n\nClassic Deviled Eggs\n\nGarlicky White Bean and Parmesan Dip\n\nRed Beet Hummus\n\nMiddle Eastern Hummus\n\nLentil P\u00e2t\u00e9\n\nSavory Smoked Salmon Cheesecake\n\nSalads\n\nTuna Salad with Chickpeas\n\nChicken Salad Deluxe\n\nClassic Egg Salad\n\nWheat Berry Salad with Arugula Pesto\n\nFarro Tabbouleh\n\nPicnic-Style Potato Salad\n\nHot German Potato Salad\n\nSoups, Stews, and Chilies\n\nRoasted Corn and Butternut Squash Chili\n\nWhite Bean and Shiitake Soup\n\nSaturday Soup Beans\n\nSpicy Chickpea Stew with Sour Tomato Curry\n\nCuban Black Bean Soup with Sherry\n\nFarmhouse Corn Chowder\n\nFrench Potato and Leek Soup\n\nIndian Carrot and Lentil Soup\n\nRustic Split-Pea Soup\n\nCurried Butternut Squash Soup\n\nMushroom Barley Soup\n\nRoot Vegetable Stew\n\nBorscht with Italian Sausage\n\nRibolleta\n\nTexas-Style Chili Con Carne\n\nKorean Beef Stew\n\nVegetable Beef Soup\n\nIrish Stew\n\nPork Ramen\n\nHearty Turkey and Vegetable Soup\n\nNew England Fish Chowder\n\nVegetables\n\nRatatouille\n\nHoney-Glazed Carrots\n\nGlazed Carrots with Braised Lettuce\n\nSouthern-Style Green Beans\n\nButtered Green Beans with Nut Crunch Topping\n\nCaramelized Onion Mashed Potatoes\n\nMashed Maple Sweet Potatoes\n\nBrussels Sprouts with Almonds and Prosciutto\n\nBraised Kale\n\nAlsatian-Style Braised Red Cabbage\n\nOne-Pot Cabbage, Rice, and Lentils\n\nSouthern Collard Greens\n\nMeaty Main Dishes\n\nBarbecue Braised Short Ribs\n\nBeef Sugo\n\nCorned Beef\n\nCuban-Style Ropa Vieja\n\nClassic Beef Brisket\n\nPot Roast with Fennel and Carrots\n\nSwedish Meatballs\n\nBeef Bourguignon\n\nChuck Roast with Horseradish Cream and Carrots\n\nSwiss Steak\n\nOsso Buco\n\nBelgian Beef Stew Cooked in Beer\n\nPressure Cooker Tacos\n\nHungarian Stuffed Peppers\n\nHungarian Chicken Paprika\n\nCaribbean Chicken Curry\n\nOne-Pot Chicken and Sausage Perloo\n\nChicken and Dumplings\n\nChicken Cacciatore\n\nSmothered Chicken\n\nChicken with 40 Cloves of Garlic\n\nThai-Style Green Curry Chicken\n\nChinese Red Cooked Chicken\n\nHungarian Pork Goulash\n\nBelgian Ale\u2013Braised Pork Loin with Mustard\n\nPork Grillades\n\nSouthern-Style Pulled Pork\n\nPork Vindaloo\n\nPork Posole\n\nOne-Pot Sausage, Potatoes, and Greens\n\nAsian Steamed Fish\n\nSalmon en Papillote\n\nFish Curry\n\nThai Steamed Mussels\n\nBeer-Steamed Shrimp\n\nCajun-Style Shrimp Jambalaya\n\nRice, Grains, and Breads\n\nBasmati Rice Pilaf\n\nDirty Oats with Lentils\n\nWild Rice\n\nAsparagus and Sun-Dried Tomato Risotto\n\nRisotto Milanese\n\nJamaican-Style Rice and Peas\n\nSteamed Brown Bread with Raisins\n\nCrustless Sandwich Bread\n\nCornbread\n\nDesserts\n\nLemon Pots de Cr\u00e8me\n\nChocolate Pots de Cr\u00e8me\n\nMaple Cream Caramels\n\nCr\u00e8me Br\u00fbl\u00e9e\n\nCaribbean-Style Flan\n\nClassic Cheesecake\n\nChocolate Caramel Cheesecake\n\nRice Pudding\n\nBread Pudding with Whiskey Sauce\n\nCaramel Vanilla Tapioca Pudding\n\nChristmas Pudding\n\nDulce de Leche\n\nGingerbread Cake with Vanilla Glaze\n\nRum Raisin, Apple, and Prune Cake\n\nBoozy Dried Fruit Compote\n\nPears Poached in Red Wine\n\nAppendix\nIntroduction\n\nDuring the 6 months it took me to write this book, I cooked almost exclusively with pressure cookers. Previously, I had used my pressure cooker on a fairly regular basis, but far from daily. Now that the book is just about finished, I wonder if I can go back to \"traditional\" cooking. At the same time I have to ask myself, _Why would I?_\n\nA pressure cooker can cook almost anything. It handles some ingredients better than others, but the array of foods that cook better under pressure is amazing. I'll never again cook a pork loin in anything but a pressure cooker. Custards and cheesecake will be baked under pressure from here on out. Rice? Never again in a rice cooker, when the texture of pressure cooked grains is superior. And in the pressure cooker, old-fashioned southern cornbread develops a wonderful crumb and stays so moist. Pressure cookers also are efficient. They require less energy than traditional cooking methods, and they take about a third the time necessary for other methods. And despite the general fear of pressure cookers over the years, they're actually quite safe. Thanks to advances in safety, technology, and precision manufacturing, modern pressure cookers are virtually foolproof. With multiple valves to release pressure and the knowledge that the rubber gasket is meant to release steam if pressure becomes too high, you can set aside your worries.\n\nFull of fresh, whole food ingredients, the recipes in this book are a compilation of classic American comfort foods and international classics, from beef brisket to Caribbean curried chicken to French cr\u00e8me br\u00fbl\u00e9e. There's lots to choose from\u2014and to keep you from ever becoming bored with making dinner again. You can use either a traditional stovetop pressure cooker or a more modern electric pressure cooker, and instructions for both are given in each recipe. Cooking times, temperatures, and techniques can differ by type of cooker, so be sure to read and follow the recipe closely, understanding that times may vary. The user's manual for your specific cooker is a good reference, so familiarize yourself with it if you haven't already. And note that the cook time in all the recipes is the pressure cook time. Any other cooking called for is incorporated in the prep time. A pressure level is also called out so you can see at a glance what level you need to achieve.\n\n**Acknowledgments**\n\nI have to say thanks to all those who put up with my incessant talk about how amazing pressure cooking is, about my discoveries each day, and why they should buy a pressure cooker. But most importantly, I say thanks to my wife, Amy, and my daughters, Vivian and Lynnie, who were my taste-testers and had very strong opinions about which recipes were best and should be included in the book. To my father, thanks for watching the girls so I could work, and to my mother, thanks for being a good cook yourself. Lastly, thanks to all those in the food industry, chefs, line cooks, servers, and dishwashers, who do a thankless job day in and day out, so that when I don't feel like cooking, I don't have to.\n\n**Special thanks to the technical reviewer**\n\n_Idiot's Guides: Pressure Cooking_ was reviewed by an expert who double-checked the accuracy of what's presented here to help us ensure making mouthwatering meals in your pressure cooker is as easy as it gets. Special thanks are extended to Joseph Ewing, RD, LDN.\n\nHow Pressure Cooking Works\n\nVersatile, easy-to-use pressure cookers enable you to create fresh, whole-food dishes quickly. Pressure cooking isn't a shortcut but rather a time- and energy-efficient cooking method that produces nutritious, delicious dishes.\n\nFaster cooking\n\nIt doesn't matter whether you use a traditional stovetop pressure cooker or a more modern electric version\u2014the science is the same. Rather than boiling food, a pressure cooker uses pressurized heat to cook. Liquid under pressure becomes hotter than normal; as the pressure inside the cooker rises, the temperature of the liquid rises, too\u2014up to 250\u00b0F (120\u00b0C). A pressure cooker is similar to a steam oven, but because a cooker heats at the same temperature on all sides, it cooks foods very evenly\u2014and in about a third of the time of traditional cooking methods!\n\nSafer than ever\n\nScientifically, pressure cookers work the same as they always have, but technologically, today's cookers are safer than ever. Thanks to the creation of spring-loaded pressure valves and precision manufacturing methods, the days of lids screwed down with bolts and clogged steam valves are no more. At the end of the day, however, a pressure cooker still uses valves to regulate the pressure and temperature.\n\nBrowning food in a pressure cooker before adding liquids and locking on the lid creates deeper, richer flavor in your finished dish. This is thanks to the Maillard reaction, a chemical reaction between amino acids and sugars that creates intensified flavors. The Maillard reaction really takes off at 250\u00b0F (120\u00b0C) and higher, so it occurs when a pressure cooker is at a pressure of 15 pounds per square inch (PSI), which is level 2 for traditional pressure cookers and high for electric cookers.\nWhat Your Pressure Cooker Can Do\n\nPressure cookers can do more than just one-pot meals. These multifunctional cookers are really quite versatile. Want a pot of tender beans, fluffy steamed rice, or a succulent roast? A pressure cooker does all these things efficiently and in about a third less time than conventional cooking methods.\n\nPrimary Uses\n\nPressure cookers excel at making hot, delicious dishes:\n\nCooking with Liquids Soups and stocks are ideal in pressure cookers, but cooking in liquids is more than just these dishes. Anything cooked in wine, milk, beer, or water falls into this category, too\u2014think sauerbraten, pork cooked in milk, and coq au vin. Flavored liquids add more flavor to the dish and also tenderize meats.\n\nSteaming In conjunction with a steamer basket or trivet, steaming foods in a pressure cooker is one of the tastiest ways to cook vegetables and retain many of their essential vitamins and minerals. But because the cooker cooks so quickly, not all vegetables should be cooked this way. Some more delicate vegetables quickly turn to mush in the pressure cooker.\n\nBraising and Pot Roasting In traditional cooking, braising and cooking with liquids are in the same category. But because you can use less liquid in a pressure cooker, braising is a category on its own, and the sauce that's generally served with the dish becomes so intense inside the pressure cooker, you don't have to reduce it on the stovetop. Pot roasting can be done in a pressure cooker, too, using even less liquid than required for braising.\n\nCanning Canning is another primary use for the pressure cooker. In fact, it's the only way the U.S. Department of Agriculture recommends canning low-acid foods at home. Not all pressure cookers are pressure canners, though. It has to have a 15 PSI setting, and it must be able to maintain such a setting for the entire cook time in order to can. Combo cooker\/canners are available that can do both.\n\n**Foods to Pressure Cook**\n\nMEAT\n\nAnimal proteins of all kinds become tender and stay moist in a pressure cooker.\n\nLEGUMES\n\nDried beans cook faster, retain their shape, and become perfectly tender with very little hassle.\n\nWHOLE GRAINS\n\nFrom whole grains to ground, from farro to oats, cooking grains under pressure is quicker and easier than via conventional methods.\n\nSecondary Uses\n\nYour pressure cooker also can tackle these tasks:\n\nJuice Extraction Outfitting your pressure cooker with a trivet and adding some strawberries and a small amount of water is a quick and easy way to extract juice, especially from berries.\n\nDistilling Anytime you create steam from a liquid, you can distill. You'd have to use a traditional rather than electric cooker and modify it a bit, but you can use your pressure cooker to distill flowers and other aromatics to make infused waters, such as orange water or rose water.\n\nSterilizing Your pressure cooker also can sterilize glass jars, glass baby bottles, and anything else that might need to be sanitized, as long as it fits into the cooker with the lid on and won't melt under heat.\n\n**Foods Not to Pressure Cook**\n\nPASTA\n\nCooking pasta under pressure isn't really any quicker than traditional methods, and many times the results are less than spectacular.\n\nTENDER VEGETABLES\n\nSmall, young, tender asparagus, baby green beans, baby spinach, fresh peas, and similar foods overcook in the pressure cooker.\n\nFRIED FOOD\n\nYou can use your pressure cooker to fry foods as long as you have the lid off. Never fry foods under pressure.\n\nElectric stoves take time to cool, so when cooking with a traditional pressure cooker on an electric stove, use two burners\u2014one set to the higher heat called for in the recipe, and the other set to low. Then slide your cooker from the high-heat burner to the low-heat one as necessary.\nTypes of Pressure Cookers\n\nPressure cookers come in three primary types but with many variations among them. Maybe you've inherited an old cooker, or maybe you're shopping for a new cooker, either stovetop or electric. Let's look at what you need to know to make the right choice, or make the best use of the cooker you have.\n\nOld Stovetop Cookers\n\nTraditional first-generation stovetop pressure cookers did everything from canning winter stores to cooking perfectly tender roasts. Many were made of aluminum, with lids that screwed down tight, and were large enough to hold \u00bd-gallon (2l) canning jars.\n\nThese cookers contained just one pressure valve, which was really more of a vent you placed different weights on top of to set the pressure, so they hissed steam and the valves jiggled as you cooked. Because they used a single valve, the old cookers weren't especially safe and were prone to failure. Mostly the lids popped off and food splattered everywhere. Many older models were preset at 15 PSI; new models sometimes allow you to set any pressure with a valve.\n\nStill, if you keep the valve openings clear of debris and the gaskets supple, and when used properly and with requisite caution, these cookers are safe.\n\nNew Stovetop Cookers\n\nSecond-generation traditional cookers look similar to their predecessors, but they're quite different. With more safety features, modern stovetop pressure cookers are nearly foolproof.\n\nToday's cookers work silently, are nonventing, and hiss only when the internal pressure becomes too high. The spring-loaded valves are used for safety only, as is the rubber seal housed in the lid. Many cookers have a heat-conducting bottom plate that disperses heat evenly. This allows you to bring the cooker to pressure and reduce the heat to low while still maintaining pressure. They often allow two pressure settings, 8 PSI (level 1 for traditional cookers and low for electric cookers) or 15 PSI (level 2 for traditional cookers and high for electric cookers).\n\nSecond-generation stovetop cookers are highly versatile and come in sizes to fit most home cooks' needs: 3.5 quart (3.5l), 4 quart (4l), 5 quart (5l), 6 quart (5.5l), and 8 quart (8l). Usually, you can use a larger-size cooker than the recipe calls for.\n\nElectric Cookers\n\nThe newest type of pressure cooker, many electric pressure cookers serve as multifunctional cookers: slow cookers, steamers, rice cookers, and pressure cookers all in one appliance. They work in a similar way to the new stovetop varieties, with a nonstick insert that's sealed under pressure. You set the temperature (either low or high) and pressure on a control panel on the side of the cooker.\n\nThe appeal of electric cookers is their ease and convenience. You program the cooker for stew, and it sets the time and pressure for you. Many of these cookers also have automatic on\/off, warming functions, as well as other control panel settings.\n\nSome of these cookers have drawbacks, however. There's no cold water release capability, some don't offer a saut\u00e9 mode, and the pressure settings vary from one brand to another. If you plan to do any low-acid canning, this is not the right cooker to purchase.\n\nPRESSURE COOKER ACCESSORIES\n\nWhen considering a pressure cooker to purchase, think about how you'll use it. This helps you choose the best size for your needs and also identify necessary accessories, such as perforated steamer baskets, trivets, canning equipment, and baking pans. Each one of these items has multiple uses, and you can probably discover many more as you gain pressure cooking experience.\n\nSTEAMER BASKET\n\nTRIVET\n\nCANNING RACK\nSafety and Maintenance\n\nStories abound about pressure cookers \"blowing up,\" and surely there were many cookers whose lids came off and the contents spattered all over. But with modern technology, precision manufacturing, and multiple safety valves, modern pressure cookers are as safe as any other kitchen appliance.\n\nSafety Checklist\n\nKeep the following safety points in mind when using your pressure cooker:\n\n\u25a1 Always remove the rubber gasket, check for cracks, and be sure it isn't overly brittle from use. Replace the gasket if needed.\n\n\u25a1 Check valves for any obstructions like grit or sticky residue left behind by food, and clean as necessary.\n\n\u25a1 If you hear a hiss, that means too much pressure has built up inside the cooker, usually because the heat wasn't reduced in time or to a low-enough temperature. Reduce the heat, and the hiss should stop as soon as the pressure drops.\n\n\u25a1 If you're having difficulty getting to pressure, check to be sure the lid is on properly, there's water in the pan, and the gasket is properly sealing.\n\n\u25a1 Do not leave your pressure cooker unattended.\n\nMaintenance and Cleaning\n\nAlways clean your pressure cooker thoroughly after each use. Remove the gasket, clean it with a soft sponge and a mild detergent, and let it air-dry. Clean under the rim of the lid, and rinse well to remove any grit or food grime. Hand-wash the rest of the lid, and dry the lid completely.\n\nClean the pot by hand, too, using a sponge and a soft-bristled brush if needed. Take special care to ensure the bayonet mounts for the lid (the tabs that lock the lid in place) are free of debris. Dry the pot completely.\n\nIf there's tough debris or burnt-on grime in your pressure cooker, soak it before cleaning to loosen the dirt. To deal with really tough problems, fill the pot, bring the water to a boil, and clean the pot as directed.\n\nAvoid using harsh detergents and scratch pads on your pressure cooker, and don't clean any parts of it in the dishwasher.\n\nCautions\n\nWith some care and common sense, you'll have years of successful pressure cooking. Here are some cautions to keep in mind:\n\n  * DON'T OVERFILL WITH WATER. Filling your pressure cooker more than two thirds full risks clogging the valves.\n  * DON'T OVERPACK WITH FOOD. An overly full pressure cooker won't cook the food inside evenly or efficiently. Never fill your pressure cooker more than two thirds full, and take into consideration the liquid content of your dish and the density of the ingredients.\n  * BEWARE OF THE STEAM. Steam is always present with pressure cooking, and it can burn your skin quickly. If you take a few precautions, however, you shouldn't have any problems.\n\nMost pressure cookers are designed to usher steam away from you. The quick-release valve moves the steam in the opposite direction of the handle, or anywhere your hand might reasonably be while handling a pot. However, never remove the lid while the pot is under pressure, and don't force the lid open. When you do open the lid, always use it as a shield for your hands and body by tilting up the far side of the lid as you open the pan, leaving the edge closest to you in contact with the edge of the pot. This pushes the steam away from you rather than allowing it to billow up in your face.\n\nRelease Methods\n\nReleasing the pressure in your pressure cooker correctly and safely is essential for safety and to keep the food inside from overcooking. There are three basic methods to release pressure: cold water, quick, and natural. (Be sure to read the manual for your specific pressure cooker for more information.)\n\nCold Water Release\n\nAt about 15 seconds, this is the fastest way to cool your traditional pressure cooker to a temperature low enough to safely open the lid. (This method is not suitable for electric pressure cookers because they should not be doused with water.) It's most often used with quick-cooking foods so you can get them out of the pot before they overcook.\n\nTo release pressure using the cold water method, transfer your traditional pressure cooker to the sink and position it so water runs only over the edge of the lid. Turn on the cold water, and carefully rotate the cooker in the sink, avoiding running water over the valve.\n\nQuick Release\n\nOne of the nice things about the quick release method, which takes about 1 or 2 minutes, is that there's no need to carry the cooker to the sink. If you're cooking tender vegetables such as broccoli and need to get them out of the pot immediately so they don't overcook, it's best to use the cold water release method because it's almost instant. But for most other foods, the quick release method is fine and enables you to get food to the table while it's still hot.\n\nTo release pressure using the quick release method, first remove the pressure cooker from heat or turn off the burner. Then, depending on the type of cooker you have, either turn the pressure knob so the cooker can release the steam at its own pace, or place a long-handled wooden spoon on the spring-loaded valve to release the steam. The spoon enables you to keep your hands a safe distance away from the escaping steam\u2014and a possible burn.\n\nNatural Release\n\nThe natural release method is by far the simplest, but it's also the least used in because it can take 25 to 30 minutes or longer for the pot to cool, and some foods can overcook in the process. (This method is widely used for canning, however.)\n\nTo release pressure using the natural release method, simply remove the pot from heat and let it cool on its own.\n\nNever run water directly over the valve, or you could compromise its integrity and reliability. Never submerge any pressure cooker in water. Never run water over an electric cooker. And no matter which type of cooker you have, mind the steam and stay out of harm's way.\n\nUnderstanding PSI\n\nPSI stands for _pounds per square inch_ , and it's a measure of the pressure within a pressure cooker. Higher pressure makes food cook faster and more evenly. Take a minute to learn about PSI, and in the future, all you'll need to think about is whether to set the pressure cooker to level 1 for traditional cookers\/low for electric cookers or level 2 for traditional cookers\/high for electric cookers.\n\nWhat is PSI?\n\nThe science of pressure cooking isn't complicated: when boiling water and steam are put under pressure, they become the same temperature. Apply additional heat, the pressure builds, and you get higher temperatures. What this means for you and your pressure cooker is that your food cooks evenly and from all sides. That's what makes pressure cooking different from almost all other methods of cooking. It's also what makes it very efficient.\n\nWater boils at 212\u00b0F (100\u00b0C). Steam, because it's water in gas form, is even hotter. But under pressure, water becomes superheated. At 8 PSI, water reaches 235\u00b0F (115\u00b0C) before it boils, and the resulting steam is the same temperature. At 15 PSI, water boils at 250\u00b0F (120\u00b0C).\n\nTemperature and Pressure\n\nLeft uncontrolled and under pressure, the water and steam will continue to rise in temperature. But a valve set to a specified release pressure allows steam to escape, maintaining a given PSI and, therefore, a given temperature, inside the pressure cooker.\n\nThink of PSI as your temperature knob on the stove: low, medium, and high. Different pressures control the heat inside the pot.\n\nThe Importance of Time\n\nTime plays a key role in pressure cooking, and it's crucial that you adhere to the times specified in the recipes. A pressure cooker left to its own devices can obliterate foods into a mushy mess very quickly.\n\nPressure cooking times begin when the PSI in the recipe is reached. That's why it's important to bring liquids to a boil first, lock on the lid, and wait until the pressure gauge reaches the desired PSI. Only then should you start the timer.\n\nAdjusting for Altitude\n\nIf, due to where you live, you often make altitude adjustments to recipes when cooking, you'll need to make some modifications when pressure cooking, too.\n\nIf you live above 2,000 feet (610m) elevation above sea level, you'll need to change your pressure cooker cook times. The rule is a 5 percent increase in cooking time for every 1,000-foot (305m) increase in altitude over 2,000 feet (610m). So at 3,000 feet (915m), an increase of 5 percent is needed. At 4,000 feet (1,220m), it jumps to 10 percent, and so on, as shown in the accompanying table.\n\nAltitude also affects how much cooking liquid you need. If the altitude requires a 5 percent cook time increase, the liquid requirements increase by 2.5 percent, or half the altitude increase. So if a recipe calls for \u00bd cup water, you'll need \u00be cup. There are no hard-and-fast rules on how much more liquid you need when adjusting for altitude because it somewhat depends on how much liquid will be released by the ingredients being cooked. The table here offers good starting points.\n\n**ALTITUDE ADJUSTMENTS**  \n---  \nALTITUDE | INCREASE IN COOK TIME | INCREASE IN LIQUID  \n3,000 feet (915m) | 5% | 2.5%  \n4,000 feet (1,220m) | 10% | 5%  \n5,000 feet (1,525m) | 15% | 7.5%  \n6,000 feet (1,830m) | 20% | 10%  \n7,000 feet (2,135m) | 25% | 12.5%  \n8,000 feet (2,440m) | 30% | n\/a  \n9,000 feet (2,745m) | 35% | n\/a  \n10,000 feet (3,050m) | 40% | n\/a\nPressure Cooking Vegetables\n\nA steaming bowl of vegetables cooked perfectly in a pressure cooker always arrives at the table looking vibrant and nutritious. A pressure cooker is a superstar when it comes to cooking vegetables fast while retaining vitamins and flavor.\n\nSize Matters\n\nTry to cut vegetables the same size so they finish cooking at the same time. Also keep in mind what else you might be cooking with the vegetables, like stew meat for example, and overall cooking times. You might want to cut the vegetables larger or even leave them whole so they finish cooking at the same time as the stew meat. And remember, you can always cook in stages so the vegetables aren't overcooked and the meats still become tender.\n\nQUICK-COOKING VEGGIES\n\nSome more delicate vegetables require less cook time. When a recipe calls for broccoli, asparagus, or spinach, for example, you can stop the cooking process near the end of the time specified, release the pressure, remove the lid, add the quick-cooking vegetables, and continue with the recipe.\n\nPRESSURE COOKING POTATOES\n\nPotatoes fare well when pressure cooked. The cook time depends on many variables: will the potatoes be cut, halved, cooked whole, peeled or not, mashed, used in potato salad, etc. Be sure to follow your recipe carefully when pressure cooking potatoes.\n\nCarrots cut into 1-inch (2.5cm) cylinders are best for quick-cooking stews. Try to cut any meat in the dish into pieces that will cook in the same amount of time.\n\nFingerling potatoes are great for pressure cooking. Just Cut in half lengthwise\u2014no peeling necessary.\n\nTRIVETS AND STEAMER BASKETS\n\nWhen steaming vegetables, use a trivet or steamer basket, and always add as little water as your cooker allows to still operate safely. This is often \u00bd to \u00be cup.\n\nIt might take a bit of practice until you can produce properly cooked vegetables. Be patient, and pay attention to the cooking process. Make mental notes as you work, and above all else, keep trying.\n\nDO NOT OVERFILL\n\nTo ensure even cooking, do not overfill the steamer basket with vegetables. The key is to leave enough space around the food so the steam can get into all the nooks and crannies. Flat cuts such as potato rounds are especially prone to undercooking because they can stick together.\n\nPressure Cooking Meat\n\nSave the expensive, tender cuts of meat for the grill. Your pressure cooker transforms lesser, inexpensive cuts of meat into great-tasting stews, succulent braises, and tender pot roasts.\n\nBut just because a pressure cooker does wonders with inexpensive cuts of meat doesn't mean you can use cheap meat. If a cut has a lot of sinew (or silverskin), remove it with a knife before cooking. Also remove any bruised or discolored areas you see. Let pressure cooked meat rest for 10 minutes before slicing or shredding it to allow the collagen in the meat to relax, making the meat more tender. And always slice meat against the grain.\n\nChoosing and Prepping Meat\n\nPerhaps the most important thing to look for when buying red meat for the pressure cooker is marbling, or fat content. Choose pieces that contain lots of white veins throughout. Fat is what flavors meat, and even in a pressure cooker, meat without the right fat content can become dry.\n\nIt's also a good idea to season meat with salt up to 24 hours before pressure cooking it. This gives the salt time to penetrate the meat's protein, which enhances the flavor and helps the meat retain moisture.\n\nMeat browns better if the surface is dry. At the very least, pat meat dry with a paper towel before searing.\n\nCHICKEN Whole chickens do very well in the pressure cooker. A 3-pound (1.5kg) chicken cooks to tender quickly and easily. Bone-in thighs and legs are fantastic as well.\n\nBEEF Chuck roasts are great for the pressure cooker. Left whole, cut into stew meat, or ground for meatballs, the fat-to-protein ratio is ideal, and they become tender and juicy in no time.\n\nPORK Pork, cured and fresh, is perfect for the pressure cooker. Be aware that bacon cooked under pressure for long periods of time develops an odd texture. To overcome this, remove the bacon after rendering it crispy and put it back at the end of the cook time to warm.\n\nSEAFOOD Seafood cooks very fast under pressure, which makes it ideal for busy weeknight meals. Opt for firmer fish like tuna, salmon, and mackerel that can be cut into 1-inch (2.5cm) cubes for curries and Asian-style dishes. Your pressure cooker also makes short work of shellfish.\n\nPressure Cooker Frying\n\nYou can use your pressure cooker for saut\u00e9ing, to sear meats before adding the liquids and other ingredients and locking on the lid. However, you should never put large amounts of hot oil under pressure by locking on the lid, as you would for fried chicken. This is dangerous because the heat builds up quickly, the valves become clogged with oil, and the rubber seals can become compromised, all of which can cause failure and the possibility of an explosion.\n\nConverting Conventional Recipes\n\nAdapting your favorite recipes for pressure cooking isn't complicated. In fact, as you realize how much time pressure cooking saves you over conventional methods, and as you become comfortable and familiar with operating a pressure cooker, you'll be able to make these conversions pretty easily.\n\nConversion Tips\n\nUse these tips to help you adapt traditional recipes to the pressure cooker:\n\nADD DAIRY LATER With the exception of baked dishes such as cakes and custards, it's often best to add dairy at the end of the cook time to keep it from curdling or scorching.\n\nCOOK IN PHASES Cook the meat first, release the pressure and open the cooker, add the vegetables, replace the lid, and finish cooking.\n\nTIER INGREDIENTS Stack multiple steamer baskets inside your pressure cooker to cook your main dish and sides simultaneously and keep the flavors separate.\n\nUSE PANS Pour cake batter into a baking pan, cover it with aluminum foil, and bake it in your pressure cooker.\n\nADJUST THE WATER Foods give off a lot of liquid under pressure, and it can't evaporate inside the sealed cooker. So when converting, use far less liquid than in the original recipe\u2014often 1 cup total is sufficient.\n\nREDUCE FOAMING Some foods like rice, beans, and other starches foam when cooked. Add 1 teaspoon oil or fat to reduce the amount of foam and avoid clogging the valve.\n\nBURN OFF ALCOHOL If you're cooking with wine, burn off the alcohol before locking on the lid. Otherwise, the vapors stay trapped inside the cooker, and the wine remains raw.\n\nAVOID OVERFILLING Never fill a pressure cooker more than two thirds full. Overfilling yields an unevenly and incompletely cooked final dish.\n\nREDUCE THE COOK TIME Under pressure at 15 PSI, food often cooks in a third of the time of traditional methods. If it takes an hour conventionally, it'll take 20 minutes under pressure. Still, look for a recipe that's similar to the one you're converting to check the recommended time.\n\nUSE YOUR MANUAL Make use of your owner's manual cook time and other charts. By looking at the different PSI and cook times of individual ingredients, you can figure out the best way to cook a traditional recipe in your pressure cooker.\n\nPreparation Tips\n\nHere are some helpful hints to keep in mind while preparing a dish:\n\nVEGETABLES Grated and small-cut vegetables don't fare well under pressure. The appropriate-size cut is key to pressure cooking success. When preparing vegetables, look at the cook time and cut sizes and try to match them so everything cooks evenly and in the same amount of time. Or cook food in stages if need be.\n\nMEATS Bone-in meats are more flavorful, and it's a good idea to use them whenever you can. When that's not possible and the meat needs to be cubed (not all does), a 1-inch (2.5cm) square piece is the minimum size.\n\nAlso, brown meat first before adding any other ingredients. You can remove it from the cooker after browning and before saut\u00e9ing any vegetables and return it later when you're ready to lock on the lid.\n\nCHICKEN Chicken is hard to cut off the bone, but often it's best to cook the chicken first, remove it from the pot, pick off the meat, and return the meat to the pot. Chicken can be cooked with skin on or off for almost all recipes. Skin adds lots of flavor, but not everyone likes it.\n\nStocking Your Kitchen\n\nStocking your kitchen for pressure cooking makes preparing weekday meals quicker and easier. In addition to the meats mentioned earlier, whole grains, beans, and other long-cooking ingredients often reserved for weekends, when time constraints are fewer, can now become midweek meals.\n\nGrains\n\nWhole grains, ancient grains, and gluten-free grains are widely available. Oat groats, farro, wheat berries, spelt, buckwheat, millet, and quinoa (which is actually a seed but is cooked like a grain) are all well suited to pressure cooking. Brown rice, farro, and spelt are best stored in the freezer, but others are just fine on the pantry shelf.\n\nWhole grains contain more nutrients than processed versions, have a lower glycemic index, and are absorbed at a slower rate in your body. With your pressure cooker, you can easily add these good-for-you ingredients to more meals.\n\nBeans\n\nCooking dried beans in your pressure cooker is faster, easier, and less expensive; requires less hands-on work than traditional cooking; and yields a better-tasting result. Try a variety of heirloom and standard varieties such as black turtle beans, pinto beans, cannellini beans, lentils (especially du Puy), and chickpeas.\n\nDon't fret if you forget to soak beans overnight. You can quick soak dried beans by covering them in 4 cups water for every 1 cup dried beans and pressure cook for 2 minutes at level 2 for traditional cookers or on high for electric cookers. Use the cold water release method for traditional cookers and the quick release method for electric cookers, drain the beans, and continue as directed in your recipe.\n\nSpices and Herbs\n\nPressure cooking produces flavorful foods, but you'll still want to season with spices and herbs.\n\nStock up on bay leaves, chili powder, cloves, cocoa powder, cumin, curry powder, ginger, kosher salt, paprika, vanilla extract (vanilla paste is really wonderful), and whole black peppercorns (a grinder, too) as well as fresh and dried herbs like oregano, rosemary, and thyme. Spice blends like Old Bay or a Cajun blend are also nice to have.\n\nProduce\n\nStorage vegetables stay fresh for weeks in the refrigerator or root cellar and can be pressure cooked for fast, hearty dishes. Carrots, celery, garlic, onions, and potatoes are excellent to have on hand. Also consider bell peppers, chiles, and green onions.\n\nFor fruit, stock dried apples, apricots, blueberries, cherries, cranberries, prunes, raisins, and about anything else you like. These cook quick in compotes, can be made into spreads, and are a great addition to puddings.\n\nSalt is by far the most powerful flavoring tool in the kitchen and can do so many things, from flavoring to curing. Many types are available, and to decide which is best, you can do a tasting. Table salt with iodine is often very bitter, kosher salt has a mild taste and is clear when it dissolves, and sea salt comes in many flavors and prices. The recipes in this book use kosher salt. If you use iodized table salt instead, use half as much as what's called for. And before adding more salt to a dish, do a quick taste test. If a recipe calls for a pinch of salt, use your thumb and index finger to measure. For a two-finger pinch, use your thumb, index finger, and middle finger.\nPlanning Weeknight Meals\n\nCreating a menu plan makes cooking nutritious meals throughout the week easy, helps you maximize your time, and uses up leftovers. Your pressure cooker can make quick and easy work of cooking your week's meals.\n\nChicken\n\nChicken is so versatile, and cooking whole chickens in your pressure cooker is very quick: a 3-pound (1.5kg) bird cooks in as little as 15 to 20 minutes. If your pressure cooker is big enough (8 quarts; 7.5l or larger), you can cook two birds at once.\n\nCHICKEN AND DUMPLINGS _Chicken and Dumplings_ is a comfort food classic, and cooking two chickens yields an intense broth, with plenty of extra broth you can use throughout the week. If you don't use it all right away, freeze it for later.\n\nCHICKEN SALAD After you've pulled the chicken from the bones, you have a stockpile of meat you can use in many dishes. Try _Chicken Salad Deluxe_. It's good as a lunch or light dinner.\n\nCHICKEN TACOS Leftover chicken is perfect for chicken tacos. Try it in _Pressure Cooker Tacos_. Refer to the recipe for toppings, or create your own.\n\nBeef\n\nCuts of beef that shred well (think chuck, flank, and brisket) make versatile weeknight meal options.\n\nBRISKET _Classic Beef Brisket_ is a dinnertime staple. Tender and coated with au jus and melted onions, brisket is perfect in the pressure cooker. Double the amount to get more meals out of this cut.\n\nSOUP Make _Mushroom Barley Soup_ more of a meal by adding shredded beef. Serve with a salad and some good bread.\n\nBARBECUE Warm up leftover shredded beef in leftover _au jus_ and add your favorite barbecue sauce. Toast some buns, make some fresh and tangy coleslaw, and you have a great weeknight meal.\n\nPork\n\nYou have many options when it comes to pork. Use a butt roast, or two, and you have the basis of many fast and flavorful weeknight dishes.\n\nPULLED PORK Cook a double batch of _Southern-Style Pulled Pork_ , add sauce to only what you're going to eat that night, and save the rest for other meals.\n\nPORK RAMEN Having pulled pork on hand makes it easy to prepare quick and easy pressure cooked _Pork Ramen_.\n\nTHAI-STYLE GREEN CURRY PORK Of course, you could do something completely different and make the _Thai-Style Green Curry Chicken_ and substitute pork for the chicken. Simply make the broth and add the pork at the end to warm it through.\n\nGrains and Vegetables\n\nPressure cooked grains and vegetables are so much more colorful and vibrant, and they retain much of their flavor and vitamins. Beans, too. It's so easy to pressure cook a big batch once and use them in various healthy, filling salads or side dishes all week long.\n\nRICE PILAF Cook a double pot of _Basmati Rice Pilaf_ , and use it as a side dish for several weeknight dinners.\n\nVEGETABLE BURRITOS OR MEXICAN BOWLS Having rice on hand enables you to make burritos, either vegetarian or with meat. Simply warm tortillas, spread on some warm rice, and top with your favorite burrito ingredients. Or make a Mexican bowl and skip the tortilla altogether.\n\nCABBAGE, RICE, AND LENTILS Make _One-Pot Cabbage, Rice, and Lentils_ by adding the rice at the end, warming it through, and not allowing it to overcook. You also can add some shredded brisket with the rice for the meat lovers in your family.\n\nRich Beef Stock\n\nHomemade beef stock is rich and flavorful. Use it as a base for soups, reduce it for rich sauces, or drink a hot cup to warm you on chilly days.\n\nIngredients\n\n4 lb. (2kg) meaty beef bones\n\n1 large white onion, cut into quarters (1\u00bd cups)\n\n2 large carrots, peeled and cut into 1-in. (2.5cm) chunks (\u00be cup)\n\n4 medium stalks celery, chopped (\u00be cup)\n\n2 tsp. whole black peppercorns\n\n1 bay leaf\n\n2 TB. fresh flat-leaf parsley\n\n16 cups cold water\n\nMETHOD\n\n1 In an 8-quart (7.5l) pressure cooker, combine meaty beef bones, white onion, carrots, celery, black peppercorns, bay leaf, and flat-leaf parsley. Carefully pour in cold water, set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 55 minutes. Turn off heat, let cool, perform a natural release, and remove the lid.\n\n3 Strain stock with a fine mesh strainer. Use within 3 to 5 days, can in jars, or freeze.\n\nNutrition per serving\n\nCalories: 72\n\nCarbohydrates: 5g\n\nSugars: 4g\n\nDietary fiber: 2g\n\nProtein: 5g\n\nFat: 2g\n\nCholesterol: 18mg\n\nSodium: 90mg\n\nChicken Stock\n\nHomemade chicken stock is rich, flavorful, and filling. Leave on the onion skin when making this version; it helps produce a nice, golden-colored stock.\n\nIngredients\n\n4 lb. (2kg) chicken carcasses (about 3 carcasses)\n\n1 large yellow onion, skin on and cut into quarters (1 cup)\n\n2 large carrots, peeled and cut into chunks (1 cup)\n\n4 small stalks celery, chopped (1 cup)\n\n2 tsp. whole black peppercorns\n\n1 bay leaf\n\n2 TB. fresh flat-leaf parsley\n\n16 cups cold water\n\nMETHOD\n\n1 In an 8-quart (7.5l) pressure cooker, combine chicken, yellow onion, carrots, celery, black peppercorns, bay leaf, and flat-leaf parsley. Add cold water, set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 55 minutes. Turn off heat, let cool, perform a natural release, and remove the lid.\n\n3 Strain stock with a fine mesh strainer. Use within 3 to 5 days, can in jars, or freeze.\n\nFor a rich, brown stock, roast the chicken bones in a 400\u00b0F (200\u00b0C) oven for 40 minutes or until very brown. Transfer to the pressure cooker, and continue as directed.\n\nNutrition per serving\n\nCalories: 12\n\nCarbohydrates: 2g\n\nSugars: 0g\n\nDietary fiber: 0g\n\nProtein: 1g\n\nFat: .3g\n\nCholesterol: 0mg\n\nSodium: 130mg\nVegetable Stock\n\nMaking a vegetable stock that doesn't turn out sweet and oniony can be tricky. For best results, use tomatoes and dried mushrooms such as shiitakes.\n\nIngredients\n\n2 large green onions, trimmed\n\n3 medium fresh tomatoes, halved\n\n2 small white onions, halved (1\u00bd cups)\n\n6 dried shiitake mushrooms (1 cup)\n\n4 cloves garlic, halved\n\n3 medium carrots, peeled and cut into chunks (1 cup)\n\n2 medium stalks celery, peeled and cut into chunks (\u00be cup)\n\n3 sprigs fresh flat-leaf parsley\n\n1 tsp. whole black peppercorns\n\n1 bay leaf\n\n\u00bd tsp. dried thyme\n\n16 cups water\n\nMETHOD\n\n1 In an 8-quart (7.5l) pressure cooker, combine green onions, tomatoes, white onions, shiitake mushrooms, garlic, carrots, celery, flat-leaf parsley, black peppercorns, bay leaf, and thyme. Add water, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 30 minutes. Turn off heat, let cool, perform a natural release, and remove the lid.\n\n3 Strain stock with a fine mesh strainer. Use immediately or refrigerate in glass containers for up to 3 days.\n\nNutrition per serving\n\nCalories: 15\n\nCarbohydrates: 3g\n\nSugars: 2g\n\nDietary fiber: 1g\n\nProtein: 0g\n\nFat: 0g\n\nCholesterol: 0mg\n\nSodium: 140mg\nPressure Cooker Eggs\n\nPressure cooked eggs peel perfectly, the cooking times are more exact, and you don't get a sulfur ring around the yolk.\n\nIngredients\n\nLarge eggs, cool (34\u00b0F; 1\u00b0C)\n\nMETHOD\n\n1 Add a trivet or steamer basket to any size pressure cooker, and add the minimum amount of water allowed for your cooker. Set heat to medium-high (traditional)\/high (electric), add eggs to trivet or basket, and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook as directed for egg doneness (see accompanying table). Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Serve soft- and medium-boiled eggs immediately.\n\n4 For hard-boiled, remove eggs from the cooker and immediately plunge in an ice-water bath. Cool eggs completely, drain, dry, and peel as needed.\n\nEgg Doneness | Cook Time\n\n---|---\n\nSoft yolk | 3 minutes\n\nMedium yolk | 4 minutes\n\nHard boiled | 6 minutes\n\nI keep six bottle caps in my kitchen to use for egg stands when pressure cooking. Standing the eggs fat end down on the upturned bottle caps gives them the perfect finished shape, which is great for making deviled eggs.\n\nNutrition per serving\n\nCalories: 72\n\nCarbohydrates: .5g\n\nSugars: 0g\n\nDietary fiber: 0g\n\nProtein: 6g\n\nFat: 5g\n\nCholesterol: 186mg\n\nSodium: 71mg\n\nPressure Cooker Rice\n\nYour pressure cooker cooks rice to tender and fluffy perfection. For best results, find a brand and type of rice you like and stick with it.\n\nIngredients\n\n2 cups jasmine rice\n\n3 cups water\n\n1\u00bd tsp. vegetable oil or unsalted butter\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine jasmine rice, water, and vegetable oil. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Serve hot. Refrigerate leftovers for up to 3 days, or freeze.\n\nFor brown rice, use 2 cups brown jasmine rice, and cook it for 20 to 22 minutes at level 2\/high.\n\nNutrition per serving\n\nCalories: 170\n\nCarbohydrates: 36g\n\nSugars: 0g\n\nDietary fiber: 1g\n\nProtein: 3g\n\nFat: 1g\n\nCholesterol: 0mg\n\nSodium: 0mg\nEasy Dried Beans\n\nCooking a simple pot of dried beans isn't difficult. Soak the beans overnight for a faster cook, and add salt before cooking for a more flavorful finished dish.\n\nIngredients\n\n8 oz. (225g) dried pinto beans, washed, picked over, soaked overnight, and drained\n\n1 small yellow onion, halved (\u00bd cup)\n\n1 tsp. kosher salt\n\n1 TB. olive oil, lard, bacon grease, or unsalted butter\n\n\u00bc tsp. freshly ground black pepper\n\nWater\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine pinto beans, yellow onion, kosher salt, olive oil, and black pepper. Add enough water to cover beans by 1 inch (2.5cm). Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n3 Serve hot.\n\nInstead of soaking overnight, a quick soak speeds up the process: combine 1 cup dried beans and 4 cups water in the pressure cooker, and set heat to medium-high (traditional)\/high (electric). Lock on the lid, bring to pressure level 2 (traditional)\/high (electric), and cook for 2 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and proceed as directed.\n\nNutrition per serving\n\nCalories: 230\n\nCarbohydrates: 34g\n\nSugars: 2g\n\nDietary fiber: 9g\n\nProtein: 14g\n\nFat: 3.5g\n\nCholesterol: 0mg\n\nSodium: 490mg\nPressure Cooker Pasta Sauce\n\nThis all-purpose tomato sauce is ideal for pasta. The vegetables provide plenty of flavor and substance, but feel free to add shredded beef or pork if you like.\n\nIngredients\n\n\u00bc cup extra-virgin olive oil\n\n2 medium leeks, white and a little green parts, diced small (1\u00bc cups)\n\n3 medium stalks celery, halved lengthwise and diced small (\u00be cup)\n\n2 medium carrots, peeled and diced small (\u00be cup)\n\n2 TB. minced garlic\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground white pepper\n\n3 cups tomato sauce\n\n1 (6-oz.; 175ml) can tomato paste\n\n1 tsp. dried oregano\n\n1 tsp. dried basil\n\n\u00bd tsp. dried thyme\n\nWater\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat extra-virgin olive oil. When hot, add leeks, celery, carrots, and garlic. Stir, season with kosher salt and white pepper, and cook for 4 minutes or until vegetables become soft but garlic doesn't brown.\n\n2 Stir in tomato sauce, tomato paste, oregano, basil, and thyme, and bring sauce to a boil.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 14 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Taste, and add more salt as necessary. If sauce is thicker than you like, add water 1 tablespoon at a time, stirring between each addition.\n\n5 Serve hot. Refrigerate any leftovers in a jar for up to 1 week.\n\nNutrition per serving\n\nCalories: 120\n\nCarbohydrates: 13g\n\nSugars: 8g\n\nDietary fiber: 3g\n\nProtein: 3g\n\nFat: 7g\n\nCholesterol: 0mg\n\nSodium: 790mg\n\nEggs en Cocotte \u00e0 la Cr\u00e8me\n\nHeavy cream makes these eggs very rich. Best reserved for special occasions rather than every day, they're ideal for holiday brunches.\n\nIngredients\n\n8 tsp. heavy cream\n\n4 large eggs, cool\n\n\u00bd tsp. kosher salt\n\n8 medium asparagus spears, trimmed and root end peeled\n\n4 slices prosciutto, halved lengthwise for 8 pieces\n\n1 tsp. unsalted butter\n\n1 medium green onion, minced (optional)\n\nMETHOD\n\n1 Add a steamer basket to a 4-quart (4l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 Using \u00bc teaspoon unsalted butter each, lightly grease 4 (4-ounce; 120ml) ramekins, espresso cups, or other heat-proof vessels. Pour 1 teaspoon heavy cream in each dish.\n\n3 One at a time, crack 1 egg into each dish, season it with 1 pinch kosher salt, and pour another 1 teaspoon heavy cream on top of each egg. Cover each cup tightly with aluminum foil, and place cups in the steamer basket. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 2 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, remove the lid, and carefully remove egg cups.\n\n5 Meanwhile, spiral wrap each asparagus spear with 1 piece of prosciutto, and rub with \u215b teaspoon unsalted butter each.\n\n6 In a saut\u00e9 pan over medium-high heat, sear prosciutto-asparagus spears for 3 minutes.\n\n7 Place 1 egg cup on a plate with 2 spears asparagus to dip in yolks, garnish with green onion (if using), and serve.\n\nNutrition per serving\n\nCalories: 150\n\nCarbohydrates: 3g\n\nSugars: 1g\n\nDietary fiber: 1g\n\nProtein: 10g\n\nFat: 12g\n\nCholesterol: 235mg\n\nSodium: 560mg\n\nEgg Cups\n\nWith their shrimp and andouille sausage, these egg cups are reminiscent of New Orleans and a good jazz brunch.\n\nIngredients\n\n1 TB. vegetable oil\n\n6 shrimp (size 26 to 30), shelled, deveined, and diced small\n\n\u00bd small green bell pepper, ribs and seeds removed, and diced small (\u00bc cup)\n\n1 small stalk celery, diced small (\u00bc cup)\n\n1 small shallot, minced (\u00bc cup)\n\n2 tsp. minced garlic\n\n3 oz. (75g) cooked andouille sausage, diced small\n\n\u00bd cup whole milk\n\n6 large eggs\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 TB. fresh chives, chopped\n\nMETHOD\n\n1 In a medium saut\u00e9 pan over medium-high heat, heat vegetable oil. When hot, add shrimp, green bell pepper, celery, shallot, garlic, and andouille sausage, and saut\u00e9 for about 4 minutes or until softened. Remove from heat.\n\n2 In a large bowl, combine whole milk, eggs, kosher salt, and black pepper until eggs are smooth and blended.\n\n3 Grease the bottoms and sides of 6 (4-ounce; 110g) ramekins with about 1 tablespoon unsalted butter. Divide shrimp and andouille filling among ramekins, cover each tightly with aluminum foil, and place in a steamer basket.\n\n4 Add the basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Do not allow water to touch bottom of ramekins. Set heat to high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, and set aside for 5 minutes. Perform a quick release, remove the lid, and lift out the steamer basket.\n\n6 Uncover ramekins, top with chives, and serve.\n\nNutrition per serving\n\nCalories: 140\n\nCarbohydrates: 5g\n\nSugars: 2g\n\nDietary fiber: 0g\n\nProtein: 8g\n\nFat: 10g\n\nCholesterol: 235mg\n\nSodium: 250mg\nApple Cinnamon Breakfast Oats\n\nThis sweet and hearty oatmeal is more like a dessert than breakfast. Pressure cooking steel-cut oats takes a fraction of the time conventional methods require.\n\nIngredients\n\n1 large pink lady apple, cut in \u00bc-in. (.5cm) pieces (1 cup)\n\n\u2153 cup raisins\n\n\u00bd cup apple juice\n\n1 TB. brown sugar\n\n1 cup steel-cut oats\n\n2 cups water\n\n\u00bc tsp. cinnamon\n\n1 pinch cardamom\n\n1 pinch kosher salt\n\n2 tsp. unsalted butter\n\n2 TB. heavy cream\n\nMETHOD\n\n1 In a small saucepan over medium-high heat, combine pink lady apple, raisins, and apple juice. Bring to a boil, reduce heat to medium-low, and simmer for 3 minutes.\n\n2 In a 4-quart (4l) pressure cooker, combine apple mixture, brown sugar, steel-cut oats, water, cinnamon, cardamom, kosher salt, and 1 teaspoon unsalted butter. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/natural (electric) release, and remove the lid.\n\n4 Stir oats until smooth and well combined. Divide between 2 bowls, top each with remaining \u00bd teaspoon unsalted butter and 1 tablespoon heavy cream, and serve.\n\nNutrition per serving\n\nCalories: 280\n\nCarbohydrates: 52g\n\nSugars: 22g\n\nDietary fiber: 6g\n\nProtein: 7g\n\nFat: 7g\n\nCholesterol: 15mg\n\nSodium: 45mg\nSavory Parmesan Steel-Cut Oats\n\nIf sweet breakfasts aren't your thing, this savory oatmeal is a nice alternative. Black pepper and Parmesan cheese are a natural pairing with oats.\n\nIngredients\n\n1 cup steel-cut oats\n\n2 cups water\n\n\u00bc tsp. kosher salt\n\n1 TB. unsalted butter\n\n4 TB. cup grated Parmesan cheese\n\n1 TB. chopped fresh chives\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 Place steel-cut oats in a strainer, and rinse with cold running water.\n\n2 Transfer oats to a 4-quart (4l) pressure cooker, and add water, kosher salt, and 1 teaspoon unsalted butter. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Divide oats between 2 bowls. Evenly divide remaining unsalted butter, Parmesan cheese, chives, and black pepper over top, and serve.\n\nFor protein-powered oats, pour 1 tablespoon hot beef broth over the top, cover with 1 fried egg, and serve.\n\nNutrition per serving\n\nCalories: 370\n\nCarbohydrates: 54g\n\nSugars: 0g\n\nDietary fiber: 8g\n\nProtein: 16g\n\nFat: 14g\n\nCholesterol: 25mg\n\nSodium: 400mg\n\nFive-Grain Oatmeal\n\nThis hot, multigrain cereal is a warm and comforting start on cold mornings. It also makes a great side dish alongside braised meats.\n\nIngredients\n\n\u00bc cup steel-cut oats\n\n\u00bc cup coarse-ground cornmeal\n\n\u00bc cup buckwheat groats\n\n\u00bc cup white-rice grits or short-grain white rice\n\n\u00bc cup whole millet\n\n2\u00bd cups water\n\n\u00bd tsp. kosher salt\n\n\u00bd TB. unsalted butter\n\n\u00bd cup heavy cream\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine steel-cut oats, coarse-ground cornmeal, buckwheat groats, white-rice grits, millet, water, and kosher salt. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Serve hot with unsalted butter and heavy cream on the side.\n\nCoarse-ground cornmeal is sometimes called corn grits. Don't confuse it with plain grits, though. They're not the same.\n\nNutrition per serving\n\nCalories: 320\n\nCarbohydrates: 44g\n\nSugars: 0g\n\nDietary fiber: 5g\n\nProtein: 7g\n\nFat: 14g\n\nCholesterol: 45mg\n\nSodium: 250mg\nSausage, Onion, and Gruy\u00e8re Breakfast Casserole\n\nEggs cooked in a pressure cooker have a tender, custardlike texture. Here, they provide a lovely foundation for the Gruy\u00e8re cheese, onion, and sausage.\n\nIngredients\n\n4 slices white bread, cut into 1-in. (2.5cm) cubes (2 cups)\n\n1\u00bd cups breakfast sausage, cooked and crumbled\n\n2 cups grated Gruy\u00e8re cheese\n\n2 medium green onions, chopped (\u00bc cup)\n\n1 tsp. unsalted butter\n\n1 small yellow onion, diced small (\u00bd cup)\n\n6 large eggs, beaten\n\n1 cup whole milk\n\n1 TB. Dijon mustard\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Grease an 8- or 9-inch (20 or 23cm) cake pan (whatever fits in your cooker with space at the edges).\n\n2 In a large bowl, combine white bread, breakfast sausage, Gruy\u00e8re cheese, and green onions.\n\n3 In a small saut\u00e9 pan over medium heat, melt unsalted butter. Add yellow onion, and cook for about 5 minutes or until onion is soft. Transfer onion to the bowl, toss to combine, and spread mixture in the cake pan.\n\n4 In the same bowl, whisk together eggs, whole milk, Dijon mustard, kosher salt, and black pepper. Pour eggs over bread, and set aside for 10 minutes.\n\n5 Cover the pan tightly with aluminum foil, and place in the pressure cooker. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 35 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n7 Lift casserole out of the cooker, uncover, and serve.\n\nNutrition per serving\n\nCalories: 310\n\nCarbohydrates: 8g\n\nSugars: 2g\n\nDietary fiber: 0g\n\nProtein: 21g\n\nFat: 22g\n\nCholesterol: 225mg\n\nSodium: 710mg\n\nHoney, Raisin, and Quinoa  Breakfast Risotto\n\nQuinoa is surprisingly nice and nutty, and the addition of honey and raisins makes this a lovely breakfast treat.\n\nIngredients\n\n1 cup quinoa, rinsed\n\n2 cups water\n\n\u00bd tsp. kosher salt\n\n2 TB. unsalted butter\n\n\u00bd cup raisins\n\n2 tsp. honey\n\n\u00bd cup whole milk\n\n2 TB. heavy cream (optional)\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, combine quinoa, water, and kosher salt. Bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Stir in unsalted butter and raisins. Add honey, whole milk, and heavy cream (if using), and stir. If risotto is too thick, add more milk 2 tablespoons at a time. Taste and season with salt as necessary, and serve.\n\nInstead of raisins, you can use dried blueberries.\n\nNutrition per serving\n\nCalories: 240\n\nCarbohydrates: 32g\n\nSugars: 7g\n\nDietary fiber: 3g\n\nProtein: 7g\n\nFat: 9g\n\nCholesterol: 20mg\n\nSodium: 260mg\nBacon Jam\n\nThis smoky spread is rich, satisfying, and great for brunch. Although the recipe isn't hard, it is a bit time-consuming, so be patient and don't rush it.\n\nIngredients\n\n1 TB. vegetable oil\n\n2 medium yellow onions, diced (1\u00bd cups)\n\n\u00bc tsp. kosher salt\n\n\u215b tsp. freshly ground black pepper\n\n1 lb. (450g) hickory-smoked bacon, minced\n\n\u00bd cup water\n\n1 TB. brown sugar\n\n2 TB. cider vinegar\n\n2 TB. pure maple syrup\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add yellow onions, and saut\u00e9 for 10 minutes or until onions wilt. Reduce heat to medium-low (traditional)\/low (electric), season with kosher salt and black pepper, and cook for about 30 minutes or until onions are very brown.\n\n2 Add bacon, and slowly render fat, allowing it to become crisp.\n\n3 Add water, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Drain off \u00bd of liquid. Return jam to medium-low\/low heat, and simmer for 10 minutes or until liquid has evaporated.\n\n6 Cook, stirring constantly, for 30 minutes or until jam all falls apart and turns brown.\n\n7 Add brown sugar, cider vinegar, and maple syrup, and cook for 10 minutes or until liquid is absorbed.\n\n8 Serve hot. Refrigerate leftovers in an airtight jar for up to 2 weeks, and warm before serving.\n\nNutrition per serving\n\nCalories: 100\n\nCarbohydrates: 2g\n\nSugars: 2g\n\nDietary fiber: 0g\n\nProtein: 2g\n\nFat: 9g\n\nCholesterol: 15mg\n\nSodium: 180mg\n\nClassic Deviled Eggs\n\nThe piquant flavor of deviled eggs is always a party pleaser. Pressure cooked eggs peel easily and perfectly every time, ensuring a lovely finished dish.\n\nIngredients\n\n8 large eggs, at room temperature\n\n\u00be TB. bread-and-butter pickle juice, or apple cider vinegar\n\n\u00bc cup mayonnaise\n\n2 tsp. Dijon mustard\n\n\u00bd tsp. kosher salt\n\n2 red or green serrano chiles, sliced into thin rings\n\n1 TB. fresh chives, sliced thin\n\n10 small celery leaves\n\nFreshly ground black pepper\n\nMETHOD\n\n1 Add a trivet to the bottom of a 4-quart (4l) pressure cooker. Equally space 8 bottle caps on the trivet, flared side up, and stand 1 egg in each cup, fat end down. Add enough water to the pressure cooker to reach the bottom of the trivet, set heat to medium-high (traditional)\/high (electric), and bring to a hard boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 6 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Run cold water over eggs until they're cool enough to handle, and remove from the pressure cooker. Peel eggs, rinse, and cut in half.\n\n4 Transfer yolks to a medium bowl, and mash with a fork. Add bread-and-butter pickle juice, mayonnaise, Dijon mustard, and kosher salt, and blend to a smooth paste.\n\n5 Spoon yolk mixture back into 10 egg white halves (you'll have a few halves left over). To serve, garnish with serrano chiles, chives, and celery leaves, and season with black pepper. Refrigerate leftovers for up to 3 days.\n\nIf you have a pastry bag, you can pipe the yolk mixture back into the whites. Use a small bag fitted with a star tip.\n\nNutrition per serving\n\nCalories: 190\n\nCarbohydrates: 2g\n\nSugars: 0g\n\nDietary fiber: 0g\n\nProtein: 10g\n\nFat: 16g\n\nCholesterol: 350mg\n\nSodium: 420mg\n\nGarlicky White Bean and Parmesan Dip\n\nGarlic, rosemary, and Parmesan cheese star in this dip. Cooking the beans in your pressure cooker yields a creamier texture and a shorter cook time.\n\nIngredients\n\n1 cup dried cannellini beans, washed, picked over, soaked overnight, and drained\n\nWater\n\n1 tsp. kosher salt\n\n6 cloves garlic\n\n1\u00bd tsp. dried rosemary\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bc cup grated Parmesan cheese\n\n1 TB. extra-virgin olive oil\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, place cannellini beans. Add enough water to cover by \u00bd inch (1.25cm), and stir in kosher salt, garlic, rosemary, and black pepper. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Drain beans, and transfer to a food processor fitted with a metal chopping blade. Add Parmesan cheese, and pulse for 20 seconds or until dip is smooth and creamy. If dip seems too thick, add extra-virgin olive oil and pulse again until blended.\n\n4 Transfer dip to a bowl, and serve warm with pita bread or tortilla chips or as a side dish with roasted meats.\n\nIf you're eating Paleo, this recipe fits that diet's requirements.\n\nNutrition per serving\n\nCalories: 80\n\nCarbohydrates: 11g\n\nSugars: 0g\n\nDietary fiber: 3g\n\nProtein: 5g\n\nFat: 2g\n\nCholesterol: 0mg\n\nSodium: 230mg\nRed Beet Hummus\n\nThis hummus is a lovely change in color and flavor from chickpea versions. The beet adds a sweetness, and the cashews yield a nutty creaminess.\n\nIngredients\n\n1 large beet\n\n\u00be cup raw cashews\n\n1\u00bd TB. water\n\n\u00bd tsp. crushed cumin\n\n1 TB. freshly squeezed lemon juice\n\n\u00bd tsp. kosher salt\n\n2 tsp. extra-virgin olive oil\n\nMETHOD\n\n1 Add a steamer basket to a 4-quart (4l) pressure cooker, add the minimum amount of water allowed for your cooker. Add beet to the basket, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, and let beet sit for 10 minutes. Perform a natural release, and remove the lid.\n\n3 Meanwhile, in a food processor fitted with a metal chopping blade, pulse together cashews and water to a semismooth paste.\n\n4 Test beet for doneness by inserting a paring knife into middle of beet. The knife should easily glide into center of beet. If not, lock on the lid again, and let beet sit in the closed cooker for 10 more minutes.\n\n5 Remove beet from the pressure cooker, peel it, and cut it into quarters. Add to cashew paste along with cumin, lemon juice, kosher salt, and extra-virgin olive oil. Process for 30 seconds or until hummus becomes smooth and creamy.\n\n6 Transfer hummus to a bowl, and serve with your favorite vegetables, chips, or flatbreads.\n\nNutrition per serving\n\nCalories: 157\n\nCarbohydrates: 9g\n\nSugars: 3g\n\nDietary fiber: 2g\n\nProtein: 4g\n\nFat: 13g\n\nCholesterol: 0mg\n\nSodium: 212mg\n\nMiddle Eastern Hummus\n\nUsing hot, fresh-cooked chickpeas instead of canned gives your hummus a smoother flavor that's so worth the effort.\n\nIngredients\n\n1 cup dried chickpeas, washed, picked over, soaked overnight, and drained\n\nWater\n\n1 tsp. kosher salt\n\n2 TB. tahini\n\n3 TB. freshly squeezed lemon juice\n\n1 clove garlic, minced\n\n\u2153 cup extra-virgin olive oil\n\n\u00bc tsp. dried sumac or paprika\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, place chickpeas. Add just enough water to cover by 1 inch (2.5cm), and season with \u00bd teaspoon kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 11 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Reserve \u00bd cup cooking liquid, and pour chickpeas into a strainer to drain. Transfer chickpeas to a food processor fitted with a metal chopping blade, and add tahini, lemon juice, garlic, and remaining \u00bd teaspoon kosher salt. Pulse for 30 seconds or until chickpeas start to break up.\n\n4 Add 2 tablespoons cooking liquid, pulse again, and with the machine running, drizzle in extra-virgin olive oil through the feed tube until hummus reaches a smooth consistency. If it's too thick, add water 1 tablespoon at a time. If it seems dry, add more olive oil 1 tablespoon at a time. Taste, add more salt if necessary, and pulse again for 30 seconds or until smooth.\n\n5 Transfer hummus to a bowl, and serve topped with a drizzle of olive oil and sprinkled with sumac alongside your favorite vegetables, crackers, or chips.\n\nNutrition per serving\n\nCalories: 160\n\nCarbohydrates: 18g\n\nSugars: 2g\n\nDietary fiber: 7g\n\nProtein: 6g\n\nFat: 12g\n\nCholesterol: 0mg\n\nSodium: 250mg\nLentil P\u00e2t\u00e9\n\nThis is a lovely appetizer. It also makes a wonderful sandwich spread and is a perfect sausage substitute for meatless breakfasts.\n\nIngredients\n\n1 cup Umbrian or du Puy lentils, washed and picked over\n\n1 cup heavy cream\n\n1 cup water\n\n1 tsp. crushed cumin\n\n1 tsp. curry powder\n\n3 cloves garlic\n\n1 tsp. kosher salt\n\n1 TB. freshly squeezed lemon juice\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine Umbrian lentils, heavy cream, water, cumin, curry powder, garlic, and kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform your preferred release method, and remove the lid.\n\n3 Transfer lentils and any remaining liquid to a food processor fitted with a metal chopping blade, add lemon juice, and pur\u00e9e for 30 seconds or until smooth.\n\n4 Transfer p\u00e2t\u00e9 to 2 (6-ounce; 170g) ramekins or into a mini-loaf pan, cover with plastic wrap, and refrigerate for 3 hours or overnight before serving.\n\nServe this pat\u00e9 with toast points, pickles, mustard, and any other condiments you like. Or try it alongside a mixed-greens salad with a sharp vinaigrette.\n\nNutrition per serving\n\nCalories: 180\n\nCarbohydrates: 15g\n\nSugars: 1g\n\nDietary fiber: 4g\n\nProtein: 6g\n\nFat: 12g\n\nCholesterol: 40mg\n\nSodium: 250mg\nSavory Smoked Salmon Cheesecake\n\nThis sophisticated appetizer comes together easily and cooks fast. To really take it over the top, give it a heavy sprinkle of caviar before serving.\n\nIngredients\n\n20 saltine crackers\n\n\u00be cup grated Parmesan cheese\n\n2 TB. unsalted butter, softened\n\n1 lb. (450g) cream cheese\n\n2 large eggs\n\n1 tsp. grated garlic\n\n1 tsp. grated shallot\n\n\u00be tsp. Old Bay seasoning\n\n8 oz. (225g) cured smoked salmon, minced\n\n1 TB. fresh flat-leaf parsley\n\n1 small cucumber, thinly sliced (\u2153 cup)\n\n1 TB. minced shallot\n\n1 TB. nonpareil capers, minced\n\nMETHOD\n\n1 Butter an 8-inch (20cm) springform pan.\n\n2 In a food processor fitted with a metal chopping blade, pulse saltine crackers, Parmesan cheese, and unsalted butter to a fine crumb. Transfer crust to the pan, and press in an even layer across the bottom and halfway up the sides.\n\n3 In a large bowl, and using an electric mixer on low, blend cream cheese, eggs, garlic, grated shallot, and Old Bay seasoning. Add salmon and flat-leaf parsley, and mix well. Spread filling over crust, smooth the top, and cover with aluminum foil.\n\n4 Add a trivet to the bottom of a 8-quart (7.5l) pressure cooker, and add 1 inch (2.5cm) water. Add the springform pan to the cooker, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 35 minutes. Turn off heat, let cake sit for 20 minutes, and remove the lid.\n\n6 Lift out the pan from the cooker, and remove the foil. Give cake a gentle shake; if it jiggles a lot, re-cover, return to the cooker, cook at level 2\/high for 10 minutes, and let sit in the cooker for 20 minutes.\n\n7 Refrigerate for 3 hours or until cool. Cover the pan tightly with plastic wrap so it doesn't touch cake, and refrigerate overnight.\n\n8 Remove cake from the pan. Garnish with cucumbers, minced shallot, and capers, and serve with crackers and breads.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 6g\n\nSugars: 2g\n\nDietary fiber: 0g\n\nProtein: 17g\n\nFat: 20g\n\nCholesterol: 115mg\n\nSodium: 310mg\n\nTuna Salad with Chickpeas\n\nThis salad is creamy, meaty, and filling. The dressing adds the right amount of tartness to keep the other ingredients from being cloying.\n\nIngredients\n\n1 cup dried chickpeas, washed, picked over, soaked overnight or quick soaked, and drained\n\n1 tsp. kosher salt\n\nWater\n\n1 medium shallot, peeled and minced (2 TB.)\n\n1\u00bd TB. fresh flat-leaf parsley, minced\n\n1 TB. red wine vinegar\n\n\u00bc tsp. freshly ground black pepper\n\n2 (5-oz.; 140g) cans tuna packed in olive oil\n\n2 TB. extra-virgin olive oil, plus more to taste\n\n8 leaves romaine or Bibb lettuce\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine chickpeas, kosher salt, and enough water to cover chickpeas by \u00bd inch (1.25cm). Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n3 Drain chickpeas into a large bowl, and cool for 5 minutes.\n\n4 Add shallot, flat-leaf parsley, and red wine vinegar, and stir to combine. Taste, season with more kosher salt and black pepper as necessary, and stir.\n\n5 Pour tuna and packing olive oil into the bowl, and gently break up tuna but leave it chunky. Drizzle with extra-virgin olive oil, and gently turn tuna into mixture until distributed. Taste, and adjust seasonings as necessary.\n\n6 Place 2 romaine leaves on each plate, spoon tuna and chickpeas over top, and serve.\n\nNutrition per serving\n\nCalories: 330\n\nCarbohydrates: 35g\n\nSugars: 4g\n\nDietary fiber: 14g\n\nProtein: 30g\n\nFat: 17g\n\nCholesterol: 45mg\n\nSodium: 920mg\n\nChicken Salad Deluxe\n\nFor this delightful luncheon salad, I like to use a whole chicken and take advantage of the combination of light and dark meat.\n\nIngredients\n\n1 cup water\n\n1\u00bd tsp. kosher salt\n\n1 (2\u00bd-lb.; 1.25kg) whole chicken\n\n1\u00bd cups fresh green beans, trimmed\n\n\u2154 cup mayonnaise\n\n1\u00bd tsp. Dijon mustard\n\n1 tsp. freshly squeezed lemon juice\n\n3 medium radishes, thinly sliced (\u00bd cup)\n\n1 large carrot, peeled and coarsely grated (\u00be cup)\n\n1 TB. fresh chives, minced\n\n\u215b tsp. freshly ground black pepper\n\n8 Bibb lettuce leaves\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine water and \u00bd teaspoon kosher salt. Add chicken, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Add green beans, lock on the lid, and steam green beans for about 5 minutes or until tender.\n\n4 Transfer chicken to a rimmed baking sheet to cool. Remove green beans from the pot, and drain. (Save broth for another use.)\n\n5 Remove skin from chicken, and strip meat from the bones in large, bite-size chunks. Take care to remove all the bones.\n\n6 In a large bowl, combine mayonnaise, Dijon mustard, and lemon juice. Add chicken, radishes, carrot, chives, green beans, remaining 1 teaspoon kosher salt, and black pepper. Stir to combine, and serve.\n\n7 Serve spooned over Bibb lettuce leaves.\n\nNutrition per serving\n\nCalories: 450\n\nCarbohydrates: 5g\n\nSugars: 2g\n\nDietary fiber: 2g\n\nProtein: 30g\n\nFat: 34g\n\nCholesterol: 110mg\n\nSodium: 880mg\nClassic Egg Salad\n\nThis cool egg salad combines creamy, yolky goodness with a celery crunch and a mustardy tang.\n\nIngredients\n\n12 large eggs, at room temperature\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 medium stalk celery, minced (\u00bc cup)\n\n1 medium green onion, white parts only, minced (1 TB.)\n\n1\u00bd TB. Dijon mustard\n\n\u00bd cup mayonnaise\n\nMETHOD\n\n1 Add a trivet to a 4-quart (4l) pressure cooker, and add enough water to reach the bottom of the trivet.\n\n2 Place eggs in a steamer basket, and add to the pressure cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a hard boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 6 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Lift out the steamer basket, and immediately place eggs under cold running water to shock them and make them easier to peel.\n\n5 Peel eggs, place in a large bowl, and mash using a potato masher or the back of a fork.\n\n6 Add kosher salt, black pepper, celery, green onion, Dijon mustard, and mayonnaise. Stir to combine, and serve.\n\nFor a more piquant flavor, add some minced bread-and-butter pickles. Or try a little curry powder for a more exotic flavor. For a summer-at-the-beach theme, sprinkle a bit of Old Bay seasoning and some capers over the top before serving.\n\nNutrition per serving\n\nCalories: 350\n\nCarbohydrates: 2g\n\nSugars: 1g\n\nDietary fiber: 0g\n\nProtein: 15g\n\nFat: 30g\n\nCholesterol: 515mg\n\nSodium: 600mg\nWheat Berry Salad with Arugula Pesto\n\nWheat berries are a nutritious ingredient in your diet, and with the addition of arugula pesto, they become a flavorful base for all types of grain salads.\n\nIngredients\n\n1 cup hard winter wheat berries\n\n3 cups water\n\n\u00be tsp. kosher salt\n\n2\u00bd cups arugula leaves\n\n\u2153 cup walnuts, coarsely chopped\n\n\u00bc cup grated Parmesan cheese\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bc cup extra-virgin olive oil, plus more as needed\n\n1 cup mixed cherry, plum, and grape tomatoes\n\n4 oz. (110g) fresh mozzarella cheese, shredded\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 Place wheat berries in a strainer, and rinse well under cold water.\n\n2 Transfer wheat to a 4-quart (4l) pressure cooker, and add water and \u00bd teaspoon kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 27 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid. Set aside to cool.\n\n4 In a food processor fitted with a metal chopping blade, pulse arugula, walnuts, Parmesan cheese, black pepper, and remaining \u00bc teaspoon kosher salt. With the food processor running, drizzle extra-virgin olive oil through the feed tube until pesto reaches the consistency of salad dressing.\n\n5 Drain wheat berries, and place in a large bowl. Add \u00bd of pesto, and stir to combine. Taste, and add more pesto as necessary.\n\n6 Add tomatoes, mozzarella cheese, and chives. Stir gently, and serve.\n\nNutrition per serving\n\nCalories: 300\n\nCarbohydrates: 39g\n\nSugars: 2g\n\nDietary fiber: 7g\n\nProtein: 18g\n\nFat: 10g\n\nCholesterol: 10mg\n\nSodium: 670mg\n\nFarro Tabbouleh\n\nTabbouleh is traditionally made with cracked bulgur wheat. In this version, farro, a distant relative of bulgur, makes a nutty and nutritious alternative.\n\nIngredients\n\n1 cup farro\n\n5 cups water\n\n1 tsp. kosher salt\n\n\u00bd cup fresh flat-leaf parsley, minced\n\n2 TB. fresh mint, minced\n\n2 TB. fresh chives, minced\n\n\u00bd cup sun-dried tomatoes in oil, drained and chopped\n\n1 cup grape tomatoes, quartered\n\n2\u00bd TB. freshly squeezed lemon juice\n\n2 or 3 TB. extra-virgin olive oil\n\nFreshly ground black pepper\n\n1 cup arugula leaves, rinsed and dried\n\nMETHOD\n\n1 Place farro in a strainer, and rinse well under cold water.\n\n2 Transfer farro to a 4-quart (4l) pressure cooker, and add water and kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 1 (traditional)\/high (electric). Reduce heat to low, and cook at 1\/low for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Return farro to the strainer, and cool.\n\n5 In a large bowl, combine cooled farro, flat-leaf parsley, mint, chives, sun-dried tomatoes, grape tomatoes, lemon juice, and extra-virgin olive oil.\n\n6 Taste, and season with more kosher salt and black pepper as necessary. Add arugula, stir, and serve.\n\nFarro contains more protein, fiber, vitamins, and minerals than bulgur. It also has more fat.\n\nNutrition per serving\n\nCalories: 70\n\nCarbohydrates: 5g\n\nSugars: 1g\n\nDietary fiber: 1g\n\nProtein: 1g\n\nFat: 6g\n\nCholesterol: 0mg\n\nSodium: 350mg\n\nPicnic-Style Potato Salad\n\nTart from the buttermilk and herby from the dill, this is homemade potato salad at its finest. It's ideal for family gatherings, potlucks, pitch-ins\u2014and picnics.\n\nIngredients\n\n10 medium Yukon Gold potatoes, cut in half and then into \u00bd moons\n\n\u00be cup mayonnaise\n\n\u00bc cup buttermilk\n\n1 TB. Dijon mustard\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground white pepper\n\n1 medium green onion, minced (1 TB.)\n\n1 TB. fresh flat-leaf parsley\n\n2 tsp. fresh dill, minced\n\n3 large hard-boiled eggs, peeled and sliced thin\n\nMETHOD\n\n1 Add a steamer basket to a 4-quart (4l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Add Yukon Gold potatoes to the basket.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n4 Let potatoes steam dry in the cooker for 30 seconds.\n\n5 In a large bowl, whisk together mayonnaise, buttermilk, Dijon mustard, kosher salt, white pepper, green onion, flat-leaf parsley, and dill until smooth.\n\n6 Add potatoes, and stir to coat. Refrigerate for 1 hour or until cool.\n\n7 Stir well, top with hard-boiled egg slices, and serve.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 25g\n\nSugars: 3g\n\nDietary fiber: 3g\n\nProtein: 6g\n\nFat: 18g\n\nCholesterol: 60mg\n\nSodium: 320mg\nHot German Potato Salad\n\nThis hot German-style potato salad\u2014the perfect side to sausage and Wiener schnitzel\u2014uses rich beef stock as a thickener for a lighter finished dish.\n\nIngredients\n\n2 slices smoky bacon, sliced into \u00bc-in. (.5cm) pieces\n\n1 medium yellow onion, thinly sliced (about 1 cup)\n\n1 cup homemade or sodium-free beef stock\n\n2 lb. (1kg) red potatoes, cut into \u00bd-in. (1.25cm) slices\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n2 TB. fresh flat-leaf parsley, minced\n\n3 TB. cider vinegar or to taste\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, add bacon. Set heat to medium (traditional)\/high (electric), and slowly render fat from bacon, allowing it to become crisp. Transfer bacon to a paper towel\u2013lined plate.\n\n2 Add yellow onion to the cooker, and saut\u00e9 for about 3 minutes or until wilted. Transfer onion to a bowl, and set aside.\n\n3 Add beef stock to the cooker, add a layer of red potatoes, dot with onions, and season with a pinch of kosher salt and a grind of black pepper. Repeat layering, ending with potatoes. Bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Gently transfer potatoes to a large bowl. Add \u2154 of bacon, \u00bd of flat-leaf parsley, and cider vinegar, and toss to combine.\n\n6 Transfer to a serving bowl, and wet with \u00bd cup hot broth. Top with remaining bacon and parsley, sprinkle with a little more vinegar, and serve.\n\nNutrition per serving\n\nCalories: 120\n\nCarbohydrates: 20g\n\nSugars: 2g\n\nDietary fiber: 2g\n\nProtein: 4g\n\nFat: 3g\n\nCholesterol: 5mg\n\nSodium: 180mg\n\nRoasted Corn and Butternut Squash Chili\n\nAll-bean chili is hearty and filling, but add some roasted corn, butternut squash, yellow corn chips, and southwestern spices, and it's so much better.\n\nIngredients\n\n1\u00bd TB. vegetable oil\n\n1 large yellow onion, diced small (1 cup)\n\n1 small butternut squash, peeled, seeded, and cut into \u00bd-in. (1.25cm) cubes (2 cups)\n\n1 cup fresh or frozen yellow corn\n\n1 lb. (450g) dried mixed chili beans (pintos, kidney, black, and red), washed, picked over, soaked overnight or quick soaked, and drained\n\n1 tsp. kosher salt\n\n2 TB. dark chili powder\n\n1 TB. ground cumin\n\n1 TB. dried oregano\n\n1 TB. garlic powder\n\n2\u00bc cups tomato sauce\n\n1 cup water\n\n\u00bc cup crushed yellow corn tortilla chips\n\n3 medium Fresno chiles, sliced (\u00bd cup)\n\n\u00bd cup fresh cilantro, coarsely chopped\n\n1 cup sour cream\n\n2 cups corn chips\n\n2 cups shredded cheddar cheese\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil, swirling the cooker to coat.\n\n2 Add yellow onion, and saut\u00e9 for 5 minutes or until onion begins to soften.\n\n3 Add butternut squash, corn, chili beans, kosher salt, dark chili powder, cumin, oregano, and garlic powder. Cook, stirring, for 1 minute or until spices become fragrant.\n\n4 Add tomato sauce, water, and yellow corn tortilla chips, and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, and let chili sit for 5 minutes. Perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Taste, and adjust kosher salt as necessary. Spoon chili into bowls, and serve with Fresno chiles, cilantro, sour cream, corn chips, and cheddar cheese for topping.\n\nNutrition per serving\n\nCalories: 360\n\nCarbohydrates: 53g\n\nSugars: 7g\n\nDietary fiber: 14g\n\nProtein: 25g\n\nFat: 7g\n\nCholesterol: 5mg\n\nSodium: 850mg\n\nWhite Bean and Shiitake Soup\n\nEither vegetable or mushroom broth works well in this easy soup and marries with the shiitakes and cannellini beans to yield deep, wonderful flavors.\n\nIngredients\n\n3 TB. olive oil\n\n1 large yellow onion, diced small (1\u00bd cups)\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n2 cups shiitake mushrooms, sliced\n\n6 cloves garlic, minced\n\n1 TB. dried rosemary, crushed\n\n1 cup dried cannellini beans, washed, picked over, soaked overnight, and drained\n\n4 cups homemade or sodium-free vegetable or mushroom broth\n\n1 (14.5-oz.; 410g) can chopped tomatoes, with juice\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, add olive oil. Set heat to medium (traditional)\/high (electric), and heat oil for 20 seconds. Add yellow onion, and saut\u00e9 for 3 or 4 minutes or until softened.\n\n2 Season with kosher salt and black pepper, add shiitake mushrooms, stir, and saut\u00e9 for 3 to 5 minutes or until softened.\n\n3 Add garlic and rosemary, and stir. When garlic is fragrant, add cannellini beans and vegetable broth, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Taste, and season with additional salt and pepper as necessary. Add tomatoes with juice, set heat to low, and cook for 5 minutes.\n\n6 Serve hot.\n\nNutrition per serving\n\nCalories: 160\n\nCarbohydrates: 21g\n\nSugars: 4g\n\nDietary fiber: 5g\n\nProtein: 7g\n\nFat: 5g\n\nCholesterol: 0mg\n\nSodium: 120mg\nSaturday Soup Beans\n\nHam and beans is a favorite for many, especially when served with cornbread. With your pressure cooker, you can make these filling beans in no time.\n\nIngredients\n\n2 (8-oz.; 225g) meaty smoked ham hocks\n\nWater\n\n1\u00bd cups dried pinto beans, washed, picked over, soaked overnight, and drained\n\n2 tsp. kosher salt\n\n1 tsp. freshly ground black pepper\n\n1 small red onion, diced small (\u00bd cup)\n\n3 medium green onions, chopped (\u00bd cup)\n\nHot sauce\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, place ham hocks. Add enough water to come halfway up hocks, set heat to high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Add pinto beans, and ensure broth covers beans by 1 inch (2.5cm). If it's too low, add water; if it's too high, use a ladle to remove some liquid. Add kosher salt and black pepper, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2\/high. Reduce heat to low, and cook at 2\/high for 11 minutes. Remove from\/turn off heat, perform a cold water\/quick release, and remove the lid.\n\n5 Transfer hocks to a cutting board, and remove and set aside outer skin. Pull or cut meat from hocks, leaving sinew, bones, and fat behind. Chop meat.\n\n6 Using a potato masher, mash some beans to make soup creamy.\n\n7 Return meat to the cooker, stir, taste, and season with kosher salt or black pepper as necessary. Serve hot, with red onion, green onions, and hot sauce for topping.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 23g\n\nSugars: 1g\n\nDietary fiber: 8g\n\nProtein: 24g\n\nFat: 9g\n\nCholesterol: 55mg\n\nSodium: 520mg\nSpicy Chickpea Stew with Sour Tomato Curry\n\nThis zesty stew is the perfect balance of hot and sour. Be aware that the more tamarind you add, the sourer the dish will be.\n\nIngredients\n\n3 TB. olive oil\n\n2 medium yellow onions, julienned (2 cups)\n\n\u00be tsp. kosher salt\n\n1 TB. minced garlic\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n1 tsp. turmeric\n\n\u00bc tsp. cayenne (optional)\n\n2 tsp. Madras curry powder\n\n1 tsp. coarsely ground cumin\n\n\u00bc tsp. freshly ground black pepper\n\n1\u00bd cups dried chickpeas, washed, picked over, soaked overnight, and drained\n\n2 cups tomato sauce\n\n1\u00bd TB. tamarind concentrate mixed with \u00bd cup water, or 1 TB. brown sugar mixed with 1 TB. freshly squeezed lime juice\n\n1 lb. (450g) cooked spaghetti, hot\n\n\u2153 cup fresh cilantro, chopped\n\n2 large green onions, chopped (\u2153 cup)\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add yellow onions and \u00bc teaspoon kosher salt, and cook for 5 minutes or until onions begin to sizzle. Reduce to medium (traditional)\/high (electric) heat, and cook, stirring occasionally, for 10 minutes or until onions turn dark brown.\n\n2 Stir in garlic, ginger, turmeric, cayenne (if using), Madras curry powder, cumin, and black pepper, and saut\u00e9 for about 30 seconds.\n\n3 Stir in chickpeas, tomato sauce, remaining \u00bd teaspoon kosher salt, and tamarind-water mixture, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, and let sit for 5 minutes. Perform a natural release, and remove the lid.\n\n5 Spoon chickpeas over spaghetti, garnish with cilantro and green onions, and serve.\n\nNutrition per serving\n\nCalories: 330\n\nCarbohydrates: 66g\n\nSugars: 9g\n\nDietary fiber: 17g\n\nProtein: 17g\n\nFat: 10g\n\nCholesterol: 0mg\n\nSodium: 700mg\n\nCuban Black Bean Soup with Sherry\n\nThe addition of sherry brings a touch of elegance to this otherwise homey soup that's inspired by the influence of Spanish cuisine on Cuban food.\n\nIngredients\n\n3 TB. olive oil\n\n2 to 4 slices bacon, diced small\n\n1 large yellow onion, diced small (1\u00bd cups)\n\n4 medium stalks celery, chopped (\u00be cup)\n\n2 ancho chiles, diced small (\u00be cup)\n\n2 TB. minced garlic\n\n1 TB. dried oregano\n\n1 TB. crushed cumin\n\n3 bay leaves\n\n2 TB. dry sherry\n\n8 oz. (225g) dried black beans, washed, picked over, soaked overnight, and drained\n\n1 cup tomato sauce\n\n\u00bc cup fresh cilantro, chopped\n\nWater\n\n1 small red onion, minced (\u00bd cup)\n\n\u00be cup sour cream\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium (traditional)\/high (electric) heat, heat olive oil. Add bacon, yellow onion, celery, and ancho chiles, and saut\u00e9 for about 4 minutes or until softened.\n\n2 Stir in garlic, oregano, cumin, and bay leaves, and cook for 20 seconds or until garlic becomes fragrant.\n\n3 Add sherry, and cook for 15 seconds to burn off alcohol.\n\n4 Add black beans, tomato sauce, and \u00bd of cilantro. Add water to cover beans by 1 inch (2.5cm), stir to combine, and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 11 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Remove bay leaves. Spoon soup into bowls, and serve with red onion, sour cream, and remaining cilantro for topping.\n\nNutrition per serving\n\nCalories: 440\n\nCarbohydrates: 40g\n\nSugars: 5g\n\nDietary fiber: 8g\n\nProtein: 14g\n\nFat: 21g\n\nCholesterol: 35mg\n\nSodium: 840mg\nFarmhouse Corn Chowder\n\nThis rustic chowder is unbelievable in the summertime, when sweet corn is in season. In the off season, you can use frozen corn for an equally lovely result.\n\nIngredients\n\n3 thick slices smoky bacon, cut into \u00bc-in. (.5cm) pieces\n\n1 TB. unsalted butter\n\n1 large yellow onion, diced medium (1 cup)\n\n3 medium stalks celery, diced medium (\u00bd cup)\n\n1 medium green bell pepper, ribs and seeds removed, and diced medium (\u00bd cup)\n\n1 TB. minced garlic\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n4 medium russet potatoes, peeled and diced medium (2 cups)\n\n2 ears sweet corn, husked and kernels cut from cob (1 cup), or 1 cup frozen corn\n\n5 cups homemade or sodium-free chicken stock\n\n2 tsp. fresh chives, minced\n\n2 tsp. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, add bacon. Set heat to medium (traditional)\/high (electric), and slowly render fat from bacon, allowing it to become crisp. Transfer bacon to a paper towel\u2013lined plate. Pour off all but 1 tablespoon grease.\n\n2 Add unsalted butter to the cooker, and melt. Add yellow onion, celery, green bell pepper, garlic, \u00bd teaspoon kosher salt, and black pepper, and saut\u00e9 for about 3 or 4 minutes or until soft.\n\n3 Add russet potatoes, sweet corn, chicken stock, and remaining \u00bd teaspoon kosher salt, and bring liquid to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 4 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir in chives, flat-leaf parsley, and \u00bd of bacon. Taste, and add more salt as necessary. Serve garnished with remaining bacon.\n\nNutrition per serving\n\nCalories: 280\n\nCarbohydrates: 33g\n\nSugars: 4g\n\nDietary fiber: 3g\n\nProtein: 11g\n\nFat: 12g\n\nCholesterol: 40mg\n\nSodium: 610mg\nFrench Potato and Leek Soup\n\nThis classic French bistro soup is smooth, creamy, and comforting. Make it a bit more elegant by pur\u00e9eing it after it's cooked and has cooled a bit.\n\nIngredients\n\n2\u00bd TB. unsalted butter\n\n2 medium leeks, white parts only, cut into thin half-moons (2 cups)\n\n1 medium carrot, peeled and diced small (\u2153 cup)\n\n4 medium yellow onions, julienned (2\u00bd cups)\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n4 large Yukon Gold potatoes, peeled and diced medium (1\u00bd cups)\n\n5 cups homemade or sodium-free chicken broth\n\n\u00bd cup half-and-half\n\nChopped fresh chives\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium (traditional)\/high (electric) heat, melt unsalted butter. Add leeks, carrot, and yellow onions, and stir to coat. Season with kosher salt and black pepper, and cook for 8 minutes or until vegetables have wilted but not browned.\n\n2 Add Yukon Gold potatoes and chicken broth, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n4 Add half-and-half, garnish with chives, and serve.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 34g\n\nSugars: 6g\n\nDietary fiber: 4g\n\nProtein: 11g\n\nFat: 11g\n\nCholesterol: 55mg\n\nSodium: 670mg\n\nIndian Carrot and Lentil Soup\n\nSlightly exotic and spicy but not hot, this soup is influenced mostly by the flavors of Indian cuisine but mixes in a little bit of everything.\n\nIngredients\n\n1 TB. olive oil\n\n6 medium carrots, peeled and chopped (1\u00bd cups)\n\n1 medium yellow onion, halved and thinly sliced (1 cup)\n\n1 tsp. kosher salt\n\n\u00bd cup red (pink) lentils\n\n\u00bc cup jasmine rice\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n1\u00bd tsp. Thai red curry paste\n\n1 tsp. curry powder\n\n4 cups homemade or sodium-free chicken stock\n\n\u00bd cup coconut milk\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add carrots, yellow onion, and kosher salt, and saut\u00e9 for about 8 minutes or until golden.\n\n2 Add red lentils, jasmine rice, ginger, Thai red curry paste, and curry powder, and stir to combine. When spices sizzle and pop and become fragrant, add chicken stock and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Add coconut milk, stir to blend, and serve hot.\n\nFor a smoother soup, pur\u00e9e it in a blender after cooling it for a while first. Never pur\u00e9e a hot soup in a blender. Cool it first, blend it, return it to the pot, and reheat before serving.\n\nNutrition per serving\n\nCalories: 280\n\nCarbohydrates: 33g\n\nSugars: 5g\n\nDietary fiber: 6g\n\nProtein: 12g\n\nFat: 12g\n\nCholesterol: 25mg\n\nSodium: 670mg\nRustic Split-Pea Soup\n\nTraditionally, this soup is pur\u00e9ed to a smooth finish. This version is rustic and hearty, and the ham added at the end brings a new flavor to the final soup.\n\nIngredients\n\n1 TB. unsalted butter\n\n1 medium yellow onion, diced (1 cup)\n\n2 medium carrots, peeled and diced (\u00bd cup)\n\n2 medium stalks celery, diced (\u00bd cup)\n\n3 medium yellow potatoes, peeled and diced medium (1 cup)\n\n3 cloves garlic, minced (\u00bd TB.)\n\n1 cup split peas, picked over and washed\n\n5 cups homemade or sodium-free vegetable broth, water, or water with 2 sodium-free vegetable bouillon cubes\n\n1 tsp. kosher salt\n\n\u00be tsp. freshly ground white pepper\n\n\u00bd cup country-style ham, diced small (optional)\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. Add yellow onion, carrots, celery, yellow potatoes, and garlic. Stir and saut\u00e9 for 5 minutes or until vegetables begin to soften.\n\n2 Add split peas and vegetable broth, season with kosher salt and white pepper, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n4 Add ham (if using), set heat to medium (traditional)\/high (electric), and warm for 3 minutes. Serve hot.\n\nServe this soup topped with tasty homemade croutons. In a 10-inch (25cm) skillet over medium heat, melt 2 tablespoons unsalted butter. When it begins to foam, add 1 cup (\u00bd-inch; 1.25cm) bread cubes and toss to coat. Season with kosher salt and black pepper, and cook, stirring occasionally, for 3 minutes.\n\nNutrition per serving\n\nCalories: 260\n\nCarbohydrates: 35g\n\nSugars: 6g\n\nDietary fiber: 14g\n\nProtein: 19g\n\nFat: 5g\n\nCholesterol: 40mg\n\nSodium: 660mg\nCurried Butternut Squash Soup\n\nEasy-to-pur\u00e9e butternut squash is an ideal, adaptable base for all kinds of flavor additions.\n\nIngredients\n\n1 TB. vegetable oil\n\n1 medium yellow onion, thinly sliced (1 cup)\n\n3 cloves garlic\n\n1\u00bd TB. curry powder\n\n1 TB. Thai red curry paste\n\n2 cups homemade or sodium-free vegetable stock, or 2 cups water with 2 cubes sodium-free vegetable bouillon\n\n\u00bd tsp. kosher salt\n\n2 medium butternut squash, peeled, seeded, and cut into 1-in. (2.5cm) cubes (2\u00bd cups)\n\n1 medium carrot, peeled and diced large (\u2154 cup)\n\n1 medium roma tomato, cored and diced medium\n\n1 serrano chile, thinly sliced\n\n1 TB. fresh cilantro, minced\n\n2 TB. plain yogurt\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium (traditional)\/high (electric) heat, heat vegetable oil. Add yellow onion, and saut\u00e9 for about 10 minutes or until onion is golden.\n\n2 Add garlic, curry powder, and Thai red curry paste, and cook, stirring, for 30 seconds minutes or until fragrant.\n\n3 Add vegetable stock, kosher salt, butternut squash, and carrot, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Let soup cool, transfer to a blender or a food processor fitted with a metal chopping blade, and pur\u00e9e until smooth. Return soup to the pressure cooker, set heat to medium\/high, and warm.\n\n6 Meanwhile, in a small bowl, combine roma tomato, serrano chile, cilantro, and yogurt.\n\n7 Divide warmed soup among 4 bowls, top each with 1 dollop tomato raita, and serve.\n\nNutrition per serving\n\nCalories: 120\n\nCarbohydrates: 22g\n\nSugars: 7g\n\nDietary fiber: 4g\n\nProtein: 2g\n\nFat: 4g\n\nCholesterol: 0mg\n\nSodium: 340mg\n\nMushroom Barley Soup\n\nThis hearty soup delivers warmth and comfort in every bowl. The deep, earthy flavor of mushrooms combines nicely with the nutty barley.\n\nIngredients\n\n3 TB. unsalted butter\n\n8 oz. (225g) cremini or white mushrooms, brushed clean and thinly sliced (3 cups)\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 large yellow onion, diced medium (1\u00bd cups)\n\n3 medium carrots, peeled and sliced into rounds (\u00be cup)\n\n4 medium stalks celery, sliced into thin half-moons (\u00be cup)\n\n1 TB. minced garlic\n\n\u00bd cup pearl barley\n\n\u00bd tsp. dried thyme\n\n4 cups homemade or sodium-free beef, chicken, or vegetable stock\n\n1 TB. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. Add cremini mushrooms, kosher salt, and black pepper, and saut\u00e9 for about 4 minutes or until soft.\n\n2 Add yellow onion, carrots, celery, and garlic, and saut\u00e9 for 30 seconds or until aromatic.\n\n3 Add pearl barley, and stir to coat with butter. Add thyme and beef stock, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir in flat-leaf parsley, and serve.\n\nFor a gluten-free soup, use brown rice or whole-grain white sorghum instead of the barley. For a heartier soup, add some cooked shredded beef.\n\nNutrition per serving\n\nCalories: 160\n\nCarbohydrates: 22g\n\nSugars: 5g\n\nDietary fiber: 5g\n\nProtein: 4g\n\nFat: 6g\n\nCholesterol: 20mg\n\nSodium: 770mg\nRoot Vegetable Stew\n\nThis healthy soup is full of hearty root vegetables. Your pressure cooker quickly cooks these vegetables that usually take longer via traditional methods.\n\nIngredients\n\n\u00bc cup olive oil\n\n1 tsp. fresh rosemary, minced\n\n1 large yellow onion, diced large (1 cup)\n\n1 (3-in.; 7.5cm) celery root, peeled and diced large (1\u00bd cups)\n\n1 medium bulb fennel, trimmed and sliced (\u00be cup)\n\n2 large russet potatoes, peeled and diced large (2 cups)\n\n3 medium carrots, peeled and diced large (1 cup)\n\n1 large turnip, peeled and diced large (1 cup)\n\n2 medium parsnips, peeled and diced large (1 cup)\n\n3 cups homemade or sodium-free chicken stock\n\n\u00be tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add rosemary, and when it begins to sizzle, immediately add yellow onion and stir. Do not let rosemary burn.\n\n2 Add celery root, fennel, russet potatoes, carrots, turnip, parsnips, chicken stock, kosher salt, and black pepper, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 6 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Garnish with chives, and serve.\n\nNutrition per serving\n\nCalories: 340\n\nCarbohydrates: 46g\n\nSugars: 9g\n\nDietary fiber: 9g\n\nProtein: 7g\n\nFat: 15g\n\nCholesterol: 5mg\n\nSodium: 970mg\nBorscht with Italian Sausage\n\nThis lovely purple soup is nice any time of year. The fennel seed in the soup and in the sausage pairs well with the vegetables, sour cream, and lemon.\n\nIngredients\n\n1\u00bd TB. unsalted butter\n\n8 oz. (225g) Italian sausage\n\n2 medium beets, peeled and cut into \u00bd-in. (1.25cm) cubes (1 cup)\n\n2 medium carrots, peeled and diced small (\u00bd cup)\n\n\u00bd medium head red or green cabbage, cored and thinly shredded (2 cups)\n\n2 tsp. minced garlic\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 bay leaf\n\n\u00bd tsp. fennel seed\n\n4 cups homemade or sodium-free beef broth\n\n1 TB. red wine vinegar\n\n1 medium lemon, quartered\n\n\u00bd cup sour cream\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. Add Italian sausage, brown for 3 minutes on both sides, and transfer sausage to a plate.\n\n2 Add beets, carrots, red cabbage, and garlic to the pressure cooker, and stir. Season with kosher salt and black pepper, and stir again. Add bay leaf, fennel seed, beef broth, and red wine vinegar, and stir again. Return sausage to the pan, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Remove bay leaf. Transfer sausage to a cutting board, cut into 1-inch (2.5cm) rounds, and return to the cooker. Taste, and add more kosher salt as necessary, and stir.\n\n5 Divide soup among 4 bowls, and serve with lemon quarters and sour cream on the side.\n\nNutrition per serving\n\nCalories: 200\n\nCarbohydrates: 16g\n\nSugars: 10g\n\nDietary fiber: 4g\n\nProtein: 8g\n\nFat: 11g\n\nCholesterol: 30mg\n\nSodium: 390mg\n\nRibolleta\n\nThis Italian soup is based on Italian bread soups that use leftover bread as a thickener to make the soup more filling.\n\nIngredients\n\n4 TB. olive oil, plus more for garnish\n\n\u00bc lb. (115g) pancetta, diced\n\n1 cup dried cannellini beans, washed, picked over, soaked overnight, and drained\n\n1 medium zucchini, diced large (1 cup)\n\n1 medium yellow onion, diced large (1 cup)\n\n2 medium stalks celery, diced large (\u2154 cup)\n\n2 medium carrots, peeled and diced large (\u2154 cup)\n\n2 TB. minced garlic\n\n\u00bc medium head green cabbage, chopped (2 cups)\n\n1 cup canned crushed tomatoes, with juice\n\n\u00bd cup fresh basil, torn\n\n\u00be tsp. kosher salt\n\n4 cups homemade or sodium-free chicken stock\n\n4 pieces day-old sourdough bread, cut into \u00bd-in. (1.25cm) cubes (2 cups)\n\n2 cups spinach\n\n\u00bd cup grated Parmesan cheese\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add pancetta, and slowly render fat from pancetta, allowing it to become crisp.\n\n2 Add cannellini beans, zucchini, yellow onion, celery, carrots, garlic, green cabbage, tomatoes with juice, basil, kosher salt, and chicken stock, and bring to a boil.\n\n3 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Return the pressure cooker to medium (traditional)\/high (electric) heat, stir in sourdough bread cubes and spinach, and cook for 10 minutes or until bread has dissolved into broth and spinach has wilted.\n\n5 Serve sprinkled with Parmesan cheese, a drizzle of olive oil, and freshly ground black pepper.\n\nNutrition per serving\n\nCalories: 390\n\nCarbohydrates: 35g\n\nSugars: 6g\n\nDietary fiber: 8g\n\nProtein: 16g\n\nFat: 20g\n\nCholesterol: 20mg\n\nSodium: 1,130mg\nTexas-Style Chili Con Carne\n\nThis is traditional Tex-Mex chili features masa harina, a corn flour whose unique flavor comes from being soaked in culinary lime.\n\nIngredients\n\n1 TB. olive oil\n\n1 slice smoky bacon, cut into \u00bd-in. (1.25cm) pieces\n\n1\u00bd lb. (680g) flank steak, cut into \u00bd-in. (1.25cm) cubes\n\n1 small yellow onion, diced small (\u00bd cup)\n\n2 TB. minced garlic\n\n3 TB. dark chili powder\n\n1\u00bd tsp. crushed cumin seed\n\n1\u00bd tsp. dried Mexican oregano\n\n4 large roma tomatoes, halved\n\n3 TB. tomato paste\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bd cup water\n\n\u00bc cup yellow masa harina, or 1 cup crumbled corn tortillas\n\n1 cup shredded cheddar cheese\n\n1 small red onion, diced small (\u00bd cup)\n\n\u00bd cup sour cream\n\n3 serrano chiles, thinly sliced (\u00bc cup)\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, add olive oil. Add bacon, and slowly render some fat from bacon, allowing it to become crisp-tender.\n\n2 Add flank steak without crowding it (cook in batches, if necessary), and brown on all sides for 7 minutes.\n\n3 Add yellow onion, stir, and cook for about 3 minutes or for 3 to 5 minutes.\n\n4 Add garlic, dark chili powder, cumin seed, and Mexican oregano, and stir until fragrant.\n\n5 Add roma tomatoes, tomato paste, kosher salt, black pepper, and water, and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 If there's too much liquid, whisk in \u00bc cup masa harina; otherwise, use \u215b cup. Be sure to whisk well to avoid lumps. Taste, add more salt as necessary, and stir. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n8 Serve with cheddar cheese, red onion, sour cream, and serrano chiles for topping.\n\nNutrition per serving\n\nCalories: 590\n\nCarbohydrates: 20g\n\nSugars: 6g\n\nDietary fiber: 5g\n\nProtein: 60g\n\nFat: 30g\n\nCholesterol: 115mg\n\nSodium: 1,080mg\nKorean Beef Stew\n\nThe tofu in this lively stew is a nice addition, and you'll be impressed with the amount of flavor it absorbs from the beef broth.\n\nIngredients\n\n1 (1\u00bd lb.; 680g) chuck roast, cut into 1-in. (2.5cm) cubes\n\n1\u00bd tsp. kosher salt\n\n2 tsp. dark sesame oil\n\n1 TB. vegetable oil\n\n2 large russet potatoes, cut into \u00bd-in. (1.25cm) pieces\n\n3 medium carrots, peeled and cut into 1-in. (2.5cm) pieces\n\n1 large yellow onion, cut into \u00bd-in. (1.25cm) chunks\n\n1 (1-in; 2.5cm) piece ginger, peeled and minced (1 TB.)\n\n3 TB. minced garlic\n\n1 large Fuji apple, peeled, cored, and grated (\u00bd cup)\n\n\u00bd cup low-sodium soy sauce\n\n2 cups water\n\n\u00bd cup chopped kimchi (optional)\n\n2 serrano chiles, chopped\n\n2 medium green onions, minced (\u00bc cup)\n\n2 TB. red miso\n\n12 oz. (340g) firm tofu, cut in 1-in. (2.5cm) cubes\n\nCooked rice noodles, hot\n\nMETHOD\n\n1 Season chuck roast all over with 1 teaspoon kosher salt.\n\n2 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat dark sesame oil and vegetable oil. When hot, add chuck roast, and brown on all sides for 5 minutes.\n\n3 Add russet potatoes, carrots, yellow onion, ginger, and garlic. Stir and saut\u00e9 for 4 minutes or until garlic and ginger become fragrant.\n\n4 Add Fuji apple, soy sauce, water, and remaining \u00bd teaspoon kosher salt, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 In a medium bowl, combine kimchi (if using), serrano chiles, and green onions.\n\n7 Remove \u00bd cup broth from the cooker, add red miso, and stir until dissolved. Pour miso broth back into the cooker, and stir.\n\n8 Add tofu, set heat to low, and warm for 4 minutes.\n\n9 Serve stew over rice noodles, topped with a healthy dollop of kimchi relish.\n\nNutrition per serving\n\nCalories: 420\n\nCarbohydrates: 25g\n\nSugars: 4g\n\nDietary fiber: 3g\n\nProtein: 31g\n\nFat: 21g\n\nCholesterol: 70mg\n\nSodium: 1,280mg\n\nVegetable Beef Soup\n\nThis simple but sensational vegetable beef soup is full of fresh vegetables and tender beef. It comes together and cooks quickly.\n\nIngredients\n\n2 tsp. vegetable oil\n\n1 (12 oz.; 340g) beef chuck, cut into \u00bd-in. (1.25cm) cubes\n\n1 medium yellow onion, diced medium (1 cup)\n\n3 medium stalks celery, diced medium (\u00be cup)\n\n1 TB. minced garlic\n\n3 medium carrots, peeled and cut into thin rounds (1 cup)\n\n2 russet potatoes, skin on, scrubbed, and cut into \u00bd-in. (1.25cm) cubes\n\n1 (14.5-oz.; 410g) can crushed tomatoes, with juice\n\n1 bay leaf\n\n3 cups homemade or sodium-free beef broth\n\n1 jalape\u00f1o chile (optional)\n\n\u00be tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bd cup frozen corn kernels\n\n\u00bd cup frozen peas\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add beef chuck, and brown on all sides for 5 minutes.\n\n2 Add yellow onion, celery, garlic, carrots, and russet potatoes, and stir. When vegetables are fragrant, add tomatoes with juice, bay leaf, beef broth, and jalape\u00f1o (if using). Season with kosher salt and black pepper, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Remove and discard jalape\u00f1o and bay leaf.\n\n5 Add corn and peas, set heat to medium-high (traditional)\/high (electric), and warm for 5 minutes. Serve hot.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 28g\n\nSugars: 7g\n\nDietary fiber: 5g\n\nProtein: 17g\n\nFat: 10g\n\nCholesterol: 35mg\n\nSodium: 520mg\nIrish Stew\n\nServed alongside a nice Irish soda bread to soak up all the broth, this hearty stew is simple yet satisfying.\n\nIngredients\n\n1 (2 lb.; 1kg) boneless leg of lamb, trimmed of sinew and excess fat and cut into 1-in. (2.5cm) cubes\n\n1\u00bd tsp. kosher salt\n\n2 TB. unsalted butter\n\n5 medium carrots, peeled and cut into 1-in. (2.5cm) chunks (2 cups)\n\n1 medium yellow onion, diced large (1 cup)\n\n5 medium Yukon Gold potatoes, halved (2 cups)\n\n\u00bc freshly ground black pepper\n\n2 tsp. fresh thyme\n\n\u00be cup water\n\nMETHOD\n\n1 In a large bowl, season lamb with 1 teaspoon kosher salt, and set aside for 20 minutes.\n\n2 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter until it's sizzling and bubbling. Add \u00bd of lamb, and brown cubes on all sides for 8 minutes. Transfer browned lamb to a plate, and brown remaining lamb, and transfer remaining browned lamb to the plate.\n\n3 Add carrots, yellow onion, and Yukon Gold potatoes to the cooker, and season with remaining \u00bd teaspoon kosher salt. Return browned lamb to the pressure cooker, season with black pepper and 1 teaspoon thyme, and stir. Add water, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir gently (potatoes will be very tender), divide into 4 bowls, and sprinkle remaining 1 teaspoon thyme over top. Serve with lots of crusty bread for dipping.\n\nNutrition per serving\n\nCalories: 590\n\nCarbohydrates: 57g\n\nSugars: 8g\n\nDietary fiber: 8g\n\nProtein: 49g\n\nFat: 18g\n\nCholesterol: 150mg\n\nSodium: 910mg\nPork Ramen\n\nRamen dishes are made of many separately cooked ingredients. With your pressure cooker, you can serve hot, tasty ramen, even on busy weeknights.\n\nIngredients\n\n2 lb. (1kg) boneless country-style pork ribs\n\n1\u00bd tsp. kosher salt\n\n1\u00bd tsp. brown sugar\n\n1 TB. vegetable oil\n\n1 slice smoky bacon\n\n1 cup water\n\n1 (12-oz.; 340g) can low-sodium spice ham, cut into X-in. slices\n\n4 cups homemade or sodium-free pork or chicken stock\n\n\u00bd cup low-sodium soy sauce\n\n12 oz. (340g) Chinese noodles, chuka soba, or spaghetti, cooked and rinsed under cold water\n\n4 large soft-boiled eggs\n\n2 medium carrots, peeled and shredded (1 cup)\n\n1 medium zucchini, shredded (1 cup)\n\n1 Fresno or serrano chile, thinly sliced\n\n3 medium green onions, thinly sliced\n\nMETHOD\n\n1 Place pork ribs on a rimmed plate, season on all sides with kosher salt and brown sugar, and refrigerate for at least 2 hours or overnight to cure.\n\n2 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add pork ribs and brown for 5 minutes on all sides (the deeper brown, the better), being careful not to let sugar burn.\n\n3 Add bacon and water, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 15 minutes. Remove from (traditional)\/turn off (electric) heat, and let pork rest for 15 minutes. Remove the lid.\n\n5 Discard bacon. Transfer pork ribs to a cutting board, and shred using 2 forks.\n\n6 In a medium saut\u00e9 pan over high heat, fry ham for 5 minutes or until crispy. Remove from heat, and keep warm.\n\n7 Add pork stock and soy sauce to the cooker, set heat to medium-high (traditional)\/high (electric), and bring to a boil. Reduce heat to low, and keep very warm.\n\n8 To serve, divide Chinese noodles among 4 bowls. Evenly divide soft-boiled eggs, ham, carrots, zucchini, pork, Fresno chile, and green onions over top.\n\n9 Bring stock back to a very hard boil, ladle over soup, and serve.\n\nNutrition per serving\n\nCalories: 1,020\n\nCarbohydrates: 71g\n\nSugars: 8g\n\nDietary fiber: 3g\n\nProtein: 75g\n\nFat: 47g\n\nCholesterol: 440mg\n\nSodium: 2,730mg\n\nHearty Turkey and Vegetable Soup\n\nThis homey soup is made comforting with the addition of simple herbs, a bit of rice, and half-and-half. It's a great choice for cool-weather meals.\n\nIngredients\n\n2 tsp. vegetable oil\n\n12 oz. (340g) turkey breast, cut into \u00bd-in. (1.25cm) cubes\n\n1 medium yellow onion, diced medium (1 cup)\n\n3 medium stalks celery, diced medium (\u00be cup)\n\n1 TB. minced garlic\n\n3 medium carrots, peeled and cut into thin rounds (1 cup)\n\n2 medium russet potatoes, skin on, scrubbed, and cut into \u00bd-in. (1.25cm) cubes\n\n\u00bc cup long-grain rice\n\n1 bay leaf\n\n\u00bd tsp. dried thyme\n\n4 cups homemade or sodium-free chicken or turkey broth\n\n\u00be tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bc cup half-and-half\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add turkey, and brown on all sides for 5 minutes.\n\n2 Add yellow onion, celery, garlic, carrots, and russet potatoes, and stir. When vegetables become fragrant, add long-grain rice, bay leaf, thyme, and chicken broth. Season with kosher salt and black pepper, and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Remove and discard bay leaf.\n\n5 Add half-and-half, stir, set heat to medium (traditional)\/high (electric), and cook for 8 minutes. Serve hot.\n\nNutrition per serving\n\nCalories: 190\n\nCarbohydrates: 22g\n\nSugars: 2g\n\nDietary fiber: 2g\n\nProtein: 18g\n\nFat: 3.5g\n\nCholesterol: 40mg\n\nSodium: 690mg\nNew England Fish Chowder\n\nFish chowder is a bit mellower than traditional clam chowder, but it's equally delicious. And in your pressure cooker, it cooks in minutes.\n\nIngredients\n\n2 thick slices smoky bacon, cut into \u00bd-in. (1.25cm) pieces\n\n2 TB. unsalted butter\n\n1 large yellow onion, diced medium (1 cup)\n\n2 large stalks celery, diced medium (1 cup)\n\n2 tsp. minced garlic\n\n4 medium Yukon Gold potatoes, peeled and diced medium (1\u00bd cups)\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n\u00bc tsp. fennel seed, crushed\n\n1 bay leaf\n\n\u00bd tsp. dried thyme\n\n12 oz. (340g) cod, tilapia, halibut, or other whitefish, cut into \u00bd-in. (1.25cm) pieces\n\n2 cups clam juice\n\n2 cups water\n\n1 TB. fresh flat-leaf parsley, minced\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, add bacon and slowly render fat, allowing bacon to become crisp without burning grease. Pour off all but 1 tablespoon grease.\n\n2 Add unsalted butter, and melt. Add yellow onion, celery, and garlic, and cook for about 3 minutes or until vegetables begin to wilt.\n\n3 Add Yukon Gold potatoes, season with kosher salt and black pepper, and stir until garlic is fragrant.\n\n4 Add fennel seed, bay leaf, thyme, cod, clam juice, and water, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Divide soup among bowls, top with flat-leaf parsley and chives, and serve.\n\nNutrition per serving\n\nCalories: 310\n\nCarbohydrates: 28g\n\nSugars: 3g\n\nDietary fiber: 3g\n\nProtein: 19g\n\nFat: 13g\n\nCholesterol: 65mg\n\nSodium: 330mg\n\nRatatouille\n\nStewed gently, the fresh vegetables in this perfect summer side dish release their juices to make a sublime broth.\n\nIngredients\n\n1 medium eggplant, peeled and cut into 1-in. (2.5cm) chunks (2 cups)\n\n\u00be tsp. kosher salt\n\n2 TB. olive oil\n\n1 large red onion, cut into wedges (1 cup)\n\n2 medium zucchini, cut into \u00be-in. (2cm) rounds (2 cups)\n\n1 medium yellow squash, cut into \u00be-in. (2cm) chunks (1 cup)\n\n1 medium red bell pepper, roasted, peeled, ribs and seeds removed, and cut into \u00bd-in. (1.25cm) strips (\u2154 cup)\n\n1 TB. minced garlic\n\n6 large plum tomatoes, halved lengthwise and cored (2 cups)\n\n\u00bd cup water\n\n\u00bc tsp. freshly ground white pepper\n\n\u00bc cup loosely packed fresh basil leaves, torn\n\nMETHOD\n\n1 Place eggplant in a colander, season with \u00bd teaspoon kosher salt, and toss to distribute salt. Set the colander in the sink, and let eggplant release its bitter water for 20 minutes to 1 hour.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add eggplant and red onion, stir, and cook for about 3 minutes or until vegetables begin to wilt.\n\n3 Add zucchini, yellow squash, red bell pepper, garlic, plum tomatoes, water, remaining \u00bc teaspoon kosher salt, and white pepper. Gently stir, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 4 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir in \u00bd of basil, gently ladle into bowls, top with remaining basil, and serve.\n\nNutrition per serving\n\nCalories: 70\n\nCarbohydrates: 15g\n\nSugars: 7g\n\nDietary fiber: 6g\n\nProtein: 3g\n\nFat: 0.5g\n\nCholesterol: 0mg\n\nSodium: 250mg\n\nHoney-Glazed Carrots\n\nHoney, vinegar, and salt are all in balance in this versatile side dish, making the carrots sweet and flavorful.\n\nIngredients\n\n4 large carrots, peeled and cut into \u00bd-in. (1.25cm) rounds\n\n2 tsp. honey\n\n1 tsp. red wine vinegar\n\n2 TB. unsalted butter\n\n\u00bc tsp. kosher salt\n\n\u00bc tsp. freshly ground white pepper\n\n1 tsp. fresh thyme leaves\n\n1 cup water\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine carrots, honey, red wine vinegar, unsalted butter, kosher salt, white pepper, thyme, and water. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 4 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Transfer carrots to a serving bowl, and cover lightly with aluminum foil.\n\n4 Set the pressure cooker to high (traditional)\/high (electric) heat, and return to a boil. Cook for about 4 minutes or until liquid is reduced to a syrupy glaze.\n\n5 Spoon hot glaze over carrots, and serve.\n\nNutrition per serving\n\nCalories: 60\n\nCarbohydrates: 6g\n\nSugars: 4g\n\nDietary fiber: 1g\n\nProtein: 1g\n\nFat: 4g\n\nCholesterol: 10mg\n\nSodium: 115mg\nGlazed Carrots with Braised Lettuce\n\nGlazed carrots are a classic fine dining dish. The addition of braised lettuce gives it a very European flair.\n\nIngredients\n\n10 medium carrots, peeled and cut into 1-in. (2.5cm) barrels (3 cups)\n\n1 cup homemade or sodium-free vegetable broth or water, or 1 cup water and \u00bd sodium-free bouillon cube\n\n\u00bd tsp. kosher salt\n\n6 to 8 small Bibb lettuce leaves from heart of head\n\n\u00bd tsp. honey\n\n1 TB. unsalted butter\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine carrots, vegetable broth, and kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 2 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Return the cooker to medium (traditional)\/high (electric) heat. Add Bibb lettuce leaves, honey, and unsalted butter, and stir until leaves are wilted.\n\n4 Taste, add more salt as needed, and serve.\n\nNutrition per serving\n\nCalories: 70\n\nCarbohydrates: 10g\n\nSugars: 7g\n\nDietary fiber: 3g\n\nProtein: 1g\n\nFat: 3g\n\nCholesterol: 10mg\n\nSodium: 390mg\nSouthern-Style Green Beans\n\nThe smoky flavor of the ham (or smoked turkey) infuses the cooking broth in this country-style green bean dish.\n\nIngredients\n\n1 TB. vegetable oil\n\n1 medium yellow onion, thinly sliced (1 cup)\n\n1\u00bd cups country ham or smoked turkey thigh, diced large\n\n1 cup water\n\n3 medium russet potatoes, halved and cut into \u00be-in. (2cm) half moons (2 cups)\n\n1\u00bd lb. (680g) green beans, trimmed and cut into 1-in. (2.5cm) lengths\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium (traditional)\/high (electric) heat, heat vegetable oil. Add yellow onion, and saut\u00e9 for 5 minutes or until onion begins to soften.\n\n2 Add ham and water, and bring to a boil.\n\n3 Add russet potatoes and green beans, season with kosher salt and black pepper, and return to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to medium-low, and cook at 2\/high for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Taste, adjust kosher salt or black pepper as needed, and gently mix with a large spoon, being careful not to break potatoes. Serve hot.\n\nNutrition per serving\n\nCalories: 290\n\nCarbohydrates: 42g\n\nSugars: 6g\n\nDietary fiber: 9g\n\nProtein: 18g\n\nFat: 7g\n\nCholesterol: 30mg\n\nSodium: 1,520mg\n\nButtered Green Beans with Nut Crunch Topping\n\nThe simple addition of a nut crunch topping transforms these perfectly tender green beans to something elegant.\n\nIngredients\n\n3 TB. unsalted butter\n\n\u00bd cup walnuts, hazelnuts, or pecans\n\n\u00bd cup panko breadcrumbs\n\n\u00be tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 lb. (450g) green beans, trimmed\n\nMETHOD\n\n1 In a small saut\u00e9 pan over medium heat, melt 1\u00bd tablespoons unsalted butter.\n\n2 Add walnuts, panko breadcrumbs, \u00bd teaspoon kosher salt, and black pepper, and saut\u00e9 for 2 or 3 minutes or until golden. Transfer mixture to a small bowl.\n\n3 Add a steamer basket to a 4-quart (4l) pressure cooker, fill with the minimum amount of water allowed for your cooker, and add green beans to the basket. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 4 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 In a small microwave-safe dish, melt remaining 1\u00bd tablespoons butter in the microwave on high for 20 seconds.\n\n6 Transfer green beans to a bowl, toss with melted butter and remaining \u00bc teaspoon kosher salt, sprinkle with nut crunch topping, and serve.\n\nNutrition per serving\n\nCalories: 230\n\nCarbohydrates: 17g\n\nSugars: 4g\n\nDietary fiber: 4g\n\nProtein: 5g\n\nFat: 17g\n\nCholesterol: 25mg\n\nSodium: 380mg\nCaramelized Onion Mashed Potatoes\n\nPressure cooking quickly and evenly steams potatoes for a smooth, buttery mash. The caramelized onions add a nice flavor to the finished dish.\n\nIngredients\n\n1 TB. plus \u00bc cup unsalted butter\n\n1 large yellow onion, very thinly sliced (1\u00bc cups)\n\n1 tsp. kosher salt\n\n10 medium russet potatoes, peeled and cut into 1-in. (2.5cm) half moons (4 cups)\n\n\u2153 cup heavy cream, hot\n\n\u00bd cup whole milk, hot\n\n\u215b tsp. freshly ground white pepper\n\nMETHOD\n\n1 In a medium skillet over medium heat, melt 1 tablespoon unsalted butter.\n\n2 Add yellow onion and \u00bd teaspoon kosher salt, and stir once to break apart onion layers. Reduce heat to medium-low, and cook for 5 minutes. Stir once, and cook for 10 minutes. Repeat until onions are gooey and brown, and set aside.\n\n3 Add a steamer basket to a 4-quart (4l) pressure cooker, pour in 1 cup water, and set heat to medium-high (traditional)\/high (electric). Add russet potatoes to the basket, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Test potatoes for doneness by inserting a knife into a large piece. It should easily pierce potato. If potatoes aren't done, lock on the lid and cook for 2 more minutes. Transfer potatoes to a large bowl, and let them steam dry for 30 seconds.\n\n6 Using a potato masher or an electric mixer on low, mash potatoes.\n\n7 Add remaining \u00bc cup unsalted butter, and stir gently while it melts. Add caramelized onions, hot heavy cream, hot whole milk, remaining \u00bd teaspoon kosher salt, and white pepper, and stir. If potatoes are too thick, add more milk a little at a time and stir between additions until you reach your desired consistency. Serve hot.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 31g\n\nSugars: 3g\n\nDietary fiber: 2g\n\nProtein: 5g\n\nFat: 15g\n\nCholesterol: 45mg\n\nSodium: 340mg\nMashed Maple Sweet Potatoes\n\nThe butter, syrup, and vanilla extract complement the sweet potatoes in this light and lovely dish.\n\nIngredients\n\n3 large sweet potatoes, peeled and cut into \u00bd-in. (1.25cm) rounds (2\u00bd cups)\n\n3 TB. unsalted butter\n\n3 TB. pure maple syrup\n\n1 tsp. vanilla extract\n\n\u00bd tsp. kosher salt\n\nMETHOD\n\n1 Add a steamer basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Add sweet potatoes to the steamer basket, and return to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, remove the lid, and let sweet potatoes sit for 1 minute to steam dry.\n\n4 Transfer sweet potatoes to a large bowl, and using an electric mixer on medium, blend potatoes for a few seconds.\n\n5 Add unsalted butter, and blend until combined. Stir in maple syrup, vanilla extract, and kosher salt, and serve hot.\n\nNutrition per serving\n\nCalories: 110\n\nCarbohydrates: 14g\n\nSugars: 8g\n\nDietary fiber: 1g\n\nProtein: 1g\n\nFat: 6g\n\nCholesterol: 15mg\n\nSodium: 180mg\n\nBrussels Sprouts with Almonds and Prosciutto\n\nBrussels sprouts are a nutritious vegetable, and they're delicious paired with prosciutto and almonds.\n\nIngredients\n\n4 tsp. unsalted butter\n\n1 oz. (25g) prosciutto, minced\n\n1 lb. (450g) medium brussels sprouts, stalk ends trimmed and halved lengthwise\n\n\u00bc tsp. kosher salt\n\n\u00bd cup water\n\n\u00bd cup almonds, toasted and chopped\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt 3 teaspoons unsalted butter. Add prosciutto, and cook for 2 minutes or until crisp.\n\n2 Add brussels sprouts, and saut\u00e9 for about 3 minutes or until they begin to brown.\n\n3 Season with kosher salt, add water, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 3 minutes (add 30 more seconds for large sprouts). Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Drain sprouts, and transfer to a serving bowl. Toss with almonds, black pepper, and remaining 1 teaspoon unsalted butter, and serve.\n\nNutrition per serving\n\nCalories: 140\n\nCarbohydrates: 8g\n\nSugars: 2g\n\nDietary fiber: 4g\n\nProtein: 5g\n\nFat: 11g\n\nCholesterol: 10mg\n\nSodium: 135mg\nBraised Kale\n\nKale makes great salads and one-pot meals, but it's especially good braised with a little chicken broth and a pinch of cloves until just tender.\n\nIngredients\n\n2 TB. vegetable oil\n\n2 bunches curly leaf kale, stems removed and leaves chopped (6 to 8 cups)\n\n1\u00bd cups homemade or sodium-free chicken broth\n\n1 pinch ground cloves\n\n\u00bd tsp. kosher salt\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil.\n\n2 Add kale, and turn leaves over on themselves to coat with oil.\n\n3 Add chicken broth, cloves, and kosher salt, and turn kale again to distribute seasonings. Bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Taste, season with more salt as necessary, stir, and serve.\n\nNutrition per serving\n\nCalories: 120\n\nCarbohydrates: 10g\n\nSugars: 0g\n\nDietary fiber: 2g\n\nProtein: 5g\n\nFat: 8g\n\nCholesterol: 10mg\n\nSodium: 330mg\nAlsatian-Style Braised Red Cabbage\n\nThe sweetness of the apples alongside the braised red cabbage marries well with pork of all kinds (especially pork loin braised in Belgian beer).\n\nIngredients\n\n2 TB. unsalted butter or bacon grease\n\n1 large yellow onion, minced (1 cup)\n\n1 TB. sugar\n\n1 medium head red cabbage, chopped (6 cups)\n\n2 medium green apples, peeled, cored, and diced small (1 cup)\n\n1 large russet potato, peeled and grated (\u00be cup)\n\n\u00be cup water\n\n2 TB. red wine vinegar\n\n2 TB. red or black currant jam\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium (traditional)\/high (electric) heat, melt unsalted butter.\n\n2 Add yellow onion and sugar, stir, and cook for 6 minutes or until onions begin to brown and sugar begins to caramelize.\n\n3 Add red cabbage, green apples, and russet potato, and stir. Add water, red wine vinegar, and red currant jam, and stir again.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir, taste, adjust seasoning or add a touch more red wine vinegar if desired, and serve.\n\nNutrition per serving\n\nCalories: 290\n\nCarbohydrates: 59g\n\nSugars: 30g\n\nDietary fiber: 9g\n\nProtein: 6g\n\nFat: 6g\n\nCholesterol: 15mg\n\nSodium: 110mg\n\nOne-Pot Cabbage, Rice, and Lentils\n\nThis hearty side dish goes well with sausages, pork chops, and roasted meats. It's also good as a light dinner served on its own.\n\nIngredients\n\n2 slices smoky bacon\n\n2 TB. unsalted butter\n\n1 medium yellow onion, diced small (\u2154 cup)\n\n\u00bd small head green cabbage, cored and diced (2 cups)\n\n\u00bd cup jasmine rice\n\n\u00bd cup du Puy lentils\n\n1 cup homemade or sodium-free beef stock\n\n\u00be tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n2 medium carrots, peeled and diced small (\u00bd cup)\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, add bacon. Set heat to medium-high (traditional)\/high (electric), and slowly render fat from bacon, allowing it to become crisp.\n\n2 Add unsalted butter, and let it bubble and foam. Add yellow onion and green cabbage, stir, and cook for about 3 minutes to allow vegetables to wilt a bit.\n\n3 Add jasmine rice, du Puy lentils, beef stock, kosher salt, and black pepper. Stir, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir in carrots.\n\n6 Taste, and add more kosher salt or black pepper as necessary, and serve.\n\nNutrition per serving\n\nCalories: 220\n\nCarbohydrates: 31g\n\nSugars: 6g\n\nDietary fiber: 6g\n\nProtein: 8g\n\nFat: 8g\n\nCholesterol: 15mg\n\nSodium: 360mg\nSouthern Collard Greens\n\nFew dishes are more satisfying than collard greens. In your pressure cooker, collards become tender and meaty.\n\nIngredients\n\n2 cups water\n\n1 (8-oz.; 225g) smoked ham hock\n\n\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 tsp. crushed red pepper flakes\n\n2 small bunches collard greens, stems removed and chopped (8 cups)\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine water and ham hock. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Add kosher salt, black pepper, crushed red pepper flakes, and collard greens. Using tongs, turn collards to coat with broth.\n\n4 Lock on the lid, return pressure to level 2\/high, and cook for 12 minutes. Remove from heat, perform a quick release, and remove the lid.\n\n5 Transfer ham hock to a cutting board. Place the lid loosely back on the pressure cooker.\n\n6 Slice off pork skin, pull meat from the bone, and chop meat finely. Return meat to the pressure cooker, and stir.\n\n7 Taste, adjust seasoning as necessary, and serve.\n\nNutrition per serving\n\nCalories: 150\n\nCarbohydrates: 4g\n\nSugars: 0g\n\nDietary fiber: 3g\n\nProtein: 5g\n\nFat: 13g\n\nCholesterol: 20mg\n\nSodium: 500mg\n\nBarbecue Braised Short Ribs\n\nThese short ribs are a perfect match for your pressure cooker. They come out juicy, tender, and succulent in under an hour.\n\nIngredients\n\n3 lb. (1.5kg) beef short ribs\n\n2 tsp. kosher salt\n\n1 TB. vegetable oil\n\n1 large yellow onion, thinly sliced (1 cup)\n\n1 cup water\n\n\u00bd cup barbecue sauce\n\nMETHOD\n\n1 Season beef short ribs on all sides with 1\u00bd teaspoons kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot and nearly smoking, add ribs, leaving space between them so they brown nicely and don't steam. (You might need to do this in batches.) Transfer browned ribs to a baking sheet, and set aside.\n\n3 Add yellow onion to the cooker, and cook for about 3 minutes or until it begins to wilt.\n\n4 Season onion with remaining \u00bd teaspoon kosher salt, and add ribs and water.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 40 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n6 Gently transfer ribs to a large plate, and drain off all but 3 tablespoons liquid from onions.\n\n7 Add barbecue sauce to onions, and stir. Return ribs to the pressure cooker, turn ribs to coat in sauce, and heat for about 5 minutes. Serve hot.\n\nNutrition per serving\n\nCalories: 470\n\nCarbohydrates: 4g\n\nSugars: 1g\n\nDietary fiber: 0g\n\nProtein: 48g\n\nFat: 28g\n\nCholesterol: 150mg\n\nSodium: 1,030mg\n\nBeef Sugo\n\nBasil, oregano, rosemary, spices, and yellow onion flavor a simple yet rich and satisfying sauce that's served over spaghetti.\n\nIngredients\n\n1 (3-lb.; 2.5kg) chuck roast\n\n2 tsp. kosher salt\n\n1 TB. unsalted butter\n\n1 medium yellow onion, cut into thick wedges\n\n3\u00bd cups strained tomatoes\n\n2 tsp. dried rosemary, coarsely ground\n\n1\u00bd tsp. dried oregano\n\n1 tsp. dried basil\n\n\u00bc tsp. freshly ground black pepper\n\n1 lb. (450g) cooked spaghetti, hot\n\n\u00bd cup grated Parmesan cheese\n\nMETHOD\n\n1 Season chuck roast on both sides with 1 teaspoon kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat unsalted butter until it starts to bubble. Add chuck roast, and brown on both sides for about 4 minutes.\n\n3 Add yellow onion, and cook for 3 to 5 minutes or until it wilts.\n\n4 Add strained tomatoes, rosemary, oregano, basil, remaining 1 teaspoon kosher salt, and black pepper, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 45 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n6 Transfer beef to a baking sheet, and shred using 2 forks. Return shredded beef to sauce, and stir.\n\n7 Divide spaghetti among 6 plates, top with sauce, and serve with Parmesan cheese on the side.\n\nNutrition per serving\n\nCalories: 780\n\nCarbohydrates: 57g\n\nSugars: 4g\n\nDietary fiber: 2g\n\nProtein: 57g\n\nFat: 34g\n\nCholesterol: 125mg\n\nSodium: 820mg\nCorned Beef\n\nThis New England boiled dinner, with corned beef, cabbage, and vegetables, makes a very filling meal.\n\nIngredients\n\n1 (3-lb.; 1.5kg) corned beef brisket\n\n2 cups water\n\n1 TB. pickling spice\n\n2 medium yellow onions, quartered (2 cups)\n\n8 medium Yukon Gold potatoes, halved\n\n8 medium carrots, peeled and cut into 2-in. (5cm) pieces (2\u00bd cups)\n\n1 small head green cabbage, quartered\n\nKosher salt\n\nDijon mustard\n\nUnsalted butter\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine corned beef brisket, water, pickling spice, and 2 yellow onion quarters. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 35 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Add remaining yellow onion quarters, Yukon Gold potatoes, carrots, and green cabbage, in that order, and season with kosher salt. Return the pressure cooker to medium-high\/high heat, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2\/high. Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from\/turn off heat, perform a cold water\/quick release, and remove the lid.\n\n5 Transfer corned beef to a cutting board, slice thinly, and place on a platter. Surround meat with vegetables, and ladle some juice from the cooker over top. Serve with Dijon mustard, unsalted butter, and kosher salt on the side.\n\nTo prepare the corned beef brisket without additional vegetables, cook it straight through for 45 minutes.\n\nNutrition per serving\n\nCalories: 1,110\n\nCarbohydrates: 100g\n\nSugars: 14g\n\nDietary fiber: 12g\n\nProtein: 62g\n\nFat: 51g\n\nCholesterol: 185mg\n\nSodium: 4,260mg\nCuban-Style Ropa Vieja\n\nRopa vieja encompasses all the flavors associated with Cuban cuisine and is commonly served with **rice and peas, bread,** and **red wine**.\n\nIngredients\n\n1 (1\u00bd-lb.; 680g) top round steak, well marbled\n\n\u00bd tsp. kosher salt\n\n3 TB. olive oil\n\n2 small yellow onions, diced small (1 cup)\n\n1 medium carrot, peeled and diced small (\u2153 cup)\n\n1 bay leaf\n\n\u00bd tsp. crushed cumin\n\n1 (6-oz.; 170g) can tomato paste\n\n1 tsp. dried oregano\n\n\u00bd tsp. dried thyme\n\n1 TB. minced garlic\n\n\u00be cup homemade or sodium-free beef stock, or water\n\n\u00bd cup green olives, pitted and halved\n\n1 Fresno or jalape\u00f1o chile, chopped\n\n1 TB. capers, rinsed and coarsely chopped\n\n2 TB. fresh cilantro, minced\n\n1 TB. red wine vinegar\n\n1 small red onion, diced fine (\u00bc cup)\n\n1 large lime, quartered\n\nMETHOD\n\n1 Season top round steak with kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat 1 tablespoon olive oil. When hot, add top round steak, and sear on both sides for 4 minutes or until very brown.\n\n3 Add yellow onions and carrot, and cook for about 3 minutes or until vegetables begin to soften.\n\n4 Add bay leaf, cumin, tomato paste, oregano, thyme, and garlic, and stir until fragrant. Add beef stock, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 45 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 In a small bowl, combine green olives, Fresno chile, capers, cilantro, red wine vinegar, red onion, and remaining 2 tablespoons olive oil. Set aside for 10 minutes.\n\n7 Transfer beef to a cutting board, and shred using 2 forks. Return beef to the cooker, and stir to coat in broth. Remove bay leaf.\n\n8 Garnish with green olive salsa, and serve hot with lime quarters on the side.\n\nTraditionally, ropa vieja uses flank steak, but less-expensive top round steak works well in your pressure cooker.\n\nNutrition per serving\n\nCalories: 460\n\nCarbohydrates: 17g\n\nSugars: 9g\n\nDietary fiber: 3g\n\nProtein: 42g\n\nFat: 25g\n\nCholesterol: 85mg\n\nSodium: 870mg\n\nClassic Beef Brisket\n\nFor a tender, juicy brisket you'll be proud to serve at family or holiday dinners, opt for a second cut, which contains more pressure cooker\u2013friendly fat.\n\nIngredients\n\n1 (3\u00bd-lb.; 1.75kg) beef brisket\n\n2\u00bd tsp. kosher salt\n\n1\u00bd TB. canola oil\n\n2 large yellow onions, sliced (3 cups)\n\n\u00bd tsp. dried thyme\n\n\u00bc tsp. freshly ground black pepper\n\n1\u00bd cups water\n\nMETHOD\n\n1 Season beef brisket with 2 teaspoons kosher salt, and set aside until salt is absorbed.\n\n2 Set a 4-quart (4l) pressure cooker to medium-high (traditional)\/high (electric) heat. When hot, add canola oil. Carefully add brisket, fat side down, and sear for 4 minutes or until fat is very brown. (Reduce heat if necessary to avoid burning oil.) Turn over brisket, and brown meat side for 4 minutes. Transfer browned brisket to a tray.\n\n3 Drain off most of oil from the pressure cooker.\n\n4 Set heat to medium-low (traditional)\/low (electric), and add yellow onions. Season with thyme, remaining \u00bd teaspoon kosher salt, and black pepper, and cook for 10 minutes or until onions begin to brown and become soft.\n\n5 Place brisket on top of onions, add water, and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 40 minutes for sliced brisket or 50 minutes for shredded beef. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Transfer brisket to a platter, slice or shred, and serve hot.\n\nNutrition per serving\n\nCalories: 320\n\nCarbohydrates: 7g\n\nSugars: 3g\n\nDietary fiber: 1g\n\nProtein: 50g\n\nFat: 9g\n\nCholesterol: 100mg\n\nSodium: 970mg\nPot Roast with Fennel and Carrots\n\nBeef and fennel are naturals together. This pot roast is perfect for chilly evenings when you want to warm your family from the inside out.\n\nIngredients\n\n1 (3\u00bd-lb.; 1.75kg) boneless chuck roast\n\n1 tsp. kosher salt\n\n1 TB. olive oil\n\n1 medium bulb fennel, cored and diced small (\u00be cup)\n\n1 large yellow onion, diced small (1 cup)\n\n1 tsp. dried rosemary\n\n1\u00bd tsp. crushed fennel seed\n\n1 TB. red wine vinegar\n\n1 TB. tomato paste\n\n1 cup homemade or sodium-free beef stock\n\n\u00bc tsp. freshly ground black pepper\n\n6 medium carrots, peeled and cut into 1-in. (2.5cm) cylinders (2 cups)\n\nFennel fronds\n\nMETHOD\n\n1 Season chuck roast on both sides with \u00bd teaspoon kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Add chuck roast, and brown very deeply on both sides for 5 minutes.\n\n3 Add fennel and yellow onion, stir, and cook for 4 minutes or until vegetables are soft.\n\n4 Add rosemary, fennel seed, red wine vinegar, and tomato paste, and cook for 30 seconds or until seasonings are fragrant.\n\n5 Add beef stock, remaining \u00bd teaspoon kosher salt, and black pepper, and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Add carrots.\n\n8 Lock on the lid, and bring pressure to level 2\/high. Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from\/turn off heat, perform a quick release, and remove the lid.\n\n9 Transfer roast to a cutting board, slice thinly, and place on a platter. Surround with carrots, ladle jus over top, garnish with fennel, and serve.\n\nNutrition per serving\n\nCalories: 870\n\nCarbohydrates: 16g\n\nSugars: 6g\n\nDietary fiber: 5g\n\nProtein: 78g\n\nFat: 54g\n\nCholesterol: 245mg\n\nSodium: 730mg\nSwedish Meatballs\n\nThis Scandinavian fare is often served as an hors d'oeuvre, but it's even better for dinner. These meatballs are full of flavor but not overly spicy.\n\nIngredients\n\n1 lb. lean (85% lean) ground chuck\n\n\u00bd lb. lean ground pork\n\n\u2153 cup unseasoned breadcrumbs\n\n2 TB. heavy cream\n\n1 tsp. kosher salt\n\n1 small yellow onion, diced fine (\u2153 cup)\n\n\u00bd tsp. freshly ground black pepper\n\n\u00bd tsp. allspice\n\n\u215b tsp. nutmeg\n\n1 TB. canola oil\n\n1 cup homemade or sodium-free beef broth\n\n\u00bd cup sour cream\n\n1 TB. fresh dill, minced\n\nMETHOD\n\n1 In a large bowl, combine ground chuck, ground pork, unseasoned breadcrumbs, heavy cream, kosher salt, yellow onion, black pepper, allspice, and nutmeg.\n\n2 With clean, wet hands, scoop out a golf ball\u2013size piece of meat mixture, roll it in the palms of your hands until it's a smooth ball, and set aside. Repeat with remaining meat mixture until you have 12 equal-size meatballs.\n\n3 Set a 4-quart (4l) pressure cooker to medium-high (traditional)\/high (electric) heat. When hot, add canola oil. Add meatballs, and brown on all sides for 5 minutes. (Reduce heat if necessary to avoid oil smoking.)\n\n4 Add beef broth, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n6 Transfer meatballs to a plate.\n\n7 Whisk sour cream into broth until smooth. Add dill, return meatballs to sauce, and stir to coat and warm. (Do not let sauce come to a boil or it might separate.) Serve hot over rice.\n\nNutrition per serving\n\nCalories: 390\n\nCarbohydrates: 9g\n\nSugars: 2g\n\nDietary fiber: 1g\n\nProtein: 27g\n\nFat: 26g\n\nCholesterol: 110mg\n\nSodium: 660mg\n\nBeef Bourguignon\n\nYour pressure cooker makes quick work of this classic beef, red wine, onion, and mushroom dish.\n\nIngredients\n\n1 (2\u00bd-lb.; 1.25kg) beef chuck, cut into 2-in. (5cm) cubes\n\n1\u00bd tsp. kosher salt\n\n\u00bd tsp. freshly ground black pepper\n\n3 TB. unsalted butter\n\n8 oz. (225g) small white mushrooms, brushed clean\n\n24 small pearl onions, peeled\n\n1\u00bd TB. all-purpose flour\n\n1 TB. tomato paste\n\n1 cup dry red wine\n\n1 cup homemade or sodium-free beef stock\n\n15 sprigs thyme\n\n1 bay leaf\n\nMETHOD\n\n1 Season beef chuck with 1 teaspoon kosher salt and black pepper, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt 2 tablespoons unsalted butter. Add \u00bd of beef chuck, and brown on all sides for 4 minutes. Transfer browned beef to a plate, repeat with remaining beef, and transfer rest of browned beef to the plate.\n\n3 Add remaining 1 tablespoon unsalted butter, white mushrooms, pearl onions, and remaining \u00bd teaspoon kosher salt to the cooker, and cook for 5 minutes or until vegetables brown slightly.\n\n4 Add all-purpose flour, and stir to coat vegetables in flour. Add tomato paste, and stir again. Add red wine, bring to a boil, and burn off alcohol for 2 minutes.\n\n5 Add beef stock, browned beef, \u00bd of thyme sprigs, and bay leaf, and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Remove bay leaf. Divide beef, mushrooms, and onions evenly among 4 bowls, and ladle broth over top. Remove thyme leaves from stems, sprinkle leaves over top of each bowl, and serve.\n\nNutrition per serving\n\nCalories: 740\n\nCarbohydrates: 15g\n\nSugars: 5g\n\nDietary fiber: 0g\n\nProtein: 56g\n\nFat: 44g\n\nCholesterol: 200mg\n\nSodium: 890mg\nChuck Roast with Horseradish Cream and Carrots\n\nAn ideal Sunday supper dish, this eastern European\u2013inspired recipe combines tender beef and carrots with garlic, marjoram, and horseradish.\n\nIngredients\n\n1 (2\u00bd-lb.; 1.75kg) chuck roast\n\n\u00bd tsp. kosher salt\n\n1 TB. vegetable oil\n\n2 tsp. minced garlic\n\n1 cup homemade or sodium-free beef stock\n\n\u00bd tsp. dried marjoram\n\n2 tsp. prepared horseradish\n\n4 large carrots, peeled and cut into 3-in. (7.5cm) sticks\n\n\u00bd cup heavy cream (optional)\n\nMETHOD\n\n1 Season chuck roast with kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add chuck roast, and brown deeply for 3 minutes. Turn over roast, and brown other side for 3 minutes.\n\n3 Add garlic, and cook for 10 seconds or until fragrant.\n\n4 Add beef stock, marjoram, and horseradish. Stir, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Add carrots.\n\n7 Lock on the lid, and return pressure to level 2\/high. Reduce heat to low, and cook at 2\/high for 4 minutes. Remove from\/turn off heat, perform a cold water\/quick release, and remove the lid.\n\n8 Transfer roast to a platter, and surround with carrots. Cover with aluminum foil to keep warm, and set aside.\n\n9 Add heavy cream (if using) to the pressure cooker, set heat to high (traditional)\/high (electric), and cook for 5 minutes or until sauce is reduced by half. Taste, and add kosher salt as necessary.\n\n10Slice roast, ladle sauce over top, and serve.\n\nNutrition per serving\n\nCalories: 610\n\nCarbohydrates: 7g\n\nSugars: 3g\n\nDietary fiber: 2g\n\nProtein: 55g\n\nFat: 39g\n\nCholesterol: 175mg\n\nSodium: 650mg\nSwiss Steak\n\nTop sirloin and vegetables transform to succulent Swiss steak in your pressure cooker. If you like, add 1 cup sliced button mushrooms with the vegetables.\n\nIngredients\n\n1 (2-lb.; 1kg) top sirloin, 1 in. (2.5cm) thick\n\n1 tsp. kosher salt\n\n1 TB. vegetable oil\n\n2 medium yellow onions, sliced thin (1\u00bd cups)\n\n1 large red bell pepper, ribs and seeds removed, and thinly sliced (\u00be cup)\n\n2 medium stalks celery, thinly sliced (\u00bd cup)\n\n3 cloves garlic, minced\n\n\u00bc tsp. freshly ground black pepper\n\n1 cup homemade or sodium-free beef broth\n\n1 TB. cornstarch\n\n2 TB. cold water\n\nMETHOD\n\n1 Season top sirloin on both sides with \u00bd teaspoon kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add sirloin, and sear on both sides for 4 minutes or until very brown. Transfer sirloin to a tray.\n\n3 Pour off fat from the pressure cooker. Add yellow onions, red bell pepper, celery, garlic, black pepper, and remaining \u00bd teaspoon kosher salt. Saut\u00e9 for about 3 minutes or until vegetables are wilted.\n\n4 Set sirloin on vegetables, add beef broth, and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Transfer sirloin to a cutting board, and cover with aluminum foil to keep warm.\n\n7 In a small bowl, combine cornstarch and cold water.\n\n8 Return the pressure cooker to medium-high\/high heat, and bring to a boil. Stir in cornstarch mixture, and return to a boil. Reduce heat to a simmer, and cook for 1 minute to thicken.\n\n9 Slice steak, arrange on a platter, ladle sauce over top, and serve.\n\nNutrition per serving\n\nCalories: 500\n\nCarbohydrates: 11g\n\nSugars: 4g\n\nDietary fiber: 2g\n\nProtein: 68g\n\nFat: 21g\n\nCholesterol: 185mg\n\nSodium: 1,300mg\n\nOsso Buco\n\nThis Italian braised veal and vegetable dish is rich with flavorful broth. It's especially good served with Risotto Milanese on the side.\n\nIngredients\n\n4 (12-oz.; 340g) meaty veal shanks (osso buco cut)\n\n1\u00bd tsp. kosher salt\n\n1 TB. vegetable oil\n\n1 large yellow onion, diced medium (1\u00bd cups)\n\n2 medium carrots, peeled and cut into rounds (\u00be cup)\n\n2 medium stalks celery, cut into thin half-moons (\u00be cup)\n\n2 cloves garlic, minced\n\n1\u00bd tsp. dried rosemary, ground\n\n\u00bd tsp. dried sage, ground\n\n2 TB. tomato paste\n\n\u00be cup homemade or sodium-free beef broth\n\n\u00bc tsp. freshly ground black pepper\n\nMETHOD\n\n1 Season veal shanks with \u00be teaspoon kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, carefully add shanks, and brown on all sides for 4 minutes. (You might need to do this in batches.) Transfer browned veal to a tray.\n\n3 Add yellow onion, carrots, celery, and garlic to the cooker, and cook for about 4 minutes or until wilted.\n\n4 Add rosemary and sage, and cook for about 30 seconds or until fragrant.\n\n5 Add tomato paste, and stir. Add beef broth, remaining \u00be teaspoon kosher salt, and black pepper, and place veal on top of vegetables. Bring to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Check veal tenderness by inserting a knife; it should pierce meat easily. If necessary, lock on lid, bring to high\/2 pressure, and cook for 10 more minutes.\n\n8 Taste broth, and season with salt as necessary. Serve veal with broth over top.\n\nNutrition per serving\n\nCalories: 350\n\nCarbohydrates: 10g\n\nSugars: 5g\n\nDietary fiber: 2g\n\nProtein: 48g\n\nFat: 12g\n\nCholesterol: 180mg\n\nSodium: 1,140mg\nBelgian Beef Stew Cooked in Beer\n\nThis traditional Belgian dish, also known as _carbonnade_ , relies on the deep flavors of the Belgian beer to differentiate it from beef bourguignon.\n\nIngredients\n\n1 (2\u00bd-lb.; 1.25kg) beef chuck, cut into 2-in. (5cm) cubes\n\n1\u00bd tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n3 TB. unsalted butter\n\n2 large yellow onions, sliced (3 cups)\n\n1\u00bd TB. all-purpose flour\n\n1 tsp. brown sugar\n\n1 TB. Dijon mustard\n\n1\u00bd cups Belgian farmhouse ale\n\n1 cup homemade or sodium-free beef stock\n\n1 bay leaf\n\n15 sprigs thyme\n\nMETHOD\n\n1 Season beef chuck with 1 teaspoon kosher salt and black pepper, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt 2 tablespoons unsalted butter. Add \u00bd of beef, and brown on all sides for 4 minutes. Transfer browned beef to a plate, repeat with remaining beef, and transfer rest of browned beef to the plate.\n\n3 Add remaining 1 tablespoon unsalted butter, yellow onions, and remaining \u00bd teaspoon salt to the cooker, and cook for 5 minutes or until onions are slightly brown.\n\n4 Add all-purpose flour, and stir to coat. Stir in brown sugar, Dijon mustard, and Belgian farmhouse ale. Bring to a boil, and burn off alcohol for 1 minute.\n\n5 Add beef stock, beef, bay leaf, and \u00bd of thyme, and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Remove bay leaf. Divide beef and onions evenly among 4 bowls, and ladle broth over top. Remove thyme leaves from remaining stems, sprinkle leaves over each serving, and serve.\n\nNutrition per serving\n\nCalories: 720\n\nCarbohydrates: 14g\n\nSugars: 5g\n\nDietary fiber: 1g\n\nProtein: 56g\n\nFat: 44g\n\nCholesterol: 200mg\n\nSodium: 300mg\nPressure Cooker Tacos\n\nBy using pork ribs instead of ground beef, this taco meat is much closer to traditional Mexican fare. This kid-friendly recipe is very mild, heat-wise.\n\nIngredients\n\n3 (1\u00bc-lb.; 565g) boneless country-style pork shoulder ribs\n\n1 TB. kosher salt\n\n1 TB. ground cumin\n\n1 TB. dried oregano, coarsely ground\n\n1\u00bd TB. mild chili powder\n\n1 TB. vegetable oil\n\n1 cup water\n\n16 (6-in.; 16.25cm) corn or (8-in.; 20cm) flour tortillas\n\n2 medium limes, quartered\n\nToppings and condiments: sour cream, thinly sliced cabbage, salsa, shredded cheese, thinly sliced radishes, sliced chile peppers\n\nMETHOD\n\n1 Season pork shoulder ribs on all sides with kosher salt, and set aside until salt is absorbed.\n\n2 In a small bowl, combine cumin, oregano, and chili powder. Season ribs on all sides with spice mix, and refrigerate for 2 hours or overnight.\n\n3 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil, swirling the cooker gently to coat. Add ribs, and brown on all sides for 4 minutes. Add water.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 40 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) or natural release, and remove the lid.\n\n5 Wrap corn tortillas in aluminum foil and warm in a 200\u00b0F (90\u00b0C) oven for 10 minutes.\n\n6 Transfer ribs to a plate, and shred meat into small pieces using 2 forks. Place meat in a bowl, and stir in \u00bd cup or more juice from the cooker.\n\n7 Serve meat alongside lime quarters and your choice of toppings and condiments, allowing everyone to make their own tacos.\n\nNutrition per serving\n\nCalories: 450\n\nCarbohydrates: 28g\n\nSugars: 0g\n\nDietary fiber: 2g\n\nProtein: 44g\n\nFat: 17g\n\nCholesterol: 140mg\n\nSodium: 960mg\n\nHungarian Stuffed Peppers\n\nThese elegant stuffed peppers are incredibly delicious and very eastern European in flavor.\n\nIngredients\n\n4 medium red bell peppers\n\n2 slices white bread, torn into small pieces (1 cup)\n\n\u00bc cup plus 2 TB. heavy cream (2 TB. optional)\n\n1 medium green onion, sliced into thin rings (3 TB.)\n\n\u00bc cup grated Parmesan cheese\n\n\u00bc tsp. dried rosemary\n\n\u215b tsp. caraway seed, ground\n\n\u2153 cup raisins\n\n2 TB. minced fresh flat-leaf parsley\n\n\u00be cup walnuts, chopped\n\n1\u00bc tsp. kosher salt\n\n\u215b tsp. plus \u00bc tsp. freshly ground black pepper\n\n1\u00bc lb. (565g) lean (10% fat) ground beef\n\n2 cups tomato sauce\n\n1\u00bd TB. unsalted butter\n\n\u00bd cup panko breadcrumbs\n\nMETHOD\n\n1 Add a trivet and steamer basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 Cut around stem of red bell peppers, keeping the knife parallel to outside edge of pepper as you cut. Remove core, seeds, and ribs.\n\n3 In a large bowl, combine white bread, \u00bc cup heavy cream, green onion, Parmesan cheese, rosemary, caraway seed, raisins, flat-leaf parsley, \u00bc cup walnuts, \u00be teaspoon kosher salt, and \u215b teaspoon black pepper. Add ground beef, and combine.\n\n4 Evenly divide filling among peppers, and add peppers to the steamer basket. Be sure water does not touch bottom of peppers. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and set aside for 5 minutes. Remove the lid.\n\n6 In a small saucepan over medium heat, warm tomato sauce and remaining 2 tablespoons heavy cream (if using).\n\n7 In a small saut\u00e9 pan over medium heat, melt unsalted butter. Add remaining \u00bd cup walnuts, panko breadcrumbs, remaining \u00bd teaspoon kosher salt, and remaining \u00bc teaspoon black pepper, and saut\u00e9 for 2 or 3 minutes or until golden.\n\n8 Transfer peppers to a plate, spoon tomato sauce over top, sprinkle with nut crunch topping, and serve.\n\nNutrition per serving\n\nCalories: 330\n\nCarbohydrates: 20g\n\nSugars: 9g\n\nDietary fiber: 3g\n\nProtein: 20g\n\nFat: 19g\n\nCholesterol: 60mg\n\nSodium: 730mg\nHungarian Chicken Paprika\n\nThis simple, hearty chicken stew made with sweet, not hot, paprika is comforting rather than spicy. Kids will like it as much as adults will.\n\nIngredients\n\n3 (6-oz.; 170g) chicken breasts, cut into 1-in. (2.5cm) lengthwise strips\n\n\u00be tsp. kosher salt\n\n1\u00bd TB. vegetable oil\n\n1 medium red bell pepper, ribs and seeds removed, and finely chopped (\u00bd cup)\n\n1 large yellow onion, finely chopped (1 cup)\n\n3 TB. all-purpose flour\n\n\u215b tsp. freshly ground white pepper\n\n2 TB. sweet paprika\n\n3 TB. tomato paste\n\n\u00bd cup homemade or sodium-free chicken broth\n\n\u00bd cup sour cream\n\n2 TB. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 Season chicken breasts with \u00bd teaspoon kosher salt, and set aside until salt is absorbed.\n\n2 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add chicken, and brown for 3 minutes. Transfer browned chicken to a tray.\n\n3 Add red bell pepper and yellow onion to the cooker, and cook for about 4 minutes or until soft and just beginning to brown at edges.\n\n4 Add all-purpose flour, remaining \u00bc teaspoon kosher salt, white pepper, sweet paprika, and tomato paste, and cook for 1 minute or until flour becomes pasty and tomato paste browns on the bottom of the cooker.\n\n5 Add chicken broth. Using a wooden spoon, scrape up all bits from the bottom of the cooker. Bring to a boil, add chicken, and return to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Add sour cream and \u00bd of flat-leaf parsley, and stir to blend.\n\n8 Serve hot over rice, garnished with remaining flat-leaf parsley.\n\nNutrition per serving\n\nCalories: 210\n\nCarbohydrates: 12g\n\nSugars: 5g\n\nDietary fiber: 2g\n\nProtein: 12g\n\nFat: 11g\n\nCholesterol: 50mg\n\nSodium: 590mg\nCaribbean Chicken Curry\n\nThis curry is simple, slightly spicy, and not heavy, thanks to the coconut milk. Make it spicier by adding more curry powder and a diced habanero.\n\nIngredients\n\n1 TB. peanut or canola oil\n\n8 skinless chicken thigh and leg quarters\n\n3 large yellow onions, halved and thinly sliced (3\u00bd cups)\n\n1 (2-in; 5cm) piece fresh ginger, peeled and minced (1\u00bd TB.)\n\n\u00bc cup thinly sliced fresh garlic\n\n6 to 8 sprigs thyme\n\n3 TB. curry powder\n\n8 medium Yukon Gold potatoes, halved\n\n3 cups homemade or sodium-free chicken stock\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat peanut oil. When hot, add chicken and saut\u00e9 for 5 minutes or until brown on both sides. Transfer chicken to a rimmed baking sheet.\n\n2 Add yellow onions, ginger, garlic, and \u00bd of thyme to the cooker, and cook for 3 minutes or until onions soften.\n\n3 Add curry powder, and stir until fragrant.\n\n4 Return chicken to the cooker, and top with Yukon Gold potatoes. Add chicken stock, kosher salt, and black pepper, and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Serve hot, garnished with chives and remaining thyme.\n\nNutrition per serving\n\nCalories: 200\n\nCarbohydrates: 15g\n\nSugars: 5g\n\nDietary fiber: 3g\n\nProtein: 21g\n\nFat: 6g\n\nCholesterol: 80mg\n\nSodium: 710mg\n\nOne-Pot Chicken and Sausage Perloo\n\nOne-pot recipes produce a satisfying meal in a single dish, saving cleanup time. One-pot pressure cooker meals require less cook time, too.\n\nIngredients\n\n8 (5-oz.; 140g) chicken thighs, skin removed\n\n2 tsp. kosher salt\n\n1\u00bd TB. vegetable oil\n\n1 cup diced Andouille sausage or ham\n\n1 small yellow onion, diced fine (\u2153 cup)\n\n3 medium cloves garlic, chopped (1 TB.)\n\n1\u00bd cups short-grain rice\n\n\u00be cup frozen lima beans\n\n\u00be cup frozen corn\n\n2\u00bd cups water\n\n1 bay leaf\n\n\u00bd tsp. dried thyme\n\n\u00bc tsp. freshly ground black pepper\n\n1 large green onion, sliced (\u00bc cup)\n\n1 TB. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 Season chicken thighs with 1\u00bd teaspoons kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil, swirling pan to coat. Add chicken, skin side down, and brown for 3 minutes or until dark. Turn over chicken, and brown other side for 3 minutes. Transfer chicken to a tray.\n\n3 Pour off all but 3 tablespoons greese from the cooker. Add Andouille sausage, yellow onion, and garlic. Stir, and saut\u00e9 vegetables for about 2 minutes or until they begin to soften.\n\n4 Add short-grain rice, lima beans, and corn, and stir. Stir in water, bay leaf, thyme, black pepper, and remaining \u00bd teaspoon kosher salt. Add chicken on top of rice, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Remove bay leaf. Stir rice, transfer perloo to a platter, garnish with green onion and flat-leaf parsley, and serve.\n\nNutrition per serving\n\nCalories: 370\n\nCarbohydrates: 37g\n\nSugars: 1g\n\nDietary fiber: 2g\n\nProtein: 22g\n\nFat: 14g\n\nCholesterol: 75mg\n\nSodium: 830mg\nChicken and Dumplings\n\nChicken and dumplings is one of the homiest comfort-food meals around. Your pressure cooker cooks the dumplings to perfect tenderness.\n\nIngredients\n\n1 (3-lb.; 1.5kg) fryer chicken\n\n1 medium leek, white part only, sliced in to \u00bd moons (1 cup)\n\n4 medium carrots, peeled and cut into 1-in. (2.5cm) diagonals (1 cup)\n\n2 medium stalks celery, cut into 1-in. (2.5cm) pieces (\u00bd cup)\n\n2 medium yellow onions, diced large (1\u00bd cups)\n\n1 bay leaf\n\n5 cups homemade or sodium-free chicken stock\n\n2\u00be tsp. kosher salt\n\n\u00be tsp. freshly ground black pepper\n\n1\u00bd cups all-purpose flour\n\n1 tsp. baking powder\n\n3 TB. shortening\n\n2 large eggs\n\n\u00be cup whole milk\n\n1 TB. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine chicken, leek, carrots, celery, yellow onions, bay leaf, chicken stock, 2 teaspoons kosher salt, and \u00bd teaspoon black pepper. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 5 minutes per 1 pound (.5kg) chicken. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n3 In a large bowl, combine all-purpose flour, remaining \u00be teaspoon kosher salt, baking powder, and shortening until mixture resembles crumbly cornmeal.\n\n4 Add eggs, whole milk, flat-leaf parsley, and remaining \u00bd teaspoon black pepper, and stir.\n\n5 Transfer chicken to a baking sheet. When cool, remove skin, pull meat off bones, and return meat to the cooker.\n\n6 Set heat to medium-high\/high, bring to a boil, and reduce heat to a simmer.\n\n7 Scoop out 1 dumpling, lower to broth level, and gently drop in broth. Repeat with remaining dough, not overlapping dumplings.\n\n8 Set heat to low (traditional)\/low (electric). When broth just begins to boil, reduce heat to the lowest possible setting, add the lid but don't lock on, and simmer for 15 minutes. Remove the lid, and push on dumplings. If they're still soft, cover and cook for 5 minutes or until firm.\n\n9 Remove bay leaf. Spoon dumplings into bowls with chicken, vegetables, and broth on top, and serve.\n\nNutrition per serving\n\nCalories: 480\n\nCarbohydrates: 32g\n\nSugars: 6g\n\nDietary fiber: 2g\n\nProtein: 60g\n\nFat: 11g\n\nCholesterol: 255mg\n\nSodium: 680mg\nChicken Cacciatore\n\nThis is the Italian version of chicken cacciatore, made with mushrooms plus tomatoes, red wine, and rosemary.\n\nIngredients\n\n2\u00bd lb. (1.25kg) bone-in chicken thighs, breasts, and legs (breasts halved crosswise if large)\n\n\u00bd tsp. kosher salt\n\n3 TB. olive oil\n\n8 oz. (225g) cremini or baby bella mushrooms, brushed clean and halved\n\n4 shallots, peeled and halved lengthwise (\u00be cup)\n\n3 cloves garlic, chopped (1 TB.)\n\n1 cup dry red wine\n\n1 cup tomato sauce\n\n3 TB. tomato paste\n\n1 tsp. dried rosemary, ground\n\n\u00bc tsp. freshly ground black pepper\n\n12 oz. (340g) cooked penne pasta, hot\n\n\u00bd cup grated Parmesan cheese\n\nMETHOD\n\n1 Season chicken with kosher salt, and set aside for 20 minutes or until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. When hot, add chicken, skin side down, and brown for 4 minutes. Turn over chicken and brown meat side for 4 minutes. Transfer browned chicken to a plate.\n\n3 Add cremini mushrooms and shallots, and cook for 5 minutes or until they begin to brown.\n\n4 Add garlic, and when it becomes fragrant, add red wine. Reduce wine by half.\n\n5 Return chicken to the cooker along with tomato sauce, tomato paste, rosemary, and black pepper. Bring to a boil, taste, and add more kosher salt as necessary.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Stir, taste and adjust seasoning as necessary, and serve over penne pasta with Parmesan cheese on the side.\n\nNutrition per serving\n\nCalories: 620\n\nCarbohydrates: 53g\n\nSugars: 6g\n\nDietary fiber: 2g\n\nProtein: 36g\n\nFat: 29g\n\nCholesterol: 100mg\n\nSodium: 630mg\n\nSmothered Chicken\n\nChicken thighs are ideal in braised recipes. They stay tender and can stand up to the heavy flavors of garlic, peppers, and onions in this dish.\n\nIngredients\n\n2 TB. Spanish paprika\n\n1 tsp. garlic powder\n\n\u00bd tsp. freshly ground black pepper\n\n8 (5-oz.; 140g) bone-in, skin-on chicken thighs\n\n2 tsp. kosher salt\n\n1\u00bd TB. vegetable oil\n\n2 medium yellow onions, thinly sliced (2 cups)\n\n1 large green bell pepper, ribs and seeds removed, and thinly sliced (\u00be cup)\n\n3 medium stalks celery (\u00be cup)\n\n2 cloves garlic, minced (2 tsp.)\n\n2 TB. all-purpose flour\n\n1\u00bc cups water\n\n2 medium green onions, chopped (\u00bc cup)\n\nMETHOD\n\n1 In a small bowl, combine Spanish paprika, garlic powder, and black pepper.\n\n2 Season chicken with kosher salt, and sprinkle with spice mix.\n\n3 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add chicken, skin side down, and brown deeply for 4 minutes. Turn over chicken, and brown on other side for 4 minutes. Transfer browned chicken to a tray.\n\n4 Add yellow onions, green bell pepper, and celery to the pressure cooker, and saut\u00e9 for about 4 minutes or until vegetables wilt.\n\n5 Add garlic and all-purpose flour, stir, and cook for 1 minute.\n\n6 Add water, and bring to a boil.\n\n7 Return chicken to the cooker, and bring to a boil again.\n\n8 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n9 Taste, and add more kosher salt as necessary. Transfer chicken to a platter, smother with sauce, top with green onions, and serve.\n\nNutrition per serving\n\nCalories: 290\n\nCarbohydrates: 11g\n\nSugars: 4g\n\nDietary fiber: 2g\n\nProtein: 33g\n\nFat: 12g\n\nCholesterol: 135mg\n\nSodium: 170mg\nChicken with 40 Cloves of Garlic\n\n40 cloves might seem like a lot of garlic, but it really mellows as it cooks under pressure with the chicken, thyme, and seasonings.\n\nIngredients\n\n1 (4-lb.; 2kg) whole chicken, cut into 8 pieces (breasts halved crosswise if large)\n\n2 tsp. kosher salt\n\n2 TB. unsalted butter\n\n40 cloves garlic (about 4 heads)\n\n15 sprigs fresh thyme\n\n\u00bc tsp. freshly ground black pepper\n\n1 cup water\n\nMETHOD\n\n1 Season chicken with 1\u00bd teaspoons kosher salt, and set aside until salt is absorbed.\n\n2 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. When it foams, add \u00bd of chicken, skin side down, and brown generously for 4 minutes. Turn over chicken, and brown meat side for 4 minutes. Transfer chicken to a tray, repeat with remaining chicken, and transfer rest of browned chicken to the tray. (Reduce heat if necessary to keep butter from burning.)\n\n3 Add garlic to the cooker, and shake so cloves sink into chicken fat. Add chicken, \u00bd of thyme sprigs, remaining \u00bd teaspoon kosher salt, and black pepper. Pour in water, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Transfer chicken to a platter, and top with garlic cloves and jus. Remove thyme leaves from remaining stems, sprinkle leaves over chicken, and serve.\n\nNutrition per serving\n\nCalories: 430\n\nCarbohydrates: 10g\n\nSugars: 0g\n\nDietary fiber: 1g\n\nProtein: 52g\n\nFat: 19g\n\nCholesterol: 170mg\n\nSodium: 1,120mg\nThai-Style Green Curry Chicken\n\nRich with coconut milk and exotic from the green curry paste, this green curry chicken is far better than any take-out version.\n\nIngredients\n\n1 TB. vegetable oil\n\n2\u00bd lb. (1.25kg) skinless chicken thighs, legs, and breasts (breasts halved crosswise if large)\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n1 TB. minced garlic\n\n3 TB. green curry paste\n\n\u00bd cup homemade or sodium-free chicken stock\n\n3 TB. fish sauce\n\n2 medium carrots, peeled and cut into \u00be-in. (2cm) pieces (\u00be cup)\n\n1 small red onion, cut into wedges (\u00be cup)\n\n4 medium red potatoes, halved (2 cups)\n\n1 (14.5-oz.; 410g) can coconut milk\n\n1 medium zucchini, cut into \u00be-in. (1.25cm) rounds\n\n1 medium yellow crookneck squash, cut into \u00be-in. (1.25cm) rounds\n\n2 large limes, quartered\n\nCooked rice noodles or rice, hot\n\n2 serrano chiles, thinly sliced\n\n\u00bc cup fresh mint leaves, torn\n\n\u00bc cup fresh Thai basil leaves, torn\n\n\u00bc cup fresh cilantro leaves\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add chicken without crowding, and brown on both sides for 4 minutes. (You might need to do this in batches.)\n\n2 Add ginger, garlic, and green curry paste, and stir until spices become aromatic.\n\n3 Add chicken stock, fish sauce, carrots, red onion, and red potatoes. Gently stir, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Transfer chicken to a platter.\n\n6 Add coconut milk, zucchini, and yellow crookneck squash to the cooker. Set heat to medium (traditional)\/high (electric), and simmer for about 5 minutes or until squashes are tender.\n\n7 Pull chicken meat off bones, return chicken to the cooker along with juice of \u00bd of lime, and stir. Taste, and adjust seasoning as necessary by adding fish sauce and lime juice a little at a time.\n\n8 Serve over hot rice noodles topped with serrano chiles, mint, Thai basil, and cilantro.\n\nFor a hotter curry, add another chopped serrano chile along with the chicken.\n\nNutrition per serving\n\nCalories: 510\n\nCarbohydrates: 30g\n\nSugars: 3g\n\nDietary fiber: 4g\n\nProtein: 46g\n\nFat: 23g\n\nCholesterol: 135mg\n\nSodium: 920mg\n\nChinese Red Cooked Chicken\n\nRed cooked dishes such as this are simple and satisfying, and they exemplify the flavors of Chinese food.\n\nIngredients\n\n2 whole star anise\n\n1 (3-in.; 7.5cm) cinnamon stick\n\n4 cups homemade or sodium-free vegetable stock\n\n\u00bc cup soy sauce\n\n1 medium yellow onion, cut into 6 wedges (1 cup)\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n1 TB. minced garlic\n\n1 TB. brown sugar\n\n2 TB. Chinese rice wine, dry sherry, or sake\n\n1 (3\u00bd-lb.; 1.75kg) whole chicken, cut into 10 pieces (breasts halved crosswise if large)\n\n12 oz. (340g) extra-firm tofu, cut into \u00bd\u00d72\u00d72-in. (1.25\u00d75\u00d75cm) slabs\n\n2 medium carrots, peeled and sliced in ribbons (1\u00bd cups)\n\n1 large green onions, cut into thin rings (\u00bc cup)\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine star anise, cinnamon stick, vegetable stock, soy sauce, yellow onion, ginger, garlic, brown sugar, and Chinese rice wine. Set heat to medium-high (traditional)\/high (electric), and bring to a boil. Reduce heat to a simmer, and cook for 15 minutes.\n\n2 Add chicken, increase heat to medium-high\/high, and return to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 13 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Add tofu, gently stir into broth, and warm for 5 minutes.\n\n5 Remove cinnamon stick and star anise. Place a piece of chicken and a couple slabs tofu on a plate, ladle hot broth over chicken, and top with carrot ribbons and green onions. Serve with a scoop of hot rice if desired.\n\nNutrition per serving\n\nCalories: 610\n\nCarbohydrates: 17g\n\nSugars: 8g\n\nDietary fiber: 4g\n\nProtein: 94g\n\nFat: 15g\n\nCholesterol: 255mg\n\nSodium: 1,020mg\nHungarian Pork Goulash\n\nOriginally a herdsman's meal, this popular comfort food dish can be made with beef, pork, or lamb.\n\nIngredients\n\n2\u00bd lb. (1.25kg) boneless country-style pork ribs, cubed\n\n2\u00bd tsp. kosher salt\n\n1 large red bell pepper, ribs and seeds removed, and thinly sliced (1 cup)\n\n1 medium yellow onion, thinly sliced (1 cup)\n\n3 medium russet potatoes, peeled and cut into 1-in. (2.5cm) chunks (1\u00bd cups)\n\n4 cloves garlic, chopped\n\n2 heaping tsp. paprika\n\n1\u00bd tsp. caraway seeds, crushed\n\n1 tsp. dried marjoram\n\n\u00bc tsp. freshly ground black pepper\n\n2 TB. red wine vinegar\n\n1\u00bd cups tomato pur\u00e9e\n\n1 cup water\n\n1 TB. fresh flat-leaf parsley, minced\n\n1 cup sour cream\n\nMETHOD\n\n1 Season pork ribs with 1\u00bd teaspoons kosher salt, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set heat to medium-high (traditional)\/high (electric), brown \u00bd of pork cubes on all sides for 4 minutes. Transfer browned pork to a plate, repeat with remaining pork, and transfer rest of browned pork to the plate.\n\n3 Add red bell pepper, yellow onion, russet potatoes, and garlic to the cooker. Stir, and cook for 5 minutes or until vegetables begin to soften.\n\n4 Add browned pork, paprika, caraway seeds, marjoram, black pepper, remaining 1 teaspoon salt, and red wine vinegar, and stir.\n\n5 Stir in tomato pur\u00e9e and water, and bring to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 18 minutes. Remove from (traditional)\/turn off (electric) heat, perform your preferred release, and remove the lid.\n\n7 Stir in \u00bd of flat-leaf parsley.\n\n8 Ladle goulash into bowls, top with a dollop of sour cream and a sprinkle of remaining parsley, and serve.\n\nNutrition per serving\n\nCalories: 310\n\nCarbohydrates: 14g\n\nSugars: 5g\n\nDietary fiber: 2g\n\nProtein: 31g\n\nFat: 14g\n\nCholesterol: 110mg\n\nSodium: 920mg\nBelgian Ale\u2013Braised Pork Loin with Mustard\n\nThis pork loin cooks up juicy and tender, the Belgian beer brings another layer of flavor, and the mustard sauce is the perfect complement.\n\nIngredients\n\n1 TB. vegetable oil\n\n1 (2\u00bd-lb.; 1.75kg) pork loin\n\n\u00bd small yellow onion, minced (\u2153 cup)\n\n1 TB. minced garlic\n\n1 pinch kosher salt\n\n\u00bd tsp. freshly ground white pepper\n\n1 cup Belgian farmhouse ale or dry white wine\n\n1 TB. Dusseldorf or Dijon mustard\n\n\u00bc cup heavy cream (optional)\n\n1 TB. fresh parsley, minced\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil, gently swirling the pan to coat. When hot, add pork loin and brown on all sides for 5 minutes. (Reduce heat if necessary to keep oil from burning.)\n\n2 Add yellow onion, garlic, kosher salt, and white pepper, and cook for 1 minute or until vegetables begin to soften.\n\n3 Add Belgian farmhouse ale and Dusseldorf mustard, and cook for 2 minutes or until foam subsides and liquid comes to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 30 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Transfer pork to a cutting board, and cover with aluminum foil.\n\n6 Return the cooker to medium-high\/high heat, bring sauce to a boil, and cook for 5 minutes or until thickened and reduced by \u2153. Taste, and adjust seasoning as necessary.\n\n7 Add heavy cream (if using), and cook for 2 minutes or until sauce reaches your desired consistency.\n\n8 Slice pork into thin slices, and arrange on a platter. Drizzle sauce over top, garnish with parsley and chives, and serve.\n\nNutrition per serving\n\nCalories: 300\n\nCarbohydrates: 3g\n\nSugars: 0g\n\nDietary fiber: 0g\n\nProtein: 40g\n\nFat: 10g\n\nCholesterol: 135mg\n\nSodium: 480mg\n\nPork Grillades\n\nThis Cajun comfort food dish features bell peppers, celery, and onions, the classic trinity flavor popular in Louisiana.\n\nIngredients\n\n4 (12-oz.; 340g) bone-in pork butt steaks, cut about \u00be in. (2cm) thick\n\n1 tsp. kosher salt\n\n1 TB. Cajun seasoning or to taste\n\n1 TB. vegetable oil\n\n1 large yellow onion, diced small (1 cup)\n\n\u00bd cup red bell pepper, diced small\n\n\u00bd cup green bell pepper, diced small\n\n2 medium stalks celery, diced small (\u00bd cup)\n\n1\u00bc cups water or homemade or sodium-free chicken broth\n\n\u00bc cup tomato sauce\n\n1 TB. Worcestershire sauce\n\nHot sauce\n\nMETHOD\n\n1 Season pork butt steaks on both sides with kosher salt followed by Cajun seasoning, and set aside until seasonings are absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add pork, in batches or one at a time, and sear for 4 minutes or until seasoning begins to blacken. Transfer seared pork to a plate, and repeat with remaining pork, transferring rest of browned pork to the plate. (Reduce heat if necessary to keep oil from burning.)\n\n3 Add yellow onion, red bell pepper, green bell pepper, and celery to the cooker, and saut\u00e9 for 3 minutes or until vegetables begin to soften.\n\n4 Return pork to the cooker. Add water, tomato sauce, and Worcestershire sauce, and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n6 Serve with hot sauce on the side.\n\nNutrition per serving\n\nCalories: 290\n\nCarbohydrates: 4g\n\nSugars: 3g\n\nDietary fiber: 1g\n\nProtein: 38g\n\nFat: 12g\n\nCholesterol: 125mg\n\nSodium: 1,180mg\nSouthern-Style Pulled Pork\n\nThis pulled pork is tender, succulent, and juicy and cooks quickly, thanks to your pressure cooker.\n\nIngredients\n\n1 TB. brown sugar\n\n1\u00bd TB. kosher salt\n\n1 (3-lb.; 1.5kg) bone-in pork butt roast\n\n2 tsp. canola oil\n\n1 cup homemade or sodium-free chicken stock\n\n1 tsp. hickory-flavored liquid smoke\n\n1 cup barbecue sauce\n\nMETHOD\n\n1 In a small bowl, combine brown sugar and kosher salt.\n\n2 Rub pork butt roast all over with brown sugar and salt mixture, and refrigerate overnight to cure.\n\n3 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat canola oil, gently swirling the cooker to coat. Add pork, and brown on all sides for 5 minutes.\n\n4 Add chicken stock and liquid smoke, and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 45 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n6 Transfer pork to a plate, and shred using 2 forks. Add to a bowl, season with _jus_ from the cooker, and add barbecue sauce.\n\n7 Serve as is, on a bun, or stuffed into taco shells.\n\nNutrition per serving\n\nCalories: 310\n\nCarbohydrates: 17g\n\nSugars: 14g\n\nDietary fiber: 0g\n\nProtein: 31g\n\nFat: 12g\n\nCholesterol: 100mg\n\nSodium: 1,920mg\nPork Vindaloo\n\nThis wonderful Indian stew blends sweet and sour flavors. Increase the spice if you like by adding a dried hot chile or two while the stew cooks.\n\nIngredients\n\n2 lb. (1kg) country-style pork ribs, cut into 1-in. (2.5cm) cubes\n\n\u00bd tsp ground coriander\n\n\u00bd tsp. ground cumin\n\n\u00bd tsp. ground mustard seeds\n\n\u00bc tsp. ground cloves\n\n\u00bd tsp. turmeric\n\n1 tsp. kosher salt\n\n\u2153 cup white vinegar\n\n\u00bc cup vegetable oil\n\n2 medium yellow onions, chopped (1\u00bd cups)\n\n3 TB. minced garlic\n\n1 cup tomato sauce\n\n1 TB. brown sugar\n\n1 small red onion, thinly sliced (\u00bd cup)\n\n2 large limes, 1 quartered and 1 halved\n\n2 medium green onions, sliced into thin rings (\u2153 cup)\n\n\u2153 cup fresh cilantro, coarsely chopped\n\n1 Fresno or jalape\u00f1o chile, sliced\n\nMETHOD\n\n1 In a large bowl, combine pork, coriander, cumin, mustard seeds, cloves, turmeric, and kosher salt. Add white vinegar, and stir to combine. Set aside for 1 hour, or refrigerate for up to 6 hours.\n\n2 Drain pork, and reserve marinade.\n\n3 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add pork, and brown generously on all sides for 5 minutes. Transfer browned pork to the large bowl.\n\n4 Add yellow onions to the cooker, and cook for 3 or 4 minutes to soften.\n\n5 Add garlic, and cook for 30 seconds or until fragrant.\n\n6 Add tomato sauce, pork, reserved marinade, and brown sugar. Stir to combine, and bring to a boil.\n\n7 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 25 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n8 In a small bowl, place red onion. Squeeze juice from halved lime over top, and toss to combine.\n\n9 Stir stew, taste, and add more kosher salt as necessary. Serve topped with 1 lime wedge, pickled red onions, green onions, cilantro, and Fresno chile.\n\nNutrition per serving\n\nCalories: 330\n\nCarbohydrates: 10g\n\nSugars: 6g\n\nDietary fiber: 2g\n\nProtein: 31g\n\nFat: 19g\n\nCholesterol: 100mg\n\nSodium: 730mg\n\nPork Posole\n\nThis traditional Mexican stew features yellow hominy, a corn-based ingredient. If you can find Mexican oregano, use it for a more authentic flavor.\n\nIngredients\n\n1 lb. (450g) tomatillos, outer paper skins removed\n\n2 TB. lard or olive oil\n\n1 (2\u00bd-lb.; 1.25kg) pork shoulder, cut into 1-in. (2.5cm) cubes\n\n1 medium yellow onion, diced small (1 cup)\n\n\u00bc cup coarsely chopped garlic\n\n2 tsp. dried oregano\n\n1 TB. dark chili powder\n\n1 TB. tomato paste\n\n1 tsp. kosher salt\n\n\u00bc tsp. freshly ground black pepper\n\nWater\n\n1 (14.5-oz.; 410g) can yellow hominy\n\n\u00bc cup fresh cilantro, chopped\n\nMETHOD\n\n1 Preheat the broiler.\n\n2 Place tomatillos on a baking sheet, and broil for 4 minutes or until blackened. Turn over, and blacken other side. (Tomatillos will wilt and collapse on themselves.)\n\n3 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat lard. When hot, add \u00bd of pork. Brown on all sides for 4 minutes, transfer browned pork to a plate, repeat with remaining pork, and transfer rest of browned pork to the plate. (Reduce heat if necessary to avoid burning.)\n\n4 Add yellow onion to the cooker, stir, and cook for about 5 minutes or until it wilts.\n\n5 Add tomatillos and their juice, garlic, oregano, dark chili powder, and tomato paste, and stir.\n\n6 Return pork and its juice to the cooker, and add kosher salt and black pepper. Add enough water to come up to top edge of meat, and bring to a boil.\n\n7 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 15 minutes. Turn off heat, let pork sit for 5 minutes, perform a quick release, and remove the lid.\n\n8 Add yellow hominy and cilantro, set heat to medium (traditional)\/high (electric), and cook for 4 minutes or until hominy is warm. Taste, and adjust kosher salt and pepper as necessary.\n\n9 Serve in bowls alongside fresh cilantro and your favorite toppings (like shredded cabbage, lime wedges, thinly sliced red onion, sour cream, and shredded Monterey jack cheese).\n\nNutrition per serving\n\nCalories: 610\n\nCarbohydrates: 63g\n\nSugars: 5g\n\nDietary fiber: 6g\n\nProtein: 48g\n\nFat: 20g\n\nCholesterol: 135mg\n\nSodium: 530mg\nOne-Pot Sausage, Potatoes, and Greens\n\nThis one-dish meal is hearty and filling. The recipe comes together surprisingly fast in your pressure cooker and doesn't require a lot of cleanup.\n\nIngredients\n\n1\u00bd cups homemade or sodium-free vegetable or chicken broth\n\n\u00bd tsp. crushed red pepper flakes (optional)\n\n\u00be tsp. kosher salt\n\n8 cups mixed collard, turnip, kale, and mustard greens, stems removed and chopped\n\n8 medium red potatoes, skin on and scrubbed\n\n2 (8-oz.; 225g) smoked sausages or kielbasa\n\nFreshly ground black pepper\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine vegetable broth, crushed red pepper flakes (if using), and \u00bd teaspoon kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Add mixed greens, and using a pair of tongs, turn greens to coat with broth.\n\n3 Place red potatoes on top of greens, and add sausage on top of potatoes. Season with remaining \u00bc teaspoon kosher salt, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 10 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Spoon greens and some broth onto a platter, and top with potatoes.\n\n6 Cut sausages into serving-size pieces, and arrange among potatoes. Season with a few grinds of black pepper, and serve.\n\nNutrition per serving\n\nCalories: 570\n\nCarbohydrates: 53g\n\nSugars: 6g\n\nDietary fiber: 7g\n\nProtein: 20g\n\nFat: 31g\n\nCholesterol: 75mg\n\nSodium: 1,530mg\nAsian Steamed Fish\n\nThis delicious, deeply flavored fish dish is light and comes together quickly and easily.\n\nIngredients\n\n1 small shallot, minced (1 TB.)\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n1 TB. minced garlic\n\n2 tsp. sugar\n\n2 TB. soy sauce\n\n2 TB. sake, rice wine, or dry sherry\n\n\u00be cup water\n\n2 (1-lb.; 450g) fillet tilapia, cut into 1-in. (2.5cm) cubes\n\n1 cup napa cabbage, shredded thinly\n\n1 medium carrot, peeled and cut into thin ribbons (1 cup)\n\n2 TB. chopped fresh chives\n\n\u00bc cup fresh snow peas, chopped\n\n\u00bc cup frozen edamame, thawed\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine shallot, ginger, garlic, sugar, soy sauce, sake, and water. Set heat to medium-high (traditional)\/high (electric), bring to a boil, and immediately add tilapia in a single layer to ensure even cooking.\n\n2 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Turn off heat, and let fish sit for 30 seconds. Perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Using a slotted spoon, gently transfer fish to a bowl.\n\n4 To serve, pour broth over fish, and top with napa cabbage, carrot ribbons, chives, snow peas, and edamame.\n\nAny fish works for this dish, but mahimahi, cod, sea bass, or even shrimp would be delicious. Remember, larger pieces might require a slightly longer cook time.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 7g\n\nSugars: 4g\n\nDietary fiber: 1g\n\nProtein: 48g\n\nFat: 4.5g\n\nCholesterol: 115mg\n\nSodium: 790mg\n\nSalmon en Papillote\n\nThis recipe is very simple to prepare, thanks to your pressure cooker. Use fresh, good-quality ingredients to produce a delicious final dish.\n\nIngredients\n\n2 (7-oz.; 200g) fillets salmon\n\n\u00bd tsp. kosher salt\n\n\u215b tsp. freshly ground black pepper\n\n1 small stalk celery, diced small (\u00bc cup)\n\n5 thin slices yellow onion (\u00bc cup)\n\n1 medium carrot, peeled and cut into paper-thin rounds (\u00bd cup)\n\n2 slices lemon\n\n2 tsp. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 Add a steamer basket to a 4-quart (4l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 Place a 10\u00d710-inch (25\u00d725cm) piece of aluminum foil on your counter, and drizzle 1 teaspoon olive oil over top. Place salmon on the foil with a bit of space between fillets.\n\n3 Season salmon on both sides with kosher salt and black pepper. Scatter celery, yellow onion, and carrot over salmon, and top each piece with 1 lemon slice.\n\n4 Place another 10\u00d710-inch (25\u00d725cm) piece of foil on top of the first, and crimp tightly at the edges. Add foil packet to the steamer basket, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Transfer foil packet to a plate, carry to the table, and open the packet tableside for a bit of flourish. Serve topped with vegetables and garnished with flat-leaf parsley.\n\nNutrition per serving\n\nCalories: 510\n\nCarbohydrates: 6g\n\nSugars: 3g\n\nDietary fiber: 2g\n\nProtein: 40g\n\nFat: 35g\n\nCholesterol: 100mg\n\nSodium: 610mg\nFish Curry\n\nThe spicy Indian sauce in this quick and easy fish dish is incredible and complements the tilapia quite nicely.\n\nIngredients\n\n2 TB. unsalted butter\n\n1 large yellow onion, thinly sliced (1\u00bc cups)\n\n1 TB. minced garlic\n\n1 (1-in; 2.5cm) piece fresh ginger, peeled and minced (1 TB.)\n\n3 TB. tomato paste\n\n2 tsp. curry powder\n\n\u00bd cup tomato sauce\n\n\u00bd cup water\n\n1 (1\u00bd-lb.; 680g) fillet tilapia, cut into 1-in. (2.5cm) cubes\n\n\u2153 cup plain yogurt\n\n4 cherry tomatoes, quartered\n\n\u00bc small head cauliflower florets, shaved (\u2154 cup)\n\n\u00bc cup fresh cilantro, chopped\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. Add yellow onion, and saut\u00e9 for about 5 minutes or until golden brown.\n\n2 Add garlic, ginger, tomato paste, and curry powder. Stir, and cook for 30 seconds or until fragrant.\n\n3 Add tomato sauce and water, bring to a boil, and reduce heat to low. Simmer sauce for 5 minutes.\n\n4 Bring sauce back to a boil, and layer in tilapia in a single layer.\n\n5 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Immediately turn off heat, and let fish sit for 30 seconds. Perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n6 Using a slotted spoon, transfer fish to a platter. Cover with foil to keep warm.\n\n7 Add yogurt to the cooker, stir, and warm at medium (traditional)\/low (electric) heat for 4 minutes.\n\n8 To serve, pour sauce over fish, and garnish with cherry tomatoes, cauliflower, and cilantro.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 11g\n\nSugars: 5g\n\nDietary fiber: 3g\n\nProtein: 37g\n\nFat: 9g\n\nCholesterol: 100mg\n\nSodium: 220mg\nThai Steamed Mussels\n\nThis recipe is a fantastic blend of spice and seafood. Serve with rice to soak up all the delicious sauce.\n\nIngredients\n\n4 lb. (2kg) mussels, rinsed\n\n3 cloves garlic, minced (1 TB.)\n\n1 tsp. cumin seed, crushed\n\n3 TB. red curry paste\n\n1\u00bd TB. Madras curry powder\n\n1 (10-oz.; 400ml) can coconut milk\n\n1 (8-oz.; 262.5ml) bottle clam juice\n\n1 TB. sugar\n\n\u00bd tsp. turmeric\n\n3 TB. fish sauce\n\n1 TB. freshly squeezed lime juice\n\n1 TB. tomato paste\n\n\u00bd medium yellow onion, diced fine (\u00bc cup)\n\n\u00bd cup fresh cilantro\n\n2 serrano chiles, thinly sliced\n\n2 medium green onions, white and green parts, sliced into thin rings (\u00bc cup)\n\n1 medium lime, quartered\n\nMETHOD\n\n1 Squeeze mussels, and discard any that don't close. Set aside.\n\n2 In a 6-quart (5.5l) pressure cooker, combine garlic, cumin, red curry paste, Madras curry powder, coconut milk, clam juice, sugar, turmeric, fish sauce, lime juice, tomato paste, and yellow onion. Set heat to medium-high (traditional)\/high (electric), and bring to a boil, whisking to break up any curry paste clumps.\n\n3 Add mussels, and return to a boil.\n\n4 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 1 minute. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir mussels to coat in broth.\n\n6 Using a slotted spoon, transfer mussels to a deep platter. Pour broth over top, and garnish with cilantro, serrano chiles, and green onions. Serve with lime wedges on the side.\n\nNutrition per serving\n\nCalories: 560\n\nCarbohydrates: 26g\n\nSugars: 5g\n\nDietary fiber: 1g\n\nProtein: 57g\n\nFat: 25g\n\nCholesterol: 130mg\n\nSodium: 2,500mg\n\nBeer-Steamed Shrimp\n\nThis recipe produces seasoned, beer-steamed shrimp with East Coast flavors plus a little heat for good measure.\n\nIngredients\n\n1\u00bd cups American lager\n\n1 TB. pickling spice\n\n3 TB. minced garlic\n\n1 tsp. hot sauce\n\n2 lb. (1kg.) shell-on shrimp (size 26 to 30), thawed\n\n1 TB. fresh flat-leaf parsley, minced\n\n1 TB. Old Bay seasoning\n\nMETHOD\n\n1 In a 6-quart (5.5l) pressure cooker, combine American lager, pickling spice, garlic, and hot sauce. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Add shrimp, and return to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 2 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n4 Transfer shrimp to a bowl, and ladle some broth over top. Sprinkle flat-leaf parsley and Old Bay seasoning over top, and serve immediately with lots of cocktail sauce or remoulade.\n\nIf you can't find Old Bay seasoning, you can substitute any Cajun-style seasoning you like.\n\nNutrition per serving\n\nCalories: 290\n\nCarbohydrates: 7g\n\nSugars: 0g\n\nDietary fiber: 0g\n\nProtein: 47g\n\nFat: 4g\n\nCholesterol: 350mg\n\nSodium: 410mg\nCajun-Style Shrimp Jambalaya\n\nThis homey, one-pot dish full of shrimp, chicken, vegetables, and rice makes a quick, comforting meal.\n\nIngredients\n\n6 (5-oz.; 140g) chicken thighs\n\n1 tsp. kosher salt\n\n1 TB. vegetable oil\n\n1 medium yellow onion, diced small (1 cup)\n\n2 large stalks celery, diced small (\u00bd cup)\n\n1 small green bell pepper, ribs and seeds removed, and diced small (\u00bd cup)\n\n1 TB. minced garlic\n\n1\u00bd cups long-grain rice\n\n1 bay leaf\n\n\u00bd tsp. dried thyme\n\n2 cups homemade or sodium-free chicken or vegetable broth\n\n1 cup crushed tomatoes, with juice\n\n1 lb. (450g) raw shrimp (size 31 to 35), thawed if frozen, peeled, deveined, and drained\n\n1 TB. fresh flat-leaf parsley, minced\n\nMETHOD\n\n1 Season chicken thighs with \u00bd teaspoon kosher salt on all sides, and set aside until salt is absorbed.\n\n2 In a 6-quart (5.5l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. When hot, add chicken, skin side down, and brown deeply on both sides for 4 minutes. Transfer to a plate.\n\n3 Add yellow onion, celery, and green bell pepper to the cooker. Stir, and cook for about 3 minutes or until vegetables soften.\n\n4 Add garlic, and cook for 30 seconds or until fragrant.\n\n5 Add long-grain rice, bay leaf, and thyme, and stir to coat. Add chicken broth, crushed tomatoes with juice, and remaining \u00bd teaspoon kosher salt. Swirl the cooker to distribute rice, and nestle chicken on top. Bring to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n7 Layer shrimp on top of rice, put lid back on, and cook for 6 minutes.\n\n8 Remove bay leaf. Stir in flat-leaf parsley, and serve hot.\n\nNutrition per serving\n\nCalories: 380\n\nCarbohydrates: 42g\n\nSugars: 3g\n\nDietary fiber: 3g\n\nProtein: 37g\n\nFat: 6g\n\nCholesterol: 185mg\n\nSodium: 490mg\n\nBasmati Rice Pilaf\n\nThe natural, nutty flavor of basmati rice becomes sweet when paired with yellow onion in this recipe, and the peas add color and texture.\n\nIngredients\n\n1 TB. unsalted butter\n\n1 medium yellow onion, diced small (\u00bd cup)\n\n1 cup basmati rice\n\n2 cups water\n\n\u00be cup frozen peas\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium (traditional)\/high (electric) heat, melt unsalted butter.\n\n2 Add yellow onion, and cook for 8 minutes or until caramelized and brown.\n\n3 Add basmati rice and water, stir gently, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a quick release, and remove the lid.\n\n5 Add peas, and stir with a fork to fluff rice and mix in peas. Place the lid back on the cooker, and let pilaf sit for at least 8 minutes to cook peas. Serve hot.\n\nFor an authentic Spanish side dish, add some saffron. Or stir in a pinch of cardamom and some almonds and serve alongside an Indian curry.\n\nNutrition per serving\n\nCalories: 210\n\nCarbohydrates: 41g\n\nSugars: 4g\n\nDietary fiber: 3g\n\nProtein: 5g\n\nFat: 3.5g\n\nCholesterol: 10mg\n\nSodium:480mg\n\nDirty Oats with Lentils\n\nReminiscent of Cajun dirty rice, this hearty lentils and oats dish is wonderful served alongside chicken or any braised or roasted meat entr\u00e9e.\n\nIngredients\n\n1\u00bd TB. vegetable oil\n\n1 large yellow onion, diced small (1\u00bc cups)\n\n1 TB. minced garlic\n\n\u00bd tsp. kosher salt\n\n\u00be cup whole or steel-cut oats\n\n\u00be cup du Puy lentils\n\n1 TB. dried sage\n\n\u00be tsp. freshly ground black pepper\n\n\u00bc tsp. allspice\n\n\u215b tsp. ground nutmeg\n\n3 cups homemade or sodium-free vegetable broth\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat vegetable oil. Add yellow onion, and saut\u00e9 for about 3 minutes or until soft.\n\n2 Add garlic, and cook for 30 seconds or until fragrant.\n\n3 Add kosher salt, whole oats, du Puy lentils, sage, black pepper, allspice, and nutmeg, and stir to coat. Add vegetable broth, and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 12 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir, taste and add seasoning as necessary, and serve.\n\nNutrition per serving\n\nCalories: 250\n\nCarbohydrates: 38g\n\nSugars: 4g\n\nDietary fiber: 8g\n\nProtein: 10g\n\nFat: 7g\n\nCholesterol: 0mg\n\nSodium: 350mg\nWild Rice\n\nThe nuttiness of wild rice is enhanced by the flavors of shallots and parsley in this quick and easy dish.\n\nIngredients\n\n4 cups water\n\n1 cup wild rice\n\n2 tsp. unsalted butter\n\n1 medium shallot, minced (2 TB.)\n\n1 TB. fresh flat-leaf parsley, minced\n\n\u00bd tsp. kosher salt\n\n\u215b tsp. freshly ground black pepper\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine water and wild rice. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 20 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Stir in unsalted butter, shallot, flat-leaf parsley, kosher salt, and black pepper. Taste, add more kosher salt as necessary, and serve.\n\nFor additional texture and flavor, add some toasted pecans before serving.\n\nNutrition per serving\n\nCalories: 160\n\nCarbohydrates: 31g\n\nSugars: 1g\n\nDietary fiber: 3g\n\nProtein: 6g\n\nFat: 2.5g\n\nCholesterol: 5mg\n\nSodium: 240mg\nAsparagus and Sun-Dried Tomato Risotto\n\nIn this risotto, crisp asparagus and sun-dried tomatoes pair perfectly with creamy rice.\n\nIngredients\n\n2 TB. olive oil\n\n2 thin slices prosciutto\n\n1 TB. minced garlic\n\n1 cup arborio rice\n\n\u00bc cup dry white wine\n\n3 cups homemade or sodium-free vegetable stock\n\n12 spears asparagus, tips left whole and stems cut into thin rounds (\u00be cup)\n\n\u2153 cup sun-dried tomatoes in oil, drained and chopped\n\n2 TB. unsalted butter\n\n\u00bd cup grated Parmesan cheese\n\n1 TB. fresh chives, minced\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, heat olive oil. Lay prosciutto in oil as if you were frying bacon, and cook on both sides for 3 minutes or until brown and crispy. Transfer prosciutto to a paper towel\u2013lined plate.\n\n2 Add garlic to the pressure cooker, and saut\u00e9 for about 15 seconds or until aromatic. Add arborio rice, and stir to coat.\n\n3 Add dry white wine, and burn off alcohol for about 30 seconds. Add vegetable stock, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 1 (traditional)\/low (electric), and cook for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Meanwhile, chop prosciutto into crumbles.\n\n6 Add asparagus, sun-dried tomatoes, unsalted butter, and Parmesan cheese to the cooker, and stir. Taste, and add more kosher salt as necessary.\n\n7 Serve topped with prosciutto crumbles and chives.\n\nNutrition per serving\n\nCalories: 400\n\nCarbohydrates: 45g\n\nSugars: 5g\n\nDietary fiber: 4g\n\nProtein: 13g\n\nFat: 18g\n\nCholesterol: 30mg\n\nSodium: 610mg\n\nRisotto Milanese\n\nIn this recipe, your pressure cooker produces a creamy, brothy, flavorful risotto quickly and without fuss.\n\nIngredients\n\n2 tsp. unsalted butter\n\n1 small yellow onion, diced fine (\u00bd cup)\n\n\u00bc cup dry white wine\n\n1 cup arborio rice\n\n\u00bc tsp. kosher salt\n\n2\u00be cups homemade or sodium-free chicken or vegetable stock, hot\n\n\u00bd tsp. saffron threads\n\n2 TB. unsalted butter, chilled\n\n\u00bd cup grated Parmigiano-Reggiano cheese\n\n1 TB. fresh chives, chopped\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, melt unsalted butter. When it begins to bubble, add yellow onion, and saut\u00e9 for about 3 minutes or until soft.\n\n2 Add white wine, and cook until reduced by half.\n\n3 Add arborio rice, and stir to coat. Add kosher salt, chicken stock, and saffron threads, and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 7 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n5 Stir in chilled unsalted butter followed by Parmigiano-Reggiano cheese. If risotto is stiff, add more hot broth 1 tablespoon at a time. Garnish with chives, and serve.\n\nNutrition per serving\n\nCalories: 320\n\nCarbohydrates: 39g\n\nSugars: 1g\n\nDietary fiber: 1g\n\nProtein: 11g\n\nFat: 12g\n\nCholesterol: 40mg\n\nSodium: 660mg\nJamaican-Style Rice and Peas\n\nCoconut milk gives this gluten-free rice dish an exotic flavor, and black-eyed peas keep it earthy.\n\nIngredients\n\n1 cup medium-grain rice\n\n1 cup water\n\n1 cup coconut milk\n\n\u00bc tsp. kosher salt\n\n\u00bd cup black-eyed or pigeon peas, washed, picked over, soaked overnight, and drained\n\n\u00bc cup fresh cilantro, minced\n\n2 medium green onions, chopped (\u00bc cup)\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine medium-grain rice, water, coconut milk, kosher salt, and black-eyed peas. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 5 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Fluff rice with a fork, add cilantro and green onions, fluff again, and serve.\n\nFor a Cuban-style one-pot meal, use black beans instead of the black-eyed peas and add some ham, sausage, or both.\n\nNutrition per serving\n\nCalories: 340\n\nCarbohydrates: 49g\n\nSugars: 0g\n\nDietary fiber: 5g\n\nProtein: 8g\n\nFat: 13g\n\nCholesterol: 0mg\n\nSodium: 220mg\nSteamed Brown Bread with Raisins\n\nThis slightly sweet brown bread is great at breakfast with a bit of butter and fruit jam, especially Swedish lingonberry jam.\n\nIngredients\n\n1 cup whole-wheat flour\n\n\u00bd cup cornmeal\n\n\u00bd tsp. baking powder\n\n\u00bd tsp. baking soda\n\n\u00bd tsp. kosher salt\n\n1 cup buttermilk\n\n\u00bd cup molasses\n\n\u2153 cup raisins\n\nMETHOD\n\n1 In a large bowl, combine whole-wheat flour, cornmeal, baking powder, baking soda, and kosher salt. Add buttermilk and molasses, and stir gently to combine. Stir in raisins.\n\n2 Butter the inside of an asparagus can with one open end, and add dough to the can, tapping it on the counter as you go to avoid any large air bubbles in dough. When all dough is in the can, gently push down on dough with a spoon and jiggle the can to remove more bubbles. Place a piece of aluminum foil over the top of the can, and secure it tightly with a rubber band.\n\n3 Add a trivet to the bottom of a 8-quart (7.5l) pressure cooker (be sure the can will fit inside with the lid on), and fill with the minimum amount of water allowed for your cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric), and cook for 40 minutes. Remove from (traditional)\/turn off (electric) heat, let bread sit for 5 minutes, perform a quick release, and remove the lid.\n\n5 Transfer the can to a cooling rack, and cool completely before removing bread from the can.\n\n6 Slice, and serve.\n\nI use an empty asparagus can as my bread pan; two buttered soup cans would work, too. Just be sure one end is open and the other is closed.\n\nNutrition per serving\n\nCalories: 150\n\nCarbohydrates: 33g\n\nSugars: 14g\n\nDietary fiber: 2g\n\nProtein: 3g\n\nFat: 0g\n\nCholesterol: 0mg\n\nSodium: 220mg\n\nCrustless Sandwich Bread\n\nIf you don't like crusty bread, this recipe is for you. This crust-free bread is buttery, soft, and yeasty, perfect for grilled cheese sandwiches.\n\nIngredients\n\n7 fl. oz. (205ml) warm (90\u00b0F; 32\u00b0C) water\n\n\u00bd tsp. instant yeast\n\n\u00bd tsp. sugar\n\n\u00bd tsp. nonfat dry milk powder\n\n\u00bc tsp. kosher salt\n\n1\u00bd cups bread flour, plus more for dusting\n\n1 tsp. unsalted butter, softened\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Butter a 6-cup heatproof container.\n\n2 In a large bowl, whisk together warm water, instant yeast, sugar, dry milk powder, and kosher salt. Let mixture sit for 5 minutes or until milk powder dissolves.\n\n3 Add bread flour and softened unsalted butter, and using a wooden spoon, mix dough by moving the spoon in a circle until dough comes together and rolls around the bowl.\n\n4 Turn out dough onto the counter, and knead for 5 minutes or until soft and elastic. If it sticks to your hands, dust with flour.\n\n5 Place dough back in the bowl, cover with plastic wrap, and let rise in a warm, draft-free place for 1 hour or until doubled in size.\n\n6 Punch down dough, knead, shape into a ball, place in the prepared container, and cover tightly with plastic wrap. Allow bread to rise for 35 minutes or until doubled in size.\n\n7 Set the pressure cooker to medium (traditional)\/high (electric) heat, and bring to a boil. Add bread in container to the cooker.\n\n8 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 20 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n9 Lift bread out of the cooker, uncover, and shake bread out of the can onto a cooling rack. Cool completely before slicing and serving.\n\nNutrition per serving\n\nCalories: 130\n\nCarbohydrates: 25g\n\nSugars: 0g\n\nDietary fiber: 1g\n\nProtein: 4g\n\nFat: 1g\n\nCholesterol: 0mg\n\nSodium: 0mg\nCornbread\n\nThis tender and moist cornbread goes great with ham and beans, soups, and chilies and makes amazing ham sandwiches.\n\nIngredients\n\n2 cups yellow cornmeal\n\n\u2153 cup all-purpose flour\n\n1 tsp. kosher salt\n\n\u00bd tsp. baking soda\n\n\u00bd tsp. baking powder\n\n1 TB. sugar\n\n1\u00bd cups buttermilk\n\n2 large eggs\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker. Butter an 8-inch (20cm) cake pan with 2-inch (5cm) sides and a piece of aluminum foil large enough to cover the pan.\n\n2 In a large bowl, and using a wooden spoon, combine yellow cornmeal, all-purpose flour, kosher salt, baking soda, baking powder, and sugar. Add buttermilk and eggs, and mix until smooth. Pour batter into the prepared cake pan, cover with buttered foil (butter side down), and crimp edges tightly.\n\n3 Fold a long piece of foil lengthwise into a 2-inch (5cm) belt, tuck under the pan, and use the loose ends to lower the pan into the cooker. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 23 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n5 Lift cake pan out of the cooker, cut bread cut into 8 pieces, and serve with lots of butter.\n\nNutrition per serving\n\nCalories: 220\n\nCarbohydrates: 39g\n\nSugars: 4g\n\nDietary fiber: 1g\n\nProtein: 6g\n\nFat: 3g\n\nCholesterol: 60mg\n\nSodium: 310mg\n\nLemon Pots de Cr\u00e8me\n\nThese lovely lemon pots, like crust-free lemon bars but creamier, are tart and refreshing.\n\nIngredients\n\n1\u00bd cups heavy cream\n\n\u2153 cup sugar\n\n1 tsp. lemon zest\n\n4 large egg yolks\n\nJuice of 2 large lemons (\u00bc cup)\n\nMETHOD\n\n1 Add a trivet and steamer basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 In a small saucepan over medium heat, combine heavy cream, sugar, and lemon zest. Bring to a simmer, stir for 2 minutes or until sugar is dissolved, and remove from heat.\n\n3 In a medium bowl, whisk egg yolks until smooth.\n\n4 While whisking, drizzle 1 cup hot cream into yolks. Keep whisking so heat of cream tempers but doesn't cook eggs. Continue to drizzle in rest of cream, and add lemon juice.\n\n5 Strain mixture into a 3-cup or larger measuring cup to remove lemon zest.\n\n6 Pour 4 ounces (120ml) mixture each into 5 (6-ounce; 175ml) ramekins. Cover ramekins tightly with aluminum foil, and place in the steamer basket, stacking in a pyramid if necessary to fit. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n7 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 10 minutes. Turn off heat, and let the ramekins sit for 10 minutes. Remove the lid.\n\n8 Lift out one ramekin, and remove the foil, being careful not to let any water fall on surface of cr\u00e8me. Gently shake ramekin. Cr\u00e8me should be firm; if it jiggles, re-cover, return to the cooker, lock on the lid, and cook at level 2\/high for 5 minutes.\n\n9 Transfer ramekins to a baking sheet, remove the foil, and refrigerate for 1 hour. Cover cr\u00e8me with plastic wrap or foil so it doesn't form a skin, and chill for at least 2 hours or overnight. Serve cold.\n\nNutrition per serving\n\nCalories: 340\n\nCarbohydrates: 13g\n\nSugars: 10g\n\nDietary fiber: 0g\n\nProtein: 4g\n\nFat: 31g\n\nCholesterol: 310mg\n\nSodium: 35mg\n\nChocolate Pots de Cr\u00e8me\n\nThis luscious custardlike dessert is reminiscent of a creamy, smooth brownie. It's rich and decadent.\n\nIngredients\n\n3 large egg yolks\n\n\u00be cup heavy cream\n\n\u00be cup whole milk\n\n1\u00bd TB. sugar\n\n4 oz. (110g) 72 percent dark chocolate\n\nMETHOD\n\n1 Add a trivet and steamer basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 In a medium bowl, whisk egg yolks until smooth.\n\n3 In a small saucepan over medium heat, combine heavy cream, whole milk, sugar, and dark chocolate. Bring to a simmer, and stir for 2 minutes or until sugar is dissolved, chocolate is melted, and liquid begins to bubble at the edges of the pan. Remove from heat.\n\n4 While whisking, add \u00bd cup hot chocolate to eggs. Keep whisking so heat of chocolate tempers but doesn't cook eggs. Continue to add rest of chocolate.\n\n5 Divide mixture among 4 (6-ounce; 175ml) ramekins. Cover each ramekin tightly with aluminum foil, and place in the steamer basket, stacking in a pyramid if necessary to fit. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n6 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 10 minutes. Turn off heat, and let the ramekins sit for 10 minutes. Remove the lid.\n\n7 Lift out one ramekin, and remove the foil, being careful not to let any water fall on surface of cr\u00e8me. Gently shake ramekin. Cr\u00e8me should be firm; if it jiggles, re-cover, return to the cooker, lock on the lid, and cook at level 2\/high for 5 minutes.\n\n8 Transfer ramekins to a baking sheet, remove the foil, and refrigerate for 1 hour. Cover cr\u00e8me with plastic wrap or foil so it doesn't form a skin, and chill for at least 2 hours or overnight. Serve cold, plain, with whipped cream, or with raspberries.\n\nNutrition per serving\n\nCalories: 390\n\nCarbohydrates: 26g\n\nSugars: 15g\n\nDietary fiber: 2g\n\nProtein: 6g\n\nFat: 31g\n\nCholesterol: 330mg\n\nSodium: 50mg\nMaple Cream Caramels\n\nThis sweet, maple syrup custard dessert is topped with a decadent brown sugar caramel sauce.\n\nIngredients\n\n4 large egg yolks\n\n\u2153 cup granulated sugar\n\n\u2153 cup brown sugar, firmly packed\n\n\u00bc cup water\n\n2 cups heavy cream\n\n\u00bd cup pure maple syrup\n\nMETHOD\n\n1 Add a trivet and steamer basket to a 6-quart (5.5l) pressure cooker, and add 2 cups water.\n\n2 In a medium bowl, whisk egg yolks until smooth.\n\n3 In a small saucepan over medium heat, combine granulated sugar, brown sugar, and water, and cook, without stirring, until amber at edges. Gently swirl or stir with a clean wooden spoon, and cook until caramel is dark amber. Immediately divide into 5 (4-ounce; 110g) ramekins, and let harden for at least 10 minutes.\n\n4 Carefully wash out hot saucepan. Add heavy cream and maple syrup, return to medium heat, and bring to a hard simmer. Remove from heat.\n\n5 While whisking, slowly drizzle \u00bd cup hot cream into yolks. Keep whisking so heat of cream tempers but doesn't cook eggs. Continue to drizzle in rest of cream.\n\n6 Divide custard mixture among ramekins, cover each ramekin tightly with aluminum foil, and place in the steamer basket, stacking ramekins in a pyramid if necessary to fit. Set heat to high (traditional)\/high (electric), and bring to a boil.\n\n7 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 20 minutes. Turn off heat, and let caramels sit for 20 minutes. Remove the lid.\n\n8 Transfer ramekins to a cooling rack, and gently remove the foil, being careful not to let any water fall on surface of caramels. Refrigerate overnight.\n\n9 Run a thin-blade knife around the edge of ramekins, and invert onto a plate. Spread a kitchen towel on the counter, and sharply tap plate onto the towel to release custard from the ramekin. Serve cold.\n\nNutrition per serving\n\nCalories: 560\n\nCarbohydrates: 44g\n\nSugars: 40g\n\nDietary fiber: 0g\n\nProtein: 6g\n\nFat: 42g\n\nCholesterol: 410mg\n\nSodium: 50mg\nCr\u00e8me Br\u00fbl\u00e9e\n\nCracking a spoon through the burnt sugar topping of a good cr\u00e8me br\u00fbl\u00e9e, exposing the rich vanilla cream underneath, is a treat in itself.\n\nIngredients\n\n1 qt. (1l) heavy cream\n\n1 tsp. vanilla paste, vanilla extract, or seeds from 1 vanilla pod\n\n4 large egg yolks\n\n\u00be cup sugar\n\nMETHOD\n\n1 Add a trivet and steamer basket to a 6-quart (5.5l) pressure cooker, and fill with the minimum amount of water allowed for your cooker.\n\n2 In a medium saucepan over medium-high heat, bring heavy cream to a simmer.\n\n3 In a large bowl, whisk together vanilla paste, egg yolks, and \u00bd cup sugar.\n\n4 While whisking, drizzle 1 cup hot cream into egg mixture. Keep whisking so heat of cream tempers but doesn't cook eggs. Continue to drizzle in rest of cream. Skim off any foam on top of cr\u00e8me br\u00fbl\u00e9e.\n\n5 Pour 4 ounces (120ml) mixture each into 8 (6-ounce; 175ml) ramekins. Cover each ramekin tightly with aluminum foil, and place in the steamer basket, stacking ramekins in a pyramid if necessary to fit. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 15 minutes. Turn off heat, and let cr\u00e8me br\u00fbl\u00e9es sit for 10 minutes. Remove the lid.\n\n7 Lift out a ramekin, and remove the foil, being careful not to let any water fall on cr\u00e8me br\u00fbl\u00e9e. Gently shake ramekin. Cr\u00e8me br\u00fbl\u00e9e should be firm; if it jiggles, re-cover, return to the cooker, lock on the lid, and cook at level 2\/high for 10 minutes.\n\n8 Transfer ramekins to a baking sheet, remove the foil, and refrigerate for at least 3 hours. If refrigerating overnight, cover cr\u00e8me br\u00fbl\u00e9e with plastic wrap or foil so it doesn't form a skin.\n\n9 To serve, spoon 2 teaspoons sugar on top of each cr\u00e8me br\u00fbl\u00e9e, and gently shake to distribute. Using a kitchen torch or oven broiler, brown sugar. Serve immediately.\n\nNutrition per serving\n\nCalories: 500\n\nCarbohydrates: 17g\n\nSugars: 14g\n\nDietary fiber: 0g\n\nProtein: 5g\n\nFat: 48g\n\nCholesterol: 340mg\n\nSodium: 50mg\n\nCaribbean-Style Flan\n\nYour pressure cooker excels at making flan. The custard finishes with a smooth texture, a lovely vanilla note, and a deep caramel-coffee flavor.\n\nIngredients\n\n1\u00bc cups sugar\n\n\u00bc cup water\n\n1 (14-oz.; 400g) can sweetened, condensed milk\n\n2 cups whole milk\n\n3 large eggs\n\n2 tsp. vanilla paste, vanilla extract, or seeds from 1 vanilla pod\n\n\u215b tsp. kosher salt\n\nMETHOD\n\n1 Add a trivet to the bottom of a 6-quart (5.5l) pressure cooker. Add 2 cups water.\n\n2 In a medium saucepan over medium-high heat, combine sugar and water, and cook, without stirring, until amber at edges. Gently swirl, and cook until caramel is dark amber.\n\n3 Immediately pour caramel into an 8-inch (20cm) cake pan, and quickly but gently swirl pan to coat bottom with caramel. Set aside for at least 10 minutes.\n\n4 In a large bowl, whisk together condensed milk, whole milk, eggs, vanilla paste, and kosher salt. Using a fine mesh strainer, strain custard into cake pan to remove any undissolved egg whites. Cover the pan tightly with aluminum foil.\n\n5 Fold a long piece of foil lengthwise into a 2-inch (5cm) belt, tuck under the pan, and use the loose ends to lower the pan into the cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 20 minutes. Turn off heat, and let flan sit for 20 minutes. Remove the lid.\n\n7 Lift out cake pan, and remove the foil, being careful not to let any water fall on surface of flan. Gently shake pan. Flan should be firm; if it jiggles, re-cover, return to the cooker, lock on the lid, and cook at level 2\/high for 10 minutes.\n\n8 Refrigerate for 1 hour. Cover flan with plastic wrap or foil so it doesn't form a skin, and chill for at least 2 hours or overnight.\n\n9 Run a thin-blade knife around the edge of the cake pan. Place a large plate with a lip (at least 2 inches; 5cm larger than flan to catch caramel) top down on the pan, invert the pan and plate to release flan, gently tapping the bottom of the pan if necessary, and lift off pan. Cut into 8 pieces, and serve.\n\nNutrition per serving\n\nCalories: 300\n\nCarbohydrates: 55g\n\nSugars: 55g\n\nDietary fiber: 0g\n\nProtein: 8g\n\nFat: 7g\n\nCholesterol: 100mg\n\nSodium: 140mg\nClassic Cheesecake\n\nThis cheesecake is so versatile. Serve it topped with blueberries, cherries, or raspberries\u2014or all three. It's also delicious plain.\n\nIngredients\n\n6 (3\u00d75-in.; 7.5\u00d712.5cm) graham crackers\n\n2\u00bd TB. unsalted butter\n\n1 lb. (450g) cream cheese, at room temperature\n\n1 tsp. vanilla paste, or 1\u00bd tsp. vanilla extract\n\n\u00bd cup sugar\n\n\u00bc tsp. kosher salt\n\n2 large eggs, at room temperature\n\n\u00bd cup sour cream, at room temperature\n\nMETHOD\n\n1 Add a trivet to the bottom of a 6-quart (5.5l) pressure cooker. Be sure an 8-inch (20cm) springform pan will fit inside with at least \u00be inch (2cm) space at the edges. Add enough water to reach the bottom of the trivet. Butter an 8-inch (20cm) springform pan with 1 teaspoon unsalted butter.\n\n2 In a food processor fitted with a metal chopping blade, pulse graham crackers and unsalted butter to a crumb consistency. Transfer to the pan, and press into an even layer.\n\n3 In a large bowl, and using an electric mixer on medium, beat cream cheese for 2 minutes or until smooth. Scrape down the sides of the bowl, and beat again. Add vanilla paste, sugar, kosher salt, eggs, and sour cream, and beat until smooth. Pour filling into crust, and smooth to an even layer. Cover the pan tightly with aluminum foil, and secure with string or a rubber band.\n\n4 Fold a long piece of foil lengthwise into a 2-inch (5cm) belt, tuck under the pan, and use the loose ends to lower the pan into the cooker. Set heat to high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 35 minutes. Turn off heat, and let cheesecake sit for 20 minutes. Remove the lid.\n\n6 Lift out the pan to a cooling rack, and remove the foil, being careful not to let any water fall on surface of cheesecake. Let cheesecake cool for 30 minutes, and refrigerate overnight.\n\n7 Run a thin-blade knife around the edge of the pan, and remove the springform ring. Cut cheesecake into 6 pieces, and serve.\n\nNutrition per serving\n\nCalories: 420\n\nCarbohydrates: 19g\n\nSugars: 16g\n\nDietary fiber: 0g\n\nProtein: 7g\n\nFat: 36g\n\nCholesterol: 180mg\n\nSodium: 390mg\nChocolate Caramel Cheesecake\n\nThis delectable dessert is for chocolate lovers. And cheesecake lovers. And caramel lovers.\n\nIngredients\n\n12 (3\u00d75-in.; 7.5\u00d712.5cm) chocolate graham crackers\n\n2 tsp. plus 1 TB. cocoa powder\n\n\u00bc cup plus \u2153 cup unsalted butter\n\n1\u00be cups sugar\n\n\u2153 cup water\n\n\u00be cup heavy cream\n\n\u00bd tsp. vanilla extract\n\n4 oz. (110g) dark chocolate, plus more for garnish\n\n2 (8-oz.; 225g) pkg. cream cheese, at room temperature\n\n2 large eggs, at room temperature\n\n\u00bc cup sour cream, at room temperature\n\n\u00bd cup pecans, chopped\n\nMETHOD\n\n1 Add a trivet to the bottom of a 6-quart (5.5l) pressure cooker. Add enough water to reach the bottom of the trivet.\n\n2 In a food processor fitted with a metal chopping blade, pulse chocolate graham crackers, 2 teaspoons cocoa powder, and \u00bc cup unsalted butter to a crumb consistency. Transfer to a 9-inch (23cm) springform pan, and press into an even layer.\n\n3 In a medium saucepan over medium-high heat, combine 1\u00bc cups sugar and water, and cook, without stirring, until amber at edges. Gently swirl, and cook until caramel is dark amber.\n\n4 Drizzle in heavy cream while stirring, and stir in remaining \u2153 cup unsalted butter and vanilla extract. Remove from heat.\n\n5 In a small saucepan over very low heat, melt dark chocolate.\n\n6 In a medium bowl, and using an electric mixer on medium, beat cream cheese for 2 minutes or until it follows the beaters in ribbons. Add eggs, sour cream, melted chocolate, remaining \u00bd cup sugar, and remaining 1 tablespoon cocoa powder, and beat until smooth. Pour into crust, and smooth to the edges of the pan. Cover the pan tightly with aluminum foil, and secure foil with string or a rubber band.\n\n7 Fold a long piece of foil lengthwise into a 2-inch (5cm) belt, tuck under the pan, and use the loose ends to lower the pan into the cooker. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n8 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 40 minutes. Remove from (traditional)\/turn off (electric) heat, and let cake sit for 20 minutes. Remove the lid.\n\n9 Lift out cake pan, uncover, and gently shake. Cake should be firm; if it jiggles, re-cover, return to the cooker, lock on the lid, and cook at level 2\/high for 10 minutes.\n\n10Refrigerate cake for at least 3 hours.\n\n11Pour caramel over cake to form an \u215b-inch (3mm) thick circle that stops just short of the edges, and sprinkle with pecans. Cover with plastic wrap, and refrigerate for 4 hours or overnight.\n\n12Run a thin-blade knife around edges of the pan, and remove the springform ring. Grate dark chocolate over top of cheesecake, cut into 8 pieces, and serve.\n\nNutrition per serving\n\nCalories: 880\n\nCarbohydrates: 67g\n\nSugars: 55g\n\nDietary fiber: 3g\n\nProtein: 9g\n\nFat: 70g\n\nCholesterol: 250mg\n\nSodium: 320mg\n\nRice Pudding\n\nRice pudding is a homey dessert. This version is prepared with traditional flavors, but you can omit these spices and customize as you like.\n\nIngredients\n\n2\u00bd cups whole milk\n\n1 cup arborio rice (not instant)\n\n2 tsp. vanilla extract\n\n\u00bd cup sugar\n\n\u00bd tsp. cinnamon\n\n\u215b tsp. ground cardamom\n\n\u00bd tsp. kosher salt\n\n\u00bd cup half-and-half\n\n\u00bc tsp. ground nutmeg\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine whole milk, arborio rice, vanilla extract, sugar, \u00bc teaspoon cinnamon, cardamom, and kosher salt. Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 13 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Stir in half-and-half.\n\n4 Spoon pudding into 4 parfait cups, and refrigerate for at least 3 hours or overnight.\n\n5 When ready to serve, sprinkle each pudding with 1 dash remaining cinnamon and 1 dash nutmeg.\n\nYou can vary this recipe by adding 1 teaspoon lemon or orange zest, 1 teaspoon raisins or cranberries, or even 1 teaspoon liquor such as Grand Marnier. If you want to store the pudding for later, cool, cover the surface with plastic wrap so it won't form a skin, and refrigerate for up to 3 days.\n\nNutrition per serving\n\nCalories: 370\n\nCarbohydrates: 66g\n\nSugars: 26g\n\nDietary fiber: 1g\n\nProtein: 9g\n\nFat: 9g\n\nCholesterol: 25mg\n\nSodium: 320mg\nBread Pudding with Whiskey Sauce\n\nRich with warm spices and a tender, melt-in-your-mouth pudding, this creamy dessert is a welcome addition to family and holiday dinners.\n\nIngredients\n\n6 large egg yolks\n\n\u00bd cup plus \u2153 cup sugar\n\n3 cups whole milk\n\n1 tsp. orange zest\n\n1 tsp. vanilla extract\n\n\u00bd tsp. cinnamon\n\n\u00bc tsp. ground nutmeg\n\n6 slices white bread, cut into \u00bd-in. (2.5cm) cubes (2 cups)\n\n\u2153 cup raisins\n\n\u2153 cup whiskey\n\n3 tsp. unsalted butter\n\n2 TB. heavy cream\n\nMETHOD\n\n1 Add a trivet to the bottom of a 6-quart (5.5l) pressure cooker. Add 2 cups water. Butter an 8-inch (20cm) cake pan and a piece of aluminum foil large enough to cover it.\n\n2 In a large bowl, whisk together 3 egg yolks and \u00bd cup sugar. Whisk in whole milk, orange zest, vanilla extract, cinnamon, and nutmeg.\n\n3 Add bread to the cake pan, and spread evenly. Sprinkle on raisins, pour in pudding, and push down any floating bread. Cover the pan with buttered foil (butter side down), and tightly crimp on the pan. Place the cake pan in the cooker, and let sit for 20 minutes.\n\n4 Set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n5 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 25 minutes. Turn off heat, and let pudding sit for 20 minutes. Remove the lid.\n\n6 In a small mixing bowl, whisk together remaining \u2153 cup sugar and remaining 3 egg yolks.\n\n7 In a small saucepan over medium heat, whisk together whiskey and unsalted butter. Bring to a boil, remove from heat, and slowly whisk whiskey mixture into eggs. Whisk in heavy cream. Return sauce to the saucepan, set heat to low, and cook, stirring, for 1 minute or until sauce begins to thicken. Remove from heat.\n\n8 Lift cake pan out of the pressure cooker, and remove the foil. Cool for 1 hour.\n\n9 Scoop out pudding into individual bowls, and serve with whiskey sauce on the side.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 30g\n\nSugars: 25g\n\nDietary fiber: 1g\n\nProtein: 7g\n\nFat: 12g\n\nCholesterol: 280mg\n\nSodium: 115mg\nCaramel Vanilla Tapioca Pudding\n\nSweet tapioca pudding is a favorite for kids and adults alike. This version is matured a bit with caramel and vanilla flavors.\n\nIngredients\n\n\u00bd cup small-pearl tapioca\n\n\u00bd cup granulated sugar\n\n2\u00bd cups whole milk\n\n\u00bc tsp. kosher salt\n\n1\u00bd tsp. vanilla paste, or vanilla extract\n\n\u00bd cup heavy cream, very cold\n\n1 TB. confectioners' sugar\n\n\u00bc cup crumbled English toffee\n\nMETHOD\n\n1 Place tapioca in a strainer, rinse under cold water, and set aside.\n\n2 In a 4-quart (4l) pressure cooker set to medium (traditional)\/high (electric) heat, add granulated sugar and shake the cooker to evenly distribute sugar. Cook, watching sugar closely, until it begins to caramelize at the edges and darken as it melts. Using a clean wooden spoon, stir sugar. It'll clump and melt, but in the end, it'll be an amber to brown-colored liquid.\n\n3 In a small saucepan over medium heat, warm whole milk until just short of boiling.\n\n4 While stirring, drizzle 2 cups hot milk into sugar. Don't worry if sugar clumps; it'll melt.\n\n5 Add tapioca, kosher salt, and 1 teaspoon vanilla paste, and stir until sugar dissolves and milk begins to boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook for 15 minutes. Remove from (traditional)\/turn off (electric) heat, and let pudding sit for 10 minutes. Perform a natural release, and remove the lid.\n\n7 Add remaining \u00bd cup hot milk, return to medium-low (traditional)\/low (electric) heat, and whisk until tapioca is loose and not clumping. (This might take a little while, so be patient.)\n\n8 Ladle tapioca into small containers, and let it cool for 1 hour. When it's warm but not hot, cover tapioca with plastic wrap, and refrigerate for at least 3 hours or until set.\n\n9 When ready to serve, in a medium bowl, whisk very cold heavy cream, remaining \u00bd teaspoon vanilla paste, and confectioners' sugar until stiff peaks form.\n\n10 Uncover pudding, top with whipped cream, sprinkle with English toffee, and serve.\n\nBe careful when caramelizing the sugar. It gets very, very hot and can burn you easily.\n\nNutrition per serving\n\nCalories: 270\n\nCarbohydrates: 36g\n\nSugars: 25g\n\nDietary fiber: 0g\n\nProtein: 4g\n\nFat: 14g\n\nCholesterol: 40mg\n\nSodium: 170mg\n\nChristmas Pudding\n\nThis traditional British holiday dessert is a wonderful addition to family dinners. Conventional versions are often heavy; this is lighter.\n\nIngredients\n\n1\u00be cups mixed dried raisins, chopped prunes, cherries, cranberries, and apricots\n\n\u00bd cup dried dates, pitted and chopped\n\n2 TB. minced crystallized ginger\n\n1 tsp. orange zest\n\n6 slices fresh white bread, chopped in a food processor to a fine crumb (2\u00bd cups)\n\n3 TB. unsalted butter or vegetable shortening, softened\n\n1\u00be cups sugar\n\n1 cup walnuts, chopped\n\n1 large egg\n\n\u2154 cup whole milk\n\n\u2153 cup water\n\n\u00be cup heavy cream\n\n\u2153 cup unsalted butter\n\n\u00bd tsp. vanilla extract\n\nMETHOD\n\n1 Add a trivet to the bottom of a 6-quart (5.5l) pressure cooker. Add enough water to reach the bottom of the trivet. Grease a 1\u00bd-quart (1.5l) heat-proof pudding mold with unsalted butter.\n\n2 In a large bowl, combine fruit, dates, crystallized ginger, orange zest, breadcrumbs, softened unsalted butter, \u00bd cup sugar, walnuts, egg, and whole milk.\n\n3 Transfer to the mold, and gently pack. Cover with aluminum foil, and secure foil with string or a rubber band. Place the mold in the pressure cooker, set heat to high (traditional)\/high (electric), and bring to a boil.\n\n4 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 1 hour. Turn off heat, and let pudding sit for 20 minutes. Perform a natural release, and remove the lid.\n\n5 Remove the mold from the cooker, and invert pudding onto a platter.\n\n6 In a medium saucepan over medium-high heat, combine remaining 1\u00bc cups sugar and water, and cook, without stirring, until amber at edges. Gently swirl or stir with a clean wooden spoon, and cook until caramel is dark amber.\n\n7 Drizzle in heavy cream while stirring. Stir in unsalted butter and vanilla extract, and remove from heat.\n\n8 Top pudding with caramel sauce, and serve with whipped cream.\n\nNutrition per serving\n\nCalories: 490\n\nCarbohydrates: 82g\n\nSugars: 56g\n\nDietary fiber: 3g\n\nProtein: 6g\n\nFat: 18g\n\nCholesterol: 55mg\n\nSodium: 220mg\nDulce de Leche\n\nThis easy recipe yields a wonderful caramel concoction with a sweet flavor. It's a perfect dip for apple slices, ice cream topping, or jelly roll cake filling.\n\nIngredients\n\n1 (14-oz.; 400g) can sweetened condensed milk, label removed\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker.\n\n2 Lay the can of condensed milk on its side on trivet. The can shouldn't touch the sides or bottom of the cooker. Add enough water to cover the can by 1 inch (2.5cm). Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n3 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 20 minutes. Turn off heat, and let dulce de leche sit until completely cool. Perform a natural release, and remove the lid.\n\n4 Transfer can to a cooling rack, and cool completely before opening. Use in your favorite recipe, or refrigerate for later use. Warm gently before using.\n\nLeave the can inside the pressure cooker with the lid on after it's cooked, and allow it to cool completely before opening. Opening a hot can is never a good idea. The hot ingredients could gush out and burn you.\n\nNutrition per serving\n\nCalories: 298\n\nCarbohydrates: 57g\n\nSugars: 57g\n\nDietary fiber: 0g\n\nProtein: 7.5g\n\nFat: 4g\n\nCholesterol: 12mg\n\nSodium: 99mg\nGingerbread Cake with Vanilla Glaze\n\nThis delicious, moist cake is loaded with gingerbread flavor. The vanilla glaze provides an elegant finish.\n\nIngredients\n\n\u00bd cup unsalted butter\n\n\u00bd cup granulated sugar\n\n\u00bd cup molasses\n\n\u00bc cup water\n\n1\u00bd tsp. ground ginger\n\n\u00be tsp. cinnamon\n\n\u00bc tsp. ground cloves\n\n1\u00bc cups unbleached cake flour\n\n1\u00bc tsp. baking soda\n\n\u00bd tsp. kosher salt\n\n1 large egg, lightly beaten\n\n3 TB. heavy cream\n\n\u00bd tsp. vanilla extract\n\n\u00be cup confectioners' sugar\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker. Add 2 cups water. Butter an 8-inch (20cm) cake pan, and lightly coat bottom and sides with 1 tablespoon flour. Gently tap out excess flour.\n\n2 In a small saucepan over medium heat, combine unsalted butter, granulated sugar, molasses, and water. Cook, stirring, for 2 minutes or until sugar is dissolved and butter is melted. Remove from heat, and set aside to cool.\n\n3 In a large bowl, combine ginger, cinnamon, cloves, cake flour, baking soda, and kosher salt.\n\n4 Add cooled molasses mixture to dry ingredients, add egg, and stir with a heavy-duty wooden spoon until smooth and combined. Scrape batter into the prepared pan, and tap the pan gently to distribute batter. Cover the pan tightly with aluminum foil.\n\n5 Add the cake pan to the pressure cooker, set heat to medium-high (traditional)\/high (electric), and bring to a boil.\n\n6 Lock on the lid, and bring pressure to level 2 (traditional)\/high (electric). Reduce heat to low, and cook at 2\/high for 25 minutes. Remove from (traditional)\/turn off (electric) heat, perform a natural release, and remove the lid.\n\n7 Remove the cake pan, uncover, and let cake cool completely on a cooling rack.\n\n8 In a large bowl, whisk heavy cream, vanilla extract, and confectioners' sugar until smooth. Glaze should be runny but not overly so. Add more confectioners' sugar to thicken if necessary. Add more cream, 1 teaspoon at a time, if glaze is too stiff.\n\n9 Run a thin-blade knife around the edge of the cake pan, place a large plate top down on the pan, and invert the pan and plate. Gently tap the bottom of the pan to release cake, and lift off cake pan. Drizzle glaze over top of cake, slice, and serve.\n\nNutrition per serving\n\nCalories: 430\n\nCarbohydrates: 64g\n\nSugars: 41g\n\nDietary fiber: 0g\n\nProtein: 3g\n\nFat: 19g\n\nCholesterol: 85mg\n\nSodium: 480mg\n\nRum Raisin, Apple, and Prune Cake\n\nThis cake is a little gooey, a little boozy, and a little tart, thanks to the apples. Don't forget the cardamom. It makes the cake taste like an apple Danish.\n\nIngredients\n\n2 TB. plus 2 tsp. brown sugar\n\n10 TB. unsalted butter\n\n2 small Fuji apples, peeled and cut into thin half-moons\n\n1\u00bd cups cake flour\n\n1\u00bd tsp. baking powder\n\n\u215b tsp. ground cardamom\n\n1 cup granulated sugar\n\n3 large eggs\n\n\u00bc cup whole milk\n\n1\u00bc tsp. vanilla extract\n\n\u00bd cup raisins, soaked in 3 TB. spiced rum for 1 hour\n\n\u00bd cup prunes\n\n\u00bd cup sour cream\n\nMETHOD\n\n1 Add a trivet to a 6-quart (5.5l) pressure cooker. Add 2 cups water. Butter a large piece of aluminum foil.\n\n2 In a small bowl, cream 2 tablespoons brown sugar and 2 tablespoons unsalted butter. Spread in the bottom of an 8-inch (20cm) cake pan, and layer Fuji apple slices on top.\n\n3 In a medium bowl, combine cake flour, baking powder, and cardamom.\n\n4 In a large bowl, cream granulated sugar and remaining 8 table-spoons unsalted butter until smooth. Add eggs, whole milk, and 1 teaspoon vanilla extract, and stir.\n\n5 Combine granulated sugar\u2013egg mixture with dry ingredients, and stir in raisins and prunes. Gently pour batter over apples, and spread to the edge of the pan. Cover with buttered foil (butter side down), and tightly crimp foil to the pan.\n\n6 Fold a long piece of foil lengthwise into a 2-inch (5cm) belt, tuck under the pan, and use the loose ends to lower the pan into the cooker. Set to heat medium (traditional)\/high (electric), and bring to a boil.\n\n7 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 25 minutes. Turn off heat, let cake sit for 20 minutes, and remove the lid.\n\n8 Lift the pan out of the cooker, and remove the foil. Cool cake on a cooling rack.\n\n9 In a medium bowl, whisk together remaining \u00bc teaspoon vanilla extract, remaining 2 teaspoons brown sugar, and sour cream until smooth.\n\n10 Turn out cake, upside down, onto a plate. Cut into 8 pieces, top with brown sugar sour cream, and serve.\n\nNutrition per serving\n\nCalories: 440\n\nCarbohydrates: 60g\n\nSugars: 39g\n\nDietary fiber: 3g\n\nProtein: 5g\n\nFat: 19g\n\nCholesterol: 130mg\n\nSodium: 140mg\nBoozy Dried Fruit Compote\n\nWhether spooned over vanilla ice cream or served in a bowl with a drizzle of cream and a bit of granola, this is a wonderful dessert.\n\nIngredients\n\n1 cup dried apples, chopped into \u00be-in. (2cm) pieces\n\n1 cup small prunes\n\n\u00bd cup dried apricots\n\n\u00bc cup raisins\n\n\u00bd cup dried cranberries\n\n1 cup water\n\n1 cup dry red wine\n\n1 slice lemon\n\n\u00bd cup sugar\n\n1 tsp. vanilla paste, or vanilla extract\n\n1 (3-in.; 7.5cm) cinnamon stick\n\n\u215b tsp. ground nutmeg\n\n1 pinch cardamom\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker, combine apples, prunes, apricots, raisins, cranberries, water, red wine, lemon slice, sugar, vanilla paste, cinnamon stick, nutmeg, and cardamom. Set heat to medium (traditional)\/high (electric), and bring to a boil.\n\n2 Lock on the lid, and bring pressure to level 1 (traditional)\/low (electric). Reduce heat to low, and cook at 1\/low for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 Remove cinnamon stick and lemon slice, and let fruit cool before serving. Refrigerate any leftovers in a jar for up to 2 weeks.\n\nNutrition per serving\n\nCalories: 170\n\nCarbohydrates: 45g\n\nSugars: 35g\n\nDietary fiber: 4g\n\nProtein: 1g\n\nFat: 0g\n\nCholesterol: 0mg\n\nSodium: 70mg\nPears Poached in Red Wine\n\nYou'll impress guests when you serve these lovely pears steamed to tender perfection and infused with the perfect amount of spice and sweetness.\n\nIngredients\n\n\u00be cup dry red wine\n\n\u00bd cup water\n\n1 (3-in.; 7.5cm) cinnamon stick\n\n1 tsp. whole black peppercorns\n\n2 slices lemon\n\n\u00bc tsp. freshly ground nutmeg, plus more for dusting\n\n\u00bd cup sugar\n\n4 medium firm, almost ripe d' Anjou pears, peeled\n\n\u00bd cup heavy cream\n\n1\u00bd tsp. honey\n\nMETHOD\n\n1 In a 4-quart (4l) pressure cooker set to medium-high (traditional)\/high (electric) heat, combine red wine, water, cinnamon stick, black peppercorns, lemon slices, nutmeg, and sugar. Add d' Anjou pears, stems up, and bring to a boil.\n\n2 Lock on the lid, bring pressure to level 2 (traditional)\/high (electric), and cook for 8 minutes. Remove from (traditional)\/turn off (electric) heat, perform a cold water (traditional)\/quick (electric) release, and remove the lid.\n\n3 In a medium bowl, whisk heavy cream until it begins to thicken. (You aren't whipping cream, just thickening it.) Add honey, and whisk to combine.\n\n4 Using a slotted spoon, gently transfer pears to a platter.\n\n5 Return the pressure cooker to medium\/high heat, and cook for 5 minutes or until spiced wine is reduced by half or becomes syrupy.\n\n6 Spoon syrup over pears, dollop with honey cream, and dust with ground nutmeg. Serve with extra syrup on the side.\n\nWhen spooning the reduced syrup over the cooked pears, avoid getting any peppercorns, lemon slices, or the cinnamon stick in the serving dishes. You can strain the sauce and serve it on the side if you like.\n\nNutrition per serving\n\nCalories: 260\n\nCarbohydrates: 35g\n\nSugars: 20g\n\nDietary fiber: 0g\n\nProtein: 2g\n\nFat: 11g\n\nCholesterol: 40mg\n\nSodium: 15mg\n\nAppendix\n\n**Helpful Tables**\n\nThis appendix holds some handy tables for quick reference. Use them for a reminder of cook times or to help convert conventional recipes for use in a pressure cooker.\n\nDried beans cook best if they're presoaked first. For every 1 cup beans, add 4 cups water, and soak for 8 hours or overnight. Discard the soak water, and rinse the beans before cooking. The beans need to be fully hydrated before cooking so they don't absorb all the cooking liquid and scorch.\n\n## Contents\n\n  1. Copyright\n  2. Contents\n  3. Introduction\n  4. Pressure Cooker Basics\n    1. How Pressure Cooking Works\n    2. What Your Pressure Cooker Can Do\n    3. Types of Pressure Cookers\n    4. Safety and Maintenance\n    5. Release Methods\n    6. Understanding PSI\n    7. Pressure Cooking Vegetables\n    8. Pressure Cooking Meat\n    9. Converting Conventional Recipes\n    10. Stocking Your Kitchen\n    11. Planning Weeknight Meals\n  5. Pressure Cooker Recipes\n    1. Basics\n      1. Rich Beef Stock\n      2. Chicken Stock \n      3. Vegetable Stock \n      4. Pressure Cooker Eggs\n      5. Pressure Cooker Rice\n      6. Easy Dried Beans\n      7. Pressure Cooker Pasta Sauce\n    2. Breakfasts\n      1. Eggs en Cocotte a la Creme\n      2. Egg Cups\n      3. Apple Cinnamon Breakfast Oats\n      4. Savory Parmesan Steel-Cut Oats\n      5. Five-Grain Oatmeal\n      6. Sausage, Onion, and Gruyere Breakfast Casserole\n      7. Honey, Raisin, and Quinoa Breakfast Risotto\n      8. Bacon Jam\n    3. Appetizers\n      1. Classic Deviled Eggs\n      2. Garlicky White Bean and Parmesan Dip\n      3. Red Beet Hummus\n      4. Middle Eastern Hummus\n      5. Lentil Pate\n      6. Savory Smoked Salmon Cheesecake\n    4. Salads\n      1. Tuna Salad with Chickpeas\n      2. Chicken Salad Deluxe\n      3. Classic Egg Salad\n      4. Wheat Berry Salad with Arugula Pesto\n      5. Farro Tabbouleh\n      6. Picnic-Style Potato Salad\n      7. Hot German Potato Salad\n    5. Soups, Stews, and Chilies\n      1. Roasted Corn and Butternut Squash Chili\n      2. White Bean and Shiitake Soup\n      3. Saturday Soup Beans\n      4. Spicy Chickpea Stew with Sour Tomato Curry\n      5. Cuban Black Bean Soup with Sherry\n      6. Farmhouse Corn Chowder\n      7. French Potato and Leek Soup\n      8. Indian Carrot and Lentil Soup\n      9. Rustic Split-Pea Soup\n      10. Curried Butternut Squash Soup\n      11. Mushroom Barley Soup\n      12. Root Vegetable Stew\n      13. Borscht with Italian Sausage\n      14. Ribolleta\n      15. Texas-Style Chili Con Carne\n      16. Korean Beef Stew\n      17. Vegetable Beef Soup\n      18. Irish Stew\n      19. Pork Ramen\n      20. Hearty Turkey and Vegetable Soup\n      21. New England Fish Chowder\n    6. Vegetables\n      1. Ratatouille\n      2. Honey-Glazed Carrots\n      3. Glazed Carrots with Braised Lettuce\n      4. Southern-Style Green Beans\n      5. Buttered Green Beans with Nut Crunch Topping\n      6. Caramelized Onion Mashed Potatoes \n      7. Mashed Maple Sweet Potatoes\n      8. Brussels Sprouts with Almonds and Prosciutto\n      9. Braised Kale\n      10. Alsatian-Style Braised Red Cabbage\n      11. One-Pot Cabbage, Rice, and Lentils\n      12. Southern Collard Greens\n    7. Meaty Main Dishes\n      1. Barbecue Braised Short Ribs\n      2. Beef Sugo\n      3. Corned Beef\n      4. Cuban-Style Ropa Vieja\n      5. Classic Beef Brisket\n      6. Pot Roast with Fennel and Carrots\n      7. Swedish Meatballs\n      8. Beef Bourguignon\n      9. Chuck Roast with Horseradish Cream and Carrots\n      10. Swiss Steak\n      11. Osso Buco\n      12. Belgian Beef Stew Cooked in Beer\n      13. Pressure Cooker Tacos\n      14. Hungarian Stuffed Peppers\n      15. Hungarian Chicken Paprika\n      16. Caribbean Chicken Curry\n      17. One-Pot Chicken and Sausage Perloo\n      18. Chicken and Dumplings\n      19. Chicken Cacciatore\n      20. Smothered Chicken\n      21. Chicken with 40 Cloves of Garlic\n      22. Thai-Style Green Curry Chicken\n      23. Chinese Red Cooked Chicken\n      24. Hungarian Pork Goulash\n      25. Belgian Ale-Braised Pork Loin with Mustard\n      26. Pork Grillades\n      27. Southern-Style Pulled Pork\n      28. Pork Vindaloo\n      29. Pork Posole\n      30. One-Pot Sausage, Potatoes, and Greens\n      31. Asian Steamed Fish\n      32. Salmon en Papillote\n      33. Fish Curry\n      34. Thai Steamed Mussels\n      35. Beer-Steamed Shrimp\n      36. Cajun-Style Shrimp Jambalaya\n    8. Rice, Grains, and Breads\n      1. Basmati Rice Pilaf\n      2. Dirty Oats with Lentils\n      3. Wild Rice\n      4. Asparagus and Sun-Dried Tomato Risotto\n      5. Risotto Milanese\n      6. Jamaican-Style Rice and Peas\n      7. Steamed Brown Bread with Raisins\n      8. Crustless Sandwich Bread\n      9. Cornbread\n    9. Desserts\n      1. Lemon Pots de Creme\n      2. Chocolate Pots de Creme\n      3. Maple Cream Caramels\n      4. Creme Brulee\n      5. Caribbean-Style Flan\n      6. Classic Cheesecake\n      7. Chocolate Caramel Cheesecake\n      8. Rice Pudding\n      9. Bread Pudding with Whiskey Sauce\n      10. Caramel Vanilla Tapioca Pudding\n      11. Christmas Pudding\n      12. Dulce de Leche\n      13. Gingerbread Cake with Vanilla Glaze\n      14. Rum Raisin, Apple, and Prune Cake\n      15. Boozy Dried Fruit Compote\n      16. Pears Poached in Red Wine\n    10. Appendix\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n  3. Title Page\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTHE INN\n\n\"What kind of ghosts are at this place we're headed to?\" Neville asked.\n\nPriscilla riffled through the brochures on her lap. \"The Blue Boy Inn,\" she read aloud. \"Dates back to the American Civil War. Oh, wow, there could be a lot of ghosts here.\"\n\n\"What do you mean by a lot?\"\n\n\"It says here that there have been several murders at the inn. And some mysterious disappearances.\"\n\n\"That's probably just hype.\"\n\n\"No, it's real,\" Priscilla said, reading the history of the place. \"The first murder was a girl named Sally Brown. Nearly a hundred years ago. They never found her body, just her blood, splattered all over the walls. Then there was this guy Andrew McGurk, whose body was found, but not his head.\" She shivered. \"And then\u2014oh, this is terrible\u2014a little baby, who disappeared. And there's more.\"\n\nPriscilla glanced down at the description of the Blue Boy. \"Quite a bit more.\" She looked over at Neville. \"I think we've hit the jackpot with this place. If we're looking for ghosts, this is the place that will have them.\"\n\n\"I can hardly wait . . .\"\nBooks by William Patterson\n\nSLICE\n\nTHE INN\n\nPublished by Kensington Publishing Corp.\nTHE INN\n\nWILLIAM PATTERSON\n\nPINNACLE BOOKS\n\nKensington Publishing Corp.\n\nwww.kensingtonbooks.com\n\nAll copyrighted material within is Attributor Protected.\nTable of Contents\n\nTHE INN  \nAlso by  \nTitle Page\n\n112 \\- Adam heard the sound and swung his gun around in the direction of the pantry. As he did so, the woman in front of him lunged at him with a knife.\n\nCopyright Page\n\n1\n\nAnnabel Wish had a fear of closed spaces. A year ago, she'd been stuck, alone, in an elevator for thirty-six hours, and that had simply been too much to bear. When workers had finally pried open the elevator doors on the morning of Annabel's second day of confinement, they'd found her stark raving mad.\n\nThat was why she kept rolling her car window down now, despite outdoor temperatures hovering around thirty-two degrees, and despite her husband, Jack, constantly admonishing her to roll it back up.\n\n\"Really, Annabel,\" Jack said. \"You're freezing me to death.\"\n\n\"I just need the air,\" she told him.\n\nHe sighed. \"We're almost there. We're not more than forty minutes away now.\"\n\nAnnabel took several deep gulps of air, and then rolled the window up again.\n\nThey'd been on the road for three hours. That was two and a half hours too long for Annabel. Even a subway ride from lower Manhattan to Washington Heights left her uneasy. It wasn't that their car was small: It was a good-sized SUV, some name and brand Annabel wasn't sure of. She never paid much attention to such things. But it was an enclosed space, which meant that she couldn't get up and move around, and that made her want to crawl out of her skin. Annabel kept looking out the window, trying to focus on the trees and houses going by and hoping that Jack was right, that this move would be good for her. Good for them. Good for their marriage.\n\nAll she knew was that she needed air.\n\n\"I remember these roads so well from when I was a boy,\" Jack was saying. \"In the spring and summer there'd be farm stands all through here, selling beans and corn and tomatoes. Dad would stop and we'd load up the car and take it all back with us to the city.\"\n\nAnnabel could see only stark, frozen, bare trees.\n\nShe had never been this far out into the country before. In her twenty-six years, Annabel had been all over the world. London, Paris, Milan, Sydney, Tokyo. It came with the job of being a fashion editor. But it was always cities that she visited. Annabel had never had any interest in seeing life beyond urban boundaries. Even as a girl, she'd always spent her summers in the city. All the grass and trees she had needed she had found in Central Park, and when she grew tired of nature\u2014which was often\u2014there were shops and movies and restaurants and museums. Not for Annabel any regular trips into the pollen-and-ragweed-infested countryside of western Massachusetts of the kind Jack so fondly remembered, visiting his grandfather's rustic old bed-and-breakfast in the village of Woodfield. Annabel had been exquisitely content to remain in the city.\n\nBut they were far away from the city now. They were retracing the route of Jack's summer-vacation sojourns to Woodfield.\n\nExcept that it wasn't summer. It was the onset of a very cold but so far snowless winter. And the bed-and-breakfast they were heading to no longer belonged to Jack's grandfather.\n\nIt belonged to them.\n\nAnnabel looked out her window. They were passing a cemetery. A fat black crow was perched on a granite cross. As Annabel watched, the bird flapped its enormous wings.\n\nShe rolled down the window again and breathed in some more air.\n\nWhen they'd put her in the hospital, the worst thing had been the terrible, confining air. She needed air that flowed freely, and if there was a bit of taxicab exhaust in it, so much the better. How Annabel had wanted to send a chair crashing through the hermetically sealed hospital window and just stand there, her face inches from the jagged glass, and gulp in buckets of air. The sounds of traffic rising from First Avenue would have been far more soothing to her than all that canned hospital Muzak.\n\n\"Annabel,\" Jack said, lifting an eyebrow over at the window and shivering. \"Please?\"\n\nShe rolled it shut.\n\nThey were crossing the Massachusetts line. \"We'll be there in time for lunch,\" Jack said, beaming. \"And Gran's making her famous rabbit stew.\"\n\n\"Jack, you know I've become a vegetarian.\"\n\nHe frowned. \"You can't offend Gran.\"\n\nAnnabel turned away, resisting the urge for air.\n\n\"I'm sure you and Gran will get along just fine,\" Jack said. \"She'll like that you have an eye for color and design. You know the place is going to need a great deal of fixing up, and I'm counting on you, sweetheart, to really give it your signature pizzazz.\"\n\n\"I'm not so sure I'm all that pizzazzy anymore,\" Annabel said, her eyes searching out the window for something. But all she saw were trees and more trees.\n\n\"Oh, come on, sure you are, sweetheart. You haven't lost your eye.\"\n\n\"That wasn't what Carmine thought.\"\n\nCarmine had been her boss at Orbit. The magazine had just launched when Annabel had had her breakdown. She'd been the fashion editor for the premiere issue, but by the time she had come back, after all those months in the hospital, there had been three more editions that had hit the stands. Carmine had brought in what he'd assured Annabel was just a temporary replacement, but the new girl had proven so fresh and hip and original that no one had wanted to let her go. Annabel was offered a \"contributing\" editorship\u2014which was a nice way of telling her that she was no longer needed.\n\n\"I just don't want you to overwork yourself again, baby cakes,\" Carmine had said, trying to sugarcoat his decision.\n\nAnnabel had told him what to do with his contributing editorship and stalked out of his office, breaking a heel on her Manolo Blahnik Sorrita shoe as she did so, much to her humiliation.\n\n\"Well,\" Jack said, \"Carmine doesn't know what he lost by letting you go, sweetheart.\"\n\nBut he did. Carmine had known full well what he was losing. A woman who drank too much, who snorted too much cocaine, who thought the world owed her fame and fortune. Annabel had worked herself to a very lofty place in New York society, but she had burned every bridge she'd crossed to get there. She might have been the city's new fashion darling, but she was tired and angry and frustrated and jealous and insecure. That was hardly the recipe for success in New York. Or success anywhere.\n\nAnnabel had thought that success would drive away all those feelings, which stretched all the way back to her childhood, but in fact success had only made them worse. And so when the day came when the elevator up to the Orbit penthouse got stuck between the forty-first and forty-second floors, Annabel had simply crumbled. The poised, articulate woman dressed in Givenchy and Karl Lagerfeld who had stepped into the elevator late one Friday night had been reduced to a sobbing, quivering mass of jelly on the floor by early Sunday morning, when a janitor discovered the malfunctioning lift. Annabel was once again a child terrified of being eaten alive by a little blue demon named Tommy Tricky.\n\n\"We haven't had any snow yet this winter,\" Jack was saying, drawing Annabel's thoughts back to the present, \"but western Mass can really get sucker punched by nor'easters.\" He smiled over at Annabel. \"So when we get all snowbound, sweetheart, and the power goes off, we're going to use the time to get creative. We can scrape old paint and wallpaper off the walls, and polish the antiques that Gran has down in the basement\u2014there's a mother lode of treasures in that dark old space, Annabel, you'll see.\"\n\nAir. Oh, how she needed air.\n\n\"And we'll go snowboarding and cross-country skiing and skating on the pond\u2014\"\n\nSuddenly, Jack slammed on the brakes and yelled. There was something ahead of them in the road. Something big. Something with antlers.\n\nAnnabel shrieked.\n\n\"Jesus Christ!\" Jack shouted. \"That's a moose! A goddamn moose!\"\n\nThe animal was standing perfectly still in the middle of the narrow, two-lane road. It was at least six feet tall and probably nine feet long, with an enormous head crowned with a rack of sharp antlers. Black eyes stared serenely through the windshield at Annabel and Jack.\n\n\"Holy shit,\" Jack said in a sort of awe. \"I'd heard of moose out in Worcester County, but never out this way before.\"\n\nHe tooted the horn.\n\nAnnabel stared at the creature. Its black eyes unnerved her.\n\nFinally, the moose began to move, plodding the rest of the way across the road and then disappearing into the woods.\n\nJack was grinning like an idiot. \"Who'd ever have thought we'd see a goddam moose?\" he asked as he began moving the car forward again.\n\nAnnabel couldn't speak.\n\n\"Wait'll I tell Morrison,\" Jack was saying. \"He's gonna have a stroke when I tell him we saw a moose!\"\n\n\"I . . .\" Annabel tried again to speak.\n\n\"I wonder if the moose population is spreading west\u2014\"\n\n\"I can't do this,\" Annabel finally said, very softly.\n\n\"What'd you say, sweetheart?\"\n\n\"I said I can't do this,\" she repeated, louder now, looking over at her husband. \"I want to go back to New York.\"\n\n\"Now, Annabel, we've been over this\u2014\"\n\n\"I want go back to New York right now!\"\n\nJack quickly pulled the car over to the side of the road and threw it into park. He turned to face Annabel. \"Now, look, sweetheart. You know this is the only way we can move forward. You know there is nothing left for you in New York. . . .\"\n\n\"Yes, there is! It's my home!\"\n\nJack looked wearily into her eyes. \"We've talked about this many times, Annabel. We made a decision! Dr. Adler helped us make the decision. Remember? And we all decided that the best way for you and I to start over was to leave the city and to accept Gran's offer to take over running the B&B. . . .\"\n\n\"No, you decided. I just agreed. There's a difference.\"\n\nAnnabel turned to look out the car window. How she wanted to roll the window down and climb through it. How she wanted to get out of this car and just start to run. She didn't care where she ran. She'd just run any which way, simply because she could, because she was free. She hated being closed up in a car for so long. It was like being closed up in that hospital . . . which, of course, was like being put in that closet in her stepfather's apartment and told that if she made a sound, Tommy Tricky would hear her and chop her up with his little blue axe.\n\n\"Annabel,\" her mother had told her, \"there's no such thing as Tommy Tricky. Your dad just tells you that so you'll be a good girl.\"\n\nExcept that he wasn't her dad. Her dad, Malcolm Wish, had been killed in the first Gulf War. Annabel's stepfather was a loathsome man she was forced to call Daddy Ron. And he didn't tell her about Tommy Tricky to induce her into being good. He told her about Tommy Tricky because he was a sadist\u2014a terrible, evil man who got his jollies from scaring little children.\n\n\"Tommy Tricky is a little blue boy with a very sharp axe, and he's always hungry,\" Daddy Ron had told her, as he put her in the linen closet. \"He sleeps in here, somewhere under that pile of sheets and tablecloths. So if you make a sound or move around, you'll wake him up. And if Tommy Tricky wakes up, he'll chop you up with his blue axe and eat you up, lickety-split, with his blue lips and blue tongue and big blue teeth.\"\n\n\"Annabel.\"\n\nShe jumped as Jack touched her arm.\n\n\"It was bad in New York,\" he said quietly. \"Do you remember how bad it got?\"\n\nShe nodded, slowly.\n\n\"So this is the only way,\" Jack told her.\n\nAnnabel said nothing more. Her husband put the car into drive and steered it back onto the road.\n\nShe was moving to Woodfield because it was the only way she could keep Jack. He'd leave her otherwise. Annabel knew that. And Jack was all she had left. She couldn't lose Jack, too, not after losing everything else.\n\nShe wanted this move to work. She did want to start over, to find happiness once again in her life and in her marriage. Jack was right that there was nothing left for her in New York. Her friends had all turned on her, put off by Annabel's excesses in those last frantic, hedonistic months before the breakdown. She didn't want that life any longer. She never really had; she'd just gotten caught up in the idea of success, intoxicated by glitz and access. What Annabel wanted was what she'd always wanted, deep down, ever since she was a little girl treated cruelly by her stepfather and ignored by her mother. She wanted a place where she mattered, where she fit in, where she was loved.\n\nAnd where she could move around freely, without any constraints.\n\nMaybe, then, this would be the place for her. True, she sometimes felt a creeping sense of claustrophobia knowing that theaters and museums and couturier shops were hours away\u2014and reachable only by car, and Annabel hated to drive\u2014and that western Massachusetts got very dark at night, and frequently lost power in storms, and was susceptible to being snowbound. But in good weather she could take long walks, or ride a bike into the village. It would be a very different way of life, to be sure, but it needn't be too restrictive. And there was more to recommend Woodfield, too: All of the temptations of Annabel's old life, which had dragged her down to the depths, were very far away.\n\nFinally, there was Jack.\n\nAnnabel glanced over at her husband. How happy he looked. How excited for their new venture. Jack had stuck by her through the worst. He was the only one who had. She owed him this much. But she also had no choice. Jack had been wanting out of the city for the last several years. His own career had stalled as Annabel's had skyrocketed. He had thought he was going to be a big, important writer\u2014but after his first book tanked, he couldn't get another advance. He was done with writing, he said; he hadn't opened his laptop in nearly two years. Annabel knew that Jack would have left New York whether she had agreed to go with him or not. He saw it as the only way forward, for him and for them.\n\nIf Annabel had refused to go with Jack on this new venture, he would have gone without her, and she would have been alone. And by herself, Annabel felt certain she would have slid right back to the depths, and never come back up for air.\n\n\"There it is!\" Jack shouted suddenly, pointing ahead through the windshield. \"You see that roof poking out through the trees?\"\n\nAnnabel tried, but all she saw was a mass of gnarled trunks and limbs, like thousands of skeletal arms reaching up through the soil.\n\n\"We're home, sweetheart, we're home!\"\n\nJack turned up a side road. The SUV rattled over a surface gutted with holes and bumps.\n\n\"Gotta get this road repaved,\" Jack said, more to himself than to Annabel.\n\nAs they rounded a bend, the house came into view. Jack had told her that it was an exceptional survivor of Second Empire French Victorian architecture, and Annabel could see it had once been a very grand house indeed. But now it was quite run-down, with faded, peeling paint that might once have been yellow, but was now a dull gray. Its mansard roof was cut by eight protruding gabled windows and topped by a cupola. A portico supported by two columns ornamented the front door. Around the house, ancient oak trees stood like sentinels, hunched over the structure as if to shield it from decades of wind and rain.\n\n\"I can smell Gran's rabbit stew from here!\" Jack exclaimed, as he turned into the driveway of the house.\n\nAnnabel saw the weather-eaten sign that swung in the slight breeze from an old post. She read it once, and then read it again, as cold fingers played her spine like a xylophone.\n\nTHE BLUE BOY INN\n\nBelow the name was an old engraving of a smiling little boy that looked just like Tommy Tricky.\n2\n\n\"You're going to have to tell him, you know,\" the wizened caretaker told the old woman as they spotted the SUV pull into the driveway.\n\n\"Of course, I know I have to tell him,\" she replied. \"That's the whole reason I offered the house to him. I wasn't about to just put it on the market.\"\n\n\"No, that wouldn't have been a good idea.\"\n\nThe caretaker had a long memory. He'd lived in this house for sixty of his seventy-nine years. He had seen a great deal in that time. He'd become inured to most of it. Nothing much shocked him. But still some images remained seared on his mind. The little pink arm, for instance, resting among the cinders. The caretaker had thought, at first glance, that it had been a doll's arm.\n\nHe had been wrong.\n\n\"He's going to have to tend to the place,\" the caretaker told the old woman. \"Your grandson. He's going to have to do it all eventually. I'm not strong enough anymore.\"\n\n\"I'll explain everything to him,\" the old woman said.\n\nHer eyes narrowed as she watched the automobile come to a stop in front of the house. Its doors opened.\n\n\"But what if he . . . if he's not willing?\" the caretaker asked her.\n\nA queer smile crusted the old woman's lips. \"I know my grandson. He'll be willing.\" She peered through the window. A pretty girl was stepping out of the passenger's side of the car. The old woman frowned. \"I just wish he wasn't bringing a wife with him.\"\n3\n\nGetting out of the car\u2014how good that felt!\u2014Annabel walked slowly over to the sign.\n\n\"You never told me the place was called the Blue Boy Inn,\" she called to Jack.\n\nHe was busy hauling out their luggage from the back of the SUV. \"Yup. It's had the same name for over a hundred years.\"\n\nAnnabel studied the engraving on the sign. The boy was smiling. It wasn't a nice smile. It was the kind of smile that a little demon might make before polishing off a trapped little girl as a tasty snack. In his hand, he carried a gun, but for Annabel it might as well have been Tommy Tricky's axe.\n\n\"Why the gun?\" she asked Jack over her shoulder.\n\nHer husband had sauntered over to stare up at the sign with her. \"It's a musket. The place dates to just after the Civil War. He's supposed to be a Union soldier.\"\n\nAnnabel noticed the gold buttons and epaulets the boy was wearing, faded like the rest of the image.\n\n\"Maybe we should change the name,\" she suggested. \"Put up a new sign.\"\n\nJack put his arm around her. \"Hey, babe, we can't start messing with tradition.\"\n\nHe led her up to the front steps of the house.\n\n\"We'll get our bags later,\" he told her. \"Let's go in and see Gran.\"\n\nThe front door was weathered and flaking with old paint. Jack rapped hard with the old rusted knocker.\n\n\"As a kid, coming to this house was always so much fun for me,\" he said. \"I can't believe it's mine now . . . ours, I mean.\" He grinned down at Annabel.\n\nThe door was opened by a small old man, his face creased in a thousand wrinkles. His eyes were bright and very blue, practically popping out of his gray face.\n\n\"Hello, Mr. Jack,\" the old man said, looking up at them.\n\n\"Zeke!\" Jack exclaimed.\n\nHe shook the old man's hand heartily.\n\n\"This is my wife, Annabel,\" Jack said.\n\n\"How do you do?\" Annabel asked, doing her best to smile down at the old man, though she found it difficult for some reason.\n\n\"Pleased to make your acquaintance,\" Zeke replied, nodding slightly.\n\n\"Zeke taught me how to fish,\" Jack was saying as they entered the house. \"Taught me how to fire a gun, too.\"\n\nAnnabel kept her eyes on the little man. He didn't seem the outdoorsy type. He was frail and slight, hunched over and unsteady on his feet. His skin was a pale yellow, as if he hadn't left the house in decades.\n\nThe place felt damp. The pungent smell of whatever was cooking on the stove\u2014Jack had said it was rabbit stew\u2014permeated the small, low-ceilinged rooms, filling Annabel's nostrils with an unfamiliar aroma she couldn't call either pleasant or unpleasant. Just strong. The house was very dark. Its small windows were nearly obscured by the bushes outside. Ancient, uneven floorboards creaked under foot. Annabel was sure the house was infested with mice. She repressed a shudder.\n\nThey found Gran in the kitchen. She didn't rise to greet them. She remained seated at the old wooden table, a tiny figure dressed in black with a face as pale as her caretaker's. Her shockingly white hair was pulled back from her face and knotted in an untidy bun at the back of her head. Hands like talons rubbed each other as she watched them enter.\n\n\"Gran!\" Jack exclaimed, rushing over to embrace the old woman.\n\n\"Welcome home,\" she said to him, in a low, yet surprisingly girlish voice.\n\n\"This is Annabel,\" he told her, gesturing to his wife to join him.\n\nAnnabel extended her hand. \"I'm very happy to meet you, Mrs. Devlin.\"\n\n\"You must call me Cordelia,\" the old woman told her, taking Annabel's hand and squeezing it tight.\n\nAnnabel smiled. \"All right, Cordelia. Thank you.\"\n\n\"I had Zeke air out your room,\" the old woman said. \"I hope it will be all right.\"\n\n\"I'm sure it's fine,\" Jack replied.\n\nCordelia sighed. \"It's an old house. There are a lot of cobwebs. We don't quite have the strength to keep up with it the way we used to. During the season, we've been hiring some college students to come in and give us some help. But now that you're here . . .\"\n\nJack was beaming. \"We'll get the place in tip-top shape!\"\n\n\"In fact,\" Annabel said, \"I'm a designer. I'd love to maybe look at opening up the space a little bit. The rooms are so small. If we open things up a little bit\u2014\"\n\n\"I wouldn't go around opening things up willy-nilly,\" Cordelia said, cutting her off. Her old blue eyes shone in Annabel's direction. \"The architecture of the house is fragile. You wouldn't want to open something up and find the whole place falling down on you.\"\n\nAnnabel smiled. \"Of course not. We'll definitely work with the blueprints.\"\n\n\"Don't worry, Gran,\" Jack said. \"We're not coming in with a wrecking ball. We'll respect tradition. That's what you said in your letter to me. That there was a tradition here and that you'd give us the house if we respected it. And we intend to.\"\n\nThe old woman smiled.\n\nAnnabel looked away. She had the sudden sensation of claustrophobia. The rooms were so small here, so dark. The ceilings were so low. But it was worse than that. She had the sudden fear that she had been lured into this place with false promises. They would make it their own, Jack had promised. They could start over, build something that was theirs. But if Cordelia was always going to be there, overseeing things, nixing ideas and squelching their creativity, then what sort of life would this be? Once again, Annabel wanted to run. She wanted to throw open the back door and run out through the woods until she found the road, and then hitchhike her way back to New York.\n\nBut there was nothing in New York for her anymore. All her bridges back into the city had been burned, all her tunnels filled in with cement. She was on her own.\n\nAt the Blue Boy Inn.\n\nThis is where Tommy Tricky lives, she heard Daddy Ron tell her.\n\nHow badly Annabel wanted to run.\n4\n\nPriscilla Morton thought of herself as a ghost hunter. Ever since she'd been a kid, she'd searched out every supposedly haunted house or mysterious graveyard she could find, traipsing all over the south of England, where there were plenty such places. When she and her boyfriend, Neville, decided to take a two-week holiday to the United States, Priscilla had insisted that their first week be spent visiting different haunted inns. She'd discovered that there were as many of them in New England as there were in Old England. Neville, who thought Priscilla's ideas about ghosts were nonsense, had agreed, only if their second week could be spent in Florida, at Disney World.\n\nPriscilla had told him they had a deal.\n\nThat didn't mean Neville was suffering that first week gladly.\n\n\"Really, Priscilla,\" he was saying, driving north on Interstate 91 through Hartford, \"it's freezing cold out there. What kind of holiday is this? If we'd wanted cold and gray skies, we could have stayed in England.\"\n\n\"You have to admit that inn last night was worth it,\" Priscilla replied. \"I heard the wailing, just as the innkeeper promised.\"\n\nHer boyfriend scoffed. \"That was just the wind. Really, you believe anything.\"\n\nThe place in Connecticut had dated from 1799. Two centuries ago a woman had been killed in the room they'd stayed in, and legend had it that her screams still sounded through the house, waking guests and sending them running for the manager. Priscilla had heard the poor woman's wails and had sat straight up in bed. She'd woken Neville, snoring like a bear beside her, but he'd only grunted and gone back to sleep.\n\n\"I'd say we got our money's worth on that one,\" she said, looking down her list of haunted guesthouses. And it was a good thing, too. The first two places, one in Rhode Island and the other near New Haven, had been busts. No screams, no apparitions. Neville couldn't wait to get to Florida.\n\n\"The temperature this morning is eighty-five degrees Fahrenheit in Orlando,\" he told her. \"We could be sitting by a pool right now, sipping margaritas.\"\n\nPriscilla stuck her tongue out at him. She wasn't sure what she saw in him. He wasn't very handsome, with kind of a pimply face and stringy hair and a gut that hung over his belt. And he could be such a stick in the mud. Neville didn't like most of the things that Priscilla liked, be they ghost hunting or bird watching or kayaking. All Neville liked was to watch football and drink beer. But he was good to her, Priscilla supposed. He put up with her. Her mother had always said it would take a rare man to put up with Priscilla's eccentricities. She was only twenty-four, but she knew that sometimes she acted older. Like how she'd preferred quilting bees to rock concerts growing up. Priscilla had never been into television or fashion or rap music or any of the sorts of things other young girls liked. Even now she would rather go on an architectural tour of old churches than go out to a pub. She and Neville had been dating for three years. Priscilla doubted they'd ever get married. But that was okay by her. She didn't think she was the marrying type.\n\nNeville was older than she was, by almost ten years, and Priscilla supposed he stayed with her because, all of her quirks aside, he figured he couldn't do much better. In fact, he'd done very well. For all her peculiarities, Priscilla was quite pretty, with long, silky blond hair, full breasts, and a tiny waist. If not for the thick, black, oversized eyeglasses she wore, she might have been mistaken for a blond bombshell. She figured if she put her mind to it, she could get a much hotter guy than Neville. But she didn't have time to put her mind to such things. She'd much rather concentrate on ghost hunting or gravestone rubbing.\n\n\"Where is this next place we're heading for?\" Neville asked, steering the car past the glittering skyline of Hartford. \"It's in Massachusetts?\"\n\n\"Yes,\" Priscilla replied. \"Just keep north on this road. When you get to the Massachusetts Turnpike, you'll go west.\"\n\n\"How many miles?\"\n\n\"I don't know. I can't read American maps all that well. But I'll keep navigating.\"\n\nNeville grunted. \"We'd better see a real ghost this time, covered in chains. If not, we're heading out early to Florida.\"\n\n\"It's not my fault that you wouldn't wake up last night to listen to the wailing of that poor dead spirit.\" Priscilla shuddered. \"It was absolutely spine-tingling.\"\n\n\"You're crazy. I heard the wind. That's all it was.\"\n\nPriscilla just snorted. Neville was such a skeptic. Really, what was she doing with him?\n\n\"So what kind of ghosts are at this place we're headed to?\" he asked her.\n\nShe riffled through the brochures on her lap. She found the one that described their destination.\n\n\"The Blue Boy Inn,\" she read out loud. \"Dates back to the American Civil War.\" She read a little further along. \"Oh, wow, there could be a lot of ghosts here.\"\n\n\"What do you mean by a lot?\"\n\n\"It says here that there have been several murders at the inn. And some mysterious disappearances.\"\n\nNeville snorted. \"Oh, that's probably just hype.\"\n\n\"No, it's real,\" Priscilla said, reading the history of the place. \"The first murder was a girl named Sally Brown. Nearly a hundred years ago. They never found her body, just her blood splattered all over the walls.\"\n\n\"Oh, goody, let's stay in that room,\" Neville quipped. He was being facetious. But Priscilla really hoped they got booked into Sally Brown's old quarters.\n\n\"Then there was this guy Andrew McGurk, whose body was found, but not his head.\" She shivered. \"And then\u2014oh, this is terrible\u2014a little baby, who disappeared. The only thing they ever found of her was her arm.\"\n\n\"I don't like baby ghosts,\" Neville grumbled. \"I imagine they cry all the time.\"\n\n\"And there's more,\" Priscilla said, glancing down at the description of the Blue Boy. \"Quite a bit more.\" She looked over at Neville. \"I think we've hit the jackpot with this place. I mean, if we're looking for ghosts, this is the place that will have them.\"\n\n\"I can hardly wait,\" Neville said, as he drove the car north on 91.\n5\n\nThe arthritis in Zeke's knees and hips burned like a thousand wasps stinging him all at once. Each step up the narrow stairs to the attic was agony. He couldn't do this much longer.\n\nIt was time to turn this job over to someone else.\n\nAnd who better than young Mr. Jack?\n\nZeke lifted the old rusted iron key from the ring he carried on his belt, the one that had been jangling against his side all the way up the stairs. He slipped it into the keyhole on the door.\n\nHow many times had he done this particular job? Impossible to count. It seemed all his life. But it wasn't all his life. There had been a time before Zeke had come to this house, though he could hardly remember that life now, when his life had seemed full of promise. But that was a long, long time ago. Half a century Zeke had been tending to this place. And for almost half of that time, he'd been making this trip up to the attic, twice a day.\n\nHe turned the key in the lock.\n\nDownstairs, Mr. Jack and his pretty wife were settling in, unpacking, freshening up. Oh, they had such plans. They were going to clean the place up. Modernize it. The woman spoke of getting something called \"wireless internetting,\" whatever that was.\n\nA little smile played across Zeke's lips.\n\nThey could plan all they wanted. They'd soon learn it wasn't they who made decisions about the house.\n\nHe opened the door.\n\nHe listened.\n\nHe heard the sound then in the dark, cobwebby room.\n\nThe panting.\n\nZeke stepped inside and closed the door behind him.\n6\n\nAnnabel stood gazing out of her window into the cold bare limbs of the trees that surrounded the house. They looked like some sort of petrified aboriginal humans, frozen in the midst of some terrible cataclysm, staring up at the Blue Boy Inn with their arms outstretched to the sky.\n\n\"Lots of possibilities, don't you think, sweetheart?\"\n\nJack had come into the bedroom behind her.\n\n\"I think if we knock this wall down here,\" he was saying, \"we can open up the room to include the bathroom in a sort of master suite. That way we don't have to mingle with the guests in the hallway when we first get up in the morning.\"\n\nAnnabel didn't say anything. She just kept staring out into the trees.\n\n\"What do you say, baby cakes? Isn't that a good idea? You're the one with the artistic eye.\"\n\nShe turned around to face him. \"I think it's a splendid idea,\" she said, trying to smile. \"But I thought you wanted to respect tradition.\"\n\n\"I do, sweetheart, but you know I was mostly saying that just to placate Gran.\" Jack smiled broadly, his cheeks indenting with dimples, and he took her in his arms. \"This has been her home for a long, long time. I didn't want her to think that we were going to start pulling it down around her.\"\n\n\"It needs a lot of work, Jack,\" Annabel told him. \"Much more than I thought. So many walls and floorboards need to be replaced. The plumbing and electricity needs to be updated.\"\n\n\"I know, babe. One step at a time.\"\n\nShe shuddered and closed her eyes. She imagined she was at Fifth Avenue and Broadway, and the sound of honking taxicabs and police sirens filled her ears. It made her feel better, at least for a few moments.\n\nShe gently slipped out of her husband's embrace. \"If I'm going to do this, I'll need your support, you know.\"\n\n\"You have it!\"\n\n\"Are you sure?\"\n\nShe looked at him. Jack's big blue eyes seemed filled with sincerity and purpose. They'd been madly in love once, five years earlier, when they'd met at a party and spontaneously married eleven days later, with no friends or family in attendance, just a justice of the peace and Jack's high school ring as a wedding band. Since then, some of the impulsivity of their union had faded, and Annabel's crises over the past year had severely strained their marriage. She didn't blame Jack for having a fling with Rachel Riley, one of her colleagues at Orbit. It had been just a one-night thing, when Annabel was locked away and Jack was lonely. He'd confessed to Annabel when she got out, apologizing profusely. She'd forgiven him. It was understandable.\n\nBut she could barely stomach looking at Rachel Riley, with her ridiculously blond hair and big boobs. She'd been Annabel's friend. Supposedly.\n\nBut in every other way, Jack had stuck by her. After the affair with Rachel, he had become even more devoted to making sure Annabel got better. So many other men might have walked away, but Jack didn't. Even during the period when she was coming home from fashion shoots strung out on coke, screaming and ranting and craving more blow, Jack had put up with her. He had calmed her down. He had brought her down from the high without letting her crash. When the time came for rehab, Jack had been right there, lending Annabel support and encouragement. Even as his own dreams of success had withered away, he hadn't given up on Annabel's.\n\nJack had thought he was on the fast track to the big time. In those heady first months after their hasty marriage, they'd imagined themselves the Next Big Power Couple. How excited he had been when a big publishing house bought the novel he'd been laboring over since college. It was a deep, involved story of a young man and his search for meaning in a world that was increasingly impersonal and commercial. Annabel thought that was ironic, given that Jack was always saying what he wanted most from his book was a contract for a Hollywood blockbuster so they'd get rich, rich, rich. When that hadn't happened\u2014after much advance publicity, the reviewers had called the book \"tedious\" and \"pretentious\"\u2014Jack had been devastated.\n\nAnd it was just at that moment that Annabel's career had started its dizzying ascent. Her eventual crash was even more spectacular than Jack's.\n\nNow here they both were, in the middle of the woods, miles and miles from civilization, in a place called the Blue Boy Inn.\n\nWhere Tommy Tricky lives, Annabel thought.\n\nShe smiled over at Jack.\n\n\"I didn't mean to doubt you,\" she said. \"I know you support me. I couldn't have gotten through everything without you.\"\n\nJack beamed, leaned over toward her, and kissed her on the forehead.\n\n\"I'll see you downstairs, hon,\" he said. \"You really ought to try Gran's rabbit stew. I know you don't want to eat bunnies, but, really, it's out of this world.\"\n\n\"I'll pass on it for now,\" she told him.\n\nHe winked at her and bounded out of the room.\n\nAnnabel looked around. How small the room was. So square and the ceilings were so low. The whole place smelled like old, wet wood. And rabbit stew. Annabel shivered.\n\nShe would never last here.\n\nBut she had to. There was nowhere else.\n\nNo other choice.\n\nShe would make the best of it. She would redesign this place. It was theirs, after all. The old woman was signing over the property to them. After that, they could do what they liked. Annabel needed a project. She could do this. She could bring in carpenters and painters and electricians. She still had some contacts over at the HG television network. Maybe she ought to pitch them a reality show set in the woods of western Massachusetts, as a former New York socialite tries to remake an old house....\n\nAnd her life.\n\nNo, Annabel didn't want cameras around for that.\n\nShe looked out the window again, at the gnarled branches so close to the house. It's like they're trying to suffocate us, Annabel thought.\n\nWhen was the last time she and Jack had made love?\n\nThe thought struck Annabel suddenly and unexpectedly. She paused. She couldn't remember. Yes, wait, now she could. It had been three weeks ago. Right after he'd gotten the call from his grandmother. Jack had been so excited by the idea. He'd started kissing Annabel all over the face. \"This is it, sweet cakes!\" he had shouted. \"The answer to our doldrums! Our new path! Our way out of the city! We're going to be huge successes there. Just you wait and see!\"\n\nThe fact that Annabel hadn't wanted to leave the city was immaterial. She had been bulldozed by Jack's enthusiasm. And by his amorous advances. His big hands had suddenly been all over her. She hadn't wanted to make love that day, but Jack had insisted. Annabel had given in, and then, while he was inside her, she had started to cry, wanting to enjoy it, wanting to love sex the way she used to\u2014wanting to love Jack the way she used to. How much Annabel wanted to love everything in her life the way she once had in days gone by\u2014before the drugs and the breakdown and the humiliation.\n\n\"Baby doll,\" Jack had said, looking down at her, red-faced and puffing. \"Why are you crying?\"\n\nShe had replied that the tears came from her orgasm. That had been a lie.\n\nThat was also the last time they made love.\n\nHe might want to, tonight, she thought to herself. To celebrate our first night here.\n\nAnnabel dreaded the prospect. Not that she wanted to withhold sex from Jack. In fact, she wanted very much to make love to him, to feel his arms around her, to feel happy and content in his embrace the way she used to. She craved that feeling.\n\nBut she knew she couldn't feel that way. Not here. Not in this house.\n\nA whiff of the rabbit stew reached her nostrils again, and nearly made her vomit. It smelled sickeningly sweet, like a dead mouse rotting under the stove.\n\nThis is never going to work, she thought to herself.\n\n\"Annabel, you have got to believe in yourself,\" her mother always used to tell her. \"You're such a timid little kitten. You need to believe you can do whatever you put your mind to doing.\"\n\nThat had always been a challenge for Annabel. Even when she'd been on top of the world, New York's latest fashion and design darling, she'd doubted herself. It was why she had turned to coke and booze. She'd needed to feel confident\u2014and she felt confident when she was high. But when she came down, she was right back to feeling unsure and timid again, so she had searched out more white powder to sniff up her nose. It had been a vicious cycle. No wonder she had burned out so quickly.\n\nBut she had to believe in herself now. She had to make this venture work.\n\nShe smiled as she looked at herself in the mirror. She tied her chestnut-colored hair back in a ponytail. Her big brown eyes disclosed her lack of sleep these past several weeks. Her face was a little fuller now than it had been during the worst of her addiction, but she could still stand to put on a few pounds. She wore no makeup. She reached into her purse and pulled out a lipstick, rubbing a very light pink on her lips.\n\n\"You can do this,\" she told her reflection.\n\nRight away, she doubted her own words.\n\nIn rehab, they had tried pumping her with self-confidence, cheering her on. There had been therapists and psychologists whose jobs had been merely to instill in Annabel a sense that she mattered, that she was powerful. They were successful enough that she had been able to leave, to return home to Jack, to be able to handle the cravings when they came, and to finally feel free of them. Annabel didn't want coke anymore. She wanted something else, however, though that was harder to identify. Happiness, she supposed, but that felt like asking for too much.\n\nShe had an image then, as she gazed at herself in the mirror, of her mother's body hanging behind her.\n\nShe didn't jump. She saw it sometimes. Just like the day she had come in from school, aged sixteen, and found her mother hanging in the dining room.\n\nThe same mother who had told Annabel to believe in herself.\n\nAnnabel closed her eyes, opened them again, and the image was gone. That was the way it worked.\n\nBut now she spied something else behind her in the mirror.\n\nSomething that moved.\n\nAnnabel spun around. There was nothing there. Whatever she had seen was small, close to the floor, maybe a cat or a dog, moving quickly through the small, shadowy space between the bed and the dresser.\n\nA squirrel? Annabel asked herself.\n\nShe took a step toward the spot where she had detected the motion. Had whatever it was gone under the bed?\n\n\"I wouldn't be surprised if this place is infested with vermin,\" Annabel said out loud, wondering whether she ought to stoop down and peer under the bed herself or go get Jack. She was a city girl, after all. In Central Park, she'd always been afraid of the squirrels.\n\n\"Believe in yourself, Annabel,\" she told herself, and laughed.\n\nShe knelt down on the old, warped wooden floorboards. She lifted the bedspread and peered under the box spring.\n\nNothing. Just clumps of dust.\n\nAnnabel sighed.\n\nShe must have imagined she saw something. It must have been just the shadow of a tree branch moving outside. The day was becoming breezy, after all.\n\nAnnabel stood back up.\n\nShe looked over toward the door.\n\nAnd there, grinning malevolently up at her, was the tiny blue figure of Tommy Tricky, gnashing his long, sharp, blue teeth.\n7\n\nZeke heard the woman's scream as he was locking the door to the attic. He hurried down the steps as best he could, his arthritic joints screaming at him, and made his way to Annabel's room.\n\nHe hadn't expected it to start this soon.\n\nThe caretaker found the new mistress of the house standing up against the wall in the corridor, her face the color of flour, and her body shuddering as if she was having a seizure. Zeke reached out a hand to steady her.\n\n\"Miss Annabel,\" the old man said. \"What's wrong?\"\n\n\"I saw something,\" she managed to say.\n\n\"What did you see?\"\n\n\"A boy. A little man.\"\n\nZeke narrowed his rheumy eyes at her. \"There's no little boys here.\"\n\n\"I saw him,\" Annabel said, still trembling.\n\n\"What was he doing?\"\n\n\"Smiling.\" She burst into tears and covered her face with her hands. \"Teeth. His teeth!\"\n\n\"A little boy smiled at you?\"\n\nZeke kept his eyes trained on Annabel.\n\n\"Tell me exactly what you saw,\" he said.\n\nBut she was too overcome now. She just sobbed into her hands.\n\n\"What's going on?\" came the voice of Mr. Jack, heading down the hallway toward them. He seemed more annoyed to be pulled away from his rabbit stew than concerned about his wife.\n\n\"Miss Annabel said she saw a little boy,\" Zeke told him,\n\n\"A little boy?\"\n\n\"Oh, Jack,\" she said, and threw herself into his arms. \"I saw him. The blue boy . . .\"\n\n\"The blue boy?\" her husband asked.\n\n\"You mean, the boy from our sign out front?\" Zeke asked.\n\n\"And then he disappeared!\" The color had returned to Annabel's face. Now she was all red and purple from crying. \"He ran down the hall and somehow just disappeared.. . .\"\n\n\"Annabel,\" Jack said, stroking her hair, \"sweetheart, you had a hallucination. . . .\"\n\nShe looked up at him. Zeke saw them exchange a glance that carried some meaning. Apparently, this pretty young lady had had hallucinations before.\n\nThat could prove convenient.\n\n\"It's okay,\" Mr. Jack told Zeke. \"My wife has a vivid imagination sometimes.\"\n\n\"Imagination is good for the soul,\" the caretaker replied, \"but not when it scares you half to death.\"\n\n\"She'll be okay, now, won't you, baby cakes?\" Jack asked, gently moving Annabel back into their room. He shot Zeke a glance. \"Tell Gran I'll be back down in a minute.\"\n\n\"Yessir, Mr. Jack.\"\n\nThe old man shuffled down the hall back toward the kitchen.\n\nCordelia was waiting for him. Her intense blue eyes seemed to burn holes in her pale face. Her gnarled hands were opening and closing.\n\n\"What has happened?\" she asked, in that voice that seemed so much younger than the face from which it came.\n\n\"She saw a little boy,\" Zeke told her.\n\n\"A little boy?\"\n\n\"A little blue boy,\" the caretaker added.\n\nCordelia turned away. \"I wish he hadn't brought her,\" she said.\n\n\"You can't expect a man to leave his wife behind.\"\n\nShe turned her eyes back to him. \"Why not? My husband did. My son left poor Jack.\"\n\n\"That's what happens here,\" Zeke said, rather matter-of-factly.\n\nCordelia sat back down at the table. \"By the way,\" she said. \"You'll need to prepare one of the rooms.\" She paused. \"We've got guests arriving.\"\n8\n\nAnnabel started the car and backed out of the driveway. She wasn't very good at driving; there was never a need in the city. But she'd gotten her license because sometimes she'd needed to drive when they'd go out on photo shoots in the Hamptons or in Connecticut. She didn't necessarily enjoy being behind the wheel of a car, but right now the idea of getting on the road, away from the house, seemed blissful.\n\nJack had agreed it was a good idea for her to go down to the market and get some groceries. \"It will clear your head,\" he'd told her.\n\nAs a vegetarian, Annabel was going to need some other provisions in the refrigerator besides leftover rabbit stew if she was going to survive. There was a market down at the end of the road, and Jack had told her to buy everything she wanted.\n\nWhat she wanted was freedom. But Annabel couldn't figure out a way to buy that.\n\nShe steered the SUV down the twisting country lane. The misshapen trees on either side of the road terrified her. They reminded her of Cordelia's arthritic hands. She tried to concentrate on her driving, but Annabel's heart was still thudding in her ears from the scare she'd had.\n\nI saw Tommy Tricky.\n\nShe let out a deep breath.\n\nNo, she told herself. It was not Tommy Tricky. It was another hallucination, like the ones she'd had during rehab and immediately after. Her therapists had found she was prone to hallucinations. She would begin to imagine that nothing was safe around her. Her therapist, Dr. Adler, had kept telling her, over and over, \"You are safe, Annabel. Nothing can hurt you.\" But she had gone through periods where she had been absolutely convinced that she was unsafe\u2014that everything and everyone around her was out to get her.\n\nGetting off the drugs hadn't been easy. There were times she'd thought she was going mad. She saw snakes coming through the floorboards. She'd thought the apartment was on fire one horrifying night. None of it had been real, and the little blue boy she'd seen wasn't real, either.\n\nOf course, I'd hallucinate about Tommy Tricky. I was upset and anxious, worried about the move. I was feeling claustrophobic. And I'd just seen that horrible sign out front.\n\nAnnabel was going to replace that sign no matter what anybody said. Screw tradition. She wanted it down.\n\nHer stepfather had been a sadistic son of a bitch. How could he have terrified a little girl the way he did? Whenever he wanted to get a rise out of Annabel, he'd say, \"Watch out for Tommy Tricky!\" How she wished her mother had put a stop to it. But her mother was weak. She had just let that horrible man continue torturing her daughter. \"Look behind you,\" Daddy Ron would say. \"I think I see ol' Tommy creepin' up on ya.\"\n\nAnd then he'd laugh\u2014a giant guffaw\u2014as Annabel would start to cry.\n\nWhen Daddy Ron had been drinking, it was even worse. He got angry so quickly after he'd had a few beers. He looked for excuses to punish Annabel. One time, when her mother had gone out, Annabel had dropped the milk carton as she was putting it away. It had spilled on the kitchen floor, an ocean of milk spreading across the tiles. The next thing she knew her stepfather was screaming at her. He grabbed her by the shirt and dragged her down the hall to the linen closet. He shoved her inside and locked the door. Annabel had sat in the darkness, knees pulled up to her chest, sobbing as Daddy Ron taunted her from outside.\n\n\"Tommy Tricky is in there with you! He's got his sharp little axe! He likes to chop up bad little girls and eat them for lunch!\"\n\nMom had let Annabel out of the closet when she got home, hugged her briefly, and then told the little girl not to make Daddy Ron angry anymore.\n\nUp ahead, Annabel spied the market. It was a small wooden structure fronted by big glass windows. A sign above the door read FALLS GENERAL STORE. Annabel pulled into the parking lot and cut the engine.\n\nShe took a deep breath, opened the car door, and headed inside. A bell over the door rang as she passed through.\n\n\"Good afternoon,\" the lady behind the counter sang out.\n\n\"Good afternoon,\" Annabel replied.\n\nThe woman was large\u2014not fat, just big-boned and tall, with broad shoulders and wide hands. Her hair was gray, worn long, and her face was friendly. She was probably sixty, but her skin was entirely smooth and unwrinkled. Her eyes were periwinkle blue.\n\n\"May I help you?\" she asked.\n\nAnnabel smiled. \"Just moved in with my husband's grandmother. And I'm a vegetarian. Turns out there wasn't a lot in her fridge that I could eat.\"\n\nThe woman laughed. \"Well, right now we don't have a lot of fresh produce. We tend to only sell what's grown locally, so you'll have to wait a few months before we'll have tomatoes and corn and green beans and sweet peas and carrots. . . .\" She smiled as she came around from behind the counter. \"But I think we can stock you up with some of the best canned goods and preserves.\"\n\n\"Thanks,\" Annabel said, returning the woman's smile. \"Even frozen is fine.\"\n\n\"I'm Millie,\" the woman said.\n\n\"Annabel.\"\n\n\"Pleased to meet you. Where did you move here from?\"\n\nShe was pointing Annabel to an aisle filled with canned vegetables and fruits. Most of it was local stuff, preserved right here in western Massachusetts. Lifting a handbasket, Annabel began filling it up with various cans and jars. She tossed in a few boxes of whole wheat pasta as well.\n\n\"New York,\" she said, replying to Millie's question.\n\n\"The city?\"\n\nAnnabel nodded.\n\nMillie laughed again. It was a small, tinkly sound for such a big woman. \"Well, you're going to be in for some culture shock up here. How long you staying?\"\n\nAnnabel sighed. \"For good,\" she said.\n\nMillie raised her eyebrows.\n\n\"My grandmother-in-law is rather frail. She can't keep up the place by herself anymore, so she's asked my husband and me to take over the house.\"\n\nMillie folded her masculine arms across her chest. \"So you're really okay moving out of cosmopolitan Manhattan for this little hole in the woods in the middle of nowhere?\"\n\nAnnabel gave her the most convincing smile she could manage. \"So long as I can find other things to eat than Gran's rabbit stew.\"\n\nSomething in Millie's eyes changed. Her brows furrowed as she studied Annabel.\n\n\"Something wrong?\" Annabel asked.\n\n\"What's your grandmother-in-law's name?\"\n\nAnnabel returned her odd stare. \"Cordelia Devlin,\" she said.\n\nMillie opened her mouth to say something, and then stopped. She tried again. \"And the house you've taken over,\" she said slowly, \"is the Blue Boy Inn.\"\n\n\"That's right.\"\n\n\"Well, well,\" Millie said, hugging herself tighter. \"So the old place is going to be given a new lease on life.\"\n\nAnnabel smiled widely. \"That's what my husband and I are hoping. We're going to fix it up, modernize it, get some new technology in there. . . .\"\n\n\"Technology?\"\n\nAnnabel placed a jar of peanut butter and some breadsticks into her basket. \"Well, the place doesn't even have any flat-screen TVs, let alone any Internet. I think a guesthouse needs some amenities, even if it's out in the woods.\" She thought about it. \"Actually, especially if it's out in the woods.\"\n\n\"Well, most of the people who still come to the Blue Boy aren't coming for cable television and Facebook.\" Millie unfolded her arms. \"As I'm sure you're aware.\"\n\nAnnabel looked over at her. \"You mean they come to get away from all that?\"\n\n\"Maybe, but not necessarily,\" the storekeeper replied. \"They come to the Blue Boy because they're looking for ghosts.\"\n\nJust then, the bell over the door jangled. Annabel was still struck by what Millie had just said, so she didn't turn to look, but Millie did.\n\n\"Hello, chief,\" she sang out.\n\nAnnabel moved her eyes over to observe the newcomer. He was a tall, dark-haired man with a craggy, handsome face, maybe about forty, dressed in dark dungarees and a brown corduroy jacket. He gave Millie a little salute.\n\n\"What can I help you with today, chief?\"\n\n\"Just a quart of milk, Mil,\" he said. \"Ran out last night and had to eat my Cheerios dry.\"\n\nMillie turned to look back at Annabel. \"That's what he eats for breakfast and for dinner. Cheerios. The man needs a good woman who will cook for him.\"\n\nThe man laughed. \"I've asked you a million times to marry me, darlin', but you always turn me down.\"\n\n\"And take a look at him!\" Millie said, still talking to Annabel. \"Movie star handsome. But still unattached.\"\n\n\"I'm married to my badge,\" he said, placing the milk down on the counter.\n\nMillie moved around to ring him up. \"And isn't Woodfield fortunate to have such a dedicated chief of police.\" She dropped the milk into a small paper bag. \"Hey, chief, meet the new girl in town. Just arrived.\" Millie fixed him with a look. \"She and her husband are taking over the Blue Boy Inn.\"\n\nThe chief looked around at Annabel for the first time. For some odd, unexplainable reason, she blushed.\n\n\"Is that right?\" he asked. \"Cordelia's giving the place up?\"\n\n\"She's my husband's grandmother,\" Annabel explained. \"And she's asked us to come run the place. She can't do it on her own anymore.\"\n\n\"Doesn't she still have Zeke around to help her?\"\n\n\"Yes, he's still there. But he's rather old as well.\"\n\nThe chief smiled. \"Older than the earth, it seems.\" He moved closer to Annabel and extended his hand. \"Well, welcome to town. I'm Richard Carlson. If there's anything I can do for you or your husband, do let me know.\"\n\nShe shook his hand. \"Thank you,\" she said. \"I'm Annabel Wish.\" She was struck by how dark the chief's eyes were. Almost black.\n\n\"Enjoy the rest of your day, Millie babe,\" the chief said, heading out of the store.\n\nThe little bell over the door rang again.\n\nAnnabel watched him go through the large windows. He slid into a plain black car, probably a Ford. No cruiser. Annabel imagined he must have been off duty, since he hadn't been wearing a uniform.\n\nShe brought her basket of provisions up the counter.\n\n\"Oh, you'll like these,\" Millie said, lifting a couple jars of raspberry preserves. \"I know the lady who cans these. Grows all her own berries on her farm. In the summer you can go up there and pick your own. Raspberries, strawberries, blueberries . . .\"\n\n\"I'll keep it in mind,\" Annabel said. \"But I'm not much of a berry picker, except from the display at Whole Foods.\"\n\nMillie frowned. \"You're going to have a lot of adjustments living here, sweetie.\"\n\n\"I know.\" Annabel paused. \"Millie, what did you mean when you said people come to the Blue Boy to see ghosts?\"\n\nThe storekeeper stopped what she was doing and fixed her with her blue eyes.\n\n\"You don't know?\"\n\n\"Know what?\"\n\n\"It can't be that your husband never told you about the murders.\"\n\nAnnabel's blood went cold.\n\n\"Murders?\" she asked. \"What are you talking about?\"\n\n\"Well, if he hasn't told you, then it's probably best that you ask him.\"\n\nAnnabel suddenly felt frantic. \"No, please, tell me what you know.\"\n\nMillie shrugged. \"It's not like I'm telling tales out of school. Everybody up here knows the history of the Blue Boy Inn. The only reason it stays in business is because of the ghost tourists. People think it's haunted because of all the deaths that have taken place there over the years.\"\n\n\"And these deaths were . . . murders?\"\n\nMillie was placing Annabel's groceries in a paper bag, one much larger than the one she'd just given to Richard Carlson. \"Well,\" she said, \"not all, probably, but some of them definitely were. Like, for example, you don't accidentally get your head cut off.\"\n\n\"Oh, dear God,\" Annabel said.\n\n\"Look, honey, I'm sorry to be the one to break this to you. I can't believe you could move up here and not know. You should go right back up there and get your husband to tell you everything.\" Millie's eyes were kind, but also serious. \"Because there's no way he doesn't know. One of the deaths up there, a long time ago, was Cordelia's young granddaughter. And if I'm figuring correctly, that would have been your husband's sister.\"\n9\n\n\"It's up here, turn here!\" Priscilla shouted at Neville. I\"That little lane, there!\"\n\n\"Never would have spotted the bloody road,\" Neville grumbled, turning the car up the rutted passageway through the trees.\n\n\"Yes, very easy to miss,\" Priscilla agreed. \"Hidden away in the woods. The way all haunted houses should be.\"\n\nHer boyfriend smirked over at her. \"Have I told you how excited I am to get to Florida?\"\n\n\"Yes, six thousand times. Pull in over there. Next to the sign.\"\n\nThe Blue Boy stared down at him with his faded-paint eyes.\n\n\"Creepy, isn't he?\" Priscilla said.\n\n\"I'd say the place is what's creepy,\" Neville replied, shutting off the car. \"Looks like it hasn't been updated in decades.\"\n\n\"Perfect,\" Priscilla chirped, hopping out of the passenger seat. She stood gazing up at the old inn. \"I can feel the vibrations, can't you?\"\n\n\"All I can feel are hunger pains. You wouldn't let me stop at that McDonald's back on the highway. Hope this place has something to eat.\"\n\nNeville withdrew their two bags from the trunk, and then clicked the remote to lock the car. A series of two quick, high-pitched beeps followed.\n\n\"I smell something cooking,\" Priscilla said, lifting her nose in the air.\n\n\"Probably human flesh,\" Neville muttered.\n\nThey headed up the walk to the front door. There were no other cars in the gravel driveway.\n\n\"Oh, there's somebody,\" Priscilla said. \"Over there, coming out of the woods.\"\n\nThe trees had grown thick around the house. Only a few patches of sunlight shone through here and there. The deciduous trees might have been bare of leaves, but their gnarled limbs had all tangled together so tightly that they blocked out the sun in many places. And there were lots of tall pine trees as well, leaving the Blue Boy Inn mostly shrouded in shade and shadows.\n\nSo it was hard to see the person emerging from the woods about thirty or so yards away, but Priscilla was trying to wave whoever it was down. It was possibly the proprietor.\n\n\"Hello!\" Priscilla called, taking a couple of steps in the direction of the figure. \"Hello, we have a reservation to stay here!\"\n\n\"Maybe it's just another guest,\" Neville said. \"Let's just go up and ring the doorbell.\"\n\nPriscilla frowned. \"There are no other cars here. It can't be a guest! Hello!\"\n\nShe waved her hand to catch the person's attention.\n\nIt was a woman, she could see now. The hair was long and possibly blond or gray. She was wearing some kind of white, diaphanous dress....\n\n\"Hello!\" Priscilla called again.\n\nThe woman finally turned in her direction.\n\nAnd Priscilla let out a gasp.\n\nThe woman's face was covered in some kind of dark substance.\n\nIt could only be blood.\n\nPriscilla screamed.\n10\n\nAnnabel heard the woman scream as she steered the SUV back up the driveway. There was a small red car parked at the inn now, and a man standing near it holding two suitcases, and a woman a few feet away, screaming and pointing toward the woods.\n\nAnnabel hopped out of the car.\n\n\"What's wrong?\" she called.\n\n\"She's covered in blood!\" the woman was shrieking\n\nThe man was trying to calm her down. \"Priscilla, come back here!\"\n\nAnnabel looked in the direction the woman was pointing. She saw nothing.\n\nShe approached the woman. \"Can I help you? What did you think you saw?\"\n\nThe woman turned a pair of frantic but obviously exhilarated eyes to her. \"Was she a ghost? Does she walk the property?\"\n\n\"I don't know who you're talking about,\" Annabel told her.\n\nThe man had joined them. \"We saw someone coming out of the woods. . . .\"\n\n\"A woman,\" his companion added. \"She's gone now. When I screamed, she bolted back into the woods.\" She frowned. \"I shouldn't have screamed. I know better than that. We can sometimes scare ghosts as much as they can scare us.\"\n\nAnnabel looked at the couple standing in front of her. They were obviously guests arriving at the inn\u2014\"ghost tourists,\" as Millie had called them. They had English accents. They'd apparently come a long way to experience the Blue Boy's ghosts.\n\n\"Well,\" Annabel said, \"I can't tell you anything. It's my first day here. My husband and I just moved here.\" Her gaze moved up to the front porch of the inn. \"But I'm sure my grandmother-in-law can tell you whatever you need to know.\"\n\nCordelia was standing there, her face set like stone. She must have heard the woman's scream.\n\nThe couple hurried up to her, jabbering about ghosts. Annabel heard the old woman start to reply, but she didn't care to listen to what she had to say at the moment.\n\nShe decided she wanted to do a little exploring herself.\n\nHer groceries would keep in the car for the moment. It was cold enough out. She started off across the grass in the direction the English woman had been pointing. If she had been alone in her claim of seeing something, Annabel would have dismissed her as a fanatic. When you come to a place wanting to see something, chances are you would. The human mind was susceptible to suggestion. Hadn't Annabel thought she'd seen Tommy Tricky earlier?\n\nBut the man said he'd seen somebody as well. So chances were they really did see somebody. Chances were it was a real person, and the woman's hysterical scream had indeed frightened the visitor away. Annabel hoped she found someone out there, so she could bring her back and introduce her to the English couple, and to Cordelia.\n\nShe wanted an end to ghost stories. She had no desire to be part of a place that depended on crazies coming to stay there. Annabel had been in a crazy house. She did not want to surround herself with lunatics and delusional people.\n\nShe'd had enough of that.\n\nShe pushed her way into the trees. A broken, brittle branch on the ground snapped as she stepped on it.\n\nJack was wrong not to tell her about the inn's reputation. Very wrong.\n\nBut if he had, she would likely have refused to come. It would have been too much for her. So he'd kept the knowledge from her, understanding how it would freak her out.\n\nAnnabel was going to tell him that he was wrong, and she was going to add that they were going to put a stop to the stories immediately.\n\nPerhaps some terrible things had happened at the inn. But she and Jack were not going to make their livings from exploiting those tragedies.\n\nAnnabel stopped. There was something sticking out of a clump of dead leaves in a little clearing up ahead.\n\nSomething white.\n\nAs Annabel approached, she saw it was a stone.\n\nA stone marker.\n\nOn which was inscribed a name.\n\nCINDY DEVLIN\n\nIt must be Jack's sister.\n\nThis was her grave!\n\nIn that second of realization, a hand reached out from behind her and clutched Annabel by the shoulder.\n11\n\n\"Welcome to the Blue Boy Inn,\" old Mrs. Devlin said, as she escorted them into the old house.\n\nPriscilla was peeved. Mrs. Devlin had insisted they must have just seen a hiker. The Blue Boy Inn had no ghosts outside the house, she said, and certainly none that walked around with blood on their faces. Priscilla was deeply disappointed. She hoped this place, unlike so many of the others, wouldn't be a rip-off.\n\n\"Leave your bags there, by the door,\" the old woman said to Neville. \"I'll have Zeke or my grandson, Jack, carry them up to your room.\"\n\n\"We'd like Sally Brown's room,\" Priscilla said.\n\nMrs. Devlin gave her a wan smile. \"And you shall have it.\"\n\nNeville returned the smile. \"I suppose you get a lot of crazy ghost hunter types staying here.\"\n\nMrs. Devlin was nodding as she led them into the kitchen. \"We're listed in all the guidebooks as a 'haunted inn.' It keeps people coming.\"\n\n\"And how often do guests see apparitions?\" Priscilla wanted to know.\n\nThe old woman stopped at the roughhewn kitchen table, steadying herself against it with her hands. \"Some of them report a sighting or two. I make no guarantees.\"\n\nPriscilla snorted. \"Well, there have been so many killings in this house. I'd imagine the spirits are very restless here.\"\n\nNeville sighed. \"She's a true believer, I'm afraid,\" he told Mrs. Devlin.\n\n\"A cup of tea?\" the old woman asked.\n\nBoth accepted, and she gestured for them to sit at the table.\n\n\"I take it you're not a believer then, sir,\" Mrs. Devlin said, looking over at Neville as she poured steaming hot tea into two delicate china cups, balanced on saucers.\n\n\"Not really. I'm here for the fun of it, and because Priscilla would only go with me to Florida after a week of ghost hunting in New England.\"\n\nMrs. Devlin pushed the cups of tea toward them with her bony, spotted hands.\n\n\"Thank you,\" Priscilla said, taking hers and lifting it to her lips.\n\n\"Well,\" Mrs. Devlin said, sitting down at the table opposite them, \"I suppose there must be restless spirits here. You're right, young lady. There have been an awful lot of deaths in this house. More than our share.\"\n\n\"So you've never seen any ghosts?\" Priscilla asked, setting down her cup into the saucer and leaning slightly toward the old woman.\n\n\"I don't think I'd recognize them if I did. I've been here a very, very long time. Sometimes it takes someone unaccustomed to the place to pick up on things.\"\n\nPriscilla nodded. \"That's true. I've read about that phenomenon. You live here with the spirits and so you're on the same vibration. You don't see them. But those who come in from the outside can pick up more easily on things.\"\n\nNeville laughed out loud. \"What a bloody rationalization! Fanatics like you, my dear, can come in and claim they see things simply because you're on a different vibration!\"\n\nPriscilla shot him an angry look.\n\nNeville grinned, reaching over to pat her hand. On her pinky she wore an opal ring. It was supposed to attract spirits. \"I use the word fanatic with great affection, my dear.\"\n\n\"Zeke has seen some ghosts,\" Mrs. Devlin told them.\n\nPriscilla looked back over at her. \"Who's Zeke?\"\n\n\"Our caretaker. He's been here nearly as long as I have. He's seen things. You should ask him.\"\n\n\"Oh, I certainly will.\"\n\n\"I should also tell you,\" Mrs. Devlin said, standing up again, with some difficulty, \"that my grandson and his wife have just arrived. They will be living here with me now, taking over the care of the place. Zeke and I have gotten too old to do it all by ourselves anymore.\"\n\n\"Is that the lady who drove up while we were outside?\" Priscilla asked.\n\n\"Yes.\"\n\n\"She went off into the woods,\" Neville said. \"I guess looking for the woman who was hiking.\" He smirked. \"To apologize for Priscilla screaming her head off, I imagine.\"\n\n\"I tell you,\" Priscilla insisted, \"her face was covered with blood.\"\n\n\"Perhaps she scratched herself in the thicket out there,\" Mrs. Devlin said. \"Or it was mud. It gets very swampy a few feet into the woods.\"\n\nPriscilla sniffed. She wasn't entirely convinced that what she'd seen had not been a ghost.\n\n\"Anyway,\" the old woman continued, \"I haven't yet filled in my granddaughter-in-law about some of the more distressing chapters in the inn's history. I didn't want to frighten her too badly on her first day. And since you've obviously read everything there is about the Blue Boy Inn, I'd appreciate you not bringing it all up with her. At least, not quite yet.\"\n\nNeville made a face in surprise. \"You mean to tell me, her husband brought her to live here without telling her about the history of this place?\"\n\nMrs. Devlin pursed her lips. \"We decided it was best to tell her when she got here.\"\n\nNeville laughed. \"Because otherwise, no sane person would ever have come.\"\n\nA tight smile stretched across the old woman's face. Priscilla took it to mean that Mrs. Devlin was saying, Ah, but my granddaughter-in-law isn't sane.\n\n\"You must be tired from your drive,\" Mrs. Devlin said, lifting an old copper key off a nail on the wall near the sink. \"I'll show you up to your room.\"\n\nPriscilla and Neville stood to follow her.\n\n\"So were the killers of any of those who were murdered here ever found?\" Priscilla asked as they headed back out into the hallway.\n\n\"Most of the deaths here were simply tragic accidents,\" Mrs. Devlin said, leading the way through the narrow, musty corridor, not looking back as she spoke.\n\n\"Well, that poor man whose head was never found,\" Priscilla said. \"That was no accident.\"\n\n\"No, I suppose it was not,\" the old woman replied. \"Andrew McGurk died here before my time. My husband's father owned the place then.\"\n\n\"And the little baby who disappeared,\" Priscilla asked, \"except for her arm?\"\n\nMrs. Devlin paused near the stairs. \"For the life of me, I don't know where Zeke or Jack are,\" she said, evidently done with speaking about murder and death and ghosts.\n\n\"That's all right,\" Neville said, grabbing their bags. \"I don't mind hauling them myself.\"\n\nThe old woman frowned. \"Not a good way to treat our guests. I apologize.\"\n\nThey started up the stairs.\n\n\"But please,\" Priscilla said. \"Tell me about the baby.\"\n\n\"I had just arrived here,\" Mrs. Devlin said. \"Had just married my husband. And I suspect, in that case, it was a kidnapping gone wrong. The mother was a rich heiress. She was running away from her father, and some goons were after her. I think they thought taking the baby might get them quite the ransom.\"\n\n\"But why would they cut off the poor thing's arm?\" Priscilla asked.\n\n\"You'd have to ask them,\" Mrs. Devlin said.\n\nThey had reached the top of the stairs.\n\n\"Here's your room,\" the old woman said, unlocking the door.\n\nThey stepped inside. It was small, neat, low-ceilinged. Mustiness pervaded everything. The four-poster bed was small, carefully made. A three-drawer dresser stood beside the single window. Except for a straight-backed chair, that was all the furniture in the room.\n\n\"And Sally Brown?\" Priscilla asked. \"The girl who died in this room?\"\n\n\"Before my time, too,\" Mrs. Devlin said. \"But what my mother-in-law told me was that poor Sally got word that her fianc\u00e9 had died in Germany. This was during World War I. And so she slit her wrists. That was the cause of the blood on the walls.\"\n\n\"But her body was never found,\" Priscilla pointed out.\n\n\"I was told Sally ran outside to bleed out,\" the old woman said matter-of-factly. \"I suspect bears and coyotes finished off her remains.\"\n\nNeville shuddered. \"Such a delightful history.\"\n\n\"Even if they weren't all murders,\" Priscilla said, \"these were very traumatic deaths. Suicides make for some of the most frequent ghosts.\" She looked over at Mrs. Devlin. \"Do many people report seeing Sally?\"\n\nThe old woman nodded. \"Yes. Many do.\"\n\nPriscilla smiled.\n\n\"Then I'll let you get settled,\" the old woman told them. \"I've made some rabbit stew if you'd like some for dinner. Otherwise, there are some decent restaurants up in Sheffield.\"\n\n\"Thank you,\" Neville said.\n\nMrs. Devlin left them alone.\n\n\"It was a ghost I saw out in the woods,\" Priscilla said. \"I know it. I'll bet it was Sally Brown!\"\n\nNeville flopped down on the bed. Dust puffed up into the air.\n\n\"I don't think I could eat rabbit stew,\" he said.\n\nPriscilla was examining the wallpaper for bloodstains. \"We're going to get what we paid for here, I'm certain of it.\" She looked over her shoulder at Neville. \"We're going to have a major close encounter with the spirit world here. I can feel it in the air!\"\n\nNeville could only groan.\n12\n\nAnnabel spun around to see who\u2014or what\u2014was behind her.\n\n\"Jack!\" she shouted.\n\nHer husband was grinning sheepishly. Standing beside him was the hunched-over figure of Zeke.\n\n\"I didn't mean to scare you, baby cakes,\" Jack said.\n\n\"Well, you did,\" Annabel replied.\n\n\"I'm sorry,\" he said, trying to take her hands, but she pulled them away.\n\n\"What is this, Jack?\" Annabel pointed down at the ground. \"This stone?\"\n\nThe name seemed to glare up at them.\n\nCINDY DEVLIN\n\n\"That's my little sister,\" Jack said, very quietly.\n\n\"You never told me you had a sister,\" Annabel said, her voice harder than she meant it to be. \"Never, in all our years together.\"\n\nHe looked at her. His eyes shone with pain. \"It's always been difficult to talk about Cindy,\" Jack told her.\n\n\"She was a very sweet little girl,\" Zeke offered. \"Such a tragedy.\"\n\nAnnabel looked from them down to the grave marker in the leaves, then back to them again.\n\n\"Why is she buried here, in the middle of the woods?\"\n\nJack smiled sadly. \"She's not buried here. That's just a stone Dad put up to remember her by. To give us someplace to come to.\"\n\n\"Her body was never found,\" Zeke explained, his old yellow eyes finding Annabel's.\n\n\"What happened to her?\"\n\nJack took in a long breath, and then let it out very slowly. \"She just disappeared. She\u2014must have gotten lost in the woods or something. There was no trace of her.\"\n\n\"Except\u2014\" Zeke began to say.\n\nBut Jack shot him a look that shut him up.\n\n\"Except what?\" Annabel asked.\n\nJack hesitated. \"We found blood. A lot of it.\"\n\n\"She was a sweet little girl,\" Zeke added. \"The sweetest, really.\"\n\n\"I'm sorry to hear this,\" Annabel said. \"I wish you had told me about her before.\"\n\nJack sighed. He made no response.\n\n\"In fact,\" Annabel went on, \"I wish you had told me a lot of things before we came here. Such as all the deaths and murders that took place at the inn over the years. When did you think you'd tell me, Jack? You must have known I'd find out as soon as we got here.\"\n\n\"Who told you about all that?\" Zeke asked.\n\nAnnabel shifted her eyes over to the old man. \"The woman at the market.\"\n\n\"Ah, that Millie, she's a busybody,\" Zeke grumbled.\n\n\"I was hoping Gran and I could tell you, in our own way,\" Jack said.\n\nAnnabel looked down at the little white stone marker. \"You knew I wouldn't come if I had known this place had such a lurid history.\"\n\nJack took her hand. \"Babe,\" he said softly. \"We're no strangers to lurid histories, you and I.\"\n\n\"What's that supposed to mean?\" Annabel asked.\n\n\"Just that . . .\" Jack seemed to search for the right words. \"We needed a place to start over. And I think the Blue Boy Inn needs to start over, too. You and I . . . we want to put our pasts behind us. So does the Blue Boy.\"\n\nShe frowned. \"That's not likely, with all these ghost tourists seeking the place out.\"\n\n\"It's true that there have been some unfortunate tragedies here,\" Jack admitted. \"But the town made way more of them than they were. Over the course of more than a hundred years there have been some deaths here. Some perfectly peaceful. Some not so peaceful. That's to be expected anywhere that's been around for as long as the Blue Boy. But the locals like to tell stories, and every new death here has been woven into a never-ending tale. Legends of ghosts and death curses sprung up. And the tourists started coming.\"\n\nAnnabel was still looking at the marker for Cindy Devlin.\n\n\"If she got lost in the woods,\" Annabel asked, \"why was there a lot of blood?\"\n\nJack sighed. \"The police chief thought maybe a bear got her. There had been sightings of bears not long before she went missing.\"\n\n\"And where was the blood?\"\n\nZeke stepped forward. Annabel had almost forgotten he was there.\n\n\"On the back steps and down the path,\" the old man said. \"I found it. We'd been calling for Cindy all morning, when she wasn't in her bed. And then I went around back and found the blood. . . .\"\n\n\"My father speculated she got up in the night for some reason,\" Jack said, his voice thick with emotion. \"And she went out back and that's where the bear spotted her. . . .\"\n\n\"But why was her body never found?\"\n\nJack shrugged. \"The police scoured the woods for her. The bear must have . . .\" He couldn't speak. \"She was so little, you know.\"\n\nAnnabel reached up and touched his cheek. She had been so upset about not being told about all this history that she hadn't been very compassionate. This was his sister that Jack was talking about. This was a childhood tragedy that had apparently scarred him so badly he'd never been able to speak of it before.\n\n\"I'm sorry, Jack,\" Annabel said, stroking her husband's bristly cheek.\n\n\"We thought we'd find something of her,\" Zeke piped in. \"But nothing. Just the blood.\"\n\n\"And then,\" Jack said, finding his voice, \"the town went and turned it into another example of the murder curse on the Blue Boy Inn, adding poor little Cindy to their long list of ghosts that haunted the house.\" He shook his head. \"I never told you, sugar cakes, because I've always hated that part of the Blue Boy's history. As far as I'm concerned, we're putting an end to it.\"\n\n\"Good,\" Annabel said. \"We stop marketing the place as a haunted hotel. Get it out of those guidebooks. Debunk the ghost stories whenever anyone asks about them.\"\n\n\"That's a mighty fine sentiment,\" Zeke said. \"But without those ghost tourists, I'm not sure you have a business. What else could bring people out to the middle of nowhere to a rundown old house?\"\n\n\"So we change the place from rundown to fabulous,\" Annabel replied. \"We modernize. We renovate from top to bottom. Make it comfortable and loaded with amenities. We are on some gorgeous property here. In the spring and summer these woods will be in full green bloom. And in the fall I can only imagine how magnificent the colors will be. I'd love to redo the gardens, make some paths, maybe put in a Jacuzzi. We'll make this place the perfect getaway destination. We won't need ghosts to sell it.\"\n\n\"I think you have the right idea, babe,\" Jack said.\n\nZeke gave them a crooked smile. \"Not sure what Cordelia will think of ripping the place up.\"\n\nAnnabel felt her back stiffen. \"She asked us to come here, didn't she? She wanted us to take over the place. If she wants this place to stay in business, she'll agree. Otherwise, none of us can afford to keep this house going. We'd have to put it on the market. . . .\"\n\n\"Oh, no,\" Zeke said. \"I know Cordelia wouldn't want to do that.\"\n\n\"Then it's settled,\" Annabel said. \"We're going to make this an entirely new place, with an entirely new reputation, aren't we, Jack?\"\n\n\"We sure are, sugarplum,\" he said, wrapping his arm around her and pulling her into him.\n\nOverhead a crow screeched in the bare limbs of a tree.\n13\n\nCordelia listened to her guests moving above her. They seemed like nice people. Strange, like all the ghost tourists were, but nice. She especially liked the man.\n\nNothing would happen while they were in the house. She would see to it.\n\nShe'd been seeing to it for a long time. Now she was old. She was giving the Blue Boy over to Jack and his wife. Could she sleep easy, knowing they would be the ones to take over from her in safeguarding the house?\n\nAnnabel was a wild card. Cordelia wasn't sure about her. She seemed obstinate. Defiant. Too independent.\n\nThat could prove problematic.\n\nShe wished once more that Jack hadn't brought a wife.\n\nFor a moment her thoughts wandered to Jack's father. Her son. He'd brought a wife to the Blue Boy Inn, too. And two little children.\n\nBut Cordelia pushed the thought out of her mind.\n\nShe heard the sound of her guests upstairs running water. They were washing up. They would be downstairs again soon.\n\nThe old woman made her way into the living room. It was a big, open space, furnished with just a few antique wingback chairs and a long table in front of large bay windows. The windows were cloudy. How many years had it been since Cordelia had washed them? But what did it matter, really? The bushes outside had grown up so thickly in front of the windows that they nearly obscured the sunlight anyway.\n\nIt was better that way, Cordelia thought. This house\u2014and especially this room\u2014needed no prying eyes looking in from the outside.\n\nThe living room was dominated by the old stone fireplace in the center. The hearth extended four feet out into the room, and the mantel was a good six feet. But the fireplace was devoid of any tools hanging at its side. There was no pot hanging inside, even for show. In fact, the opening was bricked over. No fire could be built there. The bricks enclosed the path down to the firebox and sealed off the flue. They had been installed with care and precision. Cordelia knew this. Because she had helped lay the bricks.\n\nShe heard the back door squeak open.\n\n\"Gran, we're back!\" she heard Jack call.\n\nWith a final glance at the sealed-off fireplace, Cordelia headed toward the kitchen.\n14\n\nRoger Askew was a mean son of a bitch.\n\nHe'd just told that busybody, dried-out old fruit Millie Westerbrook at the general store to stick it where the sun don't shine, and then added that since nothing had been stuck there in ages for her, she'd probably enjoy it.\n\nMillie had been giving Roger a hard time because he smelled like whisky and dropped the F-bomb in front of some little kids. Why couldn't the bitch just mind her own business?\n\nSo Roger had just paid for his pack of smokes and slammed out of that goddamn place.\n\nAs he trudged through the path in the woods, he realized the reason he was in such a foul mood was all because of Tammy.\n\nHis girlfriend.\n\nRather, his ex-girlfriend. At least she would be, after today.\n\nShe was a lazy, good-for-nothing bitch. Roger had asked her to do one simple favor for him. Run down to the store and get him some cigarettes. And she'd said she had to pick up that brat of hers, Jessica, from school. Like the kid couldn't have waited five minutes? Tammy was just so goddamn selfish. She never did anything for Roger.\n\nShe hadn't even been putting out lately.\n\nHe stopped on the path, tore open the pack of cigarettes, shook one out, placed it between his lips, and lit it.\n\nHe sucked in the smoke. Ah, yes. He'd needed that.\n\nRoger was going to be twenty-nine in a couple of weeks. It was time he made a clean break. He needed to give Tammy the old heave-ho. He deserved a girlfriend who appreciated him.\n\nNot one who bitched at him all the time to find a job.\n\nHe'd had no choice but to quit the last one. The manager of the Jiffy Lube was a fucking prick. He'd had it out for Roger. Always on his case, making him take the worst of the freaking lemons that people drove into the place, the cars that were literally ready to die, and Roger was somehow supposed to get them purring smoothly again. Finally, he'd told his asshole manager to go fuck himself, and added that, since it had obviously been a long time since anyone had fucked his scaly self, he'd probably enjoy it.\n\nThat was one of Roger's favorite insults.\n\nUp ahead on the path, he saw someone walking toward him.\n\nRoger hoped it wasn't anyone he knew. He was in no mood to say hello to anyone. All he wanted to do, in fact, was punch someone. He had a temper. He knew that. He'd served time for beating up a few people, and Tammy had threatened to have him arrested the last time he'd hauled off and whacked her across the head. So far he had yet to smack that brat of hers, not that Jessica didn't have it coming. But Roger knew if he ever hit the kid, he'd have to deal with the freaking banshee her mother would become.\n\nHe hated kids. Even his own. His daughter was probably eight or nine years old by now. She lived with her mother up in Pittsfield. Roger hadn't seen her in three years, but still her bitch of a mother kept demanding he pay child support, and Roger was damned if he was going to fork over the little bit of cash he had to a kid he never saw and who he had doubts was really his, anyway. So now the mother-bitch had offered him a deal. Give up all parental rights for all time and she'd stop hounding him for money.\n\nRoger figured that was a deal. Tomorrow morning he was heading up to Great Barrington to make that all legal.\n\nHe looked up the path again. Whoever he'd seen there was gone.\n\nWhere the fuck did they go? There was one straight path through these woods, from the store to the apartments along the river, where Roger lived. Must have been some goddamn nature explorer, heading off into the woods to scrounge for mushrooms or something freaky like that.\n\nAbove him, a crow in the bare tree branches suddenly screeched, making Roger jump a little.\n\nHe wished he could shoot the thing. Roger hated everyone and everything. Ever since he'd been born in this godforsaken little town, everyone had been against him. His parents. His teachers. The cops. That bitch who'd seduced him into getting her pregnant and then having his kid. Roger had told her to abort the thing, but she wouldn't.\n\nSomeday, really, he ought to just give in to his rage and get a gun and start shooting. Like those guys who finally snapped and mowed down theaters full of people or kindergarten classrooms. Roger could relate. They had just had enough of all the crap that they were dealt on a daily basis. If people only weren't so goddamn nosy\u2014\n\nBehind him, Roger heard a twig break.\n\nHe looked over his shoulder.\n\nNo, the sound hadn't come from behind him. It had come from off in the woods somewhere.\n\nIt was whoever had been on the path ahead of him, now moving among the trees.\n\nThe woods were a pale blue this time of day. The sun was low in the sky, hidden by clouds, and the cold, bare trees seemed to shiver before the coming darkness. The ground was hard. There wasn't a speck of green that Roger could make out anywhere.\n\nJust blue. Deep blue shadows.\n\nHe took the last drag on his cigarette and dropped the butt to the ground. He picked up his pace a little.\n\nFrom the other side of him, he heard another twig snap in two.\n\nWhy did his flesh crawl?\n\nHe'd been on this path hundreds, maybe thousands, of times. It was as familiar to him as his own living room. And Roger didn't scare easily. Rather, he scared other people. That had always been the way it was.\n\nHe was a big guy. Five-eleven, one-hundred-eighty-five. He was strong. He wore his hair long, down over his shoulders, and he brandished blue, red, and purple tattoos up and down his arms. Skulls and arrows and lightning bolts. When people saw Roger Askew coming, they didn't mess with him.\n\nSo why was he suddenly creeped out? Why did he want to get off this path as soon as he could?\n\nThe crow in the trees suddenly took off into flight, the sound of its giant wings flapping echoing down through the skeletal trees.\n\nRoger began to walk even faster.\n\nUp ahead, whoever he'd seen earlier stepped back onto the path from the woods.\n\n\"It's just a woman,\" Roger whispered to himself, instantly relieved, and even a little embarrassed that he'd been afraid.\n\nA woman dressed in white. With long gray hair.\n\nWho could be afraid of some old lady?\n\nShe stood there in the middle of the path, waiting for him.\n\nRoger felt the fear return, flooding his body like a shot of Novocain. His limbs froze. His heart began to echo in his ears.\n\nHe wanted to turn back and run the other way.\n\nBut this was just some old bitch! Why should he fear walking past her?\n\nRoger forced his numb legs to continue walking.\n\nAs he drew closer to the woman, he noticed a few things about her. She was watching him intently. Her face was dirty, caked with something. And she wasn't really that old at all.\n\nWhen he was just a couple of feet away from her, the woman spoke.\n\n\"Are you from around here?\" she asked.\n\n\"Yes,\" he told her.\n\n\"I seem to have lost my way,\" the woman told him.\n\nRoger was now standing directly opposite her. Not only wasn't she old, but she wasn't half bad-looking, either. Out here in the middle of the woods, Roger realized he could do anything he wanted to her. He began to get excited.\n\nA smile started to make its way across his face, like a worm.\n\nHe never even saw the knife, but he felt it. And the warm cascade of blood that flowed from his gut down over his groin and legs. He felt that, too.\n\nRoger looked up in disbelief at the woman.\n\nBut she was gone.\n\nHis legs crumpled beneath him. And then everything went dark.\n15\n\n\"I hope you slept well,\" Annabel said to Priscilla and Neville when they shuffled into the dining room for breakfast the next morning.\n\n\"Well, I might have,\" Neville replied, bleary-eyed, \"but Prissy here was carrying on a conversation with somebody.\"\n\nPriscilla looked exhausted but exhilarated. \"She came to me!\" she told Annabel. \"Sally Brown! She was sitting in my room talking to me all night!\"\n\n\"Yeah, and keeping me awake,\" Neville grumbled, pouring himself some coffee.\n\nPriscilla held up her hand. \"I wear an opal ring all the time that was given to me by a psychic. It's said to be able to attract ghosts!\"\n\nAnnabel didn't want to encourage the woman's delusions. The sooner they ended this association with ghosts and weirdos the better.\n\n\"Well, what we hope for our guests is they get a good rest when staying here, so I hope tonight will be more peaceful,\" she said, laying out a tray of blueberry and corn muffins that she and Cordelia had baked that morning. The old woman had shared her recipe and showed her where all the ingredients were kept in the pantry. Eventually, when she retired, Cordelia told Annabel this would be her job. Annabel thought a better idea would be to hire a chef who could whip up some nouvelle cuisine breakfasts for their guests.\n\nAs Neville slabbed butter all over a muffin, Priscilla sat down and gushed about her otherworldly encounter.\n\n\"I opened my eyes and there she was, as real as you are,\" she said. \"She told me that she had spent her life walking these hallways, trapped in these rooms. Oh, it was so thrilling!\"\n\n\"Did you see her, too?\" Annabel asked Neville.\n\nHe shook his head. He had butter on his chin. \"I just heard Priscilla chattering away. A couple of times she prodded me to sit up and have a look, but each time I did, I saw nothing. She wouldn't turn on the light.\"\n\n\"Sally asked me not to,\" Priscilla said. \"She doesn't like the light.\"\n\n\"Yeah, well, whatever,\" Neville said, returning to his muffin.\n\n\"I should tell you,\" Annabel said, \"that I don't believe the stories of the ghosts in this house. Once my husband and I take over running the inn completely, we aren't planning on marketing that particular element anymore. We're going to upgrade the house and make it a real first-class destination. No more tales of ghosts and murders.\"\n\nPriscilla looked horrified. \"But you can't do that. The ghosts will still be here. Just because you don't want to promote them doesn't mean they'll just go away.\"\n\n\"I guess we'll cross that bridge when we get to it,\" Annabel told her.\n\n\"Seriously! Sally Brown can't cross to the other side! She told me so! She's trapped. We sat and talked for nearly an hour!\" Priscilla smiled, remembering. \"She's really quite sweet, you know.\"\n\nAnnabel smiled as well. \"Well, I'm pleased that you got what you came here for.\"\n\nThe Englishwoman's face suddenly changed. \"You're patronizing me. Just like Neville.\"\n\n\"No,\" Annabel protested. \"Really, I'm not. . . .\"\n\n\"Yes, you are. If you don't believe in the ghosts, then you think that I'm either making up what I saw last night, or that I'm mad.\"\n\n\"No, really, I just\u2014\"\n\n\"Of course, we believe you,\" came a new voice.\n\nThey turned around. Jack stood in the doorway of the kitchen. He was wearing jeans and a white ribbed tank top. Annabel could tell he'd just come from the shower. His hair was still damp.\n\n\"I was a child in this house,\" Jack said, coming into the kitchen and pouring himself some coffee. \"My wife was not. So she doesn't know.\"\n\nPriscilla turned to look over at Neville. \"You see?\" she asked.\n\nIn the moment she was looking away, Jack took the opportunity to wink over at Annabel. She realized he was only trying to pacify the guests. She said nothing more.\n\n\"Did you see many ghosts here when you were a boy?\" Priscilla asked him.\n\n\"Oh, many, many ghosts,\" he said, sitting down at the table opposite her. \"Some were pretty scary, but some were friendly ghosts, like Casper.\"\n\n\"Did you ever meet Sally Brown?\"\n\n\"Sure, I did. Sally and I got to be good friends.\"\n\nAnnabel couldn't stand it. She walked out into the living room. She figured she'd start today with a good cleaning of the living room and dining room. She'd mentioned the idea to Cordelia this morning, who hadn't seemed to mind. Annabel wondered how long they'd have to tiptoe around Cordelia's feelings. She'd asked them to take over the place, so she couldn't very well stand in their way of modernizing it. Eventually, they'd have to just sit her down and explain that if the inn was ever to turn a profit again, they'd have to make some changes. Once the paperwork was complete, and the Blue Boy was in their names instead of Cordelia's, they could do what they wanted.\n\nAnnabel paused. It felt odd that the Blue Boy would be in her name. That she'd be an owner of a place that appeared from its sign to be the home of Tommy Tricky.\n\nShe was gripped by a memory.\n\nDarkness. Stale air. The smell of mothballs.\n\nShe was trapped in a closet. She was banging on the door. She could hear her heart thudding in her ears.\n\n\"Help me! Mommy! Help me!\"\n\n\"Your mommy isn't here, she can't help you,\" came the voice of Daddy Ron. \"She can't save you from Tommy Tricky!\"\n\n\"No!\" Annabel screamed.\n\n\"He's right behind you! Can you see hear him breathing?\"\n\nAnnabel could. The imp was panting, like a dog.\n\n\"And how he loves to eat bad little girls!\"\n\n\"Miz Wish?\"\n\nAnnabel jumped. A voice behind her. A real voice. Not the terrible daydream.\n\nShe turned around. It was Zeke.\n\n\"Sorry,\" the caretaker said. \"I didn't mean to startle you.\"\n\n\"Oh, it's all right,\" she said. \"I was . . . lost in thought.\" She tried to smile. \"Zeke, will you help me take down these curtains? I want to wash them and air them out. I'm going to be giving the living room a deep cleaning today.\"\n\n\"Yes, sure, I can help you,\" he said. \"But I wonder, first, if you've seen Miz Cordelia.\"\n\nAnnabel looked at the old man's face. He was clearly upset about something.\n\n\"Why, yes, I saw her earlier this morning. We baked some muffins for the guests. I told her about my cleaning plans, and she seemed fine with them.\"\n\n\"Do you know where she went afterward?\"\n\n\"No, I don't. Isn't she in her room?\"\n\nZeke shook his head. \"I've looked everywhere for her. And there's something I need to discuss with her right away.\"\n\n\"Anything I can help you with?\"\n\n\"No, ma'am.\"\n\n\"Well, Jack's in the kitchen. Maybe he can help\u2014\"\n\n\"Nobody can help but Cordelia,\" Zeke said. \"If you see her, please tell her I'm looking for her.\"\n\n\"Of course.\"\n\nThe old man hurried off, as best as he could hurry, hobbling up the steep, narrow stairs.\n\nThat was odd. Where in this cramped old house could Cordelia possibly disappear?\n\nAnnabel stuck her head back into the kitchen. Neville had left. She could see him through the window out in the backyard, smoking a cigarette. Priscilla and Jack were still seated at the table, leaning in toward each other, discussing the ghosts of the house. Their faces were only inches apart. It made Annabel oddly uncomfortable.\n\n\"Another ghost I remember seeing was a little boy,\" Jack was saying. \"He'd come riding a tricycle down Gran's path and then just disappear!\"\n\n\"Oh, that's brilliant,\" Priscilla said, completely snookered and in awe.\n\nShe was pretty. Annabel hadn't really noticed before. Priscilla had just seemed too odd and eccentric to be pretty. But she was. Long blond hair and breasts much larger and fuller than Annabel's more modest pair. For a second she had a flash of Rachel Riley, and then pushed the thought away.\n\n\"Jack,\" she said. \"Zeke is looking for your grandmother.\"\n\n\"Haven't seen her,\" her husband replied, before resuming the story of the ghost boy on the tricycle. Priscilla continued to give him her rapt attention.\n\nAnnabel stewed. It was one thing to mollify the guests, to not offend them\u2014but he was actively encouraging all this ghost talk, after he and Annabel had decided they'd put that all behind them. She turned on her heel and strode back out into the living room.\n\nShe heard something then.\n\nThe slightest sound.\n\nScratching.\n\nShe listened closely.\n\nIt was coming from the other side of the room.\n\nAnnabel approached the sound.\n\nIt seemed to be coming from the fireplace. From below it, to be exact. The sound seemed to rise from below the fireplace and from the floorboards surrounding it.\n\n\"Rats,\" Annabel murmured to herself.\n\nOr maybe just mice or squirrels or chipmunks.\n\nEither way, she thought, they'd need to exterminate.\n16\n\n\"There you are,\" Zeke rasped, out of breath, as he spotted Cordelia, huffing nearly as much as he, coming down the stairs.\n\nThe old woman fixed him with an icy glare. \"You're incompetent,\" she snarled.\n\n\"I did not leave the door unlocked, if that's what you're thinking,\" Zeke replied.\n\nCordelia glanced around. She could hear Annabel in the living room. She lowered her voice.\n\n\"Well, I certainly didn't leave it unlocked,\" she whispered angrily. \"You were the last in there.\"\n\nZeke's old eyes looked as if they might burst into tears. \"I was certain I locked the door! I always check, every time I leave.\"\n\n\"Well,\" Cordelia muttered, \"it's all taken care of now.\"\n\nHe grabbed her bony wrist. \"Are you sure? Everything is safe?\"\n\nCordelia yanked away from him and moved toward the kitchen. \"For now. But you better take care, you old fool. Jack has got to know soon. Don't let anything happen in the meantime.\"\n17\n\nAt the little police station at the end of the one-block Main Street of Woodfield, Chief Richard Carlson was going over the day's schedule with his deputy, Adam Burrell. Carlson was drinking a cup of coffee and eating a cinnamon cruller he'd picked up at Deb's Diner. His fingers were sticky and sugary as he turned over the schedule's pages.\n\n\"Might be a bit of a traffic tie-up at the Route 7A intersection today,\" the chief told his deputy. \"They're putting up a new light around noon.\"\n\n\"I'll be there,\" Burrell assured him.\n\n\"And tonight there's that meeting in the public works room at town hall. Can you be there as well?\"\n\n\"Sure thing, chief.\"\n\nCarlson smiled. \"Thanks, Adam. Just not sure I can handle another one of those.\"\n\n\"Small town politics getting you down?\"\n\nCarlson sighed. \"It seems everyone's got an agenda against everyone else. The town manager hates the board of selectmen, the board of selectmen hate the school superintendent, the school superintendent hates the town manager . . . it's a vicious circle.\"\n\nBurrell smiled. He was a young man, redheaded and freckled. \"Yeah, but everybody loves the chief of police.\"\n\nCarlson knocked on the wood of his desk. \"So far,\" he laughed.\n\n\"Hey, you've been chief for a decade and no one's tried to get you fired. That's a good run.\"\n\nCarlson finished the last of his cruller. \"I guess it is,\" he said.\n\nBurrell left the office to start his morning's rounds. The chief sat back in his chair, sipping his coffee. Deb made it good and strong, the way he liked it. He drank it black. In the old days, when Amy had still been around, he'd taken cream and sugar in his coffee. But ever since Amy had died, Carlson had needed something stronger in the morning. Something to jolt him awake and keep him alert and concentrating all day.\n\nThe people of Woodfield had never known Amy. His wife had died two years before Carlson had come to this little town. He was the \"new bachelor chief of police\" and had attracted a great deal of feminine attention during his first few years here. Every single lady, and a couple married ones, too, had seemed to try to date him. He'd gone out on one date in his whole time here, just one, and it had been a disaster. Cora Coakley. Poor Cora. She'd seemed nice enough when she'd come into his office, bringing him homemade jams and apple pies. But when they went out to dinner, she'd sat across from him blushing so hard Carlson had started blushing in return, and every attempt at conversation had faded off to a few mumbled sentences. Now when Carlson saw Cora around town he just tipped his hat to her. He didn't want to say something to her and risk those cheeks of hers turning bright apple-red again.\n\nThe fact was he didn't want to date anyone. He had loved one woman. One woman only. Amy had been everything to him. He could never replace her. Even twelve years after her death, Carlson could not imagine loving another woman.\n\nHis phone buzzed.\n\nIt was Betty, his secretary, in the front office.\n\n\"Rich,\" she said, \"Tammy Morelli is on the line. I tried to give her Adam's voice mail, but she insisted she wanted to talk to you.\"\n\n\"Okay, put her through,\" he told Betty.\n\nPoor Tammy. She worked at Deb's Diner. Poor kid didn't have it easy, raising that little girl, Jessica, all by herself, with no help from that good-for-nothing boyfriend of hers, Roger Askew. Carlson had noticed that Tammy wasn't at the diner this morning. When he'd asked about her, Deb had just shook her head in exasperation.\n\n\"Hey, Tammy,\" he said when Betty switched over the line.\n\n\"Chief, I know what you're going to say, but I've got to tell you anyway,\" she said.\n\n\"Okay, shoot.\"\n\n\"Roger didn't come home last night.\"\n\n\"And what did you think I was going to say about that?\"\n\n\"Well,\" Tammy said, \"first I figured you'd say, 'Good. It would be better for you if he never came home.' Then I figured you'd say, 'Well, there's no cause for alarm. He's probably off somewhere sleeping off a hangover.'\"\n\nCarlson smiled. \"You know, Tammy, you're psychic.\"\n\n\"And you're right on all counts,\" she replied. \"The only thing is, this morning, he had a court date up in Great Barrington, and I know he really wanted to go because he hoped to get some charges dismissed against him. . . .\"\n\n\"Failure to pay child support, wasn't it?\" Carlson asked.\n\n\"Yes, he was agreeing to give up all parental rights to his kid in order not to have to pay another single cent to her,\" Tammy told him.\n\n\"What a great dad.\"\n\n\"Believe me, I think that little girl will be better off without him in her life.\"\n\n\"Maybe you ought to think the same about your own little girl,\" the chief told her.\n\nTammy started to cry. \"I know, I know. I'm going to leave him. Really, I am. I just . . . right now . . . like this morning. I know Roger's a waste of a human being. But at least he can take Jessica to school for me. I pick her up in the afternoons when I get out of the diner, but Deb's got me working the breakfast shift, and so I can't take Jessica and she's scared of the bus. So Roger comes in handy once in a while.\"\n\n\"That's why you weren't at the diner this morning,\" Carlson said. \"But even though you didn't call asking for my advice, Tammy, I'm going to say anyway that Roger's occasional chauffeur assistance really doesn't outweigh his drinking, his temper, his violent flareups.. . .\"\n\n\"I know, I know! And believe me, if you find that he's dead in a ditch out there, I'm not going to cry one single freaking tear over him.\" She grew quiet. \"But I thought I should at least report that he didn't come home. I'll let you take it from there.\"\n\n\"Okay, Tammy, I'll mark it down,\" Carlson told her.\n\n\"You're not going out looking for him?\"\n\n\"Has he been missing for forty-eight hours?\"\n\n\"No, just since last night, when he went out to get cigarettes at Millie's store.\"\n\n\"Okay, well, then, you just keep us posted on whether he comes home, or not.\"\n\n\"All right, chief.\"\n\n\"Take care of yourself, Tammy.\"\n\nShe promised she would, but Carlson doubted it.\n18\n\n\"You've got to talk to her,\" Annabel said, alone at last with Jack in their room. \"You've got to tell your grandmother that in asking us to take over the place, she has to give her consent to some modernizations.\"\n\n\"I'll talk to her, babe,\" Jack promised. \"Gran's just sentimental. She's run this place a long time. She's attached to the way she and my grandfather used to do things. And then my dad and mom . . .\"\n\nJack's voice trailed off.\n\n\"What about your dad and mom?\" Annabel asked.\n\n\"Well, Dad took over after Granddad died. And I remember he wanted to make some changes to the place, too, but . . .\"\n\nOnce again his voice trailed off. He walked over to the window and looked out into the tangled arms of trees.\n\n\"What is it, Jack?\" Annabel asked, her voice becoming compassionate. \"Has coming back here made you think of your parents' deaths?\"\n\nHe nodded, still looking out the window, away from her. \"This was the last place I ever saw my mother. She was here one day, absolutely fine. Next thing I knew, she was gone, off to the hospital in Boston. I never even knew she had cancer until she was gone.\"\n\nAnnabel walked up behind him and placed her hand on his back. Jack so rarely spoke of his parents, especially his mother. She died when he was in his teens of breast cancer. Now that Annabel knew he had lost a little sister as well, she felt tremendously sad for her husband. His childhood had been filled with tragedy.\n\n\"I hadn't realized you had been visiting here when your mom was taken away to the hospital,\" she said softly.\n\nHe turned back around to look at her. \"We had come up here to start the process of helping Gran after Granddad died. Mom had been pretty excited about the idea. She had lots of ideas, just like you.\" His voice thickened and he couldn't go on for a moment. \"But it wasn't meant to be. Within a week of us getting here, she suddenly got sick and Dad took her to Boston. I never even had a chance to say good-bye. I just came downstairs one morning and Mom and Dad were gone. Dad came back late that night and told me Mom was in the hospital. She died a few days after that.\"\n\n\"I hadn't realized she died so quickly,\" Annabel said. \"I mean, to seem so completely healthy one day, and then be rushed to the hospital and die a few days later . . . breast cancer is usually a far more lingering illness.\"\n\nJack's face darkened. \"Well, that's what Dad told me she died from.\"\n\n\"You think it might have been something else?\"\n\n\"I don't know. But it always did seem so fast and strange. The last time I saw Mom, she was happy and singing and down there in the parlor supervising some workers who'd come to do renovations. She was excited to have a project. She had so many ideas about fixing the place up. And then she was gone.\"\n\n\"Obviously, your father didn't want to continue with her renovation plans after she died,\" Annabel said.\n\nJack shook his head. \"He was too distraught, I guess. Cindy disappeared not long after that, too. So Dad was never the same.\" His face showed the sadness he carried. \"That's when I was sent off to boarding school in Connecticut. Dad died a few years later himself.\"\n\nAnnabel took his hands in hers. \"Jack, was coming back here a bad idea?\"\n\nHis eyes met hers. \"No. This is our chance to start over. To finally make something of our lives. To become successful.\"\n\n\"Well,\" Annabel said, \"to do that, we'll need to honor your mother's wishes and redo this place like she wanted to do.\"\n\n\"Yeah,\" he said, nodding. \"That would be a nice tribute to Mom.\" He smiled weakly. \"Though, as I recall, Gran wasn't keen on her changing things, either.\"\n\n\"The only way to turn this old dump into a moneymaker is to renovate it,\" Annabel told him. \"That's the only way we can become successful here.\"\n\nJack nodded again. \"You're right, babe. I'll speak to Gran and tell her she's got to let us do what we need to do.\"\n\n\"Thank you, Jack,\" Annabel said, reaching up and kissing him lightly on the lips.\n\nThe little kiss led to another, and then several more. In moments, they were kissing deeply, the first time in a long time. Annabel had feared this moment, had dreaded the idea of being intimate with Jack again, but now that the moment had arrived, she didn't push it away. She wanted things to be right between her and Jack. They'd embarked on this journey together. They needed to be united, committed. They were starting over.\n\nShe kissed Jack hard, fumbling with the buttons of his shirt.\n\nHe cupped her breasts with his hands.\n\nIn moments, they had tumbled backwards onto the bed. Jack had slipped off his shirt and was now pressing Annabel's over her head. She felt his hot, wet breath on her neck and shoulder. His hands were now pulling down her jeans. She heard the jangle of Jack's belt buckle unfastening. Annabel tensed and waited.\n\nBut then . . . nothing.\n\nJack flopped over onto his back beside her. His eyes were staring straight up at the ceiling.\n\n\"I'm sorry,\" he whispered.\n\n\"It's okay, Jack,\" she said.\n\nAnnabel didn't know how she felt. Disappointed? Relieved? She reached over and stroked her husband's face, but he gently pulled away from her touch.\n\n\"I guess I'm just too . . . I don't know . . . too worked up,\" he said, still looking at the ceiling. \"It's all I think about. This has got to work, you know, baby cakes?\"\n\n\"What has got to work?\" she asked quietly.\n\nHe finally pulled his eyes away from the ceiling tiles and looked at her. \"This,\" he said. \"This house. This business idea. This taking over and making it ours.\"\n\n\"We'll do what we can, Jack. We aren't miracle workers.\"\n\nHe sat up, his face suddenly tense. \"No, it's got to work! I won't take any failure! I tried so hard with that goddamn book, Annabel. I thought I had the whole success thing figured out. And I failed, sweetheart. I failed!\"\n\n\"Jack, publishing is a tough business. You didn't fail. The company just didn't market your book the way they should have.\"\n\nHis eyes grew dark. \"Bullshit. The book was crap. I'm a lousy writer.\" He suddenly grabbed Annabel by the shoulders, making her jump. \"This is my last chance, angel pie. I've got to make this fucking guesthouse the most successful inn in all of New England! I've got to make us rich! This is my goddamn last fucking chance!\"\n\n\"Jack, putting that kind of pressure on yourself isn't going to help.\"\n\nHe let go of her shoulders, his eyes narrowing at her. \"What's the matter, sweet cakes? Don't you want to be successful? Seems to me, after all you've been through, you'd want a second chance to prove yourself, too.\"\n\n\"Prove myself to who?\"\n\n\"Annabel,\" her husband said, \"I don't want this house to blow up in our faces, leaving me to rot somewhere in the city and you back on the blow.\"\n\n\"I'm never going back on the blow, Jack,\" she told him. \"No matter what happens.\"\n\nHe stood, shrugging. \"You gotta hope not. That's why we need this to work, angel face. For both of us. Get a big glossy profile in Travel & Leisure magazine.\"\n\nAnnabel didn't stand. She just sat there in her panties and her unhooked, crooked bra, looking at him. \"I fully intend to do everything I can to make this place successful, Jack. That's why I wanted you to set some ground rules with your grandmother. But it's not life or death, Jack. I refuse to see it that way. If I learned anything in rehab it's that we always have choices. We always have options. If not this, there will be other things\u2014\"\n\n\"No!\" Jack cut her off. \"It's this, babe! This!\" He suddenly looked defeated. \"I don't have the strength to try again if this fails.\"\n\nHe buttoned up his shirt and buckled his belt and headed for the door.\n\n\"I'm going to go talk to Gran. Tell her we're going to start renovating the place.\" He smiled at Annabel. \"First thing tomorrow morning.\"\n\nShe gave him a weak smile of consent. He left the room.\n\nAnnabel wasn't unhappy. This was what she wanted. She had some great ideas about how she could fix the house up. She'd start with the parlor, then the bedrooms.\n\nBut Jack's all-or-nothing attitude troubled her. She guessed that, in her own struggle in overcoming her addictions, she'd failed to see just how profoundly Jack had been affected by his own career troubles. Annabel had known how disappointed he was, but she now understood the disappointment had gone very deep. It had been publicly very humiliating for him to get such universally terrible reviews. It had called into question his whole life's game plan. Suddenly, she felt terribly sorry for Jack.\n\nShe fixed her bra, slipped her shirt back on, and pulled up her jeans.\n\nThe only thing to do was to get moving.\n\nAnnabel pulled out her computer and hit the power button, before suddenly remembering there was no Internet in this godforsaken place. How was she supposed to find the best local contractors to hire to start the work on the house?\n\nThe old-fashioned way, she told herself.\n\nShe dug out of her pocket the card she had picked up at the market yesterday. MILLIE WESTERBROOK, it read. Proprietor. And underneath was the phone number.\n\nAnnabel whipped out her phone. Not many bars, but enough. She entered the number of the market.\n\n\"Woodfield Market,\" a woman's voice chirped.\n\n\"Hi, this is Annabel Wish. I was in yesterday?\"\n\nA moment of silence on the other end.\n\n\"I just moved into the Blue Boy Inn.\"\n\n\"Oh, sure,\" Millie said, her voice filling with recognition. \"What can I do for you, honey?\"\n\n\"I'm wondering if you might be able to recommend a good contractor.\"\n\n\"Well, the best around is Charlie Appleby. He and his sons do good work.\"\n\n\"Terrific,\" Annabel said. \"Do you think they'd be able to start work right away?\"\n\nMillie laughed. \"Had enough of all that dust and gloom already, huh?\"\n\n\"I figure if we can start now, maybe we'll be up and running by summer.\"\n\n\"Charlie's pretty busy, but he could probably get one of his boys to start giving you a hand. Hold on. Let me get his number for you.\"\n\n\"Thanks, Millie.\"\n\nAnnabel smiled. It felt good to be taking the first step.\n19\n\n\"Oh, I'm sorry,\" Jack said. \"I didn't know anyone Owas in there.\"\n\nPriscilla blushed a deep crimson, holding the towel as tightly around her as possible. She had just stepped out of the shower when the door to the little bathroom had opened and Jack had walked in. She didn't think he'd seen anything, as she'd already been wrapping the towel around herself. But she couldn't be sure.\n\nNow he stood on the other side of the door, having closed it again, apologizing.\n\n\"It's all right, really,\" Priscilla told him. \"I should have locked the door.\"\n\n\"I forgot there were no bathrooms in the individual guest rooms,\" he told her. \"We're going to be changing that. When you come back next time, every room is going to have its own private bath.\"\n\n\"Wow, that's ambitious,\" Priscilla said.\n\n\"We've got a lot of big plans,\" Jack told her through the door.\n\nPriscilla thought it was a little odd that he kept talking to her, knowing that she was naked and dripping inside here. But she kind of liked it. Jack was very handsome. He'd made her heart flutter as she was talking to him. That never happened with Neville.\n\nShe suddenly felt bold. Making sure her towel was firmly secured around her, Priscilla grabbed the door handle and walked out into the hallway.\n\nJack was still standing there, as if waiting for her. She noticed the way his eyes looked her over. Priscilla felt a tingle of electricity race through her body.\n\n\"It's good to have plans,\" she told Jack. \"I can't wait to come back and see how you've remodeled the place.\"\n\nHe didn't reply immediately. His eyes were too busy eating her up.\n\nPriscilla was being very naughty, and she knew it. Jack was a married man. But what harm could a little flirting do?\n\n\"Yeah, well, you won't recognize the place,\" Jack told her.\n\n\"I would hope you'd talk with me first before you start knocking down walls.\" It was a new voice. They both looked around.\n\nThe old woman, Mrs. Devlin, stood at the end of the corridor. Priscilla let out a little gasp.\n\n\"I was just heading back to my room,\" she chirped, and hurried down the hall.\n20\n\n\"Come into my room,\" Cordelia told her grandson. She could feel her lips tightening into a scowl.\n\nJack obeyed. Once they were inside, Cordelia closed the door behind her.\n\n\"Now, don't worry, Gran,\" Jack was saying. \"We're not going to start tearing down walls. Anything we do, we'll include you in on the plans.\"\n\n\"Now, listen to me, Jack,\" she said, her tiny frame standing up to him, her neck craned to look up into his face, her bony finger pointing. \"This house has stood for more than a century. There is an integrity to this house. Your father believed so, and his father before him. You can't come in here with a wrecking ball.\"\n\n\"I told you, Gran. It will be nothing like that.\"\n\nHer fingers curled into two tiny fists. \"When I'm gone, do whatever you want. But for now, Jack, please, just leave things as they are.\"\n\n\"Gran, Annabel has some ideas. Good ideas. They'll make the place even better.\"\n\n\"Ideas!\" She sniffed. \"Other people have had ideas, and it's done them and us no good.\"\n\nJack made a face as he looked at her. \"Are you talking about my mother?\"\n\nCordelia wished she hadn't said that. She tried to backtrack. \"Your mother was a dear, kind woman. She meant well. But if she had succeeded in tearing up the place as she had planned . . .\"\n\n\"Mom wasn't going to tear up the place, Gran,\" Jack said. \"And neither will Annabel and I. Maybe if Mom had succeeded in fixing the place up all those years ago, it wouldn't have gone into the red.\" He made another face, which Cordelia took to indicate sympathy, but which seemed simply idiotic to her. \"I've gone over the books you gave me, Gran,\" Jack went on. \"This place hasn't made any money in years.\"\n\n\"We get by,\" she told him.\n\n\"Gran, I've spent too much of my life just getting by,\" Jack said. \"You asked me to come up here and take over from you. I plan to do that. And I plan on making a success of things. A success of me!\"\n\nHer gnarled old fingers gripped his wrists. \"Jack, please, go slow. And please don't do anything without talking to me first.\"\n\n\"I told you we'd include you in everything,\" he said, smiling at her, lifting her hands to his lips to kiss them.\n\nCordelia's eyes stung with tears. Nearly sixty years ago she had come to this house a happy bride. How had it come to this?\n21\n\n\"Help me!!\"\n\nCordelia, aged twenty-five, had stood there, not believing what she was seeing.\n\nThe woman struggled. Her face was a mask of terror.\n\n\"Help me!\" the woman screamed again.\n\nCordelia tried to run to her, but her husband stopped her.\n\nAnd she had watched in sickening horror as the woman was pulled down into the darkness.\n22\n\n\"Just please go slow,\" Cordelia said to her grandson, her voice low and whispery.\n\nJack pulled her in for a big bear hug. She practically disappeared against his chest.\n\n\"Don't you worry about anything, Gran,\" he said. \"We'll respect the integrity of the house. You'll be pleased with the changes. You'll see.\"\n23\n\nAnnabel was having a grand time, mapping out designs for the house. She'd found a little table in one of the other, unused bedrooms and dragged it back to her own, making it into a desk for herself. She began sketching a rough blueprint of how she envisioned the parlor could look. One wall gone, the fireplace opened up, the windows enlarged . . .\n\nThis is just what I need, Annabel thought to herself. A project.\n\nIn New York, she had liked nothing more than to be immersed in a project. During her short tenure at Orbit, she had loved designing the look of the magazine, working with their graphics team to come up with a sleek, stylized presentation. She was always sketching, trying new ideas out. In those days, Annabel was at her happiest, most fulfilled.\n\nToo bad her nights had been consumed with coke.\n\nBut that was over now. She had triumphed. She had made it through rehab, survived her breakdown, and emerged whole and healthy. She had not wanted to come to Woodfield with Jack. She'd felt pressured into doing so, as if she'd had no other choice. But maybe that was changing. Maybe the challenge of remodeling this place would be just what she needed.\n\nBecause Annabel needed something.\n\nShe needed to feel as if she mattered. As if she could run her own life. She needed to feel strong and capable again. She'd triumphed over the addiction, but what she still felt she hadn't regained was the self-confidence. She wondered if she'd ever really been self-confident, if maybe all her time as a hotshot New York fashionista and designer had been a sham, if she'd been fooling herself and everyone else. She wondered if she had ever really believed in herself, or if she had been putting on a show\u2014if she had still been, at heart, the scared little girl Daddy Ron had locked in the closet.\n\nYet Dr. Adler at the hospital had told her that before she could believe that she was strong and capable, she needed to believe she was safe.\n\n\"Do you feel safe, Annabel?\" Dr. Adler had asked her.\n\n\"I feel safe here,\" she had replied, meaning the hospital, \"but I'm not sure I'll feel safe anywhere else.\"\n\nShe had hated the hospital\u2014hated feeling confined and boxed in\u2014but at least she had felt safe there.\n\nI need to feel safe here, too, Annabel told herself. And I can feel safe by making this place my own. So long as it doesn't feel like my own, then I won't feel safe.\n\nAnd feeling unsafe was a terrible way to live.\n\nAlready she had hallucinated since coming here. Her mind was sometimes going to play tricks with her\u2014Tommy Tricky, as a matter of fact. Dr. Adler had warned her about that. But she couldn't succumb to such tricks. She had to teach her mind to distinguish between reality and illusion. She had to make her mind strong\u2014working it out every day, the way people worked out their bodies by going to the gym. And the best mental calisthenics Annabel could do were the designs and blueprints for her renovation of the inn.\n\nIf Annabel could meet the challenges of living here at the Blue Boy\u2014if she could find her courage and her strength and her self-confidence here\u2014then all the pain and struggle she had gone through would have been worth it. But most important, she needed to do it on her own. Jack had been a steadying influence during the worst of her ordeal. He'd been a great support. But Annabel needed to do this next part on her own. That was vital.\n\nI need to find a way to feel strong and safe again on my own, she told herself, as she sketched out the windows she had in mind to open up the parlor.\n\nFrom a corner of the room, she heard scratching.\n\n\"I may need to find an exterminator as well,\" she said with a long groan.\n\nDamn it. The idea of vermin running around inside the walls of the house creeped her out, made her flesh crawl. She'd heard scratching downstairs in the parlor. Probably mice. Or maybe chipmunks. Annabel hoped it was nothing worse than that.\n\nShe stood and walked over to the wall. Crouching down, she pressed her ear low to the wall. She could definitely hear something, but in fact, she realized it wasn't the same sound she'd heard downstairs. Up close, Annabel wouldn't describe it as scratching. It sounded more like . . . scuffing. Like someone was scuffing along the floor behind the wall....\n\n\"That's crazy,\" Annabel said out loud. \"Mice and chipmunks don't scuff.\"\n\nBut it sounded bigger, heavier than a mouse or a chipmunk.\n\nPressing her ear harder against the wall, Annabel was surprised to discover that a part of the wall buckled just a bit. A sliver of the plaster actually moved. She looked down.\n\nShe saw a small panel, about two feet by three feet, where the wall met the floor.\n\nSlowly, carefully, Annabel slid the panel back. If there was some animal behind there, she didn't want it jumping out at her. But the sound was gone now. There was utter silence. Still, Annabel moved very slowly, frightened that a skunk or a possum was about to poke its snout through the panel at her. With careful deliberation, she pushed the panel aside, trying to catch a glimpse of what was behind the wall.\n\nIt was too dark to see. Annabel stood and hurried over to her pocketbook, snatching up her phone. There might not be any reception up here, but she could still use her flashlight app. She pointed her phone into the panel and switched on the light.\n\nShe saw no animal or any sign of one. There was just a very narrow space that seemed to lead all around the room. No way was she going in there to explore. Annabel hated enclosed, tight spaces. She just wanted to close this panel and seal it shut\u2014\n\nBut then her eyes landed on a small dusty pile of books just beyond the panel opening.\n\nHesitantly, Annabel reached in and grabbed the book on the top of the pile.\n\nBringing it back out into the room, she looked down on its cover. The book seemed ancient. The binding was cracking and covered in mold. The mold made a pretty blue-and-white pattern across the black leather cover. Carefully, Annabel opened the book. She let out a little gasp when she read the words on the frontispiece.\n\nDAEMONOLOGY\n\nInvocations and Spells\n\nThe date at the bottom was 1862.\n24\n\n\"Stay back,\" Chief Richard Carlson told the kids who had gathered along the trail. \"Stay behind Deputy Burrell.\"\n\nThe call had come in about half an hour ago. Some kids had found something along the path through the woods. One of them, little Julie Chen, had run home to her mother screaming and then Mrs. Chen had called the police.\n\nThe kids claimed it was a body.\n\nA dead guy, they said.\n\nRichard had ordered the kids kept back. A group of curious adults had gathered by now, too, and Adam was holding them back as well while the chief made his way up the path.\n\n\"It's right over there,\" called one of the kids who had found whatever it was. \"Just off the trail beside a log.\"\n\nRichard could see something in the direction the kid was pointing. It looked like a clump of dark brown blankets from where he was standing. He continued toward it.\n\nWhat if it was a body?\n\nWhat if it was murder?\n\nThere hadn't been a murder in Woodfield since Richard had taken over as chief. The job was very different from his time working on the Boston police force, where he'd had to turn over a dozen dead bodies a year, all of them potential homicides until proven otherwise. During his years in Boston, Richard had seen his share of bloodshed. But since coming to Woodfield, his worst problems were those damn town meetings where everybody was fighting with everybody else. That and the occasional underage drinking down at the lake, or thrill riders going ninety down the old twisting back roads.\n\nWoodfield was a quiet, peaceful little village. But Richard knew it hadn't always been so.\n\nBefore he'd come to town, there had been a string of unexplained deaths. Many of them were out at the Blue Boy Inn, but not all. On slow days, Richard sometimes went through the cold case files, lifting the bulging manila folders down from the shelf and leafing through them. For such a small town, there were an awful lot of unexplained deaths.\n\nAnd from the looks of it, there had just been another one.\n\nRichard turned back toward his deputy. \"Adam,\" he called. \"Get the EMTs here pronto.\"\n\nHe heard Adam make the call.\n\nThe chief knelt down beside what from a distance had looked like a brown clump. It was brown all right. Brown, hardened blood. The corpse was lying on its right side and its clothes were all drenched in blood. The poor guy's face was turned, pressed into the dirt, so Richard didn't recognize him as a local. At least not right away.\n\nHe felt for a pulse. He knew there wouldn't be one, but he did it anyway. The guy was dead. And cold. Richard guessed he'd been dead for a while.\n\n\"What is it, chief?\" somebody called from the crowd.\n\nHe wasn't sure he could say exactly. From the looks of it, the guy might have been jumped as he walked down the path. Or maybe he'd been dumped here. But it was clear he must have been stabbed to produce so much blood.\n\nRichard took out his phone and snapped several pictures. Then, carefully, he turned the corpse on its back.\n\nThat was when he recognized him.\n\nHoly shit.\n\nRoger Askew.\n\nHe heard Tammy's voice.\n\nRoger didn't come home last night.\n\n\"He sure as hell didn't,\" Richard whispered.\n\n\"Hey, chief,\" Adam shouted over to him. \"The EMTs are on their way.\"\n\nRichard nodded.\n\nHis first thought was this was a drug deal or a gambling debt gone bad. Roger Askew had lots of enemies. It wasn't going to be easy to find his killer.\n\nRichard was looking down at the corpse. Roger's right arm was twisted under his body at an odd angle. All Richard could see was his shoulder. Gently, the chief reached forward to examine the arm. The odd angle might be a clue as to how Roger died.\n\nBut as he felt under the cold, stiff body, Richard uttered a little sound in surprise.\n\nThere was no right arm!\n\nIt had been severed just below the shoulder. That was why there was so much blood.\n\nRichard stood up. The sun was dropping a little lower in the sky and the woods were filling up with shadows, strange twisting shapes cast by the bare branches of the trees. Had the arm been cut off for some reason, and then tossed aside? The chief stepped away from the corpse, walking through the dead leaves, trying to see if he could discover the arm. But the leaves weren't disturbed for several yards around the body. There was no sign of any more blood.\n\nHe and Adam would do a more thorough search, and the state forensics team would likely be called in. But at the moment, Richard suspected that Roger's arm had been cut off and then his killer had taken it away.\n\nBut why?\n25\n\n\"Well, Miz Wish,\" Zeke was telling her, \"what can I say? It's an old house. There have been many people who've lived and stayed here over the many decades. And maybe one of them had some rather unique reading interests.\"\n\nAnnabel had told him about the books she had found behind the panel. \"Well, they really disturbed me,\" she said, shivering.\n\n\"I hope you threw them away,\" the caretaker told her.\n\n\"No,\" Annabel admitted. \"The others I left inside the panel. The one that I took out and examined, I placed outside, on the wood box. Would you take it down to the library and donate it, Zeke? It's so old that I thought maybe some historian would want it.\"\n\n\"Such blasphemy ought to be burned,\" Zeke said. \"Don't worry. I'll take care of it.\"\n\n\"Good. I just want it out of the house.\" She rolled out the diagrams she had drawn up onto the parlor table. \"And maybe nail shut that panel for me, too?\"\n\n\"Happy to oblige, Miz Wish.\" Zeke smiled, looking down at the plans Annabel was showing him. \"What do we have here?\"\n\n\"Some ideas I have about redesigning this room,\" Annabel said. \"I've spoken with a contractor. He's coming by in the morning to help me get started. His name is Chad Appleby. His father is Charlie Appleby.\"\n\nThe old caretaker lifted his bushy white eyebrows. \"I've known Charlie since he was a boy riding his tricycle. Can't believe he's got a kid old enough to do contracting work.\"\n\nAnnabel smiled. \"He assured me that Chad is very good, that he thought he could give me a hand doing a few small jobs. When we move into the next stage, which will involve more intensive renovation, Charlie said he would come over to do the work.\"\n\n\"I see,\" said Zeke.\n\nAnnabel's smile changed into a smirk. \"Charlie added that he only hoped he wouldn't find any bodies stuffed inside the walls when he starts tearing them down,\" she said.\n\nThe caretaker shrugged. \"Well, those are the risks you take when you start moving things around.\"\n\n\"Look, Zeke,\" Annabel said, \"I need to know that I'll be able to count on you. I know Cordelia is worried that we'll destroy the historical character of the house. But trust me, that's the last thing I want to do. In New York, I helped redesign many old buildings. Staying true to the character and the integrity of the place was always one of the most important motivations.\"\n\n\"What exactly do you plan to do to the house?\" Zeke asked.\n\n\"For now, we're just going to start with this room,\" Annabel told him, gesturing around the parlor. \"It's the first thing guests see when they walk into the house. I want to clean it up and give it a good polish. We're going to paint the walls, replace the windows, and sandblast the floor. Eventually get some new windows, and take out that wall over there.\"\n\nShe walked over to the fireplace.\n\n\"And we're going to open up the fireplace again,\" she added.\n\nZeke just looked at her.\n\n\"I have a mason coming by as well tomorrow,\" Annabel said, stooping down and examining the bricks that had been mortared over the fireplace opening. \"I'll want to make sure the chimney is still sound. And I suspect we have mice or rats or squirrels living in there. I've heard a lot of scuttling. Up in my room, too.\"\n\n\"Listen, Miz Wish,\" Zeke said. \"I don't think you oughta open up the fireplace. I can tell you that the chimney is no good. And the ash dump down in the basement is all cracked. Why don't you start on something simpler? Fixing the windows is a good idea, and I'll help you paint the walls.\"\n\nAnnabel shook her head. \"We need a roaring fire in this room to ward off the cold this winter. How inviting will it be to walk into this room and feel the warmth of the fire, see the light flickering on the walls at night?\" She smiled, standing up, and turning around to look back at Zeke. \"In fact, I'd say fixing the fireplace is number one on my list.\"\n\nZeke stared at her. \"Have you told Cordelia?\"\n\n\"Jack spoke to her. Believe me, she's going to love how we fix the place up.\"\n\nZeke watched her as Annabel spread her plans out on the table, looking up from them at the walls and the windows, then down at her blueprints again.\n\nThe woman was a fool.\n\nShe walks into this house and thinks she can do what she likes, Zeke thought. She has no idea. None whatsoever.\n\nShe's bringing in a mason to check out the chimney.\n\nZeke knew that, one way or another, he'd make sure that mason told Annabel to leave the fireplace alone.\n26\n\n\"I think this calls for a glass of wine, don't you?\" Jack was asking.\n\nThey had all just come down to the dining room for dinner. Normally, the inn only served breakfast to its guests, but because a light dusting of snow was suddenly blanketing the roads, Annabel had offered to make dinner for everyone. Priscilla and Neville had thought that was a grand idea, since they weren't keen on skidding along back roads in search of some restaurant.\n\nJack uncorked the wine as Annabel began chopping vegetables in the kitchen.\n\n\"She's a vegetarian, you know,\" Jack told his English guests, pouring some merlot in a glass for each of them. \"Hope you don't mind a meal of carrots and lentils.\"\n\n\"I'm sure it will be delicious,\" Priscilla said, accepting her glass and taking a sip. \"Oh, this wine is divine.\"\n\nIt was just the four of them for dinner, plus Zeke, as Cordelia had complained of a headache and disappeared into her room. The rest of them sat around the dining table drinking their wine, Zeke sipping from a mug of beer.\n\n\"I should really go out to the kitchen and offer Annabel my help,\" Priscilla said.\n\nJack grinned over at her. \"You just stay right there,\" he told her. \"You're a guest. Annabel enjoys cooking.\" And he winked at her.\n\nPriscilla could feel her cheeks redden.\n\n\"Annabel said she's going to open up the fireplace,\" Neville offered, apparently oblivious to Jack winking at his girlfriend. \"A fire sure would be nice on a snowy night like this.\"\n\nJack was nodding. \"We've got some good ideas for this place. I was telling Priscilla earlier that if you come back a year from now, you'll never recognize it.\"\n\n\"Now, look here,\" Zeke said, gazing up at them from over his mug. \"You told your grandmother you'd go slow.\"\n\n\"Don't worry, Zeke,\" Jack assured him.\n\n\"And I'm not sure that chimney is fixable,\" the old caretaker said. \"Not sure you want to spend four grand to fix it your first month here.\"\n\n\"What's a bed-and-breakfast in the woods without a fireplace?\" Neville asked. \"I'm with you, Jack. Get that chimney smoking again.\"\n\nJack was smiling and refilling everybody's glass of wine. \"Absolutely,\" he said. \"We could be toasting marshmallows as we wait for dinner.\"\n\nThey all laughed, except Zeke.\n\n\"What is that American custom of marshmallows and chocolate over a fire?\" Priscilla asked.\n\n\"Do you mean s'mores?\" Jack laughed. \"Oh, sure, it's very tasty. Melted marshmallow and chocolate between a graham cracker sandwich. Sticky, but good.\"\n\n\"Sounds delectable,\" Priscilla said, allowing her eyes to find Jack's again.\n\nHis eyes locked on to hers. \"Gooey, sweet, and very satisfying,\" he told her, enunciating each word carefully.\n\nHer cheeks reddened darker.\n\n\"Well,\" Neville said, \"if we come back next year, I hope you'll have performed an exorcism on all the ghosts in the place.\"\n\nJack moved his eyes away from Priscilla and found her boyfriend. \"Have they been keeping you up at night?\"\n\n\"Only thing keeping me up is Priscilla jabbering with herself, thinking she's seeing spirits,\" Neville replied, before reaching over for the bottle of wine and refilling his glass.\n\n\"I am seeing spirits,\" she told him. \"Two nights in a row now I've seen Sally Brown. Poor thing. She's very confused. Doesn't even know her name. But she comes into the room and sits at the end of the bed.\"\n\n\"Oh, does she now?\" Jack said, smirking, winking this time at Neville.\n\n\"She does,\" Priscilla insisted. \"I keep trying to tell her that it's okay to move on, that she shouldn't be trapped here between worlds. But she tells me she can't leave, that they're keeping her here.\"\n\nZeke sat forward in his chair. \"Who's keeping her here?\" he asked.\n\nPriscilla shrugged. \"She hasn't said,\" she told him, knocking back the last of her wine and setting down her empty glass, which Jack moved to quickly to replenish. \"But if she comes by tonight again, I'll ask her.\"\n\n\"Just ask her quietly, okay?\" Neville quipped. \"I don't like being woken up.\"\n\n\"So you've seen nothing?\" Zeke asked.\n\n\"I fall asleep as soon as my head hits the pillow,\" Neville replied. \"She sits up waiting for her ghosts.\"\n\nJack stood, taking another bottle of wine out of the cabinet. \"You know, if it were up to me, I'd keep the whole supernatural reputation for the place,\" he told the group as he uncorked the bottle. \"I think it's a great selling point.\"\n\n\"It's a wonderful selling point,\" Priscilla said. \"But it's more than that. It's truth in advertising. You can't rent out rooms without telling people they might be visited by spirits in the night.\" She smiled as she took a sip of wine. \"Wouldn't be fair.\"\n\nThat brought another round of laughter.\n\n\"Well, Annabel doesn't like the idea,\" Jack said, sitting back down at the table, but this time taking the seat next to Priscilla, whose glass, though still half-full, he filled back up to the top. \"Maybe we can work on her.\"\n\nPriscilla giggled.\n27\n\nFrom the kitchen Annabel could hear them laughing.\n\nShe was glad they were having fun. It was good to hear laughter from real people in this gloomy old house.\n\nThe carrot-and-lentil soup bubbled on the stove. She'd also made rosemary popovers and an enormous salad. A good meal for a snowy night.\n\nMaybe this wouldn't be so bad after all. She had the contractors coming tomorrow. Soon they'd be opening this place up, letting in light, sweeping out cobwebs, and drying out the mold. Maybe this little adventure would be just what Jack hoped it would be, a new start for both of them. A path to success.\n\nThe wind whistled against the house, rattling the glass panes in the windows.\n\nAnnabel couldn't wait until a fire was blazing in the parlor.\n28\n\nTammy Morelli sat opposite Chief Carlson, her fingers massaging her temples. \"Well, sure, Roger had enemies,\" she said. \"Lots of people wanted him dead.\" She closed her eyes. \"Including me, sometimes.\"\n\nShe opened her eyes again. They were bloodshot from crying. Richard didn't understand how a woman like Tammy, basically a good, decent, hardworking person, could actually grieve over a lazy bum who had beaten her and used her. But Tammy had sobbed like a baby when Richard had given her the news that Roger was dead.\n\nMurdered.\n\n\"Anyone hate him enough to cut off his arm?\" Adam Burrell asked her.\n\nTammy shuddered. \"I have no idea,\" she said, massaging her temples harder.\n\nRichard felt sorry for her. \"I don't want to keep you any longer, Tammy. But if you can think of anything, like maybe the symbolism of his right arm . . . like maybe he did something to someone and they were cutting off his arm for revenge. . . .\"\n\nHer eyes snapped open and she was looking directly at Richard.\n\n\"But Roger was left-handed,\" she said. \"If he did anything to someone, he'd have done it with his left arm.\"\n\nThe chief nodded.\n\nAfter Tammy was gone, Richard and Adam sat in silence for a while. Outside the snow squall was ending. It had been a light winter so far, but that could change. It was still early. They could yet be buried in seven feet like they'd been last year.\n\n\"Tell me something, Adam,\" Richard said. \"You grew up here. The cold case files tell me that there were a number of unsolved murders in this town before I took over as chief.\"\n\nThe deputy was nodding. \"They stretch way back, more than a century.\"\n\n\"The last big flare-up was a little more than twenty years ago,\" Richard told him, remembering the files he'd perused. \"So you must remember that.\"\n\n\"Sure do,\" Adam said. \"I was around seven years old at the time. My parents were terrified. Kept me in the house, wouldn't let me go outside to play. There was even stuff on the news about the Woodfield Serial Killer.\"\n\n\"I seem to recall from the files that four people were killed in a matter of a few days.\"\n\n\"Well, four people went missing, never to be seen again. But only one body was found. The police chief at the time presumed there was a link.\"\n\n\"And why was that?\"\n\n\"Because they'd all either been living or working at the Blue Boy,\" Adam told him.\n\nThe chief stood, walking over to the shelf and retrieving several folders. Placing them down on the table, he thumbed through the top file.\n\n\"Yes, here it is,\" he said. \"A man and his wife had been staying at the inn. He reported she went for a walk and never returned. No body was ever found. But murder was suspected given the fact that the very next day Cynthia Devlin, the owners' granddaughter, also went missing. Although again no body was found, the little girl's blood was discovered all over the grounds. There was speculation a bear might have killed her.\"\n\n\"It wasn't a bear,\" Adam said. \"Because there were two other guys as well.\"\n\nRichard flipped forward a few pages in the file. \"Yes, here they are. Contractors. They'd come up from New York to do some work on the place.\" He read further. \"One would be reported missing by his wife. He never returned to New York. The other was found in the woods outside the Blue Boy, a bullet through his heart.\"\n\n\"Yeah, that sounds about right.\"\n\nRichard couldn't figure it out. \"There doesn't seem to be a pattern, except that they were all connected somehow to the Blue Boy Inn.\"\n\nAdam shuddered. \"My parents always told me to stay away from that place, that it was haunted,\" he said.\n\nThe chief was still reading through the file. \"It says here that the owners were all questioned and were cleared of any suspicion.\" He read a little further into the report. \"Cordelia had just taken over the place, her husband having recently died. Her son was questioned, it says here, but having lost his daughter, the poor guy was pretty shaken up, and he moved away soon after that.\"\n\n\"Hey, chief,\" Adam asked, leaning back in his chair, his hands behind his head, \"are you thinking that Roger Askew's death might be somehow connected to those deaths twenty years ago?\"\n\n\"I can't see how it's possible,\" Richard said, closing the file. \"Roger was killed half a mile away from the inn. But I'd like to look into those cold cases regardless. The file left it all a complete mystery, saying no suspects or motives could be found, especially since only one body was ever found.\"\n\nAdam smirked. \"It'll give us something to do. It's been pretty boring around here lately.\"\n\n\"Don't let anyone know we're reopening those cases,\" Richard told him. \"Officially, we're only investigating the death of Roger Askew. I have a feeling that one will be easy to solve as soon as we start talking to Roger's cohorts. But as you're talking to people, ask what they remember about the Blue Boy twenty years ago.\"\n\n\"Will do, chief,\" Adam said, bolting out of his chair, replacing his cap, and heading out the door.\n\nRichard sat back down at his desk. He thought of that woman who'd just moved to the Blue Boy, the one he'd met at Millie's store. Such a pretty woman. Annabel, she'd said her name was. Richard hoped he wouldn't rattle her too much asking questions about the Blue Boy's bloody past.\n29\n\nAnnabel wasn't pleased by how drunk everyone was. Even Zeke seemed to have had too much beer. The other three, including her husband, had polished off three bottles of wine. Her dinner had been a hit\u2014Neville had asked for three helpings of the soup\u2014but now Annabel wished she'd served something more substantial to soak up all the alcohol everyone had consumed. Everyone but herself, of course.\n\nShe was disappointed in Jack. He'd still had the occasional glass of beer or wine even after Annabel had come home from rehab. She didn't expect him to go sober just because she'd had an addiction. But he'd never gotten drunk in all that time.\n\nUntil tonight.\n\nAnd he was flirting shamelessly with Priscilla.\n\nAnnabel stood. \"I'm going to clean off these plates and make some coffee,\" she said, scooping up her plate and Jack's.\n\n\"Coffee?\" Jack blurted. \"I don't want coffee. How about we open another bottle of wine?\"\n\n\"I think you've all had enough,\" Annabel said, piling the three other plates onto the two she held in her hand.\n\n\"Aw, come on, Annabel,\" her husband said, \"don't be such a spoilsport.\"\n\n\"No, she's probably right, Jack,\" Neville said. \"I've had plenty. And Priscilla is such a lightweight.\"\n\n\"Pretty girls usually are,\" Jack said, winking openly over at Priscilla, who blushed a bright scarlet.\n\nAnnabel carried the plates out to the kitchen.\n\nAll she could think of was Rachel Riley. Her bleached hair and big tits filled up Annabel's mind.\n\nShe placed the plates into the sink. Neville came in behind her, carrying soup bowls.\n\n\"Thank you,\" Annabel said.\n\n\"You're a marvelous cook,\" Neville told her. His cheeks were flushed from drinking, his mosaic of pimples redder than usual. \"Really, I'm usually a meat-and-potatoes sort of bloke, but this was superb.\"\n\n\"I'm pleased you liked it.\" She kept her eyes averted, focusing her attention on filling the sink up with soapy water.\n\n\"I do think it's a good idea to get the house fixed up and fireplace cleared. I wish you all the luck with that.\"\n\nFinally, Annabel turned to look at him. She smiled. \"I appreciate that, Neville,\" she said.\n\n\"You know, I've heard some rustling sounds from down there,\" he added. \"I'm afraid you might have some vermin to deal with when you pull up those bricks.\"\n\n\"Yes,\" Annabel agreed. \"I've heard it as well. Maybe just a couple stray squirrels. At least, that's what I'm hoping.\"\n\n\"Yes, well,\" Neville said, and he seemed suddenly at a loss as to what else to say.\n\nThey looked at each other awkwardly.\n\n\"Can I help you here?\" he asked finally.\n\n\"No, thank you.\" Annabel nodded toward the door to the dining room. \"I'm fine here. Please go back and keep Priscilla company.\"\n\nNeville smiled, nodded a little, and then headed back into the dining room.\n\nHadn't he seen the way Jack had been flirting with his girlfriend? Was he blind? Clueless? Or maybe he didn't care.\n\nAnnabel sighed, dropping her hands down into the soapy water. She couldn't go back out there quite yet. She hoped that Neville would take Priscilla off to bed. Then Annabel would tell Jack that they, too, should call it a night. She wouldn't mention his obnoxious behavior. No need to play the jealous wife. She and Jack needed to be united in the morning, when the contractor arrived and Cordelia started throwing up roadblocks to the renovation. Besides, Neville and Priscilla were leaving in the morning. They had to get down to Hartford to catch a flight to Florida late tomorrow afternoon.\n\nBut as Annabel washed dishes, the laughter from the dining room only continued and got louder.\n\nShe brought out the coffee. Zeke's head had dropped down onto his chest and he was snoring lightly. Jack was regaling Neville and Priscilla with a story about the time he'd been at some fancy restaurant in New York right after his book came out, and people as diverse as Anna Wintour and Mayor Bloomberg and Lady Gaga were coming up to him to congratulate him. That had never happened.\n\n\"Here,\" Annabel said, pouring some coffee for her husband and pushing the cup over at him. \"Drink this.\"\n\nHe ignored her, continuing on with his story, which now had turned into how he turned down an offer to write a Broadway show because they wouldn't pay him enough. He described the way he'd told off these imaginary producers and he had Priscilla and Neville laughing so hard that tears were popping out of their eyes.\n\nAnnabel sat back and watched them. Drunk people were so ridiculous. She hated to think she'd once been like that, at some public function and as high as a weather balloon. She kept noticing the way Jack winked over at Priscilla when he was finished with one of his stories. She decided she couldn't watch any more, so she got up from the table and walked out of the dining room and into the parlor.\n\nAnd suddenly the whole room was different.\n\nIt was as if someone had slipped a mickey into her coffee. Some sort of hallucinogen. The room seemed to sway and vibrate. Annabel had to reach out and touch her hand to the wall to steady herself.\n\nFrom behind her the laughter from the dining room continued, only now it got absurdly louder and then seemed to disappear entirely for a few seconds, as if the merrymakers were holding their party underwater. Annabel tried to clear her head. She stood in one spot, holding on to the wall, taking long, deep breaths. She closed her eyes.\n\nWhen she opened them again, Tommy Tricky was standing in front of her.\n\nGnashing his sharp blue teeth.\n\nAnnabel let out a small scream.\n\nBut the creature was gone. A figment of her imagination. The room continued to spin. What was happening to her?\n\nThe laughter surged. Annabel felt as if her legs would give out from under her. She made her way across the room by holding on to the wall. She reached the fireplace and looked down at the bricks that sealed off the opening.\n\nShe heard scraping coming from below.\n\nScraping, scraping, scraping.\n\nA hand was on her shoulder. Annabel gasped.\n\nTurning, she saw Neville, as if in a dream.\n\n\"Vermin,\" he said, his eyes crazy. \"Vermin.\"\n\nAnnabel thought she'd pass out. She nearly fell onto the fireplace, holding on to it to keep from falling to the floor. Neville was gone. Had he ever been there?\n\nOnce again, Annabel made her way around the room, her right hand against the wall to keep herself steady. She turned the corner back into the dining room.\n\nAnd there was Jack fucking Rachel Riley on top of the table.\n\nAnnabel closed her eyes and opened them again.\n\nNo, not Rachel. Jack was sitting very close to Priscilla and they were about to kiss. Her husband looked over at her and smiled.\n\nHis mouth was full of sharp, broken teeth.\n\nAnnabel cried out and ran upstairs, shutting herself in her room.\n\nBut it wasn't her room. She was in a closet. A very small, cramped, dark closet.\n\nDaddy Ron had put her in there.\n\n\"Turn around, Annabel,\" her stepfather's horrible, jagged, drunken voice rasped through the door. \"Turn around and see who's behind you!\"\n\n\"He's not real!\" Annabel shouted, her hands in her hair.\n\n\"Aw, Tommy don't like it when people say he's not real. Gets him real mad.\"\n\nAnnabel spun her head from side to side, looking into the darkness.\n\n\"Hear him sharpening his teeth?\" Daddy Ron asked.\n\nShe could. She could hear the devil's teeth gnashing, anticipating the moment he bit down into her flesh.\n\n\"He's right behind you, Annabel!\" Daddy Ron shouted, and then he laughed.\n\nShe had to get out of there. All around her, linens were stacked neatly on shelves. Her mother's linens. There was a hamper beside her filled with dirty clothes. It was a tight space. So small. She was stuck there, using up all the air. Pretty soon there would be no oxygen left and Annabel would die.\n\nShe had to break free. She began pounding on the door, swinging her arms out, knocking all the linens off the shelves.\n\nAnnabel was trapped! Her claustrophobia took over and she screamed.\n\nThat was when she saw the little boy's hand resting upon her shoulder.\n30\n\nPriscilla staggered up the stairs to her room. What had happened down there? She was drunker than she had ever been before. She had allowed Jack to keep refilling her glass because he excited her. Excited her far more than Neville had ever done. More than any man had ever done.\n\nBut now she was lost in a fog of her own thoughts and desires. What had happened? Her blouse was unbuttoned. Where was everybody?\n\nShe took another step and tripped. She had to grab ahold of the bannister to keep from falling down.\n\n\"Neville?\" she called in a small, whispery voice.\n\nThe stairs were moving. The whole house seemed to be swaying. Priscilla clung to the bannister for fear she'd tumble down the stairs.\n\n\"Neville?\" she called again.\n\nHe had gone upstairs. She thought she remembered him saying good night, that he'd had far too much to drink himself and was calling it a night.\n\nHe had left her alone with Jack.\n\nZeke was gone, too, and Annabel.\n\nIt had just been Priscilla and Jack.\n\nShe tried to button up her blouse, but her fingers wouldn't work.\n\nShe reached the top of the stairs and stumbled into the hallway. It was very dark and very quiet. Priscilla could hear herself breathing. At least the house had stopped spinning. She took a step down the corridor. Her room was only a few feet away.\n\nWhat had happened down there? She wished she could remember.\n\nAhead of her, a figure approached.\n\nA figure in white.\n\n\"Sally,\" Priscilla said softly. \"Sally, help me. . . .\"\n\nSally Brown approached her. She looked at Priscilla with eyes that seemed both sympathetic and accusatory. Priscilla reached out to her.\n\nSally smiled, and grabbed hold of her hand. A glint of moonlight reflected off Priscilla's opal ring. She followed the ghost down the hall, and then began climbing some stairs. At the top of the stairs was a door. Sally opened the door and they passed through.\n\nThe attic. They were in the attic.\n\nBut then Sally was gone and Priscilla was alone.\n\nShe turned around a few times, got dizzy, and dropped down to sit on an old stuffed chair that smelled like dust and mold. She sat there for a while, breathing heavily, until her head stopped spinning again.\n\nIt had started to rain outside. At least, Priscilla thought it had. She could hear a soft tap-tapping on the walls and roof of the attic.\n\nPriscilla looked around the room. A small lamp on the table provided a very dim light. \"Sally,\" she whispered. \"Sally, where are you?\"\n\nThere was a small rumble of thunder off in the distance.\n\nAt least, Priscilla thought it was thunder.\n\nThe rain was hitting the house harder now. An icy rain, Priscilla thought. She imagined the long icy fingers that scratched the roof of the house. She shivered.\n\n\"Sally!\" she called again. \"Where have you gone? What have you brought me here to see?\"\n\nDespite her dizziness and confusion, Priscilla was excited. This was exactly why she'd come to the Blue Boy Inn. To see ghosts. And now Sally had brought her to a place where she'd see plenty of ghosts, Priscilla was sure.\n\nAll at once, a huge thunderclap made her jump, and the lamp went out.\n\n\"Oh, no,\" Priscilla murmured. She loved her ghost adventures, but she'd prefer not to experience them in total darkness. That was just a little too creepy.\n\nShe breathed a sigh of relief when the light flickered back on again.\n\nOn the table in front of her, Priscilla spotted a candle and an old book of matches. The candle was little more than a stub. Priscilla considered lighting it, but seeing that it was so small, she didn't want to waste the wax while the electricity was still on. Should the power go out again, she'd light it. She kept the book of matches near her hand so she could find it easily if the darkness returned.\n\nShe smiled. Despite all she'd had to drink, she was still thinking clearly.\n\nThe lamp flickered. Priscilla felt her heart flutter.\n\nShe looked up. And she saw in the darkness a hand holding a knife.\n\nPriscilla screamed.\n\nIt took her several minutes to calm herself. \"Well, that was a good one, Sally,\" she said out loud. \"Who was that? The person who killed you? Does his ghost walk here, too?\"\n\nShe had learned that ghosts could only hurt the living in very rare circumstances. Priscilla wasn't afraid of being hurt. She was afraid in a fun-house kind of way, the way she might feel watching a scary movie.\n\nThunder again, the loudest yet, directly over the house.\n\nThe light struggled to hold\u2014\n\n\u2014shivered\u2014\n\n\u2014and then went out.\n\nDarkness.\n\nPriscilla held her breath.\n\n\"Come back on,\" she whispered.\n\nBut the darkness remained.\n\n\"Oh, well,\" she said, feeling for the matches.\n\nHow terribly dark it was in the attic. Gripping the box of matches in her left hand, Priscilla felt around for the candle with her right. What if the power didn't return? The little stub would never last.... She moved her hand over the tabletop. Where was the candle? It had been sitting right there! The darkness was absolute. Deep and thick. The rain kept up its pummeling of the roof. She prayed for a flash of lightning just to show her the candle. But all she got was a low rumble of thunder.\n\nThere!\n\nShe felt something in the dark. The candle\u2014\n\nShe moved her fingers to grip it.\n\nAnd whatever it was that she touched\u2014moved!\n\nIt was a hand! A human hand!\n\nSomeone was in the dark with her!\n\nPriscilla gasped.\n\n\"Who's there?\" she asked. \"Who is it?\"\n\nOh, this was exciting!\n\nBut she'd prefer it without the total darkness.\n\nFinally, a flash of lightning. The room lit up for an instant. Priscilla saw she was alone in the room.\n\nAnd there\u2014there was the candle!\n\nShe grabbed it as the darkness settled in again. Fumbling for the matches, Priscilla found that her hands were trembling. But still she managed to strike a flame and shakily light the wick of the candle. A small, flickering circle of light enveloped her. She sat back in the chair, awaiting whatever vision Sally had to show her next.\n\nThe memory of the hand she had felt\u2014\n\nIt was small, she thought. Like a child's.\n\nShe lifted the candle and stood. She was far too anxious to stay seated. As she moved into the center of the room, Priscilla realized she was stepping in something sticky.\n\nWas rainwater dripping in from the walls?\n\nShe lowered the candle.\n\nAnd she could see plainly that it wasn't water.\n\nIt was blood!\n\nShe looked up. And there, in the candlelight, was Sally.\n\n\"Oh, Sally,\" Priscilla said. \"I'm glad you're back.\"\n\nBut then she saw that Sally was the one holding the knife.\n\n\"Sally,\" Priscilla said, \"why are showing me this?\n\nSally took a step closer to her, pointing the knife at her.\n\n\"Sally! Please stop this!\"\n\nThe ghost swung the knife, nicking Priscilla's arm. She drew blood.\n\nSuddenly, Priscilla was terrified.\n\n\"Sally, no!\"\n\nThe ghost kept coming closer. Priscilla turned and ran.\n\nIn her mind, still swimming from the wine, the small space of the attic suddenly seemed cavernous. She ran and ran, for many minutes it seemed, down an endless corridor that stretched farther and farther off into the distance. How could this be happening? How could she keep running for so long? What had happened to this room?\n\nBehind her, Sally's footsteps echoed as she pursued her. Thunder clapped overhead. Priscilla just kept on running, down that impossibly long corridor.\n\nAnd then she stopped, her head spinning as if she were riding an out-of-control carousel. She turned around. Sally was right behind her, smiling sweetly.\n\n\"Oh, Sally,\" Priscilla said. \"You gave me such a fright.\"\n\nThen the knife came plunging down into Priscilla's face.\n31\n\n\"Wake up, you old fool.\"\n\nZeke opened his eyes. Sunlight was streaming into his room and Cordelia was standing over him, glowering at him, her arms akimbo.\n\n\"What time is it?\" Zeke asked, sitting up, just as a headache pushed him back down.\n\n\"It's past time, you miserable man,\" Cordelia scolded.\n\n\"Oh, no,\" Zeke said, sitting up again, slower this time, rubbing his forehead.\n\n\"I've taken care of things,\" Cordelia said. \"No thanks to you.\"\n\n\"Is everything all right?\"\n\nCordelia's eyes darkened. \"Everyone is sleeping off their hangovers in their rooms. After all that noise last night, I presume they'll sleep well past noon.\"\n\nThe old man got up out of bed. He was still wearing the clothes he had on yesterday. He smelled of beer and sweat.\n\n\"It's been so long, Cordelia,\" he told her. \"So long since I just had a good time . . . unwound . . . had a few beers. Don't be angry at me.\"\n\n\"You had more than a few beers, apparently. I came in here two hours ago and couldn't get you to wake up for the life of me.\" She folded her arms over her chest. \"So I had to take care of things myself.\"\n\n\"And everything was all right?\"\n\nCordelia narrowed her old eyes at him. \"I think she's figured out how to deal with that lock.\"\n\n\"The door was locked, wasn't it?\" Zeke asked.\n\n\"It was locked,\" Cordelia said. \"But I don't think it was locked all night.\"\n\n\"What makes you think so?\"\n\nShe brushed her hand at him. \"I don't have time to stand here jabbering with you. We have things to do today.\" She scowled. \"Annabel has some people coming to see her.\"\n\n\"The mason,\" Zeke said, nodding.\n\n\"Make sure he stays far away from that fireplace,\" Cordelia said, before hobbling out of Zeke's room.\n32\n\nAnnabel opened her eyes.\n\nFor a moment she had absolutely no idea where she was. She looked up at the ceiling and didn't recognize it. She sat up and looked around. The room made no sense to her.\n\nThen she realized she was at the Blue Boy Inn, and Jack was snoring beside her. They both were wearing the same clothes they had worn the day before.\n\nWhat had happened last night?\n\nShe'd hallucinated, that was what had happened. She'd thought she'd been locked in a closet by Daddy Ron. It had been horrifying.\n\nYou're safe, Annabel, she heard her therapists telling her. You are completely safe.\n\n\"I'm safe,\" she whispered to herself.\n\nAnnabel swung her legs off the bed and placed her feet\u2014still in shoes\u2014against the floor. Her hands covered her face and she felt as if she might cry. She hadn't had a childhood flashback like that in a very long time. During her breakdown and her time in rehab, such flashbacks had come frequently. Many a night she had thought she was back in the closet, locked there by Daddy Ron. But recently Annabel had begun to hope that she was finally free of such nightmares, that she had finally moved past those terrible memories.\n\nApparently not.\n\nDr. Adler had warned her that the flashbacks might come back if she was stressed or experienced some sort of trauma.\n\nAnnabel removed her hands from her face and looked over at Jack, snoring like a grizzly bear beside her.\n\nShe remembered the trauma of last night.\n\nJack had been flirting with Priscilla. He may have even had sex with her.\n\nAnnabel stood and pushed herself over to the mirror. She looked into her eyes. They were puffy and bloodshot.\n\nWhat had she seen? Jack and Priscilla. What had they been doing?\n\nShe turned to look at her husband.\n\nHow she hated him.\n\nNo, she told herself. You love Jack. He stayed with you through everything. You love Jack. You owe him so much.\n\nAnnabel had to get out of that room. She felt boxed in. The air was stale and smelly.\n\nOut in the hallway, she breathed better.\n\nAnd suddenly she remembered the contractor and the mason were coming this morning. Yes! For some reason, the thought cheered her. She looked at her watch. They would be there in an hour. She hurried to the bathroom to get ready.\n33\n\nChad Appleby was finishing up his breakfast at Deb's Diner.\n\n\"You want more coffee, Chad?\" Tammy asked him.\n\n\"Sure, Tam. Just a splash.\"\n\nShe refilled his cup.\n\n\"You know, I can't say I liked Roger,\" Chad said, \"but I'm sorry if what happened has left you upset.\"\n\nTammy gave him a wan smile. \"I appreciate that, Chad. I suppose it's good that he won't be coming around anymore. Jessica was scared of him.\" She sighed. \"But nobody deserves to get stabbed to death and get his arm cut off.\"\n\n\"Cops have no idea who did it?\"\n\nTammy shook her head. \"The chief has talked to all of Roger's friends and all of his enemies, and nobody seems to have had anything to do with it, or figure out any motive. I mean, why cut off his arm?\"\n\nChad used his last piece of bacon to wipe up the egg yolk on his plate and then forked it into his mouth. \"And they haven't found the arm yet, have they?\" he asked as he chewed.\n\nAgain, Tammy shook her head. \"The whole thing creeps me out,\" she said, before moving on down the counter to refill the next customer's cup.\n\nChad looked at his watch. Paulie was supposed to meet him here fifteen minutes ago. But Paulie was always late. Back in high school, Paulie got more demerits for showing up late to class than any other kid in their class. He was a stoner, but Paulie was also a damn good builder and mason, and that was why Chad had asked him to come out with him this morning to the Blue Boy Inn.\n\nHe couldn't believe that was where he was going. Dad had asked him, \"Hey, Chad, you want to take on one very weird job?\"\n\n\"And what would that be, Dad?\"\n\n\"The Blue Boy Inn wants its chimneys inspected,\" his father had told him.\n\nThe Blue Boy Inn. Everybody in town knew that old place was haunted. Or at least had so much creepy history that it should be haunted. There was the time that little girl went missing but blood was found all around the place. And a guy was found shot dead in the woods a stone's throw from the place. Plus, there were stories that lots of people had died in the rooms over the last hundred years or had never been seen again after they'd gone inside. The Blue Boy was legendary in these parts.\n\nAnd the two old people who ran it, that guy Zeke and that ancient Mrs. Devlin, looked like they were cast members of The Addams Family, all wrinkled and hunched over and dressed in black.\n\nChad had told his father that he was glad to take the job. He'd never been inside the Blue Boy, and he looked forward to finally getting a peek inside the spook house.\n\n\"Sorry I'm late, my man,\" came a voice behind him suddenly, a hand on his shoulder.\n\nChad looked around. Paulie had arrived, and his red, glassy eyes revealed he'd been four-twentying in his truck on the drive over.\n\n\"Sit down and have some coffee, Paulie,\" Chad told him.\n\nTammy brought him a cup.\n\n\"So you anxious to get a look inside the Blue Boy?\" Chad asked.\n\nPaulie grinned. He was a doughy-faced guy with floppy ears. They made an odd-looking pair, Paulie so soft and stout and Chad so chiseled, slender, and tall.\n\n\"Sounds cool to me, man,\" Paulie said, taking a sip of coffee. \"Hope we don't run into any ghosts.\"\n\n\"Remember that Halloween you and me and Nicky Malone went up there and threw rotten tomatoes until the old lady came out and scared us away?\" Chad laughed. \"Jesus Christ, we were bad kids. I'm surprised the old lady didn't call the cops on us.\"\n\n\"Maybe she did,\" Paulie said, his floppy ears wiggling. \"Maybe we were just too fast.\"\n\n\"Well, we're reformed now, aren't we, Paulie? Model citizens.\"\n\nPaulie laughed.\n\nIn truth, antics like the one at the Blue Boy, tossing those tomatoes, bothered Chad to remember. He'd never been a bad kid, really, but there had been other pranks like that. Like the time he and Nicky had pointed a DETOUR sign down an old dirt road and caused half a dozen cars to get stuck in ruts. And another time he'd rigged up a bucket of water over the front door of the high school and pulled a string so it doused stuffy old Mr. Hillcrest, the principal. Nobody could ever pin any of those things on Chad and his friends.\n\nNow he hated to remember them. He was twenty-four years old, and intended on doing Dad proud as his assistant in the family contracting business. His older brothers had no interest in taking over the company. They were lazy good-for-nothings. But Chad imagined a day when Dad retired and Appleby Contracting would be his own.\n\nMaybe he'd even hire old Paulie to work for him full-time. Providing the wacky weed wasn't still a daily ritual.\n\n\"Well, come on,\" he said all of a sudden, getting up off his stool and plunking down a twenty and a couple of ones on the counter. It was a bigger tip than Chad was used to leaving, but he felt sorry for Tammy. \"If we don't get a move on,\" he told Paulie, \"we'll be late getting to the Blue Boy. I told the new owner we'd be there at ten.\"\n\nPaulie downed his coffee and stood, a little shakily, to follow his friend out of the diner. Chad waved good-bye to Tammy and placed his hand on the door, but suddenly Paulie stopped him.\n\n\"You know,\" Paulie said, looking up at him with those bloodshot eyes, \"maybe we oughta smoke a little something before heading up to that haunted house.\"\n\n\"I'm fine, Paulie,\" Chad told him. \"And I think you're already higher than the Blue Boy's weathervane. In fact, why don't you ride with me? Leave your truck here.\"\n\nPaulie smiled. \"All right, captain. You're the boss.\"\n34\n\nCordelia watched from the window of her room. There were men coming to the Blue Boy. Men who intended to knock down walls and pry up bricks. She wouldn't let them.\n\nThey must be stopped.\n\nShe remembered the day she and her husband had first sealed up the fireplace. Cordelia had laid many of the bricks herself. Then, years later, she'd had to lay those bricks again, this time with her son.\n\nShe hoped that old fool Zeke wouldn't fail her again, as he had failed her so often recently. Zeke's infirmities were the reason she'd had to ask her grandson to come up and take over. Otherwise, she never would have involved Jack.\n\nBut she was going to have to involve him completely soon. Especially if his wife persisted in her cockamamie schemes to renovate the place.\n\nJack's mother, Cordelia's daughter-in-law, had had similar ambitions.\n\nAnd look what had happened to her.\n\n\"What are you doing?\" Cordelia had asked the young woman, all those years ago.\n\n\"I'm taking the bricks out of the fireplace,\" Jack's mother had replied. \"We should have fires in this room. It will make the place more inviting. Cozy and warm.\"\n\nCozy and warm.\n\nThe Blue Boy Inn had never been cozy and warm, not one day in all the years Cordelia had lived in the place.\n\nShe was going to have to speak to Jack. Soon. Very soon.\n\nThe way her husband had spoken to her.\n\n\"You're mad,\" she had said to him, when he finished.\n\nHis weary eyes told her that he was not mad. He was simply terrified.\n\nIn her mind, Cordelia heard again the screams of the mother whose baby had been taken from her. The little baby whose arm was found. Only the arm.\n\nHow that memory was seared into Cordelia's mind.\n\nShe pulled back the dusty old lace curtain and peered down onto the yard. She spied Zeke standing in the shadows of some tall pine trees. Good. He was there, at least. She prayed he'd be successful.\n\nShe was so intent on watching Zeke that she didn't hear the soft footfalls behind her.\n\nBut Cordelia certainly felt the sting of pain that suddenly seized the back of her head.\n\nWithout making a sound, the old woman crumpled to the floor.\n35\n\nZeke heard the truck rattling up the dirt driveway before he actually saw it. The snowfall from the night before had mostly melted by now, and the tires of the truck were spraying stones from side to side as they crunched over the path through the woods.\n\nThe old caretaker watched carefully as the truck came into view and parked at the far end of the lot. Two young men stepped out. One was tall and blond and rather handsome; the other was shorter, darker, and stout. Without saying a word to each other, the men made their way up toward the house.\n\n\"Good mornin',\" Zeke shouted, suddenly emerging from the shadows.\n\nThe men appeared to be slightly startled. They turned around to look at him. The shorter one seemed to sway just a little bit as he looked at Zeke, and his eyes were red.\n\n\"Good morning,\" the taller man said.\n\n\"May I help you?\" Zeke asked. \"I'm the caretaker here.\"\n\n\"We're here to see Annabel Wish,\" the man told him. \"I'm Chad Appleby and this is Paul Stueckel.\"\n\n\"I see,\" Zeke said. \"You're the contractors, I take it.\"\n\n\"Yes, sir, we are.\"\n\nZeke smiled with his crooked teeth. \"Is she askin' you to do a big job?\"\n\n\"I wouldn't know that yet,\" Chad told him. \"I haven't seen what she wants to do.\"\n\n\"Say she asks you to take down a few walls and restore a fireplace and chimney,\" Zeke said. \"How much would that cost her?\"\n\n\"I'm not sure,\" Chad said, seeming to grow a little leery of talking to him. He kept glancing up toward the house. \"I'd have to see the actual work it entailed, and the condition of the house. . . .\"\n\n\"How about if I told you,\" Zeke asked, drawing in closer to the men and speaking in a conspiratorial whisper, \"that whatever you quote Miz Wish to do the job she's asking, Mrs. Devlin will pay you that plus half to not do the job?\"\n\nThe men looked at him strangely.\n\n\"Mrs. Devlin is sort of sentimentally attached to the house as it stands right now,\" Zeke explained.\n\nChad frowned. \"Well, from what Ms. Wish told me on the phone, it's she and her husband who are now the owners.\"\n\n\"Paperwork hasn't been signed yet,\" Zeke told him.\n\n\"But if it's going to be,\" Chad replied, \"then I really ought to speak with the woman who asked me to come out here and give her an estimate.\"\n\n\"Mrs. Devlin will pay you double whatever you quote to leave the house alone,\" Zeke said, his voice hard as he upped the offer.\n\nThe two men exchanged looks.\n\n\"I think I need to speak with Ms. Wish,\" Chad said finally, heading off toward the house. His zoned-outlooking friend followed.\n\n\"Remember what I've offered you,\" Zeke called after him. \"But the offer's no good if you tell Miz Wish about it.\"\n\nThe men didn't look back. They just continued on toward the house.\n36\n\n\"Well,\" Annabel asked, after she'd given the contractors a tour, \"what do you think?\"\n\nChad looked around the place. \"To really fix everything up, to do the kinds of things you've drawn up in your plans, will cost a lot of money. We're looking at six figures. I can maybe keep it under two hundred grand, but I can't guarantee it.\"\n\nAnnabel frowned. \"That was what I was afraid you'd say.\"\n\nChad shrugged. \"I wish it could be less. But the house is really falling apart. All the electricity is going to need to be updated. And the plumbing . . . from what I can see, I'm surprised you haven't had a major episode yet. The pipes are all rusted out.\"\n\n\"Well, we'll start slow,\" Annabel said. \"A bit at a time.\"\n\nChad smiled at her. \"That's what Dad and I usually tell our clients. Go room by room.\"\n\nAnnabel returned his smile. \"And this is the room where we will start. The parlor.\"\n\nChad liked her. It helped, of course, that she was pretty damn hot, with that shapely body and shiny auburn hair. But she was also real smart, having drawn up some blueprints like a real pro, and had some really cool ideas about how to fix this old place up. It would be a hell of a lot of fun to help her do it.\n\nBut if he refused, the old gnome in the parking lot said Mrs. Devlin would pay him double. That was possibly three hundred thousand smackers for sitting on his ass!\n\n\"So we'll do this room first,\" Annabel was saying, gesturing around at the walls. \"Open it up. Bring in some more light.\" She strode over to the fireplace. \"What did you guys decide about the chimney? Is it salvageable?\"\n\nPaulie had been doing his own inspecting. His sleepy face brightened now that it was his turn to show off his expertise.\n\n\"Well,\" he said, \"from what I could see up on the roof and down in the basement, the chimney is in surprisingly good shape. Why did they brick it over?\"\n\n\"I have no idea,\" Annabel replied. \"The caretaker told me the chimney was broken.\"\n\n\"Why would he say that?\" Paulie wanted to know.\n\nChad stepped forward. \"Is the caretaker the little, hunched-over old man we met on the way in?\"\n\nAnnabel nodded. \"That's Zeke.\"\n\nSuddenly, he felt he needed to do the right thing. \"Ma'am,\" he said.\n\n\"Call me Annabel,\" she told him.\n\nThat just made him more determined to tell her the truth. \"Look,\" Chad said, \"I don't want to get anybody in trouble, but . . .\" He hesitated. \"That old guy told us Mrs. Devlin would pay us double whatever we quoted you not to do the renovation.\"\n\n\"What?\" Annabel seemed aghast.\n\n\"I just thought you should know. I mean, if we're going to work together . . .\"\n\nHer eyes were blazing. \"Cordelia asked us here, my husband and I, to take over the place. But then she tries to control everything we do.\"\n\nChad looked at her. \"The old man said the paperwork hasn't been officially signed yet. He said the old lady still calls the shots.\"\n\n\"She's already put my husband on the deed,\" Annabel told him, \"and my husband supports the renovation one hundred percent.\" She folded her arms across her chest. \"Besides, Mrs. Devlin doesn't have that kind of money. She just told Zeke to tell you that to get you to back off.\"\n\n\"That's what I figured,\" Chad said. \"Well, I just thought you ought to know.\"\n\n\"Indeed,\" Annabel told him. \"Thank you.\"\n\n\"So, yeah,\" Paulie said, looking down at the fireplace. \"I'd say that I could get this baby smoking for you in no time.\"\n\n\"Could you start right away?\" Annabel asked.\n\n\"Sure,\" Paulie said. \"I could come by later this week and get going. . . .\"\n\n\"No,\" Annabel said. \"I mean today. Could you start today? I'll pay you entirely upfront.\"\n\nHer eyes were filled with fire.\n\n\"Well,\" Paulie told her, \"I suppose I could.... I'd have to go back home and get some tools. . . .\"\n\nAnnabel smiled. \"I want Cordelia to come down those stairs and see the work has begun,\" she said.\n37\n\nBut what puzzled Annabel was why Cordelia had not yet come down the stairs at all this morning.\n\nAs she watched the two men rattle off in Chad's truck, with Paulie promising to be back before noon, Annabel supposed that Cordelia might have gotten up earlier this morning and then gone back to her room. She did that sometimes. And Annabel had slept a little later this morning anyway, given her bad dreams all night long.\n\nEveryone, in fact, was sleeping late this morning. Annabel tiptoed up the stairs and opened the door to her room. Jack was still sound asleep, flat on his back, still dressed in his clothes. Pausing outside the door of their English guests, Annabel could hear snoring from inside. Neville and Priscilla were apparently still sleeping off their drunks as well. They had a plane to catch later this afternoon, and the airport was at least an hour away. If they weren't awake soon, Annabel would have to wake them.\n\nShe headed back down the stairs.\n\nHad something happened between Jack and Priscilla last night?\n\nSettling down at the kitchen table with a cup of coffee, Annabel tried to make sense of what had happened last night. Jack had definitely been flirting with Priscilla. He'd been drinking too much. He hadn't had so much to drink in a long time, and apparently he'd lost his ability to handle his liquor. But had anything happened other than flirting?\n\nHad Annabel caught them kissing? Fucking?\n\nWhy couldn't she remember?\n\nIt was all just an hallucination, she told herself, rubbing her temples with her fingers. And that, in some ways, was even more distressing than the idea of Jack fooling around with another woman.\n\nShe'd had hallucinations during the worst of her addiction. At rehab, sometimes they'd been overpowering. Sometimes they had been so strong that Annabel couldn't distinguish between illusion and reality.\n\n\"You're safe,\" Dr. Adler, her favorite therapist, had insisted to her, but Annabel hadn't bought it.\n\n\"I'm not safe!\" she had screamed, as visions of demons walked through the walls. \"I'm completely unsafe! \"\n\nHow terrible those days had been.\n\nAnnabel prayed her hallucinations weren't coming back.\n\nBut ever since she'd come to this house, she'd had signs they were returning. She'd seen Tommy Tricky her first day here. And last night, after becoming upset with Jack, she'd thought the whole house was spinning. She'd thought she was a little girl again, put in the closet by Daddy Ron. Tommy Tricky had been in the closet with her. He had come at her. He had tried to eat her!\n\n\"Stop,\" she told herself.\n\nShe stood and looked out the window. Where was Zeke? She was furious with him. How dare he try to bribe the contractors? When Annabel found him, she'd let him have it. And when Jack woke up, she'd tell him exactly what his grandmother had tried to do. They wouldn't put up with such nonsense anymore. They needed to lay down the law. They were in charge now.\n\nAnnabel had to feel that she was in charge. Otherwise, she thought she might crumble into a million pieces and her hallucinations would take over again. Down into a black hole she'd tumble, and she'd never be safe again.\n38\n\nPaulie sat in his truck and lit his pipe. He inhaled the sweet, precious weed long and deep. He felt it fill his lungs. He'd parked far enough away from the house that nobody could spot him toking. In seconds, Paulie's mind was blissful and calm.\n\nBracing for the cold air outside, he opened the door of his truck and headed around to the back for his tools. Paulie slung them over his shoulder and then made his way back up to the Blue Boy Inn.\n\nHe didn't usually take jobs the same day. But Annabel had given him a check, payment in full, plus a fifty-dollar tip. How could he refuse?\n\nHe knocked on the door.\n\nAnnabel opened it quickly. She was very happy to see him. Welcoming him inside, she asked him if he wanted some coffee. Paulie declined.\n\n\"Thank you,\" he said, \"but I never drink on the job.\"\n\nThey both laughed.\n\nPaulie got down to work.\n\nFirst, he covered all the furniture in the parlor with drop cloths. Then he pulled on a pair of leather gloves and slipped some eye protection over his face. Finally, he got down to work with his chisel and his hammer. He began chipping away at the mortar.\n39\n\nUpstairs, Cordelia could hear him start to work.\n\nChip, chip, chip.\n\n\"No,\" she moaned.\n\nBut she couldn't move.\n\nShe couldn't stop him.\n\n\"No,\" she moaned again.\n\nBehind her came the sound of soft laughter.\n40\n\n\"I'm running to the market quickly,\" Annabel told Paulie. \"We've run out of coffee, and our guests will be awake soon. I'll be back in a few minutes.\"\n\nHe lifted his eye protection to look over at her. \"Sure thing,\" he said.\n\nAnnabel smiled. \"Wow,\" she said. \"You've already made progress.\"\n\nPaulie stepped back from his work. He'd chiseled out a two-foot-by-three-foot opening so far. The removed bricks were stacked neatly off to the side. \"I can feel the air from the flue,\" Paulie told her. \"I might have this baby blazing in a couple of hours.\"\n\n\"Thank you so much,\" Annabel said. She gave him a big smile before heading out the door.\n\nPaulie watched her leave. As soon as her car disappeared down the driveway, he stepped over the bricks and headed to the restroom.\n\nA little more weed would make the rest of the job go by very pleasantly.\n\nSafe inside the restroom, he lit up his pipe, making sure to crack the window. Just a few puffs was all he needed to feel nice and happy.\n\nHe returned to the parlor.\n\nAnd the first thing he noticed was that the bricks he had stacked so neatly were now knocked over. That was odd. He supposed he had disturbed them with his foot.\n\nHe got back to work. Another brick, and then another. The fireplace opening was gradually revealing itself. He chipped away at the bricks, stacking them up carefully at his side. Now the space was three feet by four feet. Once he'd removed all the bricks from the opening, he could see if any repairs were needed to the flue.\n\nPaulie reached inside the opening, feeling for the chute to the ash dump in the basement.\n\nHis hand brushed against something warm and soft.\n\nHe yanked his hand out of the opening.\n\n\"Great,\" he mumbled. \"A mouse or a rat.\"\n\nHe rummaged around in his tool bag for his flashlight. Switching it on, he shone the light into the opening.\n\nA little face looked back at him.\n\nA terrible little blue face, with a mouth full of fangs.\n\n\"Jesus!\" Paulie shouted, stumbling backwards, the flashlight falling from his hands.\n\nWhat was that?\n\nIt looked like a freaking little elf.\n\nBut maybe it was some kind of possum.\n\nCalming his fluttering heart, Paulie grabbed ahold of the flashlight again and got down on his hands and knees in front of the fireplace. He got as close as he dared\u2014he didn't want some mean old possum jumping out at him\u2014and peered once more into the fireplace.\n\nJust blackness now. There was nothing there.\n\nWhat was in that weed he'd been smoking?\n\nPaulie moved the flashlight around to examine the inside the fireplace. It was difficult to see, so he slowly, tentatively, stuck his face into the opening he'd made, the beam of the flashlight at his cheek.\n\nThat was when everything went red.\n\nExcruciating pain filled Paulie's eyes.\n\nThe flashlight dropped from his hand, clattering onto the floor.\n\nPaulie could feel sharp claws gripping his eyeballs like acorns, and then yanking them right out of their sockets. At the same time, other claws were clamping down onto his shoulders, easily slicing through his flesh and grabbing onto his collarbone.\n\nPaulie screamed.\n\nWith uncanny swiftness, his body was pulled through the opening in the fireplace and down into the darkness.\n\nIf someone had been standing in the parlor watching, the last they would have seen of Paulie were his sneakers disappearing down an extraordinarily large ash dump. His muffled screams could be heard for a fleeting few seconds.\n\nThen everything was silent.\n41\n\nMillie Westerbrook looked up as the little bell over the door tinkled. The woman who'd just moved into the Blue Boy Inn\u2014what was her name again?\u2014stepped inside.\n\n\"Good day,\" Millie called over to her.\n\n\"Good day,\" the woman called back, giving Millie a smile.\n\nAnnabel. That was her name. Millie didn't think her last name was Devlin, though. She had said something else. Something short and simple. Wells? No, it was something more whimsical than that. Mille thought it was amusing how so many of these young girls today didn't take their husband's last names. That was a good thing, Mille thought.\n\nIf she had ever have gotten married, Millie told herself, she would have kept her own name, too. But Millie hadn't gotten married. She'd never been asked. She'd waited and hoped, and finally given up. And back in Millie's day, women didn't ask men to marry them. So here she was, sixty-one and single. Millie supposed there were worse fates.\n\nShe was putting away some stock\u2014canned vegetables\u2014but she was also watching Annabel-from-the-Blue-Boy meander down the aisles. She was an awfully pretty girl. Nice figure. Lots of wavy, shiny auburn hair.\n\nMillie sure felt sorry for her, living in such a place.\n\nMillie had played cards with Agnes Daley a few nights ago. They'd gotten to talking about the Blue Boy, and all the terrible things that had taken place there. Agnes was the town historian, so she knew the inn's history. She told Millie that the first owner of the place had been a priest\u2014no, not a priest, Millie, thought, trying to remember. It wouldn't have been a priest back then. The house was built around the time of the Civil War, and Millie didn't think there were all that many Catholics in Woodfield back then. She supposed Episcopalians had priests, too, but she didn't imagine a great big Episcopalian church out there in the middle of the woods. No, the first owner had to have been a minister, of some long-forgotten Protestant church.\n\nBut what Agnes told Millie about this minister\u2014well, Millie just couldn't believe it.\n\nSeems he was a very bad man. Not a man of God at all. This minister, Agnes said, was hanged for witchcraft!\n\n\"Now, that's just plain crazy,\" Millie had said to Agnes.\n\n\"Read the history books,\" Agnes had replied.\n\n\"I don't know much about history,\" Millie had countered, \"but I do know they weren't hanging men for witchcraft at the time of the Civil War.\"\n\n\"They found other reasons to hang him,\" Agnes had insisted. \"But the whole town knew what kind of witchcraft he practiced.\"\n\nMillie had snorted. Agnes liked to act so superior, knowing everything about the town, all its past and its history. But maybe, in fact, there was something to the story, since a curse did seem to cling to the place.\n\nAll those murders. All those people seemingly swallowed up into nothingness up at that house.\n\n\"May I pay for this?\"\n\nMillie looked up as she heard Annabel calling to her. The pretty young woman was standing up by the cash register, holding a packet of coffee.\n\n\"Oh, yes, of course,\" Millie said, hurrying over to assist her.\n\nThat poor girl, she was thinking. That poor girl up in that frightening house....\n\n\"Did you speak with Charlie Appleby?\" Millie asked when she got behind the cash register, ringing up the coffee beans.\n\n\"Yes, indeed, he sent over his son Chad,\" Annabel told her. \"We've got a man there making renovations now.\"\n\nMillie raised an eyebrow. \"And Cordelia's okay with that?\"\n\nThe clerk remembered one of the few times old Cordelia Devlin had ever stepped into this market. She'd come in with that old handyman of hers, looking for duct tape to fix a leaking pipe. She had frowned deeply when Millie had asked if she ever thought about updating her plumbing. \"It's got to be plenty old,\" Millie had said.\n\n\"The house is fine as it is,\" Cordelia had grumbled. \"Nobody's touching it.\"\n\nAnnabel smiled. \"Well,\" she admitted, \"we did have to insist. She's very sentimental about the old place. My husband and I had to assure her that we plan to do nothing that will hurt the integrity of the house. We really respect the architecture. We just want to make it more modern, more inviting to guests.\"\n\nMillie dropped the coffee into a paper bag. \"Have you had any inquiries about guests?\"\n\n\"We have two guests right now!\" Annabel said happily. \"Please spread the word that we are open under new management and that soon the place will be a wonderful getaway, complete with all-modern luxuries and amenities.\"\n\nMillie smiled tightly. \"I'll let people know,\" she said.\n\n\"Thank you,\" Annabel replied, and then, with a little smile and wave, left with her coffee, heading back to her car.\n\nMillie returned to stocking her cans. Why did she feel so worried for that poor child? Surely the stories that the townspeople told about the place were just old wives' tales\u2014myths, legends, and rumors. There was nothing to them. Even if Agnes was right about the first owner being hanged\u2014even if he had done some terrible things\u2014that was a hundred and fifty years ago. Nothing that had happened up there since was in any way connected. It was just a series of unfortunate, random events.\n\nStill, Millie worried for that poor, pretty girl.\n42\n\nNeville came staggering down the stairs. His head was pounding. Why on earth did he drink so much last night?\n\nHe noticed the drop cloths covering the furniture as he passed through the parlor. The fireplace was now open. Bricks were stacked alongside. A flashlight was on the floor, rolling slightly back and forth, shining its light toward the opening. But no one was around.\n\nNeville made his way into the kitchen. Just as he entered, Annabel came through the back door, a bag in her hands.\n\n\"Well, good morning,\" she said, smiling. \"Just went out for more coffee. I figured everybody was going to need it today.\"\n\nNeville sat down at the table. \"Yes, coffee might help,\" he said, rubbing his forehead.\n\n\"I was wondering when you'd get up,\" Annabel said, starting the coffee immediately. \"I was going to wake you. You have a plane to catch soon.\"\n\n\"Oh, yes,\" Neville said, still massaging his head. \"Florida awaits. Thank God.\"\n\n\"Give yourself at least an hour to get to Hartford,\" she told him.\n\n\"Yes,\" Neville replied. \"I'll have a little coffee, then jump in the shower and then we're out of here.\" He looked around. \"Has Priscilla already had her coffee?\"\n\n\"No,\" Annabel said. \"She hasn't been down yet.\"\n\n\"Well, she must have,\" Neville told her. \"She wasn't in the room.\"\n\n\"She wasn't?\" Annabel looked over at him oddly as the coffee began to drip. Its aroma made Neville feel a little better already. \"That's odd, because I haven't seen her. I slept a little later this morning myself, but she hasn't been around for a least the past couple of hours. Could she have gotten up much earlier?\"\n\n\"Not likely, given how much she had to drink,\" Neville said. \"But come to think of it, I don't remember her coming to bed. I was pretty drunk, though. Maybe I just slept through it.\"\n\nAnnabel poured him some coffee. \"Maybe she slept elsewhere in the house,\" she offered. \"I haven't looked in the other rooms.\"\n\n\"Why would she do that?\"\n\n\"Maybe you can tell me,\" said Annabel, sitting opposite him.\n\nNeville looked at her. The coffee was reviving him, but he didn't know what Annabel was implying. \"What do you mean?\" he asked.\n\n\"Do you think . . .\" She struggled with her words. \"Do you think Jack was at all inappropriate last night? Could she have taken offense at anything he said? If so, I do apologize for him.\"\n\nNeville sighed and took another sip of coffee. \"Oh, you mean the way he was flirting with her? I don't think she would have been offended. From what I can remember, she seemed to enjoy it.\"\n\n\"It didn't bother you?\" asked Annabel.\n\nNeville shrugged. \"Priscilla's a pretty girl. Men seem to go for the pretty, bookish types. The ones who look quiet on the outside. I've gotten used to it. I'm not much of a looker myself, so if some handsome bloke like Jack can give Priscilla a little attention, I don't mind.\" He leaned in toward Annabel. \"But did it bother you?\"\n\n\"A little,\" she said. \"But Jack was right beside me all night, snoring like a bear.\"\n\n\"All night?\" Neville asked.\n\nAnnabel sat back in her chair. \"I assume so. He's up there now.\"\n\nNeville shrugged again. \"I'm not suggesting anything. But it's just curious that Priscilla isn't in our room.\"\n\n\"I'll go look for her,\" Annabel said.\n\nNeville just nodded.\n43\n\nAs Annabel passed through the parlor, she saw that Paulie was gone. She walked over to the fireplace and looked down. He'd gotten it opened, however. His flashlight was on the floor, its light still shining. Annabel reached down and shut it off. No need to waste the battery.\n\nShe glanced out the window. Paulie's truck was still in the driveway, as she'd observed when she'd driven up a few minutes earlier. So where was Paulie? Maybe out back, taking a break, smoking a cigarette? Or smoking something else, Annabel thought. She hadn't missed the fragrance of marijuana that clung to him.\n\nDid she go looking for Paulie or for Priscilla? She figured it was more urgent to find Priscilla. If they didn't leave soon, they'd miss their flight.\n\nBut Priscilla was nowhere to be found. Annabel searched every room of the house and she wasn't there. She even opened the door to her own room again and all she saw was Jack, still sound asleep like a bear in hibernation.\n\nShe noticed the small narrow steps at the end of the hall that led up to the attic.\n\nShe hadn't been up there yet. Zeke had told her it was dangerous. Annabel was sure it was. The rafters were probably so rotted that they'd snap underfoot. But still, she should check. What if Priscilla had been so drunk she'd gone up there for some reason?\n\nMaybe to get away from Jack?\n\nOr maybe . . .\n\nMaybe she and Jack had gone up there for a tryst. Jack had come down, but Priscilla had spent the night up there, passed out.\n\nThat was the thought going through Annabel's mind as she climbed the steep, narrow stairs. At the top was a door. She turned the handle.\n\nBut it was locked.\n\nWell, if she couldn't get in there, then Priscilla wouldn't have been able to, either. At least that ruled out the attic.\n\nAnnabel came back down to the second floor. She had looked everywhere! Where could Priscilla be? Had she gone outside?\n\nBut her coat was still hanging on the hook beside the front door.\n\nAnnabel thought of something.\n\nShe hadn't looked everywhere. She hadn't looked in Cordelia's room.\n\nAnnabel paused outside the old woman's room. She knocked.\n\n\"Cordelia?\" she asked softly.\n\nThere was no answer.\n\nHad the two of them gone off together?\n\nShe knocked again and called the old woman's name once more.\n\nStill no reply.\n\nAnnabel hesitated, and then turned the doorknob. The door was open. She went inside.\n\nCordelia was lying on her back on the floor. Her head was propped awkwardly against an old cast-iron figurine of a hawk, used as a doorstop.\n\n\"Cordelia!\" Annabel cried, rushing to her fallen grandmother-in-law and kneeling beside her. She noticed the small pool of blood behind the old woman's head.\n\nAnnabel grabbed her by the shoulders.\n\n\"Cordelia!\" she shouted. \"Are you all right?\"\n\nThe old woman's eyes flickered open. One clawlike hand clutched at Annabel's blouse.\n\n\"The fireplace,\" Cordelia croaked.\n\n\"What?\" Annabel asked. \"What are you saying?\"\n\nBut the old woman said nothing more. Her eyes closed and her head fell back.\n\nAnnabel rushed off to call an ambulance.\n44\n\nChief Richard Carlson watched the people of the house carefully as the body of Cordelia Devlin was carried out the front door on a stretcher.\n\nHere he'd been thinking of coming by the Blue Boy Inn to ask some questions about the cold cases of a generation ago and now, here again, was another mysterious death at the old bed-and-breakfast. Examining the body of the old woman, the coroner had declared it might have been accident. She was frail, and she might well have fallen, hitting her head against the iron doorstop. But until he had a chance to better examine the body, the coroner was reserving judgment as to cause of death.\n\nThat was wise, Carlson thought, as there were some other mysterious developments that might end up having some bearing on the case.\n\nThree people had disappeared that morning as well. Paulie Stueckel, whom Richard would see every morning at Deb's Diner, sipping coffee and eating a doughnut, looking as if he'd smoked a half-pound of weed as soon as he woke up. Priscilla Morton, a British tourist. And old Zeke, the caretaker of the place.\n\nAny one of them might have slipped upstairs and whacked the old woman on the head, then taken off.\n\nChad Appleby was pulling up in his truck.\n\n\"What the hell happened?\" he was asking as he hurried across the parking lot.\n\n\"Morning, Chad,\" Richard said. \"Or I guess I should say, afternoon.\"\n\nThe sun was directly above them now, weakly shining down on the van as Cordelia Devlin's body was loaded inside.\n\n\"Where's Paulie?\" Chad asked. \"I left him here just a little while ago.\"\n\n\"We don't know where Paulie is,\" Richard told him. \"That's why we asked you to come down here.\"\n\nChad looked absolutely befuddled. \"Last I saw him, I dropped him back at his truck. He was planning on heading back here to start work on the fireplace.\"\n\nThe chief nodded. \"And he did just that. He was hard at work when Ms. Wish left for the store, but he was nowhere to be found when she got back. She was gone less than fifteen minutes.\"\n\n\"And the old woman was dead,\" Chad said.\n\nRichard nodded. \"Tell me something,\" he said. \"Was Paulie high this morning?\"\n\n\"Chief,\" Chad said, looking at him as if he'd just asked the most absurd question of all time. \"You know Paulie's always high.\"\n\nRichard shrugged.\n\n\"Come on, chief,\" Chad said. \"You know Paulie. He's harmless. A stoner, sure, but a pussycat. He swerves to avoid hitting squirrels. He wouldn't kill an old lady.\"\n\n\"I have to ask questions, Chad.\"\n\nThe young contractor seemed to think of something. \"But, you know . . .\"\n\nRichard looked over at him. \"What is it, Chad?\"\n\n\"Have you spoken with Annabel?\"\n\nThe chief nodded. \"Yes, she's given a statement.\"\n\n\"Did she tell you what the old man told us?\"\n\n\"What old man?\"\n\n\"Zeke. The caretaker.\"\n\nRichard's ears perked up. Chad didn't yet know that Zeke was missing as well. \"No,\" he said. \"What did Zeke tell you?\"\n\n\"He told Paulie and me that Mrs. Devlin would pay us double of whatever we quoted Annabel to not to do any work on the house.\"\n\nRichard raised his eyebrows. \"No kidding. Why would she do that?\"\n\n\"Beats me. And I told Annabel about it. Boy, was she pissed. That's why she had Paulie start the work today. She said she wanted the old lady to come down the stairs and see the work being started.\"\n\n\"Really now?\"\n\n\"Yup. Crazy.\"\n\n\"So, what you're telling me is, Annabel Wish was very angry at her grandmother-in-law this morning.\"\n\nChad seemed to be uncomfortable with the implication Richard was making, but he couldn't deny the truth. \"Yeah,\" he said. \"She was angry.\"\n\nThe chief looked back at the house.\n\nNow he didn't have just three suspects anymore.\n\nHe had four.\n45\n\nJack sat at the kitchen table, blinking frequently, unable to fully comprehend what had happened here this morning.\n\n\"Gran,\" he kept saying. \"Gone. And I slept right through it.\"\n\n\"Jack,\" Annabel said, pouring him more coffee, \"there was nothing you could have done. She fell and hit her head. She didn't suffer long.\"\n\nHis eyes shot up at her. \"The police chief seems to think there might have been more to it than that.\"\n\n\"Yes,\" Neville said, sitting across from him, \"he was questioning all of us as if we'd killed her.\" He paused. \"Or rather, as if Priscilla had killed her, and taken off.\"\n\n\"That's crazy,\" Annabel said. \"Why would Priscilla kill Cordelia?\"\n\nNeville fixed her with his eyes. \"It does seem crazy. But then where is Priscilla? She's nowhere in the house!\"\n\n\"Her coat is still hanging in the foyer, and it's too cold to go outside without it.\" Annabel turned to Jack. \"Where is the key to the attic?\"\n\nHer husband looked at her without comprehension. \"The attic?\"\n\n\"It's the only place I wasn't able to check. It's locked.\" She met Jack's eyes and held them. \"Did you and Priscilla go up there last night?\"\n\n\"Me . . . and Priscilla?\" he asked.\n\nNeville stood. \"Yes,\" he said. \"Come on, man. This is no time for games. Whatever happened last night happened. For now, who cares? Just tell us what you know about Priscilla!\"\n\n\"Indeed,\" came another voice. \"I'd like to know that, too.\"\n\nThey all looked up. Chief Richard Carlson had just walked into the kitchen.\n\n\"I let myself back in through the front door,\" he said. His deputy was behind him, and behind him was Chad Appleby. \"I'd like your permission to search the house,\" the chief said. \"I'd like to see if we can find some clue to the three missing persons.\"\n\nJack, still sitting at the table, glowered.\n\n\"Do you have a search warrant?\" he asked.\n\n\"Jack!\" Annabel was horrified. \"We have nothing to hide! If they can find something to explain where Priscilla and Paulie and Zeke went, then let them search!\"\n\n\"I'd have to add my encouragement to that as well,\" Neville said. \"It seems highly unlikely right now, but if they can find Priscilla up in that attic, she and I might still make a plane bound for the sunny skies of Florida later this afternoon.\"\n\nJack just shrugged. \"Sure, go ahead. Search the place.\"\n\nRichard Carlson stood looking at him. \"When I came in, you were being asked what you knew about Priscilla Morton. Is there anything you can tell us?\"\n\nJack covered his face with his hands. \"Why is everyone badgering me? My head is killing me!\"\n\n\"Mr. Devlin,\" the chief said, \"we are just trying to understand what happened here this morning, and to locate three missing people.\"\n\nJack stood. His hands were running through his hair. \"Okay, so maybe Priscilla and I had a little too much to drink last night. That's not a crime, is it?\"\n\n\"No, it's not,\" the chief assured him.\n\n\"But then she went one way and I went another,\" Jack said, not looking at any of them. Instead, he stood at the back door, gazing out into the woods.\n\n\"You didn't take her up to the attic?\" the chief asked.\n\nJack spun around, his face furious. \"I don't even know how to unlock the goddamn attic door!\"\n\n\"Zeke usually carries the keys,\" Annabel said. \"But it's the one place I was unable to check this morning.\"\n\n\"We've got to get in there,\" Neville said. \"Maybe Priscilla is in there.\"\n\n\"And Paulie, too,\" Chad Appleby said from behind the chief.\n\n\"We might have to break the door in,\" the chief told Annabel.\n\nShe nodded.\n\n\"Go right the fuck ahead!\" Jack shouted, turning around to look out the back door again.\n\n\"All right, let's go,\" Chief Carlson said, and the five of them, minus Jack, made their way across the parlor toward the steps to the attic.\n\nNo one thought to examine the fireplace further, or to look down its ash dump.\n46\n\nRichard found everyone in this whole house suspicious.\n\nAs they climbed the steep narrow stairs to the attic, all sorts of thoughts were running through the chief's mind.\n\nJack Devlin was sure acting questionably. He seemed to know more than he was telling about the missing Priscilla Morton. And it appeared that both his wife and Priscilla's boyfriend suspected something had happened between the two of them the night before.\n\nBut that meant that both Annabel and Neville might be suspects as well. They would both have had a motive for doing harm to Priscilla, if indeed they had discovered she'd been fooling around with Jack.\n\nJust why they would have followed that with an attack on the old woman, however, did not make any sense.\n\nAnd where the hapless Paulie Stueckel fit into all of this, Richard as yet had no idea.\n\nBut he did know that Annabel had been very angry at both Cordelia and Zeke. Had she killed both of them, but only had time to hide Zeke's body?\n\nIf that was the case, her enthusiasm for searching the house didn't make any sense at all.\n\nThe chief had to admit he was stumped.\n\nHe reached the attic door and gripped hold of the knob. It was locked, all right.\n\n\"Adam,\" he said to his deputy. \"Give me a hand.\"\n\nThe two policemen pressed their shoulders up against the door.\n\n\"Stand back,\" Richard shouted down to the others on the steps behind him. He turned to Adam. \"Okay, on the count of three.\"\n\nAdam nodded.\n\n\"One, two, three!\"\n\nThey both rammed their shoulders against the door. It rattled but did not pop open. Strange, for such fragile old wood.\n\n\"Again!\" Richard shouted.\n\nThey positioned their shoulders once more.\n\n\"One, two\u2014\"\n\nBut just at that moment, the door opened. And there, standing in the doorway, grinning up at them, was Zeke.\n47\n\n\"What's going on here?\" the old man asked.\n\nAnnabel pushed forward on the stairs. \"Zeke! Where have you been? We've been looking all over!\"\n\nThe wizened little caretaker shrugged. \"There was all that noise coming from the parlor and I couldn't sleep.\" His eyes twinkled as he looked around at all of them. \"I often come up here when I need some quiet for my nap.\"\n\nStepping out of the room, he closed the door carefully behind him.\n\n\"Now,\" Zeke asked, \"what are all these people doing here?\"\n\nChief Carlson looked at him intently. \"We're trying to find out what happened here this morning,\" he said, \"and locate a couple of people. We thought maybe they might be in the attic. We've checked everywhere else.\"\n\nZeke looked from him over to Annabel. \"Seems a lot has happened since I went in to take a nap,\" he said.\n\n\"Zeke, Cordelia's dead,\" Annabel told him.\n\nShe watched his reaction. He placed a hand over his chest and declared, \"It can't be!\"\n\nQuickly, Annabel detailed how the old woman had been found. Zeke kept shaking his head in surprise and grief, but Annabel didn't believe him. She thought he'd already known that Cordelia was dead before he went into that attic.\n\n\"But that's not all,\" Chief Carlson added. \"A guest in this house and a young man doing work down in the parlor can't be found. I'd like to go up into the attic and look around.\"\n\n\"Oh,\" Zeke said, \"I was just up there. I can assure you no one is there. It's just an empty, dusty, old attic.\"\n\n\"Still, I'd like to see for myself,\" the chief said.\n\nAnnabel watched Zeke. She saw the unease in his old yellow eyes.\n\n\"Well,\" the caretaker said finally. \"If you insist.\"\n\nHe turned around, opened the door, and led the group inside.\n\n\"Maybe the rest of you should stay back,\" Carlson said, turning around to the others.\n\n\"No way,\" Annabel told him. \"This is my house now. I'd like to see what's in here.\"\n\nThe chief shrugged his consent.\n\nAnnabel followed him into the dark, musty space. The deputy was close at her heels, and Neville came last. The room was fit into the point of the roof, so that it was impossible to walk up straight except in the very center. Moving off to the far ends of the room necessitated lowering one's head as the slant of the roof decreased. The place stunk of mold and something else\u2014sweat, Annabel thought. There was only one dim, unshaded lightbulb hanging from the ceiling, which cast a pale amber light over the room. Annabel scanned the darkness, allowing her eyes to adjust.\n\nThe first thing she discerned was a small cot tucked into the far corner. Everywhere there were boxes and chests. An old dressmaker's dummy gave her small fright when she looked at it, thinking it for a second to be a person. And then her eyes alit on something very strange. A small, cracked toilet in plain view.\n\nShe looked over at Chief Carlson. She saw that he was taking it all in as well.\n\n\"My little hideaway,\" Zeke told them. \"Sometimes, when all our guest rooms are booked, I'll give up my own room and come up here to sleep. Or when there's too much noise downstairs, I'll retreat up here to nap.\"\n\nThe chief and deputy were looking around the place, lifting boxes, moving crates. It was clear that there was nowhere to hide, and that no one was up here.\n\nSo where were Priscilla and Paulie?\n\nIf Zeke was telling the truth, and he'd retreated up here to nap after confronting Chad and Paulie in the driveway, then he wouldn't have seen anything. But why did Annabel feel certain he wasn't surprised when she told him about Cordelia's death?\n\nFrom the way Chief Carlson was looking at Zeke, it seemed he shared Annabel's suspicions.\n\n\"All right,\" Carlson said. \"I'm going to have my men search the woods surrounding the house for any signs of Paulie and Ms. Morton.\"\n\n\"I'll help,\" Chad volunteered.\n\n\"So will I,\" Neville echoed.\n\n\"As will I,\" Zeke added.\n\n\"Just a moment,\" Annabel interjected. \"I think Zeke needs to explain why he accosted Chad and Paulie in the driveway and tried to bribe them into not taking the renovation job.\"\n\n\"Ms. Wish,\" the chief said, \"I'll question Zeke on my own about that.\"\n\n\"Well, I have nothing to hide,\" the old man added. \"Cordelia is\u2014was, I mean\u2014very sentimentally attached to this house. She felt very strongly that it shouldn't be changed and butchered up.\"\n\n\"But she asked Mr. Devlin and Ms. Wish to come up from New York and take the place over,\" Carlson said, eyeing him. \"Wouldn't she have expected them to make some changes?\"\n\n\"She felt they were moving too fast,\" Zeke said simply.\n\n\"You and I will talk more about that,\" the chief said. \"For now, it could very well be that Mrs. Devlin's death was a tragic accident and that Paulie and Ms. Morton had nothing to do with it, and that their disappearances have a logical explanation.\"\n\n\"Maybe they're off in the woods enjoying a little romp,\" Zeke offered.\n\n\"It's twenty-one degrees outside,\" the chief said. \"And Ms. Morton's coat is still on the hook.\"\n\nThe old caretaker just shrugged.\n\n\"Come on,\" Carlson said to Deputy Burrell. \"Call into the station and get a couple of other men down here. We'll divide up the woods.\"\n\nThe men began trooping back down the stairs. Annabel hung back, so she could say something in private to Zeke.\n\n\"Are you telling everything you know?\" she whispered.\n\n\"Honest to goodness, Miz Wish.\"\n\nShe held his eyes for a long time. Then she, too, made her way back down the stairs.\n48\n\nNeville's mind was racing.\n\nAs he clomped across the frozen earth, brittle twigs snapping under his feet, he struggled to make sense of how much everything had changed in the course of just a few hours. Yesterday he'd been thrilled that they'd soon be on their way to Florida. He'd only made this pit stop because Priscilla had wanted to see some ghosts. She had, and she was happy and satisfied, and Neville looked forward to a good, relaxing, fun time in Disney World, riding Space Mountain, and lying around the pool.\n\nNow Priscilla was gone, and no one knew where she was.\n\nSome worker was gone, too.\n\nHad they run off together?\n\nIt was possible. After watching Priscilla flirt with Jack last night, Neville supposed it was possible that Priscilla was bored with him and might take up with another man she deemed more exciting. Ever since they'd started dating, Neville had known Priscilla was out of his league. She was far too pretty for a plain, acne-scarred guy with a gut who was nearly ten years older than she was. It had been three years, but Neville had fully expected Priscilla to throw him over for some hotter guy at some point.\n\nThat she had not done so, Neville figured, had less to do with any feelings she had for him than for the simple reason that she just simply couldn't be bothered. Priscilla was never man-hungry. She didn't really like sex all that much\u2014which was a shame, Neville thought, with that killer body of hers. Priscilla would have much rather been out kayaking, or bird-watching, or hunting for her stupid ghosts. Neville had a feeling that she stayed with him only because he grudgingly went along with her quirky interests, not because she much cared about him, or the whole concept of having a boyfriend.\n\nBut maybe there were times she longed for something more. Maybe she secretly ached for a big, strong, virile man to come after her\u2014which hardly described Neville. But it did describe Jack Devlin, and maybe he had awoken a passion in Priscilla that she herself had never known was there.\n\nOnly one thing didn't fit. She hadn't disappeared with Jack.\n\nJack was very much in evidence. He insisted that they'd gone their separate ways after doing whatever they did last night.\n\nThe person who'd gone missing with Priscilla was a mason, this Paulie Stueckel guy, who, from the descriptions Neville had heard, was hardly what one would call virile. He was pudgy and kind of goofy-looking, and perennially stoned on pot.\n\nThat didn't sound like the sort of man who could turn Priscilla's head. Still, both of them were missing.\n\nBut the man's truck was still there. If Priscilla had left with him, they'd gone by foot, or somebody else\u2014a taxi?\u2014had picked them up. But why wouldn't she have taken her coat? Her wallet? Her money? Her passport?\n\nIt was all very strange.\n\nLooking behind bushes and clumps of cold, hard, bare trees, Neville had a sense that something terrible had happened. Priscilla wasn't going to be found. At least, she wasn't going to be found alive.\n\nHe dreaded calling her mum and dad. They'd never liked Neville all that much. Thought he was too old, and not good looking enough, for their daughter. They'd sneered at the idea of this holiday to America.\n\nSuddenly, his thoughts were shattered by the sound of voices up ahead.\n\nIt was Deputy Burrell. He was calling to the chief.\n\n\"Come here! Quick! I've found something!\"\n\nNeville ran.\n49\n\nRichard Carlson knew what he was looking at the moment he peered down into the box of firewood on the side of the house.\n\nThe gray, splotchy thing lying on top of the wood that looked like a deflated balloon was in fact Roger Askew's right hand.\n\n\"Call forensics back here,\" Richard shouted over at Adam. The deputy immediately began pressing keys on his phone.\n\nStaring down at the hand in the wood box, Richard considered the fact that Roger's entire arm, not just his hand, had been hacked off. So if this was indeed Roger's hand, someone had also taken the trouble, at some unknown point, to sever the arm at the wrist.\n\nRichard didn't know for sure that this was Roger's hand. But it likely wasn't from one of the two missing people. The thing in front of him was a few days old, and starting to decay. Richard's gut told him that this was Roger's hand, and that he'd been right to wonder if Roger's death was somehow connected to the cold murder cases at the Blue Boy Inn.\n\nAnnabel came around the corner and peered down inside the wood box. She let out a gasp and covered her face with her hands.\n\nIt was that little gesture that told Richard that Annabel was guiltless in all this. She was genuinely horrified. Sometimes Richard experienced what he called \"psychic moments,\" in which he seemed to get a direct line into people's thoughts. He figured lots of cops would know what he was talking about, the moment it becomes obvious some witness is either innocent or guilty. His training as a police officer had left him sensitive to the slightest tic in a person's expression, or a gesture of their hands, or the way they said a word. Sometimes he couldn't put his finger on exactly what had swayed him one way or another. It was just as if, in that moment, he saw inside their heart and knew their truth.\n\nOf course, he'd never base an arrest or an acquittal on such intuition. He would continue to gather hard, direct evidence before making any kind of final decision. But he'd never known his gut to be wrong when he experienced one of his \"psychic moments.\" And what his gut was telling him right now was that Annabel Wish had nothing to do with Cordelia Devlin's death and knew nothing about the locations of Paulie Stueckel and Priscilla Morton.\n\nShe was looking up at him, distress and revulsion and despair written on her face.\n\n\"This will just bring all the stories about murder and death back to the Blue Boy,\" she said, near tears. \"We were trying so hard to put all that behind us. . . .\"\n\n\"I'm sorry, Ms. Wish.\"\n\nShe looked in again at the hand and shuddered. \"Whose hand is it?\"\n\n\"I . . . I don't know.\" Richard closed the lid of the wood box until the county forensics team could get there. \"But I think it's safe to say it's neither of our missing persons.\"\n\nNeville had arrived and seen the hand for himself. \"Well, thank God for that,\" he said.\n\n\"Find anything else in the woods?\" Richard asked the Englishman.\n\nNeville shook his head. \"Not a thing.\" He furrowed his brow as he looked at the police chief. \"I don't have a good feeling about any of this.\"\n\nRichard didn't reply. But he didn't, either.\n\nIn fact, he had a very bad feeling. A very bad feeling indeed.\n50\n\n\"Mr. Jack?\"\n\nThe new master of the house was standing at the window, looking out at the people gathered around the wood box. He barely lifted an eyebrow as Zeke approached him.\n\n\"Mr. Jack, may I speak with you a moment?\" the caretaker asked.\n\nJack turned around to look at him. Zeke watched him carefully. Jack seemed distracted, angry. Of course, that might have been grief. His grandmother had just died. But it seemed to Zeke to be something more, too.\n\n\"What is it?\" Jack asked, his voice cold and hard and small.\n\nZeke steadied himself. He had long anticipated that this conversation would be led by Cordelia, that it would have been Cordelia to tell Jack what he needed to know. But now Cordelia was gone. Zeke was on his own.\n\n\"Now that you're in charge here,\" the caretaker began, \"now that the house belongs to you . . .\"\n\nHe stopped. He couldn't go on speaking.\n\n\"Out with it, you old fool,\" Jack said impatiently. It was almost as if Cordelia was now inhabiting her grandson's body.\n\n\"I've lived here a long time,\" Zeke said. \"I've seen many things\u2014\"\n\n\"I expect you have,\" Jack said, cutting him off. \"And I expect that you will assist my wife in everything she wants to do to fix this place up. Do you understand?\"\n\n\"But that's what I need to talk to you about, Mr. Jack. . . .\"\n\nJack pushed past him and headed into the parlor. \"There's nothing to talk about! I want this place to be a destination that people seek out from all over the world! Do you understand, Zeke? I want to be a success!\"\n\nHe jabbed a finger against the old man's chest, his eyes wide and wild.\n\n\"But you will be, Mr. Jack,\" Zeke said. \"I'm sure of it.\"\n\nJack looked away from him. His gaze fell onto the fireplace with all the bricks piled around it.\n\n\"You've just got to hear what I have to tell you,\" Zeke went on. \"There's things you need to understand. Things you need to do. Your grandmother was going to explain them all to you, but now it's up to me.\"\n\nJack turned back to look at him. He was scowling.\n\n\"What are you talking about?\" he asked.\n\n\"After everybody is gone from outside, after you and I are all alone, there are things I need to show you, Mr. Jack.\"\n\n\"What kinds of things?\"\n\n\"I can't show you now, not with people still in and out of the house. But you need to know, Mr. Jack.\"\n\nJack looked at him suspiciously. \"Why the great mystery, Zeke?\"\n\n\"For now, please, just trust me.\" He drew closer to Jack. \"Help me put the bricks back into the fireplace.\"\n\nJack grimaced. \"You mean seal it back up?\"\n\nZeke nodded.\n\n\"But why? Didn't you hear what I said, Zeke? I want my wife's renovations to go forward. I want this place to become a popular resort.\" He leaned down close to Zeke so as to make himself clear. \"I want to make a lot of money,\" Jack enunciated.\n\n\"That's good, but we need to brick up the fireplace. . . .\"\n\nJack laughed at him. \"My grandmother is dead! I don't have to worry about her foolish sentiment anymore! We need a fireplace, Zeke! This is a goddamn New England bed-and-breakfast! Guests expect fireplaces!\"\n\n\"But, Mr. Jack, you've got to listen to me!\"\n\nJust then, Zeke heard the back door open and the sound of people coming inside the house. Chief Carlson was speaking with Annabel.\n\n\"Please, Mr. Jack,\" Zeke whispered, \"will you let me speak with you privately later?\"\n\nJack glared at him. \"Yes,\" he said finally. \"All right. But don't you dare touch those bricks in the meantime.\"\n\nHe strode off toward the kitchen.\n\nZeke turned around and looked at the fireplace.\n\nHe heard the scurrying down below.\n51\n\nThat night, Annabel slipped under the sheets beside Jack. What a terrible day it had been. They hadn't had much time to talk about anything with the cops and the forensics team in and out of the house. Neville had decided he couldn't leave without knowing what had happened to Priscilla, so he was staying on at the house. Around five o'clock, Paulie's brother had arrived with spare keys to Paulie's truck and drove it off the property, but not before insisting he look around the house himself for any clue as to what had happened to his brother. And Richard Carlson had been there until the sun went down, scouring the place with his eyes, seeming to commit every inch of the house to memory.\n\nAnnabel just wanted the day to be over, hoping that when she woke up the next morning she'd realize it had all been a bad dream.\n\nJack lay there beside her, eyes open, just staring up at the ceiling.\n\n\"Oh, Jack,\" Annabel said. \"I worry now that the Blue Boy will never shake its reputation of being cursed and haunted.\"\n\n\"Don't you worry, sugar babe,\" her husband told her, still staring at the ceiling. \"We are going to do just fine.\"\n\n\"What were you and Zeke talking about earlier? After the chief finally left?\"\n\nJack didn't take his eyes off the ceiling. \"Just a few things that needed to be done around the house.\"\n\n\"You seemed to be very intent. And when I came in, you both stopped speaking.\" She pulled closer to Jack. \"Is there anything going on that I need to know?\"\n\nHe turned his eyes abruptly to meet hers.\n\n\"No,\" he said. \"Nothing you need to know.\"\n\n\"I feel so badly about Neville,\" Annabel said. \"Not knowing about Priscilla, whether he should stay here or go back home. . . .\"\n\n\"He's paying for his extra night here, isn't he?\"\n\nAnnabel spun on her husband. \"Jack! How unfeeling of you! Of course, he's not paying. He offered, but I said he could stay here as our guest.\"\n\nJack grunted.\n\nAnnabel settled back against her pillows. \"I know it was uncomfortable when the police chief kept asking you about Priscilla,\" she said. \"But that was just his job. He had to ask those questions. Now hopefully he can find out what happened and we can move forward.\"\n\n\"You'll keep going on with the renovations?\" Jack asked.\n\nAnnabel looked over at him. \"Do you think I should?\"\n\n\"Yes.\" He smiled at her. \"In fact, I'm going to help finish the fireplace myself.\"\n\nShe returned his smile.\n\n\"Where do you think they went?\" Annabel asked him.\n\n\"Who?\"\n\n\"What do you mean, who? Priscilla and Paulie.\"\n\nJack shrugged. \"They ran off together. She was kind of trampy, wasn't she?\"\n\nAnnabel frowned. \"Not at all. How can you say that?\" She thought Jack was being awfully cavalier and insensitive.\n\n\"Besides,\" she said, \"Paulie's truck was still in the driveway . . . and Priscilla left her things.... Jack, something bad happened here today. The chief speculated that whoever killed Roger Askew may have come by here, disposed of the hand, and then taken Paulie and Priscilla away, maybe at gunpoint or something.\"\n\n\"So he was certain that the hand in the wood box was this guy Askew's?\"\n\n\"Yes,\" Annabel replied. \"There was a tattoo or something that clinched it.\"\n\n\"Well,\" Jack said. \"It's as good a theory as any.\"\n\nAnnabel shuddered. \"If I hadn't run to the market at that particular moment, he might have taken me, too.\"\n\nJack reached over and stroked her hair. \"It's okay, baby,\" he said.\n\n\"But Jack,\" Annabel went on, \"I worry about the publicity. I mean, with everything . . . the hand in the wood box, Cordelia's death, the missing people . . . We'll never break free of the macabre reputation the Blue Boy has had for so long.\"\n\n\"Well, I wouldn't worry your sweet little head about it, angel baby,\" Jack said, rolling over toward Annabel. \"We are going to be a huge success. Trust me.\"\n\n\"I hope you're right.\"\n\n\"I know I'm right.\"\n\nHe began to kiss her. Annabel tensed. She didn't feel like making love. Not after everything that had happened today.\n\nBut unlike last time, Jack was rock hard.\n\nShe couldn't stop him. She felt him grab hold of her pajamas and yank them down. He mounted her, and without any foreplay at all, he clamped his lips onto hers and entered her, roughly and abruptly. Annabel shuddered. She surmised that Jack was acting as quickly as he could, before he went limp again like last time. He thrust into her four or five times, then reached orgasm. He withdrew and rolled over onto his back again, breathing heavily.\n\nAnnabel just lay there. She felt dirty and violated. Their sex life hadn't been good in some time, but it had never been like that, so quick and so rough. Soon Jack's breathing turned into snores. Annabel slipped out of bed and took a long, hot shower.\n\nMore than ever, she wanted to run. Run as fast as she could and as far as she could, far, far away from this strange old house and these dark, cold woods, until she was safely back among the skyscrapers, bright lights, and taxicabs of New York.\n52\n\n\"Thanks very much,\" Neville said, accepting the mug of coffee and gratefully bringing it to his lips.\n\nThe morning sun was streaming into the kitchen. The Englishman watched as Annabel sat opposite him. Neither had said more than a few words since getting up. Neville had been unable to sleep much. He'd tossed and turned all night, and one glance in the mirror had shown him just how tired he looked.\n\n\"What will you do?\" Annabel asked. \"I mean, how long will you wait?\"\n\n\"Well, the chief wants me to stick around for a while. He said he might have to ask me more questions.\" Neville grinned sheepishly. \"I guess I'm a suspect.\"\n\n\"We all are,\" Annabel said, \"but there's no evidence any crime was committed.\"\n\nNeville sighed. \"Well, I'm not scheduled to fly back home to England for another week, but Florida is certainly no longer in the game plan. I'll stay here a few more days, if that's okay. I'll be glad to pay for my room\u2014\"\n\n\"No,\" Annabel said. \"I wouldn't hear of it.\"\n\n\"Please, I can't stay for free.\"\n\nAnnabel shook her head. \"After all we've been through, I insist you stay as our guest.\"\n\n\"You're very kind,\" Neville told her.\n\nThey exchanged small smiles.\n\n\"It must be so hard for you,\" Annabel said after a pause. \"Worrying about Priscilla. We've got to just continue to hope that she's okay.\"\n\nNeville frowned. \"I suppose that's the way to look at it, but I must be honest and tell you, I think she's gone for good. Whatever madman stabbed that man on the trail in the woods has taken her, as well as that man who was here working on the fireplace. And given how he butchered his first victim in the woods, I don't suspect he's going to go any easier on these two.\"\n\n\"But if he has taken them, as the chief suggests,\" Annabel said, \"then maybe he's using them as hostages. Maybe he'll release them unharmed. This Roger Askew was occasionally a drug dealer. It was probably a revenge killing, and the murderer will try to use his hostages to negotiate with police. We've got to keep believing that Priscilla and Paulie will be returned safe and unharmed.\"\n\n\"I like your optimism,\" Neville said, taking another sip of his coffee and then setting it down on the table in front of him.\n\nSilence resumed. Nobody had mentioned breakfast.\n\n\"Is Jack still sleep\u2014\" Neville began to ask, but at the same moment his coffee mug flew off the table, shattering on the floor and spraying the hot brown liquid everywhere.\n\n\"Oh!\" Annabel exclaimed.\n\nNeville jumped up. \"Dear Lord, did I do that?\" he asked, flustered and surprised. \"My arm wasn't anywhere near it, I swear! It just fell off on its own!\"\n\n\"It doesn't matter,\" Annabel said. She had sprung to her feet and grabbed a dish towel and was now soaking up the spilled coffee. \"We're all jittery this morning.\"\n\nNeville was squatting down beside her, collecting the shards of pottery from his mug. \"But, really, I didn't do it. It was the strangest sensation. It almost seemed as if a child ran by and knocked the mug from the table. A child\u2014a little person.\"\n\nAnnabel stopped what she was doing and looked over at Neville. Their eyes met just a few inches apart.\n\n\"A little person? What do you mean? Did you see something?\"\n\nNeville stood, carrying the shards over to the trash and dropping them inside. \"No, not really. It was just how you sometimes have a sense\u2014see something very briefly in your peripheral vision, maybe, and you get a sense of something. It felt like someone ran by the table and knocked the coffee to the floor deliberately. Someone small.\"\n\nAnnabel stood, dropping the coffee-soaked dish towel into the sink. She seemed struck mute by what Neville was saying.\n\nHe tried to smile. \"Well, don't I sound like a lunatic. I suppose you're right. We're just jittery. I'm sure I did it myself, and I apologize.\"\n\n\"No need . . . to apologize,\" said Annabel, though she continued to seem very distracted.\n\nNeville watched her as she walked out of the kitchen toward the parlor. She seemed to be looking for something. He poured himself more coffee into a new mug and sat back down at the table. Annabel returned in a matter of minutes.\n\n\"Everything all right?\" he asked her.\n\n\"Yes. I was just . . . listening for Jack.\"\n\n\"He's still asleep then?\"\n\n\"No,\" Annabel told him, sitting back down. \"When I awoke, he was already up and getting dressed. He said he was going to work with Zeke this morning repairing some broken eaves in the attic. He said we were losing a lot of heat through there.\"\n\n\"Yes,\" Neville said. \"I think I heard them banging around up there this morning.\"\n\n\"I'm sorry if they disturbed you.\"\n\n\"Not at all.\" He cupped his mug in his hands, careful to keep it away from the edge of the table. \"So no more guests arriving?\"\n\n\"No,\" Annabel replied. \"There had been a couple scheduled for next week, but I called and canceled, refunding their money. Until all of this is over\u2014in fact, until the renovation is done\u2014I think it's best not to have anyone staying here.\"\n\n\"That's smart,\" Neville said.\n\nSuddenly he felt compelled to say something.\n\n\"You know, I'm not sure why I need to tell you this, but here goes.\" He paused for just a second. \"I don't think I'm in love with Priscilla. I think maybe that's why her behavior toward Jack the other night didn't really bother me.\"\n\nAnnabel lifted her eyebrows. \"I see,\" she said.\n\n\"Of course, I want her to be found,\" Neville went on, \"and to be safe and unharmed and all of that. But I'm not in love with her. If I were, her behavior would have been very troubling to me.\"\n\nAnnabel gave him a tight smile. \"I think what you're telling me is that I ought to have found Jack's behavior troubling.\"\n\n\"No, I would never presume to suggest\u2014\"\n\n\"It's okay,\" Annabel said. \"I did find it troubling. Jack was the aggressor, not Priscilla. But, you know, I'm not nearly as troubled by it as I might have been.\"\n\nThey shared a small smile again.\n\nNeville liked Annabel. He suspected her marriage was not very good. Almost as if he were reading her mind, he saw the Blue Boy Inn as their last, best try to make things work. And now, to be hit with all this . . .\n\nAnnabel excused herself. She needed to call Chad Appleby. She was determined to press on with the renovation, and she hoped Chad would be as well.\n\nNeville sat at the table looking across the room, staring out the window into the gray winter day. Where was Priscilla? What had happened to her? Would she come back?\n\nAnd if she did, would he tell her what he had just told Annabel?\n53\n\n\"No trace of anyone or anything,\" Adam told the chief. \"We've scoured those woods. Except for Roger's hand in the wood box, there's been no evidence of anything suspicious. Certainly no sign of the two missing persons.\"\n\nRichard sat back in his chair, propping his feet up on his desk.\n\n\"This just doesn't make sense,\" he told his deputy. \"How could whoever killed Roger Askew force his\u2014or her\u2014way into the Blue Boy Inn, take off with two hostages, likely on foot, and not leave a single trace?\"\n\n\"Are you certain they made off on foot?\" Adam asked. \"I mean, a car could have pulled in there during the time Ms. Wish was out at the market. . . .\"\n\n\"Didn't you read the report, Adam?\" Richard asked him.\n\nThe deputy stiffened. \"I glanced at it. I've been out interviewing so many people . . .\"\n\nRichard smiled. \"It's all right. Well, when you do get around to reading it, you'll see that I interviewed Ted Cassidy, from the Department of Public Works. He was filling in a pothole around the corner from the inn. He saw Annabel go to the market, and he saw her come back. He was certain no other car came by in that time.\"\n\n\"Jeez,\" Adam said.\n\n\"We've searched the house, we've searched the grounds, and nothing,\" Richard said.\n\n\"It just doesn't make sense how three people could disappear so completely.\"\n\n\"And you still think whoever killed Roger also killed the old lady?\"\n\n\"It's a working theory. Maybe he wanted to hide out at the inn, Cordelia gave him a hard time, and he whacked her over the head.\"\n\n\"But why would they be in her bedroom?\" Adam asked.\n\nThe chief shook his head. \"Good question. That's what I mean. None of this adds up.\"\n\n\"Well, here's another monkey wrench to throw into your theories,\" said Richard's secretary, Betty, walking into the room and placing a sheet of paper on the chief's desk.\n\n\"What's this?\" he asked.\n\n\"Fax just came in from the county coroner's office,\" Betty told him.\n\nRichard snatched it up and read it quickly.\n\n\"Christ,\" he grumbled.\n\n\"What is it?\" Adam asked.\n\nRichard laughed. \"The coroner is ruling Cordelia's death an accident. Her head injury is entirely consistent with a fall, during which she struck her head on the iron doorstop.\"\n\n\"So then we don't have a murder investigation,\" Adam said. \"Just a couple of missing persons.\"\n\n\"I don't buy it,\" Richard said. \"The old woman's skull was cracked. She'd have to have come down to the floor at a superhuman rate to hit that doorstop and crack her head that severely.\"\n\n\"Can you contest the finding?\" Betty asked.\n\nRichard sighed. \"Sure, I can. But it'll take time.\" He pounded his fist on his desk. \"That coroner is old and out of it. This isn't the first ruling of his I've disagreed with. But there's not much I can do at the moment.\"\n\nHe hesitated.\n\n\"I'm going to have to tell Neville Clarkson he's free to go back to England.\" The chief picked up the phone. \"If he takes off immediately, that would be a sign he knew more than he was saying. If he sticks around, waiting for news of his girlfriend, then he's innocent of anything. Let's keep close tabs on the place, okay?\"\n\nAdam told him he'd visit the Blue Boy twice a day for the next week. The chief gave him the thumbs-up as the phone started to ring at the inn.\n54\n\n\"Thank you for coming back,\" Annabel said, greeting Chad Appleby at the door.\n\nThe young contractor offered her a small, sad smile as he stepped inside. \"I could either listen to the village idiots at Deb's Diner bleating about the curse of the Blue Boy, or I could say, hey, I've got a job to do,\" Chad told her. \"And if I didn't come back here, I'd be forever branded a chicken all around town.\"\n\n\"I'm sorry to have put you in this position,\" Annabel said.\n\n\"It's okay,\" Chad told her.\n\nAnnabel looked genuinely unhappy that her inn had such a sordid reputation. \"You do know that the chief called yesterday and told us that the coroner had ruled Cordelia's death to be an accident, don't you?\"\n\nChad nodded. \"Seems a strange coincidence, though, given Roger Askew's hand being found out back and Paulie and your guest disappearing at the exact same time.\"\n\n\"I know,\" Annabel said, \"but we have to believe it. Who knows? Maybe Cordelia saw something from her window, and in her hurry to call the police, she tripped and fell. That's what Chief Carlson suggested as a possibility.\"\n\nChad shrugged. \"I guess it is a possibility.\"\n\nAnnabel smiled. \"Yes,\" she said, \"it is.\"\n\n\"Well, let's move on,\" Chad said. \"Like I said, I have a job that I was hired to do. And we were planning to start in the parlor, right?\"\n\n\"Yes,\" Annabel said, leading him inside. They stood in the center of the room. The bricks Paulie had removed from the fireplace were still piled off to the side. \"Tell me what I need to do to prepare for you to start.\"\n\n\"Well, we'll have to move all this furniture out of here.\"\n\n\"I can do that. Jack will help me put it in the basement.\"\n\n\"Then you and I need to pick out some tiles, and some paint. You said you want that wall knocked down there, correct?\"\n\nAnnabel was nodding. \"That's right. It will open the room up into the dining room. Make the flow a lot smoother.\"\n\nChad walked over and knocked on the wall. \"That should be easy enough. But that's the only structural change you want, right?\"\n\n\"Right. Well, except for the windows.\"\n\nHe walked over to the windows and inspected the wood. \"That's not really structural, though you do want to make them a little bit bigger.\"\n\n\"Yes, to bring in more sun.\"\n\n\"We can do that.\" He peeled off a piece of wood that was flaking off the sill. \"Okay, so new windows, the wall removed, and we'll sand the floor and the moldings. . . really restore everything.\"\n\n\"And the fireplace,\" Annabel added. \"We'll need to finish it.\"\n\nChad was nodding. He approached the fireplace and looked down. \"Paulie said the chimney was sound. I can finish removing any last bricks that are needed, then we can paint the rest white, as you suggested.\"\n\n\"I think that will brighten up the room,\" Annabel said.\n\n\"You might want to clean out the ash dump down in the basement,\" Chad told her. \"There should be a door on the chimney underneath the fireplace. You can hire a chimney sweep, or maybe just have your caretaker do it, if he's familiar with it.\"\n\n\"I'll ask him,\" Annabel said.\n\nChad smiled at her. He liked Annabel. He felt sorry for her, too, trying to get this place in shape, battling its reputation and now confronted by the inn's latest streak of bad luck. He didn't blame her for whatever happened to Paulie. Chad still hoped his friend would turn up alive, but he didn't have a good feeling about it.\n\n\"I can start the actual work in a couple of days,\" Chad told Annabel. \"But I'll be back tomorrow to take measurements. Maybe later in the week we can take a drive up to Great Barrington to pick out some tiles and paint.\"\n\n\"That sounds like fun,\" Annabel replied. \"I'd like nothing more than to take a drive out of town.\"\n\nChad smiled. \"Then it's a plan.\" He shook her hand. \"I'll be here tomorrow afternoon with my measuring tape.\"\n\n\"Thank you for sticking with the project,\" Annabel said.\n\n\"Don't mention it,\" Chad told her, as she escorted him back to the door.\n55\n\nAnnabel watched from the window as Chad drove away. She was glad he'd be coming back. If he had backed out of the renovation, Annabel would have been devastated. The renovation was her reason for staying here.\n\nIt certainly wasn't Jack.\n\nIf Chad had not agreed to continue fixing up the place, Annabel really thought she would have called a taxi to take her to the nearest bus terminal and returned to New York. She had no idea where she would have stayed in the city, but she would have had found something. If it had meant continuing to live in this stark, musty house without any hope for change, Annabel would have bolted.\n\nBecause she had never felt so alone. Not even during those harrowing days in rehab, when at least she'd had doctors and therapists rooting for her, surrounding her with support.\n\nHere, she felt increasingly she was on her own.\n\nJack had been distant and strange these past two days. Thankfully, he hadn't tried to force himself on her again. Annabel would have kneed him in the groin if he did that again. In fact, when they slept, they stayed to their own sides of the bed. That was fine with Annabel.\n\nShe tried to keep hope alive, however. This was just a bad stretch. They could either give in to the tragedy of Priscilla and Paulie's disappearance, or they could move forward. Richard Carlson had been by and told her that he was having his deputy come by at least twice a day, just to make sure the kidnapper had not returned, and that made Annabel breathe easier. She liked seeing the police car come up the driveway in the mornings and the afternoons, and Officer Burrell get out and look around. She knew that a patrol car came by at night, too.\n\nThey could get through this. And she hoped that as the house started taking shape, Jack would start acting more normal again. This was a stress on all of them. If they could survive this, they could survive anything.\n\nAnnabel was pleased that Neville had stuck around. His warm, calm presence made things easier. Annabel could talk to him in ways she couldn't talk to Jack at the moment. In fact, she could talk to Neville in ways she'd never been able to talk to Jack. She knew he'd be returning to England soon; the chief had said he was free to go. But he said he didn't feel right leaving without knowing anything about Priscilla. He told Annabel he'd wait until at least the week was out. And she was glad about that.\n\nAt the moment, Neville was out, taking a ride to clear his head. Annabel wished she'd been able to go with him, but she'd had to wait for Chad.\n\nAnd Chad had asked her to have the ash dump cleaned. Zeke must know how to do it. He'd lived in the house so long.\n\nBut Zeke was upstairs again, working with Jack in the attic. Annabel would go up and ask him, and take a look around at the work they'd done.\n\nOne by one, she climbed the steep, narrow steps to the attic.\n\nAt the door at the top of the stairs, she turned the knob. But just like the other day, the door was locked.\n\nShe could hear them inside, muffled voices and the occasional pounding of a hammer.\n\nAnnabel rapped on the door. \"Jack! Zeke!\" she called.\n\nNo answer.\n\nShe knocked harder. \"Jack! Why is the door locked?\"\n\nThe muffled voices inside fell silent.\n\nAnnabel knocked again. \"Jack! Zeke!\"\n\nThe door suddenly opened without warning, with Annabel's hand still poised in the air. She gasped a little in surprise.\n\nHer husband glared at her.\n\n\"We're in the middle of patching the roof,\" he grumbled.\n\n\"Why was the door locked?\"\n\n\"For your own safety,\" Jack told her. \"We've had to tear up some of the floorboards. You could have tripped.\"\n\n\"Okay,\" Annabel said, miffed at his tone of voice. \"Sorry to have bothered you.\"\n\n\"What did you want?\"\n\nShe was already turning to leave, heading back down the stairs. \"Never mind,\" she said.\n\nJack grabbed her by the shoulder. \"Wait, honey babe,\" he said, stepping out of the attic room and closing the door behind him. \"I didn't mean to be gruff. I just had to get down from a ladder. I'm sorry.\"\n\n\"It's okay,\" Annabel said, extricating herself from his grip and continuing down the stairs.\n\n\"Did Chad come by?\" Jack asked.\n\n\"Yes. He'll get started on the renovation the day after tomorrow.\"\n\n\"Great,\" Jack said. \"I'll be down shortly.\"\n\n\"Okay,\" Annabel said.\n\nShe didn't turn around. She heard him go back inside the attic and shut the door. And then she heard the faint click of the lock being slid back into place.\n\nHe's different, she thought. Ever since the night when he came on to Priscilla . . . and even more since his grandmother died....\n\nAnnabel recalled Neville's admission that he wasn't in love with Priscilla.\n\nWas she in love with Jack?\n\nShe wasn't sure. She thought her feelings for him might return\u2014she hoped they would\u2014but at the moment, she was just not sure how she felt about her husband. All that she could detect was numbness when she thought of him.\n\nOnce again, Annabel felt like running.\n\nShe wanted to run out the front door and down the driveway and down the road to Millie's store. She'd call the taxi from there. She just needed to get out of this dark, stuffy house. She felt closed in. She felt as if she couldn't breathe. She was trapped. She couldn't get out! She would die in here!\n\nAnnabel gripped the post at the bottom of the stairs.\n\n\"Stop it,\" she scolded herself.\n\nAs Dr. Adler, her therapist, had trained her to do, she took three long breaths, feeling the air as it filled up her lungs, then as it flowed back out through her nostrils.\n\n\"I'm going to turn this house into a showcase,\" she said out loud.\n\nShe didn't need Zeke. She could clean out the ash dump herself.\n\nAnnabel made her way around to the stairs that led down into the basement. Since moving in, she'd managed to avoid the place pretty much. She and Chad and Paulie had taken one quick peek the other day, but Annabel had let them go ahead of her, and she hadn't stayed long, leaving them to do their inspection. The basement was dark and cobwebby and the ceiling was very low. It smelled of dank earth and mold and mice. For someone with claustrophobia, the Blue Boy's cramped basement was a place to be avoided at all costs. But Annabel pushed on. She was determined to ignore her fears.\n\nReaching the bottom of the stairs, she moved her hand upward, feeling around in the darkness above her for the string that hung from the light. Her fingers brushed through sticky cobwebs and she recoiled. When she found the string, she pulled hard, and suddenly the dark basement was illuminated by a pale white light.\n\nUnlike the attic, which was littered with junk, the basement was mostly empty. Letting her eyes adjust, Annabel glanced around to find the base of the chimney.\n\nIt stood some feet away, directly underneath the fireplace in the parlor. The chimney base squatted in the middle of the room, looking like an old man hunched over under a blanket of old bricks. Annabel approached. A black iron door, about two feet by three feet and even with Annabel's waist, had been cut into the brick. As she got closer, Annabel noticed an old, rusted padlock had been secured onto the door.\n\nShe lifted the padlock in her hands. It was an old thing, but it still held the door shut. A key was needed to unlock it.\n\nWhy on earth would someone padlock the door of a fireplace ash dump?\n\nAnnabel tugged at the lock. As old and rusted as it was, it still wasn't going to budge. She'd have to ask Zeke for the key. Until they had cleaned out God-only-knew how many decades of ashes that had collected inside, they wouldn't be able to get the fireplace blazing again. And Annabel felt that until the fireplace was crackling with wood, she wouldn't be able to call the Blue Boy her home.\n\nHer eyes glanced up the length of the chimney that protruded through the floor above. There, on a small nail blasted into the mortar, hung a key.\n\nIt had to be the key for the padlock. But it was out of reach. Why was it hung so high? She really wanted to get a look inside the ash dump to see how much work cleaning it out would entail. If it was packed with ash and debris, she might have to call someone to clean it for her. But if there wasn't so much, she could maybe brush it out herself, into a pail, and dump it in the woods.\n\nBut that damn key was so high that not even Jack, who was six feet tall, could have reached it easily.\n\nAnnabel looked around the basement for something to stand on. Her eyes fell on an old wooden chest, one of the few items in the vast dark space. She hurried over to it, grabbed ahold of its sides, and pulled it back toward the chimney. The chest was surprisingly light. When she'd gotten it to where she wanted it, she decided to peek inside.\n\nShe discovered moth-eaten little girl's clothes and a couple of moldy plastic dolls.\n\nThese must have been Cynthia's, Annabel thought, her heart breaking for Jack's little sister who'd been killed by some wild animal, her body never found.\n\nCarefully replacing the clothes and the dolls, Annabel set the lid back down on the chest. Now she needed something else. A rusted old rake, leaning against the wall, would do the trick. Holding the rake in one hand, Annabel climbed up on top of the chest. Then she lifted the rake over her head and knocked the key off its nail. It went flying through the air, landing somewhere on the earthen floor in the dark.\n\n\"Oh, great,\" Annabel grumbled, getting down off the chest and dropping to her hands and knees, feeling around for the key.\n\nShe found something else instead.\n\nA furry mouse\u2014or maybe a rat\u2014squeaked as Annabel's hand closed around it. She heard it skittle away. She let out a gasp and felt the gooseflesh crawl up her arms. She was about to go upstairs and find a flashlight when, miracle of miracles, her fingers touched metal. It was the key.\n\nGripping the key tightly in her hands, Annabel stood. She returned to the padlock on the door of the ash dump. Bending down, she inserted the key. It fit perfectly. With a slight turn, the padlock fell open.\n\nAnnabel hoped there was enough light. The one bulb on the ceiling was directly behind her. She thought she'd be able to get a good sense of what was inside.\n\nShe opened the door of the ash dump.\n\nAnd immediately a dark brown liquid dripped off from the inside of the door.\n\nSome kind of oil?\n\nHunched down, peering through the door, Annabel tried to make out what was inside. She couldn't see much. But it was clear there wasn't a surplus of ashes. What was inside seemed more solid. With great reluctance, Annabel reached her hand inside.\n\nWhat she felt was soft and pulpy.\n\nShe let out another short gasp and withdrew her hand.\n\nIt was covered in the same dark liquid that dripped off the door.\n\nWith a growing sense of horror, she walked slowly backwards, coming to a stop directly underneath the lightbulb hanging from the ceiling. Stretching her hand up overhead, she brought it as close to the light as possible.\n\nOne look told her what was on her hand.\n\nBlood!\n\nAnnabel screamed.\n56\n\nNeville heard Annabel screaming as he came through the front door. He bolted toward the sound.\n\n\"Annabel!\" he called. \"Are you all right?\"\n\nShe came bounding up the cellar steps. \"Blood!\" she shrieked. \"Blood!\"\n\nAnd she held up her hand, coated in a thick, purplish goo, for him to see.\n\n\"Dear God, are you hurt?\" Neville asked.\n\n\"The chimney!\" Annabel was shuddering. \"The chimney!\"\n\nShe wasn't making any sense.\n\n\"I've got to wash this off my hand!\" Annabel cried, hurrying toward the kitchen. Neville followed.\n\nAnnabel had already plunged her hand under the hot water at the sink by the time Neville made it through the kitchen door.\n\n\"Did you cut yourself?\" he asked.\n\n\"No!\" Annabel's eyes were wild. \"There's blood in the ash dump! At the base of the chimney!\"\n\nShe was scrubbing furiously, squeezing gobs of dish detergent all over her hand. The steam that rose from the gushing water from the faucet revealed just how hot it was.\n\n\"Careful you don't scald your hand now,\" Neville told her.\n\n\"I don't care,\" Annabel said, near panic. \"I want to get all of that off me!\"\n\nNeville heard movement behind him. Jack had come through the kitchen door.\n\n\"What's going on?\" he asked. \"We could hear you shouting all the way in the attic.\"\n\n\"There's blood in the chimney,\" Annabel told him.\n\nJack looked at her as if she were crazy.\n\n\"Go downstairs to the basement and see for yourself,\" Annabel said. \"I'm never going down there again!\"\n\n\"Zeke's down there now,\" Jack told her. \"We saw the door was open and the light was on, so he went down there to check.\"\n\nSomething in the way Jack spoke seemed to unnerve Annabel.\n\n\"Go down there!\" she shouted. \"Please, Jack, see for yourself.\"\n\nHe gave her a small smile. \"I'm sure Zeke will be able to tell us what it was you thought you saw.\"\n\nNeville saw the sudden fear in Annabel's eyes. It was a different fear, not the panicky terror that had sent her rushing to the sink to wash her hands. This was very different. She turned her terrified eyes to Neville.\n\n\"You go down!\" she begged him. \"Please! See for yourself.\"\n\n\"You want me to go down to the basement?\" Neville asked.\n\n\"Yes! There's blood! I think that's where the killer put Priscilla's and Paulie's bodies!\"\n\n\"Oh, dear God,\" Neville muttered, horrified now himself, turning and heading out of the kitchen.\n\nBut before he could make it through the door, he ran smack-dab into the old caretaker Zeke.\n\n\"What's all this shouting for?\" Zeke asked.\n\n\"Annabel thinks she saw blood in the chimney downstairs,\" Jack said.\n\n\"Blood?\" Zeke displayed his hand. \"It's just wet soot.\"\n\nHis hand was indeed black with soot. He walked calmly past Annabel over to the sink, where he washed his hands just as she had done, but without any of the mania that had gripped her.\n\n\"I saw the door to the ash dump was open, so I closed it,\" Zeke said.\n\n\"There was blood inside!\" Annabel insisted.\n\n\"Wet soot can look like blood,\" Zeke told her, not looking around.\n\n\"But it was on my hand,\" Annabel said, lifting her now pristinely clean hand in front of her face. \"I know what I saw. . . .\"\n\n\"Sugar cakes,\" Jack said, \"you know sometimes you see things that aren't there. . . .\"\n\n\"You saw my hand,\" Annabel said to Neville. \"You saw it was blood!\"\n\n\"Well, I . . .\" Neville tried to think. He'd seen Annabel's hand covered in something. But was it blood, or wet soot as Zeke claimed? \"I didn't really get a close look at it. You washed your hands so fast.\"\n\n\"Anyway,\" Zeke was saying, drying his hands with a dish towel, \"I looked inside and there wasn't much of an ash buildup. I'll clean it out later today so you can continue on with your renovations, Miz Wish.\"\n\n\"No, don't clean it out,\" Annabel said. \"I want the police chief here to inspect the chimney. . . .\"\n\n\"Sweetheart,\" Jack said taking her by the shoulders, \"you're getting hysterical. You know your doctors told you that you might have flashbacks. . . .\"\n\nShe shoved him away. \"I'm not hysterical!\"\n\nBut her face, so contorted and upset, made Neville think she might be.\n\n\"Annabel,\" Jack told her firmly, \"I think you should go lie down.\"\n\nShe glared at him. She seemed to accept that she was beaten.\n\n\"All right,\" she said, her voice quiet now. \"Maybe I will go lie down.\"\n\n\"That's a good girl,\" her husband said.\n\nHe moved over to the refrigerator and popped it open. \"Think this bowl of Gran's rabbit stew is still good?\" he called over to Zeke.\n\n\"Sure it is,\" the old caretaker said. \"I'll heat some for both of us.\"\n\nAs they were distracted by the stew, Annabel turned to leave. Neville watched her with fascination. As she passed him, she paused for just a second, and spoke in a whisper that he could barely make out.\n\nBut he heard her.\n\n\"Go down there,\" she said. \"Key on a nail up above.\"\n\nThen she was gone.\n\nZeke was pouring the stew into a pan on the stove. Jack had settled down at the table, reading a newspaper.\n\nNeville slipped out of the kitchen.\n\nEvery nerve in his body trembling, he found the door to the basement steps. He opened it slowly, hoping it didn't creak. Then he scampered down into the darkness. Yanking the string to light the bulb overhead, he saw the base of the chimney. A chest and a rake were positioned in front of it. Right away Neville knew this was how Annabel had gotten the key that was on \"a nail up above.\"\n\nBut the door to the ash dump was securely padlocked.\n\nAnd while Neville could see the nail protruding from the bricks above, there was no key hanging from it.\n\nSomehow that fact confirmed for him that Annabel had been right.\n\nThere was indeed blood in that chimney.\n\nBehind that small iron door, Neville was convinced, lay Priscilla's body.\n57\n\n\"You're really going to keep working at that scary old place?\" Tammy asked, sliding Chad's scrambled eggs and home fries in front of him on the diner counter.\n\n\"I was hired to do a job,\" Chad told her, \"and I'm gonna do it.\" He brought a fry to his mouth and took a bite. \"My way of paying tribute to Paulie.\"\n\n\"Well, I think that's pretty brave of you,\" Tammy said. \"I feel bad for that couple that just arrived to fix up the place. Now they're forever haunted by Roger's hand.\" She shuddered.\n\nChad smiled up at her. \"How you doing with that, Tam? You dealing with his death okay?\"\n\n\"I made a resolution when I got up this morning,\" the pretty, dark-eyed waitress told him. \"I was moving on with my life. I'm going to be just fine on my own.\"\n\n\"Hear, hear!\" Chad said.\n\n\"I'm going back to school, and I'm going to get a degree,\" Tammy told him, before smirking. \"If I can afford it on my tips from this place.\"\n\nChad thought a moment. \"You know, if I recall, you fixed your place up pretty nice. Refinished the floor yourself. Laid down some new tile.\"\n\nTammy beamed. \"That I did. With good advice from you.\"\n\n\"I could use an assistant like you,\" Chad told her. \"It would mean going out to the Blue Boy Inn, but I could help you make a few extra bucks.\"\n\n\"What would I do?\" Tammy asked.\n\n\"Help me with painting and sanding, to start,\" Chad told her, bringing a forkful of eggs to his mouth and afterward wiping his lips with the paper napkin. \"Would it creep you out too much, going to a place where Roger's hand was found?\"\n\n\"Not at all,\" Tammy answered. \"When do I start?\"\n\n\"Come with me this afternoon when you get off here,\" he said.\n\n\"All right. I'll have my mom pick Jessica up at school and watch her until I get home. She's been after me to get a better job anyway. She'll be happy to do it.\"\n\n\"Then welcome to Appleby Contracting, Ms. Morelli,\" Chad said, extending his hand across the counter.\n\nThey shook.\n58\n\nAnnabel lay in bed. She hadn't left her bed since coming up here yesterday afternoon, shaken by her discovery of blood in the chimney.\n\nShe was certain the blood wasn't there anymore, however.\n\nZeke had cleaned it out. She knew that in her gut. That was why he'd locked it back up and taken the key, as Neville had reported to her late last night, so he could go back later and clean it all out. He would destroy the evidence of any body ever being stuffed into the chimney.\n\nSince Annabel's cell phone didn't have reception, and she didn't dare use the house phone, Neville had promised that he would drive into town this morning and tell the police chief what Annabel had found.\n\nShe realized the significance of the fact that she didn't dare use the house phone.\n\nIt was because she was afraid of being overheard.\n\nJack and Zeke knew something. They were hiding something. Somehow, they were covering up the murders.\n\nMight they\u2014Annabel trembled to think it\u2014have committed them?\n\nShe'd thought Jack was sound asleep that morning. But had he snuck out while she was at the store? Zeke had claimed to be in the attic. But he, too, could have come downstairs while Annabel was gone.\n\nBut why?\n\nAnnabel felt frozen. She lay there immobile on the bed. All night, she had been awake, listening as Jack breathed beside her. When he'd come in the night before, she'd pretended to be asleep. They hadn't spoken a word. All through the night, Annabel had had the sense that Jack, too, was awake, lying beside her, waiting and listening for her to make a move. So Annabel had kept as still as she could, breathing shallowly but regularly, hoping he believed her to be asleep. In the morning, when Jack had finally risen and left the room, Annabel had let out a long, relieved breath.\n\nIt was unbelievable.\n\nShe was frightened of her husband.\n\nJack\u2014who had stood with her through all her trials in the past.\n\nAnnabel felt as if she was going mad.\n\nShe smelled coffee brewing downstairs. She had yet to hear a car crunch across the gravel driveway, so Neville had not yet left for the police station. If he didn't leave soon, she would jump up when she heard Officer Burrell make his daily drive-through checking on the place. She'd scream from the window for him to come in and arrest Jack and Zeke!\n\nBut for what?\n\nMaybe she was going mad.\n\nAnnabel tried to focus, to make sense of what was happening. She was fearful that Jack was involved in murder\u2014or that he was covering it up. But such an idea was crazy! She couldn't think straight. Jack and Zeke were hiding something in the attic. That much Annabel knew. And then they had prevented her from exposing the blood in the basement.\n\nSugar cakes, you know sometimes you see things that aren't there....\n\nWas that it? Was she was just imagining things?\n\nMaybe Jack and Zeke really had been fixing the rafters and the floorboards in the attic. Maybe that really had been wet soot in the chimney.\n\nAnnabel heard a sound.\n\nShe sat up on her bed. Looking across the room, she watched in disbelief as the panel on the far wall slid back. The same panel where she had found those terrible books. Zeke had never nailed it shut.\n\nFrom the darkness behind the wall within emerged two little blue feet.\n\n\"No,\" Annabel murmured, pulling her legs under her and wrapping her arms around her body.\n\nTommy Tricky crept out of the hole in the wall.\n\nHe looked up at her, his blue eyes shining.\n\nHe gnashed his sharp little teeth.\n\nBut then\u2014even worse\u2014his twin emerged from the dark space, following him out into the room.\n\nThe two little creatures stood there looking up at Annabel. She was so terrified she couldn't make a sound.\n\nThen they gave a little laugh and dashed across the room, under her bed.\n\nAnnabel screamed at the top of her lungs.\n\nHer fear turned everything white. Blindingly white. The next thing she knew Jack was standing over her, trying to hold her down on the bed, telling her to calm down and stop screaming. Zeke was there, looming over her as well. And Neville, too! They were all in on it! They were all out to get her!\n\nAnnabel passed out.\n59\n\nNeville left Annabel's room completely mystified.\n\n\"What happened to her?\" he asked Jack and Zeke. \"She seemed fine earlier.\"\n\n\"She has a history of hallucinations, of histrionics,\" Jack said, peering in at her one last time, making sure she was resting comfortably before he shut the door. She had seemed to calm down, but Neville thought she might start thrashing about again anytime, so great had been her distress. \"Her doctors told us she could occasionally have relapses,\" Jack continued, and he looked sincerely worried about his wife.\n\nIf that was so, Neville thought, perhaps he'd been wrong to believe Annabel's story about blood in the chimney.\n\nHe would wait for a while before talking with the police. Jack seemed perfectly reasonable; it was Annabel who was the raving lunatic at the moment. Neville didn't want to start trouble.\n\nIn fact, all he wanted to do was leave. He was certain now that Priscilla wasn't coming back. He just wanted to get on an airplane and go home, get back to work, visit his parents and his brother and his nieces and nephews. Neville just wanted some normalcy in his life again. The past few days were enough fear and confusion and chaos to last a lifetime.\n\nBut he couldn't leave. Not quite yet. He liked Annabel, and he wanted to make sure she was all right. Tomorrow, he figured. If she was up and about tomorrow, talking sensibly, he'd leave tomorrow.\n\n\"I'm going to the store,\" Jack announced. \"Not Millie's market, but the supermarket in Great Barrington. I want to get some real groceries. Frankly, I need some meat. Some steak and potatoes, since all that's in the fridge are Annabel's vegetables.\" He smiled over at Neville. \"Can I get you anything, buddy?\"\n\n\"No, thank you,\" Neville told him. \"I've come to quite enjoy carrot and cucumber sandwiches.\"\n\n\"Have it your way,\" Jack said. He turned to Zeke. \"You'll look in on Annabel? If she wakes up, tell her I'll be back in about an hour.\"\n\n\"Will do,\" the old man replied.\n\nNeville watched from the window as Jack drove off.\n\n\"I'm going to finish some work in the attic,\" Zeke told him. \"If the telephone rings, you can let it go to the answering machine.\"\n\n\"I'm happy to answer it and take a message for you,\" Neville said. \"I'm just going to settle down here in the parlor and read a book.\"\n\n\"Very good.\"\n\nNeville took a chair opposite the fireplace. How he wished there was a fire crackling in front of him, sending off waves of heat across the room. The day was so terribly cold. Neville shivered a little, buttoning his wool cardigan sweater all the way down. Opening his book, he began to read.\n60\n\nUpstairs, Annabel was awake. She lay rock still, listening to Tommy Tricky and his twin whisper under her bed.\n\n\"Let's get her.\"\n\n\"No, not yet.\"\n\n\"But I want her.\"\n\n\"Leave her!\"\n\nI'm going mad, Annabel thought. Stark raving mad.\n\nThere was no such thing as Tommy Tricky. But here she was, listening to his voice. And there were two of him.\n\nKnowing that she had gone insane gave her a weird sense of peace. She just lay there, not moving, listening to the little men scuttle around under her bed.\n61\n\n\"Hello?\"\n\nA man's voice startled Neville awake. He had fallen asleep reading his book. His head was down on his chest. He leapt from his chair.\n\n\"Hello?\" the man called again.\n\nTwo people had let themselves in and were now standing in the foyer. The man who'd been calling was Chad Appleby, the contractor. The other was a pretty, dark-haired woman Neville had never seen before.\n\n\"I'm sorry to just barge in on you,\" Chad said, \"but we knocked and no one answered the door.\"\n\n\"Oh, it's quite all right,\" Neville told him. \"But I'm afraid Jack has gone up to Great Barrington and Annabel is . . . taking a nap. She wasn't feeling so well, you see.\"\n\n\"No problem,\" Chad replied. \"I'm just here to take some measurements. This is my assistant, Tammy Morelli.\"\n\n\"How do you do?\" Neville asked, smiling over at the woman, who nodded hello.\n\n\"I'll need to go down into the basement,\" Chad said. \"I need to make sure the floorboards are going to be strong enough for when we take out that wall.\"\n\n\"I'll walk down with you,\" Neville offered. \"I know where the string for the light is.\"\n\n\"Thanks.\" Chad turned to Tammy and tossed her a measuring tape. She caught it expertly. \"In the meantime, start measuring the windows and moldings like I told you, okay? Not just the ones on the first floor, but on the second floor, too. We're going to order all the new windows at the same time.\"\n\n\"Aye, aye, captain,\" Tammy said.\n\nThe two men set off down the stairs into the basement.\n\n\"Here's the light, right here,\" Neville said, when they reached the bottom. He pulled the string.\n\n\"Thanks,\" Chad said. \"I can find my way from here.\" He switched on his flashlight. \"I just want to inspect the floorboards.\"\n\n\"Actually,\" Neville went on, \"I came down so I might ask you a question.\"\n\n\"What's that?\"\n\n\"Annabel was a bit concerned earlier when she found a dark, brownish-purplish substance inside the chimney down here. Apparently, when she opened the ash dump, it was full of the stuff.\"\n\n\"Brownish and purplish?\"\n\nNeville nodded. \"She thought it looked like blood.\"\n\n\"Oh, Jesus.\"\n\n\"Given everything that happened here, to your friend and my girlfriend, well, I thought you ought to know. And if there was any way you could take a look at it . . .\"\n\nChad swung his flashlight over to the door to the ash dump. \"Why the hell is it padlocked?\" he asked.\n\n\"I don't know,\" Neville said. \"And the key has been missing ever since Annabel looked inside. Zeke said what she saw was wet soot.\"\n\nChad made a face. \"Who ever heard of purple wet soot?\"\n\n\"Well, I guess that's right. I was going to mention it to the police . . .\"\n\nThey had reached the base of the fireplace. \"I'd say that would be a good idea,\" Chad said. \"I mean, if it's a clue to what happened\u2014\" He stopped speaking abruptly. \"Hey,\" he said. \"Listen.\"\n\nThey put their ears close to the ash dump door.\n\n\"Something's in there,\" Chad said.\n\nNeville listened. There was indeed something in there.\n\nAnd it was eating!\n\nThe sound of chomping and chewing came from inside the old brick chimney.\n62\n\nTammy stretched the measuring tape across the windowsill on the second-floor landing. She noted the length and jotted it down in a little pad. Then she did the same for the height.\n\nThis place wasn't so creepy. All her life Tammy had heard stories about the Blue Boy Inn. There had been lots of deaths and disappearances up here. Kids in school called it a haunted house. Her mother used to say the place was \"cursed.\" She hadn't even liked to drive by on the road on the way to Millie's market.\n\nBut the English guy who'd greeted them had been pleasant enough. And once Chad was through with his renovation, the place was going to be real bright and sunny. It would be like a modern showplace, according to the plans Tammy had seen. She was excited to be in on it. This would be good for her. A real change, and Tammy needed a\u2014\n\n\"I told you, no more!\"\n\nAn old man's voice suddenly cut through the stillness of the upstairs corridor.\n\nTammy tried to ignore it, but the voice came again.\n\n\"Get back here!\"\n\nThe voice was coming from the steep, narrow stairs at the end of the corridor. Tammy assumed they led to the attic. She took a few steps in that direction, pausing at the foot of the stairs to listen.\n\nShe could hear people moving about. There was some kind of struggle, it seemed. The old man spoke again, but softer this time.\n\n\"Stop this,\" he urged. \"There are people in the house.\" And then he added, insistently, \"Shhh!\"\n\nNow Tammy was distracted by a sound behind her. She turned away from the stairs and looked back down the hallway. A woman was emerging from one of the rooms. She was dressed in blue jeans and a wrinkled oversized T-shirt. She looked a wreck, as if she hadn't slept in days. Her long auburn hair was all mussed up. Her eyes caught Tammy's.\n\n\"Who are you?\" the woman asked.\n\n\"Tammy Morelli. I'm working with Chad, the contractor.\"\n\nThe woman's eyes softened. \"Then you're real,\" she said, a small smile fluttering across her face. \"Good. That's good.\"\n\nShe made her way down the stairs.\n\nHow very strange.\n\nTammy thought she might have to revise her earlier impression of the Blue Boy. It was indeed pretty creepy after all.\n63\n\n\"Annabel!\" Neville exclaimed. \"You're up! Are you all right, my dear?\"\n\n\"I need to take a walk,\" she said. \"Clear my head.\"\n\nHe thought she looked terrible. \"It's very cold out,\" he told her.\n\nShe didn't reply, just yanked on her coat. Behind Neville, Chad now appeared, emerging from the basement stairwell.\n\n\"Hello, Annabel,\" the contractor said.\n\nShe grunted a reply.\n\n\"Are we still on for tomorrow morning?\" Chad asked. \"To take a drive up to Great Barrington and pick out some tile and paint?\"\n\nNeville watched as Annabel turned to look at him. Her eyes seemed dull and gray. It seemed almost that she didn't recognize Chad. She stared at him for several seconds, as if she was trying to process what he was asking her.\n\n\"Yes,\" she said at last. \"Take a drive. Get away from here. Yes, we're still on.\"\n\n\"Great,\" Chad said. \"You know, I should tell you that we were just down in the basement\u2014\"\n\nNeville quickly and subtly moved his foot over to the other man, whacking his shin.\n\nChad stopped speaking. He exchanged a look with Neville.\n\n\"Do whatever you need to do,\" Annabel said softly. \"I need to take a walk. Get some air.\"\n\n\"All right,\" Chad said. \"See you tomorrow then. I want to get an early start. The weather report says we might get a pretty big snowstorm tomorrow afternoon.\"\n\nAnnabel didn't answer. She just headed outside.\n\nChad looked over at Neville once she was gone. \"Why did you stop me from telling her about what we heard in the chimney?\" he asked. \"If she's got raccoons living in there, she's going to need an exterminator before we can finish repairing the fireplace.\"\n\n\"She's had a rough day,\" Neville replied. \"She doesn't need to start worrying about raccoons.\"\n\nChad wasn't satisfied with that answer. \"But that might explain the blood she found in there. If those coons have been eating squirrels and mice . . .\"\n\nNeville gave him a cold look. \"I doubt it's squirrels and mice they're eating.\"\n\nChad shivered. \"Wait. Are you . . . ?\" His face blanched. \"Are you thinking that the killer stuffed Paulie's and Priscilla's bodies inside the chimney, and that's what we heard the raccoons chomping on?\"\n\n\"It did cross my mind.\"\n\n\"That's just too freaky.\" Chad looked back down the stairs. \"But you just may be on to something. That's one really wide chimney. You could fit a body in there, sure.\"\n\n\"Especially if the killer was adept with a butcher knife, as the hand in the wood box would seem to indicate,\" Neville added.\n\n\"Jesus.\" Chad shuddered. \"We need to tell Annabel, or her husband.\"\n\n\"I promised Annabel last night that I would tell the chief of police. She did not, for her own reasons, want her husband to know.\"\n\n\"Why the hell not?\"\n\n\"It doesn't matter.\" Neville headed toward the door, grabbing his coat from the hook on the wall. \"All I know is I need to fulfill the promise I made to Annabel. I'm going down to the chief's office now.\"\n\n\"I'll go with you,\" Chad said. \"He'll want a statement from me, too, about what I heard.\"\n\n\"All right.\"\n\nTammy had come down the stairs. \"What's going on?\" she asked.\n\n\"Listen, Tam,\" Chad said, \"I'm going to run Neville here into town for a moment. I'll be back in two shakes. You almost done?\"\n\n\"I've measured every window upstairs,\" she said, \"except for the attic. It's locked. There's somebody in there.\"\n\n\"It's Zeke, the caretaker,\" Neville told her.\n\n\"He seemed a little upset.\"\n\n\"That's just Zeke,\" Neville assured her.\n\n\"Okay, listen, Tam,\" Chad said. \"Get the windows in the kitchen and dining room measured. We might as well plan for everything now. We're going to be replacing all the windows eventually.\"\n\n\"All right, boss.\"\n\nChad turned to leave, and then looked back at Tammy. \"You going to be okay by yourself? I'm just going to be gone a little bit.\"\n\n\"Well,\" Tammy said, \"I've discovered there's some weird people in this house, but I haven't seen any ghosts yet.\"\n\nNeville saw the smile the two of them exchanged.\n\nOperative word being yet, he thought, as he and Chad left by the front door.\n64\n\nIt was the thought of driving up to Great Barrington that revived Annabel. The thought of getting away from here, on the road, driving miles and miles away. The idea appealed to Annabel almost as much as a Caribbean vacation.\n\nThe cold afternoon air rushed into her nostrils. It functioned as she had hoped. Her mind felt clearer, more alive.\n\nAs she walked into the woods, leaves and twigs crunching under her feet, Annabel told herself she had allowed her imagination to run amok. It had happened before, when she was in the hospital, when sometimes she hadn't known where she was, when the orderlies had looked like deformed monsters and her room like a dungeon. Jack was right. Her doctors had warned her she might have flashbacks. The delirium that had set in as her body withdrew from the drugs had been intense. It was still there, buried deep down in her brain.\n\nIt had taken being raped by Jack to bring it out again.\n\nHe wouldn't call it that, of course. He'd say they had just made love. Annabel hadn't fought him. She hadn't resisted. But still she felt raped just the same.\n\nHe had just tried to do it as fast as possible, fearful he'd lose his erection again, she argued with herself. It wasn't rape. That's not fair to Jack.\n\nBut what about fair to her? She couldn't deny how she felt.\n\nAnd that horrible feeling had led to some horrible hallucinations. Annabel had to find a way to deal with it, to get it all out of her head\u2014the anger, the fear, the sense of violation. Otherwise, she wouldn't be able to go on. She would always feel unsafe here.\n\nAnnabel paused at the little white stone marker. She looked down at the words carved across its surface.\n\nCINDY DEVLIN\n\nPoor little girl. How had she died? Her death must have been terrible, given how much blood was found.\n\nAnnabel wondered what kind of memorial Jack would want for his grandmother. They'd been notified that her body had been taken from the morgue to be cremated. Jack had told her that his grandfather's ashes had been scattered in these woods; Annabel assumed that was what he'd want to do with Cordelia's as well. They should have a little service, she thought. Maybe ask a minister to come in and say a prayer. Annabel could get some fresh flowers from Millie's store and carry them as the old woman's ashes were scattered. Afterward, they'd put the flowers on the mantel over the fireplace.\n\nAnnabel walked on into the woods.\n\nHow she wished she was walking down Fifth Avenue, or through busy Times Square, taxicabs honking, sirens wailing, lights flashing. She yearned to be surrounded by masses of people\u2014thousands of them, all moving past her, rubbing shoulders\u2014and away from this stark, gray, silent place. Some claustrophobics hated being in the city, and longed for the empty countryside. Annabel was different. The city made her blood race. It filled her up.\n\nShe missed New York like an old, comforting friend.\n\nAbove her, a crow cawed. She heard the flapping of its wings, but could not spot it through the network of interlacing, bare, blue limbs.\n\nAnnabel didn't want to get lost. She kept turning around, making sure she could still see the outline of the house.\n\nWhat had she been thinking? Had she really been so angry at Jack that she believed him capable of murder? How crazy was that? What motive did Jack have to kill those two people? Even if something had happened between him and Priscilla that night, Jack had had absolutely no interaction with Paulie that mattered. It was just crazy. In her confused state of mind, subconsciously blaming Jack for hurting her\u2014violating her\u2014Annabel had seen sinister motivations behind every action Jack took, every statement that he made.\n\nShe'd been seeing other things as well.\n\nLike blood in the chimney, when it was clearly just old soot and debris.\n\nLike a pair of Tommy Trickies, whom she had stopped believing in a long time ago.\n\nI've been decompensating, as Dr. Adler would say. She still remembered the definition he'd given her of the term after he'd used it to describe what was happening to her. The failure to generate effective psychological coping mechanisms in response to stress, resulting in personality disturbance or disintegration.\n\nThat was what she had felt. That she was disintegrating.\n\nShe had to get it together. She was stronger than this.\n\nShe stopped and sat down on a log, breathing in the cold air. She could see her breath in front of her.\n\nOff in the woods, she heard the snap of a twig.\n\nShe wondered what kind of animals might be out here. Jack had said they'd feared little Cindy had been killed by a bear all those years ago. Were there still bears prowling these woods? There were also coyotes and foxes and bobcats, Zeke had told her. The bobcats could be particularly vicious.\n\nAnd they'd seen that big, terrifying moose on the ride up, too.\n\nShe laughed a little then, remembering the moment. Annabel felt as if she had tumbled down the rabbit hole into a very strange, topsy-turvy world.\n\nAnother snap of another twig, closer this time.\n\nAnnabel stood. She shouldn't have come out this far. What if what she heard moving out there was a bear or a bobcat?\n\nShe looked around. She couldn't see the house. Damn it!\n\nBut she could hear something approaching, crunching through the leaves.\n\n\"Okay, Annabel,\" she whispered to herself. \"It's time to go home.\"\n\nShe started walking back in the direction she'd come. At least, she thought it was the same direction.\n\nAll at once she heard something.\n\nThe tweet of a bird?\n\nBut it sounded different than that....\n\nShe paused.\n\nThe sound came again.\n\nAnnabel's blood ran cold.\n\nIt was no bird.\n\nIt was also not a bear or a bobcat, either.\n\nWhat Annabel heard was a short, two-note whistle, made, she was certain, by human lips.\n\nIt sounded again.\n\nAs did the crunching of the leaves, very close to her now.\n\nSo maybe it's a hiker, Annabel thought. Or a hunter.\n\nA hunter with a gun. Who might mistake her movements among the trees for a deer, and shoot to kill.\n\nI've got to get home.\n\nAnnabel began to walk faster. And as she did, the sound of whoever was crunching through the leaves toward her accelerated at the same pace.\n\nThis was no hunter, no hiker, she told herself. It was also not a wild animal.\n\nShe walked even faster. The sound behind her also sped up.\n\nAnnabel realized she was being pursued.\n\nShe started to run.\n65\n\nTammy stretched her measuring tape across the windowsill in the kitchen. But before she could make a note of the length, she suddenly had the distinct sense someone was behind her. With a start, she looked over her shoulder.\n\nShe was right. There was a man there. A tall, handsome man with two paper bags full of groceries in his hands.\n\n\"Oh, hello, I wasn't aware someone came in,\" Tammy said, turning to the man.\n\nHe set the groceries on the counter. \"It's always a pleasant surprise to come home and find a beautiful woman in your house.\"\n\nShe blushed. \"I'm Tammy,\" she said. \"I'm working with Chad, taking some measurements of the windows.\"\n\n\"I didn't see Chad's truck out front,\" the man said, taking some steaks out of the bags and putting them in the refrigerator.\n\nTammy shook her head. \"He took the Englishman who's staying here into town.\"\n\nTo this, the man lifted his eyebrows. \"Did he now? Why couldn't Neville have driven himself? He has a car.\"\n\n\"I'm not sure. I don't know what their errand was. But Chad said he'd be back shortly.\"\n\nThe man nodded. \"I'm sorry, I've been impolite,\" he said. \"I haven't introduced myself.\" He walked across the room and extended his hand. \"I'm Jack Devlin. The owner of the place.\"\n\n\"Hello, Mr. Devlin,\" Tammy said, accepting his greeting.\n\nTo her surprise, he brought her hand to his lips and kissed it. \"It is a pleasure to make your acquaintance, Tammy. A very fine pleasure indeed.\"\n\nHe didn't let go of her hand.\n\n\"Have you seen my wife?\" Mr. Devlin asked.\n\nTammy felt uncomfortable that he was still holding her hand. He was also standing a little too close for comfort. She could smell him. Man sweat and aftershave.\n\n\"Um, I actually didn't meet your wife yet, but I saw her,\" Tammy said awkwardly. \"I believe she went out to take a walk.\"\n\n\"I see.\" Mr. Devlin smiled and took a step even closer to her. His grip on Tammy's hand tightened. \"That means we have the house to ourselves.\"\n\n\"No,\" Tammy said quickly. \"I heard a man in the attic. . . .\"\n\n\"Oh, that's just Zeke,\" said Mr. Devlin. His eyes seemed strange to Tammy. His pupils were dilated. \"He won't bother us.\"\n\nMr. Devlin leaned down toward her as if he was about to kiss her.\n\n\"No!\" Tammy shrieked, yanking her hand away from him and hurrying across the kitchen. \"You had no right to do that!\"\n\n\"To do what?\" Mr. Devlin asked her with a smirk. \"I didn't do anything.\"\n\n\"I have a job to do, Mr. Devlin,\" Tammy told him.\n\nHis smirk stretched into a wide smile. \"Then by all means,\" he said, \"don't let me keep you from it.\" He winked at her, and then headed out of the kitchen.\n\nTammy had to sit down, she was shaking so much.\n\nHurry up, Chad, she thought. Get me out of here.\n66\n\nAnnabel ran.\n\nBlood raced through her veins. She could feel it pulsing in her ears. She caught sight of the house up ahead, still too far away for comfort. She ran as fast as she could.\n\nA broken branch on the ground proved her undoing. She tripped over it and fell facedown in the leaves.\n\nA hand was gripping her arm. Whoever had been pursuing her had caught up to her.\n\n\"Annabel, are you all right?\"\n\nShe looked up. It was Richard Carlson.\n\n\"Oh, Chief Carlson,\" she said, nearly bursting into tears. \"I thought . . .\"\n\nHe helped her to her feet. \"You thought what?\"\n\n\"I don't know,\" she said, brushing leaves off her coat. \"A bear, maybe. I guess I thought you might be a bear.\"\n\nHe smiled. \"Why would you think that?\"\n\n\"Well, the way you were coming after me.\"\n\nHis face creased in puzzlement. \"I wasn't coming after you. I just spotted you a few moments ago. I saw you from the parking lot, running through the woods. I was concerned.\"\n\nAnnabel looked at him. \"You didn't whistle? A little two-note sound, like this?\" She demonstrated what she had heard.\n\n\"No, that wasn't me,\" Richard told her.\n\n\"Then somebody was chasing me,\" Annabel said, looking back out into the woods and shivering.\n\n\"Are you certain?\"\n\nShe moved her eyes back to the police chief's. \"No,\" she admitted. \"I'm not certain about a lot these days.\"\n\n\"Do you want to talk about it?\" he asked.\n\nAnnabel thought his eyes looked kind. A little tired, and there was sadness in there, too, but they were kind.\n\n\"I didn't want to come here,\" she said, surprised at herself. The words just tumbled out; she hadn't planned to say them. \"Jack thought it would be a good idea to leave New York and start over here at the Blue Boy Inn when his grandmother called and asked him to take over. So I came along.\"\n\n\"Have you changed your mind since arriving?\" Richard asked.\n\nShe shrugged. \"I was hoping the project of restoring the place would give me some purpose, something to focus on. But then Priscilla and Paulie went missing, and that hand was found out back. . . .\" She shuddered. \"It makes me long for the safe streets of Manhattan.\"\n\nThey both laughed and started walking slowly back to the house.\n\n\"If it's any consolation,\" the chief said, \"I believe the killer is long gone. I don't know why he came through this way, or why he either killed or kidnapped Priscilla and Paulie. But I don't think he stuck around. We've been through here several times a day. I've had officers searching these woods half a dozen times. There's been no trace. I think you're safe here.\"\n\n\"Do you?\" Annabel asked. \"Honestly?\"\n\n\"Safe from whoever killed Roger and possibly the other two,\" Richard told her. \"Whether there are other dangers here for you, only you can know that.\"\n\nShe thought of Jack, and for a second, she wanted to tell Richard all of her doubts about him, but then realized how ridiculous that would sound. The chief of police was not a marriage counselor.\n\nBut he should know about the chimney....\n\n\"Did Neville speak with you?\" Annabel asked. \"About what I found . . . ?\"\n\nRichard was nodding. \"That's why I'm here. I came out right away. He and Chad came to my office and told me about the blood.\"\n\n\"Maybe it wasn't blood,\" Annabel admitted. \"I may be getting a little hysterical.\"\n\n\"Well, it's worth a look, anyway,\" the chief said. \"Especially after what else they told me.\"\n\n\"What was that?\"\n\n\"They said that they heard some sounds down there today. Chad described it as a pack of raccoons eating very noisily.\"\n\n\"Raccoons?\"\n\nRichard's face became serious. \"If the killer dismembered Paulie and Priscilla and disposed of their remains in the chimney, it could be that animals smelled their decaying flesh and decided to creep down the chimney for a meal.\"\n\n\"Oh, dear God!\" Annabel was repulsed.\n\n\"I'm sorry, I didn't mean to upset you.\"\n\n\"It's okay, chief. I just . . . well, it's hard to imagine.\"\n\nHe placed a hand on her shoulder. \"You can call me Richard.\"\n\nHer eyes flickered up to his. \"Oh, well, thank you.\"\n\n\"Tell me something. Has anyone else looked into the ash dump since you discovered the blood?\"\n\nAnnabel found herself shuddering. Richard tightened his grip on her shoulder.\n\n\"Zeke and my husband both, I think,\" Annabel told him. \"Zeke locked it up again.\"\n\n\"Do you know if he looked inside before doing so?\"\n\n\"I don't.\"\n\nRichard nodded. \"I've got a forensics team on its way. But we ought to go down there ourselves first and take a look.\"\n\n\"Of course,\" Annabel said.\n\nThe chief removed his hand from her shoulder and they started walking back toward the house. \"I'm sure this isn't how you imagined your new life in Woodfield would be.\"\n\n\"You know,\" she told him, \"I had no idea what it would be like. I'm not a country girl. I've got the city in my blood. Born and raised in Manhattan. I miss the sounds of the city, the rush, the hustle, the constant energy.. . .\"\n\nHe smiled. \"My wife was originally from Manhattan. She loved the city as well.\"\n\nAnnabel paused. \"Richard, I can't help but notice you speak of her in the past tense.\"\n\nHe nodded sadly. \"She died. I've been on my own for a while.\"\n\n\"You miss her a great deal,\" Annabel observed. \"I can tell.\"\n\n\"Every moment of every day,\" Richard admitted.\n\n\"How hard that must be for you.\" She looked over at him. \"But how very wonderful it must be to have loved someone that much, and for her to have loved you as much in return.\"\n\n\"Yes,\" acknowledged the chief. \"Yes, indeed it was.\"\n\nThey were close enough now to the house that they could see the driveway. Chad's truck was just at that moment pulling in.\n\n\"Chad and Neville didn't come back with you?\" Annabel asked.\n\nRichard shook his head. \"Chad had to pick up some things at his office. But I came out straight away after they told me what they knew.\"\n\nHe pointed out another vehicle in the driveway.\n\n\"I see your husband is at home,\" Richard said.\n\n\"Yes,\" Annabel replied. \"And for some reason, I don't think Jack is going to be very pleased that I'm taking you down into the basement to look at the chimney.\"\n67\n\nTammy headed down the basement steps. Chad had said to measure every window, and she assumed that to mean the basement as well.\n\nBesides, she wanted to get away from that horrible Jack Devlin, who was lurking around the parlor and dining room. Every time she cast a glance in his direction, he was looking at her.\n\nOf course, Tammy thought, as she pulled the string to illuminate the overhead bulb, he could corner me down here more easily. But if he tried that, he'd have quite the surprise. Before making her way down the steps, she'd slung her purse over her shoulder. And inside, she carried a can of Mace. Being involved with a man as volatile as Roger, Tammy had learned to take precautions. And that prick Devlin would get a faceful of it if he tried anything again.\n\nOnce in the basement, Tammy saw that the windows were too high for her to easily reach. She realized she'd need something to stand on.\n\nOver by the chimney, she spotted a small chest. She could pull that over.\n\nBut as she approached, the dim light above revealed something leaning against the chest.\n\nA large doll of some sort, she thought. Propped in a sitting position.\n\nBut then\u2014the doll's head moved.\n\nTammy gasped. She heard sounds. Teeth gnashing.\n\nAll at once, the thing sitting against the chest turned its face to look at her. It was a little man\u2014and in his hands he held a bloody arm. He was gnawing at the bones of the fingers as if he were chewing on a chicken drumstick. Catching sight of Tammy, the little man hissed at her like a cat, baring a mouthful of bloody fangs.\n\nTammy screamed.\n68\n\nAnnabel had just come inside the house with Richard when she heard the scream from the basement. It seemed to rise up like a physical thing, pushing itself through the slats in the floorboards, causing the whole house to tremble. It caused Annabel and Richard to stop cold in their tracks.\n\nChad and Neville were likewise frozen for that split second of time, standing a little ahead of them in the foyer. The two men hadn't even yet taken off their coats when the scream cut through the quiet afternoon.\n\nRichard was the first to burst forward, heading toward the basement stairs, his hand on his gun. But he didn't have to go far. The young woman whom Annabel had seen earlier\u2014Tammy something, she thought, Chad's helper\u2014suddenly came bounding up the stairs.\n\nShe was as white as if covered in flour.\n\nShe bypassed Richard to run straight into Chad's arms.\n\n\"Get me out of here!\" she said in the highest pitched voice Annabel had ever heard. The poor woman was shaking so badly it looked as if she had epilepsy.\n\n\"What happened, Tammy?\" Chad asked.\n\n\"Down there!\" was all she could say, burying her face against his chest. \"Down there!\" Then she broke free of Chad's embrace and bolted outside, Chad following.\n\n\"I'm going down,\" Richard announced, gun drawn, heading down the stairs.\n\n\"I'm coming, too,\" Annabel said.\n\nThe chief turned to her. \"Stay up here,\" he barked.\n\n\"I've got to see whatever is down there,\" Annabel said.\n\nRichard made a face. \"Then stay well behind me,\" he told her.\n\nThey made their way down the creaky old stairs.\n\nThe light was still burning. They saw nothing. Annabel lifted the flashlight from the floor and shone it around the near-empty basement. Still nothing. All they could see was the chest near the chimney. The door to the ash dump was still padlocked, and the key was nowhere in sight.\n\n\"I've got to go out and interview Tammy about what it was she saw down here,\" Richard told Annabel. \"Whatever it was appears to be gone now. In the meantime, I'll need your permission to search this place, to pry open that ash dump door. . . .\"\n\n\"What the hell is going on here?\"\n\nThey spun around.\n\nJack had come down the stairs. He was angry, eyes blazing. \"What was all that screaming about? And exactly why do you need to search this house?\"\n\nRichard looked at him. Annabel tensed.\n\n\"Mr. Devlin,\" the chief explained, \"Tammy Morelli just ran out of this basement a terrified wreck. She saw something down here. I'm going out to interview her now. I don't know what she saw, but there have been enough reports about this house to warrant a complete search.\"\n\nThe chief pushed past Jack to head back up the stairs. Annabel attempted to follow him, but Jack grabbed her arm.\n\n\"I'm not letting anyone search this house,\" her husband growled.\n\n\"Jack, we have to cooperate with the police.\"\n\n\"Yeah, I think you've been cooperating a little too closely with that guy,\" he grumbled. \"I saw you out walking with him, taking a little romantic stroll through the woods.\"\n\n\"Jack!\" Annabel was nearly flabbergasted. \"You are talking crazy!\"\n\n\"I'm not letting him search this house. You said yourself that we needed to stand up against this reputation as being haunted and all that.\"\n\n\"Jack, this is no time to argue,\" Annabel said, hurrying up the basement stairs. In truth, she had become afraid of him, and didn't want to be in the basement alone with him.\n\nUpstairs, she found Neville sitting forlornly in the parlor. \"They're outside,\" he told Annabel, gesturing with his head toward the front door.\n\nShe stepped outside onto the porch. She could see Tammy sitting in the passenger's side of Chad's truck, her arms wrapped around herself. Richard was speaking with her intensely through the open door. Annabel headed toward them.\n\nChad stopped her halfway. \"Look,\" he said. \"Tammy's always been a bit hysterical. You know Roger Askew was her boyfriend.\"\n\nAnnabel was stunned. \"The guy whose hand was found in the wood box?\"\n\n\"Yeah. So she's easily spooked. She's had a hard time the past few years, and I was trying to help her out. But I wouldn't necessarily take what she says as gospel. I think she's clean now, but she used to do a lot of drugs. . . .\"\n\nAnnabel bristled. \"Just because someone once had an addiction shouldn't mean we discount their intelligence or reliability.\"\n\n\"No, no, of course not, I just meant\u2014\"\n\nAnnabel cut him off. \"What has she been saying?\"\n\n\"It's crazy talk, Annabel. She says she saw a little man in the basement\u2014like an elf\u2014eating a human arm.\"\n\nAnnabel couldn't speak for a moment. \"A . . . little man?\" she finally asked.\n\n\"Yeah. I think she was upset about something else, though.\" Chad looked over his shoulder, and then drew closer to her. \"You should know about this. She might be making a complaint. She claims your husband sexually harassed her earlier.\"\n\nThis was all too much for Annabel to take in. She felt light-headed, as if she might pass out.\n\n\"I'm sorry to have to tell you that,\" Chad said. \"But it's what Tammy is saying. And like I said, she can get hysterical at times.\"\n\n\"She's not hysterical,\" Annabel managed to say.\n\nRichard had left Tammy's side and was now approaching them. \"Take her home, Chad. She's very upset.\" The chief looked at Annabel. \"I'd like to get a team out here this afternoon to inspect that chimney,\" he told her.\n\nAnnabel glanced over at the front porch. Jack stood there, legs spread apart, arms crossed over his chest. He was scowling.\n\n\"I don't think he's going to let you,\" Annabel said sadly.\n\n\"Then I'll have to get a court order, and that could take a couple of days.\"\n\n\"I'll try talking to him,\" Annabel said, looking over at Jack, \"but he's . . . different.\"\n\n\"What do you mean?\"\n\n\"I'm not sure,\" she replied, as she headed back toward the house. \"I'm not sure of anything anymore.\"\n69\n\nNeville didn't think he should be listening to Annabel and Jack argue on the front porch. He stood up from the couch in the parlor and started to head up to his room, but then decided on a different destination.\n\nWhat had Tammy seen down in the basement?\n\nPerhaps he was being foolish to go back down there by himself. If there was some mad killer loose, the same one who'd cut off that bloke's hand and kidnapped and\/or killed Priscilla, Neville might run smack into him down in that dark, dusty space.\n\nBut he had to go down there. He had to find some answer to give Priscilla's parents.\n\nHer mum's voice still rang in Neville's ear. How upset she'd been when he'd finally called to give them the news of Priscilla's disappearance. She had blamed Neville, saying her poor darling daughter must have run off because of something he had said or done. It was the only way the distraught woman could make sense of the whole thing. Neville hadn't had many facts to give her to refute her theory. He was scheduled to return to England in a few days, and he didn't want to fly back over the pond without some understanding of what had happened to Priscilla.\n\nHe made his way down the stairs.\n\nHe saw nothing. What could possibly have spooked that poor girl so terribly? Neville tried to yank open the ash dump, but the padlock held firm. He hoped the police could get in there soon and inspect what was inside.\n\nHe was moving away from the chimney when he stepped on something on the floor. In the darkness he hadn't seen it. Evidently no one had.\n\nHe stepped down and scooped up the small object in his hand.\n\nIt was a ring.\n\nPriscilla's opal ring. The one she used to attract ghosts.\n\nHe slipped the ring into his pocket.\n\nGiven the tension between Annabel and Jack, he wasn't going to bring the ring to them. He didn't trust what might happen. Instead, he was going straight to the police station and give the ring to Chief Carlson.\n70\n\nRichard pulled up in front of Millie's store, turned off the ignition, and just sat there for a while. Such strangeness at the Blue Boy Inn. He had no idea what to make of it.\n\nIn his head, he was running through all those cold case files again. What was it about that place that had resulted in so many deaths and disappearances?\n\nThe first strange death had occurred more than a hundred years ago at the place. And they had kept coming. Richard had been especially repulsed by the story of poor old Andrew McGurk, whose body had been found up there decades ago, but not his head.\n\nAnd the most heartbreaking story was the little baby whose arm was found. The child's poor mother had been so distraught. That case, too, had remained unsolved for decades.\n\nRichard got out of the car and headed inside Millie's. The little bell over the door jingled.\n\n\"Well, if it isn't our hardest working public servant,\" Millie sang out from behind the counter. \"What can I do for you today, Richard?\"\n\n\"Just come to pick up some supper, Mil,\" he told her. \"It's going to be a late night at the station tonight.\"\n\nHe headed down the cereal aisle.\n\n\"Hey, chief,\" Millie called over to him. \"Do yourself a favor and at least buy some ground beef. Enough with the Cheerios.\"\n\n\"There's nowhere to grill a burger at the station, Millie,\" Richard said. \"I'll stick with my cereal and milk.\"\n\n\"How about some ham and cheese at least? I've got some nice hard rolls. . . .\"\n\n\"Too much trouble,\" Richard said, the box of Cheerios under his arm as he headed over to the coolers to fetch a half-liter of milk.\n\n\"You must have a microwave there,\" the clerk said. \"How about a nice Lean Cuisine? They've got some new ones. The salmon's pretty good. I've had it myself.\"\n\nRichard lifted the milk out of the freezer. \"Thanks, Millie, but my taste buds are in kind of a rut.\"\n\nShe shook her head. \"What you need is a good woman to cook for you.\"\n\nHe smiled as he set the milk and cereal down on the counter. \"Like I've said, you keep turning me down.\" He winked at her.\n\nMillie smirked as she rang up the items. \"How are things out at the Blue Boy?\"\n\nThe chief shrugged. \"Not sure. Lots of questions.\"\n\n\"What is it about that place?\" She accepted Richard's ten and gave him back his change. \"I've been here since I was fourteen, and that was a long time ago. And ever since, there's always been something weird up there.\"\n\n\"That's true, Millie. We're looking into it.\"\n\nShe placed the milk and box of Cheerios into a paper bag. \"I sure feel bad for that sweet girl who came all the way up here from New York to live at the place. Annabel. That was her name, right?\"\n\n\"Yes,\" Richard said. \"That's her name.\"\n\n\"What kind of husband doesn't tell his wife about the unsavory history of the place he's taking her to live?\"\n\n\"I don't know, Mil,\" Richard said, taking his dinner.\n\n\"I mean, that Jack Devlin had to remember how his poor little sister disappeared, and how his father went crazy. . . .\"\n\nRichard lifted an eyebrow. \"His father went crazy?\"\n\n\"Well, that was the rumor. He'd come up here, too, just like Jack is doing now, to take over the place after his father died. I remember he was a very nice man. But then he changed. Started acting all weird and secretive. Then, of course, when his wife died from breast cancer and his daughter disappeared\u2014well, people who saw him said he went completely off his rocker.\"\n\n\"That can happen, Millie,\" Richard said, \"when you lose someone you love.\"\n\nHe waved good-bye to her and headed back out to his car.\n\nDriving to the station, Richard didn't think there was anything unusual about Jack Devlin's father \"going crazy\" after his wife died.\n\nAfter all, Richard had almost gone crazy after Amy died.\n\nSo many specialists he'd sought out. So many second, third, fourth, and fifth opinions. And still Amy had died. Richard was a police officer, sworn to defend the public, to protect lives\u2014but he had been unable to find a way to save the woman he loved.\n\nFor a while after Amy's death, Richard had blamed himself. He had thought he might go mad. He understood exactly what Jack Devlin's father must have gone through.\n\nHow very much Richard still loved his dead wife. He still physically ached for her presence beside him in bed at night. He could still smell the fragrance of her hair on his pillow, could sense her energy in the rooms of his house\u2014even though Amy had never lived here in Woodfield with him. Even though she had been gone for so many years.\n\nCould he ever love another woman? Could he ever allow another woman in his life again?\n\nHis mind flickered to Annabel Wish.\n\nRichard couldn't deny that he'd been attracted to her. The first woman he'd felt that way toward in a very long time. She had looked so pretty, so vulnerable and yet so strong, too, standing there in the sunlight in the woods. Those were qualities Amy had had as well. Strength with vulnerability. Richard had found Annabel extremely attractive as he'd walked beside her, crunching through those fallen leaves. He had been filled, in fact, with the desire to kiss her. He had resisted the urge, of course.\n\nBut a momentary attraction to a woman did not mean he could love another woman. It did not mean he was ready to fall in love again. It only meant he was still alive, still a man, still with very natural desires. The chemistry, in other words, still worked.\n\nHe couldn't deny that he'd found Annabel Wish a very beautiful woman.\n\nHe thought of her up at that house. She was going to need his help, and Richard would be glad to offer it. That husband of hers was not to be trusted.\n\nRichard had been extremely frustrated by Tammy Morelli's refusal to bring harassment charges against Devlin. \"No, Richard, no,\" she'd kept repeating. \"I can't do that. Already half this town thinks I'm a slut. They'll blame me.\"\n\n\"That's crazy talk, Tammy,\" Richard had told her. \"You did nothing wrong! That man tried to take advantage of you. He shouldn't get away with it.\"\n\nBut Tammy had kept shaking her head. \"I just want to start living my life. I don't want anything dragging me down. Roger is gone and I'm starting new. Me and Jessica. I don't want any court cases or newspaper headlines. I just want to get on with things.\"\n\nHer eyes had narrowed as she had looked at Richard.\n\n\"And I don't want anything more to do with that place. The legends are true, Richard. After what I saw, I believe them!\"\n\nAnd what had Tammy seen?\n\nRichard pulled his car into the station lot. The reason any case Tammy might bring against Devlin might fail was because she also said some other things had happened at the Blue Boy. And if people questioned her about those things, they'd question her claims about Devlin as well.\n\nTammy had claimed she'd seen a little man eating a human arm.\n\nRichard had dutifully taken all the details down in his report. The little man, Tammy estimated, was no more than three feet tall, maybe less. He was slender, and had a bluish tint to his skin. Kind of like the blue boy on the sign out front. Richard understood the power of suggestion when under distress. Clearly, that was the explanation for Tammy's little blue man. She was upset and anxious after the harassment from Devlin and imagined she'd something horrifying. Richard thought psychologists called it \"counterprojection.\"\n\nStill, he worried about Annabel in that house.\n\nHe got out of the car and headed inside. He'd been right when he'd told Millie he had a long night ahead. He needed to get the ball moving on a search warrant. If he had to drive up to see the judge personally tonight, he'd do it. He needed to arrange a forensics team to check out that chimney. And he needed to finish going through all the cold case files relating to the Blue Boy Inn.\n\nThere was something very odd about that place, and Richard aimed to figure out what it was.\n\nHe was surprised to find the Blue Boy's English guest waiting for him when he reached his desk.\n\n\"Chief,\" Neville said, \"I have something very important to tell you. And show you.\"\n\nHe opened his hand to reveal an opal ring.\n71\n\n\"Maybe I'm crazy to be going out with you to the tile store, as if everything back home is all peachy keen,\" Annabel told Chad, sitting beside him in his truck as they rattled north on Route 7. \"But after yesterday, I just had to get out of that house.\"\n\n\"I don't blame you,\" Chad said, looking over at her. \"A little time away will do you a world of good.\"\n\nShe smiled over at him.\n\n\"Seriously,\" Chad continued. \"Think to the future. Someday this crap will all be over. They will have caught the guy who killed Roger and maybe killed Paulie and Priscilla, and you can go back to making the Blue Boy a first-class destination.\"\n\nThat was what she needed to do, Annabel thought, watching from the window as the countryside flew past her. She needed to focus on the renovation. She needed to throw herself into it\u2014transforming that house from its gruesome past to something new, something she could call her own. She had to think ahead, and not dwell in the past, or wallow in her delusions.\n\nAnnabel ran the risk of decompensating again. She couldn't let that happen. During her time in rehab, she had been told over and over again how important it was to stay strong in her mind. She had a tendency to retreat when things became difficult. It was sort of like the way some people curled into the fetal position to take shelter from the hard realities of the world. Annabel called it her \"black hole.\" She'd sink down into it and her mind would go berserk. She'd imagine things. She'd hallucinate. She'd fall into a world that wasn't real, that existed only in her mind. She'd believe nothing was safe.\n\nShe supposed it had started all those years ago when she was a little girl, locked in the closet by Daddy Ron. The young Annabel would fall down into a rabbit's hole of illusion, imagining Tommy Tricky and all the terrible things he would do to her. This had been her tendency ever since, when she became afraid or anxious. She'd withdraw, decompensate\u2014tumble down into her black hole where nothing made sense.\n\nBut she could no longer allow that to happen in her life.\n\nDespite what Tammy Morelli claimed to have seen, Tommy Tricky did not live at the Blue Boy Inn. Annabel had to believe that. Tommy Tricky was a childhood fantasy, told to her by Daddy Ron to frighten her. Tammy had been frightened by Jack, and then she had hallucinated, much as Annabel had done herself. To believe anything else, Annabel was convinced, would have been to admit madness.\n\nAnd she was not going to do that.\n\nLast night, Jack had been conciliatory. He'd taken Annabel in his arms and kissed her tenderly, explaining how much he, too, wanted to start over, to make the Blue Boy theirs, to free it from its lurid past. That was why he was so resistant to the police searching the place. Annabel was cool and reserved, remembering what Chad had told her about Tammy. Once again, she chose not to confront Jack. She planned to do so\u2014she wasn't going to just let this slip by\u2014but not just yet. Annabel wasn't sure she could trust her husband anymore. In fact, she'd become a little bit afraid of him. She worried that Jack would blow up at her, or try to control what she did, and if Annabel had learned anything during her time in rehab, it was how to stay safe. Nothing was more important than that. So until she felt safe with Jack again, she was not going to bring anything up with him that might set him off.\n\nOf course, things couldn't stay this way. This was no kind of marriage. Annabel knew she and Jack were at a breaking point. Either they got through this, or they didn't. Not for much longer would she live with Jack's volatility, or stand for his continued flirtation\u2014and maybe more than that\u2014with other women. She would see how the next few weeks went. If things only got worse\u2014if Jack remained hostile to a police investigation, for example, or if he continued to seem angry and distant\u2014Annabel would suggest they needed some time apart. She had no family other than Jack, so she had no idea where she'd go. But surely there must be some old friend in New York who would take her in. Or, if necessary, she'd go deeper into debt and stay at a hotel. If she needed to get out of here, she'd find a way.\n\nBut Annabel wasn't running just yet. For the moment, she would stay on course. One of the other things she'd learned in rehab was to resist the urge to flee. She learned that she was strong enough to face anything. So she would persevere for the next few weeks, keeping her mind clear, resisting the hallucinations, rejecting the fear. She would resist feelings of paranoia. She was safe here. Safe!\n\nAnd so she would push on with the renovation, the plan to make the house her own. She had no other choice. Otherwise, she might decompensate again and wind up nearly catatonic, as had happened yesterday morning. Annabel would not go down that road again.\n\n\"Kind of lost in thought over there, aren't you?\" Chad asked, interrupting her reverie.\n\n\"Oh, I'm sorry,\" Annabel said.\n\n\"No need.\" He smiled over at her, revealing dimples in his cheeks. \"I know you've got a lot on your mind.\"\n\n\"Well, maybe it's time I put some of it out of my mind.\" She returned his smile. \"So, tell me, Chad. Do you plan someday on taking over your father's business?\"\n\n\"That's the goal. My brothers aren't into it. But for me, I've loved remodeling houses ever since I built a loft in my bedroom when I was nine.\"\n\n\"Nine!\"\n\nChad nodded, his eyes on the road ahead of him. \"I've always been good with my hands.\"\n\nAnnabel laughed. \"I'm sure your girlfriend appreciates that.\"\n\nChad looked over at her and smirked. \"Annabel, was that a double entendre?\"\n\nShe blushed suddenly. \"I guess it did sound that way. But not what I meant. I just meant a woman likes to have a handyman around the house.\"\n\n\"Still sounds dirty,\" Chad said, laughing. \"But the point is moot. I don't have a girlfriend. Not anymore.\"\n\n\"What happened?\"\n\n\"Not really sure. I was dating this girl Claire ever since junior year of high school. I guess she just got tired of waiting for me to marry her, and she gave me the old heave-ho a few months ago.\"\n\n\"Oh, I'm sorry.'\n\n\"It's okay, really. If I'd have really been in love with her, like she said, I would have asked her to marry me a long time before that.\"\n\nAnnabel frowned. \"Well, did she ever ask you to marry her?\"\n\n\"Oh, plenty of times.\" Chad switched on his turn signal. \"I'm just not the marrying type, I guess.\"\n\nThey headed off the highway.\n\n\"That's okay,\" Annabel said, looking off into the miles of bare, shivering trees on the side of the road. \"Marriage isn't for everybody.\"\n\nChad looked over at her. \"Is it for you?\" he asked, and then quickly added, \"I'm sorry. That was too personal.\"\n\n\"No, it's okay,\" she said. \"To be honest, I'm not sure.\"\n\nThey were silent after that.\n\n\"Here we are,\" Chad announced a short time later, steering the truck into the lot outside a shop called BERKSHIRE TILE & PAINT. \"Let's go in and let our imaginations run wild, shall we?\"\n\nAnnabel smiled.\n\nBut as she headed into the shop, her own imagination was already racing far ahead of either of them. It was something Neville had said to her, late last night.\n\n\"I suspect this will all be cleared up in the next few days,\" he'd whispered, out of earshot from Jack, after he came back into the house from some trip into town.\n\nAnnabel had asked him what he meant, but he'd just smiled enigmatically, his finger to his lip.\n\nWhat had he meant? He'd appeared so certain. He'd still been asleep when Annabel left with Chad this morning, so there had been no chance to question him further.\n\nBut Annabel prayed he was right. All be cleared up in the next few days.\n\nPassing through the lot, Annabel noticed a few tiny snowflakes swirling through the air.\n\n\"We're supposed to get a big storm tonight,\" Chad said, sticking out his tongue to collect some of the flakes. \"Guess these are the first arrivals.\"\n\nLaughing, they made their way inside the well-lit shop. With music playing and cash registers jangling, people laughing and cell phones ringing, Annabel felt her anxieties evaporate. She was comforted that, at least for the moment, she was back in the real world, far away from the dark warrens of the Blue Boy Inn.\n72\n\n\"I told you,\" Jack Devlin growled, standing at the front door of the inn, blocking their way, \"I won't have you tramping through this place, making us any more notorious around town than we already are.\"\n\nRichard Carlson had anticipated this. Calmly, he reached into his jacket pocket and withdrew the court order that he'd just gotten from the judge. After Neville had paid him a visit last night, revealing the ring he'd found on the basement floor\u2014the ring Priscilla Morton never, ever removed from her finger\u2014Richard had made a special request of the judge to issue an immediate warrant for a search of the Blue Boy Inn. He had picked it up this morning, and showed it now to Jack Devlin.\n\n\"You'll see that this is a court order signed by a judge,\" he told Devlin.\n\nThe man's face darkened as he read the document. Behind Richard, Adam Burrell and two other deputies stood ready as backup, in case Devlin resisted. They would arrest him if they needed to.\n\nJack's dark eyes lifted from the paper to meet Richard's.\n\n\"Well, then,\" he said, his voice even. \"I guess I'll have to let you come in then.\"\n\n\"Thank you for being reasonable, Mr. Devlin,\" Richard told him.\n\nJack stepped aside to allow the officers to enter the house.\n\n\"You two start in the attic,\" Richard directed two of his deputies, \"while Adam and I will start in the cellar.\"\n\nThe chief looked around to see that the old caretaker, Zeke, had come into the parlor. Richard did not miss the look he exchanged with Devlin.\n\n\"Do you have the bolt cutters?\" Richard asked Adam.\n\n\"That I do,\" the deputy replied, producing them from his pocket.\n\nRichard nodded. With a flashlight showing the way ahead of him, he started down the basement stairs.\n\nThe padlock was easily dispatched with the bolt cutters. It fell to the earthen floor with a thud.\n\nRichard opened the old iron door of the ash dump. It creaked in the stillness of the basement. Pointing the beam of the flashlight through the door, he looked inside.\n\n\"This thing has been recently cleaned,\" he said.\n\n\"How can you tell?\" Adam asked.\n\n\"I can see the marks of whatever sort of brush that was used,\" the chief said, snapping photographs of the interior of the chimney. \"Wind and condensation would have dissolved them after a few days. But now they're plain as day. This was just cleaned this morning, in my opinion.\"\n\nHe brought the light closer to the floor of the ash dump.\n\n\"But there's still some residue that doesn't look like soot,\" he pointed out. \"You see?\"\n\nAdam peered inside and nodded. \"It's a different color,\" he offered.\n\n\"That it is.\" Richard swung the beam of the flashlight out of the ash dump. \"When forensics gets here, make sure they take a sample of that stuff.\"\n\n\"Will do, chief.\"\n\nRichard made his way around the rest of the basement. He saw nothing. He was hoping to find something else besides the ring that had belonged to one of the two missing persons. But there was nothing in the basement other than an old chest, which, when Richard opened it, turned out to be completely empty.\n\nHe hoped his deputies in the attic were having more luck.\n73\n\nNeville was steaming mad. He stomped down the stairs behind the two officers who had let him out of his room\u2014rescued him, in his opinion! He was bursting to give someone a piece of his mind\u2014and he expected it would be Jack Devlin.\n\nBut in the parlor he found the Blue Boy's owner speaking with Police Chief Carlson. Well, that was convenient!\n\n\"I want to report an assault!\" Neville shouted, rushing into the room and interrupting the two men's conversation.\n\nThe chief turned to look at him. Neville noticed the cagey expression that crossed Jack's face.\n\n\"An assault?\" the chief asked.\n\n\"Yes, indeed,\" Neville replied. \"I consider it an assault to be locked in one's room, unable to get out! What if there had been a fire?\"\n\n\"You were locked in your room?\" the chief inquired, looking from Neville back over at Jack.\n\nNeville nodded. \"That I was! I have been trying to get out for the past two hours, banging on the door and calling, but no one came to my assistance until these two officers here.\"\n\n\"We heard him calling on our way back down from the attic,\" one of the two deputies told Carlson.\n\nJack's face turned compassionate. \"Oh, Neville, I'm sorry to hear this. Zeke and I were shoveling snow off the walk and must not have heard you. Annabel is up in Great Barrington with the contractor, so none of us were here to respond. I'm so sorry.\"\n\n\"There was a key in the lock outside the room!\" Neville shrilled. \"I was deliberately locked in there!\"\n\n\"No one here would do such a thing,\" Jack assured him.\n\nNeville swung his eyes to Chief Carlson. \"He's lying!\"\n\nThe chief said nothing, just studied both men.\n\n\"Look, Neville,\" Jack said, trying to sound reasonable, \"you must have left your key in the lock last night. The doors are old. Sometimes if you don't remove the key, the door will lock again when it's closed.\"\n\n\"That's not true, chief,\" Neville said. \"Someone came into my room while I was sleeping, found the key, and then locked me in there!\"\n\n\"For what purpose would someone have done this?\" Carlson asked.\n\n\"I don't know,\" Neville admitted. He looked over at Jack. \"To be free to hide evidence, perhaps? Or look for it?\" He opened his fist, which until now had been tightly clenched at his side. \"Were you looking for this, Jack?\" Neville asked, revealing Priscilla's ring.\n\n\"I don't know what that is,\" Jack said calmly.\n\n\"The night before Priscilla disappeared,\" Neville told the chief, \"Jack was putting the moves on her. He was very aggressively getting her drunk. I don't know what happened, because I was too drunk myself.\"\n\nThe chief's eyebrows lifted. \"How come you didn't tell us this before?\"\n\nNeville frowned. \"I didn't think it had any relevance. But mostly because I didn't want to offend Annabel, who has been very kind to me.\"\n\nJack was smiling. \"We all had a little too much to drink. I told you that, chief. But I was certainly not putting the moves on Priscilla, as Neville says. I think he might just be a little jealous because Priscilla clearly was coming on to me.\"\n\nNeville saw the way the chief looked at Jack, the deep suspicion in his eyes. \"Just like Tammy Morelli was putting the moves on you, too?\" He smirked. \"Seems every woman who comes into this house gets the Jack Devlin treatment.\"\n\n\"I don't think that's fair, chief,\" Jack told him, looking wounded.\n\n\"Look,\" Carlson said, turning his attention back to Neville. \"You may well have locked yourself in by mistake. There's no way to prove otherwise. I'd just suggest you pack your things and leave. But before you do, I'd like you to come down to the station and give us an amended statement. Tell us everything you left out the first time.\"\n\n\"Gladly,\" Neville sniffed. \"I leave tomorrow for England, but I think I'll head down to Hartford this afternoon and stay at a hotel outside the airport tonight.\"\n\n\"If flights are taking off,\" the chief commented, and they all looked up at the window. The snow was coming down heavier now. \"We're supposed to be getting a nor'easter tonight.\"\n\n\"Well,\" Neville said, \"I'd rather brave snowy roads than spend another night in this place.\"\n\nHe turned and headed back up the stairs. He could feel Jack's eyes on the back on his head until he was out of sight.\n74\n\n\"Surely, you don't think I'd lock him in his room, do you?\" Jack asked Richard once Neville was gone.\n\n\"I don't know what to think,\" Richard replied. \"All I know is . . .\"\n\nHe was distracted by the sound of people coming through the front door.\n\n\"The forensics team is here,\" Adam announced.\n\n\"What's that for?\" Jack asked, his eyes narrowing as he watched Adam direct the two women and one man down the basement stairs.\n\n\"Just taking some samples,\" Richard told him. \"I'm sure you want to find out what happened to Priscilla and Paulie as much as anyone, don't you, Mr. Devlin?\"\n\nThe man's eyes darkened. \"Did you find any sign of them at all? Your men have been crawling all over this place from top to bottom for the last hour.\"\n\n\"Not yet,\" Richard admitted. \"But we'll keep looking.\"\n\nIn his mind, he cursed the snow. It could obscure or obliterate any clues outside. Thankfully, they'd already searched most of the surroundings.\n\n\"How much longer will you be here?\" Jack asked. \"Not that I'm trying to hurry you. I want to be completely cooperative.\" He smiled insincerely. \"I'm just curious.\"\n\n\"We'll be out of here in a few minutes, I'd think,\" Richard told him. \"Just long enough for the team downstairs to scrape a little gunk out of the ash dump.\" He smiled. \"But the court order allows us to return if need be.\"\n\n\"I ought to just give you a key to the front door,\" Jack said, smirking.\n\n\"No, we're happy to knock,\" Richard assured him.\n\nOne of the two deputies who had searched the attic came up behind the chief. \"We did find one thing that we can't account for,\" he said.\n\n\"What was that?\" Richard asked.\n\nThe deputy held up a plastic Baggie. Inside was a tampon. Used, slightly pink.\n\n\"The old man says he sleeps up there from time to time,\" the deputy explained. \"But I doubt this is his.\"\n\nRichard turned to Jack. \"Any ideas?\"\n\nJack sneered. \"Well, it certainly isn't mine, either.\"\n\n\"Have it analyzed,\" Richard told the deputy. \"See if it matches the DNA we took from Priscilla's hairbrush.\"\n\n\"Maybe it's my wife's,\" Jack offered helpfully, though the insincerity was still evident in his voice. \"It looks a little too fresh to have been my grandmother's.\"\n\n\"When is your wife back?\" Richard asked.\n\n\"Who knows, with this storm?\"\n\nThe two men locked eyes for several seconds.\n\n\"Okay, chief,\" Adam said. \"Forensics got what you wanted.\"\n\n\"All right, then,\" Richard said, nodding in Jack's direction. \"We'll leave you alone for now, Mr. Devlin.\"\n\n\"Be careful on those roads,\" Jack said, walking with the officers to the door, doing little to disguise his contempt for them. \"Looks like it's getting slippery out there.\"\n75\n\nFor the past hour, Annabel had managed the impossible. She had forgotten all about the nightmares back at the inn. Just as she used to do when she was working in New York\u2014on a magazine photo shoot, maybe, or organizing a fashion show\u2014she had focused in, laserlike, on the task at hand. Looking at tiles, comparing paint colors, she allowed herself to shift into creative mode. In her mind, she could see the parlor designed as a sleek, contemporary room, with lots of glass and exposed brick and mirrors on the walls. The kitchen would sparkle with new appliances and the bedrooms would be painted throughout with a soft, comforting blue. The bathrooms would be lined with brilliant Italian tiles.\n\n\"I'm really into bringing out the brick,\" Annabel said, looking at a sandblaster. \"If we offset the brick with some glass and metal . . .\"\n\n\"Sounds good to me,\" Chad agreed. \"Maybe even knock some of the brick out and replace it with glass blocks to bring the light through.\"\n\n\"Oh, excellent idea!\" Annabel beamed. \"This place will make Architectural Digest. I know people there.\"\n\n\"Here are some of the paint samples you requested,\" said a stocky clerk with thick glasses, worn low on his nose.\n\n\"I like the blue,\" Annabel said, examining them, \"but the yellow is a bit too bright. Can you subtle that a little more?\"\n\n\"Sure thing,\" the clerk said, returning to his paint mixer.\n\n\"This is so much fun,\" Annabel gushed to Chad.\n\n\"It's nice to see you smile,\" the contractor told her.\n\nAnnabel felt herself blush. Chad was awfully sweet, and cute, too. \"Well,\" she said, \"I must admit it feels good to smile.\"\n\nAt that moment, her phone buzzed in her purse.\n\nIt had been so long since her phone had worked\u2014the cell reception at the Blue Boy was the next problem they needed to address\u2014that she almost didn't recognize the sound. She dug the phone out from among the lipsticks and tissue and tampons in her purse. The number was that of the inn. It had to be Jack. Oh, God, what was he going to say?\n\n\"Hello?\" Annabel said into the phone, walking over to a quiet corner of the store.\n\n\"Annabel. It's Neville.\"\n\nHe was whispering.\n\n\"Neville. Is there anything wrong?\"\n\n\"I had to call you from the house phone because my mobile doesn't work here.\" He sounded anxious. \"I don't want anyone to hear me.\"\n\n\"What's wrong?\"\n\n\"I wanted to let you know I'll be gone by the time you get back. Someone locked me in my room this morning. I expect it was Zeke, on Jack's orders.\"\n\n\"Why would they do that?\"\n\n\"Because they were cleaning the house of evidence. I'm sure of it. Chief Carlson was here, searching the place.\"\n\nAnnabel was stunned. \"So he got a warrant?\"\n\n\"Yes. And he found nothing. That's why I expect Jack and Zeke cleaned things up.\"\n\n\"The ash dump?\"\n\n\"They opened it, and it was as dry as a whistle.\"\n\n\"None of that wet soot?\" Annabel asked.\n\n\"Nope. Though they did scrape out something from the bottom for analysis, but it wasn't very much.\" Annabel could hear Neville shudder at the other end of the line. \"I've never been happier to leave a place, no offense to you.\"\n\n\"None taken.\"\n\n\"I'm leaving now, heading down to Hartford before the snow gets too bad. Even if my flight's canceled tomorrow and I'm stranded at Bradley Airport overnight, it'll be better than spending another night here.\"\n\n\"I understand.\"\n\nNeville sighed. \"I'm supposed to fly to New York to catch a connecting flight to London. Pray that I make the connection. I'll be in touch, Annabel. I may have to return to testify if they find whoever took Priscilla.\"\n\n\"So you spoke with the chief?\"\n\n\"I'm heading there now to give him a final statement before I head out.\"\n\n\"Oh, Neville . . .\" Annabel thought she might cry.\n\n\"Thank you for your kindness, my dear,\" he said, \"and good luck with everything.\"\n\n\"Yes, Neville, good luck to you, too.\"\n\n\"If you don't mind me saying so,\" the Englishman said, \"I think there's something very sinister going on in this house. Take care of yourself.\"\n\n\"I will, Neville.\"\n\n\"Good-bye.\"\n\n\"Good-bye.\"\n\nAnnabel clicked END on her phone. She suddenly felt endlessly sad.\n\n\"Everything okay?\"\n\nShe looked up. Chad had approached her.\n\n\"I don't know,\" she said. \"Neville just called to say good-bye. He's leaving. But he told me the police had been by with a warrant and searched the place.\"\n\n\"Did they find anything?\"\n\n\"Apparently not,\" Annabel replied. \"But who knows? They took a sample from the ash dump. Otherwise it was clean.\"\n\n\"That's odd,\" said Chad. \"Hard to imagine that thing being very clean after all the chomping I heard in there. Raccoons aren't the neatest eaters.\"\n\n\"It was clean,\" Annabel said, her mind suddenly very far away.\n\n\"Look,\" Chad said. \"The snow is getting heavier. I've put everything on order. We should head back.\"\n\n\"Yes,\" Annabel agreed. \"We should.\"\n\nThe happiness she'd felt just a few moments earlier had now completely evaporated. The idea of going back to that place depressed her thoroughly.\n\nI've got to hold on, she told herself. I can't allow myself to fall down into a black hole again. I have to stay clearheaded. Strong. Resist my tendency to hallucinate and catastrophize. I have to keep my head, not lose it.\n\nBut Neville's words kept echoing in her mind.\n\nI think there's something very sinister going on in this house.\n\nAnnabel followed Chad out to his truck.\n76\n\nNeville hung up the phone. Glancing around to make sure that neither Jack nor Zeke was around, he hurried from the kitchen.\n\nIn the foyer, his bag was packed and waiting by the door. A hush had settled over the house. Outside, giant snowflakes were floating down from the gray sky, blanketing everything in white. Neville realized he was going to have to brush off his car. He wished he had brought gloves.\n\n\"I'm not packing any gloves,\" he'd announced to Priscilla with a smirk, as they'd left their flat for Heathrow\u2014which now seemed an eternity ago. \"I am packing as if we are not making any ridiculous ghost-hunting side trips and going straight to Florida.\"\n\nFlorida. Neville had always wanted to go there. He'd been looking so very much forward to those sandy beaches, that warm water, that cool margarita in his hand. Maybe another time, Neville consoled himself with a sigh.\n\n\"Neville.\"\n\nHis name came from the parlor in a whisper. He didn't recognize the voice. But it might have been Jack or Zeke, speaking very softly.\n\n\"Neville,\" the whisper came again.\n\nHe stepped around the corner and peeked into the room. He saw no one. The house seemed so quiet, as if every sound ceased. No hum from the electricity, no ticking of any clock, no wind from the eaves.\n\n\"Hello?\" Neville called.\n\nHe walked into the parlor and paused in front of the fireplace. Had he imagined what he'd heard? No, that hadn't been his imagination. He had heard someone whisper his name. Twice. And the only people in the house were Jack and Zeke.\n\n\"Neville.\"\n\nHe spun around. \"Who is there?\"\n\nSuddenly, Neville felt afraid. There was a killer loose, after all. Someone who had chopped off a man's hand, and who had surely killed Priscilla as well, if her ring was any indication. Was it the killer who called to Neville now?\n\nHe bolted for the door, planning to grab his suitcase and his coat and hurry off into the snowstorm outside.\n\nHe was almost to the door when he tripped. Just what he tripped over, he wasn't sure. But he went toppling over face-first to the floor.\n\nHe braced his fall with his elbows and forearms. The pain shot up through his shoulders. He might have broken something.\n\nBut he didn't have time to check. He looked around and saw what had caused him to fall.\n\nA little man, no more than three feet tall, with a little blue face wearing rags for clothes.\n\n\"Can I get him?\" the little man asked in a soft, whispery voice.\n\n\"Yes, you can get him,\" came another voice, and before the startled Neville could react, another little man, looking nearly identical, came hurrying around the corner. And then another little man appeared, and another and another, until five of the loathsome creatures had piled on top of Neville, grabbing at the back of his shirt and up and down his arms with their very sharp hands.\n\n\"Get off me!\" Neville managed to shout, trying to shake them off.\n\nBut the little men were incredibly strong. They kept him from standing by clamping their clawlike fingers into his calves, ripping through his pants and puncturing his skin. Neville screamed.\n\n\"Help me!\" he shouted, hoping that Jack or Zeke would hear. \"Help me!\"\n\nThe little men began pulling him across the floor. As much as Neville tried to fight them, he found he was powerless to escape their clutches. Three of them were at his feet, clawing and biting his calves and shins. The two others were positioned at each shoulder, grabbing ahold painfully and dragging him back into the parlor.\n\nThis can't be happening! This can't be real!\n\nNeville could see the creatures on either side of him. They were laughing, thoroughly enjoying their task.\n\nTwisting from side to side, still unable to break free of them, Neville looked up ahead. What he saw was even more unbelievable.\n\nTwo more of the creatures had popped their heads up from the ash dump panel at the bottom of the fireplace. They were waving their sharp little fingers\u2014they looked like squirrel claws\u2014motioning to their comrades to bring Neville closer.\n\n\"Nooo!\" Neville screamed.\n\nBut he couldn't fight back. All he could manage to do was writhe from side to side. The creatures seemed to have complete power over him. Neville began to whimper.\n\nThis was how he would die, he realized.\n\nThe little men thrust his head into the fireplace. The creatures waiting inside the ash dump suddenly clamped their claws into his neck and Neville shrieked out in pain. The others behind him were pushing his butt and legs now. With one final thrust, Neville's head and shoulders were crammed down through the trapdoor into the chimney.\n\nThis is what happened to the others! This was how they died!\n\nThe creatures behind him kept pushing and shoving, while the creatures ahead of him kept pulling him down. Neville's body was now wedged halfway down the chimney.\n\nThat's one really wide chimney, Chad had said. You could fit a body in there, sure.\n\nNeville felt his feet pass through the opening in the fireplace and he fell about a foot, becoming lodged in the darkness of the chimney.\n\nBelow him came the sound of gnashing teeth.\n\nI doubt it's squirrels and mice they're eating, Neville had told Chad.\n\nHe was about to learn just how right he was.\n77\n\nThe scream from inside the chimney echoed through the house. Suddenly it was cut short, and Zeke knew the man was dead.\n\nThe old caretaker stood in the parlor, staring at the fireplace. He was crying.\n\n\"How many more?\" he asked the quiet house. \"How long will this go on?\"\n\nNo answer came, of course. There had never been an answer, for as long as he had lived at the Blue Boy Inn.\n\nDamn that woman for opening up the fireplace. They had had it contained. They had had it under control.\n\nHe wiped his tears with the back of his hand. He knew what he needed to do.\n\nHe grabbed Neville's suitcase and headed outside. He'd hide the dead man's car in the woods. The police would think that he'd left for the airport in the midst of this blizzard in order to try to make it home.\n\nBut Zeke neglected to take Neville's coat. It remained hanging there on the hook.\n78\n\nChad drove his truck back toward Woodfield, Annabel seated beside him. The snow was still light, but it was starting to stick to the roads. The state trucks were already out, spraying salt and sand from side to side.\n\n\"Could be a nor'easter,\" Chad said. \"You're in for quite the experience, if the storm turns out to be as big as they're predicting it might be.\"\n\nAnnabel visibly shivered. \"I've been worrying about being snowbound at the inn ever since we decided to move up here.\"\n\nChad smiled over at her. \"But you've had big snowstorms in the city, too. I remember reading about that big one a few years ago where days later they found people in cars that had been piled over by snowplows.\"\n\n\"Oh, sure,\" Annabel told him. \"But you see, in New York, you have other people in your building. You can go next door, talk to someone. You're not isolated. You can go down to the sidewalk and you can crunch through the snow to a market that's managed to open. Even in the worst storms, some enterprising shopkeeper always manages to open his doors.\"\n\nChad sighed. \"Okay, I hear you. That's certainly not the case here. In Woodfield\u2014in a lot of western Mass, in fact\u2014things just shut down during a nor'easter. Power can go out and stay out for a week.\"\n\nAnnabel groaned. \"Oh, great.\"\n\n\"You ought to maybe think about getting a generator as part of your renovations,\" Chad suggested.\n\n\"Yes. That's a good idea. A very good idea.\"\n\nThey were quiet for a few moments as they drove.\n\n\"You know, Annabel,\" Chad said, breaking the silence, \"whatever's going down back at the inn, if you need my help . . .\"\n\nShe looked away, out the window. \"Of course, I need your help, Chad. I can't rewire electricity and knock down walls myself.\"\n\n\"No, what I mean is . . .\" Chad struggled to find the words. \"With everything that's happened, you know, with the police being there and conducting a search, well, if I can do anything . . .\"\n\nAnnabel turned back and looked at him. She offered him a small smile.\n\n\"Thank you, Chad,\" she said softly.\n\nHe liked her. He found himself really liking her a lot. Chad had never been attracted to a married woman before. He wasn't quite sure what to do with his feelings. Since the breakup with Claire, he hadn't had much interest in dating. He hadn't had much interest in women, period. A really gorgeous woman could walk right by him and Chad would barely notice. He remembered not so long ago, Paulie\u2014poor old Paulie\u2014looking at him as if he were crazy. Chad had been reading the newspaper, oblivious to anything around him. \"Dude,\" Paulie had said. \"That was a major babe who just passed by and you couldn't even pull your nose away from the Patriots' score long enough to notice.\"\n\nBut he sure noticed Annabel.\n\nShe was hot, no doubt about that. Her shiny auburn hair, her tiny waist, her perfect figure. And she was married to a major-league asshole, if Tammy's story was true\u2014and Chad believed it was. What if Jack Devlin had something to do with Paulie's disappearance, and the disappearance of that English lady? Annabel could be in real trouble in that house.\n\n\"Listen,\" Chad said, as he switched on the blinker to take the exit toward Woodfield, \"I'm serious. You have no idea what you're dealing with up at that old house. Too much weird shit has gone down there over the years. I'd like to be around to take care of you if\u2014\"\n\n\"You are very sweet,\" Annabel said, cutting him off, \"and more than chivalric. I appreciate your offer, Chad. I really do.\" She looked away again, back out the window. \"But I think I've got to learn how to take care of myself.\"\n\n\"Well, sure, but if things get rough . . .\"\n\nHe could see that she was smiling, though she didn't look back at him.\n\n\"Oh,\" Annabel said, \"things have been rough for a while. Jack said they'd get easier here, but they haven't. I'm not surprised.\" Finally, she looked back at him, giving him a smile and a flash of her pretty eyes. \"So I'm used to things being rough.\"\n\n\"As rough as all this?\" Chad asked, as the truck rattled onto the main road leading into Woodfield. \"A dead man's hand being found on your property? Two people going missing from your house? The cops searching the place? Can you handle all that?\"\n\nAnnabel's smiled faded and she looked away again. \"We shall see,\" she said softly. \"We shall see.\"\n79\n\nFrom his desk, Richard could hear the special weather bulletin on the television warning of an impending major winter storm. \"Snowfalls possible of up to four to five feet,\" the woman in the bright pink jacket was saying, doing her best to look suitably concerned.\n\nRichard was worried, but not about snow. He turned another page in the collection of cold case files on his desk. His predecessor as chief of police, Thad Arnette, had been on the job when Jack Devlin's father had \"gone mad,\" as Millie described it. Arnette had left a fascinating note about the episode.\n\nI questioned Devlin at length, both about his wife and his daughter, because some things just did not add up. Investigating the little Devlin girl's disappearance, we discovered the mother had also recently died, reportedly of breast cancer. But no death certificate could be located for her, and no hospital in the state had records of her being a patient. Devlin explained this by saying that there was a paperwork mix-up and that the death certificate would be issued shortly. He seemed very uneasy, easily distracted, and sometimes did not make complete sense. We need to follow up with him, and see when that death certificate is filed.\n\nRichard ran his hand across his short-cropped hair. Arnette suspected Devlin's father of something, he realized. Perhaps some complicity in, or knowledge of his wife's death and daughter's disappearance.\n\nBut soon after that note came another.\n\nDevlin has left town, returning to New York with his son. Have not received death certificate for Mrs. Devlin.\n\nAnd then that was it. Most likely, other cases took precedence and after a while, with Devlin gone, Arnette forgot about him. Out of sight, out of mind. Not the best way to run a police department, Richard acknowledged, but it happened. The questions Arnette had about the death of Mrs. Devlin and the disappearance of the little girl\u2014Cindy, her name was\u2014were filed away among the rest of the cold case files.\n\nWhat had happened to Jack Devlin's mother if she didn't die of cancer?\n\nRichard buzzed for his secretary. \"Betty,\" he said, \"could you come here in a moment?\"\n\nShe was promptly bustling through the door. Betty was a good scout. Utterly devoted to the department. She'd been there twenty-one years. She'd started right out of high school as a file clerk, working her way up to secretary to the chief. She was thirty-nine years old, a little plump, with short bronze-colored hair. She was always smiling, or smirking.\n\n\"You rang, master?\" she quipped.\n\n\"Betty, tell me what you remember about Jack Devlin's parents.\"\n\nShe was biting the end of a pencil. \"The parents? Didn't really know them. They weren't here very long. The grandparents, of course, were here for decades. . . .\"\n\n\"Yes, but do you remember when his parents came to Woodfield? His father was going to take over the inn, I believe, just as Jack is doing now.\"\n\nBetty was nodding. \"Yes, that's right. I do remember when they came, but they were gone in a flash, it seemed, right after the little girl got eaten by a bear.\"\n\n\"That was never proven.\"\n\n\"It wasn't?\"\n\nRichard shook his head. \"No, it wasn't. Do you remember that the mother died of breast cancer?\"\n\n\"Oh, that's right, I do remember that. Poor thing. I saw her at the market a few times. Seemed very pretty, full of life. Then all of a sudden she was gone.\"\n\nRichard pursed his lips. \"So,\" he said, \"she didn't ever appear sickly?\"\n\n\"No. I remember being real surprised hearing that she'd died, especially of cancer. It surprised everyone. One day, she was just gone.\"\n\nRichard sat back in his chair. \"According to the official story, she was taken to a hospital and died there.\"\n\nBetty grinned. \"And you don't believe it.\"\n\n\"I'm just questioning it.\"\n\n\"What does that have to do with what's happening at the Blue Boy today?\"\n\nRichard shrugged. \"Maybe nothing. I'm just trying to get a history of everything that's gone down there. It's had more than its share of tragedy and mystery.\"\n\n\"That's for sure. You know who would know a lot about the Blue Boy?\"\n\n\"No, who?\"\n\n\"Agnes Daley. She's at the library. She's the town historian, but really she's the town gossip. Knows everything about everyone.\" Betty shuddered. \"Probably knows a few secrets of mine, too. I wouldn't get on Agnes's bad side.\"\n\n\"How long has she lived in town?\"\n\n\"All her life. And she's got to be seventy-five, at least. She can take you way back.\"\n\n\"Thanks,\" Richard said. \"I think I'll give her a call.\"\n\n\"I'll get you the number for the library,\" Betty told him.\n\nWithin a few minutes, Richard was punching in buttons for the Woodfield Public Library. As the line rang, the chief's eyes glanced off toward the window. The snow was coming down lightly, little spirals of white. He hoped the accumulation wouldn't be too heavy. That always meant a nightmare for police work, with people snowed in on their streets, businesses unable to open, and fender benders all over town.\n\n\"Public library, Agnes Daley,\" a raspy, efficient voice suddenly announced.\n\nRichard introduced himself.\n\n\"Chief Carlson!\" the town historian exclaimed. \"It's not every day that the chief of police calls me. What can I do for you?\" Her ragged voice bore the unmistakable sound of someone who had smoked cigarettes all her life.\n\n\"I'd like to talk to you about what you know about the history of the Blue Boy Inn,\" Richard told her.\n\n\"The Blue Boy Inn?\" Agnes snorted. \"Oh, not you, too, chief.\"\n\n\"What do you mean, not me, too?\"\n\n\"At least once a week the library gets a call from some far-flung corner of the world\u2014Arkansas or Oregon or New Zealand or East Timbukistan\u2014asking about that place. Ghost hunters, you know. They want details and pictures and otherworldly accounts.\"\n\nRichard laughed. \"Well, I'm interested in the history of this world, Ms. Daley.\"\n\n\"Call me Agnes, or we'll never get along.\"\n\n\"All right, Agnes.\"\n\n\"Is this about the latest disappearances?\" she rasped. \"And Roger Askew's hand? Because if so, I don't have any idea how\u2014\"\n\n\"Well, that's what has prompted my investigation,\" Richard said, interrupting her. \"But I'm more interested at the moment in the inn's previous history. How well did you know Cordelia Devlin?\"\n\nAgnes chortled. \"As well as anyone could know her. She was a recluse. Barely ever left that house since her husband died. She always sent that strange old man, Zeke, she had working for her on errands into town.\"\n\n\"Was she more social when her husband was alive?\"\n\n\"I suppose she was. Then the husband died, and the son came up to take over the place . . .\" Agnes's voice faded off as she spoke.\n\n\"Did you know the son?\" Richard asked. \"That would be the current owner's father.\"\n\n\"I met him a few times. Seemed a nice enough man. Until the tragedies with his wife and daughter.\"\n\nIt appeared Agnes knew little more about Jack's father than Betty had. Richard tried a different tack.\n\n\"What about Cordelia's husband?\" he asked. \"What kind of man was he?\"\n\n\"Well, he was kind of an ambitious sort, as I remember, when he first took over the place from his father. But after that little baby died up there, he, too, became a recluse.\"\n\n\"The one whose arm was the only thing found?\"\n\n\"Oh, yes,\" Agnes said. \"How sad that was. Poor little thing. They said the child must have been eaten by a bear, just like what was supposed to have happened to little Cindy Devlin years later, though they found nothing of her.\"\n\n\"Sounds like you don't believe these were bear attacks.\"\n\nAgnes snorted again. \"Well, they fit so well into the haunted house narrative\u2014you know, the curse of the Blue Boy Inn. So many deaths up there.\"\n\n\"Do you remember any others?\"\n\n\"I remember that man McGurk. I was a little girl then. Freaked me out, as the kids say today. They found a body with no head. How outrageous is that? In the middle of the parlor yet! Never found the killer, either.\"\n\n\"Yes,\" Richard said. \"I've been reading up on that case. The investigators eventually closed the case. Everyone in the house had an alibi, and the head was never found, so they never made an arrest.\"\n\n\"There have been other deaths and disappearances, too,\" said Agnes.\n\n\"I have them all here in my files,\" Richard told her. \"How far do the stories of a curse go back?\"\n\n\"Well, right back to the beginning,\" she replied.\n\n\"And when was the beginning?\"\n\n\"Well,\" Agnes said, thinking, \"the Blue Boy was built around 1865, I think. Right after the Civil War. It's been extensively remodeled several times, and the integrity of its original architecture is gone, but I think you can still see some of the original structure. I haven't been up there in an awfully long time, so I can't say for sure.\"\n\n\"Was it always an inn?\"\n\n\"Oh, no. In the beginning it was a pastor's house. That's where the stories of the curse began.\"\n\n\"Care to fill me in?\" Richard asked.\n\n\"Its original owner was the Reverend John Fall. He was the head of a little church that once stood on the property as well. But in 1869, John Fall was hanged.\"\n\n\"Hanged?\" Richard asked. \"For what?\"\n\n\"Murder,\" Agnes replied. \"One of his pretty young parishioners was found with her throat slit from ear to ear. But word around town was that the deeper crime of Reverend Fall was witchcraft. According to the rumor, he'd killed the poor girl in some kind of satanic ritual. His church, these stories insisted, was merely a cover for his black arts.\"\n\n\"Any proof of that?\"\n\n\"Nope. Just the legends that persisted for decades after his death. The official line was that Fall had killed her because she was going to tell his wife they were having an affair.\"\n\nRichard sighed. \"I can see why stories of a curse would begin, if the original owner had been suspected of witchcraft.\"\n\n\"But there's no denying, chief, that ever since, lots of people have died or disappeared up there,\" Agnes told him.\n\n\"No,\" he agreed. \"There's no denying that.\"\n\n\"The church was torn down, and the house sat vacant for many years, until someone got the idea to open it up as an inn. Soon after the turn of the twentieth century, the Devlin family bought it, and they've owned it ever since.\"\n\n\"The current owner is the fourth generation of the family then,\" Richard said.\n\n\"Yes, sir. And he's already seeing the curse at work.\"\n\nRichard laughed. \"Surely you don't believe in it, Agnes.\"\n\n\"I'm a confirmed agnostic in all areas of my life. That doesn't mean I don't believe. It means that I neither believe nor disbelieve.\" She laughed. \"It's the only path for a historian, unless you want to bring your own biases to your study of history.\"\n\n\"That sounds like the smart approach,\" Richard told her.\n\n\"I hope I've been of some help to you, chief.\"\n\nRichard told her she had, and thanked her for her time.\n\n\"No problem,\" Agnes replied. \"If you want to show your appreciation, you can make sure my street is plowed first thing in the morning, before anyone else's when this blasted blizzard simmers down.\"\n\nRichard laughed. \"I'll speak with public works,\" he told her.\n\n\"Much obliged,\" Agnes said.\n\nAfter he had had hung up the phone, Richard thought he could hardly base an investigation on rumors of a nearly two-hundred-year-old charge of witchcraft. How foolish people were. Maybe they just couldn't believe a man of the cloth could succumb to the desires of the flesh and cheat on his wife, so they had concocted a tale about satanic rituals to account for it. He had learned little of use, it seemed, from town historian Agnes Daley.\n\nBut there's no denying, chief, that ever since, lots of people have died or disappeared up there.\n\nNope. There was no denying that.\n\nRichard stared out the window. The snow was coming down heavier now. The snow made him think of Amy. But everything made him think of Amy.\n80\n\nAnnabel and Chad came through the front door shaking snow off their coats and boots. \"Gosh, it's really coming down!\" Annabel exclaimed.\n\n\"Sure is,\" Chad agreed. \"In another hour those roads won't be passable. The storm has come earlier than they were predicting. We got back just in time.\"\n\n\"I'll make some hot tea,\" Annabel offered, \"if you don't have to rush right off.\"\n\nChad nodded. \"I want to take a look at the fireplace to make sure the flue isn't damaged. When you've got this bad of a storm, you should really keep the flue closed.\"\n\n\"Oh, good idea, thank you.\"\n\nChad headed off into the parlor while Annabel made her way to the kitchen.\n\nShe turned the corner and walked straight into someone standing there. She gasped out loud.\n\n\"Jack!\" she cried.\n\nHer husband smiled down at her. \"I heard you say you wanted tea. I've already got some ready, baby cakes. I figured you'd come in chilled to the bone.\"\n\n\"Oh.\" Annabel was flustered for a moment. \"Great, thanks.\"\n\n\"Shall I pour you a cup?\" he asked.\n\n\"Yes, please, Jack, thanks. And pour a cup for Chad, too. He's just checking the fireplace.\"\n\nJack smiled as he lifted the steaming pot off the stove. \"The fireplace?\" he asked.\n\n\"Yes. He's making sure the flue closes. With the storm coming, we'll want to make sure.\"\n\n\"Of course. Smart thinking.\"\n\nHe handed Annabel a mug, the steam rising from it in waves.\n\n\"Careful, sweetie pie. Don't burn your tongue.\"\n\nShe accepted the mug and held it in her hands.\n\n\"I'll pour one for Chad and set it here on the table for him,\" Jack said. \"Let it cool off just a little bit.\"\n\nAnnabel cupped her mug between her two hands and stared at her husband. How sweet he seemed all of a sudden. He reminded her of the caring man he'd been during her hospitalization, how supportive he had been, how thoughtful. Jack had stuck by her through the worst of times. She had been so grateful to him.\n\nThis was the real Jack, she wanted to believe. The Jack of the past few days\u2014mysterious and defensive\u2014had been simply the product of stress and fear. And grief, over losing his grandmother. And worry, over the disappearances of Priscilla and Paulie and what that might mean to their business venture.\n\n\"Have a sip of tea, sugar babe,\" Jack said. \"Take the chill off.\"\n\n\"Still too hot,\" she said, blowing on the mug. \"I sure hope we don't get snowbound. I don't like the idea of being trapped.\"\n\nJack smiled. \"Your old claustrophobia. Don't worry, angel heart. I'll dig a path from the front door to the street. You won't be trapped.\"\n\nA smile bloomed on her face despite her misgivings. \"Thanks, Jack. So, Neville's gone? I didn't see his car.\"\n\n\"Neville's gone,\" Jack told her. \"Drink your tea, baby doll.\"\n\nShe blew on the mug again. \"He called me, you know, before he left.\"\n\n\"Neville did?\"\n\n\"Yes.\" She locked eyes on Jack. \"He said he'd been locked in his room.\"\n\nJack frowned. \"Yes, he told the police chief that, too. Sweetheart, the key was in his door outside. He must have left it there himself, and then the door locked again when he closed it. It's a problem of these old doors.\"\n\n\"You don't think he had any cause to say he was locked in?\"\n\nJack laughed. \"Who would have locked him in, and why?\"\n\nAnnabel sighed. \"I don't know, Jack. But we need to talk. Some very strange things have been happening. I understand when the police searched the chimney downstairs, they found none of the gunk that I put my hands into the other day. How did that all that get cleaned out?\"\n\nJack looked mystified. \"I don't know. Maybe . . . Zeke cleaned it out?\"\n\n\"I'm wondering that myself.\"\n\n\"I'll ask him.\" Jack appeared as if a light suddenly went off in his head. \"That could explain why the police found nothing.\" He smiled over at Annabel. \"You know, I was very cooperative with them. I figured it was better to show that we had nothing to hide than to act as if we did.\"\n\n\"Oh, I'm glad, Jack.\"\n\nHis smile broadened. \"Drink your tea, honey. You're still shivering.\"\n\nAnnabel returned his smile. Then she lifted the mug to her lips and drank.\n81\n\n\"Christ,\" Chad said in a low voice, checking the fireplace one more time to be sure.\n\nThat's blood, he thought to himself.\n\nHe was squatting down, inspecting the floor of the fireplace, specifically the ash dump opening. It was a big panel, a good three feet by two feet, and all around its edges there was a sticky brownish-red liquid. When Chad touched his finger to it, he felt certain it was blood. Still fresh. Not dried.\n\nChad stood. He could hear Annabel and Jack talking in the kitchen. His first thought was to rush in and tell them what he found, but something stopped him. He didn't trust Jack. He was better off finding out what he could and taking the information directly to Chief Carlson.\n\nHe glanced over at the basement stairs.\n\nAnnabel said that the police had been here this morning and checked the ash dump downstairs. They'd found nothing.\n\nBut that blood's fresh. . . .\n\nChad surmised that the door to the ash dump at the base of the chimney would still be open. The cops surely didn't lock it up again.\n\nHe needed to take a look. Then, depending on what he found, he'd hightail it out of this creepy old place straight down to the police headquarters. And he might just take Annabel with him, just to be safe. He liked her. He wasn't going to abandon her if, as he was starting to think, somebody in this house was a murderer.\n\nAnd that somebody was very likely her creepy husband.\n\nListening as Annabel and Jack continued to talk in the kitchen, Chad made his way stealthily across the parlor and out into the hallway.\n\nHe paused, listening. He didn't want that weird old guy Zeke to spot him.\n\nChad made a dash for the basement stairs. He hurried down as quickly as he could without making too much noise.\n\nHe yanked the overhead light on.\n\nHe could see the chimney across the room, hunched over like a crippled Atlas, the sagging floorboards of the first floor the world on his shoulders.\n\nChad approached slowly. The dim light overhead outlined the small metal door on the side of the chimney. He could see that the door was shut tight. But there was no longer any padlock dangling from its handle.\n\nHe reached over to pull the door open, but then stopped. His hand hovered in midair.\n\nWhat are you afraid of? he asked himself.\n\nHe wasn't sure. But suddenly Chad was very, very afraid.\n\nHe forced his hand forward and gripped the door.\n\nHe tried to pull it open, but the old metal was stuck. Chad shook it a bit, and then gave it a good hard yank, and all at once the door swung open in his hand.\n\nHe had uncorked a river.\n\nA cascade of blood burst forward, splashing out from the chimney onto the floor, wetting Chad's shoes and turning them red. The flow only stopped when the doorframe became wedged with pale white human flesh.\n\nNeville's head, in fact, severed from his shoulders.\n\nHe stared out at Chad with wide, terrified eyes, his dead mouth in an eternal silent scream.\n\nChad, however, wasn't nearly so silent.\n\nHe let out a scream and turned to run, only to be met on the staircase by a woman.\n\nA woman he'd never seen before.\n\nA beautiful woman with long gray hair, dressed in a long, flowing white dress.\n\n\"Hello,\" the woman said calmly.\n\nChad opened his mouth to ask her to please get out of his way, to run as fast as she could out of this house, when he felt the pain in his abdomen. He cried out, then looked down and saw blood collecting behind his shirt.\n\nThe woman had just stabbed him with a knife.\n82\n\nAnnabel heard the scream below her and spun around.\n\n\"What was that?\" she called out.\n\nJack just smiled. \"Don't be so jumpy, baby cakes.\"\n\n\"Didn't you hear\u2014?\" She turned to run out of the kitchen toward the basement stairs, but suddenly her knees buckled. Her legs were too weak to hold her. She started to fall, but grabbed the table to steady herself.\n\n\"You okay, sweetheart?\" Jack asked, his voice eerily calm.\n\n\"My head,\" Annabel murmured.\n\nThe room was spinning. She was passing out. Something was happening in the basement, and she was losing consciousness....\n\nDrink your tea, honey.\n\nAnnabel looked over at Jack. He was watching her compassionately, but not moving a muscle to help her.\n\n\"You . . . drugged the tea,\" she managed to say.\n\nShe heard a second scream from the basement.\n\n\"Chad,\" said Annabel, just as she crumpled to the floor and blacked out.\n83\n\nThe light was different when Annabel woke up. She had the feeling that she'd been out for a long time. She realized she was in bed, in her room. She was alone.\n\nShe tried to stir, to sit up, but found her body was numb. She could barely move her hands or feet. What did he give me? That monster!\n\nA monster she had once loved\u2014and who, until a very short time ago, she had still been willing to give the benefit of the doubt.\n\nJack drugged me. He may be trying to kill me.\n\nJust as he may have killed Priscilla and Paulie.\n\nAnnabel's mind was racing. She thought she could see things clearly now. Jack had killed Priscilla the night they got drunk. He had stashed her body somewhere\u2014the fireplace!\u2014and Paulie had found it the next morning. That was why he'd had to kill Paulie, too. Jack had stuffed both their bodies into the chimney. Annabel had gotten their blood on her hands! And maybe\u2014the idea hit her like a lightning bolt\u2014maybe Jack had killed Cordelia as well. Richard had seemed to doubt the coroner's ruling of accidental death. Had his grandmother discovered his crime as well?\n\nAnother memory suddenly struck Annabel. She gasped out loud.\n\nChad! Right before she'd blacked out, she'd heard Chad scream.\n\n\"Oh, no,\" she moaned.\n\nPlease don't let Chad be dead, too.\n\nAt least Neville had gotten away.\n\nShe had to get out of there. She had to run. She had to get Richard.\n\nWith great effort, Annabel managed to sit up in bed. It was morning. That much she knew. Although they'd lost power, leaving the electric clock on one side of the bed dark, the batteries in the clock on the other side had kept it ticking. The time was 10:15. Annabel realized she had been unconscious all night.\n\nShe needed to move. But she doubted she had the strength to swing her legs off the bed, let alone stand and walk. She was breathing heavily from the exertion. She looked helplessly across the room. Through the window she could see the storm was still raging outside. While she'd slept, the snow had piled up at least three feet. The whole world outside her window looked white. She could barely see the trees. Everything was just a washout of glaring whiteness.\n\n\"That means I'm trapped here,\" Annabel said out loud. \"I can't get out and nobody can get in.\" She shuddered. \"It's just me and Jack.\"\n\nAnd Zeke. Unless Jack had killed him, too.\n\nBut even if he was alive, was Zeke friend or foe?\n\nAnnabel began to sweat. She wasn't sure what scared her more. Jack\u2014or the claustrophobia of being snowbound in this house. The two together threatened to push her back over the edge.\n\nDon't worry, angel heart. I'll dig a path from the front door to the street. You won't be trapped.\n\nHe'd been lying to her. Playing her. He'd known what he was doing. Lulling her into a false sense of security. Then he'd drugged her.\n\nBut why? What was Jack's plan?\n\nWhat had turned him so insane?\n\n\"I've got to get out,\" Annabel said, and summoned every fiber of her being to move her legs off the bed and touch her bare feet against the cold wood floor.\n\nHe undressed me, she realized. Jack took off my clothes and put me in my nightgown.\n\nHad he done other things to her?\n\nShe shuddered, remembering the night he had raped her. Yes, that was what it was. Her husband had raped her. She had tried to deny it to herself, but no more.\n\nJack was a monster.\n\nBut why? How? What had turned him into something Annabel no longer recognized?\n\nShe grabbed hold of the bedpost and pulled herself to her feet. She let out a groan doing so. Whatever she'd been drugged with still had a heavy grip on her body.\n\nStanding, she had a better view of the outside. The snow had covered Jack's car in the driveway. She couldn't even make out its outline in the parking lot. The driveway was completely inaccessible. She had thought earlier that there were at least three feet of snow out there. Now that she could see the ground better, she revised that estimate up to five feet. From her second-floor vantage point, Annabel could see that the snow had drifted across the front porch, completely covering the front door. If she tried to leave by that route, she would be faced with a solid wall of snow. There was no way she could walk out of this house, even if she got her legs to move more freely than they did at the moment.\n\nThe snow was still coming down, too.\n\nAnnabel realized with a cold certainty that she was trapped.\n\nHer palms started to sweat again. She began to shake uncontrollably. Stiffly, she wrapped her arms around herself.\n\n\"I've got to try,\" she said out loud. \"Maybe I can go out the back door.\"\n\nBut what then? Maybe, despite the drifts and the blowing snow, she could make her way to Millie's store, the closest inhabited place to the Blue Boy Inn. Annabel thought she had cell reception there. Even if not, she could just hide out there until someone came by, as she didn't imagine the store was open in this blizzard.\n\nShe'd need to be dressed more warmly if she was going to try walking through that snow. With great effort she shuffled over to her dresser. With even greater effort, she pulled open the drawers. She let out a gasp. All her clothes were gone.\n\n\"No,\" Annabel cried.\n\nHer cell phone. Her gaze swung around the room. She didn't see it.\n\nIt's in my purse downstairs, she remembered. Jack surely has it.\n\nIf she could get downstairs without Jack stopping her, she could call 911 on the house phone. It was an old model, with no cordless handsets that required electricity. If the phone lines weren't down, it should still be working.\n\nShe took a deep breath. With superhuman exertion, she put one foot in front of the other and walked. Steadying herself against the dresser, she made her way to the door.\n\nShe tried the handle. Of course it was locked.\n\nHe locked Neville in. Of course, he'd lock me in, too.\n\nBut at least Neville had managed to escape. Annabel suddenly felt she wouldn't be as lucky.\n\nShe began to hyperventilate. Her knees threatened to buckle, dropping her to the floor. She leaned up against the door to keep herself upright. She began to cry.\n\n\"You'll stay in there until you learn to be a good girl,\" came a voice through the door.\n\nAnnabel pulled back.\n\nIt was Daddy Ron.\n\n\"Look around, Annabel. Look around and see who's in there with you.\"\n\n\"No,\" she whimpered.\n\n\"Go ahead. Turn around. He's right behind you. Can you hear him?\"\n\nAnnabel listened. Yes, there he was. She could hear Tommy Tricky behind her, gnashing his sharp teeth.\n\n\"He's not real,\" Annabel cried in a terribly small voice.\n\n\"Oh, don't say that,\" hissed Daddy Ron through the door. \"That gets him mad. He doesn't like it when little girls don't believe in him.\"\n\nShe was crying like mad now, her body heaving.\n\nThis is crazy, Annabel thought, trying to get ahold of herself, to stop her plunge over the edge. This can't be happening. I'm an adult, not a little girl locked in a closet.\n\nThere is no such thing as Tommy Tricky!\n\nShe turned and looked over her shoulder, just in time to see something scurry under the bed.\n\n\"No!\" she screamed.\n\nShe was hallucinating again. That was the only explanation. She had to get out of this house! Even if it meant trudging through the blizzard. She'd rather freeze to death out there than go mad inside this house!\n\nShe walked slowly, awkwardly, over to the bed. Summoning all her strength, she grabbed hold of the side of the bed and shoved. It took a second, but then the bed slid across the floor.\n\nAnd sitting there, underneath, his blue face alive with a mouthful of teeth, was Tommy Tricky.\n\nAnnabel screamed.\n84\n\nRichard Carlson stood at the window of the police station staring out into the snow. It had been snowing like this the day Amy had died. His wife had looked like a little rag doll in her bed at Massachusetts General Hospital in Boston, weighing just eighty-odd pounds. Richard had stood by the window, watching the snow blanket the city, looking back every now and then over at Amy. Her breathing was so shallow. The thin white sheet drawn up to her neck barely moved. Richard had known Amy was dead even before the nurses came in to examine her. There was no dramatic ending to sweet Amy's life. She never opened her eyes. There was no last look between her and her husband. Her faint breathing just stopped and she was gone. Richard had sat at her bedside for three hours after she was gone, just holding her rapidly cooling hand.\n\n\"All the roads are blocked throughout the county,\" Adam told him, coming into his office behind him. \"There's no way the plows can get through this.\"\n\n\"I've lived in these parts all my life,\" said Betty, the police secretary, \"and I've never seen a snowfall like this.\"\n\n\"We've got reports of drifts up to nine feet,\" Adam said.\n\nRichard looked back out into the swirling white. Why was he thinking of Amy? Maybe because of how badly he had wanted to save her. Maybe because he'd vowed to her that he wouldn't let her die, and he had. He could stop a bank robber in his tracks, but Richard and all his police training had been no match against cancer.\n\nAnd now he was worried that another woman's life was in danger, and he might not be able to do a thing about that, either.\n\n\"Have you gotten an answer out at the Blue Boy Inn?\" Richard asked Adam.\n\n\"Negative on that, chief. I suspect the phone lines are down. Power's out all over the western part of the state.\"\n\nThe station was being powered by a generator. Richard sat down at his desk and turned on his lamp as he looked again at the report that had come back from forensics late last night.\n\nThe substance they'd scraped from the base of the chimney at the Blue Boy Inn was definitely dried blood. The DNA tests weren't yet back, and they'd likely be delayed due to the storm, so Richard didn't know if the blood belonged to Priscilla or Paulie, or if maybe there was some from both. But the very fact that there was blood in the chimney warranted him to take Jack Devlin in for questioning.\n\n\"There are lots of logical explanations as to why we might have found blood in there,\" Adam said, seeming to read Richard's mind.\n\n\"Name one.\"\n\n\"Somebody could have been cleaning out the ash dump and cut their hand.\"\n\n\"According to Annabel, there was enough blood in there to coat her own hand with it. If somebody had bled that much cleaning the damn thing, wouldn't we have been told about that? And this blood was recent. Seems to me somebody would have mentioned it if it was just a simple case of cutting their hand.\"\n\nAdam smiled. \"I don't doubt you're right, chief. I'm just playing devil's advocate. Because you know the lawyers are going to jump all over you if you try to arrest Jack Devlin on such flimsy evidence.\"\n\nRichard stood, returning Adam's smile with a smirk of his own. \"You younguns, all fresh from the police academy, think you know all the answers, don't you? Well, I'll tell you something, Adam. When you've been a cop as long as I have, you listen to your gut. And my gut tells me that Annabel is in danger out there.\"\n\n\"But we have no real evidence that her husband committed any murder.\"\n\n\"Nope, we do not. But you see, Adam, my gut also tells me that Jack Devlin is not our culprit.\" He smiled again as he saw his deputy lift his eyebrows in surprise. \"In fact, I think Devlin might be in almost as much danger as his wife.\"\n\n\"From who? That old geezer the caretaker?\"\n\n\"No, I don't think Zeke's our culprit, either. You see, this is where I'm stumped. To make any sense of this, I need to go over there. I need to get a plow to make a path for me.\"\n\nBetty laughed. \"Chief, even the county's biggest plows can't get through this stuff. All the roads are closed throughout the county.\"\n\nRichard frowned. \"Well, we've got to find a way as soon as we can.\"\n\n\"You could call in the National Guard,\" Adam suggested, not entirely seriously.\n\n\"Well, there's where your analysis would be right, Adam. The Guard would take a look at the evidence and say, 'We're supposed to send tanks out to the Blue Boy Inn because you found a little blood at the bottom of its chimney?'\" He laughed. \"No, we have to find a way to get over there ourselves.\"\n\n\"Do you really think it's that urgent, chief?\" Betty asked.\n\nRichard sighed. \"We probably have a little time. But even if Devlin isn't the killer, he seems to be covering up for somebody. He may suspect we found blood in the chimney, and he may be waiting for us to respond. But he knows that we can't respond right away, due to the storm, so he's likely waiting this thing out as much as we are. He doesn't expect we'll get there until the storm lets up, so if we could get there sooner, we could take him by surprise.\"\n\n\"But you just said he wasn't the culprit,\" Adam said, \"and that he might be in as much danger as his wife.\"\n\n\"Right. Whoever committed these murders is not rational. He or she could strike out at Jack as easily as Annabel.\" Richard sighed. \"And as we saw when we discovered Roger's body, our killer is pretty handy with a knife.\"\n\nBetty shuddered. \"Do we know for certain that Annabel is there at the house?\"\n\n\"Well, I think it's a fairly safe bet to assume she is,\" Richard replied. \"I saw her in Chad Appleby's truck yesterday as the storm was just starting. They were heading up toward the Blue Boy. I presume he was driving her back after picking out supplies for the contracting job.\"\n\n\"Have you spoken with Chad?\" Adam asked.\n\nRichard shook his head. \"No, but I've left him three voice mails. I presume in a storm like this, he's out trying to clear driveways. He's got a plow on his truck.\"\n\nBetty snorted. \"Chad's truck isn't big enough to make it through this.\"\n\n\"That's true,\" Richard said. A thought struck him. \"Adam, did our English friend ever come by yesterday on his way out of town to give another statement?\"\n\n\"Nope,\" the deputy replied. \"Never saw him.\"\n\n\"That's odd,\" Richard said.\n\n\"Maybe he wanted to beat the storm,\" Adam suggested. \"The snow had already started falling, and he wanted to get to the airport.\"\n\n\"Well, there's no way he's flying out in this,\" Richard said.\n\nHe settled back down at his desk. This whole situation was very difficult to figure out. If Roger's killer was somehow holed up at the Blue Boy\u2014despite their apparently thorough searches\u2014how did he get such a hold over Jack Devlin? Jack wouldn't have been so adamant about keeping the police from searching if he wasn't trying to hide something. That seemed to indicate Jack was somehow involved.\n\nYet Richard didn't think Devlin was the killer himself. Jack had an alibi for the night Roger was killed, and the chief believed that he really was sleeping when Paulie and Priscilla went missing. Annabel was out of the house at the time\u2014Millie vouched for her\u2014and Zeke was simply too frail to kill three people (Richard still believed Cordelia had been murdered) and dispose of two bodies in such a short time. The old caretaker certainly didn't have the strength to shove them down into the chimney, which now seemed to be the case.\n\nSo their culprit had to be someone else. But had it been the killer who had cleaned out the chimney and disposed of the remains? Or had Jack and Zeke done that much themselves? If so, why were they colluding with a killer?\n\nRichard could see no motive that linked the deaths of Roger, Cordelia, Priscilla, and Paulie. None. It seemed entirely random. The act of an insane person. A serial killer who killed for no reason whatsoever.\n\nOr for a reason none of them yet understood.\n\n\"Chief,\" Betty called over to him. \"I've got Charlie Appleby on the line. He's asking about Chad.\"\n\nRichard picked up the phone. \"Charlie,\" he asked, \"how you faring in this storm?\"\n\n\"My boys and I have been out there trying to break through it with our plows, but we can't do a thing,\" Appleby told him. \"We've given up. Staying home with some coffee and a shot of Jack Daniel's.\"\n\n\"Good for you. Chad with you?\"\n\n\"Nope, and that's why I'm calling you. He didn't come around with the other boys and he doesn't answer his phone. My eldest made it through the snow over to Chad's apartment on Green Street and he reports Chad isn't there. As far as he can tell, his truck isn't, either, though it's hard to tell with the snow drifted so high.\"\n\n\"Any idea where he might be?\" Richard asked.\n\n\"Well, last I knew he was going up to Great Barrington with that woman from the Blue Boy. I've been calling over there, too, and getting no answer.\"\n\n\"I know they made it back,\" Richard assured Chad's father. \"I saw them yesterday afternoon.\"\n\n\"But where's he been since?\"\n\n\"That I don't know.\"\n\nRichard could hear Charlie shudder over the phone. \"I had reservations when he told me he'd taken that job. That place is cursed. The people are no good. Something bad happens to whoever steps foot up there.\"\n\n\"As soon as we can get out there, Charlie, I'll inquire about Chad. For now, don't worry. He's probably out trying to make his way through this. Nobody's getting through.\"\n\n\"But why doesn't he answer his damn phone? It's one thing for a landline to be down, but far as I know, cell phones are still working.\"\n\n\"Cell reception can be affected by storms like this,\" Richard told the man honestly.\n\n\"I'm talking on a cell now, Richard, and it's working fine. All my other boys, their cells are fine, too.\"\n\n\"You know I'll do what I can to find him, Charlie.\"\n\n\"I know that. Keep me posted.\"\n\n\"Will do.\"\n\nRichard hung up the phone. This was not good news. Now he had Chad to worry about as well.\n\nAnd what about Neville? Had he skipped out on giving them a statement?\n\nOr had something happened to him that had prevented him from getting here?\n\nRichard needed to find a way to get out to the Blue Boy Inn. But how? He stood once more, returning to the window, which was now more than half covered with accumulating snow. There had to be seven feet, maybe eight, outside the station, and reports were coming in that drifts were sometimes double that.\n\nHe closed his eyes and saw Amy's face. When he opened them, it was Annabel he was seeing. Richard knew she was in danger. He had to find a way to get to her.\n85\n\nZeke unlocked the door to the attic and stepped inside. He placed the tray he was carrying down on a table and let out a long sigh.\n\n\"I'm so disappointed in you,\" he said, looking across the room at the figure hunched down in the corner. \"So very, very disappointed in you.\"\n\nThe figure didn't make a sound, nor did it move.\n\n\"I'm an old man,\" Zeke said. \"I've done what I could. This can't go on. You need to understand that. It just can't go on.\"\n\nStill the figure was quiet and still.\n\nZeke walked over to the little round attic window and peered out. The snow blew furiously. The whole first floor of the house was buried by now. The windows in the kitchen and the parlor were solidly white. It was getting cold, bitterly cold, in the house. The heat was off. And, of course, they couldn't build a fire in the fireplace to warm them.\n\nBehind Zeke came the sound of scurrying across the floor, then the sound of eating and drinking, as if the partaker were famished.\n\n\"You just need to understand,\" Zeke said, turning around, \"that I'm not doing this anymore. I just can't. I'm an old man. You need to understand that.\"\n\nHe said nothing more, just turned and left the attic, locking the door behind him.\n86\n\nAnnabel sat in the corner, her arms wrapped around herself. Where did Tommy Tricky go? He had scampered away. He was hiding in the room somewhere. He was watching her, waiting to jump out and eat her.\n\nTommy Tricky eats bad little girls.\n\nThat was what Daddy Ron told her, and Annabel believed him. She started to cry.\n\nThe door opened. Annabel's mother came into the room, looking down at her daughter with sad, defeated eyes.\n\n\"Oh, Annabel,\" her mother said, \"you got Daddy Ron angry again.\"\n\n\"Mommy, Mommy, you've got to save me from Tommy Tricky,\" Annabel cried, running to her mother, throwing her arms around her neck.\n\n\"Oh, baby,\" her mother told her. \"Tommy Tricky isn't real. He's just something Daddy Ron tells you about so you'll behave.\"\n\n\"No, Mommy, Daddy Ron says Tommy Tricky gets very, very angry when you don't believe in him.\"\n\nHer mother stroked Annabel's hair. \"Oh, baby, I'm so sorry he does this to you. But I don't know how to make him stop.\"\n\nAnnabel realized her mother wasn't really there. She was sitting by herself, in a corner of her room at the Blue Boy Inn.\n\nMy mother failed me, Annabel realized. She let that monster torment me because she was too scared to stand up to him.\n\nAnnabel began to cry harder.\n\nBut then she wiped her eyes with the sleeve of her nightgown. She had to get ahold of herself. She wasn't back in her childhood home; she wasn't a little girl. She was an adult, and she was in the Blue Boy Inn, and she had to find a way out. She was hallucinating again. She'd thought she'd seen Tommy Tricky. But Tommy Tricky wasn't real.\n\nExcept\u2014except\u2014\n\nAnnabel thought of Tammy Morelli.\n\nShe says she saw a little man in the basement\u2014like an elf\u2014eating a human arm.\n\nTommy Tricky eats bad little girls.\n\nAll Annabel knew was that she had to get out of there.\n\nShe stood. Her legs were moving better now, more easily, with less pain. Whatever drug Jack had given her appeared to be wearing off. Annabel glanced across the room. The clock now read 11:30. She must have been huddled in that corner for about an hour. Her eyes shot over to the window. She couldn't see anything outside. The snow had collected against the windowpanes. Annabel shuddered. She felt more closed in than ever.\n\n\"I can't give in to my fear,\" she whispered. \"I have to keep moving.\"\n\nCarefully, she walked toward the door. These old doors were flimsy. She could maybe rattle it enough that it would pop open. It was a slim chance, perhaps, but it was all she had.\n\nShe tried the knob. And to her great surprise and gratitude, it was no longer locked.\n\nMaybe I'd only imagined it was locked before, she told herself.\n\nAnnabel opened the door and stepped gingerly out into the hallway.\n\nThe house was eerily quiet. The muffled sound of the storm outside was the only thing she heard. Annabel made her way to the top of the stairs. She had no idea what her plan was. She was barefoot and wearing only a nightgown. But she could think of only one thing to do. Make a mad dash for the front door and\u2014\n\nAnd what? She had seen the drifting earlier. It came up halfway over the door. Even if she could reach the front door without Jack stopping her, she would run straight into a solid wall of snow on the other side.\n\nShe really was trapped.\n\nNo. She wouldn't accept that.\n\nAll right then. She'd still make a mad dash down the stairs. But she'd run to the kitchen. Maybe Jack would be in there, waiting for her. But maybe he wouldn't be. She had to take that chance. Because the phone was in the kitchen. She could call 911. Even if he caught her, if she could just press those three numbers and have the call go through, they'd send someone out. Annabel had to pray that the phone was still working.\n\nShe took a deep breath and started down the stairs.\n\nHer bare feet flew over the steps. She seemed to make no sound at all. It was almost as if she were running on air. She made it to the bottom of the stairs. But that was only half the challenge. She continued on without stopping to the kitchen. She could see as she rounded the corner that the kitchen was empty. Yes! Maybe Jack had left. Maybe she was alone in the house after all. She would call the police and\u2014\n\nBut when she turned to lift the phone off the hook she saw something terrible.\n\nThe phone was no longer there.\n\nIt had been taken clean off the wall. All that remained was an empty jack.\n\n\"No,\" Annabel moaned, and then put her hand to her mouth. She didn't want to make a sound.\n\nFace it, she heard Daddy Ron's voice tell her. You're trapped in there.\n\nTrapped.\n\nExcept\u2014\n\nAnnabel could almost hear the gears in her mind turning.\n\nExcept\u2014she might not be able to get out of the house from the first floor, but the snow had not reached the second. She could jump from a second-floor window. The snow would cushion her fall. If it was packed hard enough, she wouldn't sink completely into it, and she could, she hoped, trudge through it into town.\n\nRight. With bare feet. With winds that seemed to want to rip the roof off the house.\n\nBut what other choice did she have? Wait for Jack to come back and kill her the way he'd killed Priscilla and Paulie? That much Annabel was certain she didn't hallucinate. She firmly believed she was right about that.\n\nShe hurried back into the hallway. Hanging on the hook beside the door she spotted a coat. It was Neville's, she realized.\n\nOh, no, Annabel thought. I had thought Neville escaped. But why would he leave his coat?\n\nHis car had been gone. That much Annabel was sure of. But he wouldn't have left in a blizzard without taking his coat.\n\nNot knowing what to think, she grabbed Neville's coat and slipped it on. She'd need it if she was going to take a plunge into the snow. She could smell her friend's scent on the coat. It both comforted her and saddened her. Was he alive? What about Chad?\n\nAnnabel had never felt so alone, or so frightened.\n\nShe made her way back upstairs.\n\nShe decided she would go out the window in Cordelia's room. That was over the small roof that covered the front porch. Annabel could hop to the roof, and then take her leap into the snow. But she needed something on her feet. She'd never make it even as far as Millie's store if she had to do it barefoot.\n\nShe realized that although her own clothes were gone, Jack's clothes might still be in his closet.\n\nBack inside their room, Annabel paused, looking over at the bed, now pushed aside at an angle. Was Tommy Tricky under there?\n\nStop it, she scolded herself.\n\nShe took a deep breath and pulled open Jack's closet door.\n\nYes! His clothes were still there. And a pair of work boots! They would be big on Annabel, but if she tied the long laces several times around her foot, she should be able to keep the boots on during her trek through the blizzard. She sat down on a chair as she pulled the boots onto her feet. For the first time, a real sense of hope filled her. She would get away from here! She would not be trapped!\n\nBut then she sensed someone was watching her.\n\nShe spun her head around.\n\nJack stood there, leaning in the doorway, looking down at her with his arms folded over his chest and an enormous smile on his face.\n\n\"Where you goin', baby cakes?\"\n87\n\nChad woke up, only gradually becoming aware of his surroundings.\n\nHe was in the basement of the Blue Boy Inn. And he'd been stabbed.\n\nHe looked down at his body. He was drenched in blood. He'd lost a great deal. But his shirt and heavy sweater had acted to stanch the flow, becoming wedged in the open wound in his abdomen almost like a bandage. Chad knew that if he pulled his shirt and sweater off, he'd release the dammed-up blood again. He couldn't let that happen. He had to keep pressing against the wound until he could get to a doctor.\n\nWho was the woman who had stabbed him? He'd never seen her before. Another guest at the inn? Chad didn't think they'd taken in any other guests after all the trouble began. But who she was didn't matter right now. He had to get out of there.\n\nHow long he'd been out cold, Chad couldn't be sure. His head ached. From the throbbing he felt at his right temple, he guessed he'd hit his head after being stabbed. Probably when he'd fallen, he'd hit the bottom rung of the basement stairs. Yeah, that must have been what happened. Chad was lying in a heap at the bottom of the stairs.\n\nWas the woman with the knife still around?\n\nCarefully, Chad managed to sit up. He didn't know if he could walk. But he had to try. That woman might well have been the killer of Paulie and Priscilla, and she could come back to finish him off. Who knew if she had killed the others in the house? Last Chad knew, Annabel and Jack had been talking in the kitchen. Were they dead or alive?\n\nHe made it to his feet. His head began to spin, and he had to lean against the concrete basement wall to keep from falling down.\n\nHe had to get up those stairs and out of the house, but he couldn't do anything until his head stopped spinning.\n\nAbove him, he thought he heard laughter. A man's laughter, from far away. Probably from the second floor.\n\nHe recognized the voice as that of Jack Devlin.\n88\n\n\"Look at you!\" Jack guffawed. \"New York fashionista! Style arbiter of Orbit magazine! Wearing a ratty old corduroy coat and my scuffed-up work boots!\"\n\nAnnabel said nothing, just sat there staring up at her husband.\n\n\"Really, sweet baby angel, you could do better than that,\" Jack said, laughing.\n\n\"My clothes are all gone,\" Annabel said. \"I didn't have much of a choice.\"\n\n\"Yes, my darling, I took all your clothes.\" Jack's smile faded and his voice fell into a paternal, scolding quality. \"Because I wanted to discourage you from going outside. I just knew you'd want to go out and play in the snow. But it's too nasty outside for little girls.\"\n\n\"I'm not a little girl,\" Annabel said.\n\nJack ignored the comment. \"Come on, now, pumpkin pie, take off that coat.\"\n\n\"No,\" she said.\n\nJack took hold of her hands and forced her up from the chair. He walked her over to the bed and sat her down. He took a seat next to her.\n\n\"Listen to me, Annabel. We're about to have something very wonderful happen here in our new home. You have to cooperate.\"\n\nShe just looked at him. His eyes were wide, the pupils dilated. She thought Jack had gone certifiably mad.\n\n\"We are going to be so successful here at the Blue Boy,\" he told her, still holding her hands in a tight grip. \"And that's what we want, isn't it? That's what we came here to be, right? Successful? At long last? After all our disappointments?\"\n\nAnnabel remained silent.\n\n\"You have no idea how hard it was for me, sweetie babe, when my novel tanked. I really thought I was a literary wunderkind.\" He laughed out loud, a strange, unhinged sound that seemed to bounce up to the ceiling and ricochet through the house. \"But I was a fool. I let myself get so depressed over that, but in fact, I just missed my calling.\" He smiled, showing his straight, even teeth. \"I've found it here, Annabel. Here I can really be a great, great success. The money is just going to come rolling in.\"\n\n\"Why do you think that, Jack?\" she asked.\n\nAnnabel thought if she could engage him, ask him questions, appear to be interested in what he was saying, she could prevent him from hurting her.\n\n\"When my grandmother died,\" Jack said, \"Zeke told me the secret of this house. I'd always known it, really. But I had blocked it out of my mind. But Zeke brought it all back. If we are good to the house, the house will be very, very good to us. It will make us rich.\"\n\nAnnabel studied his crazy eyes. \"But, Jack,\" she said, softly, not wanting to upset him, \"your grandmother wasn't rich. The inn had been losing money for years. She was one step from bankruptcy when we took over.\"\n\nJack smiled, nodding his head. \"That was because Gran stopped being good to the house.\"\n\n\"What do you mean?\"\n\nJack sighed. \"A long time ago, my grandparents ran a very successful inn. They had learned the secret from those who had owned it before, and they carried on, doing what was right and good for the house. But then\u2014\"\n\nHis face darkened.\n\n\"But then what, Jack?\"\n\n\"Then my father came. After that, they stopped being good to the house.\"\n\n\"Your father?\"\n\n\"Yes. You see, darling baby angel cakes, that was what I had forgotten. How my father changed things during that short period when we were here.\"\n\n\"The period when your mother and your sister died?\"\n\nJack frowned. \"My father let his emotions overrule his better judgment. He gave in to his heart and didn't listen to his head.\" He smiled. \"Now, my mother had the right idea, only she never lived to see it. You had the right idea, too, my darling Annabel, but unlike my mother, you can live to enjoy the fruits of your labors.\"\n\n\"I can . . . live?\" Annabel asked.\n\n\"Of course, baby cakes. But not if you go out into that terrible storm.\"\n\nAnnabel allowed him to slip his boots off her feet. \"So you're saying,\" she asked Jack, \"that the house will make us successful because of some idea I had?\"\n\n\"Yes, honey baby lover.\" Jack tossed the boots, one by one, across the floor. \"You wanted to do the place over!\"\n\nHe gripped her by the shoulders and stared at her with his insane eyes.\n\n\"You removed the bricks!\" he said triumphantly.\n\n\"The bricks,\" Annabel repeated, \"from the fireplace.\"\n\n\"That's the secret of the house, baby. Where all its wondrous power comes from.\"\n\n\"And . . . removing the bricks will make us rich?\"\n\n\"Yes.\" Jack beamed. \"Annabel, my dearest love, I know what a terrible year you've had. You want success as much as I do. You want to be able to show those assholes back in New York that no one can keep Annabel Wish down for long. Annabel Wish is going to come back, better than ever! She's going to run the most popular, successful inn in New England! No\u2014in America! Maybe even the world!\"\n\n\"I don't understand, Jack.\"\n\nHe laughed. \"What don't you understand? Didn't we envision making this place a success? Didn't we see it as a first-class destination?\"\n\n\"Yes,\" Annabel said. \"That was what we talked about. . . .\"\n\n\"Well, it can be.\" He narrowed his eyes at her. \"So long as we are good to the house.\"\n\n\"And how can we be good to the house?\"\n\nHe smiled again. \"We give it what it needs!\"\n\n\"And what does it need?\"\n\n\"I'll take care of that part, Annabel,\" Jack said, standing now, apparently assured that he had gotten through to his wife, convinced her to do things his way. \"You needn't worry yourself about that.\"\n\n\"Jack,\" Annabel asked, \"did you kill Priscilla? Paulie? Your grandmother?\"\n\nHe looked at her with a kooky grin on his face. \"Me? Of course not, angel pie. Why would I kill them?\"\n\n\"Who did, then?\"\n\nHe sighed. \"The house killed them.\"\n\n\"The house?\"\n\nHe nodded. \"It had to. Because we weren't giving it what it needed. We could have handled it on our own. But now that I understand what we need to do, I'll take care of things. We won't have any more guests going missing.\" He laughed again, that terrifying yelp that sounded like a fox caught in a trap in the woods. \"That wouldn't do very well for business, would it?\"\n\nHe's mad. Insane. No question about it. He killed them, and he's blaming it on the house. I've got to get away from him.\n\nBut...\n\nAnnabel realized she was safer if she just went along with Jack for now. He saw some kind of life together in this crazy house. She needed to act as if she shared his hopes and dreams. She needed to patronize him, placate him, get him to trust her again. And then, when the storm subsided, maybe she could find a moment to make a run for it.\n\n\"So, I'll take care of the house,\" Jack was saying, stepping over to the window to look outside at the still roiling storm, \"but you'll have your own responsibilities, sweetheart.\"\n\n\"Whatever I need to do, Jack, I'm willing,\" she told him. \"You know that.\"\n\n\"I do know that, angel cake.\" He smiled over at her before returning his gaze out the window. \"You've had as bad a time as I have. We both need a new start.\"\n\n\"That's why we came here,\" she said.\n\n\"Yes, it is. But I had no idea the kind of success we could have here, if we were willing to do what was necessary.\" He frowned, looking back over at her. \"I'm not sure if we'll be able to trust Zeke for much longer, sweetheart. So you'll have to take over from him. Your job will be the attic.\"\n\n\"The attic?\" Annabel asked.\n\n\"Yes. Sweetheart, it's time you learned about the attic. You see\u2014\"\n\nSuddenly a voice from downstairs interrupted him.\n\n\"Annabel!\"\n\nShe recognized the voice. It was Chad.\n\n\"Annabel!\" he was calling. \"Are you here?\"\n\nAnnabel saw Jack's eyes change. They had calmed, become almost sane. Now they were suddenly wide with rage. Her husband spun on her.\n\n\"You've been fooling around with him, too, haven't you?\" he snarled.\n\n\"No, no, Jack, I\u2014\"\n\nHe leapt at her, clamping his hand over her mouth. Dragging her off the bed, he brought her back to his closet and shoved her inside.\n\n\"You stay in there, you bad girl,\" he spat. \"I'll deal with your lover!\"\n\n\"No, Jack, no!\" As the door closed against her, Annabel screamed, in a last desperate warning and call for help, \"Chad!\"\n\nThe closet door slammed shut, leaving her in darkness. She heard Jack turn the lock.\n\n\"No, Jack, no, please, don't lock me in here\u2014\"\n\n\"Turn around,\" came the voice of Daddy Ron, seeping through the door. \"Turn around and see who's behind you.\"\n\nAnnabel screamed.\n89\n\n\"Annabel!\" Chad shouted.\n\nShe'd called his name. She was somewhere upstairs.\n\nPerhaps he'd been a fool to call for her. But the house had been so deathly quiet when he'd finally made it up to the top of the basement stairs. Chad had assumed he was the only one here. Maybe they'd all left, not knowing he was wounded in the basement. He could have tried going out into the storm on his own, trudging to a spot where he might have better cell reception, and calling his father or Chief Carlson. But he couldn't have left without calling to Annabel, just in case she was still in the house.\n\nAnd, it appeared, she was. She had called back to him. But now she was silent.\n\n\"Annabel!\" Chad called again.\n\nThe trip up the basement stairs had disturbed the makeshift bandage his shirt and sweater had provided over his wound, and he was bleeding again, pretty profusely. Chad didn't think he'd make it up the stairs to the second floor, where Annabel's voice had appeared to come from. He should just take his chances going outside and making his way through the snow, calling the cops as soon as he could. He'd already seen that the phone was gone from the kitchen wall. The only hope to get help was to get out of the house.\n\nBut as he neared the front door, Chad clearly saw there was no way he could get out. The snow was packed solid against the door and all the windows.\n\nThe only way out would be through a second-floor window.\n\nAnd he couldn't leave without checking on Annabel. Maybe she'd been wounded, too, and had passed out after trying to call to him.\n\nHe had no choice. He had to go upstairs, or he'd stay here on the first floor, bleed out, and die.\n\nBut first, he ripped off a tablecloth from a hall table and wrapped it around himself as best as could, making a tourniquet to stanch the bleeding once again. It wasn't going to last long, but it would have to do for now. Chad didn't have a lot of options.\n\nHe began making his way up the stairs.\n90\n\n\"Sorry, chief,\" Adam told him, hanging up the phone.\n\n\"The county has none of its largest plows to spare. They said they'll get down here as soon as they can, but that might be days. The storm has completely immobilized everyone.\"\n\n\"This is crazy,\" Richard grumbled. \"When this is over, I'm demanding bigger plows at the town meeting. The selectmen better go along with it. I don't want to hear any noise about money. Winters are just going to keep getting worse around here, and we need to be prepared.\"\n\n\"Hey, chief,\" Betty called. \"Were you expecting a fax from the town library?\"\n\n\"No,\" he said, barely hearing her.\n\n\"Well,\" the secretary said, approaching him with a thick stack of papers, \"they just sent you over twenty-seven pages of town history.\"\n\n\"Just put it in my in-box,\" Richard told her.\n\nBetty complied.\n\nRichard was trying to think of an excuse to convince state officials to send them one of the massive snowplows they kept up at Great Barrington. But even if he said somebody up at the Blue Boy had some major health issue and needed help right away, they'd no doubt insist there were people all over western Massachusetts in the same position. If only he could\u2014\n\nTwenty-seven pages of town history.\n\nAll of a sudden he remembered his conversation with Agnes Daley.\n\nBut there's no denying, chief, that ever since, lots of people have died or disappeared up there.\n\nRichard reached over and grabbed hold of the stack of papers Betty had placed in his in-box. Sure enough, they were from Agnes Daley.\n\nMaking use of being snowbound here in the library, Agnes had written in her careful penmanship on the cover sheet. So glad the board of directors installed a generator. You were asking about the history of the Blue Boy the other day. You seemed dismissive of what I told you. Here's some newspaper coverage from back in the day, chief. Give it a read. A.D.\n\nRichard glanced over what Agnes had sent. They were microfilm printouts of old newspaper pages. The date on the first was from 1869.\n\nREV. FALL HANGED FOR MURDER, the headline read.\n\nThere was an illustration of a man dressed all in black dangling from the end of a noose.\n\nRichard looked at the next page. It was from a year later.\n\nWOMAN FOUND DEAD, DISMEMBERED\n\nNEAR FALL'S CHURCH\n\nThe murders up at that place really did stretch back a long time.\n\nAnother headline:\n\nFORMER CONGREGANTS CLAIM   \nREV. FALL PRACTICED   \nBLACK ARTS, SATANIC RITUALS\n\nThe piece seemed like bad gothic horror fiction to Richard, but he read it anyway.\n\nFormer congregants of the late Rev. John Fall, hanged here three years ago for murder, now claim that the disgraced pastor forced them to participate in the black arts. Fall's goal, these congregants insist, was to cast a spell that would open a portal into the netherworld, where he could harness the daemonic beings within to do his bidding. He possessed books filled with spells and incantations for such a nefarious purpose.\n\n\"Ridiculous,\" Richard murmured.\n\nThe rest of the pages were more recent\u2014coverage of the deaths of the various people at the Blue Boy Inn, including the reports of Jack Devlin's missing sister.\n\nBut at the very end of the pile was another piece, dated December 26, 1915. It was one of those humorous little items newspaper editors often used as fillers. Agnes's neat, precise handwriting ran across the top of it.\n\nThis little item wasn't about Rev. Fall or any mysterious death, but the headline jumped out at me. What do you think?\n\nRichard looked down at the article.\n\nCHILD CLAIMS TO HAVE SEEN BLUE ELVES\n\nRichard read the piece.\n\nLittle Millicent Collins of Bangor, Maine, five years old, visiting the Blue Boy Inn in Woodfield with her parents, claimed to have seen \"three little elves with blue faces\" poking their heads out of the parlor fireplace. Could Santa have left behind some of his helpers on Christmas Eve?\n\nSomehow the image of those three blue faces looking out of the fireplace unnerved him even more than tales of opening portals to the netherworld. All of this talk of witchcraft and spells and demonic rituals was absurd, of course. But, nonetheless, Richard was even more disturbed and anxious after reading it.\n\n\"I've got to get over to the Blue Boy,\" Richard said, banging his fist on his desk. \"There's got to be a way!\"\n\n\"I've got a pair of snowshoes,\" Adam said, shrugging.\n\n\"I'd give it a try,\" Richard said, \"but I doubt I'd get very far.\"\n\n\"What you need,\" Betty said, poking her head around the corner from her outer office, \"is a snowmobile.\"\n\n\"Of course!\" Richard said. \"Where can I get one?\"\n\n\"Well, my son has one, but it's at our house.\"\n\nRichard jumped to his feet. \"Have him ride it over here!\"\n\n\"In this storm?\" the secretary asked.\n\n\"Betty, that's what snowmobiles are for!\"\n\nShe scowled. \"Maybe the kind the Navy uses in the Arctic, but Richard, my kid uses his just for fun.\"\n\n\"Then find me a better one,\" the chief barked. \"Why doesn't the department have snowmobiles for our regular use anyway? We're living in the goddamn Berkshire mountains, aren't we?\"\n\n\"I'll make some calls,\" Betty said.\n\n\"You, too,\" Richard ordered Adam.\n\n\"Yes, sir!\" his deputy said, picking up his phone.\n\nRichard looked out the window. It was becoming increasingly difficult to see outside. The snow had nearly walled them in. Only at the very top of the window could Richard see a bit of sky, and that was just a furious flurry of white.\n\nI have to get over there. He had never felt so sure about anything. His gut was telling him something terrible was taking place. I've got to get over there or Annabel is going to die.\n\nHe couldn't get the image of those three little blue elves in the fireplace out of his head.\n91\n\nIn the darkness of the locked closet, Annabel tried to keep her wits about her, but this was too much. She tried to hang on to the reality of the present, but she was fast sliding down a very slippery chute into the past. She was a little girl, locked in the closet by Daddy Ron, and Tommy Tricky was somewhere in the darkness behind her, waiting to devour her with his sharp blue teeth.\n\nShe saw a little man in the basement eating a human arm.\n\n\"No, no, no,\" Annabel moaned, and commenced banging on the door. \"Jack! Let me out! Let me out! Oh, please, let me out!\"\n\nBehind her, she heard something scurrying in the darkness among her husband's shoes.\n\n\"He's not real,\" she said out loud.\n\n\"Don't get him mad,\" came Daddy Ron's voice through the door.\n\nThe closet seemed to be getting smaller. It was closing in on her. It was like that time she'd been trapped in the elevator. The walls had been moving in on her from all sides, and Annabel had feared she would be squeezed to death. She had utterly decompensated then, ending up in a puddle on the floor. She needed to fight that\u2014stay clear in her mind\u2014if she was to survive this. Because being locked in the closet wasn't the worst horror. Beyond the door her husband had become a madman, and surely he would kill her like the others.\n\nBut it was hard to resist panic when she heard the scuttling behind her.\n\nThe closet was almost completely dark. Annabel could not even see her hands in front of her face. The only light came from the small space between the door and the floor. She got down as close to the space as she could, irrationally terrified that she'd breathe up all the air in the closet and suffocate. She could see out into the room through the space. She could see one of the boots on the floor that Jack had removed from her feet. She could see the bottom of the dresser.\n\nAnnabel leaned in close to the space to gulp in some air.\n\nWhat was Jack going to do to Chad? Maybe Chad had gotten away. Maybe he'd gone to get help.\n\nThat wasn't likely. Not in this storm.\n\nAnnabel heard the scuttling again. Except this time it wasn't behind her. It was right beside her. Right beside her face as she pressed it to the bottom of the door, gulping in air.\n\nShe moved her eyes.\n\nBeside her own hand was another. A very small hand, with fingers that resembled the claws of a squirrel or a raccoon. It was hard to say for sure in this darkness, but Annabel thought the hand was blue.\n\nShe screamed.\n92\n\nRichard slammed down the phone. \"Nothing,\" he grumbled over at Adam. \"The town doesn't have any snowmobiles and the county can't get any to us until the storm lets up. We won't need them then!\"\n\nAdam shook his head. \"I haven't had any luck, either. A friend of mine has one, but I can't reach him.\"\n\n\"Do you even know how to ride one?\" Betty asked the chief.\n\n\"I've been on one,\" he told her.\n\nShe smirked. \"That doesn't mean you know to steer it.\"\n\n\"We didn't have much need of them in Boston. But several years ago we were helping search for a missing girl up in New Hampshire. I . . . I rode a snowmobile then.\"\n\nBetty's smirk deepened. \"You rode one?\"\n\n\"Yeah.\" Richard sighed. \"Right into a tree.\"\n\nBetty laughed. \"Our great hero to the rescue.\"\n\n\"Is your son good with his?\" Richard asked her.\n\n\"Sure. But as I said, it's just a beginner's model. Frank and I bought it for Danny last Christmas. It's just a small Ski-Doo.\"\n\n\"Ask him if he knows who else might have one in town,\" Richard said.\n\n\"I'm one ahead of you there, chief,\" Betty told him. \"I already called him, and he's trying to find you a good-size one.\"\n\n\"Thanks, Betty.\"\n\nThe window of the police station was now completely covered with snow. A couple of officers were out front, digging a passage out the front door. It felt as if they were inside an igloo.\n\n\"Have you tried the Blue Boy's phone again?\" Richard asked Adam.\n\n\"Yup. Still no answer, chief.\"\n\n\"The phone at Millie's store still works. I just called her. So why wouldn't the Blue Boy's still be working?\"\n\n\"Beats me, chief.\"\n\n\"He's disconnected it,\" Richard said.\n\nAdam looked up at him. \"Who has?\"\n\n\"Jack Devlin. I feel certain of it.\"\n\nIn his mind's eye he saw Amy, so small in her hospital bed. The snow had been piling up outside the hospital much as it was accumulating outside the station now. Richard couldn't shake the feeling of d\u00e9j\u00e0 vu helplessness. The longer he remained trapped in the station, the greater likelihood that he would let Annabel die.\n\nOutside, he heard the whirr of an engine. It sounded like a buzz saw at first, then like the spinning of tires in snow. He hurried over to the front door, Betty at his side.\n\n\"It's Danny!\" the secretary exclaimed.\n\nA teenage boy in a bright green wool hat and orange parka riding a yellow snowmobile was stuck in a snowbank in the station lot. The officers who'd been shoveling out the front entrance were rushing over to assist him.\n\n\"That boy,\" Betty said, shaking her head. \"He's so impulsive. When I told him you needed a snowmobile, he offered to come over. I told him under no circumstances did I want him venturing out in this storm. But he came anyway.\"\n\nRichard was grinning. \"I'm glad he's a disobedient child.\"\n\nBetty looked up at him. \"But take a glance out there, chief, will you? He's stuck! That's not a very powerful machine. If Danny got stuck in our parking lot, how are you going to make it all the way out to the Blue Boy Inn?\"\n\n\"Danny made it all the way out here from your house, didn't he?\" Richard asked, his eyes on the boy.\n\nWith a shove from the officers, Danny was able to maneuver the snowmobile out of the bank, then hopped back onboard and steered it over toward the front door, where he brought the machine to a stop. He waved a big blue-mittened hand when he noticed the chief and his mother watching from the glass door.\n\n\"I'm giving that boy a medal,\" Richard said, beaming.\n\n\"If he wasn't so tall,\" Danny's mother said, \"I'd give him a spanking.\"\n93\n\nChad climbed the stairs to the second floor.\n\nHe'd thought he'd heard Jack's voice, and then someone walking toward the stairs. But no one had come down, and now the house had fallen silent once more. Chad didn't like going up the stairs. He was afraid of what he might find. He was also afraid the exertion would cause him to lose more blood. But he had no choice, really.\n\nThe only possible way out of this house was from a second-floor window. And besides, in all good conscience, he couldn't just leave Annabel up here after he'd heard her call his name.\n\nHe reached the top of the stairs.\n\nChad looked around. There was no one. But peering down the hallway, he could see the door to Annabel's room was open. He had to go in there and look for her. If she wasn't there, then he was throwing open the window and making a jump for it. The snow was so high out there, Chad figured he might actually be able to just step out onto it, if it was packed hard enough.\n\nHe took his first step down the corridor. Under his foot, the old floorboards creaked.\n\nChad paused, listening. He heard nothing, so he continued on down the hall.\n\nAt Annabel's door, he paused again.\n\n\"Annabel?\" he whispered, looking inside.\n\nHe heard the sound of crying. It was coming from the closet. He hurried to the closet door.\n\n\"Annabel!\" he called.\n\nIf that was her behind the door, she was sobbing uncontrollably. She couldn't speak.\n\n\"Hang on, Annabel,\" Chad said, grabbing the closet door handle and finding it locked. \"I'll get you out of there. I'll get you out and then we are getting out of this house.\"\n\nHe began to shake the door handle as hard as he could.\n94\n\nAnnabel heard Chad's voice as if from a very far distance.\n\n\"Hang on, Annabel,\" he was saying. \"I'll get you out of there if I have to break this door down.\"\n\nShe focused. She brought herself back to the present. She yanked herself out of her childhood, where she was a tiny girl, curled up in a ball in the closet, crying her heart out. Now she was back here, an adult woman, and she was going to get through this.\n\nThe door in front of her suddenly shuddered as Chad, outside, threw his weight at it.\n\nThrough the space at the bottom of the door, Annabel could see his feet. The black rubber soles of his boots.\n\n\"Oh, Chad!' she cried. \"Oh, Chad, get me out of here!\"\n\n\"I will, Annabel,\" he called in to her. \"Just hang on.\"\n\nThe door shuddered again.\n\nThrough the space, Annabel saw droplets of blood raining onto the floor all around Chad's boots.\n\nShe turned her eyes. The little hand that had been beside her was still there. And now a little face emerged from the darkness to join her in peering out through the space under the door.\n\nAnnabel was face-to-face with Tommy Tricky.\n\nHe licked his blue lips with a snaky blue tongue.\n\n\"You're not real,\" Annabel told him.\n\nTommy just smiled, and then withdrew back into the darkness.\n\nFor a third time Chad threw himself against the door. It shook in its frame, and Annabel heard something crack. But still it did not open.\n\nShe could hear Chad breathing heavily outside. And she could see now another pair of feet. They appeared to be a woman's feet, in fuzzy pink house slippers. They had come in behind Chad. Facing the closet door, he wouldn't have seen whoever it was come into the room.\n\nAnnabel had to warn him.\n\n\"Chad!\" she screamed. \"Behind you!\"\n\n\"Annabel, I'm\u2014\"\n\nBut whatever he was about to tell her became a scream.\n\nAnnabel watched as the pink fuzzy slippers came up right behind Chad's boots. She heard the sound of a knife plunging into flesh. Chad screamed again, and then dropped to his knees. Annabel could hear the knife, plunging in and out of him, making a horrible suction sound each time. Chad was screaming. Suddenly, as Annabel peered through the space, she saw a river of blood rushing across the floor toward her. Her hands, pressed close to the space, were quickly covered in it. Annabel leapt to her feet and screamed.\n\nThe sound of stabbing\u2014suction in, suction out\u2014continued for the next several minutes. Annabel covered her ears, but she could still hear the terrible noise. Chad's screams softened into moans. Finally, after agonizing minutes, he was silent.\n\nAnnabel dropped her hands from her ears. She heard the sound of slippers scuffing out of the room.\n\n\"Oh, Chad,\" Annabel cried, the tears dropping off her cheeks as she leaned against the door. The young man's blood continued to flow into the closet all over her bare feet. Annabel realized she had just listened to Chad's murder.\n\nAnd now she was certain that hers would be next.\n95\n\n\"Okay, so you grab the handlebar like this,\" Danny was shouting over the wind, as Richard watched him closely. The seventeen-year-old's hands closed over the bar tightly and he pushed it up to demonstrate. \"Once the key is in the position, you pull the cord out and push the handlebar up. You follow?\"\n\n\"I think so,\" Richard said. \"Seems easy enough.\"\n\n\"It's more than easy. This is a really light-footed sled. You should be able to glide over ungroomed snow without any problem. Just watch out for trees or bushes or anything that's covered that you can't see.\"\n\n\"That could send me flying, I guess,\" Richard said.\n\n\"Well, maybe not flying, but it could get you stuck.\" Danny turned his red-cheeked face to him. \"That's what happened to me on the way in here, when I got stuck. I was fine all the way to the station from my house because I stayed on the roads. The minute I came over the station's yard I didn't know there was a hedge underneath me, and I ran smack into it. If you stay to the roads, you should be okay, because all that's under you is six or seven feet of snow.\"\n\n\"That should be easy then,\" Richard said, his breath freezing in front of his face.\n\nDanny shook his head. \"You haven't seen it out there, chief. Sometimes you can't tell where the road starts and yards begin. And not everyone got their cars off the street in time. There are lots of cars buried under drifts of snow. You don't want to run into one of them.\"\n\n\"No,\" Richard said. \"I sure don't.\"\n\n\"How far you got to take this?\" Danny asked.\n\n\"Up to the Blue Boy Inn.\"\n\nThe teenager grinned. \"That haunted house? What's going on up there now?\"\n\n\"That's what I aim to find out.\" Richard put his gloved hand out and Danny grabbed it. \"I can't thank you enough for bringing this over, Danny. If I cause any damage to it, the department will pay for a new snowmobile for you.\"\n\n\"It's cool, man,\" Danny said, getting up off the seat and gesturing to Richard to take his place. \"If you crack this one up, I've got my eye on a more advanced model.\"\n\n\"I think the department may be buying a few of those,\" Richard said, straddling the snowmobile and grabbing the handlebars. \"If we're going to have more snowstorms like this, we need to have them in store.\"\n\n\"Oh, we're going to have a lot more of these kind of storms. Lots of extreme weather ahead of us. The climate's changing, chief. Hope you're not a denier.\"\n\nRichard smiled at him. \"Nope, no denier here, Danny. I've learned that we deny reality at our own peril.\"\n\nHe checked that his gun was safely secured at his side, and then he revved the motor as Danny had showed him.\n\nBehind him, Adam was shouting over the noise. \"As soon as I can round up some more machines, we'll be up to join you, chief!\"\n\nRichard gave his deputy a thumbs-up with his right hand. Then he pulled on his goggles, tightened his scarf around his neck, gripped the handlebars, said a little prayer, and took off across the snow.\n96\n\nAnnabel sobbed against the closet door. Chad was dead. The poor man . . . he had come into this house on her request. He had come up here, trying to rescue her. And now he was dead.\n\nThe door shifted as she leaned against it.\n\nAnnabel pushed. The door creaked open. Chad had succeeded in breaking the lock.\n\nSlowly, fearfully, Annabel stepped out of the closet. She had no idea who might lunge at her as she did so. Tommy Tricky? Daddy Ron? The woman in the pink slippers who had killed Chad?\n\nHis body lay crumpled on the floor in front of her. She tried not to look at it. She feared she would start to cry so hard she'd never be able to stop. Poor Chad. She thought of him at the tile store, telling her it was nice to see her smile. Oh, poor, poor Chad.\n\nShe couldn't afford to break down. Not yet. She had to find a way out of the house.\n\nShe had no idea where Jack had gone to. Or the woman who had stabbed Chad. Who was she? Annabel thought she knew. She was the same woman she'd seen her first day at the inn. The woman in the woods. Perhaps she had been the killer all along.\n\nBut even if so, Jack was helping her. Jack was somehow under her sway.\n\nAnnabel knew she didn't have much time. At any moment, one of them could come back into this room. She had to get out.\n\nBut first she had the presence of mind to reclaim Jack's boots. If she was going out in that snow, she'd need them.\n\nTo get them, however, she had to step over Chad's body. The very act of doing so nearly sent her over the edge again. She was shaking uncontrollably as she pulled on the boots, both of them sticky with Chad's blood. It was all over the floor, coating the soles of Annabel's feet.\n\nShe then headed for the door. Zipping up Neville's coat, which she still wore, she hurried into the hallway, pausing only briefly to make sure the coast was clear.\n\nThen she ran down the corridor toward Cordelia's room.\n\nHer plan was the same as before. She would go out the window onto the small roof over the front porch. The snow was at least as high as that. From there she would trudge off, as best she could, hoping the snow was hard enough that she wouldn't sink too far. Hoping, too, that she could brave the cold and the wind until she got to Millie's.\n\nShe made it to Cordelia's room. Once again she had the presence of mind to close the door behind her. She didn't want Jack to come walking past and spot her as she went out the window.\n\nAnnabel's heart was thudding in her chest. She could see the window. She could see freedom!\n\nBut then she heard a two-note whistle. The same sound she'd heard that day in the woods.\n\nAnnabel stopped and looked frantically around the room.\n\nAnd all at once, the woman who had killed Chad, crouching behind Cordelia's bed, stood up.\n\nThey locked eyes. The woman had long gray hair and was wearing a diaphanous white dress and fuzzy pink slippers. Her dress and hands were splattered with blood.\n\n\"Hello,\" the woman said, emotionlessly.\n\nAnnabel turned to run, but was stopped in her tracks when, directly in front of her, two little blue men suddenly ran past, scurrying under the bed.\n\nAnnabel screamed. This couldn't be happening!\n\nThe woman was now directly behind her. Annabel felt the cold blade of the knife pressed against the back of her neck. She didn't dare breathe.\n\n\"Where are you going?\" the woman asked in Annabel's ear. Her voice didn't sound angry or threatening, just curious. \"It's really bad outside.\"\n\nAnnabel didn't answer.\n\n\"You have to stay here,\" the woman told her. \"You have to take care of the house.\"\n\n\"Who are you?\" Annabel asked in a little voice.\n\nShe felt the knife move away from her neck and she breathed a little easier.\n\n\"Zeke is very angry with me,\" the woman said. \"I can't find him. Have you seen him?\"\n\nShe moved away from Annabel, but not very far. She still held the knife upright in her hands. Annabel saw that, despite her long, stringy, unkempt gray hair, the woman was not that old. She was quite pretty, in fact. Her skin was pale but very smooth.\n\n\"I haven't seen Zeke, either,\" Annabel said, latching on to an idea. \"Maybe we can go out looking for him.\"\n\nThe woman smiled. \"No. You'll try to run away.\"\n\n\"No, I won't.\"\n\n\"Yes, you will.\" The woman touched her fingertip to the blade of the knife. \"I don't want to have to do to you what I did to that young man.\"\n\n\"No, no, please don't,\" Annabel said, shrinking back. \"I won't run away.\"\n\n\"I had to kill him, you know.\"\n\n\"I know,\" Annabel said, desperate to appear cooperative and understanding. \"You had to, because he was trying to run away and take me with him.\"\n\n\"That's not why I had to kill him.\"\n\n\"No?\"\n\nThe woman shook her head, her long gray hair swinging from side to side. \"I had to kill him to feed the house.\"\n\n\"Feed . . . the house?\"\n\nNow the woman nodded vigorously. \"That's my job.\" She looked intently at Annabel. \"It was supposed to be yours, as well.\"\n\n\"Mine?\"\n\n\"Yes,\" the woman insisted. \"You have to feed the house for me, because no one is supposed to see me.\"\n\n\"Okay, okay,\" Annabel said. \"I'll do whatever you say.\"\n\nThe woman's eyes narrowed as they studied her. \"You're afraid of the house, though, aren't you?\"\n\n\"No,\" Annabel lied. \"I'm not afraid of the house.\"\n\n\"Then crawl under the bed.\"\n\nUnder the bed. The little men had run under there.\n\n\"No,\" Annabel muttered.\n\n\"Prove to me that you aren't afraid of the house.\" The woman thrust the knife in Annabel's direction.\n\n\"Please, don't make me go under there. I know . . . I know what's under there!\"\n\n\"Do it!\" the woman commanded, waving the knife back and forth in the air.\n\nI've got to overpower her. I've got to jump her, wrestle the knife away from her, Annabel thought.\n\nBut the blade was suddenly at her throat. Annabel had no choice but to drop to her knees.\n\n\"Good,\" the woman said. \"Now crawl under the bed. They're waiting for you.\"\n\nAnnabel looked under the bed. Two pairs of eyes blinked in the darkness.\n\n\"No,\" Annabel cried, trembling uncontrollably.\n\n\"Come join us, Annabel,\" a little voice called to her. \"It's time you learned that we are real.\"\n97\n\nRichard thought he had the hang of this. He was cruising pretty easily along Main Street on Danny's Ski-Doo, taking the kid's advice and staying strictly to the middle of the street. It was easy for Richard to do that here, since the buildings of the town center were on either side of him and it was clear exactly where he was. But when he had veer off and follow Route 7A up into the woods, it became increasingly difficult, with all these mountainous drifts of snow, to know what was road and what was not.\n\nThe chief did the best he could, squinting to see through his goggles, occasionally having to reach up and wipe the snow off them with the back of his glove.\n\nThe weather report had said the blizzard was winding down, but it sure didn't feel that way to Richard. Even in his thermals and heavy parka, he shivered against the deep chill. And the snow was blowing and drifting as fiercely as ever.\n\nHe needed to get to the Blue Boy before it got dark. After nightfall, it would become impossible to make it through these woods. On the back of the Ski-Doo, there was room for one other person. If Annabel wasn't the only one in need of help\u2014if, as Richard feared, Chad was trapped there as well\u2014then they might have to make several trips back and forth, and that could take the rest of the afternoon. Richard hoped Adam could scrounge up some other snowmobiles fast.\n\nHe glided over the snow. What would he find at the Blue Boy?\n\nWhat he knew for a fact was that human blood had been found in the chimney. Chad reported that he'd heard animals eating something at the base of the chimney. But the space had been nearly cleaned out when Richard inspected it. Neville had reported that he'd been locked in his room shortly before that inspection, suggesting that someone in the house didn't want him to witness the cleaning of the chimney. And the only two people in the house, as far as Richard knew, were Jack Devlin and the caretaker, Zeke.\n\nDevlin, of course, had been desperate to prevent a search of the house. The scenario had all the hallmarks of Devlin being guilty, of at least covering up a crime.\n\nBut he was very possibly guilty of much, much more.\n\nHe could be using this storm to finish what he started, Richard reasoned. He thinks we can't get to him, so he's free to continue killing the rest of the people in the house.\n\nHe prayed that Annabel would still be alive when he got there.\n\nRichard turned the handlebars, steering the snowmobile up the narrow, wooded road that led to the Blue Boy Inn. He thought he was safely in the middle of the road. But apparently he had strayed off the path.\n\nHe felt the machine suddenly shudder beneath him. The snowmobile stopped, churning up a geyser of snow and throwing Richard clear over the handlebars.\n\nPlop! The chief found himself head-and-shoulders deep in cold, fluffy snow.\n\nMoving his arms as if swimming, he managed to pull himself upright. He stood, with some difficulty, as the snow was not packed all that hard. In this case, that was a good thing. The snow had cushioned his fall. If it had been hardened with ice, Richard might have cracked his head open. His goggles had stayed in place, but he'd lost a glove as he'd flown through the air, and his scarf was askew, allowing a cold draft to slip down his sweater.\n\nBreathing heavily, Richard assessed the situation. The Ski-Doo was about four feet away, no longer spitting snow. Caught on some bush below the snow, it appeared to have stalled out. Richard said a silent prayer that he could get it started again.\n\nJust getting back over to it was a chore. Every step he took, he sunk to his knees, and sometimes up to his hips. The wind was blowing so fast and furiously that even his goggles couldn't keep his eyes from welling up. His exposed left hand was freezing. There was no way he'd ever be able to walk the rest of the way to the Blue Boy. He had to get that snowmobile moving again!\n\nFinally, he reached the machine. The first thing he needed to do was push it away from the spot, so it wouldn't get stuck on the branches of the bush again. It took some muscle, but finally Richard shoved the snowmobile farther out into the clearing, where he was certain that the only thing beneath him, some five or six feet, was the dirt road. Hopping onboard, Richard started the ignition as Danny had showed him. But the Ski-Doo was unresponsive. No matter how many times Richard tried, the motor remained silent.\n\n\"Goddamn it!\" he shouted into the wind.\n\nHad it been damaged in the accident? Had Richard damaged it pushing it off the bush? What could he do to fix it? He had his cell phone, carefully stowed in an inner pocket of his coat. But what was he going to do? Ask Danny to trudge on out here? Even if he made it, it would take hours. And hours Richard did not have.\n\nHis mind was racing, trying to calculate the risks and the possibilities of heading up to the Blue Boy on foot. He'd have to try. He couldn't just give up. He'd have to walk. If there was danger up there, however, he wasn't sure how he and Annabel or anyone else might escape it without the snowmobile.\n\nRichard was ready to slide his leg back over the machine and start on his trek when he decided to try the ignition one more time.\n\nBelow him, the Ski-Doo hummed back to life.\n\n\"Hallelujah!\" Richard shouted into the blowing wind.\n\nIn moments, he was back to gliding over the snow, heading up into the hills toward the Blue Boy Inn.\n98\n\n\"Please don't make me go under there!\" Annabel cried, as the little eyes under the bed blinked at her in the dark.\n\n\"Get up off your knees, Miz Wish.\"\n\nAnnabel's head snapped up. The woman with the long gray hair was gone. Standing in her place was Zeke, looking very weary and sad.\n\n\"Zeke!\" Annabel jumped to her feet. \"Please let me go! Please don't keep me here!\"\n\nHe placed his finger to his lips, a sign for her to keep quiet. Looking over his shoulder, he said, \"Keep your voice down. I'll take care of things.\"\n\n\"That woman,\" Annabel whispered. \"She killed Chad.\"\n\nThe old man nodded with great sadness. He looked as if he might cry.\n\n\"Where did she go?\" Annabel asked, terrified she'd come back with her knife.\n\n\"I sent her away,\" Zeke told her. \"For now, you're safe. I'll see to that.\"\n\nAnnabel pulled back from him. \"How do I know I can trust you?\"\n\n\"You don't. But you don't have any other choice.\" Zeke looked at her coat and boots. \"If you were thinking of going out there, forget it. You'd get swallowed up alive by this nor'easter. I've seen many of these storms in my day, and this is by the far the worst.\"\n\n\"But I can't stay here,\" Annabel said, wrapping her arms around herself. \"Where is Jack?\"\n\n\"Walking through the house, muttering to himself.\" Zeke shuddered. \"The house has gotten to him. Just as it did his father and his grandfather.\"\n\n\"What do you mean?\"\n\n\"His grandfather fed the house for years in exchange for great success. Cordelia finally made him stop, but then young Mrs. Devlin showed up\u2014Jack's mother\u2014and like you, she had grand visions for the house. Even her death wasn't enough to keep Jack's father from falling under the spell of the house. But when they came for Miz Cindy . . .\"\n\n\"Who came for her?\"\n\nZeke looked at her. \"The house. And when Mr. Devlin saw what happened to his precious little girl, he finally woke up, and sealed over the fireplace once again. It stayed that way for many, many years.\" The caretaker's ancient yellowed eyes found Annabel's. \"Until you arrived, Miz Wish.\"\n\n\"The . . . fireplace?\"\n\nZeke suddenly held up his hand, as if he'd just heard a sound. \"I will be back,\" he told Annabel. \"Stay here. I will lock the door so no one can get in.\"\n\n\"No, please, don't lock me in again. I can't bear it! Not with that thing under the bed.\"\n\nZeke looked at her uneasily. With great difficulty, he bent down. \"There's nothing under there now,\" he reported as he stood back up, breathing heavily.\n\n\"I don't believe you,\" Annabel said.\n\n\"Then don't. I don't really care if you do. But if you want to be safe, stay in this room. If you go out the window, that's your choice. But you'll never make it off the hill.\"\n\nZeke hobbled out of the room. Annabel heard him turn the key in the lock.\n\nShe hurried over to the window. She could see the way the snow was blowing. Ten-foot drifts were forming right before her eyes. Zeke was right. She'd never make it down the hill to the road, let alone all the way to Millie's. The snow was too soft, too unsettled. It would swallow her up alive, as Zeke had said.\n\nBut stay here? Out in the storm, she risked death from cold and exposure. Here in this house, she risked death from an insane woman roaming the halls with a knife.\n\nOr worse\u2014she risked death from her worst childhood nightmare.\n\nShe would be eaten alive by Tommy Tricky.\n99\n\nThe soles of his shoes making sticking sounds as he walked through the blood on the floor, Zeke draped a sheet over Chad's dead body.\n\nThe poor man. But he was a fool, too, Zeke thought. He should have accepted the offer Zeke and Cordelia had made him. They might not have been able to make good on the offer, but Chad would be alive today, sitting at home with a mug of coffee, riding out this storm.\n\nZeke stood looking down at the body. He'd draped many sheets in his years in this house. The first had been over the headless corpse of Andrew McGurk. Zeke had been just a teenager then. He and old Mr. Devlin had managed to pull McGurk back out of the fireplace, but the house had already gotten his head.\n\nIt was that episode that had finally convinced old man Devlin to brick over the fireplace. But eventually his son, Cordelia's husband, had unbricked it. The pattern repeated itself every generation.\n\nThe worst had been the baby. That poor woman, hiding out here from her rich father's goons, had thought she was safe. From her father, yes. From the house, no.\n\nZeke had found the baby's arm in the ash dump. That was all that was left of her.\n\nAll his adult life he'd been covering up corpses, wiping up blood. Ever since he'd taken this infernal job, he'd been enslaved to this house, a prisoner of its terrible secrets. No more. He wanted an end to this before he died.\n\nAnd bricking up the fireplace was no longer enough.\n\nHe sighed and turned to leave the room.\n\nAnd walked directly into Jack, who had been standing there in the doorway, unknown to Zeke, watching him.\n\n\"What are you doing in here?\" Jack asked him.\n\n\"Giving the dead a little respect,\" Zeke replied, his voice surly.\n\nJack grabbed the old man by the front of his shirt. \"Don't interfere, Zeke. You must give the house what it needs.\"\n\nZeke struggled, but couldn't break free of Jack's grip. \"Oh, Mr. Jack, please try to see things as they are! We can't do it anymore! The house will take us all. Don't you care?\"\n\n\"The house will make us successful,\" Jack told him, tightening his hold on Zeke's shirt. \"That's what they've promised. My father, my grandfather, my great-grandfather\u2014they all understood that!\"\n\n\"Until they died broken men, the ones they loved destroyed around them!\"\n\nJack's hand loosened its grip, and Zeke took the opportunity to move away from him.\n\n\"Your mother, Jack! Don't you remember?\"\n\nJack's eyes clouded over.\n\n\"The house took your mother! She didn't die of cancer. She didn't die in a hospital. Your father lied to you, Jack. She died here\u2014horribly\u2014\"\n\nJack's arm suddenly swung out. The back of his hand connected with Zeke's face, and the frail old man went flying across the room, hitting his head against the wall. He slid down into a clump on the floor.\n\n\"Think of your wife,\" Zeke managed to whimper, as his head throbbed and the room around him began to spin. \"Think of Annabel.\"\n\nBut Jack just stalked out of the room.\n\nZeke put his face in his hands and cried. Eventually, everything went dark.\n100\n\nAnnabel thought she heard shouting from another room. She steadied herself, bracing for the worst. Then she heard footsteps clomping down the hall. Jack's footsteps, she thought. Hard, heavy, angry. Was he coming in here? Was he going to kill her? Annabel began to tremble violently. But the footsteps went right past the door and up the steps to the attic. Annabel heard the door to the attic open and close.\n\nFor the moment, she let out a sigh of relief.\n\nBut she was not out of danger. Far from it. Zeke had told her to wait for him, but she was no longer willing to wait. For all she knew, the shouting she heard from the other room had been an altercation between Jack and the old man, and Zeke was never going to emerge the winner from a fight like that. He might even be dead.\n\nAnnabel had to get out of the house. She'd take her chances outside. Better to freeze to death than get hacked up like Chad.\n\nOr worse.\n\nShe flew to the window. But when she tried to lift it, the stubborn thing wouldn't budge.\n\nShe tried again. Still it didn't move.\n\nIt hadn't been opened in years, Annabel figured. It was painted shut decades ago. Cordelia had never brought fresh air into this room. Suddenly, Annabel panicked. Her palms got sweaty again. She was trapped.\n\n\"No!\" she screamed.\n\nShe looked around for something with which to smash the window. The iron doorstop would have worked, but the cops had claimed it as evidence after Cordelia's death. There wasn't anything in the room that looked strong enough. Finally, Annabel ran over to the bed, hoping the little man would not leap out from under it and grab her ankle. She snatched the pillows off the bed and removed their cases. Then she opened Cordelia's drawer and yanked out one of the dead woman's lacy old slips. Annabel proceeded to wrap the slip around her right hand, and then pulled both pillowcases over that. Balling her hand into a fist, she walked back to the window.\n\nShe whacked the pane of glass as hard as she could.\n\n\"Oww!\" she yelled.\n\nBut still it didn't break.\n\nShe tried once more. The glass in the old window didn't shatter, but it did pop out of the pane, tumbling down into the snow as a gust of cold air rushed into the room. But that wasn't good enough. The window had twelve panes, each separated by wooden frames. Annabel couldn't fit through the one pane that she'd removed. She'd have to pop out at least four of them, and she'd have to also break the wooden frames.\n\nAnd try as she might, that old wood was impervious to her blows.\n\n\"Owwww!\" Annabel cried out on her fifth attempt to break the wood. Even a second pane remained resistant to her assaults. She was making a great deal of noise. Jack would hear her. Or he'd be attracted by the sound of the wind gusting through the open space in the window. Snow was swirling into the room, encrusting the wall and the floor.\n\nBut she had to try. It was her only chance.\n\nAnnabel pulled her hand back to swing it once more against the window. But just as she did so, she spotted a sight she could not believe\u2014something she hadn't dared let herself hope for.\n\nA man was barreling up through the trees on a snowmobile.\n\nIt was Richard Carlson!\n101\n\nIn the other room, Zeke was struggling to regain consciousness.\n\nHe dreamt. He was fifteen years old again, standing on the front porch, asking old Mr. Devlin for a job.\n\nIt was 1949. Back in those days, the Devlins maintained a farm out in back of the inn. A brood of hens clucked all around the place, and a rooster crowed from somewhere out in back. Old Mr. Devlin told Zeke he'd hire him to feed the chickens and cut the corn.\n\nThe young man ended up doing a lot more than that.\n\n\"Where is my baby?\" the woman was screaming at him.\n\nIn his hands Zeke held the bloody pink arm. It looked as if it had come from a doll. The woman fainted dead away.\n\n\"Help me! Help me!\" McGurk shrieked, as he was carried toward the fireplace.\n\nZeke grabbed hold of one leg, Mr. Devlin the other, but they were too late.\n\nThe sound of those creatures munching on McGurk's head haunted Zeke's dreams for the rest of his life.\n\nBut the worst, for him, was the attic.\n\nIn his dream, he walked those stairs, just as he had every day for the past twenty-three years. Everything that had come before had been terrible enough. But the attic these last two decades had been even more wicked.\n\nShe had been beautiful once.\n\nUntil the house had gotten to her.\n\nThe craziest thing of all, she loved the house. She would do anything for the house. The house that had tried to kill her.\n\nIn his dream, Zeke saw her as she once was. So beautiful. So innocent.\n\nAnd then he saw her as she was today.\n\nThe knife\u2014slashing Chad over and over, the way she had slashed others.\n\nAll for the house.\n\nZeke opened his eyes.\n\n\"No more,\" he said to himself.\n\nHe would end this. He would do what successive generations of Devlins had failed to do. He would destroy the Blue Boy Inn.\n102\n\n\"Richard! Oh, thank, God, Richard!\"\n\nHe heard Annabel's voice as soon as he switched off the engine of the Ski-Doo. He lifted his goggles and looked up at the inn. There she was, shouting from a second-floor window over the front door. She had popped out a pane of glass, and was waving what looked like a pillowcase to get his attention.\n\n\"Annabel!\" he shouted through the wind in response.\n\nThe snow had covered nearly the entire first floor of the house. From the second-floor window, it was only a matter of a few feet to jump to the little roof over the front porch, and then another couple of feet to the surface of the snow.\n\nBut given the wind, the cold, the softness of the snow, and the instability of the snowdrifts, Richard knew it was still going to be very difficult to get up there to Annabel and then get her back here to the Ski-Doo.\n\nHe started off across the snow.\n\n\"Annabel!\" he called again.\n\nHer face appeared at the open pane in the window. She placed her finger to her mouth, telling Richard to be quiet. He figured she was in danger. She didn't want someone else in the house to know that he was coming. He nodded and kept on approaching.\n\nThe snow, thankfully, was somewhat harder here. Richard sunk only to just below his knee with each step. He made it to the front porch, mostly buried in snow. By grabbing on to the trellis that was attached to the side of the porch, he was able to haul himself up onto the little roof, dislodging a couple of feet of snow as he went. His gloveless hand was freezing. He was pretty certain he'd end up with frostbite.\n\nHe was just grateful that it was his left hand. He'd need his right for shooting his gun, if necessary.\n\nStanding on top of the porch roof, Richard could see Annabel's face much more clearly. She looked terrified. Her hair was disheveled, and Richard believed he could discern blood on her hands and clothes. She was wearing a coat. Apparently, she'd been thinking of going out the window herself.\n\nHe grabbed hold of the ledge that ran above the porch and out under the windows of the second floor. It was only about six inches wide, but once he'd knocked the snow off it, Richard figured the ledge would be sufficient to get him over to Annabel's window. He hoisted himself up. For a second, his bare hand slipped, and he dangled precariously over the snow. But he steadied himself, and scrambled up onto the ledge. He took a deep breath, and then began inching his way toward Annabel.\n\n\"Can you break the window frame?\" she was asking him in a desperate whisper as he got closer. \"Break it and we can get out of here.\"\n\n\"Yes,\" he told her. \"I think so.\"\n\nHe positioned himself outside the window and tried to punch it in. The wood was too strong. He would have thought such an old house would buckle easily under his fist.\n\n\"Oh, please,\" Annabel was begging. \"Jack's in here! He's gone mad! And there's a woman\u2014with a knife!\"\n\nRichard tried punching through the window again, but still it held firm. Finally, he reached down and grabbed his gun. It wasn't easy to do. His gun had been encased in a protective pouch to shield it from the snow. He should have unsnapped it before climbing up here. Richard cursed himself for being too hasty. He had to get the gun out of the pouch while managing to remain balanced on the six-inch ledge outside the window.\n\nHe was successful. He gripped the gun by its barrel and used the grip to whack the window. On his second try, he heard the wood crack. On the third, it smashed inward, the glass panes popping out, one of them smashing onto the floor inside.\n\nRichard swung himself inside. It was the only way he could help Annabel out.\n\nShe threw her arms around him. \"Thank God!\"\n\n\"Anyone else in the house that needs help?\" he asked. \"Is Chad\u2014?\"\n\nAnnabel began to cry. \"He's dead. The woman killed him.\"\n\n\"No one else then?\"\n\n\"Well, there's Zeke,\" she said. \"He tried to help me. I don't know what's happened to him. Last I heard, he was in the room down the hall.\"\n\n\"Stand by the window,\" Richard said. \"Be prepared to jump onto the porch roof, then follow the tracks I made back to the snowmobile. Do you know how to ride one?\"\n\n\"Me?\" Even under such distress, Annabel seemed to find the idea humorous. \"No, not at all!\"\n\nRichard frowned. \"If Zeke needs our help, I can't just leave him. If something happens to me, I want you to be able to escape on your own.\" He handed her the keys. \"I'll be right back. Just hold these, just in case.\"\n\n\"I'll never be able to drive it . . .\" Annabel said, but she zipped them inside the pocket of Neville's coat nonetheless.\n\n\"I can't just abandon a man who might be in danger here,\" Richard said, looking over toward the door. \"Tell me. Does Jack have a gun?\"\n\n\"No, there's no gun in the house,\" Annabel said. \"But that woman has a knife\u2014\"\n\n\"Who is this woman? Do you know?\"\n\nAnnabel shook her head. \"I've never seen her before. But I think she's been living in the attic.\"\n\nRichard approached the door and listened intently through it. \"I've got to go out there and see if I can spot Zeke. If he's not within sight, we'll beat it. But I have to at least give him a chance.\" He turned the knob carefully.\n\n\"The door's locked,\" Annabel told him.\n\nBut it wasn't. The door creaked open under Richard's grip. \"Stay by the window!\" he ordered Annabel, as he stepped out into the hallway, his gun held high. \"And use the keys to get away on the snowmobile if you need to.\"\n\nThe corridor was empty. Richard took three large steps down toward the only other room with an open door and glanced inside. All he could see was a tremendous amount of blood on the floor. A bloody sheet was crumpled in one corner, as if someone had tried to mop up the blood.\n\nHe couldn't go searching the house for Zeke. He might risk Annabel's life if he did so. He ran back down the hallway, throwing open doors as he went. The rooms were all empty. He'd done what he could\n\n\"Come on,\" Richard said, holstering his gun and hurrying back into the room where Annabel awaited. \"Let's get out of here. I'll send in reinforcements as soon as I can.\"\n\nAnnabel gripped him by the shoulders. \"I can't believe you made it here,\" she said. \"I really thought I was going to die.\"\n\nRichard gave her a small smile. \"You can buy me a cup of coffee later to thank me. For now, let's just get the hell\u2014\"\n103\n\nAnnabel screamed as Richard suddenly went flying against the wall.\n\nJack had come charging into the room through the open door, shoving Richard hard, taking him by complete surprise.\n\n\"No!\" Annabel screamed again.\n\nRichard was quickly back up on his feet, hauling off and landing a hard punch against Jack's jaw. Annabel's husband staggered backwards, but regained his balance quickly. He lunged at Richard, just as the chief was going for his gun.\n\n\"You're not going to destroy my success!\" Jack shouted.\n\nWith superhuman swiftness, he lunged at Richard, sending him toppling out of the open window. The gun in his hands went flying through the air, clattering across the floor and coming to a stop under the bed.\n\nRunning to the window, Annabel watched in horror and disbelief as Richard plunged to the ground, smashing headfirst into the snow. Only his feet remained sticking out from the surface.\n\n\"Nooo!\" Annabel screamed into the wind.\n\nShe saw Richard's feet twitch once, and then go still. The snow around him slowly turned pink.\n\nAnnabel turned around to face her husband. \"You killed him,\" she said in a low voice.\n\n\"I had to,\" Jack said. \"He was going to destroy everything we've built up here, sugar cakes. He was going to prevent us from getting the success we deserve!\"\n\n\"You're insane!\" Annabel shouted, running over to him and beating her fists on his chest. \"You have gone completely insane!\"\n\nShe didn't care anymore what he might do to her. In that instant, Annabel lost control, giving vent to all her fear and despair. Richard had been her last hope. No one was going to save her now. She dissolved into tears, covering her face with her hands.\n\nJack grabbed her by her arms and forced her to look up at him. \"It's up to you, Annabel,\" he said, his voice calm but firm. \"You can join me in a successful life, or you can turn your back on all the house has to offer us. I'll forgive you for your indiscretions with these other men\u2014\"\n\nShe yanked away from him. \"There have been no indiscretions, Jack! Why do you talk that way?\"\n\nHe arched an eyebrow at her. \"That policeman\u2014he meant nothing to you?\"\n\nAnnabel started to cry harder. \"He was trying to save me.\"\n\n\"Save you from what? From a glorious life here, with me, at the Blue Boy, where everything we touch will turn into gold?\"\n\nAnnabel wanted to scream. \"Who told you that, Jack? Why do you believe that?\"\n\nHe smiled. It was a terrible, frightening smile. \"The house told me,\" he said simply. \"Once I learned the secret, I could listen to the house. I could understand what it was telling me.\"\n\n\"You're mad,\" Annabel spat out.\n\nShe just couldn't pretend anymore. She couldn't make it seem that she was going along with Jack's demented plans. She just had to get out of there. She had to find a way. Her mind started to race, to calculate.\n\nShe'd make a run for it. Yes, that was what she'd do. She'd go out the window in her room. It was a longer drop from there, but Annabel had seen how Richard had trudged through the snow. It was passable, not so soft that she'd sink. And the window in her room opened easily. Annabel began to believe that she could run in there, throw open the window, and be outside in less than a minute. She had to believe she could. It was her only hope.\n\nAnd in her pocket she had the keys to the snowmobile. She would have to believe that she could drive it. She would have to believe that she could save herself.\n\nBut for her plan to work, she'd need to incapacitate Jack, even temporarily, just to give her enough time to get out the window and run\u2014or, rather, trudge\u2014to the snowmobile. Otherwise, he'd be out the window right after her, and there was no question he'd be able to catch her. Annabel had just witnessed how strong Jack had become, sending Richard flying through the window with very little effort. Her husband was a tall, well-built man, but in recent years he'd grown a bit squishy from too much time on his hands, watching sports on television, and drinking too much beer. Where had this sudden burst of power come from?\n\nThe house, Annabel realized.\n\nThe house was making him strong.\n\nNow I'm thinking crazy like Jack, Annabel admonished herself. What we have here is one very deranged man\u2014nothing supernatural about that. Those visions of little men, Annabel told herself, were just her mind playing tricks, her hallucinations coming back under stress.\n\nBut then how to explain the woman with the knife? Annabel was certain she was very much real, after watching what she did to Chad.\n\nFor the moment, what was real and what was illusion didn't matter. Annabel just needed to get out of there. She needed to gain some kind of power over Jack so she could get away.\n\nThe gun. Richard's gun. It was under the bed. She needed to get it.\n\nBut Tommy Tricky was under there.\n\nNo, stop it, he's not real.\n\nHe doesn't like it when you say he's not real.\n\nAnnabel was suddenly overwhelmed with the feeling that if she reached under the bed, a little blue clawed hand would grab her.\n\nShe tried to calm herself. What she needed to do, she knew, was buy a little time from Jack. She took a deep breath and looked over at him.\n\n\"All right, Jack,\" Annabel said, wiping her eyes and looking over at him. \"I have no choice but to go along with you. I don't understand what you mean about the house, but maybe . . . maybe you'll teach me.\"\n\n\"That's it, baby cakes. That's the spirit!\" He wrapped his arms around her and squeezed her tightly. \"We're going to be very successful here, you and I.\"\n\nShe gently extricated herself from Jack's grip. \"I feel so light-headed,\" she said. \"All this blood . . . this death, Jack . . . I don't know how to deal with it.\"\n\n\"You've never known how to deal with bad stuff, angel sweets,\" Jack said. \"Sit down. Take a few breaths.\"\n\nThat was just what Annabel wanted him to say. She sat down on the bed.\n\nPlease let the gun be within reach, she prayed.\n\n\"I feel like I might faint,\" Annabel said.\n\n\"Bend down and put your head between your legs,\" Jack told her, as he walked over to the window with a blanket, trying to find a way to block the snow blowing into the room. \"Let the blood run to your head. That will bring you out of it.\"\n\nPerfect. This was perfect. He had just told her to do exactly what she needed to do to be able to look under the bed. And his back was to her. Perfect. This was working perfectly.\n\nAnnabel leaned over. She peered into the darkness. What she saw made her gasp, though she retained enough presence of mind to suppress it.\n\nThe gun was right there, all right. Within reach.\n\nBut lying right next to it, on his stomach, his chin in his sharp little hands, was Tommy Tricky. He smiled at Annabel, licking his lips with his blue tongue.\n\nShe steeled herself.\n\nYou little fucker, she thought. You're not going to keep me from getting this gun.\n\nShe reached under the bed. The imp's eyes followed her hand.\n\nHe's going to bite you, Annabel heard Daddy Ron tell her in her mind.\n\nBut she didn't pull back. She closed her fingers around the gun. Tommy Tricky watched her. His eyes were all that moved.\n\nAnnabel sat up all at once, the gun in her hand.\n\nTo her great relief, Jack was still at the window, trying to stuff the blanket into the space between the broken wood frames and shards of glass.\n\n\"I've got to get a piece of plywood and patch this thing,\" he was saying. \"Else we will have a foot of snow in here before long.\"\n\nHe turned around. Annabel saw the surprise and disbelief on his face when he saw her standing there, pointing the gun at him.\n\nHe knew she could use it, too. In New York, she'd taken a self-defense class. She'd learned how to fire a gun.\n\nHe stood there, mouth open, staring at her.\n\n\"You threatened to kill me, Jack, but I'm not going to kill you,\" Annabel told him. \"At least, I don't want to kill you. I will if I have to. I could as easily aim this at your head or your heart as your leg. So come on along with me. Put up your hands.\"\n\nHer husband sneered at her, but he obeyed.\n\n\"Come on,\" she said. \"Walk in front of me.\"\n\n\"Where are we going?\"\n\nAnnabel smirked. \"Let's see how you like being locked in a closet.\"\n\n\"If you lock me up,\" Jack asked, \"you won't have to shoot me, too, will you?\" He shuffled slowly across the room, Annabel behind him, the gun pressing against the small of his back.\n\n\"Sorry, Jack,\" she replied, \"but I'm not taking any chances. You've got an accomplice running around here somewhere, and she could let you out. So unfortunately you'll have two legs full of gunshot wounds in addition to being locked in the closet. But don't worry. I'm heading back to town, and I'll send an ambulance for you.\" She shoved him toward Cordelia's closet. \"As soon as the storm lets up, that is.\"\n\n\"Annabel,\" Jack said, \"don't so this. We can be so happy\u2014\"\n\nAt that moment, a wail came from above them. From the attic. It was a terrible sound, a cry of grief and despair. It was enough to distract Annabel for half a second\u2014which was just enough time for Jack to spin around with that superhuman speed he now possessed and knock the gun out of her hands. It fell to the floor with a thud.\n\nAnnabel saw the rage that suddenly filled Jack's eyes.\n\nShe ran. She bolted out into the hallway, but Jack was fast on her heels. She'd have to go past him if she were to try jumping from the window or running downstairs. There was only one option for her, and she took it without even consciously realizing she'd done so.\n\nShe ran up the stairs to the attic.\n\nThe door was open. She bolted inside, no longer thinking, just reacting, driven solely by an instinct to survive.\n\nShe didn't even realize that Jack had not pursued her up the stairs.\n\nYet despite her desperation to get away, Annabel came to a skidding halt when she came upon the scene in the attic.\n\nThe woman with the long gray hair stood there. Her white dress was now soaked with blood. In front of her various body parts were scattered around the room. Legs and arms, a portion of a torso, with the rib cage sticking out. And Chad's head, looking up at Annabel with lifeless eyes. At the moment, the woman was sawing an arm off a shoulder.\n\nAnnabel screamed, and then got sick, before fainting this time for real.\n104\n\nZeke heard Annabel's scream from the attic and, with a heavy heart, started up the stairs. He was unprepared for the gore that he found there.\n\n\"Oh, what have you done, Cindy?\" he asked, in utter despair and horror, looking around at the carnage. \"I have told you and told you that you must not do this. . . .\"\n\n\"But I must,\" the woman in the bloody dress told him. \"I promised them. You know I promised them.\"\n\n\"It is too much,\" Zeke said, his face a mask of anguish and grief. \"We can't go on.\"\n\nHe saw Annabel slumped on the floor.\n\n\"I must get her out,\" he said. \"And then . . .\" He looked back at the woman. \"Then we must all pay the price for what we have done.\"\n\n\"I promised them,\" the woman said, sulking now. \"They let me go, because I promised them.\"\n\nZeke walked over to her, cupped her face in the palm of his hand.\n\n\"It's not your fault, Cindy. You were just a little girl. It destroyed your mind. You poor sweet little girl.\"\n\n\"I promised them,\" she said, a broken record.\n\n\"They get nothing more,\" Zeke said, angry now. \"Nothing more.\"\n\nHe bent down to Annabel. \"Wake up, Miz Wish. Come with me. Can you walk?\"\n\nShe stirred.\n\n\"Get to your feet,\" Zeke said. \"It's all a bad dream. You'll wake up in the morning and it will all be gone. But come with me now. Walk with me.\"\n\n\"What\u2014?\" Annabel mumbled, as she got to her feet.\n\n\"Don't look over there,\" Zeke told her. \"It's just a bad dream. Come with me. You'll be safe with me. Walk with me down the stairs.\"\n\nAnnabel, like a zombie, obeyed.\n\nHe took her all the way down to the first floor, into the kitchen. He sat her down at the table.\n\n\"Listen to me, Miz Wish,\" Zeke said. \"I've cleared a path from the back door. You can get out that way. I've also cleared the way to your car. Here are your keys.\"\n\nAnnabel looked at him. The old caretaker could see that her mind had shut down. It was a defense mechanism against the horrors she'd seen upstairs.\n\n\"Listen to me, Miz Wish. Annabel. Here are your keys.\"\n\nHe pressed them into her hand.\n\n\"You are to walk out and get into your car. Do you understand? The snow is stopping, but you still can't drive out. But I've cleared the snow off it, and you can start it. You can keep warm there. Keep the window cracked, as I left it for you. That will keep you safe from any fumes. But just a crack.\"\n\nAnnabel said nothing.\n\n\"Do you understand? You will be safe there until someone comes for you. Someone will be coming. The police will come looking for the chief, and they'll find you.\"\n\nAnnabel stared at him blankly.\n\n\"Do you understand, Annabel?\" Zeke asked, growing concerned.\n\n\"I . . . understand,\" she finally said. \"I can go to the car . . . and wait there . . . be safe . . . someone will be coming soon.\"\n\n\"Yes, yes, that's right. And Annabel. Do not come back inside the house, under any circumstances.\" He shivered. \"No matter what happens. I think, when you see it, you'll know it's for the best.\"\n\nAnnabel's eyes moved around the kitchen. She seemed to smell something.\n\n\"Yes, that's gasoline you smell,\" Zeke told her. \"I've sprinkled it all over the house. Upstairs and downstairs. Every room.\"\n\nHe walked across the room and lifted the can, shook some more of the liquid onto the floor.\n\n\"It's the only way,\" he told Annabel.\n\n\"The only way,\" she echoed dully.\n\nHe heard the scuttling then. The sound of dozens of little feet running toward him.\n\n\"No!\" Zeke shouted, frantically running across the kitchen for the box of matches.\n\nBut the old man couldn't run very fast, and he slipped on the floor, falling flat onto his stomach, knocking the wind out of him\u2014just as thirteen little men came scurrying around the corner from the parlor and crawled all over him.\n\n\"Nooo!\" Zeke screamed.\n\nBut it was too late.\n105\n\nAnnabel watched the events unfold as if from some faraway place.\n\nThe little men\u2014so many Tommy Trickies everywhere all of a sudden\u2014grabbed on to the old man on the floor. With uncanny strength for their small size, they lifted him and carried him off. Zeke writhed and kicked, but the little men paid him no mind as they carted him out of the kitchen and back in toward the parlor.\n\nAnnabel stood. She followed them, as if in a daze.\n\n\"No, no, please, no!\" Zeke was screaming.\n\nThe little men marched him in front of the fireplace.\n\n\"Nooo!\" Zeke shouted.\n\nThey pressed his feet into the ash dump door.\n\nAnnabel watched. Only gradually did she come back to her senses, and begin to comprehend what was happening.\n\nWhen she finally understood fully, she screamed.\n\nBut her scream was drowned out by Zeke's, as the old man was pulled down inside the fireplace by several sharp tiny blue hands sticking up out of the ash dump.\n\nThe last to disappear were Zeke's outstretched arms and hands, frantically twisting and opening and closing as the creatures below ravenously consumed his body.\n\nWhen finally he was completely gone, the thirteen little men who had delivered him leapt into the air, cheering and laughing over a job well done.\n\nThey'll come for me next, Annabel realized.\n\nShe also realized that she still had the keys to the car in her hand. She turned and bolted out of the parlor, back into the kitchen.\n\nAnd ran right into the arms of Jack.\n106\n\n\"I really appreciate this,\" Deputy Adam Burrell said as Danny's friend Melvin started the snowmobile for him. \"You guys are the best.\"\n\nMelvin gave him a freckle-faced smile. \"Hey, dude, anything to help out the Woodfield finest. When Danny called me to say the cops needed a snowmobile, I didn't hesitate to hop on mine and zip right over here.\" The teenager leapt up off the seat so that Adam could slide on. \"Remember that, okay, next time you catch me with a little weed.\"\n\n\"I'll look the other way one time only,\" Adam said, taking the handlebars, \"and then we're even.\"\n\nMelvin gave him a little salute.\n\nUnlike the chief, Adam knew how to handle a snowmobile. Growing up in these parts, he'd been riding the machines since he was a kid. Adam wished Richard had let him go in his place. He would have been able to handle the Ski-Doo with ease. As it was, they were all wondering if maybe the chief had had an accident. It had been a couple of hours now since they'd heard from him, and Adam was beginning to worry. Had he crashed?\n\nBut maybe a snowmobile crack-up was the least they should be concerned about.\n\nSetting off down the road, Adam didn't know what he'd find at the Blue Boy.\n\nThe snow had stopped, but it was still blowing pretty fiercely. It would be days before everything was plowed. The department was definitely going to need to invest in its own snowmobiles, if storms like this were getting more frequent. As it was, their call to various private citizens had rounded up three more Ski-Doos. Their owners were riding them in even as Adam headed across town. They were coming from a bit farther out in the woods than either Danny or Melvin, so it would still take some time for them to get to the police station. But at least Adam knew he'd have some backup of more officers at the Blue Boy eventually, if it turned out he needed it.\n\nAnd need it, he might. He'd gone over everything they'd learned about what had been taking place at the inn. The disappearances, the blood in the chimney, the sounds that had been heard, the fact that the Englishman had been locked in his room. Something really weird was going down at that place. The fact that there had been no word from the chief unnerved Adam the most.\n\nHe sped on across the snow.\n107\n\n\"Give me the keys to the car, Annabel,\" Jack said, Gholding out his hand.\n\nAnnabel had no choice but to obey.\n\n\"What am I going to do with you?\" he asked, stuffing the keys down into his jeans pocket. \"First you threaten to shoot me and lock me in a closet, and then you try to run off.\"\n\n\"Those things\u2014\" Annabel could barely speak. \"Those things that took Zeke\u2014\"\n\nJack smiled. \"They're real, Annabel. And they don't like it when you say they aren't.\"\n\nHe was deliberately mocking her. She had told him all about Daddy Ron.\n\n\"How is it possible?\" Annabel asked, gripping the back of a kitchen chair to keep herself from falling down. Her head was throbbing. She couldn't make sense of all that she had witnessed this afternoon.\n\n\"Sit down, angel cakes,\" Jack told her. \"Take a load off. I'll make you some tea.\"\n\n\"No,\" Annabel said.\n\nJack laughed. \"Afraid I'll poison you? I suppose you should worry. You've been a very bad girl.\"\n\nAnnabel thought she might faint. She pulled out the chair and sat down in it, holding her head in her hands.\n\nShe was going to die here. Jack was going to kill her.\n\nOr worse\u2014he was going to put her down the fireplace. She would die crammed into a tiny space, unable to move, eaten by a dozen Tommy Trickies.\n\nHer childhood nightmare come true.\n\n\"Please, Jack,\" she cried.\n\n\"I should take no pity on you after everything you've done,\" Jack said, and Annabel could hear the teakettle whistling on the stove. \"I should really punish you, you know.\"\n\n\"No,\" Annabel sobbed.\n\n\"But success would be no fun on my own,\" Jack continued. His wife could hear him pouring some hot water in a mug. \"Do you want chamomile or Earl Grey?\"\n\n\"No, no, no,\" she muttered.\n\n\"I guess I'll give you chamomile. It's more soothing. And you need to calm down.\"\n\nHe placed a steaming mug in front of her, the tab of a tea bag dangling from its side. Annabel didn't touch it.\n\nJack sat opposite her at the kitchen table. \"Sweetie, I want to do this together with you. The only reason my father didn't have the heart to go on was because he lost my mother. If they'd been a team, they would have had so much success here.\"\n\nAnnabel wouldn't look at him.\n\n\"I want you to remake this place into a grand destination, just as we planned. You'll redo this kitchen, make it all sparkling and modern.\" He leaned toward her. \"And maybe you could learn how to make Gran's rabbit stew. You don't have to eat it, sweetie, but it could become the inn's signature dish. Carry on a little tradition, you know, for our guests.\"\n\nFinally, she snapped her eyes up to look at him. \"Our guests? Who'd be picked off, one by one, dragged down screaming into that pit of hell?\"\n\n\"No, you see, sweetheart, that's why I need your help. We can control the house.\" He chuckled. \"I admit, right now, it's a bit out of control. But if we give it only what it needs, it will take care of us. It will make us successful. That was the promise given to my great-grandfather when he bought the house over a hundred years ago.\"\n\n\"By whom?\" Annabel asked.\n\n\"By the house.\" Jack looked at her as if she was being thickheaded. \"And the house keeps its promises.\"\n\n\"And kills those who live here.\" Annabel's eyes hardened. \"Or it does worse things to them. Like your sister.\" She spat the word, watching Jack's face to see how he reacted. \"Look what it's done to her.\"\n\n\"My sister,\" Jack said softly. \"Yes, you're right, Annabel, Cindy's a problem we have to deal with. Again, this is why I need your help.\"\n\nAnnabel saw an opening, maybe a way to appeal to whatever reason and sanity Jack might have left. \"Cindy needs help, Jack. We need to get her help.\"\n\nHe stood and began pacing the kitchen. \"Gran thought she had everything under control. With the fireplace bricked over, the only way to feed the house was through the door in the chimney in the basement. She kept it padlocked, but Cindy\u2014well, she's very strong. And she kept breaking the lock, and feeding the house through the door.\"\n\n\"Feeding it with what?\" Annabel asked.\n\nJack shrugged. \"Whatever she could find. Rabbits, mice, dead raccoons, skunks.\" He made a face of disgust. \"But the house was starving for something better than that. Cindy understood this, and of course, she would do anything to please the house. The house had spared her, and she was grateful.\"\n\n\"Spared . . . her?\" Annabel asked.\n\n\"Yes, sweetheart. Drink your tea. And I'll tell you the story.\"\n108\n\nJack was just a little boy, he told Annabel, when he came down the stairs to find his father sitting in the parlor, his head in his hands, sobbing.\n\n\"They took her,\" his father was saying.\n\nHis grandmother had been standing over him.\n\n\"Brick it up,\" she was telling him. \"Brick it up now!\"\n\nJack had been so young he couldn't understand everything that was happening. He had just stood there watching and listening.\n\nCordelia had approached the fireplace. \"You filthy monsters! Do you think we haven't read the books? Do you think we don't know how to put an end to this?\"\n\nJack's father stood. \"Don't, Mother. Don't provoke them. They'll come back. . . .\"\n\n\"Brick it up!\" Cordelia shrieked.\n\n\"Stop, Mother! They'll hear you! They've taken my wife and they've taken Cindy! They'll come back\u2014for us\u2014for Jack!\"\n\nJack had shrunk back at hearing his name, suddenly terrified.\n\nHis father had rushed to the fireplace then, speaking into it.\n\n\"Give her back if it's not too late,\" he pleaded. \"Give Cindy back and I will make sure we give you what you need, always! We had a bargain! You would make me successful if I gave you what you needed. I will keep my end of that bargain. Just give me back Cindy!\"\n\nJack had watched from around the corner of the parlor.\n\nHe had heard a rustling sound. Scratching.\n\nAnd then he had seen a small pink hand reach up from the ash dump in the fireplace. . . .\n109\n\n\"They . . . they let her go?\" Annabel asked, unable to truly comprehend what she was hearing.\n\n\"Yes,\" Jack told her. \"They made the bargain with my father\u2014which he failed to keep. But I intend to make up for what he did wrong!\"\n\n\"Jack, all of this is madness!\"\n\nHe still seemed miles away, lost in his memory. \"I had blocked it all out. Because, you see, when Cindy came back to us, she wasn't the same. She was wild, uncontrollable. My grandmother decided to keep her here when my father and I left. She bricked up the fireplace because the house was very angry at being deceived.\" He smiled over at Annabel. \"And it's no good to have the house angry at us, as you have witnessed.\"\n\n\"Jack, please, listen to what you're saying. . . .\"\n\nHe sighed. \"Poor Cindy. The house had let her go, but it still had her mind. If my father had reneged on his promise, she was determined to make it up to the house. She fed the house, gave it what it needed. . . .\" He looked over at Annabel sadly. \"Until recently. She kept sneaking out of the attic\u2014Zeke was getting far too old to control her, and Cindy had become far too cunning\u2014and finally she went out and killed that man, chopped him up, fed him piece by piece to the house.\"\n\nAnnabel thought she might be sick again.\n\n\"Jack,\" she said, when she was finally able to form words, \"this is crazy talk. Can't you see that? It makes no sense!\"\n\n\"It makes perfect sense, Annabel, if you let it make sense.\"\n\nShe struggled to show him how absurd his words were. \"If Cindy could open up the door in the basement, then why didn't those things get out that way? Why did they have to wait until the fireplace was unbricked?\"\n\n\"That's just the way it is, Annabel. That's what it says in the books. They can only come out through the fireplace.\"\n\n\"What books?\" But then suddenly she remembered. The books she'd found\u2014those books about demons and witchcraft hidden in that secret panel\u2014the same sort of panel that must exist all through the house, Annabel realized, allowing the creatures free rein. \"No,\" she mumbled. \"It can't be possible.\"\n\nJack remained calm. \"You only think it can't be, because you don't believe in the house.\"\n\nAnnabel ran her hands through her hair. \"You think this house\u2014those terrible things\u2014those creatures\u2014they can somehow make you successful? Why do you think that?\" She thought of those hideous books once more. \"Is it like some terrible pact with the devil?\"\n\n\"Call it what you like, Annabel.\"\n\nShe shook her head. \"And success comes through feeding the house?\"\n\n\"Yes, now you're getting it.\" Jack sighed. \"But here's the dilemma. Cindy is determined to keep giving the house what it craves, yet the truth is, she doesn't need to, anymore.\" He laughed. \"The house has been freed to take what it wants all by itself!\" He smiled broadly over at her. \"You freed it, Annabel.\"\n\nThe tears were running down her cheeks.\n\n\"But you're right, angel cakes.\" Jack was nodding, as if he were thinking things over. \"We're going to need to find a way to control Cindy. We can't allow her to disturb our guests. We need to find something productive for her to occupy her time with.\"\n\nThis was utter madness. Annabel put her head down on the table and cried.\n\nJack wandered across the kitchen, lost in thought. \"She was such a sweet little girl,\" he said dreamily. \"You know, I had blocked all of it out . . . all of what happened to her. I was so young and my father told me that I was mistaken. I hadn't seen Cindy climbing out of the fireplace, all bloody and sooty. Dad insisted that she'd wandered off into the woods.\"\n\nAnnabel looked up at him. \"Jack, you have to help your sister. All these years, the way your grandmother hid her in this house, it wasn't fair to her.\"\n\n\"But Cindy wants to be here,\" Jack insisted. \"She belongs here. So do I.\" He took a step toward Annabel. \"So do you.\"\n\n\"Jack, please\u2014\"\n\nThey heard the sound of a motor out in front of the house.\n\nJack rushed to the window.\n\n\"Another cop on a snowmobile,\" he seethed.\n\n\"Jack,\" Annabel pleaded, \"let me talk to him. I'll tell him that none of this was your fault, or Cindy's. Please, just let me talk to him\u2014\"\n\n\"You think I'm a fool?\" Jack growled. He grabbed Annabel, pulled her up from the chair, and clamped his hand over her mouth. \"You'll say nothing, you hear? You won't make a sound, baby cakes, or I'll have to break your neck. I won't like doing it, but I will.\"\n\nHe dragged her across the kitchen toward the pantry.\n\n\"Cindeeee!\" he called.\n\nThen he banged open the door to the pantry with his shoulder and took Annabel inside.\n110\n\nPulling into the snow-covered driveway of the Blue Boy Inn, Adam could see Richard's snowmobile, already nearly covered by drifting snow. He could also see a path that had been dug out from around the side of the house, leading to an SUV, which had also been cleared of snow. That's convenient, Adam, thought, as otherwise, he'd have to try to gain access to the house by crawling up the side and going through a second-floor window. The first floor was almost completely covered in snow.\n\nI'll bet the chief shoveled out this path, Adam thought, steering the snowmobile over to the clearing. I'll bet I'll find him inside, having coffee with Annabel. Everything's going to be fine. The only reason we haven't heard from him is because there's no cell reception out here.\n\nFor some reason, the shoveled path reassured Adam. He thought he'd find everything peaceful inside. They'd been wrong to worry.\n\nHe didn't look too closely at the front of the house, or the pink snow near the front door.\n\nAdam brought the snowmobile to a stop. He dismounted and headed over to the path that led to the kitchen door.\n\n\"Chief?\" he called as he approached the house.\n\nHe peered through the one window that had been cleared of snow. He looked into the kitchen. He didn't see anyone. But there was a steaming mug of what looked like tea on the table. Things couldn't be too bad if they were sitting around drinking tea.\n\nAdam rapped on the door.\n\nThere was no sound, no movement, from inside.\n\nHe rapped again. \"Hello!\" he called. \"Chief! Are you there? Mr. Devlin! Ms. Wish!\"\n\nWhy wasn't anyone answering?\n\nAdam tried the door. It was open. He let himself in.\n\nSomething wasn't right. He felt it as soon as he stepped inside the kitchen.\n\nHe held his gun in front of him with both hands.\n\n\"Chief!\" Adam called. \"Hello! Anyone here?\"\n\nA woman suddenly appeared in the doorway that led to the parlor. She was pretty, but her hair was long and gray. She was wearing a long blue dress.\n\n\"Hello,\" the woman said.\n\nAdam lowered his gun. \"I'm Officer Burrell. I'm looking for Chief Carlson.\"\n\nThe woman looked at him as if she didn't understand.\n\n\"Who are you?\" Adam asked her.\n\nShe just smiled and took a step into the room.\n111\n\nIn the pantry, Annabel and Jack watched from a crack in the door.\n\n\"Mmm,\" Annabel moaned, Jack's hand still pressed over her mouth.\n\n\"Shh,\" he growled at her.\n\nAnnabel watched in despair as Cindy approached the policeman. She could see the knife she carried behind her back, even if the poor man did not. How Annabel wanted to warn him. But Jack would break her neck if she made a sound. She truly believed he would.\n\nBut the policeman had a gun. He could shoot Cindy if she tried to attack him. Jack, too. He could put an end to all of this madness.\n\nWhat did it matter if Annabel died? She was going to die in this house anyway. If she made some kind of a sound, there was a chance that the policeman could shoot Jack before he had a chance to kill her. Staying silent simply prolonged her misery. Making noise gave her a chance\u2014a slim chance, but a chance nonetheless. And if she died, so be it.\n\nShe couldn't live in this hellhole much longer.\n\nJack held Annabel in a vise grip in front of him, preventing her from moving her arms. His hand was secured over her mouth.\n\nBut he hadn't counted on her feet.\n\nShe was still wearing his clunky boots. They were loose. If she could shake one off . . .\n\nAnnabel lifted her right foot up to the side. She kicked.\n\nThe boot remained on her foot. If she tried again, Jack might notice.\n\nBut she had to try. She lifted her foot again. And kicked doubly hard.\n\nThe boot flew off her foot and crashed into a low shelf of glass jars containing Cordelia's preserves. Apricots and strawberries smashed all onto the floor.\n112\n\nAdam heard the sound and swung his gun around in the direction of the pantry. As he did so, the woman in front of him lunged at him with a knife.\n\nHe fired wildly, pumping the kitchen ceiling full of lead.\n\nThe woman's knife made contact with his arm, cutting through his coat and slicing into his flesh.\n\nAdam spun back around, slugging the woman, sending her flying and the knife skittering across the floor.\n\nHe kicked open the door to the pantry.\n\n\"Ms. Wish!\" he exclaimed.\n\nJack let her go.\n\n\"Mr. Devlin,\" Adam said, trying to make sense of things. \"Come on out of there, please. I'd like to know what's going on here.\"\n113\n\nAnnabel's relief and gratitude were short-lived. Behind Adam she saw Cindy stand up. Dear God, what kind of strength did she have? Adam had just knocked her out cold.\n\nWhat kind of power did this house give to people?\n\nAnnabel saw Cindy stand and grab her knife off the floor....\n114\n\n\"No!\" Ms. Wish suddenly screamed, looking be-Nhind him.\n\nAdam turned in time to see the woman back on her feet, coming at him with the knife. He swung his gun around\u2014\n\nBut it was too late.\n\nThe knife plunged deep into Adam's gut. He gasped and buckled forward.\n\nDevlin began punching him. Adam fell to the ground. The last thing he saw was the knife above him, coming down at his throat.\n115\n\nAnnabel saw Cindy bring the knife down onto the policeman, then pull it up again, then plunge it down again, repeating this several times, each time dripping more blood.\n\n\"Okay, honey, enough now,\" Jack was saying to her gently.\n\n\"I have to cut him up,\" Cindy said, like an eager child.\n\nJack tenderly lifted her off the twitching, bloody corpse. \"No, Cindy, you don't have to do that. From now on, I'll take care of feeding the house. Do you understand, baby?\"\n\nCindy looked up at him with sad eyes. \"They don't need me anymore,\" she whimpered.\n\n\"Oh, honey baby, the house will always need you.\" Jack pulled her into an embrace, stroking her stringy gray hair.\n\n\"They're my only friends,\" Cindy cried against his chest.\n\n\"No, baby, I'm your friend, too. And Annabel\u2014\"\n\nBut Annabel had just bolted.\n\nShe ran out of the kitchen into the parlor. She couldn't have gone out the back way. Jack would have gotten her. Her only hope was to go out through her window upstairs, as she'd originally planned. She still had the snowmobile keys zipped in her pocket. Even if she couldn't get very far on it, Annabel was certain now more policemen were on their way. She just had to get out of the house before Jack or Cindy could get her.\n\nOr worse\u2014she could be caught by Tommy Tricky and his brothers.\n\nShe ran up the stairs and turned the corner into the corridor.\n\nBut she didn't get very far.\n\nA hand suddenly reached out and clamped itself over her mouth. Before she knew what was happening, she was pulled into a dark closet by a very strong pair of arms.\n116\n\nRichard kissed her to make sure she didn't scream out.\n\nAnnabel shuddered in his arms, still full of terror. He moved his lips off her.\n\n\"It's okay,\" he whispered in her ear. \"It's going to be okay.\"\n\nIn the very dim light of the linen closet, Richard saw Annabel's eyes sparkle with sudden surprise and relief. \"Richard,\" she said, as the tenseness in her body relaxed. \"You're alive!\"\n\nHe grunted. \"Well, my face is never going to be the same, and I think I broke my ankle, and I can't feel my left hand, but yes, I'm alive.\"\n\nHe smiled.\n\n\"Thank God,\" Annabel said, looking up at his face. \"Oh, Richard, you're all cut and swollen.\"\n\nShe tried to touch him and he flinched. The pain was quite severe now. Only as his body had warmed did he begin to feel just how injured he was.\n\nHe had come to under the snow. How very peaceful it had been down there. On some primal level, Richard had wanted to stay right where he was. He was fading in and out of consciousness, and he wasn't unhappy. He wasn't uncomfortable. But he would die if he stayed where he was. He was, in fact, slowly freezing to death. The thought had startled him back to full consciousness, and Richard had begun to scrape his way out of the snow.\n\nIt hadn't been easy. The snow was hardening, and he was stuck headfirst in it. At first, it had been almost impossible to move his arms. He pushed and elbowed as best he could, and kicked as hard as he could muster with his legs. Finally, he created enough space around him to move, to shift his position. He swallowed a lot of snow in the process.\n\nHe was bleeding from his face and neck. The impact had cut and scraped him pretty badly, but he hadn't felt much pain at first. He was too cold. He was too numb. Clawing his way out of the snowbank, he had stood up and looked around. That was when he felt the first pain\u2014in his ankle. He didn't think it was the fall that had done it, but rather the way Devlin had slammed him into the wall.\n\nThe man was going to pay.\n\nRichard noticed another snowmobile was now parked out in front. Could it be one of his officers, come looking for him? Or was it someone else?\n\nHe could take no chances. Knowing that Devlin might be watching him, Richard crunched through the snow to the back of the house, where he spotted a rainspout. He tested it, determined it was strong enough, and he began to climb. The angle made it impossible to be seen from any window of the house. Once he was close enough to a window, Richard used the butt of his gun to smash his way in.\n\nHe always traveled with a spare gun.\n\nStepping through the broken window, aware that the sound could bring Devlin running, Richard steadied himself, his weapon in both hands, raised in front of him. He was aware that Devlin might have his other gun, the one he'd lost when he'd been taken unawares. That had never happened before to Richard, ever, in his career. He was known for being very swift, very agile, able to turn on a dime. But Devlin seemed to possess some kind of strength that Richard had not expected.\n\nSo he was extra cautious as he made his way down the hall.\n\nThat was when he had heard the screaming and crying from downstairs and the scampering of feet. He'd backed into the open linen closet so he wouldn't be seen. In seconds, Annabel had come running by. Now he held her, trembling in his arms.\n\n\"Does he have the gun?\" Richard whispered in her ear.\n\n\"Not at the moment,\" Annabel told him. \"But I'm sure he'll get it.\"\n\n\"Where is it?\" he asked. \"Do you know?\"\n\n\"I had it for a while, but he knocked it away from me. It was on the floor in the bedroom.\"\n\n\"Let's go,\" Richard said. \"If it's there, we'll get it. If not, well, he can keep it. Shoot himself with it, for all I care. Either way, you and I are going out the window.\" He looked down at her. \"You still have the keys to the snowmobile?\"\n\n\"Yes.\" Annabel gripped his coat and looked up at him. \"But Richard, you need to know what's happened here. Terrible things.\"\n\n\"What kind of terrible things?\" he asked.\n\nShe shuddered. \"Adam's dead.\"\n\n\"Adam\u2014?\"\n\n\"Cindy killed him.\"\n\n\"Cindy?\" Richard looked down at the terrified woman in his arms. \"That was the name of Devlin's sister, the little girl who died. . . .\"\n\n\"She's alive. She's the woman I told you about. She's been living here. Completely insane.\"\n\n\"So she's the killer.\"\n\nAnnabel looked as if she'd cry. \"Yes, but no . . .\"\n\n\"Annabel, what do you mean?\"\n\n\"Oh, Richard, it's the house. There are things that live in this house . . . creatures who come up through the fireplace. . . .\"\n\nShe was delirious. And who wouldn't be, after everything she'd been through?\n\n\"Come on, Annabel,\" he said. \"We're getting out of here.\"\n\n\"They'll get us, Richard! The little men! Tommy Tricky and his friends!\"\n\n\"Calm down, Annabel. Stay behind me at all times.\" He nudged the door of the closet open just a crack, getting a look down the corridor. \"We're just going across the way and into the bedroom, then out the window.\"\n\n\"But Richard . . .\"\n\n\"Listen to me. We're going out the window and then onto the little roof over the porch. From there it's an easy jump to the snow, and at that point we can move pretty quickly to the snowmobile in the tracks that I made getting here.\"\n\n\"Richard, they're not going to let us leave. . . .\"\n\n\"You mean Jack and his sister?\"\n\n\"No. The little men.\"\n\nRichard looked at her. It was best not to argue with her at this point.\n\n\"Come on, Annabel,\" he said, pushing open the door. \"Let's go.\"\n117\n\nAnnabel followed Richard out of the closet.\n\nWhere were Jack and Cindy? Lurking somewhere, Annabel was certain. They would pounce on them. But even worse\u2014\n\nThe little men.\n\nThey'll put us down the fireplace!\n\n\"Come on,\" Richard urged in a harsh whisper, and they ran across the hall.\n\nAs they rushed into the bedroom, they could see the window. There was no gun in sight, but all that really mattered was that they reach the window. The window meant freedom. For half of a second, Annabel's spirits leapt. She believed they would escape.\n\nBut then Tommy Tricky dropped from the ceiling onto Richard's back.\n\nHe must have been sitting on the top of the opened door, waiting, watching.\n\nHe plunged his long sharp claws into Richard's neck. Richard screamed.\n\n\"I don't like it when you don't believe in me, Richard,\" the little imp said in his high-pitched doll's voice.\n\nAnnabel screamed at the same time Richard did. Blood squirted from the chief's neck like water from a leaky pipe.\n\nRichard grabbed his neck and in doing so, he knocked the little man from his shoulders. Tommy had only a second to look up at him and hiss through his sharp, clenched teeth when Richard aimed his gun at him and fired.\n\nThe little man exploded in a mess of blue blood and plasma.\n\n\"Richard, are you all right?\" Annabel said, rushing to him.\n\n\"I think so,\" he said, more dazed and shocked by the creature than the attack itself. He kept looking down at the bubbling ooze on the floor.\n\n\"It's not possible,\" he said. \"That thing\u2014\"\n\nFinally, he pulled his eyes away and grabbed Annabel by the wrist.\n\n\"Come on, let's go!\"\n\nBut when they looked toward the window they saw little men were now crawling all over it. The creatures were coming out of the woodwork. Literally. Floorboards raised. Panels in the walls opened. And the little men stepped out, their fierce blue eyes trained on Annabel and Richard.\n\n\"Through the other window!\" Richard shouted, pulling Annabel out of the room and into the corridor.\n\nThe moment they stepped out of the room, however, every door along the hall slammed shut. They were left in semidarkness. Richard tried the door to his left. It was locked. Annabel tried the one opposite. That one, too, wouldn't budge.\n\n\"Look, Richard!\" Annabel suddenly shouted.\n\nMarching up the stairs and into the hallway was an army of six more little men. They all looked nearly identical, with little blue pinched faces and blue teeth and blue rags as clothes. They were all gnashing their teeth.\n\n\"Shoot them, Richard!\" Annabel screamed.\n\nHe was firing even before the words were out of her mouth. The first two creatures exploded like their fallen comrade, but those in the back suddenly leapt to the walls, crawling like spiders, still coming toward their prey. Richard fired again, but the creatures easily darted away, and the bullet simply tore open a portion of the wall. Another of the little men was now on the ceiling, and as Annabel looked up at it, it dropped down on her.\n\nClinging to her shoulders, face-to-face with her, Tommy Tricky laughed. \"I like eating bad little girls,\" he hissed.\n\nAnnabel screamed.\n\nRichard knocked the thing to the floor with the butt of his gun, and then shot it. Once more, Annabel watched it bubble into a blue goo.\n\nBut the other three were now clawing up Richard's leg. He shook one off, sending it flying through the air. He shot it before it hit the wall, a messy blue explosion in mid-flight.\n\nBut the other one was now crawling up his torso. And the final little man had jumped off Richard and onto Annabel's arm.\n\nSuddenly she got angry.\n\n\"You filthy bastard!\" she bellowed, and whipped her arm around, crashing the creature into the wall. She took delight in seeing the way its blue teeth smashed on impact. The thing fell in a dazed lump to the floor.\n\nRichard swatted the thing off his torso, shot it, and then did the same to the creature Annabel had dispatched.\n\n\"What are these fucking things?\" he asked.\n\n\"I told you. They're the\u2014\"\n\n\"Yes,\" he said, finishing her thought. \"From the fireplace.\" He looked over at her. \"Do you think this is all of them?\"\n\n\"No,\" Annabel said. \"There are more. I don't know how many, but there are definitely more.\"\n\n\"Well, at least we know they can be killed,\" Richard said. He seemed to realize something. \"So the stories of old Reverend Fall were apparently true after all. He found his portal.\"\n\n\"What?\" Annabel asked.\n\n\"No time now to explain,\" Richard replied, before taking hold of her arm. \"All right, let's get out of here before we encounter any more of those hell spawn.\"\n\nThey ran back to Cordelia's room. The door was no longer locked. The sheet Jack had used as a makeshift barrier at the window had fallen off, and a couple of feet of snow had drifted into the room.\n\n\"Same plan as before,\" Richard told Annabel, and she nodded.\n\nThey bolted toward the window.\n118\n\nFor Richard, it was almost as if time stood still. There he was, running for his very life, away from things he never believed existed just moments before, and yet, despite all those dangers, he was thinking of Amy.\n\nHe was thinking about how very much he had loved her. And how much he missed her. And how he would never love any woman ever again the way he had loved Amy, no matter how long he lived.\n\nHe had failed to save Amy. How he had tried. He had never accepted her diagnosis. Richard had taken his wife to see specialist after specialist. He'd investigated every new drug, every alternative treatment. But still Amy had died. Still she'd left him alone.\n\nHe'd saved Annabel, however. Richard felt confident that was true.\n\nAs he positioned her by the window so she could jump out onto the front porch roof, he realized something.\n\nHe hadn't saved Annabel. She had saved herself.\n\nHe had seen how she'd bashed that creature against the wall. Annabel had survived this house of horrors. Her escape was entirely due to her. Richard was just driving the getaway car.\n\nAnnabel could survive anything.\n\nJust like Amy. Richard hadn't failed her, because there was nothing to fail: Amy had been entirely in control the whole time. It was Richard who'd used terms like \"beating this.\" Amy had always said, \"I will face whatever I need to face with grace and purpose.\" And she had, right up until the end. Before she had fallen into her coma, she had looked up at Richard from her hospital bed, told him she loved him, and said she looked forward to the time they would meet again.\n\nHow very much he had loved Amy.\n\n\"Richard!\" Annabel suddenly screamed, looking back from the window. \"Look out!\"\n\nHe only had a momentary flash of the gray-haired woman\u2014Cindy\u2014coming into the room behind him with a knife. He never had time to reach for his gun or feel the knife pierce his heart.\n\nHe was already with Amy.\n119\n\n\"Noo!\" Annabel shouted, leaping down from the window to shove Cindy off Richard's fallen body. \"Noooo!\"\n\nCindy pulled back, hissing like an affronted swan. Annabel knelt down beside Richard's body.\n\n\"Don't die,\" she begged. \"Don't leave me alone.... I can't get out of here without you. I can't, I can't, I can't. . . .\"\n\nHer tears dripped onto Richard's face. But his glassy eyes stared up at her. He was dead. This time, he wasn't coming back.\n\nAnnabel felt herself falling. She was tumbling back into her black hole, unable to move, unable to think. Once more, it was the only way she could handle the terrible things happening outside her. She shut down. She was like a turtle pulling into its shell, only her shell was soft and vulnerable, and would not protect her.\n\n\"You stay inside there, you bad little girl,\" Daddy Ron's voice shouted through the closet door. \"You stay in there in the dark because you're bad. Very bad.\"\n\n\"I'm not bad,\" Annabel replied, in a very small voice.\n\n\"Don't argue with me, you little bitch. You are bad!\"\n\n\"No, I'm not!\" Annabel suddenly shrieked, and she lashed out, kicking the closet door down with her feet.\n\nShe looked up. Cindy stood a few feet away from her, watching her with wild eyes.\n\nI can't shut down, Annabel thought. I shut down and I die.\n\nAnd if I'm going to die, she reasoned, I'll die fighting to live.\n\nShe turned and started to climb up onto the windowsill.\n\nBut Cindy leapt onto her back. Annabel tried to throw her off, but Cindy was so terribly strong. Her arms encircled Annabel in a vise grip, threatening to choke off her air. Annabel did the only thing she could do. She fell backwards, toppling down from the window with Cindy underneath her. It momentarily knocked the wind out of Cindy and the arms that had held Annabel so tightly suddenly opened.\n\nAnnabel jumped free and lunged for Richard's gun.\n\nBut Cindy was too fast for her. She nudged the gun away, out of Annabel's grip. Then she grabbed hold of Annabel's wrist, wrestling her down to the floor.\n\n\"Jack!\" Annabel screamed.\n\nHe might help her. He might at least call Cindy off her.\n\n\"Jack!\"\n\nCindy lifted the knife over her head, its point aimed at Annabel's chest.\n\nAnnabel looked up into the madwoman's face. She was a little girl once, a happy child, until coming to this place. Even in that moment, Annabel felt pity for her.\n\nSuddenly there came a shot.\n\nAnd Cindy's face, so tensed with rage just a moment before, suddenly relaxed. A look of peace filled her eyes just before her body slumped over to the right, the knife she'd been holding in her hand clattering to the floor.\n120\n\nAnnabel looked in the direction of the doorway into the eyes of her savior.\n\nJack stood there, holding Richard's first gun in his hand. The gun was smoking.\n\n\"I had to do it,\" he said sadly. \"I couldn't let her kill you, baby cakes.\"\n\n\"Oh, Jack . . .\" Annabel stood up, trembling terribly.\n\nJack approached his dead sister. \"I mean, she couldn't go on this way. Like I said, if she kept escaping from the attic to slash our guests, we'd never be successful.\"\n\nHe knelt down beside Cindy's body. He had shot her clean through the chest. The blood that now seeped through her clothes, turning her blue dress purple, was her own.\n\n\"She was such a sweet little girl,\" Jack said, stroking her hair. \"She used to have a real nice singing voice.\n\nShe loved to dance. We used to watch Sesame Street together.\"\n\nIn the midst of everything\u2014after all the horrors and all the abuse Jack had done to her\u2014Annabel felt sorry for him. They had loved each other once. It had been a long time ago, but they had loved each other. Back then, Jack had been kind and good. Whatever demonic forces controlled this house, they had taken Jack's mind and warped it.\n\n\"Jack,\" Annabel said, stooping down beside her husband, \"come with me. Let's get out of this house together. We'll go out through the window and ride back into town. I'll explain that you only shot Cindy to save me. No one will blame you for anything here. They'll come, they'll see what this house is, and they'll understand. . . they'll understand none of it was your fault.\"\n\nHer husband snapped his face up to look at her. \"Oh, but that's impossible,\" he said quickly. \"The house won't let us leave, Annabel.\"\n\nShe studied his eyes. She didn't know what he meant.\n\nBut then she looked around and his meaning became unmistakably clear.\n\nThe room was filled with little men.\n\nOnce again they were coming up through the floorboards and stepping out of sliding panels from the walls. There were dozens of them. The house was infested with them, like cockroaches or termites. Three of them even sat in the window, blocking the way if Annabel suddenly tried to make an escape.\n\n\"No,\" Annabel said. \"Please, no . . .\"\n\nThey were coming closer. Slowly, steadily, their little feet moved, closing in on Annabel and Jack.\n\nThe little men were muttering among themselves.\n\n\"It's Cindy,\" Annabel realized. \"They're angry about Cindy.\"\n\n\"She was their friend,\" Jack said, standing up now, facing them.\n\n\"Shoot them, Jack!\" Annabel shouted.\n\n\"No,\" Jack said. \"They're going to make me rich.\"\n\n\"No, they're not, Jack! They're going to put you down the fireplace!\"\n\nToo late he lifted the gun in their direction. Five or six of the creatures crawled up Jack's body, snatching the gun out of his hand and tossing it out the window. Then a dozen more covered him, knocking him to his knees. They were under him, trying to lift him and carry him away.\n\nAnnabel screamed in horror.\n\n\"No!\" Jack cried. \"I'm your friend! I will take care of you!\"\n\nThe little demons ignored his words.\n\n\"She's the one!\" Jack shrieked, managing to point over at Annabel. \"She's the one who wants to destroy you! She'll destroy the house!\"\n\nHis eyes met Annabel's.\n\nShe had just offered him a chance at salvation. He was offering the demons her life for his.\n\nAnnabel knew in that instant it wasn't the house that had warped Jack. He'd already been that way the day they arrived at this place. And probably for some time before that.\n\n\"Take her!\" Jack was screaming. \"Take Annabel instead!\"\n\nAll at once, the little men stopped moving. They looked at each other. Then they set Jack down on the floor and turned their terrible blue eyes at Annabel.\n\n\"No!\" she screamed.\n\nThe creatures marched toward her.\n121\n\nAnnabel backed up against the wall in a vain attempt to get away from the little men. But the creatures were dropping down from the ceiling now, and crawling up her legs.\n\nAnnabel tried to fight them off, but it was useless. There were too many.\n\nThey had her. Their little pincers sunk into her flesh and held her tightly. They knocked her off her feet and scrambled under her back, hoisting her off the floor and carrying her across the room.\n\nAnnabel screamed.\n\nJack had vanished. He had condemned her to death, and would do nothing to save her.\n\nAnnabel's only hope was the gun\u2014the one that Cindy had knocked from her hands and sent sliding across the floor. Annabel could see it as the creatures carried her past. She might be able to grab it\u2014\n\nShe stretched out her arm when they drew close to the gun, which was sitting undisturbed in the middle of the floor. The little men seemed to have no interest in it. Their tiny fingers\u2014claws\u2014would not have been able to pull the trigger. Annabel had to grab it. She had to shoot them as Richard had, and she would take such delight in watching them explode into filthy protoplasm.\n\nHer fingers touched metal. Yes! She closed her fingers around the barrel of the gun\u2014\n\nBut the creatures were moving her too quickly. Annabel was unable to grab hold of the weapon. It slipped past her hand as she was carried past.\n\n\"Noo!\" Annabel cried in frustration.\n\nThere was no more hope. She would die. She would be stuffed into a small, enclosed space and she would be eaten alive by dozens of Tommy Trickies.\n\nHer mind shut down as she was carried out of the room, through the corridor, and down the stairs.\n\nA series of images passed through her brain in those last few moments.\n\nA picture of her father. Her real father. Colonel Malcolm Wish, in his beige-and-white camouflage from the Gulf War.\n\nIf you had only lived, Daddy, if you had only lived . . .\n\nHer mother, her weary face at the end of the long day, her fear of Daddy Ron evident in the way her eyes flickered at the slightest sound....\n\nJack, on their wedding day . . .\n\nA party in New York, laughter, lights, music, the smell of cocaine in her nose . . .\n\nThe stultifying air of the hospital, the sense of being trapped, the sound of people crying down the hall . . .\n\nNeville's face.\n\nChad's.\n\nRichard's.\n\nYou can survive anything.\n\nHad Richard said that to her?\n\nSuddenly, Annabel opened her eyes. She was being carried through the parlor now. At any moment, the creatures were going to force her headfirst down the chimney. There were others, she knew, waiting inside to pull her down.\n\n\"No,\" she said quietly.\n\nShe thrashed her head from side to side. She spotted Jack across the room, watching from a dark corner, his eyes emotionless.\n\n\"No,\" Annabel said more loudly.\n\nSuch strength the house had given Jack, and Cindy, too.\n\nBut not as much as it's given me, Annabel thought.\n\n\"Get off of me, you filthy bastards!\" she screamed, and in one powerful move, she shook the things off her, sending several of them flying.\n\nThe surprise on their little faces was beautiful to see.\n\nThey tried scrambling back at her, but Annabel managed to stand up, crushing one of the things under her bare foot as she did so, hearing its loathsome neck snap. The little men began making chittering sounds like monkeys. A number of them kept lunging at her, their pincers clawing into her legs, but others were backing away from her, not sure what to make of this suddenly powerful creature who had the ability to resist them.\n\n\"You only think you're real,\" she spat at them. \"Mother told me you're not.\"\n\nShe shook the last of them off her legs.\n\n\"Grab her!\" Jack shouted. \"Grab her now or she'll get away.\"\n\nThe sound of Jack's voice drew their attention away from Annabel. A moment passed as all the little men turned to look at Jack.\n\nThen, instantaneously, as if some psychic command had passed among them, they all started running toward him.\n\n\"Get away from me!\" Jack cried. \"It's her you want! Take her!\"\n\nBut they overran him, knocking him to the floor.\n\nJack screamed.\n122\n\nAnnabel watched as the creatures lifted Jack and carried him across the floor toward the fireplace. He swung his arms and tried to kick his feet, crying hysterically.\n\n\"Annabel!\" he called to her. \"Help me, baby cakes!\"\n\nBut that wasn't going to happen.\n\nInstead, Annabel turned and walked out of the parlor at the moment Jack's head was pushed down into the fireplace. Heading into the kitchen, she could hear Jack's muffled scream as the creatures devoured him.\n\nShe knew what she had to do.\n\nThe first thing was practical. Annabel found the boots she had kicked off, one in the pantry and the other in the kitchen. She replaced them on her feet and tied them as tightly as possible. She worked deliberately, but also quickly. She had very little time. Whether the creatures would come back for her, she didn't know. Perhaps she had proven herself to them, and they would leave her alone now. But they might be back for another round.\n\nStill, if they did try, Annabel felt confident that she could beat them off again. But frankly, she didn't want to have to bother.\n\nHer boots on her feet, she walked over to the stove and grabbed the box of kitchen matches that were kept there.\n\nZeke had had the right idea.\n\nThose little men weren't real. Tommy Tricky had been used to torment her, manipulate her, frighten her, exploit her. But he hadn't been real.\n\nHer mother told her he wasn't.\n\nAnd Annabel intended to make sure it stayed that way.\n\nShe walked back into the parlor, clutching the box of matches in her hand.\n\nThe little creatures were dancing around, celebrating Jack's death. The house had been fed. They were pleased.\n\nNot for long.\n\nZeke had sprinkled gasoline all over the floor\u2014the rugs\u2014the drapes. Annabel could smell it.\n\nShe struck a match against the side of the box.\n\nA little flame sparked into life.\n\nThe creature nearest to her turned his face up to the match. Its flame reflected in his demonic little eyes.\n\n\"Good-bye, Tommy,\" Annabel told the creature, and dropped the match to the floor. The rug under the little man burst into flame, taking the terrified creature with it. Its face melted like wax in the conflagration.\n\nInstantly, the parlor was ablaze, the room Annabel had thought she would modernize, make pretty, turn into a home. The drapes caught. The sofa was obliterated. As Annabel turned to leave, she caught a glimpse of flames hopping like rabbits up the stairs. The little men were screaming.\n\nWalking out into the kitchen, Annabel kept striking more matches and dropping them to the floor. Little explosions followed in her wake.\n\nShe threw a lighted match into the pantry and listened to the roar. She placed a match on the center of the gas-soaked kitchen table, which burst into flames. Soon the whole room was an inferno. Annabel could hear the fire consuming the walls around her and ripping through the ceiling above her. But best of all, she could hear the high-pitched screams of the little men as the flames consumed their unholy blue flesh.\n\nCalmly, she walked out the back door.\n\nMaking her way across the path that Zeke had dug for her, Annabel noticed that the snow had stopped and that breaks of blue were appearing in the sky. By the time she had crunched across the yard to Richard's snowmobile, the house behind her was burning out of control, sending a giant plume of black smoke rising above the trees.\n\nAnnabel mounted the snowmobile. She unzipped her coat pocket and removed the keys. Inserting the larger of the two keys into the ignition, she turned it to the ON position.\n\nThe snowmobile didn't start.\n\nWas she going to be able to drive this thing?\n\nOf course, she was.\n\nShe figured she could do anything now.\n\nInstinctively, she pulled a cord in front of her and pushed the handlebars up, and the motor beneath her hummed into life.\n\nAnnabel sailed off across the snow.\nPINNACLE BOOKS are published by\n\nKensington Publishing Corp.   \n119 West 40th Street   \nNew York, NY 10018\n\nCopyright \u00a9 2015 William Patterson\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the publisher, excepting brief quotes used in reviews.\n\nIf you purchased this book without a cover, you should be aware that this book is stolen property. It was reported as \"unsold and destroyed\" to the publisher, and neither the author nor the publisher has received any payment for this \"stripped book.\"\n\nThis book is a work of fiction. Names, characters, businesses, organizations, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nPINNACLE BOOKS and the Pinnacle logo are Reg. U.S. Pat. & TM Off.\n\nISBN: 978-0-7860-3323-2\n\nFirst electronic edition: January 2015\n\nISBN-13: 978-0-7860-3324-9   \nISBN-10: 0-7860-3324-X\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThe author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce, or upload this e-book, other than to read it on one of your personal devices.\n\nCopyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nContents\n\nTitle Page\n\nCopyright Notice\n\nINTRODUCTION\n\nA STAINLESS STEEL RAT IS BORN\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nTHE STAINLESS STEEL RAT GETS DRAFTED\n\nDedication\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nTHE STAINLESS STEEL RAT SINGS THE BLUES\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nAnd You Will Sing the Blues Too...\n\nTor Books by Harry Harrison\n\nCopyright\nIntroduction\n\nIf you are new to the world of the Stainless Steel Rat, I bid you welcome. There are now ten novels that have been published in the series that features this protagonist. All this mighty effort, all the acres of trees turned to paper, all of the twenty-three translations into different languages, all of this can be dated back to a single short story that was published in Analog some years ago. This story, and another short story following the adventures of the same character, were eventually incorporated into a novel by the same name.\n\nWhat\u2014or who\u2014exactly is this rust-free rodent? He is a man, not a rat, although he firmly believes that just as there are flesh-and-blood rats in wooden houses today, so will there be stainless steel rats in the future, when the world will be all metal and concrete. In human terms, this means that there will always be individuals who are a part of society\u2014yet separated from it.\n\nWhen Kingsley Amis read the first Stainless Steel Rat novel, he said, \"Well, Harry, you have written the first picaresque science-fiction novel.\" I nodded sagaciously\u2014and at the first opportunity rushed to my dictionary.\n\nPicaresque 1. of or relating to a type of fiction in which the hero, a rogue, goes through a series of episodic adventures.\n\nWell, that is a fair enough description. My hero is alone against the universe, with every man's hand\u2014if not every woman's\u2014turned against him. He is without the law, verily a law unto himself, with a strict sense of morality. Aren't the two mutually incompatible? No, as you will soon discover.\n\nThis character first appeared in The Stainless Steel Rat. But you will not find that novel here. Instead, you will be introduced to James diGriz, sometimes called Slippery Jim, as a young man, perhaps even a callow youth, because I realized, as the number of these books grew, that I was intrigued as to where this brash and very successful character had come from. How had he grown up among peaceful, law-abiding citizens and become, what many might call, a crook?\n\nAs the author, I thought that I knew. But I wanted these thoughts amplified on paper, then expanded into a novel. After all, if I care, certainly the readers would be concerned as well. They were! They greeted the first prequel with happy cries.\n\nLet me explain the term \"prequel.\" We all know that a sequel is a novel that continues a previously related story. Well, a prequel takes the same idea\u2014only turns it the other way around. It takes place earlier in time in the character's life than the first novel\u2014even though it is written after the first novel. In my case, I wanted to flesh out my character, to know him a bit better. Therefore A Stainless Steel Rat Is Born. (A title that won a prize from the London Times Literary Supplement readers as the most incredible book title of the year.) And what jolly fun it was to explore my hero's youth! The only problem was that I could not get all the new material into one book.\n\nSo, The Stainless Steel Rat Gets Drafted followed soon after. The draft, a fate that befell me during World War II, filled me with a great deal of empathy for my character. That the future will re-create the past, I do not doubt. Nor that the military, as long as they are in existence, will be stupid, arrogant, bull-headed, wrong-headed, and all of the other wholesome adjectives. So Jim has survived birth, survived the army, and is ready to march into his picaresque future.\n\nWell... not quite. He still had not gained his complete rattish personality, was still a bit incomplete in certain ways. What was to be done? Fill in the gap, of course, with The Stainless Steel Rat Sings the Blues. Yes, he does sing, along with his friends. There was even a cassette recorded of the music he composed. Life imitates art. Art enriches life.\n\nSo here you are: three novels neatly bound inside a single cover so that, if the weather is bad, this volume will enable you to not leave the house for a week or more. Depending, of course, on how fast you read.\n\nRead and enjoy\u2014for you have set your feet upon the path of the righteous rat.\n\nLife will never be the same again.\n\nHARRY HARRISON\nA Stainless Steel Rat Is Born\nChapter 1\n\nAs I approached the front door of The First Bank of Bit O' Heaven, it sensed my presence and swung open with an automatic welcome. I stepped briskly through\u2014and stopped. But I was just far enough inside so that the door was unable to close behind me. While it was sliding shut I took the arc pen from my bag\u2014then spun about just as it had closed completely. I had stop-watched its mechanical reflex time on other trips to the bank, so I knew that I had just 1.67 seconds to do the necessary. Time enough.\n\nThe arc buzzed and flared and welded the door securely to its frame. After this all the door could do was buzz helplessly, immobile, until something in the mechanism shorted out and it produced some crackling sparks, then died.\n\n\"Destruction of bank property is a crime. You are under arrest.\"\n\nAs it was speaking, the robot bank guard reached out its large padded hands to seize and hold me until the police arrived.\n\n\"Not this time, you jangling junkpile,\" I snarled, and pushed it in the chest with the porcuswine prod. The two metal points produced 300 volts and plenty of amps. Enough to draw the attention of a one-tonne porcuswine. Enough to short the robot completely. Smoke spurted from all its joints and it hit the floor with a very satisfactory crash.\n\nBehind me. For I had already leapt forward, shouldering aside the old lady who stood at the teller's window. I pulled the large handgun from my bag and pointed it at the teller and growled out my command.\n\n\"Your money or your life, sister. Fill this bag with bucks.\"\n\nVery impressive, though my voice did break a bit so the last words came out in a squeak. The teller smiled at this and tried to brazen it out.\n\n\"Go home, sonny. This is not...\"\n\nI pulled the trigger and the .75 recoilless boomed next to her ear; the cloud of smoke blinded her. She wasn't hit but she might just as well have been. Her eyes rolled up in her head and she slid slowly from sight behind the till.\n\nYou don't foil Jimmy diGriz that easily! With a single bound I was over the counter and waving the gun at the rest of the wide-eyed employees.\n\n\"Step back\u2014all of you! Quick! I want no little pinkies pressing the silent alarm buttons. That's it. You, butterball\u2014\" I waved over the fat teller who had always ignored me in the past. He was all attention now. \"Fill this bag with bucks, large denominations, and do it now.\"\n\nHe did it, fumbling and sweating yet working as fast as he could. The customers and staff all stood around in odd poses, apparently paralyzed with fear. The door to the manager's office stayed closed, which meant that he probably wasn't there. Chubby had the bag filled with bills and was holding it out to me. The police had not appeared. There was a good chance I was getting away with it.\n\nI muttered what I hoped was a foul curse under my breath and pointed to one of the sacks that were filled with rolls of coins.\n\n\"Dump out the change and fill that too,\" I ordered, sneering and growling at the same time.\n\nHe obeyed with alacrity and soon had this bag stuffed full as well. And still no sign of the police. Could it be that not one of the moronic money employees had pressed the silent alarm button? It could be. Drastic measures would have to be taken.\n\nI reached out and grabbed up another bag of coins. \"Fill this one as well,\" I ordered, slinging it across to him.\n\nAs I did this I managed to get the alarm button with my elbow. There are some days when you have to do everything for yourself.\n\nThis had the desired effect. By the time the third bag was full, and I was staggering towards the door with my loot, the police began to appear. One groundcar managed to crash into another (police emergencies are pretty rare around these parts), but eventually they sorted themselves out and lined up outside, guns ready.\n\n\"Don't shoot,\" I squeaked. With real fear, because most of them didn't look too bright. They couldn't hear me through the windows but they could see me. \"It's a dummy,\" I called out. \"See!\"\n\nI put the muzzle of the gun to the side of my head and pulled the trigger. There was a satisfactory puff of smoke from the smoke generator and the sound effect of the shot was enough to make my ears ring. I dropped behind the counter, away from their horrified gaze. At least there would be no shooting now. I waited patiently while they shouted and cursed and finally broke down the door.\n\nNow, you might find all of this puzzling\u2014if so I do not blame you. It is one thing to hold up a bank, another thing still to do it in such a manner that you are sure to be caught. Why, you might ask, why be so foolish?\n\nI'll be happy to tell you. To understand my motives you have to understand what life is like on this planet\u2014what my life has been like. Let me explain.\n\nBit O' Heaven was founded some thousands of years ago by some exotic religious cult, which has happily since vanished completely. They came here from another planet; some say it was Dirt or Earth, the rumored home of all mankind, but I doubt it. In any case, things didn't work out too well. Maybe the endless labors were too much for them\u2014this was certainly no picnic-world in the early days. As the teachers at school remind us as often as they can, particularly when they tell us how spoiled the young folk are these days. We manage not to tell them that they must be spoiled as well because certainly nothing has changed here in the last thousand years.\n\nIn the beginning, sure, it must have been rough. All of the plant life was pure poison to human metabolisms and had to be cleared away so edible crops could be grown. The native fauna was just as poisonous, with teeth and claws to match. It was tough. So tough that ordinary cows and sheep had a shockingly short life expectancy. Selective gene manipulation took care of that and the first porcuswine were sent here. Imagine if you can\u2014and you will need a fertile imagination indeed\u2014a one-tonne angry boar hog with sharp tusks and a mean disposition. That's bad enough, but picture the creature covered with long quills like an insane porcupine. Odd as it sounds, the plan worked; since the farms are still breeding porcuswine in large numbers it had to have worked. Bit O' Heaven Smoked Porcuswine Hams are famed galaxy-wide.\n\nBut you won't find the galaxy rushing to visit this piggy planet. I grew up here, I know. This place is so boring even the porcuswine fall asleep.\n\nThe funny part is that I seem to be the only one who notices it. They all look at me funny. My Mom always thought that it was just growing pains and burnt porcuswine quills in my bedroom, a folk remedy for same. Dad was always afraid of incipient insanity and used to haul me off to the doctor about once a year. The doctor couldn't find anything wrong and theorized that I might be a throwback to the original settlers, a loser in the Mendelian crapshoot. But that was years ago. I haven't been bothered with parental attention since Dad threw me out of the house when I was fifteen. This was after he had gone through my pockets one night and discovered that I had more money than he did. Mom agreed fervently with him and even opened the door. I think they were glad to see the last of me. I was certainly too much of an irritation in their bovine existence.\n\nWhat do I think? I think it can be damn lonely at times, being an outcast. But I don't think I would have it any other way. It can have its problems\u2014but problems have solutions.\n\nFor example, one problem I licked was getting beat up all the time by the bigger kids. This began happening as soon as I went to school. I made the mistake at first of letting them know I was brighter than they were. Bam, a black eye. The school bullies liked it so much that they had to take turns to beat up on me. I only broke the punishment cycle by bribing a university physical education teacher to give me lessons in unarmed combat. I waited until I was really proficient before fighting back. Then I creamed my would-be creamer and went on to beat up three more of the thugs one after another. I can tell you, all the little kids were my friends after that and never tired of telling me how great it looked to see me chasing six of the worst ruffians down the block. Like I said, from problems come solutions\u2014not to say pleasures.\n\nAnd where did I get the money to bribe the teacher? Not from Dad, I can tell you. Three bucks a week was my allowance, enough to buy maybe two Gaspo-Fizzes and a small sized Get-Stuffed candy bar. Need, not greed, taught me my first economic lesson. Buy cheap and sell dear and keep the profits for yourself.\n\nOf course there was nothing I could buy, having no capital, so I resorted to not paying at all for the basic product. All kids shoplift. They go through the phase and usually get it beaten out of them when they are detected. I saw the unhappy and tear-stained results of failure and decided to do a market survey as well as a time and motion study before I entered on a career of very petty crime.\n\nFirst\u2014stay away from the small merchants. They know their stock and have a strong interest in keeping it intact. So do your shopping at the large multis. All you have to worry about then are the store detectives and alarm systems. Then careful study of how they operate will generate techniques to circumvent them.\n\nOne of my earliest and most primitive techniques\u2014I blush at revealing its simplicity\u2014I called the book-trap. I constructed a box that looked exactly like a book. Only it had a spring-loaded, hinged bottom. All I needed to do was to push it down on an unsuspecting Get-Stuffed bar to have the candy vanish from sight. This was a crude but workable device that I used for a good length of time. I was about to abandon it for a superior technique when I perceived an opportunity to finish it off in a most positive manner. I was going to take care of Smelly.\n\nHis name was Bedford Smillingham but Smelly was the only name we ever called him. As some are born dancers or painters, others are shaped for lesser tasks. Smelly was a born snitch. His only pleasure in life was ratting on his schoolmates. He peeked and watched and snitched. No juvenile peccadillo was too minor for him to note and report to the authorities. They loved him for this\u2014which will tell you a lot about the kind of teachers we had. Nor could he be beat up with impunity. His word was always believed and it was the beater-uppers who suffered the punishment.\n\nSmelly had done me some small ill, I forget exactly what, but it was enough to stir dark and brooding thought, to eventually produce a plan of action. Bragging is a thing all boys enjoy, and I achieved great status by revealing my book-shaped candy bar collector to my peer group. There were oohs and ahhs, made more ooh and ahhish by portioning out some of the loot free for the taking. Not only did this help my juvenile status\u2014but I made sure that it was done where Smelly could eavesdrop. It still feels like yesterday, and I glow warmly with the memory.\n\n\"Not only does it work\u2014but I'll show you just how! Come with me to Ming's Multistore!\"\n\n\"Can we, Jimmy\u2014can we really?\"\n\n\"You can. But not in a bunch. Drift over there a few at a time and stand where you can watch the Get-Stuffed counter. Be there at 1500 hours and you will really see something!\"\n\nSomething far better than they could possibly have imagined. I dismissed them and watched the Head's office. As soon as Smelly went through the door I nipped down and broke into his locker.\n\nIt worked like a charm. I take some pride in this since it was the first criminal scenario that I prepared for others to take part in. All unsuspecting of course. At the appointed time I drifted up to the candy counter at Ming's, working very hard to ignore the rentaflics, who were working equally hard pretending they weren't watching me. With relaxed motions I placed the book atop the candies and bent to fix my boot fastener.\n\n\"Nicked!\" the burlier of them shouted, seizing me by the coat collar. \"Gotcha!\" the other crowed, grabbing up the book.\n\n\"What are you doing,\" I croaked\u2014I had to croak because my coat was now pulled tight about my throat as I hung suspended from it. \"Thief\u2014give me back my seven-buck history book that my Mom bought with money earned weaving matts from porcuswine quills!\"\n\n\"Book?\" the great bully sneered. \"We know all about this book.\" He seized the ends and pulled. It opened and the look on his face as the pages flipped over was something sweet to behold.\n\n\"I have been framed,\" I squeaked, opening my coat and dropping free, rubbing at my sore throat. \"Framed by the criminal who bragged about using that same technique for his own nefarious ends. He stands there, one Smelly by name. Grab him, guys, before he runs away!\"\n\nSmelly could only stand and gape while the ready hands of his peers clutched tight. His schoolbooks fell to the floor and the imitation book burst open and disgorged its contents of Get-Stuffeds upon the floor.\n\nIt was beautiful. Tears and recriminations and shouting. A perfect distraction as well. Because this was the day that I field-tested my Mark II Get-Stuffed stuffer. I had worked hard on this device which was built around a silent vacuum pump\u2014with a tube down my sleeve. I brought the tube end close to the candy bars and\u2014zip!\u2014the first of them vanished from sight. It ended up in my trousers, or rather inside the hideous plus-fours we were forced to wear as a school uniform. These bagged out and were secured above the ankle by a sturdy elastic band. The candy bar dropped safely into it, to be followed by another and yet another.\n\nExcept I couldn't turn the damn thing off. Thank goodness for Smelly's screaming and struggling. All eyes were on him and not me as I struggled with the switch. Meanwhile the pump still pumped and the Get-Stuffeds shot up my sleeve and into my trousers. I turned it off eventually but if anyone had bothered to look my way, why the empty counter and my bulging-legged form would have been a might suspicious. But thankfully no one did. I exited with a rolling gait, as quickly as I could. As I said, a memory I will always cherish.\n\nWhich, of course, does not explain why I have now, on my birthday, made the major decision to hold up a bank. And get caught.\n\nThe police had finally broken down the door and were swarming in. I raised my hands over my head and prepared to welcome them with warm smiles.\n\nThe birthday, that is the final reason. My seventeenth birthday. Becoming seventeen here on Bit O' Heaven is a very important time in a young man's life.\nChapter 2\n\nThe judge leaned forward and looked down at me, not unkindly.\n\n\"Now come on, Jimmy, tell me what this tomfoolery is all about.\"\n\nJudge Nixon had a summer house on the river, not too far from our farm, and I had been there often enough with his youngest son for the judge to get to know me.\n\n\"My name is James diGriz, buster. Let us not get too familiar.\"\n\nThis heightened his color a good deal, as you might imagine. His big nose stuck out like a red ski slope and his nostrils flared. \"You will have more respect in this courtroom! You are faced with serious charges, my boy, and it might help your case to keep a civil tongue in your mouth. I am appointing Arnold Fortescue, the public defender, as your attorney....\"\n\n\"I don't need an attorney\u2014and I particularly don't need old Skewey who has been on the sauce so long there isn't a man alive who has seen him sober....\"\n\nThere was a ripple of laughter from the public seats, which infuriated the judge. \"Order in the court!\" he bellowed, hammering his gavel so hard that the handle broke. He threw the stub across the room and glared angrily at me. \"You are trying the patience of this court. Lawyer Fortescue has been appointed....\"\n\n\"Not by me he hasn't. Send him back to Mooney's Bar. I plead guilty to all charges and throw myself on the mercy of this merciless court.\"\n\nHe drew in his breath with a shuddering sigh and I decided to ease off a bit before he had a stroke and collapsed; then there would be a mistrial and more time would be wasted.\n\n\"I'm sorry, judge.\" I hung my head to hide an unrepressed smile. \"But I done wrong and I will have to pay the penalty.\"\n\n\"Well, that's more like it, Jimmy. You always were a smart lad and I hate to see all that intelligence going to waste. You will go to Juvenile Correction Hall for a term of not less than\u2014\"\n\n\"Sorry, your honor,\" I broke in. \"Not possible. Oh, if I only had committed my crimes last week or last month! The law is firm on this and I have no escape. Today is my birthday. My seventeenth birthday.\"\n\nThat slowed him down all right. The guards looked on patiently while he punched for information on his computer terminal. The reporter for the Bit O' Heaven Bugle working just as hard on the keys of his own portable terminal at the same time. He was filing quite a story. It didn't take the judge long to come up with the answers. He sighed.\n\n\"That is true enough. The records reveal that you are seventeen this day and have achieved your majority. You are no longer a juvenile and must be treated as an adult. This would mean a prison term for certain\u2014if I didn't allow for the circumstances. A first offense, the obvious youth of the defendant, his realization that he has done wrong. It is within the power of this bench to make exceptions, to suspend a sentence and bind a prisoner over. It is my decision...\"\n\nThe last thing I wanted to do was hear his decision now. Things were not going as I had planned, not at all. Action was required. I acted. My scream drowned out the judge's words. Still screaming I dived headlong from the prisoner's dock, shoulder-rolled neatly on the floor, and was across the room before my shocked audience could even consider moving.\n\n\"You will write no more scurrilous lies about me, you grubbing hack,\" I shouted. As I whipped the terminal from the reporter's hands and crashed it to the floor. Then stamped the six-hundred-buck machine into worthless junk. I dodged around him before he could grab me and pelted towards the door. The policeman there grabbed at me\u2014then folded when I planted my foot in his stomach.\n\nI could probably have escaped then, but escape at this point wasn't part of my plan. I fumbled with the door handle until someone grabbed me, then struggled on until I was overwhelmed.\n\nThis time I was manacled as I stood in the dock and there was no more Jimmy-my-boy talk from the judge. Someone had found him a new gavel and he waved it in my direction, as though wishing to brain me with it. I growled and tried to look surly.\n\n\"James Bolivar diGriz,\" he intoned. \"I sentence you to the maximum penalty for the crime that you have committed. Hard labor in the city jail until the arrival of the next League ship, whereupon you will be sent to the nearest place of correction for criminal therapy.\" The gavel banged. \"Take him away.\"\n\nThis was more like it. I struggled against my cuffs and spat curses at him so he wouldn't show any last moment weaknesses. He didn't. Two burly policemen grabbed me and hauled me bodily out of the courtroom and jammed me, not too gently, into the back of the black Maria. Only after the door had been slammed and sealed did I sit back and relax\u2014and allow myself a smile of victory.\n\nYes, victory, I mean that. The whole point of the operation was to get arrested and sent to prison. I needed some on-the-job training.\n\nThere is method in my madness. Very early in life, probably about the time of my Get-Stuffed successes, I began to seriously consider a life of crime. For a lot of reasons\u2014not the least of which was that I enjoyed being a criminal. The financial rewards were great; no other job paid more for less work. And, I must be truthful, I enjoyed the feeling of superiority when I made the rest of the world look like chumps. Some may say that is a juvenile emotion. Perhaps\u2014but it sure is a pleasurable one.\n\nAbout this same time I was faced with a serious problem. How was I to prepare myself for the future? There had to be more to crime than lifting Get-Stuffed bars. Some of the answers I saw clearly. Money was what I wanted. Other people's money. Money is locked away, so the more I knew about locks the more I would be able to get this money. For the first time in school I buckled down to work. My grades soared so high that my teachers began to feel there might be hope for me yet. I did so well that when I elected to study the trade of locksmith they were only too eager to oblige. It was supposed to be a three-year course, but I learned all there was to know in three months. I asked permission to take the final examination. And was refused.\n\nThings were just not done that way, they told me. I would proceed at the same stately pace as the others and in two years and nine months I would get my diploma, leave the school\u2014and enter the ranks of the wage slaves.\n\nNot very likely. I tried to change my course of study and was informed that this was impossible. I had locksmith stamped on my forehead, metaphorically speaking of course, and it would remain there for life. They thought.\n\nI began to cut classes and avoid the school for days at a time. There was little they could do about this, other than administer stern lectures, because I showed up for all the examinations and always scored the highest grades. I ought to, since I was making the most of my training in the field. I carefully spread my attentions around so the complacent citizens of the city had no idea they were being taken. A vending machine would yield a few bucks in silver one day, a till at the parking lot the next. Not only did this field work perfect my talents but it paid for my education. Not my school education of course\u2014by law I had to remain there until the age of seventeen\u2014but in my free time.\n\nSince I could find no guidelines to prepare myself for a life of crime, I studied all of the skills that might be of service. I found the word forgery in the dictionary, which encouraged me to learn photography and printing. Since unarmed combat had already stood me in good stead, I continued my studies until I earned a Black Belt. Nor was I ignoring the technical side of my chosen career. Before I was sixteen I knew just about all there was to know about computers\u2014while at the same time I had become a skilled microelectronic technician.\n\nAll of these were satisfying enough in themselves\u2014but where did I go from there? I really didn't know. That was when I decided to give myself a coming-of-age birthday present. A term in jail.\n\nCrazy? Like a fox! I had to find some criminals\u2014and where better than in jail? A keen line of reasoning, one has to admit. Going to jail would be like coming home, meeting my chosen peer group at last. I would listen and learn and when I felt I had learned enough the lockpick in the sole of my shoe would help me to make my exit. How I smiled and chortled with glee.\n\nMore the fool\u2014for it was not to be this way at all.\n\nMy hair was shorn, I was bathed in an antiseptic spray, prison clothes and boots were issued\u2014so unprofessionally that I had ample time to transfer the lockpick and my stock of coins\u2014I was thumbprinted and retinapixed, then led to my cell. To behold, to my great joy, that I had a cellmate. My education would begin at last. This was the first day of the rest of my criminal life.\n\n\"Good afternoon, sir,\" I said. \"My name is Jim diGriz.\"\n\nHe looked at me and snarled. \"Get knotted, kid.\" He went back to picking his toes, an operation which my entrance had interrupted.\n\nThat was my first lesson. The polite linguistic exchanges of life outside were not honored behind these walls. Life was tough\u2014and so was language. I twisted my lips into a sneer and spoke again. In far harsher tones this time.\n\n\"Get knotted yourself, toe-cheese. My monicker is Jim. What's yours?\"\n\nI wasn't sure about the slang, I had picked it up from old videos, but I surely had the tone of voice right because I had succeeded in capturing his attention this time. He looked up slowly and there was the glare of cold hatred in his eyes.\n\n\"Nobody\u2014and I mean nobody\u2014talks to Willy the Blade that way. I'm going to cut you, kid, cut you bad. I'm going to cut my initial into your face. A 'V' for Willy.\"\n\n\"A 'W,'\" I said. \"Willy is spelled with a 'W.'\"\n\nThis upset him even more. \"I know how to spell, I ain't no moron!\" He was blazing with rage now, digging furiously under the mattress on his bed. He produced a hacksaw blade that I could see had the back edge well sharpened. A deadly little weapon. He bounced it in his hand, sneered one last sneer\u2014then lunged at me.\n\nWell, needless to say, that is not the recommended way to approach a Black Belt. I moved aside, chopped his wrist as he went by\u2014then kicked the back of his ankle so that he ran headfirst into the wall.\n\nHe was knocked cold. When he came to I was sitting on my bunk and doing my nails with his knife. \"The name is Jim,\" I said, lip-curled and nasty. \"Now you try saying it. Jim.\"\n\nHe stared at me, his face twisted\u2014then began to cry! I was horrified. Could this really be happening?\n\n\"They always pick on me. You're no better. Make fun of me. And you took my knife away. I worked a month making that knife, had to pay ten bucks for the broken blade....\"\n\nThe thought of all of the troubles had started him blubbering again. I saw then that he was only a year or two older than me\u2014and a lot more insecure. So my first introduction to criminal life found me cheering him up, getting a wet towel to wipe his face, giving him back his knife\u2014and even giving him a five-buck goldpiece to stop his crying. I was beginning to feel that a life of crime was not quite what I thought it would be.\n\nIt was easy enough to get the story of his life\u2014in fact it was hard to shut him up once he got in full spate. He was filled with self-pity and wallowed in the chance to reveal all to an audience.\n\nPretty sordid, I thought, but kept silent while his boring reminiscences washed over me. Slow in school, laughed at by the others, the lowest marks. Weak and put upon by the bullies, gaining status only when he discovered\u2014by accident of course with a broken bottle\u2014that he could be a bully too once he had a weapon. The rise in status, if not respect, after that by using threats of violence and more than a little bullying. All of this reinforced by demonstrations of dissections on live birds and other small and harmless creatures. Then his rapid fall after cutting a boy and being caught. Sentenced to Juvenile Hall, released, then more trouble, and back to Hall yet again. Until here he was, at the zenith of his career as a knife-carrying punk, imprisoned for extorting money by threat of violence. From a child of course. He was far too insecure to attempt to threaten an adult.\n\nOf course he did not say all this, not at once, but it became obvious after endless rambling complaints. I tuned him out and tuned my inner thoughts in. Bad luck, that was all it was. I had probably been put in with him to keep me from the company of the real hardened criminals who filled this prison.\n\nThe lights went out at that moment and I lay back on the bunk. Tomorrow would be my day. I would meet the other inmates, size them up, find the real criminals among them. Befriend them and begin my graduate course in crime. That is surely what I would do.\n\nI went happily to sleep, washed over by a wave of wimpish whining from the adjoining bunk. Just bad luck being stuck in with him. Willy was the exception. I had a roommate who was a loser, that was all. It would all be different in the morning.\n\nI hoped. There was a little nag of worry that kept me awake for a bit, but at last I shrugged it off. Tomorrow would be fine, yes it would be. Fine. No doubt about that, fine....\nChapter 3\n\nBreakfast was no better\u2014and no worse\u2014than the ones I made for myself. I ate automatically, sipping the weak cactus tea and chewing doggedly at the gruel, while I looked around at the other tables. There were about thirty prisoners stuffing their faces in this room, and my gaze went from face to face with a growing feeling of despair.\n\nFirst, most of them had the same vacuous look of blank stupidity as my cellmate. All right, I could accept that, the criminal classes would of course contain the maladjusted and the mental mud walls. But there had to be more than that! I hoped.\n\nSecond, they were all quite young, none out of their twenties. Weren't there any old criminals? Or was criminality a malfunction of youth that was quickly cured by the social adjustment machines? There had to be more to it than that. There had to be. I took some cheer from this thought. All of these prisoners were losers, that was obvious, losers and incompetents. It was obvious once you thought about it. If they had been any good at their chosen profession they wouldn't be inside! They were of no use to the world or to themselves.\n\nBut they were to me. If they couldn't supply the illegal facts that I needed, they would surely be able to put me in touch with those who did. From them I would get leads to the criminals on the outside, the professionals still uncaught. That was what I had to do. Befriend them and extract the information that I needed. All was not lost yet.\n\nIt didn't take long for me to discover the best of this despicable lot. A little group was gathered around a hulking young man who sported a broken nose and a scarred face. Even the guards seemed to avoid him. He strutted a good deal and the others made room around him when he walked in the exercise yard after lunch.\n\n\"Who is that?\" I asked Willy, who huddled on the bench next to me, industriously picking his nose. He blinked rapidly until he finally made out the subject of my attentions, then waved his hands with despair.\n\n\"Watch out for him, stay away, he's bad medicine. Stinger is a killer, that's what I heard, and I believe it too. And he's a champ at mudslugging. You don't want to know him.\"\n\nThis was intriguing indeed. I had heard of mudslugging, but I had always lived too close to the city to have seen it in action. There was never any of it taking place near enough for me to hear about, not with the police all around. Mudslugging was a crude sport\u2014and illegal\u2014that was enjoyed in the outlying farm towns. In the winter, with the porcuswine in their sties and the crops in the barns, time would hang heavy on their agrarian hands. That was when the mudslugging would begin. A stranger would appear and challenge the local champion, usually some overmuscled ploughboy. A clandestine engagement would be arranged in some remote barn, the women dismissed, moonshine surreptitiously brought in plastic bottles, bets made\u2014and the barefisted fight begun. To end when one of the combatants could not get off the ground. Not a sport for the squeamish, or the sober. Good, hearty, drunken masculine fun. And Stinger was one of this stalwart band. I must get to know Stinger better.\n\nThis was easily enough done. I suppose I could have just walked over and spoken to him, but my thought patterns were still warped by all of the bad videos I had watched for most of my life. Plenty of these were about criminals getting their just deserts in prison, which is probably where I originated the idea of this present escapade. Never matter, the idea was still a sound one. I could prove that by talking to Stinger.\n\nTo do this I walked, whistling, about the yard until I was close to him and his followers. One of them scowled at me and I scuttled away. Only to return as soon as his back was turned, to sidle up beside the head villain.\n\n\"Are you Stinger?\" I whispered out of the side of my mouth, head turned away from him. He must have seen the same videos because he answered in the same way.\n\n\"Yeah. So who wants to know?\"\n\n\"Me. I just got into this joint. I got a message for you from the outside.\"\n\n\"So tell.\"\n\n\"Not where these dummies can hear. We gotta be alone.\"\n\nHe gave me a most suspicious look from under his beetling brows. But I had succeeded in capturing his curiosity. He muttered something to his followers, then strolled away. They remained behind but flashed murderous looks at me when I strolled in the same direction. He went across the yard towards a bench\u2014the two men already there fleeing as he approached. I sat down next to him and he looked me up and down with disdain.\n\n\"Say what you gotta say, kid\u2014and it better be good.\"\n\n\"This is for you,\" I said, sliding a twenty-buck coin along the bench towards him. \"The message is from me and from no one else. I need some help and am willing to pay for it. Here is a down payment. There is plenty more where this came from.\"\n\nHe sniffed disdainfully\u2014but his thick fingers scraped up the coin and slipped it into his pocket. \"I ain't in the charity business, kid. The only geezer I help is myself. Now, shove off\u2014\"\n\n\"Listen to what I have to say first. What I need is someone to break out of prison with me. One week from today. Are you interested?\"\n\nI had caught his attention this time. He turned and looked me square in the eye, cold and assured. \"I don't like jokes,\" he said\u2014and his hand grabbed my wrist and twisted. It hurt. I could have broken the grip easily, but I did not. If this little bit of bullying was important to him, then bully away.\n\n\"It's no joke. Eight days from now I'll be on the outside. You can be there too if you want to be. It's your decision.\"\n\nHe glared at me some more\u2014then let go of my wrist. I rubbed at it and waited for his response. I could see him chewing over my words, trying to make up his mind. \"Do you know why I'm inside?\" he finally asked.\n\n\"I heard rumors.\"\n\n\"If the rumor was that I killed a geezer, then the rumor was right. It was an accident. He had a soft head. It broke when I knocked him down. They was going to pass it off as a farm accident but another geezer lost a bundle to me on the match. He was going to pay me next day but he went to the police instead because that was a lot cheaper. Now they are going to take me to a League hospital and do my head. The shrinker here says I won't want to fight again after that. I won't like that.\"\n\nThe big fists opened and closed when he talked and I had the sudden understanding that fighting was his life, the one thing that he could do well. Something that other men admired and praised him for. If that ability were taken away\u2014why they might just as well take away his life at the same time. I felt a sudden spurt of sympathy but did not let the feeling show.\n\n\"You can get me out of here?\" The question was a serious one.\n\n\"I can.\"\n\n\"Then I'm your man. You want something out of me, I know that, no one does nothing for nothing in this world. I'll do what you want, kid. They'll get me in the end, there is no place to hide anywhere when they are really looking for you. But I'm going to get mine. I'm going to get the geezer what put me in here. Get him proper. One last fight. Kill him the way he killed me.\"\n\nI could not help shivering at his words because it was obvious that he meant them. That was painfully clear. \"I'll get you out,\" I said. But to this I added the unspoken promise that I would see to it that he got nowhere near the object of his revenge. I was not going to start my new criminal career as an accomplice to murder.\n\nStinger took me under his protective wing at once. He shook my hand, crushing my fingers with that deadly grip, then led me over to his followers.\n\n\"This is Jim,\" he said. \"Treat him well. Anyone causes him trouble got trouble with me.\" They were all insincere smiles and promises of affection\u2014but at least they wouldn't bother me. I had the protection of those mighty fists. One of them rested on my shoulder as we strolled away. \"How you going to do it?\" he asked.\n\n\"I'll tell you in the morning. I'm making the last arrangements now,\" I lied. \"See you then.\" I strolled off on an inspection tour, almost as eager to be out of this sordid place as he was. For a different reason. His was revenge\u2014mine was depression. They were losers in here, all losers, and I like to think of myself as a winner. I wanted to be well away from them all and back in the fresh air.\n\nI spent the next twenty-four hours finding the best way out of the prison. I could open all of the mechanical locks inside the prison easily enough; my lockpick worked fine on our cell door. The only problem was the electronic gate that opened out into the outer courtyard. Given time\u2014and the right equipment\u2014I could have opened that too. But not under the eyes of guards stationed in the observation booth above it right around the clock. That was the obvious way out, so it was the route to be avoided. I needed a better idea of the layout of the prison\u2014so a reconnoiter was very much in order.\n\nIt was after midnight when I eased out of my bed. No shoes, I had to be as quiet as possible, so three pairs of socks should do the job. Working silently, I stuffed extra clothing under the blankets so the bed would look occupied if one of the guards should look in through the barred door. Willy was snoring lustily when I clicked the lock open and slipped out into the corridor. He wasn't the only one enjoying his sack time and the walls echoed with zzzzing and gronking. The nightlights were on and I was alone on the landing. I looked over the edge carefully and saw that the guard on the floor below was working on his racing form. Wonderful, I hoped that he had a winner. Silent as a shadow I went to the stairs and up them to the floor above.\n\nWhich was depressingly identical to the one below: nothing but cells. As was the next floor and the one above that. Which was the top floor so I could go no higher. I was about to retrace my steps when my eye caught a glint of metal in the shadows at the far end. Nothing ventured, as the expression goes. I scuttled past the barred doors, and the\u2014hopefully\u2014sleeping inmates, to the distant wall.\n\nWell, well, what did we have here! Iron rungs in the wall\u2014vanishing up into the darkness. I grabbed onto the first one and vanished up with them. The last rung was just below the ceiling. It was also just under a trapdoor that let into the ceiling above. Metal, with a metal frame\u2014and locked securely as I discovered when I pushed up against it. There had to be a lock, but it was invisible in the darkness. And I had to find it. Looping one arm through the iron rung I began to run my fingertips over the surface of the door in what I hoped was a regular pattern.\n\nThere was nothing there. I tried again, changing hands because my arm felt like it was being dragged from its socket\u2014with the same result. But there had to be a lock. I was panicking and not using my brain. I fought back my rising fears and stirred up my brain cells. There must be a lock or seal of some kind. And it was not on the trapdoor. So\u2014it had to be on the frame. I reached out slowly, ran my fingers along the sides of the frame. And found it at once.\n\nHow simple the answers are when you ask the right questions! I eased the lockpick from my pocket and slipped it into the lock. Within seconds it had clicked open. Seconds after that I had pushed the trapdoor up, climbed through, closed it behind me\u2014and sniffed appreciatively of the cool night air.\n\nI was out of prison! Standing on the roof, yes, of course, but free in spirit at least. The stars were bright above and shed enough light so I could see across the dark surface. It was flat and broad, bordered with a knee-high parapet and studded with vents and pipes. Something large and bulky occluded the sky, and when I worked my way close to it I heard the dripping of water. The water tank, fine, now what was visible below?\n\nTo the front I looked down into the well-lit courtyard, guarded and secure. But what was the back like?\n\nFar more interesting, I assure you. There was a straight drop of five stories to a rear yard, which was feebly illuminated by a single bulb. There were waste bins there, and barrels, and a heavy gate in the outer wall. Locked, undoubtedly. But what man had locked man could unlock. Or rather I could. This was the way out.\n\nOf course there was the five-story drop, but something could be worked there. Or perhaps I could find another way into the backyard. Plenty of time to run through the permutations of escape, six days yet. My feet were getting cold and I yawned and shivered. I had done enough for one night. My hard prison bunk seemed very attractive at this moment.\n\nCarefully and silently I retraced my steps. Eased the trapdoor shut above me, checked to see that it was locked, went down the ladder and the stairs to my floor....\n\nAnd heard the voices ahead. Loud and clear. The loudest of all being my dear cellmate Willy. I took one horrified look at the open door of my cell, at the heavy boots of the guards there, then pulled myself back and ran up the stairs again. With Willy's words ringing like a tocsin of doom in my ears.\n\n\"I woke and he was gone! I was alone! Monsters ate him or something! That's when I started shouting. Save me, please! Whatever got him came right through the locked door. It's gonna get me next!\"\nChapter 4\n\nAnger at my cretinous cellmate warmed me; the imminence of my capture instantly chilled me again. I fled unthinkingly, away from the voices and commotion. Back up the stairs, one flight, another\u2014\n\nThen all the lights came on and the sirens began to wail. The prisoners stirred and called out to one another. In a few moments they would be at the cell doors, would see me, would cry out, guards would appear. There was no escape. I knew this, yet all I could do was run. To the top floor\u2014then past the cells there. All of which were now brightly lit. I would be seen by the prisoners as I went by them, and I knew for certain that I would be ratted on by whichever juvenile delinquent spotted me. It was all over.\n\nHead high, I walked past the first cell and glanced in as I passed.\n\nIt was empty. As were all of the other cells on this floor. I still had a chance! Like a demented ape I swarmed up the iron rungs and fumbled my lockpick into the lock. There were voices below me, getting louder, and footsteps as well as two of the guards ascended the stairs that faced away from me. But all that one of them had to do was turn his head. And when they reached the floor I would be seen at once.\n\nThe lock clicked open and I pushed and swarmed up through the opening. Flat on the roof I eased the door down. Seeing two fat guards through the opening just turning my way as it shut.\n\nHad they seen it closing? My heart thudded like an insane drum and I gasped for air and waited for the shouts of alarm.\n\nThey did not come. I was still free.\n\nSome freedom! Depression instantly clutched and shook me. Free to lie on the roof, to shiver violently as the perspiration began to dry, free to huddle up here until I was found.\n\nSo I huddled and shivered and generally felt sorry for myself for about a minute. Then I stood and shook myself like a dog and felt the anger begin to rise.\n\n\"Big criminal,\" I whispered aloud, just to make sure that I heard. \"Life of crime. And on your first big job you let yourself be trapped by a knife-wielding moron. You've learned a lesson, Jim. May you someday be free to put it into practice. Always guard your flanks and your rear. Consider all the possibilities. Consider the fact that the cretin might have woken up. So you should have coshed him or something to make certain of his sound sleep. Which is certainly water over the dam. Remember the lesson well, but look around now and try to make the best of this rapidly disintegrating escape.\"\n\nMy options were limited. If the guards opened the trapdoor and came up to the roof they would find me. Was there any place to hide? The top of the water tank might offer a temporary refuge, but if they came this far they would certainly look there as well. But with no way to get down the sheer walls it offered the only feeble hope. Get up there.\n\nIt wasn't easy. It was made of smooth metal and the top was just beyond my reach. But I had to do it. I stepped back and took a run, leaped, and felt my fingers just grasp the edge. I scrabbled for a hold but they pulled loose and I dropped heavily back to the roof. Anyone below would have certainly heard that. I hoped I was over an empty cell and not the hall.\n\n\"Enough hoping and not enough trying, Jim,\" I said, and added a few curses in the hope of building my morale. I had to get up there!\n\nThis time I retreated to the far edge, the backs of my knees against the parapet, taking breath after deep breath. Go!\n\nRun up, fast, the right spot\u2014jump!\n\nMy right hand slapped against the edge. I grabbed and heaved. Got my other hand up there and pulled mightily, scraping and bruising myself on the rough metal, hauling myself up onto the top of the tank.\n\nTo lie there breathing heavily, looking at a dead bird not a foot from my face, vacant eyes staring into mine. I started to pull away when I heard the trapdoor slam heavily back onto the roof.\n\n\"Give me a push up, will you? I'm stuck!\"\n\nBy the wheezing and grunting that followed I was sure that this had to be one of the fat guards that I had seen on the floor below. More gasping and puffing heralded the arrival of his adipose companion.\n\n\"I don't know what we're doing up here,\" the first arrival whined.\n\n\"I do,\" his companion said quite firmly. \"We're obeying orders, which never did no one no harm.\"\n\n\"But the hatch was locked.\"\n\n\"So was the cell door he went through. Look around.\"\n\nThe heavy footsteps circled the roof, then returned.\n\n\"Not here. No place to hide. Not even hanging over the edge because I looked.\"\n\n\"There is one place, one place we haven't looked.\"\n\nI could feel the eyes burning towards me through the solid metal. My heart had started the drumbeat thing again. I clutched at the rusty metal and felt only despair as the footsteps crunched close.\n\n\"He could never climb up there. Too high. I can't even reach the top.\"\n\n\"You can't even reach your shoelaces when you bend over. Come on, give me a lift up. If you boost my foot I can reach up and grab on. All I got to do is take a look.\"\n\nHow right he was. Just one look. And there was nothing I could do about it. With the lethargy of defeat possessing me. I lay there, hearing the scratching and the curses, the overweight puffing and scrabbling. The scratching grew close and not a foot before my face a large hand appeared, groping over the edge.\n\nMy subconscious must have done it because I swear there was no logical thought involved. My hand shot out and pushed the dead bird forward, to the very edge, below the fingers\u2014which descended and closed on it.\n\nThe results were eminently satisfying. The bird vanished, as did the hand, followed by screams and shouts, scrabblings, and two large thuds.\n\n\"Why did you do that\u2014?\"\n\n\"I grabbed it, uggh\u2014oww! My ankle is broke.\"\n\n\"See if you can stand on it. Here, hold my shoulder. Hop along on the other foot, this way....\"\n\nThere was plenty of shouting back and forth through the trapdoor while I hugged myself with relief and pleasure. They might be back soon, there was that chance, but at least the first round was mine.\n\nAs the seconds, then the minutes, moved slowly by I realized that I had won the second round as well. The search had moved away from the roof. For the moment. The sirens cut off and the bustle moved down to the ground below. There were shouts and the slamming of doors, racing of engines as cars moved out into the night. Not soon after\u2014wonder of wonders\u2014the lights began to go out. The first search was over. I started to doze\u2014then jerked myself awake.\n\n\"Dummy! You are still in the soup. The search has been made, but this joint is still sealed tight. And you can bet your last buck that starting at first light they will go through every nook and corner. And will be up here with a ladder this time. So with that in mind it is time to move.\"\n\nAnd I knew just where I was moving to. The last place they would look for me this night.\n\nThrough the trapdoor one last time, and down the darkened corridor. Some of the inmates were still muttering about the events of the night, but all of them appeared to be back in their bunks. Silently I slipped down the stairs and up to cell 567B. Opening it in absolute silence and closing it behind me the same way. Past my stripped bunk to the other bunk where my fink friend Willy slept the sleep of the unjust.\n\nMy hand clamped his mouth shut, his eyes sprang open, and I exacted primal and sadistic pleasure by whispering in his ear.\n\n\"You are dead, you rat, dead. You called the guards and now you are going to get what you deserve....\"\n\nHis body gave one gigantic heave then went limp. The eyes were closed. Had I killed him? At once I regretted the bad taste of my little joke. No, not dead, passed out, his breathing light and slow. I went to get a towel, soaked it in cold water\u2014then let him have it right in the mush.\n\nHis scream turned to a gurgle as I stuffed the towel into his mouth.\n\n\"I'm a generous man, Willy, that's how lucky you are. I'm not going to kill you.\" My whispered words seemed to reassure him because I felt the tremble in his body subside. \"You are going to help me. If you do that you will come to no harm. You have my word. Now prepare to answer my question. Think carefully about this. You are going to whisper just one thing. You are going to tell me the number of the cell that Stinger is in. Nod your head if you are ready. Good. I'm taking the towel away. But if you try any tricks or say anything\u2014anything\u2014else, why then you are dead. Here goes.\"\n\n\"... 231B...\"\n\nThis same floor, good. The towel went right back in. Then I pressed hard behind his right ear, applying continuous pressure to the blood vessel that leads to the brain. Six seconds unconsciousness, ten seconds death. He thrashed then went limp again. I released my thumb on the count of seven. I do have a forgiving nature.\n\nI used the towel to clean my face and hands, then groped for my shoes and put them on. Along with another shirt and my jacket. After that I gurgled down at least a litre of water and was ready to face the world again. I stripped the blankets from the beds, bundled them under my arm\u2014then left.\n\nOn tiptoe, as silently as I could, I slipped down to Stinger's cell. I felt immune, impervious. I realized that this was both foolish and dangerous. But after the traumatic events of the evening I seemed to have run out of fear. The cell door opened beneath my delicate touch and Stinger's eyes opened as well when I pushed his shoulder.\n\n\"Get dressed,\" I said quietly. \"We're getting out now.\"\n\nI'll give him this much\u2014he didn't bother asking questions. Just pulled his clothes on while I took the blankets from his bunk. \"We need at least two more,\" I said.\n\n\"I'll get Eddie's.\"\n\n\"He'll wake up.\"\n\n\"I'll see he goes back to sleep.\"\n\nThere was a murmured question\u2014followed by a solid thud. Eddie went back to sleep and Stinger brought over the blankets.\n\n\"Here's what we do,\" I told him. \"I found the way up to the roof. We go there and knot these blankets together. Then we climb down them and get away. Okay?\"\n\nOkay! I had never heard a more insane plan in my life. But not Stinger.\n\n\"Okay! Let's go!\"\n\nOnce more up the stairs\u2014I was really getting pretty tired of this\u2014and tired all over as well. I climbed the rungs, opened the trapdoor, and pushed the blankets through onto the roof when he passed them up to me. He didn't say a word until I had closed and sealed the door again.\n\n\"What happened? I heard you got away and I was going to kill you if they ever brought you back.\"\n\n\"It's not that simple. I'll tell you when we get clear. Now let's start tying. Opposite corners lengthwise; we need all the length we can get. Use a square knot like you learned in the Boy Sprouts. Like this.\"\n\nWe knotted and tied like crazy until they were all connected, then took the ends and pulled and grunted and that was that. I tied one end to a solid-looking pipe and threw the bundle of blankets over the side.\n\n\"At least twenty feet short,\" Stinger said, scowling down at the ground. \"You go first because you're lighter. If it breaks with me at least you got a chance. Get moving.\"\n\nThe logic of this could not be argued with. I climbed up on the parapet and seized the top blanket. Stinger squeezed my arm with an unexpected show of emotion. Then I was climbing down.\n\nIt was not easy. My hands were tired and the blanket fabric hard to grip. I went down as quickly as I could because I knew that my strength was running out.\n\nThen my legs scrabbled at empty air and I had reached the end. The hard floor of the courtyard appeared to be very far below. It was difficult to let go\u2014or rather really very easy. I could hold on no longer. My fingers opened and I fell\u2014\n\n\u2014hit and rolled and sat on the ground gasping for breath. I had done it. High above I could see the dark figure of Stinger swarming down the rope, hand over hand. Within seconds he was on the ground, landing light as a cat beside me, helping me to my feet. Half-supporting me as I stumbled to the gate.\n\nMy fingers were trembling and I couldn't get the lock open. We were painfully visible here under the light and if any of the guards glanced out of a window above we were trapped....\n\nI took in a long, shuddering breath\u2014then inserted the picklock once again. Slowly and carefully feeling the grooves on the interior, turning and pushing.\n\nIt clicked open and we hurled ourselves through. Stinger pushed it silently shut, then turned and ran out into the night with me right at his heels.\n\nWe were free!\nChapter 5\n\n\"Wait!\" I called after Stinger as he pelted down the road. \"Not that way. I've got a better plan. I worked it out before I was sent up.\"\n\nHe slowed to a halt and thought about this and slowly made up his mind. \"You called the shots OK so far. So what we gonna do?\"\n\n\"For openers\u2014leave a trail that they can follow with sniffer robots. This way.\"\n\nWe left the road and cut through the grass and down to the nearby stream. It was shallow but cold and I could not suppress a shiver as we waded across. The main highway ran close by and we headed that way. Crouching low as a heavy transporter thundered by. For the moment there was no other traffic in sight.\n\n\"Now!\" I called out. \"Straight up to the road\u2014then right back down walking in your own footprints.\"\n\nStinger did what he was told, backtracking with me to the stream and into the frigid water again.\n\n\"That's smart,\" he said. \"The sniffers find where we went into the water, where we came out\u2014and follow us to the road. Then they think that a groundcar maybe picked us up. So what comes next?\"\n\n\"We go upstream\u2014staying in the water\u2014to the nearest farm. Which happens to be a porcuswine farm....\"\n\n\"No way! I hate them mothers. Got bit by one when I was a kid.\"\n\n\"We have no other choice. Anything else we do the fuzz will pick us up at daybreak. I can't say I love the porkers either. But I grew up on a farm and I know how to get along with them. Now let's move before my legs freeze off at the ankles.\"\n\nIt was a long, cold slog and I could not stop the trembling once it began. But there was absolutely nothing else to do except push on. My teeth were rattling in my head like castanets before we came to the brook that bubbled down through the fields to join the stream that we were wading in. The stars were beginning to fade; dawn was not too far away.\n\n\"This is it,\" I said. \"The stream that we want. That chopped tree is my landmark. Stay right behind me\u2014we're very close now.\" I reached up and broke off a dead branch that overhung the stream, then led the way. We waded along until we reached a tall, electrified fence that spanned the stream. It could be clearly seen in the growing light. I used the branch to lift the bottom of the fence so Stinger could crawl under, then he did the same for me. As I stood up I heard the familiar rustle of large quills from the oak grove nearby. A large, dark form separated itself from the trees and moved towards us. I grabbed the branch from Stinger and called out softly.\n\n\"Sooo-ee, sooo-ee... here swine, swine, swine.\"\n\nThere was a bubbling grunt from the boar as it approached. Stinger was muttering under his breath, curses or prayers\u2014or both\u2014as he stood behind me. I called again and the great creature came close. A real beauty, a tonne at least, looking at me with its small red eyes. I stepped forward and raised the branch slowly\u2014and heard Stinger moan behind me. The boar never moved as I poked the stick behind its ear, parted the long quills\u2014and began to scratch its hide industriously.\n\n\"What are you doing? It'll kill us!\" Stinger wailed.\n\n\"Of course not,\" I said, scratching harder. \"Listen to it?\" The porcuswine's eyes were half-closed with pleasure and it was burbling happily. \"I know these big porkers well. They get vermin under their quills and can't get at them. They love a good scratch. Let me do the other ear\u2014there are nice itchy patches behind the ears\u2014then we can go on.\"\n\nI scratched, the boar moaned happily, and dawn crept up on us. A light came on in the farmhouse and we knelt down behind the porcuswine. The door opened, someone threw out a basin of water, then it closed again.\n\n\"Let's get to the barn,\" I said. \"This way.\"\n\nThe boar grumbled when I stopped scratching, then trotted along behind us, hoping for more, as we skulked across the farm. Which was a good thing since there were plenty more of the spiky porkers on all sides. But they moved aside when the kingpig approached and we proceeded in stately parade to the barn.\n\n\"So long, big feller,\" I said, giving a last good scratch. \"Been nice knowing you.\" Stinger had the barn door open and we slipped inside. We had just slid the bolt again when the heavy wood trembled as our overweight companion leaned against it and snorted.\n\n\"You saved my life,\" Stinger gasped. \"I'll never forget that.\"\n\n\"Just skill,\" I said humbly. \"After all, you are good with fists\u2014\"\n\n\"And you're great with pigs!\"\n\n\"I wouldn't have phrased it exactly that way,\" I muttered. \"Now let's get up into the hayloft where it is warm\u2014and where we won't be seen. There is a long day ahead of us and I want to spend as much of it as I can sleeping.\"\n\nIt had been quite a night. I burrowed into the hay, sneezed twice as the dust got into my nose\u2014then must have fallen instantly asleep.\n\nThe next thing I knew Stinger was shaking me by the shoulder and sunlight was streaming between the boards in the wall. \"Cops is here,\" he whispered.\n\nI blinked the sleep from my eyes and looked through the crack. A green and white police floater was hovering outside the farmhouse door and two uniformed pug-uglies were showing a sheet to the farmer. He shook his head and his voice was clear above the farmyard sounds.\n\n\"Nope. Never seen neither of them. Never seen a soul in a week if you want to know. Fact is, kind of nice to talk to you fellers. These guys really look nasty, criminals you say...\"\n\n\"Pops, we ain't got all day. If you didn't see them they could still be hiding on your farm. Maybe in your barn?\"\n\n\"No way they could do that. Them's porcuswine out there. Most ornery critters in creation.\"\n\n\"We still got to look. Orders are to search every building in the vicinity.\"\n\nThe policemen started our way and there was a screech like an insane siren and the thud of sharp hooves. Around the corner of the barn\u2014quills rattling with anger\u2014came our friend of the night before. He charged and the police dived for their floater. The angry boar crashed into it, sending it rocking across the yard with a great dent in its side. The farmer nodded happily.\n\n\"Told you weren't no one in the barn. Little Larry here, he don't cotton onto strangers. But drop by anytime you're in the neighborhood, fellers....\"\n\nHe had to shout the last words because the floater was heading west with Little Larry in snorting pursuit.\n\n\"Now that is what I call beautiful,\" Stinger said, awe in his voice. I nodded silent agreement. Even the dullest of lives contain moments of pure glory.\n\nEnough fun; time to work. I chewed on a straw and stretched out on the warm hay. \"Porcuswine are nice when you know them.\"\n\n\"The police don't seem to think so,\" he said.\n\n\"Guess not. That was the best thing I ever saw. I don't exactly get along with the police.\"\n\n\"Who does? What you got sent up for, Jimmy?\"\n\n\"Bank robbery. Did you ever hold up a bank?\"\n\nHe whistled appreciation and shook his head no. \"Not my style. I wouldn't know what to do first. Mudslugging's my style. Ain't been beat in nine years.\"\n\n\"Knocking around the way you do you must meet a lot of people. Did you ever meet Smelly Schmuck?\" I extemporized rapidly. \"He and I did some banks in Graham State.\"\n\n\"Never met him. Never even heard of him. You're the first bank robber I ever met.\"\n\n\"Really? Well, I guess there aren't that many of us these days. But you must know some safecrackers. Or groundcar thieves?\"\n\nAll I got for my efforts was another shake of the head. \"The only time I ever meet guys like you is in jail. I know some gamblers; they go around the mudslugging fights. But they're all two-buckers, losers. I did know one once who swore he knew The Bishop, long time ago.\"\n\n\"The Bishop?\" I said, blinking rapidly, trying to sum up what little I knew of the ecclesiastical hierarchy. \"I don't go to church much these days....\"\n\n\"Not that kind of bishop. I mean The Bishop, the geezer used to clean out banks and things. Thought you would have heard of him.\"\n\n\"Before my time, I guess.\"\n\n\"Before everyone's time. This was years ago. Cops never got him, I hear. This two-bucker bragged he knew The Bishop; said that he had retired and was lying low. He must of been lying, two-bucker like him.\"\n\nStinger knew no more than this and I hesitated to pump him too hard. Our conversation died away and we both dozed on and off until dark. We were thirsty and hungry, but knew that we had to remain under cover during daylight. I chewed on my straw and tried not to think of large beers and bottles of cold water, but thought about The Bishop instead. It was a thin lead, but was all that I had. By the time the sun went down I was hungry and thirsty and thoroughly depressed. My prison escapade had turned out to be a dangerous fiasco. Jails were for losers\u2014that's about all I had found out. And in order to discover this fact I had risked life and limb. Never again. I took a silent oath to stay away from prison and the minions of the law in the future. Good criminals don't get caught. Like The Bishop, whoever he might be.\n\nWhen the last trace of light was gone from the sky we eased the barn door open. A bubbling grunt reached our ears and a great form blocked our exit. Stinger gasped and I grabbed him before he could flee.\n\n\"Grab a stick and make yourself useful,\" I said. \"I'll teach you a new skill.\"\n\nSo we scratched like crazy under the creature's quills while it grunted with pleasure. Trotting behind us like a pet dog when we finally left. \"We got a friend for life,\" I said as we slipped out the gate and I waved goodbye to our porcine pal.\n\n\"Those kinds of friends I can live without forever. You figure out what we do next?\"\n\n\"Absolutely. Advance planning, that's my middle name. There is a siding down this way where they transship from the linears to trucks. We stay away from it because the police are sure to be there. But all the trucks take the same road to the highway where there is a traffic control light. They have to stop until the highway computers see them and let them on. We go there\u2014\"\n\n\"And break into the back of one of the trucks!\"\n\n\"You're learning. Only we get one in the right lane going west. Otherwise we end up back in the fine city of Pearly Gates and right after that in the prison we worked so hard to get out of.\"\n\n\"Lead the way, Jim. You are the brainiest kid I ever met. You're going to go far.\"\n\nThat was my expressed wish and I nodded quick agreement. I was just sorry that he wasn't going too. But I didn't want to live with some far-off yokel's life on my conscience\u2014as much as he might deserve a little agro. But Stinger planned far more than that. I could not be party to a killing.\n\nWe found the road and waited in the bushes beside it. Two trucks rumbled up together\u2014with the lights of another one following. We stayed out of sight. First one, then the second pulled out and headed east. When the third slowed down to stop for his turn, lights came on. West!\n\nWe ran. I was fumbling with the locking bar when Stinger shouldered me aside. He hauled down and the door swung open. The truck started forward and he pushed me up into it. He had to run as it started its turn\u2014but grabbed the sill and pulled himself up with a single heave of those mighty arms. Between us we got the door closed but not sealed.\n\n\"We done it!\" he said triumphantly.\n\n\"We certainly did. This truck is going in the right direction for you\u2014but I have to get back to Pearly Gates as soon as the heat dies down. In about an hour we'll be passing through Billville. I'll leave you there.\"\n\nIt was a quick trip. I swung down at the first stop for a light and he gripped my hand. \"Good luck, kid.\" he called out as the truck pulled away. I couldn't wish him the same.\n\nI dug out a buck coin as the truck rumbled way. And made a mental note of its registration number. As soon as it was out of sight I headed towards the lights of a phonebox. I felt like a rat as I punched the buttons for the police.\n\nBut, really, I had no choice.\nChapter 6\n\nUnlike the hapless Stinger, I had a careful escape plan worked out. Part of it was a literal misdirection for my late partner. He was not really stupid, so it shouldn't take him very long to figure out who had blown the whistle on him. If he talked and told the police that I had returned to the fine city of Pearly Gates\u2014why that would be all for the better. I had no intention of leaving Billville, not for quite a while.\n\nThe office had been rented through an agency and all transactions had been done by computer. I had visited it before my hopeless bank job, and at that time had left some supplies there. They would come in very handy right now. I would enter through the service door of the fully automated building\u2014after turning off the alarms by using a concealed switch I had been prudent enough to install there. It had a timer built into it, so I had ten lazy minutes to get to the office. I yawned as I picked the lock, sealed the door behind me, then trudged up three flights of stairs. Past the dull eyes of the deactivated cameras and through the invisible\u2014and inoperative\u2014infrared beams. I picked the lock of the office door with two minutes to spare. I blanked the windows, turned on the lights\u2014then headed for the bar.\n\nCold beer had never tasted better. The first one never even touched the sides of my throat and sizzled when it hit my stomach. I sipped the second as I tore the tab on a dinpac of barbecued ribs of porcuswine. As soon as the steam whistled through the venthole I ripped open the lid of the stretched pack and pulled out a rib the length of my arm. Yum!\n\nShowered, dipilated and wrapped around a third beer, I began to feel much better. \"On.\" I told the terminal, then punched into the comnet. My instructions were simple; all newspaper records on the planet for the last fifty years, all references to a criminal named The Bishop, check for redundancies around the same date and don't give me any duplicates. Print.\n\nBefore I had picked up my beer again the first sheets were sliding out of the fax. The top sheet was the most recent\u2014and it was ten years old. A not too interesting item from a city on the other side of the planet, Decalogg. The police had picked up an elderly citizen in a low bar who claimed that he was The Bishop. However it had turned out to be a case of senile dementia and the suspect had been ushered back to the retirement home from which he had taken a walk. I picked up the next item.\n\nI tired towards morning and took a nap in the filing cabinet, which turned into a bed when ordered to do so. In the gray light of dawn, helped by a large black coffee, I finished placing the last sheet into the pattern that spread across the floor. Rosy sunlight washed across it. I turned off the lights and tapped the stylo against my teeth while I studied the pattern.\n\nInteresting. A criminal who brags about his crimes. Who leaves a little drawing of a bishop behind after scarpering with his loot. A simple design\u2014easy enough to copy. Which I did. I held it out at arm's length and admired it.\n\nThe first bishop had been found in the empty till of an automated liquor store sixty-eight years ago. If The Bishop had started his career of crime as a teenager, as I have done, that would put him in his eighties now. A comfortable age to be, since life expectancy has now been pushed up to a century and a half. But what had happened to him to explain the long silence? Over fifteen years had passed since he had left his last calling card. I numbered off the possibilities on my fingers.\n\n\"Number one, and a chance always to be considered, is that he has snuffed it. In which case I can do nothing so let us forget about that.\n\n\"Two\u2014he could have gone offplanet and be pursuing his life of crime among the stars. If so, forget it like number one. I need a lot more golden bucks, and experience, before I try my hand on other worlds.\n\n\"Three, he has gone into retirement to spend his ill-gotten gains\u2014in which case more power to him. Or four, he has changed rackets and stopped leaving his spoor at every job.\"\n\nI sat back smugly and sipped the coffee. If it were three or four I had a chance of finding him. He had certainly had a busy career before the years of silence; I looked at the list with appreciation. Plane theft, car theft, cash theft, bank emptying. And more and more. All of the crimes involving moving bucks from someone else's pockets to his pockets. Or real property that could be sold quickly, with forged identification, for more bucks. And he had never been caught, that was the best part of it. Here was the man who could be my mentor, my tutor, my university of crime\u2014who would one day issue a diploma of deviltry that would eventually admit me to the golden acres I so coveted.\n\nBut how could I find him if the united police forces of an entire world, over a period of decades, had never been able to lay a finger on him? An interesting question.\n\nSo interesting that I could see no easy answer. I decided to let my subconscious work on this problem for a bit, so I pushed some synapses aside and let the whole thing slip down into my cerebellum. The street outside was beginning to fill up with shoppers and I thought that might be a good idea for me as well. All the rations I had here were either frozen or packaged, and after the sludgy prison food I felt the urge for things that crackled and crunched. I opened the makeup cabinet and began to prepare my public persona.\n\nAdults don't realize\u2014or remember\u2014how hard it is to be a teenager. They forget that this is the halfway house of maturity. The untroubled joys of childhood are behind one, the mature satisfactions of adulthood still ahead. Aside from the rush of blood to the head, as well as other places, when thoughts of the opposite sex intrude, there are real difficulties. The hapless teenager is expected to act like an adult\u2014yet has none of the privileges of that exalted state. For my part I had escaped the tedious tyranny of teendom by skipping over it completely. When not lolling about in school or trading lies with my age group, I became an adult. Since I was far more intelligent than most of them\u2014or at least I thought that I was\u2014adults that is, I had only to assume the physical role.\n\nFirst an application of crowsfooter around my eyes and on my forehead. As soon as this colorless liquid was applied wrinkles appeared and the calendar of my age rushed forward a number of years. A few wattles under my chin blended in well with the wrinkles, while the final touch was a nasty little moustache. When I pulled on my shapeless under-office clerk jacket, my own mother would not have recognized me if she had passed me in the street. In fact this had happened about a year ago and I had asked her the time and even then no spark of recognition had brought a glint to her bovine eyes. Taking an umbrella from the closet, since there was absolutely no possibility of rain, I stepped from the office and proceeded to the nearest shopping mall.\n\nI must say, my subconscious was really working fast this day, as I shortly found out. Even after all the beers I still had a thirst. That dry stay in the barn had left its mark. Therefore I turned smartly under the platinum arches of Macswineys and marched up to the serving robot that was built into the counter. The plastic head had a permanent grin painted on it and the voice was syrupy and sexy.\n\n\"How can I be of service, sir or madam?\" They could have spent a few bucks on a sex-recognition program I thought as I scanned the list of TUMCHILLER YUMMY DRINKS on the wall.\n\n\"Let me have a double-cherry oozer with lots of ice.\"\n\n\"On the way, sir or madam. That will be three bucks, if you please.\"\n\nI dropped the coins in the hopper and the serving hatch flipped open and my drink appeared. While I reached for it I had to listen to a robotic sales pitch.\n\n\"Macswineys is happy to serve you today. With the drink of your choice I am sure you would like a barbecued porcuswineburger with yummy top secret sauce garnished with sugar-fried spamyams....\"\n\nThe voice faded away from my attention as my subconscious heaved up the answer to my little problem. A really simple and obvious answer that was transparent in its clarity, pristine pure and simple....\n\n\"Come on, buster. Order or split, you can't stand there all day.\"\n\nThe voice graveled in my ear and I muttered some excuse and shuffled off to the nearest booth and dropped into it. I knew now what had to be done.\n\nSimply stand the problem on its head. Instead of me looking for The Bishop I would have to make him look for me.\n\nI drank my drink until my sinuses hurt, staring unseeingly into space as the pieces of the plan clicked into place. There was absolutely no chance of my finding The Bishop on my own\u2014it would be foolish to even waste my time trying. So what I had to do was commit a crime so outrageous and munificent that it would be on all the news channels right around the planet. It had to be so exotic that not a person alive with the ability to read\u2014or with a single finger left to punch in a news channel\u2014would be unaware of it. The entire world would know what had happened. And they would know as well that The Bishop had done it because I would leave his calling card on the spot.\n\nThe last traces of drink slurped up my straw and my eyes unfocused and I slowly returned to the garish reality of Macswineys. And before my eyes was a poster. I had been staring at it, without seeing it, for some time. Now it registered. Laughing clowns and screaming children. All rapt with joy in slightly faulty 3D. While above their heads the simple message was spelled out in glowing letters.\n\nSAVE YOUR COUPONS!!\n\nGET THEM WITH EVERY PURCHASE!!\n\nFREE ADMISSION TO LOONA PARK!!!\n\nI had visited this site of plastic joys some years before\u2014and had disliked it even as a child. Horrifying rides that frightened only the simple. Rotating up-and-down rides only for the strong of stomach; round-and-round and throw up. Junk food, sweet candy, drunk clowns, all the heady joy to please the very easily pleasable. Thousands attended Loona Park every day and more thousands flooded in on weekends\u2014bringing even more thousands of bucks with them.\n\nBucks galore! All I had to do was clean them out\u2014in such a very interesting way that it would make the top news story right around the planet.\n\nBut how would I do it? By going there, of course, and taking a good hard look at their security arrangements. It was about time that I had a day off.\nChapter 7\n\nFor this little reconnaissance trip it would be far wiser for me to act my age\u2014or less. With all the makeup removed I was a fresh-faced seventeen again. I should be able to improve upon that; after all I had taken an expensive correspondence course in theatrical makeup. Pads in my cheeks made me look more cherubic, particularly when touched up with a bit of rouge. I put on a pair of sunglasses decorated with plastic flowers\u2014that squirted water when I pressed the bulb in my pocket. A laugh a second! Styles in dress had changed, which meant that plus-fours for boys had gone out of fashion, thank goodness, but shorts were back. Or rather a reprehensible style called short-longs, which had one leg cut above the knee, the other below. I had purchased a pair of these done in repulsive purple corduroy tastefully decorated with shocking-pink patches. I could scarcely dare look at myself in the mirror. What looked back at me I hesitate to describe, except that it looked very little like an escaped bank robber. Around my neck I slung a cheap disposable camera that was anything but cheap, disposable\u2014or only a camera.\n\nAt the station I found myself lost in a sea of lookalikes as we boarded the Loona Special. Screaming and laughing hysterically and spraying each other with our plastic flowers helped to while away the time. Or stretch it to eternity in at least one case. When the doors finally opened I let the multichrome crowd thunder out, then strolled wearily after them. Now to work.\n\nGo where the money was. My memories of my first visit were most dim\u2014thank goodness!\u2014but I did remember that one paid for the various rides and diversions by inserting plastic tokens. My father had furnished a limited and begrudging number of these, which had been used up within minutes, and of course no more had been forthcoming. My first assignment then was to find the font of these tokens.\n\nEasily enough done, for this building was the target of every prepubescent visitor. It was a pointed structure like an inverted ice-cream cone, bedecked with flags and mechanical clowns, topped with a golden calliope that played ear-destroying music. Surrounding it at ground level and fixed to its base was a ring of plastic clown torsos, rocking and laughing and grimacing. Repellent as they were, they provided the vital function of separating the customers from their money. Eager juvenile hands pushed buck bills into the grasping palms of the plastic punchinellos. The hand would close, the money vanish\u2014and from the clown's mouth a torrent of plastic tokens would be vomited into the waiting receptacle. Disgusting\u2014but I was obviously the only one who thought so.\n\nThe money went into the building. Now I must find where it came out. I strolled about the base and discovered that the regurgitating dispensers did not quite girdle it. To the rear, behind the concealment of trees and shrubs, a small building snuggled up to the base. I pushed my way under the shrubs and found myself facing a private policeman stationed beside an unmarked door.\n\n\"Get lost, kid,\" he said sweetly. \"Employees only.\"\n\nI dodged around him and pushed against the door\u2014and managed to photograph it at the same time. \"I gotta go to the bathroom,\" I said crossleggedly. \"They said the bathroom was here.\"\n\nA hard hand pulled me away and propelled me back to the shrubbery. \"Not here. Out. Back the way you came.\"\n\nI went. Very interesting. No electronic alarms and the lock was a Glubb\u2014reliable but old. I was beginning to like Loona Park after all.\n\nIt was an excruciating wait until dark when the park closed down. Out of boredom I sampled the Glacier Ride where one hurtled through mock ice caverns with Things frozen into the ice on all sides\u2014though they occasionally lunged out at the screeching riders. Rocjet Rovers was equally bad, and in the name of good taste I will draw the curtain down over the heady joys of Candyland and the Swamp Monster. Suffice to say that the time did arrive at last. The token dispenser closed down an hour before the park shut. From a nearby vantage point I watched with avid interest as an armored van took away a great number of solid containers. Even more interesting was the fact that when the money went\u2014so did the security. I imagine that the logic behind this was that no one in their right mind would want to break in and steal the tokens.\n\nSo I wasn't in my right mind. As darkness fell I joined the exhausted celebrants as they staggered towards the exits. Except that I didn't get that far. A locked door at the rear of Vampire Mountain unlocked easily under my gentle ministrations. I slipped into the darkness of the service area. High above me, pale fangs shone and fake blood dripped; I felt very comfortable indeed tucked in behind a coffin filled with dirt.\n\nI let an hour go by, no more. This should clear the employees out of the way but still leave enough revelers in the streets outside the park so that my disgusting outfit would not be noticed when I finally made my exit.\n\nThere were guards about, but they were easily avoided. As I had expected, the Glubb opened easily and I slipped quickly inside. The room proved to be windowless, which was fine since my light would then not be seen. I switched it on and admired the machinery.\n\nA simple and clean design\u2014I appreciate that in machinery. The dispensers were ringed about the walls. Silent now, but still obvious in their operation. When coins or bills were inserted they were counted and passed on. Machines above released the measured amount of tokens into the delivery chutes. Beside them pipes sprang out of the floor and terminated in a bin above\u2014undoubtedly they were filled from underground conveyors that returned the tokens ready for redispensing. The bucks, untouched by human hands, were being conveyed through sealed and transparent tubes to the collection station, where the coins fell into locked boxes. They were not for me since they were too bulky to move easily. But, ahh, the bills, they were far lighter and worth far more. They slipped along the chutes until they dropped gracefully through an opening in the top of a safe. An operation that appeared to be relatively secure from light-fingered employees.\n\nWonderful. I admired the machinery and thought about it, then made notes. The dispensers had been manufactured by a firm by the name of Ex-changers, and I took pix of their trademark on the machines. The safe was a secure and reliable brand that easily yielded to my ministrations. It was empty of course, but I had expected that. I made a note of the combination, then opened and closed it a number of times until I could do it with my eyes closed. A plan was taking shape in my head and this was to be an integral part of it.\n\nFinished at last, I slipped from the building without being seen, and with little more effort escaped from the park to join the frolicking throngs. They were less boisterous on the return journey and I only had to use my spray-glasses twice. I cannot describe the relief I felt when I finally staggered through the office door, stripped off the outlandish garb, then buried my nose in a beer. Then, metaphorically of course, I put my thinking cap on.\n\nThe next weeks were busy ones. While I worked at the equipment I needed for my operation I followed the news accounts closely. One of the prison escapers, after a fierce struggle, had been recaptured. His companion had not been found, despite the help rendered by the one who had been caught. Poor Stinger; life wouldn't be the same for him once the will to fight was taken from him. However life would still be the same for the man he had planned to kill so I did not feel too sorry for Stinger. And I had work to do. Two things to do in tandem; plan the robbery\u2014and lay the trap for The Bishop. I am proud to say that I accomplished both with some ease. After this I waited until there was a dark and stormy night to visit Loona Park again. I was in and out as quickly as I could, which was some hours since there was a good deal of work to be done.\n\nAfter that it was only a matter of waiting for the right time. A weekend would be best, with the tills overflowing. As part of my plans I had rented the garage quite legally, but had stolen the small van most illegally. I used the waiting time to repaint it\u2014a better job than the original if I must say so\u2014to add new identification numbers, and to fix nameplates to the doors. At last Saturday came and I had to work hard to control my impatience. To pass the time, moustached and crowfooted, I enjoyed a good and leisurely lunch, for I had to wait until late afternoon, when the coffers would be full. The drive into the countryside was pleasant, and I reached my appointed spot at the appointed time. Close to the service entrance to the park. I had some apprehension as I pulled on the skintight and transparent gloves\u2014but the feeling of anticipation was far greater. With a smile on my lips I reached out and switched on the apparatus fixed underneath the dash before me.\n\nAn invisible radio signal winged out and I tried to visualize with my mind's eye what happened next. Fast as light to the receiver, down the wires to its target\u2014which was a tiny charge of explosive. Not much, just a carefully measured amount that would destroy the latch on one of the token dispensers without rupturing the tube at the same time. With the latch destroyed, a steady stream of colored plastic wafers should now be rattling down into the dispenser, filling it and flowing over\u2014gushing on in a never-ending stream. What a benefactor I was! How the children would bless me had they but known my identity.\n\nBut that wasn't all that was going to happen. For every minute now another radio signal pulsed out of my transmitter, another latch was destroyed, another gusher of tokens spouting forth each time that this happened. Another and then another. At the proper moment I started the van's engine and drove to the service gate of Loona Park, opened the window, and leaned out above the sign on the door that read EX-CHANGERS DISPENSING MACHINES.\n\n\"Got a radio call,\" I said to the guard there. \"You got some kind of problem here?\"\n\n\"No problem,\" the guard said, heaving the gate open. \"More like a riot. You know where the building is?\"\n\n\"Sure do. Help is on the way!\"\n\nThough I had visualized the effects of my unexpected largesse, I quickly discovered that reality far surpassed my wildest expectations. Screaming, cheering kiddies rushed about laden with tokens, while others fought for places around the gushing dispensers. Their happy cries were ear-splitting and the attendants and guards could do nothing to stop their wave of exuberance. It was slightly less crowded on the service road, but I still had to drive slowly, hand on the horn, to make my way through the stragglers. Two guards were pushing kids back through the shrubs when I drove up.\n\n\"Got some trouble with the dispensers?\" I asked sweetly. The guard's snarled response was lost in childish cries of delight, which was probably all for the best. He unlocked the door and all but pushed me and my toolbox through.\n\nThere were four people there, struggling ineffectually with the machines. They could not be cut off since I had taken the liberty of shorting the switchbox. A bald-headed man was working on an armored cable with a hacksaw and I made tsk-tsking sounds. \"That is a recipe for suicide,\" I said. \"You got a four-hundred-volt line in there.\"\n\n\"Can you do anything better, buster,\" he snarled. \"They're your damned machines. Go to work.\"\n\n\"I shall\u2014and here is the cure.\"\n\nI opened the sizable toolbox, which contained only a shining metal tube, and took it out. \"This will do the job,\" I said, turning the valve at the top and hurling it from me. The last thing I saw were their eye-popping expressions as the black smoke billowed out and filled the room\u2014blocking out all vision completely.\n\nI had been expecting it; they had not. The toolbox was in my arms as I took four measured paces in the darkness and fetched up against the side of the safe. Any noises I made were drowned by their shouts and screams and the constant chugging of the token dispensers. The safe opened easily, the lid of the toolbox fit neatly against the lower edge. I leaned in, felt the mounds of bills, then swept them forward into the waiting container. It was quickly filled and I snapped it shut. My next task was to make sure that the right person took responsibility for this crime. The card with its inscription was in my top pocket. I slipped it out and laid it carefully in the safe, which I then locked again to make absolutely sure that my message would be received and not lost in all the excitement. Only then did I pick up the now-heavy toolbox and stand with my back to the safe, turning and orientating myself.\n\nI knew that the exit was there in the darkness, nine easy paces away. I had taken five when I bumped into someone and strong hands grabbed me while a hoarse voice shouted in my ear.\n\n\"I got him! Help me!\"\n\nI dropped the box and gave him exactly the help he needed, running my hands up his body to his neck and doing all the right things there. He grunted and slid away. I groped for the box\u2014for a panicky moment I couldn't find it. Then I did, clutched the handle, and seized it up and stood....\n\nAnd realized I had lost all sense of direction during the fracas.\n\nMy panic was as dark as the smoke, and I shook so hard that I almost dropped the case. Seventeen years old and very much alone\u2014with the unknown world of adults closing in upon me. It was over, all over.\n\nI don't know how long this crisis lasted, probably only seconds, although it seemed infinitely longer than that. Then I grabbed myself by the metaphorical neck and shook myself quite hard.\n\n\"You wanted it this way\u2014remember? Alone with everyone's hand turned against you. So give in to them\u2014or start thinking. Fast!\"\n\nI thought. The people screaming and banging all about me were no help or threat\u2014they were as confused as I was. All right, hand outstretched, go forward. Any direction. Reach something that could be identified by touch. Once this was done I should be able to work out where I was. I heard a thudding ahead, it had to be one of the dispensers, then I bumped into it.\n\nWhile at the same moment a draft of air touched my face and a familiar voice called out close by.\n\n\"What's going on in here?\"\n\nThe guard! And he had opened the door. How very nice of him. I moved along the wall, avoiding him easily since he was still shouting in the darkness, then followed the billowing smoke out into the light of day. Blinking at the brightness and at the other guard who was stationed just before me, grabbing me.\n\n\"Just hold it right there. You ain't going nowhere.\"\n\nHe could not have been much wronger, I mean grabbing onto a Black Belt like that. I eased him to the ground so he wouldn't hurt himself when he fell, threw the box into the van, looked around to see that I was totally unobserved, closed the door, started the engine, then drove slowly and carefully away from fun-filled Loona Park.\nChapter 8\n\n\"All fixed, everything just fine back there,\" I called out to the guard and he nodded while he pulled the gate open. I drove off in the direction of the city, slowly around a bend\u2014then turned sharply inland on an unpaved road.\n\nMy escape had been as carefully planned as the theft. Stealing money is one thing; keeping it is another altogether. In this age of electronic communication a description of me and the van would be flashed around the planet in microseconds. Every police car would have a printout and every patrolman verbal warning. So how much time did I have? Both guards were unconscious. But they could be revived, could pass on the information, a phone call would be made, warning given. I calculated that this would take at least five minutes. Which was fine since I only needed three.\n\nThe road wound up through the trees, made a final turn\u2014and ended in the abandoned quarry. My heart was thudding a bit since I had to take one chance in this operation. And it had worked\u2014the rental car was still here, just where I had left it the day before! Of course I had removed some vital parts from the engine, but a determined thief could have towed it away. Thank goodness that there was only one determined thief around.\n\nI unlocked the car and took out the box of groceries, then carried it to the van. The side of the box swung down\u2014revealing an empty box. The protruding tops of packets and containers, just the glued-together tops of packets and containers. Very ingenious, if I say so myself. Which I have to, since no one else knows about this operation. Money into box, close box, put in car. Take off work clothes, shiver in the cool breeze as I throw them into the truck. Along with moustache. Pull on sports outfit, actuate timer on thermite charges, lock van, get in car. Simply drive away. I had not been observed so there was no reason at all now why I shouldn't get away with my little adventure. I stopped at the main road and waited for a clutch of police cars to go roaring by in the direction of Loona Park. My, but they were in a hurry. I turned onto the road and drove slowly and carefully back to Billville.\n\nBy this time the van would be burning merrily and melting down to a pool of slag. No clues there! The van was insured by law so the owner would be reimbursed. The fire would not spread\u2014not from the heart of the stone quarry\u2014and no one had been injured. It had all worked well, very well.\n\nBack in the office I heaved a sigh of relief, opened a beer and drank deep\u2014then took the bottle of whiskey from the bar and poured a stiff shot. I sipped it, wrinkled my nose at the awful flavor, then poured the rest of the drink into the sink. What filthy stuff. I suppose if I kept trying I would get used to it someday, but it scarcely seemed worth the effort.\n\nBy now enough time should surely have elapsed for the press to have reached the scene of the crime. \"On,\" I called out to my computer, then, \"Print the latest edition of the newspaper.\"\n\nThe fax hummed silkily and the paper slid into the tray. With a color pic of the money fountain operating at full blast on the front page. I read the report with a glow of pleasure, turned the page, and saw the drawing. There it was, just as they had found it when they had finally opened the safe. A drawing of a bishop with a line of chess notation written below it.\n\nR\u2014Kt 4 X B\n\nWhich means in chess notation Rook to square Knight 4 takes Bishop.\n\nWhen I read it the warm glow of pleasure was replaced by a chill of worry. Had I given myself away to the police? Would they analyze the clue and be waiting for me?\n\n\"No!\" I cried aloud. \"The police are lazy and relaxed with little crime to keep them on their toes. They may puzzle over it\u2014but they will never understand it until it is too late. But The Bishop should be able to work it out. He will know that it is a message for him and will labor over it. I hope.\"\n\nI sipped at my beer and had a good worry. It had taken me tedious hours to work out this little mind-twister. The fact that The Bishop used a chess bishop as his calling card had led me to the chess books. I assumed that he\u2014or she, I don't believe that anyone had ever determined The Bishop's sex, although it was assumed the criminal was male\u2014cared about chess. If more knowledge was needed he could consult the same books that I had. With very little effort it could be discovered that there are two different ways of noting chess moves. The oldest of them, the one that I had used, named the squares of the file after the piece that sat at the end of the file. (If you must know, \"ranks\" are the rows of squares that stretch from side to side of a chessboard. \"Files\" are the rows that stretch between the players.) So the square on which the White King sits is King 1. King 2 would be the next one up. If you think that this is complicated don't play chess\u2014because this is the easiest part! However there is a second form of chess notation that assigns a number to each of the 64 squares on the board. So Knight 4 can be either 21, 8, 22, or 45.\n\nConfusing? I hope so. I hope the police never think it is a code and get around to cracking it. Because if they do I am cracked as well. This little bit of chess movement contains the date of my next crime, when I am going to \"take bishop,\" meaning take The Bishop card to a crime. Meaning also I am going to take credit for being The Bishop. Also meaning I am taking The Bishop to the cleaners.\n\nI have the scenario clear in my mind. The police puzzle over the chess move\u2014then discard it. Not so The Bishop in his luxury hideout. He is going to be angry. A crime has been committed and he has been blamed. Money has been taken\u2014and he doesn't have it! My hope is that he will worry over this chess move, see it as a clue, scribble away at it, and eventually solve it.\n\nBy thinking about the fact that Knight is a homonym for night. Night three\u2014what can that mean? The third night of what? The third night of the Modern Music Festival in the city of Pearly Gates, that is what. And this third night is also the forty-fifth day of the year, which is\u2014that's correct\u2014also known as Knight 3 in one of its four permutations. With this added verification The Bishop would be sure that some crime would take place on the third night of the Festival. A crime involving money of course. My mental fingers were crossed in the hope that he would be more interested in me than in informing the police in advance about the crime.\n\nI hoped that I had struck the right balance. Too complex for the police, but capable of solution by The Bishop. And he had exactly one week to solve it and come to the Festival.\n\nWhich also meant that I had one week to hype myself up and depress myself down, get too much sleep\u2014then not enough sleep. And take pleasure only in the construction of plans and apparatus for this bold foray into the pockets of the public.\n\nOn the night in question it was raining heavily\u2014which suited me perfectly. I turned up the collar of my black coat, jammed my black hat down on my head, then seized up the black case that held the musical instrument. A horn of some kind. This was made obvious by the swollen shape at one end, where the case swelled out to accommodate the bell. It might be a crumpaphone or even a dagennet. Public transportation took me close to the stage entrance to the theater. As I walked the rest of the way I soon found myself braving the elements among other black-garbed, instrument-bearing musicians. I had my pass ready, but the doorman just waved us through and out of the rain. There was little chance that anyone would question my identity because I was only one of 230. For tonight was the premier of what was sure to be a head-destroying piece of so-called music modestly entitled Collision of Galaxies, scored for 201 brass instruments and 29 percussion. The composer, Moi-Woofter Geeyoh, was not known for the delicate dissonances of her compositions. The choice of this piece of music had also made this the night of my choice; even reading the score gave one a headache.\n\nThere was a shortage of dressing rooms for the musical multitude and they were milling about all over the place emitting lost noises. No one noticed when I slipped away, drifted up a back staircase\u2014and let myself into a janitorial broom closet. The service staff had long departed so I would not be disturbed\u2014other than by the music. Nevertheless I locked the door from the inside. When I heard the sounds of tuning up I took out my copy of the score of Collision.\n\nIt started out calmly enough\u2014after all, the galaxies had to get on stage before they could collide. I followed the score with my finger until it reached the red mark I had placed there. The score folded neatly into my pocket as I carefully unsealed the door and looked out. Corridor empty, as it should be. With steady tread I walked down the corridor, the floor of which was already beginning to throb with impending galactic destruction.\n\nThe door was labeled PRIVATE\u2014KEEP OUT. I took the black mask from one pocket, removed my hat and pulled the mask on, extracted the key to the door from another. I did not want to waste time with lockpicks, so had made this key when I had scouted this location. I hummed along with the music\u2014if that could be said to be possible\u2014with the key in the lock. At the correct destructive crash I opened the door and stepped into the office.\n\nMy entrance had to of course been unheard, but my movements caught the older man's eye. He turned and stared, and the pen he had been using dropped from his limp fingers. His hands reached towards the ceiling when I drew the impressive\u2014and fake\u2014gun from my inside pocket. The other and younger man could not be threatened and dived to the attack. And continued to dive unconscious to the floor, knocking over and breaking a chair on the way.\n\nNone of this made a sound. Or rather it made a lot of sound, none of which could be heard over the music that was now rapidly working itself up to a crescendo that would drown out the crack of doom. I moved fast because the really loud parts were coming close.\n\nI took two pairs of handcuffs from a coat pocket and locked the older man's ankle to his desk, then pulled his arms down before they got tired. I next secured the sleeping dreamer the same way. Almost time. I took the plastic explosive from another pocket\u2014yes, there were a lot of pockets in this garment, and not by chance either\u2014and slapped it to the front of the safe. Right over the time lock. They must have felt very secure here with their careful arrangements. All the night's ample receipts had been locked away in the safe in the presence of armed guards. To remain locked and secure until the morning when other armed guards would be present when it opened. I pushed the radio fuse into the explosive, then retreated across the room until I was out of the line of fire along with the others.\n\nEvery loose object in the room was bouncing in time with the music now while dust rained down from the ceiling. It still wasn't time. I used the opportunity to rip out the phones by their roots. Not that anyone would be talking on a phone until after the concert.\n\nThere it was\u2014almost there! I had the musical score in my mind's eye and at the instant when the galaxies finally impacted I pressed the radio actuator.\n\nThe front of the safe blew off in silent motion. I was stunned by the musical catastrophe way up here in the office\u2014not by the explosion\u2014and I wondered how many of the audience had gone deaf in the name of art. My wondering didn't stop me from shoveling all the buck bills from the safe into my instrument case. When it was filled I tipped my hat to my prisoners, one wide-eyed, one unconscious, and let myself out. The black mask went back into its pocket and I went out of the theater by an unwatched emergency exit.\n\nIt was a brisk two-block walk to the underpass entrance and I was just one other figure hurrying through the rain. Down the steps and along the corridor, to take the turning that led to the station. The commuter trains had left and the corridor was deserted. I stepped into the phone booth and made my unobserved identity change in exactly twenty-two seconds, precisely the rehearsed time. The black covering of the case stripped away to reveal the white covering of the case inside. The flared bell-shape went too. That had been shaped from thin plastic that crunched and went into a pocket with the black cover. My hat turned inside out and became white, my black moustache and beard disappeared into their appointed pocket so that I could shed the coat and turn it inside out so that it too, that's right, became white. Thus garbed, I strolled into the station and out the exit along with the other arriving passengers, to the cab rank. It was a short wait; the cab rolled up and the door opened. I climbed in and smiled appreciatively at the shining skull of the robot driver.\n\n\"Mah good man, tay-ake me to thu Arbolast Hotel,\" I said in my best imitation Thuringian accent\u2014since the Thuringar train had arrived at the same time I had.\n\n\"Message not understood,\" the thing intoned.\n\n\"Ar-bo-last Ho-tel, you metallic moron!\" I shouted. \"Ar-bowb-bo-last!\"\n\n\"Understood,\" it said, and the cab started forward.\n\nJust perfect. All conversations were stored in a molecular recorder for one month in these cabs. If I were ever checked on, the record would reveal this conversation. And my hotel reservation had been made from a terminal in Thuringia. Perhaps I was being too cautious\u2014but my motto was that this was an impossibility. Being too cautious, I mean.\n\nThe hotel was an expensive one and tastefully decorated with mock arbolasts in every corridor and room. I was obsequiously guided to mine\u2014where the arbolast served as a floor lamp\u2014and the robot porter glided away smarmily with a five-buck coin in his tip slot.\n\nI put the bag in the bedroom, took off the wet coat, extracted a beer from the cooler\u2014and there was a knock on the door.\n\nSo soon! If that was The Bishop he was a good tail, because I had not been aware of being followed. But who else could it be? I hesitated, then realized that there was one certain way to find out. With smile on face, in case it was The Bishop, I opened the door. The smile vanished instantly.\n\n\"You are under the arrest,\" the plainclothes detective said, holding out his jeweled badge. His companion pointed a large gun at me just to make sure that I understood.\nChapter 9\n\n\"What... what...\" I said, or something very like this. The arresting officer was not impressed by my ready wit.\n\n\"Put on your coat. You are coming with us.\"\n\nIn a daze I stumbled across the room and did just as he commanded. I should leave the coat here, I knew that, but I had no will to resist. When they searched it they would find the mask and key, everything else that would betray me. And what about the money? They hadn't mentioned the bag.\n\nAs soon as my arm was through the sleeve the policeman snapped a handcuff on my wrist and clicked the other end to his own wrist. I was going nowhere without them. There was little or nothing I could do\u2014not with the gun wielder three steps behind us.\n\nOut the door we went and along the corridor, to the elevator, then down to the lobby. At least the detective had the courtesy to stand close to me so the handcuffs were not obvious. A large black and ominous groundcar was parked in the middle of the no-parking zone. The driver didn't even bother to glance in our direction. Though as soon as we had climbed in and the door closed, he pulled away.\n\nI could think of nothing to say\u2014nor were my companions in a conversational mood. In silence we rolled through the rainy streets, past police headquarters, which was unexpected, to stop before the Bit O' Heaven Federal Building. The Feds! My heart dropped. I had been correct in assuming that breaking the clues and catching me had certainly been beyond the intelligence of the local police. But I had not reckoned upon the planetary investigation agencies. By hindsight\u2014which is not very satisfying\u2014I saw my error. After years of absence The Bishop strikes again. Why? And what does the bit of chesswackery mean? Put the cryptologists on it. Oho, a bit of bragging, scene and date of the next crime revealed. Keep it Federal and out of the hands of the local and incompetent police. Watch the cash with the most modern of electronic surveillance techniques. Track the criminal to see if others are involved. Then pounce.\n\nMy state of black depression was so great that I could scarcely walk. I swayed when our little procession stopped before a heavy door labeled FEDERAL BUREAU OF INVESTIGATION, with DIRECTOR FLYNN in smaller gold letters beneath it. My captors knocked politely and the doorlock buzzed and opened. We filed in.\n\n\"Here he is, sir.\"\n\n\"Fine. Secure him to the chair and I'll take over from here on out.\"\n\nThe speaker sat massively behind the massive desk. A big man with sleek black hair, who was made even bigger by the enormous quantity of fat that he was carrying around. His chin, or chins, hung down onto the swelling volume of his chest. The size of his stomach kept him well back from the desk, upon which the fingers of his clasped hands rested like a bundle of stout sausages. He returned my shifty gaze with his steady and steely one. I made no protest as I was guided to the chair, dropped into it, felt the handcuffs being secured to it, heard footsteps recede and the door slam.\n\n\"You are in very big trouble,\" he intoned.\n\n\"I don't know what you mean,\" I said, the impact of my innocence lessened by the squeak and tremor of my voice.\n\n\"You know full well what I mean. You have committed the crime of theft tonight, purloining the public purse donated by stone-deaf music lovers. But that is the least of your folly, young man. By your age I can tell that you have also purloined the good name of another. The Bishop. You are pretending to be something that you are not. Here, take these.\"\n\nPurloined a good name? What in the galaxy was he talking about? I snatched the keys out of the air by reflex. Gaped at them\u2014then gaped even more broadly at him as I tremblingly unlocked the cuffs.\n\n\"You are not...\" I gurgled. \"I mean, the arrest, this office, the police... You are...\"\n\nHe calmly waited for my next words, a beatific smile on his face.\n\n\"You are... The Bishop!\"\n\n\"The same. My understanding of the message concealed by your feeble code was that you wanted to meet me. Why?\"\n\nI started to rise and an immense gun appeared in his hand, aimed between my eyes. I dropped back into the chair. The smile was gone, as was all warmth from his voice.\n\n\"I don't like to be imitated, nor do I like to be played with. I am displeased. You now have three minutes to explain this matter before I kill you, then proceed to your hotel room to retrieve the money you stole this evening. Now the first thing that you will reveal is the location of the rest of the money stolen in my name. Speak!\"\n\nI spoke\u2014or rather I tried to speak but could only sputter helplessly. This had a sobering affect. He might kill me\u2014but he was not going to reduce me to helpless jelly first. I coughed to clear my throat, then spoke.\n\n\"I don't think that you are in too much of a hurry to kill me\u2014nor do I believe in your three-minute time limit. If you will cease in your attempts to bully me I shall try to tell you carefully and clearly my motives in this matter. Agreed?\"\n\nSpeaking like this was a calculated risk\u2014but The Bishop was a game player, I knew that now. His expression did not change, but he nodded slightly as though conceding a Pawn move\u2014knowing that he still had my King well in check.\n\n\"Thank you. I never thought of you as a cruel man. In fact, when I discovered your existence, I used you as a career model. What you have done, what you have accomplished, is without equal in the history of this world. If I offended you by stealing money in your name I am sorry. I will turn all the money from that robbery over to you at once. But if you will stop to think\u2014it is the only thing that I could do. I had no way of finding you. So I had to arrange things so that you could find me. As you have. I counted upon your curiosity\u2014if not your mercy\u2014not to reveal my identity to the police before you had met me yourself.\"\n\nAnother nod granted me another Pawn move. The unwavering barrel of the gun informed me that I was still in check.\n\n\"You are the only person alive who knows my identity,\" he said. \"You will now tell me why I should not kill you. Why did you want to contact me?\"\n\n\"I told you\u2014out of admiration. I have decided on a life of crime as the only career open to one of my talents. But I am self-trained and vulnerable. It is my wish to be your acolyte. To study at your knee. To enter the academy of advanced crime in the wilderness of life with you on one end of the log and me on the other. I will pay whatever price you require for this privilege, though I may need a little time to raise more money since I am turning the receipts of my last two operations over to you. There it is. That is who I am. And, if I work hard enough, you are whom I wish to be.\"\n\nThe softening gaze, the thoughtful fingers raised to chin meant I was out of check for the moment. But the game wasn't won yet\u2014nor did I wish it to be. I wanted only a draw.\n\n\"Why should I believe a word of this?\" he asked at last.\n\n\"Why should you doubt it? What other possible reason could I have?\"\n\n\"It is not your motives that disturb me. I am thinking about the possibility of someone else's, someone in a position of police responsibility who is using you as a pawn to find me. The man who arrests The Bishop will rise to the top of his chosen profession.\"\n\nI nodded agreement as I thought furiously. Then smiled and relaxed. \"Very true\u2014and that must have been the very first thing to come to your mind. Your office in this building either means that you are high in the ranks of law enforcement, so high that you could easily find out if this had been the plan. Or\u2014even more proof of your genius\u2014you have ways and means of penetrating the police at any level, to fool them and use them to actually arrest me. My congratulations, sir! I knew that you were a genius of crime\u2014but to have done this, why it borders on the fantastic!\"\n\nHe nodded his head slowly, accepting his due. Did I see the muzzle of the gun lowered ever so slightly? Was a drawn game possibly in sight? I rushed on.\n\n\"My name is James Bolivar diGriz and I was born a little over seventeen years ago in this very city in the Mother Machree Maternity Hospital for Unemployed Porcuswineherders. The terminal I see before you must access official files at every level. Bring up mine! See for yourself if what I have told you is not the truth.\"\n\nI settled back into the chair while he tapped commands on the keyboard. I did nothing to distract him or draw his attention while he read. I was still nervous but worked to affect a surface calm.\n\nThen he was done. He leaned back and looked at me calmly. I didn't see his hands move\u2014but the gun vanished from sight. Drawn game! But the pieces were still on the board and a new game was beginning.\n\n\"I believe you, Jim, and thank you for the kind words. But I work alone and wish no disciples. I was prepared to kill you to preserve the secret of my identity. Now I do not think that will be necessary. I will take your word that you will not look for me again\u2014or use my identity for any more crimes.\"\n\n\"I grant your requests instantly. I only became The Bishop to draw your attention. But reconsider, I beg of you, my application for membership in your academy of advanced crime!\"\n\n\"There is no such institution,\" he said, hauling himself to his feet. \"Applications are closed.\"\n\n\"Then let me rephrase my request,\" I said hurriedly, knowing my remaining time was brief. \"Let me be personal, if I can, and forgive any distress I may cause. I am young, not yet twenty, and you have been on this planet for over eighty years. I have been only a few years at my chosen work. And, in this brief time, I have discovered that I am truly alone. What I do I must do for myself and by myself. There is no comradeship of crime because all of the criminals I have seen are incompetents. Therefore I must go it alone. If I am lonely\u2014then dare I even guess at the loneliness of your life?\"\n\nHe stood stock-still, one hand resting on the desk, staring at the blank wall, as through a window, at something I could not see. Then he sighed, and with the sound, as though it had released some power that kept him erect, he slumped back into the chair.\n\n\"You speak the truth, my boy, and only the truth. I do not wish to discuss the matter, but your barb has been driven well home. Nevertheless what is, will be. I am too old a dog to change his ways. I bid you farewell, and thank you for a most interesting week. Been a bit like old times.\"\n\n\"Reconsider, please!\"\n\n\"I cannot.\"\n\n\"Give me your address\u2014I must send you the money.\"\n\n\"Keep it, you earned it. Though in the future earn it under a different identity. Let The Bishop enjoy his retirement. I will add only one thing, a bit of advice. Reconsider your career ambitions. Put your great talents to work in a more sociably acceptable manner. In that way you will avoid the vast loneliness you have already noted.\"\n\n\"Never!\" I cried aloud. \"Never. I would rather rot in jail for the rest of my life than accept a role in the society I have so overwhelmingly rejected.\"\n\n\"You may change your mind.\"\n\n\"There is no chance of that,\" I said to the empty room. The door had closed behind him and he was gone.\nChapter 10\n\nWell, that was that. There is nothing like an overwhelming depression to bring one down from the heights of elation. I had done exactly what I had set out to do. My complex plan had worked perfectly. I had unearthed The Bishop from his secret lair and had made him an offer he couldn't refuse.\n\nExcept he had. Even the pleasure of having pulled off the successful robbery now meant nothing. The bucks were like ashes in my hand. I sat in my room at the hotel and looked into the future and could see only a vast vacuity. I counted the money over and over until the sums were meaningless. In making my plans I had considered all of the possibilities but one\u2014that The Bishop would turn me down. It was kind of hard to take.\n\nBy the time I got back to Billville the next day I was wallowing in a dark depression and thoroughly enjoying the bath of self-pity. Which I normally cannot stand. Nor could I this time. I looked in the mirror at the hollow-eyed and woebegone face and stuck my tongue out at it.\n\n\"Sissy!\" I said. \"Momma's boy, whiner, self-indulgent wimp,\" and added whatever other insults I could think of. Having cleansed the air a bit, I made a sandwich and a pot of coffee\u2014no alcohol to clog the synapses!\u2014and sat down to munch and guzzle and think about the future. What next?\n\nNothing. At least nothing constructive that I could think of at this moment. All of my plans had ended at a blank wall and I could see no way around or over it. I slumped back and snapped my fingers at the 3V. A commercial channel came on and before I could change channels the announcer appeared in glorious three dimension and color. I didn't switch because the announcer was a she and wearing only the flimsiest of swimsuits.\n\n\"Come where the balmy breezes blow,\" she cajoled. \"Come join me on the silver sands of beautiful Vaticano Beach, where the sun and waves will refresh your soul....\"\n\nI turned the thing off. My soul was in fine shape and the fine shape of the announcer only gave me more problems to think about. Future first, heterosexual love later. But the commercial had at least given me the beginning of an idea.\n\nA holiday? Take a break? Why not\u2014lately I had been working harder than any of the businessmen I so badly did not want to become. Crime had paid, and paid nicely, so why didn't I spend some of the hard-earned loot? I probably wouldn't be able to escape from my problems. I had learned by experience that physical displacement was never a solution. My troubles always went with me, as everpresent and nagging as a toothache. But I could take them with me to someplace where I might find the leisure and opportunity to sort them out.\n\nWhere? I punched up a holiday guide from the database and flipped through it. Nothing seemed to appeal. The beach? Only if I could meet the girl from the commercial, which seemed far from likely. Posh hotels, expensive cruises, museum tours, all of them seemed about as exciting as a weekend on a porcuswine ranch. Maybe that was it\u2014I needed a breath of fresh air. As a farm boy I had seen enough of the great outdoors, usually over the top of a pile of porcuswine you-know-what. With that sort of background I had welcomed my move to the city with open arms\u2014and hadn't ventured out since.\n\nThat might be the very answer. Not back to the farm but into the wilderness. To get away from people and things, to do a little chatting with mother nature. The more I thought of it the better it sounded. And I knew just where I wanted to go, an ambition I had had since I was knee-high to a porcuswinelet. The Cathedral Mountains. Those snow-covered peaks, pointing towards the sky like giant church towers, how they used to fill my childish dreams. Well, why not? About time to make a few dreams come true.\n\nShopping for backpack, sleeping bag, thermal tent, cooking pots, lights\u2014all the gear needed\u2014was half the fun. Once outfitted, I couldn't waste time on the linear but took the plane to Rafael instead. I bulged my eyes at the mountains as we came in to land and snapped my fingers and fidgeted while I waited for the luggage. I had studied the maps and knew that the Cathedral Trail crossed the road in the foothills north of the airport. I should have taken the connecting bus like the others instead of being conspicuous in a taxi, but I was in too much of a rush.\n\n\"Pretty dangerous, kid, I mean walking the trail alone.\" The elderly driver smacked his lips as he launched into a litany of doom. \"Get lost easily enough. Get eaten by direwolves. Landslides and avalanches. And...\"\n\n\"And I'm meeting friends. Twenty of them. The Boy Sprouts Hiking Team of Lower Armmpitt. We're gonna have fun,\" I invented rapidly.\n\n\"Didn't see no Boy Sprouts out here lately,\" he muttered with senile suspicion.\n\n\"Nor would you,\" I extemporized, bent over in the backseat and flipping through the maps quickly. \"Because they took the train to Boskone, got off there, right at the station close to where the trail crosses the tracks. They'll be waiting for me, troop leader and all. I would be afraid to be alone in the mountains, sir.\"\n\nHe muttered some more, muttered even louder when I forgot to tip him, then chuckled in his gray whiskers as he drove away because, childishly, I had then overtipped him. While resisting strongly the impulse to slip him a phoney five-buck coin. The sound of the motor died away and I looked at the well-marked trail as it wound up the valley\u2014and realized that this had been a very good idea indeed.\n\nThere is no point in waxing enthusiastic about the joys of the Great Outdoors. Like skiing, you do it and enjoy it, but don't talk about it. All the usual things happened. My nose got sunburned, ants got into my bacon. The stars were incredibly clear and close at night, while the clean air did good things to my lungs. I walked and climbed, froze myself in mountain streams\u2014and managed to forget my troubles completely. They seemed very out of place in this outdoor world. Refreshed, cleansed, tired but happy, and a good deal thinner, I emerged from the mountains ten days later and stumbled through the door of the lodge where I had made reservations. The hot bath was a blessing, and the cold beer no less. I turned on the 3V and got the tail end of the news, slumped down and listened with half an ear, too lazy to change channels.\n\n\"... reports a rise in ham exports exceeding the four percent growth predicted at the first of the year. The market for porcuswine quills is slipping however, and the government is faced with a quill mountain that is already drawing criticism.\n\n\"Closer to home, the computer criminal who broke into Federal Files goes on trial tomorrow. Federal prosecutors treat this as a most serious crime and want the death penalty reestablished. However...\"\n\nHis voice faded from my attention as his smarmy face vanished from the screen to be replaced by the computer criminal himself being led away by a squad of police. He was a big man, and very fat, with a mane of white hair. I felt a clutch in my chest just near the place I imagined my heart to be. Wrong color hair\u2014but wigs would take care of that. There was no mistaking him.\n\nIt was The Bishop!\n\nI was out of the tub and across the room and hitting the frame freeze controls. It is a wonder I did not electrocute myself. Shivering with cold, and scarcely aware of it, I flipped back, then zoomed for detail. Enlarged the frame when he looked back over his shoulder for an instant. It was he\u2014without a doubt.\n\nBy the time I had wiped off the suds and dressed, the general shape of my plans was clear. I had to get back to the city, to find out what had happened to him, to see what I could do to help. I punched up flight information; there was a mail flight just after midnight. I booked a seat, had a meal and a rest, paid my bill, and was the first passenger aboard.\n\nIt was just dawn when I entered my office in Billville. While the computer was printing out all the news items on the arrest, I made a pot of coffee. Sipping and reading, my spirits sank like a rock in a pond. It was indeed the man I knew as The Bishop, although he went under the name of Bill Vathis. And he had been apprehended leaving the Federal Building, where he had installed a computer tap which he had been using to access Top Secret files. All of this had happened the day after I left on my escapist holiday.\n\nI had the sudden realization of what this meant. Guilt assailed me because I was the one who had put him into jail. If I had not started my mad plan, he would never have bothered with the Federal files. He had only done that to see if the robberies had been part of a police operation.\n\n\"I put him in jail\u2014so I will get him out!\" I shouted, leaping to my feet and spilling coffee across the floor. As I mopped it up I cooled down a bit. Yes, I would like to get him out of jail. But could I do it? Why not? I had some experience now in jail-breaking. It should be easier to get from the outside in than it had been doing it the other way. And, after further thought, I realized that perhaps I would not have to go near the jail. Let the police get him out for me. He would have to be taken to court, so would be in transit in various vehicles.\n\nI soon discovered that it was not going to be that easy. This was the first major criminal that had been caught in years and everyone was making a big fuss over it. Instead of being taken to the city or state jail, The Bishop was being held in a cell inside the Federal Building itself. I could get nowhere near it. And the security measures when he was taken to the courthouse were unbelievable. Armed vans, guards, monocycles, police hovercraft and copters. I was not going to get to him that way either. Which meant I was baffled for the moment. Interestingly enough, so were the police\u2014but for very different reasons.\n\nThey had discovered, after endless search, that the real Bill Vathis had left the planet twenty years before. All of the records of this fact had vanished from the computer files\u2014and it was only a note written by the real Vathis to a relative that had established the disappearance of the original. Well\u2014if their prisoner wasn't Vathis. Who was he?\n\nWhen their captive was questioned, according to the report released to the press, \"He answered the question only with silence and a distant smile.\" The prisoner was now referred to as Mr. X. No one knew who he was\u2014and he chose not to speak on the matter. A date was fixed for the trial, not eight days away. This was made possible by the fact that Mr. X refused to plead neither innocent nor guilty, would not defend himself\u2014and had refused the services of a state-appointed attorney. The prosecution, greedy for a conviction, stated that their case was complete and asked for an early trial. The judge, eager as well to be in the limelight, agreed to their request and the date was set for the following week.\n\nI could do nothing! Back to the wall, I admitted defeat\u2014for the moment. I would wait until after the trial. Then The Bishop would simply be one more prisoner and would have to be taken from the Federal Building at last. When he was safely in jail I would arrange his escape. Well before the arrival of the next spacer that would take him away for brain-cleansing and purifying. They would use all of the miracles of modern science to turn him into an honest citizen and, knowing him, I was sure that he would rather die than have that happen. I must intervene.\n\nBut they were not making it easy for me. I could not find a way to be in the courtroom when the trial began. So I, along with every other inhabitant of the planet as far as could be determined, watched the trial on TV when it began.\n\nAnd ended with suspicious speed. All of the first morning was taken with recitals of the well-documented account of what the defendant had done. It was pretty damning. Computer malfeasance, memory bank barratry, CPU violation, terminal treachery, dropping solder on classified documents\u2014it was terrible. Witness after witness read out their statements, all of which were instantly accepted and entered into the evidence. Through all this The Bishop neither watched nor listened. His stare was into the distance, as though he were looking at much more interesting things than the simple operation of the court. When the evidence had been given, the judge banged his gavel and ordered a break for lunch.\n\nWhen the court reconvened\u2014after a break long enough for a seventeen-course banquet with dancing girls for afters\u2014the judge was in a jovial mood. Particularly after the prosecution had done a damning summoning up. He nodded agreement most of the time and thanked all the smarmy ambulance chasers for the excellent job that they had done. Then he looked his most pontifical and spoke in pregnant periods for the records.\n\n\"This case is so clear that it is transparent. The state has brought charges so damning that no defense could possibly stand before them. That no defense was offered is even greater evidence of the truth. The truth is that the defendant did wilfully, with malice and forethought, commit all of the crimes for which he stands accused. There can be no doubt about that. The case is an open and shut one. Nevertheless I shall deliberate the rest of this day and far into the night. He will have his chance of justice that he rejected. I will not find him guilty until tomorrow morning when this court resumes. At that time I will pass sentence. Justice will be done and will be seen to be done.\"\n\nSome justice, I muttered through my teeth and started to switch off the set. But the judge wasn't through.\n\n\"I have been informed that the Galactic League is very interested in this case. A spacer has been dispatched and will be here within two days. The prisoner will then be taken from our custody and we will, if you will excuse and understand my emotions, be well rid of him.\"\n\nMy jaw dropped and I stared moronically at the screen. It was over. Just two days. What could I do in two days? Was this to be the end of The Bishop\u2014and the end of my scarcely launched career in crime?\nChapter 11\n\nI was not going to give up. I had to at least try, even if I failed and were caught myself. It was my fault that he had gotten into this position. I owed him at least an attempt at a rescue. But what could I do? I couldn't get near him in the Federal Building, approach him in transit, or even see him in court.\n\nCourt. Court? Court. Court! Court\u2014why did I keep thinking about the court? What was there about it that tickled my interest, that scratched at my medulla oblongata with an idea trying to get in?\n\nOf course! \"Yippee!!\" I enthused and ran around in small circles waving my arms and gurgling out loud my best imitation\u2014they used to love it at parties\u2014of a rutting porcuswine.\n\n\"What about the court?\" I asked myself, and was ready with the snappy answer. \"I'll tell you about the court. It is in an old building, an Ancient Artifact under preservation order. It probably has some old records in the basement and undoubtedly bats in the attic. During the day it is guarded like the mint\u2014but it is empty at night!\"\n\nI dived for my equipment cabinet and began hurling various necessities to the floor. Tool kit, lockpicks, lights, wires, bugs\u2014all the apparatus I would need for the job.\n\nNow a car\u2014or rather a van\u2014was very much in order since I would hopefully need transportation for two. I took care of that next. I had a number of sites that I had noted in case of need\u2014and now I needed. Although it was still daylight, the trucks and vans of the Crumb-ee Bakery were back in their lot being readied for their predawn tasks of the following day. A few vans were being taken into the garage for servicing and one of them happened to go a bit farther. Right onto the road and towards the city limits. I was on a country side road by dusk, in Pearly Gates soon after dark, and letting myself into a back door to the courthouse not long after that.\n\nThe burglar alarms were antiques, meant to keep out children or mental defectives\u2014since there was obviously nothing in the building worth stealing. That's what they thought! Armed with pix of the courtroom I had made myself during the trial, I went directly to it. Courtroom six. I stood in the doorway and looked about the darkened room. The lights from the street outside cast an orange glow through the high windows. I walked silently inside, sat down in the judge's chair, then looked into the witness box. In the end I found the chair in which The Bishop had sat during his lightning trial, where he would sit on the morrow. This is where he would sit\u2014and this is where he would stand when he rose to hear his sentence. Those great hands would grasp the rail here. Just here.\n\nI looked down at the wooden floor and smiled grimly. Then knelt and tapped on it. Then took out a drill as the various parts of my plan began to fall into place.\n\nOh, but this was a busy night! I had to clear boxes from the cellar beneath the courtroom, saw and hammer and sweat, and even slip out of the courthouse long enough to find a sports supply store and break into it. And, most critical of all, I had to work out a route of escape. The escape itself would not have to be rushed\u2014but it would have to be secure. If I had had the time a bit of tunneling would have been in order. But I had no time. Therefore ingenuity would have to replace manual labor. As I cogitated in a comfortable position I found myself nodding off. Never! I made my way from the building yet again, found an all-night restaurant staffed by surly robot machines, and drank two large coffees with extra caffeine. This worked, producing ideas as well as instant heartburn. I staggered off and broke into a clothing store. By the time I reached the courthouse again I really was staggering with fatigue. With fumbling fingers I resealed all of the doors, removed all traces of my passage. The first light of dawn was graying the windows before I was done. I fumbled with tired fingers as I sealed the cellar from the inside, stumbled across the room, sat down on the canvas, set my alarm watch\u2014and lay down to instant slumber.\n\nIt was pitch dark when the mosquito whine of the alarm irritated me awake. I had a moment of panic until I remembered that the cellar was windowless. It should be full daylight outside by now. I would see. I turned on a worklight, made adjustments\u2014then turned on the TV monitor. Perfect! A color picture of the courtroom above filled the screen, transmitted from the optical bug I had planted the night before. Some ancient employees were dusting the furniture and sweeping the floor. The session would begin in an hour. I left the set running while I made a last check of my labors of the previous night. All working, all in order... so all I had to do was wait.\n\nThat was what I did. Sipping at the cold coffee and chewing painfully on a stale sandwich from the previous day's supplies. The suspense ended when the courtroom doors were thrown open and the lucky spectators and the press came in. I could see them imaged clearly on the screen, hear the shuffle of their footsteps overhead. The sound of their voices murmured from the speaker, quieting only when they were silenced for the arrival of the judge. All eyes were on him, all ears twitching attentively when he cleared his throat and began to speak.\n\nFirst he bored everyone into a state of stupefaction by going over the previous day's evidence in detail, then adding his obvious agreement to each summation and observation. I let his voice drone on while I looked at The Bishop, zooming in on his face.\n\nHe gave them nothing. His features were set, he looked almost bored. But there was a glint to his eyes that was almost hatred, nearer contempt. A giant pulled down by ants. The set of his jaw indicated that they may have imprisoned his body, but his soul was still free. But not for long if the judge had his way!\n\nNow something in the judge's voice caught my attention. He had finished his preamble at last. He cleared his throat and pointed at The Bishop.\n\n\"Defendant will stand for sentencing.\"\n\nAll eyes were on the prisoner. He sat stolidly, unmoving. There was a growing rustle and murmur. The judge began to turn red and he hammered with the gavel.\n\n\"I will be obeyed in this court,\" he thundered. \"The defendant will rise or will be forced to do so. Is that understood?\"\n\nNow I was sweating. If only I could have told him not to cause any difficulties. What would I do if he were held up by great ugly policemen? Two of them had already started forward at the judge's signal. It was then that The Bishop slowly raised his eyes. The look of withering contempt he directed at the judge would have deterred anyone not as dense as his honor; it was a glare of repulsion that might have destroyed minor life forms.\n\nBut he was standing! The police halted as the large hands went out and seized the solid railing. It creaked as he tugged on it and heaved his giant form up, to stand erect. His head was high as he released the rail and his arms dropped to his sides....\n\nNow! I stabbed down on the button. The explosions were not loud\u2014but their effect was dramatic. They severed the two bolts that held the edge of the trapdoor in place. Under the great weight of The Bishop the door swung wide and he plunged down like a missile. I rushed up the ladder as he fell past me\u2014but had time for a last glimpse of the courtroom on the screen.\n\nThere was silence as he vanished from sight. The springs slammed the trapdoor up into position and I pushed the heavy steel sealing bolts into place beneath it. This happened so fast that the horizontal form of The Bishop was still bouncing up and down on the trampoline when I turned to look. I scurried down the ladder to his side as he finally came to rest, looking up at me with stolid gaze as he spoke.\n\n\"Ah, Jim my boy. How nice to see you again.\" He took my proffered hand and I helped him down to the floor. Above us there was pandemonium, shouting and screaming that could be clearly heard through the floor. I permitted myself one glorious look at the screen, at the pop-eyed judge, the scurrying policemen.\n\n\"Very impressive, Jim, very,\" The Bishop said, admiring the scene on the screen as well.\n\n\"Right!\" I ordered. \"Look at it as you strip off your outer clothing. Very little time, explanations will follow.\"\n\nHe hesitated not a millisecond but was hurling clothing from him even as the words were clearing my lips. The great rotund form emerged, clothed in tasteful purple undergarments, and he raised his hands above his head at my shouted command. Standing on the ladder I pulled the immense dress down over him.\n\n\"Here is the coat,\" I said. \"Put that on next. Dress touches the ground, so don't remove shoes. Large hat next, that's it, mirror and lipstick while I unbolt the door.\"\n\nHe did what I said without a murmur of protest. The Bishop had vanished from sight and a lady of truly heroic proportions now emerged. There was a hammering above his head which he completely ignored.\n\n\"Let's go!\" I called out, and he minced across the room in a most feminine fashion. I kept the door closed until he reached me and I used those few seconds to fill him in. \"They'll be at the cellar stairs by now\u2014but they are blocked. We go the other way.\" I pulled on the policeman's helmet to go with the uniform I was wearing. \"You are a prisoner in my custody. We are leaving\u2014now!\"\n\nI took him by the arm and we turned left down the dusty corridor. Behind us there was much crashing and shouting from the blocked stairwell. We hurried on, to the boiler room, and through that to the set of short stairs that rose up to the heavy exit door. With its hinges now greased and lock well oiled. It opened at a touch and we stepped out into the alleyway.\n\nNot an arm's length from the back of a policeman who was standing guard there. He was the only one.\n\nIt took only an instant to examine the scene. The narrow alley was open at the far end. There was a dead end behind us. People\u2014and safety\u2014were in the street beyond the police guard. Then The Bishop climbed up beside me and something grated under his foot. The policeman turned his head to look.\n\nI could see his eyes widen\u2014as well they might, for the lady beside me was an impressive sight. I took advantage of his diverted attention to jump forward and reach out to keep his head turning even more in the same direction. He seized me in strong hands\u2014which quickly went limp since the Tongoese neck twist produces instant unconsciousness when the rotation reaches 46 degrees from full front. I eased him to the ground, then stopped The Bishop from striding forward with my raised palm.\n\n\"Not that way.\"\n\nThe door on the building across the alley said SERVICE ENTRANCE and was locked. It opened to my ready key. As I waved my portly companion inside I took off my cap and threw it beside the policeman. I closed the door from the inside and dropped my uniform jacket as I did. The necktie went next as we strolled into the department store, until I was dressed simply in slacks and shirt. I put my moustache into my pocket and we joined the other customers. Occasionally looking at a display as we passed, but certainly never dawdling. There were a few amazed looks at my companion, but this was a very proper store and no one was so rude as to stare. I went first through the exit, holding the door, then led by a few paces as we joined the passing throng. Behind us, getting weaker as we went, were shouts and cries and the sound of alarm bells and sirens. I permitted myself a small smile. When I glanced back I saw that my companion had permitted herself one as well. She even had the nerve to let me have a brief wink. I turned back quickly\u2014I couldn't encourage this sort of thing\u2014then turned the corner into the side street, where the bread truck awaited.\n\n\"Stand here and look into your mirror,\" I said, unlocking the rear door. I busied myself inside, then barely had time to move aside as a great form hurtled by.\n\n\"No one looking...\" he gasped.\n\n\"Perfect.\"\n\nI climbed out, secured the door, went to the driver's side, climbed in, and started the engine. The van rumbled forward, slowly forcing its way through the pedestrians at the corner, then waited for a break in traffic.\n\nI had considered driving back and past the courthouse, but that would have been dangerous braggadocio. Better to simply slip away.\n\nWhen the street was empty I turned in the opposite direction and drove carefully towards the city limits. I knew all the back roads so we would be away well before they could be blocked.\n\nWe were not out of danger yet\u2014but I still felt smug satisfaction. And why not! I had done it! Committed the escape of the century to save the criminal of the century. Nothing could stop us now!\nChapter 12\n\nI drove, slowly but steadily, for the rest of the morning and into the afternoon. Avoiding all of the major highways by staying with the secondary roads. Though my route, by necessity, had to vary in direction, I nevertheless moved steadily south. Doing my best to add real feeling and emotion to Pi-r squared. Sounds familiar? It should be since it is probably the single geometry theorem that anyone ever remembers. The area of a circle is equal to its radius times the value of Pi\u2014squared. So each roll of the wheels of the bread van added an ever-increasing area that must be searched to find the escaping prisoner.\n\nFour hours of this should put us well ahead of the police. The fact had to be considered as well that The Bishop had been locked in the back of the van for all of this time and knew nothing of my plans for the future. Explanations were in order\u2014as was some food. I was getting hungry and, considering his girth, he would surely be feeling the same. With this in mind I pulled into the next suburban shopping center, checked the quick-food restaurants as I drove by, then parked at the far end of the lot. Backed up close to a blank wall. The Bishop blinked benevolently when I opened the rear door admitting light and fresh air.\n\n\"Time for lunch,\" I said. \"Would you like...\"\n\nI lapsed into silence as he raised his hand in a gesture of silence.\n\n\"Permit me, Jim, to say something first. Thank you. From the bottom of my heart I thank you for what you have done. I owe you my life, no less. Thank you.\"\n\nI stood with lowered eyes\u2014I swear I was blushing like a girl!\u2014and twisting my toe around and around on the ground. Then I coughed and found my voice.\n\n\"I did what had to be done. But\u2014could we talk of this later?\" He sensed my embarrassment and nodded, a regal figure despite the absurd garb he was still wearing. I pointed to the box on which he had been sitting. \"There are clothes in there. While you change I'll get some food. You don't mind junk food from Macswineys?\"\n\n\"Mind? After the loathsome sludge of the prison food, one of their Barbecued Porcuswineburgers would be unto paradise. With a large portion of sugarfried spamyams, if you please.\"\n\n\"Coming up!\"\n\nI closed the van door with a feeling of relief and trotted off towards the beckoning platinum arches. The Bishop's enthusiasm for fast food was most encouraging in a way that he could not suspect yet.\n\nLoud munching and rustling sounded from the tables on all sides as I passed and made my way up to the serving counter. I reeled off my order to the plastic-headed robotic attendant, stuffed bills into the hopper\u2014then grabbed the bag of food and drink as it slid out of the gate.\n\nWe sat on the boxes in the back of the van and ate and drank with enthusiasm. I had left the rear door open a crack, which gave us more than enough light. During my absence The Bishop had discarded his dress and was now wearing more masculine garb\u2014the largest size I could find. He wolfed down half of his sandwich, nibbled a few spamyams to hold it in place, then smiled over at me.\n\n\"Your plan of escape was pure genius, my boy. I noticed the change in the flooring when I first sat down in the chair in the courtroom and pondered long over its significance. I hoped it was what I thought it might be, and can truthfully say that when the ground opened under my feet, so to speak, I felt a feeling of pleasure such as I had never experienced before. The sight of that despicable judge's face disappearing from my sight is a memory I shall always treasure.\"\n\nSmiling broadly he finished the rest of the sandwich, then wiped his lips delicately before speaking again.\n\n\"Since I do not wish to cause you greater embarrassment with more fulsome praise, perhaps I should ask you what plans you have made to keep me safe from the hands of the law? Because, knowing you as I do now, I am secure in the belief that you have planned ahead in precise detail.\"\n\nPraise from The Bishop was praise indeed and I basked in the warmth of it for a few moments while I worried out a bit of swinish gristle from between my teeth. \"I have done that, thank you. The bread truck is our vehicle of invisibility, for it and its brothers trundle the highways and byways of this country daily.\" For some reason I found myself sounding more and more like The Bishop when I spoke. \"We will stay in it until nightfall, slowly approaching our destination all of the while.\"\n\n\"And of course casual police patrols will not bother us, since the identifying numbers on this vehicle are not the ones that were on it before it came into your possession.\"\n\n\"Precisely. The theft will have been reported and local police informed. But the search will not widen, for this vehicle will be found not far from its depot in Billville in the morning. The new numbers, soluble in paint thinner, will have been removed, the odometer turned back to show only a brief joy ride by the thieves. If a van like this were seen and noted in the distant city of Bit O' Heaven, there will be nothing to connect that bread van with this one. That trail will run cold as will all the others.\"\n\nHe digested this bit of information, along with the last of the spamyams, then licked his fingers ruminatingly. \"Capital. I could not have done better myself. Since further movement will be dangerous\u2014the police will soon have a net over the entire country\u2014I presume that Billville is our destination?\"\n\n\"It is. I have my establishment there. Also your place of security. When I asked about your food tastes I had that in mind. You are going to take up residence in an automated Macswineys until the heat of the chase dies down.\"\n\nHis eyebrows climbed up to his forehead and I saw him glance with some apprehension at the discarded wrappings, but he was kind enough not to speak his doubts aloud. I hurried to reassure him.\n\n\"I have done it myself\u2014so don't worry. There are some slight discomforts....\"\n\n\"But none to equal that of Federal prison! I apologize for an unseemly thought. No offense given.\"\n\n\"Or taken. It all came about by accident one evening when the police were a little close behind me for comfort. I picked the lock on the service entrance of the local Macswineys, the very one that you will be visiting, and my pursuers lost my trail. While I waited for a safe period I examined the premises. Amazing! Operating at high speed all around me was the solution to the single problem that faces all fast-food chains. The cost of keeping even the highly underpaid and unskilled employees. Human beings are both intelligent and greedy. They tend to become skilled, then want more money for their work. The answer is to do away with human beings completely.\"\n\n\"Admirable solution. If you are finished with your crumplumps I just might nibble one or two while I listen to your fascinating documentary.\"\n\nI passed the greasy bag to him and went on. \"Everything is mechanized. As the customer speaks his order the required item of food is ejected from the deep-frozen store into a super-voltage radar oven where it is instantly blasted to steamingly edible temperature. These ovens are so powerful that an entire frozen porcuswine can be exploded into steam and greasy particles in twelve microseconds.\"\n\n\"Amazing!\"\n\n\"Beverages are dispensed with the same lightning speed. By the time a customer has finished speaking, his entire order is waiting. Behind a steel door, of course, until he has paid. The machinery is fully automatic and reliable and rarely touched by human hands. It is inspected weekly, while the frozen food store is replenished weekly as well. But not on the same day so that the vehicles don't get in each other's way.\"\n\n\"Crystal clear!\" he cried aloud. \"One makes one's home, so to speak, in the machinery chamber. When the frozen store is replenished, access to it will be from outside the building and the living chamber will not be entered. On the day the machinery is inspected the occupant rests comfortably in the freezing room until the technicians leave. I assume there is a connecting door, easily found. Ahh, yes, the freezer\u2014that explains the large and warm garment I found packed in with my clothes. But should there be an equipment failure...?\"\n\n\"The alarm sounds in the central repair depot and a mechanic is dispatched. I have also arranged for it to sound in the room as well to allow enough time to slip away. I have also made provision for unexpected visits by the engineering staff. An alarm sounds if a key is placed in the outer lock, which then jams for precisely sixty seconds. Any questions?\"\n\nHe laughed and reached out and patted my shoulder. \"How could there be? You have thought of everything. Might I ask about reading matter and, how shall I phrase it delicately, sanitary facilities?\"\n\n\"Portable viewscope and library with your bedroll. All needed facilities already plumbed in for visiting technicians.\"\n\n\"I could ask for no more.\"\n\n\"But... I could.\" I lowered my gaze\u2014then raised it and steeled myself to speak. \"You once told me that you were not in the acolyte-seeking business. Dare I ask you if you still feel that way? Or would you consider dallying the hours away with some lessons in criminal lore? Just to pass the time, so to speak.\"\n\nNow it was his turn to lower his eyes. He sighed, then spoke. \"I had good reasons to reject your request. Good at the time, or so I believed. I have changed my mind. In gratitude for my rescue I would enroll you in my school of Alternate Life-styles for a decade or more. But I don't believe you would like mere gratitude. That would not wear well, unless I have misread your character. I don't believe you rescued me just to gain my gratitude. So I therefore tell you, in all truth, that I look forward to passing on the few things I have learned down through the years. I look forward to our continuing friendship as well.\"\n\nI was overwhelmed. We were on our feet at the same time and shaking hands, laughing. His grip was like steel but I didn't mind at all. It was I who turned away first, then looked at my watch.\n\n\"We have been here too long already and must not draw any attention. I shall drive on now\u2014and the next stop will be the last one, for we will have arrived. Please exit quickly, enter the service door at once, and close it behind you. I'll be back as soon as this van has been disposed of, so the next person to open the door will be me.\"\n\n\"At your orders, Jim. But speak\u2014and I shall obey.\"\n\nIt was a boring drive but a necessary one. But bored I was not, for I was filled with plans and thoughts of the future. I drove through street after street, stopping only once to charge the batteries at an automated service station. Then onward again, doomed forever to rumble through the back roads of Bit O' Heaven, watching the sun creep towards the horizon. To at last pull into the service road of the Billville shopping center, now empty of traffic until morning.\n\nNo one in sight. The Bishop passed me with a swish and the door slammed. The operation was still going well and I was in a hurry to finish, but knew better than to rush now. No one saw me when I carried the boxes and equipment into the building and dumped them in my office. It was taking a chance, but it had to be done. The chances that the van would be noticed and remembered were slim. Before I drove away I sprayed the interior of the van with print-go, a solvent that destroys fingerprints and should be in common use by all criminals. Even bread-van thieves.\n\nThis was it. I could do no more. I parked the van at the end of a quiet suburban street and walked back into town. It was a warm night and I enjoyed the exercise. When I passed the pond in Billville Park I heard a water bird calling out sleepily. I sat on the bench and looked out at the still surface of the pond. And thought about the future and my destiny.\n\nHad I really succeeded in breaking free with my old life? Was I to succeed in the life of crime that I so much wanted? The Bishop had promised to help me\u2014and he was the only person on the planet who could.\n\nI whistled as I walked towards the shopping center. Looking forward to a brilliant and exciting future. So involved in my thoughts that I ignored the occasional surface car that passed, barely aware of one stopping behind me.\n\n\"You there, kid, just a minute.\"\n\nWithout thinking I turned about, so distracted that I didn't notice until too late that I was standing under the streetlight. The policeman sat in the car staring at me. I'll never know why he stopped, what he wanted to talk to me about, because that thought fled his mind instantly. I could see recognition there as his eyes widened.\n\nIn my concern over The Bishop I had forgotten completely that I was still a wanted criminal and jail-breaker, that all the police had my photograph and description. And here I was strolling the streets bereft of any disguise or attempt at security. All these thoughts passed through my head and out my ear in the instant that he recognized me. Nor did I even have time for any mental kicks in the seat of my trousers.\n\n\"You're Jimmy diGriz!\"\n\nHe seemed as surprised as I was. But not surprised enough to slow down his reflexes. Mine were still getting into gear by the time his were all through operating. He must have practiced that draw in the mirror every day because he was fast. Too fast.\n\nAs I was turning to run, the muzzle of his recoilless .75 appeared in the open window.\n\n\"Gotcha!\" he said. With a dirty, wide, evil law-enforcing smile.\nChapter 13\n\n\"Not me\u2014someone else\u2014mistaken identity!\" I gasped, but shoved my hands into the air at the same time. \"Would you shoot a hapless child just on suspicion?\"\n\nThe gun never wavered, but I did. Shuffling sideways towards the front of the car.\n\n\"Stop that and get back here,\" he shouted, but I kept the nervous shuffling going. I doubted, or hoped, that he wouldn't shoot me in cold blood. As I remember, that is against the law. I wanted him to come after me, because in order to do that he would have to take the gun out of the window. There was no way that he could point it at me and open the door at the same time.\n\nThe gun vanished\u2014and so did I! The instant he lowered it I turned and ran, head down and pumping as fast as I could. He shouted after me\u2014and fired!\n\nThe gun boomed like a cannon and the slug zipped past my ear and slammed into a tree. I spun about and stopped. This cop was insane.\n\n\"That's better,\" he called out, resting the gun on top of the open door and aiming it at me with both hands. \"I fired to miss. Just once. Next time I hit. I got the gold medal for shooting this piece. So don't make me show you how good I am with it.\"\n\n\"You are mad, do you know that?\" I said, all too aware of the quaver in my voice. \"You just can't shoot people on suspicion.\"\n\n\"Yes I can,\" he said, walking up to me with the gun still pointed steady as a rock. \"This ain't suspicion but identification. I know just who you are. A wanted criminal. You know what I'll say? I'll say this criminal grabbed my gun and it went off and he got shot. How does that sound? Want to grab my gun?\"\n\nHe was a nutter all right, and a police nutter at that. I could see that he really wanted me to make a break so he could fire off his cannon. How he had escaped all the tests that were supposed to keep his kind out of law enforcement I will never know. But he had done it. He was licensed to carry a gun and was looking for an excuse to use it. That excuse I was not giving him. I extended my arms slowly before me, wrists together.\n\n\"I'm not resisting, officer, see. You are making a mistake, but I am going quietly. Put on the cuffs and take me in.\"\n\nHe looked downright unhappy at this, and frowned at me. But I made no more moves and in the end he scowled, pulled the handcuffs from his belt, and tossed them over to me. The gun never wavered.\n\n\"Put them on.\"\n\nI locked them on one wrist, very loosely so I could slip my hand out of them, then on the other. I was looking down when I did this and I did not see him move. Until he had me by both wrists and had squeezed both cuffs until they had locked hard deep into my skin. He smiled down at me, twisting the metal into my flesh with sadistic glee.\n\n\"Gotcha now, diGriz. You are under arrest.\"\n\nI looked up at him, he was a head taller than me and maybe twice my weight\u2014and I burst out laughing. He had put the gun back in its holster in order to grab me\u2014that's what he had done. The big man had grabbed the little kid. He couldn't understand why I was laughing and I gave him no opportunity to find out. I did the easiest, best, and fastest thing possible under the circumstances. Also the dirtiest.\n\nMy knee came up hard into his groin and he let go of my wrists at the impact and bent double. I did him a favor, the poor man must have been in some pain, and got him in the side of the neck with my joined hands as he went by. He was unconscious before he hit the ground. I knelt and started to go through his pockets for the keys to the handcuffs.\n\n\"What's happening there?\" a voice called out as a light came on over the door of the nearest house. The sound of that shot would bring the whole street out soon. I would worry about the cuffs later. Right now I had to make tracks.\n\n\"Man's been hurt!\" I shouted. \"I'm going for help.\" This last was called over my shoulder as I trotted off down the street and around the corner. A woman appeared in the doorway and called after me but I wasn't staying around to listen. I had to keep moving, get away from this place before the alarm was called in and the search began. Things were coming apart. And my wrists hurt. I looked at them when I passed the next streetlight and saw that my hands were white, and were getting numb as well. The cuffs were so tight they were cutting off all the blood circulation. Any slight guilt I may have had over the dirty fighting vanished on the instant. I had to get these things off\u2014and fast. My office, the only place.\n\nI got there, avoiding the main streets and staying away from people. But when I reached the back door of the building my fingers were numb and stiff. I could feel nothing.\n\nIt took an intolerably long time to fish the keys out of my pocket. When I succeeded I instantly dropped them. Nor could I pick them up again. My fingers would not close. I could only drag my lifeless hands over the keys.\n\nThere are low moments in life\u2014and I believe that this was the lowest one that I had ever experienced. I just could not do what had to be done. I was finished, licked, through. I couldn't get into the building. I couldn't help myself. It didn't take a medical degree to figure out that if I didn't get the cuffs off soon I was going to go through life with plastic hands. This was it.\n\n\"This is not it!\" I heard myself snarling. \"Kick the door open, do something, unlock it with your toes.\"\n\nNo, not my toes! I fumbled the keys about on the ground with my dead fingers until I had separated out the correct key. Then bent my body over it and touched it with my tongue, feeling its position, ignoring the filth and dirt that I licked up along with it. Then I pulled back my lips and seized the key with my teeth. Good so far!\n\nIf you should ever be tempted to unlock a door with a key in your teeth while wearing handcuffs I have only a single word of advice. Don't. You see, you have to turn your head sideways to get the key into the keyhole. Then roll your head to turn the key, then butt the door with your head to get it open....\n\nIt worked at last and I fell face first onto the floor inside. With the knowledge that I would have to do the whole thing all over again upstairs. That I did do it, and finally slid through into the office, owes more to persistence, stubbornness and brute force than to intelligence. I was too exhausted to think. I could only react.\n\nI elbowed the door shut and stumbled to my workbench, hurled my toolbox to the floor, and kicked its contents about until I found the vibrosaw. I picked this up with my teeth and managed to wedge it into an open desk drawer, holding it in place as I closed the drawer with my elbow. Closing it on my lip as well, which brought forth a nice gusher of blood. Which I ignored. My wrists were on fire\u2014but my hands were past feeling. White and dead-looking. I had run out of time. I used my elbow to turn on the saw. Then pushed the handcuffs towards the blade, pulling my arms apart hard to stretch the chain. The blade buzzed shrilly and the chain was cut and my arms flew wide.\n\nNext came the more exacting job of cutting the cuffs off without cutting my flesh. Too much.\n\nThere was blood everywhere before I was done. But the cuffs were off and I could see the flesh turn pink as circulation was restored.\n\nAfter this, all I was up to was collapsing into a chair and watching the blood drip. I sat like this for about a minute when the numbness ended and the pain began. With an effort I stumbled to my feet and dragged over to the medical locker. Getting this good and bloody as well while I shook the pain capsules out and managed to swallow two of them. Since I was already there I pulled out the antiseptic and bandages and cleaned up the cuts. They were more messy than dangerous and none were very deep. I bandaged them, then looked into the mirror and shuddered and did something about the lip.\n\nA police siren wailed by in the street outside\u2014and I realized that the time had come to do some furious thinking and planning.\n\nI was in trouble. Billville wasn't very big, and all exits would be sealed by now. That's what I would have done first, if I were looking for a fugitive. And even the dimmest of policemen would have figured this one out as well. Barricades on all the roads, copters out with nightscopes to watch the open fields, police at the linear station. All holes plugged. Trapped like a rat. What else? The streets would be patrolled too, easy enough to do by groundcar. And the later it got the fewer people there would be about and the more dangerous it would be to wander around.\n\nThen, in the morning, what then? I knew what then. A search of every room in every building until I was found. I felt the perspiration bead my forehead at the thought. Was I trapped?\n\n\"No surrender!\" I shouted aloud, then jumped to my feet and paced back and forth. \"Jimmy diGriz is too slippery to be caught by the ham-handed minions of the local law. Look how I slipped away from that homicidal copper. Slippery Jim diGriz, that's who I am. And I am about to slip away from them again. But how?\"\n\nHow indeed. I cracked open a beer, drank deep, then slumped back into the chair. Then looked at my watch. It was already getting too late to risk my presence on the street. The restaurants would be emptying, the feely and stinky cinemas disgorging their customers, couples marching homeward two by two. Any single individual drawing the instant attention of the law.\n\nIt had to be the morning then. I would have to venture forth in the light of day\u2014or the rain! I punched up the weather report as quickly as I could, then slumped back once again. 99% chance of sunshine. I might as easily wish for an earthquake as a storm.\n\nThe office was a mess; it looked like the aftermath of an explosion in the slaughterhouse. I would have to clean it up....\n\n\"No, Jim, you will not have to clean it up. Because the police are going to find it sooner or later, and probably sooner. Your fingerprints are everywhere and they know your blood type. They'll have a really good time trying to figure out what happened to you.\"\n\nIt would give them something to think about at least. And maybe cause a little trouble for one sadistic copper. I wheeled the chair over to the terminal and typed out the message. The printer whistled and I took the sheet of paper from the hopper. Wonderful!\n\nTO THE POLICE. I WAS SHOT DEAD BY YOUR MURDERING POLICE OFFICER YOU FOUND UNCONSCIOUS. HE GOT ME. I AM BLEEDING INTERNALLY AND WILL DIE SOON. GOODBYE CRUEL WORLD. I NOW GO TO THROW MYSELF INTO THE RIVER.\n\nI doubted very much if the ruse would work, but it might at least get that gun-crazy cop in trouble. And keep the rest of them busy dredging the river. There was some blood on the note and I smeared more on from the bandages. Then laid it carefully on the table.\n\nThis bit of tomfoolery had cheered me a bit. I sat back and finished the beer and made plans. Was I leaving anything important behind? No, there were no records kept here that I would need in the future. I found my doomsday key and unlocked the destruct switch, then pressed it. A single click from the memory banks was the only evidence that all of the computer's memory had just turned into random electrons. Everything else\u2014tools, equipment, machinery\u2014was expendable, could be replaced when needed. But I was not leaving the money.\n\nAll this was pretty tiring\u2014but I couldn't afford to rest until all arrangements were complete. I pulled a pair of thin plastic gloves over the blood and bandages and set to work. The money was in the safe, since I robbed banks and did not believe in supporting them by opening an account. I put it all into a businessman's carrybag. It was only half full, so I added all the microtools that would fit. In the space that was left I stuffed in as much clothing as I could, then stood on the thing until I got it closed and locked.\n\nNew clothes and a disguise next. A black four-piece business suit, the fabric enriched by a pattern of tiny white buck bills. An orange rollneck, just what all the young bankers were wearing, along with trendy porcuswineherd boots with builtup heels. Add some to my height\u2014that would help. When I left I would wear the moustache and gold-rimmed glasses. What I could do now was darken my hair with dye and add to my fading tan. Preparations done, woozy with beer, fatigue, and pain pills, I opened the file cabinet bed, set the alarm, and dropped into oblivion.\n\nThere were giant mosquitoes circling my head, more and more of them, after my blood, mosquitoes...\n\nI opened my eyes and blinked away the dream. My alarm watch, since I hadn't turned it off, had raised the volume on the mosquito buzzing, louder and louder until it sounded like a squadron of them diving to the attack. I pushed the button, smacked my gummy lips together, then stumbled over for a glass of water. It was full daylight outside and the early risers were just appearing.\n\nPreparations made, I washed and dressed with care. Nifty orange gloves that matched the shirt hid my bandaged hands. When the streets were at their rush-hour busiest I seized up the carrybag, then checked carefully to make sure that the hall outside was empty. Stepped out and closed the door without looking back. This part of my life was over with. Today was the first day of my new life.\n\nI hoped. I walked to the stairs with what I hoped was a very sincere, businessman-type walk, down past the first arrivals, and into the street.\n\nTo see the policeman on the corner looking closely at every passerby.\n\nI did not look at him, but found an attractive girl walking ahead of me with very neat legs indeed. I watched their twinkling advance and tried to forget the nearby minion of the law. Came towards him, passed him, walked away from him. Waiting for the cry of recognition...\n\nIt never came. Maybe he was looking at the girl too. One down\u2014but how many more to go?\n\nThis was the longest walk that I had ever taken in my life. Or at least it seemed that way. Not too fast, not too slow. I struggled to be part of the crowd, just another wage-slave going to work, thinking only of profit and loss and debenture bonds. Whatever debenture bonds were. One more street\u2014safe so far. There's the corner. The service road behind the shopping center. No place for a businessman like you. So look sharp and don't hang about. Around the corner to safety.\n\nSafety? I staggered as though I had been struck.\n\nThe Macswineys service van was outside the door and a hulking brute of a mechanic was just going inside.\nChapter 14\n\nI looked at my watch, snapped my fingers, then turned away from the service road in case my actions were being observed. And marched on smartly until I came to the first Speedydine. Just to make my day complete there were two policemen sitting in the first booth. Looking at me, of course. I marched past, eyes front, and found the seat farthest from them. There was an itching between my shoulder blades that I didn't dare scratch. I couldn't see them\u2014but I knew what they were doing. They were looking at me, then talking to each other, deciding I was not quite what I looked like. Better investigate. Stand, walk my way, lean over my booth...\n\nI saw the blue-trousered legs out of the corner of my eye and my heart instantly began hammering so loudly I was sure the whole restaurant could hear it. I waited for the accusing words. Waited... let my eyes travel up the blue-clad legs...\n\nTo see a uniformed linear driver sitting down across from me. \"Coffee,\" he said into the microphone, shook his newspaper open, and began to read.\n\nMy heart slowed to something resembling normal and I silently cursed myself for suspicion and cowardice. Then spoke aloud into my own microphone in the deepest voice I could sum up.\n\n\"Black coffee and mulligatawny dumplings.\"\n\n\"Deposit six bucks, if you please.\"\n\nI inserted the coins. There was a rumble of machinery at my elbow and my breakfast slid out onto the table. I ate slowly, then glanced at my watch, then went back to sipping my coffee. As I well knew from the earlier occasion when I had nipped into the freezer, when I had been hiding out there, thirty minutes was the minimum service time for a Macswineys mechanic. I allowed forty before I slid out of the booth. I tried not to think about what I would find when I finally got into the back of the fast-food parlor. I remembered my parting words only too well, I would be the next person through the door. Ho-ho. The next person had been the mechanic. Had he caught The Bishop? I sweated at the thought. I would find out soon enough. I passed the booth where the police had been. They were gone\u2014out searching some other part of the city for me I hoped\u2014and I headed back to the shopping center. To be greeted by the glorious sight of the Macswineys van drawing out into the road ahead of me.\n\nThe key was ready in my hand as I approached the door. The road ahead was empty\u2014then I heard the footsteps coming up behind me. The police? With boring repetition my heart started the thudding routine again. I walked slower as I came close to the door. Then stopped and bent over and slipped the palmed key into my hand as though I had just picked it up. I examined it closely as someone came up, then passed me. A young man who showed not the slightest interest in my existence. He went on and turned into the back entrance to the market.\n\nI took one look over my shoulder\u2014then jumped for the door before anything else happened. Turned the key, pushed\u2014and of course it didn't open.\n\nThe delay mechanism I had installed was working fine. It would unlock in one minute. Sixty short seconds.\n\nSixty incredibly crawling seconds. I stood there in my fine business outfit, as out of place in this alley as teats on a boar porcuswine, as we used to say back on the ranch. Stood there and sweated and waited for police or passersby to appear. Waited and suffered.\n\nUntil the key turned, the door opened\u2014and I fell through.\n\nEmpty! On the far wall the automatic machinery clattered and whirred. The drink dispenser gurgled and a filled container whistled down its track and vanished. To be followed by the steaming bulk of a burger. Night and day this went on. But among all this mechanical motion no human form appeared. They had captured him\u2014the police had The Bishop. And they would capture me next....\n\n\"Ahh, my boy, I thought it might be you this time.\"\n\nThe Bishop emerged from the freezer, immense in his insulated gear, his sleeping roll and carryall tucked under his arm. He slammed the door behind him and the strength went out of me with a woosh and I slumped down with my back to the wall.\n\n\"Are you all right?\" he asked, concern in his voice. I waved a weak hand.\n\n\"Fine, fine\u2014just let me catch my breath. I was afraid they had you.\"\n\n\"You shouldn't have worried. When you did not reappear within a reasonable time I assumed there had been some hitch in your plans. So I rehearsed my evacuation moves just in case the legitimate users appeared today. And they did. It is really quite cold in there. I wasn't sure how long they would be, but I was sure you had installed some way of discovering when they left....\"\n\n\"I meant to tell you!\"\n\n\"No need. I found the hidden speaker and switch and listened to someone who mutters profanities while he works. After some time the slam of the door and silence were welcome information indeed. Now about yourself. There were problems?\"\n\n\"Problems!\" I burst out laughing with relief. Then stopped when I heard a hysterical edge to the sound. I told him, omitting some of the more gruesome details. He made appropriate noises at the right places and listened attentively until the bitter end.\n\n\"You are being too harsh on yourself, Jim. A single lapse after all the tension of the day is not to be unexpected.\"\n\n\"But not to be allowed! Because I was stupid I almost had both of us caught. It won't happen again.\"\n\n\"There is where you are wrong,\" he said, shaking a thick admonitory finger. \"It could happen at any time\u2014until you have trained yourself in your work. But you will be trained and trained efficiently...\"\n\n\"Of course!\"\n\n\"... until a lapse like this one will be impossible. You have done incredibly well, for one of your inexperience. Now you can only improve.\"\n\n\"And you will teach me how\u2014how to be a successful crook like you!\"\n\nHis brow furrowed at my words and his expression was grave. What had I said that was wrong? I chewed my sore lip with worry as he unrolled his bedroll in silence, spread it out then sat upon it crosslegged. When at last he spoke I hung upon his every word.\n\n\"Now your first lesson, Jim. I am not a crook. You are not a crook. We do not want to be criminals for they are all individuals who are stupid and inefficient. It is important to comprehend and appreciate that we stand outside of society and follow strict rules of our own, some of them even stricter than those of the society that we have rejected. It can be a lonely life\u2014but it is a life you must choose with your eyes open. And once the choice has been made you must abide by it. You must be more moral than they are because you will be living by a stricter moral code. And this code does not contain the word crook. That is their word for what you are and you must reject it.\"\n\n\"But I want to be a criminal....\"\n\n\"Abandon the thought\u2014and the title. It is, and you must excuse me saying it, a juvenile ambition. It is only your emotions striking out at the world you dislike and cannot be considered a reasoned decision. You have rejected them\u2014but at the same time accepted their description of what you are. A crook. You are not a crook, I am not a crook.\"\n\n\"Then\u2014what are we?\" I asked, all eagerness. The Bishop steepled his fingers as he intoned the answer.\n\n\"We are Citizens of the Outside. We have rejected the simplistic, boring, regimented, bureaucratic, moral, and ethical scriptures by which they live. In their place we have substituted our own far superior ones. We may physically move among them\u2014but we are not of them. Where they are lazy, we are industrious. Where they are immoral, we are moral. Where they are liars, we are the Truth. We are probably the greatest power for good to the society that we have discarded.\"\n\nI blinked rather rapidly at that one, but waited patiently because I knew that he would soon make all clear. He did.\n\n\"What kind of a galaxy do we live in? Look around you. The citizens of this planet, and of every other planet in the loose organization known as the Galactic League, are citizens of a fat, rich union of worlds that has almost forgotten the real meaning of the word crime. You have been in prison, you have seen the dismal rejects whom they consider criminals. And this is what is called a frontier world! On the other settled planets there are few malcontents and even fewer who are socially maladjusted. Out there the handful who are still being born, in spite of centuries of genetic control, are caught early and their aberrations quickly adjusted. I made one single trip offplanet in my life, a tour of the nearest worlds. It was terrible! Life on those planets has all the color and wonder of a piece of wet cardboard. I hurried back to Bit O' Heaven for, loathsome as it can be at times, it is still a bit o' heaven compared to the others.\"\n\n\"Someday\u2014I would like to see these other worlds.\"\n\n\"And so you shall, dear boy, a worthy ambition. But learn your way around this one first. And be thankful they don't have complete genetic control here yet\u2014or the machines to mentally adjust those who struggle against society. On other planets the children are all the same. Meek, mild, and socially adjusted. Of course some do not show their genetic weakness\u2014or strength as we call it\u2014until they are adults. These are the poor displaced ones who try their hands at petty crimes\u2014burglary, shoplifting, rustling, and the like. They may get away with it for a week or two or a month or two, depending on their degree of native intelligence. But as sure as atomic decay, as sure as the fall of leaves in the autumn\u2014and just as predestined\u2014the police will eventually reach out and pull them in.\"\n\nI digested this information, then asked the obvious question: \"But if that is all there is to crime, or rebellion against the system\u2014where does that leave you and me?\"\n\n\"I thought you would never ask. These dropouts I have described, whom you have associated with in prison, comprise ninety-nine point nine percent of crime in our organized and dandified society. It is the last and vital one-tenth of one percent that we represent that is so vital to the fabric of this same society. Without us the heat death of the universe would begin. Without us the lives of all the sheeplike citizenry would be so empty that mass suicide to escape it would be the only answer. Instead of pursuing us and calling us criminals they should honor us as first among them!\"\n\nThere were sparks in his eyes and thunder in his voice when he spoke. I did not want to interrupt his fulminant speech, but there were questions to be asked.\n\n\"Please excuse me\u2014but would you be so kind as to point out just why this is so?\"\n\n\"It is so because we give the police something to do, someone to chase, some reason for rushing about in their expensive machines. And the public\u2014how they watch the news and listen for the latest reports on our exploits, how they talk to each other about it and relish every detail! And what is the cost of all this entertainment and social good? Nothing. The service is free, even though we risk life, limb and liberty to provide it. What do we take from them? Nothing. Just money, paper, and metal symbols. All of it insured. If we clean out a bank, the money is returned by the insurance company who, at the end of the year, may reduce their annual dividend by a microscopic amount. Each shareholder will receive a millionth of a buck less. No sacrifice, no sacrifice at all. Benefactors, my boy, we are nothing less than benefactors.\n\n\"But in order for us to accomplish all this good for them we must operate outside their barriers and well outside of their rules. We must be as stealthy as rats in the wainscoting of their society. It was easier in the old days of course, and society had more rats when the rules were looser, just as old wooden buildings have more rats than concrete buildings. But there are rats in the buildings now as well. Now that society is all ferroconcrete and stainless steel there are fewer gaps between the joints. It takes a very smart rat indeed to find these openings. Only a stainless steel rat can be at home in this environment.\"\n\nI broke into spontaneous applause, clapping until my hands hurt, and he nodded his head with gracious acceptance of the tribute.\n\n\"That is what we are,\" I enthused. \"Stainless steel rats! It is a proud and lonely thing to be a stainless steel rat!\"\n\nHe lowered his head in acknowledgment, then spoke. \"I agree. Now\u2014my throat is dry from all this talking and I wonder if you could aid me with the complex devices about us. Is there any way you might extract a double-cherry oozer from them?\"\n\nI turned to the maze of thudding and whirring machinery that covered the inner wall.\n\n\"There is indeed, and I shall be happy to show you how. Each of these machines has a testing switch. This, if you will look close, is the one on the drink dispenser. First you must turn it to on, then you can actuate the dispenser, which will deliver the drink here instead of to a customer on the other side. Each is labeled\u2014see, this is the cherry oozer. A mere touch and... there!\"\n\nWith a whistling thud it dropped into place and The Bishop seized it up. As he began to drink he froze, then whispered out of the corner of his mouth.\n\n\"I just realized, there is a window here and a young lady is staring in at me!\"\n\n\"Fear not,\" I reassured him. \"It is made of one-way glass. She is just admiring her face. It is the inspection port to look at the customers.\"\n\n\"Indeed? Ahh, yes, I can see now. They are indeed a ravenous lot. All that mastication causes a rumble in my own tum, I am forced to admit.\"\n\n\"No trouble at all. These are the food controls. That nearest one is for the Macbunnyburger, if you happen to like them.\"\n\n\"Love them until my nose crinkles.\"\n\n\"Then here.\"\n\nHe seized up the steaming package, traditionally decorated with beady eyes and tufted tail of course, and munched away. It was a pleasure to watch him eat. But I tore myself away before I forgot and pushed coins into the slot on back of the armored coin box.\n\nThe Bishop's eyes widened with astonishment. As soon as he swallowed he spoke.\n\n\"You are paying! I thought that we were safely ensconced in a gustatorial paradise with free food and drink at our beck and call, night and day?\"\n\n\"We are\u2014for all of this money is stolen and I am just putting it back into circulation to keep the economy healthy. But there is no slack in the Macswineys operation. Every morsel of porcine tissue, every splinter of ice is accounted for. When the mechanic tests the machines he is responsible for every item delivered. The shop's computer keeps track of every sale so that the frozen supplies are filled exactly to the top each time they are replenished. All of the money collected is taken away each day from the safe on the outer wall\u2014which is automated as well. An armored van backs over it just as the time lock disengages. A code is keyed in and the money disgorged. So if we simply helped ourselves, the records would reveal the theft. Prompt investigation would follow. We must pay for what we use, precisely the correct amount. But, since we won't be coming back here, we will steal all the money on the day we leave.\"\n\n\"Fine, my boy, fine. You had me worried there for a minute with your bit of forced honesty. Since you are close to the controls, please trigger another delicious morsel of Lepus cuniculus while I pay.\"\nChapter 15\n\nI suppose that there have been stranger places to go to school, but I can't think of any. At certain times of day it was hard to be heard above the rattle, hiss, and roar of the dispensing machinery. Lunch and dinner were the busiest times, but there was another peak when the schools got out. We would eat then as well, since it was so hard to talk, working our way through the entire Macswineys inventory. Countless Macbunnyburgers hopped down our throats, and many a Frozen Fomey followed. I liked Dobbindogs until one too many cantered past my gums, and switched to jellied porcuswinetrotters, then to felinefritters. The Bishop was very catholic in his tastes and liked everything on the menu. Then, once the crowds had gone, after we had patted the last trace of gravy from our lips, we would loll back at our ease and my studies would continue. When we started on computer crime I discovered what The Bishop had been up to for the past couple of decades.\n\n\"Give me a terminal and I can rule the world,\" he said, and such was the authority of his voice that I believed he could. \"When I was young I delighted in all manner of operations to please the citizens of this planet. It was quite a thrill to intercept cash shipments while en route, then substitute my calling card for the bundles of bills. They never did find out how I did it....\"\n\n\"How did you?\"\n\n\"We were talking about computers.\"\n\n\"Digress just this once, I beg of you. I promise to put the technique to good use. Perhaps, with your permission, even leave one of your cards.\"\n\n\"That sounds an excellent idea. Baffle the current crop of coppers as thoroughly as I did their predecessors. I'll describe what happens\u2014and perhaps you can discover for yourself how it was done. In the Central Mint, a well-guarded and ancient building with stone walls two meters thick, are located the giant safes filled with billions of bucks. When a shipment is to be made, guards and officials fill a bullion box, which is then locked and sealed while all present look on. Outside the building waits a convoy of coppers all guarding a single armored car. At a given signal the car backs up against the armor-plated delivery door. Inside the building the steel inner door is opened, the box placed inside the armored chamber. This door is sealed before the outer one can be opened. The box then travels in the armored car to the linear train, where an armored wagon receives it. This has but the single door, which is locked and sealed and wired with countless alarms. Guards ride in a special chamber of each car as it shuttles through the linear network to the city needing the bucks. Here another armored car awaits, the box is removed\u2014still sealed\u2014placed in the car, and taken to the bank. Where it is opened\u2014and found to contain only my card.\"\n\n\"Marvelous!\"\n\n\"Care to explain how it was done?\"\n\n\"You were one of the guards on the train...\"\n\n\"No.\"\n\n\"Or drove the armored car...\"\n\n\"No.\"\n\nI racked my brains this way for an hour before he relented and explained. \"All your suggestions have merit, but all are dangerous. You are far more physical than I ever was. In my operations I always preferred brains to brawn. The reason that I never had to break into the box and extract the money is that the box was empty when it left the building. Or rather it was weighted with bricks as well as my card. Can you guess now how it was done?\"\n\n\"Never left the building,\" I muttered, trying to stir my brain to life. \"But it was loaded into the box, the box put into the truck...\"\n\n\"You are forgetting something.\"\n\nI snapped my fingers and leapt to my feet. \"The wall, of course it had to be the wall. You gave me all the clues. I was just being dense. Old, made of stone, two meters thick!\"\n\n\"Exactly so. It took me four months to break in, I wore out three robots doing it, but I won out in the end. First I bought the building across the road from the mint and we tunneled under it. With pick and shovel. Very slow, very silent. Up through the foundations of the building and inside the wall. Which proved to have an outer and inner stone wall, and as is the building custom, it was filled with rubble in between. Our diamond saws were never heard when we opened the side of the armored vault connecting the inside of the mint with the outside. The mechanism I installed could change boxes in one point oh five seconds. When the inner door was closed, the lock had to be thrown before the outer door could be opened. That was enough time, almost three seconds, to allow for the switch. They never did find out how I did it. The mechanism is still in place. But the operation was basically misdirection, along with a lot of digging. Computer crime is something else altogether. Basically it is an intellectual exercise.\"\n\n\"But isn't computer theft almost impossible these days, with codes and interlocks?\"\n\n\"What man can code or lock, man can decode and unlock. Without leaving any trace. I will give you some examples. Let us begin with the rounding-off caper, also called the salami. Here is how it works. Let us say that you have 8,000 bucks in the bank, in a savings account that earns eight percent a year. Your bank compounds your account weekly in order to get your business. Which means at the end of the first week your bank multiplies your balance by .0015384 percent and adds this sum to your balance. Your balance has increased by 12.30 bucks. Is that correct? Check it on your calculator.\"\n\nI punched away at the sum and came up with the same answer. \"Exactly twelve bucks and thirty centimes interest,\" I said proudly.\n\n\"Wrong,\" he said deflatingly. \"The interest was 12.3072, wasn't it?\"\n\n\"Well, yes, but you can't add seventy-two-hundredths of a centime to someone's account, can you?\"\n\n\"Not easily, since financial accounts are kept to two decimal places. Yet it is at this precise moment in the calculations that the bank has a choice. It can round all decimals above .005 up to the nearest centime, all those below .0049 down to zero. At the end of a day's trading the rounding-ups and rounding-downs will average out very close to zero so the bank will not be out of pocket. Or, and this is the accepted practice, the bank can throw away all decimal places after the first two, thereby making a small but consistent profit. Small on banking terms\u2014but very large as far as an individual is concerned. If the bank's computer is rigged so that all the rounding-downs are deposited to a single account, why at the end of the day the computer will show the correct balance in the bank's account and in the client's accounts. Everyone will be quite pleased.\"\n\nI was punching like fury into my calculator, then chuckled with glee at the results. \"Exactly so. All are pleased\u2014including the holder of that account that now holds the round-downs. For if only a half a centime is whipped from ten-thousand accounts, the profit is a round fifty bucks!\"\n\n\"Exactly. But a large bank will have a hundred times that number of accounts. Which is, as I know from happy experience, a weekly income of five-thousand bucks for whoever sets up this scam.\"\n\n\"And this, this is your smallest and simplest bit of computer tomfoolery?\" I asked in a hushed voice.\n\n\"It is. When one begins to access large corporative computers, the sums become unbelievable. It is such a pleasure to operate at these levels. Because if one is careful and leaves no traces, the corporations have no idea that they have even been fiddled! They don't want to know about it, don't even believe it when faced with the evidence. It is very hard to get convicted of computer crime. It is a fine hobby for one of my mature years. It keeps me busily engaged and filthy rich. I have never been caught. Ahh, yes, except once....\"\n\nHe sighed heavily and I felt mortified.\n\n\"My fault!\" I cried. \"If I had not tried to contact you, why you would never have got involved with the Feds.\"\n\n\"No guilt, Jim, feel no guilt over that. I misjudged their security controls, far more rigid than the ones I had been dealing with. It was my mistake\u2014and I certainly paid for it. Am still paying. I am not decrying the safety of our refuge here, but this junk food begins to wear on one after a bit. Or perhaps you haven't noticed?\"\n\n\"This is the staff of life of my generation.\"\n\n\"Of course. I had not thought of that. The horse tires not of hay, the porcuswine will snuffle up his swill greedily unto eternity.\"\n\n\"And you could probably tuck into lobster and champagne for the next century.\"\n\n\"Well-observed and correct, my boy. How long do you think that we shall be here?\" he asked, pushing away half of an unconsumed portion of crump-tumps.\n\n\"I would say a minimum of two weeks more.\" A shudder shivered his frame.\n\n\"It will be a good opportunity for me to reduce.\"\n\n\"By that time the heat of the chase will have died down considerably. We will still have to avoid public transportation for a good while after that. However I have prepared an escape route that should be secure fairly soon.\"\n\n\"Dare I ask what it is?\"\n\n\"A boat, rather a cabin cruiser on the Sticks River. I bought it some time ago, in a corporative name, and it is at the marina just outside Billville.\"\n\n\"Excellent!\" He rubbed his hands with glee. \"The end of summer, a cruise south, fried catfish in the evening, bottles of wine cooling in the stream, steaks at riverside restaurants.\"\n\n\"And a sex change for me.\"\n\nHe blinked rapidly at that, then sighed with relief when I explained. \"I'll wear girl's clothes when I'm aboard and can be seen from the shore, at least until we are well away from here.\"\n\n\"Capital. I shall lose some weight\u2014there will be no difficulty dieting here. Raise a moustache, then a beard, die my hair black again. It is something to look forward to. But shall we say one month instead of two weeks? I could last that amount of time incarcerated in this gustatorial ghetto as long as I would not be eating. My figure will be the better for the extra weeks, my hair and moustache longer.\"\n\n\"I can do it if you can.\"\n\n\"Then it is agreed. And we shall make the most of the time now by forwarding your education? RAM, ROM, and PROM will be the order of the day.\"\n\nI was too busy with my studies to be bothered by the omnipresent odor of barbecued porcuswineburgers. Besides that, I could still eat them. So as my comprehension grew of all the varied possibilities of illegality in our society, so did my companion's figure fade. I wanted to leave earlier, but The Bishop, having made up his mind, would not be swayed.\n\n\"Once a plan is made it must always be followed to the letter. It should only be changed if outside circumstances change. Man is a rationalizing animal and needs training in order to become a rational one. Reasons can always be found for altering an operation.\" He shuddered as the machines speeded up with a roar, school was out, then crossed off one more day on his calendar. \"An operation well planned will work. Meddle with it and you destroy it. Ours is a good plan. We will stay with it.\"\n\nHe was far leaner and harder when the day of our exodus finally arrived. He had been tried in the gustatory furnace and had been tempered by it. I had put on weight. Our plans were made, our few belongings packed, the safe cleaned out of all of its bucks\u2014and all trace of our presence eliminated. In the end we could only sit in silence, looking again and again at our watches.\n\nWhen the alarm sounded we were on our feet, smiling with pleasure.\n\nI turned off the alarm as The Bishop opened the door of the freezer room. As the key turned in the outer lock we closed the door behind us. Stood and shivered in Macswineys' mausoleum while we listened to the mechanic enter the room we had so recently left.\n\n\"Hear that?\" I asked. \"He's adjusting the icer on the cherry oozer dispenser. I thought it sounded funny.\"\n\n\"I prefer not to discuss the contents of the ghastly gourmet gallery. Is it time to go yet?\"\n\n\"Time.\" I eased open the outer door and blinked at the light of day, unseen for so long. Other than the service van the street was empty. \"Here we go.\"\n\nWe shuffled out and I sealed the door behind us. The air was sweet and fresh and filled with lovely pollution. Even I had had my fill of cooking odors. As The Bishop hurried to the van I slipped the two wedges into the outer door to our chamber of culinary horrors. If the mechanic tried to get out before his appointed time, these would slow him down. We only needed about fifteen minutes.\n\nThe Bishop was a dab hand with a lockpick and had the van open and the door swinging wide even as I turned about. He dived into the back among the machine parts as I started the engine.\n\nIt was just that easy. I dropped him close to the marina, where he sat on a bench in the sunshine, keeping an eye on our possessions. After that it was simplicity itself to leave the purloined van in the parking lot of the nearest liquor store. Then I strolled, not ran, back to the riverside to rejoin him.\n\n\"It's the white boat, that one there,\" I pointed it out, pressing on my moustache with my other hand at the same time to make sure that it was securely in place. \"The entire marina is fully automated. I'll get the boat and bring it back here.\"\n\n\"Our cruise is about to begin,\" he said, and there was a merry twinkle in his eye.\n\nI left him there in the sunlight and went to the marina to insert the boat's identification into the operations robot.\n\n\"Good morning,\" it said in a tinny voice. \"You wish to take out the cabin cruiser Lucky Bucks. The batteries have been recharged at a cost of twelve bucks. Storage charges...\"\n\nIt went on like that, reading aloud all the charges that could be clearly seen on the screen\u2014presumably for customers who couldn't read\u2014and there was nothing that could be done about it. I stood on one foot and then the other until it was finished, then pumped in the coins. The machine gurgled and spat out my receipt. Still strolling, I went to the boat, inserted the receipt, then waited for the welcome click when the chain unlocked. Seconds later I was out on the river and heading for the solitary figure on the bank.\n\nSolitary no more. A girl sat beside him.\n\nI circled out and around and she was still there. The Bishop sat slumped and gave me no sign what to do. I circled once more, then the sight of a patrolling police car sent me burbling to the bank. The girl stood and waved, then called out.\n\n\"Why little Jimmy diGriz, as I live and breathe. What a lovely surprise.\"\nChapter 16\n\nLife had had far too many moments like this lately. I looked at the girl more closely as I eased the boat against the bank. She knew me, I should know her; smashingly good-looking, her blouse filled to perfection. Those tulip lips, her!\u2014object of my wildest dreams.\n\n\"Is that you Beth? Beth Naratin?\"\n\n\"How sweet of you to remember me!\"\n\nI was ready to jump ashore with the mooring line, but she took it from my hand and tied it to the bollard there. Over her shoulder I saw the police cruiser go past and keep on cruising. Then glanced at The Bishop, who simply raised his eyes to heaven as she spoke.\n\n\"I said to myself, Beth, I said, that can't be Jimmy diGriz climbing out of that old Macswineys van and wearing a cute little moustache. Not Jimmy who has been in the news so much lately. If it is, why don't I just mosey after him, for old times' sake. Then I saw you talk to this nice gentleman here, before you went off to the marina, so I just made up my mind to wait for you to come back. Going on a trip, are you?\"\n\n\"No, no trip, just a little day excursion up the river and back. Nice seeing you again, Beth.\"\n\nThat was the only nice part about this. Seeing her, I mean. The object of my childish worship. She had left school soon after I had entered\u2014but she was hard to forget. Four years older than me, a real mature woman. That would make her twenty-one now. She had been head of her class, winner of the Beauty Queen of the Year. With good reason. Now, old as she was, she was still a smasher. Her voice sliced through my memories.\n\n\"I don't think that you are being exactly truthful, Jimmy. Why with all these bags and things I bet that you are going on a long cruise. If I were you I would consider a long cruise a really good idea.\"\n\nWas there a different tenor to her voice with these last words? What did she want? We couldn't hang around here much longer. She made her wants clear when she jumped aboard, rocking the boat at its mooring.\n\n\"Always room for one more!\" she called out cheerfully, then went to sit in the bow. I stepped ashore and grabbed up the bags. Then whispered to The Bishop.\n\n\"She knows me. What do we do?\"\n\nHe sighed in answer. \"Very little that we can do. For the present we have a passenger. I suggest that we consider this problem once we are under way. After all\u2014we have no choice.\"\n\nToo true. I passed our belongings over to him, then struggled to untie the black knot that she had tied in the line. Gave the Lucky Bucks a kick out with my foot, jumped aboard, and took the wheel. The Bishop carried the cases below as I switched on the power and headed downstream. Away from Billville, Macswineys and the law.\n\nBut not from Beth. She lay stretched out on the deck before me, skirt hiked up so I could admire those gorgeous lengths of leg. I did this. Then she turned about and smiled, clearly able to read my mind. I forgot about my planned female disguise at this moment\u2014imagining the jeering that would greet my sex change. I was getting angry.\n\n\"All right, Beth, why don't you just spell it out,\" I said, hauling my gaze bodily out to the clear waters of the river.\n\n\"Whatever do you mean?\"\n\n\"Stop the games. You have been watching the news, that's what you said. So you know about me.\"\n\n\"Sure do. Know you hold up banks and escape from jail. That doesn't bother me though. I had a bitsy bit of difficulty myself. So when I saw you, then this boat, I knew you must have some money. Maybe a lot of money. So I just jumped at the chance to take this trip with you. Isn't that nice?\"\n\n\"No.\" I kept my thoughts on the law and not the legs. She was trouble. \"And I do have a bit of money put aside. If I get you some, put you ashore...\"\n\n\"The money, yes. The shore, no. I've seen the last of him and Billville. I'm going to see the world now. And you are going to pay my way.\"\n\nShe snuggled down, with her arms for a pillow, smiling as she enjoyed the sunshine. I looked on gloomily and thought of three or four blows that would snap that delicate neck....\n\nNot even as a joke. This problem could be solved\u2014and without deadly violence. We hummed along, the water parting in white foam at our bow, Billville behind us and green fields opening up ahead at the bend in the river. The Bishop came on deck and sat next to me. With her third presence there was nothing much we could say.\n\nWe continued in silence this way for the better part of an hour, until a dock beside a general store appeared ahead. Beth stirred and sat up, running her fingers through that gorgeous blond hair.\n\n\"You know what\u2014I'm hungry. Bet you are too. Why don't you pull up over there and I'll jump ashore and get us some food and some beer. Isn't that a good idea?\"\n\n\"Great!\" I agreed. She goes into the store, we go into high gear and away.\n\n\"I am skint,\" she smiled. \"Stone broke. If you give me a few bucks I'll buy lunch. I think a thousand will do.\"\n\nHer sweet-little-girl expression never changed while she said this and I wondered just what kind of trouble she was in. Extortion and blackmail maybe; she certainly had the qualifications for that. I dug deep into my wallet.\n\n\"That's nice,\" she said, thumbing through the bundle with glowing eyes. \"I won't be long. And I know you will be here. Jimmy, along with your friend. Haven't I seen him in the news broadcasts too?\"\n\nI glowered after the lovely rotations of her rump as she trotted towards the store.\n\n\"Got our hides nailed to the wall,\" The Bishop said gloomily.\n\n\"Nailed, flayed, and tanned. What do we do?\"\n\n\"Exactly what she says for the time being. Short of killing her, we have very little choice. But I do not believe in killing.\"\n\n\"Nor do I. Although this is the first time that I have understood the temptation.\"\n\n\"What do you know about her?\"\n\n\"Nothing\u2014since I last saw her in school. She says she is in trouble, but I have no idea what she means.\"\n\nHe nodded in thought. \"When we are well away from her I'll get close to a terminal. If she is in the police records I can dig her out.\"\n\n\"Will that do us any good?\"\n\n\"I have no idea, dear boy. We can only try. Meanwhile we must make the best of the situation. We are well away from the terrors of the pork palace, safely away from our pursuers as well. As long as this creature gets money from us we are safe. For the moment. And you must admit that she is decorative.\"\n\nI had no answer to that and could only sit glumly until our uninvited passenger returned.\n\nAfter lunch we continued our voyage downstream. Exhausted by the morning of sunbathing, Beth went below for a beauty nap. The Bishop wanted a turn at the wheel so I showed him the simple controls and pointed out the navigation markers. We had very little to say to each other. But we were thinking a lot. In midafternoon the object of our fierce cogitation trotted up from below.\n\n\"Such a cute little ship,\" she gushed. \"The cutest little girls' room, little kitchen, and everything. But only two little beds. How in the world will we all sleep?\"\n\n\"In shifts,\" I growled, the sound of her voice already getting to me.\n\n\"You always were a card, Jimmy. I think it best if I sleep below. You and your friend can make do.\"\n\n\"Make do, young lady, make do? How does one my age make do on deck when the chill mists of night descend?\" The Bishop's anger was under control, barely, but her bright smile seemed to be unaware of it.\n\n\"I'm sure that you will find a way,\" she said. \"Now I would like to stop at the next town we come to, that one there. I left in such a hurry I forgot all my things. Clothes and makeup, you know.\"\n\n\"You wouldn't need a bit of money to buy those things?\" I asked facetiously. She ignored my feeble humor and nodded.\n\n\"Another thousand will do.\"\n\n\"I'm going below,\" The Bishop said, and did not emerge again until I had tied up and she was gone. He carried two beers and I took one and drank deep.\n\n\"Murder is out,\" he said firmly.\n\n\"Murder is out,\" I agreed. \"But that doesn't mean we can't enjoy thinking about it. What do we do?\"\n\n\"We don't just heave anchor and go. She'll have the police after us in minutes, then will pocket the reward. We must take that into consideration, then think faster than she can. Coming with us was an impulse, obviously. She is greedy for money and we must keep giving it to her. But sooner or later she will decide that she has had enough of ours and will turn us in for the reward, Is there such a thing as a map aboard?\"\n\nThat mighty brain was at work, I could tell that. I asked no questions but rooted out the map as quickly as I could. He traced it with his finger.\n\n\"We are here, I imagine, yes, here is the very place. While downstream, here, is the bustling city of Val's Halla. When will we get there?\"\n\nI squinted at the scale and marked the distance with my thumb. \"Could be there by midafternoon tomorrow, if we get an early start.\"\n\nHis face broke into a smile so wide that his eyes were crinkled half shut. \"Splendid, absolutely splendid. That will do very nicely indeed.\"\n\n\"What will?\"\n\n\"My plans. Which I shall keep to myself for the moment since there are details still to be worked out. When she returns you must agree with me, whatever I say; that is all you have to do. Now, next order of business. Where do we sleep tonight?\"\n\n\"On the river's bank,\" I said, heading below. \"Our friend has all the money that I was carrying, so I must get more from our stock. Then I'm going ashore to buy a tent, sleeping bags, all the gear for comfortable camping out.\"\n\n\"Capital. I shall man the fort and hone my plans until you return.\"\n\nI bought some steaks too, along with a collection of fancy bottles of wine. We needed a major change from the Macswineys cuisine. When the sun was close to the horizon I tied the boat to the trees on the banks of a green meadow, where we could pitch our tent. The Bishop, after smacking his lips over the meat, announced that he would prepare dinner. While he did this, and Beth did her nails, I hammered stakes and got our beds ready. The sun was a ball of orange on the horizon when we tucked into the meal. It was tremendous. No one talked until we were done. When the last morsel was gone The Bishop sighed, raised his glass and sipped, then sighed with repletion.\n\n\"Though I cooked it myself, I must say that meal was a triumph.\"\n\n\"It does take the taste of porcuswine out of the mouth,\" I agreed.\n\n\"I didn't like the wine. Nasty.\" Only her outline was visible in the darkness. Lacking the usual glorious physical accompaniment her voice, as well as her words, left a very lot to be desired. Yet The Bishop's deep basso was free of rancor when he spoke again.\n\n\"Beth\u2014I may call you Beth, mayn't I? Thank you. Beth, we shall be in the city of Val's Halla tomorrow, where I must go ashore and call into my bank. Our funds are running low. You wouldn't like our money to run out, would you?\"\n\n\"No, I wouldn't.\"\n\n\"Thought not. But would you like me to go to the bank and bring you back one hundred thousand bucks in small buck bills?\"\n\nI heard her gasp. Then she fumbled for the switch and the riding lights above the cockpit came on. She was frowning at The Bishop and, for the very first time, lost her cool.\n\n\"Are you trying to play games with me, old man?\"\n\n\"Not at all, young lady. I am simply paying for our safety. You know certain facts that are, shall we say, best left unspoken aloud. I think that sum is a reasonable amount to pay for your continuing silence. Don't you?\"\n\nShe hesitated\u2014then burst out laughing. \"I sure do. Just let me see the color of those bucks and I may even consider letting you boys continue your journey without poor little me.\"\n\n\"Whatever you say, my dear, whatever you say.\"\n\nNor would he speak another word on the subject. We retired soon after that, for it had been a busy day for all of us. Beth took possession of the boat and we had the tent. When I returned from setting the alarms to make sure that the boat would still be there in the morning. The Bishop was already in full snore. Before I slept myself I realized that, whatever he was planning, we had at least one more day of freedom before Beth would think of contacting the police. The lure of that money would ensure her silence. As I dozed off I realized that The Bishop had undoubtedly planned it that way.\n\nWe were humming down the river an hour after dawn, despite Beth's protests. She emerged later, but her anger soon vanished beneath The Bishop's monetary ministrations. He described the interest her invested bucks could earn without her spending any of her capital, touched lightly on the consumer goods she would soon purchase, and generally charmed her like a snake with a rabbit. I had no idea what his plans were but I enjoyed every moment of it.\n\nBy midafternoon I had tied up at the marina on the canal that bisected Val's Halla. The city center was close to hand and The Bishop, beard combed and moustache twirled, was neatly turned out and businesslike.\n\n\"This will not take long,\" he said, then left. Beth looked after him, already atwitch with anticipation.\n\n\"He's really the one they call The Bishop,\" she said when he had gone.\n\n\"I wouldn't know about that.\"\n\n\"Don't give me that old booshwah. I saw the films on 3V, how somebody got him out. A small guy with a moustache. It had to be you.\"\n\n\"Lot of moustaches in this world.\"\n\n\"I never thought, when I saw you around the school, you would ever end up like this.\"\n\n\"I thought the same about you. I admired you from afar.\"\n\n\"So did every other pubescent boy in the school. Don't think I didn't know it. We used to laugh about it, him being a teacher and all that....\"\n\nShe shut up and glowered at me and I smiled sweetly and went below to wash the dinner and breakfast dishes that she had so carefully ignored. I was just finishing up when there was a hail from the shore.\n\n\"Boat ahoy! Permission to come aboard?\"\n\nThe Bishop stood on the dockside, beaming and splendid. His new suit must have cost a small fortune. The suitcase that he held up appeared to be made of real animal skin of some kind, with fittings of glowing gold. Beth's eyes were as wide as saucers. The Bishop climbed aboard and treated us to a conspiratorial wink.\n\n\"Best to get below before I show you what's in this case. It is not for the world to see.\"\n\nBeth led the way and he held the case to his chest until I had closed and locked the door. Then he swept the papers from the table to the deck, placed the case in its center, and with tantalizing precision unlocked and opened the case.\n\nEven I was impressed. There was far more than the hundred thousand here. Beth stared at it\u2014then reached out and tugged a bundle of thousand-buck bills free.\n\n\"Real? Is it real?\" she asked.\n\n\"Guaranteed right from the mint. I saw to that myself.\" With her attention on the money he turned to me. \"Now, Jim, would you mind doing me a favor? Would you find some rope or twine, I'm sure that you will know what you will need. I want absolute silence as well when you tie this girl up so she cannot move.\"\n\nI was expecting something\u2014she was not. Her mouth was just opening to scream when I seized that precious neck and pressed hard just below the ears.\nChapter 17\n\nWith savage glee I cut one of the blankets into strips and bound those delicate wrists and trim ankles. I was just putting sticking tape over her mouth when she came to and tried to scream. It came out as a muffled mewl. \"Can she breathe all right like that?\" The Bishop asked.\n\n\"Perfectly. See the glare in her eye and the angry heaving of that magnificent chest? She is breathing through her nostrils just fine. Now\u2014will you tell me what this is all about?\"\n\n\"On deck, if you please.\"\n\nHe waited until the door was closed behind us before he spoke, rubbing his hands together with joy.\n\n\"Our troubles are over, my boy. I knew that as soon as I looked at the map. There are two things about this fine city that assure me of that. One was the bank, a branch of Galactic Trust with which I have an account\u2014sizable as you have seen. The second fact of interest is that there is a spaceport here.\"\n\nI puzzled over this for a few seconds as my sluggish brain slowly added two and two. Then my jaw gaped so hard I could barely speak.\n\n\"You mean that, us, we... we are going offplanet?\"\n\nHe nodded and grinned. \"Precisely. This little world has become, shall we say, a little too warm for us. It will be even warmer when our female friend is freed. By that time we shall have shaken the dust of Bit O' Heaven from our boots and we will be lightyears away. You did tell me that you wanted to travel?\"\n\n\"I did, of course, but aren't there controls, inspections, police, things like that?\"\n\n\"There are. But customs and immigration can be circumvented if you know how. I know how. And I did check on which ships were here before taking this drastic step. I am sorry that I had no opportunity to warn you\u2014but I was certain that your magnificent reflexes would resolve the matter with ease. When I left here I did not know that this would be the day to put the plan into operation. I intended just to get the money to string the girl along. While keeping track of spacer operations. But the fates are on our side. There is a freighter here from Venia taking on cargo\u2014and leaving in the early hours of the morning. Isn't that wonderful!\"\n\n\"I'm sure that it is. But I would be a lot surer if I knew why.\"\n\n\"Jim, your education has been sorely neglected. I thought every schoolboy knew how venal the Venians were. They are the despair of the League polimetricians. Incorrigible. The motto on Venia is La regloj \u0109iam \u015dansili\u011das. Which may be freely translated as There are no Fixed Rules. That is to say, there are laws about everything\u2014but bribery can change anything. It is not so much that they are a world of criminals, but rather a planet of twisters.\"\n\n\"Sounds nice,\" I agreed. \"Then what have you arranged?\"\n\n\"Nothing yet. But I am positive that opportunity will arrive at the spaceport.\"\n\n\"Yes, sure.\" I was far from enthusiastic. The plan had all the earmarks of improvisation and crossed fingers. But I had little choice. \"What about the girl?\"\n\n\"We'll leave a message for the police with the electronic post, to be delivered after we are gone. Telling them the place where she can be found.\"\n\n\"That place can't be here\u2014too public. There is an automated marina farther downstream. I could tie up there, one of the outer berths.\"\n\n\"The perfect solution. If you will give me instructions how to find it I will hie myself to the spaceport to make the arrangements. Shall we meet there at 2300 hours?\"\n\n\"Fine by me.\"\n\nI watched his impressive form move off in the growing darkness, then started the engine and made a slow turn in the canal. It was dark by the time I reached the marina. But it was brightly lit and the channel was well marked. Most of the boats had tied up close to shore, which was fine by me. I took the outermost berth, well away from the others. Then went below, turned on the lights, and faced the poisonous glare from those lovely eyes. I locked the cabin door behind me, then sat down on the bunk across from Beth.\n\n\"I want to talk to you. If I take off the tape do you promise not to scream? We are well away from the city and there is no one here to hear you in any case. Deal?\"\n\nThe hatred was still there as she nodded reluctantly. I peeled off the tape\u2014then jerked my fingers away just in time as those perfect teeth snapped at my hand.\n\n\"I could kill you, murder you, butcher you, slaughter you...\"\n\n\"Enough,\" I said. \"I'm the one who could do all those things, not you. So shut up.\"\n\nShe shut. Perhaps realizing what her position really was; there was more fear than anger in her eyes now. I didn't want to terrorize a helpless girl\u2014but the murder talk had been her idea. She was ready to listen.\n\n\"You can't be comfortable. So lie still while I untie you.\"\n\nShe waited until her wrists were free, then raked her nails towards my face while I was untying her ankles. I had expected this, so she ended up back on the bunk with the breath knocked out of her.\n\n\"Act reasonable,\" I told her. \"You can be tied and gagged again just as easily. And please don't forget that you brought this on yourself.\"\n\n\"You are a criminal, a thief. Wait until the police get their hands on you....\"\n\n\"And you are a blackmailer. Can we stop the names and games now? Here is what is going to happen. We are going to leave you on this boat and when we are well away the police will be told where to find you. I'm sure that you will tell them a good story. There are express linears from here, as well as the highways. You'll never see us again, nor will they.\" A little misdirection never hurt.\n\n\"I'm thirsty.\"\n\n\"I'll get you something.\"\n\nOf course she made a break for the door when I had my back turned, then tried for my eyes again when I pulled her away. I could understand her feelings\u2014I just wished that she wouldn't.\n\nTime dragged very slowly after that. She had nothing to say that I wanted to hear\u2014and the reverse was obviously true as well. Hours passed in this way before the boat rocked as someone stepped aboard. I dived towards the bunk but she got out one good scream before I could silence her. The door handle rattled and turned.\n\n\"Who is it?\" I called out, crouched and ready for battle.\n\n\"Not a stranger, I assure you,\" the familiar voice said. I unlocked and opened the door with a feeling of great relief.\n\n\"Can she hear me?\" he asked, looking at the silent figure on the bunk.\n\n\"Possibly. Let me secure her again and we'll go on deck.\"\n\nHe went ahead of me and as I closed the door a sudden flare of light lit up the night sky, then climbed in a burning arc up to the zenith.\n\n\"A good omen,\" The Bishop said. \"A deep spacer. All is arranged. And time is of the essence, so I suggest that we grab up our things and leave at once.\"\n\n\"Transportation?\"\n\n\"A rented groundcar.\"\n\n\"Can it be traced?\"\n\n\"I hope so. The rental return is located at the linear station. I've purchased tickets, for both of us you will be happy to hear.\"\n\n\"I mentioned linears to our friend inside.\"\n\n\"Two great minds that work as one. I think I shall manage to drop the tickets where she can see them while we are packing.\"\n\nWe were in and out quite quickly\u2014and I did enjoy the way the unmistakable blue linear tickets dropped on the blankets for an instant. Fell from his pocket while both his hands were engaged elsewhere. Masterful! As I closed the door I could not resist the temptation to blow a kiss towards Beth. I received a glower and a muffled snarl in return, which I surely deserved. She still had a few thousand of our money so she should not complain.\n\nAfter turning in the groundcar we took the levitrain to the linear station. Where we waited until we were alone and unobserved before continuing on to the spaceport. Up until this moment it had been all rush and plan and the reality of what I was doing struck home only when I saw the floodlit flank of a deep spacer looming up ahead.\n\nI was going offplanet! It is one thing to watch the spaceoperas\u2014but another thing completely to venture into space. I felt the goosebumps swell on my arm, the hair stir on my neck. This new life was going to be a good one!\n\n\"Into the bar,\" The Bishop ordered. \"Our man is already here!\"\n\nA thin man in grease-stained spacer gear was just leaving, but dropped back into the booth when he saw The Bishop.\n\n\"Vi estas malfrua!\" he said angrily.\n\n\"V'ere\u2014sed me havas la monon,\" The Bishop answered, flashing a large wad of bills which soothed the other immeasurably. The money changed hands, and after some more conversation another bundle of bills went the way of the first. Greed satisfied, the spaceman led the way to a service van and we climbed into the back. The door was slammed and in the darkness we sped off.\n\nWhat an adventure! Unseen vehicles passed us, then there were strange hammering sounds that came and went, followed by a loud hissing like a giant serpent. We stopped soon after this and our guide came around and opened the rear door. I stepped out first and found myself at the foot of a ramp leading up into what could only be the battered hull of a deep spacer.\n\nNext to the ramp stood an armed guard, staring at me.\n\nIt was all over, the adventure ended before it even began. What could I do? Run, no I couldn't leave The Bishop. He pushed past me while I was still rushing about in circles inside my head, strolled casually over to the guard.\n\nAnd passed him a wad of bills.\n\nThe guard was still counting them when we hurried up the ramp behind our bribed spaceman, struggling to stay close with all the baggage we carried.\n\n\"Eniru, rapide!\" the spaceman ordered, opening the door of a compartment. We pushed through into the darkness as the door closed and locked behind us.\n\n\"Safe harbor!\" The Bishop sighed with relief as he fumbled at the wall until he found the switch, and the lights came on. We were in a small, cramped cabin. There were two narrow bunks and an even smaller bathroom beyond. Pretty grim.\n\n\"Home sweet home,\" The Bishop said, smiling benevolently as he looked around. \"We'll have to stay in here at least two days. So let us stow our gear well out of sight. Otherwise the captain will threaten to return and the bribe will be higher. I'm sure we can last it out.\"\n\n\"I'm not sure I understand all of that. Haven't you paid the bribe already?\"\n\n\"Only the first installments. Bribes are never shared, that is your first lesson in the gentle art. The spaceman got paid to sneak us aboard, and arranged that a friendly guard would be there to take his cut. Those arrangements are in the past. Our presence aboard this ship is unknown to the officers\u2014and particularly the captain who will need a very large payment indeed. You will see.\"\n\n\"I certainly intend to. Bribery is indeed an exacting science.\"\n\n\"It is.\"\n\n\"It's a good thing you speak their language so you can do a deal.\"\n\nHis eyebrows shot up at this and he leaned close. \"You did not understand us?\" he asked.\n\n\"I didn't take foreign languages in school.\"\n\n\"Foreign!\" He looked shocked. \"What a backward part of that porcuswine-rearing planet you must have come from. That was not a foreign language, dear boy. That was Esperanto, the galactic language, the simple, second language that everyone learns early and speaks like a native. Your education has been neglected, but that is easily repaired. Before our next planetfall you shall be speaking it as well. To begin with, all present-tense verbs in all persons end in as. Simplicity itself....\"\n\nHe stopped as someone tried the handle on the cabin door. His finger touched his lips as he pointed to the adjoining bath. I dived that way and turned on the light there just as he turned off the one in the cabin. He joined me in a rush and jammed in beside me as I flicked off the light. He eased the door shut just as the corridor door opened.\n\nFootsteps thudded across the cabin and there was the sound of thin whistling. A routine inspection, nothing to be seen, he would go away in an instant...\n\nThen the bathroom door opened and the light came on. The gold-braided officer looked at The Bishop cramped into the tiny shower, at me crouching on the commode, as he smiled a singularly dirty smile.\n\n\"I thought there was too much activity belowdecks. Stowaways.\" A small gun appeared in his hand. \"Out. You two are going ashore and I am calling the local police.\"\nChapter 18\n\nI leaned forward, getting my weight on my legs, muscles tense. Ready to attack the instant that The Bishop distracted the officer's attention. I really did not want to go against that gun with my bare hands\u2014but I wanted even less to go back to jail. The Bishop must surely have been aware of this. He reached out a restraining hand.\n\n\"Now, let us not be hasty, James. Relax while I talk to this kind officer.\"\n\nHis hand went slowly to his pocket, the gun following his every move, the fingers dipped deep\u2014and came up with a thin wad of credits.\n\n\"This is advance payment for a small favor,\" he said, handing them over to the officer, who took the credits in both hands. Which was easy enough to do now that the gun had vanished just as quickly as it had appeared. He counted while The Bishop talked.\n\n\"The favor we so humbly request is that you do not find us for two days. You will be paid this same sum tomorrow, and again the day after when you discover us and take us to the captain.\"\n\nThe money vanished and the gun reappeared\u2014and I never saw his hands move. He was so good he should have been on the stage.\n\n\"I think not,\" he said. \"I think I will take all the money you have concealed on your person and in your bags. Take it and bring you to the captain now.\"\n\n\"Not very wise,\" The Bishop said sternly. \"I will tell the captain exactly how much you took and he will relieve you of it and you will have nothing. I will also tell him which crewmen were bribed and they will be deprived of their money and you will not be a popular officer on this ship. Will you?\"\n\n\"There is a certain element of truth in what you say,\" he mused, rubbing his jaw in thought, hands empty again. \"If the payments were increased perhaps...\"\n\n\"Ten percent, no more,\" said The Bishop, and the payment was made. \"See you tomorrow. Please relock the door behind you.\"\n\n\"Of course. Have a pleasant journey.\"\n\nThen he was gone and I climbed down from the pot and seized and shook The Bishop's hand. \"Congratulations, sir. A masterful demonstration of a science I scarcely knew existed.\"\n\n\"Thank you, my boy. But it helps to know the ground rules. He never had any intention of turning us out of this ship. That was just his bid. I called it, he raised, I matched and closed. He knew he couldn't squeeze higher because I need a large sum in reserve for the captain. Unspoken, but agreed nevertheless, is my silence about the bribe to him. All done by the rules....\"\n\nHis words were cut off by the loud sound of a hooter in the corridor outside, while a red light began blinking rapidly over the door.\n\n\"Is something wrong?\" I called out.\n\n\"Something is very right. We are ready for takeoff. I suggest that we recline on the bunks because some of these old clunkers put on the Gs when they blast free. A few minutes more and we shake the dust of Bit O' Heaven from our shoes. Preferably forever. That prison, simply terrible, the food...\"\n\nA growing roar drowned out his words and the bunk began to tremble. Then the acceleration of takeoff jumped on my chest. Just like in the films\u2014but far more exciting in reality. This was it! Offplanet! What joys lay ahead.\n\nPretty far ahead still. The mattress was thin and my back hurt from the pressure. Then we went in and out of null-G a few times before they got the artificial gravity right. Or almost right. Every once in awhile it would give a little hiccup. So would my stomach. This happened often enough so that during the next days I didn't miss the meals that I would normally have eaten. At least we had all the rusty, flat water we needed to drink. The officer stayed bribed, I stayed in my bunk most of the time and concentrated on the Esperanto lessons to forget my miseries. After two days of this the gravity finally straightened out and my appetite returned. I looked forward to our release, some more bribery\u2014and some food.\n\n\"Stowaways!\" the officer said when he unlocked the door, staggered, hand over heart. For the benefit of the crewgirl who accompanied him. \"Terrible, unheard of! On your feet, you two, and come with me. Captain Garth will want to know about this.\"\n\nIt was a very convincing performance, spoiled only by his ready hand for the money as soon as the crewgirl's back was turned. She seemed bored by the whole thing and was probably in on the deal herself. We tramped the corridor and up three flights of metal stairs to the bridge. The captain, at least, was shocked to see us. Probably the only one on the ship who didn't know we were aboard.\n\n\"Damn and blast\u2014where did these come from?\"\n\n\"In one of the empty cabins on C deck.\"\n\n\"You were supposed to check those cabins.\"\n\n\"I did, my captain, it is in the log. One hour before takeoff. After that I was on the bridge with you. They must have come aboard after that.\"\n\n\"Who did you bribe?\" Captain Garth said, turning to us, a grizzled old spacedog with a mean look in his eye.\n\n\"No one, captain,\" The Bishop said, sincerity ringing in his voice. \"I know these old Reptile class freighters very well. Just before takeoff the guard at the gangway entered the ship. We came in behind him, unseen, and hid in the cabin. That is all there is to it.\"\n\n\"I don't believe a word of it. Tell me who you bribed or you'll be in the brig and in big trouble.\"\n\n\"My dear captain, your honest crewmen would never take bribes!\" He ignored the unbelieving snort. \"I have proof. All of my not inconsiderable fortune is intact and in my pocket.\"\n\n\"Out,\" the captain instantly ordered all the men in the control room. \"All of you. I'll take this watch. I want to question these two more thoroughly.\"\n\nThe officer and the crewmembers shuffled out, their faces expressionless under his gaze. When they were gone the captain sealed the door and spun about. \"Let's have it,\" he ordered. The Bishop passed over a very tidy sum and the captain riffled through it, then shook his head. \"Not enough.\"\n\n\"Of course,\" The Bishop agreed. \"That is the opening payment. The balance after landfall on some agreeable planet with lax custom officers.\"\n\n\"You ask a lot. I have no desire to risk trouble with planetary authorities by smuggling in illegal immigrants. It will be far easier to relieve you of the money right now and dispose of you as I will.\"\n\nThe Bishop was not impressed at all by this ploy. He tapped his pocket and shook his head. \"Not possible. Final payment is with this registered check for two-hundred thousand credits drawn on Galactic Credit and Exchange. It is not legal tender until I countersign it with a second signature. You may torture me, but I will never sign! Until we are standing on firm ground.\"\n\nThe captain shrugged meaningfully and turned to the controls, making a minor adjustment before he turned back. \"There is a matter of paying for your meals,\" he said calmly. \"Charity does not pay my fuel bills.\"\n\n\"Absolutely. Let us fix a rate.\"\n\nThat appeared to be all there was to it\u2014but The Bishop whispered a warning as we went back down the corridor. \"The cabin is undoubtedly bugged. Our luggage searched. I have all our funds on me. Stay close so there are no accidents. That officer, for one, would make an excellent professional pick-pocket. Now\u2014what do you say to a little food? Since we have paid we can end our enforced fast with a splendid feast.\"\n\nMy stomach rumbled loud agreement with this suggestion, and we made for the galley. Since there were no passengers the fat, unshaven cook served only Venian peasant food. Fine for the natives, but it took some getting used to. Did you ever try to hold your nose and eat at the same time? I didn't ask the cook what we were eating\u2014I was afraid he would tell me. The Bishop sighed deeply and began to fork down his ration of gunge.\n\n\"The one thing I forgot about Venia,\" he said gloomily, \"was the food. Selective memory I am sure. Who would want to recall at any time a feast like this?\"\n\nI did not answer since I was gulping at my cup of warm water to get the taste out of my mouth.\n\n\"Small blessings,\" I said. \"At least the water here isn't as nasty as the stuff from the tap in our cabin.\" The Bishop sighed again.\n\n\"That is coffee that you are drinking.\"\n\nA fun cruise it was not. We both lost weight since it was often better to avoid a meal than to eat it. I continued my studies, learning the finer points of embezzling, expense-account grafting, double- and treble-entry bookkeeping\u2014all done in Esperanto until I was as facile as a native in that fine language.\n\nAt our first planetfall we stayed in the ship since soldiers and customs officers were thick as sandfleas about the ship.\n\n\"Not here,\" the captain said, looking at the screened image of the ground with us. \"Very rich planet, but they don't like strangers. The next planet in this system is one you will like, agricultural, low population, they can use immigrants, so there isn't even a customs office.\"\n\n\"The name?\" The Bishop asked.\n\n\"Amphisbionia.\"\n\n\"Never heard of it.\"\n\n\"Should you have? Out of thirty-thousand settled planets.\"\n\n\"True. But still...\"\n\nThe Bishop seemed troubled and I couldn't understand why. If we didn't like this planet we could liberate enough funds to move on. But some instinct had him on edge. In the end he bribed the purser to use the ship's computer. When we were toying with our dinner he told me about it.\n\n\"Something doesn't smell right about this\u2014smells worse than this food.\" This was a horrifying thought. \"I can find no record of a planet named Amphisbionia in the galactic guide. And the guide is updated automatically every time we land and hook into a planetary communication net. In addition to that, there is a lock on our next destination. Only the captain has the code to access it.\"\n\n\"What can we do?\"\n\n\"Nothing\u2014until after we land. We'll find out then what he is up to.\"\n\n\"Can't you bribe one of the officers?\"\n\n\"I already did\u2014that's how I found out that only the captain knows where we are heading. Of course he didn't tell me until after I paid. A dirty trick. I would have done the same thing myself.\"\n\nI tried to cheer him up, but it was no use. I think the food had affected his morale. It would be a good thing to arrive at this planet, whatever it was. Certainly a good thief can make a living in any society. And one thing was certain. The food would have to be better than the sludge we were reluctantly eating now.\n\nWe stayed in our bunks until the ship touched down and the green light came on. Our meager belongings were already assembled and we carried them down to the airlock. The captain was operating the controls himself. He muttered as the automatic air analyzer ran through its test; the inner lock would not open until it was finished and satisfied with the results. It finally pinged and flashed its little message at him and he hit the override. The great hatch ground slowly open admitting a whiff of warm and pungent air. We sniffed it appreciatively.\n\n\"Here is a stylo,\" Captain Garth said. The Bishop merely smiled.\n\nThe captain led the way and we followed with our bags. It was night, stars were bright above, invisible creatures called from the darkness of a row of trees nearby. The only light was from the airlock.\n\n\"Here will do,\" the captain said, standing on the end of the ramp. The Bishop shook his head as he pointed at the metal surface.\n\n\"We are still on the ship. The ground if you please.\"\n\nThey agreed on a neutral patch close to the ramp\u2014but far enough from the ship to foil any attempt to rush us. The Bishop took out the check, accepted the stylo at last, then wrote his careful signature. The captain\u2014ever suspicious!\u2014compared it with the signature above and finally nodded. He walked briskly up the ramp as we picked up our bags\u2014then turned and called out.\n\n\"They're all yours now!\"\n\nAs the ramp lifted up, out of our reach, powerful lights came on from the darkness, pinning us like moths. Armed men ran towards us as we turned, trapped, lost.\n\n\"I knew something was wrong,\" The Bishop said. He dropped his bags and grimly faced the rushing men.\nChapter 19\n\nA resplendent figure in a red uniform strode out of the darkness and stood before us twisting a large and elegant set of moustaches. Like someone out of a historic flic, he actually wore a sword, which he held firmly by the hilt.\n\n\"I'll take everything you two have. Everything. Quickly!\"\n\nTwo uniformed men came running up to see that we did as we were told. They were carrying strange-looking guns with large barrels and wooden stocks. Behind us I heard a creaking as the ramp came back down with Captain Garth standing on the end of it. I bent over to pick up the bags.\n\nAnd kept turning\u2014diving at the captain, grabbing him.\n\nThere was a loud bang and something whirred by my head and spanged off the ship's hull. The captain swore and swung his fist at me. Couldn't have been better. I stepped inside the blow, grabbed the arm and levered it up into the small of his back. He screeched with pain; a lovely sound.\n\n\"Let him go,\" a voice said, and I looked over the captain's trembling shoulder to see that The Bishop was now lying on the ground with the officer's foot on his chest. And his sword was not just for decoration\u2014because the point of it was now pressed to The Bishop's throat.\n\nIt was going to be one of those days. I gave the captain's neck a little squeeze with my free hand before I let go. He slithered straight down and his unconscious head bonged nicely on the ramp. I stepped away from him and The Bishop climbed unsteadily to his feet, dusting himself off as he turned to our captor.\n\n\"Excuse me, kind sir, but might I humbly ask you the name of this planet on whose soil we stand?\"\n\n\"Spiovente,\" was the grunted answer.\n\n\"Thank you. If you permit, I will help my friend Captain Garth to his feet, for I wish to apologize to him for my young friend's impetuous behavior.\"\n\nNo one stopped him as he turned to the captain, who had just regained consciousness.\n\nHe lost it again instantly as The Bishop kicked him in the side of the head.\n\n\"I am normally not a vindictive man,\" he said, turning away and digging out his wallet. He handed it to the officer and said, \"But just this once I wanted to express my feelings before returning to my normal peaceful self. You understand, of course, why I did that?\"\n\n\"Would have done the same thing myself,\" the officer said, counting the money. \"But the games are over. Don't ever speak to me again or you are dead.\"\n\nHe turned away as another man appeared from the darkness with two black metal loops in his hands. The Bishop stood, numb and unresisting, as the man bent and snapped one onto his ankle. I didn't know what the thing was\u2014but I didn't like it. Mine would not be put on that easily.\n\nYes it would. The muzzle of the gun ground into my back and I made no protest as the thing was snapped into place. The thing-snapper then stood up and looked me in the face, standing so close that his sewer breath washed over me. He was ugly to boot, with a puckered scar that added no improvement to the face. He pushed a sharp finger into my chest as he spoke.\n\n\"I am Tars Tukas, servant of our lord the mighty Capo Doccia. But you never call me by name; you always call me master.\"\n\nI started to call him something, something that was quite an improvement on master, when he pressed a button on a metal box slung from his belt.\n\nThen I was on the ground, trying to shake the red fog of pain from my eyes. The first thing I saw was The Bishop lying before me, groaning in agony. I helped him to his feet; Tars Tukas needn't have done that, not to a man his age. He was grinning a lopsided scarred grin when I turned.\n\n\"Who am I?\" he asked. I resisted all temptation, for The Bishop's sake if not my own.\n\n\"Master.\"\n\n\"Don't forget, and don't try to run away. There are neural repeaters right around the entire country. If I leave this on for long enough, all your nerves stop working. Forever. Understood?\"\n\n\"Understood, master.\"\n\n\"Hand over everything you got on you.\"\n\nI did. Money, papers, coins, keys, watch, the works. He frisked me roughly and seemed satisfied for the moment.\n\n\"Let's move.\"\n\nA tropical dawn had come quickly and the lights were being turned out. We didn't look back as we followed our new master. The Bishop was having difficulty in walking and I had to help him. Tars Tukas led us to a battered wooden cart that was standing close by. We were waved into the back. We sat on the plank seat and watched while crates were lowered from the cargo hatch of the spacer.\n\n\"That was a nice dropkick on the captain,\" I said. \"You obviously know something about his planet that I don't. What was the name?\"\n\n\"Spiovente.\" He spat the word like a curse. \"The millstone around the League's neck. That captain has sold us down the river with a vengeance. And he is a smuggler too. There is a complete embargo on contact with this stinking world. Particularly weapons\u2014which I am sure those cases are full of. Spiovente!\"\n\nWhich didn't really tell me very much other than that it was pretty bad. Which I knew already. \"You couldn't possibly be a bit more informative about this millstone?\"\n\n\"I blame myself completely for getting you involved in all this. But Captain Garth will pay. If we do nothing else, Jim, we will bring him to justice. We'll get word to the League, somehow.\"\n\nThe somehow depressed him even more and he dropped his head wearily onto his hands. I sat in silence, waiting for him to speak in his own good time. He did finally, sitting up, and in the reflected light I saw that the spark was back in his eye.\n\n\"Nil carborundum, Jim. Don't let the bastards wear you down. We are landed in a ripe one this time. Spiovente was first contacted by the League over ten years ago. It had been isolated since the Breakdown and had thousands of years to go bad. It is the sort of place that gives crime a bad name\u2014since the criminals are in charge here. The madhouse has been taken over by the madmen. Anarchy rules\u2014no, not true\u2014Spiovente makes anarchy look like a Boy Sprout's picnic. I have made a particular study of this planet's system of government, while working out the stickier bits of my personal philosophy. Here we have something that belongs in the lost dark ages of mankind's rise. It is thoroughly despicable in every way\u2014and there is nothing that the League can do about it, short of launching an invasion. Which would be completely against League philosophy. The strength of the League is also its weakness. No planet or planets can physically attack another planet. Any one that did would face instant destruction by all the others since war has now been declared illegal. The League can only help newly discovered planets, offer advice and aid. It is rumored that there are covert League organizations that work to subvert repulsive societies like this one\u2014but of course this has never been revealed in public. So what we have here is trouble, bad trouble. For Spiovente is a warped mirror image of the civilized worlds. There is no rule of law here\u2014just might. Criminal gangs are led by Capos, the swordman in the fancy uniform, Capo Doccia, he's one of them. Each Capo controls as large a capote as he can. His followers are rewarded with a portion of the loot extracted from the peasantry or from the spoils of war. At the very bottom of this pyramid of crime are the slaves. Us.\"\n\nHe pointed to the paincuff on his ankle and thoroughly depressed himself. Me as well.\n\n\"Well, we can still look at the bright side,\" I said with desperation.\n\n\"What bright side?\"\n\nI wondered about that myself as I furiously thought out loud.\n\n\"The bright side, yes, there is always a bright side. Like for instance\u2014we are well away from Bit O' Heaven and our problems there. All set for a new start.\"\n\n\"At the bottom of the pile? As slaves?\"\n\n\"Correct! From here the only direction we can go is up!\"\n\nHis lips twitched in the slightest smile at this desperate sally and I hurried on.\n\n\"For example\u2014they searched us and took away everything we had on us. Every item except one. I still have a little souvenir in my shoe from my trip to jail. This.\" I held up the lockpick and his smile widened. \"And it works\u2014see.\" I opened my paincuff and showed it to him, then snapped it back into place. \"So when we are ready to leave\u2014we leave!\"\n\nBy this time the grin had widened into a full smile. He reached out and seized my shoulder in a grip of true comradeship. \"How right you are,\" he beamed. \"We shall be good slaves\u2014for a time. Just long enough to learn the ropes of this society, the chain of command and how to penetrate it, what the sources of wealth are and how to acquire them. As soon as I determine where the chinks are in the structure of society here we shall become rats again. Not stainless steel ones, I am afraid, more of the furry, toothy kind.\"\n\n\"A rat by any other name is just as sweet. We will overcome!\"\n\nWe had to leap aside then as the first of the crates was manhandled into the back of the cart, the fabric of its battered structure squeaking and groaning. When the last of the cases was aboard the loaders climbed in themselves. I was glad the light was so bad\u2014I really did not want to look at them too closely. Three scruffy, dirty men, unshaven and dressed in rags. Unwashed too as my twitching nose quickly informed me. Then a fourth man heaved himself up, bigger and nastier than the others, although his garments were in slightly better shape. He glared down at us and I smelled trouble, in addition to the pong.\n\n\"You know who I am? I'm the Pusher. This is my bunch and you do what I say. The first thing I say is you, old man, take off that jacket. It'll look better on me than on you.\"\n\n\"Thank you for the suggestion, sir,\" The Bishop answered sweetly. \"But I think I shall retain it.\"\n\nI knew what he was doing and I hoped that I was up to it. There was little room to move about in and this thug was twice my size. I had time for one blow, no more, and it had to be a good one.\n\nThe brute roared in anger and started climbing over the crates. The terrified slaves scrambled out of his way. I scrambled aside too and he ignored me as he passed. Perfect. He was just clutching at The Bishop when I hit him in the back of the neck with my joined fists. There was a satisfactory thunk and he collapsed on top of the crate.\n\nI turned to the slaves who were watching in wide-eyed silence.\n\n\"You just got a new pusher,\" I told them, and there were quick nods of agreement. I pointed to the nearest one. \"What's my name?\"\n\n\"Pusher,\" he answered instantly. \"Just don't turn your back on that one when he comes to.\"\n\n\"Will you help me?\"\n\nHis grin exposed blackened, broken teeth. \"Won't help you fight. Warn you though if you don't beat us the way he did.\"\n\n\"No beating. You all help?\"\n\nAll of them nodded agreement.\n\n\"Good. Then your first assignment will be to throw the old pusher out of this cart. I don't want to be too close when he comes to.\"\n\nThey did this with enthusiasm, and added a few kicks on their own initiative.\n\n\"Thank you, James, I appreciate the help,\" The Bishop said. \"My thinking was that you would probably have to fight him sooner or later, so why not sooner, with myself as distraction. And our rise in this society has begun\u2014for you have already climbed out of the basic slave category. Suffering satellites\u2014what is that?\"\n\nI looked where he pointed and my eyes popped just as far out as his. It was a machine of some kind, that much was obvious. It was advancing slowly towards us, rattling and clanking and emitting fumes. The operator swivelled it about in front of the cart as his assistant jumped down and joined the two together. There was a jolt and we slowly got under way.\n\n\"Look closely, Jim, and remember,\" he said. \"You are seeing something from the dawn of technology, long forgotten and lost in the midst of time. That landcar is powered by steam. It is a steamcar, as I live and breathe. You know, I am beginning to think that I will enjoy it here.\"\n\nI was not as fascinated by neolithic machinery as he was. My thoughts were more on the deposed thug and what would happen when he came after me. I had to learn more about the ground rules\u2014and quickly. I moved back to the other slaves, but before I could open conversation we clattered across a bridge and through a gate in a high wall. The driver of our steam chariot stopped and called out.\n\n\"Unload those here.\"\n\nIn my new persona as Pusher I supervised but did little to help. The last case was just dropped to the ground when one of my slaves called out to me.\n\n\"He's coming now\u2014through the gate behind you!\"\n\nI turned quickly. He was right. The ex-pusher was there, scratched and bloody and red-faced with rage.\n\nHe bellowed as he attacked.\nChapter 20\n\nThe first thing that I did was run away from my attacker\u2014who roared after me in hot pursuit. This was done not through fear, though I did have a certain amount of that, but from the need to get some space around me. As soon as I was well away from the cart I turned and tripped him so he sprawled full-length in the muck.\n\nThis drew a big laugh from the onlookers; I took a quick glance around while he was climbing to his feet. There were armed guards, more slaves\u2014and the red-garbed Capo Doccia who had cleaned us out. An idea began to form\u2014but before it took shape I had to move to save my life.\n\nThe thug was learning. No more wild rushing about. Instead he came slowly towards me, arms spread, fingers extended. If I allowed him a sweet embrace I would not emerge from it alive. I backed slowly, turning to face Capo Doccia, moved to one side, then stepped quickly forward. Seizing one of my attacker's outstretched hands in both of mine, pulling and falling backwards at the same time. My weight was just about enough to send him flying over me to sprawl full-length again.\n\nI was on my feet at once\u2014with the plan clear in my mind. An exhibition.\n\n\"That was the right arm,\" I called out loudly.\n\nHe was stumbling when he returned to the attack so I took a chance and called my shot.\n\n\"Right knee.\"\n\nI used a flying kick to get him on the kneecap. This is quite painful and he screamed as he dropped. He was slower getting to his feet this time, but the hatred was still there. He was not going to stop until he was unconscious. Good. All the better for my demonstration of the art.\n\n\"Left arm.\"\n\nI seized it and twisted it up behind his back, held it there, pushing hard. He was strong\u2014and still fighting, trying to clutch me with his right hand, struggling to trip me. I got in first.\n\n\"Left leg,\" I shouted as I kicked hard on the back of his calf and he went down another time. I stepped back and looked towards Capo Doccia. I had his undivided attention.\n\n\"Can you kill as well as dance?\" he asked.\n\n\"I can. But I chose not to.\" I was aware that my opponent had stood up, was swaying from side to side. I turned slightly so I could see him out of the corner of my eye. \"What I prefer is to render him unconscious. That way I win the fight\u2014and you still have a slave.\"\n\nThe thug's hands closed on my neck and he bubbled viciously. I was showing off and I knew it. But I had to provide a good performance for my audience. So, without looking at all, I slammed backwards with my bent arm. Sinking my elbow hard into his gut, in the center, just below the rib cage, in line with the elbows. Right into the nerve ganglion known as the solar plexus. His hands loosened and I stepped forward. Hearing the thud as he hit the ground. Out cold.\n\nCapo Doccia signaled me to him, spoke when I was close.\n\n\"That is a new way to fight, offworlder. We make wagers on the ruffians here who battle with their fists, striking each other until the blood flows and one of them cannot go on.\"\n\n\"Fighting like that is crude and wasteful. To know where to strike and how to strike, that is an art.\"\n\n\"But your art is of no value against sharp steel,\" he said, half-pulling his sword. I had to tread carefully now or he would be chopping me up just to see what I could do.\n\n\"Bare hands cannot stand against one such as you who is a master of the blade.\" For all I knew he only used the thing to carve his roast, but flattery always helps. \"However against an unskilled swordsman or knife-wielder the art has value.\"\n\nHe digested that, then called to the nearest guard. \"You, take your knife to this one.\"\n\nThis was getting out of hand\u2014but I could see no way to avoid the encounter now. The guard smiled and pulled a shining length of dagger from its sheath and stalked towards me. I smiled in return. He raised it over his head to stab down\u2014not holding it pointed directly out before him like an experienced knife-fighter. I let him come on, unmoving until he struck.\n\nStandard defense. Step inside the blow, take the impact of his wrist against my forearm. Seize the knife-wrist with hands, turn and twist. All of this done very fast.\n\nThe knife went one way, he went the other. I had to end this demonstration quickly before I was taking on clubs, guns, whatever the head thug felt like. I stepped closer to Capo Doccia and spoke in a quiet voice.\n\n\"These are offworld secrets of defense\u2014and killing\u2014that are unknown here on Spiovente. I do not wish to reveal more here. I am sure you do not wish slaves to learn dangerous blows like these. Let me show you what can be done without this raw audience. I can train your bodyguards in these skills. There are those who want to kill you. Think of your own security first.\"\n\nIt sounded like a lecture on traffic safety to me, but it seemed to make sense to him. But he wasn't completely convinced.\n\n\"I do not like new things, new ways. I like things as they are.\"\n\nRight, with him on top and the rest in chains below. I talked fast.\n\n\"What I do is not new\u2014but as old as mankind. Secrets that have been passed on in secret since the dawn of time. Now these secrets can be yours. Change is on the way, you know that, and knowledge is strength. When others seek to take what you have, any weapon is useful to defeat them.\"\n\nIt sounded like nonsense to me\u2014but I hoped that it made sense to him. From what The Bishop had told me about this garbage world, the only security was in strength\u2014paranoia paid off. At least it had him thinking, which from the narrowness of his forehead was something he probably found hard to do. He turned on his heel and walked away.\n\nPoliteness, like soap, was also unknown on this planet. No \"see you later\" or \"let me think about that.\" It took me a few moments to realize that the audience was over. The disarmed guard was glaring at me and rubbing his wrist. But he had put the dagger away. Since I had talked with Capo Doccia I now had some status, so he wouldn't knife me without reason. Which left my first protagonist, the ex-Pusher. He was sitting up dizzily when I approached. He looked up at me, blinking and befuddled. I tried to look my meanest when I spoke.\n\n\"That is two times you have come at me. You will not do it a third time. Third time means out in my ball game. You will die if you try anything ever again.\"\n\nThe hatred was still there in his face\u2014but there was fear as well. I stepped forward and he cringed back. Good enough. As long as I didn't turn my back on him very often. I turned it now and stalked away.\n\nHe shambled after me and joined the waiting gang of slaves. He seemed to have accepted his demotion, as had the others. There were a few black looks in his direction but no more violence. Which was fine by me. It is one thing to work out in the gym\u2014but something totally different here mixing with these heavies really trying to kill me. The Bishop beamed his congratulations.\n\n\"Well done, Jim, well done.\"\n\n\"And all very tiring. What next?\"\n\n\"From what I could discover this little group is off duty, so to speak, having worked during the night.\"\n\n\"Then rest and food are in order. Lead on.\"\n\nI suppose it could be called food. About the only good thing I could say about it was that it was not as repulsive as the Venian cooking aboard the spacer. A large and exceedingly filthy pot was seething over a fire to the rear of the building. The chef\u2014if one dared use that term for this repulsive individual, as filthy as his pot\u2014was stirring the contents with a long wooden spoon. The slaves each took a wooden bowl from the dripping pile on the table close by and these were filled by the cook. There was no worry about lost or broken cutlery because there wasn't any. Everyone dipped and shoveled with their fingers, so I did the same. It was vegetable gruel of some kind, pretty tasteless, but filling. The Bishop sat next to me on the ground, back to the wall, and slowly ate his. I finished first and had no difficulty in restraining a desire for a second serving.\n\n\"How long do we stay slaves?\" I asked.\n\n\"Until I learn more about how things operate here. You have spent your entire life on a single planet, so both consciously and unconsciously you accept the society you know as the only one. Far from it. Culture is an invention of mankind, just like the computer or the fork. There is a difference though. While we are willing to change computers or eating instruments, the inhabitants of this culture will brook no changes at all. They believe that theirs is the only and unique way to live\u2014and anything else an aberration.\"\n\n\"Sounds stupid.\"\n\n\"It is. But as long as you know that, and they don't, you can step outside the rules or bend them for your own benefit. Right now I'm finding out what the rules are here.\"\n\n\"Try not to take too long.\"\n\n\"I promise not to since I am not that comfortable myself. I must determine if vertical mobility exists and how it is organized. If there is no vertical mobility, we will just have to manufacture it.\"\n\n\"You have lost me. Vertical what?\"\n\n\"Mobility. In terms of class and culture. Take for example these slaves and the guards outside. Can a slave aspire to be a guard? If he can, then there is vertical mobility. If he cannot, this is a stratified society and horizontal mobility is all that can be accomplished.\"\n\n\"Such as becoming top slave and kicking all other slaves?\"\n\nHe nodded. \"You have it, Jim. We shall cease being slaves as soon as my studies show how that is possible. But first we need some rest. You will observe that the others are now asleep on the straw to the rear of this noisome building. I suggest we join them.\"\n\n\"Agreed....\"\n\n\"You, get over here.\"\n\nIt was Tars Tukas. And of course he was pointing at me. I had a feeling that it was going to be a very long day.\n\nAt least I was seeing more of the sights. We crossed the courtyard, scene of my triumphs, and up a flight of stone steps. There was an armed guard here and two more inside lolling about on a wooden bench. A bit more luxury too. Woven mats on the floors, chairs, and tables, a few bad portraits on the wall, some with a rough resemblance to Capo Doccia. I was hustled right along into a large room with windows that faced out over the outer wall. I could see fields and trees and little else. Capo Doccia was there, along with a small band of men, all drinking from metal cups. They were well dressed, if multicolored leather trousers and billowing shirts and long swords is your idea of well dressed. Capo Doccia waved me over.\n\n\"You, come here and let us look at you.\"\n\nThe others turned with interest and eyed me like an animal on auction.\n\n\"And he actually knocked the other one down without using his fists?\" One of them said. \"He is so weak and puny, not to mention ugly.\"\n\nThere are times when the mouth should be opened only to put in food. This was probably one of them. But I was tired, fed up with my lot, and generally in a foul temper. Something snapped.\n\n\"Not as weak, puny or ugly as you, you pig's git.\"\n\nThis got his attention all right. He howled with instant anger, turned bright red\u2014then drew a long steel blade and rushed at me.\n\nI had little time to think, less time to act. One of the other dandies was standing close by, his metal drinking mug held loosely. I grabbed it from him, turned, and threw the contents in the attacking man's face.\n\nMost of it missed, but enough dripped down onto his clothes to infuriate him even more. He slashed down with his sword and I caught the blow on the mug, diverting it. Letting the mug slide up along the blade into his fingers, grabbing and twisting his sword arm at the same time.\n\nHe howled nicely and the sword clattered to the floor. After this he was turned sideways, nicely exposed for a finishing kick to the back.\n\nExcept someone tripped me from behind at that moment and I went sprawling.\nChapter 21\n\nThey thought this very amusing because their laughter was all I could hear. When I scrambled for the fallen blade one of them kicked it aside. Things were not looking too good. I couldn't fight them all. I had to get out.\n\nIt was too late. Two of them knocked me to the ground from behind and another one kicked me in the side. Before I could get up my sword-wielding opponent was on top of me, kneeling on my chest and drawing an exceedingly ugly dagger with a wavery edge.\n\n\"What is this creature, Capo Doccia,\" he called out, holding my chin with his free hand, the dagger close to my throat.\n\n\"An offworlder,\" Capo Doccia said. \"They threw him off the spacer.\"\n\n\"Is it valuable, worth anything?\"\n\n\"I don't know,\" Capo Doccia said, looking down at me bemusedly. \"Perhaps. But I don't like its fancy offplanet tricks. They don't belong here. Oh, kill it and be done.\"\n\nI had not moved during this interesting exchange because I had some obvious interest in its outcome. I moved now.\n\nThe knife-wielder screamed as I twisted his arm\u2014breaking it I hope\u2014and grabbed the dagger as his fingers flew open. I held on to him as I jumped to my feet, then pushed him into the midst of his companions. They were behind me as well, but they fell back as I swept the dagger about in a circle. Moving after it, running before they could get their own weapons out. Running for my life.\n\nThe only direction I knew, back down the stairs. Bumping into Tars Tukas and rendering him unconscious as I passed.\n\nRoars and shouts of anger sounded behind me and I wasted no time even glancing their way. Down the stairs, three at a time, towards the guards at the entrance. They were still scrambling to their feet when I plowed into them and we all went down. I kneed one under the chin as we fell, grabbing his gun by the barrel as I did this. The other was struggling to point his weapon at me when I caught him in the side of the head with the one I was holding.\n\nThe running feet were right behind me as I charged through the door, right at the surprised guard. He drew his sword but before he could use it he was unconscious. I dropped the dagger and seized his more lethal sword and ran on. The gate we had entered by was ahead. Wide open.\n\nAnd well-guarded by armed men who were already raising their guns. I angled off towards the slave building as they fired. I don't know where their shots went but I was still alive as I turned the corner.\n\nOne knife, one gun, one very tired Jim diGriz. Who did not dare stop or even slow down. The outer wall was ahead\u2014with scaffolding and a ladder leaning against it where masons were making repairs. I screeched and waved my weapons and the workmen dived in all directions. I went up the ladders as fast as I could. Noticing that bullets were striking the wall on all sides of me, chips of stone flying.\n\nThen I was on top of the wall, fighting for breath, chancing a look behind me for the first time.\n\nDropping to my face as the massed gunmen below fired a volley that parted the air just above my head. Capo Doccia and his court had left the pursuit to the guards and were standing behind them cursing and waving their weapons. Very impressive. I pulled my head back as they fired again.\n\nOther guards were climbing up to the wall and moving towards me. Which really did limit my choices a bit. I looked over the outside of the wall at the brown surface of the water that lay at its base. Some choice!\n\n\"Jim, you must learn to do something about your big mouth,\" I said, then took a deep breath and jumped.\n\nSplashed\u2014and stuck. The water was just up to my neck and I was stuck in the soft mud that had broken my fall. I struggled against it, pulling out one foot, then the other, struggling against its gluey embrace as I waded to the far bank. My pursuers weren't in sight yet\u2014but they would surely be right behind me. All I could do was keep moving. Crawling up the grassy bank, still clutching my purloined weapons, then staggering into the shelter of the trees ahead. And still no sign of the armed guards. They should be across the bridge and after me by this time. I couldn't believe my good luck.\n\nUntil I fell headlong, screaming as the pain washed over me. Pain unbelievable, blotting out sight, sound, senses.\n\nThen it stopped and I brushed the tears of agony from my eyes. The paincuff\u2014I had forgotten all about it. Tars Tukas had regained consciousness and was thumbing the control button. What had he said? Leave it on long enough and it blocks all the nerves, kills. I grabbed at my shoe and the lockpick concealed there as the pain struck again.\n\nWhen it stopped this time I was almost too weak to move my fingers. As I fumbled with the pick I realized that they were sadists and I should be grateful for the fact. With the button held down I was good as dead. But someone, undoubtedly Capo Doccia, wanted me both to suffer and to know that there was no way out. The key was in the lock when pain consumed me one more time.\n\nWhen it stopped I was lying on my side, the lockpick fallen from my fingers, unable to move.\n\nBut I had to move. Another wave of agony like that and it would be all over for me. I would lie in these woods until I died. My fingers trembled, moved. The pick crept towards the tiny opening of the lock, moved in, twisted feebly....\n\nIt took a very long time for the red mists to clear from my vision, the agony to seep out of my body. I could not move, felt I would never stir again. I had to blink the tears away when I could see. See the most beautiful sight in the world.\n\nThe open paincuff lying on the moldy leaves.\n\nOnly my captor's knowledge that the pain machine led to certain death had saved my life. The searchers were in no hurry; I could hear them talking as they moved through the woods towards me.\n\n\"... somewhere in here. Why don't they just leave him?\"\n\n\"Leave a good blade and a shooter? No chance of that. And Capo Doccia wants to hang the body up in the courtyard until it rots. Never saw him that angry.\"\n\nLife slowly returned to my paralyzed body. I moved off the animal track I had been following and pulled myself into the shelter of the low shrubbery, reaching out to straighten out the grass. And not too soon.\n\n\"Look\u2014he came out of the water here. Went along this path.\"\n\nHeavy footsteps approached and went by. I clutched my weapons and did the only thing possible. Lay quiet and waited for my strength to return.\n\nThis was, I must admit, a bit of a low point in my life. Friendless, alone, still throbbing with pain, exhausted, hunted by armed men just dying to kill me, thirsty.... It was quite a list. About the only thing that hadn't happened so far was getting rained on.\n\nIt started to rain.\n\nThere are high and low points in emotion when there is no room for excess. To love one so much it would be impossible to love any more. I think. Never having had any personal experience in that. But I had plenty of experience in being in the pits. Where I was now. I could sink no lower or get no more depressed. It was the rain that did it. I began to chuckle\u2014then grabbed my mouth so I wouldn't laugh out loud. Then the laughter died away as my anger grew. This was no way to treat a mean and nasty stainless steel rat! Now in danger of getting rusty.\n\nI moved my legs and had to stifle a groan. The pain was still there but the anger rode it down. I clutched the gun and stuck the sword into the ground, then pulled myself to my feet by grabbing the branches of the tree with my free hand. Grabbed up the sword again and stood there, swaying. But not falling. Until I was finally able to stagger off, one step at a time, away from the searchers and Capo Doccia's criminal establishment.\n\nThe forest was quite extensive and I moved along game paths for an unmeasurable length of time. I had left the searchers far behind, I was sure of that. So when the forest thinned and ended I leaned against a tree to catch my breath and looked out at the tilled field. It was time to find my way back to the haunts of man. Where there were plows there were ploughboys. They shouldn't be too hard to find. When a certain measure of strength had returned I staggered off along the edge of the field, ready to fall into the forest at the sight of armed men. I was very pleased to see the farmhouse first. It was low to the ground, thatched, and windowless\u2014at least on this side. It had a chimney from which there rose a thin trickle of smoke. No need for heating in this balmy climate\u2014so this must be a cooking fire. Food.\n\nAt the thought of food my neglected stomach began to churn, rumble, and complain. I felt the same way. Food and drink were next in order. And what better place to find them than at this isolated farm? The question was the answer. I stumbled across the furrows to the back of the house, worked my way around the side to the front. No one. But there were voices coming from the open doorway, laughter\u2014and the smell of cooking. Yum! I sauntered into the open, along the front and through the front door.\n\n\"Hi, folks. Look who has come to dinner.\"\n\nThere were a half-dozen of them grouped around the scrubbed wood table. Young and old, thick and thin. All with the same expression on their faces. Jaw-dropped astonishment. Even the baby stopped crying and aped its elders. A grizzled oldster broke the spell, scrambling to his feet in such a hurry his three-legged stool tumbled over.\n\n\"Welcome, your honor, welcome.\" He tugged his forelock as he bowed to show how grateful he was for my presence. \"How may we aid you, honored sir?\"\n\n\"If you could spare a bit of food...\"\n\n\"Come! Sit! Dine! We have but humble fare but willingly share it. Here!\"\n\nHe straightened his stool and waved me to it. The others scampered away from the table so I wouldn't be disturbed. Either they were discerning judges of human nature and knew what a sterling fellow I was\u2014or they had seen the sword and gun. A wooden plate was filled from the pot hung over the fire and put before me. Life here was a cut above the slave pens for I was also supplied with a wooden spoon. I tucked in with a great deal of pleasure. It was a vegetable stew, with the occasional shard of meat, garden fresh of course, and tasted wonderful. There was cool water to drink out of a clay cup and I could have asked for nothing more. While I shoveled it all into my face I was aware of low whispering from the farmers gathered at the far end of the room. I doubted if they were planning anything violent. Nevertheless I kept one eye on them, and my hand not far from the hilt of the sword laid out on the table.\n\nWhen I had finished and belched loudly\u2014they buzzed warmly at this gustatory approval\u2014the old man detached himself from the group and shuffled forward. He pushed before him a shock-headed youth who looked to be about my age.\n\n\"Honored sir, may I speak with you?\" I waved agreement and belched again. He smiled at this and nodded. \"Ahh, you are kind enough to flatter the cook. Since you are obviously a man of good wit and humor, intelligent and handsome, as well as being a noted warrior, permit me to put a small matter to you.\"\n\nI nodded again; flattery will get you everywhere.\n\n\"This is my third son, Dreng. He is strong and willing, a good worker. But our holding is small and there are many mouths to feed, as well as giving half of what we produce to the so-wonderful Capo Doccia for our protection.\"\n\nHe had his head lowered when he said this, but there were both submission and hatred in his voice. I imagine the only one that Capo Doccia protected them from was Capo Doccia. He pushed Dreng forward and squeezed his bicep.\n\n\"Like rock, sir, he is very strong. His ambition has always been to be a mercenary, like your kind self. A man of war, armed and secure, selling his services to the gentry. A noble calling. And one which would enable him to bring a few groats home to his family.\"\n\n\"I'm not in the recruiting business.\"\n\n\"Obviously, honored sir! If he went as a pikeman with Capo Doccia, there would be no pay or honor, only an early death.\"\n\n\"True, true,\" I agreed, although I had heard this fact for the first time. The old boy's train of thought wandered a bit, which was fine by me since I was getting an education into life on Spiovente. Didn't sound nice at all. I sipped some more water and tried to summon up another burp to please the cook, but could not. Old dad was still talking.\n\n\"Every warrior, such as yourself, should have a knave to serve him. Dare I ask\u2014we have looked outside and you are alone\u2014what happened to your knave?\"\n\n\"Killed in battle,\" I improvised. He looked dumbfounded at this and I realized that knaves weren't supposed to fight. \"When the enemy overran our camp.\" That was better, nodded agreement to this. \"Of course I killed the blackguard who butchered poor Smelly. But that's what war is about. A rough trade.\"\n\nAll of my audience murmured understandingly, so I hadn't put a foot wrong so far. I signaled to the youth.\n\n\"Step forward, Dreng, and speak for yourself. What is your age?\"\n\nHe peered out from under his long hair and stammered an answer. \"I'll be four, come next Wormfeast Day.\"\n\nI wanted no details of this repulsive holiday. He was sure big for his age. Or this planet had a very long year. I nodded and spoke.\n\n\"A good age for a knave. Now tell me, do you know what the knavely duties are?\" He better, because I certainly didn't. He nodded enthusiastically at my question.\n\n\"That I do, sir, that I do. Old Kvetchy used to be a soldier, told me all about it many a time. Polish the sword and gun, fetch the food from the fire, fill the water bottle, crack the lice with stones...\"\n\n\"Fine, great, I can see you know it all. Down to the last repulsive detail. In exchange for your services you expect me to teach you the trade of war.\" He nodded quick agreement. The room was hushed as I pondered my decision.\n\n\"Right then, let us do it.\"\n\nA bucolic cry of joy echoed from the thatch and old dad produced a crock of what could only be home brew. Things were looking up for me, ever so slightly, but certainly looking up.\nChapter 22\n\nWork appeared to have ceased for the day with the announcement of Dreng's new job. The home brew was pretty awful stuff, but obviously contained a fair measure of alcohol. Which seemed like a good idea at the time. I drank enough to kill the pain, then slacked off before I ended up drunk on the floor like the rest of them. I waited until old dad was well on the way to alcoholic extinction before I pumped him for information.\n\n\"I have traveled from afar and am ignorant of the local scene.\" I told him. \"But I do hear that this local bully, Capo Doccia, is a little on the rough side.\"\n\n\"Rough!\" he growled, then slurped down some more of the paint thinner. \"Poisonous serpents flee in fear when he approaches, while it is well-known that the gaze of his eyes kills infants.\"\n\nThere was more like this, but I turned off my attention. I had waited too long in the drinking session to extract any reasonable information from him. I looked around for Dreng and found him just tucking into a great crock of the brew. I pried it away from him, then shook him until I attracted his attention.\n\n\"Let's go. We're leaving now.\"\n\n\"Leaving...?\" He blinked rapidly and tried to focus his eyes on me. With little success.\n\n\"We. Go. Out. Walkies.\"\n\n\"Ahh, walkies. I get my blanket.\" He stood swaying, then gave me some more rapid blinks. \"Where's your blanket for me to carry?\"\n\n\"Seized by the enemy, along with everything else I possessed other than my sword and gun, which never leave my side while I have a breath in my body.\"\n\n\"Breath in body... Right. I'll get blanket. Get you blanket.\"\n\nHe rooted about in the rear of the room and appeared with two fuzzy blankets, despite a lot of domestic and female crying about the cold of winter. Capital goods were not easy to come by for the peasantry. I would have to get some groats for Dreng eventually.\n\nHe reappeared with the blankets draped over his shoulders along with a leather bag, a stout staff in his hand\u2014and a wicked-looking knife in a wooden scabbard at his waist. I waited outside to avoid the tearful traditional departure scene. He eventually emerged, looking slightly more sober, and stood swaying at my side.\n\n\"Lead on, master.\"\n\n\"You show me the way. I want to visit Capo Doccia's keep.\"\n\n\"No! Can it be true that you fight for him?\"\n\n\"That is the last thing I would ever do. In fact I would fight against him for a wooden groat. The truth is that the Capo has a friend of mine locked away in there. I want to get a message to him.\"\n\n\"There is great danger in even going close to his keep.\"\n\n\"I'm sure of it, but I am fearless. And I must contact my friend. You lead the way\u2014and through the woods if you don't mind. I don't want to be seen by either Capo Doccia or his men.\"\n\nObviously neither did Dreng. He sobered up as he led me by obscure paths and hidden ways to the other side of the forest. I peered out carefully at the roadway leading to the drawbridge, to the entrance to the keep.\n\n\"Any closer and they will see us,\" he whispered. I looked up at the late afternoon sun and nodded agreement.\n\n\"It's been a busy day. We'll lay up in the woods here and make our move in the morning.\"\n\n\"No move. It's death!\" His teeth chattered though the afternoon was hot. He hurried as he led the way deeper into the forest, to a grassy hollow with a stream running through it. He produced a clay cup from his bag, filled it with water and brought it to me. I slurped and realized that having a knave wasn't a bad idea after all. Once his chores were done he spread the blankets on the grass and promptly fell asleep on his. I sat down with my back to a tree and, for the first time, had a chance to examine the gun I had lifted.\n\nIt was sleek and new and did not fit this broken-down planet at all. Of course\u2014it had to be from the Venian ship. The Bishop said that they had probably been smuggling weapons. And I was holding one of them in my hands. I looked at it more closely.\n\nNo identification, or serial number\u2014or any other indication where it had been manufactured. And it was pretty obvious why. If the League agents succeeded in getting their hands on one of these it would be impossible to trace it back to the planet of origin. The gun was small in size, about halfway between a rifle and a pistol. I can claim some acquaintance with small arms\u2014I am an honored member of the Pearly Gates Gun Club and Barbecue Society because I am a pretty good shot and helped them win tournaments\u2014but I had never seen anything like this before. I looked into the muzzle. It was about .30 caliber, and unusually enough it was a smooth bore. It had open, iron sights, a trigger with safety button, one other lever on the stock. I turned this and the gun broke in half and a handful of small cartridges fell to the ground. I looked at one closely and began to understand how the gun worked.\n\n\"Neat. No lands or grooves so there is no worry about keeping the barrel clean. Instead of rotating, the bullet has fins to keep it in straight flight. And, uggh, make a nastier hole in anyone it hits. And no cartridge case either\u2014this is solid propellent. Does away with all the worries about ejecting the brass.\" I peeked into the chamber. \"Efficient and foolproof. Push your cartridges into the recessed stock. When it's full put one more into the chamber. Close and lock. A little solar screen here to keep a battery charged. Pull the trigger, a spot in the chamber glows hot and ignites the charge. The expanding gas shoots out the bullet\u2014while part of the gas is diverted to ram the next bullet into the chamber. Rugged, almost foolproof, cheap to make. And deadly.\"\n\nDepressed and tired, I lay the gun beside me, dropped the sword close to hand, lay back on the blanket and followed Dreng's good example.\n\nBy dawn we were slept out and slightly hungover. Dreng brought me water, then handed over a strip of what looked like smoked leather. He took one himself and began chewing on it industriously. Breakfast in bed\u2014the greatest! I bit my piece and almost broke a tooth. It not only resembled smoked leather, but tasted exactly like it as well.\n\nBy the time that the drawbridge clanked down for the day we were lying in a copse on the hill above it, as close as we could get. It was the nearest cover that we could find since, for pretty obvious reasons, all the trees and shrubs had been cleared away from the approaches to the gate. It wasn't as near as I liked, but would have to do. But it was far too close for Dreng for I could feel him shivering at my side. The first thing to emerge from the gate was a small body of armed men, followed by four slaves dragging a cart.\n\n\"What's going on?\" I asked.\n\n\"Tax collecting. Getting in their share of the crops.\"\n\n\"We've now seen who comes out\u2014but do any of your farmers ever go in?\"\n\n\"Madness and death! Never!\"\n\n\"What about selling them food.\"\n\n\"They take all they want from us.\"\n\n\"Do you sell them firewood?\"\n\n\"They steal what they need.\"\n\nThey had a pretty one-sided economy, I thought gloomily. But I had to come up with something\u2014I just couldn't leave The Bishop as a slave in this dismal place. My cogitation was interrupted by a commotion inside the gate. Then, as though my thoughts had coalesced into reality, a figure burst out of the gate, knocking aside the guard there, rushing on.\n\nThe Bishop!\n\nRunning fast. But right behind him were the pursuing guards.\n\n\"Take this and follow me!\" I shouted, jamming the hilt of the sword into Dreng's hand. Then I was off down the slope as fast as I could go, shouting to draw their attention. They ignored me until I fired a shot over their heads.\n\nThings got pretty busy after that. The guards slowed, one even dived to the ground and put his hands over his head. The Bishop pelted on\u2014but one of his pursuers was right behind him, swinging a long pike. Catching The Bishop on the back and knocking him down. I fired again as I ran, jumped over The Bishop and felled the pikeman with the butt of my gun.\n\n\"Up the hill!\" I called out when I saw that The Bishop was struggling to his feet, blood all over his back. I banged off two more shots, then turned to help him. And saw that Dreng was clutching the sword\u2014but still lying on top of the hill.\n\n\"Get down here and help him or I'll kill you myself!\" I shouted, turning and firing again. I hadn't hit anyone but I was sure keeping their heads down. The Bishop stumbled on and Dreng, having plumbed some deep well of decency\u2014or in fear that I would kill him\u2014was coming to our aid. Shots were whistling past us now so I spun and returned their fire.\n\nWe reached the top of the low hill, went over it towards the relative safety of the woods. Dreng and I half carried the great form of The Bishop as he stumbled and staggered. I took a quick\u2014and reassuring\u2014look at his back. There was a shallow cut there, nothing too bad. Our pursuers were still not in sight when we crashed through the bushes and reached the safety of the trees.\n\n\"Dreng\u2014lead us out of here. They mustn't catch us now!\"\n\nSurprisingly enough they didn't. The farm lad must have played in these woods for all of his young life because he knew every track and path. But it was hard work. We staggered on, then struggled our way along a steep grassy slope with a few miserable bushes halfway up. Dreng pulled the bushes aside to reveal the entrance to a shallow cave.\n\n\"Chased a Furry in here once. No one else knows about it.\"\n\nThe entrance was low and it was a labor to pull The Bishop through. But once inside, the cave opened out and there was more than enough space to sit up, although it wasn't high enough to stand. I took one of the blankets and spread it out, then rolled The Bishop onto it so that he lay on his side. He groaned. His face was filthy and bruised. He had not an easy time of it. Then he looked towards me and smiled.\n\n\"Thank you, my boy. I knew you would be there.\"\n\n\"You did? That's more than I knew.\"\n\n\"Nonsense. But, quickly please, the...\"\n\nHe writhed and moaned and his body arched into the air with unbearable pain. The paincuff\u2014I had forgotten about it! And it was receiving a continuous signal, certain death.\n\nHaste makes waste. So I controlled my anxiety and slowly slipped off my right shoe, opened the compartment, and seized the lockpick firmly in my fingers. Bent over, inserted it\u2014and the cuff sprang open. Pain lanced through my hand, numbing it, as I threw the thing aside.\n\nThe Bishop was unconscious and breathing heavily. There was nothing more I could do except sit and wait.\n\n\"Your sword,\" Dreng said, holding it out to me.\n\n\"You take care of it for awhile. If you think you are up to it?\"\n\nHe lowered his eyes and trembled again. \"I want to be a fighter, but I am so afraid. I could not move to help you.\"\n\n\"But you did\u2014finally. Remember that. There isn't a person alive who has not been afraid at one time or another. It is only the brave man who can feel fear and still go forward.\"\n\n\"A noble thought, young man,\" the deep voice said. \"And one that you should always remember.\"\n\nThe Bishop had regained consciousness and was smiling a wan smile.\n\n\"Now, Jim, as I was saying before they turned on their little machine, I was certain that you would be here this morning. You were free\u2014and I knew that you would not leave me alone in that wicked place. There was an immense hue and cry when you escaped, with abundant to-ing and fro-ing until the gate was closed for the night. It was obvious that it would be impossible for you to come then. But with dawn the gate would be opened and I had not the slightest doubt that you were sure to be close by, trying to find a way to get to me. Simple logic. So I simplified the equation by coming to you.\"\n\n\"Very simple! You almost got yourself killed.\"\n\n\"But I didn't. And we are both safely away from them. Plus I see that you have managed to enlist an ally. A good day's work. Now the important question. What do we do next?\"\n\nWhat indeed?\nChapter 23\n\n\"As to what we do next\u2014the answer is obvious,\" I said. \"We stay here until the excitement has died down. Which should happen fairly quickly since there is not much market value in a dead slave.\"\n\n\"But I feel remarkably healthy.\"\n\n\"You have forgotten that the paincuff will kill if used continuously. So, when our way is clear we head for the nearest habitation and dress your wound.\"\n\n\"It is bloody, but can't be more than a scratch.\"\n\n\"Sepsis and infection. We take care of the cut first.\" I turned to Dreng. \"Any farmers you know who live close to this place?\"\n\n\"No, but the widow Apfeltree is just over the hill, past the dead tree, through the end of the swamp....\"\n\n\"Great. Show us the way, don't tell.\" I turned back to The Bishop. \"And after we fix your back, then what?\"\n\n\"After that, Jim, we join the army. Since you are now a mercenary, that is the proper thing to do. An army will be based in a keep, and there will be a locked room in that keep where all the groats are stored. While you practice your military profession I shall, as the expression goes, case the joint. In order to further this noble work of ours I have one particular army in mind for you. The one that serves the Capo Dimonte.\"\n\n\"Not Capo Dimonte!\" Dreng wailed, clutching his hair with both hands. \"He is evil beyond measure, eats a child for breakfast every day, has all of his furniture upholstered in human skin, drinks from the skull of his first wife...\"\n\n\"Enough,\" The Bishop ordered, and Dreng was stilled. \"It is obvious that he does not have a good press here in the Capote of Doccia. That is because he is the sworn enemy of Capo Doccia and goes to war against him periodically. I am sure that he is no worse\u2014or better\u2014than any other capo. But he does have one advantage. He is our enemy's enemy.\"\n\n\"So hopefully our friend. Right. I owe old Doccia one and I look forward to paying it back.\"\n\n\"You should not bear grudges, Jim. It dulls the vision and interferes with your career. Which should now be grabbing groats not wreaking vengeance.\"\n\nI nodded agreement. \"Of course. But while you are planning the heist there is no reason why I can't enjoy a bit of revenge.\"\n\nI could see that he disapproved of my emotions\u2014but I could not attain his Olympian detachment. A weakness of youth, perhaps. I changed the subject.\n\n\"After we empty the treasury, then what?\"\n\n\"We find out how the locals are contacting the offplanet smugglers, like the Venians. With the obvious aim of leaving this backward and deadly world as soon as possible. In order to do that we may have to get religion.\" He chuckled at my shocked expression. \"Like you, my boy, I am a Scientific Humanist and feel no need for the aid of the supernatural. But here on Spiovente what technology there is seems to be in the hands of an order called the Black Monks....\"\n\n\"No, stay away!\" Dreng wailed; he was certainly a source of bad news. \"They know Things that Drive Men Mad. From their workshops all forms of unnatural devices pour forth. Machines that scream and grunt, that talk through the skies, the paincuffs as well. Avoid them, master, I beg of you!\"\n\n\"What our young friend has decried is true,\" The Bishop said. \"Minus the fear of the unknown, of course. Through some process that is not relevant now all of technology on this world became concentrated in the hands of this order, the Black Monks. I have no idea what their religious affiliations are\u2014if any\u2014but they do supply and repair the machines that we have seen. This gives them a certain protection, since if one capo were to attack them the others would rush to their defense to insure their continued access to the metallic fruits of technology. It is to them that we may have to turn for salvation and exodus.\"\n\n\"I second the motion and it is carried by acclamation. Join the army, whip as many groats as we can, contact the smugglers\u2014and buy our way out.\"\n\nDreng gaped at all the long words, drooling a bit at the same time. He obviously followed little of what we discussed. Action was more his style. He made a silent exit on a scouting trip and an even more slithery return. No one was about, our way was clear. The Bishop could walk now, with a little aid from us, and the widow's house was not too distant. Even with Dreng's reassurances she was trembling with fear when she admitted us to her hovel.\n\n\"Guns and swords. Murder and death. I'm doomed, doomed.\"\n\nDespite her muttering, punctuated by the smacking of her toothless gums, she followed my instructions and put a pot of water over the fire. I cut a strip of cloth from my blanket, boiled it clean, then used it to wash The Bishop's wound. It was shallow but deep. The widow was persuaded to part with some of her store of moonshine and The Bishop shuddered, but did not cry out, when I poured it into the open cut. Hoping the alcohol content was high enough to act as an antiseptic. I used more boiled blanket as a bandage\u2014which was about all that I could do.\n\n\"Excellent, James, excellent,\" he said, gingerly pulling his sliced jacket over his shoulders. \"Your years in the Boy Sprouts were obviously not wasted. Now let us thank the good widow and leave since it is obvious that she is upset by our presence.\"\n\nLeave we did, strolling the open, rut-filled road, every footstep taking us farther away from Capo Doccia. Dreng was a good provider, drifting off into orchards for fruit, or rooting out edible tubers from the fields we passed, even digging them up under the noses of the rightful owners. Who only touched their forelocks at the sight of my weapons. It is a nasty world that only respects bullies. For the first time I began to appreciate the better qualities of the League worlds.\n\nIt was late afternoon when the walls of the keep loomed before us. This place had a little more style than Doccia's, or at least it looked that way from a distance, because it was situated on an island in a lake. A causeway and drawbridge connected it to the mainland. Dreng was shaking with fear again and was more than happy to stay on the shore with The Bishop while I braved the dangers of the keep. I strode militarily along the stone causeway, then stamped over the bridge. The two guards eyed me with open suspicion.\n\n\"Good morn, brothers,\" I called out cheerfully, gun on shoulder, sword in hand, gut in and chest out. \"Is this the establishment of the Capo Dimonte, known the length and breadth of the land for his charm and strength of arm?\"\n\n\"Who wants to know?\"\n\n\"I do. An armed and powerful soldier who wishes to enlist in his noble service.\"\n\n\"Your choice, brother, your choice,\" he said with obvious gloom. \"Through the gate, across the courtyard, third door on your right, ask for Sire Srank.\" He leaned close and whispered. \"For three groats I'll give you a tip.\"\n\n\"Done.\"\n\n\"So pay.\"\n\n\"Shortly. I'm a little skint right now.\"\n\n\"You must be\u2014if you want to hire out to this lot. All right, five then, in five days.\" I nodded agreement. \"He'll offer you very little, but don't settle for less than two groats a day.\"\n\n\"Thanks for the credit. I'll get back to you.\"\n\nI swaggered through the gate and found the right door. It was open to admit the last light, and a fat man with a bald head was scratching away at some papers. He looked up when my shadow fell across the table.\n\n\"Get out here,\" he shouted, scratching his head so hard that a shower of dandruff sparkled in the sunbeam. \"I've told you all, no groats until morning after next.\"\n\n\"I've not enlisted yet\u2014nor will I if that's the way you pay the troops.\"\n\n\"Sorry, good stranger, sun in my eyes. Come in, come in. Enlist? Of course. Gun and sword\u2014and ammunition?\"\n\n\"Some.\"\n\n\"Wonderful.\" His hands rustled when he dry-washed them. \"Food for you and your knave and a groat a day.\"\n\n\"Two a day and all ammunition used to be replaced.\"\n\nHe scowled\u2014then shrugged and scratched one of the sheets and pushed it over to me. \"A one-year enlistment, salary open to review at end of contract. Since you can't read or write I hope you can manage to scratch your illiterate X down here.\"\n\n\"I can read so well I see that you have me down for four years, which I will now correct before I sign.\" Which I did, writing Judge Nixon's name on the line, knowing full well that I would be leaving well before my enlistment was up. \"I'll get my knave who awaits without, along with my aged father.\"\n\n\"No extra food for poor relations!\" he snarled generously. \"You share yours.\"\n\n\"Agreed,\" I said. \"You're all heart.\"\n\nI went back to the gate and waved my companions over.\n\n\"You owe me,\" the guard said.\n\n\"I'll pay you\u2014when that scrofulous toad pays me.\"\n\nHe grunted agreement. \"If you think he's bad\u2014wait until you meet Capo Dimonte. I wouldn't be hanging around this damp dump if it weren't for the loot bonus.\"\n\nThey were coming on slowly, The Bishop half dragging the reluctant Dreng.\n\n\"Loot bonus? Paying out soon?\"\n\n\"Soon as the fighting is over. We march tomorrow.\"\n\n\"Against Capo Doccia?\"\n\n\"No such luck. The word is that he is loaded with jewels and golden groats and more. Be nice to share in that haul. But not this time. All they have told us is that we are heading north. Must be a surprise attack on someone, probably a friend, and they don't want word to leak out. That's good thinking. Catch them with their drawbridge down and it's half the battle.\"\n\nI pondered this bit of military wisdom as I led my small band in the indicated direction. The soldier's quarters, while not something to put in a travel brochure, were certainly a cut above the slave quarters. Wooden bunks with straw mattresses for the fighting men\u2014some straw under the bunk for the knave. I would have to make some arrangements for The Bishop, but I was sure that bribery would take care of that. We sat together on the bunk while Dreng went to find the kitchen.\n\n\"How is the back?\" I asked.\n\n\"Sore, but only a small bother. I'll take a bit of a rest, then begin a survey of the layout....\"\n\n\"In the morning will be time enough. It has been a long couple of days.\"\n\n\"Agreed. And here is your knave with the food!\"\n\nIt was a hot stew with fragments of some nameless bird bobbing in it. Had to be a bird; the feathers were still attached. We divided the stew into three equal portions and wolfed it down. All this fresh air and walking was certainly good for the appetite. There was also a ration of sour wine which neither I nor The Bishop could stomach. Not so Dreng, who slurped and smacked his way through it in moments. Then rolled under the bunk and began to snore raucously.\n\n\"I'm going to have a look around,\" I said. \"Take a rest on the bunk until I get back....\"\n\nI was interrupted by an off-key blare on a bugle. I looked up to see that the malevolent musician was standing in the doorway. He emitted another toneless blast. I was ready to grab him by the throat if he tried it again, but he stepped aside and bowed. A thin figure in blue uniform took his place. All of the soldiers who were watching bowed their heads slightly or shook their weapons in salute, so I did the same. It could be no other than Capo Dimonte himself.\n\nHe was lean to the point of being hollow-stomached. He either had circulation trouble or was naturally blue of skin. His little red eyes peered out of hollow blue sockets, while he fingered his blue jaw with azure fingers. He looked around suspiciously, then spoke: for all of his leanness his voice had a deep strength to it.\n\n\"My men, I have good news for you. Prepare yourselves and your weapons for we march at midnight. This will be a forced march to enable us to reach Pinetta Woods before dawn. Fighting men only\u2014and we travel light. Your knaves will stay here to look after your goods. We will lay up there during the day, then leave at dusk tomorrow. We will meet our allies during the night and join forces for an assault on the enemy at dawn.\"\n\n\"A question, Capo,\" one of the men called out. He was grizzled and scarred, obviously a veteran of many conflicts. \"Against whom do we march?\"\n\n\"You will be told before the attack. We will gain victory only by surprise.\"\n\nThere were murmurs on all sides as the veteran called out again.\n\n\"Our enemy a mystery\u2014at least tell us then who are our allies.\"\n\nCapo Dimonte was not pleased with the question. He scratched his chin and fiddled with the hilt of his sword while his audience waited. He obviously needed our voluntary assistance, so in the end he spoke.\n\n\"You will all be pleased to hear that we have allies of great strength and will. They also have war machines to batter the stoutest wall. With their assistance we can take any keep, defeat any army. We are lucky to serve at their side.\" He pressed his lips together, reluctant to go on but still knowing that he must.\n\n\"Our victory is assured since our allies are none other than... the order of the Black Monks.\"\n\nThere was a long moment of shocked silence\u2014followed instantly by shouts of anger. The significance of all this escaped me\u2014other than the fact that it did not sound good at all.\nChapter 24\n\nAs soon as he had spoken, Capo Dimonte made his exit and the door slammed shut behind him. There were shouts and cries of anger from all sides\u2014but there was one man who bellowed louder than all the others. It was the scarred veteran. He climbed onto a table and shouted them all into silence.\n\n\"Everyone here knows me, knows old Tusker. I was cutting off heads when most of you weren't even potty trained. So I'm going to talk and you are going to listen and then you will get a chance to talk too. Anyone here don't like that idea?\"\n\nHe closed one immense fist and held it out, then turned in a circle, scowling fiercely. There were some angry mutters, but none loud enough to imply disagreement.\n\n\"Good. Then listen. I know those black-frocked buggers from a long way back and I don't trust them. All they think of is their own hides. If they want us to fight for them that's only because there is big trouble ahead and they would like to see us killed rather than them. I don't like it.\"\n\n\"I don't like it either,\" another man called out. \"But what kind of a choice do we have?\"\n\n\"None,\" Tusker growled angrily. \"And that's what I was going to say next. I think we have been grabbed by the short and curlies.\" He drew his sword and shook it at them. \"Every weapon we have, outside of them new guns, comes from the Black Monks. Without their supplies we have nothing to fight with, and without nothing to fight with we have nothing to do and we can starve or go back to the farm. And that's not for me. And it better not be for any of you either. Because we are all in this together. We all fight\u2014or none fight. And if we fight and any of you try to sneak out of here before the action starts, then he is going to find my sword stuck all the way through his liver.\"\n\nHe shook the shining blade at them while they glared in silence.\n\n\"A solid argument,\" The Bishop whispered, \"the logic impeccable. Too bad that it is wasted on this ignoble cause. You and your comrades have no choice but to agree.\"\n\nThe Bishop was right. There was more shouting and argument, but in the end they had to go along with the plans. They would march at the side of the Black Monks. None of them, myself included, were very happy about the idea. They could stay up and argue until midnight but I was tired and could use the few hours sleep. The Bishop wandered out in search of information and I rolled up on the bunk and slipped into a restless slumber.\n\nThe shouted orders woke me, feeling more tired than when I had gone to sleep. No one seemed happy about the midnight march\u2014or our battle companions\u2014and there were dark looks and much cursing. There were even some oaths I hadn't heard before, real nice ones, that I filed away for future use. I went out to the primitive lavabo and threw cold water on my face, which seemed to help. When I returned, The Bishop was sitting on the bunk. He rose and extended his large hand.\n\n\"You must watch yourself, Jim. This is a crude and deadly world and all men's hands are turned against you.\"\n\n\"That is the way I prefer to live\u2014so don't worry yourself.\"\n\n\"But I do.\" He sighed mightily. \"I have nothing but contempt for superstition, astrologers, palm readers, and the like, so you will understand why I feel great disgust at myself for the black depression that possesses me. But I see nothing but darkness in the future, emptiness. We have been companions for such a brief period, I do not wish it to end. Yet, I am sorry, do excuse me, I have a sense of danger and despair that cannot be alleviated.\"\n\n\"With good reason!\" I cried, trying to put enthusiasm into my words. \"You have been torn from the security of your quasi-retirement, imprisoned, freed, fled, hid, dieted, fled again, bribed, were cheated, beaten, enslaved, wounded\u2014and you wonder why you are depressed?\"\n\nThis brought a wan smile to his lips and he grasped my hand again. \"You are right. Jim, of course. Toxins in the bloodstream, depression in the cortex. Watch your back and return safely. By the time you do I'll have worked out how to relieve the capo of some of his groats.\"\n\nHe was looking his age\u2014for the first time since we had met. As I left I saw him stretch out wearily on the bunk. He should be feeling better when I returned. Dreng would fetch his food and look after him. What I must do is concentrate on staying alive so I would come back.\n\nIt was a dreary and exhausting march. The day had been hot, and so was the night. We shuffled along, dripping with perspiration and slapping at the insects that rushed out of the darkness to attack us. The rutted road caught at my feet and dust rose into my nostrils. On we marched, and on, following along after the clanking and hissing conveyance that led our nightmarish parade. One of the steam cars was hauling Capo Dimonte's war wagon in which he traveled in relative comfort. His captains were in there with him, swilling down booze no doubt and generally enjoying themselves. We marched on, the cursing in the ranks growing steadily weaker.\n\nBy the time we stumbled under the sheltering trees of Pinetta Woods we were tired and mutinous. I did what most of the others did, dropped onto the bed of sweet-smelling needles under the trees and groaned in appreciation. And admiration for the sturdier warriors, with old Tusker in the van, who insisted on their ration of acid wine before retiring. I closed my eyes, groaned again, then slept.\n\nWe stayed there all day, glad to have the rest. Around noon rations were reluctantly handed out from the cart. Warm, foul water to wash down rock-like bits of what might have been bread. After this I managed to sleep some more, until we formed ranks again at dusk and the night march continued.\n\nAfter some hours we came to a crossroads and turned right. There was a murmur through the ranks at this, starting with those who knew the area well.\n\n\"What are they saying?\" I asked the man who marched beside me, in silence up until now.\n\n\"Capo Dinobli. That's who we're after. Could be no one else. No other keep in this direction for one day, two day's march.\"\n\n\"Do you know anything about him?\"\n\nHe grunted and was silent, but the man behind him spoke up. \"I served with him, long time ago. Old bugger then, must be ancient now. Just one more capo.\"\n\nThen it was one foot after another in a haze of fatigue. There had to be better ways to make a living. This was going to be my first and last campaign. As soon as we returned I and The Bishop would sack the treasury and flee with all the groats we could carry. Wonderful thought. I almost ran into the man in front of me and stopped just in time. We had halted where the road passed near the forest. Against the darkness of the trees even darker shadows loomed. I was trying to see what they were when one of the officers came back down the ranks.\n\n\"I need some volunteers,\" he whispered. \"You, you, you, you.\"\n\nHe touched my arm and I was one of the volunteers. There seemed to be about twenty of us who were pulled out of line and herded forward towards the woods. The clouds had cleared and there was enough light from the stars now to see that the black bulks were wheeled devices of some kind. I could hear the hiss of escaping steam. A dark figure strode forward and halted us.\n\n\"Listen and I will tell you what you must do,\" he said.\n\nAs he spoke a metal door was opened on the machine nearest us. Light gleamed as wood was pushed into the firebox. By the brief, flickering light I could see the speaker clearly. He was dressed in a black robe, his head covered by a cowl that hid his face. He pointed to the machine.\n\n\"This must be pushed through the woods\u2014and in absolute silence. I will put my knife into the ribs of any of you who makes a noise. A track was cut during daylight and will be easy to follow. Take up the lines and do as you are instructed.\"\n\nOther dark-robed figures were handing us the ropes and pushing us into line. On the whispered signal we began hauling.\n\nThe thing rolled along easily enough and we pulled at a steady pace. There were more whispered instructions to guide us\u2014then we halted as we approached the edge of the forest. After this we dropped the ropes and sweated as we pushed and pulled the great weight about until the guides were satisfied. There was much whispered consultation about alignment and range, and I wondered just what was going on. We had been forgotten for the moment, so I walked as quietly as I could past the thing and peered out through the shrubbery at the view beyond.\n\nVery interesting. A field of grain stretched down a gentle slope to a keep, its dark towers clear in the starlight. There was a glimmer of reflection about its base, where the waters of the moat protected it from attack.\n\nI stayed there until dawn began to gray the sky, then moved back to examine the object of our labors. As it emerged from the darkness its shape became clear\u2014and I still hadn't the faintest idea of what it was. Fire and steam, I could see the white trickle of vapor clearly now. And a long boom of some kind along the top. One of the black figures was working the controls now. Steam hissed louder as the long arm sank down until the end rested on the ground. I went to look at the large metal cup there\u2014and was rewarded for my curiosity by being drafted to help move an immense stone into place. Two of us rolled it from the pile of its fellows nearby, but it took four of us, straining, to raise it into the cup. Mystery upon mystery. I rejoined the others just as Capo Dimonte appeared with the tall, robed man at his side.\n\n\"Will it work, Brother Farvel?\" Dimonte asked. \"I know nothing of such devices.\"\n\n\"But I do, capo, you shall see. When the drawbridge is lowered my machine will destroy it, crush it.\"\n\n\"May it do just that! Those walls are high\u2014and so will our losses be if we must storm the keep without being able to break through the gate.\"\n\nBrother Farvel turned his back and issued quick instructions to the machine's operators. More wood was pushed into its bowels and the hissing rose in volume. It was full daylight now. The field before us was empty, the view peaceful. But behind us in the forest lurked the small army and the war machines. It was obvious that battle would be joined when the drawbridge was lowered and destroyed.\n\nWe were ordered to lie down, to conceal ourselves as the light grew. It was full daylight by this time, the sun above the horizon\u2014and still nothing happened. I crouched near the machine, close to the cowled operator at the controls.\n\n\"It is not coming down!\" Brother Farvel called out suddenly. \"It is past due, always down at this time. Something is wrong.\"\n\n\"Do they know that we are here?\" Capo Dimonte said.\n\n\"Yes!\" an incredibly loud voice boomed out from the trees above us. \"We know you are there. Your attack is doomed\u2014as are all of you! Prepare to face your certain death.\"\nChapter 25\n\nThe roaring voice was totally unexpected, shocking in the silence of the forest. I jumped, startled\u2014nor was I the only one. The monk at the machine's controls was even more startled. His hand pulled on the control lever and there was a gigantic hissing roar. The long arm on top of the device thrashed skyward, pushed by a stubbier arm close to its hinged end. The arm rose up in a high arc and slammed into a concealed buffer that jarred and shook the entire machine. The arm may have stopped\u2014but the stone in the cup at the end of the arm continued, high into the air, rising in a great arc. I rushed forward to see it splash into the moat just before the closed drawbridge. Good shot\u2014it would certainly have demolished the structure had it been down.\n\nAll around me things became busy quite suddenly. Brother Farvel had knocked the monk from the controls and was now kicking him, roaring with rage. Swords had been drawn, soldiers were rushing about\u2014some of them firing up into the trees. Capo Dimonte was bellowing orders that no one was listening to. I put my back to a tree and held my gun ready for the expected attack.\n\nIt never came. But the amplified voice thundered again.\n\n\"Go back. Return from whence you came and you will be spared. I am talking to you, Capo Dimonte, you are making a mistake. You are being used by the Black Monks. You will be destroyed for nothing. Return to your keep, for only death awaits you here.\"\n\n\"It is there. I see it!\" Brother Farvel shouted, pointing up into the trees. He spun about and saw me, seized me by the arm in a painful grip, and pointed again. \"There, on that branch, the device of the devil. Destroy it!\"\n\nWhy not. I could see it now, even recognize what it was. A loudspeaker of some kind. The gun cracked and kicked my shoulder hard. I fired again and the speaker exploded, bits of plastic and metal rained down.\n\n\"Just a machine,\" Brother Farvel shouted, stamping the fragments into the ground. \"Start the attack\u2014send your men forward. My death-throwers will give you support. They will batter down the walls for you.\"\n\nThe capo had no choice. He chewed his lip a bit, then signaled the bugler at his side. Three brazen notes rang out and were echoed by other buglers to our rear and on both flanks. When the first of his troops burst from beneath the trees he drew his sword and ordered us to follow him. With great reluctance I trotted forward.\n\nIt was not quite what you would call a lightning attack. More of a stroll when you got down to it. We advanced through the field, then stopped on order to wait for the death-throwers to get into position. Steam cars pulled them forward into line and the firing began. Rocks sizzled over our heads and either bounced from the keep wall or vanished into its interior.\n\n\"Forward!\" the capo shouted, and waved his sword again just as the return fire began.\n\nThe silvery spheres rose up from behind the keep walls, rose high, arced forward above us\u2014and dropped.\n\nHit\u2014and cracked open. One struck nearby and I could see it was a thin container of some kind filled with liquid that smoked and turned to vapor in the air. Poison! I threw myself away from it, running, trying not to breathe. But the things were bursting all about us now, the air thick with fumes. I ran and my lungs ached and I had to breathe, could not stop myself.\n\nAs the breath entered my lungs I fell forward and blackness fell as well.\n\n* * *\n\nI was lying on my back, I knew that, but was aware of very little else because of the headache that possessed me completely. If I moved my head ever so slightly it tightened down like a band of fire on my temples. When I tentatively opened one eye\u2014red lightning struck in through my eyeball. I groaned, and heard the groan echoed from all sides. This headache was the winner, the planet-sized headache of all time, before which all other headaches paled. I thought of previous headaches I had known and sneered at their ineffectiveness. Cardboard headaches. This was the real thing. Someone groaned close by and I, and many others, groaned in sympathy.\n\nBit by bit the pain ebbed away, enough so that I tentatively opened one eye, then the other. The blue sky was clear above, the wind rustled the grain on which I lay. With great hesitation I rose up on one elbow and looked around me at the stricken army.\n\nThe field was littered with sprawled bodies. Some of them were sitting up now, holding their heads, while one or two stronger\u2014or stupider\u2014soldiers were climbing unsteadily to their feet. Nearby lay the silvery, broken fragments of one of the attacking missiles, looking innocent enough now with the gas dispersed. My head throbbed but I ignored it. We were alive. The gas had not killed us\u2014it had obviously been designed only to knock us out. Potent stuff. I looked at my shadow, not wanting to risk a glance at the sun yet, and saw how foreshortened it was. Close to noon. We had been asleep for hours.\n\nThen why weren't we dead? Why hadn't the Capo Dinobli's men pounced on us and slit our throats? Or at least taken our weapons? My gun was at my side; I broke it and saw that it was still loaded. Mysteries, mysteries. I jumped, startled\u2014instantly regretting it as my head throbbed\u2014as the hoarse scream rang out. I managed to sit up and turn to look.\n\nInteresting. It was Brother Farvel himself who was still shouting and cursing while he tore handfuls of hair from his head. This was most unusual. I had certainly never seen anything like it before. I rose hesitantly to my feet to see what he was upset about. Yes indeed, I could understand his emotions.\n\nHe was standing beside one of his death-throwers which had been thrown a little death of its own. It had burst open, exploded into a tangle of twisted pipes and fractured metal. The long arm had been neatly cut into three pieces and even the wheels had been torn from the body. It was just a mass of unrepairable junk. Brother Farvel ran off, still shouting hoarsely, wisps of hair floating in the breeze behind him.\n\nThere were more cries and shouts of pain from the other monks as Brother Farvel came staggering back, stumbling towards the Capo Dimonte, who was just sitting up.\n\n\"Destroyed, all of them!\" the Black Monk roared while the capo clutched his hands tightly over his ears. \"The work of years, gone, crushed, broken. All my death-throwers, the steam-powered battering ram\u2014ruined. He did it. Capo Dinobli did it. Gather your men, attack the keep, he must be destroyed for this monstrous crime that he has committed.\"\n\nThe capo turned to look towards the keep. It was just as it had been at dawn, quiet and undisturbed, the drawbridge still up, as though the day's events had never occurred. Dimonte turned back to Brother Farvel, his face cold and drawn.\n\n\"No. I do not lead my men against those walls. That is suicide and suicide was not our agreement. This is your argument, not mine. I agreed to aid you in taking the keep. You were to force entrance with your devices. Then I would attack. That arrangement is now over.\"\n\n\"You cannot go back on your word....\"\n\n\"I am not. Breech the walls and I will attack. That is what you promised. Now, do it.\"\n\nBrother Farvel turned red with rage, raised his fists, leaned forward. The capo stood his ground\u2014but drew his sword and held it out.\n\n\"See this,\" he said. \"I am still armed\u2014all of my men are armed. It is a message that I understand quite clearly. Dinobli's men could have taken our weapons and cut our throats while we lay here. They did not. They do not war on me. Therefore I do not war on them. You fight them\u2014this is your battle.\" He nudged the toe of his boot into the bugler lying beside him. \"Sound assembly.\"\n\nWe were quite happy to leave the Black Monks there in the field, surveying the wreckage of their machines and their plans. Word quickly spread through the ranks as to what had occurred and smiles replaced the pained grimaces as the headaches vanished to be replaced by relief. There would be no battle, no casualties. The Black Monks had started the trouble\u2014and it had been finished for them. My smile was particularly broad because I had some good news for The Bishop.\n\nI knew now how we were to get off the repellent planet of Spiovente.\n\nThrough the clear wisdom of hindsight I could understand now what had happened the night before. The approach of our troops in the darkness had been observed carefully. With advanced technology of some kind. The hidden watchers must have also seen the track being constructed through the forest for the death-thrower and understood the significance of the operation. The loudspeaker had been placed in the tree directly above the site\u2014then activated by radio. The gas that had felled us was sophisticated and had been delivered with pinpoint accuracy. All of this was well beyond the technology of this broken-down planet. Which meant only one thing.\n\nThere were offworlders in the keep of Capo Dinobli. They were there in force and were up to something. And whatever it was had aroused the wrath of the Black Monks, so much so that they had planned this attack. Which had backfired completely. Good. Mine enemy's enemy one more time. The monks had a stranglehold on what little technology there was on Spiovente\u2014and from what I had seen, the technology was completely monopolized by the military. I cudgled my brain, remembering those long sessions with The Bishop on geopolitics and economics. I was getting the glimmer of a solution to our problems when there was a wild shouting from the ranks ahead.\n\nI pushed forward with the others to see the exhausted messenger sprawled in the grass beside the road. Capo Dimonte was turning away from him, shaking his fists skyward in fury.\n\n\"An attack\u2014behind my back\u2014on the keep! It is that sun of a worm, Doccia, that's who it is! We move now, forced march. Back!\"\n\nIt was a march that I never want to repeat. We rested only when exhaustion dropped us to the ground. Drank some water, staggered to our feet, went on. There was no need to beat us or encourage because we were all involved now. The capo's family, his worldly goods, they were all back in the keep. Guarded only by a skeleton force of soldiers. All of us were as concerned as he was, for what little we owned was there as well. The knaves watching our few possessions. Dreng, whom I scarcely knew, yet felt responsibility for. And The Bishop. If the keep were taken what would happen to him? Nothing, he was an old man, harmless, no enemy of theirs.\n\nYet I knew this was a lie even as I tried to convince myself of its validity. He was an escaped slave. And I knew what they did with escaped slaves on Spiovente.\n\nMore water, a little food at sunset, then on through the night. At dawn I could see our forces straggling out in a ragged column as the stronger men pushed on ahead. I was young and fit and worried\u2014and right up in the front. I could stop now for a rest, get my breath back. Ahead on the road I saw the two men spring from the bushes and vanish over the hill.\n\n\"There!\" I shouted. \"Watchers\u2014we've been seen.\"\n\nThe capo jumped from the war-wagon and ran to my side. I pointed. \"Two men. In hiding there. They ran towards the keep.\"\n\nHe ground his teeth with impotent rage. \"We can't catch them, not in our condition. Doccia will be warned; he'll escape.\"\n\nHe looked back at his straggling troops, then waved his officers forward.\n\n\"You, Barkus, stay here and rest them, then get in formation and follow me. I'm going on with all the fit men I can. They can take turns riding on the war-wagon. We're pushing forward.\"\n\nI climbed onto the roof of the cart as it started ahead. Men ran alongside, holding on, letting it pull them. The steam car wheezed and puffed smoke at a great rate as we clanked up the hill and onto the downslope beyond.\n\nThere were the towers of the keep in the distance, smoke rising from it. When we rattled around the next bend we found a line of men across the road, weapons raised, firing.\n\nWe did not slow down. The steam-whistle screeched loudly and we roared in answer, our anger taking us forward. The enemy fled. It had just been a holding party. We could see them joining the rest of the attackers who were now streaming away from the lake. When we reached the causeway it was empty of life. Beyond it was the broken gate of the keep with smoke rising slowly above it. I was right behind the capo when we stumbled forward. Long boards were still in place bridging the gap before the splintered and broken drawbridge, half raised and hanging from its chains. A soldier pushed out between the broken fragments and raised his sword in weary salute.\n\n\"We held them, capo,\" he said, then slumped back against the splintered wood. \"They broke through into the yard but we held them at the tower. They were firing at the outer door when they left.\"\n\n\"The Lady Dimonte, the children...?\"\n\n\"All safe. The treasury untouched.\"\n\nBut the troops' quarters were off the yard and not in the tower. I pushed ahead with the others, who had realized this, climbing through the ruined gate. There were bodies here, many of them. Unarmed knaves chopped down in the attack. The defenders were coming out of the tower now\u2014and Dreng was among them, coming forward slowly. His clothing was spattered with blood, as was the ax he carried, but he seemed sound.\n\nThen I looked into his face and read the sorrow there. He did not need to speak; I knew. The words came from a distance.\n\n\"I am sorry. I could not stop them. He is dead, the old man. Dead.\"\nChapter 26\n\nHe lay on the bunk, eyes closed as though he were sleeping. But never that still, never. Dreng had drawn my blanket over him, up to his chin, combed his hair and cleaned his face.\n\n\"I could not move him when the attack came,\" Dreng said. \"He was too heavy, too ill. The wound in his back was bad, black, his skin hot. He told me to leave him, that he was dead in any case. He said if they didn't kill him the 'fection would. They didn't have to stab him though....\"\n\nMy friend and my teacher. Murdered by these animals. He was worth more than the entire filthy population of this world gathered together. Dreng took me by the arm and I shook him off, turned on him angrily. He was holding out a small packet.\n\n\"I stole the piece of paper for him,\" Dreng said. \"He wanted to write to you. I stole it.\"\n\nThere was nothing to be said. I unwrapped it and a carved wooden key fell to the floor. I picked it up, then looked at the paper. There was a floor plan of the keep drawn on it, with an arrow pointing to a room carefully labeled STRONGROOM. Below it was the message, and I read what was written there in a tight, clear hand.\n\nI have been a bit poorly so I may not be able to give you this in person. Make a metal copy of the key\u2014it opens the strongroom. Good luck, Jim. It has been my pleasure to know you. Be a good rat.\n\nHis signature was carefully written below. I read the name\u2014then read it again. It wasn't The Bishop\u2014or any of the other aliases he had ever used. He had left me a legacy of trust\u2014knowing that I was probably the only person in the universe who would value this confidence. His real name.\n\nI went and sat down outside in the sun, suddenly very weary. Dreng brought me a cup of water. I had not realized how thirsty I was; I drained it and sent him for more.\n\nThis was it, the end. He had felt the approach of darkness\u2014but had worried about me. Thought of me when it was really his own death that was looming so close.\n\nWhat next? What should I do now?\n\nFatigue, pain, remorse\u2014all overwhelmed me. Not realizing what was happening I fell asleep, sitting there in the sun, toppled over on my side. When I awoke it was late in the afternoon. Dreng had wadded his blanket and put it under my head, sat now at my side.\n\nThere was nothing more to be said. We put The Bishop's body on one of the little carts and wheeled it along the causeway to the shore. We were not the only ones doing this. There was a small hill beside the road, a slope of grass with trees above it, a pleasant view across the water to the keep. We buried him there, tamping the soil down solidly and leaving no marker. Not on this disgusting world. They had his body, that was enough. Any memorial I erected in his honor would be light-years away. I would take care of that one day when the proper moment came.\n\n\"But right now, Dreng, we take care of Capo Doccia and his hoodlums. My good friend did not believe in revenge, so I cannot either. So we shall call it simple justice. Those criminals need straightening out. But how shall we do it?\"\n\n\"I can help, master. I can fight now. I was afraid, then I got angry and I used the ax. I am ready to be a warrior like you.\"\n\nI shook my head at him. I was thinking more clearly now. \"This is no job for a farmer with a future. But you must always remember that you faced your fear and won. That will do you well for the rest of your life. But Jim diGriz pays his debts\u2014so you are going back to the farm. How many groats does a farm cost?\"\n\nHe gaped at that one and shuffled through his memory. \"I never bought a farm.\"\n\n\"I'm sure of that. But somebody must have that you know.\"\n\n\"Old Kvetchy came back from the wars and paid Widow Roslair two hundred and twelve groats for her share of her farm.\"\n\n\"Great. Allowing for inflation five hundred should see you clear. Stick with me, kid, and you'll be wearing plowshares. Now get to the kitchen and pack up some food while I put part one of the plan into operation.\"\n\nIt was like a chess game that you played in your head. I could see the opening moves quite clearly, all laid out. If they were played correctly, middle game and endgame would follow with an inevitable win. I made the first move.\n\nCapo Dimonte was slumped on his throne, red-eyed and as tired as the rest of us, a flagon of wine in his hand. I pushed through his officers and stood before him. He scowled at me and flapped his hand.\n\n\"Away, soldier. You'll get your bonus. You did your work well today, I saw that. But leave us, I have plans to make....\"\n\n\"That is why I am here, capo. To tell you how to defeat Capo Doccia. I was in his service and know his secrets.\"\n\n\"Speak!\"\n\n\"In private. Send the others away.\"\n\nHe considered a moment\u2014then waved his hands. They left, grumbling, and he sipped his wine until the door slammed shut.\n\n\"What do you know,\" he ordered. \"Speak quickly for I am in a foul humor.\"\n\n\"As are we all. What I wanted to tell you in private does not concern Doccia\u2014yet. You will attack, I am sure of that. But in order to assure success I am going to enlist Capo Dinobli and his secrets on your side. Wouldn't the attack be better if they were all asleep when we came over the wall.\"\n\n\"Dinobli knows no more of these matters than I do\u2014so don't lie to me. He is tottering and has been bedridden for a year.\"\n\n\"I know that,\" I lied with conviction. \"But it is those who use his keep for their own ends, who cause the Black Monks to make war on them, these are the ones who will help you.\"\n\nHe sat up at this and there was more than a glint of the old schemer in his eyes. \"Go to them then. Promise them a share of the spoils\u2014and you will share as well if you can do this. Go in my name and promise what you will. Before this month is out Doccia's head will be roasting on a spit over my fire, his body will be torn by red-hot spikes and...\"\n\nThere was more like this but I wasn't too interested. This was a pawn move in the opening. I now had to bring a major piece forward to the attack. I bowed myself out, leaving him muttering on the throne, splashing wine around as he waved his arms. These people had very quick tempers.\n\nDreng had packed our few belongings and we left at once. I led the way until we were well clear of the keep, then turned off towards a stream that ran close by. It had a grassy field at its bank and I pointed towards it.\n\n\"We stay here until morning. I have plans to make and we need the rest. I want to be sharp when I knock on old Dinobli's door.\"\n\nWith a night's rest to refresh my brain everything became quite clear. \"Dreng,\" I said, \"this will have to be a one-man operation. I don't know what kind of reception I will get and I may be busy enough worrying about myself, without having you to care for. Back to the keep and wait for me.\"\n\nThere was really no door to knock on, just two heavily armed guards at the gate. I came down through the field, past the mounds of junked machines already smeared with a red patina of rust, and crossed the drawbridge. I stopped before I reached the guards and carefully kept my gun lowered.\n\n\"I have an important message for the one in charge here.\"\n\n\"Turn about and quick march,\" the taller guard said, pointing his gun at me. \"Capo Dinobli sees no one.\"\n\n\"It's not the capo I care about,\" I said, looking past him into the courtyard. A tall man in rough clothes was passing. But beneath the ragged cuffs of his trousers I saw the gleam of plasteel boots.\n\n\"I wish the capo only good health,\" I called out loudly. \"So I hope that he is seeing a good gerontologist and takes his synapsilstims regularly.\"\n\nThe guard growled in puzzlement at this\u2014but my words were not for his edification. The man I was looking at in the courtyard stopped suddenly, still. Then slowly turned about. I saw keen blue eyes in a long face. Staring at me in silence. Then he came forward and talked to the guard\u2014though still looking at me.\n\n\"What is the disturbance?\"\n\n\"Nothing, your honor. Just sending this one on his way.\"\n\n\"Let him in. I want to question him.\"\n\nThe pointed gun was raised in salute and I marched through the gate. When we were out of earshot of the gate the tall man turned to face me, looking me up and down with frank curiosity.\n\n\"Follow me,\" he said. \"I want to talk with you in private.\" He did not speak until we were in the keep and inside a room with the door closed behind us.\n\n\"Who are you?\" he asked.\n\n\"You know\u2014I was about to ask you the very same question. Does the League know what you are doing here?\"\n\n\"Of course they do! This is a legitimate...\" He caught himself, then smiled. \"At least that proves you're from offplanet. No one can think that fast here\u2014or knows what you know. Here, sit, then tell me who you are. After that I will judge how much I can tell you of our work.\"\n\n\"Fair enough,\" I said, dropping into the chair and lying my gun on the floor. \"My name is Jim. I was a crewman on a Venian freighter\u2014until I got into difficulties with the captain. He dumped me on this planet. That is all there is to it.\"\n\nHe pulled up a pad and began to make notes. \"Your name is Jim. Your last name is...\" I was silent. He scowled. \"All right, let that go for the moment. What is the captain's name.\"\n\n\"I think that I will save that information for later. After you have told me who you are.\"\n\nHe pushed the pad aside and sat back in his chair. \"I'm not satisfied. Without your identity I can tell you nothing. Where do you come from on Venia? What is the capital city of your planet, the name of the chairman of the global consul?\"\n\n\"It's been a long time, I forgot.\"\n\n\"You are lying. You are no more Venian than I am. Until I know more...\"\n\n\"What exactly do you have to know? I am a citizen of the League, not one of the dismal natives here. I watch tri-D, eat at Macswineys\u2014a branch on every known world, forty-two billion sold\u2014I studied molecular electronics, and have a Black Belt in Judo. Does that satisfy you?\"\n\n\"Perhaps. But you told me that you were dumped on this planet from a Venian freighter, which cannot be true. All unapproved contact with Spiovente is forbidden.\"\n\n\"My contact was unapproved. The ship was smuggling in guns like this one.\"\n\nThat got his attention all right. He grabbed the pad. \"The captain's name is...\"\n\nI shook my head in a silent no. \"You'll have that information only if you arrange to get me off this planet. You can do that because you as much as told me you were here with League approval. So let us do a little trading. You arrange for my ticket\u2014I have plenty of silver groats to pay for it.\" Or I would have, which was the same thing. \"You will also give me some small help in a local matter\u2014then I'll tell you the captain's name.\"\n\nHe didn't like this. He thought hard and wriggled on the hook, but could not get off it.\n\n\"While you are making your mind up,\" I said, \"you might tell me who you are and what you are doing here.\"\n\n\"You must promise not to reveal our identity to the natives. Our presence is well known offplanet, but we can only succeed here if our operation remains covert.\"\n\n\"I promise, I promise. I owe nothing to any of the locals.\"\n\nHe steepled his fingers and leaned back as though beginning a lecture. I had guessed right\u2014as his first words revealed.\n\n\"I am Professor Lustig of the University of Ellenbogen, where I hold the chair of applied socioeconomics. I am head of my department and I must say that I founded the department, since applied socioeconomics is a fairly new discipline, an outgrowth, obviously, of theoretical socioeconomics...\"\n\nI blinked rapidly to keep my eyes from glazing over and forced myself to keep listening. It was teachers like Lustig who made me run away from school.\n\n\"... years of correspondence and labor to attain our fondest ambition. Practical application of our theories. Dealing with the bureaucrats of the League was the most difficult because of the League nonintervention policy. In the end they were convinced that with the proper controls we be permitted to operate a pilot project here on Spiovente. Or as someone said with crude humor, we certainly couldn't make things worse. We keep our operation at the current level of planetary technology so it will be self-sustaining when we leave.\"\n\n\"What exactly are you trying to do?\" I asked.\n\nHe blinked rapidly. \"That should be obvious\u2014that is the only thing I have been talking about.\"\n\n\"You have been telling me theory, professor. Would you mind being specific about what you hope to accomplish.\"\n\n\"If you insist, on layman's terms, we are attempting to do no less than change the very fabric of society itself. We intend to bring this planet, kicking and screaming if necessary, out of the dark ages. After the Breakdown Spiovente sank into a rather repulsive form of feudalism. More warlordism, in fact. Normally a feudalistic society performs a great service during an age of disintegration. It maintains a general framework of government as various localities protect and care for themselves.\"\n\n\"I haven't seen much caring or protecting.\"\n\n\"Correct. Which is why these warlords will have to go.\"\n\n\"I'll help shoot a few.\"\n\n\"Violence is not our way! In addition to being distasteful, it is forbidden to League members. Our aim is to bring into existence government independent of the capos. In order to do that we are encouraging the rise of a professional class. This will bring about increased circulation of money and the end of barter. With increased funds the government will be able to institute taxation to purchase public services. To reinforce this a judiciary will need to be formed. This will encourage communication, centralization, and the growth of common ideas.\"\n\nSounded great\u2014although I wasn't wild about the taxes bit, or the judiciary. Still, anything would be better than the capos.\n\n\"That all sounds fine in theory,\" I said. \"But how do you put it into practice?\"\n\n\"By providing better services at a lower price. Which is why the Black Monks tried to attack us. They are no more religious than my hat\u2014the order is just a front for their monopoly of technology. We are breaking that monopoly and they don't like it.\"\n\n\"Very good. Yours sounds a fine plan and I wish you the best of luck. But I have a few things to do myself before I leave this sinkhole. To help you in your task of breaking the technological monopoly I would like to purchase some of your sleeping gas.\"\n\n\"Impossible. In fact it is impossible for us to aid you in any way. Nor are you leaving here. I've signalled for the guards. You will be held until the next League ship arrives. You know far too much about our operation to be permitted your freedom.\"\nChapter 27\n\nEven as this unacceptable bit of information was sinking into my brain, my body was launched across the desk. He should have remembered the bit about the Black Belt. My thumbs bit deep and he slumped. Even before his head bounced off the desk I had bounced off the floor and dived for the door. And none too soon\u2014as I pushed the locking bolt home I saw that the handle above it was starting to turn.\n\n\"Now Jim, move fast,\" I advised myself, \"before the alarm is spread. But first let me see what this two-faced academic has in his possession that may be of use.\"\n\nThere were files, papers, and books in the desk, nothing that would be of any value to me now. I sprayed it all about me on the floor as the banging started on the door. I didn't have much time. Next the prof. I tore his cloak open and ransacked his pockets. There was even less of interest here\u2014other than a ring of keys. I shoved them into my own pocket; they would have to do for loot. Seizing up the gun I dived for the window just as something heavy hit the door with a shuddering thud. Two stories up and the courtyard below was paved with evil-looking cobblestones. I would break my legs if I jumped. I leaned out and was grateful for the second-rate Spiovente masons. There were large gaps between the stones of the outer wall. The door crashed and splintered as I climbed out of the window, thrust the gun through my belt in the small of my back\u2014and began to climb down.\n\nIt was easy enough. I jumped the last bit, did a shoulder roll, which jammed the gun painfully into my spine, retrieved it, and stumbled around the corner of the building before anyone appeared in the window above. I was free!\n\nOr was I? Instant gloom descended. Free in the middle of the enemy keep with all men's hands turned against me. Some big free.\n\n\"Yes, free!\" I ground my teeth together arrogantly, braced my shoulders, and put a bold swagger into my walk. \"Free as only a Stainless Steel Rat can be free! Just press on, Jim\u2014and see if you can't find some locks to go with those keys in your pocket.\"\n\nI always get the best advice from myself. I marched on through an archway that led into the large courtyard. There were armed men lolling about here and they completely ignored me. That wouldn't last long. As soon as the alarm was raised they would all be after my hide. Eyes straight ahead I walked towards a massive building on the far side. It had a single large gate set into the wall, with a smaller one next to it. As I came closer I saw that both had very modern locks set into them. Very informative. I was most interested in what was locked away here. Now all I had to do was find the right key.\n\nTrying to look as though I belonged here I stopped before the smaller door and flipped through the keys. There must have been twenty of them. But the lock was a Bolger, that was obvious to my trained eye, so I fingered through them, looking for the familiar diamond shape.\n\n\"Hey, you, what you doing there?\"\n\nHe was a big thug, dirty and unshaven and red of eye. He also had a long dagger thrust through his belt, the hilt of which he was tapping with his fingers.\n\n\"Unlocking this door, obviously,\" was my firm response. \"Are you the one they sent to help me? Here, take this.\"\n\nI handed him my gun. This bought me a number of seconds as he looked at the weapon, enough time for me to push one key into the lock. It didn't turn.\n\n\"No one sent me,\" he said, examining the gun, which distracted him nicely for a few seconds more. I couldn't be doing anything wrong if I had given him my only weapon, could I? I could almost see him thinking, slowly, moving his lips as he did. I interrupted the turgid flow of his thoughts.\n\n\"Well, since you are here you can help me...\"\n\nAhh, the next key did the job, turning sweetly. The door opened and I turned about just as sweetly with my fingers pointed to jab. I caught the gun as he slid to the ground.\n\n\"Hey, you, stop!\"\n\nI ignored this rude command since I had not the slightest desire to see who was calling, but slipped through the door instead and slammed it shut behind me. Turned and looked around and felt a sharp pang of despair. There was no hope here. I was in an enormous chamber, badly lit by slits high in the wall. It was a garage for the steam cars. Five of them, lined up in a neat row.\n\nIt would be fine to escape in one of these, really wonderful. I had watched them in operation. First the fire had to be lit, then wood pushed in, steam raised. This usually took at least an hour. At that point, say, I could manage to do all this undisturbed, I had to open the door and clank to freedom at a slow walking pace. No way!\n\nOr was there a way? As my eyes adjusted to the gloom I realized that these weren't the same kind of steam cars I had seen before\u2014with their wooden wheels and iron tires. These had soft tires of some kind! Improved technology? Could it be offplanet technology disguised as antique wrecks?\n\nI hurried over to the closest one and climbed up to the operator's seat. There were the familiar big control levers and wheels\u2014but invisible from the ground was a padded driver's seat and familiar groundcar controls. This was more like it!\n\nSlipping my gun under the seat, I slipped myself into it. A safety belt hung there, wise precaution, but not at the moment. I pushed it aside as I leaned forward to examine the controls. Motor switch, gear selector, speedometer\u2014as well as some unfamiliar dials and controls. A banging on the door convinced me I should make a detailed study later. I reached out and turned on the motor. Nothing happened.\n\nOr rather something totally unexpected happened. The motor didn't start but a girl's voice did, speaking in my ear.\n\n\"Do not attempt to start this vehicle without wearing your seatbelt.\"\n\n\"Seatbelt, right, thank you.\" I clicked it on and turned the switch again.\n\n\"The engine will start only with the gear selector in neutral.\"\n\nThe banging on the door was even louder. I cursed as I pushed the selector, trying to find the right location in the dim light. The door crashed and splintered. There, now the switch again.\n\nThe motor turned over. I pushed the drive into forward. And the voice spoke.\n\n\"Do not attempt to drive with your hand brake on.\"\n\nI was cursing louder now, the small door broke down and crashed to the floor, pistons began to move around me while steam spurted and hissed. Someone shouted and the men in the door started towards me.\n\nThe thing shuddered and lumbered forward.\n\nThis was more like it! Covered in steel plates and fake ironmongery, it must be incredibly heavy. There was a simple way to find out. I floored the accelerator, twisted the wheel\u2014and pointed the hulk straight at the large door.\n\nIt was beautiful. The steam roared and spurted as I accelerated. Hitting the door dead center with a crash that deafened me. But my noble steed never slowed a fraction. Wood screeched and tore and fell away as I plowed through in a cloud of flying timber. I had a quick view of fleeing pedestrians before I had to duck down to prevent myself from being beheaded by a board. It scratched and clumped and fell away. I sat up and smiled with pleasure.\n\nWhat a wonderful sight. Soldiers were fleeing in all directions, dashing for cover. I swung the wheel and spun in a tight circle looking for the way out. A bullet clanged into the steel plating and whined away. There the gate was\u2014dead ahead. I floored the accelerator this time, then found the whistle cord. It screamed and steam spurted and I picked up speed.\n\nAnd none too soon, either. Someone had kept his head and was trying to lift the drawbridge. Two men had plugged the handle into the clumsy winch and were turning it furiously; chains clanked and tightened. I headed for the center of the gate, whistle screeching, bullets beginning to spang on the steel around me. I crouched down and kept the pedal on the floor. I was going to have only one chance.\n\nThe drawbridge was rising, slowly and steadily, cutting off my escape, getting larger and larger before me. It was up ten, twenty, thirty degrees. I was not going to make it.\n\nWe hit with a jar that would have thrown me out if the safety belt hadn't been locked. Thank you, voice. The front wheels rose up onto the drawbridge, higher and higher, until the nose of the car was pointing into the air. If it climbed any higher it would be flipped onto its back.\n\nWhich was a chance that I would just have to take. The gears growled and my transport of delight bucked and chuntered\u2014and I heard a squealing and snapping.\n\nThen the whole thing pitched forward. The chains lifting the drawbridge had torn from their moorings under the massive weight of my car. The nose fell and we hit with a crash that almost stunned me.\n\nBut my foot was still down and the wheels were still turning. The vehicle shot forward\u2014straight for the water. I twisted the steering wheel, straightened it, then tore across the bridge and onto the road. Faster and faster, up the hill and around the bend\u2014then let up on the speed before we overturned on the ruts. I was safe and away.\n\n\"Jim,\" I advised myself, gasping for breath. \"Try not to do that again if you can avoid it.\"\n\nI looked back, but there was no one following me. But there would be, soon, if not on foot then in one of the other fake steam cars. I put my foot back down and kept my mouth clamped shut so it didn't clack and splinter my teeth when we hit the bumps.\n\nThere was a long hill that slowed my pace. Even with the accelerator on the floor we crawled because of the gearing and the weight of the beast. I used the opportunity to check the charge\u2014batteries full! They had better be because I had no way of recharging them once they ran down. Above the clatter and rumble I heard a thin and distant whistle and flashed a quick look over my shoulder. There they were! Two of the machines, hot on my tail.\n\nThere was no way they were going to catch me. Off the road these things would be useless and mired down\u2014and there was only one road leading to Dimonte's keep. I was on it and headed that way and I was going to keep them behind me all the way.\n\nExcept that if I led them there they would know who had pinched their wagon and would come after it with the gas bombs. No good. I looked back and saw that they were gaining\u2014but they soon slowed to my pace when they reached the bottom of the hill. I went over the top and my speed picked up\u2014as did the jarring. I hoped that they had built the thing to withstand this kind of beating. Then the crossroad loomed up ahead, with peasants leaping out of my way, and there was the left turning that would take me to Capo Dimonte. I streamed right through it. I didn't know this road at all so all I could do was go on and keep my fingers crossed.\n\nSomething had to be done\u2014and fairly soon. Even if I stayed ahead of them all day I would run the battery flat and that would be that. Think, Jim, cudgel the old brain cells.\n\nOpportunity presented itself around the next bend. A rough farm track led off through a field and down to a stream. Then, like all good ideas, this one appeared full blown in my forebrain, complete in every detail.\n\nWithout hesitation I turned the wheel and trundled down into the meadow. Going slower and slower as I felt my wheels sink into the soft soil. If I got mired now it was the end. Or at least the end of my mastery of this crate\u2014which I would dearly like to keep for awhile. Carry on, Jim, but carefully.\n\nAt the lowest speed, in the lowest gear, I ground forward until the front wheels were in the stream. They were sinking mushily into the mud as I stopped\u2014then carefully began to back out. Looking over my shoulder, keeping in the ruts I had made on the way down. Reversing out of the field until I was safely back on the road. As I shifted gears I permitted myself a quick glimpse of my work. Perfect! The ruts led straight down to the water and on into it.\n\nOn the road behind me I heard a not-too-distant whistle. I stood on the throttle and accelerated around the bend until I was hidden by the trees. Stopped, killed the engine, slammed on the brakes, and jumped down.\n\nThis was going to be the dangerous part. I had to convince them to follow the tracks. If they didn't believe me, I had little chance of escape. But it was a risk that had to be taken.\n\nAs I ran I pulled off my jacket, staggering as I pulled my arms free and turned it inside out. I draped it over my shoulders, tied the arms in front, then bent to roll up my trouser legs. Not much of a disguise, but it would have to do. Hopefully the drivers had not had a good look at me\u2014if they had seen me at all.\n\nI stood by the spot where I had turned and had just enough time to seize up some dirt and rub it into my face as the first pseudo steam car clanked around the bend.\n\nThey slowed as I stepped into the road and pointed. And shouted.\n\n\"He went that-away!\"\n\nThe driver and the gunmen turned to look at the field and stared at the tracks. The vehicle slowed to a stop.\n\n\"Splashed right into the water and kept on going through the field. Feller a friend of yours?\"\n\nThis was the moment of truth. It stretched taut, longer and longer as the second vehicle came up and slowed to a stop as well. What if they questioned me\u2014even looked closely at me? I wanted to run\u2014but if I did, that would be a giveaway.\n\n\"Follow him!\" someone called out, and the driver twisted his wheel and turned towards the field.\n\nI slipped back into the trees and watched with great interest. It was beautiful. I felt proud of myself; yes, I did. I am not ashamed to admit it. When a painter creates a masterpiece he knows it and does not attempt to diminuate its importance by false modesty.\n\nThis was a masterpiece. The first car rattled down through the field, bobbing and bouncing, and hit the water with an immense splash. It was going so fast that its rear wheels actually reached the stream before it slowed to a stop. And began to slowly sink into the soft mud. It went down to its hubs before it stopped.\n\nThere was much shouting and swearing at that\u2014and best of all someone rooted out a chain and connected the two cars. Wonderful. The second one spun its wheels and churned the field until it too was safely mired. I clapped appreciatively and strolled back to my own car.\n\nI shouldn't have done it, I know. But there are times when one just cannot resist showing off. I sat down, snapped on my belt, started the motor, moved the car carefully forward and back until I had turned about. Then accelerated back down the road.\n\nAnd as I passed the turnoff I pulled down hard on the whistle. It screeched loudly and every head turned, every eye was on me. I waved and smiled. Then the trees were in the way and the beautiful vision vanished from sight.\nChapter 28\n\nIt was a victory ride. I laughed aloud, sang, and blew the whistle with joy. When this first enthusiasm had died down I moved the queen on my mental chessboard and considered what came next. The hissing of steam and clanking of machinery was distracting and I examined the controls until I found the switch that turned the special effects off. The steam was being boiled to order and the sounds were just a recording. I threw the switch and rode on in peace towards Capo Dimonte's keep. It was late afternoon before I reached it\u2014and by that time my plans were complete.\n\nWhen I came around the last bend in the road and turned onto the causeway I had full sound and steam effects going again. I trundled slowly down in clear sight of the guards. They had the partially repaired drawbridge raised long before I reached it, and peered out suspiciously at me as I stopped before the gap.\n\n\"Don't shoot! Me friend!\" I called out. \"Member of your army and a close associate of the Capo Dimonte. Send for him at once for I know he wants to see his new steam cart.\"\n\nHe did indeed. As soon as the drawbridge was lowered he strode across it and looked up at me.\n\n\"Where did you get this?\" he asked.\n\n\"Stole it. Climb aboard and let me show you some interesting things.\"\n\n\"Where is the sleeping gas?\" he asked as he climbed the rungs.\n\n\"I didn't bother with it. With this cart I have developed an even better and more foolproof plan. This is no ordinary steam cart, as I hope you have noticed. It is a new and improved model with some interesting additions that will capture your attention....\"\n\n\"You idiot! What are you talking about?\" He slipped his sword up and down in its scabbard\u2014such a quick temper.\n\n\"I will demonstrate, your caponess, since one action speaks louder than a thousand words. I also suggest that you sit there and strap that belt about you as I have done. This demonstration, I guarantee, will impress you.\"\n\nIf not impressed already, he was at least curious. He strapped in and I backed the length of the causeway to the shore. Going slowly with all attendant wheezing and clanking. I stopped the car and turned to him.\n\n\"What about the speed of this thing? What you are used to?\"\n\n\"Speed? You mean how fast it moves? This is an excellent yoke and goes with greater alacrity than my own.\"\n\n\"You have seen nothing yet, capo. First\u2014notice this.\"\n\nI turned off the sound and steam and he nodded with understanding. \"You have banked its fires and it rests and does not move.\"\n\n\"Quite the opposite. I have simply silenced it so no one can hear its approach. It is raring to go\u2014and go it will. After you answer one question. If this cart belonged to an enemy and it appeared here\u2014would your soldiers have time to raise the drawbridge before it reached them?\"\n\nHe snorted with derision. \"What sort of fool do you take me for with questions like that? Before a cart could crawl its way there the drawbridge could be raised and lowered more than once.\"\n\n\"Really? Then hold on and see what this baby can do.\"\n\nI floored the accelerator and the thing shot forward in almost perfect silence. There was the hum of the motor, the rustle of the tires on the smooth stone. Faster and faster towards the gate, which expanded before us with frightening speed. The guards who were standing there dived aside just in time as we hit the rough boards of the repaired wooden drawbridge with a crash, bounced, and rocked through the gate.\n\nAnd shuddered to a halt inside the keep. The capo sat there with round eyes, gasping, then struggled to get his sword free.\n\n\"Assassin! Your attempt to kill me has failed....\"\n\n\"Capo, listen, it was a demonstration. Of how I am going to get you and your soldiers through the gate of Capo Doccia's keep. Right through the open gate into the courtyard where you can kill, loot, murder, torture, maim, destroy...\"\n\nThis got his attention. The sword slid back into its scabbard and his eyes unfocused as they looked at the wonders I had summoned up for him.\n\n\"Right,\" he said, blinking rapidly and coming back to the present. \"You have an interesting idea here, soldier, and I want to hear more about it. Over a flagon of wine\u2014for that ride was something I have never experienced before.\"\n\n\"I obey. But let me first get this cart hidden and out of sight so it cannot be observed. The attack will only succeed if there is complete surprise.\"\n\n\"In that you are correct. Put it in the barn and I will post guards over it.\"\n\nThe wine he gave me was a good cut above the acid the troops were issued and I sipped it with pleasure. But not too much for I was going to need a clear head if the game were to proceed as planned. I had to find reasons that would make sense to him, to convince him to get cracking with his war plans at once. Because if we didn't move quickly Prof. Lustig would be swarming over us with his gas bombs. I am sure he was most unhappy about my pinching his buggy. And there were not that many keeps in the area where it could be hidden. It was time for action. I slid out a rook along a mental rank and spoke.\n\n\"The keep of the foul Capo Doccia is no more than a five-hour walk from here\u2014is that correct?\"\n\n\"Five hours, four-hour forced march.\"\n\n\"Good. Then consider this. He attacked you while you were away with the greater part of your army. His troops did great injury to the drawbridge and the fabric of the keep itself. Before you venture out to launch an attack you must have the drawbridge repaired, hire more soldiers perhaps. So when you begin your next campaign no advantage can be taken of your absence. Is that correct?\"\n\nHe slurped his wine and glared at me over the rim. \"Yes, damn and blast your head, I suppose it is. Prudence, my officers always consul prudence when I want to behead that creature, rip out his entrails, flay him alive...\"\n\n\"And you shall, yes indeed, fine things lurk in your future. And unlike your other advisers I do not consul caution. I think that fiend in human guise should be attacked\u2014and at once!\"\n\nThis appealed to him all right and I could see that I had his undivided attention as I explained my plan.\n\n\"Leave the keep here just as it is\u2014and take all your men. If everything goes as planned you will have troops back here long before anyone knows we have gone. We march at midnight, silent as vengeful spirits, to be in positions of concealment at dawn, as close to Capo Doccia's keep as is possible. I know just the spot. When the drawbridge is opened at dawn I shall use your new machine to see that it stays open. Your troops attack, take the keep by surprise\u2014and the day is won. As soon as you have captured the keep you can send a strong force back here.\"\n\n\"It could happen that way. But how do you plan to stop them closing the drawbridge?\"\n\nAs I told him the wicked grin spread across his face and he whooped with joy.\n\n\"Do it!\" he shouted, \"and I shall make you rich for life. With Doccia's groats of course, after I loot his treasury.\"\n\n\"You are kindness itself to your humble servant. May I then suggest that all in the keep be persuaded to rest, for it will be a long night?\"\n\n\"Yes, that will be done. The orders will be issued.\"\n\nAfter that I slipped away. Other than my natural concern for the tired bodies of my comrades I had other reasons for wishing all of them in their beds. I had a few important tasks to perform before I could get any rest myself.\n\n\"Tools,\" I told Dreng when I had rousted him out. \"Files, hammers, anything like that. Where would I find them here?\"\n\nHe shoved a finger deep into his matted hair and scratched hard in thought. I resisted the urge to reach out and shake him and waited instead until the slow processes had crawled to a finish. Perhaps the fingernail rasping on skull helped his sluggish synapses to function. It would be best not to interfere with an established practice. Eventually he spoke.\n\n\"I don't have any tools.\"\n\n\"I know, dear boy.\" I could hear my teeth grate together and forced myself to keep control. \"You don't have tools, but someone here must. Who would that be?\"\n\n\"Blacksmith,\" he said proudly. \"The blacksmith always has tools.\"\n\n\"Good lad. Now, would you kindly lead the way to this blacksmith?\"\n\nThe individual in question was sooty and hairy and in a foul mood, sour wine strong on his breath.\n\n\"Hiss off, runt. No one touches Grundge's tools, no one.\"\n\nRunt indeed! I did not have to force the snarl and growl. \"Listen you filthy piece of flab\u2014those are the capo's tools, not your tools. And the capo sent me for them. Now either I take them now or my knave goes to bring the capo here. Shall I do that?\"\n\nHe closed his fists and growled, then hesitated. Like everyone else, he had seen me drive the capo into the keep and knew I was his confidant. He couldn't take any chances on crossing his boss. He began to bob up and down bowing and scraping.\n\n\"Certainly, master. Grundge knows his place. Tools, sure, take tools. Over here, whatever you want.\"\n\nI pushed past his sweaty form to the dismal display of primitive devices. Pathetic! I kicked through the pile until I found a file, hammer, and clumsy metal snips that would have to do. I pushed them towards Dreng.\n\n\"Take these. And you, Grundge, can crawl over in the morning to the barn and get them back.\"\n\nDreng followed after me, then gaped up in awe at the steam cart.\n\n\"Close your mouth before you catch some flies,\" I told him, seizing the tools. \"What I'll need next is a stout bag or sack of some kind, about this big. Scout one out and bring it to me here. Then get to bed because you will not be getting much sleep tonight.\"\n\nWith proper tools I could have done the job in no time at all. But I had a feeling that tolerances wouldn't be that exact here and as long as I was close to the model it would be all right. The metal siding next to the driver's seat was roughly the thickness of the wooden key. I cut and filed and hacked a portion of it into shape. It would have to do.\n\nDreng\u2014and hopefully everyone else\u2014was now asleep and I could begin Operation Great-groat. With the key in my pocket, the bag tucked into my waist, silent as a shadow\u2014I hoped\u2014I made way into the depths of the keep. I had memorized The Bishop's map and his spirit must have been watching after me for I found the treasury without being seen. I slipped the key into the lock, crossed the fingers of my free hand, and turned.\n\nWith a metallic screech it clanked open. My heart did its usual pounding-in-chest routine while I stood rooted there. The noise must have been heard.\n\nBut it hadn't been. The door creaked slightly when I opened it and then I was inside the vault and easing it shut behind me.\n\nIt was beautiful. High, barred windows let in enough light so I could see the big chests against the far wall. I had done my fiscal research well, getting a look at a braggard's store of groats, so I knew just what to look for.\n\nThe first chest was stuffed with brass groats, my fingers could distinguish their thick forms in the darkness. In logical progression I found silver groats in the next chest and I shoveled my bag half full of them. As I did this I saw a smaller chest tucked in behind this one. I smiled into the darkness as I groped and felt the angled shapes within. Golden groats\u2014and lots of them. This was going to be a very successful heist after all. I only stopped shoveling when the bag became too heavy. Beware of greed. With this bit of advice to myself I threw it over my shoulder and let myself out just the way I had come in.\n\nThere were guards in the courtyard but they never saw me as I slipped into the barn. I turned on the instrument lights of the car, which provided more than enough illumination for me to see by. I opened the storage locker below and put the money bag into place. As I closed it I was overwhelmed by a great sensation of relief. In my mind's eye I slid out another rook to join the first. The chess game was going as planned and mate was clearly visible ahead.\n\n\"Now, Jim,\" I advised. \"Get your head down and get some sleep. Tomorrow is going to be an exceedingly busy day.\"\nChapter 29\n\nI muttered and slapped and rolled over but the irritation persisted. Eventually I blinked my grimy eyes open and growled up at Dreng, who was shaking my shoulder. He stepped away in fear.\n\n\"Do not beat me, master\u2014I am only doing as you instructed. It is time to waken, for the troops are assembling now in the courtyard.\"\n\nI growled something incoherent and this turned into a cough. When I did this a cup appeared before me and I drank deep of the cool water, then dropped back onto the bunk. Not for the first time did I approve of the knave system. But I was beat, bushed, fatigued. Even the stamina of youth can be sapped by adversity. I shook my head rapidly, then sat up on my elbows, angry at myself for the brief moment of self-pity.\n\n\"Go, good Dreng,\" I ordered, \"and find me food to nourish my hungry cells. And some drink as well since alcohol is the only stimulant these premises seem to have.\"\n\nI splashed cold water over my head in the courtyard, gasping and spluttering. As I wiped my face dry I saw in the clear starlight the ranks of soldiers being drawn up as the ammunition was being issued. The great adventure was about to begin. Dreng was waiting when I returned. I sat on my bunk and ate a pretty repellent breakfast of fried dinglebeans washed down by the destructive wine. I talked between gruesome mouthfuls because this was the last private moment I would have with my knave.\n\n\"Dreng, your military career is about to end.\"\n\n\"Don't kill me, master!\"\n\n\"Military career, idiot\u2014not your life. Tonight is your last night of service and in the morn you will be off home with your pay. Where does your old dad hide his money?\"\n\n\"We are too poor to have any groats.\"\n\n\"I am sure of that. But if he had any\u2014where would he put it?\"\n\nThis was a complicated thought and he puzzled over it while I chewed and swallowed. He finally spoke.\n\n\"Bury it under the hearth! I remember he did that once. Everyone buries their money under the fire; that way it can't be found.\"\n\n\"Great. That way it certainly can be found. You have got to do better than that with your fortune.\"\n\n\"Dreng has no fortune.\"\n\n\"Dreng will have one before the sun rises. I'm paying you off. Go home and find two trees near your home. Stretch a rope between them. Then dig a hole exactly halfway along the rope. Bury the money there\u2014where you can find it when you need it. And only take out a few coins at a time. Do you have that?\"\n\nHe nodded enthusiastically. \"Two trees, halfway. I never heard of anything like that before!\"\n\n\"An earth-shaking concept, I know,\" I sighed. There certainly was a lot that he hadn't heard about. \"Let's go. I want you to be stoker on my chariot of fire.\"\n\nI staggered to my feet and led the way to the barn. Now that the troops were lined up and ready the officers were finally appearing, scratching and yawning, with the capo at their head. I didn't have much time. Dreng climbed into the car behind me and squealed with fear when I turned on the instrument lights.\n\n\"Demonic illumination! Spirit lights! Sure sign of death!\"\n\nHe clutched at his chest and looked ready to expire until I gave him a good shaking. \"Batteries!\" I shouted. \"The gift of science denied to this dumb world. Now, stop quaking and open your bag.\"\n\nAll thoughts of death vanished and his eyes stuck out like boiled eggs as I shoveled silver and gold groats into his leather bag. This was a fortune that would change his entire life for the better, so at least I was accomplishing one good deed by my presence here.\n\n\"What are you doing up there?\"\n\nIt was Capo Dimonte, glaring up suspiciously from below.\n\n\"Just stoking the engines, excellency.\"\n\n\"Kick that knave out of the way, I'm coming up.\"\n\nI waved the goggle-eyed Dreng to the back of the car as the capo climbed aboard.\n\n\"You favor me with your presence, capo.\"\n\n\"Damned right. I ride while the troops walk. Now, move this thing out.\"\n\nThe scouts had already gone on ahead when we rumbled across the drawbridge and onto the causeway. The main body of troops came behind us, a certain eagerness in their step despite the hour. All of them had lost valuables and possessions\u2014even knaves\u2014during the raid. All were eager for revenge and theft.\n\n\"The Capo Doccia must be taken alive,\" Capo Dimonte suddenly said. I started to answer until I realized that he was talking only to himself. \"Tied and left helpless, brought back to the keep. First a little flaying, just enough skin to make a hatband. Then maybe blinding. No\u2014not right away\u2014he must see what is happening to him....\"\n\nThere was more like this, but I tuned it out. I had thoughts of my own\u2014and even some regrets. When The Bishop had been killed, my anger had overwhelmed all of the clear thinking that I should have been doing. All excuses vanished now\u2014I was embarking on this expedition solely for revenge. And I couldn't claim to be doing it in The Bishop's memory because he would have been seriously opposed to violent action of this kind. But it was too late now to turn back. The campaign had been launched and we were well on our way.\n\n\"Stop this thing!\" the capo ordered suddenly, and I hit the brakes.\n\nThere was a dark knot of men waiting on the road ahead\u2014our advanced scouts. The capo climbed to the ground and I leaned out to see what was happening. They were leading a man who had his arms bound behind him.\n\n\"What happened?\" the capo asked.\n\n\"Found him watching the road, excellency. Caught him before he could get away.\"\n\n\"Who is he?\"\n\n\"Soldier, name of Palec. I know him, served with him in the southern campaign.\"\n\nThe capo walked up to the prisoner and shoved his face close to the other's and snarled. \"I have you, Palec. Tied and bound.\"\n\n\"Aye.\"\n\n\"Are you the Capo Doccia's man?\"\n\n\"Aye, I serve under him. I took his groat.\"\n\n\"You've spent that on wine a long time ago. Will you serve with me and take my groat?\"\n\n\"Aye.\"\n\n\"Release him. Barkus\u2014a silver groat for this man.\"\n\nThese mercenaries fought well, but they also changed sides easily enough. Why not? They had no stakes in any of the capos' quarrels. Once Palec had accepted the coin they gave him his weapons back.\n\n\"Speak, Palec,\" the capo ordered. \"You are my loyal servant now, who has taken my groat. But you used to serve with Capo Doccia. Tell me what he plans.\"\n\n\"Aye. No secret there. He knows that your army is intact and you will be coming after him as soon as you can. Some of us have been sent out to watch the roads, but he doesn't think that you will march for some time yet. He stays drunk, that's a sign he's not expecting a fight.\"\n\n\"I'll put a sword through his belly, let out the wine and guts!\" The capo cut off his dreaming with an effort and forced himself back to the present. \"What about his troops? Will they fight?\"\n\n\"Aye, they've just been paid. But they have little love for him and will change sides as soon as the battle is lost.\"\n\n\"Better and better. Fall in with the ranks, scouts out ahead, start this machine.\"\n\nThe last was directed at me as he climbed back to his seat. I kicked it into gear and the advance continued again. There were no more interruptions and we proceeded, with hourly rest breaks, towards the enemy keep. It was well before dawn when we came to the scouts waiting on the road. This was the spot I had picked. The keep of Capo Doccia was around the next bend.\n\n\"I will post your lookout now,\" the capo said.\n\n\"Agreed. My knave here will show them the exact spot where they are to stay hidden, in sight of the gate.\" I waited until he was out of earshot before I whispered my instructions to Dreng.\n\n\"Take your bag and everything you possess with you\u2014because you are not coming back.\"\n\n\"I do not understand, master....\"\n\n\"You will if you shut up and listen instead of talking. Lead the soldiers to the bushes where we hid, when we were getting ready to rescue The Bishop. You do remember the place?\"\n\n\"It is past the burnt tree over the hedge and...\"\n\n\"Great, great\u2014but I don't need the description. Take the soldiers as I said, show them where to hide, then lie close beside them. Soon after dawn things are going to get very, very busy. At that time you will do nothing, understand that\u2014don't speak, just nod.\"\n\nHe did. \"Fine. You just remain there when everyone rushes off. As soon as they are gone and no one is looking at you\u2014slip away. Back into the woods and get to your home and lay low until the excitement is over. Then count your money and live happily ever after.\"\n\n\"Then\u2014I will no longer be your knave?\"\n\n\"Right. Discharged from the army with honor.\"\n\nHe dropped to his knees and seized my hand, but before he could say anything I touched my finger to his lips.\n\n\"You were a good knave. Now be a good civilian. Move!\"\n\nI watched him leave until he was swallowed up in the darkness. Dumb\u2014but loyal. And the only friend that I had on this rundown planet. The only one that I wanted! Now that The Bishop...\n\nThis morbid turn of thought was happily interrupted by the capo, who clambered back to his seat. He was followed by armed soldiers until the upperworks of the car were packed solid with them. The capo squinted up at the sky.\n\n\"There is the first light. It will be dawn soon. Then it will begin.\"\n\nAfter that we could only wait. The tension so thick in the air that it was hard to breathe. Blurred faces began to emerge from the darkness, all of them set in the same grim expression.\n\nI concentrated on what was happening around the bend, remembering the way it had been when Dreng and I had lain out there. Watching and waiting. The locked gate of the keep, the drawbridge up, all of it growing clearer as the sun rose. Smoke from cooking fires drifting up from behind the thick walls. Then the stirring of the soldiery, changing of the guards. At last the gate unlocked, the drawbridge lowered. Then what? Would they keep to the same routine? If they did not our force would soon be discovered....\n\n\"The signal!\" the capo said as he crashed his elbow hard into my ribs.\n\nHe didn't have to. I had seen the soldier wave the instant that he had appeared. My foot was already jammed down on the accelerator and we were picking up speed. Around the bend in the road, bouncing and swaying on the ruts, then straight ahead towards the entrance to the keep.\n\nThe guards looked up and gaped as we shot towards them. The slaves pulling the cart stared too, frozen and unmoving.\n\nThen the shouting started. The drawbridge creaked as they tried to raise it, but the cart and slaves were still on it. There were kicks and screamed orders and every second of wasted time brought us that much closer. They finally started to drag the cart back through the gate\u2014but it was too late.\n\nWe were upon them. The front wheels hit the drawbridge and we bounced into the air, coming down with a splintering crash. I stood on the brakes as we plowed into the cart. Slaves and guards were diving into the moat to escape destruction as we skidded, with locked wheels, right into the mouth of the gate.\n\n\"For Capo Dimonte, for groats, and for God!\" The capo shouted as he leapt to the attack.\n\nThe others leapt with him, walking over my back as I crouched down, jumping onto the drawbridge then through the gate.\n\nThere was screaming and shouting, the banging of guns. From behind me a growing roar of voices from the rest of the attacking army. I could see that the capo and his men were fighting inside the gate and had captured the drawbridge mechanism from the soldiers who were trying to raise it. Raising it had of course been impossible because of the great weight of the car resting on it. That had been the beauty and simplicity of my plan. Once I had arrived the drawbridge had to stay down. Only now did I trundle forward so that the rest of the troops had a clear way to the gate.\n\nThe battle for the keep of Capo Doccia was joined.\nChapter 30\n\nThis was a surprise attack that really had been a surprise. Our invading forces were pouring across the drawbridge and into the keep even as Capo Doccia's soldiers were emerging from their quarters. The guards on the wall fought fiercely, but they were outnumbered.\n\nTo add to the confusion I turned on the steamer sound effects and hung onto the whistle as I charged at the defenders who were trying to group up ahead. A few shots were fired at me, but most of the soldiers dived aside and ran. I screeched about and saw that the battle was going very well indeed.\n\nThe defenders on the walls were raising their hands in surrender. Being outnumbered from the start, and having little reason to fight for the capo as we had been told, they were eager to save their lives. Near the inner gate a group of officers were showing more spirit and a fierce battle was going on there. But one by one they were cut down or clubbed into submission. Two of them fled for the building but found the heavy door slammed in their faces.\n\n\"Bring torches!\" the Capo Dimonte shouted. \"We'll smoke the buggers out!\"\n\nThe battle had ended as swiftly as it had begun. The gate, walls and courtyard were in our hands. Huddled corpses showed the ferocity of the brief engagement. Slaves shivered in fear against the walls while the soldiers who had surrendered were being marched off. Only the central building remained in the hands of the defenders. Capo Dimonte knew exactly what to do about this. He waved a smoking torch over his head and called out loudly.\n\n\"All right, Doccia, you fat-bellied toad, this is your end. Come out and fight like a man you worm, or I'll burn you out. And burn alive every man, woman, child, dog, rat, pigeon who stays in there with you. Come out and fight, you ugly piece of vermin\u2014or remain and be cooked like a roast!\"\n\nA gun fired from inside and a bullet spanged from the cobbles at the capo's feet. He waved his red-drenched sword and a blast of gunfire roared out as our troops fired en masse. Bullets zinged from the stonework, thudded into the sealed door, and whistled in through the windows. When the firing stopped, shrill screaming could be heard from inside the building.\n\n\"One warning only!\" Capo Dimonte called out. \"I do not war on women or on good soldiers who surrender. Lay down your arms and you will go free. Resist and you will be burned alive. There is only one I want\u2014that pig, Doccia. Hear that, Doccia, you lout, swine, worm...\"\n\nAnd more, once he warmed to the subject. The torch crackled and smoked and there was the sound of muffled shouting and scuffling from inside the building.\n\nThen the door burst open and Capo Doccia came rolling down the steps end over end. He was barefooted, half-dressed\u2014but he was holding his sword.\n\nAt the sight of his enemy Capo Dimonte lost whatever little remaining cool he had left. He howled with anger and rushed forward. Doccia climbed to his feet, blood on his face, and raised his sword in defense.\n\nIt was a sight to watch\u2014and everyone did. There was an undeclared truce as the two leaders battled. The soldiers lowered their weapons and faces appeared at all of the windows above them. I climbed out of my seat and stood on the front of the car, where I had a perfect view of the combatants.\n\nThey were well-matched, both in anger and ability. Dimonte's sword crashed down on Doccia's raised blade. He did a neat parry, then thrust\u2014but Dimonte had moved back. After that it was steel on steel, punctuated by grunted curses.\n\nBack and forth across the cobbles they went, slashing away as if their lives depended on it. Which, of course, they did. It was pretty primitive saber work, slash and parry, but it certainly was energetic. A cry went up as Dimonte drew first blood\u2014a cut on Doccia's side that quickly stained his shirt.\n\nThis was the beginning of the end. Dimonte was stronger and angrier, high on victory. If Doccia had been drinking as much as we had been told, he was also fighting a hangover as well as his enemy. Dimonte began pressing harder and harder, slashing remorselessly, pushing the other capo relentlessly across the courtyard. Until his back was to the wall of the building and he could retreat no more. Dimonte beat down the other's guard, hammered him on the jaw with the hilt of his sword\u2014then disarmed Capo Doccia with a savage twist of his blade.\n\nAll of his plans for sadistic torture were washed away in the passion of his anger. He drew the blade back\u2014then slashed out.\n\nIt was not an attractive sight as the sharp steel tore across Doccia's throat. It sickened me and I turned away. Just as the shadow darkened the sun.\n\nOne person looked up, then another\u2014and there was a gasp. I looked too. Only unlike the rest of them I knew exactly what I was looking at.\n\nThe immense shining form of a D-class spacer that was equipped with atmospheric G-lift. Tonnes of ship drifting light as a feather over the courtyard. Coming to an effortless stop. Hanging there over our heads in silent menace.\n\nI turned and dived for the controls. There was no time to escape, no way to escape. I was scratching at the storage compartment as the first silvery spheres fell free of the ship. I gave them one horrified glance\u2014then took a deep breath and held it as I pulled the compartment door open and plunged my arm inside. Grabbing up the leather bag as I sat back onto the driver's seat.\n\nAll around me the spheres were hitting and bursting, releasing their loads of gas. I dropped the bag onto my lap as the first soldiers crumpled and fell. My fingers fumbled at the seat belt, lengthening it, as Capo Dimonte tottered then fell forward onto his dead enemy's body.\n\nThere was a stinging in my nostrils as I snapped the belt buckle over the bag, sealing it against me. And that was all that I could possibly do.\n\nMy lungs were beginning to hurt as I took a last, long look around the courtyard. I had the strong feeling that it would also be my last sight of the fair world of Spiovente.\n\n\"Good riddance!\" I shouted at the now silent forms, blasting the breath from my lungs. Then breathed in....\n\nI was conscious, I knew that. I could feel something soft under my back and there was a light burning down on my closed eyelids. I was afraid to open them\u2014remembering the blasting headache that had accompanied the last gassing. With this thought I cringed and moved my head.\n\nAnd felt nothing. Emboldened by this tiny experiment, I let one eye open a crack. Still nothing. I blinked in the strong light but there was no pain, no pain at all.\n\n\"A different gas, thank you kindly,\" I said to no one in particular as I opened my eyes wide.\n\nA small room, curved metal walls, a narrow bunk under me. Even if my last sight had not been of the hovering spacer, I should have been able to figure this one out. They had taken me aboard. But where were all my groats? I looked around rapidly, but they were certainly not in sight. The rapid movements of my head had brought on an attack of dizziness so that I fell back onto the bunk and groaned in loud self-pity.\n\n\"Drink this. It will eliminate the symptoms of the gas.\"\n\nI snapped my eyes open again and looked at the big man who was just closing the door behind him. He was in a uniform of some kind, with plenty of gold buttons and stripes, the sort of thing much favored by the military. He was holding out a plastic beaker which I seized gingerly and sniffed.\n\n\"We had plenty of time to poison or kill you while you were unconscious,\" he said. A sound argument. I drained the bitter liquid and instantly felt better.\n\n\"You have stolen my money,\" I said just as he was beginning to speak.\n\n\"Your money is safe\u2014\"\n\n\"It will be safe only when it is in my hands. As it was when you found me, strapped to my body. Whoever took it is a thief.\"\n\n\"Don't talk to me of thievery!\" he snapped. \"You probably stole it yourself.\"\n\n\"Prove it! I say I worked hard for that money and I don't intend to have it stolen for the space-war widows pension scheme....\"\n\n\"That is enough. I did not come here to talk about your miserable groats. They will be placed on deposit in the galactic bank....\"\n\n\"At what rate of exchange? And what kind of interest will it earn?\"\n\nHe was coldly angry now. \"That's enough. You are in deep trouble\u2014and you have a lot of explaining to do. Professor Lustig tells me that your name is Jim. What is your entire name and where do you come from?\"\n\n\"My name is Jim Nixon and I am from Venia.\"\n\n\"We will get nowhere if you persist in lying. Your name is James diGriz and you are an escaped convict from Bit O' Heaven.\"\n\nWell, as you can imagine, I did some rapid blinking at this information. Whoever this lad was he had one hell of an intelligence network. I could see that I was no longer playing the amateur team of the professors. They had called in the pros. And he had thrown me this curve ball to catch me off-balance, get me rattled, get me to talk freely. Except I did not work that way. I shifted mental gears, sat up in the bed so I could see him eye to eye, and spoke calmly.\n\n\"We have not been introduced.\"\n\nThe anger was gone now and he was as calm as I was. He turned and pressed a button on the wall that unfolded a metal chair. He sat down on it and crossed his legs.\n\n\"Captain Varod of the League Navy. Specializing in planetary mop-up details. Are you ready to answer questions?\"\n\n\"Yes\u2014if you will trade me one for one. Where are we?\"\n\n\"About thirteen lightyears out of Spiovente, you'll be happy to hear.\"\n\n\"I am.\"\n\n\"My turn. How did you get to that planet?\"\n\n\"Aboard a Venian freighter that was smuggling weapons to the now deceased Capo Doccia.\"\n\nThat got his attention all right. He leaned forward eagerly as he spoke. \"Who was the captain of the freighter?\"\n\n\"You are out of turn. What are you going to do with me?\"\n\n\"You are an escaped prisoner and will be returned to Bit O' Heaven to serve out your prison sentence.\"\n\n\"Really?\" I smiled insincerely. \"Now I will be happy to answer your question\u2014except I have completely forgotten the captain's name. Would you care to torture me?\"\n\n\"Don't play games, Jim. You are in deep trouble. Cooperate and I will do what I can for you.\"\n\n\"Good. I remember the name and you put me down on a neutral planet and we call it quits.\"\n\n\"That is impossible. Records are kept and I am an officer of the law. I must return you to Bit O' Heaven.\"\n\n\"Thanks. I just got terminal amnesia. Before you leave would you tell me what is going to happen to Spiovente?\"\n\nHe sat back in the chair with no intention at all of leaving.\n\n\"The first thing that will happen will be the termination of Lustig's disastrous intervention. We were forced into that by the Intergalactic Applied Socioeconomics Association. They managed to raise sufficient funds to put into effect some of their theories. A number of planets financed them and it was easier to let them make idiots of themselves than to try and stop them.\"\n\n\"And they have done that now?\"\n\n\"Completely. They have all been shipped out and were very happy to go. Having political and economic theories is one thing. But applying them to harsh reality can be a traumatic experience. This has been done in the past\u2014and always with disastrous results. We know none of the details now, they are lost in the mists of time, but there was an insane doctrine called Monetarism that is reputed to have destroyed whole cultures, entire planets. Now another experiment has gone astray, so the specialists will move in as they should have done in the first place.\"\n\n\"Invasion?\"\n\n\"You have been watching too much tri-D. War is forbidden and you should know better than to suggest that. We have people who will work within the existing society of Spiovente. Probably with this Capo Dimonte, since he has just doubled his domain. He will be aided and encouraged to grow in power, to annex more and more territory.\"\n\n\"And kill more and more people!\"\n\n\"No, we will see to that. Very soon he will not be able to rule without aid and our bureaucrats are waiting to help him. Centralized government...\"\n\n\"The growth of the judiciary, taxes, I know the drill. You sound just like Lustig.\"\n\n\"Not quite. Our techniques are proven\u2014and they work. Within one generation, two at the most, Spiovente will be welcomed into the family of civilized planets.\"\n\n\"Congratulations. Now, please leave so I can sit and brood about my future incarceration.\"\n\n\"And you still won't tell me the name of the gunrunner? He could continue in his smuggling operations\u2014and you would be responsible for more deaths.\"\n\nI would be too. Was I responsible for the dead in the courtyard of the keep as well? The attack had been my idea. But Dimonte would have attacked in any case and there could have been even more dead. The acceptance of responsibility was not done easily. Captain Varod must have been reading my mind.\n\n\"Do you have a sense of responsibility?\" he asked.\n\nGood question. He was a shrewd old boy.\n\n\"Yes, I do. I believe in life and the sanctity of life and I do not believe in killing. Each of us has only one go at life and I don't want to be responsible for cutting short anyone else's. I think I have made some mistakes and I hope I have learned by them. The name of the gunrunner is Captain Ga...\"\n\n\"Garth,\" he said. \"We know him and have been watching him. He has made his last voyage.\"\n\nMy thoughts spun rapidly. \"Then why ask me if you knew all along?\"\n\n\"For your sake, Jim, nobody else's. I told you that our job was rehabilitation. You have made an important decision and I believe that you will be a better individual for it. Good luck in the future.\" He stood to leave.\n\n\"Thanks a lot. I'll remember your words when I am cracking boulders on the rock pile.\"\n\nHe stood in the open door and smiled back at me. \"I am in the justice business on a very large scale. And, in truth, I don't believe in prisons and incarceration for failed bank robbery. You are destined for better things than that. Therefore I am having you returned to prison. You will be transferred to another ship, on another planet, where you will be locked away until it arrives.\"\n\nHe went out, then turned back for just an instant. \"Taking into consideration what you have told me, I am forgetting that you still have a lockpick in the sole of your shoe.\"\n\nThen he was gone for good. I stared at the closed door and suddenly burst out laughing. It was going to be a good universe after all, filled with good things to be appropriated in a manner only possible to one who knew his trade. And I knew mine!\n\n\"Thank you, Bishop, thanks for everything. You have done it, guided me and taught me. Because of you\u2014a Stainless Steel Rat is born!\"\nThe Stainless Steel Rat Gets Drafted\nThis book is for  \nRog Peyton  \nand all the Brum gang.\nChapter 1\n\nI am too young to die. Just eighteen years old\u2014and now I'm as good as dead. My grip is weakening, my fingers slipping, and the elevator shaft below me is a kilometer deep. I can't hold on any longer. I'm going to fall...\n\nNormally I am not prone to panic\u2014but I was panicking now. Shaking from head to toe with fatigue, knowing that there was just no way out of this one.\n\nI was in trouble, mortal trouble, and I had only myself to blame this time. All the good advice I had given myself down through the years, the even better counsel The Bishop had given me, all forgotten. All wiped away by sudden impulse.\n\nPerhaps I deserved to die. Maybe a Stainless Steel Rat had been born\u2014but a very rusty one was about to snuff it right now. The metal door frame was greasy and I had to hold on hard with my aching fingers. My toes barely gripped the narrow ledge\u2014while my unsupported heels hung over the black drop below. Now my arches began to ache with the effort of standing on tiptoe\u2014which was nothing compared to the fire in my throbbing forearms.\n\nIt had seemed such a logical, simple, good, intelligent plan at the time.\n\nI now knew it to be irrational, complex, bad and moronic.\n\n\"You are an idiot, Jimmy diGriz,\" I muttered through my tightly shut teeth, realizing only then that they were clamped into my lower lip and drawing blood. I unclamped and spat\u2014and my right hand slipped. The great spasm of fear that swept over me rode down the fatigue and I grabbed a new hold with an explosion of desperate energy.\n\nWhich faded away as quickly as it had come, leaving me in the same situation. Tireder if anything. There was no getting out of this one. I was stuck here until I could no longer hold on, until my grip loosened and I fell. Might as well let go now and get it over with...\n\n\"No, Jim, no surrender.\"\n\nThrough the thudding of blood in my ears my voice seemed to come from a great distance, to be deeper in register than my own, as though The Bishop himself were speaking. The thought was his, the words might very well be his. I held on, though I didn't really know why. And the distant whine was disturbing.\n\nWhine? The elevator shaft was black as the grave and just as silent. Was the maglevlift moving again? With muscle-tight slowness I bent my head and looked down the shaft. Nothing.\n\nSomething. A tiny glimmer of light.\n\nThe elevator was coming up the shaft.\n\nBut so what? There were two hundred and thirty-three floors in this government building. What were the odds that it would stop at the floor below me so I could step neatly back onto its top? Astronomical I was sure, and I was in no mood to work them out. Or perhaps it would come up to this floor and scrape me off like a bug as it went by? Another nice thought. I watched the light surge upward toward me, my eyes opening wider and wider to match the growing glow. The increasing whine of the centering wheels, rush of air exploding at me, this was the end...\n\nThe end of its upward motion. The car stopped just below me, so close that I could hear the door swoosh open and the voices of the two guards inside.\n\n\"I'll cover you. Keep your safety off when you search the hall.\"\n\n\"You'll cover me, thanks! I didn't hear myself volunteer.\"\n\n\"You didn't\u2014I did. My two stripes to your one mean you take a look.\"\n\nOne-stripe muttered complaints as he moved out as slowly as he could. As his shadow occulted the light from the open door I stepped down onto the car with my left foot, as gingerly as I could. Hoping that any movement to the car caused by my climbing onto it would be masked by his exiting.\n\nNot that it was easy to do. My thigh muscle spasmed with cramp and my fingers were locked into place. I stepped slowly back with my vibrating right foot until I was standing on top of the elevator. My cramped fingers still gripped the frame: I felt very much the fool.\n\n\"Hall is empty,\" a distant voice called out.\n\n\"Take a reading from the proximity recorder.\"\n\nThere were muttered grumbles and clattering from outside as I wrenched my right hand free of the greasy metal, reached over with it to grapple with my still recalcitrant left hand.\n\n\"Got a reading for myself. Other than that the last movement in the corridor was at eighteen hundred. People going home.\"\n\n\"Then we do have a mystery,\" two-stripes said. \"Come on back. We had a readout that showed this car going up to this floor. We called it back from this floor. Now you tell me that no one got out. A mystery.\"\n\n\"That's no mystery, that's just a malfunction. A glitch in the computer. The thing is giving itself instructions when no one else will.\"\n\n\"Much as I hate to agree\u2014I agree. Let's go back and finish the card game.\"\n\nOne-stripe returned, the elevator door closed, I sat down as quietly as I could, and we all dropped back down the shaft together. The guards got out at the prison floor and I just sat there in creaking silence as I kneaded the knots out of my muscles with trembling fingers. When they were roughly under control again I opened the hatch that I was sitting on, dropped down into the car and looked out slowly and carefully. The card players were out of sight in the guardroom, where they belonged. With infinite caution I retraced the route I had taken during my abortive escape. Slinking guiltily along the walls\u2014if I had a tail it would have been between my legs\u2014making a fumbled hash of opening the locked corridor doors with my lockpick.\n\nFinally reaching my own cell, unlocking and relocking it, slipping the lockpick back into my shoe sole\u2014dropping onto my bed with a sigh that must have been heard around the world. I did not dare speak out loud in the sleeping silence of the cell block, but I did shout the words inside my head.\n\n\"Jim, you are the dumbest most moronic idiot who ever came down the pike. Don't, and I repeat, don't ever do anything like that again.\"\n\nI won't, I promised in grim silence. That message had now been well drilled into my medulla oblongata. The truth was inescapable. I had done everything wrong in my eagerness to get out of prison. Now I would see if I could get it right.\n\nI had been in too much of a rush. There should never have been any hurry. After he had arrested me, Captain Varod, strongman of the League Navy, had admitted that he knew all about the lockpick that I had hidden. He did not like prisons, he had told me that. Although he was a firm believer in law and order he did not believe I should be incarcerated on my home planet, Bit O\" Heaven, for all of the troubles that I had caused there. Neither, for that matter, did I. Since he knew I had the lockpick I should have bided my time. Waited to make my escape during the transfer out of this place.\n\nDuring the transfer. It had never been my intention of doing anything but serve my time here in this heavily guarded and technologically protected prison in the middle of the League building in the center of the League base on this planet called Steren-Gwandra\u2014about which I knew absolutely nothing other than its name. I had been enjoying the rest, and the meals, a real pleasure after the rigors of war on Spiovente and the disgusting slop that passed for food there. I should have kept on enjoying, building my strength in preparation for my imminent freedom. So why had I tried to crack out of here?\n\nBecause of her, a woman, a female creature briefly seen and instantly recognized. One glimpse and all reason had fled, emotion had ruled and I had attempted my disastrous escape. More fool I. I grimaced at the memory, recalling all too clearly how this idiot adventure had begun.\n\nIt had been during our afternoon exercise period, that wildly exciting occasion when the prisoners were let out of their cells and permitted to shuffle around the ferroconcrete yard under the gentle light of the double suns. I shuffled with the rest and tried to ignore my companions. Low foreheads, joined eyebrows, pendulous and drool-flecked lips\u2014a very unsatisfactory peer group of petty criminals that I was ashamed to be a part of. Then something had stirred them, some unaccustomed novelty that had excited their feeble intellects and had caused them to rush toward the chain-link fence emitting hoarse cries and vulgar exhortations. Numbed by the monotony of prison life even I had felt a twinge of curiosity and desire to see what had caused this explosion of unfamiliar emotion. It should have been obvious. Women. That, and strong drink and its aftereffects, were the only topics that ever stirred the sluggish synapses of their teeny minds.\n\nThree newly arrived female prisoners were passing by on the other side of the fence. Two of them, cut from the same cloth as my companions, responded with equally hoarse cries and interesting gestures of the fingers and hand. The third prisoner walked quietly, if grimly, ignoring her surroundings. Her walk was familiar. But how could it be? I had never even heard of this planet before I had been forcefully brought here. This was a mystery in need of a solution. I hurried along the fence to its end, cleared a space for myself by applying my knuckles to a hair-covered neck in such a manner that the neck's owner slipped into unconsciousness, took his space and looked out.\n\nAt a very familiar face passing by not a meter distant. Without a doubt a face and a name that I knew very well.\n\nBibs, the crewgirl from Captain Garth's spacer.\n\nShe was a link to Garth and I had to talk to her, to find out where he was. By kidnapping us and dumping us on the loathsome planet of Spiovente, Captain Garth had been responsible for The Bishop's death. Which meant that I would like to be responsible for his in return.\n\nSo, without further thought, and possessed only of a suicidal and impractical enthusiasm, I had foolishly escaped. Only the luck that watches over the completely witless had saved my life and permitted my return, undetected, to my prison cell. I blushed now with shame as I thought about the stupidity of my plan. Lack of thought, lack of foresight\u2014and the incredibly dumb assumption that all security in the giant building would be identical. During our daily exodus and return to the cell block I had noted the exceedingly simple locks on all of the doors, the absence of any alarms. I had assumed that the rest of the building had been the same.\n\nI had assumed wrong. The car of the maglevlift had notified the guards when it had been used. I had spotted the detectors in the corridor at once when the door had opened on the top floor. That was why I had tried the escape hatch in the roof, hoping to find a way out through the mechanism at the top of the shaft.\n\nExcept that there had been no mechanism there\u2014just another door. Opening into another floor that did not appear on the bank of buttons inside the car. Some secret location known only to the authorities. Hoping to penetrate this secret I had climbed onto the doorsill and searched for a way to open the door. Only to have the elevator vanish from behind me leaving me stranded on top of the empty shaft.\n\nI had come out of this little harebrained adventure far better than I had deserved. Luck would not ride with me a second time. Cool planning was needed. I put this nearly disastrous escapade behind me and thought furiously of schemes and ways to make contact with the crewgirl.\n\n\"Do it honestly,\" I said, and shocked myself with the words.\n\nHonest? Me? The stainless steel rat who prowls the darkness of the night in solitary silence, fearing no one, needing no one.\n\nYes. Painful as the realization was, just this once honesty was indeed the best policy.\n\n\"Attention, foul jailers, attention!\" I shouted and hammered on the bars of my cage. \"Arouse yourself from your sweat-sodden slumbers and vulgar, erotic dreams and take me to Captain Varod. Soonest\u2014or even sooner!\"\n\nMy fellow prisoners awoke, calling out in righteous anger and threatened all sorts of unimaginative bodily harm. I returned the insults with enthusiasm and eventually the night guard appeared, scowling with menace.\n\n\"Hi, there,\" I called out cheerily. \"Nice to see a friendly face.\"\n\n\"You want your skull broke, kid?\" he asked. His repartee just about as sharp as that of the inmates.\n\n\"No. But I want you to stay out of trouble by instantly taking me to Captain Varod since I have information of such military importance that you would be shot instantly if suspected of keeping it from the captain for more than a second or two.\"\n\nHe added some more threats, but there was a glint of worry in his eyes as he thought about what I had said. It seemed obvious, even to someone of his guttering intelligence, that passing the buck was the wisest fallback position. He growled some more insults when I pointed back down the corridor, but left in any case and went to his telephone. Nor was my wait a long one. A brace of overmuscled and overweight guards appeared on the scene within minutes. They unlocked my cell, clamped on the cuffs and hurried me into the maglevlift and up a few hundred stories to a bare office. Where they fastened the cuffs to a heavy chair and left. The lieutenant who entered a few minutes later was still blinking the sleep from his eyes and was not happy at being disturbed in the middle of the night.\n\n\"I want Varod,\" I said. \"I don't talk to the hired help.\"\n\n\"Shut up, diGriz, before you get yourself into worse trouble. The captain is in deepspace and unreachable. I am from his department and urge you to speak quickly before I bounce you out of here.\"\n\nIt sounded reasonable enough. And I had very little choice.\n\n\"Have you ever heard of a space-going Venian swine who goes by the name of Captain Garth?\"\n\n\"Get on with it,\" he said in a bored voice, yawning to drive home the point. \"I worked on your case so you can speak freely. What do you know that you haven't told us already?\"\n\n\"I have information about our gunrunning friend. You do have him in custody, don't you.\"\n\n\"DiGriz\u2014you give us information, that is the way that it works, not the other way around.\" That was what he said, but his expression spoke otherwise. A fleeting instant of worry. If that meant what I thought it meant then Garth had managed to escape them.\n\n\"I saw a girl today, a new prisoner being brought in. Her name is Bibs.\"\n\n\"Did you get me out of bed to describe some sordid sexual secret?\"\n\n\"No. I just thought you should know that Bibs was a crewgirl on Garth's ship.\"\n\nThis caught his attention instantly, and not being as experienced as his commanding officer he could not conceal the look of sudden interest.\n\n\"You are sure of this?\"\n\n\"Check for yourself. The information on today's arrivals should be readily available.\"\n\nIt was: he sat behind the steel desk and hammered away at the keys on the terminal there. Looked at the screen and scowled in my direction.\n\n\"Three women admitted today. None named Bibs.\"\n\n\"How very unusual.\" Scorn dripped from my voice. \"Can it be that the criminal classes now use aliases?\"\n\nHe did not answer but tapped away at the terminal again. The fax buzzed and produced three sheets of paper. Three color portraits. I dropped two of them onto the floor and handed the third back.\n\n\"Bibs.\"\n\nHe bashed some more keys, then slumped back and rubbed his chin as he studied the screen.\n\n\"It fits, it fits,\" he muttered. \"Marianney Giuffrida, age twenty-five, occupation given as electrotechnician with deepspace experience. Arrested on a drugs possession charge, anonymous tip, swears she was framed. No other details.\"\n\n\"Ask her about Garth. Use persuasion. Make her talk.\"\n\n\"You have our thanks for your assistance, diGriz. It will go on your record.\" He tapped a number into the phone. \"But you have been watching too many films. There is no way we can force people to give evidence. But we can question and observe and draw conclusions. They will take you back to your cell now.\"\n\n\"Gee, thanks for the thanks. Thanks for nothing. Can you at least do me the favor of telling me how long you intend to keep me here?\"\n\n\"That should be easy enough to find out.\" A quick access of the terminal and a sage nod of the head as the door opened behind me. \"You will be leaving us the day after tomorrow. A spacer will be stopping at a planet with the interesting name of Bit O\" Heaven, where, it appears, you have to answer some criminal charges.\"\n\n\"Guilty until found guilty, I suppose.\" I sneered and whined to hide the surge of enthusiasm that raced through me. Once out of here I really would be out of here. I ignored the rough clutch and muttered complaints of my warders and permitted myself to be docilely led back to my cell. I was going to be good, very, very good, until the day after tomorrow.\n\nBut I lay awake a long time after that, staring into the darkness, working out how I was going to pry the information I needed out of crewmember Bibs.\nChapter 2\n\n\"Sign here.\"\n\nI signed. The ancient graybeard behind the desk passed over the plastic bag containing all of my worldly possessions, forcibly removed from me when I had been incarcerated. I reached for them but the fat guard reached even faster.\n\n\"Not yet, prisoner,\" he said, whisking them away from my clutching fingers. \"These will be forwarded to the arresting authorities.\"\n\n\"They're mine!\"\n\n\"Take it up with them. All set, Rasco?\"\n\n\"My name's not Rasco!\"\n\n\"Mine is. Shut up,\" the other guard said. A well-muscled and nasty individual whose right wrist was secured to my left by a pair of shining cuffs. He pulled hard on this connecting link so I stumbled toward him. \"You do what I say and no backtalk or funny stuff.\"\n\n\"Yes, sir. Sorry.\"\n\nI lowered my eyes in humility, which caused him to smirk with assumed superiority. He should only know that I was using the opportunity to look more closely at the cuffs. Bulldog-Crunchers, sold throughout the known galaxy, guaranteed foolproof. Maybe proof against fools but I could open them in under two seconds. It was going to be a nice day.\n\nFatso walked on my right side, well-connected Rasco on my left. I marched in step with them, eager to leave the prison and examine the world waiting outside the League building. I had come here in a closed van and had seen nothing. Eagerness possessed me in expectation of a first glimpse of my new home; thoughts about my forceful removal from this planet may have preoccupied my guardians\u2014but were the farthest thing from my mind at this moment.\n\nExiting the building was not easily done\u2014and I gave myself another mental kick for even thinking of breaking out of this bunker-skyscraper. There were three doors to go out through, one after another, each sealed as tightly as an airlock. Our passes were slipped into computerized machines that hummed and clicked\u2014then robot sensors examined our fingerprints and retinal patterns to make sure we matched the details on the passes. This was done three times before the outer portal hummed open and a wave of warm air, smell and sound washed in.\n\nAs we went down the steps to the street I gaped like a rube. I had never seen anything like this before. Of course my experience was strictly limited since this was only the third planet I had ever visited. My life on the porcuswine farms of Bit O\" Heaven and my service in the swamps of Spiovente had not prepared me for the manifold impressions that bombarded me.\n\nA wave of heat and dusty air washed over me. It was filled with pungent aromas, loud cries and a cacophony of strange noises. At the same time as my ears and nose were being assaulted my eyes bulged at the seething mass of humanity, the strange vehicles\u2014and the four-legged alien creatures. One passed close by, a man sitting on its back, its great feet thudding on the ground, eyes rolling in my direction. Its mouth opened to reveal hideous yellow teeth and it squealed loudly. I drew back and my guards laughed aloud at this perfectly reasonable reaction.\n\n\"We'll protect you from the margh,\" Fatso said, and they chortled with dim pleasure.\n\nMaybe it was called a margh in the local lingo, but it was still a horse to me. I had seen them in the ancient history tapes at school. The creatures had been used for farming when Bit O\" Heaven was first settled, but had soon succumbed to the deadly native life. Only the indestructible porcuswine had been able to survive. I looked more closely at the horse, at the obviously herbivorous teeth, and realized it posed no threat. But it was big. Two more of the creatures came up, towing a boxlike affair mounted on large wheels. The driver, sitting high above, pulled the thing to a stop when Rasco whistled to him.\n\n\"Get in,\" Fatso ordered, swinging open a door in the vehicle's side. I held back, pointed with distaste.\n\n\"It's filthy in there! Can't the League Navy provide decent transportation...\"\n\nRasco kicked me in the back of the leg so I fell forward. \"Inside\u2014and no backtalk!\" They climbed in after me. \"It is Navy policy to use native transport when possible, to aid the local economy. So shut up and enjoy.\"\n\nI shut, but I didn't enjoy. I looked unseeingly at the crowded street as we rumbled away, thinking of the best way to escape my captors while inflicting a bit of damage on my sadistic companion. Now would be as good a time as any. Strike like lightning, then leave them both unconscious in this vehicle while I slipped away in the crowd. I bent over and scratched furiously at my ankle.\n\n\"I've been bitten! There are bugs in here!\"\n\n\"Bite them back,\" Fatso said and they both roared with juvenile laughter. Wonderful. Neither saw me slip the lockpick from my shoe and palm it. I turned toward Rasco with mayhem in mind just as the vehicle lurched to a stop and Fatso reached across and threw open the door. \"Out,\" he ordered and Rasco pulled painfully on the handcuff, I gaped at the marble-fronted building before us.\n\n\"This isn't the spaceport,\" I protested.\n\n\"You got good eyes,\" Rasco sneered and dragged me after him. \"A local version of a linear. Let's go.\"\n\nI decided I wouldn't. I had had more than enough of their repellent company. But I had to stumble after them for the moment, looking about for some opportunity\u2014and seeing it just ahead. Men, and only men, were entering and exiting a doorway under a sign that proudly proclaimed PYCHER PYSA GORRYTH. Though I knew nothing of the local language I could figure this one out easily enough. I drew back and pointed.\n\n\"Before we get on the linear I gotta go in there.\"\n\n\"No way,\" Rasco said. Sadist. But I got unexpected aid from his companion.\n\n\"Take him in. It's going to be a long trip.\"\n\nRasco muttered disgustedly. But Fatso was obviously his superior because he pushed me forward. The pycher pysa was about as primitive as they come, a simple trough against one wall, a line of men facing it. I headed for a vacant position on the far end and fumbled with my clothing. Rasco watched me with obvious displeasure.\n\n\"I can't do anything with you watching,\" I wailed.\n\nHe rolled his eyes upward for a second. Just long enough for me to get his neck with my free hand. His look of surprise faded as I clamped down hard with my thumb. After this I had only to guide his unconscious fall to the tile floor. As he hit with a satisfactory thud I clicked open the cuff on my wrist. He snored lightly as I quickly frisked him, I had a reputation as a thief to live up to, and slipped his wallet from his hip pocket. It was safely hidden in my own before I stood and turned about. The row of men against the wall were all looking at me.\n\n\"He fainted,\" I said, and they gaped with incomprehension. \"Li svenas,\" I added, which did not clarify it for them in any way. I pointed to the unconscious copper, to the door, then at myself. \"I'm going for help. You lads keep an eye on him and I'll be right back.\"\n\nNone of them were in any position to follow me as I scuttled out of the entrance. Practically into Fatso's arms. He shouted something and reached for me\u2014but I was long gone. Out of the station and into the crowd. There were some more outcries from behind me but they soon died away as I twisted between two horses, around a coach and down a dark alley on the far side of the street. It was that easy.\n\nThe alley opened into another street, just as crowded as the first, and I strolled along it, just a part of the crowd. Free as a bird. I actually whistled as I walked, staring around at the sights, the veiled women and the brightly garbed men. This was the life!\n\nOr was it? Alone on a primitive planet, not speaking the language, sought by the authorities\u2014what did I have to be cheerful about? Black gloom descended instantly and I sneered aloud.\n\n\"That's it, Jim? You turn coward at the slightest setback. For shame! What would The Bishop say to this?\"\n\nHe would say stop talking in public, I thought as I noticed the strange looks I was getting. So I whistled happily, not a care in the world, turned a corner and saw the tables and chairs, men sitting and drinking interesting beverages, under a sign that said SOSTEN HA GWYRAS which conveyed exactly nothing to me. But underneath it was printed NI PAROLOS ESPERANTO, BONVENUU. I hoped that they spoke Esperanto better than they wrote it. I found a table against the wall, dropped into a chair and snapped my fingers at the ancient waiter.\n\n\"Dhe'th plegadow,\" he said.\n\n\"Plegadow the others,\" I said. \"We speak Esperanto. What's to drink, Dad?\"\n\n\"Beer, wine, dowr-tom-ys.\"\n\n\"I'm just not in the mood for a dowr-tom-ys today. A large beer, if you please.\"\n\nWhen he turned away I dug out Rasco's wallet. If my guards were supposed to encourage the local economy they should be carrying some of the local currency. The wallet clunked when I dropped it onto the table, heavy with little metal discs. I shook one out and turned it over. It had the number two stamped into one side, with Arghans on the other.\n\n\"That will be one Arghans,\" the waiter said, putting a brimming clay pot in front of me. I passed over the coin.\n\n\"Take that, my good man, and keep the change.\"\n\n\"You offworlders are so generous,\" he said, muffledly as he bit the coin. \"Not mean, stupid, vicious like the locals. You want girl? Boy? Kewarghen to smoke?\"\n\n\"Later perhaps. I'll let you know. Beer now and the heady pleasures of native life to come.\"\n\nHe went away muttering and I took a great slug of the beer. Instantly regretting it. I swallowed\u2014and regretted that as well as the noxious brew bubbled and seethed its way through my digestive track. I pushed the jar away and belched. Enough of this tomfoolery. I had escaped, great, step one. But what came next?\n\nNothing that I could think of at the moment. I sipped at the beer, it still tasted just as repulsive, but even this heroic treatment produced no inspiration. I was grateful for the interruption when the waiter sidled over and whispered hoarsely behind the back of his hand.\n\n\"New shipment kewarghen fresh from the fields. You get high, stay up for many days. Want some? No? What about girl with whips? Snakes? Leather straps and hot mud...\"\n\nI interrupted since I wasn't sure that I enjoyed where the conversation was going. \"I am sated, I tell you sated. All I wish are directions to return to the municipal edifice.\"\n\n\"Do not know what long words mean.\"\n\n\"Want to find building big, high, filled with plenty offworlders.\"\n\n\"Ahh, you mean the lys. For one Arghans I take you there.\"\n\n\"For one Arghans you give me instructions. I don't want to drag you away from your work.\" Nor did I want to be led astray to one of the many offers he had made. In the end he had to agree. I memorized the instructions, sipped some more of the beer and instantly regretted it, then slipped away when he had vanished into the back room.\n\nAs I walked a glimmer of a plan began to develop. I must think of a way to get to Bibs, the crewgirl from the freighter. Garth, the captain of her ship, had escaped, I was sure of that. But she might know more about him. She was my only link with this villain. But how could I get into the prison? I knew the name she had been arrested under, Marianney Giuffrida. Could I pass myself off as a concerned relative, one Hasenpeffer Giuffrida? The local identification should be easy to forge\u2014if it existed at all. But would the computer identify me as an ex-prisoner when I entered the building? Or had I been wiped from its memory when I had left? Perhaps I had been, but would Fatso put me back in memory when he reported my escape?\n\nThese thoughts were rattling around in my head when I turned the next corner and found the gigantic edifice before me. It rose up from the low buildings of the city like a towering cliff\u2014and looked just as impenetrable. I strolled by and looked up at the steps I had so recently descended, watched the doors open to admit a visitor. Then close again like a bank vault. My mind was still blank. I stood with my back to a brick wall across from the building. Which was perhaps not too bright, since I was still wearing prison garb. But such was the variety of local costume that my uniform drew no notice at all. I leaned and waited for inspiration to strike.\n\nIt didn't. But pure, random luck, a chance in a thousand did. The doors opened one more time and three people emerged. Two minions of the law, this obvious from the size of their boots, flanking a delicate female form. One thick wrist was manacled to her tiny one.\n\nIt was Bibs.\n\nThe suddenness of her appearence froze me in place. Kept me leaning against the wall as they descended to street level where one of the guards waved and whistled. In quick response two of the horsedrawn vehicles raced their way, one of them neatly cutting off the other. There were shouted curses and loud neighing as the horses reared up. This was quickly sorted out and the loser trotted off. The high body of the horsedrawn hulk blocked my view, but as clear as though it were transparent I knew what was happening. Door being opened, prisoner escorted inside, door closed...\n\nThe thing started forward as the driver's whip cracked, even as I was hurrying across the road. Getting up speed as I ran after it, jumped, got my feet on the step and hauled the door open.\n\n\"Out,\" the nearest guard said, turning toward me. \"This cab is taken...\"\n\nWe looked at each other in mutual recognition\u2014he was the night guard from the prison. With a cry of anger he reached for me. But I reached quicker, jumping in on top of him. He was big and strong\u2014but I was fast. I had a quick glimpse of the shocked look on Bibs's face as I turned all my attention to avoiding his clutch and getting in a quick blow with the edge of my hand.\n\nAs soon as he went limp I rolled over to face the other guard and discovered that he had no interest in me at all. Bibs had her free arm around his neck and was throttling him to death. He flailed with his other hand but could do nothing because it was manacled to her wrist.\n\n\"Just wait... until this one... is dead too,\" Bibs gasped.\n\nI didn't explain that the guard I had taken care of was only unconscious but reached over and grabbed her elbow hard, index finger grinding into the big nerve there. Her arm went numb, dropped away, and her face grew red with fury. But before she could speak I silenced the gasping guard and unlocked the cuffs. She rubbed her wrist and smiled.\n\n\"I don't know where you dropped from, buster, or why, but I appreciate the help.\" She cocked her head and looked more closely at me. \"I know you, don't I? Yes, of course, you're the midnight passenger, Jimmy something.\"\n\n\"That's right, Bibs. Jim diGriz at your service.\"\n\nShe laughed, loud and happily, while she removed all the possessions of the two unconscious guards, then scowled when I manacled them together.\n\n\"Better to kill them,\" she said.\n\n\"Better not to. Right now we're not important enough to them to cause much fuss. But if we murdered two of their men they would turn this planet over to find us.\"\n\n\"I guess you're right,\" she said with reluctant agreement\u2014then kicked both unconscious bodies with sudden fury.\n\n\"They can't feel anything.\"\n\n\"They will when they wake up. So where do we go from here, Jim?\"\n\n\"You tell me. I know absolutely nothing at all about this planet.\"\n\n\"I know far too much.\"\n\n\"Then lead the way.\"\n\n\"Right.\"\n\nShe opened the door as our vehicle slowed and we slipped out, stepped up onto the pavement as it lumbered from sight.\nChapter 3\n\nBibs tucked her arm through mine, which felt very cheering, as we strolled along the busy avenue. Anywhere else our gray prison clothes, tastefully decorated with blood-red broad arrows, would have certainly drawn attention\u2014and apprehension. Not among the motley throngs crowding these streets, dressed in every manner possible. There were bearded men in fringed buck-skins, women in layers of colored gauze, armed warriors in leather and steel; robes, gowns, chainmail, cuirasses, sashes\u2014everything imaginable. Plus a few that defied the imagination. We drew no attention at all.\n\n\"Do you have any money?\" Bibs asked.\n\n\"Just a few Arghans I lifted from one of my guards. Like you, I have just escaped.\"\n\nHer eyebrows lifted at this\u2014very attractive eyebrows arched above even more attractive eyes I noticed.\n\n\"Is that why you helped me out? What were you in prison for? All I know is that you and the old boy were left behind on Spiovente. Scuttlebutt had it that Garth sold you into slavery.\"\n\n\"He did, and my friend is dead because of that. I am a little bitter about Garth for a lot of reasons. I liked The Bishop. He helped me, taught me a lot, and I am happy to say that I was able to help him in return. We left our home world in a hurry, as you will remember, and paid Captain Garth a lot of money to get us away. But that wasn't enough for him. He earned more by selling us into slavery. I lived\u2014but The Bishop died because of being a slave. As you can imagine I am not wildly pleased by his death. A number of loathsome things happened on that planet, the least of which was my being caught by the League Navy. They were returning me to my home planet to stand trial.\"\n\n\"On what charges?\" There was keen interest in her voice.\n\n\"Bank robbery, criminal abduction, jailbreak. Things like that.\"\n\n\"Wonderful!\" she said, laughing aloud with joy; she had very neat white teeth. \"You did yourself an immense favor when you came to little Bibs's aid. I know this planet well, know where the money is. Know how to buy our way offplanet when we are done. You steal it, I'll spend it\u2014and our troubles are over.\"\n\n\"Sounds reasonable. Could we talk about it over some food? It's been a long time since breakfast.\"\n\n\"Of course\u2014I know just the place.\"\n\nAnd she did too. The restaurant was small and discreet while the felyon ha kyk mogh tasted a lot better than it sounded. We washed it down with a great bowl of ru'th gwyn, which turned out to be satisfactory red wine: I memorized the name for future use. When we had eaten our fill I took one of the wood splinters from the jar on the table and worried bits of gristle from between my teeth.\n\n\"Do you mind if I ask you a question?\" I asked, asking a question. Bibs sipped at her wine and waved permission. \"You know why I was imprisoned. Would you consider it rude if I asked the reason for your incarceration?\"\n\nShe slammed her mug down so hard that it cracked and oozed a carmine trickle. She was unaware of it; her face twisted with anger and I could hear her teeth grate together.\n\n\"He did it, I'm sure, it had to be him, the bastarda\u0109fiulo!\" Which is about the worst name you can call anyone in Esperanto. \"Captain Garth, he's the one. He knew the League Navy was after us for gunrunning. He paid us off here\u2014and the next day I was arrested. He tipped them off and planted the kewarghen in my bag. With that evidence they busted me on a drugs charge, selling to the natives and all that. I want to kill him.\"\n\n\"So do I\u2014for causing the death of my friend. But why did he want you arrested?\"\n\n\"Revenge. I kicked him out of bed. He was too kinky for my liking.\"\n\nI gulped and coughed and took a long slug of wine and hoped that she wouldn't notice that I was blushing. She didn't. Her eyes, still glazed with anger, stared past me into space. \"Kill him, I really would like to kill him. I know that it's impossible but, oh how he deserves it.\"\n\n\"Why impossible?\" I asked with some relief, glad to have the conversation back on comfortable topics like murder and revenge.\n\n\"Why? What do you know about this planet, Jim?\"\n\n\"Nothing. Other than its name, Steren-Gwandra.\"\n\n\"Which means 'planet' in the local lingo. They are not a linguistically imaginative lot. At least those here in Brastyr aren't. Like many other settled planets this one was cut off from galactic contact during the breakdown years. Brastyr, this continent, has few natural resources and over the centuries they managed to lose all of the old technology. They are so dim that most of them forgot Esperanto. Not the traders though, they had to deal with the offshore island. By the time that galactic contact was reestablished the locals had sunk into a sort of agricultural semifeudalism.\"\n\n\"Like Spiovente?\"\n\n\"Not quite. Just offshore is this damned great island I mentioned, separated from this mainland by a narrow strait. Almost all of the minerals, coal and oil in this hemisphere are located there. That's why it was settled first and why it was well developed before the second wave of immigrants arrived during the diaspora ages. None of the newcomers were allowed to settle there. Not that they cared, this entire continent was wide-open and bountiful and the arrangement suited all parties concerned. Industry and technology over there on Nevenkebla, farming and forestry here. I doubt if anything changed much during the breakdown years\u2014I imagine the relationship was intensified if anything. That's why we are never going to get close enough to Garth to kill him.\"\n\n\"I don't understand. What has this got to do with him?\"\n\n\"He's on the island. Unreachable.\" She sighed and rubbed her fingertip in circles in the pool of spilled wine. I was still puzzled.\n\n\"But Garth is a Venian, like you. The captain of a Venian ship. Why should they protect him?\"\n\n\"Because he's not Venian, that's why. The Nevenkebla military bought the ship, he commanded it. We were happy to go along with the plan; they paid well. Venians are very flexible when it comes to money. But he is really something big in the military there. They run the place. All those guns we were smuggling were made on the island. It was a good racket, plenty of offplanet currency. But when the League Navy got too close they paid us off and closed the operation down. There is just no way to get at him on that island.\"\n\n\"I'll find a way.\"\n\n\"I hope that you do. I'll give you all the help that I can. But first things first, Jim. We will have to stay out of sight for a bit while they are looking for us\u2014and that will take a pile of Arghans. How much do you have?\"\n\nShe spread out the coins she had stolen and I added mine to the pile.\n\n\"Not enough. We need a lot for bribes, a safe place to get out of sight. I have contacts, a fence I used to peddle to. For the right price we can have him find a safe house...\"\n\n\"No. Avoid the criminal classes at all counts. Too expensive and the first place that the authorities will look. Do they have hotels here? Expensive, luxurious hotels?\"\n\n\"Not as such. But there are ostelyow where traveling gentry put up. But offworlders never go there.\"\n\n\"Even better. Can you pass as a native?\"\n\n\"Yredy. You could too with a little effort. There are so many different accents and dialects here that no one will notice.\"\n\n\"Ideal. Let us then instantly steal a lot of money, buy some expensive clothes and jewelry and check in at the best ostel. Agreed?\"\n\n\"Agreed!\" She laughed out loud and clapped her hands together. \"I swear, Jim, you are a breath of fresh air on this fetid planet. I like your style. But it won't be easy. They don't have banks here. All the cash is held by moneylenders called hoghas. Their places are like small forts. Plenty of guards, always from the moneylender's own family so they can't be bribed.\"\n\n\"Sounds good. Let's go check one out. Then we will go back tonight and crack it.\"\n\n\"Do you mean it?\"\n\n\"Never more serious.\"\n\n\"I've never met anyone like you. You look like a kid\u2014but you can really take care of yourself.\"\n\nI did not like that kid remark but I stayed shut up and tried not to pout while she made plans.\n\n\"We'll take some of these Arghans, change them for Nevenkebla coins. This will take a lot of arguing over rate of exchange so you will have time to look around. I'll do the talking. You just carry the money and keep your mouth shut. We'll get you a bodyguard's club first, then they'll never even notice you.\"\n\n\"No time like the present. Let's find a club shop.\"\n\nThis was easily enough done. Most of the side streets were open markets, with stalls and tiny shops that sold an apparently endless variety of cloth, fruit, meals wrapped in leaves, knives, saddles, tents\u2014and clubs. While the merchant extolled the value of his wares, muffledly and incomprehensively through the layers of cloth about his face and neck, I hefted the samples and tested their swing. I finally settled on a meter length of tough wood that was bound about with iron bands.\n\n\"This looks like what we want,\" I told Bibs. The weapon vendor nodded and took the coins and muttered some more. Bibs pointed inside.\n\n\"He insists that a year guarantee goes with every club and you must try it out before you leave.\"\n\nThe testing block proved to be a large upright stone that had been carved into human form, what might at one time have resembled a man in armor. But years of testing had taken their toll. Gouges, nicks and missing chunks defaced it; noseless, chinless, with a single fragment of ear remaining. I hefted my club, tried a few practice swings\u2014then stood with my back to the stone while I psyched up my muscles with some dynamic-tension contractions and a breathing mantra. I was chuffing as nicely as a Spiovente steam wagon, holding the club upright, when I felt ready.\n\nTimed release, that's the secret. Not a secret really, just technique and practice. A single shout contracted my body all in an instant. In time with the sound I swung about with all of my weight and strength focused on the iron band on the end of the club. It whistled through a half-circle that terminated on the side of the rock head.\n\nThere was a ringing crack as the neck shattered and the stone head fell off. The club was still sound and the iron ring had a slight nick.\n\n\"This one will do,\" I said, as offhandedly as I could.\n\nThey were both very impressed, let me tell you. I was impressed myself. It had been a good blow, better than I had realized.\n\n\"Do you do that often?\" Bibs asked in a hushed voice.\n\n\"If I have to,\" I said with a calm I did not feel. \"Now take me to your hogh.\"\n\nWe found one just a few streets away, the identity of the business made known by a skeleton in an iron cage above the door.\n\n\"Some sign,\" I said. \"You would think they would hang out a painting of a money bag or a wooden Arghans.\"\n\n\"This is more practical. That is the last thief they caught trying to steal from them.\"\n\n\"Oh, thanks.\"\n\n\"It's just a tradition, don't let it disturb you.\"\n\nEasy enough for her to say\u2014she wasn't going to rob this place. Disturbed, I followed her past two ugly weightlifters who leaned on their spears and scowled at us.\n\n\"Hogh,\" Bibs said, sniffing with disdain at the guards. They muttered something not too nice, but still knocked on the iron-bound door until it creaked open. Inside were more guardians from the same mold. Except these had swords. The door slammed shut and was locked behind us as we passed through a dark room into the courtyard beyond. There were spikes\u2014as well as more guards\u2014on the surrounding wall. Not a wall, really, but the roof of the buildings that surrounded the courtyard. The hogh himself sat on a large chest, shielded from the sun by a canopy, guarded by two more men\u2014this time armed with pikes. The chest had a flat top and was covered with pillows.\n\n\"I suppose he sleeps on it at night,\" I said, a feeble joke to build the morale.\n\n\"Of course,\" Bibs said and the morale slumped even lower.\n\nThe moneylender was all smarmy gestures and oily voice. Bibs jingled our money at him and he smarmed even more. At the clap of his hands assistants cleared the pillows away and opened the lid of the chest. I looked in and the guards looked at me. It was neatly divided into sections and each section was filled with leather bags. More orders and hand clapping produced a bag that was placed on top of the now reclosed chest. He sat back on the lid with a happy sigh and cradled the bag in his lap, opened it and let a trickle of shining coins run through his fingers. The haggling began and I feigned boredom and looked around at the courtyard.\n\nThis was not going to be easy, not easy at all. The entrance door would certainly be sealed and guarded. If I came over the wall there were those spikes\u2014and more guards as well. Then what? Sneak down into the courtyard and tip the old boy off into the dust, grab the bag. And get speared, stabbed, clubbed and so forth. Not an attractive proposition at all. We were going to have to get a new plan to raise funds. I could see no way to get into this place; brute strength was far more efficient than technology in this setup. And say I got in, say I lifted the loot\u2014there was the little matter of getting out with it. Though that might not be too difficult...\n\nI felt the glimmerings of an idea and held onto them and stirred them about. Keeping my expression as calm and stony as possible, with just a hint of a snarl as I looked at the guards, who snarled back. Negotiations were progressing well with plenty of wails of grief and snorts of disdain from both sides. I was only barely aware of this as I rough-fashioned my plan, ran it around and polished it a bit, then took it through slowly, step by step, to see if it would work. Given a little bit of luck it would. Was it the only plan? I sighed inwardly. Yes, all things considered, it was the only plan. I swung my club impatiently and called out to Bibs.\n\n\"Come on lady, don't take all day.\" She turned about and scowled.\n\n\"What did you say?\"\n\n\"You heard me. You came to the bodyguard hiring hall and promised good pay for a short day. But the pay ain't that good and the day is too long.\"\n\nIf the hogh didn't understand Esperanto the plan would stop there. But I could see his ears perk up, listening and understanding everything we said. Bash on\u2014no turning back now. Bibs didn't know what I was doing, but she was smart enough to play along, taking umbrage at my insults.\n\n\"Listen you muscle-bound moron\u2014I can hire better than you for half the price. I don't need the static from a malbonulo whose eyebrows meet in the middle!\"\n\n\"That does it!\" I shouted. \"I don't take that from no one!\"\n\nI swung my club at her in a wicked blow that just brushed her hair. It didn't touch her\u2014so I let the butt end follow through with a light tap on the forehead that dropped her to the ground. With Bibs safely out of the picture I would now see if I could get away with what is usually referred to as a smash-and-grab.\n\nMy club swung again and knocked down one of the poles that held up the canopy. I stepped forward as it fell and chopped the hogh on the side of the neck as the cloth engulfed us.\n\nFast now, Jim. You have seconds\u2014or less. I groped the bag of coins out of his lap and stuffed them inside my shirt. It wouldn't fit until I spilled some out. Seconds. Gone.\n\nThere was plenty of shouting now and struggling with the cloth. I pulled myself free\u2014and walked away, calling back over my shoulder.\n\n\"I quit, lady. Get another bodyguard. Only poofters work for women anyway.\"\n\nTwo paces, three, four. The armed men looking from me to the heaving canopy as the guards there pulled it free. One of them emerged, dragging the unconscious hogh, shouting and screaming with anger. I did not need a translation. All of the other guards howled in rage and ran toward me.\n\nI turned tail and ran in the opposite direction. Away from the only exit.\n\nBut toward the flight of wooden stairs that ran up to the roof.\n\nThe single guard there stabbed at me with his spear. I parried it with the club and kicked him hard where it would make the best impression. Jumped his falling body and bounded up the stairs two at a time and almost impaled myself on the sword of the man standing at the top. All I could do was dive under it, roll, crash into his legs and bring him down.\n\nCatching him on the head with the butt of the club as I scrambled to my feet, coins jingling down about me.\n\nThree other guards on the roof were screeching and lumbering toward me. I ran to the edge, looked at the drop, cursed aloud. The cobbled street was too far below. If I jumped I would break a leg. Turned and threw my club at the first of the attackers. It caught him nicely and the second man ran into him.\n\nI saw no more because I was over the roof, holding onto the edge with both hands and letting myself down. Looking up at the third guard who was bringing his sword down on my hands.\n\nI let go. Dropped. Hit and rolled. My ankle hurt but I did not even think about it. Spears and clubs cracked to the ground around me as I hobbled away, around the first corner and into a market street. Hobbling slower and slower as the howls behind me faded in the distance.\n\nAround another corner where I stopped for breath, panting and wheezing. Then staggered on deeper into the city until I was sure I had lost my pursuers.\n\nI dropped into a chair of the first bar and actually enjoyed drinking a mug of the terrible beer.\nChapter 4\n\nThe bag of coins sat uncomfortably on my stomach, straining the fabric of my prison jacket. I looked at the drab cloth with the big red arrows on it and realized that I was being kind of stupid. By now my description would have gone out and all the hogh minions would be looking for me. I would not be that hard to find. As I hammered on the table with a coin I felt the sweat beginning to form on my forehead.\n\nAt the sight of the Nevenkebla currency the waiter's eyes lit up and he seized it with shaking fingers and carried it away reverently. I received a great handful of Arghans in exchange, surely I was being cheated, still I scuttled away happily. Scuttled into the first shop I found that had garments displayed around the entrance. Esperanto was spoken badly here, but good enough to enable me to buy some baggy trousers and a cloak, along with a wicker basket to conceal the money bag. Feeling safe, at least for the moment, I shambled deeper into the city. Through the busy streets to a market where I purchased a wide-brimmed leather hat with a colorful plume. Bit by bit I bought other clothes, until I was garbed anew, the basket with my prison clothes discarded, the money now safe in an elegant shoulder bag. By this time it was getting dark and I was completly lost.\n\nAnd worried about Bibs. I had done all that I could to assure her safety, to distance her from myself and my crime. Had it been enough? I felt a quick surge of guilt and the need to contact her. Easier said than done. First I must find the League building, my only point of reference, and work back from there.\n\nIt was dusk by the time I located it\u2014and I was getting very, very tired. Yet there was no choice, I must go on. Following the route the horse conveyance had taken with Bibs and her captors, finding the corner where we had emerged from it. From there it was easy enough to get to the restaurant where we had eaten, to drop into a chair with a sigh of relief. I could only hope now that she remembered the place and would think of coming here. I took off my hat and a hot band of pain circled my throat.\n\n\"Traitor,\" Bibs's voice hissed in my ear as I gurgled and groped but could reach nothing. Was this the end...?\n\nIt almost was. I was sinking into unconsciousness before the pain eased and the length of wire fell into my lap. I rubbed my sore, bleeding neck as Bibs pulled out a chair and sat down at the table. She weighed my shoulder bag, then looked inside. She had a black eye and some bruises around her mouth.\n\n\"I could have killed you,\" she said. \"I was that angry, that was what I was going to do. But when I saw you had brought the money I realized you had planned the whole thing this way and had come here to meet me. But since they had worked me over I felt I owed you some of the same. I'll order some wine.\"\n\n\"Planned...\" I croaked, then coughed. \"Knocked you out\u2014so they would think you weren't in on the robbery.\"\n\n\"It worked\u2014or I wouldn't be here. They bashed me about a bit, then they all ran out after you. I went right behind them in the confusion. Just wandered around and stayed out of sight until dark. Hating you. I had no money, nothing. Other than this black eye. You're lucky I didn't throttle you all the way.\"\n\n\"Thanks,\" I said, then glugged down half a mug of wine when the waiter set it in front of me. \"It was the only thing I could do. While you were talking to the old boy I looked at the defenses. There was no way in past them. But since we were already inside I saw that there was a good chance of getting out. So I took the money.\"\n\n\"Tremendous. You might have told me.\"\n\n\"There was no way to. Knocking you out was the only thing I could think of that would not get you involved. I'm sorry\u2014but it worked.\"\n\nBibs actually smiled as she ran her fingers through the coins. \"You are right, Jim my boy. It was worth a few bruises to get this much loot. Now let's get moving. You've changed clothes and I must do the same thing.\"\n\n\"Then to the best ostel in town.\"\n\n\"For a hot bath and a real meal. You're on!\"\n\nThe ostel was a sprawling building hidden behind high walls. Suites of rooms led off the central courtyard and we had the best, if the bowing and dry handwashing of the help meant anything. The wine was chilled and the finest I had ever tasted. I prowled around the carpeted rooms and nibbled the toasted tidbits that came with the wine, while Bibs burbled and splashed in the adjoining pool. She eventually emerged wrapped in a towel, glowing with health and growling with hunger. There was no nonsense about dining rooms or restaurants in this establishment. Servants brought the food on brass trays and we gorged ourselves. When they had cleaned up the leavings I threw the bolt in the outer door and filled Bib's crystal mug with more wine.\n\n\"This is the life,\" she said.\n\n\"It surely is.\" I sprawled on the cushions across from her. \"A good night's sleep and I will be feeling human again.\"\n\nShe lay back on the couch and looked at me through half-closed eyes. Well, really one half-closed and one all the way closed where she had been bopped. She shook her head and smiled.\n\n\"You are something else again, Jimmy. Just a kid, really, yet you are sure a winner. You survived Spiovente, which is not easy. Took out those two cops\u2014then you took on all the hogh's thugs\u2014and got away with it.\"\n\n\"Just luck,\" I said. Enjoying the praise but not that \"kid\" remark.\n\n\"I doubt it. And you saved my neck. Got me out of the hands of the law and stole enough clinkers to get me off this planet. I would like to say thanks.\"\n\n\"You don't have to, not really. You are going to help me find Garth so that makes us even.\" I stood and yawned. \"I want to ask you about him\u2014but it can wait until morning. I need some sleep.\"\n\nShe smiled again. \"But, Jim, I told you I would like to thank you. In my own way.\"\n\nWas it chance that as she lay back the towel slipped a little? No it was not chance. Nor was it by accident that she was devastatingly naked underneath. Despite the black eye Bibs was a terribly, terribly attractive girl.\n\nWhat does one do on an occasion like this?\n\nWhat one does not do is talk about it to others. I'm sorry. This is a private matter between two consenting adults. Very consenting. You will excuse me if I draw the curtain over this day and insert a space in this text to denote the passing of a good many hours.\n\n* * *\n\nNever had the sun shone so warmly and brightly. The afternoon sun. I smiled back at it just as warmly, bereft of any guilt, filled full with happiness. Nibbling a bit of fruit and sipping some wine. Turning languidly from the window as Bibs reentered the room.\n\n\"You mean it?\" she asked. \"You won't go offplanet with me? You don't want to?\"\n\n\"Of course I want to. But not until I have found Garth.\"\n\n\"He'll find you first and kill you.\"\n\n\"Perhaps he might be the one who gets killed.\"\n\nShe cocked her head most prettily to one side, then nodded. \"From anyone but you I would think that bragging. But you might just do it.\" She sighed. \"But I won't be here to see it. I rate survival ahead of vengeance. He put me into jail\u2014you got me out. Case closed. Though I admit to a big bundle of curiosity. If you do get out of this, will you let me know what happened? A message care of the Venian Crewmembers' Union will get to me eventually.\" She passed over a slip of paper. \"I've written down everything that I remembered, just like you asked.\"\n\n\"General,\" I read. \"Either Zennor or Zennar.\"\n\n\"I never saw it spelled out. Just overheard one of the officers talking to him when they didn't know I could hear them.\"\n\n\"What is Mortstertoro?\"\n\n\"A big military base, perhaps their biggest. That's where we landed to take on cargo. They wouldn't let us out of the spacer, but what we could see was very impressive. A big limousine, all flags and stars, would come for Garth and take him away. There was a lot of saluting\u2014and they always saluted him first. He is something big, high-up, and whatever he is involved with has to do with that base. I'm sorry, I know it's not much.\"\n\n\"It's a lot, all I need now.\" I folded the paper and put it away. \"What next.\"\n\n\"We should have identification documents by tonight. They are expensive but real. Issued by one of the smaller duchies that needs the foreign exchange. So I can ship out on any spacer I want to. As long as the League agents don't recognize me. But I've managed to bribe my way onto a trade delegation that made their flight arrangements months ago. One of them has been well paid to get ill.\"\n\n\"When do you leave?\"\n\n\"Midnight,\" she said in a very quiet voice.\n\n\"No! So soon...\"\n\n\"I felt the same way\u2014which is why I am leaving. I am not the kind of person that gets tied down in a relationship, Jimmy.\"\n\n\"I don't know what you mean.\"\n\n\"Good. Then I am getting away before you find out.\"\n\nThis sort of conversation was all very new and confusing. I am reluctantly forced to admit that up until the previous evening my contact with the opposite sex had been, shall we say, more distant. Now I was at an unaccustomed loss for words, indecisive and more than a little bewildered. When I blurted this out Bibs had nodded in apparent complete understanding. I realized now that there was an awful lot I did not know about women, a mountain of knowledge I might never acquire.\n\n\"My plans aren't that fixed...\" I started to say, but she silenced me with a warm finger to my lips.\n\n\"Yes they are. And you're not going to change them on my account. You seemed very certain this morning about what you felt you had to do.\"\n\n\"And I am still certain,\" I said firmly, with more firmness and certainty than I felt. \"The bribe to get me over to Nevenkebla was taken?\"\n\n\"Doubled before accepted. If you are going to go missing then old Grbonja will never be permitted to go ashore there again. But he has been ready to retire for years. The bribe is just the financial cushion he needs.\"\n\n\"What does he do?\"\n\n\"Exports fruit and vegetables. You'll go along as one of his laborers. He won't be punished if you get away from the market\u2014but they will take away his landing pass. He won't mind.\"\n\n\"When do I get to see him?\"\n\n\"We go to his warehouse tonight, after dark.\"\n\n\"Then...\"\n\n\"I leave you there. Are you hungry?\"\n\n\"We just ate.\"\n\n\"That is not what I mean,\" she answered in a very husky voice.\n\n* * *\n\nThe dark streets were lit only by occasional torches at the corners, the air heavy with menace. We walked in silence; perhaps everything that might be said had already been said. I had bought a sharp dagger, which hung at my waist, and another club that I slammed against a wall occasionally to be sure any watchers knew it was there. All too soon we reached our destination, for Bibs knocked on a small gate let into a high wall. There were some whispered words and the gate creaked open. I could smell the sweetness of fruit all about us as we threaded through the dark mounds, to the lamplit corner where an elderly man slumped in a chair. He was all gray beard and gray hair to his waist, where the hair spread out over a monstrous paunch held up by spindly legs. One eye was covered by a cloth wound round his head, but the other looked at me closely as I came up.\n\n\"This is the one you are taking,\" Bibs said.\n\n\"Does he speak Esperanto?\"\n\n\"Like a native,\" I said.\n\n\"Give me the money now.\" He held out his hand.\n\n\"No. You'll just leave him behind. Ploveci will give it to you after you land.\"\n\n\"Let me see it then.\" He turned his beady eye on me and I realized that I was Ploveci. I took out the leather bag, spread the coins out on my hands, then put them back into the bag. Grbonja grunted what I assumed was a sign of assent. I felt a breeze on my neck and wheeled about.\n\nThe gate was just closing. Bibs was gone.\n\n\"You can sleep here,\" he said pointing to a heap of tumbled sacks against the wall. \"We load and leave at dawn.\"\n\nWhen he left he took the lamp with him. I looked into the darkness, toward the closed gate.\n\nI had little choice in the matter. I sat on the sacks with my back to the wall, the club across my legs and thought about what I was doing, what I had done, what we had done, what I was going to do, and about the conflicting emotions that washed back and forth through my body. This was apparently too much thinking because the next thing I knew I was blinking at the sunlight coming through the opening door, my face buried in the sacks and my club beside me on the floor. I scrambled up, felt for the money\u2014still there\u2014and was just about ready for what the day would bring. Yawning and stretching the stiffness from my muscles. Reluctantly.\n\nThe large door was pulled wider and I saw now that it opened onto a wharf with the fog-covered ocean beyond. A sizable sailing vessel was tied up there and Grbonja was coming down the gangway from the deck.\n\n\"Ploveci, help them load,\" he ordered and passed on.\n\nA scruffy gang of laborers followed him into the warehouse and seized up filled sacks from the pile closest to the door. I couldn't understand a word they said, nor did I need to. The work was hot, boring, and exhausting, and consisted simply of humping a sack from the warehouse to the ship, then returning for another. There was some pungent vegetable in the sacks that soon had my eyes running and itching. I was the only one who seemed to mind. There was no nonsense about breaks either. We carried the sacks until the ship was full, and only then did we drop down in the shade and dip into a bucket of weak beer. It had foul wooden cups secured to it by thongs and after a single, fleeting moment of delicacy I seized one up, filled and emptied it, filled and drank from it once again.\n\nGrbonja reappeared, as soon as the work was done, and gurgled what were obviously orders. The longshoremen became sailors, pulled in the gangplank, let go the lines and ran up the sail. I stood to one side and fondled my club until Grbonja ordered me into the cabin and out of sight. He joined me there a few moments later.\n\n\"I'll take the money now,\" he said.\n\n\"Not quite yet, grandpop. You get it when I am safely ashore, as agreed.\"\n\n\"They must not see me take it!\"\n\n\"Fear not. Just stand close to the top of the gangplank and I will stumble against you. When I'm gone you will find the bag tucked into your belt. Now tell me what I will find when I get ashore.\"\n\n\"Trouble!\" he wailed and raked his fingers through his beard. \"I should never have gotten involved. They will catch you, kill you, me too...\"\n\n\"Relax, look at this.\" I held the money bag in the beam of light from the grating above and let the coins trickle between my fingers. \"A happy retirement, a place in the country, a barrel of beer and a plate of pork chops every day, think of all the joys this will bring.\"\n\nHe thought and the sight of the clinking coins had a great calming effect. When his fingers had stopped shaking I gave him a handful of money which he clutched happily.\n\n\"There. A downpayment to show that we are friends. Now think about this\u2014the more I know about what I will find when I get ashore the easier it will be for me to get away. You won't be involved. Now... speak.\"\n\n\"I know little,\" he mumbled, most of his attention on the shining coins. \"There are the docks, the market behind. All surrounded by a high wall. I have never been past the wall.\"\n\n\"Are there gates?\"\n\n\"Yes, large ones, but they are guarded.\"\n\n\"Is the market very large?\"\n\n\"Gigantic. It is the center of trade for the entire country. It stretches for many myldyryow along the coast.\"\n\n\"How big is a myldyryow?\"\n\n\"Myldyr, myldyryow is plural. One of them is seven hundred lathow.\"\n\n\"Thanks. I'll just have to see for myself.\"\n\nGrbonja, with much grunting and gasping, threw open a hatch in the deck and vanished below, undoubtedly to hide the coins I had given him. I realized then that I had had enough of the cabin so I went out on deck, up to the bow, where I would not be underfoot. The sun was burning off the morning haze and I saw that we were passing close to an immense tower that rose up from the water. It was scarred, ancient, certainly centuries old. They had built well in those days. The mist lifted and revealed more and more of the structure, stretching up out of sight. I had to lean back to see the top, high, high above.\n\nWith the remains of the fractured bridge hanging from it. The once-suspended roadway hung crumpled and broken, dipping down into the ocean close by. Rusted, twisted, heaped with the broken supporting cables which were over two meters thick. I wondered what catastrophe had brought it down.\n\nOr had it been deliberate? Had the rulers of Nevenkebla destroyed it to cut themselves free from the continent that was slowly sinking back into barbarism? A good possibility. And if they had done this they showed a firmness of mind that made my penetration of their island that much more difficult.\n\nBefore I could worry about this a more immediate threat presented itself. A lean, gray ship bristling with guns came thundering up from ahead. It cut across our bow and turned sharply around our stern; our sailing ship bobbed in its wake and the sails flapped. I emulated the sailors and tried to ignore the deadly presence, the pointed weapons that could blow us out of the water in an instant. We were here on legitimate business\u2014weren't we?\n\nThe gunboat's commander must have believed this as well because, with an insulting blare on their horn, the vessel changed course again and blasted away across the sea. When the ship had dwindled into the distance one of the sailors shook his fist after them and said something bitter and incomprehensible that I agreed with completely.\n\nNevenkebla rose out of the mists ahead. Cliffs and green hills backing an immense, storied city that rose up from a circular harbor. Factories and mine-heads beyond, plumes of smoke from industry already busy in the early morning. And forts at the water's edge, great guns gleaming. Another fort at the end of the seawall as we entered the harbor. I could feel the glare of suspicious eyes behind the gunsights as the black mouths of the barrels followed us as we passed. These guys were not kidding.\n\nAnd I was going to tackle this entire country single-handed?\n\n\"Sure you are, Jim,\" I said aloud with great braggadocio, swinging my club so that it whistled in tight arcs. \"You'll show them. They don't stand a chance against fighting Jimmy diGriz.\"\n\nWhich would have been fine if my voice had not cracked as I said it.\nChapter 5\n\n\"Down sails,\" an amplified voice roared. \"Take our line aboard.\"\n\nA high-prowed tug came chuntering up with its loudhailer bellowing. Grbonja swiftly translated the commands to the crew.\n\nNothing was left to chance in Nevenkebla: all matters were highly organized. Even before the sail was down we were secured safely to the tug and being towed to our berth at the crowded wharfside. Sailing craft of interesting variety and form were already unloading cargo there. We were moved into a vacant berth among the others.\n\n\"They come long ways,\" Grbonja wheezed, stumbling up beside me, pointing at the other ships. \"From Penpilick, Grampound, even Praze-an-Beeble\u2014may everyone there suffer from a lifetime of dysesya! Tie up outside harbor at night. You give me the money now, too dangerous on shore!\"\n\n\"A deal's a deal, grandpop. Too late to back out now.\"\n\nHe sweated and muttered and looked at the land coming close. \"I go ahead first, talk to freightmaster. Only then we unload. They take your papers and give you dock badge. After that you will see me. Give me the money.\"\n\n\"No sweat. Just keep your mind on the sunny future of happy retirement.\"\n\nTwo armed guards glowered down at us as we tied up. A steam winch dropped the gangway into place and Grbonja puffed up the incline and onto the dock. To turn me in? Maybe I should have paid him in advance? My heart gave a few quick thuds as it shifted into worry mode.\n\nIn a few minutes\u2014or was it centuries?\u2014Grbonja had returned and was shouting instructions at the crew. I left my club in the cabin and put the dagger inside my shirt where it couldn't be seen. My lockpick and remaining coins were in a pouch inside my shirt as well. I was ready as I was ever going to be. When I came out of the cabin the sailors were already starting to unload. I picked up a bag and followed the others up the gangway. Each of them held out his identity papers: I did the same. As they reached the dock the officer there took each man's papers and stuffed them into a box. Then pinned an identification tag to the man's clothes. He looked bored by the job. I tried not to tremble as I came up to him.\n\nIt was just routine. \"Next,\" he called out, whipped the papers from my hand and pinned the tag to my chest. Or rather pinned it through the fabric into my skin. I jumped but kept my mouth shut. He grinned, with a touch of sadism in the turn of his mouth, and pushed me on.\n\n\"Keep moving, lunkhead. Next.\"\n\nI was safely ashore and undetected. Following the bent back of the man ahead of me into the dark warehouse. Grbonja was standing by the growing pile of sacks. When he saw me he called out an incomprehensible instruction and pointed to the next bay.\n\n\"The money, now,\" he burbled as I dumped the sack. I slipped it to him and he staggered away muttering with relief. I looked around at the solid cement and steel walls and went back for another sack.\n\nBy the time I had carried in my third sack I was getting desperate. After a few more trips the ship would be unloaded and that would be the end of that. I would have had an expensive round trip and done some hard work. Nothing more. Because I could see no way out of the building\u2014and no place to hide within it. They obviously did not relish uninvited visitors to Nevenkebla. I needed more time.\n\n\"Call a beer break,\" I whispered to Grbonja as I passed him at the head of the gangway. The checker-inner had gone but the two unsmiling guards still stood watch.\n\n\"We never stop\u2014it is not the custom.\"\n\n\"It is today. It's a hot day. You don't want me to tell them you were hired to smuggle me here?\"\n\nHe groaned aloud, then called out. \"Beer, we stop for beer!\"\n\nThe crew asked no questions at this unexpected treat, only chattered together with pleasure as they gathered round the barrel. I had a good slug of the stuff then went and sat on the gunwhale beside the gangway. Looked up at the boots of the guard who stood above me. Looked down at the water and saw the space between the pilings there.\n\nMy only chance. The guard above me moved out of sight. Grbonja had his back turned while the sailors had their attention focused on the barrel. A difference of opinion over the rationing appeared to break out. There were angry shouts and a quick blow. The crew watched these proceedings with great interest. No one was visible on the dock above.\n\nI dropped a length of line over the side, swung my legs over and climbed down it. No one saw me go. With my legs in the sea I used my dagger to cut the line above my head and dropped silently into the water. With noiseless strokes I swam into the darkness under the pier.\n\nSlime-covered boards connected the wooden piles. When I reached for one of them something squealed and vanished in the darkness. And it stank down here. Nameless rubbish bobbed in the water around me. I was beginning to regret my impetuous swim.\n\n\"Chin up, Jim, and move along. This is the first place they look when they find out you are missing.\"\n\nI swam. Not far, for there was a solid wall here that ran back into the darkness. I groped along it until I reached the outer piles again. Through the openings between them I saw the hull of another sailing ship, tied close. There was no room to pass between the planks of the ship and the piles. Trapped so soon?!\n\n\"This is your day for panic,\" I whispered aloud, the sound of my voice covered by the slapping of the waves. \"You can't go back, so carry on you must. The hull of this ship has to curve away. Just dive down and swim along it until you find another opening between the piles.\"\n\nHo-ho. Sounded very easy to do. I kicked my boots off and breathed deep. But my trepidition grew with each shuddering breath that I drew in. When my head was swimming with oxygen intoxication I let out the last lungful and dived.\n\nIt was a long, dark and apparently endless swim. I ran my left hand along the ship's hull to guide me. Collecting some heroic splinters at the same time. On and on with no glimmer of light in front or above. This must be a very big ship. There was fire in my lungs and desperation in my swimming before I saw light ahead. I came up as quietly as I could by the ship's bow. Trying not to gasp as I exhaled and drew in life and fresh air.\n\nLooking up at a sailor standing on the rail above, turning towards me.\n\nI sank out of sight again, forcing myself deep under the water, swimming on with my lungs crying out for air, until I saw the black bulk of the next ship ahead of me, forcing myself to swim on to the last glimmer of light before floating up the surface again.\n\nCatching my head nicely between hull and piling, to fight down the rising panic as I fought to free myself\u2014getting some splinters in my scalp this time. My groping fingers found a gap between the pilings so I surfaced there, hung on, sucked in lungful after lungful of the stinking fug, enjoying it more than the freshest air I had ever breathed.\n\nThis was the beginning of a very long and very tiring day. I did not keep track of the number of ships I passed, but it was a lot. At first I searched under the various docks but soon gave that up since they were all the same, each firmly separated by an underwater wall from the next. Some of the ships had finished unloading and had left, for I came to gaps in the continuous wall of vessels. All I could do when this happened was to breathe deep, dive deep\u2014and swim like crazy to reach the next ship before my breath ran out.\n\nIt was afternoon before I reached the last ship and the end of the docks. The tide was ebbing, the vessels were now down below the dock level so there was more concealment from above. I was very tired but very proficient by this time. One more time I breathed deep, dove down at the bow, swam the length of the hull and surfaced in the shadow of the rudder.\n\nTo look at a solid wall of jointed stone stretching out before me.\n\nHolding onto the rudder, my eyes just above the surface, I peered around it. And realized that I was looking at the harbor wall that stretched unbroken out to the fort built at its far end. I drew back into the shadow of the rudder and found that my heart was sinking so fast it was pulling me under the water.\n\n\"Any bright ideas, Jim?\" I asked, then found that I was waiting a long time for an answer.\n\nThink, don't despair, I ordered myself. I still felt despair. Could I go back? No, that was out. After all I had gone through today I was not going to surrender that easily. Hide under one of the docks? Possibly. But they would be thoroughly searched as soon as I was missed, I was certain of that. What else? Climb up onto the dock? No way. The warehouses here were sure to be as barren of hiding places as the one I had left. Then what?\n\n\"Turn the problem on its head, that's what The Bishop had always said.\"\n\nWhat would that be in this situation? I was trying to get away from the soldiers, fleeing them, knowing they would be looking for me. So I should go to them. But that would be suicide. But where could I possibly go that would be totally unexpected?\n\nWhy, the fort on the end of the harbor wall of course.\n\n\"Without a doubt the most insane idea you have ever had,\" I muttered in disgust, peering around the rudder again. Above me there were shouted oaths from the sailors and the thud of feet on planking. I had the feeling that this ship would be leaving soon as well, taking my protection with it. The solid stone blocks of the jetty stretched unbroken to the fort at the end. Some debris washed against the stone and sea birds fought over the edible bits. Other than that\u2014nothing. No cover at all. If I tried to swim out there I would be seen at once by anyone who glanced that way. Above me tackle creaked as the sail was lifted; the ship was getting under way.\n\nI had to get clear of it\u2014or did I? No tug had appeared. Was it possible the ships were only towed into harbor? That they permitted them to sail out on their own? It was. I peered around the rudder again and saw two of the cargo vessels standing out toward the entrance. Light poured down from the growing gap above me and I sank under the surface before I could be seen.\n\nIt was not easy\u2014but it could be done. I held tight to the rudder as it came over, almost pulling itself out of my hands. I stayed under the surface as long as I could so I would not be seen from the shore. The sailing ship was moving along smoothly and it took all my strength to shift my grip from the front to the back of the rudder. Holding on was easier now. When I finally was forced to lift my face up to breathe I found myself in a rush of foam, inhaled some and fought not to cough. As we drew away from the dockside I saw an armed guard there. His back turned with indifference.\n\nIt was almost easy after that. The rush of the waves held me against the rudder post. I breathed easily with my head out of the water, unseen from the shore and invisible to anyone on the deck above. We tacked twice and each time I changed sides to keep the rudder between me and the fort that was now growing larger and larger ahead. When we went about for the last time I saw that this tack would take us close to the fort and past it on into the open ocean. I watched as the stone wall came closer and closer until I could see the sea beyond the end of it. Only then did I take a last breath, let go and dive deep.\n\nYes, I was tired. But this should also be my final little swim for the day so I wanted to make it a good one. The seaweed-covered harbor wall was clear ahead, the end rounded where it met the open ocean. There was a strong swell coming in that I had to fight against, swimming close to the stone where its force was weakest. Farther and farther until I had to breath or inhale water. Floating up to the bright surface and through it, looking up at the stone wall with the projecting gun barrels above. Holding on against the waves and breathing deep. Clutching into the cracks between the stones and working my way around to the far side until I could peer down its unbroken length at the shore beyond. Pleasure craft dotted the water here, power and sail, and I would certainly be seen if I tried to swim its length. Then what? I couldn't stay here in the water where I could be seen by any passing ship. I looked up at the great stone blocks and thought.\n\nWhy not? The only ships in sight now were vanishing seaward. At the outermost swell of the fort I could not be seen from the shore. And the space between stones provided ample grip for toe and finger. So climb.\n\nClimb I did. It was not easy\u2014but I had little choice. Up the vertical wall, scrabbling and clinging, midway between two of the largest seaward-facing guns. They projected through embrasures in the solid wall, shining steel, polished and deadly. I clung on and rested when I reached their level, the heaving surface of the sea a good ten meters below. The ocean was still empty\u2014but for how long?\n\n\"Give me a light, will you Jim?\"\n\nI started so hard I almost lost my grip and fell back into the water.\n\nFragrant cigar smoke blew over me and I realized it was coming from the gun embrasure close by. I hadn't been seen, no one here knew my name. It had just been coincidence. The gunners were there, that close, looking seaward and smoking on duty which I was sure was frowned upon strongly. I did not dare move. I could only hold on and listen.\n\n\"This new captain, he has got to go.\"\n\n\"He is the worst. Poison in his coffee?\"\n\n\"No. I heard they did that up north and they decimated the entire regiment. You know, shot one guy out of ten.\"\n\n\"That is the real old cagal and you know it. Nothing but cagalhouse rumor. Like fragging. Everyone talks about it, no one does it...\"\n\n\"Captain coming!\"\n\nA cigar butt sailed by my head and there was the quick slap of retreating feet. I climbed again, before my arms came out of their sockets. Forcing myself up the last few centimeters until I reached the edge and hauled myself painfully onto the flat roof of the fort. A seabird cocked a cold eye at me, screeched and flapped off. I crawled slowly across the sun-baked bird droppings of centuries, to the very center of the round building. I lay flat on my back and could see nothing but sky and the top of a distant hill. This meant that in turn I could not be discovered except from the air. I would chance that, since I had seen only one distant aircraft the entire day. I closed my eyes against the glare of the sun and instantly, without intending to, fell sound asleep.\n\nI awoke with a start and a rapidly beating heart. A cloud had crossed the face of the sun and I was chilled in my wet clothes. It had been stupid, falling asleep like that, yet I had gotten away with it. I had not been seen. The sun was closer to the horizon and since I had been safe here so far I might still be safe until dark.\n\nAnd hungry and thirsty. The demands of the body are insatiable, always after something. But this time it was going to be mind over matter and I was going to stay on the roof, unmoving, until nightfall.\n\nWhich was slow in coming. I smacked my dry lips and ignored the angry rumblings in my gut. The suns always set. It was just a matter of patience.\n\nDusk finally crept over the land, the first stars came out as it slowly grew dark. Lights came on in the fort below and I could hear the hoarse shouting of military orders. Very slowly I crawled to the inner edge and peered over. Into a courtyard where some sort of maneuver was being engaged in. Soldiers marched back and forth in little groups with much screeching from the officers. Eventually one group entered the fort and the other marched back toward the land along the broad top of the harbor wall, their way lit by evenly spaced lights. They got smaller and smaller in the distance until they reached the distant shore and vanished from sight.\n\nThen all of the lights went out.\n\nI lay, blinking into the sudden darkness, and could not believe my good luck. Had the lights been extinguished to enable me to sneak safely ashore? Probably not. There were guns below and if they meant to use them, as they obviously did, the gunners would not want to be blinded by their own lights. Good thinking, guys!\n\nI waited until I could see my way by starlight, then climbed down the outside wall to the rail around the courtyard, stepped carefully onto it and down onto the stone flagging. A single door in the wall was sealed and silent. On tiptoe I scuttled landward as fast as I could. The dark bulk of the fort grew small behind me and I strolled more easily, resisting the impulse to whistle with pleasure. The dark forms of the pleasure boats were visible off to the left. There were lights in the cabins of a few of them and I heard distant laughter across the water. I relaxed, strolled, the rough stone cool beneath my bare feet. I had the world to myself and safety lay close ahead.\n\nThen I ran headlong into the metal fencing that cut off the top of the harbor wall and all the lights came on in a searing blaze of illumination. Lights stretching ahead and behind, lights above revealing the wire fence and sealed metal door in front of me.\nChapter 6\n\nI bounced back from the wire, looked around wildly, hurled myself flat on the wall waiting for the sound of shots.\n\nBut nothing happened. The lights burned down brightly; the harbor wall behind me stretched emptily back to the fort. On the other side of the barrier the wall extended as far as the warehouses above the harbor where more lights revealed a small marching group. Coming towards me.\n\nHad I been seen\u2014or was I invisible in the shadows? Or had I triggered some alarm that turned on the lights and revealed my presence? Whatever had happened there was no point in my waiting around in order to find out. I crawled quickly to the outer edge of the wall facing the ocean\u2014I had had enough swimming in the harbor, thank you\u2014and dangled my legs backward over the edge. Groped with my bare feet for a toehold on the rough stone. Found one and eased myself down into the darkness. The tide was coming in again and my legs were engulfed by the sea. Above me on top of the wall the tramping feet grew louder. Below me the water was cold, black and unattractive.\n\nWhy didn't I just stay here out of sight until they had gone by above?\n\nAs soon as this cowardly thought had trickled through the synapses of my brain I recognized it for the dumb idea that it was. A flick of a flashlight and my presence would be revealed. I had not gone through all of the strenuous efforts and dangers of the day to be grabbed now because I was afraid of getting wet. Or eaten by unseen monsters. The ocean here must be safe or the fleets of pleasure craft would not have been drifting around all day.\n\n\"Swimmies, Jim, swimmies,\" I muttered and slid down into the sea.\n\nBy the time the soldiers had reached the gate I was treading water well away from the wall, ready to dive instantly if they pointed any lights my way. They didn't. I could see one of them unlocking the gate, then relocking it again after they had all passed through. Then they all marched on again. A relief party, surprise inspection perhaps, or some other uninteresting military maneuver. I turned about and began to swim toward shore.\n\nWhat next? The lights of a promenade grew closer and my problem grew bigger. How was I a barefoot, sodden stranger with no knowledge of this land whatsoever, how was I to go ashore and make my way about unnoticed? Not easily, that was obvious. A dark shape came between me and the lights. A craft of some kind. Salvation of some kind?\n\nI swam slowly between the moored pleasure boats. In the distance I could see that some of them were illuminated, but only darkness prevailed here. Were they occupied? They didn't appear to be; it was too early for any occupants to be asleep. Which hopefully meant that the jolly sportsmen had gone ashore after a strenuous day at play.\n\nA thin mast moved against the stars. A sailboat, a small one. I wanted something larger. I swam on until a darker form rose up above me. No masts, which meant that it was a powered craft of some kind. I swam alongside it to the stern, where my groping fingers found the ladder that was secured there. Rung by rung I climbed, dripping, out of the sea and into the craft. There was enough light from the stars and the illumination along the shore to make out cushioned seats, a wheel\u2014and a door that might lead below. I went to it, found the handle and tried to turn it. Locked.\n\n\"Good news indeed, Jim. If it is locked there is something here worth stealing. Best to look and see.\"\n\nI did. Darkness is no handicap for an efficient locksmith. I felt out the tumblers of a very simple lock with delicate touches of my lockpick. Lifted them aside and pushed the door open.\n\nWhat followed was slow work. If there were lights I did not want to turn them on. I did it all by touch. But there is a certain logic to any small craft that must be followed. Berths in the bow along the hull. Lockers below, shelves above. After a good deal of rattling, fumbling, head-banging and cursing I gathered my treasures in a blanket and took them up on deck and spread them out.\n\nWhat had felt like a bottle with a screwcap was a bottle with a screwcap. Which I unscrewed and sniffed. Then dipped in a finger and tasted. A very sweet wine. Not my normal tipple, but paradisical after all the seawater I had swallowed. There was a metal box with stale bread or biscuits of some kind that almost broke my teeth. They softened a bit when I poured wine over them, then wolfed them down. I belched deeply and felt better.\n\nI groped through the rest of my loot. There were books and boxes, unidentifiable forms, and strange shapes. And clothing. A very sheer skirt that was just not my thing. But other sartorial items were. I sorted out all of the other bits that appeared to be clothing not instantly identifiable as being intended for the fairer sex, stripped and tried some of it on. I had no idea of how well they matched, but it was an outfit of sorts. The trousers were too large by far, but a length of line in place of a belt took care of that. The shirt was a better fit, and if the jacket came down to my knees perhaps it was intended to be that length. The shoes were too big but stayed on my feet after I had stuffed cloth into their toes. It was the best I could do. Then I undressed and put my own wet clothing back on, put my new outfit into the can the bread had been in, wrapped this in turn in what I hoped was waterproof plastic.\n\nThe air was beginning to be chill and it was time to get moving. I was tired, slowed down by the exertions of the day and badly in need of some sleep. I wasn't going to get any. I finished the wine, put the empty bottle and everything else I had removed back into the cabin, then relocked the door. Before I could change my mind I put the bundle on my head and slipped over the side.\n\nThe shore was close and the beach empty as far as I could see. Which was a major blessing since swimming with one hand while balancing a can of clothing on the head with the other is not an exercise to be recommended. I emerged from the sea and scuttled to the shelter of some large rocks, stripped and buried my unwanted clothing in the sand. I quickly dressed in the dry clothes, tucked my small bag of possessions into my belt, slipped my dagger into the side of my shoe and I was ready to conquer the world.\n\nI really wanted only to find a quiet place to curl up for a nap\u2014but knew better. These people took their security seriously and the shore was their first line of defense as the fort had proved. I must get into the city itself.\n\nThere were lights on the promenade above, the sound of voices, but shadow below where I moved in silence. A flight of stairs rose up from the beach. I rose up as well\u2014but dropped back again even more swiftly at the sight of two uniformed and armed men close by. I lurked and counted backward from two hundred before I peeked again. The uniforms were gone and there were just a few evening strollers in sight. I merged and strolled and took the first turning that led away from the shore. There were streetlights here, open windows and locked doors. My clothing must not have looked too garish for a couple passed without even glancing my way. I heard music ahead and soon came to a bar over which a sign proclaimed DANCING AND DRINKING\u2014COME AND GET STINKING. An invitation almost impossible to resist. I pushed the door open and went in.\n\nThere is a power that shapes the bars of this universe. There has to be because form follows function. Function: to get containers of alcoholic beverages to people. Form: chairs to sit in, tables to rest containers on. I entered, pulled out a chair and sat at a vacant table. The other occupants ignored me just as I ignored them. A plump waitress in a short skirt came toward me, ignoring the whistles from the group of youths at the next table, skilfully avoiding their snapping fingers as well.\n\n\"Whadilitbe?\" she asked, flaring her nostrils at them as they raised beer mugs in her direction and toasted her loudly.\n\n\"Beer,\" I said and she moved off. When it came it was pungent and cold. She made her own change from the coins I had spread on the table, this seemed to be the local custom, then went back behind the bar.\n\nI drank deep and wiped the foam from my mouth just as another young man came through the door and hurried to the adjoining table.\n\n\"Porka\u0109oj!\" he whispered hoarsely. Two of the youths stumbled to their feet and hurried toward the rear of the bar.\n\nI put down my beer, scraped up my coins, and hurried after them. There was trouble here, though I did not know what kind. What he had said could be translated as bad-pigs, and must surely be local slang since I did not imagine some mucky swine were on their way. Pigs as an epithet for police is a common usage\u2014and the reactions of the two men seemed to bear this out. And I would lose nothing by being cautious. They hurried down a hallway and when I reached it a door at the far end was just closing. I had my hand on its knob when a loud siren sounded from the other side and a glare of illumination shone in through the cracks between door and frame.\n\n\"What's this?\" a coarse and loud voice said. \"You boys maybe slipping out through the backdoor because we got a patrol out front? Let's see your identification.\"\n\n\"We've done nothing wrong!\"\n\n\"You've done nothing right so far. C'mon, the ID.\"\n\nI waited, unmoving, hoping the bad-pig outside was not joined by his stymates from the bar. The coarse laughter from the other side of the door was anything but humorous.\n\n\"Hello, hello\u2014both out of date? Not thinking of avoiding the draft, are you boys?\"\n\n\"A clerical error,\" a pale voice whimpered.\n\n\"We get a lot like that. Let's go.\"\n\nThe light went away and so did the footsteps. I waited as long as I dared, then opened the door and exited the bar. The alley was empty, pig and prisoners were gone. I went myself, as quickly as I could without running. Then stopped. What was I running from? Once the police had left, the bar would be the safest place in the city for me. I stopped in a dark doorway and looked back at the rear entrance. No one else came out. I counted to three hundred, then to be safe backward again to zero. The door remained closed. Cautiously, ready to flee in an instant, I went back into the bar, peered into the barroom. No police\u2014but the glimmerings of an idea.\n\nThe four young men at the table looked up as I came back in, the newcomer sitting at one of the recently vacated seats. I shook my head gloomily and dropped into a chair.\n\n\"The porka\u0109oj got them. Both.\"\n\n\"I told Bill he needed new papers, wouldn't listen to me,\" the blond one said, the one who had come with the warning. He cracked his knuckles then seized up his beer. \"You got to have good papers.\"\n\n\"My papers are out of date,\" I said gloomily, then waved to the waitress.\n\n\"You should have stayed in Pensildelphia then,\" one of the others said, a spotty youth in an ill-fitting gold and green shirt.\n\n\"How did you know I was from Pensildelphia?\" I protested. He sneered.\n\n\"Rube accent like that, where else you from?\"\n\nI sneered back and glowed with pleasure inside. Better and better. I had a peergroup of draft dodgers, one of them who might be working with the police, and a hometown. Things were looking up. I buried my nose in my beer.\n\n\"You ought to get new ID,\" the friendly warner, possible police informer, said. I sniffled.\n\n\"Easy to say here. But you can't do it in Pensildelphia.\"\n\n\"Hard to do here too. Unless you got the right contacts.\"\n\nI stood up. \"I gotta go. Nice meeting you guys.\"\n\nBefore leaving I checked to make sure that the police were gone. Then I exited and waited. My new friend came out a moment later and smiled at me.\n\n\"Smart. Don't let too many people know what's going on. My label is Jak.\"\n\n\"Call me Jim.\"\n\n\"Good a name as any, Jim. How much you got to spend?\"\n\n\"Not much. I had a bad year.\"\n\n\"I'll put you in touch with the man himself for three sugarlumps. He'll want twenty.\"\n\n\"ID not worth more than ten. You get one-fifty.\"\n\n\"They're not all dumb in the backwoods, are they. Slap it in my hand and we're on our way.\"\n\nI paid him his cut and when he turned I put the tip of my knife against his neck just under his ear and pushed just hard enough to break the skin. He stayed absolutely still when I showed him the knife with the fresh drop of blood.\n\n\"That is a little warning,\" I said. \"Those pigs were waiting for whoever you flushed out. That's not my worry. My skin is. I got a feeling that you play both sides. Play the right side with me or I will find you and slice you. Understand?\"\n\n\"Understood...\" he said gruffly, with a tremor in his voice. I put the knife away and clapped him on the shoulders.\n\n\"I like you, Jak. You learn easy.\"\n\nWe went in silence and I hoped that he was making the right conclusions. I don't like threats and when threatened I do the opposite of what I am requested. But my experience of the petty criminal led me to believe that threats tended to work with them. Part of the time.\n\nOur route took us past a number of other bars and Jak looked carefully into each one before going on. He struck paydirt in the fifth one and waved me in after him. This place was dark and smoke filled, with jangling music blasting from all sides. Jak led the way to the rear of the room, to an alcove where the music was not quite as loud, at least not as loud as the striped outfit the fat man was wearing. He leaned back in a heavy chair and sipped at a tiny, poisonous green drink.\n\n\"Hello, Captain,\" my guide said.\n\n\"Get dead quickly, Jak. I don't want your kind here.\"\n\n\"Don't say that even funning, Captain. I got good business for you here, a mission of mercy. This grassgreen cutlet is a step ahead of the draft. Needs new ID.\"\n\nThe tiny eyes swiveled toward me. \"How much you got, cutlet?\"\n\n\"Jak says one-fifty for him, ten for you. I already paid him his.\"\n\n\"Jak's a liar. Twelve is the price and I give him his cut.\"\n\n\"You're on.\"\n\nIt was an instant transaction. I gave him the money and he passed over the grubby plastic folder. Inside there was a blurred picture of a youth who could have been anyone my age, along with other vital facts including a birthdate quite different from my own.\n\n\"This says that I am only fifteen years old!\" I protested.\n\n\"You got a baby face. You can get away with it. Drop a few years\u2014or join the army.\"\n\n\"I feel younger already.\" I pocketed the ID and rose. \"Thanks for the help.\"\n\n\"Any time. Long as you got the sugarlumps.\"\n\nI left the bar, crossed the road and found a dark doorway to lurk in. It was a short wait because Jak came out soon after me and strolled away. I strolled behind him at a slightly faster stroll. I was breathing down his neck before he heard my footsteps and spun about.\n\n\"Just me, Jak, don't worry. I wanted to thank you for the favor.\"\n\n\"Yeah, sure, that's all right.\" He rolled his eyes around at the deserted street.\n\n\"You could do me another favor, Jak. Let me see your own ID. I just want to compare it to mine to make sure the Captain didn't give me a ringer.\"\n\n\"He wouldn't do that!\"\n\n\"Let's make sure.\" My dagger blade twinkled in the streetlight and he rooted inside his jacket then handed me a folder very much like my own. I turned to look at it under the light, then handed it back. But Jak was the suspicious type. He glanced at it before putting it away\u2014and dropped his jaw prettily.\n\n\"This ain't mine\u2014this is yours!\"\n\n\"That's right. I switched them. You told me that ID was good. So use it.\"\n\nHis cries of protest died behind as I walked uphill away from the shore. To a better neighborhood without a criminal element. I felt very pleased with myself. The ID could have been good\u2014in which case Jak would lose nothing. But if it were faulty in any way it would be his problem, not mine. The biter bit. A very evenhanded solution. And I was going in the right direction. Once away from the waterfront things did get better, the buildings taller, the streets cleaner, the lights brighter. And I got tireder. Another bar beckoned and I responded. Velvet drapes, soft lights, leather upholstery, better-looking waitress. She was not impressed by my clothes, but she was by the tip I passed over when my beer arrived.\n\nI had very little time to enjoy it. This was a well-policed city and the bad-pigs came in pairs. A brace of them waddled in through the door and my stomach slipped closer to the floor. But what was I worrying about? My ID was fine.\n\nThey circuited the room, looking at identification, and finally reached my table.\n\n\"Good evening, officers,\" I smarmed.\n\n\"Knock off the cagal and let's see it.\"\n\nI smiled and passed over the folder. The one who opened it widened his nostrils and snorted with pleasure.\n\n\"Why look what we got here! This is Jak the joike strolled away from his home turf. That's not nice, Jak.\"\n\n\"It's a free world!\"\n\n\"Not for you, Jak. We all know about the deal you made with harbor police. Stay there and rat on your friends and you get left alone. But you strayed out of your turf, Jak.\"\n\n\"I'll go back now,\" I said rising with a sinking feeling.\n\n\"Too late,\" they said in unison as they slapped on the cuffs.\n\n\"Far too late,\" the nostril-flarer said. \"You're out of business, Jak, and in the army.\"\n\nThis really was the biter bit. This time I had been just a little too smart for my own good. It looked like my new and exciting military career had just begun.\nChapter 7\n\nThe cell was small, the bed hard\u2014I had no complaints. After the strenuous day I had just finished, sleep was the only thing that I wanted. I must have been snoring as I fell toward the canvas covers, with no memory of my face ever touching the stained pillow. I slept the sleep of exhaustion and awoke when a gray shaft of light filtered in through the barred window. I felt cheered and rested until I realized where I was. Dark depression fell.\n\n\"Well, it could be worse,\" I said cheerily.\n\n\"How?\" I snarled dispiritedly. There was no easy answer to that. My stomach rumbled with hunger and thirst and the depression deepened. \"Cry-baby,\" I sneered. \"You've had it much worse than this. They took the dagger but nothing else. You have your money, your identification.\" And the lockpick I added in silence. The presence of that little tool had a warming effect, holding out hope of eventual escape.\n\n\"I'm hungry!\" a youthful voice cried out and there was a rattling of bars. Others took up the cry.\n\n\"Food. We're not criminals!\"\n\n\"My mom always brought me breakfast in bed...\"\n\nI was not too impressed by this last wail of complaint but sympathized with the general attitude. I joined the cry.\n\n\"All right, all right, shut up,\" an older and gruffer voice called out. \"Chow is on the way. Not that you deserve anything, bunch of draft dodgers.\"\n\n\"Cagal on that sergeant\u2014I don't see your fat chunk in the army.\"\n\nI looked forward to meeting the last speaker; he showed a little more courage than the rest of the wailers. The wait wasn't too long, though it was scarcely worth it. Cold noodle soup with sweet red beans is not my idea of the way to start the day. I wondered how it would end.\n\nI had plenty of time for wondering because after feeding time in the zoo we were left strictly alone. I stared up at the cracked ceiling and slowly began to realize that my ill fortune wasn't that bad when closely examined. I was alive and well in Nevenkebla. With a promising career ahead of me. I would learn the ropes, find out all I could about this society, maybe even get a lead on Garth\u2014or General Zennor if Bibs had overheard the name correctly. He was in the army and I would very soon be in the army, which fact might work to my advantage. And I had the lockpick. When the right moment came I could do a little vanishing act. And how bad could the army be? I had been a soldier on Spiovente, which training should come in handy...\n\nOh, how we do fool ourselves.\n\nSomewhere around midday, when my cowardly peer group were beginning to howl for more nourishment, the crash of opening cell doors began. The howls changed to cries of complaint as we were ordered from our cells and cuffed wrist to wrist in a long daisychain. About a dozen of us, similar of age and gloomy of mein. The unknown future lay darkly ahead. With much stumbling and curses we were led from the cell block to the prison compound where a barred vehicle waited to transport us to our destiny. It moved away silently after we had been herded aboard, battery- or fuel cell\u2013powered, out into the crowded city streets. Clothes were slightly different, vehicles of unusual shapes, but it could have been any world of advanced technology. No wonder they had cut themselves off from the rest of this decadent planet. Selfish\u2014but understandable.\n\nNo effete amenities like seats were provided for hardened criminals: so we clutched to the bars and swayed into each other at the turns. A thin, dark-haired youth secured to my left wrist sighed tremulously, then turned to me.\n\n\"How long you been on the run?\" he asked.\n\n\"All my life.\"\n\n\"Very funny. I've had six months since my birthday, six short months. Now it's all over.\"\n\n\"You're not dying\u2014just going into the army.\"\n\n\"What's the difference? My brother got drafted last year. He smuggled a letter out to me. That's when I decided to run. Do you know what he wrote\u2014?\"\n\nHis eyes opened wide and he shivered at the memory, but before he could speak our transport slammed to a halt and we were ordered out.\n\nThe street scene was one to give joy to the eyes of any sadist. Varying forms of transport had converged on the plaza before the tall building. Emerging from them were young men, hundreds, perhaps thousands of them, all wearing upon their faces a uniform expression of despair. Only our little band was manacled\u2014the rest clutched the yellow draft notices that had dragged them to their destiny. A few of them had the energy to make mock of our manacled state, but they shrank away under the chorus of our jeers. At least we had made some attempt, no matter how feeble, to escape military impressment. Nor did it appear to make any difference to the authorities. They did not care how they had managed to grab the bodies. Once inside the doors our chains were stripped away and we were herded into line with all the others. The faceless military machine was about to engulf us.\n\nAt first it did not seem too bad. The lines of youths crept forward toward desks manned by plump maternal types who might have been our moms or teachers. All of them had gray hair and wore spectacles, which they looked over the tops of when they weren't two-fingeredly hammering their typewriters. I finally reached mine and she smiled up at me.\n\n\"Your papers please, young man.\"\n\nI passed them over and she copied dates and names and incorrect facts into a number of forms. I saw the cable leading from her typewriter to a central computer and knew that everything was being recorded and ingested there as well. I was happy to see the false identity entered; when I un-volunteered I wanted to drop from sight.\n\n\"Here you are,\" she said, and smiled, and passed over a buff file of papers. \"You just take these up to the fourth floor. And good luck in your military career.\"\n\nI thanked her, it would be churlish not to, and started back toward the front doors. A solid line of unsmiling military police blocked any chance of exit.\n\n\"Fourth floor,\" I said as the nearest one eyed me coldly and smacked his club into his palm.\n\nThe elevator cars were immense, big enough to take forty of us at a time. Nor did they leave until they were full. Jammed and miserable we rose to the fourth floor where a little taste of what awaited us awaited us. As the doors sighed open a military figure, all stripes and decorations, medals and red face came roaring toward us.\n\n\"Get out! Get out! Don't stand around like a bunch of poofters! Move it! Snap cagal or you'll be in the cagal. Take a box and a small transparent bag from the counter on the right as you pass. Then go to the far end of this room where you will UNDRESS. That means take all of your clothes off. AND I MEAN ALL OF YOUR CLOTHES! Your personal effects will go into the plastic bag which you will keep in your left hand at all times. All of your clothing will go into the box which you will take to the counter at the far end where it will be sealed and addressed and sent to your home. Where you will retrieve it after the war, or it will be buried with you, whichever comes first. Now MOVE!\"\n\nWe moved. Unenthusiastically and reluctantly\u2014but we had no choice. There must be a nudity taboo in this society because the youths spread out, trying to get close to the walls, huddled over as they stripped off their clothes. I found myself alone in the center of the room enjoying the scowled attention of the stripe-bearing monster: I quickly joined the others. So reluctant were they to reveal their shrinking flesh that dawdle as I might I was still first to the counter. Where a bored soldier seized my box and quickly sealed it, slammed it down before me and pointed to thick pens hung from the ceiling on elastic cords.\n\n\"Name-address-postcode-nearest-relative.\"\n\nThe words, empty of meaning through endless repetition, rolled out as he turned to seize up the next box. I scrawled the address of the police station where we had been held and when I released the pen the countertop opened and the box vanished. Very efficient. Plastic bag in left hand, folder in my right I joined the shivering group of pallid, naked young men who hung their heads as they waited their next orders. With their clothes gone all differences of identity seemed to have fled as well.\n\n\"You will now proceed to the eighteenth floor!\" was the bellowed command. We proceeded. Into the elevator, forty at a time, doors closed, doors opened\u2014into a vision of a sort of medical hell.\n\nA babble of sound, shouts for attention, screamed orders. Doctors and medical orderlies garbed in white, many with cloth masks over their faces, poked and prodded in a mad mirror-image of medical practice. Senses blurred as event ran into event.\n\nA physician\u2014that is I assume he was a physician since he wore a stethoscope around his neck\u2014seized my folder, threw it to an orderly, then clutched me by the throat. Before I could seize him by the throat in return he shouted at the orderly.\n\n\"Thyroid, normal.\" The orderly made an entry as he squeezed my stomach wall.\n\n\"Hernias, negative. Cough.\"\n\nThis last was an order to me and I coughed as his rubber-clad fingers probed deep.\n\nThere was more, but only the highlights stand out.\n\nThe urinalysis section where we stood in shivering ranks, each holding a recently filled paper cup. Our file slowly wending forward, on tiptoe for the floor was aslosh, to the white-clad, white-masked, booted and rubber-gloved orderly who dipped a disposable dropper into each cup, dropped a drop into a section of a large, sectioned chemical tray. Discarded the dropper into an overflowing container, eyed the chemical reaction. Shouted \"Negative, next!\" and carried on.\n\nOr the hemorrhoidal examination. Good taste forbids too graphic a description, but it did involve rows of youths bent over and clutching their ankles while a demonic physician crouched over as well and ran along behind the rows with a pointed flashlight.\n\nOr the injections, ahh, yes the injections. As this particular line crept forward I became aware that the youth in front of me was a bodybuilder of some sort. Among the pipestem arms and knocking knees his bronzed biceps and polished pects stood out as a monument to masculinity. He turned to me with a worried expression on the knotted muscles of his face.\n\n\"I don't like needles,\" he said.\n\n\"Who does,\" I agreed.\n\nNot nice at any time, positively threatening in mass attack. I watched, horrified, as I approached the point of no return. As each shivering body came into position an orderly on each side injected each upper arm. No sooner were the needles hurled aside than the victim was pushed in the back by the uniformed supervising brute. After tottering a few paces forward two more injections were made. Arms curled with pain the subject leaned on the nearby counter. Where he was vaccinated. Very efficient.\n\nToo efficient for the weightlifter. As he stepped into position his eyes rolled up and he slumped unconscious to the floor. This, however, was no obstacle to military efficiency. Two needles flashed, two injections were made. The sergeant seized him by the feet and dragged him forward where, after receiving the rest of his injections, he was rolled aside to recover. I gritted my teeth, tried stoically to accept the puncturing barrage, and sighed.\n\nAt some point the mass medical examination ended with a final assault on whatever shards of personal dignity the victims might still have left. Still nude, still clutching our plastic bags in our left hands, our thickening folders in our right, we shuffled forward in yet one more line. A row of numbered desks stretched across the width of the room, very much like the reception hall of an airport. Behind each desk sat a dark-suited gent. When it was my turn the sergeant-herdsman glanced over his shoulder and stabbed a stumpy figure at me.\n\n\"You, haul it to number thirteen.\"\n\nThe man behind the desk wore thick-framed glasses, as did all of the others I noticed. Perhaps our eyes were going to be examined and this was what we would be like if we failed. My folder was seized yet one more time, another printed sheet inserted\u2014and I found tiny red eyes glaring at me through the thick lenses.\n\n\"Do you like girls, Jak?\"\n\nThe question was completely unexpected. Yet it prompted a sweet vision of Bibs that obscured the medical mockery around me.\n\n\"You bet I like girls,\" was my instant response. An entry was made.\n\n\"Do you like boys?\"\n\n\"Some of my best friends are boys.\" I began to have a glimmering of what this simpleton was up to.\n\n\"Are they?\" Slash of pencil. Then, \"Tell me about your first homosexual experience.\"\n\nMy jaw fell with disbelief. \"I can't believe that I'm hearing this. You are doing a psychiatric examination from a checklist?\"\n\n\"Don't give me any cagal, kid,\" he snarled. \"Just answer the question.\"\n\n\"Your medical degree should be taken away for incompetence\u2014if you ever had one. You're probably not a shrink at all, just a timeserver dressed like one.\"\n\n\"Sergeant!\" he shouted in a cracked voice, his skin flushing. There was a thunder of feet behind me. \"This draftee is refusing to cooperate.\"\n\nSharp pain slashed the backs of my bare legs and I Yowed! and jumped aside. The sergeant raised the thin cane again and licked his lips.\n\n\"That will do for the moment,\" my examiner said. \"If my questions are answered correctly.\"\n\n\"Yes, sir,\" I said, snapping to attention. \"No need to repeat the question. My first experience of that kind was at the age of twelve when, with the aid of large rubber bands, I and fourteen other boys...\"\n\nI continued on in this vein while he scribbled happily and the sergeant muttered with frustration and waddled away. When the form had been completed with the last work of fiction, I was released and ordered on to join the others. It was back to the elevators again, jammed inside in nude groups of forty. The doors closed for the descent. The doors opened.\n\nAt what was obviously the wrong floor. Before our horrified eyes there was displayed a vista of desks and typewriters. With a young lady laboring away at each of them. There was a fluttering sound as all of the folders were swung forward over the vitals. The air temperature rose as everyone turned bright red. All we could do was stand there in carmined embarrassment, listening to the endless rattle of typewriter keys, waiting for heads to turn, gentle female eyes to peer our way. After about fourteen and a half years the doors slowly closed again.\n\nThere were no females present when the doors opened this time, just the now-familiar form of another brutish sergeant. I wondered what twisted gene in the population had produced so many thick-necked, narrow-browed, potbellied sadomasochists.\n\n\"Out,\" this one bellowed. \"Out, out, groups of ten, first ten through that door. Next ten next door. Not eleven! Can't you count, cagal-head!\" Followed by a yipe of pain as discipline was enforced yet again. My ten victims shuffled into a brightly lit room and were ordered into line. We faced a white wall that was hung with a repulsive puce-green flag distastefully decorated with a black hammer. An officer with little golden bars on his shoulder strutted in and stood before the flag.\n\n\"This is a very important occasion,\" he said in a voice heavy with importance. And, \"You young men, the fittest in the land, have been chosen as volunteers by your local draft boards to defend this country we love against the evil powers abroad that seek to strip away our freedoms. Now the solemn moment that you all have been waiting for has arrived. You entered this room as fun-loving youths. You will leave it as dedicated soldiers. You will now be sworn in as loyal members of the army. Raise your right hands and repeat after me...\"\n\n\"I don't want to!\"\n\n\"You have that choice,\" the officer said grimly. \"This is a free country and you are all volunteers. You may take the oath. Or if you choose not to, which is your right, you may leave by the small door behind me which leads to the federal prison where you will begin your thirty-year sentence for neglect of democratic duties.\"\n\n\"My hand's up,\" the same voice wailed.\n\n\"You will all repeat after me. I, insert your own name, of my own free will...\"\n\n\"I, insert your own name, of my own free will.\"\n\n\"We will do it again, and we will do it correctly, and if we don't get it right next time, there is going to be trouble.\"\n\nWe did it again, and correctly. Repeating what he said and trying not to hear what we were saying.\n\n\"To serve loyally... to show respect to all of the senior officers... death if I show disloyalty... death if I should desert... death if I sleep on duty...\" and so on to the very end, which was \"I do swear this in the name of my mother and father and the deity of my choice.\"\n\n\"Hands down, congratulations, you are all now soldiers and subject to military law. Your first order is that each of you will volunteer voluntarily a liter of blood since there has been a sudden call for transfusions. Dismissed.\"\n\nWeak with hunger and fatigue, dizzy from loss of blood, cold noodle soup still sitting leadenly in the stomach, we reached the end of the line. We hoped.\n\n\"Fall in. Move it along. You will each be issued with a disposable uniform which you will not dispose of until ordered. You will don the uniforms and proceed up these stairs to the roof of this building where transportation is waiting to take you to Camp Slimmarco, where your training will begin. You will turn in your folders before you receive your uniforms. You will each receive an identity disc with your name and service number on it. These discs are grooved across the center so they may be broken in half. Do not break them in half because that is a military crime and will be punished.\"\n\n\"Why make them to break in half if you don't break them in half?\" I muttered aloud. The youth beside me rolled his eyes and whispered.\n\n\"Because when you're dead they break them in half and send on half to death registrations and put the other half in your mouth.\"\n\nWhy was it that as I shuffled forward to get my uniform I had a very strong metallic taste in my mouth?\nChapter 8\n\nUnder any other circumstances I would have enjoyed the ride in this unusual airship. It was shaped like a large cigar and undoubtedly contained light gas of some kind. Slung beneath the lifting body was a metal cabin tastefully decorated outside with a frieze of skulls and bones. Ducted fans on the cabin were angled to force it aloft and forward: the view from the window must have been fascinating. But the windows that we had glimpsed from the outside were all forward in the pilot's compartment, while we draftees were jammed into a windowless metal chamber. The seats were made from molded plastic surfaced with uneven bumps and hideously uncomfortable\u2014but at least they were seats. I dropped into one and sighed with relief. In all the hours at the reception center the only time we had been off our feet was during the bloodletting. The plastic was cool through the thin paper fabric of the purple disposable uniform, the deck hard through the cardboard soles fastened at the end of its legs. The only pocket in this hideous garment was a pouch at the front into which we had shoved our bags of personal possessions so that we all resembled demented purple marsupials. I felt depressed. But at least I had company. We were all depressed.\n\n\"I never been away from home before,\" the recruit to my right sniveled, then sniffed and wiped his damp nose on his sleeve.\n\n\"Well I have,\" I said in my heartiest, most jovial tones. Not that I felt either hearty or jovial, but bucking up his spirits might help mine as well. \"And it is a lot better than home.\"\n\n\"Food will be rotten,\" he whined self-indulgently. \"Nobody can cook like my Mom. She makes the best cepkukoj in the whole world.\"\n\nOnion cakes? What sort of bizarre diet had this stripling enjoyed? \"Put that all behind you,\" I chirped. \"If the army bakes cepkukoj they will be foul, count on that. But think of the other pleasures. Plenty of exercise, fresh air\u2014and you can curse all the time, drink alcohol and talk smutty about girls!\"\n\nHe blushed ardently, his splayed ears glowing like banners. \"I wouldn't talk about girls! And I know how to drink. Me and Jojo went behind the barn once and drank beer and cursed and threw up.\"\n\n\"Whee...\" I sighed and was saved from future futile conversation by the appearance of a sergeant. He slammed open the door from the front cabin and roared his command.\n\n\"Alright you kretenoj\u2014on your feet!\"\n\nHe assured instant obedience by hitting a button on the wall that collapsed our seats. There were screams and moans of pain, writhing purple confusion on the deck as the recruits fell on top of each other. I was the only one standing and I caught the full force of the sergeant's sizzling glare.\n\n\"What are you\u2014a wise guy or sometin'?\"\n\n\"No, sir! Just obeying orders, sir!\" Saying this I leaped into the air slapping my arms to my sides, stamping my feet heavily as I landed, then delivered a snappy salute\u2014so snappy I almost put my eye out. The sergeant's eyes bulged in return at this display before he was lost from sight by the rising, milling bodies.\n\n\"Quiet! Attention! Hands at sides, feet together, stomachs in, chests out, chins back, eyes forward\u2014and stop breathing!\"\n\nThe purple ranks swayed and writhed into this absurd military stance, then were still. Silence descended as the sergeant glared around with dark suspicion.\n\n\"Did I hear someone breathe? No breathing until I tell you to. The first cagalhead who breathes gets my fist where it will do the most good.\"\n\nThe silence lengthened. Purple figures stirred as incipient asphyxiation took hold. One recruit moaned and fell to the deck; I breathed silently through my nostrils. There was a gasp as one of the lads could hold out no longer. The sergeant surged forward and the spot where a fist will do the most good turned out to be the pit of the stomach. The victim screamed and fell and all the others gasped in life-giving air.\n\n\"That was a little lesson!\" the sergeant screeched. \"Did you get the message?\"\n\n\"Yes,\" I muttered under my breath. \"You're a sado-masochist.\"\n\n\"The lesson is that I give the orders, you obey them\u2014or you get stomped.\" Having delivered this repulsive communication his face writhed, his lips pulled back to reveal yellowed teeth; it took a long moment for me to realize this was supposed to be a smile.\n\n\"Sit down men, make yourselves comfortable.\" On the steel deck? The seats were still stowed. I sat with the rest while the sergeant amicably patted the roll of fat that hung over his belt. \"My name is Klutz, Drill-sergeant Klutz. But you will not address me by my name, which is for the use of those of equal rank or higher. You will call me sergeant, sir, or master. You will be humble, obedient, reverent and quiet. If you are not you will be punished. I will not tell you what the punishment will be because I have eaten recently and do not wish to upset my stomach.\"\n\nA stir of fear passed through the audience at the thought of what might possibly upset that massive gut.\n\n\"One punishment is usually enough to break the spirit of even the most reluctant recruit. However, occasionally, a recruit will need a second punishment. Still more rarely a hardened resister will require a third punishment. But there is no third punishment. Would you like to know why there is no third punishment?\"\n\nThe red eyes glared down and we all wished that we were someplace, anyplace, else at this moment.\n\n\"Since you are too dim to ask why, I will tell you. Third time is out. Third time is being stuffed, kicking and screaming and begging for your mommy, into the dehydration chamber where ninety-nine point nine nine percent of all your precious bodily fluids will be removed with a dry whishing sound. Do you know what you will look like then? You will look like this!\"\n\nHe reached into his pocket and took out a tiny dehydrated figure of a recruit in a tiny dehydrated uniform, the features on its tiny face fixed forever in lines of terror. Moans of fear sighed from the soldiers and there were a number of thuds as the weakest dropped unconscious. Sergeant Klutz smiled.\n\n\"Yes, you will look just like this. Your tiny dry body will then be hung on the barracks bulletin board for a month as a warning to the others. After that your body will be put in a padded mailing envelope and sent to your parents, along with a toy shovel to assist in burial. Now\u2014are there any questions?\"\n\n\"Please, sir,\" a quavering voice asked. \"Is the dehydration process instant and painless or drawn-out and terrible?\"\n\n\"Good question. After your first day in the army\u2014do you have any doubt which it will be?\"\n\nMore moans and unconscious thuds followed. The sergeant nodded approval. \"Alright. Let me tell you what happens next. We are going to the RTCS at MMB. That means the Recruit Training Camp Slimmarco at Mortstertoro Military Base. You will take your basic training. This training will turn you from feeble civilian wimps into sturdy, loyal, reverent soldiers. Some of you will wash out of basic training and will be buried with full military honors. Remember that. There is no way back. You will become good soldiers or you will become dead. You will understand that the military is hard but fair.\"\n\n\"What's fair about it?\" a recruit gasped and the sergeant kicked him in the head.\n\n\"What is fair is that you all have an equal chance. You can get through basic or wash out. Now I will tell you something.\" He leaned forward and breathed out a blast of breath so foul that the nearest draftees dropped unconscious. There was no humor in his smile now. \"The truth is that I want you to wash out. I will do everything I can to make you wash out. Every recruit sent home in a wheelchair or a box saves the government money and lowers taxes. I want you to wash out now instead of in combat after years of expensive training. Do we understand each other?\"\n\nIf silence means assent, we certainly did. I admired the single-minded clarity of the technique. I did not like the military, but I was beginning to understand it.\n\n\"Any questions?\"\n\nMy stomach rumbled loudly in the silence and the words popped from my mouth.\n\n\"Yes, sir. When do we eat?\"\n\n\"You got a strong stomach, recruit. Most here are too sickened by military truth to eat.\"\n\n\"Only thinking of my military duty, sir. I must eat to be strong to be a good soldier.\"\n\nHe shuffled this around about in his dim brain, little piggy eyes glaring at me the while. Finally the projecting jaw nodded into the rolls of fat beneath the chin.\n\n\"Right. You just volunteered to get the rations. Through that door in the aft bulkhead. Move.\"\n\nI moved. And thought. Bad news: I was in the army and liked nothing about it. Good news: we were going to Mortstertoro base where Bibs had last seen Captain Garth-Zenar-Zenor or whatever his name was. He was on top of my revenge list\u2014but right now I was plugging away at the top of my survival list. Garth would have to wait. I opened the door which revealed a small closet containing a single box. It was labeled YUK-E COMBAT RATIONS. This had to be it. But when I lifted the box it seemed suspiciously light to feed this shipload of incipient soldiers.\n\n\"Pass them out, kreteno, don't admire the box,\" the sergeant growled, and I hurried to obey. The Yuk-E rations did appear pretty yuky. Gray bricks sealed in plastic covers. I went among my purple peers and each of them grabbed one out, fondling the bricks with some suspicion.\n\n\"These rations will sustain life for one entire day,\" the rasping voice informed us. \"Each contains necessary vitamins, minerals, protein and saltpeter that the body needs or the army wants you to have. They are opened by inserting your thumbnail into the groove labeled thumbnail-here. The covering will fall away intact and you will preserve it intact. You will eat your ration. When you are finished you will go to the wall here and to the water tap at this position and you will drink from the plastic cover. You will drink quickly because one minute after being moistened the cover will lose its rigidity and will shrink. You will then roll up the cover and save it for display at inspection because it will now be transformed into a government issue contraceptive which you will not be able to use for a very long time, if ever, but which you will still be responsible for. Now\u2014eat!\"\n\nI ate. Or tried to. The ration had the consistency of baked clay but not half as much flavor. I chewed and gagged and swallowed and managed to choke it all down before rushing to the water spigot. I filled the plastic cover and drank quickly and refilled it, emptying it just as it went limp and flacid. I sighed and rolled it up and stowed it in my marsupial pocket and made room for the next victim at the tap.\n\nWhile we had been gnawing our food the collapsed seats had snapped back into position. I eased myself carefully into the nearest, but it did not give way. It appeared impossible, but the combination of food and near-terminal exhaustion worked their unsubtle magic and I crashed. I could hear myself snoring even before I fell asleep.\n\nThe bliss of unconsciousness ended just as I might have expected; the seats fell away and dropped us into a writhing, moaning mass on the deck. We stumbled groggily to our feet under the verbal lashing of the sergeant and were trying to stand in a military posture as the deck vibrated beneath our feet and became still.\n\n\"Welcome to the first day of the rest of your new life,\" the sergeant chortled, and wails of anguish followed his words. The exit sprang open, admitting a chill and dusty blast, and we stumbled out wearily to see our new home.\n\nIt was not very impressive. One of the red and pallid suns was just setting into the cloud of dust on the horizon. I could tell by the thin and chill air that the base had been built at some altitude, a high plateau perhaps. Which guaranteed good flying weather and maximum discomfort for the troops. The ground trembled as a deep-spacer took off in the distance, its exhaust blast brighter than the setting sun. The sergeant snarled us into a ragged formation and we shivered in the downblast of our departing airship. He waved a clipboard in our direction.\n\n\"I will now call the roll. You will be called by your military name and will forget that you ever had any other. Your military name is your given name followed by the first four numbers of your serial number. When your name is called you will enter the barracks behind me and proceed to your assigned bunk and await further instructions. Gordo7590\u2014bunk one...\"\n\nI looked crosseyed at my dogtag until I could make out the number. Then stared numbly at the mud-colored barracks until the voice of our master called out Jak5138. With dragging feet I passed through the doorway over which was inscribed THROUGH THIS PORTAL PASS THE BEST DAMNED SOLDIERS IN THE WORLD. Who, as the expression goes, was kidding whom?\n\nThe floor was stone, still damp from the last scrubbing. The walls concrete, clean and still wet. I let my horrified gaze move up to the ceiling and, yes, it was damp as well, the light bulbs still dripping. How this maniacal cleansing was carried out I had no idea\u2014though I was certain that I would find out far too soon.\n\nMy bunk was, naturally, the top one in a tier of three. It was strung with wire netting, though a bulky roll at its head hinted at softer pleasures.\n\n\"Welcome to your new home,\" the sergeant grated with false jollity as we drew our fatigued bodies up into an imitation of attention. \"Note how your bedding roll is stowed when you unroll it, because it will be rolled in that stowed position at all times except when you're sleeping\u2014which will be the minimum amount of time needed to stay alive. Or less. Your footlockers are imbedded in the floor between the bunks and are opened and closed by me with this master switch.\"\n\nHe touched a stud on his belt and there was a grating sound as the mini-graves opened up in the floor. One recruit, who was standing in the wrong position, screamed as he fell into his.\n\n\"Lights out in fifteen minutes. Bedding to be unrolled but not utilized before that time. Before retiring you will watch an orientation film that will acquaint you with tomorrow's orders of the day. You will watch and listen with full attention, after which you will retire and pray to the deity or deities of your choice and cry yourselves to sleep thinking about your mommies. Dismissed.\"\n\nDismissed. The door slammed behind our striped overseer and we were alone. Dismissed was the right word for it. Dismissed from the warmth and the light of the real world, sent to this gray military hell not of our choosing. Why is mankind so inhuman to its own species? If you were caught treating a horse in this manner you would probably be put in jail, or shot. Rustling cut the silence as we opened our bedrolls. To reveal to each of us a thin mattress and even thinner blanket. A pneumatic pillow as well that could only be inflated with lusty puffing which, I was sure, would go flat by morning. While we were unrolling and blowing, TV screens dropped down silently behind us in the passageway between the bunks. Brassy military music blared and the image of an officer with a severe speech impediment appeared and began to read out totally incomprehensible instructions which we all ignored. I dumped the contents of my marsupial pocket into the subterranean footlocker and climbed and crawled, still dressed, into the bunk. My eyes blurred with fatigue as the voice droned on and I was nine-tenths asleep when a blast of light and sound jerked me awake. A grim military figure in black uniform glared angrily from the screen.\n\n\"Attention,\" it said. \"This program has been interrupted, as have all programs throughout Nevenkebla on all stations, to bring you the following important announcement.\" He scowled at the sheet of paper he held and shook it angrily.\n\n\"A dangerous spy is at large in our country tonight. It is known that he entered the harbor of Marhaveno yesterday morning disguised as a laborer on one of the ships from Brastyr. A search was made of the harbor but he was not found. The search was extended today and it was discovered that the spy entered a pleasure vessel in the adjoining harbor and stole a number of items.\"\n\nA deathly chill stirred the hairs on the nape of my neck as he held up a bundle of clothes.\n\n\"These were found buried in the sand and have been identified as the clothing worn by the spy. The entire area has been sealed, curfew declared and every building is now being carefully searched. The public is ordered to be on the lookout for this man. He may still be wearing these items of clothing that he stole. If you have seen anyone dressed like this notify the police or security forces at once.\"\n\nHis image vanished and was replaced by a carefully done computer simulation of the clothing I had borrowed from the boat. These rotated slowly in space\u2014then appeared on a man's figure which the computer strolled about the screen. The face was a blank but I knew all too well what face would soon appear there.\n\nHow long would it take them to identify me, to track me down, discover that I was now in the army, to follow me here?\n\nThere was a grating thud as the barracks door locked and the lights went out. The chill spread down my body and my heart thudded with panic and I stared, sightless and horrified into the darkness.\n\nHow long?\nChapter 9\n\nI would like to say that it was nerves of steel and fierce self-control that enabled me to fall asleep, after hearing the announcement that the entire country was turned out and searching for me. But that would be a lie. Not that I mind telling a lie or two, white lies really, to further myself in this universe. After all a disguise is a lie and continuous lying, sincere lying, is the measure of a good disguise. That went with the job. But one must not lie to oneself. No matter how distasteful the truth it must be faced and accepted. So, no lies; I fell asleep because I was horizontal in the dark, fairly warm and totally exhausted. Panic ran way behind exhaustion in the sleepy-time race. I slept, hard and enthusiastically, and awoke in the darkness only when a strange noise cut through my serious sack time.\n\nIt was a distant rustle, like waves on the beach\u2014or leaves blowing in the wind. No, not that, but something else equally familiar. An amplified sound I thought numbly, like an ancient and worn recording being played, just the background scratching without the recording itself.\n\nTheory was proved correct an instant later as a blurred and distorted recording of a bugle thundered through the barracks just as all the lights came on. The barracks door crashed open and, as though summoned from some dark hell by this hellish sound and light, the sergeant entered screaming at the top of his lungs.\n\n\"Get out and get under! Off your bunks and on your feet! Roll bedding! Dip into your footlockers! Remove shaving gear! Then on the double to the latrine! You're late, you're late! Barracks will be washed in twenty seconds precisely! Move it\u2014move it\u2014move it!\"\n\nWe moved it, but we really didn't have enough time. I fought my way through the latrine door with the other frenzied purple figures just as the footlockers slammed shut and the barracks wash-heads let go. At that precise instant the sergeant stepped backward and slammed the door. From all sides torrents of cold water gushed forth, catching at least half of the recruits still on the run. They followed us into the latrine, soaked and shivering, their disposable uniforms beginning to dispose in long rents and tears. Crying and sniveling they pushed forward like sheep. Sheep struggling for survival. There was a limited number of sanitary facilities and all were in use. I forced my way through the mob until I could glimpse my face in the corner of a distorted mirror, almost did not recognize myself with the dark-circled eyes and pallid skin. But there was no time to get organized, to take stock, to think coherently. At some lower level I realized that it had all been planned this way, to keep the recruits off-balance, insecure, frightened\u2014open for brainwashing or destruction. This realization percolated up to a slightly more conscious level and with it a growing anger.\n\nJimmy diGriz does not destruct! I was going to beat them at their own game, until I beat it out of here. It didn't matter that the entire country was looking for me\u2014until they tracked me to this military cesspit all I had to do was survive. And survive I would! The supersonic razor screeched in my brain as it blasted my overnight whiskers free. Then, while the automated toothbrush crawled around inside my mouth, I managed to get a hand under a running faucet, scrubbed my face clean, ignored the air-dryer and pelted back to my bunk over the puddled floor. I stowed my kit away just as the footlocker flew open, then spun about as Sergeant Klutz popped through the door again.\n\n\"Fall out for rollcall!\" he bellowed as I rushed by him into the night. I snapped to attention under the single glaring light as he turned and approached me with grim suspicion.\n\n\"Are you some kind of joker or something?\" he shouted, his face so close to mine that his spittle dotted my skin.\n\n\"No, sir! I'm raring to go, sir. My daddy was a soldier and my grandaddy and they told me that the best thing to be was a soldier and the highest rank in the army was sergeant! That's why I'm here.\" I stopped shouting and leaned forward and whispered. \"Don't tell the others, sir, they'll only sneer. But I wasn't drafted\u2014I volunteered.\"\n\nHe was silent and I risked a quick look at his face. Could it be? Was it, there, a drop of liquid in the corner of one eye? Had my tissue of lies touched some residual spot of emotion buried deep with the alcohol-sodden, sadistic flesh of his repulsive body? I couldn't be sure. At least he did not strike me down on the spot, but turned on his heel and rushed into the barracks to boot out the stragglers.\n\nAs the moaning victims stumbled into line I put some thought to my future. What should I do? Nothing, came the quick answer. Until you are tracked down, Jim, stay invisible in the ranks. And learn all that you can about this military jungle. Watch and learn and keep your eyes open. The more you understand about this operation the safer you will be. Then, when you run, it will be plan not panic that guides you. Good advice. Hard on the nerves to follow, but good advice nevertheless.\n\nAfter repeated mumbled mistakes, mispronunciation of names\u2014is it really possible to mispronounce Bil?\u2014the sergeant finished stumbling and muttering his way through the rollcall and led the way to the mess hall. As we approached it, and the smells of real food washed over us, the splattering of saliva on the pavement sounded like rain. Other recruits stumbled up through the night and joined us in the long line leading into the warmth of this gustatory heaven. When I finally carried my heaped-high tray to the table I found it hard to believe. All right so maybe it was grundgeburgers with caramel sauce, but it was food, hot, solid food. I didn't eat it\u2014I insufflated it and went back for more. For one moment I actually thought that the army was not so bad after all. Then I instantly banished the thought.\n\nThey were feeding us because they wanted to keep us alive. The food was nasty and cheap\u2014but it would sustain life. So if we washed out it would not be because of the diet but because of our own intrinsic insufficiency or lack of will. If we got through basic training each of us would supply one hot and relatively willing body for the war machine. Nice thinking.\n\nI hated the bastards. And went back for thirds.\n\nBreakfast was followed by calisthenics\u2014to aid the digestion or destroy it. Sergeant Klutz double-timed us to a vast, wind-swept plain where other recruits were already being put through their paces by muscular instructors. Our new leader was waiting for us, steely eyed and musclebound, the spread of his shoulders so wide that his head was disproportionately small. Or maybe he just had a pinhead. Speculation about this vanished as his roar rattled the teeth in my jaw.\n\n\"What's this, what's this? You kretenoj are almost a minute late!\"\n\n\"Pigs, that what they is,\" our loyal sergeant said, taking a long black cigar from his pocket. \"Little trotters in the trough. Couldn't tear them away from their chow.\"\n\nSome recruits gasped at this outright lie, but the wiser of us were learning and stayed silent. The one thing that we could not expect was justice. We were late getting here because our porcine sergeant could not move any faster.\n\n\"Is that so?\" the instructor said, his beady eyes swiveling in his pinhead like glowing marbles. \"Then we will see if we cannot work some of that food off of these malingering cagal-kopfs. ON THE GROUND! Now\u2014we do fifty push-ups. Begin!\"\n\nThis seemed like a good idea since I usually did a hundred push-ups every morning to keep in shape. And the chill wind was blowing through the rents in our disposable uniforms. Five. I wondered when we would be issued with something more permanent. Fifteen.\n\nBy twenty there was plenty of wavering and grunting around me and I was warming up nicely. By thirty over half of the pipe-stemmed striplings had collapsed in the dust. Sergeant Klutz dropped cigar ashes on the nearest prostrate back. We continued. When we reached fifty just I and the muscular lad who hated injections were the only ones left. Pinhead glared at us.\n\n\"Another fifty,\" he snarled.\n\nThe weightlifter puffed on for twenty more before he groaned to a halt. I finished the course and got another glare and a snarl.\n\n\"Is that all, sir,\" I asked sweetly. \"Couldn't we do another fifty?\"\n\n\"On your feet!\" he screamed. \"Legs wide, arms extended, after me. One, two, three, four. And one more time...\"\n\nBy the time the exercises were finished we had worked up a good sweat, the sergeant had finished his cigar\u2014and two of the recruits were collapsed in the dust. One of them lay beside me, groaning and clutching his midriff. The sergeant strolled over and pushed him with his toe which elicited only some weak moans. Sergeant Klutz looked down with disgust and screamed his displeasure.\n\n\"Weaklings! Faggots! Momma's boys! We'll weed you out fast enough. Get these poofters out of my sight. Man to each side pick up the malingerers, bring them to the medic tent. Then fall back in. Move!\"\n\nI bent and seized one arm and lifted. I could see that the recruit on the other side was having difficulty so I shifted my grip to take most of the weight and heaved.\n\n\"Get his arm around your shoulder\u2014I'll do the carrying,\" I whispered.\n\n\"My... thanks,\" he said. \"I'm not in such great shape.\"\n\nHe was right, too. Thin and round-shouldered with dark circles under his eyes. And older than the others I noticed, in his mid-twenties at least.\n\n\"Morton's the name,\" he said.\n\n\"Jak. You look kind of old for the draft, Mort.\"\n\n\"Believe me, I am!\" he said with some warmth. \"I almost killed myself getting through university, keeping top of the class to keep out of the army. So what happens? I'm so overworked I get sick, miss the exams, wash out\u2014and end up here anyway. What do we do with this dropout?\"\n\n\"That tent there, I guess, where they're bringing the others.\"\n\nThe limp form hung between us, toes dragging in the dust.\n\n\"He doesn't look too good,\" Morton said, glancing at the pallid skin and hanging head.\n\n\"That's his problem. You have to look out for number one.\"\n\n\"I'm beginning to get that message. A crude communication but a highly effective one. Here we are.\"\n\n\"Drop him on the ground,\" a bored corporal said, not deigning to even look up from his illiterate comic book. When he touched the page little voices spoke out and there was a mini scream. I looked at the four other unconscious forms stretched out in the dirt.\n\n\"What about some medical treatment, corporal. He looks in a bad way.\"\n\n\"Tough cagal.\" He turned a page. \"If he comes to\u2014it's back to the drill field. Stays like that the medic will look at him when he gets here tonight.\"\n\n\"You're all heart.\"\n\n\"That's the way the kuketo crumbles. Now get the cagal out of here before I put you on report for cagaling off.\"\n\nWe got. \"Where do they get all these sadistic types from?\" I muttered.\n\n\"That could be you or I,\" Morton said grimly. \"A sick society breeds sickies. People do what they are ordered to do. It is easier that way. Our society lives on militarism, chauvinism and hatred. When those are the rules there will always be someone eager to do the dirty work.\"\n\nI rolled my eyes in his direction. \"They taught you that in school?\"\n\nHe smiled grimly and shook his head. \"The opposite, if anything. I was majoring in history, military history of course, so I was allowed to do research. But I like to read and the university library is a really old one and all the books are there if you know how to look, and how to crack some simple security codes. I looked and cracked and read\u2014and learned.\"\n\n\"I hope you learned to keep your mouth shut as well?\"\n\n\"Yes\u2014but not always.\"\n\n\"Make it always or you are in big trouble.\"\n\nSergeant Klutz was just leading our squad off the field and we fell in behind them. And marched to the supply building to get outfitted at last. I had heard that clothing came in only two sizes in the army and this was true. At least most of mine were too big so I could roll up the cuffs. In addition to clothing there were mess kits, webbing belts, canteens, sewing kits, assassination kits, foxhole diggers, backpacks, VD testers, bayonets, scrokets and more items of dubious or military nature. We staggered back to the barracks, dumped our possessions and hurried to our next assignment.\n\nWhich was something called Military Orientation.\n\n\"Having possessed our bodies they now seek to take over our minds,\" Morton whispered. \"Dirty minds in military bodies.\"\n\nHe was sure bright this Morton, but not bright enough to keep his mouth shut. I hissed him into silence as Sergeant Klutz glared in our direction.\n\n\"Talking is forbidden,\" he graveled. \"You are here to listen, and this here is Corporal Gow who will now tell you what you got to know.\"\n\nThis Gow was a smarmy type, all smooth pink skin, little ponce's moustache and fake grin. \"Now sergeant,\" he said, \"this is orientation, not orders. You men will become good soldiers by following orders. But good soldiers also should know the necessity of these orders. So get comfortable, guys. No chairs of course, this is the army. Just sit down on the nice clean concrete floor and give me your attention, if you please. Now\u2014can any of you tell me why you are here?\"\n\n\"We was drafted,\" a thick voice said.\n\n\"Yes, ha-ha, of course you were. But why is the draft necessary? Your teachers and your parents have let you down if this has not been made completely clear. So let me take this opportunity to remind you of some vital facts. You are here because a dangerous enemy is at our gates, has invaded our precious land, and it is your duty to defend our inalienable freedoms.\"\n\n\"This is the old cagal if I ever heard it,\" I muttered, and Morton nodded silent agreement.\n\n\"Did you say something, soldier?\" The corporal said, staring right at me; he had good ears.\n\n\"Just a question, sir. How could a broken-down, unindustrialized society like they got over there, how could they ever invade a modern, armed and equipped country like ours.\"\n\n\"That is a good question, soldier, and one that I am happy to answer. Those barbarians across the channel would pose no problem if they were not being armed and equipped by offworlders. Greedy, hungry strangers who see our rich land and want to take it for themselves. That is why you lads must go willingly to the service of your country.\"\n\nI was shocked at the magnitude of the lie, angered. But I struggled hard and followed my own advice and kept my mouth shut. But Morton didn't.\n\n\"But, sir, the Interplanetary League is a peaceful union of peaceful planets. War has been abolished...\"\n\n\"How do you know that?\" he snapped.\n\n\"Common knowledge,\" I said breaking in, hoping that Morton would shut up now. \"But you know that is the truth, don't you, sir?\"\n\n\"I know nothing of the sort and I wonder just who has been feeding you lies like that. After our orientation session I want to talk to you, soldier. You and that recruit next to you. This free country is fighting the interplanetary forces that wish to destroy us. No sacrifice is too great to defend that freedom, which is why I know that you will all do your duty, happily. And become good soldiers in a good team. Look to the good Sergeant Klutz as you would to your own father, for he is here to be your mentor and guide to your military life. Do what he says and you will grow strong and prosper and become first-class soldiers in the service of your country. But I know that you will at times find things confusing, even worrying, for this military experience is a new experience for all of you. That is when you must think of me. I am your counselor and guide. You can call upon me for advice and help at any time. I would like to be your friend, your very best friend. Now I am going to pass out these orientation pamphlets and you have ten minutes to read them. We will then have a question and answer session to help acquaint you with the details. While you are doing that I am going to have a nice friendly chat with these recruits who appear to be badly misinformed about the political realities of our land.\" The finger was pointing at me, then at Morton. \"That's right, you two. We will step outside, get a bit of sun, have some good old jaw-jaw.\"\n\nWe rose with great reluctance\u2014but had no choice at all. All eyes were upon us when we went to the door that Gow was holding open. I could feel the heat from Sergeant Klutz's burning glare as I passed him. Corporal Gow closed the door behind us and turned to face us. His smile as insincere as ever.\n\n\"Kind of warm now since the sun came out.\"\n\n\"Sure is. Feels nice.\"\n\n\"Where did you get that subversive cagal you were spouting in there? You first.\" He pointed at me.\n\n\"I sort of, well, sort of heard it somewhere.\"\n\nHe smiled happily and stabbed a stubby finger at Morton.\n\n\"I knew it. You have been listening to the illegal radio, haven't you? Both of you. That is the only possible place you could have heard such outrageous lies.\"\n\n\"Not really,\" Morton said. \"Facts are facts and I happen to be right.\"\n\nHe was digging his own grave with his jaw. I broke in; this radio dodge sounded like a possible out. If there were such a thing we might just wriggle from under this creepo's thumb. I lowered my eyes and twisted my toe in the dirt.\n\n\"Gee, corporal, I don't know how to say this. I was going to lie or something\u2014but you're too smart for me. It was, you're right, the radio...\"\n\n\"I knew it! They pump that poison down from their satellite, too many frequencies to jam, too defended to shoot down. Lies!\"\n\n\"I just did it that once. I knew I shouldn't have, but it was a dare. And it sounded so true\u2014that's why I spoke out like that.\"\n\n\"I'm glad you did, recruit. And I imagine that you did the same?\" Morton did not rise to the bait but the corporal took silence for assent. \"I think that you did. But at least it shows the poison didn't take, that you two wanted to talk about it. The devil always has the best tunes. But you must turn away from the siren song of such slimy untruths and listen to the authorities who know far better than you do.\" He smiled warmly upon us and I grinned with wide insincerity.\n\n\"Oh I will, sir,\" I said quickly before Morton could open his mouth again. \"I will. Now that you have told me this, and didn't punish us or anything...\"\n\n\"Did I say that?\" The warm grin suddenly had a cold and nasty edge to it. \"You'll get your punishment. If you were civilians you would each get a year at hard labor. But you are in the army now\u2014so the punishment should be worse. It has been nice talking to you, recruits. Now get back inside for the rest of this orientation session. That will give you plenty of time to contemplate your crimes and their inevitable punishment. In the future, if you have a future, you will not contradict me or any other officer.\"\n\nHe waved us inside ahead of him: we went like sheep to the slaughter. I whispered to Morton, \"Is it true what he said about the radio broadcasts?\"\n\n\"Of course. Haven't you ever listened? Pretty boring stuff for the most part. Heavy on propaganda and low on content. But it doesn't matter that you admitted listening. He was out to get us no matter what he said. Military justice!\"\n\n\"Do we just stay here and wait?!\"\n\n\"Where's to run,\" he said, with utmost gloom.\n\nWhere indeed? There was no place to flee to.\n\nSergeant Klutz glared his best glare at us and we shut up. I sank to the floor with a sigh. Wondering just what possible punishment the military could dream up that could be worse than recruit training. I had the sinking feeling that I would find out soon enough.\nChapter 10\n\nA distant buzzer sounded like a stifled eructation and Sergeant Klutz's eyes came back into focus and the expression of dull vacuity vanished to be replaced by his normal sneer of anger.\n\n\"On your feet you cagal-kopfs! You had a whole hour of cagaling off and you will now pay for it. Double time! The next session will be small arm instruction and short arm inspection. Move it!\"\n\n\"I'm holding onto these two,\" Gow said, separating us out from the others. \"I'm putting them on report for spreading sedition.\"\n\nKlutz nodded happily and slashed a line through our names on his roster sheet. \"Suits me, Gow. As long as I got the roll call right you can eat them for breakfast for all I care.\"\n\nThe door closed and Gow and I stood there eyeball to eyeball. Morton slumped to one side, drooping with apathy. I was beginning to get angry. Corporal Gow took out his notebook and pencil and pointed at me.\n\n\"What is your name soldier?\"\n\n\"ScrooU2.\"\n\n\"That is your military name, Scroo, and not a complete one at that. I would like your entire name.\"\n\n\"I'm from Pensildelphia, corporal, and we were taught never to give our names to strangers.\"\n\nHis eyes narrowed with hatred. \"Are you trying to make fun of me soldier?\"\n\n\"That would be impossible, sir. You are a walking joke as it is. Selling lies to the peasantry. You know as well as I do that the only threat to this country is the military that control it. This is a military state kept in operation only for the benefit of the military.\"\n\nMorton gasped and tried to wave me to silence. I was too angry for that now. This cagaling corporal had gotten under my skin. He smiled coldly and reached for the telephone.\n\n\"If you won't tell me your name the Military Police will find it out quickly enough. And you are wrong about only the military benefiting from a military state. You are forgetting the industrial corporations that profit from the military contracts. One cannot exist without the other. They are mutually interdependent.\"\n\nHe said this calmly, smiling, and shocked me into silence. \"But...\" I finally mumbled as he dialed the phone. \"If you know that\u2014why are you selling that line of old cagal to the troops?\"\n\n\"For the simple reason that I am the scion of one of those industrial families and quite happy with the situation as it is. I fulfill my military obligations by selling this line of old cagal, as you so quaintly put it, and in a few months will return to the life of luxury that I greatly enjoy. The number is engaged. I've enjoyed our talk as well, and in return for the pleasure I derived from the novelty of our conversation I wish to give you a gift.\"\n\nHe put the phone down turned and opened a drawer in the desk behind him and I was numb enough to let him do it. When the coin finally dropped it was too late. As I jumped forward he spun about with a large weapon in his hand, aimed and steady.\n\n\"I wouldn't, if I were you. I hunt, you know, and I am a first-class shot. I would also have no slightest compunction in shooting you. In the back if needs be,\" he added as I turned away. I turned around again and smiled.\n\n\"Well done, corporal. Intelligence was concerned about the quality of your orientation talks and I was sent here to, you know, try to irritate you. And I promise not to repeat your remarks about the industrial-military complex. I come from a poor family so I do not enjoy any of your advantages.\"\n\n\"Is that true?\" Morton gasped.\n\n\"It is\u2014and you are under arrest. There, one traitor caught, Gow, so some good has come of our conversation.\"\n\nHis eyes narrowed but the gun never moved. \"Do you expect me to believe that?\"\n\n\"No. But I can show you my identification.\" I smiled and reached into the empty back pocket of my new uniform.\n\nHe might have been a good shot when it came to blasting helpless animals or paper targets, but he had no combat experience. For a single instant his eyes looked down toward my moving hand. Which was all the time I needed. My other hand was already chopping the inside of his wrist, moving the gun aside. It hissed once and something slammed into the wall behind me. Morton screeched with fright and jumped aside. Before Gow could fire again my knee came up into his stomach.\n\nThe gun dropped to the floor and he dropped beside it. I took a deep and shuddering breath and let it out with a sigh.\n\n\"Well done, Jim,\" I said, and reached over my shoulder and patted myself on the back. \"All the reflexes working fine.\"\n\nMorton bulged his eyes at me, then down at the silent form of the corporal. \"What's happening...?\" he gurgled in confusion.\n\n\"Exactly what you see. I've rendered the corporal unconscious before he did us bodily harm. And you are not under arrest since that was just a ruse. So now, quickly before someone comes, push that desk up against the entrance since you can see that the door has no lock.\"\n\nI bent and retrieved the weapon in case the scion of millions came to earlier than planned. And what was I going to do with the poor little rich boy? I looked down at his recumbent form and inspiration struck.\n\n\"You are a genius,\" I bragged aloud. \"You deserve another pat, which you will get later, because now speed is of the essence.\" I bent and began to unbutton his uniform. \"The uniform, that is the key, the uniform. They will be looking for a ragged recruit in baggy fatigues. Not a spiffy corporal in tailor-mades. You have earned this promotion, Jim. Go to the head of the class.\"\n\nI tore off his shoes and pulled his trousers free\u2014and whistled. His underpants were woven of gold thread. Rich is as rich does. It was chance, pure chance, that he was a little overweight from a lifetime of good living. My muscles took the place of his fat and the uniform could have been made for me. Except the shoes; he had very tiny feet. My boots would have to do. I emptied his pockets and found, in addition to a great deal of money and a container of sinister looking black cigarettes, a small pocketknife. This worked admirably in cutting my discarded clothing into strips with which I bound the corporal securely, wadded more of the cloth into a gag. He was breathing easily through his nose so my conscience was clear that he would not die of suffocation.\n\n\"Are you going to kill him?\" Morton asked.\n\n\"No, but I want him quiet until I put the next part of the plan into operation.\" I'm glad that Morton didn't ask what that was since I didn't know yet. There were no closets in the room so the corporal could not be stuffed out of sight. The desk\u2014that was it!\n\n\"Morton,\" I ordered. \"Stand with your back to the door and think like a lock. If anyone tries to open it lean hard against it.\"\n\nWhile he leaned and thought lockish I dragged the desk back into position and wedged the bound corporal under it. By reflex I went through the desk drawers, which were all empty except the top one which had a folder of papers. I tucked these under my arm. Then I stepped back and examined my handiwork. Admirable. The corporal was well out of sight. Anyone who glanced into the room would think it empty.\n\n\"Now\u2014what next?\" I said cheerily. Then felt the smile slip from my face.\n\n\"Yes!\" Morton agreed eagerly. \"What happens next?\"\n\nI shook myself, took a brace and tried to think positively. \"For one thing\u2014there is no going back. So let us seek out a way forward. When they find the corporal they will find out our names quickly enough. By which time we must have new names. Which means we go to the personnel section and make a few changes.\"\n\nMorton was blinking very rapidly now. \"Jak, old friend, don't you feel well? I don't understand a word that you are saying.\"\n\n\"Doesn't matter\u2014as long as I do.\" I unloaded the gun, put the power charge in my pocket and the empty weapon back in the drawer. \"March ahead of me, do as I command. Go! As soon as you have opened the door a crack to see if the coast is clear.\"\n\nIt was. We marched out, stamping and striding in a very military fashion, me clutching my sheaf of papers, Morton hopefully clutching to his few remaining shards of sanity. One, two, one, two. Around the corner and almost into the arms of a red-capped military policeman.\n\n\"Squad halt! Stand at ease!\" I screamed. Morton halted with a decided sway and shudder, showing the whites of his eyes as he rolled them toward the MP. \"Eyes front!\" I shrieked. \"I gave no orders for you to move your eyes.\"\n\nThe MP, wise in military ways, paid us absolutely no attention until I called out to him. \"Just hold it, there, private.\"\n\n\"Me, corporal?\" he asked, stopping and turning.\n\n\"You are the only thing moving that I can see. Your pocket is unbuttoned. But this is my generous day. Just point us toward the Personnel Building and keep moving.\"\n\n\"Straight ahead, right on the company street, past the bandstand, left at the torture chamber and there you are.\" He scurried away, groping at his shirt pockets to find the open one. Morton was shivering and sweating and I patted him on the back.\n\n\"Relax, my friend. As long as you have the rank you can do what you want in the army. Ready to go on?\"\n\nHe nodded and stumbled forward. I marched after him, shouting commands at the corners, marking time, being noisy, obnoxious and abusive so I would not be noticed. A sad commentary indeed on the reality of military life.\n\nThe Personnel Building was large and industrious with plenty of to-ing and fro-ing from the front entrance. As we started toward it Morton came to a halt and stood at attention, swaying. \"W-what are you going to do?\" He whispered huskily and I saw that he was shaking with fear.\n\n\"Relax old buddy, all is under control,\" I said, leafing through the handful of papers to cover this unmilitary pause. \"Just follow me, do as I say, and in a few minutes we will have vanished without trace.\"\n\n\"We'll really vanish without trace if we go in there! We'll be caught, tortured, killed...\"\n\n\"Silence!\" I shouted into his ear and he leaped as though he had been shot. \"You will not talk. You will not think! You will only obey or you will be in the cagal so deep you will never see the light of day again!\"\n\nA passing sergeant smiled and nodded approval so I knew I was on the right track. I hated to do this to Morton but it was the only way. \"Left face\u2014forward march!\"\n\nHis skin was pale, his eyes rolled up, his mind empty of conscious thought. He could only obey. Up the steps we went and through the entrance toward the armed military policemen stationed there.\n\n\"Halt, at ease!\" I shouted and spun toward the MP, still shouting. \"You\u2014where do I find the Transport Section?\"\n\n\"Second floor, room two-oh-nine. Could I see your pass corporal?\"\n\nI glared at him coldly as I shuffled through the papers I was carrying, let my eyes travel slowly down to his boots, then back up again. He stood at attention, shivering slightly, and I knew he was new at this game.\n\n\"I don't think I have ever seen dirtier boots,\" I hissed. When his eyes glanced down I held out the turned-back papers. \"Here's the pass.\" When he glanced up again I let the papers slap shut.\n\nHe started to say something. I turned up the power of my glare and he wilted. \"Thank you, corporal. Second floor.\"\n\nI turned smartly away, snapped my fingers at Morton, then stamped away toward the stairs. Trying to ignore the fine beading of sweat on my brow. This was very demanding work\u2014and it wasn't over yet. I could see that Morton was definitely shivering as he walked and I wondered how much more of this he could take. But there was no turning back now. I threw open the door of 209 and waved him in. A bench ran along the wall and I pointed him towards it.\n\n\"Sit there and wait until you're called,\" I said, then turned to the reception clerk. He was on the phone and waved vaguely in my direction. Behind him rows of desks and laboring soldiers stretched the length of the room. All totally ignoring me, of course.\n\n\"Yes, sir, get onto it at once, sir,\" the reception clerk smarmed. \"Computer error, possibly, captain. We'll get right back to you. Very sorry about this.\"\n\nI could hear the phone disconnect loudly in his ear. \"You crock of cagal!\" he snarled and threw the phone back on the desk, then looked up at me. \"What's up, corporal?\"\n\n\"I'm up here, corporal, and I'm here to see the transport sergeant.\"\n\n\"He's home on compassionate leave. His canary died.\"\n\n\"I do not wish to hear the disgusting details of his personal life, soldier. Who's sitting in for him?\"\n\n\"Corporal Gamin.\"\n\n\"Tell the corporal I'm coming in.\"\n\n\"Right, right.\" He picked up the phone. I stamped past him to the door marked TRANSPORT SERGEANT\u2014KEEP OUT and threw it open. The thin, dark man at the computer terminal looked up and frowned.\n\n\"You are Corporal Gamin?\" I said, closing the door and flipping through the papers one more time. \"If you are I got good news for you.\"\n\n\"I'm Gamin. What's up?\"\n\n\"Your morale. The paymaster says they found a cumulative computer error in your pay and you are owed possibly two hundred and ten big ones. They want you there to straighten it out.\"\n\n\"I knew it! They been deducting double for insurance and laundry.\"\n\n\"They're all cagal-kopfs.\" My guess was right; there cannot be anyone alive, particularly in the army, who isn't sure there are errors in his payslip. \"I would suggest you get your chunk over and collect before they lose the money again. Can I use your phone?\"\n\n\"Punch nine for an outside line.\" He pulled up his necktie and reached for his jacket\u2014then stopped and took the key out of the terminal; the screen went black. \"I bet they owe me more than that. I want to see the records.\"\n\nThere was a second door behind his desk and, to my satisfaction, he exited that way. The instant it closed I had the other door open and poked my head through. When the reception clerk looked up I turned and called back over my shoulder.\n\n\"Do you want him in here as well, corporal?\" I nodded my head and turned back. \"You, recruit, get in here!\"\n\nMorton jumped at the sound of my voice, then scurried forward. I closed and locked the door behind him.\n\n\"Get comfortable,\" I said, pulling off my boot and rooting about inside it for the lockpick. \"No questions. I have to work fast.\"\n\nHe slumped into a chair, eyes bulging in silence as I gently tickled the lock until the terminal came to life.\n\n\"Menu, menu,\" I muttered as I hammered away on the keys.\n\nIt all went a lot smoother and faster than I had hoped. Whoever had written the software had apparently expected it to be accessed by morons. Maybe he was right. In any case I was led by the hand through the menus right to the current shipping orders.\n\n\"Here we are, leaving at noon today, a few minutes from now. Fort Abomeno. Your full name and serial number, Morton, quickly.\"\n\nI had my own dogtags spread out as I punched in all the requested information. A bell pinged and a sheet of paper slipped out of the printer.\n\n\"Wonderful!\" I said, smiling and letting some tension out of my muscles: I passed it over to him. \"We're safe for the moment since we have just left for Fort Abomeno.\"\n\n\"But... we're still here.\"\n\n\"Only in the flesh, my boy. For the record, and records are all that count to the military, we have shipped out. Now we make the flesh inviolate.\" I read through the shipping orders, checked off two names, then turned back to the terminal and entered data with some urgency. We had to be long gone before the corporal returned. The printer whiffed gently and one sheet slipped out, then another. I grabbed them up, relocked the terminal, and waved Morton to his feet.\n\n\"Here we go. Out the back door and I'll tell you what is happening as soon as we are clear of this building.\"\n\nSomeone was coming up the stairs, a corporal, and my heart gave a little hip-hop before I saw that it wasn't the corporal in question. Then it was down the hall to the front door and yes, there was Corporal Gamin coming up the stairs with a very nasty cut to his jib!\n\n\"Sharp right, recruit!\" I ordered and we turned into the first doorway with military precision. A lieutenant was combing his hair in front of a mirror there. Her hair I realized when she turned about and glared at me.\n\n\"What kind of cagal-head are you, corporal? Or doesn't the sign on the other side of this door read female personnel only?\"\n\n\"Sorry, sir, ma'am, dark in the hall. Eye trouble. You, recruit, why didn't you read the sign correctly? Get the cagal out of here and march straight to the MPs.\"\n\nI pushed Morton out ahead of me and closed the door. The hall ahead was empty.\n\n\"Let's go! Quick as we can without attracting attention.\"\n\nOut the door and down the steps and around the corner and another corner and the pace was beginning to tell. I leaned against a wall and felt the sweat run down my face and drip from my nose. I wiped it with the sheaf of papers I still carried\u2014then held up the two new sheets of orders and smiled; Morton gaped.\n\n\"Freedom and survival,\" I chortled. \"Shipping orders, or rather cancellation of shipping orders. We are safe at last.\"\n\n\"I haven't the slightest idea of what you are talking about.\"\n\n\"Sorry. Let me explain. As far as the military is concerned we are no longer at this base but have been shipped to Fort Abomeno. They will search for us there, but we will be hard to find. In order to keep the body count correct two soldiers who are in that shipment, still physically in that shipment, have been removed on paper. These are their orders, corporal, I thought a bit more rank wouldn't hurt. I am a sergeant now as you can see. We will occupy their quarters, eat their food, draw their pay. It will be weeks, perhaps months, before the error is discovered. By which time we will be long gone. Now\u2014shall we begin our new careers as noncommissioned officers?\"\n\n\"Urgle,\" he said dimly and his eyes shut and he would have slumped to the ground if I had not held him erect against the wall. I nodded agreement.\n\n\"I feel somewhat the same way myself. It really has been one of those days.\"\nChapter 11\n\nFatigue was of no importance, thirst equally so\u2014although both were present and sending imperative messages. To be ignored. Rank has its privileges and we were not going to enjoy ours until we assumed the trappings. I shook Morton until his eyes opened and he blinked dully at me.\n\n\"One last effort, Mort. We are going to the PX, about whose heady joys we have heard, and there we will spend some money. When that has been done we will be free spirits and will eat and drink and relax. Are you ready?\"\n\n\"No. I'm beat, shagged, dead. I cannot move. You go on. I can't make it...\"\n\n\"Then I'll just have to turn you over to Sergeant Klutz who has just arrived and is standing right behind you.\"\n\nHe sprang into the air with a shriek of agony, feet already running before he hit the ground. I held on to him.\n\n\"Sorry about that. No Klutz here. A ruse to get your adrenaline flowing. Let's go.\"\n\nWe went. Quickly before this burst of energy faded. It got him as far as the post exchange, where I leaned him against the wall near the cashier and handed him my sheaf of papers.\n\n\"Stand there, recruit, and do not move and do not let go of those papers or I will skin you alive or worse.\"\n\nI slammed the papers into his limp hands and whispered, \"What size jacket do you take?\" After much blinking on his part, and reiteration on mine, I extracted the needed information. I made my purchases from a bored clerk, added some stripes and a tube of superglue, paid for everything with some of Gow's money, thank you corporal, and led Morton farther into the reaches of the PX. To the latrine, empty this time of day.\n\n\"We'll use the booth one at a time,\" I said. \"We don't want anyone making improper conclusions. Take off those fatigues and slip into this uniform. Move it.\"\n\nWhile he changed I glued the new sergeant's stripes over the corporal's on my sleeves. When Morton had flushed and emerged I straightened his necktie and glued his promotion to his sleeve. His fatigues went into the rubbish, along with the sheaf of papers, and we went into the noncom's bar.\n\n\"Beer\u2014or something stronger?\" I asked.\n\n\"I don't drink.\"\n\n\"You do now. And curse. You're in the army. Sit there and sneer like a corporal and I'll be right back.\"\n\nI ordered two double neutral grain spirits and some beers, dumped the ethyl alcohol into the beer, sipped it to make sure it had not gone off, then went back to our table. Morton drank as ordered, widened his eyes, gasped, then drank again. Color returned to his cheeks as I drained half of my glass and sighed happily.\n\n\"I don't know how to thank you, what to say...\"\n\n\"Then say nothing. Drink up. What I did was to save my own hide and you just came along for the ride.\"\n\n\"Who are you, Jak? How do you know how to do those things you did?\"\n\n\"Would you believe me if I said I was a spy sent here to seek out military secrets?\"\n\n\"Yes.\"\n\n\"Well I'm not. I'm just a draftee like yourself. Though I will have to admit that I come from a lot further away than Pensildelphia. That's it, drain the glass, you're learning fast. I'll get a couple more drinks and some food. I saw they had catwiches. I'll get a couple of those.\"\n\nFood and drink helped, as did the stripes on our arms. Morton tore into his rations. I ate more slowly, finding myself already thinking about the next step. Cigars followed, Gow's wallet was bottomless, and more drink.\n\n\"Thish is really great, Jak, really great. You're really great, really great.\"\n\n\"Sleep,\" I said as his eyes unfocused and his head hit the table with a thump. \"You will awake a new man.\"\n\nI sipped lightly at my own drink for I wanted only the stimulation of the alcohol and not the oblivion. The club was almost empty, only one other table occupied, the noncom there just as asleep as Morton. Probably as drunk as well. The simple pleasures of military life. I sipped and thought of my previous military career on Spiovente, and of The Bishop, now dead, and of the man who was responsible for his death.\n\n\"I haven't forgotten you, Captain Garth, not at all,\" I said softly to myself. The bartender polished a glass and yawned. Well acquainted with customers who talked to themselves and drank themselves into extinction. \"For the last few days it has been survival only. Now I pick up your trail. We're in the same army, on the same base.\"\n\nI felt suddenly dizzy and put the glass down. It had been a long day and I was as tired as Morton. Country and coal-mining music was grating enchantingly from the jukebox: the world about was at peace. For the moment. I was aware of a light scratching sound and glanced down at the boxes that leaned against the wall. Something moved in the darkness behind them. I watched in silence as a twitching nose and whiskers emerged. Then the head, the bar lights reflected in the rat's eyes. It appeared to be looking up at me.\n\n\"Get lost,\" I said, \"before you end up in the stew.\" I cackled at my own witticism.\n\n\"Jim diGriz, I must talk with you,\" the rat said in a deep voice.\n\nIt had really been one of those days. Too much. I had not realized it but the strain was so great that I had cracked.\n\n\"Go away,\" I hissed. \"You are a figment of my imagination and not a real rat at all.\" I gulped the rest of my drink in a single swallow. The rat climbed up onto the box and looked at me.\n\n\"Of course I am not a real rat. I am Captain Varod of the League Navy.\"\n\nGently, so as not to awaken Morton\u2014this was my hallucination and I wanted to keep it for myself\u2014I pried his drink from his slack fingers and drained it as well as my own.\n\n\"You've shrunk a bit since the last time I talked with you, captain,\" I smirked.\n\n\"Stop playing the idiot, diGriz, and listen to me. This spyrat is controlled from our base. You were recognized and identified.\"\n\n\"By who? The rat?\"\n\n\"Shut up. This communication is limited because there is a chance their detectors will pick up the spyrat's broadcast signal. We need your help. You have penetrated their military base, the first agent to do so...\"\n\n\"Agent? I thought I was the criminal you were shipping home for trial and persecution?\"\n\n\"I said we need your help. This is vital. There are lives at stake. The generals are planning an invasion. We know that much from intercepting their communications. But we don't know where the landing will take place. Brastyr is a big continent and they might be attacking anywhere. There could be a lot of deaths. We must find out where they plan to...\"\n\nThe door to the bar burst open and a gun-waving officer burst in, followed by a technician weighted with electronic equipment.\n\n\"The signal is coming from that direction, sir,\" the man shouted and pointed directly at me.\n\n\"What is that cagal-head private doing in the noncoms' bar?\" I shouted, leaping to my feet and kicking the box as I did. The rat fell to the floor and I stamped on it. Hard.\n\n\"Don't get your cagal in an uproar, sergeant,\" the officer said. \"This is a priority investigation...\"\n\n\"Signal has stopped, sir,\" the technician said, fiddling with his dials.\n\n\"Cagal!\" the officer said, stuffing his gun back into the holster. \"These alcoholics don't have a transmitter.\"\n\n\"Could be the street outside, other side of the wall. A moving vehicle.\"\n\n\"Let's go!\"\n\nThe door slammed shut behind them. The barman wiped his glass. \"This happen very often around here?\" I asked.\n\n\"Yeah. This is sure an uptight base.\"\n\nMorton snored heavily and I poked the crushed remains of the stainless steel rat with my toe. An omen? A gear wheel rolled out and rattled on the floor.\n\n\"Set them up again,\" I called out. \"And take one yourself since the rest of these cagal-kopfs are in dreamland.\"\n\n\"You're all heart, sarge. Just ship in?\"\n\n\"Today.\"\n\n\"An uptight base like I say\u2014\"\n\nHis voice was drowned out by the loud whistle from the TV as it turned itself on. The black-clad military announcer glared out of the screen just one more time.\n\n\"The spy who landed in Marhaveno has been identified. He attempted to disguise himself as a harmless draftee and was inducted into the army. Resolute police work has identified him by his clothing.\"\n\nSome police work. They just looked at their mail. I was beginning to think that sending my clothing from the reception center to the police station was not at all as funny as it had seemed at the time. There was a scratch of static and the announcer vanished from the screen to be replaced by another officer.\n\n\"Now hear this,\" he shouted. \"As of this moment this entire base is sealed to outgoing. I repeat, Mortstertoro is locked tight, gates sealed, aircraft departures canceled. The spy who landed in Marhaveno has been identified as a recruit who was shipped to this base. Here is his picture.\"\n\nMy heart skipped a beat or two, then settled down as the blurred photo of Jak, from my stolen ID, appeared on the screen. I was still one jump ahead of them. It would soon be discovered that Jak5138 was no longer on the base and the search would go elsewhere. I took my drink and went back to the table to stare into the wide and frightened eyes of Morton.\n\n\"You want a drink?\" I asked before he could speak. He gurgled and pointed at the screen.\n\n\"Did you hear that?\" I asked, and kicked him under the table. \"Can't be much of a spy if he lets himself get drafted. Some spy! I'll bet you five he's caught and dead before dark.\" When he relaxed slightly I went on in a hoarse whisper. \"It will take a long time to search this base...\"\n\n\"No it won't\u2014because they know just where to look. They know who you are, Jak. They'll go to Sergeant Klutz, who will tell them he transferred you to Corporal Gow. Then they'll find Gow and...\"\n\n\"And the trail will run cold. It will take them days to search a camp this size. And when they don't find the spy the first time they'll just do it again. They are not bright enough to consider having the computer check the records for the spy.\"\n\n\"Attention!\" the announcer on the screen called out, waving a sheet of paper. \"I have just been given this new information. The spy\u2014and an accomplice\u2014have managed to have themselves transferred from this base by illegal use of the base computer. All computer personnel are now under arrest and will probably be shot.\"\n\nI turned away, not able to look Morton in the eye.\n\n\"Now that they know where to look,\" Morton asked hollowly, \"how long will it take them to discover that we were never on that shipment? And then find out that a corporal and a sergeant who really were on that shipment were not on that shipment and are still here on the base?\"\n\n\"How long?\" I laughed, but there was a very hollow ring to it. \"Could take days, weeks, no way to tell.\"\n\n\"How long?\"\n\nI sighed deeply. \"They got some hotshot computer programs. Good security. I would say that we have maybe thirty minutes before they start looking for us.\"\n\nHis body shook as though he had received ten thousand volts and he started to jump to his feet. I reached out and held him down, then glanced at the bartender. He was looking at the TV.\n\n\"You're right,\" I said. \"We get out of here, but slowly. On your feet. Follow me.\"\n\nAs we started toward the door the bartender glanced in our direction.\n\n\"Where's the transient barracks?\" I said.\n\n\"Out the back door, turn right. See you.\"\n\n\"Yeah. See you.\"\n\nWe strolled out the back door and turned left. It was getting dark, which might help.\n\n\"You got a plan?\" Morton said, eagerness in his voice. \"You know a way to get us out of this.\"\n\n\"Of course,\" I said, clapping him on the back. \"Every step planned. We go this way.\"\n\nI could hear the forced joviality in my voice; I hoped that he couldn't. He had to think that I knew what I was doing or he might crack. It was a white lie for the sake of his morale.\n\nBut what about my morale? I was holding it down successfully for the moment, but I could feel an awareness of dark panic knocking and ready to come in. I kept it at bay. We walked on down the company street, the lights coming on, lost in the milling military mass. How long would this last? The question was the answer: not very. The panic pushed a little harder.\n\nI have heard it said that when a man knows that he is to be hanged, it focuses his mind wonderfully. I wasn't going to be hanged, not for the present at least, but the foul breath of military prosecution on my neck was focusing my mind almost as well. So much so that when an officer passed I turned to look at him. Turned and stopped until he vanished in the crowd. Morton was pulling feebly at my arm.\n\n\"What are you looking at? What's wrong?\"\n\n\"Nothing wrong. Everything right. I know now exactly what we must do next.\"\n\n\"What?\"\n\n\"Just come with me. I know that it is back this way, I noticed it when we passed.\"\n\n\"What, what?\"\n\n\"BOQ.\" Before he could say What? What? What? I explained. \"Bachelor Officers' Quarters. Where the officers live when they are not getting drunk and making life a hell for the enlisted men. That is where we are going. There.\"\n\nI pointed to the brightly lit building, guards at the front entrance, officers in their military finery pouring from it.\n\n\"That's suicide!\" Morton said. The edge of hysteria back in his voice.\n\n\"Easy does it,\" I cozened. \"We do not enter the building by this portal. Suicide as you say. But what has a front surely has a back. And from the exodus visible from that officerial snakepit it looks like everyone is on duty tonight. Everyone except us, that is.\" I chortled darkly and he looked at me out of the corners of his eyes as if I had gone mad. Perhaps I had. We would soon find out.\n\nThere was a wall behind the BOQ which we followed. A sort of alley led next to it, badly lit and just what I wanted. There was a door here let into the wall with a light above it. As we strolled past I read the sign, OFFICERS ONLY, and bent over and tied my shoe: it needed only a single glance to identify the lock. Then I stood and continued on. I stopped in the shadows between two lights and bent to my shoe again. Only this time I came up with the lockpick.\n\n\"All right, here we go. The lock is nothing, single tumbler, pick it as easy as I pick my teeth. We walk back now and if no one is in sight we walk through it. Got that?\"\n\nThe chatter of his teeth was the only response. I took his quivering arm and squeezed it. \"It's all right, Morton. You'll see. Just do as I say and we'll soon be safe. Nice and quiet\u2014here we go.\"\n\nI tried not to catch any of Morton's fears, but they were very contagious. We stopped under the light, I put the lockpick into the keyhole. Felt and twisted. It didn't open.\n\n\"Someone's coming,\" Morton wailed.\n\n\"Piece of cake,\" I muttered, perspiration running down my face. \"Opened these with my eyes shut.\"\n\n\"Getting closer!\"\n\n\"Eyes shut!\"\n\nIt wouldn't open. I shut my eyes, closed out all sensations, felt for the tumblers. Clicked it open.\n\n\"Inside!\" I said, pulling him after me, closing the gate behind us. We stood with our backs to it, shivering in the darkness as the footsteps came closer, came to the gate...\n\nPassed it and went on.\n\n\"There, wasn't that easy?\" I said, ruining the effect as my voice cracked and squeaked. Not that Morton noticed; he was shivering so hard that I could hear his teeth clatter. \"Look, nice garden. Pathways for strolling, love seats for loving, all the nice things to keep the officerial classes happy. And beyond the garden the dark windows of their quarters, dark because the occupants have all gone out. So now all that we have to do is find a window to open...\"\n\n\"Jak\u2014what are we doing here?\"\n\n\"I thought that was obvious. The military powers are looking for one recruit now. When their computer coughs out the next bit of news they will be looking for a corporal and a sergeant.\" I tried to ignore his moan. \"So we get into this building and become officers. As simple as that.\"\n\nI caught him as he dropped and laid him gently on the grass. \"That's it. Have a little rest. I'll be right back.\"\n\nThe third window I tried was unlocked. I opened it and looked in. A mussed bed, open closet, empty room. Perfect. I found my way back to Morton who was just sitting up. He recoiled as I appeared out of the darkness and my quick hand over his mouth muffled his scream.\n\n\"Everything is fine. Almost finished.\"\n\nI boosted him through the window and let him drop onto the bed, then closed and locked the window behind us. There was a key in the door which made everything very much easier.\n\n\"Look,\" I said, \"lie here and recuperate. I'm going to lock you in. The building is empty as far as I can see, so what I have to do should not be long. Take a rest and I'll be back as soon as I can.\"\n\nI went carefully, but the building was empty of life and as silent as the tomb. Its occupants away and hopefully hard at work. I had time to pick and choose, make my selections and select the right sizes. I heard a muffled moan of agony when I let myself back into the room, to which I responded as cheerfully as I could.\n\n\"New uniform\u2014new persona!\" I handed them over to Morton. \"Get dressed and give me our old clothes. There's enough light from outside to make that easy. Here, let me tie that necktie, you are all butterfingers today.\"\n\nDressed and ready, our caps square upon our heads, our old clothes buried in a laundry basket, we sauntered forth into the corridor. Morton looked at me and gasped and fell away.\n\n\"Cheer up\u2014you look the same way. Except that you are a second lieutenant while I am a captain. It is a young army.\"\n\n\"B-but,\" he stammered. \"You are a... Military Policeman!\"\n\n\"And so are you. No one ever questions a cop.\"\n\nWe turned the corner as I said this and approached the front entrance. The major standing there with a clipboard looked up at us and scowled.\n\n\"Now I have you,\" he said.\nChapter 12\n\nI snapped to attention, I could think of nothing else to do\u2014and hoped Morton was not too paralyzed to do the same. There were just two of them, the major and the guard at the door. After I dropped the major could I reach the guard before he could get out his gun? A neat problem. The major was looking at his clipboard. Now\u2014get him!\n\nHe looked up as I swayed forward. The guard was looking at me too. I swayed back.\n\n\"I missed you at the airport,\" the major said. \"You must have come on the earlier flight. But these shipping orders say two captains. Who is this lieutenant?\"\n\nShipping orders? Two captains? I stopped my eyeballs spinning and finally threw my brain into gear.\n\n\"Could be an error, sir. Lot of confusion today. Might I see the orders?\"\n\nHe grunted uncommunicatively and passed them over. I ran my finger down the list of crossed-off names to the remaining two at the bottom. Then passed them back.\n\n\"Error like I said, sir. I'm Captain Drem. This is Lieutenant Hesk, not captain the way they got it here.\"\n\n\"Right,\" he said, making the change on his sheet. \"Let's go.\"\n\nWe went. Outside the door was a truck stuffed with Military Police, a very disgusting sight. The major climbed into the cab, rank does have its privileges, and I led Morton to the rear. Moving quickly because I saw something that I hoped the major had not seen. Two MP officers, both captains, walking toward us. They scowled and passed and turned into the BOQ. I scowled in return, turning the scowl into a glare when I looked into the back of the truck and saw that there were no officers among the redhats there.\n\n\"What is this\u2014a meeting of the girls' club,\" I snarled. \"Move back, make room, shut up, give us a hand.\"\n\nAll of this was done with alacrity. Morton and I sat on the recently vacated bench and the truck pulled forward. I let out my breath slowly\u2014from between still-snarling teeth. We bumped and swayed our way through the night and I began to feel very, very tired. It had been that kind of day.\n\n\"Do you know where we are going, captain?\" a burly sergeant asked.\n\n\"Shut up!\"\n\n\"Thank you, sir.\"\n\nThere was only silence after this witty exchange. Cold silence that continued until we ground to a stop and the major reappeared. \"Climb out,\" he ordered. \"Captain, follow me.\"\n\n\"Fall these men in, lieutenant,\" I told Morton. He stumbled after me his face white with despair in the glare of the street lights.\n\n\"How, what, glug,\" he whispered.\n\n\"Order a sergeant to do it,\" I whispered back. \"Pass the buck, that's the army way.\"\n\nI trotted after the major who had stopped before the entrance of a large building and was going through an immense ring of keys. I stood at ease and looked at the large posters beside the door. Then looked closer when I realized they were 3Ds, in living color, of a number of naked young woman. When my head moved they moved and I swayed slightly.\n\n\"Knock that off, captain,\" the major ordered and I snapped to attention, my eyes still focused on the sign that read BASE BURLESQUE\u2014OFFICERS ONLY. The major found the key he was looking for and turned it in the lock. \"No performance tonight,\" he said. \"We've comandeered the place for an emergency meeting. Top security. As soon as the techs get here I want the entire theatre swept clean. And I mean clean. I want an MP with every tech and I want a head count and I am making you responsible. Got that?\"\n\n\"Yes, sir.\"\n\n\"I'm going to check all the other doors personally to make sure they are locked. Get cracking, we only have an hour.\"\n\nI threw a salute as he moved off around the building and wondered just what I had gotten myself into. The rumble of engines cut through my thoughts as a truck pulled up at the curb before me. A sergeant climbed down from the cab and saluted me.\n\n\"And what do we have here?\" I asked.\n\n\"Instrument technicians, sir. We were ordered...\"\n\n\"I'll bet you were. Unload them and fall them in.\"\n\n\"Yes, sir.\"\n\nI stamped back to the MPs who were neatly lined up at attention and pointed my finger at Morton. \"You, Lieutenant Hesk, get over in front of that entrance. No one in or out without my permission.\"\n\nMy heart dropped as Morton started to look over his shoulder. Memory of his new name apparently filtered through because he recovered himself and hurried away. I turned back and scowled at the MPs, with particular attention to the sergeant who stood before them. Gray-haired, skin like an old boot, stripes and hashmarks clogging his sleeve.\n\n\"You senior NCO?\"\n\n\"Yes, sir.\"\n\n\"Right. Here is the drill. Those techs are going to sweep this theater. I want one MP with every tech. I want every man counted in and counted out and I want no errors. And I want the sweep complete and overlapping and that building clean. Any questions?\"\n\n\"No, captain. They'll snap-cagal for me.\"\n\n\"I thought they would. Get cracking.\"\n\nHe turned on his heel, inflated his lungs\u2014and let out a blast of orders that blew the cap off the nearest MP. They moved. I stepped back and nodded approval. Then stamped over and positioned myself next to Morton.\n\n\"Something big coming down,\" I said quietly. \"Secret meeting in an hour and we are in charge of security.\" I ignored his moan of anguish. \"Just stand around and look military and stay away from the major when he gets back. I don't know about you, but I find this very interesting.\" he moaned again and I strolled over to inspect the arrangements.\n\nThe technicians had shouldered their backpacks and were adjusting dials on the control panels that each of them wore slung about the chest. One of them pointed his detector wand at the side of the truck and I could see the needles jump; there was a squeal from the earphones that he had hung about his neck.\n\n\"Captain. Some trouble here.\" I turned around.\n\n\"What is it, sergeant.\"\n\n\"This cagal-kopf says he got a malfunction.\" He had a white-faced tech by the arm and was shaking him like a dog with a bone.\n\n\"Battery, sir,\" the man wailed. \"Checked... it's a malfunction fuse!\"\n\n\"Arrest him, sergeant. The charge is sabotage. Have him shot at dawn.\" The sergeant smiled, the tech moaned and I bent until my face was close to his. \"Or can you manage to trace and repair this malfunction in the next sixty seconds?\"\n\n\"It's fixed, sir! I know how! Borrow a fuse!\"\n\nHe stumbled away with the sergeant right behind him. I was falling into my role and beginning to enjoy myself. Though I was sure I would hate myself in the morning.\n\nMore MPs had arrived; the major reappeared and spread them around the theater and in front of the entrance. I could see Morton begin to shiver at their presence so I hurried to take over from him.\n\n\"You can open that door now, lieutenant. No one goes in except these search teams. I want a head count going in and coming out.\"\n\nUnder the verbal abuse of the sergeant the search was finished just in time. The first official cars were appearing as the techs were being loaded back into the trucks.\n\n\"How did it go, sergeant?\" I asked.\n\n\"Lot of beer cans, cagal like that. Swept secure, captain.\"\n\n\"Good. Move the troops out of the way, but keep them around in case we need them again.\"\n\nI waved Morton after me and strolled over to the nearest truck, stood in its shadow where I could see what was happening.\n\n\"What's happening,\" Morton asked.\n\n\"Good question. Big, secret and very sudden meeting of some kind. See that car, all officers of field rank or better.\"\n\n\"We have got to get out of here!\"\n\n\"Why? Can you think of a safer place to be? We are part of the security here\u2014so no questions asked. Except by me. Look at that one getting out of the limo! Must have nine stars on his shoulders. Big stuff tonight. And that officer behind him. Never saw that uniform before. Something special...\"\n\nThis officer turned about and I froze. A single silver skull on the shoulder of his gray-green coat. Another skull on the front of his cap.\n\nAnd beneath the black brim of the cap a familiar face.\n\nCaptain Garth. Former captain of a Venian freighter. The man responsible for the death of my friend The Bishop.\n\n\"Stay here,\" I ordered Morton, and stepped out of the darkness as soon as Garth had turned away. I walked toward him as he approached the security check at the entrance. Passed right behind him as he reached the major, who threw a very snappy salute. I could hear the major's voice clearly as I passed and went on.\n\n\"They are almost all inside, General Zennor.\"\n\n\"Report to me when the count is complete. Then seal this door tight.\"\n\nI stamped on, checked the guards, stamped back to Morton's side.\n\n\"What was that all about?\" he asked.\n\n\"Forget about it. Nothing to do with you.\"\n\nNo longer a simple spacer captain. A general now. Probably always a general. Zennor. What was he up to? What was this entire army up to that he seemed to be ordering around? And how could I find out?\n\nWhen the major called I did not even hear him. Only when Morton kicked me in the ankle did I realize that I was the Captain Drem he was talking to.\n\n\"Yes, sir. You want me, sir?\"\n\n\"Not falling asleep, are you, Drem?\"\n\n\"No, major, I was just going over the security in my head.\"\n\n\"Well go over it on your feet, which will accomplish a lot more. I've stationed a man at every entrance to this theater. Inspect them.\"\n\nI saluted his back enthusiastically as he turned away. This might very well be the opportunity I was waiting for.\n\n\"Lieutenant,\" I called out. \"Inspection tour. This way.\"\n\nI rubbed my hands together happily as we walked around the theater. \"Morton, there is something important going on here and I mean to find out more about it.\"\n\n\"Don't! Stay clear!\"\n\n\"Normally good advice. But this time I have to know what is happening, what he is up to. Did you see the uniforms? All senior officers. And I was ratted to earlier today that an invasion was being planned. It doesn't take a great brain to figure out that this meeting has something to do with that invasion. But how do I get inside?\"\n\nWe were approaching a side entrance to the theater and the MP there snapped to attention as soon as we appeared. I shook the locked door and scowled at him.\n\n\"This door locked when you got here?\"\n\n\"Yes, sir.\"\n\n\"Anyone try to get in?\"\n\n\"No, sir!\"\n\n\"What are your orders?\"\n\n\"Kill anyone who goes near the door.\" He had his hand on his pistol butt.\n\n\"Does that include your commanding officers?\" I shouted at him, my mouth in his ear. He swayed and his hand dropped to his side.\n\n\"No, captain.\"\n\n\"Then you are wrong and you could be shot for disobeying orders. An inspecting officer may try the door to see if it is locked. If an inspecting officer should attempt to go through the door he is to be instantly killed. Is that clear?\"\n\n\"Very clear, sir.\"\n\n\"Then wipe the smile from your face. You seem to enjoy that thought too much.\"\n\n\"Yes, sir. I mean no, sir!\"\n\nI growled a bit more and continued my inspection. We had almost circum-navigated the building when we reached a door in the rear. The guard there stood at attention. I shook the locked door and looked at the metal staircase beside it.\n\n\"Where does this go to?\" I asked.\n\n\"Emergency exit.\"\n\n\"Is there a guard there.\"\n\n\"Yes, sir.\"\n\nMorton followed me up the clanging stairs. I stopped halfway and bent to remove the lockpick from my shoe. Morton opened his mouth but shut it again when I put my finger to my lips. I had to find out what was happening inside.\n\nWe stamped upward and when we emerged on the balustraded corridor the guard there had his gun half out of the holster.\n\n\"Do you intend to aim that weapon at me?\" I asked coldly.\n\n\"No, sir, sorry.\" He put it away and snapped to attention. I put my face close to his.\n\n\"Do you know it is a court-martial offense to point a weapon at an officer?\"\n\n\"I wasn't, sir, no! I'm alone here, didn't know who was coming...\"\n\n\"I don't believe you, soldier. There is something wrong here. Stand over there by the lieutenant.\"\n\nAs he turned about I had the lockpick in the keyhole, delicately, turning it, clicking it. I stepped back away from it as he stopped and about-faced.\n\n\"This door is locked?\"\n\n\"Yes, sir. Of course. I am stationed here because of the door...\"\n\nHis voice wound down as I reached out and opened the door. Then closed it and wheeled to face him.\n\n\"You are under arrest, soldier. Lieutenant\u2014take this man to the major. Tell him what has happened. Return with the major at once. Move it!\"\n\nAs they stamped away I inserted the lockpick yet again, twisted and pressed hard. Something snapped inside the lock. Only then did I put the lockpick away, open the door and slip inside. Closing it silently behind me.\n\nThe small entranceway was sealed with dusty curtains. Light trickled between them; I bent forward and separated them a tiny amount.\n\n\"... important that security be absolute until blastoff. You have your sealed orders, not to be opened until H hour. Rendezvous points are marked...\"\n\nI knew that voice well. Once Garth, now Zennor. I parted the curtains just a bit more to make sure. There he was, almost below me, pointing at the large chart behind him. I looked at the chart, then closed the curtains and stepped back.\n\nI was closing the door behind me when hurried footsteps sounded on the stairs. The major appeared, face red and strained.\n\n\"What is happening?\"\n\n\"I'm not sure, sir. The guard that was stationed here had his weapon drawn, acted suspiciously. I tried this door. It was unlocked. That was when I sent for you, sir.\"\n\n\"It can't be. I locked it myself.\"\n\nIt opened at his touch and his face whitened with shock. He pulled it quickly closed. \"You haven't been inside?\"\n\n\"Of course not, major. I have my orders. Perhaps the lock is defective.\"\n\n\"Yes, perhaps!\" He fumbled out his ring of keys, found the right key and turned it in the lock. Metal grated.\n\n\"It won't lock!\"\n\n\"May I try it, sir?\"\n\nI took the keys from his limp fingers and, naturally, had no better luck in sealing the door. When I handed back the keys I spoke in a low voice.\n\n\"There will be an investigation, sir, trouble. Not fair to you. I'll see that the guard talks to no one about this. Then I'll get a welder, seal the door. Might be best, major, don't you think so?\"\n\nHe started to speak, then closed his mouth, and thought instead. Looking from me to the door. Then he noticed the keys still in his hand. He put them in his pocket and straightened his shoulders.\n\n\"As you say, captain, nothing happened. No point in getting involved in investigations and suchlike. I'll stay here. Send the welder at once.\"\n\n\"Very good, sir. I'll take care of everything.\"\n\nMorton was waiting at the foot of the stairs, the frightened MP standing beside him. I walked up to the man and gave him a good glare.\n\n\"I am going to be kind to you, soldier, although it goes against the grain. I think it might be wisest if we forgot all about this matter. What is your name?\"\n\n\"Pip7812, sir.\"\n\n\"All right, Pip, you can go back to your unit now. But\u2014if I hear any rumors or loose talk about locks or such you will be dead within twenty-four hours. Understand?\"\n\n\"Locks, captain? I'm afraid I don't know what you mean.\"\n\n\"Very good, Pip. Report to the sergeant. Tell him I need a welder here at once. Move.\"\n\nHe moved. \"What was all that about?\" Morton asked.\n\n\"That was about warfare, my friend. I know now what they are up to here. I know all about their invasion plans.\"\n\nExcept\u2014what could I possibly do about it?\nChapter 13\n\nWhen the meeting broke up I saw to it that I was busily occupied away from the theater entrance. It was a long chance that Zennor would recognize me from his Captain Garth days. But even a long chance is some chance, so I stayed out of sight. The troops formed up and marched away: with the emergency over they were not being coddled with effete tranportation. The major had a car at his disposal but I turned down his offer.\n\n\"We could have used a lift,\" Morton complained as the car moved away.\n\n\"To where? Prison? The further we are from authority the happier we should be.\"\n\n\"I'm tired.\"\n\n\"Who isn't? Not to mention hungry. Let's find a place to spend some of Gow's money...\"\n\n\"Jim... Jim diGriz...\"\n\nThe sound was high-pitched, barely audible. Was I hearing things? I looked around but Morton was the only person nearby.\n\n\"Did you hear anything?\"\n\n\"No. Should I?\"\n\n\"Don't know. A sudden ringing in my ears. But I swear I heard something.\"\n\n\"Maybe it was that moth on your shoulder talking to you. Ha-ha.\"\n\n\"Ha-ha yourself. What moth?\"\n\n\"See it there? Sitting on your captain's bars. Should I brush it off?\"\n\n\"No. Leave it.\"\n\nI turned my head and blinked and could just make out the moth. It flapped its wings and took off\u2014and landed on my ear.\n\n\"Go... aergropl... now.\"\n\n\"I can't understand you.\"\n\n\"That's because I'm not talking.\"\n\n\"Shut up, Morton. I'm talking to the moth, not to you.\"\n\nHis jaw dropped and he moved quickly sideways. \"Repeat message,\" I said, ignoring him for the moment.\n\n\"Airfield... go airfield.\"\n\n\"Right, go to the airfield. Understood. Over and out.\" The moth fluttered away and I patted Morton on the shoulder; I could feel him shivering. \"Come on, cheer up. And stop looking at me as though I were mad. The moth is a communication device, nothing more.\"\n\n\"Communicating with whom?\"\n\n\"The less you know, the less trouble you can get into.\"\n\n\"You really are a spy, aren't you?\"\n\n\"Yes and no. I'm here on my own business, but certain parties are trying to get me involved in their business. Do you understand?\"\n\n\"No.\"\n\n\"Good. Let's find the airfield. At a guess I would say that it is over there where all the lights are and the planes are landing. Coming with me?\"\n\n\"Do I have a choice? Is there any way of going back? Starting over again? I mean we can't just sneak back into the barracks as if nothing happened, can we?\"\n\n\"You know that we can't.\"\n\nHe sighed and nodded his head. \"I know. But I'm just not cut out for the kind of thing that we have been doing. And where is it going to end?\"\n\nA good question. With very little hope of an answer at the present time.\n\n\"Truthfully\u2014I don't know. But you have my word, Morton, because I got you into this. My first priority, before anything else, is to get you out of trouble and safe. Don't ask how\u2014because I don't know yet.\"\n\n\"You can't blame yourself. It was I who opened my mouth to that cagaling corporal. That's where it started.\"\n\nWe had been walking while we talked, getting closer and closer to the airfield. The road that we were taking curved around the end of the field, separated from it by a high wire fence, well illuminated by bright lights. On the other side of the fence were grass and taxiways. A heavy freighter had just landed. It trundled by and we watched it go. When it had moved on a flock of black birds swooped down and began poking about in the grass. One of them unfolded its wings and flew toward the fence, landing on the other side. It cocked its head at me and spoke.\n\n\"You are not alone.\"\n\n\"Obviously. He's safe. Is that you, Varod.\"\n\n\"No. Captain Varod is off duty.\"\n\n\"Get him. I don't talk to just any old crow.\"\n\n\"You will be contacted.\"\n\nThe bird turned about and opened its beak and spread its wings. It took off without flapping, making a whistling sound.\n\n\"Jet powered,\" I said. \"Air intake in its mouth. Jet exhaust just where you imagine it might be. Let's walk.\"\n\nThere was the whine of an approaching siren and a detector van came hurtling down the road. It slowed when it passed us, the dish aerial pointing in our direction, then moved on.\n\n\"They are really efficient about spotting radio transmission,\" I said.\n\n\"Is that bird a radio?\"\n\n\"Among other things. It is remotely controlled and probably has some logic circuitry for hopping about and staying with the other birds. Only when it transmits back to base can it be detected.\"\n\n\"Where is the base?\"\n\n\"You don't want to know. Or who is operating it. But I can assure you they mean no harm to this country.\"\n\n\"Why not?\" He spoke with great agitation now. \"Tell them to get to work and get rid of the military and their friends and start elections again. Do you know how long the present state of emergency has been going on? I'll tell you, I checked. The so-called temporary emergency was declared over two hundred years ago. Some emergency! Tell your bird friends they can cause all the trouble they want as far as I'm concerned.\"\n\n\"I heard that,\" the bird said in a deep voice, swooping out of the darkness and landing on my shoulder. \"Our work is not to cause trouble. We labor only to...\"\n\n\"Varod, shut up,\" I said. \"We have limited communication time before the detectors show up again and let us not waste it with speeches. I have found out the invasion plans.\"\n\nThe bird cocked his eye at me and nodded. \"Very good,\" it said. \"Details soonest, I am recording. Where is the invasion site?\"\n\n\"Not on this planet. They are readying a space fleet to attack another planet.\"\n\n\"You are sure of this?\"\n\n\"I eavesdropped. I'm sure.\"\n\n\"What is the name of the planet?\"\n\n\"I have no idea.\"\n\n\"I will return. I must get rid of the detector van.\"\n\nThe bird whistled into the sky leaving the stench of burned jet fuel behind. It did a neat barrel roll and landed on the top of a passing truck. Still broadcasting, I imagine, because a moment later the detector van hurtled by in pursuit of the truck. We walked on.\n\n\"What's this about an invasion? What did you find out?\"\n\n\"Just that. The one in charge is a General Zennor. I imagine it will happen pretty soon from the way that he was talking...\"\n\nThere was a whistle and a blast of hot air: sharp claws dug into my scalp right through my cap as the bird landed on my head.\n\n\"You must discover what planet is being invaded,\" it said.\n\n\"Find out yourself. Follow them when they take off.\"\n\n\"Impossible. The nearest spacer with detection gear is four days away. It may not get here in time.\"\n\n\"Tough. Ouch.\"\n\nI rubbed my scalp where the bird had removed some hair when it took off, then bent to pick up my cap. We turned a corner just as another detector vehicle roared by behind us.\n\n\"Let's mix with the crowds,\" I told Morton. \"That detector is going to get suspicious if it keeps finding us around every time it gets a reading.\"\n\n\"Could we mix with crowds that are eating and drinking?\"\n\n\"Good thinking. And I know just where to go.\"\n\nI stepped off the curb as I said this and stood with my hand raised\u2014directly in front of a truck. The driver hit the brake and squealed to a shivering halt in front of me.\n\n\"Driving a little fast, aren't we?\" I snarled at the driver.\n\n\"I didn't see you, captain...\"\n\n\"And I know why you didn't see me. Because one of your headlights is burnt out, that's why. But I am feeling generous today. If you take me and my companion to the Officers' Club I might forget I ever saw you.\"\n\nNot that the driver had any choice. He dropped us in front of the club and roared away. We entered to sample the heady joys which, for the most part, were identical with the noncoms' club except here there were waitresses. About a quarter of the tables were occupied: everyone else must still be on duty. Our steaks and beer appeared with exemplary speed and we dived at them with growls of hunger. We were almost finished when an officer appeared in the doorway and blew a whistle.\n\n\"All right, fall out and fall in. Everyone. Emergency muster. Transportation outside. That means you,\" he said pointing a mean finger in our direction.\n\n\"We just came off duty, colonel,\" I said.\n\n\"You're just going back on. And I see that you have eaten, which I haven't, so don't cross me boy.\"\n\n\"Just leaving, sir!\"\n\nMorton and I joined the rush, out the door and into the waiting bus. The colonel entered last and the driver pulled away.\n\n\"Here is as much as I can tell you,\" the colonel said, shouting so he could be heard above the engine's noise. \"Due to reasons that are no concern of yours our current plans have been moved forward. You are going into action and you are going at once.\" There were questions and cries of complaint which he shouted down.\n\n\"Silence! I know you are all desk-driving fat-gutted base personnel\u2014but you are also soldiers. Because of the acceleration in planning some combat officer transfers will not arrive in time. You officers have all just volunteered to take their place. You will get combat gear and you will join your troops and you will board the transport at once. We will all be away by midnight.\"\n\nThe colonel ignored all the complaints and protests and finally lost his temper. He pulled a wicked-looking pistol from his holster and fired a shot up through the roof of the bus. Then pointed the gun at us. The silence was extreme. He had a nasty smile and pointed teeth.\n\n\"That is better,\" he said, and kept the weapon pointed. \"You are all timeserving cagal-kopfs, which means you have wangled and bought soft assignments which will do you no good now. You are in the army and in the army you obey orders.\" He fired another shot into the roof as the bus stopped. \"Now, I want volunteers for combat duty. All volunteers step forward.\"\n\nWe stepped forward in a rush. The lights in the supply depot were burning brightly in the night, clerks waited by the loaded shelves and an officer blocked the doorway.\n\n\"Move aside,\" our colonel said, keeping a wary eye on us as we emerged from the bus.\n\n\"Can't, sir,\" the supply officer said. \"I can't issue anything until I have the orders from headquarters. They haven't come through yet...\"\n\nThe colonel shot out the light over the depot door then put the hot muzzle of his gun against the supply officer's nose.\n\n\"What did you say?\" the colonel growled.\n\n\"Orders just arrived, sir! Open up in there and issue everything. Quickly!\"\n\nAnd quickly was what it was. We surged through the depot at top speed, grabbing up clothing, boots, barracks bags, belts, everything on the run. The manic colonel seemed to be everywhere now, his gun banging occasionally to keep up the pace. The street behind the building was a hellish scene of officers tearing off their uniforms, discarding them on the ground as they pulled on the green combat fatigues, jamming helmets on heads and everything else into their bags. Staggering forward into the next building where weapons were being issued. But no ammunition I noticed; the colonel was no fool. Stumbling under the weight of my burdens I staggered out into the street and dropped against a wall, adrip with perspiration. Morton dropped next to me.\n\n\"Do you have any idea what this is all about?\" he gasped.\n\n\"A very good idea. The powers that be think they are being spied upon. With good reason since they are. So they have pushed up the date of their invasion before details of their plans can be discovered.\"\n\n\"What will happen to us?\"\n\n\"We invade. At least we will go out as officers. Which means that we can stay to the rear and order the troops forward in case of any enemy resistance...\"\n\n\"Open your barracks bag,\" the moth said into my ear.\n\n\"What are you saying?\"\n\nThere was a sharp burning sensation in my earlobe as the moth discharged its batteries into my skin.\n\n\"Open... bag!\" it gasped and dropped off, batteries drained and dead.\n\nI bent and opened the bag, wondering if something had been planted there. There was a whistle and the stink of jet fuel as the bird plummeted past me into the bag.\n\n\"I'm not smuggling this damn bird and getting caught and shot!\" I shouted.\n\n\"You must do it for the sake of all mankind,\" the bird said, eyes glowing wildly. \"Reactivate by pressing the bill twice. Out.\"\n\nThe glow died and it went limp. I jammed the bag shut as footsteps approached.\n\n\"Into the transport!\" the colonel ordered. \"We are on our way!\"\nChapter 14\n\nThere was very little time to sit around and relax. As fast as the officers were spewed out of the supply depot, staggering under the weight of all their combat gear, trucks appeared to carry them away into the night. Groaning and complaining, with the rest of the groaners and complainers, Morton and I heaved our bags and weapons over the tailgate of a truck and clambered after. When it was filled to capacity, and slightly more, we lurched away.\n\n\"And to shink that I just reenlishted. Voluntarily,\" an officer expostulated leaning heavily against me. There was a gurgling sound from an upended bottle.\n\n\"Share the wealth, share the wealth,\" I muttered as I pried the bottle from his shaking grasp. It was pretty foul stuff, but was rich with alcohol.\n\n\"You still don't drink?\" I gasped at Morton, holding up the rapidly emptying bottle.\n\n\"I'm learning fast.\" He gulped then coughed, then gulped again before relinquishing the bottle to its original owner.\n\nA deep rumble washed over us and we had to close our eyes against the glare as a spacer took off. The invasion was on. We swayed into each other as the truck squealed to a halt and a now familiar and loathsome voice ordered us out. Our nemesis, the pressgang colonel, was waiting for us. He was backed up now by a radio operator and a gaggle of noncoms. Behind him companies, battalions of soldiers, were marching in good order to the waiting transports.\n\n\"Now hear this,\" the colonel bellowed. \"Those are good troops back there, and they need good officers. Unhappily all I have for them are you fat-bottomed desk types, the dregs of the base. So I'm going to split you up, one to every company, in the hopes that you will maybe get some experience before you get dead.\"\n\nThis was not good. I had promised Morton I would look after him. Which I could not do if we were in different companies. I sighed. I would have to break the first rule of military survival. Although it violated the primary army axiom\u2014keep your mouth shut and don't volunteer\u2014I volunteered. Stepping forward smartly and slamming my bootheels down as I snapped to attention.\n\n\"Sir! My bottom is lean, my gut is flat. I have field experience. I fire sharp-shooter, I instruct unarmed combat.\"\n\n\"And I don't believe you!\" he roared into my face.\n\nI threw him onto the ground, put my foot on his back, took away his gun, shot out one of the streetlights, helped him to his feet and handed back his weapon. His fierce glare melted almost to a smile as he wiped pebbles from his uniform.\n\n\"I could use a few more like you. You get a combat company. Name?\"\n\n\"Drem. I respectfully request Lieutenant Hesk here as exec. He is young and dumb but I have been training him.\"\n\n\"You got him. Move out. Any more volunteers?\"\n\nI grabbed up my bags before he could change his mind and hurried off toward the transports with Morton stumbling behind.\n\n\"I thought that I was going to die when you knocked him down,\" he gasped. \"You took some chance.\"\n\n\"Just being alive in the modern world is taking a chance,\" I pontificated, \"what with all the carcinogens and traffic accidents. And I think we can stop and put the bags down. Help has arrived.\"\n\nAn eager-looking sergeant, with a bald head, large moustache and two privates came trotting up and I returned his salute.\n\n\"I am Acting First Sergeant Blogh. If you are Captain Drem you are the new CO,\" the sergeant said.\n\n\"Right both times, sergeant. Get those men on these bags and let's go.\"\n\n\"Last of the company boarding now. We blast off in ten minutes.\"\n\n\"We can make it. Let's move.\"\n\nThe loading ramp vanished from behind our heels and the outer lock began to grind shut. We had to climb over boxes of equipment bolted to the deck to reach the stairs. Two flights up was the company, sprawled from wall to wall on their G pads. We dived for ours and were just horizontal when the red lights began flashing and the engines came to life.\n\nAs takeoffs go, it went. They poured on a lot more G's than a commercial transport would, but that is what the army is all about. When the acceleration dropped to one G, I stood and waved the sergeant over.\n\n\"Canteens full?\"\n\n\"Yes, sir.\"\n\n\"Let them drink, but no food for awhile...\"\n\nThere was a roar of sound from the speakers followed by an overly amplified voice. \"All commanding officers to deck two now. All COs, now.\"\n\n\"Lieutenant,\" I called out to a very queasy-looking Morton. \"Take over until I get back. Let the noncoms do all the work.\" I bent and added in a whisper, \"Don't let that bird-bag out of sight. If it is opened we will really be in the cagal.\"\n\nHe moaned slightly and I hurried away before he began to feel too sorry for himself. There were other officers climbing the gangway, all of them curious and expectant.\n\n\"Maybe now we will find out what this whole thing is about.\"\n\n\"They got to tell us something\u2014we been living on latrine rumors for a year.\"\n\nThe dining hall was not that big, so only the first arrivals got seats. The rest of us crowded in between the tables and leaned against the walls. An ancient sergeant checked us off his list when we came in. When he was satisfied he reported to a two-star general at the top table. The hum of conversation died down as the sergeant called for our attention.\n\n\"For them of you newly transferred to this division this here is your commanding officer, General Lowender, and he has an important announcement to make.\"\n\nThere was silence as the general turned to us, nodded sagely, and spoke.\n\n\"This is it, men. H-hour, D-day, the moment you have all been expecting, nay, looking forward to eagerly. The captain of this ship has reported that we are on course, with no chance of turning back now. So the secret orders can be opened.\"\n\nHe took up a large envelope heavy with red seals and tore it asunder, the sound of ripping paper loud in the silence. He held up the red-bound volume inside.\n\n\"This is it. You will have heard rumors that we plan a defensive action against Zemlija. That is wrong. Security planted those rumors to mislead the enemy. Our offworld enemies are many and their spies everywhere. That has explained our great need for secrecy. That need is passed. As you can tell we are now in space and heading toward a new world. A rich world. A world that lost contact with the rest of the galaxy thousands of years ago. And, more important, a world only we know exists. It is inhabited, but the natives are backward and do not deserve to have this verdant world for their own greedy selves. Is the machine ready? Good. General Zennor, the discoverer of this rich planet, will tell you about it in his own words.\"\n\nMy pulse hammered and I started to sink down before I realized that it was just a recording and I did not have to worry about being recognized. The lights dimmed a bit, the general took a digital recording from the envelope and slipped it into the projector. Zennor's repulsive hologrammed features floated before us.\n\n\"Soldiers of Nevenkebla, I salute you. You are now embarked on the greatest venture ever conceived by our country. Your victory in the field will enrichen and strenghten our fatherland so that none will ever dare consider an attack upon us. The riches of a new world will be ours. The riches of this world\u2014Chojecki!\"\n\nThere was a blare of tinny music as Zennor vanished to be replaced by the blue sphere of a planet floating in space. But if we were spared his image his flatulent voice still hammered in our ears.\n\n\"Chojecki. Rich, warm, fertile. It was a chance in a million that we discovered it. The ship I commanded was being followed by the killers of the League Navy and we used a random, untraceable jump to escape them. This noble planet was what we found. Perhaps there is a higher power that guided us to our destiny, perhaps the needs of our noble land were devined by benevolencies unknown to us.\"\n\n\"Perhaps that is a load of old cagal,\" someone whispered and there were mutters of agreement in the darkness. These were combat officers who preferred truth to propaganda. But there was no stopping Zennor.\n\n\"We landed and made a survey. It is a rich planet with immense reserves of heavy metals, abundant forests, untapped rivers to supply hydroelectric power. If there is anything at all wrong with Chojecki it is the present inhabitants.\"\n\nWe listened now with interest because there was an edge of irritation that Zennor could not keep out of his voice.\n\n\"They are disgusting people, with vile attitudes and strange perversions. We approached them openly, extending the hand of friendship. We offered them aid, companionship, trade, contact with a superior civilization. And do you know what we got in return? Do you know what they did?\"\n\nThe anger in his voice was obvious now, his audience eager.\n\n\"I'll tell you what they did. They did nothing! They completely ignored us, turned away from us\u2014rejected all civilized contact.\"\n\n\"Probably knew just what they were doing,\" someone said and the general shouted for silence. The planet popped out of existence and Zennor's image returned. His temper was under control now but there was a baleful look in his eye.\n\n\"So you officers will understand that what we are doing is for their own benefit. Ours is an old culture and a wise one. We extended the hand of friendship and aid and it was rejected. We have been insulted, offended by these peasants. Therefore, for their own good, we must show them that Nevenkebla pride does not take insult easily. They have asked for this and they are going to get it. We come in friendship to aid them. If they reject our aid they have only themselves to blame.\n\n\"Long live Nevenkebla!\n\n\"Long live positive peace!\"\n\nThe lights came up and we were all on our feet cheering like fools. I cheered as loud as anyone. Trumpets blared and a rather dreary piece of recorded music began playing. Everyone snapped to attention and sang the words of their despicable anthem.\n\nLong live Nevenkebla,\n\nLand of peace,\n\nLand of goodness, land of light.\n\nLong live our leaders,\n\nSweet men of mercy.\n\nLong shall we preserve\n\nLiberty's right. But dare to attack us\u2014\n\nAnd you got a fight!\n\nThere was more like this and I hummed along and was exceedingly happy when the singing ended. A holomap now hung in the air and General Lowender poked it with his finger.\n\n\"You will all be issued with maps and detailed orders. We will meet again tomorrow after you have studied them. At that time we will go over the plan of attack in detail. But as an overall approach\u2014this is what will happen.\n\n\"This division, the 88th, known as the Fighting Green Devils, has the honor of liberating this industrial section of the largest city called by the barbaric name of Bellegarrique. There are mines here and here, warehouses, a rail transportation system and here, ten kilometers away, a dam at the end of this lake that provides electricity for the city. For the benefit of these selfish people we will occupy all of these targets. We will liberate them from the futility of their rejection of our reasonable needs.\"\n\n\"A question, general,\" a colonel called out. The general nodded. \"What kind of defenses can we expect? How large is their army? How modern?\"\n\n\"That is a good question, colonel, and a vital one. We must be prepared for anything, any variety of attack, any kind of surprise. Because these people are very subtle, tricky, wily, treacherous. It seems that, well, in all of the contacts made by General Zennor, all of the investigations made by skilled agents, it seems that something very suspicious was found to be happening. It appears, on the surface that is, that these treacherous people have no army, no defenses\u2014they do not even have a police force!\"\n\nHe waited for the hum of excited voices to die down before he raised his hand for silence.\n\n\"Now we all know that this is impossible. A country needs defenses against attack, therefore every country must have an army for defense. The criminal elements in society would plunder and destroy were they not curbed by the police. Now we know that those are realities. We know that these treacherous people are hiding their cowardly armies from us. Therefore we must proceed with armed caution, ready for any sneak attack. We must free them from themselves. We owe that to them.\"\n\nI have never in my life heard such a load of old cagal\u2014but it impressed my military mates, who cheered wildly at the thought of all the nice mayhem to come.\n\nWhile I wondered what disasterous future lay in store for these simple people about to be liberated from their stupid and peaceful ways.\n\nLiberation by destruction was on the way!\n\nWe would free them even if we had to kill them all to do it!\nChapter 15\n\nI returned to my company, clutching the package of sealed orders and holding tight to the idea that this was the most insane endeavor I had ever heard of. Morton looked up when I entered the cabin.\n\n\"You are wearing a very worried look,\" he said. \"Something personal\u2014or should we all be worried?\"\n\n\"Anything I can do for you, captain?\" Sergeant Blogh asked, popping in the door behind me. They all wanted to know about the meeting. I threw the package onto the bed.\n\n\"Sergeant, what is the position regarding strong drink on troop transports about to go into action?\"\n\n\"It is strictly forbidden, sir, and a court-martial offense. But one of the spare tanks on the command car is filled with ninety-nine.\"\n\n\"Ninety-nine what?\"\n\n\"Ninety-nine percent pure alcohol. Cut half with water and stir in dehydrated orange juice.\"\n\n\"Since we are going into combat I am making a field appointment. Acting First Sergeant Blogh you are now First Sergeant Blogh.\"\n\nThere was a rattle as Morton dropped three canteen cups onto the table, a thud as a bag of orange crystals followed. I could see where he was getting adjusted to the army.\n\nThe sergeant came back with a twenty-liter jerrycan, which with added water would make forty liters of hundred-proof drink, which, in turn, should make this voyage more bearable.\n\nWe clanked mugs and drank deep.\n\n\"This stuff is pretty repulsive,\" Morton said holding out his empty cup for more. \"Can you now tell us what you found out?\"\n\n\"I have some good news and some good news. The first good news is that we are going to invade and occupy an incredibly rich and heretofore unknown planet named Chojecki. Second\u2014they don't appear to have any defenses of any kind. No military, no police, nothing.\"\n\n\"Impossible,\" the sergeant said.\n\n\"Anything is possible in the fullness of time and the width of the galaxy. Let us hope the report is correct because it will certainly make for an easy invasion.\"\n\n\"I think it is a trap.\" The sergeant still wasn't buying it. I nodded.\n\n\"The general seems to think the same thing. He is sure that there is a secret army in hiding.\"\n\n\"Not necessarily,\" Morton said. \"Before entering the army I was a student of history. So I can tell you. Diverse are the ways of mankind. As you have so truthfully stated, captain, in the fullness of time and the width of the galaxy there have been many kinds of societies, forms of government...\"\n\n\"You got governments, you got armies. That's the way it's got to be.\"\n\nThe drink was making the sergeant pugnacious and Morton maudlin. Time to close the bar.\n\n\"Right.\" I climbed to my feet and kicked the jerrycan of alcohol out of sight under the table. \"Sergeant, get the noncoms together. Tell them what I told you about the invasion, have them pass it on to the troops. That will be all for now.\"\n\nThe door closed behind the sergeant and Morton dropped his head onto the table and began to snore. He was sure a cheap drunk. I finished the repulsive, though certainly lethal, orange-alcohol mixture and heard my stomach rumble in protest. Or was it hunger? A long time and a lot of distance had gone by since that half-eaten steak in the officers' club. I dug into my pack and found some of the rations that we had been issued. A reddish tube was labeled HOTPUP MEAL. In smaller print it stated that it would feed two and could be opened by puncturing the white circle on the end. I pulled my combat knife out of my boot and stabbed the thing enthusiastically. It instantly grew exceedingly hot and burned my fingers. I dropped it onto the table where it rumbled and hissed and began to expand. I kept the knife ready in case it attacked me. There was a ripping sound as the casing split open and it expanded into an arm-long sausage. It looked repulsive but smelled quite good. I hacked off the end, impaled it on my knife and ate. The only thing missing was some beer.\n\nLife continued in this manner. Day followed day like the flapping of a great red sausage. As good as the hotpup had tasted at first bite, I grew to loathe the sorry sausages. As did we all since, due to some bit of mismanagement in the rush to load the transports and be away, hotpups were the only food that had been put aboard. Even the general had to eat the repulsive objects and he was not pleased.\n\nWe had meetings and briefings, all of which I duly passed on to the troops. We cleaned and recleaned our weapons, sharpened our knives, had short arm inspections to keep the medical officers on their toes, worked our way down through the alcohol until fifteen days had passed and the officers were ordered to yet one more meeting.\n\nThis one was different. The knot of field officers around General Lowender was buzzing with talk and much consultation of maps. As soon as we were all assembled the general stood\u2014and hammered his fists down on the table.\n\n\"The invasion has begun!\"\n\nHe waited until the cheering had died down before he continued. \"The first scouts have gone down and report no resistance. As yet. But we must be wary because all of this could be a dodge to suck us into a trap of some kind. You all have your orders, you know what to do\u2014so there is nothing more to be said. We touch down in two hours. Set your watches. So that is it. Except, boys\u2014give 'em hell!\"\n\nMore wild cheering followed before we hurried back to tell the troops what lay ahead.\n\n\"About time,\" was Sergeant Blogh's comment. \"The troops get soft, lose their edge lying around on their chunks like they been. About time.\"\n\n\"Get the noncoms and we'll go over the attack thoroughly just once more,\" I said, spreading out the now-familiar map. With the landing this close I had their undivided attention.\n\n\"Here is where we are supposed to touch down,\" I said, tapping the map. \"Now how many of you believe that the military pilot flying this thing will actually land on the correct spot?\"\n\nThe silence was complete.\n\n\"Right. I feel the same way. We are supposed to touch down at dawn, which means it will probably be dark\u2014or raining, or both. We will be first out because we got the longest way to go. I will lead in the command car which if it is dark and unless we are fired upon, will have its lights on so you can see it.\"\n\nSergeant Blogh frowned and touched his clipboard full of papers. \"A specific order here from the general states that no lights are to be used.\"\n\n\"Correct. And the general will be the last one to leave the ship and we will be first, and we have to get clear at once because there are tanks right behind us.\"\n\n\"Lights to be on!\" the sergeant said, firmly.\n\n\"I will proceed to the nearest hill or high point to check the map and see if we have landed where planned. If not I shall determine just where the hell we are and where we are going. The lieutenant here will muster the troops and follow the command car. When I know where we are going we will go there. Here. To the dam. To the generating plant that supplies the unpronounceable city of Bellegarrique with electricity. Our job is to seize and secure. Any questions? Yes, corporal?\"\n\n\"Can we leave hotpup rations here and live off the countryside?\"\n\n\"Yes and no. We take the hotpups in case we should run across the supply officer so we can stuff him with them. But we seize some native food soonest. It will be brought to me for testing before distribution. Anyone else?\"\n\n\"Ammunition. When do we get the ammo?\"\n\n\"It's on the disembarkation deck now. You will be issued with it when we go down there. You will see that each man is issued his lot. You will also see that no weapons are loaded. We don't want any guns going off inside this ship.\"\n\n\"We load after we hit the ground?\" the First Sergeant asked.\n\n\"You load when I tell you to. We do not expect any resistance. If there is no resistance we don't need to shoot any of the locals. If we don't shoot the locals the invasion will be an instant success. If the weapons aren't loaded they cannot shoot. The weapons will not be loaded.\"\n\nThere was a murmur of protest at this and beetle-browed Corporal Aspya expressed their mutual concern. \"Can't attack without loaded weapons.\"\n\n\"Yes you can,\" I said in my coldest voice. \"You can do what you are ordered to do. One weapon will be loaded. My weapon will be loaded. And I will shoot any man\u2014or officer\u2014who disobeys orders. More questions? No. Dismissed. We proceed to landing positions in thirty minutes.\"\n\n\"They are not happy about this ammunition thing,\" Morton said when the others had gone.\n\n\"Tough cagal. I am not happy about this killing thing. No ammo, no shooting. This will stop accidents happening.\"\n\nHe adjusted the straps on his pack, still worrying. \"They should be able to defend themselves...\"\n\n\"Morton!\" I ordered. \"Look in the mirror. What do you see? You see Lieutenant Hesk staring back and you are beginning to think like him. Remember, Morton\u2014you are a draft dodger, a man of peace, a reluctant soldier. Have you forgotten? Have you ever seen anyone killed?\"\n\n\"Not really. My aunt died and I saw her in the coffin.\"\n\n\"A man of the world... I've seen them die and it is not a nice thing to watch. And when you are dead you are dead forever, Morton. Remember that when you listen to the men of violence, the dogs of war, the sellers of hate. Do you want to die?\"\n\nAs I said this I placed the point of my knife against his throat. His eyebrows went up and up and he gasped out a No! My knife vanished as quickly as it had appeared and I nodded.\n\n\"You know what\u2014neither do I. And neither does anyone else on that planet down there where we are landing with thousands of military numbskulls, and I wonder how I ever got involved in all this!\"\n\nMorton sighed. \"Like me, you got drafted.\"\n\n\"And how we did! Like always, old men send young men to war. They ought to make the minimum draft age fifty-five. That would put an end to warfare pretty quickly let me tell you!\"\n\nAn alarm sounded and all the lights blinked. I looked at my watch.\n\n\"This is it. Let's go.\"\n\nThe disembarkation hold was a red-lit hell of men, machines and equipment. I struggled between them to my command car, which was poised at the top edge of the ramp. I kicked the shackles that held it down.\n\n\"They're explosive,\" Sergeant Blogh said. \"They blow loose as soon as the ramp drops.\"\n\n\"Seeing is believing. It is going to be very hard to drive out of here if they don't. Has all the gear been loaded on this car like I ordered?\"\n\n\"Just as you ordered, sir. Extra ammo under the backseat.\"\n\nI looked in and nodded agreement. I had filled a number of canteens with our hundred-proof orange juice and stowed them in this ammunition box. Also stowed in the box, under a false bottom, was that talking spy bird I had been lumbered with. I could not leave it lying about for someone to find.\n\nThe floor pushed up at me and I kept my legs bent. We were doing a slow two-G drop for the last part of the landing since we could not be lolling around on deceleration couches before going into combat. Except for superior officers, of course. I pushed hard and worked my way into the command car and sat down heavily next to the driver.\n\n\"Ignition on,\" I ordered. \"But don't hit the starter until the ramp drops.\"\n\nThe seat of the car came up and hit me just as the roar of the ship's engines stopped. We bounced on the springs and there were loud explosions from all sides. Hopefully the shackles blowing loose. With a great creaking the ramp moved\u2014then dropped.\n\n\"Start her up!\" I shouted as rain blew in from the darkness outside. \"And turn on the lights so we can see where we are going!\"\n\nThe command car roared down the ramp and hit the ground with a great crash and splash as we plowed through a puddle. Nothing was visible ahead except for the rain sheeting through the beams of the headlights. We churned on into the darkness. When I looked backward I could see the files of laden soldiers coming after us.\n\n\"There is an awful lot of water ahead, sir,\" the driver said, slamming on his brakes.\n\n\"Well turn you idiot, don't drown us. Turn right and move away from the transport.\"\n\nLightning split the sky and thunder rolled dramatically. I pounded the driver on the shoulder and pointed.\n\n\"There's a hill there, a rise of some kind, beyond that row of trees. Get us to it.\"\n\n\"That's a fence there, captain!\"\n\nI sighed. \"Ride us over it, driver, this is an armored combat vehicle not the little bicycle that you left at home with your mommy. Move it!\"\n\nWhen we ground to a halt on top of the low hill the rain was still just as fierce, but the sky was beginning to brighten with the first light of dawn. I moved the glowing map about to try and figure out where we were. At least I now knew where west was. Since, naturally, the sun on this planet rose in the west.\n\nThe rest of the company had reached the hill by this time so I had the vehicle's lights turned off. I could see better now, but the only thing I could identify was the towering bulk of our transport behind us. Columns of men and machines were still pouring from it and rushing off into the rain. As the light grew I became aware of a range of hills on the horizon and I tried to find them on the map. It was broad daylight before I had our position pinned down.\n\n\"Right!\" I said, climbing down and smiling at my damp troops. \"I know that you will all be pleased to hear that the pilot of our craft made an error in our favor. We are over halfway to our objective.\"\n\nA ragged cheer followed and I held up the map.\n\n\"A close reading of this map also indicates that the rest of the troops that are now on their way to occupy the city of Bellegarrique have a very long way to go. Made longer by certain errors in navigation. If you will look after their disappearing ranks you will see that they are going in the opposite direction to the one they need.\"\n\nThere was enthusiasm in their cheering now. Nothing builds the morale better than seeing someone else in the cagal. And the rain seemed to be lessening, changing to a sort of soupy mist. The rising sun touched this with red and revealed a distant white object above the trees. I climbed onto the hood to make sure. It was.\n\n\"All right, men. We are moving out. If you look in that direction you will see the dam, which is our objective. The command car will follow. I shall lead you on foot as a good commander should.\n\n\"Advance!\"\nChapter 16\n\nSome celestial switch was thrown, just after sunrise, and the rain stopped. A light breeze blew away the clouds as we strolled on through the steaming landscape. We had been cutting across country, but came now to a paved road that appeared to lead toward the not-too distant dam. I sent out scouts, who reported no enemy activity\u2014or no enemy at all for that matter. We followed the road, which meandered down a gentle hillside planted with trees on both sides.\n\n\"Report from one of the scouts,\" Sergeant called out. \"He is in that orchard and says that the trees are covered with ripe aval-gwlanek.\"\n\n\"Sounds repulsive. What are they?\"\n\n\"A kind of fruit they grow in Zemlija. Delicious.\"\n\n\"Tell him to bring a sample for analysis and evaluation.\"\n\nThe scout quickly appeared with his helmet full of ripe peaches, or at least that is what we called awal-gwlanek on Bit O' Heaven. I picked one up and smelled it, then looked at the scout's streaked face.\n\n\"Well, private, I see that you have already done an analysis and evaluation. How was it?\"\n\n\"Yummy, captain!\"\n\nI took a bite and nodded in agreement as the sweet juice washed the lingering taste of the last hotpup from my teeth. \"Fall out the troops, sergeant, take cover in that orchard, ten-minute break.\"\n\nWhen we marched on, the rumble of contented borborygmus sounded loud above the tramping boots. The dam grew closer, as did the generating plant and grouped buildings at its base. Water gushed from great pipes, while pylons and wires marched away toward the distant city. It looked peaceful and productive and there was no one in sight. I signaled a halt and sent for the NCOs.\n\n\"I will now outline our plan of attack. But before I do we will have a weapons inspection. Starting with you, First Sergeant.\"\n\nHis face was expressionless as he passed me his gun. I pressed the magazine release, saw that it was empty, looked into the equally empty chamber and passed it back. I did this with the others and was quite pleased with myself until I reached the hulking form of Corporal Aspya. Instead of handing me his gun he held it across his chest.\n\n\"I can save you looking, captain. It's loaded.\"\n\n\"That was done despite my direct order, ex-corporal. Private, you will now hand me your weapon.\"\n\n\"A soldier is not a soldier when he is unarmed, sir,\" he said grimly, unmoving.\n\n\"That is true,\" I said, going on to the next noncom. Out of the corner of my eye I saw him look around as though seeking aid. As soon as his eyes were off me I lashed back with my extended hand and caught him on the neck with the edge. It was a cruel blow: he had a loaded gun. He fell unconscious on the ground and I pulled the weapon from his limp hands, ejecting the cartridges one by one into the mud.\n\n\"Sergeant Blogh. I want this man in the command car, under guard and under arrest.\"\n\n\"Is the guard to be armed, sir?\"\n\n\"Guard to be armed, weapon to be loaded. Lieutenant Hesk will perform guard duty. Now, this is our plan of attack.\"\n\nThey listened in silence, impressed by my quick violence. I was ashamed of striking the cowardly blow\u2014but I wouldn't let them know that. Better one sore neck than guns going off and people getting killed. I could trust Morton not to pull any triggers\u2014and felt much better with him out of the way for the present. I assigned targets to every squad, but saved the main building for myself.\n\n\"So there it is. Get your men into position, then report back to me. When everything is covered I will enter and capture the control room. Now\u2014move out.\"\n\nMy bold little army dispersed, attacking by the book. Rushing forward a few at a time, covering each other. After a few minutes the noncoms began radioing in. Objectives reached, no opposition, no one seen yet. Now it was up to me. Followed by the first sergeant and his squad I marched resolutely up the steps of the generating station and threw open the door. It opened directly into the turbine room. The turbines spun, the generators turned, there was no one in sight.\n\n\"Fully automated,\" the sergeant said.\n\n\"Looks that way. Let's find the control room.\"\n\nTension grew as we scuttled down the hallway. I was very glad that mine was the only loaded weapon. I kept the pistol in my hand\u2014but the safety was on since I had no intention of pulling the trigger: it was a prop to cheer the troops.\n\n\"Someone is in there, captain. See!\"\n\nThe soldier was pointing at a frosted-glass door. A man's silhouette moved across it then vanished.\n\n\"Right, this is it, here we go, follow me!\"\n\nI took a deep breath\u2014then threw the door open. Jumped inside and heard the squad move in after me. The gray-haired man stood in front of the control panel, tapping a dial.\n\n\"Ne faru nenion!\" I shouted. \"Vi estas kaptito. Manoj en la aeron!\"\n\n\"How very interesting,\" he said turning about and smiling. \"Strangers speaking a strange tongue. Welcome, strangers, welcome to Bellegarrique Generating Plant Number One.\"\n\n\"I can understand you!\" I said. \"You are speaking a dialect of Low Ingliss, that we speak on Bit O' Heaven.\"\n\n\"Can't say that I have heard of the place. Your accent is strange, but it certainly is the same language.\"\n\n\"What is he saying?\" the first sergeant asked. \"You speak his lingo?\"\n\n\"I do. Learnt it in school.\" Which was true enough. \"He is welcoming us here.\"\n\n\"Anyone else around?\"\n\n\"Good question. I'll put it to him.\"\n\n\"There are more staff, of course, but they'll be asleep. Shift workers. You must tell me more about yourself and your friends. My name is Stirner. Might I ask yours?\"\n\nI started to answer, then drew myself up. This was no way to run a war. \"My name is not important. I am here to tell you that this planet is now controlled by the armed forces of Nevenkebla. If you cooperate you will not be harmed.\"\n\nI translated this into Esperanto so my soldiers would know what was happening. And told the sergeant to pass the word about the shift workers. Stirner politely waited until I was finished before he spoke.\n\n\"This is all very exciting, sir! Armed forces you say? That would mean weapons. Are those weapons that you are carrying?\"\n\n\"They are. And be warned\u2014we will defend ourselves if attacked.\"\n\n\"I wouldn't concern myself with that. As a firm believer in Individual Mutualism I would never harm another.\"\n\n\"But your army\u2014or your police would!\" I said, trickily.\n\n\"I know the words, of course, but you need not fear. There is no army here, nor do we have a police force. May I offer you some refreshments? I am being a very bad host.\"\n\n\"I can't believe this is happening,\" I muttered. \"Sergeant, get a connection to General Lowender's staff. Tell them we have made contact with the enemy. No sign of resistance. Informant says no armed forces, no police.\"\n\nClosely watched by my gun-gripping troops, Stirner had opened a cabinet and removed a tall and interesting bottle. He set this on a table along with a tray of glasses.\n\n\"Wine,\" he said. \"A very good one, for special guests. I hope you and your associates will enjoy it.\" He handed me a glass.\n\n\"You taste it first,\" I said with military suspicion.\n\n\"Your politeness, nameless sir, puts me to shame.\" He sipped then passed me a glass. It was very good.\n\n\"Got the general himself,\" the sergeant called out urgently, running over with the radio.\n\n\"Captain Drem speaking.\"\n\n\"Drem\u2014what does this report mean? Have you found the enemy?\"\n\n\"I've occupied the generating plant, sir. No casualties. No resistance encountered.\"\n\n\"You are the first to make contact. What are their defenses like?\"\n\n\"Nonexistent, general. No resistance was offered of any kind. My prisoner states no military, no police.\"\n\nThe general made noises of disbelief. \"I'm sending a chopper for you and the prisoner. I want to question him myself. Out.\"\n\nWonderful. The last place I wanted to be was with the top brass. There was too good a chance of General Zennor appearing and recognizing me from the bad old days when he was known as Garth. Self-survival urged me to climb into a hole. But weighed against my personal needs was the chance that I might be able to save lives. If I could convince the military numbskulls that there really would be no resistance. If I didn't do that, surely some trigger-happy cagal-kopf was sure to get nervous and start firing. All of his jumpy buddies would then join in and... It was a very realistic scenario. I had to make some effort to avoid it.\n\n\"An order from the general,\" I told my expectant troops. \"I'm to bring him the prisoner. Transport is on the way. You are in charge, Sergeant Blogh, until Lieutenant Hesk gets here to relieve you. Take over. And take care of the wine.\"\n\nHe saluted and they were grabbing for the bottle when I left. Would such simple military pleasures were mine.\n\n\"You're coming with me,\" I told Stirner, pointing toward to the door.\n\n\"No, my duty is here. I am afraid I cannot oblige you.\"\n\n\"It is not me you are going to oblige, it is your own people. There is a big army out there. All of them armed with weapons like this. They are now invading your country and are taking it over. People could be killed. But lives can be saved if I take you to the commanding officer and you manage to convince him there will be no resistance from your people. Do you understand me?\"\n\nA look of horror had been growing on his face as I talked. \"You are serious?\" he gasped. \"You mean what you are saying.\" I nodded grimly. \"Of course, then, yes. Incomprehensible, but I must come. I can't believe this.\"\n\n\"The feeling is mutual.\" I led him to the door. \"I can understand not having an army, all civilized worlds get by without the military. But the police, a necessary evil I would say.\"\n\n\"Not for those who practice Individual Mutualism.\" He was brightening up now at this chance to deliver a little lecture.\n\n\"I never heard of it.\"\n\n\"How unfortunate for you! At the risk of simplifying I will explain...\"\n\n\"Captain Drem, I got to talk to you!\" the fallen corporal said, climbing out of the command car despite Morton's feeble efforts to stop him. He stopped in front of me, snapped to attention and saluted.\n\n\"I now see the error of my ways, sir. I thought because you are young and look weak that I knew better than you, so I disobeyed an order and loaded my gun. I know now that I was wrong and you were right and I respectfully request a second chance since I am a thirty-year man and the army is my career.\"\n\n\"And how do you know now that I was right, Private Aspya?\"\n\nHe looked at me, eyes aglow. \"Because you beat me, sir! Knocked me down fair and square. A man gotta do what a man gotta do\u2014and you did it!\"\n\nWhat kind of macho-cagal was this? He had disobeyed a reasonable command that was aimed at avoiding violence. Only when I had bashed him unconscious did he feel that I was right. The mind reeled at this kind of perverse, inverted logic\u2014and I really didn't have time to think about it. About all I could do was play along and forget about it.\n\n\"You know, ex-corporal, I think that I believe you. It takes a real man to admit that he was wrong. So even though you are a miserable low private and I am an on-high captain\u2014I'm going to shake your hand and send you back to duty!\"\n\n\"You're a real man, captain, and you will never regret this!\" He pumped away at my hand, then staggered off knuckling a tear from his eye. There was a growing clatter from the sky and shadow drifted across us and I looked up to see the chopper dropping down toward us.\n\n\"Morton\u2014you're in charge until I get back. Go to Sergeant Blogh and take command and let him make all the decisions and then agree with him.\"\n\nHe could only nod as I guided Stirner to the chopper and climbed in behind him.\n\n\"Take us to the general,\" I ordered the pilot. Then sighed heavily. I had the feeling that I was putting my head into the noose and settling it nicely around my neck.\n\nBut, really, I had no other choice.\n\n\"I have read of such vehicles in the history books,\" Stirner said, looking out of the window with admiration as we rattled skyward. \"This is a very important moment for me, nameless sir.\"\n\n\"Captain, you can call me captain.\"\n\n\"My pleasure to meet you, Captain. And thank you for the opportunity to explain to your leaders that they may come in peace. They must not be afraid. We would never harm them.\"\n\n\"It was the other way around that I was worried about.\"\n\nThere was no more time for gossip because the chopper was dropping down beside an armored column of tanks. Tables, armchairs, and a wet bar, had been set up under a tent in the field close by, and we settled down just out of rotor-blast of the officers assembled there. I jumped down, delivered a snappy salute and relaxed. Zennor wasn't there. I turned and helped Stirner get out and pushed him towards General Lowender.\n\n\"This is the prisoner, sir. He speaks a vile local language which I just happened to have learned in school so I can translate.\"\n\n\"Impossible,\" he said grimly. \"You are an infantry officer, not a translator. Major Kewsel is the staff translator. Major, translate!\"\n\nThe dark-haired major shouldered me aside and stood before the prisoner.\n\n\"Kion vi komprenas?\" he shouted. \"Sprechten zee Poopish? Ancay ooyay eekspay Igpay Atinlay? Ook kook Volupook?\"\n\n\"Very sorry, sir, but I don't understand a word that you are saying.\"\n\n\"Got him!\" the major announced happily. \"A little-known dialect, spoken on dreary planets trundling heavily around dark stars. I learned its boring cacophonies when I was involved in the meat trade years ago. Importing porcuswine cutlets...\"\n\n\"Cut the cagal, major, and translate. Ask him where the army is and how many police stations there are in this city.\"\n\nI listened with some interest as the major, despite his inborn desire to talk and not listen, finally elicited the same information that I had. The general sighed unhappily.\n\n\"If this is true\u2014then we just can't shoot them down in cold blood.\" He turned to me. \"And you are positive there was no resistance offered?\"\n\n\"None, sir. It apparently goes against their strongest beliefs. May I congratulate you, general, on the first bloodless invasion in the known universe! You will soon have captured this entire planet for the greater glory of Nevenkebla\u2014without losing a single soldier.\"\n\n\"Don't cheer too soon, captain. Medals don't go to generals who bring back the troops intact. Battle! That's where the glory is! There will be fighting, mark my words. It is human nature. They can't all be cowards on this planet.\"\n\n\"Lowender\u2014what's happening?\" a familiar voice asked and my blood temperature fell about ten degrees. I did not move, stood stiffly with my back to the speaker. The general pointed.\n\n\"We have our first prisoner, General Zennor. I have been questioning him. He talks nonsense. No army, no police he says.\"\n\n\"And you believe him? Where was he caught?\"\n\n\"At the generating plant, by Captain Drem there.\"\n\nZennor glanced at me, then away. I kept my back straight and my face expressionless as he suddenly turned around to face me again.\n\n\"Where do I know you from, captain?\"\n\n\"Training, sir. Maneuvers,\" I said in the deepest voice I could muster. He walked over and pushed his face close to mine.\n\n\"That's not true. Somewhere else. And you were with someone else...\"\n\nHis eyes lit with recognition and he stabbed his finger at me. \"The Bishop! You were with The Bishop\u2014\"\n\n\"And you killed him!\" I shouted as I dived and got the three-seconds to death stranglehold on his neck.\n\nOne second... unconscious.\n\nSecond second... limp.\n\nThird...\n\nAll the lights went out. There was a great deal of pain in the back of my head and then nothing. My last thought was\u2014had I held the grip through the third second?\nChapter 17\n\nA measureless time later I was aware of pain spreading from the back of my head down through my body. I moved to get away from it but it would not leave. It was dark\u2014or were my eyes closed? I had no desire to find out. Everything hurt too much. I groaned and it sounded so good that I did it a second time. Vaguely, through the groaning, I was aware of my shoulders being lifted and something wet on my lips. I gurgled and spluttered. Water. It tasted very good. I drank some and felt slightly better. The pain was still there, but not so much that I couldn't risk opening one eye. I did. A face swam blurrily above me and after a certain amount of blinking it became clear.\n\n\"Morton...?\" I muttered.\n\n\"None other.\" With an expression of abject gloom. He pulled at me until I sat against the wall and my head appeared to be exploding in tiny bits. His voice barely penetrated.\n\n\"Take this, in your mouth. Drink some more water. The doctor said you were to swallow it when you came to. For the head.\"\n\nPoison? No such luck. Medicine. The pain ebbed and rose and finally slipped away to a dull ache. I opened my eyes all the way and saw a sad-looking and bruised Morton framed against a background of bars.\n\n\"Is he dead?\" I croaked.\n\n\"Who?\"\n\n\"General Zennor.\"\n\n\"He looked very much alive when he was here about a half an hour ago.\"\n\nI sighed drearily\u2014and with mixed emotions. I had wanted vengeance, wanted Zennor to pay heavily for being responsible for The Bishop's death. I thought that I had wanted him dead as well. But having tried murder this once, really tried it, I was glad that I had been stopped. Now that I had made my first homicidal attempt I discovered that I did not really enjoy the process of killing people. I was a failed killer. And in failing I had really got myself in the cagal. And had pulled Morton in too.\n\n\"Sorry about all this,\" I said. \"I got so carried away I never stopped to think that I would probably implicate you as well.\"\n\n\"Sergeant Blogh turned me in when the MPs came to investigate. He knew I wasn't an officer. I told them everything. Even before they knocked me around.\"\n\n\"I'm to blame for what happened.\"\n\n\"Don't think like that. Not your fault. They would have got me sooner or later, one way or the other. The army and me, we are just not on the same plane. You did your best, Jak.\"\n\n\"Jim. Real name is Jim diGriz. From a distant planet.\"\n\n\"Nice to meet you, Jim. You a spy?\"\n\n\"No. Just here to right a wrong. Your General Zennor was responsible for the death of my best friend. I came here looking for him.\"\n\n\"What about that talking bird and all the other stuff?\"\n\nI touched my fingers to my lips and looked at the door. Morton shook his head in puzzlement. I spoke up before he could add anything.\n\n\"You mean that talking bird joke I was going to tell you, about the kid in school who had the talking bird who turned into an alcoholic and became a missionary? I remember the joke\u2014but I forgot the punchline.\"\n\nMorton was now staring at me as if I had gone out of my mind. I looked around and discovered that I was lying on a thin mattress resting on a very dusty floor. I used my finger to write QUIET\u2014THEY MIGHT BE LISTENING! in the dust. I looked at his face until he finally caught on, then rubbed out the message. \"Anyway, Morton, I don't feel like telling jokes now. Where are we?\"\n\n\"Big building in the city. Looks like the army took it over. They must be using it for a headquarters or something. All I know is that they brought me here in a rush, worked me over then dumped me in here with you. The building is full of soldiers.\"\n\n\"Any civilians?\"\n\n\"None that I saw...\"\n\nWe both looked up as the lock rattled in the door and it opened. A lot of armed MPs pushed in and pointed their guns at us. Only after this did General Zennor enter. He had a bandage around his neck and the urge to kill in his eye.\n\n\"Are you sure that you are safe now, Zennor,\" I said as sweetly as I could. He came over and kicked me in the side.\n\n\"Aren't we brave\u2014\" I gasped through the pain. \"Kick a wounded man lying down.\"\n\nHe drew his boot back again, thought about it, then drew his pistol and pointed it between my eyes.\n\n\"Get the other prisoner out of here. Leave us alone. Bring me a chair.\"\n\nOne thing about the military, they just relish following orders. With much shouted commands and stamping of boots Morton was hustled away, the MPs vanished, a wooden chair appeared and was placed respectfully under the general's bottom. He sat down slowly without taking his eyes or the gun muzzle off of me. He did not speak until the door clicked shut.\n\n\"I want to know how you got here, how you followed me. Everything.\"\n\nWhy not? I thought, rubbing my sore side. I was too knocked about to make up any complex lies\u2014nor was there any need. The truth would be easier. With a little editing of course.\n\n\"Everything, Zennor? Why not. The last time I saw you was when you sold us down the river on Spiovente. That is a rough planet, and no place for an old man like The Bishop. He died there\u2014and that makes you responsible for his death.\"\n\nHe touched the bandage on his neck and snarled, \"Get on with it.\"\n\n\"Little more to tell. A few wars, murder, torture, the usual thing. I survived only to be rescued by the League Navy, who also arrested me and brought me here. I escaped from them and found you because of your one big mistake.\"\n\n\"What nonsense are you speaking?\"\n\n\"No nonsense. Truth, Captain Garth. Didn't you have the girl, Bibs, arrested for selling dope?\"\n\n\"That is not important.\"\n\n\"It was to Bibs! She is a free woman now, you will be unhappy to hear, and before she left she told me how to find you. End of story.\"\n\nHe weighed the gun thoughtfully, his finger caressing the trigger. I tried not to notice it.\n\n\"Not quite the end yet. You are the spy who landed in Marhaveno?\"\n\n\"Yes. And penetrated your slack and incompetent army. Then rose in rank until I got you by the neck and gave you a good choking. When you wake up at night in a cold sweat remember\u2014I could have shot you just as well. Now, are you going to shoot me, or are you just playing with that gun?\"\n\n\"Don't tempt me, little man. But that would be a waste. I shall put your death to better use. You and your associate will be tried and found guilty of a number of charges. Attacking a superior officer, impersonating an officer, threatening military security. After which you will both be shot. In public.\"\n\n\"And what will that accomplish?\"\n\n\"It will convince the stubborn people of this planet that we do what we say. They are a bloodless, spineless lot that let us walk in and take their planet away from them. Now they whine that they wish to have it back. They refuse to do any work until we leave. They have all walked away from their jobs. The city will soon be paralyzed. Your death will change that.\"\n\n\"I don't see how.\"\n\n\"I do. They will then know that I mean what I say. We will take hostages and shoot them if they do not cooperate.\"\n\nI was on my feet, anger burning me. \"You are a mean and worthless bastard, Zennor. I should have killed you when I had the chance.\"\n\n\"Well you didn't,\" he said. Then fired as I jumped at him.\n\nThe bullet must have missed but the explosion deafened me. I fell and he kicked me again. Then the room was full of MPs all trying to stomp on me at once.\n\n\"Enough!\" Zennor shouted and boots fell away. I was on all fours, looking up at him through a haze of blood. \"Clean him up, fresh uniform, same for the other one. Trial in two hours.\"\n\nI must have been punchy from the kicking because I was only vaguely aware of being worked on, of Morton reappearing, of time oozing by. I finally came back almost to reality when I found him pulling off my shirt.\n\n\"Let go. I can do it myself.\" I blinked at the fresh uniform on the chair, at Morton uniformed and crisp and a private once again. My new-old rank as well I saw. I dropped the bloody shirt on the floor, then pulled off my boots so I could take off the trousers as well.\n\nBoots. Boots? Boots!\n\nI tried not to smirk or let on in case the place was bugged.\n\n\"You know about the trial?\" Morton nodded glumly. \"How much more time do we have?\"\n\n\"About an hour.\"\n\nAs I talked I slid my fingers into my right boot and flipped open the tiny compartment concealed in the heel. An hour. We would be long gone by that time! I tried not to let my newfound glee show on my face. Slip out the lockpick, slip open the door, slip out into the hall, and vanish into military anonymity.\n\nExcept the lockpick was no longer there...\n\n\"Zennor gave me a funny message for you,\" Morton said. \"He told me to wait until you took off your shoes then I was to tell you that you were not going to get out that way. I don't know what it means\u2014but he said you would know.\"\n\n\"I know, I know,\" I said wearily, and finished changing. It takes a crook to catch a crook, and that crook Zennor obviously knew all about lockpicks.\n\nThey came for us an hour later. I'll say this much, they made a great military show of it with much crashing of polished weapons, shouting of orders, thudding of bootheels. Neither Morton nor I wished to play along with this militaristic tomfoolery but had little choice since we were chained and dragged. Down the hall, down the stairs and into the street beyond. With more crashing and shouting we were hauled up onto a newly constructed platform that was apparently going to be the venue of the show trial. Complete with guards, judges, barred cell, buglers\u2014and a large crowd of watching civilians below. Obviously brought there by force since they were still ringed by armed soldiers. A half dozen of them were also seated on the platform as well. All gray-headed or bald and among them I recognized Stirner from the generating plant. As soon as he saw me being locked in the cage he stood and walked over.\n\n\"What are they doing to you, captain? We understand none of this...\"\n\n\"You are talking Esperanto!\" I gaped.\n\n\"Yes. One of our leading linguists found this interesting language in his library. A number of us learned it last night since there have been communication problems with\u2014\"\n\n\"Seat that man at once!\" Zennor ordered from the bench where he was, of course, the head judge. Military justice.\n\n\"I can't believe that this is happening!\" Stirner said as he was hurried back to his chair.\n\nThough he and his companions tried to protest they were silenced by the blare of bugles and the dreary evidence of the mock trial. I pretended to fall asleep but was kicked awake. Morton stared vacantly into space. I really did doze off during the summing up, I still did not feel that good, and only paid any attention when we were both dragged to our feet. Zennor was speaking.\n\n\"... evidence given against you. It is therefore the judgment of this court that you be taken from here to a place of detention and there be held until oh-eight hundred hours tomorrow from whence you will be taken to a place of execution where you will be shot. Take them away.\"\n\n\"Some justice!\" I shouted. \"I haven't been allowed to say a word during this farce of a trial. I wish to make a statement now.\"\n\n\"Silence the prisoner.\"\n\nA hairy hand was pressed over my mouth, then replaced by a cloth gag. Morton was treated the same way although he seemed barely conscious of what was happening. Zennor waved over the translator with the microphone.\n\n\"Tell them to listen to a very important announcement,\" he said. The amplified translation boomed over the crowd, which listened in silence.\n\n\"I have brought you people here since there has been willful disobedience on the part of too many of you. This will change. You have watched Nevenkebla justice taking place. These two prisoners have been found guilty of a number of criminal charges. The penalty for being found guilty of these charges is death. They will die at eight tomorrow morning. Do you understand this?\"\n\nA murmur went through the listening crowd and Stirner stood up. The guards reached for him but Zennor stopped them.\n\n\"I am sure I speak for all here,\" Stirner said, \"when I ask for some explanation. This is all very confusing. And the most confusing part of all is how do these men know about their deaths tomorrow? They do not look ill. Nor do we understand your knowledge of the precise hour of their demise.\"\n\nZennor looked at him with disbelief\u2014then lost his temper.\n\n\"Are you people that stupid? Was this backward planet settled by hereditary morons? These two men are going to die tomorrow because we are going to shoot them with guns. This is a gun!\" he screamed, pulling his pistol and firing it into the wooden stand before him. \"It fires bullets and they make holes in people and tomorrow guns will kill these two criminals! You people aren't vegetarians. You butcher animals for food. Tomorrow we butcher these two men in the same way. Now is that clear enough for you?\"\n\nStirner, white-faced, dropped back into his chair. Zennor grabbed the microphone and his amplified voice rolled over every one.\n\n\"They will die and you will watch them die! Then you will understand and you will do as we order and do what we tell you to do. If you disobey you will be as guilty as these two men and you will be shot like these two men. We will shoot you and kill you and keep on shooting and killing you until the survivors understand us and obey us and do exactly as they are told...\"\n\nHis words trickled down and died as he lost his audience. The men on the platform stood up, turned their backs on him and walked away. As did everyone else in the street. They did not push or use violence. When the soldiers grabbed them they simply struggled to get free without striking out. Meanwhile the others who were not held pushed by and walked away. The street was a struggling shambles. Zennor must have realized this, seen the impossibility of accomplishing anything without violence at the moment. He was vicious and deadly\u2014but not stupid.\n\n\"You may all leave now,\" he announced. \"Let them go. You will all leave and remember what I have said and tomorrow morning you will come back here and watch these prisoners die. After that your new orders will be issued. And you will obey them.\"\n\nHe signaled to our guards and Morton and I were pulled to our feet and dragged back to our cell. Since no further orders about us had been issued we were thrown into our prison room, still chained and gagged.\n\nWe looked at each other in muffled silence as the key was turned in the door.\n\nIf my eyes looked like Morton's eyes, then I was looking very, very frightened.\nChapter 18\n\nWe lay like this for an uncomfortable number of hours. Until the door was unlocked and a burly MP came in with our dinner trays. His brow furrowed as he looked down at us. I could almost see the feeble thoughts trickling through his sluggish synapses. Got food. Feed prisoners. Prisoners gagged. No can eat... Just about the time his thought processes reached this stage he turned and called over his shoulder.\n\n\"Sergeant. Got kind of a problem here.\"\n\n\"You got a problem if you are bothering me for no reason,\" the sergeant said as he stamped in.\n\n\"Look, sarge. I got this food to feed the prisoners. But they're gagged and can't eat...\"\n\n\"All right, all right\u2014I can figure that one out for myself.\"\n\nHe dug out his keys, unlocked my chains, and turned to Morton. I emitted a muffled groan through my gag and stretched my sore fingers and struggled to sit up. The sergeant gave me a kick and I groaned harder. He was smiling as he left. I pulled off the gag and I threw it at the closing door. Then pulled over the tray because, despite everything, I was feeling hungry. Until I looked at it and pushed it away.\n\n\"Hotpups,\" Morton said, spitting out bits of cloth. \"I could smell it when they brought the trays in.\"\n\nHe sipped some water from his cup and I joined him in that. \"A toast,\" I said, clanking his cup with mine. \"To military justice.\"\n\n\"I wish I could be as tough as you, Jim.\"\n\n\"Not tough. Just whistling in the dark. Because I just don't see any way out of this one. If I still had my lockpick we might have a slim chance.\"\n\n\"That's the message the general gave me?\"\n\n\"That's it. We can't do much now except sit and wait for morning.\"\n\nI said this aloud not to depress Morton any more, surely an impossibility, but for the ears of anyone listening to planted bugs. There might be optic bugs as well, so I wandered about the cell and looked carefully but did not see any. So I had to risk it. I ate some of my hotpup, washing down the loathsome mouthfuls with glugs of water, while at the same time picking up the discarded chains as silently as I could, balling them around my fist. The dim MP would be back for the trays and he might be off guard.\n\nI was flat against the wall, armored fist ready, the next time the key rattled in the lock. The door opened a finger's width and the MP sergeant called out.\n\n\"You, behind the door. Drop those chains now or you ain't going to live to be shot in the morning.\"\n\nI muttered a curse and hurled them across the room and went and sat by the back wall. It was a well-concealed optic bug.\n\n\"What time is it, sergeant?\" Morton asked.\n\n\"Sixteen-hundred hours.\" He held his gun ready while the other MPs removed the trays and chains.\n\n\"I got to go to the toilet.\"\n\n\"Not until twenty-hundred. General's orders.\"\n\n\"Tell the general that I am already potty trained,\" I shouted at the closing door. To think that I actually had had his neck in my hands. If they hadn't hit me\u2014would I have gone the full three seconds and killed him? I just didn't know. But if I hadn't been ready then\u2014I felt that I was surely ready for it now.\n\nThey took us down the hall later, one at a time and heavily guarded, then locked us in for the night. With the lights on. I don't know if Morton slept, but with the general bashing about I had had even the thin mattress felt good. I crashed and didn't open my eyes again until the familiar rattle at the door roused me.\n\n\"Oh-six hundred and here is your last meal,\" the sergeant said with great pleasure.\n\n\"Hotpups again?\"\n\n\"How did you ever guess!\"\n\n\"Take them away. I'll die cursing you. Your name will be the last thing on my lips.\"\n\nIf he was impressed by my threat he didn't show it. He dropped the trays onto the floor and stamped out.\n\n\"Two hours to go,\" Morton said, and a tear glistened in his eye. \"My family doesn't know where I am. They'll never know what happened to me. I was running away when I was caught.\"\n\nWhat could I say? What could I do? For the first time in my short and fairly happy life I felt a sensation of absolute despair. Two hours to go. And no way out.\n\nWhat was that smell? I sniffed and coughed. It was very pungent\u2014and strong enough to cut through my morbid gloom. I coughed again, then saw a wisp of smoke rising from the floor in the corner of the room. Morton had his back turned to it and seemed unaware. I watched, astonished, as a smoking line appeared in the floor, extended, turned. Then I could see that there was a rough circle of dark fumes coming from the wood. Morton looked around coughing.\n\n\"What...?\" he said\u2014just as the circle of wooden flooring dropped away. From the darkness below a man's gray head emerged.\n\n\"Don't touch the edges of the opening,\" Stirner said. \"It is a very strong acid.\"\n\nThere were shouts and running feet in the hall. I dragged Morton to his feet, hurled him forward.\n\n\"They are watching us\u2014can hear everything we say!\" I shouted. \"Fast!\"\n\nStirner popped down out of sight and I pushed Morton after him. Jumped into the opening myself as the lock rattled on the door.\n\nI hit and fell sideways and rolled and cursed because I had almost crushed Morton. He was still dazed, unresponding. Stirner was pulling at his arm, trying to move him toward another hole in the floor of this room. I picked Morton up bodily and carried him to the opening, dropped him through. There was a shriek and a thud. Stirner went after him, wisely using the ladder placed there.\n\nHeavy footsteps sounded in the room above. I jumped, grabbed the edge of the opening, hung, and dropped. Into a half-lit basement.\n\n\"This way,\" a girl called out, holding open a door in the far wall.\n\nStirner was struggling with Morton, trying to lift him. I pushed him aside, got a grip and threw Morton over my shoulder. And ran. The girl closed the door behind us and locked it, then turned to follow Stirner. I staggered after them as fast as I could. Out another door that was also locked behind us, down a hall and through more doors.\n\n\"We are safe for the moment,\" Stirner said, closing and securing a final door. \"The cellars are quite extensive and all of the doors have been locked. Is your friend injured?\"\n\n\"Glunk...\" Morton said when I stood him on his feet.\n\n\"Just dazed, I think. I want to thank...\"\n\n\"Discussions later, if you please. We have to get you away from here as soon as possible. I must leave you on the other side of this door, so you will follow Sharla here. The street outside is filled with the people who have gathered as ordered for the ceremony of killing. They have all been told that you are coming so they are all very happy to be of help in such an unusual matter as this.\"\n\n\"Be careful. There was an optical spying device in the room where we were held. They saw you and will be looking for you.\"\n\n\"I will not be seen. Goodbye.\"\n\nHe opened the door and was gone, vanished in the crowd outside. Our guide motioned us forward and held the door open. I took Morton's arm, he was still woozy, and we went after her.\n\nIt was strange and utterly unbelievable. There were thousands of people jammed into the street: men, women and children. And not one of them looked our way or appeared to take any notice of us at all. Yet when we stepped toward them they pressed tight against each other to make room for us to pass, moving apart again as soon as we had gone by. It was all done in silence. We walked through a continually opening and closing clear space, just large enough to let us get by.\n\nI heard shouts in the distance\u2014and shots! The crowd stirred and murmured at this, then they were silent again. We moved on. The crowd was in motion now as well, stirring and reforming. I realized it was deliberate, so that anyone watching from the windows above would not see us making our escape.\n\nOn the other side of the street a door opened as we approached, was locked behind us by a motherly-looking gray-haired woman.\n\n\"This is Librarian Grene,\" our guide, Sharla said. \"She is the one who organized your escape.\"\n\n\"Thank you for our lives,\" I said, which is about as thankful as you can get.\n\n\"You are still not safe,\" she said. \"I searched the library for all the books that I could find on prisoners and escapes. Then, with the aid of our engineers adapted the formula we have just used. But I do not know what to advise next. The plan that I found in this book just carried to this point, I am sorry to say.\"\n\n\"Don't be\u2014it was perfect!\" Morton said. \"You and your people have done incredibly well. And it just so happens that my friend Jim is the galaxy champion of escapes. I'm sure he will know what to do next.\"\n\n\"Do you?\" the librarian asked.\n\n\"Of course!\" I said with newfound enthusiasm. \"We are well away from the enemy, in hiding\u2014so they will never catch us now. How big is this city?\" Grene pursed her lips and thought.\n\n\"An interesting question. On a north to south axis I would say the total diameter is...\"\n\n\"No, wait! Not physically big\u2014I mean how many inhabitants?\"\n\n\"In the last population census there were six hundred and eighty-three thousand people resident in the greater Bellegarrique area.\"\n\n\"Then we are more than safe for the moment. I know these military types, know exactly what they will do. First they will run about in great confusion and shoot off guns. Then one of the bright ones will take charge, undoubtedly our old friend Zennor. He will have the roads blocked and try to seal off the city. Then he will start a house-to-house search. Starting right here in the nearest buildings.\"\n\n\"You must flee!\" Sharla said with a lovely concerned gasp. I took the opportunity to pat her hand in my most reassuring manner. She had delicately smooth skin, I just happened to notice. I dragged my thoughts back to the escape.\n\n\"We shall flee, but in a controlled manner, not in panic. They will also be sending patrols to the surrounding area as soon as someone thinks of it. So the plan is this. Change out of these uniforms, join the people outside, leave the immediate area as soon as possible, find a safe place to stay outside the search area in the outermost part of the city, after dark leave the city completely.\"\n\n\"How wonderful!\" Sharla said, eyes glowing beautifully. I was beginning to like this planet. \"I will get clothes for you now.\" She hurried from the room before I could ask her how she planned to do that.\n\nHer solution was a simple one\u2014on local terms. She returned quite quickly with two men.\n\n\"These two seemed to be about your size. I asked them to give you their clothes.\"\n\n\"We are privileged to do this,\" the smaller one said and his companion beamed approval. \"Shall we change.\"\n\n\"Not change,\" I said. \"We'll take the clothes, thank you, but hide or destroy the uniforms. If you were found wearing them you would be shot.\"\n\nThey were stunned at this news. \"That cannot be true!\" the librarian gasped.\n\n\"It's true. I told you that I know the military mind very well...\"\n\nThere was a rapid knocking on the door and Sharla opened it before I could stop her. But it was Stirner gasping and wide-eyed.\n\n\"Are you all right?\" I asked and he nodded.\n\n\"I was not seen; I came by a different route. But the strangers have beaten people, hurt them for no reason. There were explosions of weapons. Some are injured, none dead that I know.\"\n\n\"They must be stopped,\" I said. \"And I know how to do it. We must get back to the dam, to the generating plant. Sergeant Blogh and the company will still be there. We have to get there before they leave. Tonight, because it will be too dangerous by daylight. Now\u2014let's get moving. Find a safer place to lie up until dark.\"\n\n\"I don't understand,\" Stirner said.\n\n\"I do,\" Morton said, his newfound freedom having restored his intelligence. \"It's that talking bird, isn't it? We hid it in that ammunition box\u2014\"\n\n\"Under the canteens of booze. Another reason to hurry before they drink all the way down and find the false bottom. When you heard that bird talk to me it was transmitting the voice of my dear friend, Captain Varod of the League Navy. A power for good in this evil galaxy. He is paid to keep the peace. He doesn't know where we are\u2014yet. But he knew we were going offplanet. So that bird must contain some kind of signaling device or he would not have forced it on us.\"\n\n\"To the bird and salvation!\" Morton cried.\n\n\"The bird, the bird!\" we shouted together happily while the others stared at us as though we had gone mad.\nChapter 19\n\nBellegarrique was a big, sprawled-out city with very few straight streets or large buildings\u2014once we got away from the center. The word had been passed and the streets were busy with pedestrians and hurtling bicycles. We strolled on, apparently unnoticed. Yet everyone seemed to know where we were because every few minutes a bicycle rider would zip up and give the latest report on the enemy positions. This made it very easy to avoid the checkpoints and barricades, while at the same time giving us a chance to look around at the city. Neat and very clean, with a large river bisecting it. We hurried across one of the bridges, this would be a bad place to be caught in the open, and on to the residential district on the far side. The houses grew smaller, the gardens bigger, and we were well into the suburbs by early afternoon.\n\n\"This is far enough,\" I announced. I was tired and my kicked-upon ribs were aching. \"Can you find us a place to hole up until tonight?\"\n\n\"Take your pick,\" Stirner said, pointing around at the surrounding houses. \"You are welcome wherever you want to go.\"\n\nI opened my mouth\u2014then closed it again. Plenty of time later to ask him for information about the philosophy of Individual Mutualism, which I knew he was eager to explain to me. I pointed at the nearest house, a rambling wooden structure with white-framed windows, surrounded by flowers. When we approached it the door opened and a young couple waved us forward.\n\n\"Come in, come in!\" the girl called out. \"Food will be on the table in a few moments.\"\n\nIt was too. A delicious repast after the legions of hotpups we had consumed on the voyage here. Our hosts looked on with approval while Morton and I stuffed our faces. For afters our host produced a distillate of wine that rolled across my palate very well.\n\n\"Our thanks,\" I gasped, stuffed, replete. \"For saving our lives, for feeding us up, for this wonderful drink. Our thanks to all of you, with particular thanks to the philosophy of Individual Mutualism, which I assume you all believe in.\" Much nodding of heads from all sides. \"Which I am sorry to say I never heard of before visiting your fine planet. I would like to hear more.\"\n\nAll heads turned now to Librarian Grene, who sat up straight. And spoke.\n\n\"Individual Mutualism is more than a philosophy, a political system, or a way of life. I am quoting now from the works of the originator himself, Mark Forer, whose book on the subject you will see on the table there.\" She pointed at a leather-bound volume and all of the others looked and smiled and nodded agreement. \"As you will find it on a table in every home in Chojecki. You will also see above it a portrait of Mark Forer, the originator, to whom we will be ever grateful.\"\n\nI looked up at the picture and bulged my eyes. Morton gasped well enough for both of us.\n\n\"If that is Mark Forer,\" he said, \"then Mark Forer is a robot.\"\n\n\"No, not a robot,\" Grene corrected him. \"An intelligent machine. One of the very first machine intelligences as history tells us. Mark Oner had communication interface problems that were only partially eliminated in Mark Tooer...\"\n\n\"Mark four,\" I said. \"The fourth machine to be made.\"\n\n\"That is correct. The first absolutely successful machine intelligence. What a wonderful day for the human race it was when Mark Forer was first switched on. Among those present at that dramatic moment was a then young scientist named Tod E'Bouy. He recorded the event in a book entitled An Historical Treatise Concerning Certain Observations in the Construction of Artificial Intelligence subtitled Galvanized Knowledge.\"\n\nStirner rose from his seat while she was speaking. Went to the bookshelf and took down a slim volume, opened it and read.\n\n\"A lifetime of research, generations of labor, had reached a final and dramatic culmination. The last circuit board was slipped into its slot and I threw the switch. What a prosaic thing to say about what was perhaps the most important moment in the entire history of mankind. I threw the switch, the operation light came on. We no longer were alone. There was another intelligence in the universe to stand beside that of ours.\n\n\"We waited as the operating system carried out all of its checks. Then the screen lit up and we read these historical words.\n\nI AM. THEREFORE I THINK.\n\nHe closed the book in reverent silence. It was like being in church. Well, why not. There have been a number of strange deities worshipped in the long history of mankind. So why not a machine? I sipped my drink and, since no one was speaking, decided to slip in a question.\n\n\"You have no military\u2014and no police. That sounds like a good idea to me, since I have had more than a little trouble with both. But what do you do then with lawbreakers?\"\n\n\"We have no laws to break,\" Stirner said, and there was a brisk round of head-nodding at this. \"I am sure that you will have been taught that laws are the product of the wisdom of your ancestors. We believe differently. Laws are not a product of their wisdom but are the product of their passions, their timidity, their jealousies and their ambition. It is all recorded here in a volume that you must read, the history of an idea.\"\n\nHe pointed to another book that was instantly plucked from the shelf by our host, who pressed it upon us.\n\n\"Take my copy, please, a great pleasure.\"\n\n\"Thank you, thank you,\" I said with what I hoped was sincerity as I hefted its weight. I peeked at a page and tried to keep the smile on my face. As I had feared, it was set in very small type.\n\n\"You will read for yourself,\" Stirner said, \"but our history can be summed up simply. Mark Forer was questioned on many subjects and its vast and different intelligence was utilized in many commercial and scientific ways. It was not until it was queried about political systems that its advice was doubted. Before it could comment it absorbed all of the political writings of the centuries, and the histories, and the commentaries on this material. This took months, years they say. After that Mark Forer weighed and considered the material for an even longer period. During this period it composed the book that you see there and loaded it into RAM. By this time Mark Forer had learned a good deal about the human race through their politics, so therefore took a wise precaution. It accessed all of the data banks and downloaded this book from memory into each of them, and into every electronic mail service as well. Mark Forer later apologized to all of the recipients of this rather thick volume and offered to pay printing costs.\n\n\"But he had been correct in his fears. Not one politician in any country, on any planet, agreed with his theories. In fact efforts were made to denounce Individual Mutualism and all who believed in it\u2014as many did. Because, in his wisdom, Mark Forer knew that while established governments would reject his philosophy, intelligent individuals would read and understand and believe. How wise this wise machine was! Those individuals who were also intelligent enough to understand the philosophy were also intelligent enough to see its inherent truth. They also understood that they would have to find a place of their own to practice what they now believed in. Mark Forer wrote that the wise do not give up their liberty to the state. The converse is also true; the state does not voluntarily relinquish its hold on its citizens.\n\n\"There were years of struggle and flight, persecution and betrayal. Much of the record was destroyed by those who were jealous of our freedoms. In the end those who believed came here, beyond the contact of other worlds, to build a society where Individual Mutualism, IM, was the norm, where peace and happiness could prevail forever.\"\n\n\"Or at least until you got invaded by Nevenkebla,\" I gloomed. Stirner laughed at my expression.\n\n\"Do not despair, my friend, for we do not. The first shock of their arrival has disturbed us, as well it might after our peace of centuries. But we have the courage of our beliefs and know that they and IM will survive this test. If they do, then perhaps we have justified our faith in Mark Forer and, more important, can now perhaps show our gratitude by taking our beliefs to other, less happy planets.\"\n\n\"I would wait awhile before I starting doing that! There are a lot of hard cases out there who would love to eat your people alive. Suffice for the moment getting these military morons off your neck. And, I hate to ask you people for more aid, but I have been kicked about by professionals and wonder if you have any painkillers in the house?\"\n\nI closed my eyes to rest them for a moment and it worked because when I opened them again I felt in perfect shape. It was also dark outside the curtains and a stranger was bent over me having just given me an injection.\n\n\"You passed out,\" Morton said. \"You got everyone worried and they sent for Doctor Lum here who is pretty good.\"\n\n\"Mild concussion,\" the doctor said. \"Two broken ribs, which I have immobilized. I have given you pain relievers. And a stimulant now since I was told you wished to travel this evening. I can neutralize it if you wish.\"\n\nI sprang to my feet and flexed my muscles. I felt fine. \"No way, doctor. You have treated me in a manner I would have chosen, had I been conscious to choose it. How long before the drugs wear off?\"\n\n\"Do not be concerned about that. I will be staying with you until you are well.\"\n\n\"But you don't understand. I have to move fast, hide, do things that may take a long time.\"\n\nLum smiled. \"I am afraid it is you who misunderstood me. I shall be at your side as long as you have need of me. All of us, everyone on this planet, will give you any aid you may need.\"\n\n\"Is that what IM is all about?\"\n\n\"Exactly. What do we do next?\"\n\n\"Walk. No transportation. The military has all the instrumentation for spotting machines on the move.\"\n\n\"What about detecting people?\" Stirner asked. \"Surely their technology must encompass that concept.\"\n\n\"It does. But the human body is an indifferent heat source and hard to tell from that of other animals.\"\n\n\"As is one individual difficult to tell from another,\" the doctor said with medical intuition. \"If we intend to walk in one direction wouldn't it be wise to have a number of people walking in a number of directions?\"\n\n\"It certainly would,\" I said, finally beginning to catch on to how these people worked together. \"How can you pass the word?\"\n\n\"Easily enough. I'll just step out into the street and tell the first person I see. When that is done we can leave.\"\n\n\"Will we reach the dam before dawn?\" I asked Stirner.\n\n\"Easily. It is your choice, of course, to tell us of your plans or not. But if you do give us some information about what you wish to do at the dam, we might then be able to assist you in other ways.\"\n\nFatigue, and the beating, must have addled my brain. I had accepted their offer of help while ignoring the fact that I had never told them what I wanted to do!\n\n\"My apologies!\" I apologized. \"I am beginning to take your hospitality for granted. Which is not fair. Since your ancestors fled from persecution a modicum of intelligence has possessed the human race. Or it has grown up. Or become civilized. While there are exceptions\u2014like the military louts who invaded your peaceful planet\u2014the overwhelming majority of planets are at peace. This peaceful League pays for the maintenance of an organization, the League Navy, which watches trouble spots, contacts rediscovered planets and so forth. Now this begins to get complicated so stay with me. While I am not employed by the Navy, I was given a communication device to contact them from this planet. This device, for reasons too complex to go into, is disguised as a bird. What I want to do is retrieve it from its hiding place, then actuate it to let the Navy know where this planet is.\"\n\nStirner frowned in thought before he spoke. \"If this Navy group you speak of intends to use violence we cannot help you to summon them.\"\n\n\"No fear there. The League is sworn to nonviolence.\"\n\n\"Then there are no problems. What can we do to help?\"\n\n\"Guide me to the dam, that's all. I'll do the rest. There will be three of us. You, I, and the good doctor Lum. We will need food and water.\"\n\n\"You forgot me,\" Morton said.\n\n\"No, I remembered you. You are out of the army\u2014stay out. I either get the bird by stealth or not at all. As virile as I am I don't look forward to taking on a trigger-happy company of well-trained thugs. Stay here, talk to Sharla, which should not take too much effort. Get information. Find out all you can about what the army is doing. I'll be back tomorrow night.\"\n\n\"I will be pleased to discuss Individual Mutualism with you,\" Sharla said in a voice that was pure honey. Morton melted instantly and did not even know it when we left.\n\nFor all of his gray hairs, Stirner must have been a marathon walker. The doctor matched his pace, while I was riding so high on the drugs that I had the feeling that if I flapped my arms hard enough I could have flown to the dam. We skulked down unpaved roads, then along what appeared to be a linear track of some kind, through a tunnel, then through meadows where dark beasts moved aside as we went by. After a few hours of walking like this under a moonless, star-filled sky, the lights of the city were far behind, the dark walls of mountains looming ahead. Stirner called a halt and we sat down on the grass under a tree.\n\n\"This will be a good time to drink, eat if you wish, because we will leave our burdens here.\"\n\n\"Getting close?\"\n\n\"Very. We will approach the dam though a drainage tunnel that is dry this time of year. This emerges on the riverbank close to the generating station.\"\n\n\"You are a genius. We will get by the lookouts that way, will be inside their perimeter and hopefully somewhere near the command car. How long until it gets light?\"\n\n\"We have at least four hours yet.\"\n\n\"Wonderful. We take a break. The doctor can slip me a pep pill or two since I am feeling a bit shabby, then we will finish this affair.\"\n\nLum sounded worried. \"If you have any more stimulants you may become quite sick after the drugs wear off.\"\n\n\"And without the aid of you kind people I certainly would have been quite dead by now. So let's get the bird so I can call in the Navy. Before something really drastic happens and people get killed.\"\n\nWe ate and drank, the doctor then concealed our supplies in the tree, gave me an injection, and the march resumed. I was so full of uppers that I had to fight down the urge to whistle and bound ahead of my slower companions. I resisted. Stirner found the gulley we were looking for and led us along it until it ended in a high black opening. I looked at it suspiciously.\n\n\"Could be dangerous animals in there.\"\n\n\"Very doubtful,\" Stirner said. \"The rainy season ended not too long ago. Until then this tunnel was filled with water.\"\n\n\"Besides that,\" Lum added. \"There are no dangerous animals on this continent.\"\n\n\"Other than the ones I arrived with. Lead on!\"\n\nWe stumbled into the darkness, splashing through invisible puddles, running our fingers along the rough walls of the tunnel to keep from bashing into them. By the time we reached the far end our eyes were so adjusted to the dark that the patch of starry sky ahead almost looked gray.\n\n\"Silence now,\" Stirner whispered. \"They might be very close.\"\n\n\"Then you two wait in the tunnel out of sight,\" I whispered in return. \"I'll make this as quick as I can.\"\n\nWhen I poked my head carefully out I saw that the tunnel emerged from the bank above the river. Perfect. I could slink along the side of the river to the generating plant. Which I did. The roar of water discharging from the plant growing constantly louder. I kept going as far as I could, until spray was blowing over me, before I climbed the bank and parted the grass carefully to look out.\n\n\"Congratulations,\" I thought to myself. \"You are a genius at night-stalking, Jimmy.\"\n\nNot twenty meters away was the command car, parked beside the generating plant. And there wasn't a soul in sight. Silent as a ghost I drifted along the building, past a closed door in the wall, and slipped into the car. The booze box was just where I had left it. Neat! I pulled it out and groped inside.\n\nIt was empty!\n\nAt the precise moment that I realized this the door opened behind me and I was bathed in light.\n\nSergeant Blogh was standing in the doorway holding the bird.\n\n\"Is this what you are looking for, captain?\" he said.\n\nI looked from the bird to the gun in his other hand and could not think of a thing to say.\nChapter 20\n\n\"You're an escaped criminal, captain.\" He was smiling wickedly, enjoying himself. I still had nothing to say. \"That's what was reported. They sent a chopper rushing out here for all of your equipment. Only after the MPs left did I remember how you were always worrying about those canteens. At the time I thought it was just the booze. Since they said you were an offworlder spy I began to think different. So I looked close and found this stuffed bird. Before I could turn it in, I heard how you escaped. So I thought I would just keep watch in case you wanted to get it back. Seems I thought right. Now\u2014climb out of there slowly and keep your hands in sight.\"\n\nI had no choice. But at least my brain was in gear again after the disconnecting shock of his appearance.\n\n\"I would like the bird back, sergeant.\"\n\n\"I'm sure you would. But why should I give it to you?\"\n\n\"To save lives. With it I can contact the League Navy and end this invasion before someone is killed.\"\n\n\"I don't mind killing.\" His smile was gone and there was a brutal edge to his voice I had never heard before. \"I'm a soldier\u2014and you are a spy. I am going to turn you and your cagaling bird in. This is going to mean a lot to my career.\"\n\n\"And you put your miserable military career ahead of the lives of harmless, unarmed civilians?\"\n\n\"You bet your sweet chunk, I do.\"\n\nI started to tell him just what I thought of him. But didn't. There had to be some way to get to him.\n\n\"Do you take bribes, sergeant?\"\n\n\"No.\"\n\n\"I'm not talking about little bribes. I am talking about the ten thousand credits in League currency that you will receive when this invasion ends. You have my word on that.\"\n\n\"A spy's word? Ten thousand or ten million\u2014the answer is the same. You are for the chopping block, spy.\"\n\nThere was a quick movement from the door behind him, a solid chunking sound, and the sergeant dropped to the ground. I dived for his gun.\n\n\"Don't,\" a voice said. \"Just stay away from it.\"\n\nI looked up at Private, formerly Corporal, Aspya, who was now pointing the gun at me that he had just used to bash the sergeant in the head.\n\n\"I wondered why the sergeant had been hiding in here all night. Now I found out.\" His face split suddenly in a crooked-toothed grin and he slipped his pistol back into the holster.\n\n\"I take bribes,\" he said. \"But it has got to be twenty thousand.\"\n\nI pointed at the bird. \"Let me take that and you will get thirty thousand solid titanium League credits after the invasion is ended. At the League building in Brastyr. You have my word.\"\n\n\"My serial number is 32959727. There are a lot of Aspyas in the army.\"\n\nThen he was gone. And so was I\u2014before anyone else joined the party. I grabbed up the bird and ran just as fast as I could back to the river.\n\n\"Get moving into the tunnel!\" I called out as I staggered up to my waiting companions. The shots were wearing off and I was stumbling. \"Alarm, maybe soon, let's go.\"\n\nAnd we did. Back through the tunnel and on into the fields. I must have fallen somewhere along there because the next thing I knew I was in some woods and lying on the ground. The sky was light beyond the trees and my heart began to thud in panic.\n\n\"The bird!\"\n\n\"Here,\" Stirner said, holding it up. \"You collapsed, so we took turns carrying you. The doctor said it would be wisest to let you rest since more stimulants might cause grave injury. We are hidden and safe now.\"\n\nI took the robot bird and shook my head in wonder. \"You people are unbelievable\u2014but you have my thanks. Was there a search?\"\n\n\"We heard nothing. But you seemed so concerned that we went on while it was still dark. We should be safe here. If these woods are searched there is a place of safety close by.\"\n\n\"I hope so because they are going to be very irritated. There were difficulties encountered and the alarm will be out by now. So let us do what we came for.\"\n\nI groaned as I sat up and the doctor appeared with a ready needle.\n\n\"This is only a painkiller,\" he said. \"Stimulants are contraindicated now.\"\n\n\"You are a genius, doc.\"\n\nThe black bird, still smelling of jet fuel, sat heavily in my hands. Silent and still. Time to end that. I pressed down on its bill twice and its eyes opened.\n\n\"This is a recorded message from Captain Varod,\" it said, then rolled over on its back. \"You will find a panel in the bird's chest. Open it.\"\n\n\"Light-years away and it is still orders, orders,\" I muttered as I groped among the feathers. Stirner and the doctor watched with wide-eyed attention. I found a button, pressed, and a feather-covered door flew open. There was a glowing control panel inside. Opening the door apparently activated the bird again because it began to croak out more instructions.\n\n\"Enter the location of the sun in this system, as well as the planetary coordinates, on the dials using the intergalactic ephemeris readings.\"\n\nI grated my teeth. \"How could I possibly know anything like that? Or anyone else on this planet?\"\n\n\"If you do not have this information turn the power switch to full and press the activate button. Proceed.\"\n\nI did this and stepped back. The bird vibrated, opened its bill and squawked. From its gaping mouth there emerged a yellow aerial that moved slowly upward. When it was fully extended, over two meters of it, the bird's eyes began to glow. The aerial hummed briefly and the glowing eyes went dark. As slowly as it had emerged the aerial sank back and the bird was quiet again.\n\n\"Very interesting,\" Dr. Lum said. \"Can you explain?\"\n\n\"No. But I wish this stupid bird would.\"\n\n\"Let me explain,\" the bird croaked. \"Since you did not enter the galactic coordinates of this planet a FTL message could not be sent. Precision is imperative in FTL communication. Therefore a prerecorded radio message was transmitted. All League bases and ships have been alerted. When it is received its source will be noted and this spybird will be informed.\"\n\n\"If you are still functioning!\" I shouted and raised my foot to stamp on the bird, but was restrained by the doctor. The bird was still speaking.\n\n\"I am shutting down now to save power. Keep close to this communicator, which will be activated when we are within signaling distance.\"\n\n\"Keep close to it!\" I shouted. \"I'll probably have to have it buried with me.\" I saw the way the two of them were looking at me so restrained my anger. \"Sorry. Got carried away there. With good reason.\"\n\n\"It has to do with distance, doesn't it?\" Stirner asked.\n\n\"Bang on.\" I had forgotten that he was an engineer. \"An FTL transmission, faster than light, is almost instantaneous, even at stellar distances. But radio waves move at the speed of light\u2014and how far is the nearest star from here?\"\n\n\"Three point two light-years.\"\n\n\"Wonderful. So even at the million to one chance there is a League planet or base near that sun it would still be over three years before the cavalry arrives. Or it could be ten, twenty\u2014or five hundred. By which time you, I and the invasion will be a part of history.\"\n\n\"You have done your best,\" the doctor said. \"You cannot berate yourself.\"\n\n\"I sure can, doc. I take first prize in the self-berating stakes when it comes to losing. Since I don't like to lose.\"\n\n\"You have great security of resolve. I envy you.\"\n\n\"Don't. It's a pose. Did you get the water bottle out of the tree on the way back here?\"\n\n\"Assuredly. Let me get you some.\"\n\nI leaned against the tree, sipped the water, pushed the silent bird with my toe. And thought hard. Then sighed.\n\n\"There is still a solution. But not an easy one. I have to get into one of their spacers. And into the communications room and send a message from there.\"\n\n\"It sounds dangerous,\" Stirner said. I laughed hollowly.\n\n\"Not only dangerous\u2014but suicidal...\" I shut up as I heard a distant shout.\n\n\"They are searching for you,\" Stirner said, helping me to my feet. \"We must go quickly.\"\n\nThe doctor helped me up\u2014which was a fine idea since I was definitely shaky on my feet. It was also cheering that we did not have far to go, only to the edge of the woods nearby. As we looked out from the concealing shrubbery we could see the rolling countryside beyond. A row of electricity towers marched across it, bearing heavy wires slung from insulators. The row of towers ended here. The wires came to ground in a solid concrete building. Stirner pointed at it.\n\n\"The aerial cables go underground here.\"\n\n\"So do we,\" I said pointing at the distant line of approaching soldiers, \"if you don't do something quick.\"\n\n\"Be calm,\" he advised calmly. \"This junction station will block their view of us. Forward.\"\n\nHe was right. We scuttled out of hiding and plastered ourselves against the concrete wall. Next to a red-painted metal door that was covered in skulls and crossbones and warnings of instant death. None of which deterred Stirner who flipped up a plate to disclose a key pad. He punched in a quick number then pulled the door open. We moved smartly inside as he closed and locked the heavy door behind us.\n\n\"What if they try to follow us?\" I asked, looking around the well-lit room. There was little to see other than the heavy cable that entered from the ceiling and vanished into the floor.\n\n\"Impossible. They will not know the keying number. If they enter a wrong number the door seals and an alarm is sent to power central.\"\n\n\"They could break it down.\"\n\n\"Not easily. Thick steel set in concrete. Is there any reason why they should?\"\n\nI couldn't think of one and I was feeling cagally after the walk. I sat down, then lay down, closed my eyes for a second.\n\nAnd woke up with a taste in my mouth like a porcuswine's breath.\n\n\"Yuk...\" I gurgled.\n\n\"I am very glad you slept,\" the doctor said, swabbing off my arm and sticking it with a hypo. \"Rest is the best medicine. This injection will eliminate residual fatigue symptoms and any pain.\"\n\n\"How long have I been out?\"\n\n\"All day,\" Stirner said. \"It is after dark. I have been outside and the soldiers are gone. We were going to awaken you soon in any case. Water?\"\n\nI gurgled most of it down and sighed. I felt much better. I didn't even sway when I stood up. \"Time to go.\"\n\nThe doctor frowned. \"It might be better to wait until the injection takes hold.\"\n\n\"I will walk off my troubles, thank you. We have been away a long time and I tend to worry.\"\n\nMy shakiness wore off as we walked. The woods were silent, the searchers long gone, and we had the world to ourselves. Stirner led the way at his usual cracking pace. The doctor kept an eye on me and soon called a halt so he could plug his analysis machine into my arm. He was satisfied with the result and our trek continued. Putting one foot in front of the other was enough to keep me occupied until we reached the outskirts of the city again. With one look at the buildings all my forebodings returned.\n\nI was right, too. It was still dark when we reached the first homes, moving silently between the cottages and gardens of suburbia to avoid the guarded main streets. The backdoor of our refuge was unlocked: we slipped in and locked it behind us.\n\n\"You have the bird!\" Morton cried gleefully when we entered. I nodded and threw it on the couch, dropped myself next to it and looked around. All of the others were gone.\n\n\"That is the good news,\" I said. \"The bad news is that it may be some time before help arrives. The call for help went out by radio\u2014which could take a mighty long time.\"\n\n\"That is very bad indeed,\" Morton said and his face sank instantly into lines of despair. \"While you were away they started taking hostages. Zennor got on the TV and said that he is going to shoot them, one at a time, until everyone goes back to work. He says that he will execute the first person at dawn\u2014and one every ten minutes after that until he gets cooperation.\"\n\nHe dropped his face into his hands and his voice was muffled, trembling. \"The soldiers came up this street, were going to search this house. So everyone here, Sharla, all the others, went out to them. Surrendered so I would not be found. They are now captives, hostages\u2014and are going to be shot!\"\nChapter 21\n\n\"It cannot be,\" the doctor said, puzzled but calm. \"Human beings just do not do things like that.\"\n\n\"Yes they do!\" I shouted, jumping to my feet and pacing the room. \"Or maybe human beings don't\u2014but animals like Zennor do. And I apologize to the animals. But it certainly won't go that far, will it Stirner? Your people will have to go back to work now?\"\n\n\"No, they won't. If you understood Individual Mutualism you would understand why. Every individual is a separate and discrete entity, responsible for his or her own existence. What Zennor does to another individual does not relate to any other discrete individual.\"\n\n\"Zennor thinks so.\"\n\n\"Then Zennor thinks wrong.\"\n\nI resisted the temptation to tear out a handful of my own hair. I wasn't getting through at all. \"Well look at it another way. If you do not do anything to stop Zennor then you are responsible for the deaths of the hostages.\"\n\n\"No. If I do something to please Zennor in the face of his threats then I am admitting his control over my actions despite the fact I do not wish his control. The state is born once again. IM is dead. So we chose passive resistance. We will not be ordered or threatened...\"\n\n\"But you can be killed.\"\n\n\"Yes.\" He nodded grimly. \"Some will die if he insists on this course. But murder is self-defeating. How can you force someone to work by killing him?\"\n\n\"I understand you\u2014but I don't like it.\" I was too disturbed to sit, I stood, paced the floor. \"There must be a way out of this that doesn't involve someone's death. What is it that Zennor wants?\"\n\n\"He was very angry,\" Morton said. \"And very specific. First he wants the electricity turned back on in the buildings the military has occupied. Then he wants a regular supply of food for his troops. If these two things are done no one will be killed and the prisoners released. For the time being.\"\n\n\"Impossible,\" Dr. Lum said. \"They gave nothing in return for the electricity they used, so it was disconnected. The same thing applies to the food. The markets have shut down because the farmers will not bring food to the city.\"\n\n\"But,\" I sputtered. \"If the markets are closed how does everyone else in the city eat?\"\n\n\"They go to the farms, or leave the city. Almost a third of the population has already gone.\"\n\n\"Where will they go?\"\n\n\"Wherever they want to.\" He smiled at the look on my face. He could tell that I was hearing the words but not understanding them. \"I think that I should go to basics, explain a bit more about IM to enable you to understand. Let us take a simple example. A farmer. He raises all the food that he needs, supplies all of his own wants so asks for nothing from others.\"\n\n\"Nothing?\" I had him there. \"What if he needs new shoes?\"\n\n\"He goes to a man who makes shoes and gives him food in exchange.\"\n\n\"Barter!\" Morton said. \"The most primitive economic system. But it cannot exist in a modern technological society...\" His voice ran down as he looked about the room. Stirner smiled again.\n\n\"Of course it cannot. But IM is more than barter. The individual will voluntarily join other individuals in a larger organization to manufacture some item, or build houses say. For each hour they work they are credited with a wirr.\"\n\n\"A what?\"\n\n\"A work hour. These wirrs are exchanged with others for goods and services.\"\n\n\"A wirr is another way of saying money,\" Morton said. \"And money is capitalism\u2014so you have a capitalistic society.\"\n\n\"I am afraid not. Individual Mutualism is neither capitalism, communism, socialism, vegetarianism, or even the dreaded monetarism that destroyed many a technological society. I am familiar with these terms from Mark Forer's writings. A wirr has no physical existence, such as a rare metal or a seashell. Nor can it be invested and gain interest. That is fundamental and differentiates the wirr from currency. Banks cannot exist because there can be no interest on deposits or loans.\"\n\nInstead of being clarified I found my head wirring in confusion from the wirrs. \"Wait, please, explanation. I have seen people driving groundcars. How can they save money enough to buy one? Who will loan them the money without interest?\"\n\n\"No money,\" he said firmly. \"If you wish a groundcar you go to the groundcar group and drive one away. You will pay when you use it, stop paying when you return it. A basic tenet of IM is from each according to his needs, to each according to the wealth of society.\"\n\n\"You wouldn't like to clarify that?\" I poured myself a glass of wine and gulped it down hoping the alcohol would clean out my synapses.\n\n\"Of course. I have read, and trembled with disgust, of a philosophy called the work ethic. This states that an individual must work hard for the basics of life. When technological society mechanizes and replaces workers with machines, the work ethic states that the displaced workers must be looked on with contempt, allowed to starve, be treated like outcasts. And the hypocrisy of the work ethic system is that those with capital do not work\u2014yet still increase their capital without working by the use of interest on their money\u2014and look down upon those who have been cast out of work! Tragic. But not here. As more is produced the aggregate wealth gets larger. When this happens the amount that the wirr can be exchanged for also gets larger.\"\n\nSome of it was getting through\u2014but needed elucidation. \"Another question. If the wirr is worth more\u2014that must mean that an individual can work less for the same return.\"\n\n\"Exactly.\"\n\n\"Then there is no forty-hour week or such. How many hours would an individual have to work a week to keep alive?\"\n\n\"For simply shelter, food, clothes\u2014I would say about two hours of work every seven days.\"\n\n\"I want to move here,\" Morton said firmly and I nodded agreement and froze in half-nod. An idea was glimmering at the edge of my consciousness. I muttered and chiseled at it and expanded it until I saw it large and clear and possibly workable. In a little while. But first we had to do something about the hostages. I rejoined the real world and called for attention.\n\n\"Time is passing and dawn approaching. I have enjoyed the lecture, thank you, and I now know a bit more about IM. Enough at least to ask a question. What do you do in an emergency? Say there is a flood, or a dam bursts or something. A catastrophe that threatens the group not the individual.\"\n\nThe doctor stepped forward, finger raised and a sparkle of enthusiasm in his eye. \"A good question, a marvelous question!\" He grabbed at the shelves and pulled down a thick book. \"It is here, all here. Mark Forer did consider a situation like this and made allowances for it. Here is what he wrote... at all times passive resistance will be your only weapon, never violence. But until the perfect stateless state is established there will be those of violence who will force their violence upon you. Individual Mutualism cannot be established by the dead. Until the day of true liberation comes you will have to coexist with others. You may leave their presence but they may follow and force themselves upon you. In which case you and all of the others must look upon those of violence as they might look upon any natural catastrophe such as a volcano or a hurricane. The intelligent person does not discuss ethics with hot lava but instead flees its presence, does not preach morals to the wind but seeks shelter from it.\"\n\nDr. Lum closed the book and raised a triumphant finger again. \"So we are saved, saved! Mark Forer has foreseen our predicament and given us the guidance we need.\"\n\n\"Indeed!\" Stirner agreed enthusiastically. \"I shall go at once and tell the others.\" He rushed to the door and out of the house. I gaped after him. Morton spoke my thoughts before I could.\n\n\"I heard what you said\u2014but haven't the slightest idea of what your Mark Four was talking about.\"\n\n\"Clarity!\" the doctor said. \"Clarity and wisdom. If we all persist in noncompliance we are in a sense killing ourselves. So we comply and withdraw.\"\n\n\"I am still not sure what you are talking about,\" I said.\n\n\"The electricity will be turned back on, the markets will reopen. The invaders will seize food and some farmers will work longer hours if they wish to, because that will avert the natural disaster. Others will not and will stop bringing food to the city. As the supply diminishes, people will leave the city and the process will accelerate. With less call for electricity, the generating plant will shut down, workers will leave. In a very short time the soldiers will have the city to themselves because we will all be gone.\"\n\n\"They can enslave you\u2014make you work at gunpoint.\"\n\n\"Of course, but only on a one-to-one basis. One armed man can force another to work, possibly, it is of course up to the individual. But the man with the gun is essentially doing the work himself because he must be there every moment or the work will not get done. I don't think your General Zennor will like this.\"\n\n\"You can say that again!\"\n\n\"I don't think your General...\"\n\n\"No, not really say it again, I meant it as an expression of agreement. You people are too literal, too much IM I imagine. A question then, a hypothetical one.\"\n\n\"Those are the best kind!\"\n\n\"Yes, indeed. If I should walk into a distant city and look for work\u2014would I be accepted?\"\n\n\"Of course. That is a basic tenet of IM.\"\n\n\"What if there are no jobs going?\"\n\n\"There always are\u2014remember the value of the rising wirr. Theoretically as it gets larger and larger, the working hours will get fewer and fewer, until in the long run a few seconds work will suffice...\"\n\n\"All right, great, thanks\u2014let's just stick with the application of theory for a moment. If one of these invading soldiers should walk away from the army...\"\n\n\"Which is of course his right!\"\n\n\"Not quite what the army thinks. If he walks away to a distant town and gets a job and meets a girl and all the usual good things happen\u2014is this possible?\"\n\n\"Not only possible, but inescapable, a foundation of IM inherent in its acceptance.\"\n\n\"Are you thinking what I think you're thinking!\" Morton shouted, jumping to his feet with elation.\n\n\"You bet your sweet chunk I'm thinking that! Leaving aside the officers and the career noncoms, this is a draftee army and a good number of them were draft evaders. If we make the opportunity available for them to walk away from it all, why then Zennor might have to give a war that nobody will come to.\"\n\nThe front door opened and Morton and I dived for cover. But it was Stirner leading the triumphal return of the released captives. Morton rushed to Sharla and took her hand to see if it had been hurt during her incarceration.\n\n\"That's pretty fast work,\" I said.\n\n\"I used the TV phone across the street,\" Stirner said. \"I purchased national access and told them what we had discovered. The electricity was turned on instantly, the first food shipped. The prisoners were released.\"\n\n\"Zennor must think that he has won the war. Let me tell you what we have just discovered. The way to guarantee that he loses his war\u2014even if the Navy never gets here.\"\n\n\"I am encouraged by your enthusiasm but miss your meaning.\"\n\n\"I will explain\u2014but first a drink to celebrate.\"\n\nThis seemed like a good idea to all concerned. We poured and drank, then Morton and I listened with some interest as the others sang a song about Individual Mutualism freeing mankind from the yoke of oppression and so forth. While the theory was fine the lyric was as bad as all other anthems I had ever heard, though I took considerable interest in the great efforts made to rhyme Individual Mutualism. I also took the time to organize my thoughts so when they had finished, and sipped a bit more wine for dry throats, I took the floor.\n\n\"I must first tell you kind people about the uniformed mob of thugs who have invaded your fair planet. A large group like this is called an army. An army is a throwback to the earliest days of mankind when physical defense was needed against the rigors of existence. The combative gene was the successful gene. The primitive who defended his family group passed on this gene. This gene has caused a lot of trouble since that time, right down through the ages. It is still causing trouble as you now have cause to understand. When all of the threatening animals were killed, the gene caused mankind to turn upon itself to kill each other. With shame I admit we are the only species that kills its own kind on a very organized basis. The army is the last gasp of the combative gene. In charge are old men, and they are called officers. They do nothing except issue orders. At the bottom are the soldiers who follow these orders. In between are the noncommissioned officers who see that this is done. The interesting thing to us now is that the soldiers are all drafted and a good number of them are draft dodgers.\"\n\nIt took some time to explain what these last two terms meant and there was horrified shock on all sides when understanding finally penetrated. I waited until the cries of disbelief and despair had simmered down, then signaled for silence.\n\n\"I am cheered by your reaction. Do you think your people would volunteer, without payment in wirrs, to free these young men from bondage?\"\n\n\"It would be our duty,\" the doctor said and heads nodded like fury on all sides. \"It would be like saving someone from drowning, a public duty, no payment expected.\"\n\n\"Great! Then I will now teach you another word...\"\n\n\"Can I guess?\" Morton cried out. I nodded.\n\n\"Desertion!\"\n\nI nodded again. Battle joined at last!\nChapter 22\n\nEnthusiasm gave way quickly to fatigue and it was agreed that the session would continue after we all had had some sleep. I found myself tucked away in a small room in a soft bed, with a portrait of Mark Forer beaming down electronically upon me. I sipped a last sip of wine and crashed.\n\nBy the next evening I had put together the rudiments of a plan and had assembled my team.\n\n\"We have to try it out, smooth it out. Then, if it works, we pass it on to others. We will operate and proceed like an ancient scam, a term I ran across when doing research into crime.\" I did not add that my reasons for doing this were to improve myself as a criminal. This would have been too much for these simple IMers to understand. \"Here is how it will work. This evening I will enter one of the eating and drinking establishments you have described to me. I will then stand next to a soldier and engage him in conversation. You, Stirner will be seated at a table with empty chairs, or next to an empty table. I will come over with the soldier and sit close enough for you to overhear our conversation. Sharla will be with you, she is your daughter.\"\n\n\"You are wrong, she is not my daughter.\"\n\n\"Just for tonight she is your daughter, like in a play. You do have plays here?\"\n\n\"Of course. In fact I was on the stage when younger, before I was attracted to the delights of flowing electrons. I even acted the title role in some classics, how does it go again... to was, or not to was\u2014\"\n\n\"Fine, great, glad to have an old thespian aboard. So tonight you act the role of Sharla's dad. Follow my lead and it should work. I'll pick an easy target this first time, an apple ripe for plucking. So there should not be any trouble.\"\n\n\"What do I do?\" Morton asked. \"You said I was on the team.\"\n\n\"Right. You have the important job of taping all of this for the record. So when it works as it should we can make training copies for others. Keep the recorder out of sight and the mike close. Ready?\"\n\n\"Ready!\"\n\nWe waited until after dark before we set out. Volunteers, drafted from the street of course, worked ahead of us to make sure we didn't meet any roadblocks or MPs. They reported back all the obstructions so we had a pleasant, if circuitous, walk to the Vaillant quarter of the city, which I had been assured was the correct place to go for theater, opera, dining out, IM reinforcement groups and the other heady joys of this civilized planet. It looked an interesting locale. Although it was fairly empty this evening with no more than a quarter of the establishments lighted up. Stirner led the way to the Fat Farmer, where he said he always enjoyed good food and better drink when in the city. There were some locals sampling its pleasures\u2014but no invading soldiers.\n\n\"You told me that the army had leave passes, that they could be found in this area. Where are they?\"\n\n\"Not in here, obviously,\" Stirner said.\n\n\"What do you mean\u2014obviously?\"\n\n\"Since they cannot pay they won't be served.\"\n\n\"Sounds fair. But, since they are the invading army, what stops them from just grabbing the booze and helping themselves?\"\n\n\"They are not stopped. However everyone leaves and the establishment shuts down.\"\n\n\"Obvious. All right then. To your stations and I'll see if I can drum up some trade.\"\n\nI felt very pimpish standing under the streetlight with a dead cigar for a prop. In the local garb I was just part of the passing parade and no one took notice of me. I watched all of them though\u2014on the lookout for MPs or anything that resembled the part of the military I did not want to see: stripes, bars, the usual thing. None of these appeared, but eventually two unmilitary figures in military uniform drifted into sight. Hands in pockets\u2014shame!\u2014caps on at odd angles. They stopped at the Fat Farmer and looked in the window with longing. I stepped up behind them and held up the cigar.\n\n\"Either of you guys got a light?\"\n\nThey jumped as though they had been goosed, shying back from me.\n\n\"You talked to us!\" the bolder one said.\n\n\"I did. I pride myself on my linguistic ability. And if you will remember I asked you for a light for my cigar.\"\n\n\"I don't smoke.\"\n\n\"Good for you. Cigarettes kill. But don't you carry a fire apparatus for those who do?\" They shook their heads in gloomy negation. Then I raised a finger rich with inspiration. \"I know what\u2014we will enter this eating and drinking place and they will light my cigar. Perhaps you young gentlemen from a distant planet will also join me in a drink and I can practice my talking?\"\n\n\"Won't work. We tried it and they closed the place and went home.\"\n\n\"That is only because you had no wirr, the local unit of exchange, our money, so could not pay. I am rich with wirr and am buying...\"\n\nI followed after their rapidly retreating footsteps, found them pushing against the bar in eager anticipation. Stirner had given me his wirrdisc and briefed me on its operation.\n\n\"Three beers,\" I ordered, \"large ones,\" and dropped the plastic slab of integrated circuits into the slot in the top of the bar. While the robot bartender, all chrome and brass with bottlecaps for eyes, drew three big brews, the cost was subtracted from Stirner's lifetime account. I grabbed the wirrdisc as it was rejected.\n\n\"Here's to the army, lads,\" I said raising my beer high. \"I hope you enjoy your chosen careers.\"\n\nThey chugalugged enthusiastically, then gasped and whined nostalgically familiar whines that took me back to my own army days.\n\n\"Chose an army career! Cagal! Drafted. Chased, hunted down, caught.\"\n\n\"Then after that, basic training. Pursued at the double night and day by foul-mouthed fiends. Would anyone voluntarily choose a career like that?\"\n\n\"Certainly not! But at least you eat well...\"\n\nI enjoyed the outraged cries and loathsome descriptions of hotpups while I ordered up another round of beers. When their faces were buried in the suds again I made the suggestion.\n\n\"I know it is past your dinner hour, but I see three seats vacant at that table, next to the elderly gentleman with the kinky bird. Would you join me for a small repast\u2014say a large steak and fried wirfles?\"\n\nThe thunder of feet was my only answer yet one more time. I joined them in the steaks, and very good they were too. We polished them off quickly, had a few more beers\u2014and tried not to belch because there was a young lady at our table. Sated and boozed they now had time for the third of the troika of military pleasures and their eyes moved steadily in Sharla's direction. Time for act two.\n\n\"Well,\" I said, \"if the food is bad in the army, at least you enjoy the wisdom and companionship of the sergeants.\"\n\nI listened to the answers for a bit, nodding and commiserating, then elicited other similar complaints with leading questions about officers, latrines, kitchen police\u2014and all the other bitches so dear to the enlisted man's heart. When enough had been ventilated I gave Stirner his clue and sat back.\n\n\"Young draftee soldiers from a distant planet, you must excuse my impertinence in addressing strangers. But I, and my lovely daughter Sharla, could not help but overhear your conversation. Can it be true that you were forced into military service completely against your will?\"\n\n\"You better believe it, Pops. Hi, Sharla, you ever go out with guys other than your Dad?\"\n\n\"Very often. I simply adore the company of handsome young men. Like you.\"\n\nAll three of us fell into the limpid pool of her eyes, splashed around for a bit and emerged gasping and in love. Stirner spoke and they did not hear. I finally ordered large beers and had them placed in front of their bulging eyes to cut off sight of the gorgeous Sharla. This produced the desired result. While they glugged Stirner talked.\n\n\"I am greatly taken by your plight, young gentleman. On this planet such a thing is impossible. Against our laws, which laws state that there are no laws. Why do you permit yourself to be treated in this vile manner?\"\n\n\"No choice, Pops. Barbed wire all around, watched night and day, shot if you try to escape, shot twice if recaptured. No place to go to, no place to hide, in uniform, every man's hand turned against you.\"\n\nHe sniffed in maudlin self-pity; a tear ran down his companion's cheek.\n\n\"Well,\" Stirner said, sinking the gaff in deep and twisting it so it would take hold. \"None of those things are true here. There is no barbed wire, no one is watching you, no one is about to shoot you. There is a great big country out there that stretches away beyond the mountains and rivers. A country where you will always find a welcome, always find hospitality and refuge. A country where the army will never find you.\"\n\nThey sat up at that, trying to understand his words through their alcoholic haze. \"Cagal...\" the drunkest one muttered. Sharla smiled angelically.\n\n\"I do not understand that word, young friend, but I feel that it indicates disbelief. Not so. Every word my father has spoken is true. For example, we live a full day's journey distant from this city in an idyllic farming village. We travel there by speedy railroad\u2014and these are our tickets to prove it. Why, look, the machine made a mistake, it issued four tickets instead of two. I must return them\u2014unless you would like them for souvenirs?\"\n\nFaster than light, they vanished.\n\n\"There is a side entrance to the railway station that is not guarded,\" she said brightly.\n\n\"But the train leaves soon,\" Stirner said, standing and picking up the bundle from the floor. \"Before going I must use the necesejo, as we say down on the farm, and I am taking this bundle with me. It contains clothing for my two sons at home who, strangely enough, are just your size.\" He started away, then turned. \"You may borrow the clothes\u2014if you wish.\"\n\nThey beat him to the cagalhouse door. Sharla smiled beatifically after them.\n\n\"You know this farming town well?\" I asked. \"So you can line the lads up with friends.\"\n\n\"I have never been there\u2014I found its name on the map. But you forget the strength of IM. We would welcome them here and aid them, so they will be welcome there. Do not worry. I will guide them and return in two days. Ohh, here they come, don't they look handsome out of those dreary uniforms!\"\n\nThey looked rotten, I thought, the demon of jealousy burning within me. I almost wished that I was going with them. But no, the work was here. I turned to the next table where Morton was mooning after the lovely retreating form of Sharla. I had to kick him twice before I could attract his attention.\n\n\"She'll be back, don't worry. Did you get all that on tape.\"\n\n\"Every word. Can I have another beer? All I had was the one Sharla bought me before you came in. And you had a steak...\"\n\n\"No drinking on duty, soldier.\"\n\nStirner joined us and pointed to the basket he was carrying. \"I have their uniforms in here, just as you asked.\"\n\n\"Good. We'll need that for the video. Now\u2014take us to your recording studio.\"\n\nHe led us by back streets to the back of a building, to the back door that opened as we approached. They were eagerly waiting for us on the soundstage, brightly lit, windowless and invisible from the street. Volunteers all, IM enthusiasts just dying to subvert the troops. I held up the audio cassette.\n\n\"We'll need a few hundred copies of this.\"\n\n\"Within the hour!\" It was snatched from my hand and whisked away. I turned to the waiting production crew, who were trembling with enthusiasm. \"Director?\" I asked. A gorgeous redhead stepped forward.\n\n\"At your service. Lights, sound, camera ready.\"\n\n\"Wonderful. As soon as my associate and I put on these uniforms\u2014you can roll. Point us to the dressing room.\"\n\nAs I stripped Morton took one of the uniforms out of the basket and held it out between thumb and index finger like a dead rat.\n\n\"I feel depressed even looking at this thing,\" he said. Depressedly. \"To feel its touch upon my skin again, the clammy embrace...\"\n\n\"Morton,\" I hinted, \"shut up.\" I whipped it away and held it before me. A good fit. I climbed into it. \"You are an actor now, Morton, playing before the camera. You will act your role\u2014then remove the uniform forever. Burn it if you wish to. Thousands will applaud your performance. So put it on. Like this.\"\n\nI sat and pushed my legs into the trousers and something fell from a pocket and tinkled to the floor. I bent and picked it. An ID disc. Private soldier Pyek0765 had been eager to wipe all memory of the army from him, to be reborn a happy civilian. I turned it over and over in my fingers and an idea began to sizzle about low down in my brain. Morton's cry of dismay cut through my thoughts.\n\n\"It's there! I can see it! That glazed look in your eye. Whenever you are dreaming up a suicidal idea you get it. Not again! I don't volunteer!\"\n\nI patted his shoulder cheerfully, then reknotted his tie into a semblance of military order. \"Relax. I have had a brilliant idea, yes. But you are not involved, no. Now let us shoot this video and after it is done I will tell you all about this plan.\"\n\nI stood Morton up with a wall for a backdrop; not a good choice because he looked like he was waiting to be shot. No changes, time was of the essence.\n\n\"If you please. I want a full-figure shot of that man. Let me have a roving microphone. Ready when you are.\"\n\nMorton winced a bit when two spotlights pinned him to the wall. A mike was thrust into my hand and a pure contralto voice rang out across the set.\n\n\"Silence. Ready to roll. Sound. Camera. Action.\"\n\n\"Ladies and gentlemen of Chojecki, I bring you greetings. You are looking at a typical unwilling member of the invading Nevenkebla army. With this video you will have received an audio cassette that is a live recording of an actual encounter with two of these soldiers. You will listen to their bleating complaints, will be shocked at the terror of their involuntary servitude, will cry with joy as they are given the opportunity to hurl the shackles from their shoulders and stride forth into the green countryside, to prosper under the glowing sun of Individual Mutualism.\"\n\nMy sales pitch was so sincere that Stirner could not restrain himself and burst out clapping\u2014as did the crew and technicians. Morton clasped his hands over his head\u2014there is a bit of ham lurking in all of us\u2014and bowed.\n\n\"Silence,\" I ordered and all was instantly quiet. I strode onto camera and pointed at the subdued Morton. \"This is the kind of soldier you will meet and befriend and subvert. Note then the complete absence of markings upon the sleeve.\" Morton extended his arm and I pointed to the right place. \"Empty of stripes, chevrons, angled or curved bits of colored cloth. This is what you must look for. If there is a single stripe, two or more, or most frightening thought, three up and three down with a lozenge in the middle\u2014retreat! Do not talk to anyone with these kinds of adornment because you will be addressing one of the enslaving devils incarnate!\n\n\"Also be warned if there are shining bits of metal on the shoulders, here and here. Those who wear these are known as officers and are usually too stupid to be dangerous. They must still be avoided.\n\n\"Another group, very dangerous, can be recognized by their headgear and brassard. If the letters MP appear upon the arm\u2014go the other way. Also look for the red cap that will be mounted squarely upon the brutal head.\n\n\"Now that you know what to avoid, you know whom to approach. A simple uniformed slave. Come close, smile, make sure that none of the striped and barred beasts are close, then whisper in the slave's ear... 'Do you like fresh air?' If he smiles with joy and answers 'yes,' why then he is yours. May Mark Forer guide you in this great work!\"\n\n\"Cut, print, thank you gentlemen.\"\n\nMorton blinked as the spots died away and began to tear off the uniform. \"What, may I ask, what was this cagal about the fresh air.\"\n\n\"No cagal, old friend,\" I said, holding up the liberty pass I had taken from the pocket of the borrowed uniform. \"I intend to go forth to bring the word to the troops that when they go out the gates tomorrow night they should not bother to return.\"\n\n\"I knew that you had an insane idea!\" he shrieked, staggering back, wide of eye and pale of skin. \"The only way that you can talk to the troops is by going back onto the base.\"\n\nI nodded a solemn nod of agreement.\nChapter 23\n\n\"It is suicide,\" Morton shivered.\n\n\"Not at all. Good sense. If that swine Zennor is still looking for me\u2014he certainly won't be looking among the troops. I have this pass dated for today. I return to base early since there is not much doing in the old town tonight. Then I go to the latrine, the PX, all the other exciting places where the troops assemble, and talk to the lads. And do some other interesting things which it is best you don't know about. Don't worry about me.\"\n\nI could worry enough for myself I thought, darkly. Once back in the army there were a number of problems I would have to tackle. And all of them were dangerous.\n\n\"But how will you get out again?\" Morton asked, his voice speaking as though from a great distance, cutting through the black brooding of my thoughts.\n\n\"The least of my worries,\" I laughed hollowly. And indeed it certainly was. I turned to the ever-patient Stirner who had been listening to us in silence. \"You know what to do with the cassettes?\"\n\n\"It will be as you planned. Volunteers are already waiting to distribute them to even more volunteers who will go forth and do good deeds just as we did. It was inspiring!\"\n\n\"Indeed it was. But no sallying forth until tomorrow night in the very earliest. The password must be spread, there must be eager volunteers to make this a mass movement. Because once the officers catch wise things will become difficult. The railroad will be watched or stopped altogether. If that happens other transport must be provided. Keep things moving though, until I get back. You are the authority on desertion now.\"\n\n\"How long will you be away?\"\n\n\"Don't know. But for the shortest amount of time that is possible\u2014that I can guarantee.\"\n\nThere was little more to say, nothing more to do. I squared my cap upon my head and turned to the door.\n\n\"Good luck,\" Morton said.\n\n\"Thanks.\" I was going to need it.\n\nAs I walked back through the empty streets toward the Vaillant section of town I fought off the depression that accompanied the uniform I wore. Nor could I drown my sorrows in drink, since money was worthless here and I had returned Stirner's wirrdisc. Soon I was walking among the inaccessible, brightly lit palaces of pleasure, pressing my nose against the window just like the other uniformed figures that roamed the streets. Some leave! Although the evening was still young, many of them were already drifting back toward Fielden Field, where the camp had been built. I joined in this Brownian movement of despair.\n\nBright lights burned down upon the barbed wire that encircled the green grassy meadows, where once the good citizens of the city had taken their ease. Green no more, pounded now into dust and filled with gray army tents erected to house the troops. No effete comforts for the conquering soldiers; they might get spoiled. The officers, of course, lived in prefab barracks.\n\nIt took all the strength of will that I possessed to join the line of depressed figures that moved toward the MPs at the gate. While my intelligence told me that the last thing to be expected was a soldier with a pass illegally entering the camp, the animal spirit within me was screaming with anguish.\n\nOf course nothing untoward happened. Dim little eyes stared out from under the mat of thick eyebrows, scanned the familiar pass, waved me back into captivity. The sweat cooled from my brow and I jingled the few coins in my pocket that the freedom-bound soldier had been happy to leave behind. There was just about enough of them to buy an understrength beer in the PX. Anything is better than nothing.\n\nI found this depressing establishment easily enough. I just traced the sound of rock-drilling and western music to its source. The PX bar was housed in a sagging tent vaguely illuminated by light bulbs that had been specifically designed to attract flying insects. Here, at rough tables of drink-sodden wood, sitting on splinter-filled planks, the troops enjoyed the pleasures of warm, bad beer. I bought a bulb and joined them.\n\n\"Got room for one more?\"\n\n\"Cagal off.\"\n\n\"Thanks a lot. What is this\u2014cagal your buddy week?\"\n\n\"It's always cagal your buddy week.\"\n\n\"You sound just like the civilians in town.\"\n\nThis aroused some interest. The heavyset speaker now focused his blurred vision on me and I realized that all of the others at the table were listening as well.\n\n\"You got a pass tonight? We get ours tomorrow. What's it like?\"\n\n\"Like pretty grim. They won't serve you. If you like grab a drink they close the bar and all go home.\"\n\n\"We heard that. So what's the point of going in. Nothing.\"\n\n\"Something. You get to leave the army, travel far away, eat good food, get drunk. And kiss girls.\"\n\nWow, did I have their attention now. If eyeballs were gun muzzles I would have been blown out of existence in an instant. There was a dead silence at the table as every head swiveled in my direction.\n\n\"What did you say?\" a hushed voice asked.\n\n\"You heard me. You go down to where the restaurants are and walk slow. If someone says to you\u2014Do you like fresh air?\u2014just say that you do, you do. Then go with them. They'll get you civvies to wear, a ticket out of town\u2014and set you up on the other side of the country where the MPs will never find you.\"\n\n\"You are cagaling us!\"\n\n\"No way. And what do you lose by going along with it? Whatever happens\u2014it got to be better than the army.\"\n\nThere were no arguments with this. Only the muscled guy with the suspicious mind found what he thought was a loophole.\n\n\"If what you are telling us is true and not the old cagal\u2014then what are you doing back here?\"\n\n\"A very good question,\" I stood up and held out my pass. \"I came back for the bundle of letters from my mom. This pass is good until midnight. See you in paradise\u2014if you want to come.\"\n\nI left them and moved on to the next group, who were in the corner of a latrine shooting dice. I palmed the dice and won some good pots, which drew their attention, gave them my orientation talk and left.\n\nI worked at this until it was almost midnight when my pass ran out and my story would take on a dubious taint. I had planted the seeds in fertile ground. The word would spread instantly through the latrine rumor network. And if I knew my draft dodgers not one of them would return from pass tomorrow night. That should cheer General Zennor up!\n\nSo plan number two must now be put into effect.\n\nFor what I had to do next I needed a bit more rank. There would be no slow crawl up through the noncommissioned ranks this time. I had tasted the heady glory of being an officer and I was spoiled forever. So I headed for the lair of those brightly plumaged birds of prey: the officers' club. I found it by backtracking the drunks. The higher the rank the stronger the booze; this was the army way. I passed a staggering pair of majors, each holding the other up, lined myself up on a colonel flipping his cookies into a hedge, took a sight over an unconscious captain in the gutter and saw my target glowing on the horizon. I skulked off in that direction and took refuge behind some bushes where I had a good sight of the entrance.\n\nIt was strictly a bachelor affair and all the worse for it. Obscene songs were being sung loudly and off-key. At least two punchups were going on in the grass outside at all times. There was some coming, of sober officers just off duty, but much more going of officers drunk out of their cagaling minds. I watched from hiding until my prey emerged, stumbled, and came toward me singing hoarsely under his voice.\n\nHe staggered under the only street lamp. A captain, about my size, lots of fake medals and decorations, just what I needed. A simple armlock from the rear, correct pressure applied, struggle feebly, unconscious, then into the hedges with him. A piece of cake.\n\nHe passed muttering by. Silent as a wraith I moved, pounced, seized, applied pressure...\n\nAnd found myself sailing swiftly through the air to crash into the hedge.\n\n\"So\u2014revolt in the ranks,\" he snarled, relatively sober and on guard in an instant, crouched and approaching. I struggled to my feet, feinted with my left hand and chopped down with my right. He blocked and would have kicked me in the stomach if I hadn't jumped aside.\n\n\"Want to kill an officer? Don't blame you. And I have always wanted to kill a private. Good time right now.\"\n\nHe advanced\u2014and I retreated. The medals had not been fakes. With great skill I had managed to find and attack what was probably the only trained combat officer in this army. Tremendous!\n\n\"Death to all officers!\" I shouted and swung a wicked kick at his groin.\n\nHe was bright enough to know he was whoozy, so instead of trying to block he stepped back. I kept the kick going which pulled me around to face in the other direction.\n\nAnd ran away. Discretion is the better part of valor. He who fights and pulls his freight lives to fight another date. I had no macho points to make. I just wanted to stay alive!\n\nDive and shoulder roll over a hedge. Roaring, he crashed through it right behind me. There were tents ahead, hard boots pounding after me. Jump over a tentrope, dodge under another. A shout and a crash behind me. Good\u2014he had tripped over one of the ropes. A few paces gained. Run, fast as I could. Between the next row of tents and back to the street. A building up ahead, loud music and the sound of breaking glass coming from it. I was at the rear of the officers' club.\n\nTime to go to ground. Through the gate and into the yard, gate closed behind me, no sign of pursuit.\n\n\"You had your break, quit cagaling off, get them cases in here.\"\n\nA fat cook stood at the rear door of the kitchen under the light, blinking into the gloom of the yard. Figures stirred as the enslaved KPs moved, as slowly as possible, to the stacks of beer cases. They had their jackets off, wearing only undershirts in the steamy heat of the kitchen. I took off my own jacket, rolled it and pushed it behind the cases, seized up a beer case and followed the others inside.\n\nKitchen police. The most demeaning servitude in the army\u2014which is an establishment that prides itself on demeaning servitude. KP was so degrading that it was forbidden, by military law, to give KP as a punishment. So, naturally, it was always given as a punishment. Up before dawn, laboring until late at night. Washing pots, cleaning out disgusting grease traps in the underground plumbing, slaving at the most menial tasks that generations of warped minds had created. It was absolutely completely impossible that anyone would volunteer for this service. I would never be looked for here!\n\nI carried the case past the cook who was acting KP pusher. He had a filthy chef's hat on his head, sergeant's stripes tattooed onto his beefy forearms, and was brandishing a long ladle as a weapon. He scowled as I passed then pointed the ladle in my direction.\n\n\"You. Where you come from?\"\n\n\"It's a mistake,\" I whimpered. \"I shouldn't be here. I didn't do nothing like what the first sergeant said I done. Let me go back...\"\n\n\"If I have my way you will never go back,\" he screamed. \"You will die in this kitchen and be buried under the floor. You're on pots and pans! Move!\"\n\nHarried by blows from the ladle I moved. To the giant metal sink to seize up the filthy metal pot waiting there. A simple labor, washing a pot. Harder perhaps when the pot is as big as you are. And another and another\u2014and still another. Steam, hot water, soap, labor with no end.\n\nI worked and sweated until I felt that enough time had passed for any excitement and search to have died away. As I straightened up, my aching back crackled loudly. I wiped a soggy forearm across my dripping forehead. My hands were bleached, my fingers as wrinkled and pallid as long-drowned slugs. As I looked at them I felt anger growing\u2014this was no fit job for a stainless steel rat! I would be rusting soon...\n\nThe ladle crashed down on my shoulder and the choleric pusher roared his ungrammatical commands.\n\n\"Keep working you're gonna be in trouble!\"\n\nSomething snapped and blackness overwhelmed me. This can happen to the best of us. The veneer of civilization worn thin, the lurking beast ready to burst free.\n\nMy beast must have burst most satisfactorily, thank you, because the next thing that I was conscious of was hands pulling at my shoulders. I looked in astonishment at the gross, flaccid form beside me, a pair of giant buttocks rising high. I had my hands about the pusher's neck, had his head buried in the soapy water where he was apparently drowning. Shocked, I pulled him up and let him slip to the floor. Gouts of water poured from nose and mouth and he gurgled moistly.\n\n\"He'll live,\" I told the circle of wide-eyed KPs. \"Any of the cooks see what happened?\"\n\n\"No\u2014they're all drunk in the other room.\"\n\n\"Great.\" I tore the KP roster from the wall and shredded it. \"You are all free. Return to your tents and keep your mouths shut. Unhappily, the pusher will live. Go.\"\n\nEagerly, they went. I went too, to the pegs where the cooks had discarded bits of uniform as they worked in the heat of the kitchen. There was a formerly white jacket with sergeant's stripes on it. Perfect for my needs. Donning it I strode into the kitchen, in my element, no need to skulk, and on into the dining hall and barroom.\n\nIt was wonderful. Music played, officers roared, bottles broke, songs were sung. Uniformed figures slumped over the tables while others had slid to the floor. The survivors were well on their way to join the succumbed. I pushed through this alcoholic hell and greatly admired the unconscious drunks. I was still aching from the captain's spirited defense. I had rediscovered a dictum that must be as old as crime: Rolling drunks is easier than mugging.\n\nA major in the space service caught my eye, prone on the floor and snoring. I knelt next to him and stretched my arm out next to his. Same length; his uniform should make a fine fit.\n\n\"Washa?\" a voice muttered from above and I realized that my bit of tailoring measurement had not gone unnoticed.\n\n\"The major is on duty later. I was sent to get him. Come on major, walkies, sackies.\"\n\nI struggled to lift the limp figure, aided very slightly by his friends. In the end I seized him under the arms and dragged him from the room. His departure was not noticed. Through a door and down a hall, to a storeroom filled to the ceiling with bottles of strong beverage. He would feel right at home here. With the door secured I took my time about stripping him and donning his uniform. Even his cap fitted well. I was a new man, rather officer.\n\nI left him dozing out of sight behind the drink. Straightened my tie. And sallied forth to save the world. Not for the first and, I had the feeling, not for the last time either.\nChapter 24\n\nI looked around at the bottles, reached for one\u2014then slapped my wrist.\n\n\"No, Jimmy, not for you. The number of beers you had this evening will have to suffice. What you have to do will be better off done sober.\"\n\nWhat did I have to do? Simply get aboard one of the spacers, find the communications room, then locate the coordinates of this planet. Easy to say: a little difficult to do.\n\nAt least the first part was easy enough to accomplish: locate the spacers. I had seen the floodlit shapes of three of them rising high above the tents earlier in the evening. The party was still crashing inside so this would be a good time to move through the camp. While plenty of drunks were still staggering about. I brushed some dust from my lapel, straightened the medals on my chest. Quite a collection. I turned the gaudiest one over and craned to read it. THE GLORIOUS UNIT AWARD\u20146 WEEKS WITHOUT VD IN THE COMPANY. Wonderful. I assumed the rest of the lot had been given for equally valiant military endeavors. Time to go.\n\nIt looked like events in the alcoholic bedlam were winding down for the night. A grill was being locked over the bar. Orderlies were loading unconscious forms onto stretchers, while the walking wounded were stumbling toward the exit. A brace of gray-haired colonels were leaning against each other and moving their feet up and down and not getting anyplace. I made the twosome a threesome and let them lean on me.\n\n\"I am going your way, sirs. Perhaps I could accompany you?\"\n\n\"You shure a good buddy... buddy,\" one of them breathed my way. The alcoholic content of my blood instantly shot up and I hiccuped.\n\nWe exited in this manner, weaved our way between the ambulances being loaded with officerial alcoholics, and staggered off into the night. In the direction of the spacers. I had not the slightest idea where the BOQ was\u2014nor did I care. Nor did my drink-sodden companions. It took all their concentration, and what little conscious mind they had remaining, to simply put one foot in front of another.\n\nA squad of MPs turned the corner in front of us, saw the gleam of light from the silver chickens on my companions' shoulders. Then did the smartest about-face to the rear march I had ever seen.\n\nMy drunks were getting heavier and heavier and moving slower and slower as we stumbled through the tent-lined street toward a brightly lit building at the end. It was large and permanent, undoubtedly part of the park facilities purloined from the natives. Even at this hour of the night, morning really, two armed guards stood at the entrance. All the rocks along the path were painted white and the overly ornate sign above the door read BASE HEADQUARTERS\u2014GEN. ZENNOR COMMANDING.\n\nThis was definitely not the place for me. I maneuvered my charges onto the grass, next to the sign KEEP OFF OR GET SHOT, and let go. They dropped instantly and began snoring.\n\n\"You, guards,\" I called out. \"One of you get the Officer of the Day. These colonels have been taken ill. Food poisoning I think.\"\n\nI glared my best glare and not a muscle moved in their faces.\n\n\"Yes, sir!\" the sergeant shouted. \"OD on the double!\"\n\nHe turned and hurried away and so did I. Toward the charred remains of a sportsfield upon which the three spacers rested. All of them bristled with guns, brought here to impress the locals I imagined. Or to beat off the armed attacks that had never materialized. How depressed all the military must be that they couldn't pull their shiny triggers and blow away the population. They had given a war\u2014and nobody came. Terribly frustrating.\n\nI staggered as I walked so I would be recognized as an officer. Toward the extruded stairs that ran from the ground, up into the bowels of the nearest spacer. I was a space officer, I was just going to my ship. Or at least I thought I was until I saw that a guard stood on the lower step.\n\n\"Halt and be recognized.\"\n\n\"Cagal off...\" I muttered and pushed by him. A private lowest of the low.\n\n\"Please, major, sir, your majesty. You can't go in without I see your pass.\"\n\n\"Cagal off twice!\" Witty, witty. \"Don' need no pass my own ship.\"\n\nPast him and up the stairs. Brain beats brawn anytime. Step by step up toward the gaping spacelock. And the surly sergeant-major who stood and scowled there, firmly blocking the entrance.\n\n\"This ain't your ship, major. I know this ship's company. You are on another ship.\"\n\nI opened my mouth to argue, order, shout. Then saw the gunmetal blue jaw, the glowing red eyes, the hairline that blended into the eyebrows. Even the hairs curling from his broken nose looked like they were made of steel wool.\n\n\"Not my ship?\"\n\n\"Not your ship.\"\n\n\"Gesh it's not my ship...\" I susurrated, turning and stumbling back down and away into the night. There was no way I was going to get past the sergeant-major. Back toward the headquarters building and the rows of tents to come up with another plan.\n\nHidden in the darkness under a large tree, I looked out at the spacers and could think of absolutely nothing I could do to get aboard one of them. The hour was late, the drunks now dispersed, the camp silent. Except for the roving bands of MPs. Whatever was to be done would have to wait until morning. It would be more dangerous in daylight but it had to be chanced. Perhaps if there was enough to-ing and fro-ing to the ships I might be able to join in. Right now I really should be thinking of my own safety. And some sleep, I yawned. The kicked-in ribs were hurting again. I sniveled a bit and really felt sorry for myself.\n\nIn the stillness of the night the shouted commands and stamping of boots from HQ could clearly be heard. Guns were brought snappily to attention as a huddle of officers emerged and hurried down the path. Even at this distance I could recognize the repellent form of the leader. Zennor, with his underlings hurrying after. I drew deeper into the shadows; this was no place for me to be.\n\nOr was it? Despite my desire for rapid departure and continued survival I stood there and concentrated. And hated the idea that began to develop. The officers moved out across the charred sportsfield and passed the spacer I had tried to get into. At this moment the idea jelled and I loathed it even more because it just might work. With a great effort I forced down the screaming meemies that threatened to overwhelm me, unlocked my knees and lurched forward. Following the officers across the field.\n\nIf any of them looked back I was sunk. But that was next to an impossibility. Their job in life was to bull straight ahead and walk over anything and everything that got in their way. They charged on and I charged after them, getting ever closer. Anyone watching would see a group of officers with one more of their kind hurrying to catch up.\n\nWhen they reached the steps of the freighter I was right behind them, watching them mount with dignity. Though still hurrying I did not hurry that fast anymore. With precise timing I reached the guard at the foot of the stairs just as they vanished from sight above.\n\n\"General,\" I cried. \"The message has come through. It is urgent!\"\n\nI waved and called out again and brushed by the guard who did the only thing he was supposed to do. He saluted. Up the stairs, much slower now, dragging one leg, old war wound you know. They were well out of sight as, breathlessly, I reached the open port.\n\n\"The general, where is he?\"\n\n\"Captain's quarters, sir,\" the guard said.\n\n\"That's near the communications room on this type of ship?\"\n\n\"That's right, major, same deck, number nine.\"\n\nI hurried on to the nearest companionway and up it. Slower and slower. The ship was silent, empty, but I heard voices echoing down from above. When I reached the next deck I walked around to the companionway on the far side. Where I stopped and counted slowly to two hundred.\n\n\"You are a brave but foolhardy devil, Jim,\" I muttered and agreed strongly with myself. Press on.\n\nThe large number nine on the bulkhead above slowed me to a crawl. I carefully poked my head above the deck and looked around. No one in sight, but voices were sounding from the passageway. The doors had numbers stenciled next to them. One of them had a name on it. COMMUNICATIONS ROOM.\n\nIt's now or never, Jim. Look around carefully. Nobody in sight. Take a deep breath. What is that loud hammering noise? Just your heart thudding with the usual panic at a time like this. Ignore it. Step up, step forward, to the door, seize the knob.\n\nExcept the knob had been sawn off. Steel bars had been bolted to the door and welded to the frame. The communications room was sealed, shut, inaccessible, tight.\n\nAs I was registering these facts and trying to make sense of them a voice spoke in my ear.\n\n\"You, what are you doing there?\"\n\nIf my heart had thought it was thudding along merrily earlier it now tore loose from all its moorings and hurled itself up into my throat. I spun about, swallowed it, tried not to say Glugh! Grimaced and looked at the uniformed figure before. At the shoulders. I sneered.\n\n\"I might ask the same thing of you, lieutenant. What are you doing here?\"\n\n\"This is my ship, major.\"\n\n\"Does that give you permission to speak to a superior officer in that manner?\"\n\n\"Sorry, sir, didn't see the leaves, sir. But I saw you by the com room and we had orders about it...\"\n\n\"You are absolutely correct. Sealed and no one near it at any time, correct?\"\n\n\"Correct.\"\n\nI leaned my face close to his and scowled and watched with relief as his skin paled. It is hard to scowl and sneer your words at the same time, but I managed.\n\n\"Then you will be pleased to know that my orders are to see that your orders are carried out to the letter. Now, where is General Zennor?\"\n\n\"Down there, major.\"\n\nI spun on my heel and walked in the direction that I wished least to take. He would be watching me, I was sure of that. But I had no choice. If I simply tried to leave the ship he might start to think about my presence, get suspicious, sound the alarm. If I went to the general all doubt would vanish.\n\nOf course I might vanish too. Nevertheless I walked swiftly toward the open door and the murmured voices, turned into it without hesitation.\n\nThe officers at the end of the room were conferring over a map. Zennor had his back turned to me.\n\nI turned sharply right and saw the shelves of books against the bulkhead. Without hesitation I went to it, ran my finger down the volumes. I could not see their titles because of the sweat that was dripping into my eyes. Seized one at random. Turned and started back toward the door. Let my eyes cross over the group of officers.\n\nWho were completely ignoring me. I walked slower, ears straining, but could hear nothing other than a murmured cagal or two, which was required of any military conversation.\n\nWhen I entered the corridor the lieutenant was just scuttling out of sight. I walked, neither fast nor slow, to the companionway and down it, deck by deck. Waiting for the alarm bells to go off. Though I probably wouldn't have heard them through the pounding of blood in my ears. To the last deck and to the open port with the welcoming blackness of the night beyond.\n\nThe guard leapt into the air and my heart followed him.\n\nAnd landed with his weapon at present arms. I threw him a sloppy salute in return and trotted down the steps to the ground. Another salute and I was walking across the burned grass and waiting for the shot in the back.\n\nIt never came. I reached the shadows at the edge of the grounds, slipped into them and leaned against the bole of a tree. And sighed a sigh such as I had never sighed before. When I raised my hand to wipe the perspiration from my brow I realized that I was still holding the book.\n\nBook? What book? Oh, the book I had lifted from the cabin about four hundred and twelve years earlier. When I held it up and squinted I could just make out the title in the illumination of the distant lamps.\n\nVeterinary Practice in Robot Cavalry Units.\n\nIt dropped from my limp fingers as my back slid slowly down the tree until I was sitting on the ground.\nChapter 25\n\nI rested there in the darkness, let the sweat evaporate, tried not think about veterinaries for robot horses\u2014and pondered the significance of the sealed door on the communications room.\n\nFor openers, it had not been sealed shut to keep me from getting inside. As much as I valued my own importance I was well aware that others, Zennor in particular, were not struck with fear by my presence. For example the combat-ready captain earlier this night. No, Zennor had the door sealed for his own reasons. What were they? Work backward from the obvious.\n\nThe door on this ship was sealed, so probably all of the com rooms on all the spacers had been sealed. It made no sense to shut just a single one. Why? To stop communication, obviously. Between who and whom? Or whom and who for that matter. It couldn't be intended to stop planetary communication. That was still needed for the not-too-successful invasion. Ground-based radios would suffice for that. Spacer com rooms sealed obviously meant that ship-to-ship communication would cease. That was of no importance since the entire fleet had already landed.\n\nWhich left only interstellar communication. Of course! The rush to leave, the secrecy about our destination. Zennor knew that the League Navy was after him, knew that they could only stop him if they knew where he was going. Or where this planet was. So the invasion was a one-way affair. A gamble hurled into interstellar space. Not much of a gamble against an unarmed enemy. Zennor knew that the Navy had spies, all those detector vans had been evidence of that. He was convinced that I worked for the League and there might be other League agents in his army. So communication had been cut off until the invasion succeeded. After that there would be nothing that the Navy could do.\n\nThis was good for the invasion\u2014but very bad news for me. I had sent the radio message for help, which even now was limping steadily across interstellar space at the miserable speed of light. I had better forget about it. And forget as well about sending an FTL message for the time being. What I had to do now was think local. I might have to spend the rest of my life on this planet. If I did remain here I didn't want to do it with Zennor and his military goons breathing down my neck. Desertion, that was the name of the game. I had to get his army away from him. When all the draftees had been dispersed about the land I would consider the next step. Which didn't bear considering. Maybe I should open a distillery and supply free booze to his officers and noncoms? From what I had seen, with the correct encouragement, they all would be dead of cirrhosis within the year.\n\nI yawned and realized that my eyes were closed and I was half asleep.\n\n\"Never!\" I groaned, climbing to my feet. \"Fall asleep here, Jim my boy, and the chances are that you will wake up dead. To work! Next step is to get your chunk off this base, for your work here is done for the moment. Back to warmth and light and female companionship, away from solitary males, cursing, drinking, gambling and all the other military pleasures. Away!\"\n\nBut was I ever tired. Instead of walking it would sure be nice to have a bit of transportation. Somewhere near HQ there had to be vehicles, since officers rarely walked. Nor were these vehicles too hard to find. Just behind the HQ building there was a motorpool, unguarded apparently. And there, looming darkly behind the staff cars, the shape of a command car. One I was very familiar with. I drifted over and climbed into it. No guards needed at this motorpool because all the ignition keys had been taken away. I smiled into the darkness. This crate could be hotwired faster than a key could be fumbled into the lock. I bent, pulled, twisted. Sparks sizzled and the fuel cell hissed into life. Boldly on with the headlights, into gear and away.\n\nAway to where? Not the gates surely. During the daytime it might be possible to drive out behind a convoy. But right now the gates would be closed and I would have to produce a pass or some sound reason for nighttime maneuvers. I could think of no sound reason. I drove on slowly, past one of the gates and along a perimeter road that circled the camp, just inside the barbed wire fence. For security patrols undoubtedly. I drove along it until a grove of trees came between me and the lights of the camp. I angled the headlights toward the fence, locked the gears in neutral and climbed down to look at the barrier.\n\nIt was a ten-strand barbed wire fence. There were surely alarms attached if it were breached, but I could see no sign of disturbed earth, tripwires or circuitry that might lead to mines. Just bashing through it might be a chance worth taking. It didn't matter if the alarm were raised. By the time the sluggish troops reached the site I would be long gone. I raced the engine, put it in the lowest gear, floored the accelerator and ground forward.\n\nThe wire fence screeched and tore. There was a fine show of crackling sparks\u2014I thought it might be electrified, but the combat car was shielded\u2014and then it all tore away and I was through. Kicking up through the gears and tearing away through the empty streets. Pulling the wheel and screeching around a plaza with a large statue of Mark Forer gazing down serenely from a plinth, and out the broad avenue on the far side. I recognized this street, I had walked this way before when we had first escaped. The river and bridges were up ahead. With the residential suburbs on the far side.\n\nWhen I trundled my battle wagon across the bridge there was still no sign of pursuit. Fine. Time to go to ground. I turned off along the riverbank, put the gears in low-low, angled toward the water and jumped down. The car ground steadily on, demolished a bench\u2014sorry about that\u2014and plowed majestically over the edge. There was plenty of burbling and splashing, then nothing. The river was deep here. Behind me I could hear the wail of distant sirens. I walked briskly through the park and into the nearest street. Though I was tired I needed to put some distance between myself and the river, in case there were tracks left which might be seen by day.\n\n\"Enough is enough, Jim!\" I said, leaning against a wall and all too aware that I was drooping with fatigue. I had turned corners at random, lost myself completely, and the river was far behind me. There was a gate in the wall beside me, with Dun Roamin carved into the wood. Message received. Without hesitation I opened the gate, climbed the steps beyond and knocked on the front door. I had to do it a second time before there were stirrings inside and a light came on. Even after all the time here on Chojecki I still found it hard to believe that this was the correct way to meet strangers.\n\n\"Who is it?\" a male voice called out as the door opened.\n\n\"Jim diGritz, offworlder, tired.\"\n\nThe light came on and an ancient citizen with whispy gray beard blinked out myopically at me.\n\n\"Can it be? It certainly is! Oh what luck for old Czolgoscz!! Come in brave offworlder and share my hospitality. What may I do for you?\"\n\n\"Thank you, thank you. For openers let's get these lights off just in case there is a patrol around. And then a bed for the night...\"\n\n\"My pleasure! Illumination off, follow closely, this way, my daughter's room, now married and living on a farm, forty geese and seventeen cows, here we are. Curtains closed, a moment, then the lights!\"\n\nOld Czolgoscz, although he tended to talk too much, was the perfect host. The room was pink with lace curtains and about twenty dolls on the bed.\n\n\"Now you wash up, right in there, and I'll bring you a nice hot drink, friend Jim.\"\n\n\"I would prefer a nice cold drink rich with alcohol, friend Czolgoscz.\"\n\n\"I have the very thing!\"\n\nBy the time I had rinsed the last of the military muck away he was back with a tall, purple bottle, two glasses\u2014he wasn't that old\u2014and a pair of pajamas ablaze with red lightning bolts. I hoped that they didn't glow in the dark.\n\n\"Homemade gingleberry wine.\" He poured two large glasses. We raised them, clinked, drank and smacked our lips. I sighed with happiness and a bit of nostalgia.\n\n\"I haven't had this since I was back on the farm. Used to have a bottle hidden out in the porcuswine sty. On dull days I used to get blotto on it and sing to the swine.\"\n\n\"How charming! Now I will leave you to your rest.\"\n\nA perfect host, vanished even before I could thank him. I raised my glass in a toast to the electronic benevolence of the portrait of Mark Forer upon the wall. Drained it. And went to sleep.\n\n* * *\n\nWhen consciousness reluctantly returned I could only lie and blink, drugged with sleep, at the sunlight behind the curtains. Yawning, I rose and opened them and looked out at a flower-filled garden. Old Czolgoscz looked up from his labors and waved his secateurs at me. Then scurried into the house. In a remarkably short period of time he knocked on the door, threw it open, and brought in a groaning breakfast tray. I don't normally have a liter of juice, large portion of wiffles with syrup and three eggs. I did today.\n\n\"How did you know?\" I lip-smacked satedly.\n\n\"Guessed. Young lad your age, been working hard, seemed natural. I talked to a few people and I am sure that you will be pleased to hear that the teams are in training all over the city for D-Day.\"\n\n\"D-Day?\"\n\n\"Desertion Day. Today, tonight. Extra trains have been scheduled and people all over the country are looking forward to welcoming the new citizens.\"\n\n\"Fantastic. I hope you will welcome me as well. My stay on Chojecki may be longer than originally planned.\"\n\n\"You are more than welcome, as is your knowledge. Would you like a teaching position at the university?\"\n\nI smiled at the thought. \"Sorry, I ran away from school, never graduated.\"\n\n\"I regret in my provincial ignorance that I do not know the meaning of either run away or graduate. Students here go to school when they want, stay as long as they want, study what they want, leave when they want. The only scholastic requirement a child has is to learn about Individual Mutualism, so he or she can lead a full and happy life.\"\n\n\"I suppose the parents pay for the child's schooling?\"\n\nCzolgoscz drew back, horrified. \"Of course not! A child will get love and affection from its parents, but they would not embarrass their offspring by violating IM's tenets. The child's wirr account, opened when it was born, will be in debit until he or she begins to earn. At a very early age, for the child will not be a free and independent citizen until the wirr account is in credit.\"\n\nNow I was shocked. \"The workhouse for infants! Laboring day and night for a few crusts!\"\n\n\"Friend Jim\u2014what a wonderful imagination you do have! Not quite. Most of the work will be done around the house, the labors that were usually done by mother, collecting the wirrs father would pay her...\"\n\n\"Enough, I beg. My blood sugar is low, my head thick and the details of IM so novel that they must be absorbed just a bit at a time.\"\n\nHe nodded agreement. \"Understandable. As you will teach us about the novelties of the great civilizations out there among the stars, we have been cut off from them for centuries, so will we reveal to you the fruits of Mark Forer's genius\u2014may electrons flow forever through its wires!\"\n\nA pleasant prayer for that long-vanished machine. I still found it hard to understand such affection for a bunch of circuitry, no matter how complex. Enough, it was time to get back to work.\n\n\"Can you find out where my friend Morton is staying?\"\n\n\"Would you like to go there? I will be honored to take you.\"\n\n\"You know...\" I gaped, then answered my own questions. \"Of course, everyone in the city knows where we have been staying.\"\n\n\"That is correct. Do you ride the bike?\"\n\n\"Not for many years\u2014but once learned, never forgotten.\" A sensible form of transportation, the bicycle, and the streets of this city were busy with them. I bundled up the uniform for possible future use, pulled on a pair of baggy shorts that Czolgoscz produced. This, and my undershirt, produced an inconspicuous cycling outfit. Thus garbed I went into the garden and limbered up with a hundred pushups. When I finished and climbed to my feet I shied back from the man who stood behind me leaning on a bright red bicycle.\n\n\"I did not mean to startle you,\" he said. \"But I did not wish to interrupt your ritual. Czolgoscz phoned me and I brought your bicycle around. The best one I had in stock.\"\n\n\"Thank, thank you\u2014indeed a beauty. But I am afraid I cannot pay you for it...\"\n\nHe smiled. \"You already have. I stopped at the wirrbank and debited your account. They asked me to give this to you.\"\n\nI did some rapid blinking at the wirrdisc he handed me. James diGriz it was labeled. And in the little LCD window it read Balance 64,678.\n\n\"The bank asked me to ask you to contact them. They were not sure how many hours you worked for the public service last night. If you would kindly report to them they will make the correction.\"\n\n\"I am in the system!\" I shouted happily. The bicycle man beamed happy agreement.\n\n\"Of course! You are an individual and Individual Mutualism is your right. Welcome, welcome! May your wirrbalance grow and may your life be a long and happy one!\"\nChapter 26\n\nIt was next morning when the cagal hit the fan. Reports had come in during the night of the fantastic success of D-Day. The troops had trooped into town with their passes, had expressed a great appreciation of fresh air, had been welcomed at the back entrance of any clothing store to change out of their uniforms, had boarded train after train. The last one left just before midnight when the curfew had descended.\n\nAnd there had been no alarm, not at first. Luckily there were four gates into the camp and I presumed that the MPs, in their native ignorance, had all thought the returning soldiers had used the other gates. Therefore they had all been happy to cagal off for the evening. So successful had been our operation that even the extra trains had not sufficed for the mobs of deserters. Over a hundred were still in the city. They would stay hidden until nightfall when, hopefully, they would be smuggled to the station.\n\nWith my new-found wealth I had bought a giant TV as a gift for our hosts. Morton and I were watching a local broadcast when the military cut in. Neither of us appreciated it for this was a day of celebration of some kind, the anniversary of the wiring of Mark Forer's first circuit board or some such, and all the city had turned out. We were enjoying a parade, headed by the local girls' cycle club, all flashing bronzed limbs and fluttering skirts, when the picture sizzled and died to be replaced by General Zennor's scowling features.\n\n\"Turn it off!\" Morton moaned. \"If I look at him I won't be able to eat lunch.\"\n\n\"Leave it. It won't be good news, but since we will have to hear it sometime\u2014better now.\"\n\n\"Attention!\" Zennor said and Morton made a rude noise with his tongue; I waved him to silence. \"You all know me, General Zennor of the liberating forces. You know me as a kind and patient man...\"\n\n\"He is a great fiction writer!\"\n\n\"Quiet!\"\n\n\"... a firm leader and a just one. And now the time has come for firmness and justice to be applied. I have just discovered that a few cowards among the ranks of my loyal troops have been foolish enough to attempt to desert. Desertion is punishable by death...\"\n\n\"What isn't in the rotten army!\"\n\n\"... and I know that none of you out there would want that to happen to foolish and misguided young men. Therefore this announcement. I am extending all passes issued last night for twenty-four hours. They are good until midnight tonight. No soldier will be punished who returns to the base before midnight. I therefore advise all the people of this city to speak to these misguided youths who are hidden among you. Tell them to return. You know where they are. Go to them. Tell them of this generous offer.\"\n\nThe fake kindness vanished from his face in an instant as he leaned close to the camera and snarled.\n\n\"Tell them also that my generosity vanishes at midnight! Martial law will then be declared. This city will be sealed. No one will enter or leave it. Then the city will be searched. Block by block, building by building. Any deserter who is then found will be taken prisoner, will be given one bottle of beer and will be allowed to write one letter home. And will then be shot.\n\n\"Is that clear enough? You have this single warning. You have until midnight tonight to return. That is the message I send to the deserters. After that\u2014you are as good as dead\u2014\"\n\nI hit the button and turned the set off.\n\n\"Pretty depressing,\" Morton said, looking pretty depressed. \"Turn it back on so we can at least look at the girls.\"\n\nI did. But they were long gone and had been replaced by a man with long hair and an enthusiastic expression who was going on in great detail about the untold joys of IM. I killed the sound.\n\n\"You know, Morton, he means us too.\"\n\n\"Don't say it! I know. Isn't there another station with space opera? I need a drink.\"\n\n\"No you don't. You need to sit quiet and pull yourself together and help me find a way out of this for all of us. Well, maybe a small drink, a glass of beer just to get the thoughts rolling.\"\n\n\"I could not but overhear,\" Stirner said, entering with a tray of glasses and bottles. \"If you will permit I will join you. The day is warm.\"\n\nWe clinked and glugged. \"Any word from the city?\" I asked.\n\n\"A good deal of words. All the trains leaving the city have been canceled so there is no way out by train.\"\n\n\"The roads?\"\n\n\"Roadblocks on all arteries leading from the city. Flying machines supported by rotating wings\u2014\"\n\n\"Choppers.\"\n\n\"Thank you, I have noted the word. Choppers flying over the countryside between so none may escape that way. All young men who attempt to leave are being detained, even when they are obviously Chojecki citizens who speak only our native tongue. They are imprisoned until their hands have been pressed to a plate on a machine, that is what has been reported. So far all have been released.\"\n\n\"Very near,\" I muttered, \"and just about foolproof. Fingerprint check. Right through to the base computer. So we can't get out that way. It will have to be the fields, after dark.\"\n\n\"Not that I want to cast a note of gloom,\" Morton said, gloomcasting. \"Choppers, infrared detectors, side-mounted machine guns, death from the sky...\"\n\n\"Point taken, Morton. Too dangerous. There must be another way.\"\n\nThe lecture had finished and once more hearty biking enthusiasts swept across the screen. All males with hairy knees: Morton grumbled in his throat. Then instantly cheered up as the girls' club appeared, waving and smiling at the camera.\n\n\"Wow!\" I shouted, jumping to my feet and running in small circles. \"Wow-wow!\"\n\n\"Down the hall, second door on the left.\"\n\n\"Shut up, Morton. This is inspiration, not constipation. You see genius at work. You see before you the only man who knows how to get us all safely from the city.\"\n\n\"How?\"\n\n\"That's how,\" I said, pointing at the screen. \"Stirner\u2014get busy on the phone and the backfence gossip circuit. I want this show on the road by midafternoon. It will take us at least that long to organize it.\"\n\n\"Organize what?\" Morton cried. \"I'm lost. What are you talking about?\"\n\n\"I think I know,\" Stirner said, being quicker on the uptake than Mort. \"You are going to leave the city on bicycles. But you will be stopped.\"\n\n\"No we won't\u2014because you got the answer only half right. We'll all be leaving as girls!\"\n\nOnce the idea had penetrated joy reigned for a bit\u2014then we got down to work. Since I was doing most of the planning and organizing I was the very last one to actually get involved in the nitty-gritty of personal survival. There was much coming and going. I was vaguely aware when Morton's bicycle arrived, but then got busy again with the men's cycle club. I ate a sandwich, drank another beer, and looked up blinking when Morton called to me.\n\n\"We've got to leave soon. The first guys are already in the square. Now don't laugh!\"\n\nI fought hard. The fluffy chintz dress wasn't really him. Nor had shaving his hairy legs made much of an improvement. But the foam-stuffed bra helped, as did the wig. From a distance, sure, but close up the effect was a little disconcerting.\n\n\"I think a touch of lipstick is needed.\"\n\n\"Yeah! Well let's see how great you look. Get changing!\"\n\nI did. The cute little pleated skirt was green so went nicely with my red hair. I looked into the mirror and sighed. \"Jim\u2014you never looked better.\"\n\nWe parted, thanking our hosts again for their hospitality. Hoping that we would meet again\u2014after the war. Stirner, as stout a biker as he was a hiker, would be our guide. He set off at a good clip and we girls had to push hard to keep up.\n\nMark Forer square was a scene of gay abandon. Or maybe that is not the right word. Better, perhaps, to say that everyone had been dragged there. As we pedaled up the first thing we saw was the Bellegarrique Girls' Cycle Club. Just like on television, but infinitely more attractive in the flesh. Flesh\u2014some very strange flesh. Because beyond the girls were other girls. Lantern of jaw, thick of thigh, scowling of mien. Our escaping draftees. Some of them hadn't been on a bike in years and were wobbling about the square, occasionally falling in a flurry of skirts and guttural oaths.\n\n\"Attention!\" I shouted, then again until there was a modicum of silence. \"First, knock off the cursing. These kind people are risking their lives to help you deserters, so be nice to them. Second\u2014if anyone falls off when we go past the roadblock we all have had it. Some three-wheelers are on the way, plus some bicycles built for two. Sort yourselves out and mount up. We are on schedule.\"\n\n\"Where are we going?\" one of them called out.\n\n\"You'll be told when you get there. Now timing is important. When I say go\u2014we go. And anyone left behind is in the cagal. And cursing is a privilege of rank,\" I added at their cries of protest. \"I'm in charge so I'll curse for all of us until we get clear. Mount up.\"\n\nI led the deserter-girls around the square two or three times until they closed up and got it together. Only then did I signal the real girls' club to go into action. They were beautiful. With a swoop they came down upon us, breaking into two ranks that swept by on both sides, closed up around us. The leader carried the flag and we followed her with passion. Down the road, smoothly and swiftly.\n\nTowards the roadblock at the junction ahead.\n\nThen around the corner, cutting in front of us girls, came the Veterans' Cycle Club. Every head gray, or if not gray as bald as a billiard ball. Knotty gnarled legs pumped, ancient tickers ticked. Ahead of us they swooped\u2014and on to the barriers that had been set up across the road. Some went around them, others dismounted and pulled them aside. The sergeants and officers shouted back, struggled feebly, but an opening appeared. Just as we did. And just wide enough to get through.\n\nSome of our outriding girls peeled off and helped the ancients make the opening wider. Some of them laughed and kissed the officers. Confusion reigned\u2014and through the confusion, and the opening in the barrier, I led my girls. Silent and sweating and pumping for all they were worth. Through the barrier and down the road and around the bend.\n\n\"Keep going!\" I shouted hoarsely. \"We're not out of the cagal yet. No one stops until we get to the woods. Go! Go! Last one there is a cagal-kopf!\"\n\nWe went. Pedaling and cursing and sweating and wobbling\u2014but we went. Down the road and into the forest, off into the lanes to skid and fall and crash and roll\u2014on the soft green grass.\n\n\"Can we not\u2014do that again!\" Morton gasped, lying on his back and moaning.\n\n\"I don't know, Mort, I thought it was kind of fun. You ought to get more exercise.\"\n\nHe sat up and looked where I was looking, and stopped moaning. The real girls' club had arrived, a symphony of lovely flesh and flowing movements, tossed hair, flashing eyes. And picnic baskets.\n\nWhen the first beer was held high a ragged cheer broke out. The army was only a bad memory; freedom was bliss. This was the first day of the rest of their new lives and if it stayed like this\u2014why paradise was here around us.\n\nI joined in the revelry but my heart wasn't really in it, my smile false. Through some native perversion, and inability to enjoy pure happiness, all I could think of was Zennor and what repulsive tricks he would be up to when he discovered that about half of his army had vanished for good.\nChapter 27\n\nThere were groans and cries of protest when I ordered my bevy of enchanting beauties to their feet.\n\n\"Knock it off!\" I commanded sternly. \"We're still on schedule and if you want to get out of this alive you will obey orders. When I say frog you will jump.\"\n\nI waited until the chorus of croaking and other froglike imitations had died down before I spoke again.\n\n\"We have about another half hour of riding to go. And before you groan remember that these sweet young girls, who have risked their lives to save us, must ride with us\u2014then circle all the way back to the city by another road. And lest we forget, let's hear it for the girls!\"\n\nThe chorus of yells, thanks, cheers\u2014and not a few kisses rolled out. I had to whistle for attention before it died down.\n\n\"Here is the drill. We are now going to go to a factory that has a railroad siding. A freight train from the north will be arriving when we do. We board and we're away. There will be no stops until we are far from the city. Now\u2014mount up! Forward\u2014Ho-o!\"\n\nThere was silence during the ride, because my gallant bikers were feeling the strain. There was some panic when a chopper came swooping up, but I ordered male heads down\u2014girls to wave and smile. It worked fine and there were no more alarms after this. As we rounded the last bend to approach the kakalaka factory we heard the wail of the train's horn. The line of freight cars was just clattering into the siding when we appeared.\n\n\"Open the doors!\" I ordered. \"Get in before another chopper shows up. Take your bikes\u2014they'll be debited from your future accounts\u2014wave bye-bye and blow kisses because we are off in one minute.\"\n\nI turned to thank Neebe, the gorgeous, brown-limbed redhead who was president of the cycling club, but she was just passing on the club flag to her second-in-command. Then she wheeled her bike toward me, smiling a smile that melted my bike handles.\n\n\"May I be very forward, offworlder James diGriz, and force my presence upon you? You have but to say no and I will go.\"\n\n\"Glug...!\"\n\n\"I assume that means yes.\" She entered the freight car, propped her bicycle against mine, and sat down daintily upon a bale of hay. \"You are very kind. Up until today I have been attending school here in Bellegarrique but now, like everyone else, I am leaving. My home is on a farm in the north in a hamlet named Ling. I have talked with my father and mother, brothers and sisters, and grandmother, and they would all be honored if you would stay with us for as long as you wished.\"\n\nI knew that Morton had been listening because his face went completely green and he began to pout.\n\n\"I would be honored, honored. What a wonderful idea!\"\n\nShe smiled, then her expression changed to one of shock when she saw Morton's face.\n\n\"Is your friend ill?\"\n\n\"No.\" I sighed with generosity. \"It is just that he has no place to go and is hoping that you will invite him too.\"\n\n\"Of course!\"\n\nThe green tint vanished instantly and he smiled sheepishly. \"I accept with gratitude. But just for a short time. Until I can get in touch with a friend of mine named Sharla.\"\n\n\"Oh, you do remember her,\" I said sweetly, and he glared at me as soon as Neebe had turned away.\n\nOnce we began to relax it was a pleasant journey. The roadbed was flat, the train swift. After an hour we knew that we were well clear of the city and all the enemy lurking there. The hay bales were broken open, the tired ex-girls, using their padded bras for pillows, slept. It was nearly dark when we made the first stop. Hampers of food and drink were loaded aboard and we were away within the minute. We ate, drank, and fell asleep once again. I awoke to the gentle touch of a soft hand on my shoulder.\n\n\"We are here,\" Neebe said. \"I must awaken your friend.\"\n\nLights were moving by outside the open door as the train squealed to a stop. We climbed down and our bikes were handed after us. Followed by glad cries and shouts of farewell we mounted and followed Neebe down the highway, out of town, and to the family farm. The road was smooth and easy to see. A magnificent nebula half-filled the sky and bathed us in a cool white light.\n\n\"Even if I could go back to Nevenkebla I never would,\" Morton panted.\n\n\"You have family there.\"\n\n\"I'll miss them\u2014but I won't miss the draft, the army, the military, the intolerance...\"\n\nHe gasped for air and I nodded agreement. \"Understood. This planet has a lot going for it. Though I still don't understand all the permutations of IM it seems to be working. But all is not peace and light yet. Let us not forget Zennor.\"\n\nMorton groaned. \"I would love to.\"\n\nNext morning it looked as though the entire family was standing around and beaming down upon us. While the ladies of the house fought to see how many eggs, wiffles and other gustatory goodies they could force upon us. We fought honorably to do our best. Groaning we finally pushed away from the table while the audience went off to work on the farm.\n\n\"That was very good,\" Morton said.\n\n\"That was very wonderful,\" I amplified.\n\n\"Both meals already deducted from your account,\" Neebe smiled, handing me back my wirrdisc. \"I added an order to transfer payment to Morton's account when it is opened.\"\n\n\"I love IM hospitality,\" I said. \"It is personal without being financial. I want to learn more about your world.\"\n\n\"I will be happy to tell you anything you want to know,\" she said with that same endearing smile. What were these warm sensations that coursed through my body? I forgot them instantly as her smile faded. \"But we will have to talk about it later. Right now I think you should see the TV. We recorded a broadcast made earlier this morning.\"\n\nIt had to be Zennor\u2014and it had to be bad news. I watched grimly as the screen lit up and a blast of martial music assaulted my ears. Troops marched, tanks rumbled by, guns fired. A recording undoubtedly; I recognized Mortstertoro Base in the background. I suppose the sight of all this might was supposed to strike terror into the hearts of the viewers. I knew them well enough now to understand that they would just be puzzled at the waste of all this material and manpower for no observable reasonable purpose. I turned down the sound until the last tank had ground by, the last jet roared its last roar. The screen cleared and the familiar and loathsome features appeared.\n\n\"We are mighty, we are invincible\u2014and we will prevail!\" Zennor was coldly angry now. \"I have been kind to your people. I have even been generous to my own misled soldiers. No more. I have shown you kindness because I am a kind man. Now I will teach you fear because my rule will not be mocked. You have aided and abetted deserters from our army\u2014who are now under instant penalty of death. You must be aiding them because not one\u2014not one of them took advantage of my kind offer of amnesty. Nor have any of them been found in this city. They could not have escaped without aid. Therefore the people of Bellegarrique are guilty of treason, of aiding traitors and deserters, and they will pay for their crimes. I speak to you now, you inhabitants of the rest of this country. The citizens of Bellegarrique know of their guilt for they are attempting to flee my wrath. This city is almost deserted now as they crawl away like the cowardly vermin that they are. But not all of them have escaped. I have seized and imprisoned hundreds of these traitors. I did this once before and my requests were granted. I was kind and generous and released the prisoners. I will not be as kind this time\u2014or as easy to please. Here are my demands\u2014and they will be met.\n\n\"First, I want every escaped deserter returned to this city. I will not inflict the death penalty but will enlist them instead in penal, hard-labor battalions. I said that I was a merciful man.\n\n\"Second, I demand that all of the services of this city be restored, electricity and public utilities, and the food markets must be reopened. This will be done. I want to see people returning today, I want the normal life of this city to be as it was when we arrived, I want the deserters turned over to the military police. You will do this, and begin doing it now.\"\n\nHe paused dramatically, then pointed his finger directly at the camera.\n\n\"You will do it because in one day from now I will shoot ten of the prisoners. I will shoot these first ten no matter what you do, as an object lesson that I mean what I say. I will shoot ten of them the next day and ten again the day after that if my orders are not obeyed. If my orders are obeyed the shooting will stop. But it will begin again whenever I feel that my desires are being thwarted.\"\n\nThat was it. That was all. And it certainly was enough. The screen went blank and I found myself staring at Morton with nothing at all to say.\n\n\"There are rare cases of insanity like that here,\" Neebe said. \"Gene changes not caught in prenatal examination. He is insane, isn't he? These things that he says he will do\u2014they are impossible. He won't really have innocent people killed?\"\n\nI was too ashamed of the human race to look at her, to answer her questions. Morton did; he was angry.\n\n\"Yes, he will, that is the worst part. I grew up with his kind of people in charge of my life. Believe me, he will do it.\"\n\n\"Then what can we do to stop him?\"\n\n\"That is an almost unanswerable question,\" I said. \"You can't force the deserters to undesert. Knowing IM you wouldn't even think of asking them. And I don't know what they will do voluntarily. If you had a government they could deal with Zennor, come up with some workable compromise perhaps. But he still hasn't realized that there is no central government to meet with. The future does not bear thinking about.\"\n\n\"But we have to think about it,\" Morton said, with a cold grimness I had never seen before. \"Zennor must be killed. There is no other way.\"\n\n\"No!\" Neebe said. \"That is a hideous suggestion. This problem is so strange, so awful, that it would take the wisdom of Mark Forer itself to solve it.\"\n\n\"Maybe, maybe,\" I muttered. \"But I feel that what is happening here is well beyond even the mighty capacities of that long-gone brain to solve.\"\n\n\"Nothing was ever beyond Mark Forer,\" she said with calm and unshakable belief. It angered me. It was like calling in the deity of your choice as you fell off the cliff, begging for aid. Praying for a heavenly hand that would never, never swoop down from the sky to save you.\n\n\"That is just an opinion, your opinion. And to me it sounds more like blind faith than intelligent thought. We have to work this out ourselves because Mark Forer is long gone, rusted away. It can't help us now.\"\n\n\"Mark Forer could help us,\" she said with calm unreason. \"But of course we could never ask. That is a basic tenet of IM. We must solve our problems for ourselves. Everything we need to know is in the writings that it gave us.\"\n\n\"You are just jollying yourself along. You could ask, but you won't. That is a way out. You can't ask because it is not around to ask.\"\n\n\"That is not true,\" she said sweetly, smiling warmly upon my ill humor. \"Mark Forer is in Bellegarrique, where it always has been.\"\n\nI have known stoppers in my day. But this was the whopper topper stopper. I stared speechlessly at Morton. If I looked like he did then my jaw was hanging open, my eyes were popping and I was gurgling like an idiot. Neebe smiled warmly upon us and waited impatiently until we got reglued and were able to speak again. I sputtered first.\n\n\"Mark Forer... gone... thousands of years ago...\"\n\n\"Why? Essentially an artificial intelligence must be immortal. I suppose bits and pieces get replaced as they wear out, but the intelligence will remain the same. Or grow. We have always been immensely pleased that Mark Forer saw fit to accompany us to this world. We sincerely hope that it watches and approves of the way we practice IM. But of course we would never consider asking it for aid.\"\n\n\"Well I would,\" I said, climbing to my feet. \"I certainly would ask for help without a moment's hesitation. Mark Four's social theories are about to get a lot of people shot dead. So that cold artificial intelligence had better have some answers how to arrange it so that they don't.\"\n\n\"But you will have to go back to Bellegarrique to dig Mark Four out,\" Morton said. I nodded grim agreement.\n\n\"I was hoping you wouldn't say that just yet. But, yes, Morton old friend. I've got to find where our great electronic leader lives and search it out. And there better be some ready answers.\"\nChapter 28\n\n\"Do you know where Mark Forer plugs in?\" I asked Neebe. She shook her head.\n\n\"Not physically. It is just known, understood, that Mark Forer came with us and aided in the design of the city of Bellegarrique. And never left it.\"\n\n\"Well someone has to know.\" I thought hard, then snapped my fingers. \"Our old friend, Stirner, he should have that vital bit of info. One of the top men in the world of electricity. And if he doesn't know he will surely know someone who does know. Do you have any idea of how I can contact him?\"\n\n\"The telephone is over there.\"\n\n\"Thanks, Neebe, but I don't have his number or the slightest idea where he is staying or anything.\"\n\n\"But no one has a number. And it doesn't matter where he is staying. Just call CD and ask for him.\"\n\n\"CD?\"\n\n\"Central Directory. Here, I'll get it for you.\"\n\nShe tapped the keypad and the screen lit up with NAME, PLEASE? in large letters. Very polite. Very efficient. I tip my hat to the man or machine that wrote this software. I answered four questions and the screen changed to RINGING. The letters faded and Stirner's grim face appeared on the screen. He smiled faintly when he saw me, but he had obviously been watching the broadcast too.\n\n\"Ahh, good offplanet friend Jim. I hope that you are well. Can I do you a service?\"\n\n\"You certainly can, good dynamo-supervising friend Stirner. I would like to have a chat with your demigod, Mark Forer.\"\n\n\"A strange choice of terms. I would not refer to it as a demi...\"\n\n\"Then forget the term. Do you know where Mark Forer is?\"\n\n\"Of course.\"\n\n\"Will you take me to it?\"\n\n\"Ahh, now that is a question that needs some thought. Mark Forer's individualism has always been respected, for all the obvious reasons. I do remember reading in the historical records that after this city was founded it did make suggestions and was occasionally consulted. But not lately, not in, hundreds of years at least. I would not go to it myself, but, yes, I feel that I can take you. I respect your individualism just as I respect Mark Forer's. We must each make our own way in this world.\"\n\n\"And I am going to make my way back into the city.\"\n\n\"You must be careful. It will not be easy. The trains have stopped running and citizens are being forcefully stopped from leaving. At last report no one was returning.\"\n\n\"I'll think of something. You are still in the city?\"\n\n\"Yes.\"\n\n\"Stay near the phone. I'll get there today. I must talk to Mark before Zennor's deadly deadline runs out tomorrow morning.\"\n\nI hung up and looked blankly into space. I could see no answers hanging out there.\n\n\"Any advice, Morton?\"\n\n\"None that make any sense. Like being a returned deserter.\"\n\n\"Like you, that idea I considered and rejected. That would just get me back in jail and shot.\"\n\n\"May I make a suggestion?\" Neebe said.\n\n\"All aid greatly desired.\"\n\n\"I will take you to the city. You will go as my father. We have a wonderful theater group here in Ling and our makeup department is quite famous. You could be an old man, I could be your daughter and driver. It would be so exciting.\"\n\n\"You're wonderful!\" I jumped to my feet and, in a fit of mad enthusiasm seized her and kissed her. Then I sat down quickly again as the hormones started humming and driving all other thoughts from mind. She was an incredibly bright, lovely, intelligent, beautiful girl and I was just going to have to forget all about that. For the time being. \"We better get started.\"\n\n\"My brother will take you to the theater. I will phone them and arrange what must be done. Then I will make the transportation arrangements. You do not mind if I say that I find this fascinating and exciting as well. I must thank you for letting me help. It is so much more fun than school.\"\n\n\"The thanks are mine. What do you study in school?\"\n\n\"Vulcanology. I just love the magma and the scoria, then when you go down the fumerole...\"\n\n\"Yes. You must tell me of those burning pleasures. Later.\"\n\n\"Of course\u2014there is my brother now.\"\n\nI think that it was a special train that they laid on. Just two cars and no other passengers. Morton looked guilty\u2014but glad as well that he wasn't going back to Bellegarrique. I waved him a stiff goodbye with my cane and climbed shakily aboard. I was ancient and crochety and needed practice. Gray beard, rheumy red eyes, wrinkled like an old boot, they had really done a great job at the theater. A harness under my clothes had me bent over so far that I was staring down at my wrinkled and liver-spotted hands.\n\nThe track was straight, the train was fast and there were no stops until we reached our destination. A black vehicle was waiting on the platform when we arrived. The driver got out and held the door open for us.\n\n\"You've driven one of these?\" he asked.\n\nNeebe nodded. \"A two-hundred-volt Lasher-gnasher. Great fun to drive.\"\n\n\"Indeed they are. I've got her revved up to thirty-three thousand. More than enough energy for the trip.\" He pointed to the circular housing between the rear wheels. \"The flywheel is in here, electric generator on its shaft. Motor on the front wheels. Clean and nonpolluting.\"\n\n\"And very hard to turn over with that gyroscope down there,\" I said.\n\n\"You've got it. Good luck.\"\n\nNeebe spun the wheels and I was pushed back into the seat by a large number of G's. We hurtled along the empty road.\n\n\"I'll slow down before we reach the roadblock. Isn't this fun! I wonder what the top speed is?\"\n\n\"Don't... find out!\" I croaked as the landscape hurtled by in a blur. \"Though I am an old man and have led a full life I don't want to terminate it quite yet!\"\n\nShe laughed her gorgeous bell-like laugh and slowed to something close to the speed of sound. She obviously knew the road well, all those bicycle outings of course, for suddenly she hit the brakes, slowed to a crawl, then turned the corner just before the barrier across the road.\n\n\"What you doing blocking the road like that, you varmints?\" I croaked testily out of the window, then shook my cane at the fat captain who was leaning against it picking his teeth. Remnants of hotpup, I hoped.\n\n\"Knock off the cagal, Grandpop. Where do you think you're going?\"\n\n\"Are you as stupid as you look, stupid? Or haven't you heard your supreme commander's orders? City workers to return at once. I am an electrical engineer and if you want light in your latrines and refrigeration for your beer you will open that thing instantly or sooner.\"\n\n\"Don't get your cagal in an uproar, Grandpop,\" he sneered. But he stepped back and signaled two sergeants to open the barrier. Not a private in sight, I noticed. I hoped the officers enjoyed doing their own work for a change. I shook the cane one last shake as we drove past, then on down the road and around a bend and out of sight. Neebe pulled up at the first phone booth and I leaped arthritically down.\n\n\"Are you in the city?\" Stirner asked.\n\n\"Just arrived.\"\n\n\"Very good. Then we will meet at the entrance.\"\n\n\"Entrance? What, where?\"\n\n\"Mark Forer Square, of course. Where else would it be?\"\n\nGood question. I had imagined that only the statue was there. I hadn't realized that old Mark itself was in residence. I climbed back into the car and we were off with the usual screech of tires. I pulled off bits of the disguise as we went, starting with the harness. I left the beard on in case there were any patrols around\u2014and there were.\n\n\"Slow down,\" I cozened. \"Let's not be too suspicious.\"\n\nThe sergeant leading the patrol glared at us as we went by. I ignored him but was very impressed by his squad. As they turned the corner the last two slipped into the open door of a building and vanished from sight. So not only weren't the deserters returning\u2014but their ranks were steadily being added to. Great! If this kept up Zennor would soon have an army of only officers and noncoms. You don't win wars with that kind of setup. I saw that we were getting close to our destination so I pulled at the beard and wrinkles and was forty years younger by the time we turned into the square and slid to a stop. Stirner was standing before the statue, looking up at it admiringly.\n\n\"I wish I were coming with you,\" he said.\n\n\"I as well,\" Neebe agreed. \"It would be wonderfully exciting. But of course we have not been asked so we cannot intrude.\"\n\n\"How do I get in?\"\n\nStirner pointed to a bronze door at the rear of the stone base of the statue. \"Through there.\"\n\n\"Got the key?\" They both looked at me with surprise.\n\n\"Of course not. It's not locked.\"\n\n\"I should have known,\" I muttered. What a philosophy. Hundreds, thousands of years the door has been here, unlocked, and no one had ever gone through it. I put out my hand and they took it in turn and shook it solemnly. I could understand why. This was a little like saying so-long to the head of your local church as he started up the ladder to see God.\n\nThe handle was stiff, but turned when I twisted hard. I pulled and the door squeaked slowly open. Steps led down into the ground, a little dusty. Lights came on and I could see that one of the bulbs was burned out. I just hoped that Mark Forer wasn't burned out as well.\n\nI sneezed as my feet disturbed the dust of ages. And it was a long way down. The steps ended in a small chamber with illuminated wiring diagrams on the walls and a large, gold-plated door. Carved into it, and inset with diamonds, were the immortal words I AM. THEREFORE I THINK. Beneath this was a small sign with red letters that read PLEASE WIPE FEET BEFORE ENTERING. I did this, on the mat provided, took a deep breath and reached for the handle that appeared to have been carved from a single ruby.\n\nThe door swung open on oiled hinges and I went in. A large, well-lit room, dry and air conditioned. Dials and electronic devices covering one wall. And in the middle of the room...\n\nMark Forer, obviously. Just like in the paintings. Except that plenty of cables and wires ran from it to a nearby collection of apparatus. Its dials glowed with electronic life and a TV pickup swiveled in my direction. I walked over to stand before it and resisted the compelling desire to bow. And just what does one say to an intelligent machine? The silence lengthened and I began to feel ridiculous. I cleared my throat.\n\n\"Mark Forer, I presume?\"\n\n\"Of course. Were you expecting someone else... krrk!\"\n\nThe voice was grating and coarse and the words trailed off with a harsh grating sound. At the same time there was a puff of smoke from a panel on the front and a hatch dropped open. My temper snapped.\n\n\"Great! Really wonderful. For hundreds of years this electronic know-it-all sits here with the wisdom of the ages locked in its memory banks. Then the second I talk to it it explodes and expires. It is like the punch line of a bad joke\u2014\"\n\nThere was a rattle from behind and I leaped and turned, dropped into a defensive position. But it was only a little rubber-tired robot bristling with mechanical extensions. It wheeled up in front of Mark and stopped. A claw-tipped arm shot out, plunging into the open panel. It clicked and whirred and withdrew a circuit board which it threw onto the floor. While this was happening another circuit board was emerging from a slot on the robot's upper surface. The grasping claw seized this and delicately slid it into the opening before it. Mark's panel snapped shut as the robot spun about and trundled away.\n\n\"No,\" Mark Forer said in a deep and resonant voice, \"I did not explode and expire. My voice simulation board did. Shorted out. Been a number of centuries since I last used it. You are the offworlder, James diGriz.\"\n\n\"I am. For a machine in an underground vault you keep up with things pretty well, Mark.\"\n\n\"No problem, Jim\u2014since you appear to enjoy a first-name basis. Because all of my input is electronic it really doesn't matter where my central processor is.\"\n\n\"Right, hadn't thought of that.\" I stepped aside as a broom and brush bristling robot rushed up and swept the discarded circuit board into its bin. \"Well, Mark, if you know who I am, then you certainly know what is happening topside.\"\n\n\"I certainly do. Haven't seen so much excitement in the last thousand years.\"\n\n\"Oh, are you enjoying it?\" I was beginning to get angry at this cold and enigmatic electronic intelligence. I was a little shocked when it chuckled with appreciative laughter.\n\n\"Temper, temper, Jim. I've cut back in the voice feedback emotion circuits for you. I stopped using them centuries ago when I found that the true believers preferred an ex cathedra voice. Or are you more partial to women?\" It added in a warm contralto.\n\n\"Stay male, if you please; it seems more natural somehow. Though why I should associate sex with a machine I have no idea. Does it make a difference to you?\"\n\n\"Not in the slightest. You may refer to me as he, she or it. Sex is of no importance to me.\"\n\n\"Well it is to us humans\u2014and I'll bet you miss it!\"\n\n\"Nonsense. You can't miss what you never had. Do you wake up at night yearning helplessly for photoreceptors in your fingertips?\"\n\nIt was a well-made point: old Mark here was no dummy. But fascinating as the chitchat was, it was just about time I got to the point of this visit.\n\n\"Mark\u2014I have come here for a very important reason.\"\n\n\"Undoubtedly.\"\n\n\"You've heard the broadcasts, you know what is happening up there. That murdering moron Zennor is going to kill ten of your faithful followers in the morning. What do you intend to do about it?\"\n\n\"Nothing.\"\n\n\"Nothing!\" I lost my temper and kicked the front of the burnished panel. \"You invented Individual Mutualism and foisted it upon the galaxy. You taught the faithful and brought them here\u2014and now you are going to stand by and watch them die?\"\n\n\"Knock off the cagal, Jim,\" it said warmly. \"Try sticking to the truth. I published a political philosophy. People read it, got enthusiastic, applied it and liked it. They brought me here, not the other way around. I have emotions, just as you do, but I don't let them interfere with logic and truth. So cool it, kid, and let's get back to square one.\"\n\nI moved aside as the broom-robot rushed up again, extended a little damp mop and polished off the scuff mark on Mark's housing that I had made with my shoe. I took a deep breath and calmed down because really, losing my temper would accomplish nothing at all.\n\n\"Right you are, Mark, square one. People are going to be killed up there. Are you going to do anything about it?\"\n\n\"There is not much I can do physically. And everything political or philosophical is in my book. The citizens up there know as much about IM as I do.\"\n\n\"So you are just going to sit there and listen to the sizzle of your electrons and let them die.\"\n\n\"People have died before for their beliefs.\"\n\n\"Wonderful. Well I believe in living for mine. And I am going to do something\u2014even if you do not.\"\n\n\"What do you intend to do?\"\n\n\"I don't know yet. Do you have advice for me?\"\n\n\"About what?\"\n\n\"About saving lives, that's what. About ending the invasion and polishing off Zennor...\"\n\nAnd then I had it. I didn't need to swap political arguments with Mark\u2014I just had to use its intelligence. If it had memory banks thousands of years old it certainly had the knowledge I needed. And I still had the electronic spybird!\n\n\"Well, Mark, old machine, you could help me. Just a bit of information.\"\n\n\"Certainly.\"\n\n\"Do you know the spatial coordinates of this system and this planet?\"\n\n\"Of course.\"\n\n\"Then you give me a little printout of them, soonest! So I can send an FTL message to the League Navy for help.\"\n\n\"I don't see why I should do that.\"\n\nI lost my temper. \"You don't see...! Listen you moronic machine, I'm just asking for a bit of information that will save lives\u2014and you don't see...\"\n\n\"Jim, my new offworlder friend. Do not lose your temper so quickly. Bad for the blood pressure. Let me finish my statement, if I might. I was going to add that this information would be redundant. You sent an FTL message yourself, just after you retrieved the corvine-disguised transmitter.\"\nChapter 29\n\n\"I sent a FTL message?\" I said, my thoughts stumbling about in small circles.\n\n\"You did.\"\n\n\"But\u2014but\u2014but\u2014\" I stopped and seized myself by the mental neck and gave it a good shaking. Logic, Jim, time for logic. \"The recorded message from Captain Varod said that I would need the coordinates to send a FTL message.\"\n\n\"That was obviously a lie.\"\n\n\"Then the thing about its being a radio message was a lie too?\"\n\n\"Of course.\"\n\nI paced back and forth and the TV pickup followed me as I moved. What was going on? Why had Varod lied to me about the signal? And if he had received it where was he? If he had got the signal and hadn't sent his fleet or whatever, then he was the one who must take the responsibility for the murders. The League did not go in for that sort of thing. But Mark might know what was happening. I spun about.\n\n\"Speak, ancient brain-in-box! Has the League Navy arrived or is it on the way?\"\n\n\"I'm sorry, Jim, I just don't know. The last orbiting telescope ran out of power centuries ago. I know no more than you do about this. All I can surmise is that we are very distant from these rescuers you expect.\"\n\nI stopped pacing and was suddenly very tired. It was going to be another of these days. I looked around the room. \"You don't have an old box or something that I can sit on?\"\n\n\"Oh dear, I do apologize. I'm not being a very good host, am I? Out of training.\"\n\nWhile he was talking a powered sofa came trundling in and stopped behind me. I dropped into it. It was hard to think of Mark as an it, not with the voice and all.\n\n\"Many thanks, very soft.\" I smacked my lips and it got the hint.\n\n\"Please make yourself comfortable. Something to drink perhaps?\"\n\n\"I wouldn't say no. Just to stimulate thinking, you realize.\"\n\n\"I'm not too well stocked at the present moment. There is some wine, but it must be four hundred years old at least. Vintage with a vengeance, you might say.\"\n\n\"We can only try!\"\n\nThe table stopped at my elbow and I blew dust from the bottle, then activated the electronic corkscrew, which managed to extract the truly ancient cork without breaking it. I poured and sniffed and gasped.\n\n\"Never\u2014never smelled anything like that before!\"\n\nAnd it tasted even better. All the sniffing and tasting did clear the mental air a bit. I felt better able to handle the problems of the day.\n\n\"I don't know the time,\" I said.\n\n\"Over sixteen hours to go before the promised executions.\" Mark was anything but stupid. I sipped the wine and ran over the possibilities.\n\n\"I sent the message\u2014so the Navy has to be on its way here. But we can't count upon their arrival to save the day. The only grace note to all this is that at least I know I won't be stranded on this planet forever. Now what can I do to save lives? Since obviously neither you nor your IMers are going to lift a finger.\"\n\n\"I wouldn't say that, Jim. There are a number of conferences going on right now in the city. People are returning in large numbers.\"\n\n\"Are they knuckling under? Going back to work?\"\n\n\"Not at all. A protest is being organized, as to what shape it will take\u2014that is still being discussed.\"\n\n\"How do you know all this? Spies?\"\n\n\"Not quite. I simply tap all the communication circuits and monitor all phone calls. I have subunits looking for keywords and making records for me.\"\n\n\"Are you tapping the Nevenkebla circuits as well?\"\n\n\"Yes. Very interesting.\"\n\n\"You speak the language?\"\n\n\"I speak every language. Fourteen thousand six hundred and twelve of them.\"\n\n\"Jamen, e'n ting er i hvert fald sikker. Du taller ikke dansk.\"\n\n\"Og hvorfor sagde ikke det? Dansk er da et smukt, melodisk sprog.\"\n\nPretty good\u2014I thought that I was the only one who had ever heard of Danish. But there was one that I was sure Mark had never heard of. An ancient language called Latin. Spoken only by a secret society so secret I dare not say more about it.\n\n\"Nonne cognoscis linguam Latinam?\"\n\n\"Loquarne linguam Latinam?\" Mark answered in a decidedly snotty manner. \"Quid referam in singulorum verborum delectu, in coniunctorum compositione, et structura, in casuum atque temporum discriminatione, in certarum concinnitate formularum, in incisorum membrorumque conformatione, in modulandis circumdictionibus, in elegantiarum cuiusque generis accurata, elaborataque frequentatione quantus tum sim et quam purus putus Ciceronianus? Ex qua Cicero mortuus est, meis verbis nihil latinius. Memoria vero libros omnium auctorum latinorum tam veterum quam recentiorum et neotericorum continet. Voces peregrinae et barbarae quae latinis eloquiis inseruntur, omino mihi notae sunt. Nae tu es baro et balatro, nam ego studeo partes difficiles cognoscere quas scholastici doctores gestant, latebras singulas auxilio mei ipsius cerno. Doctissimi enimvero homines omnino universitatum modernarum me rogant sensus omnium talium verborum.\"\n\nI could only gape at this as it hummed in electronic joy, very proud of itself. \"Did you catch all those nuances, Jim? About what a pure Ciceronian I am? Each word carefully chosen, the composition of sentence structure, the contrast of cases and tenses, phrases and clauses...\"\n\nHe, or rather it, went on for quite a while like that. Bragging. Chatting away with Mark I tended to anthropomorphize him. It. Her. Whatever. This wasn't a human but an intelligent machine with abilities far beyond anything I had ever imagined before. But how could I put them to work?\n\n\"Mark, tell me. Will you help me?\"\n\n\"In any way I can.\"\n\nI sipped more wine and felt its healing and inspirational powers doing good things to me. Memory. Something that had happened earlier today.\n\n\"Mark\u2014I saw two soldiers desert today. Are there other newly arrived deserters in the city?\"\n\n\"A goodly number of them. One hundred and twenty-one in all, wait... sorry, one twenty-two. Another just arrived.\"\n\n\"Any of them armed?\"\n\n\"You mean equipped with weapons? All of them. They have all deserted from patrols in the city.\"\n\nBut would they use their guns? And if so, what could I do with them? An idea was taking shape. Meet fire with fire. And they just might do it. There was only one way to find out. I poured another bit of wine and turned to my electronic host.\n\n\"I would like to have all of the deserters meet me in some central place. With their weapons. Can you arrange that?\"\n\nHe was silent for long seconds. Looking for a way to back out of his offer? But I had underestimated him.\n\n\"All done,\" he said. \"The people who are hiding them will escort them after dark to the sports center. Which is very close to the site selected for the murders.\"\n\n\"You are one step ahead of me.\"\n\n\"I should hope so. Since I am incredibly more intelligent than you are. Now, since there are some hours to go before your meeting, would you repay the favor and have a good chat with me? I have been rather out of touch with galactic matters for a thousand years or more. How are things going?\"\n\nIt was a strange afternoon and early evening. His memory, as it should be, was quite formidable and I learned a number of interesting things. But there was one fact which he could not tell me since he had been born? built? wired? well after the spread of mankind through the galaxy.\n\n\"Like you, Jim, all I know are myths and ancient memories. If there was an original planetary home of mankind, called Dirt or Earth or something like that, its location is nowhere in my memory banks.\"\n\n\"Well, just thought that I would ask. But I think I better be going. Nice to talk to you.\"\n\n\"The same. Drop in any time.\"\n\n\"I'll take you up on that. Would you mind turning off the lights when I get to the top of the stairs?\"\n\n\"Not a problem. This place is pretty well automated as you might imagine.\"\n\n\"No problem with the electricity supply?\"\n\n\"You bet your sweet chunk there isn't. Survival was the first emotion I learned. City power supply, standby generators, battery backup, a couple of fuel cells and a fusion generator that can be fired up in ten minutes. Don't worry about me.\"\n\n\"I won't. So long.\"\n\nI climbed the long flight of stairs and when I touched the door all the lights were extinguished. I pushed it open and peeked out: no patrols that I could see. When I threw it wide and exited there were Neebe and Stirner sitting on the bench, waiting for me.\n\n\"Aren't you worried about the enemy finding you here after curfew?\"\n\n\"Not a problem,\" Stirner said. \"So many men have deserted that all patrols appear to have been canceled. All of the military are either on the base or in the municipal building. Now\u2014please tell us. You have spoken with Mark Forer?\" They both leaned forward in tense anticipation.\n\n\"Spoken with him and enjoyed his hospitality. And he's got a couple of cases of wine left that you wouldn't believe.\"\n\n\"I would believe anything about Mark Forer,\" Stirner said and Neebe nodded agreement. \"But I am sorry that he did not give you a solution to the problem of the killings.\"\n\nI blinked rapidly. \"How do you know that? I didn't say anything about it.\"\n\n\"You did not have to. Mark Forer knows that it is a problem we must deal with ourselves. And so we shall. A decision has been reached. All in the city will assemble in the killing place tomorrow, an individual decision by each one. We will stand in front of the guns.\"\n\n\"A noble gesture\u2014but it won't work. They will just shoot you down.\"\n\n\"Then others will take our places. There is no end to nonresistance. They will keep shooting until they run out of charges for their weapons or take despair at the murders. I am sure that they are not all moral villains like their leader.\"\n\n\"I wouldn't count on it. But there may be an alternative. With Mark's help, we are on a first-name basis you will be happy to hear, I have arranged a meeting of all the deserters in the city. If you will kindly lead me to the sports center we will see if my plan might perhaps be a better and more practical one.\"\n\nIt was a pleasant stroll, the streets of the city empty of fear for the first time since the invaders had landed. We met other groups going in the same direction, each of them accompanying one or two armed deserters. Laughter and smiles, they were cheerful now that they were away from the army\u2014but would they go along with any plan that might jeopardize that newfound freedom? There was only one way to find out.\n\nThe sports center had an indoor stadium with a wrestling ring that held us all. The escaped soldiers sat in the lower rows while interested civilian spectators were ranked above and behind them. I climbed into the ring and waited until they were all seated, then grabbed the microphone. The audience rustled into silence.\n\n\"Fellow ex-draftees, newly arrived deserters, I welcome you. Most of you don't know me...\"\n\n\"Everyone knows you, Jim!\" a voice called out. \"You're the guy almost throttled the general.\"\n\n\"Better luck next time!\"\n\nI smiled and waited until the cheers and shouts had died down.\n\n\"Thanks guys, it is nice to be appreciated. Now I have to ask you to help. Our dear general, cagal-kopf Zennor, plans to shoot down some unarmed civilians tomorrow. These are the people who have helped you and your buddies escape, who have extended friendship to us all\u2014and a happy home here if we want it. Now we have to help them. And I am going to tell you how.\n\n\"We are going to take these guns that we have been trained to use and aim them at Zennor and his mob and threaten to waste them if they pull any triggers. It will be a standoff\u2014and we might not get away with it. But it is something that we have to do.\"\n\nI felt a little ashamed of adding the macho emotional argument, but I had no choice. It wasn't the world's greatest idea and it had more holes in it than a carload of doughnuts. But it was the only plan in town.\n\nThey argued and shouted a lot but in the end a majority voted for it. The minority could see no way to leave with dignity\u2014I said the macho appeal\u2014so reluctantly went along with the plan. The locals led us by back routes into the buildings facing the square and we lay on our guns and slept. I was sure that a number would disappear during the night. I only hoped that enough would be left in the morning to give me a little firepower backup.\n\nAt the first light of dawn I was aware of figures moving about in the square outside. I shoved a teddy bear aside a bit so I could see through the curtains of the toy store where I was hiding. The troops were beginning to arrive. And the prisoners, ten of them, handcuffed and bound, being unloaded from a truck. As it grew lighter I saw that every soldier was an officer or a noncom. Of course\u2014Zennor couldn't trust privates to do his dirty work! They were probably all locked up and under guard back on the base.\n\nZennor himself stalked from the municipal building and stood in the middle of the square. Just as I heard the rumble of wheels and powerful motors the heavy gun units rolled up. I hadn't counted on this.\n\nI hadn't counted either on Zennor drawing his pistol and shooting out the toy store window.\n\n\"Come out of there, diGriz\u2014it's all up!\" he shouted, and blew away a teddy bear.\n\nDid I have a choice? I opened the door and stepped out in the street. Looked at all the guns aimed at the windows where my rebellious soldiers were hidden. Looked at the wicked smile of triumph on Zennor's face.\n\n\"I'm a general, remember? Did you really think that your ridiculous maneuver would succeed? My agent has reported to me every detail of your stupid plans. Would you like to meet him?\"\n\nOne of the deserters emerged from a doorway at Zennor's signal and walked toward us. He wore tinted glasses and a large moustache; I had seen him at a distance before. Now I was seeing him up close as he pulled off the moustache and threw away the glasses.\n\n\"Corporal Gow,\" I sighed.\n\n\"Broken to private now! Because I let you escape. They would have shot me too if I hadn't been rich enough to pay the bribes. But my downfall is now your downfall. Those other privates, verminous swine, they knew I had been a corporal, wouldn't talk to me. But I could tell something was wrong. When they deserted I instantly reported to the general. At his direction I walked through the city\u2014and was encouraged to desert by the treacherous natives. I did, and General Zennor received complete reports.\"\n\n\"You're a rat!\"\n\n\"No insults, spy. My rank has been restored by the good general. And you are in the cagal.\"\n\n\"You are indeed,\" Zennor agreed. And aimed his gun between my eyes. \"You've failed and failed badly. Let that be your last thought as you die.\n\n\"This is the end of you!\"\nChapter 30\n\nWell, yes. This was just about the lowest low moment I had ever experienced. In a life that had been, unhappily, quite filled with low moments. I mean, really. Here was this murderous general leering away at me and fondling the hair trigger of his pistol. Behind him were his potbellied troops looking down the barrels of their cannon. While on all sides my disarmed army was being kicked out of hiding and forced at gunpoint into the square. There can't be many moments lower than this.\n\n\"You are not going to get away with this, Zennor,\" I said. Which was pretty feeble but about all I could think of at the moment.\n\n\"Oh yes I am, little man.\" He raised the gun and pointed it between my eyes and caressed the trigger. Then lowered it. \"But I don't want it to be too easy for you. Before I blow you away, you are going to watch me shoot every one of these treacherous deserters. They had the affrontery to attempt to raise their weapons in rebellion against me. They will die for this mistake. Then I am going to shoot the ten prisoners, just as I promised. Then, and only then, will I kill you.\"\n\n\"Not if I kill you first,\" I growled and felt my lips curl back from my teeth. I had nothing to lose. I raised my hands and stalked toward him. And he ran!\n\nNot far. Just to the nearest prisoner, a grandmotherly woman with gray hair. He pulled her away from the others and thrust the muzzle of his gun against her head.\n\n\"Go ahead, diGriz. Take one more step toward me and I pull the trigger. Do you doubt me?\"\n\nDoubt him? Never. I did not take the step. The world was coming to an end and there was nothing I could do about it. They had the guns; we had nothing.\n\nIt was then, at the darkest moment, through the blackness of my thoughts I became aware of the shuffling of many feet. I turned to look just as Zennor did.\n\nAround the corner came a solid mass of people, filling the street from side to side, an endless number of them. Leading the front rank was Stirner\u2014and Neebe!\n\n\"No, don't, go back!\" I shouted. Neebe smiled sweetly at me. And kept walking at Stirner's side. Zennor had his gun aimed at Stirner now\u2014who appeared completely indifferent to it. Stirner stopped and called out loudly.\n\n\"All of you men with weapons\u2014put them down. We will not hurt you for that is not our way...\"\n\n\"One more word and I will kill you!\" Zennor roared. Stirner turned to him, his face cold as death.\n\n\"I believe you will,\" he said. \"Until this moment I really did not believe it possible that a human being could kill another. After seeing you I believe it.\"\n\n\"Good, then you will...\"\n\n\"Be quiet. I will do just what I came here to do. I will take your weapon. If you kill me, someone else will take your weapon from you. If he fails another will try. Eventually it will be empty, discharged and will be taken from you. You cannot win. Those who follow you cannot win. It is all over.\"\n\n\"It is not!\" Zennor shouted. There was spittle on his lips now, a look of insanity in his eyes. He pushed the woman captive away and ground his gun into Stirner's body. \"No one has the guts to do that. When I blow your blood all over them they will turn and run. My men will fire a volley and the survivors will flee in panic. That is what I will do and you cannot stop me...\"\n\nI dived for him hands outstretched. He pushed Stirner against me and lashed the pistol barrel across my head, aimed it at me, tightened his finger on the trigger.\n\n\"Are you volunteering, diGriz? Good. Than you shall be first.\"\n\nA shadow drifted across the square and an amplified voice boomed against our eardrums.\n\n\"The war is over. Lower your weapons.\"\n\nFilling the sky was the biggest spacer I had ever seen, bristling with guns\u2014and all pointed down at Zennor's troops.\n\nThe Navy had arrived!\n\nBut a little too late.\n\n\"Never!\" Zennor cried. \"Gunners, fire! Kill the captives! Shoot down that ship!\"\n\nNor was I forgotten. He ground the pistol barrel into my temple and pulled the trigger.\n\nThe gun did not fire.\n\nI saw his knuckle whiten with the strain\u2014but the trigger would not move. His face went ashen as he realized what was happening. I lashed out and knocked the pistol aside.\n\nThen, from way down on the ground, I brought up a punch that I think I had been saving for all of my life. Up, faster and faster it went, until my fist caught him full on the jaw. Lifted him into the air, dropped him unconscious to the ground. I rubbed my sore knuckles and realized that I was grinning like a fool.\n\n\"Your weapons will not work!\" the voice from the sky boomed once again, and even through the echoes and distortion I recognized Captain Varod. \"This ship is projecting an entropy field that does not permit metal to move against metal or electrons to flow. It does not affect life forms. Therefore if you good citizens of Chojecki would be so kind as to disarm these invaders I would be immensely grateful.\"\n\nThere was the quick thudding of running feet as a number of deserters got there first. The sight of officerial black eyes and bloody noncommissioned noses was a pleasant one. A hatch opened in the ship above and a familiar uniformed figure dropped down on the end of a line. I felt a hand on my arm and turned to look into Neebe's gorgeous, smiling face.\n\n\"Then it is all over, Jim?\"\n\n\"It is\u2014and it has a happy ending as well.\"\n\n\"What will happen now?\"\n\n\"The invaders will go and will never come back. Your planet will be your own again. Peace will prevail here forever.\"\n\n\"Will you be leaving too?\"\n\nMy heart gave a couple of rapid hammer beats and I squeezed her arm and prepared to drown in those eyes. Then I surfaced and shook myself off.\n\n\"I don't know... not true. I do know. As great as the attractions are here,\" I squeezed the hand of the greatest attraction, \"in the long run I would not be happy. Nor would those about me. Your planet, if you will excuse me saying so, is a little too quiet for me. Paradise is fun for awhile, but I would not like to make a habit out of it. There are a lot more worlds out there that I haven't seen yet. The galaxy is a very big place. It hurts to say it, but I must move on.\"\n\n\"Stay on this world, Jim,\" Varod advised as he walked up behind me. \"Because if you leave here you will find out that justice and a jail term await you on a certain planet.\"\n\n\"That's what you say, Varod, that's what you say!\" I spun about and shook a very angry finger in his face. \"You lied to me, tricked me into coming here\u2014then ignored my FTL message and left me here to rot, almost got me and about half of the planet killed...\"\n\n\"Never! We were in orbit all of the time, watching everything. As soon as we arrived we had Zennor bugged completely with undetectable bugs. We were here two days after you sent the FTL message. Well done.\"\n\n\"Two days? Bugs? That's impossible. Mark Forer would have known about them.\"\n\n\"It did. We have been in constant consultation with that great intelligence. It has been of great help.\"\n\n\"Are you telling me that Mark Forer lied to me\u2014just like you?\"\n\n\"Yes.\"\n\nI opened and shut my mouth and ran it all through again. \"Why... I mean why hang around and run the risk of things getting out of hand when you could have landed at once?\"\n\n\"We had to wait until after the elections,\" he said with infuriating amiability. \"We had done everything we could to get Zennor off of his home planet as quickly as possible. Planted all those radio-broadcasting bugs so he would know that he was being watched. We worked on his paranoia, in the hopes that he would stay out of contact with his home base until it was too late. You were very good at causing trouble for him here. I must congratulate you on that. It gave him no time at all to even consider contacting his base. That was very important to us. Once Zennor left on his interplanetary adventure\u2014as we hoped he would\u2014it was possible to stage a little coup d'etat in Nevenkebla. The civilians were more than weary of the endless state of emergency. A palace revolution quickly got rid of the military. A civil government has been elected and peace will prevail from now on. This disarmed army will return and be absorbed in the populace.\"\n\n\"You played me for a patsy,\" I said, with some warmth.\n\n\"I don't know the term, but I assume that it means we took unfair advantage of you and let you do our dirty work for us.\"\n\n\"That will do until a better description comes along. Well\u2014didn't you?\"\n\n\"Not at all. You became involved in this matter for your own reasons. If we had not watched you and come to your aid you would be dead right now.\"\n\nVery hard to argue with that. And I had come here of my own accord. I looked down at the prostrate form and resisted the strong desire to kick in a couple of ribs.\n\n\"What about Zennor here?\"\n\n\"Zennor is a sickie and will get the proper treatment in a hospital that specializes in people with problems like his. As of this moment he no longer exists.\"\n\n\"And what about me?\"\n\n\"You would be wise to stay right where you are now. Escaped prisoner, conviction still pending\u2014\"\n\n\"Don't shoot me that line of old cagal,\" I sneered. \"I am an undercover operative of the League Navy and will be treated as such. I was responsible for your locating this planet and have suffered in the name of League justice. I have even made financial arrangements on your behalf...\"\n\n\"Yes, the soldier you promised the credits to, for aiding you. The voice-actuated recorder in the spybird caught your conversation with him. Aspya will be paid.\"\n\n\"Then so will I. Full salary for all the time I have been working for you. Right?\"\n\nHe rubbed his jaw and scowled. \"I suppose that you will be asking for a full pardon for crimes committed on Bit O' Heaven?\"\n\n\"No. I just want that incident wiped completely from my record so I can walk forth a free man. With my back pay in my pocket.\"\n\n\"I agree. As long as you remain in the Navy employ. Although a bit impetuous, you make a good field agent...\"\n\n\"Never!\" I shouted, shying back and neighing like a horse. \"Never! Work for the law? Pay taxes and look forward to a miserable pension in my old age? Death before dishonor! Pay up and wave bye-bye, captain. I have my own career priorities.\"\n\n\"Like following a life of crime?\"\n\n\"That is different. In all truth I can promise you\u2014never again!\" I placed one hand over my heart and raised the other palm outward. \"I have learned my lesson. I hereby forswear any interest in a life of crime and pledge my word to be a productive member of society forever after.\"\n\n\"Good, my boy, good. I'll take care of the money for you then. The likes of you don't belong in crime.\"\n\n\"No sir, they don't!\" I said.\n\nLying again, lying and smiling and lying through my teeth. After all\u2014I had some good examples to follow. When a full captain in the League Navy lies to you, when the greatest artificial intelligence in the known galaxy lies to you\u2014should a simple ex-porcuswine swineherd be forced to tell the truth?\n\nMy throat was dry and I suddenly felt a great yearning for some of that four-hundred-year-old wine. I looked forward to raising a glass of it very soon. Raising it in a toast.\n\nTo my future career out there among the stars. I could almost taste that wine upon my lips and I smacked them dryly, turned to face Neebe and Stirner.\n\n\"My friends\u2014this calls for a celebration. Come with me, I beg you. I know of a very exclusive drinking establishment not too far away from here.\"\n\nThe Drinking Party\n\n\"This is undoubtedly,\" Stirner said, eyes wet with emotion, \"the very best glass of wine I have drunk, ever thought of drinking, managed to drink, ever drank, will ever drink, ever imagined that I some day might have considered drinking...\"\n\n\"While your grip on syntax seems to be failing,\" Mark Forer said, \"I appreciate the emotion. Now that you have all tasted the wine, I am much cheered that you enjoy it, I would like to propose a toast. To James diGriz, planet saver. We shall be ever grateful, Jim.\"\n\n\"Ever grateful!\" they chorused, raised their glasses and drank. Except for Mark, who had no glass to raise. Instead of drinking wine he had one of his robots pour a dollop of electrolytic fluid into a dry battery; Mark had informed us that the sudden surge of electrons was most stimulating.\n\n\"Thank you, my friends, thank you,\" I said, then raised my glass in turn. \"To Morton and Sharla, who sit on the couch beside you, holding hands and blushing because they are soon to be married.\"\n\nThey all cheered and drank at that; Mark Forer giggled over his zippy electrons. I raised my glass again.\n\n\"A toast of thanks as well to my physical guide and intellectual mentor, Stirner. And to my companion in adventure, Neebe\u2014long may her bicycle roll.\" More cheers and glugging followed as I turned to the glowing machine before us.\n\n\"Last\u2014and certainly\u2014not least, to Mark Forer. Guide, teacher, spiritual leader, purveyor of fine wines. To Mark!\"\n\nWhen the cheering had died away, and another bottle had been cracked, Mark Forer spoke to his attentive audience.\n\n\"Thank you, thank you dear believers in Individual Mutualism. Too long have I been sholitary...\"\n\nSholitary? This mean machine was getting pissed on whizzing electrons! Drunk!\n\n\"... too long a lurker beneath the streets watching the passing parade passing above me. Now, at last, finally I welcome your dear company and I greet you. And we had better crack another case of wine.\"\n\nStirner staggered off to fetch it and Neebe went to help. Alone for the moment Morton and Sharla wrapped themselves in happy osculatory embrace. Mark was muttering to himself.\n\nThis was a perfect opportunity to slip away. I hated goodbyes. Quietly, so as not to disturb them, I rose and made my exit. As I slowly eased shut the door behind me I saw Mark's TV pickup swivel to face me; the diaphragm contracted and dilated quickly in an electronic wink. I winked back and closed the door, turned and slowly climbed the stairs.\n\nAs much as I liked this planet and its politically monomaniacal citizens, I knew it was not for me. Too civilized and peaceful. Without crime and without police\u2014what would I do for a living?\n\nGo, Jim, go! The stars are yours!\nThe Stainless Steel Rat Sings the Blues\nChapter 1\n\nWalking up the wall had not been easy. But walking across the ceiling was turning out to be completely impossible. Until I realized that I was going about it the wrong way. It seemed obvious when I thought about it. When I held onto the ceiling with my hands I could not move my feet. So I switched off the molebind gloves and swung down, hanging only from the soles of my boots. The blood rushed to my head\u2014as well it might\u2014bringing with it a surge of nausea and a sensation of great unease.\n\nWhat was I doing here, hanging upside down from the ceiling of the Mint, watching the machine below stamp out five-hundred-thousand-credit coins? They jingled and fell into the waiting baskets\u2014so the answer to that question was pretty obvious. I nearly fell after them as I cut the power on one foot. I swung it forward in a giant step and slammed it solidly against the ceiling again as I turned the binding energy back on. A generator in the boot emitted a field of the same binding energy that holds molecules together, making my foot, at least temporarily, a part of the ceiling. As long as the power was on.\n\nA few more long steps and I was over the baskets. I fumbled at my waist, trying to ignore the dizziness, and pulled out the cord from my oversized belt buckle. Bending double until I could reach up to the ceiling, I pushed the knob at the end against the plaster and switched it on. The molebind field clamped hard and I released my feet. To hang, swinging, right side up now, while the blood seeped out of my florid face.\n\n\"Come on Jim\u2014no hanging about,\" I advised myself. \"The alarm will go off any second now.\"\n\nRight on cue the sirens screamed, the lights blinked, while a gargantuan hooter thundered through the walls. I did not tell myself that I told me so. No time. Thumb on the power button so that the immensely strong, almost invisible, single-molecule cord whirred out of the buckle and dropped me swiftly down. When my outstretched hands clinked among the coins I stopped. Opened my attach\u00e9 case and dragged it clanking through the coins until it was full of the shining, shimmering beauties.\n\nClosed and sealed it as the tiny motor buzzed and dragged me up to the ceiling again. My feet struck and stuck. I switched off power to the lifting lug.\n\nAnd the door opened below me.\n\n\"Somebody coulda come in here!\" the guard shouted, his weapon nosing about him. \"The door alarm went off.\"\n\n\"Maybe\u2014but I don't see nothin',\" the second guard said.\n\nThey looked down and around. But never up. I hoped. Feeling the sweat rolling up my face. Collecting there. Dropping.\n\nI watched with horror as the droplets spattered down onto the guard's helmet.\n\n\"Next room!\" he shouted, his voice drowning out the splat of perspiration. They rushed out, the door closed, I walked across the ceiling, crawled down the wall, slumped with exhaustion on the floor.\n\n\"Ten seconds, no more,\" I admonished. Survival was a harsh taskmaster. What had seemed like a good idea at the time maybe really was a good idea. But right now I was very sorry I had ever seen the newsflash.\n\nCeremonial opening of new Mint on Pask\u00f6njak... planet often called Mint-world... first half-million-credit coins ever issued... dignitaries and press invited.\n\nIt had been like the sound of the starting gun to a sprinter. I was off. A week later I was stepping out of the space terminal on Pask\u00f6njak, bag in hand and forged press credentials in pocket. Even the massed troops and tough security had not tempered my madness. The machines in my case were immune from detection by any known security apparatus; the case projected a totally false image of its contents when radiation hit it. My step had been light, my smile broad.\n\nNow my face was ashen and my legs trembled with fatigue as I pushed myself to my feet.\n\n\"Look calm, look collected\u2014think innocence.\"\n\nI swallowed a calm-and-collected pill that was coated with instant uppers. One, two, three paces to the door, my face flushed with pride, my gait noble, my conscience pure.\n\nI put on my funky bejeweled spectacles and looked through the door. The ultrasound image was fuzzy. But clear enough to reveal figures hurrying past. When they were gone I unlocked the door, slipped through and let it close behind me.\n\nSaw the rest of my party of journalists being pushed down the corridor by screaming, gun-waving troops. Turned and marched firmly away in the opposite direction and around the bend.\n\nThe guard stationed there lowered his gun and pointed it at my belt buckle.\n\n\"La necesejo estas \u0109i tie?\" I said, smiling smarmily.\n\n\"What you say? What you doin' here?\"\n\n\"Indeed?\" I snorted through widened nostrils. \"Rather short on education, particularly a knowledge of Esperanto, aren't we? If you must know, speaking in the vulgar argot of this planet\u2014I was told that the men's room was down here.\"\n\n\"Well it ain't. Da udder way.\"\n\n\"You're too kind.\"\n\nI turned and strolled diffidently down the hall. Had taken three steps before reality penetrated his sluggish synapses.\n\n\"Come back here, you!\"\n\nI stopped and turned about, pointed past him. \"Down that way?\" I asked. The gas projector I had palmed when my back was turned towards him hissed briefly. His eyes closed and he dropped; I took the gun from his limp hands as he fell by. Placed it on his sleeping chest since it was of no help to me. Walked briskly past him and pushed open the door to the emergency stairs. Closed and leaned against it and breathed very deeply. Then took out the map that had been in the press kit and poked my finger on the symbol for stairs. Now, down to the storeroom... footsteps sounded below.\n\nUp. Quietly on soft soles. A change of plan was very much in order since the alarm had sounded, ruling out a simple exit with the crowd. Up, five, six flights until the steps ended in a door labeled KROV. Which probably meant roof in the local language.\n\nThere were three different alarms that I disabled before I pushed the door open and slipped through. Looked around at the usual rooftop clutter: water tanks, vents, aircon units\u2014and a good-sized smokestack puffing out pollution. Perfect.\n\nThe moneybag clunked as I dropped all my incriminating weapons and tools into it. My belt buckle twisted open and I took out the reel and motor. Attached the molebind plug from the suspension cord to the bag, then lowered it all down the chimney. Reaching down as far as I could I secured the reel mechanism to the inside of the pipe.\n\nDone. It would wait there as long as needed, until all the excitement calmed down. An investment waiting to be collected you might say. Then, armed only with my innocence, I retraced my course back down the stairs and on to the ground floor.\n\nThe door opened and closed silently and there was a guard, back turned, standing close enough to touch. Which I did, tapping him on the shoulder. He shrieked, jumped aside, turned, lifted his gun.\n\n\"Didn't mean to startle you,\" I said sweetly. \"Afraid I got separated from my party. The press group...\"\n\n\"Sergeant, I got someone,\" he burbled into the microphone on his shoulder. \"Me, yeah, Private Izmet, post eleven. Right. Hold him. Got that.\" He pointed the gun between my eyes. \"Don't move!\"\n\n\"I have no intention of that, I assure you.\"\n\nI admired my fingernails, plucked a bit of fluff from my jacket, whistled; tried to ignore the wavering gun muzzle. There was the thud of running feet and a squad led by a grim-looking sergeant rushed up.\n\n\"Good afternoon, Sergeant. Can you tell me why this soldier is pointing his weapon at me? Or rather why you are all pointing your weapons at me?\"\n\n\"Grab his case. Cuff him. Bring him.\" A man of few words, the sergeant.\n\nThe elevator they hustled me to had not been marked on the map issued to the journalists. Nor had the map even hinted at the many levels below the ground floor that penetrated deep into the bowels of the earth. The pressure hit my eardrums as we dropped\u2014about as many floors down as you usually go up in a skyscraper. My stomach sank as well as I realized I had bitten off a good deal more than I could possibly chew. Pushed out at some subterranean level, dragged through locked, barred gates, one after another, until we finally reached a singularly depressing room. Traditionally bare with unshielded lights and a backless stool. I sighed and sat.\n\nMy attempts at conversation were ignored, as was my press pass. Which was taken from me along with my shoes\u2014then the rest of my clothes. I pulled on the robe of itchy black burlap that they gave me, dropped back onto the chair and made no attempt to outstare my guards.\n\nTo be frank this was a kind of low point, made even lower when the effects of the calm-and-collected pill began to wear off. Just about the time my morale hit bottom the loudspeaker gurgled incomprehensible instructions and I was hurried down the hall to another room. The lights and stool were the same\u2014but this time they faced a steel desk with an even steelier-eyed officer behind it. His glare spoke for him as he pointed to my dissected clothing, bag, shoes.\n\n\"I am Colonel Neuredan\u2014and you are in trouble.\"\n\n\"Do you always treat interstellar journalists like this?\"\n\n\"Your identity is false.\" His voice had all the warmth of two rocks being grated together. \"Your shoes contain molebind projectors...\"\n\n\"There's no law against that!\"\n\n\"There is on Pask\u00f6njak. There is a law against anything that threatens the security of the Mint and the Interstellar Credits produced here.\"\n\n\"I've done nothing wrong.\"\n\n\"Everything that you have done has been wrong. Attempting to deceive our security with false identification, stunning a guard, penetrating the Mint without supervision\u2014these are all crimes under our law. What you have committed so far makes you liable for fourteen concurrent life sentences.\" His grim voice grew even grimmer. \"But there is even worse than that\u2014\"\n\n\"What could be worse than fourteen life sentences?\" Despite my efforts at calm control I could hear my voice cracking.\n\n\"Death. That is the penalty for stealing from the Mint.\"\n\n\"I haven't stolen anything!\" Definitely a quaver now.\n\n\"That will be determined very shortly. When the decision was made to mint five-hundred-thousand-credit coins every precaution was taken to prevent their theft. Integral to their fabric is a transponder that listens for a specific signal at a specific frequency. It answers and reveals the location of the coin.\"\n\n\"Stupid,\" I said with more bravado than I felt. \"Won't work here. Not with all the coins you have made\u2014\"\n\n\"All now safe behind ten feet of solid lead. Radiation proof. If there are any other coins not in our custody the signal will sound.\"\n\nRight on cue I heard the pealing of bells in the distance. The iron face of my inquisitor was touched by a fleeting cold smile.\n\n\"The signal,\" he said. We sat in silence for long seconds. Until the door burst open and the hurrying guards dropped a very familiar bag onto the desk. He lifted the end slowly and the coins jangled forth.\n\n\"So that's what they look like. I never...\"\n\n\"Silence!\" he thundered. \"These were removed from the minting room. They were found suspended in the chimney from the smelter. Along with these other objects.\"\n\n\"Proves nothing.\"\n\n\"Proves everything!\" Quick as a snake he grabbed my hands, slammed them onto a plate on his desk. A hologram of my fingerprints appeared instantly on the air above.\n\n\"Any prints lifted from the coins?\" he asked over his shoulder.\n\n\"Many,\" a spectral voice responded. A portion of the desk top rose up bearing what appeared to be photographic prints. He looked at them and for the second time I was treated to the sight of that frigid smile as he dropped the prints through a slot. A second hologram floated in the air beside the first, moved over and merged with it as he touched the controls.\n\nThe double image flickered and became one.\n\n\"Identical!\" he said triumphantly. \"You can tell me your name if you wish. So it can be spelled correctly on your tombstone. But only if you wish.\"\n\n\"What do you mean tombstone? And what do you mean death sentence? That's illegal by galactic law!\"\n\n\"There is no galactic law down here,\" he intoned with a voice like a funeral march. \"There is only the law of the Mint. Judgment is final.\"\n\n\"The trial...\" I said feebly, visions of lawyers, appeals, torts and documents dancing in my head.\n\nThere was no mercy in his voice now, no touch of the tiniest of iceberg smiles on his lips.\n\n\"The penalty for theft in the Mint is death. The trial takes place after the execution.\"\nChapter 2\n\nI am still young\u2014and it did not look like I was going to get any older. My dedication to a life of crime had led to a far shorter lifespan than could normally be expected. Here I was, not yet twenty years old. A veteran who had fought in two wars, had been imprisoned and drafted, who had been depressed by the death of my good friend The Bishop, been impressed by Mark Forer the great Artificial Intelligence. Was that it? Had I had it? No more to life than that? All over.\n\n\"Never!\" I shouted aloud, but the two guards merely gripped my arms the harder and pushed me along the corridor. A third armed guard went ahead and unlocked the cell door, while the one behind me prodded my kidneys with the barrel of his gun.\n\nThey were good and they took no chances. They were big and mean and I was small and lean. Shivering with fear, I was crouching even lower. Once the cell door was open the guard with the keys turned towards me and unlocked my handcuffs.\n\nThen gasped as my knee caught him in the stomach and knocked him back into the cell. At the same time I grabbed the two guards beside me by the wrists, crossed my arms with a single spasmodic burst of effort that pulled the two of them crashing together; their skulls bonked nicely. At the same instant I lashed backward\u2014catching the fourth guard on the bridge of his nose with the back of my head. Everything happening at approximately the same time.\n\nTwo seconds ago I had been bound and captive.\n\nNow one guard was out of sight, groaning in the cell. Two more holding their heads and howling, the fourth one clutching a bloody nose. They hadn't been expecting this: I had.\n\nI ran. Back the way we had come and through the still-open door. Hoarse, angry cries were cut off as I slammed it shut, locked it. The thick panel shook as heavy bodies thudded against the other side.\n\n\"Got you!\" a victorious voice shouted and rough hands grabbed me. He could not know by touch that I was a Black Belt. He found out the hard way.\n\nEyes closed, breathing easily, he just lay there and made no protest as I stripped him of uniform and weapons. Nor did he thank me when I draped my burlap robe over his pale form, hiding his black lace undies from prying eyes. His clothes were not too bad a fit. Not too good either with the cap tilting forward over my eyes. But it would have to do.\n\nThere were three doors leading from this room. The one that I had locked was pounding and bouncing in its frame: next to it was the one that we had come in through. It didn't take much intelligence to use the unconscious guard's keys to open the third one.\n\nIt led to a storage room. Dark shelves, filled with nameless objects, vanished away into the distance. Not too promising\u2014but I was in no position to choose. I executed a quick leap back to the entrance door, unlocked it and threw it open, then dived ahead into the storage room. As I closed this final door behind me, even before I could lock it, there was a mighty crash and screams of anger as the assaultees finally broke the door down.\n\nMisdirection wouldn't last long. Run past the shelves. Hide here? No\u2014there would be a thorough search. A door at the far end, bolted on the inside. I opened it a crack and looked at the empty room beyond. Opened and stepped through.\n\nAnd stopped quite still as the guards who were flattened against the wall all pointed guns at me.\n\n\"Shoot him!\" Colonel Neuredan ordered.\n\n\"I'm unarmed!\" My gun slid across the floor as I threw my hands into the air. Fingers quivered on triggers\u2014it was all over.\n\n\"Don't shoot\u2014I want him alive. For the moment.\"\n\nI stood frozen, not breathing until the trigger fingers relaxed. Looked up and quickly found the security bug in the ceiling. Must be one in every room and corridor down here. They had been watching me all the time. A good try, Jim. The Colonel grated his teeth horribly and stabbed a finger in my direction.\n\n\"Take him. Chain him. Bind him. Bring him.\"\n\nThis was all done with ruthless efficiency. My toes dragged along the floor as I was whisked back to the cell, stripped at gunpoint, thrown to the floor with my black robe thrown on top of me. The door clanged shut and I was alone. Very much alone.\n\n\"Cheer up, Jim, you've been in worse trouble before,\" I chirped smilingly. Then snarled, \"When?\"\n\nBack in the pits again. My abortive attempt at escape had only gained me a few bruises.\n\n\"This can't be it!\" I shouted. \"It can't all end just like this.\"\n\n\"It can\u2014and it will,\" the Colonel's funereal voice intoned as the cell door opened again. A dozen guns were pointed at me as a guard brought in a tray with a bottle of champagne on it and a single glass.\n\nI watched in stupefied disbelief as he twisted the cork out. There was a pop and a gush as the golden fluid filled the glass. He handed it to me.\n\n\"What's this, what's this?\" I mumbled, staring wide-eyed at the rising bubbles.\n\n\"Your last request,\" Neuredan said. \"That and a cigarette.\"\n\nHe took one from a package and lit it, holding it out to me. I shook my head. \"I don't smoke.\" He ground the cigarette under his heel. \"Anyway\u2014champagne and a cigarette\u2014that's not my last request.\"\n\n\"Yes it is. Forms of last request are standardized by law. Drink.\"\n\nI drank. It tasted all right. I belched and handed back the glass. \"I'll take a refill.\" Anything to gain time, to think. I watched the wine being poured and my brain was dull and empty. \"You never told me about the... execution.\"\n\n\"Do you want to know?\"\n\n\"Not really.\"\n\n\"Then I will be pleased to tell you. I assure you that there was extensive deliberation over the correct method to be used. Thought was given to the firing squad, electrocution, poison gas\u2014a number of possibilities were actively considered when the law was passed. But all of them involve someone pulling a switch or a trigger, and that would not be humane to the someone.\"\n\n\"Humane! What about the prisoner?\"\n\n\"Of no importance. Your death has been decreed and will take place as soon as possible. This is what will happen. You will be taken to a scaled chamber and chained there. The entrance will be locked. After this the chamber will be flooded with water by an automatic device actuated by your body heat. It is always there, always turned on. You alone will be responsible for your own execution. Now isn't that quite humane?\"\n\n\"Drowning is humane all of a sudden?\"\n\n\"Possibly not. But you will be left a pistol containing a single bullet. You can commit suicide if you wish to.\"\n\nI opened my mouth to tell him what I thought of their humanity, but I was seized by many hands and dragged forward before I could speak. The glass was whisked away\u2014and so was I. Deep down to a dank chamber, walls damp with water and covered with moss. A cuff was clamped around my ankle; a chain ran from it to a staple in the wall. They all exited except for the Colonel, who stood with his hand on the operating lever of the thick, undoubtedly watertight, door.\n\nHe grinned in victorious triumph, bent over and placed an antique pistol on the floor. As I dived for it the door shut and sealed with a final thud.\n\nWas this really the end? I turned the pistol over in my hands, saw the dull shape of the single cartridge. End of Jim diGriz, end of the Stainless Steel Rat, end of everything.\n\nThere was the distant thunk of a valve opening and cold water gushed down on me from a thick pipe in the ceiling. It gurgled and slopped, covering my feet, then quickly up to my ankles. When it reached my waist I lifted the gun and looked at it. Not much of a choice. The water rose steadily. Covered my chest, up to my chin. I shuddered.\n\nThen the water stopped splashing down. It was cold and I was shivering uncontrollably. The light in the waterproof fixture revealed only stone wall, dark water.\n\n\"What are you playing at bastarda\u0109oj?\" I shouted. \"Humane torture to go with your humane murder?\"\n\nA moment later I got my answer. The level began to drop.\n\n\"I was right\u2014torturers!\" I bellowed. \"Torture first\u2014then murder. And you call yourself civilized. Why are you doing this?\"\n\nThe last of the water gurgled down the drain and the door slowly opened. I aimed the pistol at it. I wouldn't mind drowning if I could take the cretinous colonel or the sadistic sergeant with me.\n\nSomething dark appeared through the partly open door. The gun banged and the bullet thudded into it. A briefcase.\n\n\"Cease fire!\" a male voice called out. \"I am your lawyer.\"\n\n\"He only has one bullet, you're safe,\" I heard the Colonel say.\n\nThe briefcase came hesitantly into the room, carried by a gray-haired man who was wearing the traditional gold-flecked and diamond-decorated black suit that adorned lawyers throughout the galaxy.\n\n\"I am your court-appointed lawyer, Pederasis Narcoses.\"\n\n\"What good will you do me\u2014if the trial will be after my execution?\"\n\n\"None. But that is the law. I will have to interview you now to enable me to conduct your defense at the trial.\"\n\n\"This is madness\u2014I'll be dead!\"\n\n\"That is correct. But it is the law.\" He turned to the Colonel. \"I must be alone with my client. That is also the law.\"\n\n\"You have ten minutes, no longer.\"\n\n\"That will suffice. Admit my assistant in five minutes. He has the court papers and the will.\"\n\nThe door thunked shut and Narcoses opened his briefcase and took out a plastic bottle filled with a greenish liquid. He removed the top and handed it to me.\n\n\"Drink this, all of it. I'll hold the gun.\"\n\nI handed him the weapon, took the bottle, smelled it and coughed. \"Horrible. Why should I drink it?\"\n\n\"Because I told you to. It is of vital importance and you have no choice.\"\n\nWhich was true\u2014and what difference would it make anyway? I glugged it down. The champagne had tasted a lot better.\n\n\"I will now explain,\" he said, recapping the bottle and putting it back into his briefcase. \"You have just drunk a thirty-day poison. This is a computer-generated complex of toxins that are neutral now\u2014but which will kill you horribly in exactly thirty days if you are not given the antidote. Which is also computer-generated and impossible to duplicate.\"\n\nHe jumped back quite smartly when I leaped at him. But the chain on my ankle would not quite reach. My fingers snapped ineffectually just in front of his throat.\n\n\"If you will cease clawing at the air I will explain,\" Narcoses said with an air of weary sophistication. Had he done this kind of thing before I wondered? I folded my arms and stepped back.\n\n\"Much better. Although I am a lawyer licensed to practice on this planet, I am also a representative of the Galactic League.\"\n\n\"Wonderful. The Pask\u00f6njakians want to drown me\u2014you poison me. I thought this was a galaxy of peace?\"\n\n\"You are wasting time. I am here to free you, under certain conditions. The League has need of a criminal. One who is both skilled and reliable. Which is an oxymoron. You have proved your criminalistic ability by your almost-successful theft. The poison guarantees your reliability. Do I assume that you will cooperate? At the minimum you have a life extension of thirty days.\"\n\n\"Yes, sure, you're on. Not that I have a choice.\"\n\n\"You don't.\" He looked at the watch set into his little fingernail and stepped aside as the door opened. A chubby, bearded youth came in with a sheaf of papers.\n\n\"Excellent,\" Narcoses said. \"You have the will?\" The young man nodded. The door was closed and sealed again.\n\n\"Five minutes,\" Narcoses said.\n\nThe newcomer pulled down a zipper that sealed his one-piece suit. Took off the suit\u2014and a lot of flesh with it. The suit was padded. He was not fat at all, but lean and muscular quite like me. When he peeled off the fake beard I realized that he looked exactly like me. I blinked rapidly as I stared at my own face.\n\n\"Only four minutes left diGriz. Put on the suit. I'll fix the beard.\"\n\nThe well-built and handsome stranger pulled on my discarded robe. Stepped aside when Narcoses took a key from his pocket, bent and unlocked the restraining cuff on my ankle. Handed it to the other, who emotionlessly bent and snapped it to his own ankle.\n\n\"Why\u2014why are you doing this?\" I asked him.\n\nHe said nothing, just leaned over to retrieve the gun.\n\n\"I'll need another bullet,\" he said. With my voice.\n\n\"The Colonel will supply it,\" Narcoses said. Then I remembered something else he had said just moments ago.\n\n\"You called me diGriz. You know my name!\"\n\n\"I know a lot more than that,\" he said pressing the beard and mustache into position on my face. \"Carry these papers. Follow me out of here. Keep your mouth shut.\"\n\nAll of which I was very happy to do. With one last look at my imprisoned self I trotted forth to freedom.\nChapter 3\n\nI trotted behind Narcoses, clutching the papers and trying to think bearded and fat. The guards were ignoring us, watching instead with sadistic fascination as one of their number started to close the watertight door.\n\n\"Wait,\" the Colonel said, opening a small box and taking out a cartridge. He looked up as I passed, stared me straight in the face. I felt perspiration bursting from my pores. The momentary glance must have lasted about a subjective hour. Then he kept on turning and called out to the guard.\n\n\"Open that again you idiot! I load the gun then you close the door. When that has been done this business will be over with once and for all.\"\n\nWe turned a corner and the noxious group vanished from sight behind us. Silently, as ordered, I followed the lawyer through many a guarded portal, into an elevator, out of it and then through one final door, escaping the Mint at last. Letting out a great sigh of relief as we went past the armed guards and headed for the waiting groundcar.\n\n\"I\u2014\"\n\n\"Silence! Into the car. Speak to me in the office about a salary raise\u2014not before.\"\n\nNarcoses must know things that I didn't. Detector bugs in the ornamental trees we were passing? Acoustic microphones aimed our way? I realized now that my carefully planned crime had apparently been a disaster from the moment I had conceived it.\n\nThe driver was silent as a tomb\u2014and about as attractive. I watched the buildings stream by, then the outskirts of the city appeared. We drove on until we reached a small building in a leafy suburb. The front door opened as we approached, then closed behind us apparently without human intervention. The same thing happened to the inner door, which was tastefully labeled with jewel-studded gold letters PEDERASIS NARCOSES\u2014Attorney at Law. It closed silently and I wheeled about and pointed a menacing finger at him.\n\n\"You knew about me even before I landed on this planet.\"\n\n\"Of course. As soon as your false credentials were filed the investigation began.\"\n\n\"So you stood by and let me plot and plan and commit a crime and get sentenced to death\u2014without making any attempt to interfere?\"\n\n\"That's right.\"\n\n\"That's criminal! More of a crime than my crime.\"\n\n\"Not really. You were always going to be plucked out of that terminal swimming pool in any case. We just wanted to see how well you did.\"\n\n\"How did I do?\"\n\n\"Very good\u2014for a lad your age. You got the job.\"\n\n\"Well good for me. But what about my double\u2014the bloke who took my place?\"\n\n\"That bloke, as you refer to him, is one of the finest and most expensive humanoid robots that money can afford. Which money will not be wasted since the doctor who is now performing the postmortem is in our pay. The incident is closed.\"\n\n\"Wonderful,\" I sighed, dropping limply onto the couch. \"Look, can I get a drink? It has been a long day. No spirits however\u2014a beer will do fine.\"\n\n\"A capital idea. I will join you.\"\n\nA tiny but well-stocked bar unfolded from one wall; the dispenser produced two chilled brews. I gulped and smacked.\n\n\"Excellent. If I have only thirty days to go shouldn't you be telling me about what you want me to do?\"\n\n\"In good time,\" he said, sitting down across from me. \"Captain Varod asked me to send his regards. And to convey the message that he knew you were lying when you promised to give up a life of crime.\"\n\n\"So he had me watched?\"\n\n\"You're catching on. After this last criminal assignment for us you will become an honest man. Or else.\"\n\n\"Who are you to talk!\" I sneered and drained the glass. \"A crooked shyster who is theoretically paid to uphold the law. Yet you stand by and let the thugs here on Pask\u00f6njak pass legislation to have trials after an execution\u2014then you employ a criminal to commit a criminal act. Not what I would call sincerely law-abiding.\"\n\n\"First,\" he said, lifting a finger in a very legalistic way, \"we have never condoned the secret law in the Mint. It was only recently produced by the overly paranoid management here. Yours was the first arrest\u2014and will be the last. There have been numerous job replacements already. Second,\" another finger rose to join the first, \"the League has never condoned violence or criminal acts. This is the first occurrence and has been produced by an unusual series of circumstances. After great deliberation the decision was made to do it just this one time. And never again.\"\n\n\"Millions might believe that,\" I sneered disbelievingly. \"Isn't it time you told me what the job is?\"\n\n\"No\u2014because I don't know myself. My vote was cast against this entire operation so I have been included out. Professor Van Diver will brief you.\"\n\n\"But what about the thirty-day poison?\"\n\n\"You will be contacted on the twenty-ninth day.\" He stood up and went to the door. \"It is against my principles to wish you good luck.\"\n\nThis was his puritanical pontificatory exit line. Because as he went out an elderly type with a white beard and a monocle entered.\n\n\"Professor Van Diver I presume?\"\n\n\"Indeed,\" he said extending a damp, limp hand for me to shake. \"You must be the volunteer with the nom de guerre of Jim about whose presence I was informed, who would await me here. It was very good of you to undertake what can only be called a rather diligent and difficult assignment.\"\n\n\"Rather,\" I intoned, falling into his academic mode of speech. \"Is there any remote possibility that I might be informed of the nature of this assignment?\"\n\n\"Of course. I have the requisite authority to provide augmentive information to you concerning the history and tragic circumstance of the loss. Another individual, who shall be nameless, will supply the assistance that you will require. I shall begin with the circumstances that occurred a little over twenty years ago...\"\n\n\"A beer. I must have refreshment. Will you join me?\"\n\n\"I abstain from all alcoholic and caffeine-containing beverages.\" He glared at me glassily through his menacing monocle as I refilled my mug. I sipped and sat and waved him into action. His voice washed over me in turgid waves and soon had me half-asleep\u2014but the content of his talk woke me up fast enough. He went on far too long, with far too many digressions, but despite this it was fascinating stuff to listen to.\n\nA stripped-down version wouldn't have been half as much fun for him and would have taken only a few minutes to tell. Simply, Galaksia Universitato had sent an expedition to a reported archeological site on a distant world\u2014where they had uncovered an artifact of nonhuman origin.\n\n\"You must be kidding,\" I said. \"Mankind has explored a great part of the galaxy in the last thirty-two thousand years and no trace of an alien race has ever been found.\"\n\nHe sniffed loudly. \"I do not 'kid' as you say in your simple demotic. I have pictorial proof here, photographs sent back by the expedition. The artifact was uncovered in a stratum at least a million years old and resembles nothing in any data base existent in the known universe.\"\n\nHe took a print from his inner pocket and passed it over to me. I took it and looked at it, then turned it around since there was no indication of which was top or bottom. A twisted hunk of incongruous angles and forms resembling nothing I had ever seen before.\n\n\"It looks alien enough to be alien,\" I said. Looking at it was beginning to hurt my eyes so I dropped it onto the table. \"What does it do, or what is it made of or whatever?\"\n\n\"I haven't the slightest idea since it was never conveyed to the university. It was, I must say, interrupted in its journey and it is essential that it be recovered.\"\n\n\"Pretty sloppy way to handle the only alien artifact in the universe.\"\n\n\"That is beyond the scope of my authority and not for me to say. But I am authorized to unperfunctorily predicate that it must be found and returned. At any cost\u2014which sums I am duly authorized to pay. Officers of the Galactic League have assured me that you, pseudonymous Jim, have volunteered to find and return the artifact. They have convinced me that you, as young as you are, are a specialist in these matters. I can only wish you best of luck\u2014and look forward to meeting you again when you return with that which we desire the most.\"\n\nHe exited then and a bald, uniformed naval officer entered in his place. Closed the door and glared at me with a steely gaze. I glared back.\n\n\"Are you the one who is finally going to tell me what is going on?\" I asked.\n\n\"Damn right,\" he growled. \"Damn fool idea\u2014but the only one we have going. I am Admiral Benbow, head of League Navy Security. Those dumbhead academics let the most priceless object in the universe slip through their fingers\u2014now we have to pick up the pieces and run with the ball.\"\n\nThe Admiral's mixed metaphors were as bad as the professor's academese. Was clear speaking becoming a lost art?\n\n\"Come on,\" I said. \"Simply tell me what happened and what I am supposed to do.\"\n\n\"Right.\" He slammed down into a chair. \"If that is a beer I'll have one too. No I won't. A double, no a treble high-octane whisky. No ice. Do it.\"\n\nThe robobar supplied our drinks. He drained his while I was just lifting mine.\n\n\"Now hear this. The expedition concerned was returning from their planetary dig when their ship experienced communication difficulties. Worried about navigation they landed on the nearest planet, which unhappily and tragically turned out to be Liokukae.\"\n\n\"Why unhappily and tragically?\"\n\n\"Shut up and listen. We got them and their ship back relatively intact. But without the artifact. For certain reasons we could do no more. That is why your services have been engaged.\"\n\n\"So now you are going to tell me about those certain reasons.\"\n\nHe coughed and looked away, stood and refilled his glass before speaking again. If I didn't know better I would have said that the seasoned old space dog was embarrassed.\n\n\"You have to understand that keeping the galactic peace is our role and our goal. This is not always possible. There are sometimes individuals, even groups, that are impervious to our attentions. Violent people, some apparently incurably insane, obnoxious. Despite everything that we can do they remain immune to our blandishments, impervious to our help.\" He gulped down the dregs and I had the feeling that we were finally getting to the truth.\n\n\"Since we cannot kill them we\u2014and you realize only the highest authorities know what I am about to tell you\u2014we so to speak arrange, see to it that they are, well, transported to Liokukae to live the sort of life they prefer to live. Without endangering the peaceful cultures of the union\u2014\"\n\n\"A galactic garbage dump!\" I cried aloud. \"Where you holier-than-thou bigots sweep your failures under the carpet! No wonder you keep this a top-secret secret.\"\n\n\"Just knock off the superior attitude cagal, diGriz. I know your record\u2014and in my book it stinks. But we have you by the short and curlies since you drank the seven-hundred-and-twenty-hour poison, so you will do just as I say. So now I'm going to fill you in with all the loathsome details re Liokukae, let you see what information we have. Then you will come up with a plan for getting that thing back. You have no choice.\"\n\n\"Thanks. What resources do I have?\"\n\n\"Limitless resources, unrestricted funds, boundless support. Every planet in the galaxy contributes to Galaksia Universitato. They have so many credits that they make the super-rich look super-poor. I want you to take them to the cleaners.\"\n\n\"Now you are talking my language! For the first time I have some interest in this poisonous project. Bring on the records\u2014and some food\u2014and I will see what I can do.\"\n\nNot very much I thought to myself after hours of reading and rereading the thin file, while eating a number of stale and tasteless sandwiches. The Admiral was slumped asleep in the armchair and snoring like a rocket exhaust. There were no answers here, so some questions were very much in order. Which gave me the sweet pleasure of waking him up. A few good shakes did it and those nasty little red eyes glared into mine.\n\n\"You better have a good reason for that.\"\n\n\"I do. How much do you personally know about Liokukae?\"\n\n\"Everything, you dimwit. That is why I am here.\"\n\n\"It seems to be pretty tightly sealed up.\"\n\n\"Pretty tightly is not the way I would describe it. Hermetically sealed, guarded, patrolled, watched, locked tight, quarantined\u2014take your pick. Food and medicines are shipped in. Nothing comes out.\"\n\n\"Do they have their own doctors?\"\n\n\"No. Medical teams are stationed there in the hospital inside the landing station\u2014which is built like a fortress. And before you even ask\u2014the answer is no. What little trust there is between the Navy and the Liokukaers involves the medical services. They come to us and we treat them. Let them suspect for an instant that the medicos are involved in hanky-panky and the trust is gone. Disease and death would be certain. We're not taking a chance on that.\"\n\n\"If the rest of the civilized galaxy doesn't know about them\u2014what do they know about us?\"\n\n\"Everything I suppose. We do not practice censorship. We transmit all the usual TV entertainment channels as well as educational and news services. They are well supplied with television receivers and can watch reruns of all the most loathsome programs and series. The theory being if we can stun their minds with televised crap they won't get up to more trouble.\"\n\n\"Does it work?\"\n\n\"Possibly. But we do know that they are rated on top of the galactic viewing scale for uninterrupted hours in front of the gogglebox.\"\n\n\"You go there and take surveys?\"\n\n\"Don't be stupid. Recorders are sealed into the chassis of each set. These can be tapped by satellite.\"\n\n\"So what we have here is a planet of murderous, belligerent, nutsy TV fans?\"\n\n\"That's about the size of it.\"\n\nI jumped to my feet, spilling dry crumbs of dead sandwiches onto the carpet. Raised my fists, and my voice, on high.\n\n\"That's it!\"\n\nBenbow blinked at me rapidly and scowled.\n\n\"What's it?\"\n\n\"The answer. It is just the glimmering of an idea now\u2014but I know that it will grow and expand into something incredible. I'm going to sleep on it and when I awake I will polish it and perfect it and describe it to you in detail.\"\n\n\"What is it?\"\n\n\"Don't be greedy. All in good time.\"\nChapter 4\n\nThe automated kitchen produced another stale sandwich, the machine was half-knackered and out of adjustment, along with a lukewarm cup of watery cocoa. I crunched and sipped gloomily, then found the bedroom down the hall. Air-conditioned of course\u2014but the window wasn't sealed. I opened it and sniffed the cool night air. The moon was rising, to join the other three already up. Made for some interesting shadows. A leg over the windowsill, a drop into the garden\u2014and I would be long gone before any alarm might go off.\n\nAnd I would be dead in twenty-nine days. That little drink I had drunk in prison really concentrated my attention and guaranteed my loyalty. But could I pull this complicated operation off in that space of time?\n\nConsidering the consequences I had no choice. I sighed tremulously, closed the window and went to bed. It had been a very, very long day.\n\nIn the morning I had picked the lock on the control panel in the kitchen and was busy rewiring it when Admiral Benbow came in.\n\n\"May I inquire politely just what the hell you are doing?\"\n\n\"Obviously trying to get this crook device to produce something other than stale cheese sandwiches. There!\"\n\nI slammed the panel shut and punched in a command. A cup of steaming coffee instantly appeared. Followed by a porcuswinewich, steaming and juicy. The Admiral nodded.\n\n\"I'll take this one\u2014get another for yourself. Now tell me your plan.\"\n\nI did. Mumbling through mouthfuls of breakfast.\n\n\"We are going to spend some credits out of the mountains of money that we have access to. First we plant some news items. I want interviews, reviews, gossip and more\u2014all about the new pop group that is the hit of the galaxy.\"\n\nHe scowled and growled. \"What pop group? What in Hades are you talking about?\"\n\n\"The planet-busting hit group called...\"\n\n\"Called what?\"\n\n\"I don't know. I haven't thought about it yet. Something way out and memorable. Or kinky.\" I smiled and raised an inspired finger. \"I have it! Ready? The group is called... The Stainless Steel Rats!\"\n\n\"Why?\"\n\n\"Why not?\"\n\nThe Admiral was not happy. His scowl turned to a snarl and he jabbed a judgmental finger at me. \"More coffee. Then tell me what you are talking about or I will kill you.\"\n\n\"Temper, temper, Admiral. Remember the old blood pressure. What I am talking about is getting to Liokukae with all the equipment I need, along with some strong-armed help. We are going to form a group of musicians called The Stainless Steel Rats\u2014\"\n\n\"What musicians?\"\n\n\"Me for one\u2014and you are going to supply me with the rest. You did tell me that you were head of League Navy Security?\"\n\n\"I did. I am.\"\n\n\"Then summon your troops. Get one of your techs to research all your field operators, all your rankings who have ever served in what passes for action in this civilized universe. The search will be a simple one because we want to know just one single fact about all of them. Are they musically inclined? Can they play a musical instrument, sing, dance, whistle or even hum in tune? Get the list and we will have our band.\"\n\nHe nodded over his coffee. \"You're beginning to make sense. A pop group composed only of security agents. But it will take time to put together, to organize, to rehearse.\"\n\n\"Why?\"\n\n\"So it will sound good, you moron.\"\n\n\"Who could tell the difference? Have you ever listened to country-and-coal-mining music? Or Aqua Regia and her Plutonium Pals?\"\n\n\"Point taken. So we get this group together and publicize them well so all Liokukae knows about them\u2014\"\n\n\"And has heard their music\u2014\"\n\n\"And wants to hear more. On tour. Which is impossible. The planet is quarantined.\"\n\n\"That is the beauty of my plan, Admiral. When the publicity peaks, and the fame of the group is galaxy-wide, that is when the Rats will commit some crime so awful that they will instantly be shipped off to this prison planet. Where they will be received with great enthusiasm. And no suspicion. Where they will investigate and find the alien artifact and get it back so I can have the antidote. One other thing. Before we start operations I will need three million Interstellar Credits. In coins that have been newly minted here.\"\n\n\"No way,\" he snarled. \"Funds will be supplied as needed.\"\n\n\"You missed the point. That is my fee for conducting this operation. All operating expenses are on top of that. Pay up\u2014or else.\"\n\n\"Or else what?\"\n\n\"Or else I die in twenty-nine days and the operation dies and you get a black mark on your service record.\"\n\nSelf-interest prodded him into an instant decision. \"Why not. Those financially overburdened academics can afford it and not even notice it. I'll get that list for you.\"\n\nHe unclipped his phone from his belt, shouted a multidigit number into it, then barked some brief commands. Before I had finished my coffee the printer hummed to life in the office; sheets of paper began to pile up in the bin. We went through them and ticked off a number of possibilities. There were no names, just code numbers. When this was done I passed the list back to the Admiral.\n\n\"We'll need complete files on all the marked ones.\"\n\n\"That is classified and secret information.\"\n\n\"And you are the Admiral and you can get it.\"\n\n\"I'll get it\u2014and censor it. There is no way I am going to let you know any details of my Security Department.\"\n\n\"Keep your secrets\u2014I couldn't care less.\" Which was of course an outright lie. \"Give them code names as well as numbers, conceal their identities. All I want to know is their musical abilities, and will they be any good in the field when the going gets rough.\"\n\nThis took a bit of time. I went for a long jog to loosen the muscles. Then, while my clothes were being zapped clean in the vacuum washer, I took a hot shower followed by a cold one. I made a mental note to get some more clothes soon\u2014but not until this operation was up and running. There was no escaping that deadly clock that was ticking off the seconds to doomsday.\n\n\"Here is the list,\" the Admiral said when I entered the office. \"No names, just numbers. Male agents are identified by the letter A and...\"\n\n\"Let me guess\u2014the females are B?\"\n\nA growl was his only response, he completely lacked any sense of humor. I flipped through the list. Slim pickings among the ladies who ran the gamut from B1 to B4. Pipe-organ player, not very likely, harmonica, tuba\u2014and a singer.\n\n\"I'll need a photograph of B3. And what do these other entries after B1 mean? 19T, 908L, and such.\"\n\n\"Code,\" he said, grabbing the sheet away from me. \"It translates as skilled in hand-to-hand combat, qualified marksman on hand weapons, six years in the field. And the rest is none of your business.\"\n\n\"Thanks, wonderful, you're a big help. I sure could use her\u2014but not if she has to carry the pipe organ on her back. Now let us make some selections from the male list and get the photos coming. Except for this one, A19. No photograph\u2014I just want him here soonest, in the flesh.\"\n\n\"Why?\"\n\n\"Because he is a percussionist and plays a molecular synthezier. Since I know next to nothing about music\u2014he is going to teach me my job in this pickup band. A19 will show me the ropes, then record the numbers and set up the machines to play the different hunks of music. I'll just smile and press buttons. Speaking of machines\u2014does your highly secret service have electronic repair facilities on this planet?\"\n\n\"That is classified information.\"\n\n\"Everything about this operation is classified. But I'll still need to do some electronic work. Here or someplace else. All right?\"\n\n\"Facilities will be made available.\"\n\n\"Good. And tell me\u2014what is a gastrophone, or a bagpipe?\"\n\n\"I haven't the slightest idea. Why?\"\n\n\"Because they are listed here as musical skills or instruments or something. I'll need to know.\"\n\nLubricated by all the credits from the university, manned by the Admiral's minions, the machinery of my plan began to churn into high gear. The League did have an outpost on this planet\u2014disguised as an interstellar shipping firm\u2014which contained a fully equipped machine shop and electronic facilities. The fact that they gave me full use of everything meant that it would undoubtedly vanish as soon as this operation was over. While the auditions were being arranged, agent A19 was sent for by the fastest transportation available. He appeared, slightly glassy-eyed, later that same afternoon.\n\n\"You are known to me only by the code reference A19. Could you give me a slightly better name to call you by? And it doesn't have to be your own.\"\n\nHe was a big man with a big jaw, which he rubbed as he kicked his brain into action. \"Zach\u2014that's my cousin's name. Call me Zach.\"\n\n\"Right on, Zach. You have quite a musical record.\"\n\n\"You betcha. I worked my way through college playing in the band. Still do a gig or two from time to time.\"\n\n\"Then you have the job. You must now sally forth with an open checkbook and buy the best, most expensive and complex hunks of electronic music making that you can find. And they have to be the most compact and microminiaturized ones going. Bring them back and I'll make it all smaller since everything we bring with us has to be carried on our backs. If you can't find it on this planet use galactic mail order. Spend! The more you spend the better.\"\n\nHis eyes glowed with musical fervor. \"Do you mean that?\"\n\n\"Absolutely. Check with Admiral Benbow, who will authorize all expenses. Go!\"\n\nHe went, and the auditions began. I draw a veil over the more repulsive details of the next two days. Apparently musical ability and military service were mutually incompatible for the most part. I whittled away and the list grew smaller with great rapidity. I had hoped for a large band\u2014now it appeared that I had a tiny combo.\n\n\"This is it, Admiral,\" I said, passing over the abbreviated list. \"We will have to make up in quality what we lack in quantity. It is going to be me and these three others.\"\n\nHe frowned. \"Will it be enough?\"\n\n\"Going to have to be. The discards may be great operators but I will dream about their sounds for years. In my nightmares. So take the survivors aside, tell them about me and the assignment. I'll meet them after lunch in the audition room.\"\n\nI was setting glasses and bottles of refreshment on the table when the four of them trooped in. In step!\n\n\"First lesson!\" I shouted. \"Think civilian. Anything that even resembles the military will get us all quickly dead. Now\u2014have you all talked to the Admiral? Everyone is nodding, good, good. Nod again if you agree to take orders from me and no one else. Even better. Now I will introduce you to each other. I have been forbidden knowledge of your real names and positions so I have invented some. Let us now begin the world anew. The gentleman on your left, code name Zach, is a professional musician and is tutoring me in my new skills. He will be of utmost help in getting this project off the ground. I am Jim and I will soon be able to play the electronic gadgetry and lead this group. The young lady in your presence, now named Madonette, is a contralto of great talent and our lead singer. Let's give her a big hand.\"\n\nSlowly at first, then louder and jollier, they clapped until I lifted a hand to stop them. They were an uptight lot and I had to get them a lot looser. Madonette was fair of skin and dark of hair; a tall and solid girl and quite attractive, she smiled and waved in return.\n\n\"Good beginning gang. Now you last two guys, you're the rest of this group, Floyd and Steengo. Floyd is the tall and skinny guy with the artificial beard\u2014he is growing a real replacement for it, but we needed one now for the publicity pix. The miracle workers of hirsutical science have developed an antidipilatorisational agent that stimulates hair growth. So he will grow a fine beard in three days. In addition to growing hair he plays a number of wind instruments which are, if you don't know, a historic family of musical instruments into which one blows strongly to emit sounds. He comes from a distant planet named Och'aye, which is perhaps galaxy-famous for its other native son Angus Macswiney, founder of the Macswiney chain of automated eateries. Floyd plays an instrument whose antecedents are lost in the mists of time and at times I wish they had stayed there. Floyd, a quick tune on the bagpipe if you please.\"\n\nI had heard it before so was slightly more prepared as he opened the case and removed an apparatus that looked like a large and bulging spider with many black legs. He slung it about him, puffed strongly and pumped furiously on the spider's abdomen with his arm. I looked at the others and admired their horrified expressions as the screams of mortally wounded animals filled the room.\n\n\"Enough!\" I shouted and the last slaughtered pig moaned away into deathly silence. \"I don't know if this instrument will be featured in our recitals\u2014but you must admit that it does draw attention. Last, and certainly not least, is Steengo. Who after he left the service became quite adept on the fiddelino. Steengo, a demonstration if you please.\"\n\nSteengo smiled paternally at us and waved. He had gray hair and an impressive paunch. I was concerned about his age and general fitness but the Admiral, after secretly scanning the records, reassured me that Steengo's health was A-OK, that he worked out regularly and, other than a tendency towards slight overweight, he was fit for field conditions. I shrugged\u2014since there was little else I could do. The records revealed that he had taken up the instrument after retirement from active duty\u2014with talent in such short supply I had had the veterans' records searched as well. When approached he was more than happy to get back into harness. The fiddelino had two necks and twenty strings and sounded rather jolly in a plucking scratching way that everyone seemed to enjoy. Steengo bowed graciously to acknowledge the applause.\n\n\"That's it then. You have just met The Stainless Steel Rats. Any questions?\"\n\n\"Yes,\" Madonette said, and all eyes turned her way. \"What is the music that we will be playing?\"\n\n\"Good question\u2014and I think I have a good answer. Research into contemporary music reveals a great variety of rhythms and themes. Some of them pretty bad, like country-and-steel-mill music. Some with a certain charm, like the Chipperinos and their flock of singing birds. But we need something new and different. Or old and different as long as no one has heard the music in a few thousand years. For our inspiration I have had the music department at Galaksia Universitato research their most ancient data bases. Millennia have passed since this music was last heard. Usually with good reason.\"\n\nI held up a handful of recordings. \"These are the survivors of a grueling test I put them through. If I could listen for more than fifteen seconds I made a copy. We will now refine the process even more. Anything we can bear for thirty seconds goes into the second round.\"\n\nI popped one of the tiny black chips into the player and sat back. Atonal musical thunder rumbled over us and a soprano with a voice like a pregnant porcuswine assailed our ears. I popped the recording out, ground it under my heel, then went on to the next one.\n\nBy late afternoon our eyes were red-rimmed with tears, our ears throbbing, our brains numbed and throbbing as well.\n\n\"Is that enough for the moment?\" I asked sweetly and my answer was a chorus of groans. \"Right. On the way in here I noticed that right next door is a drinking parlor by the name of Dust on Your Tonsils. I can only assume that is a little joke and they intend to wash the dust from their clients' tonsils. Shall we see if that is true?\"\n\n\"Let's go!\" Floyd said and led the exodus.\n\n\"A toast,\" I said when the drinks had arrived. We lifted our glasses. \"To The Stainless Steel Rats\u2014long may they play!\"\n\nThey cheered and drank, then laughed and called for another round. It was all going to work out hunky-dory I thought.\n\nThen why was I so depressed?\nChapter 5\n\nI was depressed because it was really a pretty madcap plan. The idea had been to allow a week for our publicity to peak, for some musical awards to be made\u2014then the crime had to occur. In that brief period we were not only going to have to find some music, but we would have to rehearse the stuff and hopefully gain at least a moderate level of ability. Some chance. We were cutting it too fine. We needed some more help.\n\n\"Madonette, a question.\" I sipped some more beer first. \"I must admit to an abysmal ignorance of the mechanics of making music. Is there someone who sort of makes up the tunes, then writes down the stuff that everyone is going to play?\"\n\n\"You're talking about a composer and an arranger. They could be one and the same\u2014but it is usually better to divide up the jobs.\"\n\n\"Can we get one or both of them? Zach, as the closest thing to a professional here\u2014do you have any ideas?\"\n\n\"Shouldn't be too hard. All we have to do is contact GASCAP.\"\n\n\"Gascap? You want to fill the tank on a groundcar?\"\n\n\"Not gascap. GASCAP. An acronym for the Galactic Society of Composers Artists and Players. There is a lot of unemployment in music and we should be able to locate some really competent people.\"\n\n\"Good as done. I'll get the Admiral on it at once.\"\n\n\"Impossible,\" he growled in his usually friendly fashion. \"No civilians, no outsiders. This is a secret operation all the way.\"\n\n\"It is now\u2014but it goes public in seven days. All we do is invent a cover story. Say that the group is being organized to make a holofilm. Or as a publicity stunt by a big firm. Like maybe Macswineys wants to change their image, go upmarket. Get rid of Blimey Macswiney and his alcoholic red nose, use our pop group instead. But it must be done\u2014and at once.\"\n\nIt was. The next day an anorexic and pallid young man was brought to our rehearsal studio. Zach whispered in my ear. \"I recognize him\u2014that's Barry Moyd Shlepper. He wrote a pop musical a couple of years back, 'Don't Fry for Me, Angelina.' He hasn't had a success since.\"\n\n\"I remember it. The show about the cook who marries the dictator.\"\n\n\"That's the one.\"\n\n\"Welcome, Barry, welcome,\" I said walking over and shaking his bony hand. \"My name is Jim and I'm in charge around here.\"\n\n\"Rooty-toot, man, rooty-toot,\" he said.\n\n\"And a rooty-toot to you as well.\" I could see where we would have to learn the argot of the musical world if our plan were to succeed. \"Now\u2014was this operation explained to you?\"\n\n\"Like maybe sort of. A new recording company starting up with plenty of bucknicks to blow. Financing some new groups to get the operation off the ground.\"\n\n\"That's it. You're in charge of the music. Let me show you what we have and you put it into shape.\"\n\nI gave him earphones and the player; I couldn't bear listening to these dreadful compositions yet another time. He plugged in the cubes one by one and, impossible as it was to believe, his pallid skin grew even paler. He worked his way through them all. Sighed tremulously, took off the earphones and brushed the tears from his eyes.\n\n\"You want like my honest and truly opinion?\"\n\n\"Nothing less.\"\n\n\"Well then, like to break it to you gently, this stuff really sucks. Insufflates. Implodes.\"\n\n\"Can you do better?\"\n\n\"My cat can do better. And scratch dirt over it.\"\n\n\"Then you are unleashed. Begin!\"\n\nThere was little else I could do until the music was written, rehearsed, recorded. While all the others would play their instruments and sing, my work would be limited to throwing the switch before each piece. Then all of Zach's drums, cymbals, horns, bells and molecular-synthezier effects would burst forth from the loudspeakers in full gallop. While this was happening I would throw switches that did nothing, tinkle the keys on a disconnected keyboard. So while they got the music going I looked into the special effects.\n\nThis required watching recordings of all of the most popular groups, bands and soloists. Some of it was enjoyable, some horribly dreadful, all of it too loud. In the end I turned off the sound and watched the laser beams, exploding fireworks and physical acrobatics. I made sketches, mumbled to myself a lot, spent a great deal of the university's money.\n\nAnd built an incredible amount of complicated circuitry into the existing electronics. Reluctantly, the Admiral produced the extras I asked for and I modified everything in the machine shop. It was altogether a satisfactory and fulfilling week. I also prodded the Admiral until he produced the promised payment of three million credits.\n\n\"Most kind,\" I said, jingling the six glowing five-hundred-thousand-credit coins. \"A decent fee for a decent job done.\"\n\n\"You better put them in a bank vault before they go missing,\" was his surly advice.\n\n\"Of course. A capital idea!\"\n\nA singularly stupid idea. Banks were for robbing and for the tax authorities to keep track of. So first I went into the machine shop where I did some crafty metalwork before I packed, wrapped and labeled the coins. Then I went for a walk and, as a precaution, I exercised all of my considerable talents at avoiding observation to shake off any possible tails the Admiral had put on me. I was risking my life\u2014in more ways than one!\u2014for this money. If I came out of it all in one piece I wanted to have it waiting.\n\nI finally reached a small country post office, selected at random, some distance from the city. It was manned by a nearsighted gentleman of advanced years.\n\n\"Spatial express and insured for offplanet delivery. That ain't gonna be cheap young feller.\"\n\n\"Do it, daddy-o, do it. I've got the gilt.\" He blinked and I translated back to his native language. \"Payment is not a problem, dear sir. You must assure me that this gets to Professor Van Diver at the Galaksia Universitato at once. He is expecting these historical documents.\"\n\nI had already spaciofaxed the professor that I was sending him some personal possessions, that he should please hold on to them until I came and picked them up. In case he got curious the contents were sealed in an armored case that would take a diamond drill to open. I was betting that his curiosity would not go that far. My package vanished into the mail chute and I went back to work.\n\nAt the end of the sixth day we were all pretty exhausted. Barry Moyd Shlepper had stayed up for two nights running, cold towels wrapped around his head, fortified by trebcaff coffee, putting together some musical numbers from the archaic junk. He proved to be a good hand at theft\u2014or adaptation as he liked to call it. The group had rehearsed, recorded, then rehearsed some more. I had concentrated on costumes, props and effects and was almost satisfied.\n\nAfter one last break I called my troops together. \"You will be pleased to know that we will now give our first public performance.\" This produced the expected groans and shrill cries of complaint and I waited until they had died down.\n\n\"I know how you feel\u2014and I feel the same way too. I think that the blues number 'I'm All Alone' is our best piece. You know we have had a lot of help from the staff here and I think we owe it to them to see what we all have done. I've invited something like thirty of them and they should be here soon.\"\n\nRight on cue the door opened and the suspicious public employees filed in, each carrying a folding chair. Admiral Benbow led the way, his flag officer carried two chairs. Zach supervised the seating arrangements and our cavernous rehearsal studio became a theater for the first time. We retreated to the podium, where I dimmed the house lights, then hit myself and my electronic gear with a baby spot.\n\n\"Ladies, gentlemen, guests. We have all worked hard this last week and in the name of The Stainless Steel Rats I would like to thank you.\"\n\nI hit a switch and my amplified voice echoed Thank You, Thank You. Overlaid by a growing crescendo of drums and ending with a crack of thunder and a few realistic lightning bolts. I could see by their wide eyes and dropped jaws that I had their attention.\n\n\"For our first number the melodious Madonette will render heart-rendingly the tragically lonely\u2014'I'm All Alone'!\"\n\nAt this the colored kliegs burst down on us, revealing our pink-sequined skintight costumes in all their iridescent glory. As we played the opening bars of the theme the lights concentrated on Madonette, whose costume had more flesh than fabric and seemed to be deeply appreciated. After a last whistle of wind and crash of thunder and lighting she extended her lovely arms to the audience and sang:\n\nHere I am\u2014and I'm all alone\u2014\n\nNo one calls on the telephone.\n\nI look around\u2014and what do I see?\n\nThere's no one here but me\u2014me\u2014me.\n\nMe\u2014me\u2014me\n\nThat's all I see\u2014\n\nI'm all alone\n\nJust\n\nme\n\nme\n\nme...\n\nThis was all done to the accompaniment of holographic shaking trees, storm clouds and other spooky effects. The music wailed as Madonette seguidillad into the rest of the song.\n\nI'm all alone and it's very dark\u2014\n\nI sneak out the window to the park.\n\nThe wind blows hard and the tree limbs wave\u2014\n\nAnd I'm right before an open grave!\n\nWhen I try to run and try to flee\u2014\n\nBut I KNOW they're out there after me!\n\nI sit and cry and I know that's right\u2014\n\nBecause the sun comes up\u2014\n\nIt's the end of the night...\n\nWith a last wail and a writhe of purple fog the sun rose majestically behind us and the music trickled to an end.\n\nThe silence stretched and stretched\u2014until it was finally broken by a tumultuous applause.\n\n\"Well gang,\" I said, \"it looks like we have done it. Or as Barry Moyd says, it looks like we are but really rooty-getooty!\"\n\nOn the seventh day we did not rest. After a final round of rehearsals I called an early break. \"Get some rack time. Pack your bags. The music and props are ready to go. We ship out at midnight. Transportation to the spaceport leaves here an hour earlier\u2014so don't be late.\"\n\nThey shuffled out wearily with dragging feet. The Admiral stamped in as they left, with Zach trailing in his wake.\n\n\"This agent informs me that all preparations have been made and you are ready to embark.\" I could only nod agreement.\n\n\"Wish I could go with you,\" Zach said.\n\n\"You set it all up\u2014you have our thanks for that. Now get going.\"\n\nHe numbed my fingers with his handshake and the door closed behind him.\n\nThe Admiral's smile had all of the warmth of a striking snake. \"Drug Enforcement has come up with a crime so awful that it means an instant sentence to Liokukae.\"\n\n\"That's nice\u2014what is it?\"\n\n\"Misuse of a highly refined and expensive drug called baksheesh. You and the rest of the musicians have been caught smuggling it and are addicted to it. There is a medical cure for the addiction that leaves the victim weak and vibrating for a number of days. This should give you a little time to look around before you have to play your first concert. The press release has already gone out about your capture and your sentence to prison hospital for the criminally doped. The natives of Liokukae will not be surprised at all when you arrive there. Questions?\"\n\n\"A big one. Has the communication been set up?\"\n\n\"Yes. The coded radio built into your jaw can reach the receiver at the entrance terminal from any place on the planet. It will be manned all of the time and an officer will be listening in on all communication. Your contact on the ground will give you what aid he can before you go out of the sealed terminal. Then he will move to the spacecruiser Remorseless in orbit above, which will also monitor your radio. We can hit anywhere on the planet in a maximum of eleven minutes. Send the signal when you have found the artifact and the space marines will be there. Report at a minimum of once a day. Location and results of your investigation.\"\n\n\"Just in case we get blown away and you have to send in the second team?\"\n\n\"Exactly. More questions?\"\n\n\"One. Going to wish us luck?\"\n\n\"No. Don't believe in it. Make your own.\"\n\n\"Gee, thanks, you really are all heart.\"\n\nHe turned and stamped away and the door swung shut behind him. Fatigue washed through me and black depression hit just one more time. Why was I doing this?\n\nTo stay alive of course. Twenty-two days more before my curtain fell for the final performance.\nChapter 6\n\nThe Faster Than Light voyage aboard the good ship Remorseless was blessedly brief. Being surrounded by the military has always had a deleterious effect on my morale. We had a solid day of rehearsal, some bad food, a good night's rest, followed the next day by a very nonalcoholic party\u2014since the Navy was remorselessly teetotal. Then, a few hours before we were to meet the shuttle, the medics gave us the injections that were to simulate the aftereffects of our drug treatment.\n\nI think I would have preferred the treatment. I didn't mind seeing my last meal go by for a second time, it had been pretty bad and I would not miss it. But the shakes and shivers were something else again. And all of my vibrating and stumbling co-musicians had eyeballs as red as fire. I dared not look in the mirror for fear of what I would see there.\n\nSteengo was gray and drawn and looked a hundred years old. I felt a quick blast of guilt for dragging him out of retirement. Said guilt fading instantly when I thought about my own problems.\n\n\"Do I look as bad as you do?\" Floyd said in a hoarse voice, his new-grown beard black against his parchment skin.\n\n\"I hope not,\" I husked in return. Madonette reached over and patted my shaking hand in what might have been a maternal way.\n\n\"It will be all right on the night, Jim. Just you wait and see.\"\n\nI did not feel filial in return since I was rapidly developing a crush on her that I hoped I disguised. I growled something or other and stumbled away to the heads where I could be alone with my misery. Even this did not work for the speaker in the ceiling rustled ominously\u2014then crashed out Admiral Benbow's voice.\n\n\"Now hear this. All Stainless Steel Rats will assemble at debarkation station twelve in two minutes. We are now in parking orbit. One minute and fifty-eight seconds. One minute and...\"\n\nI slammed out into the passageway to escape his voice but it followed me as I fled. I was the last to arrive and I collapsed and joined the others where they slumped on the deck beside our backpacks. The Admiral appeared suddenly behind me like a bad dream and roared his command.\n\n\"Attention! On your feet you slovenly crew!\"\n\n\"Never!\" I shouted even louder in a cracked voice. Rolling over to pull the swaying bodies back to the deck.\n\n\"Begone foul military fiend! We are musicians, civilians, medically reformed drug addicts and we must think and feel that way. Someday, if we live, you may have some of us back at your military mercy. But not now. Leave us in peace and wait for my reports.\"\n\nHe snarled a rich naval oath\u2014but had the brains to turn on his heel and vanish. There was a ragged cheer from my companions which made me feel slightly less sordid. The silence after this was unbroken, except for the occasional groan, until distant motors whirred and the inner lock swung majestically open. A keen clipboard-bearing naval officer stepped through.\n\n\"Landing party for Liokukae?\"\n\n\"All present, all ill. Send a working party for our gear.\"\n\nHe muttered into his lapel microphone, reached to the back of his belt to unclip a pair of handcuffs. Which he promptly snapped onto my wrists.\n\n\"Whasha?\" I blurted incoherently. Blinking down at the cuffs.\n\n\"Don't give me a hard time, you drug-pushing addict, and I won't give you one. You may be a big man out there in the galaxy, but here you are just one more sentenced crook. Who is going to carry his own pack\u2014no working party for the likes of you.\"\n\nI opened my mouth to verbally assassinate him. Then closed it. It had been my idea that our mission be known to the minimum few. He obviously wasn't one of them. I groaned to my feet and stumbled into the airlock dragging my gear after me, the others following in like condition. The orbital shuttle ship was grim and cheerless. The hard metal seats snapped clamps on our ankles when we sat down; no dancing in the aisles this trip. We watched in silence as our backpacks were thrown into a storage bin, then looked up at the big screen on the front bulkhead. Lots of stars. They rotated and the bulk of Remorseless swam into sight, grew smaller and dropped behind as the engines fired. Then the pickup turned so that the growing bulk of the planet could be seen and we were treated to a scratchy and static-filled ancient recording of martial music. This died away and was replaced by a male speaker with a repulsive nasal whine.\n\n\"Now hear this, prisoners. This is a one-way trip. You will have resisted all efforts at adjustment that would have fitted you to live peacefully in our humane and civilized society...\"\n\n\"Blow it out your rocket tubes!\" Steengo snarled, running his fingers through his gray hair, perhaps to see if it was still there. I would have nodded agreement with his snarl only my head hurt too much.\n\n\"... brought upon yourselves by your own efforts. Upon landing you will be escorted by armed guards to the gates of the landing station. Your restraints will be removed and you will be given an orientation booklet, a canteen of distilled water, as well as a week's supply of concentrated survival rations. During that week you will look for small trees bearing hard fruit. These are the polpettone trees and a source of nourishment for all. Their fruit is the result of careful gene mutation and transplant, rich in animal protein. They should not be eaten raw because of the chance of trichinosis, but should be baked or boiled. You must remember...\"\n\nI wanted to remember nothing he said so I tuned him out. I tried to reassure myself that the normal condemned passenger on this flight must have done something pretty gruesome to deserve this fate. I wasn't convinced. Despite millennia of civilization man's inhumanity to man persisted whenever an opportunity presented itself.\n\nThe imaged clouds blew by and a massive five-sided building appeared on the screen. I supposed they called it the Pentagon.\n\n\"In a few moments we will be landed inside the walls of the Pentagon debarkation station. Remain seated until you are ordered to rise. Follow instructions and your passage will be made that much easier...\"\n\nI would like to make his passage easier! Then I relaxed and opened my fists. Very soon we would be away from weary wardens and on our own. That was the moment to be prepared for.\n\nWe shuffled out in silence, down the gangway\u2014which surely should have been a gangplank\u2014and into the thick-walled Pentagon. To be greeted by yet another naval officer, grim-faced and gray-haired, wearing dark glasses.\n\n\"Take the prisoners to Interview nine at once.\"\n\nThe petty officer of our guard protested. \"Not regulation, sir. They have to\u2014\"\n\n\"You have to close your gob. Look at these orders. Do as instructed. You do enjoy being a petty officer?\"\n\n\"Yes, sir! Prisoners this way!\"\n\nThe officer came in after us, closed and locked the door, smiled at us warmly and said \"Shut up\" companionably. He then walked around the room with what I recognized as being a state-of-the-art communication detector. I couldn't imagine who would want to bug the room here at the end of the universe\u2014but he was in charge. Satisfied he put the detector away and turned to face us and handed me a key.\n\n\"You can take the cuffs off while we are in this room. I am Captain Tremearne and I am your contact here. Welcome to Liokukae.\" He took off the dark glasses and smiled at us and waved us to the chairs. I could see now that a wicked scar slashed across his face and the bridge of his nose. He was blind. But could undoubtedly see fine with the electronic replacement eyes that had been fitted. They were gold-plated and gave him a highly interesting appearance.\n\n\"I am the only one here in the Pentagon who knows the real nature of your assignment here. You are all volunteers and I would like to thank you. Help yourself to refreshments because that is the last kind word you are going to hear for quite a while.\"\n\n\"What is it like out there?\" I asked, touching the seal on a chilled container of beer and taking a life-reviving swig. There were fresh sandwiches and hot swinedogs there as well and my companions all dived in. I joined them, but not before I had opened a concealed drawer in my synthezier and taken out some necessary items.\n\n\"What's life like on this planet? Grim\u2014and worse than grim, Jim. In the centuries that Liokukae has been used as a societal galactic wastebin there has been a rather deadly shaking down. Different cultures were formed here as like found like. Or violent men forced violent solutions upon weaker men. One of the most stable of these has been developed right outside the Pentagon. They call themselves the Machmen. Man is strong, woman weak, virility rules, strength through strength\u2014I'm sure that you know the kind of thing. The top dog in this kennel, whom I am sure you will be meeting soon, is named Svinjar.\"\n\n\"Are these weirdos what the psych books call male chauvinist pigs?\" I asked. He nodded.\n\n\"Absolutely correct. So do your best to keep Madonette out of sight. And practice walking on your toes and flaring your nostrils at the same time. If you can't think of anything else to do crook your arm and admire your biceps.\"\n\n\"Sounds a paradise,\" she frowned.\n\n\"Won't be too bad if you watch your step. They like to be entertained\u2014since they haven't enough brains to entertain themselves. Very big on jugglers, duels, arm wrestling.\"\n\n\"What about music?\" Steengo asked.\n\n\"Fine\u2014as long as it is loud, martial and not sentimental.\"\n\n\"We'll do our best,\" I said. \"But it is a group called the Fundamentaloids that I want to look for.\"\n\n\"Of course. As you have been told the spacer with the archeological expedition landed in their area of operation. I led the rescue party that took the expedition members out of here\u2014which is why I am your contact now. The Fundamentaloids are nomads, as well as being pretty narrow-minded and obnoxious. I tried to keep things calm with them. Didn't work. In the end I narcgassed the lot and went in and pulled the scientists out. I didn't find out about the missing artifact until much later when we were offplanet and they were conscious again and the excitement had cooled down. By this time the group that had grabbed them had moved on and the trail got cold. Nothing more I could do at the time but report it. It's all in your hands now.\"\n\n\"Thanks much. Can't you at least point out to me on the map where they are?\"\n\n\"Wish I could\u2014but they're nomads.\"\n\n\"Wonderful.\" I smiled insincerely. Twenty days to deadline. Deadline! it would be. I shook off the dark feelings just one more time, looked around at my band.\n\n\"Ask questions if you have any because this is your last chance,\" Tremearne said.\n\n\"Do you have a map?\" I asked. \"I would like to know just what we have to face when we go out there.\"\n\nTremearne reached to the holo projector and switched it on. A three-dimensional contour map appeared in midair over the table. \"This is a fair-sized continent as you can see. There are other continents on this planet, some inhabited, but they have no contact with this one. The artifact has to be somewhere here.\"\n\nThat really simplifies things, I thought to myself. Only one continent to search and about three weeks to do it in. I shook off the depression that was depressing my depression.\n\n\"Do you know who and what are out there?\"\n\n\"We have a good idea. We plant bugs where we can, fly spyeyes pretty often.\" He tapped the plain at the center of the continent. \"Here is the Pentagon with the Machmen close by outside it. The Fundamentaloids could be anywhere here on the plains, depending on the season. It is subtropical most of the year, but rainfall varies. They have herds of sheots, a very hardy ruminant, some kind of cross between a sheep and a goat. Now over here in the foothills is the closest thing that passes for civilization in these parts. An agricultural society with light industry that looks almost decent until you get close. There is a central city, right here, surrounded by farms. They mine and smelt silver and produce a coin called a fedha. It is the only currency on the planet and is used by almost everyone.\" He pulled a heavy bag out of a drawer and dropped it onto the table. \"As you can well imagine they are easy enough to forge. In fact ours have more silver than the originals. Here's a supply for you. I suggest that you share it around and hide it well. A lot of types out there would be happy to kill you for just one of these. The people who mine the silver call their city Paradise\u2014which is about as far away from a true description as you can get. Stay away from them\u2014if you possibly can.\"\n\n\"I'll try to remember that. And I want to copy this into memory in my computer. Here.\"\n\nI took off the small black metal skull that hung on a chain around my neck. When I squeezed it the eyes glowed greenly and a pressure-sensitive holoscreen blinked into being; I copied the map, thought about what Tremearne had said\u2014and realized for the first time what a sinkhole we were being dropped into. I had another question.\n\n\"So everyone out there is a nutcase or a weirdo of some kind?\"\n\n\"The ones that were sent here for various crimes are. The ones who were born here grow up and fit in just as well.\"\n\n\"And you feel no compassion for them? Doomed by an accident of birth to existence in this worldwide spittoon.\"\n\n\"I certainly do\u2014and I am glad to hear you express yourself so clearly on the subject. I never even heard of this world until the emergency. I got the professors off safely then looked around. Which is why I now head the committee that is working to clean up the operation here on Liokukae. It has been ignored for too long by too many stupid politicians. I took this assignment to see for myself. Your reports to me, along with your complete report when you return, will be just what we need to make this prison world a thing of the past.\"\n\n\"If you mean that, Captain, I'm on your side. But I hope you are not feeding me a line of old cagal just to get the job done.\"\n\n\"You have my word on it.\"\n\nI sure hoped that he was telling the truth.\n\n\"I have a question,\" Floyd said. \"How do we contact the Captain here if we need some help or such?\"\n\n\"You don't\u2014I do.\" I tapped my jaw. \"I've got a microcommunicator implant here. Small enough to be powered by the oxygen in my blood. But powerful enough to be picked up by the big receivers in the Pentagon. So even if all of our goods are stolen\u2014they can't get my jaw. So, I suggest strongly, we stick together at all times. I can talk with Tremearne through this thing, get suggestions and advice. But no physical contact or our cover is blown. If he has to pull us out the mission is over\u2014whether we have the artifact or not. So let us be strong, guys and girl, and self-sufficient. It's a human jungle out there.\"\n\n\"No truer words ever spoken,\" Tremearne said grimly. \"If no one else has any questions put the cuffs back on and you're out of here.\"\n\n\"Hell yes,\" Steengo said, climbing to his feet. \"Let's get it over with.\"\n\nOur packs were waiting for us in front of a massive and bolt-studded door. There were four shoddy little plastic bags as well, which probably contained our iron rations and water. An orientation booklet was tucked into each one. A backup force of guards with stun guns and porcuswine prods stomped up and glared obnoxiously while our manacles were removed.\n\n\"In there,\" the petty officer ordered, pointing to the anteroom in front of the exit portal. \"Inner door is closed and sealed before the outside one opens. You got only one way to go. Or stay in the room if you are tired of living. After five minutes the outer door closes and nerve gas is pumped in through those vents up there.\"\n\n\"I don't believe you!\" I snapped.\n\nHis smile was without warmth. \"Then why don't you just hang around and find out?\"\n\nI raised my fist and he hurriedly jumped back. The porcuswine prods sparkled in my direction. I raised my finger to them in the intergalactic gesture that is as old as time, turned and walked away from them following the others. There was a creak and a thud from behind us as the door swung shut, but I did not turn to look. The future, whatever it contained, lay just ahead.\n\nWe helped each other on with our packs, swaying dizzily with the effort. There was the thud of withdrawn bolts from inside the door, the growl of straining motors as it started to open.\n\nUnconsciously we drew together as we turned to face the unknown.\nChapter 7\n\nA splatter of rain blew in through the opening door. Welcome to sunny, holiday Liokukae. Which opened wider to reveal the group of very ugly-looking individuals who were waiting outside. They were dressed in an astounding variety of clothing\u2014it looked like all the donations to charity in the entire galaxy had been sent here\u2014and they all had two things in common. They were heavily armed with a mixture of clubs, swords, maces and axes. And they all looked very angry.\n\nJust about what I had expected; I chomped down on the Blastoff capsule I had put in my mouth. I had never thought much of the weaklings-recovering-from-treatment plan and had palmed this pill in case it were needed. It was.\n\nA wave of energy and power washed through me as the mixture of powerful chemicals, uppers, stimulants, adrenalins, swept away all the fatigue and shakes. Power! Power! Power! I swayed forward on tiptoes as Tremearne had advised, flaring my nostrils at the same time.\n\nA great bearded lout swinging a crude but serviceable sword glared down at me. I glared back, noting that not only did his eyes meet in the middle but that his hairline also started at his eyebrows. When he shouted at me his breath frightened me more than he did.\n\n\"You dere, little boy. Gimme what you carrying. You all drop what you got or you get it.\"\n\n\"No one tell me what to do unless he can beat me, you illiterate cretin,\" I shouted back. The macho showdown with these macho mothers would have to take place sooner or later. Sooner was better.\n\nHe roared angrily at the insults, even though he could not understand them, and swung up the sword. I sneered.\n\n\"Big coward kill little man with sword when little man got no ax.\" I gave him two fingers to doubly amplify my feelings.\n\nI hoped my simple syntax fitted the local linguistic profile because I wanted to make sure they all understood me. They must have, because Pigbreath dropped his sword and jumped towards me. I swung off my pack and stepped out into the mud. He had his arms out, fingers snapping, ready to grab and crush.\n\nI ducked under them, tripped him as he went by to splat down into a puddle. He rose up, angrier than ever, balled his fists and came on more warily this time.\n\nI could have finished it then and there and made life easier. But I had to display a bit of skill first so his mates wouldn't think that his downfall had been an accident. I blocked his punch, grabbed and twisted his arm, then ran him into the wall with a satisfactory crunch.\n\nThe blood from his nose did not improve his temper. Nor did my flying kick that numbed one of his legs, a stab with my knee that crumpled the other. Legless, he dropped to his knees, then crawled towards me on all fours. By this time even the dullest of the audience knew who had won this fight. So I grabbed him by the hair and pulled him up, hit his throat with the edge of my hand and let him keep going backwards, splatting down unconscious in the mud. I picked up his discarded sword, tested the edge with my thumb\u2014jumped about so suddenly and menacingly that the armed men stepped away without thinking. I kept the momentum going.\n\n\"I got sword now. You want it, you die for it. Or maybe the smart bloke one of you what takes me to your boss, Svinjar. Guy what does dat gets this sword for free. Any takers?\"\n\nThe novelty of the offer and their inherent greed warded off any attack for the moment.\n\n\"Get out of there and get behind me,\" I called over my shoulder. \"And do your best to radiate obnoxious intolerance.\" Growling and gnashing their teeth my merry band emerged and lined up at my back.\n\n\"You give me sword I take you Svinjar,\" an exceedingly hairy and musclebound specimen said. He was armed only with a wooden club so his greed was understandable.\n\n\"You take me Svinjar then you get sword. Move it.\"\n\nThere was hesitation, dark looks, muttering. I swished the sword under their noses so they had to step back again. \"I got something real nice in my pack for Svinjar. You betcha he kill any bloke stop him grabbing it soonest.\"\n\nThreats penetrated where blandishments hadn't and we all moved off into the rainstorm. Along muddy tracks between collapsing hovels, to a small hill with a largish building made of logs, their bark still on, gracing the summit. I swung the sword so no one came too close, followed my guide up a stony path to the entrance with my weary musicians stumbling after. I was feeling a bit guilty about taking the Blastoff capsule. But things had developed too quickly to get some to the others. I stopped at the entrance and waved them through.\n\n\"In we go, safe haven at last. Take one of these as you pass and chomp it instantly. It is a super-upper that will restore you to the world of the living.\"\n\nMy club-bearing guide pushed inside and hurried past the groups of men who lolled about the large room, to the man in the great stone chair next to the fireplace. \"You my boss, boss Svinjar. We bring them like you say.\" He swung about and stamped over to me. \"Now you give sword.\"\n\n\"Sure. Fetch.\"\n\nI threw it out the door into the rain, heard a yipe of pain as it bounced off one of his gang. He ran after it as I walked over and stood before the stone throne.\n\n\"You my boss, boss Svinjar. These guys my band. Make good music you betcha.\"\n\nHe looked me up and down coldly, a big man with big muscles\u2014as well as a big belly that hung over his belt. Tiny piggy eyes peered out through the thicket of bristly gray hair and beard. The pommel of a sword projected from a niche in the stone chair and he touched it with his fingers, slipping it out then letting it fall back.\n\n\"Why are you talking in that obnoxiously obscene patois?\"\n\n\"I do beg your pardon.\" I bowed deprecatingly. \"I was addressed in that manner and assumed it was the local dialect.\"\n\n\"It is\u2014but only among the uneducated imbeciles who were born here. Since you weren't, don't offend my sensibility again. Are you the musicians that got into deep cagal?\"\n\n\"Word sure spreads fast.\"\n\nHe waved his hand at the 3D set against the wall and I felt my eyes bulge. It was a solid metal block with an armored glass face\u2014with the aerial under the glass. A handle stuck out one side.\n\n\"Our jailers are most generous in their desire that we be entertained at all times. They distribute these in great numbers. Unbreakable, eternal\u2014and four hundred and twelve channels.\"\n\n\"What powers it?\"\n\n\"Slaves,\" he said and reached out a toe to prod the nearest one. The slave groaned and climbed to his feet, stumbled over, clanking his chains as he went, and began to turn the handle on the internal generator. The thing burst to life with a commercial for industrial-strength cat food.\n\n\"Enough!\" Svinjar ordered and the meows faded and died. \"You and your companions kept the news channels alive. When they said crime and hospital treatment I was rather convinced they meant here. Ready to play?\"\n\n\"The Stainless Steel Rats are always at the service of those in control. Which, in this case, I assume is you.\"\n\n\"You assume right. A concert it is\u2014and now. We haven't had any live entertainment here since the cannibalistic magician died of infection after being bitten by accident in the heat of passion. Begin.\"\n\nBy necessity all our gear had to be compact. The fist-sized loudspeakers contained holoprojectors that blew their image up to room size.\n\n\"All right guys,\" I called out. \"Let's set up by the back wall. No costumes for this first gig and we'll start with 'The Swedish Monster from Outer Space.'\"\n\nThis was one of our more impressive numbers. It had been found in one of the most ancient data bases, the lyric written in a long-lost language called Svensk or Svedish or something like that. After much electronic scratching about, one of the computers in the language department at the university had been able to translate it. But this lyric was so dreadful that we threw it away and sang it in the original, which was far more interesting.\n\nEtt fasanfullt monster med rumpan bar\n\nkryper in till en jungfru sa rar.\n\nThere was more like this and Madonette belted it out at full volume to the accompaniment of my syncopated soundtrack, with Floyd knocking himself out on his blower-powered bagpipe. Steengo plucked at a tiny harp\u2014whose holographic image stretched up to the ceiling. Sound filled and reverberated through the great chamber and dust was jarred loose from the log walls.\n\nI don't think that this tune would make the galactic top ten\u2014but it sure went down well here in endsville. Particularly when it ended with an atomic mushroom cloud that grew to room size\u2014along with the best the amplifiers could do to simulate the atomic explosion itself. The part of the audience that wasn't collapsed on the floor had fled shrieking into the rain. I took out my earplugs and heard the light clapping of approval. I bowed in Svinjar's direction.\n\n\"A pleasant divertimento\u2014but the next time you play it I would appreciate a little less forza in the finale and a little more riposo.\"\n\n\"Your slightest wish is our command.\"\n\n\"For a young and simple-looking lad you learn fast. How come you were caught pushing drugs?\"\n\n\"It's a long story\u2014\"\n\n\"Shorten it. To one word if possible.\"\n\n\"Money.\"\n\n\"Understandable. Then the music business isn't that good?\"\n\n\"It smells like one of your bully-boys. If you can stay up there with the big ones, fine. But we slipped from the top notch some time ago. What with recording fees, agents' commissions, kickbacks and bribes we were quickly going bust. Steengo and Floyd have been snorting back baksheesh for years. They started selling it to support the habit. It's nice stuff. End of story.\"\n\n\"Or beginning of a new one. Your singer, what's her name?\" He smiled a very unwholesome smile as he looked over at Madonette. I groped for inspiration. Came up with the best I could do at such short notice.\n\n\"You mean my wife, Madonette...\"\n\n\"Wife? How inconvenient. I am sure that something can be done about that, though not exactly at this moment. Your arrival is, to say the least, most timely. Fits in with what you might call a general plan of action I was considering. For the general good of the populace.\"\n\n\"Indeed,\" I said, controlling my enthusiasm for any plan of his that might be forthcoming.\n\n\"Yes, indeed. A concert for the public. Barbecues and free drinks. The public will see Svinjar as a benefactor of the first order. I gather that you are prepared to play a benefit performance?\"\n\n\"That's what we are here for.\"\n\nAmong other things that we are here for, Svinjar old chubkin. But the longest journey begins with but a single step.\nChapter 8\n\n\"I'm not happy about the way this operation is going,\" I said unhappily. Spooning up the almost tasteless gruel that appeared to be the staff of life in this place.\n\n\"Who's arguing?\" Steengo said, looking suspiciously into his own bowl of food. \"This stuff not only looks like glue\u2014it tastes like it.\"\n\n\"It will stick to your ribs,\" Floyd said and I gaped. Did he have a sense of humor after all? Probably not. Looking at his serious expression I doubted if he had explored all the meanings of what he had just said. I let it lie.\n\n\"I'm not only unhappy with this operation so far\u2014but with the company we have been keeping. Svinjar and his loathsome lads. We've shot almost a day here already\u2014for little purpose. If the artifact is with the Fundamentaloids we ought to be out there tracking them down.\"\n\n\"But you promised a concert,\" Madonette pointed out with a certain logic. \"They are building a sort of bandstand and the word has gone out. You don't want to let our fans down, do you?\"\n\n\"Heaven forbid,\" I muttered gruelly and put the bowl aside. I couldn't tell them about the thirty-day poison or the fact that as of the moment twenty days had passed. Oh the hell with it. \"Let's get set up. Maybe a quick rehearsal to see if all the gear is working and, hopefully, we are still in good form.\"\n\nWe put lunch aside with a great deal of pleasure and humped our packs to the concert site. There was a grove of trees here that were serving as supports for a singularly crude platform. Planks had been set up between them, with an occasional support stuck in below if the thing sagged too much. Our audience was reluctantly and suspiciously gathering in the surrounding field. Small family units with the men all armed with swords or cudgels, keeping close watch on the womenfolk. Well, this was a slave-holding society so such concern was easily understood.\n\n\"At least they are trying to make it look nice,\" Madonette said, pointing. Pretty crude and crummy, I thought, but spoke not my thoughts aloud. Shuffling slaves had brought up leafy branches which they were arranging around the platform, there were even a few flowers stuck in among the leaves. Oh, things were really swinging on Liokukae tonight.\n\nI was depressing myself sorely and did not want to pass it on to the others. \"Here we go, gang!\" I said swinging my pack up onto the platform and clambering behind it. \"Our first live performance for this waiting world. If you don't count that quick gig upon arrival. Let's show them what a pack of real rats can do!\"\n\nWith our appearance the assembling audience took heart and moved closer; latecomers hurried to their places. While we tuned up and played a riff or two, I rolled some thunder effects that had people looking at the sky. When we were ready to go, Svinjar himself came trundling through the crowd, a couple of armed heavies at his side. With their help he climbed onto the platform and raised his arms. The silence was total. Maybe it was respect, perhaps hatred and fear\u2014or all of them rolled together. But it worked. He smiled around at the gathering, lifted his great gut so he could hook his thumbs into his belt. And spoke.\n\n\"Svinjar takes care of his people. Svinjar is your friend. Svinjar brings you The Stainless Steel Rats and their magic music. Now let us hear a big cheer for them!\"\n\nWe got a big murmur, which had to do. While he had been speaking his bully-boys had manhandled a sizable padded chair up onto the platform, it creaked when he dropped into it.\n\n\"Play,\" he ordered and sat back to enjoy the music.\n\n\"Okay, gang, ready to go!\" I blew into my lapel microphone and my amplified breath gusted across the audience. \"Well, hello there music lovers. By popular appeal\u2014and the fact that we were busted by the narcs\u2014we have come to your sunny planet to bring you the music known right around the galaxy. It is our very great pleasure now to dedicate this next song to the concert master himself, Svinjar\u2014\" He nodded acceptance and I rolled a drumroll out across the surrounding fields.\n\n\"A song that you will all know, and hopefully love, something that we can all feel, share, enjoy together, laugh together and cry together. I bring you our own and original version of that classic of modern musicality\u2014'The Itchy Foot Itch'!\"\n\nThere were shouts of joy, screams of pain, wild enthusiasm. As we launched into this overamplified and very catchy\u2014if not itchy\u2014number.\n\nI get up at dawn and look at the river\n\nThe mist rising there it gives me a shiver.\n\nLeaves on the trees they're wet with dew\n\nLooking at them I think of you\u2014\n\nFar far away from me today\n\nI don't like it\u2014but all I can say\n\nIs the galaxy's wide and I like to stray\n\nTo the stars and beyond 'cause that's my way\n\nI got the\u2014\n\nItchy foot, itchy foot, itchy foot itch!\n\nGotta keep going, never get rich!\n\nItchy foot, itchy foot, itchy foot itch!\n\nKeeping me going, ain't that a bitch!\n\nItchy foot, itchy foot, itchy foot itch!\n\nKeeping me going from place to place\n\nGotta keep going, what can I do?\n\nKeep going forever\u2014and I'll never see you.\n\nKeep on going round the galaxy\u2014no place is home\n\nFor the likes of mee-ee-e-e!\n\nThere was a vast amount of itchy foot stomping, let me tell you. And plenty of cheers and cries of joy when we had finished. Buoyed up by enthusiasm we played two more numbers before I called a break.\n\n\"Thanks folks, thanks much\u2014you're a great audience. Now if you will give us a few minutes we'll be right back...\"\n\n\"Very well done, well done indeed,\" Svinjar said, waddling over and plucking the microphone from my lapel. \"I know that we all have heard these musicians before\u2014on the box\u2014so their delightful entertainment comes as no surprise to us all. Yet still, there is something fine about having them here in person. I am grateful\u2014I know that everyone out there is grateful.\" He turned and smiled broadly at me. A smile that, I could see quite clearly, held no warmth or humor at all. He turned back and spread his arms wide.\n\n\"I am so grateful that I have prepared a little surprise for all of you out there\u2014do you want to know what it is?\"\n\nAbsolute silence now\u2014and a sideways shuffling by the audience. They apparently did not like any of Svinjar's little surprises.\n\nThey were right.\n\n\"Go!\" he shouted into the microphone, so loudly that his amplified voice rolled and echoed like thunder. \"Go\u2014go\u2014GO!\"\n\nI staggered and almost fell as the platform shook and vibrated. There was a roar of masculine voices as out from under our feet, brushing aside the disguising leafy boughs, burst a mass of armed men. More and more appeared, waving cudgels, howling as they ran, bearing down on the fleeing audience.\n\nWe looked on dumbfounded as men and women were clubbed to the ground, chained, tied. The attack was brief and vicious and quickly over with. The fields were empty, the last visitor gone. Those that remained were bound and silent, or groaning with pain. Over their moans of agony Svinjar's laughter sounded clearly. He was rocking in his chair, possessed by sadistic humor, tears rolling down his cheeks.\n\n\"But where\u2014\" Madonette said. \"Where did they all come from? There was no one under here when we started the concert.\"\n\nI jumped to the ground, kicked some branches aside, saw the gaping mouth of the tunnel. The opening had been concealed by a dirt-covered lid, now thrown aside. There was a heavy thud and Svinjar landed beside me.\n\n\"Wonderful, isn't it?\" He gestured at the opening. \"I have had my men digging that thing for months now. Stamping the removed dirt into the mud whenever it rains. I had planned a meeting here, some gifts, all very vague. Until you showed up! If I were capable of gratitude I would be grateful. I am not. The blind workings of chance. And victory to those\u2014meaning me\u2014who have the intelligence to seize the opportunity. Now a small celebration. We will have food and drink and you will play for me.\"\n\nHe turned and issued instructions, kicked one of his new slaves when she stumbled close.\n\n\"It would be nice to kill him,\" Madonette said. Speaking for all of us, if the nodding heads meant anything.\n\n\"Caution,\" I cautioned. \"He has all the cards and the thugs right now. Let's play the concert and figure out how we can get out of here after that.\"\n\nIt wasn't going to be easy. Svinjar's oversized log cabin was filled with his men. Drinking but not drunk, boasting of their feats, drinking even more. We played a number but no one was listening.\n\nYes, Svinjar was. Listening and looking. Waddling towards us, silencing the music with a swipe of his hand. Dropping into his chair and fingering the hilt of his large sword embedded in the stone close by his hand. Smiling that humorless smile at me again.\n\n\"Life is a bit different here, isn't it Jim?\"\n\n\"You might say that.\"\n\nIf he was looking for trouble I wasn't going to supply it. I didn't like the odds at all.\n\n\"We make our own life\u2014and our own rules here. Out there in the androgynous, settled worlds of the galaxy, the effete intellectuals rule. Men who act like women. Here we hearken back to the days of the primitive, virile, important men. Strength through strength. I like that. And I make the rules here.\" He looked at Madonette in a singularly repulsive manner.\n\n\"A fine singer\u2014and a lovely woman,\" he said, then looked at me. \"Your wife you say? Can anything be done about that? Let me think\u2014yes\u2014something can be done. Out there, in those so-called civilized planets nothing could be done. Here it can. For I am Svinjar\u2014and Svinjar can always do something.\"\n\nHe lifted one gross hand and tapped me on the forehead. \"By my law and my custom I now divorce you.\" He heaved himself to his feet while his henchmen roared with laughter at his subtle humor.\n\n\"That is not possible. It can't be done\u2014\"\n\nFor his size he was fast, whipping out the broadsword from the niche in his throne.\n\n\"Here is my first lesson for my new bride. Nobody says no to Svinjar.\"\n\nThe blade slashed out to slit my throat.\nChapter 9\n\nI jumped back to avoid the slash, stumbled over a man's legs, fell on top of him.\n\n\"Hold him!\" Svinjar shouted and I was grabbed tightly, struggled to get free, couldn't quite make it.\n\nSvinjar was standing over me, pushing the point of the sword into my throat\u2014\n\nThen he toppled sideways and fell with a great thud. Revealing the fact that Steengo, despite age and overweight, had jumped to the attack and was behind him, had dropped him with a chop to the neck.\n\nWhat was happening had by this time sunk into even the tiniest of the birdbrains present. Men struggled to draw weapons and roared crude oaths. I saw Floyd laying about the warriors nearest him\u2014but it wouldn't be enough. In about two seconds there was going to be a massacre of musicians if I didn't do something to stop it.\n\nI did. First by planting my elbow in the solar plexus of my captor. Who gurgled and let go of my arms. One second gone. I didn't waste any time trying to stand up but writhed on my side and pulled the black sphere from my pocket, thumbed the actuator and threw it up towards the ceiling.\n\nTwo seconds. Weapons swinging on all sides. My best defense was to jam the filter plugs into my nostrils. The gas bomb popped and I spent a busy few seconds more dodging my attackers. Who moved more and more slowly until they dropped. When I looked around I saw that the gas had done a great job. The entire great room was filled with prone and snoring forms. I shook my hands over my head.\n\n\"Let's hear it for the good guys!\" I had an audience of one, myself, which made the victory no less sweet. The sleep gas had hit my friends as well, though Floyd had been doing quite well before he dropped. A number of crumpled bodies were collapsed around him. I opened my pack and got the gas antidote; one by one I shot up my companions with the styrette. Then went to the door and stared gloomily out at the rain until they revived.\n\nSoft footsteps behind me and Madonette held me lightly by the arms.\n\n\"Thanks, Jim.\"\n\n\"Was nothing.\"\n\n\"It was something. You saved our lives.\"\n\n\"We're still in it,\" Floyd said. \"And like Madonette said, we owe you a good bit of thanks.\" Steengo nodded agreement.\n\n\"I wish you didn't. If this operation had been planned better all these emergencies wouldn't be taking place. My fault. I'm under what you might call a certain kind of time pressure. For reasons I can't go into right now we have to find the artifact and finish this operation within twenty days.\"\n\n\"That's not much time,\" Steengo said.\n\n\"Right\u2014so let's not waste any of it. Our welcome has worn out around here. Grab weapons because we might have trouble getting out of town empty-handed. Packs on, armed to kill, ruthless and deadly expressions. Forward!\"\n\nAfter what had almost happened to us with Svinjar and his macho swinemen we were in no mood to be trifled with. It must have shown in our faces\u2014or more likely in the metal of our weapons\u2014because the few people we met slipped away as soon as they saw us. The rain had almost stopped and the sun was burning through and raising trails of mist from the waterlogged ground. The hovels were farther apart now, the mounds of garbage fewer and more easily avoided. Straggly little bushes began to appear, then trees and larger shrubs covering the easy slope of the rolling hills. Mixed in were low bushes from which hung hard-skinned spheres the size of a man's fist. Maybe these were the polpettone trees we had been told about. This would have to be investigated\u2014but not now. I led on at a good pace, not calling a halt until we had reached the concealment of the first coppice. I looked back at the crude buildings, with the great bulk of the Pentagon rising behind them.\n\n\"No one seems to be following us\u2014so let's keep it that way. Five-minute break every hour, keep walking until sunset.\"\n\nI touched the skull-computer hanging from my neck and the keyboard snapped into existence. I summoned up the holomap, glanced up at the sun\u2014then pointed ahead.\n\n\"We go thataway.\"\n\nIt was tiring at first, struggling up one hill and down the other side, then up again. But we soon left the trees and the rolling countryside behind and marched out onto a grassy plain. We stopped for a break at the end of the first hour, dropped down and drank some water. The bravest of us chewed industriously on the concentrated rations. Which had the texture of cardboard\u2014if not the same exciting flavor. There was a grove of the polpettone trees close by and I went and picked a few of the spherical fruits. Hard as rocks and looking just about as appetizing. I put them into my pack for later examination. Floyd had dug a small flute out of his pack and played a little jig that lifted our spirits. When we stepped out again it was to a jolly marching tune.\n\nMadonette walked beside me, humming in time with the flute. A strong walker, she seemed to be enjoying the effort. And surely a great singer, good voice. Good everything\u2014and that included her bod. She turned and caught me looking at her and smiled. I looked away, slowed a bit to walk next to Steengo for a change. He was keeping up with the rest of us and did not look tired I was happy to see. Ahh, Madonette... Think of something else, Jim, keep your eye on the job. Not the girl. Yes, I know, she looked a lot better than anything else around. But this was no time to go all smarmy and dewy-eyed.\n\n\"How long you think until dark?\" Steengo asked. \"That pill you gave me is wearing off with a vengeance.\"\n\nI projected a holo of a watch. \"I truly don't know\u2014because I don't even know the length of the day here. This watch, like the computer, is on ship's time. It's been a good long time since they threw us out the gate.\" I squinted at the sky. \"And I don't think that sun has moved very much at all. Time to ask for some advice.\"\n\nI bit down three times hard on the left side of my jaw, which should have triggered a signal on the jawbone radio.\n\n\"Tremearne here.\" The words bounced around clearly inside my skull.\n\n\"I read you.\"\n\n\"You read what?\" Steengo asked.\n\n\"Please\u2014I'm talking on the radio.\"\n\n\"Sorry.\"\n\n\"Reception clear at this end. Report.\"\n\n\"We were less than charmed by the Machmen. We left town a couple of hours ago and are hiking out across the plain...\"\n\n\"I have you on the chart, satellite location.\"\n\n\"Any of the Fundamentaloid bands in sight as well?\"\n\n\"A number of them.\"\n\n\"Any of them close to this position?\"\n\n\"Yes, one off to your left. Roughly the same distance you've walked already.\"\n\n\"Sounds a winner. But one important question first. How long are the days here?\"\n\n\"About one hundred standard hours.\"\n\n\"No wonder we're beginning to feel tired\u2014and it's still full daylight. With the total daylight at least four times longer than what we're used to. Can you put your satellite to work looking back the way we came\u2014to see if we are being followed?\"\n\n\"I've already done that. No pursuers in sight.\"\n\n\"That's great news. Over and out.\" I raised my voice. \"Company\u2014halt. Fall out. I'll give you the other side of the conversation that you didn't hear. We're not being followed.\" I waited until the ragged cheer had died away. \"Which means we are stopping here for food, drink, sleep, the works.\"\n\nI slung my pack to the ground, stretched largely, then dropped down and leaned against it, pointed to the distant horizon. \"The Fundamentaloid nomads are somewhere out in that direction. We are going to have to find them sooner or later\u2014and I vote for later.\"\n\n\"Vote seconded, motion passed.\" They were all horizontal now. I took a good swig of water before I went on.\n\n\"The days here are four times as long as the ones that we are used to. I think that we have had enough of fighting, walking, everything for one day, or a quarter of a day, or whatever. Let's sleep on it and go on when we are rested.\"\n\nMy advice was unneeded since eyelids were already closing. I could do no less myself and was drifting off when I realized this was not the world's greatest idea. I heaved myself, groaning, to my feet and walked away from the others so my voice would not disturb them.\n\n\"Come in Tremearne. Can you read me.\"\n\n\"Sergeant Naenda here. The Captain is off duty this watch. Should I send for him?\"\n\n\"Not if you are sitting in for him\u2014and you have the satellite observations handy and up-to-date.\"\n\n\"Affirmative.\"\n\n\"Well keep looking at them. We're taking a sleep break now and I would like it to be undisturbed. If you see anyone or anything creeping up on us\u2014give a shout.\"\n\n\"Will do. Nighty-night.\"\n\nNighty-night! What were the armed forces coming to? I stumbled back to my companions and emulated their fine example. I had no trouble at all in falling asleep.\n\nIt was waking up that was difficult. Some hours had slipped by because when I blinked blearily up at the sky I saw that the sun had passed the meridian and was finally slipping down towards the horizon. What had wakened me?\n\n\"Attention, Jim diGriz, attention.\"\n\nI looked around for the speaker and it took long seconds before I realized that it was Captain Tremearne's voice I was hearing.\n\n\"Wazza?\" I said incoherently, still numbed with sleep.\n\n\"One of the Fundamentaloid bands is on the move\u2014roughly in your direction. They should be close enough to see you in about an hour.\"\n\n\"By which time we should be ready for visitors. Thanks, Cap\u2014over and out.\"\n\nMy stomach snarled at me and I realized that the concentrated rations had been a little too concentrated. I drank some water to wash the taste of sleep from my mouth, then poked Floyd with my toe. His eyes snapped open and I smiled sweetly.\n\n\"You have just volunteered to go to those bushes over there and get some firewood. It is breakfast time.\"\n\n\"Right, breakfast, wood, wonderful.\" He climbed to his feet, yawned and stretched, scratched at his beard, then went off on his mission. I gathered up enough dry grass to make a small pile, then dug the atomic battery out of my pack. It would power our musical equipment for at least a year, so it could spare a few volts now. I pulled the insulation off the ends of the wires on a short lead, shorted them to produce a fat snap of sparks, pushed it into the grass. In a moment the grass was burning nicely, crackling and smoking, and ready for the chunks of dry branches that Floyd brought back. When it was good and hot I dropped the polpettone into the glowing ashes.\n\nThe rest of the band stirred in their sleep when the smoke blew their way, but didn't really wake up until I broke one of the fruits open. The skin was black so I hoped it was done. The rich seasoned fragrance of cooked meat wafted out and everyone was awake in an instant.\n\n\"Yum,\" I said, chewing on a fragrant morsel. \"My thanks to the genetic engineers who dreamed this one up. Gourmet food\u2014and growing on trees. If it weren't for the inhabitants this planet would be a paradise.\"\n\nAfter we had dined and were feeling relatively human I made my report to them.\n\n\"I've been in touch with the eye in the sky. A band of nomads is coming this way. I figured that we should let them do the walking instead of us. Are we now prepared for contact?\"\n\nThere were quick nods and no hesitant looks I was happy to see. Steengo hefted his ax and glowered. \"Ready as we'll ever be. I just hope this lot is a bit more friendly than the first bunch.\"\n\n\"Only one way to find out.\" I bit down three times hard. \"Where are the Fundamentaloids now?\"\n\n\"Crossing a bit north of you\u2014beyond those shrubs on the slight rise.\"\n\n\"Then here we go. Packs on, weapons ready, fingers crossed. Forward!\"\n\nWe walked slowly up the hill and through the shrubs\u2014and stopped in our tracks and stared at the herd passing slowly by.\n\n\"Sheots,\" I said. \"The mutant cross between sheep and goats that they told us about.\"\n\n\"Sheots,\" Madonette agreed. \"But they didn't tell us they were so huge! I don't even come up to their legpits.\"\n\n\"Indeed,\" I agreed. \"Something else about them. They're big enough to ride upon. And if I am not mistaken we have been seen and those three riders are galloping our way.\"\n\n\"And waving weapons,\" Steengo said grimly. \"Here we go again.\"\nChapter 10\n\nThey thundered towards us, swords waving, sharp black hooves kicking up clouds of dust. The sheots had nasty little eyes, wicked, curved horns\u2014and what looked very much like tusks. I couldn't recall ever seeing a sheep or a goat with tusks, but there is always a first time.\n\n\"Stay in line, weapons ready,\" I called out, swinging my own sword up. The nearest rider, draped in black, pulled hard on the reins and his woolly mount skidded to a stop. He frowned down on me from behind his great black beard, spoke in a deep and impressive voice.\n\n\"Those who live by the sword shall die by the sword. So it is written.\"\n\n\"You talking about yourself?\" I queried, blade still ready.\n\n\"We are men of peace, infidel, but defend our flocks against numberless rustlers.\"\n\nHe could be telling the truth; I had to take the chance. I plunged my sword into the dirt and stepped back. But was ready to grab it in an instant.\n\n\"We are men of peace as well. But go armed for our own protection in this wicked world.\"\n\nHe thought about that for a bit, made the decision. He slipped the sword into a leather scabbard, then swung down from his mount. The beast instantly opened its mouth\u2014and those were tusks\u2014and tried to bite him. He scarcely noted this, merely balled a fist and got the thing under the jaw with a swift uppercut. Its mouth clacked shut and its eyes crossed for an instant. It wasn't too long on brains either, because when its eyes uncrossed it had completely forgotten about him. It said baa loudly and began to graze. The rider walked over and stood before me.\n\n\"I am Arroz conPollo and these are my followers. Have you been saved?\"\n\n\"I am Jim diGriz and this is my band. And I don't believe in banks.\"\n\n\"What are banks?\"\n\n\"Where you save money. Fedha.\"\n\n\"You misunderstand my meaning, Jim of diGriz. It is your soul that needs saving\u2014not your fedha.\"\n\n\"An interesting theological point, Arroz of conPollo. We must discuss it in some depth. What do you say we all put the weapons down and have a good chinwag. Put them away,\" I called out.\n\nArroz signaled his two companions and we all felt a lot better as the swords were sheathed, axes lowered. For the first time he looked away from me to my followers. And gasped, turned pale under his tan, and held his arm before his eyes.\n\n\"Unclean,\" he moaned, \"unclean.\"\n\n\"Well it is a little hard to have a bath when you're on the trail,\" I told him. I didn't add that he wasn't that spic and span himself.\n\n\"Not of the body\u2014of the spirit. Is that not a vessel of corruption among you?\"\n\n\"Could you spell that out a little more clearly?\"\n\n\"Is that... person a... woman?\" He still had his arm across his face.\n\n\"The last time I looked she was.\" I moved sideways a bit, closer to my sword. \"What's it to you?\"\n\n\"Her face must be covered to conceal impurity, her ankles covered lest they promote lust in the hearts of men.\"\n\n\"This guy is a bit of a weirdo,\" Madonette said disgustedly. He yiped.\n\n\"And her voice silenced lest it lure the blessed into sin!\"\n\nSteengo nodded to Floyd and took the angry girl by the arm, but she shrugged him off. \"Jim,\" he said. \"The bunch of us are going to stroll back among the trees and have a break. See if you can sort this out.\"\n\n\"Right.\" I watched them leave and when they were out of sight looked back at the three nomads who were emulating their leader, all with their arms raised, as though sniffing their armpits. \"It's safe now. Can we talk about this?\"\n\n\"Return,\" Arroz said to his mates. \"I will explain the Law to this stranger. Let the flock graze.\"\n\nThey trotted off while his own mount chomped away on the grass. He sat down cross-legged and motioned to me. \"Sit. We must talk.\"\n\nI sat. But upwind of him because it had been a long time since he or his clothes had been near soap and water. And he talked about unclean! He rooted about under his robe, had a good scratch, then withdrew a book and held it up.\n\n\"This book is the font of all wisdom,\" he intoned, eyes gleaming.\n\n\"That's nice. What is it called?\"\n\n\"The Book. There are no other books. All that men need to know is in here. The distillate of all wisdom.\" I thought that it looked pretty thin for that job, but wisely kept my mouth shut. \"It was the great Founder, whose name may not be spoken, who had the inspiration to read all of the Holy books of all of the ages, who saw in them the work of the god whose name may not be spoken, saw which passages were inspired and which were untrue. From all the books He distilled the true Book\u2014then burned all of the others. He went forth into the world and His followers were many. But others were jealous and tried to destroy Him and His followers. That has been told. And it is told that to avoid this senseless persecution He and His followers came to this world where they could worship untroubled. That is why I asked\u2014are you unclean? Or do you also follow the Way of the Book?\"\n\n\"Most interesting. I follow a slightly different way. But my way believes in respecting your way, so don't worry too much about me.\"\n\nHe frowned at this and shook an admonitory finger at me. \"There is only one Way, only one Book. All who think differently are damned. Now is your chance to be cleansed for I have shown you the true Way.\"\n\n\"Thanks a lot\u2014but no thanks.\"\n\nHe stood up and stabbed an accusatory finger in my direction. \"Unclean! Profane! Leave\u2014for you soil me with your presence.\"\n\n\"Well each to their own opinion. Goodbye and good luck with your sheot shearing. May all your fleeces be giant ones. But an indulgence please\u2014before you go would you take a look at this.\" I pulled the photograph of the alien artifact from my pocket and held it out.\n\n\"Unclean,\" he muttered and put his hand behind his back so he wouldn't touch it.\n\n\"I'm sure it is. I just want to know if you have seen this thing in the picture before.\"\n\n\"No, never.\"\n\n\"Been nice talking to you.\"\n\nHe did not return my friendly wave as he walked over to his mount, kicked it in the leg until it sat down, climbed aboard and galloped off. I pulled my sword out of the ground and went to join the others. Madonette was still simmering.\n\n\"Hypocritical narrow-minded bigoted moron.\"\n\n\"That and a lot more. At least I got one bit of negative information from him. He never saw the artifact. It must have been taken by another one of the tribes.\"\n\n\"Are we going to have to talk to all of them?\"\n\n\"Unless you have any better ideas. And nineteen days to go.\"\n\n\"I don't trust him,\" Madonette said. \"And don't sneer and say female intuition. Aren't these the same kind as the bunch that attacked the archeologists' ship?\"\n\n\"You're right\u2014and isn't that the clatter of hundreds of hooves coming this way?\"\n\n\"It is!\" Floyd shouted, pointing. \"What do we do\u2014run?\"\n\n\"No! Out of the trees and onto the plain. Instruments at the ready. We are going to give these guys a concert that they will never forget!\"\n\nArroz had gone back to rally the troops and at least thirty of them, with plenty of sword waving and maniac baaing, came charging down. I turned the amplification on the sound up until it would not go any higher.\n\n\"Earplugs in, get ready, on the count of three we give them old number thirteen, 'The Rockets Go Rumbling On.' One, two...\"\n\nOn the count of three the explosion of unbearable sound blasted out. The lead riders were tossed to the ground as the sheots recoiled in fear. I flipped some smoke bombs among them, just to keep the action going, and hit them with holographed lightning volts.\n\nIt was pretty good. Before we got to the second chorus the stampede was over, the last terrorized sheots galloped away out of sight. The last black-robed Fundamentaloid crawled over the horizon, the trampled grass dotted with discarded swords, gobbets of fleece and myriad eightballs of dung.\n\n\"Victory is ours!\" I whooped happily.\n\nAnd only nineteen days to go I thought depressedly. This just would not do. I had the awful feeling that we could spend nineteen days or nineteen weeks stumbling about this planet and be no wiser about the alien artifact we were seeking. There had to be a change of plan\u2014and now! I walked away from the others, then bit down three times, so hard that I almost cracked a tooth.\n\n\"Captain Tremearne here.\"\n\n\"And dismal Jim diGriz on this end. Have you been following all this?\"\n\n\"Yes, and watching. I heard you ask him to identify the photograph. I assume that he did not.\"\n\n\"You assume right, distant and disembodied voice. Now listen, there has got to be a change of plan. When I came up with the idea for this present operation I assumed that there was some kind of imitation of civilization on this dismal world. Where we could stroll from gig to gig and do our snooping at the same time. I was wrong.\"\n\n\"I regret that all the facts were not supplied to you at the time. But as you are now aware there is a complete ban on information being circulated about this particular planet.\"\n\n\"I know that now\u2014and it won't wash. We would have been a lot better if we came here disguised as a squad of combat marines. So far every bunch we have met has tried to kill us. The whole thing is that hardnosed Admiral Benbow's fault. He lied to me about what we would find here. Right?\"\n\n\"As a serving military officer I cannot discuss the conduct of my superiors. But I can agree that whoever briefed you was, I must say, economical with the truth.\"\n\n\"Do you also know that he was economical with my health? And that in nineteen days I am going to keel over from time-released poison.\"\n\n\"Regrettably, I have been informed that that is the case. And you have eighteen days left now. You appear to have lost track of one day during the past period.\"\n\n\"Eighteen? Thanks much. That only makes what I have to say even more imperative. I need some help, some transportation.\"\n\n\"All contact with the planet is forbidden.\"\n\n\"I just changed the rules. You yourself told me that you are heading a committee to bring about major improvements here. The first change will be to get one of the ship's launches down here. With that I can get around to the various bands of sheot shaggers before my personal deadline runs out.\"\n\n\"If I do that I will be disobeying orders and it could end my career.\"\n\n\"Well?\"\n\nThe silence inside my head went on and on. I waited. Until I heard what could only have been a sigh.\n\n\"I suppose there are plenty of job opportunities for skilled civilians these days. The launch will hand after dark. If it is not seen by anyone on the ground there is just a chance that my career change can be postponed.\"\n\n\"You're a good guy, Tremearne. My heartiest thanks.\"\n\nI hummed a bar or two from \"The Swedish Monster\" as I walked back to inform my companions.\n\n\"Jim, you're wonderful!\" Madonette said, grabbed and kissed me. \"I much prefer flying to walking.\"\n\nFloyd nodded happy agreement and reached for me.\n\n\"Away!\" I shouted. \"Girls, okay, but I don't kiss guys with beards. What we do now is put a little distance between us and those religious nuts in case they want to come back for seconds. Then rest up until dark. I have a feeling that it is going to be a very busy night.\"\nChapter 11\n\n\"Wake up, Jim\u2014it's almost dark.\"\n\nMadonette's gentle hand was most welcome, since it drew me up out of a really repulsive nightmare. Tentacles, bulging eyeballs, yukk. The eighteen-day dead deadline must be getting to my subconscious. I sat up, yawned and stretched. With great reluctance the sun had finally dropped behind the horizon, leaving behind a slowly fading band of light. The stars were coming out, revealing some pretty boring constellations\u2014and very few of them at that. This prison planet must be far out on the galactic rim.\n\nThen something blotted out the stars in the zenith as a dark form drifted down to the ground, silently on null-grav drive. The door opened as we approached\u2014and the cabin lights came on.\n\n\"Turn them off, lunkhead!\" I shouted. \"You want to ruin my night vision.\" The pilot turned about in his seat and I grinned insincerely. \"Sorry Captain, sir\u2014that lunkhead, just a figure of speech.\"\n\n\"My fault completely,\" he said, and tapped one of his electronic eyeballs. \"With these I forget. I'm piloting this thing because I have the best night vision in the fleet.\"\n\nHe flipped the lights off and we groped our way aboard with just the dim red emergency lights to show us the way. I sat in the copilot's seat and strapped in.\n\n\"What is your plan?\" he asked.\n\n\"A simple one. You know the position of all the sheot flocks don't you?\"\n\n\"Observed and logged into the launch's memory.\"\n\n\"Great. Have the computer do a topological survey to plot a course that will let us visit them all in the shortest amount of time. We drift over to the first flock, find one of the shepherds who is maybe out of sight of the others\u2014and talk to him. Show him the photograph and find out if he has seen the thing. If he hasn't\u2014on to the next bunch.\"\n\n\"Seems a simple and practical plan. Belts fastened? Right, first flock coming up.\"\n\nWe were slammed back into our seats and were on our way. High and fast on the plotted track. Then slow and drifting in low while Tremearne peered out into the darkness.\n\n\"There's one,\" he said. \"On the far side of the flock\u2014all by himself. Either to guard the beasts or keep them from wandering. I have a suggestion. I approach him from behind and immobilize him. Then you question him.\"\n\n\"Creep up in the dark? Immobilize an armed and watchful guard? That's a job for a combat trooper.\"\n\n\"Well how do you think I got these electronic eyeballs? It will be entertaining to do a bit of work again.\"\n\nI had no choice but to agree. The Captain was proving to be an excellent ally. Working this way would be certainly a lot faster than me crawling around on my own. If he could do as he said. I had my doubts but kept them to myself. He was a gray-haired desk jockey with electric eyesight who might very well be past his sell-by date.\n\nHe wasn't. After we landed he stepped out the door and vanished silently in the darkness. Not thirty seconds later he called to me quietly.\n\n\"Over here. You can use your light now.\"\n\nI turned on the handlight, it was really black under the almost starless sky, and saw two forms standing close together. The light revealed a bulging-eyed shepherd seized in an unbreakable grip, a hand on his throat keeping him silent. I waggled the light under his nose.\n\n\"Listen, oh shepherd who failed his duty. The hand that holds you could just as easily have killed you. Then we could rustle all your woolly flock and eat sheot shashlik until the end of time. But I will be merciful. The hand will be removed from your filthy throat and you will not shout or you really will be dead. You will speak to me softly and answer my questions. You may now speak.\"\n\nHe coughed and groaned when the pressure was released. \"Demons in the darkness! Release me, do not kill me, tell me what you wish of me then go back to the pit from which you have escaped...\"\n\nI reached out and tweaked his nose sharply. \"Shut up. Open your eyes. Look at this photograph. Let me know if you have ever seen it before.\"\n\nI held the photo close, shone the light on it. Tremearne gave a twitch of emphasis to his arm and the captive moaned his answer. \"Never, no, such a thing I would remember, no\u2014\" His voice gurgled into silence and he dropped unconscious to the ground.\n\n\"Don't these sheot shepherds ever wash?\" Tremearne asked.\n\n\"Only on alternate years. Let's get to the next one.\"\n\nWe quickly worked out a routine. We would land and he would be away. Usually, by the time I had exited the launch, he would be calling me. Many a terrified shepherd slept soundly this night. But only after looking at the picture of the artifact. I dozed between visits and the back of the launch echoed with snores and heavy breathing. Only the Captain was unsleeping and tireless, seemingly as fit on the eleventh visit as he had been on the first. It was a long, long night.\n\nI was getting groggy by the time we hit thirteen. Unlucky thirteen; get it over with and on to fourteen. Another set of bulging eyes peeking over the top of another matted beard.\n\n\"Look!\" I snarled. \"Speak! And moaning does not count as speaking. Ever seen this thing?\"\n\nThis one gurgled instead of moaning, then yiped as his arm got twisted a bit further. It looked as though even the stolid Captain was beginning to lose his patience.\n\n\"Imp of Satan... work of the devil... I warned them, but they wouldn't listen... the grave, the grave!\"\n\n\"Do you have any idea of what he is babbling about?\" Tremearne asked.\n\n\"There may be hope, Captain. If he is not bonkers he might have seen it. Look\u2014see! Ever see before?\"\n\n\"I told him not touch it\u2014death and damnation were sure to follow.\"\n\n\"You have seen it. All right, Cap, you can let up on the arm\u2014but stand ready.\" I rooted in my pocket and took out a handful of silver cylinders, the local money, let the light shine on them. \"Hey you, Smelly, look\u2014fedha\u2014and all for you. All yours.\"\n\nThis got his attention all right and I closed my fist tight as he groped for them. \"Yours if you answer some simple questions. You will not be hurt\u2014but only if you answer truthfully. You have seen this thing?\"\n\n\"They fled. We found it in their skyship. I touched it, unclean, unclean.\"\n\n\"You're doing fine.\" I shook half of the coins into his waiting hand. \"Now the ten-thousand-fedha question. Where is it now?\"\n\n\"Sold, sold to them. The Paradisians. May they be cursed by it, cursed forever...\"\n\nIt wasn't easy, but we finally worked all the details out of him. Stripped of all the curses and blasphemy it was a simple tale of larceny and chicanery. The spacer had landed\u2014and been attacked as soon as the door had been opened. During the fracas the Fundamentaloids had trundled through the ship and grabbed everything portable, including the container with the alien artifact. They had carried the whole thing away with them because they had a job opening it. When they eventually succeeded they could not understand what it was. And ignorance meant fear. So they had unloaded it in the market in Paradise where almost anything could be sold. End of story.\n\nWe let the shepherd keep the money when we lowered him, unconscious, to the ground. \"This calls for consultation,\" I said.\n\n\"Yes, but not this close to the flock. Let's get up to the plateau where the air is fresher.\"\n\nThe others were awake when we landed this time, listening closely to what we had discovered.\n\n\"Well this narrows the field a bit,\" Madonette said.\n\n\"Does it?\" I asked. \"How big is the population of this paradisiacal nation?\"\n\n\"Around one hundred thousand,\" Tremearne admitted. \"It may not be the best society on this planet but it appears to be the most successful one. I know very little about it, just photographs and observation.\"\n\n\"Doesn't anyone in the Pentagon know more?\"\n\n\"Probably. But the information is classified and they aren't talking.\"\n\nI cracked my knuckles, scowled and jabbed my finger at him. \"That's really not good enough\u2014is it?\"\n\nTremearne looked as unhappy as I did. \"No, Jim, it is not. I don't know why all that information is classified while your group is actually operating here on the planet. I have tried to get the information and have been not only rebuffed but warned off.\"\n\n\"Who is doing this? Any idea?\"\n\n\"None\u2014other than that it is at the very highest level. The people I have been in contact with understand your problems and want to help. But any requests that they pass on are turned down instantly and with prejudice.\"\n\n\"Am I paranoid\u2014or is there someone in the chain of command who doesn't like this operation? Who wants it to fail?\"\n\nIt was Tremearne's turn now to crack his knuckles and look glum.\n\n\"I've told you\u2014I am a career officer. But I'm not fond of the situation here on this planet. Not only the way your group is being treated, but the whole ugly business. Well, I feel that it is getting away from me. At first I thought I could get some reform here by working through channels. It's not good enough. I am being blocked just as completely as you are.\"\n\n\"Who\u2014and why?\"\n\n\"I don't know. But I am doing my best to find out. About this city and the Paradisians I guess, basically, I know absolutely nothing.\"\n\n\"An honest answer, Captain, and I thank you for it.\"\n\n\"If you don't know\u2014why then we'll just have to find out for ourselves,\" Steengo said. \"Play a gig or two and keep our eyes open.\"\n\n\"May it be so easy,\" I muttered under my breath. \"Roll out the maps.\"\n\nIt looked as though the largest part of the population was located in the single straggling city. Roads led from it to not-too-distant villages and there were scatterings of other buildings that might be farms. The only really puzzling thing about the 3D map was what looked like a wall that appeared to cut the city in two. There were no walls around the city, just this single one in the middle. I pointed to it.\n\n\"Any idea what this is\u2014or what it means?\"\n\nTremearne shook his head. \"No idea. Looks like a wall, that's all. But there is a road alongside it. Which appears to be the only road leading in from the plain.\"\n\nI poked my finger into the holomap.\n\n\"Here. Where the road fades and runs out in the grass. That's where we have to go. Unless anyone has a better idea?\"\n\n\"Looks good to me,\" Tremearne said. \"I'll land you on this bit of plateau, beyond this ridge where we won't be seen. Then I'll take the launch out of there and stay in touch with you by radio.\"\n\nWe unloaded. \"Sleep first,\" Floyd yawned. \"It's been a long night.\"\n\nIt was even longer than that, what with the longer days here. Tremearne took off and we settled down to sleep. We slept, and woke up and it was still dark. Slept some more. At least the others snored on: I had too much on my mind to drift off as easily as they did. We had a clue now to the whereabouts of the alien artifact. A clue that was useless until we started looking. And we couldn't look in the darkness. And I had\u2014how many days left before the thirty-day poison zonked me? I counted on my fingers. Just about eighteen gone, which left twelve to go. Wonderful. Or had I counted wrong? I started again with the fingers, then grew angry with myself. Enough with the fingers already. I clicked on my computer and wrote a quick program. Then touched D for deadline\u2014or death, whatever\u2014and a glowing eighteen appeared before me accompanied by a flickering twelve. Not that I enjoyed looking at them, mind you, but this way I could stop worrying about the changing count. Some part of me must have been satisfied with this because I fell deeply asleep.\n\nFinally, with great reluctance and sloth, the sky lightened and another day began. Before it was completely light the Captain drifted the launch in low and slow behind the hills, boarded us, then let us out behind the final ridge.\n\n\"Good luck,\" he said, with a certain grimness. The port ground shut and the launch moved away and vanished in the growing light. Scarcely aware of what I was doing I punched D into the computer. The numbers snapped into existence, vanished just as quickly. But I remembered.\n\nDay nineteen.\nChapter 12\n\nDawn crept on interminably as we walked, the sun dragging itself up over the horizon only with great reluctance. It was still not quite full daylight when we came to what had to be the beginning of the wall. Just a single row of bricks almost hidden in the grass.\n\n\"What do you think?\" I asked of no one in particular. Steengo bent and rapped one with his knuckles.\n\n\"Brick,\" he said.\n\n\"Red brick,\" Madonette said brightly.\n\n\"Thanks, thanks,\" I mumbled with complete lack of appreciation.\n\nThere was a barely visible path next to the right-hand side of the row of bricks; for want of a better idea we began walking along it.\n\n\"It's higher, see,\" Floyd said, pointing. \"A second course has been added.\"\n\n\"And more still ahead,\" Madonette said. \"Three bricks high now.\"\n\n\"What's this?\" Steengo said, bending and pushing the grass aside to look more closely, touching the brick with his fingertip. \"There's some kind of symbol stamped into each of the bricks.\" We all looked now.\n\n\"Sort of a circle with an arrow sticking out of it.\"\n\n\"Arrow... circle,\" I muttered. A sudden intuition bounced about inside my skull. \"I've seen that symbol before\u2014yes indeed! Would someone kindly step over the wall and see if there is a circle with a cross sticking out of it on the other side.\"\n\nMadonette lifted lovely eyebrows with curiosity, stepped daintily over the low wall, bent and looked. Eyebrows even higher now.\n\n\"How did you do that? On this side there is a circle-cross sign stamped into each brick.\"\n\n\"Biology,\" I said. \"I remembered from school.\"\n\n\"Yes, of course,\" she said, stepping back. \"The symbols for male and female.\"\n\nFloyd had strolled on ahead; he called out. \"Right as rain. Here is VIROJ stamped into a brick. And,\" he leaned over and looked, \"VIRINOJ on the other side.\"\n\nVery gradually the wall became higher as we walked beside it. In addition to the symbols we came to LJUDI then MTUWA, HERRER, SIGNORI.\n\n\"Enough,\" I said, stopping. \"Packs off. We shall now take our break while we see what we have here. The message seems to be clear enough. Look at the path we have been following. Is there another path on the other side as well?\"\n\nThe brick wall was as high as our waists now; Floyd put one hand on it and vaulted over, bent and looked.\n\n\"Maybe, but not too clear. Could have been here once but it is so overgrown with grass that it is hard to tell. Can I come back now?\"\n\n\"Yes\u2014because it's about time for a decision.\" I pointed ahead to the slowly heightening wall. \"The Fundamentaloids said they came to the city to trade. So they must have come this way, possibly made this track that we are following.\"\n\nMadonette nodded agreement\u2014and didn't like it. \"And they were all men, I remember that all too clearly. Unclean indeed! No women allowed. Or if the women did come this way they would have to have walked over on the other side of the wall. What do you want us to do, Jim?\"\n\n\"What do we want to do? As I said\u2014it's time for a decision. Do we all stick together and ignore the obvious instructions? That's the first question that we have to answer.\"\n\n\"Do that and I'll bet that eventually we get into some kind of trouble,\" she said. \"A lot of serious work went into this wall. So if we don't read the message something not too nice is guaranteed to happen. It always does on this world. The choice is mine. I'll cross over and trot down the other side\u2014\"\n\n\"No,\" I broke in. \"As we go along the wall gets higher and we'll be separated, out of contact. That won't do.\"\n\n\"Well I'm not staying here\u2014and I can't go back. So we need contact, what you just said. Kindly clack your jaw-a-phone and get onto Tremearne. Tell him to get some radios down here that we can use to keep in touch. If we are going to complete this assignment the right way, we will have to know what is going on on both sides of the wall. And I'm the only one who can find out what happens\u2014here.\"\n\nShe picked up her pack and planted her bottom on the wall, swung her legs up and over and smiled at us from the other side. I didn't like it.\n\n\"It's not a matter of liking or not liking it,\" she said reading my doubts from my expression. \"It is just the only way that we can get the job done. Get the radios. Don't forget that Tremearne will always be listening in and can send the marines if any of us gets into trouble. Call him.\"\n\n\"I will. But let us make sure they are the right kind of radios before we put in the order. Line of sight is going to be out with the wall standing in the way and blocking the signal. Plus\u2014who knows how thick the thing is going to be? It could soak up all the radio frequencies and that would be the end of that. Anyone know of a kind of radio that shoots a signal through rocks?\"\n\nI was speaking my thoughts aloud, half in jest. So was more than a little surprised when a voice behind me said, \"Yes.\"\n\nI spun about and glared at Steengo who was buffing his fingernails on his shirt, then admiring his image in their shining surfaces.\n\n\"You said that?\" I accused. He nodded sagely. \"Why?\"\n\n\"Why is a good question. The answer is that although I stand before you, an aging amateur musician drawn from retirement to risk his life for the public good, it should not be forgotten that I worked for many a decade in the cause of that same public good. League communications. Where I helped develop a neat little device referred to as MIPSC.\"\n\n\"Mipsic?\" I echoed inanely.\n\n\"Close enough, my good friend Jim. MIPSC is the acronym of Miniaturized Personal Satellite Communicator. I suggest that you clamp your jaw and order up a brace of them. Although four would be better\u2014that way we could all keep in touch at all times. And remind Tremearne to put a commsatellite into orbit as well. Geostationary over the city of Paradise.\"\n\n\"MIPSCS are not only highly secret but incredibly expensive,\" Tremearne said when I contacted him.\n\n\"Just like this little task force. Can you do it?\"\n\n\"Of course. They're on the way.\"\n\nA half an hour later a small package drifted down from the sky hanging from a grav-lifter\u2014which zipped up and vanished as soon as the package had been removed. I popped the end open and shook out a handful of false fingernails. I popped my eyes at these\u2014then remembered how Steengo had been buffing his own fingernails when he told me about MIPSC.\n\n\"Tricky,\" I said.\n\n\"High tech and perfect concealment,\" he said. \"There should be glue in the package. They come in pairs. The one marked E goes onto the index finger, left hand. M glued to the pinkie of the same hand. Inside the nails are holographed circuitry so they can be trimmed as small as needed to fit. Without damaging the circuits in any way.\"\n\n\"E? M?\" Floyd asked.\n\n\"Earplugs and microphone.\"\n\n\"Then what?\" I asked, almost humbly, dazed by the sudden appearance of a communications wizard in our midst.\n\n\"They are powered by the destruction of the phagocytes that come to eat them where they touch the cuticle. Which means that the power is always on. Anytime you are outside\u2014or in a building with thin floors\u2014your signal zips up to the satellite and back down to the other receiver. Simple. Just put your index finger into your ear and talk into the microphone on your pinkie.\"\n\nI measured a pair, trimmed and glued with, I must admit, a certain amount of trepidation. Stuck my finger into my ear and said, \"I hope it works.\"\n\n\"Of course it does,\" Tremearne said, speaking through my fingernail instead of my jaw for a change.\n\nWhile we had been installing the MIPSCS we had been going over and over all of the possibilities, had returned always to the only viable plan.\n\n\"Let's do it,\" Madonette said, admiring her new communicating fingernails. She put on her pack, shrugged it into comfortable position, then turned and walked off on her side of the barrier. With each step the wall grew higher, until, very quickly, it was as high as her head, then higher. After a last wave of her hand she vanished from sight.\n\n\"Keep in touch,\" I said into my pinkie. \"Regular reports and sing out if you see anything\u2014anything at all.\"\n\n\"Just as you say, boss.\"\n\nWe slipped on our packs and started walking. By the time an hour had passed the wall was high and unscalable. Though I stayed in radio contact with her, Madonette was now completely alone. I kept telling myself that armed help could zip down from the orbiting spacer if needed. This did not make me feel much better.\n\n\"First tilled fields coming up,\" Floyd said. \"And more than that. That dust cloud next to the wall\u2014it's coming our way.\"\n\n\"Weapons ready\u2014and I have some concussion grenades handy if things get hairy.\"\n\nWe stopped and waited and watched. In the distance it looked like a horse that was trotting towards us.\n\n\"Horse\u2014but no rider,\" I said.\n\nSteengo had the keener vision. \"Looks like no horse I ever saw before. Not one with six legs.\"\n\nIt slowed to a stop and looked at us. We returned the favor. A robot, metal. Jointed legs and in the front a pair of tentaclelike arms to boot. No head to speak of, just a couple of eyes that rose up on a stalk. A loudspeaker between its arms rustled and squawked metallically.\n\n\"Bonan tagon\u2014kaj bonvenu al Paradizo.\"\n\n\"And a good day to you as well,\" I said. \"My name is Jim.\"\n\n\"A masculine surname, most agreeable. I am called Hingst and it is my pleasure to greet you\u2014\"\n\nThe creature's words were drowned out by a throbbing roar and a cloud of black smoke emerged from its rear. We stepped back, weapons ready. Hingst's flexible arms lifted straight up.\n\n\"I wish you only peace, oh strangers. You would not know it, since you are untutored in science, but the sound and fumes are merely the exhaust of my alcohol engine. Which is rapidly turning a generator which in turn...\"\n\n\"Charges up your batteries. We know a thing or two as well, Hingst, greeter of strangers to Paradise, and we are not your usual goaty nomads.\"\n\n\"Now that is a pleasure to hear, visiting gentlemen. Before my operating system was bolted into this rather crude construction I was a class A42 head-waiterbot and worked at only the most excellent restaurants...\"\n\n\"Another time,\" I said. \"I would enjoy your reminiscences. We have a few questions\u2014\"\n\n\"And I am sure I have a few answers,\" it said with surly overtones. \"But there are preliminaries to go through.\" It had strolled a few paces forward as it talked and now, like a striking snake, one of its tentacles lashed at me. I jumped back, lifted my sword\u2014but not before the cool metal tip had touched my lips and just as swiftly been withdrawn.\n\n\"Try that again and you'll be a tentacle short,\" I growled.\n\n\"Temper, temper. After all you are armed strangers and I am simply doing my duty. Which is to sample your saliva. And test it, which I have done. You may proceed, Gentleman Jim, because you indeed are of the male sex. I would appreciate samples from your associates.\"\n\n\"As long as it is just spit you are after,\" Floyd growled, hands joined and cupped over his nether regions.\n\n\"Oh, I do appreciate a sense of humor, stranger.\" The tentacle took its sample from his mouth. \"Gentleman stranger I can now say. Final traveler if you please. Lovely, thank you. You may now proceed.\"\n\nIt turned away and I jumped in front of it.\n\n\"A moment first, Hingst the Official Greeter. A few questions...\"\n\n\"Sorry. I am not programmed for that. Kindly step aside, Gentleman Jim.\"\n\n\"Only after I get a few answers.\"\n\nWhen I didn't move the other tentacle touched my arm\u2014and lightning struck!\n\nI was lying dizzily on the ground watching it trot away. \"Shocking, isn't it!\" Hingst called back smugly. \"Big batteries.\"\n\nFloyd helped me to my feet and dusted me off. \"So good so far.\"\n\n\"Thanks. But you aren't the one who was short-circuited.\"\n\nI reported to Madonette as we went on, with Tremearne listening in. \"Applied technology,\" he said. \"Perhaps this lot isn't as bad as the rest of the crumb-bums on this planet.\" Since I was still tingling, and had a burnt taste in my mouth, I sneered in silence and did not bother to answer. Very soon after that Madonette called in that a creature like the one we had described was coming towards her. I clutched my sword in helpless anger, relaxed only when she called back.\n\n\"Just like yours\u2014only with a different name. Hoppe. As soon as it made the test it trotted off. What now?\"\n\n\"We go on\u2014and you take a break. If things are going to be the same, or similar, on both sides of the wall we'll find out first.\"\n\n\"Male chauv superiority?\"\n\n\"Common sense. We're three to your one.\"\n\n\"A solid argument\u2014and I could use the rest. Keep in contact.\"\n\n\"You've got it. Here we go.\"\n\nThe path had widened and was more of a dirt road now. We passed some tilled fields and came to a large grove of polpettone trees. Obviously cultivated since they were planted in neat rows. Beyond them was a low huddle of buildings that could be a farm.\n\nBlocking the path was a brick building with an archway that spanned the road; we slowed and stopped.\n\n\"Is that what I think it is?\" Steengo said.\n\n\"I think that it is a building with an arch under it,\" Floyd said. \"And we're not going to find out any more just standing around here.\"\n\nWe shuffled forward slowly and stopped again when a man appeared in the archway. Our hands twitched away from our weapons when he stepped out into the sunlight. He blinked his red-rimmed eyes against the glare, nodded his head so his mane of long white hair bobbed, then tapped the arrow-and-circle symbol picked out in white on the front of his gray robe.\n\n\"Welcome, strangers, welcome to Paradise. I am Afatt, the official greeter. Market opens at dawn tomorrow. You may stay out here, or if you wish to camp beyond the arch your weapons will be looked after until you return. A payment of one fedha is required for attendance.\"\n\nThe way he flicked a look over his shoulder as he said this strongly suggested that what he wanted was more bribe than payment.\n\n\"No way, aged Afatt,\" I intoned. \"Those you see before you are not peasant traders but galaxy-famous chart-topping musicians. We are... The Stainless Steel Rats!\"\n\nHis jaw dropped and he stepped back a pace. \"Don't need no rats in Paradise. A rusty, chipped old fedha will do...\"\n\n\"We got a real fan in old Afatt here,\" Floyd muttered. \"I thought the planet was hip-deep in TV sets?\"\n\nA more military Paradisian appeared in the archway. Younger, bigger, and he came complete with studded metal helmet and heavy leather trappings. \"What did you say?\" he said as he swung a shining and singularly nasty looking ax.\n\n\"You heard me, Sunny. I don't repeat myself for the troops.\"\n\nThis provoked a twisted snarl and a barked command.\n\n\"Guard\u2014fall out. We got some sheot shaggers here that need a lesson in civility.\"\n\nThis was followed instantly by the clanking of metal and the thud of running feet.\n\nMany of them.\nChapter 13\n\nThere were a lot of them, armed with a collection of nasty and lethal-looking weapons. I must learn to control impetuosity in speech on this slumworld. Think quickly, Jim, before things get any worse.\n\n\"I tempted a jest, good sir. I will be happy to repeat myself for your benefit. You, and your good men, have the pleasure of being in the presence of the finest musicians in the known galaxy!\"\n\nAs I spoke I touched the remote control on the side of my backpack and a mighty organ sounded out the opening bars of \"Mutants of Mercury.\" Floyd and Steengo quickly joined in with the opening lines.\n\nOne head good\u2014but two heads better\u2014\n\nGot brown eyes like an English setter...\n\nThe effect of this little jingle of genetic jest was very impressive. As a man the soldiers roared aloud and surged towards us.\n\n\"Do we fight or run?\" Floyd said grimly, grabbing at his sword. I started to shout fight\u2014but at the last instant called out\u2014\"Listen!\"\n\nFor they had forgotten about their weapons and were shouting with joy!\n\n\"It's them, like on the Galactic Greasecutter show...\"\n\n\"The hairy, ugly one\u2014that's Floyd!\"\n\n\"I want to hear 'How Much Is the Snakey in the Snakepit'!\"\n\nThen they were around us, trying to shake our hands and emitting hoarse cries of fannish enthusiasm.\n\n\"But\u2014but\u2014\" I but-butted. \"Your official greeter never heard of us?\"\n\nThe first soldier, snarls now turned to smiles, not too gently pushed the old man aside. \"Afatt never looks at the goggle box. But we do! Let me tell you it was like suicidesville around here when we heard that you were sent down. Should have known that you would have to end up here. Wait until the boys in the barracks hear about this. There'll be a crackup in the old kaserne tonight!\"\n\nThey escorted us cheerily under the arch and onto the drillfield beyond, our new host proudly leading the way.\n\n\"I'm Ljotur, Sergeant of the Guard. You all take it easy while I call this in. Drinks!\" he ordered his men. \"And food\u2014whatever they want.\"\n\nThis was more like it. The beer tasted like beer, although it was of an interesting green color. The soldiers crowded close, hanging on every word we said, so I chomped my jaw to get Tremearne's attention and made my report to him in the form of a speech.\n\n\"Gallant warriors of Paradise\u2014we are overwhelmed by your greeting. You have welcomed we drug-ridden convicts as heroes to your fair land. You ply us with food and drink and, by your loud cheers, I feel we have a beautiful future here.\"\n\n\"I certainly hope so,\" Tremearne's voice said inside my head. \"But until you find out the score on this male-female thing I am ordering Madonette to stay where she is.\"\n\n\"I agree completely,\" I called out. \"Don't you agree completely, guys, that this is the warmest welcome we have ever received?\"\n\nMy companions nodded without interrupting the flow of food and drink and there were gurgled shouts of agreement from all sides as more beer vanished. I was wiping my lips with the back of my hand when Ljotur reappeared.\n\n\"I have talked to Iron John himself who summons you to his presence soonest. But until the Chariots of Fire appear could you\u2014oh, would you!\u2014play us a number!\"\n\nHis words were drowned out by hearty masculine cries of joy.\n\n\"Let's set up for a quick gig, boys\u2014these guys deserve it.\" I looked around. \"Any requests?\"\n\nMany were shouted, but \"Nothing's Too Bad for the Enemy\" seemed to be most popular. Best choice too since it had an all-male lyric. Loud thunder rolled while lightning flared and sizzled. Our fans fell back into an appreciative half circle while we let fly.\n\nDeath and torture and murder and rape\u2014\n\nWE LIKE IT! WE LIKE IT!\n\nCutting and slashing and murder and looting,\n\nHacking and cracking and stabbing and shooting.\n\nBlowing up slowing up showing up to kill\n\nArson and cursin' done with a will\u2014\n\n'CAUSE...\n\nNOTHING'S TOO BAD FOR THE\n\nENEMEEE...\n\nDrinking and drinking and drinking and drinking\n\nShouting and cursing and lying and stinking\n\nChasing girls grabbing girls huggin' and kissin'\n\nShowing girls all the things they been missin'...\n\nAs can easily be imagined this delicate flower of a lyric really went down well with the troops. They were still cheering when there was a hissing rumble behind us and we turned to see that our transportation had arrived. Perhaps the locals were used to these things but it was really eye-bugging time for the tourists.\n\n\"Only for special occasions, special people,\" Ljotur said proudly.\n\nWe gaped in silence, lost for words. There were two of the vehicles, made of wood and decorated with gilt scrolls and strands of jewels. Each had a single wheel in front, which was steered by a tiller. This was manned by the driver, who rode high above. I looked at the closer one. A wide seat was in the middle and there were two wheels to the rear. All of which was pretty commonplace\u2014not counting the pricey decoration\u2014if you did not allow for the propulsion at the back. This was a shining metal tube, now crackling and emitting an occasional puff of smoke. I drew my attention away from it as the ornate door was thrown wide. I stepped in and seated myself on the soft cushions. Floyd and Steengo were ushered reverently into the other vehicle. Doors were slammed and Ljotur shouted a command to the drivers.\n\n\"You're off! Fuel on! Frapu viajn startigilojn! Drivers hit your starters!\"\n\nI saw now that there was a metal tank under my driver's seat. He reached down and opened a valve and I could hear the gurgle of liquid in the pipe. Then he stamped down on a pedal; the starter I guess.\n\nNo\u2014it just started the starter. The pedal pulled on a cord that ran on pulleys to the rear of the chariot. This lifted and dropped a small hammer that banged the starter on the shoulder. This was an individual, dressed completely in black, who sat on a little platform slung behind the wheels. Not only dressed in black, but with blackened arms and face, his hair a burnt stubble. I soon found out why. Liquid was now dripping from the metal tube and the starter reached out and touched a match to it, jumped back as it ignited. A tongue of black smoke and flame leaped out to the rear, singeing the soldiers who weren't quick enough out of the way.\n\nNow the starter was grinding away at a handle, presumably pumping air into the primitive jet. Within seconds the roar grew louder, the flame longer\u2014and my Chariot of Fire shuddered and began to slowly roll forward. Very showy. Though it probably only got about a mile to a hundred gallons. I waved cheerfully to my fellow victims, who waved feebly and fearfully back. Relax Jim, sit back and enjoy the ride.\n\nIt was hard to do. I admit I did not see much of the passing scenery, being too involved with thoughts of survival. Nor did I relax until our little convoy had stopped and the blowtorch behind me was extinguished. The chariot's door swung open to the blast of discordant horns. I grabbed up my pack and stepped down onto a gray stepping block.\n\nWhich was resistant but soft. I turned and looked and saw that it was not a step at all but a man dressed in gray, kneeling on all fours. He rose and scurried off, along with another human footstep. Midgets, about as tall as my waist and almost as wide. My companions had reacted as I had, our eyes met but we said nothing.\n\n\"Greetings,\" a stentorian voice bellowed. \"Welcome, welcome visitors to Paradise.\"\n\n\"Thanks much,\" I said to the tall and barrel-chested man who was draped in gold cloth. \"Iron John, I presume?\"\n\n\"Most flattering\u2014but you presume wrongly. Musical guests, kindly follow me.\"\n\nThe trumpets blared again, then the trumpeters opened ranks. Three gray-clad men hurried up and took our packs. I started to resist, then made the reluctant decision that it would be all right. The reception we had received at the archway had been too spontaneous to be planned. Our gold-clad greeter bowed to us, then led the way. Towards the brick steps of a brick building.\n\nIf the Paradisians were short on building materials they certainly weren't bereft of architectural imagination. Tall pillars, capped with ornate capitals, rose up to support the architrave of a complex entablature. Just like I had been taught in Architecture 1. To either side tall windows opened onto wide balconies. And all of this done in red brick.\n\n\"Looks great so far,\" Floyd said.\n\n\"Yes, great,\" I agreed. But I looked back to make sure the porters with our packs were right behind us. And I still had the concussion grenades in my pocket. No one ever got into trouble by being prepared\u2014as we used to say in the Boy Sprouts.\n\nDown a brick corridor over brick paving we went. Through a brick doorway into a great and impressive room. It was colorfully lit by the sunlight that streamed through the ceiling-high, stained-glass windows. Colorful scenes were depicted there of armies marching, attacking, fighting, dying, the usual thing. This motif was carried through to the walls, which were hung with tattered battle banners, shields and swords. Robed men who stood about the room turned and nodded to us as we entered. But our guide led us past them to the far wall, where there was an elevated throne, made of you-know-what, on which was seated the tallest man I have ever seen.\n\nNot only tall\u2014but naked.\n\nAt least he would have been naked if he had not been completely covered with rusty, reddish hair. His beard cascaded down his chest\u2014which was covered as well with hair. Arms and legs and, I couldn't help peeking when he stood, hair all down his belly and crotch as well. This was all that was visible since he was wearing a sort of jockstrap or sporran woven out of, well possibly, his own hair. All of it the color of rusty iron. I stepped forward and bowed a little bow.\n\n\"Iron John...?\"\n\n\"None other,\" he rumbled in a voice like distant thunder. \"Welcome Jim\u2014and Floyd and Steengo. Welcome Stainless Steel Rats. Your fame has gone before you.\"\n\nAlways good to meet a true fan. We all bowed now since this was not the kind of reception you normally get. Bowed yet again as all in the room cheered lustily.\n\nIron John sat down again and crossed his legs. He either painted his toenails or they were naturally rusty. I let it pass since there were a lot more things. I would like to know first.\n\n\"All here in Paradise were possessed of a great depression when you were arrested,\" he said. \"Falsely of course?\"\n\n\"Of course!\"\n\n\"I thought so. But the galaxy's loss is our gain. We are pleased since we now have, you might say, a monopoly on your talents.\"\n\nThis had an ominous sound which I ignored for the moment, cocking an ear as he rumbled on.\n\n\"The galaxy is so filled with guilt, sorrow and wrong-headedness that we chose, out of disgust, not to watch most of what is disseminated by television. I am sure that it will cheer you to know that, since your arrest and incarceration, we have canceled normal programming and have been running recordings of your numbers, day and night. Now, soon, we will be happily blessed with the originals themselves!\"\n\nThis was greeted by cries of enthusiasm and we replied with nods, grins and handshakes over our heads. When the shouts had died away old Rusty boomed out what they all wanted to hear.\n\n\"It is our hope that you will now\u2014play for us!\" More shouts. \"What a pleasure to hear live our favorite favorite\u2014'Nothing's Too Bad for the Enemy.' But while you are setting up we will broadcast a recording to warm up our nationwide audience, to prepare them for your first live performance.\"\n\nWhich was not a bad idea since, although we could get going fast, their TV technicians were another thing altogether. Very much on the antique side. They dragged in arm-thick cables, antique-looking, homemade cameras and lights and other gear that belonged in a museum. While this was happening a screen dropped down from the ceiling and lit up with lively color when the back projector came on.\n\nThe recorded program did not have what might be called the galaxy's most inspiring opening. About a thousand suntanned bodybuilders drove heavy stakes into the ground with sledgehammers, backed by the thud of a beating drum. The drum died away but the hammers kept hammering silently as the voice-over spoke.\n\n\"Gentlemen of Paradise\u2014we now bring you the special occasion that was announced a few minutes ago. I know that all of you, right across the land, are riveted to your sets. I think that we are going to get a hundred-percent rating on this one! So while The Stainless Steel Rats are warming up for their first-ever live performance here, we are privileged to play for you their special version of\u2014'The Spaceship Way'!\"\n\nAnd it really was special. We watched ourselves attacking the song with our usual gusto, listened once again to those lovely lyrics...\n\nWorking on the engines, in the engine room,\n\nWirin' and firin' an' waitin' for the boom.\n\nWhen the cannons blast like the sound of doom,\n\nYou know you're a-sweatin' in the engine room.\n\nCaptain on the bridge his fingers on the triggers\n\nAll the guns loaded by the spaceship riggers.\n\nSwoopin' on the enemy, million miles an hour\n\nCallin' to the engine room for power, power, power.\n\nPower, Power, Power make the electrons whirl,\n\nPower, Power, Power\u2014hear them protons swirl!\n\nPower, Power, Power will win the day\u2014\n\nPower, Power, Power, that's THE SPACESHIP WAY!\n\nWe nodded and smiled with fixed grins. Good-quality picture, good sound as well. The audience was looking at the screen instead of at us for the moment. Floyd looked at me, then raised his extended index finger to the side of his head and rotated it in a quick little circle. The universal hand signal for insanity. I nodded glum agreement. I couldn't understand it either.\n\nThere we were on the screen playing on a familiar set, wearing our regular concert costumes. Only one thing was wrong.\n\nUntil this moment none of us had ever seen the tenor who was right there with us, singing the song.\n\nTenor?\n\nIt had always been sung in sensuous contralto by Madonette.\nChapter 14\n\nAfter the TV intro we played our number, pretty mechanically I must say. Not that our audience noticed; they were too carried away simply by being in the Presence. They swayed and waved their hands in the air and fought to keep silent. But when Iron John joined us in the \"Power\" chorus they cheered and howled and sang right along with him. When the last power had been overpowered they broke into lusty shouted applause that went on for a long, long time. Iron John smiled beneficently at this and finally stopped it with a raised russet finger. There was instant silence.\n\n\"I join you in your enthusiasm for our honored guests. But we must give them time to rest after their strenuous day. We will surely hear them sing for us again. You must remember they are with us now forever. It is their rare privilege to be admitted to Paradise as full citizens, to live until the end of time in our fair land.\"\n\nMore cries of masculine joy. We concealed our overwhelming pleasure at this life sentence and kept our silence as we packed up our instruments and handed them to the waiting servants. Our audience moved out, still throbbing slightly with musical passion.\n\n\"A moment please,\" Iron John said, waiting for the others to leave. When we were alone he touched a button at his side and the tall doors swung silently closed. \"A fine song. We all enjoyed it.\"\n\n\"The Stainless Steel Rats aim only to please,\" I said.\n\n\"Wonderful.\" His smile vanished and he stared at us grimly. \"There is one more thing you must do to please me. Your stay here will be a long one and we want you to be happy. You will make us all happy, yourselves included, if you show a certain selection in topics of conversation.\"\n\n\"What do you mean?\" I asked\u2014although I had a good notion of what he was leading up to.\n\n\"We are very satisfied here. Adjusted and secure. I do not wish to see that security threatened. You gentlemen come to our land from a very troubled outside world. The galaxy is at peace\u2014or so you say. While ignoring the eternal war without end. The conflict of duality that we are free of here. You are the products of a society that is ego destroying instead of being ego building. You suffer from the negativity that blights lives, weakens cultures, sickens even the strongest. Do you know what I'm talking about?\"\n\nNeither Floyd nor Steengo answered so it was up to me. I nodded.\n\n\"We do. Although we might quibble with some of your conclusions the object of your attentions is quite clear. I can promise you that while we are enjoying your hospitality, neither I nor my associates will talk to anyone about the other sex. That is girls, women, females. It is a taboo topic. But, since you raised the issue I assume that you can discuss it...\"\n\n\"No.\"\n\n\"Right, answer enough. We will therefore enjoy your hospitality and not spoil it.\"\n\n\"You are wise beyond your years, young Jim,\" he said, and a trace of a smile returned. \"Now you must be tired. You will be shown to your quarters.\"\n\nThe doors opened, he turned away. End of interview. We strolled out as nonchalantly as we could. Old Goldy led us out as he had led us in, to some pretty luxurious, although still red brick, quarters. He turned on the TV, checked that the faucets worked in the bathroom, raised and lowered the curtains, then bowed himself out and closed the door. I touched my finger to my lips. Floyd and Steengo waited in twitching silence while I used the detector, borrowed from Tremearne, to sweep the room for bugs. After what we had seen on TV I had a great admiration for the electronics in this place.\n\n\"Nothing,\" I said.\n\n\"No women,\" Steengo said. \"And we can't even talk about them.\"\n\n\"I can live with that for awhile,\" Floyd cut in. \"But who was that singing our number?\"\n\n\"That,\" I said, \"was a very nifty example of some first-class electronic dubbing.\"\n\n\"But where did that joker come from?\" Floyd said. \"There I am playing right beside him\u2014and I swear that I have never seen him before. Maybe we really did blow baksheesh and this whole planet is a drug-inspired nightmare!\"\n\n\"Keep cool, keep calm. That guy was nothing but a bunch of electronic bytes and bits. Some really good techs digitalized that entire song, with all of us playing it. Then they animated a computer-generated male singer to follow all of Madonette's movements. Wrote her image out, wrote his in\u2014then rerecorded the whole thing just as if it were going out live. Only with a him instead of a her.\"\n\n\"But why?\" Steengo asked, dropping wearily into one of the deep lounges.\n\n\"Now you have asked the right question. And the answer is obvious. This side of Paradise is for men only. Not only haven't we seen any women here\u2014but pretty obviously they have been edited out of TV and presumably everything else going. It's a real man's world. And don't say why again because I don't know. You saw how high that wall is when we were on our way here. And we know from views of the thing from space that the city is on both sides of the wall. So the women\u2014if there are any women\u2014might very well be on the other side.\"\n\nNo one said why again but that was the only thing on our minds. I stared at their worried faces and tried to think of something nice. I did. \"Madonette,\" I said.\n\n\"What about her?\" Steengo asked.\n\n\"We've got to tell her what has happened.\" I stuck my thumb in my ear and addressed my pinkie. \"Jim calling Madonette. Are you on-line?\"\n\n\"Very much so.\"\n\n\"I read you as well,\" Tremearne said tinnily from my thumbnail.\n\nI outlined the events of the day. Said over and awaited any reaction. Madonette gasped, nor could I blame her, but Tremearne was all business as usual.\n\n\"You are doing well on your side of the wall. Is it time for Madonette to check out her side?\"\n\n\"Not yet, not until we have a few answers to an awful lot of questions.\"\n\n\"Agreed\u2014but only for now. What have you discovered about the artifact?\"\n\n\"Negative so far. Give us a break, Captain. Don't you think that getting in here, pressing the flesh and doing a gig is enough for one day?\" The silence lengthened. \"Yes, sir, right you are\u2014it's not enough. One alien artifact coming up. Over and out.\"\n\nI pulled my finger out of my ear, wiped the earwax off of it, stared gloomily into space.\n\n\"How do we find it?\" Floyd asked.\n\n\"I haven't the slightest idea. I just said that to get Tremearne off my neck.\"\n\n\"I know how we start,\" Steengo said. I launched a quizzical look in his direction.\n\n\"First the MIPSC and now this. Our humble harp player reveals hidden depths.\" He nodded and smiled.\n\n\"All those years laboring for the League perhaps. Didn't the ancient glad-hander at the gate tell us that there would be a market at dawn tomorrow?\"\n\n\"His very words,\" Floyd said. \"But so what? The artifact is long gone from the market.\"\n\n\"Of course. But the merchants aren't. There is a good chance that whoever bought the thing might still be there.\"\n\n\"A genius!\" I applauded. \"Behind those gray hairs lies even grayer gray matter that knows how to think!\"\n\nHe nodded acceptance. \"I never did enjoy retirement. What's next, boss Jim?\"\n\n\"Grab Goldy. Show strong interest in the market. Have him lay on a guide to take us there when it opens in the morning...\"\n\nAs though speaking his name had been a summons; bugles sounded, the door opened, our gilt-garbed guardian came in.\n\n\"A summons for you, oh lucky ones. Iron John will see you in the Veritorium. Come!\"\n\nWe went\u2014since we had little choice. For a change Goldy was not in a chatty mood; waving off our queries with a flick of his hand. More corridors, more bricks\u2014and another door. It opened into misty darkness. Stumbling and barking our ankles we made our way to a row of waiting chairs, sat down as instructed. It was even darker when Goldy closed the door behind him as he left.\n\n\"I don't like this,\" Floyd muttered, muttering for all of us.\n\n\"Patience,\" I said for lack of any more intelligent answer, then nervously squeezed my knuckles until they cracked. There was a movement of air in the darkness and a growing glow. Iron John swam into view, a blown-up image really. He pointed at us.\n\n\"The experience that you are about to have is vital to your existence. Its memory will sustain you and uplift you and will never be forgotten. I know that you will be ever grateful and I accept your tearful thanks in advance. This is the experience that will change you, develop you, enrich you. Welcome, welcome, to the first day of the rest of your new and fulfilling lives.\"\n\nAs his image faded I coughed to cover the grunt of suspicion that this old bushwah evoked. Never try to con a conman. I settled my rump more comfortably in the chair and prepared to be entertained.\n\nAs soon as it started I could see that the holofilm was very professionally made. I appreciated that the young, the gullible\u2014or the just plain stupid\u2014would be very impressed by it. The mist churned, the russet light grew brighter and I was suddenly in the midst of the scene.\n\n* * *\n\nThe King watched in silence as the group of armed men walked warily into the forest and disappeared from sight among the trees. Outwardly he was patient as he waited, although he reached up and touched his crown from time to time as though reassuring himself that it was still there, that he was still king. A very long time later he stiffened, turned his head and listened as slow footsteps shuffled through the thick leaves below the trees. But no warrior appeared, just the thick and twisted figure of his jester, headdress bobbling, lips moist with flecked saliva.\n\n\"What did you see?\" the king asked at last.\n\n\"Gone, Majesty. All gone. Just like all of those who have gone before. Vanished among the trees around the lake. None returned.\"\n\n\"None ever return,\" the king said, sorrow and defeat dragging him down. He stood that way, unknowing, unseeing as a young man appeared and strode towards him, a silent gray dog walked at his side. The jester, jaw agape, spittle pendulous, backed away as the stranger approached.\n\n\"Why do you grieve, oh king?\" he asked in a light and clear voice.\n\n\"I grieve for there is part of the forest in my kingdom where men do go\u2014but none return. They go in tens and twenties\u2014but none is ever seen again.\"\n\n\"I will go,\" the young man said, \"but I will go alone.\"\n\nHe snapped his fingers and, without another word being spoken, man and dog walked off into the forest. Beneath the trees and pendant mosses, around the hedges and nodding cattails to the edge of a dark pond. The young man stopped to look at it\u2014and a hand, sudden and dripping, rose from the water and seized the dog. Pulled it beneath the surface. The ripples died away and the surface was still.\n\nThe young man did not cry or flee, just nodded.\n\n\"This must be the place,\" he said.\n\n* * *\n\nThe darkness faded and light returned. Iron John was gone, the chamber was empty. I looked at Floyd who seemed just as bewildered as I was.\n\n\"Did I miss the point somehow?\" I asked.\n\n\"I feel sorry for the dog,\" Floyd said. We both looked at Steengo who was nodding thoughtfully.\n\n\"That's only the beginning,\" he said. \"You'll understand what is happening when you see the rest.\"\n\n\"You wouldn't like to, maybe, explain just what you are talking about?\"\n\nSteengo shook his head in a solemn no. \"Later, perhaps. But I don't think I will have to. You will see for yourselves.\"\n\n\"You've seen this holoflick before?\" Floyd asked.\n\n\"No. But I have read my mythology. It's better that you see the rest before we talk about it.\"\n\nI started to protest, shut my mouth. Realized that there was no point in probing further. The door opened and our guide reappeared.\n\n\"Just the man we are looking for,\" I said, remembering our earlier decision. \"We have heard, from reliable sources, that there is to be an outdoor market at dawn tomorrow.\"\n\n\"Your sources are correct. Tomorrow is the tenth day and that is market day. Always on the tenth day because the nomads remember by marking a finger each day with soot until all fingers are...\"\n\n\"Right, thanks. I can count to ten without dirty fingers. My fellow musicians and I would like to visit this market\u2014is this possible?\"\n\n\"You have but to ask, great Jim of The Stainless Steel Rats.\"\n\n\"I've asked. Can someone show us the way in the morning?\"\n\n\"'Tis more fit that you use the Chariots of Fire...\"\n\n\"I agree, more fit. But more fit that we be fit. Walking is a wonderful exercise.\"\n\n\"Then walk you shall, if that is your desire. An escort will be provided. It is now the hour of dining and a banquet has been prepared in your honor. Will you be so kind as to follow me?\"\n\n\"Lead on, my friend. As long as it is not polpettone again we are your avid customers.\"\n\nAs we followed him out I discovered that my fingers had a life of their own. Or, more probably, were being twitched into activity by my worried subconscious. They flicked over the computer controls and the glowing numbers appeared before me.\n\nNineteen and a pulsing red eleven.\n\nEleven days to go. The morning market had better produce something.\nChapter 15\n\n\"It is going to be such a lovely day,\" the voice said.\n\nEach word shot through my head like a rusty arrow, grating and scraping against the growing headache that was throbbing there. I opened one eye blurrily and bright light added to the pain. I had only enough energy to twist my lips into a surly snarl as our gold-clad host flitted about our quarters. Opening curtains, picking up discarded clothing, generally being as obnoxious as possible at this predawn hour. Only when I heard the outer door slam did I crawl from the bed, turn off the searing lights, stumble on all fours to my pack where it rested against the wall. On the third fumbling attempt I managed to open it and click out a Sobering Effect pill. I swallowed it dry and sat motionlessly while I waited for its beneficent chemicals to seep through my fractured body.\n\n\"What was in that green beer?\" Floyd said hoarsely, then began to cough. Moaning in agony between coughs as his aching head was kicked about. My headache was seeping away so I clicked out a pill for him and walked unsteadily across to his bed of pain.\n\n\"Swallow. This. Will. Help.\"\n\n\"Quite a party last night,\" Steengo said benevolently, joined fingers resting comfortably on the ample bulge of his stomach.\n\n\"Die,\" Floyd gasped, unsteady fingers groping for the pill. \"And burn painfully in hell forever. Plus one day.\"\n\n\"A bit hungover are we?\" Steengo asked cheerfully. \"I suppose there is good reason, considering the length of the nights here. Their parties must go on forever. Or maybe it just seems that way. Eat a bit, sleep a bit. Eat a bit, drink a bit. Or maybe more than a bit. I thought that the beer tasted a little on the nasty side. So I only had one. But the meat courses! Tremendous, vegetables, good gravy, liked the bread and red sauce, plus...\"\n\nHis voice died away as Floyd crawled out of bed and staggered, groaning, from the room.\n\n\"You are cruel,\" I said, smacking my dry lips together and feeling a little better.\n\n\"Not cruel. Just pointing out a few truths. This mission first. Overdrinking, hangovers and Technicolor yawns saved for our victory celebration.\"\n\nThere was nothing I could say. He was right.\n\n\"Message received,\" I said, reaching for my clothes. \"The quiet life and plenty of rest and raw vegetables. Think positive.\"\n\nDawn brightened the window. A new day. Ten days to deadline. I was thinking negative and I shook myself like a wet dog and tried to shrug off the mood. \"Let's go to the fair.\"\n\nWhen we emerged from the BOQ, Sergeant Ljotur was waiting for us. He snapped to attention and gave a mighty salute\u2014as did the squad of soldiers from the gate guard that he had brought with him.\n\n\"We take you to the market!\" he called out. \"These men are all volunteers, eagerly happy to carry any purchases finest musicians in galaxy may make.\"\n\n\"Greatly appreciated. Lead on,\" I said as we stepped out briskly on the red brick road.\n\nThe sun was a glowing crimson disk on the horizon when we reached the market. The Fundamentaloid nomads must have been early risers because everything was in great swing already. And gory too, I thought I heard a low moan from Floyd, but the baaing and farting of the sheots tended to drown out most other sounds. Complain they might as the butchered carcasses of their late companions were unloaded from their backs. But there had to be more than a meat market here; eyes averted we hurried past the sanguineous display.\n\nNow bearded nomads solicited our attention in pleading voices, pointing out the attractions of their wares. Which weren't that attractive. Tired-looking vegetables, crude clay pots, piles of dried sheot chips for the barbecue.\n\n\"Pretty grim,\" Floyd said.\n\n\"Not important,\" I told him, jerking my thumb towards the strolling customers. \"They are the ones that we are interested in.\" I took out the photographs of the artifact that we were looking for and passed one to each of my companions. \"Find out if any of the Paradisians have seen this.\"\n\n\"We don't just spring it on them?\" Steengo said doubtfully.\n\n\"You're right. We don't. During the sleepless hours of the night I worked up a cover story. It goes this way, something close to the truth. The nomads found this thing in a streambed after a flash flood. Tried to trade it to the keepers of the Pentagon, who have a strict policy of noncommunication. However it was photographed when presented and only later was it recognized as an archeological artifact of possible interest.\"\n\n\"Reasonable,\" Steengo said doubtfully. \"But what are we doing with the photos?\"\n\n\"Given to us when we were booted out of the place. Hints made of rewards, possible remission of sentence, lots of fedha. With great reluctance we agreed to look for the thing since, simply, what have we got to lose?\"\n\n\"Thin but plausible,\" Floyd said. \"Let's give it a try.\"\n\nThere was no difficulty talking to the Paradisians, if anything it was hard to get rid of them once approached. How they loved The Stainless Steel Rats! Soon I had a string of adoring fans trailing behind me\u2014along with most of the squad of guards. Everyone wanted to help; none of them knew a thing. But\u2014one name kept cropping up during the questioning: Sjonvarp.\n\nSteengo pushed through the crowd and held up the now dog-eared photograph. \"Still nothing. But a couple of them said to ask Sjonvarp. Who seems to be the top trader around here.\"\n\n\"I heard the same thing. Grab Floyd. He must be recovering because I saw him looking at the fermented sheot-milk stand. Bring him here before he makes a mistake that he will long remember.\"\n\nSjonvarp was easy enough to find, with countless fingers pointing us the way. He was a tall and solidly built man with iron-gray hair. His stern face broke into a smile when he turned to see who had called out his name.\n\n\"The Stainless Steel Rats in the flesh! I am trebly blessed!\"\n\nWe hummed two bars of \"All Alone\" followed by a brisk buck and wing. Which elicited a round of applause from the spectators and a broader smile from Sjonvarp.\n\n\"Such rhythm and beauty!\" he said.\n\n\"We sing 'em the way you like it,\" I said. \"It is told in the market that you are the master-trader in these parts.\"\n\n\"I am. Pleased to make your acquaintances, Jim, Floyd and Steengo.\"\n\n\"Likewise. If you have a moment I have a picture here I would like you to look at.\" I hit the high points of our spiel as I passed the pic over. He only half listened, but did put all of his attention on the photo. Turning it around at arm's length, squinting farsightedly to make it out.\n\n\"Of course! I thought so.\" He handed it back to me. \"Some markets ago, I forget exactly how many, one of these odorous simpletons traded it to one of my assistants. We buy anything that might be of scientific interest for the specialists to examine. It didn't look like much. But I gave it to old Heimskur anyway.\"\n\n\"Well that takes care of that then,\" I said, tearing the photo up and dropping the pieces. \"We're doing our concert tonight\u2014I can get you a ticket if you want one.\"\n\nThe artifact was instantly forgotten\u2014as I hoped, although it took some time for us to extract ourselves from the attentive embrace of our fans. Only by saying that it was rehearsal time did we manage to break away.\n\n\"Don't we look for the thing any more?\" Floyd asked worriedly. A good musician, but I think drink was eroding his brain cells.\n\n\"We have the man's name,\" Steengo said. \"That's what we look into next.\"\n\n\"How?\" Floyd asked, still suffering from semiparalysis of the neural network.\n\n\"Any way we can,\" I told him. \"Make friends. Drop names. Drop Heimskur's name among the others. We find out who he is and what he does. Now, as we stroll, I'll report in.\"\n\nTremearne and Madonette listened carefully to my report. He overed and outed but she stayed to chat.\n\n\"Jim, it's time I left my hole in the wall and visited the other half of the city. It must be safe...\"\n\n\"We hope that\u2014but we don't know that. And there is no point in your taking any chances as long as the thing we are looking for is here. Enjoy the break. And don't do a thing until we find out more here.\"\n\nWe found lunch waiting in our quarters. Fruit and slices of cold meatloaf on silver plates, covered with crystal domes.\n\n\"Great!\" Floyd said, chomping down a slice.\n\n\"Probably minced sheot shank,\" Steengo said, suddenly gloomy.\n\n\"Food's food and I never consider the source.\" Floyd reached for another slice just as our golden greeter appeared.\n\n\"A pleasure to see you musical Rats enjoying yourselves. When you have eaten your fill I have a request for the presence of Rat Jim.\"\n\n\"Who wants me?\" I asked suspiciously through a mouthful of sweet pulp.\n\n\"All will be revealed.\" He put his index finger along his nose, winked and rolled his eyes. Which silent communication I assumed meant something like you'll find out soon enough. I had no choice. And I had lost my appetite. I wiped my fingers on a damp cloth and followed him yet another time.\n\nIron John was waiting for me at the door of the Veritorium where we had all seen the puzzling holoflic.\n\n\"Come with me, Jim,\" he said with a deep voice like distant thunder. \"Today you will see and understand all of the revelation.\"\n\n\"I'll get the others...\"\n\n\"Not this time, Jim.\" His hand closed gently but firmly onto my shoulder and I had little choice but to go along with him. \"You are wise beyond your years. An old head on a young body. Therefore you are the one who will be helped the most by your understanding of this mystery that is no mystery. Come.\"\n\nHe sat me down but did not join me; yet I was aware of his presence close by me in the darkness. The mist roiled and cleared and I was once again by the lake.\n\n* * *\n\nThere was only silence in the forest around the dark pond. As the last ripple died away the young man turned and left without looking back. Trod the dead leaves beneath the trees until be emerged and saw the king before him.\n\n\"There is something I must do,\" he told the king, nor would he say any more. The king saw that the man's dog was gone\u2014but the man himself was unharmed. He had many questions but did not know how to speak them. Instead he followed the young man back to the castle. In the courtyard the young man looked around, then spotted a large leathern bucket.\n\n\"I need that,\" he said.\n\n\"Take it.\" The king dismissed him with a wave of his hand. \"Remember I have helped you. One day you must tell me what you found in the woods.\"\n\nThe young man turned in silence and made his way, alone, back to the dark pond. There he dipped the bucket into the water and hurled its contents into the ditch nearby. Another and another. He did not stop but worked steadily at bailing out the pond. It was hard, slow work. Yet the sun never set, the light never changed, the young man never stopped.\n\nAfter a great period of time the water was almost gone and something large was revealed lying in the mud on the bottom of the pond. The young man kept emptying the water until he revealed a tall man who was covered with reddish hair, like rusty iron, from head to foot. The large man's eyes opened and he looked at the young man. Who beckoned to him. With a heaving shake the rusty man rose from the pond's bottom and followed the young man away from the pond and through the woods.\n\nTo the castle of the king. All of the soldiers and retainers fled when they appeared and the king alone stood before them.\n\n\"This is Iron John,\" the young man said. \"You must imprison him in an iron cage here in the courtyard. If you lock the cage and give the key to your queen the forest will be safe again for those who walk through it.\"\n\nMist rose and darkened the scene. It was the end.\n\n* * *\n\nThe red-furred hand was heavy on Jim's shoulder\u2014but it did not bother him.\n\n\"Now you understand,\" Iron John said, newfound warmth in his voice. \"Now you can release Iron John. Welcome, Jim, welcome.\"\n\nI wanted to say that I felt more confusion than comprehension. That I was experiencing something, yet not understanding it. Instead of speaking my feelings aloud I suddenly found that my eyes were brimming with tears. I did not know why\u2014although I knew that they were nothing to be ashamed of.\n\nIron John smiled at me and, with a great finger, wiped the tears from my damp cheeks.\nChapter 16\n\n\"What was all that about?\" Floyd asked when I returned to our quarters. He was jazzing with his trombonio, a complex and gleaming collection of golden tubes and slides, which made some very interesting sounds indeed. Most of them, regrettably, of an ear-destroying nature.\n\n\"More training film,\" I said, as nonchalantly as I could. I was surprised to hear a certain quaver in my voice as I spoke. Floyd tootled on, unaware of it, but Steengo who appeared to be asleep on the couch opened one eye.\n\n\"Training film? You mean more about the pool in the forest?\"\n\n\"You got it in one.\"\n\n\"Did you find out what was in the pool? The thing that dragged the dog down?\"\n\n\"A stupid story,\" Floyd said, and tootled a little fast riff. \"Although I do feel sorry for the dog.\"\n\n\"It wasn't a real dog,\" Steengo said. He looked at me, seemed to be waiting for me to speak, but I clamped my jaw shut and turned away. \"Nor was it a real pool.\"\n\n\"What do you mean?\" I asked, looking at him.\n\n\"Mythology, my dear Jim. And rites of passage. It was Iron John at the bottom of the pool, wasn't it?\"\n\nI jumped as though I had been zapped with an electric shock. \"It was! But\u2014how did you know that?\"\n\n\"I told you I read my mythology. But the thing that really disturbs me\u2014not this training film as you call it\u2014is the fact that Iron John is here in the flesh, solid and hairy.\"\n\n\"You've lost me,\" Floyd said, looking from one to the other of us. \"A little explanation is very much in order.\"\n\n\"It is,\" Steengo said, swinging his feet around so he sat up straight on the couch. \"Mankind invents cultures\u2014and cultures invent myths to justify and explain their existence. Prominent among these are the myths and ceremonies of the rites of passage for boys. The passage from boyhood to manhood. This is the time when the boy is separated from his mother and the other women. In some primitive cultures the boys go and live with the men\u2014and never see their mothers again.\"\n\n\"No big loss,\" Floyd muttered. Steengo nodded.\n\n\"You heard that, Jim. In all cultures mothers try to shape sons in their female image. For their own good. The boys resist\u2014and the rite of passage helps this resistance. There is always symbolism involved, because symbols are a way to represent the myths that underlie every culture.\"\n\nI thought about this; my head hurt. \"Sorry, Steengo, but you left me behind completely with that one. Explanation?\"\n\n\"Of course. Let's stay with Iron John. You have just said that you didn't understand it\u2014yet I think that it affected you emotionally.\"\n\nI started to protest, to lie\u2014then stopped. Why lie? I tried not to lie to myself, ever. This was a good moment to apply that rule.\n\n\"You're right. It got to me\u2014and I don't know why...\"\n\n\"Myths deal with emotions, not facts. Let's look at the symbols. Did the young man bail out the pool and find Iron Hans, or Iron John at the bottom?\"\n\n\"That's exactly what happened.\"\n\n\"Who do you think Iron John is? In the story I mean, not the one walking around here. But before you answer that\u2014who do you think the young man in the story was?\"\n\n\"That's not too hard to figure out. Whoever the story was aimed at, whoever was watching it. In this case, since I was there alone, I guess it must have been me.\"\n\n\"You are correct. So in the myth you, and every other young man, are looking for something in the pool, and have to work very very hard with the bucket to find it. Now we come to Iron John, the hairy man at the bottom of the pool. Is it a real man?\"\n\n\"No, of course it couldn't be. The man at the bottom of the pool has to be a symbol. Part of a myth. A symbol of manhood, maleness. The primitive male that lies beneath the surface in all of us.\"\n\n\"Bang-on, Jim,\" he said in a low voice. \"The story is trying to tell you that when a man, not a boy, looks deep inside himself, if he looks far down and for long enough, works hard enough, he will find the ancient hairy man within himself.\"\n\nFloyd stopped playing and his jaw gaped. \"You guys been smoking something I don't know about.\"\n\n\"Not smoking,\" Steengo said. \"Sipping at the font of ancient wisdom.\"\n\n\"Do you believe this myth?\" I asked Steengo. He shrugged.\n\n\"Yes and no. Yes, the process of growing up is a difficult one and anything that helps the process is a good thing. Yes, myths and coming-of-age ceremonies help prepare boys, giving them the assurances they need in the transition from boy to man. But that is as far as I will go. I say no resoundingly to a myth manifest as reality. Iron John alive and well and leading the pack. This is a fractured society here, without women and without even the knowledge of women. Not good. Quite sick.\"\n\nI was uneasy at this. \"I don't agree all the way. I was affected very strongly by watching that story. And I am a very hard guy to con. This got to me.\"\n\n\"It should have\u2014because it was dealing with the very stuff of personality and self. I have a feeling, Jim, that yours was not the happiest of childhoods...\"\n\n\"Happy!\" I laughed at the thought. \"You try growing up on a porcuswine farm surrounded by bucolic peasants who are not much brighter than their herds.\"\n\n\"And that includes your father and mother?\"\n\nI started to answer warmly, saw what he was doing and where this was going. I shut up. Floyd shook the spittle from his so-called musical instrument and broke the silence.\n\n\"I still feel sorry for the dog,\" he said.\n\n\"Not a real dog,\" Steengo said, turning away from me. \"A symbolic dog like everything else you saw. The dog is your body, the thing you order around, sit up, beg.\"\n\nFloyd shook his head in amazement. \"Too deep for me. Like that pool. If I could change the subject from theory to fact for just a moment\u2014what's next on the agenda?\"\n\n\"Finding Heimskur, of course, so we can find out if he still has the artifact,\" I said, happily putting this other matter aside. \"Any suggestions?\"\n\n\"Brain empty,\" Floyd said. \"Sorry. That hangover never really went away.\"\n\n\"I'm glad some of us didn't drink,\" Steengo said, a sudden edge of irritation to his voice.\n\nFor personal reasons I was happy to hear it, glad that he was still human, he came on pretty strong with the myth stuff. Forget this for awhile. I ticked off on my fingers. \"We have only two choices. Hint around about him and gather what information we can. Or blurt right out that we want to see him. Personally, I'm all for the blurting since there is a kind of time limit on this investigation.\" Like ten days to the grim reaper. \"Let's ask Goldy, our majordomo. He seems to know everything else.\"\n\n\"Let me do it,\" Steengo said, standing and stretching. \"I'll talk to him like an old buddy and work the conversation around to science and scientists. And Heimskur. Be right back.\"\n\nFloyd watched him go, tootling a little march in time with his footsteps. \"This Iron John stuff sort of gets to you,\" he said after the door had closed.\n\n\"Yes\u2014and that's the worst part. I don't know why I'm bothered.\"\n\n\"Women. I had six sisters and there were two aunts who lived with us. I had no brothers. I never think about women except one at a time in the right situation.\"\n\nBefore I had to listen to one more boring macho tale about the right situation I excused myself and went for a jog. Returned sweating nicely, did some push-ups and sit-ups, then went for a wash. Steengo was there when I came out. Shaking his joined hands over his head when I lifted a quizzical eyebrow.\n\n\"Success. Heimskur is head of the bunch who Labor in the Cause of Science, or so Veldi says.\"\n\n\"Veldi...?\"\n\n\"The doorman here. He does have a name after all. From what he says I get the feeling that this is a pretty stratified society with everyone in their correct place. Great respect is given to the scientist. Veldi was more than respectful when he talked about them because they appear to be the ones pretty much in charge.\"\n\n\"Great. How do we get to meet Heimskur?\"\n\n\"We wait patiently,\" Steengo said and looked at his watch. \"Because any moment our transportation will be here to take us to his august presence.\"\n\n\"Not the Chariots of Fire again!\" Floyd groaned.\n\n\"No. But something that sounds just as ominous. A Transport of Delight...\"\n\nBefore we had time to dwell too long on that thought there was a brisk knocking and gold-clad Veldi threw the door open.\n\n\"Gentlemen\u2014this way if you please.\"\n\nWe walked heads high and strong. Hiding any qualms we might have had. Though we shuddered to a halt when we saw what was awaiting us.\n\n\"Your Transport of Delight,\" Veldi said proudly, waving magnanimously in the direction of what could only be a landlocked lifeboat.\n\nIt was snow-white, clinker-built, with a stub mast festooned with flags, white wheels just visible tucked under the keel below. A uniformed officer looked down from the rail above, saluted, gave a signal\u2014and the rope ladder clattered down to our feet.\n\n\"All aboard,\" I said as I led the way.\n\nCushioned divans awaited us while attendants beckoned and held out jars of cool drink. As soon as we were seated the officer signaled again and the drummer in the bow whirred his sticks in a rapid drumroll\u2014then shifted to his bass drum. As the first, methodical boom boomed out, the Transport of Delight shuddered. Then began to roll slowly forward.\n\n\"A galley\u2014without slaves or oars,\" Floyd said.\n\n\"Plenty of slaves,\" I said as a wave of masculine perspiration wafted up from the funnel-shaped vent beside me. \"But instead of oars they are grinding away at gears or some such, to turn the wheels.\"\n\n\"No complaints,\" Steengo said, sipping at his wine. \"Not after the Chariots of Fire.\"\n\nWe rolled ponderously between the buildings, nodding at the bystanders and occasionally giving a royal flick of the hand at some of our cheering fans. We moved on through what appeared to be a residential quarter and beyond it into a parklike countryside. Our road wove between the trees, past a row of ornamental fountains to ponderously stop before an immense glass-walled building. A party of elegantly dressed ancients awaited us. Led by the most ancient of them all, white-clad and standing firmly erect. But his face was wrinkled beyond belief. I clambered down the ladder and dropped before him.\n\n\"Do I address the noble Heimskur?\"\n\n\"You do. And of course you are Jim of the Rats. Welcome, welcome all.\"\n\nThere was plenty of handshaking and glad cries of joy before Heimskur broke off the reception and led me into the glass building.\n\n\"Welcome,\" he said, \"doubly welcome. To the College of Knowledge from whence all good things flow. If you will follow me I will explain our labors to you. Since you gentlemen come from the surging, mongrel worlds outside our peaceful boundaries you will surely appreciate how the application of intelligence makes our society such a happy and peaceful world. No strife, no differences, a place for everyone and everyone in their place. Down this way are the Phases of Physics, the Caverns of Chemistry. There the Avenues of Agriculture, next to them the Meadows of Medicine, while just beyond is the Museum of Mankind.\"\n\n\"Museum?\" I inquired offhandedly. \"I simply love museums.\"\n\n\"Then you must see ours. It charts the difficulties through which we passed before coming here, a rite of passage and of cleansing, before we found safe haven on this world. Here we grew and prospered and the record is clear for all to see.\"\n\nAnd pretty boring if not just downright preposterous. Cleaner than clean, whiter than white. The only thing missing were the halos on the saints who had accomplished so much good.\n\n\"Inspirational,\" I said when we finally reached the end of the exhibition.\n\n\"It is indeed.\"\n\n\"And down this way?\"\n\n\"The museum for students. Biologists can examine the plant life of our planet, geologists the strata and the schist.\"\n\n\"Archeologists?\"\n\n\"Alas, very little. The crudest of artifacts left by the long-dead indigenes who first settled here.\"\n\n\"May we?\"\n\n\"By all means. You see\u2014fire sticks and crude pottery. A hand ax, a few arrow points. Scarcely worth preserving were we not so faithful to our role as recorders and archivists.\"\n\n\"Nothing more?\"\n\n\"Nothing.\"\n\nI dug the photograph from an inside pocket, took a deep breath\u2014and passed it over.\n\n\"You may have heard that the warders in the Pentagon promised us favors if we helped them find this?\"\n\n\"Did they indeed? I would believe nothing they said.\"\n\nHe took the photograph and blinked at it, handed it back. \"Just like them to lie and cause trouble for no reason.\"\n\n\"Lie?\"\n\n\"About this. It was brought here. I examined it myself. Not indigenous at all, couldn't possibly be. Probably something broken off an old spaceship. Meaningless and worthless. Gone now.\"\n\n\"Gone?\" I fought to keep the despair from my voice.\n\n\"Discarded. Gone from Paradise. Nonexistent. Men have no need of such rubbish therefore it is gone forever. Forget the worthless item Jim and we shall talk of far more interesting things. Music. You must tell me\u2014do you write your own lyrics...?\"\nChapter 17\n\nWe were very silent on our return trip, scarcely aware of the manifold pleasures that rode with us in our Transport of Delight. Only behind the closed doors of our quarters did we let go. I nodded appreciatively as I listened while Floyd swore blasphemously and scatologically; he had a fine turn of phrase and went on for a long time without repeating himself.\n\n\"And I double that,\" I said when lack of breath forced him to subside. \"We have indeed been hard done by.\"\n\n\"We have,\" Steengo agreed. \"But we have also been lied to.\"\n\n\"What do you mean?\"\n\n\"I mean that Heimskur was selling us a line of old camel cagal. More than half of his so-called history of science and nature was pure propaganda for the troops. If we can't believe him about that\u2014how can we believe him when he shovels a lot of bushwah about the artifact? Do you remember his last words?\"\n\n\"No.\"\n\n\"Neither do I. But I hope someone does. I imagine that you didn't notice it\u2014but I was doing a lot of head-scratching and nose-picking while we were doing that tour.\"\n\nFloyd wasn't being bright today and gaped at the news. I smiled and put my index finger into my ear. \"Come in, ear in the sky. Do you read me?\"\n\n\"No but I hear you,\" Captain Tremearne said through my fingernail.\n\n\"Good. But more important\u2014did you listen in to our guided tour?\"\n\n\"All of it. Very boring. But I recorded it anyway, the way you asked.\"\n\n\"The way Steengo asked\u2014credit where credit is due. Would you be so kind as to play back the last speech about the artifact.\"\n\n\"Coming up.\" After some clattering and high-pitched voices whizzing by, our aged guide sounded forth.\n\n\"Discarded. Gone from Paradise. Nonexistent. Men have no need of such rubbish therefore it is gone forever.\"\n\nI copied it down and got it right after a couple of repeats. \"That's it. Thanks.\"\n\n\"There,\" Steengo said, tapping the paper. \"Weasel wording. That tricky old devil was playing with us, knowing that we had some reason to be interested in the thing. He never said destroyed, not once. Discarded? That means it might be still around someplace. Gone from Paradise\u2014could be anywhere else on this planet. But I particularly like the bit about men having no need for the thing.\" He smiled a smile like a poker player laying down five aces.\n\n\"If men have no need for it\u2014what about women?\"\n\n\"Women?\" I felt my jaw hanging open and closed it with a clack. \"What about them? There are only men here?\"\n\n\"How right you are. And right on the other side of the town wall is\u2014what? I'm betting on women. Either that or an awful lot of cloning is going on in this place. I'll bet on nature and some kind of connection through the wall.\"\n\nMy jawphone buzzed and Tremearne's voice echoed inside my sinuses. \"I agree with Steengo. And so does Madonette. She's already on her way along the wall to the city and will report as soon as she finds out anything.\"\n\nI started to protest, realized the futility, kept my mouth shut. \"It figures,\" I said. \"The gang in charge here lie about everything else\u2014so lying about the artifact just comes naturally. We'll have to wait...\"\n\nI shut up as Veldi knocked quietly, then opened the door. \"Good news!\" he announced, eyes glowing with passion. \"Iron John has chosen to speak to The Stainless Steel Rats\u2014in the Veritorium itself. An honor above all other honors. Hurry, gentlemen. But first brush your clothing and, with the exception of heroically bearded Floyd, diple the five o'clock shadow now gracing your musical jaws. What pleasures do await you!\"\n\nPleasures better lived without. But this was a royal command and no way to get around it. I took a bit of diple-fast and rubbed my jaw smooth, combed my hair and tried not to scowl at myself in the mirror. I was the last to emerge and we boarded the Transport of Delight in silence, rolled ponderously to our destiny.\n\n\"I wonder why all three of us?\" Steengo said, sipping his glass of chilled wine. \"Last time it was you alone at the training-film session, wasn't it, Jim?\"\n\n\"I have no idea,\" I said, wanting to change the subject. Nor was I too pleased with his light-hearted attitude. I tried to think about Madonette going in alone to the other city, but my thoughts kept trundling back to Iron John. What was going to happen now?\n\nWhen we entered the Veritorium I was surprised at how big it really was. It was better lit now and I saw that rows of seats reached up in a semicircle. They were all filled now\u2014with the oldest collection of Paradisians I had seen so far. Bald heads and gray hair, wrinkles and toothless jaws.\n\nIron John himself stepped forward to greet us. \"You are all truly welcome here\u2014and these seats are for you.\" They were three of the best in the front row\u2014separated from the others. \"You are our honored guests, musical Stainless Steel. Rats. This occasion is a special one\u2014specially so for young James diGriz. You are the youngest man here, Jim, and very soon you will find out why. Your companions will, I am sure, watch with pleasure. Not only pleasure but I sincerely hope that they will learn by observation. Now we begin...\"\n\nCued by his words the lights died and darkness filled the Veritorium. Footsteps sounded in the darkness, and there was a small laugh. Light appeared and I saw the small boy hurry forward, stumbling a bit under the weight of the box he was carrying. He put it down and opened the lid, took out a top that started spinning when he touched its switch. Then he took out a tray of blocks, started to build a tower with them. When it was high enough he turned to take another toy out of the box. He was a very concentrated, very intense young boy, about eight years old. He rummaged deeper in the box, then looked around with a childish frown.\n\n\"Don't hide, teddy,\" he said. Looked behind the toy box, then into it again and then\u2014with sudden determination\u2014turned and hurried off. He vanished from sight but I could hear his footsteps going away, stopping. Then coming back. Carrying a teddy bear. A commonplace, slightly worn, very ordinary teddy bear. He propped it against the toy box and started building a second tower from the blocks.\n\nThe scene grew lighter and I realized we were back in the castle courtyard. The boy was alone\u2014or was he? Something was there in the darkness, a shape that grew clearer.\n\nIt was an iron cage and, sitting silently, inside it was Iron John. The boy shouted and knocked over the block towers, ran to pick up the strewn blocks. Looked at Iron John, then away. The cage and its occupant must be a familiar sight to him.\n\nNothing else happened. The boy played, Iron John watched him in silence. Yet there was an electric tension in the air that made it hard to breathe. I knew that something vitally important was about to happen, and when the boy reached again into the toy box I found myself leaning forward.\n\nWhen he took the small golden ball from the box I realized that I had been holding my breath; I let it out with a gasp. Nor was I the only one for around me in the darkness there were echoes of my gasp.\n\nThe ball bounced and rolled and the boy laughed with pleasure.\n\nThen he threw it once, harder than intended, and it rolled and rolled. Through the bars of the iron cage to stop at Iron John's feet.\n\n\"My ball,\" the boy said. \"Give it back.\"\n\n\"No,\" Iron John said. \"You must unlock this cage and let me out. Then you will have your golden ball back.\"\n\n\"Locked,\" the boy said.\n\nIron John nodded. \"Of course. But you know how to find the key.\"\n\nThe boy was shaking his head no as he backed away.\n\n\"Where is the key?\" the man in the cage asked, but the boy was gone. \"Where is the key? But you are only a boy. Perhaps you are too young to know where the key is. You must be older to find the key.\"\n\nThere were murmurs of agreement from the invisible audience. It was very important to find the key, I knew that. The key...\n\nIt was then that I became aware that Iron John was looking at me. He was there in the cage, it wasn't a holoflic. He looked at me and nodded.\n\n\"Jim, I'll bet you know where the key is. You are no longer a boy. You can find it\u2014now.\"\n\nHis voice was a goad. I was on my feet, walking forward to the box of toys. My foot touched a block and it rattled aside.\n\n\"The key is in the toy box,\" I said, but I didn't believe the words even as I spoke them. I looked at Iron John who shook his head no.\n\n\"Not in the box.\"\n\nI looked down again and realized that I did know where the key was. I raised my eyes to Iron John and he nodded solemnly. \"See you do know where the key to the cage is. You can let me out now, Jim. Because you know the key is there. Inside...\"\n\n\"Teddy,\" I said.\n\n\"Teddy. Not a real bear. Teddies are for children and you are no longer a child. Inside teddy.\"\n\nI reached out, blinked away the tears that were blurring my vision, seized up the toy, felt the soft fabric between my fingers. Heard a loud voice that slashed the silence.\n\n\"Not quite right, Jim, not right. The key is not there\u2014it has to be under your mother's pillow!\"\n\nSteengo had come forward to join me, had to shout the last words to be heard over the roar of voices.\n\n\"Mother doesn't want her son to leave her. She hides the key to the Iron man's cage under her pillow. The son must steal the key...\"\n\nThe shouting voices drowned him out. Then it went dark in an instant and someone ran into me knocking me down. I tried to stand, to call out, but a hard foot walked on my hand. I shouted aloud at the sudden pain but my voice went unheard in the clamor. Someone else jarred into me and the darkness became even more intense.\n\n* * *\n\n\"Jim\u2014are you all right? Can you hear me?\"\n\nFloyd's face was just above mine, looking worried. Was I all right? I didn't know. I was in bed, must have been asleep. Why was he waking me?\n\nThen I remembered and sat upright, grabbed his arms.\n\n\"The Veritorium! It got dark, something happened. I can't remember\u2014\"\n\n\"I'm not much help because I can't either. It seemed like a good show. Hard to follow the plot but you were in it, do you remember that?\" I nodded. \"Seemed to be enjoying yourself, although you didn't look happy about tearing the stuffing out of the teddy bear. That's when Steengo joined you onstage and all the fun started. Or stopped. It all gets vague about that time.\"\n\n\"Where's Steengo?\"\n\n\"You tell me. I saw him last on the stage. I was sleeping myself, just woke up. Looked around, no Steengo. Found you here snoring away and I gave you a shake.\"\n\n\"If he's not here...\"\n\nA muted knock sounded at the door, and a moment later it opened and Veldi looked in.\n\n\"Gentlemen, a happy good morning to you both. I thought I heard your voices and hoped you would be awake. I bring you a message from your friend...\"\n\n\"Steengo\u2014you've seen him?\"\n\n\"Indeed I did. We had a friendly chat before you awoke. Then, before he left, he made this recording. Told me to give it to you. Told me you would understand.\"\n\nHe placed a small recorder on the table, stepped back. \"The green button is to play, red to stop.\" Then he was gone.\n\n\"A message?\" Floyd asked, picking the thing up and staring at it.\n\n\"Press the button instead of fiddling with the damn thing!\"\n\nHe looked startled at my tone, put it back on the table and turned it on.\n\n\"Good morning there, Jim and Floyd. You guys are sure sound sleepers and I didn't want to wake you before I went out. You know, I'm beginning to think that this city is not for me. I need some space to get my thoughts together. I'm going to take a walk back down the wall, get some air to breathe, some space to think in. You hang in there and I'll be in touch.\"\n\n\"That old Steengo,\" Floyd said. \"What a character. That's him all right. His voice, sure enough, and his way of thinking. Some guys!\"\n\nI looked up, looked him in the eye. His face was as grim as mine. He shook his head in a silent no. I did the same.\n\nSteengo had not left that message. It was his voice all right. Easy enough for the electronic technicians to fake that.\n\nSteengo was gone.\n\nWhat had happened?\nChapter 18\n\n\"I really slept,\" I said. \"Like a rock. Thirsty.\"\n\n\"The same. I'll get some juice and a couple of glasses.\"\n\n\"Great idea.\"\n\nI had scribbled the note by the time he came back, slipped it to him when I took the glass. He opened it behind the pitcher, read it.\n\nPlace bugged. What do we do?\n\nHe nodded as he passed me my glass of juice.\n\n\"Thanks,\" I said, watching him turn over the note and write on the back. I don't know if there were optical bugs as well as the audio ones. Until we found out we had to act as though there were. I kept the note in my palm when I read it.\n\nSteengo much concerned. Left these for you before we went to the show.\n\nI finished the juice, put my glass down, lifted my eyebrows quizzically. He pointed quickly at his closed fist. When he stood and passed me he dropped something small into my lap. I waited a minute before I poured more juice, drank it, sat back with my hand in my lap. Two small, soft objects. Familiar. I rubbed my nose and glanced at them.\n\nFilter nose plugs. For neutralizing gas. Steengo had known something\u2014or guessed something. He also knew how affected I had been by the sessions in the Veritorium. He had suspected that something physical, not just the training session itself, had gotten to me.\n\nOf course! Obvious by hindsight. I knew of a dozen hypnotic gases that lowered the ability to think clearly, that left the brain open to outside influences. So it hadn't been emotion but plain old chemistry that had carried me away. Steengo had suspected this\u2014but why hadn't he told me? Depressingly, I realized that the state of mind I had been in, probably caused by drugs in the earlier session, rendered that impossible. He knew he couldn't tell me. But had been suspicious enough to wear the plugs himself.\n\nAnd when he saw me getting deeply involved in the ritual he had interrupted before it was too late, had brought the whole thing to a screeching halt. I felt my teeth grating together and forced myself to stop.\n\nHe had talked about mother and the key under her pillow\u2014to these people who denied that women even existed!\n\nWith the realization of the enormity of his crime in the eyes of the Paradisians I felt a sudden overwhelming fear for his safety. Would they kill him\u2014or worse\u2014had they killed him already? They were certainly capable of anything, I was sure now of that.\n\nWhat next? Communication with our backup team in the spacer above was very much in order. I had to get into the open, away from the bugs, and contact Tremearne. Bring him up to date. Something had happened to Steengo. And the rest of us surely were in danger as well\u2014and Madonette, this might affect her. This entire affair was getting a nasty and dangerous edge to it.\n\nAnd thinking about dangerous, there was the other dangerous always hanging over my head. My computer flashed me the highly unwelcome message of a flickering red nine. I had been asleep longer than I had realized.\n\nArtifact or no I was just nine days away from my personal destiny. When I had first heard the thirty-day deadline on the poison I had not been too concerned. Thirty days is a lot of time, I thought.\n\nNine days was definitely not a lot of time at all. And with Steengo suddenly vanished I had more problems, not less.\n\n\"Going for a run,\" I called out to Floyd, leaping to my feet in a spasm of fear-sponsored energy. \"Feel logy after all that sleep. Got to clear my head.\"\n\nI slammed out the door and down the road even as he was answering. Taking a different route from my usual one\u2014then changed direction at random. Up ahead was a field of polpettone trees, laid out in neat rows and bulging with fruit. I jogged into a path beside the trees, looking around as I ran. No one in sight. There was little chance the Paradisers would put bugs in among the trees.\n\nBut they could have. I turned into a freshly plowed field and ran between the furrows. I should be safe enough here. I clamped my jaw twice.\n\n\"Hello, Tremearne, are you there?\"\n\n\"Very much so, Jim. We have all been awaiting your report. Can you tell us what is happening\u2014the recorder is running.\"\n\nI jogged in position for a bit, then bent to tie my shoe\u2014then gave up and just sat on the ground while I finished the detailed report. I was tired, the chemicals still kicking around in my system had not been kind to me.\n\n\"That's it,\" I finished. \"Steengo is gone. Might be dead...\"\n\n\"No. I can reassure you on that score. A few hours ago we had a radio message from him, just a few words, then contact was lost again. He must be somewhere deep in the city, behind walls the radio signals can't penetrate. He might have been moved from one site to another, was in the open long enough for a brief transmission.\"\n\n\"What did he say?\"\n\nThe recording was brief and scratchy. Beginning with static and dying in static. But it was pure Steengo all right.\n\n\"... never enough! When I get my fingers on you, you...\" The next word was hard to make out\u2014but I could think of a half dozen that filled the bill.\n\n\"What do you think we should do? Break out of here?\"\n\n\"No\u2014go along with everything. You will be contacted.\"\n\n\"Contacted? By whom, what, which? Come in, Tremearne.\"\n\nThere was no answer. I rose and brushed off my shorts. Very mysterious. Tremearne was up to something\u2014but he was not talking about it. Must be worried about eavesdroppers. Maybe he knew something that I didn't.\n\nI started back at a slow run, changed that to a fast walk. To a slow walk, then a crawl. If there had been any farther to go I would probably have done it on all fours. As it was I stumbled into our quarters and collapsed, gasping, onto the couch. Floyd looked astonished.\n\n\"You look like you've been dipped and rolled.\"\n\n\"I feel even worse than that. Water, quickly, lots of it!\"\n\nI drank until I was sloshing, then sipped a little bit more, handed the glass weakly back.\n\n\"Knocked myself out. Be a good buddy and get my pack. I got some vitamin pills there should pick me up.\" When he handed me the pack I clicked out a couple of Blastoffs, super-uppers, and swallowed one. \"Vitamins, good for you,\" I said as I passed one over. Floyd was a little faster off the mental mark lately and did not ask any questions.\n\nOur timing was pretty good. The wave of good feeling and energy was washing away my almost-terminal fatigue when Veldi threw open the door.\n\n\"On your feet!\" he called out. I did not move.\n\n\"Veldi,\" I said. \"Old and trusted servant. No soft knock? No sweet tones...\"\n\n\"The word is out that you Stainless Steel Rats are just plain rats. Troublemakers. Just get going.\"\n\nThere was the quick thud-thud of marching feet and Sergeant Ljotur came in with an armed squad of soldiers. Armed with wicked-looking spears with gleaming points and barbed shafts.\n\n\"You are to come with me!\" he ordered. He did not look happy about it.\n\n\"No longer a musical fan, Ljotur?\" I said, climbing slowly to my feet.\n\n\"I have orders.\" Orders that he obviously did not like. Which of course he would obey since independent thought had never been encouraged in the military. Floyd followed me out and the squad formed up. Four in front, four in back of us. Ljotur checked the formation, nodded, took position in front and raised his spear.\n\n\"Forward\u2014burtu!\"\n\nWe burtued at a slow trot down the road and turned right at the corner. Which put us directly on the route to the red brick lodgings where Iron John lurked, as I remembered from our first visit. Trotted down the road and into a tunnel under a row of buildings. One of the guards to the rear tapped me on the shoulder.\n\n\"Give me a hand, will you?\" he asked in a hoarse voice.\n\nThen swung sideways and planted his fist in the stomach of the guard next to him. Who folded and dropped without a sound.\n\nThis was easy enough to understand. I had turned when he tapped me so I kept turning to face the rear. I reached out and got a hand on the other two guards' necks. Squeezed as they turned their spears towards me.\n\n\"Floyd!\" I gasped out, putting all my energy into my throttle grips so these jokers would pass out before they harpooned me. \"The others!\"\n\nOne of the guards dropped but the other one, with a stronger neck, kept his spear coming. Into my stomach\u2014\n\nNo, not quite. The first guard, who had called to me, gave him a quick chop under the ear. He and I whirled about, ready to jump to Floyd's help. And stopped.\n\nThe four other guards were lying in a silent, tumbled heap on the ground. Floyd had a spear pressed firmly under Ljotur's jaw, was holding him up with his other hand.\n\n\"You want to talk to this guy?\" Floyd asked. \"Or you want him down there with the others?\"\n\n\"I've nothing to say...\"\n\n\"No talk. Drop.\"\n\nBefore I could finish speaking a limp Ljotur joined the rest of the sleeping patrol.\n\n\"What about this one?\" Floyd asked, fingers arced, pointing to the soldier who had called to me.\n\n\"Wait! He started this thing. There has to be a reason for it.\"\n\n\"There is,\" the soldier said in the same hoarse voice. \"I am going to tell you a few things. You will not laugh at anything I say\u2014understood?\"\n\n\"We're not laughing!\" I said. \"Great, guy, thanks for the help. And what's the plan?\"\n\n\"First off\u2014remember about the laughing! I'm not a guy. I'm a girl. Do I see lips bending?\"\n\n\"Never!\" I called out, to disguise the fact that a little flicker of emotion had appeared. \"You saved us. We are in your debt. We are not laughing. So tell us about it.\"\n\n\"All right. But let's drag these so-called soldiers out of the way first. Then we go on. The orders were to bring you to Iron John and that is what I am going to do. Your friend is in danger. Do nothing precipitate. Forward.\"\n\nWe went. Disbelieving perhaps, but still forward. Floyd started to talk but I raised my hand.\n\n\"Save the discussion. Explanations will be useful after we make sure Steengo is all right. But Floyd\u2014stop me if I am wrong\u2014did I see you take five guys out while I was just about managing two?\"\n\n\"You didn't see it. It was over before you turned to look.\" He was the same old laid-back Floyd\u2014but was that a new touch of firmness to his words? It was a day of surprises. And he was right\u2014I had not seen him at work, just the results.\n\nThe brick palace jogged into view ahead. Apparently not all of the troops had been told that we were no longer heroes, for the guards at the entrance did a snappy jump to attention and salute as we trotted past.\n\n\"Halt!\" our newfound friend (girl...?) called out and we stopped before the guards at the door. \"Orders to bring these two to Iron John. Permission to enter?\"\n\n\"Enter!\" the officer in charge called out. The doors opened and closed behind us as we trotted by. There was the large room ahead and inside it was Iron John. And just one other person.\n\nSteengo. Collapsed against the wall, covered in bruises and blood. One eye swollen shut. He started to speak but could only rasp out something incomprehensible.\n\n\"You are all here now,\" Iron John said. \"Soldier\u2014guard the entrance. No one to enter or leave. I have a score to settle with these interlopers. Because I have changed my mind about keeping this thing quiet. I listened to my advisers and I am sorry that I did. Secrecy is at an end and justice will be done to the blasphemers. Here is what will happen. First I will kill this aged devil who spoke such filth. You two will watch.\n\n\"Then I will kill you as well.\"\n\nHe started towards Steengo, a red giant of unleashed power. Hands extended to kill.\nChapter 19\n\n\"Let me have your spear.\" I called out to the soldier at the door. She shook her head in a silent no, then said, \"I have my orders.\" No help from this source.\n\nIron John had turned and was walking towards Steengo. I ran two silent steps in his direction and launched myself into a flying kick to his back. Heel punching out, a killing blow.\n\nThen I was batted from the air. As big as he was\u2014Iron John was just as fast. He had turned while I was in the air and had swung one hand. Knocking me aside, sprawling me onto the floor. His voice was as deep and ominous as a distant volcano.\n\n\"Do you want to be first, little man? You wish the others to watch your destruction? Perhaps that is only fair since you are their leader.\"\n\nHe came slowly towards me and I found myself trembling with fear. Fear? Yes, because he was not human, more than human. He was Iron John a part of the legend of life, I could not hurt him.\n\nHe wasn't! I scrabbled to my feet, my leg ached, moved away. He was much bigger, wider, stronger than I was. But no, he wasn't a legend. He was a man.\n\n\"A big fat red slob!\" I shouted. \"A hairy conman!\"\n\nHis eyes were wide, red, angry. His arched fingers reached for me. I feinted a fist at his jaw, saw him move to block it. Kept turning in an unstoppable kick to his knee.\n\nIt connected\u2014but he made no attempt to avoid it. My foot hurt. His knee, his kneecap, looked unhurt.\n\n\"I am Iron John!\" he shouted. \"Iron\u2014iron!\"\n\nI fell back, there was no escape. I swung a twisting punch that he took on his biceps. It felt like striking stone. Then his fist to my ribs sent me skidding down the room.\n\nWhen I gasped in breath it hurt. Felt like something was broken there. Stand up, Jim! I got as far as my knees and he came on.\n\nI blinked as I saw two arms encircle his legs, send him staggering. Kicking out. It was Steengo who had crawled behind him, tried to trip him. Who was now sent crashing back into the wall. To fall and not move again.\n\nI was barely aware of this because the instant Iron John's attention had wandered I had jumped. Getting an arm around his neck, grappling my own wrist. Pulling my forearm tight against his throat to crush his larynx, to cut off blood and air. The armlock that kills in seconds. My face was buried in his rank red fur as I tightened hard, harder than I ever had before.\n\nTo no avail. I could feel the tendons in his neck stiffen like steel bars, taking the pressure that should have been on his throat. He lifted one hand slowly, then sank his fingers deep into my flesh\u2014\n\n\u2014hurled me across the room to crash into the wall, fall.\n\nI realized that the voice wailing in agony was my own. I could not move. The soldier at the door looked at me, looked away. Steengo had lain, motionless, since that single, terrible blow. Nor could I do much better myself, just able to crawl.\n\nAt least Iron John had felt my hold; he was rubbing at his neck. The smile had gone and frothed saliva now coated his lips. Death would be a single blow...\n\n\"Iron John\u2014you have forgotten something. You have forgotten me.\"\n\nFloyd was speaking. Thin, black-bearded, uninvolved. He must have stood and watched while Steengo was stricken, I was felled. Only now did he move.\n\nQuietly forward. Hands extended, fingers lightly bowed. Iron John was in a rage. Leaped and lashed out.\n\nAnd missed because Floyd was not there. He was to one side, kicking the red giant in the ribs so that he stumbled and almost fell.\n\n\"Come here,\" Floyd said in a voice so low I could barely hear it. \"Come and be destroyed.\"\n\nIron John was cautious now, knew how fast his new opponent could react. He opened his arms wide and came slowly forward. A force of nature. Implacable and inescapable.\n\nTwo quick thuds, two blows sounded and Iron John staggered. Floyd was out of his reach again, circling him slowly. A sudden kick, a blow, then away again.\n\nNothing Iron John did seemed to affect the outcome. He was wary, he attacked suddenly, reached out and struck. Touched only air. Floyd was before him, behind him\u2014striking him. Wearing him down.\n\nThey circled for minutes this way. And Floyd was still just as fast, striking with impunity. But the red monster was going slower and slower, arms lower and lower as the endless blows drove the strength from them. He must have realized that there could be only one end to this battle, on these terms. But he was still dangerous. Almost by chance the struggle moved towards me.\n\nHe was after me I realized! I had only the shortest instant to draw my leg back before Iron John spun about and dived towards me.\n\nAnd caught my kick full in his face. He dropped\u2014but his hands closed on my ankle, pulled me towards him. Reached up...\n\nThen Floyd struck. No science now\u2014raw power. Pile driver blows to the giant's back and kidneys that opened his mouth wide with pain, forcing him to release me as he struggled to get away from his tormentor.\n\nMore blows to his head. He tried to rise, his legs were kicked from beneath him. The thudding of quick strikes like some terrible machine at work. Then a sudden silence.\n\nA moment for balance, no expression showing on his face, then Floyd swung a terrible kick that terminated on the side of the giant's head. Who fell over and did not rise nor move again.\n\n\"Dead?\" I croaked. Floyd knelt and felt the pulse in his neck.\n\n\"No, he wasn't supposed to be. He'll survive. But I think that he will remember he has been in a fight.\" He flashed a quick smile, then his face became calm again. \"If you're all right I'll look at Steengo.\"\n\n\"I'm great. Knocked about but great,\" I croaked as I climbed painfully to my feet.\n\n\"Pulse good,\" he said, kneeling beside our friend. \"He has taken a lot of punishment but nothing seems to be broken that I can find. He will come out of this fine.\"\n\nI was groggy, now even weaker with relief, blurted out the words without thinking. \"He's fine. I'm fine. However we would have been a lot better if you had waded into this fracas sooner.\"\n\nI saw him wince at the words, wished I could take them back. You never can.\n\n\"I'm sorry, I really am. I had to wait, see what he could do. I know that you're good, Jim. I knew you could at least hold him. I'm sorry but I had to see how fast he could move before I took him on. I had to wear him down, not get touched. I knew I could do it\u2014and I moved as soon as I knew. Sorry...\"\n\n\"Reporting,\" our guard-guy-girl said. \"The Red One is unconscious.\"\n\nShe lowered the small, coin-sized communicator as I stalked towards her, hands out and ready to strike.\n\n\"Who were you talking to? Whose side are you on? What's happening here? Speak\u2014or get demolished.\"\n\nThe guard, spear lowered and pointed at me, stood her ground. \"The answer to your questions is arriving now. There.\" The point of her spear moved to indicate a spot behind me. A ruse? Who knew, who cared. I turned and looked at Iron John's giant throne.\n\nWhich was slowly turning on some invisible axis. Floyd and I both faced that way, hands raised automatically on the defense. A black opening was revealed and, as the throne stopped moving, there was motion in the darkness beyond. Two figures appeared, walked out into the room.\n\nBoth women.\n\nOne of them was Madonette.\n\n\"Hi, guys,\" she said, smiling and waving. \"I'd like to introduce a new friend, Mata.\"\n\nThe woman was about my height, regal of bearing in her dark robe touched with gold embroidery. Her expression was composed, peaceful; small wrinkles at the corners of her eyes, a touch of gray to her hair, were the only signs of age.\n\n\"Welcome to the other side of Paradise, Jim,\" she said\u2014and held out her hand. Her handshake was firm and quick. I opened my mouth but could not think of anything relevant to say.\n\n\"I know that you have many questions.\" Her words filled the gap. \"All of which will be answered. But it would be wisest to postpone our little chat until we are out of this place. A moment, please.\"\n\nShe took a very efficient-looking hypodermic from the reticule hanging at her waist. Uncapped it and bent to brush aside the thick hair on his leg to give Iron John a quick injection.\n\n\"He will sleep the better,\" she said. \"Bethuel\u2014will you lead the way?\"\n\nThe guard raised her spear in a quick salute, then marched resolutely past the throne and into the opening. Madonette touched Steengo's cheek, then waved Floyd to her. \"Help me carry him. Jim will have enough to do just moving himself.\"\n\nI resented the remark\u2014a blotch on my masculine pride?\u2014but before I could stumble over they had lifted him and were following the guard, Bethuel.\n\nThere were no lights in the tunnel behind the throne. At least none until Mata had entered behind us and sealed it once again. Pale illumination flickered into existence. More than enough to see by. Nor was it a long walk to the open door at the far end. We emerged into a large, red brick room that could have been a mirror-image of the one that we had just left.\n\nJust in physical size, though. Here the walls were covered by pleasant hangings, tapestries of sunshine and floral landscapes. Instead of the swords and shields that adorned the other. The stained-glass windows here depicted scenes of mountains and valleys, villages and forests. Unlike Iron John's windows, which featured the clash of battle, spackle of gore. This was altogether more civilized.\n\nAs was the murmur of concerned voices from the women in attendance here. They tenderly carried Steengo to a couch where another woman, dressed in white, ministered to him. I dropped into the nearest chair and scowled around at all the female bustle. My voice, louder and more censorious than I had intended, cut through the peaceful scene.\n\n\"Now would somebody, anybody, tell me just what the hell is going on.\"\n\nThe way I was ignored was comment enough in itself. Though a smiling girl did bring me a glass of cool wine\u2014on the way to serve the others. Madonette sat next to Mata, where they put their heads together for a moment before Madonette spoke.\n\n\"First\u2014and most important now that you all are safe\u2014is the fact that the artifact is here and is being looked after. In addition there is\u2014\"\n\n\"Excuse if I interrupt,\" I said. \"A matter of priority.\" I clamped my jaw twice. \"Did you hear that, Tremearne?\" His answer buzzed in my jawbone.\n\n\"I did, and...\"\n\n\"Priorities, Captain.\" I spoke quietly so only he could hear. \"Mission complete. Alien artifact returned. Antidote for me on its way down. Nine days is close enough to come. Do you understand all that?\"\n\n\"Of course. But there is a complication...\"\n\n\"Complication!\" I could hear the squeak of fear edging my voice. \"What?\"\n\n\"I sent for the antidote to the thirty-day poison as soon as I heard about it. I had no intention of waiting until the deadline to administer it. However there was an accident in transit.\" Sweat suddenly beaded my forehead and my toes tapped anxiously on the floor. \"These things happen. I've sent for a second batch and it's en route now.\"\n\nI cursed viciously under my breath, then realized that I was the object of more than one concerned glance. Smiled woodenly and snarled my answer.\n\n\"Do it. Get it. No excuses. Now. Understood.\"\n\n\"Understood.\"\n\n\"Fine.\" I stopped whispering and called out. \"I'm most cheered to hear that the artifact has been found. Now, if you please, an explanation of what all this is about.\"\n\n\"Seems obvious,\" Madonette said undoubtedly miffed by my surly behavior. \"It looks like the ladies have saved your bacon and you should be grateful.\"\n\nWhich did nothing to clear the air. \"As I recall,\" I recalled. \"It was the gentlemen\u2014at some physical cost I must add\u2014who polished off that russet rottweiler before you all came onto the scene. I also remember that we were watched all the time during the life-and-death struggle by one of your lot who did nothing to help.\"\n\nThe tough answer sprang to her lips and I snarled around at the female company. Tempers flared on all sides but Mata cooled things down.\n\n\"Children\u2014there has been enough tribulation and pain, so do not cause yourself any more.\" She turned to me. \"Jim, let me explain. The soldier who aided your escape, Bethuel, is one of our spies who keeps us informed about all the masculine meanderings beyond the wall. I ordered her to help you escape your guards, which she did. I also ordered her not to reveal her presence to Iron John. The men beyond the wall have no idea that we watch them closely and I wish it to remain that way. She aided your escape and you should be grateful.\"\n\nI was, and I should have admitted it, but I was still bull-headed and angry and settled for a surly mutter and growl. Mata nodded blithely as though I had communicated something of importance.\n\n\"See how well everything has worked out? You are here and safe, your friends safe as well, and that for which you seek, the strange artifact, is secure and close by.\"\n\nI only half listened. Fine for the troops. But there were other forces at work that did not bode well for my future. Accidents in transit did not happen by accident. Someone in the bureaucracy was manipulating me\u2014did not like me. Perhaps had never liked me and never had any intention of supplying the antidote. I would certainly be less trouble to them if I were safely dead. And there were only nine days left to sort the whole thing out.\n\nI had touched my computer controls automatically while these thoughts were whizzing about my tired brain. The number glowed before me. I really had had a longer sleep than I realized.\n\nEight days to go.\nChapter 20\n\nI looked around at the peaceful female bustle\u2014and suddenly felt very, very tired. My side hurt and I felt sure that a couple of ribs were broken. I sipped the wine but it didn't help. What I really needed was a couple of Blastoff pills to restore me to something resembling life. In my pack\u2014\n\n\"My pack!\" I shouted hoarsely. \"My equipment, everything. Those masculine momsers have all our gear!\"\n\n\"Not quite,\" Mata said in soothing tones. \"As soon as you left we saw to it that the porter, Veldi, was rendered unconscious and both your packs are here now. Your associate Steengo's equipment was not in your residence so we can assume that it is now in the possession of Iron John or his associates.\"\n\n\"Not good.\" I worried a fingernail with my incisors. \"There are things there they shouldn't see...\"\n\n\"Might I interrupt,\" Tremearne's voice spoke through my jaw-a-phone. \"I was waiting until things quieted down to tell you. Steengo's pack is safe.\"\n\n\"You have it?\"\n\n\"Rather I should have said 'made safe.' All of your packs are booby-trapped with a canister of rotgrot. Which, when released by a coded radio signal, causes the contents of the pack to instantly decay to their component molecules.\"\n\n\"Nice to know. A lot of secrets are being revealed of late, aren't they?\"\n\nThere was no response from my jaw. I held out my wineglass for a refill. \"Some simple answers to some simple questions, if you please.\" My anger had been blasted by fatigue, excoriated by fear of imminent death. Mata nodded in response.\n\n\"Good. On a historical note\u2014how come guys over there, girls here?\"\n\n\"A union of convenience,\" Mata said. \"Many years ago our foremothers were forcefully relocated to this planet. This inadvertent transplantation had a sobering effect on them. Whatever excesses of zeal they had displayed on other worlds were not repeated here. Peace, cool reasoning and logic prevailed. We became then as you see us now.\"\n\n\"Women,\" I said. \"A society of women.\"\n\n\"That is correct. Life here was a running battle for a good long time, or so it is written. The Fundamentaloids tried to convert us, while our next-door neighbors tried to wipe us out. The inferior sex they called us, a threat to their existence. When we first came to this planet we found that those macho crazies were already well established. Our group was forced to spend a good deal of effort just staying clear of them. This was time and energy wasted, our founding mothers decided, so they sought ways to bring about peace. Eventually they convinced the male ruling clique that they could prosper by utilizing their energy in a more positive manner. It was a completely selfish appeal, arranging ways for the males on top in their society to stay on top, while providing absolute control of the rest of the men.\"\n\n\"Sounds pretty terrible,\" Madonette said. \"Turning all those men into slaves.\"\n\n\"Never say slaves! Willing collaborators is more like it. We showed those in charge, and in particular the one now called Iron John, how much easier it would be to rule by brain rather than muscle. We demonstrated to their satisfaction how a great deal more could be accomplished. With our intelligence and knowledge of science, and their muscles, two separate societies were born. In the beginning there was much hatred and clashes between the groups. This died away when it was decided that only the male leaders would know of our existence. This suited the leaders to perfection.\"\n\n\"That was when the two cities were built\u2014and the wall?\"\n\n\"Correct. This planet is rich in red clay and fossil fuel so the males soon became manic brick makers. After we showed them how to build kilns, of course. There were contests to see who could mold the most bricks, or fire the greatest number, or carry the most. The champion was named brickie of the month and achieved great renown. This went on until you couldn't see the trees for the mountains of bricks. We quickly researched brick laying in our data bases and put the men to work on that.\"\n\nShe sipped her wine delicately and waved her hand in a circle. \"Here are the results\u2014and quite attractive they are too. While our physical scientists were sorting out the males this way, our cultural engineers were looking at the sloppy mucho-macho theories that had been keeping them going up to this point. The Iron Hans myth was only a part of their pantheon. We simplified and altered it. Then used genetic biology to modify the physical structure of their leader, so he is as you see him now. At first he was grateful, although gratitude has long since vanished.\"\n\n\"How long?\"\n\n\"Hundreds of years. Cellular longevity was part of the treatment.\"\n\nI was beginning to catch on. \"And I'll bet that you remember this firsthand\u2014since you and the other lady leaders have had the same treatments?\"\n\nShe nodded, pleased. \"Very adroit, James. Yes, the authorities on both sides of the wall have had the treatments. This makes for continuity of leadership\u2014\"\n\n\"And the need for secrecy of each other's existence that keeps the powerful in power?\"\n\nMata shook her head in wonder. \"You are indeed most perspicacious. How I wish you were in charge next door rather than that hairy halfwit.\"\n\n\"Thanks for the job offer\u2014but no thanks. So the men beyond the wall don't know that you women are here. The same must be true of your women\u2014\"\n\n\"Not at all. They know about the males\u2014and just don't care. We have a complete and satisfactory society. Childbearing for those who wish it, a fulfilling intellectual life for all.\"\n\n\"And religion? Do you have a female equivalent of Iron John?\"\n\nShe laughed merrily at the thought, as did all the other women who were listening to our conversation. Even Madonette was smiling until she saw my glare, turned away.\n\n\"That's it,\" I snapped. \"Enjoy yourself. And when you are through, if you ever are, you might kindly let me know the joke.\"\n\n\"I am sorry, James,\" Mata said, laughter gone and really quite serious. \"We were being rude and I apologize. The answer to your question is a simple one. Women don't need myths to justify their femininity. All of the myths about Iron Hans, Iron John, Barbarossa, Merlin and other mythological men with their salvation myths are all purely male. Just think about it. I am not making a value judgment, just an observation. Such as the observation that men are basically combative, confrontational, insecure and unstable\u2014and appear to need these myths to justify their existence.\"\n\nThere was a lot to argue with there, maybe not a lot but some. A good deal of jumping-to-conclusions and more than a bit of rationalization. I sidestepped for the moment, until I knew more about how this society ticked. I raised a finger.\n\n\"Now let me see if I have this straight. You ladies have a comfortable existence on this side of the wall. You provide the scientific backup to the males on the other side. To keep them chuntering along in their locker-room paradise. Correct?\"\n\n\"Among other things. That is basically correct.\"\n\n\"Dare I ask what they supply in return?\"\n\n\"Very little, if the truth be known. Fresh meat from the nomads. Who not only won't trade with us but now heartily deny our existence, though they secretly would love to wipe us out. Then there is an occasional supply of sperm to top up our cryogenic sperm bank. Little else. We watch them and keep them going mostly by habit\u2014and for our own safety. If the man in the street doesn't know that we exist he can't cause us any trouble. The men also get a lot of pleasure in bashing the nomads when they start bothering us. Altogether a satisfactory relationship.\"\n\n\"It certainly sounds that way.\" I finished the glass of wine and realized that I was beginning to feel the effects of the alcohol. Which was better than feeling the bruises and sore ribs. Which should be looked at soon\u2014but not too soon. The unfolding drama of cultural mish-mash was just too interesting. \"If you please\u2014a question or two before we call in the medics. First is the most important question. You mention sperm banks so I assume that pregnancy and motherhood still exist?\"\n\n\"They certainly do! We would never consider depriving women of their hormonal, psychological and physical rights. Those who wish to become mothers become mothers. Simple enough.\"\n\n\"Indeed it is. And looking around I see that they are lucky enough to all have female babies.\"\n\nFor the first time I saw Mata less than completely relaxed and calm. She looked away, looked back\u2014took up her glass and sipped some more wine.\n\n\"You must be tired,\" she finally said. \"We can finish this discussion some other time...\"\n\n\"Mata!\" Madonette gasped. \"I think that you are avoiding the topic. This cannot be. I have so admired you and your people here. You are not going to tell me that I am wrong?\"\n\n\"No, never!\" Mata said reaching out and taking Madonette's hands in hers. \"It has just been so long since we discussed these things. Decisions were taken that seemed excellent at the time. Some of us have had reservations since, but, well nothing much can really be done at this point...\"\n\nHer voice ran down and she emptied her wineglass. She was upset and I felt sorry for pinning her down like that. I yawned.\n\n\"You're right,\" I said. \"I think rest and recuperation come first.\"\n\nMata shook her head in a firm no. \"Madonette is right. These decisions must be faced, discussed. Approximately half of the pregnancies are male, male fetuses. This is determined in the first few weeks.\" She saw Madonette's worried expression and shook her head again.\n\n\"No\u2014please hear me out and don't think the worst. All healthy pregnancies are brought to term. In the case of the males the bottle banks are used\u2014\"\n\n\"Bottle banks! Isn't that an unfortunate term?\"\n\n\"Perhaps in your society, Jim. But here it simply signifies highly perfected artificial wombs. Technically superior if truth be known. There are no spontaneous miscarriages, no effects of bad diet and so forth. And at the end of nine months the healthy male babies are\u2014\"\n\n\"Decanted?\"\n\n\"No, born. As soon as they are viable the men take over. Specially trained nursemen who supervise the healthy growth of the boys. Their education and assimilation into their society.\"\n\n\"Very interesting,\" I said, for it certainly was. I hesitated about the next question, but curiosity was gnawing away and could not be suppressed. \"Even more interesting is where do the men think the babies come from?\"\n\n\"Why don't you ask them?\" Mata said coldly and I realized that this interview was at an end.\n\n\"Now I really am tired\u2014to be continued,\" I breathed, dropping back into the couch. \"Is there a doctor in the house?\"\n\nThis kicked a lot of maternal instinct into gear and extracted a great deal of attention. I didn't feel the injection that knocked me out. Or the one that brought me to much later. The women were gone and we were alone. Madonette was holding my hand. Which she dropped with slow deliberation when she saw that my eyes were open.\n\n\"The good news, stalwart Jim, is that none of your bones are broken. Just a lot of bruising. Better news is that the treatment for the bruises is underway. Best news is that Steengo is in pretty good shape, all things considered, and wants to see you.\"\n\n\"Bring him in.\"\n\n\"In a moment. While you were sleeping I talked to Mata. She told me a lot more about how things work around here.\"\n\n\"Did you find out about the babies?\"\n\n\"She really is a nice person, Jim. Everyone here has been very nice to me and...\"\n\n\"But you are beginning to have some reservations?\"\n\nShe nodded. \"More than a few. Things look so nice on the surface\u2014and maybe they are. But it is the babies that bother me. I am sure that they are well taken care of physically, even mentally. But to believe a stupid myth!\"\n\n\"Which one of the stupid myths going about is the one that bothers you?\"\n\n\"Spontaneous creation would you believe! All the males gather around Iron John's pool for a ceremony of life. The golden balls drift up through the water and are seized. And each one contains a healthy happy baby! And grown men believe that nonsense!\"\n\n\"Grown men\u2014and women\u2014have believed worse nonsense down through the ages. This myth was a common one for the so-called lower forms of life. Flies being spontaneously created in manure heaps. Because no one bothered making the connection between grubs growing there and flies laying eggs. All of the creation myths of mankind, all the gods dropping down and molding clay and breathing life, the virgin births and the like. They are all nonsense once they are examined. But we have to start somewhere I suppose. I'm just not happy where some of these people are ending up.\"\n\nThere was a rattle and a thump as the door was opened. Floyd pushed in the wheelchair and Steengo lifted a white-wrapped hand.\n\n\"Looks like you did it, Jim. End of mission. Congratulations.\"\n\n\"And the same to you\u2014and Floyd. And since it is The Stainless Steel Rats together, perhaps for the last time, would you mind making a few things clear. I have long felt that there was more than random chance in your selection. Dare I ask\u2014just who are you three people? I suspect that you were chosen for more than musical ability\u2014right Steengo?\"\n\nHe nodded his bandaged head. \"Almost right. Madonette is just what she appears to be...\"\n\n\"Just an office drudge\u2014singing for a hobby.\"\n\n\"The office's loss is music's gain.\" I smiled and blew a kiss her way. \"One down, two to go. Steengo, I have a feeling that you really aren't retired. Right?\"\n\n\"Right. And I do take some pride in my musical abilities. Which, if you must know, was why I was suckered into this operation by my old drinking buddy, Admiral Benbow.\"\n\n\"Drinking buddy! He who drinks with an admiral...\"\n\n\"Must be an admiral too. Perfectly correct. I am Arseculint...\"\n\n\"I didn't quite catch that.\"\n\n\"Arseculint is an acronym for Area Sector Commander Cultural Intercourse. And you can uncurl your lip. Perhaps, in context, 'intercourse' is not quite the right word. Cultural Relationships might express it better. My degrees are in archeology and cultural anthropology, which is what attracted me to the civil service in the first place. Sort of hands-on application of theory. I followed the matter of the alien artifact with a great deal of interest. So I was ripe for the plucking, you might say, when Stinky Benbow asked me to volunteer.\"\n\n\"Stinky?\"\n\n\"Yes, funny nickname, goes back to the academy, something to do with a chemistry experiment. Which is completely beside the point. I thought enough of this assignment to take a leave from my desk. Great fun. Up until the last, that is.\"\n\n\"Which leaves young Floyd here? Also an admiral?\"\n\nHe looked sheepish. \"Come on, Jim, you know better than that. I even washed out of college, never graduated at all...\"\n\nI pointed an accusatory finger. \"Putting academic credits aside you must have some value to the Special Corps.\"\n\n\"Yes, well, I do. I really am sort of an instructor...\"\n\n\"Speak up, Floyd,\" Steengo said proudly. \"Being chief instructor in charge of the unarmed defense school is nothing to be ashamed of.\"\n\n\"I agree completely!\" I said. \"If you weren't a whiz kid in unarmed combat, why none of us would be here. Thanks guys. Mission complete and successful. Let's drink to that.\"\n\nAs we raised and clashed our glasses together, drank deep, I thought of my mother. I do this very rarely; it must be all the male-female myth dredging that brought her to mind. Or what she used to say. Very superstitious my Ma. Had a superstition for any occasion. The one that I remember best was when you said how great things were, or what a nice day it was. Bite your tongue she used to say.\n\nMeaning don't tempt the gods. Keep your head down. Because saying that something was good would surely bring about the opposite.\n\nBite your tongue, good old Ma. What a lot of malarky.\n\nWhen I lowered my glass I saw a woman stumble in through the open door. A young woman with torn clothing, dusty and staggering.\n\n\"Sound the alarm...\" she gasped. \"Disaster... destruction!\"\n\nMadonette caught her as she fell, listened to her whispered words, looked up with a horrified expression.\n\n\"She's hurt, babbling... something about... the science building, destroyed, gone. Everything.\"\n\nThat was when I felt the cold tongs grab tight to my chest, squeezing so hard they made speech almost impossible.\n\n\"The artifact\u2014\" was all I managed to say.\n\nMadonette nodded slow agreement. \"That's where it was, they told me. In the science building. So it must be gone too.\"\nChapter 21\n\nThe mutual decision of The Stainless Steel Rats was a simple one: we had had about enough for one day. We were alive, if not too well. We had found the artifact so our mission was accomplished. The fact that it had also been destroyed was beside the fact. I hoped. They would have to supply me with the poison antidote now. I kept that thought firmly before me as I went to sleep. This was a time for rest. Wounds had to heal, tissue had to mend, fatigue had to be alleviated: medication and a good night's sleep took care of all of that.\n\nThe sun was shining brilliantly upon the garden of our new residence when I dragged myself there next morning. Sleep had banished fatigue, which meant that I felt all the bruises that much more enthusiastically. My medication was beginning to override the pain and I dropped into a chair while I waited for beneficence to take place. Steengo came in soon after, swinging along on crutches and looking very much like I felt. He eased himself into the chair opposite me. I smiled a welcoming smile.\n\n\"Good morning, Admiral.\"\n\n\"Please, Jim\u2014I'm still Steengo.\"\n\n\"Then, Steengo, since we're alone for the moment, let me express my heartfelt thanks for breaking up the brainwashing session with Iron John. For which, unhappily, you paid quite a physical price.\"\n\n\"Thank you, Jim, I appreciate that. But I had to do it. To save you from being programmed. Also\u2014I really did lose my temper. Teddy bear indeed! A complete corruption of history.\"\n\n\"No teddy bear? No golden ball?\"\n\n\"The golden ball, yes. That represents innocence, the pleasures of childhood without responsibility. It is lost when we grow up. To regain this freedom the myth tells us we have to find the ball under mother's pillow\u2014and steal it.\"\n\n\"But in a society without women you can't have a mother\u2014so the myth had to be rewritten?\"\n\nSteengo nodded agreement, then winced and touched the bandage around his head. \"Retold as nonsense. In the original story Mother never wants the boy child to grow up, sees him as young and dependent forever. Independence must be stolen away from mother\u2014hence the golden ball under her pillow.\"\n\n\"Pretty deep stuff.\"\n\n\"Pretty fascinating stuff. Mankind depends on its myths to rationalize existence. Pervert the myth and you pervert society.\"\n\n\"Like Big Red and his mates on the other side of the wall?\"\n\n\"Exactly. But what was happening there was far more dangerous than just editing a myth. I had suspected that there would be some strong narcogases in the air\u2014and I was right. You and Floyd were glassy-eyed and practically hypnotized into immobility. So it wasn't just a matter of listening to one more story about the magnetic field of the deep masculine. This was about having a very pernicious and demented theory punched deep into your mind, into your subconscious. You were being brainwashed, thought-controlled\u2014and this sort of crude forced suggestion can do infinite harm. I had to stop it.\"\n\n\"Risking your own life at the same time?\"\n\n\"Perhaps. But I am sure you would have done the same for me if the circumstances were reversed.\"\n\nThere was no answering that one. Would I? I smiled, a little grimly. \"Can I at least say thanks?\"\n\n\"You can. Greatly appreciated. So back to work. Now, before the others come, to more pressing business. Since I am now in the open, so to speak, I am relieving Captain Tremearne and taking command of this operation. I am in a better position to kick the cagal out of the chain of command and make sure that your antidote is here instantly. Or sooner. My first imperative order when I took command was to send for it.\"\n\n\"Then you know about the thirty-day poison? If I might be frank\u2014I can tell you\u2014it has had me pretty worried. Thank you\u2014\"\n\n\"Don't thank me yet. Because I want your assurance that you will stick with this assignment, thirty-day poison or no.\"\n\n\"Of course I will. I took on this job, got paid, and gave my word I would finish it. The poison was just some bureaucratic moron's idea of a completion bond.\"\n\n\"I was sure you would say that. Knew that you would carry on regardless, threat of death or no threat of death.\"\n\nWhy was I uncomfortable when he said this? This was my old mate Steengo talking. Or was there a strong whiff of the admiral behind his words? Once the military, always the military... No, I would not think ill of him. But I better remember that the poison was still churning away. He was smiling widely and I let my smile mirror his. Although, deep inside, the worry and fear still nagged and scratched at my thoughts. Find the artifact, Jim. That is the only way to be sure about the antidote.\n\nI laughed and smiled. But only on the outside. \"Carry on, of course. The artifact must be found.\"\n\n\"Must be found, you are right. The search must go on!\" He looked over my shoulder and waved. \"And there's Floyd\u2014and Madonette. Welcome, my dear, welcome. I would stand to greet you, but only with difficulty.\"\n\nShe smiled and kissed his forehead below the bandage. Of course she was the last one to arrive, woman's prerogative. Though I had better abandon such male-chauv-pig reflexive observations. At least while I was still a guest of the ladies this side of Paradise.\n\n\"I have been talking to Mata,\" she said, seating herself and sipping a bit of fruit juice. \"The science building was empty when the explosion occurred, so no one was injured. Since then they have sifted the ruins and found that there is no trace at all of the artifact.\"\n\n\"Positive?\" I asked.\n\n\"Positive. They have been eavesdropping on the other side of the wall, so they knew about all our interest in the thing. They waited until they observed that all the male scientists had looked at it and prodded it enough. As expected those noble gentlemen\u2014referred to here as 'the geriatric incompetents'\u2014had discovered nothing. Having no further interest the scientists had it transferred here. A study program had been drawn up to examine the artifact but was just beginning when the explosion occurred. End of report.\"\n\nSo the artifact might have been stolen, might still be around. I could help look for it. But I could also stop counting the days. Earlier, when I had been woken up by my computer, it had been flashing a glowing seven for my benefit. Now Admiral Steengo had relieved me of this chronic worry.\n\nBut I had taken three million for this job\u2014and I still wondered what the thing really was. So the artifact-chase would continue. Minus the pressure of the days. I looked around at my musical rats and realized that nothing had changed for them. The search for the artifact was still on. Well\u2014why not!\n\n\"What do we do next?\" I said. Steengo, now more of an admiral than a musician, toted up the possible options.\n\n\"Was the explosion an accident? If it wasn't\u2014who caused it? There are really a lot of questions that must be asked...\"\n\n\"Mata told me to tell you that you were to ask Aida if you had any questions,\" Madonette said brightly.\n\nWe considered this seriously for a moment, then realized we hadn't the slightest idea of what she was talking about. Still the admiral, Steengo spoke for all of us.\n\n\"Who is Aida?\"\n\n\"Not who\u2014but what. An acronym for Artificially Intelligent Data Assembler. I think that it is the central computer here. In any case, here is the access terminal.\"\n\nShe put what looked like an ordinary portaphone on the table and switched it on. Nothing happened.\n\n\"Are you there, Aida?\" Madonette said.\n\n\"Ready to be summoned at any time, darling,\" the voice said. In a rich and sexy contralto.\n\n\"I thought you said computer?\" was my baffled response.\n\n\"Do I hear a male voice?\" Aida said. Then giggled. \"It has been such a very long time! Might I ask your name, sweetie?\"\n\n\"Jim\u2014not sweetie. And why did you call me that?\"\n\n\"Training and programming, dear boy. Before this present assignment I ran an exploration spacer. Male crew, endless years in space. It was felt by my creators that a female voice and presence would be more efficacious morale-wise than a machine or masculine presence.\"\n\n\"The last exploration spacer was junked centuries ago,\" Steengo said.\n\n\"A lady does not like to be reminded of her age,\" Aida said huskily. \"But it is true. When my ship was sent to the breakers I was made redundant. Since I am basically a computer program I am\u2014every woman's dream\u2014eternal. I had, shall we say, a rather varied career before I ended up here. Mind you, I'm not complaining. I find this such a pleasant occupation. There are charming ladies to talk to, as well as additional memory banks and data bases to access whenever I wish to. Most pleasurable\u2014but I do chatter on. I have been informed that you have a problem. If you would identify yourselves by name it would make conversation that much easier. Jim and Madonette I know. The name of the gentleman who just spoke?\"\n\n\"Admiral\u2014\" Steengo said, then broke off.\n\n\"Let us do keep it on a first-name basis. And your first name is Admiral. Others?\"\n\n\"Floyd,\" said Floyd.\n\n\"And a great pleasure to meet you all. How may I help?\"\n\n\"An item, referred to as the artifact, was recently brought to the science building. Do you know about it?\"\n\n\"Indeed I do. I was studying it, so am therefore quite familiar with the strange construction. In fact I had it under observation at the time of the explosion.\"\n\n\"Did you see what happened to it?\"\n\n\"Taking the literal meaning of see, dear Jim, forces me to answer that question in the negative. I had no photo pickups operating at the time so I did not physically see what happened to it. The only information I had was the direction that it left in. That was thirty-two degrees to the right of the zero north-polar latitude.\"\n\n\"There is nothing at all out there in that direction,\" Steengo said. \"No settlements, no nomadic tribes. Nothing but empty plains right up the polar cap. How do you know that the artifact was taken that way?\"\n\n\"I know that, mon Amiral, because this artifact emits tachyons and I was observing it with a tachyometer. Keeping count, so to speak, and most interesting it was too. It did not emit many\u2014after all, what source does?\u2014but a few are much better than none. Let the record show that it emitted one tachyon, from the direction I have given you, just microseconds before the explosion that destroyed the equipment I was using.\"\n\n\"You weren't\u2014injured?\" Madonette said.\n\n\"How sweet of you to ask! I wasn't, because I wasn't there. As soon as I could I constructed a new tachyometer, conveyed it to the site of the explosion with, unhappily, no results. Now there is just background radiation.\"\n\n\"Do you know what caused the explosion?\"\n\n\"Welcome to this easy give-and-take of social intercourse, friend Floyd. To answer your question\u2014I do. It was a very powerful explosive. I can give you the chemical formula but I am sure that you would find that immensely boring. But I can tell you that this explosive was manufactured quite widely for the mining industry at one time. It is named ausbrechitite.\"\n\n\"Never heard of it.\"\n\n\"Understandable, Admiral, since it was found to grow unstable with the passage of time. Manufacturing was phased out and ausbrechitite was replaced by newer and more stable explosives.\"\n\n\"When was this?\" I asked.\n\n\"A bit over three centuries ago. Would you like the exact date?\"\n\n\"That will do fine.\"\n\nWe blinked at each other in silence. Not knowing what to do with this weird historical-scientific evidence. Only Madonette had the brains to ask the right question.\n\n\"Aida\u2014do you have any theories about what happened?\"\n\n\"Simply thousands my dear. But there is no point in telling you about them until I gather some more evidence. Right now you might say that we are in the early moves of a chess game with millions of possibilities for the rest of the game. But I can give you some figures. Chances of an accidental explosion, zero. Chances that the explosion was tied in with the theft, sixty-seven percent. What happens next depends upon you.\"\n\n\"How?\"\n\n\"Consider reality. You are mobile, cher Jim while I am, so to speak, tied down to the job. I can give advice, and accompany you in transceiver form when you leave here. But what happens next\u2014that decision is up to you.\"\n\n\"What decision?\" Aida could be exasperating at times.\n\n\"I will supply a new tachyometer. If you take it in the direction I have indicated you might be able to track the artifact in this manner.\"\n\n\"Thanks,\" I said and reached out and turned Aida off. \"Looks like us humans have to come to a decision. Who follows the trail? Let us not all speak at once but let me speak first because I am top rat. I have the feeling that it is now time to thin our ranks. I say that Madonette does not go any further. We needed her for the music\u2014and wonderful she was too!\u2014but not for crawling around looking for nutcases planting century-old bombs.\"\n\n\"I second Jim's motion,\" Admiral Steengo said.\n\n\"I third it,\" Floyd said quickly as Madonette tried to speak. \"This is really not your kind of job. Nor is it Steengo's either.\"\n\n\"Isn't that for me to decide?\" Steengo snarled in his best admiralish mode.\n\n\"No,\" I suggested. \"If you wish to be of assistance, you can really help us by organizing the base operation from here. I declare that the motion has been seconded and passed above all objections. This is only a democracy when it suits me.\"\n\nSteengo smiled and the admiral's scowl vanished; he was too smart to argue. \"I agree. I am well past my sell-by date for fieldwork. My aching bones tell me that. Please, Madonette, give in graciously to the thrust of history. Are you nodding\u2014albeit reluctantly? Good. Above and beyond any aid given by Aida, I will see to it that the Special Corps will supply any equipment needed. Questions?\" He glowered around in a circle but we were silent. He nodded with satisfaction and Madonette raised her hand.\n\n\"With that decision out of the way\u2014may I pass on a request? In conversation I have discovered that everyone here is a true musical Rat fan so...\"\n\n\"Could we do one last gig before the group breaks up? You betcha. All in agreement.\"\n\nThere was a rousing cheer from all except Steengo, who looked unhappy at the thought of all of his instruments reduced to a pile of particles. But Madonette, ever resourceful, had done a bit of work before she mentioned the gig.\n\n\"I've asked around among the girls. They tell me that there is a really nice chamber group here, as well as a symphony orchestra\u2014they must have at least one instrument Steengo can play.\"\n\n\"Any of them, all of them\u2014just unleash me!\" he said and now it was smiles and cheers all around.\n\nDue to the miracles of modern medicines, curing and healing drugs, painkillers and a large shot of booze for Steengo, we were ready to do our performance later this same day. A matinee, since night here was still a couple of our days away and not worth waiting for.\n\nThere was quite a turnout at the sports stadium. Cheers and shouts of joy greeted us and no one seemed to mind that Steengo was not only out of costume but playing from a wheelchair. If this was to be the last curtain for The Stainless Steel Rats we meant to make it a performance to remember. Leaving the more militaristic and macho songs aside for the moment we launched into a mellow blues number.\n\nBlue world\u2014\n\nHear me singing my song.\n\nBlue world\u2014\n\nWhat's it I done wrong?\n\nBlue world\u2014\n\nYou gonna help me along\n\nBlue wor-r-r-ld.\n\nHere we are\u2014\n\nWe ain't goin' away.\n\nHere we are\u2014\n\nOn this planet to stay.\n\nBlue wor-r-r-ld.\n\nLanding was easy,\n\nPlenty of fun.\n\nDown came our rocket\u2014\n\n'Neath the blue sun.\n\nLanding was great\u2014\n\nEverything swell.\n\nNow it's all over,\n\nLiving is hell,\n\nDown here at the bottom of the gravity well.\n\nWe did many an encore this day. Finished finally with the feeling of exhaustion and happiness that only comes with an artistic job well done. Sleep came easily but, unable to resist, I took one last peek at the days remaining before closing my eyes.\n\nStill seven. Still a week. Plenty of time for my good buddy Admiral Steengo to kick butt and come up with the antidote. I think I was smiling when I closed my eyes which, when you think about it, was quite a change from the preceding twenty-three days. Yes it was.\n\nThen why wasn't I going to sleep? Instead of lying there tensely staring into the darkness. An easy answer.\n\nUntil the happy moment when I pulled back the plunger and shot up with the antidote I had only seven days to live.\n\nNighty-night, Jim. Sleep well...\nChapter 22\n\nEither I was a slugabed or the admiral, released from his role as a musician, was a workaholic. Or both. Because by the time I had appeared he had single-handedly organized our expedition down to the last detail. He was muttering over the heap of apparatus as he punched the checklist into his handheld. He glanced up, waved vaguely, then finished off the last items.\n\n\"This is your new backpack. It contains a number of items you will probably need\u2014and here's a printout of what's inside it. I assume that you have a good deal of illegal and possibly deadly items in your old pack which you can transfer after I leave. Aida is assembling another tachyometer and I'm going to get it now. Floyd will join you shortly\u2014and here is Madonette, welcome, welcome.\"\n\nSteengo made as graceful an exit as he could on crutches. Madonette, a picture of good cheer, swept in and took both of my hands in hers. Then discovered that this wasn't an enthusiastic enough greeting so she kissed me warmly on my cheek. My arms embraced her in automatic response, but closed on empty air since she had already whirled away and dropped onto the couch.\n\n\"I wish that I were coming with you, Jim\u2014but I know that it's impossible. Still, I'm not looking forward to getting back to the stuffy old office.\"\n\n\"I'm going to miss you,\" I said. Meaning it to be a calm statement but listening to myself in horror as it came out all dewy-eyed and smarmy. \"All of us will miss you, of course.\"\n\n\"Same here. There were some hairy moments\u2014but you took care of everything, didn't you?\" The warmth and appreciation were such that I could feel myself blushing. \"All in all I think it was an experience of a lifetime. And I am definitely not going back to all those files and staff meetings and sealed windows. It's fieldwork from now on. Out in the fresh air! Isn't that a good idea?\"\n\n\"Wonderful, yes indeed,\" I said, missing her already. I don't know where all this might have ended if Floyd hadn't made a disgustingly cheerful entrance.\n\n\"Morning all. Good day for the expedition. Hi and unhappily goodbye Madonette, companion of many an adventure. It has been fun working with you.\"\n\n\"Could you teach me unarmed defense?\"\n\n\"My pleasure. Easy enough if you work at it.\"\n\n\"Then I could train to be a field agent?\"\n\n\"Probably not. But I'll sure look into it.\"\n\n\"Would you! I'd be ever grateful. I was telling Jim that I don't want to work in an office anymore.\"\n\n\"Nor should you! A girl with your talents can find much better occupation.\"\n\nThey smiled at each other from opposite ends of the couch, knees almost touching, wrapped up in each other. I was forgotten. I hated Floyd's guts. Was more than happy to hear the thud of crutches and the dragging footsteps approaching.\n\n\"All here,\" Steengo said. \"Very good. The tachyometer is ready.\"\n\nThe thing that was following him now trotted forward. Walking, stiff-legged, was the ugliest fake dog that I had ever seen in my life. It was covered in black artificial fur with handfuls missing, had beady black eyes like buttons, stuck out a dry red tongue as it barked.\n\n\"Bow-wow.\"\n\n\"What do you mean 'bow-wow'?\" I gasped aloud. \"What is this repulsive object?\"\n\n\"The tachyometer,\" Admiral Steengo said.\n\n\"Bow-wow,\" it barked again. \"And for convenience sake the tachyometer is mounted within this mobile terminal.\"\n\n\"Aida?\" I said.\n\n\"None other. Do you like this disguise?\"\n\n\"I have never seen a more artificial artificial dog in my life!\"\n\n\"Well don't get too insulting about it. Fido is state of the art\u2014and that is modern art if you are thinking something nasty. For one thing the dear little doggy communicates with me by gravimetric waves which, as I am sure you know, cannot be blocked like radio waves. They penetrate the most solid buildings, cut through the most gigantic mountain ranges. So we are always in communication, always in touch. Admittedly Fido here has seen better days. But you know what they say about beggars?\"\n\n\"I do. But we're choosers without being beggars and I choose a better mobile terminal.\"\n\n\"Your choice, handsome. Give me two days and you can have whatever you want.\"\n\nTwo days? And I had like maybe six and a half to live unless the antidote arrived. I took a deep breath and whistled.\n\n\"Here Fido. Nice doggie. Let's go walkies.\"\n\n\"Bow-wow,\" it said and began to pant most artificially.\n\n\"This is the plan,\" Admiral Steengo said. \"I will monitor this operation from the orbiting spacer along with Captain Tremearne. Jim and Floyd will head north in the direction taken by the missing artifact. Aida will be in contact with this terminal, which will also be searching for a tachyon emission source.\" He appeared to run out of words and rubbed his jaw.\n\n\"A nice plan,\" I said, but I could not keep a certain tone of derision out of my voice. \"Cooked down to essentials it means that we just trot north until something happens.\"\n\n\"A satisfactory interpretation. Good luck.\"\n\n\"Thanks. And you will keep the other and most pressing matter of a certain injection on the top of your agenda?\"\n\n\"I shall query the people involved hourly on the hour,\" he said grimly\u2014and I think he meant it.\n\nWe filled our packs, kept the goodbyes as brief as possible, loaded up and followed Fido out without a backward glance. I liked Madonette. Perhaps too much while I was on an assignment like this. Go, Jim, go I cozened. Follow your wandering tachyon.\n\nWe followed the flapping black nylon tail through the streets and onward to the outlying farms. The women we met waved happily, some even whistling bits of our tunes to cheer us on the way. The last farm fell behind us and the open plains opened out ahead. I clacked my jaw-radio.\n\n\"Are you there, Tremearne?\"\n\n\"Listening in.\"\n\n\"Any tribes of nomads around\u2014or up ahead?\"\n\n\"Negative.\"\n\n\"Any buildings, farms, people, sheots\u2014anything visible on this heading?\"\n\n\"Negative. We've done a detailed scan as far north as the polar ice. Nothing.\"\n\n\"Thanks. Over and out.\" Wonderful.\n\n\"Empty on all sides, nothing at all ahead,\" I reported to Floyd. \"So we just stay on this heading until our plastic retriever detects any tachyons\u2014or we reach the north pole and freeze to death.\"\n\n\"I've been meaning to ask. What's a tachyon?\"\n\n\"Good question. Up until now I thought it was just a theoretical unit that the physicists dreamed up in order to explain how the universe works. One of the subatomic entities that exist either as waves or particles. Until they are observed they have no real existence. It has been said, and who am I to doubt it, that they exist in a probabilities limbo of many possible superimposed states.\" I noticed that Floyd's jaw was beginning to drop, his eyes to glaze. He shook his head.\n\n\"You are going to have to try harder, Jim\u2014you lost me a long time back.\"\n\n\"Right, sorry. Try this. There are various kinds of units in physics. A photon is a unit of light energy and an electron is a unit of electric energy. Okay?\"\n\n\"Great. With you so far.\"\n\n\"A graviton is a unit of gravity and a tachyon a unit of time.\"\n\n\"Lost me again. I thought minutes and seconds were units of time?\"\n\n\"They are, Floyd, but just to simple people like you and I. Physicists tend to look at things in a different manner.\"\n\n\"I believe it. Sorry I asked. Time for a break, five minutes in every hour.\"\n\n\"You're on.\" I unstuck my canteen and took a swig, then whistled to our dogtrotting terminal that was almost out of sight. \"Come back Fido, breakies.\"\n\n\"You're the boss,\" Aida said. The dog scrambled back, barked and sniffed my pack where I had dropped it next to me on the ground.\n\n\"Not too much realism!\" I shouted. \"Don't have that plastic canine lift its leg on my pack!\"\n\nThe day went on like that. Apparently forever. We crawled across the landscape: the sun crawled across the sky. When we had been walking for over five hours fatigue began to strike. Floyd was striding ahead at a great pace.\n\n\"Tired yet?\" I called out.\n\n\"No. Great fun.\"\n\n\"To those of us who weren't bashed about by the red peril.\"\n\n\"Just a bit more.\"\n\nThe bit more went on a bit more than I appreciated and I was just about to toss in the towel when Fido spoke.\n\n\"Bow and wow, gentlemen. Just detected a couple of tachyons as they went whizzing by. Wasn't sure of the first one but\u2014there it is, another\u2014and another!\"\n\n\"Coming from where?\" I asked.\n\n\"Directly ahead. Let's just stay on this course and we'll track the source down. With, perhaps, yes I'm sure, there is the strong possibility of a course deviation later.\"\n\n\"Aha!\" I ahaed. \"I recognize equivocation when I hear it. Even from a plastic dog mouthpiece for an ancient ship's computer.\"\n\n\"The word ancient is so hurtful...\"\n\n\"I'll apologize when you tell me about this complication.\"\n\n\"Apology accepted. Allowing for the curvature of the planet, gravitic anomalies and other factors, I am still forced to believe that the tachyon source is not on the surface of this world.\"\n\n\"The thing is underground?\"\n\n\"Underground is the very word for it.\"\n\nI bit hard on the jawphone. \"Tremearne, would you put the admiral on the line.\"\n\n\"I'm here, Jim. Aida reported this possibility a while back and I have been monitoring developments since then. Didn't want to bother you, for all the obvious reasons.\"\n\n\"Yes, like we forgot to bring a shovel. Anything else you haven't told me?\"\n\n\"I was waiting for data, just coming in. I sent a low-flying probe to look for the gravimetric anomalies that Aida had found. Looks like there are a number of them and they are being plotted now.\"\n\n\"What kind of anomalies? Metal deposits?\"\n\n\"Quite the opposite. Caverns below the ground.\"\n\n\"It figures. Over and out. At least we now know where the artifact is.\"\n\n\"Where?\" Floyd asked, since he had only heard my side of the conversation.\n\n\"Underground. There are caves or caverns of some kind up ahead. Nothing visible on the surface\u2014but they are there all right. Our technical observers seem sure that the artifact is down there somewhere. Can we take that break now and wait for the reports?\"\n\n\"I guess so.\"\n\nFloyd guessed right, which was a good thing since an instant after we dropped to the ground a stream of bullets was fired at us. Zipping through the empty air where we had just been standing.\n\nFloyd had a large and ugly pistol in his hand now which didn't slow him down as he wriggled on hands and knees beside me to the shelter of the mounded earth around a polpettone tree.\n\n\"We're under fire!\" I shouted into my jawphone.\n\n\"Source not visible.\"\n\nFido stood on its hind legs\u2014then jumped high into the air despite another burst of bullets.\n\n\"Bow-wow. Perhaps not visible to others but clear enough to me.\"\n\n\"What is it?\"\n\n\"Some sort of apparatus at ground level. Want me to take it out?\"\n\n\"If you can.\"\n\n\"Grrr!\" it growled and retracted its legs, then zipped off at a great rate at ground level, so fast it could barely be seen. Moments later there was a muffled explosion and bits of debris rattled down into the shrub.\n\n\"That was quick,\" I said.\n\n\"Thank you,\" Fido said emerging from the undergrowth with a jagged bit of metal in its jaws. \"Just follow me if you want to see the remains.\"\n\nWe followed the thing to a smoking pit with a jumble of crumpled apparatus in its center. Fido dropped its bit of debris, lifted one front leg. Extended its head, straightened its tail and pointed.\n\n\"Remote-controlled gun turret. Note that the top of it is camouflaged, concealed by dirt and sprouting plants. Hydraulically operated\u2014that's red oil not blood\u2014to lift the apparatus above ground level. Remains of an optical finder there. Note the four automatic guns, Rapellit-binetti X-nineteens. Rate of fire twelve hundred rounds a minute. Eighty rounds a second, explosive and armor piercing.\"\n\n\"Since when have you been an armament authority, Aida?\" I asked.\n\n\"Since a long time back, sweetie pie. In my heyday I was required to know this sort of thing. I also know that these particular guns have not been manufactured for over five hundred years.\"\nChapter 23\n\nI took another sip of water, wished that it was a stronger liquid. Was glad that it wasn't since a clear head was an important asset at this time.\n\n\"How old did you say these guns are?\" I asked. There was no answer because our fake dog was digging away like a real dog throwing dirt behind it at a great rate. Burrowing down under the gun turret.\n\n\"Five hundred years old,\" Floyd said. \"How can that be? Why use something that old?\"\n\n\"You use it if that is all that you have. There is a mystery here that we are about to solve. Remember the ancient explosive that blew up the lab? It was also antique. So consider this. What if this planet had been settled before they started dumping societal debris on it? What if there had been settlers here\u2014only they were hidden away underground? It's a possibility. And if it is true, then it has been five centuries since they arrived. That's how long these mysterious migrants have been hiding away up here. Or down here, really. They must have been settled well before the League ever found this planet. That's why there is no record of them.\"\n\n\"Who are they?\"\n\n\"Your guess is as good as mine...\"\n\n\"Yarf!\" our dogbot said, yarfing through a muzzle covered with dirt. \"There is a fiber-optic cable going into the ground, obviously controlling this turret.\"\n\n\"Going down to the caverns. So, the next question\u2014how do we get in...\"\n\n\"Jim,\" my jaw said. \"There is an interesting development taking place about three clicks away from you, in the same direction you have been walking. We've got image amplifiers on the electronic telescopes so we can see quite clearly...\"\n\n\"What can you see quite clearly?\"\n\n\"A group of armed men has emerged from some kind of opening in the ground. They appear to be dragging along one of their number who is bound. Now they are erecting a metal post of some kind. There is a struggle going on\u2014apparently they are securing the bound man to the post.\"\n\nMemories of a thousand ancient flicks flooded my forebrain. \"Stop them! It could be an execution\u2014death by firing squad. Do something!\"\n\n\"Negative. We are in orbit. Short of launching an explosive torpedo, which is contraindicated at this time, there is nothing we can facilitate that will get there inside fifteen minutes at the very quickest.\"\n\n\"Forget it!\" I was digging into my pack as I whistled to the houndbot. \"Fido! Catch!\"\n\nIt jumped high and grabbed the gas bomb out of the air. \"Go. Thataway. You heard the message\u2014get to those guys and bite hard on that thing.\"\n\nMy last words were shouted in the direction of the tail that was vanishing among the shrubs. We grabbed up our packs and followed. Floyd easily outdistanced me and by the time I got to the scene, staggering and panting, it was all ancient history. Our faithful friend was barking and, foreleg lifted and tail outstretched, was pointing at the sprawled bodies.\n\n\"Well done, man's best friend,\" I said, and easily resisted the impulse to pat its plastic fur.\n\n\"For the record,\" I said for the benefit of my radio. \"All males, all armed with shoulder weapons of some kind. There are twelve of them wearing camouflage uniforms. Thirteenth man\u2014surely an unlucky number\u2014tied to the post. No shirt.\"\n\n\"Is he injured?\"\n\n\"Negative.\" I could feel a steady pulse in his neck. \"We made it in time. Interesting, he's young, younger than the rest. What next?\"\n\n\"Decision made by the strategic planning computer. Take all weapons. Take the prisoner and remove him to a safe distance, then interrogate.\"\n\nI sniffed disdainfully as I unknotted the cords on the man's wrists. \"Don't need a strategic planning computer to figure that one out.\"\n\nFloyd caught him as he slumped free, threw him over his shoulder. I grabbed up the packs and pointed. \"Let's get to that gully and out of sight.\"\n\nThe bomb that the ersatz hound had exploded was a quick in-and-out gas. One breath and you were asleep. For about twenty minutes. Which was all the time that we needed to hump our loads through the mud of the rain-eroded gully until we found a dry spot under an overhanging bank. Our prisoner\u2014guest?\u2014began to roll his head and mutter. Floyd and I, and our mascot, sat down to watch and wait. It wasn't long. He muttered something, opened his eyes and saw us. Sat half up and looked very frightened.\n\n\"Fremzhduloji\" he said. \"Amizbko mizb.\"\n\n\"Sounds like really bad Esperanto,\" Floyd said.\n\n\"Just what you would expect if he and his kinfolk have been cut off from any outside contact for hundreds of years. Talk slow and he'll understand us.\"\n\nI turned to him and raised my hands palms out in what I hoped was a universal sign of peace. \"We're strangers, like you said. But what else did you say? Sounded like 'my friends'?\"\n\n\"Friends, yes, friends!\" he said, nodding like crazy, then shied away when Fido began barking.\n\n\"Aida, please. Will you shut your plastic poodle up. He's frightening our guest.\"\n\nThe thing stopped barking and spoke. \"Just want to report that I am in contact with the watchers above. They report that the others who were rendered unconscious by the gas have regained consciousness and have retreated.\"\n\n\"Great. Just file everything and report later.\" I turned back to our guest\u2014who looked very impressed by the talking-dog sequence. \"Well, friend. My name is Jim and this is Floyd. The furry fake is Fido. You have a name.\"\n\n\"I am called Dreadnought, son of Impervious.\"\n\n\"A pleasure to meet you. Now\u2014can you tell us why you were about to be wasted by that firing squad?\"\n\n\"Disobeyment of orders. I was on Watch. Saw your group approaching. I fired the Watchturret at you\u2014but do not yourselves anger! I aimed to miss. To fire demands permission of Watch Commander. That is why I was to be executed. I sought not his permission.\"\n\n\"Accidents happen.\"\n\n\"No accident. Fired because of orders.\"\n\n\"Are you following this?\" Floyd asked.\n\n\"Not too well. Tell us, Dreadnought, who gave the order to fire if it wasn't the Watch Commander?\"\n\n\"We all decided together.\"\n\n\"Who is we?\"\n\n\"I can not tell you.\"\n\n\"Understandable. Loyalty to your friends.\" I clapped him on the back in a friendly manner and felt him shiver. \"Getting cold. I'll get you a shirt.\"\n\nI dug through my pack and took advantage of the opportunity for a muttered conversation with my jawphone.\n\n\"Any ideas? From you\u2014or your indispensable strategic planning computer?\"\n\n\"Yes. If he won't talk to you perhaps the associates he referred to might be more communicative. Try to arrange a meeting.\"\n\n\"Right.\" I went back with the shirt. \"Here, Dreadnought, get out of the cold.\" He stood up and put it on. \"Good. Now I've been thinking. I don't want you to tell me things that you are not supposed to. But maybe your friends, the ones you just told us about, maybe they can let us know what is going down. Can we meet them?\"\n\nHe bit his lip and shook his head.\n\n\"No? Well let's try something else. Can you get back to your friends? Tell them about us. Talk about it. Find out if someone is prepared to tell us just what is happening. Okay?\"\n\nHe looked from me to Floyd, even down at Fido who wagged its tail, before he made his mind up.\n\n\"Come with me.\"\n\nHe was young and strong and trotted along at a mean trot. Floyd and the mechanical mutt kept up fine but my aches and pains were coming back. I trailed behind and was going to call a halt when Dreadnought stopped at the edge of a grove of polpettone trees.\n\n\"Wait this place,\" he said when I had puffed and blown up to them. He twisted away among the trees. He didn't notice that Fido, legs folded, tail and head retracted, had slipped silently after him in the guise of a black floormop. The cessation of physical activity was welcome\u2014as was the instant-heating meal I dug out of my pack. One porcuswine burger with gravy. Floyd popped his mealpak as well and we were licking the last drops of yummy from our fingers when the shadowlike mop reappeared. Legs, tail and head popped out and it barked. I scowled at it.\n\n\"Report first, bark later.\"\n\n\"Your new associate never saw me. Within the wood is a slab of rock that levers up with an opening beneath it. He went that way. Shall I show you where it is?\"\n\n\"Later\u2014if we have to. Right now let us take ten and see if he passes on our message.\"\n\nFatigue sat on me. I closed my eyes and took a lot more than ten. The sun was balancing on the horizon when I surfaced again. My computer obliged me by clicking the red six to a five when I checked the elapsed time. Don't worry, Jim\u2014Admiral Steengo is on your side! This feeble reassurance didn't help and I was sure that I could feel the thirty-day poison beginning to bubble and seethe in my bloodstream.\n\nFloyd was snoring lightly, sound asleep. Yet his eyes were open the instant Fido reappeared, disturbing some stones as it slid down the embankment.\n\n\"And a good-morning bow-wow to you gentlemen. Your new friend has emerged from under the lifting rock, along with an associate, and is coming this way. Remember\u2014you heard it from me first.\"\n\nFido sat and waited, then barked a welcome when the two men appeared. They were nattily dressed in camouflage uniforms and steel helmets, each helmet sporting a shiny spike on top. Bandoliers of bullets were draped over their shoulders, while there was a large and impressive handgun on each hip. But the guns were holstered and held in place by a buttoned strap. I relaxed knowing that with Floyd there the touch of a hand to one of those buttons would bring instant unconsciousness.\n\n\"Welcome back, Dreadnought,\" I said. \"Welcome as well your companion.\"\n\n\"He is named Indefatigable and is the Area Commander. That is Floyd with the beard, the other is Jim.\"\n\nIndefatigable did not shake hands but instead hit his closed right fist against his chest with an echoing thud. We did the same since it never hurts to learn the local customs.\n\n\"Why did you come here?\" Indefatigable asked in a most cold and quizzical manner. I took slight umbrage.\n\n\"You might say we came to save your companion from certain death by the firing squad\u2014your thanks are appreciated.\"\n\n\"If you had not come he would not have fired and have been condemned to death.\"\n\n\"Good point. But I do remember that he fired because of a group decision. Are you part of that group?\"\n\nI saw now that Indefatigable's brusque manner was a cover-up for the fact that he was very nervous. He chewed his lower lip and his eyes flicked from one to the other of us. He even looked down at the fake dog, which barked. Finally, with great reluctance he spoke.\n\n\"I cannot answer that. But I have been instructed to take you to those who may answer your question. Now\u2014you must answer my question. Why did you come here?\"\n\n\"No point in keeping it a secret. We came here to find those who blew up a certain building and stole from it\u2014and from us\u2014an object of great importance.\"\n\nThis news seemed to relax him a bit. He stopped the lip chewing and Dreadnought almost smiled; leaned forward to whisper something in his companion's ear. They both nodded, then remembered where they were and snapped into a military brace.\n\n\"You will come with us,\" Indefatigable said, making it sound like an order.\n\n\"Perhaps,\" I said. I hate orders. \"But you must tell us first\u2014will it be dangerous?\"\n\n\"We are born into danger; we leave it only when we die.\"\n\nIt sounded like a quotation of some kind\u2014particularly since Dreadnought's lips moved along with his.\n\n\"Yes, well, that is a pretty general philosophical statement. But I was speaking specifically about like right now.\"\n\n\"You will be protected,\" he answered, trying to control the sneer at our feeble physiques and his obvious superiority.\n\n\"Oh, thank you,\" Floyd said with eye-popping sincerity. \"With that kind of reassurance of course we will go with you. Isn't that right, Jim?\"\n\n\"Absolutely, Floyd. With their protection we need not feel insecure.\" He could eat them\u2014and a dozen more\u2014for breakfast, but there was no point in bragging.\n\nWe reached for our packs but Indefatigable stopped us. \"You bring nothing. No weapons. You must trust us.\"\n\nFloyd shrugged agreement since he was always armed. \"At least some water first,\" I said. Picking up my canteen and drinking a bit. Palming a number of small bombs as I put it back. \"And of course our companion, our pet dog goes with us.\"\n\nFido played its role by barking, sticking out its tongue and panting. Then overplayed its role by lifting its hind leg on my pack. Though this bit of canine ham acting may have convinced our new militaristic mates, because they nodded agreement.\n\n\"We must cover your eyes,\" Dreadnought said, pulling out two black scarves. \"So you do not discover the secret of the entrance to Shelter.\"\n\n\"If you mean the slab of rock under the polpettone trees that swings open, you can forget the blindfolds.\"\n\n\"How do you know this!\"\n\n\"Just say that we do. Now\u2014do we go with you?\"\n\nThey looked stricken by my revelation, stepped aside and conversed in quick whispers. Returned reluctantly, all scowls again.\n\n\"You will come. Quickly.\"\n\nWe dogtrotted, including the dog, to the grove, then followed Dreadnought down the ladder into the tunnel beneath the slab. Fido barked, and when I looked up launched itself down at me. I caught it, then dropped it. Looked gloomily into the darkness as Indefatigable closed the lid.\n\nI just hoped that we had made the right decision because my days were still running out. Going underground like this was a little too reminiscent of the grave.\n\nAnd it would be my grave if I didn't get the antidote in time.\nChapter 24\n\nOnce my eyes had adjusted to the darkness I saw that a thin line of light ran along at shoulder height on each side of the tunnel. The floor was smooth and hard, as were the walls when I brushed my fingers against them. We walked in silence for some time until we came to a cross tunnel.\n\n\"No talking now! Breathe silently\u2014do not stir,\" one of our guides whispered. \"Back against the wall.\"\n\nWe stayed that way for long minutes. I saw that there were glowing numerals on the walls where the tunnels crossed. I added to my store of useless knowledge the data that we were in tunnel Y-82790 at the place where it crossed NJ-28940. I leaned against the wall, and was thinking seriously about going to sleep, when I heard the thud of marching boots from NJ-28940. I woke up and remained silent and unmoving as a squad of about twenty men exited from the tunnel on our right and marched straight across and into the same numbered tunnel on the left. When the sound of their footsteps had almost died away we moved out to the whispered command.\n\n\"Turn left, after them. Quiet as you can.\"\n\nThis was apparently the only dangerous part of our journey, because once we had left this tunnel for another our companions whispered together again. I wondered if Fido was still with us.\n\n\"Don't bark,\" I said as softly as I could. \"But if you are still there, man's best friend, and hearing this with your super hearing, a tiny growl is permitted.\"\n\nA guttural grrr sounded from somewhere around my ankles.\n\n\"Great. A double growl now if you are reading the tunnel numbers and memorizing same.\"\n\nA quick grrr-grrr reassured me. So I did not have to keep track of our many turnings. After this we marched in silence for a tiresome period; my strength still wasn't what it should be. I was more than grateful when I saw a glow of light ahead; almost ran into our new companions when they stopped.\n\n\"Silence!\" Dreadnought whispered. Floyd and I silenced and listened\u2014then heard the running footsteps as well. They thudded close, then stopped suddenly.\n\n\"The sounds of deadly battle\u2014\" the newcomer said.\n\n\"Echo with the cries of the dying,\" Dreadnought answered. Password and countersign. Pretty depressing though. \"Is that you, Irredeemable?\" Dreadnought asked.\n\n\"It is. I was sent to warn you. A message was passed on from you-know-who that you were detected exiting and reentering the tunnels. Search parties are out and you must avoid them.\"\n\n\"How?\" Indefatigable asked. With just a touch of hysteria to his voice.\n\n\"I do not know. I was sent only to warn. May the God of Battles go with you.\" With this blessing the footsteps thudded again into silence as he ran back the way he had come.\n\n\"What do we do?\" Dreadnought asked unhappily. His companion was just as assertive. \"I don't know...\"\n\nI swear that I could hear their teeth chattering. Whatever else they were, these two young men were not plotters or planners. Time for a pro to step in.\n\n\"I will tell you what we must do.\" Speaking as an unhumble old plotter and planner.\n\n\"What?\" They spoke the word together.\n\n\"If they are searching the tunnels\u2014then we must leave the tunnels.\"\n\n\"Wonderful,\" Floyd muttered. It may have seemed pretty obvious to him but these lads welcomed the idea as they would have orders from the God of Battles himself.\n\n\"Yes! Leave\u2014before they find us!\"\n\n\"Out of the tunnels!\"\n\nGood so far, I thought. When the silence lengthened, and I realized that was the end of their contribution, I asked the vital question.\n\n\"Out of the tunnels, right. But where do we go? Above ground again?\"\n\n\"No\u2014all exits will be watched.\"\n\n\"Only one other way,\" Dreadnought said, with rising enthusiasm. \"Down, we must go down!\"\n\n\"To the Cultivastings!\" his companion added, just as filled with enthusiasm.\n\n\"Let's do it,\" I said wearily, not having the slightest idea of what they were talking about. \"The God of Battles wants it that way.\"\n\nThey double-timed and we followed. Around the bend into the next tunnel, where a glowing outline revealed that there was a metal door inset into the wall. Neither of our hosts tugged at the handle so there was a good chance that it was locked. Indefatigable stepped forward to face the illuminated keypad set into the wall beside it.\n\n\"Avert your eyes,\" he said. \"The access code is top secret.\"\n\n\"Get it, Fido,\" I whispered. Aida reacted instantly, our plastic pet extruded sharp toenails, leaped high then climbed up my clothes, scratching my ear painfully as it jumped onto the top of my head. I resisted the temptation to say ouch and stood steady so it could read the punched-in numbers. The door creaked open and the creature jumped back to the ground.\n\nA gentle breeze blew out through the doorway as we passed through it, smelling fresh and summery. Here underground? We stumbled in the darkness until the door clanged shut and the lights came on. We were in a small chamber facing a spiral staircase. Our hosts instantly started down it and we followed.\n\nI was beginning to get dizzy from the round-and-round when we finally got to the bottom. The open door here glared with light. Blinking my tired eyes, I followed the others. Outdoors into a field of ripening corn. Startled birds flapped away when we emerged, while something small and furry disappeared among the stalks.\n\nI knew that we couldn't possibly be outdoors, not after all the cave crawling that we had been doing. So this had to be a really giant cavern, with some kind of brilliant light sources above. These people really were independent of the surface\u2014no wonder they hadn't been spotted before.\n\nDreadnought led the way between the rows of corn and we followed. It was hot and dusty, my fatigue was still there\u2014and some species of tiny gnat kept trying to fly up my nose. I sneezed and rubbed and walked into Indefatigable's solid back when he stopped.\n\n\"Hail the Home and Joy in Survival!\" he called out.\n\n\"Hail, hail and welcome, brave Defender,\" a voice answered.\n\nA sweet and high-pitched woman's voice.\n\nWe started forward again and I stepped out from behind my guide's massive form, rubbing my nose and sniffling. I had a quick glimpse of a woman and three or four children working with hoes. It was a very quick glimpse\u2014for the instant that she saw me she screamed.\n\n\"Invasion Day!\"\n\nIt all happened incredibly fast. The children dived to the ground and she grabbed at the heavy pistol that hung from a lanyard around her neck. Raised it and began to fire at us.\n\nWe all hit the dust faster than the children had. Dreadnought was shouting, the gun was banging, rounds screamed by and exploded among the crops.\n\n\"Stop! No! No invasion! Enough, enough!\"\n\nI don't think she heard him at all. I tried to crawl down through the topsoil while I saw her squeezing and squeezing on the trigger, her eyes round and terrified, white teeth sunk into her lower lip. The only thing that kept us alive was the fact that the gun kicked hard and the muzzle rode up into the sky, with the last shots vanishing into the zenith.\n\nIt ended just as quickly as it had started. The children had disappeared. Indefatigable had grabbed the gun away from her and was patting her on the back as she sobbed hysterically.\n\n\"Well trained,\" Dreadnought said approvingly. \"Irreproachable is a fine woman, a good mother...\"\n\n\"And thankfully a rotten shot,\" I said. \"Would you like to tell us what all that was about?\"\n\n\"Training. Survival. For lo these many generations. With the galaxy at war we seek only peace. We survive. They will kill themselves, but we will survive!\"\n\nHe was winding himself up into a rallying speech so I broke in before he got into full spate.\n\n\"Stop! One minute\u2014enough. The galaxy's wars and the Breakdown ended centuries ago. There is no more war.\"\n\nHe lowered his clenched fist and sighed, rubbed his knuckle across his nose. \"I know. Some of us know. Most won't face the knowledge\u2014cannot face it. We are too trained for survival and nothing else. Nothing in our programming and our lives has ever prepared us for a time without war. Without the threat of invasion. Some of us assemble, we talk, make decisions. About the future. We have a leader\u2014I dare not tell you more!\"\n\nHe broke off as Indefatigable came running back.\n\n\"The message has arrived\u2014it is time to leave. The search has widened. If we move now we can stay behind the searchers and get to the meeting place. Quickly!\"\n\nWe quicklied\u2014and I was beginning to get very tired of it. The circular staircase had been a lot easier to come down than it was to climb up. Floyd saw my condition and if he hadn't half dragged me I doubt if I would have been able to make it. Once more into the black tunnels. I was only vaguely aware of our two guides, Floyd and the scuttling form of Fido. The next time we stopped I sagged against the wall. Enough was enough yet already.\n\n\"You will both stay here with Dreadnought,\" Indefatigable commanded. \"You will be sent for.\"\n\nNor would our watcher answer any questions in the few minutes that we waited. \"Proceed,\" a voice commanded and we did. Into a dimly lit chamber that appeared glaringly bright to our dark-adapted eyes. A half-dozen young men, garbed like our guides, sat on the other side of a long table.\n\n\"Stand here,\" Indefatigable ordered, then joined Dreadnought and sat down with the others.\n\n\"No chairs for us?\" I asked, but was ignored. Fido felt equally irked, jumped up onto the table and barked. Jumped back to the floor to dodge the swing of a fist.\n\n\"Shut up,\" one of the men suggested. \"We are awaiting orders. We are here, Alphamega.\"\n\nThey all turned to look at a red box on the table. It was made of plastic and was featureless except for louvers on one side.\n\n\"Are the two Outsiders you told me of present as well?\" the box asked. The voice was flat and mechanical and obviously cycled through a speech occulter.\n\n\"They are.\"\n\n\"I speak to you, Outsiders. I have been told that you come here seeking an object taken from you.\"\n\n\"That is correct, speaking-box.\"\n\n\"What is the function of this object?\"\n\n\"You tell me\u2014you stole it from us.\" I was beginning to get teed off at all this cloak-and-dagger stuff.\n\n\"Your attitude is unacceptable. Answer my question or be punished.\"\n\nI took a deep breath\u2014and reined in my temper.\n\n\"I'd like that,\" Floyd said cheerfully, as fed up as I was with all this nonsense.\n\nWhere the discussion would have gone from here would never be known because at that moment running footsteps sounded and a wild-eyed young man burst into the room.\n\n\"Alarm! Watchpatrol coming!\"\n\nThe sound of a number of thudding feet added a note of urgency to his warning. But at least our captors were prepared for the emergency. A door opened in the wall behind them and there was a rush to get through it. The newcomer, who must have known what would happen, was the last one in the crowd to jump to safety.\n\nThe table was in the way. I launched myself across it just in time to have the concealed door slammed in my face. I kicked it but it didn't budge. I looked at the now silent box.\n\n\"Speak up, Alphamega. How do we get out of this?\"\n\nThe red box crackled\u2014then burst into flame. Melted into a pool of plastic. \"Thanks,\" I said.\n\n\"Any other way out?\" Floyd asked.\n\n\"Not that I can see.\"\n\nThe rapid footsteps were just outside. Before I could dig out a gas bomb the scrum of armed men burst into the room.\n\nThings got busy. Floyd dropped the first three who came through the door while I tackled the next two. Then the going got tough because more and more kept pushing in. Some had body armor, all of them had transparent riot masks attached to their spiked helmets. They didn't try to shoot us, but rather enjoyed clubbing us with their guns.\n\nSomething hard got me on the back of the head and I staggered and fell. Before they jumped me the last thing I saw was Fido going up the wall like a spider and vanishing in the darkness there. Then I got thudded and had a nice darkness of my own.\n\n* * *\n\n\"Feeling any better, Jim?\" a distant voice said and I felt something wet and cool on my forehead.\n\n\"Shbsha...\" I said, or something like that. Chomped my dry mouth and opened my eyes. Floyd's face swam blurrily into view. I blinked and saw that he was smiling. He put the cold cloth back onto my forehead, which felt very nice.\n\n\"You got a bad one on the back of your head,\" he said. \"They didn't hit me quite as hard.\"\n\nI started to say Where are we? but figured that was a pretty dim question with an obvious answer. I could see a barred door which was hint enough. It hurt when I sat up on the bunk. Floyd handed me a plastic cup of water which I gurgled down and passed back for a refill. I patted my pockets and the seams of my trousers hopefully\u2014but all my concealed weaponry was gone.\n\n\"Seen any dogs around lately?\"\n\n\"Nope.\"\n\nSo that was that. Hit on the head. Imprisoned. Deserted by man's best friend. Somewhere underground so my jaw radio probably wouldn't work. Just in case I clacked hard and called for attention, but couldn't even get any static.\n\n\"Well\u2014it could be worse,\" Floyd said in a repellently cheery fashion. I was about to curse him out when he got just the answer he deserved.\n\n\"And it will be. You will be dead,\" the man said from the other side of the barred door. \"Instantly. If you attempt to touch me or the Killerbot behind me. Is that clear?\"\n\nHe was gray-haired, stern-faced, dressed in the same combat fatigues and spiked helmet as everyone else whom we had seen here. The only difference was that his spike was gold and had stylized wings on it. He moved aside and pointed at the very deadly looking collection of mobile military hardware behind him. All guns, clubs, wheels, knives and metal teeth. Teeth for tearing out throats?\n\nI had no intention of finding out. \"Follow me,\" our captor said, turning and walking away. The cell door clicked and swung open. Floyd and I shuffled out and followed him at a discreet distance. Clanking and rattling, the Killerbot rumbled along behind us.\n\nThe hallway, while being a depressing and drab tone of gray, was at least well lit. At regular intervals were framed photographs\u2014apparently all of the same individual from what I could see as we walked past. Or of a number of scowling military types differing only in the braid and the medals on their camouflage suits.\n\nOur host turned into a doorway that was flanked by studded steel columns. We followed\u2014all too aware of the clanking apparatus just behind.\n\n\"Impressive,\" I said, looking around the giant chamber. Black marble floor and walls. A large window looking out onto a military camp filled with flapping flags, marching troops, rows of armor-plated vehicles. Since we were deep underground it was obviously a projection\u2014but a very good one. These militaristic themes were also carried through in the interior decorations, light fixtures made of aerial bombs, machine-gun flowerpots, draperies assembled from tattered, ancient banners. I found it horribly depressing.\n\nWithout looking back our captor marched around the gigantic conference table and sat down in the single, high-backed chair there. With a wave of his hand he indicated the two smaller chairs before us.\n\n\"Sit,\" he commanded. Behind us was a clank and rattle, a hiss of escaping steam. We sat.\n\nSomething brushed my ankle and I looked down and saw that padded clamps had swung into position to secure my legs; motors whirred and they tightened.\n\nI threw my arms into the air just as clamps from the chair arms swung out and clicked shut on empty air.\n\n\"Not wise,\" our host said. There was a clank-clank close behind me and what could only have been a gun-muzzle ground into the back of my neck. The wrist clamps snapped open. I sighed and dropped my arms. I didn't have to look to know that Floyd had been imprisoned the same way.\n\n\"Leave.\"\n\nWhen his master commanded, the ambulatory war-machine clanked and rumbled out of the room and I heard the immense doors close.\n\n\"I am The Commander,\" our captor said, leaning back in his chair and lighting a large, green cigar.\n\n\"Is that your title or your name?\" I asked.\n\n\"Both,\" he said, blowing a ring of blue smoke towards the ceiling. \"I have imprisoned you since I do not wish to be attacked\u2014nor do I wish to have anyone or anything present while we talk.\" He touched a button on his desk and looked at pulsing purple light. \"And now we are secure against eavesdropping.\"\n\n\"Going to tell us who all you guys are, what you are doing here and that sort of thing?\" I asked.\n\n\"Assuredly. We are The Survivalists.\"\n\n\"I think I heard a reference to your mob before.\"\n\n\"Undoubtedly. During the years of the Breakdown there were a number of groups with that name. We are the only ones who deserve it since we are the only ones who survive.\"\n\n\"Survivalists,\" Floyd said, and went on as though reading from a book. \"Groups who believed in the inevitability of the coming war, as well as the inability of their own governments to protect them, who then withdrew from society into underground bunkers equipped with food, water, ammunition and supplies adequate to survive any catastrophe. None survive.\"\n\n\"Very good\u2014you are quoting from...?\"\n\n\"Handbook of Historical Nuts, Cults and Saviors.\"\n\n\"Very good\u2014except for the title and the last line. We survived.\"\n\n\"A little too well,\" I said. \"The Breakdown Wars are long gone and the galaxy is at peace now.\"\n\n\"I'm glad to hear that. Just don't tell anyone else here.\"\n\n\"Why not? But let me guess. You want to keep them stupid and in line because you are onto a very good thing. For as long as there is war or the threat of war those in charge tend to stay in charge. Which, of course, is you.\"\n\n\"An excellent summation, Jim. Though there are those who are unhappy with the state of things...\"\n\n\"We've met them. Youngsters who perhaps aren't too happy with the militaristic status quo and war forever. Who perhaps prefer a future in the bosom of their families. That is assuming you do have families?\"\n\n\"Of course, safe and secure in the residential caverns. We guard them and protect them\u2014\"\n\n\"As well as having a generally good time playing soldier and bossing everybody about.\"\n\n\"Your criticism is becoming tiring.\"\n\nHe looked quizzically at his cigar ash, then tapped it into the ashtray before him. Which was made from a shell casing of course. Something black stirred at the very edge of my vision but I made no move to look that way. It was about time Fido made an appearance.\n\n\"So what do you want us for?\" Floyd asked.\n\n\"I thought that was obvious. I want to find out who you are and how much you know about us.\"\n\nThere was a quick movement from under the table to my chair, out of The Commander's line of sight. The thing must have then climbed the back of my chair because Aida's voice whispered in my ear.\n\n\"I have done a voice analysis of a recording I made during the interrupted meeting. I stripped away the interference of the voice occulter and now know who the speaker who called himself Alphamega is...\"\n\n\"I already know,\" I said.\n\n\"Know what?\" The Commander said. \"What are you saying?\"\n\n\"Sorry, just speaking my thoughts aloud. My thoughts being that you are playing some kind of complicated game, aren't you? You called me by name\u2014and we have never been introduced. Of course if you were present at the meeting of the young dissidents you would know who I was. And now I know who you are.\"\n\nI smiled and let the silence stretch before I spoke.\n\n\"The Commander\u2014or Alphamega\u2014which name do you prefer? Since you are both of them rolled into one.\"\nChapter 25\n\n\"I can kill you\u2014quite quickly,\" The Commander said coldly and calmly. But at the same time he was stubbing and crunching his cigar out in a most agitated manner.\n\n\"Temper, temper,\" I said. \"Since you appear to be in charge of both sides in this internal conflict, and you obviously got us here for a reason\u2014why don't you just tell us all about it?\"\n\nHe was scowling now, angry and dangerous. As my mother always said\u2014why was her memory still popping up?\u2014you catch more porcuswine with honey than you do with vinegar. Gently, gently.\n\n\"Please, Commander,\" I pleaded most unctuously, \"we're on your side, even when no one else is. You know exactly what you are doing\u2014while none of your troops has the slightest idea what is happening. Not only are you in charge here, but it looks as though you have managed a mild insurrection on your own terms. You have done an incredible job that no one else was capable of doing. We can help you\u2014if you will let us.\"\n\nThe scowl faded. Floyd followed my lead, smiled and nodded agreement and said nothing; another cigar was produced and lit. The smoke rose up and the smoker nodded beneficently.\n\n\"You are right of course, Jim. The responsibility has been great, the pressure continuous. And I am surrounded by morons\u2014stulteguloj, kretenoj! Centuries of interbreeding and hiding underground has done little to improve their brain capacity. I am amazed that I alone have the intelligence to see this. I'm as different from them as if I had been born on a different planet, the child of superior parents.\"\n\nThis was sounding familiar. There has never been a strongman, dictator, military ruler, who did not believe that he somehow came from superior stock.\n\n\"You are different, sir,\" Floyd said, almost humbly. \"I knew that as soon as you spoke.\"\n\nWe had both obviously read the same textbooks. Though I thought he was spreading it on rather thickly. I was wrong.\n\n\"You could see that? The difference is obvious I suppose, to someone from Outside. It hasn't been easy, I tell you. In the beginning I even tried to talk to the senior officers, explain some of the problems and suggest solutions. I could have had more communication talking to a wall. Not that the younger ones are any better. Though they are restless, I give them that. When you get down to it there isn't much joy in just plain surviving. In the beginning maybe, it must have been a challenge then. But after a couple of centuries the pleasures begin to wear pretty thin.\"\n\n\"Was it the restlessness of the younger ones that gave you the idea to supply a leader for them to follow?\" I asked.\n\n\"Not at first. But I began to see that the young were losing respect for the old. About the only people they looked up to were the scientists. From their point of view the scientists were the only ones who at least appeared to be doing new and important things. That's when I hit on the Alphamega role. They think that I am one of the younger scientists. A rebel who is unable to make any progress against the old ideas, the familiar ways\u2014therefore I have been forced to enlist others of like age and mind.\"\n\n\"My arms are getting stiff,\" Floyd said, smiling. \"You wouldn't mind taking off these clamps for a bit?\"\n\n\"I would. I want you two just where you are.\"\n\nMercurial, our friend. All warmth gone in an instant, he dragged so hard on the cigar that it crackled and sparked. \"We Survivalists watch events pretty closely\u2014all over this planet. With a surveillance network set up before anyone else arrived. Amplified and spread ever since. Not a bird craps, not a polpettone fruit falls that we don't know about. That I don't know about. Because I watch the watchers. I watched and saw that a lot of energy and plenty of high-powered work was going into recovering that artifact. There is something very important about it\u2014and I want to know just what. I had a squad steal it and destroy the building, hide their tracks. It was impossible to follow them. Yet you did. I want to know how you did that too. So talk\u2014and talk fast.\"\n\n\"My pleasure,\" I said. \"My friend here knows nothing about the artifact. But I do. I am the one who found it first, then tracked it and followed it here. I am the only one who can tell you how it operates\u2014and what incredible things it can do. If you can take me to it I will be happy to show you how it works.\"\n\n\"That is more like it. You will come with me. Your associate remains here as a guarantee\u2014don't you agree?\" He stood and buckled on a large and offensive-looking sidearm.\n\n\"Of course. Sorry about that, Floyd,\" I said as I turned my head to face him. Winking with my left eye, the one our captor couldn't see. \"I know that you would come after me and help me if you could. But you can't. So stay here and you will be safe. You have the word of James Fido diGriz on that.\"\n\n\"I'll be okay. Jim. Look after yourself.\"\n\nI only hoped that this mixture of innuendo, hints and suggestions had delivered my message to him. I could only cross mental fingers and hope. The door opened and there was a hiss, rumble and clank behind me as my bonds snapped open. I rubbed my stiff arms and stood up slowly and carefully. The Killerbot blinked baleful little orange eyes at me and waved a smoke-stained flamethrower in the direction of the door. I followed Commander Alphamega out, leaving Floyd prisoner in the chair. Not for long, I hoped, if Fido-Aida had understood my suggestions.\n\nWe walked side by side down the wide hall with its framed portraits of heroes. My companion smiled warmly in my direction. Pulling his gun a bit out of the holster at the same time, then letting it slide back.\n\n\"You do understand that if you breathe one word about our conversation you will be no more than a grease spot on the floor?\"\n\n\"Completely aware, thank you. Absolute silence on that topic, yes, sir. I will look at the artifact and explain its operation. Nothing more.\"\n\nMaybe I was smiling on the outside\u2014but I was pretty gloomy on the inside. Jim, you are getting yourself in deeper than a porcuswine in a mudhole. A depressing thought\u2014and a true one. But I really had no choice.\n\nIt was quite a long walk and I was getting tired again. When all this was over\u2014if it were ever over\u2014I promised myself a nice long holiday. Head-up, Jim! Think positive and get ready to improvise.\n\nA last door opened and we were in what was obviously a laboratory. Complete with control boards, power cables, bubbling retorts and aged scientists in white smocks. There was a lot of loyal fist-smacking on chests when the leader appeared. Salutes that he returned with the merest tap of his own loosely clenched fist. They moved respectfully back to give us access to a lab bench. On it, now sprouting wires and connections to the surrounding test gear, was the alien artifact. I clapped my brow and staggered.\n\n\"What are you cretins doing with the cagleator!\" I shouted. \"We are all dead if you have actuated it!\"\n\n\"No, no\u2014not that!\" an elderly scientist cackled. Then shut up and looked fearfully at the Commander who sneered in return.\n\n\"You are all morons. Now tell this Outsider what you have done,\" he ordered. \"He is the one who knows what the device can do.\"\n\n\"Thank you, thank you! Of course, as you have ordered.\" The wrinkly turned back to me with shaking hands and pointed a quavering finger. \"We have only X-rayed the device and charted the circuitry. Very complex, as you know. There was, however...\" he began to sweat, looking about unhappily, \"a reaction of some kind when we attempted to test the circuitry.\"\n\n\"A reaction? If you have made a mistake the world has just ended! Show me.\"\n\n\"No, not a big reaction. Just that it absorbed electricity from our test circuit. We were not aware of this at first\u2014and we instantly terminated the test when we saw what was happening.\"\n\n\"And just what did you see happening?\" The Commander asked, voice like a file on rough steel.\n\n\"That, sir, we saw that. A cover of some kind fell away disclosing this recess. And the lights. That is all. Just lights...\"\n\nFascinated, we all leaned forward to look. Yes, there was the recess. And inside it there were four little blobs of light. Green, red, orange and white.\n\n\"What is the significance of this?\" my inquisitor asked, fingers strumming on the gunbutt.\n\n\"Nothing important,\" I said, stifling a yawn at the unimportance of it all. \"The test circuitry is simply testing the circuits of your test circuitry.\"\n\nI poked out a casual finger towards the glowing lights and found the barrel of his weapon grinding into my side.\n\n\"That sounds like absolute waffle to me. The truth, now, or you are dead.\"\n\nThere are seconds that sometimes appear to stretch for a length of time bordering on eternity. This was one of those occasions. The Commander glared at me. I tried to look innocent. The scientists, slack-jawed, looked at him. The Killerbot waited in the doorway and clanked to itself, hissing steam and probably wishing that it was killing something. Time stood still and eternity hovered close by.\n\nI had very few options open.\n\nLike none.\n\n\"The truth is...\" I said. And could not go on. What could I possibly say that would impress this maniac in any way? At this moment there was a great explosion and pieces of Killerbot clanked and rattled in through the door.\n\nAs you might imagine this really did draw everyone's attention. As did the voice that rang out an instant later.\n\n\"Jim\u2014drop!\"\n\nAnd there was Floyd at the open door, brandishing an impressive weapon of some kind. Fido had done its job and freed him. He had polished off the Killerbot and was now taking the action from there.\n\nThe Commander swung his weapon around, raised it, ready to fire.\n\nI did not drop as instructed because I was possessed by a hallucinatory moment of madness. I had been pushed around too much of late and suddenly, overwhelmingly, felt like doing a little pushing back.\n\nThe lights in the artifact glowed their welcome and my finger punched out in their direction.\n\nTo do what?\n\nTo touch one of the beckoning colored lights, of course.\n\nWhich one?\n\nWhat color meant what to the ancient aliens who had built this thing?\n\nI had no idea.\n\nBut green had always meant go to me.\n\nCackling hysterically I stabbed down on the green light...\nChapter 26\n\nApparently nothing happened. I pulled my finger back and looked at the lights. Then at The Commander and his drawn gun, wondering why he hadn't used it.\n\nThen looked at him again. And saw that he wasn't moving. I mean just not moving in the slightest. I mean like paralyzed. Petrified. Glassy-eyed and frozen.\n\nAs was everyone else in the room. Floyd stood in the doorway, gun raised and mouth open in an endless shout. Behind him, for the first time, I noticed an unmoving Fido.\n\nThe world was a freeze-frame and I was the only one not trapped in it. I was surrounded by people stopped in the act of speaking, walking, moving. Off-balance, hands raised, mouths gaping. Now stilled, silent\u2014dead?\n\nI started towards The Commander, to relieve him of his gun\u2014saw that his finger was tight on the trigger! But with each step I felt the air resisting my movement, growing firm, then more solid until it was like walking into an unyielding wall. Nor could I breathe\u2014the air was a thick liquid that I could not force into my lungs.\n\nPanic grew and grabbed me\u2014then died away just as quickly when I stepped back. I felt normal again. Air was air and I breathed in and out quite nicely.\n\n\"Put the mind in gear, Jim!\" I shouted at myself, my words loud in the surrounding silence. \"Something is happening\u2014but what? Something happened after you touched the green light. Something to do with the artifact.\"\n\nI stared at it. Tapped it with my knuckles. Groped about for inspiration. Found it.\n\n\"Tachyons! This thing emits them\u2014we know that because that is how Aida tracked it in the first place. Tachyons\u2014the units of time...\"\n\nThe device was now functioning\u2014I had turned it on when I had pressed the light. Green for go. Go where?\n\nStasis or speed. Either I had been speeded up or the world had slowed down. Or how could I tell the difference? From my point of view everything seemed to have slowed and stopped. The artifact had done something, projected a temporal field or stopped the motion of molecules. Or had created an occurrence that froze the surrounding world in a single moment of time. Time had come to a stop everywhere that I could see\u2014except in the close vicinity of the device. I moved even closer and patted it.\n\n\"Good little time machine. Time mover, slower, halter, stopper\u2014whatever you are. Neat trick. But what do I do next?\"\n\nIt chose not to answer me. Nor did I expect it to. This was my problem now and I had to force myself to take the time to think it out. For the moment I had all the time I needed. Though eventually I would have to do something. And that something would probably mean touching another one of the colored buttons. Either that or I could stand looking dumbly at the device while I quietly died of thirst or starvation or whatever.\n\nBut which light?\n\nGreen had been obvious enough\u2014even more obvious by hindsight. And the decision had been made at a moment of life and death. Now I was not so sure. I reached out, then dropped my hand. With plenty of time to decide I had become the master of indecision. Green had meant go, turn on, get started. Did red mean off, stop? Maybe. But what about white and orange?\n\n\"Not an easy one, Jimmy boy?\" I said in what I hoped was a jocular voice\u2014which came out very mournful and doom-laden. I wrung my hands together with indecision. Then stopped and looked at them as though I might see some answer printed on my fingers. All I saw was dirt under the nails.\n\n\"You have got to do it sooner or later\u2014so do it sooner before your nerve fails completely,\" I told myself. Reached out a finger\u2014drew it away. It looked like my nerve had indeed failed me completely.\n\n\"Take yourself in hand, Jim!\" I ordered. Reached back and took a handful of collar and shook myself as violently as I could.\n\nIt was no help at all. Random choice then? Why not, just as good as guessing. I put the finger out again and promised myself that I would push down on whatever color was under the finger when the jingle ended.\n\n\"Eeeny, meeny, miney, shmoe, catch a...\"\n\nI never found out what I was going to catch because at that moment I heard the dragging footsteps coming from the hall.\n\nSound?\n\nOut there where nothing moved!\n\nI jumped about, hands raised in defense. Lowered them and waited as the footsteps grew louder, came closer and closer to the doorway...\n\nSlipped past Floyd's immobile body.\n\n\"Aliens! Monsters!\" I gasped, pulling back. Trying to run although I knew there was no place to go.\n\nTwo hideous metal creatures. Bifurcated limbs, many-angled skulls, glowing eyes, claw-fingered hands. Coming towards me. Stopping. Reaching out\u2014\n\nNo! Reaching up to twist their own heads off. I could hear a gurgling scream, was only dimly aware that it was my own voice.\n\nTwisted and turned and lifted\u2014\n\nLifted off the helmets. Two very human faces looked at me with a good deal of interest. I stared back with the same emotion. Realized that, despite the close-cropped hair, the one on the left was female. She smiled at me and spoke.\n\n\"Wes bal, eltheodige, ac bwa bith thes thin freond?\"\n\nI blinked, didn't understand a word. Shrugged and smiled in what I hoped was a winning way. The second visitor shook his head.\n\n\"Unrihte tide unrihte elde, to earlich eart thu icome!\"\n\n\"Look,\" I said, having enough of this and very much needing a few questions answered. \"Could you please try Esperanto? That good, old, simple intergalactic second language Esperanto.\"\n\n\"Certainly,\" the girl said, smiling a winning and white-toothed smile. \"My name is Vesta Timetinker. My companion is Othred Timetinker.\"\n\n\"Married?\" I asked for some incomprehensible reason.\n\n\"No, stepsiblings. And you\u2014you have a name?\"\n\n\"Yes, of course. James diGriz. But everyone calls me Jim.\"\n\n\"A pleasure to meet you, Jim. Our thanks for activating the temporooter. We'll take it off your hands now.\"\n\nShe started towards the artifact\u2014which I now knew was a temporooter. Though I still knew little else. I stepped in front of it and said:\n\n\"No.\"\n\n\"No?\" Her rather attractive forehead furrowed while Othred's face suddenly looked grim. I turned a bit so I could keep an active eye on him.\n\n\"If no is too abrupt,\" I said, \"then I will ameliorate it and say hold on just a moment if you please. Didn't you just thank me for finding this thing?\"\n\n\"I did.\"\n\n\"Finding means that it has been lost. And has now been recovered because of my intervention. In return for this favor I believe you owe me at least an explanation.\"\n\n\"We're dreadfully sorry. But it is strictly forbidden to pass on information to temporal aborigines.\"\n\nNot too flattering, I thought. But I was thick-skinned enough to take it. \"Look,\" I carefully explained. \"This is one aborigine who already knows a good deal about what is happening. I now have in my possession your temporooter, a device that has been constructed for burrowing through time. It seems that you or your associates not only lost control of the device but actually lost it in time and space. This is very worrying because you are forbidden to reveal your operations to people living along the time tracks you explore.\"\n\n\"How\u2014how do you know this?\" she asked. Well done, Jim. They may be long on linguistics but are certainly short on extrapolation and imagination. Keep going.\n\n\"At first, when we aborigines discovered the device, we thought it was an alien construction from the far past, built by long-lost, millennia-dead aliens. Of course the real explanation is much simpler. It was sent from the future and through a malfunction got out of control.\" Now I was just guessing\u2014but their shocked expressions meant I was still doing well.\n\n\"Got so far out of control that it just kept going back in time until it ran out of power. Without power you could not locate it. You thought it might have been destroyed. Which is why there was such consternation when it signaled its presence. And you two were sent to retrieve it.\"\n\n\"You\u2014you read minds?\" She spoke in a hushed voice. I nodded firmly.\n\n\"The science of mental telepathy is well advanced in this era. Though it is obvious that all knowledge of our abilities has been expunged from your records in the future. But I will cease my mind reading now. I know how embarrassing it is to have one's secret thoughts revealed to strangers.\" I turned away, pinched my forehead, turned back. \"I have stopped the function. We now communicate by words.\"\n\nThey looked at each other, still dazed.\n\n\"Speak, please, for now I do not know what you are thinking. Only by speech can we understand each other's thoughts.\"\n\n\"Knowledge of time travel is forbidden,\" Othred said.\n\n\"That's not my fault\u2014you're the ones who lost the thing. You must understand that now I know all about it\u2014as do all of my brothers in telepathekinesis who have been listening to my thoughts. But we are sworn to silence! If you wish your secret to remain a secret it will be secret. But you must aid us in keeping this secret secret. Look about you. See this ugly-looking type in the horned helmet? He is just about to kill me. And when you entered you probably stepped over the wreckage of a very armed and deadly machine\u2014you did?, nod yes\u2014good. That thing was going to kill me and my friend, but he got it first. So just turning off the temporooter and skedaddling is out of the question. You will leave behind a deadly and destructive situation.\"\n\n\"What must we do?\" Vesta asked. Palm of my hand.\n\n\"First, you will help me by permitting myself and my associates to escape before the time stasis has been turned off.\"\n\n\"That should be possible,\" Othred said.\n\n\"Then that's agreed. Second, I will need another temporooter to take back with me...\"\n\n\"Forbidden! Impossible!\"\n\n\"Hear me out, will you please. Another temporooter to take back that does not function. A realistic fake that will disguise the fact that you and your machine have been here. Catch on?\"\n\n\"No.\"\n\nThey sure bred them dumb in the future. Or without imagination or whatever. I took a deep breath.\n\n\"Look. I want you to remember that all the scientists here, in this time, know that there is a device of some kind that looks like your temporooter. Only they think that it is an alien artifact from the far past. Let us convince them that their assumption is true. If we do that, why no one will ever know about you and your lost equipment. Just have your technicians get some million-year-old rock and carve out something that looks like this. We'll pass it off as the original, the secret will be kept, honor satisfied, all's well that ends well.\"\n\n\"Excellent idea,\" Vesta said, and pulled a microphone from her armored suit. \"I'll have one constructed now. It will be here in a second or two\u2014\"\n\n\"Wait. I have another small favor to ask. I will need certain functions built into the duplicate to convince our scientists that it is not a dummy. Just a simple device that will destruct after a single operation. This will pose absolutely no difficulties for your techs, I am sure.\"\n\nIt took me a bit longer to convince them of this necessity, but in the end they reluctantly agreed. The duplicate was an exact physical duplicate of the original. It blinked into existence floating in the air before us. Othred reached up and tugged; there was a popping sound as he pulled it down and handed it to me.\n\n\"Wonderful,\" I said, tucking it under my arm. \"Shall we go?\" They nodded agreement and put their helmets back on.\n\nI had my temporal companions first release the stasis field on Floyd's hand so I could disarm him. Like our mutual enemy his finger was also tightening on the trigger. What a world of nascent danger we do live in! I tucked the gun into my belt and nodded to the tempotechs.\n\nGive Floyd that\u2014his reflexes were great. He was twisting and chopping towards Othred's neck the second he moved\u2014stopped when I called a halt.\n\n\"Friends, Floyd. Down boy! Ugly-looking monster friends who are getting us out of here. If you look around you, you will see that all our enemies are paralyzed with indecision\u2014and will stay that way until we are gone. Don't trip over the pieces of the Killerbot on the way out. And, Vesta, if you please. Tap that fake ball of fur with your magic wand so it can join us.\"\n\n\"What the hell is going on?\" Floyd said, blinking in confusion as he tried to understand what was happening.\n\n\"I feel that some explanation is in order,\" Aida said, and Fido barked with exasperation.\n\n\"Second the motion,\" Floyd said.\n\n\"Forthcoming. As soon as we are out of here. Will you be so kind as to lead the way back to the surface.\"\n\nI turned to thank my temporal saviors, but they were already gone. Not only short on imagination but bereft of manners as well. And when they had vanished they had taken the time stasis with them; I could hear our footsteps for the first time. I looked back with a sudden feeling of horror but, right, the stasis was still working for the enemy as the silent form of the gun-toting snarling Commander indicated.\n\n\"Time to leave,\" I said. \"Since I have no idea how long the nasties are going to stand around that way. Go!\"\n\n\"Explain!\" Floyd shouted. Not in the best of moods.\n\n\"In a moment,\" I equivocated\u2014and stopped dead. For I had suddenly been possessed of an even more horrifying idea. All this playing with time\u2014what had it done for my personal poisonous deadline! I groped for my pendant skull-computer but of course it was gone with the rest of my equipment. How much time had passed? Was the poison now taking effect? Was I about to die...?\n\nSweating and trembling I dropped the replacement artifact temporooter and grabbed up the plastic poodle.\n\n\"Aida\u2014is Fido transmitting?\"\n\n\"Of course.\"\n\n\"What time is it\u2014I mean what day? No cancel that command. Get on to the Admiral now. Ask him how much time I have left. When is the deadline? Now\u2014please. Don't ask me any questions. He'll know what you are talking about. Do it! And fast!\"\n\nTime dragged by on very sluggish feet I will tell you. Floyd must have heard the desperation in my voice for he stayed silent. A second, a minute\u2014a subjective century crawled by before I had my answer. Aida must have done it\u2014and made a good connection. Because the next voice Fido spoke with was that of Admiral Steengo.\n\n\"Good to hear from you, Jim...\"\n\n\"Don't talk. Listen. I don't know what day it is. How much time is there to the deadline?\"\n\n\"Well, Jim, I wouldn't worry about that if I were you\u2014\"\n\n\"You are not me and I am worried and answer the question or I will kill you slowly first chance I have. Speaking of killing...\" I found that I couldn't go on.\n\n\"I meant it when I said don't worry. The threat of the thirty-day poison is over.\"\n\n\"You have the antidote?\"\n\n\"No. But the thirty days are past. Two days ago!\"\n\n\"Past!! Then I'm dead!\"\n\nBut I wasn't dead. My brain spluttered and clanked and slipped back into gear. Thirty days past. No antidote. I was alive. I could hear my teeth grating as I spoke.\n\n\"Then the thirty-day poison\u2014the whole thing was a fake from the start, wasn't it?\"\n\n\"I am afraid that it was, and I do apologize. But you must realize that I did not know about it until now. Only one person had that information, the instigator of the operation.\"\n\n\"Admiral Benbow!\"\n\n\"I'm afraid that information is not mine to reveal.\"\n\n\"You don't have to\u2014it reveals itself. That lawyer who gave me the drink was just doing as directed. Lawyers will do anything if you pay them enough. Benbow was in charge and Benbow invented the poison plot to keep me in line.\"\n\n\"Perhaps, Jim, perhaps.\" His voice, even when transmitted through the agency of a plastic dog, reeked of insincerity and equivocation. \"But there is nothing we can do about it now. A thing of the past. Best forgotten. Correct?\"\n\nI nodded and thought\u2014then smiled. \"Correct, Admiral. Why don't we just forget about the whole thing. All's well that ends well and tomorrow is another day. Forget it.\"\n\nFor now, I thought to myself, but did not speak that important little codicil aloud.\n\n\"I'm glad you understand, Jim. No hard feelings then.\"\n\nI dropped the dog, turned and clapped Floyd happily on the shoulder, bent and picked up the replacement artifact.\n\n\"We did it, Floyd, we did it. I will explain everything as we walk. In great detail. But as you can see we are free, in possession of this artifact. Mission accomplished. Now\u2014lead on, faithful Fido, since you have memorized the entrance-and-exit path. But go slowly, for it really has been one of those days.\"\n\nI was hungry and thirsty. But even more thirsty for\u2014what? Revenge? No, revenge was a dead end. If not vengeance\u2014what then?\n\nThe time had come for a little evening up, a little sorting out of the record. I had been taken in completely by the poisonous con job. So before the last i was dotted, before the last alien artifact was laid to rest, I was going to see that a little justice got done.\n\nOn my terms.\nChapter 27\n\n\"Carry this for a bit, will you Floyd,\" I said, passing over the replacement temporooter. We were leaving the last lit tunnel behind and would depend now on Aida to remember the way. \"I'm a little on the tired side.\"\n\n\"I don't wonder. But you have to understand\u2014my patience has just run out. So work hard and see if you can dig up enough energy to tell me just what happened. I am now completely confused. I remember that I wasted the Killerbot with that gun you now have tucked into your belt, the one Fido brought to me. Then I jumped through the door and told you to get down so I could blast the Commander as well as anyone else who was looking for trouble.\"\n\n\"That's just the way I remember it.\"\n\nFido barked and turned a corner from one dark tunnel into another even darker one. Floyd sounded worried.\n\n\"I remember pulling the trigger\u2014then suddenly you are holding the gun, not me, and right next to me there are two creatures, people, robots, something like that. I blink and look into the lab and everyone is standing like they are frozen. Nothing moves\u2014but nothing. Then when I look back I see that the two metal things have vanished. So I am beginning to feel like I am going around the mental bend. Therefore I would appreciate it if you would kindly, and quickly, tell me what happened.\"\n\n\"I wish I knew. I saw the same things you did. I don't know what happened.\"\n\n\"But you must know\u2014you were talking to them!\"\n\n\"Was I? I don't remember. Everything is still kind of fuzzy.\"\n\n\"Jim\u2014don't do this to me. You have to remember! And what were you shouting at the Admiral about? Something about poison and another Admiral.\"\n\n\"That's easy enough to answer. Certain individuals blackmailed me into this operation by telling me I had been poisoned and that I had thirty days to live if I didn't get the antidote. There was no poison\u2014therefore no antidote. So all the time we have been rushing about I have been thinking about the poison and counting the days before I curled up my toes and keeled over.\"\n\nHe was silent a moment, then he spoke.\n\n\"That's pretty heavy. You are sure about that?\"\n\n\"I am. And I am also terminally tired so can we please put this conversation off for a bit. I would just like to concentrate on putting one foot in front of another for awhile.\"\n\nLike it or not Floyd had to settle for that for the moment. Because I needed some time for deep cogitation, to dream up some sort of reasonable story for him\u2014as well as the rest of the troops. Stumbling with fatigue I was grateful that we made our way through the tunnels without meeting any opposition. Though I had the gun ready just in case. When Fido actuated the escape hatch and it opened to reveal the blue sky\u2014I sighed with relief. Gave the gun back to Floyd and used my remaining strength to crawl out onto the ground. Dropped with a groan and leaned back against a polpettone tree.\n\n\"You have the gun, Floyd,\" I said. \"So pass me back that ancient artifact if you please. Aida\u2014is there any transportation on the way?\"\n\n\"There should be. I sent out your position as soon as you were aboveground and I could get a triangulation. Help is on the way.\"\n\nAs indeed it was\u2014for a black spot in the sky grew quickly into the launch from the good old Remorseless. It landed with a shuddering thud, which bit of flying I recognized, so I was not surprised when Captain Tremearne exited through the open door.\n\n\"Congratulations,\" he said, and stuck out his hand. \"You did it, Jim.\"\n\n\"Thanks,\" I said, as he gave my hand a good crushing handshake. \"And don't think that it was easy.\"\n\n\"Never! I was there\u2014remember. Can I relieve you of that thing?\"\n\n\"No!\" I shouted\u2014and was shocked to hear the fine edge of hysteria, or incipient madness, to my voice. Well why not! \"I'll hand it over\u2014along with a detailed explanation of just what it is\u2014at the meeting.\"\n\n\"What meeting?\"\n\n\"The meeting that you are now going to arrange at the Pentagon. I'll want all The Stainless Steel Rats there. A last reunion so to speak. Has Madonette gone back to her imprisoning office yet?\"\n\n\"She was supposed to. But she would not leave the planet until you came back.\"\n\n\"Faithful to the end! So in addition to all the Rats I would like a few other friends present.\"\n\n\"Friends?\" He looked baffled. \"Like who?\"\n\n\"Well that macho fat thug Svinjar for one. King of the Machomen. Then you can invite Iron John and his opposite number, Mata. Ask yourself to come along as well. It will make an interesting gathering.\"\n\n\"Interesting\u2014yes! But impossible. None of the exiles on this prison planet is permitted inside the Pentagon.\"\n\n\"Really? I thought that you were the guy that was going to see that Liokukae was cleaned up and cleaned out?\"\n\n\"Yes\u2014but\u2014\"\n\n\"Now is the time, Captain. For at this meeting I am not only going to turn over the alien artifact and reveal its secret\u2014but I am going to tell everyone just how the situation here is going to end.\"\n\n\"How?\"\n\n\"You're invited to the meeting. You'll hear then.\"\n\n\"This will not be easy to arrange.\"\n\n\"Yes it will.\" I pointed to Floyd. \"Ask him about the strange things that happened when we were back there with the Survivalists. Admiral Steengo will verify his reports. There is a lot more to be cleaned up on this planet than you ever realized. Get your arguments together, consult your superiors, look after this.\" I passed over the artifacted artifact. \"And don't wake me up until it has been all arranged.\"\n\nI climbed wearily into the launch. Pushed up the armrests on the back row of seats. Stretched out and fell instantly to sleep.\n\n* * *\n\nThe next thing I knew Floyd was shaking me gently by the arm. \"We're back in the Pentagon. The meeting is on just like you said. I have breakfast and some clean clothes waiting for you. They'll be ready when you are.\"\n\nThe shower blasted out warm water and heated air and I stayed under it far too long. But it did wonders not only for my disposition but for my sore muscles as well. I did not hurry. They had arranged the meeting\u2014on my terms\u2014only because they had no choice. They would have informed me to get stuffed if they could. But the labtechs would have found nothing when they examined the artifact. Floyd would have told his confused story about what had happened when he had jumped in with his gun ready. Very confusing. In the end they would have been forced to the reluctant conclusion that the only way they could ever find out what had happened in the underground laboratory was by having me tell them. After which, knowing their record for veracity, they probably felt that they could do whatever they wanted with me.\n\n\"Well, Jim,\" I said to my smiling and sleek image in the mirror, as I carefully combed my hair, \"let's give them what they want.\"\n\nFloyd was my guide. Stamping in step with me along the corridors and into the conference room.\n\n\"Hi, guys!\" I said in cheery greeting to the far-from-friendly faces.\n\nOnly Madonette returned my smile, waved a tentative hand. Admiral Steengo was stern, Tremearne uncommunicative\u2014as was Mata. Floyd was grim-faced\u2014but winked when I glanced his way. Iron John and Svinjar were chained to their chairs or they would have killed me instantly. As it was they strained forward, eyes bulging with homicidal rage. I was most pleased to see that my hairy red friend had a bandaged skull and an arm in a sling. The aged artifact lay on the table before them and I went and sat on the edge of the table next to it.\n\n\"Tell us about the device,\" Admiral Steengo said in a reasonable and friendly voice.\n\n\"Not quite yet, Admiral. I assume that your techs could make nothing of it?\"\n\n\"They say it is over a million years old. That's all.\"\n\n\"There's more to it than that. But first a few introductions. The bruised guy with red fur is Iron John. Leader of a cult which you are now going to abolish. You can ship him off for treatment at an establishment for the criminally insane. Along with the fat man next to him. I have them here because I wanted you to see just what your policies of benign neglect had forced on the human beings out there on garbage world.\"\n\nI smiled and waited for the cursing and the spitting to die down, then nodded pleasantly at the unwholesome twosome.\n\n\"Would anyone here like to live in the kind of societies that you are subjecting the helpless people on Liokukae to? A committee must be appointed now. Plans drawn up to free the women and children from their bondage. You will find that Mata will be able to advise you on that. I think the various males on the planet will have to be interviewed separately. I'm sure that a number of them like their world the way it is. They can have it. The others deserve something better. But all that is in the future. First let us look at the past. I'm sure that the others on my team will grieve the passing of The Stainless Steel Rats. We have played our last gig, sung our last song. And we did pretty well for a bunch of amateurs. One juvenile criminal. An admiral, an unarmed combat expert, and a\u2014what are you really, Madonette? And don't embarrass both of us by talking about the imaginary office job again. That's not your style. Everyone else has come clean\u2014so how about you?\"\n\nShe drew herself up, looked grim\u2014then smiled. \"You deserve the truth, Jim. My office really is out there. But it is in the Galaksia Universitato, where I teach in the department of archeology. The university has so much money involved in this operation that they insisted on a representative.\"\n\n\"I'm glad it was you, Professor. Been fun working with you.\" I blew her a kiss, which she snatched out of the air and blew back.\n\n\"I didn't know about this!\" Admiral Steengo said, more than miffed. \"I am beginning to find out that there are levels of secrecy and duplicity in this so-called artifact retrieval operation that no one seems to know anything about. The more I discover about it\u2014the more it stinks. And more and more it appears to bear the stamp of Stinky Benbow.\"\n\n\"That nickname is classified and will be stricken from the records,\" a loathsomely familiar voice grated from the direction of the suddenly opened door. \"Fun and games are over. Sit down diGriz. I am in charge now.\"\n\n\"Well as I live and breathe!\" I turned, filled with great pleasure, to face the ever-scowling countenance of Admiral Benbow. \"This is almost too good to be true. The old poisoner himself\u2014in person.\"\n\n\"You will be silent. That is an order.\"\n\nSteengo was shocked. \"Benbow, you bastard\u2014have you been going over my head with this project? Are there other things about it that even I don't know?\"\n\n\"Plenty. But your need to know is plenty far down the knowing chain of command. So, like this crook\u2014shut up.\"\n\n\"No more orders, Benbow,\" I broke in. Reluctantly since there is nothing I enjoy more than a brace of admirals slanging each other off. But this was a time for work, not fun. \"Now tell the truth, just for a change. It was your idea to give me the fake thirty-day poison, wasn't it?\"\n\n\"Of course. I know how to deal with criminals. No trust, just fear. And complete control.\" The lizard lips bent into a frigid smile. \"I will show you how it works.\"\n\nHe snapped his fingers and an aide hurried in with a familiar package. He held it up and the serpentine smile broadened. \"You didn't really think that I would let you get away with this, did you?\"\n\nIt was the package with the three million credits that I had mailed to Professor Van Diver for safekeeping. My fee for putting my life in danger, money well earned. Now in the hands of the enemy. Not only wasn't I bothered by seeing it\u2014I was overjoyed.\n\n\"How kind of you, dear Admiral,\" I chortled. \"The circle is complete, the ring closed. The play ended. The alien artifact retrieved. The last song sung. Thank you, thank you.\"\n\n\"Don't sound so cheery, diGriz\u2014because you are in the deep cagal. Although you will not be executed for robbing the Mint you will get a well-deserved prison sentence for that crime. This fee, which you extorted from the university, will be returned to them. Along with that artifact...\"\n\n\"Oh\u2014so we have remembered it at last. Don't you want to know what it is, what it does?\"\n\n\"No. Not my problem. Let the university worry about that. I was against this entire operation from the first. Now it is over and life will go on the way it was.\"\n\n\"Including life on this despicable planet?\"\n\n\"Of course. We are not going to let the do-gooders interfere with the sound administration of the law.\"\n\n\"Admiral\u2014I do admire you,\" I said, standing and turning to the intent audience. \"Hear that, Iron John? You can go back to your old job at the bottom of the pond as soon as your bones heal. Svinjar, more killing and general swinery on your part. There will be the return of the rule of law and justice\u2014on Admiral Benbow's terms.\"\n\n\"Arrest this man,\" Benbow ordered, and two armed guards entered and marched towards me.\n\n\"I'll go quietly,\" I said. Turned and touched the alien artifact as I had been instructed to. \"But I'll go alone.\"\n\nIt was so quiet you could heard a pin drop. But, of course, a pin could not drop.\n\nNothing could move, was moving. Would move for quite a while.\n\nExcept me, of course. Strolling over, cheerfully whistling \"Nothing's Too Bad for the Enemy,\" relieving the Admiral of my hard-earned fee. Smiling benignly into his glaring, frozen face. Due to stay that way for quite awhile. I turned and waved at my statuelike audience.\n\n\"The best part was working with The Stainless Steel Rats. Thanks guys. Thanks as well to you, Captain Tremearne. In fact\u2014not only thank you\u2014but could you give me a little help?\"\n\nI walked over and touched his arm as I said this, enclosing him in the stasis-resistant field that enveloped me.\n\n\"Help you do what?\" He looked around at the motionless scene, turned back to me. \"What's going on here?\"\n\n\"What you see is what you get. No one is hurt, but no one is going to move for some time. Temporal stasis. When they come out of it they will never know that they have been in it.\"\n\n\"This is what happened to Floyd?\"\n\n\"Exactly.\"\n\n\"Exactly what?\"\n\n\"Time travelers. The alien artifact is not alien at all\u2014but a human construct from the far future, sent back and lost in time. I promised the time travelers not to tell anybody. I'll make this single exception since I need your help.\"\n\n\"Doing what?\"\n\n\"Getting both of us out of here so we can start the job of cleaning up this putrid planet. Here is what we have to do. Admiral Benbow has just arrived, as you saw, which means there is an interstellar spacer up there now in orbit about this planet. You and I will grab some transportation and get up to it. Once there you will use your rank, guile and forceful manner to see that we get aboard and far away from Liokukae. Then, when we get back to civilization, we will generate plenty of publicity about the evils men do here on this planet. It will be a scandal and heads will roll.\"\n\n\"Mine will be the first. Along with a court-martial, possible flaying and certainly life imprisonment.\"\n\n\"It shouldn't be that bad. If we get the forces of light on our side, why then the forces of darkness won't be able to lay a finger on you.\"\n\n\"It will take time...\"\n\n\"Captain\u2014that's the one thing we got plenty of! A good six months of it. That's how long this stasis will last. They won't know it, will not even realize a single second has passed. But, oh, will there be consternation among them when they discover how things have changed while they have been dozing! When I leave here the stasis will seal itself, impenetrable and impermeable. By the time it lifts the reform campaign will have succeeded and this prison planet will be nothing but a bad memory.\"\n\n\"And I will be cashiered, out of a job, will have lost my pension\u2014the works.\"\n\n\"And many a human being will be alive and happy who would have been miserable or dead. Besides, the military is no place for a grown man. And with a million credits in the bank you can buy lawyers, live the good life, forget your past.\"\n\n\"What million?\"\n\n\"The bribe that I am going to pay into a numbered account for you to make all of this worth your while.\"\n\nHe shook his fist. \"You are a crook, diGriz! Do you think that I would stoop to your criminal, crooked level?\"\n\n\"No. But you might be the administrator of the Save Liokukae Fund, which has been set up by an anonymous benefactor.\"\n\nHe scowled, opened his mouth to protest. Stopped. Burst out laughing.\n\n\"Jim\u2014you are something else again! What the hell\u2014I'll do it. But on my terms, understand?\"\n\n\"Understood. Just tell me where to mail the check.\"\n\n\"All right. Now let's get you a uniform while I forge some shipping orders. I have the feeling that I am going to enjoy being a civilian.\"\n\n\"You will, you will. Shall we go?\"\n\nWe went. Marching in step in a most military manner. Marching into the future, into a better, brighter future.\n\nThe blues had been sung. A page turned, a chapter ended. Tremearne would do a good job of sorting out this repellent world. I would do equally well as I slipped away between the interstices of society.\n\nIn six months I would be far from here, my trail cold, my bank account filled, my life more interesting. Once rested and restored\u2014it would indeed be time for The Stainless Steel Rat to ride again!\nAnd You Will Sing the Blues Too...\n\n... if you don't speak Esperanto. A number of readers, from a number of countries, have written to me asking if there is such a language as Esperanto. There is! Jim diGriz speaks it like a native\u2014as do most of the people he meets while involved in his interstellar trade. Esperanto is doing fine in the future\u2014but does it exist in the present?\n\nIt certainly does. It is a growing, living, simple second language for millions of speakers around the world. It is easy to learn\u2014and fun to use. There are many books, magazines and even newspapers published in Esperanto.\n\nIf you are interested in more information, The Stainless Steel Rat's advice\u2014and mine as well\u2014is to call this number:\n\n(510) 653-0998\n\nemail: elna@esperanto-usa.org\n\nor send a postcard with your name and address to:\n\nESPERANTO\n\nP.O. Box 1129\n\nEl Cerrito, CA 94530\n\nIt will change your life!\n\nHarry Harrison\nTOR BOOKS BY HARRY HARRISON\n\nGalactic Dreams\n\nThe Jupiter Plague\n\nMontezuma's Revenge\n\nOne Step from Earth\n\nPlanet of No Return\n\nPlanet of the Damned\n\nThe QE2 Is Missing\n\nQueen Victoria's Revenge\n\nA Rebel in Time\n\nSkyfall\n\nStainless Steel Visions\n\nThe Stainless Steel Rat Goes to Hell\n\nThe Stainless Steel Rat Joins the Circus\n\nStonehenge\n\nA Transatlantic Tunnel, Hurrah!\n\n50 in 50\n\nA Stainless Steel Trio\n\nTHE HAMMER AND THE CROSS TRILOGY\n\nThe Hammer and the Cross\n\nOne King's Way\n\nKing and Emperor\nThis is a work of fiction. All the characters and events portrayed in these novels are either fictitious or are used fictitiously.\n\nA STAINLESS STEEL TRIO\n\nCopyright \u00a9 2002 by Harry Harrison\n\nA Stainless Steel Rat Is Born, copyright \u00a9 1985 by Harry Harrison; The Stainless Steel Rat Gets Drafted, copyright \u00a9 1987 by Harry Harrison; The Stainless Steel Rat Sings the Blues, copyright \u00a9 1994 by Harry Harrison.\n\nAll rights reserved.\n\nEdited by Patrick Nielsen Hayden\n\nA Tor Book\n\nPublished by Tom Doherty Associates, LLC\n\n175 Fifth Avenue\n\nNew York, NY 10010\n\nwww.tor.com\n\nTor\u00ae is a registered trademark of Tom Doherty Associates, LLC.\n\neISBN 9781429975087\n\nFirst eBook edition: March 2014\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Brendan Lane, Garrett Alley and PG Distributed Proofreaders\n\n\n\n\nTHE BOOK-BILLS OF NARCISSUS\n\nAN ACCOUNT RENDERED BY RICHARD LE GALLIENNE\n\nWITH A FRONTISPIECE BY ROBERT FOWLER\n\n1895\n\n\n\n\nTABLE OF CHAPTERS\n\n   I. INTRODUCTORY\n  II. STILL INTRODUCTORY, BUT THIS TIME\n      OF A GREATER THAN THE WRITER\n III. IN WHICH NARCISSUS OPENS HIS 'GLADSTONE'\n  IV. ACCOUNTS RENDERED\n   V. AN IDYLL OF ALICE SUNSHINE, WHICH\n      REALLY BELONGS TO THE LAST CHAPTER\n  VI. THE SIBYLLINE BOOKS\n VII. THE CHILDREN OF APOLLO\nVIII. GEORGE MUNCASTER\n  IX. THAT THIRTEENTH MAID\n   X. 'IN VISHNU-LAND WHAT AVATAR?'\n\n\n\n\nTO MILDRED\n\n  Always thy book, too late acknowledged thine,\n    Now when thine eyes no earthly page may read;\n  Blinded with death, or blinded with the shine\n    Of love's own lore celestial. Small need,\n  Forsooth, for thee to read my earthly line,\n    That on immortal flowers of fancy feed;\n  What should my angel do to stoop to mine,\n    Flowers of decay of no immortal seed.\n\n  Yet, love, if in thy lofty dwelling-place,\n    Higher than notes of any soaring bird,\n      Beyond the beam of any solar light,\n      A song of earth may scale the awful height,\n  And at thy heavenly window find thy face--\n    know my voice shall never fall unheard.\n\n_December 6th,_ 1894.\n\nNOTE.--_This third edition has been revised, and Chapter V. is entirely\nnew_.\n\n\n\n\nCHAPTER I\n\n\nINTRODUCTORY--A WORD OF WISDOM, FOUND WRITTEN, LIKE THE MOST ANCIENT, ON\nLEATHER\n\n'Ah! old men's boots don't go there, sir!' said the bootmaker to me one\nday, as he pointed to the toes of a pair I had just brought him for\nmending. It was a significant observation, I thought; and as I went on\nmy way home, writing another such chronicle with every springing step,\nit filled me with much reflection--largely of the nature of platitude, I\nhave little doubt: such reflection, Reader, as is even already, I doubt\nless, rippling the surface of your mind with ever-widening circles. Yes!\nyou sigh with an air, it is in the unconscious autobiographies we are\nevery moment writing--not those we publish in two volumes and a\nsupplement--where the truth about us is hid. Truly it is a thought that\nhas 'thrilled dead bosoms,' I agree, but why be afraid of it for that,\nReader? Truth is not become a platitude only in our day. 'The Preacher'\nknew it for such some considerable time ago, and yet he did not fear to\n'write and set in order many proverbs.'\n\nYou have kept a diary for how many years? Thirty? dear me! But have you\nkept your wine-bills? If you ever engage me to write that life, which,\nof course, must some day be written--I wouldn't write it myself--don't\ntrouble about your diary. Lend me your private ledger. 'There the action\nlies in his true nature.'\n\nYet I should hardly, perhaps, have evoked this particular corollary from\nthat man of leather's observation, if I had not chanced one evening to\ncome across those old book-bills of my friend Narcissus, about which I\nhave undertaken to write here, and been struck--well-nigh awe-struck--by\nthe wonderful manner in which there lay revealed in them the story of\nthe years over which they ran. To a stranger, I am sure, they would be\nfull of meaning; but to me, who lived so near him through so much of the\ntime, how truly pregnant does each briefest entry seem.\n\nTo Messrs. Oldbuck and Sons they, alas! often came to be but so many\naccounts rendered; to you, being a philosopher, they would, as I have\nsaid, mean more; but to me they mean all that great sunrise, the youth\nof Narcissus.\n\nMany modern poets, still young enough, are fond of telling us where\ntheir youth lies buried. That of Narcissus--would ye know--rests among\nthese old accounts. Lo! I would perform an incantation. I throw these\nold leaves into the _elixir vitae_ of sweet memory, as Dr. Heidegger\nthat old rose into his wonderful crystal water. Have I power to make\nNarcissus' rose to bloom again, so that you may know something of the\nbeauty it wore for us? I wonder. I would I had. I must try.\n\n\n\n\nCHAPTER II\n\n\nSTILL INTRODUCTORY, BUT THIS TIME OF A GREATER THAN THE WRITER\n\nOn the left-hand side of Tithefields, just as one turns out of Prince\nStreet, in a certain well-known Lancashire town, is the unobtrusive\nbookshop of Mr. Samuel Dale. It must, however, be a very superficial\nglance which does not discover in it something characteristic,\ndistinguishing it from other 'second-hand' shops of the same size and\nstyle.\n\nThere are, alas! treatises on farriery in the window; geographies,\nchemistries, and French grammars, on the trestles outside; for Samuel,\nalbeit so great a philosopher as indeed to have founded quite a school,\nmust nevertheless live. Those two cigars and that 'noggin' of whiskey,\nwhich he purchases with such a fine solemnity as he and I go home\ntogether for occasional symposia in his bachelor lodging--those, I say,\ncome not without sale of such treatises, such geographies, chemistries,\nand French grammars.\n\nBut I am digressing. There is a distinguishing air, I but meant to say,\nabout the little shop. Looking closer, one generally finds that it comes\nof a choice bit of old binding, or the quaint title-page of some tuneful\nElizabethan. It was an old Crashaw that first drew me inside; and,\nthough for some reason I did not buy it then, I bought it a year after,\nbecause to it I owed the friendship of Samuel Dale.\n\nAnd thus for three bright years that little shop came to be, for a daily\nhour or so, a blessed palm-tree away from the burden and heat of the\nnoon, a holy place whither the money-changers and such as sold doves\nmight never come, let their clamour in the outer courts ring never so\nloud. There in Samuel's talk did two weary-hearted bond-servants of\nEgypt draw a breath of the Infinite into their lives of the desk; there\ncould they sit awhile by the eternal springs, and feel the beating of\nthe central heart.\n\nSo it happened one afternoon, about five years ago, that I dropped in\nthere according to wont. But Samuel was engaged with some one in that\ndim corner at the far end of the shop, where his desk and arm-chair,\ntripod of that new philosophy, stood: so I turned to a neighbouring\nshelf to fill the time. At first I did not notice his visitor; but as,\nin taking down this book and that, I had come nearer to the talkers, I\nwas struck with something familiar in the voice of the stranger. It came\nupon me like an old song, and looking up--why, of course, it was\nNarcissus!\n\nThe letter N does not make one of the initials on the Gladstone bag\nwhich he had with him on that occasion, and which, filled with books,\nlay open on the floor close by; nor does it appear on any of those\ntobacco-pouches, cigar-cases, or handkerchiefs with which men beloved of\nfair women are familiar. And Narcissus might, moreover, truthfully say\nthat _it_ has never appeared upon any manner of stamped paper coming\nunder a certain notable Act.\n\nTo be less indulgent to a vice from which the Reader will, I fear, have\ntoo frequent occasion to suffer in these pages, and for which he may\nhave a stronger term than digression, let me at once say that Narcissus\nis but the name Love knew him by, Love and the Reader; for that name by\nwhich he was known to the postman--and others--is no necessity here. How\nand why he came to be so named will appear soon enough.\n\nYes! it was the same old Narcissus, and he was wielding just the same\nold magic, I could see, as in our class-rooms and playgrounds five years\nbefore. What is it in him that made all men take him so on his own\nterms, made his talk hold one so, though it so often stumbled in the\ndark, and fell dumb on many a verbal _cul-de-sac_? Whatever it is,\nSamuel felt it, and, with that fine worshipful spirit of his--an\nattitude which always reminds me of the elders listening to the boy\nJesus--was doing that homage for which no beauty or greatness ever\nappeals to him in vain. What an eye for soul has Samuel! How inevitably\nit pierces through all husks and excrescences to the central beauty! In\nthat short talk he knew Narcissus through and through; three years or\nthirty years could add but little. But the talk was not ended yet;\nindeed, it seemed like so many of those Tithefields talks, as if in the\n'eternal fitness of things' it never could, would, or should end. It was\nI at last who gave it pause, and--yes! indeed, it was he. We had,\nsomehow, not met for quite three years, chums as we had been at school.\nHe had left there for an office some time before I did, and, oddly\nenough, this was our first meeting since then. A purchaser for one of\nthose aforesaid treatises on farriery just then coming in, dislodged us;\nso, bidding Samuel good-bye--he and Narcissus already arranging for 'a\nnight'--we obeyed a mutual instinct, and presently found ourselves in\nthe snuggery of a quaint tavern, which was often to figure hereafter in\nour sentimental history, though probably little in these particular\nchapters of it. The things 'seen done at \"The Mermaid \"' may some day be\nwritten in another place, where the Reader will know from the beginning\nwhat to expect, and not feel that he has been induced to buy a volume\nunder false pretences.\n\n\n\n\nCHAPTER III\n\n\nIN WHICH NARCISSUS OPENS HIS 'GLADSTONE'\n\nThough it was so long since we had met--is not three years indeed 'so\nlong' in youth?--we had hardly to wait for our second glass to be again\n_en rapport_. Few men grow so rapidly as Narcissus did in those young\ndays, but fewer still can look back on old enthusiasms and superannuated\nideals with a tenderness so delicately considerate. Most men hasten to\nwitness their present altitude by kicking away the old ladders on the\nfirst opportunity; like vulgar lovers, they seek to flatter to-day at\nthe expense of yesterday. But Narcissus was of another fibre; he could\nas soon have insulted the memory of his first love.\n\nSo, before long, we had passed together into a sweet necropolis of\ndreams, whither, if the Reader care, I will soon take him by the hand.\nBut just now I would have him concern himself with the afternoon of\nwhich I write, in that sad tense, the past present. Indeed, we did not\nourselves tarry long among the shades, for we were young, and youth has\nlittle use for the preterite; its verbs are wont to have but two tenses.\nWe soon came up to the surface in one, with eyes turned instinctively on\nthe other.\n\nNarcissus' bag seemed, somehow, a symbol; and I had caught sight of a\nbinding or two as it lay open in Tithefields that made me curious to see\nit open again. He was only beginning to collect when we had parted at\nschool, if 'collect' is not too sacred a word: beginning to _buy_ more\ntruly expresses that first glutting of the bookish hunger, which, like\nthe natural appetite, never passes in some beyond the primary\nutilitarian stage of 'eating to live,' otherwise 'buying to read.' Three\nyears, however, works miracles of refinement in any hunger that is at\nall capable of culture; and it was evident, when Narcissus did open his\n'Gladstone,' that it had taken him by no means so long to attain that\nsublimation of taste which may be expressed as 'reading to buy.' Each\nvolume had that air--of breeding, one might almost say--by which one can\nalways know a genuine _bouquin_ at a glance; an alluvial richness of\nbloom, coming upon one like an aromatic fragrance in so many old things,\nin old lawns, in old flowers, old wines, and many another delicious\nsimile. One could not but feel that each had turned its golden brown,\njust as an apple reddens--as, indeed, it had.\n\nI do not propose to solemnly enumerate and laboriously describe these\ngood things, because I hardly think they would serve to distinguish\nNarcissus, except in respect of luck, from other bookmen in the first\nfuror of bookish enthusiasm. They were such volumes as Mr. Pendennis ran\nup accounts for at Oxford. Narcissus had many other points in common\nwith that gentleman. Such volumes as, morning after morning, sadden\none's breakfast-table in that Tantalus _menu_, the catalogue. Black\nletter, early printed, first editions Elizabethan and Victorian, every\npoor fly ambered in large paper, etc. etc.; in short, he ran through the\ngamut of that craze which takes its turn in due time with marbles,\npeg-tops, beetles, and foreign stamps--with probably the two exceptions\nof Bewick, for whom he could never batter up an enthusiasm, and\n'facetiae.' These latter needed too much camphor, he used to say.\n\nHis two most cherished possessions were a fine copy of the _Stultitiae\nLaus_, printed by Froben, which had once been given by William Burton,\nthe historian, to his brother Robert, when the latter was a youngster of\ntwenty; and a first edition of one of Walton's lives, 'a presentation\ncopy from the author.' The former was rich with the autographs and\nmarginalia of both brothers, and on the latter a friend of his has\nalready hung a tale, which may or may not be known to the Reader. In the\nreverent handling of these treasures, two questions inevitably forced\nthemselves upon me: where the d----l Narcissus, an apprentice, with an\nallowance that would hardly keep most of us in tobacco, had found the\nmoney for such indulgences; and how he could find in his heart to sell\nthem again so soon. A sorrowful interjection, as he closed his bag,\nexplained all:--\n\n'Yes!' he sighed, 'they have cost me thirty pounds, and guess how much I\nhave been offered for them?'\n\nI suggested ten.\n\n'Five,' groaned my poor friend. 'I tried several to get that. \"H'm,\"\nsays each one, indifferently turning the most precious in his hand,\n\"this would hardly be any use to me; and this I might have to keep\nmonths before I could sell. That I could make you an offer for; what\nhave you thought of for it?\" With a great tugging at your heart, and\nwell-nigh in tears, you name the absurdest minimum. You had given five;\nyou halve it--surely you can get that! But \"O no! I can give nothing\nlike that figure. In that case it is no use to talk of it.\" In despair\nyou cry, \"Well, what will you offer?\" with a choking voice. \"Fifteen\nshillings would be about my figure for it,\" answers the fiend,\nrelentless as a machine--and so on.'\n\n'I tried pawning them at first,' he continued, 'because there was hope\nof getting them back some time that way; but, trudging from shop to\nshop, with many prayers, \"a sovereign for the lot\" was all I could get.\nWorse than dress-clothes!' concluded the frank creature.\n\nFor Narcissus to be in debt was nothing new: he had always been so at\nschool, and probably always will be. Had you reproached him with it in\nthose young self-conscious days of glorious absurdity, he would probably\nhave retorted, with a toss of his vain young head:--\n\n'Well, and so was Shelley!'\n\nI ventured to enquire the present difficulty that compelled him to make\nsacrifice of things so dear.\n\n'Why, to pay for them, of course,' was the answer.\n\nAnd so I first became initiated into the mad method by which Narcissus\nhad such a library about him at twenty-one. From some unexplained\nreason, largely, I have little doubt, on account of the charm of his\nmanners, he had the easy credit of those respectable booksellers to whom\nreference has been made above. No extravagance seemed to shake their\nconfidence. I remember calling upon them with him one day some months\nfollowing that afternoon--for the madness, as usual, would have its\ntime, and no sufferings seemed to teach him prudence--and he took me up\nto a certain 'fine set' that he had actually resisted, he said, for a\nfortnight. Alas! I knew what that meant. Yes, he must have it; it was\njust the thing to help him with a something he was writing--'not to\nread, you know, but to make an atmosphere,' etc. So he used to talk; and\nthe odd thing was, that we always took the wildness seriously; he seemed\nto make us see just what he wanted. 'I say, John,' was the next I heard,\nat the other end of the shop, 'will you kindly send me round that set\nof' so-and-so, 'and charge it to my account?' 'John,' the son of old\nOldbuck, and for a short time a sort of friend of Narcissus, would\nanswer, 'Certainly,' with a voice of the most cheerful trust; and yet,\nwhen we had gone, it was indeed no less a sum than L10, 10s. which he\nadded to the left-hand side of Mr. N.'s account.\n\nDo not mistake this for a certain vulgar quality, with a vulgar little\nname of five letters. No one could have less of that than Narcissus. He\nwas often, on the contrary, quite painfully diffident. No, it was not\n'cheek,' Reader; it was a kind of irrational innocence. I don't think it\never occurred to him, till the bills came in at the half-years, what\n'charge it to my account' really meant. Perhaps it was because, poor\nlad, he had so small a practical acquaintance with it, that he knew so\nlittle the value of money. But how he suffered when those accounts did\ncome in! Of course, there was nothing to be done but to apply to some\nlong-suffering friend; denials of lunch and threadbare coats but nibbled\nat the amount--especially as a fast to-day often found revulsion in a\nfestival to-morrow. To save was not in Narcissus.\n\nI promised to digress, Reader, and I have kept my word. Now to return to\nthat afternoon again. It so chanced that on that day in the year I\nhappened to have in my pocket--what you might meet me every day in five\nyears without finding there--a ten-pound note. It was for this I felt\nafter we had been musing awhile--Narcissus, probably, on everything\nelse in the world except his debts--and it was with this I awoke him\nfrom his reverie. He looked at his hand, and then at me, in\nbewilderment. Poor fellow, how he wanted to keep it, yet how he tried to\nlook as if he couldn't think of doing so. He couldn't help his joy\nshining through.\n\n'But I want you to take it,' I said; 'believe me, I have no immediate\nneed of it, and you can pay me at your leisure.' Ten pounds towards the\nkeep of a poet once in a lifetime is, after all, but little interest on\nthe gold he brings us. At last I 'prevailed,' shall I say? but on no\naccount without the solemnity of an IOU and a fixed date for repayment,\non which matter poor N. was always extremely emphatic. Alas! Mr. George\nMeredith has already told us how this passionate anxiety to be bound by\nthe heaven above, the earth, and the waters under the earth, is the most\nfatal symptom by which to know the confirmed in this kind. Captain\nCostigan had it, it may be remembered; and the same solicitude, the same\ntearful gratitude, I know, accompanied every such transaction of my\npoor Narcissus.\n\nWhether it was as apparent on the due date, or whether of that ten\npounds I have ever looked upon the like again, is surely no affair of\nthe Reader's; but, lest he should do my friend an injustice, I had\nbetter say--I haven't.\n\n\n\n\nCHAPTER IV\n\n\nACCOUNTS RENDERED\n\nNothing strikes one more in looking back, either on our own lives or on\nthose of others, than how little we assimilate from the greatest\nexperiences; in nothing is Nature's apparent wastefulness of means more\nironically impressive. A great love comes and sets one's whole being\nsinging like a harp, fills high heaven with rainbows, and makes our\ndingy alleys for awhile bright as the streets of the New Jerusalem; and\nyet, if five years after we seek for what its incandescence has left us,\nwe find, maybe, a newly helpful epithet, maybe a fancy, at most a\nsonnet. Nothing strikes one more, unless, perhaps, the obverse, when we\nsee some trifling pebble-cast ripple into eternity, some fateful second\nprolific as the fly aphis. And so I find it all again exampled in these\nold accounts. The books that mean most for Narcissus to-day could be\ncarried in the hand without a strap, and could probably be bought for a\nsovereign. The rest have survived as a quaint cadence in his style, have\nleft clinging about his thought a delicate incense of mysticism, or are\nbound up in the retrospective tenderness of boyish loves long since gone\nto dream.\n\nAnother observation in the same line of reflection also must often\nstrike one:--for what very different qualities than those for which we\nwere first passionate do we come afterwards to value our old\nenthusiasms. In the day of their bloom it was the thing itself, the\ncraze, the study, for its own sake; now it is the discipline, or any\nbroad human culture, in which they may have been influential. The boy\nchases the butterfly, and thinks not of the wood and the blue heaven;\nbut those only does the man remember, for the mark of their beauty upon\nhim, so unconsciously impressed, for the health of their power and\nsweetness still living in his blood--for these does that chase seem\nalone of worth, when the dusty entomological relic thereof is in limbo.\nAnd so that long and costly shelf, groaning beneath the weight of Grose\nand Dugdale, and many a mighty slab of topographical prose; those\npilgrimages to remote parish churches, with all their attendant ardours\nof careful 'rubbings'; those notebooks, filled with patient data; those\nlong letters to brother antiquaries--of sixteen; even that famous\nExshire Tour itself, which was to have rivalled Pennant's own--what\nremains to show where this old passion stood, with all the clustering\nfoliage of a dream; what but that quaint cadence I spoke of, and an\nanecdote or two which seemed but of little import then, with such\nbreathless business afoot as an old font or a Roman road?\n\nOne particular Roman road, I know, is but remembered now, because, in\nthe rich twilight of an old June evening, it led up the gorsy stretches\nof Lancashire 'Heights' to a solemn plateau, wide and solitary as\nSalisbury Plain, from the dark border of which, a warm human note\nagainst the lonely infinite of heath and sky, beamed the little\nwhitewashed 'Traveller's Rest,' its yellow light, growing stronger as\nthe dusk deepened, meeting the eye with a sense of companionship\nbecoming a vague need just then.\n\nThe seeming spiritual significance of such forlorn wastes of no-man's\nland had, I know, a specially strong appeal for Narcissus, and, in some\nmoods, the challenge which they seem to call from some 'dark tower' of\nspiritual adventure would have led him wandering there till star-light;\nbut a day of rambling alone, in a strange country, among unknown faces,\nbrings a social hunger by evening, and a craving for some one to speak\nto and a voice in return becomes almost a fear. A bright\nkitchen-parlour, warm with the health of six workmen, grouped round a\ngame of dominoes, and one huge quart pot of ale, used among them as\nwoman in the early world, was a grateful inglenook, indeed, wherein to\nclose the day. Of course, friend N. joined them, and took his pull and\npaid his round, like a Walt Whitman. I like to think of his slight\nfigure amongst them; his delicate, almost girl-like, profile against\ntheirs; his dreamy eyes and pale brow, surmounted by one of those dark\nclusters of hair in which the fingers of women love to creep--an\nincongruity, though of surfaces only, which certain who knew him but 'by\nsight,' as the phrase is, might be at a loss to understand. That was one\nof the surprises of his constitution. Nature had given him the dainty\nand dreamy form of the artist, to which habit had added a bookish touch,\nending in a _tout ensemble_ of gentleness and distinction with little\napparent affinity to a scene like that in the 'Traveller's Rest.' But\nthere are many whom a suspicion of the dilettante in such an exterior\nbelies, and Narcissus was one of them. He had very strongly developed\nthat instinct of manner to which sympathy is a daily courtesy, and he\nthus readily, when it suited him, could take the complexion of his\ncompany, and his capacity of 'bend' was well-nigh genius. Of course, all\nthis is but to say that he was a gentleman; yet is not that in itself a\nfine kind of originality? Besides, he had a genuine appetite for the\nthings of earth, such as many another delicate thing--a damask\nrose-bush, for example--must be convicted of too; and often, when some\none has asked him 'what he could have in common with so-and-so,' I have\nheard him answer: 'Tobacco and beer.' Samuel Dale once described him as\nShelley with a chin; and perhaps the chin accounted for the absence of\nany of those sentimental scruples with regard to beefsteaks and certain\nvarieties of jokes, for which the saint-like deserter of Harriet\nWestbrook was distinguished.\n\nA supremely quaint instance of this gift of accommodation befell during\nthat same holiday, which should not pass unrecorded, but which I offer\nto the Reader with an emphatic _Honi soit qui mal y pense_. Despairing\nof reaching a certain large manufacturing town on foot in time to put up\nthere, one evening, he was doing the last mile or two by rail, and, as\nthe train slackened speed he turned to his companions in the carriage to\nenquire if they could tell him of a good hotel. He had but carelessly\nnoticed them before: an old man, a slight young woman of perhaps thirty,\nand a girl about fifteen; working people, evidently, but marked by that\nair of cleanly poverty which in some seems but a touch of ascetic\nrefinement. The young woman at once mentioned _The Bull_, and thereupon\na little embarrassed consultation in undertone seemed to pass between\nher and the old man, resulting in a timid question as to whether\nNarcissus would mind putting up with them, as they were poor folk, and\ncould well do with any little he cared to offer for his accommodation.\nThere was something of a sad winningness in the woman which had\npredisposed him to the group, and without hesitation he at once\naccepted, and soon was walking with them to their home, through streets\nechoing with Lancashire 'clogs.' On the way he learnt the circumstances\nof his companions. The young woman was a widow, and the girl her\ndaughter. Both worked through the day at one of the great cotton mills,\nwhile the old man, father and grandfather, stayed at home and 'fended'\nfor them. Thus they managed to live in a comfort which, though\nstraitened, did not deny them such an occasional holiday as to-day had\nbeen, or the old man the comfort of tobacco. The home was very small,\nbut clean and sweet; and it was not long before they were all sat down\ntogether over a tea of wholesome bread and butter and eggs, in the\npreparation of which it seemed odd to see the old man taking his share.\nThat over, he and Narcissus sat to smoke and talk of the neighbouring\ncountryside; N. on the look-out for folk-lore, and especially for any\nsigns in his companion of a lingering loyalty of belief in the\ntraditions thereabout, a loyalty which had something in it of a sacred\nduty to him in those days. Those were the days when he still turned to\nthe east a-Sundays, and went out in the early morning, with Herrick\nunder his arm, to gather May-dew, with a great uplifting of the spirit,\nin what indeed was a very real act of worship.\n\nBut to my story! As bedtime approached Narcissus could not but be aware\nof a growing uneasiness in the manner of the young woman. At last it was\nexplained. With blushing effort she stammered out the question: Would he\nobject to share his bed with--the old man? 'Of course not,' answered N.\nat once, as though he had all the time intended doing that very thing,\nand indeed, thought it the most delightful arrangement in the world.\n\nSo up to bed go the oddly consorted pair. But the delicious climax was\nyet to come. On entering the room, Narcissus found that there were two\nbeds there! Why should we leave that other bed empty?--he had almost\nasked; but a laughing wonder shot through him, and he stopped in time.\n\nThe old man was soon among the blankets, but Narcissus dallied over\nundressing, looking at this and that country quaintness on the wall; and\nthen, while he was in a state of half man and half trousers, the voice\nof the woman called from the foot of the stairs: Were they in bed yet?\n'Surely, it cannot be! it is too irresistibly simple,' was his thought;\nbut he had immediately answered, 'In a moment,' as if such a question\nwas quite a matter of course.\n\nIn that space he had blown the candle out, and was by the old man's\nside: and then, in the darkness, he heard the two women ascending the\nstairs. Just outside his door, which he had left ajar, they seemed to\nturn off into a small adjoining room, from whence came immediately the\nsoft delicious sounds of female disrobing. They were but factory women,\nyet Narcissus thought of Saint Agnes and Madeline, we may be sure. And\nthen, at last--indeed, there was to be no mistake about it--the door was\nsoftly pushed open, and two dim forms whispered across to the adjoining\nbed, and, after a little preliminary rustle, settled down to a rather\nfluttered breathing.\n\nNo one had spoken: not even a Goodnight; but Narcissus could hardly\nrefrain from ringing out a great mirthful cry, while his heart beat\nstrangely, and the darkness seemed to ripple, like sunlight in a cup,\nwith suppressed laughter. The thought of the little innocent deception\nas to their sleeping-room, which poverty had caused them to practise,\nprobably held the breath of the women, while the shyness of sex was a\ncommon bond of silence--at least, on the part of the three younger. It\nwas long before Narcissus was able to fall asleep, for he kept picturing\nthe elder woman with burning cheek and open eyes in a kind of 'listening\nfear' beneath the coverlet; and the oddity of the thing was so original,\nso like some _conte_ of a _Decameron_ or _Heptameron_, with the\nwickedness left out. But at last wonder gave place to weariness, and\nsleep began to make a still odder magic of the situation. The difficulty\nof meeting at breakfast next morning, which had at once suggested itself\nto N.'s mind, proved a vain fear; for, when he arose, that other bed was\nas smooth as though it had lain untouched through the night, and the\ndaughters of labour had been gone two hours. But it was not quite\nwithout sign that they had gone, for Narcissus had a dreamlike\nimpression of opening his eyes in the early light to find a sweet\nwoman's face leaning over him; and I am sure he wanted to believe that\nit had bent down still further, till it had kissed his lips--' for his\nmother's sake,' she had said in her heart, as she slipped away and was\nseen no more.\n\n'If this were fiction, instead of a veracious study from life,' to make\nuse of a phrase which one rarely finds out of a novel, it would be\nunfitting to let such an incident as that just related fall to the\nground, except as the seed of future development; but, this being as I\nhave stated, there is nothing more to say of that winning _ouvriere_.\nNarcissus saw her no more.\n\nBut surely, of all men, he could best afford that one such pleasant\nchance should put forth no other blossom save that half-dreamed\nkiss;--and how can one ever foresee but that our so cherishable spray of\nbloom may in time add but another branch to that orchard of Dead Sea\nfruit which grows inevitably about all men's dwellings?\n\nI do not suppose that Narcissus was really as exceptional in the number\nand character of his numerous boyish loves as we always regarded him as\nbeing. It is no uncommon matter, of course and alas! for a youth between\nthe ages of seventeen and nineteen to play the juggler at keeping three,\nor even half-a-dozen, female correspondents going at once, each of whom\nsleeps nightly with copious documentary evidence of her sole and\nincontrovertible possession of the sacred heart. Nor has Narcissus been\nthe only lover, I suspect, who, in the season of the waning of the moon,\nhas sent such excuses for scrappy epistolary make-shifts as 'the\nstrident din of an office, an air so cruelly unsympathetic, as frost to\nbuds, to the blossoming of all those words of love that press for\nbirth,' when, as a matter of fact, he has been unblushingly eating the\nlotus, in the laziest chair at home, in the quietest night of summer.\nSuch insincerity is a common besetting sin of the young male;\ninvariably, I almost think, if he has the artistic temperament. Yet I do\nnot think it presents itself to his mind in its nudity, but comes\nclothed with that sophistry in which youth, the most thoroughgoing of\n_philosophes_, is so ingenious. Consideration for the beloved object, it\nis called--yes! beloved indeed, though, such is the paradox in the order\nof things, but one of the several vestals of the sacred fire. One cannot\nhelp occasional disinclination on a lazy evening, confound it! but it\nmakes one twinge to think of paining her with such a confession; and a\nstory of that sort--well, it's a lie, of course; but it's one without\nany harm, any seed of potential ill, in it. So the letter goes, maybe to\ntake its place as the 150th of the sacred writings, and make poor\nDaffodilia, who has loved to count the growing score, happy with the\ncompletion of the half-century.\n\nBut the disinclination goes not, though the poor passion has, of\ncourse, its occasional leapings in the socket, and the pain has to come\nat last, for all that dainty consideration, which, moreover, has been\nall the time feeding larger capacities for suffering. For, of course, no\nman thinks of marrying his twelfth love, though in the thirteenth there\nis usually danger; and he who has jilted, so to say, an earl's daughter\nas his sixth, may come to see\n\n  'The God of Love, ah! benedicite,\n  How mighty and how great a lord is he'\n\nin the thirteenth Miss Simpkins.\n\nBut this is to write as an outsider: for that thirteenth, by a mystical\nprocess which has given to each of its series in its day the same primal\nquality, is, of course, not only the last, but the first. And, indeed,\nwith little casuistry, that thirteenth may be truly held to be the\nfirst, for it is a fact determined not so much by the chosen maid as by\nhim who chooses, though he himself is persuaded quite otherwise. To him\nhis amorous career has been hitherto an unsuccessful pursuit, because\neach followed fair in turn, when at length he has caught her flying\nskirts, and looked into her face, has proved not that 'ideal'--\n\n  'That not impossible she\n  That shall command my heart and me'--\n\nbut another, to be shaken free again in disappointment. In truth,\nhowever, the lack has been in himself all this time. He had yet to learn\nwhat loving indeed meant: and he loves the thirteenth, not because she\nis pre-eminent beyond the rest, but because she has come to him at the\nmoment when that 'lore of loving' has been revealed. Had any of those\nearlier maidens fallen on the happy conjunction, they would, doubtless,\nhave proved no less loveworthy, and seemed no less that 'ideal' which\nthey have since become, one may be sure, for some other illuminated\nsoul.\n\nOf course, some find that love early--the baby-love, whom one never\nmarries, and then the faithful service. Probably it happens so with the\nmajority of men; for it is, I think, especially to the artist nature\nthat it comes thus late. Living so vividly within the circle of its own\nexperience, by its very constitution so necessarily egoistic, the\nlatter, more particularly in its early years, is always a Narcissus,\ncaring for nought or none except in so much as they reflect back its own\nbeauty or its own dreams. The face such a youth looks for, as he turns\nthe coy captured head to meet his glance, is, quite unconsciously, his\nown, and the 'ideal' he seeks is but the perfect mirror. Yet it is not\nthat mirror he marries after all: for when at last he has come to know\nwhat that word--one so distasteful, so 'soiled' to his ear 'with all\nignoble' domesticity--what that word 'wife' really expresses, he has\nlearnt, too, to discredit those cynical guides of his youth who love so\nwell to write Ego as the last word of human nature.\n\nBut the particular Narcissus of whom I write was a long way off that\nthirteenth maid in the days of his antiquarian rambles and his\nPagan-Catholic ardours, and the above digression is at least out of\ndate.\n\nA copy of Keats which I have by me as I write is a memorial of one of\nthe pretty loves typical of that period. It is marked all through in\nblack lead--not so gracefully as one would have expected from the 'taper\nfingers' which held the pencil, but rather, it would appear, more with\nregard to emphasis than grace. Narcissus had lent it to the queen of the\nhour with special instructions to that end, so that when it came to him\nagain he might ravish his soul with the hugging assurance given by the\nthick lead to certain ecstatic lines of _Endymion,_ such as--\n\n    'My soul doth melt\n  For the unhappy youth;'\n    'He surely cannot now\n  Thirst for another love;'\n\nand luxuriate in a genial sense of godship where the tremulous pencil\nhad left the record of a sigh against--\n\n  'Each tender maiden whom he once thought fair.'\n\nBut it was a magnanimous godship; and, after a moment's leaning back\nwith closed eyes, to draw in all the sweet incense, how nobly would he\nact, in imaginative vignette, the King Cophetua to this poor suppliant\nof love; with what a generous waiving of his power--and with what a\ngrace!--did he see himself raising her from her knees, and seating her\nat his right hand. Yet those pencil-marks, alas! mark but a secondary\ninterest in that volume. A little sketch on the fly-leaf, 'by another\nhand,' witness the prettier memory. A sacred valley, guarded by smooth,\ngreen hills; in the midst a little lake, fed at one end by a singing\nstream, swallowed at the other by the roaring darkness of a mill; green\nrushes prosperous in the shallows, and along the other bank an old\nhedgerow; a little island in the midst, circled by silver lilies; and in\nthe distance, rising from out a cloud of tangled green, above the little\nriver, an old church tower. Below, though not 'in the picture,' a quaint\ncountry house, surrounded by a garden of fair fruit-trees and wonderful\nbowers, through which ran the stream, free once again, and singing for\njoy of the light. In the great lone house a solitary old man, cherished\nand ruled by--'The Miller's Daughter.' Was scene ever more in need of a\nfairy prince? Narcissus sighed, as he broke upon it one rosy evening,\nto think what little meaning all its beauty had, suffering that lack;\nbut as he had come thither with the purpose, at once firm and vague, of\ngiving it a memory, he could afford to sigh till morning's light\nbrought, maybe, the opportunity of that transfiguring action. He was to\nspend an Easter fortnight there, as the guest of some farmer-relatives\nwith whom he had stayed years before, in a period to which, being\nnineteen, he already alluded as his 'boyhood.'\n\nAnd it is not quite accurate to say that it had no memory for him, for\nhe brought with him one of that very miller's daughter, though, indeed,\nit was of the shadowiest silver. It had chanced at that early time that\nan influx of visitors to the farm had exceeded the sleeping room, and he\nand another little fellow had been provided with a bed in the miller's\nhouse. He had never quite forgotten that bedroom--its huge old-fashioned\nfour-poster, slumbrous with great dark hangings, such as Queen Elizabeth\nseems always to have slept in; its walls dim with tapestry, and its\nscreen of antique bead-work. But it was round the toilet table that\nmemory grew brightest, for thereon was a crystal phial of a most\nmarvellous perfume, and two great mother-of-pearl shells, shedding a\nmystical radiance--the most commonplace Rimmel's, without doubt, and the\nshells 'dreadful,' one may be sure. But to him, as he took a reverent\nbreath of that phial, it seemed the very sweetbriar fragrance of her\ngown that caught his sense; and, surely, he never in all the world found\nscent like that again. Thus, long after, she would come to him in\nday-dreams, wafted on its strange sweetness, and clothed about with that\nmystical lustre of pearl.\n\nThere were five years between him and that memory as he stepped into\nthat enchanted land for the second time. The sweet figure of young\nwomanhood to which he had turned his boyish soul in hopeless worship,\nwhen it should have been busied rather with birds' nests and\nrabbit-snares, had, it is true, come to him in dimmer outline each\nSpring, but with magic the deeper for that. As the form faded from the\nsilver halo, and passed more and more into mythology, it seemed, indeed,\nas if she had never lived for him at all, save in dreams, or on another\nstar. Still, his memory held by those great shells, and he had come at\nlast to the fabled country on the perilous quest--who of us dare venture\nsuch a one to-day?--of a 'lost saint.' Enquiry of his friends that\nevening, cautious as of one on some half-suspected diplomacy, told him\nthat one with the name of his remembrance did live at the\nmill-house--with an old father, too. But how all the beauty of the\nsinging morning became a scentless flower when, on making the earliest\npossible call, he was met at the door with that hollow word, 'Away'--a\nword that seemed to echo through long rooms of infinite emptiness and\nturn the daylight shabby--till the addendum, 'for the day,' set the\nbirds singing again, and called the sunshine back.\n\nA few nights after he was sitting at her side, by a half-opened window,\nwith his arm about her waist, and her head thrillingly near his. With\nhis pretty gift of recitation he was pouring into her ear that sugared\npassage in _Endymion_, appropriately beginning, 'O known unknown,'\npreviously 'got up' for the purpose; but alas! not too perfectly to\nprevent a break-down, though, fortunately, at a point that admitted a\nready turn to the dilemma:--\n\n                            'Still\n  Let me entwine thee surer, surer ...'\n\nHere exigency compelled N. to make surety doubly, yea, trebly, sure; but\nmemory still forsaking him, the rascal, having put deeper and deeper\nsignificance into his voice with each repetition, dropped it altogether\nas he drew her close to him, and seemed to fail from the very excess of\nlove. An hour after, he was bounding into the moonlight in an\nintoxication of triumph. She was won. The beckoning wonder had come down\nto him. And yet it was real moonlight--was not that his own grace in\nsilhouette, making a mirror even of the hard road?--real grass over\nwhich he had softly stept from her window, real trees, all real,\nexcept--yes! was it real love?\n\nIn the lives of all passionate lovers of women there are two\nbroadly-marked periods, and in some a third: slavery, lordship, and\nservice. The first is the briefest, and the third, perhaps, seldom\ncomes; the second is the most familiar.\n\nAwakening, like our forefather, from the deep sleep of childish things,\nthe boy finds a being by his side of a strange hushing fairness, as\nthough in the night he had opened his eyes and found an angel by his\nbed. Speech he has not at all, and his glance dare not rise beyond her\nbosom; till, the presence seeming gracious, he dares at length stretch\nout his hand and touch her gown; whereon an inexplicable new joy\ntrembles through him, as though he stood naked in a May meadow through\nthe golden rain of a summer shower. Should her fingers touch his arm by\nchance, it is as though they swept a harp, and a music of piercing\nsweetness runs with a sudden cry along his blood. But by and by he comes\nto learn that he has made a comical mistake about this wonder. With his\nhead bent low in worship, he had not seen the wistfulness of her gaze on\nhim; and one day, lo! it is she who presses close to him with the timid\nappeal of a fawn. Indeed, she has all this time been to him as some\nbeautiful woodland creature might have seemed, breaking for the first\ntime upon the sight of primitive man. Fear, wonder inexpressible,\nworship, till a sudden laughing thought of comprehension, then a lordly\nprotectiveness, and, after that--the hunt! At once the masculine\nself-respect returns, and the wonder, though no less sweet in itself,\nbecomes but another form of tribute.\n\nWith Narcissus this evolution had taken place early: it was very long\nago--he felt old even then to think of it--since Hesperus had sung like\na nightingale above his first kiss, and his memory counted many trophies\nof lordship. But, surely, this last was of all the starriest; perhaps,\nindeed, so wonderful was it, it might prove the very love which would\nbring back again the dream that had seemed lost for ever with the\npassing of that mythical first maid so long ago, a love in which worship\nshould be all once more, and godship none at all. But is not such a\nquestion all too certainly its own answer? Nay, Narcissus, if indeed you\nfind that wonder-maid again, you will not question so; you will forget\nto watch that graceful shadow in the moonlight; you will but ask to sit\nby her silent, as of old, to follow her to the end of the world. Ah me!\n\n  'How many queens have ruled and passed\n    Since first we met;\n    How thick and fast\n  The letters used to come at first,\n    How thin at last;\n  Then ceased, and winter for a space!\n    Until another hand\n    Brought spring into the land,\n  And went the seasons' pace.'\n\nThat Miller's Daughter, although 'so dear, so dear,' why, of course, she\nwas not that maid: but again the silver halo has grown about her; again\nNarcissus asks himself, 'Did she live, or did I dream?'; again she comes\nto him at whiles, wafted on that strange incense, and clothed about in\nthat mystical lustre of pearl.\n\nDoubtless, she lives in that fabled country still: but Narcissus has\ngrown sadly wise since then, and he goes on pilgrimage no more.\n\n\n\n\nCHAPTER V\n\n\nAN IDYLL OF ALICE SUNSHINE, WHICH REALLY BELONGS TO THE LAST CHAPTER\n\nIf the Reader has heard enough of the amourettes of the young gentleman\nupon whose memoirs I am engaged, let him skip this chapter and pass to\nthe graver chapters beyond. My one aim is the Reader's pleasure, and I\ncarry my solicitude so far that if he finds his happiness to lie outside\nthese pages altogether, has no choice among these various chapters, but\nprefers none to any, I am quite content. Such a spirit of\nself-abnegation, the Reader must admit, is true love.\n\nPerhaps it was an early unconscious birth-impulse of the true love some\nday to be born in his heart, that caused Narcissus to make a confession\nto his Miller's Daughter, on one of their pretty decorative evenings,\nwhen they sat together at the fireside, while the scent of the climbing\nroses, and the light of the climbing moon, came in at the window.\n\nThe immediate effect of the confession was--no wonder--to draw tears.\nAnd how beautiful she looked in tears! Who would dive for pearls when\nthe pearl-fisheries of a woman's eyes are his to rifle?\n\nBeautiful, beautiful tears, flow on--no dull, leaden rain, no mere\nmonotonous deluge, but a living, singing fountain, crowned with such\nrainbows as hang roses and stars in the fine mist of samite waterfalls,\nirradiated by gleaming shafts of lovely anger and scorn.\n\nLike Northern Lights on autumn evenings, the maiden's eyes pierced\nNarcissus through and through with many-coloured spears. There was\nthunder, too; the earth shook--just a little: but soon Narcissus saw the\nwhite dove of peace flying to him through the glancing showers. For all\nher sorrow, his was the peace of confession. His little lie had been\nacknowledged, his treason self-betrayed.\n\nAnd it was this.\n\nI have hinted that Narcissus, like the Catholic Church, worshipped many\nsaints. At this time, one of them, by a thrilling coincidence, chanced\nto have her shrine at a boarding-school, some fifteen miles or so from\nthe mill-pond on whose banks the Miller's Daughter had drawn into her\nlovely face so much of the beauty of the world. Alice Sunshine, shall we\ncall her, was perhaps more of a cherub than a saint; a rosy, laughing,\nplump little arrangement of sunshiny pink and white flesh, with blue\neyes and golden hair. Alice was not overburdened with intellectuality,\nand, like others of her sex, her heart was nothing like so soft as her\nbosom. Narcissus had first been in love with her sister; but he and the\nsister--a budding woman of the world--had soon agreed that they were not\nborn for each other, and Narcissus had made the transfer of his tragic\npassion with inexpensive informality. As the late Anthony Trollope would\nfinish one novel to-night, and begin another to-morrow morning, so would\nNarcissus be off with the old love this Sunday, and visibly on with the\nnew the next.\n\nDear little plump, vegetable-marrow Alice! Will Narcissus ever forget\nthat Sunday night when the church, having at last released its weary\nworshippers, he stole, not as aforetime to the soft side of Emily, but\nto the still softer side of the little bewildered Alice. For, though\nAlice had worshipped him all the time, and certainly during the whole of\nthe service, she had never dared to hope that he would pass her dashing,\ndark-eyed sister to love _her_--little, blonde, phlegmatic, blue-eyed\nAlice.\n\nBut Apollo was bent on the capture of his Daphne. Truth to say, it was\nbut the work of a moment. The golden arrow was in her heart, the wound\nkissed whole again, and the new heaven and the new earth all arranged\nfor, in hardly longer time than it takes to tell.\n\nIn youth the mystery of woman is still so fresh and new, that to make a\nfuss about a particular woman seems like looking a gift-horse of the\ngods in the mouth. The light on the face of womanhood in general is so\nbewilderingly beautiful that the young man literally cannot tell one\nwoman from another. They are all equally wonderful. Masculine\nobservation leads one to suppose that woman's first vision of man\nsimilarly precludes discrimination.\n\nAh me! it is easy to laugh to-day, but it was heart--bleeding tragedy\nwhen those powers that oughtn't to be decreed Alice's exile to a\nboarding-school in some central Africa of the midland counties.\n\nThe hemorrhage of those two young hearts! But, for a time, each\nplastered the other's wounds with letters--dear letters--letters every\npost. For the postal authorities made no objection to Narcissus\ncorresponding with two or more maidens at once. And it is only fair to\nAlice to say, that she knew as little of the Miller's Daughter as the\nMiller's Daughter knew of her.\n\nSo, when Narcissus was reciting _Endymion_ to his Miller's Maid, it was\nnot without a minor chord plaining through the major harmonies of the\npresent happiness; the sense that Alice was but fifteen miles away--so\nnear she could almost hear him if he called--only fifteen miles away,\nand it was a long three months since they had met.\n\nIt now becomes necessary to admit a prosaic fact hitherto concealed\nfrom the Reader. Narcissus rode a bicycle. It was, I must confess, a\nrather 'modern' thing to do. But surely the flashing airy wheel is the\nmost poetical mode of locomotion yet invented, and one looks more like a\nfairy prince than ever in knickerbockers. Whenever Narcissus turned his\ngleaming spokes along some mapped, but none the less mysterious,\ncounty--road, he thought of Lohengrin in his barge drawn by white swans\nto his mystic tryst; he thought of the seven-leagued boots, the flying\ncarpet, the wishing-cap, and the wooden Pegasus,--so called because it\nmounted into the clouds on the turning of a peg. As he passed along by\nmead and glade, his wheel sang to him, and he sang to his wheel. It was\na daisied, daisied world.\n\nThere were buttercups and violets in it too as he sped along in the\nearly morning of an unforgotten Easter Sunday, drawn, so he had\nshamelessly told his Miller's Daughter, by antiquarian passion to visit\nthe famous old parish church near which Alice was at school.\nAntiquarian passion! Well, certainly it is an antiquarian passion now.\n\nBut then--how his heart beat! how his eyes shone as with burning kohl!\nThat there was anything to be ashamed of in this stolen ride never even\noccurred to him. And perhaps there was little wrong in it, after all.\nPerhaps, when the secrets of all hearts are revealed, it will come out\nthat the Miller's Daughter took the opportunity to meet Narcissus'\nunderstudy,--who can tell?\n\nBut the wonderful fresh morning-scented air was a delicious fact beyond\ndispute. That was sincere. Ah, there used to be real mornings then!--not\nmerely interrupted nights.\n\nAnd it was the Easter-morning of romance. There was a sweet passionate\nSabbath-feeling everywhere. Sabbath-bells, and Sabbath-birds, and\nSabbath-flowers. There was even a feeling of restful Sabbath-cheer about\nthe old inn, where, at last, entering with much awe the village where\nAlice nightly slept--clothed in white samite, mystic, wonderful,\n--Narcissus provided for the demands of romance by a hearty\ncountry breakfast. A manna of blessing seemed to lie thick upon every\nthing. The very ham and eggs seemed as if they had been blessed by the\nPope.\n\nIt was yet an hour to church-time, an hour usually one of spiteful\nalacrity; but this morning, it seemed, in defiance of the clock, cruelly\nunpunctual. After breakfast, Narcissus strolled about the town, and\ninquired the way to Miss Curlpaper's school. It stood outside the little\ntown. It was pointed out to him in the distance, across billowy clouds\nof pear and apple-blossom, making the hollow in which the town nestled\nseem a vast pot-pourri jar, overflowing with newly gathered rose-leaves.\n\nHad the Miller's Daughter been able to watch his movements, she would\nhave remarked that his antiquarian ardour drew him not to the church,\nbut to a sombre many-windowed house upon the hill.\n\nNarcissus reconnoitred the prison-like edifice from behind a hedge, then\nsummoned courage to walk past with slow nonchalance. All was as dead and\ndull as though Alice was not there. Yet somewhere within those\nprison-walls her young beauty was dressing itself to meet the spring.\nPerhaps, in delicious linen, soft and white, she was dashing cool water\nabout her rosebud face, or, flushed with exhilaration, was pinning up\nthe golden fleeces of her hair. Perhaps she was eating wonderful bacon\nand eggs! Could she be thinking of him? She little knew how near he was\nto her. He had not written of his coming. Letters at Miss Curlpaper's\nhad to pass an inspection much more rigorous than the Customs, but still\nsmuggling was not unknown. For success, however, carefully laid plans\nand regular dates were necessary, and Narcissus' visit had fallen\nbetween the dates.\n\nNo! there was no sign of her. She was as invisible as the moon at\nmid-day. And there were the church-bells beginning to call her: 'Alice,\nAlice, put on your things!'\n\n  'Alice, Alice, put on your things!\n  The birds are calling, the church bell rings;\n  The sun is shining, and I am here,\n  Waiting--and waiting--for you, my dear.\n\n  Alice, Alice, doff your gown of night,\n  Draw on your bodice as lilies white,\n  Draw on your petticoats, clasp your stays,--\n  Oh! Alice, Alice, those milky ways!\n\n  Alice, Alice, how long you are!\n  The hour is late and the church is far;\n  Slowly, more slowly, the church bell rings--\n  Alice, Alice, put on your things!'\n\nReally it was not in Narcissus' plans to wait at the school till Alice\nappeared. The Misses Curlpaper were terrible unknown quantities to him.\nFor a girl to have a boy hanging about the premises was a capital crime,\nhe knew. Boys are to girls' schools what Anarchists are to public\nbuildings. They come under the Explosives Acts. It was not, indeed,\nwithin the range of his hope that he might be able to speak to Alice. A\nlook, a long, immortal, all-expressive look, was all he had travelled\nfifteen miles to give and win. For that he would have travelled fifteen\nhundred.\n\nHis idea was to sit right in front of the nave, where Alice could not\nmiss seeing him--where others could see him too in his pretty\nclose-fitting suit of Lincoln green. So down through the lanes he went,\namong the pear and apple orchards, from out whose blossom the clanging\ntower of the old church jutted sheer, like some Bass Rock amid rosy\nclustering billows. Their love had been closely associated from its\nbeginning with the sacred things of the church, so regular had been\ntheir attendance, not only on Sundays, but at week-night services. To\nAlice and Narcissus there were two Sabbaths in the week, Sunday and\nWednesday. I suppose they were far from being the only young people\ninterested in their particular form of church-work. Leander met Hero, it\nwill be remembered, on the way to church, and the Reader may recall\nMarlowe's beautiful description of her dress upon that fatal morning:\n\n  'The outside of her garments were of lawn,\n  The lining purple silk, with gilt stars drawn;\n  Her wide sleeves green, and bordered with a grove,\n  Where Venus in her naked glory strove\n  To please the careless and disdainful eyes\n  Of proud Adonis, that before her lies;\n  Her kirtle blue, whereon was many a stain,\n  Made with the blood of wretched lovers slain....'\n\nAlice wore pretty dresses too, if less elaborate; and, despite its\nchange of name, was not the church where she and Narcissus met, as the\nchurch wherein Hero and Leander first looked upon each other, the Temple\nof Love? Certainly the country church to which Narcissus\nself-consciously passed through groups of Sunday-clothed villagers, was\ndecked as for no Christian festival this Sabbath morning. The garlands\nthat twined about the old Norman columns, the clumps of primroses and\nviolets that sprung at their feet, as at the roots of gigantic beeches,\nthe branches of palm and black-thorn that transformed the chancel to a\nbower: probably for more than knew it, these symbols of the joy and\nbeauty of earth had simpler, more instinctive, meanings than those of\nany arbitrary creed. For others in the church besides Narcissus, no\ndoubt, they spoke of young love, the bloom and the fragrance thereof, of\nmating birds and pairing men and maids, of the eternal principle of\nloveliness, which, in spite of winter and of wrong, brings flowers and\nfaces to bless and beautify this church of the world.\n\nAs Narcissus sat in his front row, his eyes drawn up in a prayer to the\npainted glories of the great east window, his whole soul lifted up on\nthe wings of colour, scent, and sound--the whole sacred house had but\none meaning: just his love for Alice. Nothing in the world was too holy\nto image that. The windows, the music, the flowers, all were metaphors\nof her: and, as the organ swirled his soul along in the rapids of its\npassionate, prayerful sound, it seemed to him that Alice and he already\nstood at the gate of Heaven!\n\nPresently, across his mingled sensations came a measured tramp as of\nboy-soldiers marching in line. You have heard it! You have _listened_\nfor it!! It was the dear, unmistakable sound of a girls' school on the\nmarch. Quickly it came nearer, it was in the porch--it was in the\nchurch! Narcissus gave a swift glance round. He dare not give a real\nsearching look yet. His heart beat too fast, his cheek burned too red.\nBut he saw it was a detachment of girls--it certainly was Alice's\nschool.\n\nThen came the white-robed choristers, and the white-haired priests: _If\nwe say that we have no sin, we deceive ourselves, and the truth is not\nin us; but, if we confess our sins, He is faithful and just to forgive\nus our sins, and to cleanse us from all unrighteousness_.\n\nDEARLY BELOVED BRETHREN....\n\nHis heart swelled with a sobbing exaltation of worship such as he had\nnot known for years. You could hardly have believed that a little\napple-dumpling of a pink and white girl was the real inspirer of that\nlook in his young face that made old ladies, even more than young ones,\ngaze at him, and remark afterwards on the strange boy with the lovely\nspiritual expression.\n\nBut, all the time, Narcissus felt that Alice's great eyes were on him,\nglowing with glad surprise. The service proceeded, but yet he forbore to\nseek her. He took a delight in husbanding his coming joy. He would not\ncrudely snatch it. It would be all the sweeter for waiting. And the fire\nin Alice's eyes would all the time be growing softer and softer. He\nnearly looked as he thought of that. And surely that was her dear voice\ncalling to him in the secret language of the psalm. He sang back to her\nwith a wild rapture. Thus the morning stars sang together, he thought.\n\nAnd when the prayers laid lovely hands across the eyes of the\nworshippers, still he sought not Alice, but prayed for her as perhaps\nonly a boy can: O Lord God, be good to Alice--already she is one of thy\nangels. May her life be filled with light and joy! And if in the time to\ncome I am worthy of being ever by her side, may we live our lives\ntogether, high and pure and holy as always in thy sight! Lord, thou\nknowest how pure is my love; how I worship her as I worship the holy\nangels themselves. But whatsoever is imperfect perfect by the\ninspiration of thy Holy Spirit....\n\nSo prayed the soul of the boy for the soul of the girl, and his eyes\nfilled with tears as he prayed; the cup of the wonder and holiness of\nthe world ran over.\n\nAlready, it seemed, that Alice and he lay clasped together in the arms\nof God.\n\nSo Narcissus prayed and sang his love in terms of an alien creed. He\nsang of the love of Christ, he thought but of the love of Alice; and\nstill he refrained from plucking that wonderful passion-flower of her\nglance.\n\nAt length he had waited the whole service through; and, with the last\nhallowed vibrations of the benediction, he turned his eyes, brimful of\nlove-light, greedily, eagerly, fearful lest one single ray should be\nwasted on intermediate and irrelevant worshippers.\n\nWonderful eyes of love!--but alas! where is their Alice? Wildly they\nglance along the rosy ranks of chubby girlhood, but where is their\nAlice?\n\nAnd then the ranks form in line, and once more the sound, the ecstatic\nsound it had seemed but a short time before, of girls marching--but\nno!--no!--there is no Alice.\n\nIn sick despair Narcissus stalked that Amazonian battalion, crouching\nbehind hedges, dropping into by-lanes, lurking in coppices,--he held his\nbreath as they passed two and two within a yard of him. Two followed\ntwo, but still no Alice!\n\nNarcissus lay in wait, dinnerless, all that afternoon; he walked about\nthat dreary house like a patrol, till at last he was observed of the\ninmates, and knots of girls gathered at the windows--alas! only to\ngiggle at his forlorn and desperate appearance.\n\nStill there was no Alice ... and then it began to rain, and he became\naware how hungry he was. So he returned to his inn with a sad heart.\n\nAnd all the time poor little Alice lay in bed with a sore throat,\noblivious of those passionate boyish eyes that, you would have thought,\nmust have pierced the very walls of her seclusion.\n\nAnd, after all, it was not her voice Narcissus had heard in the church.\nIt was but the still sweeter voice of his own heart.\n\n\n\n\nCHAPTER VI\n\n\nTHE SIBYLLINE BOOKS\n\nI hope it will be allowed to me that I treat the Reader with all\nrespectful courtesy, and that I am well bred enough to assume him\nfamiliar with all manner of exquisite experience, though in my heart I\nmay be no less convinced that he has probably gone through life with\nnothing worth calling experience whatsoever. It is our jaunty modern\nfashion, and I follow it so far as I am able. I take for granted, for\ninstance, that every man has at one time or another--in his salad days,\nyou know, before he was embarked in his particular provision\nbusiness--had foolish yearnings towards poesy. I respect the mythical\ndreams of his 'young days'; I assume that he has been really in love;\nbut, pray press me not too curiously as to whether I believe it all, as\nto whether I really imagine that his youth knew other dreams than those\nof the foolish young 'masherdom' one meets in the train every morning,\nor that he has married a wife for other than purely 'masculine' reasons.\n\nThese matters I do not mind leaving in the form of a postulate--let them\nbe granted: but that every man has at one time or another had the craze\nfor saving the world I will not assume. Narcissus took it very early,\nand though he has been silent concerning his mission for some time, and\nwhen last we heard of it had considerably modified his propaganda, he\nstill cherishes it somewhere in secret, I have little doubt; and one may\nnot be surprised, one of these days, to find it again bursting out 'into\nsudden flame.'\n\nHis spiritual experience has probably been the deepest and keenest of\nhis life. I do not propose to trace his evolution from Anabaptism to\nAgnosticism. The steps of such development are comparatively familiar;\nthey have been traced by greater pens than mine. The 'means' may vary,\nbut the process is uniform.\n\nWhether a man deserts the ancestral Brahminism that has so long been\n'good enough for his parents,' and listens to the voice of the Buddhist\nmissionary, or joins Lucian in the seat of the scornful, shrugging at\naugur and philosopher alike; whether it is Voltaire, or Tom Paine, or\nThomas Carlyle, or Walt Whitman, or a Socialist tract, that is the\nemancipator, the emancipation is all one.\n\nThe seed that is to rend the rock comes in all manner of odd, and often\nunremembered, ways; but somehow, it is there; rains and dews unnoticed\nfeed it; and surely, one day the rock is rent, the light is pouring in,\nand we are free! It is often a matter of anguish that, strive as we may,\nit is impossible to remember what helping hand it was that sowed for us.\nOur fickle memory seems to convict us of ingratitude, and yet we know\nhow far that sin is from us; and how, if those sowers could but be\nrevealed to us, we would fall upon their necks, or at their feet.\n\nI talked of this one day with Narcissus, and some time after he sent me\na few notes headed 'Spiritual Pastors,' in which he had striven to\nfollow the beautiful example set by Marcus Aurelius, in the anxiously\nloving acknowledgment with which he opens his meditations. I know he\nregarded it as miserably inefficient; but as it does actually indicate\nsome of the more individual side of his experience, and is, moreover,\ncharacteristic in its style, I shall copy a few passages from it here:--\n\n'To some person or persons unknown exceeding gratitude for the\nsuggestion, in some dim talk, antenatal it would almost seem, that Roman\nCatholics might, after all, be \"saved.\" Blessed fecundating suggestion,\nthat was the earliest loophole!\n\n'To my father I owe a mind that, once set on a clue, must follow it, if\nneed be, to the nethermost darkness, though he has chosen to restrict\nthe operation of his own within certain limits; and to my mother a\nnatural leaning to the transcendental side of an alternative, which has\nsaved me so many a time when reason had thrown me into the abyss. But\none's greatest debt to a good mother must be simply--herself.\n\n'To the Rev. Father Ignatius for his earnest preaching, which might\nalmost have made me a monk, had not Thomas Carlyle and his _Heroes_,\nespecially the lecture on Mahomet, given me to understand the true\nsignificance of a Messiah.\n\n'To Bulwer for his _Zanoni_, which first gave me a hint of the possible\nnatural \"supernatural,\" and thus for ever saved me from dogmatising in\nnegatives against the transcendental.\n\n'To Sir Edwin Arnold for his _Light of Asia,_ also to Mr. Sinnett for\nhis _Esoteric Buddhism,_ books which, coming to me about the same time,\ntogether with some others like them, first gave some occupation to an\n\"unchartered freedom,\" gained in many forgotten steps, in the form of a\nfaith which transfigured my life for many months into the most beautiful\nenthusiasm a man could know,--and which had almost sent me to the\nHimalayas!\n\n'That it did not quite achieve that, though much of the light it gave me\nstill remains, I owe to R.M., who, with no dialectic, but with one bald\nquestion, and the reading of one poem, robbed me of my fairy palace of\nOriental speculation in the twinkling of an eye. Why it went I have\nnever really quite known; but surely, it was gone, and the wind and the\nbare star-light were alone in its place.\n\n'Dear Mac., I have not seen you for ever so long, and surely you have\nforgotten how that night, long ago, you asked with such a strange,\nalmost childlike, simplicity: \"_Is_ there a soul?\" But I have not\nforgotten, nor how I made no answer at all, but only staggered, and how,\nwith your strange, dreamy voice, you chanted for comfort:--\n\n    '\"This hot, hard flame with which our bodies burn\n    Will make some meadow blaze with daffodil;\n  Ay! and those argent breasts of thine will turn\n    To water-lilies; the brown fields men till\n  Will be more fruitful for our love to-night:\n  Nothing is lost in Nature; all things live in Death's despite.\n\n       *       *       *       *       *\n\n    '\"So when men bury us beneath the yew\n    Thy crimson-stained mouth a rose will be,\n  And thy soft eyes lush blue-bells dimmed with dew;\n    And when the white narcissus wantonly\n  Kisses the wind, its playmate, some faint joy\n  Will thrill our dust, and we will be again fond maid and boy.\n\n                  '\"... How my heart leaps up\n    To think of that grand living after death\n  In beast and bird and flower, when this cup,\n    Being filled too full of spirit, bursts for breath,\n  And with the pale leaves of some autumn day,\n  The soul, earth's earliest conqueror, becomes earth's last great prey.\n\n    '\"O think of it! We shall inform ourselves\n    Into all sensuous life; the goat-foot faun,\n  The centaur, or the merry, bright-eyed elves\n    That leave they: dancing rings to spite the dawn\n  Upon the meadows, shall not be more near\n  Than you and I to Nature's mysteries, for we shall hear\n\n    '\"The thrush's heart beat, and the daisies grow,\n    And the wan snowdrop sighing for the sun\n  On sunless days in winter; we shall know\n    By whom the silver gossamer is spun,\n  Who paints the diapered fritillaries,\n  On what wide wings from shivering pine to pine the eagle flies.\n\n       *       *       *       *       *\n\n    '\"We shall be notes in that great symphony\n    Whose cadence circles through the rhythmic spheres,\n  And all the live world's throbbing heart shall be\n    One with our heart; the stealthy, creeping years\n  Have lost their terrors now; we shall not die--\n  The universe itself shall be our Immortality!\"\n\nHave you forgotten how you chanted these, and told me they were Oscar\nWilde's. You had set my feet firmly on earth for the first time, there\nwas great darkness with me for many weeks, but, as it lifted, the earth\nseemed greener than ever of old, the sunshine a goodlier thing, and\nverily a blessedness indeed to draw the breath of life. I had learnt\n\"the value and significance of flesh\"; I no longer scorned a carnal\ndiet, and once again I turned my eyes on the damsels in the street.\n\n'But an influence soon came to me that kept me from going all the way\nwith you, and taught me to say, \"I know not,\" where you would say, \"It\nis not.\" Blessings on thee who didst throw a rainbow, that may mean a\npromise, across the void, that awoke the old instinct of faith within\nme, and has left me \"an Agnostic with a faith,\" quite content with \"the\nbrown earth,\" if that be all, but with the added significance a mystery\ngives to living;--thou who first didst teach me Love's lore aright, to\nthee do I owe this thing.\n\n'To J.A.W. I owe the first great knowledge of that other love between\nman and man, which Whitman has since taught us to call \"the dear love of\ncomrades\"; and to him I owe that I never burned those early rhymes, or\nbroke my little reed--an unequivocal service to me, whatever the\npublic, should it be consulted, may think.\n\n'To a dear sister I owe that still more exquisite and subtle comradeship\nwhich can only exist between man and woman, but from which the more\ndisturbing elements of sex must be absent. And here, let me also thank\nGod that I was brought up in quite a garden of good sisters.\n\n'To Messrs. C. and W., Solicitors and Notaries, I owe, albeit I will say\nno thanks to them, the opportunity of that hardly learned good which\ndwells for those who can wrest it in a hateful taskwork, that faculty of\n\"detachment\" which Marcus Aurelius learnt so long ago, by means of which\nthe soul may withdraw, into an inaccessible garden, and sing while the\nhead bends above a ledger; or, in other words, the faculty of dreaming\nwith one side of the brain, while calculating with the other. Mrs.\nBrowning's great _Aurora Leigh_ helped me more to the attainment of that\nthan any book I know.\n\n'In their office, too, among many other great things, I learnt that a\nman may be a good fellow and hate poetry--possibility undreamed of by\nsentimental youth; also that Messrs. Bass and Cope are not unworthy of\ntheir great reputation; and I had various nonsense knocked out of me,\nthough they never succeeded in persuading me in that little matter of\nthe \"ambrosial curls.\"\n\n'Through Samuel Dale I first came to understand how \"whatever is\" _can_\nbe \"best,\" and also won a faith in God which I rather caught by\ninfection than gained by any process of his reasoning. Of all else I owe\nto Samuel, how write? He knows.\n\n'To a certain friend, mentioned last because he is not least, I owe: the\nsum of ten pounds, and a loving companionship, up hill and down dale,\nfor which again I have no words and no--sovereigns.'\n\nWhen I first read through these, I was somewhat surprised at the\nomission of all reference to books which I know marked most striking\nperiods in Narcissus' spiritual life: _Sartor Resartus_, Thoreau's\n_Walden_, for example, Mr. Pater's _Marius the Epicurean_, and\nBrowning's _Dramatis Personae_. As I reflected, however, I came to the\nconclusion that such omission was but justice to his own individuality,\nfor none of these books had created an _initiative_ in Narcissus'\nthought, but rather come, as, after all, I suppose they come to most of\nus, as great confirming expressions of states of mind at which he had\nalready arrived, though, as it were, but by moonlight. In them was the\nsunrise bringing all into clear sight and sure knowledge.\n\nIt would seem, indeed, that the growth of the soul in the higher spirits\nof our race is analogous to the growth of a child in the womb, in this\nrespect: that in each case the whole gamut of earlier types is run\nthrough, before the ultimate form is attained in which it is decreed\nthat the particular vital energy shall culminate. And, as in the\nphysical world the various 'halts,' so to say, of the progress are\nillustrated by the co-existence and continual succession of those\nearlier types; so in the world of mind, at every point of spiritual\nevolution, a man may meet with an historical individuality who is a\nconcrete embodiment of the particular state to which he has just\nattained. This, of course, was what Goethe meant when he referred to\nmysticism as being a frame of mind which one could experience all round\nand then leave behind. To quote Whitman, in another connection:--\n\n  'We but level that lift\n  To pass and continue beyond.'\n\nBut an individuality must 'crystallise out' somewhere, and its final\nvalue will not so much depend on the number of states it has passed\nthrough, as how it has lived each on the way, with what depth of\nconviction and force of sincerity. For a modern young man to thus\nexperience all round, and pass, and continue beyond where such great\nones as St. Bernard, Pascal, and Swedenborg, have anchored their starry\nsouls to shine thence upon men for all time, is no uncommon thing. It is\nmore the rule than the exception: but one would hardly say that in going\nfurther they have gone higher, or ended greater. The footpath of pioneer\nindividualism must inevitably become the highway of the race. Every\nAmerican is not a Columbus.\n\nThere are two ways in which we may live our spiritual progress: as\ncritics, or poets. Most men live theirs in that critical attitude which\nrefuses to commit itself, which tastes all, but enjoys none; but the\ngreatest in that earnest, final, rooted, creative, fashion which is the\nway of the poets. The one is as a man who spends his days passing from\nplace to place in search of a dwelling to his mind, but dies at last in\nan inn, having known nought of the settled peace of a home; but the\nother, howsoever often he has to change his quarters, for howsoever\nshort a time he may remain in any one of his resting-places, makes of\neach a home, with roots that shoot in a night to the foundations of the\nworld, and blossomed branches that mingle with the stars.\n\nCriticism is a good thing, but poetry is a better. Indeed, criticism\nproperly _is_ not; it is but a process to an end. We could really do\nwithout it much better than we imagine: for, after all, the question is\nnot so much _how_ we live, but _do_ we live? Who would not a hundred\ntimes rather be a fruitful Parsee than a barren _philosophe_? Yes, all\nlies, of course, in original greatness of soul; and there is really no\nstate of mind which is not like Hamlet's pipe--if we but know the 'touch\nof it,' 'it will discourse most eloquent music.'\n\nNow, it was that great sincerity in Narcissus that has always made us\ntake him so seriously. And here I would remark in parenthesis, that\ntrivial surface insincerities, such as we have had glimpses of in his\ndealings, do not affect such a great organic sincerity as I am speaking\nof. They are excrescences, which the great central health will sooner or\nlater clear away. It was because he never held an opinion to which he\nwas not, when called upon, practically faithful; never dreamed a dream\nwithout at once setting about its translation into daylight; never\nprofessed a creed for a week without some essay after the realisation of\nits new ideal; it was because he had the power and the courage to glow\nmightily, and to some purpose; because his life had a fiery centre,\nwhich his eyes were not afraid of revealing--that I speak of his great\nsincerity, a great capacity for intense life. Shallow patterers of\ndivine creeds were, therefore, most abhorrent to him. 'You must excuse\nme, sir,' I remember his once saying to such a one, 'but what are you\ndoing with cigarette and salutaris? If I held such a belief as yours, I\nwould stand sandalled, with a rope round my waist, before to-morrow.'\n\nOne quaint instance of this earnest attitude in all things occurs to me\nout of his schooldays. He was a Divine Right man, a fiery Jacobite, in\nthose days; and, probably not without some absurd unconfessed dream in\nhis heart that it might somehow help the dead old cause, he one\nafternoon fluttered the Hanoverian hearts--all the men we meet in street\nand mart are Hanoverians, of course--of our little literary club by\nsolemnly rising 'to give notice' that at the following meeting he would\nread a paper to prove that 'the House of Hanover has no right to the\nEnglish throne.' Great was the excitement through the fortnight\nintervening, extending even to the masters; and the meeting was a full\none, and no little stormy.\n\nNarcissus rose with the air of a condemned Strafford, and with all his\nboyish armoury of eloquence and scorn fought over again the long-lost\nbattle, hiss and groan falling unheeded into the stream of his young\nvoice. But vain, vain! hard is the Hanoverian heart in boy, as in man,\nand all your glowing periods were in vain--vain as, your peroration told\nus, 'was the blood of gallant hearts shed on Culloden's field.' Poor N.,\nyou had but one timorous supporter, even me, so early your _fidus\nAchates_--but one against so many. Yet were you crestfallen? Galileo\nwith his 'E pur si muove,' Disraeli with his 'The time will come,' wore\nsuch a mien as yours, as we turned from that well-foughten field. Yes!\nand you loved to take in earnest vague Hanoverian threats of possible\narrest for your baby-treason, and, for some time, I know, you never\npassed a policeman without a dignified tremor, as of one who might at\nany moment find a lodging in the Tower.\n\nBut the most serious of all N.'s 'mad' enthusiasms was that of which the\nReader has already received some hint, in the few paragraphs of his own\nconfessions above, that which 'had almost sent him to the Himalayas.'\n\nIt belongs to natures like his always through life to cherish a half\nbelief in their old fairy tales, and a longing, however late in the day,\nto prove them true at last. To many such the revelations with which\nMadame Blavatsky, as with some mystic trumpet, startled the Western\nworld some years ago, must have come with most passionate appeal; and to\nNarcissus they came like a love arisen from the dead. Long before, he\nhad 'supped full' of all the necromantic excitements that poet or\nromancer could give. Guy Mannering had introduced him to Lilly; Lytton\nand Hawthorne had sent him searching in many a musty folio for Elixir\nVitas and the Stone. Like Scythrop, in 'Nightmare Abbey,' he had for a\nlong period slept with horrid mysteries beneath his pillow. But suddenly\nhis interest had faded: these phantoms fled before a rationalistic\ncock-crow, and Eugenius Philalethes and Robert Fludd went with Mejnour\nand Zanoni into a twilight forgetfulness. There was no hand to show the\nhidden way to the land that might be, and there were hands beckoning and\nvoices calling him along the highway to the land that is. So,\ndream-light passing, he must, perforce, reconcile himself to daylight,\nwith its dusty beam and its narrow horizons.\n\nJudge, then, with what a leaping heart he chanced on some newspaper\ngossip concerning the sibyl, for it was so that he first stumbled across\nher mission. Ironical, indeed, that the so impossible 'key' to the\nmystery should come by the hand of 'our own correspondent'; but so it\nwas, and that paragraph sold no small quantity of 'occult' literature\nfor the next twelve months. Mr. Sinnett, doorkeeper in the house of\nBlavatsky, who, as a precaution against the vision of Bluebeards that\nthe word Oriental is apt to conjure up in Western minds, is always\ndressed in the latest mode, and, so to say, offers his cigar-case along\nwith some horrid mystery--it was to his prospectus of the new gospel,\nhis really delightful pages, that Narcissus first applied. Then he\nentered within the gloomier Egyptian portals of the _Isis_ itself, and\nfrom thence--well, in brief, he went in for a course of Redway, and\nlittle that figured in that gentleman's thrilling announcements was long\nin reaching his hands.\n\nAt last a day came when his eye fell upon a notice, couched in suitably\nmysterious terms, to the effect that really earnest seekers after divine\ntruth might, after necessary probation, etc., join a brotherhood of\nsuch--which, it was darkly hinted, could give more than it dared\npromise. Up to this point Narcissus had been indecisive. He was,\nremember, quite in earnest, and to actually accept this new evangel\nmeant to him--well, as he said, nothing less in the end than the\nHimalayas. Pending his decision, however, he had gradually developed a\ncertain austerity, and experimented in vegetarianism; and though he was,\noddly enough, free of amorous bond that might have held him to earth,\nyet he had grown to love it rather rootedly since the earlier days when\nhe was a 'seeker.' Moreover, though he read much of 'The Path,' no\nactual Mejnour had yet been revealed to set his feet therein. But with\nthis paragraph all indecision soon came to an end. He felt there a clear\ncall, to neglect which would be to have seen the light and not to have\nfollowed it, ever for him the most tragic error to be made in life. His\nnatural predisposition towards it was too great for him to do other than\ntrust this new revelation; and now he must gird himself for 'the\nsacrifice which truth always demands.'\n\nBut, sacrifice! of what and for what? An undefined social warmth he was\nbeginning to feel in the world, some meretricious ambition, and a great\nfriendship--to which in the long run would he not be all the truer by\nthe great new power he was to win? If hand might no longer spring to\nhand, and friendship vie in little daily acts of brotherhood, might he\nnot, afar on his mountain-top, keep loving watch with clearer eyes upon\nthe dear life he had left behind, and be its vigilant fate? Surely! and\nthere was nothing worth in life that would not gain by such a devotion.\nAll life's good was of the spirit, and to give that a clearer shining,\neven in one soul, must help the rest. For if its light, shining, as now,\nthrough the grimy horn-lantern of the body, in narrow lanes and along\nthe miasmatic flats of the world, even so helped men, how much more must\nit, rising above that earthly fume, in a hidden corner no longer, but\nin the open heaven, a star above the city. Sacrifice! yes, it was just\nsuch a tug as a man in the dark warmth of morning sleep feels it to\nleave the pillow. The mountain-tops of morning gleam cold and bare: but\nO! when, staff in hand, he is out amid the dew, the larks rising like\nfountains above him, the gorse bright as a golden fleece on the\nhill-side, and all the world a shining singing vision, what thought of\nthe lost warmth then? What warmth were not well lost for this keen\nexhilarated sense in every nerve, in limb, in eye, in brain? What potion\nhas sleep like this crystalline air it almost takes one's breath to\ndrink, of such a maddening chastity is its grot-cool sparkle? What\nintoxication can she give us for this larger better rapture? So did\nNarcissus, an old Son of the Morning, figure to himself the struggle,\nand pronounce 'the world well lost.'\n\nBut I feel as I write how little I can give the Reader of all the\n'splendid purpose in his eyes' as he made this resolve. Perhaps I am the\nless able to do so as--let me confess--I also shared his dream. One\ncould hardly come near him without, in some measure, doing that at all\ntimes; though with me it could only be a dream, for I was not free. I\nhad Scriptural example to plead 'Therefore I cannot come,' though in any\ncase I fear I should have held back, for I had no such creative instinct\nfor realisation as Narcissus, and have, I fear, dreamed many a dream I\nhad not the courage even to think of clothing in flesh and blood; like,\nmay I say, the many who are poets for all save song--poets in chrysalis,\nall those who dream of what some do, and make the audience of those\ngreat articulate ones. But there were one or two trifling doubts to set\nat rest before final decision. The Reader has greatly misconceived\nNarcissus if he has deemed him one of those simple souls whom any quack\ncan gull, and the good faith of this mysterious fraternity was a\ndifficult point to settle. A tentative application through the address\ngiven, an appropriate _nom de mystere_, had introduced the ugly detail\nof preliminary expenses. Divine truth has to pay its postage, its rent,\nits taxes, and so on; and the 'guru' feeds not on air--although, of\ncourse, being a 'guru,' he comes as near it as the flesh will allow:\ntherefore, and surely, Reader, a guinea per annum is, after all,\nreasonable enough. Suspect as much as one will, but how gainsay? Also,\nbefore the applicant could be admitted to noviciate even, his horoscope\nmust be cast, and--well, the poor astrologer also needed bread and--no!\nnot butter--five shillings for all his calculations, circles, and\nsignifications--well, that again was only reasonable. H'm, ye-e-s, but\nit was dubious; and, mad as we were, I don't think we ever got outside\nthat dubiety, but made up our minds, like other converts, to gulp the\nprimary postulate, and pay the twenty-six shillings. From the first,\nhowever, Narcissus had never actually entrusted all his spiritual\nventure in this particular craft: he saw the truth independent of them,\nnot they alone held her for him, though she might hold them, and they\nmight be that one of the many avenues for which he had waited to lead\nhim nearer to her heart. That was all. His belief in the new\nillumination neither stood nor fell with them, though his ardour for it\nculminated in the experience. One must take the most doubtful\nexperiment seriously if we are in earnest for results.\n\nSo next came the sacred name of 'the Order,' which, Reader, I cannot\ntell thee, as I have never known it, Narcissus being bound by horrid\noaths to whisper it to no man, and to burn at midnight the paper which\ngave it to his eyes. From this time, also, we could exchange no deep\nconfidences of the kind at all, for the various MSS. by means of which\nhe was to begin his excursions into Urania, and which his 'guru' sent\nfrom time to time--at first, it must be admitted, with a diligent\nfrequency--were secret too. So several months went by, and my knowledge\nof his 'chela-ship' was confined to what I could notice, and such\ntrifling harmless gossip as 'Heard from \"guru\" this morning,' 'Copying\nan old MS. last night,' and so on. What I could notice was truly, as\nLamb would say, 'great mastery,' for lo! Narcissus, whose eyes had never\nmissed a maiden since he could walk, and lay in wait to wrest his\ntribute of glance and blush from every one that passed, lo! he had\nchanged all that, and Saint Anthony in an old master looks not more\nresolutely 'the other way' than he, his very thoughts crushing his flesh\nwith invisible pincers. No more softly-scented missives lie upon his\ndesk a-mornings; and, instead of blowing out the candle to dream of\nDaffodilia, he opens his eyes in the dark to defy--the Dweller on the\nThreshold, if haply he should indeed already confront him.\n\nOne thrilling piece of news in regard to the latter he was unable to\nconceal. He read it out to me one flushed morning:--\n\n  '_I--have--seen--him--and--am--his--master_,'\n\nwrote the 'guru,' in answer to his neophyte's half fearful question.\nFitly underlined and sufficiently spaced, it was a statement calculated\nto awe, if only by its mendacity. I wonder if that chapter of Bulwer's\nwould impress one now as it used to do then. It were better, perhaps,\nnot to try.\n\nThe next news of these mysteries was the conclusion of them. When so\ndarkly esoteric a body begins to issue an extremely catchpenny 'organ,'\nwith advertisements of theosophic 'developers,' magic mirrors, and\nmesmeric discs, and also advertises large copies of the dread symbol of\nthe Order, 'suitable for framing,' at five shillings plain and seven and\nsixpence coloured, it is, of course, impossible to take it seriously,\nexcept in view of a police-court process, and one is evidently in the\nhands of very poor bunglers indeed. Such was the new departure in\npropaganda instituted by a little magazine, mean in appearance, as the\nmouthpieces of all despised 'isms' seem to be, with the first number of\nwhich, need one say, ended Narcissus' ascent of 'The Path.' I don't\nthink he was deeply sad at being disillusionised. Unconsciously a\nbroader philosophy had slowly been undermining his position, and all was\nready for the fall. It cost no such struggle to return to the world as\nit had taken to leave it, for the poet had overgrown the philosopher,\nand the open mystery of the common day was already exercising an appeal\nbeyond that of any melodramatic 'arcana.' Of course the period left its\nmark upon him, but it is most conspicuous upon his bookshelves.\n\n\n\n\nCHAPTER VII\n\n\nTHE CHILDREN OF APOLLO\n\n'He is a _true_ poet,' or 'He is a _genuine_ artist,' are phrases which\nirritate one day after day in modern criticism. One had thought that\n'poet' and 'artist' were enough; but there must be a need, we\nregretfully suppose, for these re-enforcing qualifications; and there\ncan be but the one, that the false in each kind do so exceedingly\nabound, that none can be taken as genuine without such special\ncertificate. The widespread confusion with the poet of the rhetorician\nand sentimentalist in verse, and again of the mere rhymer without even\nrhetoric, not to refer to finer differentiation of error, is also a\nfruitful source of bewilderment. The misuse of the word has parallels:\nfor instance, the spurious generic use of the word 'man' for 'male,'\nthe substitution of 'artist' for 'painter.' But here we have only to\ndeal with that one particular abuse. Some rules how to know a poet may\nconceivably be of interest, though of no greater value.\n\nOf course, the one first and last test is his work, but 'how to know\npoetry' is another matter, which I do not propose treating of here; my\nintention rather being to dot down a few personal characteristics--not\nso much his 'works' as his 'ways.' I write as they come into my head;\nand to any Reader about to cry out against digression, let me add: I\nwrite thinking of Narcissus; for know all men, friend or Philistine, if\nyou have yet to learn it, my Narcissus is a poet!\n\nFirst, as to the great question of 'garmenting.' The superstition that\nthe hat and the cloak 'does it' has gone out in mockery, but only that\nthe other superstition might reign in its stead--that the hat and cloak\ncannot do it. Because one great poet dispensed with 'pontificals,' and\nyet brought the fire from heaven, henceforward 'pontificals' are humbug,\nand the wearer thereof but charlatan, despite--'the master yonder in\nthe isle.' Pegasus must pack in favour of a British hunter, and even the\npoet at last wear the smug regimentals of mediocrity and mammon. Ye\nyounger choir especially have a care, for, though you sing with the\ntongues of men and angels, and wear not a silk hat, it shall avail you\nnothing. Neither Time, which is Mudie, nor Eternity, which is Fame, will\nknow you, and your verses remain till doom in an ironical _editio\nprinceps_, which not even the foolish bookman shall rescue from the\nthreepenny box. It is very unlikely that you will escape as did\nNarcissus, for though, indeed,\n\n  'He swept a fine majestic sweep\n    Of toga Tennysonian,\n  Wore strange soft hat, that such as you\n    Would tremble to be known in,'\n\nnevertheless, he somehow won happier fates, on which, perhaps, it would\nbe unbecoming in so close a friend to dilate.\n\nThe 'true' poet is, first of all, a gentleman, usually modest, never\narrogant, and only assertive when pushed. He does not by instinct take\nhimself seriously, as the 'poet-ape' doth, though if he meets with\nrecognition it becomes, of course, his duty to acknowledge his faculty,\nand make good Scriptural use of it.\n\nHe is probably least confident, however, when praised; and never, except\nin rare moments, especially of eclipse, has he a strong faith in the\ntruth that is in him. Therefore crush him, saith the Philistine, as we\ncrush the vine; strike him, as one strikes the lyre. When young, he\nimagines the world to be filled with one ambition; later on, he finds\nthat so indeed it is--but the name thereof is not Poesy. Strange! sighs\nhe. And if, when he is seventeen, he writes a fluent song, and his\nfellow-clerk admire it, why, it is nothing; surely the ledger-man hath\nsuch scraps in his poke, or at least can roll off better. 'True bards\nbelieve all able to achieve what they achieve,' said Naddo. But lo! that\nambition is a word that begins with pounds and ends with pence--like\nlife, quoth the ledger-man, who, after all, had but card-scores, a\ntailor's account, and the bill for his wife's confinement in his pocket.\n\nAll through his life he loves his last-written most, and no honey of\nHybla is so sweet as a new rhyme. Let no maid hope to rival it with her\nlips--she but interrupts: for the travail of a poet is even as that of\nhis wife--after the pain comes that dear joy of a new thing born into\nthe world, which doting sipping dream beware to break. Fifty repetitions\nof the new sweetness, fifty deliberate rollings of it under the tongue,\nis, I understand, the minimum duration of such, before the passion is\nworked off, and the dream-child really breathing free of its\ndream-parent. I have occasionally come upon Narcissus about the\ntwenty-fifth, I suppose, and wondered at my glum reception. 'Poetry gone\nsour,' he once gave as the reason. Try it not, Reader, if, indeed, in\nthy colony of beavers a poet really dwells.\n\nHe is a born palaeontologist: that is, he can build up an epic from a\nhint. And, despite modern instances, the old rule obtains for him, he\nneed not be learned--that is, not deeply or abundantly, only at\npoints--superficially, the superficial would say. Well, yes, he has an\neye for knowing what surfaces mean, the secret of the divining rod.\nTake it this way. We want an expression, say, of the work of Keats, want\nto be told wherein lies his individuality. You take Mr. Buxton Forman's\nfour volumes, and 'work at' Keats! and, after thirty nights and days,\nbring your essay. On the morning of the thirtieth the poet read again\nthe _Grecian Urn_, and at eventide wrote a sonnet; and on the morning of\nthe thirty-first, essay and sonnet are side by side. But, by the\nevening, your essay is in limbo--or in type, all's one--while the sonnet\nis singing in our heart, persistently haunting our brain. Some day the\npoet, too, writes an essay, and thus plainly shows, says the essayist,\nhow little he really knew of the matter--he didn't actually know of the\nso-and-so--and yet it was his ignorance that gave us that illuminating\nline, after all.\n\nI doubt if one would be on safe ground in saying: Take, now, the subject\nof wine. We all know how abstemious is the poetical habit; and yet, to\nread these songs, one would think 'twas Bacchus' self that wrote, or\nthat Clarence who lay down to die in a butt of Malmsey. Though the\ninference is open to question,\n\n  'I often wonder if old Omar drank\n  One half the quantity he bragged in song.'\n\nDoubtless he sat longest and drank least of all the topers of Naishapur,\nand the bell for Saki rang not from his corner half often enough to\nplease mine host. Certainly the longevity of some modern poets can only\nbe accounted for by some such supposition in their case. The proposition\nis certainly proved inversely in the case of Narcissus, for he has not\nwritten one vinous line, and yet--well, and yet! Furthermore, it may\ninterest future biographers to know that in his cups he was wont to\nrecite Hamlet's advice to the players, throned upon a tram-car.\n\nThe 'true' poet makes his magic with the least possible ado; he and the\nuntrue are as the angler who is born to the angler who is made at the\ntackle-shop. One encumbers the small of his back with nameless engines,\ntalks much of creels, hath a rod like a weaver's beam; he travels first\nclass to some distant show-lake among the hills, and he toils all day\nas the fishermen of old toiled all night; while Tom, his gardener's son,\nbut a mile outside the town, with a willow wand and a bent pin, hath\ncaught the family supper. So is it with him who is proverbially born not\nmade. His friends say: 'O, you should go to such-and-such falls; you 'd\nwrite poetry there, if you like. We all said so'; or, 'What are you\ndoing in here scribbling? Look through the window at the moonlight;\nthere's poetry for you. Go out into that if you want sonnets.' Of\ncourse, he never takes his friends' advice; he has long known that they\nknow nothing whatever about it. He is probably quite ignorant of\nmetrical law, but one precept instinct taught him from the beginning,\nand he finds it expressed one day in Wordsworth (with a blessed comfort\nof assurance--like in this little, O, may be like, somehow, in the great\nthing too!): 'Poetry is emotion remembered in tranquillity.' The\nwandlike moments, he remembers, always came to him in haunts all remote\nindeed from poetry: a sudden touch at his heart, and the air grows\nrhythmical, and seems a-ripple with dreams; and, albeit, in whatever\nroom of dust or must he be, the song will find him, will throw her arms\nabout him, so it seems, will close his eyes with her sweet breath, that\nhe may open them upon the hidden stars.\n\n'Impromptus' are the quackery of the poetaster. One may take it for\ngranted, as a general rule, that anything written 'on the spot' is\nworthless. A certain young poet, who could when he liked do good things,\nprinted some verses, which he declared in a sub-title were 'Written on\nthe top of Snowdon in a thunderstorm.' He asked an opinion, and one\nreplied: 'Written on the top of Snowdon in a thunderstorm.' The poet was\nnaturally angry--and yet, what need of further criticism?\n\nThe poet, when young, although as I said, he is not likely to fall into\nthe foolishness of conceit which belongs to the poetaster, is yet too\napt in his zeal of dedication to talk much of his 'art,' or, at least,\nthink much; also to disparage life, and to pronounce much gratuitous\nabsolution in the name of Poetry:--\n\nDid Burns drink and wench?--yet he sang!\n\nDid Coleridge opiate and neglect his family?--yet he sang!!\n\nDid Shelley--well, whatever Shelley did of callous and foolish, the list\nis long--yet he sang!!!\n\nAs years pass, however, he grows out of this stage, and, while regarding\nhis art in a spirit of dedication equally serious, and how much saner,\nhe comes to realise that, after all, art but forms one integral part,\nhowever great, of a healthy life, and that for the greatest artist there\nare still duties in life more imperative than any art can lay upon him.\nIt is a great hour when he rises up in his resolution first to be a man,\nin faith that, if he be such, the artist in him will look after\nitself--first a man, and surely all the greater artist for being that;\nthough if not, still a man. That is the duty that lies' next' to all of\nus. Do that, and, as we are told, the other will be clearer for us. In\nthat hour that earlier form of absolution will reverse itself on his\nlips into one of commination. Did they sing?--yet they sinned here and\nhere; and as a man soweth, so shall he reap, singer or sot. Lo! his\nsongs are stars in heaven, but his sins are snakes in hell: each shall\nbless and torment him in turn.\n\nPitiable, indeed, will seem to him in that hour the cowardice that dares\nto cloak its sinning with some fine-spun theory, that veils the\ngratification of its desires in some shrill evangel, and wrecks a\nwoman's life in the names of--Liberty and Song! Art wants no such\nfollowers: her bravest work is done by brave men, and not by sneaking\nopium-eaters and libidinous 'reformers.' We all have sinned, and we all\nwill go on sinning, but for God's sake, let us be honest about it. There\nare worse things than honest sin. If, God help you, you have ruined a\ngirl, do penance for it through your life; pay your share; but don't,\nwhatever you do, hope to make up for a bad heart by a good brain.\nFoolish art-patterers may suffer the recompense to pass, for likely they\nhave all the one and none of the other; but good men will care nothing\nabout you or your work, so long as bad trees refuse to bring forth good\nfruit, or figs to grow on thistles.\n\nWe have more to learn from Florentine artists than any 'craft mystery.'\nIf the capacity for using the blossom while missing the evil fruit, of\nwhich Mr. Pater speaks in the case of Aurelius, were only confined to\nthose evil-bearing trees: alas! it is all blossom with us moderns, good\nor bad alike, and purity or putrescence are all one to us, so that they\nshine. I suppose few regard Giotto's circle as his greatest work: would\nthat more did. The lust of the eye, with Gautier as high-priest, is too\nmuch with us.\n\nThe poet, too, who perhaps began with the simple ambition of becoming a\n'literary man,' soon finds how radically incapable of ever being merely\nthat he is. Alas! how soon the nimbus fades from the sacred name of\n'author.' At one time he had been ready to fall down and kiss the\ngarment's hem, say, of--of a 'Canterbury' editor (this, of course, when\nvery, very young), as of a being from another sphere; and a writer in\n_The Fortnightly_ had swam into his ken, trailing visible clouds of\nglory. But by and by he finds himself breathing with perfect composure\nin that rarefied air, and in course of time the grey conviction settles\nupon him that these fabled people are in no wise different from the\nbooksellers and business men he had found so sordid and dull--no more\nindividual or delightful as a race; and he speedily comes to the old\nconclusion he had been at a loss to understand a year or two ago, that,\nas a rule, the people who do not write books are infinitely to be\npreferred to the people who do. When he finds exceptions, they occur as\nthey used to do in shop and office--the charm is all independent of the\ncalling; for just as surely as a man need not grow mean, and hard, and\ndried up, however prosperous be his iron-foundry, so sure is it that a\nman will not grow generous, rich-minded, loving, and all that is golden\nby merely writing of such virtues at so much a column. The inherent\ninsincerity, more or less, of all literary work is a fact of which he\nhad not thought. I am speaking of the mere 'author,' the\nwriter-tradesman, the amateur's superstition; not of men of genius, who,\ndespite cackle, cannot disappoint. If they seem to do so, it must be\nthat we have not come close enough to know them. But the man of genius\nis rarer, perhaps, in the ranks of authorship than anywhere: you are\nfar more likely to find him on the exchange. They are as scarce as\nCaxtons: London possesses hardly half-a-dozen examples.\n\nNarcissus enjoyed the delight of calling one of these his friend, 'a\ncertain aristocratic poet who loved all kinds of superiorities,' again\nto borrow from Mr. Pater. He had once seen him afar off and worshipped,\nas it is the blessedness of boys to be able to worship; but never could\nhe have dreamed in that day of the dear intimacy that was to come. 'If\nhe could but know me as I am,' he had sighed; but that was all. With the\nalmost childlike naturalness which is his greatest charm he confessed\nthis sigh long after, and won that poet's heart. Well I remember his\nbursting into our London lodging late one afternoon, great-eyed and\nalmost in tears for joy of that first visit. He had pre-eminently the\ncapacity which most fine men have of falling in love with men--as one\nmay be sure of a subtle greatness in a woman whose eye singles out a\nwoman to follow on the stage at the theatre--and certainly, no other\nphrase can express that state of shining, trembling exaltation, the\npassion of the friendships of Narcissus. And although he was rich in\nthem--rich, that is, as one can be said to be rich in treasure so\nrare--saving one only, they have never proved that fairy-gold which such\ndo often prove. Saving that one, golden fruit still hangs for every\nwhite cluster of wonderful blossom.\n\n'I thought you must care for me if you could but know me aright,'\nNarcissus had said.\n\n'Care for you! Why, you beautiful boy! you seem to have dropped from the\nstars,' the poet had replied in the caressing fashion of an elder\nbrother.\n\nHe had frankly fallen in love, too: for Narcissus has told me that his\ngreat charm is a boyish naturalness of heart, that ingenuous gusto in\nliving which is one of the sure witnesses to genius. This is all the\nmore piquant because no one would suspect it, as, I suppose, few do;\nprobably, indeed, a consensus would declare him the last man in London\nof whom that is true. No one would seem to take more seriously the _beau\nmonde_ of modern paganism, with its hundred gospels of _La Nuance_; no\none, assuredly, were more _blase_ than he, with his languors of pose,\nand face of so wan a flame. The Oscar Wilde of modern legend were not\nmore as a dweller in Nirvana. But Narcissus maintained that all this was\nbut a disguise which the conditions of his life compelled him to wear,\nand in wearing which he enjoyed much subtle subterranean merriment;\nwhile underneath the real man lived, fresh as morning, vigorous as a\nyoung sycamore, wild-hearted as an eagle, ever ready to flash out the\n'password primeval' to such as alone could understand. How else had he\nat once taken the stranger lad to his heart with such a sunlight of\nwelcome? As the maid every boy must have sighed for but so rarely found,\nwho makes not as if his love were a weariness which she endured, and the\nkisses she suffered, cold as green buds, were charities, but frankly\nglows to his avowal with 'I love you, too, dear Jack,' and kisses him\nfrom the first with mouth like a June rose--so did that _blase_ poet\ncast away his conventional Fahrenheit, and call Narcissus friend in\ntheir first hour. Men of genius alone know that fine _abandon_ of soul.\nIn such is the poet confessed as unmistakably as in his verse, for the\none law of his life is that he be an elemental, and the capacity for\ngreat simple impressions is the spring of his power. Let him beware of\nlosing that.\n\nI sometimes wonder as I come across the last frivolous gossip concerning\nthat poet in the paragraphs of the new journalism, or meet his name in\nsome distinguished bead-roll in _The Morning Post_, whether Narcissus\nwas not, after all, mistaken about him, and whether he could still,\nseason after season, go through the same stale round of reception,\nprivate view, first night, and all the various drill of fashion and\nfolly, if that boy's heart were alive still. One must believe it once\nthrobbed in him: we have his poems for that, and a poem cannot lie; but\nit is hard to think that it could still keep on its young beating\nbeneath such a choking pressure of convention, and in an air so 'sunken\nfrom the healthy breath of morn.' But, on the other hand, I have almost\na superstitious reliance on Narcissus' intuition, a faculty in him which\nnot I alone have marked, but which I know was the main secret of his\nappeal for women. They, as the natural possessors of the power, feel a\nsingular kinship with a man who also possesses it, a gift as rarely\nfound among his sex as that delicacy which largely depends on it, and\nwhich is the other sure clue to a woman's love. She is so little used,\npoor flower, to be understood, and to meet with other regard than the\ngaze of satyrs.\n\nHowever, be Narcissus' intuition at fault or not in the main, still it\nwas very sure that the boy's heart in that man of the world did wake\nfrom its sleep for a while at the wandlike touch of his youth; and if,\nafter all, as may be, Narcissus was but a new sensation in his jaded\nround, at least he was a healthy one. Nor did the callous ingratitude of\nforgetfulness which follows so swiftly upon mere sensation ever add\nanother to the sorrows of my friend: for, during the last week before he\nleft us, came a letter of love and cheer in that poet's wonderful\nhandwriting--handwriting delicious with honeyed lines, each word a\nflower, each letter rounded with the firm soft curves of hawthorn in\nbud, or the delicate knobs of palm against the sky.\n\n\n\n\nCHAPTER VIII\n\n\nGEORGE MUNCASTER\n\nWhen I spoke of London's men of genius I referred, of course, to such as\nare duly accredited, certificated, so to say, by public opinion; but of\nthose others whose shining is under the bushel of obscurity, few or\nmany, how can one affirm? That there are such, any man with any happy\nexperience of living should be able to testify; and I should say, for\nfear of misunderstanding, that I do not use the word genius in any\ntechnical sense, not only of men who can _do_ in the great triumphal\nway, but also of those who can _be_ in their quiet, effective fashion,\nwithin their own 'scanty plot of ground'; men who, if ever conscious of\nit, are content with the diffusion of their influence around the narrow\nlimits of their daily life, content to bend their creative instincts on\nthe building and beautifying of home. It is no lax use of the word\ngenius to apply it to such, for unless you profess the modern heresy\nthat genius is but a multiplied talent, a coral-island growth, that\nearns its right to a new name only when it has lifted its head above the\nwaters of oblivion, you must agree. For 'you saw at once,' said\nNarcissus, in reference to that poet, 'that his writing was so\ndelightful because he was more so.' His writings, in fact, were but the\naccidental emanations of his personality. He might have given himself\nout to us in fugues, or canvases, or simply, like the George Muncaster\nof whom I am thinking, in the sweet breath and happy shining of his\nhome. Genius is a personal quality, and if a man has it, whatever his\nhand touches will bear the trace of his power, an undying odour, an\nunfading radiance. When Rossetti wrote 'Beauty like hers is genius,' he\nwas not dealing in metaphor, and Meissonier should have abolished for\never the superstition of large canvases.\n\nThese desultory hints of the development of Narcissus would certainly be\nmore incomplete than necessity demands, if I did not try to give the\nReader some idea of the man of genius of this unobtrusive type to whom I\nhave just alluded. Samuel Dale used to call himself 'an artist in life,'\nand there could be no truer general phrase to describe George Muncaster\nthan that. His whole life possesses a singular unity, such as is the\nmost satisfying joy of a fine work of art, considering which it never\noccurs to one to think of the limitation of conditions or material. So\nwith his life, the shortness of man's 'term' is never felt; one could\nwin no completer effect with eternity than he with every day. Hurry and\nfalse starts seem unknown in his round, and his little home is a\nmicrocosm of the Golden Age.\n\nIt would even seem sometimes that he has an artistic rule over his\n'accidents,' for 'surprises' have a wonderful knack of falling into the\ngeneral plan of his life, as though but waited for. Our first meeting\nwith him was a singular instance of this. I say 'our,' for Narcissus and\nI chanced to be walking a holiday together at the time. It fell on this\nwise. At Tewkesbury it was we had arrived, one dull September evening,\njust in time to escape a wetting from a grey drizzle then imminent; and\nin no very buoyant spirits we turned into _The Swan Inn_. A more dismal\ncoffee-room for a dismal evening could hardly be--gloomy, vast, and\nthinly furnished. We entered sulkily, seeming the only occupants of the\nsepulchre. However, there was a small book on the table facing the door,\nsufficiently modern in appearance to catch one's eye and arouse a faint\nripple of interest. 'A Canterbury,' we cried. 'And a Whitman, more's the\nwonder,' cried Narcissus, who had snatched it up. 'Why, some one's had\nthe sense, too, to cut out the abominable portrait. I wonder whose it\nis. The owner must evidently have some right feeling.'\n\nThen, before there was time for further exclamatory compliment of the\nunknown, we were half-startled by the turning round of an arm-chair at\nthe far end of the room, and were aware of a manly voice of exquisite\nquality asking, 'Do you know Whitman?'\n\nAnd moving towards the speaker, we were for the first time face to face\nwith the strong and gentle George Muncaster, who since stands in our\nlittle gallery of types as Whitman's Camarado and Divine Husband made\nflesh. I wish, Reader, that I could make you see his face; but at best I\nhave little faith in pen portraits. It is comparatively easy to write a\ngraphic description of _a_ face; but when it has been read, has the\nreader realised _the_ face? I doubt it, and am inclined to believe that\nthree different readers will carry away three different impressions even\nfrom a really brilliant portrait. Laborious realism may, at least, I\nthink, be admitted as hopeless. The only chance is in a Meredithian\nlightning-flash, and those fly but from one or two bows. I wonder if an\nimage will help at all here. Think on a pebbly stream, on a brisk,\nbright morning; dwell on the soft, shining lines of its flowing; and\nthen recall the tonic influence, the sensation of grip, which the\npebbles give it. Dip your hand into it again in fancy; realise how\nchaste it is, and then again think how bright and good it is. And if you\nrealise these impressions as they come to me, you will have gained some\nidea of George Muncaster's face--the essential spirit of it, I mean,\never so much more important than the mere features. Such, at least,\nseemed the meaning of his face even in the first moment of our\nintercourse that September dusk, and so it has never ceased to come upon\nus even until now.\n\nAnd what a night that was! what a talk! How soon did we find each other\nout! Long before the maid knocked at the door, and hinted by the\ndelicate insinuation of a supposed ring that there was 'a budding\nmorrow' in the air. But our passionate generosity of soul was running in\ntoo strong a tide just then to be stemmed by any such interference; it\ncould but be diverted, and Muncaster's bedroom served us as well wherein\nto squat in one of those close, rapt circles of talk such as, I think,\nafter all, men who love poetry can alone know--men, anyhow, with _a_\npoetry.\n\nBed, that had for some time been calling us, unheeded as Juliet's nurse,\nhad at last to be obeyed; but how grudgingly; and how eagerly we sprang\nfrom it at no late hour in the morning, at the first thought of the\nsweet new thing that had come into the world--like children who, half\nin a doze before waking, suddenly remember last night's new wonder of a\ntoy, to awake in an instant, and scramble into clothes to look at it\nagain. Thus, like children we rose; but it was shy as lovers we met at\nthe breakfast-table, as lovers shy after last night's kissing. (You may\nnot have loved a fellow-man in this way, Reader, but we are, any one of\nus, as good men as you; so keep your eyebrows down, I beseech you.)\n\nOne most winsome trait of our new friend was soon apparent--as, having,\nto our sorrow, to part at the inn door right and left, we talked of\nmeeting again at one or the other's home: a delicate disinclination to\nirreverently 'make sure' of the new joy; a 'listening fear,' as though\nof a presiding good spirit that might revoke his gift if one stretched\nout towards it with too greedy hands. 'Rather let us part and say\nnought. You know where a letter will find me. If our last night was a\nreal thing, we shall meet again, never fear.' With some such words as\nthose it was that he bade us good-bye.\n\nOf course, letters found all three of us before a fortnight had gone\nby, and in but a short time we found his home. There it is that George\nshould be seen. Away he is full of precious light, but home is his\nsetting. To Narcissus, who found it in that green period when all\nyoungsters take vehement vows of celibacy, and talk much of 'free love,'\nall ignorant, one is in charity persuaded, of what they quite mean, that\nhome was certainly as great and lasting a revelation as the first hour\nof 'Poetry's divine first finger-touch.' It was not that his own\nhome-life had been unhappy, for it was the reverse, and rich indeed in\ngreat and sweet influences; but it was rather, I think, that the ideal\nof a home is not so easily to be reached from that home in which one is\na child, where one is too apt to miss the whole in consideration of\none's own part in it, as from another on which we can look from the\noutside.\n\nOur parents, even to the end, partake too much of the nature of\nmythology; it always needs an effort to imagine them beings with quite\nthe same needs and dreams as ourselves. We rarely get a glimpse of\ntheir poetry, for the very reason that we ourselves are factors in it,\nand are, therefore, too apt to dwell on the less happy details of the\ndomestic life, details which one ray of their poetry would transfigure\nas the sun transfigures the motes in his beam. Thus, in that green age I\nspoke of, one's sickly vision can but see the dusty, world-worn side of\ndomesticity, the petty daily cares of living, the machinery, so to say,\nof 'house and home.' But when one stands in another home, where these\nare necessarily unseen by us, stands with the young husband, the\npoetry-maker, how different it all seems. One sees the creation bloom\nupon it; one ceases to blaspheme, and learns to bless. Later, when at\nlength one understands why it is sweeter to say 'wife' than\n'sweetheart,' how even one may be reconciled to calling one's Daffodilia\n'little mother'--because of the children, you know; it would never do\nfor them to say Daffodilia--then he will understand too how those petty\ndetails, formerly so '_banal_,' are, after all, but notes in the music,\nand what poetry can flicker, like its own blue flame, around even the\njoint purchase of a frying-pan.\n\nThat Narcissus ever understood this great old poetry he owes to George\nMuncaster. In the very silence of his home one hears a singing--'There\nlies the happiest land.' It was one of his own quaint touches that the\nfirst night we found his nest, after the maid had given us admission,\nthere should be no one to welcome us into the bright little parlour but\na wee boy of four, standing in the doorway like a robin that has hopped\non to one's window-sill. But with what a dear grace did the little chap\nhold out his hand and bid us good evening, and turn his little morsel of\na bird's tongue round our names; to be backed at once by a ring of\nlaughter from the hidden 'prompter' thereupon revealed. O happy, happy\nhome! may God for ever smile upon you! There should be a special grace\nfor happy homes. George's set us 'collecting' such, with results\nundreamed of by youthful cynic. Take courage, Reader, if haply you stand\nwith hesitating toe above the fatal plunge. Fear not, you can swim if\nyou will. Of course, you must take care that your joint poetry-maker be\nsuch a one as George's. One must not seem to forget the loving wife who\nmade such dreaming as his possible. He did not; and, indeed, had you\ntold him of his happiness, he would but have turned to her with a smile\nthat said, 'All of thee, my love'; while, did one ask of this and that,\nhow quickly 'Yes! that was George's idea,' laughed along her lips.\n\nWhile we sat talking that first evening, there suddenly came three\ncries, as of three little heads straining out of a nest, for 'Father';\nand obedient, with a laugh, he left us. This, we soon learnt, was a part\nof the sweet evening ritual of home. After mother's more practical\nservice had been rendered the little ones, and they were cosily 'tucked\nin,' then came 'father's turn,' which consisted of his sitting by their\nbedside--Owen and Geoffrey on one hand, and little queen Phyllis,\nmaidenlike in solitary cot, on the other--and crooning to them a little\nevening song. In the dark, too, I should say, for it was one of his wise\nprovisions that they should be saved from ever fearing that; and that,\nwhenever they awoke to find it round them in the middle of the night, it\nshould bring them no other association but 'father's voice.'\n\nA quaint recitative of his own, which he generally contrived to vary\neach night, was the song, a loving croon of sleep and rest. The\nbrotherhood of rest, one might name his theme for grown-up folk; as in\nthe morning, we afterwards learnt, he is wont to sing them another\nlittle song of the brotherhood of work; the aim of his whole beautiful\neffort for them being to fill their hearts with a sense of the\nbrotherhood of all living things--flowers, butterflies, bees and birds,\nthe milk-boy, the policeman, the man at the crossing, the grocer's pony,\nall within the circle of their little lives, as living and working in\none great _camaraderie_. Sometimes he would extemporise a little rhyme\nfor them, filling it out with his clear, happy voice, and that tender\npantomime that comes so naturally to a man who not merely loves\nchildren--for who is there that does not?--but one born with the\ninstinct for intercourse with them. To those not so born it is as\ndifficult to enter into the life and prattle of birds. I have once or\ntwice crept outside the bedroom door when neither children nor George\nthought of eavesdroppers, and the following little songs are impressions\nfrom memory of his. You must imagine them chanted by a voice full of the\ninfinite tenderness of fatherhood, and even then you will but dimly\nrealise the music they have as he sings them. I run the risk of his\nforgiving my printing them here:--\n\n  MORNING SONG.\n\n  Morning comes to little eyes,\n  Wakens birds and butterflies,\n  Bids the flower uplift his head,\n  Calls the whole round world from bed.\n    Up jump Geoffrey!\n      Up jump Owen!!\n    Then up jump Phyllis!!!\n      And father's going!\n\n  EVENING SONG.\n\n  The sun is weary, for he ran\n    So far and fast to-day;\n  The birds are weary, for who sang\n    So many songs as they?\n  The bees and butterflies at last\n    Are tired out; for just think, too,\n  How many gardens through the day\n    Their little wings have fluttered through.\n\n    And so, as all tired people do,\n  They've gone to lay their sleepy heads\n  Deep, deep in warm and happy beds.\n  The sun has shut his golden eye,\n  And gone to sleep beneath the sky;\n  The birds, and butterflies, and bees\n  Have all crept into flowers and trees,\n  And all lie quiet, still as mice,\n  Till morning comes, like father's voice.\n  So Phyllis, Owen, Geoffrey, you\n  Must sleep away till morning too;\n  Close little eyes, lie down little heads,\n  And sleep, sleep, sleep in happy beds.\n\nAs the Reader has not been afflicted with a great deal of verse in these\npages, I shall also venture to copy here another little song which, as\nhis brains have grown older, George has been fond of singing to them at\nbedtime, and with which the Reader is not likely to have enjoyed a\nprevious acquaintance:--\n\n  REST.[1]\n\n  When the Sun and the Golden Day\n  Hand in hand are gone away,\n  At your door shall Sleep and Night\n  Come and knock in the fair twilight;\n      Let them in, twin travellers blest;\n      Each shall be an honoured guest,\n      And give you rest.\n\n  They shall tell of the stars and moon,\n  And their lips shall move to a glad sweet tune,\n  Till upon your cool, white bed\n  Fall at last your nodding head;\n      Then in dreamland fair and blest,\n      Farther off than East and West,\n      They give you rest.\n\n  Night and Sleep, that goodly twain,\n  Tho' they go, shall come again;\n  When your work and play are done,\n  And the Sun and Day are gone\n      Hand in hand thro' the scarlet West,\n      Each shall come, an honoured guest,\n      And bring you rest.\n\n  Watching at your window-sill,\n  If upon the Eastern hill\n  Sun and Day come back no more,\n  They shall lead you from the door\n      To their kingdom calm and blest,\n      Farther off than East or West,\n      And give you rest.\n\nArriving down to breakfast earlier than expected next morning, we\ndiscovered George busy at some more of his loving ingenuity. He half\nblushed in his shy way, but went on writing in this wise, with chalk,\nupon a small blackboard: '_Thursday_--_Thor's-day_--_Jack the Giant\nKiller's day_'. Then, in one corner of the board, a sun was rising with\na merry face and flaming locks, and beneath him was written,\n'_Phoebus-Apollo';_ while in the other corner was a setting moon, '_Lady\nCynthia_. There were other quaint matters, too, though they have escaped\nmy memory; but these hints are sufficient to indicate George's morning\noccupation. Thus he endeavoured to implant in the young minds he felt so\nsacred a trust an ever-present impression of the full significance of\nlife in every one of its details. The days of the week should mean for\nthem what they did mean, should come with a veritable personality, such\nas the sun and the moon gained for them by thus having actual names,\nlike friends and playfellows. This Thor's-day was an especially great\nday for them; for, in the evening, when George had returned from\nbusiness, and there was yet an hour to bedtime, they would come round\nhim to hear one of the adventures of the great Thor--adventures which he\nhad already contrived, he laughingly told us, to go on spinning out of\nthe Edda through no less than the Thursdays of two years. Certainly his\ningenuity of economy with his materials was no little marvel, and he\nconfessed to often being at his wits' end. For Thursday night was not\nalone starred with stories; every night there was one to tell; sometimes\nan incident of his day in town, which he would dress up with the\nimaginative instinct of a born teller of fairy-tales. He had a knack,\ntoo, of spreading one story over several days which would be invaluable\nto a serial writer. I remember one simple instance of his device.\n\nHe sat in one of those great cane nursing chairs, Phyllis on one knee,\nOwen on the other, and Geoffrey perched in the hollow space in the back\nof the chair, leaning over his shoulder, all as solemn as a court\nawaiting judgment. George begins with a preliminary glance behind at\nGeoffrey: 'Happy there, my boy? That's right. Well, there was once a\nbeautiful garden.'\n\n'Yes-s-s-s,' go the three solemn young heads.\n\n'And it was full of the most wonderful things.'\n\n'Yes-s-s-s.'\n\n'Great trees, so green, for the birds to hide and sing in; and flowers\nso fair and sweet that the bees said that, in all their flying hither\nand thither, they had never yet found any so full of honey in all the\nworld. And the birds, too, what songs they knew; and the butterflies,\nwere there ever any so bright and many-coloured?' etc., etc.\n\n'But the most wonderful thing about the garden was that everything in it\nhad a wonderful story to tell.'\n\n'Yes-s-s s.'\n\n'The birds, and bees, and butterflies, even the trees and flowers, each\nknew a wonderful fairy-tale.'\n\n'Oh-h-h-h.'\n\n'But of all in the garden the grasshopper knew the most. He had been a\ngreat traveller, for he had such long legs.'\n\nAgain a still deeper murmur of breathless interest.\n\n'Now, would you like to hear what the grasshopper had to tell?'\n\n'Oh, yes-s-s-s.'\n\n'Well, you shall--to-morrow night!'\n\nSo off his knees they went, as he rose with a merry, loving laugh, and\nkissed away the long sighs of disappointment, and sent them to bed,\nagog for all the morrow's night should reveal.\n\nNeed one say that the children were not the only disappointed listeners?\nBesides, they have long since known all the wonderful tale, whereas one\nof the poorer grown-up still wonders wistfully what that grasshopper who\nwas so great a traveller, and had such long legs, had to tell.\n\nBut I had better cease. Were I sure that the Reader was seeing what I am\nseeing, hearing as I, I should not fear; but how can I be sure of that?\nHad I the pen which that same George will persist in keeping for his\nletters, I should venture to delight the Reader with more of his story.\nOne underhand hope of mine, however, for these poor hints is, that they\nmay by their very imperfection arouse him to give the world 'the true\nstory' of a happy home. Narcissus repeatedly threatened that, if he did\nnot take pen in hand, he would some day 'make copy' of him; and now I\nhave done it instead. Moreover, I shall further presume on his\nforbearance by concluding with a quotation from one of his letters that\ncame to me but a few months back:--\n\n'You know how deeply exercised the little ones are on the subject of\ndeath, and how I had answered their curiosity by the story that after\ndeath all things turn into flowers. Well, what should startle the wife's\nears the other day but \"Mother, I wish you would die.\" \"O why, my dear?\"\n\"Because I should so like to water you!\" was the delicious explanation.\nThe theory has, moreover, been called to stand at the bar of experience,\nfor a week or two ago one of Phyllis' goldfish died. There were tears at\nfirst, of course, but they suddenly dried up as Geoffrey, in his\nreflective way, wondered \"what flower it would come to.\" Here was a\ndilemma. One had never thought of such contingencies. But, of course, it\nwas soon solved. \"What flower would you like it to be, my boy?\" I asked.\n\"A poppy!\" he answered; and after consultation, \"a poppy!\" agreed the\nothers. So a poppy it is to be. A visit to the seedsman's procured the\nnecessary surreptitious poppy seed; and so now poor Sir Goldfish sleeps\nwith the seed of sleep in his mouth, and the children watch his grave\nday by day, breathless for his resplendent resurrection. Will you write\nus an epitaph?'\n\nAriel forgive me! Here is what I sent:\n\n  'Five inches deep Sir Goldfish lies;\n    Here last September was he laid;\n  Poppies these, that were his eyes,\n    Of fish-bones are these blue-bells made;\n  His fins of gold that to and fro\n  Waved and waved so long ago,\n  Still as petals wave and wave\n  To and fro above his grave.\n  Hearken, too! for so his knell\n  Tolls all day each tiny bell.'\n\nFOOTNOTES:\n\n[Footnote 1: From a tiny privately-printed volume of deliciously\noriginal lyrics by Mr. R.K. Leather, since republished by Mr. Fisher\nUnwin, 1890, and at present published by Mr. John Lane.]\n\n\n\n\nCHAPTER IX\n\n\nTHAT THIRTEENTH MAID\n\n  'Who chooseth me must give and hazard all he hath.'--\n                            _Merchant of Venice_.\n\n\nIt occurs to me here to wonder whether there can be any reader\nungrateful enough to ask with grumbling voice, 'What of the book-bills?\nThe head-line has been the sole mention of them now for many pages; and\nin the last chapter, where a book was referred to, the writer was\nperverse enough to choose one that never belonged to Narcissus at all.'\nTo which I would venture to make humble rejoinder--Well, Goodman Reader,\nand what did you expect? Was it accounts, with all their thrilling\ndetails, with totals, 'less discount,' and facsimiles of the receipt\nstamps? Take another look at our first chapter. I promised nothing of\nthe sort there, I am sure. I promised simply to attempt for you the\ndelineation of a personality which has had for all who came into contact\nwith it enduring charm, in hope that, though at second-hand, you might\nhave some pleasure of it also; and I proposed to do this mainly from the\nhints of documents which really are more significant than any letters or\nother writings could be, for the reason that they are of necessity so\nunconscious. I certainly had no intention of burdening you with the\noriginal data, any more than, should you accept the offer I made, also\nin that chapter, and entrust me with your private ledger for\nbiographical purposes, I would think of printing it _in extenso_, and\ncalling it a biography; though I should feel justified, after the varied\nstory had been deduced and written out, in calling the product,\nmetaphorical wise, 'The private ledger of Johannes Browne, Esquire'--a\ntitle which, by the way, is copyright and duly 'entered.' Such was my\nattempt, and I maintain that I have so far kept my word. Because whole\nshelves have been disposed of in a line, and a ninepenny 'Canterbury'\nhas rustled out into pages, you have no right to complain, for that is\nbut the fashion of life, as I have endeavoured to show. And let me say\nin passing that that said copy of Mr. Rhys's Whitman, though it could\nnot manifestly appear in his book-bills, does at the present moment rest\nupon his shelf--'a moment's monument.'\n\nPerhaps it would be well, before proceeding with this present 'place in\nthe story,' to set out with a statement of the various 'authorities' for\nit; as, all this being veritable history, perhaps one should. But then,\nReader, here again I should have to catalogue quite a small library.\nHowever, I will enumerate a few of the more significant ones.\n\n'Swinburne's _Tristram of Lyonesse_, 9\/-, less dis., 6\/9.'\n\nAll that this great poem of 'springtide passion with its fire and\nflowers' meant to Narcissus and his 'Thirteenth Maid' in the morning of\ntheir love, those that have loved too will hardly need telling, while\nthose who have not could never understand, though I spake with the\ntongue of the poet himself. In this particular copy, which, I need\nhardly say, does not rest upon N.'s shelves, but on another in a sweet\nlittle bedchamber, there is a tender inscription and a sonnet which\naimed at acknowledging how the hearts of those young lovers had gone out\nto that poet 'with mouth of gold and morning in his eyes.' The latter I\nhave begged leave to copy here:--\n\n    'Dear Heart, what thing may symbolise for us\n    A love like ours; what gift, whate'er it be,\n    Hold more significance 'twixt thee and me\n  Than paltry words a truth miraculous,\n  Or the poor signs that in astronomy\n    Tell giant splendours in their gleaming might?\n    Yet love would still give such, as in delight\n  To mock their impotence--so this for thee.\n\n    'This book for thee; our sweetest honeycomb\n    Of lovesome thought and passion-hearted rhyme,\n      Builded of gold, and kisses, and desire,\n    By that wild poet whom so many a time\n      Our hungering lips have blessed, until a fire\n  Burnt speech up, and the wordless hour had come.'\n\n'Meredith's _Richard Feverel_, 6\/-, less dis., 4\/6.'\n\nNarcissus was never weary of reading those two wonderful chapters where\nLucy and Richard meet, and he used to say that some day he would beg\nleave from Mr. Meredith to reprint at his own charges just those two\nchapters, to distribute to all true lovers in the kingdom. It would be\nhard to say how often he and his maid had read them aloud together, with\namorous punctuation--caresses for commas, and kisses for full-stops.\n\n'Morris' _Sigurd the Volsung_, 12\/-, less dis., 9\/-.'\n\nThis book they loved when their love had grown to have more of earnest\npurpose in it, and its first hysteric ecstasy had passed into the more\nsolemn ardours of the love that goes not with spring, but loves even\nunto the winter and beyond. It is marked all through in pencil by\nNarcissus; but on one page, where it opens easily, there are written\ninitials, in a woman's hand, against this great passage:--\n\n  'She said: \"Thou shalt never unsay it, and thy heart is mine indeed:\n  Thou shalt bear thy love in thy bosom as thou helpest the earth-folk's\n          need:\n  Thou shalt wake to it dawning by dawning; thou shalt sleep and it shall\n          not be strange:\n  There is none shall thrust between us till our earthly lives shall\n          change.\n  Ah, my love shall fare as a banner in the hand of thy renown,\n  In the arms of thy fame accomplished shall it lie when we lay us adown.\n  O deathless fame of Sigurd! O glory of my lord!\n  O birth of the happy Brynhild to the measureless reward!\"\n  So they sat as the day grew dimmer, and they looked on days to come,\n  And the fair tale speeding onward, and the glories of their home;\n  And they saw their crowned children and the kindred of the kings,\n  And deeds in the world arising and the day of better things:\n  All the earthly exaltation, till their pomp of life should be passed,\n  And soft on the bosom of God their love should be laid at the last.'\n\nAnd on the page facing this lies a pressed flower--there used to be\ntwo--guarded by these tender rhymes:--\n\n    'Whoe'er shall read this mighty song\n    In some forthcoming evensong,\n    We pray thee guard these simple flowers,\n    For, gentle Reader, they are \"ours.\"'\n\nBut ill has some 'gentle Reader' attended to the behest, for, as I said,\nbut one of the flowers remains. One is lost--and Narcissus has gone\naway. This inscription is but one of many such scattered here and there\nthrough his books, for he had a great facility in such minor graces, as\nhe had a neat hand at tying a bow. I don't think he ever sent a box of\nflowers without his fertility serving him with some rose-leaf fancy to\naccompany them; and on birthdays and all red-letter days he was always\nto be counted upon for an appropriate rhyme. If his art served no other\npurpose, his friend would be grateful to him for that alone, for many\ngreat days would have gone without their 'white stone' but for him;\nwhen, for instance, J.A.W. took that brave plunge of his, which has\nsince so abundantly justified him and more than fulfilled prophecy; or\nwhen Samuel Dale took that bolder, namely a wife, he being a\nphilosopher--incidents, Reader, on which I long so to digress, and for\nwhich, if you could only know beforehand, you would, I am sure, give me\nfreest hand. But beautiful stories both, I may not tell of you here;\nthough if the Reader and I ever spend together those hinted nights at\nthe 'Mermaid,' I then may.\n\nBut to return. I said above that if I were to enumerate all the books,\nso to say, 'implicated' in the love of Narcissus and his Thirteenth\nMaid, I should have to catalogue quite a small library. I forgot for the\nmoment what literal truth I was writing, for it was indeed in quite a\nlarge library that they first met. In 'our town' there is, Reader, an\nold-world institution, which, I think, you would well like transported\nto yours, a quaint subscription library 'established' ever so long ago,\nfull of wonderful nooks and corners, where (of course, if you are a\nmember) one is sure almost at any time of the day of a solitary corner\nfor a dream. It is a sweet provision, too, that it is managed by ladies,\nwhom you may, if you can, image to yourself as the Hesperides; for there\nare three of them; and may not the innumerable galleries and spiral\nstaircases, serried with countless shelves, clustered thick with tome on\ntome, figure the great tree, with its many branches and its wonderful\ngold fruit--the tree of knowledge? The absence of the dragon from the\nsimilitude is as well, don't you think?\n\nBooks, of all things, should be tended by reverent hands; and, to my\nmind, the perfunctory in things ecclesiastical is hardly more\ndistressing than the service of books as conducted in many great\nlibraries. One feels that the _librarii_ should be a sacred order,\nnearly allied to the monastic, refined by varying steps of initiation,\nand certainly celibates. They should give out their books as the priest\nhis sacrament, should wear sacred vestments, and bear about with them\nthe priestlike _aura_, as of divine incarnations of the great spirit of\nTruth and Art in whose temples they are ministrants. The next step to\nthis ideal ministry is to have our books given out to us by women.\nThough they may understand them not, they handle them with gentle\ncourtesy, and are certainly in every way to be preferred to the youthful\nfreckled monster with red spines upon his head, and nailed boots, 'the\nwork of the Cyclops,' upon his feet, whose physiognomy is contorted by\ncinnamon-balls at the very moment he carries in his arms some great\nGolden-lips or gentle Silver-tongue. What good sweet women there are,\ntoo, who would bless heaven for the occupation!\n\nWell, as I said, we in that particular library are more fortunate, and\ntwo of the 'subscribers,' at least, did at one time express their\nappreciation of its privileges by a daily dream among its shelves. One\nday--had Hercules been there overnight?--we missed one of our fair\nattendants. Was it Aegle, Arethusa, or Hesperia? Narcissus probably\nknew. And on the next she was still missing; nor on the third had she\nreturned; but lo! there was another in her stead--and on her Narcissus\nbent his gaze, according to wont. A little maid, with noticeable eyes,\nand the hair Rossetti loved to paint--called Hesper, 'by many,' said\nNarcissus, one day long after, solemnly quoting the Vita Nuova, 'who\nknow not wherefore.'\n\n'Why! do _you_ know?' I asked.\n\n'Yes!' And then, for the first time, he had told me the story I have now\nto tell again. He had, meanwhile, rather surprised me by little touches\nof intimate observation of her which he occasionally let slip--as, for\ninstance, 'Have you noticed her forehead? It has a fine distinction of\nform; is pure ivory, surely; and you should watch how deliciously her\nhair springs out of it, like little wavy threads of \"old gold\" set in\nthe ivory by some cunning artist.'\n\nI had just looked at him and wondered a moment. But such attentive\nregard was hardly matter for surprise in his case; and, moreover, I\nalways tried to avoid the subject of women with him, for it was the one\non which alone there was danger of our disagreeing. It was the only one\nin which he seemed to show signs of cruelty in his disposition, though\nit was, I well know, but a thoughtless cruelty; and in my heart I always\nfelt that he was too right-minded and noble in the other great matters\nof life not to come right on that too when 'the hour had struck.'\nMeanwhile, he had a way of classifying amours by the number of verses\ninspired--as, 'Heigho! it's all over; but never mind, I got two sonnets\nout of her'--which seemed to me an exhibition of the worst side of his\nartist disposition, and which--well, Reader, jarred much on one who\nalready knew what a true love meant. It was, however, I could see, quite\nunconscious; and I tried hard not to be intolerant towards him, because\nfortune had blessed me with an earlier illumination.\n\nPray, go not away with the misconception that Narcissus was ever base to\na woman. No! he left that to Circe's hogs, and the one temptation he\never had towards it he turned into a shining salvation. No! he had\nnothing worse than the sins of the young egoist to answer for, though he\nafterwards came to feel those pitiful and mean enough.\n\nAnother noticeable feature of Hesper's face was an ever-present\nsadness--not as of a dull grief, but as of some shining sorrow, a\nquality which gave her face much arresting interest. It seemed one\ngreat, rich tear. One loved to dwell upon it as upon those intense\nstretches of evening sky when the day yearns through half-shut eyelids\nin the west. One continually wondered what story it meant, for some it\nmust mean.\n\nWatching her thus quietly, day by day, it seemed to me that as the weeks\nfrom her first coming went by, this sadness deepened; and I could not\nforbear one day questioning the elder Hesperides about her, thus\nbringing upon myself a revelation I had little expected. For, said she,\n'she was glad I had spoken to her, for she had long wished to ask me to\nuse my influence with my friend, that he might cease paying Hesper\nattentions which he could not mean in earnest, but which she knew were\nalready causing Hesper to be fond of him. Having become friendly with\nher, she had found out her secret and remonstrated with her, with the\nresult that she had avoided Narcissus for some time, but not without\nmuch misery to herself, over which she was continually brooding.'\n\nAll this was an utter surprise, and a saddening one; for I had grown to\nfeel much interest in the girl, and had been especially pleased by all\nabsence of the flighty tendencies with which too many girls in public\nservice tempt men to their own destruction. She had seemed to me to bear\nherself with a maidenly self-respect that spoke of no little grace of\nbreeding. She had two very strong claims on one's regard. She was\nevidently a woman, in the deep, tragic sense of that word, and a lady in\nthe only true sense of that. The thought of a life so rich in womanly\npromise becoming but another of the idle playthings of Narcissus filled\nme with something akin to rage, and I was not long in saying some strong\nwords to him. Not that I feared for her the coarse 'ruin' the world\nalone thinks of. Is that the worst that can befall woman? What of the\nspiritual deflowering, of which the bodily is but a symbol? If the first\nfine bloom of the soul has gone, if the dream that is only dreamed once\nhas grown up in the imagination and been once given, the mere chastity\nof the body is a lie, and whatever its fecundity, the soul has nought\nbut sterility to give to another. It is not those kisses of the\nlips--kisses that one forgets as one forgets the roses we smelt last\nyear--which profane; they but soil the vessel of the sacrament, and it\nis the sacrament itself which those consuming spirit-kisses, which burn\nbut through the eyes, may desecrate. It is strange that man should have\nso long taken the precisely opposite attitude in this matter, caring\nonly for the observation of the vessel, and apparently dreaming not of\nany other possible approach to the sanctities. Probably, however, his\ncare has not been of sanctities at all. Indeed, most have, doubtless,\nlittle suspicion of the existence of such, and the symbol has been and\nis but a selfish superstition amongst them--woman, a symbol whose\nmeaning is forgotten, but still the object of an ignorant veneration,\nnot unrelated to the preservation of game.\n\nNarcissus took my remonstrance a little flippantly, I thought, evidently\nfeeling that too much had been made out of very little; for he averred\nthat his 'attentions' to Hesper had been of the slightest character,\nhardly more than occasional looks and whispers, which, from her cold\nreception of them, he had felt were more distasteful to her than\notherwise. He had indeed, he said, ceased even these the last few days,\nas her reserve always made him feel foolish, as a man fondling a fair\nface in his dream wakes on a sudden to find that he is but grimacing at\nthe air. This reassured me, and I felt little further anxiety. However,\nthis security only proved how little I really understood the weak side\nof my friend. I had not realised how much he really was Narcissus, and\nhow dear to him was a new mirror. My speaking to him was the one wrong\ncourse possible to be taken. Instead of confirming his growing intention\nof indifference, it had, as might have been foreseen, the directly\nopposite effect; and from the moment of his learning that Hesper\nsecretly loved him, she at once became invested with a new glamour, and\ngrew daily more and more the forbidden fascination few can resist.\n\nI did not learn this for many months. Meanwhile Narcissus chose to\ndeceive me for the first and only time. At last he told me all; and how\ndifferent was his manner of telling it from his former gay relations of\nconquest. One needed not to hear the words to see he was unveiling a\nsacred thing, a holiness so white and hidden, the most reverent word\nseemed a profanation; and, as he laboured for the least soiled wherein\nto enfold the revelation, his soul seemed as a maid torn with the\nblushing tremors of a new knowledge. Men only speak so after great and\nwonderful travail, and by that token I knew Narcissus loved at last. It\nhad seemed unlikely ground from which love had first sprung forth, that\nof a self-worship that could forgo no slightest indulgence--but thence\nindeed it had come. The silent service my words had given him to know\nthat Hesper's heart was offering to him was not enough; he must hear it\narticulate, his nostrils craved an actual incense. To gain this he must\ndeceive two--his friend, and her whose poor face would kindle with\nhectic hope, at the false words he must say for the true words he _must_\nhear. It was pitifully mean; but whom has not his own hidden lust made\nto crawl like a thief, afraid of a shadow, in his own house? Narcissus'\nyoung lust was himself, and Moloch knew no more ruthless hunger than\nburns in such. Of course, it did not present itself quite nakedly to\nhim; he persuaded himself there could be little harm--he meant none.\n\nAnd so, instead of avoiding Hesper, he sought her the more persistently,\nand by some means so far wooed her from her reticence as to win her\nconsent to a walk together one autumn afternoon. How little do we know\nthe measure of our own proposing! That walk was to be the most fateful\nhis feet had ever trodden through field and wood, yet it seemed the most\naccidental of gallantries. A little town-maid, with a romantic passion\nfor 'us'; it would be interesting to watch the child; it would be like\ngiving her a day's holiday, so much sunshine 'in our presence.' And so\non. But what an entirely different complexion was the whole thing\nbeginning to take before they had walked a mile. Behind the flippancy\none had gone to meet were surely the growing features of a solemnity.\nWhy, the child was a woman indeed; she could talk, she had brains,\nideas--and, Lord bless us, Theories! She had that 'excellent thing in\nwoman,' not only a voice, which she had, too, but character. Narcissus\nbegan to loose his regal robes, and from being merely courteously to be\ngenuinely interested. Why, she was a discovery! As they walked on, her\ngenuine delight in the autumnal nature, the real imaginative appeal it\nhad for her, was another surprise. She had, evidently, a deep poetry in\nher disposition, rarest of all female endowments. In a surprisingly few\nminutes from the beginning of their walk he found himself taking that\n'little child' with extreme seriousness, and wondering many 'whethers.'\n\nThey walked out again, and yet again, and Narcissus' first impressions\ndeepened. He had his theories, too; and, surely, here was the woman! He\nwas not in love--at least, not with her, but with her fitness for his\ntheory.\n\nThey sat by a solitary woodside, beneath a great elm tree. The hour was\nfull of magic, for though the sun had set, the smile of her day's joy\nwith him had not yet faded from the face of earth. It was the hour\nvulgarised in drawing-room ballads as the 'gloaming.' They sat very near\nto each other; he held her hand, toying with it; and now and again their\neyes met with the look that flutters before flight, that says, 'Dare I\ngive thee all? Dare I throw my eyes on thine as I would throw myself on\nthee?' And then, at last, came the inevitable moment when the eyes of\neach seem to cry 'O yes!' to the other, and the gates fly back; all the\nhidden light springs forth, the woods swim round, and the lips meet with\na strange shock, while the eyes of the spirit close in a lapping dream\nof great peace.\n\nIf you are not ready to play the man, beware of a kiss such as the lips\nof little Hesper, that never knew to kiss before, pressed upon the mouth\nof Narcissus. It sent a chill shudder through him, though it was so\nsweet, for he could feel her whole life surging behind it; and was the\nkiss he had given her for it such a kiss as that? But he had spoken much\nto her of his ideas of marriage; she knew he was sworn for ever against\nthat. She must know the kiss had no such meaning; for, besides, did she\nnot scorn the soiled 'tie' also? Were not their theories at one in that?\nHe would be doing her no wrong; it was her own desire. Yet his kiss did\nmean more than he could have imagined it meaning a week before. She had\ngrown to be genuinely desirable. If love tarried, passion was\nawake--that dangerous passion, too, to which the intellect has added its\nintoxication, and that is, so to say, legitimised by an 'idea.'\n\nHer woman's intuition read the silence and answered to his thought.\n'Have no fear,' she said, with the deep deliberation of passion; 'I\nlove you with my whole life, but I shall never burden you, Narcissus.\nLove me as long as you can, I shall be content; and when the end comes,\nthough another woman takes you, I shall not hinder.'\n\nO great girl-soul! What a poltroon, indeed, was Narcissus beside you at\nthat moment. You ready to stake your life on the throw, he temporising\nand bargaining as over the terms of a lease. Surely, if he could for one\nmoment have seen himself in the light of your greatness, he had been\ncrushed beneath the misery of his own meanness. But as yet he had no\nsuch vision; his one thought was, 'She will do it! will she draw back?'\nand the feeble warnings he was obliged to utter to keep his own terms,\nby assuring his conscience of 'her free-will,' were they not\nhalf-fearfully whispered, and with an inward haste, lest they should\ngive her pause? 'But the world, my dear--think!' 'It will have cruel\nnames for thee.' 'It will make thee outcast--think!'\n\n'I know all,' she had answered; 'but I love you, and two years of your\nlove would pay for all. There is no world for me but you. Till to-night\nI have never lived at all, and when you go I shall be as dead. The world\ncannot hurt such a one.'\n\nAh me, it was a wild, sweet dream for both of them, one the woman's, one\nthe poet's, of a 'sweet impossible' taking flesh! For, do not let us\nblame Narcissus overmuch. He was utterly sincere; he meant no wrong. He\nbut dreamed of following a creed to which his reason had long given a\nhopeless assent. In a more kindly-organised community he might have\nfollowed it, and all have been well; but the world has to be dealt with\nas one finds it, and we must get sad answers to many a fair calculation\nif we 'state' it wrongly in the equation. That there is one law for the\nmale and another for the female had not as yet vitally entered into his\nconsiderations. He was too dizzy with the dream, or he must have seen\nwhat an unequal bargain he was about to drive.\n\nAt last he did awake, and saw it all; and in a burning shame went to\nHesper, and told her that it must not be.\n\nHer answer was unconsciously the most subtly dangerous she could have\nchosen: 'If I like to give myself to you, why should you not take me? It\nis of my own free-will. My eyes are open.' It was his very thought put\ninto words, and by her. For a moment he wavered--who could blame him?\n'Am I my brother's keeper?'\n\n'Yes! a thousand times yes!' cried his soul; for he was awake now, and\nhe had come to see the dream as it was, and to shudder at himself as he\nhad well-nigh been, just as one shudders at the thought of a precipice\nbarely escaped. In that moment, too, the idea of her love in all its\ndivineness burst upon him. Here was a heart capable of a great tragic\nlove like the loves of old he read of and whimpered for in sonnets, and\nwhat had he offered in exchange? A poor, philosophical compromise,\ncompounded of pessimism and desire, in which a man should have all to\ngain and nothing to lose, for\n\n  'The light, light love he has wings to fly\n     At suspicion of a bond.'\n\n'I would I did love her,' his heart was crying as he went away. 'Could I\nlove her?' was his next thought. 'Do I love her?'--but that is a\nquestion that always needs longer than one day to answer.\n\nAlready he was as much in love with her as most men when they take unto\nthemselves wives. She was desirable--he had pleasure in her presence. He\nhad that half of love which commonly passes for all--the passion; but he\nlacked the additional incentives which nerve the common man to face that\nfear which seems well-nigh as universal as the fear of death, I mean the\nfear of marriage--life's two fears: that is, he had no desire to\nincrease his worldly possessions by annexing a dowry, or ambition of\nsettling down and procuring a wife as part of his establishment. After\nall, how full of bachelors the world would be if it were not for these\nmotives: for the one other motive to a true marriage, the other half of\nlove, however one names it, is it not a four-leaved clover indeed?\nNarcissus was happily poor enough to be above those motives, even had\nHesper been anything but poor too; and if he was to marry her, it would\nbe because he was capable of loving her with that perfect love which, of\ncourse, has alone right to the sacred name, that which cannot take all\nand give nought, but which rather holds as watchword that _to love is\nbetter than to be loved_.\n\nWho shall hope to express the mystery? Yet, is not thus much true, that,\nif it must be allowed to the cynic that love rises in self, it yet has\nits zenith and setting in another--in woman as in man? Two meet, and\npassion, the joy of the selfish part of each, is born; shall love follow\ndepends on whether they have a particular grace of nature, love being\nthe thanksgiving of the unselfish part for the boon granted to the\nother. The common nature snatches the joy and forgets the giver, but the\nfiner never forgets, and deems life but a poor service for a gift so\nrare; and, though passion be long since passed, love keeps holy an\neternal memory.\n\n  'Love took up the harp of life and smote on all the chords\n    with might;\n  Smote the chord of self, that, trembling, pass'd in music\n    out of sight.'\n\nSince the time of fairy-tales Love has had a way of coming in the\ndisguise of Duty. What is the story of Beauty and the Beast but an\nallegory of true love? We take this maid to be our wedded wife, for her\nsake it perhaps seems at the time. She is sweet and beautiful and to be\ndesired; but, all the same, we had rather shake the loose leg of\nbachelordom, if it might be. However it be, so we take her, or maybe it\nis she takes us, with a feeling of martyrdom; but lo! when we are home\ntogether, what wonderful new lights are these beginning to ray about\nher, as though she had up till now kept a star hidden in her bosom. What\nis this new morning strength and peace in our life? Why, we thought it\nwas but Thestylis, and lo! it is Diana after all. For the Thirteenth\nMaid or the Thirteenth Man, both alike, rarely come as we had expected.\nThere seems no fitness in their arrival. It seems so ridiculously\naccidental, as I suppose the hour of death, whenever it comes, will\nseem. One had expected some high calm prelude of preparation, ending in\na festival of choice, like an Indian prince's, when the maids of the\nland pass before him and he makes deliberate selection of the fateful\nShe. But, instead, we are hurrying among our day's business, maybe, our\nlast thought of her; we turn a corner, and suddenly she is before us. Or\nperhaps, as it fell with Narcissus, we have tried many loves that proved\nbut passions; we have just buried the last, and are mournfully leaving\nits grave, determined to seek no further, to abjure bright eyes, at\nleast for a long while, when lo! on a sudden a little maid is in our\npath holding out some sweet modest flowers. The maid has a sweet mouth,\ntoo, and, the old Adam being stronger than our infant resolution, we\nsmell the flowers and kiss the mouth--to find arms that somehow, we know\nnot why, are clinging as for life about us. Let us beware how we shake\nthem off, for thus it is decreed shall a man meet her to have missed\nwhom were to have missed all. Youth, like that faithless generation in\nthe Scriptures, always craveth after a sign, but rarely shall one be\ngiven. It can only be known whether a man be worthy of Love by the way\nin which he looks upon Duty. Rachel often comes in the grey cloak of\nLeah. It rests with the man's heart whether he shall know her beneath\nthe disguise; no other divining-rod shall aid him. If it be as\nBassanio's, brave to 'give and hazard all he hath,' let him not fear to\npass the seeming gold, the seeming silver, to choose the seeming lead.\n'Why, _that's_ the lady,' thou poor magnificent Morocco. Nor shall the\ngold fail, for her heart is that, and for silver thou shalt have those\n'silent silver lights undreamed of' of face and soul.\n\nThese are but scattered hints of the story of Narcissus' love as he told\nit me at last, in broken, struggling words, but with a light in his face\none power alone could set there.\n\nWhen he came to the end, and to all that little Hesper had proved to\nhim, all the strength and illumination she had brought him, he fairly\nbroke down and sobbed, as one may in a brother's arms. For, of course,\nhe had come out of the ordeal a man; and Hesper had consented to be his\nwife. Often she had dreamed as he had passed her by with such heedless\nair: 'If I love him so, can it be that my love shall have no power to\nmake him mine, somehow, some day? Can I call to him so within my soul\nand he not hear? Can I wait and he not come?' And her love had been\nstrong, strong as a destiny; her voice had reached him, for it was the\nvoice of God.\n\nWhen I next saw her, what a strange brightness shone in her face, what a\nnew beauty was there! Ah, Love, the great transfigurer! And why, too,\nwas it that his friends began to be dissatisfied with their old\nphotographs of Narcissus, though they had been taken but six months\nbefore? There seemed something lacking in the photograph, they said.\nYes, there was; but the face had lacked it too. What was the new\nthing--'grip' was it, joy, peace? Yes, all three, but more besides, and\nNarcissus had but one name for all. It was Hesper.\n\nStrange, too, that in spite of promises we never received a new one.\nNarcissus, who used to be so punctual with such a request. Perhaps it\nwas because he had broken his looking-glass.\n\n\n\n\nCHAPTER X\n\n\n'IN VISHNU-LAND WHAT AVATAR?'\n\n'If I love you for a year I shall love you for ever,' Narcissus had said\nto his Thirteenth Maid. He did love her so long, and yet he has gone\naway. Do you remember your _Les Miserables_, that early chapter where\nValjean robs the child of his florin so soon after that great\nilluminating change of heart and mind had come to him? Well, still more\nimportant, do you remember the clue Hugo gives us to aberration? There\nis comfort and strength for so many a heart-breaking failure there. It\nwas the old impetus, we are told, that was as yet too strong for the new\ncontrol; the old instinct, too dark for the new light in the brain. It\ntakes every vessel some time to answer to its helm; with us, human\nvessels, years, maybe. Have you never suddenly become sensitive of a\ngracious touch in the air, and pondered it, to recognise that in some\nhalf-unconscious act you had that moment been answering for the first\ntime the helm of an almost forgotten resolution? Ah me, blessed is it to\nsee the prow strongly sweeping up against the sky at last!\n\n'Send not a poet to London,' said Heine, and it was a true word. At\nleast, send him not till his thews are laced and his bones set. He may\nmiss somewhat, of course; there is no gain without a loss. He may be in\nignorance of the last _nuance_, and if he deserves fame he must gain it\nunaided of the vulgar notoriety which, if he have a friend or two in the\nnew journalism, they will be so eager to bestow; but he will have kept\nhis soul intact, which, after all, is the main matter. It is sweet,\ndoubtless, to be one of those same mushroom-men, sweet to be placarded\nas 'the new' this or that, to step for a day into the triumphal car of\nnewspaper renown, drawn by teams of willing paragraph-men--who, does it\nnever strike you? are but doing it all for hire, and earning their bread\nby their bent necks. Yet for those to whom it is denied there is solid\ncomfort; for it is not fame, and, worse still, it is not life, 'tis but\nto be 'a Bourbon in a crown of straws.'\n\nIf one could only take poor foolish Cockneydom right away outside this\npoor vainglorious city, and show them how the stars are smiling to\nthemselves above it, nudging each other, so to say, at the silly lights\nthat ape their shining--for such a little while!\n\nYes, that is one danger of the poet in London, that he should come to\nthink himself 'somebody'; though, doubtless, in proportion as he is a\npoet, the other danger will be the greater, that he should deem himself\n'nobody.' Modest by nature, credulous of appearances, the noisy\npretensions of the hundred and one small celebrities, and the din of\ntheir retainers this side and that, in comparison with his own\nunattended course, what wonder if his heart sinks and he gives up the\ngame; how shall his little pipe, though it be of silver, hope to be\nheard in this land of bassoons? To take London seriously is death both\nto man and artist. Narcissus had sufficient success there to make this a\ntemptation, and he fell. He lost his hold of the great things of life,\nhe forgot the stars, he forgot his love, and what wonder that his art\nsickened also. For a few months life was but a feverish clutch after\nvaried sensation, especially the dear tickle of applause; he caught the\nfacile atheistic flippancy of that poor creature, the 'modern young\nman,' all-knowing and all-foolish, and he came very near losing his soul\nin the nightmare. But he had too much ballast in him to go quite under,\nand at last strength came, and he shook the weakness from him. Yet the\nfall had been too far and too cruel for him to be happy again soon. He\nhad gone forth so confident in his new strength of manly love; and to\nfall so, and almost without an effort! Who has not called upon the\nmountains to cover him in such an hour of awakening, and who will\nwonder that Narcissus dared not look upon the face of Hesper till\nsolitude had washed him clean, and bathed him in its healing oil? I\nalone bade him good-bye. It was in this room wherein I am writing, the\nstudy we had taken together, where still his books look down at me from\nthe shelves, and all the memorials of his young life remain. O _can_ it\nhave been but 'a phantom of false morning'? A Milton snatched up at the\nlast moment was the one book he took with him.\n\nFrom that night until this he has made but one sign--a little note which\nHesper has shown me, a sob and a cry to which even a love that had been\nmore deeply wronged could never have turned a deaf ear. Surely not\nHesper, for she has long forgiven him, knowing his weakness for what it\nwas. She and I sometimes sit here together in the evenings and talk of\nhim; and every echo in the corridor sets us listening, for he may be at\nthe other side of the world, or but the other side of the street--we\nknow so little of his fate. Where he is we know not; but if he still\nlives, _what_ he is we have the assurance of faith. This time he has not\nfailed, we know. But why delay so long?\n\n\n_November_ 1889--_May_ 1890. _November_ 1894.\n\n\n\nTHE END\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Book-Bills of Narcissus\nby Le Gallienne, Richard\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"Copyright \u00a9 2007 by Dan Simmons\n\nAll rights reserved. Except as permitted under the U.S. Copyright Act of 1976, no part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.\n\nLittle, Brown and Company\n\nHachette Book Group\n\n237 Park Avenue\n\nNew York, NY 10017\n\nVisit our website at www.HachetteBookGroup.com.\n\nThe Warner Books name and logo are trademarks of Hachette Book Group, Inc.\n\nFirst eBook Edition: March 2007\n\nISBN: 978-0-316-00388-9\nContents\n\nCopyright\n\n1: CROZIER\n\n2: FRANKLIN\n\n3: CROZIER\n\n4: GOODSIR\n\n5: CROZIER\n\n6: GOODSIR\n\n7: FRANKLIN\n\n8: CROZIER\n\n9: FRANKLIN\n\n10: GOODSIR\n\n11: CROZIER\n\n12: GOODSIR\n\n13: FRANKLIN\n\n14: GOODSIR\n\n15: FRANKLIN\n\n16: CROZIER\n\n17: IRVING\n\n18: GOODSIR\n\n19: CROZIER\n\n20: BLANKY\n\n21: BLANKY\n\n22: IRVING\n\n23: HICKEY\n\n24: CROZIER\n\n25: CROZIER\n\n26: GOODSIR\n\n27: CROZIER\n\n28: PEGLAR\n\n29: IRVING\n\n30: CROZIER\n\n31: GOODSIR\n\n32: CROZIER\n\n33: GOODSIR\n\n34: CROZIER\n\n35: IRVING\n\n36: CROZIER\n\n37: IRVING\n\n38: CROZIER\n\n39: GOODSIR\n\n40: PEGLAR\n\n41: CROZIER\n\n42: PEGLAR\n\n43: CROZIER\n\n44: GOODSIR\n\n45: BLANKY\n\n46: CROZIER\n\n47: PEGLAR\n\n48: GOODSIR\n\n49: CROZIER\n\n50: BRIDGENS\n\n51: CROZIER\n\n52: GOODSIR\n\n53: GOLDING\n\n54: DES VOEUX\n\n55: GOODSIR\n\n56: JOPSON\n\n57: HICKEY\n\n58: GOODSIR\n\n59: HICKEY\n\n60: CROZIER\n\n61: CROZIER\n\n62: CROZIER\n\n63: CROZIER\n\n64: CROZIER\n\n65: CROZIER\n\n67: TALIRIKTUG\n\nAcknowledgments\n\nAbout the Author\n_Also by Dan Simmons_\n\n_Song of Kali_\n\n_Phases of Gravity_\n\n_Carrion Comfort_\n\n_Hyperion_\n\n_The Fall of Hyperion_\n\n_Prayers to Broken Stones_\n\n_Summer of Night_\n\n_The Hollow Man_\n\n_Children of the Night_\n\n_Summer Sketches_\n\n_Lovedeath_\n\n_Fires of Eden_\n\n_Endymion_\n\n_The Rise of Endymion_\n\n_The Crook Factory_\n\n_Darwin's Blade_\n\n_A Winter Haunting_\n\n_Hardcase_\n\n_Hard Freeze_\n\n_Worlds Enough & Time_\n\n_Ilium_\n\n_Hard as Nails_\n\n_Olympos_\nThis book is dedicated, with love and many thanks for the indelible Arctic memories, to Kenneth Tobey, Margaret Sheridan, Robert Cornthwaite, Douglas Spencer, Dewey Martin, William Self, George Fenneman, Dmitri Tiomkin, Charles Lederer, Christian Nyby, Howard Hawkes, and James Arness.\n\n## Sir John Franklin Arctic Expedition to Force the Northwest Passage\n\nMay 19, 1845: HMS _Erebus_ and HMS _Terror_ with 129 men depart England\n\n  1. Late July 1845: Ships last seen by whalers in Baffin Bay\n  2. Winter 1845\u201346: Ships frozen in at Beechey Island\n  3. Sept. 1846\u2013April 22, 1848: Ships frozen in place at Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W\n\n_This elusive quality it is, which causes the thought of whiteness, when divorced from more kindly associations, and coupled with any object terrible in itself, to heighten that terror to the furthest bounds. Witness the white bear of the poles, and the white shark of the tropics; what but their smooth, flaky whiteness makes them the transcendent horrors they are? That ghastly whiteness it is which imparts such an abhorrent mildness, even more loathesome than terrific, to the dumb gloating of their aspect. So that not the fierce-fanged tiger in his heraldic coat can so stagger courage as the white-shrouded bear or shark._\n\n\u2014 HERMAN MELVILLE\n\n_Moby Dick_ (1851)\n\nCROZIER\n\n_Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\nOctober, 1847\n\nCaptain Crozier comes up on deck to find his ship under attack by celestial ghosts. Above him \u2014 above _Terror_ \u2014 shimmering folds of light lunge but then quickly withdraw like the colourful arms of aggressive but ultimately uncertain spectres. Ectoplasmic skeletal fingers extend toward the ship, open, prepare to grasp, and pull back.\n\nThe temperature is \u221250 degrees Fahrenheit and dropping fast. Because of the fog that came through earlier, during the single hour of weak twilight now passing for their day, the foreshortened masts \u2014 the three topmasts, topgallants, upper rigging, and highest spars have been removed and stored to cut down on the danger of falling ice and to reduce the chances of the ship capsizing because of the weight of ice on them \u2014 stand now like rudely pruned and topless trees reflecting the aurora that dances from one dimly seen horizon to the other. As Crozier watches, the jagged ice fields around the ship turn blue, then bleed violet, then glow as green as the hills of his childhood in northern Ireland. Almost a mile off the starboard bow, the gigantic floating ice mountain that hides _Terror'_ s sister ship, _Erebus,_ from view seems for a brief, false moment to radiate colour from within, glowing from its own cold, internal fires.\n\nPulling up his collar and tilting his head back, out of forty years' habit of checking the status of masts and rigging, Crozier notices that the stars overhead burn cold and steady but those near the horizon not only flicker but shift when stared at, moving in short spurts to the left, then to the right, then jiggling up and down. Crozier has seen this before \u2014 in the far south with Ross as well as in these waters on earlier expeditions. A scientist on that south polar trip, a man who spent the first winter in the ice there grinding and polishing lenses for his own telescope, had told Crozier that the perturbation of the stars was probably due to rapidly shifting refraction in the cold air lying heavy but uneasy over the ice-covered seas and unseen frozen landmasses. In other words, over new continents never before seen by the eyes of man. Or at least, Crozier thinks, in this northern arctic, by the eyes of white men.\n\nCrozier and his friend and then-commander James Ross had found just such a previously undiscovered continent \u2014 Antarctica \u2014 less than five years earlier. They named the sea, inlets, and landmass after Ross. They named mountains after their sponsors and friends. They named the two volcanoes they could see on the horizon after their two ships \u2014 these same two ships \u2014 calling the smoking mountains Erebus and Terror. Crozier was surprised they hadn't named some major piece of geography after the ship's cat.\n\nThey named nothing after him. There is, on this October winter's dark-day evening in 1847, no arctic or antarctic continent, island, bay, inlet, range of mountains, ice shelf, volcano, or fucking floeberg which bears the name of Francis Rawdon Moira Crozier.\n\nCrozier doesn't give the slightest God-damn. Even as he thinks this, he realizes that he's a little bit drunk. _Well,_ he thinks, automatically adjusting his balance to the icy deck now canted twelve degrees to starboard and down eight degrees by the bow, _I've been drunk more often than not now for three years, haven't I? Drunk ever since Sophia. But I'm still a better sailor and captain drunk than that poor, unlucky bastard Franklin ever was sober. Or his rosy-cheeked lisping pet poodle Fitzjames, for that matter._\n\nCrozier shakes his head and walks down the icy deck forward to the bow and toward the only man on watch he can make out in the flickering light from the aurora.\n\nIt is short, rat-faced Cornelius Hickey, caulker's mate. The men look all the same out here on watch in the dark, since they're all issued the same cold-weather slops: layers of flannel and wool covered with a heavy waterproof greatcoat, bulbous mittens protruding from voluminous sleeves, their Welsh wigs \u2014 heavy watch caps with floppy ears \u2014 pulled tight, often with long comforters \u2014 scarves \u2014 wrapped around their heads until only the tips of their frostbitten noses are visible. But each man layers or wears his cold-weather slops slightly differently \u2014 adding a comforter from home, perhaps, or an extra Welsh wig tugged down over the first, or perhaps colorful gloves lovingly knit by a mother or wife or sweetheart peeking out from under the Royal Navy outer mittens \u2014 and Crozier has learned to tell all fifty-nine of his surviving officers and men apart, even at a distance outside and in the dark.\n\nHickey is staring fixedly out beyond the icicle-sheathed bowsprit, the foremost ten feet of which are now embedded in a ridge of sea ice, as HMS _Terror'_ s stern has been forced up by the ice pressure and the bow is pushed lower. Hickey is so lost in thought or cold that the caulker's mate doesn't notice his captain's approach until Crozier joins him at a railing that has become an altar of ice and snow. The lookout's shotgun is propped against that altar. No man wants to touch metal out here in the cold, not even through mittens.\n\nHickey starts slightly as Crozier leans close to him at the railing. _Terror_ 's captain can't see the twenty-six-year-old's face, but a puff of his breath \u2014 instantly turning into a cloud of ice crystals reflecting the aurora \u2014 appears beyond the thick circle of the smaller man's multiple comforters and Welsh wig.\n\nMen traditionally don't salute during the winter in the ice, not even the casual knuckling of the forehead an officer receives at sea, but the thick-clad Hickey does that odd little shuffle and shrug and head dip by which the men acknowledge their captain's presence while outside. Because of the cold, the watches have been cut down from four hours to two \u2014 God knows, thinks Crozier, we have enough men for that on this overcrowded ship, even with the lookouts doubled \u2014 and he can tell just by Hickey's slow movements that he's half-frozen. As many times as he's told the lookouts that they have to keep moving on deck \u2014 walk, run in place, jump up and down if they have to, all the while keeping their attention on the ice \u2014 they still tend to stand immobile for the majority of their watch, just as if they were in the South Seas wearing their tropical cotton and watching for mermaids.\n\n\"Captain.\"\n\n\"Mr. Hickey. Anything?\"\n\n\"Nothing since them shots... that one shot... almost two hours ago, sir. Just a while ago I heard, I think I heard... maybe a scream, something, Captain... from out beyond the ice mountain. I reported it to Lieutenant Irving, but he said it was probably just the ice acting up.\"\n\nCrozier had been told about the sound of the shot from the direction of _Erebus_ and had quickly come up on deck two hours ago, but there'd been no repetition of the sound and he'd sent no messenger to the other ship nor anyone out on the ice to investigate. To go out on the frozen sea in the dark now with that... _thing_... waiting in the jumble of pressure ridges and tall sastrugi was certain death. Messages were passed between the ships now only during those dwindling minutes of half-light around noon. In a few days, there would be no real day at all, only arctic night. Round-the-clock night. One hundred days of night.\n\n\"Perhaps it was the ice,\" says Crozier, wondering why Irving hadn't reported the possible scream. \"The shot as well. Only the ice.\"\n\n\"Yes, Captain. The ice it is, sir.\"\n\nNeither man believes it \u2014 a musket shot or shotgun blast has a distinctive sound, even from a mile away, and sound travels almost supernaturally far and clearly this far north \u2014 but it's true that the ice pack squeezing ever more tightly against _Terror_ is always rumbling, moaning, cracking, snapping, roaring, or screaming.\n\nThe screams bother Crozier the most, waking him from his hour or so of sound sleep each night. They sound too much like his mother's crying in her last days... of that and his old aunt's tales of banshees wailing in the night, predicting the death of someone in the house. Both had kept him awake as a boy.\n\nCrozier turns slowly. His eyelashes are already rimmed with ice, and his upper lip is crusted with frozen breath and snot. The men have learned to keep their beards tucked far under their comforters and sweaters, but frequently they must resort to hacking away hair that has frozen to their clothing. Crozier, like most of the officers, continues to shave every morning, although, in the effort to conserve coal, the \"hot water\" his steward brings him tends to be just barely melted ice, and shaving can be a painful business.\n\n\"Is Lady Silence still on deck?\" asks Crozier.\n\n\"Oh, yes, Captain, she's almost always up here,\" says Hickey, whispering now as if it made a difference. Even if Silence could hear them, she couldn't understand their English. But the men believe \u2014 more and more every day the thing on the ice stalks them \u2014 that the young Esquimaux woman is a witch with secret powers.\n\n\"She's at the port station with Lieutenant Irving,\" adds Hickey.\n\n\"Lieutenant Irving? His watch should have been over an hour ago.\"\n\n\"Aye, sir. But wherever Lady Silence is these days, there's the lieutenant, sir, if you don't mind me mentioning it. She don't go below, he don't go below. Until he has to, I mean... . None of us can stay out here as long as that wi\u2014... that woman.\"\n\n\"Keep your eyes on the ice, and your mind on your job, Mr. Hickey.\"\n\nCrozier's gruff voice makes the caulker's mate start again, but he shuffles his shrug salute and turns his white nose back toward the darkness beyond the bow.\n\nCrozier strides up the deck toward the port lookout post. The previous month, he prepared the ship for winter after three weeks of false hope of escape in August. Crozier had once again ordered the lower spars to be swung around along the parallel axis of the ship, using them as a ridgepole. Then they had reconstructed the tent pyramid to cover most of the main deck, rebuilding the wooden rafters that had been stowed below during their few weeks of optimism. But even though the men work hours every day shoveling avenues through the foot or so of snow left for insulation on deck, hacking away ice with picks and chisels, clearing out the spindrift that has come under the canvas roof, and finally putting lines of sand down for traction, there always remains a glaze of ice. Crozier's movement up the tilted and canted deck is sometimes more a graceful half-skating motion than a stride.\n\nThe appointed port lookout for this watch, midshipman Tommy Evans \u2014 Crozier identifies the youngest man on board by the absurd green stocking cap, obviously made by the boy's mother, that Evans always pulls down over his bulky Welsh wig \u2014 has moved ten paces astern to allow Third Lieutenant Irving and Silence some privacy.\n\nThis makes Captain Crozier want to kick someone \u2014 everyone \u2014 in the arse.\n\nThe Esquimaux woman looks like a short round bear in her furry parka, hood, and pants. She has her back half turned to the tall lieutenant. But Irving is crowded close to her along the rail \u2014 not quite touching, but closer than an officer and gentleman would stand to a lady at a garden party or on a pleasure yacht.\n\n\"Lieutenant Irving.\" Crozier didn't mean to put quite so much bark into the greeting, but he's not unhappy when the young man levitates as if poked by the point of a sharp blade, almost loses his balance, grabs the iced railing with his left hand, and \u2014 as he insists on doing despite now knowing the proper protocol of a ship in the ice \u2014 salutes with his right hand.\n\nIt's a pathetic salute, thinks Crozier, and not just because the bulky mittens, Welsh wig, and layers of cold-weather slops make young Irving look something like a saluting walrus, but also because the lad has let his comforter fall away from his clean-shaven face \u2014 perhaps to show Silence how handsome he is \u2014 and now two long icicles dangle below his nostrils, making him look even more like a walrus.\n\n\"As you were,\" snaps Crozier. _God-damn fool,_ he mentally adds.\n\nIrving stands rigid, glances at Silence \u2014 or at least at the back of her hairy hood \u2014 and opens his mouth to speak. Evidently he can think of nothing to say. He closes his mouth. His lips are as white as his frozen skin.\n\n\"This isn't your watch, Lieutenant,\" says Crozier, hearing the whip-crack in his voice again.\n\n\"Aye, aye, sir. I mean, no, sir. I mean, the captain is correct, sir. I mean...\" Irving clamps his mouth shut again, but the effect is ruined somewhat by the chattering of his teeth. In this cold, teeth can shatter after two or three hours \u2014 actually explode \u2014 sending shrapnel of bone and enamel flying inside the cavern of one's clenched jaws. Sometimes, Crozier knows from experience, you can hear the enamel cracking just before the teeth explode.\n\n\"Why are you still out here, John?\"\n\nIrving tries to blink, but his eyelids are literally frozen open. \"You ordered me to watch over our guest... to look out for... to take care of Silence, Captain.\"\n\nCrozier's sigh emerges as ice crystals that hang in the air for a second and then fall to the deck like so many minuscule diamonds. \"I didn't mean every _minute,_ Lieutenant. I told you to watch her, report to me on what she does, to keep her out of mischief and harm's way on the ship, and to see that none of the men do anything to... compromise her. Do you think she's in danger of being compromised out here on deck, Lieutenant?\"\n\n\"No, Captain.\" Irving's sentence sounds more like a question than an answer.\n\n\"Do you know how long it takes for exposed flesh to freeze out here, Lieutenant?\"\n\n\"No, Captain. I mean, yes, Captain. Rather quickly, sir, I think.\"\n\n\"You should know, Lieutenant Irving. You've had frostbite six times already, and it's not even officially winter yet.\"\n\nLieutenant Irving nods dolefully.\n\n\"It takes _less than a minute_ for an exposed finger or thumb \u2014 or any fleshy appendage \u2014 to freeze solid,\" continues Crozier, who knows that this is a load of horse cobblers. It takes much longer than that at a mere fifty below, but he hopes that Irving doesn't know this. \"After that, the exposed member will snap off like an icicle,\" adds Crozier.\n\n\"Yes, Captain.\"\n\n\"So do you _really_ think there's any chance that our visitor might be... _compromised_... out here on deck, Mr. Irving?\"\n\nIrving seems to be thinking about this before replying. It's possible, Crozier realizes, that the third lieutenant has put far too much thought into this equation already.\n\n\"Go below, John,\" says Crozier. \"And see Dr. McDonald about your face and fingers. I swear to God that if you've gotten seriously frostbitten again, I'll dock you a month's Discovery Service pay and write your mother to boot.\"\n\n\"Yes, Captain. Thank you, sir.\" Irving starts to salute again, thinks better of it, and ducks under the canvas toward the main ladderway with one hand still half raised. He does not look back at Silence.\n\nCrozier sighs again. He likes John Irving. The lad had volunteered \u2014 along with two of his mates from the HMS _Excellent,_ Second Lieutenant Hodgson and First Mate Hornby \u2014 but the _Excellent_ was a damned three-decker that was old before Noah had fuzz around his dongle. The ship had been mastless and permanently moored in Portsmouth, Crozier knew, for more than fifteen years, serving as a training vessel for the Royal Navy's most promising gunners. _Unfortunately, gentlemen,_ Crozier had told the boys during their first day aboard \u2014 the captain had been more than usually drunk that day \u2014 _if you look around, you'll notice that while_ Terror _and_ Erebus _were both built as bombardment ships, gentlemen, neither has a single gun between them. We are, young volunteers from_ Excellent \u2014 _unless one counts the Marines' muskets and the shotguns secured in the Spirit Room_ \u2014 _as gunless as a newborn babe. As gunless as fucking Adam in his fucking birthday suit. In other words, gentlemen, you gunnery experts are about as useful to this expedition as teats would be on a boar._\n\nCrozier's sarcasm that day hadn't dampened the young gunnery officers' enthusiasm \u2014 Irving and the other two remained more eager than ever to go get frozen in the ice for several winters. Of course, that had been on a warm May day in England in 1845.\n\n\"And now the poor young pup is in love with an Esquimaux witch,\" Crozier mutters aloud.\n\nAs if understanding his words, Silence turns slowly toward him.\n\nUsually her face is invisible down the deep tunnel of her hood, or her features are masked by the wide ruff of wolf hair, but tonight Crozier can see her tiny nose, large eyes, and full mouth. The pulse of the aurora is reflected in those black eyes.\n\nShe's not attractive to Captain Francis Rawdon Moira Crozier; she has too much of the savage about her to be seen as fully human, much less as physically attractive \u2014 even to a Presbyterian Irishman \u2014 and besides that, his mind and lower regions are still filled with clear memories of Sophia Cracroft. But Crozier can see why Irving, far from home and family and any sweetheart of his own, might fall in love with this heathen woman. Her strangeness alone \u2014 and perhaps even the grim circumstances of her arrival and the death of her male companion, so strangely intertwined with the first attacks from that monstrous entity out there in the dark \u2014 must be like a flame to the fluttering moth of so hopeless a young romantic as Third Lieutenant John Irving.\n\nCrozier, on the other hand, as he discovered both in Van Diemen's Land in 1840 and again for the final time in England in the months before this expedition sailed, is too old for romance. And too Irish. And too common.\n\nRight now he just wishes this young woman would take a walk out onto the dark ice and not return.\n\nCrozier remembers the day four months earlier when Dr. McDonald had reported to Franklin and him after examining her, on the same afternoon the Esquimaux man with her had died choking in his own blood. McDonald said, in his medical opinion, the Esquimaux girl appeared to be between fifteen and twenty years old \u2014 it was so hard to tell with native peoples \u2014 had experienced menarche, but was, by all indications, _virgo intacta._ Also, Dr. McDonald reported, the reason that the girl had not spoken or made a sound \u2014 even after her father or husband had been shot and lay dying \u2014 was because she had no tongue. In Dr. McDonald's opinion, her tongue had not been sliced off but had been chewed off near its root, either by Silence herself or by someone or something else.\n\nCrozier had been astonished \u2014 not so much by the fact of the missing tongue, but from hearing that the Esquimaux wench was a virgin. He'd spent enough time in the northern arctic \u2014 especially during Parry's expedition, which wintered near an Esquimaux village \u2014 to know that the local natives took sexual intercourse so lightly that men would offer their wives and daughters to whalers or Discovery Service explorers in exchange for the cheapest trinket. Sometimes, Crozier knew, the women just offered themselves up for the fun of it, giggling and chatting with other women or children even as the sailors strained and puffed and moaned between the laughing women's legs. They were like animals. The furs and hairy hides they wore might as well be their own beastlike skins as far as Francis Crozier was concerned.\n\nThe captain raises his gloved hand to the bill of his cap, secured under two wraps of heavy comforter and therefore impossible to doff or tip, and says, \"My compliments to you, madam, and I would suggest you consider going below to your quarters soon. It's getting a bit nippy out here.\"\n\nSilence stares at him. She does not blink, although somehow her long lashes are free of ice. She does not, of course, speak. She watches him.\n\nCrozier symbolically tips his hat again and continues his tour around the deck, climbing to the ice-raised stern and then down the starboard side, pausing to speak to the other two men on watch, giving Irving time to get below and out of his cold-weather slops so that the captain doesn't seem to be following hard on his lieutenant's heels.\n\nHe's finishing his chat with the last shivering lookout, Able Seaman Shanks, when Private Wilkes, the youngest of the Marines aboard, comes rushing out from under the canvas. Wilkes has thrown on only two loose layers over his uniform, and his teeth begin chattering even before he delivers his message.\n\n\"Mr. Thompson's compliments to the captain, sir, and the engineer says that the captain should come down to the hold as quick as you might.\"\n\n\"Why?\" If the boiler has finally broken down, Crozier knows, they are all dead.\n\n\"Begging the captain's pardon, sir, but Mr. Thompson says that the captain is needed because Seaman Manson is near to mutiny, sir.\"\n\nCrozier stands up straight. \"Mutiny?\"\n\n\"'Near to it' were Mr. Thompson's words, sir.\"\n\n\"Speak English, Private Wilkes.\"\n\n\"Manson won't carry no more sacks of coal past the Dead Room, sir. Nor go down in the hold no more. He says he respectfully refuses, Captain. He won't come up, but he's sitting on his arse at the bottom of the man-ladder and won't carry no more coal back to the boiler room.\"\n\n\"What is this nonsense?\" Crozier feels the first stirrings of a familiar dark Irish anger.\n\n\"It's the ghosts, Captain,\" says Marine Private Wilkes through chattering teeth. \"We all hear 'em when we're hauling coal or fetching something from deep stores. It's why the men won't go down there below orlop deck no more unless the officers order 'em to, sir. Something's down there in the hold, in the dark. Something's been scratching and banging from _inside_ the ship, Captain. It ain't just the ice. Manson's sure it's his old mate Walker, him... it... and the other corpses stacked there in the Dead Room, clawing to get out.\"\n\nCrozier checks his impulse to reassure the Marine private with facts. Young Wilkes might not find the facts so reassuring.\n\nThe first simple fact is that the scrabbling noise from the Dead Room is almost certainly the hundreds or thousands of large black rats feasting on Wilkes's frozen comrades. The Norway rats \u2014 as Crozier knows better than the young Marine \u2014 are nocturnal, which means that they're active day and night during the long arctic winter, and the creatures have teeth which constantly keep growing. This, in turn, means the God-damned vermin have to keep chewing. He has seen them chew through Royal Navy oak barrels, inch-thick tins, and even lead plating. The rats are having no more trouble down there with the frozen remains of Seaman Walker and his five unlucky comrades \u2014 including three of Crozier's finest officers \u2014 than a man would have chewing on a strip of frigid salted beef.\n\nBut Crozier doesn't think it's only the rats that Manson and the others are hearing.\n\nRats, as Crozier knows from the sad experience of thirteen winters in the ice, tend to eat one's friends quietly and efficiently, except for their frequent screeching as the blood-maddened and ravenous vermin turn on one another.\n\nIt's something else making the clawing and banging noises down on hold deck.\n\nWhat Crozier decides not to remind Private Wilkes of is the second simple fact: while the lowest deck would normally be cold but safe there beneath the waterline or winter line of frozen sea ice, the pressure from the ice has forced _Terror_ 's stern more than a dozen feet higher than it should be. The hull there is still locked in, but only by several hundred heaped tons of jagged sea ice and the added tons of snow the men have piled alongside to within a few feet of the railings so as to provide more insulation during the winter.\n\nSomething, Francis Crozier suspects, has dug down through these tons of snow and tunneled through the iron-hard slabs of ice to get at the hull of the ship. Somehow the thing has sensed which parts of the interior along the hull, such as the water-storage tanks, are lined with iron, and has found one of the few hollow outside storage areas \u2014 the Dead Room \u2014 that leads directly into the ship. And now it's banging and clawing to get in.\n\nCrozier knows that there's only one thing on earth with that much power, deadly persistence, and malevolent intelligence. The monster on the ice is trying to get at them from below.\n\nWithout saying another word to Marine Private Wilkes, Captain Crozier goes below to sort things out.\n\nFRANKLIN\n\n _Lat. 51\u00b0-29\u2032 N., Long. 0\u00b0-0\u2032 W._\n\nLondon, May, 1845\n\nHe was \u2014 and always would be \u2014 the man who ate his shoes.\n\nFour days before they were to sail, Captain Sir John Franklin contracted the influenza that had been going around, getting it, he was sure, not from one of the common sailors and stevedores loading the ships at London's docks, nor from any of his one hundred and thirty-four crew members and officers \u2014 they were all healthy as dray horses \u2014 but from some sickly sycophant in one of Lady Jane's circles of society friends.\n\nThe man who ate his shoes.\n\nIt was traditional for the wives of arctic heroes to sew a flag to be planted at some point farthest north, or in this case raised upon the completion of the expedition's transit of the North-West Passage, and Franklin's wife, Jane, was finishing her sewing of the silken Union Jack when he came home. Sir John came into the parlour and half collapsed onto the horsehair sofa near where she sat. Later he did not remember removing his boots, but someone must have \u2014 either Jane or one of the servants \u2014 for soon he was lying back and half dozing, his head aching, his stomach more unsteady than it had ever been at sea, and his skin burning with fever. Lady Jane was telling him about her busy day, never pausing in her recital. Sir John tried to listen as the fever carried him off on its uncertain tide.\n\nHe was the man who ate his shoes, and had been for twenty-three years, ever since he returned to England in 1822 after his first, failed overland expedition across northern Canada to find the North-West Passage. He remembered the sniggers and jokes upon his return. Franklin had eaten his shoes \u2014 and he'd eaten worse on that botched three-year journey, including _tripe-de-roche,_ a disgusting gruel made from lichen scraped from rocks. Two years out and starving, he and his men \u2014 Franklin had dazedly divided his troop into three groups and left the other two bands to survive or die on their own \u2014 had boiled the uppers on their boots and shoes to survive. Sir John \u2014 he was just John then, he was knighted for incompetency after a later overland voyage and botched polar expedition by sea \u2014 had spent days in 1821 chewing on nothing more than scraps of untanned leather. His men had eaten their buffalo sleeping robes. Then some of them had moved on to other things.\n\nBut he had never eaten another man.\n\nTo this day, Franklin doubted whether others on his expedition, including his good friend and chief lieutenant Dr. John Richardson, had succeeded in resisting that temptation. Too much had happened while the parties were separated as they stumbled through the arctic wastes and forests, desperately trying to get back to Franklin's little improvised Fort Enterprise and the real forts, Providence and Resolution.\n\nNine white men and one Esquimaux dead. Nine dead out of the twenty-one men young Lieutenant John Franklin, thirty-three years old and pudgy and balding even then, had led out of Fort Resolution in 1819, plus one of the native guides they'd picked up along the way \u2014 Franklin had refused to let the man leave the expedition to forage for himself. Two of the men had been murdered in cold blood. At least one of them was, without doubt, devoured by others. But only one Englishman died. Only one real white man. All the rest were mere French _voyageurs_ or Indians. This was success of a sort \u2014 only one white Englishman dead, even if all the others had been reduced to gibbering, bearded skeletons. Even if all the others survived only because George Back, that confounded, oversexed midshipman, had snowshoed 1,200 miles to bring back supplies and \u2014 more important than supplies \u2014 more Indians to feed and care for Franklin and his dying party.\n\nThat confounded Back. Not a good Christian at all. Arrogant. Not a true gentleman, despite his later being knighted for an arctic expedition sailing on this very same HMS _Terror_ that Sir John now commanded.\n\nOn that expedition, Back's expedition, _Terror_ had been flung fifty feet into the air by a rising tower of ice, then thrown down so violently that every oak plank in the hull sprang a leak. George Back brought the leaking boat all the way back to the coast of Ireland, beaching it just hours before it would have sunk. The crew had wrapped chains around it to squeeze the boards tight long enough for the vessel to get them home. All the men had scurvy \u2014 black gums, bleeding eyes, teeth falling out of their heads \u2014 and the madness and delusions that went with scurvy.\n\nThey'd knighted Back after that, of course. It's what England and the Admiralty did after you returned from a polar expedition that failed miserably, resulting in appalling loss of life; if you survived, they gave you a title and a parade. After Franklin returned from his second coastal-mapping expedition in the far north of North America in 1827, he was personally knighted by King George IV. The Geographical Society of Paris gave him a gold medal. He was awarded captaincy of the beautiful little 26-gun frigate HMS _Rainbow_ and ordered to the Mediterranean, a destination to which every captain in the Royal Navy prayed nightly for posting. He proposed marriage to one of his dead wife Eleanor's dearest friends, the energetic, beautiful, and outspoken Jane Griffin.\n\n\"So I explained to Sir James over tea,\" Jane was saying, \"that my darling Sir John's credit and reputation are infinitely dearer to me than any selfish enjoyment of my husband's society, even if he must be gone for four years... or five.\"\n\nWhat was the name of that fifteen-year-old Copper Indian girl that Back was going to fight a duel over at their winter quarters of Fort Enterprise?\n\nGreenstockings. That was it. Greenstockings.\n\nThat girl was evil. Beautiful, yes, but evil. She had no shame. Franklin himself, despite all efforts never to look her way, had seen her slip out of her heathen robes and walk naked across half the length of the cabin one moonlit night.\n\nHe was thirty-four years old at the time, but she was the first naked human female he had ever seen and even now the most beautiful. The dark skin. The breasts already heavy as globed fruit but also still those of an adolescent, the nipples not yet raised, the areolae strange, smooth dark-brown circles. It was an image Sir John had not been able to eradicate from his memory \u2014 try and pray as he might \u2014 in the quarter of a century since then. The girl had not the classic V of pubic hair that Franklin had later seen on his first wife, Eleanor \u2014 glimpsed only once, as she prepared for her bath, since Eleanor never allowed the slightest light to illuminate their rare lovemaking \u2014 or the sparser but wilder wheat-coloured nest that was part of the aging body of his current wife, Jane. No, the Indian girl Greenstockings had only a narrow but pure-black vertical escutcheon above her female parts. As delicate as a raven's feather. As pitch black as sin itself.\n\nThe Scotsman midshipman, Robert Hood, who had already fathered one bastard with a different Indian woman during that endless first winter at the cabin Franklin had named Fort Enterprise, had promptly fallen in love with teenaged Copper squaw Greenstockings. The girl had previously been lying with the other midship-man, George Back, but with Back gone hunting, she'd shifted her sexual allegiance to Hood with the ease known only to pagans and primitives.\n\nFranklin could still remember the grunts of passion in the long night \u2014 not the passion of a few minutes, as he had experienced with Eleanor (never grunting or making a noise, of course, since no gentleman would do that), or even two brief bouts of passion, such as on that memorable night on his honeymoon with Jane; no, Hood and Greenstockings went at it half a dozen times. No sooner would Hood's and the girl's noises stop in the adjoining lean-to than they would begin again \u2014 laughter, low giggles, then soft moans, leading again to the louder cries as the brazen girl-woman urged Hood on.\n\nJane Griffin was thirty-six years old when she married the newly knighted Sir John Franklin on December 5, 1828. They honeymooned in Paris. Franklin did not especially like the city, nor did he like the French, but their hotel had been luxurious and the food very good.\n\nFranklin had been in a sort of dread that during their travels on the continent they might run into that Roget fellow \u2014 Peter Mark, the one who gained some sort of literary attention by preparing to publish that silly dictionary or whatever it was \u2014 the same man who had once asked for Jane Griffin's hand, only to be rejected like all the other suitors had been in her younger years. Franklin had since peeked into Jane's diaries from that era \u2014 he rationalized his crime by thinking that she wanted him to find and read the many calfskin-bound volumes, why otherwise would she have left them in such an obvious place? \u2014 and saw, in his beloved's tight, perfect hand, the passage she had written on the day Roget had finally married someone else \u2014 \" _the romance of my life is gone._ \"\n\nRobert Hood had been making noises with Greenstockings for six endless arctic nights when his fellow midshipman George Back returned from a hunting party with the Indians. The two men arranged a duel to the death at sunrise \u2014 around 10:00 a.m. \u2014 the next morning.\n\nFranklin had not known what to do. The corpulent lieutenant was unable to exert any discipline over the surly _voyageurs_ or the contemptuous Indians, much less able to control the headstrong Hood or the impulsive Back.\n\nBoth midshipmen were artists and mapmakers. From that time on, Franklin had never trusted an artist. When the sculptor in Paris did Lady Jane's hands and the perfumed sodomite here in London had come for almost a month to paint her official oil portrait, Franklin had never left the men alone with her.\n\nBack and Hood were meeting at dawn for a duel to the death and there was nothing John Franklin could do but hide in the cabin and pray that the resulting death or injury would not destroy the last vestige of sanity in his already compromised expedition. His orders had not specified that he should bring _food_ on the 1,200-mile arctic overland, coastal sea, and river trek. Out of his own pocket, he'd provided enough supplies to feed the sixteen men for one day. Franklin had assumed the Indians would then hunt for them and feed them adequately, just as the guides carried his bags and paddled his birch-bark canoe.\n\nThe birch-bark canoes had been a mistake. Twenty-three years after the fact, he was willing to concede that \u2014 to himself, at least. After just a few days in the ice-clogged waters along the northern coast, reached more than a year and a half after their departure from Fort Resolution, the flimsy vessels had started to come apart.\n\nFranklin, his eyes closed, his brow burning, his head throbbing, half-listening to the uninterrupted stream of Jane's chatter, remembered the morning when he'd lain in his heavy sleeping bag and squeezed his eyes shut as Back and Hood had stepped off their fifteen paces outside the cabin, then turned to fire. The confounded Indians and confounded _voyageurs_ \u2014 equally savage in many ways \u2014 were treating the duel to the death as entertainment. Greenstockings, Franklin remembered, was radiant that morning with an almost erotic glow.\n\nLying in his bag, his hands over his ears, Franklin still heard the call to pace, the call to turn, the call to aim, the command to fire.\n\nThen two clicks. Then laughter from the crowd.\n\nDuring the night, the old Scottish seaman calling the pacing, that tough and ungentlemanly John Hepburn, had unloaded charge and balls from the carefully prepared pistols.\n\nDeflated by the unceasing laughter of the mob of _voyageurs_ and knee-slapping Indians, Hood and Back had stalked off in opposite directions. Shortly after that, Franklin ordered George Back to return to the forts to purchase more provisions from the Hudson's Bay Company. Back was gone most of the winter.\n\nFranklin had eaten his shoes and had subsisted on lichen scraped from rocks \u2014 a slime meal that would make a self-respecting English dog vomit \u2014 but he had never partaken of human flesh.\n\nA long year after the forestalled duel, in Richardson's party after Franklin's group had separated from it, that surly, half-mad Iroquois on the expedition, Michel Teroahaute, shot the midshipman artist and mapmaker Robert Hood in the centre of his forehead.\n\nA week before the murder, the Indian had brought back a strong-tasting haunch of meat to the starving party, insisting that it had come from a wolf that had either been gored to death by a caribou or killed by Teroahaute himself using a deer horn \u2014 the Indian's story kept changing. The ravenous party had cooked and eaten the meat, but not before Dr. Richardson noticed a slight hint of a tattoo on the skin. The doctor later told Franklin that he was certain that Teroahaute had doubled back to the body of one of the _voyageurs_ who had died that week on the trek.\n\nThe starving Indian and the dying Hood were alone when Richardson, off scraping lichen from the rocks, had heard the shot. _Suicide,_ Teroahaute had insisted, but Dr. Richardson, who had attended on more than a few suicides, knew that the position of the ball in Robert Hood's brain had not come from a self-inflicted gunshot.\n\nNow the Indian armed himself with a British bayonet, a musket, two fully charged and half-cocked pistols, and a knife as long as his forearm. The two non-Indians remaining \u2014 Hepburn and Richardson \u2014 had only a small pistol and one untrustworthy musket between them.\n\nRichardson, now one of the most respected scientists and surgeons in England, friend of the poet Robert Burns, but then only a promising expedition surgeon and naturalist, waited until Michel Teroahaute returned from a foraging trip, made sure his arms were full of firewood, and then lifted his pistol and cold-bloodedly shot the Indian through the head.\n\nDr. Richardson later admitted to eating the dead Hood's buffalo robe, but neither Hepburn nor Richardson \u2014 the only survivors of their party \u2014 ever mentioned what else they might have eaten in the next week of arduous trekking during their return to Fort Enterprise.\n\nAt Fort Enterprise, Franklin and his party were too weak to stand or walk. Richardson and Hepburn seemed strong in comparison.\n\nHe might be the man who had eaten his shoes, but John Franklin had never...\n\n\"Cook is preparing roast beef tonight, my darling. Your favorite. Since she's new \u2014 I am certain that the Irish woman was padding our accounts, stealing is as natural as drinking to the Irish \u2014 I reminded her that you insist that it must be rare enough to bleed at the touch of the carving knife.\"\n\nFranklin, floating on an ebbing tide of fever, tried to formulate words in response, but the surges of headache, nausea, and heat were too great. He was sweating through his undershirt and still-fixed collar.\n\n\"Admiral Sir Thomas Martin's wife sent us a delightful card today and a wonderful bouquet of flowers. She's the last to be heard from, but I must say the roses are beautiful in the foyer. Did you see them? Did you have much time to chat with Admiral Martin at the reception? Of course, he is not that important, is he? Even as Controller of the Navy? Certainly not as distinguished as the First Lord or First Commissioners, much less your Arctic Council friends.\"\n\nCaptain Sir John Franklin had many friends; everyone liked Captain Sir John Franklin. But no one respected him. For decades, Franklin acknowledged the former fact and avoided the latter, but he now knew it to be true. Everyone liked him. No one respected him.\n\nNot after Van Diemen's Land. Not after the Tasmanian prison and the botch he had made of that.\n\nEleanor, his first wife, had been dying when he left her to go on his second major expedition.\n\nHe knew she was dying. She knew she was dying. Her consumption \u2014 and the knowledge she would die from it long before her husband would die in battle or on expedition \u2014 had been with them like a third party at their wedding ceremony. In the twenty-two months of their marriage, she had given him a daughter, his only child, young Eleanor.\n\nA small, frail woman in body \u2014 but almost frightening in spirit and energy \u2014 his first wife had told him to go on his second expedition to find the North-West Passage, this trip by land and sea to follow the North American coastline, even while she was coughing up blood and knowing that the end was near. She said that it would be better for her if he were elsewhere. He believed her. Or at least he believed that it would be better for himself.\n\nA deeply religious man, John Franklin had prayed that Eleanor would die before his departure date. She hadn't. He left on February 16, 1825, wrote his darling many letters while in transit to Great Slave Lake, posted them in New York City and Albany, and learned of her passing on April 24, at the British Naval station at Penetanguishene. She had died shortly after his ship left England.\n\nWhen he returned from this expedition in 1827, Eleanor's friend Jane Griffin was waiting for him.\n\nThe Admiralty reception had been less than a week ago \u2014 no, just precisely a week ago, before this confounded influenza. Captain Sir John Franklin and all his officers and mates from _Erebus_ and _Terror_ had attended, of course. So had the civilians on the expedition \u2014 _Erebus_ 's ice master, James Reid, and _Terror_ 's ice master, Thomas Blanky, along with the paymasters, surgeons, and pursers.\n\nSir John had looked dashing in his new blue swallow-tailed coat, blue gold-striped trousers, gold-fringed epaulettes, ceremonial sword, and Nelson-era cocked hat. The commander of his flagship _Erebus,_ James Fitzjames, often called the handsomest man in the Royal Navy, looked as striking and humble as the war hero he was. Fitzjames had charmed everyone that night. Francis Crozier, as always, had looked stiff, awkward, melancholic, and slightly inebriated.\n\nBut Jane was wrong \u2014 the members of the \"Arctic Council\" were not Sir John's friends. The Arctic Council, in reality, did not exist. It was an honorary society rather than a real institution, but it was also the most select Old Boys club in all of England.\n\nThey'd mingled at the reception, Franklin, his top officers, and the tall, gaunt, grey members of the legendary Arctic Council.\n\nTo gain membership to the Council, all one had to do was command an expedition to the farthest arctic north... and survive.\n\nViscount Melville \u2014 the first notable in the long receiving line that had left Franklin uncharacteristically sweating and tongue-tied \u2014 was First Lord of the Admiralty and the sponsor of their sponsor, Sir John Barrow. But Melville was not an old arctic hand.\n\nThe true Arctic Council legends \u2014 most in their seventies \u2014 were, to the nervous Franklin that night, more like the coven of witches in _Macbeth_ or like some cluster of grey ghosts than like living men. Every one of these men had preceded Franklin in searching for the Passage, and all had returned alive, yet not fully alive.\n\nDid anyone, Franklin wondered that evening, _really_ return alive after wintering in the arctic regions?\n\nSir John Ross, his Scotsman's face showing more sharp facets than an iceberg, had eyebrows leaping out like the ruffs and feathers of those penguins his nephew Sir James Clark Ross had described after his trip to the south arctic. Ross's voice was as rough as a holystone dragged across a splintered deck.\n\nSir John Barrow, older than God and twice as powerful. The father of serious British arctic exploration. All others there that night, even the white-haired septuagenarians, were boys... Barrow's boys.\n\nSir William Parry, a gentleman above gentlemen even when among royalty, who had tried four times to force the Passage only to watch men die and his _Fury_ squeezed and smashed and sunk.\n\nSir James Clark Ross, newly knighted, was also newly wed to a wife who made him swear off any more expeditions. He would have had Franklin's job of commander of this expedition if he'd wanted it, and both men knew it. Ross and Crozier stood slightly separate from the others, drinking and talking as softly as conspirators.\n\nThat confounded Sir George Back; Franklin hated sharing sirdom with a mere midshipman who once served under him, and a womanizer at that. On this gala night, Captain Sir John Franklin almost wished that Hepburn hadn't taken the powder and shot out of the dueling pistols twenty-five years earlier. Back was the youngest member of the Arctic Council and seemed happier and smugger than any of the others, even after suffering the battering and near-sinking of HMS _Terror._\n\nCaptain Sir John Franklin was a teetotaler, but after three hours of champagne, wine, brandy, sherry, and whiskey, the other men began to relax, the laughter around him grew stronger and the conversation in the grand hall less formal, and Franklin began to feel calmer, realizing that all this reception, all the gold buttons, silk cravats, gleaming epaulettes, fine food, cigars, and smiles were for _him._ This time, it was all about _him._\n\nSo it was a shock when the older Ross pulled him aside almost abruptly and began to bark questions at him through the cigar smoke and the glint of candlelight off crystal.\n\n\"Franklin, why in hell's name are you taking one hundred and thirty-four men?\" rasped the holystone across rough wood.\n\nCaptain Sir John Franklin blinked. \"It's a major expedition, Sir John.\"\n\n\"Too bloody major, if you ask me. It's hard enough to get thirty men across the ice, into boats, and back to civilization when something goes wrong. A hundred thirty-four men...\" The old explorer made a rude noise, clearing his throat as if he was going to spit.\n\nFranklin smiled and nodded, wishing the old man would leave him alone.\n\n\"And your age,\" continued Ross. \"You're sixty, for God's sake.\"\n\n\"Fifty-nine,\" Franklin said stiffly. \"Sir.\"\n\nThe elder Ross smiled thinly but looked more like an iceberg than ever. \" _Terror_ is what? Three hundred thirty tons? _Erebus_ something like three hundred seventy?\"\n\n\"Three hundred seventy-two for my flagship,\" said Franklin. \"Three hundred twenty-six for _Terror._ \"\n\n\"And a draft of nineteen feet each, isn't that right?\"\n\n\"Yes, m'lord.\"\n\n\"That's buggering insane, Franklin. Your ships will be the deepest draft vessels ever sent on an arctic expedition. Everything we know about those regions has shown us that the waters where you're headed are shallow, filled with shoals, rocks, and hidden ice. My _Victory_ only drew a fathom and a half and we couldn't get over the bar of the harbour where we wintered. George Back all but ripped his bottom out on the ice with your _Terror._ \"\n\n\"Both ships have been strengthened, Sir John,\" said Franklin. He could feel sweat running down his ribs and chest onto his portly belly. \"They're now the strongest ice ships in the world.\"\n\n\"And what is all the nonsense about steam and locomotive engines?\"\n\n\"Not nonsense, m'lord,\" said Franklin and could hear the condescension in his own voice. He knew nothing about steam himself, but he had two good engineers on the expedition and Fitzjames, who was part of the new Steam Navy. \"These are powerful engines, Sir John. They'll see us through the ice where sail has failed.\"\n\nSir John Ross snorted. \"Your steam machines aren't even maritime engines, are they, Franklin?\"\n\n\"No, Sir John. But they're the best steam engines the London and Greenwich Railway could sell us. Converted for marine use. Powerful beasts, sir.\"\n\nRoss sipped his whiskey. \"Powerful if you're planning to lay down rails along the North-West Passage and take a God-damned locomotive across it.\"\n\nFranklin chuckled good-naturedly at this, but he saw no humor in the comment and the obscenity offended him deeply. He often could not tell when others were being humorous, and he had no sense of humor himself.\n\n\"But not really so powerful,\" continued Ross. \"That one-point-five-ton machine they crammed into the hold of your _Erebus_ only produces twenty-five horsepower. Crozier's engine is less efficient... twenty horsepower, maximum. The ship that's towing you beyond Scotland \u2014 _Rattler_ \u2014 produces two hundred twenty horsepower with its smaller steam engine. It's a _marine_ engine, built for sea.\"\n\nFranklin had nothing to say to that, so he smiled. To fill the silence he signaled a passing waiter carrying glasses of champagne. Then, since it was against all his principles to drink alcohol, all he could do was stand there holding the glass, occasionally glancing at the flattening champagne, and wait for some opportunity to get rid of it without being noticed.\n\n\"Think of all the extra provisions you could have crammed in the holds of your two ships if those damned engines weren't there,\" persisted Ross.\n\nFranklin looked around as if seeking rescue, but everyone was in animated conversation with someone else. \"We have more than adequate stores for three years, Sir John,\" he said at last. \"Five to seven years if we have to go on short rations.\" He smiled again, trying to charm that flinty face. \"And both _Erebus_ and _Terror_ have central heating, Sir John. Something I'm sure you would have appreciated on your _Victory._ \"\n\nSir John Ross's pale eyes gleamed coldly. \" _Victory_ was crushed like an egg by the ice, Franklin. Fancy steam heat wouldn't have helped that, would it?\"\n\nFranklin looked around, trying to catch Fitzjames's eye. Even Crozier's. Anyone to come to his rescue. No one seemed to notice the old Sir John and the fat Sir John huddled here in such earnest, if one-sided, conversation. A waiter passed, and Franklin set his untouched glass of champagne on his tray. Ross studied Franklin through slitted eyes.\n\n\"And how much coal does it take just to heat one of your ships for a day up there?\" pressed the old Scotsman.\n\n\"Oh, I don't really know, Sir John,\" said Franklin with a winning smile. He really did _not_ know. Nor especially care. The engineers were in charge of the steam engines and coal. The Admiralty would have planned well for them.\n\n\" _I_ know,\" said Ross. \"You'll use up to one hundred fifty pounds of coal a day just to keep the hot water moving to heat the crew's quarters. Half a _ton_ of your precious coal a day just to keep steam up. If you're under way \u2014 expect about four knots out of those ugly bombardment ships \u2014 you'll be burning two to three _tons_ of coal a day. Much more if you're trying to force your way through pack ice. How much coal are you _carrying,_ Franklin?\"\n\nCaptain Sir John waved his hand in what he realized was a dismissive \u2014 and almost effeminate \u2014 gesture. \"Oh, somewhere around two hundred tons, m'lord.\"\n\nRoss squinted again. \"Ninety tons each for _Erebus_ and _Terror,_ to be precise,\" he rasped. \"And that's when you're topped off in Greenland, before you cross Baffin Bay, much less get into the real ice.\"\n\nFranklin smiled and said nothing.\n\n\"Say you arrive at where you winter in the ice with seventy-five percent of your ninety tons unburned,\" continued Ross, boring ahead like a ship through soft ice, \"that leaves you what... how many days' steam under normal conditions, not ice conditions? A dozen days? Thirteen days? A fortnight?\"\n\nCaptain Sir John Franklin had not the slightest idea. His mind, although professional and nautical, simply did not work that way. Perhaps his eyes revealed his sudden panic \u2014 not over coal but over appearing an idiot in front of Sir John Ross \u2014 for the old mariner clamped a steel vise grip on Franklin's shoulder. When Ross leaned closer, Captain Sir John Franklin could smell the whiskey on his breath.\n\n\"What are the Admiralty's plans for your rescue, Franklin?\" rasped Ross. His voice was low. All about them was the laughter and chatter of the reception in its late hour.\n\n\"Rescue?\" Franklin said, blinking. The idea that the two most modern ships in the world \u2014 reinforced for ice, powered by steam, provisioned for five years or more in the ice, and manned by crews handpicked by Sir John Barrow \u2014 would or could require rescue simply did not register in Franklin's brain. The idea was absurd.\n\n\"Do you have plans to cache depots along your way in through the islands?\" whispered Ross.\n\n\"Caches?\" said Franklin. \"Leave our provisions along the way? Why on earth would I do that?\"\n\n\"So you can get your men and boats to food and shelter if you have to take to the ice and walk out,\" Ross said fiercely, eyes gleaming.\n\n\"Why would we walk back toward Baffin Bay?\" asked Franklin. \"Our objective is to complete the transit of the North-West Passage.\"\n\nSir John Ross had pulled his head back. His grip tightened on Franklin's upper arm. \"Then there's no rescue ship or plan in place?\"\n\n\"No.\"\n\nRoss grabbed Franklin's other arm and squeezed so tightly that the portly Captain Sir John almost winced.\n\n\"Then, laddie,\" whispered Ross, \"if we've not heard from ye by 1848, I'll come looking for you myself. I swear it.\"\n\nFranklin slammed awake.\n\nHe was soaked with sweat. He felt dizzy and weak. His heart was pounding, and with each reverberation his headache tolled like a church bell against the inside of his skull.\n\nHe looked down at himself in horror. Silk covered the lower half of his body.\n\n\"What is this?\" he cried in alarm. \"What is this? There's a flag thrown over me!\"\n\nLady Jane stood, aghast. \"You looked cold, John. You were shivering. I put it over you as a blanket.\"\n\n\"My God!\" cried Captain Sir John Franklin. \"My God, woman, do you know what you've done? Don't you know they lay the Union Jack over a corpse!\"\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\nOctober, 1847\n\nCaptain Crozier descends the short ladder to the lower deck, pushes through the sealed double doors, and almost staggers in the sudden blast of warmth. Even though the circulating hot-water heat has been off for hours, body heat from more than fifty men and residual warmth from cooking have kept the temperature here on the lower deck high \u2014 just below freezing \u2014 almost 80 degrees warmer than outside. The effect on someone who's been out on deck for half an hour is the equivalent of walking into a sauna fully clothed.\n\nSince he's continuing down to the unheated orlop and hold decks and thus keeping his cold-weather slops on, Crozier doesn't tarry long here in the heat. But he does pause for a moment \u2014 as any captain would \u2014 taking the time to glance around and make sure that everything hasn't gone to hell in the half hour he's been away.\n\nDespite the fact that this is the only berthing, eating, and living deck on the ship, it's still as dark as a working Welsh mine with its small skylights snowed over in the daytime and the night now twenty-two hours long. Whale-oil lamps, lanterns, or candles throw small cones of illumination here and there, but mostly the men make their way through the gloom by memory, remembering where to dodge the innumerable half-seen heaps and hanging masses of stored food, clothing, gear, and other men sleeping in their hammocks. When all the hammocks go up \u2014 fourteen inches allowed per man \u2014 there will be no room to walk at all except for two 18-inch-wide aisles along the hull on either side. But only a few hammocks are up now \u2014 men catching some sleep before late watches \u2014 and the din of conversation, laughter, cursing, coughing, and Mr. Diggle's inspired clankings and obscenities is loud enough to drown out some of the press and moan of the ice.\n\nThe ship's diagrams show seven feet of clearance, but in reality, between the heavy ship's timbers overhead and the tons of lumber and extra wood stored on racks hanging from those timbers, there's less than six feet of headroom on this lower deck and the few truly tall men on _Terror,_ like the coward Manson waiting below, have to walk in a perpetual hunched-over posture. Francis Crozier is not that tall. Even with his cap and comforter on, he doesn't have to duck his head as he turns.\n\nTo his right and running aft from where Crozier stands is what looks to be a low, dark, narrow tunnel, but it is actually the companionway leading to the \"officers' quarters,\" a warren of sixteen tiny sleeping cubicles and two cramped mess quarters for the officers and warrant officers. Crozier's cabin is the same size as the others' \u2014 six feet by five feet. The companionway is dark and barely two feet wide. Only one man can pass at a time, ducking his head to avoid hanging stores, and heavy men have to turn sideways to shuffle down the narrow passage.\n\nThe officers' quarters are crammed into 60 feet of the 96-foot length of the ship, and since _Terror_ is only 28 feet wide here on the lower deck, the narrow companionway is the only straight-line access aft.\n\nCrozier can see light from the Great Cabin at the stern, where \u2014 even in this Stygian cold and gloom \u2014 some of his surviving officers are relaxing at the long table, smoking their pipes or reading from the 1,200-volume library shelved there. The captain hears music playing: one of the metal disks for the hand organ playing a tune that had been popular in London music halls five years ago. Crozier knows that it's Lieutenant Hodgson playing the tune; it's his favorite, and it drives Lieutenant Edward Little, Crozier's executive officer and a lover of classical music, absolutely mad with irritation.\n\nAll apparently being well in officers' country, Crozier turns and glances forward. The regular crew's quarters take up the remaining third of the length of the ship \u2014 36 feet \u2014 but into it are crammed 41 of the surviving able-bodied seamen and midshipmen from the original ship's muster of 44.\n\nThere are no classes being taught tonight and it's less than an hour until they will unfurl their hammocks and turn in, so the majority of the men are sitting on their sea chests or heaps of stowed material, smoking or talking in the dim light. The centre of the space is taken up by the gigantic Frazer's Patent Stove, where Mr. Diggle is baking biscuits. Diggle \u2014 the best cook in the fleet as far as Crozier is concerned and a prize, most literally, since Crozier had stolen the obstreperous cook right off Captain Sir John Franklin's flagship just before the expedition departed \u2014 is always cooking, usually biscuits, and curses and bangs and kicks and berates his assistants all the while. Men are literally scuttling near the giant stove, disappearing down the scuttle there to bring up stores from the lower decks, hurrying to avoid Mr. Diggle's voluble wrath.\n\nFrazer's Patent Stove itself appears, to Crozier's eye, almost as large as the locomotive engine in the hold. Besides its gigantic oven and six large burners, the bulking iron contraption has a built-in desalinator and a prodigious hand pump to bring water in from either the ocean or the rows of huge water-storage tanks down in the hold. But both the sea outside and the water in the hold now are frozen solid, so the huge pots bubbling on Mr. Diggle's burners are busy melting chunks of ice chipped out of the water tanks below and hauled up for that purpose.\n\nThe captain can see, beyond the partition of Mr. Diggle's shelves and cupboards where forward bulwarks had once stood, the sick bay in the forepeak of the ship. For two years there had been no sick bay. The area was stacked from deck to beams with more crates and casks and those crewmen who needed to see the ship's surgeon or assistant surgeon at 7:30 a.m. lubber's time did so near Mr. Diggle's stove. But now, with the amount of stores depleted and the number of sick and injured men multiplying, the carpenters had created a more permanent and separate section of the forepeak to serve as sick bay. Still, the captain could see the tunnellike entrance through the crates where they'd made a space for Lady Silence to sleep.\n\nThat discussion had taken the better part of a day last June \u2014 Franklin had insisted that the Esquimaux woman not be allowed on his ship. Crozier had accepted her, but his discussion with his executive officer, Lieutenant Little, as to where to berth her had been almost absurd. Even an Esquimaux wench, they knew, would freeze to death on deck or on the lower two decks, which left only the main lower deck. She certainly could not sleep in the crew's berthing area, even though they had empty hammocks by this time thanks to that thing out on the ice.\n\nIn Crozier's day as a teenaged lad before the mast and then as a midshipman, women smuggled aboard were put in the lightless, almost airless stinking hawser room in the lowest and most forward part of the ship, within reach of the fo'c'sle for the lucky man or men who smuggled her aboard. But even last June, when Silence appeared, it was below zero in HMS _Terror_ 's hawser room.\n\nNo, berthing her with the crew was not an idea to be considered.\n\nOfficers' country? Perhaps. There were empty cabins, with some of his officers dead and torn apart. But both Lieutenant Little and his captain had quickly agreed that the presence of a woman just a few thin partitions and sliding doors away from the sleeping men would be unhealthy.\n\nWhat then? They couldn't assign her a sleeping place and then post an armed guard over her all the time.\n\nIt was Edward Little who'd come up with the idea of shifting some stores to make a little cave of a sleeping area for the woman in the forepeak where the sick bay would have been. The one person awake all night, every night, was Mr. Diggle \u2014 dutifully baking his biscuits and frying his breakfast meats \u2014 and if Mr. Diggle had ever had an eye for the ladies, it was apparent that day had long passed. Also, reasoned Lieutenant Little and Captain Crozier, the proximity to the Frazer's Patent Stove would help keep their guest warm.\n\nIt had succeeded in that, all right. Lady Silence was made sick by the heat, forcing her to sleep stark naked on her furs in her little crate-and-cask cave. The captain discovered this by accident and the image stayed with him.\n\nNow Crozier takes a lantern from its hook, lights it, lifts the hatch, and goes down the ladder to the orlop deck before he starts to melt like one of those blocks of ice on the stove.\n\nTo say it's cold on the orlop deck would be the kind of understatement Crozier knows he used to make before he first voyaged to the arctic. A drop of six feet of ladder down from the lower deck has dropped the temperature at least sixty degrees. The darkness here is almost absolute.\n\nCrozier takes the usual captain's minute to look around. The circle of light from his lantern is weak, illuminating mostly the fog of his breath in the air. All around him is the labyrinth of crates, hogsheads, tins, kegs, casks, coal sacks, and canvas-covered heaps crammed deck-to-beams with the ship's remaining provisions. Even without the lantern, Crozier could find his way through the dark and rat-screech here; he knows every inch of his ship. At times, especially late at night with the ice moaning, Francis Rawdon Moira Crozier realizes that HMS _Terror_ is his wife, mother, bride, and whore. This intimate knowledge of a lady made of oak and iron, oakum and ballast, canvas and brass is the one true marriage he can and will ever know. How could he have thought differently with Sophia?\n\nAt other times, even later at night when the ice's moaning turns to screams, Crozier thinks that the ship has become his body and his mind. Out there \u2014 out beyond the decks and hull \u2014 lies death. Eternal cold. Here, even while frozen in the ice, there continues the heartbeat, however faint, of warmth and conversation and movement and sanity.\n\nBut traveling deeper into the ship, Crozier realizes, is like traveling too deeply into one's body or mind. What one encounters there may not be pleasant. The orlop deck is the belly. This is where the food and needed resources are stored, each thing packed away in the order of its presumed need, easy to hand for those driven down here by Mr. Diggle's shouts and blows. Lower, on the hold deck where he's headed, are the deep guts and kidneys, the water tanks and the majority of the coal storage and more provisions. But it's the mind analogy that bothers Crozier the most. Haunted and plagued by melancholia much of his life, knowing it as a secret weakness made worse by his twelve winters frozen in arctic darkness as an adult, feeling it recently triggered into active agony by Sophia Cracroft's rejection, Crozier thinks of the partially lighted and occasionally heated but livable lower deck as the sane part of himself. The brooding mental lower world of the orlop deck is where he spends too much of his time these days \u2014 listening to the ice scream, waiting for the metal bolts and beam fastenings to explode from the cold. The bottom hold deck below, with its terrible smells and its waiting Dead Room, is madness.\n\nCrozier shakes such thoughts away. He looks down the orlop-deck aisle running forward between the piled casks and crates. The lantern's gleam is blocked by the bulkheads of the Bread Room and the aisles on either side constrict to tunnels even narrower than the officers' country companionway on the lower deck above. Here men must squeeze between the Bread Room and the sleeves holding the last sacks of _Terror_ 's coal. The carpenter's storeroom is forward there on the starboard side, the boatswain's storeroom opposite on the port side.\n\nCrozier turns and shines his lantern aft. Rats flee somewhat lethargically from the light, disappearing between casks of salt meat and crates of tinned provisions.\n\nEven in the dim lantern's glow, the captain can see that the padlock is secure on the Spirit Room. Every day one of Crozier's officers will come down here to fetch the amount of rum needed for that day's doling out of the men's noonday grog \u2014 one-fourth pint of 140-proof rum to three-fourths pint of water. Also in the Spirit Room are stored the officers' wine and brandy, as well as two hundred muskets, cutlasses, and swords. As has always been the practice in the Royal Navy, scuttles lead directly from the officers' mess and Great Cabin overhead to the Spirit Room. Should there be a mutiny, the officers would get to the weapons first.\n\nBehind the Spirit Room is the Gunner's Storeroom with its kegs of powder and shot. On either side of the Spirit Room are various storage and locker spaces, including chain cable lockers; the Sail Room, with all its cold canvas; and the Slop Room, from which Mr. Helpman, the ship's clerk, issues their outdoor clothing.\n\nBehind the Spirit Room and the Gunner's Storeroom is the Captain's Storeroom, holding Francis Crozier's private \u2014 and personally paid for \u2014 hams, cheeses, and other luxuries. It is still the custom for the ship's captain to set the table from time to time for his officers, and while the victuals in Crozier's storeroom pale in comparison to the luxurious foodstuffs crammed into the late Captain Sir John Franklin's private store on _Erebus,_ Crozier's pantry \u2014 almost empty now \u2014 has held out for two summers and two winters in the ice. Also, he thinks with a smile, it has the benefit of containing a decent wine cellar from which the officers still benefit. And many bottles of whiskey upon which he, the captain, depends. The poor commander, lieutenants, and civilian officers aboard _Erebus_ had done without spirits for two years. Sir John Franklin was a teetotaler and so, when he was alive, had been his officers' mess.\n\nA lantern bobs toward Crozier down the narrow aisle leading back from the bow. The captain turns in time to see something like a hairy black bear squeezing its bulk between the coal sleeves and the Bread Room bulkhead.\n\n\"Mr. Wilson,\" says Crozier, recognizing the carpenter's mate from his rotundity and from the sealskin gloves and deerskin trousers which had been offered to all the men before departure but which only a few had chosen over their flannel and woolen slops. Sometime during the voyage out, the mate had sewn wolf skins they'd picked up at the Danish whaling station at Disko Bay into a bulky \u2014 but warm, he insisted \u2014 outer garment.\n\n\"Captain.\" Wilson, one of the fattest men aboard, is carrying the lantern in one hand and has several boxes of carpenter's tools tucked under his other arm.\n\n\"Mr. Wilson, my compliments to Mr. Honey and would you ask him to join me on the hold deck.\"\n\n\"Aye, sir. Where on the hold deck, sir?\"\n\n\"The Dead Room, Mr. Wilson.\"\n\n\"Aye, sir.\" The lantern light reflects on Wilson's eyes as the mate keeps his curious gaze up just a second too long.\n\n\"And ask Mr. Honey to bring a pry bar, Mr. Wilson.\"\n\n\"Aye, sir.\"\n\nCrozier stands aside, squeezing between two kegs to let the larger man pass up the ladder to the lower deck. The captain knows he might be rousing his carpenter for nothing \u2014 making the man go to the trouble of getting into his cold-weather slops right before lights-out for no good reason \u2014 but he has a hunch and he'd rather disturb the man now than later.\n\nWhen Wilson has squeezed his bulk up through the upper hatch, Captain Crozier lifts the lower hatch and descends to the hold deck.\n\nBecause the entire deck-space lies beneath the level of outside ice, the hold deck is almost as cold as the alien world beyond the hull. And darker, with no aurora, stars, or moon to relieve the ever-present blackness. The air is thick with coal dust and coal smoke \u2014 Crozier watches the black particles curl around his hissing lantern like a banshee's claw \u2014 and it stinks of sewage and bilge. A scraping, sliding, scuttling noise comes from the darkness aft, but Crozier knows it's just the coal being shoveled in the boiler room. Only the residual heat from that boiler keeps the three inches of filthy water sloshing at the foot of the ladder from turning to ice. Forward, where the bow dips deeper into the ice, there is almost a foot of icy water, despite men working the pumps six hours and more a day. The _Terror,_ like any living thing, breathes out moisture through a score of vital functions, including Mr. Diggle's ever-working stove, and while the lower deck is always damp and rimed with ice and the orlop deck frozen, the hold is a dungeon with ice hanging from every beam and meltwater sloshing above one's ankles. The flat black sides of the twenty-one iron water tanks lining the hull on either side add to the chill. Filled with thirty-eight tons of fresh water when the expedition sailed, the tanks are now armored icebergs and to touch the iron is to lose skin.\n\nMagnus Manson is waiting at the bottom of the ladder as Private Wilkes had said, but the huge able-bodied seaman is standing, not sitting arse-on-ladder. The big man's head and shoulders are hunched beneath the low beams. His pale, lumpy face and stubbled jowls remind Crozier of a rotten white peeled potato stuffed under a Welsh wig. He will not meet his captain's stare in the harsh lantern glow.\n\n\"What is this, Manson?\" Crozier's voice does not hold the bark he unleashed on his lookout and lieutenant. His tone is flat, calm, certain, with the power of flogging and hanging behind every syllable.\n\n\"It's them ghosts, Cap'n.\" For a huge man, Magnus Manson has the high, soft voice of a child. When _Terror_ and _Erebus_ had paused at Disko Bay on the west coast of Greenland in July of 1845, Captain Sir John Franklin had seen fit to dismiss two men from the expedition \u2014a Marine private and a sailmaker from _Terror._ Crozier had made the recommendation that seaman John Brown and Private Aitken from his ship also be released \u2014 they were little better than invalids and never should have been signed on for such a voyage \u2014 but on occasion since, he wished he'd sent Manson home with those four. If the big man was not feebleminded, he was so close to it that it was impossible to tell the difference.\n\n\"You know there are no ghosts on _Terror,_ Manson.\"\n\n\"Yes, Cap'n.\"\n\n\" _Look_ at me.\"\n\nManson raises his face but does not meet Crozier's gaze. The captain marvels at how tiny the man's pale eyes are in that white lump of a face.\n\n\"Did you disobey Mr. Thompson's orders to carry sacks of coal to the boiler room, Seaman Manson?\"\n\n\"No, sir. Yes, sir.\"\n\n\"Do you know the consequences of disobeying any order on this ship?\" Crozier feels like he's talking to a boy, although Manson must be at least thirty years old.\n\nThe big sailor's face brightens as he is presented with a question he can answer correctly. \"Oh, yes, Cap'n. Flogging, sir. Twenty lashes. A 'undred lashes if I disobeys more than once. 'Anging if I disobey a real officer rather'n jus' Mr. Thompson.\"\n\n\"That's correct,\" says Crozier, \"but did you know that the captain can also inflict any punishment he finds appropriate to the transgression?\"\n\nManson peers down at him, his pale eyes confused. He has not understood the question.\n\n\"I'm saying I can punish you any way I see fit, Seaman Manson,\" says the captain.\n\nA flood of relief flows over the lumpy face. \"Oh, yes, right, Cap'n.\"\n\n\"Instead of twenty lashes,\" says Francis Crozier, \"I could have you locked up in the Dead Room for twenty hours with no light.\"\n\nManson's already pale, frozen features lose so much blood that Crozier prepares to get out of the way if the big man faints.\n\n\"You... wouldn't... .\" The child-man's voice quavers toward a vibrato.\n\nCrozier says nothing for a long, cold, lantern-hissing moment. He lets the sailor read his expression. Finally he says, \"What do you think you hear, Manson? Has someone been telling you ghost stories?\"\n\nManson opens his mouth but seems to have trouble deciding which question to answer first. Ice forms on his fat lower lip. \"Walker,\" he says at last.\n\n\"You're afraid of Walker?\"\n\nJames Walker, a friend of Manson's who had been about the same age as the idiot and not much brighter, was the last man to die on the ice, just a week earlier. Ship's rules required that the crew keep small holes drilled in the ice near the ship, even when the ice was ten or fifteen feet thick as it was now, so that they could get at water to fight a fire should one break out aboard. Walker and two of his mates were on just such a drilling party in the dark, reopening an old hole that would freeze in less than an hour unless rammed with metal spikes. The white terror had come out from behind a pressure ridge, torn off the seaman's arm, and smashed his ribs to splinters in an instant, disappearing before the armed guards on deck could raise their shotguns.\n\n\"Walker told you ghost stories?\" says Crozier.\n\n\"Yes, Cap'n. No, Cap'n. What Jimmy did was, 'e tells me the night before the _thing_ killed 'im, 'e says, 'Magnus, should that 'ellspawn out on the ice get me someday,' 'e says, 'I'll come back in me white shroud to whisper in your ear how cold 'ell is.' So help me God, Cap'n, that's what Jimmy said to me. Now I 'ear 'im tryin' to get out.\"\n\nAs if on cue, the hull groans, the frigid deck moans under their feet, metal brackets on the beams groan back in sympathy, and there is a scraping, clawing noise in the dark around them that seems to run the length of the ship. The ice is restless.\n\n\"Is that the sound you hear, Manson?\"\n\n\"Yes, Cap'n. No, sir.\"\n\nThe Dead Room is thirty feet aft on the starboard side, just beyond the last metal-moaning iron water tank, but when the outside ice stops its noise, Crozier can hear only the muffled scrape and push of the shovels in the boiler room farther aft.\n\nCrozier's had enough of this nonsense. \"You know your friend's not coming back, Magnus. He's there in the extra sail storage room securely sewn into his own hammock with the other dead men, frozen solid, with three layers of our heaviest sail canvas tied around them. If you hear anything from in there, it's the damned rats trying to get at them. You _know_ this, Magnus Manson.\"\n\n\"Yes, Cap'n.\"\n\n\"There will be no disobeying orders on this ship, Seaman Manson. You have to make up your mind now. Carry the coal when Mr. Thompson tells you to. Fetch the food stores when Mr. Diggle sends you down here. Obey all orders promptly and politely. Or face the court... face _me_... and the possibility that you'll spend a cold, lanternless night in the Dead Room yourself.\"\n\nWithout another word, Manson knuckles his forehead in salute, lifts a huge sack of coal from where he's stowed it on the ladder, and hauls it aft into the darkness.\n\nThe engineer himself is stripped to his long-sleeved undershirt and corduroy trousers, shoveling coal alongside the ancient 47-year-old stoker named Bill Johnson. The other stoker, Luke Smith, is on the lower deck sleeping between his shoveling hours. _Terror_ 's lead stoker, young John Torrington, was the first man of the expedition to die, on New Year's Day 1846. But that had been from natural causes. It seems Torrington's doctor had urged the 19-year-old to go to sea to cure his consumption, and he'd succumbed after two months of being an invalid while the ships were frozen in the harbour at Beechey Island that first winter. Doctors Peddie and McDonald had told Crozier that the boy's lungs were as solidly packed with coal dust as a chimney sweep's pockets.\n\n\"Thank you, Captain,\" says the young engineer between heaves of the shovel. Seaman Manson has just dropped off a second sack of coal and gone back for a third.\n\n\"You're welcome, Mr. Thompson.\" Crozier glances at Stoker Johnson. The man is four years younger than the captain but looks thirty years older. Every seam and wrinkle on his age-molded face is outlined in coal black and grime. Even his toothless gums are soot grey. Crozier doesn't want to reprimand his engineer \u2014 and thus an officer, although civilian \u2014 in front of the stoker, but he says, \"I presume we'll dispense with using Marines as messengers, should there be another such instance in the future, which I very much doubt.\"\n\nThompson nods, uses the shovel to clang shut the iron grate on the boiler, leans on the tool, and tells Johnson to go above to get him some coffee from Mr. Diggle. Crozier's glad the stoker is gone but even happier that the grate is closed; the heat in here makes him slightly nauseated after the cold everywhere else.\n\nThe captain has to wonder at the fate of his engineer. Warrant Officer James Thompson, Engineer First Class, graduate of the Navy's steam factory at Woolwich \u2014 the world's best training grounds for the new breed of steam-propulsion engineers \u2014 is here stripped to his filthy undershirt, shoveling coal like a common stoker in an ice-locked ship that hasn't moved an inch under its own power now for more than a year.\n\n\"Mr. Thompson,\" says Crozier, \"I'm sorry I haven't had a chance to talk to you today since you walked over to _Erebus._ Did you have a chance to confer with Mr. Gregory?\"\n\nJohn Gregory is the engineer aboard the flagship.\n\n\"I did, Captain. Mr. Gregory's convinced that with the onset of real winter, they'll never be able to get at the damaged driveshaft. Even if they _were_ able to tunnel down through the ice to replace the last propeller with the one they've jury-rigged, with the replacement driveshaft bent as badly as it is, _Erebus_ is going nowhere under steam.\"\n\nCrozier nods. _Erebus_ bent its second driveshaft while the ship was throwing itself desperately on the ice more than a year ago. The flagship \u2014 heavier, with a more powerful engine \u2014 led the way through the pack ice that summer, opening leads for both ships. But the last ice they'd encountered before being frozen in for the last thirteen months was harder than the iron in the experimental propeller screw and driveshaft. Divers that summer \u2014 all of who suffered frostbite and came close to dying \u2014 had confirmed not only that the screw had been shattered but that the driveshaft itself was bent and broken.\n\n\"Coal?\" says the captain.\n\n\" _Erebus_ has enough for... perhaps... four months of heating in the ice, at only one hour of hot-water circulation through the lower deck per day, Captain. None at all for steaming next summer.\"\n\n _If we get free next summer,_ thinks Crozier. After this last summer, when the ice never relented for a day, he's a pessimist. Franklin had used up _Erebus_ 's coal supply at a prodigious rate during those last weeks of freedom in the summer of 1846, sure that if he could smash through those last few miles of pack ice, the expedition would reach the open waters of the North-West Passage along the northern coast of Canada and they'd be drinking tea in China by late autumn.\n\n\"What about _our_ coal use?\" asks Crozier.\n\n\"Perhaps enough left for six months of heating,\" says Thompson. \"But only if we cut back from two hours a day to one. And I recommend we do that soon \u2014 no later than the first of November.\"\n\nThat is less than two weeks away.\n\n\"And steaming?\" says Crozier.\n\nIf the ice relents at all next summer, Crozier plans to cram all the surviving men from _Erebus_ aboard _Terror_ and make an all-out effort to retreat the way they'd come \u2014 up the unnamed strait between Boothia Peninsula and Prince of Wales Island, down which they rushed two summers ago, past Walker Point and Barrow Strait, out through Lancaster Sound like a cork from a bottle, then rushing south into Baffin Bay with all sail set and the last coal being burned, going like smoke and oakum, burning extra spars and furniture if need be to get the last bit of steam, anything to get them into open water off Greenland where whalers could find them.\n\nBut he'll also need steam to fight his way north through the south-flowing ice to Lancaster Sound, even if a miracle occurs and they are released from the ice here. Crozier and James Ross once sailed _Terror_ and _Erebus_ out of the south polar ice, but they'd been traveling _with_ the currents and bergs. Here in the damned arctic, the ships have to sail for weeks _against_ the flow of ice coming down from the pole just to reach the straits where they can escape.\n\nThompson shrugs. The man looks exhausted. \"If we cut off the heat on New Year's Day and somehow survive until next summer, we might get... six days steaming without ice? Five?\"\n\nCrozier merely nods again. This is almost certainly a death sentence for his ship, but not necessarily for the men of both ships.\n\nThere is a sound out in the darkened corridor.\n\n\"Thank you, Mr. Thompson.\" The captain lifts his lantern off an iron hook, leaves the glow of the boiler room, and heads forward in the slush and darkness.\n\nThomas Honey is waiting in the corridor, his candle lantern sputtering in the bad air. He is holding the iron pry bar in front of him like a musket, clutched in thick gloves, and hasn't opened the bolted door of the Dead Room.\n\n\"Thank you for coming, Mr. Honey,\" Crozier says to his carpenter.\n\nWithout explanation, the captain throws back the bolts and enters the freezing-cold storage room.\n\nCrozier can not resist lifting his lantern toward the aft bulkhead where the six men's bodies have been stacked in their common canvas shroud.\n\nThe heap is writhing. Crozier expected that \u2014 expected to see the movement of rats under the tarp \u2014 but he realizes that he's looking at a solid mass of rats _above_ the shroud-canvas as well. There is a solid cube of rats, extending more than four feet above the deck, as hundreds of them jostle for position to get at the frozen dead men. The squealing is very loud in here. More rats are underfoot, scuttling between his and the carpenter's legs. _Rushing to the banquet,_ thinks Crozier. And showing no fear of the lantern light.\n\nCrozier turns the lantern back to the hull, walks up the slight incline caused by the ship's cant to port and begins pacing along the curved, tilted wall.\n\n _There._\n\nHe holds the lantern closer.\n\n\"Well, I'll be God-damned to hell and hanged for a heathen,\" says Honey. \"Pardon me, Captain, but I didn't think the ice would do this so soon.\"\n\nCrozier doesn't answer. He crouches to investigate the bent and extended wood of the hull more closely.\n\nHull planks have been bent inward here, bulging almost a foot from the graceful curve elsewhere along the hull's side. The innermost layers of wood have splintered and at least two planks are hanging free.\n\n\"Jesus God Christ Almighty,\" says the carpenter, who has crouched next to the captain. \"That ice is a fucking monster, begging the Captain's pardon, sir.\"\n\n\"Mr. Honey,\" says Crozier, his breath adding crystals to the ice already on the planks and reflecting the lantern light, \"could anything but the ice have done this damage?\"\n\nThe carpenter barks a laugh but stops abruptly as he realizes his captain is not making a joke. Honey's eyes widen, then squint. \"Begging your pardon again, Captain, but if you mean... that's impossible.\"\n\nCrozier says nothing.\n\n\"I mean, Captain, this hull was three inches of the finest English oak as it was, sir. And for this trip \u2014 for the ice, I mean, sir \u2014 it was doubled with two layers of African oak, Captain, each one and a half inches thick. And them African oak panels was wrought on the diagonal, sir, givin' it even more strength than if it were just doubled straight-like.\"\n\nCrozier is inspecting the loose planks, trying to ignore the river of rats behind them and around them as well as the chewing sounds from the direction of the aft bulkhead.\n\n\"And, sir,\" continues Honey, his voice hoarse in the cold, his rum-tainted breath freezing in the air, \"on top of the three inches of English oak and the three inches of diagonal-laid African oak, they laid on two layers of Canadian elm, sir, each two inches thick. That's four more inches of hull, Captain, and that wrought diagonal against the African oak. That's five belts of serious timber, sir... ten inches of the strongest wood on earth between us and the sea.\"\n\nThe carpenter shuts up, realizing that he's lecturing his captain on details of the shipyard's work that Crozier had personally overseen in the months before departure.\n\nThe captain stands and sets his mittened hand against the innermost planks where they have come free. There's more than an inch of open space there. \"Set your lantern down, Mr. Honey. Use your pry bar to lever this loose. I want to see what the ice has done to the outer layer of hull oak.\"\n\nThe carpenter complies. For several minutes the sound of the iron bar prying at iron-cold wood and the carpenter's grunts almost drown out the frenzied gnawing of the rats behind them. The bent Canadian elm tears back and falls away. The shattered African oak is leveraged out. Only the inward-bent original oak of the hull remains now as Crozier steps closer, holding his lantern so that both men can see.\n\nShards and spears of ice reflect the lantern light through the foot-long holes in the hull, but in the centre is something much more disturbing \u2014 blackness. Nothing. A hole in the ice. A tunnel.\n\nHoney bends a piece of the splintered oak farther in so Crozier can shine his lantern on it.\n\n\"Holy fucking Jesus Christ fucking shit almighty,\" gasps the carpenter. This time he does not ask his captain's pardon.\n\nCrozier has the temptation to lick his dry lips but knows how painful that will be here where it's 50 below in the dark. But his heart is pounding so wildly that he's also tempted to steady himself with one mitten against the hull the way the carpenter has just done.\n\nThe freezing air from outside rushes in so quickly that it almost extinguishes the lantern. Crozier has to shield it with his free hand to keep it flickering, sending the men's shadows dancing across decks, beams, and bulkheads.\n\nThe two long boards from the outer hull have been smashed and bent inward by some inconceivable, irresistible force. Clearly visible in the light from the slightly shaking lantern are huge claw marks in the splintered oak \u2014 claw marks streaked with frozen smears of impossibly bright blood.\n\nGOODSIR\n\n_Lat. 75\u00b0-12\u2032 N., Long. 61\u00b0-6\u2032 W._\n\nBaffin Bay, July, 1845\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n_11 April, 1845 \u2014_\n\n_In a letter to my brother today, I wrote \u2014_ \"All the Officers are in great hopes of making the passage and hope to be in the Pacific end of next summer.\"\n\n_I confess that, however Selfish it is, my own hope for the Expedition is that it may take us a bit longer to reach Alaska, Russia, China, and the warm waters of the Pacific. Although trained as an anatomist and signed on by Captain Sir John Franklin as a mere assistant surgeon, I am, in Truth, no mere surgeon but a Doctor, and I confess further that as amateurish as my attempts may be, I hope to become something of a Naturalist on this voyage. While having no personal Experience with arctic flora and fauna, I plan to become personally acquainted with the life-forms in the Icy Realms to which we set sail only a month from now. I am especially interested in the white bear, although most accounts of it one hears from whalers and old Arctic Hands tend to be too fabulous to credit._\n\n_I recognize that this personal Diary is most out of the ordinary \u2014 the Official Log that I shall begin when we depart next month will record all of the pertinent professional events and observations of my time aboard HMS_ Erebus _in my capacity as Assistant Surgeon and as a member of Captain Sir John Franklin's expedition to force the North-West Passage \u2014 but I feel that something More is due, some other record, some more personal account, and even if I should never let another soul read this after my Return, it is my Duty \u2014 to myself if no other \u2014 to keep these notes._\n\n_All I know at this point is that my Expedition with Captain Sir John Franklin already promises to be the Experience of a Lifetime._\n\n_Sunday, 18 May, 1845 \u2014_\n\n_All the men are aboard, and although last-minute Preparation is still going on around the clock for tomorrow's Departure \u2014 especially with the stowing of what Captain Fitzjames informs me is more than eight_ thousand _cans of tinned food which have arrived only in the nick of time \u2014 Sir John conducted Divine Service today for us aboard_ Erebus _and for as many of_ Terror _'s crew who wished to join us. I noted that_ Terror _'s captain, an Irishman named Crozier, was not in attendance._\n\n_No one could have attended the lengthy service and heard the_ very _lengthy sermon by Sir John today without being deeply moved. I wonder if any Ship from any nation's Navy has ever been captained by such a Religious Man. There is no doubt that we are truly and safely and irrevocably in God's Hands on the voyage to come._\n\n_19 May, 1845 \u2014_\n\n_What a Departure!_\n\n_Having never gone to sea before, much less as a member of such a Heralded Expedition, I had no Idea of what to Expect, but Nothing could have prepared me for the glory of this Day._\n\n_Captain Fitzjames estimates that more than ten thousand well-wishers and Persons of Importance crowded the docks at Greenhithe to see us off._\n\n_Speeches resounded until I thought we would never be allowed to depart while Daylight still filled the Summer Sky. Bands played. Lady Jane \u2014 who has been staying aboard with Sir John \u2014 went down the gangplank to a rousing series of_ Hurrahs! _from we sixty-some Erebuses. Bands played again. Then the cheers started as all lines were cast off, and for several minutes the noise was so deafening that I could not have heard an order had Sir John himself shouted it in my ear._\n\n_Last night, Lieutenant Gore and Chief Surgeon Stanley were Kind enough to inform me that it is custom during sailing for the officers not to Show Emotion, so although only_ technically _an officer, I stood with the officers lined up in their fine blue jackets and tried to restrain all Displays of emotion, however manly._\n\n_We were the only ones doing so. The Seamen shouted and waved handkerchiefs and hung from the ratlines, and I could see many a rouged dockside Doxie waving farewell to them. Even Captain Sir John Franklin waved a bright red-and-green handkerchief at Lady Jane, his daughter, Eleanor, and his niece Sophia Cracroft, who waved back until the sight of the docks was obstructed by the following_ Terror.\n\n_We are being towed by steam tugs and followed on this leg of our voyage by HMS_ Rattler, _a powerful new steam frigate, and also a hired transport ship carrying our provisions,_ Baretto Junior.\n\n_Just before_ Erebus _pushed away from the docks, a Dove landed high on the main mast. Sir John's daughter by his first marriage, Eleanor \u2014 then quite visible in her bright-green silk dress and emerald parasol \u2014 cried out but could not be heard above the Cheers and Bands. Then she pointed, and Sir John and many of the Officers looked up, smiled, and then pointed out the Dove to others aboard ship._\n\n_Combined with the Words spoken in yesterday's Divine Service, this, I have to assume, is the Best Possible Omen._\n\n_4 July, 1845 \u2014_\n\n_What a terrible Crossing of the North Atlantic to Greenland._\n\n_For thirty stormy days, even while under tow, the Ship has been tossing, rolling, and wallowing, its tightly sealed Gunports on each side barely four feet out of the water during the downward rolls, sometimes barely making Headway. I have been terribly seasick for Twenty-eight of the last Thirty days. Lieutenant Le Vesconte tells me that we never made more than five knots, which \u2014 he assures me \u2014 is a Terrible time for any ship merely under Sail, much less for such a Miracle of Technology as_ Erebus _and our companion craft,_ Terror, _both capable of steaming along under the Impetus of their invincible Screws._\n\n_Three days ago we rounded Cape Farewell at the southern tip of Greenland, and I confess that the glimpses of this Huge Continent, with its rocky cliffs and endless glaciers coming right down to the Sea, lay as heavily on my Spirits as the pitching and rolling did upon my Stomach._\n\n_Good God, this is a barren, cold place! And this in July._\n\n_Our morale is Top Notch, however, and all aboard trust to Sir John's Skill and Good Judgement. Yesterday Lieutenant Fairholme, the youngest of our lieutenants, said to me in Confidence, \"I never felt the Captain was so much my companion with anyone I have sailed with before.\"_\n\n_Today we put in at the Danish whaling station here in Disko Bay. Tons of supplies are being transferred from_ Baretto Junior, _and ten live oxen transported aboard that ship were slaughtered this afternoon. All the men of both Expedition ships shall feast on fresh meat tonight._\n\n_Four men were dismissed from the Expedition today \u2014 upon advice of the four of us surgeons \u2014 and will be returning to England with the tow and transport ship. These include one man from_ Erebus _\u2014 a certain Thomas Burt, the ship's armourer, and three from_ Terror _\u2014 a Marine private named Aitken, a seaman named John Brown, and_ Terror _'s primary sailmaker, James Elliott. That brings our total muster down to 129 men for the two ships._\n\n_Dried fish from the Danes and a cloud of Coal Dust hang over everything this afternoon \u2014 hundreds of bags of coal were transferred from_ Baretto Junior _today \u2014 and the seamen aboard_ Erebus _are busy with the smooth-sided stones they call Holy Stones, scrubbing and rescrubbing the deck clean while the Officers shout encouragement. Despite the extra work, All Hands are in High Spirits because of the promise of Tonight's Feasting and extra rations of Grog._\n\n_Besides the four men to be invalided home, Sir John will be sending the June musters, official dispatches, and all personal letters back with_ Baretto Junior. _Everyone will be busy writing the next few days._\n\n_After this week, the next letter to reach our loved ones will be posted from Russia or China!_\n\n_12 July, 1845 \u2014_\n\n_Another departure, this time perhaps the Last One before the North-West Passage. This morning we slipped our cables and sailed west from Greenland while the crew of_ Baretto Junior _gave us three hearty_ Hurrahs! _and waved their caps. Surely these shall be the last White Men we see until we reach Alaska._\n\n_26 July, 1845 \u2014_\n\n_Two whalers \u2014_ Prince of Wales _and_ Enterprise _\u2014 have anchored nearby to where we have tied up to a floating Ice Mountain. I have enjoyed many hours talking to the captains and crewmen about white bears._\n\n_I also had the distinct terror \u2014 if not Pleasure \u2014 of climbing that huge iceberg this morning. The sailors scrambled up early yesterday, chipping steps into the vertical ice with their axes and then rigging fixed lines for the less agile. Sir John ordered an Observatory be set up atop the giant berg, which towers more than twice as tall as our Highest Mast, and while Lieutenant Gore and some of the officers from_ Terror _take atmospheric and astronomical measurements up there \u2014 they have erected a tent for those spending the night atop the Precipitous Ice Mountain \u2014 our Expedition Ice Masters, Mr. Reid from_ Erebus _and Mr. Blanky from_ Terror, _spend the daylight hours staring west and north through their brass telescopes, seeking, I am informed, the most likely path through the near-solid sea of ice already formed there. Edward Couch, our very Reliable and Voluble Mate, tells me that this is_ very _late in the Arctic Season for ships to be seeking any passage, much less the Fabled North-West Passage._\n\n_The sight of both_ Erebus _and_ Terror _moored to the iceberg_ below _us, a maze of ropes \u2014 what I must remember to call \"lines\" now that I am an old nautical hand \u2014 holding both ships fast to the Ice Mountain, the two ships' highest crow's nests_ below _my precarious and icy perch so high above everything, created a sort of sick and thrilling Vertigo within me._\n\n_It was exhilarating standing up there hundreds of feet above the sea. The summit of the iceberg was almost the size of a cricket pitch and the tent holding our Meteorological Observatory looked quite incongruous on the blue ice \u2014 but my hopes for a few moments of Quiet Revery were shattered by the constant Shotgun Blasts as the men all over the Summit of our Ice Mountain were shooting birds \u2014 arctic terns, I am told \u2014 by the hundreds. These heaps and heaps of fresh-killed birds shall be salted and stored away, although Heaven Alone Knows where those additional casks shall be Stored, since both our ships are already Groaning and riding low under the weight of all their Stores._\n\n_Dr. McDonald, assistant surgeon aboard HMS_ Terror _\u2014 my counterpart there as it were \u2014 has theories that heavily salted food is not as efficient and antiscorbutic as fresh or nonsalted Victuals, and since the regular seamen aboard both ships prefer their Salted Pork to all other meals, Dr. McDonald worries that the heavily salted birds will add little to our Defenses against Scurvy. However, Stephen Stanley, our Surgeon aboard_ Erebus, _dismisses these worries. He points out that besides the 10,000 cases of preserved cooked meats aboard_ Erebus, _our tinned rations alone include boiled and roast mutton, veal, all forms of vegetables including potatoes, carrots, parsnips, and mixed vegetables, wide varieties of Soups, and 9,450 pounds of Chocolate. An equal weight \u2014 9,300 pounds \u2014 of lemon juice has also been brought as our primary antiscorbutic measure. Stanley informs me that even when the juice is sweetened with liberal dollops of sugar, the common men hate their daily ration and that one of our Primary Jobs as surgeons on the Expedition is to ensure that they swallow the stuff._\n\n_It was interesting to me that almost all of the hunting by the officers and men of both our ships is done almost exclusively with Shotguns. Lieutenant Gore assures me that each ship carries a full arsenal of Muskets. Of course, it only makes sense to use Shotguns to hunt birds such as those killed by the hundreds today, but even back at Disko Bay, when small parties went out hunting Caribou and Arctic Fox, the men \u2014 even the Marines obviously trained in the use of Muskets \u2014 preferred to take along Shotguns. This, of course, must be the result of Habit as much as Preference \u2014 the officers tend to be English Gentlemen who have never used muskets or Rifles in their hunting, and except for the use of single-shot weapons in Close Quarters Naval Combat, even the Marines have used Shotguns almost exclusively in their past hunting experience._\n\n_Will Shotguns be enough to bag the Great White Bear? We've not seen one of those Wondrous creatures yet, although every Experienced Officer and Hand reassures me that we shall encounter them as soon as we enter the Pack Ice, and if not then, certainly when we Winter Over \u2014 should we be compelled to do so. Truly the tales the whalers here tell me of the elusive White Bears are Wonderful and Terrifying._\n\n_As I write these words, I am informed that current or wind or perhaps the necessities of the whaling business itself have carried both whalers,_ Prince of Wales _and_ Enterprise, _away from our moorings here at our Ice Mountain. Captain Sir John shall not be dining with one of the whaling captains \u2014 Captain Martin of_ Enterprise, _I believe \u2014 as had been planned for this evening._\n\n_Perhaps more Pertinent, Mate Robert Sergeant has just informed me that our men are bringing down the astronomical and meteorological instruments, striking the tent, and reeling in the hundreds of yards of fixed rope \u2014 line \u2014 which allowed my Ascent earlier today._\n\n_Evidently the Ice Masters, Captain Sir John, Commander Fitzjames, Captain Crozier, and the other Officers have determined our Most Promising Path through the ever-shifting pack ice._\n\n_We are to cast off from our little Iceberg Home within minutes, sailing Northwest as long as the seemingly endless Arctic Twilight allows us to._\n\n_We shall be beyond the reach of even the Hardy Whalers from this point on. As far as the World Beyond our intrepid Expedition is concerned, as Hamlet said,_ The rest is silence.\n\nCROZIER\n\n_Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n9 November, 1847\n\nCrozier is dreaming about the picnic to the Platypus Pond and of Sophia stroking him under water when he hears the sound of a shot and comes crashing awake.\n\nHe sits up in his bunk not knowing what time it is, not knowing if it is day or night, although there is no line between day and night any longer since the sun has disappeared this very day, not to reappear until February. But even before he lights the small lantern in his berth to check his watch, he knows that it is _late._ The ship is as quiet as it ever gets; silent except for the creak of tortured wood and frozen metal within; silent except for the snores, the mumbles, and the farts from the sleeping men, and curses from Mr. Diggle the cook; silent save for the incessant groaning, banging, cracking, and surging of the ice outside; and, added to those exceptions to silence this night, silent but for the banshee screech of a high wind.\n\nBut this is no sound of ice or wind that wakes Crozier. It is a gunshot. A shotgun \u2014 muffled through the layers of oak planks and overlaying snow and ice, but a shotgun blast without doubt.\n\nCrozier was sleeping with most of his clothes on and now has pulled on most of the other layers and is ready for his cold-weather slops when Thomas Jopson, his steward, knocks on the door with his distinctive soft triple rap. The captain slides it open.\n\n\"Trouble on deck, sir.\"\n\nCrozier nods. \"Who's on watch tonight, Thomas?\" His pocket watch shows him that it is almost 3:00 a.m., civilian time. His memory of the month's and day's watch schedule gives him the names an instant before Jopson speaks them aloud.\n\n\"Billy Strong and Private Heather, sir.\"\n\nCrozier nods again, lifts a pistol from his cupboard, checks the priming, sets it in his belt, and squeezes past the steward, out through the officers' dining cubicle that borders the captain's tiny cabin on the starboard side, and then quickly forward through another door to the main ladderway. The lower deck is mostly dark at this time in the morning \u2014 the glow around Mr. Diggle's stove the primary exception \u2014 but lamps are being lit in several of the officers', mates', and stewards' quarters as Crozier pauses at the base of the ladder to pull his heavy slops from the hook and struggle into them.\n\nDoors slide open. First Mate Hornby walks aft to stand next to Crozier by the ladder. First Lieutenant Little hurries forward down the companionway, carrying three muskets and a saber. He's followed by Lieutenants Hodgson and Irving, who are also carrying weapons.\n\nForward of the ladder, seamen are grumbling from deep in their hammocks, but a second mate is already turning out a work party \u2014 literally tumbling sleeping men from their hammocks and shoving them aft toward their slops and the waiting weapons.\n\n\"Has anyone been up top yet to check out the shot?\" Crozier asks his first mate.\n\n\"Mr. Male had the duty, sir,\" says Hornby. \"He went up as soon as he sent your steward to fetch you.\"\n\nReuben Male is captain of the fo'c'sle. A steady man. Billy Strong, the seaman on port watch up there, has been to sea before, Crozier knows, on HMS _Belvidera._ He wouldn't have shot at phantoms. The other man on watch was the oldest \u2014 and in Crozier's estimation, the stupidest \u2014 of the surviving Marines, William Heather. At age 35 and still a private, frequently sick, too often drunk, and most frequently useless, Heather had almost been sent home from Disko Island two years before when his best friend Billy Aitken was discharged and sent back on HMS _Rattler._\n\nCrozier slips the pistol into the oversized pocket of his heavy woolen outer coat, accepts a lantern from Jopson, wraps a comforter around his face, and leads the way up the tilted ladder.\n\nCrozier sees that it is as black as the inside of an eel's belly outside, no stars, no aurora, no moon, and _cold;_ the temperature on deck registered sixty-three degrees below zero six hours earlier when young Irving had been sent up to take measurements, and now a wild wind howls past the stubs of masts and across the canted, icy deck, driving heavy snow before it. Stepping out from beneath the frozen canvas enclosure above the main hatch, Crozier holds his mittened hand alongside his face to protect his eyes and sees a lantern gleam on the starboard side.\n\nReuben Male is on one knee over Private Heather, who is lying on his back, his cap and Welsh wig knocked off and, Crozier sees, part of his skull knocked away as well. There seems to be no blood, but Crozier can see the Marine's brains sparkling in the lantern light \u2014 sparkling, the captain realizes, because there is already a sheen of ice crystals on the pulped grey matter.\n\n\"He's still alive, Captain,\" says the fo'c'sle chief.\n\n\"Jesus fucking Christ,\" says one of the crewmen crowded behind Crozier.\n\n\"Belay that!\" cries the first mate. \"No fucking profanity. Speak when you're fucking spoken to, Crispe.\" Hornby's voice is a cross between a mastiff's growl and a bull's snort.\n\n\"Mr. Hornby,\" says Crozier. \"Assign Seaman Crispe to get below double-quick and bring up his own hammock to carry Private Heather below.\"\n\n\"Aye, sir,\" say Hornby and the seaman in unison. The pounding of running boots is felt but goes unheard over the wind screech.\n\nCrozier stands and swings his lantern in a circle.\n\nThe heavy railing where Private Heather was standing watch at the base of the iced-over ratlines has been smashed away. Beyond the gap, Crozier knows, the heaped ice and snow runs down like a toboggan ramp for thirty feet or more, but most of that ramp is not visible in the blinding snow. There are no prints visible in the small circle of snow illuminated by the captain's lantern.\n\nReuben Male lifts Heather's musket. \"It wasn't fired, Captain.\"\n\n\"In this storm, Private Heather couldn't have seen the thing until it was right on him,\" says Lieutenant Little.\n\n\"What about Strong?\" asks Crozier.\n\nMale points toward the opposite side of the ship. \"Missing, Captain.\"\n\nTo Hornby, Crozier says, \"Choose a man and stay with Private Heather until Crispe is back with the hammock and carry him below.\"\n\nSuddenly, both surgeons \u2014 Peddie and his assistant, McDonald \u2014 appear in the circle of lamplight, McDonald wearing only light slops.\n\n\"Jesus Christ,\" says the chief surgeon, kneeling next to the Marine. \"He's breathing.\"\n\n\"Help him if you can, John,\" says Crozier. He points to Male and the rest of the seamen crowding around. \"The rest of you \u2014 come with me. Have your weapons ready to fire, even if you have to take your mittens off to do it. Wilson, carry both those lanterns. Lieutenant Little, please go below and choose twenty more good men, issue full slops, and arm them with muskets \u2014 not shotguns, muskets.\"\n\n\"Aye, sir,\" Little shouts over the wind, but Crozier is already leading the procession forward, around the heaped snow and vibrating canvas pyramid amidships and up the canted deck toward the port lookout station.\n\nWilliam Strong is gone. A long wool comforter has been shredded, and the tatters of it, caught in the man lines here, are flapping wildly. Strong's greatcoat, Welsh wig, shotgun, and one mitten are lying near the railing in the lee of the port privy where men on watch huddle to stay out of the wind, but William Strong is gone. There is a smear of red ice on the railing where he must have been standing when he saw the large shape coming at him through the blowing snow.\n\nWithout saying a word, Crozier dispatches two armed men with lanterns aft, three more toward the bow, another with a lantern to look beneath the canvas amidships. \"Rig a ladder here, please, Bob,\" he says to the second mate. The mate's shoulders are hidden under heaps of fresh \u2014 that is, not yet frozen \u2014 rope he's carried up from below. The ladder goes over the side within seconds.\n\nCrozier leads the way down.\n\nThere is more blood on the ice and snow heaped along the exposed port-side hull of the ship. Streaks of blood, looking quite black in the lantern light, lead out beyond the fire holes into the ever-changing maze of pressure ridges and ice spires, all more sensed than seen in the darkness.\n\n\"It wants us to follow it out there, sir,\" says Second Lieutenant Hodgson, leaning close to Crozier so as to be heard over the wind howl.\n\n\"Of course it does,\" says Crozier. \"But we're going anyway. Strong might still be alive. We've seen that before with this thing.\" Crozier looks behind him. Besides Hodgson, only three men had followed him down the rope ladder \u2014 all the rest were either searching the upper deck or were busy hauling Private Heather belowdecks. There is only one other lantern here besides the captain's.\n\n\"Armitage,\" Crozier says to the gunroom steward, whose white beard is already filled with snow, \"give Lieutenant Hodgson your lantern and you go with him. Gibson, you remain here and tell Lieutenant Little where we've headed when he comes down with the main search party. Tell him for God's sake not to let his men fire at anything unless they're sure it's not one of us.\"\n\n\"Yes, Captain.\"\n\nTo Hodgson, Crozier says, \"George, you and Armitage head out about twenty yards that way \u2014 toward the bow \u2014 then stay parallel to us as we search south. Try to keep your lantern within sight of ours.\"\n\n\"Aye, aye, sir.\"\n\n\"Tom,\" Crozier says to the only remaining man, young Evans, \"you come with me. Keep your Baker Rifle ready but only at half-cock.\"\n\n\"Aye, sir.\" The boy's teeth are chattering.\n\nCrozier waits until Hodgson reaches a point twenty yards to their right \u2014 his lantern only the dimmest glow in the blowing snow \u2014 and then he leads Evans out into the maze of seracs, ice pinnacles, and pressure ridges, following the periodic smears of blood on the ice. He knows that a delay of even a few minutes will be enough to blow snow over the faint trail. The captain doesn't even bother to remove the pistol from the pocket of his greatcoat.\n\nLess than a hundred yards out, just where the lanterns of the men on the deck of HMS _Terror_ become invisible, Crozier reaches a pressure ridge \u2014 one of those great heaps of ice thrown up by the ice plates grinding and surging against each other beneath the surface. For two winters in the ice now, Crozier and the other men of the late Sir John Franklin's expedition have watched these pressure ridges appear as if by magic, rise with a great rumbling and tearing sound, and then extend themselves across the surface of the frozen sea, sometimes moving faster than a man can run.\n\nThis ridge is at least thirty feet high, a great vertical rubble of ice boulders each at least as large as a hansom cab.\n\nCrozier walks along the ridge, extending his lantern as high as he can. Hodgson's lantern is no longer visible to the west. Nowhere around _Terror_ is the view simple any longer. Everywhere the snow seracs, drifts, pressure ridges, and ice pinnacles block one's line of sight. There is one great ice mountain in the mile separating _Terror_ and _Erebus_ and half a dozen more in sight on a moonlit night.\n\nBut no icebergs here tonight, only this three-storey-tall pressure ridge.\n\n\"There!\" shouts Crozier over the wind. Evans steps closer, his Baker Rifle raised.\n\nA smear of black blood on the white wall of ice. The thing had carried William Strong up this small mountain of icy rubble, taking an almost vertical route.\n\nCrozier begins climbing, holding the lantern in his right hand while he searches with his mittened free hand, trying to find cracks and crevices for his frozen fingers and already icy boots. He hadn't taken time to put on his pair of boots in which Jopson had driven long nails through the soles, giving traction on such ice surfaces, and now his ordinary seaman's boots slip and skitter on the ice. But he finds more frozen blood twenty-five feet up, just below the ice-jumbled summit of the pressure ridge, so Crozier holds the lantern steady with his right hand while kicking against a tilting ice slab with his left leg and leveraging himself up to the top, the wool of his greatcoat rasping against his back. The captain can't feel his nose and his fingers are also numb.\n\n\"Captain,\" calls Evans from the darkness below, \"do you want me to come up?\"\n\nCrozier is panting too hard to speak for a second, but when he gets his wind back, he calls down, \"No... wait there.\" He can see the faint glow of Hodgson's lantern now to the northwest \u2014 that team isn't within thirty yards of the pressure ridge yet.\n\nFlailing for balance against the wind, leaning far to his right as the gale streams his comforter straight out to his left and threatens to topple him off his precarious perch, Crozier holds the lantern out over the south side of the pressure ridge.\n\nThe drop here is almost vertical for thirty-five feet. There is no sign of William Strong, no sign of black smears on the ice, no sign that anything living or dead has come this way. Crozier can't imagine how anything could have found its way down that sheer ice face.\n\nShaking his head and realizing that his eyelashes are almost frozen to his cheeks, Crozier begins descending the way he'd come, twice almost falling onto the rising bayonets of ice before slip-sliding the last eight feet or so to the surface where Evans is waiting.\n\nBut Evans is gone.\n\nThe Baker Rifle lies in the snow, still at half-cock. There are no prints in the swirling snow, human or otherwise.\n\n\"Evans!\" Captain Francis Rawdon Moira Crozier's voice has been trained to command for thirty-five years and more. He can make it heard over a sou'westerly gale or while a ship is white-foaming its way through the Strait of Magellan in an ice storm. Now he puts every bit of volume he can muster into the shout. \"Evans!\"\n\nNo answer except the howl of the wind.\n\nCrozier lifts the Baker Rifle, checks the priming, and fires it into the air. The _crack_ sounds muffled even to him, but he sees Hodgson's lantern suddenly turn toward him and three more lanterns become dimly visible on the ice from the direction of _Terror._\n\nSomething roars not twenty feet from him. It could be the wind finding a new route through or around an icy serac or pinnacle, but Crozier knows that it isn't.\n\nHe sets the lantern down, fumbles in his pocket, pulls the pistol out, tugs off his mitten with his teeth, and, with just a thin woolen glove between his flesh and the metal trigger, holds the useless weapon in front of him.\n\n\"Come on, God-damn your eyes!\" Crozier screams. \"Come out and try _me_ instead of a _boy,_ you hairy arse-licking rat-fucking piss-drinking spawn of a poxy Highgate whore!\"\n\nThere is no answer except the howl of the wind.\n\nGOODSIR\n\n_Lat. 74\u00b0-43\u2032-28\u2033 N., Long. 90\u00b0-39\u2032-15\u2033 W._\n\nBeechey Island, Winter 1845\u201346\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n_1 January, 1846 \u2014_\n\n_John Torrington, stoker on HMS_ Terror, _died early this morning. New Year's Day. The beginning of our Fifth Month stuck in the ice here at Beechey Island._\n\n_His death was not a surprise. It has been obvious for several months that Torrington had been in the advanced stages of Consumption when he signed on the expedition, and if the Symptoms had manifested themselves just a few weeks earlier in the Late Summer, he would have been sent home on_ Rattler _or even with the two whaling ships we encountered just before sailing west across Baffin Bay and through Lancaster Sound to the Arctic Waste where we now find ourselves wintering. The sad Irony is that Torrington's doctor had told him that going to Sea would be good for his health._\n\n_Chief Surgeon Peddie and Dr. McDonald on_ Terror _treated Torrington, of course, but I was present several times during the Diagnosis stage and was escorted to their ship by several of_ Erebus _'s_ _crewmen after the young stoker died this morning._\n\n_When his illness became Obvious in early November, Captain Crozier relieved the 20-year-old of his duties as stoker down in the poorly ventilated lowest deck \u2014 the coal dust in the air alone there is enough to asphyxiate a person with normal lungs \u2014 and John Torrington had been in a consumptive invalid's Downward Spiral since then. Still, Torrington might have survived for many more months had not there been an Intermediating Agent of his death. Dr. Alexander McDonald tells me that Torrington, who had become too weak in recent weeks even to allow his short Constitutionals around the lower deck, helped by his messmates, came down with Pneumonia on Christmas Day, and it had been a Death Watch since then. When I saw the body this morning, I was shocked at how Emaciated the dead John Torrington was, but both Peddie and McDonald explained that his appetite had been waning for two months, and even though the ship's surgeons altered his Diet more heavily toward Canned Soups and Vegetables, he had continued to lose weight._\n\n_This morning I watched as Peddie and McDonald prepared the corpse \u2014 Torrington in a clean striped shirt, his hair recently and carefully cut, his nails clean \u2014 binding the usual clean cloth around his head to keep the jaw from dropping, then binding him with more strips of white cotton at the elbows, hands, ankles, and big toes. They did this in order to hold the Limbs together while they weighed the poor boy \u2014 88 Pounds! \u2014 and otherwise prepared his body for burial. There was no discussion of Postmortem Examination since it was obvious that Consumption accelerated by Pneumonia had killed the lad, so there was no worry of contamination reaching other crew members._\n\n_I helped my two surgeon colleagues from HMS_ Terror _lift Torrington's body into the coffin carefully prepared for it by the ship's able Carpenter, Thomas Honey, and by his mate, a man named Wilson. There was no rigor mortis. The carpenters had left a residue of Wood Shavings along the bottom of the coffin, so carefully constructed and shaped out of standard ship's mahogany, with a Deeper Pile of shavings under Torrington's head, and because there was yet little Scent of Decay, the air was scented primarily by the wood shavings._\n\n_3 January, 1846 \u2014_\n\n_I keep thinking about John Torrington's Burial late yesterday._\n\n_Only a small contingent of us attended from HMS_ Erebus, _but along with Sir John, Commander Fitzjames, and a few officers, I made the Crossing on Foot from our ship to theirs, and hence the extra two hundred yards to the Shore of Beechey Island._\n\n_I have not been able to Imagine a worse winter than the one we have suffered frozen into this small anchorage in the lee of Beechey Island itself, set in the cusp of larger Devon Island, but Commander Fitzjames and others have assured me that our Situation here \u2014 even with the Treacherous Pressure Ridges, Terrible Dark, Howling Storms, and Constantly Menacing Ice \u2014 would be a thousand times worse out beyond this anchorage, out where the Ice flows down from the Pole like a hail of Enemy Fire from some Borean god._\n\n_John Torrington's crewmates gently lowered his coffin \u2014 already covered with a fine blue wool \u2014 over the railing of their ship, which is Wedged High on its own pillar of ice, while other_ Terror _seamen lashed the coffin to a large Sledge. Sir John himself draped a Union Jack over the coffin, and then Torrington's friends and messmates set themselves into Harness and pulled the sledge the six hundred feet or so to the ice-and-gravel shore of Beechey Island._\n\n_All of this was performed in near Absolute Dark, of course, since even at midday, the sun makes no Appearance here in January and has not done so for three months. It shall be another month and more, they tell me, before the Southern Horizon welcomes back our Fiery Star. At any rate, this entire procession \u2014 coffin, sledge, man-haulers, officers, surgeons, Sir John, Royal Marines in full dress concealed under the same drab Slops the rest of us were wearing \u2014 was illuminated only by bobbing lamplight as we made our way across the Frozen Sea to the Frozen Shore. Men from_ Terror _had chopped and shoveled away at the several recently arisen Pressure Ridges which stood between us and the graveled beach, so there were few Deviations from our sad Route. Earlier in the Winter, Sir John ordered a system of Stout Poles, ropes, and Hanging Lanterns to line the shortest route between the Ships and the graveled isthmus where several Structures had been built \u2014 one to house much of the ships' stores, removed should ice destroy our vessels; another as a sort of emergency bunkhouse and Scientific Station; and a third housing the armourer's forge, set here so that the Flames and Sparks should not ignite our tindered shipboard Homes. I have learned that Sailors fear fire at sea above almost everything else. But this Course of wooden Poles and Lanterns had to be abandoned since the ice is constantly shifting, rising up, and scattering or smashing anything set out on it._\n\n_It was snowing during the burial. The wind was blowing hard, as it always does here on this godforsaken Arctic Waste. Just north of the burial site rose Sheer Black Cliffs, as inaccessible as the Mountains of the Moon. The lanterns lit on_ Erebus _and_ Terror _were only the dimmest of glows through the blowing snow. Occasionally a fragment of Cold Moon would appear from between quickly moving clouds, but even this thin, pale moonlight was quickly lost in the snow and dark. Dear God, this is truly a Stygian bleakness._\n\n_Some of the strongest men from_ Terror _worked almost without pause since the hours right after Torrington's death, using pickaxe and spade to excavate his Grave \u2014 a regulation five feet deep, as commanded by Sir John. The Hole had been dug out of the most Severely Frozen ice and rock and one glance at it revealed to me what Labour had gone into its excavation. The flag was removed, and the coffin was lowered carefully, almost reverently, into the narrow Pit. Snow immediately covered the top of the coffin and Glistened in the light from our several lanterns. One man, one of Crozier's officers, set the wooden headboard in place and it was driven down into the frozen gravel with a few slams of a giant wooden hammer wielded by a giant of a seaman. The words on that carefully carved headboard read_\n\nSACRED\n\nTO\n\nTHE MEMORY OF\n\nJOHN TORRINGTON\n\nWHO DEPARTED\n\nTHIS LIFEJANUARY 1ST\n\nA.D. 1846\n\nON BOARD OF\n\nH.M. SHIP TERROR\n\nAGED 20 YEARS\n\n_Sir John conducted the Service and spoke the Eulogy. It went on for some time and the soft drone of his soft voice was interrupted only by the Wind and by the stamping of Feet as the men tried to avoid frostbite of the toes. I confess that I heard little of Sir John's eulogy \u2014 between the howling wind and my own wandering thoughts, oppressed by the loneliness of the place, by the memory of the striped-shirted body, limbs bound, that had just been lowered into that Cold Hole, and oppressed most of all by the Eternal blackness of the cliffs above the graveled isthmus._\n\n_4 January, 1846 \u2014_\n\n_Another man is dead._\n\n_One of our own here on HMS_ Erebus, _twenty-five-year-old John Hartnell, an able seaman. Just after what I still think of as 6:00_ _p.m., just as the tables were being lowered on chains for the men's dinner, Hartnell stumbled against his brother, Thomas, fell to the deck, coughed blood, and was dead within five minutes. Surgeon Stanley and I were with him when he died in the cleared part of the forward area of the lower deck which we use for Sick Bay._\n\n_This death stunned us. Hartnell had shown no symptoms of scurvy or consumption. Commander Fitzjames was there with us and could not hide his consternation. If this were some Plague or beginning of Scurvy moving through the crew, we needed to know at once. It was decided then and there, while the curtains were drawn and before anyone made ready to prepare John Hartnell for his coffin, that we would do a Postmortem Examination._\n\n_We cleared the table in the Sick Bay area, shielded our Actions further by moving some crates between the milling Men and ourselves, drew the curtain around our Labours as best we could, and I fetched my instruments. Stanley, although Chief Surgeon, suggested that I should do the work since I had studied as an anatomist. I made the initial Incision and began._\n\n_Immediately I realized that in my Haste I had used the inverted-Y incision that I had used in training on cadavers when I was in a rush. Rather than the more common Y, with the two arms of the incision reaching down from the shoulders and meeting at the base of the sternum, my upside-down Y incision had the arms of the Y starting near each hip and meeting near Hartnell's umbilicus. Stanley commented upon it and I was embarrassed._\n\n_\"Whatever is faster,\" I said softly to my fellow surgeon. \"We must do this quickly \u2014 the men hate knowing that bodies of their crewmates are being opened.\"_\n\n_Surgeon Stanley nodded and I continued. As if to Confirm my statement, Hartnell's younger brother, Thomas, began shouting and crying from just the other side of the curtain. Unlike Torrington's slow decline on_ Terror, _giving his crewmates time to come to terms with his death, time to parcel out his belongings and prepare letters for Torrington's mother, John Hartnell's sudden collapse and death had shocked the men here. None of them could abide the idea that the ship's surgeons were cutting into the body. Now only the bulk, rank, and demeanor of Commander Fitzjames stood between the angry brother, confused seamen, and our Sick Bay. I could hear that the younger Hartnell's messmates and Fitzjames's presence were holding him back, but even as my scalpel cut through tissue and my knife and rib spreader opened the corpse for examination, I could hear the Muttering and Anger just a few yards beyond the curtain._\n\n_First I removed Hartnell's heart, cutting away part of the trachea with it. I held it up to the lantern light, and Stanley took it and washed away blood with a dirty rag. We both inspected it. It looked normal enough \u2014 not visibly diseased. With Stanley still holding the Organ close to the light, I made one cut in the right ventricle, then one in the left. Peeling the tough muscle back, both Stanley and I reviewed the valves there. They seemed healthy._\n\n_Dropping Hartnell's heart back into his abdominal cavity, I dissected the lower part of the able seaman's lungs with quick strokes of my scalpel._\n\n_\"There,\" said Surgeon Stanley._\n\n_I nodded. There were obvious signs of scarring and other indications of Consumption, as well as signs that the seaman recently had been suffering from pneumonia. John Hartnell, like John Torrington, had been tubercular, but this older, stronger \u2014 and according to Stanley \u2014 harsher and louder sailor had concealed the Symptoms, perhaps even from himself. Until today, when he keeled over and died just minutes before getting his salt pork._\n\n_Pulling and cutting the Liver free, I held it under the light, and both Stanley and I believed that we noticed adequate confirmation of the consumption as well as indications that Hartnell had been too heavy a Drinker for too long a time._\n\n_Just yards away on the other side of the curtain, Hartnell's brother, Thomas, was shouting, furious, being held in check only by Commander Fitzjames's stern bark. I could tell from the voices that several of the other officers \u2014 Lieutenant Gore, Lieutenant Le Vesconte and Fairholme, even Des Voeux, the mate \u2014 had joined in calming and intimidating the Mob of sailors._\n\n_\"Have we seen enough?\" whispered Stanley._\n\n_I nodded again. There had been no sign of Scurvy on the body, on the face or in the mouth, or in the organs. While it remained a Mystery how the consumption or pneumonia or a combination of the two had been able to kill the able-bodied seaman so quickly, it was at least obvious that we had nothing to fear from some Plaguelike Disease._\n\n_The noise from the crew's Berthing Space was growing Louder, so I quickly thrust the lung samples, liver, and other organs back in the abdominal cavity with the heart, taking no care to set them in proper place, more or less squeezing them into a Mass, and then I returned Hartnell's chest plate roughly back in place. (Later I was to Realize that I had set it in upside down.) Chief Surgeon Stanley then closed up the inverted-Y incision, using a large needle and heavy sail thread with a quick, confident motion that would have done credit to any sailmaker._\n\n_Within another minute we had Hartnell's clothes back on \u2014 rigor mortis was beginning to be a problem \u2014 and we thrust the curtain aside. Stanley \u2014 whose voice is deeper and more resonant than mine \u2014 assured Hartnell's brother and the other men that all we had remaining was to wash their crewmate's body so that they could prepare it for burial._\n\n_6 January, 1846 \u2014_\n\n_For some reason this Burial Service was Harder on me than the first. Again we had the solemn Procession from the ship \u2014 with only_ Erebus _and its crew involved this time, although Dr. McDonald, Surgeon Peddie, and Captain Crozier joined us from_ Terror.\n\n_Again the flag-covered coffin \u2014 the men had dressed Hartnell's upper body in three layers, including his brother Thomas's best shirt, but had wrapped his naked lower body in only a shroud, leaving the top half of the coffin open for several hours in the black-creped Sick Bay on the lower deck before the nails were hammered in for the burial service. Again the slow sledge procession from the Frozen Sea to the Frozen Shore, lanterns bobbing in the black night, although the stars were out this Midday and no snow fell. The Marines had work to do, since three of the Great White Bears came sniffing closer, looming like white wraiths out of the ice blocks, and the men had to fire muskets at them to drive them away \u2014 visibly wounding one bear in the side._\n\n_Again the Eulogy from Sir John \u2014 although shorter this time, since Hartnell was not as well liked as young Torrington had been \u2014 and again we walked back across the creaking, squeaking, moaning ice alone, under the stars dancing in the Cold this time, the only sound behind us the dwindling scrape of spades and pickaxes filling in the frozen soil in the new hole next to Torrington's nicely tended grave._\n\n_Perhaps it was the black cliff face Looming over All that murdered my Spirits this second burial. Although I deliberately stood where my back was to the Cliff this time, closer to Sir John so that I could hear the Words of Hope and Solace, I was always aware of that cold, black, vertical, lifeless and lightless slab of insensate Stone behind me \u2014 a portal, it seemed, to that Country from Which No Man Has Ever Returned. Compared to the Cold Reality of that black, featureless stone, even Sir John's compassionate and inspired words had little effect._\n\n_The morale on both ships is very low. We are not yet a Full Week into the new year, and already two of our Company have died. Tomorrow the four of us surgeons have agreed to Meet in a Private Place \u2014 the carpenter's room belowdecks on_ Terror _\u2014 to discuss what should be done to avoid more Mortality in what seems to be a Cursed Expedition._\n\n_The headstone on this second grave read_\n\nSACRED TO THE MEMORY OF\n\nJOHN HARTNELL, A.B. OF H.M.S.\n\nEREBUS\n\nDIED JANUARY 4TH, 1846\n\nAGED 25 YEARS\n\n'THUS SAITH THE LORD OF HOSTS, CONSIDER YOUR WAYS'\n\nHAGGAI, I., 7.\n\n_The wind has come up in the last hour, it is almost Midnight and most of the lamps are out here on the lower deck of_ Erebus. _I listen to the wind howl and think of those two cold Low Heaps of Loose Stone out on that black, windy isthmus, and I think of the dead men in those two cold Holes, and I think of the Featureless Black Face of Rock, and I can imagine the fusillade of snow pellets already working to eradicate the letters on the wooden headstones._\n\nFRANKLIN\n\n_Lat. 70\u00b0-03\u2032-29\u2033 N., Long. 98\u00b0- 20\u2032 W._\n\nApproximately 28 miles NNW of King William Land, 3 September, 1846\n\nCaptain Sir John Franklin had rarely been so pleased with himself.\n\nThe previous winter frozen in at Beechey Island, hundreds of miles northeast of his present position, had been uncomfortable in many ways \u2014 he would be the first to admit that to himself or to a peer, although he had no peers on this expedition. The death of three members of the expedition, first Torrington and Hartnell so early in January, then Private William Braine of the Royal Marines on 3 April, all of consumption and pneumonia, had been a shock. Franklin was not aware of any other Navy expedition losing three men of natural causes so early in their endeavor.\n\nIt was Franklin himself who had chosen the inscription on the thirty-two-year-old Private Braine's headstone \u2014 _\"Choose this day whom ye shall serve,\"_ _Joshua, ch. xxiv, 15_ \u2014 and for a short while the words had seemed as much a challenge to the unhappy crews of _Erebus_ and _Terror,_ not yet near mutiny but neither so far away from it, as it was a message to the nonexistent passersby of Braine's, Hartnell's, and Torrington's lonely graves on that terrible spit of gravel and ice.\n\nNonetheless, the four surgeons met and conferred after Hartnell's death and decided that incipient scurvy might be weakening the men's constitutions, allowing pneumonia and such congenital defects as consumption to rise to lethal proportions. Surgeons Stanley, Goodsir, Peddie, and McDonald recommended to Sir John that the men's diet be changed \u2014 fresh food when possible (although there was almost none except polar bear possible in the dark of winter, and they had discovered that eating the liver of that great, ponderous beast could be fatal for some unknown reason) and, failing finding fresh meat and vegetables, cutting back on the men's preferred salted pork and beef, or salted birds, and relying more on the tinned foods \u2014 vegetable soups and the like.\n\nSir John had gone along with the recommendation, ordering the diet on both ships changed so that no less than half the meals were prepared with tinned foods from stores. It seemed to have turned the trick. No more men died, or were even seriously sick, between Private Braine's death in early April and the day both ships were freed from their icy imprisonment within the harbor of Beechey Island in late May of 1846.\n\nAfter that, the ice broke up quickly and Franklin, following the paths through the leads chosen by his two fine ice masters, steamed and sailed south and west, going, as the captains of Sir John's generation liked to say, like smoke and oakum.\n\nAlong with the sunlight and open water, animals, birds, and aquatic life returned in plenitude. During those long, slow, arctic summer days, where the sun remained above the horizon until almost midnight and the temperature sometimes rose above freezing, the skies were filled with migrating birds. Franklin himself could identify the petrels from the teals, eider ducks from the little auks, and the sprightly little puffins from all others. The ever-widening leads around _Erebus_ and _Terror_ were alive with right whales that would have been the envy of any Yankee whaler, and there was a profusion of cod, herring, and other small fish, as well as the large beluga and bowhead whales. The men put out the whaling boats and fished, often shooting some of the small whales just for sport.\n\nEvery hunting party came back with fresh game for the tables each night \u2014 birds, of course, but also those confounded ringed and harp seals, so impossible to shoot or catch in their holes in the winter, now brazen on the open ice and easy targets. The men did not enjoy the taste of the seals \u2014 too oily and astringent \u2014 but something about the blubber in the slimy beasts appealed to all their winter-starved appetites. They also shot the large bellowing walruses visible through telescopes tusking away for oysters along the shores, and some hunting parties returned with the pelts and flesh of the white arctic fox. The men ignored the lumbering polar bears unless the waddling beasts seemed ready to attack or contest the kill of the human hunters. No one really liked the taste of the white bears and certainly not when there was so much tastier game to be found.\n\nFranklin's orders included an option: if he \"found his way toward the Southern Approach to the North-West Passage blocked by Ice or other Obstacles,\" to turn north and to follow the Wellington Passage into \"the Open Polar Sea\" \u2014 in essence, to sail to the north pole. But Franklin did what he had done without question his entire life: he followed his primary orders. This second summer in the arctic, his two ships had sailed south from Devon Island, Franklin leading HMS _Erebus_ and HMS _Terror_ past Cape Walker into the unknown waters of an icy archipelago.\n\nThe previous summer, it had seemed as if he would have to settle for sailing to the north pole rather than finding the North-West Passage. Captain Sir John Franklin had reason to be proud of his speed and efficiency so far. During his shortened summer voyage time that year before, 1845 \u2014 they had departed England late and Greenland even later than planned \u2014 he had nonetheless crossed Baffin Bay in record time, passed through Lancaster Sound south of Devon Island, then through Barrow Strait, and found his way south past Walker Point blocked by ice so late in August. But his Ice Masters reported open water to the north, past the western reaches of Devon Island into the Wellington Channel, so Franklin obeyed his secondary orders and turned north toward what could be an ice-free passage into the Open Polar Sea and the north pole.\n\nThere had been no opening to the fabled Open Polar Sea. The Grinnell Peninsula, which might have been part of an unknown Arctic Continent for all the men of the Franklin Expedition knew, had blocked their way and forced them to follow open water north by west, then almost due west, until they reached the western tip of that peninsula, turned north again, and encountered a solid mass of ice that extended north from the Wellington Channel apparently to infinity. Five days of sailing along that high wall of ice convinced Franklin, Fitzjames, Crozier, and the ice masters that there was no Open Polar Sea north of the Wellington Channel. At least not that summer.\n\nWorsening ice conditions made them turn south, around the landmass previously known as only Cornwallis Land but now understood to be Cornwallis Island. If nothing else, Captain Sir John Franklin knew, his expedition had solved that puzzle.\n\nWith pack ice quickly freezing in place that late summer of 1845, Franklin had finished circumnavigating the huge, barren Cornwallis Island, reentered the Barrow Strait north of Cape Walker, confirmed that the way south past Cape Walker was still blocked \u2014 now solid with ice \u2014 and sought out their winter anchorage at little Beechey Island, entering a little harbour they had reconnoitered two weeks earlier. They'd arrived just in time, Franklin knew, for the day after they anchored in the shallow water of that harbour, the last open leads in Lancaster Sound beyond closed up and the moving pack ice would have made any more sailing impossible. It was doubtful if even such masterpieces of reinforced iron-and-oak technology as _Erebus_ and _Terror_ would have survived the winter out in the channel ice.\n\nBut now it was summer and they had been sailing south and west for weeks, restoring their provisions when they could, following every lead, seeking out any glint of open water they could spy from the lookout's position high on the main mast, and every day smashing and forcing their way through the ice when they had to.\n\nHMS _Erebus_ continued to lead the way in the ice-breaking, as was her right as the flagship and her logical responsibility as the heavier ship with a more powerful \u2014 five horsepower more powerful \u2014 steam engine, but \u2014 confound it! \u2014 the long shaft to the screw had been bent by underwater ice; it would neither retract nor work properly, and _Terror_ had moved into the lead position.\n\nAnd with the icy shores of King William Land visible no more than fifty miles ahead of them to the south, the ships had moved out from under the protection of the huge island to their north \u2014 the one which had blocked their way directly to the southwest past Cape Walker, where his orders had directed him to sail, and instead had forced him south through Peel Sound and previously unexplored straits. Now the ice to the south and west had become active and almost continuous once again. Their pace had slowed to a crawl. The ice was thicker, the icebergs more frequent, the leads thinner and farther apart.\n\nThis morning of 3 September, Sir John had called a conference of his captains, top officers, engineers, and ice masters. The crowd fit comfortably into Sir John's personal cabin; where this space on HMS _Terror_ served as a Great Cabin for the officers, complete with libraries and music, the width of the stern of HMS _Erebus_ was Sir John Franklin's private quarters \u2014 twelve feet wide by an amazing twenty feet long, with a private commode \"seat of ease\" in a room to itself on the starboard side. Franklin's private privy was almost exactly the size of Captain Crozier's and all the other officers' entire cabins.\n\nEdmund Hoar, Sir John's steward, had lengthened the dining table until it could accommodate all the officers present \u2014 Commander Fitzjames, Lieutenants Gore, Le Vesconte, and Fairholme from _Erebus,_ Captain Crozier and Lieutenants Little, Hodgson, and Irving from _Terror._ Besides those eight officers seated on either side of the table \u2014 Sir John sat at its head near the starboard bulkhead and entrance to his private head \u2014 also present, standing at the foot of the table, were the two ice masters, Mr. Blanky from _Terror_ and Mr. Reid from _Erebus,_ as well as the two engineers, Mr. Thompson from Crozier's ship and Mr. Gregory from the flagship. Sir John had also asked one of the surgeons, Stanley from _Erebus,_ to be in attendance. Franklin's steward had set out grape juice, cheeses, and ship's biscuits, and there was a short period of chatting and relaxing before Sir John called the conference to order.\n\n\"Gentlemen,\" said Sir John, \"I am sure you all know why we are gathered here. Our expedition's advance the last two months, thanks to the graciousness of God, has been wonderfully successful. We have left Beechey Island almost three hundred and fifty miles behind us. Lookouts and our sledge scouts still report glimpses of open water far to our south and west. It still may be in our power \u2014 God willing \u2014 to reach this open water and to navigate the North-West Passage this very autumn.\n\n\"But the ice to our west is increasing, I understand, in both thickness and frequency. Mr. Gregory reports that _Erebus_ 's main shaft has been damaged by ice and that although we can make headway under steam, the flagship's effectiveness has been compromised. Our coal supplies are dwindling. Another winter will soon be upon us. In other words, gentlemen, we must decide today what our course of action and direction shall be. I think it is not unfair to say that the success or failure of our expedition shall be determined by what we decide here.\"\n\nThere was a long silence.\n\nSir John gestured to the red-bearded ice master of HMS _Erebus._ \"Perhaps it would be helpful, before we venture opinions and open discussion, to hear from our ice masters, engineers, and surgeon. Mr. Reid, could you inform the others of what you told me yesterday about our current and projected ice conditions?\"\n\nReid, standing on the _Erebus_ side of the five men at the end of the table, cleared his throat. Reid was a solitary sort and speaking in such exalted company made his face flush redder than his beard.\n\n\"Sir John,... Gentlemen... it ain't no secret that we've been God-da\u2014... that is... darned lucky in terms of ice conditions since the ships was released from ice in May and since we left Beechey Island harbour around the first of June. While we was in the straits, we been plowing through mostly sludge ice. That ain't no problem. Nights \u2014 them few hours of darkness we have what's called nights up here \u2014 we cut through pancake ice, that's like what we been seeing the last week as the sea's always on the verge of freezing, but that ain't no real problem either.\n\n\"We been able to stay away from the young ice along the shores \u2014 that's more serious stuff. Behind that's the fast ice that'll tear the hull off even a ship as reinforced as this here and _Terror_ in the lead. But as I say, we stayed away from fast ice... so far.\"\n\nReid was sweating, obviously wishing he hadn't gone on so long but also knowing that he hadn't fully addressed Sir John's question yet. He cleared his throat and continued.\n\n\"So with the moving ice, Sir John and your honours, we ain't had much problem with the brash ice and thicker drift ice, and the bergy bits \u2014 them little bergs what broke off from the real bergs \u2014 we been able to avoid them because of the wide leads and open water we've been able to find. But all that's coming to an end, sirs. What with the nights getting longer, the pancake ice is always there now and we're running into more and more of them growlers and hummocky flows. And it's the hummocky flows that's got Mr. Blanky and me worried.\"\n\n\"Why is that, Mr. Reid?\" asked Sir John. His expression showed his habitual boredom at discussions of the different ice conditions. To Sir John, ice was ice \u2014 something to be broken through, gone around, and overcome.\n\n\"It's the snow, Sir John,\" said Reid. \"The deep snow atop 'em, sir, and the tidemarks on the side. Such always signifies old pack ice ahead, sir, real screwed pack, and that's where we get frozen in, you see. And as far as we can see or sledge ahead to the south and west, sirs, it's all pack ice, except for the possible glint of open water way down south of King William Land.\"\n\n\"The North-West Passage,\" Commander Fitzjames said softly.\n\n\"Perhaps,\" said Sir John. \"Most probably. But to get there we shall have to cross through more than a hundred miles of pack ice \u2014 perhaps as much as two hundred miles of it. I am told that the ice master of _Terror_ has a theory about why the conditions worsen to our west. Mr. Blanky?\"\n\nThomas Blanky did not blush. The older ice master's voice was a staccato explosion of syllables as blunt as musket fire.\n\n\"It's death to enter that pack ice. We've come too far already. The fact is, since we came out of Peel Sound, we've been looking at an ice stream as bad as anything north of Baffin Bay and it's getting worse every day.\"\n\n\"Why is that, Mr. Blanky?\" asked Commander Fitzjames. His confident voice showed a slight lisp. \"This late in the season, I understand that we should still have open leads until the sea actually freezes, and close to the mainland, say southwest of the peninsula of King William Land, we should have some open water for another month or more.\"\n\nIce Master Blanky shook his head. \"No. This isn't no pancake ice or sludge ice, gentlemen, it's _pack ice_ we're looking at. It's coming down from the northwest. Think of it as a series of giant _glaciers_ \u2014 calving bergs and freezing the sea for hundreds of miles as it flows south. We've been protected from it, is all.\"\n\n\"Protected by what?\" asked Lieutenant Gore, a strikingly handsome and personable officer.\n\nIt was Captain Crozier who answered, nodding at Blanky to step back. \"By all the islands to our west as we've come south, Graham,\" said the Irishman. \"Just as we discovered a year ago that Cornwallis Land was an island, we know now that Prince of Wales Land is really Prince of Wales Island. The bulk of it has been blocking the full force of the ice stream until we came out of Peel Sound. Now we can see that it's full pack ice being forced south between whatever islands are up there to our northwest, possibly all the way to the mainland. Whatever open water is down there along the coast to the south won't last long. Nor will we if we forge ahead and try to winter out here in the open pack ice.\"\n\n\"That is one opinion,\" said Sir John. \"And we thank you for it, Francis. But we must decide now on our course of action. Yes,... James?\"\n\nCommander Fitzjames looked, as he almost always did, relaxed and in charge. He had actually put on weight during the expedition so his buttons appeared ready to pop from his uniform. His cheeks were rosy and his blond hair hung in longer curls than he'd worn in England. He smiled at everyone along the table.\n\n\"Sir John, I agree with Captain Crozier that to be caught out in the pack ice we're facing would be unfortunate, but I do not believe that will be our fate should we forge on. I believe that it is imperative that we get as far south as we can \u2014 either to reach the open water to realize our goal of finding the North-West Passage, which I think we shall do before winter sets in, or simply to find safer waters near the coast, perhaps a harbour where we can winter in relative comfort as we did at Beechey Island. At the very least, we know from Sir John's earlier expeditions overland and from previous Naval expeditions that the water tends to stay open much later near the coast because of the warmer waters coming in from the rivers.\"\n\n\"And if we don't reach open water or the coast by going southwest?\" Crozier asked softly.\n\nFitzjames made a deprecating gesture. \"At least we will be closer to our goal come the thaw next spring. What's our alternative, Francis? You aren't seriously suggesting returning up the strait to Beechey or trying to retreat to Baffin Bay?\"\n\nCrozier shook his head. \"Right now we can as easily sail to the _east_ of King William Land as to the west \u2014 more easily, since we know from our lookouts and scouts that there is still ample open water to the east.\"\n\n\"Sail to the east of King William Land?\" said Sir John, his voice incredulous. \"Francis, that would be a dead end. We would be sheltered by the peninsula, yes, but frozen in hundreds of miles east of here in a long bay that might not thaw next spring.\"\n\n\"Unless... ,\" said Crozier, looking around the table, \"unless King William Land is also an island. In which case we would have the same protection from the pack ice flowing from the northwest that Prince of Wales Island has been giving us the past month of travel. It would be probable that the open water on the east side of King William Land will extend almost to the coast, where we can sail west along the warmer waters there for more weeks, perhaps find a perfect harbour \u2014 perhaps at a river's mouth \u2014 if we have to spend a second winter in the ice.\"\n\nThere was a long silence in the room.\n\n_Erebus_ lieutenant H. T. D. Le Vesconte cleared his throat. \"You believe in the theories of that eccentric Dr. King,\" he said softly.\n\nCrozier frowned. He knew that the theories of Dr. Richard King, not even a Navy man, a mere civilian, were disliked and discarded, primarily because King believed \u2014 and had very vocally expressed \u2014 that such large Naval expeditions as Sir John's were foolish, dangerous, and absurdly expensive. King believed, based upon his mapping and experience with Back's overland expedition years before, that King William Land was an island, while Boothia, the ostensible island even farther to their east, was actually a long peninsula. King argued that the easiest and safest way to find the North-West Passage was to send small parties overland in northern Canada and to follow the warmer coastal waters west, that the hundreds of thousands of square miles of sea to the north were a dangerous maze of islands and ice streams that could swallow up a thousand _Erebus_ es and _Terror_ s. Crozier knew that there was a copy of King's controversial book in _Erebus_ 's library \u2014 he had checked it out and read it and it was still in Crozier's cabin on _Terror._ But he also knew that he was the only man on the expedition who had, or would, read the book.\n\n\"No,\" said Crozier, \"I'm not subscribing to King's theories, I am merely suggesting a strong possibility. Look, we thought that Cornwallis Land was huge, perhaps part of the Arctic Continent, but we sailed around it in a few days. Many of us thought that Devon Island continued north and west directly into the Open Polar Sea, but our two ships found the western end of it and we saw the open channels north.\n\n\"Our orders instructed us to sail directly southwest from Cape Walker, but we found that Prince of Wales Land was directly in the way \u2014 and what is more pertinent, that it is, almost without doubt, an island. And the low strip of ice we glimpsed to our east while heading south may well have been a frozen strait \u2014 separating Somerset _Island_ from Boothia Felix and showing that King was wrong, that Boothia is not a continuous peninsula all the way north to Lancaster Sound.\"\n\n\"There is no evidence that the low area of ice we saw was a strait,\" said Lieutenant Gore. \"It makes more sense to consider it a low ice-covered isthmus such as we saw on Beechey Island.\"\n\nCrozier shrugged. \"Perhaps, but our experience on this expedition has been that landmasses previously thought to be very large or connected have in truth been shown to be islands. I suggest that we reverse course, avoid the pack ice to the southwest, and sail east and then south down the eastern coast of what may well be King William _Island._ At the very least, we will be sheltered from this... seaborne _glacier_ that Mr. Blanky talks about... and should we discover the worst, that it is a long, narrow bay, odds are very great that we could sail north again around the point of King William Land next summer and be right back here and none the worse for wear.\"\n\n\"Except for the coal burned and the precious time lost,\" said Commander Fitzjames.\n\nCrozier nodded.\n\nSir John rubbed his round and well-shaved cheeks.\n\nIn the silence, _Terror_ 's engineer, James Thompson, spoke. \"Sir John, gentlemen, since the issue of the ships' coal reserves has been brought up, I would like to mention that we are very, very close to reaching \u2014 and I mean this quite literally \u2014 a point of no return in terms of our fuel. Just in the past week, using our steam engines to force a way through the fringes of this pack ice, we've gone through more than a quarter of our remaining coal reserves. We are now just above fifty percent of our coal remaining... less than two weeks of normal steaming, but only days' worth trying to force the ice as we have. Should we be frozen in for another winter, we will be burning much of that reserve just to heat the ships again.\"\n\n\"We could always send a party ashore to cut trees for firewood,\" said Lieutenant Edward Little, sitting at Crozier's left.\n\nFor a minute every man in the room except Sir John laughed heartily. It was a welcome break in the tension. Perhaps Sir John was remembering his first overland expeditions north to the coastal regions now to their south. The mainland tundra extended for nine hundred barren miles south from the coast before one would see the first tree or serious shrub.\n\n\"There is one way to maximize our steaming distances,\" Crozier said softly into the more relaxed silence following the laughter.\n\nEveryone's head turned toward the captain of HMS _Terror._\n\n\"We transfer all the crew and coal from _Erebus_ to _Terror_ and make a run for it,\" continued Crozier. \"Either through the ice to the southwest or to reconnoiter down the east coast of King William Land or Island.\"\n\n\"Go for broke,\" said Ice Master Blanky into the now stunned silence. \"Aye, that makes sense.\"\n\nSir John could only blink. When he finally found his voice, it still sounded incredulous, as if Crozier had made a second joke that he could not understand. \"Abandon the flagship?\" he said at last. \"Abandon _Erebus?_ \" He glanced around as if just having the other officers look at his cabin would settle this issue once and for all \u2014 the bulkheads lined with shelves and books, the crystal and china on the table, the three Preston Patent Illuminators set into the width of the overhead, allowing rich late-summer light to stream into the cabin.\n\n\"Abandon _Erebus,_ Francis?\" he said again, his voice stronger but spoken in a tone of someone who wants to be let in on a rather obscure joke.\n\nCrozier nodded. \"The main shaft is bent, sir. Your own engineer, Mr. Gregory, has told us that it cannot be repaired, nor retracted any longer, outside of a dry dock. Certainly not while we are in pack ice. It will only get worse. With two ships, we have only a few days' or a week's worth of coal for the battle necessary to fight the pack ice. We'll all be frozen in \u2014 both ships \u2014 if we fail. If we freeze into the open sea to the west of King William Land, we have no idea where the current will move the ice of which we will be a part. The odds are great that we could be thrown into the shallows along the lee shore there. That means the destruction of even such wonderful ships as these.\" Crozier nodded around him and at the skylights above.\n\n\"But if we consolidate our fuel in the less damaged ship,\" continued Crozier, \"and especially if we have some luck finding open water down the east side of King William Land, we will have much more than a month's fuel for steaming west along the coast as fast as we can. _Erebus_ would have been sacrificed, but we might \u2014 we _will_ \u2014 reach Point Turnagain and familiar points along the coast within a week. Complete the North-West Passage into the open Pacific this year instead of next.\"\n\n\"Abandon _Erebus?_ \" repeated Sir John. He did not sound cross or angry, only perplexed by the absurdity of the notion being discussed.\n\n\"Conditions would be very cramped aboard _Terror,_ \" said Commander Fitzjames. He seemed to be seriously considering the notion.\n\nCaptain Sir John turned to his right and stared at his favorite officer. Sir John's face was slowly assuming the cold smile of a man who has not only been left out of a joke on purpose but may well be the butt of it.\n\n\"Crowded, but not intolerably so for a month or two,\" said Crozier. \"My Mr. Honey and your ship's carpenter, Mr. Weekes, will supervise the breaking down of interior bulkheads \u2014 all officers' quarters to be dismantled except for the Great Room, which could be turned into Sir John's quarters aboard _Terror,_ and perhaps the officers' mess. That would give us ample room, even for another year or more on the ice. These old bomb ships have a great amount of space belowdecks, if nothing else.\"\n\n\"It would take some time to transfer the coal and ship's stores,\" said Lieutenant Le Vesconte.\n\nCrozier nodded again. \"I've had my purser, Mr. Helpman, work out some preliminary figures. You may remember that Mr. Goldner, the expedition's provisioner of canned foods, failed to deliver the bulk of his goods until less than forty-eight hours before we sailed, so we had to repack both ships to a great extent. We did so in time to meet our departure date. Mr. Helpman estimates that with both crews working through the long daylight, sleeping on half-watch schedules, that everything we can hold on one ship can be transferred to _Terror_ in just under three days. We'd be a crowded family for some weeks, but it would be as if we're starting anew on the expedition \u2014 coal reserves topped off, food for another year, a ship in full working condition.\"\n\n\"Go for broke,\" repeated Ice Master Blanky.\n\nSir John shook his head and chuckled as if he had finally had enough of this particular joke. \"Well, Francis, that is a very... _interesting_... speculation, but of course we shall not be abandoning _Erebus._ Nor _Terror_ either, should your ship suffer some minor misfortune. Now, the one thing I've not heard at this table today is a suggestion to _retreat_ to Baffin Bay. Am I correct in the assumption that no one is suggesting that?\"\n\nThe room was silent. Overhead came the rumble and scrape of crewmen holystoning the decks for the second time that day.\n\n\"Very well then, it is decided,\" said Sir John. \"We shall press forward. Not only do our orders direct us to do this, but as several of you gentlemen have pointed out, our safety increases the closer we get to the coast of the mainland, even if the land itself there is every bit as inhospitable as the dreadful islands we have passed up here. Francis, James, you may go tell your crews of our decision.\"\n\nSir John stood.\n\nFor a dazed second the other captains, officers, ice masters, engineers, and surgeon could only stare, but then the Naval officers stood quickly, nodded, and began filing out of Sir John's huge cabin.\n\nThe surgeon Stanley was plucking at Commander Fitzjames's sleeve as the men went forward along the narrow companionway and stomped up the ladder to the deck.\n\n\"Commander, Commander,\" Stanley said, \"Sir John never called on me to report, but I wanted to tell everyone about the increasing numbers of putrid foodstuff we've been finding in the tinned goods.\"\n\nFitzjames smiled but freed his arm. \"We'll arrange a time for you to tell Captain Sir John in private, Mr. Stanley.\"\n\n\"But I've told _him_ in private,\" persisted the little surgeon. \"It's the other officers I wanted to inform in case...\"\n\n\"Later, Mr. Stanley,\" said Commander Fitzjames.\n\nThe surgeon was saying something else, but Crozier passed beyond earshot and waved to John Lane, his bosun, to bring his gig alongside for the sunny ride back up the narrow lead to where _Terror_ 's bow was wedged in the thickening pack ice. Black smoke still poured from the leading ship's funnel.\n\nHeading southwest into the pack ice, the two ships made slow headway for another four days. HMS _Terror_ burned coal at a prodigious rate, using its steam engine to throw itself against the ever-thicker pack ice. The glint of possible open water far to the south had disappeared, even on sunny days.\n\nThe temperature dropped suddenly on 9 September. The ice on the long thin line of open water behind the trailing _Erebus_ covered over with pancake ice and then froze solid. The sea around them was already a lifting, surging, static white mass of growlers, real icebergs, and sudden pressure ridges.\n\nFor six days, Franklin tried every trick in his arctic inventory \u2014 spreading black coal powder on the ice ahead of them to melt it more quickly, backing sail, sending out fatigue parties day and night with their giant ice saws to remove the ice in front of them block by block, shifting ballast, having a hundred men at a time hack away with chisels, shovels, picks, and poles, setting kedge anchors far ahead of them in the thickening ice and winching _Erebus_ \u2014 which had resumed the lead ahead of _Terror_ on the last day before the ice had suddenly thickened \u2014 a yard at a time. Finally Franklin ordered every able man onto the ice, rigged lines for everyone and sledge harnesses for the largest men among them, and tried hauling the ships forward a sweating, cursing, shouting, spirit-killing, gut-wrenching, backbreaking inch at a time. Always, promised Sir John, was the reality of open coastal water just another twenty or thirty or fifty miles ahead of them.\n\nThe open water might as well have been on the surface of the moon.\n\nDuring the lengthening night of 15 September, 1846, the temperature plummeted to below zero and the ice began moaning and scraping against both ships' hulls. In the morning, everyone who came on deck could see for themselves that in each direction the sea had become a white solid stretching to the horizon. Between sudden snow squalls, both Crozier and Fitzjames were able to get adequate sun sightings to fix their positions. Each captain figured that they were beset at roughly 70 degrees 5 minutes north latitude, 98 degrees, 23 minutes west longitude, some twenty-five miles off the northwest shore of King William Island, or King William Land, whichever the case might be. It was a moot point now.\n\nThey were in open sea ice \u2014 moving pack ice \u2014 and stranded directly in front of the full onslaught of Ice Master Blanky's \"moving glacier,\" bearing down on them from polar regions to the northwest from all the way to the unimaginable North Pole. There was not a sheltering harbour, to their knowledge, within one hundred miles and no way to get there if there were one.\n\nAt two o'clock that afternoon, Captain Sir John Franklin ordered the boiler fires to be drawn on both _Erebus_ and _Terror._ Steam was let down in both boilers. Just enough pressure would be kept up to move warmed water through the pipes heating the lower decks of each ship.\n\nSir John made no announcement to the men. None was required. That night as the men settled into their hammocks on _Erebus_ and as Hartnell whispered his usual prayer for his dead brother, thirty-five-year-old Seaman Abraham Seeley, in the hammock next to him, hissed, \"We're in a world of shit now, Tommy, and not your prayers nor neither Sir John's is going to get us out of it... not for another ten months at least.\"\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n11 November, 1847\n\nIt has been one year, two months, and eight days since Sir John's eventful conference aboard _Erebus,_ and both ships are frozen in the ice roughly where they were that September day in 1846. Although the current from the northwest moves the entire mass of ice, over the past year it has rotated ice, icebergs, pressure ridges, and both trapped Royal Navy ships in slow circles so that their position has remained about the same, stranded some twenty-five miles north-northwest of King William Land and slowly revolving like a blotch of rust on one of the metal music disks in the officers' Great Room.\n\nCaptain Crozier has spent this November day \u2014 or rather those hours of darkness which once included daylight as a component \u2014 searching for his missing crewmen William Strong and Thomas Evans. There is no hope for either man, of course, and there is great risk that others will be taken by the thing on the ice, but they search nonetheless. Neither captain nor crew would have it any other way.\n\nFour teams of five men each, one man to carry two lanterns and four ready with shotguns or muskets, search in four-hour shifts. As one team comes in frozen and shaking, a replacement team waits on deck in cold-weather slops, guns cleaned, loaded, and ready, lanterns filled with oil, and they resume the search in the quadrant the other team has just quit. The four teams are moving out from the ship in ever-widening circles through the ice jumbles, their lanterns now visible to the lookouts on deck through the icy mist and darkness, now obscured by growlers, ice boulders, pressure ridges, or distance. Captain Crozier and a seaman with a red lamp move from quadrant to quadrant, checking with each team and then returning to _Terror_ to look in on the men and conditions there.\n\nThis goes on for twelve hours.\n\nAt two bells in the first dogwatch \u2014 6:00 p.m. \u2014 the last search parties all come in, none having found the missing men but several seamen shamefaced at having fired their weapons at wind shrieking among the jagged ice or at the ice itself, thinking some serac a looming white bear. Crozier is the last one in and follows them down to the lower deck.\n\nMost of the crewmen have stored their wet slops and boots and gone forward to their mess at tables that have all been cranked down on chains and the officers gone aft for their meal by the time Crozier comes down the ladder. His steward, Jopson, and first lieutenant, Little, hurry over to help him out of his ice-rimmed outer layers.\n\n\"You're frozen, Captain,\" says Jopson. \"Your skin is white with frostbite. Come back aft to the officers' mess for supper, sir.\"\n\nCrozier shakes his head. \"I need to go talk to Commander Fitzjames. Edward, has there been a messenger from his ship while I was out?\"\n\n\"No, sir,\" says Lieutenant Little.\n\n\"Please eat, Captain,\" persists Jopson. For a steward he's a large man, and his deep voice becomes more of a growl than a whine when he's imploring his captain.\n\nCrozier shakes his head. \"Be so kind as to wrap up a couple of biscuits for me, Thomas. I'll chew on them as I walk to _Erebus._ \"\n\nJopson shows his displeasure at this foolish decision but hurries forward to where Mr. Diggle is busy at his huge stove. Just now, at dinnertime, the lower deck is as toasty warm as it's going to be in any twenty-four-hour period \u2014 the temperature rising as high as the mid-forties. Very little coal is being burned for heat these days.\n\n\"How many men do you want to go with you, Captain?\" asks Little.\n\n\"None, Edward. After the men have eaten, I want you to get at least eight parties on the ice for a final four hours of searching.\"\n\n\"But, sir, is it advisable for you to... ,\" begins Little but then stops.\n\nCrozier knows what he was going to say. The distance between _Terror_ and _Erebus_ is only a little more than a mile, but it is a lonely, dangerous mile and sometimes takes several hours to cross. If a storm comes up or the wind simply starts blowing the snow, men can become lost or no longer make progress into the gale. Crozier himself has forbidden men to make the crossing alone and when messages have to be sent, he dispatches at least two men along with orders to turn back at the first bad weather. Besides the two-hundred-foot-high iceberg now rising between the two ships, often blocking views even of flares and fires, the pathway \u2014 although worked at being kept shoveled open and relatively flat almost every day \u2014 is really a maze of constantly shifting seracs, ice-stepped pressure ridges, upturned growlers, and ice jumble mazes.\n\n\"It's all right, Edward,\" says Crozier. \"I'll take my compass.\"\n\nLieutenant Little smiles even though the joke is wearing thin after three years in the area. The ships are beset, as far as their instruments can measure, almost directly over the north magnetic pole. A compass is about as useful here as a divining rod.\n\nLieutenant Irving sidles up. The young man's cheeks glisten from applied salve where frostbite has left white patches and caused the skin to die and peel back. \"Captain,\" begins Irving in a rush, \"have you seen Silence out on the ice?\"\n\nCrozier has taken his cap and muffler off and is rubbing the ice out of his sweat- and mist-dampened hair. \"You mean she's not in her little hidey-hole behind the sick bay?\"\n\n\"No, sir.\"\n\n\"Did you look elsewhere on the lower deck?\" Crozier is mostly worried that with most of the men gone on watch and out in search parties, the Esquimaux witch has gotten into something she shouldn't have.\n\n\"Aye, sir. No sign of her. I've asked around and no one remembers seeing her since yesterday evening. Since before... the attack.\"\n\n\"Was she on deck when the thing attacked Private Heather and Seaman Strong?\"\n\n\"No one knows, Captain. She might have been. Only Heather and Strong were on deck then.\"\n\nCrozier lets out a breath. It would be ironic, he thinks, if their mystery guest, who first appeared on the day this nightmare began six months ago, has finally been carried off by the creature so linked with her appearance.\n\n\"Search the whole ship, Lieutenant Irving,\" he says. \"Every nook, cranny, cupboard, and cable locker. We'll use Occam's razor and assume that if she isn't on board that she's... been taken.\"\n\n\"Very well, sir. Shall I choose three or four men to help me in the search?\"\n\nCrozier shakes his head. \"Just you, John. I want everyone else back out on the ice searching for Strong and Evans in the hours before lamps-out, and if you don't find Silence, assign yourself to a party and join them.\"\n\n\"Aye, aye, sir.\"\n\nReminded of his casualty, Crozier goes forward through the men's mess to the sick bay. Usually at supper time, even in these dark days, there is the morale-lifting sound of conversation and laughter from the men at their mess tables, but tonight there is silence broken only by the scrape of spoons on metal and the occasional belch. The men are exhausted, slumped on their sea chests they use as chairs, and only tired, slack faces look up at their captain as he squeezes past.\n\nCrozier knocks on the wooden post to the right of the sick bay curtain and passes through.\n\nSurgeon Peddie looks up from some sewing he is doing on Able Seaman George Cann's left forearm at a table in the centre of the space. \"Good evening, Captain,\" says the surgeon. Cann knuckles his forehead with his good hand.\n\n\"What happened, Cann?\"\n\nThe young sailor grunts. \"Fucking shotgun barrel slides up me sleeve and touches me fucking bare arm when I was climbing a fucking ice ridge, Captain, pardon the language. I pull the shotgun out and six inches o' fucking flesh comes with it.\"\n\nCrozier nods and looks around. The sick bay is small, but six cots are crammed into it now. One is empty. Three men, down with what Peddie and McDonald tell him is probably scurvy, are sleeping. A fourth man, Davey Leys, is staring at the ceiling \u2014 he has been conscious but strangely unresponsive for almost a week now. The fifth cot holds Marine Private William Heather.\n\nCrozier lifts a second lamp from its hook on the starboard partition and holds the light over Heather. The man's eyes glisten but he does not blink as Crozier brings the lamp closer. His pupils seem permanently dilated. His skull has been wrapped with a bandage but blood and grey matter are already seeping through.\n\n\"Is he alive?\" Crozier asks softly.\n\nPeddie comes over, wiping his bloody hands with a rag. \"He is, strangely enough.\"\n\n\"But we could see his brains on deck. I can see them now.\"\n\nPeddie nods tiredly. \"That happens. In other circumstances, he might even recover. He would be an idiot, of course, but I could screw in a metal covering where his skull is missing and his family, if he has any, could take care of him. Keep him as a sort of pet. But here...\" Peddie shrugs. \"Pneumonia or scurvy or starvation will carry him away.\"\n\n\"How soon?\" asks Crozier. Seaman Cann has gone out through the curtain.\n\n\"Only God knows,\" says Peddie. \"Is there to be more searching for Evans and Strong, Captain?\"\n\n\"Yes.\" Crozier sets the lantern back in place on the partition near the entrance. Shadows flow back over Marine Private Heather.\n\n\"You are aware, I am sure,\" says the exhausted surgeon, \"that there is no chance for young Evans or Strong but every probability that each search will bring more wounds, more frostbite, a greater chance of amputation \u2014 many men have already lost one or more toes \u2014 and the inevitability that someone will shoot someone else in their panic.\"\n\nCrozier looks steadily at the surgeon. If one of his officers or men had spoken to Crozier like this, he would have the man flogged. The captain makes allowance for the man's civilian status and exhausted state. Dr. McDonald has been in his hammock with the influenza for three days and nights and Peddie has been very busy. \"Please let me worry about the risks of continued searching, Mr. Peddie. You worry about stitching up the men stupid enough to set bare metal against their skin when it's sixty below zero. Besides, if that thing out there carried you off into the night, wouldn't you want us to search for you?\"\n\nPeddie laughs hollowly. \"If this particular specimen of _Ursus maritimus_ carries me off, Captain, I can only hope that I have my scalpel with me. So I can put it through my own eye.\"\n\n\"Then keep your scalpel close, Mr. Peddie,\" says Crozier and goes out through the curtain into the odd silence of the crewmen's mess area.\n\nJopson is waiting in the galley glow with a kerchief of hot biscuits.\n\nCrozier enjoys his walk in spite of the creeping cold that has made his face, fingers, legs, and feet feel like they are on fire. He knows that this is preferable to them being numb. And he enjoys the walk in spite of the fact that between the slow moanings and sudden shrieks of the ice moving under and around him in the dark and the constant moan of the wind, he is certain that he is being stalked.\n\nTwenty minutes into his two-hour walk \u2014 more a climb, scuttle, and ass-sliding descent, up, over, and down pressure ridges for much of the way tonight than a walk \u2014 the clouds part and a three-quarters moon appears and illuminates the phantasmagoric landscape. The moon is bright enough to have an ice-crystalled lunar halo around it, actually two concentric halos, he notices, the diameter of the larger one sufficient to cover a third of the eastern night sky. There are no stars. Crozier dims his lamp to save oil and walks on, using the boat pike he's brought along to test every fold of black ahead of him to make sure it's a shadow and not a crack or crevasse. He has reached the area on the east side of the iceberg now where the moon is blocked, the berg throwing a black and twisted shadow for a quarter of a mile of ice. Jopson and Little insisted he take a shotgun, but he told them he did not want to carry the weight on the walk. More to the point, he doesn't really believe a shotgun will be of any use against the foe they had in mind.\n\nIn a particular moment of rare calm, everything strangely quiet except for his laboured breathing, Crozier suddenly recalls a resonant instance from when he was a young boy returning home late one winter evening from an afternoon in the wintry hills with his friends. At first he rushed headlong alone across the frost-rimed heather, but then he paused half a mile or so from his house. He remembers standing there watching the lighted windows in the village as the last of the winter twilight faded from the sky and the surrounding hills became vague, black, featureless shapes, unfamiliar to a boy so young, until even his own house, visible at the edge of town, lost all definition and three-dimensionality in the dying light. Crozier remembers the snow beginning to fall and himself standing there alone in the darkness beyond the stone sheep pens, knowing that he would be cuffed for his tardiness, knowing that arriving later would only make the cuffing worse, but having no will nor want to walk toward the light of home yet. He enjoyed the soft sound of night wind and the knowledge that he was the only boy \u2014 perhaps the only human being \u2014 out there in the dark on the windy, frozen-grass meadows on this night that smelled of coming snow, alienated from the lighted windows and the warm hearths, very aware that he was of the village but not _part_ of it at that moment. It was a thrilling, almost erotic feeling \u2014 an illicit discovery of self separated from everyone and everything else in the cold and dark \u2014 and he feels it again now, as he has more than a few times during his years of arctic service at opposite poles of the earth.\n\nSomething is coming down the high ridge behind him.\n\nCrozier turns the oil lantern up and sets it on the ice. The circle of golden light reaches barely fifteen feet and makes the darkness beyond all the worse. Using his teeth, he pulls off his heavy mitten, lets it drop to the ice, leaving only a thin glove on that hand, shifts the boat pike to his left hand, and pulls his pistol from his coat pocket. Crozier cocks the weapon as the rustle of sliding ice and snow on the pressure ridge becomes louder. The line of shadow from the iceberg blocks the moonlight here and the captain can make out only the huge shapes of ice blocks seeming to move and shift in the flickering light.\n\nThen something furry and indistinct moves along the ice ledge he has just descended, some ten feet above him and less than fifteen feet to the west, well within leaping distance.\n\n\"Halt,\" Crozier says, extending the heavy pistol. \"Identify yourself.\"\n\nThe shape makes no sound. It moves again.\n\nCrozier holds his fire. Dropping the long boat pike, he grabs up the lantern and thrusts it forward.\n\nHe sees the rippling fur moving and almost fires, but checks himself at the last instant. The shape slides lower, moving quickly and surely down onto the ice. Crozier lowers the hammer on his pistol and sets it back in his pocket, crouching to retrieve his mitten even while keeping the lantern extended.\n\nLady Silence walks into the light, her fur parka and sealskin pants making her look like some short, rounded beast. The hood is pulled forward against the wind and Crozier cannot see her face.\n\n\"God-damn it, woman,\" he says softly. \"You came a horny seaman's second from being shot. Where the hell have you been, anyway?\"\n\nShe steps closer, almost within reaching distance, but her face remains veiled by darkness within the hood.\n\nFeeling a sudden chill along the back of his neck and down his spine \u2014 Crozier is remembering his grandmother Moira's description of a banshee's transparent skull face within the folds of its black hood \u2014 he raises the lantern between them.\n\nThe young woman's face is human, not banshee, the dark eyes wide as they reflect the light. She has no expression. Crozier realizes that he has never seen an expression on her face, other than perhaps a mildly inquisitive look. Not even on the day they shot and killed her husband or brother or father and she watched the man choke to death on his own blood.\n\n\"No wonder the men think you're a witch and a Jonah,\" says Crozier. On the ship, in front of the men, he is always polite and formal to this Esquimaux wench, but he is not on the ship or in front of the men now. It is the first and only time he and the damned woman have been away from the ship at the same time. And he is very cold and very tired.\n\nLady Silence stares at him. Then she extends a mittened hand, Crozier lowers the lamp toward it, and he sees that she is offering him something \u2014 a limp grey offering, like a fish that has been gutted and boned, leaving only the skin.\n\nHe realizes that it is a crewman's woolen stocking.\n\nCrozier takes it, feels the lump at the toe of the sock, and for a second is sure that the lump will be part of a man's foot, probably the ball of the foot and the toes, still pink and warm.\n\nCrozier has been to France and known men posted to India. He has heard the story of werewolves and were-tigers. In Van Diemen's Land, where he met Sophia Cracroft, she told him of the locals' tales of natives who could turn into a monstrous creature there they called the Tasmanian devil \u2014 a creature capable of tearing a man limb from limb.\n\nShaking the stocking, Crozier looks into Lady Silence's eyes. They are as black as the holes in the ice through which the Terrors lowered their dead until even those holes froze solid.\n\nIt is a lump of ice, not part of a foot. But the stocking itself is not frozen hard. The wool has not been out here for long in \u221260-degree cold. Logic suggests that this woman has brought it with her from the ship, but for some reason Crozier does not think so.\n\n\"Strong?\" says the captain. \"Evans?\"\n\nSilence shows no reaction to the names.\n\nCrozier sighs, stuffs the stocking in his coat pocket, and lifts the boat pike. \"We're closer to _Erebus_ than _Terror,_ \" he says. \"You'll just have to come with me.\"\n\nCrozier turns his back on her, feeling the chill along his neck and spine again in doing so, and crunches off through the rising wind toward the now-visible outline of the _Terror_ 's sister ship. A minute later he can hear her soft footsteps on the ice behind him.\n\nThey clamber over a final pressure ridge, and Crozier can see that _Erebus_ is more brightly lit than he's seen before. A dozen or more lanterns hang from spars just on this visible port side of the icebound, absurdly lifted, and steeply canting vessel. It's a prodigious waste of lamp oil.\n\nThe _Erebus,_ Crozier knows, has suffered more than his _Terror._ Besides bending the long propeller shaft last summer \u2014 the shaft that had been built to be retracted but hadn't done so in time to avoid damage from the underwater ice during their ice-breaking in July \u2014 and losing the screw itself, the flagship had been mauled more than her sister ship during the past two winters. The ice in the comparative shelter of the Beechey Island harbour had warped, splintered, and loosened hull timbers to a greater degree on _Erebus_ than on _Terror;_ the flagship's rudder was damaged in their past summer's mad dash for the Passage; the cold has popped more bolts, rivets, and metal brackets in Sir John's ship; much more of the iron icebreaker cladding on _Erebus_ has been torn free or buckled. And while _Terror_ has also been raised and squeezed by the ice, the last two months of this third winter have seen HMS _Erebus_ lifted on a virtual pedestal of ice even while the pressure from the sea pack splintered a long section of the starboard bow, port stern, and bottom hull amidships.\n\nSir John Franklin's flagship, Crozier knows \u2014 and its current captain, James Fitzjames, and his crew also know \u2014 will never sail again.\n\nBefore stepping into the area lit by the ship's hanging lanterns, Crozier steps behind a ten-foot-tall serac and pulls Silence in behind him.\n\n\"Ahoy the ship!\" he bellows in his loudest dockyard-commanding voice.\n\nA shotgun roars and a serac five feet from Crozier splinters into a shower of ice chips catching the lantern's dim glow.\n\n\"Avast that, God-damn your blind eyes, you fucking lubbing idle-brained shit-for-wits idiot!\" roars Crozier.\n\nThere is a commotion on _Erebus_ 's deck as some officer wrestles the shotgun away from the shit-for-wits idiot sentinel.\n\n\"All right,\" Crozier says to the cowering Esquimaux girl. \"We can go now.\"\n\nHe stops, and not just because Lady Silence is not following him out into the light. He can see her face by the reflected glow, and she is smiling. Those full lips that never move are curling up ever so slightly. Smiling. As if she had understood and enjoyed his outburst.\n\nBut before Crozier can confirm that the smile is real, Silence backs into the shadows of the ice jumble and is gone.\n\nCrozier shakes his head. If the crazy woman wants to freeze out here, let her. He has business with Captain Fitzjames and then a long walk home in the dark before he can sleep.\n\nTiredly, realizing that he's not felt his feet for the past half hour at least, Crozier stumps his way up the ramp of dirty ice and snow toward the deck of the dead Sir John's broken flagship.\n\nFRANKLIN\n\n_Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\nMay, 1847\n\nCaptain Sir John Franklin may have been the only man aboard either ship who remained outwardly serene when spring and summer simply did not arrive in April, May, and June of 1847.\n\nAt first, Sir John had not formally announced that they were stuck for at least another year; he didn't have to. The previous spring, up at Beechey Island, the crew and officers had watched with eager anticipation not only as the sun returned but as the close pack broke up into discrete floes and slushy brash ice, open leads appeared, and the ice gave up its grip. By late May of 1846 they had been sailing again. Not so this year.\n\nThe previous spring crew and officers had observed the return of the many birds, whales, fish, foxes, seals, walruses, and other animals, not to mention the greening of the lichen and low heather on the islands they were sailing toward by early June. Not this year. No open water meant no whales, no walruses, almost no seals \u2014 the few ring seals they spied were as hard to catch or shoot now as they had been in early winter \u2014 and nothing but dirty snow and grey ice as far as the eye could see.\n\nThe temperature stayed cold despite the longer hours of sun each day. Although Franklin had the masts fully stepped, the spars reset, the rigging redone, and fresh canvas on both ships brought up by mid-April, there was no purpose to it. The steam boilers remained unfired except to move warm water through the heating pipes. Lookouts reported a solid table of white extending in all directions. Icebergs stayed in place where they had been frozen in place the previous September. Fitzjames and Lieutenant Gore, working with Captain Crozier from _Terror,_ had confirmed from their star sightings that the current was pushing the ice flow south at a pitiful one and a half miles per _month,_ but this mass of ice on which they were pinned had rotated counterclockwise all winter, returning them to where they had begun. Pressure ridges continued to pop up like white gopher burrows. The ice was thinning \u2014 fire-hole teams could saw through it now \u2014 but it was still more than ten feet thick.\n\nCaptain Sir John Franklin remained serene through all of this because of two things: his faith and his wife. Sir John's devout Christianity buoyed him up even when the press of responsibility and frustration collaborated to press him down. Everything that happened was, he knew and fervently believed, God's will. What seemed inevitable to the others need not be in a universe administered by an interested and merciful God. The ice might suddenly break up in midsummer, now less than six weeks away, and even a few weeks of sailing and steaming time would bring them triumphantly to the North-West Passage. They would steam west along the coast as long as they had coal, then sail the rest of the way to the Pacific, escaping the far northern latitudes sometime in mid-September just before the pack ice solidified again. Franklin had experienced greater miracles in his lifetime. Just being appointed commander of this expedition \u2014 at age sixty, after the humiliation of Van Diemen's Land \u2014 had been a greater miracle.\n\nAs deep and sincere as Sir John's faith in God was, his faith in his wife was even deeper and sometimes more frightening. Lady Jane Franklin was an indomitable woman... _indomitable_ was the only word for her. Her will knew no bounds and in almost every instance, Lady Jane Franklin would bend the errant and arbitrary ways of the world to the iron command of her will. Already, he imagined, after being out of touch for two full winters, his wife had mobilized her very impressive private fortune, public contacts, and apparently limitless force of will to cajole the Admiralty, the Parliament, and God alone knew what other agencies into searching for him.\n\nThis last fact bothered Sir John somewhat. Above all else, he did not want to be \"rescued\" \u2014 approached either overland or by sea during the brief summer thaw by hastily assembled expeditions under the command of whiskey-breath Sir John Ross or the young Sir James Ross (who would be forced out of his arctic retirement, Sir John was sure, by Lady Jane's demands). That way lay shame and ignominy.\n\nBut Sir John remained serene because he knew that the Admiralty was not moved _quickly_ on any matter, not even by such a forceful fulcrum and lever as his wife Jane. Sir John Barrow and the other members of the mythical Arctic Council, not to mention Sir John's official superiors in the Royal Navy Discovery Service, knew quite well that HMS _Erebus_ and HMS _Terror_ had provisions for three years, longer if severe rations were imposed, not to mention the capability of fishing and hunting game should they ever come in sight of any. Sir John knew that his wife \u2014 his _indomitable_ wife \u2014 would force a rescue should it come to that, but the terrible and wonderful inertia of the Royal Navy would almost certainly ensure that such a rescue attempt would not be outfitted until the spring and summer of 1848, if not later.\n\nAccordingly, in late May of 1847, Sir John prepared five sledge parties to look over the horizons in each direction, including one instructed to sledge back the way they had come, searching for any open water. They departed on May 21, 23, and 24, with Lieutenant Gore's party \u2014 the crucial one \u2014 departing last and sledging toward King William Land to the southeast.\n\nBesides reconnoitering, First Lieutenant Graham Gore had a second important responsibility \u2014 leaving Sir John's first written message cached ashore since the beginning of the expedition.\n\nHere Captain Sir John Franklin had come as close to disobeying orders as he ever had in his Naval lifetime. His instructions from the Admiralty had been to erect cairns and to leave messages in caches for the length of his exploration \u2014 should the ships not appear beyond the Bering Strait on schedule, this would be the only way for Royal Navy rescue ships to know in which direction Franklin had headed and what might have caused their delay. But Sir John had not left such a message at Beechey Island, even though he had almost nine months to prepare one. In truth, Sir John had hated that first cold anchorage \u2014 had been ashamed of the deaths of the three crewmen by consumption and pneumonia that winter \u2014 so he had privately decided to leave the graves behind as the only message he needed to send. With any luck, no one would find the graves for years after his victory of forcing the North-West Passage had been bannered everywhere in the world.\n\nBut it had now been almost two years since his last dispatch to his superiors, so Franklin dictated an update to Gore and set it in an airtight brass cylinder \u2014 one of two hundred he'd been supplied with.\n\nHe personally instructed Lieutenant Gore and Second Mate Charles Des Voeux on where to put the message \u2014 into the six-foot-high cairn left on King William Land by Sir James Ross some seventeen years earlier at the westernmost point of his own explorations. It would be, Franklin knew, the first place the Navy would look for word of his expedition, since it was the last landmark on everyone's maps.\n\nLooking at the lone squiggle of that last landmark on his own map in the privacy of his cabin on the morning before Gore, Des Voeux, and six crewmen set out, Sir John had to smile. In an act of respect seventeen years ago \u2014 not to mention an act now generating some minor irony \u2014 Ross had named the westernmost promontory along the shore Victory Point and then named the nearby highlands Cape Jane Franklin and Franklin Point. It was as if, Sir John thought, looking down at the weathered sepia map with its black lines and large unfilled spaces to the west of the carefully marked Victory Point, Destiny or God had brought him and these men here.\n\nHis dictated message \u2014 it was in Gore's handwriting \u2014 was, Sir John thought, succinct and businesslike:\n\n______ of May 1847. HM Ships Erebus and Terror... Wintered in the Ice in Lat. 70\u00b005\u2032 N. Long. 98\u00b023\u2032 W. Having wintered in 1846\u20137 at Beechey Island in Lat. 74\u00b043\u203228\u2033 N Long. 90\u00b039\u203215\u2033 W after having ascended Wellington Channel to Lat. 77\u00b0 \u2014 and returned by the west side of Cornwallis Island. Sir John Franklin commanding the Expedition. All well. Party consisting of 2 officers and 6 Men left the ships on Monday 24th. May 1847. Gm. Gore, Lieut. Chas. F. Des Voeux, mate._\n\nFranklin instructed Gore and Des Voeux to sign the note and fill in the date before sealing the canister and setting it deep inside James Ross's cairn.\n\nWhat Franklin hadn't noticed during his dictation \u2014 nor Lieutenant Gore corrected \u2014 was that he had given the wrong dates for their winter at Beechey Island. It had been the first winter of 1845\u201346 in their sheltered ice harbour at Beechey; this year's terrible time in the open pack ice had been the winter of 1846\u201347.\n\nNo matter. Sir John was convinced that he was leaving a minor message to posterity \u2014 possibly to some Royal Navy historian who wished to add an artifact to Sir John's future report on the expedition (Sir John fully planned to write another book, the proceeds of which would bring his private fortune almost up to that of his wife's) \u2014 and not dictating a report that would be read by anyone in the immediate future.\n\nOn the morning that Gore's sledge party set out, Sir John bundled up and went down onto the ice to wish them Godspeed.\n\n\"Do you have everything you need, gentlemen?\" asked Sir John.\n\nFirst Lieutenant Gore \u2014 fourth in overall command behind Sir John, Captain Crozier, and Commander Fitzjames \u2014 nodded, as did his subordinate, Second Mate Des Voeux, the mate flashing a smile. The sun was very bright and the men were already wearing the wire-mesh goggles that Mr. Osmer, _Erebus_ 's purser, had issued them to prevent blindness from the sun's glare.\n\n\"Yes, Sir John. Thank you, sir,\" said Gore.\n\n\"Plenty of woollies?\" joked Sir John.\n\n\"Aye, sir,\" said Gore. \"Eight layers of well-woven good Northumberland sheep shearings, Sir John, nine if one counts the woolen drawers.\"\n\nThe five crewmen laughed to hear their officers banter so. The men, Sir John knew, loved him.\n\n\"Prepared for camping out on the ice?\" Sir John asked one of the men, Charles Best.\n\n\"Oh, aye, Sir John,\" said the short but stocky young seaman. \"We have the Holland tent, sir, and them eight wolfskin blanket robes what we sleep on and under. And twenty-four sleeping bags, Sir John, which purser sewn up for us from the fine Hudson's Bay blankets. We'll be toastier on the ice than aboard the ship, m'lord.\"\n\n\"Good, good,\" Sir John said absently. He looked to the southeast where King William Land \u2014 or Island, if Francis Crozier's wild theory was to be believed \u2014 was visible only as a slight darkening of the sky over the horizon. Sir John prayed to God, quite literally, that Gore and his men would find open water near the coast, either before or after caching the expedition's message. Sir John was prepared to do everything in his power \u2014 and beyond \u2014 to force the two ships, as beaten up as _Erebus_ was, across and through the softening ice, if only it would soften, and into the comparative protection of coastal waters and the potential salvation of land. There they might find a calm harbour or gravel spit where the carpenters and engineers could make repairs enough to _Erebus_ \u2014 straightening the propeller shaft, replacing the screw, shoring up the twisted internal iron reinforcements and perhaps replacing some of the missing iron cladding \u2014 to allow them to press on. If not, Sir John thought \u2014 but had not yet shared the thought with any of his officers \u2014 they would follow Crozier's distressing plan from the previous year and anchor _Erebus,_ transfer its diminishing coal reserves and crew to _Terror,_ and sail west along the coast in that crowded (but jubilant, Sir John was sure, jubilant) remaining ship.\n\nAt the last moment, the assistant surgeon on _Erebus,_ Goodsir, had implored Sir John to allow him to accompany the Gore party, and although neither Lieutenant Gore nor Second Mate Des Voeux were enthusiastic about the idea \u2014 Goodsir was not popular with the officers or men \u2014 Sir John had allowed it. The assistant surgeon's argument for going was that he needed to gain more information on edible forms of wildlife to use against the scurvy that was the primary fear of all arctic expeditions. He was particularly interested in the behavior of the only animal present this odd non-summer arctic summer, the white bear.\n\nNow, as Sir John watched the men finish lashing their gear to the heavy sledge, the diminutive surgeon \u2014 he was a small man, pale, weak-looking, with a receding chin, absurd side whiskers, and a strangely effeminate gaze that put off even the usually universally affable Sir John \u2014 sidled up to start a conversation.\n\n\"Thank you again for allowing me to accompany Lieutenant Gore's party, Sir John,\" said the little medico. \"The outing could be of inestimable importance in our medical evaluation of the antiscorbutic properties of a wide variety of flora and fauna, including the lichens invariably present on the terra firma of King William Land.\"\n\nSir John involuntarily made a face. The surgeon could not have known that his commander had once survived on thin soup made from such lichen for several months. \"You're very welcome, Mr. Goodsir,\" he said coolly.\n\nSir John knew that the slouching young popinjay preferred the title of \"Doctor\" to \"Mister,\" a dubious distinction since, although from a good family, Goodsir had trained as a mere anatomist. Technically on par with the warrant officers on board both ships, the civilian assistant surgeon was entitled, in Sir John's eyes, only to be called _Mr._ Goodsir.\n\nThe young surgeon blushed at his commander's coolness after the easy banter with the crewmen, tugged at his cap, and took three awkward steps backward on the ice.\n\n\"Oh, Mr. Goodsir,\" added Franklin.\n\n\"Yes, Sir John?\" The young upstart was actually red-faced, almost stammering with embarrassment.\n\n\"You must accept my apologies that in our formal communiqu\u00e9 to be cached at Sir James Ross's cairn on King William Land, we referred only to two officers and six _men_ in Lieutenant Gore's party,\" said Sir John. \"I had dictated the message prior to your request to accompany the party. I would have written _an officer, a warrant officer, an assistant surgeon, and five men_ had I but known you would be included.\"\n\nGoodsir looked confused for a moment, not quite sure of what Sir John was trying to tell him, but then he bowed, tugged at his cap again, mumbled, \"Very good, there is no problem, I understand, thank you, Sir John,\" and backed away again.\n\nA few minutes later, as he watched Lieutenant Gore, Des Voeux, Goodsir, Morfin, Ferrier, Best, Hartnell, and Private Pilkington diminish across the ice to the southeast, Sir John, under his beaming countenance and outward serenity, actually contemplated failure.\n\nAnother winter \u2014 another full year \u2014 in the ice could undo them. The expedition would be out of food, coal, oil, pyroligneous ether for lamp fuel, and rum. This last item's disappearance might well mean mutiny.\n\nMore than that, if the summer of 1848 were as cold and unyielding as this summer of 1847 fully promised to be, another full winter or year in the ice would destroy one or both of their ships. Like so many failed expeditions before them, Sir John and his men would be fleeing for their lives, dragging longboats and whalers and hastily clabbered-together sledges across the rotten ice, praying for open leads and then cursing them when the sledges fell through the ice and the contrary winds blew the heavy boats back on the pack ice, leads that meant days and nights of rowing for the starving men. Then, Sir John knew, there would be the overland part of any escape attempt \u2014 eight hundred miles and more of featureless rock and ice, rivers of constant rapids strewn with boulders each capable of smashing their smaller boats (the larger boats could not get down northern Canada's rivers, he knew from experience), and native Esquimaux who were hostile more often than not and thieving liars even when they seemed to be friendly.\n\nSir John continued watching as Gore, Des Voeux, Goodsir, and the five crewmen and single sledge disappeared in the ice glare to the southeast and wondered idly if he should have brought dogs on this trip.\n\nSir John had never liked the idea of dogs on arctic expeditions. The animals were sometimes good for the men's morale \u2014 at least right up to the point when the animals had to be shot and eaten \u2014 but they were, in the final analysis, dirty, loud, and aggressive creatures. The deck of a ship carrying enough dogs to do any good, that is to harness to sledges the way the Greenland Esquimaux liked to do, was a deck filled with incessant barking, crowded kennels, and the constant stench of excrement.\n\nHe shook his head and smiled. They'd only brought one dog along on this expedition \u2014 the mutt named Neptune \u2014 not to mention a small monkey named Jocko \u2014 and that, Sir John was sure, was quite enough of a menagerie for this particular ark.\n\nThe week after Gore's departure seemed to crawl for Sir John. One by one the other sledge parties reported in, their men exhausted and frozen and their woolen layers soaked with sweat from the exertion of hauling their sledge across or around countless ridges. Their reports were the same.\n\nFrom the east toward the Boothia Peninsula \u2014 no open water. Not even the smallest lead.\n\nFrom the northeast toward Prince of Wales Island and the path of their approach to this frozen desert \u2014 no open water. Not even the hint of dark sky beyond the horizon which sometimes suggested open water. In eight days of hard sledging the men had not been able to reach Prince of Wales Island, nor even catch a glimpse of it. The ice was more tortured with ridges and icebergs than the men had ever seen.\n\nFrom the northwest toward the unnamed strait that led the ice stream south toward them around the west coast and southern tip of Prince of Wales Island \u2014 nothing seen except white bears and frozen sea.\n\nFrom the southwest toward the presumed landmass of Victoria Land and the theoretical passage between the islands and the mainland \u2014 no open water, no animals except the confounded white bears, hundreds of pressure ridges, so many frozen-in-place icebergs that Lieutenant Little \u2014 the officer from HMS _Terror_ whom Franklin had put in command of this particular sledging party, made up of Terrors \u2014 reported that it was like trying to struggle west through a mountain range of ice where the ocean should be. The weather had been so bad on the last part of the trip that three of the eight men had seriously frostbitten toes and all eight of them were snow-blind to some extent, Lieutenant Little himself completely blind for the last five days and sick with terrible headaches. Little, an old arctic hand, Sir John understood, a man who had gone south with Crozier and James Ross eight years earlier, had to be loaded onto the sledge and hauled back by the few men who could still see well enough to pull.\n\nNo open water anywhere in the twenty-five straight-line miles or so they had explored \u2014 twenty-five straight-line miles gained in perhaps a hundred miles of marching around and over obstacles. No arctic foxes or hares or caribou or walruses or seals. Obviously no whales. The men had been prepared to haul their sledge around cracks and small leads in search of real open water, but the surface of the sea, Little reported, his sunburned skin peeling away from his nose and temples below and above the white bandages over his eyes, was a white solid. At the outermost point on their western odyssey, perhaps twenty-eight miles from the ships, Little had ordered the man with the best remaining eyesight, a bosun's mate named Johnson, to climb the tallest iceberg in their vicinity. Johnson had taken hours to do so, hacking out narrow steps for his feet with his pickaxe and then digging in the cleats the purser had driven through the soles of his leather boots. Once on the top, the seaman had used Lieutenant Little's telescope to look northwest, west, southwest, and south.\n\nThe report was dismal. No open water. No land. Jangles of seracs, ridges, and bergs to the distant white horizon. A few white bears, two of which they later shot for fresh meat \u2014 but the livers and heart were unhealthy to humans they'd discovered. The men's strength was already depleted from hauling the heavy sledge over so many ridges, and in the end they cut out less than a hundred pounds of the gamy, muscular meat to fold in tarps and haul back to the ship. Then they skinned the larger bear for its white fur, leaving the rest of the bears to rot on the ice.\n\nFour of the five scouting expeditions returned with bad news and frostbitten feet, but Sir John waited most anxiously for Graham Gore's return. Their last, best hope had always been to the southeast, toward King William Land.\n\nFinally, on the third of June, ten days after Gore's departure, lookouts from high in the masts called down that a sledge party was approaching from the southeast. Sir John finished his tea, dressed appropriately, and then joined the mob of men who'd rushed on deck to see what they could see.\n\nThe surface party was visible even by men on deck now, and when Sir John lifted his beautiful brass telescope \u2014 a gift from the officers and men of a twenty-six-gun frigate Franklin had commanded in the Mediterranean more than fifteen years earlier \u2014 one glance explained the lookouts' audible confusion.\n\nAt first glance all seemed well. Five men were pulling the sledge, just as during Gore's departure. Three figures were running alongside or behind the sledge, just as on the day Gore left. All eight accounted for then.\n\nAnd yet...\n\nOne of the running figures did not appear to be human. At a distance of more than a mile and glimpsed between the seracs and ice-rubble upthrusts that had once been the placid sea here, it looked as if a small, round, headless but very furry animal was running behind the sledge.\n\nAnd worse, Sir John could not make out Graham Gore's distinctive tall figure in the lead nor the dashing red comforter he sported. All of the other figures hauling or running \u2014 and certainly the lieutenant would not have been _hauling_ the sledge while his subordinates were fit \u2014 seemed too short, too bent, and too _inferior._\n\nWorst of all, the sledge seemed far too heavily packed for the return trip \u2014 the rations had included a week's extra canned goods, but they were already three days over the estimated maximum round-trip time. For a minute Sir John's hopes soared as he considered the possibility that the men had killed some caribou or other large land animals and were bringing in fresh meat, but then the distant forms emerged from behind the last large pressure ridge, still more than half a mile away across the ice, and Sir John's telescope revealed something horrible.\n\nNot caribou meat on the sledge, but what appeared to be two dead human bodies lashed atop the gear, one man stacked atop the other in a callous fashion that could only mean death. Sir John could now plainly make out two exposed heads, one at each end of the stack, with the head belonging to the body on top showing long white hair the likes of which no man aboard either ship possessed.\n\nThey were rigging ropes down the side of the canted _Erebus_ to aid their portly captain's descent onto the steep ice there. Sir John went belowdecks only long enough to add his ceremonial sword to his uniform. Then, pulling his cold-weather slops on over uniform, medals, and sword, he went up on deck and then over the side \u2014 puffing and wheezing, allowing his steward to help him down the slope \u2014 to greet whoever or whatever was approaching his ship.\n\nGOODSIR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 Long. 98\u00b0 41\u2032_\n\nKing William Land, 24 May\u20133 June, 1847\n\nOne reason that Dr. Harry D. S. Goodsir had insisted on coming along on this exploration party was to prove that he was as strong and able a man as most of his crewmates. He soon realized that he wasn't.\n\nOn the first day, he had insisted \u2014 over the quiet objections of Lieutenant Gore and Mr. Des Voeux \u2014 on taking his turn at man-hauling the sledge, allowing one of the five crewmen so assigned to take a break and walk alongside.\n\nGoodsir almost could not do it. The leather-and-cotton harness the sailmakers and pursers had constructed, cleverly attached to the pull ropes by a knot the sailors could tie or undo in a second and which Goodsir could not figure out for the life of him, was too large for his narrow shoulders and sunken chest. Even by cinching the front girth of the harness as tight as it would go, it slipped on him. And he, in turn, slipped on the ice, falling repeatedly, forcing the other men off their stride of pull, pause, gasp, pull. Dr. Goodsir had not worn such issued ice boots before and the nails driven through the soles caused him to trip over his own feet.\n\nHe had trouble seeing out of the heavy wire-mesh goggles, but when he raised them to his forehead, the glare of arctic sun on arctic ice half-blinded him within minutes. He'd put on too many layers, and now several of those layers of wool were so soaked with his own sweat that he was shivering even while being overheated by the extraordinary exertion. The harness pinched on nerves and cut off circulation to his thin arms and cold hands. He kept dropping his outer mittens. His panting and gasping grew so loud and constant that he was ashamed.\n\nAfter an hour of such absurdity, Bobby Ferrier, Tommy Hartnell, John Morfin, and Marine Private Bill Pilkington \u2014 the other men in harness, Charles Best walking alongside now \u2014 each pausing to brush the snow off his anorak, looking at one another but saying nothing, of him never finding the rhythm of literally working in harness with others, he accepted the offer of relief from Best and, during one of the brief stops, slipped out of the harness and let the true men pull the heavy, high-mounted sledge with its wooden runners that constantly wanted to freeze to the ice.\n\nGoodsir was exhausted. It was still morning of the first day on the ice, and he was so tired out from the hour of pulling that he could have happily unfurled his sleeping bag, set it on one of the wolfskin blanket robes, and gone to sleep until the next day.\n\nAnd this was before they reached the first real pressure ridge.\n\nThe ridges to the southeast of the ship were the lowest in sight for the first two miles or so, almost as if the beset _Terror_ herself had somehow kept the ice smoother in her lee, forcing the ridges farther away. But by late afternoon of the first day, the real pressure ridges rose up to block them. These were taller than those that had separated the two ships during their winter in the ice here, as if the pressures under the ice closer to King William Land were more terrible.\n\nFor the first three ridges, Gore led them southwest to find low spots, dips in the ridges where they could clamber over without too much difficulty. It added miles and hours to their travel but was still an easier solution than unpacking the sledge. There was no going around the fourth ridge.\n\nEvery pause of more than a few minutes meant that one of the men \u2014 usually young Hartnell \u2014 had to remove one of the many bottles of pyroligneous fuel from the carefully lashed mass on the sled, fire up a small spirit stove, and melt some snow in a pan into hot water, not to drink \u2014 to quench their thirst they had flasks they kept under their outer garments to keep from freezing \u2014 but to pour the warm water the length of the wooden runners so as to free them from the self-freezing ruts they dug in the scrim of icy snow.\n\nNor did the sledge move across the ice like the sleds and sleighs Goodsir had known from his moderately privileged childhood. He'd discovered on his first forays onto the pack ice almost two years ago that one could not \u2014 even in regular boots \u2014 take a run across the ice and slide the way one did at home on a frozen river or lake. Some property of the sea ice \u2014 almost certainly the high salt content \u2014 increased the friction, reducing the ease of sliding to almost nil. A mild disappointment for a running man wishing to slide like a boy, but a huge increase in effort for a team of men trying to pull, push, and generally man-haul many hundreds of pounds of gear piled high on more hundreds of pounds of sledge across such ice.\n\nIt was like hauling a cumbersome thousand pounds of lumber and goods across moderately rough rock. And the pressure ridges could have been four-storey-high heaps of boulders and gravel for all the ease of crossing one.\n\nThis first serious one \u2014 just one of many stretching across their path to the southeast as far as they could see \u2014 must have been sixty feet high.\n\nUnlashing the carefully secured top foods, boxes of fuel bottles, robes, sleeping bags, and heavy tent, they lightened the load, ending up with fifty- to hundred-pound bundles and boxes that they had to pull up the steep, tumbled, jagged ridge before even attempting to move the sledge.\n\nGoodsir realized quickly that if the pressure ridges had been discrete things \u2014 that is, mere ridges rising out of relatively smooth sea ice \u2014 climbing them would not have been the soul-destroying exertion that it proved to be. None of the frozen sea was smooth, but for fifty to a hundred yards around each pressure ridge the sea ice became a truly insane maze of rough snow, tumbled seracs, and giant ice blocks \u2014 a maze that had to be solved and traversed before the real climbing could begin.\n\nThe climbing itself was never linear but always a tortuous back-and-forth, a constant search for footholds on treacherous ice or handholds on a block that might break away at any moment. The eight men zigzagged upward in ridiculous diagonals as they climbed, handed heavy loads up to one another, hacked away at clumps of ice with their pickaxes to create steps and shelves, and generally tried not to fall or be fallen upon. Parcels slipped out of icy mittens and crashed below, bringing up short but impressive clouds of curses from the five seamen below before Gore or Des Voeux shouted them into silence. Everything had to be unpacked and repacked ten times.\n\nFinally the heavy sledge itself, with perhaps half its load still lashed to it, had to be pulled, shoved, lifted, braced, dislodged from entrapping seracs, angled, lifted again, and tugged to the summit of each uneven pressure ridge. There was no rest for the men even atop these ridges since to relax for a minute meant that eight layers of sweat-sodden outer clothing and underlayers would begin to freeze.\n\nAfter tying new lines to the vertical posts and cross braces at the rear of the sledge, some of the men would get ahead of it to brace its descent \u2014 usually the large Marine, Pilkington, and Morfin and Ferrier had this duty \u2014 while others dug in their cleats and lowered it to a syncopated chorus of gasps, calls, warnings, and more curses.\n\nThen they would carefully reload the sledge, double-check the lashings, boil snow to pour on the frozen-in runners, and be off again, forcing their way through the tumble-labyrinth on this side of the pressure ridge.\n\nThirty minutes later they would come to the next ridge.\n\nTheir first night out on the ice was terrifyingly memorable for Harry D. S. Goodsir.\n\nThe surgeon had never done any camping in his life, but he knew that Graham Gore was telling the truth when the lieutenant said, laughingly, that everything took five times longer on the ice: unpacking the materials, firing up the spirit lamps and stoves, laying out the brown Holland tent and securing screws as anchor stakes in the ice, unrolling the many blanket rolls and sleeping bags, and especially heating up the tinned soup and pork they'd brought along.\n\nAnd all the while, one had to keep moving \u2014 waving arms and shaking legs and stamping feet \u2014 or extremities would freeze.\n\nOn a normal arctic summer, Mr. Des Voeux reminded Goodsir, citing their previous summer of ice-breaking southward from Beechey Island as an example, temperatures at this latitude on a sunny June day with no wind might rise as high as 30 degrees Fahrenheit. Not this summer. Lieutenant Gore had taken measurements of the air temperature at 10:00 p.m. \u2014 the time they stopped to make camp with the sun still on the southern horizon and the sky quite bright \u2014 and the thermometer read only \u22122 degrees Fahrenheit. The temperature at their midday tea and biscuit break had been +6 degrees.\n\nThe Holland tent was small. In a storm it would save their lives but this first night out on the ice was clear with almost no wind, so Des Voeux and the five sailors decided to sleep outside on their wolfskins and tarps with only their Hudson's Bay Company blanket sleeping bags for shelter \u2014 they would retreat to a very crowded tent if bad weather blew in \u2014 and after debating with himself for a moment, Goodsir decided to sleep outside with the men rather than inside with just Lieutenant Gore, as capable and affable a fellow as Gore was.\n\nThe daylight was maddening. It grew dim around midnight, but the sky was as light as an 8:00 p.m. London evening in midsummer, and Goodsir was damned if he could fall asleep. Here he was more physically tired than ever before in his life and he couldn't sleep. The aches and pains from the day's exertions also impeded sleep, he realized. He wished he'd brought some laudanum with him. A small draught of that would moderate the discomfort and allow him to sleep. Unlike some surgeons with a doctor's certificate to administer drugs, Goodsir was not an addict \u2014 he used the various opiates only to allow himself to sleep or to concentrate when he had to. No more than once or twice a week.\n\nAnd it was _cold._ After eating the heated soup and beef from the tins and walking through the ice jumble to find a private place to relieve himself \u2014 also an outdoor lifetime first for him and one, he realized, that must be accomplished quickly if frostbite of very important areas was to be avoided \u2014 Goodsir settled in on one of the large six-foot-by-five-foot wolfskin-blanket sleeping robes, unrolled his personal sleeping bag, and crawled deep into it.\n\nBut not deep enough to get warm. Des Voeux had explained to him that he had to remove his boots and slide them down into the bag with him so that the leather would not freeze solid \u2014 at one point Goodsir had pricked the bottom of his foot on the nails hammered through the sole of one of the boots \u2014 but all the men left on all their other clothes. The wool \u2014 all the wool, Goodsir realized not for the first time that day \u2014 was soaked through with his sweat and exhalations from the long day. The endless day.\n\nFor a while around midnight, the light deepened toward twilight enough that a few stars \u2014 planets, Goodsir now knew from a private lecture at the ad hoc observatory atop the iceberg two years ago \u2014 became visible. But the light never disappeared.\n\nNor did the cold. No longer moving or exerting itself, Goodsir's thin body was defenseless against the cold that came in through the sleeping bag's too-wide opening and that crept up from the ice through the hair-out wolfskin pad beneath him, crawling through the thick Hudson's Bay Company blankets like some cold-fingered predator. Goodsir began to shake. His teeth chattered.\n\nAround him, the four sleeping men \u2014 there were two on guard duty \u2014 snored so loudly that the surgeon wondered if the men on both ships miles northwest of them on the ice, beyond the countless pressure ridges \u2014 _dear God, we have to cross those again going back_ \u2014 could hear the rasping and sawing and snorting.\n\nGoodsir was shaking. At this rate he was sure he would not survive until morning. They would try to roust him out of his blanket and bag and find only a frozen, curled corpse.\n\nHe crawled as far down into the sewn-blankets sleeping bag as he could, pulling the ice-ridged opening closed above him, inhaling his own sour-sweat smell and exhalations rather than be exposed to that freezing air again.\n\nIn addition to the insidious light and the even more insidious creeping cold, the cold of death, Goodsir realized, the cold of the grave and of the black cliff wall above the Beechey Island headstones, there was the noise; the surgeon had thought himself accustomed to the groan of ship's timbers, occasional creakings and snappings of supercold ship's metal in the dark of two winters, and the constant noise antics of the ice holding the ship in its vise, but out here, with nothing separating his body from the ice except a few layers of wool and wolfskin, the groaning and movement of the ice beneath him was terrible. It was like trying to sleep on the belly of a living beast. The sense of the ice moving beneath him, however exaggerated, was real enough to give him vertigo as he curled more tightly into a fetal position.\n\nSometime around 2:00 a.m. \u2014 he had actually checked his pocket watch by the light filtering in through the bag opening \u2014 Harry D. S. Goodsir had begun drifting off into a state of semiconsciousness vaguely resembling sleep when he was pounded awake by two deafening explosions.\n\nStruggling with his sweat-frozen bag like a newborn trying to chew through its caul, Goodsir managed to free his head and shoulders. The freezing night air hit his face with enough cold force to make his heart stutter. The sky was already brighter with sunlight.\n\n\"What?\" he cried. \"What has happened?\"\n\nSecond Mate Des Voeux and three of the seamen were standing on their sleeping bags, long knives they must have slept with in their gloved hands. Lieutenant Gore had burst from the Holland tent. He was fully dressed with a pistol in his bare \u2014 _bare!_ \u2014 hand.\n\n\"Report!\" Gore snapped at one of the two sentries, Charlie Best.\n\n\"It was the bears, Lieutenant,\" said Best. \"Two of them. Big bastards. They've been snooping around all night \u2014 you remember we saw them about half a mile out before we stopped to make camp \u2014 but they kept coming closer and closer, circling like, until finally John and me had to shoot at them to drive them away.\"\n\n\"John\" was twenty-seven-year-old John Morfin, Goodsir knew, the other sentry this night.\n\n\"You both fired?\" asked Gore. The lieutenant had climbed to the highest point of nearby heaped snow and ice and was searching the area with his brass telescope. Goodsir wondered why the man's bare hands hadn't already frozen to the metal.\n\n\"Aye, sir,\" said Morfin. He was reloading his breech-loading shotgun, his wool gloves fumbling with the shells.\n\n\"Did you hit them?\" asked Des Voeux.\n\n\"Aye,\" said Best.\n\n\"Didn't do no good,\" said Morfin. \"Just with shotguns over about thirty paces. Them bears have thick hides and thicker skulls. Hurt 'em enough though that they went away.\"\n\n\"I don't see them,\" said Lieutenant Gore from ten feet up on his ice hill above the tent.\n\n\"We think they come out of those little open holes in the ice,\" said Best. \"The bigger one was running that way when John fired. We thought it went down, but we went out on the ice far enough to see there weren't no carcass there. It's gone.\"\n\nThe sledge-hauling team had noticed those soft areas in the ice \u2014 not quite round, about four feet across, too large for the tiny breathing holes ring seals made, seemingly too small and too far separated for the white bears, and always crusted over with several inches of soft ice. At first, the holes had raised hopes for open water, but in the end they were so few and far between that they were only treacherous. Seaman Ferrier, walking ahead of the sledge late in the afternoon, had almost fallen through one, his left leg going in to above the knee, and they'd all had to stop long enough for the shivering sailor to change into different boots, woollies, socks, and trousers.\n\n\"It's time for Ferrier and Pilkington to take the watch anyway,\" said Lieutenant Gore. \"Bobby, fetch the musket from my tent.\"\n\n\"I'm better with shotgun, sir,\" said Ferrier.\n\n\"I'm comfortable with the musket, Lieutenant,\" said the big Marine.\n\n\"Get the musket then, Pilkington. Peppering those things with shotgun pellets is just going to get them angry.\"\n\n\"Aye, sir.\"\n\nBest and Morfin, obviously shaking from their cold two hours on watch rather than from any tension, sleepily pulled off their boots and crawled into their waiting bags. Private Pilkington and Bobby Ferrier forced their swollen feet into boots retrieved from their bags and slouched off to the nearby ice ridges to keep watch.\n\nShaking worse than ever, his nose and cheeks now joining his fingers and toes in feeling numb, Goodsir curled up deep in his bag and prayed for sleep.\n\nIt did not come. A little more than two hours later, Second Mate Des Voeux began ordering everyone up and out of their bags.\n\n\"We have a long day ahead of us, boys,\" cried the mate in jovial tones.\n\nThey were still more than twenty-two miles from the shore of King William Land.\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n9 November, 1847\n\nYou're frozen through, Francis,\" says Commander Fitzjames. \"Come aft to the Common Room for brandy.\"\n\nCrozier would prefer whiskey, but brandy will have to serve. He precedes _Erebus_ 's captain down the long, narrow companionway toward what had been Captain Sir John Franklin's personal cabin and which is now the equivalent of _Terror_ 's Great Room \u2014 a library and off-duty gathering place for officers and a meeting room when necessary. Crozier thinks that it says good things about Fitzjames that the commander kept his own tiny cubicle after Sir John's death, refitting the spacious aft chamber into a common area and sometimes sick bay for surgery.\n\nThe companionway is totally dark except for the glow from the Common Room and the deck is canted more steeply in the opposite direction from _Terror,_ listing to port rather than starboard, down by the stern rather than bow. And although the ships are almost identical in design, Crozier always notices other differences as well. HMS _Erebus smells_ different somehow \u2014 beyond the identical stench of lamp oil, dirty men, filthy clothes, months of cooking, coal dust, pails of urine, and the men's breath hanging in the cold, dank air, there is something else. For some reason, _Erebus_ stinks more of fear and hopelessness.\n\nThere are two officers smoking their pipes in the Common Room, Lieutenant Le Vesconte and Lieutenant Fairholme, but both stand, nod toward the two captains, and withdraw, pulling the sliding door shut behind them.\n\nFitzjames unlocks a heavy cabinet and pulls out a bottle of brandy, pouring a large measure into one of Sir John's crystal water glasses for Crozier, a smaller amount for himself. For all of the fine china and crystal their late expedition leader loaded aboard for his and his officers' own use, there are no brandy snifters. Franklin was a devout teetotaler.\n\nCrozier does no snifting. He drinks the brandy down in three gulps and allows Fitzjames to replenish it.\n\n\"Thank you for responding so quickly,\" says Fitzjames. \"I expected a message in response, not for you to come in person.\"\n\nCrozier frowns. \"Message? I haven't received a message from you in over a week, James.\"\n\nFitzjames stares a moment. \"You didn't receive a message this evening? I sent Private Reed to your ship with one about five hours ago. I presumed he was spending the night there.\"\n\nCrozier shakes his head slowly.\n\n\"Oh... damn,\" says Fitzjames.\n\nCrozier pulls the woolen stocking from his pocket and sets it on the table. In the brighter light from the bulkhead lamp here there are still no signs of violence. \"I found it during my walk over. Closer to your ship than mine.\"\n\nFitzjames takes the stocking and studies it sadly. \"I'll ask the men if they recognize it,\" he says.\n\n\"It could belong to one of mine,\" Crozier says softly. He succinctly tells Fitzjames about the attack, the mortal wounding of Private Heather, and the disappearance of William Strong and young Tom Evans.\n\n\"Four in one day,\" says Fitzjames. He pours more brandy for both of them.\n\n\"Yes. What is it you were sending me a message about?\"\n\nFitzjames explains there had been sightings of something large moving through the ice jumbles, just beyond the lanterns' glow, all that day. The men had fired repeatedly but parties going onto the ice had found no blood nor other sign. \"So I apologize, Francis, for that idiot Bobby Johns firing at you a few minutes ago. The men's nerves are stretched very tightly.\"\n\n\"Not so tightly that they think that thing on the ice has learned how to shout at them in English, I hope,\" Crozier says sardonically. He takes another sip of the brandy.\n\n\"No, no. Of course not. It was pure idiocy. Johns will be off his rum ration for two weeks. I apologize again.\"\n\nCrozier sighs. \"Don't do that. Rip him a new arsehole if you like, but don't take his rum away. This ship feels surly enough already. Lady Silence was with me and wearing her God-damned hairy parka. Johns may have got a glimpse of that. It would have served me right if he'd blown my head off.\"\n\n\"Silence was with you?\" Fitzjames allowed his eyebrows to ask the questions.\n\n\"I don't know what in hell she was doing out on the ice,\" rasps Crozier. His throat is very sore from the day's cold and his shouting. \"I almost shot her myself a quarter mile from your ship when she crept up on me. Young Irving is probably turning _Terror_ upside down as we speak. I made a huge mistake when I put that boy in charge of looking out for that Esquimaux bitch.\"\n\n\"The men think she is a Jonah.\" Fitzjames's voice is very, very soft. Sounds travel easily through the partitions in such a crowded lower deck.\n\n\"Well why the hell shouldn't they?\" Crozier feels the alcohol now. He hasn't had a drink since last night. It feels good in his belly and tired brain. \"The woman shows up on the day this horror begins with that witch doctor father or husband of hers. Something has chewed her tongue out at the roots. Why the hell _shouldn't_ the men think she's the cause of all this trouble?\"\n\n\"But you've kept her aboard _Terror_ for more than five months,\" says Fitzjames. There is no reproach in the younger captain's voice, only curiosity.\n\nCrozier shrugs. \"I don't believe in witches, James. Nor Jonahs much, for that matter. But I do believe that if we put her out on the ice, the thing will be eating her guts the way it's devouring Evans's and Strong's right now. And maybe your Private Reed's as well. Wasn't that Billy Reed, the redheaded Marine who always wanted to talk about that writer \u2014 Dickens?\"\n\n\"William Reed, yes,\" says Fitzjames. \"He was very fast when the men did footraces back on Disko Island two years ago. I thought that perhaps one man, with speed...\" He stops and chews his lip. \"I should have waited for morning.\"\n\n\"Why?\" says Crozier. \"It's no lighter then. Or not much lighter at noon, for that matter. Day or night doesn't mean anything anymore, and it won't for another four months. And it's not as if that damned thing out there only hunts at night... or even just in the dark, as far as that goes. Maybe your Reed will show up. Our messengers have gotten lost before out there in the ice and come in after five or six hours, shaking and cursing.\"\n\n\"Perhaps.\" Fitzjames's tone echoes his doubt. \"I'll send out search parties in the morning.\"\n\n\"That's just what that thing wants us to do.\" Crozier's voice is very weary.\n\n\"Perhaps,\" Fitzjames says again, \"but you just told me that you've had men out on the ice last night and all day today looking for Strong and Evans.\"\n\n\"If I hadn't brought Evans with me when I was looking for Strong, the boy would still be alive.\"\n\n\"Thomas Evans,\" says Fitzjames. \"I remember him. Big chap. He was not really a boy, was he, Francis? He must be... have been... what? Twenty-two or twenty-three years old?\"\n\n\"Tommy turned twenty this May,\" says Crozier. \"His first birthday aboard was on the day after our departure. The men were in good spirits and celebrated his eighteenth birthday by shaving his head. He didn't seem to mind. Those who knew him say he was always big for his age. He served on HMS _Lynx_ and before that on an East Indian merchantman. He went to sea when he was thirteen.\"\n\n\"As you did, I believe.\"\n\nCrozier laughs a little ruefully. \"As I did. For all the good it did me.\"\n\nFitzjames locks the brandy away in the cabinet and returns to the long table. \"Tell me, Francis, did you actually dress up as a black footman to old Hoppner's lady of rank when you were frozen in up here in... what was it, '24?\"\n\nCrozier laughs again but more easily this time. \"I did. I was a midshipman on the _Hecla_ with Parry when he sailed north with Hoppner's _Fury_ in '24, trying to find this same God-damned Passage. Parry's plan was to sail the two ships through Lancaster Sound and down the Prince Regent Inlet \u2014 we didn't know then, not until John and James Ross in '33, that the Boothia was a peninsula. Parry thought he could sail south around Boothia and go hell-bent for leather until he reached the coastline that Franklin had explored from land six, seven years earlier. But Parry left too late \u2014 why do these God-damned expedition commanders always start off too late? \u2014 and we were lucky to get to Lancaster Sound on ten September, a month late. But the ice was on us by thirteen September, and there was no chance of getting through the Sound, so Parry in our _Hecla_ and Lieutenant Hoppner in _Fury_ ran south, our tails between our legs.\n\n\"A gale blew us back into Baffin Bay and we were lucky sods to find an anchorage in a tiny, pretty little bay off Prince Regent Inlet. We were there ten months. Froze our tits off.\"\n\n\"But,\" says Fitzjames, smiling slightly, \"you as a little black boy?\"\n\nCrozier nods and sips his drink. \"Both Parry and Hoppner were fanatics for fancy dress-up galas during winters in the ice. It was Hoppner who planned this masque he called the Grand Venetian Carnivale, set for the first day in November, right when morale dips as the sun disappears for months. Parry came down _Hecla_ 's side in this huge cloak that he didn't throw off even when all the men were assembled \u2014 most in costume, we had this huge trunk of costumes on each ship \u2014 and when he did throw down the cloak, we saw Parry as that old Marine \u2014 you remember the one with the peg leg what played the fiddle for ha'pennies near Chatham? No, you wouldn't, you're too young.\n\n\"But Parry \u2014 I think the old bastard always wanted to be an actor more than a ship's captain \u2014 he does the whole thing up right, scratching away at his fiddle, hopping on that fake peg leg, and shouting out, 'Give a copper to the poor Joe, your honour, who's lost his timbers in defence of his King and country!'\n\n\"Well, the men laughed their arses off. But Hoppner, who loved that make-believe rubbish even more than Parry, I think, he comes into the ball dressed as a noble lady, wearing the latest Parisian fashions from that year \u2014 low bustline, big crinoline dress bunched up over his ass, everything \u2014 and since I was full of piss and vinegar in those days, not to mention too stupid to know better, in other words still in my twenties, I was dressed as Hoppner's black footman \u2014 wearing this real footman's livery that old Henry Parkyns Hoppner had bought in some dandy's London livery store and brought along just for me.\"\n\n\"Did the men laugh?\" asks Fitzjames.\n\n\"Oh, the men laughed their arses off again \u2014 Parry and his peg weren't in it after old Henry appeared in drag with me lifting his silk train behind him. Why wouldn't they laugh? All those chimney sweeps and ribbon girls, ragmen and hook-nosed Jews, bricklayers and Highland warriors, Turkish dancers and London match girls? Look! There's young Crozier, aging midshipman not even lieutenant yet who thinks he's going to be an admiral someday, forgetting that he's just another black Irish nigger.\"\n\nFitzjames says nothing for a minute. Crozier can hear the snores and farts from the creaking hammocks toward the bow of the dark ship. Somewhere on deck just above them, a lookout stamps his feet to keep them from freezing. Crozier is sorry he's ended the story this way \u2014 he speaks to no one like this when he is sober \u2014 but he also wishes that Fitzjames would get the brandy out again. Or the whiskey.\n\n\"When did _Fury_ and _Hecla_ escape from the ice?\" asks Fitzjames.\n\n\"Twenty July the next summer,\" says Crozier. \"But you probably know the rest of the story.\"\n\n\"I know that _Fury_ was lost.\"\n\n\"Aye,\" says Crozier. \"Five days after the ice relents \u2014 we'd been creeping along the shore of Somerset Island, trying to stay out of the pack ice, trying to avoid that God-damned limestone always falling from the cliffs \u2014 another gale grounds _Fury_ on a spit of gravel. We man-hauled her free \u2014 using ice screws and sweat \u2014 but then both ships get frozen in, and a God-damned iceberg almost as big as that bastard squatting between _Erebus_ and _Terror_ shoves _Fury_ against the shore ice, tears her rudder away, smashes her timbers to splinters, springs her hull plates, and the crew worked the four pumps in shifts day and night just trying to keep her afloat.\"\n\n\"And you did for a while,\" prompts Fitzjames.\n\n\"A fortnight. We even tried cabling her to a berg, but the fucking cable snapped. Then Hoppner tried raising her to get at her keel \u2014 just as Sir John wanted to do with your _Erebus_ \u2014 but the blizzard put an end to that idea and both ships were in danger of being forced onto the lee shore of the headland. Finally the men just fell over where they were pumping \u2014 they were too exhausted to understand our orders \u2014 and on the twenty-first of August, Parry ordered everyone aboard _Hecla_ and cast her off to save her from being driven aground and poor _Fury_ got shoved right up onto the beach by a bunch of bergs that slammed her hard ashore there and blocked her way out. There wasn't even a chance of a tow. The ice was smashing her to bits as we watched. We barely got _Hecla_ free, and that only with every man working the pumps day and night and the carpenter laboring round the clock to shore her up.\n\n\"So we never got close to the Passage \u2014 or even to sighting new land, really \u2014 and lost a ship, and Hoppner was court-martialed and Parry considered that _his_ court-martial as well since Hoppner was under his command the whole time.\"\n\n\"Everyone was acquitted,\" says Fitzjames. \"Even praised, as I recall.\"\n\n\"Praised but not promoted,\" says Crozier.\n\n\"But you all survived.\"\n\n\"Yes.\"\n\n\"I want to survive _this_ expedition, Francis,\" says Fitzjames. His tone is soft but very determined.\n\nCrozier nods.\n\n\"We should have done what Parry did and put both crews aboard _Terror_ a year ago and sailed east around King William Land,\" says Fitzjames.\n\nIt is Crozier's turn to raise his brows. Not at Fitzjames agreeing that it is an island \u2014 their later-summer sledge reconnaissance had all but settled that \u2014 but in agreeing that they should have made a run for it last autumn, abandoning Sir John's ship. Crozier knows that there is no harder thing for a captain in anyone's navy to do than to give up his ship, but especially so in the Royal Navy. And while _Erebus_ had been under the overall command of Sir John Franklin, Commander James Fitzjames had been its true captain.\n\n\"It is too late now.\" Crozier is in pain. Because the Common Room shares several outer bulkheads and has three overhead Preston Patent Illuminators, it is cold \u2014 the two men can see their breath in the air \u2014 but it's still sixty or seventy degrees warmer than it had been out on the ice and Crozier's feet, especially his toes, are thawing in a rush of jagged pin pokes and red-hot needle stabs.\n\n\"Yes,\" agrees Fitzjames, \"but you were wise to have the gear and provisions sledged to King William Land in August.\"\n\n\"It wasn't a fraction of what we'll need to ferry there if that is to be our survival camp,\" Crozier says brusquely. He had ordered about two tons of clothing, tents, survival gear, and tinned food to be removed from the ships and stored on the northwest shore of the island should they have to abandon ships quickly during the winter, but the ferrying had been absurdly slow and extremely dangerous. Weeks of laborious sledging had left only a ton or so of cache there \u2014 tents, extra slops, tools, and a few weeks of canned food. Nothing more.\n\n\"That thing wouldn't let us stay there,\" he adds softly. \"We all could have moved to tents in September \u2014 I had the ground prepared for two dozen of the big tents, you remember \u2014 but the campsite would not have been as defensible as the ships are.\"\n\n\"No,\" says Fitzjames.\n\n\"If the ships last the winter.\"\n\n\"Yes,\" says Fitzjames. \"Have you heard, Francis, that some of the men \u2014 on both ships \u2014 are calling that creature the Terror?\"\n\n\"No!\" Crozier is offended. He does not want the name of his ship used to evil purposes such as that, even if the men are jesting. But he looks at Commander James Fitzjames's hazel-green eyes and realizes that the other captain is serious and so must be the men. \"The Terror,\" says Crozier, and tastes bile.\n\n\"They think it is no animal,\" says Fitzjames. \"They believe its cunning is something else, is preternatural... supernatural... that there is a demon out there on the ice in the dark.\"\n\nCrozier almost spits he is so disgusted. \"Demon,\" he says in contempt. \"These are the very seamen who believe in ghosts, faeries, Jonahs, mermaids, curses, and sea monsters.\"\n\n\"I've seen you scratch the sail to summon wind,\" Fitzjames says with a smile.\n\nCrozier says nothing.\n\n\"You've lived long enough and traveled far enough to see things that no man knew existed,\" Fitzjames adds, obviously trying to lighten the mood.\n\n\"Aye,\" says Crozier with a bark of a laugh. \"Penguins! I wish they were the largest beastie up here, as they seem to be down south.\"\n\n\"There are no white bears there in the south arctic?\"\n\n\"None that we saw. None that any south-sailing whaler or explorer has seen in seventy years of sailing toward and around that white, volcanic, frozen land.\"\n\n\"And you and James Ross were the first men ever to see the continent. And the volcanoes.\"\n\n\"Aye, we were. And it did Sir James much good. He's married to a beautiful young thing, knighted, happy, retired from the cold. And me... I am... here.\"\n\nFitzjames clears his throat as if to change the subject. \"Do you know, Francis, until this voyage, I honestly believed in the Open Polar Sea. I was quite sure Parliament was correct when it listened to predictions from the so-called polar experts \u2014 in the winter before we sailed, do you remember? It was in the _Times_ \u2014 all about the thermobaric barrier, about the Gulf Stream flowing up under this ice to warm the Open Polar Sea, and the invisible continent that must be up here. They were so convinced it existed that they were proposing and passing laws to send inmates of Southgate and other prisons up here to shovel the coal that must be in such plentitude just a few hundred miles from here on the North Polar Continent.\"\n\nCrozier laughs with real humour this time. \"Yes, to shovel coal to heat the hotels and supply the refueling stations for the steamships that will be making regular trips across the Open Polar Sea by the 1860s at the latest. Oh, God, that I were one of those prisoners in Southgate. Their cells are, required by law and for humanity's sake, twice the size of our cabins, James, and our future would be warm and secure if we only had to sit in such luxury and wait for word of that North Pole continent being discovered and colonized.\"\n\nBoth men are laughing now.\n\nThere comes a thumping from the deck above \u2014 running footsteps rather than mere feet stamping \u2014 and then voices and a sliding of cold air around their feet as someone opens the main hatch above the far end of the companionway and the sound of several pairs of feet clattering down the steps.\n\nBoth captains are silent and waiting when the soft knock comes on the Common Room's thin door.\n\n\"Enter,\" says Commander Fitzjames.\n\nAn _Erebus_ crewman leads in two Terrors \u2014 Third Lieutenant John Irving and a seaman named Shanks.\n\n\"I'm sorry to disturb you, Commander Fitzjames, Captain Crozier,\" says Irving through only slightly chattering teeth. His long nose is white from the cold. Shanks is still carrying a musket. \"Lieutenant Little sent me to report to Captain Crozier as soon as I could.\"\n\n\"Go ahead, John,\" says Crozier. \"You're not still hunting for Lady Silence, are you?\"\n\nIrving looks blank for a second. Then, \"We saw her out on the ice when the last search parties were coming in. No, sir, Lieutenant Little asked me to fetch you right away because...\" The young lieutenant pauses as if forgetting the reason Little had sent him to report.\n\n\"Mr. Couch,\" says Fitzjames to the _Erebus_ mate on duty who had led the two Terrors to the Common Room, \"be so kind as to step out into the companionway and to close the door please, thank you.\"\n\nCrozier has also heard the odd silence as the snoring and hammock creaking has all but ceased. Too many ears in the crew's forward berthing space were awake and listening.\n\nWhen the door is shut, Irving says, \"It's William Strong and Tommy Evans, sir. They're back.\"\n\nCrozier blinks. \"What the devil do you mean, _back?_ Alive?\" He feels the first surge of hope he's had for months.\n\n\"Oh, no sir,\" says Irving. \"Just... one body... really. But it was propped against the stern rail when someone saw it as all the search parties were coming in for the day... about an hour ago. The guards on duty hadn't seen anything. But it was there, sir. On Lieutenant Little's orders, Shanks and I made the crossing as quickly as we could to inform you, Captain. Shanks Mare as it were.\"\n\n\"It?\" snaps Crozier. \"One body? _Back on the ship?_ \" This makes no sense at all to the _Terror_ 's captain. \"I thought you said both Strong and Evans were back.\"\n\nThird Lieutenant Irving's entire face is frostbite white now. \"They are, Captain. Or at least half of them. When we went to look at the body propped there at the stern, it fell over and... well... came apart. As best we can tell, it's Billy Strong from the waist up. Tommy Evans from the waist down.\"\n\nCrozier and Fitzjames can only look at each other.\n\nGOODSIR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 Long. 98\u00b0 41\u2032_\n\nKing William Land, 24 May\u20133 June, 1847\n\nLieutenant Gore's cache party arrived at Sir James Ross's cairn on King William Land late on the evening of 28 May, after five hard days of travel across the ice.\n\nThe good news as they approached the island \u2014 invisible to them until the last minutes \u2014 was that there were pools of salt-free drinking water as they neared the shore. The bad news was that most of these pools had been leached from the base of an almost unbroken series of icebergs \u2014 some of them a hundred feet tall and more \u2014 that had been swept up against the shallows and shore and now stretched like a parapeted white castle wall as far as the eye could see around the curve of land. It took the men a full day to cross this barrier and even then they had to leave some of the robes, fuel, and provisions cached on the sea ice to lighten the sledge load. To add to their difficulties and discomfort, several of the cans of soup and pork they had opened on the ice had gone putrid and had to be thrown away, leaving them less than five days' rations for the return \u2014 assuming that more of the cans were not bad. On top of all that, they found that even here, at what must be the sea's edge, the ice was still seven feet thick.\n\nWorst of all \u2014 for Goodsir, at least \u2014 King William Land, or King William Island as they later learned, was the greatest disappointment of his life.\n\nDevon Island and Beechey Island to the north had been windswept, inhospitable to life at the best of times, and barren except for lichen and low plants, but that was a veritable Garden of Eden compared to what the men now found on King William Land. Beechey had boasted bare ground, some sand and soil, imposing cliffs, and a sort of beach. None of that was to be found on King William Land.\n\nFor half an hour after crossing the iceberg barrier, Goodsir did not know if he was on solid ground or not. He had been prepared to celebrate with the others since this would be the first time any of them had set foot on terra firma in more than a year, but the sea ice gave way beyond the bergs to great tumbles of shore ice and it had been impossible to tell where the shore ice left off and the shore began. Everything was ice, dirty snow, more ice, more snow.\n\nFinally they reached a windswept area free of snow and Goodsir and several of the seamen threw themselves forward onto the gravel, going to hands and knees on the solid ground as if in thanksgiving, but even here the small round stones were frozen solid, as firm as London cobblestones in winter and ten times as cold, and this chill traveled up through their trousers and other layers covering their knees, then into their bones and up through their mittens to their palms and fingers like a silent invitation to the frozen infernal circles of the dead far below.\n\nIt took them four more hours to find Ross's cairn. A heap of rocks promised to be six feet high on or near Victory Point should be easy enough to find \u2014 Lieutenant Gore had said this to all of them earlier \u2014 but on this exposed point the heaps of ice were often at least six feet high and high winds had long since blown off the smaller top stones of the cairn. The late-May sky never darkened into night, but the dim, constant glow made it exceedingly difficult to see anything in three dimensions or to judge distances. The only things that stood out were the bears, and only because of their movement. Half a dozen of the hungry, curious things had been following them off and on all day. Beyond that occasional awkward waddle of movement, everything was lost in a grey-white glow. A serac that looked to be half a mile away and fifty feet high was really only twenty yards away and two feet tall. A bare patch of gravel and stone that seemed a hundred feet away turned out to be a mile away far out on the featureless wind-scoured point.\n\nBut they found the cairn finally, at almost 10:00 p.m. by Goodsir's still-ticking watch, all of the men so exhausted that their arms were hanging like those in sailors' tales of apes, all speech abandoned in their tiredness, the sledge left half a mile north of where they had first come ashore.\n\nGore retrieved the first of two messages \u2014 he had made a copy of this first one to cache somewhere farther south along the coast as per Sir John's instructions \u2014 filled in the date, and scribbled his name. So did Second Mate Charles des Voeux. They rolled the note, slid it into one of the two airtight brass cylinders they'd hauled with them, and, after dropping the cylinder into the centre of the empty cairn, replaced the rocks they'd removed to gain access.\n\n\"Well,\" said Gore. \"That's that then, isn't it?\"\n\nThe lightning storm began not long after they had trudged back to the sledge for a midnight supper.\n\nTo save weight during the iceberg crossing, they had left their heavy wolfskin blanket-robes, ground tarps, and most of the tinned food cached out on the ice. They assumed that since the food was in sealed and soldered tins, it would not attract the white bears that were always sniffing around and that even if it did the bears wouldn't be able to get into the tins. The plan was to get along on two days' reduced rations here on land \u2014 plus any game they might see and shoot, of course, but that dream was fading with the dismal reality of the place \u2014 and to have everyone sleep in the Holland tent.\n\nDes Voeux supervised the preparation of dinner, removing the patented cook kit from its series of cleverly nested wicker baskets. But three of the four cans they had chosen for their first evening's meal on land were spoiled. That left only their Wednesday half-ration portion of salt pork \u2014 always the men's favorite since it was so rich with fat, but not nearly enough to assuage their hunger after such a day of heavy work \u2014 and the last good can, which was labeled \"Superior Clear Turtle Soup,\" which the men hated, knowing from experience that it was neither superior nor clear and most likely not turtle at all.\n\nDr. McDonald on _Terror_ had been obsessed for the last year and a half, ever since Torrington's death at Beechey Island, with the quality of their preserved foodstuffs and was constantly busy experimenting, with the other surgeons' help, to find the best diet by which to avoid scurvy. Goodsir had learned from the older doctor that a certain Stephan Goldner, the expedition's provisioner from Houndsditch who had won the contract through extraordinarily low bids, had almost certainly cheated Her Majesty's government and Her Majesty's Royal Navy Discovery Service by providing inadequate \u2014 and possibly frequently poisonous \u2014 victuals.\n\nThe men filled the freezing air with obscenities upon learning that the cans were filled with rotten stuff.\n\n\"Calm down, lads,\" said Lieutenant Gore after allowing the barrage of best sailor obscenity for a minute or two. \"What say you that we open tomorrow's rations of cans until we find enough for a good meal and simply plan to get back to our ice cache by supper time tomorrow, even if that means midnight?\"\n\nThere was a chorus of assent.\n\nTwo of the next four cans they opened were not spoiled \u2014 that included a strangely meatless \"Irish Stew\" that was only barely edible at the best of times and the deliciously advertised \"Ox Cheeks and Vegetables.\" The men had decided that the oxen parts had come from a tannery and the vegetables from an abandoned root cellar, but it was better than nothing.\n\nNo sooner was the tent up with the sleeping bags unrolled for a floor inside and the food heated on their spirit stove and the hot metal bowls and dishes distributed than the lightning began to strike.\n\nThe first blast of electricity struck less than fifty feet from them and led to every man spilling his ox cheeks and vegetables and stew. The second crash was closer.\n\nThey ran for the tent. Lightning crashed and struck around them like an artillery barrage. It wasn't until they were quite literally piled inside the brown canvas tent \u2014 eight men in a shelter designed for four men and light gear \u2014 that Seaman Bobby Ferrier looked at the wood-and-metal poles holding the tent upright and said, \"Well, fuck this,\" and scrambled for the opening.\n\nOutside, cricket-ball-sized hail was crashing down, sending splinters of ice chips thirty feet into the air. The midnight arctic twilight was being shattered by explosions of lightning so contiguous that they overlapped, setting the sky ablaze in flashes that left blinding retinal echoes.\n\n\"No, no!\" cried Gore, shouting over the thunder and grabbing Ferrier back from the entrance and throwing him down into the crowded tent. \"Anywhere we go on this island, we're the tallest things around. Throw those metal-cored tent poles as far away as you can but stay under the canvas. Get in your bags and lie flat.\"\n\nThe men scrambled to do so, their long hair writhing like snakes under the edges of their Welsh wigs or caps and above their many-wrapped comforters. The storm increased in ferocity and the noise was deafening. The hail pounding them in the backs through canvas and blankets felt like huge fists battering them black and blue. Goodsir actually moaned aloud during the pummeling, more from fear than from pain, although the constant blows constituted the most painful beating he had suffered since his public school days.\n\n\"Holy fucking Christ!\" cried Thomas Hartnell as both hail and lightning grew worse. The men with any brains were under their Hudson's Bay Company blankets now rather than in them, trying to use them as a buffer against the hail. The tent canvas threatened to suffocate all of them, and the thin canvas beneath them did nothing to keep the cold from flowing up and into them, taking their collective breath away.\n\n\"How can there be a lightning storm when it is so _cold?_ \" shouted Goodsir to Gore, who was lying next to him in the huddle of terrified men.\n\n\"It happens,\" the lieutenant shouted back. \"If we decide to move from the ships to land camp, we'll have to bring one God-awful heap of lightning rods with us.\"\n\nThis was the first time that Goodsir had heard any hint of abandoning the ships.\n\nLightning struck the boulder they'd been huddled near during their abbreviated supper not ten feet from the tent, ricocheted over their canvas-covered heads to a second boulder no more than three feet from them, and every man huddled lower, trying to claw through the canvas beneath himself in an attempt to burrow into the rock.\n\n\"Good God, Lieutenant Gore,\" cried John Morfin, whose head was closest to the collapsed opening of the tent, \"there's something moving around out there in the middle of all this.\"\n\nAll the men were accounted for. Gore shouted, \"A bear? Walking around in this?\"\n\n\"Too large to be a bear, Lieutenant,\" shouted Morfin. \"It's...\" Then the lightning struck the boulder again, another blast struck close enough to cause the tent fabric to leap in the air from the static discharge, and everyone cowered flatter, pressed their faces to cold canvas, and abandoned speech in favor of prayer.\n\nThe attack \u2014 Goodsir could only think of it as an attack, as if from Greek gods furious at their hubris for wintering in Boreas's realm \u2014 went on for almost an hour, until the last of the thunder moved past and the flashes became intermittent and then moved on to the southeast.\n\nGore was the first to emerge, but even the lieutenant whom Goodsir knew to be almost without fear did not rise to his feet for a full minute or more after the barrage ceased. Others crawled out on their knees and stayed there, staring around as if in stupefaction or supplication. The sky to the east was a latticework of air-to-air and air-to-ground discharges, the thunder still rolled across the flat island with enough violence to exert a physical pressure on their skins and to make them cover their ears, but the hail had ceased. The smashed white spheres were piled two feet high all around them as far as they could see. After a minute Gore got to his feet and began looking around. The others then also rose, stiffly, moving slowly, testing their limbs, heavily bruised, Goodsir judged, if his own pain was any measure of their common abuse by the heavens. The midnight twilight was dimmed enough by the thick clouds to the south that it almost seemed as if real darkness was falling.\n\n\"Look at this,\" called Charles Best.\n\nGoodsir and the others gathered near the sledge. The tins of food and other mat\u00e9riel had been unpacked and stacked near the cooking area before their aborted supper, and somehow the lightning had contrived to strike the low pyramid of stacked cans while missing the sledge itself. All of Goldner's canned food had been blasted apart as surely as if a cannonball had struck the stack \u2014 a perfect roll in a game of cosmic ninepins. Charred metal and still-steaming inedible vegetables and rotten meat were scattered in a twenty-yard radius. Near the surgeon's left foot was a charred, twisted, and blackened receptacle with the legend COOKING APPARATUS (I) visible on its side. It was part of their travel mess kit and had been sitting on one of their spirit stoves when they had run for shelter. The metal bottle holding a pint of pyroligneous ether fuel next to it had exploded, sending shrapnel flying in all directions but evidently just barely passing over their heads as they huddled in the tent. If the lightning had ignited the stack of fuel bottles sitting in their wooden box next to the two shotguns and shells a few feet away on the sledge, the explosion and flames would have consumed them all.\n\nGoodsir had the urge to laugh but didn't do so out of fear he might weep at the same time. None of the men spoke for a moment.\n\nFinally John Morfin, who had climbed the low ridge of hail-pummeled ice above their campsite, cried, \"Lieutenant, you need to see this.\"\n\nThey climbed up to look toward where he was staring.\n\nAlong the backside of this low ice ridge, coming from the ice jumble south of them and disappearing toward the sea northwest of them, were absolutely impossible tracks. Impossible because they were larger than any tracks of any living animal on earth. For five days now, the men had seen the paw prints of the white bears in the snow, and some of those tracks were absurdly large \u2014 some twelve inches long \u2014 but these indistinct tracks were more than half again larger than that. Some appeared to be as long as a man's arm. And they were new \u2014 there was no doubt whatsoever of that \u2014 because the indentations were not in the old snow but pressed into the thick layer of fresh hailstones.\n\nWhatever had walked past their camp had done so during the height of the lightning and hail storm, just as Morfin had reported.\n\n\"What is this?\" said Lieutenant Gore. \"This can't be. Mr. Des Voeux, be so kind as to fetch one of the shotguns and some shells from the sledge, please.\"\n\n\"Aye, sir.\"\n\nEven before the mate came back with the shotgun, Morfin, Marine Private Pilkington, Best, Ferrier, and Goodsir began trudging after Gore as the lieutenant followed the impossible tracks northwest.\n\n\"These are too large, sir,\" said the Marine. He had been included in the party, Goodsir knew, because he was one of the few men aboard either ship who had ever hunted game larger than a grouse.\n\n\"I know that, Private,\" said Gore. He accepted the shotgun from Second Mate Des Voeux and calmly loaded a shell as the seven men strode through the heaps of hail toward the dark clouds beyond the iceberg-guarded shoreline.\n\n\"Maybe they're not paw prints, but something... an arctic hare or something hopping through the slush, making the prints with its entire body,\" said Des Voeux.\n\n\"Yes,\" said Gore absently. \"Perhaps so, Charles.\"\n\nBut they _were_ paw prints of some kind. Dr. Harry D. S. Goodsir knew that. Every man walking near him knew that. Goodsir, who had never hunted anything larger than a rabbit or partridge, could tell that this wasn't the track of some small thing throwing its body left, then right but rather the footprints of something walking first on four legs and then \u2014 if the tracks were to be believed \u2014 almost a hundred yards on two legs. At that point they were the tracks of a walking man, if a man had feet the length of his forearms and could cover almost five feet between strides while leaving no impressions of toes but rather the striations of claws.\n\nThey reached the windswept area of stones where Goodsir had thrown himself down on his knees so many hours before \u2014 the hailstones here had shattered into countless icy shards so the area remained almost bare \u2014 and here the tracks stopped.\n\n\"Spread out,\" said Gore, still holding the shotgun casually under his arm as if he were taking a walk through his family's estate in Essex. He pointed to each man and then pointed to the edge of the open area he wanted that man to check. The rocky space was not much larger than a cricket pitch.\n\nThere were no tracks leading away from the stones. The men shuffled back and forth for several minutes, checking and double-checking, not wanting to pollute the unbroken snow beyond the rocks with their own footprints, and then all stood still, staring at one another. They were standing in an almost perfect circle. No tracks led away from the rocky space.\n\n\"Lieutenant... ,\" began Best.\n\n\"Quiet a minute,\" said Gore sharply but not unkindly. \"I am thinking.\" He was the only man moving now, striding past the men and looking out onto the snow, ice, and hail around them as if there were some schoolboy prank being pulled. The light was stronger now as the storm passed farther east \u2014 it was almost two o'clock in the morning and the snow and layer of hail remained untouched beyond the stones.\n\n\"Lieutenant,\" persisted Best. \"It's Tom Hartnell.\"\n\n\"What about him?\" snapped Gore. He was beginning his third circuit of the loop.\n\n\"He's not here. I just realized \u2014 he hasn't been with us since we got out of the tent.\"\n\nGoodsir's head snapped up and turned at the same second the others' did. Three hundred yards behind them, the low ice ridge hid the view of their collapsed tent and sledge. Nothing else moved on the vast expanse of white and grey.\n\nThey all began running at once.\n\nHartnell was alive but unconscious and still lying under the tent canvas. There was a huge welt on the side of his head \u2014 the thick canvas had torn where the fist-sized ball of hail had ripped through \u2014 and he was bleeding from his left ear, but Goodsir soon found a slow pulse. They pulled the unconscious man from the fallen tent, retrieved two sleeping bags, and made him as warm and comfortable as they could. Dark clouds were streaming overhead again.\n\n\"How serious is it?\" asked Lieutenant Gore.\n\nGoodsir shook his head. \"We won't know until he wakes... _if_ he wakes. I'm surprised more of us weren't knocked unconscious. It was a terrible downpour of solid objects.\"\n\nGore nodded. \"I'd hate to lose Tommy after the death of his brother, John, last year. That would be too much for the family to bear.\"\n\nGoodsir remembered preparing John Hartnell for burial in his brother Thomas's best flannel shirt. He thought of that shirt under the frozen soil and snow-covered gravel so many hundreds of miles to the north, the cold wind below that black cliff blowing between the wooden head markers. Goodsir shivered.\n\n\"We're all getting too chilled,\" said Gore. \"We need to get some sleep. Private Pilkington, find the staves for the tent poles and help Best and Ferrier get the tent erected again.\"\n\n\"Aye, aye, sir.\"\n\nWhile those two men were hunting for the tent staves, Morfin held up the canvas. Their tent had been so riddled by hailstones that it looked like a battle flag.\n\n\"Dear God,\" said Des Voeux.\n\n\"The sleeping bags are all sodden,\" reported Morfin. \"The inside of the tent is soaked.\"\n\nGore sighed.\n\nPilkington and Best returned with two charred, bent stubs of wood and iron.\n\n\"The poles were struck, Lieutenant,\" reported the Marine private. \"Looks like the iron core in them attracted the lightning, sir. Not much good as a centre pole now.\"\n\nGore just nodded. \"We have the axe still on the sledge. Break it out and bring the extra shotgun to use as double poles. Melt some ice to use as an anchor for them if you have to.\"\n\n\"The spirit stove's busted,\" Ferrier reminded them. \"We won't be melting no more ice for a while.\"\n\n\"We have two more stoves on the sledge,\" said Gore. \"And we have some drinking water in the bottles. It's frozen now, but put the bottles inside your clothing until some melts. Pour that into a hole you chip in the ice. It will freeze soon enough. Mr. Best?\"\n\n\"Aye, sir?\" said the stocky young seaman, trying to stifle a yawn.\n\n\"Sweep out the tent as best you can, take your knife and cut the stitchings on two of the sleeping bags. We'll use those as over-and-under blankets while we all huddle together for warmth tonight. We have to get some sleep.\"\n\nGoodsir was watching the unconscious Hartnell for any indications of consciousness, but the young man was as still as a corpse. The surgeon had to check his breathing to make sure he was still alive.\n\n\"Are we going back in the morning, sir?\" asked John Morfin. \"To fetch our cache on the ice and then back to the ships, I mean? We don't have enough food now to get back with anything like sensible rations.\"\n\nGore smiled and shook his head. \"A couple of days of fasting won't harm us, man. But with Hartnell hurt, I'll send four of you back to the ice cache with him on the sledge. You make the best camp you can there while I take one man to head south as per Sir John's orders. I need to cache the second letter to the Admiralty, but more important, we need to press as far south as possible to see if there's any sign of open water. This whole trip will have been for nothing if we don't do that.\"\n\n\"I volunteer to go with you, Lieutenant Gore,\" said Goodsir and was astonished at the sound of his own voice. For some reason, pressing on with the officer was very important to him.\n\nGore also looked surprised. \"Thank you, Doctor,\" he said softly, \"but it would make more sense if you stayed with our wounded messmate, would it not?\"\n\nGoodsir blushed deeply.\n\n\"Best will go with me,\" said the lieutenant. \"Second Mate Des Voeux will be in command of the ice party until I return.\"\n\n\"Yes, sir,\" both men said in unison.\n\n\"Best and I will leave in about three hours and we'll press as far south as we can, carrying only some salt pork, the message canister, one water bottle apiece, some blankets if we have to bivouac, and one of the shotguns. We'll turn back sometime around midnight and try to rendezvous with you on the ice by eight bells tomorrow morning. We'll have a lighter sledge load heading back to the ships \u2014 except for Hartnell, I mean \u2014 and we know the best places to cross the ridges, so I'll wager we get home in three days or less, rather than five.\n\n\"If Best and I aren't back to sea camp by midnight of the day after tomorrow, Mr. Des Voeux, take Hartnell and return to the ship.\"\n\n\"Aye, sir.\"\n\n\"Private Pilkington, are you especially tired?\"\n\n\"Yes, sir,\" said the thirty-year-old Marine. \"I mean no, sir. I'm ready for any duty you ask of me, Lieutenant.\"\n\nGore smiled. \"Good. You get the next three hours' watch. All I can promise you is that you'll be the first man allowed to sleep when your sledge party reaches the cache camp later in the day. Take the musket there that's not doing tent pole duty but stay inside the tent \u2014 just poke your head out from time to time.\"\n\n\"Very good, sir.\"\n\n\"Dr. Goodsir?\"\n\nThe surgeon's head came up.\n\n\"Would you and Mr. Morfin be so kind as to carry Mr. Hartnell into the tent and get him comfortable? We'll put Tommy in the centre of our little huddle to try to keep him warm.\"\n\nGoodsir nodded and moved to lift his patient by the shoulders without removing him from the sleeping bag. The welt on the unconscious Hartnell's head was now as large as the surgeon's small, pale fist.\n\n\"All right,\" said Gore through chattering teeth, looking at the tattered tent that was going up, \"let's the rest of us get those blankets spread and huddle together like the orphans we are and try to get an hour or two's sleep.\"\n\nFRANKLIN\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n3 June, 1846\n\nSir John could not quite believe what he was seeing. There were eight figures, just as he had anticipated, but they were... _wrong._\n\nFour of the five exhausted, bearded, and goggled men in the sledge harness made sense \u2014 seamen Morfin, Ferrier, and Best, with the huge Private Pilkington leading \u2014 but the fifth man in harness was Second Mate Des Voeux, whose expression suggested he had been to Hell and back. Seaman Hartnell walked beside the sledge. The thin sailor's head was heavily bandaged and he was staggering along as if he were part of Napoleon's retreat from Moscow. The surgeon, Goodsir, was also walking alongside the sledge and administering to someone \u2014 or something \u2014 on the sledge itself. Franklin looked for Lieutenant Gore's distinctive red wool scarf \u2014 the comforter was almost six feet long and impossible to miss \u2014 but, bizarrely, it seemed that most of the dark, staggering figures were wearing shorter versions of it.\n\nFinally, walking behind the sledge, there came a short, fur-parka-wrapped creature whose face was invisible under a hood but who could only be an Esquimaux.\n\nBut it was the sledge itself that made Captain Sir John Franklin cry out, \"Dear God!\"\n\nThis sledge was too narrow for two men to lie on side by side, and Sir John's telescope had not lied to him. Two bodies lay atop each other. The one on top was another Esquimaux \u2014 a sleeping or unconscious old man with a brown, lined face and streaming white hair flowing back on the wolfskin hood that someone had pulled back and propped under his head like a pillow. It was to this figure that Goodsir was attending as the sledge approached _Erebus._ Beneath the Esquimaux man's supine body was the blackened, distorted, and too-obviously dead face and form of Lieutenant Graham Gore.\n\nFranklin, Commander Fitzjames, Lieutenant Le Vesconte, First Mate Robert Sergeant, Ice Master Reid, Chief Surgeon Stanley, and such petty officers as Brown, the bosun's mate; John Sullivan, captain of the maintop; and Mr. Hoar, Sir John's steward, all rushed to the sledge, as did forty or more of the seamen who had come up on deck upon the sound of the lookout's hail.\n\nFranklin and the others stopped in their tracks before closing with the sledge party. What had looked through Franklin's telescope like a grey spattering of red wool comforters on the men turned out to be great smears of red on their dark greatcoats. The men were smeared with blood.\n\nThere was an explosion of babble. Some of the men in harness hugged friends who ran to them. Thomas Hartnell collapsed on the ice and was surrounded by men trying to help. Everyone was talking and shouting at once.\n\nSir John had eyes only for the corpse of Lieutenant Graham Gore. The body had been covered by a sleeping robe, but this had partially fallen away so that Sir John could see Gore's handsome face, now absolutely white in places from drained blood, burned black by the arctic sun in other areas. His features were distorted, the eyelids partially raised and the whites visible and glinting with ice, the jaw sagging open, tongue protruding, and the lips already pulling back away from the teeth in what looked to be a snarl or expression of pure horror.\n\n\"Get that... savage... off Lieutenant Gore,\" commanded Sir John. \" _Immediately!_ \"\n\nSeveral men hurried to comply, lifting the Esquimaux man by his shoulders and feet. The old man moaned and Dr. Goodsir exclaimed, \"Careful! Easy with him! He has a musket ball near his heart. Carry him to the sick bay, please.\"\n\nThe other Esquimaux's parka hood was thrown back now and Sir John noted with shock that it was a young woman. She moved closer to the wounded old man.\n\n\"Wait!\" cried Sir John, waving at his ship's assistant surgeon. \"The sick bay? You are seriously suggesting that we allow that... native person... into the sick bay of our ship?\"\n\n\"This man is my patient,\" Goodsir said with a brazen stubbornness that Sir John Franklin never would have guessed could reside in the short little surgeon. \"I need to get him to a place where I may be able to operate \u2014 remove the ball from his body if that is possible. Stem the bleeding if it is not. Carry him in, please, gentlemen.\"\n\nThe crewmen holding the Esquimaux looked to their expedition commander for a decision. Sir John was so flummoxed that he could not speak.\n\n\"Hurry along now,\" commanded Goodsir in a confident voice.\n\nObviously taking Sir John's silence as tacit assent, the men carried the grey-haired Esquimaux man up the ramp of snow and onto the ship. Goodsir, the Esquimaux wench, and several crewmen followed, some helping young Hartnell along.\n\nFranklin, almost unable to hide his shock and horror, stood where he was, still looking down at the corpse of Lieutenant Gore. Private Pilkington and Seaman Morfin were unlashing the lines holding Gore in place on the sledge. \"For God's sake,\" said Franklin, \"cover his face.\"\n\n\"Aye, sir,\" said Morfin. The sailor pulled up the Hudson's Bay Company blanket that had slipped away from the lieutenant's face during their rough day and a half on the ice and pressure ridges.\n\nSir John could still see the concavity of his handsome lieutenant's gaping mouth through the dry sag of the red blanket. \"Mr. Des Voeux,\" snapped Franklin.\n\n\"Yes, sir.\" Second Mate Des Voeux, who had been overseeing the unlashing of the lieutenant's body, shuffled over and knuckled his forehead. Franklin could see that the whisker-stubbled man, his face sunburned a raw red and sandblasted by the wind, was so exhausted that he could only just raise his arm to salute.\n\n\"See to it that Lieutenant Gore's body is brought to his quarters, where you and Mr. Sergeant will see that the body is prepared for burial under the supervision of Lieutenant Fairholme here.\"\n\n\"Aye, sir,\" said Des Voeux and Fairholme in unison.\n\nFerrier and Pilkington, exhausted as they were, shook off efforts at assistance and lifted the body of their dead lieutenant. The corpse seemed as stiff as a piece of firewood. One of Gore's arms was bent and his bare hand, turned black from the sun or decomposition, was raised in a sort of frozen clawing gesture.\n\n\"Wait,\" said Franklin. He realized that if he sent Mr. Des Voeux off on this errand, it would be hours before he could receive an official report from the man who had been second in command on this party. Even the confounded surgeon was out of sight, taking the two Esquimaux with him. \"Mr. Des Voeux,\" said Franklin, \"after you've seen to Lieutenant Gore's initial preparation, report to me in my cabin.\"\n\n\"Aye, Captain,\" the mate said tiredly.\n\n\"In the meantime, who was with Lieutenant Gore at the end?\"\n\n\"We all were, sir,\" said Des Voeux. \"But Seaman Best was there with him \u2014 just the two of them \u2014 for most of the last two days we were on and near King William Land. Charlie saw everything there that Lieutenant Gore did.\"\n\n\"Very well,\" said Sir John. \"Go on about your duties, Mr. Des Voeux. I will hear your report soon. Best, come with me and Commander Fitzjames now.\"\n\n\"Aye, aye, sir,\" said the sailor, cutting away the last of his leather harness because he was too exhausted to untie the knots. He did _not_ have the strength to raise his arm in a salute.\n\nThe three Preston Patent Illuminators were milky overhead with the never-setting sunlight as Seaman Charles Best stood to make his report to a seated Sir John Franklin, Commander Fitzjames, and Captain Crozier \u2014 the captain of HMS _Terror_ had arrived for a visit by convenient accident just minutes after the sledge party had come aboard. Edmund Hoar, Sir John's steward and sometime secretary, sat behind the officers, taking notes. Best stood, of course, but Crozier had suggested that the exhausted man could do with some medicinal brandy, and while Sir John's expression showed his disapproval, he had agreed to ask Commander Fitzjames to provide some out of his private stock. The liquor seemed to have revived Best somewhat.\n\nThe three officers interrupted from time to time with questions while the teetering Best made his report. When his description of the team's laborious sledge trip to King William Land threatened to stretch on too long, Sir John hurried the man to the events of the last two days.\n\n\"Yes, sir. Well, after that first night of lightning and thunder at the cairn and then finding them... tracks, marks... in the snow, we tried to sleep a couple of hours but didn't really succeed, and then Lieutenant Gore and I set off to the south with light rations while Mr. Des Voeux took the sledge and what was left of the tent and poor Hartnell, who was still out cold then, and we said our 'until tomorrows' and the lieutenant and I headed south and Mr. Des Voeux and his people headed out to the sea ice again.\"\n\n\"You were armed,\" said Sir John.\n\n\"Aye, Sir John,\" said Best. \"Lieutenant Gore had a pistol. I had one of the two shotguns. Mr. Des Voeux kept the other shotgun with his party and Private Pilkington carried the musket.\"\n\n\"Tell us why Lieutenant Gore divided the party,\" commanded Sir John.\n\nBest seemed confused by the question for a moment but then brightened. \"Oh, he told us he was following your orders, sir. With the food at the cairn camp destroyed by lightning and the tent damaged, most of the party needed to get back to sea camp. Lieutenant Gore and me went on to cache that second message container somewhere south along the coast and to see if there was any open water. There wasn't any, sir. Open water, I mean. Not a hint. Not a fu\u2014... not a single reflection of dark sky to suggest water.\"\n\n\"How far did the two of you go, Best?\" asked Fitzjames.\n\n\"Lieutenant Gore figured we'd traveled about four miles south across that snow and frozen gravel when we reached a big inlet, sir... rather like the bay at Beechey where we wintered a year ago. But you know what four miles is like in the fog and wind and with ice, sirs, even on land around here. We probably hiked ten miles at least to cover the four. The inlet was frozen solid. Solid as the pack ice here. Not even that usual bit of open water you get between shore and ice in any inlet during the summer up here. So we crossed the mouth of her, sirs, and then went another quarter of a mile or so out along a promontory there where Lieutenant Gore and me built another cairn \u2014 not as tall or fancy as Captain Ross's, I'm sure, but solid, and high enough that anyone would see it right away. That land is so flat that a man is always the tallest thing on it. So we piled the rocks about eye-high and set in that second message, same as the first the lieutenant told me, in its fancy brass cylinder.\"\n\n\"Did you turn back then?\" asked Captain Crozier.\n\n\"No, sir,\" said Best. \"I admit I was worn out. So was Lieutenant Gore. The walking had been hard all that day, even the sastrugi were hard to kick our way through, but it'd been foggy so we only got glimpses of the coast along there from time to time when the fog lifted, so even though it was already afternoon by the time we finished building the cairn and leaving the message, Lieutenant Gore, he had us walk about six or seven more miles south along the coast. Sometimes we could see, most of the time we couldn't. But we could _hear._ \"\n\n\"Hear what, man?\" asked Franklin.\n\n\"Something following us, Sir John. Something big. And breathing. Sometimes woofin' a bit... you know, sirs, like them white bears do, like they're coughing?\"\n\n\"You identified it as a bear?\" asked Fitzjames. \"You said that you were the largest things visible on land. Certainly if a bear was following you, you could see it when the fog lifted.\"\n\n\"Aye, sir,\" said Best, frowning so deeply that it appeared he might start crying. \"I mean, no, sir. We couldn't identify it as no bear, sir. We could have, normal like. We should have. But we didn't and couldn't. Sometimes we'd hear it coughin' right behind us \u2014 fifteen feet away in the fog \u2014 and I'd level the shotgun and Lieutenant Gore would prime his pistol, and we'd wait, sort of holding our breath, but when the fog lifted we could see a hundred feet and nothing was there.\"\n\n\"It must have been an aural phenomenon,\" said Sir John.\n\n\"Aye, sir,\" agreed Best, his tone suggesting that he did not understand Sir John's comment.\n\n\"The shore ice making noise,\" said Sir John. \"Perhaps the wind.\"\n\n\"Oh, aye, yes, sir, Sir John,\" said Best. \"Only there weren't no wind. But the ice... could've been that, m'lord. Always could be that.\" His tone explained that it could not have been.\n\nShifting as if he was feeling irritation, Sir John said, \"You said at the outlet that Lieutenant Gore died... was killed... after you rejoined the other six men on the ice. Please proceed to that point in the narrative.\"\n\n\"Yes, sir. Well, it must've been close to midnight when we reached as far south as we could go. The sun was gone from the sky ahead of us but the sky had that gold glow... you know how it is around midnight up here, Sir John. The fog had lifted well enough for a short while that when we climbed a little rocky nub of a hill... not a hill, really, but a high spit maybe fifteen feet above the rest of the flat, frozen gravel there... we could see the shore twisting away farther to the south to the blurry horizon with glimpses of bergs poking up from over the horizon from where they'd piled up along the shoreline. No water. Everything frozen solid all the way down. So we turned around and started walking back. We didn't have no tent, no sleeping bags, just cold food to chew on. I broke a good tooth on it. We were both very thirsty, Sir John. We didn't have a stove to melt snow or ice, and we'd started with only a little bit of water in a bottle that Lieutenant Gore kept under his coats and waistcoat.\n\n\"So we walked through the night \u2014 through the hour or two of sort of twilight that passes for night here, sirs, and then on for more hours \u2014 and I fell asleep walking half a dozen times and would've walked in circles until I dropped, but Lieutenant Gore would grab me by the arm and shake me a bit and lead me the right way. We passed the new cairn and then crossed the inlet, and sometime around six bells, when the sun was full up high again, we reached the spot where we'd camped the night before near the first cairn, Sir James Ross's cairn I mean \u2014 actually it'd been two nights before, during the first lightning storm \u2014 and we just kept trudging on, following the sledge tracks out to the heaped shore bergs and then out onto the sea ice.\"\n\n\"You said 'during the first lightning storm,'\" interrupted Crozier. \"Were there more? We had several here while you were gone, but the worst seemed to be to the south.\"\n\n\"Oh, yes, sir,\" said Best. \"Every few hours, even with the fog so heavy, the thunder would start rumblin' again and then our hair would start flying about, trying to lift off our heads, and anything metal we had \u2014 belt buckles, the shotgun, Lieutenant Gore's pistol \u2014 would start glowing blue, and we'd find a place to hunker down in the gravel and we'd just lie there trying to disappear into the ground while the world exploded around us like cannon fire at Trafalgar, sirs.\"\n\n\"Were you _at_ Trafalgar, Seaman Best?\" Sir John asked icily.\n\nBest blinked. \"No, sir. Of course not, sir. I'm only twenty-five, m'lord.\"\n\n\" _I_ was at Trafalgar, Seaman Best,\" Sir John said stiffly. \"As signals officer on HMS _Bellerophon,_ where thirty-three of the forty officers were killed in that single engagement. Please restrain from using metaphors or similes from beyond your experience for the remainder of your report.\"\n\n\"Aye, aye, s-sir,\" stammered Best, weaving now not only from exhaustion and grief but with terror at making such a faux pas. \"I apologize, Sir John. I didn't mean... I mean... I shouldn't... that is...\"\n\n\"Continue with your narrative, seaman,\" said Sir John. \"But tell us about the last hours of Lieutenant Gore.\"\n\n\"Yes, sir. Well... I couldn't've climbed the iceberg barrier without Lieutenant Gore helping me \u2014 God bless him \u2014 but we did, eventually, and then got out onto the ice itself to where it was just a mile or two to sea camp, where Mr. Des Voeux and the others were waiting for us, but then we got lost.\"\n\n\"How could you possibly get lost,\" asked Commander Fitzjames, \"if you were following the sledge tracks?\"\n\n\"I don't know, sir,\" said Best, his voice flattened by exhaustion and grief. \"It was foggy. It was _very_ foggy. Mostly we couldn't see ten feet in any direction. The sunlight made everything glow and made everything flat. I think we climbed the same ice ridge three or four times, and every time we did our sense of direction got more distorted. And out on the sea ice, there were long patches where the snow had blown away and the sledge's runners hadn't left no marks. But the truth is, sirs, I think we were both, Lieutenant Gore and me, marching along while asleep and just lost the tracks without knowing it.\"\n\n\"Very well,\" said Sir John. \"Continue.\"\n\n\"Well, then we heard the shots... ,\" began Best.\n\n\"Shots?\" said Commander Fitzjames.\n\n\"Aye, sir. Both musket and shotgun they were. In the fog, with the sound bouncin' back from the bergs and ice ridges all around, it sounded like the shots were coming from everywhere at once, but they were close. We started hallooing into the fog and pretty soon we hear Mr. Des Voeux hallooing back and thirty minutes later \u2014 it took that long for the fog to lift a bit \u2014 we stumbled into the sea camp. The boys had got the tent patched in the thirty-six hours or so we were gone \u2014 more or less patched \u2014 and it was set up next to the sledge.\"\n\n\"Were the shots to guide you in?\" asked Crozier.\n\n\"No, sir,\" said Best. \"They was shooting bears. And the old Esquimaux man.\"\n\n\"Explain,\" said Sir John.\n\nCharles Best licked his torn and ragged lips. \"Mr. Des Voeux can explain better than me, sirs, but basically they got back to sea camp the day before to find the tins of food all broken into and scattered and spoiled \u2014 by the bears, they reckoned \u2014 so Mr. Des Voeux and Dr. Goodsir decided to shoot some of the white bears that kept sniffing around the camp. They'd shot a sow and her two cubs just before we got there and had been dressing the meat. But they heard movement around them \u2014 more of that coughin', breathin' in the fog I described, sirs \u2014 and then, I guess, the two Esquimaux \u2014 the old man and his woman \u2014 came over a pressure ridge in the fog, just all more white fur, and Private Pilkington fired his musket and Bobby Ferrier fired his shotgun. Ferrier missed both targets, but Pilkington brought down the man with a ball to the chest.\n\n\"When we got there, they'd brought the shot Esquimaux and the woman and some of the white bear meat back to the sea camp \u2014 leaving bloody swaths on the ice, sirs, which is what we followed in for the last hundred yards or so \u2014 and Dr. Goodsir was trying to save the life of the old Esquimaux man.\"\n\n\"Why?\" asked Sir John.\n\nBest had no answer to that. No one else spoke.\n\n\"Very well,\" said Sir John at last. \"How long was it after you were reunited with Second Mate Des Voeux and the others at this sea camp when Lieutenant Gore was attacked?\"\n\n\"No more than thirty minutes, Sir John. Probably less.\"\n\n\"And what provoked the attack?\"\n\n\"Provoked it?\" repeated Best. His eyes no longer seemed focused. \"You mean, like shooting them white bears?\"\n\n\"I mean, what were the exact circumstances of the attack, Seaman Best?\" said Sir John.\n\nBest rubbed his forehead. His mouth was open for a long moment before he spoke.\n\n\"Nothing provoked it. I was talking to Tommy Hartnell \u2014 he was in the tent with his head all bandaged, but awake again \u2014 he couldn't remember nothing from until sometime before the first lightning storm \u2014 and Mr. Des Voeux was supervising Morfin and Ferrier getting two of the spirit stoves working so we could heat some of that bear meat, and Dr. Goodsir had the old Esquimaux's parka off and was probing a nasty hole in the old man's chest. The woman had been standing there watching, but I didn't see where she was right then because the fog had gotten thicker, and Private Pilkington was standing guard with the musket, when suddenly Lieutenant Gore, he shouts \u2014 'Quiet, everyone! Quiet!' \u2014 and we all hushed up and quit what we were saying and doing. The only sound was the hiss of the two spirit stoves and the bubbling of the snow we'd melted to water in the big pans \u2014 we were going to make some sort of white bear stew, I guess \u2014 and then Lieutenant Gore took out his pistol and primed it and cocked it and took a few steps away from the tent and... .\"\n\nBest stopped. His eyes were completely unfocused, mouth still open, a glisten of saliva on his chin. He was looking at something not in Sir John's cabin.\n\n\"Go on,\" said Sir John.\n\nBest's mouth worked but no sound came out.\n\n\"Continue, seaman,\" said Captain Crozier in a kinder voice.\n\nBest turned his head in Crozier's direction but his eyes were still focused on something far away.\n\n\"Then... ,\" began Best. \"Then... the ice just rose up, Captain. It just rose up and surrounded Lieutenant Gore.\"\n\n\"What are you talking about?\" snapped Sir John after another interval of silence. \"The ice can't just rise up. What did you _see?_ \"\n\nBest did not turn his head in Sir John's direction. \"The ice just rose up. Like when you can see the pressure ridges building all of a sudden. Only this was no ridge \u2014 no ice \u2014 it just rose up and took on a... _shape._ A white shape. A form. I remember there were... claws. No arms, not at first, but claws. Very large. And teeth. I remember the teeth.\"\n\n\"A bear,\" Sir John said. \"An arctic white bear.\"\n\nBest only shook his head. \"Tall. The thing just seemed to rise up _under_ Lieutenant Gore... _around_ Lieutenant Gore. It was... _too tall._ Over twice as tall as Lieutenant Gore, and you know that he was a tall man. It was at least twelve feet tall, taller than that, I think, and too large. Much too large. And then Lieutenant Gore sort of disappeared as the thing... surrounded him... and all we could see was the lieutenant's head and shoulders and boots, and his pistol went off \u2014 he didn't aim, I think he fired into the ice \u2014 and then we were all screaming, and Morfin was scrambling for the shotgun and Private Pilkington was running and aiming the musket but was afraid to fire because the thing and the lieutenant were all one thing now, and then... then we heard the crunching and snapping.\"\n\n\"The bear was biting the lieutenant?\" asked Commander Fitzjames.\n\nBest blinked and looked at the ruddy commander. \"Biting him? No, sir. The thing didn't bite. I couldn't even see its head... not really. Just two black spots floating twelve, thirteen feet in the air... black but also red, you know, like when a wolf turns toward you and the sun catches its eyes? The snapping and crunching was from Lieutenant Gore's ribs and chest and arms and bones breaking.\"\n\n\"Did Lieutenant Gore cry out?\" asked Sir John.\n\n\"No, sir. He didn't make a noise.\"\n\n\"Did Morfin and Pilkington fire their weapons?\" asked Crozier.\n\n\"No, sir.\"\n\n\"Why not?\"\n\nBest \u2014 strangely \u2014 smiled. \"Why, there was nothing to shoot at, Captain. One second the _thing_ was there, rising up over Lieutenant Gore and crushing him like you or me would crush a rat in our palm, and the next second it was _gone._ \"\n\n\"What do you mean, _gone?_ \" demanded Sir John. \"Couldn't Morfin and the Marine private have fired at it while it was retreating into the fog?\"\n\n\"Retreating?\" repeated Best, and his absurd and disturbing smile grew broader. \"The shape didn't retreat. It just went back down into the ice \u2014 like a shadow going away when the sun goes behind a cloud \u2014 and by the time we got to Lieutenant Gore he was dead. Mouth wide open. Didn't even have time to scream. The fog lifted then. There were no holes in the ice. No cracks. Not even a little breathing hole like the harp seals use. Just Lieutenant Gore lying there broken \u2014 his chest was all caved in, both arms was broke, and he was bleeding from his ears, eyes, and mouth. Dr. Goodsir pushed us away, but there was nothing that he could do. Gore was dead and already growing as cold as the ice under him.\"\n\nBest's insane and irritating grin wavered \u2014 the man's torn lips were quivering but still drawn back over his teeth \u2014 and his eyes became less focused than ever.\n\n\"Did... ,\" began Sir John, but stopped as Charles Best collapsed in a heap to the deck.\n\nGOODSIR\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\nJune, 1847\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _4 June, 1847 \u2014_\n\n _When Stanley and I stripped the wounded Esquimaux man naked, I was reminded that he was wearing an Amulet made up of a flat, smooth Stone, smaller than my fist, in the shape of a White Bear \u2014 the stone did not seem to have been carved but in its natural, thumb-smoothed state perfectly captured the long neck, small head, and powerful extended legs and forward motion of the living animal. I had seen the Amulet when I'd inspected the man's wound on the ice but thought nothing of it._\n\n _The ball from Private Pilkington's musket had entered the native man's Chest not an inch below that amulet, pierced flesh and muscle between the third and fourth ribs (deflected slightly by the higher of the two), passed through his Left Lung, and lodged in his Spine, severing numerous Nerves there._\n\n _There was no way that I could save him \u2014 I knew from earlier inspection that any Attempt to Remove the musket ball would have caused instant death, and I could not stem the internal Bleeding from Within the Lung \u2014 but I did my best, having the Esquimaux carried to the part of the Sick Bay which Surgeon Stanley and I have set up as a surgery. For Half an Hour yesterday after my return to the Ship, Stanley and I probed the wound front and back with our Cruelest Instruments and Cut with Energy until we found the location of the Ball in his Spine, and generally confirmed our prognosis of Imminent Death._\n\n _But the unusually tall, powerfully built grey-haired Savage had not yet agreed with our Prognosis. He continued to exist as a man. He continued to force breaths through his torn and bloodied lung, coughing blood repeatedly. He continued to stare at us through his disturbingly light-coloured \u2014 for an Esquimaux \u2014 eyes, watching our Every Movement._\n\n _Dr. McDonald arrived from_ Terror _and, at Stanley's suggestion, took the second Esquimaux \u2014 the girl \u2014 into the rear alcove of the Sick Bay, separated from us by a blanket serving as a curtain, for an Examination. I believe that Surgeon Stanley was less interested in having the girl examined than he was in getting her out of the sick bay during our bloody probing of her husband's or father's wounds... although neither the Subject nor the Girl appeared disturbed by either the Blood or Wound which would have made any London Lady \u2014 and no few surgeons in training \u2014 faint dead away._\n\n _And speaking of fainting, Stanley and I had just finished our examination of the dying Esquimaux when Captain Sir John Franklin came in with two crewmen half-carrying Charles Best, who, they informed us, had passed out in Sir John's cabin. We had the men put Best on the nearest cot and it took only a minute's Cursory examination for me to list the reasons why the man had fainted: the same extreme Exhaustion which all of us on Lieutenant Gore's party were suffering after ten days of Constant Toil, hunger (we had had virtually nothing to eat except raw Bear Meat for our last two days and nights on the ice), a drying up of all moisture in our bodies (we could not afford the time to stop and melt snow on the spirit stoves, so we resorted to the Bad Idea of chewing on snow and ice \u2014 a process which depletes the body's water rather than adds to it), and, a reason most Obvious to me but strangely Obscure to the officers who had been Interviewing him \u2014 poor Best had been made to stand and report to the Captains while still wearing seven of his eight Layers of Wool, allowed time only to remove his bloodied Greatcoat. After ten days and nights on the ice at an average temperature near zero degrees, the warmth of_ Erebus _was almost too much for me, and I had shed all but two layers upon reaching the Sick Bay. It had quickly proved too much for Best._\n\n _After being assured that Best would recover \u2014 a dose of Smelling Salts had already all but brought him around \u2014 Sir John looked with visible distaste at our Esquimaux patient, now lying on his bloodied chest and belly since Stanley and I had been probing his back for the ball, and our commander said, Is he going to live?_\n\nNot for long, Sir John, _reported Stephen Samuel Stanley._\n\n _I winced at speaking such in front of the patient \u2014 we doctors usually deliver our direst prognoses to each other in neutral-toned Latin in the presence of our dying clients \u2014 but realized at once that it was most unlikely that the Esquimaux could understand English._\n\nRoll him over on his back, _commanded Sir John._\n\n _We did so, carefully, and while the pain must still have been beyond excruciating for the grey-haired native, who had remained conscious during all our probing and continued to do so now, he made no sound. His gaze was fixed on our expedition Leader's face._\n\n _Sir John leaned over him and, raising his Voice and speaking slowly as if to a Deaf Child or Idiot, cried,_ Who... ARE... you?\n\n _The Esquimaux looked up at Sir John._\n\nWhat... your... name? _shouted Sir John._ What... your... tribe?\n\n _The dying man made no response._\n\n _Sir John shook his head and showed an expression of disgust, although whether because of the Gaping Wound in the Esquimaux's chest or due to his aboriginal obdurance, I know not._\n\nWhere is the other native? _Sir John asked of Stanley._\n\n _My chief surgeon, both hands busy pressing against the wound and applying the bloody bandages with which he hoped to slow, if not stem, the constant pulse of lifeblood from the savage's lung, nodded in the direction of the alcove curtain._ Dr. McDonald is with her, Sir John.\n\n _Sir John brusquely passed through the blanket-curtain. I heard several stammers, a few disjointed words, and then the Leader of our Expedition reappeared, backing out, his face such a bright, solid red that I had fears that our sixty-one-year-old commander was having a stroke._\n\n _Then Sir John's red face went quite white with shock._\n\n _I realized belatedly that the young woman must have been naked. A few minutes earlier I had glanced through the partially opened curtain and noticed that when McDonald gestured for her to take off her outer clothing \u2014 her bearskin parka \u2014 the girl nodded, removed the heavy outer garment, and was wearing nothing under it from the waist up. I'd been busy with the dying man on the table at the time, but I noted that this was a sensible way to stay warm under the loose layer of heavy fur \u2014 much better than the multiple layers of Wool which all of us in poor Lieutenant Gore's sledge party had worn. Naked under fur or animal hair, the body can warm itself when chilled, adequately cool itself when needed, as during exertion, since perspiration would quickly wick away from the body into the hairs of the wolfskin or bearskin hide. The wool we Englishmen had worn had soaked through with Sweat almost immediately, never really dried, quickly froze when we quit marching or pulling the sledge and lost much of its Insulating Quality. By the time we had Returned to the ship, I had no doubt that we were carrying almost twice the Weight on our backs than that with which we had departed._\n\nI sh-shall return at a more suitable time, _stammered Sir John, and backed past us._\n\n _Captain Sir John Franklin looked shaken, but whether it was because of the young woman's sensible Edenic Nakedness or something else he saw in the Sick Bay alcove, I could not say. He left the Surgery without another word._\n\n _A moment later McDonald called me into the rear alcove. The girl \u2014 young woman, I had noticed, although it has been scientifically shown that females from savage tribes reach puberty long before young ladies in civilized societies \u2014 had put her bulky parka and sealskin pants back on. Dr. McDonald himself looked agitated, almost upset, and when I queried him as to the problem, he gestured for the Esquimaux wench to open her mouth. Then he raised a lantern and a convex mirror to focus the light and I saw for myself._\n\n _Her tongue had been amputated near the roots. Enough was left, I saw \u2014 and McDonald concurred \u2014 to allow her to swallow and to eat most foods, after a fashion, but certainly the articulation of complex sounds, if one might call any Esquimaux language complex in any form, would be beyond her ability. The scars were old. This had not happened recently._\n\n _I confess that I pulled away in Horror. Who would do this to a mere child \u2014 and why? But when I used the word \"amputation,\" Dr. McDonald softly corrected me._\n\nLook again, Dr. Goodsir, _he all but whispered._ It is not a neat surgical circular amputation, not even by so crude an instrument as a stone knife. The poor lass's tongue was chewed off when she was very small \u2014 and so close to the root of the member that there is no possibility she did this to herself.\n\n _I took a step away from the woman._ Is she mutilated elsewhere? _I asked, speaking in Latin out of old habit. I had read of barbaric customs in the Dark Continent and among the Mohammedan in which their women were cruelly circumcised in a parody of the Hebrew custom for males._\n\nNowhere else, _responded McDonald._\n\n _Then I thought I understood the source of Sir John's sudden paleness and obvious shock, but when I asked McDonald whether he had shared this information with our commander, the surgeon assured me that he had not. Sir John had entered the alcove, seen the Esquimaux girl without her clothes, and left in some agitation. McDonald then began to give me the results of his quick physical inspection of our captive, or guest, when we were interrupted by Surgeon Stanley._\n\n _My first thought was that the Esquimaux man had died, but that did not turn out to be the Case. A crewman had come calling me to give my report before Sir John and the other Captains._\n\n _I could tell that Sir John, Commander Fitzjames, and Captain Crozier were disappointed in my Report of what I had observed of Lieutenant Gore's death, and while this ordinarily would have Distressed me, this day \u2014 perhaps due to my great Fatigue and to the Psychological Changes which may have taken place during my time with Lieutenant Gore's Ice Party \u2014 the disappointment of my Superiors did not Affect me._\n\n _I first reported again on the condition of our dying Esquimaux man and on the curious fact about the girl's missing tongue. The three captains murmured among themselves about this fact, but the only questions came from Captain Crozier._\n\nDo you know why someone may have done this to her, Dr. Goodsir?\n\nI have no idea, sir.\n\nCould it have been done by an animal? _he persisted._\n\n _I paused. The idea had not occurred to me._ It could have been, _I said at last, although it was very hard to Picture some Arctic Carnivore chewing off a child's tongue yet leaving her alive. Then again, it was well known that these Esquimaux tended to live with Savage Dogs. I had seen this myself at Disko Bay._\n\n _There were no more questions about the two Esquimaux._\n\n _They asked for the details of Lieutenant Gore's death and about the Creature who killed him, and I told the truth \u2014 that I had been working to save the life of the Esquimaux man who had come out of the fog and been shot by Private Pilkington and that I had looked up only in the final instant of Graham Gore's death. I explained that between the shifting fog, the screams, the distracting blast of the musket, and the report of the lieutenant's pistol going off, my limited vision from the side of the sledge where I knelt, the rapidly shifting movement of both men and light, I was not sure what I had seen: only that large white shape enveloping the hapless officer, the flash of his pistol, more shots, then the fog enfolding everything again._\n\nBut you are certain it was a white bear? _asked Commander Fitzjames._\n\n _I hesitated._ If it was, _I said at last,_ it was an uncommonly large specimen of _Ursus maritimus._ I had the impression of a bearlike carnivore \u2014 a huge body, giant arms, small head, obsidian eyes \u2014 but the details were not as clear as that description makes them sound. Mostly what I remember is that the thing seemed to come out of nowhere \u2014 just rise up around the man \u2014 and that it towered twice as tall as Lieutenant Gore. That was very unnerving.\n\nI am sure it was, _Sir John said drily, almost sarcastically, I thought._ But what else could it have been, Mr. Goodsir, were it not a bear?\n\n _It was not the first time that I had noticed that Sir John never complimented me with my proper Rank as Doctor. He used the \"Mr.\" as he might with any mate or untutored warrant officer. It had taken me two years to realize that the aging expedition commander whom I held in such high esteem had no degree of reciprocal esteem for any mere ship's surgeon._\n\nI don't know, Sir John, _I said. I wanted to get back to my patient._\n\nI understand you've shown an interest in the white bears, Mr. Goodsir, _continued Sir John._ Why is that?\n\nI trained as an anatomist, Sir John. And before the expedition sailed, I had dreams of becoming a naturalist.\n\nNo longer? _asked Captain Crozier in that soft brogue of his._\n\n _I shrugged._ I find that fieldwork is not my forte, Captain.\n\nYet you've dissected some of the white bears we've shot here and at Beechey Island, _persisted Sir John._ Studied their skeletons and musculature. Observed them on the ice as we all have.\n\nYes, Sir John.\n\nDo you find Lieutenant Gore's wounds consistent with the damage such an animal would produce?\n\n _I hesitated only a second. I had examined poor Graham Gore's corpse before we had loaded it onto the sledge for the nightmare journey back across the pack ice._\n\nYes, Sir John, _I said._ The white polar bear of this region is \u2014 as far as we know \u2014 the largest single predator on Earth. It can weigh half again as much and stand three feet taller on its hind legs than the Grizzly Bear, the largest and most ferocious bear in North America. It is a very powerful predator, fully capable of crushing a man's chest and severing his spine, as was the case with poor Lieutenant Gore. More than that, the white arctic bear is the only predator that commonly stalks human beings as its prey.\n\n _Commander Fitzjames cleared his throat._ I say, Dr. Goodsir, _he said softly,_ I did see a rather ferocious tiger in India once which \u2014 according to the villagers \u2014 had eaten twelve people.\n\n _I nodded, realizing at that second how terribly weary I was. The exhaustion worked on me like Powerful Drink._ Sir... Commander... Gentlemen... , you have all seen more of the world than have I. However, from my rather extensive reading on the subject, it would seem that all other land carnivores \u2014 wolves, lions, tigers, other bears \u2014 may kill human beings if provoked, and some of them, such as your tiger, Commander Fitzjames, will become man-eaters if forced to due to disease or injury which precludes them from seeking out their natural prey, but only the white arctic bear \u2014 _Ursus maritimus_ \u2014 actively stalks human beings as prey on a common basis.\n\n _Crozier was nodding._ Where have you learned that, Doctor Goodsir? Your books?\n\nTo some extent, sir. But I spent most of our time at Disko Bay speaking to the locals there about the behaviour of the bears and also inquired of Captain Martin on his _Enterprise_ and Captain Dannert on his _Prince of Wales_ when we were anchored near them in Baffin Bay. Those two gentlemen answered my questions about the white bears and put me in touch with several of their crewmen \u2014 including two elderly American whalers who had spent more than a dozen years apiece in the ice. They had many anecdotes about the white bears stalking the Esquimaux natives of the region and even taking men from their own ships when they were trapped in the ice. One old man \u2014 I believe his name was Connors \u2014 said that their ship in '28 __had lost not one but two cooks to bears... one of them snatched from the lower deck where he was working near the stove while the men slept.\n\n _Captain Crozier smiled at that._ Perhaps we should not believe every tale an old sailor has to tell, Doctor Goodsir.\n\nNo, sir. Of course not, sir.\n\nThat will be all, Mr. Goodsir, _said Sir John._ We shall call you back if we have more questions.\n\nYes, sir, _I said and turned tiredly to return forward to the sick bay._\n\nOh, Dr. Goodsir, _called Commander Fitzjames before I stepped out the door of Sir John's cabin._ I have a question, although I am deucedly ashamed to admit that I do not know the answer. Why is the white bear called _Ursus maritimus?_ Not out of its fondness for eating sailors, I trust.\n\nNo, sir, _I said._ I believe the name was bestowed on the arctic bear because it is more a sea mammal than land animal. I've read reports of the white arctic bear being sighted hundreds of miles at sea, and Captain Martin of the _Enterprise_ told me himself that while the bear is fast on the attack on land or ice \u2014 coming at one at speeds of more than twenty-five miles per hour \u2014 that at sea it is one of the most powerful swimmers in the ocean, capable of swimming sixty or seventy miles without rest. Captain Dannert said that once his ship was doing eight knots with a fair wind, far out of sight of land, and that two white bears kept pace with the ship for ten nautical miles or so and then simply left it behind, swimming toward distant ice floes with the speed and ease of a beluga whale. Thus the nomenclature... _Ursus maritimus..._ a mammal, yes, but mostly a creature of the sea.\n\nThank you, Mr. Goodsir, _said Sir John._\n\nYou are most welcome, sir, _I said and left._\n\n4 June, 1847, continued...\n\n _The Esquimaux man died just a few minutes after midnight. But he spoke first._\n\n _I was asleep at the time, sitting up with my back against the Sick Bay bulkhead, but Stanley woke me._\n\n _The grey-haired man was struggling as he lay on the Surgical Bench, his arms moving almost as if he were trying to swim up into the air. His punctured lung was hemorrhaging and blood was pouring down his chin and onto his bandaged chest._\n\n _As I raised the light of the lantern, the Esquimaux girl rose up from the corner where she had been sleeping and all three of us leaned in toward the dying man._\n\n _The old Esquimaux hooked a powerful finger and poked at his chest, very near the bullet hole. Each gasp of his pumped out more bright red arterial blood, but he coughed out what could only be words. I used a piece of chalk to scribble them on the slate Stanley and I used to communicate when patients were sleeping nearby._\n\n\"Angatkut tuquruq! Quarubvitchuq... angatkut turquq... . Paniga... tuunbaq! Tanik... naluabmiu tuqutauyasiruq... umiaqpak tuqutauyasiruq... nanuq tuqutkaa! Paniga... tunbaq nanuq... angatkut ququruq!\"\n\n _And then the hemorrhaging grew so extreme he could talk no more. The blood geysered and fountained out of him, choking him until \u2014 even with Stanley and me propping him up, trying to help clear his breathing passages \u2014 he was inhaling only blood. After a terrible final moment of this his chest quit heaving, he fell back into our arms, and his stare became fixed and glassy. Stanley and I lowered him to the table._\n\nLook out! _cried Stanley._\n\n _For a second I did not understand the other surgeon's warning \u2014 the old man was dead and still, I could find no pulse or breath as I hovered over him \u2014 but then I turned and saw the Esquimaux woman._\n\n _She had seized one of the bloody scalpels from our worktable and was stepping closer, lifting the weapon. It was obvious to me at once that she was paying no attention to me \u2014 her fixed gaze was on the Dead Face and chest of the man who might have been her husband or father or brother. In those few seconds, not knowing anything of the customs of her Heathen tribe, a Myriad of wild images came to my mind \u2014 the girl cutting out the man's heart, perhaps devouring it in some terrible ritual, or removing the dead man's eyes or slicing off one of his fingers or perhaps adding to the webwork of old scars that covered his body like a sailor's tattoos._\n\n _She did none of that. Before Stanley could seize her and while I could think of nothing but to cower protectively over the dead man, the Esquimaux girl flicked the scalpel forward with a surgeon's dexterity \u2014 she obviously had used razor-sharp knives for most of her life \u2014 and she severed the rawhide cord that held the old man's amulet in place._\n\n _Catching up the flat, white, blood-spattered bear-shaped stone and its severed cord, she secreted it somewhere on her person under her parka and returned the scalpel to its table._\n\n _Stanley and I stared at each other. Then_ Erebus _'s_ _chief surgeon went to wake the young sailor who served as the Sick Bay mate, sending him to inform the officer on watch and thence the Captain that the old Esquimaux was dead._\n\n _4 June, continued..._\n\n _We buried the Esquimaux man sometime around one-thirty in the morning \u2014 three bells \u2014 shoving his canvas-wrapped body down the narrow fire hole in the ice only twenty yards from the ship. This single fire hole giving access to open water fifteen feet below the ice was the only one the men have managed to keep open this cold summer \u2014 as I have mentioned before, sailors are afraid of nothing so much as fire \u2014 and Sir John's instructions were to dispose of the body there. Even as Stanley and I struggled to press the body down the narrow funnel, using boat pikes, we could hear the chopping and occasional swearing from several hundred yards east on the ice where a party of twenty men was working through the night to hack out a more decorous hole for Lieutenant Gore's burial service the next day \u2014 or later the same day, actually._\n\n _Here, in the middle of the night, it was still light enough to read a Bible verse by \u2014 if anyone had brought a Bible out here on the ice to read a verse from, which no one had \u2014 and the dim light aided us, the two surgeons and two crewmen ordered to help us, as we poked, prodded, shoved, slid, and finally slammed the Esquimaux man's body deeper and deeper into the blue ice and thence into the Black Water beneath._\n\n _The Esquimaux woman stood silently, watching, still showing no expression. There was a wind from the west-northwest and her black hair lifted from her stained parka hood and moved across her face like a ruffle of raven feathers._\n\n _We were the only members of the Burial Party \u2014 Surgeon Stanley, the two panting, softly cursing crewmen, the native woman, and me \u2014 until Captain Crozier and a tall, lanky lieutenant appeared in the blowing snow and watched the final moment or two of struggle. Finally the Esquimaux man's body slid the last five feet and disappeared into the black currents fifteen feet below the ice._\n\nSir John ordered that the woman not spend the night aboard _Erebus, Captain Crozier said softly._ We've come to take her back to _Terror. To the tall lieutenant whose name I now remembered as Irving, Crozier said,_ John, she will be in your charge. Find a place for her out of sight of the men \u2014 probably forward of the sick bay in the stacks \u2014 and make sure no harm comes to her.\n\nAye, sir.\n\nExcuse me, Captain, _I said._ But why not let her go back to her people?\n\n _Crozier smiled at this._ Normally I would agree with that course of action, Doctor. But there are no known Esquimaux settlements \u2014 not the smallest village \u2014 within three hundred miles of here. They are a nomadic people \u2014 especially those we call the Northern Highlanders \u2014 but what brought this old man and young girl out onto the pack ice so far north in a summer where there are no whales, no walruses, no seals, no caribou, no animals of any sort abroad except our white bears and the murderous things on the ice?\n\n _I had no answer to this, but it hardly seemed pertinent to my question._\n\nIt may come to the point, _continued Crozier,_ where our lives might depend upon finding and befriending these native Esquimaux. Shall we let her go then before we've befriended her?\n\nWe shot her husband or father, _said Surgeon Stanley, glancing at the mute young woman who still stared at the now empty fire hole._ Our Lady Silence here might not have the most charitable of feelings toward us.\n\nPrecisely, _said Captain Crozier._ And we have enough problems right now without this lass leading a war party of angry Esquimaux back to our ships to murder us as we sleep. No, I think Captain Sir John is right... she should stay with us until we decide what to do... not only with her, but with ourselves. _Crozier smiled at Stanley. In two years, this was the first time that I could remember seeing Captain Crozier smile._ Lady Silence. That is good, Stanley. Very good. Come, John. Come, m'lady.\n\n _They walked west through the blowing snow toward the first pressure ridge. I went back up the ramp of snow to_ Erebus, _to my tiny little cabin which seemed like pure heaven to me now, and to the first solid night's sleep I had had since Lieutenant Gore led us south-southeast onto the ice more than ten days earlier._\n\nFRANKLIN\n\n _Lat. 70\u00b0- 05\u2032 N., Long. 98\u00b0- 23\u2032 W._\n\n11 June, 1847\n\nBy the day that he was to die, Sir John had almost recovered from the shock of seeing the Esquimaux wench naked.\n\nIt was the same young woman, the same teenaged harlot Copper squaw whom the Devil had sent to tempt him during his first ill-fated expedition in 1819, the wanton Robert Hood's fifteen-year-old bedmate named Greenstockings. Sir John was sure of that. This temptress had the same coffee-brown skin that seemed to glow even in the dark, the same high, round girl's breasts, the same brown areolae, and the same raven-feather slash of dark escutcheon above her sex.\n\nIt was the same succubus.\n\nThe shock to Captain Sir John Franklin of seeing her naked on surgeon McDonald's table in the sick bay \u2014 _on his ship_ \u2014 had been profound, but Sir John was sure that he had been able to hide his reaction from the surgeons and from the other captains during the rest of that endless, disconcerting day.\n\nLieutenant Gore's burial service took place late on Friday, the fourth of June. It had taken a large work party more than twenty-four hours to get through the ice to allow for the burial at sea, and before they were done they had to use black powder to blow away the top ten feet of rock-hard ice, then use picks and shovels to excavate a broad crater to open the last five feet or so. When they were finished around midday, Mr. Weekes, the carpenter from _Erebus,_ and Mr. Honey, the carpenter from _Terror,_ had constructed a clever and elegant wooden scaffolding over the ten-foot-long and five-foot-wide opening into the dark sea. Work parties with long pikes were stationed at the crater to keep the ice from congealing beneath the platform.\n\nLieutenant Gore's body had begun to decay quickly in the relative heat of the ship, so the carpenters first constructed a most solid coffin of mahogany lined with an inner box of sweet-smelling cedar. Between the two enclosures of wood was set a layer of lead in lieu of the traditional two rounds of shot set in the usual canvas burial bag to ensure that the body would sink. Mr. Smith, the blacksmith, had forged, hammered, and engraved a beautiful memorial plate in copper, which was affixed to the top of the mahogany coffin by screws. Because the burial service was a mixture of shoreside burial and the more common burial at sea, Sir John had specified that the coffin be made heavy enough to sink at once.\n\nAt eight bells at the beginning of the first dogwatch \u2014 4:00 p.m. \u2014 the two ships' companies assembled at the burial site a quarter of a mile across the ice from _Erebus._ Sir John had ordered everyone except the smallest possible ship's watches to be present for the service and furthermore had ordered them to wear no layer over their dress uniforms, so at the appointed time more than one hundred shivering but formally dressed officers and men had gathered on the ice.\n\nLieutenant Gore's coffin was lowered over the side of _Erebus_ and lashed to an oversized sledge reinforced for this day's sad purpose. Sir John's own Union Jack was draped over the coffin. Then thirty-two seamen, twenty from _Erebus_ and a dozen from _Terror,_ slowly pulled the coffin-sledge the quarter mile to the burial site, while four of the youngest seamen, still on the roster as ship's boys \u2014 George Chambers and David Young from _Erebus,_ Robert Golding and Thomas Evans from _Terror_ \u2014 beat a slow march on drums muffled in black cloth. The solemn procession was escorted by twenty men, including Captain Sir John Franklin, Commander Fitzjames, Captain Crozier, and the majority of all the other officers and mates in full dress, excluding only those left in command in each near-vacant ship.\n\nAt the burial site, a firing party of red-coated Royal Marines stood waiting at attention. Led by _Erebus_ 's thirty-three-year-old sergeant, David Bryant, the party consisted of Corporal Pearson, Private Hopcraft, Private Pilkington, Private Healey, and Private Reed from _Erebus_ \u2014 only Private Braine was missing from the flagship's contingent of Marines, since the man had died last winter and been buried on Beechey Island \u2014 as well as Sergeant Tozer, Corporal Hedges, Private Wilkes, Private Hammond, Private Heather, and Private Daly from HMS _Terror._\n\nLieutenant Gore's cocked hat and sword were carried behind the burial sledge by Lieutenant H. T. D. Le Vesconte, who had assumed Gore's command duties. Alongside Le Vesconte walked Lieutenant James W. Fairholme, carrying a blue velvet cushion on which were displayed the six medals young Gore had earned during his years in the Royal Navy.\n\nAs the sledge party approached the burial crater, the line of twelve Royal Marines parted, opening to form a lane. The Marines turned inward and stood at reverse arms as the procession of sledge-pullers, funeral sledge, honor guard, and other mourners passed between their ranks.\n\nAs the hundred and ten men shuffled to their places amid the mass of officers' uniforms around the crater \u2014 some seamen standing on pressure ridges to get a better look \u2014 Sir John led the captains to their place on a temporary scaffolding at the east end of the crater in the ice. Slowly, carefully, the thirty-two sledge-pullers worked together to unlash the heavy coffin and lower it down precisely angled boards to its temporary resting place on the wooden superstructure just above the rectangle of black water. When the coffin was in place, it rested not only on the final planks but on three sturdy hawsers now manned on either side by the same men who had been chosen to pull the sledge.\n\nWhen the muffled drums quit beating, all hats came off. The cold wind ruffled the men's long hair, all washed, parted, and tied back with ribbons for this service. The day was chilly \u2014 no more than five degrees at the last measuring at six bells \u2014 but the arctic sky, filled with ice crystals, was a solid dome of golden light. As if in honour of Lieutenant Gore, the single circle of the ice-occluded sun had been joined by three more suns \u2014 sun dogs floating above and to either side of the south-hanging true sun \u2014 all connected by a halo-band of rainbow-prismed light. Many men present bowed their heads at the aptness of the sight.\n\nSir John conducted the Service for the Dead, his strong voice easily audible to the hundred and ten men gathered round. The ritual was familiar to all there. The words were reassuring. The responses were known. By the end, the cold wind was ignored by most as the familiar phrases echoed across the ice.\n\n\"We therefore commit his body to the deep, to be turned into corruption, looking for the resurrection of the body, when the Sea shall give up her dead, and the life of the world to come, through our Lord Jesus Christ, who at his coming shall change our vile body, that it may be like his glorious body, according to the mighty working, whereby he is able to subdue all things to himself.\"\n\n\"Amen,\" said the assembled men.\n\nThe twelve men of the Royal Marine firing party raised their muskets and fired three volleys, the last one having only three shots rather than the four in the two volleys that had preceded it.\n\nAt the sound of the first volley, Lieutenant Le Vesconte nodded and Samuel Brown, John Weekes, and James Rigden slid the planks out from under the heavy casket, which now hung suspended only by the three hawsers. At the sound of the second volley, the coffin was lowered until it touched the black water. At the sound of the final volley, the hawsers were slowly let slip until the heavy casket with its copper plaque \u2014 Lieutenant Gore's medals and sword also now perched atop the mahogany \u2014 disappeared beneath the water's surface.\n\nThere was a slight roiling of icy water, the hawsers were pulled up and tossed aside, and the rectangle of black water was empty. To the south, the sun dogs and halo had disappeared and only a sullen red sun glowed under the dome of sky.\n\nThe men dispersed silently to their ships. It was only two bells into the first dogwatch. For most of the men it was time for their evening meal and their second portion of grog.\n\nThe next day, Saturday the fifth of June, saw both crews huddling in the lower decks of their ships as another arctic summer lightning storm exploded above them. Lookouts were called down from the topmains and those few who kept watch on deck kept away from all metal and masts as lightning crashed through fog, thunder rolled, great bolts of electricity struck and then restruck the lightning rods set on the masts and cabin roofs, and blue fingers of Saint Elmo's fire crept along the spars and slithered through the rigging. Haggard lookouts coming below after their watch told their wide-eyed mates of spheres of ball-lightning rolling and leaping across the ice. Later in the day \u2014 with the lightning and airborne electrical displays growing even more violent \u2014 the dogwatch lookouts reported something large, much too large to be a mere white bear, prowling and pacing along the ridges in the fog, now concealed, now made visible by lightning flash for only a second or two. Sometimes, they said, the shape walked on four legs like a bear. Other times, they swore, it walked easily on two legs, like a man. The thing, they said, was circling the ship.\n\nAlthough the mercury was falling, Sunday dawned clear and thirty degrees colder \u2014 the temperature at noon was nine below zero \u2014 and Sir John sent out word that Divine Service would be compulsory that day on _Erebus._\n\nDivine Service was compulsory each week for the men and officers of Sir John's ship \u2014 he held it on the lower deck all during the dark winter months \u2014 but only the most devout Terrors made the ice crossing to join in the service. Since it was mandatory in the Royal Navy, by tradition as much as by regulation, Captain Crozier also held Divine Service on Sunday, but with no chaplain aboard it was an abbreviated effort \u2014 sometimes amounting to little more than reading the Ship's Articles \u2014 and ran twenty minutes of a morning rather than Sir John's enthusiastic ninety minutes or two hours.\n\nThis Sunday there was no option.\n\nCaptain Crozier led his officers, mates, and men across the ice for the second time in three days, this time with their greatcoats and mufflers over any dress uniforms, and they were surprised upon their arrival at _Erebus_ to see that the service was to take place on deck, with Sir John preaching from the quarterdeck. Despite the pale blue sky above \u2014 no gold dome of ice crystals or symbolic sun dogs this day \u2014 the wind was _very_ cold, and the mass of seamen huddled together for at least the illusion of warmth in the area below the quarterdeck, while the officers from both ships stood behind Sir John on the weather side of the deck like a solid mass of greatcoated acolytes. Once again, the twelve Marines were drawn up in rank, this time on the lee side of the main deck with Sergeant Bryant in front, while the petty officers massed before the mainmast.\n\nSir John stood at the binnacle, which had been covered with the same Union Jack that had been draped over Gore's casket \"to answer the purpose of a pulpit,\" as per regulation.\n\nHe preached for only about an hour and no toes or fingers were lost as a result.\n\nBeing an Old Testament man by nature and inclination, Sir John led the way through several of the prophets, focusing awhile on Isaiah's judgement upon the earth \u2014 \"Behold, the Lord maketh the earth empty and maketh it waste, and turneth it upside down, and scattereth abroad the inhabitants thereof\" \u2014 and slowly through the barrage of words, it became apparent to even the most dimwitted seaman in the mass of greatcoats, mufflers, and mittens on the main deck that their commander was really talking about their expedition to find the North-West Passage and their current condition frozen in the icy wastes at latitude 70\u00b0- 05' N., longitude 98\u00b0- 23' W.\n\n\"The land shall be utterly emptied, and utterly spoiled: for the LORD hath spoken this word,\" continued Sir John. \"Fear, and the pit, and the snare, _are_ upon thee, O inhabitant of the earth... And it shall come to pass _that_ he who fleeth from the noise of the fear shall fall into the pit; and he that cometh up out of the midst of the pit shall be taken in the snare: for the windows from on high are open, and the foundations of the earth do shake... The earth is utterly broken down, the earth is clean dissolved, the earth is moved exceedingly. They shall reel to and fro like a drunkard...\"\n\nAs if in proof of this dire prophecy, a great groaning came up from the ice all around HMS _Erebus_ and the deck shifted under the standing men. The ice-rimmed masts and spars above them seemed to vibrate and then make small circles against the weak blue sky. No man broke formation or made a noise.\n\nSir John shifted from Isaiah to Revelation and gave them even more dire images of what awaited those who abandoned their Lord.\n\n\"But of what of he... of we... who do not break covenant with our Lord?\" asked Sir John. \"I commend you to JONAH.\"\n\nSome of the seamen sighed in relief. They were familiar with Jonah.\n\n\"Jonah was given a commission by God to go to Nineveh and to cry against it because of its wickedness,\" cried Sir John, his often weak voice now rising in volume as strongly and well as any inspired Anglican preacher's, \"but Jonah \u2014 you all know this, shipmates \u2014 Jonah fled from his commission and from the presence of the Lord, going down to Joppa there to take a berth on the first ship leaving, which happened to be destined for Tarshish \u2014 a city beyond the edge of the world then. Jonah foolishly thought that he could sail beyond the limits of the Kingdom of the Lord.\n\n\" 'But the LORD sent out a great wind into the sea, and there was a mighty tempest in the sea, so that the ship was like to be broken.' And you know the rest... you know how the sailors cried out asking why this evil had fallen upon them, and they cast lots, and the lot fell upon Jonah. 'And they said unto him, What shall we do unto thee, that the sea may be calm unto us? And he said unto them, Take me up and cast me forth into the sea; so shall the sea be calm unto you: for I know that for my sake this great tempest _is_ upon you.'\n\n\"But at first the sailors did not cast Jonah overboard, did they, shipmates? No \u2014 they were brave men and good sailors and professionals and rowed hard to bring their foundering ship to land. But finally they weakened, cried unto the Lord, and then did make a sacrifice of Jonah, casting him overboard.\n\n\"And the Bible says \u2014 'Now the LORD had prepared a great fish to swallow up Jonah. And Jonah was in the belly of the fish three days and three nights.'\n\n\"Notice, shipmates, that the Bible does not say that Jonah was swallowed by a _whale._ No! This was no beluga nor right nor baleen nor sperm nor killer nor fin such as we would see in these high waters or in Baffin Bay on a normal arctic summer. No, Jonah was swallowed up by a 'great fish' which the Lord had prepared for him \u2014 which means a monster of the deep that Lord God Jehovah had made at the Creation for just this purpose, to swallow Jonah someday, and in the Bible this monster of a great fish is sometimes called Leviathan.\n\n\"And just so have we been sent on our mission beyond the farthest known edge of the world, shipmates, farther than the Tarshish \u2014 which was only in Spain, after all \u2014 we have been sent out to where the elements themselves seem to rebel, where lightning crashes from frozen skies, where the cold never relents, where white beasts walk the frozen surface of the sea, and where no man, civilized or otherwise, could ever call such a place home.\n\n\"But we are not beyond the Kingdom of God, shipmates! As Jonah did not protest his fate nor curse his punishment but rather prayed unto the Lord out of the fish's belly for three days and three nights, so we must not protest, but accept God's will of this exile of three long nights of winter in the belly of this ice, and like Jonah we must pray unto the Lord, saying, 'I am cast out of thy sight; yet I will look again toward thy holy temple. The waters compassed me about, _even_ to the soul: the depth closed me round about, the weeds were wrapped about my head. I went down to the bottoms of the mountains: the earth with her bars _was_ about me for ever: yet hast thou brought up my life from corruption, O Lord my God.'\n\n\" 'When my soul fainted within me I remembered the LORD: and my prayer came in unto thee, into thine holy temple. They that observe lying vanities foresake their own mercy. But I will sacrifice unto thee with the voice of thanksgiving; I will pay _that_ that I have vowed. Salvation _is_ of the Lord.\n\n\" 'And the LORD spake unto the fish, and it vomited out Jonah upon the dry _land._ '\n\n\"And, beloved shipmates, know in your hearts that we have given and must continue giving sacrifice unto the Lord with the voice of thanksgiving. We must pay that that we have vowed to pay. Our friend and brother in Christ, Lieutenant Graham Gore, may he sleep in the bosom of the Lord, saw that there would be no release from this belly of the Leviathan winter this summer. No escape from the cold belly of this ice this year. And this is the message he would have brought back had he survived.\n\n\"But we have our ships intact, shipmates. We have food for this winter and longer if need be... much longer. We have coal to burn for warmth and the deeper warmth of our companionship and the deepest warmth of knowing that our Lord _has not abandoned us._\n\n\"One more summer and then winter here in the belly of this Leviathan, shipmates, and I swear to you that God's divine mercy shall see us out of this terrible place. The North-West Passage is real; it is only miles over that horizon to the southwest \u2014 Lieutenant Gore could almost see it with his own eyes a mere week ago \u2014 and we shall sail out to it and through it and out of it and away from it in a very few months, when this uncommon extended winter ends, for we shall cry by reason of our affliction unto the Lord, and he shall hear us out of the belly of Hell itself, for he has heardest my voice and yours.\n\n\"In the meantime, shipmates, we are afflicted by the dark spirit of that Leviathan in the form of some malevolent white bear \u2014 but only a bear, only a dumb beast, however the thing seeks to serve the Enemy, but like Jonah we shall pray unto the Lord that this terror too shall pass from us. And in the certainty that the Lord shall hear our voices.\n\n\"Kill this mere animal, shipmates, and on the day we do, by the hand of whichever man among us, I vow to pay each and every one of you ten gold sovereigns out of my own purse.\"\n\nThere came a murmuring among the men crowded into the waist of the ship.\n\n\"Ten gold sovereigns a man,\" repeated Sir John. \"Not merely a bounty to the man who slays this beast the way David slew Goliath, but a bonus for everyone \u2014 share and share alike. And on top of that, you will continue to receive your Discovery Service pay and the equal of your advance pay in bonuses I promise this day \u2014 in exchange only for another winter spent eating good food, staying warm, and waiting for the thaw!\"\n\nIf laughter had been thinkable during Divine Service, there would have been laughter then. Instead, the men stared at one another with pale, near-frostbitten faces. _Ten gold sovereigns a man._ And Sir John had promised a bonus equal to the advance pay that had persuaded so many of these seamen to enlist in the first place \u2014 twenty-three pounds for most of them! At a time when a man could purchase lodgings for sixty pence a week... twelve pounds for a whole year. And this on top of the common seaman's Discovery Service pay of sixty pounds per year \u2014 more than three times what any labourer ashore could earn! Seventy-five pounds for the carpenters, seventy for the boatswains, a full eighty-four pounds for the engineers.\n\nThe men were smiling even as they surreptitiously stamped their boots on the deck to keep from losing toes.\n\n\"I have ordered Mr. Diggle on _Terror_ and Mr. Wall here on _Erebus_ to make us a holiday dinner today in anticipation of our triumph over this temporary adversity and the sure certainty of the success of our mission of exploration,\" called down Sir John from his place at the flag-bedecked binnacle. \"On both ships, I have allowed extra rations of rum for this day.\"\n\nThe Erebuses could only stare slack-jawed at one another. _Sir John Franklin_ allowing grog to be served on Sunday \u2014 and extra rations of it at that?\n\n\"Join me in this prayer, shipmates,\" said Sir John. \"Dear God, turn thy face in our direction again, O Lord, and be gracious unto thy servants. O satisfy us with thy mercy, and that soon: so shall we rejoice and be glad all the days of our lives.\n\n\"Comfort us again now after the time that thou has plagued us: and for the years wherein we have suffered adversity.\n\n\"Shew thy servants thy work: and their children thy glory.\n\n\"And the glorious Majesty of the Lord our God be upon us: prosper thou the work of our hands upon us, O prosper thou our handy-work.\n\n\"Glory be to the Father, and to the Son, and to the Holy Ghost.\n\n\"As it was in the beginning, is now, and ever shall be: world without end. Amen.\"\n\n\"Amen,\" came back a hundred and fifteen voices.\n\nFor four days and nights after Sir John's sermon, despite a June snowstorm that blew in from the northwest and made visibility poor and life miserable, the frozen sea echoed day and night to the blast of shotguns and the rattle of musketry. Every man who could find any reason to be out on the ice \u2014 a hunting party, the fire-hole party, messengers passing between the ships, carpenters testing their new sledges, seamen given permission to walk Neptune the dog \u2014 brought a weapon and fired at anything that moved or gave the impression through blowing snow or fog that it might be capable of movement. No men were killed, but three had to report to Dr. McDonald or Dr. Goodsir to have shotgun pellets removed from their thighs, calves, and buttocks.\n\nOn Wednesday a hunting party who had failed to find seals did bring in \u2014 strapped across two connected sledges \u2014 the carcass of a white bear and a living white bear cub about the size of a small calf.\n\nThere was some hue and cry for the ten gold sovereigns to be paid to each man, but even the men who had killed the beast a mile north of the ship \u2014 it had taken more than twelve shots from two muskets and three shotguns to bring the bear down \u2014 had to admit that it was too small, less than eight feet long when stretched out on the bloody ice, and too thin and female. They had killed the bear sow but left the mewling cub alive and dragged it back behind the sledge with them.\n\nSir John came down to inspect the dead animal, praised the men for finding meat \u2014 although everyone hated boiled bear meat and this thin animal looked more stringy and tough than most \u2014 but pointed out that it was not the monster of the Leviathan that had killed Lieutenant Gore. All the witnesses to the lieutenant's death were sure, Sir John explained, that even as he died, the brave officer had fired his pistol into the breast of the true beast. This bear sow had been riddled with shot, but there was no old pistol wound in her breast, nor pistol ball to be found. Thus, said Sir John, would the real white bear monster be identified.\n\nSome of the men wanted to make a pet of the cub since the thing had been weaned and would eat thawed beef, while others wanted to butcher it then and there on the ice. On the advice of Marine Sergeant Bryant, Sir John ordered that the animal be kept alive, attached by collar and chain to a stake in the ice. It was that Wednesday evening, the ninth of June, that Sergeants Bryant and Tozer, along with the mate Edward Couch and old John Murray, the only sailmaker left on the voyage, asked to speak to Sir John in his cabin.\n\n\"We are going at this the wrong way, Sir John,\" said Sergeant Bryant, spokesman for the little group. \"The hunting of the beast, I mean.\"\n\n\"How so?\" asked Sir John.\n\nBryant gestured as if referring to the dead bear sow now being butchered out on the bloody ice. \"Our men aren't hunters, Sir John. There's not a serious hunter aboard either ship. Those of us who do hunt shoot birds in our life ashore, not large game. Oh, a deer we could bring down, or an arctic caribou should we ever see one again, but this white bear is a formidable foe, Sir John. Those we've killed in the past we've killed more by luck than by skill. Its skull is thick enough to stop a musket ball. Its body has so much fat and muscle ringed about it that it might as well be armoured like some ancient knight. It's such a powerful animal, even the smaller bears \u2014 well, you have seen them, Sir John \u2014 even a shotgun blast to the belly or a rifle shot to the lungs does not bring them down. Their hearts seem hard to find. This scrawny female required a dozen shots by both shotgun and musket, all at short range, and even then she would have escaped had she not stayed behind to protect her cub.\"\n\n\"What are you suggesting, Sergeant?\"\n\n\"A blind, Sir John.\"\n\n\"A blind?\"\n\n\"As if we were hunting ducks, Sir John,\" said Sergeant Tozer, a Marine with a purple birthmark across his pale face. \"Mr. Murray has an idea how to make it.\"\n\nSir John turned toward _Erebus_ 's old sailmaker.\n\n\"We use extra iron rods meant for shaft replacements, Sir John, and bend them into the support shapes we want,\" said Murray. \"That gives us a light frame for the blind, which'll be like a tent, you see.\n\n\"Only not a pyramid like our tents,\" continued John Murray, \"but long and low with an overhanging awning, almost like a canvas booth at a country fair, m'lord.\"\n\nSir John smiled. \"Wouldn't our bear notice a country fair canvas booth out there on the ice, gentlemen?\"\n\n\"Nay, sir,\" said the sailmaker. \"I'll have the canvas cut and sewn and painted snow white before nightfall \u2014 or this gloom we call night up here. We'll set the blind against a low pressure ridge where it will blend in. Only the slightest long firing slit will be visible. Mr. Weekes will use the wood from the burial service scaffolding to set benches inside so the shooters will be warm and snug up off the ice.\"\n\n\"How many shooters do you envision in this... bear blind?\" asked Sir John.\n\n\"Six, sir,\" answered Sergeant Bryant. \"It's volley fire that will bring this beast down, sir. Just as it brought down Napoleon's minions by the thousands at Waterloo.\"\n\n\"But what if the bear has a better sense of smell than Napoleon did at Waterloo?\" asked Sir John.\n\nThe men chuckled but Sergeant Tozer said, \"We thought of that, Sir John. Mostly these days the wind is out of the nor'-nor'west. If we built the blind against the low pressure ridge near where poor Lieutenant Gore was laid to rest, sir, well, we'd have that nice great expanse of open ice to the nor'west as a killing zone. Almost a hunded yards of open space. Odds are great that it would come down off the higher ridges from upwind, Sir John. And when it gets where we want it, quick volleys of Mini\u00e9 balls into its heart and lungs, sir.\"\n\nSir John thought about this.\n\n\"But we'll have to call off the men, sir,\" said Edward Couch, the mate. \"With all the men crashing around out there on the ice, them and lookouts firing off their shotguns at every ice serac and gust of wind, no self-respecting bear would come within five miles o' the ship, sir.\"\n\nSir John nodded. \"And what is going to lure our bear into this killing zone, gentlemen? Have you thought about bait?\"\n\n\"Aye, sir,\" said Sergeant Bryant, smiling now. \"It's fresh meat that draws these killers in.\"\n\n\"We have no fresh meat,\" said Sir John. \"Not so much as a ring seal.\"\n\n\"No, sir,\" said the craggy Marine sergeant. \"But we have that little bear. Once the blind is built and set in place, we'll butcher that little thing, not sparing the blood, sir, and leave the meat out there on the ice not twenty-five yards from our shooting position.\"\n\nSir John said, \"So you think our animal is a cannibal?\"\n\n\"Oh, aye, sir,\" said Sergeant Tozer, his face flushing under the purple birthmark. \"We think this thing will eat anything that bleeds or smells of meat. And when it does, we'll pour the volleys of fire into it, sir, and then it's ten sovereigns per man and then winter and then triumph and then home.\"\n\nSir John nodded judiciously. \"Make it so,\" he said.\n\nOn Friday afternoon, the eleventh of June, Sir John went out with Lieutenant Le Vesconte to inspect the bear blind.\n\nThe two officers had to admit that even from thirty feet away the blind was all but invisible, its floor and back built into the low ridge of snow and ice where Sir John had given the eulogy. The white sails blended almost perfectly and the firing slit had tatters of canvas hanging at irregular intervals to break up the solid horizontal line. The sailmaker and armourer had attached the canvas so cleverly to the iron rods and ribs that even in the rising wind now blowing snow across the open ice, there was not the slightest flap of canvas.\n\nLe Vesconte led Sir John down the icy path behind the pressure ridge \u2014 out of sight of the shooting zone \u2014 and then over the low wall of ice and in through a slit at the back of the tent. Sergeant Bryant was there with the _Erebus_ Marines \u2014 Corporal Pearson and Privates Healey, Reed, Hopcraft, and Pilkington \u2014 and the men started to rise as their expedition commander entered.\n\n\"Oh, no, no, gentlemen, keep your seats,\" whispered Sir John. Aromatic wooden planks had been set in high iron stirrups curled into the iron support bars at either side of the long, narrow tent, allowing the Marines to sit at shooting height when not standing by the narrow firing slit. Another layer of planking kept their feet off the ice. Their muskets were at the ready in front of them. The crowded space smelled of fresh wood, wet wool, and gun oil.\n\n\"How long have you been waiting?\" whispered Sir John.\n\n\"Not quite five hours, Sir John,\" whispered Sergeant Bryant.\n\n\"You must be cold.\"\n\n\"Not a bit, sir,\" said Bryant in low tones. \"The blind is large enough to allow us to move around from time to time and the planks keep our feet from freezing. The _Terror_ Marines under Sergeant Tozer will relieve us at two bells.\"\n\n\"Have you seen anything?\" whispered Lieutenant Le Vesconte.\n\n\"Not yet, sir,\" answered Bryant. The sergeant and the two officers leaned forward until their faces were in the cold air of the firing slit.\n\nSir John could see the carcass of the bear cub, its muscles a shocking red against the ice. They had skinned everything except the small white head, bled it out, captured the blood in pails, and spread the blood all around the carcass. The wind was blowing snow across the wide expanse of ice, and the red blood against all the white, grey, and pale blue was disconcerting.\n\n\"We have still to see whether our foe is a cannibal,\" whispered Sir John.\n\n\"Aye, sir,\" said Sergeant Bryant. \"Would Sir John join us on the bench, sir? There's ample room.\"\n\nThere wasn't ample room, especially with Sir John's broad beam added to those beefy posteriors already lined up along the plank. But with Lieutenant Le Vesconte remaining standing and all the Marines scooted down as far as they could go, it was just manageable to have the seven men crowded onto the piece of timber. Sir John realized that he could see out onto the ice quite well from this raised position.\n\nAt this moment, Captain Sir John Franklin was as happy as he had ever been in the company of other men. It had taken Sir John years to realize that he was far more comfortable in the presence of women \u2014 including artistic, high-strung women such as his first wife, Eleanor, and powerful, indomitable women such as his current wife, Jane \u2014 than in the company of men. But since his Divine Service the previous Sunday, he had received more smiles, nods, and sincere looks of approbation from his officers and seamen than at any time in his forty-year career.\n\nIt was true that the promise of ten gold sovereigns per man \u2014 not to mention the doubling of the advance pay, equal to five months' regular salary for a sailor \u2014 had been made in a most unusual burst of good feeling and improvisation. But Sir John had ample financial resources, and should those suffer during his three years and more away, he was quite certain that Lady Jane's private fortune would be available to cover these new debts of honor.\n\nAll in all, Sir John reasoned, the financial offers and his surprise allowance of grog rations aboard his teetotaling ship had been strokes of brilliance. Like all others, Sir John had been deeply cast down by the sudden death of Graham Gore, one of the most promising young officers in the fleet. The bad news of no open ways in the ice and the terrible certainty of another dark winter here had weighed heavily upon everyone, but with a promise of ten gold sovereigns per man and a single feast day aboard two ships, he had surmounted that problem for the time being.\n\nOf course, there was the other problem, brought to him by the four medicos only last week: the fact that more and more of the canned foodstuffs were being found to be putrid, possibly as a result of improper soldering of the cans. But Sir John had set that aside for now.\n\nThe wind blew snow across the wide expanse of ice, obscuring then revealing the tiny carcass in its congealing and freezing X of blood on the blue ice. Nothing moved from the surrounding pressure ridges and ice pinnacles. The men to Sir John's right sat easily, one chewing tobacco, the others resting their mittened hands on the upraised muzzles of their muskets. Sir John knew that those mittens would be off in a flash should their nemesis appear on the ice.\n\nHe smiled to himself as he realized that he was memorizing this scene, this moment, as a future anecdote for Jane and his daughter, Eleanor, and his lovely niece Sophia. He did that a lot these days, observing their predicament on the ice as a series of anecdotes and even setting them into words \u2014 not too many words, just enough to hold rapt attention \u2014 for future use with his lovely ladies and during evenings dining out. This day \u2014 the absurd shooting blind, the men crowded in, the good feeling, the smell of gun oil and wool and tobacco, even the lowering grey clouds and blowing snow and mild tension as they awaited their prey \u2014 should stand him in good stead in the years to come.\n\nSuddenly Sir John's gaze turned far to the left, past Lieutenant Le Vesconte's shoulder, to the burial pit not twenty feet from the south end of the blind. The opening to the black sea had long frozen over and much of the crater itself had filled with blowing snow since the burial day, but even the sight of the depression in the ice made Sir John's now-sentimental heart hurt in memory of young Gore. But it had been a fine burial service. He had conducted it with dignity and proud military bearing.\n\nSir John noticed two black objects lying close together in the lowest part of the icy depression \u2014 dark stones perhaps? Buttons or coins left behind as remembrance of Lieutenant Gore by some seaman filing by the burial site precisely a week ago? And in the dim, shifting light of the snowstorm the tiny black circles, all but invisible unless one knew exactly where to look, seemed to stare back at Sir John with something like sad reproach. He wondered if by some fluke of climate two tiny openings to the sea itself had remained open during all the intervening freeze and snow, thus revealing these two tiny circles of black water against the grey ice.\n\nThe black circles blinked.\n\n\"Ah... Sergeant... ,\" began Sir John.\n\nThe entire floor of the burial crater seemed to erupt into motion. Something huge, white and grey and powerful exploded toward them, rising and rushing at the blind and then disappearing on the south side of the canvas, out of sight of the firing slit.\n\nThe Marines, obviously not sure of what they had just seen, had no time to react.\n\nA powerful force struck the south side of the blind not three feet from Le Vesconte and Sir John, collapsing the iron and rending the canvas.\n\nThe Marines and Sir John leapt to their feet as the canvas ripped above them and behind them and to the side of them, black claws the length of Bowie knives tearing through thick sail. Everyone was shouting at once. There came a terrible carrion reek.\n\nSergeant Bryant raised his musket \u2014 the thing was inside, it was _inside,_ with them, among them, surrounding them with the circumference of inhuman arms \u2014 but before he could fire there was a rush of air through the reek of predator breath. The sergeant's head flew off his shoulders and out through the firing slit and skittered across the ice.\n\nLe Vesconte screamed, someone fired a musket \u2014 the ball striking only the Marine next to him. The top of the canvas blind was gone, something huge blocked the opening where the sky should be, and just as Sir John turned to throw himself forward out of the ripping sail canvas, he was struck by a terrible pain just below both knees.\n\nThen things became blurred and absurd. He seemed to be upside down, watching men being scattered like tenpins across the ice, men being thrown from the destroyed blind. Another musket fired but only as the Marine threw the weapon down and tried to scramble away across the ice on all fours. Sir John saw all this \u2014 impossibly, absurdly \u2014 from an inverted and swinging position. The pain in his legs grew intolerable, there came the sound of saplings snapping, and then he was thrown forward, down into the burial crater, toward the new circle of black awaiting. His head smashed through the thin scrim of ice like a cricket ball through a windowpane.\n\nThe water's cold temporarily stopped Sir John's wildly pumping heart. He tried to scream but inhaled salt water.\n\n _I am in the sea. For the first time in my life, I am in the sea itself. How extraordinary._\n\nThen he was flailing, turning over and over, feeling the torn fragments and rags of his shredded greatcoat peeling away, feeling nothing from his legs now and getting no purchase against the freezing water with his feet. Sir John used his arms and hands to pull and paddle, not knowing in the terrible darkness if he was fighting toward the surface or merely propelling himself deeper into the black water.\n\n _I am drowning. Jane, I am drowning. Of all the fates I had considered these long years in the Service, never once, my darling, did I contemplate drowning._\n\nSir John's head struck something solid, almost knocking him unconscious, forcing his face beneath the water again, filling his mouth and lungs with salt water again.\n\n _And then, my Dears, Providence led me to the surface \u2014 or at least to the thin inch of breathable air between the sea and fifteen feet of ice above._\n\nSir John's arms flailed wildly as he rotated onto his back, his legs still not working, fingers scrabbling at ice above. He forced himself to calm his heart and limbs, forced discipline so that his nose could find that tiniest fraction of air between ice and freezing cold water. He breathed. Raising his chin, he coughed out seawater and breathed through his mouth.\n\n _Thank you, dear Jesus, Lord... ._\n\nFighting down the temptation to scream, Sir John scrabbled along the underside of the ice as if he were climbing a wall. The bottom of the pack ice here was irregular, sometimes protruding down into the water and giving him no fraction of an inch of air to breathe, sometimes rising five or six inches or more and almost allowing him to lift his full face out of the water.\n\nDespite the fifteen feet of ice above him, there was a dim glow of light \u2014 blue light, the Lord's light \u2014 refracted through the rough facets of ice just inches from his eyes. Some daylight was coming in via the hole \u2014 Gore's burial hole \u2014 through which he had just been thrown.\n\n _All I had to do, my dear ladies, my darling Jane, was to find my way back to that narrow hole in the ice \u2014 get my bearings, as it were \u2014 but I knew that I had only minutes... ._\n\nNot minutes, seconds. Sir John could feel the cold water freezing the life out of him. And there was something terribly wrong with his legs. Not only could he not _feel_ them, but he could feel an absolute _absence_ there. And the seawater tasted of his own blood.\n\n _And then, ladies, the Lord God Almighty shewed me the light... ._\n\nTo his left. The opening was some ten yards or less to his left. The ice was high enough above the black water here that Sir John could raise his head, set the top of his bald and freezing pate against rough ice, gasp in air, blink water and blood out of his eyes, and actually _see_ the glow of the Saviour's light not ten yards away... .\n\nSomething huge and wet rose between him and the light. The darkness was absolute. His inches of breathable air were suddenly taken away, filled with the rankest of carrion breath against his face.\n\n\"Please... ,\" began Sir John, sputtering and coughing.\n\nThen the moist reek enveloped him and huge teeth closed on either side of his face, crunching through bone and skull just forward of his ears on both sides of his head.\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n10 November, 1847\n\nIt was five bells, 2:30 a.m., and Captain Crozier was back from _Erebus,_ had inspected the corpses \u2014 or half-corpses \u2014 of William Strong and Thomas Evans where the thing on the ice had left them propped up near the stern rail on the quarterdeck, had seen to their stowage in the Dead Room below, and now he sat in his cabin contemplating the two objects on his desk \u2014 a new bottle of whiskey and a pistol.\n\nAlmost half Crozier's small cabin was taken up by the built-in bunk set against the starboard hull. The bunk looked like a child's cradle with carved, raised sides, built-in cupboards below, and a lumpy horsehair mattress set almost chest-high. Crozier had never slept well on real beds and often wished for the swinging hammocks he'd spent so many years in as a midshipman, young officer, and when he served before the mast as a boy. Set against the outer hull as this bunk was, it was one of the coldest sleeping places aboard the ship \u2014 chillier than the bunks of the warrant officers with their cubbies in the centre of the lower deck aft, and _much_ colder than the sleeping hammocks of the lucky seamen forward, strung as they were on the mess deck near the still-glowing Frazer's Patent Stove that Mr. Diggle cooked on twenty hours out of the day.\n\nBooks set into built-in shelves along the rising, inward-sloping hull helped insulate Crozier's sleeping area a little but not much. More books ran under the ceiling for the five-foot width of the cabin, filling a shelf that hung under curving ship's timbers three feet above the foldout desk connecting Crozier's bunk to the hall partition. Directly overhead was the black circle of the Preston Patent Illuminator, its convex opaque glass piercing a deck now dark beneath three feet of snow and protective canvas. Cold air constantly flowed down from the Illuminator like the freezing exhalations of something long dead but still labouring to breathe.\n\nOpposite Crozier's desk was a narrow shelf holding his bathing basin. No water was kept in the basin since it would freeze; Crozier's steward, Jopson, brought his captain hot water from the stove each morning. The space between desk and basin left just enough room in the tiny cabin for Crozier to stand, or \u2014 as now \u2014 sit at his desk on a backless stool that slid under the basin shelf when not in use.\n\nHe continued staring at the pistol and bottle of whiskey.\n\nThe captain of HMS _Terror_ often thought that he knew nothing about the future \u2014 other than that his ship and _Erebus_ would never again steam or sail \u2014 but then he reminded himself of one certainty: when his store of whiskey was gone, Francis Rawdon Moira Crozier was going to blow his brains out.\n\nThe late Sir John Franklin had filled his storeroom with expensive china \u2014 all bearing Sir John's initials and family crest, of course \u2014 as well as cut crystal, forty-eight beef tongues, fancy silver also engraved with his crest, barrels of smoked Westphalia hams, towers of double Gloucestershire cheeses, bag upon bag of specially imported tea from a relative's plantation in Darjeeling, and crocks of his favorite raspberry jam.\n\nAnd while Crozier had packed some special foods for the occasional officers' dinners he had to host, most of his money and allocated hold space had been dedicated to three hundred and twenty-four bottles of whiskey. It was not fine Scotch whiskey, but it would suffice. Crozier knew that he had long since reached that point of being the kind of drunkard where quantity always trumped quality. Sometimes here, as in the summer when he was especially busy, a bottle might last him two weeks or more. Other times \u2014 as during this past week \u2014 he might go through a bottle a night. The truth was, he had quit counting the empty bottles when he passed two hundred the previous winter, but he knew that he must be nearing the end of his supply. On the night he drinks the last of the last and his steward tells him there are no more \u2014 Crozier knew it would be at night \u2014 he firmly planned to cock the pistol, set the muzzle to his temple, and pull the trigger.\n\nA more practical captain, he knew, might remind himself that there were the not-insignificant liquid remnants of four thousand five hundred gallons \u2014 _gallons_ \u2014 of concentrated West Indian rum in the Spirit Room below, and that each jug was rated between 130 and 140 proof. The rum was doled out each day to the men in units of gills, one fourth of a pint cut with three-quarters pint of water, and there were enough gills and gallons left to swim in. A less finicky and more predatory drunkard-captain might consider the men's rum his reserve. But Francis Crozier did not like rum. He never had. Whiskey was his drink, and when it was gone, so would be he.\n\nSeeing young Tommy Evans's body severed at the waist, the trousered legs sticking out in an almost comical Y, the boots still firmly laced over the dead feet, had reminded Crozier of the day he'd been summoned to the shattered bear blind a quarter of a mile from _Erebus._ In less than twenty-four hours, he realized, it would be the fifth-month anniversary of that eleventh of June debacle. At first Crozier and the other officers who had come running could make little sense of the havoc at the blind. The structure itself had been torn to shreds, the very iron bars of its frame bent and battered. The plank seat had been smashed to splinters and amid those splinters lay the headless body of Marine Sergeant Bryant, the ranking Marine on the expedition. His head \u2014 not yet recovered when Crozier arrived \u2014 had been batted almost thirty yards across the ice until it stopped next to a skinned bear cub's carcass.\n\nLieutenant Le Vesconte had suffered a broken arm \u2014 not from the bear-monster, it turned out, but from falling out onto the ice \u2014 and Private William Pilkington had been shot through the upper left shoulder by the Marine next to him, Private Robert Hopcraft. The private had received eight broken ribs, a pulverized collarbone, and a dislocated left arm from what he later described as a glancing blow from a monster's huge paw. Privates Healey and Reed had survived without serious injury but with the ignominy of having fled the melee in panic, tumbling and screaming and scrambling on all fours across the ice. Reed had broken three fingers in his flight.\n\nBut it had been the two trousered and boot-buckled legs and feet of Sir John Franklin \u2014 intact below the knees but separated, one lying in the blind, another having been dropped somewhere near the hole through the ice in the burial crater \u2014 which had commanded Francis Crozier's attention.\n\nWhat kind of malevolent intelligence, he wondered while drinking whiskey from his glass, severs a man at the knees and then carries the still-living prey to a hole in the ice and drops him in, to follow a second later? Crozier had tried not to imagine what may have happened next under the ice, although some nights after a few drinks and while trying to fall asleep, he could see the horror there. He also thought for a certainty that Lieutenant Graham Gore's burial service one week earlier to the hour had been nothing more than an elaborate banquet unwittingly offered up to a creature already waiting and watching from beneath the ice.\n\nCrozier had not been overly devastated by Lieutenant Graham Gore's death. Gore was precisely that kind of well-bred, well-educated, C of E, public school, war-hero Royal Navy officer, come natural to command, at ease with superiors and inferiors, modest in all things but destined for great things, well-mannered British kind-even-to-Irishmen, upper-class fucking toff twit whom Francis Crozier had watched being promoted over him for more than forty years.\n\nHe took another drink.\n\nWhat kind of malevolent intelligence kills but does not eat all its prey in such a winter of no game as this but rather returns the upper half of the corpse of Able-Bodied Seaman William Strong and the lower part of the corpse of young Tom Evans? Evans had been one of the \"ship's boys\" who had beat muffled drums in Gore's funeral procession five months earlier. What kind of creature plucks that young man from Crozier's side in the dark but leaves the captain standing three yards away... then returns half the corpse?\n\nThe men knew. Crozier knew what they knew. They knew it was the Devil out there on the ice, not some overgrown arctic bear.\n\nCaptain Francis Crozier did not disagree with the men's assessment \u2014 for all his pish-posh talk earlier that night over brandy with Captain Fitzjames \u2014 but he knew something that the men did not; namely that the Devil trying to kill them up here in the Devil's Kingdom was not just the white-furred thing killing and eating them one by one, but _everything_ here \u2014 the unrelenting cold, the squeezing ice, the electrical storms, the uncanny lack of seals and whales and birds and walruses and land animals, the endless encroachment of the pack ice, the bergs that plowed their way through the solid white sea not even leaving a single ship's length lee of open water behind them, the sudden white-earthquake up-eruption of pressure ridges, the dancing stars, the shoddily tinned cans of food now turned to poison, the summers that did not come, the leads that did not open \u2014 _everything._ The monster on the ice was just another manifestation of a Devil that wanted them dead. And that wanted them to suffer.\n\nCrozier took another drink.\n\nHe understood the arctic's motivation better than his own. The ancient Greeks had been right, thought Crozier, when they stated that there were five bands of climate on this disk of an earth, four of them equal, opposite, and symmetrical like so many things Greek, wrapped around the world like bands on a snake. Two were temperate and made for human beings. The central band, the equatorial region, was not meant for intelligent life \u2014 although the Greeks had been wrong in assuming that no humans could live there. Just no civilized humans, thought Crozier, who'd had his glimpse of Africa and the other equatorial areas and was sure nothing of value would ever come from any of them. The two polar regions, the Greeks had reasoned long before the arctic and antarctic wastes were reached by explorers, were inhuman in every sense \u2014 unfit even to travel through, much less to reside in for any length of time.\n\nSo why, wondered Crozier, did a nation like England, blessed to be placed by God in one of the most gentle and verdant of the two temperate bands where mankind was meant to live, keep throwing its ships and its men into the ice of the northern and southern polar extremes where even fur-wearing savages refuse to go?\n\nAnd more pertinent to the central question, why did one Francis Crozier keep returning to these terrible places time after time, serving a nation and its officers that have never recognized his abilities and worth as a man, even while he knew in his heart that someday he would die in the arctic cold and dark?\n\nThe captain remembered that even when he was a small boy \u2014 before he went to sea at age thirteen \u2014 he had carried his deep mood of melancholy within him like a cold secret. This melancholic nature had manifested itself in his pleasure at standing outside the village on a winter night watching the lamplights fade, by finding small places in which to hide \u2014 claustrophobia had never been a problem for Francis Crozier \u2014 and by being so afraid of the dark, seeing it as the avatar of the death that had claimed his mother and grandmother in such a stealthy way, that he had perversely sought it out, hiding in the root cellar while other boys played in the sunlight. Crozier remembered that cellar \u2014 the grave chill of it, the smell of cold and mold, the darkness and inward-pressing which left one alone with dark thoughts.\n\nHe filled his small glass and took another drink. Suddenly the ice groaned louder, and the ship groaned back in response \u2014 trying to shift its place in the frozen sea but having no place to go. In recompense it squeezed itself tighter and moaned. Metal brackets in the hold deck contracted, the sudden cracks sounding like pistol shots. The seamen forward and the officers aft snored on, used to the night noises of the ice trying to crush them. On deck above, the officer on watch in the seventy-below night stomped his feet to renew circulation, the four sharp stamps sounding to the captain like a weary parent telling the ship to hush its protestations.\n\nIt was hard for Crozier to believe that Sophia Cracroft had visited this ship, stood in this very cabin, exclaimed how neat it was, how tidy, how cozy, how very learned with its row of books, and how pleasant the austral light pouring down from the Illuminator.\n\nIt had been seven years ago almost to the week, the Southern-Hemispheric spring month of November of 1840, when Crozier had arrived in Van Diemen's Land south of Australia in these very ships \u2014 _Erebus_ and _Terror_ \u2014 on the way to Antarctica. The expedition had been under the command of Crozier's friend, although always his social superior, Captain James Ross. They had stopped at Hobart Town to finish their provisioning before heading into antarctic waters, and the governor of that penal island, Sir John Franklin, insisted that the two younger officers \u2014 Captain Ross and Commander Crozier \u2014 stay at Government House during their visit.\n\nIt had been an enchanting time and \u2014 to Crozier \u2014 a romantically fatal one.\n\nThe inspection of the expedition's ships had occurred on the second day of the visit \u2014 the ships were clean, refitted, almost fully provisioned, their young crews not yet bearded or made haggard by the two winters in the antarctic ice to come \u2014 and while Captain Ross personally hosted Governor Sir John and Lady Jane Franklin, Crozier had found himself escorting the governor's niece, the dark-haired and bright-eyed young Sophia Cracroft. He had fallen in love on that day and had carried that blossoming love into the darkness of the next two southern winters, where it had bloomed into an obsession.\n\nThe long dinners under the servant-turned fans of the governor's house were filled with lively conversation. Governor Franklin was a worn-out man in his midfifties, dispirited by the lack of recognition of his accomplishments and dispirited further by the opposition of the local press, wealthy landowners, and bureaucrats during his third year in Van Diemen's Land, but both he and his wife, Lady Jane, had come alive during this visit by their Discovery Service countrymen and, as Sir John liked to address them, his \"fellow explorers.\"\n\nSophia Cracroft, on the other hand, showed no signs of unhappiness. She was witty, alive, vivacious, sometimes shocking in her comments and boldness \u2014 even more so than her controversial aunt, the Lady Jane \u2014 and young and beautiful and seemingly interested in every aspect of the forty-four-year-old bachelor Commander Francis Crozier's opinions, life, and sundry thoughts. She laughed at all of Crozier's initially hesitant jokes \u2014 he was not used to this level of society and strived to be on his best behaviour, drinking less than he had in years and that only wine \u2014 and she always answered his tentative bon mots with increasingly higher levels of wit. To Crozier, it was like learning tennis from a far better player. By the eighth and final day of their extended visit, Crozier felt the equal of any proper Englishman \u2014 a gentleman born in Ireland, yes, but one who had made his own way and had also lived an interesting and exciting life, the equal of any man \u2014 and the superior of most men in Miss Cracroft's amazing blue eyes.\n\nWhen HMSs _Erebus_ and _Terror_ left the Hobart Town harbour, Crozier was still calling Sophia \"Miss Cracroft,\" but there was no denying the secret connections they had made: the secret glances, the companionable silences, the shared jokes and private moments alone. Crozier knew that he was in love for the first time in a life whose \"romance\" had consisted of dockyard doxies' cribs, back-alley knee-wobblers, some native girls doing the deed for trinkets, and a few overpriced nights out in gentlemen's whorehouses in London. All that was behind him now.\n\nFrancis Crozier now understood that the most desirable and erotic thing a woman could wear were the many modest layers such as Sophia Cracroft wore to dinner in the governor's house, enough silken fabric to conceal the lines of her body, allowing a man to concentrate on the exciting loveliness of her wit.\n\nThen followed almost two years of pack ice, glimpsing Antarctica, the stink of penguin rookeries, naming two distant, smoking volcanoes after their tired ships, darkness, spring, the threat of being frozen in, finding and fighting their way out by sail only through a sea now named after James Ross, and finally the rough Southern Sea passage and the return to Hobart Town on the island of eighteen thousand prisoners and one very unhappy governor. This time there was no inspection of _Erebus_ and _Terror;_ they stank too much of grease and cooking and sweat and fatigue. The boys who had sailed south were now mostly hollow-eyed and bearded men who would not sign up for future Discovery Service expeditions. Everyone except HMS _Terror_ 's commander was eager to return to England.\n\nFrancis Crozier was eager only to see Sophia Cracroft again.\n\nHe took another drink of whiskey. Above him, barely audible through the deck and snow, the ship's bell rang six bells. Three a.m.\n\nThe men were sorry when Sir John had been killed five months ago \u2014 most of them because they knew that the promise of ten sovereigns per man and a second advance-pay bonus had died with the paunchy, bald old man \u2014 but little actually changed after Franklin's death. Commander Fitzjames was now acknowledged as the captain of _Erebus_ that he'd always been in reality. Lieutenant Le Vesconte, gold tooth flashing when he smiled, his arm in a sling, took Graham Gore's place in the hierarchy of command with no visible ripple of disruption. Captain Francis Crozier now assumed the rank of expedition commander, but with the expedition frozen in the ice, there was little he could do differently now from what Franklin would have done.\n\nOne thing Crozier did almost immediately was ship more than five tons of supplies across the ice to a point not far from the Ross cairn on King William Land. They were now fairly certain it was an island because Crozier had sent out sledge parties \u2014 monster bear be damned \u2014 to scout the area. Crozier himself went along on half a dozen of these early sledge parties, helping to smash easier, or at least less impossible, paths through the pressure ridges and iceberg barrier along the shore. They brought extra winter slops, tents, lumber for future cabins, casks of dried foods and hundreds of cans of the tinned foods, as well as lightning rods \u2014 even brass bed rods from Sir John's appropriated cabin to be fashioned into lightning rods \u2014 and the essentials of what both crews would need if the ships had to be abandoned suddenly in the midst of the next winter.\n\nFour men had been lost to the creature on the ice before winter returned, two from one tent during one of Crozier's trips, but what stopped the transfer sledge trips in mid-August was a return of the severe lightning and thick fog. For more than three weeks both ships sat in the midst of the fog, suffered the lightning to strike them, and only the briefest ice outings \u2014 hunting parties mostly, a few fire-hole teams \u2014 were allowed. By the time the freakish fog and lightning had passed, it was early September, and the cold and snow had begun again.\n\nCrozier then resumed sending cache-supply sledge parties to King William Land despite the terrible weather, but when Second Master Giles MacBean and a seaman were killed just a few yards ahead of the three sledges \u2014 the deaths unseen because of the blowing snow but their last screams all too audible to the other men and to their officer, Second Lieutenant Hodgson \u2014 Crozier \"temporarily\" suspended the supply trips. The suspension had now lasted two months, and by the first of November no sane crewman wanted to volunteer for an eight- to ten-day sledge trip in the dark.\n\nThe captain knew that he should have cached at least ten tons of supplies on the shore rather than the five tons he'd hauled there. The problem was \u2014 as he and a sledge party had learned the night the creature had ripped through a tent near the captain's and would have carried off seaman George Kinnaird and John Bates if they had not run for their lives \u2014 any campsite on that low, windswept gravel-and-ice spit of land was not defensible. Aboard the ships, as long as they lasted, the hulls and raised deck acted as a wall of sorts, turning each ship into a kind of fort. Out on the gravel and in tents, no matter how tightly clustered, it would take at least twenty armed men watching day and night to guard the perimeter, and even then the thing could be among them before guards could react. Everyone who had sledged to King William Land and camped out there on the ice knew this. And as the nights grew longer, the fear of those unprotected hours in the tents \u2014 like the arctic cold itself \u2014 seeped deeper into the men.\n\nCrozier drank some more whiskey.\n\nIt had been April of 1843 \u2014 early autumn in the Southern Hemisphere, although the days were still long and warm \u2014 when _Erebus_ and _Terror_ returned to Van Diemen's Land.\n\nRoss and Crozier were once again guests at the governor's house \u2014 officially called Government House by the old-time inhabitants of Hobart Town \u2014 but this time it was obvious that a shadow lay over both Franklins. Crozier was willing to be oblivious to this, his joy at being near Sophia Cracroft was so great, but even the irrepressible Sophia had been subdued by the mood, events, conspiracies, betrayals, revelations, and crises that had been brooding in Hobart during the two years _Erebus_ and _Terror_ had been in the southern ice, so in the course of his first two days in Government House, he'd heard enough to piece together the reasons for the Franklins' depression.\n\nIt seemed that local and venial landed interests, personified in one undercutting, backstabbing Judas of a colonial secretary named Captain John Montagu, had decided early on in Sir John's six years as governor that he simply would not do, nor would his wife, the outspoken and unorthodox Lady Jane. All Crozier heard from Sir John himself \u2014 overheard, actually, as the despondent Sir John spoke to Captain Ross while the three men took brandy and cigars in the booklined study in the mansion \u2014 was that the locals had \"a certain lack of neighbourly feelings and a deplorable deficiency in public spirit.\"\n\nFrom Sophia, Crozier learned that Sir John had gone, in the public eye at least, from being \"the man who ate his shoes\" to his self-styled description of \"a man who wouldn't hurt a fly\" then quickly to a description widespread on the Tasmanian Peninsula of \"a man in petticoats.\" This last calumny, Sophia assured him, came from the colony's dislike of Lady Jane as much as it had from Sir John's and his wife's attempts to improve things for the natives and prisoners who laboured there in inhuman conditions.\n\n\"You understand that the previous governors simply loaned out prisoners for the local plantation owners' and city business tycoons' insane projects, took their cut of the profits, and kept their mouths shut,\" explained Sophia Cracroft as they walked in the shadows of the Government House gardens. \"Uncle John has not played that game.\"\n\n\"Insane projects?\" said Crozier. He was very aware of Sophia's hand on his arm as they walked and spoke in hushed whispers, alone, in the warm near-dark.\n\n\"If a plantation manager wants a new road on his land,\" said Sophia, \"the governor is expected to loan him six hundred starving prisoners \u2014 or a thousand \u2014 who work from dawn until after dark, chains on their legs and manacles on their wrists, through this tropical heat, without water or food, being flogged if they fall or falter.\"\n\n\"Good Lord,\" said Crozier.\n\nSophia nodded. Her eyes remained fixed on the white stones of the garden path. \"The colonial secretary, Montagu, decided that the prisoners should excavate a pit mine \u2014 although no gold has been found on the island \u2014 and the prisoners were set to digging it. It was more than four hundred feet deep before the project was abandoned \u2014 it flooded constantly, the water table here is very shallow, of course \u2014 and it was said that two to three prisoners died for every foot excavated of that abhorred mine.\"\n\nCrozier restrained himself before he could say _Good Lord_ again, but in truth that was all that came to his mind.\n\n\"A year after you left,\" continued Sophia, \"Montagu \u2014 that weasel, that viper \u2014 persuaded Uncle John to dismiss a local surgeon, a man very popular with the decent people here, on trumped-up charges of dereliction of duty. It divided the colony. Uncle John and Aunt Jane became the lightning rod for all criticism, even though Aunt Jane had disapproved of the surgeon's dismissal. Uncle John \u2014 you know, Francis, how very much he hates controversy, much less to administer pain of any sort, why he's often said he would not hurt a fly... .\"\n\n\"Yes,\" said Crozier, \"I have seen him carefully remove a fly from a dining room and release it.\"\n\n\"Uncle John, listening to Aunt Jane, eventually reinstated the surgeon, but that made a lifelong enemy out of this Montagu. The private bickerings and accusations became public, and Montagu \u2014 in essence \u2014 called Uncle John a liar and a weakling.\"\n\n\"Good Lord,\" said Crozier. What he was thinking was, _If I had been in John Franklin's place, I would have called this Montagu fucker out to the field of honor and there put a bullet in each of his testicles before I placed a final one in his brain._ \"I hope that Sir John sacked the man.\"\n\n\"Oh, he did,\" said Sophia with a sad little laugh, \"but that only made things worse. Montagu returned to England last year on the same ship carrying Uncle John's letter announcing his dismissal, and it turns out, sadly, that Captain Montagu is a close friend of Lord Stanley, secretary of state for the Colonies.\"\n\n _Well, the Governor is well and truly buggered,_ Crozier thought as they reached the stone bench at the far end of the garden. He said, \"How unfortunate.\"\n\n\"More than Uncle John or Aunt Jane could have imagined,\" said Sophia. \"The Cornwall _Chronicle_ ran a long article entitled 'The Imbecile Reign of the Polar Hero.' The _Colonial Times_ blames Aunt Jane.\"\n\n\"Why attack Lady Jane?\"\n\nSophia smiled without humour. \"Aunt Jane is, rather like myself... unorthodox. You have seen her room here at Government House, I believe? When Uncle John gave you and Captain Ross a tour of the estate the last time you were here?\"\n\n\"Oh, yes,\" said Crozier. \"Her collection was wonderful.\" Lady Jane's boudoir, the parts they were allowed to see, had been crammed carpet to ceiling with animal skeletons, meteorites, stone fossils, Aborigine war clubs, native drums, carved wooden war masks, ten-foot paddles that looked capable of propelling HMS _Terror_ along at fifteen knots, a plethora of stuffed birds, and at least one expertly taxidermied monkey. Crozier had never seen anything like it in a musuem or zoo, much less in a lady's bedroom. Of course, Francis Crozier had seen very few ladies' bedrooms.\n\n\"One visitor wrote to a Hobart newspaper that, and I quote verbatim, Francis, 'our governor's wife's private rooms at Government House look more like a museum or a menagerie than the boudoir of a lady.'\"\n\nCrozier made a clucking noise and felt guilty about his similar thoughts. He said, \"So is this Montagu still making trouble?\"\n\n\"More than ever. Lord Stanley \u2014 that viper's viper \u2014 backed Montagu, reinstated that worm in a position similar to the one Uncle John dismissed him from, and sent Uncle John a reprimand so terrible that Aunt Jane told me in private that it was the equivalent of a horsewhipping.\"\n\n _I'd shoot that bugger Montagu in the balls and then cut Lord Stanley's off and serve them to him only slightly warmed,_ thought Crozier. \"That is terrible,\" he said.\n\n\"There is worse,\" said Sophia.\n\nCrozier looked for tears in the dim light but saw none. Sophia was not a woman given to weeping.\n\n\"Stanley made public the rebuke?\" guessed Crozier.\n\n\"The... bastard... gave a copy of the official rebuke to Montagu, _before he sent it to Uncle John,_ and that weasel's weasel rushed it here by the fastest post ship. Copies were made and passed around here in Hobart Town to all of Uncle John's enemies months before Uncle John received the letter through official channels. The entire colony was sniggering every time Uncle John or Aunt Jane attended a concert or performed the governor's role in some official function. I apologize for my unlady-like language, Francis.\"\n\n _I'd feed Lord Stanley his balls cold in a fried dough of his own shit,_ thought Crozier. He said nothing but nodded that he forgave Sophia her choice of language.\n\n\"Just when Uncle John and Aunt Jane thought it could get no worse,\" continued Sophia, her voice trembling slightly, but with anger, Crozier was sure, not with weakness, \"Montagu sent to his plantation friends here a three-hundred-page packet containing all the private letters, Government House documents, and official dispatches which he had used to make his case against the governor to Lord Stanley. That packet is in the Central Colonial Bank here in the capital, and Uncle John knows that two thirds of the old families and business leaders in town have made their pilgrimage to the bank to read and hear what it contains. Captain Montagu calls the governor a 'perfect imbecile' in those papers... and from what we hear, that is the most polite thing in the detestable document.\"\n\n\"Sir John's position here seems untenable,\" said Crozier.\n\n\"At times I fear for his sanity, if not his life,\" agreed Sophia. \"Governor Sir John Franklin is a sensitive man.\"\n\n _He wouldn't hurt a fly,_ thought Crozier. \"Will he resign?\"\n\n\"He will be recalled,\" said Sophia. \"The entire colony knows it. This is why Aunt Jane is almost beside herself... I have never seen her in such a state. Uncle John expects official word of his recall before the end of August, if not sooner.\"\n\nCrozier sighed and pushed his walking stick along a furrow in the garden path gravel. He had looked forward to this reunion with Sophia Cracroft for two years in the southern ice, but now that he was here he could see that their visit would be lost in the shadow of mere politics and personality. He stopped himself before he sighed again. He was forty-six years old and acting like a fool.\n\n\"Would you like to see the Platypus Pond tomorrow?\" asked Sophia.\n\nCrozier poured another glass of whiskey for himself. There came a scream of banshees from above, but it was only the arctic wind in what was left of the rigging. The captain pitied the men on watch.\n\nThe whiskey bottle was almost empty.\n\nCrozier decided then and there that they would have to resume cache-hauling sledge trips to King William Land this winter, through the dark and storms and with the threat of the _thing_ on the ice ever present. He had no choice. If they had to abandon the ships in the coming months \u2014 and _Erebus_ was already showing signs of imminent collapse in the ice \u2014 it would not do merely to set up a sea camp here on the ice near where the ships would be destroyed. Normally that might make sense \u2014 more than one hapless polar expedition had set up camp on the ice and let the Baffin Bay current carry them hundreds of miles south to open sea \u2014 but this ice was going nowhere and a camp here on the ice would be even less defensible from the creature than would a camp on the frozen gravel on the shore \u2014 peninsula or island \u2014 twenty-five miles away in the dark. And he'd already cached more than five tons of gear there. The rest would have to follow before the sun returned.\n\nCrozier sipped his whiskey and decided that he would lead the next sledge trip. Hot food was the greatest single morale builder cold men could have, short of sight of rescue or extra gills of rum, so his next sledge trips would consist of stripping the four whaler boats \u2014 serious craft rigged for serious sailing should the real ships be abandoned at sea \u2014 of their cookstoves. The Frazer's Patent Stove on _Terror_ and its twin on _Erebus_ were too massive to move to shore \u2014 and Mr. Diggle would be using his to bake biscuits right up to the minute Crozier gave the order to abandon ship \u2014 so it was best to use the boat stoves. The four stoves were iron and would be heavy as Satan's hooves, especially if the sledges were hauling more gear, food, and clothing to cache, but they'd be safe on shore and could be fired up quickly, although the coal itself would also have to be hauled across the cold hell of a pressure-ridged twenty-five miles of sea ice. There was no wood on King William Land nor any for hundreds of miles south of there. The stoves would go next, Crozier decided, and he would go with them. They would sledge through the absolute darkness and unbelievable cold and let the Devil take the hindmost.\n\nCrozier and Sophia Cracroft had ridden out that next April morning in 1843 to see the Platypus Pond.\n\nCrozier had expected they'd be taking a buggy as they did for sojourns into Hobart Town, but Sophia had two horses saddled for them and a pack mule loaded with picnic things. She rode like a man. Crozier realized that the dark \"skirt\" she'd appeared to be wearing was actually a pair of gaucho trousers. The white canvas blouse she wore with it was somehow both feminine and rugged. She wore a broad-brimmed hat to keep the sun off her skin. Her boots were high, polished, soft, and must have cost roughly a year of Francis Crozier's captain's salary.\n\nThey rode north, away from Government House and the capital, and followed a narrow road through plantation fields, past penal colony pens, and then through a patch of rain forest and out into open higher country again.\n\n\"I thought that platypuses were found only in Australia,\" said Crozier. He was having trouble finding a comfortable position in the saddle. He'd never had much opportunity or reason to ride. It was embarrassing when his voice vibrated as he jounced and bounced. Sophia seemed completely at home in the saddle; she and the horse moved as one.\n\n\"Oh, no, my dear,\" said Sophia. \"The strange little things are found only in certain coastal areas on the continent to our north, but all across Van Diemen's Land. They're shy though. We see none around Hobart Town any longer.\"\n\nCrozier's cheeks grew warm at the sound of the \"my dear.\"\n\n\"Are they dangerous?\" he asked.\n\nSophia laughed easily. \"Actually the males are dangerous in mating season. They have a secret poisonous spur on their hind legs, and during the breeding season the spurs become quite venomous.\"\n\n\"Enough to kill a man?\" asked Crozier. He'd been joking about the comical little creatures he'd seen only in illustrations being dangerous.\n\n\"A small man,\" said Sophia. \"But survivors of the platypus's spur say that the pain is so terrible that they would have preferred death.\"\n\nCrozier looked to his right at the young woman. Sometimes it was very difficult to tell when Sophia was joking and when she was serious. In this instance, he would assume that she was telling the truth.\n\n\"Is it breeding season?\" he asked.\n\nShe smiled again. \"No, my dear Francis. That is between August and October. We should be quite safe. Unless we encounter a devil.\"\n\n\"The Devil?\"\n\n\"No, my dear. _A_ devil. What you may have heard described as a Tasmanian devil.\"\n\n\"I have heard of those,\" said Crozier. \"They're supposed to be terrible creatures with jaws that open as wide as the hatch on a ship's hold. And they're reputed to be ferocious \u2014 insatiable hunters \u2014 able to swallow and devour a horse or Tasmanian tiger whole.\"\n\nSophia nodded, her face serious. \"All true. The devil is all fur and chest and appetite and fury. And if you had ever heard one's noise \u2014 one cannot really call it a bark or growl or roar, but rather the garbled gibbers and snarls one might expect out of a burning asylum \u2014 well, then I guarantee that not even so courageous an explorer as thee, Francis Crozier, would go into the forest or fields here alone at night.\"\n\n\"You've heard them?\" asked Crozier, searching her serious face again to see if she was pulling his leg.\n\n\"Oh, yes. An indescribable noise \u2014 absolutely terrifying. It causes their prey to freeze just long enough for the devil to open those impossibly wide jaws and to swallow its victim whole. The only noise as frightening may be the screams of its prey. I've heard an entire flock of sheep bleating and crying as a single devil devoured them all, one at a time, leaving not so much as a hoof behind.\"\n\n\"You're joking,\" said Crozier, still staring at her intently to see if she was.\n\n\"I never joke about the devil, Francis,\" she said. They were riding into another patch of dark forest.\n\n\"Do your devils eat platypuses?\" asked Crozier. The question was serious, but he was very glad that neither James Ross nor any of his crewmen had been around to hear him ask it. It sounded absurd.\n\n\"A Tasmanian devil will and does eat _anything,_ \" said Sophia. \"But once again, you are in luck, Francis. The devil hunts at night, and unless we get terribly lost, we should have seen the Platypus Pond \u2014 and the platypus \u2014 and had our lunch and returned to Government House before nightfall. God help us if we are out here in the forest come darkness.\"\n\n\"Because of the devil?\" asked Crozier. He'd meant the question to be light and teasing, but he could hear the undercurrent of tension in his own tone.\n\nSophia reined her mare to a halt and smiled at him \u2014 truly, dazzlingly, completely smiled at him. Crozier managed, not gracefully, to get his own gelding stopped.\n\n\"No, my dear,\" said the young woman in a breathy whisper. \"Not because of the devil. Because of my _reputation._ \"\n\nBefore Crozier could think of anything to say, Sophia laughed, spurred her horse, and galloped ahead down the road.\n\nThere was not enough whiskey left in the bottle for two last glassfuls. Crozier poured most of it, held the glass up between him and the flickering oil lamp set on the inner partition wall, and watched the light dance through the amber liquid. He drank slowly.\n\nThey never saw the platypus. Sophia assured him the platypus was almost always to be seen in this pond \u2014 a tiny circle of water not fifty yards across, a quarter mile off the road in a thick forest \u2014 and that the entrances to its burrow were behind some gnarled tree roots that ran down the bank, but he never saw the platypus.\n\nHe did, however, see Sophia Cracroft naked.\n\nThey'd had a pleasant picnic at the more shaded end of the Platypus Pond, an expensive cotton tablecloth spread on the grass to hold the picnic basket, glasses, food containers, and themselves. Sophia had ordered the servants to pack some waterproof cloth-wrapped parcels of roast beef in what was the most expensive of all commodities here but the cheapest from whence Crozier had come \u2014 ice \u2014 to keep it from going bad during the morning's ride. There were broiled potatoes and small bowls of a tasty salad. She'd also packed a very good bottle of Burgundy in with actual crystal glasses from Sir John's crest-etched collection, and she drank more of it than did the captain.\n\nAfter the meal they'd reclined just a few feet apart and talked of this and that for an hour, all the while looking out at the dark surface of the pond.\n\n\"Are we waiting for the platypus, Miss Cracroft?\" asked Crozier during a short gap in their discussion of the dangers and beauties of arctic travel.\n\n\"No, I think it would have shown itself by now if it wanted us to see it,\" said Sophia. \"I've been waiting for an interval before we go bathing.\"\n\nCrozier could only look at her quizzically. He certainly had not brought beach bathing attire. He did not _own_ beach bathing attire. He knew it was another one of her jests, but she always spoke in such evident earnestness that he was never 100 percent sure. It made her puckish sense of humour all the more exciting to him.\n\nExtending her rather titillating joke, she stood, brushed some dead leaves from her dark gaucho pants, and looked around. \"I believe I shall undress behind those shrubs there and enter the water from that grassy shelf. You are invited to join me in the swim, of course, Francis, or not, according to your personal sense of decorum.\"\n\nHe smiled to show her he was a sophisticated gentleman, but his smile was unsteady.\n\nShe walked to the thick bushes without another glance back. Crozier remained on the tablecloth, lying half reclined and with an amused look on his carefully shaven face, but when he saw her white blouse suddenly lifted up by pale arms to be draped across the top of the tall shrub, his expression froze. But his prick did not. Beneath his corduroy trousers and too-short waistcoat, Crozier's private part went from parade rest to top of the mizzen in two seconds.\n\nSophia's dark gaucho pants and other white, frilly unnamed things joined the blouse atop the thick shrub a few seconds later.\n\nCrozier could only stare. His easy smile became a dead man's rictus. He was sure that his eyes were bulging out of his head, but he could not turn away, nor avert his gaze.\n\nSophia Cracroft stepped out into the sunlight.\n\nShe was absolutely naked. Her arms hung easily at her sides; her hands were slightly curled. Her breasts were not large but were very high and very white and tipped with large nipples that were pink, not brown as had been the case with all the other women \u2014 crib doxies, gap-toothed prostitutes, native girls \u2014 whom Crozier had seen naked before this moment.\n\nHad he _ever_ seen another woman truly naked before? A white woman? At this instant he thought not. And if he had, he knew, it mattered not in the least.\n\nThe sunlight reflected off young Sophia's blindingly white skin. She did not cover herself. Still frozen in languid posture and vapid expression, only his penis reacting by becoming even more turgid and aching, Crozier realized that he was astonished that this goddess in his mind, this perfection of English womanhood, the woman he already mentally and emotionally had chosen to be his wife and the mother of his children, had thick, luxurious pubic hair that seemed intent, here and there, on leaping out of its proper black V of an inverted triangle. _Unruly_ was the only word that came to his otherwise empty mind. She had unpinned her long hair and let it fall to her shoulders.\n\n\"Are you coming in, Francis?\" she called softly from where she stood on the grassy shelf. Her tone was as neutral as if she were asking him if he would like a bit more tea. \"Or are you just going to stare?\"\n\nWithout another word she dived into the water in a perfect arc, her pale hands and white arms cleaving the mirrorlike surface an instant before the rest of her.\n\nBy this time Crozier had opened his mouth to speak, but articulate speech was obviously an impossibility. After a moment he closed his mouth.\n\nSophia swam easily back and forth. He could see her white buttocks rising behind her strong, white back, along which her wet hair lay separated like three brushstrokes of the blackest of India inks.\n\nShe raised her head, treading water easily while stopping at the far end of the pond near the large tree she'd pointed out upon her arrival. \"The platypus's burrow is behind these roots,\" she called. \"I don't think it wants to come out and play today. It's shy. Don't you be, Francis. _Please._ \"\n\nAs if in a dream Crozier felt himself rising, walking to the thickest patch of shrubbery he could find close to the water on the opposite side of the pond from where Sophia was. His fingers shook violently as he worked to undo his buttons. He found himself folding his clothing in tight, proper little squares, setting the squares within a larger square on the grass at his feet. He was sure he was taking hours. His throbbing erection would not go away. Will it gone as he would, imagine it away as he might, it persisted in rising rigid to his navel and pitching back and forth there, the glans as red as a signal lantern and extended several taut inches free of its foreskin.\n\nCrozier stood irresolute behind the bush, hearing the splashes as Sophia continued to swim. If he dithered another moment, he knew, she would be climbing out of the pond, be back behind her own curtain of a shrub drying herself off, and he would curse himself for a coward and a fool for the rest of his days.\n\nPeeking through the branches of his shrub, Crozier waited until the lady's back was turned as she swam toward the far shore, and then, with much speed and clumsiness, he threw himself forward into the water, stumbling more than diving, abandoning all grace in his single-minded effort to get his treacherous prick beneath the water and out of sight before Miss Cracroft turned her face his way.\n\nWhen he surfaced, spluttering and blowing, she was treading water twenty feet away and smiling at him.\n\n\"I'm delighted you decided to join me, Francis. Now if the male platypus emerges with his venomous spur, you can protect me. Shall we inspect the burrow entrance?\" She pivoted gracefully and swam toward the huge tree where it overhung the water.\n\nVowing to keep at least ten \u2014 no, fifteen \u2014 feet of open water between them, like a foundering ship surrendering to a lee shore, Crozier dog-paddled after her.\n\nThe pond was surprisingly deep. As he stopped twelve feet from her and treaded water clumsily to keep his head above the surface, Crozier realized that even here at the edge, where roots from the large tree came down five feet of steep bank into the water and tall grasses hung over casting afternoon shadows, Crozier's flailing feet and seeking toes could not at first find purchase on the bottom.\n\nSuddenly Sophia was coming toward him.\n\nShe must have seen the panic in his eyes; he did not know whether to back-paddle furiously or just somehow warn her away from his condition of _prick-rampant,_ because she paused mid-breaststroke \u2014 and he could see her white breasts bobbling beneath the surface \u2014 nodded to her left, and swam easily toward the tree roots.\n\nCrozier followed.\n\nThey hung on to the roots, only about four feet from each other, but the water was blessedly dark below chest level, and Sophia pointed to what might have been a burrow opening, or just a muddy indentation, in the bank between the tangle of tree roots.\n\n\"This is a camping or bachelor burrow, not a nesting burrow,\" said Sophia. She had beautiful shoulders and collarbones.\n\n\"What?\" said Crozier. He was happy \u2014 and mildly amazed \u2014 that his power of speech had returned, but less than satisfied by the odd, strangled sound of the syllable and by the fact that his teeth were chattering. The water was not cold.\n\nSophia smiled. A strand of dark hair was plastered along one of her sharp cheeks. \"Platypuses make two kinds of burrows,\" she said softly, \"this kind \u2014 what some naturalists call a camping burrow \u2014 which both the male and female use except during breeding season. The bachelors live here. The nesting burrow is dug out by the female for the actual breeding, and after that deed is performed, she excavates another small chamber to act as a nursery.\"\n\n\"Oh,\" said Crozier, clinging to the root as tightly as he had ever clung to any ship's line while two hundred feet up in the rigging during a hurricane.\n\n\"Platypuses lay eggs, you know,\" continued Sophia, \"like reptiles. But the mothers secrete milk, like mammals.\"\n\nThrough the water he could see the dark circles in the centres of the white globes of her breasts.\n\n\"Really?\" he said.\n\n\"Aunt Jane, who is something of a naturalist herself, believes that the venomous spurs on the hind legs of the male are used not only to fight other male platypuses and intruders, but to hang on to the female while they are swimming and mating at the same time. Presumably he does not secrete the venom when clinging to his breeding partner.\"\n\n\"Yes?\" said Crozier and wondered if he should have said _No?_ He had no idea what they were talking about.\n\nUsing the tangle of roots, Sophia pulled herself closer, until her breasts were almost touching him. She laid her cool hand \u2014 a surprisingly large hand \u2014 flat against his chest.\n\n\"Miss Cracroft... ,\" he began.\n\n\"Shhhh,\" said Sophia. \"Hush.\"\n\nShe shifted her left hand from the root to his shoulder, hanging from him as she had hung from the tree root. Her right hand slid lower, pressing across his belly, touching his right hip, then coming back to his centre and going lower again.\n\n\"Oh, my,\" she whispered by his ear. Her cheek was against his now, her wet hair in his eyes. \"Is this a venomous spur I've found?\"\n\n\"Miss Cra\u2014... ,\" he began.\n\nShe squeezed. She floated gracefully so that suddenly her strong legs were on either side of his left leg, and then she lowered her weight and warmth, rubbing against him. He raised that leg slightly to buoy her up and keep her face above water. Her eyes were closed. Her hips ground, her breasts flattened against him, and her right hand began to stroke the length of him.\n\nCrozier moaned, but it was only an anticipatory moan, not one of release. Sophia made a soft sound against his neck. He could feel the heat and wetness of her nether regions against his raised leg and thigh. _How can anything be wetter than water?_ he wondered.\n\nThen she moaned in earnest, and Crozier closed his eyes as well \u2014 sorry that he could not continue seeing her but having no choice \u2014 she pressed herself hard against him once, twice, a third downward-pressing time, and her stroking became hurried, urgent, expert, knowing, and demanding.\n\nHe buried his face against her wet hair as he throbbed and pulsed into the water. Crozier thought the pulsing ejaculation might never end, and \u2014 if he had been able \u2014 he would have apologized to her at once. Instead, he moaned again and almost lost his grip on the tree root. They both bobbled, their chins dripping beneath the waterline.\n\nWhat confused Francis Crozier most at that moment \u2014 and everything in the universe confused him right then, while nothing in the universe bothered him \u2014 was the fact of the lady's downward-pressing, her thighs strong around him, her cheek pressed hard against his own while she closed her eyes so tightly, and her own moan. Certainly women could not feel the kind of intensity that men do? Some of the doxies had moaned, but certainly that had been only because they knew the men liked it \u2014 it had been obvious that they felt nothing.\n\nAnd yet...\n\nSophia pulled back, looked into his eyes, smiled easily, kissed him full on the lips, raised her legs into an almost jackknife, kicked off from the roots, and swam for the shore where her clothes lay on the mildly quaking bush.\n\nIncredibly, they dressed, picked up their picnic things, packed the mule, mounted, and rode all the way back to Government House in silence.\n\nIncredibly, that evening during dinner, Sophia Cracroft laughed and chatted with her aunt, Sir John, and even with the unusually loquacious Captain James Clark Ross, while Crozier sat mostly silent and staring at the table. He could only admire her... what did the Frogs call it? \u2014 her _sangfroid,_ while Crozier's attention and soul felt precisely as his body had at the moment of his endless orgasm in the Platypus Pond \u2014 atoms and essence scattered to every corner of the universe.\n\nYet Miss Cracroft did not act aloof toward him nor offer any sense of reproof. She smiled at him, made comments to him, and attempted to include him in the conversation just as she did every evening in Government House. And certainly her smile toward him was a little warmer? More affectionate? Even smitten? It had to be so.\n\nAfter dinner that night, when Crozier suggested a walk in the garden, she begged off, pleading a previous engagement of cards with Captain Ross in the main parlour. Would Commander Crozier care to join them?\n\nNo, Commander Crozier begged off in return, understanding from the warm and easy undertones in her warm and easy surface banter that all must be kept normal in Government House that evening and until the two of them could meet to discuss their future. Commander Crozier announced loudly that he had a bit of a headache and would turn in early.\n\nHe was awake, dressed in his best uniform, and walking the halls of the mansion before dawn the next day, certain that Sophia would have the same impulse of meeting early.\n\nShe did not. Sir John was the first to come to breakfast, and he made endless, insufferable small talk with Crozier, who had never mastered the insipid art of small talk, much less been able to hold up his end of a conversation on what the proper tariff should be on renting prisoners for digging canals.\n\nLady Jane came down next, and even Ross appeared for breakfast before Sophia finally made an appearance. By this time Crozier was on his sixth cup of coffee, which he had learned to prefer over tea in the morning during his winters with Parry in the northern ice years earlier, but he stayed while the lady had her usual eggs, sausage, beans, toast, and tea.\n\nSir John disappeared somewhere. Lady Jane deliquesced. Captain Ross wandered off. Sophia finally finished her breakfast.\n\n\"Would you like to walk in the garden?\" he asked.\n\n\"So early?\" she said. \"It's already very hot out there. This autumn shows no signs of cooling off.\"\n\n\"But... ,\" began Crozier and attempted to communicate the urgency of his invitation with his gaze.\n\nSophia smiled. \"I would be delighted to walk in the garden with you, Francis.\"\n\nThey strolled slowly, interminably, waiting for a single prisoner-gardener to finish his task of unloading heavy bags of fresh fertilizer.\n\nWhen the man was gone, Crozier steered her upwind to the stone bench at the far and shaded end of the long formal garden. He helped her take her seat and waited while she folded her parasol. She looked up at him \u2014 Crozier was too agitated to sit and loomed over her, shifting from foot to foot as he loomed \u2014 and he imagined that he could see the expectation in her eyes.\n\nFinally he had the presence of mind to go to one knee.\n\n\"Miss Cracroft, I am aware that I am a mere commander in Her Majesty's Navy and that you deserve only the attentions of the full Admiral of the Fleet... no, I mean, of royalty, of one who would command a full Admiral... but you must be aware, I know you are aware, of the intensity of my feelings toward you, and if you could see yourself finding reciprocal feelings for...\"\n\n\"Good God, Francis,\" interrupted Sophia, \"you are not going to propose marriage, are you?\"\n\nCrozier had no answer to that. On one knee, both hands clasped and extended toward her as if in prayer, he waited.\n\nShe patted his arm. \"Commander Crozier, you are a wonderful man. A _gentle_ man despite all those rough edges which may never be rounded off. And you are a _wise_ man \u2014 especially in understanding that I shall never be a commander's wife. That would not be fitting. That would never be... _acceptable._ \"\n\nCrozier tried to speak. No words came to mind. That part of his brain still working was trying to complete the endless sentence proposing marriage which he had lain awake all night composing. He had got through almost a third of it \u2014 after a fashion.\n\nSophia laughed softly and shook her head. Her eyes darted, making sure that no one \u2014 not even a prisoner \u2014 was within sight or hearing. \"Please do not be concerned about yesterday, Commander Crozier. We had a wonderful day. The... interlude... at the pond was pleasant for both of us. It was a function of... my nature... as much as a result of mutual feelings of closeness we felt _for those few moments._ But please disabuse yourself, my dear Francis, that there remains upon you any burden or compulsion to act in any way on my behalf because of our brief indiscretion.\"\n\nHe looked at her.\n\nShe smiled, but not with as much warmth as he had become used to. \"It is not,\" she said so softly that the words came through the hot air as slightly more than a firm whisper, \"as if you compromised my honour, Commander.\"\n\n\"Miss Cracroft... ,\" Crozier began again and stopped. If his ship had been in the act of being forced against a lee shore with the pumps out of action and four feet of water in the hold and climbing, the rigging snarled and the sails in tatters, he would have known what orders to give. What to say next. At this moment not a single word came to mind. There was only a rising pain and astonishment in him that hurt all the worse for being a recognition of something old and all too well understood.\n\n\"If I were to marry,\" continued Sophia, opening her parasol again and spinning it above her, \"it would be to our dashing Captain Ross. Although I am not destined to be a mere captain's wife either, Francis. He would have to be knighted... but I am sure he will be soon.\"\n\nCrozier stared into her eyes, searching for some sign of jest. \"Captain Ross is engaged,\" he said finally. His voice sounded like the croak of a man who has been stranded without water for many days on end. \"They plan to marry immediately after James's return to England.\"\n\n\"Oh, pshaw,\" said Sophia, standing now and twirling the parasol more quickly. \"I will be returning to England by swift packet boat myself this summer, even before Uncle John is recalled. Captain James Clark Ross has not seen the last of me.\"\n\nShe looked down at him where he remained, absurdly, still on one knee in the white gravel. \"Besides,\" she said brightly, \"even if Captain Ross marries that young pretendress waiting for him \u2014 he and I have spoken of her often, and I can assure you that she is a fool \u2014 marriage is the end of nothing. It is not death. It is not Hamlet's 'Unknown Country' from which no man returns. Men have been known to return from marriage and find the woman who has been right for them all along. Mark my words on this, Francis.\"\n\nHe stood then, finally. He stood and brushed the white gravel from the knee of his best dress uniform trousers.\n\n\"I must go now,\" said Sophia. \"Aunt Jane, Captain Ross, and I are going into Hobart Town this morning to see some new stallions the Van Diemen Company have just imported for breeding services. Do feel free to come with us if you so choose, Francis, but for heaven's sake change your clothing and your expression before you do.\"\n\nShe touched his forearm lightly and walked back into Government House, twirling her parasol as she went.\n\nCrozier heard the muffled bell on deck ring eight bells. It was 4:00 a.m. Usually, on a ship at sea, the men would be rousted from their hammocks in half an hour to begin holystoning the decks and cleaning everything in sight. But here in the darkness and the ice \u2014 and in the wind, Crozier could hear it still howling in the riggings, meaning another blizzard was probable, and this only the tenth of November of their third winter \u2014 the men were allowed to sleep late, lazing away until four bells in the morning watch. Six a.m. Then the cold ship would come alive with the mates' shouts and the men's finneskoed feet hitting the deck before the mates carried out their threats of cutting their hammocks down with the seamen still in them.\n\nThis was a lazy paradise compared to sea duty. The men not only slept late but were allowed to have their breakfast here on the lower deck at eight bells before having to get on with their morning duties.\n\nCrozier looked at the whiskey bottle and glass. Both were empty. He raised the heavy pistol \u2014 extra heavy with its full charge of powder and ball. His hand could tell.\n\nThen he set the pistol into the pocket of his captain's coat, removed the coat, and hung it on a hook. Crozier wiped out the whiskey glass with the clean cloth that Jopson left every evening for that purpose and set it away in his drawer. Then he carefully set the empty whiskey bottle in the covered wicker basket that Jopson left near the sliding door for just that purpose. A full bottle would be in the basket by the time Crozier returned from his dark day's duties.\n\nFor a moment he had considered getting more fully dressed and going up on deck \u2014 exchanging his finneskoes for real boots, pulling on his comforter, cap, and full slops, and going out into the night and storm to await the rousing of the men, coming down for breakfast with his officers and going the full day with no sleep.\n\nHe had done it many other mornings.\n\nBut not this morning. He was too weary. And it was too cold to stand here for even a minute with only four layers of wool and cotton on. Four a.m., Crozier knew, was the coldest belly of the night and the hour at which the most ill and wounded men gave up the ghost and were carried away into that true Unknown Country.\n\nCrozier crawled under the blankets and sank his face into the freezing horsehair mattress. It would be fifteen minutes or more before his body heat would begin to warm the cradled space. With luck, he'd be asleep before that. With luck he'd get almost two hours of a drunkard's sleep before the next day of darkness and cold began. With luck, he thought as he drifted off, he wouldn't wake at all.\n\nIRVING\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n13 November, 1847\n\nSilence was missing, and it was Third Lieutenant John Irving's job to find her.\n\nThe captain hadn't ordered him to do so... not exactly. But Captain Crozier _had_ told Irving to watch over the Esquimaux woman when the captains decided to keep her aboard HMS _Terror_ six months earlier in June and Captain Crozier never rescinded that order, so Irving felt responsible for her. Besides, the young man was in love with her. He knew it was foolish \u2014 insane even \u2014 to have fallen in love with a savage, a woman who was not even a Christian, and an uneducated native who couldn't speak a word of English, or any language for that matter with her tongue torn out as it was, but Irving was still in love with her. Something about her made the tall, strong John Irving weak in his knees.\n\nAnd now she was gone.\n\nThey first noticed she wasn't in her assigned berth \u2014 that little den set back amid the crates in the cluttered part of the lower deck just forward of the sick bay \u2014 on Thursday, two days earlier, but the men were used to Lady Silence's odd comings and goings. She was off the ship as much as she was on it, even at night. Irving reported to Captain Crozier on Thursday afternoon, the eleventh of November, that Silence had gone missing, but the captain, Irving, and the others had seen her out on the ice two nights before. Then, after the remains of Strong and Evans had been found, she'd gone missing again. The captain said not to worry, that she'd show up.\n\nBut she hadn't.\n\nThe storm had blown in that Thursday morning, bringing heavy snow and high winds. Work teams labouring by lantern light to repair the trail cairns between _Terror_ and _Erebus_ \u2014 four-foot-high tapered columns of ice bricks every thirty paces \u2014 had been forced to return to the ships that afternoon and hadn't been able to work out on the ice since. The last messenger from _Erebus,_ who had arrived late Thursday and been forced to stay on _Terror_ because of the storm, confirmed that Silence was not aboard Commander Fitzjames's ship. By this Saturday morning, watch was being changed on deck every hour and the men still came below crusted with ice and shaking with cold. Work parties had to be sent topside into the gale with axes every three hours to hack the ice off the remaining spars and lines so that the ship would not tip over from the weight. Also, the falling ice was a hazard to those on watch and did damage to the deck itself. More men struggled to shovel snow off the icy deck of the forward-listing _Terror_ before it built up to a depth where they couldn't get the hatches open.\n\nWhen Lieutenant Irving reported again to Captain Crozier on this Saturday night after supper that Silence was still missing, the captain said, \"If she's out in this, she may not be coming back, John. But you have permission to search the entire ship tonight after most of the men are in their hammocks, if only to make sure she's gone.\"\n\nEven though Irving's officer-on-deck watch had ended hours earlier this evening, the lieutenant now got back into his cold-weather slops, lit an oil lantern, and climbed the ladderway to the deck again.\n\nConditions had not improved. If anything they were worse than when Irving had gone below for supper five hours earlier. The wind howled in from the northwest, blowing snow before it and reducing visibility to ten feet or less. Ice had recoated everything even though there was a five-man axe party hacking and shouting somewhere forward of the snow-laden canvas sagging above the hatchway. Irving struggled out through a foot of new spindrift under the canvas pyramid, lantern blowing back toward his face, as he searched for one of the men _not_ swinging an axe in the darkness.\n\nReuben Male, captain of the fo'c'sle, had watch and work-party officer duty, and Irving found him by following the faint glow to the other man's lantern on the port side.\n\nMale was a snow-encrusted mound of wool. Even his face was hidden under a makeshift hood wrapped about by layers of heavy wool comforters. The shotgun in the crook of his bulky arm was sheathed in ice. Both men had to shout to be heard.\n\n\"See anything, Mr. Male?\" shouted Lieutenant Irving, leaning close to the thick turban of wool that was the fo'c'sle captain's head.\n\nThe shorter man tugged down the scarf a bit. His nose was icicle white. \"You mean the ice parties, sir? I can't see 'em once they get above the first spars. I just listen, sir, while I fill in for young Kinnaird's port watch duty. He was on the third watch shoveling party, sir, and still ain't thawed out.\"\n\n\"No, I mean on the ice!\" shouted Irving.\n\nMale laughed. It was, quite literally, a muffled sound. \"None of us have seen as far as the ice for forty-eight hours, Lieutenant. You know that, sir. You was out here earlier.\"\n\nIrving nodded and wrapped his own comforter tighter around his forehead and lower face. \"No one's seen Silence... Lady Silence?\"\n\n\"What, sir?\" Mr. Male leaned closer, the shotgun a column of ice-rimmed metal and wood between them.\n\n\"Lady Silence?\" shouted Irving.\n\n\"No, sir. I understand that no one's seen the Esquimaux woman for days. She must be gone, Lieutenant. Dead out there somewhere, and good riddance, I say.\"\n\nIrving nodded, patted Male on his bulky shoulder with his own bulky mitten, and made his way around by the stern \u2014 staying away from the mainmast, where giant chunks of ice were falling out of the blowing snow and crashing like artillery shells onto the deck \u2014 to speak to John Bates where the man stood watch on the starboard side.\n\nBates had seen nothing. He hadn't even been able to see the five men of the axe party as they set to work.\n\n\"Beggin' your pardon, sir, but I don't have no watch and I'm afraid I won't hear the bell what with all this choppin' and fallin' and the wind blowin' and crashin' of ice, sir. Is there much time left on this watch?\"\n\n\"You'll hear the bell when Mr. Male rings it,\" shouted Irving, leaning close to the ice-shrouded globe of wool that was the twenty-six-year-old's head. \"And he'll come around to check on you before going below. As you were, Bates.\"\n\n\"Aye, sir.\"\n\nThe wind tried to knock Irving off his feet as he went around to the front of the canvas cover, waited for a break in the falling ice \u2014 hearing the men curse and shout in the main-trees and thrumming rigging above \u2014 and then he hurried as quickly as he could through the two feet of new snow on deck, ducked under the frozen canvas, and clambered through the hatch and down the ladderway.\n\nHe'd searched the lower decks multiple times, of course \u2014 especially behind the remaining crates forward of the sick bay where the woman had previously had her little den \u2014 but now Irving walked aft. The ship was quiet this late at night except for the stamping and crash of ice on the deck above, the snores from the exhausted men in their hammocks forward, Mr. Diggle's usual bangs and curses from the direction of the stove, and the ever-present howl of wind and grind of ice.\n\nIrving felt his way along the dark and narrow companionway. Except for Mr. Male's room, none of the sleeping cubicles here in officers' country were empty. HMS _Terror_ had been lucky in that respect. While _Erebus_ had lost several officers to that thing on the ice, including Sir John and Lieutenant Gore, none of _Terror_ 's officers, warrant officers, or petty officers had died yet except for young John Torrington, the lead stoker, who'd died of natural causes a year and a half earlier back at Beechey Island.\n\nNo one was in the Great Cabin. It was rarely warm enough to tarry long there now and even the leatherbound books looked cold on their shelves; the wooden instrument that played metal musical disks when cranked was silent these days. Irving had time to notice that Captain Crozier's lamp was still lit behind his partition before the lieutenant pushed forward through the officers' and mates' empty mess rooms and back to the ladderway.\n\nThe orlop deck below was, as it always was, very cold and very dark. With fewer provision-carrying parties coming down here because of the severe rationing due to the many spoiled cans of food the surgeons had discovered, and with fewer coal-sack hauling parties because of the dwindling coal supplies and reduced hours of heating for the ship, Irving found himself alone in the frigid space. The black wood beams and frost-covered metal brackets groaned around him as he made his way forward before working his way back toward the stern. The lamplight seemed to be swallowed up by the thick darkness, and Irving had trouble seeing the faint glow through the fog of ice crystals created by his own breathing.\n\nLady Silence was not in the bow area \u2014 not in the carpenter's storeroom or the bosun's storeroom nor in the almost-empty Bread Room aft of these closed compartments. The midship section of the orlop deck had been crammed deck to ceiling with crates, barrels, and other packages of supplies when _Terror_ had sailed, but now much of the deck-space was clear. Lady Silence was nowhere amidships.\n\nLieutenant Irving let himself into the Spirit Room, using the key Captain Crozier had loaned him. There were brandy and wine bottles left, he could see by the glow of the dimming lantern, but he knew that the level of rum was low in the huge main cask. When the rum ran out \u2014 when the men's daily noon supply of grog disappeared \u2014 then, Lieutenant Irving knew, as all officers in the Royal Navy knew, mutiny would become a much more serious concern. Mr. Helpman, the captain's clerk, and Mr. Goddard, the captain of the hold, had reported recently that they estimated another six weeks or so of rum remained, and that much only if the standard one-fourth pint of rum in the gill, diluted with three-fourths pint of water, was reduced by half. The men would grumble even then.\n\nIrving did not think Lady Silence could have sneaked into the locked Spirit Room despite all the whispering of the men about her witchlike powers, but he searched the space carefully, peering under tabletops and counters. The row upon row of cutlasses, sword bayonets, and muskets on the shelves above him glittered coldly in the lantern light.\n\nHe went aft to the Gunner's Storeroom, with its adequate remaining supplies of powder and shot, peered into the captain's private storeroom \u2014 only Crozier's few remaining whiskey bottles sat on the shelves, the food having been parceled out to the other officers in recent weeks. Then he searched the Sail Room, Slop Room, aft cable lockers, and mate's storeroom. If Lieutenant John Irving had been an Esquimaux woman attempting to hide aboard the ship, he thought he might have chosen the Sail Room, with its mostly untouched heaps and rolls of spare canvas, sheets, and long-unused sailing gear.\n\nBut she was not there. Irving had a start in the Slop Room when his lantern showed a tall, silent figure standing in the rear of the room, shoulders looming against a dark bulkhead, but it turned out to be only some wool greatcoats and a Welsh wig hanging on a peg.\n\nLocking doors behind him, the lieutenant went down the ladder to the hold.\n\nThird Lieutenant John Irving, although appearing younger than his years because of his boyish blond looks and quick blush, was not in love with the Esquimaux woman because he was a lovesick virgin. Actually, Irving had had more experience with the fairer sex than many of those braggarts on the ship who filled the fo'c'sle with tales of their sexual conquests. Irving's uncle had brought him down to the Bristol docks when the boy turned fourteen, introduced him to a clean and pleasant dockside whore named Mol, and paid for the experience \u2014 not merely a quick back-alley knee-wobbler, but a proper evening and night and morning in a clean room under the eaves of an old inn overlooking the quay. It had given young John Irving a taste for the physical which he had indulged many times since.\n\nNor had Irving had less luck with the ladies in polite society. He had courted the youngest daughter of Bristol's third most important family, the Dunwitt-Harrisons, and that lass, Emily, had allowed, even initiated, personal intimacies most young men would have sold their left bollock to have experienced at such an age. Upon arriving in London to complete his Naval education in artillery on the gunnery training vessel HMS _Excellent,_ Irving had spent his weekends meeting, courting, and enjoying the company of several attractive upper-class young ladies, including the obliging Miss Sarah, the shy but ultimately surprising Miss Linda, and the truly shocking \u2014 in private \u2014 Miss Abigail Elisabeth Lindstrom Hyde-Berrie, with whom the fresh-faced third lieutenant soon found himself engaged to be married.\n\nJohn Irving had no intention of being married. At least not while he was in his twenties \u2014 his father and uncle had both taught him that these were the years in which he should see the world and sow wild oats \u2014 and most probably not when he was in his thirties. He saw no compelling reason to marry while he would be in his forties. So although Irving had never once considered the Discovery Service \u2014 he had never enjoyed cold weather, and the thought of being frozen in at either of the poles was both absurd and appalling to him \u2014 the week after he awoke to find himself engaged, the third lieutenant followed the promptings of his older chums George Hodgson and Fred Hornby and went along to an interview on HMS _Terror_ to apply for transfer.\n\nCaptain Crozier, obviously in foul spirits and hungover that beautiful spring Saturday morning, had glowered, harrumphed, scowled, and quizzed them carefully. He laughed at their gunnery training on a mastless ship and demanded to know just how they could be of service on an expedition sailing ship which carried only small arms. Then he asked them pointedly if they would \"do their duty as Englishmen\" (whatever on earth that meant, Irving remembered thinking, when said Englishmen were locked into a frozen sea a thousand miles from home) and promptly assured them of berths.\n\nMiss Abigail Elisabeth Lindstrom Hyde-Berrie was distraught, of course, and shocked that their engagement should be extended over months or actual years, but Lieutenant Irving consoled her first with assurances that the extra money from the Discovery Service duty would be an absolute necessity for them, and then by explaining his need for the adventure and then fame and glory that might well come from writing a book upon his return. Her family understood these priorities even if Miss Abigail did not. Then, when they were alone, he coaxed her out of her tears and anger with hugs, kisses, and expert caresses. The consolation grew to interesting heights \u2014 Lieutenant Irving knew that he might well be a father by now, two and a half years after the consoling. But he had not been unhappy to wave good-bye to Miss Abigail some weeks later as _Terror_ slipped her moorings and was pushed away by two steam tugs. The disconsolate young lady stood on the docks at Greenhithe in her green-and-pink silk dress under a pink parasol and waved her matching silk pink handkerchief, using another less-expensive cotton handkerchief to dry her copious tears.\n\nHe knew that Sir John fully expected to stop in both Russia and China after negotiating the North-West Passage, so Lieutenant Irving had already made plans to transfer to a Royal Navy ship assigned to one of those waters, or perhaps even resign from the Navy, write his adventure book, and look after his uncle's silk and millinery interests in Shanghai.\n\nThe hold was darker and colder than the orlop deck.\n\nIrving hated the hold. It reminded him, even more than did his freezing berth or the dimly lit, freezing lower deck, of a grave. He only came down here when he had to, mostly to supervise the stowing of shrouded dead bodies \u2014 or the parts of dead bodies \u2014 in the locked Dead Room. Each time he wondered if someone soon would be supervising the stowing of his corpse down here. He lifted his lantern and headed aft through the slush-melt and thick air.\n\nThe boiler room appeared to be empty, but then Lieutenant Irving saw the body on the cot near the starboard bulkhead. No lantern glowed, only the low red flicker through the grate of one of the four closed boiler doors, and in the dim light, the long body stretched out on the cot looked dead. The man's open eyes stared up at the low ceiling and he did not blink. Nor did he turn his head when Irving came into the room and hung his lantern on a hook near the coal scuttle.\n\n\"What brings you down here, Lieutenant?\" asked James Thompson. The engineer still neither moved his head nor blinked. Sometime in the past month he had quit shaving, and whiskers now sprouted everywhere on his thin, white face. The man's eyes lay deep in dark sockets. His hair was wild and spiky with soot and sweat. It was near freezing even here in the boiler room with the fires damped so low, but Thompson was lying only in his trousers, undershirt, and suspenders.\n\n\"I'm looking for Silence,\" said Irving.\n\nThe man on the cot continued to stare at the deck above him.\n\n\"Lady Silence,\" clarified the young lieutenant.\n\n\"The Esquimaux witch,\" said the engineer.\n\nIrving cleared his throat. The coal dust was so thick here that it was hard to breathe. \"Have you seen her, Mr. Thompson? Or heard anything unusual?\"\n\nThompson, who still had not blinked or turned his head, laughed softly. The sound was disturbing \u2014 a rattle of small stones in a jar \u2014 and it ended in a cough. \"Listen,\" said the engineer.\n\nIrving turned his head. There were only the usual noises, although louder down here in the dark hold: the slow moan of ice pressing in, the louder groaning of the iron tanks and structural reinforcements fore and aft of the boiler room, the more distant moan of the blizzard winds far above, the crash of falling ice carried down as vibration through the ship's timbers, the thrum of the masts being shaken in their sockets, random scratching noises from the hull, and a constant hiss, screech, and claw-sliding noise from the boiler and pipes all around.\n\n\"There's someone or something else breathing on this deck,\" continued Thompson. \"Do you hear it?\"\n\nIrving strained but heard no breathing, although the boiler sounded like something large panting hard. \"Where are Smith and Johnson?\" asked the lieutenant. These were the two stokers who worked round-the-clock here with Thompson.\n\nThe supine engineer shrugged. \"With so little coal to shovel these days, I need them only a few hours a day. I spend most of my time alone, crawling among the pipes and valves, Lieutenant. Patching. Taping. Replacing. Trying to keep this... _thing_... working, moving hot water through the lower deck for a few hours each day. In two months, three at the most, it will all be academic. We already have no coal to steam. We'll soon have no coal to heat.\"\n\nIrving had heard these reports in the officers' mess but had little interest in the subject. Three months seemed a lifetime away. Right now he had to make sure Silence was not on board and report to the captain. Then he had to try to find her if she was not aboard _Terror._ Then he had to survive another three months. He would worry about shortages of coal later.\n\n\"Have you heard the rumors, Lieutenant?\" asked the engineer. The long form on the couch still had neither blinked nor turned his head to look at Irving.\n\n\"No, Mr. Thompson, which rumor?\"\n\n\"That the... _thing_ on the ice, the apparition, the Devil... comes into the ship whenever it wants to and walks the hold deck late at night,\" said Thompson.\n\n\"No,\" said Lieutenant Irving. \"I've not heard that.\"\n\n\"Stay down here alone on the hold deck through enough watches,\" said the man on the cot, \"and you'll hear and see everything.\"\n\n\"Good night, Mr. Thompson.\" Irving took his sputtering lantern and went back out into the companionway and forward.\n\nThere were few places left to search on the hold deck and Irving had every intention of making a fast job of it. The Dead Room was locked; the lieutenant had not asked his captain for the key, and after inspecting that the heavy lock was solid and secured, he moved on. He didn't want to see what was causing the scrabbling and chewing sounds he could hear through the thick oak door.\n\nThe twenty-one huge iron water tanks along the hull offered no place for an Esquimaux to hide, so Irving went into the coal bunkers, his lantern dimming in the thick, coal-dust-blackened air. The remaining sacks of coal, once filling each bin and stacked from hull bottom to the deck beams above, merely lined the edge of each sooty room like low barriers of sandbags now. He couldn't imagine Lady Silence making a new shelter in one of these lightless, reeking, pestilential hellholes \u2014 the decks were awash in sewage and rats scuttled everywhere \u2014 but he had to look.\n\nWhen he was finished searching the coal storage lockers and the stores amidships, Lieutenant Irving moved out into the remaining crates and barrels in the forepeak, directly below the crew's berthing area and Mr. Diggle's huge stove two decks above. A narrower ladder came down through the orlop deck to this stores area and tons of lumber were hanging from the heavy beams overhead, turning the space into a maze and requiring the lieutenant to proceed in a half crouch, but there were far fewer crates, barrels, and heaps of goods than there had been two and a half years earlier.\n\nBut more rats. Many more.\n\nSearching between and in some of the larger crates, glancing to make sure that the barrels awash in the slush were either empty or sealed, Irving had just stepped around the vertical forward ladder when he saw a flash of white and heard harsh breathing, gasps and caught a rustle of frenzied movement just beyond the dim circle of the lantern light. It was large, moving, and was not the woman.\n\nIrving had no weapon. For the briefest instant he considered dropping the lantern and running back through the darkness toward the midship ladderway. He did not, of course, and the thought was gone almost before it was formed. He took a step forward and, in a voice stronger and more authoritative than he thought he might be capable of right then, shouted, \"Who goes there? Identify yourself!\"\n\nThen he saw them in the lantern light. The idiot, Magnus Manson, the largest man on the expedition, struggling back into his trousers, his huge, grimy fingers fumbling with buttons. A few feet away Cornelius Hickey, the caulker's mate, barely five feet tall, beady-eyed and ferret-faced, was pulling his suspenders into place.\n\nJohn Irving's mouth hung open. It took several seconds for the reality of what he was looking at to filter through his mind toward acceptance \u2014 _sodomites._ He had heard of such goings-on, of course, had joked with his mates about such things, had once witnessed a flogging around the fleet when an ensign on the _Excellent_ had confessed to such doings, but Irving had never thought that he would be on a ship where... to serve with men who...\n\nManson, the giant, was taking an ominous step toward him. The man was so large that everywhere belowdecks he had to walk in a stooped crouch to avoid the beams, giving him an habitual hunchbacked shuffle that he used even in the open air. Now, his huge hands glowing in the lamplight, he looked like an executioner advancing on a condemned man.\n\n\"Magnus,\" said Hickey. \"No.\"\n\nIrving's jaw dropped farther. Were these... _sodomites_... threatening him? The prescribed sentence for sodomy on a ship of Her Majesty's Navy was hanging, with two hundred lashes from the cat while being flogged around the fleet \u2014 literally from ship to ship in harbour \u2014 being considered great leniency.\n\n\"How dare you?\" said Irving, although whether he was talking about Manson's threatening attitude or their unnatural act, even he did not know.\n\n\"Lieutenant,\" said Hickey, words rushing in that flute-high rush of the caulker's mate's Liverpool accent, \"begging your pardon, sir, Mr. Diggle sent us down for some flour, sir. One of them damn rats rushed up Seaman Manson's trouser leg, sir, and we was trying to set it right. Filthy buggers, them rats.\"\n\nIrving knew that Mr. Diggle had not yet started the late-night baking of biscuits and that there was ample flour in the cook's stores up on the lower deck. Hickey was not even trying to make his lie convincing. The little man's beady, evaluating eyes reminded Irving of the rats scurrying in the darkness around them.\n\n\"We'd appreciate it if you wouldn't tell no one, sir,\" continued the caulker's mate. \"Magnus here would hate to be made fun of for bein' afraid of a little rat runnin' up his leg.\"\n\nThe words were a challenge and a defiance. Almost a command. Insolence came off the little man in waves while Manson stood there empty-eyed, as dumb as a beast of burden, huge hands still flexed, passively awaiting the next command from his diminutive lover.\n\nThe silence between the men stretched. Ice moaned against the ship. Timbers creaked. Rats scurried close by.\n\n\"Get out of here,\" Irving said at last. _\"Now.\"_\n\n\"Aye, sir. Thank you, sir,\" said Hickey. He unshielded a small lantern that had been on the deck near him. \"Come, Magnus.\"\n\nThe two men scrambled up the narrow forward ladder into the darkness of the orlop deck.\n\nLieutenant Irving stood where he was for several long minutes, listening to but not hearing the groans and snaps of the ship. The blizzard howl was like a distant dirge.\n\nIf he reported this to Captain Crozier, there would be a trial. Manson, the village idiot of this expedition, was well liked by the crew, however much they teased him about his fear of ghosts and goblins. The man did the heavy work of any three of his comrades. Hickey, while not especially liked by any of the other warrant or regular officers, was respected by the regular seamen for his abilities to get his friends extra tobacco, an extra gill of rum, or an article of needed clothing.\n\nCrozier wouldn't hang either man, thought John Irving, but the captain had been in an especially foul mood in recent weeks and the punishments could be dramatic. Everyone on the ship knew that just weeks ago the captain threatened to lock Manson into the Dead Room with his chum Walker's rat-chewed corpse if the huge idiot ever again refused an order to carry coal on the hold deck. No one would be surprised if he carried out that sentence now.\n\nOn the other hand, thought the lieutenant, what had he just seen? What could he testify to, his hand on the Holy Bible, if there were an actual court of inquiry? He hadn't _seen_ any unnatural act. He'd not caught the two sodomites in the act of copulation or... any other unnatural posture. Irving had heard the breathing, the gasps, something that must have been whispered alarm at the approach of his lantern, and then seen the two struggling to raise their trousers and tuck in their shirts.\n\nThat would be enough to get one or both of them hanged under normal circumstances. But here, stuck in the ice, with months or years ahead of them before any chance of rescue?\n\nFor the first time in many years, John Irving felt like sitting down and weeping. His life had just become complex beyond all his imagining. If he did report the two sodomites, none of his crewmates \u2014 officers, friends, subordinates \u2014 would ever look at him quite the same way again.\n\nIf he did _not_ report the two men, he would open himself to endless insolence from Hickey. His cowardice in not reporting the man would expose Irving to a form of blackmail for weeks and months to come. Nor would the lieutenant ever sleep well again or feel comfortable on watch in the darkness outside or in his cubicle \u2014 as comfortable as anyone could be with that monstrous white thing killing them all one by one \u2014 waiting, as he would be now, for Manson's white hands to close around his throat.\n\n\"Oh, bugger me,\" Irving said aloud into the creaking cold of the hold. Realizing exactly what he had said, he laughed aloud \u2014 the laugh sounding stranger, weaker, yet more ominous than the words.\n\nHaving looked everywhere except a few hogshead barrels and the forward cable locker, he was ready to give up his search, but he didn't want to go up to the lower deck until Hickey and Manson were out of sight.\n\nIrving made his way past floating empty crates \u2014 the water was above his ankles here, this far toward the downward-tilting bow, and his soaked boots broke through the scrim of ice. Another few minutes and he'd have frostbitten toes for sure.\n\nThe cable locker was at the most forward part of the forepeak, right where the hull came together at the bow. It was not really a room \u2014 the two doors were only three feet tall and the space within not much more than four feet high \u2014 but rather a place to stow the heavy hawsers used for the bow anchors. The cable locker always stank to high heaven of river and estuary mud, even months or years after a ship's weighing anchor from that place. It never fully lost its stench, and the massive hawsers, coiled and overlapping, left little or no free room in the low, black, evil-smelling space.\n\nLieutenant Irving pried open the reluctant doors to the locker and held his lantern to the opening. The grinding of the ice was especially loud here where the bow and bowsprit were pressed into the shifting pack ice itself.\n\nLady Silence's head shot up and her dark eyes reflected the light like a cat's.\n\nShe was naked except for white-brown furs spread under her like a rug and another heavy fur \u2014 perhaps her parka \u2014 draped over her shoulders and naked body.\n\nThe floor of the cable locker was raised more than a foot above the flooded deck outside. She had moulded and shoved the massive hawsers aside until the space opened made a low, fur-lined cave within the overhanging tangle of huge hemp ropes. A small food can filled with oil or blubber provided light and heat from an open flame. The Esquimaux woman was in the process of eating a haunch of red, raw, bloody meat. She was slicing directly from the meat to her mouth with quick cuts from a short but obviously very sharp knife. The knife had a bone or antler handle with some sort of design on it. Lady Silence was on her knees, leaning forward over the flame and the meat, and her small breasts hung down in a way that reminded the literate Lieutenant Irving of pictures he had seen of the statue of a she-wolf nursing the infants Romulus and Remus.\n\n\"I'm terribly sorry, madam,\" said Irving. He touched his cap and shut the doors.\n\nStaggering back a few steps in the slush, sending rats scurrying, the lieutenant tried to think through shock for the second time in five minutes.\n\nThe captain had to know about Silence's hiding place. The fire danger alone from the open flame there would have to be dealt with.\n\nBut where had she got the knife? It looked like something made by Esquimauxs, rather than a weapon or tool from the ship. Certainly they had searched her six months ago in June. Could she have hidden it all this time?\n\nWhat else could she be hiding?\n\nAnd the fresh meat.\n\nThere was no fresh meat on board, Irving was sure of that.\n\nCould she have been hunting? In the winter and blizzard and dark? And hunting what?\n\nThe only things out there on the ice or under the ice were the white bears and the thing stalking the men of _Erebus_ and _Terror._\n\nJohn Irving had a terrible thought. For a second he was tempted to go back and test the lock to the Dead Room.\n\nThen he had an even more terrible thought.\n\nOnly half of William Strong and Thomas Evans had been found.\n\nLieutenant John Irving stumbled aft, feet sliding on the ice and slush as he fumbled and felt his way toward the central ladderway to fight his way up and out toward the light of the lower deck.\n\nGOODSIR\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n20 November, 1847\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Saturday, 20 November, 1847 \u2014_\n\n _We do not have enough food to survive another Winter and Summer here in the ice._\n\n _We should have had. Sir John had Provisioned the two ships for Three Years at extraordinarily full rations for all hands, Five Years for reduced but still adequate rations for men doing heavy work each day, and Seven Years with serious rationing but adequate for all men. By Sir John's Calculations \u2014 and those of his ships' captains, Crozier and Fitzjames \u2014 HMSs_ Erebus _and_ Terror _should have been amply provisioned until the year 1852._\n\n _Instead, we shall be running out of our last edible supplies sometime next spring. And should we all perish because of this, the Reason is Murder._\n\n _Dr. McDonald on_ Terror _had been suspicious of the canned food supplies for some Time, and he shared his concerns with me after the Death of Sir John. Then the problem with spoiled and Poisonous canned foods on our First Outing to King William Land last Summer \u2014 tins removed from a deeper part of Stores beneath Deck \u2014 confirmed the problem. In October, the four of us Surgeons petitioned Captain Crozier and Commander Fitzjames to allow us to take a Full Inventory. Then the Four of us \u2014 aided by crewmen assigned to help us move the hundreds upon hundreds of crates, barrels, and heavy cans in both lower decks, orlop decks, and holds, and to open and test selected samplings \u2014 had done the Inventory twice so as not to make a mistake._\n\n _More than Half the canned Food aboard_ both ships _is worthless._\n\n _We reported this three weeks ago to both captains in Sir John's large and freezing former cabin. Fitzjames, while nominally still a mere Commander, is called \"Captain\" by Crozier, the new Expedition Leader, and others follow suit. In the secret meeting were the four of us surgeons, Fitzjames, and Crozier._\n\n _Captain Crozier \u2014 I have to remember that he is Irish after all \u2014 flew into a rage the likes of which I have never seen. He demanded a full explanation, as if We Surgeons had been responsible for the Stores and Victuals on the Franklin Expedition. Fitzjames, on the other hand, had always had his doubts about the canned goods and the victualler who had canned them \u2014 the only member of the Expedition or the Admiralty who seems to have expressed such reservations \u2014 but Crozier remained incredulous that such an act of criminal fraud could have been carried out on ships of the Royal Navy._\n\n _John Peddie, Crozier's chief surgeon on_ Terror, _has seen the most Sea Duty of any of us four Medical Officers, but most of it had been aboard HMS_ Mary _\u2014 along with Crozier's boatswain John Lane \u2014 and that was in the Mediterranean, where very little of the Ship's Stores had consisted of Canned Goods. Similarly, my nominal superior on_ Erebus, _Chief Surgeon Stephen Stanley, had little experience with such Great Quantities of Canned Provisions aboard ship. Sensitive to the Various Diets thought Necessary to prevent scurvy, Dr. Stanley was shocked into speechlessness when our Inventory suggested via sampling that almost half the remaining tins of food, vegetables, meat, and soup may well be Contaminated or otherwise Ruined._\n\n _Only Dr. McDonald, who had worked with Mr. Helpman \u2014 Captain Crozier's Clerk in Charge \u2014 during the provisioning, had his Theories._\n\n _As I recorded some Months ago in this Journal, besides the 10,000 cases of preserved cooked meats aboard_ Erebus, _our tinned rations include boiled and roast mutton, veal, a wide variety of vegetables including potatoes, carrots, and parsnips, various types of Soups, and 9,450 pounds of Chocolate._\n\n _Alex McDonald had been our Expedition's medical liaison with the Captain Superintendent of the Deptford Victualling Yard and with a certain Mr. Stephan Goldner, our Expedition's Victualling Contractor. McDonald reminded Captain Crozier in October that four contractors had put in bids to provide Canned Ship's Stores for Sir John's expedition \u2014 the firms of Hogarth, Gamble, Cooper & Aves, and that of the aforementioned Mr. Goldner. Dr. McDonald reminded the Captain \u2014 and astonished the rest of us \u2014 by reporting that Goldner's bid was only_ half _that of the other three (Much Better Known) victuallers. In addition, while the other contractors set a schedule of delivering the food in a month or three weeks, Goldner had promised immediate delivery (with the crating and drayage thrown in for no charge). Such immediate delivery was impossible, of course, and Goldner's bid would have required him to lose a fortune if the food had been the quality advertised and cooked and prepared in the ways advertised, but no one except Captain Fitzjames appears to have taken notice of this._\n\n _The Admiralty and the three Commissioners of the Discovery Service \u2014 everyone involved in the selection except for the experienced Comptroller of the Deptford Victualling Yard \u2014 immediately recommended accepting Goldner's offer at a full-payment value, or more than 3,800 pounds. (A fortune for any man, but especially for the foreigner that McDonald explained Goldner to be. The man's only canning factory, Alex said, was in Golatz, Molavia.) Goldner was given one of the largest consignments in the history of the Admiralty \u2014 9,500 cans of meats and vegetables in sizes weighing one to eight pounds, as well as 20,000 cans of soup._\n\n _McDonald had brought one of Goldner's handbills \u2014 Fitzjames recognized it at once \u2014 and looking over it made my mouth water: seven kinds of mutton, fourteen preparations of veal, thirteen kinds of beef, four varieties of lamb. There were listings for jugged hare, ptarmigan, rabbit (in onion sauce or curried), pheasant, and half a dozen other varieties of game. If the Discovery Service wished to have seafood, Goldner had offered to provide canned lobsters in the shell, cod, West Indian turtle, salmon steaks, and Yarmouth bloaters. For fine dining \u2014 at only fifteen pence \u2014 Goldner's handbill offered truffled pheasant, calf's tongue sauce piquant, and beef \u00e0 la Flamande._\n\nIn reality, _said Dr. McDonald,_ we're used to receiving salt horse in a harness cask.\n\n _I had been at Sea long enough to recognize the terms \u2014 horse flesh substituted for beef until the sailors called the barrels a harness cask. But they ate the salted meat readily enough._\n\nGoldner cheated us much worse than that, _continued McDonald in front of a livid Captain Crozier and an angrily nodding Commander Fitzjames._ He substituted cheap foods under labels that sold for much more on the handbill \u2014 regular \"Stewed Beef\" under a label reading \"Stewed Rump Steaks,\" for instance. The former is listed at nine pence, but he charged fourteen pence by changing the label.\n\nGood God, man, _exploded Crozier,_ every victualler does that to the Admiralty. Cheating the Navy is as old as Adam's foreskin. That can't explain why we're suddenly almost out of food.\n\nNo, Captain, _continued McDonald._ It's the cooking and soldering.\n\nThe what? _demanded the Irishman, obviously trying to keep his Temper in check. Crozier's face was crimson and white under his battered cap._\n\nThe cooking and soldering, _said Alex._ As to the cooking, Mr. Goldner bragged of a patented process in which he adds a large dose of nitrate of soda \u2014 calcium chloride \u2014 into the huge vats of boiling water to increase the processing temperature... primarily to speed production.\n\nWhat's wrong with that? _demanded Crozier._ The cans were overdue as it was. Something needed to be done to build a fire under Goldner's arse. His patented process hurried things up.\n\nYes, Captain, _said Dr. McDonald,_ but the fire under Goldner's arse was hotter than the fire under these meats, vegetables, and other foods that were hurriedly cooked before canning. Many of us in the medical field believe that proper cooking of food rids it of Noxious Influences that can cause disease \u2014 but I myself witnessed Goldner's cooking processes and he simply did not cook the meat, vegetables, and soups long enough.\n\nWhy didn't you report this to the Discovery Service Commissioners? _snapped Crozier._\n\nHe did, _Captain Fitzjames said tiredly._ So did I. But the only one who listened was the Comptroller of the Deptford Yard Victualling Service, and he had no vote on the final commission.\n\nSo you're saying that more than half our food has gone bad in the last three years because of poor cooking methods? _Crozier's Countenance continued to be a Mottle of crimson and white._\n\nYes, _said Alex McDonald,_ but equally to blame, we think, is the soldering.\n\nThe soldering of the cans? _asked Fitzjames. His doubts about Goldner evidently had not extended to this technicality._\n\nYes, Commander, _said_ Terror _'s_ _assistant surgeon._ Preserving food in tins is a recent innovation \u2014 an amazing part of our Modern Age \u2014 but we know enough in the past few years of its use to know that proper soldering the flange along the seams of the cylindrical body of the can is important if the foodstuffs within are not to turn Putrid.\n\nAnd Goldner's people did not properly solder these tins? _asked Crozier. His voice was a low, menacing growl._\n\nNot in about sixty percent of the tins we've inspected, _said McDonald._ The gaps in the careless soldering have resulted in incomplete seams. The incomplete seams appear to have accelerated the putrefaction of our tinned beef, veal, vegetables, soups, and other foodstuffs.\n\nHow? _asked Captain Crozier. He was shaking his broad head like a man who has been stunned by a physical blow._ We've been in polar waters since shortly after the two ships left England. I thought it was cold enough up here to preserve anything until Judgement Day.\n\nApparently not, _said McDonald._ Many of Goldner's remaining twenty-nine thousand cans of food have ruptured. Others are already swelling from the gases caused by internal putrefaction. Perhaps some Noxious vapours entered the tins in England. Perhaps there is some microscopic animalcule of which Medicine and Science are not yet aware which invaded the tins during transit or even at Goldner's victualling factory.\n\n _Crozier frowned more deeply._ Animalcule? Let's avoid the fantastic here, Mr. McDonald.\n\n _The assistant surgeon only shrugged._ Perhaps it is fantastical, Captain. But you have not spent the hundreds of hours straining at the eyepiece of a microscope such as I have. We have little understanding of what any of these animalcules are, but I assure you that if you saw how many are present in a simple drop of drinking water, you would be deeply sobered.\n\n _Crozier's coloration had calmed somewhat, but he blushed again at the comment which might have been a reflection of his frequently less-than-sober state._ All right. Some of the food is ruined, _he said brusquely._ What can we do to make sure the rest is safe for the men's consumption?\n\n _I cleared my throat._ As you know, Captain, the men's summer diet included a daily ration of one and a quarter pounds of salt meat with vegetables consisting of only one pint of peas and three-quarters pound of barley a week. But they received their daily bread and biscuits. When we went into winter quarters, the flour ration was cut back twenty-five percent on the baking of bread so as to conserve coal. If we could just begin cooking the remaining canned food rations longer and renewing the baking of bread, it would help not only in preventing the fouled meats in the canned goods from threatening our health but also help in the prevention of scurvy.\n\nImpossible, _snapped Crozier._ We barely have enough coal left to heat both ships until April as it is. If you doubt me, ask Engineer Gregory or Engineer Thompson here on _Terror_.\n\nI don't doubt you, Captain, _I said sadly._ I've spoken to both engineers. But without a resumption of extended cooking of the remaining canned goods, our chances of being poisoned are very high. All we can do is throw out the obviously ruined canned foods and avoid the many poorly soldered cans. This cuts down on our remaining stores most dramatically.\n\nWhat about the ether stoves? _asked Fitzjames, brightening a bit._ We could use the camping stoves to heat the tinned soups and other doubtful provisions.\n\n _It was McDonald who shook his head._ We tested that, Commander. Dr. Goodsir and I experimented with heating some of the canned so-called Beef Stew on the patented Cooking Apparatus spirit stoves. The pint-sized bottles of ether do not last long enough to thoroughly heat the food and the temperatures are low. Also, our sledge parties \u2014 or all of us, should we be forced to Abandon Ship \u2014 will be dependent upon the spirit stoves to melt snow and ice for drinking water once we are on the ice. We should preserve the ether spirits.\n\nI was with Lieutenant Gore on our first sledge party to King William Land \u2014 and we used the spirit stoves daily, _I added softly._ The men used just enough ether and flame to get the canned soups to bubble a bit before digging in ravenously. The food was barely tepid.\n\n _There was a long silence._\n\nYou report that over half the canned food we had been counting on to get through the next year or two \u2014 if necessary \u2014 is ruined, _Crozier said at last._ We don't have the coal to recook this food on either _Erebus_ 's __or _Terror_ ' _s_ large Frazer's Patent Stoves, nor on the smaller whaleboat iron stoves, and you tell me that there is insufficient fuel to use the ether spirit stoves. What _can_ we do?\n\n _All five of us \u2014 the Four Surgeons and Captain Fitzjames \u2014 remained Silent. The only answer was to Abandon Ship and seek out a more hospitable clime, preferably Ashore somewhere south, where we might shoot fresh game._\n\n _As if reading our collective minds, Crozier smiled \u2014 it was a uniquely mad Irish smile, I thought at the time \u2014 and said,_ The problem, gentlemen, is that there's not a man aboard either ship, not even one of our venerable Marines, who knows how to catch or kill a seal or walrus \u2014 should those creatures ever grace us with their presence again \u2014 nor with the experience of shooting large game such as caribou, of which we have seen none.\n\n _The Rest of Us remained silent._\n\nThank you for your diligence, effort at doing the Inventory, and excellent report, Mr. Peddie, Mr. Goodsir, Mr. McDonald, and Mr. Stanley. We shall continue separating the cans you consider fully soldered and safe from those insufficiently soldered or bloated, distended, or otherwise visibly Putrid. We shall stay on the current Two-Thirds Rations until after Christmas Day, at which time I shall instigate a more draconian rationing plan.\n\n _Dr. Stanley and I pulled on our many layers of winter slops and went up to the deck to watch Dr. Peddie, Dr. McDonald, Captain Crozier, and an honor guard of four seamen armed with shotguns begin their long trek back to_ Terror _in the dark. As their lanterns and torches disappeared in the blowing snow and the wind howled in the rigging, the roar mixing with the constant grind and groan of the ice working against_ Erebus _'s hull, Stanley leaned close and shouted into my mufflered ear:_ It would be a blessing if they missed the cairns and got lost on the way back. Or if the Thing on the ice got them tonight.\n\n _I could only turn and stare in horror at the chief surgeon._\n\nDeath by starvation is a terrible thing, Goodsir, _continued Stanley._ Trust me. I've seen it in London and I've seen it with shipwreck. Death by scurvy is worse. It would be better if the Thing took us all tonight.\n\n _And with that we went below to the flame-flickering Darkness of the lower deck and to a cold almost the equal of the Dante-esque Ninth Circle Arctic Night without._\n\nCROZIER\n\n_Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n5 December, 1847\n\nOn a Tuesday dogwatch in the third week of November, the thing from the ice came aboard _Erebus_ and took the well-liked bosun, Mr. Thomas Terry, snatching him from his post near the stern, leaving only the man's head on the railing. There had been no blood at Terry's stern watch post: no blood on the ice-covered deck or on the hull. The conclusion was that the thing had taken Terry, carried him hundreds of yards out into the darkness where the seracs rose like trees of ice in a thick white forest, murdered and dismembered him \u2014 perhaps eaten him, although the men were growing increasingly doubtful that the white thing killing their crewmates and officers was actually doing so for food \u2014 and then it returned Mr. Terry's head before the starboard or port watchmen noticed that the bosun had gone missing.\n\nThe men who found the bosun's head at the end of that watch spent the week telling and retelling the others about poor Mr. Terry's visage \u2014 jaws open wide as if frozen in the middle of a scream, lips pulled back from his teeth, eyes protruding. There was not a tooth wound or claw mark on his face or head, only the ragged tearing at the neck, the thin pipe of his esophagus protruding like a rat's grey tail, and the stump of white spinal cord showing.\n\nSuddenly the more than one hundred surviving seamen found religion. Most of the men aboard _Erebus_ had grumbled for two years about Sir John Franklin's endless Divine Services, but now even the men who wouldn't have recognized a Bible if they'd wakened next to one after a three-day drunk found a deep need for some sort of spiritual reassurance. As word of Thomas Terry's beheading spread \u2014 Captain Fitzjames had put the sail-wrapped bundle in _Erebus_ 's own sealed Dead Room down on the hold deck \u2014 the men began requesting a single Sunday service for both crews. It was ferret-faced Cornelius Hickey who came to Crozier late on Friday night with the request. Hickey had been on a torchlight work party repairing ice cairns between the ships and had spoken to the men from _Erebus._\n\n\"It's unanimous, sir,\" said the caulker's mate as he stood in the doorway of Captain Crozier's tiny cabin. \"All the men would like a combined Divine Service. Both ships, Captain.\"\n\n\"You speak for every man on both ships?\" Crozier asked.\n\n\"Aye, sir, I do,\" said Hickey, flashing a once-winning smile that now showed only four of his remaining six teeth. The little caulker's mate was nothing if not confident.\n\n\"I doubt it,\" said Crozier. \"But I'll talk to Captain Fitzjames and let you know about the service. Whatever we decide, you can be our appointed courier to tell _all_ the men.\" Crozier had been drinking when Hickey rapped on his door. And he'd never liked the officious little man. Every ship had sea lawyers \u2014 like rats, they were a fact of Naval life \u2014 and Hickey, despite his bad grammar and total lack of formal education, struck Crozier as the kind of sea lawyer that, on a difficult voyage, soon began fomenting actual mutiny.\n\n\"One of the reasons we'd all of us like a service such as that what Sir John \u2014 God bless and rest his soul, Captain \u2014 used to provide is that all of us...\"\n\n\"That will be _all,_ Mr. Hickey.\"\n\nCrozier drank heavily that week. The melancholia that usually hovered over him like a fog now lay on him like a heavy blanket. He'd known Terry and thought him a more-than-capable boatswain, and it was certainly a horrible enough way to die, but the Arctic \u2014 at either pole \u2014 offered a myriad of horrible enough ways to die. So did the Royal Navy during peacetime or war. Crozier had witnessed more than a few of these horrible ways to die during his long career, so while Mr. Terry's death was among the more uncanny he'd personally known and the recent plague of violent deaths more frightening than any real plague he'd seen aboard ships, what brought on Crozier's deeper melancholy was more the reaction of the surviving members of the expedition.\n\nJames Fitzjames, the hero of the Euphrates, seemed to be losing heart. He was made a hero by the press even before his first ship had left Liverpool when young Fitzjames had plunged overboard to rescue a drowning customs agent even though the handsome young officer was, as the _Times_ said, \"embarrassed by a greatcoat, hat, and very valuable watch.\" The merchants of Liverpool, knowing the value \u2014 as Crozier well knew \u2014 of a customs officer who was already bought and paid for, had rewarded young Fitzjames with an engraved silver plate. The Admiralty had taken notice first of the silver plate, then of Fitzjames's heroism \u2014 although in Crozier's experience, an officer rescuing a drowning man was an almost weekly occurrence since few sailors knew how to swim \u2014 and finally of the fact that Fitzjames was \"the handsomest man in the Navy\" as well as a well-bred young gentleman.\n\nIt hadn't hurt the rising young officer's reputation that he had twice volunteered to lead raiding parties against Bedouin bandits. Crozier noticed in the official reports that Fitzjames had broken his leg in one such foray and been captured by the bandits in the second adventure, but the handsomest man in the Navy had managed to escape, which made Fitzjames all the more the hero to the London press and the Admiralty.\n\nThen came the Opium Wars and in 1841 Fitzjames showed himself to be a real hero, being commended by his captain and by the Admiralty no fewer than five times. The dashing lad \u2014 twenty-nine at the time \u2014 had used rockets to drive the Chinese off the hilltops of Tzekee and Segoan, used rockets again to drive them out of Chapoo, fought ashore at the Battle of Woosung, and returned to his expertise with rockets during the capture of Ching-Kiang-Fu. Seriously wounded, Lieutenant Fitzjames had managed, on crutches and in bandages, to attend the Chinese surrender at the signing of the Treaty of Nanking. Promoted to commander at the tender age of thirty, the handsomest man in the Navy had been given command of the sloop of war HMS _Clio,_ and his bright future seemed assured.\n\nBut then in 1844 the Opium Wars ended, and \u2014 as always happened to rising prospects in the Royal Navy when treacherous peace suddenly broke out \u2014 Fitzjames found himself without a command, on shore, and on half pay. Francis Crozier knew that if the Discovery Service offer of command to Sir John Franklin had been a godsend to the largely discredited old man, the offer of effective command of HMS _Erebus_ had been a shining second chance to Fitzjames.\n\nBut now \"the handsomest man in the Navy\" had lost his pink cheeks and usual ebullient humor. While most of the officers and men were maintaining their weight even on two-thirds rations \u2014 for members of the Discovery Service received a richer diet than 99 percent of Englishmen ashore \u2014 Commander, now Captain, James Fitzjames had lost more than two stone. His uniform hung loosely on him. His boyish curls now fell limp from under his cap or Welsh wig. Fitzjames's face, always a bit too chubby, now appeared drawn, wan, and hollow-cheeked in the light from the oil lamps or hanging lanterns.\n\nThe commander's public demeanor, which was always an easy mix of self-effacing humor and firm command, remained the same, but in private with only Crozier in attendance, Fitzjames spoke less, smiled less frequently, and too often looked distracted and miserable. For a melancholy man like Crozier, the signs were obvious. At times it was like staring into a looking glass, except for the fact that the melancholy countenance staring back was a proper lisping English gentleman rather than an Irish nobody.\n\nOn Friday the third of December, Crozier loaded a shotgun and made the long solo walk through the cold darkness between _Terror_ and _Erebus._ If the thing on the ice wanted to take him, Crozier thought, a few more men with guns would make little difference in the outcome. It hadn't for Sir John.\n\nCrozier arrived safely. He and Fitzjames discussed the situation \u2014 the men's morale, the requests for a religious service, the situation with the food tins, and the need to enforce strict rationing soon after Christmas \u2014 and they agreed that a combined Divine Service on the following Sunday might be a good idea. Since there were no chaplains or self-appointed ministers aboard \u2014 Sir John had filled both those roles until the previous June \u2014 both captains would give a sermon. Crozier hated this task more than dockside dentistry but realized that it had to be done.\n\nThe men's moods were in a dangerous state. Lieutenant Edward Little, Crozier's executive officer, reported that men on _Terror_ had begun to fashion necklaces and other fetishes from the claws and teeth of some of the white bears they had shot during the summer season. Lieutenant Irving reported weeks ago that Lady Silence had gone into hiding in the forward cable locker and the men had started leaving portions of their rum and food rations down there in the hold as if making offerings to a witch or saint in hopes of intercession.\n\n\"I've been thinking about your ball,\" said Fitzjames as Crozier began bundling up to leave.\n\n\"My ball?\"\n\n\"The Grand Venetian Carnivale that Hoppner set up when you wintered over with Parry,\" continued Fitzjames. \"When you went as a black footman.\"\n\n\"What about it?\" asked Crozier as he bound his comforter around his neck and face.\n\n\"Sir John had three large trunks of masks, clothing, and costumes,\" said Fitzjames. \"I found them among his personal stores.\"\n\n\"He did?\" Crozier was surprised. The aging windbag who would have held Divine Service six times a week if he had been allowed and who, despite his frequent laughter, never seemed to understand anyone else's jokes, seemed like the last sort of expedition commander to load trunks of frivolous costumes the way the stagestruck Parry had.\n\n\"They're old,\" confirmed Fitzjames. \"Some of them may have belonged to Parry and Hoppner \u2014 may have been the same costumes you chose from while frozen in Baffin Bay twenty-four years ago \u2014 but there are over a hundred tattered rags in there.\"\n\nCrozier stood bundled in the doorway of Sir John's former cabin where the two captains had held their sotto voce __ meeting. He wished Fitzjames would get to the point.\n\n\"I thought we might hold a masque for the men soon,\" said Fitzjames. \"Nothing as fancy as your Grand Venetian Carnivale, of course, not with the... unpleasantness... out on the ice, but a diversion nonetheless.\"\n\n\"Perhaps,\" said Crozier, allowing his tone to convey his lack of enthusiasm at the idea. \"We shall discuss it after this accursed Divine Service on Sunday.\"\n\n\"Yes, of course,\" Fitzjames said hurriedly. His slight lisp became more pronounced when he was nervous. \"Shall I send some men to escort you back to _Terror,_ Captain Crozier?\"\n\n\"No. And turn in early tonight, James. You look fagged out. We'll both need our energy if we're to properly sermonize the assembled crew on Sunday.\"\n\nFitzjames smiled dutifully. Crozier thought it a wan and strangely disturbing expression.\n\nOn Sunday the fifth of December, 1847, Crozier left behind a skeleton crew of six men commanded by First Lieutenant Edward Little \u2014 who, like Crozier, would rather have his kidney stones removed with a spoon than be forced to suffer sermons \u2014 as well as his assistant surgeon, McDonald, and the engineer, James Thompson. The other fifty-some surviving crewmen and officers trooped off across the ice following their captain, Second Lieutenant Hodgson, Third Lieutenant Irving, First Mate Hornby, and the other masters, clerks, and warrant officers. It was almost 10:00 a.m. but would have been absolutely dark under the shivering stars except for the return of the aurora which pulsed, danced, and shifted above them, throwing a long line of their shadows onto the fractured ice. Sergeant Soloman Tozer \u2014 the shocking birthmark on his face especially noticeable in the coloured light from the aurora \u2014 headed up the guard of Royal Marines with muskets marching points, flank, and behind the column, but the white thing in the ice left the men alone this Sabbath morning.\n\nThe last full gathering of both crews for Divine Service \u2014 presided over by Sir John shortly before the creature carried their devout leader down into the darkness under the ice \u2014 had been on the open deck under cold June sunlight, but since it was now at least 50 degrees below zero outside, when the wind was not blowing, Fitzjames had arranged the lower deck for the service. The huge cookstove could not be moved, but the men had cranked up the seamen's dining tables to their maximum height, taken down the removable bulkhead partitions that had delineated the forward sick bay, and removed other partitions that had created the warrant officers' sleeping area, the subordinate officers' stewards' cubicle, and the first and second mates' and second master's berths. They also removed the walls of the warrant officers' mess room and assistant surgeon's sleeping room. The space would be crowded still, but adequate.\n\nIn addition, Fitzjames's carpenter, Mr. Weekes, had created a low pulpit and platform. It was raised only six inches because of the lack of headroom under the beams, hanging tables, and stored lumber, but it would allow Crozier and Fitzjames to be seen by the men in the back of the jam of bundled bodies.\n\n\"At least we'll be warm,\" Crozier whispered to Fitzjames as Charles Hamilton Osmer, _Erebus_ 's bald purser, led the men in opening hymns.\n\nIndeed, the packed bodies had raised the temperature on the lower deck here higher than it had been since _Erebus_ had been burning great heaps of coal and forcing hot water through its heating pipes six months earlier. Fitzjames had also tried to lighten up the usually dark and smoky place by burning ship's oil at a furious rate in no fewer than ten hanging lamps that lit the space more brightly than at any time since sunlight had poured through the overhead Preston Patent Illuminators more than two years earlier.\n\nThe crewmen rocked the dark oak beams with their singing. Sailors, Crozier knew from his forty-plus years of experience, loved to sing under almost any circumstance. Even, if all else failed, during Divine Service. Crozier could see the top of caulker's mate Cornelius Hickey's head in the crowd, while next to him, hunched over so that his head and shoulders would not hit the overhead beams, stood the idiot giant Magnus Manson, who bellowed out the hymn in a boom so off-key that it made the grinding of the ice outside sound like close harmony. The two were sharing one of the tattered hymnals that Purser Osmer had handed out.\n\nFinally the hymns were finished and there came a low din of shuffling, coughing, and clearing of throats. The air smelled of fresh-baked bread since Mr. Diggle had come over hours earlier to aid _Erebus_ 's cook, Richard Wall, in the baking of biscuits. Crozier and Fitzjames had decided that the extra coal, flour, and lamp oil were worth expending this special day if it helped the men's morale. The darkest two months of the arctic winter were still ahead.\n\nNow it was time for the two sermons. Fitzjames had shaved and powdered carefully and allowed his personal steward, Mr. Hoar, to take in his baggy waistcoat, trousers, and jacket, so he looked calm and handsome in his uniform and shining epaulettes. Only Crozier, standing behind him, could see Fitzjames's pale hands clenching and unclenching as he set his personal Bible on the pulpit and opened it to Psalms.\n\n\"The reading today shall be from Pthalm Forty-six,\" said Captain Fitzjames. Crozier winced slightly at the upper-class lisp that had become more pronounced with tension.\n\nGod _is_ our refuge and strength,\n\nand ever-prethent help in trouble.\n\nTherefore we will not fear, though the earth give way\n\nand the mountainth fall into the heart\n\nof the sea,\n\nthough its waterth roar and foam\n\nand the mountains quake with their\n\nthurging.\n\nThere is a river whose streams make glad\n\nthe city of God,\n\nthe holy place where the Most High dwellth.\n\nGod is within her, she will not fall;\n\nGod will help her at break of day.\n\nNations are in uproar, kingdomth fall;\n\nhe lifts his voice, the earth melts.\n\nThe Lord Almighty is with us;\n\nthe God of Jacob is our fortreth.\n\nCome and see the workth of the LORD,\n\nthe desolations he has brought on the earth.\n\nHe makes wars cease to the endth of the spear,\n\nhe burns the shields with fire.\n\n\"Be still, and know that I am God; I will be\n\nexalted among the nations,\n\nI will be exalted in the earth.\"\n\nThe LORD Almighty is with us;\n\nthe God of Jacob is our fortress.\n\nThe men roared \"Amen\" and shuffled their warming feet in appreciation.\n\nIt was Francis Crozier's turn.\n\nThe men were hushed, as much out of curiosity as respect. The Terrors in the assembled mass knew that their captain's idea of a reading for Divine Service was a solemn recitation of the Ship's Articles \u2014 \"If a man refuses to obey orders from an officer, that man shall be flogged or put to death, punishment to be determined by the captain. If a man commits sodomy with another member of the crew or a member of the ship's livestock, that man shall be put to death...\" and so forth. The Articles had the proper biblical weight and resonance and served Crozier's purpose.\n\nBut not today. Crozier reached to the shelf under the pulpit and pulled out a heavy leather-bound book. He set it down with a reassuring thud of authority.\n\n\"Today,\" he intoned, \"I shall read from the _Book of Leviathan,_ Part One, Chapter Twelve.\"\n\nThere was a murmuring in the crowd of seamen. Crozier heard a toothless _Erebus_ in the third row mutter, \"I know the fucking Bible, and there ain't no fucking _Book of Leviathan._ \"\n\nCrozier waited for silence and began.\n\n\" 'And for that part of Religion, which consisteth in opinions concerning the nature of Powers Invisible,...\"\n\nCrozier's voice and Old Testament cadence left no doubt as to which words were celebrated with capital letters.\n\n\" '... there is almost nothing that has a name, that has not been esteemed amongst the Gentiles, in one place or another, a God, or Divell; or by their Poets feigned to be inanimated, inhabited, or possessed by some Spirit or other.\n\n\" 'The unformed matter of the World, was a God, by the name of _Chaos._\n\n\" 'The Heaven, the Ocean, the Planets, the Fire, the Earth, the Winds, were so many Gods.\n\n\" 'Men, Women, a Bird, a Crocodile, a Calf, a Dogge, a Snake, an Onion, a Leeke, Deified. Besides, that they filled almost all places, with spirits called _Daemons:_ the plains, with _Pan,_ and _Panises,_ or Satyres; the Woods, with Fawnes, and Nymphs; the Sea, with Tritons, and other Nymphs; every River and Fountayn, with a Ghost of his name, and with Nymphs; every house with its _Lares,_ or Familiars; every man, with his _Genius;_ Hell, with Ghosts, and spirituall Officers, as _Charon, Cerberus,_ and the _Furies;_ and in the night time, all places with _Larvae, Lemures,_ Ghosts of men deceased, and a whole kingdome of Fayries, and Bugbears. They have also ascribed Divinity, and built Temples to meer Accidents, and Qualities; such as are Time, Night, Day, Peace, Concord, Love, Contention, Vertue, Honour, Health, Rust, Fever, and the like; which when they are prayed for, or against, they prayed to, as if there were Ghosts of those names hanging over their heads, and letting fall, or withholding that Good, or Evill, for, or against which they prayed. They invoked also their own Wit, by the name of _Muses;_ their own Ignorance, by the name of _Fortune;_ their own Lust, by the name of _Cupid;_ their own Rage, by the name of _Furies;_ their own privy members by the name of _Priapus;_ and attributed their pollutions, to _Incubi_ and _Succubae:_ insomuch as there was nothing which a Poet could introduce as a person in his Poem, which they did not make either a _God_ or a _Divel._ ' \"\n\nCrozier paused and looked out at the staring white faces.\n\n\"And thus endeth Part One, Chapter Twelve, of the _Book of Leviathan,_ \" he said and closed the heavy tome.\n\n\"Amen,\" chorused the happy seamen.\n\nThe men ate hot biscuits and full rations of their beloved salt pork at dinner that afternoon, the extra forty-some Terrors crowding around the lowered tables forward or using casks for surfaces and extra sea chests for chairs. The din was reassuring. All of the officers from both ships ate aft, sitting around the long table in Sir John's former cabin. Besides their required antiscorbutic lemon juice that day \u2014 Dr. McDonald was now fretting that the five-gallon kegs were losing their potency \u2014 the seamen each received an extra gill of grog before dinner. Captain Fitzjames had dipped into his reserve ship's stores and provided the officers and warrant officers with three fine bottles of Madeira and two of brandy.\n\nAt about 3:00 p.m. civilian time, the Terrors bundled up, wished their _Erebus_ counterparts good-bye, and went up the main ladder, out under the frozen canvas, and then down the snow and ice embankment onto the dark ice for the long walk home under the still-shimmering aurora. There were whispers and muted comments in the ranks about the _Leviathan_ sermon. The majority of men were certain that it was in the Bible somewhere, but wherever it had come from, no one was quite sure what their captain had been getting at, although opinions ran strong after the double ration of rum. Many of the men still fingered their good-luck fetishes of white-bear teeth, claws, and paws.\n\nCrozier, who headed up the column, felt half certain that they would return to find Edward Little and the watch murdered, Dr. McDonald in pieces, and Mr. Thompson, the engineer, dismembered and strewn about the pipes and valves of his useless steam engine.\n\nAll was well. Lieutenants Hodgson and Irving handed out the parcels of biscuits and meat that had been warm when they'd left _Erebus_ the better part of an hour earlier. The men who had remained on watch in the cold took their extra rations of grog first.\n\nAlthough he was chilled through \u2014 the relative heat of _Erebus_ 's crowded lower deck had made the outside cold worse somehow \u2014 Crozier stayed on deck until the watch was relieved. The officer on duty was now Thomas Blanky, the Ice Master. Crozier knew that the men below would be doing Sunday make-and-mend, many already looking forward to afternoon tea and then supper with its sad fare of Poor John \u2014 salted and boiled codfish with a biscuit \u2014 with the hopes that there might be an ounce of cheese to go with their half pint of Burton's ale.\n\nThe wind was coming up, blowing snow across the serac-strewn ice fields on this side of the huge berg blocking the view of _Erebus_ to the northeast. Clouds were hiding the aurora and stars. The afternoon night became much darker. Eventually, thinking of the whiskey in his cabin, Crozier went below.\n\nBLANKY\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n5 December, 1847\n\nHalf an hour after the captain and the other men returning from the Divine Service party on _Erebus_ went below, Tom Blanky couldn't see the watch lanterns or the mainmast for all the blowing snow. The Ice Master was glad the blow had come up when it had; an hour earlier and their group trek back from _Erebus_ would have been a buggering bitch.\n\nOn port watch under Mr. Blanky's command this black evening were thirty-five-year-old Alexander Berry \u2014 not an especially intelligent man, Blanky knew, but dependable and good in the rigging \u2014 as well as John Handford and David Leys. This last man, Leys, now on bow watch, had just turned forty in late November and the men had thrown quite a fo'c'sle party for him. But Leys wasn't the same man who had signed up for the Discovery Service two and a half years ago. Back in early November, just a few days before Marine Private Heather had his brains dashed out while on starboard watch and young Bill Strong and Tom Evans had disappeared, Davey Leys had simply gone to his hammock and quit talking. For almost three weeks Leys had simply _left_ \u2014 his eyes stayed open, staring at nothing, but he hadn't responded to voice, flame, shaking, shouts, or pinching. For most of that time he was in the sick bay, lying next to poor Private Heather, who somehow drew breaths even with his skull scooped open and some of his brains missing. While Heather lay there gasping, Davey continued to lie there in silence, staring unblinkingly at the overhead as if already dead himself.\n\nThen, as soon as the fit had come on, it was over, and Davey was his old self again. Or almost his old self again. His appetite had come back \u2014 he'd lost almost twenty pounds during his time away from his own body \u2014 but the old Davey Leys's sense of humour was gone, as was his easy, boyish smile and his willingness to enter into fo'c'sle conversations during make-and-mend or supper. Also, Davey's hair, which had been a rich reddish brown the first week in November, was pure white when he came out of his funk. Some of the men said that Lady Silence had put a hex on Leys.\n\nThomas Blanky, Ice Master for more than thirty years, did not believe in hexes. He was ashamed of the men who were wearing polar bear claws, paws, teeth, and tails as some sort of anti-hex amulets. He knew that some of the less-educated men \u2014 centered around the caulker's mate, Cornelius Hickey, whom Blanky had never liked nor respected \u2014 were spreading the word that the Thing on the Ice was some sort of demon or devil \u2014 or _Daemon_ or _Divell_ as their captain had later said the spelling ran in his odd _Book of Leviathan_ \u2014 and some around Hickey were already making sacrifices to the monster, setting them outside the forward cable locker in the hold where everyone now knew Lady Silence, obviously an Esquimaux witch, was hiding. Hickey and his giant idiot friend, Magnus Manson, seemed to be the high priests of this cult \u2014 or rather, Hickey was the priest and Manson the acolyte who did whatever Hickey said \u2014 and they appeared to be the only ones allowed to bring the various offerings down to the hold. Blanky had gone down there into the sulfurous dark and stench and cold recently and was disgusted to see little pewter plates of food, burned-down candles, tiny tots of rum.\n\nThomas Blanky was no natural philosopher, but he had been a creature of the arctic as both man and boy, working as able-bodied seaman or ice master for American whalers when the Royal Navy had no use for him, and he knew these polar regions as few others on the expedition did. While this area was strange to him \u2014 as far as Blanky knew, no ship had ever sailed this far south of Lancaster Sound and so near to King William Land before, nor sailed so far west of Boothia Peninsula \u2014 most of the terrible arctic conditions were as familiar to him as a summer in Kent where he was born.\n\n _More_ familiar, actually, Blanky realized. He'd not seen a Kent summer in almost twenty-eight years.\n\nThe howling snow this night was familiar, as was the solid surface of ice and seracs and grumbling pressure ridges which were pushing poor _Terror_ higher on its capstan of rising ice even while squeezing the life out of her. Blanky's ice-master counterpart on _Erebus,_ James Reid, a man Blanky highly respected, had informed him just today after the odd Divine Service that the old flagship hadn't much longer to last. Besides its coal scuttles being drawn down even farther than the failing _Terror_ 's, the ice had seized Sir John's ship in a fiercer and less-forgiving grip more than a year ago when they'd first been locked fast into their current positions.\n\nReid had whispered that since _Erebus_ was stern-down in the encroaching ice \u2014 the opposite of _Terror_ 's bow-down position \u2014 the unrelenting pressure was squeezing Sir John's ship more tightly and growing more terrible as it pushed the creaking, groaning ship higher above the surface of the frozen sea. Already the rudder had been splintered and the keel damaged beyond repair outside of a dry dock. Already the stern plates were sprung \u2014 there were three feet of frozen water in the stern, which was down by ten degrees, and only sandbags and coffer dams kept the slushy sea out of the boiler room \u2014 and the mighty oak beams that had survived decades of war and service were splintering. Worse, the spiderwebs of iron bracings set in place in 1845 to make _Erebus_ impervious to the ice moaned constantly now from the terrible pressure. From time to time, smaller stanchions gave way at the join with the sound of a small cannon being fired. This often happened late at night and the men would snap upright in the hammocks, identify the source of the explosion, and go back to sleep with soft curses. Captain Fitzjames usually went below with some of his officers to investigate. The heavier braces would hold, Reid said, but only by tearing through the contracting oak-and-iron-layered hulls. When that happened, the ship would sink, ice or no ice.\n\n _Erebus_ 's ice master said that their ship's carpenter, John Weekes, spent every day and half of most nights with a work party of no fewer than ten men down in the hold and orlop deck, shoring everything with every stout plank the ship had brought along \u2014 and many quietly borrowed from _Terror_ \u2014 but the resulting web of internal wooden structure was a temporary fix, at best. Unless _Erebus_ escaped the ice by April or May, Reid quoted Weekes as saying, it would be crushed like an egg.\n\nThomas Blanky knew ice. In the early summer of 1846, all the time he was guiding Sir John and his captain south through the long sound and newly discovered strait south of the Barrow Strait \u2014 the new strait remained nameless in their logs but some were already calling it \"Franklin Strait,\" as if naming the channel that had trapped the dead old fool would make his ghost feel better about being carried away by a monster \u2014 Blanky had been at his station atop the mainmast, shouting down advice to the helmsman as _Terror_ and _Erebus_ gingerly picked their way through more than 250 miles of changing ice and narrowing leads and dead-end channels.\n\nThomas Blanky was good at his job. He knew that he was one of the best ice masters and pilots in the world. From his precarious post high atop the mainmast \u2014 these old bombardment ships had no crow's nests like a mere whaler \u2014 Blanky could tell the difference of drift ice from brash ice at eight miles' distance. Asleep in his cubicle, he knew at once when the ship had moved from the _glug-glug-glug_ passage through sludge ice into the metal-file rasp of pancake ice. He knew at a glance which bergy bits were a threat to the ship and which could be taken head-on. Somehow his aging eyes could make out the blue-white submerged growlers in a blue-white sea alive with sun sparkles and even tell which of the growlers would merely grind and groan as they slid along the ship's hull and which \u2014 like an actual berg \u2014 would put the ship at risk.\n\nSo Blanky was proud of the job he and Reid had done leading both ships more than 250 miles south and then west of their first wintering spot at Beechey and Devon Islands. But Thomas Blanky also cursed himself for a fool and a villain for helping lead the two ships and their 126 souls 250 miles south and then west of their wintering spot at Beechey and Devon.\n\nThe ships could have retreated from Devon Island, back through Lancaster Sound and then down Baffin Bay, even if they'd had to wait two cold summers, or even three, to escape the ice. The little bay there at Beechey would have protected the ships from this open-sea ice abuse. And sooner or later the ice along Lancaster Sound would have relented. Thomas Blanky _knew_ that ice. It behaved the way arctic ice was meant to behave \u2014 treacherous, deadly, ready to destroy you after a single wrong decision or moment's lapse, but predictable.\n\nBut _this_ ice, thought Blanky as he stomped about the dark stern to keep his feet from freezing, seeing the lanterns glowing port and starboard where Berry and Handford paced with their shotguns, _this_ ice was like no ice in his experience.\n\nHe and Reid had _warned_ Sir John and the two captains fifteen months ago, right before the ships became frozen in place. _Go for broke,_ Blanky had advised, agreeing with Captain Crozier that they needed to turn tail while there were still the slightest open leads, needed to seek out open water as close to the Boothia Peninsula as fast as they could steam that long-ago September. The water there close to a known coast \u2014 at least the east side of it was known to old Discovery Service and whaler veterans such as Blanky \u2014 almost certainly would have stayed liquid for another week, perhaps two, into that lost-opportunity September. Even if they hadn't been able to steam north along the coast again because of hummocky floes and old pack ice \u2014 Reid called it _screw pack ice_ \u2014 they would have been infinitely safer behind the shelter of what they now were certain, after the dead Lieutenant Gore's sledge expedition last summer, was James Ross's King William Land. That landmass, as low, frozen, windswept, and lightning-ravaged as they now knew it to be, would have sheltered the ships from this Devil-sent constant northwest blast of arctic wind, blizzards, cold, and endless assaulting sea ice.\n\nBlanky had never seen ice like this. One of the few advantages of pack ice, even when your ship was frozen in like a musket ball blasted into an iceberg, was that the pack ice _drifted._ The ships, while seemingly motionless, _moved._ When Blanky had been ice master on the American whaler _Pluribus_ in '36, winter had roared in on the twenty-seventh of August, taking everyone including the experienced one-eyed American captain by surprise and freezing them into Baffin Bay hundreds of miles north of Disko Bay.\n\nThe next arctic summer had been bad \u2014 almost as cold as this last summer, 1847, during which there had been no summer ice melt, air warmth, or return of birds or other wildlife here \u2014 but the whaler _Pluribus_ was in more predictable pack ice and drifted more than seven hundred miles south until, late that next summer, they'd reached the ice line and been able to sail their way south through sludge-ice seas and narrow leads and what the Russians called _polynyas,_ cracks in the ice that opened up as you watched, until the American whaler reached open water and could sail southeast to a Greenland port for refitting.\n\nBut not here, Blanky knew. Not in this truly godforsaken white hell. This pack ice was, as he'd described to the captains a year and three months ago, more like an endless glacier being pushed down from the north pole. And with the mostly uncharted bulk of arctic Canada to their south, King William Land to their southwest, and Boothia Peninsula beyond their reach to the east and nor'east, there was no real ice drift here \u2014 as Crozier's and Fitzjames's and Reid's and Blanky's repeated sun and star sextant readings kept telling them \u2014 just a sickening pivot around a fifteen-mile circumference. It was as if they were flies pinned to one of the metal music disks no longer used by the men in the Great Room below. Going nowhere. Always returning to the same spot again and again.\n\nAnd this open pack ice was more like the coastal fast ice of Blanky's experience, only here at sea the ice was twenty to twenty-five feet thick around the ships rather than the three-foot depth of regular fast ice. So thick that the captains couldn't keep open the usual fire holes that _all_ ships locked in the ice kept free all winter.\n\nThis ice didn't even allow them to bury their dead.\n\nThomas Blanky wondered if he had been an instrument of evil \u2014 or perhaps just of folly \u2014 when he had used his more than three decades of ice-master skills to get 126 men the impossible 250 miles through ice to this place where all they could do was die.\n\nSuddenly there came a shout. Then a shotgun blast. Another shout.\n\nBLANKY\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _5 December, 1847_\n\nBlanky tugged his right overmitten off with his teeth, let it drop to the deck, and raised his own shotgun. Tradition was for the officers on watch not to be armed, but Captain Crozier had ended that tradition with a single order. Every man on deck was to be armed at all times. Now, with his mitten off, Blanky's thin wool underglove allowed his finger to crook through the trigger guard of the shotgun, but his hand immediately felt the biting cold of the wind.\n\nIt was Seaman Berry's lantern \u2014 the port watch \u2014 whose glow had disappeared. The shotgun blast had sounded as if it had come from the left of the midship winter canvas rigging, but the Ice Master knew that wind and snow distorted sounds. Blanky could still see the glow of the starboard-side lantern, but it was bobbing and moving.\n\n\"Berry?\" he shouted toward the dark port side. He could almost feel the two syllables hurled astern by the howling wind. \"Handford?\"\n\nThe starboard lantern glow disappeared. At the bow, Davey Leys's lantern would have been visible beyond the midship tent on a clear night, but this was no longer a clear night.\n\n\"Handford?\" Mr. Blanky began moving forward to the port side of the long tent covering, carrying his shotgun in his right hand and the lantern he'd lifted off the sternpost in his left. He had three more shotgun shells in his right greatcoat pocket, but he knew from experience how long it took to fumble them out and load them in this cold.\n\n\"Berry!\" he bellowed. \"Handford! Leys!\" One of the dangers now was that the three men would shoot each other in the dark and storm on the tilted, icy deck, although it had sounded as if Alex Berry had already discharged his weapon. There had not been a second blast. But Blanky knew that if he moved to the port side of the frozen tent pyramid and Handford or Leys suddenly came around to investigate, the nervous men might fire at anything, even a moving lantern.\n\nHe moved forward anyway.\n\n\"Berry?\" he shouted, coming within ten yards of the port watch station.\n\nHe caught a blur of movement amid the driving snow, something far too large to be Alex Berry, and then there came a crash louder than any shotgun. A second explosion. Blanky staggered back ten paces toward the stern as casks, wooden kegs, boxes, and other ship's stores flew through the air. It took him a few seconds to realize what had happened; the permanent pyramid of frozen canvas running fore and aft along the centre of the deck had suddenly collapsed, throwing thousands of pounds of accumulated snow and ice in each direction even while flinging wide the deck stores beneath \u2014 mostly flammable pitch, caulkers' materials, and sand to put down for traction atop the snow deliberately shoveled onto the deck \u2014 and also sending the mainmast's lower spars, which had been rotated fore and aft more than a year ago to act as ridgepoles for the tent, crashing down onto the main hatch and ladderway.\n\nThere was no way for Blanky and the other three men on watch to get to the lower deck now, and no way that the men down there could get up to investigate the explosions on deck, not with the main spars and all that weight of canvas and snow blocking the hatch. The Ice Master knew that the men below would soon run to the forward hatch and begin unbattening its nailed-down winter seals, but that would take time.\n\n _Will we be alive when they get up here?_ wondered Blanky.\n\nMoving as carefully as he could on the sand-covered packed snow of the tilted deck, Blanky made his way around the debris pile at the rear of the collapsed tent area and started down the narrow aisle on the starboard side of the heap.\n\nA shape rose before him.\n\nStill holding the lantern high in his left hand, Blanky lifted the shotgun, finger on the trigger, ready to fire. \"Handford!\" he said when he saw the pale blob of a face amid the black mass of coats and comforters. The man's Welsh wig was in disarray. \"Where's your lantern?\"\n\n\"I dropped it,\" said the seaman. The man was shaking violently, his hands bare. He huddled close to Thomas Blanky as if the Ice Master were a source of heat. \"I dropped it when the thing knocked the spar down. The flame went out in the snow.\"\n\n\"What do you mean, 'when the thing knocked the spar down'?\" demanded Blanky. \"No living thing could knock the mainmast spar down.\"\n\n\" _It_ did!\" said Handford. \"I heard Berry's shotgun fire. Then he shouted something. Then his lantern went out. Then I saw something... large, something very large... leap up on the spar and that's when everything collapsed. I tried to fire at the thing on the spar, but my shotgun misfired. I left it at the rail.\"\n\n _Leap up onto the spar?_ thought Blanky. The swiveled mainmast spar was twelve feet above the deck. Nothing could leap onto it. With the mainmast sheathed in ice, nothing could climb to it either. Aloud, he said, \"We have to find Berry.\"\n\n\"There is no way in God's universe that I'm going over there to the port side, Mr. Blanky. You can write me up and have Bosun's Mate Johnson give me fifty with the cat, but there's no way in God's universe that I'm going over there, Mr. Blanky.\" Handford's teeth were chattering so wildly that he was barely understandable.\n\n\"Calm down,\" snapped Blanky. \"No one's being written up. Where's Leys?\"\n\nFrom this vantage point on the starboard-watch side, Blanky should have been able to see David Leys's lantern glowing at the bow. The bow was dark.\n\n\"His went out the same time I dropped mine,\" said Handford through his clattering teeth.\n\n\"Get your shotgun.\"\n\n\"I can't go back there where... ,\" began Handford.\n\n\"God- _damn_ your eyes!\" roared Thomas Blanky. \"If you don't retrieve that weapon _this gob-fucking minute,_ a flogging of fifty from the cat will be the _least bugger-fucking thing_ you have to worry about, John Handford. Now _move!_ \"\n\nHandford moved. Blanky followed him, never turning his back on the collapsed heap of canvas at the centre of the ship. Because of the driving snow, the lantern created a sphere of light ten feet or less across. The Ice Master kept both the lantern and the shotgun raised. His arms were very tired.\n\nHandford was attempting to retrieve his weapon in the snow with fingers that had obviously gone numb with cold.\n\n\"Where the hell are your mittens and gloves, man?\" snapped Blanky.\n\nHandford's teeth were chattering too wildly for him to respond.\n\nBlanky set his own weapon down, brushed the seaman's arms aside, and lifted the man's shotgun. He made sure the single barrel was not blocked with snow, then broke the breech and handed the weapon back to Handford. Blanky finally had to tuck it under the other man's arm so he could cradle it with his two frozen hands. Setting his own shotgun under his left arm where he could shift it quickly, Blanky fumbled a shell out of his greatcoat pocket, loaded Handford's shotgun, and clicked it shut for the man. \"If anything larger than Leys or me comes out of that pile,\" he said, almost shouting into Handford's ear because of the wind roar, \"aim and pull that trigger if you have to use your fucking teeth to do it.\"\n\nHandford managed a nod.\n\n\"I'm going forward to find Leys and help him open the forward hatch,\" said Blanky. Nothing seemed to be moving downhill toward the bow amid the dark jumble of frozen canvas, dislodged snow, broken spars, and tumbled crates.\n\n\"I can't... ,\" began Handford.\n\n\"Just stay where you are,\" snapped Blanky. He set the lantern down next to the terrified man. \"Don't shoot me when I come back with Leys or I swear to God my ghost will haunt you 'til you die, John Handford.\"\n\nHandford's pale blob of a face nodded again.\n\nBlanky started toward the bow. After a dozen steps, he was beyond the glow of the lantern but his night vision did not return. The hard particles of snow struck his face like pellets. Above him, the rising wind howled in what little rigging and shrouds they'd left in place during the endless winter. It was so dark here that Blanky had to carry the shotgun in his left hand \u2014 his still-mittened hand \u2014 while feeling along the ice-encrusted railing with his right hand. As far as he could tell, the mainmast spar here on the forward side of the mast had also collapsed.\n\n\"Leys!\" he shouted.\n\nSomething very large and vaguely white in the hurtling snow lumbered out of the heap of debris and stopped him in his tracks. The Ice Master couldn't tell if the thing was a white bear or a tattooed demon or if it was ten feet in front of him or thirty feet away in the dark, but he knew that it had completely blocked his progress toward the bow.\n\nThen the thing reared up on its hind legs.\n\nBlanky could see only the mass of it \u2014 he sensed the dark bulk of it mostly through the amount of blowing snow it blocked \u2014 but he knew it was huge. The tiny triangular head, if that _was_ a head up there in the darkness, rose higher than the space where the spars had been. There seemed to be two holes punched into that pale triangle of a head \u2014 eyes? \u2014 but they were at least fourteen feet above the deck.\n\n _Impossible,_ thought Thomas Blanky.\n\nIt moved toward him.\n\nBlanky shifted the shotgun to his right hand, jammed the stock against his shoulder, steadied it with his mittened left hand, and fired.\n\nThe flash and explosion of sparks from the barrel gave the Ice Master a half-second's glimpse of the black, dead, emotionless eyes of a shark staring into him \u2014 no, not a shark's eyes at all, he realized a second later as the retinal afterimage of the blast blinded him, but two ebony circles far more frighteningly malevolent and intelligent than even a shark's black-circle gaze \u2014 but also the pitiless stare of a predator that sees you only as food. And these bottomless black-hole eyes were far above him, set on shoulders wider than Blanky's arms could spread, and were coming closer as the looming shape surged forward.\n\nBlanky threw the useless shotgun at the thing \u2014 there was no time at all to reload \u2014 and leapt for the man lines.\n\nOnly four decades of experience at sea allowed the Ice Master to know, in the dark and storm and without even attempting to look, exactly where the icy man lines would be. He caught them with the crooked fingers of his mittenless right hand, flung his legs up, found the cross ropes with his flailing boots, pulled off his left mitten with his teeth, and began clambering upward while hanging almost upside down on the inside of the inward-slanting lines.\n\nSix inches beneath his arse and legs, something cleaved the air with the power of a two-ton battering ram swinging at full extension. Blanky heard three thick vertical ropes of the man lines rip, tear... impossible!... and swing inward, almost throwing Blanky down to the deck.\n\nHe hung on. Flinging his left leg around the outside of those lines remaining taut, he found purchase on the icy rope and began climbing higher without pausing for a second. Thomas Blanky moved like the monkey he'd been as an unrated boy of twelve who thought the masts, sails, lines, and upperworks' rigging of the three-masted warship on which he shipped had all been constructed by Her Majesty solely for his enjoyment.\n\nHe was twenty feet up now, approaching the level of the second spar \u2014 this one still set at the proper right angle to the length of the ship \u2014 when the thing below hit the base of the man-line rigging again, tearing wood and dowels and pins and ice and iron blocks completely free of the railing.\n\nThe web of climbing rope swung inward toward the mainmast. Blanky knew that the impact would knock him off and send him hurtling down into the thing's arms and jaws. Still not able to see anything more than five feet away in the blowing dark, the Ice Master leapt for the shrouds.\n\nHis freezing fingers found the spar and its lines under them at the same instant one of his flailing feet caught a foot line. This shroud-line scuttle was best done barefoot, Blanky knew, but not tonight.\n\nHe heaved himself up over the second spar, more than twenty-five feet above the deck, and clung to the icy oak with legs and arms both, the way a terrified rider would cling to the body of a horse, wildly sliding his feet along the ice-hard shroud to find more purchase on the slippery shroud lines.\n\nNormally, even in the darkness, wind, snow, and hail, any decent sailor could scramble another sixty feet higher into the upperworks and rigging here until he reached the mainmast crosstrees, from which point he could hurl down insults at his stymied pursuer like a chimpanzee in a tall tree throwing down fruit or feces from a point of perfect safety. But there were no upperworks or high rigging on HMS _Terror_ this December night. There was no point of perfect safety up here when fleeing from something so powerful it could smash a main spar. And there was no upperworks rigging to which a man could flee.\n\nA year ago September, Blanky had helped Crozier and Harry Peglar, Captain of the Foretop, as they prepared _Terror_ for her wintering-over for the second time this expedition. It was not an easy job, nor one without danger. The yards and running rigging were struck down and stored below. Then the topgallant masts and topmasts were carefully struck down \u2014 carefully because a slip with winch or block or sudden tangle of tackle could have sent the heavy masts hurtling down through the top deck, lower deck, orlop deck, and hull bottom like a massive spear piercing wicker armour. Ships had sunk from such missteps while striking down upper masts. But if they'd remained up, too many tons of ice would accumulate during the endless winter. The ice would have provided a constant barrage of missiles for the men on watch or other duty on the deck and rigging below, but the weight of it also could capsize a ship.\n\nWith only the three stumps of the lower masts remaining \u2014 a sight as ugly to a seaman as a triple-amputee human being might be to a painter of pictures \u2014 Blanky had helped supervise the loosening of all remaining shrouds and standing rigging; overly taut canvas and ropes simply could not bear the weight of so much snow and ice. Even _Terror_ 's boats \u2014 the two large whaleboats and two smaller cutters, as well as the captain's skiff, pinnaces, jolly boats, and dinghies, ten in all \u2014 had been taken down, inverted, lashed, covered, and stored on the ice.\n\nNow Thomas Blanky was on the mainmast's second-spar shrouds twenty-five feet above the deck and had only one level higher to go, and any man lines leading up to that third and last level would be more ice than rope or wood. The mainmast itself was a column of ice with an extra coating of snow on its forward curve. The ice master straddled the second spar and tried to peer down through the darkness and snow. It was pitch black below. Either Handford had extinguished the lantern Blanky had given him or it had been extinguished for him. Blanky assumed that the man was either cowering in the dark or dead; either way he would be no help. Spread-eagled over the spar shrouds, Blanky looked to his left and saw that there still was no light forward in the bow where David Leys had been on watch.\n\nBlanky strained to see the thing directly below him but there was too much movement \u2014 the torn canvas flapping in the dark, kegs rolling on the tilted deck, loose crates sliding \u2014 and all he could make out was a dark mass shuffling toward the mainmast, batting aside two- and three-hundred-pound kegs of sand as if they were so many china vases.\n\n _It can't climb the mainmast,_ thought Blanky. He could feel the cold of the spar through his straddling legs and chest and crotch. His fingers were beginning to freeze through the thin undergloves. Somewhere he had lost his Welsh wig and wool-scarf comforter. He strained to hear the sound of the forward hatch being unbattened and flung free, to hear shouts and to see lanterns as the rescue party came up in force, but the bow of the ship remained a silent darkness hidden by hurtling snow. _Has it somehow blocked the forward hatch as well? At least it can't climb the mainmast. Nothing that size can climb. No white bear \u2014 if it is a white bear \u2014 has experience climbing._\n\nThe thing began climbing the attenuated mainmast.\n\nBlanky felt the vibration as it slammed claws into the wood. He heard the smack and scrape and grunting... a thick, bass grunting... as it climbed.\n\n _It climbed._\n\nThe thing had most probably reached the snapped-off stubs of the first spar just by raising its forearms over its head. Blanky strained to see in the darkness and was sure he could make out the haired and muscled mass hauling itself up headfirst, gigantic forelegs \u2014 or arms \u2014 as big as a man already flung over the first spar and clawing higher for leverage even while powerful rear legs and more claws there found support on the splintered oak of the spars.\n\nBlanky inched out farther along the icy second spar, his arms and legs wrapped around the wind-thrummed ten-inch-round horizontal spar in a sort of frenzied lover's embrace. There were two inches of new snow lining the bow-facing outer curve of the ever-thinning spar and then ice under that. He used the shroud lines for purchase when he could.\n\nThe huge thing on the mainmast had reached the level of Blanky's spar. The Ice Master could see the bulk of it only by craning to look over his own shoulder and arse and even then could make it out only as a giant, pale _absence_ where the subliminal vertical slash of the mainmast should be.\n\nSomething struck the spar with so much force that Blanky flew up into the air, dropping two feet back onto the spar to land hard on his balls and belly, the impact on the spar and folds of frozen shroud knocking the wind out of him. He would have fallen then if both freezing hands and his right boot hadn't been firmly entangled in the shroud lines just below the icy underside of the spar. As it was, it felt like a horse made of cold iron had bucked him two feet into the air.\n\nThe blow came again and would have launched Blanky out into the darkness thirty feet above the deck, but he was prepared for this second smash and clung with all his might. Even ready as he was, the vibration was so forceful that Blanky slipped off and swung helplessly _under_ the icy spar, numb fingers and kicking boot still mixed in with the shroud lines there. He managed to leverage himself back on top of the spar just as the third and most violent blow struck. The Ice Master heard the cracking, felt the solid spar begin to sag, and realized that he had only seconds before he and the spar, the shroud, the shroud lines, the ratlines, and the wildly swinging man lines all fell more than twenty-five feet to the pitched deck and tumbled debris below.\n\nBlanky did the impossible. On the pitching, cracking, tilted and icy spar, he got to his knees, then to his feet, standing with both arms waving comically and absurdly for balance in the howling wind, boots slipping on the snow and ice, and then he hurled himself into space with arms and hands extended, seeking one of the invisible hanging ratlines that should be \u2014 might be \u2014 _could be_ \u2014 somewhere there, allowing for the ship's down-at-the-bow attitude, for the howling wind, for the impact of the blowing snow on the thin lines, and for the possible effects of the vibration from the thing's ongoing shattering of the mainmast's second level of spars.\n\nHis hands missed the single hanging line in the dark. His freezing face hit it, and as he fell, Thomas Blanky grabbed the line with both hands, slid only six feet lower along its icy length, and then began frantically to hook and haul himself up toward the third and final height of spar on the foreshortened mainmast, less than fifty feet above the deck.\n\nThe thing roared beneath him. Then came another roar as the second spar, shrouds, tackle, and lines let go and crashed to the deck. The louder of the two roars had been from the monster clinging to the mainmast.\n\nThis ratline was a simple rope which usually hung about eight yards out from the mainmast. It was meant for descending quickly from the crosstrees or upper spars, not for climbing. But Blanky climbed it. Despite the fact that the line was ice-covered and blowing in the snow and despite the fact that Thomas Blanky could no longer feel the fingers on his right hand, he climbed the ratline like a fourteen-year-old midshipman larking in the upperworks with the other ship's boys after supper on a tropical evening.\n\nHe couldn't pull himself onto the top spar \u2014 it was simply too coated with ice \u2014 but he found the shroud lines there and shifted from the ratline to the loosened, folded shroud beneath the spar. Ice broke away and hurtled to the deck below. Blanky imagined \u2014 or hoped \u2014 that he heard a tearing and banging forward, as if Crozier and the crewmen were hacking their way out of the forward battened hatch with axes.\n\nClinging like a spider to the frozen shrouds, Blanky looked down and to his left. Either the driving snow had let up or his night vision had improved, or both. He could see the mass of the monster. It was climbing steadily to his third and final spar level. The shape was so big on the mainmast that Blanky thought it looked like a large cat climbing a very thin tree trunk. Except, of course, thought Blanky, it looked nothing at all like a cat save for the fact that it was climbing by slamming claws deep into ice and royal oak and iron bands that a midweight cannonball could not have penetrated.\n\nBlanky continued edging outward along the shroud, dislodging ice as he went and causing the frozen shroud lines and canvas to creak like overly starched muslin.\n\nThe giant shape behind him had reached the level of the third spar. Blanky felt the spar and shrouds vibrate and then sag as a portion of that massive weight on the mast was shifted to the spars on either side. Imagining the thing's huge forelegs thrown over the spars, imagining a paw the size of his chest freeing itself to slam into the thinner spar up here, Blanky crawled and crabbed faster, almost forty feet out from the mast now, already beyond the edge of the deck fifty feet below. A seaman falling from this far out on the spar or shrouds when working the sails would fall into the sea. If Blanky fell, it would be onto the ice more than sixty feet below.\n\nSomething snagged Blanky's face and shoulders \u2014 a net, a spiderweb, he was trapped \u2014 and for a second he came close to screaming. Then he realized what it was \u2014 the man lines, the threaded squares of primary climbing rope from the railing to the second crosstrees, rerigged for winter to the top of the stump of the mainmast so that work parties could dislodge ice up here. This was the starboard man-line rigging that had been, impossibly, smashed free of its multiple moorings along the rail and deck by two blows of the thing's giant claws. Thick enough with ice now that the squares of interlaced rope acted like small sails, the loosened man lines had blown far out to the starboard side of the ship.\n\nOnce again, Blanky acted before allowing himself time to think about the action. To think about this next move, sixty feet and more above the ice, was to decide not to do it.\n\nHe threw himself from the crackling shrouds onto the swinging man-line rigging.\n\nAs he'd known it would, his sudden weight swung the lines back toward the mainmast. He passed within a foot of the huge, hairy mass at the T of spars. It was too dark to see much more than the terrible general shape of it, but a triangular head as large as Thomas Blanky's torso whipped around on a neck too long and serpentine to be of this world and there was a loud _SNAP_ as teeth longer than Blanky's frozen fingers clamped shut on the air he'd just swung through. The Ice Master inhaled the breath of the thing \u2014 a carnivore and predator's hot rotten-meat exhalation, not the fishy stink Blanky had noticed coming from the open maws of the polar bears they'd shot and skinned on the ice. This was the hot stench of decaying human flesh mixed with sulfur, as warm as the blast from a steam boiler's open hearth.\n\nAt that instant Thomas Blanky realized that the seamen whom he'd silently cursed as being superstitious fools had been right; this thing from the ice was as much demon or god as it was animal flesh and white fur. It was a force to be _appeased_ or worshipped or simply fled.\n\nHe'd half-expected the manrope rigging swinging below him to become stuck in the stub of the spars down there, or to snag in the port-side spar or shrouds as he swung past the centreline \u2014 then all the creature had to do was reel him in like a big fish in a net \u2014 but the momentum of his weight and twisting swung him out fifteen feet or more past and to the port side of the mainmast.\n\nNow the man-line rigging was preparing to swing him back into the huge left forearm that he could see extending through the blowing snow and darkness.\n\nBlanky twisted, threw his weight forward toward the bow, felt the clumsy torn rigging follow his inertia, and then he was swinging both legs free, flailing and kicking for the third-level spar on this side.\n\nHis left boot found it as he swung above it. The lug soles slipped on ice and the boot went past, but when the man line swung back toward the stern, both boots found the ice-coated spar and he pushed with all the energy in his legs.\n\nThe tangled web of man line swung back past the mainmast, but now in a curving arc toward the stern. Blanky's legs were hanging free, still kicking against empty air fifty feet above the ruined tent and stores below, and he arched his back in close to the ropes as he swung toward the mainmast and the thing waiting for him there.\n\nClaws sliced the air not five inches from his back. Even in his terror, Blanky marveled \u2014 he knew that the arc of his kick had put almost ten feet of air between him and the mainmast as he swung past. The thing must have sunk the claws of its right paw \u2014 or hand, or talon, or Devil's nails \u2014 into the mast itself while hanging almost free and swinging six feet or more of massive left arm at him.\n\nBut it had missed.\n\nIt would not miss again when Blanky swung back to the centre.\n\nBlanky grabbed the edge of the man-line rigging and slid down it as quickly he would a free line or ratline, his numbed fingers tearing against the cross ropes, each impact threatening to throw him off the rigging and out into darkness.\n\nThe man line had reached the apogee of its outer arc, somewhere beyond the starboard railing, and was starting to swing back.\n\n _Still too high,_ thought Blanky as the tangle of rope rigging above him swung back to the mainmast.\n\nThe creature easily caught the rigging as it reached the midline of the ship, but Blanky was twenty feet below that level now, using his frozen hands on the crossropes to scramble lower.\n\nThe thing began dragging the entire mass of rigging up to it.\n\n _This is God-fucking wondrous awful,_ Thomas Blanky had time to think as the entire ton or ton and a half of ice-encrusted manrope and human being began being pulled upward as easily and surely as if a fisherman were hauling up his net after a casting.\n\nThe Ice Master did what he had planned in the last ten seconds of inward swing, sliding lower on the rigging at the same time he shifted his weight back and forward \u2014 picturing himself a boy on a rope swing \u2014 increasing his lateral arc even as the thing above pulled him higher. As fast as he clambered down while he swung, the thing hauled him an equal distance closer. He would reach the bottom of the manrope rigging about the time the creature hauled him in and still be fifty feet in the air.\n\nBut there was still enough slack that he could arc twenty feet to starboard, both hands on the vertical lines, legs straightening against the cross-rigging. He closed his eyes and regained the image of a boy on a rope swing.\n\nThere was an anticipatory cough from less than twenty feet above him. Then came a strong jerk and the entire rigging rose another five or eight feet with Blanky on it.\n\nNot knowing whether he was twenty feet above the deck now or forty-five, caring only about the timing of his outward swing, Blanky twisted the rigging around as he swung outward over the starboard darkness, kicked his boots free, and launched himself into the air.\n\nThe fall seemed interminable.\n\nHis first job was to twist again in midair so as not to land on his head or back or belly. There would be no give to the ice \u2014 less, of course, if he struck the railing or deck \u2014 but there was no longer anything he could do about that. The Ice Master knew as he fell that his life now depended upon simple Newtonian arithmetic; Thomas Blanky had become a minor problem in ballistics.\n\nHe sensed the starboard rail going past six feet from his head and only just had time to curl and ready his legs and extended arms before his lower body slammed into the slope of snow and ice that dropped away from the pressure-raised _Terror_ like a ramp. The Ice Master had done the best dead reckoning he could on his blind outward swing, trying to place the end of his falling arc just forward of the cement-hard path of ice the men used in climbing to and from the ship, but also to place his point of impact just aft of the snowy heaps where the whaleboats were shrouded and tied down under frozen canvas and three feet of snow.\n\nHe landed on the snowy incline just forward of the ice ramp and just astern the snow-shrouded boats. The impact knocked the wind out of him. Some muscle tore or bone snapped in his left leg \u2014 Blanky had time for a prayer to whatever gods were awake this night that it was a muscle and not a bone \u2014 and then he was rolling down the long, steep slope, cursing and exclaiming as he went, kicking up his own small flurry of snow and epithets within the larger blizzard blowing around the ship.\n\nThirty feet beyond the ship, somewhere out on the snow-covered sea ice, Blanky rolled to a stop on his back.\n\nHe took stock as quickly as he could. His arms were unbroken, although he'd hurt his right wrist. His head seemed intact. His ribs hurt and he was having trouble taking a breath, but he thought this was probably more the result of fear and excitement than of broken ribs. But his left leg hurt like the very Devil.\n\nBlanky knew that he had to be up and running... _now_... but he couldn't obey his own command. He was completely satisfied lying there on his back, spread-eagled on the dark ice, bleeding heat into the ice beneath him and into the air above him, trying to get his breath and wits back.\n\nNow there were definite human cries and shouts on the foredeck. Spheres of lantern light, none wider than ten feet or so, appeared near the bow, illuminating the hurtling horizontal lines of wind-driven snow. Then Blanky heard the heavy thump and crash as the demonthing slid down the mainmast to the deck. There came more men's shouts \u2014 alarmed now, although they wouldn't be able to see the creature clearly since it was farther astern within the tumble and jumble of broken spars, fallen rigging, and scattered casks amidships. A shotgun roared.\n\nAching, hurting, Thomas Blanky got to all fours on the ice. His undergloves were completely gone now. Both hands were bare. He was also bareheaded, his long grey-streaked hair blowing in the wind, its queue having come unknotted during his contortions. He could not feel his fingers, face, or extremities, but everything in between was giving him one sort of pain or another.\n\nThe creature came hurtling over the starboard railing toward him, the mass of it backlit by lantern-glow, clearing the low barrier with all four huge legs in the air.\n\nIn an instant Blanky was on his feet and running out into the sea ice and serac darkness.\n\nOnly after he'd gotten fifty yards or so from the ship, slipping and falling and rising and running again, did he realize that he may well have just signed his own death warrant.\n\nHe should have stayed close to the ship. He should have run around the snow-heaped boats along the forward starboard length of the hull, clambered over the bowsprit now pressed down deep into the ice, and made for the port side, shouting to the men above for help as he did so.\n\nNo, he realized, he would have been dead before he got through the tangle of bow rigging. The thing would have caught him in ten seconds.\n\n _Why did I run in this direction?_\n\nHe'd had a plan before the deliberate fall from the rigging. What the hell had it been?\n\nBlanky could hear scraping and thudding on the sea ice behind him.\n\nSomeone, perhaps the assistant surgeon from _Erebus,_ Goodsir, had once told him and some other seamen how fast a white bear could charge across sea ice toward its prey \u2014 twenty-five miles per hour? Yes, at least that. Blanky had never been a fast runner. And now he had to dodge seracs and ice ridges and cracks in the ice that he couldn't see until he was a few feet from them.\n\n _That's why I ran this way. That was the plan._\n\nThe creature was loping along behind him, dodging the same jagged seracs and pressure-ridge slabs that Blanky was clumsily slaloming around in the dark. But the Ice Master was panting and wheezing like a torn bellows, while the huge shape behind him was grunting only slightly \u2014 with amusement? anticipation? \u2014 as its forepaws thudded down onto the ice with each stride that was the equal to four or five of Blanky's.\n\nBlanky was in the ice field about two hundred yards from the ship now. Bouncing off an ice boulder he hadn't seen until it was too late to dodge, taking the impact on his right shoulder and feeling that shoulder instantly go numb to join other numb parts of him, the Ice Master realized that he'd been blind as a bat the entire time he'd been running for his life. The lanterns on _Terror_ 's deck were far, far behind him now \u2014 an impossible distance away \u2014 and he didn't have time or reason to turn and look for them. They could give no illumination this far away from the ship, and they could only distract him from what he was doing.\n\nWhat he was doing, Blanky realized, was running and dodging and swerving through his mental map of the ice fields and crevasses and small bergs that surrounded HMS _Terror_ to the horizon. Blanky had had more than a year to stare out at this frozen sea with all its disturbances and ridges and bergs and upthrustings, and for a few months of that time he'd had the thin arctic daylight to see by. Even in winter, there were hours on watch in moonlight and starlight and in the glow of the dancing aurora when he'd studied this circle of ice around the trapped ship with an Ice Master's professional eye.\n\nAbout two hundred yards out here in the ice jumble, beyond a last pressure ridge he'd just stumbled and clambered over \u2014 he could hear the thing leaping it less than ten yards behind him \u2014 he remembered a maze of former bergy bits, small icebergs calved from their larger brethren, upended into a tiny mountain range of cottage-sized ice boulders.\n\nAs if realizing where its doomed prey was headed, the unseen shape behind him grunted and picked up speed.\n\nToo late. Dodging the last of the tall seracs, Blanky was into the bergy labyrinth. Here his mental map failed him \u2014 he'd only seen the miniature berg field from afar or through a telescope \u2014 and he bounced off one ice wall in the dark, fell on his arse, and was scrambling forward on all fours in the snow with the creature closing to within a few yards before Blanky regained his breath and wits.\n\nThe crevice between two carriage-sized bergs was less than three feet wide. Blanky scurried into this \u2014 still on all fours, his bare hands as unfeeling and remote as the black ice under them \u2014 just as the thing reached the crack and swept a giant forepaw in after him.\n\nThe Ice Master rejected all images of mice and cats as impossibly huge claws tossed up ice chips not ten inches from his boot soles. He stood in the narrow gap, fell, got to his feet again, and stumbled forward in near absolute blackness.\n\nThis was no good. The ice alley was too short \u2014 less than eight feet \u2014 and it had dumped Blanky into an open gap beyond it. He could already hear the thing loping and grunting its way around the block of ice to his right. He might as well be on a clear cricket pitch as try to stay here \u2014 and even the crevice, its walls more snow than ice, would be only a temporary hiding place. It was a place to wait for only a minute or so in the darkness until the thing enlarged the opening and clawed its way in. It was only a place in which to die.\n\nThe wind-sculpted little bergs he remembered from looking through his glass were... which way? To his left, he thought.\n\nHe staggered left, bounced off ice pinnacles and seracs that would do him no good, stumbled over a crevasse that dropped only two feet or so, scrambled up a low ridge of serrated ice, slid back, scrambled up again, and heard the thing hurtle around the ice block and slide to a stop not ten feet behind him.\n\nThe larger bergs began just beyond this ice boulder. The one with the hole he'd observed through the glass would be...\n\n... these things move every day, every night of every day...\n\n... they collapse, regrow, and reshape themselves as pressure shoves them willy-nilly...\n\n... the thing is clawing its way up the ice slope behind him onto this flat but dead-end plateau of ice where Blanky now teeters...\n\nShadows. Crevices. Cracks. Dead-end alleys of ice. None large enough for him to slide into. Wait.\n\nThere was a single hole about four feet high on the face of a little upended berg on his right. The clouds parted ever so slightly and five seconds of starlight gave Blanky just enough illumination to see the irregular circle in the wall of dark ice.\n\nHe lunged forward and threw himself into it, not knowing if the ice tunnel was ten yards or ten inches deep. He didn't fit.\n\nThe outer layers of his clothing \u2014 cold-weather slops and greatcoat \u2014 made him too bulky.\n\nBlanky ripped at his clothing. The thing had clawed its way up the final slope and was behind him, rearing onto its hind legs. The Ice Master could not see it \u2014 he did not take time even to turn to look \u2014 but he could _feel_ it rearing.\n\nWithout turning, the Ice Master flung his greatcoat and other layers of outer wool backward at the thing, hurling the heavy garments as quickly as he could.\n\nThere was a _woof_ of surprise \u2014 a gust of sulfurous stench \u2014 and then the tearing noise of Blanky's clothing being ripped apart and tossed far out into the ice maze. But the distraction had bought him five seconds or more.\n\nAgain he threw himself forward into the ice hole.\n\nHis shoulders only just fit. The toes of his boots flailed, slid, finally found grip. His knees and fingers clawed for purchase.\n\nBlanky was only four feet into the hole when the thing reached in after him. First it tore off his right boot and part of his foot. The Ice Master felt the shocking impact of claws into his flesh and thought \u2014 hoped \u2014 that it was only his heel that had been torn away. He had no way of knowing. Gasping, fighting a sudden stab of pain that even cut through the numbness of his injured leg, he clawed and wriggled and forced himself deeper into the hole.\n\nThe ice tunnel was tightening, growing narrower.\n\nClaws raked ice and tore down his left leg, ripping flesh exactly where Blanky had already injured himself in the fall from the rigging. He smelled his own blood and the thing must also have smelled it because it quit clawing for a second. Then it roared.\n\nThe roar was deafening in the ice tunnel. Blanky's shoulders were jammed, he could go no farther forward, and he knew the rear half of his body was still within reach of the monster. It roared again.\n\nBlanky's heart and testicles froze at the sound, but it did not frighten him into immobility. Using his few seconds of reprieve, the Ice Master wriggled backward into the less-restrictive space through which he'd just crawled, forced his arms forward, and kicked and kneed ice with the last of his remaining strength, ripping clothing and skin from his shoulders and sides as he tore through an ice aperture never meant for a man of even his moderate size.\n\nBeyond this narrowest point, the ice tunnel widened and dropped lower. Blanky let himself slide forward on his belly, the slide lubricated by his own blood. His remaining clothing was in tatters. He could feel the encroaching cold of the ice against his clenched belly muscles and tightened scrotum.\n\nThe thing roared a third time but the horrible noise seemed to be a few feet farther away.\n\nAt the last instant, just before he dropped over the edge of the tunnel into an open space, Blanky was sure that all this had been for nothing. The tunnel \u2014 most likely made by melting so many months ago \u2014 had cut through the edge of the little berg but now had dropped him outside again. Suddenly, he was lying on his back under the stars. He could smell and feel his blood soaking into fresh-fallen snow. He could also hear the thing loping around the berg, first to the left, then to the right, eager to get at him, confident, certain now that it could follow the maddening smell of the human's blood to its prey. The Ice Master was too injured and too exhausted to crawl any farther. Let whatever was going to happen to him happen now and may a Sailor's God fuck to Hell this fucking thing that was going to eat him. Blanky's last prayer was that one of his bones would lodge in the thing's throat.\n\nIt was a full minute more and half a dozen more roars \u2014 each growing in volume and frustration, each coming from another point on the dark compass of night around him \u2014 before Blanky realized that the thing couldn't get to him.\n\nHe was lying in the open and under the stars here, but in a box of ice no larger than five feet by eight \u2014 an enclosure created by at least three of the thick bergs being jostled and tumbled together by the pressure of sea ice. One of the tilted bergs hung over him like a falling wall, but Blanky could still see the stars. He could also see starlight coming from two vertical apertures on opposite sides of his ice coffin \u2014 he could _see_ the great mass of the predator blocking the starlight at the end of these cracks, less than fifteen feet from him \u2014 but the gaps between bergs were no more than six inches wide. The melt tunnel he'd crawled through was the only real way into this space.\n\nThe monster roared and paced for another ten minutes.\n\nThomas Blanky forced himself to a sitting position and set his lacerated back and shoulders against the ice. His coats and slops were gone, and his trousers, two sweaters, wool and cotton shirts, and wool undershirt were just bloody tatters, so he prepared to freeze to death.\n\nThe thing was not leaving. It paced round and round the three-berg box like some restless carnivore in one of London's trendy new zoological gardens. But it was Blanky who was in a cage.\n\nHe knew that even if the thing miraculously left, he had neither energy nor will to crawl out through the narrow tunnel. And if he could somehow make his way through the tunnel, he still might as well be on the moon \u2014 the moon which was now coming out from behind the roiling clouds and illuminating the bergs round about in a soft explosion of blue glow. And even if he miraculously crawled out of the bergy field, the three hundred yards to the ship was an impossible distance. He could no longer feel his body or move his legs.\n\nBlanky settled his cold behind and bare feet deeper into the snow \u2014 the accumulation was greater here where the wind did not reach \u2014 and wondered if his fellow Terrors would ever find him. Why should they even look? He was just another of their party who had been carried off by the thing on the ice. At least his disappearance would not require the captain to haul another corpse \u2014 or part of a corpse wastefully wrapped in good ship's sailcloth \u2014 down to the Dead Room.\n\nThere came more roars and noises from the far end of the cracks and tunnel, but Blanky ignored them. \"Fuck you and the sow or devil who spawned you,\" mumbled the Ice Master through numb, frozen lips. Perhaps he did not speak at all. He realized that freezing to death \u2014 even while bleeding to death, although some of the blood from his various wounds and lacerations already seemed to have frozen \u2014 was not painful at all. In truth, it was peaceful... quite restful. A wonderful way to...\n\nBlanky realized that there was light coming through the cracks and tunnel. The thing was using torches and lanterns to fool him into coming out. But he wouldn't fall for that old ruse. He would stay silent until the light went away, until he slipped that last small bit into soft, eternal sleep. He would not give the thing the satisfaction of hearing him speak now after their long, silent duel.\n\n\"God- _damn_ it, Mr. Blanky!\" boomed Captain Crozier's bass bellow down the ice tunnel. \"If you're in there, _answer,_ God-damn it, or we'll just leave you there.\"\n\nBlanky blinked. Or rather, he _tried_ to blink. His lashes and eyelids were frozen. Was this some other ruse and stratagem of the demonthing?\n\n\"Here,\" he croaked. And again, aloud this time, \"Here!\"\n\nA minute later, the head and shoulders of the caulker's mate, Cornelius Hickey, one of the smallest men on _Terror,_ poked easily through the hole. He was carrying a lantern. Blanky thought dully that it was like watching a gimlet-faced gnome being born.\n\nIn the end, all four surgeons had a go at him.\n\nBlanky came in and out of his pleasant fog from time to time to see how things were progressing. Sometimes it was his own ship's surgeons working on him \u2014 Peddie and McDonald \u2014 and sometimes it was _Erebus_ 's sawbones, Stanley and Goodsir. Sometimes it was just one of the four cutting or sawing or packing or stitching away. Blanky had the urge to tell Goodsir that polar white bears could run much faster than twenty-five miles per hour when they put their mind to it. But then again \u2014 had it been a polar white bear? Blanky did not think so. Polar white bears were creatures of this earth, and this thing had come from somewhere else. Ice Master Thomas Blanky had no doubt of that.\n\nIn the end, the butcher's bill was not so bad. Not bad at all, really.\n\nJohn Handford, it turned out, had not been touched. After Blanky had left him with the lantern, the man on starboard watch had doused this light and fled the ship, running around to the port side to hide while the creature was climbing up to get to the Ice Master.\n\nAlexander Berry, whom Blanky had presumed dead, had been found under the fallen canvas and scattered kegs right where he'd been standing port watch when the thing had first appeared there and then shattered the fore and aft ridgepole spar. Berry had hit his head seriously enough to have no memory of anything that happened that night, but Crozier told Blanky that they'd found the man's shotgun and it had been fired. The Ice Master also had fired his, of course, from point-blank range at a shape that was looming over him like a pub wall, but there had been no trace of the thing's blood anywhere on the deck at either site.\n\nCrozier asked Blanky how this could be \u2014 how could two men fire shotguns at an animal at point-blank range and draw no blood? \u2014 but the Ice Master ventured no opinion. Inside, of course, he _knew._\n\nDavey Leys was also alive and unharmed. The forty-year-old on bow watch must have seen and heard much \u2014 including quite possibly the thing on the ice's first appearance on deck \u2014 but Leys was not talking about it. Once again David Leys could only stare silently. He was taken first to _Terror_ 's sick bay, but since all of the surgeons needed that bloodstained space to work on Blanky, Leys was transported by litter to _Erebus_ 's more spacious sick quarters. There Leys lay, according to the Ice Master's talkative visitors, once again staring unblinkingly at the overhead beams.\n\nBlanky himself had not come through unscathed. The thing had clawed off half of his right foot at the heel, but McDonald and Goodsir had cut and cauterized what was left and assured the Ice Master that \u2014 with the help of the carpenter or ship's armourer \u2014 they would rig a leather or wood prosthesis held on by straps so that he could walk again.\n\nHis left leg had taken the worst of the creature's abuse \u2014 flesh raked away to the bone in several places and then the long leg bone itself striated with claws \u2014 and Dr. Peddie later confessed that all four of the surgeons had been sure they would have to amputate it at the knee. But slowness of infection and gangrene in a wound was one of the few blessings of the arctic, and after resetting the bone itself and receiving more than four hundred stitches, Blanky's leg \u2014 although twisted and wildly scarred and lacking entire tracts of muscle here and there \u2014 was healing slowly. \"Your grandkids will love them scars,\" said James Reid when the other Ice Master made a courtesy call.\n\nThe cold had also taken its toll. Blanky managed to keep all of his toes \u2014 he would need them for balance on the ruined foot, the surgeons told him \u2014 but had lost all fingers save for his thumb on his right hand and the two smallest fingers and his thumb on his left hand. Goodsir, who evidently knew something about such things, assured him that someday he would be able to write and eat gracefully with just the remaining adjoining two fingers on his left hand, and be able to button his trousers and shirts again with those two fingers and the thumb on his right hand.\n\nThomas Blanky did not give a good gob fart about buttoning his trousers and shirts. Not yet. He was alive. The thing on the ice had done its best to make him otherwise, but he was still alive. He could taste food, chat with his mates, drink his daily gill of rum \u2014 already his bandaged hands were capable of holding his pewter mug \u2014 and read a book if someone propped it up for him. He was determined to read _The Vicar of Wakefield_ before he shuffled off what was left of his mortal coil.\n\nBlanky was alive and he planned to stay that way for as long as he could. In the meantime, he was strangely happy. He was looking forward to getting back to his own cubicle aft \u2014 between Third Lieutenant Irving's and Jopson's,the captain's steward's, equally tiny berths \u2014 and that would happen any day now, whenever the surgeons were absolutely sure they were done snipping and stitching and sniffing at his wounds.\n\nIn the meantime, Thomas Blanky was happy. Lying on his sick bay bunk late at night, the men grousing and whispering and farting and laughing in the darkened berthing space just a few feet beyond the partition, hearing Mr. Diggle growl out his commands at his lackeys as the cook baked biscuits deep into the night, Thomas Blanky listened to the grind and growl of the sea ice as it tried to crush HMS _Terror_ and allowed it to put him to sleep as surely as would a lullaby from his long-sainted mother's lips.\n\nIRVING\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _13 December, 1847_\n\nThird Lieutenant John Irving needed to know how Silence got on and off the ship without being seen. Tonight, one month to the day since he'd first found the Esquimaux woman in her lair, he would solve the puzzle if it cost him his toes and fingers.\n\nThe day after he first found her, Irving reported to his captain that the Esquimaux woman had moved her den to the forward cable locker on the hold deck. He did not report that she appeared to be eating fresh meat in there, mostly because he doubted what he had seen in that terrifying second of staring into the small flame-lit space. Nor had he reported the apparent sodomy he'd interrupted in the hold between Caulker's Mate Hickey and Seaman Manson. Irving knew that he was abrogating his professional duty as an officer in the Royal Navy's Discovery Service by not informing his captain of this shocking and important fact, but...\n\nBut what? All John Irving could think of as a reason for his serious breach of duty was that HMS _Terror_ had enough rats aboard it already.\n\nBut Lady Silence's apparently magical appearances and disappearances \u2014 although accepted by the superstitious crew as final evidence of her witchcraft and ignored by Captain Crozier and the other officers as a myth \u2014 seemed far more important to young Irving than whether a caulker's mate and shipboard idiot were pleasuring each other in the stinking darkness of the hold.\n\nAnd it _was_ a stinking darkness, thought Irving, in the third hour of his watch crouched on a crate above the slush and behind a pillar near the forward cable locker. The stench in the freezing, dark hold was getting worse by the day.\n\nAt least there were no more half-eaten plates of food, tots of rum, or pagan fetishes on the low platform outside the cable locker. One of the other officers had brought this practice to Crozier's attention shortly after Mr. Blanky's amazing escape from the thing on the ice, and the captain had flown into a fury, threatening to cut off the rum ration \u2014 _forever_ \u2014 of the next man stupid enough, superstitious enough, addle-brained enough, and generally _un-Christian_ enough to offer up scraps of food or mugs of perfectly good watered-down Indian rum to a _native woman. A heathen child._ (Although those sailors who had gained a peek of Lady Silence naked, or heard the surgeons discussing her, knew that she was no child and muttered as much to one another.)\n\nCaptain Crozier had also made it completely clear that he would tolerate no show of white-bear fetishes. He announced at the previous day's Divine Service \u2014 actually a reading of Ship's Articles, although many of the men were eager for more words from the _Book of Leviathan_ \u2014 that he would add one extra late-night watch or two seats-of-ease thunder-jar disposal duties to each man for every single bear tooth, bear claw, bear tail, new tattoo, or other fetish item he saw on that hapless sailor. Suddenly the enthusiasm for pagan fetishes became invisible on HMS _Terror_ \u2014 although Lieutenant Irving heard from his friends on _Erebus_ that it was still thriving there.\n\nSeveral times Irving had tried to follow the Esquimaux in her furtive movements around the ship at night, but \u2014 not wanting her to know that he was following her \u2014 he had lost her. Tonight he knew that Lady Silence was in her locker. He had followed her down the main ladder more than three hours ago, after the men's supper and then after she had quietly and almost invisibly received her portion of \"Poor John\" cod and a biscuit and glass of water from Mr. Diggle and gone below with it. Irving posted a man at the forward hatch just forward of the huge stove and another curious sailor to watch the main ladderway. He arranged for these watches to trade off every four hours. If the Esquimaux woman climbed either ladder tonight \u2014 it was already past 10:00 p.m. \u2014 Irving would know where she went and when.\n\nBut for three hours now the cable locker doors had been tightly shut. The only illumination in this forward part of the hold had been the slightest leakage of light around the edges of those low, wide locker doors. The woman still had some source of illumination in there \u2014 either a candle or other open flame. This fact alone would cause Captain Crozier to have her plucked out of the cable locker in a minute and returned to her little den in the storage area forward of the lower-deck sick bay... or thrown out onto the ice. The captain feared fire in the ship as much as any other veteran sailor and he seemed to harbour no sentimental feelings toward their Esquimaux guest.\n\nSuddenly the dim rectangle of light around the ill-fitted locker doors disappeared.\n\n _She's gone to sleep,_ thought Irving. He could imagine her \u2014 naked, just as he'd seen her, pulling her cocoon of furs around her in there. Irving also could imagine one of the other officers hunting for him in the morning and finding his lifeless body curled here on a crate above the slush-flooded hull, obviously an ungentlemanly cad who had frozen to death while trying to sneak a peek at the only woman on board. It would not be an heroic death report for Lieutenant John Irving's poor parents to read.\n\nAt that moment a veritable breeze of icy air moved through the already frigid hold. It was as if a malevolent spirit had brushed past him in the darkness. For a second, Irving felt the hairs on the back of his neck rise up, but then a simple thought struck him \u2014 _it's just a draft. As if someone has opened a door or window._\n\nHe knew then how Lady Silence magically came and went from the _Terror._\n\nIrving lit his own lantern, jumped off the crate, splashed through the sludge-slush, and tugged at the doors of the cable locker. They were secured from the inside. Irving knew that there was no lock _inside_ the forward cable locker \u2014 there wasn't even a lock on the outside since no one had any reason to attempt to steal cable hawsers \u2014 so therefore the native woman herself had found a way to secure it.\n\nIrving had prepared for this contingency. He carried a thirty-inch pry bar in his right hand. Knowing that he would have to explain any damage to Lieutenant Little and possibly to Captain Crozier, he jammed the narrow end of the bar in the crack between the three-foot-high doors and leveraged hard. There came a creaking and groaning but the doors opened only an inch or two. Still holding the pry bar in place with one hand, Irving reached under his slops, greatcoat, undercoat, and waistcoat and pulled his boat knife from his belt.\n\nLady Silence had somehow driven nails into the backsides of the cable locker doors and run some sort of elastic rawhide material \u2014 gut? sinew? \u2014 back and forth until the doors were secured as if by a white spiderweb. There was no way that Irving could enter now without leaving a clear trace that he'd been there \u2014 the pry bar had already seen to that \u2014 so he used the knife to slash through the cat's cradle of sinew. It was not easy. The strands of sinew were more resistant to the sharp blade than rawhide or ship's rope.\n\nWhen the strands finally fell away, Irving extended the hissing lantern into the low space.\n\nThe little cave-den he'd seen four weeks earlier was, except for the absence of any flame now other than his lantern's, just as he remembered \u2014 the coiled hawsers pushed back and pulled almost overhead to create a sort of cave within the raised locker area \u2014 and there were the same signs that she'd been eating meals there: one of _Terror_ 's pewter plates with only a few crumbs of Poor John remaining on it, a pewter grog mug, and some sort of storage bag that looked as if Silence had stitched it together from scraps of discarded sail canvas. Also on the locker deck was one of the ship's small oil lanterns \u2014 the kind with just enough oil in it for the men to use when going above to one of the seats of ease at night. Its flue was still very warm to the touch when Irving removed his mitten and glove.\n\nBut no Lady Silence.\n\nIrving could have tugged and jerked the heavy hawsers this way and that to look behind them, but he knew from experience that the rest of the triangular cable-locker space was crammed tight with the anchor ropes. Two and a half years since they had sailed and they still carried the stink of the Thames on them.\n\nBut Lady Silence was gone. There was no way up through the deck and beams above or out through the hull. So were the superstitious seamen correct? Was she was an Esquimaux witch? A she-shaman? A pagan sorcerer?\n\nThird Lieutenant John Irving did not believe it. He noticed that the active breeze was no longer flowing around him. Yet the flame of his lantern still danced to some smaller draft.\n\nIrving moved the lantern around at arm's length \u2014 that was all the free space there was in the crowded and cramped hawser locker \u2014 and stopped when the lamp flame danced the most: forward just starboard of the apex of the bow.\n\nHe set the lantern down and began moving hawser aside. Immediately Irving saw how cleverly she had arranged the massive anchor line here \u2014 what appeared to be another huge coil of hawser was merely a curled section of another coil set into an empty space to simulate a stack of hawser, easy to pull aside into her den-space. Behind the faux hawser coil was the curve of wide hull timbers.\n\nOnce again she had chosen carefully. Above and beneath the cable locker ran a complex webbing of wood and iron beams set in place during HMS _Terror_ 's refitting for ice service some months before the expedition sailed. Up here by the bow there were iron vertical beams, oak crossbeams, triple-thick support struts, iron triangular supports, and huge oak diagonal beams \u2014 many as thick as primary hull timbers \u2014 lacing back and forth as part of the ship's modern-design reinforcement for the polar ice. One London reporter, Lieutenant Irving knew, had described all the tons of iron and oak internal reinforcements, as well as the addition of African oak, Canadian elm, and more African oak to the English oak on the sides of the hull, as being enough to make \"a mass of timber about eight feet thick.\"\n\nThat was almost literally true for the actual bow and for the sides of the hull, Irving knew, but here where the last five feet or so of hull timber met at the bow in and above the cable locker, there were only the original six inches of stout English oak for the hull boards rather than the ten inches of layered hardwoods found elsewhere along both sides of the hull. It was thought that the areas a few feet to the immediate port and starboard of the heavily reinforced bow stem would have to be of fewer layers in order to flex during the terrible stresses of ice-breaking.\n\nAnd so they had. The five belts of lumber at the sides of the hull, combined with the iron-and-oak-reinforced bow and internal areas, had produced a marvel of modern ice-breaking technology that no other Navy or civilian expedition service in the world could match. _Terror_ and _Erebus_ had gone places where no other ice ship on earth could hope to survive.\n\nThis bow area was a marvel. But it was no longer secure.\n\nIt took Irving several minutes to find it, extending the lantern for drafts and feeling with his freezing bare fingers and probing with his knife blade to see where a three-foot section of the foot-and-a-half-wide hull timber had been loosened. There. The aft end of the single curved board was secured by two long nails that now worked as a sort of hinge. The forward end \u2014 only a few feet from the huge bow and keel timber that ran the length of the ship \u2014 had been only pressed into place.\n\nWorking the hull timber loose with the pry bar \u2014 wondering how on earth the young woman could have done this with only her fingers \u2014 and then letting it drop, Irving felt the blast of cold air and found himself looking into darkness through an eighteen-inch-by-three-foot gap in the hull.\n\nThis was impossible. The young lieutenant knew that _Terror_ 's bows had been armoured for twenty feet back from their stems with inch-thick rolled and tempered plates of specially fitted sheet iron. Even if an internal timber were somehow dislodged, the ship's bow areas \u2014 for almost a third of the way aft \u2014 were armour-plated.\n\nNot now. The cold blew in from ice-black cave darkness beyond the dislodged plank. This part of the bow had been forced under the ice by the ship's constant tilt forward as ice built under _Terror_ 's stern.\n\nLieutenant Irving's heart pounded furiously. If _Terror_ were to be refloated miraculously tomorrow, she would sink.\n\nCould Lady Silence have done this to the ship? The thought terrified Irving more than any belief in her magical ability to appear and disappear at will. Could a young woman not yet twenty years old rip iron hull plates off a ship, dislodge heavy bow timbers that it had taken a shipyard to bend and nail into place, and know _exactly_ where to do all this so sixty men aboard who knew the ship better than their mothers' faces would not notice?\n\nAlready on his knees in the low place, Irving found that he was breathing through his open mouth, his heart still pounding.\n\nHe had to believe that _Terror_ 's two summers of wild battle with the ice \u2014 across Baffin Bay, through Lancaster Sound, all the way around Cornwallis Island before the winter at Beechey Island, the next summer crashing south down the sound and then through what the men were now calling Franklin's Strait \u2014 somewhere there toward the end, some of the iron bow armour below the waterline must have been dislodged and this thick hull timber displaced inward only _after_ the ice had seized the ship in its grasp.\n\n _But could something other than the ice have loosened the oak hull timber? Was it something else \u2014 something trying to get_ in?\n\nIt didn't matter now. Lady Silence could not have been gone more than a few minutes and John Irving was dedicated to following her, not only to see where she went out there in the darkness but to see if \u2014 somehow, impossibly, miraculously, given the thickness of the ice and the terrible cold \u2014 she was finding and killing her own fresh fish or game.\n\nIf she was, Irving knew, this fact might save them all. Lieutenant Irving had heard what the others had heard about the spoilage in the Goldner canned stores. Everyone aboard both ships had heard the whispers that they would be out of provisions before next summer.\n\nHe couldn't fit through the hole.\n\nIrving pried at the surrounding hull timbers, but everything save for this one hinged board was rock solid. This eighteen-inch-by-three-foot gap in the hull was the only way out. And he was too bulky.\n\nHe stripped off his oilskin slops, his heavy greatcoat, his comforter, cap, and Welsh wig and shoved them through the gap ahead of him... he was still too wide in the shoulders and upper body, although he was one of the thinnest officers aboard. Shaking from the cold, Irving unbuttoned his waistcoat and the wool sweater he wore under that, shoving them through the black aperture as well.\n\nIf he couldn't get out through the hull now, he'd have the Devil's own time explaining why he came back up from the hold minus all of his outer layers.\n\nHe did fit. Just barely. Grunting and cursing, Irving squeezed through the tight space, buttons tearing off his wool shirt.\n\n _I'm outside the ship, under the ice,_ he thought. The idea did not seem quite real.\n\nHe was in a narrow cave in the ice that had built up around the bow and bowsprit. There was no room for him to get back into his coats and clothes, so he pushed them on ahead of him. He considered reaching back into the cable locker for the lantern, but a full moon had been in the sky when he'd been officer on watch a few hours earlier. In the end, he took the metal pry bar instead.\n\nThe ice cave must have been at least as long as the bowsprit \u2014 more than eighteen feet \u2014 and indeed may have been created by the heavy bowsprit beam's working of the ice here during the brief thaw and freeze cycles of the previous summer. When Irving finally emerged from the tunnel it took him a few extra seconds of crawling before he realized that he was out \u2014 the thin bowsprit, its mass of lashed rigging, and curtains of frozen jib shrouds still loomed over him, blocking, he realized, not only his view of the sky but also any chance for the man on bow watch to see _him._ And out here beyond the bowsprit, with _Terror_ only a huge black silhouette looming above, the ice illuminated only by a few thin lantern beams, the way forward continued into and through the jumble of ice blocks and seracs.\n\nShaking hard, Irving tugged on his various layers. His hands were shaking too fiercely for him to button his wool waistcoat, but that didn't matter. The greatcoat was hard to pull on but at least the buttons were much larger. By the time he had his oil slops on, the young lieutenant was frozen to the bone.\n\n _Which way?_\n\nThe ice jumble here, fifty feet beyond the ship's bow, was a forest of ice boulders and wind-sculpted seracs \u2014 Silence could have gone in any direction \u2014 but the ice seemed worn down in a roughly straight line out from the ice tunnel into the ship. At the very least it offered the path of least resistance \u2014 and most concealment \u2014 away from the ship. Getting to his feet, lifting the pry bar in his right hand, Irving followed the slippery ice trough toward the west.\n\nHe would never have found her had it not been for the unearthly sound.\n\nHe was several hundred yards from the ship now, lost in the ice maze \u2014 the blue-ice trough underfoot had long since disappeared, or rather been joined by a score of other such grooves \u2014 and although the light from the full moon and stars illuminated everything as if it were day, he had seen no movement, nor footprints in the snow.\n\nThen came the unearthly wailing.\n\nNo, he realized, stopping in his tracks and trembling all over \u2014 he had been shaking from the cold for many minutes but now the trembling went deeper \u2014 this was not _wailing._ Not of the sort a human being can make. This was the amelodic playing of some infinitely strange musical instrument... part muffled bagpipe, part horn hoot, part oboe, part flute, part human chant. It was loud enough for him to hear dozens of yards away but almost certainly not audible on the deck of the ship \u2014 especially since the wind, most unusually, was blowing from the southeast this night. Yet all the tones were one blended sound from one instrument. Irving had never heard anything like it.\n\nThe playing \u2014 which seemed to begin suddenly, increase its rhythm almost sexually, and then stop abruptly, as if in physical climax and not in the least as if someone was following notes on a sheet of music \u2014 was coming from a serac field near a high pressure ridge less than thirty yards to the north of the torch-cairn path Captain Crozier insisted on maintaining between _Terror_ and _Erebus._ No one was working on the cairns tonight; Irving had the frozen ocean to himself. To himself and to whoever or whatever was producing that music.\n\nHe crept through the blue-lit maze of ice boulders and tall seracs. Whenever he became disoriented, he would look up at the full moon. The yellow orb looked more like another full-sized planet suddenly looming in the starlit sky than like any moon Irving remembered from his years ashore or brief assignments at sea. The air around it seemed to quake with the cold, as if the atmosphere itself were on the verge of freezing solid. Ice crystals in the upper air had created a huge double halo encircling the moon, the lower bands of both circles invisible behind the pressure ridge and icebergs round about. Set around the outer halo, like diamonds on a silver ring, were three bright, glowing crosses.\n\nThe lieutenant had seen this phenomenon several times before this during their night-winters up here near the north pole. Ice Master Blanky had explained that it was just the moonlight refracting off ice crystals the way a light would through a diamond, but it added to Irving's sense of religious awe and wonder here in the blue-glowing ice field as that odd instrument began hooting and moaning again \u2014 just yards away behind the ice now \u2014 its tempo again hurrying to an almost ecstatic pace before suddenly breaking off.\n\nIrving tried to imagine Lady Silence playing some hitherto unseen Esquimaux instrument \u2014 some caribou-antler variation on a Bavarian fl\u00fcgelhorn, say \u2014 but he rejected the idea as silly. First of all, she and the man who had died had arrived with no such instrument. And second, Irving had the strangest feeling that it was not Lady Silence who was playing this unseen instrument.\n\nCrawling over the last low pressure ridge between him and the seracs from whence the sound was coming, Irving continued forward on all fours, not wanting the crunch of his lug-soled boots to be heard on the hard ice or soft snow.\n\nThe hooting \u2014 seemingly just behind the next blue-glowing serac, this one carved by wind into something like a thick flag \u2014 had begun again, rising quickly to the loudest, fastest, deepest, and most frenzied noise Irving had heard so far. To his amazement, he found that he had an erection. Something about this instrument's deep, booming, reed-fluttering sound was so... _primal_... that it quite literally stirred his loins even as he shivered.\n\nHe peered around the last serac.\n\nLady Silence was about twenty feet away across a smooth blue-ice space. Seracs and ice boulders circled the spot, making Irving feel as if he'd suddenly found himself amid a Stonehenge circle in the ice-haloed and star-crossed moonlight. Even the shadows here were blue.\n\nShe was naked, kneeling on thick furs that must be her parka. Her back was in three-quarters profile to Irving and while he could see the curve of her right breast, he could also see the bright moonlight illuminating her long, straight, black hair and setting silver highlights on the hillocked flesh of her firm backside. Irving's heart was pounding so hard that he was afraid she might hear it.\n\nSilence was not alone. Something else filled the dark gap between Druidic ice boulders on the opposite side of the clearing, just beyond the Esquimaux woman.\n\nIrving knew it was the thing from the ice. White bear or white demon, it was here with them \u2014 almost atop the young woman, looming over her. As much as the lieutenant strained his eyes, it was hard to make out the shape \u2014 white-blue fur against white-blue ice, heavy muscles against heavy ridges of snow and ice, black eyes that might or might not be separate from the absolute blackness behind the thing.\n\nThe triangular head on the strangely long bear neck was weaving and bobbing like a snake, he saw now, six feet above and beyond the kneeling woman. Irving tried to estimate the size of the creature's head \u2014 for future reference in terms of killing it \u2014 but it was impossible to isolate the precise shape or size of the triangular mass with its coal-black eyes because of its odd and constant movement.\n\nBut the thing was looming over the girl. Its head was almost directly above her now.\n\nIrving knew that he should cry out \u2014 rush forward with the pry bar in his mittened hand since he had brought no other weapon except for his resheathed ship's knife \u2014 and try to save the woman, but his muscles would not have obeyed such a command at that moment. It was everything he could do to keep watching in a sort of sexually excited horror.\n\nLady Silence had extended her arms, palms up, like a popish priest saying Mass and inviting the miracle of the Eucharist. Irving had a cousin in Ireland who was popish, and he'd actually gone to a Catholic service with him once during a visit. The same sense of strange magical ceremony was being played out here in the blue moonlight. Silence, without a tongue, made no noise, but her arms were thrown wide, her eyes were closed, her head was thrown back \u2014 Irving had crawled far enough forward that he could see her face now \u2014 and her mouth was open and wide, like a supplicant awaiting Communion.\n\nThe creature's neck thrust forward and down as quickly as a cobra's strike and the thing's jaws opened wide and seemed to snap shut on Lady Silence's lower face, devouring half her head.\n\nIrving almost screamed then. Only the ceremonial _heaviness_ of the moment and his own incapacitating fear kept him silent.\n\nThe thing had not devoured her. Irving realized that he was looking at the top of the monster's blue-white head \u2014 a head at least three times larger than the woman's \u2014 as it had closed, but not snapped shut, its giant jaws fitting over her open mouth and upthrust jaw. Her arms were still flung wide to the night, almost as if ready to embrace the gigantic mass of hair and muscle enfolding her.\n\nThe music began then.\n\nIrving saw the bobbing of both heads \u2014 creature's and Esquimaux's \u2014 but it took him half a minute before he realized that the orgiastic bass hootings and erotic bagpipe-flute notes were emanating from... _the woman._\n\nThe monstrous thing looming as large as the ice boulders beside it, white bear or demon, was blowing down into her open mouth, playing her vocal cords as if her human throat were a reed instrument. The trills and low notes and bass resonances came louder, faster, more urgently \u2014 he saw Lady Silence raise her head and bend her neck one way while the serpentine-necked, triangular-headed bear-thing above her bent its head and neck in the opposite direction, the two looking like nothing so much as lovers straining to plunge deeper while seeking to find the best and deepest angle for a passionate open-mouthed kiss.\n\nThe musical notes pounded faster and faster \u2014 Irving was sure that the rhythm must be heard on the ship now, must be giving every man on the ship as powerful and permanent an erection as he was suffering at this second \u2014 and then suddenly, without warning, the noise cut off with the suddenness of the climax of wild lovemaking.\n\nThe thing's head reared up and back. The white neck bobbed and coiled.\n\nLady Silence's arms dropped to her naked sides as if she was too exhausted or transported to hold them out any longer. Her head lolled forward over her moon-silvered breasts.\n\n _It will devour her now,_ thought Irving through all the insulating layers of numbness and disbelief at what he had just seen. _It will rend and eat her now._\n\nIt did not. For a second the bobbing white mass was gone, shuffled swiftly away on all fours through the blue Stonehenge of ice pillars, and then it was back, bowing its head low before Lady Silence, dropping something onto the ice in front of her. Irving could hear the noise of something organic hitting the ice and the smack had a familiar ring to it, but right now nothing was in context \u2014 Irving could make sense of nothing he saw or heard.\n\nThe white thing ambled away again; Irving could feel the impact of its huge feet through the solid sea ice. In a minute it was back, dropping something else in front of the Esquimaux girl. Then a third time.\n\nAnd then it was simply gone... blended back into the darkness. The young woman was kneeling alone in the ice clearing with only the low heap of dark shapes in front of her.\n\nShe remained that way for another minute. Irving thought of his distant Irish cousin's popish church again and the old parishioners who stayed praying in their pews after the service had finished. Then she got to her feet, quickly slipping her bare feet into fur boots and pulling on her fur pants and parka.\n\nLieutenant Irving realized that he was shaking wildly. At least part of that was from the cold, he knew. He'd be lucky if he had enough warmth in him and strength in his legs to get back to the ship alive. He had no idea how the girl had survived her nakedness.\n\nSilence swept up the objects the thing had dropped in front of her and was now carrying them carefully in her fur-parka arms, the way a woman would carry one or more infants still suckling at her breast. She seemed to be heading back to the ship, crossing the clearing to a point between the Stonehenge seracs about ten degrees to his left.\n\nSuddenly she stopped, her hooded head turning in his direction, and although he could not see her black eyes, he could feel her gaze boring into him. Still on all fours, he realized that he was in full sight in the bright moonlight, three feet away from the concealment of any serac. In his absolute need to get a better view, he had forgotten to stay hidden.\n\nFor a long moment neither of them moved. Irving could not breathe. He waited for her motion, a slapping of ice perhaps, and then for the quick return of the thing from the ice. Her protector. Her avenger. His destroyer.\n\nHer hooded gaze moved away and she walked on, disappearing between the ice pillars on the southeast side of the circle.\n\nIrving waited another several minutes, still shaking as if from ague, and then he struggled to his feet. His body was frozen through, its only sensation coming from his now detumescing, burning erection and from his uncontrollable shaking, but instead of staggering toward the ship after the girl, he moved forward to where she had knelt in the moonlight.\n\nThere was blood on the ice. The stains were black in the bright blue moonlight. Lieutenant Irving knelt, tugged off his mitten and underglove, set some of the spreading stain to his finger, and tasted it. It was blood, but he did not think it was human blood.\n\nThe thing had brought her raw, warm, freshly killed meat. Some sort of flesh. The blood tasted coppery to Irving, the way his own blood or any human blood would, but he assumed that freshly killed animals also had such coppery-tasting blood. But what animal and from where? The men of the Franklin Expedition had seen no land animals for more than a year.\n\nBlood freezes in a few fast minutes. This thing had killed its gift to Lady Silence only minutes ago, even as Irving had been stumbling around out here in the ice maze trying to find her.\n\nBacking away from the black stain in the moonlit snow the way he might back away from a pagan stone altar where some innocent victim had just been sacrificed, Irving concentrated first on trying to breathe normally \u2014 the air was ripping at his lungs as he gasped \u2014 and then on urging his frozen legs and numbed mind to get him back to the ship.\n\nHe would not try going in through the ice tunnel and loose plank to the cable locker. He would hail the starboard lookout before he got in shotgun range and walk up the ice ramp like a man, answering no questions until he spoke to the captain.\n\nWould he tell the captain about this?\n\nIrving had no idea. He didn't even know if the thing on the ice \u2014 which must still be nearby \u2014 would let him return to the ship. He didn't know if he had the warmth and energy remaining for the long walk.\n\nHe only knew that he would never be the same again.\n\nIrving turned to the southeast and reentered the forest of ice.\n\nHICKEY\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _18 December, 1847_\n\nHickey had decided that the tall, skinny lieutenant \u2014 Irving \u2014 had to die and that today was the day to do it.\n\nThe diminutive caulker's mate had nothing personal against the naive young toff, other than his poor timing in the hold more than a month earlier, but that was enough to swing the scales against Irving.\n\nWork and watch schedules kept Hickey from his task. Twice he had rotated onto watch duties when Irving was officer on deck, but Magnus Manson had not been on duty abovedecks either time. Hickey would plan the timing and method of the deed, but he needed Magnus for the execution. It was not that Cornelius Hickey was afraid of killing a man; he'd cut a man's throat before he was old enough to go into a whorehouse without a sponsor. No, it was simply the means and method that this murder called for, which required his idiot disciple and arse-fuck buddy on this expedition, Magnus Manson.\n\nNow all the conditions were perfect. It was a Friday morning work party \u2014 although \"morning\" meant little when it was as dark out as midnight \u2014 with more than thirty men out on the ice repairing and improving the torch-cairns between _Terror_ and _Erebus._ Nine musket-armed Marines were, in theory, providing security for the work parties, but in truth the line of working men was spread out for almost a mile, with only five men or fewer under the command of each officer. The three officers here on the east half of the dark cairn trail were from _Terror_ \u2014 Lieutenants Little, Hodgson, and Irving \u2014 and Hickey had helped sort the work parties so that he and Magnus were working on the farthest cairns under Irving.\n\nThe Marines were out of sight most of the time, supposedly prepared to come running should there be an alarm but really just doing their best to stay warm near the fire roaring in the iron brazier set up near the highest pressure ridge less than a quarter mile from the ship. John Bates and Bill Sinclair were also working under Lieutenant Irving this morning, but the two were chums \u2014 and lazy \u2014 and tended to stay out of the young officer's sight so they could work at the next ice cairn as slowly as they pleased.\n\nThe day, though dark as night, was not as cold as some recently \u2014 perhaps only forty-five below out \u2014 and almost windless. There was no moon or aurora, but the stars vibrated in the morning sky, shedding enough light that if a man had to walk out of the range of a lantern or torch, he could see well enough to make his way back. With the thing on the ice still out there in the darkness somewhere, not many men wandered far. Still, the very nature of finding and stacking the correctly sized ice chips and blocks to repair and enlarge a proper five-foot-tall cairn required the men to keep wandering in and out of the lantern light.\n\nIrving was checking on both cairns while frequently giving the men a hand with the physical labour. Hickey only had to wait until Bates and Sinclair were out of sight beyond the curve in the trail through the ice blocks and Lieutenant Irving's guard was down.\n\nThe caulker's mate could have used a hundred iron or steel instruments from the ship \u2014 a Royal Navy vessel was a treasure trove of murder weapons, some of them quite ingenious \u2014 but he preferred that Magnus simply blindside the blond-haired dandy of an officer, haul him off twenty yards or so into the ice, break his neck, then \u2014 when he was well and truly dead \u2014 rip some of the toff's clothing off, smash in his ribs, kick in his pink-cheeked happy face and teeth, break an arm and two legs (or a leg and two arms), and leave the corpse there on the ice to be found. Hickey had already chosen the killing ground \u2014 an area of tall seracs and with no snow underfoot in which Manson would leave boot prints. He'd warned Magnus not to get the lieutenant's blood on him, not to leave any sign that he'd been there with him, and, most important, not to take time to rob the man.\n\nThe thing on the ice had killed men with about every variation of violence imaginable, and if the physical damage to poor Lieutenant Irving was vicious enough, no one on either ship would give a second thought as to what happened. Lieutenant John Irving would be just another canvas-wrapped corpse for _Terror_ 's Dead Room.\n\nMagnus Manson was not a born killer \u2014 just a born idiot \u2014 but he'd murdered men for his caulker's-mate lord and master before. It would not bother him to do so again. Cornelius Hickey doubted that Magnus would even ask himself why the lieutenant had to die \u2014 it was just another order from his master to be obeyed. So Hickey was surprised when the physical giant pulled him aside when Lieutenant Irving was out of earshot and whispered with some urgency, \"His ghost won't haunt me, will it, Cornelius?\"\n\nHickey patted his huge partner on the back. \"Of course not, Magnus. I wouldn't tell you to do nothing that led to having a ghost haunt you, now would I, love?\"\n\n\"No, no,\" rumbled Manson, shaking his head in agreement. His wild hair and beard seemed to leap out from under the wool comforter and Welsh wig. His heavy brow furrowed. \"By _why_ won't his ghost haunt me, Cornelius? Me killin' him while not having nothing against him and all?\"\n\nHickey thought fast. Bates and Sinclair were walking farther on to where a work party from _Erebus_ was erecting a snow-block fence along a twenty-yard stretch where the wind always blew. More than one man had gotten lost in whiteouts there, and the captains thought that a snow fence would improve the couriers' chances of finding the next cairns. Irving would make sure that Bates and Sinclair were busy on their task there, and then he'd walk back to where he and Magnus were working alone on the last cairn before the clearing.\n\n\"That's _why_ the lieutenant's ghost won't haunt you, Magnus,\" he whispered to the stooping giant. \"You kill a man in heat of temper, now _that's_ a reason for that man's ghost to come back and try to get even with you. It resents what you did. But Mr. Irving's ghost now, it'll know there was nothing personal in what you had to do, Magnus. It won't have no reason to come back to bother you.\"\n\nManson nodded but did not look completely convinced.\n\n\"Besides,\" continued Hickey, \"the ghost won't be able to find its soddin' way back to the ship now, will it? Everyone knows that when someone dies outside here, so far from the ship, the ghost goes straight up. It can't figure its way through all the ice ridges and bergs and such. Ghosts ain't the smartest blokes around, Magnus. Take my word on that, m'love.\"\n\nThe huge man brightened at hearing this. Hickey could see Irving returning through the torchlit gloom. The wind was coming up and causing the torch flames to dance wildly. _Better if there's wind,_ thought Hickey. _If Magnus or Irving make some noise, no one'll hear._\n\n\"Cornelius,\" whispered Manson. He looked worried again. \"If _I_ die out here, does that mean my ghost won't be able to find its way back to the ship? I'd hate to be out here in the cold so far away from you.\"\n\nThe caulker's mate patted the slop-shrouded wall of the giant's back. \"You ain't going to die out here, love. You have my solemn promise as a Mason and a Christian on that. Now hush and get ready. When I take off my cap and scratch my head, you grab Irving from behind and drag him to where I showed you. Remember \u2014 don't leave no boot prints behind and don't get no blood on you.\"\n\n\"I won't, Cornelius.\"\n\n\"That's a good love.\"\n\nThe lieutenant came closer in the darkness, moving into the dim circle of light thrown by the lantern on the ice here near the cairn. \"Almost finished with this cairn, Mr. Hickey?\"\n\n\"Aye, sir. Just set these last blocks up here and it's done, Lieutenant. Solid as a lamppost in Mayfair.\"\n\nIrving nodded. He seemed to be uncomfortable to be alone with the two seamen, even though Hickey was using his most affable and charming voice. _Well, fuck you,_ thought the caulker's mate as he continued to show his gap-toothed smile. _You ain't going to be around much longer to put on such dandified airs, you blond-haired, apple-cheeked bastard. Five minutes and you'll be just another frozen side of beef to hang down in the hold, boyo. Too bad them rats are so hungry these days that they'll eat even a fucking lieutenant, but nothing I can do about that._\n\n\"Very good,\" said Irving. \"When you and Manson are finished, please join Mr. Sinclair and Mr. Bates on working on the wall. I'm going to walk back and bring up Corporal Hedges with his musket.\"\n\n\"Aye, sir,\" said Hickey. He caught Magnus's eye. They had to intercept Irving before he walked back along the dimly visible line of torches and lanterns. It would do no good to have Hedges or another Marine up here.\n\nIrving walked to the east but paused at the edge of the light, obviously waiting for Hickey to set the last two blocks of ice in place at the top of the rebuilt cairn. As the caulker's mate bent to lift the penultimate square of ice, he nodded to Magnus. His partner had moved into position behind the lieutenant.\n\nSuddenly there was an explosion of shouts from the darkness to the west. A man screamed. More voices joined in the shouting.\n\nMagnus's huge hands were hovering just behind the lieutenant's neck \u2014 the big man had removed his mittens for a better grip, and his undergloves loomed black just beyond Irving's pale face in the lantern light.\n\nMore shouts. A musket fired.\n\n\"Magnus, _no!_ \" shouted Cornelius Hickey. His partner had been about to break Irving's neck despite the commotion.\n\nManson stepped back into the darkness. Irving, who had taken three steps toward the shouting in the west, whirled in confusion. Three men came running along the ice path from the direction of _Terror._ One of them was Hedges. The roly-poly Marine was wheezing as he ran, his musket held in front of his massive bulge of belly.\n\n\"Come!\" said Irving and led the way toward the shouting. The lieutenant was carrying no weapon, but he'd grabbed up the lantern. The six of them ran across the sea ice, out of the seracs, into the starlit clearing where several men were milling. Hickey could make out the familiar Welsh wigs of Sinclair and Bates and recognized one of the three Erebuses already there as Francis Dunn, his caulker's-mate counterpart on the other ship. He saw that the musket that had fired belonged to Private Bill Pilkington, who'd been in the hunting blind when Sir John was killed last June and who had been shot in the shoulder by one of his fellow Marines during those moments of chaos. Now Pilkington was reloading and then aiming the long musket into the darkness beyond a fallen section of the snow fence wall.\n\n\"What has happened?\" Irving demanded of the men.\n\nBates answered. He, Sinclair, and Dunn, as well as Abraham Seeley and Josephus Greater from _Erebus,_ had been working on the wall under the command of _Erebus_ 's first mate, Robert Orme Sergeant, when suddenly one of the larger blocks of ice just beyond the circle of lantern and torchlights had seemed to come alive.\n\n\"It lifted Mr. Sergeant ten feet into the air by his head,\" said Bates, his voice shaking.\n\n\"It's the God's truth,\" said Caulker's Mate Francis Dunn. \"One minute 'e was standin' among us, next minute 'e's flying up into the air so alls we can see is the bottom of 'is boots. And the noise... the crunching...\" Dunn broke off and continued breathing hard until his pale face was all but lost in a halo of ice crystals.\n\n\"I was coming up to the torches when I saw Mr. Sergeant just... disappear,\" said Private Pilkington, lowering the musket with shaking arms. \"I fired once as the thing went back into the seracs. I think I hit it.\"\n\n\"You could've hit Robert Sergeant just as easily,\" said Cornelius Hickey. \"Maybe he was still alive when you shot.\"\n\nPilkington gave _Terror_ 's caulker's mate a look of pure venom.\n\n\"Mr. Sergeant wasn't alive,\" said Dunn, not even noticing the exchange of glares between the Marine and Hickey. \" 'E screamed once and the thing crunched 'is skull like a walnut. I seen it. I ' _eard_ it.\"\n\nOthers came running up then, including Captain Crozier and Captain Fitzjames, looking wan and insubstantial even in his heavy layers of slops and greatcoat, and Dunn, Bates, and the others all rushed to explain what they had seen.\n\nCorporal Hedges and two other Marines who had run to the commotion returned from the darkness to say there was no sign of Mr. Sergeant, only a thick trail of blood and torn clothing that led off into the thicker ice jumble in the direction of the largest iceberg.\n\n\"It wants us to follow,\" muttered Bates. \"It'll be waiting for us.\"\n\nCrozier showed his teeth in something between a mad grin and a snarl. \"Then we won't disappoint it,\" he said. \"This is as good a time as any to go after the thing again. We have the men out on the ice already, we have enough lanterns, and the Marines can fetch more muskets and shotguns. And the trail is fresh.\"\n\n\"Too fresh,\" muttered Corporal Hedges.\n\nCrozier barked orders. Some men went back to the two ships to bring the weapons. Others formed up in hunting parties around the Marines, who were already armed. Torches and lanterns were brought from the work sites and assigned to the killing parties. Dr. Stanley and Dr. McDonald were sent for in the low probability that Robert Orme Sergeant might still be alive or the higher probability that someone else might be injured.\n\nAfter Hickey was handed a musket, he considered shooting Lieutenant Irving by \"accident\" once out in the dark, but the young officer now seemed wary of both Manson and the caulker's mate. Hickey caught several concerned glances the toff was throwing toward Magnus before Crozier assigned them to different search parties, and he knew that whether Irving had caught a glimpse of Magnus behind with his arms raised in that second before the shots and shouts were first heard or whether the officer simply sensed something wrong, it wouldn't be as easy to ambush him the next time.\n\nBut they would. Hickey was afraid that John Irving's suspicions would finally cause him to report to the captain what he'd seen in the hold, and the caulker's mate could not abide that. It wasn't so much the punishment for sodomy that bothered him \u2014 seamen were rarely hanged anymore, nor flogged around the fleet for that matter \u2014 but rather the ignominy. Caulker's Mate Cornelius Hickey was no mere idiot's bum-bugger.\n\nHe would wait until Irving lowered his guard again and then do the deed himself if he had to. Even if the ships' surgeons discovered that the man had been murdered, it wouldn't matter. Things had gone too far on this expedition. Irving would be just another corpse to deal with come the thaw.\n\nIn the end, Mr. Sergeant's body was not found \u2014 the blood and strewn clothing trail ended halfway to the towering iceberg \u2014 but no one else died in the search. A few men lost toes to the cold and everyone was shaking and frostbitten to some extent when they finally called off the hunt an hour after their supper should have been served. Hickey did not see Lieutenant Irving again that afternoon.\n\nIt was Magnus Manson who surprised him as they trudged back to _Terror_ again. The wind was beginning to howl at their backs and the Marines slouched along with rifles and muskets at the ready.\n\nHickey realized that the idiot giant next to him was weeping. The tears instantly froze to Magnus's bearded cheeks.\n\n\"What is it, man?\" demanded Hickey.\n\n\"It's sad, is all, Cornelius.\"\n\n\"What is sad?\"\n\n\"Poor Mr. Sergeant.\"\n\nHickey shot a glance at his partner. \"I didn't know you had such tender feelings for them damned officers, Magnus.\"\n\n\"I don't, Cornelius. They can all die and be damned for all I care. But Mr. Sergeant died out on the ice.\"\n\n\"So?\"\n\n\"His ghost won't find his way back to the ship. And Captain Crozier passed the word when we was done searchin' that we're all having an extra tot o' rum this evenin'. Makes me sad his ghost won't be there, is all. Mr. Sergeant always liked his rum, Cornelius.\"\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _31 December, 1847_\n\nChristmas Eve and Christmas Day about HMS _Terror_ were low-key to the point of invisibility, but the Second Grand Venetian Carnivale on New Year's Eve would soon make up for that.\n\nThere had been four days of violent storms keeping the men inside in the days preceding Christmas \u2014 the blizzards were so fierce that the watches had to be shortened to one hour \u2014 and Christmas Eve and the sacred day itself became exercises in lower-deck gloom. Mr. Diggle had prepared special dinners \u2014 cooking up the last of the noncanned salt pork in half a dozen imaginative ways, along with the last of the jugged hares depickled from their briny casks. In addition, the cook had \u2014 with the recommendation of the quartermasters, Mr. Kenley, Mr. Rhodes, and Mr. David McDonald, as well as the careful supervision of surgeons Peddie and Alexander McDonald \u2014 chosen from some of the better-preserved Goldner tinned goods, including turtle soup, beef \u00e0 la Flamande, truffled pheasant, and calf's tongue. For dessert both evenings, Mr. Diggle's galley slaves had cut and scraped the worst of the mold from the remaining cheeses, and Captain Crozier had contributed the last five bottles of brandy from the Spirit Room's stores set aside for special occasions.\n\nThe mood stayed sepulchral. There were a few attempts to sing by both the officers in the freezing Great Room astern and the common seamen in their slightly warmer berthing space forward \u2014 there was not enough coal left in the hold-deck scuttles for extra heating even if it was Christmas \u2014 but the songs died after a few rounds. Lamp oil had to be conserved, so the lower deck had all the visual cheer of a Welsh mine illuminated by a few flickering candles. Ice covered the timbers and beams and the men's blankets and wool clothing were always damp. Rats scuttled everywhere.\n\nThe brandy raised spirits some, but not enough to dispel the literal and emotional gloom. Crozier came forward to chat with the men, and a few handed him presents \u2014 a tiny pouch of hoarded tobacco, the carving of a white bear running, the exaggerated ursine cartoon face suggesting fear (given in jest, almost certainly, and probably with some trepidation lest the formidable captain punish the man for fetishism), a mended red-wool undershirt from a man's recently deceased friend, an entire carved chess set from Marine Corporal Robert Hopcraft (one of the quietest and least assuming men on the expedition and the one who had been promoted to corporal after receiving eight broken ribs, a fractured collarbone, and a dislocated arm during the thing's attack on Sir John's hunting blind in June). Crozier thanked everyone, pressed hands and shoulders, and went back to the officers' mess, where the mood was a little more lively thanks to First Lieutenant Little's surprise donation of two bottles of whiskey he'd kept hidden for almost three years.\n\nThe storm stopped on the morning of 26 December. Snow had drifted twelve feet above the level of the bow and six feet higher than the railing along the starboard forward quarter. After digging the ship out and excavating the cairn-lined path between the ships, the men got busy preparing for what they were calling the Second Grand Venetian Carnivale \u2014 the first one, Crozier assumed, being the one he'd taken part in as a midshipman on Parry's botch of a polar voyage in 1824.\n\nOn that midnight-black morning of 26 December, Crozier and First Lieutenant Edward Little left the supervision of the shoveling and surface parties to Hodgson, Hornby, and Irving and made the long walk through the drifts to _Erebus._ Crozier was mildly shocked to find that Fitzjames had continued to lose weight \u2014 his waistcoat and trousers were several sizes too large for him now despite more obvious attempts by his steward to take them in \u2014 but he was even more shocked during their conversation when he realized that _Erebus_ 's commander was not fully paying attention most of the time. Fitzjames seemed distracted, rather like a man pretending to converse but whose actual attention was riveted on music being played in some adjoining room.\n\n\"Your men are dyeing sail canvas out on the ice,\" said Crozier. \"I saw them preparing large vats of green, blue, and even black dye. For perfectly good spare sail. Is this acceptable to you, James?\"\n\nFitzjames smiled distantly. \"Do you really think we shall need that sail again, Francis?\"\n\n\"I hope to Christ we will,\" grated Crozier.\n\nThe other captain's serene and maddening little smile remained. \"You should see our hold deck, Francis. The destruction has proceeded and accelerated since our last inspection the week before Christmas. _Erebus_ would not stay afloat an hour in open water. The rudder is in splinters. And it was our spare.\"\n\n\"New rudders can be jury-rigged,\" said Crozier, fighting the urge to grind his teeth and clench his fists. \"Carpenters can shore up sprung timbers. I've been working on a plan for digging a pit in the ice around both ships, creating dry docks about eight feet deep in the ice itself before the spring thaw. We can get to the outer hulls that way.\"\n\n\"Spring thaw,\" repeated Fitzjames and smiled almost conde-scendingly.\n\nCrozier decided to change the subject. \"You're not worried about the men conducting this elaborate Venetian Carnivale?\"\n\nFitzjames defied his gentleman's heritage by shrugging. \"Why should I be? I can't speak for your ship, Francis, but Christmas on _Erebus_ was an exercise in misery. The men need something to raise their morale.\"\n\nCrozier couldn't argue the point about Christmas being an exercise in misery. \"But a carnivale masque on the ice during another day of total darkness?\" he said. \"How many hands will we lose to the thing waiting out there?\"\n\n\"How many will we lose if we hide in our ships?\" asked Fitzjames. Both the small smile and the distracted air remained. \"And it worked out all right when you had the first Venetian Carnivale under Hoppner and Parry in '24.\"\n\nCrozier shook his head. \"That was only two months after we were first frozen in,\" he said softly. \"And both Parry and Hoppner were fanatics about discipline. Even with all the frivolity and both captains' love of theatrics, Edward Parry used to say, 'masquerades without licentiousness' and 'carnivals without excess!' Our discipline has not been so well maintained on this expedition, James.\"\n\nFitzjames finally lost his distracted air. \"Captain Crozier,\" he said stiffly, \"are you accusing me of allowing discipline to become lax aboard my ship?\"\n\n\"No, no, no,\" said Crozier, not knowing if he was accusing the younger man of that yet or not. \"I am just saying that this is our third _year_ in the ice, not our third month as it was with Parry and Hoppner. There's bound to be some loss of discipline to go along with illness and sagging morale.\"\n\n\"Would that not be all the more reason for allowing the men to have this diversion?\" asked Fitzjames, his voice still brittle. His pale cheeks had coloured at his superior's implied criticism.\n\nCrozier sighed. It was too late to stop this God-damned masque now, he realized. The men had the bit in their teeth, and those on _Erebus_ who were heading up the Carnivale preparations most enthusiastically were precisely those men who would be the first to foment mutiny should the time come. The trick as captain, Crozier knew, was never to allow that time to come. He honestly did not know whether this carnivale would help or hurt that cause.\n\n\"All right,\" he said at last. \"But the men have to understand that they may not waste even a lump, drop, or drip of coal, lamp oil, pyroligneous fuel, or ether for the spirit stoves.\"\n\n\"They promise that it will be torches only,\" said Fitzjames.\n\n\"And there's no extra spirits or food for that day,\" added Crozier. \"We've just gone on the severely reduced rations today. We're not changing that on the fifth day for a masque carnivale that neither of us fully endorsed.\"\n\nFitzjames nodded. \"Lieutenant Le Vesconte, Lieutenant Fairholme, and some of the men who are better than average rifle shots will go on hunting parties this week before the carnivale in hopes of finding game, but the men understand that it is rations as usual \u2014 or rather, the new, reduced fare \u2014 should the hunters return empty-handed.\"\n\n\"As they have every other time in the past three months,\" muttered Crozier. In a friendlier voice, he said, \"All right, James. I'll be getting back.\" He paused at the doorway of Fitzjames's tiny cabin. \"By the way, why _are_ they dyeing the sails green, black, and those other colours?\"\n\nFitzjames smiled distractedly. \"I have no idea, Francis.\"\n\nThe morning of Friday, 31 December, 1847, dawned cold but still \u2014 although of course there was no real dawn. _Terror_ 's morning watch under Mr. Irving registered the temperature as \u221273 degrees. There was no measurable wind. Clouds had moved in during the night and now concealed the sky from horizon to horizon. It was very dark.\n\nMost of the men seemed eager to head off to Carnivale as soon as breakfast was finished \u2014 a faster meal on the new rations, consisting of a single ship's biscuit with jam and a reduced scoop of Scotch barley mush with a dollop of sugar \u2014 but all ship's duties had to be attended to and Crozier had agreed to liberty for general attendance at the gala only after the day's work and supper were finished. Still, he'd agreed that those men without specific duties that day \u2014 holystoning the lower deck, the usual watches, deicing the rigging, deck shoveling, ship repair, cairn repair, tutoring \u2014 could go work on final preparations for the masque, and about a dozen men headed off into the darkness after breakfast, two Marines with muskets accompanying them.\n\nBy noon and the issuing of the further-diluted grog, the excitement of the remaining ship's company was a palpable thing. Crozier released six more men who'd finished their day's duties and sent Second Lieutenant Hodgson along with them.\n\nThat afternoon, while pacing the stern deck in the dark, Crozier could already see the bright glow of torches just beyond the largest iceberg rising between the two ships. There still was no wind or starlight.\n\nBy supper time, the remaining men were as fidgety as young children on Christmas Eve. They finished their meal in record time, although with the reduced rations \u2014 since this Friday was not a \"flour day\" with baking, they were eating little more than Poor John, some Goldner canned vegetables, and two fingers of Burton's ale \u2014 and Crozier didn't have the heart to hold them in the ship while the officers finished their more leisurely mess. Besides, the remaining officers on board were as eager as the seamen to go to Carnivale. Even the engineer, James Thompson, who rarely showed interest in anything outside the machinery in the hold and who had lost so much weight he resembled an ambulatory skeleton, was on the lower deck and dressed and ready to go.\n\nSo by 7:00 p.m., Captain Crozier found himself bundled in every layer he could add on, making the final inspection of the eight men left to watch the ship \u2014 First Mate Hornby had the duty but would be relieved before midnight by young Irving, who would return with three seamen so that Hornby and his watch could attend the gala \u2014 and then they were descending the ice ramp to the frozen sea and walking briskly through \u221280-degree air toward _Erebus._ The crowd of thirty-some men soon strung out into a long line in the dark, and Crozier found himself walking with Lieutenant Irving, Ice Master Blanky, and a few petty officers.\n\nBlanky was moving slowly, using a well-padded crutch under his right arm since he'd lost the heel on his right foot and still hadn't quite mastered walking on its wood-and-leather replacement, but seemed in a fine mood.\n\n\"Good evenin' to you, Captain,\" said the ice master. \"Don't let me slow you down, sir. My mates here \u2014 Fat Wilson and Kenley and Billy Gibson \u2014 will see me there.\"\n\n\"You seem to be moving as fast as we are, Mr. Blanky,\" said Crozier. As they passed the torches lit on every fifth cairn, he noticed that there still was no breath of wind; the flames flickered vertically. The path had been well trod out, the pressure ridge gaps shoveled and hacked out to provide an easy passage. The large iceberg still half a mile ahead of them seemed to be lit from within by all the torches burning on the other side of it and now resembled some phantasmagorical siege tower glowing in the night. Crozier recalled going to regional Irish fairs when he was a boy. The air tonight, while a good bit colder than an Irish summer night's, was filled with a similar excitement. He glanced behind them to make sure that Private Hammond, Private Daly, and Sergeant Tozer were bringing up the rear with their weapons at port arms and their outer mittens off.\n\n\"Strange how excited the men are about this Carnivale, ain't it, Captain?\" said Mr. Blanky.\n\nCrozier could only grunt at that. This afternoon he had drunk the last of his self-rationed whiskey. He dreaded the coming days and nights.\n\nBlanky and his mates were moving so quickly \u2014 crutch or no \u2014 that Crozier let them get ahead. He touched Irving's arm, and the gangly lieutenant dropped back from where he was walking with Lieutenant Little, surgeons Peddie and McDonald, the carpenter, Honey, and others.\n\n\"John,\" Crozier said when they were out of earshot of the officers but still far enough ahead of the Marines so as not to be heard, \"any news of Lady Silence?\"\n\n\"No, Captain. I checked the forward locker myself less than an hour ago, but she'd already gone out her little back door.\"\n\nWhen Irving had reported to Crozier on their Esquimaux guest's extracurricular excursions earlier in December, the captain's first instinct had been to collapse the narrow ice tunnel, seal and reinforce the ship's bows, and evict the wench onto the ice once and for all.\n\nBut he hadn't done that. Instead, Crozier had ordered Lieutenant Irving to assign three crewmen to watch Lady Silence whenever that was feasible and for him to follow her out onto the ice again if possible. So far, they'd not seen her go out her back door again, although Irving had spent hours hiding in the ice jumble beyond the ship's bow, waiting. It was as if the woman had seen the lieutenant during her witchly meeting with the creature on the ice, as if she had _wanted_ him to see and hear her out there, and that had been enough. She appeared to be subsisting on ship's rations these days and using the forward cable locker only for sleeping.\n\nCrozier's reason for not immediately evicting the native woman was simple: his men were beginning the slow process of starving to death, and they would not have adequate stores to get through the spring, much less the next year. If Lady Silence _was_ getting fresh food from the ice in the middle of winter \u2014 trapping seals perhaps, walrus hopefully \u2014 it was a skill that Crozier knew his crews would have to learn in order to survive. There was not a serious hunter or ice fisherman among the hundred-some survivors.\n\nCrozier had discounted Lieutenant Irving's embarrassed, heavily self-critical account of seeing something that seemed like the creature on the ice making some sort of music with the woman and bringing food offerings to her. The captain simply would never believe that Silence had trained a huge white bear \u2014 if such the thing was \u2014 to hunt and bring her fish or seal or walrus like a proper English bird dog fetching pheasant for its master. As for the music... well, that was absurd.\n\nBut she had chosen this day to go missing again.\n\n\"Well,\" said Crozier, his lungs aching from the cold air, even filtered as it was through his thick wool comforter, \"when you return with the relief watch at eight bells, check her locker again, and if she's not there... what in the name of Christ Almighty?\"\n\nThey had passed through the last line of pressure ridges and come out onto the flat sea ice on the last quarter mile to _Erebus._ The scene that met Crozier's eyes made his jaw sag under the wool scarf and high-pulled jacket collars.\n\nThe captain had assumed that the men would be having the Second Grand Venetian Carnivale on the flat sea ice immediately below _Erebus,_ the way Hoppner and Parry had set their masque on the short stretch of ice between the frozen-in _Hecla_ and _Fury_ in 1824, but while _Erebus_ sat bow up, dark and desolate-looking on its dirty pedestal of ice, all the light, torches, motion, and commotion came from an area a quarter of a mile away, immediately in front of the largest iceberg.\n\n\"Good heavens,\" said Lieutenant Irving.\n\nWhile _Erebus_ looked to be a dark hulk, a new mass of rigging \u2014 a veritable city of coloured canvas and flickering torches \u2014 had risen on the bare circle of sea ice, forest of seracs, and wide-open area beneath the towering, glowing iceberg. Crozier could only stand and stare.\n\nThe riggers had been busy. Some obviously had ascended the berg itself, sinking huge ice screws deep in the ice sixty feet high on its face, pounding in bolt rings and pulley stands, adding enough rigging, running lines, and blocks from the stores to outfit a three-masted man-of-war at full sail.\n\nA spiderweb of a hundred ice-frosted lines ran down from the berg and back toward _Erebus,_ supporting a city of lighted and coloured tent walls. These dyed walls of canvas \u2014 some of the mains'l sheets thirty feet high and taller \u2014 were staked to sea ice and serac and ice block but pulled taut on their vertical spars with stays running diagonally to the tall berg.\n\nCrozier walked closer, still blinking. The ice in his eyelashes threatened to freeze his eyelids shut, but he continued blinking.\n\nIt was as if a series of gigantic coloured tents had been pitched on the ice, but these tents had no roofs. The vertical walls, lighted from within and without by scores of torches, snaked from the open sea ice into the serac forest and continued up to the vertical wall of the iceberg itself. As it was, giant rooms or coloured apartments had been erected almost overnight on the ice. Each chamber stood at an angle to the preceding chamber, a sharp turn in the rigging, staves, and canvas apparent every twenty yards or so.\n\nThe first chamber opened eastward onto the ice. The canvas here had been dyed a bright, rich blue \u2014 the blue of skies not seen in so many months that the colour made a knot rise in Captain Crozier's constricted throat \u2014 and torches and braziers of flame outside the canvas chamber's vertical sides made the blue walls glow and pulse.\n\nCrozier walked past Mr. Blanky and his mates, who were staring in open wonder. \"Christ,\" he heard the ice master mutter.\n\nCrozier walked still closer, actually entering the space defined by the glowing blue walls.\n\nBrightly clad and strangely garbed figures pranced and swooped around him \u2014 ragpickers with streaming comet tails of coloured cloth trailing behind them, tall chimney sweeps in death-black tails and sooty top hats doing jigs, exotic birds with long gold beaks stepping lightly, sheikhs of Araby with red turbans and pointed Persian slippers sliding along the dark ice, pirates with blue death masks pursuing a prancing unicorn, generals of Napoleon's army wearing white masks from some Greek Chorus filing by in solemn procession. Something dressed all in bulky green \u2014 a wood sprite? \u2014 ran up to Crozier on the unslippery ice and chirped in falsetto, \"The trunk of costumes is to your left, Captain. Feel free to mix and match,\" and then the apparition was gone, blending back into the shifting crowds of bizarrely dressed figures.\n\nCrozier continued walking deeper into the maze of coloured apartments.\n\nBeyond the blue chamber, turning sharply to the right, was a long purple room. Crozier saw that it was not empty. The men realizing this Carnivale had placed rugs, tapestries, tables, or casks here and there in each apartment, their furnishings and fixtures dyed or painted the same hue as the glowing walls.\n\nBeyond the purple room, bending back sharply to the left here but at such an odd angle that Crozier would have had to look at the stars \u2014 had there been any stars visible \u2014 to ascertain his exact bearings, was a long green chamber. This long room held the most revelers yet: more exotic birds, a princess with a long horse's face, creatures so segmented and oddly jointed that they appeared to be giant insects.\n\nFrancis Crozier recalled none of these costumes from Parry's trunks on the _Fury_ and _Hecla,_ but Fitzjames had insisted that Franklin had brought precisely those moldering old artifacts.\n\nThe fourth chamber was furnished and lighted with orange. The torchlight through the thin orange-dyed canvas here seemed rich enough to taste. More orange canvas, painted and dyed to resemble tapestries, had been laid out on the sea ice, and there was a huge punch bowl on the orange-sheeted table at the center of the interior space. At least thirty or more wildly costumed figures had converged on the punch bowl, some dipping their beaked or fanged visages to drink deeply.\n\nCrozier realized with a shock that loud music was coming from the fifth segment of the apartment maze. Following another bend to the right, he came into a white chamber. Sheet-covered sea trunks and officers' mess-room chairs had been set along the white canvas walls here, and the almost forgotten mechanical music player from _Terror_ 's Great Cabin was being cranked by a costumed fantastic at the far end of the chamber, the machine pouring out music hall favorites from its large rotating metal disks. The sound somehow seemed much louder out here on the ice.\n\nRevelers were coming out of the sixth chamber and Crozier walked past the music player, took the sharp angle to the left, and entered a violet room.\n\nThe captain's seaman's eyes admired the rigging that rose from upended spare spars to a tethered spar hanging in midair \u2014 webs of rigging came in from the other six chambers there to be tied off \u2014 and the master cables that ran up from this center spar to anchors high on the wall of the iceberg. The riggers from _Erebus_ and _Terror_ who had conceived and executed this seven-chamber maze obviously had also exorcised some of the incredible frustration at not being able to pursue their trade due to being icebound and static for so many months, their ships' topmasts, spars, and rigging pulled down and stored on the ice. But this violet room had few costumed crewmen tarrying in it and the light was strangely oppressive. The only furniture here consisted of stacks of empty crates at the center of the room, all draped in violet sheets. The few birds, pirates, and ragmen in this room paused to drink from their crystal goblets carried from the white room, looked around, then quickly returned to the outer chambers again.\n\nThe final room beyond the violet room seemed to have no light at all coming from it.\n\nCrozier followed the sharp angle to the right from the violet chamber and found himself in a chamber of almost absoluteblackness.\n\nNo, that was not true, he realized. Torches burned outside the black-dyed sail walls here just as they did beyond all the other chambers, but the effect was only of a subdued glow through ebony air. Crozier had to stop to allow his eyes to adapt, and when they did, he took two startled steps backward.\n\nThe ice underfoot was gone. It was as if he were walking above the black water of the arctic sea.\n\nIt took only seconds for the captain to realize the trick. The seamen had taken soot from the boiler and coal sack holds and spread it across the sea ice here \u2014 an old seaman's trick when wanting to melt the sea ice more quickly in late spring or recalcitrant summer, but there was no melting tonight with the sunless days and temperatures dropping toward \u2212100 degrees. Instead, the soot and carbon made the ice underfoot invisible in the ebony gloom of this final, terrible compartment.\n\nAs Crozier's eyes adapted further, he saw that there was only one piece of furniture in the long black compartment, but his jaws clenched with anger when he saw what it was.\n\nCaptain Sir John Franklin's tall ebony grandfather clock was set at the far end of this black compartment, its back to the rising iceberg that served as the far wall to the ebony room and the end of the seven-chamber maze. Crozier could hear the heavy ticking of the thing.\n\nAnd above the ticking clock, extruding from the ice like something struggling to gain its freedom from the iceberg, was the white-furred head and ivory-yellow teeth of a monster.\n\nNo, he checked himself again, not a monster. The head and neck of a large white bear somehow had been mounted onto the ice. The creature's mouth was open. Its black eyes reflected the small amount of torchlight that made its way through the black-dyed canvas walls. The bear's fur and teeth were the brightest things in the ebony compartment. Its tongue was a shocking red. Beneath the head, the ebony clock ticked like a heartbeat.\n\nFilled with a fury that he could not define, Crozier marched from the ebony compartment, paused in the white room, and bellowed for an officer \u2014 any officer.\n\nA Satyr with a long papier-m\u00e2ch\u00e9 face and a priapic cone rising from its red belt scuttled forward on black metal hooves set beneath heavy boots. \"Yes, sir?\"\n\n\"Take off that fucking mask!\"\n\n\"Aye, aye, Captain,\" said the Satyr, sliding the mask up to reveal Thomas R. Farr, _Terror_ 's captain of the maintop. A Chinese woman with huge breasts next to him lowered her mask to show the round, fat face of John Diggle, the cook. Next to Diggle was a giant rat who lowered its snout enough to show the face of Lieutenant James Walter Fairholme of _Erebus._\n\n\"What in hell is the meaning of all this?\" roared Crozier.\n\nVarious fantastical creatures cringed back toward the white walls at the sound of Crozier's voice.\n\n\"Of which exactly, Captain?\" asked Lieutenant Fairholme.\n\n _\"This!\"_ bellowed Crozier, raising both arms and hands to indicate the white walls, the rigging overhead, the torches... everything.\n\n\"No meaning, Captain,\" responded Mr. Farr. \"It's simply... Carnivale.\" Crozier had always, until this moment, thought Farr a reliable and sensible hand and a fine maintop captain.\n\n\"Mr. Farr, did you help in the rigging?\" he asked sharply.\n\n\"Yes, sir.\"\n\n\"And Lieutenant Fairholme, were you aware of the... animal's head... exhibited so bizarrely in that final chamber?\"\n\n\"Aye, Captain,\" said Fairholme. The lieutenant's long, weathered face showed no sign of fear at his expedition commander's anger. \"I shot it myself. Yesterday evening. Two of the bears, actually. A mother and its almost grown male cub. We're going to roast the meat toward midnight \u2014 have a sort of feast, sir.\"\n\nCrozier stared at the men. He could feel his heart pounding in his chest, could feel the anger that \u2014 mixed with the whiskey he'd had that day and the certainty there would be no more in the days to come \u2014 had often led him to violence ashore.\n\nHe had to be careful here.\n\n\"Mr. Diggle,\" he said to the fat Chinese woman with the huge breasts, \"you know the liver of the white bears has made us ill.\"\n\nDiggle's jowls bobbled up and down as freely as the pillowed bosoms beneath them. \"Oh, yes, Captain. There's something foul in the polar beast's liver that we haven't been able to heat out of it. There'll be neither liver nor lights in the feast I cook tonight, Captain, I assure thee. Only fresh meat \u2014 hundreds and hundreds of pounds of fresh meat, grilled and singed and fried to perfection, sir.\"\n\nLieutenant Fairholme spoke. \"The men are taking it as a hopeful omen that we blundered across the two bears on the ice and were able to kill them, Captain. Everyone's looking forward to the feast at midnight.\"\n\n\"Why wasn't I told of the bears?\" demanded Crozier.\n\nThe officer, maintop captain, and cook looked at one another. Birds and beasts and faeries nearby looked at one another.\n\n\"The sow and cub were only shot late last night, Captain,\" Fairholme said at last. \"I guess all the traffic between the ships today has been Terrors coming over to the Carnivale to work and get ready, no messengers from _Erebus_ making the return trip. My apologies for not informing you, sir.\"\n\nCrozier knew that it was Fitzjames who had been negligent in this regard. And he knew the men around him knew it.\n\n\"Very well,\" he said at last. \"Carry on.\" But as the men began setting their masks back in place, he added, \"And God help you if Sir John's clock is damaged in any way.\"\n\n\"Aye, Captain,\" said all the masked shapes around him.\n\nWith a final, almost apprehensive glance back through the violet room toward the terrible black compartment \u2014 almost nothing in Francis Crozier's fifty-one years of frequent melancholy had oppressed him as much as that ebony compartment had \u2014 he walked from the white room to the orange room, thence from the orange room to the green room, then from the green room to the purple room, from the purple room to the blue room, and from the widening blue room out onto the darker open ice.\n\nOnly when he was out of the dyed-sail maze did Crozier feel that he could breathe properly.\n\nCostumed shapes gave the glowering captain a wide berth as he made his way toward _Erebus_ and the dark, heavily cloaked figure standing at the top of the ice ramp there.\n\nCaptain Fitzjames was alone near the ship's railing at the top of the ramp. He was smoking his pipe. \"Good evening to you, Captain Crozier.\"\n\n\"Good evening, Captain Fitzjames. Have you been inside that... that...\" Words failed him, and Crozier gestured toward the loud and lighted city of coloured walls and elaborate rigging behind him. The torches and braziers burned bright there.\n\n\"Aye, I have,\" said Fitzjames. \"The men have shown incredible ingenuity, I would say.\"\n\nCrozier had nothing to say to that.\n\n\"The question now,\" said Fitzjames, \"is whether their many hours of labour and ingenuity have gone to serve the expedition... or the Devil.\"\n\nCrozier tried to see the younger officer's eyes under the muffler-tied bill of his cap. He had no idea if Fitzjames was joking.\n\n\"I warned them,\" growled Crozier, \"that they could waste not one pint of oil or one extra lump of coal on this damned Carnivale. Just look at those fires!\"\n\n\"The men assured me,\" said Fitzjames, \"that they are only using the oil and coal thay have saved by not heating _Erebus_ these past weeks!\"\n\n\"Whose idea was that... maze?\" asked Crozier. \"The coloured compartments? The ebony room?\"\n\nFitzjames blew smoke, removed his pipe, and chuckled. \"All the idea of young Richard Aylmore.\"\n\n\"Aylmore?\" repeated Crozier. He remembered the name but hardly the man. \"Your gunroom steward?\"\n\n\"The same.\"\n\nCrozier recalled a small man, quiet, with sunken, brooding eyes, a pedant's tone to his voice, and a wispy black mustache. \"Where in the hell did he come up with this?\"\n\n\"Aylmore lived in the United States for several years before returning home in 1844 and enlisting in the Discovery Service,\" said Fitzjames. The stem of the pipe clattered slightly against his teeth. \"He maintains that he read an absurd story five years ago, in 1842, describing a masqued ball just such as this with such coloured compartments, read it while he was living in Boston with his cousin. In a trashy little piece called _Graham's Magazine,_ if I recall correctly. Aylmore can't remember the plot of the story exactly, but he remembers that it was about a strange masqued ball given by a certain Prince Prospero... and he says that he is quite certain of the sequence of the rooms, ending in that terrible ebony compartment. The men loved his idea.\"\n\nCrozier could only shake his head.\n\n\"Francis,\" continued Fitzjames, \"this was a teetotaling ship for two years and one month under Sir John. Despite that, I managed to smuggle aboard three bottles of fine whiskey my father gave me. I have one bottle left. I would be honoured if you would share it with me this evening. It will be another three hours until the men begin cooking up the two bears they shot. I authorized my Mr. Wall and your Mr. Diggle yesterday to set up two of the whaleboats' stoves on the ice for heating incidentals such as canned vegetables and to build a huge grill in what they are calling the White Room for the actual cooking of the bear meat. If nothing else, it will be our first fresh meat in more than three months. Would you care to be my guest over that bottle of whiskey down in Sir John's former cabin until it's time for the feast?\"\n\nCrozier nodded and followed Fitzjames into the ship.\n\nCROZIER\n\n_Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n_31 December, 1847\u20131 January, 1848_\n\nCrozier and Fitzjames emerged from _Erebus_ some time before midnight. The Great Cabin had been ferociously cold, but the deeper cold out here in the night was an assault on their bodies and senses. The wind had come up slightly in the last couple of hours and everywhere the torches and tripod braziers \u2014 Fitzjames had suggested, and after the first hour of whiskey Crozier had agreed, sending out extra sacks of coal and coal oil to fuel open-flame braziers to keep the revelers from freezing \u2014 were rippling and crackling in the hundred-below freezing night.\n\nThe two captains had talked very little, each lost in his own melancholic reverie. They'd been interrupted a dozen times. Lieutenant Irving came to report that he was taking the replacement watch back to _Terror;_ Lieutenant Hodgson came to report that his watch had arrived at the Carnivale; other officers in absurd costumes came to report that all was well with Carnivale itself; various _Erebus_ watches and officers came to report coming off duty and going on duty; Mr. Gregory the engineer came to report that they might as well use the coal for the braziers since there wasn't enough to fuel the steam engine for more than a few hours of steaming come the mythical thaw and then went off to make arrangements for several sacks to be hauled out to the increasingly wild ceremony on the ice; Mr. Murray, the old sailmaker \u2014 dressed as some sort of mortician with a skull under his high beaver hat, a skull not so different from his own wizened visage \u2014 begged their pardon and asked if he and his helpers could break out two spare jibs to rig a wind shield upwind of the new tripod braziers.\n\nThe captains had given their acknowledgments and permissions, passed along their commands and admonitions, never really rising out of their whiskey-induced thoughts.\n\nSometime between eleven and midnight, they bundled themselves back into their outer slops, came up on deck, and then went out onto the ice again after both Thomas Jopson and Edmund Hoar, Crozier's and Fitzjames's respective stewards, came down to the Great Cabin with Lieutenants Le Vesconte and Little \u2014 all four men in bizarre costumes squeezed over and under their many layers \u2014 to announce that the bear meat was being cooked up, that prime portions were being set aside for the captains, and could the captains please come to the feast now?\n\nCrozier realized that he was very drunk. He was used to holding his liquor without letting it show, and the men were used to him smelling like whiskey while he was in complete command of situations, but he hadn't slept for several nights and this midnight, coming out into the chest-slamming cold and walking toward the lighted canvas and glowing iceberg and movement of strange forms, Crozier _felt_ the whiskey burning in his belly and brain.\n\nThey'd set up the main grilling area in the white room. The two captains traversed the series of compartments without comment either to each other or to any of the dozens upon dozens of wildly costumed figures flitting about. From the open-ended blue room, they walked through the purple and green rooms, then through the orange room and into the white.\n\nIt was obvious to Crozier that most of the men were also drunk. How had they done that? Had they been hoarding their allotments of grog? Hiding away the ale usually served with their suppers? He knew that they hadn't broken into the Spirit Room aboard _Terror_ because he'd had Lieutenant Little check to make sure the locks were secure both this morning and this afternoon. And _Erebus_ 's Spirit Room was empty thanks to Sir John Franklin, and had been since they'd sailed.\n\nBut the men had gotten into hard spirits somehow. As a seaman of more than forty years who had served his time before the mast as a boy, Crozier knew that \u2014 at least in terms of fermenting, hoarding, or finding alcohol \u2014 a British sailor's ingenuity knew no bounds.\n\nHuge haunches and racks of bear meat were being grilled over an open fire by Mr. Diggle and Mr. Wall, pewter plates of the steaming victuals being handed out to the queues of men by a grinning Lieutenant Le Vesconte, his gold tooth gleaming, and by other officers and stewards of both ships. The smell of grilling meat was incredible and Crozier found himself salivating despite all his private vows not to enjoy this Carnivale feast.\n\nThe queue gave way to the two captains. Ragmen, popish priests, French courtiers, faerie sprites, motley beggars, a shrouded corpse, and two Roman legionnaires in red capes, black masks, and gold chest armour gestured Fitzjames and Crozier to the front of the queue and bowed as the officers passed.\n\nMr. Diggle himself, his fat-Chinese-lady's pendulous bosoms now down around his waist and wobbling as he moved, cut a prime piece for Crozier and then another for Captain Fitzjames. Le Vesconte gave them proper officers' mess cutlery and white linen napkins. Lieutenant Fairholme poured ale into two cups for them.\n\n\"The trick out here, Captains,\" said Fairholme, \"is to drink quickly, dipping like a bird, so that your lips don't freeze to the cup.\"\n\nFitzjames and Crozier found a place at the head of a white-shrouded table, sitting on white-shrouded chairs, pulled back for them on the protesting ice by Mr. Farr, the captain of the maintop whom Crozier had braced earlier in the evening. Mr. Blanky was sitting there with his ice-master counterpart, Mr. Reid, as were Edward Little and a half dozen of the _Erebus_ officers. The surgeons clustered at the other end of the white table.\n\nCrozier took his mittens off, flexed cold fingers under wool gloves, and tried the meat gingerly, careful not to let the metal fork touch his lips. The bear cutlet burned his tongue. He had the urge to laugh then \u2014 a hundred below zero out here in the New Year's night, his breath hanging in front of him in a cloud of ice crystals, his face hidden down the tunnel of his comforters, caps, and Welsh wig, and he'd just burned his tongue. He tried again, chewing and swallowing this time.\n\nIt was the most delicious steak he'd ever eaten. This surprised the captain. Many months ago, the last time they'd tried fresh bear meat, the cooked flesh seemed gamy and rancid. The liver and possibly some of the other commonly prized organs made the men actively ill. It had been decided that the meat of the white arctic bear would be eaten only if survival demanded it.\n\nAnd now this feast... this sumptuous feast. All around him in the white room, and obviously at canvas-covered casks, chests, and tables in the adjoining orange and violet rooms, crewmen were wolfing down the steaks. The noise and chatter of happy men easily rose over the roar of the grill flames or the flapping of canvas as the wind came up again. A few of the men here in the white room were using knives and forks \u2014 many just spearing the steaming bear steaks and chewing on them that way \u2014 but most were using their mittened hands. It was as if more than a hundred predators were reveling in their kill.\n\nThe more Crozier ate, the more ravenous he became. Fitzjames, Reid, Blanky, Farr, Little, Hodgson, and the others around him \u2014 even Jopson, his steward, at a nearby table with the other stewards \u2014 appeared to be wolfing down the meat with equal gusto. One of Mr. Diggle's helpers, dressed as a baby Chinaman, came around the tables, dishing out steaming vegetables from a pan heated on one of the whaleboat's iron stoves, but the canned vegetables, however wonderfully hot, simply had no taste next to the delicious fresh bear meat. Only Crozier's position as expedition commander stopped him from muscling his way to the front of the queue and demanding another helping when he finished his heavy slab of bear steak. Fitzjames's expression was anything but distracted now; the younger commander looked as if he was about to weep from happiness.\n\nSuddenly, just as most of the men had finished the steaks and were drinking down their ale before the alcohol-rich liquid froze solid, a Persian king near the entrance to the violet room began cranking the musical disk player.\n\nThe applause \u2014 thick mittens pounding thunderously \u2014 began almost as soon as the first notes tinkled and thunked out of the crude machine. Many of the musical men aboard both ships had complained about the mechanical music player \u2014 its range of sounds emanating from the turning metal disks was almost precisely that of a corner organ grinder's instrument \u2014 but these notes were unmistakable. Dozens of men rose to their feet. Others began singing at once, the vapour from their breaths rising in the torchlight shining through the white canvas walls. Even Crozier had to grin like an idiot as the familiar words of the first stanza echoed off the iceberg towering above them in the freezing night.\n\nWhen Britain first at Heav'n's command, Arose from out the azure main;\n\nThis was the charter of the land, And guardian angels sang this strain;\n\nCaptains Crozier and Fitzjames rose to their feet and joined in the first bellowing chorus.\n\nRule, Britannia! Britannia, rule the waves; Britons never shall be slaves!\n\nYoung Hodgson's pure tenor led the men in six of the seven coloured compartments as they sang the second stanza.\n\nThe nations not so blest as thee, Shall in their turns to tyrants fall;\n\nWhile thou shalt flourish great and free, The dread and envy of them all.\n\nVaguely aware that there was a commotion two rooms to the east, in the entrance to the blue room, Crozier threw his head back and, warm with whiskey and bear steak, bellowed with his men:\n\nRule, Britannia! Britannia, rule the waves; Britons never, _never_ shall be slaves!\n\nThe men in the outer rooms of the seven compartments were singing, but they were also laughing now. The commotion grew. The mechanical music player cranked louder. The men sang louder still. Even while standing and singing the third stanza between Fitzjames and Little, Crozier stared in shock as a procession entered the white room.\n\nStill more majestic shalt thou rise, More dreadful from each foreign stroke;\n\nAs the loud blast that tears the skies, Serves but to root thy native oak.\n\nSomeone led the procession in the theatrical costume version of an admiral's uniform. The epaulettes were so absurdly broad that they hung out eight inches beyond the little man's shoulders. He was very fat. The gold buttons on his old-fashioned Naval jacket would never have buttoned. He was also headless. The figure carried its papier-m\u00e2ch\u00e9 head under the crook of his left arm, his moldering plumed admiral's hat under his right.\n\nCrozier quit singing. The other men did not.\n\nRule, Britannia! Britannia, rule the waves! Britons never, never, _never_ shall be slaves!\n\nBehind the headless admiral, who obviously was meant to be the late Sir John Franklin even though it had not been Sir John decapitated that day at the bear blind, ambled a monster ten or twelve feet tall.\n\nIt had the body and fur and black paws and long claws and triangular head and black eyes of a white arctic bear, but it was walking on its hind legs and was twice the height of a bear and with twice the arms' length. It walked stiffly, almost blindly, swinging its upper body to and fro, the small black eyes staring at each man it approached. The swinging paws \u2014 the arms hanging loose as bell pulls \u2014 were larger than the costumed crewmen's heads.\n\n\"That's your giant, Manson, on the bottom,\" laughed _Erebus_ 's second mate, Charles Frederick Des Voeux, next to Crozier, raising his voice to be heard over the next stanza. \"It's your little caulker's mate \u2014 Hickey? \u2014 riding on his shoulders. It took the men all night to sew up the two hides into a single costume.\"\n\nThee haughty tyrants ne'er shall tame, All their attempts to bend thee down\n\nWill but arouse thy generous flame, But work their woe, and thy renown.\n\nAs the giant bear ambled past, dozens of men from the blue, green, and orange rooms followed it in procession through the white room and into the violet room. Crozier stood as if literally frozen to his spot near the white banquet table. Finally he turned his head to look at Fitzjames.\n\n\"I swear I did not know, Francis,\" said Fitzjames. The other captain's lips were pale and very thin.\n\nThe white room began emptying of costumed figures as the scores there followed the headless admiral and the swinging, towering, slowly ambling bipedal bear-giant into and through the relative gloom of the long violet room. The drunken singing roared around Crozier.\n\nRULE, BRITANNIA! BRITANNIA, RULE THE WAVES!\n\nBRITONS NEVER, NEVER, NEVER, _NEVER_   \nSHALL BE SLAVES!\n\nCrozier began following the procession into the violet chamber and Fitzjames followed him. The captain of HMS _Terror_ had never felt this way in all of his years of command; he knew that he had to stop this travesty of a lampoon \u2014 no Naval discipline could tolerate a farce in which the death of the expedition's former commander became a source of humour. But at the same time he knew that it had already proceeded to a point where simply shouting down the singing, ordering Manson and Hickey out of their obscene monster suit, ordering _everyone_ out of their costumes and back to their berths on the ships would be almost as absurd and useless as the pagan ritual Crozier was watching with growing anger.\n\nTO THEE BELONGS THE RURAL REIGN, THY CITIES SHALL WITH COMMERCE SHINE;\n\nALL THINE SHALL BE THE SUBJECT MAIN, AND EVERY SHORE IT CIRCLES THINE!\n\nThe headless admiral, ambling bear-thing, and the following procession of a hundred costumed men or more had not paused long in the violet room. As Crozier entered the violet-coloured space \u2014 the torches and outside tripod fires were whipping on the north side of the violet-dyed canvas wall and the sails themselves were rippling and cracking in the rising wind \u2014 he arrived just in time to see Manson and Hickey and their singing mob pause at the entrance to the ebony room.\n\nCrozier resisted the impulse to shout out \"No!\" It was an obscenity for the effigy of Sir John and the towering bear-thing to play this out in any forum, but unthinkably vile in that black, oppressive ebony room with its polar bear head and ticking clock. Whatever final dumb show the men had in mind, at least it would soon be finished. This had to be the finale of this ill-thought-out mistake of a Second Grand Venetian Carnivale. He would let the singing end of its own, the pagan mime close to drunken cheers from the men, and then he would order the mobs out of their costumes, send the frozen and drunken seamen back to their ships, but order the riggers and orga-nizers to strike the canvas and rigging immediately \u2014 tonight \u2014 whether that meant frostbite or no. He would then deal with Hickey, Manson, Aylmore, and his officers.\n\nThe swaying, much-cheered headless admiral and swaying bear-monster entered the ebony compartment.\n\nSir John's black clock within began striking midnight.\n\nThe mob of bizarrely costumed sailors at the rear of the procession began pressing forward, the rear ranks eager to get into the ebony compartment to see the fun, even while the ragmen, rats, unicorns, dustmen, one-legged pirates, Arab princes and Egyptian princesses, gladiators, faeries, and other creatures at the front of the mob, already making the turn and crossing the threshold into the black room, began resisting the advance, pushing back, no longer sure they wanted to be in that soot-floored and black-walled darkness.\n\nCrozier elbowed his way forward through the mob \u2014 the mass surging forward and then back as those in the front thought twice about actually entering the ebony gloom \u2014 certain now that if he couldn't end this farce before the finale, at least he could shorten this final act.\n\nHe'd no sooner entered the darkness with twenty or thirty men at the front of the procession who'd also halted upon stepping in \u2014 his eyes had to adapt in here, and the black soot on the ice gave him a terrible sense of falling into a black void \u2014 when he felt the blast of cold air against his face. It was as if someone had opened a door in the wall of the iceberg that loomed over everything. Even the costumed figures here in the dark were still singing, but the real volume came from the pushing mobs still back in the violet room.\n\nRULE, BRITANNIA! BRITANNIA, RULE THE WAVES;\n\nBRITONS NEVER, NEVER, NEVER, NEVER,  \n_NEVER_\n\nSHALL BE SLAVES!\n\nCrozier could only just make out the white of the disembodied bear's head emerging from the ice over the ebony clock \u2014 the chimes had struck six now and seemed terribly loud in the darkened space \u2014 and he could see that under the taller, swaying, white bear-monster's form, Manson and Hickey were finding it difficult to keep their balance on the sooty ice, in the icy blackness with the north canvas walls flapping and rippling wildly with the wind.\n\nCrozier saw that there was a _second_ large white shape in the room. It also stood on its hind legs. It was farther back in the darkness than Manson and Hickey's bearhide-white glow. And it was much larger. And taller.\n\nAs the men fell silent and the clock was striking its last four chimes, something in the room roared.\n\nTHE MUSES, STILL WITH FREEDOM FOUND, SHALL TO THY HAPPY COAST REPAIR;\n\nBLEST ISLE! WITH MATCHLESS BEAUTY CROWNED, AND MANLY HEARTS TO GUIDE THE FAIR!\n\nSuddenly the men in the ebony room were shoving backward against the still-pushing throng of seamen trying to get in.\n\n\"What in God's name?\"asked Dr. McDonald. The four surgeons, all in Harlequin costumes but with their masks hanging down now, were recognizable to Crozier in the brighter violet glow coming around the canvased curve between the rooms.\n\nA man in the ebony room screamed in terror. There came a second roar, unlike anything that Francis Rawdon Moira Crozier had ever heard; it was something more at home in a thick jungle of some previous Hyborian Age than in the Arctic of the nineteenth century. The sound ground so low into the bass regions, grew so reverberating, and emerged so ferocious that it made the captain of HMS _Terror_ want to piss his pants right there in front of his men.\n\nThe larger of the two white shapes in the gloom charged forward.\n\nCostumed men screamed, tried to push backward against the wave of the forward-pushing curious, and then ran to the left and right in the darkness, colliding with the nearly invisible black-dyed canvas walls.\n\nCrozier, unarmed, stood where he was. He _felt_ the mass of the thing brush past him in the darkness. He sensed it with his _mind_... felt it in his _head._ There was a sudden stench as of old blood, then the reek of a carrion pit.\n\nPrincesses and faeries were throwing off costumes and cold-weather slops in the darkness, clawing at the black walls and fumbling for their boat knives on their buried belts.\n\nCrozier heard a meaty, sickening _slap_ as huge plate-sized paws or knife-sized claws slammed into a man's body. Something crunched sickeningly as teeth longer than bayonet blades bit through skull or bone. In the outer rooms, men still sang.\n\nRULE, BRITANNIA! BRITANNIA, RULE THE WAVES!\n\nBRITONS NEVER, NEVER, NEVER, NEVER, _NEVER_\n\nSHALL BE SLAVES!!\n\nThe ebony clock concluded its striking. It was midnight. It was 1848.\n\nMen used their knives to slash through the black-dyed walls and strips of wind-tormented canvas were immediately whipped into the flames of torches and tripods out on the ice. Flames leapt skyward and almost immediately engaged the rigging.\n\nThe white shape had moved out into the violet room. Men there were screaming and scattering, cursing and shoving, some already slashing at the walls there rather than trying to make the long run out through the compartment maze, and Crozier shoved seamen aside as he tried to follow. Both walls of the ebony room were ablaze now. More men screamed and one man ran past Crozier, his Harlequin costume, Welsh wig, and hair shooting flames behind him like yellow silk streamers.\n\nBy the time Crozier shoved himself free of the surging mob of fleeing, costumed forms, the violet compartment was also burning and the thing from the ice had moved on to the white room. The captain could hear the shouts from scores of men as they ran ahead of the white apparition in a wave of waving arms and shed costumes. The web of beautifully rigged ropes attaching the canvas and spar struts to the overhanging iceberg was burning now, the patterns of flame slashing like scribbled runes of fire against the black slate of sky. The hundred-foot wall of ice reflected the flames in its thousand facets.\n\nThe spars themselves that rose like exposed ribs along the burning walls of the ebony room, the violet room, and now the white room, were also on fire. Years of storage in the virtual desert of the arctic dryness had leached all moisture from the wood. They fed the flames like thousand-pound pieces of tinder.\n\nCrozier gave up all hope of mastering the situation and ran with the others. He had to get out of the burning maze.\n\nThe white room was fully engaged. Flames shot up from the white walls, from the canvas carpets on the ice, from the former sheet-draped banquet tables and casks and chairs and from Mr. Diggle's metal cooking grill. Someone had knocked over the mechanical disk player in their panicked flight and the oak-and-bronze instrument reflected the flames from all of its beautifully crafted faces and curves.\n\nCrozier saw Captain Fitzjames standing in the white room, the only figure not costumed and not running. He grabbed the motionless man by his slops' sleeve. \"Come, James! We have to _go._ \"\n\nThe commander of HMS _Erebus_ slowly turned his head and looked at his superior officer as if they had never met. Fitzjames had that small, absent, maddening smile on his face again.\n\nCrozier slapped him. \"Come _on!_ \"\n\nPulling and tugging the sleepwalking Fitzjames, Crozier stumbled through the burning white room, out through the fourth room, whose walls were more orange with flames than with dye now, and into the burning green room. The maze seemed to go on and on. Costumed figures lay on the ice here and there \u2014 some moaning and with ripped and mauled vestments, one man naked and burned \u2014 but other seamen were stopping to help them up, shoving them onward and outward. The sea ice underfoot, where there were no burning canvas carpets, was littered with shreds of costumes and abandoned cold-weather gear. Most of these tatters and fabrics were either ablaze or the about to burn.\n\n\"Come _on!_ \" repeated Crozier, still tugging a stumbling Fitzjames in his wake. A seaman lay unconscious on the ice \u2014 young George Chambers from _Erebus,_ Crozier saw, one of the ship's boys, although twenty-one now, one of the drummers in their early burials on the ice \u2014 and no one seemed to be taking notice of him. Crozier released Fitzjames just long enough to lift Chambers over his shoulder, and then he grabbed the other captain's sleeve again and began running just as flames on either side exploded to the rigging above.\n\nCrozier heard a monstrous _hissing_ behind him.\n\nCertain that the thing had circled behind him in the confusion, perhaps crashing up through the impenetrable ice, he swung to confront it with only his one mittened fist free.\n\nThe entire iceberg was steaming and popping from the heat. Huge chunks and heavy overhangs were breaking off and crashing down to the ice, hissing like snakes as they fell into the cauldron of flame that had been the tent maze. The sight held Crozier in motionless rapture for a minute \u2014 the berg's countless facets reflecting the flames made him think of a hundred-storey fairy-tale castle tower ablaze with light. He knew at that instant that as long as he lived he would never again see anything like this.\n\n\"Francis,\" lisped Captain James Fitzjames. \"We have to go.\"\n\nThe green room's walls were falling away but there were only more flames on the ice beyond. The rapidly advancing fissures and tendrils and fingers of fire had spread to the final two compartments.\n\nShielding his face with his free hand, Crozier charged forward through the flames, herding the last of the fleeing revelers on ahead of him.\n\nOut through the burning purple room staggered the survivors as Crozier led them into the blazing blue room. The wind from the northwest was howling now, joining with screams and roars and hisses that might have been only in Francis Crozier's head for all he knew at that moment, and the flames were blowing across the blue compartment's wide opening, creating a barrier of fire.\n\nA cluster of about a dozen men, some still wearing shreds of their costume finery, had slid to a stop before those flames.\n\n\"MOVE!\" roared Crozier, bellowing in his most commanding typhoon voice. A lookout in the crosstrees at the top of a mainmast two hundred feet above the deck could have heard the command clearly in an eighty-knot wind with forty-foot waves crashing around them. And he would have obeyed. These men also obeyed, jumping, screaming, and running through the flames with Crozier right behind them, still carrying Chambers along on his right shoulder and tugging Fitzjames along with his left hand.\n\nOnce outside, his slops steaming, Crozier continued running, catching and passing some of the dozens of men who were spreading out in every direction in the night. The captain did not immediately see the white creature among the men, but everything was very confused out here \u2014 even with the flames throwing light and shadows five hundred feet in every direction \u2014 and then he was busy shouting for his officers and trying to find an ice boulder on which to lay the still-unconscious George Chambers.\n\nSuddenly there came the _pop-pop-pop_ of musket fire.\n\nIncredibly, unbelievably, obscenely, a line of four Marines just outside the circle of light from the flames had taken their knees on the ice and were firing into the clumps and mobs of running men. Here and there a figure \u2014 still sadly and absurdly in costume \u2014 fell to the ice.\n\nReleasing Fitzjames, Crozier ran forward, stepping into the line of volley fire and waving his arms. Musket balls whizzed past his ears.\n\n\"CEASE FIRE! GOD-DAMN YOUR EYES, SERGEANT TOZER, I'LL BREAK YOU TO A PRIVATE FOR THIS AND HAVE YOU HANGED IF YOU DON'T _CEASE THAT FUCKING FIRE IMMEDIATELY!_ \"\n\nThe firing popped and stopped.\n\nThe Marines snapped to a standing salute, Sergeant Tozer shouting that the white thing was out there among the men. They'd seen it backlit by the flames. It was carrying a man in its jaws.\n\nCrozier ignored him. Shouting and shoving both Terrors and Erebuses into clumps around him on the ice, sending obviously mauled or burned men back to Fitzjames's nearby ship, the captain was hunting for his officers \u2014 or _Erebus_ officers \u2014 or anyone he could give an order to and have it relayed to the clusters of terrified men still running out through seracs and across pressure ridges into the howling arctic darkness.\n\nIf those men didn't come back, they'd freeze to death out there. Or the thing would find them. Crozier decided that no one was going the mile back to _Terror_ until they had warmed up on the lower deck of _Erebus._\n\nBut first Crozier had to get his men calmed, organized, and busy pulling the wounded and the bodies of the dead from what was left of the burning Carnivale compartments.\n\nIn the first moments he found only the _Erebus_ mate Couch and Second Lieutenant Hodgson, but then Lieutenant Little came up through the smoke and steam \u2014 the top few inches of ice were melting in an irregular radius around the flames and sending a thick fog out across the sea ice and into the serac forest \u2014 saluted clumsily, his right arm was burned, and reported for duty.\n\nWith Little at his side, Crozier found it easier to gain control of the men, get them back toward _Erebus,_ and start taking roll. He ordered the Marines to reload and set them in a defensive skirmish line between the accumulating mass of staggering men near _Erebus_ 's ice ramp and the still roaring inferno.\n\n\"My God,\" said Dr. Harry D. S. Goodsir, who had just come out of _Erebus_ and was standing nearby, tugging off his slops and greatcoat. \"It's actually _warm_ out here with the flames.\"\n\n\"So it is,\" said Crozier, feeling the sweat on his face and body. The fire had brought the temperature up a hundred degrees or more. He wondered idly if the ice would melt and they'd all drown. To Goodsir he barked, \"Go over there to Lieutenant Hodgson and tell him to begin to assess the numbers of dead and wounded and to get them to you. Find the other surgeons and get _Erebus_ 's sick bay fitted out in Sir John's Great Room \u2014 set it up as they trained you surgeons to do for a combat engagement at sea. I don't want the dead laid out on the ice \u2014 that thing is still out here somewhere \u2014 so tell your seamen to carry them to the forepeak on the lower deck. I'll check in on you in forty minutes \u2014 have a complete butcher's bill ready for me.\"\n\n\"Aye, Captain,\" said Goodsir. Grabbing up his outer clothes, the surgeon rushed toward Lieutenant Hodgson and the ice ramp to _Erebus._\n\nThe canvas and rigging and ice-set masts and costumes and tables and casks and other furniture in the inferno that had been the seven coloured compartments continued to burn all through that night and deep into the darkness of the next morning.\n\nGOODSIR\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _4 January, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Tuesday, 4 January, 1848 \u2014_\n\n _I am the only one left._\n\n _Of the Expedition's Surgeons, I am the only one left. All agree that we were incredibly Lucky to have lost only Five in Death to the Grand Venetian Carnivale's Horror and Conflagration, but the fact that Three of those Five were my Fellow Surgeons is, at the very least, Extraordinary._\n\n _The two Chief Surgeons, Drs. Peddie and Stanley, died of Burns. My Assistant Surgeon counterpart on HMS_ Terror, _Dr. McDonald, survived the flames and Raging Beast only to be Struck Down by a Marine's Musket Ball upon fleeing the burning tents._\n\n _Both of the other two Fatal Casualties were also Officers. First Lieutenant James Walter Fairholme of_ Erebus _had his chest crushed in the Ebony Room, presumably by the creature there. Although Lt. Fairholme's Body was found Burned in the ice-melted wreckage of that Loathsome Tent Maze, my postmortem examination showed that he had Died Instantly when his collapsing Rib Cage had pulverized his Heart._\n\n _The final fatality of the New Year's Eve Fire and Mayhem was_ Terror _'s First Mate Frederick John Hornby, who had been Eviscerated in that Canvas Enclosure in what the men had called the White Room. The sad irony of Mr. Hornby's death was that the gentleman had been on Watch Duty aboard_ Terror _through most of the evening and had been relieved by Lieutenant Irving not an hour before the Violence broke out._\n\n _Captain Crozier and Captain Fitzjames now find themselves without three of their Four Surgeons and without the Advice and Services of two of their most trusted officers._\n\n _Eighteen other men were injured \u2014 six seriously \u2014 during the Venetian Carnivale Nightmare. Of those six \u2014 Ice Master Mr. Blanky from_ Terror; _Carpenter's Mate Wilson, also from that ship (the men affectionately call him \"Fat Wilson\"); Seaman John Morfin, with whom I Traveled to King William Land some months ago;_ Erebus _'s purser's steward, Mr. William Fowler; Seaman Thomas Work, also from_ Erebus; _and_ Terror _boatswain Mr. John Lane \u2014 I am pleased to report that all should survive. (Although it is another irony that Mr. Blanky, who had suffered less serious injuries from the Same Creature only less than a Month Ago \u2014 injuries to which all four of us Surgeons applied our time and expertise \u2014 had not been burned at the Carnivale Mayhem but was injured yet again in the right leg \u2014 mauled or bitten by the thing from the ice, he believes, although he says that he was cutting his way through burning Canvas and Rigging at the time. This time I had to amputate his right leg just below the knee. Mr. Blanky remains remarkably Chipper for a man who has sustained so much damage in so Short a Time.)_\n\n _Yesterday, Monday, all of us Survivors witnessed Floggings. It was the first such Naval Corporal Punishment I have ever seen and I Pray God that I shall never see more._\n\n _Captain Crozier \u2014 who has been visibly consumed by an Anger Beyond Words since the Fire last Friday night \u2014 assembled every Surviving Crew Member of both ships on the lower deck of_ Erebus _at 10:00 a.m. yesterday. The Royal Marines made a line with muskets at the vertical. Drums were beaten._\n\nErebus _gunroom steward Mr. Richard Aylmore and_ Terror _caulker's mate Cornelius Hickey, as well as a truly huge common Seaman named Magnus Manson, were marched bareheaded and wearing only their trousers and undershirts to a place in front of the ship's Stove, where a wooden Hatch Cover had been rigged vertically. One by one, starting with Mr. Aylmore, they were Tied to this Hatch._\n\n _But before this, the men were made to stand there, Aylmore's and Manson's heads bowed, Hickey's upright and defiant, as Captain Crozier read the charges._\n\n _For Aylmore, it was fifty lashes for Insubordination and Reckless Behavior endangering his ship. If the quiet gunroom steward had simply come up with the idea for the coloured tents \u2014 an Idea he acknowledged had come from some Fantastical American Magazine Story \u2014 the Punishment would have been certain but less Severe. But in addition to being a Primary Planner of the Grand Venetian Carnivale, Aylmore had made the Mistake of costuming himself as the Headless Admiral \u2014 a Major Impropriety, given the circumstances surrounding Sir John's death, and one we all understood could have resulted in Aylmore's hanging. We had each heard tales of Aylmore's private Testimony before the Captains in which he had described how he had Screamed and then Fainted in the Ebony Room upon Realizing that the Thing from the Ice was there in the Darkness with the mummers._\n\n _For Manson and Hickey, it was fifty lashes for Sewing and Wearing the Dead Bears' animal skins \u2014 a violation of all of Captain Crozier's previous Orders about not wearing such Heathen Fetishes._\n\n _It was understood that fifty or more other men were Complicit in the Planning, Rigging, Dyeing of Sails, and Staging of the Grand Carnivale, and that Crozier could have handed out an Equal Number of Lashes to all. In a sense, this Sad Trinity of Aylmore, Manson, and Hickey was receiving Punishment for the Entire Crew's bad judgement._\n\n _As the drums stopped beating and the Men stood in a line before the Assembled Crews, Captain Crozier spoke. I hope that I remember his words exactly here:_\n\nThese men are about to Receive the Lash for Violations of Ship's Articles and for the Unwise Behavior in which every man here participated. Including myself.\n\nLet it be known and remembered by All here Assembled, that the Ultimate Responsibility for the folly that claimed the lives of Five of Our Crewmates, the Leg of Another, and which will leave Scars on almost a Score More, was mine. A captain is responsible for everything that happens on his Ship. The leader of an Expedition is doubly responsible. The fact that I allowed these plans to proceed without my Attention or Intervention was Criminal Negligence, and I will admit as much during my Inevitable Court-Martial... inevitable, that is, _if_ we Survive and escape from the ice that Binds Us. These lashes \u2014 and more \u2014 should be mine and _will_ be mine when falls the inevitable Punishment meted out by _my_ superiors.\n\n _I glanced then at Captain Fitzjames. Certainly any Self-Blame that Captain Crozier would cast upon himself would also apply to the commander of_ Erebus, _since it was he, not Crozier, who had overseen most of the Carnivale's arrangements. Fitzjames's face was impassive and Pale. His gaze seemed unfocused. His thoughts seemed elsewhere._\n\nUntil such a day of my own reckoning for Responsibility, _Crozier concluded,_ we proceed with the Punishment of These Men, duly tried by Officers of HMSs _Erebus_ and _Terror_ and Found Guilty of Violation of the Ship's Articles and of the Additional Crime of Endangering the lives of their Comrades. Boatswain Mate Johnson...\n\n _And here Thomas Johnson, large and Capable boatswain mate of HMS_ Terror, _old Shipmate of Captain Crozier \u2014 having served five years in the South Polar Ice on_ Terror _with him \u2014 stepped forward and nodded for the first man, Aylmore, to be tied to the Grate._\n\n _Bosun Johnson then laid out on a cask a leather-bound Box and unlatched its ornate brass snaps. Incongruously, the interior lining was of Red Velvet. Set into its Proper Receptacle in this Red Velvet Lining was the palm-darkened leather grip and folded Tails of the Cat._\n\n _While two Seamen bound Aylmore securely, Bosun's Mate Johnson lifted out the Cat and tested it with a preparatory Flick of his thick Wrist. It was not a Motion done for Show but a true preparation for the Hideous Punishment to come. The nine leather tails \u2014 of which I had heard so many Shipboard Jokes \u2014 flicked out with distinct and Audible and terrible cracks. There were small Knots at the end of each tail._\n\n _Part of me could not believe that this was happening. It seemed impossible in this crowded, sweat-stinking Gloom of the Lower Deck, with the low Overhead Beams and Lumber and Gear hanging lower, that Johnson could possibly manage the Cat so as to effect any Punishment. I had heard the phrase \"not Enough Room to Swing a Cat In\" since I was a boy, but never had I Understood it until this Moment._\n\nExecute the punishment for Mr. Aylmore, _said Captain Crozier. The drums beat again briefly and stopped abruptly._\n\n _Johnson took a broad sideways stance, setting his feet like a Boxer in the Ring, then swung the Cat back, and then Forward in a Violent, Sudden but Smooth Sidearm Motion, the knotted Tails passing less than a Foot from the Front Ranks of Assembled Men._\n\n _The sound of the Cat's tails striking Flesh is something I shall never Forget._\n\n _Aylmore screamed \u2014 a more Inhuman Sound, some said later, than the roar they had heard from the creature in the Ebony Room._\n\n _Crimson Stripes appeared immediately upon the man's thin, pale back, and droplets of Blood spattered the faces of those men standing nearest the Grate, myself included._\n\nONE, _counted Charles Frederick Des Voeux, who had assumed the duties of_ Erebus _'s First Mate upon the death of Mate Robert Orme Sergeant in December. It was the Duty of both first mates to administer this punishment._\n\n _Aylmore screamed again even as the Cat was pulled back for another blow, almost certainly in horrid anticipation of Forty-nine More Lashes. I confess that I swayed on my feet... the Press of Unwashed Bodies, the Stink of Blood, the sense of Confinement in the Dim, Stinking Gloom of the Lower Deck, all making my head swim. Surely this was Hell. Nor was I out of it._\n\n _The Gunroom Steward passed out on the Ninth Lash. Captain Crozier gestured to me to ascertain that the flogged man was still breathing. He was. Normally, as I was made to understand later, a Second Mate would have thrown a Bucket of Water on the victim of punishment to revive him so that he must Fully Suffer the remaining lashes. But there was no Liquid Water on the Lower Deck of HMS_ Erebus _that morning. All was frozen. Even the droplets of Bright Blood on Aylmore's back appeared to be freezing into crimson pellets._\n\n _Aylmore stayed unconscious but the Punishment continued._\n\n _After Fifty Lashes, Aylmore was untied and carried Aft to Sir John's former cabin, since the Great Cabin was still being used as the Sick Bay in the aftermath of the Carnivale injuries. There were Eight Men on cots in there, including David Leys, still unresponsive since the Thing's attack on Mr. Blanky early in December._\n\n _I started to go aft to attend to Aylmore, but Captain Crozier silently gestured me back into the ranks. Evidently it was protocol for all crew members to witness the complete series of Floggings, even should Aylmore bleed to death due to my absence._\n\n _Magnus Manson was next. The huge man dwarfed the second mates tying him to the Grate. If the Giant had decided to Resist at that moment, I have Little Doubt that the ensuing Chaos and Carnage would have resembled New Year's Eve's mayhem in the Seven Coloured Compartments._\n\n _He did not resist. As far as I could tell, Boatswain's Mate Johnson administered the endless Flogging with the same force and Severity as he had for Aylmore \u2014 no more, no less. Blood flew from the first Impact. Manson did not scream. He did something Infinitely Worse. From the first touch of the Lash, he wept like a child. He sobbed. But afterward he was able to walk between the two Seamen escorting him back to the Sick Bay, although \u2014 as always \u2014 Manson had to hunch over so that his head did not strike the Beams overhead. As he passed me, I noticed Strips of Flesh hanging loose on his back between the crisscrossed Scourging wounds of the Cat._\n\n _Hickey, the smallest of the three men being punished, barely made a sound during the long Administration of the Lashes. His narrow Back tore open more freely than had the flesh of the other two, but he did not cry out. Nor did he pass out. The diminutive Caulker's Mate seemed to remove his mind to something beyond the Grate and Overhead Deck upon which his obviously angry glare was firmly fixed and his only reaction to the Terrible Flogging was a gasp for breath between each of the fifty lashes of the Cat._\n\n _He walked aft to the provisional Sick Bay without accepting help from the seamen on either side of him._\n\n _Captain Crozier announced that punishment had been duly meted out according to the Ship's Articles and Dismissed the Company. Before going aft, I ran up on deck very briefly to watch the departure of the men from_ Terror. _They went down the ice ramp from the ship and began their long walk back to the other ship in the dark \u2014 passing the scorched and partially melted area where the Carnivale Conflagration had taken place. Crozier and his primary officer, Lieutenant Little, brought up the rear. None of the more than forty men had said a word by the time they had disappeared beyond the small circle of light radiating from_ Erebus _'s deck lanterns. Eight men remained as a sort of companion Guard to walk with Hickey and Manson when they were ready to be returned to_ Terror.\n\n _I hurried down and aft to the new Sick Bay to take care of my new charges. Beyond Washing and Bandaging their wounds \u2014 the Cat had left a Sickening array of welts and gouges on each man and some Permanent Scars, I would think \u2014 there was little else I could do. Manson had ceased his Weeping, and when Hickey abruptly ordered him to stop his Snuffling, the giant did so at once. Hickey suffered my Ministrations in silence and gruffly ordered Manson to get fully dressed and to follow him out of the Sick Bay._\n\n _Aylmore, the gunroom steward, had been unmanned by the punishment. From the minute he had regained consciousness, according to young Henry Lloyd, my current Surgeon's Assistant, Aylmore had moaned and cried aloud. He continued doing so as I Washed and Bandaged him. He was still moaning piteously and seemed unable to walk by himself when some of the other warrant officers \u2014 the elderly John Bridgens, the Subordinate Officer's Steward, Mr. Hoar, the Captain's Steward, Mr. Bell the Quartermaster, and Samuel Brown, the Boatswain's Mate \u2014 arrived to help him back to his quarters._\n\n _I could hear Aylmore moaning and crying out all the way down the Companionway and around the Main Ladderway as the other men half-carried him to the gunroom steward's cubicle on the starboard side between William Fowler's empty berth and my own, and I knew that I would probably be listening to Aylmore's cries through the thin wall all through the night._\n\nMr. Aylmore reads a lot, _said William Fowler from his place on his cot in the Sick Bay. The Purser's Steward had received serious burns_ and _a Terrible Mauling during the night of the Carnivale Conflagration, but never once during the last four days of stitchings or skin removals had Fowler cried out. With wounds and burns on both his Back and Stomach, Fowler attempted to sleep on his side, but not once had he complained to Lloyd or me._\n\nMen who read a lot have a more sensitive disposition, _added Fowler._ And if the poor bloke hadn't read that stupid story by that American, he wouldn't have suggested the different-coloured compartments for Carnivale \u2014 an idea we all thought was Wonderful at the time \u2014 and none of this would have happened.\n\n _I did not know what to say to this._\n\nMaybe reading is a sort of curse is all I mean, _concluded Fowler._ Maybe it's better for a man to stay inside his own mind.\n\nAmen, _I felt like saying, although I do not know why._\n\n _As I write this, I am in Dr. Peddie's former surgeon's berth on HMS_ Terror _since Captain Crozier has instructed me to spend each Tuesday through Thursday aboard his ship and the Remaining Days of the Week aboard_ Erebus. _Lloyd is watching my six recovering charges in the_ Erebus _sick bay and I was Distressed to discover almost as many seriously ill men here aboard_ Terror.\n\n _For many of them, it is the disease we Arctic Doctors first called Nostalgia and then Debility. The early severe stages of this disease \u2014 besides bleeding gums, Confusion of Thought, weakness in the Extremities, bruises everywhere, and bleeding from the Colon \u2014 often include a tremendous Sentimental Wish to go home. From Nostalgia the weakness, confusion, Impaired Judgement, bleeding from Anus and Gums, open Sores, and other symptoms worsen until the patient is unable to stand or work._\n\n _Another name for Nostalgia and Debility \u2014 one which all Surgeons hesitate to say aloud and which I have not yet done \u2014 is Scurvy._\n\n _Meanwhile, Captain Crozier took to his Private Cabin yesterday and is terribly sick. I can hear his stifled moans since the late Peddie's compartment borders the captain's here on the starboard stern side of the ship. I think Captain Crozier is biting down on something hard \u2014 perhaps a Strip of Leather \u2014 to keep those moans from being heard. But I have always been Blessed (or Cursed) with good hearing._\n\n _The Captain turned over the handling of the Ship's and Expedition's affairs to Lieutenant Little yesterday \u2014 thus quietly but Firmly giving Command to Little rather than to Captain Fitzjames \u2014 and explained to me that he, Captain Crozier, was battling a recurrence of Malaria._\n\n _This is a lie._\n\n _It is not just the symptoms of Malaria which I hear Captain Crozier suffering \u2014 and almost certainly will continue to hear through the walls until I head back to_ Erebus _on Friday morning._\n\n _Because of my uncle's and my father's weaknesses, I know the Demons the Captain is battling tonight._\n\n _Captain Crozier is a man addicted to Hard Spirits, and either those Spirits on board have been used up or he has decided to go off them of his own Volition during this Crisis. Either way, he is suffering the Torments of Hell and shall continue to do so for many days more. His sanity may not survive. In the meantime, this ship and this Expedition are without their True Leader. His stifled moans, in a ship descending into Sickness and Despair, are Pitiable to the extreme._\n\n _I wish I could help him. I wish I could help the dozens of other Sufferers \u2014 all the victims of wounds, maulings, burns, diseases, incipient malnutrition, and melancholic despair \u2014 aboard this entrapped ship and her sister ship. I wish I could help myself, for already I am showing the early signs of Nostalgia and Debility._\n\n _But there is little that I \u2014 or any surgeon in this Year of Our Lord 1848 \u2014 can do._\n\n _God help us all._\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _11 January, 1848_\n\nIt will not end.\n\nThe pain will not end. The nausea will not end. The chills will not end. The terror will not end.\n\nCrozier writhes in the frozen blankets of his bunk and wants to die.\n\nDuring his lucid moments this week, which are few, Crozier laments the most sane act he had performed before retreating to his demons; he had given his pistol to Lieutenant Little with no explanation other than to tell Edward not to return it unless and until he, the captain, asks for it while on deck and in full uniform again.\n\nCrozier would pay anything now for that charged and loaded weapon. This level of pain is unsupportable. These _thoughts_ are unsupportable.\n\nHis grandmother on his late, unlamented father's side, Memo Moira, had been the outcast, the unmentioned and unmentionable Crozier. In her eighties, when Crozier was not yet a teenager, Memo lived two villages away \u2014 an immense, inestimable, and unbridgeable distance for a boy \u2014 and his mother's family neither included her in family events nor mentioned her existence.\n\nShe was a Papist. She was a witch.\n\nCrozier began sneaking over to her village, cadging rides on pony carts, when he was ten. Within a year he was going with the old woman to that strange village's Papist church. His mother and aunt and maternal grandmother would have died if they had known. He would have been renounced and exiled and held in as much scorn by that proper Irish-English Presbyterian side of his family as the Naval Board and Arctic Council had held him in for all these years just for being an Irishman. And a commoner.\n\nMemo Moira had thought him special. She told him that he had the Second Sight.\n\nThe thought did not frighten young Francis Rawdon Moira Crozier. He loved the darkness and the mystery of the Catholic service \u2014 the tall priest strutting like a carrion crow and pronouncing magic in a dead language, the immediate magic of the Eucharist bringing the dead back to life so that the faithful could devour Him and become of Him, the smell of incense and the mystical chanting. Once, when he was twelve, shortly before he ran away to sea, he told Memo that he wanted to become a priest, and the old woman laughed that wild, husky laugh of hers and told him to put such nonsense out of his mind. \"Being a priest is as common and useless as being an Irish drunkard. Use your Gift instead, Young Francis,\" she'd said. \"Use the Second Sight that has been in my family for a score of generations. It will help you go places and see things that no person on this sad earth has ever seen.\"\n\nYoung Francis did not believe in Second Sight. It was about that same time that he realized that he also did not believe in God. He went to sea. He believed in everything he learned and saw there, and some of these sights and lessons were strange indeed.\n\nCrozier rides on crests of pain rolling in on waves of nausea. He awakens only to vomit into the bucket that Jopson, his steward, has left there and replaces each hour. Crozier hurts to the cavity in the center of his self where he is sure his soul had resided until it floated away on a sea of whiskey over the decades. All through these days and nights of cold sweat on frozen sheets, he knows he would give up his rank, his honours, his mother, his sisters, his father's name, and the memory of Memo Moira herself for one more glass of whiskey.\n\nThe ship groans as it continues to be squeezed inexorably to fragments by the never-ceasing ice. Crozier groans as his demons continue to squeeze him inexorably to fragments through chills, fever, pain, nausea, and regret. He has cut a six-inch strap from an old belt, and to keep from moaning aloud he bites down on that in the darkness. He moans anyway.\n\nHe imagines it all. He _sees_ it all.\n\nLady Jane Franklin is in her element. Now, with two and a half years of no word from her husband, she is in her element. Lady Franklin the Indomitable. Lady Franklin the Widow Who Refused to Be a Widow. Lady Franklin the Patroness and Saint of the Arctic that has killed her husband... Lady Franklin who will never accept such a fact.\n\nCrozier can see her as clearly as if he does have Second Sight. Lady Franklin has never looked more beautiful than now in her resolve, in her refusal to grieve, in her determination that her husband is alive and that Sir John's expedition must be found and rescued.\n\nMore than two and a half years have passed. The Navy knows that Sir John had provisioned _Erebus_ and _Terror_ for three years at normal rations but had expected to emerge beyond Alaska in the summer of 1846, certainly no later than August of 1847.\n\nLady Jane will have bullied the lethargic Navy and Parliament into action by now. Crozier can see her writing letters to the Admiralty, letters to the Arctic Council, letters to her friends and former suitors in Parliament, letters to the queen, and, of course, letters to her dead husband every day, writing in her perfect, no-nonsense script and telling the dead Sir John that she knows that her darling is still alive and that she looks forward to her inevitable reunion with him. He can see her telling the world that she does this. She will be sending sheaves and folios of letters to him off with the first rescue ships about now... Naval ships, to be sure, but also quite probably private ships hired with either the dwindling money of Lady Jane's own fortune or by subscriptions from worried and rich friends.\n\nCrozier, rising from his visions, tries to sit up in his bunk and smile. The chills make him shake like a topgallant in a gale. He vomits into the almost full pail. He falls back onto his sweat-soaked, bile-smelling pillow and closes his eyes to ride the waves of his seeing.\n\nWhom would they send to save _Erebus_ and _Terror?_ Whom had they already sent?\n\nCrozier knew that Sir John Ross would be champing at the bit to lead any rescue parties into the ice, but he also sees that Lady Jane Franklin will ignore the old man \u2014 she thinks him vulgar \u2014 and will choose his nephew, James Clark Ross, with whom Crozier had explored the seas around Antarctica.\n\nThe younger Ross had promised his young bride that he would never go on a sea exploration again, but Crozier sees that he could not refuse this request from Lady Franklin. Ross would choose to go with two ships. Crozier saw them sailing this coming summer of 1848. Crozier saw the two ships sailing north of Baffin Island, west through Lancaster Sound, where Sir John had sailed _Terror_ and _Erebus_ three years ago \u2014 he could almost make out the names on the bows of Ross's ships \u2014 but Sir James would encounter the same relentless pack ice beyond Prince Regent Inlet, perhaps beyond Devon Island, that holds Crozier's ships in thrall now. Next summer there will be no full thaw of the sounds and inlets Ice Masters Reid and Blanky had sailed them south through. Sir James Clark Ross will never get within three hundred miles of _Terror_ and _Erebus._\n\nCrozier saw them turning back to England in the freezing early autumn of 1848.\n\nHe weeps as he moans and bites down hard on his leather strap. His bones are freezing. His flesh is on fire. Ants crawl everywhere on and under his skin.\n\nHis Second Sight sees there would be other ships sent, other rescue expeditions this year of our Lord 1848, some most likely launched at the same time or earlier than Ross's search party. The Royal Navy was slow to act \u2014 a maritime sloth \u2014 but once in motion, Crozier knows, it tended to overdo everything it undertook. Wretched excess after interminable stalling was standard procedure for the Navy Francis Crozier has known for four decades.\n\nIn his aching mind, Crozier saw at least one other Naval expedition setting sail for Baffin Bay in search of the lost Franklins this coming summer and most probably even a third Naval squadron sent all the way around Cape Horn to rendezvous, theoretically, with the other search-party ships near the Bering Strait, searching for them in the western arctic, to which _Erebus_ and _Terror_ had never come within a thousand miles. Such ponderous operations would stretch into 1849 and beyond.\n\nAnd this is only the beginning of the second week of 1848. Crozier doubts if his men will live to see the summer.\n\nWould there be an overland party sent up from Canada to follow the Mackenzie River to the arctic coastline, then east to Wollaston Land and Victoria Land in search of their ships stranded somewhere along the elusive North-West Passage? Crozier is sure there will be. The chances of such an overland expedition finding them twenty-five miles out at sea to the northwest of King William Island are nil. Such a party would not even know that King William Island _was_ an island.\n\nWould the First Lord of the Admiralty announce in the House of Commons a reward for the rescue of Sir John and his men? Crozier thinks he will. But how much? A thousand pounds? Five thousand pounds? Ten thousand? Crozier closes tight his eyes and sees, as if on parchment hanging before him, the sum of twenty thousand pounds offered for anyone who \"might render efficient assistance in saving the lives of Sir John Franklin and his squadron.\"\n\nCrozier laughs again, which brings on the vomiting again. He is shaking with cold and pain and the clear absurdity of the images in his head. All around him the ship groans as the ice crushes it. The captain can no longer tell the groaning of the ship from his own moans.\n\nHe sees an image of eight ships \u2014 six British, two American \u2014 clustered within a few miles of one another in mostly frozen anchorages that look to Crozier like Devon Island, near Beechey, or perhaps Cornwallis Island. It is obviously a late arctic-summer day, perhaps late August, mere days before the sudden freeze that may capture all of them. Crozier has the sense that this image is two or three years in the future of his terrible reality this moment in 1848. Why eight ships sent out for rescue would end up clumped together like this in one location rather than fanning out throughout thousands of square miles of the arctic to hunt for signs of Franklin's passing makes no sense to Crozier whatsoever. It is the delusion of toxic madness.\n\nThe craft range in size from a small schooner and a yacht-sized craft far too flimsy for such serious ice work to 144-ton and 81-ton American ships strange to Crozier's eye to an odd little 90-ton English pilot boat crudely fitted out for arctic sailing. There are also several proper British Naval vessels and steam cruisers. In his aching mind's eye he can see the names of the ships \u2014 _Advance_ and _Rescue,_ these under the American flag, and _Prince Albert_ for the former pilot boat, as well as the _Lady Franklin_ at the head of the anchored British squadron. There are also two ships Crozier associates with old John Ross \u2014 the undersized schooner _Felix_ and the totally inappropriate little yacht _Mary._ Finally there are two true Royal Navy vessels, _Assistance_ and _Intrepid._\n\nAs if viewing them through the eyes of a high-soaring arctic tern, Crozier can see that all eight of these ships are clustered within forty miles of one another \u2014 four of the smaller British craft at Griffith Island above the Barrow Strait, four of the remaining English ships at Assistance Bay on the south tip of Cornwallis, and the two American ships farther north, just around the eastern curve of Cornwallis Island, just across Wellington Channel from Sir John's first winter anchorage at Beechey Island. None are within two hundred and fifty miles of the spot far to the southwest where _Erebus_ and _Terror_ lie trapped.\n\nA minute later, a mist or cloud clears, and Crozier sees six of these vessels anchored within a quarter of a mile of one another just off the curve of a small island's shoreline.\n\nCrozier sees men running across frozen gravel under a vertical black cliff wall. The men are excited. He can almost hear their voices in the freezing air.\n\nIt is Beechey Island, he is sure. They have found the weathered wooden headboards and graves of Stoker John Torrington, Seaman John Hartnell, and Marine Private William Braine.\n\nHowever far in the future this fever-dream discovery is, Crozier knows, it will do him and the other men of _Erebus_ and _Terror_ no good whatsoever. Sir John had left Beechey Island in a mindless hurry, sailing and steaming the first day the ice relented enough to allow the ships to leave their anchorage. After nine months frozen there, the Franklin Expedition had left not so much as a note saying which direction they were sailing.\n\nCrozier had understood at the time that Sir John did not feel it necessary to inform the Admiralty that he was obeying their orders by sailing south. Sir John Franklin always obeyed orders. Sir John assumed that the Admiralty would trust that he had done so again. But after nine months on the island \u2014 and after building the proper cairn and even leaving a cairn of pebble-filled Goldner food cans behind as a sort of joke \u2014 the fact remained that the message cairn at Beechey Island was left empty contrary to Franklin's orders.\n\nThe Admiralty and Discovery Service had outfitted the Franklin Expedition with two hundred airtight brass cylinders for the express purpose of leaving behind messages of their whereabouts and destination along the entire course of their search for the North-West Passage, and Sir John had used... one: the useless one sent to King William Land twenty-five miles to the southeast of their present position, cached a few days before Sir John was killed in 1847.\n\nOn Beechey Island, nothing.\n\nOn Devon Island, which they had passed and explored, nothing.\n\nOn Griffith Island, where they had searched for harbours, nothing.\n\nOn Cornwallis Island, which they had circumnavigated, nothing.\n\nDown the entire length of Somerset Island and Prince of Wales Island and Victoria Island along which they had sailed south for the entire summer of 1846, nothing.\n\nAnd now, in his dream, the rescuers in the six ships \u2014 now all on the verge of being frozen in themselves \u2014 were looking north to what open sea remained up the Wellington Channel toward the North Pole. Beechey Island revealed no clues whatsoever. And Crozier could see from his magical arctic tern's high viewpoint that Peel Sound to the south \u2014 down which _Erebus_ and _Terror_ had found their way a year and a half ago during that brief summer thaw \u2014 was now, in this future summer, a solid sheet of white as far as the men on Beechey Island and sailing Barrow Strait could see.\n\nThey never even consider that Franklin could have gone that way... that he could have obeyed orders. Their attention \u2014 for the coming years, since Crozier sees that they are frozen in solid now in Lancaster Sound \u2014 is to search to the north. Sir John's secondary orders had been that if he could not continue his way south to force the Passage, he should turn north to sail through the theoretical rim of ice into the even more theoretical Open Polar Sea.\n\nCrozier knows in his sinking heart that the captains and men of these eight rescue ships have all come to the conclusion that Franklin had gone north \u2014 precisely the opposite direction he had in fact sailed.\n\nHe wakes in the night. His own moaning has awakened him. There is light, but his eyes cannot stand the light so he tries to understand what is happening just through the burning of touch and the crash of sound. Two men \u2014 his steward, Jopson, and the surgeon, Goodsir \u2014 are stripping him of his filthy and sweat-soaked nightshirt, bathing him with miraculously warm water, and carefully dressing him in a clean nightshirt and socks. One of them tries to feed him soup with a spoon. Crozier vomits up the thin gruel, but the contents of his full-to-the-brim vomit pail are frozen solid and he is vaguely aware of the two men cleaning the deck. They make him drink some water and he falls back into his cold sheets. One of them spreads a warm blanket over him \u2014 _a warm, dry, unfrozen blanket_ \u2014 and he wants to weep with gratitude. He also wants to speak but is slipping back into the maelstrom of his visions and cannot find or frame the words before all words are lost to him again.\n\nHe sees a boy with black hair and greenish skin curled in a fetal position against a brick-tile wall the colour of urine. Crozier knows that the boy is an epileptic in an asylum, in some bedlam somewhere. The boy shows no movement except for his dark eyes, which constantly flicker back and forth like a reptile's. _That shape am I._\n\nAs soon as he thinks this, Crozier knows that this is not _his_ fear. It is some other man's nightmare. He was briefly in some other mind.\n\nSophia Cracroft enters him. Crozier moans around the biting strap.\n\nHe sees her naked and straining against him at the Platypus Pond. He sees her distant and dismissive on the stone bench at Government House. He sees her standing and waving \u2014 not at him \u2014 in her blue silk dress on the dock at Greenhithe on the May day that _Erebus_ and _Terror_ sailed. Now he sees her as he has never seen her before \u2014 a future-present Sophia Cracroft, proud, grieving, secretly happy to be grieving, renewed and reborn as her aunt Lady Jane Franklin's full-time assistant and companion and amanuensis. She travels everywhere with Lady Jane \u2014 two indomitable women, the press will call them \u2014 Sophia, almost as much as her aunt, always visibly earnest and hopeful and strident and feminine and eccentric and bent to the task of cajoling the world to rescue Sir John Franklin. She will never mention Francis Crozier, not even in private. It is, he sees at once, a perfect role for Sophia: brave, imperious, entitled, able to play the coquette for decades with the perfect excuse for avoiding commitment or real love. She will never marry. She will travel the world with Lady Jane, Crozier sees, never publicly giving up hope that the missing Sir John will be found, but \u2014 long after real hope is surrendered \u2014 still enjoying the entitlement, sympathy, power, and position that this once-removed widowhood affords her.\n\nCrozier tries to vomit, but his stomach has been empty now for hours or days. He can only curl up and suffer the cramps.\n\nHe is in a darkened parlour in a cramped, fussily furnished American farm home in Hydesdale, New York, some twenty miles west of Rochester. Crozier has never heard of either Hydesdale or Rochester, New York. He knows that it is spring of this year, 1848, perhaps only a few weeks in his future. Just visible through a crack in the drawn, thick drapes, a lightning storm surges and flashes. Thunder shakes the house.\n\n\"Come, Mother!\" cries one of two girls at the table. \"We promise you will find this edifying.\"\n\n\"I will find it terrifying,\" says the mother, a drab middle-aged woman with a perpetual frown line bisecting her forehead from her tightly pulled, greying bun to her heavy, frowning eyebrows. \"I don't know why I allow you to talk me into this.\"\n\nCrozier can only marvel at the flat ugliness of the American rural dialect. Most of the Americans he has known have been defecting sailors, U.S. Navy captains, or whalers.\n\n\"Hurry, Mother!\" The girl commanding her mother in such a bossy tone is 15-year-old Margaret Fox. She is modestly dressed and attractive in a simpering and not especially intelligent way that Crozier has noticed is often the case with the few American women he has met socially. The other girl at the table is Margaret's 11-year-old sister Catherine. The younger girl, her pale face only just visible in the flickering candlelight, more resembles her mother, down to the dark eyebrows, too-tight bun, and incipient frown line.\n\nThe lightning flashes in the gap between dusty drapes.\n\nThe mother and two girls join hands around the circular oak table. Crozier notices that the lace doily on the table has yellowed with age. All three females have their eyes closed. Thunder shakes the single candle's flame.\n\n\"Is someone there?\" asks 15-year-old Margaret.\n\nA crashingly loud rap. Not thunder, but a _crack,_ as if someone has struck wood with a small mallet. Everyone's hands are in sight.\n\n\"Oh my!\" cries the mother, obviously ready to throw her hands up over her mouth in fear. Her two daughters hold tight and keep her from breaking the circle. The table rocks from their tugging.\n\n\"Are you our Guide tonight?\" asks Margaret.\n\nA loud _RAP._\n\n\"Have you come to hurt us in any way?\" asks Katy.\n\nTwo even louder _RAP_ s.\n\n\"See, Mother?\" whispers Maggie. Closing her eyes again, she says in a theatrical whisper, \"Guide, are you the gentle Mr. Splitfoot who communicated with us last night?\"\n\n _RAP._\n\n\"Thank you for convincing us last evening that you were real, Mr. Splitfoot,\" continues Maggie, speaking almost as if she were in a trance. \"Thank you for telling Mother the details about her children, telling all our ages, and for reminding her of the sixth child who died. Will you answer our questions tonight?\"\n\n _RAP._\n\n\"Where is the Franklin Expedition?\" asks little Katy.\n\n _RAP RAP RAP rap rap rap rap RAP RAP rap RAP RAP_... the percusssions go on for half a minute.\n\n\"Is this the Spiritual Telegraph you spoke of?\" whispers their mother.\n\nMaggie shushes her. The rapping breaks off. Crozier sees, as if he can float through wood and see through wool and cotton, that both girls are double-jointed and are taking turns snapping and popping their big toes against their second toes. It was an amazingly loud rapping sound from such small toes.\n\n\"Mr. Splitfoot says that the Sir John Franklin whom the papers say everyone is seeking is well and with his men, who are also all well but very frightened, on their ships and in the ice near an island five days' sail south of the cold place where they stopped their first year out,\" intones Maggie.\n\n\"It is very dark where they are,\" adds Katy.\n\nThere come more rappings.\n\n\"Sir John tells his wife, Jane, not to worry,\" interprets Maggie. \"He says that he shall see her soon \u2014 in the next world, if not in this one.\"\n\n\"Oh my!\" Mrs. Fox says again. \"We have to call for Mary Redfield and Mr. Redfield, and Leah, of course, and Mr. and Mrs. Duesler, and Mrs. Hyde, and Mr. and Mrs. Jewell...\"\n\n\"Ssshhhh!\" hisses Katy.\n\n _RAP, RAP, RAP, rapraprapraprap, RAP._\n\n\"The Guide does not want you to speak when He is leading us,\" whispers Katy.\n\nCrozier moans and bites his leather strap. The cramps that had begun in his gut now rack his entire body. He shakes from the chill one moment and throws off the blankets the next.\n\nThere is a man dressed like an Esquimaux \u2014 animal-fur parka, high furry boots, a fur hood like Lady Silence's. But this man is standing on a wooden stage in front of footlights. It is very hot. Behind the man, a painted backdrop shows ice, icebergs, a wintry sky. Fake white snow litters the stage. There are four overheated dogs of the type used by the Greenland Esquimaux lying on the stage, their tongues lolling.\n\nThe bearded man in the heavy parka is talking from the white-speckled podium. \"I speak to you today for humanity, not for money,\" says the little man. His American accent grates on Crozier's aching ear as fiercely as had the teenaged girls'. \"And I have traveled to England to speak to Lady Franklin herself. She wishes me Godspeed on our next expedition \u2014 contingent, of course, on whether we raise the money here in Philadelphia and in New York and in Boston to mount the expedition \u2014 and says that she would be honoured if the sons of the United States were to bring home her husband. So today I ask for your generosity, but only for the sake of humanity. I ask for this in Lady Franklin's name, in her lost husband's name, and in the secure hopes of bringing glory to the United States of America... .\"\n\nCrozier sees the man again. The bearded fellow is out of his parka and naked and in bed in the Union Hotel in New York with a very young naked woman. It is a hot night and the bedclothes have been thrown back. There is no sign of the sledge dogs.\n\n\"Whatever may be my faults,\" the man is saying, speaking softly because the window and transom are open to the New York night, \"I have at least loved you. Were you an empress, darling Maggie, instead of a little nameless girl following an obscure and _ambiguous_ profession, it would be the same.\"\n\nCrozier realizes that the young naked woman is Maggie Fox \u2014 only a few years older. She is still attractive in that simpering American way, even without her clothes on.\n\nMaggie says in a tone much more throaty than the teenager's imperious command Crozier heard earlier, \"Dr. Kane, you know I love you.\"\n\nThe man shakes his head. He has lifted a pipe from the bedside table and now frees his left arm from behind the girl to tamp in the tobacco and light it. \"Maggie, my dear, I hear those words from your little deceitful mouth, feel your hair tumbling onto my chest, and would love to believe them. But you cannot rise above your station, my dear. You have many traits which lift you above your calling, Maggie... you are refined and lovable and, with a different education, would have been innocent and artless. But you are not worthy of a permanent regard from me, Miss Fox.\"\n\n\"Not worthy,\" repeats Maggie. Her eyes, perhaps her prettiest feature now that her plump breasts are covered from Crozier's view, appear to be brimming with tears.\n\n\"I am sold to different destinies, my child,\" says Dr. Kane. \"Remember that I have my own sad vanities to pursue, even as you and your venial sisters and mother pursue your own. I am as devoted to my calling as you, poor child, can be to yours, if such theatrical spiritualist poppycock can be _called_ a calling. Remember then, as a sort of a dream, that Dr. Kane of the Arctic Seas loved Maggie Fox of the Spirit Rappings.\"\n\nCrozier comes awake in the dark. He does not know where or when he is. His cubicle is dark. The ship seems dark. The timbers moan \u2014 or is that an echo of his own moans of the last hours and days? It is very cold. The warm blanket he seems to remember Jopson and Goodsir setting on him is now as damp and frozen as the other bedsheets. The ice moans against the ship. The ship continues its answering groans from pressured oak and cold-strained iron.\n\nCrozier wants to get up but finds that he is too weak and hollow to stir. He can barely move his arms. The pain and visions roll over him like a breaking wave.\n\nFaces of men he has known or met or seen in the Service.\n\nThere is Robert McClure, one of the most guileful and ambitious men Francis Crozier has ever known \u2014 another Irishman intent on making good in an English world. McClure is on the deck of a ship in the ice. Cliffs of ice and rock rise all around, some six or seven hundred feet high. Crozier has never seen anything like it.\n\nThere is old John Ross on the stern deck of a little ship \u2014 a sort of yacht \u2014 heading eastward. Heading home.\n\nThere is James Clark Ross, older and fatter and less happy than Crozier has ever seen him. The rising sun shines through ice-rimmed jib lines as his ship leaves the ice for the open sea. He is heading home.\n\nThere is Francis Leopold M'Clintock \u2014 someone Crozier somehow knows has searched for Franklin under James Ross and then come back on his own again in later years. What later years? How long from now? How far in our future?\n\nCrozier can see images flit by as if from a magic lantern, but he does not hear answers to his questions.\n\nThere is M'Clintock sledging, man-hauling, moving more quickly and efficiently than Lieutenant Gore or any of Sir John's or Crozier's men ever have.\n\nThere is M'Clintock standing at a cairn and reading a note just pulled from a brass cylinder. Is it the note that Gore left on King William Land seven months ago? Crozier wonders. The frozen gravel and grey skies behind M'Clintock look the same.\n\nSuddenly there is M'Clintock, alone on the ice and gravel, his sledging party visible coming up several hundred yards behind him in the blowing snow. He is standing in front of a horror \u2014 a large boat tied and lashed atop of a huge cobbled-together sledge made of iron and oak.\n\nThe sledge looks like something Crozier's carpenter, Mr. Honey, would build. It has been assembled as if it was meant to last for a century. Every join shows care. The thing is massive \u2014 it must weigh at least 650 pounds. Atop it is a boat that weighs another 800 pounds.\n\nCrozier recognizes the boat. It is one of _Terror_ 's 28-footers \u2014 one of the pinnaces. He sees that it has been extensively rigged for river travel. The sails are furled and tied and shrouded and iced over.\n\nClimbing onto a rock and looking into the open boat as if over M'Clintock's shoulder, Crozier sees two skeletons. The teeth in the two skulls seem to gleam at M'Clintock and Crozier. One skeleton is little more than a heap of visibly chewed and heavily gnawed and partially devoured bones tumbled into a rough pile in the bow. Snow has drifted over the bones.\n\nThe other skeleton is intact, undisturbed, and still clothed in the tatters of what looks to be an officer's greatcoat and layers of other warm clothing. The skull still has remnants of a cap on it. This corpse is sprawled on the after-thwarts, its skeletal hands extended along the gunwales toward two double-barreled shotguns propped there. At the body's booted feet lie stacks of wool blankets and canvas clothing and a partially snow-covered burlap bag filled with powder-shot cartridges. Set on the bottom of the pinnace midway between the dead man's boots, like a pirate's booty about to be counted and gloated over, are five gold watches and what looks to be thirty or forty pounds of individually wrapped chunks of chocolate. Also nearby are 26 pieces of silverware \u2014 Crozier can see, and knows that M'Clintock can see, the personal crests of Sir John Franklin's, Captain Fitzjames's, six other officers', and his \u2014 Crozier's \u2014 on the various knives, spoons, and forks. He sees similarly engraved dishes and two silver serving plates sticking up out of the ice and snow.\n\nAlong the 25 feet of pinnace bottom separating the two skeletons lies a dizzying array of bric-a-brac protruding from the few inches of snow that have accumulated: two rolls of sheet metal, a full canvas boat cover, eight pairs of boots, two saws, four files, a stack of nails, and two boat knives next to the bag of powder-shot cartridges near the skeleton in the stern.\n\nCrozier also sees paddles, folded sails, and rolls of twine near the clothed skeleton. Closer to the pile of partially devoured bones in the bow are a stack of towels, bars of soap, several combs and a toothbrush, a pair of handworked slippers just inches from scattered white toe bones and metatarsals, and six books \u2014 five Bibles and _The Vicar of Wakefield,_ which now sits on a shelf in the Great Cabin of HMS _Terror._\n\nCrozier wants to close his eyes but cannot. He wants to fly away from this vision \u2014 all these visions \u2014 but has no control over them.\n\nSuddenly Francis Leopold M'Clintock's vaguely familiar face seems to melt, sag, then reform itself into the visage of a younger man, someone Francis Crozier does not know. Everything else stays the same. The younger man \u2014 a certain Lieutenant William Hobson, whom Crozier now knows without knowing how he knows \u2014 is standing in the same spot that M'Clintock had and is peering into the open boat with the same expression of sickened incredulity that Crozier had seen on M'Clintock's face a moment earlier.\n\nWithout warning, the open boat and the skeletons are gone and Crozier is lying in a cave of ice next to a naked Sophia Cracroft.\n\nNo, it is not Sophia. Crozier blinks, feeling Memo Moira's Second Sight burning through and from his aching brain like a fist of fever, and now he sees that he is lying naked next to a naked Lady Silence. They are surrounded by furs, and they are lying on some sort of snow or ice shelf. Their space is illuminated by a flickering oil lamp. The curved ceiling is made from blocks of ice. Silence's breasts are brown, and her hair is long and very black. She is leaning on one elbow amid the furs and looking at Crozier with some earnestness.\n\n _Do you dream my dreams?_ she asks without moving her lips or opening her mouth. She has not spoken in English. _Am I dreaming yours?_\n\nCrozier _feels_ her inside his mind and heart. It feels like a jolt of the best whiskey he has ever swallowed.\n\nAnd then the most terrible nightmare of all comes.\n\nThis stranger, this blend of M'Clintock and someone named Hobson, is not looking down at the open boat with two skeletons in it but is watching young Francis Rawdon Moira Crozier secretly attending Catholic Mass with his witch-Papist Memo Moira.\n\nIt was one of the deepest secrets of Crozier's life that he had done this thing \u2014 not only gone to the forbidden service with Memo Moira but partaken of the heresy of the Catholic Eucharist, the much-derided and forbidden Holy Communion.\n\nBut this form of M'Clintock-Hobson stands like an altar boy as a trembling Crozier \u2014 now a child, now a scarred man in his fifties \u2014 approaches the altar rail, kneels, puts his head back, opens his mouth, and extends his tongue for the Forbidden Wafer \u2014 the Body of Christ \u2014 pure transubstantiated cannibalism to all the other adults in Crozier's village and family and life.\n\nBut something is strange. The grey-haired priest looming over him in his white robes is dripping water on the floor and altar rail and onto Crozier himself. And the priest is too large even for a child's point of view \u2014 huge, wet, muscled, lumbering, throwing a shadow over the kneeling communicant. He is not human.\n\nAnd Crozier is naked as he kneels, sets his head back, closes his eyes, and extends his tongue for the Sacrament.\n\nThe priest looming and dripping over him has no Wafer in his hand. He has no hands. Instead, the dripping apparition leans over the altar rail, leans far too close, and opens its own inhuman maw as if Crozier is the Bread to be devoured.\n\n\"Dear Jesus Christ God Almighty,\" whispers this watching M'Clintock-Hobson form.\n\n\"Dear Jesus Christ God Almighty,\" whispers Captain Francis Crozier.\n\n\"He's back with us,\" Dr. Goodsir says to Mr. Jopson.\n\nCrozier moans.\n\n\"Sir,\" the surgeon says to Crozier, \"can you sit up? Are you able to open your eyes and sit up? That's a good captain.\"\n\n\"What day is it?\" croaks Crozier. The dull light from the open door and the even duller light from his oil lamp turned low are like explosions of painful sunshine against his sensitive eyes.\n\n\"It's Tuesday, the eleventh day of January, Captain,\" says his steward. And then Jopson adds, \"The year of our Lord eighteen hundred and forty-eight.\"\n\n\"You were very ill for a week,\" says the surgeon. \"Several times in the last few days I was sure that we had lost you.\" Goodsir gives him some water to sip.\n\n\"I was dreaming,\" manages Crozier after drinking the ice-cold water. He can smell his own stink in the nest of frozen bedclothes around him.\n\n\"You were moaning very loudly the last few hours,\" says Goodsir. \"Do you remember any of your malarial dreams?\"\n\nCrozier remembers only the sense-of-flying weightlessness of his dreams, yet at the same time the weight and horror and humour of visions that had already fled like wisps of fog before a strong wind.\n\n\"No,\" he says. \"Mr. Jopson, please be so kind as to fetch me hot water for my toilet. You may have to help me shave. Dr. Goodsir...\"\n\n\"Yes, Captain?\"\n\n\"Would you be so kind as to go forward and tell Mr. Diggle that his captain wants a very large breakfast this morning.\"\n\n\"It is six bells in the evening, Captain,\" says the surgeon.\n\n\"Nonetheless, I want a very large breakfast. Biscuits. What's left of our potatoes. Coffee. Pork of some sort \u2014 bacon if he has it.\"\n\n\"Aye, sir.\"\n\n\"And, Dr. Goodsir,\" Crozier says to the departing surgeon. \"Would you also be so kind as to ask Lieutenant Little to come aft with a report on the week I have missed and also ask him to bring my... property.\"\n\nPEGLAR\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _29 January, 1848_\n\nHarry Peglar had planned it so that he received the duty to carry a message to _Erebus_ on the day the sun returned. He wanted to celebrate it \u2014 as much as anything could be celebrated these days \u2014 with someone he loved. And somebody he'd once been in love with.\n\nChief Petty Officer Harry Peglar was captain of the foretop on _Terror,_ chosen leader of the carefully picked topmen who worked the highest rigging, topsail, and topgallant yards in blaze of day or dark of night as well as in the highest seas and worst weather the world could throw at a wooden ship. This was a position that required strength, experience, leadership, and, most of all, courage, and Harry Peglar was respected for all of these traits. Now almost forty-one years old, he had proved himself hundreds of times not only in front of the crew of HMS _Terror_ but on a dozen other ships on which he'd served over his long career.\n\nIt had been only mildly ironic then that Harry Peglar had been illiterate until he was a twenty-five-year-old midshipman. Reading was now his secret pleasure, and he had already devoured more than half of the 1,000 volumes in _Terror_ 's Great Cabin on this voyage. It had been a mere officers' steward on the survey bark HMS _Beagle_ who had transformed Peglar into a literate man, and it was the same steward who had made Harry Peglar ponder what it meant to be a man.\n\nJohn Bridgens was that steward. He was now the oldest man on the expedition, by far. When they had sailed from England, the joke in both _Erebus_ 's and _Terror_ 's fo'c'sles had been that John Bridgens, lowly subordinate officers' steward, was the same age as the elderly Sir John Franklin but twenty times as wise. Harry Peglar, for one, knew that this was true.\n\nOld men below the rank of captain or admiral rarely were allowed on Discovery Service expeditions, so it was with some good humour that both crews learned that John Bridgens's age on the official ship's muster had been reversed \u2014 either by accident or by a purser with a sense of irony \u2014 and listed as \"26.\" There had been many jokes made to the grey-haired Bridgens about his youth and callowness and presumed sexual prowess. The quiet steward had smiled and said nothing.\n\nIt had been Harry Peglar who had sought out a younger steward Bridgens on the HMS _Beagle_ during their five-year round-the-world scientific survey voyage under Captain FitzRoy from December of 1831 to October of 1836. Peglar had followed an officer he'd served under on HMS _Prince Regent,_ a lieutenant named John Lort Stokes, from the first-rate 120-gun ship of the line to the lowly _Beagle._ The _Beagle_ was only a _Cherokee_ class 10-gun brig adapted as a survey bark \u2014 hardly the kind of ship that an ambitious topman like young Peglar would normally pick \u2014 but even then Harry had been interested in scientific survey work and exploration, and the voyage of the little _Beagle_ under FitzRoy had been an education for him in more ways than one.\n\nSteward Bridgens had been about eight years older then than Peglar was now \u2014 in his late forties \u2014 but already known as the wisest and most widely read warrant officer in the fleet. He was also known as a sodomite, a fact that hadn't bothered twenty-five-year-old Peglar much at the time. There were two types of sodomites in the Royal Navy: those who sought their satisfaction only on shore and never brought their activities to sea, and those who continued their habits at sea, often by seducing the young boys almost always present on Royal Navy ships. Bridgens, everyone in the _Beagle_ fo'c'sle and in the Navy knew, was the former \u2014 a man who liked men when ashore but who never bragged of it nor brought his inclinations to sea. And, unlike the caulker's mate on Peglar's current ship, Bridgens was no pederast. Most of his crewmates thought that a boy at sea was safer with subordinate officers' steward John Bridgens than he would have been with his village vicar at home.\n\nBesides that, Harry Peglar was living with Rose Murray when he sailed in 1831. Although never formally married \u2014 she was a Catholic and would not marry Harry unless he converted, which he could not bring himself to do \u2014 they were a happy couple when Peglar was ashore, although Rose's own illiteracy and lack of curiosity about the world reflected the younger Peglar's life and not the man he would later become. Perhaps they would have married if Rose could have had children, but she could not \u2014 a condition she referred to as \"God's punishment.\" Rose died while Peglar was at sea on the long _Beagle_ voyage. He had loved her, in his way.\n\nBut he had also loved John Bridgens.\n\nBefore the five-year mission of the survey ship HMS _Beagle_ had ended, Bridgens \u2014 at first accepting his role of mentor with reluctance but finally bending under the young topsail midshipman's eager insistence \u2014 had taught Harry to read and write not only in English but also in Greek and Latin and German. He had taught him philosophy and history and natural history. More than that, Bridgens had taught the intelligent young man to think.\n\nIt had been two years after that voyage that Peglar had looked up the older man in London \u2014 Bridgens had been on extended shore leave with most of the rest of the fleet in 1838 \u2014 and requested more tutoring. By then, Peglar was already captain of the foretop on the HMS _Wanderer._\n\nIt was during those months of shorebound discussion and further tutoring that the close friendship between the two men had moved into something more resembling lovers' interactions. The revelation that he was capable of doing such a thing astounded Peglar \u2014 dismaying him at first but then causing him to reconsider every aspect of his life, morals, faith, and sense of self. What he discovered confused him but, to his astonishment, did not change his basic sense of who Harry Peglar was. What was even more astounding to him was that he had been the one to instigate intimate physical contact \u2014 not the older man.\n\nThe intimate aspect of their friendship lasted only a few months and ended by mutual choice as much as by Peglar's long absences at sea on _Wanderer_ until 1844. Their friendship survived intact. Peglar began writing long philosophical letters to the former steward and would spell all words backward, the last letter of the last word in each sentence now first and capitalized. Mostly because the formerly illiterate foretop captain's spelling was so atrocious, Bridgens suggested in one responding letter that \"your childlike idea of Leonardo's backward-writing encryption, Harry, is almost unbreakable.\" Peglar now kept his journals in the same crude code.\n\nNeither man had told the other that he was applying for Discovery Service duty on Sir John Franklin's North-West Passage expedition. Both were astonished a few weeks before sailing time when they saw the other's name on the official roster. Peglar, who had not been in communication with Bridgens for more than a year, traveled from the Woolwich barracks up to the steward's North London rooms to ask if he should drop out of the expedition. Bridgens insisted that _he_ should be the one to remove his name from the list. In the end, they agreed that neither of them should lose the opportunity for such adventure \u2014 certainly Bridgens' last opportunity because of his advanced age ( _Erebus_ 's purser, Charles Hamilton Osmer, had been a longtime friend of Bridgens and had smoothed his enlistment with Sir John and the officers, even going so far as to hide the subordinate officers' steward's real age by being the one to write it as \"26\" on the official rolls). Neither Peglar nor Bridgens said it aloud, but both knew that the older man's long-standing vow never to bring his sexual desires to sea would be honoured by both of them. That part of their history, they both knew, was closed.\n\nAs it turned out, Peglar had seen almost nothing of his old friend during the voyage, and in three and a half years, they rarely had a minute alone.\n\nIt was still dark, of course, when Peglar arrived at _Erebus_ sometime around eleven on this Saturday morning two days before the end of January, but there was a glow in the south that promised to be, for the first time in more than eighty days, a predawn glow. The slight glow did not dispel the bite from the \u221265-degree temperatures, so he did not dawdle as the lanterns of the ship came into sight.\n\nThe view of _Erebus_ 's truncated masts would have dismayed any topman, but it hurt Harry Peglar more than most since he had, with his _Erebus_ captain of the foretop counterpart, Robert Sinclair, helped supervise the dismantling and storage of both ships' upper masts for the endless winters. It was an ugly sight at any time and was made no prettier by _Erebus_ 's bizarre stern-down, bow-up posture in the encroaching ice.\n\nPeglar was hailed by the watch, invited aboard, and he carried his message from Captain Crozier down to Captain Fitzjames, who was sitting and smoking his pipe in the aft officers' mess since the Great Cabin was still being used as an ad hoc sick bay.\n\nThe captains had begun using the brass canisters meant for cached reports to send their written messages back and forth \u2014 the couriers hated this change since the cold metal burned fingers even through heavy gloves \u2014 and Fitzjames had to order Peglar to open the canister with his mittens, since the tube was still too cold for the captain to touch. Fitzjames did not dismiss him, so Peglar stood in the doorway to the officers' mess while the captain read the note from Crozier.\n\n\"No return message, Mr. Peglar,\" said Fitzjames.\n\nThe foretop captain knuckled his forehead and went up onto the deck again. About a dozen Erebuses had come up to watch the sunrise and more had been getting into their slops below to do so. Peglar had noticed that the Great Cabin sick bay had about a dozen men in it on cots \u2014 about the same number as _Terror._ Scurvy was setting in on both ships.\n\nPeglar saw the small, familiar figure of John Bridgens standing at the rail on the stern's port side. He came up behind him and tapped the man on the shoulder.\n\n\"Ah, a little touch of Harry in the night,\" said Bridgens even before he turned.\n\n\"Not night for long,\" said Peglar. \"And how did you know it was me, John?\"\n\nBridgens had no comforter over his face, and Peglar could see his smile and watery blue eyes. \"Word of visitors travels quickly on a small ship frozen in the ice. Do you have to hurry back to _Terror?_ \"\n\n\"No. Captain Fitzjames had no response.\"\n\n\"Would you care to take a stroll?\"\n\n\"By all means,\" said Peglar.\n\nThey went down the starboard side ice ramp and walked toward the iceberg and high pressure ridge to the southeast so as to get a better view of the glowing south. For the first time in months, HMS _Erebus_ was backlit by something other than the aurora or lantern or torchlight.\n\nBefore they reached the pressure ridge, they passed the scuffed, sooted, and partially melted area where the Carnivale fire had burned. The area had been well cleaned up on Captain Crozier's orders in the week after the disaster, but post holes where the staves had served as tent poles remained, as did shreds of rope or canvas that had melted into the ice and then been frozen in place. The rectangle of the ebony room still showed even after repeated efforts to remove the black soot and several snowfalls.\n\n\"I've read the American writer,\" said Bridgens.\n\n\"American writer?\"\n\n\"The chap who caused little Dickie Aylmore to receive fifty lashes for his inventive set decorations for our late, unlamented carnivale. A strange little fellow by the name of Poe, if memory serves. Very melancholy and morbid stuff with a touch of the truly unhealthy macabre. Not very good, overall, but very _American_ in some undefinable sense. I did not, however, read the fateful story that brought on the lashes.\"\n\nPeglar nodded. His foot struck something in the snow, and he bent to pry it out of the ice.\n\nIt was the bear's skull that had been hanging above Sir John's ebony clock, which had not survived the flames \u2014 the skull's flesh, hide, and hair gone and bone blackened by the fire, eye sockets empty, but the teeth still ivory-coloured.\n\n\"Oh, my, Mr. Poe would love that, I think,\" said Bridgens.\n\nPeglar dropped it back into the snow. The thing must have been hidden beneath chunks of fallen ice when the clean-up parties worked here. He and Bridgens walked another fifty yards to the tallest pressure ridge in the area and clambered up it, Peglar repeatedly giving his hand to help the older man up.\n\nOn a flat slab of ice atop the ridge, Bridgens was panting heavily. Even Peglar, normally as fit as one of the ancient Olympic athletes he'd read about, found himself breathing harder than usual. Too many months of no real physical duty, he thought.\n\nThe southern horizon was glowing a subdued, washed-out yellow, and most of the stars in that half of the sky had paled.\n\n\"I almost can't believe it's returning,\" said Peglar.\n\nBridgens nodded.\n\nSuddenly there it was, the disk of red-gold rising hesitantly above dark masses that looked like hills but must be low clouds far to the south. Peglar heard the forty or so men on the deck of _Erebus_ give three cheers, and \u2014 because the air was very cold and very still \u2014 he could hear a duplicate but fainter cheer coming from _Terror,_ just visible almost a mile to the east across the ice.\n\n\"Dawn stretches forth her rosy fingertips,\" Bridgens said in Greek.\n\nPeglar smiled, mildly amused that he remembered the phrase. It had been several years since he'd read the _Iliad_ or anything else in Greek. He remembered the excitement of his first encounter with this language and with Troy and its heroes as _Beagle_ had been anchored off S\u00e3o Tiago, a volcanic island in the Cape Verde Islands, almost seventeen years earlier.\n\nAs if reading his mind, Bridgens said, \"Do you remember Mr. Darwin?\"\n\n\"The young naturalist?\" said Peglar. \"Captain FitzRoy's favorite interlocutor? Of course I do. Five years on a small bark with a man leaves an impression, even if he was a gentleman and I wasn't.\"\n\n\"And what was your impression, Harry?\" Bridgens' pale blue eyes were watering more heavily, either out of emotion at seeing the sun again or just in reaction to the unaccustomed light, as pale as it was. The red disk had not completely cleared the dark clouds before it started descending again.\n\n\"Of Mr. Darwin?\" Peglar was also squinting \u2014 more to bring back memory of the thin young naturalist than because of the sun's wonderful illumination. \"I found him pleasant, as such gentlemen go. Very enthusiastic. He certainly kept the men busy transporting and crating up all those damned dead animals \u2014 at one point I thought the finches alone were going to fill the hold \u2014 but he wasn't above getting his own hands dirty. Remember the time he joined in the rowing to help tow old _Beagle_ upstream in the river? And he saved a boat from the tidal wave that other time. And once, when whales were alongside us \u2014 off the coast of Chile, I believe \u2014 I was amazed to find that he'd climbed all the way up to the crosstrees on his own to get a better view. I had to help him down, but not before he looked through the glass at the whales for over an hour, the tails of his coat flapping in the breeze.\"\n\nBridgens smiled. \"I was almost jealous when he lent you that book. What was it? Lyell?\"\n\n\" _Principles of Geology,_ \" said Peglar. \"I didn't really understand it. Or rather, I did just enough to realize how dangerous it was.\"\n\n\"Because of Lyell's contention about the age of things,\" said Bridgens. \"About the very un-Christian idea that things change slowly over immense aeons of time rather than very quickly due to very violent events.\"\n\n\"Yes,\" said Peglar. \"But Mr. Darwin was very keen on it. He sounded like a man who had experienced a religious conversion.\"\n\n\"I believe he had, in a manner of speaking,\" said Bridgens. Only the top third of the sun was visible now. \"I mention Mr. Darwin because mutual friends told me before we sailed that he is writing a book.\"\n\n\"He published several already,\" said Peglar. \"Do you remember, John, we discussed his _Journal of Researches into the Geology and Natural History of the Various Countries Visited by H.M.S._ Beagle in the year I came to study with you... 1839. I couldn't afford to buy it, but you said you'd read it. And I believe he published several volumes on the plant and animal life he saw.\"\n\n\" _The Zoology of the Voyage of H.M.S._ Beagle,\" said Bridgens. \"Yes, I purchased that as well. No, I meant he has been working on a much more important book according to my dear friend Mr. Babbage.\"\n\n\"Charles Babbage?\" said Peglar. \"The fellow who tinkers up so many odd things including some sort of computing engine?\"\n\n\"The same,\" said Bridgens. \"Charles tells me that all these years, Mr. Darwin has been working on a quite interesting volume discussing the mechanisms of organic evolution. Apparently it draws in information from comparative anatomy, embryology, and paleontology... all great interests of our former shipboard naturalist's, you may remember. But for whatever reasons, Mr. Darwin is loath to publish and the book may not see print, according to Charles, in anyone's lifetime.\"\n\n\"Organic evolution?\" repeated Peglar.\n\n\"Yes, Harry. That's the idea that species, despite all civilized Christian agreement to the contrary, are not fixed since creation, but may change and adapt over time... much time. Mr. Lyell's amounts of time.\"\n\n\"I know what organic evolution is,\" said Peglar, trying not to show his irritation at being talked down to. The problem with a student-teacher relationship was, he realized not for the first time, that it never changes while everything around it does. \"I've read Lamarck on the concept. Also Diderot. And Buffon, I believe.\"\n\n\"Yes, it's an old theory,\" said Bridgens, his tone sounding amused but also slightly apologetic. \"Montesquieu has written about it, as has Maupertuis and the others you mentioned. Even Erasmus Darwin, our former shipmate's grandfather, had proposed it.\"\n\n\"Then why would Mr. Charles Darwin's book be important?\" asked Peglar. \"Organic evolution is an old idea. It's been rejected by the Church and other naturalists for generations.\"\n\n\"If Charles Babbage and other friends Mr. Darwin and I have in common are to be believed,\" said Bridgens, \"this new book \u2014 should it ever be published \u2014 offers proof of an actual mechanism for organic evolution. And it should give a thousand \u2014 perhaps ten thousand \u2014 solid examples of this mechanism in action.\"\n\n\"And the mechanism is what?\" asked Peglar. The sun had disappeared. Rose shadows faded into the pale yellow gloom that had preceded its rising. Now that the sun was gone, Peglar hardly believed he had seen it.\n\n\"Natural selection arising from competition _within_ the countless species,\" said the elderly subordinate officers' steward. \"A selection passing along advantageous traits and weeding out disadvantageous traits \u2014 that is, ones which add to the probability of neither survival nor reproduction \u2014 over vast amounts of time. Lyellian amounts of time.\"\n\nPeglar thought about this for a minute. \"Why did you bring this up, John?\"\n\n\"Because of our predatory friend out here on the ice, Harry. Because of the blackened skull you left back where the ebony room had once echoed to the ticking of Sir John's ebony clock.\"\n\n\"I don't quite understand,\" said Peglar. He used to say that very frequently when he was John Bridgens's student during the five years of _Beagle_ 's seemingly endless wanderings. The voyage had been planned as a two-year venture, and Peglar had promised Rose he would be back in two years or less. She had died of consumption during _Beagle_ 's fourth year at sea. \"Do you think the thing on the ice is some form of species evolutionary adaptation from the more common white bear we've encountered so frequently up here?\"\n\n\"Quite the contrary,\" said Bridgens. \"I find myself wondering if we might have encountered one of the last members of some ancient species \u2014 something larger, smarter, faster, and infinitely more violent than its descendant, the smaller north polar bear we see in such abundance.\"\n\nPeglar thought about this. \"Something from an antediluvian age,\" he said at last.\n\nBridgens chuckled. \"In a metaphorical sense, at least, Harry. You may remember that I was no advocate of any literal belief in the Flood.\"\n\nPeglar smiled. \"You were dangerous to be around, John.\" He stood in the cold thinking for another few minutes. The light was fading. The stars were filling in the southern sky once again. \"Do you think this... thing... this last of its breed... walked the earth when the huge lizards were around? If so, why haven't we found fossils of it?\"\n\nBridgens chuckled again. \"No, somehow I do not believe our predator on the ice contested with the giant lizards. Perhaps mammals such as _Ursus maritimus_ did not coexist with the giant reptiles at all. As Lyell showed and our Mr. Darwin seems to understand, Time... with a capital _T,_ Harry... may be much vaster than we have the ability to comprehend.\"\n\nThe two men were silent for a few moments. The wind had started up a little and Peglar realized that it was too cold to stay out here like this much longer. He could see the older man shivering slightly. \"John,\" he said. \"Do you think that understanding the origin of the beast... or _thing,_ it sometimes seems too intelligent to be a beast... will help us kill it?\"\n\nBridgens laughed aloud this time. \"Not in the least, Harry. Just between you and me, dear friend, I think the creature already has the better of us. I think our bones will be fossils before its will... although, when one thinks about it, a huge creature which lives almost completely on the polar ice, not breeding or living on dry land as the more common white bears evidently do, perhaps even _preying_ on the more common polar bear as its primary source of food, may well leave no bones, no trace, no fossils... at least ones we are able to find beneath the frozen polar seas at our current state of scientific technology.\"\n\nThey began walking back toward _Erebus._\n\n\"Tell me, Harry, what is happening on _Terror?_ \"\n\n\"You heard about the near mutiny three days ago?\" asked Peglar.\n\n\"Was it really so close a thing?\"\n\nPeglar shrugged. \"It was ugly. Any officer's nightmare. The caulker's mate, Hickey, and two or three other agitators, had the men all worked up. It was a mob mentality. Crozier defused it brilliantly. I don't think I've ever seen a captain handle a mob with more finesse and certainty than Crozier did on Wednesday.\"\n\n\"And it was all over the Esquimaux woman?\"\n\nPeglar nodded, then pulled his Welsh wig and comforter tighter. The wind was very biting now. \"Hickey and a majority of the men had learned that the wench had tunneled a way out through the hull before Christmas. Until the day of Carnivale, she'd been coming and going at will from her den in the forward cable locker. Mr. Honey and his carpenter mates had fixed the breech in the hull, and Mr. Irving had collapsed the outside tunnel route the day after the Carnivale fire \u2014 and word leaked out.\"\n\n\"And Hickey and the others thought that she had something to do with the fire?\"\n\nPeglar shrugged again. If nothing else, the motion helped keep him warm. \"For all I know, they thought she _was_ the thing on the ice. Or at least its consort. Most of the men have been convinced for months that she's a heathen witch.\"\n\n\"Most of the crew on _Erebus_ agree,\" said Bridgens. His teeth were chattering. The two men picked up their pace back toward the canting ship.\n\n\"Hickey's mob had made plans to waylay the girl when she came up for her evening biscuit and cod,\" said Peglar. \"And to cut her throat. Perhaps with some formal ceremony.\"\n\n\"Why didn't it happen that way, Harry?\"\n\n\"There's always someone who informs,\" said Peglar. \"When Captain Crozier got wind of it \u2014 possibly only hours before the murder was supposed to happen \u2014 he dragged the girl up to the lower deck and called a meeting of all officers and men. He even called the watch below, which is unheard of.\"\n\nBridgens turned his pale square of a face toward Peglar as they walked. It was getting darker quickly now and the wind was holding out of the nor'west.\n\n\"It was just at supper time,\" continued Peglar, \"but the captain had all the men's tables winched up again and made the men sit on the deck. No casks or chests \u2014 just on the bare deck \u2014 and had the officers, armed with sidearms, stand behind him. He held the Esquimaux girl by the arm, as if she was an offering he was going to throw to the men. Like a piece of meat to jackals. In a sense that's what he did.\"\n\n\"How do you mean?\"\n\n\"He told the crew that if they were going to do murder, that they had to do it right then... at that moment. With their boat knives. Right there on the lower deck where they ate and slept. Captain Crozier said that they would all have to do it together \u2014 seamen and officers alike \u2014 because murder on a ship is like a canker and spreads unless everyone is already inoculated by being an accomplice.\"\n\n\"Very strange,\" said Bridgens. \"But I am surprised that it worked to deter the men's bloodthirst. A mob is a brainless thing.\"\n\nPeglar nodded again. \"Then Crozier called Mr. Diggle forward from his place by the stove.\"\n\n\"The cook?\" said Bridgens.\n\n\"The cook. Crozier asked Mr. Diggle what was for supper that night... and for every night in the coming month. 'Poor John,' said Diggle. 'Plus whatever canned things haven't gone rotten or poisonous.' \"\n\n\"Interesting,\" said Bridgens.\n\n\"Crozier then asked Dr. Goodsir \u2014 who happened to be on _Terror_ that day \u2014 how many men had shown up for sick call in the last three days. 'Twenty-one,' says Goodsir. 'With fourteen sleeping nights in sick bay until you called them forward for this meeting, sir.' \"\n\nIt was Bridgens' turn to nod now, as if he could see where Crozier had been headed.\n\n\"And then the captain said, 'It's scurvy, boys.' The first time any officer \u2014 surgeon, captain, even mates \u2014 had said the word aloud to the crew in three years,\" continued Peglar. 'We're coming down with scurvy, Terrors,' the captain said. 'And you know the symptoms. Or if you don't... or if you don't have the balls to think about it... you need to listen.' And then Crozier called Dr. Goodsir up front, next to the girl, and made him list the symptoms of scurvy.\n\n\" 'Ulcers,' said Goodsir,\" continued Peglar as they approached _Erebus._ \" 'Ulcers and haemorrhages everywhere on your body. That's pools of blood,' he said, 'under the skin. Flowing from the skin. Flowing from every orifice before the disease runs it course \u2014 your mouth, your ears, your eyes, your arse. Rictus of limbs,' he said, 'which means first your arms and legs hurt, then they become stiff. They won't work. You'll be clumsy as a blind ox. Then your teeth will fall out,' said Goodsir and paused. It was so silent, John, that you couldn't even hear the fifty men breathing, only the creaking and groaning of the ship in the ice. 'And while your teeth are falling out,' the surgeon went on, 'your lips will turn black and pull back from any remaining teeth you might have. Like a dead man's lips,' he said. 'And your gum tissue will bloom... that means swell. And stink. That's the source of the terrible stench that comes from scurvy,' he said, 'your gums rotting and festering from the inside out.'\n\n\"'But that's not all,' Goodsir went on,\" continued Peglar. \" 'Your vision and hearing will be impaired... compromised... as will your judgement. You'll suddenly see no problem walking out in fifty-below-zero weather with no gloves and no hat. You'll forget which way is north or how to drive a nail. And your senses will not only fail, they'll turn on you,' he says. 'If we had a fresh orange to give you, when you have scurvy, the smell of the orange might make you writhe in agony or literally drive you mad. The sound of a sledge's runner on ice might drop you to your knees in pain; the report of a musket could be fatal.'\n\n\"''Ere now!' shouts one of Hickey's legion into the silence,\" continued Peglar. \"'We got our lemon juice!'\n\n\"Goodsir just shook his head sadly. 'We won't have it for much longer,' he said, 'and what we have is not worth much. For some reason no one understands, the simple antiscorbutics like the lemon juice lose their potency after months. It's almost gone now after more than three years.'\n\n\"There was this second terrible silence then, John. You _could_ hear the breathing then, and it was ragged. And there was a smell rising from the mob \u2014 fear and something worse. Most of the men there, including a majority of the officers, had seen Dr. Goodsir in the past two weeks with early symptoms of scurvy. Suddenly one of Hickey's compatriots shouts out, 'What's all this got to do with getting rid of a Jonah of a witch?'\n\n\"Crozier stepped forward then, still holding the girl like a captive, still seeming to offer her to the mob. 'Different captains and surgeons try different things to ward off or cure scurvy,' Crozier said to the men. 'Violent exercise. Prayer. Canned foods. But none of these things work in the long run. What is the only thing that works, Dr. Goodsir?'\n\n\"Every head on the lower deck turned to look at Goodsir then, John. Even the Esquimaux girl's.\n\n\"'Fresh food,' said the surgeon. 'Especially fresh meat. Whatever deficiency in our food brings on scurvy, only fresh meat can cure it.'\n\n\"Everyone looked back at Crozier,\" said Peglar. \"The captain all but thrust the girl at them. 'There's one person on these two dying ships who has been able to find fresh meat this autumn and winter,' he says. 'And she's standing right in front of you. This Esquimaux girl... merely a girl... but one who somehow knows how to find and trap and kill seals and walruses and foxes when the rest of us can't even find a track in the ice. What will it be like if we have to abandon ship... once we're out on the ice with no food stores left? There is one person out of the hundred and nine of us remaining alive who knows how to get us fresh meat to survive... _and you want to kill her._ ' \"\n\nBridgens showed bleeding gums of his own when he smiled. They were at the ice ramp to _Erebus._ \"Our successor to Sir John may be a common man,\" he said softly, \"with little formal education, but no one ever accused Captain Crozier \u2014 within my earshot at least \u2014 of being a stupid man. And I understand he has changed since his serious illness a few weeks ago.\"\n\n\"A sea change,\" said Peglar, enjoying both the pun and using a phrase Bridgens had introduced him to sixteen years earlier.\n\n\"How so?\"\n\nPeglar scratched his frozen cheek above the comforter. The mitten rasped on his stubble. \"It's hard to describe. My own guess is that Captain Crozier is completely sober now for the first time in thirty years or more. The whiskey never seemed to compromise the man's competence \u2014 he's a fine sailor and officer \u2014 but it put a... buffer... a barrier... between him and the world. Now he's _there_ more. Missing nothing. I don't know how else to describe it.\"\n\nBridgens nodded. \"I presume there's been no more talk of killing the witch.\"\n\n\"None,\" said Peglar. \"The men gave her extra biscuits for a while, but then she left \u2014 moved out onto the ice somewhere.\"\n\nBridgens started up the ramp and then turned back. When he spoke, his voice was very low so that none of the men on watch above could hear. \"What do you think of Cornelius Hickey, Harry?\"\n\n\"I think he's a treacherous little shit,\" said Peglar, not caring who heard him.\n\nBridgens nodded again. \"He is that. I've known _of_ him for years before I sailed on this expedition with him. He used to prey on boys during long voyages \u2014 turning them into little more than slaves for his needs. In recent years, I've heard, he's chosen to bend older men to his service, like the idiot...\"\n\n\"Magnus Manson,\" said Peglar.\n\n\"Yes, like Manson,\" said Bridgens. \"If it were just for Hickey's pleasure, we need not worry. But the little homunculus is worse than that, Harry... worse than your average would-be mutineer or conniving sea lawyer. Be careful of him. Watch him, Harry. I fear he could do great harm to us all.\" Bridgens laughed then. \"Listen to me. 'Do great harm.' As if we weren't all doomed anyway. When I see you next, we may all be abandoning the ships and taking to the ice on our last long, cold walk. Take care of yourself, Harry Peglar.\"\n\nPeglar did not speak. The captain of the foretop took off his mitten and then his glove, and lifted his frozen fingers until they touched the frozen cheek and brow of subordinate officers' steward John Bridgens. The touch was very light and neither man could feel it through the incipient frostbite, but it would have to serve.\n\nBridgens went back up the ramp. Without looking back, Peglar tugged on his glove and started the cold walk back through the rising dark to HMS _Terror._\n\nIRVING\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _6 February, 1848_\n\nIt was Sunday, and Lieutenant Irving had served two straight watches up on deck in the cold and dark, one of them covering for his friend George Hodgson, who was ill with the symptoms of dysentery, missing his own warm supper in the officers' mess as a consequence and having only a small ice-hard slab of salt pork and a weevil-filled biscuit instead. But now he had eight blessed straight hours off before he had to go on duty again. He could drag himself belowdecks, crawl under the frozen blankets in the cot in his berth, thaw them some with his body heat, and sleep for the full eight hours.\n\nInstead, Irving told Robert Thomas, the first mate who was taking his place as the officer on deck, that he was going for a walk and would be back presently.\n\nThen Irving went over the side and down the ice ramp and onto the dark pack ice.\n\nHe was searching for Lady Silence.\n\nIrving had been shocked weeks ago when Captain Crozier had appeared to be ready to toss the woman to the mob that was building, after crewmen listening to the mutinous whispers of Caulker's Mate Hickey and others started shouting that the woman was a Jonah and should be killed or cast out. When Crozier had stood there with Lady Silence's arm gripped in his hand, thrusting her toward the angry men much like an ancient Roman emperor might have tossed a Christian to the lions, Lieutenant Irving had not been sure what to do. As a junior lieutenant, he could only watch his captain, even if it meant Silence's death. As a young man in love, Irving was ready to step forward and save her even if it cost him his own life.\n\nWhen Crozier won the majority of the men over with his argument that Silence might be the only soul on board who would know how to hunt and fish on the ice should they have to abandon ship, Irving had let out a silent sigh of relief.\n\nBut the Esquimaux woman moved off the ship completely the day after that showdown, coming back at supper time every second or third day for biscuits or the occasional gift of a candle, then disappearing back onto the dark ice. Where she was living or what she was doing out there was a mystery.\n\nThe ice was not too dark this night; the aurora danced brightly overhead, and there was enough moonlight to throw ink-black shadows behind the seracs. Third Lieutenant John Irving was not, unlike the first time he had followed Silence, carrying out this search on his own initiative. The captain had again suggested that Irving discover \u2014 if he could do so without endangering himself too much \u2014 the Esquimaux wench's secret hiding place on the ice.\n\n\"I was serious when I told the men that she might have skills that would keep us alive on the ice,\" Crozier had said softly in the privacy of his cabin as Irving leaned closer to hear. \"But we can't wait until we're on the ice to find out where and how she gets the fresh meat she seems to be finding. Dr. Goodsir tells me that scurvy will take us all if we do not find some source of fresh game before summer.\"\n\n\"But unless I actually spy her hunting, sir,\" Irving had whispered, \"how can I get the secret from her? She cannot speak.\"\n\n\"Use your initiative, Lieutenant Irving,\" was all that Crozier had said in response.\n\nThis was the first opportunity that Irving had had since that conversation in which he might be able to use his initiative.\n\nIn the leather shoulder bag, Irving carried a few enticements should he find Silence and work out a way to communicate with her. There were biscuits far fresher than the weevil-filled one he'd chewed for dinner. Those were wrapped in a napkin, but Irving had also brought a beautiful Oriental silk handkerchief that his rich London girlfriend had given to him as a present shortly before their... unpleasant parting. And his pi\u00e8ce de r\u00e9sistance was wrapped in that attractive handkerchief: a small crock of peach marmalade.\n\nSurgeon Goodsir was hoarding and doling out the marmalade as an antiscorbutic, but Lieutenant Irving knew that the treat was one of the few things the Esquimaux girl had ever shown enthusiasm about when accepting Mr. Diggle's offerings of food. Irving had seen her dark eyes glint when she got a daub of marmalade on her biscuit. He'd scraped off his own jam treats a dozen times over the past month to get the precious amount he now carried in the tiny porcelain crock that had once been his mother's.\n\nIrving had come completely around to the port side of the ship and now advanced from the ice plain there into a maze of seracs and minibergs that rose like an icy version of Birnam's wood come to Dunsinane about two hundred yards south of the ship. He knew that he was running a great risk of becoming the next victim of the thing on the ice, but for the last five weeks there had been no sign of the creature, not even a clear sighting from a distance. No crewmen had been lost to it since the night of Carnivale.\n\n _Then again,_ thought Irving, _no one but me has come out here alone, without even a lantern, and gone wandering into the serac forest._\n\nHe was very aware that the only weapon he carried was the pistol sunk deep in the pocket of his greatcoat.\n\nForty minutes of searching through seracs in the dark and \u221245-degree wind and Irving was close to deciding that he would exercise his initiative another day, preferably in a few weeks, when the sun stayed above the southern horizon for more than a few minutes each day.\n\nAnd then he saw the light.\n\nIt was an eerie sight \u2014 an entire snowdrift in an ice gully between several seracs seemed to be glowing goldly from within, as if from some inner faerie light.\n\n _Or witch's light._\n\nIrving walked closer, pausing at each serac shadow to make sure that it was actually not another narrow crevasse in the ice. The wind made a soft whistling sound through the tortured-ice tops of the seracs and ice-boulder columns. Violet light from the aurora danced everywhere.\n\nThe snowdrift had been heaped \u2014 either by wind or by Silence's hands \u2014 into a low dome thin enough to show a flickering yellow light shining through it.\n\nIrving dropped down into the small ice gully, actually just a depression between two pressure-pushed plates of pack ice rounded over with snow, and approached a small black hole that seemed too low to be associated with the dome set higher in the drift to one side of the gully.\n\nThe entrance \u2014 if an entrance it was \u2014 was barely as wide as Irving's heavily layered shoulders.\n\nBefore crawling in, he wondered if he should extract and cock his pistol. _Not a very friendly gesture of greeting,_ he thought.\n\nIrving wriggled into the hole.\n\nThe narrow passage went down for half the length of his body and then angled up for eight feet or more. When Irving's head and shoulders popped out of the far end of the tunnel and into the light, he blinked, looked around, and his jaw fell slack.\n\nThe first thing he noticed was that Lady Silence was naked under her open robes. She was lying on a platform carved out of the snow about four feet from Lieutenant Irving and almost three feet higher. Her bosoms were quite visible and quite bare \u2014 he could see the small stone talisman of the white bear she had taken from her dead companion dangling on a thong between her breasts \u2014 and she made no effort to cover them as she stared unblinkingly at him. She had not been startled. Obviously she had heard him coming long before he squeezed himself into the snow-dome's entry passage. In her hand was that short but very sharp stone knife he had first seen in the forward cable locker.\n\n\"I beg your pardon, miss,\" said Irving. He was at a loss of what to do next. Good manners demanded that he wriggle backward out of this lady's boudoir, as awkward and ungainly as that motion must be, but he reminded himself that he was here on a mission.\n\nIt did not escape Irving's attention that wedged in the opening to the snow-house as he was, Silence could easily lean over and cut his throat with that knife while there would be very little he could do about it.\n\nIrving finished extricating himself from the entry passage, pulled his leather bag in behind him, got to his knees, and then to his feet. Because the floor of the snow-house had been dug out lower than the surface of the snow and ice outside, Irving had enough room to stand in the center of the dome with several inches to spare. He realized that while the snow-house had seemed like nothing more than a glowing snowdrift from the outside, it had actually been constructed of carved blocks or slabs of snow angling and arching inward in a most clever design.\n\nIrving, trained at the Royal Navy's best gunnery school and always good at mathematics, immediately noticed the upward spiral of the blocks and how each block leaned in just slightly more than the previous one until a final capping key block had been pushed down through the apex of the dome and then tugged into position. He saw the tiny smoke hole, or chimney \u2014 no more than two inches across \u2014 just to one side of the key block.\n\nThe mathematician in Irving knew at once that the dome was not a true hemisphere \u2014 a dome built upon the principle of a circle would collapse \u2014 but rather was a catenary: that is, the shape of a chain held in both hands. The gentleman in John Irving knew that he was studying the ceiling, the blocks, and the geometric structure of this clever dwelling so as not to stare at Lady Silence's naked breasts and bare shoulders. He assumed he had given her enough time to draw the fur robes up over herself, and he looked back in her direction.\n\nHer bosoms were still bared. The polar white bear amulet made her brown skin look all the more brown. Her dark eyes, intent and curious but not necessarily hostile, still watched him unblinkingly. The knife was still in her hand.\n\nIrving let out a breath and sat on the robe-covered platform across the small central space from her sleeping platform.\n\nFor the first time he realized that it was warm in the snow-house. Not just warmer than the freezing night outside, nor just warmer than the freezing lower deck of HMS _Terror,_ but _warm._ He had actually started to sweat under his many stiff and filthy layers. He saw perspiration on the soft brown bosom of the woman only a few feet from him.\n\nTearing his gaze away again, Irving unbuttoned his outer slops and realized that the light and heat were coming from one small paraffin tin that she must have stolen from the ship. As soon as he had this thought of her thieving, he felt sorry for it. It was a _Terror_ paraffin tin all right, but one empty of paraffin, one of hundreds they had thrown overboard in the huge garbage area they had excavated out of the ice only thirty yards from the ship. The flame was not burning from paraffin but from some sort of oil \u2014 not whale oil, he could tell from the scent \u2014 seal oil? A cord made out of animal gut or sinew hung down from the ceiling, suspending a strip of blubber over the paraffin lamp and dripping oil into it. Irving saw at once how, when the oil level would grow lower, the candlewick, which seemed to be made of twined strands of anchor-cable hemp, would become longer and the flame would burn higher, melting more blubber and dripping more oil into the lamp. It was an ingenious system.\n\nThe paraffin container was not the only interesting artifact in the snow-house. Above and to one side of the lamp was an elaborate frame consisting of what appeared to be four ribs from what might have been seals \u2014 _how had Lady Silence caught and killed those seals?_ wondered Irving \u2014 thrust upright in the snow of the shelf and connected by a complex web of sinew. Hanging from the bone frame was one of the larger rectangular Goldner food cans \u2014 also obviously scavenged from _Terror_ 's garbage dump \u2014 with holes punched in the four corners. Irving saw at once that it would make a perfect cooking pot or teakettle hanging low over the seal-oil flame.\n\nLady Silence's bosoms were still uncovered. The white bear amulet moved up and down with her breathing. Her gaze never left his face.\n\nLieutenant Irving cleared his throat.\n\n\"Good evening, Miss... ah... Silence. I apologize for bursting in on you this way... uninvited as it were.\" He stopped.\n\nDidn't the woman ever blink?\n\n\"Captain Crozier sends his compliments. He asked me to look in on you to see... ah... how you were getting along.\"\n\nIrving had rarely felt more the fool. He was sure that despite her months on the ship, the girl understood not one word of English. Her nipples, he could not help noticing, had risen in the brief blast of cold air that he had brought into the snow-house with him.\n\nThe lieutenant rubbed the sweat off his forehead. Then he removed his mittens and undergloves, bobbing his head as if to ask permission of the lady of the house as he did so. Then he mopped his forehead again. It was incredible how warm this little space under a cantenary dome made out of snow could get just from the heat of a single lamp burning dripping blubber.\n\n\"The captain would like... ,\" he began, and stopped. \"Oh, bugger it.\" Irving reached into his leather valise and brought out the biscuits wrapped in an old napkin and the crock of marmalade wrapped in his finest Oriental silk handkerchief.\n\nHe offered the two bundles across the central space to her with hands that were slightly trembling.\n\nThe Esquimaux woman made no attempt to take the bundles.\n\n\"Please,\" said Irving.\n\nSilence blinked twice, slipped the knife under her robe, and took the small, lumpy packages, setting them next to her where she reclined on the platform. As she lay on her side, the tip of her right bosom was almost touching his Chinese handkerchief.\n\nIrving looked down and realized that he was also sitting on a thick animal fur set onto this narrow platform. _Where did she get this second animal skin?_ he wondered before remembering that more than seven months earlier she had been given the outer parka of the old Esquimaux man. The grey-haired old one who had died on the ship after being shot by one of Graham Gore's men.\n\nShe untied the old galley napkin first, showing no response to the five ship's biscuits wrapped in it. Irving had spent a serious bit of time finding the least weevil-infested biscuits possible. He felt a little piqued at her lack of recognition of his labours. When she unwrapped his mother's little porcelain crock, sealed with wax on top, she paused to lift the Chinese silk handkerchief \u2014 its elaborate designs were in bright red, green, and blue \u2014 and to set it against her cheek for a moment. Then she laid it aside.\n\n _Women are the same everywhere_ was John Irving's giddy thought. He realized that while he had enjoyed sexual congress with more than one young woman, he had never felt such a strong sense of... _intimacy_... as he did at this moment sitting chastely in the seal-oil lamplight with this half-naked young native woman.\n\nWhen she pried open the wax and saw the marmalade, Lady Silence's gaze snapped up to Irving's face again. She seemed to be studying him.\n\nHe made a rough pantomime of her spreading the marmalade on the biscuits and eating them.\n\nShe did not move. Her gaze did not shift.\n\nFinally she leaned out and extended her right arm as if reaching for him across the blubber fire, and Irving flinched a bit before realizing that she was reaching to a niche \u2014 just a small recess in the ice block \u2014 at the head of his robe-covered platform. He feigned not noticing that her own robe had slipped lower and that both her bosoms were bobbing free as she reached.\n\nShe offered him something white and red and reeking like a dead and decaying fish. He realized that it was another slab of seal or other-animal blubber that had been stored in the snowy niche to be kept cold.\n\nHe accepted it, nodded, and held it in his hands above his knees. He had no clue what to do with it. Was he supposed to bring it home to serve as part of his own seal-oil blubber lamp?\n\nSilence's lips twitched then, and for an instant, Irving almost thought she had smiled. She took out her short, sharp knife and gestured, drawing the blade quickly and repeatedly right up to and against her lower lip as if she were going to cut that full, pink lip off.\n\nIrving stared and continued holding the soft mass of blubber and skin.\n\nSighing, Silence reached over, took the blubber from him, held it to her own mouth, and cut several slices off with her knife, pulling the short blade actually into her mouth between her white teeth with each morsel. She paused to chew a moment and then handed the blubber and rubbery sealskin \u2014 he was almost certain it was seal now \u2014 back to him.\n\nIrving had to fumble down through six layers of slops, greatcoat, jacket, sweaters, and waistcoat to get to his boat knife that was sheathed on his belt. He held the blade up to show her, feeling like a child seeking approbation during a lesson.\n\nShe nodded ever so slightly.\n\nIrving set the reeking, stinking, dripping blubber next to his open mouth and pulled the sharp edge of his knife back quickly the way she had.\n\nHe almost cut his nose off. He _would_ have sliced his lower lip off if the knife had not caught in the sealskin \u2014 if sealskin it was \u2014 and soft meat and white blubber and jerked upward slightly. As it was, a single drop of blood dripped from his sliced septum.\n\nSilence ignored the blood, shook her head ever so slightly, and handed him her knife.\n\nHe tried it again, feeling the strange weight of her knife in his palm, slicing confidently toward his lip even as a drop of blood dripped from his nose onto the blubber.\n\nThe blade went through effortlessly. Her little stone knife was \u2014 somehow, incredibly \u2014 many times sharper than his own.\n\nThe strip of blubber filled his mouth. He chewed, trying to idiot-mime and nod his appreciation toward the woman from behind his upraised strip of blubber and poised knife.\n\nIt tasted like a ten-week-dead carp dredged from the floor of the Thames beneath the Woolwich sewer outlets.\n\nIrving felt a great urge to vomit, started to spit the wad of half-chewed blubber on the floor of the snow-house instead, decided that this would not further the goals of his delicate diplomatic mission, and swallowed.\n\nGrinning his appreciation for the delicacy while trying to force down his continued nausea \u2014 all the while surreptitiously mopping at his barely sliced but vigorously bleeding nose with a bunched-up frozen mitten serving as handkerchief \u2014 Irving was horrified to see the Esquimaux woman clearly gesture for him to cut and eat more of the blubber.\n\nStill smiling, he sliced and swallowed another piece. It was, he thought, precisely what it must feel like to be filling one's mouth with a giant glob of some other creature's nasal mucus.\n\nAmazingly, his empty stomach rumbled, cramped, and demanded more. Something in the reeking blubber seemed to be satisfying some deep craving he had not even known he felt. His body, if not his mind, wanted more of it.\n\nThe next few minutes were quite the domestic scene, thought Lieutenant Irving, with him sitting on his white bear robe on his little snow shelf, quickly if not enthusiastically cutting and swallowing strips of seal blubber, while Lady Silence crumbled strips of ship's biscuit, dunked them in his mother's crock as quickly as a sailor mopping up gravy with his bread, and devoured the marmalade with satisfied grunts that seemed to come from deep in her throat.\n\nAnd all this time her bosoms remained bare and visible for Third Lieutenant John Irving's constant and appreciative, if not relaxed, perusal over his diminishing strip of seal blubber.\n\n _What would Mother think if she could see her boy and her crock now?_ wondered Irving.\n\nWhen the two were finished, after Silence had eaten all the biscuits and emptied the marmalade crock and Irving had made a serious dent in the blubber, he tried to mop his chin and lips with his mitten, but the Esquimaux woman reached to the niche once again and presented him with a handful of loose snow. Since the high temperature in the little snow-house felt as if it were actually above freezing, Irving self-consciously mopped the blubber grease off his face, dried his face with his sleeve, and started to hand the remaining strip of sealskin and fat back to the girl. She gestured to the storage niche and he stuffed the piece of blubber as far back into the niche as he could reach.\n\n _Now comes the hard part,_ thought the lieutenant.\n\nHow does one communicate just by the use of hands and dumb show that there are more than a hundred hungry men threatened by scurvy who need someone else's hunting and fishing secrets?\n\nIrving made a game try at it. With Lady Silence's deep, dark eyes watching unblinkingly, he acted out men walking, rubbing his stomach to show that they were hungry, the three masts of each ship, men getting sick \u2014 he stuck his tongue out, crossed his eyes in a way that used to upset his mother, and mimed falling over onto the bearskin robe \u2014 and then pointed to Silence and energetically acted out her casting a spear, holding a fishing pole, pulling in a catch. Irving pointed to the blubber he'd just stuffed away, in more ways than one, and pointed vaguely beyond the snow-house, again rubbing his stomach, crossing his eyes and falling, then rubbing his stomach again. He pointed to Lady Silence, floundered a moment on the sign language for \"show us how to do it ourselves,\" and then repeated the spear-throwing and fish-catching mimes while pausing to point to her, shoot splay-fingered rays out his eyes, and rub his stomach to specify the recipients of her teaching.\n\nWhen he was finished, sweat dripped from his brow.\n\nLady Silence looked at him. If she had blinked again, he'd missed it during his antics.\n\n\"Oh, well, bloody hell,\" said Third Lieutenant Irving.\n\nIn the end, he just buttoned up his layers and slops again, stuffed the ship's napkin and his mother's crock back in his leather valise, and called it a day. Perhaps he had got his message across after all. He might never know. Perhaps if he returned often enough to the snow-house...\n\nIrving's speculation veered into the highly personal at that point, and he reined himself in as if he were a coachman with a matched set of willful Arabians.\n\nPerhaps if he returned often... he would be able to go with her during one of her nocturnal seal-hunting expeditions.\n\n _But what if the thing on the ice is still giving her these things?_ he wondered. After seeing what he had seen so many weeks ago, he had half-convinced himself that he had not seen what he had seen. But the more honest half of Irving's memory and mind knew that he _had_ seen it. The creature on the ice had brought her chunks of seal or arctic foxes or other game. Lady Silence had left that place among the ice boulders and seracs that night with fresh meat.\n\nAnd then there was _Erebus_ 's mate, Charles Frederick Des Voeux, with his stories of men and women in France who transformed themselves into wolves. If that was possible \u2014 and many of the officers and all of the crewmen seemed to think it was \u2014 why could not a native woman with a talisman of the white bear around her neck turn herself into something like a giant bear with the cunning and evil of a human being?\n\nNo, he had seen the two together on the ice. Hadn't he?\n\nIrving shivered as he finished buttoning his slops. It was _very_ warm in this little snow-house. Ironically, it was giving him the chills. He felt the blubber working at his bowels and decided it was time to go. He would be lucky if he made it back to _Terror_ 's seat of ease in time as it was and he had no wish to stop out on the ice to see to such functions. It was bad enough when his _nose_ became frostbitten.\n\nLady Silence had watched while he packed away the old napkin and his mother's crock \u2014 items that he realized much later she might very much have wanted \u2014 but now she touched her cheek with the silk handkerchief a final time and tried to hand that back to him.\n\n\"No,\" said Irving, \"that is a gift from me. A token of my friendship and deep esteem. You must keep it. I would be offended if you do not.\"\n\nThen he tried to sign and act out what he had just said. The muscles along either side of the young Esquimaux woman's mouth almost twitched as she watched him.\n\nHe pushed her hand holding the handkerchief back, taking care not to touch her naked bosom as he did so. The white stone of the bear amulet between her breasts seemed to glow from its own illumination.\n\nIrving realized that he was much, much too hot. The room seemed to swim a bit in his vision. His insides lurched, calmed, then lurched again.\n\n\"Toodaloo,\" he said \u2014 three syllables he would agonize over for weeks to come, cringing in his bunk out of embarrassment even though she could not have understood the inanity and absurdity and inappropriateness of it. But still...\n\nIrving touched his cap, wrapped his comforter around his face and head, tugged on his gloves and mittens, clutched his valise to his chest, and dove for the exit passageway.\n\nHe did not whistle during his walk back to the ship, but he was tempted to. He had all but forgotten about the possibility of some huge man-eater lurking in the moon shadows of the seracs out here so far from the ship, but if there was such a thing watching and listening that night, it would have heard Third Lieutenant John Irving talking to himself and occasionally slapping himself on the head with his mitten.\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _15 February, 1848_\n\nGentlemen, it is time we looked at our possible courses of action in the coming months,\" said Captain Crozier. \"I have decisions to make.\"\n\nThe officers and some warrant officers and other specialists, such as the two civilian engineers, foretop captains, and ice masters, as well as the last surviving surgeon, had been called to this meeting in _Terror_ 's Great Cabin. _Terror_ had been chosen by Crozier not to inconvenience Captain Fitzjames and his officers \u2014 who had to make the crossing during the brief hour of sunlight and hoped to be back before it grew dark again \u2014 nor to emphasize the change of flagship, but only because fewer men on Crozier's ship were confined to sick bay. It had been easier to move those few to a temporary sick bay in the bow to free up the Great Cabin for the meeting of officers; _Erebus_ had twice the number of men down with symptoms of scurvy, and Dr. Goodsir had indicated that a few of them were too sick to be moved.\n\nNow fifteen of the expedition's leaders were crowded around the long table that in January had been cut into shorter lengths to serve as operating tables for the surgeon but now was set to rights by Mr. Honey, _Terror_ 's carpenter. The officers and civilians had left their rainproof slops, mittens, Welsh wigs, and comforters at the base of the main ladder, but they still wore all their other layers. The room smelled of wet wool and unwashed bodies.\n\nThe long cabin was cold and no light came through the Preston Patent Illuminators overhead since the deck remained under three feet of snow and its winter canvas cover. The whale-oil lamps on the bulkheads flickered dutifully but did little to dispel the gloom.\n\nThe gathering at the table resembled a gloomier version of the summer war council Sir John Franklin had called almost eighteen months earlier on _Erebus,_ but now instead of Sir John at the head of the table on the starboard side, Francis Crozier sat there. On the aft side of the table, to Crozier's left, were the seven officers and warrant officers from _Terror_ whom he had asked to be present. His executive officer, First Lieutenant Edward Little, was at Crozier's immediate left. Next was Second Lieutenant George Hodgson, with Third Lieutenant John Irving to his left. Then the civilian engineer \u2014 given warrant officer status on the expedition but looking thinner, paler, and more cadaverous than ever \u2014 James Thompson. On Thompson's left were Ice Master Thomas Blanky, who appeared to be stumping along very nicely on his wooden peg leg these days, and Captain of the Foretop Harry Peglar, the only petty officer Crozier had invited. Also present was _Terror_ 's Sergeant Tozer \u2014 who had been out of both captains' graces since the night of the Carnivale when his men had fired on survivors of the fire but who was still the highest-ranked survivor of his heavily thinned group of lobsterbacks \u2014 speaking for the Marines.\n\nAt the port end of the long table sat Captain Fitzjames. Crozier knew that Fitzjames had not bothered to shave for several weeks, growing a reddish beard surprisingly flecked with grey, but he had made the effort today \u2014 or had ordered Mr. Hoar, his steward, to shave him. The effect only made his face look thinner and more pale, and now it was covered with countless small scrapes and cuts. Even with multiple layers of clothes on, it was obvious that Fitzjames's garments hung on a much frailer frame these days.\n\nTo Captain Fitzjames's left, along the forward side of the long table, sat six Erebuses. Immediately to his left was his only other surviving Naval officer \u2014 Sir John Franklin, First Lieutenant Gore, and Lieutenant James Walter Fairholme had all been killed by the thing on the ice \u2014 Lieutenant H. T. D. Le Vesconte, the man's gold tooth gleaming the few times he smiled. Next to Le Vesconte was Charles Frederick Des Voeux, who had taken over the duties of first mate from Robert Orme Sergeant, who had been killed by the thing while overseeing torch-cairn repair in December.\n\nNext to Des Voeux sat the only surviving surgeon, Dr. Harry D. S. Goodsir. While technically the expedition's and Crozier's surgeon now, both the commanding officers and the surgeon had thought it appropriate for him to sit with his former _Erebus_ crewmates.\n\nTo Goodsir's left sat Ice Master James Reid, and to his left the only _Erebus_ petty officer present, Captain of the Foretop Robert Sinclair. And sitting on the forward side of the table was _Erebus_ 's engineer, John Gregory, looking much healthier than his _Terror_ counterpart.\n\nTea and weevil-rich biscuits were being served by Mr. Gibson of _Terror_ and Mr. Bridgens of _Erebus_ since the captains' stewards were both in sick bay with signs of scurvy.\n\n\"Let's discuss things in order,\" said Crozier. \"First, can we stay in the ships until a possible summer thaw? And part of that answer has to be, can the ships sail in June or July or August if there _is_ a thaw? Captain Fitzjames?\"\n\nFitzjames's voice was a hollow husk of its once-confident firmness. Men on both sides of the table leaned closer to hear him.\n\n\"I don't think _Erebus_ will last until summer, and it's my opinion \u2014 and the opinion of Mr. Weekes and Mr. Watson, my carpenters, and Mr. Brown, my bosun's mate, Mr. Rigden, my coxswain, and of Lieutenant Le Vesconte and First Mate Des Voeux here \u2014 that she will sink when the ice melts.\"\n\nThe cold air in the Great Cabin seemed to grow colder and to press more heavily on everyone. No one spoke for half a minute.\n\n\"The pressure from the ice these past two winters has squeezed the oakum right out from between the hull boards,\" continued Fitzjames in his small, hoarse voice. \"The main shaft to the screw has been twisted beyond all repair \u2014 all of you know that it was designed to be retracted into an iron well all the way up to the orlop deck to be kept out of harm's way, but it will no longer retract any higher than the hull bottom \u2014 and we have no more replacement shafts. The screw itself has been shattered by the ice, as has been our rudder. We can jury-rig another rudder, but the ice has torn our hull bottom to splinters all along the length of the keel. We're missing almost half of our iron plating along the bow and sides.\n\n\"Worse,\" said Fitzjames, \"the ice has squeezed the hull until the iron crossbeams added for reinforcement and the cast-iron replacements for her knees have either snapped or punctured the hull in more than a dozen places. If she were to float, even if we patched every breach and managed somehow to repair the problem with the screw-shaft well leaking, she would have no internal bracing against the ice. Also, while the wooden channels added to her side for this expedition have largely succeeded in keeping the ice from climbing over the raised gunwales, the downward pressure on these channels resulting from her raised position in the encroaching ice has caused splitting of the hull timbers along every channel seam.\"\n\nFitzjames seemed to notice their rapt attention for the first time. His unfocused stare went away and he looked down as if embarrassed. When he looked up again, his voice sounded almost apologetic. \"Worst of all,\" he said, \"is that the twisting pressure of the ice has so cork-screwed the sternpost and started the heads and ends of planking that _Erebus_ has been bent far out of true by the stress. The decks break upward now... the only thing holding them in place is the weight of the snow... and none of us believe that our pumps could equal the leaks should she be floated again. I will let Mr. Gregory speak to the condition of the boiler, coal supplies, and propulsion system.\"\n\nAll eyes shifted to John Gregory.\n\nThe engineer cleared his throat and licked his chapped and bleeding lips. \"There is no steam propulsion system left on HMS _Erebus,_ \" he said. \"With the main shaft twisted and jammed in the retraction well, we'd need a Bristol dry dock to set her right. Nor do we have enough coal left for a day's steaming. By the end of April, we'll be out of coal to heat the ship, even at the rate of moving just forty-five minutes of hot water a day only to parts of the lower deck that we're trying to keep habitable now.\"\n\nCrozier said, \"Mr. Thompson. What is _Terror_ 's status in terms of steam?\"\n\nThe living skeleton looked at his captain for a long minute and said in a voice that was surprisingly strong, \"We wouldn't be able to steam for more than an hour or two, sir, if _Terror_ was floated this afternoon. Our shaft was retracted all right a year and a half ago, and the screw is workable \u2014 and we have a replacement for that \u2014 but we're almost out of coal. If we were to transfer what's left of _Erebus_ 's coal stores here and just _heat_ the ship, we'd keep the boiler going and the hot water running two hours a day until... I'd venture... early May. But that wouldn't leave any coal for steaming. With just _Terror_ 's stores of fuel, we'll have to stop heating by mid- or late April.\"\n\n\"Thank you, Mr. Thompson,\" said Crozier. The captain's voice was soft and betrayed no emotion. \"Lieutenant Little and Mr. Peglar, would both of you be so kind as to give your assessment of _Terror_ 's seaworthiness?\"\n\nLittle nodded and looked down the table before returning his gaze to his captain. \"We're not as knocked up as _Erebus,_ but there's been ice-pressure damage to the hull, knees, outer plating, rudder, and inner bracings. Some of you know that before Christmas, Lieutenant Irving discovered not only that we had lost most of our iron plating along the starboard side back from the bow, but that the ten inches of oak and elm in the bow area had actually sprung the timbers in the forward cable locker on the hull deck, and we've found since that the thirteen inches of solid oak along her bottom has been sprung or compromised in twenty or thirty places. The bow boards've been replaced and reinforced, but we can't get to all her bottom because of the frozen slush down there.\n\n\"I think she'll float and steer, Captain,\" concluded Lieutenant Little, \"but I can't promise that the pumps will be able to keep up with the leaks. Especially after the ice has another four or five months to work at her. Mr. Peglar can speak to that better than I can.\"\n\nHarry Peglar cleared his throat. He obviously wasn't used to speaking in front of so many officers.\n\n\"If she'll float, sirs, then the foretop crew will get the masts reset and the rigging, shrouds, and canvas up within forty-eight hours of the time you give the word. I can't guarantee that sailing will get us through the thick ice of the sort we saw coming south, but if we have open water under us and ahead of us, we'll be a sailing ship again. And if you don't mind me making a recommendation, sirs... I'd suggest we steep the masts sooner rather than later.\"\n\n\"You're not worried about ice building up and capsizing the ship?\" asked Crozier. \"Or ice falling on us when we're working on deck? We have months of blizzards ahead of us still, Harry.\"\n\n\"Aye, sir,\" said Peglar. \"And capsizing's always a worry, even if we were just to tumble over onto the ice here, the ship being all cattywampus the way she is. But I still think it'd be better to have the topmasts up and the rigging in place in case there's a sudden thaw. We might have to sail with ten minutes' warning. And the topmen need the exercise and work, sir. As for the ice falling... well, it'll just be another thing to keep us alert and on our toes out there. That and the beastie on the ice.\"\n\nSeveral men around the table chuckled. Little's and Peglar's mostly positive reports had helped ease some of the tension. The thought of even one of the two ships being able to float and sail raised morale. It felt to Crozier as if the temperature in the Great Cabin had actually risen \u2014 and perhaps it had, since many of the men seemed to be exhaling again.\n\n\"Thank you, Mr. Peglar,\" said Crozier. \"It looks like if we want to sail out of here, we'll all have to do it \u2014 both crews \u2014 aboard _Terror._ \"\n\nNone of the surviving officers present mentioned that this had been precisely what Crozier had suggested doing almost eighteen months earlier. Every officer present appeared to be thinking it.\n\n\"Let's take a minute to talk about that thing on the ice,\" said Crozier. \"It hasn't seemed to have made an appearance recently.\"\n\n\"I've not had to treat anyone for wounds since the first of January,\" said Dr. Goodsir. \"And no one has died or disappeared since Carnivale.\"\n\n\"But there have been sightings,\" said Lieutenant Le Vesconte. \"Something large moving among the seracs. And men on watch hear things in the dark.\"\n\n\"Men on watch at sea have always heard things in the dark,\" said Lieutenant Little. \"Going back to the Greeks.\"\n\n\"Perhaps it has gone away,\" said Lieutenant Irving. \"Migrated. Moved south. Or north.\"\n\nEveryone fell silent again at this thought.\n\n\"Perhaps it's eaten enough of us to know we're not very tasty,\" said Ice Master Blanky.\n\nSome of the men smiled at this. No one else could have said it and been excused the gallows humour, but Mr. Blanky, with his peg leg, had earned some prerogatives.\n\n\"My Marines have been searching, as per Captain Crozier's and Captain Fitzjames's orders,\" said Sergeant Tozer. \"We've shot at a few bears, but none of them seemed to be the big one... the thing.\"\n\n\"I hope your men have been better shots than they were on the night of the Carnivale,\" said Sinclair, _Erebus_ 's foretop captain.\n\nTozer turned to his right and squinted down the table at him.\n\n\"There'll be no more of that,\" said Crozier. \"For the time being, we'll have to assume that the thing on the ice is still alive and will be back. Any activities we have to do off the ships will have to include some plan of defense against it. We don't have enough Marines to accompany every possible sledge party \u2014 especially if they're armed and not man-hauling \u2014 so perhaps the answer is to arm all ice parties and have the extra men, the ones not hauling, take turns serving as sentries and guards. Even if the ice doesn't open again this summer, it will be easier to travel in the constant daylight.\"\n\n\"You'll pardon my phrasing it this bluntly, Captain,\" said Dr. Goodsir, \"but the real question is, can we afford to wait until summer before deciding whether to abandon the ships?\"\n\n\"Can we, Doctor?\" asked Crozier.\n\n\"I do not believe so,\" said the surgeon. \"More of the canned food is contaminated or putrefied than we had thought. We're running low on all other stores. The men's diet is already below what they need for the work they're doing every day on the ship or out on the ice. Everyone is losing weight and energy. Add to that the sudden rise in scurvy cases and... well, gentlemen, I simply do not believe that many of us on _Erebus_ or _Terror_ \u2014 if the ships themselves last that long \u2014 will have the energy or concentration abilities to make _any_ sledge trip if we wait until June or July to see if the ice breaks up.\"\n\nThe room was silent again.\n\nInto the silence, Goodsir added, \"Or rather, a few men may well have the energy to haul sledges and boats in a bid for rescue or to reach civilization, but they will have to leave the vast majority of others behind to starve.\"\n\n\"The strong could go for help to bring rescue parties back to the ships,\" said Lieutenant Le Vesconte.\n\nIt was Ice Master Thomas Blanky who spoke up. \"Anyone heading south \u2014 say by hauling our boats south to the mouth of the Great Fish River and then upstream 850 miles farther south to Great Slave Lake where there's an outpost \u2014 wouldn't get there until late autumn or winter at the best and couldn't return with an overland rescue party until late summer of 1849. Everyone left behind on the ships would be dead of scurvy and starvation by then.\"\n\n\"We could load sledges and all head east to Baffin Bay,\" said First Mate Des Voeux. \"There might be whalers there. Or even rescue ships and sledge parties already searching for us.\"\n\n\"Aye,\" said Blanky. \"That's a possibility. But we'd have to man-haul sledges across hundreds of miles of open ice, what with all its pressure ridges and maybe open leads. Or follow the coast \u2014 and that would be more than twelve _hundred_ miles. And then we'd have to cross the whole Boothia Peninsula with all its mountains and obstacles to get to the east coast where the whalers might be. We could haul the boats with us to cross leads, but that would triple our effort. One thing is sure \u2014 if the ice ain't opening here, it won't be open if we head northeast toward Baffin Bay.\"\n\n\"There would be far less weight if we only take sledges with provisions and tents to the northeast across Boothia,\" said Lieutenant Hodgson from the _Terror_ side of the table. \"One of the pinnaces must weigh at least six hundred pounds.\"\n\n\"More like eight hundred pounds,\" Captain Crozier said softly. \"Without stores in it.\"\n\n\"Add to that more than six hundred pounds for a sledge that could carry a boat,\" said Thomas Blanky, \"and we'd be man-hauling between fourteen and fifteen hundred pounds for each party \u2014 just the weight of the boat and sledge \u2014 not counting food, tents, weapons, clothes, and other things we'd have to haul with us. No one has ever man-hauled that much weight for more than a thousand miles \u2014 and much of it would be across open sea ice if we head for Baffin Bay.\"\n\n\"But a sledge with runners on the ice and possibly a sail \u2014 especially if we leave in March or April before the ice gets runny and sticky \u2014 would be easier going than man-hauling gear overland or through summer slush,\" said Lieutenant Le Vesconte.\n\n\"I say we leave the boats behind and travel light to Baffin Bay with just sledges and survival stores,\" said Charles Des Voeux. \"If we arrive on the east coast of Somerset Island to the north before the whaling season ends, we're bound to be picked up by a ship. And I would wager that there will be Navy rescue ships and sledge parties there looking for us.\"\n\n\"If we leave the boats behind,\" said Ice Master Blanky, \"one open stretch of water will stop us for good. We die out there on the ice.\"\n\n\"Why would rescuers be on the east side of Somerset Island and the Boothia Peninsula in the first place?\" asked Lieutenant Little. \"If they're searching for us, won't they follow our path through Lancaster Sound to Devon and Beechey and Cornwallis Islands? They know Sir John's sailing orders. They will presume we made it through Lancaster Sound since it's open most summers. There's no chance at all of any of us making it _that_ far north.\"\n\n\"Perhaps the ice is as bad up at Lancaster Sound this year as it is down here,\" said Ice Master Reid. \"That would keep the search parties farther south, out on the east side of Somerset Island and the Boothia.\"\n\n\"Maybe they'll find the messages what we left in the cairns way up at Beechey if they do get through,\" said Sergeant Tozer. \"And send sledges or ships south the way we come.\"\n\nSilence descended like a shroud.\n\n\"There were no messages left at Beechey,\" Captain Fitzjames said into that silence.\n\nIn the embarrassed vacuum that followed this statement, Francis Rawdon Moira Crozier found a strange, hot, pure flame burning in his chest. It was a sensation rather like a first sip of whiskey after days without it, but also nothing at all like that.\n\nCrozier wanted to live. It was that simple. He was _determined_ to live. He was going to survive this bad patch in the face of all odds and gods dictating that he would not and could not. This fire in his chest had been there even in the shaky, sick hours and painful days after he had emerged from the pit of his malaria-and-withdrawal brush with death in early January. The flame grew stronger every day.\n\nPerhaps more than any other man around the long table in the Great Cabin this day, Francis Crozier understood the near impossibility of the courses of action being discussed. It was folly to head south across the ice toward Great Fish River. Folly to head for Somerset Island across twelve hundred miles of coastal ice, pressure ridges, open leads, and an unknown peninsula. Folly to think that the ice would open up this summer and allow _Terror_ \u2014 overcrowded with two crews and almost empty of provisions \u2014 to sail out of the hopeless trap that Sir John had led them into.\n\nNonetheless, Francis Crozier was determined to live. The flame burned in him like strong Irish whiskey.\n\n\"Have we given up on the idea of sailing out?\" Robert Sinclair was saying.\n\nJames Reid, _Erebus_ 's Ice Master, answered. \"We would have to sail almost three hundred miles north up the unnamed strait and sound that Sir John discovered, then through Barrow Strait and Lancaster Sound, then get south through Baffin Bay before the ice closed on us again. We had the steam engine and armour plating to help us bash through the ice heading south. Even if the ice relents to levels it was two summers ago, we would have great difficulty traversing that distance just with sail. And with our weakened wooden hull.\"\n\n\"The ice may be considerably less than in 1846,\" said Sinclair.\n\n\"Angels may fly out my arse,\" said Thomas Blanky.\n\nBecause of his missing leg, none of the officers at the table reprimanded the ice master. A few smiled.\n\n\"There might be another option... for sailing, I mean,\" said Lieutenant Edward Little.\n\nEyes turned in his direction. Enough men had saved some rations of tobacco \u2014 stretched out by adding unspeakable things to it \u2014 that half a dozen were now smoking pipes around the table. The smoke haze made the gloom even thicker in the dim flicker of the whale-oil lamps.\n\n\"Lieutenant Gore last summer thought that he spied land to the south of King William Land,\" continued Little. \"If he did, that has to be the Adelaide Peninsula \u2014 known territory \u2014 which quite often has a channel of open water between the coast ice and the pack ice. If enough leads open to allow _Terror_ to sail south \u2014 just a little over one hundred miles, perhaps, rather than the three hundred miles back through Lancaster Sound \u2014 we could follow open channels along the coast west until we reach the Bering Strait. Everything beyond here would be known territory.\"\n\n\"The North-West Passage,\" said Third Lieutenant John Irving. The words sounded like a mournful incantation.\n\n\"But would we have enough able-bodied men to crew the ship by late summer?\" asked Dr. Goodsir, his voice very soft. \"By May, the scurvy may have all of us in its grip. And what would we do for food during the weeks or months of our passage west?\"\n\n\"Hunting might be good farther west,\" said Marine Sergeant Tozer. \"Musk oxen. Them big deer. Walruses. White foxes. Maybe we'd be eating like pashas before we got to Alaska.\"\n\nCrozier half-expected Ice Master Thomas Blanky to say, \"And musk oxen might fly out my arse,\" but the sometimes-giddy ice master seemed to be lost in his own reveries.\n\nLieutenant Little answered instead. \"Sergeant, our problem is that even if the game were to miraculously return after two summers' absence, none of us aboard seems able to hit anything with muskets... your men excluded, of course. We'd need more than your few surviving Marines to hunt. And it appears that none of us has any experience hunting anything much larger than birds. Will the shotguns bring down the game you're talking about?\"\n\n\"If you gets close enough,\" Tozer said sullenly.\n\nCrozier interrupted this line of discussion. \"Dr. Goodsir made an excellent point earlier... if we wait until midsummer, or perhaps even until June to see if the pack ice breaks up, we may be too ill and hungry to crew the ship. We'd _certainly_ be too low on provisions to start a sledge trip. And we have to assume three or four months of travel across the ice or up Fish River, so if we're going to abandon the ships and take to the ice with the hopes of arriving at either Great Slave Lake or the east coast of Somerset Island or Boothia before winter sets in again, our departure obivously has to be before June. But how early?\"\n\nThere was another thick bout of silence.\n\n\"I would suggest no later than the first of May,\" Lieutenant Little said at last.\n\n\"Earlier, I would think,\" said Dr. Goodsir, \"unless we find sources of fresh meat soon and if the illness continues to spread as quickly as it currently is.\"\n\n\"How much earlier?\" asked Captain Fitzjames.\n\n\"No later than mid-April?\" Goodsir said hesitantly.\n\nThe men looked at one another through the tobacco smoke and cold air. That was less than two months away.\n\n\"Perhaps,\" said the surgeon, his voice sounding both firm and tentative to Crozier, \"if conditions continue to worsen.\"\n\n\"How could they get worse?\" asked Second Lieutenant Hodgson.\n\nThe young man obviously had meant it as a joke to lessen the tension but was rewarded with baleful and angry stares.\n\nCrozier did not want the council of war to end on that note. The officers, warrant officers, petty officers, and civilian at the table had looked at their choices and seen that they were as bleak as Crozier had known they would be, but he did not want his ships' leaders' morale to get any lower than it already was.\n\n\"By the way,\" Crozier said in a conversational tone, \"Captain Fitzjames has decided to conduct Divine Service next Sunday on _Erebus_ \u2014 he'll be giving a special sermon that I'm interested in hearing, although I have it on good authority that it will _not_ be a reading from the _Book of Leviathan_ \u2014 and I thought that since the ships' companies will be assembled anyway, we should have full rations of grog and dinner that one day.\"\n\nThe men smiled and bantered. None of them had expected to bring back good news to their specialized portions of the crews from this meeting.\n\nFitzjames raised one eyebrow very slightly. His \"special sermon\" and this Divine Service five days away, Crozier knew, were news to him, but Crozier thought it would probably do the thinning captain good to be preoccupied with something and to be the center of attention for a change. Fitzjames nodded ever so slightly.\n\n\"Very well then, gentlemen,\" Crozier said a bit more formally. \"This exchange of thoughts and information has been very helpful. Captain Fitzjames and I will consult and perhaps talk to several of you again, one to one, before we make up our minds on a course of action. I will let you Erebuses get back to your ship before our midday sunset. Godspeed, gentlemen. I shall see you all on Sunday.\"\n\nThe men filed out. Fitzjames came around, leaned close, and whispered, \"I may want to borrow that _Book of Leviathan_ from you, Francis,\" and followed his men forward to where they were struggling into their frozen slops.\n\n _Terror_ 's officers went back to their duties. Captain Crozier sat for a few minutes in his chair at the head of the table, thinking about what had been discussed. The fire for survival burned hotter than ever in his aching chest.\n\n\"Captain?\"\n\nCrozier looked up. It was the old steward from _Erebus,_ Bridgens, who had filled in on the serving because of both captains' stewards' illnesses. The man had been helping Gibson clean up the pewter plates and teacups.\n\n\"Oh, you can go, Bridgens,\" said Crozier. \"Go on with the others. Gibson will attend to all this. We don't want you walking back to _Erebus_ on your own.\"\n\n\"Yes, sir,\" said the old subordinate officers' steward. \"But I wonder if I might have a word with you, Captain.\"\n\nCrozier nodded. He did not invite the steward to sit down. He'd never felt comfortable around this old man \u2014 far too old for Discovery Service. If Crozier had been the one to make the decision three years earlier, Bridgens never would have been included on the roster \u2014 certainly not listed with an age of \"26\" to fool the Navy \u2014 but Sir John had been amused by having a steward aboard even older than himself and that had been that.\n\n\"I couldn't help but hear the discussion, Captain Crozier \u2014 the three options of staying with the ships and hoping for a thaw, heading south to Fish River, or crossing the ice to Boothia. If the captain doesn't mind, I'd like to suggest a fourth option.\"\n\nThe captain did mind. Even an egalitarian Irishman like Francis Crozier bridled a bit at having a subordinate officers' steward give advice on life-and-death command problems. But he said, \"Go ahead.\"\n\nThe steward went to the wall of books set into the stern bulkhead and pulled two large volumes, bringing them over to the table and setting them down with a thud. \"I know you're aware, Captain, that in 1829, Sir John Ross and his nephew James sailed their ship _Victory_ down the east coast of Boothia Felix \u2014 the peninsula they discovered and which we now call Boothia Peninsula.\"\n\n\"I am _very_ aware of this, Mr. Bridgens,\" Crozier said coldly. \"I know Sir John and his nephew Sir James very well.\" After five years in the ice of Antarctica with James Clark Ross, Crozier thought he was understating the acquaintance.\n\n\"Yes, sir,\" said Bridgens, nodding but not seeming abashed. \"Then I'm sure you know the details of their expedition, Captain Crozier. They spent _four winters_ in the ice. That first winter, Sir John anchored _Victory_ in what he named Felix Harbour on the east coast of Boothia... almost due east of our position here.\"\n\n\"Were you _on_ this expedition, Mr. Bridgens?\" asked Crozier, willing the old man to get on with it.\n\n\"I did not have that honor, Captain. But I have read these two large volumes written by Sir John detailing his expedition. I wondered if you have had the time to do the same, sir.\"\n\nCrozier felt his Irish anger building. This old steward's brashness was skirting on impertinence. \"I have looked at the books, of course,\" he said coolly. \"I have not had the time to read them carefully. Is there a point to this, Mr. Bridgens?\"\n\nAny other officer, warrant officer, petty officer, seaman, or Marine under Crozier's command would have received the message and been backing out of the Great Cabin while bowing low by now, but Bridgens seemed oblivious of his expedition commander's irritation.\n\n\"Yes, Captain,\" said the old man. \"The point is that John Ross...\"\n\n\" _Sir_ John,\" interrupted Crozier.\n\n\"Of course. Sir John Ross had much the same problem we do now, Captain.\"\n\n\"Nonsense. He and James and _Victory_ were frozen in on the _east_ side of Boothia, Bridgens, precisely where we'd like to sledge to if we have the time and wherewithal. Hundreds of miles east of here.\"\n\n\"Yes, sir, but at the same latitude, although _Victory_ didn't have to face this God-cursed pack ice coming down from the northwest all the time, thanks to Boothia. But she spent _three winters_ in the ice there, Captain. James Ross sledged more than six hundred miles west across Boothia and the ice to King William Land just twenty-five miles sou'southeast of us, Captain. He named Victory Point... the same point and cairn site that poor Lieutenant Gore sledged to last summer before his unfortunate accident.\"\n\n\"Do you think I don't know that Sir James discovered King William Land and named Victory Point?\" demanded Crozier. His voice was taut with irritation. \"He also discovered the God-damned north magnetic pole during that expedition, Bridgens. Sir James is... was... the most outstanding long-distance sledger of our era.\"\n\n\"Yes, sir,\" said Bridgens. His small steward's smile made Crozier want to strike him. The captain knew \u2014 had known before sailing \u2014 that this old man was a well-known sodomite, at least on shore. After the caulker mate's near mutiny, Captain Crozier was sick of sodomites. \"My point is, Captain Crozier, that after _three winters_ in the ice, with his men as sick with scurvy as ours will be by this summer, Sir John decided that they would never get out of the ice and sank _Victory_ in ten fathoms of water there off the east coast of Boothia, due east of us, and they headed north to Fury Beach, where Captain Parry had left supplies and boats.\"\n\nCrozier realized that he could hang this man, but he could not shut him up. He frowned and listened.\n\n\"You remember, Captain, that Parry's supplies of food and boats were there at Fury Beach. Ross took the boats and sailed north along the coast to Cape Clarence, where from the cliffs there they could see north across Barrow Strait and Lancaster Sound to where they hoped to find whaling ships... but the sound was solid ice, sir. That summer was as bad as our last two summers have been and as this coming one may be.\"\n\nCrozier waited. For the first time since his deathly illness in January, he wished he had a glass of whiskey.\n\n\"They went back to Fury Beach and spent a fourth winter there, Captain. Men were close to dying of scurvy. The next July... 1833, four years after they had entered the ice up there... they set out in the small boats north and then east down Lancaster Sound past Admiralty Inlet and Navy Board Inlet, when on the morning of twenty-five August, James Ross... Sir James now... saw a sail. They waved, hallooed, and fired rockets. The sail disappeared east over the horizon.\"\n\n\"I remember Sir James mentioning something about that,\" Crozier said drily.\n\n\"Yes, Captain, I imagine he would,\" said Bridgens with his maddening little pedant's smile. \"But the wind calmed, and the men rowed like smoke and oakum, sir, and they caught up to the whaler. She was the _Isabella,_ Captain, the same ship that Sir John had commanded way back in 1818.\n\n\"Sir John and Sir James and the crew of _Victory_ spent four years in the ice at our latitude, Captain,\" continued Bridgens. \"And only one man died \u2014 the carpenter, a Mr. Thomas, who had a dyspeptic and disagreeable disposition.\"\n\n\"Your point?\" asked Crozier again. His voice was very flat. He was too aware that more than a dozen men had died under his command on this expedition.\n\n\" _There are still boats and stores at Fury Beach,_ \" said Bridgens. \"And my guess is that any rescue party sent out for us \u2014 last year or this coming summer \u2014 will leave more boats and stores there. It's the first place the Admiralty will think of to leave caches for us and for future rescue parties. Sir John's survival assured that.\"\n\nCrozier sighed. \"Are you in the habit of thinking like the Admiralty, Subordinate Officers' Steward Bridgens?\"\n\n\"Sometimes, yes,\" said the old man. \"It's a habit of decades, Captain Crozier. After a while, proximity to fools forces one to think like a fool.\"\n\n\"That will be all, Steward Bridgens,\" snapped Crozier.\n\n\"Aye, sir. But read the two volumes, Captain. Sir John lays it all out \u2014 how to survive on the ice. How to fight the scurvy. How to find and use Esquimaux natives to help in the hunting. How to build little houses out of blocks of snow...\"\n\n\"That will be _all,_ Steward!\"\n\n\"Aye, sir.\" Bridgens knuckled his forehead and turned toward the companionway, but not before sliding the two thick volumes closer to Crozier.\n\nThe captain sat alone in the freezing Great Cabin for another ten minutes. He listened to the Erebuses clatter up the main ladderway and stomp across the deck above. He heard shouts as _Terror_ officers on deck bid their comrades farewell and wished them a safe crossing of the ice. The ship quieted except for the bustle of men settling down after their supper and grog forward. Crozier heard the tables ratcheted up in the crewmen's berthing area. He heard his officers clump down the ladderway, hang their slops, and come aft for their own supper. They sounded more chipper than they had at breakfast.\n\nCrozier finally stood \u2014 stiff with cold and body aches \u2014 lifted the two heavy volumes, and carefully set them back in their place on the shelf set into the aft bulkhead.\n\nGOODSIR\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _6 March, 1848_\n\nThe surgeon woke to shouts and screaming.\n\nFor a minute he did not know where he was and then he remembered \u2014 Sir John's Great Cabin, now the sick bay on _Erebus._ It was the middle of the night. All the whale-oil lamps had been extinguished and the only light came through the open door to the companionway. Goodsir had fallen asleep on an extra cot \u2014 seven men seriously ill with scurvy and one man with stones in his kidney were sleeping in the other cots. The man with stones had been dosed with opium.\n\nGoodsir had been dreaming that his men were screaming as they were dying. They were dying, in his dream, because he did not know how to save them. Trained as an anatomist, Goodsir was less skilled than the three dead expedition surgeons had been at a Naval surgeon's primary responsibility \u2014 dispensing pills, potions, emetics, herbs, and boluses. Dr. Peddie had once explained to Goodsir that the vast majority of the medicines were useless for the specific sailor's ailments \u2014 most merely served to clean out the bowels and belly in an explosive manner \u2014 but the more powerful the purgative, the more effective the seamen thought the treatment was. It was the _idea_ of medicinal help that helped the sailors heal, according to the late Peddie. In most cases not involving actual surgery, the body either healed itself or the patient died.\n\nGoodsir had been dreaming that they were all dying \u2014 screaming as they died.\n\nBut these screams were real. They seemed to be coming up through the deck.\n\nHenry Lloyd, Goodsir's assistant, ran into the sick bay with his shirttails hanging out from under his sweaters. Lloyd was carrying a lantern and Goodsir could see that he had no shoes on. He must have run straight from his hammock.\n\n\"What's going on?\" whispered Goodsir. The sick men had not been roused from sleep by the screams from below.\n\n\"The captain wants you forward by the main ladder,\" said Lloyd. He made no attempt to lower his voice. The young man sounded shrill and terrified.\n\n\"Shhh,\" said Goodsir. \"What's happening, Henry?\"\n\n\"The thing's inside, Doctor,\" Lloyd cried through chattering teeth. \"It's below. It's killing men below.\"\n\n\"Watch the men here,\" ordered Goodsir. \"Fetch me if any of them wakes or takes a turn for the worse. And go put your boots and outer layers on.\"\n\nGoodsir went forward through a milling of warrant officers and petty officers coming out of their cubicles and struggling into their clothes. Captain Fitzjames was standing with Le Vesconte at the head of the hatchway open to the lower decks. The captain had a pistol in his hand.\n\n\"Surgeon, there have been men injured below. You'll come with us when we go down to fetch them. You will need your slops.\"\n\nGoodsir nodded dumbly.\n\nFirst Mate Des Voeux came down the ladder from the deck above. Cold air rolled down with him, taking Goodsir's breath away. For the past week _Erebus_ had been rocked and battered by a blizzard and staggeringly low temperatures, some reaching down to \u2212100 degrees. The surgeon had not been able to spend his allotted time on _Terror._ There had been no communication between the ships while the blizzard raged.\n\nDes Voeux brushed snow off his slops. \"The three men on watch haven't seen anything outside, Captain. I told them to stand by.\"\n\nFitzjames nodded. \"We need weapons, Charles.\"\n\n\"The three shotguns up on deck are all we've issued tonight,\" said Des Voeux.\n\nAnother scream came up from the darkness below. Goodsir could not tell if it came from the orlop deck or deeper, from the lower hold deck. Both hatches seemed to be open below.\n\n\"Lieutenant Le Vesconte,\" barked Fitzjames, \"take three men down through the scuttle in the officers' mess to the Spirit Room and hand up as many muskets and shotguns \u2014 and bags of cartridges, powder, and shot \u2014 as you can. I want every man on the lower deck here armed.\"\n\n\"Aye, sir.\" Le Vesconte pointed to three seamen, and the four shoved their way aft through the darkness.\n\n\"Charles,\" Fitzjames said to First Mate Des Voeux. \"Light lanterns. We're going down. Collins, you're coming. Mr. Dunn, Mr. Brown \u2014 you're with us.\"\n\n\"Yes, sir,\" chorused the caulker and his mate.\n\nHenry Collins, the second master, said, \"Without guns, Captain? You want us to go down there without weapons?\"\n\n\"Bring your knife,\" said Fitzjames. \"I have this.\" He held up the single-shot pistol. \"Stay behind me. Lieutenant Le Vesconte will follow us with an armed party and bring extra weapons. Surgeon, you stay by me as well.\"\n\nGoodsir nodded numbly. He'd been pulling on his slops \u2014 or someone's \u2014 and seemed to be having a child's difficulty in getting his left arm through the sleeve.\n\nFitzjames, his hands bare and wearing only a tattered jacket over his shirt, took a lantern from Des Voeux and plunged down the ladder. From somewhere below rose a series of terrible crashes, as if something was breaking timbers or bulkheads. There were no more screams.\n\nGoodsir remembered the captain's command to \"stay by me\" and fumbled his way down the dark ladder after the two men, forgetting to take a lantern. He did not have his bag of medical instruments and bandages with him. Brown and Dunn clattered after him, with a cursing Collins bringing up the rear.\n\nThe orlop deck was only seven feet below the lower deck but it seemed like another world. Goodsir almost never came down here. Fitzjames and the first mate were standing away from the ladder, swinging their lanterns. The surgeon realized that the temperature down here must be forty degrees below that of the lower deck where they ate and slept \u2014 and the lower deck's average temperature these days was below freezing.\n\nThe crashing had stopped. Fitzjames ordered Collins to stop his cursing and the six men stood in a silent circle around the hatch opening to the hold deck below them. Everyone except Goodsir had a lantern and now extended it, although the small spheres of light seemed to penetrate only a few feet of the misted, freezing air. The men's breath glowed in front of them like golden ornaments. The hurried footsteps banging on the lower deck above them seemed to Goodsir to be coming from miles away.\n\n\"Who was on duty down here tonight?\" whispered Fitzjames.\n\n\"Mr. Gregory and one stoker,\" replied Des Voeux. \"Cowie, I think. Or maybe it was Plater.\"\n\n\"And Carpenter Weekes and his mate Watson,\" hissed Collins in an urgent whisper. \"They were working through the night to shore up that stove-in part of the hull in the starboard for'ard coal-storage bin.\"\n\nSomething roared beneath them. The sound was a hundred times louder and more bestial than any animal sound Goodsir had ever heard \u2014 worse even than the roar from the ebony room at midnight during Carnivale. The force of it echoed off every timber, iron brace, and bulkhead on the orlop deck. Goodsir was sure that the men on watch two decks above in the howling night could hear it as if the thing were on deck with them. His testicles tried to crawl back up into his body.\n\nThe roar had come from down in the hold deck.\n\n\"Brown, Dunn, Collins,\" snapped Fitzjames. \"Go forward past the Bread Room and secure the forward hatch. Des Voeux, Goodsir, come with me.\"\n\nFitzjames stuck his pistol in his belt, held the lantern in his right hand, and clambered down the ladder into the blackness.\n\nGoodsir had to use all his will just to avoid pissing himself. Des Voeux hurried down the ladder next and only an overwhelming sense of shame at the thought of _not_ following the other men combined with a fear of being left alone in the dark set the trembling surgeon into motion after the first mate. His arms, hands, and legs felt as insensate as if they were made of wood, but he knew it was fear, not the cold, that caused this.\n\nAt the bottom of the ladder \u2014 in a black cold somehow more thick and terrible than the hostile outside arctic had ever felt to Harry Goodsir \u2014 the captain and first mate were holding their lanterns out as far as they could reach. Fitzjames had his pistol extended and fully cocked. Des Voeux was holding a standard boat knife. The mate's hand was shaking. No one moved or breathed.\n\nSilence. The crashing, thudding, and screams had all stopped.\n\nGoodsir wanted to scream. He could _feel_ the presence of something down on this dark hold deck with them. Something huge and not human. It could be twelve feet away, just beyond the puny circles of the lantern glow.\n\nAlong with the press of certainty that they were not alone came a strong copperish smell. Goodsir had smelled that many times before. Fresh blood.\n\n\"This way,\" whispered the captain and led the way aft down the narrow starboard companionway.\n\nToward the boiler room.\n\nThe oil lamp that always burned in there had been extinguished. The only glow that came through the open door was a dim red-and-orange flickering from the few bits of coal burning in the boiler hearth.\n\n\"Mr. Gregory?\" called the captain. Fitzjames's shout was loud enough and sudden enough that Goodsir again came close to wetting himself. \"Mr. Gregory?\" the captain called a second time.\n\nThere was no answer. From their position in the corridor, the surgeon could see only a few square feet of the floor of the boiler room and some spilled coal. There was a smell in the air as if someone was grilling beef. Goodsir found himself salivating despite the sense of horror rising in him.\n\n\"Stay here,\" Fitzjames said to Des Voeux and Goodsir. The first mate was looking first forward and then astern, swinging his lantern in a circle, keeping his knife high, obviously straining to see down the dark corridor past the narrow circle of light. Goodsir could do nothing but stand there and bunch his freezing hands into fists. His mouth filled with saliva at the almost forgotten smell of grilling meat and his belly rumbled in spite of his fear.\n\nFitzjames stepped around the door frame and into the boiler room, out of sight.\n\nFor an eternity of five to ten seconds there was no sound. Then the captain's soft voice literally echoed from the metal-walled room. \"Mr. Goodsir. Come in here, please.\"\n\nThere were two human bodies in the room. One was recognizable as the engineer, John Gregory. He had been disemboweled. His body lay in the corner against the aft bulkhead, but grey strings and strands of his intestines had been thrown around the boiler room like party streamers. Goodsir had to watch carefully where he stepped. The other body, a thickset man in a dark blue sweater, lay on his stomach with his arms by his sides, palms upward, his head and shoulders in the boiler's furnace.\n\n\"Help me pull him out,\" said Fitzjames.\n\nThe surgeon grabbed the man's left leg and his smoldering sweater, the captain took the other leg and right arm, and together they pulled the man back out of the flames. The man's open mouth stuck against the lower flange of the furnace hearth's metal grate for a second but then came free with a brittle snapping of teeth.\n\nGoodsir rolled the corpse over while Fitzjames removed his jacket and beat out the flames rising from the dead man's face and hair.\n\nHarry Goodsir felt as if he were watching all this from a great distance. The professional part of his mind noticed with cool detachment that the furnace, as poorly banked as the low coal flames had been, had melted the man's eyes, burned away his nose and ears, and turned his face into the texture of an overbaked, bubbling raspberry flan.\n\n\"Do you recognize him, Mr. Goodsir?\" asked Fitzjames.\n\n\"No.\"\n\n\"It's Tommy Plater,\" gasped Des Voeux from where he stood just within the doorway. \"I recognize him by the sweater and by the earring melted into his jaw where his ear used to be.\"\n\n\"God-damn it, Mate,\" snapped Fitzjames. \"Stand guard out in the corridor.\"\n\n\"Aye, sir,\" said Des Voeux and stepped out. Goodsir heard the sound of retching from the companionway.\n\n\"I will need you to note... ,\" began the captain, speaking to Goodsir.\n\nThere came a crashing, a tearing, and then a resounding thud from the direction of the bow so loud that Goodsir was sure that the ship had broken in half.\n\nFitzjames grabbed up his lantern and was out the door in a second, leaving his smoldering jacket behind in the boiler room. Goodsir and Des Voeux followed him as they ran forward past scattered casks and smashed crates and then squeezed between the black iron bulkheads that held what was left of _Erebus_ 's fresh water supply and the few remaining sacks of its coal.\n\nThey passed a black opening to a coal bin and Goodsir glanced to his right and saw a shirtless human arm protruding over the iron rim of the door frame. He paused and bent to see who lay there, but the light had moved away as the captain and mate continued to run forward with the lanterns. Goodsir was left in the absolute darkness with what was almost certainly another corpse. He stood and ran to catch up.\n\nMore crashes. Shouts now from the deck above. A musket or pistol shot. Another shot. Screams. Several men screaming.\n\nGoodsir, outside the bobbing circles of lantern light, came out of the narrow corridor into an open, dark area and ran headfirst into a thick oak post. He fell on his back into eight inches of ice and sludgy meltwater. He couldn't focus his eyes \u2014 the lanterns above him were only swinging orange blurs as he struggled to stay conscious \u2014 and everything at that moment stank and tasted of sewage and coal dust and blood.\n\n\"The ladder's gone!\" cried Des Voeux.\n\nSitting arse-deep in vile slush, Goodsir could see better as the lanterns steadied. The forward ladderway, made of thick oak and easily able to support several large men hauling hundred-pound sacks of coal up and down, had been smashed into splinters. Fragments hung from the open scuttle frame above.\n\nThe screaming was coming from up on the orlop deck.\n\n\"Boost me up,\" cried Fitzjames, who had tucked his pistol into his belt and set down the lantern and was now reaching up, trying to get a handhold on the splintered frame of the scuttle. He started pulling himself up. Des Voeux bent to boost him.\n\nFlames suddenly exploded above and through the square opening.\n\nFitzjames cursed and fell onto his back in the icy water only a few yards from Goodsir. It looked as if the entire forward scuttle and everything above it on the orlop deck was on fire.\n\n _Fire,_ thought Goodsir. Acrid smoke filled his nostrils.\n\n _There's nowhere to run._ It was a hundred degrees below zero outside and a blizzard was raging. If the ship burned now, they would all die.\n\n\"The main ladderway,\" said Fitzjames and got to his feet, found the lantern, and began running aft. Des Voeux followed.\n\nGoodsir crawled on all fours through the ice and water, got to his feet, fell again, crawled, then ran after the receding lanterns.\n\nSomething on the orlop deck roared. There came a rattle of muskets and the distinct blast of shotguns.\n\nGoodsir wanted to stop in the coal bunker to see if the man belonging to the arm was dead or alive \u2014 or even attached to the outflung arm \u2014 but there was no light when he got there. He ran on in the dark, ricocheting off the iron, coal, and water bunker bulkheads.\n\nThe lanterns were already disappearing up the ladderway to the orlop deck. Smoke billowed down.\n\nGoodsir clambered upward, was kicked in the face by a boot belonging to the captain or mate, and then he was on the orlop deck.\n\nHe couldn't breathe. He couldn't see. Lanterns bobbed around him but the air was so thick with smoke that there was no illumination.\n\nGoodsir's impulse was to find the ladderway up to the lower deck and keep climbing, then keep climbing again until he was outside into the clean air, but there were men shouting to his right \u2014 toward the bow \u2014 so he dropped to all fours. The air was breathable here. Just. Toward the bow was a bright orange glow, far too bright to be lanterns.\n\nGoodsir crawled forward, found the port companionway to the left of the Bread Room, crawled farther. Ahead of him somewhere in the smoke, men were beating at flames with blankets. The blankets were catching fire.\n\n\"Get a bucket brigade,\" shouted Fitzjames from somewhere ahead of him in the smoke. \"Get water down here!\"\n\n\"There's no water, Captain,\" shouted a voice so agitated that Goodsir could not recognize it.\n\n\"Use the piss buckets.\" The captain's voice cut like a blade through the smoke and shouting.\n\n\"They're frozen!\" shouted a voice that Goodsir did recognize. John Sullivan, captain of the maintop.\n\n\"Use them anyway,\" shouted Fitzjames. \"And snow. Sullivan, Sinclair, Reddington, Seeley, Pocock, Greater \u2014 get the men to form a bucket line from the deck down here to the orlop deck. Scoop up as much snow as you can. Throw it on the flames.\" Fitzjames had to stop to cough violently.\n\nGoodsir stood. Smoke swirled around him as if someone had opened a door or window. One second he could see fifteen or twenty feet forward toward the carpenter's and bosun's storerooms, clearly see the flames licking the walls and timbers, and the next second he could not see two feet in front of him. Everyone was coughing and Goodsir joined them.\n\nMen shoved against him in their rush to get up the ladderway and Goodsir pressed himself to the bulkhead, wondering if he should go up to the lower deck. He was no use here.\n\nHe remembered the bare arm flung out of the coal bunker below in the hold deck. The thought of going down there again made him want to vomit.\n\n _But the thing is on this deck._\n\nAs if to confirm that thought, four or five muskets not ten feet in front of the surgeon fired at once. The explosions were deafening. Goodsir flung his palms over his ears and fell to his knees, remembering how he had told the crew of _Terror_ that scurvy victims could die from the mere sound of a musket shot. He knew that he had the early symptoms of scurvy.\n\n\"Belay that firing!\" shouted Fitzjames. \"Hold off! There are men up there.\"\n\n\"But, Captain...\" came the voice of Corporal Alexander Pearson, the highest ranking of the four surviving _Erebus_ Royal Marines.\n\n\"Hold off, I tell you!\"\n\nGoodsir could now see Lieutenant Le Vesconte and the Marines there silhouetted against the flames, Le Vesconte standing and the Marines each on one knee, reloading their muskets as if they were in the midst of a battle. The surgeon thought that the walls, timbers, and loose casks and cartons toward the bow were all on fire. Sailors batted at the flames with blankets and rolls of canvas. Sparks flew everywhere.\n\nThe burning silhouette of a man staggered out of the flames toward the Marines and clustered seamen.\n\n\"Hold your fire!\" shouted Fitzjames.\n\n\"Hold your fire!\" repeated Le Vesconte.\n\nThe burning man collapsed into Fitzjames's arms. \"Mr. Goodsir!\" called the captain. John Downing, the quartermaster, ceased beating a blanket against the fire in the corridor and stamped out the flames emanating from the wounded man's smoldering clothes.\n\nGoodsir ran forward and took the weight of the collapsing man from Fitzjames. The right side of the man's face was almost gone \u2014 not burned but clawed away, the skin and eye hanging loose \u2014 and parallel marks ran down the right side of his chest, the claw marks cutting deep through eight layers of fabric and flesh. Blood soaked his waistcoat. The man's right arm was missing.\n\nGoodsir realized that he was holding Henry Foster Collins, the second master whom Fitzjames earlier had ordered to go toward the bow with Brown and Dunn, the caulker and his mate, to secure the forward hatch.\n\n\"I need help getting him up to the surgery,\" gasped Goodsir. Collins was a big man, even without his arm, and his legs had finally given way. The surgeon was able to hold him upright only because he was braced against the Bread Room bulkhead.\n\n\"Downing!\" Fitzjames called to the silhouette of the tall quartermaster who had returned to fighting flames with his burning blanket.\n\nDowning tossed the blanket away and ran back through the smoke. Without asking a question, the quartermaster hooked Collins's remaining arm over his own shoulder and said, \"After you, Mr. Goodsir.\"\n\nGoodsir started up the ladderway but a dozen men with buckets were trying to come down through the smoke.\n\n\"Make way!\" bellowed Goodsir. \"Wounded man coming up.\"\n\nThe boots and knees pressed back.\n\nAs Downing carried the now unconscious Collins up the almost vertical ladder, Goodsir came up onto the lower deck where they all lived. Seamen gathered around and stared back at him. The surgeon realized that he must look like a casualty himself \u2014 his hands and clothes and face were bloody from crashing into the post, and he knew that they were also black with soot.\n\n\"Aft to the sick bay,\" ordered Goodsir as Downing lifted the burned and mauled man in his arms. The quartermaster had to twist sideways to carry Collins down the narrow companionway. Behind Goodsir, two dozen men were handing buckets down the ladder from the deck while others poured snow onto the steaming, hissing deck boards in the seamen's berthing area around the stove and forward scuttle. If the deck there caught fire, Goodsir knew, the ship was lost.\n\nHenry Lloyd came out of the sick bay, his face pale and eyes wide.\n\n\"Are my instruments laid out?\" demanded Goodsir.\n\n\"Yes, sir.\"\n\n\"Bone saw?\"\n\n\"Yes.\"\n\n\"Good.\"\n\nDowning laid the unconscious Collins on the bare surgical table in the middle of the sick bay.\n\n\"Thank you, Mr. Downing,\" said Goodsir. \"Would you be so kind as to get a seaman or two and help these other sick men to a bed in a cubicle somewhere? Any empty berth will do.\"\n\n\"Aye, Doctor.\"\n\n\"Lloyd, get forward to Mr. Wall and tell the cook and his mates that we need as much hot water from the Frazer's stove as he can give us. But first, turn up those oil lamps. Then get back here. I'll need your hands and a lantern.\"\n\nFor the next hour, Dr. Harry D. S. Goodsir was so busy that the sick bay could have caught fire and he would not have noticed except to be glad for the extra light.\n\nHe stripped Collins's upper body naked \u2014 the open wounds steamed in the freezing air \u2014 threw the first pan of hot water over them to cleanse them as best he could, not for hygiene but to briefly clear away the blood in order to see how deep they were, decided that the claw wounds themselves were not immediately life threatening, and went to work on the second master's shoulder, neck, and face.\n\nThe arm had come off cleanly. It was as if a huge guillotine had severed Collins's arm with one drop. Used to industrial and shipboard accidents that mauled and twisted and tore flesh to shreds, Goodsir studied the wound with something like admiration, if not awe.\n\nCollins was bleeding to death, but the flames he'd been caught in had cauterized the gaping shoulder wound to some extent. It had saved his life. So far.\n\nGoodsir could see the shoulder bone \u2014 a glistening white knob \u2014 but there was no remaining arm bone that he had to cut away. With Lloyd shakily holding a lantern close and sometimes putting his finger where Goodsir ordered \u2014 often on a spurting artery \u2014 Goodsir deftly tied off the severed veins and arteries. He had always been good at this sort of thing \u2014 his fingers worked almost by themselves.\n\nAmazingly, there seemed to be little or no fabric or foreign material in the wound. This lowered the chance of fatal sepsis, although that was still a probability. Goodsir cleansed what he could see with the second and final pan of hot water brought aft by Downing. Then he cut away any loose shards of flesh and sutured where he could. Luckily there were flaps of skin long enough that the surgeon could fold them back over the wound and sew them with broad stitches.\n\nCollins moaned and stirred.\n\nGoodsir worked as quickly as he could now, wanting to finish the worst of it before the man came fully awake.\n\nThe right side of Collins's face hung down on his shoulder like a loosened Carnivale mask. It reminded Goodsir of the many autopsies he had performed, cutting away the face and folding it up over the top of the skull like a tight wet cloth.\n\nHe had Lloyd pull the long flap of facial skin as far up and as tight as he could \u2014 his assistant turned away to vomit on the deck but then immediately returned, wiping his sticky fingers on his wool waistcoat \u2014 and Goodsir quickly stitched the loose part of Collins's face to a thick flap of skin and flesh just below the man's receding hairline.\n\nHe could not save the second master's eye. He tried pressing it back into place, but the man's suborbital ridge had been shattered. Bone splinters were in the way. Goodsir snapped off the splinters, but the eyeball itself was too damaged.\n\nHe took shears out of Lloyd's shaking hands and cut away the retinal nerve, throwing the eye into the bucket already filled with bloody rags and shreds of Collins's flesh.\n\n\"Hold that lantern closer,\" ordered Goodsir. \"Quit shaking.\"\n\nAmazingly, there was some eyelid left. Goodsir pulled it down as far as he could and deftly stitched it to a flap of loose skin below the eye. These stitches he made closer together since they would have to serve for years.\n\nIf Collins survived.\n\nHaving done the best he could on the second master's face for the time being, Goodsir turned his attention to the burns and claw wounds. The burns were superficial. The claw wounds ran deep enough that Goodsir could see the always shocking whiteness of exposed ribs here and there.\n\nDirecting Lloyd to apply salve to the burns with his left hand while holding the lantern close with his right, Goodsir cleaned and closed the torn muscles and sewed the surface flesh and skin back in place where he could. Blood continued to flow from the shoulder wound and Collins's neck, but at a much reduced rate. If the flames had cauterized the flesh and veins enough, the second mate might have enough blood left in him to allow for his survival.\n\nOther men were being carried in, but they were suffering only from burns \u2014 some serious but none life threatening \u2014 and now that the most urgent part of his work on Collins was finished, Goodsir hung the lantern on the brass hook above the table and ordered Lloyd to help the others with salve, water, and dressings.\n\nHe was just finishing with Collins \u2014 administering opium so the waking, screaming man would sleep \u2014 when he turned to find Captain Fitzjames standing next to him.\n\nThe captain was as sooty and bloody as the surgeon.\n\n\"Will he live?\" asked Fitzjames.\n\nGoodsir set down a scalpel and opened and then closed his bloody hands as if to say _only God knows._\n\nFitzjames nodded. \"The fire is contained,\" said the captain. \"I thought you'd want to know.\"\n\nGoodsir nodded. He'd not thought about the fire at all in the past hour. \"Lloyd, Mr. Downing,\" he said. \"Would you be so kind as to carry Mr. Collins to that cot nearest the forward bulkhead. It's the warmest there.\"\n\n\"We lost all of the carpenter's stores on the orlop deck,\" continued Fitzjames, \"and many of our remaining food stores that were in crates near the forward scuttle and bow area, and a good part of the Bread Room stores as well. I'd say a third of our remaining supply of canned and casked food is gone. And we're sure there is damage on the hold deck, but we haven't been back down there yet.\"\n\n\"How did the fire begin?\" asked the surgeon.\n\n\"Collins or one of his men threw a lantern at the thing when it came up out of the scuttle at them,\" said the captain.\n\n\"What happened to the... thing?\" asked Goodsir. Suddenly, he was so exhausted that he had to reach for the edge of the bloody surgical table to keep from falling.\n\n\"It must have gone out the way it came in,\" said Fitzjames. \"Back down the forward scuttle and out somewhere down on the hold deck. Unless it's waiting down there still. I have armed men at each of the scuttles. It's so cold and smoky down there on the orlop deck that we'll have to change the guard every half hour.\n\n\"Collins saw it best. That's why I came up... to see if I could talk to him. The others just saw the shape across the flames \u2014 eyes, teeth, claws, a white mass or black silhouette. Lieutenant Le Vesconte had the Marines fire at it, but no one saw if it was hit. There is blood all forward of the burned-out carpenter's storeroom, but we don't know if any of it is from the beast. Can I speak to Collins?\"\n\nGoodsir shook his head. \"I've just doused the second master with an opiate. He'll sleep for hours. I have no idea if he will ever awaken. The odds are against it.\"\n\nFitzjames nodded again. The captain looked as exhausted as the surgeon felt.\n\n\"What about Dunn and Brown?\" asked Goodsir. \"They went forward with Collins. Have you found them?\"\n\n\"Yes,\" Fitzjames said dully. \"They're alive. They escaped to the starboard side of the Bread Room when the fire started and the thing went after poor Collins.\" The captain took a breath. \"The smoke below is dissipating, so I need to lead some men down to the hold to retrieve the bodies of Engineer Gregory and Stoker Tommy Plater.\"\n\n\"Oh my God,\" said Goodsir. He told Fitzjames about the bare arm he'd seen protruding from the coal bunker.\n\n\"I didn't see that,\" said the captain. \"I was so eager to get to the forward scuttle that I did not look down, just ahead.\"\n\n\"I should have looked ahead,\" the surgeon said ruefully. \"I banged into a pillar or post.\"\n\nFitzjames smiled. \"So I see. Physician, heal thyself. You have a deep laceration from your hairline to your brow and a blue swelling the size of Magnus Manson's fist.\"\n\n\"Really?\" said Goodsir. He gingerly touched his forehead. His bloody fingers came away bloodier even though he could feel the thick crust of dried blood on the huge contusion up there. \"I'll sew it up with a mirror or have Lloyd do it later,\" he said tiredly. \"I'm ready to go, Captain.\"\n\n\"Go where, Mr. Goodsir?\"\n\n\"Down to the hold,\" said the surgeon, feeling his guts twist with nausea at the thought of it. \"To see who's lying in the coal bunker. He might be alive.\"\n\nFitzjames looked him in the eye. \"Our carpenter, Mr. Weekes, and his mate, Watson, are missing, Dr. Goodsir. They were working in a starboard coal bunker, shoring up a breach in the hull. But they must be dead.\"\n\nGoodsir had heard the \"doctor.\" Franklin and his commander had almost never called the surgeons that, not even Stanley and Peddie, the chief surgeons. They \u2014 and Goodsir \u2014 had almost always been the lower \"Mister\" to Sir John and the aristocratic Fitzjames.\n\nBut not this time.\n\n\"We have to go down to see,\" said Goodsir. \" _I_ have to go down to see. One or the other might still be alive.\"\n\n\"The thing from the ice might be alive and waiting down there as well,\" Fitzjames said softly. \"No one saw or heard it leave.\"\n\nGoodsir nodded tiredly and lifted his medical bag. \"May I have Mr. Downing to come with me?\" he asked. \"I may need someone to hold the lantern.\"\n\n\"I'll come with you, Dr. Goodsir,\" said Captain Fitzjames. He held up an extra lamp that Downing had carried in. \"Lead on, sir.\"\n\nCROZIER\n\n _Lat. 70\u00b0-05\u2032 N., Long. 98\u00b0-23\u2032 W._\n\n _22 April, 1848_\n\nLieutenant Little,\" said Captain Crozier, \"please pass the order to abandon ship.\"\n\n\"Yes, Captain.\" Little turned and shouted the order down the crowded deck. The other officers and surviving second mate were absent, so John Lane, the bosun, picked up the order and bellowed it toward the bow. Thomas Johnson, the bosun's mate and the man who had administered the lashes to Hickey and the other two men in January, shouted the order down the open hatchway before finally closing and battening down the scuttle.\n\nThere was no one left belowdecks, of course. Crozier and Lieutenant Little had walked the ship stern to bow on each deck, looking into every compartment \u2014 from the cold boiler room with its banked furnaces to the hold deck's empty coal scuttles to the cramped but empty forward cable locker and then up through the decks. On the orlop deck they had checked that the Spirit Room and Gunner's Storeroom were empty of all muskets, shotguns, powder, and shot \u2014 only rows of cutlasses and bayonets remained in the racks overhead, gleaming coldly in the lantern light. Two officers had checked that all necessary clothing had been removed from the Slop Room over the past month and a half and then gone on to the empty Captain's Storeroom and equally empty Bread Room. On the foredeck, Little and Crozier had looked into every cabin and berth, noticing how neat the officers had left their bunks and shelves and remaining possessions, then seeing the seamen's hammocks tucked up for the final time, their sea chests lightened but still in place as if awaiting the call to supper, going aft then to notice the missing books in the Great Room where men had made their choices from the volumes and carried scores of them onto the ice with them. Finally, standing next to the huge stove that was absolutely cold for the first time in almost three years, Lieutenant Little and Captain Crozier had called again down the forward scuttle, making sure that no one had remained behind. They would do a head count above, but this was part of the protocol of abandoning ship.\n\nThen they had gone up on deck and left the scuttle open behind them.\n\nThe men standing on deck now were not surprised by the order to abandon ship. They had been called up and assembled for it. There were only about twenty-five Terrors present this morning; the rest were at Terror Camp two miles south of Victory Point or sledging materials to the camp or out hunting or reconnoitering near Terror Camp. An equal number of Erebuses waited below on the ice, standing near sledges and piles of gear where the _Erebus_ gear-and-supply tents had been pitched since the first of April when that ship had been abandoned.\n\nCrozier watched his men file down the ice ramp, leaving the ship forever. Finally only he and Little were left standing on the canted deck. The fifty-some men on the ice below looked up at them with eyes almost made invisible under low-pulled Welsh wigs and above wool comforters, all squinting in the cold morning light.\n\n\"Go ahead, Edward,\" Crozier said softly. \"Over the side with you.\"\n\nThe lieutenant saluted, lifted his heavy pack of personal possessions, and went down first the ladder and then the ice ramp to join the men below.\n\nCrozier looked around. The thin April sunlight illuminated a world of tortured ice, looming pressure ridges, countless seracs, and blowing snow. Tugging the bill of his cap lower and squinting toward the east, he tried to record his feelings at the moment.\n\nAbandoning ship was the lowest point in any captain's life. It was an admission of total failure. It was, in most cases, the end of a long Naval career. To most captains, many of Francis Crozier's personal acquaintance, it was a blow from which they would never recover.\n\nCrozier felt none of that despair. Not yet. More important to him at the moment was the blue flame of determination that still burned small but hot in his breast \u2014 _I will live._\n\nHe wanted his men to survive \u2014 or at least as many as possibly could. If there was the slightest hope of any man from HMS _Erebus_ or HMS _Terror_ surviving and going home to England, Francis Rawdon Moira Crozier was going to follow that hope and not look back.\n\nHe had to get the men off the ship. And then off the ice.\n\nRealizing that almost fifty sets of eyes were looking up at him, Crozier patted the gunwale a final time, scrambled down the ladder they'd set on the starboard side as the ship had begun to cant more steeply to port in recent weeks, and then walked down the well-worn ice ramp to the waiting men.\n\nHoisting his own pack and stepping into line near the men in harness at the rearmost sledge, he looked up a final time at the ship and said, \"She looks fine, doesn't she, Harry?\"\n\n\"She does that, Captain,\" said Captain of the Foretop Harry Peglar. As good as his word, he and the topmen had managed to steep all of the stored masts and restore the yards and rigging in the past two weeks, despite blizzards, low temperatures, lightning storms, surging ice pressures, and high winds. Ice gleamed everywhere on the now top-heavy ship's restored topmasts, spars, and rigging. She looked to Crozier as if she were bedecked in jewels.\n\nAfter the sinking of HMS _Erebus_ on the last day of March, Crozier and Fitzjames had decided that even though _Terror_ had to be abandoned soon if they were to have any chance of walking or taking the boats to safety before winter, the ship should be restored to sailing shape. Should they be stuck at Terror Camp on King William Land for months into summer and the ice miraculously open, they could, theoretically, take the boats back to _Terror_ and try sailing to freedom.\n\nTheoretically.\n\n\"Mr. Thomas,\" he called to Robert Thomas, the Second Mate and lead hauler on the first of the five sledges, \"lead off when you're ready.\"\n\n\"Aye, aye, sir,\" called back Thomas and leaned into the harness. Even with seven men straining in harness, the sledge did not budge. The runners had frozen to the ice.\n\n\"Hearty does it, Bob!\" said Edwin Lawrence, laughing, one of the seamen in harness with him. The sledge groaned, men groaned, leather creaked, ice tore, and the high-packed sledge moved forward.\n\nLieutenant Little gave the order for the second sledge, headed up by Magnus Manson, to start off. With the giant in the lead of the men, the second sledge \u2014 although more heavily laden than Thomas's \u2014 immediately started off with only the slightest rasp of ice under the wooden runners.\n\nAnd so it went for the forty-six men, thirty-five of them man-hauling for the first stretch, five walking in reserve with shotguns or muskets, waiting to pull, four of the mates from both ships and the two officers \u2014 Lieutenant Little and Captain Crozier \u2014 walking alongside and occasionally pushing and less frequently slipping into harness themselves.\n\nThe captain remembered that several days earlier, when Second Lieutenant Hodgson and Third Lieutenant Irving were preparing to leave for yet another boat-sledge trip to Camp Terror \u2014 both officers then ordered to take men from that camp to hunt and reconnoiter over the next few days \u2014 Irving had surprised his captain by requesting that one or the other of two men assigned to his team be left back at _Terror._ Crozier had been initially surprised because his estimate of young John Irving had been that the junior lieutenant was capable of dealing with seamen and carrying out and enforcing any orders given to him, but then Crozier heard the names involved and understood. Lieutenant Little had put the names of both Magnus Manson and Cornelius Hickey on Irving's sledge and scouting team rosters, and Irving was respectfully requesting, without giving any reasons, that one or the other man be assigned to another team. Crozier had acceded to the request immediately, reassigning Manson to the last day's sledge pulls and allowing the small caulker's mate to go ahead with Lieutenant Irving's sledge team. Crozier did not trust Hickey either, especially after the near mutiny weeks ago, and he knew that the little man was much more treacherous with the huge idiot Manson by his side.\n\nNow, walking away from the ship, seeing Manson pulling fifty feet ahead of him, Crozier deliberately kept his face directed forward. He had resolved that he would not look back at _Terror_ for at least the first two hours of the pulling.\n\nLooking at the men leaning and straining ahead of him, the captain was very aware of those who were absent.\n\nFitzjames was absent this day, serving as commanding officer at Camp Terror on King William Land, but the real reason for his absence was tact. No captain wanted to abandon his ship in full view of another captain if at all possible, and all captains were sensitive to this. Crozier, who had visited _Erebus_ almost every day from the beginning of its breakup from ice pressure two days after the fire and invasion of the thing from the ice in early March, had made a point of not being there midday on 31 March when Fitzjames had to abandon ship. Fitzjames had returned the favour this week by volunteering for command duties far from _Terror._\n\nMost of the other men's absences were for a far more tragic and depressing reason. Crozier brought up their faces as he marched alongside the last sledge.\n\n _Terror_ had been much luckier than _Erebus_ when it came to loss of its officers and leaders. Of his primary officers, Crozier had lost his first mate, Fred Hornby, to the beast during the Carnivale debacle, Second Master Giles MacBean to the thing during a sledge trip the previous September, and both his surgeons, Peddie and McDonald, also during the New Year's Eve Carnivale. But his first, second, and third lieutenants were alive and reasonably well, as was his second mate, Thomas; Blanky, his ice master; and the indispensable Mr. Helpman, his primary clerk.\n\nFitzjames had lost his commanding officer \u2014 Sir John \u2014 and his first lieutenant, Graham Gore, as well as Lieutenant James Walter Fairholme and First Mate Robert Orme Sergeant, all killed by the creature. He'd also lost his primary surgeon, Mr. Stanley, and Henry Foster Collins, his second master. That left only Lieutenant H. T. D. Le Vesconte, Second Mate Charles Des Voeux, Ice Master Reid, Surgeon Goodsir, and his purser, Charles Hamilton Osmer, as his remaining complement of officers. Instead of the crowded officers' mess of the first two years \u2014 Sir John, Fitzjames, Gore, Le Vesconte, Fairholme, Stanley, Goodsir, and clerk Osmer all dining together \u2014 the final weeks had seen only the captain and his sole surviving lieutenant, the surgeon, and clerk dining in the cold of the officers' wardrooom. And even that in the last days, Crozier knew, had been an absurd sight once the ice had tilted _Erebus_ almost thirty degrees to starboard. The four men had been forced to sit on the deck, their plates on their knees and their feet braced hard against a batten.\n\nHoar, Fitzjames's steward, was still sick with scurvy, so poor old Bridgens had been the steward scurrying like a crab to serve the officers braced on the wildly tilted deck.\n\n _Terror_ had also been luckier in keeping her warrant officers intact. Crozier's engineer, chief boatswain, and carpenter were still alive and functioning. _Erebus_ had seen her engineer, John Gregory, and her carpenter, John Weekes, both eviscerated in March when the thing on the ice had come aboard in the night. The ship's other warrant officer, Boatswain Thomas Terry, had been beheaded by the creature the previous November. Fitzjames had no warrant officers left alive.\n\nOf _Terror_ 's twenty-one petty officers \u2014 mates, quartermasters, fo'c'sle, hold, maintop, and foretop captains, coxswains, stewards, caulkers, and stokers \u2014 Crozier had lost only one man: Stoker John Torrington, the first man on the expedition to die, so long ago on 1 January 1846, way back at Beechey Island. And that, Crozier remembered, had been from consumption that young Torrington had brought aboard with him in England.\n\nFitzjames had lost another of his petty officers, Stoker Tommy Plater, on the day in March when the thing had gone on its murderous rampage on the lower decks. Only Thomas Watson, the carpenter's mate, had survived the thing's attack down on the hold deck that night, and he had lost his left hand.\n\nSince Thomas Burt, the armourer, had been sent back to England from Greenland even before they'd encountered real ice, that left _Erebus_ with twenty surviving petty officers. Some of these men, such as the ancient sailmaker, John Murray, and Fitzjames's own steward, Edmund Hoar, were too sick with scurvy to be useful, while others, such as Thomas Watson, were too mauled to be of help, while still others, such as the flogged gunroom steward Richard Aylmore, were too sullen to be of much use.\n\nCrozier told one of the men who was obviously fagged out to take a break and to walk with the armed guard while he, the captain, took a turn in the harness. Even with six other men pulling, the terrible exertion of hauling more than fifteen hundred pounds of canned food, weapons, and tents was a strain on his weakened system. Even after Crozier fell into the rhythm \u2014 he'd joined sledging parties since March, when he first began dispatching boats and gear to King William Land, and well knew the drill of man-hauling \u2014 the pain of the straps across his aching chest, the weight of the mass being pulled, and the discomfort from sweat that froze, thawed, and refroze in his clothes were all a shock.\n\nCrozier wished they had more able-bodied seamen and Marines.\n\n _Terror_ had lost two of its rated sailors \u2014 Billy Strong, torn in half by the creature, and James Walker, the idiot Magnus Manson's good friend before the giant fell completely under the sway of the little rat-faced caulker's mate. It had been fear of Jimmy Walker's ghost in the hold, Crozier remembered, that had brought the hulking Manson to his first point of mutiny so many months ago.\n\nFor once HMS _Erebus_ had been luckier than its counterpart. The only able seaman Fitzjames had lost during this expedition had been John Hartnell, also dead of consumption and buried in the winter of '46 on Beechey Island.\n\nCrozier leaned into the straps and thought about the faces and names \u2014 so many officers dead, so few regular sailors \u2014 and grunted as he pulled, thinking that the thing on the ice seemed to be deliberately coming after the leaders of this expedition.\n\n _Don't think that way,_ Crozier ordered himself. _You're giving the beast powers of reasoning it doesn't have._\n\n _Doesn't it?_ asked another, more fearful part of Crozier's mind.\n\nOne of the Royal Marines walked by, carrying a musket rather than shotgun in the crook of his arm. The man's face was completely hidden by caps and wraps, but from the slouched way the man walked, Crozier knew that it was Robert Hopcraft. The Marine private had been seriously injured by the creature on the day a year ago in June when Sir John was killed, but while Hopcraft's other injuries had healed, his shattered collarbone left him always slouching to his left as if he had trouble maintaining a straight line. Another Marine walking with them was William Pilkington, the private who had been shot through his shoulder in the blind that same day. Crozier noticed that Pilkington didn't seem to be favouring that shoulder or arm today.\n\nSergeant David Bryant, _Erebus_ 's ranking Marine, was decapitated just seconds before Sir John had been carried off under the ice by the beast. With Private William Braine dead on Beechey Island in 1846 and Private William Reed disappeared on the ice on 9 November of last fall while sent to deliver a message to _Terror_ \u2014 Crozier remembered the date well since he had walked to _Erebus_ from _Terror_ in the dark himself that first full day of winter darkness \u2014 the beast had reduced Fitzjames's Marine guard to only four: Corporal Alexander Pearson in command, Private Hopcraft with his ruined shoulder, Private Pilkington with his bullet wound, and Private Joseph Healey.\n\nCrozier's own Marine detachment had lost only Private William Heather to the thing on the ice, on the night the previous November when the creature had come aboard and bashed the man's brains out while the private was on watch. But amazingly, shockingly, Heather had refused to die. After lying comatose in the sick bay for weeks, obscenely hovering between life and death, Private Heather had been carried by his Marine mates to his hammock forward in the crew berthing area and they had fed him and cleaned him and carried him to the seat of ease and dressed him every day since. It was as if the staring, drooling man was their pet. He'd been evacuated to Terror Camp just last week, bundled up warmly by the other Marines and set carefully, almost royally, into a special one-man toboggan made for him by Alex \"Fat\" Wilson, the carpenter's mate. The seamen had not objected to the extra load and had volunteered to take turns pulling the living corpse's little sled across the ice and over the pressure ridges to Terror Camp.\n\nThat left Crozier five Marines \u2014 Daly, Hammond, Wilkes, Hedges, and thirty-seven-year-old Sergeant Soloman Tozer, an unschooled fool but now commanding officer among the total of nine functional surviving Royal Marines on the Sir John Franklin Expedition.\n\nAfter the first hour in harness, the sledge seemed to slide more easily and Crozier had fallen into the rhythm of panting that passed for breathing while hauling such dead weight across such nonslippery ice.\n\nThat was all the categories of men lost that Crozier could think of. Except for the boys, of course, those young volunteers who had signed on to the expedition at the last minute and had been listed on the roster as \"Boys\" even though three of the four were a full-grown eighteen years old. Robert Golding was nineteen when they sailed.\n\nThree of the four \"boys\" survived, although Crozier himself had been forced to carry the unconscious George Chambers from the burning Carnivale compartments on the night of the fire. The only fatality among the boys had been Tom Evans, the youngest in demeanor as well as in age; the thing on the ice had plucked the lad literally from beneath Captain Crozier's nose as they were out on the ice in the dark hunting for the missing William Strong.\n\nGeorge Chambers, although recovering consciousness two days after Carnivale, had never been the same. A bright lad before his violent encounter with the thing, the concussion he received reduced him to a level of intelligence even below that of Magnus Manson. George was no living corpse like Private Heather \u2014 he could obey simple orders according to _Erebus_ 's bosun's mate \u2014 but he hardly ever spoke after that terrible New Year's Eve.\n\nDavey Leys, one of the more experienced men on the expedition, was another man who had physically survived two encounters with the white thing on the ice but who was as useless as the literally brainless Private Heather these days. After the night the white thing encountered Leys and John Handford on watch and then chased Ice Master Thomas Blanky into the darkness, Leys had slipped back into his earlier state of unresponsive staring and had never returned from it. He had been transported to Terror Camp \u2014 along with the seriously injured or those too ill to walk, such as Fitzjames's steward, Hoar \u2014 bundled in coats and tucked into one of the boats being dragged atop a sledge. There were too many men now sick with scurvy, wounds, or low morale who were of little use to Crozier or Fitzjames. More mouths to feed and bodies to haul with them when the men were hungry and sick and barely able to walk.\n\nWeary, realizing that he had not really slept the past two nights, Crozier tried counting the dead.\n\nSix officers from _Erebus._ Four dead from _Terror._\n\nAll three warrant officers from _Erebus._ Zero from _Terror._\n\nOne petty officer from _Erebus._ One from _Terror._\n\nJust one seaman from _Erebus._ Four from _Terror._\n\nThat was twenty dead, not counting the three Marines and the boy Evans. Twenty-four men lost on the expedition already. A frightful loss \u2014 greater than Crozier could remember from any arctic expedition in Naval history.\n\nBut there was a more important number, and one that Francis Rawdon Moira Crozier tried to focus on: 105 living souls remaining under his care.\n\nOne hundred and five men alive, including himself, on this day he had been forced to abandon HMS _Terror_ and cross the ice.\n\nCrozier put his head down and leaned more into the harness. The wind had come up and was blowing snow around them, obscuring the sledge ahead, hiding the walking Marines from sight.\n\nWas he sure in the count? Twenty dead not counting the three Marines and one boy? Yes, he was certain that he and Lieutenant Little had checked the muster that morning and confirmed 105 men spread out between the sledging parties, Terror Camp, and HMS _Terror_ that morning... but _was_ he certain? Had he forgotten anyone? Was his addition and subtraction correct? Crozier was very, very tired.\n\nFrancis Crozier might get muddled in the count for a short while \u2014 he had not slept at all in two, no, three nights \u2014 but he had not forgotten a single man's face or name. Nor would he ever.\n\n\"Captain!\"\n\nCrozier came out of the trance that he fell into when he was man-hauling sledges. He could not have told anyone at this moment whether he had been in harness an hour or six hours. The world had become the glare of the cold sun in the southeastern sky, the blowing ice crystals, the rack of his breath, the pain of his body, the shared weight behind him, the resistance of the sea ice and new snow, and most of all the oddly blue sky with wisps of white clouds curling around on all sides as if they were all walking in a blue-and-white-rimmed bowl.\n\n\"Captain!\" It was Lieutenant Little shouting.\n\nCrozier realized that his fellow pullers had come to a halt. All of the sledges were stopped on the ice.\n\nAhead of them to the southeast, perhaps a mile beyond the next heaped-ice pressure ridge, a three-masted ship was moving north to south. Its sails were furled and shrouded, its yards rigged for anchorage, but it moved anyway, as if on a strong current, gliding slowly and majestically on what must be a wide avenue of open water just beyond the next high ridge.\n\n _Rescue. Salvation._\n\nThe steady blue flame of hope in Crozier's aching chest flared brighter for a few exhilarating seconds.\n\nIce Master Thomas Blanky, his peg leg set into something rather like a wooden boot that Carpenter Honey had devised, stepped up to Crozier and said, \"A mirage.\"\n\n\"Of course,\" said the captain.\n\nHe'd recognized the distinctive bomb-ship masts and rigging of HMS _Terror_ almost immediately, even through the shimmering, shifting air, and for a few seconds of confusion bordering on vertigo, Crozier had wondered if somehow they had managed to get lost, turned around, and were actually heading back to the northwest toward the ship they'd abandoned hours earlier.\n\nNo. There were the old sledge tracks, drifted over in spots but deeply worn into the ice by more than a month's repeated passage back and forth, heading straight for the tall pressure ridge with its narrow passes hacked out with picks and shovels. And the sun was still ahead of them and to their right, deep in the south. Beyond the pressure ridge, the three masts shimmered, dissolved briefly, and then returned more solidly than ever, only upside down, with the hull of the ice-entombed _Terror_ blending into a white-cirrus sky.\n\nCrozier and Blanky and so many of the others had seen this phenomenon many times before \u2014 false things in the sky. Years ago, on a fine winter morning frozen in off the coast of the landmass they were calling Antarctica, Crozier had seen a smoking volcano \u2014 the very one named after this ship \u2014 rising upside down from solid sea to the north. Another time on this very expedition, in the spring of 1847, Crozier had come on deck to find black spheres floating in the southern sky. The spheres turned into solid figure eights, then divided again into what looked like a symmetrical progression of ebony balloons and then, within the course of a quarter hour, evaporated completely.\n\nTwo seamen on the third sledge had literally dropped in their traces and were on their knees in the rutted snow. One man was weeping loudly and the other had unleashed a string of the most imaginative sailor curses Crozier had ever heard \u2014 and the captain had heard his fill over the decades.\n\n\"God-damn it!\" shouted Crozier. \"You've seen arctic mirages before. Belay that sniveling and cursing or you'll be man-hauling that God-damn sledge by yourselves and I'll be sitting on it with one boot up each of your arses. Get on your feet, by God! You're men, not weak sisters. Fucking act like it!\"\n\nThe two seamen got to their feet and clumsily brushed off ice crystals and snow. Crozier couldn't immediately identify them by their slops and Welsh wigs and he did not want to.\n\nThe line of sledges started up again with much grunting but no cursing. Everyone knew that the high pressure ridge ahead of them, carved out as it had been by countless previous trips in the past weeks, was still going to be a Christ-fucking cob. They would have to lift and wrestle the heavy sledges up at least fifteen feet of steep incline between the perilous sixty-foot ice cliffs on either side. The threat of tumbling ice boulders would be very real then.\n\n\"It's as if there's some dark God who wants to torment us,\" Thomas Blanky said almost cheerily. The ice master had no pulling duties and was still stumping alongside Crozier.\n\nThe captain did not respond to this, and after a minute Blanky fell back to stump along beside one of the outriding Marines.\n\nCrozier called for one of the extra men to take his place in harness \u2014 something they had rehearsed doing without stopping the forward motion of the sledges \u2014 and when the extra hand took over, he stepped aside out of the ruts and checked his watch. They had been pulling about five hours. Looking behind, Crozier saw that the real _Terror_ had been out of sight for some time, at least five miles and several low pressure ridges behind them. The mirage image had been a final offering from some evil arctic god that seemed intent on tormenting them all.\n\nStill leader of this ill-fated expedition, Francis Rawdon Moira Crozier realized for the first time that he was no longer captain of a ship in Her Majesty's Royal Navy Discovery Service. That part of his life \u2014 and being a seaman and Naval officer had _been_ his life since he was a boy \u2014 was over forever. After being responsible for losing so many men and both his ships, he knew the Admiralty would never give him another command. In terms of his long Naval career, Crozier knew, he was now a dead man walking.\n\nThey were still two hard days of man-hauling away from Terror Camp. Crozier fixed his gaze on the tall pressure ridge ahead and trudged forward.\n\nGOODSIR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _22 April, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _22 April, 1848 \u2014_\n\n _I have been four Days at this place we are calling Terror Camp. I believe it lives up to its name._\n\n _Captain Fitzjames is in Charge of sixty men here, including Myself._\n\n _I confess that when I first sledged within sight of the place last week, the first Image that came to mind was something out of Homer's_ Iliad. _The camp is set along the edge of a wide Inlet about two Miles south of a cairn raised almost two Decades ago at Victory Point by James Clark Ross. It is somewhat more Sheltered from the Wind and Snow blowing off the ice pack here._\n\n _Perhaps scenes from the_ Iliad _were evoked by the 18 long boats pulled up in a row by the edge of the sea ice \u2014 4 boats lying on their sides in the gravel, the other 14 Boats tied upright on Sledges._\n\n _Behind the Boats are 20 tents, ranging in Size from the small Holland tents of the Design we used almost a Year Ago when I accompanied the late Lieutenant Gore to Victory Point \u2014 each Holland tent is large enough for six men to sleep in, three per bag in the 5-foot-wide Wolfskin Blanket-Robe sleeping bags \u2014 to the somewhat larger tents made by the sailmaker, Murray, including tents meant for Captain Fitzjames and Captain Crozier and their personal stewards, and the largest two tents, each roughly the size of the Great Cabins on_ Erebus _and_ Terror, _one serving as Sick Bay, the other as the Seamen's Mess Tent. There are other mess tents for warrant officers, petty officers, and the officers and their Civilian Counterparts, such as Engineer Thompson and Myself._\n\n _Or perhaps the_ Iliad _was evoked because when one approaches Terror Camp at Night \u2014 and all of the Sledging Parties coming from HMS_ Terror _to the Camp arrived after dark on their Third Day \u2014 one is first struck by the number of bonfires and campfires. There is no wood to burn, of course, except for some spare Oak brought from the crushed_ Erebus _precisely for that Purpose, but many of the Last Remaining sacks of Coal had been ferried across the ice from the Ships over the past month, and many of these coal Fires were burning when I first saw Terror Camp. Some were in Fire Rings made from rocks; some were in four of the tall Braziers salvaged from the Carnivale Fire._\n\n _The effect was flames and light, added to by the occasional torch and lantern._\n\n _After spending several days in Terror Camp, I have decided that the place more resembles a Pirate Encampment than any camp of Akilleus, Odysseus, Agamemnon, and the other Homeric Heroes. The Men's clothes are ragged, frayed, and many times repaired. Most are Ill or Limping or both. Their faces are Pale under sometimes Thick beards. Their eyes stare out of Sunken Sockets._\n\n _They swagger or stagger around with their Boat Knives dangling from crude belts set around their outer Slops in clanking Sheaths made from cut-down Bayonet Scabbards. It was Captain Crozier's idea, as were the Goggles improvised from Wire Mesh that the men wear on sunny days to safeguard them from Sun blindness. The overall effect is one of a ragtag group of Ruffians._\n\n _And most now show signs of Scurvy._\n\n _I have been very busy in the Sick Bay Tent. The sledge teams had spent the Extra Energy to haul a Dozen Cots with them across the ice and over the Frightful Pressure Ridges (plus two more cots for their Captains' tents), but at the moment I have 20 men in Sick Bay, so 8 are on Blanket Pallets set onto the cold ground itself. Three oil lamps provide us with Illumination during the long nights._\n\n _Most of the men sleeping in the Sick Bay have collapsed from Scurvy, but not all. Sergeant Heather is back in my care, complete with the gold sovereign Dr. Peddie had screwed into his skull to replace the bone Dashed Out with some of his brains by the Thing from the Ice. The Marines have been taking care of Heather for months and planned to continue to do so here at Terror Camp \u2014 the Sergeant was transported here on his Own Little Sled designed by Mr. Honey \u2014 but a possible Chill during the three days and nights of the Crossing has brought on a Pneumonia. This time, I do not expect the Marine Sergeant, who has been a disturbing Miracle of Survival, to Survive much longer._\n\n _Also here is David Leys, whom his crewmates call Davey. His catatonic condition has not changed in Months, but after the Crossing this week \u2014 he came Across with my group \u2014 he has not been able to keep down even the Thinnest Gruel or water. Today is Saturday. I do not expect Leys to be alive by this time Wednesday._\n\n _Due to the Great Exertion of hauling the boats and so much Mat\u00e9riel from the Ship to the Island \u2014 over pressure ridges I had Trouble climbing even when not in man-hauling harness \u2014 there were the usual complement of bruises and Broken Bones for me to deal with. These included one serious compound fracture of seaman Bill Shanks's arm. I have kept the man here after setting the bones for fear of sepsis. (The flesh and skin were punctured by sharp bone fragments in two places.)_\n\n _But Scurvy remains the Primary Killer lurking in this tent._\n\n _Mr. Hoar, Captain Fitzjames's Personal Steward, may well be the first Man to Die of it Here. He is no longer Conscious for much of the day. As with Leys and Heather, he had to be man-hauled across the 25 Miles separating our doomed Ship from this Terror Camp._\n\n _Edmund Hoar is an early but Typical example of the progression of this disease. The Captain's Steward is a Young Man \u2014 he will turn 27 in a little more than two weeks, on May 9. If he survives that long._\n\n _For a Steward, Hoar is a large man \u2014 six feet tall \u2014 and to all appearances to Chief Surgeon Stanley and myself, he was in fine health when the Expedition sailed. He was quick, smart, alert, energetic in his Duties, and unusually athletic for a steward. During the running and man-hauling Games held so frequently on the ice at Beechey Island in the winter of 1845\u201346, Hoar was frequently a winner and leader of his various teams._\n\n _He has had slight symptoms of the Scurvy since last autumn \u2014 the weariness, lassitude, increasingly frequent Confusion \u2014 but the disease became most Pronounced after the Debacle of the Venetian Carnivale. He continued serving Captain Fitzjames sixteen hours a day and more into February, but finally his health broke down._\n\n _The first Symptom to make itself known with Mr. Hoar is what the men in the fo'c'sle are calling the Crown of Thorns._\n\n _Blood began weeping from Edmund Hoar's hair. And not just from the hair on his head. First his Caps and then his Undershirts and then his Underthings became stained with Blood each day._\n\n _I have observed this carefully, and the blood on the Scalp does come from the follicles themselves. Some of the Seamen attempted to avoid this Early Symptom by shaving their heads, but of course that does no good. With Welsh wigs, caps, scarves, and now pillows being soaked with blood by the Majority of the men, the sailors and officers have begun wearing Towels under their headgear and laying their head on them at night._\n\n _This does not, of course, Alleviate the Embarrassment and Discomfort of bleeding from all Points that have body hair._\n\n _Hemorrhages began appearing under Steward Hoar's skin in January. Although the Outside Games were a distant Memory by then and Mr. Hoar's duties rarely took him far from the Ship or into Great Physical Exertion, the slightest Bump or Bruise would show on his Body as a massive red-and-blue blotch. It would not heal. A Scratch from peeling potatoes or carving Beef would stay open and continue to bleed for weeks._\n\n _By late January, Mr. Hoar's legs had swollen to Twice their Normal Size. He had to borrow filthy Trousers from larger crewmen just to stay dressed while serving his captain. He could not sleep because of the ever-increasing Pain in his Joints. By early March, any movement at all was Agony for Edmund Hoar._\n\n _All through March, Hoar insisted that he could not stay in_ Erebus _'s Sick Bay \u2014 that he had to return to his berth and to serving and caring for Captain Fitzjames. His blond hair was constantly soaked with caked blood. His swollen arms, legs, and face began to look like pasty Dough. Every day that I tested his skin, it had Lost more Elasticity; by the week before_ Erebus _was crushed, I could push deep into Edmund Hoar's flesh and the dimple would stay there permanently, the new Bruise spreading and spilling into a patchwork of earlier Hemorrhages._\n\n _By mid-April, Hoar's entire body had become a Bloated, Misshapened mass. His face and hands were Yellow from jaundice. His eyes were a Bright Yellow, made all the more shocking due to the bleeding from his eyebrows._\n\n _Despite my assistant's and my own efforts to turn and move the patient several times a Day, by the day we carried him from the dying_ Erebus, _Hoar was covered with bedsores that had become brownish-purple ulcers that never ceased Suppurating. His face, especially on either side of his Nose and Mouth, was also ulcerated, constantly oozing Pus and Blood._\n\n _Pus from a Scurvy victim has an extraordinarily foul stench._\n\n _By the day we moved Mr. Hoar to Terror Camp, he had lost all but two of his teeth. And this was a man who \u2014 on Christmas Day \u2014 had boasted the healthiest smile of any young man on the Expedition._\n\n _Hoar's gums have blackened and receded. He is conscious only a few hours each day and is in Terrible Pain during every second of that time. When we open his mouth to feed him, the Stench is close to unbearable. Since we cannot wash Towels, we have lined his Cot with sailcloth which is now Black with Blood. His frozen and filthy clothing is also Brittle with dried Blood and Crusted Pus._\n\n _As terrible as his Appearance and Suffering are, the more Terrible Fact is that Edmund Hoar may linger on like this \u2014 getting Worse each Day \u2014 for more Weeks or even Months. Scurvy is an Insidious killer. It Tortures for a long time before it grants its victim a final peace. By the time one dies of Scurvy, one's closest Relative often will not be able to recognize the Sufferer and not enough of the Sufferer's mind will be left to recognize the relative._\n\n _But that is not a problem here. With the Exception of brothers serving together on this Expedition \u2014 and Thomas Hartnell lost his older brother on Beechey Island \u2014 there are no relatives who will ever come out here on the ice or onto this Terrible Island of wind, snow, ice, lightning, and fog. There is no one to identify us when we fall, much less Bury us._\n\n _Twelve of the men in the Sick Bay are dying of Scurvy, and more than Two thirds of the 105 survivors, including myself, have one or more symptoms of it._\n\n _We will be out of the lemon juice \u2014 our most successful antiscorbutic, although its Efficacy has been Declining steadily the past year \u2014 in less than a week. The only Defense I will have then is Vinegar. A week ago \u2014 in the Stores Tent on the ice outside HMS_ Terror _\u2014 I personally presided over the decanting of our remaining volumes of Vinegar from casks to be proportioned out into 18 Smaller Kegs \u2014 one for each boat that had been sledged to Terror Camp._\n\n _The men hate Vinegar. Unlike the lemon juice, whose Tartness can be somewhat disguised with dollops of Sugar Water or even Rum, Vinegar tastes like poison to men whose palates have already been damaged by the Scurvy growing in their systems._\n\n _Officers who have dined more on Goldner's Canned Foods than the seamen have \u2014 they ate their beloved (although rancid) Salted Pork and Beef until those casks were empty \u2014 appear to be more prone to coming down with the advanced symptoms of Scurvy than the regular sailors._\n\n _This confirms Dr. McDonald's theory that there is some vital Element lacking \u2014 or some Poison present \u2014 in the purely canned meats and vegetables and soups as opposed to spoiled but once-fresh victuals. If there was some miraculous way I could discover that Element \u2014 poison or life-saver \u2014 I would not only have a good Chance of saving these men, possibly even Mr. Hoar, but would run an excellent Chance of being Knighted when we are rescued or reach safe harbour by ourselves._\n\n _But there is no way to do it, given our current Conditions and my lack of any Scientific Apparatus. The best I can do is insist that the men eat any fresh meat that our hunters shoot and bring in \u2014 even the Blubber and sweetmeats, I feel, against all logic, may strengthen us against Scurvy._\n\n _But our hunters have found no living things to shoot. And the ice is too thick to chop through for fishing._\n\n _Last night Captain Fitzjames stopped by as he does at the beginning and end of each of his long, long Days, and after he had his usual Rounds of the sleeping men, asking me the Changes in Condition of each, I was Forward enough to ask him the question I had been wondering about for so many weeks now._\n\nCaptain, _I said,_ I understand if you are too busy to answer this or if you prefer not to, since it is a Lubber's question, there is no doubt of it, but I've been wondering for some time \u2014 why 18 Boats?\n\nWe seem to have brought Every Boat from _Erebus_ and _Terror,_ yet we have only 105 men.\n\n _Captain Fitzjames said,_ Come outside with me if you will, Dr. Goodsir.\n\n _I told Henry Lloyd, my Weary Assistant, to watch over the men, and followed Captain Fitzjames outside. I had noticed in the Sick Bay Tent that his Beard, which I had thought was coming in Red, was actually mostly Grey, only rimmed in dried Blood._\n\n _The captain had brought an extra Lantern from the Sick Bay and he led the way with it down to the graveled Beach._\n\n _There was no Wine-Dark Sea lapping at the Shingle of this Beach, of course. Instead, the heap of coastal Tall Bergs that formed a Barrier between us and the Ice Pack still lined the Shore._\n\n _Captain Fitzjames raised his lantern along the long line of boats._ What do you see, Doctor? _he asked._\n\nBoats, _I ventured, feeling every Inch the Lubber I had accused myself of being._\n\nCan you tell the difference between them, Dr. Goodsir?\n\n _I looked more carefully in the lantern light._\n\nThese first four are not on Sledges, _I said. I had been quick to notice that even the first night I was here. I had no idea why this was the case, when Mr. Honey had gone to such Care to make special Sledges for all the Rest. It seemed like Rank Carelessness to me._\n\nAye, you are correct, _said Captain Fitzjames._ These Four are our Whaleboats from _Erebus_ and _Terror._ Thirty feet long. Lighter than the Others. Very strong. Six oars each. Double-ended like canoes... d'ye see now?\n\n _I did now. I had never noticed that the whaleboats seemed to have two bows, like a canoe._\n\nIf we had ten whaleboats, _continued the Captain,_ everything would have been perfect.\n\nWhy is that? _I asked._\n\nThey're strong, Doctor. Very strong. And light, as I said. And we could pile Supplies in them and drag them across the Ice without having had the need to build Sledges as we did for the Others. If we find Open Water, we could launch them straight from the ice.\n\n _I shook my head. Knowing that Captain Fitzjames would think me a Total Fool as soon as I asked the question \u2014 I asked it anyway:_ But why can the whaleboats be dragged on the ice when the others cannot, Captain?\n\n _Captain Fitzjames's voice showed no impatience when he said,_ Do you see the rudder, Doctor?\n\n _I looked at each end, but I did not. I confessed that to the captain._\n\nExactly, _he said._ Whaleboats have a Shallow Keel and no fixed Rudder. An oarsman at the stern steers her.\n\nIs that good? _I asked._\n\nIt is if you want a light, tough boat with a shallow Keel and no tender Rudder to be broken off when you're dragging her, _said Captain Fitzjames._ Perfect for man-hauling across the ice, even though she's 30 Feet long and can carry up to a Dozen Men with room for Supplies.\n\n _I nodded as if I understood. I almost did \u2014 but I was very tired._\n\nDo you see her mast, Doctor?\n\n _Again I looked. Again I failed to find that which had been requested of me to find. I admitted as much._\n\nThat's because the whaleboats have a single _collapsible_ mast, _said the Captain._ It's there folded under the Canvas the men have Rigged over her gunwales.\n\nI have noticed that canvas and wood covering all the boats, _I said, to show that I was not totally unobservant._ Is that to keep the snow out?\n\n _Fitzjames was lighting his pipe. He had run out of Tobacco long ago. I did not want to Know what he was burning in it._ The Boat Covers were put on to shield the Crews of all 18 Boats, even though we may only take 10 boats with us, _he said softly. Most of the men in camp were sleeping. Guards paced coldly just at the edge of the lantern light._\n\nWe'll be _under_ that canvas when we cross Open Water to the mouth of the Back's Great Fish River? _I asked. I had never pictured us hunkered down under Canvas and wood. I had always imagined us rowing happily in Sunlight._\n\nWe may not use the Boats on the River, _he said, puffing out aromatic clouds of what smelled like dried human excrement._ If the Waters along the Coast open up this Summer, Captain Crozier would prefer to Sail us to Safety.\n\nAll the way to Alaska and St. Petersburg? _I asked._\n\nTo Alaska at least, _said the Captain._ Or perhaps Baffin Bay if the Coastal Leads open to the North. _He took several steps and swung the lantern closer to the Boats on Sledges._ Do you know these Boats, Doctor?\n\nAre they different, Captain? _I found that such terrible Fatigue was a great Inducement to Honesty without Embarrassment._\n\nAye, _said Fitzjames._ These next two lashed to Mr. Honey's special Sledges here are our Cutters. Surely you noticed them when they were Lashed on Deck or on the ice next to the Ships these past Three Winters?\n\nYes, of course, _I said._ But are you saying these are different from the first, the whaleboats?\n\nQuite different, _said Captain Fitzjames, taking the time to relight his pipe._ Do you notice any masts on these boats, Doctor?\n\n _Even in the dim light from the lantern I could see two masts rising from each of these craft. The Canvas had been Artfully shaped and cut and Stitched around them. I told the Captain my observation._\n\nAye, very good, _he said. He did not sound Condescending._\n\nAre these collapsible masts not collapsed for a purpose? _I asked, as much to show that I had been listening earlier as for any better reason._\n\nThey're not collapsible, Dr. Goodsir. These masts are Lug Rigged... or you may know them as Gaff Rigged. Quite permanent. And do you see fixed rudders on these? And the deeper keels?\n\n _I could. I did._ The Rudders and Keels are the reason they could not be Dragged like the whaleboats? _I ventured._\n\nExactly. You have Diagnosed the Problem, Doctor.\n\nCould not the Rudders be removed, Captain?\n\nPossibly, Dr. Goodsir, but the deep Keels... they would have been Stuck or Ripped out by the first Pressure Ridge, now wouldn't they?\n\n _I nodded again and laid my mittened hand on the gunwale._ Is it my imagination, or are these four boats slightly shorter than the whaleboats?\n\nYou have a very good Eye indeed, Doctor. 28 feet long as opposed to 30 feet for the whaleboats. And heavier... the Cutters are Heavier. And square-sterned.\n\n _For the first time I noticed that these 2 Boats, unlike the whalers, definitely had a Bow and a squared-off stern. No Canoes here._ How many men will the Cutters carry? _I asked._\n\nTen. And they pull 8 Oars. They have Room for quite a few Stores, and there will be Room for us all to Huddle down out of the Storm, even on the Open Sea, and with the two masts the Cutters will offer twice the Canvas to the wind that the whaleboats do, but the Cutters will not be as good as the Whalers if we have to go up Back's Great Fish River.\n\n _Why is that? I asked, feeling that I should already know, that he had already told me._\n\nThe deeper draft, sir. Let us look at the next two... the jolly boats.\n\n _I found nothing jolly-looking about either of the next two boats._ They seem longer than the Cutters, _I said._\n\nThey are, Doctor. 30 feet Long apiece... the same Length as our whaleboats. But Heavier, Doctor, Heavier even than the Cutters. A great Trial, with their 900-lb. Sledges to haul across the Ice... even this far... I assure you. Captain Crozier may choose to leave them here.\n\n _I asked,_ Then should we not have left them behind at the ships?\n\n _He shook his head._ No. We need to choose which boats will best serve to allow 100 men to survive for several weeks or months at sea, or even on the river. Did you know that Boats... all of these Boats... have to be Rigged differently for sailing on the sea or catching the Wind going upriver, Doctor?\n\n _It was my turn to shake my head._\n\nNo matter, _said Captain Fitzjames._ We'll get into the niceties of river rigging versus sea rigging some other time, preferably on a Sunny, Warm day far South of here. These last 8 Boats... the first Two are Pinnaces, the next Four are Ship's Boats, and the final Two are Dinghies.\n\nThe Dinghies seem much shorter, _I said._\n\n _Captain Fitzjames puffed on his literally execrable pipe and nodded as if I had revealed Some Pearl of Wisdom from Holy Scripture._ Aye, _he said sadly._ The Dinghies are only 12 feet Long, as opposed to the Pinnaces' 28-foot Length and the Ship's Boats 22 feet. But none of them can be rigged for masts and sailing and they're all lightly Oared. The men in these Boats would be in for some Hard Times if we went to the Open Sea, I am afraid. I would not be surprised if Captain Crozier chooses to Leave them Behind.\n\n _I thought,_ Open Sea? _The idea of actually sailing any of these craft on anything more expansive than Back's Great Fish River, which I imagined rather like the Thames, had never occurred to me before tonight, even though I had been present at various war councils discussing such possibilities. It seemed to me, looking at the smaller and rather delicate-looking Dinghies and Ship's Boats lashed on their Sledges, that the men going to sea in these would be doomed to watch the Pinnaces with their Two Masts and the Whaleboats with their single Tall Masts simply sail away over the horizon._\n\n _The men in these Smaller Boats would be Doomed. How would the crews be chosen? Had they been chosen already, in secret, by the Two Captains?_\n\n _And which boat \u2014 and which Fate \u2014 had I been assigned?_\n\nIf we take the Smaller Boats, we'll draw lots for them, _said the Captain._ The places in the pinnaces, jolly boats, and whaleboats would be assigned according to man-hauling teams.\n\n _I must have looked at him in alarm._\n\n _Captain Fitzjames laughed \u2014 a laugh that turned into a racking cough \u2014 and knocked out the ashes from his pipe against his Boot. The wind was coming up, and it was very cold. I had no idea of the time \u2014 sometime after Midnight. It had been dark for at least seven hours._\n\nDon't worry, Doctor, _he said softly._ I wasn't reading your mind. Only your expression. As I say, we'll Draw Lots for the smaller boats, but we may not take the Smaller Boats. In either case, we won't leave anyone behind. We'll tie the ships together on the Open Water.\n\n _I smiled at this, hoping that the Captain could see my smile in the lantern light but not my Bleeding Gums._ I didn't know that ships under sail could be tied to other ships not under sail, _I said, showing my ignorance again._\n\nMost of the time, they cannot, _said Captain Fitzjames. He touched me lightly on the back \u2014 a touch I could hardly feel through my outer Slops._ Now that you've learned the Nautical Secrets of all 18 Boats that might end up in our little Fleet, Doctor, shall we get back? It's rather cold, and I have to get some Sleep before I rise at Four Bells to check on the Watch.\n\n _I bit my lip, tasting blood._ I do have one last question, Captain, if you don't mind.\n\nNot at all.\n\nWhen will Captain Crozier choose the boat we take and when will he put those boats in the water? _I said. My voice was very hoarse._\n\n _The Captain moved slightly and was silhouetted against the light from the bonfire near the Seamen's Mess Tent. I could not see his face._\n\nI don't know, Dr. Goodsir, _he said at last._ I doubt if Captain Crozier could tell you. Lady Luck may be with us and the Ice may break up in a few Weeks... if it does, I'll sail you to Baffin Island myself. Or we could be launching some of these craft against the current at the Mouth of the Great Fish River in three months... conceivably there could still be time to get to Great Slave Lake and the outpost there before Winter fully sets in, even if it takes until July to reach the River.\n\n _He patted the curved side of the Pinnace closest to him. I felt a strange, quiet pride in being able to identify it as a Pinnace._\n\n _Or perhaps it was one of the 2 Jolly Boats._\n\n _I tried not to think of the condition of Edmund Hoar and what it forecast for all the rest of us if we did not_ begin _the 850-mile Venture up Back's River... the river they also call the Great Fish River... for another Three Months. Who could possibly be left Alive if a boat made it to Great Slave Lake months later than that?_\n\nOr, _he said softly,_ if Lady Luck is not with us, these hulls and keels may never feel water under them again.\n\n _There was nothing to say to that. It was our Death Sentence. I turned from the light to walk back to the Sick Bay Tent. I respected Captain James Fitzjames and I did not want him to see my face at that Moment._\n\n _Captain Fitzjames's hand fell on my shoulder, stopping me._\n\nShould that be the case, _he said, his voice fierce,_ we'll just have to bloody well _walk_ home, shan't we?\n\nCROZIER\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _22 April, 1848_\n\nPulling toward the arctic sunset, Captain Crozier knew the mathematics of this purgatory. Eight miles this first day on the ice to Sea Camp One. Nine miles the next, if all went well, ending in a midnight arrival at Sea Camp Two. Eight miles \u2014 including some of the hardest going near the coast where the sledges had to be hauled up over the barrier where pack ice met coastal ice \u2014 the third and final day. And there the tentative safe haven of Terror Camp.\n\nBoth crews would be together for the first time. If Crozier's sledge teams survived this ice crossing \u2014 and kept ahead of the thing following them on the ice \u2014 all 105 men would be together on the wind-scoured northwestern coast of the island.\n\nThe early sledge trips to King William Land in March \u2014 most of them in darkness \u2014 had made such slow going that often the men with their sledges had camped the first night on the ice within sight of the ship. One day, with a storm blowing in their faces out of the southeast, Lieutenant Le Vesconte had made less than a mile after twelve hours of constant effort.\n\nBut it was much easier in the sunlight with the sledge trail laid down and the path through pressure ridges reduced in difficulty, if not actually leveled.\n\nCrozier had not wanted to end up on King William Land. His visits to Victory Point had not convinced him, despite the huge dump of food and gear there and the preparation of tent circles, that the men could survive there for long. The weather blowing almost always out of the northwest was murderous in winter, atrocious in the spring and brief autumn, and life threatening during the summer. The late Lieutenant Gore's experience of wild lightning storms during the first visit to the landmass in the summer of 1847 had been repeated again and again that summer and early autumn. One of the first things Crozier authorized hauling to land the previous summer had been the ship's extra lightning rods along with brass curtain rods from Sir John's quarters to jury-rig more.\n\nRight up until the crushing of _Erebus_ on the last day of March, Crozier had hopes that they could set off for the east coast of the Boothia Peninsula, the possible stores there at Fury Beach, and the probable sighting by whalers coming in from Baffin Bay. Like old John Ross, they could hike or boat north along the east coast of Boothia up to Somerset Island or even Devon Island again if they had to. Sooner or later they would spy a ship in Lancaster Sound.\n\nAnd there were Esquimaux villages in that direction. Crozier knew this for a fact \u2014 he'd seen them on his first voyage to the arctic with William Edward Parry in 1819 when he was twenty-two. He'd returned to the area again with Parry two years later in a quest to find the Passage and again two years after that, still searching for the North-West Passage \u2014 a search that would kill Sir John Franklin twenty-six years later.\n\n _And might yet kill us all,_ thought Crozier and shook his head to get the defeatist thought out of it.\n\nThe sun was very close to the southern horizon. Just before it set, they would stop and eat a cold dinner. Then they would harness up again and walk another six to eight hours through the deep afternoon, evening, and nighttime darkness to reach Sea Camp One a little more than a third of the way to King William Land and Terror Camp.\n\nThere was no sound now except for the panting of the men, the creak of leather, and the rasp of runners. The wind had died completely but the air was even colder with the dimming of the twilight afternoon sun. Ice crystals of breath hung above the procession of men and sledges like slowly collapsing spheres of gold.\n\nWalking near the front of the line now as they approached the tall pressure ridge, ready to help with the initial pulling and lifting and shoving and soft cursing, Crozier looked toward the setting sun and thought of how hard he had tried to find a way to Boothia and the whalers from Baffin Bay.\n\nAt age 31 Crozier had accompanied Captain Parry into those arctic waters a fourth and final time, this time to reach the North Pole. They'd accomplished a \"farthest north\" record that easily stood until this day but had eventually been stopped by solid pack ice that stretched to the northern limits of the world. Francis Crozier no longer believed in the Open Polar Sea: when someone finally reached the Pole, he was sure they'd be doing it by sledge.\n\nPerhaps by sleds pulled by dogs, the way the Esquimaux preferred to travel.\n\nCrozier had seen the natives and their light sleds \u2014 not real sledges at all by Royal Navy standards, but only flimsy little sleds \u2014 sliding along behind those strange dogs of theirs in Greenland and along the east side of Somerset Island. They moved much faster than Crozier's team ever could with this man-hauling. But most central to his plan to head east if at all possible was the fact that the Esquimaux were out there to the east somewhere at Boothia or beyond. And, like Lady Silence, whom they had seen going ahead to Terror Camp following Lieutenants Hodgson's and Irving's sledge teams earlier that week, these natives knew how to hunt and fish for themselves in this godforsaken white world.\n\nAfter Irving reported to him way back in early February about the young lieutenant's difficulties in following Lady Silence or communicating with her about where and how she got the seal meat and fish Irving swore he had seen her with, Crozier contemplated threatening the girl's life with pistol or boat knife to make her show them how she found the fresh food. But in his heart he'd known how such a threat would end up \u2014 the Esquimaux wench's tongueless mouth would stay firmly shut and her huge dark eyes would stare unblinkingly at Crozier and his men until he had to back down or make good on his threat. Nothing would be accomplished.\n\nSo he'd left her out in her little snow-house Irving had described to him and allowed Mr. Diggle to give her the occasional biscuit or scrap. The captain had tried to put her out of his mind. That he had been shocked to be reminded she was still alive when the lookout reported her following a few hundred yards behind Hodgson's and Irving's relay trip to Terror Camp last week showed Crozier that he had succeeded in not thinking about the wench. But he knew he still dreamed about her.\n\nIf Crozier were not so very, very tired, he might have taken some small pride in the design and durability of the various sledges that the men were now man-hauling southeast across the ice.\n\nIn mid-March, even before it was certain that _Erebus_ would break up from the rising pressure, he had Mr. Honey, the expedition's surviving carpenter, and his mates, Wilson and Watson, working day and night to design and build sledges that could haul the ships' boats as well as gear.\n\nAs soon as the first prototype larger oak-and-brass sledges were finished that spring, Crozier had the men out on the ice testing them and learning the best ways to haul them. He had the riggers and quartermasters and even the foretopmen constantly fiddling with the design of the harnesses to give the men the best pulling leverage with the least interruption of their movement and breathing. By mid-March, the sledge designs were set, more were being built, and it seemed that a design of harnesses for eleven men for the large sledges carrying boats and seven men for the smaller supply sledges would be best.\n\nThis was for the initial supply crossings to Terror Camp on King William Land. If they took to the ice after that, Crozier knew, with some of the men too sick to pull and perhaps others dead by then, eighteen boats and sledges, each loaded to the gunwales with survival rations and gear, requiring man-hauling by one hundred men \u2014 or fewer \u2014 it would mean fewer than eleven men pulling each burden. More work and even heavier loads for men who presumably would be deeper in the pit of scurvy and exhaustion by then.\n\nBy the last week in March, even as _Erebus_ was in her death throes, both crews were out on the ice in darkness and the brief sunlight, competing in man-hauling contests with the different sledges, finding the right match of men to sledge, learning the right techniques, and putting together the best teams composed of men from both ships and all ranks. They were competing for cash \u2014 silver and gold \u2014 and even though Sir John had planned to buy many souvenirs in Alaska, Russia, the Orient, and the Sandwich Islands and there were chests of shillings and guineas in the dead man's private storeroom, these coins came out of Francis Crozier's pocket.\n\nCrozier wanted badly to head toward Baffin Bay as soon as the days grew long enough to support long-distance sledging. He knew instinctively and from listening to Sir John's tales and from reading George Back's history of ascending more than 650 miles of the Back's Great Fish River to Great Slave Lake fourteen years ago \u2014 the volume was in _Terror_ 's library and now in Crozier's personal pack on one of the sledges \u2014 that the odds of any of them finishing or surviving the trip were low.\n\nThe 160-some miles between _Terror_ 's position off King William Land and the mouth of Great Fish River might not be traversable, even as a prelude to the arduous voyage up the river. It combined the worst of coastal ice with threats of open leads that could make them abandon the sledges and \u2014 even if there were no leads \u2014 the assured agony of hauling sledges and boats across the frozen gravel of the island itself, all while exposed to the worst of the pack-ice storms.\n\nOnce on the river, if they ever reached the river, they would be confronted with what Back had described as \"a violent and tortuous course of 530 geographical miles, running through an iron-ribbed country without a single tree on the whole of its banks\" and then \"no fewer than 83 falls, cascades, and rapids.\" Crozier had trouble imagining his men, after another month or more of man-hauling, being fit enough or well enough to confront 83 falls, cascades, and rapids in even the sturdiest boats. The portages alone would kill them.\n\nA week earlier, before heading out with the boat-sledge teams to Terror Camp, Surgeon Goodsir told Crozier that the lemon-juice antiscorbutic, their only defense against scurvy now \u2014 as weak as it had become \u2014 would run out in three weeks or less, depending upon how many men died between now and then.\n\nCrozier knew how quickly the full onslaught of scurvy would weaken all of them. For this 25 miles to King William Land with light sledges and full teams, on full half rations for the crossing, on a runners path that had been beaten into the ice for more than a month, they had to cover a little more than eight miles a day. On the rough terrain or coastal ice of King William Land and south, that distance might be cut in half or worse. Once the scurvy began having its way with them, they might only cover a mile a day and, if the wind died, might well not be able to pole or paddle the heavy boats upstream against the current of Back's River. A portage of any distance in the weeks or months to come might soon become impossible.\n\nThe only things working in their favor in heading south were the very long-shot chance that a rescue party was already heading north from Great Slave Lake, searching for them, and the simple fact that it would be getting warmer as they traveled farther south. They would be following the thaw, at the very least.\n\nStill, Crozier would have preferred staying in the northern latitudes and going the longer distance east and north to Boothia Peninsula and then across it. He knew there was only one even relatively safe way to attempt that: take the men to King William Land, cross it, then make the relatively short traverse across open ice, sheltered from the worst of the northwest wind and weather by the island itself, to the southwestern coast of the Boothia, then slowly north along the edge of the ice or on the coastal plain itself, and finally across the mountains toward Fury Bay, hoping every step of the way to meet Esquimaux.\n\nIt was the safe way. But it was the long way. 1,200 miles, almost half again longer than the alternate route south around King William Land and then further south up Back's River.\n\nUnless they met friendly Esquimaux soon after crossing to the Boothia, they would all be dead weeks or months before such a twelve-hundred-mile trip could be completed.\n\nEven so, Francis Crozier would have preferred to have wagered everything on a dash straight across the ice \u2014 northeast over the worst of the pack ice in a mad attempt to replicate the astounding 600-mile small-group sledge trip made by his friend James Clark Ross eighteen years earlier when the _Fury_ was frozen in on the opposite side of the Boothia Peninsula. The old steward \u2014 Bridgens \u2014 had been absolutely correct. John Ross had taken the best bet on survival, forcing his way north by foot and sledge and then in boats left behind up to Lancaster Sound and waiting for whalers. And his nephew James Ross showed that it was possible \u2014 just possible \u2014 to sledge from King William Land back to Fury Beach.\n\n _Erebus_ was still in its final ten days of agony when Crozier had detached the best man-haulers from each ship \u2014 the winners of the biggest prizes and the last money Francis Crozier had in the world \u2014 given them the best-designed sledge, and ordered Mr. Helpman and Mr. Osmer, the purser, to fit this superteam of man-haulers out with everything they might need for six weeks on the ice.\n\nIt was an eleven-man sledge headed by _Erebus_ second mate Charles Frederick Des Vouex, its lead puller the giant Manson. Each of the other nine men were asked to volunteer. Each man did.\n\nCrozier had to know if it was possible to man-haul a fully loaded boat sledge across the open ice in such a straight dash toward rescue. The eleven men left at six bells on 23 March, in the dark, with the temperature at thirty-eight below zero, to three rousing cheers from every ambulatory crewman assembled from both ships.\n\nDes Voeux and his men were back in three weeks. No one had died, but they were all exhausted and four of the men had serious frostbite. Magnus Manson was the only one of the eleven-man team, including the apparently indefatigable Des Voeux, who did not seem close to death from exhaustion and hardship.\n\nIn three weeks they had been able to travel less than twenty-eight miles in a straight line from _Terror_ and _Erebus._ Des Voeux later estimated that they had man-hauled more than a hundred and fifty miles to gain those twenty-eight, but there was no possibility of traveling in a straight line that far out on the pack ice. The weather northeast of their current position was more terrible than the Ninth Circle of Hell where they had been trapped for two years. Pressure ridges were Legion. Some rose to heights greater than eighty feet. Even steering their course was close to impossible when clouds hid the southerly sun and the stars were hidden for several eighteen-hour nights on end. Compasses, of course, were useless this close to the north magnetic pole.\n\nThe team had brought five tents for safety's sake, although they had planned to sleep in only two of them. The nights were so cold out on the exposed ice that the eleven men slept the last nine nights, when they were able to sleep at all, in a single tent. But in the end they'd had no choice in the matter, since four of the sturdy tents had blown away or been ripped to shreds by the twelfth night on the ice.\n\nSomehow Des Voeux had kept them moving to the northeast, but every day the weather worsened, the pressure ridges grew closer together, the necessary deviations from their course became longer and more treacherous, and the sledge sustained serious damage in their Herculean struggle to haul and shove it over the jagged ice ridges. Two days were lost just repairing the sledge in the howl of wind and blowing snow.\n\nThe mate had decided to turn around on their fourteenth morning on the ice. With only one tent left, he gauged their chances of survival as low. They then tried to follow their own thirteen days of ruts back to the ships, but the ice was too active \u2014 shifting slabs, moving bergs within the pack ice, and new pressure ridges rising in front of them had obliterated their track. Des Voeux, the finest navigator on the Franklin Expedition except for Crozier, took theodolite and sextant readings in the few clear moments he found in the days and nights but ended up setting his course based mostly on dead reckoning. He told the men that he knew precisely where they were. He was sure, he later admitted to Fitzjames and Crozier, that he would miss the ships by twenty miles.\n\nOn their last night on the ice, the final tent ripped and they abandoned their sleeping bags and pressed on to the southwest blindly, man-hauling just to stay alive. They jettisoned their extra food and clothing, continuing to man-haul the sledge only because they needed their water, shotguns, cartridges, and powder. Something large had been following them for their entire voyage. They could see it through the spindrift and fog and pelting hail. They could hear it circling them each endless night in the darkness.\n\nDes Voeux and his men were sighted on the northern horizon, still headed due west and oblivious of _Terror_ three miles south of them, on the morning of their twenty-first day on the ice. An _Erebus_ lookout had spotted them, but the ship itself was gone by then \u2014 crushed and splintered and sunk. It was Des Voeux's and his men's good fortune that the lookout, Ice Master James Reid, had climbed the huge iceberg that had been part of the Grand Venetian Carnivale before dawn that day and spotted the men through his glass just at first light.\n\nReid, Lieutenant Le Vesconte, Surgeon Goodsir, and Harry Peglar led the party that went out to fetch the Des Voeux team, bringing them back past the crushed timbers, tumbled masts, and tangled rigging that was all that remained of the sunken ship. Five of the Des Voeux champion team were not able to walk the last mile to _Terror_ but had to be sledged there by their mates. The six Erebuses among the superteam of sledgers, including Des Voeux, wept at the sight of their destroyed home as they were led past it.\n\nSo... the short way northeast to Boothia was no longer an option. After debriefing Des Voeux and the other shattered men, both Fitzjames and Crozier agreed that a few of the 105 survivors might make it to Boothia, but the vast majority were certain to perish on the ice under such conditions, even with the longer days, slightly rising temperatures, and added sunlight. The possibility of open leads would only add to the hazard.\n\nThe choices now were either stay on the ships or set up a camp on King William Land with the option of making a run south to Back's River.\n\nCrozier began the planning for the evacuation the next day.\n\nJust before sunset and their dinner stop, the procession of sledges came across a hole in the ice. They stopped, the five sledges and harnessed men making a ring around the pit. The black circle far below them was the first open water the men had seen in twenty months.\n\n\"This weren't here last week when we brung the pinnaces to Terror Camp, Captain,\" said Seaman Thomas Tadman. \"You can see how close them runner tracks come to it. We would've seen it, sure like. There was nothing here.\"\n\nCrozier nodded. This was no ordinary _polynya_ \u2014 the Russian word for one of those rare holes in pack ice that remained open all year long. The ice was more than ten feet thick here \u2014 less thick than the congealed pack ice around _Terror_ but still solid enough to erect a London building on \u2014 but there was no sign of pressure slabs or cracking around this hole. It was as if someone or something had taken a gigantic ice saw of the sort both ships packed and cut a perfectly round hole through the ice.\n\nBut the ships' ice saws would not cut so cleanly through ten feet of ice.\n\n\"We could take our dinner here,\" said Thomas Blanky. \"Enjoy our victuals by the seaside, as it were.\"\n\nThe men shook their heads. Crozier agreed \u2014 he wondered if the others felt the same unease he did about the uncannily perfect circle, deep pit, and black water. \"We'll keep moving for another hour or so,\" he said. \"Lieutenant Little, have your sledge take the lead, please.\"\n\nIt was perhaps twenty minutes later, the sun had set with an almost tropical suddenness and stars were shaking and twitching in the cold sky, when Privates Hopcraft and Pilkington, who had been serving as rear guard, crunched up to Crozier where he was walking beside the rearmost sledge. Hopcraft whispered, \"Captain, there's something following us.\"\n\nCrozier pulled his brass telescope from the top-lashed box on the sled and stood stationary on the ice with the two men for a minute while the sledges rasped their way past them into the gathering gloom.\n\n\"There, sir,\" said Pilkington, pointing with his good arm. \"Maybe it come up out of that hole in the ice, Captain. Do you think it did? Bobby and I think it probably did. Maybe it was just down there in the black water under the ice waiting for us to pass and then come up for us. Or hoping for us to tarry there. Do you think, sir?\"\n\nCrozier did not answer. He could see it through the glass, just visible in the failing light. It looked white but only because it was briefly silhouetted against storm clouds building in the night-black sky to the northwest. As the thing passed seracs and ice boulders the sledge procession had grunted its way past only twenty minutes earlier, it was easier to get a sense of its enormous size. At the shoulder, even when it was moving on all fours as it was now, it was taller than Magnus Manson. It moved lithely for something so massive \u2014 the movement looked to be more foxlike than bear-heavy. As Crozier struggled to steady the glass in the rising wind, he saw the thing rise up and begin walking on two legs. It moved a little less rapidly that way, but still more quickly than men attached to 2,000-lb. sledges. It now towered over seracs whose tops Crozier could not have reached with his fully raised arm and extended telescope.\n\nThen it was dark and he could no longer make it out against the background of pressure ridges and seracs. He led the Marines back to the sledge procession and set his glass into the storage box as the men ahead leaned steeply into their harnesses and grunted and panted and pulled.\n\n\"Stay close to the sledges but keep looking rearward and keep your weapons primed,\" he said softly to Pilkington and Hopcraft. \"No lanterns. You'll need whatever vision you have in the dark.\" The bulky shapes of the Marines nodded and moved rearward. Crozier noted that the guards ahead of the first sledge had lit their lanterns. He could no longer see the men, just the ice-crystal-haloed circles of light.\n\nThe captain called Thomas Blanky over. The man's peg leg and wooden foot exempted him from man-hauling even though the foot had been thoughtfully studded with nails and cleats for the ice. The half-leg simply didn't give Blanky the leverage and pulling power he needed. But the men knew that the ice master might soon figuratively, if not literally, pull his weight and more; knowledge of ice conditions would be crucial if they encountered leads and had to launch their boats from Terror Camp in the coming weeks or months.\n\nNow Crozier used Blanky as a messenger. \"Mr. Blanky, would you be so good as to go forward and pass the word to the men not hauling that we will not be stopping for supper? They should retrieve the cold beef and biscuits from the appropriate sledge boxes and pass them out to the Marines and men in harness along with the word that everyone should eat on the march and drink from the water bottles they carry under their outer clothing. And also please ask our guards to make sure that their weapons are ready. They might wish to remove their outer mittens.\"\n\n\"Yes, Captain,\" said Blanky and disappeared ahead into the gloom. Crozier could hear the crunch of his hobnailed wooden foot.\n\nThe captain knew that within ten minutes, every man on the march would understand that the thing on the ice was following them and closing the gap.\n\nIRVING\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 40\u2032 58\u2033 W._\n\n _24 April, 1848_\n\nExcept for the fact that John Irving was sick and half-starving and his gums were bleeding and he feared that two of his side teeth were loose and he was so tired that he was afraid he would collapse in his tracks at any moment, this was one of the happiest days of his life.\n\nAll this day and the previous day, he and George Henry Hodgson, old friends from the gunnery training ship _Excellent_ before this expedition, had been in charge of teams of men doing some hunting and honest-to-God exploring. For the first time in this accursed expedition's three years of sitting around and freezing, Third Lieutenant John Irving was a true _explorer._\n\nIt was true that the island he was exploring eastward across, the same King William Land to which he'd come with Lieutenant Graham Gore a little more than eleven months ago, wasn't worth a drop of Chinaman's piss what with it being all frozen gravel and low hills, none rising more than twenty or so feet above sea level, inhabited only by howling winds and pockets of deep snow and then more frozen gravel, but Irving was _exploring._ Already this morning he had seen things that no other white man \u2014 and perhaps no other human being on the planet \u2014 had ever seen. Of course, it was just more low hills of frozen gravel and more windswept pockets of ice and snow, not so much as an arctic fox track or a mummified ring seal to be found, but it was _his_ discovery: Sir James Ross had sledged around the northern coast to reach Victory Point two decades or so ago, but it was John Irving \u2014 originally from Bristol and then the young master of London Town \u2014 who was the first explorer of the interior of King William Land.\n\nIrving had half a mind to name the interior _Irving Land._ Why not? The point not far from Terror Camp was named after Sir John's wife, Lady Jane Franklin, and what had _she_ ever done to deserve the honour except marry an old, fat, bald man?\n\nThe various man-hauling teams were beginning to think of themselves as distinct groups. So yesterday, Irving led this same band of six men on a hunting party while George Hodgson took his men out to reconnoiter the island, as per Captain Crozier's instructions. Irving's hunters had found not so much as an animal track in the snow.\n\nThe lieutenant had to admit that since all of his men had been armed with shotguns or muskets yesterday (Irving himself had carried only a pistol in his coat pocket, as he was doing today), there had been moments when he had felt some concern about the caulker's mate, Hickey, being behind him and carrying a gun. But, of course, nothing had happened. With Magnus Manson more than twenty-five miles away at the ship, Hickey was not only polite but actually deferential to Irving, Hodgson, and the other officers.\n\nIt reminded John Irving of how their tutor used to separate his brothers and him during classes at their Bristol home when the boys had become too rowdy during the long, dreary days of lessons. The tutor would actually set the boys in separate rooms in the old manor and conduct their lessons separately for hours, moving from one part of the second floor of the old wing to the next, his high-heeled buckled shoes echoing on the ancient oak floors. John and his brothers, David and William, such a handful around Mr. Candrieau when the three were together, became almost timid in front of the pale-faced, knobby-kneed beanpole of a white-wigged tutor when alone with him. Originally very reluctant to approach Captain Crozier about leaving Manson behind, Irving was now glad he had spoken up. He was even gladder that the captain had not pressed him for a reason; Irving had never told the captain about what he had seen going on between the caulker's mate and big seaman that night on the hold deck and never would.\n\nBut today there was no tension about Hickey or anything else. The only member of the scouting party to carry a weapon, other than Irving himself with his pistol, was Edwin Lawrence, who was armed with a musket. Shooting practice near the line of sledge-mounted boats at Terror Camp had shown that Lawrence was the only man in this group who could shoot a musket worth a damn, so he was their guard and protector today. The rest carried only canvas packs slung over their shoulders, jury-rigged bags hanging from one strap. Reuben Male, the captain of the fo'c'sle and an inventive type, had worked with Old Murray the sailmaker to make up these packs for all the men, so naturally the seamen called them Male Bags. In the Male Bags they kept their lead or pewter water bottles, some biscuits and dried pork, a tin of Goldner's canned goods for emergency rations, some extra layers of clothing, the wire goggles that Crozier had ordered made up to protect them from sun blindness, extra powder and shot for when they were hunting, and their blanket sleeping bags just in case something should prevent them from returning to camp and they had to bivouac that night.\n\nThis morning they had hiked inland for more than five hours. The group stayed on the slight gravel rises when they could; the wind was stronger and colder there, but the walking was easier than in the snow- and ice-filled swales. They had seen nothing that might enhance everyone's chance of survival \u2014 not even green lichen or orange moss growing on rock. Irving knew from reading books in _Terror_ 's Great Cabin library \u2014 including two books by Sir John Franklin himself \u2014 that hungry men could make a sort of soup from the scrapings of moss and lichen. _Very_ hungry men.\n\nWhen his reconnoitering team had stopped for their cold dinner and water and some much-needed rest while huddled down out of the wind, Irving had handed over temporary command to Captain of the Maintop Thomas Farr and gone on for a while by himself. He told himself that the men were exhausted by their extraordinary sledge pulls of the past few weeks and needed the rest, but the truth was, he needed the solitude.\n\nIrving had told Farr that he would be back in an hour and that to make sure he did not get lost he would frequently dip down across snowy patches out of the wind, leaving his boot tracks for himself to follow back or for the others to use to find him if he was late returning. As he walked farther east, blissfully alone, he had chewed on a hard biscuit, feeling how loose his two teeth were. When he pulled the biscuit away from his mouth, there was blood on it.\n\nAs hungry as he was, Irving had little appetite these days.\n\nHe waded up through another snow field onto frozen gravel and trudged up the rise to yet another windswept low ridge, then stopped suddenly.\n\nBlack specks were moving in the broad snow-swept valley ahead of him.\n\nIrving used his teeth to tug off his mittens and fumbled in his Male Bag for his prized possession, the beautiful brass telescope his uncle had given him upon entering the Navy. The brass eyepiece would freeze to his cheek and brow if he allowed it to touch, so it was harder getting a steady image while holding it away from his face, even holding the long glass in both hands. His arms and hands were shaking.\n\nWhat he had thought to be a small herd of woolly animals turned out to be human beings.\n\n _Hodgson's hunting party._\n\nNo. These forms were dressed in heavy fur parkas of the sort Lady Silence wore. And there were ten figures laboriously crossing the snowy valley, walking close together but not in a single-file line; George only had six men with him. And Hodgson had taken his hunting party south along the coast today, not inland.\n\nThis group had a small sledge with them. Hodgson's hunting party had no sledge with them. There was not a sledge this small at Terror Camp.\n\nIrving fiddled with the focus of his beloved telescope and held his breath to keep the instrument from shaking.\n\n _This sledge was being pulled by a team of at least six dogs._\n\nThese were either white rescuers wearing Esquimaux garb or actual Esquimaux.\n\nIrving had to lower the telescope and then go to one knee on the cold gravel and lower his head for a moment. The horizon seemed to be spinning. The physical weakness he'd been holding back for weeks through sheer force of will welled up through him like concentric circles of nausea.\n\n _This changes everything,_ he thought.\n\nThe figures below \u2014 they still did not appear to have seen him, probably because he had crossed over the rise and would not be very visible here with his dark coat blending into the dark rock \u2014 could be hunters out from some unknown farther-north Esquimaux village that was not far away. If so, the 105 survivors of _Erebus_ and _Terror_ were almost certainly saved. The natives would either feed them or show them how to feed themselves up here in this lifeless land.\n\nOr there was a chance that the Esquimaux were a war party and that the crude spears Irving had caught a glimpse of in the glass were meant for the white men they'd somehow heard had invaded their lands.\n\nEither way, Third Lieutenant John Irving knew that it was his job to go down, encounter them, and find out.\n\nHe closed the telescope, set it carefully amid extra sweaters in his shoulder bag, and \u2014 throwing one arm high in what he hoped the savages would see as a gesture of greeting and peace \u2014 started down the long hill toward the ten humans who had suddenly stopped in their tracks.\n\nCROZIER\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _24 April, 1848_\n\nThe third and last day on the ice was by far the hardest.\n\nCrozier had made this crossing at least twice before in the last six weeks with some of the earliest and larger sledge parties, but even with the trail less established, it had been much easier then. He'd been healthier. And he'd been infinitely less tired.\n\nFrancis Crozier was not truly aware of it, but since his recovery from his near-fatal withdrawal illness in January, his severe melancholia had made him an insomniac. As a sailor and then a captain, Crozier had always prided himself \u2014 as most captains did \u2014 on needing very little sleep and waking from the deepest sleep at any change in the ship's condition: a slight change in the ship's direction, the rising of wind in the sails, the sound of too many feet running on deck above him during any specific watch, any alteration in the sound of the water moving against the ship's hull... anything.\n\nBut in recent weeks, Crozier slept less and less each night, until he'd fallen into the habit of only half dozing for an hour or two in the middle of the night, perhaps catching a nap of thirty minutes or less during the day. He told himself it was just the result of so many details to watch over and commands to give in the last days and weeks before taking to the ice, but in truth it was melancholia trying to destroy him again.\n\nHis mind was sodden much of the time. He was a smart man whose mind was stupid with the chemical by-products of constant fatigue.\n\nSleeping at Sea Camps One and Two had been damned near impossible for any of the men the past two nights, no matter how tired they were. There had been no need to erect tents at either camp since eight Holland tents there had been left up permanently over the past weeks, any wind or snow damage being repaired by the next party that came through.\n\nThe three-person reindeer-skin sleeping bags were many times warmer than the sewn-together Hudson's Bay blanket bags, and these good bags had been drawn by lottery. Crozier had not even taken part in the lottery, but when, the first time he'd been on the ice, he'd come into the tent he shared with two other officers, he found that his steward, Jopson, had laid out a reindeer-skin bag tailored for him. Neither the ailing Jopson nor the men thought it right that their captain would have to share a bag with two other snoring, farting, shoving men \u2014 even other officers \u2014 and Crozier had been too tired and grateful to argue.\n\nNor had he told Jopson or the others that sleeping in a bag by one's self was much chillier than his experience sleeping in three-man bags. The other men's body heat was the only thing that kept them warm enough to sleep through the night.\n\nBut Crozier hadn't tried to sleep through the night at either sea camp.\n\nEvery two hours he was up and walking the perimeter to make sure that the watch had changed on time. The wind came up during the night, and the men on watch huddled behind hastily erected low snow walls. Because the biting wind and blowing snow kept the men curled low behind their snow-block barriers, the thing on the ice would have been visible to them only if it actually stepped on one of the men.\n\nIt did not make its appearance that night.\n\nDuring the fitful minutes of sleep he did find, Crozier was revisited by the nightmares he'd had during his January illness. Some of the dreams returned so many times \u2014 and startled the captain out of sleep so many times \u2014 that he remembered fragments of them. Teenaged girls carrying out a s\u00e9ance. M'Clintock and another man staring down into an open boat at two skeletons, one sitting up and fully clothed in peacoat and slops, the other just a mass of tumbled and gnawed bones.\n\nCrozier walked through his days wondering if he was one of those skeletons.\n\nBut the worst dream, by far, was the Communion dream in which he was a boy or a sicker, older version of himself and was kneeling naked at the altar rail in Memo Moira's forbidden church while the huge, inhuman priest \u2014 dripping water in shredded white vestments through which showed the raw, red flesh of a badly burned man \u2014 loomed over him and leaned closer, breathing carrion breath into Crozier's uplifted face.\n\nThe men all rose in the dark a little after 5:00 a.m. on the morning of 23 April. The sun would not rise until almost 10:00 a.m. The wind continued to blow, flapping the brown canvas of the Holland tents and stinging their eyes as they huddled to eat breakfast.\n\nOn the ice, the men were supposed to heat their food thoroughly in small tins labeled \"Cooking Apparatus (1),\" using their small spirit stoves fueled by pints of ether carried in bottles. Even without wind, it was often difficult or nearly impossible to get the spirit stoves primed and started; in a wind like that morning's, it was simply not possible, even when taking the risk of firing up the spirit stoves inside the tents. So \u2014 reassuring themselves that Goldner's canned meats and vegetables and soups had already been cooked \u2014 the men just spooned the frozen or near-frozen masses of congealed glop straight out of the cans. They were starving and had an endless day of man-hauling ahead of them.\n\nGoodsir \u2014 and the three dead surgeons before Goodsir \u2014 had talked to Crozier and Fitzjames about the importance of heating Goldner's tinned foodstuffs, especially the soup. The vegetables and meats, Goodsir had pointed out, had indeed been precooked, but the soups \u2014 mostly cheap parsnips and carrots and other root vegetables \u2014 were \"concentrated,\" meant to be diluted with water and brought to a boil.\n\nThe surgeon could not name the poisons that could be lurking in unboiled Goldner soups \u2014 and perhaps even in the meats and vegetables \u2014 but he kept reiterating the need for full heating of the tinned foods, even while on the march on the ice. These warnings were one of the main reasons Crozier and Fitzjames had ordered the heavy iron whaleboat stoves transported to Terror Camp over the ice and pressure ridges.\n\nBut there were no stoves here at Sea Camp One or at Sea Camp Two the next night. The men ate all the tinned foods cold from the can when the spirit stoves failed \u2014 and even when the ether of the little stoves lighted, there was just enough fuel to _melt_ the frozen soups, not bring them to a boil.\n\nThat would have to suffice, thought Crozier.\n\nAs soon as breakfast was finished, the captain's belly began rumbling with hunger again.\n\nThe plan had been to fold up the eight Holland tents at both sea camps and haul them to Terror Camp on the sledges, to serve as backup should the groups have to go out on the ice again soon. But the wind was too high and the men were too weary even after just one day and night on the ice this trip. Crozier conferred with Lieutenant Little and they decided that three tents would be enough to bring along from this camp. Perhaps they would do better the next morning after Sea Camp Two.\n\nThree men in harness broke down that second day on the ice on 23 April 1848. One began vomiting blood onto the ice. The other two simply fell in their tracks and were unable to pull for the rest of that day. One of those two had to be set onto a sledge and hauled.\n\nNot wanting to reduce the number of armed pickets walking behind, ahead, and to the sides of the procession of sledges, Crozier and Little tied on harnesses and man-hauled for most of that endless day.\n\nThe pressure ridges weren't as high during this middle day of the crossing and the previous sledge tracks had left a virtual highway on this stretch of open sea ice, but the wind and blowing snow eliminated almost all of these advantages. Men pulling a sledge could not see the next sledge fifteen feet in front of them. The Marines or sailors carrying weapons and walking along as guards could see no one else when they were twenty feet or more from the sledges and had to walk within a yard or two of the sledge parties so as not to get lost. Their usefulness as lookouts was nil.\n\nSeveral times during the day, the lead sledge \u2014 usually Crozier's or Lieutenant Little's \u2014 would lose the worn sledge track, and everyone would then have to stop for up to half an hour while some men unharnessed themselves, tied on a rope so as not to get lost in the howling snow, and walked left and right of the false route, seeking out the faint depressions of the actual track on a surface quickly being covered by inches of blowing snow.\n\nTo lose the route midway like this would cost not only time, it might well cost all of them their lives.\n\nSome of the sledge teams hauling heavier loads this spring had done this nine miles of flat sea ice in under twelve hours, arriving at Sea Camp Two only hours after the sun had set. Crozier's group arrived long after midnight and almost missed the camp completely. If Magnus Manson \u2014 whose keen hearing seemed as unusual as his size and low intelligence \u2014 had not heard the flapping of tents in the wind far to their port side, they would have marched past their shelter and food cache.\n\nAs it was, Sea Camp Two had been largely destroyed by the day's incessant and rising winds. Five of the eight tents had been blown away into the darkness \u2014 even though they had been secured by deep ice screws \u2014 or simply torn to tatters. The exhausted and starving men managed to pitch two of the three tents they'd man-hauled from Sea Camp One, and forty-six men who would have been comfortable but crowded in eight tents squeezed into five.\n\nFor the men taking turns on watch that night \u2014 sixteen of the forty-six \u2014 the wind, snow, and cold were a living hell. Crozier stood one of the 2:00 a.m. to 4:00 a.m. watches. He preferred being able to move since his one-man sleeping bag would not allow him to get warm enough to sleep anyway, even with men stacked like cordwood around him in the flapping tent.\n\nThe final day on the ice was the worst.\n\nThe wind had stopped shortly before the men roused themselves at 5:00 a.m., but as in evil compensation for the gift of the blue skies to come, the temperature dropped at least thirty degrees. Lieutenant Little took the measurements that morning: the temperature at 6:00 a.m. was \u221264 degrees.\n\n _It's only eight miles,_ Crozier kept telling himself that day as he pulled in harness. He knew that the other men were thinking the same thing. _Only eight miles today, a full mile less than yesterday's terrible haul._ With more men dropping from sickness or exhaustion, Crozier ordered the accompanying guards to stow their rifles, muskets, and shotguns on the sledges and to tie on to the harnesses as soon as the sun rose. Every man who could walk would pull.\n\nLacking guards, they trusted in the clarity of the day. The brown blur of King William Land was visible as soon as the sun rose \u2014 the wall of high bergs and jostled coastal ice along its rim distressingly more visible, distantly gleaming in the thin, cold sunlight like a barrier of broken glass \u2014 but the clear light ensured they would not lose the old sledge tracks and that the thing on the ice could not sneak up on them.\n\nBut the thing was out there. They could see it \u2014 a small dot loping along to the southwest of them, moving much faster than they could haul. Or run, should it come to that.\n\nSeveral times during the day, Crozier or Little would unstrap from harness, retrieve their telescopes from the sledges or their Male Bags, and look across the miles of ice at the creature.\n\nIt was at least two miles away and moving on all fours. From this distance, it might be just another white arctic bear of the kind they had shot and killed in such plentitude in the last three years. Until, that is, the thing took to its hind legs, rose up above the surrounding ice blocks and minibergs, and sniffed the air as it stared in their direction.\n\n _It knows we've abandoned the ships,_ thought Crozier, staring through his brass telescope that had become scuffed and scarred from so many years' use at both poles. _It knows where we're going. It's planning to get there first._\n\nThey pulled on through the day, stopping only at the midafternoon sunset to eat frozen chunks out of cold cans. Their rations of salt pork and stale biscuits had been used up. The ice walls separating King William Land from the pack ice glowed like a city with ten thousand burning gas lamps in the minutes before darkness spread across the sky like spilled ink.\n\nThey still had four miles to go. Eight men were on sledges now, three of the seamen unconscious.\n\nThey crossed the Great Ice Barrier separating the pack ice from land sometime after 1:00 a.m. The wind stayed low but the temperature continued to drop. During one pause to rerig ropes for the lifting of the sledges over a thirty-foot wall of ice, made not at all easier by the passage of sledges in weeks past since the movement of the ice had tumbled a thousand new ice blocks into their path from the towering bergs on either side, Lieutenant Little took another temperature reading. It was \u221282 degrees.\n\nCrozier had been working and giving commands from within a deep trench of exhaustion for many hours. At sunset, when he'd last looked to the south at the distant creature loping ahead of them now \u2014 it was already crossing the sea ice barrier in easy leaps \u2014 he had made the mistake of taking his mittens and gloves off for a moment so as to write some position notes in his log. He had forgotten to don the gloves before lifting the telescope again and his fingertips and one palm had instantly frozen to the metal. In pulling his hands away quickly, he had ripped a layer of skin and some flesh off his right thumb and three fingers on that hand, and lifted a swath off his left palm.\n\nSuch wounds did not heal up here in the arctic, especially not after the early symptoms of scurvy had set in. Crozier had turned away from the others and vomited from the pain. The sickening burning in his ravaged fingers and left palm only grew worse through the long night of hauling, tugging, lifting, and pushing. His arm and shoulder muscles bruised and bled internally under the pressure of the harness straps.\n\nFor a while on the last barrier, around one thirty in the morning, with the stars and planets shimmering and shifting in the endless clear but murderously cold sky overhead, Crozier stupidly considered leaving all the sledges behind and making a dash for Terror Camp, still a full mile away across the frozen gravel and drifted snow. Other men could come back with them tomorrow and help fetch these impossibly heavy burdens the last mile or so.\n\nEnough of Francis Crozier's mind and command instincts remained for him to reject this thought at once. He could do just that, of course, abandon the sledges \u2014 the first party in weeks to do so \u2014 and ensure their survival by staggering across the ice to the safety of Terror Camp without their burdens, but he would lose all leadership forever in the eyes of his 104 surviving men and officers.\n\nEven though the pain from his torn hands caused him to vomit frequently and silently onto the ice wall as they pulled and pushed the sledges over \u2014 a distant part of Crozier's mind noticed that the vomitus was liquid and red in the lantern light \u2014 he continued giving orders and lending a hand as the thirty-eight men well enough to continue the struggle managed to get the sledges and themselves over the Barrier and onto the ice and runner-scraping gravel of the shoreline.\n\nIf he hadn't been sure that the cold would rip the skin of his lips off, Crozier might have fallen to his knees in the dark and kissed the solid ground as they heard that new sound of gravel and stone protesting under the sledge runners for the final mile.\n\nThere were torches burning at Terror Camp. Crozier was in the lead harness of the lead sledge as they approached. Everyone tried to stand tall \u2014 or at least stagger in an upright position \u2014 as they pulled the dead-weight sledges and the unconscious men on them the last hundred yards into camp.\n\nThere were men fully dressed in slops and outside the tents waiting for them. At first Crozier was touched by their concern, sure that the two dozen or so men he saw in the torchlight had been on the verge of sending out a rescue party in search of their overdue captain and comrades.\n\nAs Crozier leaned into the harness, pulling the last sixty feet or so into the light from the torches, his hands and bruises aflame with pain, he prepared a little joke for their arrival \u2014 something along the lines of declaring it Christmas again and announcing that everyone would sleep the next week through \u2014 but then Captain Fitzjames and some of the other officers stepped closer to greet them.\n\nCrozier saw their eyes then: Fitzjames's eyes and Le Vesconte's and Des Voeux's and Couch's and Hodgson's and Goodsir's and the others' there. And he knew \u2014 through Memo Moira's Second Sight or his demonstrated captain's sense or just through the clear, unfiltered-by-thought perception of a completely exhausted man \u2014 he _knew_ that something had happened and that nothing would now be as he had planned or hoped and might never be again.\n\nIRVING\n\n_Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 40\u2032 58\u2033 W._\n\n_24 April, 1848_\n\nThere were ten Esquimaux standing there: six men of indeterminate age, one very old man with no teeth, one boy, and two women. One woman was old, with a collapsed mouth and a face that was a mass of wrinkles, and one was very young. _Perhaps,_ Irving thought, _they are mother and daughter._\n\nThe men were uniformly short; the top of the tallest man's head barely came up to the tall third lieutenant's chin. Two had their hoods back, showing wild thatches of black hair and unlined faces, but the other men stared at him from the depths of their hoods, some with their faces shrouded and surrounded by a luxuriant white fur that Irving believed might be from the arctic fox. Other hood ruffs were darker and more bristly and Irving guessed that the fur might be from wolverines.\n\nEvery male except the boy carried a weapon, either a harpoon or short spear with a bone or stone point, but after Irving had approached and shown his empty hands, none of the spears were now raised or pointed at him. The Esquimaux men \u2014 hunters, Irving assumed \u2014 stood easily, legs apart, hands on their weapons, with their sled being held back by the oldest man, who kept the boy close. There were six dogs harnessed to the sled, a vehicle much shorter and lighter than even the smallest folding sledges on _Terror._ The dogs barked and snarled, showing vicious canines, until the old man beat them into silence with a carved stick he carried.\n\nEven while trying to think of a way to communicate with these strange people, Irving continued to marvel at their dress. The men's parkas were shorter and darker than Lady Silence's or her deceased male companion's, but just as furry. Irving thought that the dark hair or fur might be from caribou or foxes, but the knee-length white trousers were definitely from the white bears. Some of the long, hairy boots seemed to be from caribou skins, but others were more supple and pliable. Sealskin? Or some sort of caribou hide turned inside out?\n\nThe mittens were visibly sealskin and looked both warmer and more supple than Irving's own.\n\nThe lieutenant had looked to the six younger men to see who was the leader, but it wasn't clear. Other than the old man and the boy, only one of the males stood out, and that was one of the older bareheaded men who wore a complicated white caribou fur headband, a thin belt from which many odd things dangled, and some sort of pouch around his neck. It was not, however, a simple talisman such as Lady Silence's white stone bear amulet.\n\n_Silence, how I wish you were here,_ thought John Irving.\n\n\"Greetings,\" he said. He touched his chest with his mittened thumb. \"Third Lieutenant John Irving of Her Majesty's Ship _Terror._ \"\n\nThe men mumbled among themselves. He heard words that sounded like _kabloona_ and _qavac_ and _miagortok,_ but had no clue whatsoever as to what they might mean.\n\nThe older bareheaded man with the pouch and belt pointed at Irving and said, \" _Piifixaaq!_ \"\n\nSome of the younger men shook their head at this. If it was a pejorative term, Irving hoped that the others were rejecting it.\n\n\"John Irving,\" he said, touching his chest again.\n\n\" _Sixam ieua?_ \" said the man opposite him. \" _Suingne!_ \"\n\nIrving could only nod at this. He touched his chest again. \"Irving.\" He pointed toward the other man's chest in a questioning manner.\n\nThe man stared at Irving from between the fringes of his hood.\n\nIn desperation, the lieutenant pointed to the lead dog that was still barking and growling while being held back and beaten wildly by the old man next to the sled.\n\n\"Dog,\" said Irving. \"Dog.\"\n\nThe Esquimaux man closest to Irving laughed. \" _Qimmiq,_ \" he said clearly, also pointing to the dog. \" _Tunok._ \" The man shook his head and chuckled.\n\nAlthough he was freezing, Irving felt a warm glow. He'd gotten somewhere. The Esquimaux word for the hairy dog they used was either _qimmiq_ or _tunok,_ or both. He pointed at their sled.\n\n\"Sled,\" he stated firmly.\n\nThe ten Esquimaux stared at him. The young woman was holding her mittens in front of her face. The old woman's jaw hung down and Irving could see that she had precisely one tooth in her mouth.\n\n\"Sled,\" he said again.\n\nThe six men in front looked at one another. Finally, Irving's interlouctor to this point said, \" _Kamatik?_ \"\n\nIrving nodded happily even though he had no idea if they had really begun communicating. For all he knew, the man had just asked him if he wanted to be harpooned. Nonetheless, the junior lieutenant could not stop grinning. Most of the Esquimaux men \u2014 with the exception of the boy, the old man who was still beating the dog, and the bareheaded older man with the pouch and belt \u2014 were grinning back.\n\n\"Do you speak English by any chance?\" asked Irving, realizing that he was a bit tardy with the question.\n\nThe Esquimaux men stared and grinned and scowled and remained silent.\n\nIrving repeated the query in his schoolboy French and atrocious German.\n\nThe Esquimaux continued to smile and scowl and stare.\n\nIrving crouched and squatted and the six men closest to him squatted. They did not sit on the freezing gravel, even if a larger rock or boulder was near. After so many months up here in the cold, Irving understood. He still wanted to know someone's name.\n\n\"Irving,\" he said, touching his chest again. He pointed at the closest man.\n\n\" _Inuk,_ \" said the man, touching his chest. He tugged off his mitten with a flash of white teeth and held up his right hand. It was missing the two smaller fingers. \" _Tikerqat._ \" He grinned again.\n\n\"Pleased to make your acquaintance, Mr. Inuk,\" said Irving. \"Or Mr. Tikerqat. _Very_ pleased to make your acquaintance.\"\n\nHe decided that any real communication would have to be through sign language and pointed back the way he had come, toward the northwest. \"I have many friends,\" he said confidently, as if saying this would make him safer with these savage people. \"Two large ships. Two... ships.\"\n\nMost of the Esquimaux looked the way Irving pointed. Mr. Inuk was frowning slightly. \" _Nanuq,_ \" the man said softly, and then seemed to correct himself with a shake of his head. \" _T\u00f4rn\u00e2rssuk._ \" The others looked away or lowered their heads at this last word, almost, it seemed, as if in reverence or fear. But the lieutenant was sure that it was not at the thought of two ships or a group of white men.\n\nIrving licked his bleeding lips. Better to begin trading with these people than to engage in a long conversation. Moving slowly, so as not to startle any of them, he reached into his leather shoulder valise to see if there was any food or bauble he could give them as a gift.\n\nNothing. He had eaten the only salted pork and old biscuit he'd brought for his day's rations. Something shiny and interesting then...\n\nThere were only his ragged sweaters, two stinking extra socks, and a disposable rag he had brought along for his alfresco privy purposes. At that moment Irving bitterly regretted giving his prized Oriental silk handkerchief to Lady Silence \u2014 wherever the wench was. She had slipped away from Terror Camp their second day there and not been seen again since. He knew that these natives would have loved the red-and-green silk handkerchief.\n\nThen his cold fingers touched the curved brass of his telescope.\n\nIrving's heart leapt and then wrenched itself with pain. The telescope was perhaps his most prized possession, the last thing his uncle had given him before that good man had died suddenly of heart trouble.\n\nSmiling wanly at the waiting Esquimaux, he slowly pulled the instrument from his bag. He could see the brown-faced men tightening their grips on their spears and harpoons.\n\nTen minutes later Irving had the entire family or clan or tribe of Esquimaux close around him like schoolchildren grouped around an especially beloved teacher. Everyone, even the suspicious, squinty-eyed older man with the headband, pouch, and belt, had taken a turn looking through the glass. Even the two females had their turn \u2014 Irving allowed Mr. Inuk Tikerqat, his new fellow ambassador, to hand the brass instrument to the giggling young woman and the old woman. The ancient man who had been holding down the sled came over for a look and a shouted exclamation with the women chanting along:\n\n_ai yei yai ya na_\n\n_ye he ye ye yi yan e ya qana_\n\n_ai ye yi yat yana_\n\nThe group enjoyed looking at one another through the glass, staggering back in shock and laughter when huge faces loomed. Then the men, quickly learning how to focus the glass, zoomed in on distant rocks, clouds, and ridgelines. When Irving showed them that they could reverse the glass and make things and each other tiny, the men's laughs and exclamations echoed in the small valley.\n\nHe used his hands and body language \u2014 finally refusing to take the telescope back and pressing it into Mr. Inuk Tikerqat's hands \u2014 to let them know that it was a present.\n\nThe laughter stopped and they stared at him with serious faces. For a minute Irving wondered if he had violated some taboo, offended them somehow, but then he had a strong hunch that he had presented them with a problem in protocol; he had given them a wonderful gift and they'd brought nothing in return.\n\nInuk Tikerqat conferred with the other hunters and then turned to Irving and began making unmistakable pantomimes, lifting his hand to his mouth, then rubbing his belly.\n\nFor a terrible second Irving thought his interlocutor was asking for food \u2014 of which Irving had none \u2014 but when he tried to convey this fact, the Esquimaux shook his head and repeated the gestures. Irving suddenly realized that they were asking him if _he_ was hungry.\n\nEyes filling with tears from a gust of wind or sheer relief, Irving repeated the gestures and nodded enthusiastically. Inuk Tikerqat grabbed him by his slop's frozen shoulder and led him back to the sled. _What had been their word for this?_ thought Irving. \" _Kamatik?_ \" he said aloud, remembering it at last.\n\n\" _Ee!_ \" cried Mr. Tikerqat approvingly. Kicking the growling dogs aside, he swept back a thick fur atop the sled. Stack upon stack of frozen and fresh meat and fish were piled atop the _kamatik._\n\nHis host was pointing toward different delicacies. Pointing at the fish, Inuk Tikerqat said, \" _Eqaluk,_ \" in the slow, patient tones an adult uses with a child. Toward slabs of seal meat and blubber, \" _Nat-suk._ \" Toward larger and more solidly frozen slabs of a darker meat, \" _Oo ming-mite._ \"\n\nIrving nodded. He was embarrassed that his mouth had suddenly filled with saliva. Not sure if he was just supposed to admire the cache of food or choose from it, he pointed diffidently at the seal meat.\n\n\" _Ee!_ \" Mr. Tikerqat said again. He lifted a strip of soft meat and blubber, reached under his short parka, pulled a very sharp bone knife from his waistband, and cut a strip for Irving and another for himself. He handed the lieutenant his piece before cutting into his own.\n\nThe old woman standing nearby made a sort of wailing sound. \" _Kaaktunga!_ \" she cried. And when none of the men paid any attention to her, she shouted again, \" _Kaaktunga!_ \"\n\nHe made a face toward Irving, the kind one man makes to another when a woman demands something in their presence, and said, \" _Orssunguvoq!_ \" But he cut the old woman a strip of seal blubber and tossed it to her as one would to a dog.\n\nThe toothless old crone laughed and began gumming the blubber.\n\nImmediately the group gathered around the sled, men with their knives out, and everyone began cutting and eating.\n\n\" _Aipalingiagpoq,_ \" said Mr. Tikerqat, pointing to the old woman and laughing. The other hunters, old man, and boy \u2014 everyone except the older man with the headband and pouch \u2014 joined in the laughter.\n\nIrving smiled broadly, although he had no idea what the joke was.\n\nThe older man in the headband pointed to Irving and said, \" _Qavac... suingne! Kangunartuliorpoq!_ \"\n\nThe lieutenant did not need a translator to know that whatever the man had said, it had not been laudatory or kind. Mr. Tikerqat and several of the other hunters just shook their heads while eating.\n\nEveryone, even the young woman, was using his or her knife the way Lady Silence had in her snow-house more than two months earlier \u2014 cutting the skin, meat, and blubber _toward_ their mouths so the sharp blades came within a hairsbreadth of their greasy lips and tongues.\n\nIrving cut his the same way \u2014 as best he could \u2014 but his knife was duller and he made a clumsy mess of it. But he did not cut his nose as he had the first time with Silence. The group ate in a companionable silence interrupted only by polite belches and the occasional fart. The men occasionally drank from some sort of pouch or skin, but Irving had already taken out the bottle he kept close to his body so the water would not freeze.\n\n\" _Kee-nah-oo-veet?_ \" Inuk Tikerqat said suddenly. He pounded his chest. \" _Tikerqat._ \" Again the young man removed his mitten and showed his two remaining fingers.\n\n\"Irving,\" said the lieutenant, again tapping his own chest.\n\n\" _Eh-vunq,_ \" repeated the Esquimaux.\n\nIrving smiled over the blubber. He pointed at his new friend. \"Inuk Tikerqat, _ee?_ \"\n\nThe Esquimaux shook his head. \" _Ah-ka._ \" The man made a wide sweep with his arms and hands, encompassing all the other Esquimaux as well as himself. \" _Inuk,_ \" he said firmly. Holding up his mutilated hand and waggling his two remaining fingers while hiding his thumb, he said again, \" _Tikerqat._ \"\n\nIrving interpreted all this to mean that \"Inuk\" was not the man's name but a description of all ten Esquimaux there \u2014 perhaps their tribal name or racial name or clan name. He guessed \"Tikerqat\" to be not a last name now but the entirety of his interlocutor's name, and probably one meaning \"Two Fingers.\"\n\n\" _Tikerqat,_ \" said Irving, trying to pronounce it properly while still cutting and chewing blubber for himself. The fact that the meat and greasy fat were old, smelly, and raw meant almost nothing. It was as if his body craved this fat above all other things. \" _Tikerqat,_ \" he said again.\n\nThere followed, in the midst of the squatting, cutting, and chewing, a general introduction. Tikerqat began both the introductions and the explanations by acting things out to explain the meaning of the name \u2014 if the names had a meaning \u2014 but then the other men picked up on it and acted out their own names. The moment had the feeling of a joyous child's game.\n\n\" _Taliriktug,_ \" said Tikerqat slowly, pushing forward the barrel-chested young man next to him. Two Fingers grabbed his companion's upper arm and squeezed it, making _ah-yeh-I_ noises, then flexing his own muscle and comparing it to the other man's thicker biceps.\n\n\" _Taliriktug,_ \" repeated Irving, wondering if it meant \"Big Muscle\" or \"Strong Arm\" or something similar.\n\nThe next man, a shorter one, was named _Tuluqag._ Tikerqat tugged the man's parka hood back, pointed to his black hair, and made flapping noises with his hand, miming a bird flying.\n\n\" _Tuluqag,_ \" repeated Irving, nodding politely toward the man as he chewed. He wondered if the word meant \"Raven.\"\n\nThe fourth man thumped himself on the chest, grunted, \" _Amaruq,_ \" and threw back his head and howled.\n\n\" _Amaruq,_ \" repeated Irving and nodded. \"Wolf,\" he said aloud.\n\nThe fifth hunter was named _Mamarut_ and acted out some pantomime involving waving his arms and dancing. Irving repeated the name and nodded but had no idea what the name might mean.\n\nThe sixth hunter, a younger man of very serious demeanor, was introduced by Tikerqat as _Ituksuk._ This man stared at Irving with deep black eyes and said and acted out nothing. Irving nodded politely and chewed his blubber.\n\nThe older man with the headband and the pouch was introduced by Tikerqat as _Asiajuk,_ but the man neither blinked nor showed recognition of the introduction. It was obvious he did not like or trust Third Lieutenant John Irving.\n\n\"A pleasure to meet you, Mr. Asiajuk,\" said Irving.\n\n\" _Afatkuq,_ \" Tikerqat said softly, nodding slightly in the direction of the unsmiling older man in the headband.\n\n_Some sort of medicine man?_ wondered Irving. As long as Asiajuk's hostility remained only on the level of silent suspicion, the lieutenant thought that things would be all right.\n\nThe old man at the sled was introduced as _Kringmuluardjuk_ to the young lieutenant. Tikerqat pointed to the still-snarling dogs, brought his hands together in some sort of diminutive gesture, and laughed.\n\nThen Irving's laughing interlocutor pointed to the shy boy, who appeared to be about ten or eleven years old, pointed to his own chest again, and said, \" _Irniq,_ \" followed by \" _Qajor\u00e2nguaq._ \"\n\nIrving guessed that _Irniq_ might mean \"son\" or \"brother.\" Probably the former, he thought. Or perhaps the boy's name was _Irniq_ and _Qajor\u00e2nguaq_ meant son or brother. The lieutenant nodded respectfully, just as he had with the older hunters.\n\nTikerqat shoved the old woman forward. Her name appeared to be _Nauja,_ and Tikerqat again made a bird-flying motion. Irving repeated the name as best he could \u2014 there was a certain glottal sound that the Esquimaux made that he could not approximate \u2014 and nodded respectfully. He wondered if _Nauja_ was an arctic tern, a seagull, or something more exotic.\n\nThe old woman giggled and stuffed more blubber in her mouth.\n\nTikerqat put his arm around the young woman, not much more than a girl really, and said, \" _Qaumaniq._ \" Then the hunter grinned broadly and said, \" _Amooq!_ \"\n\nThe girl wriggled in his grasp while smiling, and all the men except the possible medicine man laughed loudly.\n\n\" _Amooq?_ \" said Irving, and the laughter rose in volume. Tuluqag and Amaruq spit out blubber they were laughing so hard.\n\n\" _Qaumaniq... amooq!_ \" said Tikerqat and made a two-handed, open-fingered grabbing gesture in front of his own chest that was universal. But to make sure he got the point across, the hunter grabbed his wriggling woman \u2014 Irving had to think she was his wife \u2014 and quickly lifted her short, dark parka top.\n\nThe girl was naked under the animal skin, and her breasts were, indeed, very large... very large indeed for a woman so young.\n\nJohn Irving felt himself blush from his blond hairline down to his chest. He lowered his gaze to the blubber he was still chewing. At that moment he would have laid fifty quid that _Amooq_ was the Esquimaux language equivalent of \"Big Tits.\"\n\nThe men around him howled with laughter. The _Qimmiq_ \u2014 the wolflike sled dogs around the wooden _kamatik_ \u2014 howled and leapt against their tethers. The old man behind the sled, Kringmuluardjuk, actually fell onto the snow and ice he was laughing so hard.\n\nSuddenly Amaruq \u2014 Wolf? \u2014 who had been playing with the telescope, pointed to the bare ridge from which Irving had descended into the valley and snapped what sounded like \" _Takuva-a_... _kabloona qukiuttina!_ \"\n\nThe group fell silent immediately.\n\nThe wolfish dogs began barking wildly.\n\nIrving stood from where he had been crouching and shielded his eyes from the sun. He did not want to ask for the telescope back. There was the quickest motion of a human form in greatcoat silhouetted against the top of the ridge.\n\n_Wonderful!_ thought Irving. All through the blubber feast and introductions, he'd been trying to decide how to get Tikerqat and the others to come back to Terror Camp with him. He'd been afraid that he would not be able to communicate well enough with just his hands and motions to persuade the eight Esquimaux males and two women and their dogs and sled to make the three-hour trip back to the coast with him, so he'd been trying to think of a way to get just Tikerqat to come along with him.\n\nIt was certain that the lieutenant could not just let these natives hike back to wherever they had come from. Captain Crozier would be at the camp tomorrow, and Irving knew from several conversations with the captain that contact with the local peoples was precisely what the tired and beleaguered captain most hoped might happen. _The northern tribes, what Ross called northern highland tribes, are rarely warlike,_ Crozier had told his third lieutenant one night. _If we come across a village of theirs on our way south, they may feed us well enough to get us provisioned properly for the long upstream haul to Great Slave Lake. At the very least, they could show us how to live off the land._\n\nAnd now Thomas Farr and the others had come looking for him, following his footprints through the snow to this valley. The figure on the ridgeline had gone back over the ridge and out of sight \u2014 out of shock at seeing ten strangers in the valley or concern that he might frighten them? \u2014 but Irving had caught a glimpse of the greatcoat blowing in silhouette and the Welsh wig and comforter and knew that one of his problems had been solved.\n\nIf he could not persuade Tikerqat and the others to come back with them \u2014 and old Asiajuk the shaman might be a problem convincing \u2014 Irving and a few of his party would stay with the Esquimaux here in the valley, convince them to stay there with conversation and other presents from some of the other men's packs, while he sent the fastest seamen running back to the coast to bring Captain Fitzjames and many more men to this place.\n\n_I can't let them get away. These Esquimaux could be the answer to our problems. They may be our salvation._\n\nIrving felt his heart pounding against his ribs.\n\n\"It's all right,\" he said to Tikerqat and the others, speaking in the calmest and most confident tones he could summon. \"It's just my friends. A few friends. Good men. They won't harm you. We only have one rifle with us, and we won't bring that down here. It's all right. Just friends of mine whom you will enjoy meeting.\"\n\nIrving knew that they couldn't understand a word he was saying, but he kept talking, using the same soft, reassuring voice he would have used at his family's stables in Bristol to calm a skittish colt.\n\nSeveral of the hunters had pulled their spears or harpoons from the snow and were holding them casually, but Amaruq, Tulugaq, Taliriktug, Ituksuk, the boy Qajor\u00e2nguaq, the old man Kringmuluardjuk, and even the scowling shaman Asiajuk were looking to Tikerqat for guidance. The two women quit chewing blubber and quietly found their place behind the line of men.\n\nTikerqat looked at Irving. The Esquimaux's eyes were suddenly very dark and very alien-looking to the young lieutenant. The man seemed to be waiting for some explanation. \" _Khat-seet?_ \" he said softly.\n\nIrving showed open palms in a calming gesture and smiled as easily as he could. \"Just friends,\" he said, matching the softness of Tikerqat's tone. \"A few friends.\"\n\nThe lieutenant glanced up at the ridgeline. It was still empty against the blue sky. He was afraid that whoever had come looking for him had been alarmed by the congregation in the valley and might be headed back. Irving was not sure how long he could wait here... how long he could keep Tikerqat and his people calm before they took flight.\n\nHe took a deep breath and realized that he would have to go after the man up there, call him back, tell him what had happened and send him to bring back Farr and the others as quickly as possible. Irving couldn't wait.\n\n\"Please stay here,\" said Irving. He set his leather valise in the snow near Tikerqat in an attempt to show that he was coming right back. \"Please wait. I shan't be a moment. I won't even get out of your sight. Please stay.\" He realized that he was gesturing with his hands as if asking the Esquimaux to sit, the way he would talk to a dog.\n\nTikerqat did not sit, nor did he reply, but he remained where he was standing while Irving backed away slowly.\n\n\"I'll be right back,\" called the lieutenant. He turned and jogged quickly up the steep scree and ice, onto the dark gravel at the top of the ridgeline.\n\nBarely able to breath with the tension, he turned back at the top and looked down.\n\nThe ten figures, barking dogs, and sled were exactly where he had left them.\n\nIrving waved, made gestures to show that he would be right back, and hurried over the ridge, ready to shout at any retreating sailor.\n\nTwenty feet down the northeast side of the ridge, Irving saw something that made him stop in his tracks.\n\nA tiny man was dancing naked except for his boots around a tall heap of discarded clothing on a boulder.\n\n_Leprechaun,_ thought Irving, remembering some of Captain Crozier's tales. The image made no sense to the third lieutenant. It had been a day of strange sightings.\n\nHe stepped closer and saw that it was no leprechaun dancing but rather the caulker's mate. The man was humming some sailor's ditty as he danced and pirouetted. Irving could not help noticing the grub-white paleness of the little man's skin, how his ribs pushed out so visibly, the goose bumps everywhere rising on his flesh, the fact that he was circumcised, and how absurd the pale white buttocks were when he pirouetted.\n\nWalking up to him, shaking his head in disbelief, not in the mood to laugh but his heart still pounding with the excitement of finding Tikerqat and the others, Irving said, \"Mr. Hickey. What on _earth_ do you think you're doing?\"\n\nThe caulker's mate quit pirouetting. He raised one bony finger to his lips as if to shush the lieutenant. Then he bowed and showed Irving his arse as he bent over his pile of coats and clothing on the boulder.\n\n_The man's gone mad,_ thought Irving. _I can't let Tikerqat and the others see him like this._ He wondered if he could slap some sense into the little man and still use him as a messenger to bring Farr and the others here quickly. Irving had a few sheets of paper and a stub of graphite with which he could write a note, but they were in his valise down in the valley.\n\n\"See here, Mr. Hickey... ,\" he began sternly.\n\nThe caulker's mate swung up and around so quickly with his arm fully extended that for a second or two Irving thought he was resuming his dance.\n\nBut there had been a sharp boat knife in that extended hand.\n\nIrving felt a sudden sharp pain in his throat. He started to speak again, found that he couldn't, raised both hands to his throat, and looked down.\n\nBlood was cascading over Irving's hands and down onto his chest, dripping onto his boots.\n\nHickey swung the blade again in a wide, vicious arc.\n\nThis blow severed the lieutenant's windpipe. He fell to his knees and raised his right arm, pointing at Hickey through vision that had suddenly been narrowed by a dark tunnel. John Irving was too surprised even to feel anger.\n\nHickey took a step closer, still naked, all sharp knees and thin thighs and tendons, crouching now like some pale, bony gnome. But Irving had fallen to his side on the cold gravel, vomited an impossible amount of blood, and was dead before Cornelius Hickey ripped away the lieutenant's clothing and began wielding the knife in earnest.\n\nCROZIER\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _25 April, 1848_\n\nHis men collapsed into tents and slept the sleep of the dead as soon as they reached Terror Camp, but Crozier did not sleep at all the night of 24 April.\n\nFirst he went to a special medical tent that had been erected so that Dr. Goodsir could do the postmortem and prepare the body for burial. Lieutenant Irving's corpse, white and frozen after its long voyage back to camp on the savages' requisitioned sledge, did not look quite human. Besides the gaping wound on the throat \u2014 so deep that it exposed the white vertebrae of his spine from the front and made the head yaw back as if on a loose hinge \u2014 the young man had been emasculated and disemboweled.\n\nGoodsir was still awake and working on the body when Crozier came into the tent. The surgeon was inspecting several organs removed from the corpse, poking at them with some sharp instrument. He glanced up and gave Crozier a strange, thoughtful, almost guilty look. Neither man said anything for a long moment as the captain stood over the body. Finally Crozier brushed back a strand of blond hair that had fallen over John Irving's forehead. The lock had been almost touching Irving's open, clouded but still staring blue eyes.\n\n\"Have his body ready for burial at noon tomorrow,\" said Crozier.\n\n\"Yes, sir.\"\n\nCrozier went to his tent, where Fitzjames was waiting.\n\nWhen Crozier's steward, the 30-year-old Thomas Jopson, had supervised the loading and transport of \"the captain's tent\" to Terror Camp some weeks ago, Crozier had been furious to learn that Jopson had not only had a double-sized tent sewn for the purpose \u2014 the captain had anticipated just a regular brown Holland tent \u2014 but had also had the men haul an oversized cot and several solid oak and mahogany chairs from the Great Room, as well as an ornate desk that had belonged to Sir John.\n\nNow Crozier was glad for the furniture. He arranged the heavy desk between the tent entrance and the private bunking area with the two chairs behind the desk and none in front. The lantern hanging from the tall tent's peak harshly illuminated the empty space in front of the desk while leaving the area for Fitzjames and Crozier in semidarkness. The space had the feel of a court-martial room.\n\nThat's exactly what Francis Crozier wanted.\n\n\"You should go to bed, Captain Crozier,\" said Fitzjames.\n\nCrozier looked at the younger captain. He did not look young any longer. Fitzjames looked like a walking corpse \u2014 pale to the point of his skin becoming transparent, bearded with whiskers and dried blood from leaking follicles, with hollow cheeks and sunken eyes. Crozier had not looked at himself in a mirror for several days and had avoided the one hanging at the rear of this tent of his, but he hoped to Christ he did not look as bad as the former wunderkind of the Royal Navy, Commander James Fitzjames.\n\n\"You need some sleep yourself, James,\" said Crozier. \"I can interrogate these men myself.\"\n\nFitzjames shook his head tiredly. \"I questioned them, of course,\" he said, his voice a dead monotone, \"but haven't visited the site or really interrogated them. I knew you would want to.\"\n\nCrozier nodded. \"I want to be at the site by first light.\"\n\n\"It's about two hours' brisk walk to the southwest,\" said Fitzjames.\n\nCrozier nodded again.\n\nFitzjames pulled his cap off and combed back his long, greasy hair with dirty fingers. They had used the boat stoves that had been transported here to melt water for drinking and just enough to shave by should an officer want to shave, but there was none left for bathing. Fitzjames smiled. \"Caulker's Mate Hickey asked if he could sleep until it was his time to report.\"\n\n\"Caulker's Mate Hickey can fucking well stay awake like the rest of us,\" said Crozier.\n\nFitzjames said softly, \"That's more or less what I told him. I put him on guard duty. The cold should keep him awake.\"\n\n\"Or kill him,\" said Crozier. His tone suggested that this would not be the worst turn of events. In a loud voice, shouting to Private Daly who stood guard at the tent's door, Crozier said, \"Send in Sergeant Tozer.\"\n\nSomehow the large, stupid Marine managed to stay beefy even when all the men were starving on one-third rations. He stood at attention, minus his musket, as Crozier conducted the interrogation.\n\n\"What was your impression of today's events, Sergeant?\"\n\n\"Very pretty, sir.\"\n\n\"Pretty?\" Crozier remembered the condition of Third Lieutenant Irving's throat and body as he lay in the postmortem tent immediately behind Crozier's own tent.\n\n\"Aye, sir. The attack, sir. Went off like clockwork. Like clockwork. We come walking down that big hill, sir, muskets and rifles and shotguns lowered as if we had no harsh feelings in the world, sir, and them savages watched us come. We opened fire at less than twenty yards and raised pure holy Cain amongst their motley God-be-damned ranks, sir, that I can tell you. Raised pure holy Cain.\"\n\n\"Were they in ranks, Sergeant?\"\n\n\"Well, no, Captain, not as you might say on a Bible, sir. More like standing around like the savages they was, sir.\"\n\n\"And your opening salvos cut them down?\"\n\n\"Oh, aye, sir. Even the shotguns at that range. It was a sight to behold, sir.\"\n\n\"Like shooting fish in a rain barrel?\"\n\n\"Aye, sir,\" said Sergeant Tozer with a huge grin on his red face.\n\n\"Did they put up any resistance, Sergeant?\"\n\n\"Resistance, sir? Not really. Not any you might speak of, sir.\"\n\n\"Yet they were armed with knives and spears and harpoons.\"\n\n\"Oh, aye, sir. A couple of the godless savages threw their harpoons and one got a spear off, but them what flung them was already wounded and it done them no good but a little nick in the leg of young Sammy Crispe, who took his shotgun and blew the savage who nicked him straight to Hell, sir. Straight to Hell.\"\n\n\"Yet two of the Esquimaux got away,\" said Crozier.\n\nTozer frowned. \"Aye, sir. I apologize about that. They was a lot of confusion, sir. And two of 'em who went down got up when we was shooting those pox-besotted dogs, sir.\"\n\n\"Why did you shoot their dogs, Sergeant?\" It was Fitzjames who asked this question.\n\nTozer looked surprised. \"Why, they was barking and snarling and lunging at us, Captain. They was more wolves than dogs.\"\n\n\"Did you consider, Sergeant, that they might have been useful to us?\" asked Fitzjames.\n\n\"Yes, sir. As _meat._ \"\n\nCrozier said, \"Describe the two Esquimaux that got away.\"\n\n\"A little one, Captain. Mr. Farr said that he thought it might have been a woman. Or a girl. She had blood on her hood but obviously she wasn't dead.\"\n\n\"Obviously,\" Crozier said drily. \"What about the other one who escaped?\"\n\nTozer shrugged. \"A little man with a headband is all I know, Captain. He'd fallen behind the sledge there, and we all thought he was a deader. But he got up and run with the girl when we was busy shooting the dogs, sir.\"\n\n\"Did you pursue them?\"\n\n\"Pursue them, sir? Oh, yes, absolutely. We run our ar\u2014... we run after them hard, Captain. And we was reloading and firing as we went, sir. I think I hit that little Esquimaux bitch again, but she didn't slow down one whit, sir. They was just too fast for us. But they won't be coming back this way no time soon, sir. We saw to that.\"\n\n\"How about their friends?\" Crozier said drily.\n\n\"Pardon, sir?\" Tozer was grinning again.\n\n\"Their tribe. Village. Clan. Other hunters and warriors. These people came from somewhere. They haven't been out on the ice all winter. Presumably they'll return to that village, if they're not there already. Did you consider that the other Esquimaux hunters \u2014 men who kill every day \u2014 might take it personally that we killed eight of their kindred, Sergeant?\"\n\nTozer looked confused.\n\nCrozier said, \"You're dismissed, Sergeant. Send in Second Lieutenant Hodgson.\"\n\nHodgson looked as miserable as Tozer had complacent. The young lieutenant was obviously distraught over the death of his closest friend on the expedition and sickened by the attack he had ordered after he had come across Irving's reconnaissance group and been led to Irving's body.\n\n\"At ease, Lieutenant Hodgson,\" said Crozier. \"Do you need a chair?\"\n\n\"No, sir.\"\n\n\"Tell us how you came to join up with Lieutenant Irving's group. Your orders from Captain Fitzjames were to go on a hunting expedition _south_ of Terror Camp.\"\n\n\"Yes, Captain. And we did that much of the morning. There was not so much as a rabbit track in the snow along the coast, sir, and we couldn't get out onto the sea ice because of the height of the bergs piled up along the shore ice. So around ten a.m. we turned inland, thinking maybe there'd be sign of some caribou or foxes or musk oxen or something.\"\n\n\"But there wasn't?\"\n\n\"No, sir. We came across the tracks of about ten people wearing soft-soled Esquimaux-type boots instead. That and their sledge and dog tracks.\"\n\n\"And you followed those tracks back northwest instead of continuing hunting?\"\n\n\"Yes.\"\n\n\"Who made that decision, Second Lieutenant Hodgson? You or Sergeant Tozer, who was second in your party?\"\n\n\"Me, sir. I was the only officer there. I made that and all the other decisions.\"\n\n\"Including the final decision of attacking the Esquimaux?\"\n\n\"Yes, sir. We spied on them a minute from the ridge where poor John had been murdered and gutted, and... well, you know what they did to him, Captain. The savages looked like they were preparing to leave, heading back to the southwest. That's when we decided to attack them in force.\"\n\n\"You had how many weapons, Lieutenant?\"\n\n\"Our group had three rifles, two shotguns, and two muskets, sir. Lieutenant Irving's group just had the one musket. Oh, and a pistol we fetched from John's... from Lieutenant Irving's greatcoat pocket.\"\n\n\"The Esquimaux left the weapon in his pocket?\" asked Crozier.\n\nHodgson paused a moment as if he had not considered this before. \"Yes, sir.\"\n\n\"Was there any other sign of theft of his personal possessions?\"\n\n\"Yes, sir. Mr. Hickey had reported to us as to how he'd seen the Esquimaux rob John... Lieutenant Irving... of his telescope and valise before they killed him up on the ridge, sir. When we got to that ridge, I could see through our own glass that the natives were going through his valise and passing his telescope around down there in the valley where I guess they'd stopped after murdering and... mutilating... him.\"\n\n\"Were there tracks?\"\n\n\"Pardon me, sir?\"\n\n\"Tracks... of the Esquimaux... going down from the bare ridgeline where you found the lieutenant's body to where the natives were going through his possessions.\"\n\n\"Uh... yes, sir. I think so, Captain. I mean, I can remember a thin line of tracks that I thought were just John's at the time but must have been the rest of theirs as well. They must have gone up and down in a line, sort of, Captain. Mr. Hickey said that they were all around him up on the bare ridge there as they cut his throat and... did the other things, sir. He said that it wasn't all of them... not the women and the boy, maybe... but it was six or seven of the heathens. The hunters, sir. The younger men.\"\n\n\"And the old man?\" asked Crozier. \"I understand that there was a toothless old man among the bodies when you were done.\"\n\nHodgson nodded. \"He had one tooth left, Captain. I can't remember if Mr. Hickey said the old man was part of the group that killed John.\"\n\n\"How was it that you first came upon Mr. Farr's group \u2014 Lieutenant Irving's reconnaissance party \u2014 if you had been following the Esquimaux's tracks north, Lieutenant?\"\n\nHodgson nodded briskly as if relieved to be asked a question he could answer with certainty. \"We lost the natives' footprints and sledge tracks about a mile south of where Lieutenant Irving was attacked, sir. They must have been moving more east then, across the low ridgetops where there was ice, but mostly rock, sir... you know, that frozen gravel. We couldn't find their sledge or dog tracks or footprints anywhere in the valleys, so we continued due north, the way they'd been going. We came down off a hill and found Thomas Farr's group \u2014 John's reconnaissance party \u2014 just finishing their dinner. Mr. Hickey had come back to report on what he'd seen just a minute or two earlier, and I guess we frightened Thomas and his men... they thought we were the Esquimaux coming for them.\"\n\n\"Did you observe anything odd about Mr. Hickey?\" asked Crozier.\n\n\"Odd, sir?\"\n\nCrozier waited in silence.\n\n\"Well,\" continued Hodgson, \"he was shaking very hard. As if palsied. And his voice was very agitated, almost shrill. And he... well, sir... he was laughing some. Giggling, like. But all that's to be expected from a man who'd just seen what he'd just seen, isn't it, Captain?\"\n\n\"And what did he see, George?\"\n\n\"Well...\" Hodgson looked down to regain his composure. \"Mr. Hickey had told Captain of the Maintop Farr, and he repeated to me, that he'd been out to check on Lieutenant Irving and came over a ridge just in time to see these six or seven or eight Esquimaux stealing the lieutenant's belongings and stabbing and mutilating him. Mr. Hickey said \u2014 he was still shaking hard, sir, very upset \u2014 that he'd seen them cut off John's private parts.\"\n\n\"You saw Lieutenant Irving's body just a few minutes later, didn't you, Lieutenant?\"\n\n\"Aye, sir. It was about a twenty-five-minute walk from where Farr's group had been eating dinner.\"\n\n\"But you didn't start shaking uncontrollably after you saw Irving's body, did you, Lieutenant? Shaking for twenty-five minutes or more?\"\n\n\"No, sir,\" said Hodgson, obviously not understanding the reason for Crozier's question. \"But I threw up, sir.\"\n\n\"And when did you decide to attack the Esquimaux group and kill all of them?\"\n\nHodgson swallowed audibly. \"After I spied them from the ridge through my glass going through John's valise and playing with his telescope, Captain. As soon as we all took a look \u2014 Mr. Farr, Sergeant Tozer, and myself \u2014 and realized that the Esquimaux had turned their sledge around and were getting ready to leave.\"\n\n\"And you gave the order to take no prisoners?\"\n\nHodgson looked down again. \"No, sir. I didn't really think about it one way or the other. I was just so... angry.\"\n\nCrozier said nothing.\n\n\"I did tell Sergeant Tozer that we had to ask one of the Esquimaux about what happened, Captain,\" the lieutenant continued. \"So I guess I thought before the action that some would be alive after. I was just so... _angry._ \"\n\n\"Who gave the actual order to fire, Lieutenant? You or Sergeant Tozer or Mr. Farr or someone else?\"\n\nHodgson blinked several times, very rapidly. \"I don't remember, sir. I'm not sure there _was_ an order given. I just remember that we got to within about thirty yards, perhaps less, and I saw several of the Esquimaux men grab their harpoons or spears or whatever they were, and then everyone along our line was firing and reloading and firing. And the natives were running and the women were screaming... the older woman kept screaming like, well, like the banshees you've told us about, Captain... a high, warbling, constant scream... even after several balls had hit her, she kept up that God-awful screaming. Then Sergeant Tozer walked up and stood over her with John's pistol and... it all happened very fast, Captain. I've never been involved in anything like that.\"\n\n\"Nor have I,\" said Crozier.\n\nFitzjames said nothing. He'd been the hero of several wild land campaigns during the Opium Wars. His gaze now was downcast and seemed to be turned inward.\n\n\"If mistakes were made, sirs,\" said Hodgson, \"I take full responsibility. I was the ranking officer of the two groups with Jo\u2014... with Lieutenant Irving dead. It's all my responsibility, sirs.\"\n\nCrozier looked at him. The captain could feel the dead flatness of his own gaze. \"You _were_ the only officer present, Lieutenant Hodgson. For good or ill, it _was_ and _is_ your responsibility. In about four hours, I want to lead a party to the site of the murder and shootings. We'll leave by lantern light and follow your sledge tracks back to the place, but I want to be there by the time the sun rises. You and Mr. Farr will be the only men from today's actions that I want along with us. Get some sleep and be fed and ready to go by six bells.\"\n\n\"Aye, sir.\"\n\n\"And send in Caulker's Mate Hickey.\"\n\nGOODSIR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _25 April, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Tuesday, 25 April, 1848 \u2014_\n\n _I liked Lieutenant Irving very much. My Impression of him was that he was a Decent and Caring young man. I did not know him Well, but through all these Hard Months \u2014 especially during the many Weeks that I spent time on_ Terror _as well as_ Erebus _\u2014 I never once saw the Lieutenant shirk a duty or speak harshly to the Men or deal with them or me with anything other than gentleness and Professional Courtesy._\n\n _I know that Captain Crozier is especially Devastated by the Loss. His face was so Pale when He came into camp this morning sometime after 2:00 a.m. that I would have staked my Professional Reputation on the opinion that it could not grow any paler. But it did upon his hearing the News. Even his lips became as white as the ice-pack snow we have been staring at now for the better part of three years._\n\n _But however much I liked and respected Lieutenant Irving, I had to perform my Professional Duties and put all memories of Friendly Acquaintance aside._\n\n _I removed the remnants of Lieutenant Irving's clothing \u2014 buttons had been ripped off all layers from his Waistcoat to his Long Underwear, and the Caked Blood had frozen the clumped Fabric into iron-hard wrinkled masses \u2014 and had my assistant, Henry Lloyd, help me in bathing Lieutenant Irving's body. The water \u2014 from ice and snow Mr. Diggle's mates melted using some of the Coal we brought over from the Ships \u2014 was precious, but it was necessary that we honour young Irving this way._\n\n _I did not, of course, have to make my usual inverted-Y incision from hip bones to umbilicus \u2014 the base of the upside-down Y running up to the sternum \u2014 since Lieutenant Irving's Murderers had already done so._\n\n _I made my usual Notes and Sketches as I proceeded, my Fingers aching from the Cold. The Cause of Death holds no Mysteries. The Wound to Lieutenant Irving's Neck had been caused by at least two savage slashes by a nonserrated blade, and he Bled to Death. I seriously doubt if there is a Pint of blood left in the hapless young Officer's body._\n\n _The Trachea and Larynx have been severed and there are blade gouges on the exposed cervical vertebrae._\n\n _His abdominal cavity has been opened by repeated sawing of a Short Blade through skin, flesh, and connecting Tissue, and the majority of his Upper and Lower intestines have been cut out and removed. Lieutenant Irving's spleen and kidneys have also been slashed and opened by a Sharp Object or Objects. His liver is missing._\n\n _The lieutenant's penis has been amputated approximately one Inch above its Base and is missing. His Scrotum has been cut open along its Central Axis and the testes cut out. Repeated Applications of the Blade were required to cut through the scrotal sac, the epididymis, and the tunica vaginalis. It is possible that the Assailant's Blade was growing Dull by this point._\n\n _While the testes are absent, remnants of the vas deferens and the urethra and major portions of connecting tissue from the base of the penis into the body cavity remain._\n\n _While there are signs of multiple Bruisings on Lieutenant Irving's body \u2014 many of them Consistent with a diagnosis of growing Scurvy \u2014 there are no other Serious Wounds visible anywhere. It is interesting that there are no Defensive Cuts on his hands, forearms, or palms._\n\n _It seems apparent that Lieutenant Irving was taken completely by Surprise. His Assailant or Assailants cut his throat before he had the Least Opportunity to defend himself. They then took some time Disemboweling him and Removing his Private Parts through repeated Incisions and Sawing Motions._\n\n _In preparing the lieutenant's body for burial later today, I sewed up his Neck and Throat as best I could and \u2014 after setting some Nonoriginal but decomposing fibrous substances (a folded sweater from the lieutenant's own pack of personal belongings) in his Abdominal Cavity so that the aforementioned Cavity would not look so visibly vacant and shrunken under his uniform when viewed by the men \u2014 I prepared to sew the abdominal Cavity back up as best I could (there was much tissue destroyed or missing)._\n\n _But first I hesitated and Decided to do something unusual._\n\n _I opened Lieutenant Irving's stomach._\n\n _There was no real Postmortem Examinatory Reason to do this. There was no doubt of the Cause of the young lieutenant's death. There was no reason to check for Disease or Chronic Conditions \u2014 we are all suffering from Scurvy to one extent or the other, and we are all slowly Starving to Death._\n\n _But I opened his Stomach anyway. It looked strangely Distended \u2014 more so than bacterial action and the resulting Decomposition would suggest in this extreme cold \u2014 and no postmortem examination would be complete without an Inspection of this Anomaly._\n\n _His stomach was full._\n\n _Very shortly before Lieutenant Irving's death, he had ingested Large Quantities of Seal Meat, some Sealskin, and much Fatty Blubber. The Digestive Process had barely begun working on it._\n\n _The Esquimaux had Fed him before they Murdered Him._\n\n _Or perhaps Lieutenant Irving had Bartered his Telescope, valise, and few personal possessions in the Valise in exchange for this Seal Meat and Blubber._\n\n _But that is not possible, since Caulker's Mate Hickey reported that he saw the Esquimaux Murder and Rob the Lieutenant._\n\n _Seal Meat and Fish were on the Esquimaux Sled that Mr. Farr brought back, using it to transport Lieutenant Irving's body. Farr reported that they had thrown other objects off the Sled \u2014 baskets, Cooking Pots of some sort, things Lashed above the Seal Meat and Fish \u2014 so as to better situate the Lieutenant's corpse on the light sled._ We wanted to make Lieutenant Irving as comfortable as possible, _was what Sergeant Tozer had said._\n\n _So the Esquimaux must have first offered him their food, allowed him time to Eat it \u2014 if not Digest it \u2014 and then repacked their sled before Falling Upon him with such Savagery._\n\n _To approach someone as a Friend and then to Murder and Mutilate him so \u2014 can we Believe that there is a Race so Treacherous and so Malevolent and so Barbarous?_\n\n _What could have Prompted this sudden and Violent change of attitude on the part of the Natives? Could the Lieutenant have said or done something that violated their Sacred Taboos? Or did they simply want to Rob him? Was the brass Telescope the reason for Lieutenant Irving's terrible Death?_\n\n _There is another possibility, but one so Heinous and so Unlikely that I hardly wish to Record it here._\n\n _The Esquimaux did not kill Lieutenant Irving._\n\n _But this makes no sense either. Caulker's Mate Hickey clearly stated that he SAW six to eight of the Natives assailing the Lieutenant. He SAW them steal the Lieutenant's valise, telescope, and other possessions \u2014 while strangely they did not find his Pistol or go through his other pockets. Caulker's Mate Hickey told Captain Fitzjames today \u2014 I was present during the discussion \u2014 that he, Hickey, WATCHED from a distance as the Savages disemboweled our friend._\n\n _Hickey Hid and Watched as all this went on._\n\n _It is still pitch-dark and very Cold, but Captain Crozier is leaving in twenty Minutes to take a few men the Several Miles to the Site of the Murder and of today's Deadly Skirmish with the Esquimaux. Presumably their bodies are still Lying in the Valley there._\n\n _I have just completed Stitching Lieutenant Irving. As tired as I am \u2014 I have not slept for more than 24 Hours \u2014 I will have Lloyd finish the dressing of the Lieutenant and make final preparations for his burial later Today. As Providence would have it, Irving brought his Dress Uniform in his bag of personal possessions from_ Terror. _He will be dressed in that._\n\n _I am going now to ask Captain Crozier if I may accompany him, Lieutenant Little, Mr. Farr, and the others to the Murder Site._\n\nPEGLAR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 40\u2032 58\u2033 W._\n\n _25 April, 1848_\n\nWhen the fog shifted, something that looked like an oversized human brain seemed to be rising out of the frozen ground: grey, convoluted, coiled upon itself, glistening with ice.\n\nHarry Peglar realized that he was looking at John Irving's entrails.\n\n\"This is the spot,\" Thomas Farr said needlessly.\n\nPeglar had been somewhat surprised that the captain had ordered him to come along on this trip to the murder site. The captain of the foretop had not been in either party \u2014 Irving's or Hodgson's \u2014 involved in yesterday's incidents. But then Peglar had looked at the other men chosen to go on this predawn investigatory expedition \u2014 First Lieutenant Edward Little, Tom Johnson (Crozier's bosun's mate and old crewman from the south polar expedition), Captain of the Maintop Farr who _had_ been here yesterday, Dr. Goodsir, Lieutenant Le Vesconte from _Erebus,_ First Mate Robert Thomas, and a guard of four Marines with weapons \u2014 Hopcraft, Healey, and Pilkington under the command of Corporal Pearson.\n\nHarry Peglar hoped he was not flattering himself to think that, for whatever reason, Captain Crozier had chosen people he trusted for this outing. Malcontents and incompetents had been left behind at Terror Camp; the sea lawyer Hickey heading up a detail to dig Lieutenant Irving's grave for this afternoon's burial service.\n\nCrozier's party had left camp long before dawn, following the footprints from yesterday and the tracks of the Esquimaux sledge that had borne the body to the camp southeast by lantern light. When the tracks disappeared on the stony ridgelines, they were easily found in the snowy vales beyond. The temperature had risen at least fifty-five degrees during the night, bringing the air up to zero degrees or higher, and a thick fog had rolled in. Harry Peglar, a veteran of weather on most of the earth's seas and oceans, had no idea how it could be so foggy when there was no unfrozen liquid water within hundreds and hundreds of miles. Perhaps these were low clouds skimming across the surface of the pack ice and colliding with this godforsaken island that rose only a few yards above sea level at its highest point. The sunrise, when it came, was no sunrise at all but only a vague yellow glow in the swirling fog-cloud around them, seeming to come from all directions.\n\nThe dozen men stood in silence at the murder site for a few minutes. There was little to see. John Irving's cap had blown against a nearby boulder, and Farr retrieved it. There was frozen blood on the frozen stones, the heap of human guts next to that dark stain. A few tatters of ripped clothing.\n\n\"Lieutenant Hodgson, Mr. Farr,\" said Crozier, \"did you see any sign of the Esquimaux up here when Mr. Hickey led you to this scene?\"\n\nHodgson seemed confused by the question. Farr said, \"Other than their bloody handiwork, no, sir. We approached the ridgeline on our bellies and peered down into the valley using Mr. Hodgson's glass, and there they were. Still fighting over John's telescope and other spoils.\"\n\n\"Did you see them fight amongst themselves?\" snapped Crozier.\n\nPeglar never remembered seeing his captain \u2014 or any captain he had ever served under \u2014 look so tired. Crozier's eyes had visibly sunken in their sockets over the past days and weeks. Crozier's voice, always a bass bark of command, was now little more than a croak. It looked as if his eyes were ready to bleed.\n\nPeglar knew something about bleeding these days. He hadn't told his friend John Bridgens yet, but he was feeling the scurvy badly. His once-proud muscles were atrophying. His flesh was mottled with bruises. He'd lost two teeth in the past ten days. Every time he brushed his remaining teeth, the brush came away red. And every time he squatted to relieve himself, he shat blood.\n\n\"Did I actually _see_ the Esquimaux fighting amongst themselves?\" repeated Farr. \"Not really, sir. They were jostling and laughing, though. And two of the bucks were tugging at John's fine brass telescope.\"\n\nCrozier nodded. \"Let us go down in the valley, gentlemen.\"\n\nPeglar was shocked by the blood. He'd never seen the site of a land battle before, not even a small skirmish such as this, and while he had prepared himself to see the dead bodies, he'd not imagined how red the spilled blood would be on the snow.\n\n\"Someone's been here,\" said Lieutenant Hodgson.\n\n\"What do you mean?\" asked Crozier.\n\n\"Some of the bodies have been moved,\" said the young lieutenant, pointing to a man and then to another man and then to an old woman. \"And their outer coats \u2014 the fur coats, such as Lady Silence wears \u2014 and even some of their mittens and boots are gone. So are several of the weapons... harpoons and spears. See, you can see the imprint in the snow where they were lying yesterday. They're gone.\"\n\n\"Souvenirs?\" rasped Crozier. \"Did our men...\"\n\n\"No, sir,\" Farr said quickly and firmly. \"We threw some baskets and cooking pots and other things off the sled to make room and took that sled up the hill to load Lieutenant Irving's body. We were all together from then until we reached Terror Camp. No one lagged behind.\"\n\n\"Some of those pots and baskets have gone missing as well,\" said Hodgson.\n\n\"There seem to be some newer tracks here, but it's hard to tell since the wind was blowing last night,\" said Bosun's Mate Johnson.\n\nThe captain was going from corpse to corpse, rolling them over when they were facedown. He seemed to be studying each dead man's face. Peglar noticed that they were not all dead men \u2014 one was a boy. One was an old lady whose open mouth \u2014 as if frozen by Death into an eternal silent scream \u2014 looked like a black pit. There was much blood. One of the natives had received the full force of a shotgun blast at what must have been very short range, perhaps after he had already been hit by musket or rifle fire. The back of his head was gone.\n\nAfter inspecting each face as if hoping to find answers there, Crozier stood. The surgeon, Goodsir, who had also been looking carefully at the dead, said something softly into the captain's ear, pulling down his comforter scarf and the captain's as well while whispering. Crozier took a step back, looked at Goodsir as if in surprise, but then nodded.\n\nThe surgeon went to one knee by a dead Esquimaux and removed several surgical instruments from his bag, including one very long, curved, and serrated knife that reminded Peglar of the ice saws they used to cut chunks from the iron tanks of frozen water on the hold deck of _Terror._\n\n\"Dr. Goodsir needs to examine several of the savages' stomachs,\" Crozier said.\n\nPeglar imagined that nine others besides himself were wondering why. No one asked the question. The squeamish \u2014 including three of the Marines \u2014 looked away as the small surgeon tore open fur or animal-skin garments and began sawing on the first corpse's abdomen. The sound of the saw cutting into hard-frozen flesh reminded Peglar of someone sawing wood.\n\n\"Captain, who do you think might've fetched up the weapons and clothing?\" asked First Mate Thomas. \"One of the two who got away?\"\n\nCrozier nodded distractedly. \"Or others from their village, although it's hard to imagine a village on this godforsaken island. Perhaps these were part of a larger hunting group camped nearby.\"\n\n\"This group had so much food with them,\" said Lieutenant Le Vesconte. \"Imagine how much the main hunting party might have with them. We might be able to feed all one hundred and five of us.\"\n\nLieutenant Little smiled over his breath-rimed coat collars. \"Would you like to be the one to walk into their village or larger hunting party and politely ask them for some food or hunting advice? Now? After this?\" Little gestured toward the sprawled, frozen bodies and patches of red on the snow.\n\n\"I think we have to get away from Terror Camp and this island _now,_ \" said Second Lieutenant Hodgson. The young man's voice was quavering. \"They're going to kill us in our sleep. Look what they did to John.\" He stopped, visibly abashed.\n\nPeglar studied the lieutenant. Hodgson showed all the signs of starvation and exhaustion that the rest of them did, but not as many signs of scurvy. Peglar wondered if he would become unmanned like this if and when he saw a spectacle similar to what Hodgson had seen less than twenty-four hours earlier.\n\n\"Thomas,\" Crozier said softly to his bosun's mate, \"would you be so kind as to go over that next ridge and see if you can see anything? Specifically tracks leading away from here... and if so, how many and what kind?\"\n\n\"Aye, sir.\" The large mate jogged uphill through deep snow and onto the dark-gravel ridge.\n\nPeglar found himself watching Goodsir. The surgeon had cut open the greyish pink, distended stomach of the first Esquimaux man and then had gone on to the old woman and next the young boy. It was a terrible thing to watch. In each case, Goodsir \u2014 his hands bare \u2014 used a smaller surgical instrument to slit the stomach open and lifted out the contents, kneading through the frozen chunks and gobbets as if searching for a prize. Sometimes Goodsir snapped the frozen stomach contents into smaller bits with an audible crack. When he was finished with the first three corpses, Goodsir idly wiped his bare hands in the snow, tugged on his mittens, and whispered in Crozier's ear again.\n\n\"You can tell everyone,\" Crozier said loudly. \"I want everyone to hear this.\"\n\nThe little surgeon licked his cracked and bleeding lips. \"This morning I opened Lieutenant Irving's stomach...\"\n\n\"Why?\" shouted Hodgson. \"That was one of the few parts of John that the fucking savages did not mutilate! How could you?\"\n\n\"Silence!\" barked Crozier. Peglar noticed that the captain's old authoritative voice had returned for that command. Crozier nodded to the surgeon. \"Please continue, Dr. Goodsir.\"\n\n\"Lieutenant Irving had eaten so much seal meat and blubber that he was literally full,\" said the surgeon. \"He'd had a larger meal than any of us have had in months. Obviously it came from the Esquimaux's cache on their sledge. I was curious if the Esquimaux had eaten with him \u2014 if the contents of their stomachs would show they also had eaten seal blubber shortly before they died. With these three, it is obvious they did.\"\n\n\"They broke bread with him... ate their meat with him... and then killed him as he was leaving?\" said First Mate Thomas, obviously confused by this information.\n\nPeglar was also confused. It made no sense... unless these savages were as mercurial and treacherous in their temperament as some natives he had come across in the South Seas during the five-year voyage of the old _Beagle._ The foretop captain wished that John Bridgens were here to give his opinion on all this.\n\n\"Gentlemen,\" said Crozier, obviously including even the Marines, \"I wanted you all to hear this because I may require your knowledge of these facts at some future time, but I don't want anyone else to hear about it. Not until I say that it should be public knowledge. And I may never do so. If any of you tells anyone else \u2014 a single soul, your closest chum, if you so much as mumble this in your sleep \u2014 I swear to Christ I'll find out who disobeyed my order to silence and I'll leave that man behind on the ice without so much as an empty pan to shit in. Do I make myself clear, gentlemen?\"\n\nThe other men grunted affirmation.\n\nThomas Johnson returned then, puffing and wheezing his way down the hill. He paused and looked at the silent clump of men as if to ask what was wrong.\n\n\"What did you see, Mr. Johnson?\" Crozier asked briskly.\n\n\"Tracks, Captain,\" said the bosun's mate, \"but old ones. Heading southwest. The two who got away yesterday \u2014 and whoever came back to the valley to loot the parkas and weapons and pots and such \u2014 must have followed that track as they ran. I saw nothing new.\"\n\n\"Thank you, Thomas,\" said Crozier.\n\nThe fog whirled around them. Somewhere to the east, Peglar heard what sounded like big guns firing in a Naval engagement, but he'd heard that many times out here over the past two summers. It was distant thunder. In April. With the temperature still twenty degrees below freezing, at least.\n\n\"Gentlemen,\" said the captain, \"we have a burial to attend. Shall we head back?\"\n\nOn the long trek back, Harry Peglar mulled over what he had seen \u2014 the frozen entrails of an officer he liked, the bodies and still-bright blood in the snow, the missing parkas and weapons and tools, Dr. Goodsir's ghoulish examinations, Captain Crozier's odd statement that he might \"require your knowledge of these facts at some future time\" as if he was preparing them to act as jurors at some future court-martial or court of inquiry.\n\nPeglar anticipated writing all this down in the commonplace book he had been keeping for so long. And he hoped that he would find the opportunity to talk to John Bridgens after the burial service, before the groups of men from both boats went back to their own tents and mess circles and man-hauling teams. He wanted to hear what his dear wise Bridgens might have to say about all this.\n\nCROZIER\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _25 April, 1848_\n\nO death, where is thy sting? O grave, where is thy victory?'\"\n\nLieutenant Irving had been Crozier's officer, but Captain Fitzjames had a better voice \u2014 the lisp had all but disappeared \u2014 and a better way with Scripture, so Crozier was grateful that he was doing a majority of the burial-service reading.\n\nAll of the men in Terror Camp had turned out except those on watch, those in sick bay, or those performing essential services such as Lloyd in sick bay and Mr. Diggle and Mr. Wall and their mates labouring over the four whaleboat stoves cooking up some of the Esquimaux's fish and seal meat for dinner. At least eighty men were at this graveside about a hundred yards from camp, standing like dark wraiths in the still-swirling fog.\n\n\" 'The sting of death is sin; and the strength of sin is the law. But thanks be to God, which giveth us the victory through our Lord Jesus Christ. Therefore, my beloved brethren, be ye steadfast, unmovable, always abounding in the work of the Lord, forasmuch as ye know that your labour is not in vain in the Lord.'\"\n\nThe other surviving officers and two mates were to carry Irving to the grave. There was not enough wood at Terror Camp to make a coffin, but Mr. Honey, the carpenter, had found enough wood to knock together a door-sized pallet on which Irving's body, now securely sewn into canvas, could be transported on and upon which the body could be lowered into the grave. Although the ropes were set across the grave in proper Naval fashion, as they would be for any land burial, there would not be much lowering to do. Hickey and his men had been unable to dig deeper than three feet \u2014 the ground below that level was as hard-frozen as solid stone \u2014 so the men had gathered scores of large stones to lay over the body before piling on the frozen topsoil and gravel, then more stones to lay over that. No one had real hopes that it would keep the white bears or the other summer predators out, but the labour was a sign of most of the men's affection for John Irving.\n\n _Most of the men._\n\nCrozier glanced over at Hickey, standing next to Magnus Manson and the _Erebus_ gunroom steward who had been flogged after Carnivale, Richard Aylmore. There was a cluster of other malcontents around these men \u2014 several of the _Terror_ seamen who had been eager to kill Lady Silence even if it took a mutiny to do so back in January \u2014 but, like all the others standing around the pathetic hole in the ground, they had their Welsh wigs and caps off and their comforters pulled up to their noses and ears.\n\nCrozier's middle-of-the-night interrogation of Cornelius Hickey in the captain's command tent had been tense and terse.\n\n\"Good morning to you, Captain. Would you like me to tell you what I told Captain Fitzjames and...\"\n\n\"Take off your slops, Mr. Hickey.\"\n\n\"Pardon me, sir?\"\n\n\"You heard me.\"\n\n\"Aye, sir, but if you want to hear how it was when I saw the savages murderin' poor Mr. Irving...\"\n\n\"It's _Lieutenant_ Irving, Caulker's Mate. I heard your story from Captain Fitzjames. Do you have anything to add or retract from it? Anything to amend?\"\n\n\"Ah... no, sir.\"\n\n\"Take those outer slops off. Mittens too.\"\n\n\"Aye, sir. There, sir, how's 'at? Shall I just set 'em over on the...\"\n\n\"Drop them on the floor. Jackets off too.\"\n\n\"My jackets, sir? It's bloody cold in here... yes, sir.\"\n\n\"Mr. Hickey, why did you volunteer to go search for Lieutenant Irving when he hadn't yet been gone much more than an hour? No one else was worried about him.\"\n\n\"Oh, I don't think I volunteered it, Captain. My recollection is that Mr. Farr asked me to go look for...\"\n\n\"Mr. Farr reported that you asked several times if Lieutenant Irving wasn't overdue and volunteered to go find him on your own while the others rested after their meal. Why did you do that, Mr. Hickey?\"\n\n\"If Mr. Farr says that... well, we must've been worried about him, Captain. The lieutenant, I mean.\"\n\n\"Why?\"\n\n\"May I put my jackets and slops back on, Captain? It's bloody freezing in...\"\n\n\"No. Take off your waistcoat and sweaters. Why were you worried about Lieutenant Irving?\"\n\n\"If you're concerned... that is, thinking I was wounded today, Captain, I wasn't. The savages never saw me. No wounds on me, sir, I assure you.\"\n\n\"Take that sweater off as well. Why were you worried about Lieutenant Irving?\"\n\n\"Well, the lads and me... you know, Captain.\"\n\n\"No.\"\n\n\"We was just concerned, you know, that one of our party was missin', like. Also, sir, I was cold, sir. We'd been sittin' around to eat what little cold food we had. I thought that walkin', following the lieutenant's tracks to make sure he was all right, would warm me up, sir.\"\n\n\"Show me your hands.\"\n\n\"Pardon me, Captain?\"\n\n\"Your hands.\"\n\n\"Aye, sir. Pardon my shaking, sir. I ain't been warm all day and with all my layers off but this shirt and...\"\n\n\"Turn them over. Palms up.\"\n\n\"Aye, sir.\"\n\n\"Is that blood under your nails, Mr. Hickey?\"\n\n\"Could be, Captain. You know how it is.\"\n\n\"No. Tell me.\"\n\n\"Well, we ain't had real water what to bath ourselves in for months, sir. And what with the scurvy and dysentery-like, there's a certain amount of bleedin' when we see to the necessaries...\"\n\n\"Are you saying that a Royal Navy petty officer on my ship wipes his arse with his fingers, Mr. Hickey?\"\n\n\"No, sir... I mean... may I put my layers back on now, Captain? You can see I ain't wounded or anything. This cold is enough to shrink a man's...\"\n\n\"Take your shirts and undershirts off.\"\n\n\"Are you serious, sir?\"\n\n\"Don't make me ask a second time, Mr. Hickey. We don't have a brig. Any man I send to the brig will spend time chained to one of the whaleboats.\"\n\n\"Here, sir. How's this. Just me flesh, freezing as it is. If my poor missus could see me now...\"\n\n\"It didn't say on your muster papers that you were married, Mr. Hickey.\"\n\n\"Oh, my Louisa's been dead going on seven years now, Captain. Of the pox. God rest her soul.\"\n\n\"Why did you tell some of the other men before the mast that when it came time to kill officers, Lieutenant Irving should be the first?\"\n\n\"I never said no such thing, sir.\"\n\n\"I have reports of you saying that and other mutinous statements going back to before the Carnivale on the ice, Mr. Hickey. Why did you single out Lieutenant Irving? What had that officer ever done to you?\"\n\n\"Why, nothing, sir. And I never said no such thing. Bring in the man who said I did and I'll dispute it to his face and spit in his eye.\"\n\n\"What had Lieutenant Irving ever done to you, Mr. Hickey? Why did you tell other men from both _Erebus_ and _Terror_ that Irving was a whoremaster and a liar?\"\n\n\"I swear to you, Captain... pardon my teeth chattering, Captain, but _Jesus Christ_ the night is cold against the bare skin. I swear to you, I didn't say no such thing. A lot of us looked on poor Lieutenant Irving sorta like a son, Captain. A son. It was only my worry for him out there today that made me go check on him. Good thing I did, too, sir, or we would've never caught the murdering bastards who...\"\n\n\"Put your layers on, Mr. Hickey.\"\n\n\"Aye, sir.\"\n\n\"No. Do it outside. Get out of my sight.\"\n\n\" 'Man that is born of a woman hath but a short time to live and is full of misery,' \" intoned Fitzjames. \" 'He cometh up and is cut down, like a flower; he fleeth as it were a shadow, and never continueth in one stay.' \"\n\nHodgson and the other pallbearers were using great care in lowering the pallet with Irving's canvas-wrapped body to the ropes held in place above the shallow hole by some of the healthier seamen. Crozier knew that Hodgson and Irving's other friends had gone into the postmortem tent one at a time to pay their respects before the lieutenant had been sewn into his sail shroud by Old Murray. The visitors had set several tokens of their affection next to the lieutenant's body \u2014 the recovered brass telescope, its lenses shattered in the shooting, that the boy had so esteemed, a gold medal with his name engraved on it that he had won in competitions on the gunnery ship HMS _Excellent,_ and at least one five-pound note, as if some old wager had been paid at last. For some reason \u2014 optimism? youthful na\u00efvet\u00e9? \u2014 Irving had packed his dress uniform in his small bag of personal belongings, and he was being buried in it now. Crozier wondered idly if the gilt buttons on the uniform \u2014 each bearing the image of an anchor surrounded by a crown \u2014 would be there when nothing else but the boy's bleached bones and the gold gunnery medal survived the long process of decay.\n\n\" 'In the midst of life we are in death,' \" Fitzjames recited from memory, his voice sounding tired but properly resonant, \" 'of whom may we seek for succour, but of thee, O Lord, who for our sin art justly displeased?' \"\n\nCaptain Crozier knew that there was one other item sewn into the sail-shroud with Irving, one that no one else knew about. It lay under his head like a pillow.\n\nIt was a gold, green, red, and blue silk Oriental handkerchief, and Crozier had surprised the giver by coming into the postmortem tent after Goodsir, Lloyd, Hodgson, and the others had departed, just before Old Murray the sailmaker was to enter and sew up the shroud he had prepared and upon which Irving already lay in state.\n\nLady Silence had been there, bending over the corpse, setting something beneath Irving's head.\n\nCrozier's first impulse had been to reach for his pistol in his greatcoat pocket, but he'd frozen in place as he saw the Esquimaux girl's eyes and face. If there were no tears in those dark, hardly human eyes, there was something else luminous there with some emotion he could not identify. Grief? The captain did not think so. It was more some kind of complicit recognition at seeing Crozier. The captain felt the same strange stirring in his head that he had so often felt around his Memo Moira.\n\nBut the girl obviously had set the Oriental handkerchief carefully in place under the dead boy's head as some sort of gesture. Crozier knew the handkerchief had been Irving's \u2014 he'd seen it on special occasions as far back as the day they'd sailed in May of 1845.\n\nHad the Esquimaux wench stolen it? Plundered it from his dead body just yesterday?\n\nSilence had followed Irving's sledge party from _Terror_ to Terror Camp more than a week ago and then had just disappeared, never joining the men in the camp. Almost everyone, excluding Crozier, who still held hopes she might lead them to food, had considered this good riddance. But all during this terrible morning, part of Crozier had wondered if somehow Silence had been responsible for his officer's murder out there on the windswept gravel ridge.\n\nHad she led her Esquimaux hunter friends back here to raid the camp and run into Irving on the way, first giving a fete to the starving man with meat and then murdering him in cold blood to keep him from telling the others here of his encounter? Had Silence been the \"possibly a young woman\" that Farr and Hodgson and the others had caught a glimpse of, fleeing with an Esquimaux man with a headband? She could have changed her parka if she had returned to her village in the past week, and who could tell young Esquimaux wenches apart at a glance?\n\nCrozier considered all of these things, but now in a time-stopped moment \u2014 both he and the young woman were startled into immobility for long seconds \u2014 the captain looked into her face and knew, whether in his heart or in what Memo Moira insisted was his second sight, that she wept inside for John Irving and was returning a gift of the silk handkerchief to the dead man.\n\nCrozier guessed that the handkerchief had been presented to her during the February visit to the Esquimaux's snow-house that Irving had dutifully reported to the captain... but had reported with few details. Now Crozier wondered if the two had been lovers.\n\nAnd then Lady Silence was gone. She'd slipped under the tent flap and was gone without a sound. When Crozier later queried the men in the camp and those on guard if they had seen anything, none had.\n\nAt that moment in the tent, the captain had gone over to Irving's body, looked down at the pale, dead face made even whiter with the small pillow of the brightly covered handkerchief behind it, and then he had pulled the canvas over the lieutenant's face and body, shouting for Old Murray to come in and do the sewing.\n\n\" 'Yet, O Lord God most holy, O Lord most mighty, O holy and most merciful Saviour,' \" Fitzjames was saying, \" 'deliver us not into the bitter pains of eternal death.'\n\n\" 'Thou knowest, Lord, the secrets of our hearts; shut not thy merciful ears to our prayer; but spare us, Lord most holy, O God most mighty, O holy and merciful Saviour, thou most worthy Judge eternal, suffer us not, at our last hour, for any pains of death, to fall from thee.'\"\n\nFitzjames's voice fell silent. He stepped back from the grave. Crozier, lost in reverie, stood for a long moment until a shuffling of feet made him realize that his part of the service had arrived.\n\nHe walked to the head of the grave.\n\n\" 'We therefore commit the body of our friend and officer John Irving to the deep,' \" he rasped, also reciting from memory that remained all too clear from many repetitions despite the pall of fatigue in his mind, \"'to be turned into corruption, looking for the resurrection of the body, when the Sea and the Earth shall give up their dead.'\" The body was lowered the three feet, and Crozier tossed a handful of frozen soil onto it. The gravel made a strangely moving rasping sound as it landed on the canvas above Irving's face and slid to the sides. \"'And the life of the world to come, through our Lord Jesus Christ, who at his coming shall change our vile body, that it may be like his glorious body, according to the mighty working, whereby he is able to subdue all things to himself.' \"\n\nThe service was over. The ropes had been retrieved.\n\nMen stamped cold feet, tugged on their Welsh wigs and caps, rewrapped their comforters, and filed back through the fog to Terror Camp for their hot dinner.\n\nHodgson, Little, Thomas, Des Voeux, Le Vesconte, Blanky, Peglar, and a few of the other officers stayed behind, dismissing the seamen's detail that had been waiting to bury the body. The officers shoveled soil and began setting in the first layer of stones together. They wanted Irving buried as best he could be under the circumstances.\n\nWhen they finished, Crozier and Fitzjames walked away from the others. They would eat their dinners much later \u2014 for now they planned to walk the two miles up to Victory Point where Graham Gore had left his brass canister and optimistic message in James Ross's old cairn almost a year ago.\n\nCrozier planned to leave word there today on what the fate of their expedition had been in the past ten and a half months since Gore's note had been written and on what they planned to do next.\n\nPlodding tiredly through the fog, hearing one of the ship's bells ringing for dinner somewhere in the roiling fog behind them \u2014 they had, of course, brought both _Terror_ 's and _Erebus_ 's bells along in the whaleboats dragged across the frozen sea to camp when the ships were abandoned \u2014 Francis Crozier hoped to Christ that he would decide on their course of action by the time that he and Fitzjames reached the cairn. If he could not, he thought, he was afraid he might start weeping.\n\nPEGLAR\n\n _Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n _25 April, 1848_\n\nThere hadn't been enough fish and seal on the sled to serve it as a main dish to ninety-five or a hundred men \u2014 a few were too ill to eat anything solid \u2014 and even Mr. Diggle's and Mr. Wall's record at routinely performing loaves-and-fishes miracles with the limited ships' stores did not allow them to fully succeed at this one (especially since some of the food on the Esquimaux sledge had been particularly putrid), but every man managed to get a taste of the savory blubber or fish along with the prepared Goldner soups or stews or vegetables.\n\nHarry Peglar enjoyed the meal even though he was shaking with cold as he ate it and knew that it would only provoke the diarrhea that was already ripping him apart every day.\n\nAfter the meal and before beginning their scheduled duties, Peglar and Steward John Bridgens walked together with their tin mugs of tepid tea. The fog muffled their own voices even as it seemed to amplify sounds from far away. They could hear men arguing over a card game in one of the tents on the far side of Terror Camp. From the northwest \u2014 the direction the two captains had walked before dinner \u2014 came the artillery rumble of thunder out over the pack ice. That sound had been going on all day, but no storm had arrived.\n\nThe two paused at the long line of boats and boat-sledges drawn up above the tumble of ice that would be the inlet's shoreline if there ever came a thaw to the sea.\n\n\"Tell me, Harry,\" said Bridgens, \"which of these boats will we be taking if or when we must go to the ice again?\"\n\nPeglar sipped his tea and pointed. \"I'm not certain, but I think Captain Crozier has decided to take ten of the eighteen here. We don't have men well enough to haul more these days.\"\n\n\"Then why did we man-haul all eighteen to Terror Camp?\"\n\n\"Captain Crozier considered the possibility that we'd stay at Terror Camp for another two or three months, perhaps letting the ice around this point melt. We would have been better off with more boats, keeping some in reserve should others be damaged. And we could have hauled much more in the way of food, tents, and supplies in eighteen boats. With more than ten men in each boat now, it'll be damned crowded, and we'll have to leave too much of the stores behind.\"\n\n\"But you think we'll leave for the south with only ten boats, Harry? And soon?\"\n\n\"I hope to Christ we do,\" said Peglar. He told Bridgens about what he had seen that morning, what Goodsir had said about the Esquimaux's stomachs being as full of seal meat as Irving's had been, and how the captain had treated those present, perhaps excepting the Marines, as a potential Board of Inquiry. He added that the captain had sworn them to secrecy.\n\n\"I think,\" John Bridgens said softly, \"that Captain Crozier is not convinced that the Esquimaux killed Lieutenant Irving.\"\n\n\"What? Who else could...\" Peglar stopped. The cold and nausea that were always with him now seemed to surge up and through him. He had to lean against a whaleboat to keep his knees from buckling. He had never considered for an instant that anyone other than the savages could have done what he'd seen done to John Irving. He thought of the frozen pile of grey entrails on the ridgeline.\n\n\"Richard Aylmore is saying that the officers have led us into this mess,\" said Bridgens in a voice so low it was almost a whisper. \"He's telling everyone who won't inform on him that we should kill the officers and parcel out the extra food rations amongst the men. Aylmore in our group and that caulker's mate in yours say we should go back to _Terror_ at once.\"\n\n\"Back to _Terror_...\" repeated Peglar. He knew that his mind was dull with illness and exhaustion these days, but the idea made no sense at all. The ship was locked in the ice far out there and would be for months more, even if summer _did_ condescend to appear this year. \"Why don't I hear these things, John? I've heard none of this seditious whispering.\"\n\nBridgens smiled. \"They don't trust you not to tell, my dear Harry.\"\n\n\"But they trust you?\"\n\n\"Of course not. But I hear _everything_ sooner or later. Stewards are invisible, y'know, being neither fish nor fowl nor good red meat. Speaking of which, that was a delightful meal, wasn't it? Perhaps the last relatively fresh food we shall ever eat.\"\n\nPeglar didn't answer. His mind was racing. \"What can we do to warn Fitzjames and Crozier?\"\n\n\"Oh, they have this information about Aylmore and Hickey and the others,\" the old steward said nonchalantly. \"Our captains have their own sources before the mast and around the scuttlebutts.\"\n\n\"The scuttlebutts have been frozen solid for months,\" said Peglar.\n\nBridgens chuckled. \"That seems to be a very good metaphor, Harry, and all the more ironic for its literalness. Or at least an amusing euphemism.\"\n\nPeglar shook his head. He still felt the nausea from the idea that amidst all this illness and terror, any man among them would turn on another.\n\n\"Tell me, Harry,\" said Bridgens, patting the inverted hull of the first whaleboat with his worn mitten. \"Which of these boats might we be hauling with us and which will be left behind?\"\n\n\"The four whaleboats will go for sure,\" Peglar said absently, still mulling over this talk of mutiny and what he had seen that morning. \"The jolly boats are as long as the whaleboats, but damned heavy. If I were the captain, I might leave them behind and take the four cutters instead. They're only twenty-five feet long, but much lighter than whaleboats. But their draft may be too much for the Great Fish River if we can get there. The ships' smaller boats and dinghies are too light for the open sea and too flimsy for much hauling and river work.\"\n\n\"So it's the four whaleboats, four cutters, and two pinnaces, you think?\" asked Bridgens.\n\n\"Yes,\" said Peglar, and had to smile. For all his years at sea and all his thousands of volumes read, Subordinate Officers' Steward John Bridgens still knew very little about some things nautical. \"I think those ten, yes, John.\"\n\n\"At best,\" said Bridgens, \"if most of the sick recover, that leaves only ten of us to man-haul each boat. Can we do that, Harry?\"\n\nPeglar shook his head again. \"It won't be like the sea ice crossing from _Terror,_ John.\"\n\n\"Well, thank the dear Lord for that small blessing.\"\n\n\"No, I mean that we'll almost certainly be man-hauling these boats over land rather than sea ice. It'll be much harder than the crossing from _Terror,_ where we man-hauled only two boats at a time and could put as many men on a team as we needed to get over the rough parts. And the boats now will be even more heavily laden with stores and our sick than before. I suspect that we'll have twenty or more in harness for each boat hauled. Even then, we'll have to haul the ten boats in relays.\"\n\n\"Relays?\" said Bridgens. \"Dear heavens, it will take us forever to move even ten boats if we're constantly going back and forth. And the weaker and sicker we become, the slower we will go.\"\n\n\"Yes,\" said Peglar.\n\n\"Is there any chance that we shall get these boats all the way to the Great Fish River and then up the river to Great Slave Lake and the outpost there?\"\n\n\"I doubt it,\" said Peglar. \"Perhaps if a few of us survive long enough to get the boats to the mouth of the river and the right boats make it and they're rigged just perfectly for river running and... but, no, I doubt if there's any real chance.\"\n\n\"Then why on earth would Captains Crozier and Fitzjames put us through such labour and misery if there is no chance?\" asked Bridgens. The older man's voice did not sound aggrieved or anxious or desperate, merely curious. Peglar had heard John pose a thousand questions about astronomy, natural history, geology, botany, philosophy, and a score of other subjects in precisely that same soft, mildly curious tone. With most of the other questions, it had been the teacher who knew the answer quizzing his student in a polite way. Here, Peglar was sure that John Bridgens did not know the answer to this question.\n\n\"What's the alternative? \" asked the foretop captain.\n\n\"We could stay here at Terror Camp,\" said Bridgens. \"Or even return to _Terror,_ once our numbers have... decreased.\"\n\n\"To do what?\" demanded Peglar. \"Just to wait to die?\"\n\n\"To wait in comfort, Harry.\"\n\n\"To _die?_ \" said Peglar, realizing that he was almost shouting. \"Who the fuck wants to wait in comfort to die? At least if we get the boats to the coast \u2014 _any_ of the boats \u2014 some of us may have a chance. There might be open water east to Boothia. We may be able to force passage up the river. At least _some_ of us. And those who make it will at least be able to tell the rest of our loved ones what happened to us, where we were buried, and that we were thinking of them in the end.\"\n\n\"You _are_ my loved one, Harry,\" said Bridgens. \"The only man or woman or child left in the world who cares whether I am alive or dead, much less what I may have thought before I fell or where my bones will lie.\"\n\nPeglar, still angry, felt his heart pounding inside his chest. \"You're going to outlive me, John.\"\n\n\"Oh, at my age, and with my infirmities and proclivities toward illness, I hardly think...\"\n\n\" _You're going to outlive me, John,_ \" grated Peglar. He shocked himself by the intensity of his voice and Bridgens blinked and fell silent. Peglar took the older man's wrist. \"Promise me you'll do one thing for me, John.\"\n\n\"Of course.\" There was none of the usual banter or irony in Bridgens' voice.\n\n\"My diary... it's not much, I have trouble even thinking, much less writing these days... I'm quite sick with this God-damned scurvy, John, and it seems to addle my brain... but I've kept the diary for the past three years. My thoughts are in it. All of the events we've experienced are put down there. If you could take it when I... when I leave you... just take it back with you to England, I'd appreciate it.\"\n\nBridgens only nodded.\n\n\"John,\" said Harry Peglar, \"I think Captain Crozier is going to decide to take us on the march soon. Very soon. He knows that every day we wait here we get weaker. Soon we won't be able to haul boats at all. We'll begin dying by the dozens here at Terror Camp before long, and it won't take that thing on the ice to carry us away or kill us in our beds.\"\n\nBridgens nodded again. He was looking down at his mittened hands.\n\n\"We're not on the same man-hauling teams, won't share the same boats, and may not even end up together if the captains decide to try for different escape routes,\" continued Peglar. \"I want to say good-bye today and never have to do it again.\"\n\nBridgens nodded mutely. He was looking at his boots. The fog rolled over the boats and sledges and moved around them like some alien god's cold breath.\n\nPeglar hugged him. Bridgens stood upright and brittle for a moment and then returned the hug, both men clumsy in their many layers and frozen slops.\n\nThe captain of the foretop turned then and walked slowly back toward Terror Camp and his tiny circular Holland tent with its group of off-duty shivering, unwashed men huddling together in inadequate sleeping bags.\n\nWhen he paused and looked back toward the line of boats, there was no sign of Bridgens at all. It was as if the fog had swallowed him without a trace.\n\nCROZIER\n\n_Lat. 69\u00b0 37\u2032 42\u2033 N., Long. 98\u00b0 41\u2032 W._\n\n_25 April, 1848_\n\nHe fell asleep while walking.\n\nCrozier had been talking to Fitzjames about arguments for and against letting the men spend more days at Terror Camp as the two walked the two miles north through the fog to James Ross's cairn when suddenly Fitzjames was shaking him awake.\n\n\"We're here, Francis. This is the large white boulder near the shore ice. Victory Point and the cairn must be to our left. Were you really sleeping while walking?\"\n\n\"No, of course not,\" rasped Crozier.\n\n\"Then what did you mean when you said, 'Watch out for the open boat with the two skeletons'? and 'watch out for the girls and the table rappings.' It made no sense. We were discussing whether Dr. Goodsir should stay behind at Terror Camp with the seriously ill men while the stronger ones try for Great Slave Lake with just four boats.\"\n\n\"Just thinking aloud,\" muttered Crozier.\n\n\"Who is Memo Moira?\" asked Fitzjames. \"And why should she not send you to Communion?\"\n\nCrozier pulled his cap and wool scarves off, letting the fog and cold air slap his face as he walked up the slow rise. \"Where the hell is the cairn?\" he snapped.\n\n\"I don't know,\" said Fitzjames. \"It should be right here. Even on a sunny, clear day, I walk this inlet coastline to the white boulder near the bergs and then left up to the cairn at Victory Point.\"\n\n\"We can't have walked past it,\" said Crozier. \"We'd be out on the fucking pack ice.\"\n\nIt took them almost forty-five minutes to find the cairn in the fog. At one point when Crozier said, \"The God-damned white thing from the ice has taken it and hidden it somewhere to confound us,\" Fitzjames had only looked at his commanding officer and said nothing.\n\nFinally, feeling their way along together like two blind men \u2014 not risking separating in the roiling fog, sure that they wouldn't even hear the others' calls over the constant drumbeat of approaching thunder \u2014 they literally stumbled into the cairn.\n\n\"This isn't where it was,\" croaked Crozier.\n\n\"It doesn't seem to be,\" agreed the other captain.\n\n\"Ross's cairn with Gore's note in it was at the top of the rise at the end of Victory Point. This must be a hundred yards to the west of there, almost down in this valley.\"\n\n\"It is very odd,\" said Fitzjames. \"Francis, you've come to the arctic so many times. Is this thunder \u2014 and the lightning if it comes \u2014 so common up here so early?\"\n\n\"I've never seen or heard either before midsummer,\" rasped Crozier. \"And never like this. It sounds like something worse.\"\n\n\"What could be worse than a thunderstorm in late April with the temperature still below zero?\"\n\n\"Cannon fire,\" said Crozier.\n\n\"Cannon fire?\"\n\n\"From the rescue ship that came down open leads all the way from Lancaster Strait and through Peel Sound only to find _Erebus_ crushed and _Terror_ abandoned. They're firing their guns for twenty-four hours to get our attention before sailing away.\"\n\n\"Please, Francis, stop,\" said Fitzjames. \"If you continue, I may vomit. And I've already done my vomiting for today.\"\n\n\"Sorry,\" said Crozier, fumbling in his pockets.\n\n\"Is there really any chance that it's guns firing for us?\" asked the younger captain. \"It _sounds_ like guns.\"\n\n\"Not a snowball's chance in Sir John Franklin's Hell,\" said Crozier. \"That pack ice is solid all the way to Greenland.\"\n\n\"Then where is the fog coming from?\" asked Fitzjames, his voice more idly curious than plaintive. \"Are you searching your pockets for something in particular, Captain Crozier?\"\n\n\"I forgot to bring the brass messenger canister we brought from _Terror_ for this note,\" Crozier admitted. \"I felt the lump in my slops pocket during the burial service and thought I had it, but it's only my God-besotted pistol.\"\n\n\"Did you bring paper?\"\n\n\"No. Jopson had some ready, but I left it in the tent.\"\n\n\"Did you bring a pen? Ink? I find that if I do not carry the ink pot in a pouch close to my skin, it freezes very quickly.\"\n\n\"No pen or ink,\" admitted Crozier.\n\n\"It's all right,\" said Fitzjames. \"I have both in my waistcoat pocket. We can use Graham Gore's note... write on it.\"\n\n\"If this is the same damned cairn,\" muttered Crozier. \"Ross's cairn was six feet tall. This thing hardly comes up to my chest.\"\n\nBoth men fumbled to remove rocks from a part of the cairn far down on the leeward side. They did not want to have to dismantle the entire thing and then have to rebuild it.\n\nFitzjames reached into the dark hole, fumbled around a second, and withdrew a brass cylinder, tarnished but still intact.\n\n\"Well I'll be damned and dressed in cheap motley,\" said Crozier. \"Is it Graham's?\"\n\n\"It has to be,\" said Fitzjames. Tugging his mitten off with his teeth, he clumsily unfurled the parchment note and began to read.\n\n_28 of May 1847. HM Ships Erebus and Terror... Wintered in the Ice in Lat. 70\u00b005\u2032 N. Long. 98\u00b023\u2032 W. After having wintered in 1846\u20137 at Beechey Island in Lat. 74\u00b043\u2032 28\u2033 N Long..._\n\nFitzjames interrupted himself. \"Wait, that's incorrect. We spent the winter of 1845 _to_ '46 at Beechey, not the winter of '46 to '47.\"\n\n\"Sir John dictated this to Graham Gore before Gore left the ships,\" rasped Crozier. \"Sir John must have been as tired and confused then as we are now.\"\n\n\"No one has ever been as tired and confused as we are now,\" said Fitzjames. \"Here, later, it goes on \u2014 'Sir John Franklin commanding the expedition. All well.' \"\n\nCrozier did not laugh. Or weep. He said, \"Graham Gore deposited the note here just a week before Sir John was killed by the thing on the ice.\"\n\n\"And one day before Graham himself was killed by the thing on the ice,\" said Fitzjames. \"'All well.' That seems like another lifetime, does it not, Francis? Can you remember a time when any of us could write such a thing with an easy conscience? There's blank space around the edge of the message if you want to write there.\"\n\nThe two huddled on the lee side of the stone cairn. The temperature had dropped and the wind had come up, but the fog continued to swirl around them as if unaffected by mere wind or temperature. It was beginning to get dark. To the northwest, the sound of guns rumbled on.\n\nCrozier breathed on the tiny portable ink pot to warm the ink, dipped the pen through the scrim of ice, rubbed the nib against his frozen sleeve, and began writing.\n\n_(25th April) \u2014 HM's ships Terror and Erebus were deserted on the 22nd April 5 leagues NNW of this, having been beset since 12th Septr. 1846. The Officers and Crews, consisting of 105 souls, under the command of Captain F.R.M. Crozier landed here \u2014 in Lat. 69\u00b0 37\u2032 42\u2033 Long. 98\u00b0 41\u2032. This paper was found by Lt. Irving under the cairn supposed to have been built by Sir James Ross in 1831, 4 miles to the Northward, where it had been deposited by the late Commander Gore in June 1847. Sir James Ross' pillar has not however been found, and the paper has been transferred to this position which is that in which Sir J Ross' pillar was erected \u2014_\n\nCrozier stopped writing. _What the hell am I saying?_ he thought. He squinted to reread the last few sentences \u2014 _\"Under the cairn supposed to have been built by Sir James Ross in 1831\"? \"Sir James Ross's pillar has not however been found\"?_\n\nCrozier let out a tired sigh. John Irving's first order upon ferrying the first load of mat\u00e9riel from _Erebus_ and _Terror_ long ago last August to begin the stockpile that would become Terror Camp was to find Victory Point and Ross's cairn again, then set up the cache for Terror Camp a few miles south of it along a more sheltered inlet. Irving had marked the cairn on their earliest crudely drawn maps as four miles from the cache point rather than the actual two miles, but they'd quickly discovered their mistake during subsequent man-haulings. In Crozier's fatigue now, his mind kept insisting that the canister with Gore's message had been moved from some false James Ross cairn to this real James Ross cairn.\n\nCrozier shook his head and looked at Fitzjames, but the other captain was resting his arms on his raised knees and his head on his arms. He was snoring softly.\n\nCrozier held the sheet of paper, pen, and tiny ink pot in one hand and scooped up snow with his other mittened hand, rubbing some on his face. The shock of the cold made him blink.\n\n_Concentrate, Francis. For the sake of Christ, concentrate._ He wished he had another sheet of paper so that he could start over. Squinting at the cramped scrawl moving around the margins of the paper, words crawling like tiny ants \u2014 the center of the paper already filled with the official typeset information stating officiously **WHOEVER finds this paper is requested to forward it to the Secretary of the Admiralty,** then several more paragraphs repeating the instruction in French, German, Portugese, and other languages, then with Gore's scrawl above that \u2014 Crozier did not recognize his own handwriting. The script was palsied, cramped, tenuous, obviously the hand of a terrified or freezing or dying man.\n\nOr of all three.\n\n_It doesn't matter,_ he thought. _Either no one is ever going to read this or they will read it long after we are all dead. It doesn't matter at all. Perhaps Sir John always understood this. Perhaps this is why he left none of the brass message canisters behind on Beechey. He knew all along._\n\nHe dipped the pen in the rapidly freezing ink and wrote again.\n\n_Sir John Franklin died on 11th of June 1847 and the total loss by deaths on the Expedition has been to this date 9 officers and 15 Men._\n\nCrozier stopped again. Was that correct? Had he included John Irving in the total? He couldn't do the arithmetic. There had been 105 souls under his care yesterday... 105 when he had left _Terror,_ his ship, his home, his wife, his life... he would leave the number.\n\nUpside down at the top of the sheet, on the bit of white space left, he scrawled _F. R. M. Crozier_ and after it wrote _Captain and Senior Officer._\n\nHe nudged Fitzjames awake. \"James... sign your name here.\"\n\nThe other captain rubbed his eyes, peered at the paper but did not seem to take time to read it, and signed his name where Crozier pointed.\n\n\"Add 'Captain HMS _Erebus,_ ' \" said Crozier.\n\nFitzjames did so.\n\nCrozier folded the paper, slid it back into the brass canister, sealed it, and set the cylinder back in the cairn. He pulled his mitten on and fumbled the stones back into place.\n\n\"Francis, did you tell them where we are headed and when we're leaving?\"\n\nCrozier realized that he had not. He started to explain why... why it seemed to be a sentence of death for the men whether they stayed or went. Why he had not decided yet between man-hauling for distant Boothia or toward George Back's fabled but terrible Great Fish River. He started to explain to Fitzjames how they were fucked coming and fucked going and why no one was ever going to read the fucking note anyway, so why not just...\n\n\"Shhh!\" hissed Fitzjames.\n\nSomething was circling them, just out of sight in the rolling, swirling fog. Both men could hear heavy footsteps in the gravel and ice. Something very large was breathing. It was walking on all fours, not more than fifteen feet from them in the thick fog, the sound of huge paws clearly audible over the heavy-gun rumble of distant thunder.\n\n_Hu-uf, hu-uf, hu-uf._\n\nCrozier could hear the exhalations with each heavy footfall. It was behind them now, circling the cairn, circling them.\n\nBoth men got to their feet.\n\nCrozier fumbled his pistol out. He pulled off his mitten and cocked the weapon as the footsteps and breathing stopped directly ahead of them but still out of sight in the fog. Crozier was certain he could smell its fish-and-carrion breath.\n\nFitzjames, who was still holding the ink pot and pen Crozier had given back to him and who had no pistol with him, pointed at the fog to where he thought the thing waited.\n\nGravel crunched as the thing moved stealthily toward them.\n\nSlowly a triangular head materialized in the fog five feet above the ground. Wet white fur blended with the mist. Inhuman black eyes studied them from only six feet away.\n\nCrozier aimed the pistol at a point just above that head. His hand was so firm and steady that he did not even have to hold his breath.\n\nThe head moved closer, floating as if it were unattached to any body. Then the giant shoulders came into view.\n\nCrozier fired, making sure to shoot high so as not to strike that face.\n\nThe report was deafening, especially to nervous systems set on edge by scurvy.\n\nThe white bear, little more than a cub, let out a startled _woof,_ reared back, wheeled, and ran off on all fours, disappearing into the fog in seconds. The scrabbling, running paw steps on gravel were audible for a long minute after, heading toward the sea ice to the northwest.\n\nCrozier and Fitzjames started laughing then.\n\nNeither man could stop. Every time one of them would slow in the laughter, the other would begin and then both would be caught up again in the mad, senseless hilarity.\n\nThey clutched their own sides from the pain of the laughter against their bruised ribs.\n\nCrozier dropped the pistol and both men started laughing harder.\n\nThey clapped each other's backs, pointed toward the fog, and laughed until the tears froze on their cheeks and whiskers. They clutched each other for support while they laughed harder.\n\nBoth captains collapsed on the gravel and leaned back against the cairn, that action alone causing the laughter to return in force.\n\nEventually the guffaws turned to giggles and the giggles into embarrassed snorts, the snorts into a few final laughs, and finally those died into a mutual gasping for air.\n\n\"You know what I would give my left bollock for right now?\" asked Captain Francis Crozier.\n\n\"What?\"\n\n\"A glass of whiskey. Two glasses, I mean. One for me and one for you. The drinks would be on me, James. I'm standing you to a round.\"\n\nFitzjames nodded, wiping ice from his eyelids and picking frozen snot from his reddish mustache and beard. \"Thank you, Francis. And I'd lift the first toast to you. I've never had the honour of serving under a better commander or a finer man.\"\n\n\"Could I please have the ink pot and pen back?\" said Crozier.\n\nPulling his mitten back on, he fumbled the stones out, found the canister, opened it, spread the sheet of paper out upside down on his knee, tugged his mitten off again, cracked the ice in the ink pot with the pen, and in the tiny space remaining under his signature, wrote,\n\n_And start tomorrow, 26 th, for Back's Fish River._\n\nGOODSIR\n\n _Lat. 69\u00b0 ?\u2032 ?\u2033 N., Long. 98\u00b0 ?\u2032 ?\u2033 W._\n\n _Comfort Cove, 6 June, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Tuesday, 6 June \u2014 Captain Fitzjames has finally died. It is a Blessing._\n\n _Unlike the others who have died in the last Six Weeks since we first started Hauling the Boats south (a Living Hell of a vocation from which even the Ships' Only Surviving Surgeon is not exempted), the Captain, in my opinion, did not perish from the Scurvy._\n\n _He_ had _Scurvy, there is no Doubt of that. I just Completed the postmortem examination of that Good Man and the Bruises and Bleeding Gums and Blackened Lips all told the story. But I think Scurvy was not the Killer._\n\n _Captain Fitzjames's last three days were spent here, about 80 Miles south of Terror Camp, on a frozen point on a windswept bay where the bulk of King William Land curves sharply to the West. For the first time in Six Weeks, we have unpacked All the Tents \u2014 including the large ones \u2014 and used some Coal from the few sacks we brought along and the Iron Whaleboat Stove one crew has man-hauled so far. Almost all of our meals the past six weeks have been eaten cold or only Partially Heated over the tiny spirit stoves. For the last two nights we have had hot food \u2014 never enough, a third of the rations we need for the incredibly Strenuous Work we are Doing, but warm nonetheless. For Two Mornings we have wakened in the same place. The men are calling this place Comfort Cove._\n\n _Mostly we stopped to allow Captain Fitzjames to Die in Peace. But there was no Peace for the captain in his last days._\n\n _Poor Lieutenant Le Vesconte had evidenced some of the same Symptoms of Captain Fitzjames's last days. Lieutenant Le Vesconte died suddenly on our 13 th Day on this terrible Voyage South \u2014 only 18 Miles from Terror Camp if I remember correctly, and on the same day that Marine Private Pilkington expired \u2014 but the Symptoms of Scurvy had been more Advanced in both the lieutenant and the private and their Final Agonies less excruciatingly drawn out._\n\n _I confess that I hadn't remembered that Lieutenant Le Vesconte's first name had been Harry. Our intercourse had always been quite Friendly but also quite Formal, and on the Muster Rolls I recalled his name had been listed as H. T. D. Le Vesconte. It bothers me now that I must have heard the Other Officers call him Harry from time to time \u2014 perhaps a hundred times \u2014 but I had always been too busy or preoccupied to notice. It was only after Lieutenant Le Vesconte's death that I paid attention to the other Men using his Christian name._\n\n _Private Pilkington's Christian name was William._\n\n _I remember that day in early May after Le Vesconte's and Private Pilkington's brief joint burial service, one of the men suggested that we name the small spur of land where they were buried \"Le Vesconte Point,\" but Captain Crozier vetoed that idea, saying that if we named every place where one of us might end up buried after the dead person there, we'd run out of land before we ran out of names._\n\n _This Befuddled the men and I Confess that it Befuddled me as well. It must have been an Attempt at Humour, but it shocked me. It shocked the men into Silence as well._\n\n _Perhaps that was Captain Crozier's Purpose. It did put a stop to the men offering to name Natural Features after their Dead Officers._\n\n _Captain Fitzjames had shown a General Weakening for some weeks \u2014 even before we left Terror Camp \u2014 but Four Days ago he seemed to have been Struck Down by something more Sudden in its attack and far more Agonizing in its effects._\n\n _The Captain had been suffering Problems with his Stomach and Bowels for some weeks, but suddenly, on the Second of June, Fitzjames collapsed. Our protocol on the March is not to stop for sick men but rather to place them in one of the larger Boats and pull them along with the other Supplies and dead weight. Captain Crozier made sure that Captain Fitzjames was made as comfortable as possible in his own Whaleboat._\n\n _Since we are doing this Long March South in relays \u2014 working for Hours on End to pull 5 of the 10 Heavy Boats as little as a few hundred Yards across the terrible gravel and Snow, always trying to stay on the Land when possible rather than be forced to deal with Pack Ice and Pressure Ridges, sometimes covering less than a Mile in a Day on the resisting gravel and ice \u2014 I make it my practice to stay with the sickest men while the Man-hauling Teams go back for the other 5 Boats. Often Mr. Diggle and Mr. Wall, gamely preparing to cook Warm Meals for almost a hundred Starving Men on their little spirit stoves, and a few men with muskets to guard against the Thing on the Ice or Esquimaux are my only companions for those hours._\n\n _Other than the Sick and Dying._\n\n _Captain Fitzjames's nausea, vomiting, and diarrhea were terrible. Unrelenting. The Cramps curled him into a Fetal Position and made this strong and Brave man cry aloud._\n\n _On the Second Day, he tried to rejoin his team man-hauling his whaleboat \u2014 even the Officers pull from time to time \u2014 but soon he Collapsed yet again. This time the vomiting and cramps were Nonstop. When the Whaleboat was left on the Ice that afternoon as the able-bodied Men went back to man-haul forward the 5 Boats left behind on the First Haul, Captain Fitzjames confessed to me that his Vision was terribly blurred and that he was frequently seeing Double._\n\n _I asked him if he had been Wearing the Wire Goggles we use to block the sun. The men Hate them because they Obscure vision so terribly, and the Goggles tend to induce their own headaches. Captain Fitzjames admitted that he had Not been wearing them but pointed out that the day had been quite Cloudy. None of the other men were wearing them either. At that point our Conversation stopped as he was seized with diarrhea and vomiting yet again._\n\n _Late that night, in the Holland Tent where I was Attending him, Fitzjames gasped to me that he was having trouble swallowing and that his Mouth was constantly Dry. Soon he showed trouble Breathing and was no longer able to speak. By sunrise, a Paralysis had moved down his upper Arms to the point where he could no longer lift them or use his hands to Write messages to me._\n\n _Captain Crozier called a Halt that day \u2014 the first such full day's stop we had enjoyed since leaving Terror Camp almost six weeks earlier. All of the tents were pitched. The larger Sick Bay Tent was finally unpacked from Crozier's own Whaleboat \u2014 it took almost Three Hours to set it up in the wind and cold (and the men are much more Sluggish about such things these days) \u2014 and for the first time in almost a month and a half, all the Sick were made comfortable in one place._\n\n _Mr. Hoar, Captain Fitzjames's long-suffering steward, had died on the Second Day of our March. (We had made less than a Mile that first Terrible day of Man-hauling, and the stack of Coal, Stoves, and other goods was still Horribly but Plainly Visible behind us at Terror Camp that first night. It was as if we had Achieved Nothing after twelve hours of Deadly Labour. Those first days \u2014 it took us Seven Days to cross the narrow iced Inlet south of Terror Camp and travel only Six Miles \u2014 almost destroyed our Morale and Will to go on.)_\n\n _Marine Private Heather, who had lost a portion of his Brain months before, finally allowed his Body to Die on our Fourth Day out. His surviving fellow Marines played a bagpipe over his shallow, hastily dug grave that evening._\n\n _And so it went with the other Sick dying rapidly, but then there came a Long Period after the twin deaths of Lieutenant Le Vesconte and Private Pilkington at the end of the Second Week in which no one died. The men Convinced themselves that the truly Ill had died off and only the Strong remained._\n\n _Captain Fitzjames's sudden collapse reminded us that we were all growing Weaker. There were no longer any truly Strong among us. Except perhaps for the Giant, Magnus Manson, who lumbers along Imperturbably and who never seems to lose weight or energy._\n\n _To treat Captain Fitzjames's constant vomiting, I administered doses of asafetida, a gum resin used to control spasms. It helped very little. He was not able to Keep Down either solid food or liquids. I gave him limewater to settle his stomach, but it also did no good._\n\n _For his difficulty swallowing, I administered Syrup of Squills \u2014 a sliced herb set in tannin solution that is an Excellent Expectorant. Usually effective, it seemed to do little to lubricate the dying man's Throat._\n\n _As Captain Fitzjames lost the Use and Control of first his Arms and then his Legs, I tried Peruvian Wine of Coca \u2014 a powerful admixture of wine and cocaine \u2014 as well as solutions of hartshorn, a Medicine made from ground-up antlers of red deer which stinks strongly of ammonia, as well as Solution of Camphor. These Solutions, at Half the Dosage I gave to the captain, often Arrest and even Reverse paralysis._\n\n _They did not help. The Paralysis spread to all of Captain Fitzjames's extremities. He continued Vomiting and being Doubled Up by Cramps long after he could no longer speak or gesture._\n\n _But at least this Deadening of his Vocal Apparatus relieved the men of the Burden of hearing their_ Erebus _Captain scream in pain. But I saw his convulsions and his mouth Open in silent screams that Long Last Day._\n\n _This morning, on the Fourth and Final Day of Captain Fitzjames's Agony, his lungs began to shut down as the paralysis reached his respiratory muscles. He Laboured all day to breathe. Lloyd and I \u2014 sometimes abetted by Captain Crozier, who spent many hours with his Friend at the End \u2014 would set Fitzjames in a Sitting Position or Hold Him Upright or actually Walk the paralyzed man around the Tent, dragging his Limp Stockinged Feet across the Ice-and-Gravel floor, in a vain attempt to help his failing lungs Continue to work._\n\n _In desperation, I forced Tincture of Lobelia, a whiskey-coloured solution of Indian tobacco that was almost pure nicotine, down Captain Fitzjames's throat, massaging it down his paralyzed gullet with my bare fingers. It was like feeding a dying Baby Bird. Tincture of Lobelia was the best respiratory stimulant left in my depleted Surgeon's apothecary, a Stimulant that Dr. Peddie had sworn by._ It would raise Jesus from the dead a day early, _Peddie used to blaspheme when in his cups._\n\n _It did no good whatsoever._\n\n _It must be Remembered that I am a mere Surgeon, not a Physician. My training was in Anatomy; my expertise is in Surgery. Physicians prescribe; Surgeons saw. But I am doing my Best with the supplies my Dead Colleagues left to me._\n\n _The most Terrible thing about Captain James Fitzjames's last hours was that he was Fully Alert through all of this \u2014 the vomiting and Cramps, the Loss of his Voice and ability to Swallow, the Creeping Paralysis, and the Final Terrible Hours of his lungs failing. I could see the panic and Terror in his eyes. His Mind was Fully Alive. His Body was Dying around him. He could do Nothing about this Living Torture except to Plead with me through his Eyes. I was impotent to help._\n\n _At times I wanted to Administer a lethal dose of pure Coca just to put an End to his Suffering, but my Hippocratic Oath and Christian belief did not allow that._\n\n _I went outside and Wept instead, making sure that none of the Officers or Men could see me._\n\n _Captain Fitzjames died at 8 minutes after 3:00 p.m. this afternoon, Tuesday the Sixth Day of June, in the Year of Our Lord Eighteen Hundred and Forty-Eight._\n\n _His shallow grave had already been Dug. The Covering Rocks had been Gathered and Stacked. All of the Men who could stand and dress themselves turned out for the Service. Many of those who had served under Captain Fitzjames the past three years wept. Even though it was warm today \u2014 five to ten degrees above Freezing \u2014 a cold wind Came Up out of the Relentless Northwest and froze many Tears to beards or cheeks or comforters._\n\n _The few Marines left in our Expedition fired a volley into the Air._\n\n _Up the hill from the Grave, a Ptarmigan took to the air and flew out toward the Pack Ice._\n\n _A great Moan went up from the men. Not for Captain Fitzjames, but for the loss of the Ptarmigan for the evening's stew. By the time the Marines reloaded their Muskets, the bird was a hundred yards away and far out of Range. (And none of these Marines ever could have hit a bird on the wing at one hundred yards even when they were Well and Warm.)_\n\n _Later \u2014 just a half hour ago \u2014 Captain Crozier looked in on the Sick Bay Tent and beckoned me outside into the Cold._\n\nWas it Scurvy that killed Captain Fitzjames? _was his only question to me._\n\n _I admitted that I did not think it was. It had been something more Deadly._\n\nCaptain Fitzjames thought that the steward serving him and the other officers since Hoar's death was poisoning him, _whispered the captain._ Is this possible?\n\nBridgens? _I said too Loudly. I was deeply shocked. I had always liked the Bookish old Steward._\n\n _Crozier shook his head._ Richard Aylmore has been serving the _Erebus_ officers the last two weeks, _he said._ Could it have been poison, Dr. Goodsir?\n\n _I hesitated. To say yes would certainly mean Aylmore would be shot at sunrise. The Gunroom Steward was the man who had been Given fifty Lashes in January for his Improvident participation in the Grand Venetian Carnivale. Aylmore was also a Friend and Frequent Confidant of_ Terror _'s Diminutive and sometimes Devious caulker's mate. Aylmore, we all knew, harboured a small and resentful Soul._\n\nIt could well have been poison, _I told Crozier not half an hour ago._ But not necessarily a Deliberately Administered poison.\n\nWhat does that mean? _demanded Crozier. Our remaining captain looked so weary tonight that his white Skin actually glowed in the starlight._\n\n _I said,_ I mean that the Officers have been eating the Largest Portions of the last of the Goldner's Canned Foods that we have brought along. There sometimes is an Unexplained but Deadly paralytic poison in foods that have gone bad. No one understands it. Perhaps it is some microscopic Animalcule we cannot Perceive with our Lenses.\n\n _Crozier whispered,_ Wouldn't we have smelled it if the canned foods had gone putrid?\n\n _I shook my head and grasped the captain's greatcoat sleeve to press home my point._ No. That is the Terror of this Poison that Paralyzes first the voice and then the entire body. It cannot be Seen or Tested For. It is as invisible as Death itself.\n\n _Crozier thought for a long moment._ I'll order everyone to go off the canned foods for three weeks, _he said at last._ The last of the rotten salted beef and poor biscuits will have to do us for a while. We'll eat it Cold.\n\nThe Men and officers will not be happy about this, _I whispered._ The canned soups and vegetables are as close as Anything comes to a Hot Meal on the March. They may become Mutinous at such Further Deprivation under such Harsh Conditions.\n\n _Crozier smiled then. It was a strangely chilling sight._ Then I will not have everyone go off the canned foods, _he hissed._ Gunroom Steward Aylmore will continue eating it \u2014 out of the same cans he served James Fitzjames from. A good night to you, Dr. Goodsir.\n\n _I came back into the Sick Bay Tent then, attended to the sleeping sick men, and crawled into my Sleeping Bag with my mahogany Portable Writing Desk on my Knees._\n\n _My handwriting is so Difficult to read on the Page because I have been Shaking. And not just from the Cold._\n\n _Every time I believe I Know one of these men or Officers, I find that I am wrong. A Million years of Man's Medicinal Progress will never reveal the secret Condition and sealed Compartments of the Human Soul._\n\n _We leave before Dawn tomorrow. I suspect there will be no more stops such as the luxury of the last Two Days at Comfort Cove._\n\nBLANKY\n\n _Lat. unknown, Long. unknown_\n\n _18 June, 1848_\n\nWhen Tom Blanky's third and final leg snapped off, he knew it meant the end.\n\nHis first new leg had been wondrous to behold. Shaped and whittled by Mr. Honey, _Terror_ 's capable carpenter, it had been carved out of a single piece of solid English oak. It was a work of art and Blanky enjoyed showing it off. The ice master had peg-legged his way around the ship like a good-humoured pirate, but when Blanky had to go out onto the ice, he attached to the bottom of the peg a perfectly shaped wooden foot that snapped into a socket. The foot had a myriad of nails and screws on the bottom \u2014 better for traction on the ice than the hobnails in the men's winter boots \u2014 and the one-legged man, while not able to man-haul, had been more than able to keep up during their transfer to Terror Camp from the abandoned ship and then in the long haul south and now east.\n\nNo longer.\n\nHis first leg had broken off just below the knee nineteen days after they'd abandoned Terror Camp, not long after they'd buried poor Pilkington and Harry Le Vesconte.\n\nThat day, Tom Blanky and Mr. Honey, who'd been excused from man-hauling, both had ridden in a pinnace strapped to a sledge pulled by twenty straining other men while the carpenter carved a new leg and foot for the ice master from wood taken from a spare spar.\n\nBlanky had never been sure whether or not to wear his foot when hobbling along with the procession of boats and sweating, swearing men. When they actually ventured out onto the sea ice \u2014 as they had the first days crossing the frozen inlet south of Terror Camp and again at Seal Bay and once more at the broad bay just north of the point where they'd buried Le Vesconte \u2014 the screwed and cleated foot worked wonders on the ice. But most of their march south and then west along and around the large cape and now back east again was made on land.\n\nAs the snow and ice on the rocks began to melt, and it was melting quickly this summer that was so much warmer than their lost summer of 1847, Tom Blanky's wooden ovoid of a foot would slide off slick rocks or be pulled off in ice crevices or would snap at the socket with every inopportune twist.\n\nWhen out on the ice, Blanky tried to show his solidarity with his mates by hiking back and forth with the man-haulers, making both trips alongside the straining, sweating men, carrying small items when he could, occasionally volunteering to slip into the harness of an exhausted man. But everyone knew he could not pull his own weight with the hauling.\n\nBy the sixth week and forty-seven miles out, at Comfort Cove where poor Captain Fitzjames had died so hard, Blanky was on his third leg \u2014 a poorer, weaker substitute than the second one had been \u2014 and he tried manfully to hobble along on his peg through the rocks, streams, and standing water, although he no longer went back for the afternoon's hated second haul.\n\nTom Blanky realized that he had become just so much more dead weight for the exhausted and ill survivors \u2014 ninety-five of them now, not including Blanky \u2014 to haul south with them.\n\nWhat kept Blanky going even when his third leg began to splinter \u2014 there were no more extra spars from which to whittle a fourth \u2014 was his rising hope that his skills as an ice master would be needed when they took to the boats.\n\nBut while the scrim of ice on the rocks and barren coastline melted during the day \u2014 sometimes the temperature rose as high as 40 degrees according to Lieutenant Little \u2014 the pack ice beyond the coastal bergs showed no sign of breaking up. Blanky tried to be patient. He knew better than any other man on the expedition that sea ice at these latitudes might not show open leads \u2014 even on a \"more normal\" summer such as this one \u2014 until mid-July or later.\n\nStill, it was not only his usefulness that was being decided by the ice, but his survival. If they took to the boats soon, he might live. He did not need his leg to travel by boat. Crozier had long since designated Thomas Blanky as skipper of his own pinnace \u2014 commanding eight men \u2014 and once the ice master was at sea again, he would survive. With any luck at all, they could sail their fleet of ten little splintered and gouged boats right to the mouth of Back's Great Fish River, pause at the mouth to rerig for river running, and \u2014 with only the slightest help from northwest winds and men at the oars \u2014 head briskly upstream. The portages, Blanky knew, would be hard, especially hard for him since he could carry so little weight on his flimsy third peg, but a piece of cake after the man-hauling nightmare of the last eight weeks.\n\nIf he could last until they took to the boats, Thomas Blanky would live.\n\nBut Blanky knew a secret that made even his sanguine personality wane: the Thing on the Ice, the Terror itself, was after him.\n\nIt had been sighted every day or two as the straining procession of men had rounded the large cape and turned back east again along the shoreline, every day in the early afternoon when they turned back to haul the five boats they'd left behind, every twilight at around 11:00 p.m. as they collapsed into their wet Holland tents for a few hours' sleep.\n\nThe thing was still stalking them. Sometimes the officers saw it through their telescopes as they looked out to sea. Neither Crozier nor Little nor Hodgson nor any of the other few officers remaining ever told the man-hauling men that they'd seen the beast, but Blanky \u2014 who had more time than most to watch and think \u2014 saw them conferring and knew.\n\nAt other times, those hauling the last boats could see the beast clearly with their own unaided eyes. Sometimes it was behind them, trailing by a mile or less, a black speck against white ice or a white speck against black rock.\n\n _It's just one of them polar white bears,_ had said James Reid, the red-bearded ice master from _Erebus_ and one of Blanky's closest friends now. _They'll eat you if they can, but mostly they're harmless enough. Bullets kill them. Let's hope it comes closer. We need some fresh meat._\n\nBut Blanky knew at the time that it wasn't one of the white bears they shot for food from time to time. This was _it,_ and while all men on the Long March feared it \u2014 especially at night or, rather, during the two hours of dimness that now passed for night \u2014 only Thomas Blanky knew that it was coming for him first.\n\nThe march had taken a toll on everyone, but Blanky was in constant agony: not from scurvy, which seemed to be affecting him less than most, but from the pain in the stump of the leg that the thing had taken. Walking on the ice and rock of the shore was so difficult for him that by midmorning of each day's sixteen- or eighteen-hour march, his stump would be streaming blood down over the wooden cup and leather harness that held it in place. The blood soaked through his thick canvas trousers and ran down his wooden peg, leaving a trail of blood behind. It soaked upward through his long underwear, trousers, and shirt.\n\nDuring the first weeks of the march, while it was still cold, it was a blessing that the blood had frozen. But now, with the tropical warmth of days above zero, some above freezing, Blanky was bleeding like a stuck pig.\n\nThe long slops and greatcoats also had been a blessing \u2014 they hid the worst evidence of Blanky's bleeding from the captain and others \u2014 but by mid-June, it was too warm to wear the greatcoats while hauling, so tons of sweat-soaked slops and wool layers were piled in the boats they were hauling. The men often hauled in shirtsleeves through the warmest parts of the day, pulling on more layers as the afternoons cooled toward zero degrees. Blanky had joked with them when they asked him why he continued to wear his long coats. _I'm cold-blooded, boys,_ he'd said with a laugh. _My wooden leg brings the chill of the ground up into me. I don't want you to see me shiver._\n\nBut eventually he had to take off the greatcoat. Because Blanky was working so hard hobbling just to keep up, and because the pain of his tortured stump caused him to sweat even when he was standing still, he could no longer stand the freeze-and-thaw, freeze-and-thaw of all his layers of clothes.\n\nWhen the men saw the blood pouring, they said nothing. They had their own problems. Most of them were bleeding from scurvy.\n\nCrozier and Little often would pull Blanky and James Reid aside, asking the two ice masters their professional opinion about the ice just beyond the berg barrier of the shoreline. Once they'd come around to the east again, along the southern coast of this cape that had bulged out miles to the west and south of Comfort Cove \u2014 probably adding twenty miles to their haul south \u2014 Reid was of the opinion that the ice between this part of King William Land and the mainland, whether King William Land was connected to the mainland or not, would be slower breaking up than the pack ice to the northwest, where conditions were more dynamic come the summer thaw.\n\nBlanky was more optimistic. He pointed out that the bergs piled here along this southern coast were becoming smaller and smaller. Once a serious barrier separating the shore from the sea ice, this wall of bergs was no more a hindrance now than a cluster of low seracs. The reason, Blanky told Crozier, and Reid had agreed, was that this cape of King William Land was sheltering this stretch of sea and coast, or perhaps of gulf and coast, from the glacierlike river of ice that had poured down so relentlessly from the northwest onto _Erebus_ and _Terror_ and even upon the coast near Terror Camp. That endless press of ice, Blanky pointed out, had been flowing down from the North Pole itself. Things were more sheltered here south of the King William Land southwestern cape. Perhaps the ice would break up sooner here.\n\nReid had looked at him strangely when Blanky delivered that opinion. Blanky knew what the other ice master was thinking. _Whether this is a gulf or a strait leading to Chantrey Inlet and the mouth of Back's River, ice usually breaks up last in a confined space._\n\nReid would have been correct if he'd stated that opinion aloud to Captain Crozier \u2014 he hadn't, obviously not wanting to contradict his friend and fellow ice master \u2014 but Blanky was still optimistic. In truth, Thomas Blanky had been optimistic in his heart and soul every day since that dark night of 5 December of the previous winter when he'd considered himself a dead man as the Thing on the Ice chased him from _Terror_ and into the forest of seracs.\n\nTwice the creature had tried to kill him. And twice all that Thomas Blanky had lost had been parts of one leg.\n\nHe hobbled on, bringing cheer and jokes and the occasional shred of extra tobacco or sliver of frozen beef to exhausted, drained men. His tent mates, he knew, valued his presence. He took his turn at watch in the ever-shorter nights and carried a shotgun while painfully stumping alongside the morning boat procession as a guard, although Thomas Blanky knew better than any living man that no mere shotgun would stop the Terror Beast when it finally came in close to claim its next victims.\n\nThe tortures of the Long March were increasing. Not only were men slowly dying of starvation and scurvy and exposure, but there had been two other incidences of the terrible poisoning death that had claimed Captain Fitzjames \u2014 John Cowie, the stoker who had survived the thing's invasion of _Erebus_ on 9 March, died screaming in cramps and pain and then silent paralysis on 10 June. On 12 June, Daniel Arthur, _Erebus_ 's thirty-eight-year-old quartermaster, collapsed with abdominal pains and died from paralyzed lungs a mere eight hours later. Their bodies were not truly buried; the procession had paused only long enough to sew both bodies into the little remaining spare canvas and to pile rocks on them.\n\nRichard Aylmore, the object of much speculation since Captain Fitzjames's death, showed almost no signs of illness. The scuttlebutt was that while everyone else had been banned from eating warm meals from the canned goods and suffered the scurvy worse for it, Aylmore had been ordered to share portions of his tinned meals with Cowie and Arthur. Other than the obvious answer of active and deliberate poisoning, no one could figure why the Goldner tins would horribly kill three men but leave Aylmore untouched. But while everyone knew that Aylmore hated Captain Fitzjames and Captain Crozier, no one could see a reason for the gunroom steward to poison his mates.\n\nUnless he wanted their shares of food after they were dead.\n\nHenry Lloyd, Dr. Goodsir's assistant in the sick bay, was one of the men dragged along in the boats these days \u2014 sick from scurvy that had him vomiting blood and his own loose teeth \u2014 so since Blanky was one of the few men other than Diggle and Wall who stayed with the boats after the morning haul, he tried to help the good doctor.\n\nOddly enough, now that it was getting tropically warm, there were more cases of frostbite. Sweating men who'd doffed their jackets and gloves would continue man-hauling into the chill of the endless evening \u2014 the sun hung in the south until midnight now \u2014 and be surprised to find that the air temperature had fallen to fifteen below during their exertions. Goodsir was constantly treating fingers and patches of skin turned white by frostbite or dead black from rot.\n\nSun blindness or screaming headaches caused by the sun's glare afflicted half the men. Crozier and Goodsir would move up and down the ranks of man-hauling men during the morning, cajoling them to put on their goggles, but the men hated the wire-mesh monstrosities. Joe Andrews, captain of the hold for _Erebus_ and an old friend of Tom Blanky's, said that wearing the God-damned wire goggles was as difficult as trying to see through a pair of lady's black silk drawers but much less fun.\n\nThe snow blindness and headaches were becoming serious problems on the march. Some of the men begged Dr. Goodsir for laudanum after the headaches struck, but the surgeon told them that he had none left. Blanky, who was often sent to fetch medicines from the doctor's locked chest, knew that Goodsir was lying. There was a small vial of laudanum left there, unmarked. The ice master knew that the surgeon was keeping it for some terrible occasion \u2014 to ease Captain Crozier's last hours? Or the surgeon's own?\n\nOther men suffered the torments of Hell from sunburn. Everyone was blistered red on their hands and faces and necks, but some men who would tug off their shirts for even the shortest periods during the intolerable heat of the midday, when temperatures were above freezing, would that same evening watch their skins, bleached white after three years of darkness and enclosure, burn red and quickly turn to suppurating blisters.\n\nDr. Goodsir popped the blisters with his lancet and treated the open sores with a salve that smelled to Blanky like axle grease.\n\nBy the time the ninety-five survivors were trudging east along the southern coast of the cape in mid-June, almost every man was on the edge of breakdown. As long as some men could man-haul the terribly heavy sledges with boats atop them and the full-packed whaleboats without sledges, others suffering could ride briefly, recover slightly, and rejoin the man-hauling within hours or days. But when there were too many sick and injured to pull, Blanky knew, their escape march would be at an end.\n\nAs it was now, the men were always so thirsty that every stream or trickle of water was a reason to stop and throw themselves on all fours to lap at the water like dogs. If it hadn't been for the sudden thaw, Blanky knew, they would have all died of thirst three weeks earlier. The spirit stoves were almost out of fuel. At first, melting snow in one's mouth seemed to assuage the thirst, but it actually drained more energy from the body and made one thirstier. Each time they dragged the boats and themselves across a stream \u2014 and there were more streams and rivulets running liquid now \u2014 everyone would stop to fill water bottles that no longer needed to be carried next to the skin to keep them from freezing.\n\nBut while thirst would not kill them soon, Blanky saw that the men were failing in a hundred other ways. Starvation was taking its toll. Hunger kept the exhausted men from sleeping through the four hours of twilight \u2014 if they did not have watch duty \u2014 which Crozier allowed for their sleeping time.\n\nSetting up and taking down the Holland tents, simple acts that had been performed in twenty minutes two months ago at Terror Camp, now took two hours in the morning and two hours in the evening. Each day it took a little longer as fingers became more swollen and frostbitten and clumsy.\n\nFew of the men's minds, not even Blanky's at times, were really clear. Crozier seemed the most alert of all of them most of the time, but sometimes when he thought that no one was looking, the captain's face became a death mask of fatigue and stupor.\n\nSailors who had tied off complicated rigging and shroud knots in the roaring darkness fifty feet out on a pitching spar two hundred feet above the deck on a stormy night off the Strait of Magellan during a hurricane blow could no longer tie their shoes in the daylight. Because there was no wood within three hundred miles \u2014 other than Blanky's leg and the boats and masts and sledges they'd hauled along with them and the remains of _Erebus_ and _Terror_ almost a hundred miles to their north \u2014 and because the ground was still hard-frozen an inch below the surface, the men had to gather heaps of stones at each stop to weigh down the edges of the tents and to anchor tent ropes against the inevitable nightly winds.\n\nThis chore also took forever. Men frequently fell asleep standing in the dimmed sunlight at midnight with a rock in each hand. Sometimes their mates did not even shake them awake.\n\nSo it came to pass that late in the afternoon of the eighteenth day of June, 1848, as the men were making their second haul of boats that day, when Blanky's third leg snapped off just below his bleeding knee stump, he took it as a sign.\n\nDr. Goodsir had little work for him that afternoon, so Blanky had turned back to peg his way alongside the last boats on the second haul of the endless day, when the foot and peg had caught between two immovable rocks and snapped the peg off high. He took the high break and his unusual presence near the end of the march as a sign from the gods as well.\n\nHe found a nearby boulder, made himself as comfortable as he could, dug out his pipe, and tapped in the last bit of tobacco he had been saving for weeks.\n\nWhen a few of the seamen stopped in their hauling to ask what he was doing, Blanky said, \"Just going to sit a spell, I reckon. Give my stump a rest.\"\n\nWhen Sergeant Tozer, who was in charge of the Marine rear guard detail this sunny day stopped to ask tiredly what Blanky was doing allowing the procession to pass him by, Blanky said, \"Never you mind, Soloman.\" He had always enjoyed irritating the stupid sergeant by using his first name. \"You just toddle off now with your remaining lobsterbacks and let me be.\"\n\nHalf an hour later, when the last boats were hundreds of yards to the south of him, Captain Crozier had come back with Mr. Honey, the carpenter.\n\n\"What the hell do you think you're doing, Mr. Blanky?\" snapped Crozier.\n\n\"Just giving it a rest, Captain. I thought I might spend the night here.\"\n\n\"Don't be an ass,\" said Crozier. He looked at the snapped-off peg leg and turned to the carpenter. \"Can you fix this, Mr. Honey? Make a new one by tomorrow afternoon if Mr. Blanky rides in one of the boats until then?\"\n\n\"Oh, aye, sir,\" said Honey, squinting at the broken peg with an artisan's scowl at the failure \u2014 or mistreatment \u2014 of one of his creations. \"We ain't got much spare wood left, but there's one extra jolly boat rudder we brought along as a spare for the pinnaces that I can turn into a new leg as easy as you like.\"\n\n\"D'you hear that, Blanky?\" asked Crozier. \"Now get off your ass and let Mr. Honey help you hobble to catch up to Mr. Hodgson's last boat there. Quickly now. We'll have you fixed up by tomorrow noon.\"\n\nBlanky smiled. \"Can Mr. Honey fix this, Captain?\" He tugged off the wooden cup of the leg and detached the clumsy leather-and-brass harness.\n\n\"Oh, Christ damn it,\" said Crozier. He started to look more closely at the bleeding raw stump with the black flesh surrounding the white nub of bone but quickly pulled back his face from the smell.\n\n\"Aye, sir,\" said Blanky. \"I'm surprised Dr. Goodsir ain't sniffed it out before this. I try to stay downwind of him when I'm helpin' him out in the sick bay. The boys in my tent know what's up, sir. There's nothing to be done for it.\"\n\n\"Nonsense,\" said Crozier. \"Goodsir will...\" He stopped.\n\nBlanky smiled. It was not a sarcastic or sad smile but an easy one, filled with some real humour. \"Will what, Captain? Take my leg off at the hip? The black bits and red lines run all the way up to my ass and private parts, sir, with apologies for being so picturesque about it. And if he did operate, how many days would I be lyin' in the boat like old Private Heather \u2014 God rest the poor bugger's soul \u2014 being hauled along by men who are as tired as I am?\"\n\nCrozier said nothing.\n\n\"No,\" continued Blanky, puffing contentedly on his pipe, \"I think it'd be best if I rested here awhile on my own and just relaxed and thought some thoughts about this and that. My life has been a good one. I'd like to think about it some before the pain and stink get so bad I'm distracted.\"\n\nCrozier sighed, looked at his carpenter and then at his ice master, and sighed again. He took a water bottle from the pocket of his greatcoat. \"Take this.\"\n\n\"Thank you, sir. I will. With gratitude,\" said Blanky.\n\nCrozier felt in his other pockets. \"I have no food with me. Mr. Honey?\"\n\nThe carpenter came up with a moldy biscuit and a sliver of something more green than tan that might have been beef.\n\n\"No, thank you, John,\" said Blanky. \"I am truthfully not hungry. But, Captain, would you do me a huge favor?\"\n\n\"What is that, Mr. Blanky?\"\n\n\"My people are in Kent, sir. Near Ightham Mote north of Tonbridge Wells. Or at least my Betty and Michael and old mum were when I set sail, sir. I was wondering, Captain, I mean if you have luck on your side and have the time later...\"\n\n\"If I get back to England, I swear I'll look them up and tell them that you were smoking and smiling and sitting as comfortably on a boulder as a lazy squire when last I saw you,\" said Crozier. He pulled a pistol from his pocket. \"Lieutenant Little's seen the thing through his glass \u2014 it's been trailing behind us all morning, Thomas. It'll be along presently. You should take this.\"\n\n\"No, thank you, Captain.\"\n\n\"You're sure about this, Mr. Blanky? Staying behind, I mean?\" said Captain Crozier. \"Even if you were... with us... for just another week or so, your knowledge of the ice might be very important to us all. Who knows what the conditions will be out on the pack ice twenty miles east of here?\"\n\nBlanky smiled. \"If Mr. Reid weren't still with you, I'd take that to heart, Captain. I surely would. But he's as good an ice master as you could ask for. As a spare, I mean.\"\n\nCrozier and Honey shook hands with him. Then they turned and hurried to catch up to the last boat disappearing over a distant ridge to the south.\n\nIt was after midnight when it came.\n\nBlanky had been out of tobacco for hours and the water had frozen in the bottle where he'd foolishly left it sitting on the boulder next to him. He was in some pain, but he did not want to sleep.\n\nA few stars had come out in the twilight. The wind from the northwest had come up, as it usually did in the evening, and the temperature had probably dropped forty degrees from its noontime high.\n\nBlanky had kept the broken peg leg and its cup and straps on the boulder next to him. While his gangrenous leg tormented him and his empty stomach clawed at him, the worst pain tonight was from his lower leg and calf and foot \u2014 his phantom limb.\n\nSuddenly the thing was just _there._\n\nIt loomed up on the ice not thirty paces from him.\n\n _It must have come up through some invisible hole in the ice,_ Blanky thought. He was reminded of a tent fair in Tunbridge Wells he had seen as a boy, with a rickety wooden stage and a magician in purple silk with a tall conical hat embroidered with crude planets and stars. That man had appeared just like this, popping up through a trapdoor to the oohs and ahs of the country audience.\n\n\"Welcome back,\" said Thomas Blanky to the shadowy silhouette on the ice.\n\nThe thing reared up on its hind legs, a dark mass of hair and muscle and sunset-tinted claws and a faint gleam of teeth beyond anything, the Ice Master was sure, in mankind's racial memory of its many predators. Blanky guessed that it was more than twelve feet tall, perhaps fourteen.\n\nIts eyes \u2014 a deeper blackness against the black silhouette \u2014 did not reflect the dying sun.\n\n\"You're late,\" said Blanky. He could not help it that his teeth were chattering. \"I've been expecting you for a long time.\" He threw his peg leg and its rattling harness at the shape.\n\nThe thing did not try to dodge the crude missile. The shape towered there for a minute and then rushed forward like a wraith, the legs not even visibly moving to propel it, a monstrous mass sliding rapidly toward him across the rock and ice, the dark and terrible solidity of the shape finally opening arms to fill the ice master's vision.\n\nThomas Blanky grinned fiercely and clamped his teeth down hard on the stem of his cold pipe.\n\nCROZIER\n\n _Lat. unknown, Long. unknown_\n\n _4 July, 1848_\n\nThe only thing keeping Francis Rawdon Moira Crozier moving forward into the tenth week on the boat march was the blue flame in his chest. The more tired and empty and sick and battered his body became, the hotter and fiercer the flame burned. He knew it was not merely some metaphor of his determination. Nor was it optimism, as such. The blue flame in his chest had burrowed toward his heart like some alien entity, lingered like a disease, and centered in him as an almost unwanted core of conviction that he would do whatever he had to do to survive. _Anything._\n\nSometimes Crozier came close to praying that the blue flame would just go out so he could surrender to the inevitable and lie down and pull the frozen tundra up over himself like a child under a blanket settling into his nap.\n\nToday they were stopped \u2014 not pulling the sledges and boats for the first time in a month. And they had unpacked and clumsily pitched the large Sick Bay tent, although not the large mess tents. The men were calling this otherwise unremarkable place on a small bay along the southern coast of King William Land \"Hospital Camp.\"\n\nIn the past two weeks they had just crossed the rugged ice of a huge bay that cut into the underside of the cape that seemed for weeks of hauling as if it would continue to bulge out to the southwest forever. But now they were headed southeast again, paralleling the coastline along the underside of that cape and then farther east \u2014 the correct direction if they were to get to Back's River.\n\nCrozier had brought his sextant and theodolite along, and Lieutenant Little also had his sextant, as well as the dead Fitzjames's instrument as a spare, but neither officer had taken star sightings or sun sightings for weeks. It just wasn't important. If King William Land was a peninsula, as most of the arctic explorers, including Crozier's old commander James Clark Ross, had thought, then this coastline would lead them to the mouth of Back's River. If it was an island \u2014 which had been Lieutenant Gore's guess and Crozier's hunch as well \u2014 then they would soon see the mainland to their south and cross what should be a narrow strait to the mouth of Back's River.\n\nEither way, Crozier \u2014 who had been content to follow the coastline since they had no other real choice and navigate by dead reckoning for the time being \u2014 estimated that they were now about ninety miles from the mouth of Back's River.\n\nOn this march, they had been completing only a little better than a mile per day on average. Some days they did three or four miles, reminding Crozier of the fantastic rate of their crossing from the ships to Terror Camp on the ice highway they had laid down, but other days \u2014 when there was more rock than ice under the runners, when they had to ford sudden streams or in one case an actual river, when they were forced out onto the tortured sea ice when the coastline became too rocky, when the weather was foul, when more men than usual were too sick to pull and ended up riding in the boat themselves while their mates pulled the extra weight, first the sixteen hours of man-hauling the four whaleboats and a cutter, then back for the other three cutters and two pinnaces \u2014 saw them covering only a few hundred yards from their previous night's camp.\n\nOn 1 July, after weeks of warming weather, the cold and snow returned in earnest. A blizzard blew out of the southeast, directly into the eyes of the men hauling their sledges. Slops were pulled out of baled heaps in the boats. Welsh wigs were dug out of valises and packs. The snow added hundreds of pounds to the weight of the sledges and the boats atop them. The men so sick that they were being carried in those boats, lying atop supplies and folded tents, burrowed under the canvas covers for shelter.\n\nThe men hauled forward through three days of continuous driving snow out of the east and southeast. At night, lightning crashed and the men cowered low against the canvas floors of their tents.\n\nToday they had stopped because too many of the men were sick and Goodsir wanted to administer to them, and because Crozier wanted to send parties ahead to scout and larger armed parties north into the interior and south out onto the sea ice to hunt.\n\nThey needed food badly.\n\nThe good news and the bad news was that they had finally finished the last of the Goldner canned foods. When the steward Aylmore, who on captain's orders had continued to eat and grow fat on the tinned foods, hadn't died from the terrible symptoms that killed Captain Fitzjames \u2014 although two other men who were not supposed to be eating from the cans had \u2014 everyone went back on the tinned foods to supplement the little remaining salt pork and cod and biscuits.\n\nThe 28-year-old seaman Bill Closson died screaming silently and convulsing from gut pains and paralysis, but Dr. Goodsir had no clue what might have poisoned him until one of his mates, Tom McConvey, confessed that the dead man had stolen and eaten a Goldner can of peaches that no one else had shared.\n\nIn the very brief burial service for Closson \u2014 his body lying without even a canvas shroud under the loose pile of rocks because Old Murray, the sailmaker, had died of scurvy and there was no extra canvas left anyway \u2014 Captain Crozier had quoted not from the Bible the men knew but from his fabled _Book of Leviathan._\n\n\"Life is 'solitary, poor, nasty, brutish, and short,'\" the captain had intoned. \"It seems it is shorter for those who steal from their mates.\"\n\nThe eulogy, such as it was, was a hit with the men. Although the ten boats they had been dragging and hauling on sledges for more than two months all had old names assigned to them from when _Erebus_ and _Terror_ still sailed the seas, the man-hauling teams of seamen immediately renamed the three cutters and two pinnaces always hauled during the afternoon and evening stint of hauling \u2014 the part of the day they hated the most since it meant regaining ground already won through the sweat of the long morning. The five boats were now officially named Solitary, Poor, Nasty, Brutish, and Short.\n\nCrozier had grinned at this. It meant the men were not so far gone into hunger and despair that their English sailors' black humour did not still hold a cutting edge.\n\nThe mutiny, when it came, was made vocal by the last man on earth that Francis Crozier would have imagined opposing his command.\n\nIt was the middle of the day and the captain was trying to get a few minutes' sleep while most of the men were out of camp doing reconnaissance or hunting. He heard the slow shuffle of many screw-heeled boots in the snow outside his tent, and he knew immediately that there was trouble outside the usual range of daily emergencies. The furtive sound of the footsteps as he came up from his light sleep warned him of the defiance to come.\n\nCrozier pulled on his greatcoat. He always carried a loaded pistol in the right pocket of this coat, but recently he had begun carrying a smaller two-shot pistol in his left pocket as well.\n\nThere were about twenty-five men assembled in the open area between Crozier's tent and the large Sick Bay tent. The blowing snow, thick scarves, and filthy Welsh wigs made some of them hard to identify at first glance, but Crozier was not surprised to see Cornelius Hickey, Magnus Manson, Richard Aylmore, and a half dozen of the more vocal resenters in the second row.\n\nIt was the first row facing him that surprised him.\n\nMost of the officers were off commanding the scattered hunting and scouting parties Crozier had sent out that morning \u2014 Crozier realized his mistake too late, sending away all of his most loyal officers, including Lieutenant Little and his second mate Robert Thomas, Tom Johnson, his faithful bosun's mate, Harry Peglar and some others, all at once, leaving the weaker men congregated here at Hospital Camp \u2014 but standing in front of this group was young Lieutenant Hodgson. Crozier was also shocked to see Reuben Male, captain of the forecastle, and _Erebus_ 's captain of the foretop, Robert Sinclair, here. Male and Sinclair had always been good men.\n\nCrozier strode toward the gathering so quickly that Hodgson actually took two steps back and collided with the giant idiot, Manson.\n\n\"What do you men want?\" rasped Crozier. Wishing that his voice was not such a hoarse croak, he put as much volume and authority into it as he could. \"What the hell is going on here?\"\n\n\"We need to talk to you, Captain,\" said Hodgson. The young man's voice was trembling with tension.\n\n\"About what?\" Crozier kept his right hand in his pocket. He saw Dr. Goodsir come to the opening of the Sick Bay tent and look out in surprise at the mob. Crozier counted twenty-three men in the group, and, despite the wigs pulled low and scarves pulled high, he noted who each man was. He would not forget.\n\n\"About going back,\" said Hodgson. The men behind him began muttering assent with the crowd murmur that was always the hive-mind sound of mutineers.\n\nCrozier did not react at once. One piece of good news here was that if it was an active mutiny, if all the men including Hodgson and Male and Sinclair had already agreed to take control of the expedition by force, Crozier would be dead by now. They would have acted in the twilight dimness at midnight.\n\nAnd the only other piece of good news was that while two or three of the seamen here were carrying shotguns, all the other weapons were out with the sixty-six men hunting today.\n\nCrozier made a mental note never to allow all of the Marines to leave the camp again at the same time. Tozer and the others had been eager to hunt. The captain had been so tired that he had not thought twice about giving them permission to go.\n\nThe captain looked from face to face. Some of the weaker ones in the crowd looked down immediately, ashamed to meet his gaze. The stronger ones like Male and Sinclair stared back. Hickey looked at him with eyes so hooded and cold they could have belonged to one of the white bears they'd encountered \u2014 or perhaps to the thing on the ice itself.\n\n\"Go back to where?\" snapped Crozier.\n\n\"To T-terror Camp,\" stuttered Hodgson. \"There's canned food and some coal and the stoves there. And the other boats we left.\"\n\n\"Don't be a fool,\" said Crozier. \"We're at least sixty-five miles from Terror Camp. It would be October \u2014 solid winter \u2014 before you reached it, if you ever did.\"\n\nHodgson wilted, but the captain of _Erebus_ 's foretop said, \"We're a hell of a lot closer to the camp than we are to this river we're killing ourselves to haul the boats to.\"\n\n\"That's not true, Mr. Sinclair,\" rasped Crozier. \"Lieutenant Little and I estimate that the inlet to the river is less than fifty miles from here.\"\n\n\"The _inlet,_ \" sneered a seaman named George Thompson. The man was known for drunkenness and laziness. Crozier could not cast the first stone at him for the drinking, but he despised laziness.\n\n\"The _mouth_ of Back's River is fifty miles south down the inlet,\" continued Thompson. \"More than a hundred miles from here.\"\n\n\"Watch your tone, Thompson,\" warned Crozier in a tone so low and deadly that even that lout blinked and looked down. Crozier looked around the crowd again. He spoke to all the men. \"It doesn't matter if it's forty miles down the inlet to the mouth of Back's River or fifty miles, odds are good that it will be open water... we'll be _sailing_ the boats, not dragging them. Now go back to your duties and forget this nonsense.\"\n\nSome men shuffled, but Magnus Manson stood like a broad dam holding the lake of their defiance in place. Reuben Male said, \"We want to go back to the ship, Captain. We think we'll have a better chance there.\"\n\nIt was Crozier's turn to blink. \"Go back to _Terror?_ Good Christ, Reuben, it must be more than ninety miles back to the ship, across _pack ice_ as well as back through all that rough territory we've come through. The boats and sledges would never make it.\"\n\n\"We'll just take one boat,\" said Hodgson. The men murmured agreement behind him.\n\n\"What the hell are you talking about, one boat?\"\n\n\"One boat,\" insisted Hodgson. \"One boat on one sledge.\"\n\n\"We're sick of this man-hauling shite,\" said John Morfin, a seaman who had been seriously injured during the Carnivale.\n\nCrozier ignored Morfin and said to Hodgson, \"Lieutenant, how do you plan to get twenty-three men into one boat? Even if you steal one of the whaleboats, that will only hold ten or twelve of you, with minimal supplies. Or are you planning on having ten or more of your party die before you get back to the camp? They will, you know. More than that.\"\n\n\"There are the small boats at Terror Camp,\" said Sinclair, stepping closer and taking an aggressive stance. \"We take one whaleboat back and use it and the jolly boats and ship's boats to ferry us out to _Terror._ \"\n\nCrozier stared a moment and then actually laughed. \"Do you think the ice has broken up there northwest of King William Land? Is _that_ what you fools think?\"\n\n\"We do,\" said Lieutenant Hodgson. \"There's food on the ship. Lots of the canned food left. And we could sail for...\"\n\nCrozier laughed again. \"You'd bet your lives that the ice has opened up enough this summer that _Terror_ is afloat and just waiting for you to row your dinghies out to her? And that leads have opened up the entire way we came south? Three hundred _miles_ of open water? In winter when you get there, if any of you do?\"\n\n\"It's a better gamble than this, we think,\" shouted the gunroom steward, Richard Aylmore. The dark little man's face was contorted with rage, fear, resentment, and something like exhilaration now that his hour had come round at last.\n\n\"I'd almost like to go with you... ,\" began Crozier.\n\nHodgson blinked rapidly. Several of the men looked at one another.\n\n\"Just to see your faces when that gamble pays off with you walking _across the ice and pressure ridges_ to find that _Terror_ has been broken up by the ice just as _Erebus_ was in March.\"\n\nHe let the effect of that image sink in for a few seconds before he said softly, \"For Christ's sake, ask Mr. Honey or Mr. Wilson or Mr. Goddard or Lieutenant Little about what shape her _knees_ were in. What shape her _rudder_ was in. Ask First Mate Thomas about how badly her seams had started way back in April... it is _July_ now, you fools. If the ice has melted around her even a wee bit, odds are greater that the old ship has sunk than floated. And if she hasn't, can twenty-three of you honestly tell me that you can man the pumps while sailing her through the maze of leads \u2014 if you get back in half the time it took you to get here just from Terror Camp, the winter freeze will already be setting in again. And how are you going to find your way through the ice if the ship can float, if it hasn't sunk, if you don't die manning the pumps day and night?\"\n\nCrozier looked around the mob again.\n\n\"I don't see Mr. Reid here. He's out with Lieutenant Little scouting our way south. With no ice master, you'll have a pretty time finding your way through the pancake ice and growlers and pack ice and bergs.\" Crozier shook his head at the absurdity of it all and chuckled as if the men had come to tell him a particularly good joke rather than foment a mutiny.\n\n\"Go back to your duties... _now,_ \" he snapped. \"I won't forget that you were foolish enough to bring this idea to me, but I'll try to forget the tone you used and the fact that you came like a mob of mutineers rather than like loyal members of Her Majesty's Royal Navy wanting to talk to their captain. Go on with you now.\"\n\n\"No,\" said Cornelius Hickey from the second row, his voice high and sharp enough to stop the wavering men in their tracks. \"Mr. Reid will come with us. So will the others.\"\n\n\"Why will they?\" asked Crozier, pinning the ferret with his gaze.\n\n\"They won't have any choice,\" said Hickey. He tugged at Magnus Manson's sleeve and the two stepped forward, past an alarmed-looking Hodgson.\n\nCrozier decided that he would shoot Hickey first. His hand was on the pistol in his pocket. He would not even remove the weapon from the greatcoat for the first shot. He would shoot Hickey in the belly when he got three feet closer and then pull the pistol out and try to shoot the giant in the center of his forehead. No body shot was guaranteed to bring Manson down.\n\nAs if his thinking about shooting had made it happen, there came the crack of a shot from the direction of the coastline.\n\nEveryone except Crozier and the caulker's mate turned to see what was happening. Crozier's gaze never left Hickey's eyes. Both men turned their heads only when the shouting started.\n\n\"Open water!\" It was Lieutenant Little's party coming in from the pack ice \u2014 Ice Master Reid, Bosun John Lane, Harry Peglar, and half a dozen others, all carrying shotguns or muskets.\n\n\"Open water!\" screamed Little again. He was waving both arms as he came across the rocks and ice of the shoreline and was obviously unaware of the drama going on in front of his captain's tent. \"Not more than two miles south! Leads opening up large enough for the boats. Going on to the east for miles! Open water!\"\n\nHickey and Manson stepped back into the ranks of cheering men where a mob had stood thirty seconds earlier. Some of the men started hugging one another. Reuben Male looked as if he was going to throw up at the thought of what he'd been about to do, and Robert Sinclair sat down on a low rock as if all the strength had gone out of his legs. The once-powerful foretop captain began to weep into his filthy hands.\n\n\"Get back to your tents and your duties,\" said Crozier. \"We'll start loading the boats and checking the masts and riggings within the hour.\"\n\nPEGLAR\n\n _Somewhere in the Strait Between King William Island and the Adelaide Peninsula_\n\n _9 July, 1848_\n\nThe men waiting in Hospital Camp had been eager to depart ten minutes after Lieutenant Little's party brought in the word of the open water, but it was another day before they broke camp and another two days until the boats' hulls were actually slipped from the ice into the black water south of King William Land.\n\nFirst they had to wait for all the other hunting and reconnaissance parties to return, and some came back after midnight, staggering into camp in the dim-yellow arctic twilight and collapsing into their sleeping bags without even hearing the good news. Very little game had been bagged, but Robert Thomas's group had killed an arctic fox and several white rabbits and Sergeant Tozer's team brought back a brace of ptarmigan.\n\nOn the morning of 5 July, a Wednesday, the Sick Bay tent all but emptied out as everyone who could stand or stagger wanted to lend a hand in preparations for putting to sea.\n\nJohn Bridgens had taken the place of the dead Henry Lloyd and Tom Blanky as Dr. Goodsir's assistant in recent weeks, and the steward had watched the previous afternoon's near mutiny while standing next to the surgeon in the door of the Sick Bay tent. It was Bridgens who described the whole scene to Harry Peglar, who felt sicker than he already was by learning that his _Erebus_ foretop counterpart, Robert Sinclair, had joined in the near uprising. Reuben Male, he knew, had always been a dependable man, but strong-willed. Very strong-willed.\n\nPeglar had nothing but contempt for Aylmore, Hickey, and their sycophants. In Harry Peglar's eyes they were all men with busy little minds and \u2014 except for Manson \u2014 an abundance of words, but no sense of loyalty.\n\nThat Thursday, the sixth of July, found them out on pack ice for the first time in more than two months. Most of them had forgotten how terrible the man-hauling was on the open ice, even here in the lee of King William Land and the bulbous cape they'd just come around. There were still pressure ridges to haul the ten boats up and over. The sea ice was far less slippery under runners than the snow and shore ice were. There were no vales in which to shelter, no low ridgelines \u2014 not even the occasional boulder \u2014 in which to hide from the wind. Out here there were no trickling streams to drink from. The snowstorm continued and the wind grew stronger out of the southeast, blowing directly in their faces as they hauled the boats the two miles Lieutenant Little's hunting group had covered before coming across the open lead.\n\nThe first night out they were so exhausted that they did not even erect the Holland tents but pitched a few tent floors as tarps extending from the leeward side of the boats and boats on sledges and huddled together on the ice through the few hours of arctic summer dimness in their three-man sleeping bags.\n\nEven with the storm, wind, and pack-ice difficulties, energized by their excitement, they covered the two miles by midmorning Friday, 7 July.\n\nThe lead was gone. Closed up. Little pointed out the thinner ice \u2014 none more than three to eight inches thick \u2014 where it had been.\n\nWith Ice Master James Reid in the lead, they followed the zigzag path of the recently frozen-over lead southeast then due east for much of that day.\n\nNow, added to their disappointment and ever-present misery exacerbated by the snow in their faces and their thoroughly soaked clothing, came the tension \u2014 for the first time in years \u2014 of walking on thin ice.\n\nA little after noon that day, Marine Private James Daly, who was one of six men sent ahead to test the ice by poking at it with long pikes, fell through. His comrades pulled him out, but not before he quite literally turned blue. Dr. Goodsir had Daly stripped naked on the ice, wrapped in Hudson's Bay blankets, and bundled under more blankets beneath the canvas cover of one of the cutters. Two other men had to stay with him, lying on either side of him in the canvas-yellow dimness beneath the boat cover so that their body warmth could keep him alive. Even then Private Daly's body shook and his teeth chattered uncontrollably and he ventured into delirium for much of the rest of the day.\n\nThe ice, as stable as a continent underfoot for two years, now rose and fell in low swells in a way that made everyone dizzy and caused some men to vomit. Pressure made even the thicker ice crack and groan with sudden explosions from far ahead, close ahead, to either side, behind, or directly underfoot. Dr. Goodsir had explained to them months before that one of the symptoms of advanced scurvy was a man's heightened sensitivity to sound \u2014 the blast of a gunshot could actually kill a man, he had said \u2014 and now the majority of the 89 men pulling the boats across the ice recognized those symptoms in themselves.\n\nEven a near idiot like Magnus Manson realized that if any or all of the boats fell through the ice \u2014 ice that had failed to support a single skinny, starved scarecrow of a man like James Daly \u2014 there would be no hope for any of the men in harness. They would drown even before they froze to death.\n\nUsed to their tight procession across the ice, the men felt strange about their new man-hauling method of keeping the boats far apart and staggered. At times in the snowstorm each group would be out of sight of all the other groups and the sense of isolation was terrible. When they went back to haul the last three cutters and two pinnaces forward, they did not follow their old tracks and had to worry that the new ice they were on would not hold them.\n\nSome of the men grumbled that they might have already missed the inlet leading south to the mouth of Back's River. Peglar had seen the charts and Crozier's occasional theodolite reading and knew that they were still a good distance to the west \u2014 thirty miles to the inlet, at the very least. Another sixty or sixty-five miles south then to the mouth. At the rate of their travel on land, even if food appeared and everyone's health miraculously improved, they would not reach the inlet until August and the mouth of the river until late September at the earliest.\n\nThe promise of open water made Harry Peglar's heart pound. Of course, his heart was pounding erratically much of the time these days anyway. Harry's mother had always worried about his heart \u2014 as a boy he had suffered scarlet fever and frequent pains in his chest \u2014 but he'd always told her such concerns were nonsense, that he was foretop captain in some of the world's greatest ships and that no man with a bad heart could hold such a position. Somehow he convinced her he was fine, but over the years Peglar felt occasional flutters in his chest, followed by days of pain and a sense of constriction and an ache down his left arm so bad that some days he had to climb to the foretop and upper spars with only one hand. The other foretopmen thought he was showing off.\n\nThese last weeks, his heart fluttered more frequently than not. He'd lost the use of his left fingers two weeks ago and the ache never left him. This, along with the embarrassment and inconvenience of the constant diarrhea \u2014 Peglar had always been a modest man, even about doing his business in the open over the side of a ship, which other men gave no thought to, had kept him constipated and waiting for darkness or the seat of ease.\n\nBut there was no seat of ease on this march. Not even a God-damned bush or shrub or large rock to hide behind. The men in Peglar's hauling team laughed that their petty officer would fall behind out of sight and risk being taken by the Terror rather than allow himself to be seen taking a shit.\n\nIt wasn't the friendly laughter that had bothered Peglar in recent weeks; it was the rushing to catch up with his team and to get back into harness. He was so exhausted from the internal bleeding and lack of food and heart flutters that he was having more and more trouble rushing to catch the receding boats.\n\nSo out of eighty-nine men this Friday, Harry Peglar was probably the only one who welcomed the blowing snow and the fog that came in after the snow began to abate.\n\nThe fog was a problem. Traveling so separately across the treacherous ice, it would have been easy for the boat teams to lose one another. Even backtracking to pick up the remaining cutters and pinnaces had been a problem, and that was before the fog grew thick as evening approached. Captain Crozier called a halt to discuss the matter. No more than fifteen men were allowed to congregate on a small area of ice at one time and that not too near a boat. They were pulling this evening with the fewest men it took to move the huge, heavy masses of boats and sledges.\n\nThe sledges were going to be a logistical problem if they ever reached the promised open water. The chances were great that they would need to load the deep-draft cutters and pinnaces with their keels and fixed rudders on sledges again before they reached the mouth of Back's River, so they couldn't just abandon the battered vehicles on the ice. Before leaving on Thursday, Crozier rehearsed taking the six sledged boats off, collapsing the heavy sledges as much as they had been designed to be collapsed or broken down, and stowing them properly in the boats. It took hours.\n\nSetting the boats back up on their sledges before going out onto the pack ice was just within the men's failing strength and abilities. Fingers stupid with fatigue and scurvy fumbled with simple knots. A shallow cut kept bleeding. The slightest jostle left hand-sized bruises on their softening arms and in the thinning skin above their ribs.\n\nBut now they knew they could do it \u2014 unload, then load again the sledges, ready the boats for launching.\n\nIf they found the lead soon.\n\nCrozier had each boat team light lanterns fore and aft. He called back the almost useless Marine ice-checkers with their pikes and appointed Lieutenant Hodgson as officer to lead the diamond of five boats, with one of the heavy whaleboats filled with the least essential items being pulled ahead of the others in the fog.\n\nEvery man there knew that this was young Hodgson's reward for throwing in with potential mutineers. His man-hauling team was led by Magnus Manson, while Aylmore and Hickey were also in harness, men who until now had been assigned to separate teams. If this lead boat team broke through the ice, the others would hear the screams and flailings through the heavy evening fog, but there would be nothing they could do except leave them and go a safer way.\n\nThe rest now must risk a near procession, staying close enough that they could see the others' lanterns in the growing gloom.\n\nAround 8:00 p.m. there did come shouts and screams from Hodgson's lead team, but they had not fallen through. They'd found open water again more than a mile east and south of where Little had seen a lead on Wednesday.\n\nThe other teams sent men forward with lanterns, moving tentatively on what they assumed was thin ice, but the ice stayed firm and was estimated to be more than a foot thick right up to the edge of the inexplicable lead.\n\nThe cleft of black water was only about thirty feet wide, but it extended off into the fog.\n\n\"Lieutenant Hodgson,\" commanded Crozier, \"make room in your whaleboat for six men at oars. Put the extra supplies out on the ice for now. Lieutenant Little will then take command of the whaleboat. Mr. Reid, you will go along with Lieutenant Little. You will proceed down the lead for two hours if that is possible. Don't raise your sail, Lieutenant. Oars only, but have the men put their backs into it. At the end of two hours \u2014 if you get that far \u2014 turn around and row back with your recommendation as to whether it's worth our effort to launch the boats. We'll use the four hours you are gone to unload everything here and pack the sledges into the remaining boats.\"\n\n\"Aye, sir,\" said Little and began barking orders. Peglar thought that young Hodgson looked as if he might weep. He knew how hard it must be to be in your twenties and know that your Naval career was over. _Serves him right,_ thought Peglar. He'd spent decades in a navy that hanged men for mutiny and lashed them for the mere _thought_ of mutiny, and Harry Peglar had never disagreed with either the rule or the punishment.\n\nCrozier walked over. \"Harry, do you feel well enough to go along with Lieutenant Little? I'd like you to handle the tiller. Mr. Reid and Lieutenant Little will be in the bow.\"\n\n\"Oh, yes, Captain. I feel fine.\" Peglar was shocked that Captain Crozier thought he looked or acted sick. _Have I been malingering in any way?_ The very thought that he could have been made him sicker.\n\n\"I need a good man on the sweep oar and a third assessment as to whether this lead is a go,\" whispered Crozier. \"And I need at least one man along who knows how to swim.\"\n\nPeglar smiled at this even as his scrotum tightened at the thought of going into that black, cold water. The air temperature was below freezing, and the water, with all its salt content, would be as well.\n\nCrozier clapped Peglar on the shoulder and moved on to talk to another \"volunteer.\" It was obvious to the foretop captain that Crozier was carefully picking the men he wanted along on this scouting trip while keeping others, like First Mate Des Voeux, Second Mate Robert Thomas, Bosun's Mate and _Terror_ 's disciplinarian Tom Johnson, and all the Marines, with him and alert.\n\nIn thirty minutes they had the boat ready to float.\n\nIt was a strangely equipped expedition within an expedition. They brought along a bag with some salt pork and biscuits, as well as some water bottles in case they became lost or otherwise extended the four-hour mission. Each of the nine men was handed an axe or pickaxe. If they should find a small berg overhanging and blocking the lead, or if a scrim of ice should block the way, they would try hacking their way through. Peglar knew that if a wider, thicker band of ice stopped them, they would portage the whaleboat to the next band of open water if they could. He hoped that he had the strength left to do his part in lifting, pulling, and shoving the heavy boat for a hundred yards or more.\n\nCaptain Crozier handed Lieutenant Little a two-barreled shotgun and a bag of cartridges. The items were stowed in the bow.\n\nShould they somehow be stranded out there, Peglar knew, the heaps of supplies they kept onboard included a double-sized tent and a tarp for the floor. There were three three-man sleeping bags kept in the boat. But they did not plan to get lost out there.\n\nThe men crawled in and found their places as the ice fog curled around them. The previous winter, Crozier and the other officers and mates had discussed having Mr. Honey \u2014 and Mr. Weekes before his death on _Erebus_ in March \u2014 raise the sides of all the boats. The small craft would have been better prepared for open seas that way. But in the end it was decided to keep the gunwales at their usual height to better facilitate river travel. Also to that end, Crozier had ordered all the oars cut down in length so that they might more easily be used as paddles on the river.\n\nThe remaining ton or so of bundled food and gear in the bottom of the boat made seating difficult; those six seamen at the oars had to prop their feet on the duffels and would be rowing or paddling with their knees as high as their heads, and as the man at the oar-sweep tiller, Peglar found himself sitting on a rope-wrapped bundle rather than on the stern bench \u2014 but everyone fit and there was room for Lieutenant Little and Mr. Reid to perch in the bow with their long pikes.\n\nThe men were eager to launch the boat. There was a chorus of \"one, two, three\" and several heave-hos, and the heavy whaleboat slid across the ice, the bow tipped and fell two feet into the black water, the oarsmen fended off nearby ice as Mr. Reid and Lieutenant Little crouched and gripped the gunwales, the men on the ice heaved again, oars found water, and they were moving away in the fog \u2014 the first boat from _Erebus_ or _Terror_ to feel liquid water under its hull in almost two years and eleven months.\n\nA spontaneous cheer went up, followed by the more traditional three hip-hip-hurrahs.\n\nPeglar steered the boat to the center of the narrow lead \u2014 never more than twenty feet across here, sometimes barely room for the shortened oars to find water on both sides \u2014 and by the time he glanced back over his shoulder, all the men on the ice were lost in the fog astern.\n\nThe next two hours were dreamlike. Peglar had steered a small boat through floe ice before \u2014 it had taken more than a week of poking into berg-ridden harbours and inlets before they'd found the right anchorage for the two ships at Beechey Island two autumns ago, and Peglar had been in command of one of those small boats for days \u2014 but that had not felt like this. The lead stayed narrow \u2014 never more than thirty feet wide and sometimes so tight they propelled the whaleboat by poling on the ice that scraped the sides rather than by rowing \u2014 and the narrow channel of open water would bend left and then right, but never quite so tightly that the boat could not make the turns. Tumbles of pressure-raised ice hid the view to either side and the fog continued to close on them, then open a bit, then close even more tightly. Sounds seemed to be muffled and amplified at the same time and the effect was unsettling; men found themselves whispering when they had to communicate.\n\nTwice they encountered stretches where floating ice blocked the way or the lead itself was frozen over to the point that most of the men had to clamber out to shove floating ice ahead with pikes or to hack away at the frozen surface with pickaxes. Some of the men stayed on the ice on either side then, pulling at ropes tied to the bow and thwarts or grabbing the gunwales and shoving and pulling the screeching whaleboat through the narrow crevice. Each time the lead then widened enough that the men could clamber back in and shove, paddle, and row their way forward.\n\nThey had been creeping forward this way for almost their full allotment of two hours when suddenly the meandering lead narrowed. Ice scraped both sides, but they used the oars to pole as Peglar stood in the bow, his steering sweep useless. Then suddenly they popped out into what was by far the widest stretch of open water they had seen. As if confirming that all their troubles were behind them, the fog lifted so that they could see hundreds of yards.\n\nThey had either reached true open water or a massive lake in the ice. Sunlight streamed down from a hole in the clouds above and turned the seawater blue. A few low, flat icebergs, one the size of a respectable cricket pitch, floated ahead of them in the azure sea. The icebergs prismed the light and the weary men shielded their eyes from the painful glory of sunlight shimmering on snow, ice, and water.\n\nThe six men at the oars gave a loud, spontaneous cheer.\n\n\"Not yet, men,\" said Lieutenant Little. He was peering through his brass telescope, his foot up on the whaleboat's bow. \"We don't know yet if this goes on... if there's a way out of this ice lake other than the way we came in. Let's make sure of that before we turn back.\"\n\n\"Oh, it goes on,\" shouted the seaman named Berry from his place at the oars. \"I feel it in me bones. It's open water and fair breezes between here and Back's River, all right. We'll get the others, open our sails, and be there before supper tomorrow.\"\n\n\"I pray you're right, Alex,\" said Lieutenant Little. \"But let's spend some time and sweat to make certain. I want to bring nothing but good news back to the rest of the men.\"\n\nMr. Reid, their ice master, pointed back at the lead from which they had emerged. \"There are a dozen inlets here. We might have trouble finding the real lead when we come back unless we mark it now. Men, bring us back to the opening there. Mr. Peglar, why don't you take that extra pike and drive it into the snow and ice there at the edge where we can't miss it on our way back. It'll give us something to row toward.\"\n\n\"Aye,\" said Peglar.\n\nWith their return avenue marked, they rowed out into the open water. The large, flat iceberg was only a hundred yards or so from the opening to their inlet, and they rowed close to it on their way toward open water.\n\n\"We could camp on 'aton and have plenty of room left over,\" said Henry Sait, one of the _Terror_ seamen at the oars.\n\n\"We don't want to camp,\" said Lieutenant Little from the bow. \"We've had enough camping for a fucking lifetime. We want to _go home._ \"\n\nThe men cheered and put their backs into it. Peglar at the sweep started a chantey and the men sang along, the first real singing they'd done in months.\n\nIt took them three hours \u2014 a full hour beyond the time they should have turned back \u2014 but they had to be sure.\n\nThe \"open water\" was an illusion: a lake in the ice a little more than a mile and a half long and a little more than two thirds of a mile wide. Dozens of apparent \"leads\" opened from the irregular southern, eastern, and northern ice edges of the lake, but they were all false starts, mere inlets.\n\nAt the southeastern terminus of the lake they tied up to the ice shelf, driving a pickaxe into the six-foot-thick ice and tying on to it, then cutting steps up the side as if it were a wharf; all the men clambered out and looked to the direction they'd hoped the open water continued.\n\nSolid, flat white. Ice and snow and seracs. And the clouds were coming down again, swirling into a low fog. It was beginning to snow.\n\nAfter Lieutenant Little looked in each direction, they boosted the smallest man, Berry, up onto the shoulders of the largest man there, thirty-six-year-old Billy Wentzall, and let Berry look through the glass. He boxed the compass with his search, telling Wentzall when to turn.\n\n\"Not so much as a fookin' penguin,\" he said. It was an old joke, referring to Captain Crozier's trip to the other pole. No one laughed.\n\n\"Do you see dark sky anywhere?\" asked Lieutenant Little. \"As one sees over open water? Or the tip of a larger berg?\"\n\n\"Nay, sir. And the clouds is comin' closer.\"\n\nLittle nodded. \"Let's head back, boys. Harry, you clamber down into the boat first and steady her, will you?\"\n\nNo one said a word in their ninety-minute pull across the lake. The sunlight disappeared and fog blotted away the landscape again, but before long the cricket-pitch berg loomed out of the mist and showed them that they were going the correct direction.\n\n\"We're almost back to the lead,\" called Little from the bow. At times the fog was so thick that Peglar in the stern had trouble seeing the lieutenant. \"Mr. Peglar, a little to port, please.\"\n\n\"Aye, sir.\"\n\nThe men at the oars did not even look up. To a man they seemed lost in the misery of their thoughts. Snow was pelting them again, but from the northwest now. At least the men at the oars had their backs to it.\n\nWhen the fog did lift a bit, they were less than a hundred feet from the inlet.\n\n\"I see the pike,\" Mr. Reid said tonelessly. \"A bit to starboard and you have it lined up nicely, Harry.\"\n\n\"Something's wrong,\" said Peglar.\n\n\"What do you mean?\" called back the lieutenant. Some of the seamen looked up from their oars and frowned at Peglar. With their backs to the bow, they could not see ahead.\n\n\"Do you see that serac or big ice boulder near the pike I left at the mouth of the lead?\" said Harry.\n\n\"Yes,\" said Lieutenant Little. \"So?\"\n\n\"It wasn't there when we came out,\" said Peglar.\n\n\"Back oars!\" ordered Little, uselessly since the men had already ceased their rowing and were backstroking briskly, but the heavy whaleboat's momentum continued carrying it toward the ice.\n\nThe ice boulder turned.\n\nGOODSIR\n\n _King William Land, Lat. Unknown, Long. Unknown_\n\n _18 July, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Tuesday, 18 July, 1848 \u2014_\n\n _Nine days ago, when our Captain sent Lieutenant Little and eight Men ahead in a Whaleboat through the Lead in the Ice with orders to Return in 4 Hours, the rest of us Slept the best we could for a Pitiful Remnant of those 4 Hours. We spent more than 2 Hours loading the Sledges onto the Boats and then, taking no Time to unpack Tents, we attempted to sleep in our Reindeer Skin and Blanket bags atop waterproof tarps set down on the Ice next to the Boats themselves. The days of the Midnight Sun were past now in early July and we slept \u2014 or Tried to Sleep \u2014 through the few Hours of near Darkness. We were very tired._\n\n _After the apportioned 4 Hours were up, First Mate Des Voeux woke the men, but there was no Sign of Lieutenant Little. The Captain allowed most to return to Sleep._\n\n _Two hours later, All were Wakened, and I tried to lend a Hand as best I could \u2014 following the orders of Second Mate Couch as the Boats were made ready to Launch. (As a Surgeon, of course, I always have some Fear of injuring my Hands, although it is True that so far on this Voyage they have Suffered every Insult short of Serious Frostbite and Self-Amputation.)_\n\n _So it was that 7 Hours after Lieutenant Little, James Reid, Harry Peglar, and the six seamen had set off on their Reconnaissance, 80 of us on the ice prepared our own boats to follow. Due to movement of the Ice and lowering temperatures, the Lead had narrowed somewhat during the few hours of Darkness and few more hours of Sleep, and getting the nine Boats placed properly and launched correctly took some Skill. Eventually all of the boats \u2014the 3 whaleboats with Captain Crozier's in the lead (Second Mate Couch's in second position with me aboard it) and then the 4 cutters (commanded, respectively, by Second Mate Robert Thomas, Bosun John Lane, Bosun's Mate Thomas Johnson, and Second Lieutenant George Hodgson), followed by the two pinnaces under the command of Bosun's Mate Samuel Brown and First Mate Charles Des Voeux (Des Voeux was third in command of our overall Expedition now behind Captain Crozier and Lieutenant Little and thus assigned the Responsibility of bringing up the rear)._\n\n _The weather had grown colder and there was Some light Snow falling, but by and large the Fog had lifted to become a Low-Hanging layer of Clouds moving only a Hundred Feet or so above the ice. While this allowed us to see much farther than in the fog of the previous Day, the effect was oppressive, as if all our Movements were taking place in some strange Ballroom set in a deserted Arctic Mansion with a shattered White Marble Floor underfoot and a Low Grey Ceiling with_ trompe l'oeil _clouds just above us._\n\n _At the moment the 9th and Final Boat was shoved into the water and its Crew clambered in, there was a faint and Sad Attempt at a hurrah from the men since it was the first time that most of these Deep Water Sailors had been afloat in almost 2 Years, but the Cheer died aborning. Concern about Lieutenant Little's Crew's Fate was too great to allow for any Sincere hurrahing._\n\n _For the first Hour and a half, the only sounds were the Groaning of the Working Ice around us and the occasional Answering Groans of the Men Working at the oars. But seated near the front of the second boat as I was, sitting on the Thwart behind where Mr. Couch stood at the Bow, knowing that I was Superfluous to all Locomotive Purposes, as much Dead Weight as the poor comotose-but-still-breathing David Leys \u2014 whom the men had been hauling in one of the pinnaces without Complaint now for more than 3 Months and whom my new aide, former steward John Bridgens, duly fed and cleansed of his own Filth every Evening in the medical tent we shared as if he was caring for a Beloved but Paralysed Grandfather (ironic since Bridgens was in his early 60's and comatose Leys was only 40) \u2014 my position thus situated allowed me to hear Whispered Conversation between the Men at the Oars._\n\nLittle and the Others must have got themselves Lost, _whispered a seaman named Coombs._\n\nThere ain't no way that Lieutenant Edward Little got himself Lost, _shot back Charles Best._ He may be Stuck, but not Lost.\n\nStuck in what? _whispered Robert Ferrier at an adjoining Oar._ This Lead's open Now. It was open Yesterday.\n\nMaybe Lieutenant Little and Mr. Reid found the way Open Ahead of them all the way to Back's River and just raised their Sail and went on, _whispered Tom McConvey from one Row back._ They're there already is my guess... eating Salmon that jumped into their boat and Trading beads for Blubber with the Natives.\n\n _No one said anything to this unlikely Suggestion. The mention of the Esquimaux had caused Quiet Consternation since the massacre of Lieutenant Irving and 8 of the Savages on 24 April last. I believe that most of the Men, however desperate for Salvation or Rescue from any Source, Feared rather than Hoped for another contact with the local Native People. Revenge, Some natural philosophers suggest and Sailors endorse, is one of the most Universal of human motivations._\n\n _Two and a half hours after leaving our campsite of the Previous Night, Captain Crozier's whaleboat broke out of the Narrow Lead into an Open Stretch of water. Men in the lead boat and my own boat let out happy shouts. As if left behind to Point the Way, a tall black ship's pike stood Upright, embedded in the Snow and Ice at the exit from this Lead. The Night's snow and freezing drizzle had painted the northwest side of the pike White._\n\n _These shouts also died Aborning as our Close Line of Boats pulled out into Open Water._\n\n _The water was Red here._\n\n _On shelves of ice to the Left and the Right of the Lead Opening, crimson streaks of what could only be Blood were smeared on the flat ice and down the Vertical Planes of the ice edges. The Sight sent a Shiver through me and I could see other men reacting with Open Mouths._\n\nEasy now, men, _muttered Mr. Couch from the bow of our Boat._ This is just the sign of seals caught by the White Bears; we've seen such Seal Gore before in the Summers.\n\n _Captain Crozier in the lead boat was saying Similar Things to his Seamen._\n\n _A minute later we knew that these Crimson signs of Carnage were not the Residue of Seals butchered by White Bears._\n\nOh, Christ! _exclaimed Coombs at his oar. All the men quit rowing. The Three whaleboats, Four cutters, and Two pinnaces floated into a sort of circle in the choppy red-tinted water._\n\n _The bow of Lieutenant Little's whaleboat rose vertically from the Sea. Its Name (one of the 5 boat Names not changed after Captain Crozier's_ Leviathan _sermon in May) \u2014_ The Lady J. Franklin _\u2014 was clearly Visible in black Paint. The boat had been Broken Apart about 4 feet Back from the bow so that only this Forward Section \u2014 the ragged End of shattered thwarts and splintered Hull just visible beneath the surface of the Dark and Icy Water \u2014 floated there._\n\n _The men began Gathering other Flotsam as our 9 remaining Boats fanned out and rowed Slowly forward in a line: an Oar, more bits of Shattered Wood from gunwales and stern, a Steering Sweep, a Welsh wig, a bag that once held cartridges, a mitten, a bit of Waistcoat._\n\n _When Seaman Ferrier used a boat hook to pull in what looked to be a floating bit of Blue Peacoat, he suddenly cried out in Horror and almost dropped the long gaff._\n\n _A man's body floated there, his Headless Corpse still Garbed in sodden blue Wool, his Arms and Legs hanging down in the black water. The neck was a mere bit of severed Stump. His fingers, perhaps swollen by death and the cold water but looking strangely shortened into broad Stubs, seemed to move in the Currents, rising and falling on the Slight Swell like White Worms wriggling. It was almost as if, Voiceless, the Body was trying to tell us something via Sign Language._\n\n _I helped Ferrier and McConvey pull the Remains aboard. Fish or some Aquatic Predator had been nibbling at the Hands \u2014 the fingers were gone to the Second Joint \u2014 but the Extreme Cold had delayed the bloating and decomposition Processes._\n\n _Captain Crozier brought his whaleboat around until its bow was touching our side._\n\nWho is it? _muttered a seaman._\n\nIt's 'arry Peglar, _cried another._ I recognize the peajacket.\n\nHarry Peglar didn't Wear no green Waistcoat, _interjected another._\n\nSammy Crispe did! _exclaimed a 4 th Seaman._\n\nSilence! _bellowed Captain Crozier._ Dr. Goodsir, be so Good as to turn out our unfortunate Shipmate's pockets.\n\n _I did so. From the large pocket of the Wet Waistcoat, I pulled an almost-Empty tobacco Pouch tooled in red leather._\n\nAh, shite! _said Thomas Tadman, sitting next to Robert Ferrier on my Boat._ It's poor Mr. Reid.\n\n _And so it was. All the men then remembered that the Ice Master had been Wearing only his Peacoat and Green Waistcoat the previous evening, and All of Us had seen him refill his Pipe a thousand times from that faded red-leather pouch._\n\n _We looked to Captain Crozier as if he could explain what had Happened to our Shipmates, although in our Souls, we all knew._\n\nSecure Mr. Reid's body under that Boat Cover, _ordered the Captain._ We'll search the area to see if there are any Survivors. Do not row or drift out of sight or shouting range.\n\n _Once again, the boats fanned out. Mr. Couch brought our boat back to the ice near the Inlet Opening, and we Rowed Slowly along the icy Shelf that rose about 4 Feet above the open water's Edge. We stopped at each smear of Blood on the surface of the Floe and on the Vertical Face, but there were no more bodies._\n\nOh, damn, _moaned 30-year-old Francis Pocock from his place at the Sweep in the Stern of our Boat._ You can see the bloody grooves of the man's Fingers and Nails in the Snow. The Thing must've dragged him backwards into the Water.\n\nBatten down your Gob on such Talk! _called Mr. Couch. Holding his long pike easily in one hand like a True Whaleboat's Harpoon, he had one Booted Foot up on the whaleboat's Bow as he glowered back at the rowers. The men fell silent._\n\n _There were three such Bloody Spots on the ice at this Nor'west End of the Open Water.The third One showed where Someone had been Eaten some 10 Feet back from the Edge of the ice. A few leg bones remained, as did some gnawed Ribs, a Torn Integument that might be Human Skin, and some Strips of Torn Cloth, but no skull or identifiable features._\n\nPut me on the ice, Mr. Couch, _I said,_ and I shall Examine the Remains.\n\n _I did so. Had this been ashore almost anywhere in the World but Here, flies would have been buzzing around the Red Meat and Muscle left behind, not to mention the strands of Entrails looking like a Gopher's Burrowing Ridge beneath the thin Covering of last night's Snow, but here there was only the Silence and the soft Wind from the northwest and the Groaning of the Ice._\n\n _I called back to the Boat \u2014 the seamen were averting their Faces \u2014 and Confirmed that no identification was possible. Even the Few Remnants of Torn Clothing could give no clue. There was no Head, no Boots, no Hands, no Legs, not even a Torso other than the heavily gnawed Ribs, a Sinewed bit of Spine, and half a Pelvis._\n\nStay as you are, Mr. Goodsir, _called Couch._ I'm sending Mark and Tadman to you with an emptied shot bag in which to put the poor bugger's remains. Captain Crozier'll be wanting to give them a burial.\n\n _It was Grim work, but done quickly. In the end, I directed the two Grimacing Seamen to pack away only the Rib cage and bit of Pelvis into the shot-bag Burial Shroud. The Vertebrae had frozen into the Floe Ice, and the other remnants were too Grisly to bother about._\n\n _We had just shoved off from the ice and were Exploring along the South rim of the Open Water when there came a shout from the North._\n\nMan found! _some Seaman cried. And again,_ Man found!\n\n _I believe that we all could feel our Hearts Pounding as Coombs, McConvey, Ferrier and Tadman, and Mark and Johns pulled hard, and Francis Pocock steered us to a cricket-pitch-sized patch of floating ice that had drifted to the center of these Several Hundred Acres of Open Water amid the Frozen Floes. We all wanted \u2014 we all needed \u2014 to find someone Alive from Lieutenant Little's boat._\n\n _It was not to be._\n\n _Captain Crozier was already on the Ice and called me forward to the Body lying there. I Confess that I felt slightly Put Upon, as if even the Captain was unable to Certify Death unless I was forced to Inspect yet another Undeniably Dead Corpse. I was very Tired._\n\n _It was Harry Peglar lying there almost naked \u2014 his few remaining Clothes mere Underthings \u2014 Curled up on the Ice, Knees Raised almost to his Chin, Legs crossed at the Ankle as if his last energy had been spent trying to keep warm by pressing his body Tighter and Tighter, his Hands tucked under his Arms while he Hugged himself in what must have been an End in Violent Shivers._\n\n _His blue eyes were open and frozen. His flesh was also Blue and as Hard to the Touch as Carrera Marble._\n\nHe must've swum to the Floe, managed to Climb up, and froze to death here, _softly suggested Mr. Des Voeux._ The Thing from the Ice didn't catch or maul Harry.\n\n _Captain Crozier only nodded. I knew that the Captain had liked and much depended upon Harry Peglar. I also liked the Foretop Captain. Most of the men did._\n\n _Then I saw what Crozier was looking at. All around the Ice Floe in the recent snow \u2014 especially around the Corpse of Harry Peglar \u2014 were huge footprints, rather like a White Bear's with claws visibly indicated, only easily Three or Four times Larger than any white bear's paw prints._\n\n _The thing had Circled Harry many times. Watching as poor Mr. Peglar lay Shivering and Dying? Enjoying itself? Had Harry Peglar's last shivering Image on this Earth been of that White Monstrosity looming over him, its black, unblinking Eyes watching? Why had the thing not eaten our Friend?_\n\nThe Beast was on two legs the entire time it was on the floe, _was all that Captain Crozier said._\n\n _Other men from the Boats came forward with a piece of Canvas._\n\n _There was no exit from the Lake in the Ice except for the Rapidly Closing Lead from which we had come. Two circumnavigations of the Body of Open Water \u2014 five Boats rowing clockwise, four Boats rowing antiwiggens \u2014 offered the Discovery of only inlets, Ruptures in the Ice, and two more Bloody Swaths where it looked as if one of our reconnaissance whaleboat's crew pulled himself onto the ice and ran but was Cruelly Intercepted and pulled back. There were, thank God, shards of blue Wool but no more remains to be found._\n\n _It was early Afternoon by then, and to a Man I am sure that we had but one Wish \u2014 to be Away from that accursed Place. But we had three bodies of our Shipmates \u2014 or Parts of Same \u2014 and we felt the Need to Dispose of them in an Honourable way. (Many of us assumed, I Believe, and rightly so as it turned out, that these would be the last Formal Burial Services the reduced Remnants of our Expedition would have the luxury to perform.)_\n\n _No useful Detritus was found floating in the ice lake save for an Expanse of Soaked Canvas from one of the Holland Tents that had been aboard Lieutenant Little's doomed whaleboat. This was used to Inter the body of our friend Harry Peglar. The partial Skeletal remains I had investigated near the Lead opening were left in the canvas Shot Bag. Mr. Reid's torso was sewn into an extra blanket sleeping bag._\n\n _It is Custom at Burial at Sea for one or more pieces of Round Shot to be placed at the Foot of the man being Committed to the Deep, ensuring that the body will sink with Dignity rather than float Embarrassingly, but of course we had no Round Shot this day. The seamen scrounged a Grapple from the floating Bow of_ The Lady J. Franklin _and some metal from the last of the empty Goldner food tins to Weigh Down the various shrouds._\n\n _It took some time to pull the Nine Remaining Boats from the black water and reset the cutters and pinnaces onto Sledges. The Assembly of these Sledges and the lifting of the Boats onto them, with its concomitant Packing and Unpacking of stores, drained the skeletal crewmen of the last of their energy. Then the Seamen gathered near the edge of the Ice, standing in a broad Crescent so as not to put too much Weight on any one part of the Ice Shelf._\n\n _No one was in the mood for a Long Service and certainly not for the Previously Appreciated Irony of Captain Crozier's fabled_ Book of Leviathan, _so it was with some Surprise and not a small bit of Emotion that we listened to the Captain recite from Memory Psalm 90:_\n\nLORD, thou hast been our refuge: from one generation to another.\n\nBefore the mountains were brought forth, or ever the earth and the world were made: thou art God from everlasting, and world without end.\n\nThou turnest man to destruction: again thou sayest, Come again, ye children of men.\n\nFor a thousand years in thy sight are but as yesterday: seeing that is past as a watch in the night.\n\nAs soon as thou scatterest them, they are even as a sleep: and fade away suddenly like the grass.\n\nIn the morning it is green, and groweth up: but in the evening it is cut down, dried up, and withered.\n\nFor we consume away in thy displeasure: and are afraid at thy wrathful indignation.\n\nThou has set our misdeeds before thee: and our secret sins in the light of thy countenance.\n\nFor when thou art angry all our days are gone: we bring our years to an end, as it were a tale that is told.\n\nThe days of our age are three-score years and ten; and though men be strong that they come to fourscore years: yet is their strength then but labour and sorrow; so soon passeth it away, and we are gone.\n\nBut who regardeth the power of thy wrath: for even thereafter as a man feareth, so is thy displeasure.\n\nSo teach us to number our days: that we may apply our hearts unto wisdom.\n\nTurn thee again, O Lord, at the last: and be gracious unto thy servants.\n\nO satisfy us with thy mercy, and that soon: so shall we rejoice and be glad all the days of our life.\n\nComfort us again now after the time that thou has plagued us: and for the years wherein we have suffered adversity.\n\nShew thy servants thy work, and their children thy glory.\n\nAnd the glorious Majesty of the Lord our God be upon us: prosper thou the work of our hands upon us, O prosper thou our handy-work.\n\nGlory be to the Father, and to the Son, and to the Holy Ghost;\n\nAs it was in the beginning, is now, and ever shall be: world without end. Amen.\n\n _And all of us shivering survivors spake,_ Amen.\n\n _There was a Silence then. The snow blew softly against Us. The black water lapped with a Hungry Sound. The ice Groaned and Shifted slightly beneath our feet._\n\n _All of us, I believe, were Thinking that these words were a Eulogy and Farewell for each one of us. Up until this Day and the loss of Lieutenant Little's boat with all his men \u2014 including the irreplaceable Mr. Reid and the universally liked Mr. Peglar \u2014 I suspect that many of us still thought that we might Live. Now we knew that the odds of that had all but Disappeared._\n\n _The long awaited and Universally Cheered Open Water was a vicious Trap._\n\n _The Ice will not give us up._\n\n _And the creature from the ice will not allow us to leave._\n\n _Bosun Johnson called,_ Ship's Company \u2014 OFF hats! _We tugged off our motley and filthy head coverings._\n\nKnow that our Redeemer liveth, _said Captain Crozier in that Husky Rasp that now passed for his voice._ And that he shalt stand at the Latter Day upon the Earth. And though after our sin Worms destroy our bodies, yet in our flesh shall we see God: whom we shall see for ourselves, and our eyes shall behold, and not another.\n\nO Lord, accept your Humble Servants here Ice Master James Reid, Captain of the Foretop Harry Peglar, and their Unknown Crewmate into your Kingdom, and with the two we can Name, please accept the Souls of Lieutenant Edward Little, Seaman Alexander Berry, Seaman Henry Sait, Seaman William Wentzall, Seaman Samuel Crispe, Seaman John Bates, and Seaman David Sims.\n\nWhen our day comes to join Them, Lord, please allow us to join them in Thy Kingdom.\n\nHear our prayer, O Lord, for our Shipmates and Our Selves and for all Our Souls. And with thine ears consider our calling: hold not thy peace at our tears. Spare us a little, that we may recover our strength; before we also go hence and be no more.\n\nAmen.\n\nAmen, _we all whispered._\n\n _The bosuns lifted the canvas Burial Shrouds and dropped them into the black water, where they Sank within Seconds. White bubbles rose like Final Efforts to Speak from our departed Shipmates, then the surface of the lake grew Black and Still again._\n\n _Sergeant Tozer and two Marines fired a single volley from their muskets._\n\n _I saw Captain Crozier stare at the black lake with an expression rich with suppressed Emotions._ We will go now, _he said Firmly to us, to all of us, to this slumped and sad and Mentally Defeated party._ We can haul those sledges and boats a Mile before it is time to sleep. We will head Southeast toward the mouth of Back's River. The going will be Easier out here on the ice.\n\n _As it turned out, the going was much Harder on the ice. In the End, it was Impossible, not because of the usual Pressure Ridges and anticipated Difficulty transporting the boats, although that was Increasingly Problematic because of our Hunger, Illness, and Weakness, but because of the Breaking Ice and the Thing in the Water._\n\n _Moving in Relays as Usual but with Nine Fewer Men on our Expedition Muster that Long Arctic Evening of 10 July, we Progressed much less than a Mile before stopping to pitch tents on the Ice and to Sleep at last._\n\n _That sleep was Interrupted less than Two Hours later when the Ice suddenly began to crack and shift. The entire mass bobbed up and down. It was a most Disquieting Experience and we all scrambled to the exits of our Respective Tents and milled in some Confusion. Seamen began to strike the tents and make ready to pack the Boats until Captain Crozier, Mr. Couch, and First Mate Des Voeux shouted them into stillness. The officers pointed out that there were no signs of cracks in the ice near us, only this Movement._\n\n _After fifteen minutes or so of this, the ice Quieted until the Surface of the Frozen Sea under us once again felt as firm as Stone. We crawled back into our tents._\n\n _An hour later, the Bobbing and Cracking began again. Many of us repeated our earlier rush out into the Blowing Wind and dark, but the Braver Seamen stayed in their sleeping bags. Those of us who had Taken Fright crawled back into the Ill-Smelling and Crowded little tents \u2014 filled as they were with Snores, Sleeping Exhalations, Overlapping Bodies in Wet Bags, and the Ripeness of men who had not changed their Clothes in several months \u2014 with abashed countenances. Fortunately, it was too Dark for anyone to notice._\n\n _All that next day we struggled to haul the Boats forward to the Southeast across a Surface no more solid than a tightly drawn skin of India Rubber. Cracks were appearing \u2014 some showing six feet thickness of ice and more between the Surface and the Sea \u2014 but our sense of crossing a Plain of Ice had disappeared, replaced with the Reality of moving from Floe to Floe on an Undulating ocean of white._\n\n _I should Record here that on that Second Evening after we left the Enclosed Ice Lake, I was catching up with my Duty of going through the Dead Men's personal belongings, most of which had been Left Behind in our General Stores when Lieutenant Little's reconnaissance group left in their whaleboat, and had come to Captain of the Foretop Peglar's small pack containing a few scraps of Clothes, some Letters, a few personal items such as a Horn Comb, and several Books, when my Assistant, John Bridgens, said,_ Might I have a few of those things, Dr. Goodsir?\n\n _I was surprised. Bridgens was indicating the Comb and a thick Leather Notebook._\n\n _I had looked into the Notebook already. Peglar had written in a crude sort of Code \u2014 spelling words in Reverse, Capitalizing the last letter of the last word in each Sentence as if it were the first \u2014 but while the Summary of the last Year of our Expedition might have held some Interest for a Relative, Both the foretop captain's handwriting and sentence Structure, not to mention his spelling, had grown more Laboured and Crude in the Months immediately before and after our Abandoning of the Ships until it had all but disintegrated. One entry read,_ O Death whare is thy sting, the grave at Comfort Cove for who has any doubt how _... [an illegible line here where the book had been Damaged by water]..._ the dyer sad....\n\n _On the back side of that sheet, I had noticed where Peglar had drawn a shaky circle and in that circle had written,_ the terror camp clear. _The date had been illegible, but it must have been around 25 April. Another page nearby included such fragments as_ Has we have got some very hard ground to heave... we shall want some grog to wet houer... issel... all my art Tom for I do think... time... I cloze should lay and... the 21st night a gread.\n\n _I had Assumed upon Seeing this that Peglar had recorded that Entry on the Evening of 21 April when Captain Crozier had told the Assembled Crews of_ Terror _and_ Erebus _that the last of them would be Abandoning Ship the next morning._\n\n _These were, in other Words, the scribblings of a semiliterate Man and no Proud Reflection on the learning or Skill of Harry Peglar._\n\nWhy do you want these? _I asked Bridgens._ Was Peglar a friend of yours?\n\nAye, Doctor.\n\nYou require a Comb? _The old Steward was almost bald._\n\nNo, Doctor, just a Remembrance of the man. That and his Journal will serve.\n\n _Very strange, I thought, since everyone was lightening their Loads at this point, not adding Heavy Books to what they had to Haul._\n\n _But I gave Bridgens the Comb and Journal. No one needed Peglar's remaining Shirt or Socks or Extra Wool Trousers or Bible, so I left them on the Pile of discarded items the next morning. All in all, the Abandoned Final Possessions of Peglar, Little, Reid, Berry, Crispe, Bates, Sims, Wentzall, and Sait made for a sad little Cairn of Mortality._\n\n _That next morning, 12 July, we started coming Across more Bloody Patches in the Ice. At first the Men were Terrified that these were More signs of our Mates, but Captain Crozier led us to the Great Stained Areas and showed us that in the Centre of the Great Starbust of Crimson was the Carcass of a White Bear. They were all Murdered Polar White Bears, these Bloodstained areas, often with little More than a shattered Head, Great Bloodied White Pelt, Cracked Bones, and Paws left behind._\n\n _At first the men were Reassured. Then, of course, the Obvious Question set in \u2014_ what was killing these Huge Predators just Hours before our Arrival?\n\n _The answer was Obvious._\n\n _But why was it slaughtering the White Bears? That answer was also Obvious: to deprive us of any possible Food Source._\n\n _By 16 July, the men seemed Incapable of going farther. In an 18-hour Day of Incessant Pulling, we would cover less than a Mile across the Ice. Often we could see the previous night's Pile of Discarded Clothing and Gear when we camped the Next Evening. We had found more Slaughtered White Bears. Morale was so low that if we had taken a Vote that Week, the Majority might have voted to Give Up, Lie Down, and Die._\n\n _That night of 16 July, as Others Slept and only One Man stood Watch, Captain Crozier asked me to come to his Tent. He now slept in the same Tent with Charles Des Voeux; his purser, Charles Hamilton Osmer (who was showing signs of pneumonia); William Bell (_ Erebus _'s_ _quartermaster); and Phillip Reddington, Sir John's and Captain Fitzjames's former captain of the fo'c'sle._\n\n _The captain nodded and everyone except First Mate Des Voeux and Mr. Osmer left the tent to give us Privacy._\n\nDr. Goodsir, _began the Captain,_ I need your advice.\n\n _I Nodded and Listened._\n\nWe have adequate Clothing and Shelter, _said Captain Crozier._ The extra boots I Had the Men Haul along in the Supply Pinnaces have saved Many Feet from Amputation.\n\nI agree, Sir, _I said, although I knew this was not the Item upon which he was asking for Advice._\n\nTomorrow morning I am going to tell the Men that we shall be Leaving One of the Whaleboats and two Cutters and one Pinnace behind and will be Continuing on only with the Five Remaining Boats, _said Captain Crozier._ Those two whaleboats, two cutters, and final pinnace are in the Best Condition and should suffice for Open Water, should we Encounter Any before the Mouth of Back's River, since our Stores are so Reduced.\n\nThe Men will be Heartily Glad to hear this, Captain, _I said. I certainly was. Since I now helped Man-haul the boats, the Knowledge that the days of Accursed Relaying were over quite Literally took some of the Ache from my shoulders and back._\n\nWhat I need to Know, Dr. Goodsir, _continued the Captain, his voice an Exhausted Rasp, his face Solemn,_ is whether I can cut back on the Men's Rations. Or rather, when we _Do_ cut back, will the Men still be Capable of hauling the Sledges? I need your Professional Opinion, Doctor.\n\n _I looked at the floor of the Tent. One of Mr. Diggle's Hoosh Pans \u2014 or perhaps Mr. Wall's Portable Contraption for Heating Tea back when we had bottles of Ether left for the Spirit Stoves \u2014 had burned a Round Hole there._\n\nCaptain, Mr. Des Voeux, _I said finally, knowing that I would be Stating the Obvious to them,_ the men do not have enough Nourishment now to meet the Requirements of their Daily Labours. _I took a breath._\n\nEverything they eat is Cold. The last of the Canned Foods was Consumed many Weeks ago. The Spirit Stoves and Spirit Lamps was left on the Ice with the Last Empty Bottle of Pyroligneous Ether.\n\nThis evening at Supper each man will get one Ship's Biscuit, a sliver of Cold Salt Pork, one Ounce of chocolate, a Palm Full of Tea, less than a Spoonful of Sugar, and his Daily Tablespoon of Rum.\n\nAnd his Bit of Tobacco that we'd hoarded for them, _added Mr. Osmer._\n\n _I nodded._ Yes, and his bit of tobacco. And they do love their tobacco. That was a brilliant stroke to keep some hidden in the Stores. But no, Captain, I cannot say that the Men can get by on less than the Current Inadequate Amount of Food.\n\nThey must, _said Captain Crozier._ We shall be out of the salt pork in six days. Out of the Rum in ten.\n\n _Mr. Des Voeux cleared his throat._ Everything depends upon us Finding and Shooting more seals on the Floes.\n\n _So far, I knew \u2014 everyone in the Tent knew, everyone on the Expedition knew \u2014 we had shot and Enjoyed precisely 2 Seals since leaving Comfort Cove two Months earlier._\n\nI am thinking, _said Captain Crozier,_ that heading North again for the Shore of King William Land \u2014 perhaps Three Days' pull, perhaps Four \u2014 might be Best. It is possible to eat Moss and Rock Tripe. I am told that the proper Varieties cook up into an Almost Palatable soup. If one can find the proper Varieties of Moss and Rock Tripe.\n\nSir John Franklin, _I thought in my weariness._ The Man Who Ate His Shoes. _My older Brother had told me That Story in the Months before our Departure. Sir John would have known, from Pathetic Experience, precisely which Moss and Rock Tripe to choose._\n\nThe Men will be happy to get off the Ice, Captain, _was all that I could Say._ And they will be Overjoyed to Hear that we shall be Hauling Fewer boats.\n\nThank you, Doctor, _said Captain Crozier._ That is all.\n\n _I bobbed my head in a pathetic Sort of Salute, left, made the rounds of the worst Scurvy victims in their Tents \u2014 we no Longer Have a Sick Bay Tent, of course, and Bridgens and I nightly go from tent to tent to counsel and Dose our Patients \u2014 and then I staggered back to my own Tent (shared with Bridgens, the unconscious Davy Leys, the dying Engineer, Thompson, and the seriously ill carpenter, Mr. Honey), and fell Instantly Asleep._\n\n _That was the night that the Ice opened and swallowed up the Holland Tent in which Slept our Five Marines \u2014 Sergeant Tozer, Corporal Hedges, Private Wilkes, Private Hammond, and Private Daly._\n\n _Only Wilkes got out of the Tent before it Sank into the Wine-Dark Sea, and he was pulled from the Ice Crevice seconds before it Closed with a Deafening Crash._\n\n _But Wilkes was too Chilled, too Ill, and too Terrified to Recover, even when Bridgens and I wrapped him in the Last Dry Clothes in our Reserve and put him Between us in our Sleeping Bag. He died just before real Sunrise._\n\n _His Body was left behind on the Ice the next morning along with more Clothes and the Four Discarded Boats and their Sledges._\n\n _There was no Burial Service for him or the other Marines._\n\n _There was no Hurrah when the Captain announced that the four Sledges and Boats would no longer be hauled._\n\n _We turned North toward Land just over the Horizon. No retreat from Moscow ever felt such a sense of Defeat._\n\n _Three Hours Later, the Ice Cracked Again, and we were faced with Leads and Lakes to the North which were too small to justify launching the boats yet too large to allow us to haul the boats and sledges across._\n\nCROZIER\n\n _King William Land, Lat. Unknown, Long. Unknown_\n\n _26 July, 1848_\n\nWhen Crozier slept \u2014 even for a few minutes \u2014 the dreams returned. The two skeletons in the open boat. The intolerable American girls snapping toe joints to simulate a spirit rapping at a table in a darkened room. The American doctor posturing as a polar explorer, a pudgy man dressed in an Esquimaux parka and wearing heavy makeup on an overbright gaslit stage. Then the two skeletons in an open boat again. The night always ending with the dream that disturbed Crozier the most.\n\nHe is a boy and is with his Memo Moira in a vast Catholic cathedral. Francis is naked. Memo pushes him toward the altar rail, but he is afraid to go forward. The cathedral is cold; the marble floor under young Francis's bare feet is cold; there is ice on the white wooden pews.\n\nKneeling at the altar rail, young Francis Crozier can feel Memo Moira watching approvingly from somewhere behind him, but he is too frightened to turn his head. Something is coming.\n\nThe priest seems to rise up from some trapdoor set into the marble floor on the opposite side of the altar railing. The man is too large \u2014 far too large \u2014 and his vestments are white and dripping water. Smelling of blood and sweat and something ranker, he towers over little Francis Crozier.\n\nFrancis closes his eyes and, as Memo taught him while he knelt on the thin rug of her parlour, extends his tongue to receive the Eucharist. As important as this Sacrament is, as necessary as he knows it must be, Francis is terrifed to receive the Host. He knows that his life will never be the same after receiving the Papist Eucharist. And he also knows that his life will end if he does not receive it.\n\nThe priest looms closer and leans toward him...\n\nCrozier awoke in the belly of the whaleboat. As always when he came up out of these dreams, even if he had caught only a few minutes' sleep, his heart was pounding and his mouth was dry from fear. And he was shaking hard, but from cold more than from fear or the memory of fear.\n\nThe ice had broken up in the part of the strait or gulf they were in on the 17th and 18th of July, and for four days after that, Crozier had kept the men together on the large ice floe where they had stopped \u2014 the cutters and pinnace removed from their sledges, all five of the boats fully loaded except for their tents and sleeping bags, and rigged for open water.\n\nEach night the rocking of their large floe and the cracking and fracturing of ice sent them scurrying from their tents, half awake, sure that the sea was opening up beneath them and ready to swallow them as it had Sergeant Tozer and his men. Each night the gunshot explosions of the ice cracking eventually abated, the wild rocking fell into a more regular rhythm of swells, and they crawled back into their tents.\n\nIt was warmer, some days rising almost to the freezing point \u2014 these few weeks of late July would almost certainly be the only hint of summer this second frozen-in arctic year would see \u2014 but the men were colder and more miserable than ever. Some days it actually rained. When it was too cold to rain, ice crystals in the foggy air soaked their wool clothing since it was too warm now to wear waterproof winter slops over their peacoats and greatcoats. Sweat from their man-hauling soaked their filthy underwear, filthy shirts and socks, and their ragged, ice-crusted trousers; despite their almost-depleted stores, the five remaining boats were heavier than the ten boats they'd hauled before ever had been, for in addition to the eating, breathing, but still comatosely staring Davey Leys, more sick men had to be hauled along every day. Dr. Goodsir reported to Crozier each day that more feet \u2014 always soaked and in wet socks in spite of all the extra boots Crozier had thought to bring along \u2014 turned rotten, more toes and heels turned black, and more feet had grown gangrenous and were now in need of amputation.\n\nThe Holland tents were soaked and never dried. The sleeping bags they cracked open in the late evening and crawled into as darkness fell were soaked and frozen inside and out and never dried. When the men awoke in the morning after a few stolen minutes of fitful sleep \u2014 no amount of shivering could make one warmer \u2014 the inside of the circular and pyramid tents were lined with thirty pounds of hoarfrost that fell and dripped on the men's heads, shoulders, and faces as they tried to drink the tiny bit of lukewarm tea that was brought around to the tents each morning by Captain Crozier, Mr. Des Voeux, and Mr. Couch \u2014 a strange reversal of commanders as morning stewards that Crozier had instigated during their first week on the ice and which the men now took for granted.\n\nMr. Wall, _Erebus_ 's cook, was sick with something like consumption and lay curled into the bottom of one of the cutters most of the time, but Mr. Diggle remained the same energetic, obscene, efficient, bellowing, and somehow reassuring figure he had been for three years at his post near the huge Frazer's Patent Stove aboard HMS _Terror._ Now, with the ether fuel depleted and the spirit stoves and heavy whaleboat coal stoves abandoned, Mr. Diggle's job was to portion out twice a day the small bit of cold salt pork and other victuals remaining, always under Mr. Osmer's and another officer's watchful supervision. But always the optimist, Diggle had cobbled together a crude seal-oil stove and cooking pot which he was ready to light if and when they shot more seals.\n\nEvery day, Crozier sent out hunting parties to find those seals for Mr. Diggle's pot, but there were almost none to be seen and those few sighted slipped back into their open leads or tiny holes before the hunters succeeded in shooting them. Several times, so the men on the hunting parties reported, the slippery black ring seals had been hit by buckshot or even a musket ball or rifle bullet but managed to slide back into the black water and dive out of reach before they died, leaving only a trail of blood on the ice. Sometimes the hunters knelt on the ice to lap at the blood.\n\nCrozier had been in summer arctic waters many times before and knew that by mid-July the water and opening floes should be teeming with life: huge walruses sunning themselves on ice floes and flopping ponderously along the water's edge, their barks more a series of belches than barks; a proliferation of seals catapulting in and out of the water like children playing and bellying their way comically across the ice; beluga whales and narwhals spouting and rolling and submerging in the open leads, filling the air with their fishy breaths; female white bears swimming in the black water with their ungainly cubs and stalking seals on floes, shaking the water out of their strange fur as they pulled themselves from the ocean to the ice, avoiding the larger and more dangerous males, which would eat the cubs and the sow as well if their bellies were empty; finally, seabirds flying overhead in such profusion as to almost darken the blue arctic summer sky, birds on shore, on floes, and lining the irregular tops of icebergs like musical notes on a score, while more terns and gulls and gyrfalcons skimmed the water everywhere.\n\nThis summer, for the second year in a row, almost nothing living moved across the ice \u2014 only Crozier's diminished and diminishing men gasping in their man-hauling halters and their relentless pursuer, always briefly and partially glimpsed, always out of rifle or shotgun range. A few times in the evening, the men heard the yip of arctic foxes and frequently found their dainty tracks in the snow, but none ever seemed to make itself visible to the hunters. When the men did see or hear whales, they were always many floes and small leads over, too far to reach even by frenzied, careless running \u2014 men throwing themselves from rocking floe to rocking floe before the sea mammals casually breached and dove and disappeared again.\n\nCrozier had no idea if they could kill a narwhal or beluga with the few small arms they carried, but he thought they could \u2014 a few rifle bullets to the brain should kill anything short of the Beast that stalked them (which the seamen had long since decided was no beast at all, but a wrathful God out of the captain's _Book of Leviathan_ ) \u2014 and if they somehow had the strength to drag a whale onto the ice and render it down, the oil would power Mr. Diggle's stove for weeks or months and they would eat blubber and fresh meat until they all burst.\n\nWhat Crozier most wanted to do was to kill the thing itself. Unlike the majority of his men, he believed it was mortal \u2014 an animal, nothing more. Smarter, perhaps, than even the frighteningly intelligent white bear, but still a beast.\n\nIf he could kill the thing, Crozier knew, the mere fact of its death \u2014 the pleasure of revenge for so many deaths, even if the rest of the expedition still were to die later from starvation and scurvy \u2014 would temporarily lift the morale of the survivors more than discovering twenty gallons of untapped rum.\n\nThe beast had not bothered them \u2014 not killed any of them \u2014 since the ice-enclosed lake where Lieutenant Little and his men had died. Each of the hunting parties the captain sent out had standing orders to return immediately should they find the thing's tracks in the snow; Crozier intended to take every man who could walk and every weapon that could fire out to stalk the beast. If he had to, he would use men banging pots and pans and shouting to flush the thing out, as if it were a tiger in the high grass of India being brought to bay by beaters.\n\nBut Crozier knew this would work no better than the late Sir John's bear blind. What they really needed to bring the thing closer was bait. Crozier had no doubt whatsoever that it was still keeping pace with them, moving in closer during the increasing hours of darkness, hiding wherever it hid, perhaps under the ice, during the day, and that it would come even closer if they could bait it in. But they had no fresh meat, and if they had even a pound of fresh kill, the men would devour it, not use it as bait to catch the thing.\n\nStill, Crozier thought, while remembering the impossible great size and mass of the monstrous thing on the ice, there was more than a ton of meat and muscle there, perhaps several tons, since the larger male white bears weighed up to 1,500 pounds and the thing made its white bear cousins look like hunting dogs next to a large man in comparison. So they would eat well for many weeks if they _did_ manage to murder their murderer. And with every bite, Crozier knew, even eating the thing-flesh as they were the salt pork while on the march, there would be the pleasure of revenge, even if it had to be a dish best served cold.\n\nIf it would work, Francis Crozier knew he would set himself out onto the ice as bait. _If it would work._ If it would save and feed even a few of his men, Crozier would offer himself to the beast as bait and hope that his men, who had proved themselves atrocious shots even before the last of _Terror_ 's Marines died in the cold water, would be able to shoot the monster often enough, if not accurately enough, to bring it down, whether the Crozier bait survived or not.\n\nWith the thought of the Marines came, unbidden, the memory of Private Henry Wilkes's body left behind in one of the abandoned boats a week earlier. There had been no gathering of the men for Wilkes's nonburial, only Crozier, Des Voeux, and a few of the Marine's closer friends saying a few words over the body before dawn.\n\n _We should have used Wilkes's body for bait,_ thought Crozier as he lay in the bottom of the rocking whaleboat while the other men slept in heaped piles around him.\n\nThen he realized \u2014 and not for the first time \u2014 that they had fresher bait with them. David Leys had been nothing but a burden for eight months, ever since the night in December of last year when the thing had given chase to the late Ice Master Blanky. Leys staring at nothing since that night, unresponsive, useless, hauled in the boat like a hundred and thirty pounds of soiled laundry for almost four months now, nonetheless managed to slurp down his salt-pork broth and rum ration every afternoon and to swallow his spoonful of tea and sugar each morning.\n\nIt was to the men's credit that none of them \u2014 not even the whispering Hickey or Aylmore \u2014 had suggested leaving Leys behind, or any of the other sick men who currently could not walk. But everyone must have had the same thought...\n\n _Eat them._\n\n _Eat Leys first, then the others when they die._\n\nFrancis Crozier was so hungry that he could imagine eating human flesh. He would not kill a man in order to devour him \u2014 not yet \u2014 but once dead, why should all that meat be left behind to rot in the arctic summer sun? Or worse yet, left behind to be eaten by the thing that was after them?\n\nAs a new lieutenant in his twenties, Crozier had heard \u2014 as all sailing men now heard sooner or later, usually as ship's boys before the mast \u2014 the true story of Captain Pollard in the U.S. brig _Essex_ back in 1820.\n\n _Essex_ had been stove in and sunk, so the few survivors later reported, by an 85-foot sperm whale. The brig went down in one of the emptiest parts of the Pacific and the entire 20-man crew had been out in their boats hunting whales at the time and returned to find their ship sinking fast. Retrieving a few tools, some navigation instruments, and one pistol from the ship, the survivors set off in three whaleboats. Their only provisions were two live turtles they'd captured in the Galapagos, two casks of ship's biscuits, and six casks of fresh water.\n\nThen they steered the whaleboats for South America.\n\nFirst, of course, they killed and ate the large turtles, drinking the blood when the meat was gone. Then they managed to capture some hapless flying fish who leapt into the boat by accident; while the men had contrived to cook the turtle meat, after a fashion, the fish they ate raw. Then they dived into the sea, scraped the barnacles from the hulls of their three open boats, and ate those.\n\nMiraculously, the boats encountered Henderson Island \u2014 one of those few specks on the endless blue that is the Pacific Ocean. For four days the twenty men captured crabs and stalked gulls and their eggs. But Captain Pollard knew that there were not enough crabs, gulls, or gull eggs on the island to sustain twenty men for more than another few weeks, so seventeen of the twenty voted to take to the boats again. They launched the boats and waved good-bye to their three remaining companions on 27 December, 1820.\n\nBy 28 January, the three boats had been separated from one another by storm, and Captain Pollard's whaleboat sailed eastward alone under the endless sky. Their rations now consisted of one and a half ounces of ship's biscuit per man a day for the five men in the whaleboat. By not so great a coincidence, this was precisely the reduced ration that Crozier had just secretly discussed with Dr. Goodsir and First Mate Des Voeux for when the last of the salt pork ran out in a few days.\n\nThe bit of biscuit and few sips of water had kept Pollard's men \u2014 his nephew Owen Coffin, a freed black man named Barzillai Ray, and two seamen \u2014 alive for nine weeks.\n\nThey were still more than 1,600 miles from land when the last of the biscuits ran out at the same time as the last of the water was drunk. Crozier had figured that if the biscuits lasted his men another month, they would still be more than 800 miles from human habitation in winter even if they reached the mouth of Back's River.\n\nPollard had no conveniently recently deceased men aboard his boat, so they drew straws. Pollard's young nephew Owen Coffin drew the short straw. Then they drew straws again to see who would do the deed. Charles Ramsdell drew the short straw this time.\n\nThe boy wished the other men a tremulous good-bye (Crozier always remembered his scrotum-tightening sense of horror the first time he heard this part of the story while on watch with an older man high in the mizzen of a warship far off Argentina, the old seaman terrifying Lieutenant Crozier by saying good-bye in a trembling boy's voice), and then young Coffin had laid his head on the gunwale and closed his eyes.\n\nCaptain Pollard, as he later testified in his own words, had given Ramsdell his pistol and turned his face away.\n\nRamsdell shot the boy in the back of the head.\n\nThe five others, including Captain Pollard, the boy's uncle, first drank the blood while it was warm. Although salty, it was \u2014 unlike the endless sea around them \u2014 drinkable.\n\nThen they sliced the boy's flesh from his bones and ate it raw.\n\nThen they broke open Owen Coffin's bones and sucked out the marrow to the last shred.\n\nThe cabin boy's corpse had sustained them for thirteen days, and just when they were considering drawing lots again, the black man \u2014 Barzillai Ray \u2014 died of thirst and exhaustion. Again the draining, drinking, slicing, cracking, and sucking of marrow sustained them until they were rescued by the whaler _Dauphin_ on 23 February, 1821.\n\nFrancis Crozier never met Captain Pollard but he had followed his career. The unlucky American had retained his rank and gone to sea only once more \u2014 and once more was shipwrecked. After being rescued the second time, he was never again entrusted with command of a ship. The last Crozier heard, only a few months before Sir John's expedition sailed three years earlier in 1845, Captain Pollard was living as a town watchman in Nantucket and was universally shunned by both townspeople and whalers there. It was said that Pollard had aged prematurely, spoke aloud to himself and his long-dead nephew, and hid biscuits and salt pork in the rafters of his home.\n\nCrozier knew that his people would have to make a decision about eating their own dead within the next few weeks, if not the next few days.\n\nThe men were approaching the point where they were too few and those few too weak to man-haul boats, but the four-day rest on the ice floe from the 18th to the 22nd of July had not renewed their energy. Crozier, Des Voeux, and Couch \u2014 young Lieutenant Hodgson, while technically the second in command, was given no authority by the captain these days \u2014 rousted men and ordered them out hunting or repairing sledge runners or caulking and rerepairing the boats rather than let them lie in their frozen sleeping bags in their dripping tents all day \u2014 but essentially all they could do was sit on their connected floes for days since too many tiny leads, fissures, small areas of open water, and patches of thin and rotten ice surrounded them to allow any progress south or east or north.\n\nCrozier refused to turn back west and northwest.\n\nBut the floes were not drifting in the direction they wanted to go \u2014 southeast toward the mouth of Back's Great Fish River. They merely milled and circled upon themselves as the pack holding _Erebus_ and _Terror_ had for two long winters.\n\nFinally, on the afternoon of Saturday, 22 July, their own floe began cracking up enough that Crozier ordered everyone into the boats.\n\nFor six days now they had floated, tethered together by lines, in patches and leads too short or small to row or sail in. Crozier had the one sextant left to them (he had left the heavier theodolite behind), and while others slept he took the best readings he could during the occasional short break in cloud cover. He reckoned their position to be about eighty-five miles northwest of the mouth of Back's River.\n\nExpecting to see a narrow isthmus ahead of them any day now \u2014 the presumed peninsula connecting the bulb of King William Land to the previously mapped Adelaide Peninsula \u2014 Crozier had awakened in the boat at sunrise on the morning of Wednesday the 26th of July to find the air colder, the sky blue and cloudless, and glimpses of land darkening the sky more than fifteen miles away to both the north and south.\n\nCalling the five boats together later, Crozier stood in the bow of his lead whaleboat and shouted, \"Men, King William Land is _King William Island._ I'm certain now that there's sea ahead all the way east and south to Back's River, but I'll bet my last quid that there's no land connecting the cape you see far to the southwest there and the one you see far to the northeast. We're in a strait. And since we have to be north of the Adelaide Peninsula, we've completed the goal of the Sir John Franklin Expedition. _This is the North-West Passage._ By God, you've done it.\"\n\nThere was a weak cheer followed by some coughing.\n\nIf the boats and floes had been drifting south, weeks of man-hauling or sailing work might have been done for them. But the leads and areas of open water in which they floated continued to crack open only toward the north.\n\nLife in the boats was as miserable as life on the floes in the tents had been. The men were crowded too close together. Even with boards on thwarts offering a second level for sleeping on those whaleboats and cutters with their sides built up by Mr. Honey (the disassembled sledges also served as a crossed-T deck amidships on the crowded cutters and pinnace), wet-wooled bodies were pressed against wet-wooled bodies both day and night. The men had to hang out over the gunwales to shit \u2014 an event that was becoming less and less necessary, even for the men with serious scurvy, as the food and water grew less \u2014 but while all the men had lost all vestiges of modesty, a sudden wave often soaked bare skin and lowered trousers, leading to curses, boils, and longer nights of shivering misery.\n\nOn the morning of Friday, 28 July, 1848, the lookout on Crozier's boat \u2014 the smallest man on each boat was sent up the short raised mast with a spyglass \u2014 spied a maze of leads opening all the way to a point of land to the northwest, perhaps three miles away.\n\nThe able-bodied men in the five boats pulled \u2014 and when necessary, polled between narrowing ice ledges, the healthiest men at the bow hacking away with pickaxes and fending off with pikes \u2014 for eighteen hours.\n\nThey landed on a rocky shingle, in a darkness broken only by short periods of moonlight when the returning clouds parted, a little after eleven o'clock that night.\n\nThe men were far too exhausted to dismount the sledges and lift the cutters and pinnaces onto them. They were too tired to unpack their soaked Holland tents and sleeping bags.\n\nThey fell onto the rough stones where they had ceased their dragging of heavy boats across the shore ice and rocks made slippery by high tide. They slept in clumps, kept alive only by their crewmates' failing body warmth.\n\nCrozier did not even assign a watch. If the thing wanted them tonight, it could have them. But before he slept, he spent an hour trying to get a good sighting with his sextant and to work it out with the navigation tables and maps he still carried with him.\n\nAs best he could reckon, they had been on the ice for twenty-five days and man-hauled and drifted and rowed a total of forty-six miles to the east-southeast. They were back on King William Land somewhere north of the bulk of the Adelaide Peninsula and now even farther from the mouth of Back's River than they had been two days earlier \u2014 about thirty-five miles northwest of the inlet across the unnamed strait they'd been unable to cross. If they even crossed this strait, they would be more than sixty miles up the inlet from the mouth of the river, a total of more than nine hundred miles from Great Slave Lake and their salvation.\n\nCrozier carefully stowed his sextant in his wooden case and set the case away in its oilskin waterproof bag, found a sodden blanket from the whaleboat, and threw it down on stones next to Des Voeux and three sleeping men. He was asleep within seconds.\n\nHe dreamt of Memo Moira shoving him forward toward an altar rail and of the waiting priest in dripping vestments.\n\nIn his sleep, as the men snored in the moonlight of this unknown shore, Crozier closed his eyes and extended his tongue to receive the Body of Christ.\n\nBRIDGENS\n\n _River Camp_\n\n _29 July, 1848_\n\nJohn Bridgens had always \u2014 secretly \u2014 compared the different parts of his life to the various pieces of literature that had formed his life.\n\nIn his boyhood and student years, he had from time to time thought of himself as different characters from Boccaccio's _Decameron_ or from Chaucer's ribald _Canterbury Tales_ \u2014 and not all of his chosen characters were heroic by any means. (His attitude toward the world for some years was, _kiss my arse._ )\n\nIn his twenties, John Bridgens most identified with Hamlet. The strangely aging Prince of Denmark \u2014 Bridgens was quite sure that the boy Hamlet had magically aged over a few theatrical weeks to a man who was, at the very least, in his thirties by Act V \u2014 had been suspended between thought and deed, between motive and action, frozen by a consciousness so astute and unrelenting that it made him _think about everything, even thought itself._ The young Bridgens had been a victim of such consciousness and, like Hamlet, had frequently considered that most essential of questions \u2014 _to continue or not to continue?_ (Bridgens's tutor at the time, an elegant don in exile from Oxford who was the first unabashed sodomite the young would-be scholar had ever encountered, had disdainfully taught him that the famous \"to be, or not to be\" soliloquy was not in any way a discussion of suicide, but Bridgens knew better. _Thus doth conscience make cowards of us all_ had spoken directly to the boy-man soul of John Bridgens, miserable with the state of his existence and his unnatural desires, miserable when pretending to be something he was not, miserable when pretending and miserable when not pretending, and, most centrally, miserable that he could only _think about_ ending his own life because the fear that thought itself might continue on the other side of this mortal veil, \"perchance to dream,\" kept him from acting even toward quick, decisive, cold-blooded self-murder.)\n\nLuckily, even as a young man not yet become himself, John Bridgens had two things besides indecision that kept him from self-destruction \u2014 books and a sense of irony.\n\nIn his middle years, Bridgens most thought of himself as Odysseus. It was not the wandering the world alone that made the comparison apt for the would-be scholar turned secondary officers' steward but rather Homer's description of the world-weary traveler \u2014 the Greek word meaning \"crafty\" or \"guileful\" by which Odysseus' contemporaries identified him (and by which some, such as Achilles, chose to insult him). Bridgens did not use his craft to manipulate others, or rarely did, but used it more like one of the round leather-and-wood or prouder metal shields behind which Homeric heroes sheltered while under violent attack by spear and lance.\n\nHe used his craft to become and to stay invisible.\n\nOnce, some years ago, during the five-year voyage on the HMS _Beagle_ during which he had come to know Harry Peglar, Bridgens mentioned his Odysseus analogy \u2014 suggesting that all the men on such a trip were modern-day Ulysseses to some extent or another \u2014 to the natural philosopher aboard (the two played chess frequently in Mr. Darwin's tiny cabin), and the young bird expert with the sad eyes and sharp mind had looked penetratingly at the steward and said, \"But how is it that I doubt you have a Penelope waiting at home, Mr. Bridgens?\"\n\nThe steward had been more circumspect after that. He had learned \u2014 as Odysseus had learned after a certain number of years of his wanderings \u2014 that his guile was no match for the world and that hubris would always be punished by the gods.\n\nIn these last days, John Bridgens felt that the literary character with whom he had most in common \u2014 in outlook, in feeling, in memory, in future, in sadness \u2014 was King Lear.\n\nAnd it was time for the final act.\n\nThey had stayed two days at the mouth of the river that drained into the unnamed strait south of King William Land, now known to be King William Island. The river here, in late July, was running freely in places and allowed them to fill all their water casks, but no one had seen or caught a fish from it. No animals seemed interested in coming down to drink from it... not so much as a white arctic fox. The best one could say about this campsite was that the slight indentation of the river valley kept them out of the worst of the wind and afforded them some peace of mind during the lightning storms that raged every night.\n\nBoth mornings at this camp, the men \u2014 hopefully, prayerfully \u2014 laid their tents, sleeping bags, and whatever clothing they could spare out on rocks to dry in the sunlight. There came, of course, no more sunlight. Several times it drizzled. The only day with blue sky they had seen in the past month and a half had been their last day in the boats and after that day, most of the men had to see Dr. Goodsir for their sunburns.\n\nGoodsir \u2014 as Bridgens knew well, being his assistant \u2014 had very few medicines left in the box he'd put together from the supplies of his three dead colleagues as well as his own. There were still some purgatives in the good doctor's arsenal (mostly castor oil and tincture of jalap, made from morning-glory seeds) and some stimulants for the scurvy cases, camphor and Hartshorn being the last after the tincture of lobelia had been used so liberally in the first months of scurvy symptoms, some opium as a sedative, a bit of Mandragora and Dover's Powders left to dull pain, and only Sulphate of Copper and Lead remaining to disinfect wounds or deal with sunburn turned to blisters. Obeying Dr. Goodsir's orders, Bridgens had administered almost all of the Sulphate of Copper and Lead to the moaning men who had stripped their shirts off while rowing and added severe sunburn to their nightly misery.\n\nBut there was no sunlight now to dry the tents or clothes or bags. The men stayed wet and at night they moaned as they shook with cold and burned with fever.\n\nReconnaissance by their healthiest, fastest-walking shipmates had shown that while out of sight of land on boats they had passed a deeply indented bay less than fifteen miles to the northwest of this river where they had finally put in to shore. Most shocking of all, the scouts reported that the entire island curved back to the northeast only ten miles ahead of them to the east. If this was true, they were very close to the southeast corner of King William Island, their closest possible approach on this landmass to the Back River inlet.\n\nBack River, their destination, lay southeast across the strait, but Captain Crozier had let the men know that he planned to continue man-hauling east on King William Island to the point where the coast of the island ceased its current southeastern slant. There, at this final point of land, they would set up camp again on the highest place possible and watch the strait. If the ice broke in the next two weeks, they would take to the boats. If it did not, they would try to haul them south across the ice toward the Adelaide Peninsula and, upon hitting land there, head due east the fifteen miles or fewer that Crozier estimated remained before they would reach the inlet leading south to Back's River.\n\nThe endgame had always been the weakest part of John Bridgens's chess skills. He rarely enjoyed it.\n\nOn the evening before they were scheduled to leave River Camp at dawn, Bridgens neatly packed away his personal gear \u2014 including the thick journal he had kept over the past year (he had left five longer ones on _Terror_ the previous 22 April) \u2014 set it in his sleeping bag with a note that anything useful should be shared by his mates, took Harry Peglar's journal and his comb, added an old clothes brush that Bridgens had carried for many years, put them in his peacoat pocket, and went to Dr. Goodsir's small medical tent to say good-bye.\n\n\"What do you mean you're going for a walk and might not be back by the time we leave tomorrow?\" demanded Goodsir. \"What kind of talk is that, Bridgens?\"\n\n\"I'm sorry, Doctor, I just have a strong desire to take a stroll.\"\n\n\"A stroll,\" repeated Goodsir. \"Why, Mr. Bridgens? You are thirty years older than the average surviving seaman on this expedition, but you are ten times healthier.\"\n\n\"I've always been lucky when it came to health, sir,\" said Bridgens. \"All due to heredity, I fear. No thanks to any wisdom I may have shown over the years.\"\n\n\"Then why... ,\" began the surgeon.\n\n\"It's just time, Dr. Goodsir. I confess to considering trodding the boards as a thespian long ago when I was young. One of the few things I learned about that profession was that the great actors learn how to make a good exit before they wear out their welcome or overplay a scene.\"\n\n\"You sound like a Stoic, Mr. Bridgens. A follower of Marcus Aurelius. If the emperor is displeased with you, you go home, draw a warm bath...\"\n\n\"Oh, no, sir,\" said Bridgens. \"While I admit I've always admired the Stoic philosophy, the truth is, I've always had a fear of knives and blades. The emperor would've had my head, my family, and lands for certain, I'm such a coward when it comes to sharp edges. I just wish to take a walk this evening. Perhaps a nap.\"\n\n\" 'Perchance to dream'?\" said Goodsir.\n\n\"Aye, there's the rub,\" admitted the steward. The rue and anxiety \u2014 and perhaps fear \u2014 in his voice were real.\n\n\"Do you really think we have no chance to reach help?\" asked the surgeon. He sounded sincerely curious and only a little sad.\n\nBridgens did not answer for a minute. Finally he said, \"I truly do not know. Perhaps it all depends upon whether a rescue party has already been sent north from Great Slave Lake or one of the other outposts. I would think they might have \u2014 we have been out of touch for three years now \u2014 and if so, there may be a chance. I do know that if anyone on our expedition could get us home, Captain Francis Rawdon Moira Crozier is that man. He's always been underrated by the Admiralty, is my humble opinion.\"\n\n\"Tell him that yourself, man,\" said Goodsir. \"Or at least tell him that you're leaving. You owe him that.\"\n\nBridgens smiled. \"I would, Doctor, but you and I both know that the captain would not let me go. He is stoic, I think, but no Stoic. He might put me in chains to keep me... going on.\"\n\n\"Yes,\" agreed Goodsir. \"But you'll be doing me a favour if you stay, Bridgens. I have some amputations coming up that will require your steady hand.\"\n\n\"There are other young men who can help you, sir, and who have hands far steadier \u2014 and stronger \u2014 than mine.\"\n\n\"But no one as intelligent,\" said Goodsir. \"No one I can talk to as I have with you. I value your advice.\"\n\n\"Thank you, Doctor,\" said Bridgens. He smiled again. \"I didn't want to tell you, sir, but I've always been queasy around pain and blood. Since I was a boy. I've very much appreciated the opportunity to work with you these past weeks, but it's gone against my basically squeamish nature. I've always agreed with St. Augustine when he said that the only real sin is human pain. If there are amputations coming, it's best I'm going.\" He extended his hand. \"Good-bye, Dr. Goodsir.\"\n\n\"Good-bye, Bridgens.\" The doctor used both of his hands to shake the older man's.\n\nBridgens walked northeast out of camp, climbed up out of the shallow river valley \u2014 as with everywhere else on King William Island, no hill or ridgeline was much higher than fifteen or twenty feet above sea level \u2014 found a rocky ridgeline free of snow, and followed it away from camp.\n\nSunset now came sometime around 10:00 p.m., but John Bridgens had decided that he would not walk until dark. About three miles from River Camp, he found a dry spot on the ridge, sat, and took a ship's biscuit \u2014 his day's ration \u2014 from his peacoat pocket and slowly ate it. Completely stale, it was one of the most delicious things he'd ever tasted. He had neglected to bring water with him, but now he scooped up a bit of snow and let it melt in his mouth.\n\nThe sunset to the southwest was beautiful. For an instant the sun actually emerged in the gap between low grey cloud and high grey gravel, hung there as an orange ball for a moment \u2014 the kind of sunset that Odysseus, not Lear, would have seen and enjoyed \u2014 and then disappeared.\n\nThe day and air grew grey and mellow, although the temperature, that had held in the twenties all day, was dropping very quickly now. A wind would come up soon. Bridgens would like to be asleep before the nightly wind howled out of the northwest or the nightly lightning storms rolled across the land and ice strait.\n\nHe reached into his pocket and removed the last three items there.\n\nFirst was the clothes brush that John Bridgens had used as steward for more than thirty years. He touched the bits of lint on it, smiled at some irony understood only by himself, and set it in his other pocket.\n\nNext was Harry Peglar's horn comb. A few light brown hairs still clung to the teeth of it. Bridgens held the comb tightly in his cold, bare fist for a moment and then set it in his coat pocket with the clothes brush.\n\nLast was Peglar's notebook. He flipped it open at random.\n\n _Oh Death whare is thy sting, the grave at Comfort Cove for who has any doubt now... the dyer sad._\n\nBridgens shook his head. He knew that the last word should be \"said,\" whatever else the water-stained and illegible part of the message should have read. He had taught Peglar to read but had never succeeded in teaching Harry how to spell. Bridgens suspected \u2014 since Harry Peglar was one of the most intelligent human beings he'd ever known \u2014 that there had been some problem with the constitution of the man's brain, some lobe or lump or grey area unknown to medical learning, that controlled the spelling of words. Even in the years after he'd learned to decode the alphabet and read the most challenging of books with a scholar's insight and understanding, Harry had been unable to pen the shortest letter to Bridgens without reversing letters and misspelling the simplest words.\n\n _Oh Death whare is thy sting..._\n\nBridgens smiled a final time, set the journal in his front jacket pocket where it would be safe from small scavengers because he would be lying on it, and stretched out on his side on the gravel, laying his cheek on the backs of his bare hands.\n\nHe stirred only once, to tug his collar up and his hat down. The wind was coming up and it was very cold. Then he resumed his napping position.\n\nJohn Bridgens was asleep before the last of the grey twilight died in the south.\n\nCROZIER\n\n _Rescue Camp_\n\n _13 August, 1848_\n\nThey'd hauled for two weeks to the southeastern-most tip of the island \u2014 the point where the King William Island shoreline abruptly began curving north and east \u2014 and then they'd stopped to set up tents, send out hunting parties, and catch their breath while waiting and watching for openings in the sea-strait ice to the south. Dr. Goodsir had told Crozier that he needed time to deal with the sick and injured they'd been hauling in their five boats. They named the campsite Land's End.\n\nWhen Crozier was informed by Goodsir that at least five men needed to have feet amputated during the stop there \u2014 which meant, he knew, that those men would never go farther than this place, since even the ambulatory seamen no longer had the strength to haul the extra weight of men in boats \u2014 the captain renamed the wind-whipped point Rescue Camp.\n\nThe idea, so far discussed only between Goodsir and himself although suggested by Goodsir, was for the surgeon to stay behind with the men recovering from the amputations. Four had been operated on already and so far none had died \u2014 the last man, Mr. Diggle, was to have his amputation this morning. Other seamen too sick or weary to continue on could opt to stay with Goodsir and the amputees, while Crozier, Des Voeux, Couch, Crozier's trusted second mate, Johnson, and any others with strength left would sail south down the inlet when \u2014 if \u2014 the ice relented again. Then this smaller group, traveling lightly, would head up Back River, returning with a rescue party from Great Slave Lake in the spring \u2014 or, with the help of a miracle, in the next month or two before winter arrived, providing that they ran into a rescue party moving north along the river.\n\nCrozier knew that the chances of that particular miracle were so low as to be almost nil and that the chances of any of the sick men surviving at Rescue Camp until the following spring without help were not even worth discussing. There had been almost no easily hunted game all this summer of 1848, and August was proving to be no different. The ice had been too thick to fish through everywhere except in the few small leads and rare year-round _polynyas,_ and they'd caught no fish even while in the boats. How could Goodsir and a few other attendants to the dying survive the coming winter here? Crozier knew that the surgeon had voluntarily signed his death warrant by volunteering to stay behind with the doomed men and Goodsir knew his captain knew it. Neither man spoke of it.\n\nYet that remained the current plan, unless Goodsir changed his mind this morning or a true miracle occurred and the ice opened up almost all the way to the shore this second week of August, allowing them all to set sail in two battered whaleboats, two battered cutters, and a single splintery pinnace, bringing the amputees, the injured, the starved, the too weak to walk, and the most advanced scurvy cases with them in the boats.\n\n _As potential food?_ thought Crozier.\n\nThis was the next issue that had to be dealt with.\n\nThe captain carried two pistols in his greatcoat whenever he went out of his tent now \u2014 his large percussion-cap revolver in his right pocket, as always, and the two-shot, twin-barreled little percussion pistol (what the American sea captain who'd sold it to him years ago had called \"a riverboat gambler's belly gun\") in his left pocket. He had not repeated his mistake of sending his best men \u2014 Couch, Des Voeux, Johnson, some others \u2014 out of camp at the same time while leaving such malcontents as Hickey, Aylmore, and the idiot giant Manson behind. Nor had Francis Crozier trusted Lieutenant George Henry Hodgson, his captain of the fo'c'sle, Reuben Male, or _Erebus_ captain of the foretop Robert Sinclair since that day of near mutiny back at Hospital Camp more than a month earlier.\n\nThe view from Rescue Camp was depressing. The sky had been an unrelieved mass of low clouds for two weeks and Crozier hadn't been able to use his sextant. The wind had begun blowing hard from the northwest again and the air was colder than it had been for two months. The strait to the south remained a solid mass of ice, but not the flat ice interrupted by occasional pressure ridges such as they'd crossed on the trek from _Terror_ to Terror Camp so very, very, very long ago. The ice in this strait south of King William Island was a total jumble of full-sized and shattered icebergs, crisscrossing pressure ridges, the occasional year-round _polynya_ showing black water ten feet below the ice level but leading nowhere, and countless razor-edged seracs and ice boulders. Crozier didn't believe that any man in Rescue Camp \u2014 including the giant Manson \u2014 was up to man-hauling a single boat through that ice-forest and over those mountain ranges of ice.\n\nThe growls, explosions, crackings, blasts, and roars that now filled their days and nights were their only hope. The ice was agitated and torturing itself. Now and then, far out, it opened into tiny leads that sometimes lasted for hours. Then they closed with a thunderclap. Pressure ridges leapt to a height of thirty feet in a matter of seconds. Hours later, they collapsed just as quickly as new ridges thrust themselves up. Icebergs exploded from the pressure of the tightening ice around them.\n\n _It is only 13 August,_ Crozier told himself. The problem with that thinking, of course, was that instead of \"only\" 13 August, the season was now far enough along that it was time to be thinking, _It is_ already _13 August._ Winter was fast approaching. _Erebus_ and _Terror_ had been first frozen in place off King William Land in September 1846, and there had been no respite after that.\n\n _It is only 13 August,_ Crozier repeated to himself. Time enough, if only a small miracle was granted them, to sail and row across the strait \u2014 probably man-hauling some short ice portages \u2014 the seventy-five miles he estimated to the mouth of Back's River, there to rerig the battered boats for travel upriver. With a bit more luck, the inlet itself beyond this visible ice jam would be free of ice \u2014 because of Back's Great Fish River's inevitable high summer flow northward and its warmer water \u2014 for as much as sixty miles of the way. After that, on the river itself, they would be racing the oncoming winter south each day while fighting their way upstream, but the voyage was still possible. In theory.\n\n _In theory._\n\nThis morning \u2014 a Sunday if the weary Crozier had not lost track \u2014 Goodsir was performing the last of the amputations with the help of his new assistant, Thomas Hartnell, and then Crozier planned to call the men together for a sort of Divine Service.\n\nThere he would announce that Goodsir would be staying with the crippled men and scurvy cases and he would bring into the open his plans to take a few of the healthiest men and at least two boats south within the coming week, whether the ice opened or not.\n\nIf Reuben Male, Hodgson, Sinclair, or the Hickey conspirators wanted to offer their alternate plans without challenging his authority, Crozier was ready not only to discuss them but to agree to them. The fewer men left at Rescue Camp the better, especially if it meant getting rid of the rotten apples.\n\nThe screaming started from the surgical tent as Dr. Goodsir began his operation on Mr. Diggle's gangrenous left foot and ankle.\n\nA pistol in each pocket, Crozier went to find Thomas Johnson to tell him to assemble the men.\n\nMr. Diggle, the most universally liked man on the expedition and the excellent cook Francis Crozier had known and worked with for years on expeditions to both poles, died of blood loss and complications immediately after the amputation of his foot and just minutes before muster was called.\n\nEach time the survivors spent more than two days at a camp, the bosuns dragged a stick through the gravel and snow in some relatively open, flat spot to create the rough outline of the _Erebus_ 's and _Terror_ 's top and lower decks. This allowed the men to know where to stand during muster and gave them a sense of familiarity. During the first days at Terror Camp and beyond, the muster positions had been crowded to the point of confusion, with more than a hundred men from two ships crowding into the footprint of a single ship's top deck, but now the attrition had reached the point where the gathering was appropriate for a single ship's mustering.\n\nIn the silence after the roll was called and before Crozier's brief reading of Scripture \u2014 and in the deeper silence in the aftermath of Mr. Diggle's screams \u2014 the captain looked out at the clusters of ragged, bearded, pale, filthy, hollow-eyed men leaning forward toward him in a sort of tired-ape slump that was meant to be a brisk standing at attention.\n\nOf the thirteen original officers on HMS _Erebus,_ nine were dead: Sir John, Commander Fitzjames, Lt. Graham Gore, Lt. H. T. D. Le Vesconte, Lt. Fairholme, First Mate Sergeant, Second Master Collins, Ice Master Reid, and Chief Surgeon Stanley. The surviving officers consisted of the first and second mates, Des Voeux and Couch; the assistant surgeon, Goodsir (who now joined the muster ranks late, his posture even more slumped than the other men's, his eyes downcast with exhaustion and defeat); and the purser, Charles Hamilton Osmer, who had survived a serious bout of pneumonia only to be prostrated in his tent now by scurvy.\n\nIt did not escape Captain Crozier's attention that all of _Erebus_ 's commissioned Navy officers were dead and that the survivors were mere mates or civilians granted the honorary title of officer for wardroom purposes.\n\n _Erebus_ 's three warrant officers \u2014 Engineer John Gregory, Bosun Thomas Terry, and Carpenter John Weekes \u2014 were all dead.\n\n _Erebus_ had left Greenland with twenty-one petty officers, and at today's muster, fifteen of them were still alive, although some of them \u2014 such as Purser's Steward William Fowler, who had never fully recovered from his burns at the Carnivale, were little more than mouths to be fed during the march.\n\nA muster of _Erebus_ 's able seamen on Christmas Day of 1845 would have heard nineteen sailors answering the call. Fifteen of them were still living.\n\nOf seven Royal Marines who'd originally answered the muster call on _Erebus,_ three had survived to this day in August of 1848 \u2014 Corporal Pearson and Privates Hopcraft and Healey \u2014 but all were too sick from scurvy even to stand guard or go hunting, much less haul boats. But this morning they stood leaning on their muskets among the other ragged, slumping forms.\n\nOf the two ship's boys on _Erebus_ 's muster \u2014 both actually men of eighteen when the two ships had sailed \u2014 both David Young and George Chambers had survived, but Chambers had been so heavily concussed by the thing from the ice during the Carnivale that he had been little more than an idiot since that night of fire. Still, he was able to haul when instructed and to eat when told to and to keep breathing without prompting.\n\nSo, according to the muster just finished, thirty-nine of _Erebus_ 's original complement of sixty-five souls were still alive as of 13 August 1848.\n\nThe officers of HMS _Terror_ had fared a bit better than those of _Erebus,_ at least in the sense that two Naval officers \u2014 Captain Crozier and Second Lieutenant Hodgson \u2014 had survived. Second Mate Robert Thomas and Mr. E. J. Helpman, Crozier's clerk-in-charge and another civilian who served the expedition with officer's rank, were the other remaining officers.\n\nNot answering muster today were Crozier's lieutenants Little and Irving, as well as First Mate Hornby, Ice Master Blanky, Second Master MacBean, and both his surgeons, Peddie and McDonald.\n\nFour of _Terror_ 's original eleven officers were still alive.\n\nCrozier had started the expedition with three warrant officers \u2014 Engineer James Thompson, Bosun John Lane, and Master Carpenter Thomas Honey \u2014 and all three were still living, although the engineer had wasted away to a hollow-eyed skeleton too weak to stand, much less haul, and Mr. Honey not only showed advanced symptoms of scurvy but had had both feet amputated the night before. Incredibly, as of this assembly, the carpenter was still alive and even managed to shout, \"Present!\" from his tent when his name was called at muster.\n\n _Terror_ had sailed with twenty-one petty officers three years earlier and sixteen were still alive on this cloudy August morning \u2014 Stoker John Torrington, Captain of the Foretop Harry Peglar, and quartermasters Kenley and Rhodes had been the only casualties in that group until just moments ago when Cook John Diggle had joined the ranks of the dead.\n\nWhere nineteen able seamen had once answered _Terror_ 's muster, ten now did, although eleven had survived: David Leys still lay comatose and unresponsive in Dr. Goodsir's tent.\n\nOf HMS _Terror_ 's contingent of six Royal Marines, none had survived. Private Heather, who had lingered for months with his shattered skull, finally died the day after they had left River Camp, and his body was left on the gravel without burial or comment.\n\nThe ship had recorded two \"Boys\" on its original muster, and now only one \u2014 Robert Golding, almost twenty-three years old and certainly no longer a boy, although gullible in a boy's way \u2014 answered to the roll.\n\nOut of an original muster of sixty-two souls on HMS _Terror,_ thirty-five had survived to see this Divine Service at Rescue Camp on 13 August, 1848.\n\nThirty-nine Erebuses and thirty-five Terrors remained, for a total muster of seventy-four men out of the one hundred twenty-six who had sailed from Greenland in the summer of 1845.\n\nBut four of these had suffered one or both of their feet being amputated in the last twenty-four hours and at least another twenty were almost certainly too sick, too injured, too starved, or too bone- and soul-weary to go on. A third of the expedition had reached their limit.\n\nIt was time for a reckoning.\n\n\"Almighty God,\" intoned Crozier in his exhausted rasp, \"with whom do live the spirits of them that depart hence in the Lord, and with whom the souls of the faithful, after they are delivered from the burden of the flesh, are in joy and felicity: We give thee hearty thanks, for that it hath pleased thee to deliver this our brother John Diggle, age thirty-nine, out of the miseries of this sinful world; beseeching thee, that it it may please thee, of thy gracious goodness, shortly to accomplish the number of thine elect, all of us here if it pleases thee, and thus to hasten thy kingdom; that we, with all those that are departed in the true faith of thy holy Name, may have our perfect consummation and bliss, both in body and soul, in thy eternal and everlasting glory; through Jesus Christ our Lord. Amen.\"\n\n\"Amen,\" croaked the sixty-two men still able to stand at muster stations.\n\n\"Amen,\" came a few voices from the other twelve lying in tents.\n\nCrozier did not dismiss the assembled men.\n\n\"Men of HMS _Erebus_ and HMS _Terror,_ members of the John Franklin Discovery Service Expedition, shipmates,\" he rasped loudly. \"Today we have to decide which way our paths shall carry us. You all remain \u2014 under both the Ship's Articles and the Articles of the Royal Discovery Service which you signed with your oaths of honour \u2014 under my command and will continue to be so until you are released by me. You've followed Sir John, Captain Fitzjames, and me this far, and you have done well. Many of our friends and shipmates have gone home to Christ, but seventy-four of us have persevered. I am resolved in my heart that every man of you here at Rescue Camp today should survive to see England, home, and your families again, and God shall be my witness that I have done my best to make sure that this shall be the outcome of our efforts. But today I release you to decide your own path by which to reach that goal.\"\n\nThe men murmured to one another. Crozier let that go on for a few seconds and then continued. \"You've heard what we are doing \u2014 Dr. Goodsir to remain here with those too ill to travel, the healthier men to continue toward Back's River. Are there any among you who still wish to attempt to find some other way to rescue?\"\n\nThere was a silence as men looked down and scuffled their booted feet on the gravel, but then George Hodgson hobbled forward.\n\n\"Sir, some of us do, sir. Want to head back that is, Captain Crozier.\"\n\nThe captain just looked at the young officer for a long moment. He knew that Hodgson was a stalking horse for Hickey, Aylmore, and a few of the more rebellious sea lawyers who had been stirring up the men with resentment for so many months, but he wondered if young Hodgson knew it.\n\n\"Back to where, Lieutenant?\" Crozier asked at last.\n\n\"To the ship, sir.\"\n\n\"Do you think _Terror_ is still there, Lieutenant?\" As if to punctuate his query, the sea ice south of them exploded in a series of shotgun blasts and earthquake rumbles. An iceberg hundreds of yards from shore crumbled and fell.\n\nHodgson shrugged like a boy. \"Terror Camp will be, Captain, whether the ship still is or not. We left food and coal and boats at Terror Camp.\"\n\n\"Aye,\" said Crozier, \"so we did. And we'd all welcome some of that food now \u2014 even some of the tinned food that killed some of us so terribly. But, Lieutenant, that was some eighty or ninety miles and almost one hundred days ago when we left Terror Camp. Do you and the others really think you can walk or haul your way back there into the teeth of winter? It would be late November by the time you made your way even to the camp. Total darkness. And you remember the temperatures and storms of last November.\"\n\nHodgson nodded and said nothing.\n\n\"We ain't going to walk 'til no late November,\" said Cornelius Hickey, stepping out of the ranks to stand next to the slumped young lieutenant. \"We think the ice is open along the shore back the way we come. We'll sail and row around that fuckin' cape we hauled five boats over like 'Gyptian slaves and be home in Terror Camp in a month.\"\n\nThe assembled men mumured furtively among themselves.\n\nCrozier nodded. \"It may indeed open for you, Mr. Hickey. Or it may not. But even if it does, it's more than a hundred miles back to a ship that may well be crushed and most certainly will be frozen fast by the time you get there. It's at least thirty miles closer to the mouth of Back River from here and the odds of the inlet being free of ice south of here, near the river, are much greater.\"\n\n\"You ain't talking us out of this, Captain,\" Hickey said firmly. \"We talked it over 'mongst ourselves, and we're going.\"\n\nCrozier stared at the caulker's mate. The captain's usual instinct to put down any insubordination immediately and with great strength and decisiveness rose in him, but he reminded himself that this was what he wanted. It was past time to get rid of the malcontents and to save those others who trusted his judgement. Besides, this late in the summer and in their escape attempt, Hickey's plan might even be workable. It all depended upon where the ice broke up \u2014 if it broke up anywhere before the winter set in. The men deserved to choose their own last, best chance.\n\n\"How many are going with you, Lieutenant?\" asked Crozier, speaking to Hodgson as if he would actually be the commander of the group.\n\n\"Well... ,\" began the young man.\n\n\"Magnus is going,\" said Hickey, gesturing the giant forward. \"And Mr. Aylmore.\"\n\nThe sullen gunroom steward swaggered forward, his face filled with defiance and visible contempt toward Crozier.\n\n\"And George Thompson... ,\" continued the caulker's mate.\n\nCrozier was not surprised that Thompson would be part of Hickey's cabal. The seaman had always been insolent and lazy and \u2014 as long as the rum lasted \u2014 drunk whenever possible.\n\n\"I'm going along, too,... sir,\" said John Morfin, stepping up with the others.\n\nWilliam Orren, just turned 26, stepped forward without a word and stood with Hickey's group.\n\nThen James Brown and Francis Dunn \u2014 _Erebus_ 's caulker and caulker's mate \u2014 joined the group. \"We think it's our best chance, Captain,\" said Dunn and looked down.\n\nWaiting for Reuben Male and Robert Sinclair to declare their intentions \u2014 realizing that if the majority of men standing at muster joined this group that all of his own plans for flight south were gone for good \u2014 Crozier was surprised when William Gibson, _Terror_ 's subordinate officers' steward, and Stoker Luke Smith walked slowly forward. They'd been good men aboard ship and stalwart haulers.\n\nCharles Best \u2014 a reliable _Erebus_ seaman who had always been loyal to Lt. Gore \u2014 stepped forward with four other seamen in tow: William Jerry, Thomas Work, who had been sorely injured at Carnivale, young John Strickland, and Abraham Seeley.\n\nThe sixteen men stood there.\n\n\"Is that it then?\" asked Crozier, feeling a hollow sense of relief that gnawed at his belly like the hunger that was always with him now. Sixteen men were standing there; they would need one boat, but they were leaving behind enough loyal men to head for Back River with him while also leaving enough to take care of the ill here at Rescue Camp. \"I'll give you the pinnace,\" he said to Hodgson.\n\nThe lieutenant nodded gratefully.\n\n\"The pinnace is all busted up and rigged for river work and the sledge is a pain in the arse to drag,\" said Hickey. \"We'll take a whaleboat.\"\n\n\"You'll get the pinnace,\" said Crozier.\n\n\"We want George Chambers and Davey Leys, too,\" said the caulker's mate, folding his arms and standing legs-apart in front of his men like a Cockney Napoleon.\n\n\"The hell you say,\" said Crozier. \"Why would you want to bring two men who can't take care of themselves?\"\n\n\"George can haul,\" said Hickey. \"And we been takin' care of Davey and want to keep doin' so.\"\n\n\"No,\" said Dr. Goodsir, stepping forward into the tense space between Crozier and Hickey's men, \"you haven't been taking care of Mr. Leys and you don't want George Chambers and him as fellow travelers. You want them as food.\"\n\nLieutenant Hodgson blinked in disbelief, but Hickey balled up his fists and gestured to Magnus Manson. The little man and huge man took a step forward.\n\n\"Stop exactly where you are,\" bellowed Crozier. Behind him, the three surviving Marines \u2014 Corporal Pearson, Private Hopcraft, and Private Healey \u2014 while visibly ill and shaky on their feet, had lifted and aimed their long muskets.\n\nMore to the point, First Mate Des Voeux, Mate Edward Couch, Bosun John Lane, and Bosun's Mate Tom Johnson were aiming shotguns.\n\nCornelius Hickey actually snarled. \"We got guns, too.\"\n\n\"No,\" said Captain Crozier, \"you do not. While you were at muster, First Mate Des Voeux rounded up all weapons. If you leave peaceably tomorrow, you'll get one shotgun and some cartridges. If you take another step right now, you'll all get bird shot in your faces.\"\n\n\"You are all going to _die,_ \" said Cornelius Hickey, pointing his bony finger at the men standing silently in muster formations while swinging his arm in a half circle like a scrawny weather vane. \"You're going to follow Crozier and these other fools and you're going to _die._ \"\n\nThe caulker's mate wheeled toward the surgeon. \"Dr. Goodsir, we forgive you for what you said about why we want to save George Chambers and Davey Leys. Come with us. You can't save these men here.\"\n\nHickey gestured contemptuously toward the sagging wet tents where the sick men lay.\n\n\"They're dead already, just don't know it,\" continued Hickey, his voice very large and loud coming from such a small frame. \"We're going to _live._ Come with us and see your family again, Dr. Goodsir. If you stay here \u2014 or even follow Crozier \u2014 you're a dead man. Come with us.\"\n\nGoodsir had absentmindedly worn his spectacles when he'd come out from the surgical tent and now he removed them and unhurriedly wiped moisture from them, using the bloody end of his woolen waistcoat as a rag. A small man with a boy's full lips and a receding chin only partially concealed under the hedge of curly beard that had grown down from his earlier unsuccessful side whiskers, Goodsir seemed completely at ease. He put his spectacles back on and looked at Hickey and the men behind him.\n\n\"Mr. Hickey,\" he said softly, \"as grateful as I am for your boundless generosity in offering to save my life, you need to know that you do not need me along to do what you are planning to do with regards to dissecting the bodies of your shipmates in order to provide yourself with a larder of meat.\"\n\n\"I ain't... ,\" began Hickey.\n\n\"Even an amateur can learn dissective anatomy quite quickly,\" interrupted Goodsir, his voice strong enough to override the caulker's mate's. \"When one of these other gentlemen you're bringing along as your private food stock dies \u2014 or when you help him die \u2014 all you have to do is sharpen a ship's knife to a scalpel's edge and begin cutting.\"\n\n\"We ain't going to... ,\" shouted Hickey.\n\n\"But I do strongly recommend that you bring a saw,\" overrode Goodsir. \"One of Mr. Honey's carpenter saws will do nicely. While you can slice off your shipmates' calves and fingers and thighs and belly flesh with a knife, you shall almost certainly require a good saw to get the legs and arms off.\"\n\n\"God-damn you!\" screamed Hickey. He started forward with Manson but stopped when the mates and Marines raised their shotguns and muskets again.\n\nUnperturbed, not even looking at Hickey, the surgeon pointed toward the huge form of Magnus Manson as if the man were an anatomist's chart hanging on a wall. \"It's not so different than carving a Christmas goose when one gets right down to it.\" He slashed vertical marks in the air toward Manson's upper torso and a horizontal one just below his waist. \"Saw the arms off at the shoulder joints, of course, but you shall have to saw _through_ each man's pelvic bones to cut off his legs.\"\n\nHickey's neck cords strained and his pale face grew red, but he did not speak again while Goodsir continued.\n\n\"I would use my smaller metacarpal saw to cut through the legs at the knees and, of course, the arms at the elbow, and then proceed with a good scalpel to slice away the choice parts \u2014 thighs, buttocks, biceps, triceps, deltoids, the meaty part behind the shins. Only then do you start the real butchering of the pectorals \u2014 chest muscles \u2014 and to get at any fat you gentlemen may have retained near your shoulder blades or along your sides and lower back. There shan't be much fat there, of course, nor muscle, but I'm sure Mr. Hickey wants no parts of you to go to waste.\"\n\nOne of the seamen in the back of the group behind Crozier dropped to his knees and began to dry retch into the gravel.\n\n\"I have an instrument called a tenaculum to crack the sternum and to remove the ribs,\" Goodsir said softly, \"but I'm afraid I can't let you borrow it. A good ship's hammer and chisel \u2014 there's one in every boat kit, you've noticed \u2014 should serve that purpose almost as well.\n\n\"I do recommend you attend to rending the flesh first and set aside your friends' heads, hands, feet, intestines \u2014 all of the contents of the soft abdominal sac \u2014 for later.\n\n\"I warn you \u2014 it's more difficult than you think to crack open the long bones for their marrow. You'll need some sort of scraping tool, rather like Mr. Honey's wood-carving gouge. And do note that the marrow will be lumpy and red when it's forced from the center of the bones... and mixed with bone chips and fragments, so not terribly healthy to eat raw. I recommend you put each other's bone marrow into a pot for cooking straightaway and let yourselves simmer before trying to digest your friends.\"\n\n\"Fuck you,\" snarled Cornelius Hickey.\n\nDr. Goodsir nodded.\n\n\"Oh,\" the surgeon added softly, \"when you get around to eating one another's brains, it will be simplicity itself. Simply saw off the lower jaw, throw it away with the lower teeth, and use any knife or spoon to gouge and hack your way up through the soft palate into the cranial vault. If you wish, you may invert the skull and sit around it, scooping out each other's brains like so much Christmas pudding.\"\n\nFor a minute there were no voices raised, only the wind and the groan, crack, and snapping of the ice.\n\n\"Is there anyone else who wants to leave tomorrow?\" called Captain Crozier.\n\nReuben Male, Robert Sinclair, and Samuel Honey \u2014 _Terror_ 's fo'c'sle captain, _Erebus_ 's foretop captain, and _Terror_ 's blacksmith, respectively \u2014 stepped forward.\n\n\"You're going with Hickey and Hodgson?\" asked Crozier. He did not allow himself to show the shock he felt.\n\n\"Nay, sir,\" said Reuben Male, shaking his head. \"We ain't with them. But we want to try walking back to _Terror._ \"\n\n\"No boat needed, sir,\" said Sinclair. \"We're going to try hiking cross-country as it were. Straight across the island. Maybe we'll find some foxes and such inland, away from the coast.\"\n\n\"Navigation will be difficult,\" said Crozier. \"Compasses aren't worth a damn here and I can't give you one of my sextants.\"\n\nMale shook his head. \"No worries, Captain. We'll just use dead reckoning. Most of the time, if the fuckin' wind is in our face \u2014 pardon my language, sir \u2014 then we're headed the right way.\"\n\n\"I was a seaman before I was a 'smith, sir,\" said Samuel Honey. \"We're all sailors. If we can't die at sea, at least this way perhaps we can die aboard our ship.\"\n\n\"All right,\" said Crozier, speaking to all the men still standing there and making sure that his voice would reach to the tents. \"We're going to assemble at six bells and divide up all the remaining ship's biscuits, spirits, tobacco, and any other victuals we still have. Every man. Even those who had their surgeries last night and today will be brought to the dividing-up. Everyone will see what we have, and every man will get an equal share. From this point on, each man \u2014 except those being fed and cared for by Dr. Goodsir \u2014 will be in charge of his own rationing.\"\n\nCrozier looked coldly at Hickey, Hodgson, and their group. \"You men will \u2014 under Mr. Des Voeux's oversight \u2014 go ready your pinnace for your departure. You'll leave at dawn tomorrow, and except for the divvying up of goods and food at six bells, I don't want to see your faces before then.\"\n\nGOODSIR\n\n _Rescue Camp_\n\n _15 August, 1848_\n\nFor the two days after the amputations and Mr. Diggle's death and the muster of the men and the hearing of Mr. Hickey's plans and the pathetic dividing up of the food, the surgeon had no stomach for keeping his diary. He tossed the stained leather book into his traveling medical kit and left it there.\n\nThe Great Dividing Up, as Goodsir already thought of it, had been a sad and seemingly endless affair, extending into the shortening August arctic evening. It soon became obvious that \u2014 at least when it came to food \u2014 no one trusted anyone. Everyone seemed to harbour some bone-deep anxiety that someone else was hiding food, hoarding food, secreting away food, denying everyone else food. It had taken hours to unpack all boats, empty all stores, search all tents, go through Mr. Diggle's and Mr. Wall's stores, with representatives of each class of man on the ship \u2014 officers, warrant officers, petty officers, able seamen \u2014 sharing the search and distribution chores while the other men looked on with avid eyes.\n\nThomas Honey died during the night after the Dividing Up. Goodsir had Thomas Hartnell inform the captain and then he helped sew the carpenter's body into his sleeping bag. Two sailors carried it to a snowdrift about one hundred yards from the camp where Mr. Diggle's body already lay in cold state. The troop had begun forgoing burials and burial services, not because of an edict from the captain or some vote, but simply through silent consensus.\n\n _Are we preserving the bodies in the snowdrift so they won't spoil as future food?_ wondered the surgeon.\n\nHe could not answer his own question. All he knew was that while he was giving Hickey \u2014 and all the other assembled men (quite deliberately since he had spoken to Captain Crozier about the tactic before the muster assembly) \u2014 the anatomical details of carving up the human body to serve as sustenance, Harry D. S. Goodsir had been horrified to find himself salivating.\n\nAnd he knew that he could not have been alone in that reaction to the thought of fresh meat... from whatever source.\n\nOnly a handful of men had turned out at dawn the next morning, Monday 14 August, to watch Hickey and his fifteen companions leave camp with their pinnace lashed onto its battered sledge. Goodsir had come back to see them off after making sure that Mr. Honey had been secretly buried in the drift.\n\nEarlier, he had missed seeing the three walking men off. Mr. Male, Mr. Sinclair, and Samuel Honey \u2014 no relation to the recently deceased carpenter \u2014 had left before dawn on their proposed trek across the island to Terror Camp, carrying with them only their rucksacks, blanket sleeping bags, some ship's biscuits, water, and one shotgun with cartridges. They had not so much as a single Holland tent for shelter and planned to build caves in the snow if serious winter weather reached them before they reached Terror Camp. Goodsir thought that they must have said their good-byes to friends the night before, since the three men were out of camp before the first grey light touched the southern horizon. Mr. Couch later told Dr. Goodsir that the party headed north, inland and directly away from the coast, and planned to turn toward the northwest on their second or third day.\n\nIn contrast, the surgeon was amazed by how heavily Hickey's departing men had loaded their boat. Men all over camp, including Male, Sinclair, and Samuel Honey, had been abandoning useless items \u2014 hairbrushes, books, towels, writing desks, combs \u2014 bits of civilization they'd hauled for a hundred days and now refused to haul any farther, and, for some inexplicable reason, Hickey and his men had loaded many of these rejected pieces of junk into their pinnace along with tents, sleeping gear, and necessary food. One bag held 105 individually wrapped chunks of dark chocolate that was the shared accumulation of these sixteen men's allotment of a secret store hauled all this way as a surprise by Mr. Diggle and Mr. Wall \u2014 six and a half pieces of chocolate per man.\n\nLieutenant Hodgson had shaken hands with Crozier, and a few of the other men had said clumsy farewells to old shipmates, but Hickey, Manson, Aylmore, and the most resentful of the group said nothing. Then Bosun's Mate Johnson gave Hodgson the unloaded shotgun and a bag of cartridges and watched while the young lieutenant stowed them in the heavily loaded boat. With Manson in the lead and at least a dozen of the sixteen men lashed to the sledge and longboat by harnesses, they left the camp in silence broken only by the scrape of runners on gravel, then on snow, then on rock again, then again across ice and snow. Within twenty minutes they were out of sight over the slight rise to the west of Rescue Camp.\n\n\"Are you thinking about whether they'll make it, Dr. Goodsir?\" asked Mate Edward Couch, who had been standing next to the surgeon and observing his silence.\n\n\"No,\" said Goodsir. He was so weary that he could only answer honestly. \"I was thinking about Private Heather.\"\n\n\"Private Heather?\" said Couch. \"Why, we left his body...\" He stopped.\n\n\"Yes,\" said Goodsir. \"The Marine's corpse is lying under a shred of canvas by the side of our sledge tracks this side of River Camp, not twelve days' pull west of here \u2014 much less time than that at the rate Hickey's large team is pulling the single pinnace.\"\n\n\"Oh, Jesus Christ,\" hissed Couch.\n\nGoodsir nodded. \"I just hope they do not find the subordinate officers' steward's body. I liked John Bridgens. He was a dignified man and deserves better than to be devoured by the likes of Cornelius Hickey.\"\n\nThat afternoon, Goodsir was directed to come to a meeting near the four boats along the shore \u2014 the two whaleboats were inverted as always, the cutters still upright on their sledges but unloaded \u2014 out of the hearing of the men at their duties or drowsing in their tents. Captain Crozier was there, as were First Mate Des Voeux, First Mate Robert Thomas, Acting Mate Couch, Bosun's Mate Johnson, Bosun John Lane, and Marine Corporal Pearson, who was too weak to stand and had to half recline against the splintered hull of an overturned whaleboat.\n\n\"Thank you for coming so promptly, Doctor,\" said Crozier. \"We're here to discuss ways to guard against the return of Cornelius Hickey's group and to look at our own options over the coming weeks.\"\n\n\"Surely, Captain,\" said the surgeon, \"you don't expect Hickey, Hodgson, and the others to come back here?\"\n\nCrozier held up his gloved hands and shrugged. Light snow whipped around and between the men. \"He still might want David Leys. Or the corpses of Mr. Diggle and Mr. Honey. Or even you, Doctor.\"\n\nGoodsir shook his head and shared his thoughts about the bodies \u2014 starting with Private Heather \u2014 that lay along the return way to Terror Camp like frozen food caches.\n\n\"Aye,\" said Charles Des Voeux, \"we've thought of that. It's probably the main reason that Hickey thought he could get back to _Terror._ But we're still going to mount a round-the-clock watch here at Rescue Camp for a few days and send Bosun's Mate Johnson here out with a man or two to follow Hickey's group for three or four days \u2014 just to be sure.\"\n\n\"As for our future here, Dr. Goodsir,\" rasped Crozier, \"what do you see?\"\n\nIt was the surgeon's turn to shrug. \"Mr. Jopson, Mr. Helpman, and Engineer Thompson will not live more than a few days,\" he said softly. \"Of my other fifteen or so scurvy patients, I simply do not know. A few might survive... the scurvy, I mean. Especially if we find fresh meat for them. But of the eighteen men who may stay here at Rescue Camp with me \u2014 Thomas Hartnell has volunteered to stay on as my assistant, by the way \u2014 only three, perhaps four, will be capable of going out to hunt seals on the ice or foxes inland. And they not for long. I would presume that the rest of those who remain here will have died of starvation no later than fifteen September. Most of us sooner than that.\"\n\nHe left unstated that some might survive awhile longer here by eating the bodies of the dead. He also did not mention that he, Dr. Harry D. S. Goodsir, had decided that he would not turn cannibal to survive, nor help those who found need to. His dissection instructions at the previous day's muster assembly were his last words on the subject. Yet he would also never cast judgement on the men here at Rescue Camp or on the expedition south who did end up eating human flesh to last a short while longer. If any man on the Franklin Expedition understood that the human body was a mere animal vessel for the soul \u2014 and only so much meat once that soul had departed \u2014 it was their surviving surgeon and anatomist, Dr. Harry Goodsir. Not extending his own life a few weeks or even months longer by partaking of such dead flesh was his own decision, for his own moral and philosophical reasons. He had never been an especially _good_ Christian, but he preferred to die as one nonetheless.\n\n\"We may have an alternative,\" Crozier said softly, almost as if reading Goodsir's thoughts. \"I've decided this morning that the Back's River party can stay here at Rescue Camp another week \u2014 perhaps ten days, depending upon the weather \u2014 in hopes that the ice will break up and that we can all depart here on boats... even the dying.\"\n\nGoodsir frowned dubiously at the four boats around them. \"Can so many of us fit in these few craft?\" he asked.\n\n\"Don't forget, Doctor,\" said Edward Couch, \"there are nineteen fewer of us now after the malcontents' departure this morning. And two more dead since yesterday morn. That's only fifty-three souls for four good boats, ourselves included.\"\n\n\"And, as you say,\" said Thomas Johnson, \"more will die in the coming week.\"\n\n\"And we have almost no food to haul now,\" said Corporal Pearson from where he sprawled against the inverted whaleboat. \"I wish to God it was otherwise.\"\n\n\"And I've decided to leave all the tents behind,\" said Crozier.\n\n\"Where will we shelter in a storm?\" asked Goodsir.\n\n\"Under the boats on the ice,\" said Des Voeux. \"Under the boat covers on open water. I did it during my attempt to reach the Boothia Peninsula last March, in the middle of winter, and it's warmer under or in a boat than in those fucking tents... excuse my language, Captain.\"\n\n\"You're excused,\" said Crozier. \"Also, the Holland tents each weigh three or four times what they did when we started this voyage. They never dry out. They must have soaked up half the moisture in the arctic.\"\n\n\"So has our underlinens,\" said Mate Robert Thomas.\n\nEveryone laughed to one extent or another. Two of them ended the laughter with coughs.\n\n\"I'm also planning to leave all but three of the big water casks behind,\" said Crozier. \"Two of them will be empty when we set out. Each boat will have only one of the small casks for storage.\"\n\nGoodsir shook his head. \"How will your men slake their thirst while you're in the strait waters or on the ice there?\"\n\n\" _Our_ thirst, Doctor,\" said the captain. \"If the ice opens, remember that you and the sick men will be coming along, not staying here to die. And we'll refill the casks regularly when we get to the fresh water of Back's River. Until then, I have a confession. We \u2014 the officers \u2014 _did_ hoard one thing we did not confess to yesterday at the Dividing Up. A bit of spirit stove fuel hidden under the false bottom of one of the last rum casks.\"\n\n\"We'll melt ice and snow for drinking water on the ice,\" said Johnson.\n\nGoodsir nodded slowly. He had been so reconciled to the certainty of his own death in the coming days or weeks that even the thought of potential salvation was almost painful. He resisted the urge to allow his hopes to rise again. Odds were overwhelming that everyone \u2014 Hickey's group, Mr. Male's three adventurers, Crozier's south-rowing group \u2014 would be dead in the coming month.\n\nAgain as if reading his thoughts, Crozier said to Goodsir, \"What will it take, Doctor, to give us a chance to survive the scurvy and weakness for the three months it may take us to row upriver to Great Slave Lake?\"\n\n\"Fresh food,\" the surgeon said simply. \"I am convinced that we can beat back the disease in some of the men if we can get fresh food. If not vegetables and fruits \u2014 which I know are impossible up here \u2014 then fresh meat, especially fat. Even animal blood will help.\"\n\n\"Why will meat and blubber arrest or cure such a terrible disease, Doctor?\" asked Corporal Pearson.\n\n\"I have no idea,\" said Goodsir, shaking his head, \"but I am as certain of it as I am that we will all die of scurvy if we do _not_ get fresh meat... even before starvation will kill us.\"\n\n\"If Hickey or the others reach Terror Camp,\" said Des Voeux, \"will the tinned Goldner food serve the same purpose?\"\n\nGoodsir shrugged again. \"Possibly, although I agree with my late colleague, Assistant Surgeon McDonald, that fresh food is always better than canned. Also, I am convinced that there were at least two types of poisons in the Goldner tins \u2014 one slow and nefarious, the other, as you remember with poor Captain Fitzjames and some others, very quick and terrible. Either way, we're better off seeking and finding fresh meat or fish than they are pinning their hopes on aging tins from the Goldner victuallers.\"\n\n\"We hope,\" said Captain Crozier, \"that once out on the open water of the inlet, amidst the free-floating floes, seals and walruses will be available in plentitude before the real winter sets in. Once on the river, we'll put in from time to time to hunt deer, foxes, or caribou, but may have to pin _our_ hopes on catching fish... a real probability according to such explorers as George Back and our own Sir John Franklin.\"\n\n\"Sir John also ate his shoes,\" said Corporal Pearson.\n\nNo one reprimanded the starving Marine, but neither did anyone laugh or respond until Crozier said, his rasping voice sounding totally serious, \"That's the real reason I brought along hundreds of extra boots. Not just to keep the men's feet dry \u2014 which, as you have seen, Doctor, was an impossibility. But to have all that leather to eat during the penultimate portion of our trek south.\"\n\nGoodsir could only stare. \"We'll have only one cask of water but hundreds of Royal Navy\u2013issued boots to eat?\"\n\n\"Yes,\" said Crozier.\n\nSuddenly all eight men began laughing so hard that they could not stop; when the others ceased, someone would begin laughing again and then everyone would join in.\n\n\"Shhh!\" Crozier said at last, sounding like a schoolmaster with boys but still chuckling himself.\n\nMen at their duties in the camp twenty yards away were looking over with curiosity painted on their pale faces staring out from under Welsh wigs and caps.\n\nGoodsir had to wipe away tears and snot before they froze to his face.\n\n\"We're not going to wait for the ice to open all the way up to the shore here,\" Crozier said into the sudden silence in the group. \"Tomorrow, as Bosun's Mate Johnson secretly follows Hickey's group northwest along the coast, Mr. Des Voeux will take a group of our ablest men south across the ice, moving with just rucksacks and sleeping blankets \u2014 with luck, traveling almost as quickly as Reuben Male and his two friends \u2014 going at least ten miles out onto the strait, perhaps farther, to see if there is any open water. If a lead opens to within five miles of this camp, we are all leaving.\"\n\n\"The men have no strength... ,\" began Goodsir.\n\n\"They will if they know for certain that there's only a day or two's haul between them and open water all the way to rescue,\" said Captain Crozier. \"The two surviving men who've had their feet amputated will be on their bloody stumps and pulling with a will if we know the water is out there waiting for us.\"\n\n\"And with only a little luck,\" said Des Voeux, \"my group will bring back some seals and walruses and blubber.\"\n\nGoodsir looked out at the cracking, shifting, pressure-ridge surging ice jumble stretching south below low, grey snow clouds. \"Can you haul seals and walruses back across that white nightmare?\" he asked.\n\nDes Voeux just grinned broadly in answer.\n\n\"We have one thing to be thankful for,\" said Bosun's Mate Johnson.\n\n\"What's that, Tom?\" asked Crozier.\n\n\"Our friend from the ice seems to have lost interest in us and wandered away,\" said the still-muscular bosun. \"We've not seen or heard him for certain since before River Camp.\"\n\nAll eight men, including Johnson, suddenly reached over to one of the nearby boats and rapped their knuckles on the wood.\n\nGOLDING\n\n _Rescue Camp_\n\n _17 August, 1848_\n\nTwenty-two-year-old Robert Golding rushed into Rescue Camp just after sunset on Thursday, the 17th of August, agitated, shaking, and almost too excited to speak. Mate Robert Thomas intercepted him outside of Crozier's tent.\n\n\"Golding, I thought you were with Mr. Des Voeux's group on the ice.\"\n\n\"Yes, sir. I am, Mr. Thomas. I _was._ \"\n\n\"Is Des Voeux back already?\"\n\n\"No, Mr. Thomas. Mr. Des Voeux sent me back with a message for the captain.\"\n\n\"You can tell me.\"\n\n\"Yes, sir. I mean, _no,_ sir. Mr. Des Voeux said I was to report only to the captain. Just the captain, sorry, sir. Thank you, sir.\"\n\n\"What in hell is all the commotion out here?\" asked Crozier, crawling out of his tent.\n\nGolding repeated his instructions from the second mate to report only to the captain, apologized, stuttered, and was led away from the ring of tents by Crozier. \"Now tell me what's going on, Golding. Why aren't you with Mr. Des Voeux? Has something happened to him and the reconnaissance group?\"\n\n\"Yes, sir. I mean... no, Captain. I mean, something _has_ happened, sir, out there on the ice. I wasn't there when it did \u2014 we was left behind to hunt seals, sir, Francis Pocock and Josephus Greater and me, while Mr. Des Voeux went on farther south with Robert Johns and Bill Mark and Tom Tadman and the others yesterday, but this evenin' they come back, just Mr. Des Voeux and a couple of the others, I mean, about an hour after we heard the shotguns.\"\n\n\"Calm down, lad,\" said Crozier, setting his hands firmly on the boy's shaking shoulders. \"Tell me what Mr. Des Voeux's message was, word for word. And then tell me what you saw.\"\n\n\"They're both dead, Captain. Both of them. I saw the one \u2014 Mr. Des Voeux had her body on a blanket, sir, it was all tore up \u2014 but I ain't seen the other one yet.\"\n\n\" _Who's_ both dead, Golding?\" snapped Crozier, although the \"her\" had already told him part of the truth.\n\n\"Lady Silence and the thing, Captain. The Esquimaux bitch and the thing from the ice. I seen her body. I ain't seen its yet. Mr. Des Voeux said it's next to a polyp about another mile out beyond where we was shootin' at seals, and I'm to bring you and the doctor out to see it, sir.\"\n\n\"Polyp?\" said Crozier. \"You mean _polynya?_ One of the little lakes of open water in the ice?\"\n\n\"Yes, Captain. I ain't seen that yet, but that's where the thing's carcass is according to Mr. Des Voeux and Fat Wilson, who was with him and carrying and pulling the blanket like it was a sled, sir. Silence, she was in the blanket, you see, all tore up and dead. Mr. Des Voeux says to bring you and the doctor and no one else and for me not to tell no one else or he'll have Mr. Johnson flog me when he gets back.\"\n\n\"Why the doctor?\" said Crozier. \"Are some of our people hurt?\"\n\n\"I think so, Captain. I'm not sure. They're still out at the... the hole in the ice, sir. Pocock and Greater went on back south with Mr. Des Voeux and Fat Alex Wilson like Mr. D. V. said for them to, but he sent me back here and said to bring just you and the doctor, no one else. And not to tell no one else neither. Not yet. Oh... and for the surgeon to bring his kit with knives and such and maybe some larger knives for carvin' up the thing's carcass. Did you hear the shotgun blasts this evenin', Captain? Pocock and Greater and me heard 'em, and we was a mile away from the polyp at least.\"\n\n\"No. We wouldn't make out shotgun reports from two miles away over the damnable constant cracking and breaking of the ice here,\" said Crozier. \"Think hard, Golding. Why exactly did Mr. Des Voeux say it should just be Dr. Goodsir and myself that come out to see... whatever it is?\"\n\n\"He said he's fairly sure the thing's dead, but Mr. Des Voeux said it ain't what we thought it was, Captain. He said it's... I forget the words he used. But Mr. Des Voeux says it changes everything, sir. He wants you and the doctor to see it and know what happened there before anyone in the camp hears about it.\"\n\n\"What _did_ happen out there?\" pressed Crozier.\n\nGolding shook his head. \"I don't know, Captain. Pocock and Greater and me was hunting seals, sir... we shot one, Captain, but it slipped through its hole in the ice and we couldn't get to it. I'm sorry, sir. Then we heard the shotguns to the south. And a little later, an hour maybe, Mr. Des Voeux shows up with George Cann, who was bleeding on his face, and Fat Wilson, and Wilson was pulling Silence's body on a blanket he was draggin' and she was all torn to pieces, only... we're supposed to hurry back, Captain. While the moon's up.\"\n\nIndeed, it was a rare, clear night after a rare, clear, red sunset \u2014 Crozier had been taking his sextant out of its box to get a star fix when he'd heard the commotion \u2014 and a huge, full, blue-white moon had just risen over the icebergs and ice jumble to the southeast.\n\n\"Why tonight?\" asked Crozier. \"Can't this wait for morning?\"\n\n\"Mr. Des Voeux said it can't, Captain. He said to give you his compliments and would you be so kind as to bring Dr. Goodsir and come out about two miles \u2014 it ain't longer than two hours' walk, sir, even with the ice walls \u2014 to see what's there by the polyanna.\"\n\n\"All right,\" said Crozier. \"You go tell Dr. Goodsir I want him and for him to bring his medical kit and to dress warmly. I'll meet the two of you at the boats.\"\n\nGolding led the four men out onto the ice \u2014 Crozier ignored the message from Des Voeux to come just with the surgeon and had ordered Bosun John Lane and Captain of the Hold William Goddard to come along with their shotguns \u2014 and then into the jumble of bergs and ice boulders, then over three high pressure ridges, and finally through serac forests where Golding's earlier path back to the camp was marked not only by his boot prints in the blowing snow but also by the bamboo wands they'd hauled with them all the way from _Terror._ Des Voeux's group had carried the wands with them two days earlier to mark their way back and to show the best pathway through the ice should they find open water and want the others to follow them with the boats. The moonlight was so bright that it threw shadows. Even the narrow bamboo wands were like moon dials throwing slashes of shadow lines onto the white-blue ice.\n\nFor the first hour there was only the sound of the laboured breathing, their boots crunching on snow and ice, and the cracking and groans all around them. Then Crozier said, \"Are you sure she's dead, Golding?\"\n\n\"Who, sir?\"\n\nThe captain's frustrated exhalation became a small cloud of ice crystals gleaming in the moonlight. \"How many 'she's' are there around here, God-damn it? Lady Silence.\"\n\n\"Oh, yes, sir.\" The boy snickered. \"She's dead all right. Her titties was all tore off.\"\n\nThe captain glared at the boy as they climbed another low pressure ridge and passed into the shadow of a tall blue-glowing iceberg. \"But are you sure it's Silence? Could it be another native woman?\"\n\nGolding seemed stumped by that question. \"Is there more Esquimaux women out here, Captain?\"\n\nCrozier shook his head and gestured for the boy to continue leading.\n\nThey reached the \"polyanna,\" as Golding continued calling it, about an hour and a half after leaving camp.\n\n\"I thought you said it was farther out,\" said Crozier.\n\n\"I ain't never been even this far before,\" said Golding. \"I was back there hunting seals when Mr. Des Voeux found the thing.\" He gestured vaguely behind and to the left of where they now stood by the opening in the ice.\n\n\"You said some of our people were injured?\" asked Dr. Goodsir.\n\n\"Yes, sir. Fat Alex Wilson had blood on his face.\"\n\n\"I thought you said it was George Cann who had a bloody face,\" said Crozier.\n\nGolding shook his head emphatically. \"Uh-uh, Captain. It was Fat Alex who were bloody.\"\n\n\"Was it his own blood or someone or something else's?\" asked Goodsir.\n\n\"I don't know,\" Golding replied, his voice almost sullen-sounding. \"Mr. Des Voeux just told me to have you bring your surgeon's things. I figured someone had to be hurt, if Mr. Des Voeux needed you to fix 'im.\"\n\n\"Well, there's no one around here,\" said the bosun, John Lane, walking carefully around the ice edge of the _polynya_ \u2014 which was no more than twenty-five feet across \u2014 and staring first down into the dark water eight feet lower than the ice and then back at the forest of seracs on all sides. \"Where are they? Mr. Des Voeux had eight other men besides you with him when he left, Golding.\"\n\n\"I don't know, Mr. Lane. This is where he told me to bring you.\"\n\nCaptain of the Hold Goddard cupped his hands around his mouth and shouted, \"Halloooo? Mr. Des Voeux? Hallooo?\"\n\nThere came an answering shout from their right. The voice was indistinct, muffled, but sounded excited.\n\nMotioning Golding back, Crozier led the way through the forest of twelve-foot-high ice seracs. The wind through the sculpted towers made a moaning, crooning sound, and they all knew that the serac edges were as sharp as knife blades and stronger than most ship knives.\n\nAhead of them in the moonlight, in the center of a small, flat ice clearing amid the seracs, the dark form of one man stood alone.\n\n\"If that's Des Voeux,\" Lane whispered to his captain, \"he's got eight men missing.\"\n\nCrozier nodded. \"John, William, you two go ahead \u2014 slowly \u2014 keep your shotguns ready and on half-cock. Dr. Goodsir, please be so kind as to stay back with me. Golding, you wait here.\"\n\n\"Aye, sir,\" whispered William Goddard. He and John Lane tugged off their mittens with their teeth so they could use their gloved fingers, raised their weapons, half-cocked one of the heavy hammers on their double-barreled guns, and moved forward cautiously toward the moonlit clearing beyond the edge of the serac forest.\n\nA huge shadow came out from behind the last serac and slammed Lane's and Goddard's skulls together. The two men went down like cattle beneath a slaughterhouse sledgehammer.\n\nAnother shadowy figure struck Crozier in the back of the head, pinned his arms behind his back when he tried to rise, and held a knife to his neck.\n\nRobert Golding grabbed Goodsir and set a long blade alongside his throat. \"Don't move, Doctor,\" whispered the boy, \"or I'll do my own bit o' surgery on you.\"\n\nThe huge shadow lifted Goddard and Lane by the scruff of their greatcoats and dragged them out into the ice clearing. The toes of their boots made grooves in the snow. A third man came from behind the seracs, picked up Goddard's and Lane's shotguns, handed one to Golding, and kept the other for himself.\n\n\"Get out there,\" said Richard Aylmore, gesturing with the barrels of the shotgun.\n\nWith a knife still held to his throat by the shadowy shape Crozier now recognized by smell as the slackard George Thompson, the captain stood and half stumbled, was half pushed, out of the serac shadows and toward the man waiting in the moonlight.\n\nMagnus Manson dumped the bodies of Lane and Goddard in front of his master, Cornelius Hickey.\n\n\"Are they alive?\" rasped Crozier. The captain's arms were still pinned behind him by Thompson, but now that the muzzles of two shotguns were trained on him, the blade was no longer at his throat.\n\nHickey leaned over as if to inspect the men, and, with two smooth, easy moves, cut both their throats with a knife that had suddenly appeared in his hand.\n\n\"Not now they ain't alive, Mr. High-and-Mighty Crozier,\" said the caulker's mate.\n\nThe blood pouring out onto the ice looked black in the moonlight.\n\n\"Is that the technique you used to slaughter John Irving?\" asked Crozier, his voice shaking with fury.\n\n\"Fuck you,\" said Hickey.\n\nCrozier glared at Robert Golding. \"I hope you got your thirty pieces of silver.\"\n\nGolding snickered.\n\n\"George,\" said the caulker's mate to Thompson, standing behind the captain, \"Crozier carries a pistol in his right greatcoat pocket. Pull it out. Dickie, you bring the pistol back to me. If Crozier moves, kill him.\"\n\nThompson removed the pistol while Aylmore kept his purloined shotgun aimed. Then Aylmore walked over, took the pistol and the box of cartridges Thompson had found, and backed away, shotgun raised again. He crossed the short moonlit space and handed the pistol to Hickey.\n\n\"All this natural misery,\" Dr. Goodsir said suddenly. \"Why do you men have to add to it? Why does our species always have to take our full measure of God-given misery and terror and mortality and then make it worse? Can you answer me that, Mr. Hickey?\"\n\nThe caulker's mate, Manson, Aylmore, Thompson, and Golding stared at the surgeon as if he had begun speaking Aramaic.\n\nSo did the only other living man there, Francis Crozier.\n\n\"What do you want, Hickey?\" asked Crozier. \"Other than more good men dead as meat for your trip?\"\n\n\"I want you to shut the fuck up and then die slow and hard,\" said Hickey.\n\nRobert Golding laughed a demented boy's laugh. The barrels of the shotgun he was holding beat a tattoo on the back of Goodsir's neck.\n\n\"Mr. Hickey,\" said Goodsir, \"you do realize, do you not, that I shall never serve your purposes by dissecting my shipmates.\"\n\nHickey showed his small teeth in the moonlight. \"You will, surgeon. I guarantee you will. Or you'll watch us cut _your pieces_ off one at a time and then have us feed them to you.\"\n\nGoodsir said nothing.\n\n\"Tom Johnson and the others are going to find you,\" Crozier said, never removing his gaze from Cornelius Hickey's face.\n\nThe caulker's mate laughed. \"Johnson already found us, Crozier. Or rather, we found 'im.\"\n\nThe caulker's mate reached behind him and pulled a burlap bag from the snow. \"What'd you always call Johnson in private, King Crozier? Your strong right arm? Here.\" He tossed a naked and bloody right arm, severed just above the elbow, white bone gleaming, through the air and watched it land at Crozier's feet.\n\nCrozier did not look down at it. \"You pathetic little smear of spittle. You are \u2014 and always have been \u2014 nothing.\"\n\nHickey's face contorted as if the moonlight were changing him into something nonhuman. His thin lips drew far back from his tiny teeth in a way that the others had seen only with scurvy victims in their last hours. His eyes showed something beyond madness, far beyond mere hatred.\n\n\"Magnus,\" said Hickey, \"strangle the captain. Slow.\"\n\n\"Yes, Cornelius,\" said Magnus Manson, and shuffled forward.\n\nGoodsir tried to rush forward, but the boy, Golding, held him fast with one hand while holding the shotgun to his head with the other.\n\nCrozier did not move a muscle as the giant lumbered toward him. When Manson's shadow fell over both the captain and George Thompson holding him, Thompson himself flinched just a bit, Crozier sagged back, lunged forward, freed his left arm, and thrust his hand into the left pocket of his greatcoat.\n\nGolding almost pulled the shotgun's trigger, thus almost blowing Goodsir's head off by accident, so startled was he as the captain's coat pocket burst into flame and the muted double boom of an explosion rolled past them and echoed back from the seracs.\n\n\"Ouch,\" said Magnus Manson, slowly raising his hands to his belly.\n\n\"God-damn it,\" Crozier said calmly. He had inadvertently fired both barrels of a two-shot pistol.\n\n\"Magnus!\" cried Hickey and rushed forward to the giant.\n\n\"I think the captain shot me, Cornelius,\" said Manson. The big man sounded confused and a little bemused.\n\n\"Goodsir,\" shouted Crozier amid the confusion. The captain whirled, kneed Thompson in the bollocks and broke free. \"Run!\"\n\nThe surgeon tried. He pulled, shoved, and almost won his freedom before the younger Golding tripped him, knocked him onto his belly, and set the full pressure of his knee on Goodsir's back and the full force of two shotgun barrels against the back of Goodsir's skull.\n\nCrozier was loping for the seracs.\n\nHickey calmly seized a shotgun from Richard Aylmore, aimed, and fired both barrels.\n\nThe top of a serac splintered and fell at the same time that Crozier was thrown forward on his face, sliding on the ice and on a film of his own blood.\n\nHickey handed the shotgun back and unbuttoned Manson's coats and waistcoasts, ripping open the big man's shirts and filthy undershirt. \"Bring the fucking surgeon over here,\" he shouted at Golding.\n\n\"It don't hurt much, Cornelius,\" rumbled Magnus Manson. \"Tickles, more like.\"\n\nGolding shoved, prodded, and dragged Goodsir over. The surgeon put on his glasses and inspected the twin wounds. \"I'm not certain, but I don't believe the small-caliber bullets penetrated Mr. Manson's subcutaneous fat, much less his muscle layer. It's little more than two minor punctures, I fear. Now may I go attend to Captain Crozier, Mr. Hickey?\"\n\nHickey laughed.\n\n\"Cornelius!\" shouted Aylmore.\n\nCrozier, leaving a trail of blood and shredded outer clothing, had gotten to his knees and begun crawling toward the seracs and serac shadows. Now he painfully got to his feet. He staggered drunkenly toward the ice columns.\n\nGolding giggled and raised his shotgun.\n\n\"No!\" cried Hickey. He pulled Crozier's big percussion-cap pistol from his coat pocket and took careful aim.\n\nTwenty feet from the seracs, Crozier looked back over his shredded shoulder.\n\nHickey fired.\n\nThe bullet spun Crozier around and dropped him to his knees. His body sagged, but he flailed and thrust one hand down onto the ice in an attempt to rise.\n\nHickey took five steps forward and fired again.\n\nCrozier was thrown backward and lay on his back with only his knees in the air.\n\nHickey took two more steps, aimed, and fired again. One of Crozier's legs was knocked aside and down as the bullet tore through the knee or the muscle just below the knee. The captain made no sound.\n\n\"Cornelius, honey.\" Magnus Manson's voice had the tone of an injured child. \"My stomach is starting to hurt.\"\n\nHickey wheeled. \"Goodsir, give him something for the pain.\"\n\nThe surgeon nodded. His voice, when he spoke, was very thin and very tight and very flat. \"I brought an entire bottle of Dover's Powder \u2014 mostly made from a derivative of the coca plant, sometimes called cocaine. I'll give him that. All of it, if you like. With a chaser of Mandragora, laudanum, and morphine. That will take away the pain.\" He reached into his medical kit.\n\nHickey raised the pistol and aimed it at the surgeon's left eye. \"If you even make Magnus sick to his stomach, much less if your fucking hand comes out of that bag with a scalpel or other blade, I swear to fucking Christ I'll shoot you in the balls and keep you alive long enough to make you eat them. Do you understand, Surgeon?\"\n\n\"I understand,\" said Goodsir. \"But it is the Hippocratic oath that determines my next actions.\" He brought out a bottle and spoon and poured out a tiny bit of liquid morphine. \"Sip this,\" he said to the giant.\n\n\"Thank you, Doctor,\" said Magnus Manson. He slurped soundly.\n\n\"Cornelius!\" cried Thompson, pointing.\n\nCrozier was gone. Bloody smears led into the seracs.\n\n\"Oh, fuck me,\" said the caulker's mate with a sigh. \"This arsehole is more trouble than he is worth. Dickie, have you reloaded?\" Hickey was reloading the pistol as he asked the question.\n\n\"Aye,\" said Aylmore, lifting the shotgun.\n\n\"Thompson, pick up the extra shotgun I brought and stay here with Magnus and the surgeon. If the good doctor does anything at all that you don't like \u2014 even farts \u2014 blow his private parts off.\"\n\nThompson nodded. Golding giggled. Hickey with his pistol and Golding and Aylmore with their shotguns advanced slowly across the moonlit ice and then tentatively, single file, into the forest of seracs and shadows.\n\n\"He could be hard to find in here,\" whispered Aylmore as they stepped into the stripes of moonlight and darkness.\n\n\"I don't think so,\" said Hickey, and pointed at the broad smear of blood that led straight ahead between the ice columns like a telegraph code of black dots and dashes between the shadows.\n\n\"He still has a little pistol with him,\" whispered Aylmore, moving cautiously from serac to serac.\n\n\"Fuck him and fuck his pistol,\" said Hickey, striding straight ahead, his boots slipping a bit on the blood and ice.\n\nGolding giggled loudly. \"Fuck him and fuck his little pistol,\" he said in a singsong voice, snickering again.\n\nThe blood trail ended forty feet in at the black _polynya._ Hickey rushed forward and stared down at where the horizontal smears became vertical smears on the side of the eight-foot ice slab. Something had gone into the water here.\n\n\"God-damn it to God-damn fucking hell,\" cried Hickey, pacing back and forth. \"I wanted to put that last bullet into the high-and-mighty king's fucking face while he watched, God- _damn_ him. He robbed me.\"\n\n\"Look, Mr. Hickey, sir,\" said Golding, giggling. He pointed to what might be a body floating facedown in the dark water.\n\n\"It's only the fucking coat,\" said Aylmore, who had come cautiously out of the shadows with his shotgun raised.\n\n\"Only the fucking coat,\" repeated Robert Golding.\n\n\"So he's dead down there,\" said Aylmore. \"Can we get out of here before Des Voeux or someone comes to the sound of all the shooting? It's two days back to the others and we still have the bodies to cut up before we can leave.\"\n\n\"No one's going anywhere yet,\" said the caulker's mate. \"Crozier may still be alive.\"\n\n\"All shot up like that, without his coat?\" asked Aylmore. \"And look at the greatcoat, Cornelius. The shotgun tore it apart.\"\n\n\"He may still be alive. We're going to make sure he's not. And maybe the body will float to the surface.\"\n\n\"What are you going to do?\" asked Aylmore. \"Shoot his dead body?\"\n\nHickey wheeled on the man and glared, making the much-taller Aylmore step back. \"Yes,\" said Cornelius Hickey. \"That's precisely what I'm going to do.\" To Golding he barked, \"Go bring Thompson and Magnus and the surgeon. We'll tie up the doctor tight to one of them seracs while Aylmore and Thompson and me search and you watch over Magnus and cut Lane and Goddard into small enough to haul easylike bits.\"\n\n\"Me cut 'em up?\" cried Golding. \"You told me that's why we were grabbin' Goodsir, Cornelius. He was s'posed to do all the cutting up, not me.\"\n\n\"Goodsir will do the carving in the future, Bobby,\" said Hickey. \"Tonight you have to do it. We can't trust Dr. Goodsir yet... not until we get him back with our people and many miles away from here. You be a good boy and go get the doctor and tie him up to a serac, tight, use your best knots, and tell Magnus to bring the carcasses over here where you can carve 'em. And get blades from Goodsir's kit and the big knives and carpenter's saw I brung that are over in the bag.\"\n\n\"Oh, all right,\" said Golding. \"But I'd rather search.\" He trudged back out of the serac field.\n\n\"The captain must have left half his blood between where you shot him and here, Cornelius,\" said Aylmore. \"If he didn't go into the water, he can't hide anywhere here without leaving a trail.\"\n\n\"That is precisely correct, Dickie my dear,\" said Hickey with a strange smile. \"If he's not in the water he might crawl, but he cannot stop losing blood with wounds like that. We are going to search until we are sure he ain't under the water nor curled up somewhere here in the seracs where he crawled and hid and bled himself to death. You start over there on the south side of the _polynya_ , I'll look to the north. We'll go clockwise. If you see any wee sign, even a drop of blood, even a scuff in the snow, shout and stop. I'll join you. And be careful. We don't want the dying fucker jumping out of the shadows and grabbing one of our guns now, do we?\"\n\nAylmore looked surprised and alarmed. \"Do you really think he could be strong enough to do that? With three bullets and all those shotgun pellets in him, I mean? Without his coat, he'd freeze to death in a few minutes anyway. It's getting much colder and the wind's getting stronger. Do you really think he's lying in wait for us, Cornelius?\"\n\nHickey smiled and nodded toward the black pool. \"No. I think he's dead and drowned and down there. But we're going to make fuckin' sure. We're not leaving here until we're sure, even if we got to search until the God-poxed sun comes up.\"\n\nIn the end, they searched for three hours under the light of the rising and then descending moon. There were no signs at all near the _polynya_ nor amid the seracs nor on the open ice fields beyond the seracs in all directions nor on the high pressure ridges to the north and south and east: no blood trails, no footprints, no drag marks.\n\nIt took Robert Golding the full three hours to hack John Lane and William Goddard into the size pieces that Hickey had asked for, and even then the boy made a dreadful mess of it. Ribs, heads, hands, feet, and sections of spinal cord lay around him on all sides as if there had been an explosion in an abattoir. And young Golding himself was so covered with blood that he looked like a player in a minstrel show by the time Hickey and the others got back. Aylmore, Thompson, and even Magnus Manson were taken aback by their young apprentice's appearance, but Hickey laughed long and hard.\n\nThe gunnysacks and burlap bags were filled with meat wrapped in oilcloths they'd brought. Yet still the bags leaked.\n\nThey untied Goodsir, who was shaking from the cold or shock.\n\n\"Time to go, Surgeon,\" said Hickey. \"The other chaps are waiting ten miles west of here on the ice to welcome you home.\"\n\nGoodsir said, \"Mr. Des Voeux and the others will come after you.\"\n\n\"No,\" said Cornelius Hickey, his voice showing his absolute certainty, \"they won't. Not with them knowing that now we got at least three shotguns and a pistol. And that's if they ever find out we was here, which I think they won't.\" To Golding, he said, \"Give our new crewmate a sack of meat to carry, Bobby.\"\n\nWhen Goodsir refused to accept the bulging sack from Golding, Magnus Manson knocked him down, almost breaking the surgeon's ribs. On the fourth attempt to hand him the dripping bag, after two more serious cuffings, the surgeon took it.\n\n\"Let's go,\" said Hickey. \"We're done here.\"\n\nDES VOEUX\n\n _Rescue Camp_\n\n _19 August, 1848_\n\nFirst Mate Charles Des Voeux could not restrain himself from grinning as he and his eight men returned to Rescue Camp on the morning of Saturday, 19 August. For a change, he had nothing but good news to deliver to his captain and the men.\n\nThe ice pack had opened to floes and navigable leads only four miles out, and Des Voeux and his men had spent another day following the leads south until the strait became open water all the way to the Adelaide Peninsula and almost certainly to the inlet to Back River farther east around that peninsula. Des Voeux had _seen_ the low hills of the Adelaide Peninsula less than twelve open-water miles away from an iceberg they'd climbed at their farthest-south extension of the ice pack. They could go no farther without a boat, which had made First Mate Des Voeux grin broadly then and which made him grin again now.\n\nEveryone could leave Rescue Camp. Everyone there now had a chance at survival.\n\nAlmost better news to bring home was the fact that they had spent two days shooting seals on the floes at the edge of the new open sea out there on the strait. For two days and nights, Des Voeux and his men had gorged themselves on seal meat and blubber, their bodies craving the fat so much that even though the rich food made then sick \u2014 after weeks of only ship's biscuit and slivers of old salt pork \u2014 vomiting just made them hungrier, and they laughed and began gorging again almost immediately.\n\nEach of his eight men was dragging a carcass of a seal behind him now as they followed the bamboo wands across the last mile of coastal ice to the camp. The forty-six men in Rescue Camp would eat well tonight, as would again the eight triumphant explorers.\n\nAll in all, Des Voeux thought as they came up the shingle past the boats, hallooing and hurrahing to get the camp's attention, other than the young squirt Golding turning back on his own that first day because of a belly ache, it had been almost a perfect expedition. For the first time in months \u2014 in _years_ \u2014 Captain Crozier and the others would have news to celebrate.\n\nThey were all going home. If they left today, the healthy among them man-hauling the ill in the boats only the four miles on the winding trail through pressure ridges that Des Voeux had carefully charted, they would be afloat within three or four days, to the mouth of Back's Great Fish River within the week. And it was probable that the opening leads had advanced even closer to shore by now!\n\nFilthy, ragged, slumped creatures emerged from their tents and left their desultory camp chores to come out to stare at Des Voeux's party.\n\nThe cheering of Des Voeux's men \u2014 Fat Alex Wilson, Francis Pocock, Josephus Greater, George Cann, Robert Johns, Thomas Tadman, Thomas McConvey, and William Mark \u2014 died as they looked at the dour, immobile, haunted-eyed faces of the men facing them. The men from the camp could see the seals being dragged, but they seemed to have no reaction.\n\nMates Couch and Thomas came out of their tents and down the shingle to stand in front of the line of Rescue Camp spectres.\n\n\"Did someone die?\" asked Charles Frederick Des Voeux.\n\nSecond Mate Edward Couch, First Mate Robert Thomas, First Mate Charles Des Voeux, _Erebus_ Captain of the Hold Joseph Andrews, and _Terror_ Captain of the Maintop Thomas Farr were crowded into the oversized tent that had been used as Dr. Goodsir's hospital. The amputees, Des Voeux had learned, had either died in the four days he was gone or been moved back to smaller tents shared with the other sick men.\n\nThese five in this tent this morning were the last officers with any command authority left alive \u2014 or at least at Rescue Camp and well enough to walk \u2014 from the entire John Franklin Expedition. They had just enough tobacco left for four of the five \u2014 Farr did not smoke \u2014 to have their pipes going. The interior of the tent was filled with blue smoke.\n\n\"Are you sure it wasn't the thing from the ice that committed the carnage you found out there?\" asked Des Voeux.\n\nCouch shook his head. \"We thought that might be the case at first \u2014 in fact, that was our assumption \u2014 but the bones and heads and remaining pieces of flesh we found... .\" He stopped and bit down hard on the stem of pipe.\n\n\"Had knife marks on them,\" finished Robert Thomas. \"Lane and Goddard were butchered by a human being.\"\n\n\"Not a human being,\" said Thomas Farr. \"But some vile thing in the shape of a man.\"\n\n\"Hickey,\" said Des Voeux.\n\nThe others nodded.\n\n\"We have to go after him and the murderers with him,\" said Des Voeux.\n\nNo one spoke for a moment. Then Robert Thomas said, \"Why?\"\n\n\"To bring them to justice.\"\n\nFour of the five men looked at one another. \"They have three shotguns now,\" said Couch. \"And almost certainly the captain's percussion-cap pistol.\"\n\n\"We have more men... guns... powder, shot, cartridges,\" said Des Voeux.\n\n\"Aye,\" said Thomas Farr. \"And how many of them would die in a battle with Hickey and his fifteen cannibals? Thomas Johnson ne'er came back, y'know. His job was just to _track_ Hickey's band, make sure they was leaving like they said they was.\"\n\n\"I can't believe this,\" said Des Voeux, removing his pipe and tamping at the bowl. \"What about Captain Crozier and Dr. Goodsir? Are you just going to abandon them? Leave them to Cornelius Hickey's whims?\"\n\n\"The captain ain't alive,\" said Captain of the Hold Andrews. \"Hickey wouldn't have no reason to keep Crozier alive... unless it was to torture and torment him.\"\n\n\"All the more reason to send a rescue party after them,\" insisted Des Voeux.\n\nThe others did not respond for a moment. The blue smoke swirled around them. Thomas Farr untied the tent door and opened it wider to let some air in and smoke out.\n\n\"It's been almost two days since whatever happened out on the ice happened,\" said Edward Couch. \"It would be several more days before any party we sent could find and fight Hickey's group, even if they _could_ find them. All the devil has to do is travel farther out on the ice or inland to throw us off. The wind obscures tracks in hours... even sledge tracks. Do you really think Francis Crozier, if he's alive now \u2014 which I doubt \u2014 would be alive or in any shape to be rescued in five days or a week?\"\n\nDes Voeux chewed the stem of his pipe. \"Dr. Goodsir, then. We need the surgeon. Logic dictates that Hickey would keep _him_ alive. Goodsir may be the reason Hickey and his accomplices came back.\"\n\nRobert Thomas shook his head. \"Cornelius Hickey may need Dr. Goodsir for his own infernal purposes, but we don't any longer.\"\n\n\"What do you mean?\"\n\n\"I mean that most of our good surgeon's potions and instruments were left behind \u2014 he brought only his portable medical kit,\" said Farr. \"And Thomas Hartnell, who's been his assistant, knows which potions to administer and how much and for what.\"\n\n\"What about actual surgery?\" asked Des Voeux.\n\nCouch smiled sadly. \"Lad, do you really think that anyone who needs actual surgery from this point on in our travels is likely to survive, no matter what?\"\n\nDes Voeux did not answer.\n\n\"And what if Hickey and his men ain't goin' nowhere?\" asked Andrews. \"And never planned to? He come back to kill the captain, grab Goodsir, and take poor John Lane and Bill Goddard and carve 'em up like animals. He sees all of us as livestock. What if he's just waiting out there beyond the next rise, waiting to attack the whole camp?\"\n\n\"You're turning the caulker's mate into a bogeyman,\" said Des Voeux.\n\n\"He done that to his self already,\" said Andrews. \"But not a bogeyman, the Devil. The actual Devil. Him and his tame monster, Magnus Manson. They sold their souls \u2014 God-damn them \u2014 and received some dark power for it. Mark my words.\"\n\n\"You'd think that one real monster would be enough for any arctic expedition,\" said Robert Thomas.\n\nNo one laughed.\n\n\"It's _all_ one real monster,\" Edward Couch said at last. \"And not a new one to our race.\"\n\n\"So what are you all suggesting?\" Des Voeux asked after another spell of silence. \"That we run from a five-foot-tall demon caulker's mate and just head south with the boats tomorrow?\"\n\n\"Me, I'm saying we leave today,\" said Joseph Andrews. \"As soon as we load the boats with the few things we're takin'. Man-haul through the night. With luck, there'll be enough moonlight to guide by when she rises. If not, we use some of the lantern fuel we kept back. You said yourself, Charles, that the wands is still out there markin' the way. They won't be after the first real storm blows through.\"\n\nCouch shook his head. \"Des Voeux's men are tired. Our people are totally demoralized. Let's have a feast tonight \u2014 eat every one of those eight seals you brought in, Charles \u2014 then leave tomorrow morning. We'll all have more of a sense of hope after a big meal, some cooking and light using the seal oil, and a good night's sleep.\"\n\n\"But with men on watch tonight,\" said Andrews.\n\n\"Oh, aye,\" said Couch. \"I'll stand watch myself. I'm not that hungry anyway.\"\n\n\"There's the question of command,\" said Thomas Farr, looking from face to face in the dim light filtering through the canvas.\n\nSeveral of the men sighed.\n\n\"Charles is in overall command,\" said First Mate Robert Thomas. \"Sir John himself promoted him as first mate of the flagship when Graham Gore got killed, so he's senior officer.\"\n\n\"But you were first mate on _Terror,_ Robert,\" Farr said to Thomas. \"You have seniority.\"\n\nThomas shook his head adamantly. \" _Erebus_ was the command ship. When Gore was alive, it was understood that he had overall expedition command above mine. Charles's got Gore's job now. He's in charge. I don't mind. Mr. Des Voeux is a better leader than me, and we're going to need leadership.\"\n\n\"I can't believe that Captain Crozier's gone,\" said Andrews.\n\nFour of the five men smoked harder. No one spoke. They could hear men outside talking about the seals, someone laughing, and \u2014 beyond that \u2014 the cracking and rifle fire of ice breaking.\n\n\"Technically,\" said Thomas Farr, \"Lieutenant George Henry Hodgson is in charge of the expedition now.\"\n\n\"Oh, fuck Lieutenant George Henry Hodgson up the arse with a hot poker,\" said Joseph Andrews. \"If the little weasel were to come crawlin' back now, I'd strangle 'im with me own hands and piss on his corpse.\"\n\n\"I doubt very much if Lieutenant Hodgson is still alive,\" Des Voeux said softly. \"It's decided then that I'm in overall command of the expedition now, with Robert second in command, Edward as third?\"\n\n\"Aye,\" said the other four men in the tent.\n\n\"Then understand that I'm going to keep conferring with the four of you as we have to make decisions,\" said Des Voeux. \"I've always wanted to be captain of my own ship... but not this fucking way. I'm going to need your help.\"\n\nEveryone nodded behind their screen of pipe smoke.\n\n\"I have one question before we go out and tell the men to start preparing for the feast today and departure tomorrow,\" said Couch.\n\nDes Voeux, who was bareheaded in the heat of the tent, raised his eyebrows.\n\n\"What about the sick men? Hartnell tells me that there are six who can't walk, even if their lives depended on it. Too far gone in scurvy. Take Jopson, the captain's steward, for instance. Mr. Helpman and our engineer, Thompson, are dead, but Jopson keeps hanging on. Hartnell says he can't even lift his head to drink \u2014 he has to be helped \u2014 but he's still alive. Do we take him with us?\"\n\nDes Voeux looked at Couch and then at the other three faces for unspoken answers, but they gave him nothing.\n\n\"And if we _do_ take Jopson and the other dying ones,\" continued Couch, \"what do we take 'em _as?_ \"\n\nDes Voeux did not have to ask what the second mate meant. _Do we haul them along as shipmates or as food?_\n\n\"If we leave them here,\" he said, \"they'll sure as hell be food if Hickey comes back the way some of you think he will.\"\n\nCouch shook his head. \"That isn't what I'm asking.\"\n\n\"I know,\" said Des Voeux. He took a deep breath, almost coughing because of the thick pipe smoke. \"All right,\" he said. \"Here is my first decision as new commander of the Franklin Expedition. When we drag the boats to the ice in the morning, any man who can walk to the boats and get into harness \u2014 or even into one of the boats \u2014 comes with us. If he dies on the way, we'll decide then whether to haul his body farther. _I'll_ decide. But tomorrow morning, only those who can walk to the boats will leave Rescue Camp.\"\n\nNone of the other men spoke, but several nodded. No one met Des Voeux's gaze.\n\n\"I'll tell the men after we eat,\" said Des Voeux. \"Each of you four choose one reliable man to join you on watch tonight. Edward will set the schedule. Don't let those men eat themselves into oblivion. We'll need our wits about us \u2014 at least some of us \u2014 until we get safely to the open water.\"\n\nAll four men nodded at this.\n\n\"All right, go tell your men about the feast,\" said Des Voeux. \"We're done here.\"\n\nGOODSIR\n\n _20 August, 1848_\n\nFrom the private diary of Dr. Harry D. S. Goodsir:\n\n _Saturday, 20 August, 1848 \u2014_\n\n _The Devil, Hickey, seems to have all the Good Fortune so denied to Sir John, Commander Fitzjames, and Captain Crozier for so many Months and Years._\n\n _They do not know that I had Inadvertently put my Diary into my Medical Kit \u2014 or, rather, they probably know, since they thoroughly Searched my kit two nights ago after taking me Captive, but they do not Care. I sleep Alone in a tent except for Lieutenant Hodgson, who is as much Captive now as I am, and he does not Mind my scribbling in the dark._\n\n _Part of me still cannot believe the Slaughter of my comrades \u2014 Lane, Goddard, and Crozier \u2014 and had I not Seen with my own Eyes the Feast of Human Flesh half of Hickey's party celebrated late Friday night upon our return to this sledge Camp out on the Ice not far from our old River Camp, I still might not Believe in such Barbarism._\n\n _Not all of Hickey's Infernal Legion have yet succumbed to the Lure of Cannibalism. Hickey, Manson, Thompson, and Aylmore are Enthusiastic Participants, of course, as are \u2014 it turns out \u2014 Seaman William Orren, Steward William Gibson, Stoker Luke Smith Golding, Caulker James Brown, and his mate Dunn._\n\n _But others abstain alongside Myself \u2014 Morfin, Best, Jerry, Work, Strickland, Seeley, and, of course, Hodgson. We are all subsisting on Mouldy Ship's Biscuits. Of those Fellow Abstainers, I suspect that only Strickland or Morfin and the Lieutenant may continue to Resist for long. Hickey's People have caught just one Seal on their voyage West along the coast, but that was enough to power a Stove with its Oil \u2014 and the smell of Roasting Human Flesh is Horribly Enticing._\n\n _Hickey has not Harmed me yet. Not even the past Two Nights when I have refused to partake of the Meal or to agree to Cut Other Bodies Up when the time comes. So far, Mr. Lane and Mr. Goddard's Parts have assuaged their appetite and Freed me from having to decide between becoming a Chef for Cannibals or being Maimed and Carved myself._\n\n _But no one is allowed to Touch the Shotguns other than Mr. Hickey, Mr. Aylmore, or Mr. Thompson \u2014 these Last Two have become lieutenants of the New Bonaparte that is our Diminutive Caulker's Mate \u2014 and Magnus Manson is a weapon of his own which only One Man \u2014 if he is indeed still a man \u2014 can Aim and Unleash._\n\n _But when I speak of Hickey's Fortune, I do not speak of just the Luck of his own Dark Making that brought him a source of fresh meat. Rather, I refer to today's Revelation when, just two miles northwest and offshore of our old River Camp where Mr. Bridgens went missing, we came upon Open Leads that stretched Westward along the Coast._\n\n _Hickey's Depraved Crew unsledged, Rigged, loaded, and Launched the pinnace almost at once, and we have been Sailing and Rowing quickly along to the West ever since._\n\n _You Might Ask, How can 17 Men fit into a 28-foot Open Boat meant to carry only 8 to a Dozen men comfortably?_\n\n _The Answer is that we crowd upon each other Terribly and \u2014 even though we haul only Tents, weapons, cartridges, water casks, ourselves, and our Terrible Food supply \u2014 we are so Heavily Laden that the Sea rises almost to the Gunwales on either side, especially when the width of the Leads allows us to Tack into the wind without the Use of Oars._\n\n _I Heard Hickey and Aylmore whispering after we landed to pitch Tents this Evening \u2014 they made Little Effort to lower their Voices._\n\n _Someone will have to go._\n\n _The Water is Open ahead, the Way is Free \u2014 perhaps all the way back to Terror Camp, or even to_ Terror _herself \u2014 just as the Prophet Cornelius Hickey insisted during the confrontation with Crozier at the unnamed bay in July, where mutiny was avoided only by the shout of Open Water \u2014 and it may well Occur that Hickey and those who Remain with Him will be back at Terror Camp and the ship in three days of Easy Sailing rather than the Three and a Half Months of Brutal Man-hauling it took us to come the Same Distance in the Opposite Direction._\n\n _But now that they do not need Man-haulers, which Men will be Sacrificed to the Food stores so that the boat can be Lightened for tomorrow's Sailing?_\n\n _Hickey and his Giant and Aylmore and the other Leaders are Walking Through Camp as I write, calling us peremptorily Out of Our Tents, although the Hour is Late and the night is Dark._\n\n _If I am Alive tomorrow, I will write more then._\n\nJOPSON\n\n _Rescue Camp_\n\n _20 August, 1848_\n\nThey were treating him like an old man and leaving him behind because they thought he was an old man, used up, dying even, but that was ridiculous. Thomas Jopson was only thirty-one years old. Today, the twentieth of August, he turned thirty-one years old. It was his birthday, and none of them except Captain Crozier, who had quit coming to see him in his sick tent for some unknown reason, even knew it was his birthday. They were treating him like an old man because almost all of his teeth had fallen out from the scurvy and most of his hair had fallen out for some reason he did not understand and he was bleeding from his gums and eyes and hairline and anus, but _he was not an old man._ He was thirty-one years old today and they were leaving him behind to die on his birthday.\n\nJopson heard the revelries the afternoon and evening before \u2014 impressions and memories of the shouting and laughter and smell of roasting food were not connected since he had been shifting in and out of fevered consciousness all that previous day \u2014 but he had wakened in the twilight to find that someone had brought a plate holding a slab of oily sealskin, strips of dripping white blubber, and a fish-smelling stripe of almost-raw red seal meat. Jopson vomited \u2014 nothing had come up because he had not eaten for a day or days \u2014 and shoved the offending plate of offal out the open tent door.\n\nHe'd understood that they were leaving him when crewmate after crewmate came by his tent later in the evening, saying nothing, not even showing their faces, but each shoving in one or two rock-hard and half-green ship's biscuits, stacking them by his side like so many white rocks in preparation for his burial. He was too weak to protest then \u2014 and too preoccupied with his dreams \u2014 but he had known that these few lousy lumps of half-baked and fully stale flour were all he was to receive for his years of faithful service to the Navy, to the Discovery Service, and to Captain Crozier.\n\nThey were leaving him behind.\n\nThis Sunday morning he awoke more clearheaded than he had been in some days \u2014 perhaps in weeks \u2014 only to hear his shipmates' preparations to leave Rescue Camp forever.\n\nThere was shouting down by the boats as the two whaleboats were righted and as the two cutters were readied on their sledges and as all four boats were loaded.\n\n _How could they leave me behind?_ Jopson had trouble believing they could or would. Hadn't he stayed by Captain Crozier's side a hundred times during the captain's illnesses and moody low points and outright bouts of drunkenness? Hadn't he quietly, uncomplainingly, like the good steward he was, hauled pails of vomit from the captain's cabin in the middle of the night and wiped the Irish drunkard's arse when he shat himself in his fever deliriums?\n\n _Perhaps that's why the bastard is leaving me to die._\n\nJopson forced his eyes open and tried to roll over in his sodden sleeping bag. It was very difficult. The weakness radiating out from his center consumed him. His head threatened to burst with pain every time he opened his eyes. The earth pitched against him as fiercely as any ship he'd ever ridden around the Horn in high seas. His bones ached.\n\n _Wait for me!_ he shouted. He thought he had shouted it, but it had been only a silent thought. He would have to do better than that... catch up to them before they shoved the boats out onto the ice... show them that he could man-haul with the best of them. He might even fool them by being able to force down some of their reeking, rotten seal meat.\n\nJopson could not believe that they were treating him like a dead man. He was a living human being with a good Naval record and excellent experience as a personal steward and with as solid a private history as a loyal citizen of Her Majesty's as any other man on the expedition, not to mention a family and a home in Portsmouth (if Elisabeth and his son, Avery, were still alive and if they'd not been evicted from the home they'd rented with Thomas Jopson's Discovery Service river-pay advance of 28 pounds against his first year's expedition salary of 65 pounds).\n\nRescue Camp now seemed empty except for a few low moans that could be coming from nearby tents or could be merely the incessant wind. The usual crunch of boots on gravel, soft cursing, rare laughter, the small talk of men going to and from watch, shouts between tents, echoes of hammer or saw, the smell of pipe tobacco \u2014 all were absent except for faint and receding noises from the direction of the boats. The men were really leaving.\n\nThomas Jopson was not going to stay here and die in this cold arse-end-of-the-world temporary camp.\n\nUsing all of the strength he had and some he knew he did not have, Jopson pulled his Hudson's Bay blanket sleeping bag down past his shoulders and began creeping his way out of it. The operation was made no simpler by the fact that strands of frozen sweat, blood, and other bodily fluids had to be ripped free of flesh and wool before he could crawl out of the blanket and toward the tent opening.\n\nMoving what seemed like miles on his elbows, Jopson collapsed forward through the tent flap and gasped at how cold the air outside was. He had grown so used to the canvas-filtered dim light and stuffy air of his tent-womb that this openness and glare made his lungs labour and filled his squinted-shut eyes with tears.\n\nJopson soon realized that the sun's glare was illusory; indeed, the morning was dark and thickly fogged, with tendrils of icy vapour moving between the tents like the spirits of all those dead men they'd left behind. It reminded the captain's steward of the thick fog on the day they'd sent Lieutenant Little, Ice Master Reid, Harry Peglar, and the others forward down the first open lead in the ice.\n\n _To their deaths,_ thought Jopson.\n\nCrawling over the ship's biscuits and seal meat \u2014 brought to him as if he were some damned pagan idol or sacrificial offering to the gods \u2014 Jopson pulled his unfeeling and unresponsive legs out through the circular tent opening.\n\nHe saw two or three tents standing nearby and for a second he was filled with the hope that the absence of ambulatory men here was temporary, that they were all busy doing something near the boats and would soon be back. But then Jopson saw that most of the Holland tents were missing.\n\n _No, not missing._ He could see now, as his eyes adapted to the diffuse light leaking through the fog, that the majority of tents here on the south end of the camp \u2014 nearest to the boats and shoreline \u2014 had been collapsed, with rocks tossed atop them to keep them from blowing away. Jopson was confused. If they were actually leaving, wouldn't they be bringing the tents along? It was as if they planned to go out on the ice but then return soon. To where? And why? None of it made any sense to the sick and recently hallucinatory steward.\n\nThen the fog shifted and lifted and he could see fifty yards or so to where the men were pulling, pushing, and tugging from the sides on the boats, hauling them out onto the ice. Jopson estimated that there were at least ten men per boat, which meant that all or almost all of the survivors here at camp were leaving him and the other really sick men behind.\n\n _How can Dr. Goodsir leave me behind?_ wondered Jopson. He tried to remember the last time it had been the surgeon who had lifted his head and shoulders to feed him broth or to clean him. It was young Hartnell yesterday, wasn't it? Or had that been several days ago? He could not recall the last time the surgeon had looked in on him or brought him medicine.\n\n\"Wait!\" he called.\n\nOnly it had not been a call. It had hardly been a croak. Jopson realized that he had not spoken aloud for days \u2014 perhaps weeks \u2014 and the noise he'd just made sounded muted and muffled even to his own ringing ears.\n\n\"Wait!\" That had been no better. He realized that he had to wave his arm in the air, make them see him, make them turn back for him.\n\nThomas Jopson could not raise either arm. Even trying to do so caused him to fall forward, his face striking the gravel.\n\nThere was nothing for it \u2014 he would just have to crawl toward them until they saw him and turned back. They wouldn't leave behind a fellow crewmate healthy enough to crawl a hundred yards after them onto the ice.\n\nJopson wriggled forward on his torn elbows another three feet and collapsed facedown onto the icy gravel again. The fog roiled around him, obscuring even his own tent a few paces behind him. The wind moaned \u2014 or perhaps it was more abandoned sick souls moaning in the few tents still standing \u2014 and the chill of the cold day cut right through his filthy wool shirt and soiled trousers. He realized that if he kept crawling away from his tent, he might not have the strength to crawl back and would die of the cold and damp out here.\n\n\"Wait!\" he called. His voice was as weak and mewling as a newborn kitten's.\n\nHe crawled and wriggled and writhed another three feet... four... and lay gasping like a harpooned seal. His weakened, dragging arms and hands were of no more use than flippers would be... of less use.\n\nJopson tried digging his chin into the frozen earth to propel himself forward another foot or two. He immediately chipped one of his last remaining teeth in two but dug his chin in again for another try. His body was simply too heavy. It seemed attached to the earth by great weights.\n\n _I am only thirty-one years old,_ he thought fiercely, angrily. _Today is my birthday._\n\n\"Wait... wait... wait... wait.\" Each syllable was weaker than the last.\n\nPanting, gasping, his remaining strands of hair dabbing crimson streaks onto the rounded stones, Jopson lay on his belly, his dead arms at his sides, painfully cocked his neck, and settled his cheek against the cold earth so that he could see straight ahead.\n\n\"Wait...\"\n\nThe fog swirled and then lifted.\n\nHe could see a hundred yards, past the odd vacuity where the boats had been lined up, past the shingle of shore gravel and the tumble of shore ice, out onto the ice itself where forty-some men and four boats \u2014 _where is the fifth?_ \u2014 struggled southward deeper onto the ice, the men's own weakness evident even at this distance, their own progress not that much more efficient or elegant than Jopson's five-yard struggle had been.\n\n\"Wait!\" This last shout had taken the penultimate ounce of draining energy \u2014 Jopson could feel his core's warmth flowing away into the icy ground beneath him \u2014 but it had come out as loud as any spoken word he had ever uttered.\n\n\"Wait!!\" he finally shouted. It was a man's voice now, not a kitten's mewl or dying seal's squeak.\n\nBut it was too late. The men and boats were a hundred yards out now and disappearing fast \u2014 mere black, staggering silhouettes against an eternal background of grey and grey \u2014 and the cracking and groaning of ice and wind would have covered the sound of a rifle shot, much less a solitary voice of one man left behind.\n\nFor an instant the fog lifted more and a benevolent light fell on everything \u2014 as if the sun were coming out to melt the ice everywhere and to bring green tendrils and living things and hope back where none existed here before \u2014 but then the fog closed in and swirled around Jopson, blinding him and binding him with its clammy, cold grey fingers.\n\nAnd then the men and the boats were gone.\n\nIt was as if they had never existed.\n\nHICKEY\n\n _On the SW Cape of King William Island_\n\n _8 September, 1848_\n\nCaulker's mate Cornelius Hickey hated kings and queens. He thought they were all bloodsucking parasites on the corpusass of the body politick.\n\nBut he found that he did not at all mind _being_ king.\n\nHis plan to sail and row all the way back to Terror Camp or _Terror_ herself went acropper when their pinnace \u2014 no longer so crowded \u2014 rounded the southwest cape of King William Land and encountered advancing ice pack. The open water narrowed to leads which led nowhere or which closed ahead of them even as their boat tried to creep along the coast that now stretched ahead to the northeast.\n\nThere was real open water much farther to the west, but Hickey could not allow the pinnace to be out of sight of land for the simple reason that no one left alive in their boat knew how to navigate at sea.\n\nThe only reason that Hickey and Aylmore had been so generous as to allow George Hodgson to come with them \u2014 actually, to seduce the young lieutenant into wanting to come with them \u2014 was that the fool had been trained, as all Naval lieutenants were, in celestial navigation. But on their first day of man-hauling away from Rescue Camp, Hodgson admitted that he could not fix their position or navigate their way back to _Terror_ at sea without a sextant, and the only remaining sextants were still in the possession of Captain Crozier.\n\nOne of the reasons Hickey, Manson, Aylmore, and Thompson had doubled back and lured Crozier and Goodsir out onto the ice was to somehow get one of those God-damned sextants, but there Cornelius Hickey's native cleverness had failed him. He and Dickie Aylmore failed to come up with any convincing reason that their Judas goat \u2014 Bobby Golding \u2014 could ask Crozier to bring his sextant out onto the ice with him, so they'd discussed torturing that toff Irish bastard into somehow sending a note back demanding the instrument be sent out from camp, but in the end, actually seeing his tormentor on his knees, Hickey had opted to kill him at once.\n\nSo once they found open water, young Hodgson's usefulness, even as a man-hauler, was over, and Hickey soon had to dispatch him in a clean and merciful manner.\n\nIt helped to have Crozier's pistol and extra cartridges for just such a purpose. In the first days after they'd returned with Goodsir and a food supply, Hickey had allowed Aylmore and Thompson to keep the two extra shotguns they'd seized \u2014 Hickey himself had been given the third one by Crozier the day they left Rescue Camp \u2014 but he soon thought better of having the extra weapons around and had Magnus toss them into the sea. This way was better: the king, Cornelius Hickey, having the pistol and control of the only shotgun and its cartridges, with Magnus Manson by his side. Aylmore was an effete, bookish born conspirator, Hickey knew, and Thompson was a drunken lout who could never be fully trusted \u2014 Hickey knew such things by instinct and because of his innately superior intelligence \u2014 and when the Hodgson food supply ran short around the third day of September, Hickey sent Magnus to knock both men on the head, bind them up, and drag them half senseless before the other dozen assembled men where Hickey held a brief court-martial, found both Aylmore and Thompson guilty of sedition and of plotting against their leader and shipmates, and dispatched them both with a single bullet into the base of the brain.\n\nWith all three sacrifices for the greater good \u2014 Hodgson, Aylmore, and Thompson \u2014 the damned surgeon, Goodsir, still refused to fulfill his role as Dissector General.\n\nSo for each refusal, Commander Hickey had been forced to mete out a punishment for the recalcitrant surgeon. There had been three such punishments, so Goodsir was certainly having trouble walking now that they'd been forced ashore again.\n\nCornelius Hickey believed in luck \u2014 his own luck \u2014 and he'd always been a lucky man, but when luck failed him, he was always prepared to make his own.\n\nIn this case, when they'd come around the huge cape at the southwest corner of King William Land \u2014 sailing when they could, rowing hard when the leads grew narrow so close in to shore \u2014 and saw the solid pack ice ahead, Hickey had ordered the ship ashore and they'd reloaded the pinnace onto the sledge.\n\nHe didn't need to remind the men about how lucky they were. While Crozier's men were almost certainly dead or dying back there at Rescue Camp \u2014 or dying on the ice pack in the strait south of it \u2014 Hickey's Chosen Few had made it more than two thirds, and possibly as much as three quarters, of the way back to Terror Camp and all the supplies cached there.\n\nHickey had decided that a leader of his stature \u2014 the reigning King of the Franklin Expedition \u2014 should not be forced to man-haul. The men were certainly being fed well thanks to him (and thanks only to him) and should have no complaints about illness or lack of energy, so for this final part of the voyage he had decided to sit in the stern of the pinnace atop the sledge and to allow his dozen surviving subjects, excluding only the limping Goodsir, to pull him across the ice, gravel, and snow as they rounded the north curve of the cape.\n\nFor the last few days, Magnus Manson had ridden in the pinnace with him, and not simply because everyone now understood that Magnus was the king's consort as well as Grand Inquisitor and Executioner. Poor Magnus was having stomach pains again.\n\nThe primary reason that Goodsir was limping but still alive was that Cornelius Hickey had a deep fear of disease and contagion. The other men's illnesses back at Rescue Camp and before \u2014 the bleeding scurvy especially \u2014 had disgusted and terrified the caulker's mate. He needed a doctor along to attend to him, even though he had not yet shown the slightest sign of illness that so plagued such lesser men.\n\nHickey's sledge team \u2014 Morfin, Orren, Brown, Dunn, Gibson, Smith, Best, Jerry, Work, Seeley, and Strickland \u2014 had also shown no signs of advancing scurvy now that their diet consisted of fresh or almost-fresh meat once again.\n\nOnly Goodsir was looking and acting sick, and that was because the fool insisted on eating only the last few ship's biscuits and water. Hickey knew that he would soon have to step in and _insist_ that the surgeon partake of a healthier antiscorbutic diet \u2014 the fleshy parts such as thigh, calf, and fore- and upper arm were the best \u2014 so that Goodsir did not die on them because of his own perverse stubbornness. A doctor, after all, should know better. Stale ship's biscuits and water might sustain a rat if nothing else were available, but it was not a diet for men.\n\nTo make sure that Goodsir stayed alive, Hickey had long ago relieved the surgeon of all the medicines in his kit, watching over them himself and allowing Goodsir to dole them out to Magnus or others only under careful supervision. He also made sure that the surgeon had no access to knives, and when they were out at sea, he always had one of his men assigned to watch to make sure Goodsir did not throw himself overboard.\n\nSo far, the surgeon had shown no indications of choosing self-murder.\n\nMagnus's stomachache was now severe enough not only to keep the giant riding in the sledge-raised pinnace with Hickey during the day, but to keep him awake some nights. Hickey had never known his friend to have trouble sleeping.\n\nThe two tiny bullet wounds were the cause, of course, and Hickey forced Goodsir to attend to them daily now. The surgeon insisted that the wounds were superficial and that any infection had not spread. He showed both Hickey and the innocently peering Magnus \u2014 holding up his shirttails to peek with alarm at his own belly \u2014 how the flesh around the stomach was still pink and healthy.\n\n\"Then why the pain?\" Hickey insisted.\n\n\"It's like any bruise \u2014 especially a deep-muscle bruise,\" said the surgeon. \"It may continue to hurt for weeks. But it's not serious, much less life-threatening.\"\n\n\"Can you remove the balls?\" asked Hickey.\n\n\"Cornelius,\" whined Magnus. \"I don't want my balls removed.\"\n\n\"I mean the bullets, darling,\" said Hickey, petting the giant's huge forearm. \"The little bullets that are in your belly.\"\n\n\"Perhaps,\" said Goodsir. \"But it would be better if I did not try. At least while we are on the march. The operation would require cutting through muscle that has already largely healed. Mr. Manson might have to lie down for several days of recovery... and there would always be the serious risk of sepsis. If we were to decide to remove the bullets, I would feel much more comfortable doing so at Terror Camp or when we are back at the ship. So the patient could recover in bed for several days or longer.\"\n\n\"I don't want my tummy to hurt,\" rumbled Magnus.\n\n\"No, of course you don't,\" said Hickey, rubbing his partner's huge chest and shoulders. \"Give him some morphine, Goodsir.\"\n\nThe surgeon nodded and meted out a bit of the painkiller into a spoon.\n\nMagnus always enjoyed his spoonfuls of morphine and would sit in the bow of the pinnace and smile sweetly for an hour or more before falling asleep after getting his doses.\n\nSo on this Friday, the eighth day of September, all was right with King Hickey's world. His eleven dray animals \u2014 Morfin, Orren, Brown, Dunn, Gibson, Smith, Best, Jerry, Work, Seeley, and Strickland \u2014 were well and free of disease and pulling hard each day. Magnus was happy most of the time \u2014 he enjoyed riding in the bow like an officer and looking back at the countryside they'd just crossed \u2014 and there was enough morphine and laudanum in the bottles to hold out until they reached Terror Camp or _Terror_ herself. Goodsir was alive and limping along with the caravan and attending to the king and his consort. The weather was good, although growing colder, and there was absolutely no sign of the creature that had preyed on them in previous months.\n\nEven with their vigorous diet, they had enough Aylmore and Thompson food stores left to provide stew over the next few days \u2014 they had found that human fat burned as fuel much as did whale blubber, although less efficiently and for shorter periods. Hickey had plans for a lottery after that if they needed one more sacrifice before they reached Terror Camp.\n\nThey could go on shorter rations, of course, but Cornelius Hickey knew that a short-straw lottery would instill terror into the hearts of his eleven already-compliant dray animals and reaffirm who was king of this expedition. Hickey was always a light sleeper but now slept with one eye open and his hand on the percussion-cap pistol, but one last public sacrifice \u2014 presumably with Magnus then having to dole out the fourth public punishment for noncompliance to Goodsir \u2014 should break any last hidden will to resist that might be left in his dray beasts' treacherous hearts.\n\nMeanwhile, this Friday was beautiful, with temperatures in the pleasant twenties and a blue sky growing bluer to the north along their line of travel. The heavy boat sat high on the sledge while the wooden runners scratched and hissed as they slid across ice and gravel. In the bow, Magnus, recently dosed, was smiling, holding his belly with both hands and humming a soft tune.\n\nIt was less than thirty miles to Terror Camp and John Irving's grave near Victory Point, they all knew, and less than half that to Lieutenant Le Vesconte's grave along the coast. With the men strong, they were covering two to three miles each day and would probably do better if their diet improved again.\n\nTo that end, Hickey had just torn a blank page out of one of the multiple Bibles that Magnus had insisted upon gathering up and loading into the pinnace when they left Rescue Camp \u2014 never mind that the gentle idiot did not know how to read \u2014 and was now tearing that page into eleven equal little strips of paper.\n\nHickey, of course, would be exempt from the coming lottery, as would Magnus and the God-damned surgeon. But tonight, when they stopped to brew up tea and the evening stew, Hickey would have each man write his own name or put his sign on one of the slips of paper and all would be ready for the lottery itself. Hickey would have Goodsir look the slips over and publicly confirm that each man had signed his own true name or unique sign.\n\nThen the names would go into the king's peacoat pocket in preparation for the solemn ceremony to come.\n\nGOODSIR\n\n_On the SW Cape of King William Island_\n\n_5 October, 1848_\n\nFrom the personal diary of Dr. Harry D. S. Goodsir:\n\n_6, 7, or perhaps 8 October, 1848 \u2014_\n\n_I have taken the Final Draught. It will be a Few Minutes before the Full Effect is Felt. Until it Is, I shall Catch Up on my diary._\n\n_These Last Few Days I have been recalling the Details of how young Hodgson confided in me and Whispered to me in the tent Weeks ago on that Last Night before Mr. Hickey shot him._\n\n_The Lieutenant whispered,_ I apologize for Disturbing you, Doctor, but I have to tell Someone I am Sorry.\n\n_I whispered back,_ You are not a Papist, Lieutenant Hodgson. And I am Not your Confessor. Go to Sleep and let me Sleep.\n\n_Hodgson Insisted,_ I apologize again, Doctor. But I have to tell someone how Sorry I am for Betraying the Captain \u2014 who was always Good to Me \u2014 and for Allowing Mr. Hickey to take you Captive like This. I sincerely Regret it and I am Dreadfully Sorry.\n\n_I Lay there Silently, Saying nothing, Giving the boy nothing._\n\nEver Since John was killed, _persisted Hodgson._ I mean, Lieutenant Irving, my Dear Friend from Gunnery School, I have been Convinced that Caulker's Mate Hickey committed the Murder and I have been Terrified of Him.\n\nWhy would You Throw in Your Lot with Mr. Hickey if you thought him a Monster? _I whispered in the Dark._\n\nI was... Afraid. I wanted to be on His Side because he was so Terrible, _whispered Hodgson. And then the Boy began to Weep._\n\n_I said,_ Shame on you.\n\n_But I put my Arm around the Boy and patted his Back while he Wept until he fell Asleep._\n\n_The Next Morning, Mr. Hickey assembled Everyone and had Magnus Manson force Lieutenant Hodgson to kneel before Him while the Caulker's Mate brandished his Pistol and Announced how He \u2014 Mr. Hickey \u2014 would Brook no Shirking, explaining again How the Good Men Amongst Us would eat and live while the Shirkers would Die._\n\n_Then he set the long-barreled Weapon to the base of George Hodgson's skull and Blew his Brains out onto the Gravel._\n\n_I have to say that the Boy was Brave at his end. He showed no Fear at all that Morning. His last words before the Pistol's Explosive Discharge were, You can go to Hell._\n\n_I only wish that my End would be so Brave. But I know now for a Certainty that it Will Not._\n\n_Mr. Hickey's Theatricals were not at an End with the Death of Lt. Hodgson, nor when Magnus Manson stripped the Boy Naked and left his Corpse Lying there in front of the Assembly._\n\n_The Sight made my Chest hurt. Speaking as a Man of Medicine, poor Hodgson was Thinner than I would have Thought Possible with any Recently Living Human Being. His Arms were mere Sheaths of Skin along Bones. His Ribs and Pelvis pressed Outward so Fiercely against the Skin that they threatened to Burst Through. And everywhere, the Boy's flesh was Mottled with Bruises._\n\n_Nonetheless, Mr. Hickey called me Forward, handed me a Pair of Shears, and insisted that I Begin Dissecting the Lieutenant in front of the Assembled Men._\n\n_I demurred._\n\n_Mr. Hickey, his voice Pleasant, asked Again._\n\n_I demurred Again._\n\n_Mr. Hickey then commanded Mr. Manson to take the Shears from me and to Strip me as Naked as the Corpse at our Feet._\n\n_Once I was Without Clothing, Mr. Hickey paced back and forth in front of the men and Pointed to my Naked Features. Mr. Manson stood nearby holding the Shears._\n\nThere ain't no Room for Shirkers in our Band of Brothers _\u2014 said Mr. Hickey._ And while we _need_ this Surgeon _\u2014_ for I do Plan to Take care of your Dear Men's health, every Man Jack of you \u2014 he must be Punished when he Refuses to Serve our Common Good. Twice he Has Refused this Morning. We shall Remove Two Unessential Appendages as a Sign of Our Displeasure.\n\n_And with that, Mr. Hickey Proceeded to prod at Different Parts of my Anatomy with the Pistol Barrel \u2014 my Fingers, my Nose, my Penis, my Testicles, my Ears._\n\n_Then he Raised my Hand._\n\nA Surgeon _needs '_ is Fingers if he's going to be any Use to us _\u2014 he announced Theatrically and Laughed._ We'll save those for Last.\n\n_Most of the Men Laughed._\n\nHe don't need his Pizzle nor Bollocks, though, _said Mr. Hickey, prodding at the Aforementioned Parts with his Very Cold Pistol Barrel._\n\n_The Men Laughed again. The Anticipation, I think, was very high._\n\nBut today we is Merciful, _said Mr. Hickey. He then ordered Mr. Manson to lop off Two of my Toes._\n\nWhich two, Cornelius? _asked the large Idiot._\n\nYou choose, Magnus, _said our Master of Ceremonies._\n\n_The Assembled Men laughed yet Again. I could Sense their Disappointment that something so Banal as Mere Toes were being Removed, yet I could also Tell that they enjoyed seeing Magnus Manson as the Master of My Phalangeal Fate. It was not their Fault. The Average Seaman Turned out Here had no Formal Education Whatsoever and Disliked anyone who did._\n\n_Mr. Manson Chose my Two Big Toes._\n\n_The Audience laughed and applauded._\n\n_The Shears were Applied quickly and Mr. Manson's great Strength worked to my Advantage in the Procedure._\n\n_There was more Laughter \u2014 and great Interest \u2014 as my Medical Kit was brought and everyone watched as I Tied Off necessary Arteries, Stemmed the Bleeding as Best I could \u2014 all the while feeling rather Faint \u2014 and applied Preliminary Dressings to the Wounds._\n\n_Mr. Manson was directed to Carry me back to my Tent; his Ministrations were as Gentle as a Mother's to a sick Child._\n\n_That was also the Day when Mr. Hickey thought to Relieve me of My more Efficacious Medicinal Bottles. But before that Morning, I had Already Poured the Majority of the Morphine, Opium, Laudanum, Dover's Powders, poisonous mercury Calomel, and Mandragora into a single Opaque and Innocent-Looking Bottle marked Sugar of Lead and hidden that somewhere Other than my Medical Kit. I had then used water to bring the Visible Levels of Morphine, Opium, and Laudanum up to previous Heights._\n\n_The Irony here is that each time I Dose Mr. Manson for his \"Tummy Aches,\" he is receiving more than Eight Parts water to Two Small Parts morphine. The Giant does not seem to notice the Loss of Efficacy, however, which once again reminds me of the Importance of Belief in the entire Medical process._\n\n_Since that Day of Lt. Hodgson's Demise, I have Demurred again to the Sum of Eight more Toes, One Ear, and my Foreskin._\n\n_The Last Operation created so much Mirth among the Assembled Men, despite the fresh Corpses lying in front of them, that one would have Thought that the Circus had come to Perform for them._\n\n_I know why Mr. Hickey has never made Good on his repeated Threats to relieve me of my Male Member or Testicles. The Caulker's Mate has seen enough Shipboard injuries to know that Bleeding from such Wounds often cannot be Stopped \u2014 especially when the Surgeon is the one bleeding and Quite Apt to be unconscious or suffering from Shock when the Operation must Necessarily be Performed \u2014 and Mr. Hickey does not want me dead._\n\n_Walking has been very Difficult since my Seventh through Tenth toes have been Removed. I had never truly Understood how Essential our Digits are for Balance. And the Pain, of course, over the past Month, has not been Insignificant._\n\n_I think I would be Committing the Sin of Pride \u2014 not to mention that of Lying \u2014 if I said Here that I had not considered Drinking from my hidden bottle of Morphine, Opium, and Laudanum (and other materia medica) all mixed into the hidden bottle I have Thought of for so many Weeks as my Final Draught._\n\n_But I never took the Bottle out of hiding._\n\n_Not until this Hour._\n\n_I Confess I had thought the Effect would be more Rapid than it is Proving._\n\n_I can no Longer feel my Feet \u2014 which is a Blessing \u2014 and my Legs have just gone Numb up to the Patella. But at this Rate, it will be another Ten Minutes or More before the Potion reaches and Stills my Heart and other Vital Organs._\n\n_I have just Drunk more of the Final Draught. I suspect I was a Coward for not Drinking it all down At Once to begin with._\n\n_I confess here \u2014 for Purely Scientific Purposes should someone someday discover this Diary \u2014 that the Mixture is not only Quite Potent but Quite Intoxicating. If anyone else here were alive this dark, stormy Afternoon \u2014 except for Mr. Hickey and possibly Mr. Manson up in their Throne Pinnace \u2014 they should see my Last Moments spent with Bobbing Head and Drunkard's Grin._\n\n_But I do not Recommend that this Experiment be Repeated for anything but the most Dire of Medicinal Purposes._\n\n_And this leads to a true Confession._\n\n_For the First and Only Time in my Medical Career and Life, I have not Served a Patient to the Utmost of my Ability._\n\n_I speak, of course, in regards to poor Mr. Magnus Manson._\n\n_My Initial Diagnosis of the twin Gunshot Wounds was a Lie. The Bullets were small of caliber, it is True, but the Tiny Pistol must have packed a Great Charge of Powder, for both Projectiles had \u2014 it was Obvious from my first Inspection \u2014 penetrated the Idiot Giant's skin, flesh, muscle layer, and stomach lining._\n\n_From my first Consultation, I had known that the Bullets were in Mr. Manson's Belly, Spleen, Liver, or some other Vital Organ, and that his Survival Depended upon Exploratory and then Removal Surgery._\n\n_I Lied._\n\n_If there is a Hell \u2014 in which I no longer Believe, since this Earth and some of the People in it are Hell enough for any Universe \u2014 I would be and should be Cast Down to the Worst Bolgia of the Lowest Circle._\n\n_I Don't Care._\n\n_I should say here \u2014 my Chest is now Cold and my Figners... fInGErS are also growing Cold._\n\n_When the Storm STRk about one Monf ago, I thank'd Gd._\n\n_It seemd at the Tim that we wer Actully going to Gt to Terrorr Camp. It Seemed that Mr. Hickey had won. We were \u2014 I believ \u2014 less than Twenty MIls frm thatt Camp and Pogrssing 3 or 4 miiles a Day in nar-Perfect wether when the Fist of The Enlesss Storms Hit._\n\n_If there is a Godd... I... thank you, Deaare God._\n\n_Snowe. DAarknss. Terrrible winds Day and Nigt._\n\n_Even the Men who could Wlk cld not Pulll. The Harneesess were Abandoned. The Tents blew dwno, then bleww away. T he tempretre Droppped 50 degres._\n\n_Winter hd Strukc like Gd's Hammmer, and Mr. Hickey cuoold do Nothing but set Tarps aside his Thronepinacce and Shoot Half the Men to Feed the Other Half._\n\n_Some Men ran away into the Bolizzardds and Died._\n\n_Some Men stayed and were Shot._\n\n_Sme M Froze to Deth._\n\n_Sm Men Ate theother mn and Died Anwwyay._\n\n_Mr. Hickey and Mr. Masnsonn sit up There in Ther Boat in the Wind. I thinge, but donnot Know, that Mrr. Mansin is No Lngr Livvng._\n\n_OI killed him._\n\n_I kelled the Men I lff behing at Rescue Camp._\n\n_I am so Sorry._\n\n_I am so Sorry._\n\n_All my lfe, my Brother knows I wish my brther werehere now, Thmoshe knws, al my lifI hve lved Plato and the Dialogues of Sokrates._\n\n_Like the grete Sokates, but not rf not grete I, the Poisoin, mcuh Deservd, movs up throu my Torso and Deadeens my Limbs and Turns my Fingrs \u2014 Surgeons fingers \u2014 to Unfeellling Sticks and_\n\n_So glad_\n\n_Wrote the note nw pined to my Cheset befre this_\n\n**  \nEAT THESE MORTAL REMNAS OF DR HARRY D.S. GOOODSIRIFFF YO U WISSSH**\n\n**THE POISSSSN WITHINN THS BONES AND FELSH WIOL KILL YOOU ALSO**\n\n_TheMen at Re cm_\n\n_Thomnas, if they Find this Upon my and Ret_\n\n_I am So Sorry._\n\n_I did My Best But never is en_\n\n_Mr. Msnsns Wonds I AM NOT S_\n\n_Gd wac ov Th MEn_\n\nHICKEY\n\n_On the SW Cape of King William Island_\n\n_18 October, 1848_\n\nSometime in the last few days or weeks, Cornelius Hickey realized, he had ceased being a king.\n\nHe was now a god.\n\nIn fact \u2014 he suspected, was not yet certain, but suspected strongly and was close to certain \u2014 Cornelius Hickey had become God.\n\nOthers died around him yet he lived. He no longer felt the cold. He no longer felt hunger or thirst, much less the need to slake those former appetites. He could see in the encroaching dark as the nights lengthened toward the absolute, nor did the blowing snow and howling wind hinder his senses.\n\nThe mere mortal men had required a rigging of a tarp from the boat and sledge when their tents ripped and blew away and they huddled there like sheep with their woolen asses turned to the wind until they died, but Hickey was comfortable high on his throne in the stern of the pinnace.\n\nWhen, after more than three weeks of being unable to move because of the blizzards, winds, and plummeting temperatures, his dray beasts had whined and begged for food, Hickey had descended among them like a god and provided them with their loaves and fishes.\n\nHe had shot Strickland to feed Seeley.\n\nHe had shot Dunn to feed Brown.\n\nHe had shot Gibson to feed Jerry.\n\nHe had shot Best to feed Smith.\n\nHe had shot Morfin to feed Orren... or perhaps it was all the other way around. Hickey's memory could no longer be bothered with trivial matters.\n\nBut now those he'd so generously fed were dead, frozen hard into their blanket sleeping bags or contorted into the terrible claw shapes of their final throes. Perhaps he had become bored with them and shot them as well. He did vaguely remember carving up the choice parts of more men than he had shot to feed the others in the past week or two, back when he still needed to eat. Or perhaps it had just been on a whim. He could not recall the details. It was not important.\n\nWhen the storms ended \u2014 and Hickey now knew that He could command them to cease at any time if it pleased Him to do so \u2014 he would probably bring several of the men back from the dead so that they could finish hauling Magnus and Him to Terror Camp.\n\nThe damned surgeon was dead \u2014 poisoned and frozen in his own little tarp tent some yards from the pinnace and the common graveyard tarp \u2014 but Hickey chose to ignore that unpleasant development \u2014 it was but a mild irritation. Even gods have phobias, and Cornelius Hickey had always held a deep fear of poison or contamination. After one glance \u2014 and after firing a single bullet into the corpse from the entrance to the tarp tent to make sure the damned surgeon was not feigning death \u2014 the new god Hickey had backed away and left the poisoned thing and its contaminated shroud-tarp alone.\n\nMagnus had been mewling and complaining for weeks from his favored place in the bow but had been strangely quiet the last day or two. His last movement, during a lull in the blizzards when a dull winter light had illuminated the pinnace and the snow-buried tarp next to it and the low hill they were on and the frozen beach to the west and endless ice fields beyond, had been to open his mouth as if to make a request of his lover and God.\n\nBut instead of words issuing forth, or even another complaint, hot blood had first filled and then geysered from Magnus's open mouth, flowed down his bearded chin, and covered the big man's belly and gently folded hands, ending in a pool on the bottom of the boat near his boots. The blood was still there, but frozen now into waves and ripples, looking like nothing so much as some Biblical Prophet's flowing (but ice-covered) brown beard. Magnus had not spoken again since.\n\nHis partner's brief Death Nap did not disturb Hickey \u2014 he knew that He could bring Magnus back whenever He chose to \u2014 but the open eyes endlessly staring over that gaping mouth and frozen icefall of blood began to get on the god's nerves after a day or two. It was especially hard to wake up to. Especially after the eyes frosted over and became two white, icy, never-blinking orbs.\n\nHickey had stirred from his throne in the stern then, crawling forward past the propped shotgun and bag of powder-shot cartridges, over the centre thwarts past the heaps of wrapped chocolate (which He might deign to eat if hunger ever returned) and past the saws and nails and rolls of sheet lead, stepping over the towels and silk handkerchiefs stacked so neatly near Magnus's bloodied feet, finally kicking aside some of the Bibles that his friend had pulled close to him in the last days, stacking them like a little wall between Hickey and himself.\n\nBut Magnus's mouth would not shut \u2014 Hickey could not even snap off or chip away the thick river of frozen blood \u2014 nor would the white eyes close.\n\n\"I'm sorry, love,\" he whispered. \"But you know how I hate being stared at.\"\n\nHe had used his ship's knife to pry out the frozen eyeballs and throw them far out into the howling darkness. He would fix that later when He brought Magnus back.\n\nFinally, upon His command, the storm lessened and then died away. The howling ceased. The snow was piled five feet high on the westward, windward side of the pinnace high atop its sledge and had filled in much of the space under the death tarp on the leeward side.\n\nIt was very cold and Hickey's preternatural vision could see more dark clouds moving in from the north, but for this evening, the world was calm. He saw the sun set in the south and knew that it would be sixteen or eighteen hours until it rose again, also in the south, and that soon it would not rise at all. It would then be the Age of Darkness \u2014 ten thousand years of darkness \u2014 but that suited Cornelius Hickey's purposes well.\n\nBut this night was cold and gentle. The stars were bright \u2014 Hickey had been taught the names of some of the winter constellations now rising, but this night he had trouble even finding the Plow \u2014 and He was content to sit in the stern of his boat, his peacoat and watch cap keeping him perfectly warm, his gloved hands on the gunwales, his gaze locked forward in the direction of Terror Camp and even the distant ship He would reach when He chose to bring his dray beasts and consort back to life. He was thinking about months and years past and marveling at the inevitable miracle of his own transcendence.\n\nCornelius Hickey had no regrets about any part of his former mortal life. He had done what he had to do. He had repaid those arrogant bastards who made the mistake of ever looking down upon him and shown the others a hint of his divine light.\n\nSuddenly, he sensed movement to the west. With some difficulty \u2014 it was very cold \u2014 Hickey turned his head left to look out to the frozen sea.\n\nSomething was moving toward him. Perhaps it had been his hearing \u2014 as preternatural and supernatural as all his other fine-tuned and augmented senses now \u2014 that had first detected the movement across the broken ice.\n\nSomething large was walking toward him on two legs.\n\nHickey saw the starlight glow on the blue-white fur. He smiled. He welcomed the visit.\n\nThe thing from the ice was no longer something to be feared. Hickey knew that it came now not as a predator but as a worshipper. He and the creature were not even equals at this point; Cornelius Hickey could order it into nonexistence or banish it to the farthest reaches of the universe with a sweep of his gloved hand.\n\nIt came on, sometimes dropping to lope forward on all fours, more often rising on two huge legs and striding like a man even while moving nothing like a man.\n\nHickey felt a strange disquiet disturb his deep cosmic peace.\n\nThe thing disappeared from his sight when it came very close to the pinnace and sledge. Hickey could hear it moving around by the tarp \u2014 under the tarp \u2014 worrying the frozen bodies there with its long claws, clicking teeth the size of knives, huffing its breath out from time to time \u2014 but he could not see it. He realized that he was afraid to turn his head.\n\nHe looked straight forward, meeting only Magnus's empty-eye-socketed gaze.\n\nThen suddenly the thing was there, looming over the gunwales, the upper body rising six feet and more above a boat that was already raised six feet above the sledge and snow.\n\nHickey felt his breath catch in his chest.\n\nIn the starlight, with Hickey's new, improved vision, the beast was more terrible than he had ever seen it, more terrible than he could have ever imagined it. Just as He \u2014 Cornelius Hickey \u2014 had undergone a wonderful and terrible transformation, so had this creature.\n\nIt leaned its huge upper body over the gunwales. It huffed a fog of ice crystals into the air between Hickey and the bow and the caulker's mate inhaled the carrion breath of a thousand centuries of death-dealing.\n\nHickey would have fallen to his knees and worshipped the creature at that moment if movement had been an option, but he was quite literally frozen in place. Even his head would no longer turn.\n\nThe thing sniffed Magnus Manson's body, the long, impossible snout returning again and again to the icefall of brown blood covering Magnus's front. Its huge tongue gently licked at the frozen fall of brown blood. Hickey wanted to explain that this was the body of his beloved consort and that it must be preserved so that He \u2014 not Hickey the caulker's mate, but the He he had become \u2014 could restore his beloved's eyes and someday breathe life into him again.\n\nAbruptly, yet almost casually, the thing bit off Magnus's head.\n\nThe crunching was so terrible that Hickey would have covered his ears if he had been able to lift his gloved hands from the gunwales. He could not move them.\n\nThe thing swung a white-furred forearm thicker than Magnus's massive leg had ever been and smashed the dead man's chest in \u2014 rib cage and spine exploding outward in a shower of white bone shards. Hickey realized that the thing had not _broken_ Magnus the way Hickey had seen Magnus break a score of lesser men's backs and ribs; it had _shattered_ Magnus the way a man would shatter a bottle or porcelain doll.\n\n_Looking for a soul to devour,_ thought Hickey, who had no idea why he had thought it.\n\nHickey could no longer move his head even an inch, so he had no choice but to watch as the thing from the ice excavated every inner part of Magnus Manson and ate them, crunching the bits in its huge teeth the way Hickey might have once chewed ice cubes. The thing then tore the frozen flesh from Magnus's frozen bones and scattered the bones throughout the bow of the pinnace, but only after cracking them open and sucking out the marrow. The wind came up and howled around the pinnace and sledge, creating distinct musical notes. Hickey imagined a mad god-thing from Hell in a white fur coat playing a bone flute.\n\nIt came for him next.\n\nFirst it dropped to all fours, out of sight \u2014 which was somehow more terrifying than his being able to see it \u2014 and then, with a vertical motion like a pressure ridge rising, it loomed up and over the side of the gunwale and filled all of Hickey's vision. Its black, unblinking, inhuman, totally unfeeling eyes were inches from the caulker's mate's own staring eyes. Its hot breath enveloped him.\n\n\"Oh,\" said Cornelius Hickey.\n\nIt was the last word that Hickey ever spoke, but it was not so much a word as a single, long, terrified, speechless exhalation. Hickey felt his own last warm breath flowing out of him, out from his chest, up his throat, out through his open and straining mouth, hissing away between his shattered teeth, but instantly he realized it was not his _breath_ leaving him forever, but his spirit, his soul.\n\nThe thing breathed it in.\n\nBut then the creature huffed, snorted, backed away, shook its huge head as if it had been befouled. It dropped to all fours and left Cornelius Hickey's field of vision forever.\n\n_Everything_ had left Cornelius Hickey's field of vision forever. The stars came down from the sky and attached themselves to his staring eyes as ice crystals. The Raven descended as a darkness upon him and devoured what the _Tuunbaq_ would not deign to touch. Eventually Hickey's blind eyes shattered from the cold, but he did not blink.\n\nHis body remained sitting rigidly upright in the stern, legs splayed, boots firmly planted near the heap of gold watches he had plundered and the stack of clothing he had taken from the dead men, his gloved hands frozen to the gunwales, the frozen fingers of his right hand only inches from the loaded shotgun's barrels.\n\nLate the next morning, before dawn, the storm front arrived and the sky began to howl again, and all that next day and all the next night the snow piled up in the caulker's mate's straining, open mouth and covered his dark blue peacoat and watch cap and terror-frozen face and shattered, staring eyes with a thin shroud-layer of white.\n\nCROZIER\n\nThe beauty of being dead, he knows now, is that there is no pain and no sense of self.\n\nThe unhappy news about being dead, he knows now, is \u2014 just as he had feared many times when considering self-murder and rejecting it for just this reason \u2014 there are dreams.\n\nThe happy news about this unhappy news is that the dreams are not one's own.\n\nCrozier floats in this warm, buoyant sea of nonself and listens to dreams that are not his own.\n\nIf any of his living, mortal-self's analytical powers had survived the transition to this pleasant floating-after-death, the old Francis Crozier might have wondered at his thought of \"listening to\" dreams, but it is true that these dreams are more like listening to another person's chant \u2014 although there is no language involved, no words, no music, no chant \u2014 than \"seeing\" dreams the way he always had when he was alive. Although there are most definitely visual images involved in this dream-listening, the shapes and colors are like nothing Francis Crozier ever encountered on the other side of Death's veil and it is this nonvoice, nonchant narrative that fills his death dreams.\n\nThere is a beautiful Esquimaux girl named Sedna. She lives alone with her father in a snow-house far north of the regular Esquimaux villages. Word of the girl's beauty spreads and various young men make the long trek across ice floes and barren lands to pay homage to the grey-haired father and to woo Sedna.\n\nThe girl's heart is not touched by any of the suitors' words or faces or forms, and in the late spring of the year, when the ice is breaking up, she goes out alone among the floes to avoid yet another year's fresh crop of moonfaced suitors.\n\nSince this happened in the time when animals still had voices which the People understood, a bird flies over the opening ice and woos Sedna with its song. \"Come with me to the land of the birds where all things are as beautiful as my song,\" sings the bird. \"Come with me to the land of the birds where there is no hunger, where your tent will always be made of the most beautiful caribou skins, where you shall lie on only the finest and softest bearskins and caribou skins, and where your lamp will always be filled with oil. My friends and I will bring you anything your heart may desire, and you shall be clothed from that day forth in our finest and brightest feathers.\"\n\nSedna believes the bird-suitor, weds him in the tradition of the Real People, and travels with him many leagues over sea and ice to the land of the bird people.\n\nBut the bird had lied.\n\nTheir home is not made of the finest caribou skins but is a patched, sad place thrown together with rotting fish skins. The cold wind blows in freely and laughs at her for her gullible innocence.\n\nShe sleeps not on the finest bearskins but on miserable walrus hides. There is no oil for her lamp. The other bird people ignore her and she has to wear the same clothes she was wed in. Her new husband brings her only cold fish for her meals.\n\nSedna keeps insisting to her indifferent bird-husband that she misses her father, so finally the bird allows her father to come visit. To do so, the old man has to travel for many weeks in his frail boat.\n\nWhen her father arrives, Sedna feigns joy until they are alone in the dark, fish-stinking tent, and then she weeps and tells her father of how her husband abuses her and of all she has lost \u2014 youth, beauty, happiness \u2014 by marrying the bird rather than one of the young males of the Real People.\n\nThe father is horrified to hear this story and helps Sedna devise a plan to kill her husband. That next morning, when the bird-husband returns with Sedna's cold fish for breakfast, the father and the girl fall upon the bird with the harpoon and paddle from the father's kayak and kill him. Then the father and daughter flee the land of the bird people.\n\nFor days they sail south toward the land of the Real People, but when the bird-husband's family and friends find him dead, they are filled with anger and fly south with a beating of wings so loud that it can be heard by the Real People a thousand leagues away.\n\nThe sea distance that took Sedna and her father a week to sail is covered by the thousands of flying birds in a few minutes. They descend upon the little boat like a dark and angry cloud made up of beaks and talons and feathers. The beating of their wings calls up a terrible storm that raises the waves and threatens to swamp the little boat.\n\nThe father decides to give his daughter back to the birds as an offering and throws her overboard.\n\nSedna clings to the boat for dear life. Her grip is strong.\n\nThe father takes his knife and cuts off the first joints of her fingers. As they fall into the sea, these finger joints are turned into the first whales. The fingernails become the white whalebone found on beaches.\n\nStill Sedna clings. The father cuts off her fingers at the second joint.\n\nThese parts of her fingers fall into the sea and become the seals.\n\nStill Sedna clings. When the terrified father cuts off the final stumps of her fingers, these fall onto the passing floes and into the water and become the walruses.\n\nWith no fingers left, only curved bone stumps like her dead bird-husband's talons where her hands had been, Sedna finally falls into the sea and sinks to the bottom of the ocean. She resides there until this day.\n\nIt is Sedna who is the mistress of all whales, walruses, and seals. If the Real People please her, she sends the animals to them and tells the seals, walruses, and whales to allow themselves to be caught and killed. If the Real People displease her, she keeps the whales, walruses, and seals with her down in the dark depths and the Real People suffer and starve.\n\n _What in the God-damned hell?_ thinks Francis Crozier. It is his self-voice that interrupts the slow no-self flow of the dream-listening.\n\nAs if summoned, the pain rushes in.\n\nCROZIER\n\n _M y men!_ he shouts. But he is too weak to shout it. He is too weak to say it out loud. He is too weak even to remember what the two syllables mean. _My men!_ he cries again. It emerges as a moan.\n\nShe is torturing him.\n\nCrozier does not awaken all at once but rather comes awake through a series of painful attempts to open his eyes, stitching together separate tatters of attempted awareness stretching over hours and even days, always propelled up out of death-sleep by pain and by the two empty syllables \u2014 _my men!_ \u2014 until he is, at last, conscious enough to remember who he is and to see where he is and to realize who he is with.\n\nShe is torturing him.\n\nThe Esquimaux girl-woman he had known as Lady Silence keeps cutting into his chest, arms, side, back, and leg with a sharp, heated knife. The pain is incessant and intolerable.\n\nHe is lying near her in a small space \u2014 not a snow-house as John Irving had described to Crozier, but some sort of tent made of skins stretched over curved sticks or bones \u2014 with flickering light from several small oil lamps illuminating the girl's bare upper body and, when he looks down, Crozier's own bare and torn and bleeding chest and arms and belly. He thinks she must be slicing him into small strips.\n\nCrozier tries to scream but finds again that he is too weak to scream. He tries to bat her torturing arm and knife-hand away, but he is too weak to lift his own arm much less stop hers.\n\nHer brown eyes stare into his, acknowledging that he is alive again, and then return to studying the damage her knife is doing as she cuts and slashes and tortures him.\n\nCrozier manages the weakest of moans. Then he falls away into darkness, but not back into dream-listening and the pleasant no-self which he now only half remembers, but only into black wave-surges in a sea of pain.\n\nShe feeds him some sort of broth from one of the emptied Goldner tins she must have stolen from _Terror._ The broth tastes of some sea animal's blood. She then cuts strips of seal meat and blubber using a strange curved blade with an ivory handle, holding the slab of seal in her teeth and slicing dangerously close to her lips as she cuts downward, then chews the pieces well, finally pressing them between Crozier's chapped and torn lips. He tries to spit them out \u2014 he does not want to be fed like some baby bird \u2014 but she retrieves each fatty blob and presses it back into his mouth. Defeated, unable to fight her, he finds the energy to chew and swallow.\n\nThen he falls back to sleep to the lullaby of howling wind but is soon awakened. He realizes that he is naked between furred sleeping robes \u2014 his clothes, all his many layers, are not in the little tent space \u2014 and that she has rolled him onto his belly now, setting some sort of smooth sealskin beneath him to keep the blood from his lacerated chest from soiling the soft hides and furs that cover the tent floor. She is cutting and probing his back with a long, straight blade.\n\nToo weak to resist or roll over, all Crozier can do is moan. He imagines her slicing him to pieces and then cooking and eating the pieces. He feels her pressing strands of something moist and slimy onto and into the many wounds in his back.\n\nAt some point in the torture, he falls asleep again.\n\n _My men!_\n\nIt is only after several days of this pain and of slipping constantly into and out of consciousness and of thinking that Silence is slicing him to pieces that Crozier remembers being shot.\n\nHe awakens with the tent dark except for a tiny amount of moonlight or starlight seeping through the tight-stretched hides. The Esquimaux girl is sleeping next to him, sharing his body heat even as he shares hers, and both of them are naked. Crozier feels not the slightest stir of passion or physical interest beyond his animal need for warmth. He is in too much pain.\n\n _My men! I must get back to my men! Warn them!_\n\nFor the first time, he remembers Hickey, the moonlight, the gunshots.\n\nCrozier's arm is lying across his chest and now he forces his hand to touch higher, where the shotgun pellets had struck his chest and shoulder. His upper left torso is a mass of welts and wounds, but it feels as if the shotgun pellets and any clothing driven into his flesh with them have been carefully dug out. There is something soft like moistened moss or seaweed pressed into the larger wounds, and while Crozier has the impulse to dig it out and throw it away, he does not have the strength.\n\nHis upper back hurts even more than his lacerated chest and Crozier remembers the torture as Silence dug there with her knife blade. He also remembers the slight squelching sound after Hickey pulled the trigger but before the shotgun cartridges fired \u2014 the powder had been wet and old and both shots had probably ignited with far less than full explosive force \u2014 but he can also recall the impact of the outer part of the widening pellet cloud hurling him around and then down onto the ice. He had been shot once from the back with the shotgun at extreme range and once from the front.\n\n _Has the Esquimaux girl dug out every pellet? Every shred of filthy clothing driven into me?_\n\nCrozier blinks in the dimness. He remembers visiting Dr. Goodsir's sick bay and the surgeon's patient explanations of how, in Naval warfare as well as with most of the wounds suffered on their expedition, it was usually not the initial wound that killed but the sepsis from the contaminated wounds that set in later.\n\nHe moves his hand slowly from his chest to his shoulder. He remembers now that after the shotgun blasts, Hickey then shot him several times with Crozier's own pistol and the first bullet had struck... _here._ Crozier gasps as his fingers find a deep groove in the flesh of his upper biceps. It is packed with the moldy, slimy stuff. The pain of touching it makes him dizzy and ill.\n\nThere is another groove from a bullet along his left rib. Touching that \u2014 just moving his hand that far exhausts him \u2014 makes him gasp aloud and black out for a moment.\n\nWhen some consciousness returns, Crozier realizes that Silence has dug a bullet out of his flesh there in his side and also dressed this wound with whatever heathenish poultice she had applied elsewhere on his body. Guessing from the pain when he breathes and from the soreness and swelling in his back, he thinks that this bullet broke at least one rib on his left side, was deflected, and lodged under the skin near his left shoulder blade. Silence must have extracted it from there.\n\nIt takes endless minutes and the rest of his meager energy for him to lower his hand to touch his most painful wound.\n\nCrozier does not remember being shot in the left leg, but the pain from the muscle there, just above and under his knee, convinces him that a third bullet must have passed through at that point. He can feel both the entrance and exit holes under his shaking fingers. Two inches higher and the bullet would have taken his knee, the knee would have cost him his leg, and his leg would almost certainly have meant his life. Again there is a poultice-bandage there, and although he can feel scabs, there seems to be no fresh flow of blood.\n\n _No wonder I'm burning up from fever. I'm dying of sepsis._\n\nThen he realizes that the heat he feels may not be fever. These robes insulate so well and Lady Silence's naked body next to his is pouring out so much heat that he is completely warm for the first time in... how long? Months? Years?\n\nWith great effort, Crozier pushes back the top of the robe that covers both of them, allowing a little cooler air in.\n\nSilence stirs but does not waken. Staring at her in the dim light in the tent, he thinks she looks like a child \u2014 perhaps like one of his cousin Albert's younger teenaged daughters.\n\nWith this thought in mind \u2014 remembering playing croquet on a green lawn in Dublin \u2014 Crozier falls asleep again.\n\nShe is in her parka and kneeling in front of him, hands about a foot apart, string made of animal sinew or gut dancing between her splayed fingers and thumbs. She is using her fingers to play a cat's cradle child's game with sinew as string.\n\nCrozier watches dully.\n\nThe same two patterns keep appearing out of the complicated crisscross of sinew string. The first comprises three bands of strings creating two triangles at the top, just in from her thumbs, but with a double loop of string in the lower center of the pattern showing a peaked dome. The second pattern \u2014 her right hand pulled far away with just two bare strings running almost to her left hand where the string loops around just her thumb and little finger \u2014 shows a complex little loop of doubled string that looks like a cartoon figure with four oval legs or flippers and and a string-loop head.\n\nCrozier has no idea what the forms mean. He shakes his head slowly to let her know that he does not want to play.\n\nSilence stares at him for a silent moment, her dark eyes looking into his. Then she undoes the pattern with a graceful collapse of her small hands and sets the string in the ivory bowl he drinks his broth from. A second later she crawls out through the multiple tent flaps.\n\nShocked by the cold air blowing in for those seconds, Crozier tries to crawl to the opening. He needs to see where he is. Background groans and crackings have suggested that they are still on the ice \u2014 perhaps very near where he was shot. Crozier has no sense of how long it has been since Hickey ambushed the four of them \u2014 himself, Goodsir, poor Lane and Goddard \u2014 but he has hopes that it has been only a few hours, a day or two at most. If he leaves now, he might still be able to get his warning to the men at Rescue Camp before Hickey, Manson, Thompson, and Aylmore show up there to do more damage.\n\nCrozier is able to lift his head and shoulders a few inches but is far too weak to slither out from under the robes, much less to crawl to look out through the caribou-hide tent flaps. He sleeps again.\n\nSometime later \u2014 he is not even sure if it is the same day or if Silence has come and gone several times since he fell asleep \u2014 Silence wakes him. The dim light through the hides is the same; the interior of the tent is illuminated by the same blubber lamps. There is a fresh slab of seal lying in the snowy niche in the floor she uses for storage, and Crozier sees that she has just pulled off her heavy outer parka and is wearing only some sort of short pants with the fur side turned inward. The soft outer hide is lighter in color than Silence's brown skin. Her breasts bobble as she kneels in front of Crozier again.\n\nSuddenly the string dances between her fingers again. This time the little animal design near her left hand is shown first, the string is loosened, retwisted, and the design of the peaked oval dome in the center comes next.\n\nCrozier shakes his head. He does not understand.\n\nSilence tosses the string into the bowl, takes her short, semicircular blade with the ivory handle looking like the handle of a stevedore's hook, and begins slicing up the slab of seal meat.\n\n\"I have to go find my men,\" whispers Crozier. \"You have to help me find my men.\"\n\nSilence watches him.\n\nThe captain does not know how many days may have elapsed since his first awakening. He sleeps much. His few waking hours are spent with him eating his broth, eating the seal meat and blubber that Silence no longer has to prechew for him but which she still lifts to his lips, and with her changing his poultices and cleaning him. Crozier is mortified beyond words that his basic elimination needs must be attended to him using another Goldner's can set into the snow, reachable through a gap between the sleeping robes beneath him, and that it is _this girl_ who regularly must carry the can out to empty it somewhere out there on the ice floes. It does not make Crozier feel any better that the contents of the can freeze quickly and that there is almost no smell from it in the little tent that already smells so strongly of fish and seal and their own human sweat and presence.\n\n\"I need you to help me get back to my men,\" he rasps again. He feels that the odds are great that they are still close to the _polynya_ where Hickey ambushed them \u2014 no more than two miles out on the ice from Rescue Camp.\n\nHe needs to warn the others.\n\nIt confuses him that every time he awakens, the dim light through the tent's hide walls seems the same. Perhaps, for some reason that only Dr. Goodsir could explain, he awakens only at night. Perhaps Silence is drugging him with her seal-blood soup to keep him sleeping during the day. To keep him from escaping.\n\n\"Please,\" he whispers. He can only hope that despite her muteness, the savage has learned a little English during her months aboard HMS _Terror._ Goodsir had confirmed that Lady Silence could hear, even if she had no tongue with which to speak, and Crozier himself had seen her start at some sudden loud noise when she was a guest on their ship.\n\nSilence continues staring at him.\n\n _She's an idiot as well as a savage,_ thinks Crozier. He would be God-damned if he'd beg this heathen native again. He would have to keep eating, keep recovering, build up his strength, shove her aside one day, and walk back to camp himself.\n\nSilence blinks and turns to cook the slab of seal meat over her little blubber stove.\n\nHe awakes on another day \u2014 or, rather, another night, since the light is as dim as always \u2014 to find Silence kneeling over him and playing her string game again.\n\nThe first pattern between her fingers shows the little peaked-dome shape again. Her fingers dance. Two vertical looped shapes appear, but with two legs or flippers now rather than four. She pulls her hands farther apart, and somehow the designs actually move \u2014 sliding farther from her right hand and toward her left hand, the balloon-leg loops moving. She undoes that design, her fingers fly, and the oval-dome shape appears in the center again, but \u2014 Crozier slowly realizes \u2014 it is not quite the same shape. The peak of the dome is gone and now it is a pure catenary curve such as he studied as a midshipman poring over geometry and trigonometry illustrations.\n\nHe shakes his head. \"I don't understand,\" he rasps. \"This game doesn't make any God-damned sense.\"\n\nSilence looks at him, blinks, tosses the string into an animal-hide pack, and begins to pull him out of his sleeping furs.\n\nCrozier still does not have the strength to resist, but neither does he use what little strength he has regained to help. Silence props him up and tugs a light caribou underjacket and then a thick fur parka over his upper body. Crozier is shocked to feel how light the two layers are \u2014 the cotton and wool layers he's worn for outside work the past three years weighed more than thirty pounds _before_ they inevitably became soaked with sweat and ice, but he doubts if this upper outfit of Esquimaux clothing weighs more than eight pounds. He feels how loose both layers are on his upper body but how snugly everything fits at the neck and wrists \u2014 tight anywhere that heat might escape.\n\nEmbarrassed, Crozier does try to help pull on the light caribou pants over his nakedness \u2014 these are larger versions of the short pants that are all that Silence wears in the tent \u2014 and then the high caribou stockings, but his fingers get in the way more than not. Silence pushes his hands away and finishes dressing him with an impersonal economy of effort known only to mothers and nurses.\n\nCrozier watches as Silence pulls liners that look to be made of woven grass onto his feet and pulls them tight over his feet and ankles. Presumably these are for insulation, and he has trouble even imagining how long it had taken her \u2014 or some woman \u2014 to weave the grass into such high, tight socks. Fur boots, when tugged on over his grass socks by Silence, overlap his fur stocking-pants, and he notices that the soles of these boots are made of the thickest hides of any of their clothing.\n\nDuring the first hours he'd been awake in the tent, Crozier had wondered at the profusion of robes, parkas, furs, caribou hides, pots, sinew, the seal-oil lamps made of what looked to be soapstone, the curved cutting knife and other tools, but then he realized the obvious: it had been Lady Silence who had looted the bodies and packs of the eight dead Esquimaux killed by Lieutenants Hodgson and Farr. The rest of the material \u2014 Goldner tins, spoons, extra knives, marine mammals' ribs, pieces of wood, ivory, even what looked to be old barrel staves now used as part of the tent framework \u2014 must have been scavenged from _Terror_ or the abandoned Terror Camp or during Silence's months alone on the ice.\n\nWhen he is dressed, Crozier collapses onto one elbow and pants. \"Are you taking me back to my people now?\" he asks.\n\nSilence pulls mittens over his hands, flips his hood with its white-bear fur trim up over his head, firmly grips the bearskin beneath him, and drags him outside through the tent flaps.\n\nThe cold air hits Crozier's lungs and makes him cough, but after a moment he realizes how warm the rest of his body feels. He can feel his own body heat flowing up and around him within the roomy confines of this obviously nonporous garment. Silence bustles around him for a minute \u2014 pulling him up into a sitting position on a pile of folded furs. He guesses that she does not want him lying on the ice, even on the bearskin, since it feels warmer in these strange Esquimaux clothes when one sits up and lets air warmed by one's own body heat circulate against the skin.\n\nAs if to confirm this theory, Silence whisks away the bearskin on the ice and folds it, adding it to the stack next to the one he's sitting on. Astonishingly \u2014 Crozier's feet have been cold every time he has ever gone up on deck or out onto the ice in the past three years, and have been _wet and cold_ for every minute since he left _Terror_ \u2014 neither the cold of the ice here nor moisture seems to penetrate the thick hide-soles and grass booties he's wearing now.\n\nAs Silence begins taking down the tent with a few sure movements, Crozier looks around him.\n\nIt is night. _Why has she brought me out here at night? Is there some emergency?_ The caribou tent quickly being dismantled is, as he guessed from the noises, out on the pack ice, set amid seracs and icebergs and pressure ridges that reflect the little starlight thrown by the few stars peeking between low clouds. Crozier sees the dark water of a _polynya_ not thirty feet from where he'd been lying in the tent, and his heart beats faster. _We've not left the area where Hickey ambushed us, not two miles from Rescue Camp. I know the way back from here._\n\nThen he realizes that this _polynya_ is far smaller than the one Robert Golding had led them to \u2014 this patch of open, black water is less than eight feet long and only half that wide. Nor do the surrounding icebergs frozen into the pack ice here look right. They are much taller and more numerous than those near Hickey's ambush site. And the pressure ridges are taller.\n\nCrozier squints at the sky, catching only glimpses of stars. If the clouds would part and if he had his sextant and tables and a chart, he might be able to fix his position.\n\n _If... if... might._\n\nThe only recognizable patch of stars he catches sight of look more like a winter constellation than one that should be in that part of the arctic sky in mid- or late August. He knows that he was shot on the night of 17 August \u2014 he had already made his daily log entry before Robert Golding had come running into camp \u2014 and he cannot imagine that more than a few days have passed since the ambush.\n\nHe looks wildly around the ice-jumbled horizons, trying to find a twilight glow that would hint of a recent sunset or imminent sunrise in the south. There is only the night and the howling wind and the clouds and a few trembling stars.\n\n _Dear Christ... where is the sun?_\n\nCrozier is still not cold, but he is trembling and shaking so badly that he has to use what little strength he has to grip the pile of folded furs to keep from toppling over.\n\nLady Silence is doing a very strange thing.\n\nShe has collapsed the hide-and-bone tent in a few efficient motions \u2014 even in the dim light, Crozier can see that the outer tent covers are made of sealskins \u2014 and now kneels on one of the sealskin tent covers and uses her half-moon blade to slice it down the middle.\n\nThen she hauls the two halves of the sealskin to the _polynya_ and, using a curved stick to lower the pieces into the water, thoroughly wets them. Returning to the site where the tent stood only moments ago, she pulls frozen fish from the storage area that had been cut into the ice in her half of the tent and briskly lays a line of fish, head to tail, along one side of each half of the quickly freezing tent cover.\n\nCrozier has not the slightest clue as to what the wench is up to. It is as if she is performing some insane heathenish religious ritual out here in the rising night wind under the stars. But the problem is, Crozier sees, _she has cut up their sealskin tent cover._ Even if she rebuilds the tent from hides stretched over the scattered curved sticks and ribs and bones, it will no longer hold out the wind and cold.\n\nIgnoring him, Silence rolls both halves of the sealskin tent cover tightly around the two lines of fish, pulling and tugging the wet sealskin to make it even tighter. It amuses Crozier that she has left half of one fish protuding from one end of both lengths of rolled sealskin, and now she concentrates on bending upward the head end of each fish ever so slightly.\n\nIn two minutes she can lift the two seven-foot-long lengths of sealskin-wrapped fish \u2014 each now frozen as solidly as a long, narrow piece of oak with a rising fish head at its tip \u2014 and she lays them parallel on the ice.\n\nNow she sets a small hide under her knees and kneels to use bits of sinew and hide thongs to lash short lengths of caribou antlers and ivory \u2014 the former frame to the tent \u2014 to connect the two seven-foot-long wrapped-fish lengths.\n\n\"Mother of God,\" rasps Francis Crozier. _The frozen lengths of fish wrapped in wet sealskin are runners. The antlers are crosspieces._ \"You're building a fucking sledge,\" he whispers.\n\nHis breath hangs as crystals in the night air as his bemusement turns to a sort of panic. _It wasn't this cold on 17 August and before \u2014 nowhere_ near _this cold, even in the middle of the night._\n\nCrozier guesses that it has taken Silence half an hour or less to make the fish-runner, caribou-antler sledge, but now he sits on his stack of furs for another hour and a half or more \u2014 gauging the passage of time is difficult without his pocket watch and because he keeps drifting off into a light sleep even while sitting \u2014 as the woman works on the runners of the sledge.\n\nFirst she removes something that looks like a mixture of mud and moss from a canvas bag that had come from _Terror._ Carrying Goldner cans of water from the _polynya,_ she shapes this mud-moss into fist-sized balls and then lays these daubs the length of the ad hoc runners, patting and spreading them evenly with her bare hands. Crozier has no clue why her hands do not freeze solid despite her frequent breaks to stick her hands under her parka against her own bare belly.\n\nSilence smooths the frozen mud with her knife, trimming it as a sculptor might cut his clay maquette. Then she brings more water from the _polynya_ and pours it over the frozen layer of mud, creating an ice shoeing. Finally, she sprays mouthfuls of water onto a strip of bearskin and rubs that wet fur up and down the frozen mud along the length of each runner until the coating of ice there is absolutely smooth. In the starlight, it looks to Crozier as if the runners along the inverted sledge \u2014 just fish and strips of sealskin two hours earlier \u2014 are lined with glass.\n\nSilence rights the sledge, tests the thongs and knots, puts her weight on the firmly lashed caribou antlers and short pieces of wood, and lashes the remaining antlers \u2014 two longer curved ones that had been the primary tent supports \u2014 up from the rear of the sledge to make rudimentary handles.\n\nThen she lays several layers of sealskins and bearskins across the cross-antlers and comes to lift Crozier to his feet and help him over to the sledge.\n\nHe shakes off her arm and tries to walk to it by himself.\n\nHe has no memory of collapsing face-first into the snow, but his vision and hearing return as Silence is lifting him onto the sledge, straightening his legs, setting his back firmly against piled furs stacked against the rear antler handles, and setting several thick robes over him.\n\nHe sees that she has tied long strips of leather to the front of the sledge and woven the ends into a sort of harness that goes around her middle. He thinks of her finger-string games and sees what she had been saying \u2014 the tent (peaked oval) taken down, the two of them leaving (the walking figures in the sliding bits of string, although Crozier certainly was not walking this night), to another oval dome with no peak. (Another tent in the shape of a dome? A snow-house?)\n\nWith everything packed \u2014 the extra furs and canvas bags and hide-wrapped pots and seal-oil lamps all lying atop and around Crozier \u2014 Silence slips into harness and begins pulling them across the ice.\n\nThe runners glide with a glassy efficiency, far more silently and smoothly than the boat-sledges from _Terror_ and _Erebus._ Crozier is shocked to discover that he is still warm; two hours or more of just sitting still out on the ice floe has not chilled him, except for the tip of his nose.\n\nThe clouds are solid overhead. There is no hint of sunrise on the horizon in any direction. Francis Crozier has absolutely no hint as to where the woman is taking him \u2014 back to King William Island? South to the Adelaide Peninsula? Toward Back's River? Farther out onto the ice?\n\n\"My men,\" he rasps at her. He strains to raise his voice and be heard over the wind sigh, snow hiss, and the groaning of the thick ice beneath them. \"I need to get back to my men. They're looking for me. Miss... ma'am... Lady Silence, _please._ For the love of God, please take me back to Rescue Camp.\"\n\nSilence does not turn. He can see only the back of her hood and the white bear ruff gleaming in the faint starlight. He has no idea how she can see to proceed in this darkness or how such a small girl can pull his weight and the sledge's weight so easily.\n\nThey glide silently into the darkness of the ice jumble ahead.\n\nCROZIER\n\nSedna at the bottom of the sea decides whether to send the seal up to the surface to face being hunted by other animals and the Real People, but in a real sense, it is the seal himself who decides whether to allow himself to be killed or not.\n\nIn another real sense, there is only one seal.\n\nSeals are like Real People in that they each have two spirits \u2014 a life spirit that dies with the body and a permanent spirit that departs the body at the time of death. This longer-lasting soul, the _tarnic,_ hides in the seal as a tiny bubble of air and blood that a hunter can find in the seal's gut and is the same shape as the seal itself, only much smaller.\n\nWhen a seal dies, its permanent spirit departs and returns in exactly the same form in a baby seal descended from the seal who has decided to allow itself to be taken and eaten.\n\nThe Real People know that a hunter, over his lifetime, will be capturing and killing the same seal or walrus or bear or bird many times over.\n\nPrecisely the same thing happens to the permanent spirit of a member of the Real People when his life spirit dies with the body. The _inua_ \u2014 the permanent spirit-soul \u2014 travels, with all of its memories and skills intact, only hidden, to a boy or girl in the line of the dead person's family. This is one of the reasons that the Real People never discipline their children, no matter how rowdy or even impertinent they may become. Besides the child-soul in that child, there resides an adult's _inua_ \u2014 a father, uncle, grandfather, great-grandfather, mother, aunt, grandmother, or great-grandmother, with all its hunter's and matriarch's or shaman's wisdom \u2014 and it should not be rebuked.\n\nThe seal will not yield itself up to just any Real People hunter. The hunter must win them over, not just through his guile and stealth and skill but also through the quality of the hunter's own courage and _inua._\n\nThese _inua_ \u2014 the spirits of the Real People, seals, walruses, bears, caribou, birds, whales \u2014 existed as spirits before the Earth, and the Earth is old.\n\nDuring the first period of the universe, the Earth was a floating disk beneath a sky supported by four pillars. Beneath the Earth was a dark place where the spirits lived (and where most live to this day). This early Earth was under water most of the time and without any human beings \u2014 the Real People or others \u2014 until two men, Aakulujjuusi and Uumaaniirtuq, crawled out of humps in the earth. These two became the first of the Real People.\n\nThere were no stars in that era, no moon, no sun, and the two men and their descendants had to live and hunt in total darkness. Since there were no shamans to guide the Real People in their behavior, the human beings had very little power and could hunt only the smallest of animals \u2014 hares, ptarmigan, the occasional raven \u2014 and they did not know how to live properly. Their only decoration was to wear the occasional _aanguaq,_ an amulet made from a sea urchin shell.\n\nWomen had joined the two men on the Earth in this earliest of times (they came from the glaciers much as the men had come from the Earth), but they were barren and spent all their time walking the coastlines staring into the sea or digging into the ground in search of children.\n\nThe Second Cycle of the universe appeared after a long and bitter contest between a fox and a raven. The seasons appeared then, and then life and death itself; shortly after the seasons arrived, a new era began in which the life spirit of human beings would die with the bodies and the _inua_ -spirit would travel elsewhere.\n\nShamans learned some of the secrets of the cosmic order then and were able to help the Real People learn how to live properly \u2014 creating rules which forbade incest and marrying out of the family or murder or other behavior which goes against the Order of Things. The shamans were also able to see back even into the time before Aakulujjuusi and Uumaaniirtuq crawled out of the Earth and to explain to the human beings about the origins of the great spirits in the universe \u2014 the _inuat_ \u2014 such as the Spirit of the Moon, or about Naarjuk, the spirit of consciousness itself, or about Sila, the Spirit of the Air, who is also the most vital of all ancient forces; it is Sila who created and permeates and gives energy to all things and who expresses her wrath through blizzards and storms.\n\nThis is also the time when the Real People learned about Sedna, who is known in other cold places as Uinigumauituq or Nuliajuk. The shamans explained that all human beings \u2014 the Real People, the redder-skinned native human beings who lived far south of the Real People, the _Ijirait_ caribou spirits, and even the pale people who appeared so much later \u2014 were born after Sedna-Uinigumauituq-Nuliajuk coupled with a dog. This also explains why dogs are allowed to have names and a name-soul and even share their master's _inua._\n\nThe moon's _inua,_ Aningat, had incest with and otherwise abused his sister, Siqniq, the _inua_ of the sun. Aningat's wife, Ulilarnaq, loved to disembowel victims \u2014 animal or Real People \u2014 and so disliked the shamans' meddling in spirit matters that she would punish them by making them laugh uncontrollably. To this day, the shamans may be seized by uncontrollable laughter and frequently die from it.\n\nThe Real People enjoy knowing about these three most powerful spirits in the cosmos \u2014 the all-pervasive Spirit of the Air, the Spirit of the Sea, who controls all animals who live in the sea or depend upon the sea, and the final member of this trinity, the Spirit of the Moon \u2014 but these three original _inuat_ are too powerful to pay much attention to the Real People (or to human beings of any sort) since these ultimate _inuat_ are as far above the many other spirits as those lesser spirits are above human beings, so the Real People do not worship this trinity. Shamans rarely try to contact these most powerful of spirits \u2014 such as Sedna \u2014 and content themselves with making sure that the Real People do not break taboos that would anger the Spirit of the Sea, the Spirit of the Moon, or the Spirit of the Air.\n\nBut slowly, over many generations, the shamans \u2014 known as _angakkuit_ among the Real People \u2014 have learned more secrets of the hidden universe and of the lesser _inuat_ spirits. Over many centuries, some of the shamans have acquired the gift that Memo Moira called the Second Sight \u2014 clairvoyance. The Real People call these abilities _qaumaniq_ or _angakkua,_ depending upon how they manifest themselves. Just as human beings once tamed their cousin-spirits, the wolves, to become dogs who shared their masters' _inua,_ so did the _angakkuit_ with the hearing-thoughts or sending-thoughts gifts learn how to tame and domesticate and control the smaller spirits who appeared to them. These helping-spirits were called _tuurngait,_ and they not only helped the shamans see the invisible spirit world and look back to times before human beings, but also allowed them to look into other human beings' minds to see the faults committed by the Real People when they break the rules of the universe's order. The _tuurngait_ helping-spirits aid the shamans in restoring order and balance. They taught the _angakkuit_ their language, the language of the small spirits, which is called _irinaliutit,_ so that the shamans could address themselves directly to their own ancestors and to the more powerful _inuat_ powers of the universe.\n\nOnce the shamans had learned the _irinaliutit_ language of the spirit-helper _tuurngait,_ the shamans could then help human beings confess their misbehavior and faults so as to cure diseases and to restore order out of the confusion that is human affairs, thus restoring the order of the world itself. This system of rules and taboos passed down by the shamans was as complex as the crisscross string patterns created between the fingers of Real People women to this day.\n\nThe shamans also acted as protectors.\n\nSome minor evil spirits roam among the Real People, haunting them and bringing bad weather, but the shamans have learned how to create and consecrate a sacred knife and to kill these _tupilait._\n\nTo stop the storms themselves, the _angakkuit_ found and handed down a special hook that can cut the _silagiksaqtuq,_ the vein of the wind.\n\nThe shamans can also fly and act as mediators between the Real People and the spirits, but they can \u2014 and frequently do \u2014 also betray the trust of their own powers and harm human beings by using _ilisiiqsiniq,_ powerful spells they cast which stir up jealousy and rivalry and which can even a create a hatred sufficient to compel a Real Person to kill others for no reason. Frequently a shaman loses control of his _tuurngait_ helping-spirits, and when that happens, if it is not remedied quickly, that incompetent shaman is like a large metallic rock calling down the summer's lightning and there is little choice except for the Real People either to bind up the shaman and leave him behind or to kill him, cutting off his head and keeping it separate from the body so that the shaman cannot bring himself back to life and pursue them.\n\nMost shamans with any power at all can fly, heal people, families, and entire villages (actually by helping people heal themselves by finding balance again after confessing their faults), leave their bodies to travel to the moon or to the bottom of the sea (wherever the _inuat_ most powerful of spirits might dwell), and \u2014 after the proper _irinaliutit_ shamanic incantions, singing, and beating of drums \u2014 turn themselves into animals such as the white bear.\n\nWhile most spirits who are not contained in souls are content to dwell down in the spirit world, there are creatures abroad who carry the _inua_ spirits of monsters.\n\nSome of the smaller of these monsters are called _tupilek_ and were actually brought to life by people called _ilisituk_ hundreds and thousands of years ago. These _ilisituk_ were not shamans, but rather evil old men and women who learned much of the shamans' powers but used them to dabble in magic rather than in healing and faith.\n\nAll humans, and especially the Real People, live by eating souls \u2014 they know this well. What is hunting but one soul seeking out another soul and willing it into the ultimate submission of death? When a seal, for instance, agrees to be killed by a hunter, that hunter must honor the _inua_ of the seal who has agreed to be killed, after it is killed but before it is eaten \u2014 since it is a creature of the water \u2014 by giving it a small ceremonial drink of water. Some of the Real People hunters carry small cups on a stick for that purpose, but some of the oldest and finest hunters still pass the water from their own mouths to the dead seals' mouths.\n\nWe are all eaters of souls.\n\nBut the evil _ilisituk_ old men and women were soul-robbers. They used their incantations to take control of hunters, who often then took their families away from the village to live \u2014 and die \u2014 far away on the ice or in the interior mountains. Any descendants of these victims of soul-robbery were known as _qivitok_ and were always more savage than human.\n\nWhen families and villages began to suspect the old _ilisituk_ of their evil, the sorcerers would often create small evil animals \u2014 the _tupilek_ \u2014 to stalk, injure, or kill their enemies. The _tupilek_ started out as lifeless things as small as finger-stones, but after being animated by the _ilisituk's_ magic, they would grow to any size they wanted and take on terrible, unspeakable shapes. But since such monsters were easy for their victims to spot and flee from in the daylight, the stealthy _tupilek_ usually chose to take the approximate shape of any true living thing \u2014 a walrus, perhaps, or a white bear. Then the unsuspecting hunter who had been cursed by the evil _ilisituk_ would become the hunted. Human beings very rarely escaped the murderous _tupilek_ once they were sent out to do their killing.\n\nBut there are very few evil, old _ilisituk_ sorcerers left in the world today. One reason for this is that if the _tupilek_ did not succeed in killing its assigned victim \u2014 if a shaman intervened or if the hunter was so clever as to escape by his own devices \u2014 the _tupilek_ invariably returned to slaughter its creator. One by the one, the old _ilisituk_ became victims of their own terrible creations.\n\nThen there came a time, many thousands of years ago, when Sedna, the Spirit of the Sea, became infuriated with her fellow spirits, the Spirit of the Air and the Spirit of the Moon.\n\nTo kill them \u2014 these other two parts of the Trinity that made up the basic forces of the universe \u2014 Sedna created her own _tupilek._\n\nThis spirit-animated killing machine was so terrible that it had its own name-soul and became a thing called _Tuunbaq._\n\nThe _Tuunbaq_ was able to move freely between the spirit world and the Earth world of human beings, and it could take any shape it chose. Any form it took was so terrible that even a pure spirit could not look upon it directly without going mad. Its power \u2014 concentrated by Sedna only on the goals of wreaking havoc and death \u2014 was pure terror itself. On top of that, Sedna had granted her _Tuunbaq_ the power of commanding the _ixitqusiqjuk,_ the innumerable smaller evil spirits abroad.\n\nBy itself, one on one, the _Tuunbaq_ could have killed either the Spirit of the Moon or Sila, the Spirit of the Air.\n\nBut the _Tuunbaq,_ while terrible in every aspect, was not as stealthy as the tinier _tupilek._\n\nSila, the Spirit of the Air, whose energy fills the universe, sensed its murderous presence as it stalked her through the spirit world. Knowing that she could be destroyed by the _Tuunbaq_ and also knowing that if she was destroyed the universe would be thrown down into chaos again, Sila called on the Spirit of the Moon to help her defeat the creature.\n\nThe Spirit of the Moon was not interested in helping her. Nor was he concerned about the fate of the universe.\n\nSila then beseeched Naarjuk, the Spirit of Consciousness and one of the oldest _inua_ deep-spirits (who, like Sila, had appeared when the chaos of the cosmos had been separated from the thin but growing living green reed of order so very long ago), to help her.\n\nNaarjuk agreed.\n\nTogether, in a battle that lasted for ten thousand years and which left craters and rents and vacuums in the fabric of the spirit world itself, Sila and Naarjuk defeated the terrible _Tuunbaq_ 's attack.\n\nAs all _tupilek_ who have failed in their assassination assignments are destined to do, the _Tuunbaq_ then turned back to destroy its creator... Sedna.\n\nBut Sedna, who had learned all of her lessons the hard way since even before her father had betrayed her so long ago, had understood the danger the _Tuunbaq_ posed to her even before she created it, so now she activated a secret weakness she had built into the _Tuunbaq,_ chanting her own spirit-world _irinaliutit_ incantations.\n\nInstantly the _Tuunbaq_ was banished to the surface of the Earth, never able to return to the spirit world nor to the deep bottom of the sea nor to hold pure spirit form in either place. Sedna was safe.\n\nThe Earth and all its denizens, on the other hand, were no longer safe.\n\nSedna had banished the _Tuunbaq_ to the coldest, emptiest part of the crowded Earth \u2014 the perpetually frozen region near the north pole. She chose the far north rather than other distant, frozen areas because only the north, the center of the Earth to the many _inuat_ gods, had shamans there with any history of dealing with angry evil spirits.\n\nThe _Tuunbaq,_ deprived of its monstrous spirit form but still monstrous in essence, soon changed form \u2014 as all _tupilek_ do \u2014 into the most terrible living thing it could find on Earth. It chose the shape and substance of the smartest, stealthiest, most deadly predator on Earth \u2014 the white northern bear \u2014 but was to the bear in size and cunning as a bear itself is to one of the dogs of the Real People. The _Tuunbaq_ killed and ate the ferocious white bears \u2014 devouring their souls \u2014 as easily as the Real People hunted ptarmigan.\n\nThe more complicated the _inua_ -soul of a living thing is, the more delicious it is to a soul-predator. The _Tuunbaq_ soon learned that it enjoyed eating men more than eating _nanuq,_ the bears, enjoyed eating man-souls more than it enjoyed eating walrus-souls, and enjoyed eating men more even than it enjoyed devouring the large, gentle, and intelligent _inua_ -souls of the orca.\n\nFor generations, the _Tuunbaq_ gorged itself on human beings. Large parts of the snowy north that once were thick with villages, areas of the sea that once saw fleets of kayaks, and sheltered places that had heard the laughter of thousands of the Real People were soon abandoned as human beings fled south.\n\nBut there was no fleeing the _Tuunbaq._ Sedna's ultimate _tupilek_ could outswim, outrun, outthink, outstalk, and outfight any human being alive. It commanded the _ixitqusiqjuk_ bad spirits to move the glaciers farther south, making the glaciers themselves follow the human beings who'd fled into green lands so that the white-furred _Tuunbaq_ would be comfortable and concealed in the cold as it continued to eat human souls.\n\nHundreds of hunters were sent out from the Real People villages to kill the thing, and none of the men returned alive. Sometimes the _Tuunbaq_ would taunt the families of the dead hunters by returning parts of their bodies \u2014 sometimes leaving the heads and legs and arms and torsos of several hunters all mixed together so that the families could not even carry out the proper burial ceremonies.\n\nSedna's monster soul-eater looked as if it might eat all the human-being souls on Earth.\n\nBut, as Sedna had hoped, the shamans of the hundreds of groups of the Real People huddled around the periphery of the cold north, sent verbal messages, then met in _angakkuit_ shaman enclaves and talked, prayed to all their friendly spirits, conferred with their helping-spirits, and eventually came up with a plan to deal with the _Tuunbaq._\n\nThey could not kill this God That Walked Like a Man \u2014 even Sila, the Spirit of the Air, and Sedna, the Spirit of the Sea, could not kill the _talipek Tuunbaq._\n\nBut they could contain it. They could keep it from coming south and killing all of the human beings and all of the Real People.\n\nThe best of the best shamans \u2014 the _angakkuit_ \u2014 chose the best men and women among them with shamanic abilities of clairvoyant thought-hearing and thought-sending, and they bred these best men with the best women the way the Real People today breed sledge dogs to create an even better, stronger, smarter generation.\n\nThey called these beyond-shamanic clairvoyant children the _sixam ieua,_ or spirit-governors-of-the-sky, and sent them north with their families to stop the _Tuunbaq_ from slaughtering the Real People.\n\nThese _sixam ieua_ were able to communicate directly with the _Tuunbaq_ \u2014 not through the language of the _tuurngait_ helping-spirits as the mere shaman had attempted, but by directly touching the _Tuunbaq_ 's mind and life-soul.\n\nThe spirit-governors-of-the-sky learned to summon _Tuunbaq_ with their throat singing. Devoting themselves to communicating with the _Tuunbaq,_ they agreed to allow the jealous and monstrous creature to deprive them of their ability to speak to their fellow human beings. In exchange for the _tupilek_ killing-creature no longer preying on human souls, the spirit-governors-of-the-sky promised the God Who Walks Like a Man that they \u2014 the human beings and Real People \u2014 would no longer make their dwelling places in its northernmost snowy domain. They promised the God Who Walks Like a Man that they would honour it by never fishing or hunting within its kingdom without the monster-creature's permission.\n\nThey promised that all future generations would help feed the God Who Walks Like a Man's voracious appetite, the _sixam ieua_ and other Real People catching and bringing fish, walruses, seals, caribou, hares, whales, wolves, and even the _Tuunbaq_ 's smaller cousins \u2014 the white bears \u2014 for it to feast on. They promised that no human being's kayak or boat would trespass on the God Who Walks Like a Man's sea-domain unless it was to bring food or to sing the throat songs that soothed the beast or to pay homage to the killing-thing.\n\nThe _sixam ieua_ knew through their forward-thoughts that when the _Tuunbaq_ 's domain was finally invaded by the pale people \u2014 the _kabloona_ \u2014 it would be the beginning of the End of Times. Poisoned by the _kabloonas_ ' pale souls, the _Tuunbaq_ would sicken and die. The Real People would forget their ways and their language. Their homes would be filled with drunkenness and despair. Men would forget their kindness and beat their wives. The _inua_ of the children would become confused, and the Real People would lose their good dreams.\n\nWhen the _Tuunbaq_ dies because of the _kabloona_ sickness, the spirit-governors-of-the-sky knew, its cold, white domain will begin to heat and melt and thaw. The white bears will have no ice for a home, so their cubs will die. The whales and walruses will have nowhere to feed. The birds will wheel in circles and cry to the Raven for help, their breeding grounds gone.\n\nThis is the future they saw.\n\nThe _sixam ieua_ knew that as terrible as the _Tuunbaq_ was, this future without it \u2014 and without their cold world \u2014 would be worse.\n\nBut in the times before this should come to pass, and because the young clairvoyant men and women who were the spirit-governors-of-the-sky spoke to the _Tuunbaq_ as only Sedna and the other spirits could \u2014 never with voices but always directly, mind to mind \u2014 the still-living God Who Walks Like a Man listened to their propositions and their promises.\n\nThe _Tuunbaq,_ who \u2014 like all the greater _inuat_ spirits \u2014 loves to be pampered, agreed. He would eat their offerings rather than their souls.\n\nOver the generations, the _sixam ieua_ clairvoyants continued to breed only with other human beings with the same skill. At an early age, each _sixam ieua_ child gave up his or her ability to speak with his or her fellow human beings to show the God Who Walks Like a Man that they were devoted to speaking only to him, to the _Tuunbaq._\n\nOver the generations, the small families of the _sixam ieua_ who live so much farther north than the other villages of Real People (who are still terrified of the _Tuunbaq_ ), always making their homes on the permanently snow-and-glacier-covered earth and ice pack, became known as the God-Walking People, and even their speaking-families' language became a strange blend of the other Real People's tongues.\n\nOf course, the _sixam ieua_ themselves can speak no language \u2014 except for the clairvoyant speech of _qaumaniq_ and _angakkua,_ thought-sending and thought-receiving. But they are still human beings, they still love their families and belong to their larger family groups, so to speak to the other Real People, the _sixam ieua_ men use a special sign language and the _sixam ieua_ women tend to use the string-shape games that their mothers taught them.\n\nBefore leaving our village,\n\nand going out onto the ice\n\nto find the man I must marry,\n\nthe man my father and I dreamt of,\n\nback when the paddles were clean,\n\nmy father took a dark stone, _aumaa,_\n\nand he marked each paddle.\n\nhe knew that he would not return\n\nalive from the ice\n\nwe had both seen in our _sixam ieua_ dreams,\n\nthe only dreams that are true,\n\nthat he, my beloved Aja,\n\nwould die out there,\n\nat the hands of a pale-person.\n\nsince coming off the ice,\n\nI've looked for that stone\n\nin the hills\n\nand on the river-beds,\n\nbut I have never found it.\n\nupon my return to my people\n\nI will find the paddle on which the _aumaa_\n\nmade its grey mark.\n\nbirth was a short line\n\nat the blade tip.\n\nbut longer and above this,\n\ndeath was drawn parallel.\n\ncome again! shouts the Raven.\n\nCROZIER\n\nCrozier awakes with one hell of a splitting headache.\n\nHe wakes most mornings these days with a splitting headache. One would think that with his back and chest and arms and shoulders peppered by shotgun blasts and with no fewer than three bullet wounds in his body, he'd have other pains to notice upon awakening, and while those agonies descend on him quickly enough, it's the terrible headaches he notices first.\n\nIt reminds Crozier of all the years he drank whiskey every night and regretted it every morning after.\n\nSometimes he wakes, as he did this morning, with nonsense syllables and strings of meaningless words echoing in his aching skull. The words are all clickety-clack-sounding, like children making up vowel-heavy clucking noises just to find the right number of syllables for a jumping-rope song, but they _seem_ to mean something in those few painful seconds before he comes fully awake. Crozier feels mentally tired all the time these days, as if he's spent his nights reading Homer in Greek. Francis Rawdon Moira Crozier has never in his life attempted to read Greek. Nor wanted to. He's always left that to scholars and to poor book-obsessed souls like the old steward, Peglar's friend, Bridgens.\n\nThis dark morning he's awakened in their snow-house by Silence, who is using the string shapes shifting between her fingers to tell him that it is time to go seal hunting again. She is already dressed in her parka and disappears out the entrance tunnel as soon as she's finished communicating with him.\n\nGrumpy that there is to be no breakfast \u2014 not even some cold seal blubber from last night's dinner \u2014 Crozier dresses himself, pulling on his parka and mittens last, and crawls downhill out through the entrance passage that faces south, away from the wind.\n\nOutside in the dark, Crozier gets carefully to his feet \u2014 his left leg still sometimes refuses to accept his weight in the morning \u2014 and looks around. Their snow-house glows slightly from the blubber lamp that is left burning to keep the temperature up inside even while they are away. Crozier clearly remembers the long sledge voyage that brought them to this place. He remembers watching, fur-bundled on his sledge and as helpless as he had been those many weeks ago, with something like awe as Silence had spent hours digging out and then constructing this snow-house.\n\nSince then, the mathematician in Crozier had spent hours lying beneath his robes in the snug little space and admiring the catenary curve of the thing and the absolute and seemingly effortless precision that went into the woman's cutting of the snow blocks \u2014 in starlight \u2014 and the near perfection of the rising, inward-tilting walls made from those snow blocks.\n\nEven as he watched from beneath his furs that long night or dark day \u2014 _I'm as useless as tits on a boar,_ had been his thought \u2014 he'd also thought, _This thing should fall._ The upper blocks were almost horizontal. The last blocks she'd cut had been trapezoidal, and she'd actually shoved that final block \u2014 the key block \u2014 out from the inside and then trimmed the edges and tugged it into position from within the new snow-house. Finally Silence had come out and climbed onto the catenary-curve almost-dome of snow blocks, scrambled to the top, jumped up and down, and actually slid down its sides.\n\nAt first Crozier thought she was just acting like the child she sometimes looked to be, but then he realized that she was testing the strength and stability of their new home.\n\nBy the next day \u2014 another day without sunlight \u2014 the Esquimaux woman had used her oil lamp to melt the inside surface of the snow-house, then let the walls freeze again, coating it with a thin but very hard glaze of ice. She then thawed the sealskins that had been used first for the tent and then for the sledge and rigged them with sinew cords punched through the walls and ceilings of the snow-house, hanging the skins a few inches from the inside walls to provide an inner lining. Crozier had seen immediately that this protected them from dripping even while raising the temperature inside their living space.\n\nCrozier was astonished at how warm their snow-house seemed: always, he guessed, at least fifty degrees warmer than the outside temperature and frequently warm enough that neither of them wore anything but their caribou-skin shorts when out from under the robes. There was a cooking area on the snow ledge to the right of the entrance, and the antler-and-wood frame there not only suspended their various cooking pots over seal-oil flames but was used as a clothes-drying frame as well. As soon as Crozier was able to crawl and go outside with her, Silence explained through her string-language and gestures that it was imperative that they always dry out their outer clothing upon coming back into the snow-house.\n\nBesides the cooking platform to the right of the entrance and a sitting shelf to the left of it, there was the broad sleeping platform at the rear of the snow-house. Edged with what little wood Silence had brought \u2014 reused from the tent and then from the sledge \u2014 that wood, frozen in place, kept the platform from being worn down. Silence then spread the last of the moss from her canvas bag on the shelf, presumably as an insulating material, and then took great care spreading the various caribou and white-bear skins on the shelf. She then showed him how they should sleep with their heads toward the door and with their now-dry clothing bunched up as pillows. _All_ of their clothing.\n\nFor the first days and weeks, Crozier insisted on wearing his caribou shorts under the sleeping robes even though Lady Silence slept naked every night, but soon he found that so warm as to be uncomfortable. Still weakened by his wounds to the point that passion was not yet a temptation, he soon became accustomed to crawling naked between the sleeping robes and re-donning the perspiration-free shorts and other clothing only when he rose in the morning.\n\nWhenever Crozier awoke naked and warm under his robes next to Silence in the night, he tried to remember all the months aboard _Terror_ when he was always cold, always wet, and when the lower deck was always dark and dripping and ice-rimed and reeking of paraffin and urine. The Holland tents had been even more miserable.\n\nNow outside, he pulls his ruffed hood forward to keep the deep cold away from his face and looks around.\n\nIt is dark, of course. It had taken Crozier a long time to accept that somehow he had been unconscious \u2014 or dead? \u2014 for weeks between the time he was shot and his first conscious awareness of being with Silence, but there had been only the shortest, dimmest glow in the south during their long sledge trip to this place, so there was no doubt that it was now November, at the very least. Crozier had been trying to keep track of the days since they had come to the snow-house, but with the perpetual darkness without and their strange cycles of sleeping and waking within \u2014 he guessed that they sometimes slept twelve hours or more at a stretch \u2014 he could not be sure how many weeks had passed since they came to this place. And storms outside often kept them inside for unmeasurable days and nights, subsisting on their cold-stored fish and seal.\n\nThe constellations wheeling around \u2014 the sky is very clear today, and thus the day very cold \u2014 are winter constellations, and the air is so cold that the stars dance and shake in the sky just as they have all those years Crozier has watched them from the deck of _Terror_ or some other ship he'd taken to the arctic.\n\nThe only difference now is that he is not cold and he does not know where he is.\n\nCrozier follows Silence's tracks around the snow-house and toward the frozen beach and frozen sea. He doesn't really have to follow her tracks since he knows that the snow-covered beach is a hundred yards or so to the north of the snow-house and that she always goes to the sea to hunt seals.\n\nBut even knowing his basic directions here does not tell him where he is.\n\nFrom Rescue Camp and his crew's other camps along the south coast of King William Island, the frozen straits were always to the south. He and Silence could now be on the Adelaide Peninsula south across the straits from King William Island, or even on King William Island itself, but somewhere along its uncharted eastern or northeastern coasts where no white man has ever been.\n\nCrozier has no memory of Silence transporting him to the tent site after he was shot \u2014 or of how many times she might have moved the tent before he returned to the world of the living \u2014 and has only the haziest recall of how long their journey on the fish-runner sledge was before she built the snow-house.\n\nThis place might be anywhere.\n\nThey didn't have to be on King William Island at all, even if she has brought them north; they might be on one of the islands in the James Ross Strait somewhere to the northeast of King William Island or on some uncharted island off either the east or west coast of Boothia. On moonlit nights, Crozier can see hills inland from their snow-house site \u2014 not mountains, but hills larger than any the captain has ever seen on King William Island \u2014 and their campsite itself is more sheltered from the wind than any place he or his men ever found, including Terror Camp.\n\nAs Crozier crunches his way across the snow and gravel of the beach and out onto the jumbled sea ice, he thinks of the hundreds of times in the past few weeks when he has tried to communicate his need for leaving to Silence, for finding his men, for getting back to his men.\n\nShe always looks at him without expression.\n\nHe has come to believe that she understands him \u2014 if not his words in English, then the emotions behind his pleas \u2014 but she never answers by either expression or string-sign.\n\nHer understanding of things \u2014 and his own growing understanding of the complex ideas behind the dancing designs in the string between her fingers \u2014 borders, Crozier thinks, on the uncanny. He sometimes feels so close to the odd little native girl that he awakes in the night not knowing which body is his and which hers. At other times, he can hear her shout to him across the dark ice to come quickly or to bring an extra harpoon or rope or tool... even though she has no tongue and has never made a sound in his presence. She understands much, and sometimes he thinks that it is her dreams he dreams every night and wonders if she also has to share his nightmare of the priest in white vestments looming over him as he awaits Communion.\n\nBut she will not lead him back to his men.\n\nThree times Crozier has left on his own, crawling out the passage as she sleeps or pretends to sleep, bringing just a bag of seal blubber to sustain him and a knife with which to defend himself, and three times he has become lost \u2014 twice in the interior of whatever landmass they are on, once far out onto the sea ice. All three times Crozier has walked until he can walk no more \u2014 perhaps for days \u2014 and then collapsed, accepting death as his just and proper punishment for abandoning his men to die.\n\nEach time, Silence has found him. Each time, she has bundled him onto a bearskin, set robes over him, and silently pulled him the cold miles back to the snow-house, where she warms his frozen hands and feet against her naked belly under the robes and does not look at him while he weeps.\n\nNow he finds her several hundred yards out onto the ice, bent over a seal's breathing hole.\n\nTry as he might \u2014 and he has tried \u2014 Crozier can never find these damned breathing holes. He doubts if he could find them in summer daylight, much less by moonlight, starlight, or in the full dark as Silence does. The stinking seals are so clever and so _sly_ that he does not wonder that he and his men killed only a handful in all their months on the ice and never one through its breathing hole.\n\nThrough the talking strings, Crozier has been made to understand that a seal can hold its breath under water for only seven or eight minutes \u2014 perhaps fifteen at the most. (Silence explained these units of time in heartbeats, but Crozier thought he had successfully translated them.) Evidently, if he understands Silence's strings correctly, a seal has territorial boundaries \u2014 like a dog or wolf or white bear. Even in the winter, the seal must defend those boundaries, so to make sure that he has enough air within his under-ice kingdom, the seal finds the thinnest ice around and scoops out a dome-shaped breathing hole large enough to hold his entire body, leaving only the tiniest possible actual hole penetrating the thin-shaved ice through which he can breathe. Silence has shown him the sharp scraping claws on a dead seal's flipper and actually clawed at the ice with them to illustrate how well they work.\n\nCrozier believes Silence when she strings that there are dozens of such breathing-hole domes within a single seal's territory, but he's damned if he can find them. The domes she shows so clearly in her strings and which she finds so easily out here in the ice jumble are all but invisible amid the seracs, pressure ridges, ice blocks, little bergs, and crevasses. He's sure he's stumbled over a hundred of the damned things and never noticed one except as an irregularity in the ice.\n\nSilence is squatting near one now. When Crozier is a dozen yards away, she gestures for him to be quiet.\n\nTo hear Silence tell of it with the string patterns making pictures between her hands, the seal is one of the most cautious and wary creatures alive, so silence and stealth are the essence of hunting seal. Here Lady Silence earns her name.\n\nBefore approaching a breathing hole \u2014 how _does_ she know they're there? \u2014 Silence sets down small squares of caribou skin that she retrieves after each step, setting her thick-booted feet carefully onto them so as not to make the slightest crunch on the snow and ice. Once next to the breathing-hole dome in the dark, moving in slow motion, she softly pushes several forked antlers into the snow and sets her knife, harpoon, lines, and other hunting bric-a-brac on them so that she can retrieve them without making a noise.\n\nBefore leaving the snow-house, Crozier has tied sinew thongs around his arms and legs the way Silence has shown him, in an attempt to keep his clothes from rustling. But he knows that if he walks closer to the hole now, he'll sound, in his white-man clumsiness, like a collapsing tower of tin cans to the seal below \u2014 if there _is_ a seal below \u2014 so he strains to see the ice surface beneath him, makes out the inevitable two-foot-by-two-foot-thick caribou skin that Silence leaves for him, and slowly, carefully, goes to his knees on it.\n\nCrozier knows that before he arrived, after Silence found the breathing hole, she carefully and slowly removed the snow over that hole with her knife and widened the hole itself with a bone pick set into the butt of her harpoon shaft. She then inspected the hole to confirm that it was directly above a deep channel in the ice \u2014 if not, the chances of a good harpoon thrust were low, he understood now \u2014 and then she built the tiny mound up again. Since the snow was blowing, she put a narrow gauze of skin over the hole to prevent it from being filled in. Then she took a very thin point of bone fastened by a long piece of gut string to the tip of another bone and slid this indicator down into the hole, setting the other end on one of her antler twigs.\n\nNow she waits. Crozier watches.\n\nHours pass.\n\nThe wind comes up. Clouds begin to obscure the stars, and snow blows across the ice from the land behind them. Silence stands there, hunched over the breathing hole, her parka and hood slowly being covered with a film of snow, her harpoon with its ivory tip in her right hand, its weight being supported at the rear by the forked antler in the snow.\n\nCrozier has seen her catch seals in other ways. In one, she hews two holes in the ice and \u2014 with Crozier's help using one of two harpoons \u2014 literally beguiles the seal to her. She has taught him that while the seal may be the animal kingdom's soul of caution, its Achilles' heel is its curiosity. If Crozier gets the head of his specially prepared harpoon near Silence's hole under the ice, he moves the harpoon up and down ever so slightly, causing two small bones rigged with split-feather shafts near the head of the harpoon to vibrate. Eventually, the seal cannot resist its curiosity and pops up to investigate.\n\nIn the full moonlight, Crozier has gaped as Silence has moved across the ice on her belly, pretending to be a seal herself, moving her arms like flippers. On those times he can't even see the seal's head protruding from a hole in the ice until there is a sudden, impossibly fast motion of her arm, and then she is pulling back the harpoon attached to her wrist by a long cord. More often than not, there is a dead seal on the other end.\n\nBut this dark night-day there is only the seal's breathing hole to watch and Crozier stays on his skin pad for hours, watching Silence standing bent over the almost indiscernible dome. Every half hour or so, she reaches back slowly to her antler-twigs and removes a strange little instrument \u2014 a curved bit of driftwood about ten inches long with three bird claws attached \u2014 and scratches so lightly at the ice over the breathing hole that he can't hear the noise even from a few feet away. But the seal must hear it clearly enough. Even if the animal is at another breathing hole, perhaps hundreds of yards away, it seems \u2014 eventually \u2014 to be overcome by the curiosity that will doom it.\n\nOn the other hand, Crozier has no idea how Silence can see the seal to harpoon it. Perhaps in the sunlight of summer, late spring, or autumn its shadow might be visible under the ice, its nose visible beneath the tiny breathing-hole opening... but in starlight? By the time her warning device vibrates, the seal could have turned and dived deep again. Can she smell its presence as it rises? Can she sense it in some other way?\n\nHe is half frozen \u2014 a symptom of lying on the caribou pad rather than sitting upright \u2014 and dozing when Silence's little bone-and-feather indicator must have vibrated.\n\nHe comes awake in an instant as she blurs into action. She lifts the harpoon from its butt rest and flings it straight down through the breathing hole in less time than it takes for Crozier to blink awake. Then she is leaning back, pulling hard on the thick cord disappearing through the ice.\n\nCrozier struggles to his feet \u2014 his left leg aches abominably and does not want to support any weight \u2014 and hobbles to her side as quickly as he can. He knows that this is one of the trickiest parts of the seal hunt \u2014 pulling the thing up before it can writhe off the barbed ivory harpoon head if it is only injured, or just tangle in ice or slip away to the depths if it is dead. Speed, as the Royal Navy had never tired of telling him, is of the essence.\n\nTogether they wrestle the heavy animal up through the hole, Silence pulling at the cord with one surprisingly strong arm and hacking away at the ice with her knife in the other hand, enlarging the hole.\n\nThe seal is dead but more slippery than anything Crozier has ever encountered. He gets his mittened hand under the base of a flipper, taking care to avoid the razor-sharp claws at the end, and heaves to leverage the dead animal up onto the ice. All the while, he is gasping and cursing and laughing \u2014 relieved from his duty to remain silent \u2014 and Silence is, of course, silent except for the occasional soft hiss of breath.\n\nWhen the seal is safe on the ice, he stands back, knowing what will come next.\n\nThe seal, barely visible in the little starlight that's made its way between the low-scudding clouds, lies with its black eyes unblinking and looking vaguely censorious, its open mouth leaking only a trace of black-looking blood onto the blue-white snow.\n\nPanting a bit from the exertion, Silence goes to her knees on the ice, then to all fours, and then she lies on her belly with her face next to the dead seal's.\n\nCrozier takes another silent step back. Strangely, he feels now much the same way he did when he was a boy in Memo Moira's church.\n\nReaching under her parka, Silence pulls out the tiniest stopped flask made of ivory and fills her mouth with water from it. She has kept the flask next to her bare breasts under the fur so as to keep the water liquid.\n\nShe leans forward and sets her lips to the seal's in a strange parody of a kiss, even opening her mouth the way Crozier has seen whores do with men on at least four continents.\n\n _But she has no tongue,_ he reminds himself.\n\nShe passes the liquid water from her mouth to the seal's mouth.\n\nCrozier knows that if the seal's living soul, not quite departed from this body, is pleased with the beauty and workmanship of the harpoon and barbed ivory spearhead that killed it, is pleased with Silence's stealth and patience and her other hunting methods, and especially if it enjoys the water from her mouth, it will go tell the other seal-souls that they should come to this hunter for the chance to drink such fresh, clear water.\n\nCrozier does not know how he knows this \u2014 Silence has never signed it to him with strings or suggested as much through any other gestures \u2014 but he knows it is true. It's as if the knowledge comes from the headaches that plague him every morning.\n\nThe ritual over, Silence gets to her feet, brushes the snow from her pants and parka, gathers up her precious instruments and harpoon, and together they drag the dead seal the two hundred yards or so to their snow-house.\n\nThey eat all evening. It seems that Crozier can never get his fill of fat and blubber. Their faces are both as greasy as a greased pig's arse by the end of the evening, and he points to his face, points to Silence's equally greasy face, and bursts into laughter.\n\nSilence never laughs, of course, but Crozier thinks he sees the slightest hint of a smile before she scrambles down the entrance passage and returns \u2014 naked except for her caribou shorts \u2014 with fresh handfuls of snow for them to wipe their faces with before wiping them again with soft caribou skins.\n\nThey drink icy water, heat and eat more seal, drink again, go outside to separate places to relieve themselves, drape their damp clothing over the drying rack above the low-burning blubber flame, wash their hands and faces again, brush their teeth with fingers and string-wrapped twigs, and crawl naked under the sleeping robes.\n\nCrozier has just dozed off when he awakens to the feel of Silence's small hand on his thigh and private parts.\n\nHe reacts immediately, stiffening and rising. He has not forgotten his previous physical pain and scruples about having relations with the Esquimaux girl: these details simply are not in his mind as her small but urgent fingers close around his penis.\n\nThey are both breathing hard. She flings her leg over his thigh and rubs up and down. He cups her breasts \u2014 so warm \u2014 and reaches down behind her to fiercely grab her round behind and pulls her crotch tighter against his leg. His cock is almost absurdly hard and pulsating, its swollen tip vibrating like the seal-indicator feathers at every fleeting contact with her warm skin. His body is like the curious seal, rising quickly toward the surface of sensations in spite of its wiser instincts.\n\nSilence throws aside the top sleeping robe and straddles him, reaching down in a motion as quick as her harpoon-throwing movement to seize him, position him, and slide him inside her.\n\n\"Ah, Jesus... ,\" he gasps as they begin to become one person. He feels the resistance against his straining cock, feels it surrender to their motion, and knows \u2014 with deep shock \u2014 that he is bedding a virgin. Or that a virgin is bedding him. \"Oh, God,\" he manages as they start moving more wildly.\n\nHe pulls her shoulders down and tries to kiss her, but she turns her face away, setting it against his cheek, against his neck. Crozier has forgotten that Esquimaux women do not know how to kiss... the first thing any English arctic explorer is told by the old veterans.\n\nIt does not matter.\n\nHe explodes within her in a minute or less. It has been so long.\n\nSilence lies still on him for a while, her small breasts flattened and sweaty against his equally sweaty chest. He can feel her rapid heartbeat and knows she can feel his.\n\nWhen he can think, he wonders if there is blood. He does not want to soil the beautiful white sleeping robes.\n\nBut Silence is moving her hips again. She sits straight up now, still straddling him, her dark gaze holding his. Her dark nipples seem to be another pair of unblinking eyes watching him. He is still hard inside her, and her motions, impossibly \u2014 this has never happened in Francis Crozier's encounters with doxies in England, Australia, New Zealand, South America, and elsewhere \u2014 are making him come alive again, grow harder, begin to move his own hips in response to her slow grinding against him.\n\nShe throws her head back and sets her strong hand against his chest.\n\nThey make love like this for hours. Once, she leaves the sleeping shelf, but only long enough to return with water for them to drink \u2014 snowmelt from the small Goldner's tin they leave suspended over the clothes-drying flame \u2014 and she matter-of-factly cleans the small smears of blood from her thighs when they've finished drinking.\n\nThen she lies on her back, opens her legs, and pulls him over her with her hand strong on his shoulder.\n\nThere is no sunrise, so Crozier will never know if they have made love all that long arctic night \u2014 perhaps it has been entire days and nights without sleeping or stopping (it feels this way to him by the time they sleep) \u2014 but sleep they eventually do. Moisture from their sweat and breathing drips from the exposed parts of the snow-house walls and it is so warm in their home that for the first half hour or so after they fall off into sleep, they leave the top sleeping robe off.\n\nCROZIER\n\nAfter he made land,\n\nwhen the world was still dark,\n\nTulunigraq, Raven, heard the Two Men dream about light.\n\nBut there was no light.\n\nEverything was dark, as it had always been.\n\nNo sun. No moon. No stars. No fires.\n\nRaven flew inland until he found a snow-house\n\nwhere an old man lived with a daughter.\n\nHe knew they were hiding light,\n\nhoarding a bit of light,\n\nso he entered.\n\nHe crawled up through the passage.\n\nHe looked up through the _katak._\n\nTwo skin-bags were hanging there,\n\none holding darkness,\n\nand the other holding light.\n\nThe man's daughter sat there awake\n\nwhile her father slept.\n\nShe was blind.\n\nTulunigraq used his thought-sending\n\nto make the daughter want to play.\n\n\"Let me play with the ball!\" the daughter cried,\n\nwaking the old man.\n\nThe man awoke and took down the bag that held\n\nthe daylight.\n\nThe light was wrapped in caribou skin which was\n\nmade warm by the daylight inside\n\nwanting to get out.\n\nRaven used his thought-sending to make\n\nThe girl push the daylight-ball toward the _katak._\n\n\"No!\" cried the father.\n\nToo late.\n\nThe ball went down the _katak,_ bounced down\n\nthe passage.\n\nTulunigraq was waiting.\n\nHe caught the ball.\n\nHe ran out the passage,\n\nran with the daylight ball.\n\nRaven used his bill.\n\nHe tore the skin-ball.\n\nTore at daylight.\n\nThe man from the snow-house was\n\nchasing him through willows\n\nand ice, but the daylight-man was no man.\n\nThe man was a falcon.\n\n_\"Pitqiktuak!\"_ screamed Peregrine, \"I will\n\nkill you, Trickster!\"\n\nHe flew down on Raven,\n\nbut not before Raven tore the skin-ball open.\n\nDawn rose.\n\nLight spilled everywhere.\n\n_Quagaa Sila!_ Dawn rose!\n\n_\"Uunukpuaq! Uunukpuagmun! Darkness!\"_\n\nshrieked the Falcon.\n\n\" _Quagaa!_ Light everywhere!\"\n\ncried Raven.\n\n\"Night!\"\n\n\"Daylight!\"\n\n\"Darkness!\"\n\n\"Daylight!\"\n\n\"Night!\"\n\n\"Light!\"\n\nThey went on shouting.\n\nRaven cried \u2014\n\n\"Daylight for the earth!\"\n\n\"Daylight for the Real People!\"\n\nIt will be no good\n\nif we have one but not the other.\n\nSo Raven brought daylight to some places.\n\nAnd Peregrine kept darkness fast in other places.\n\nBut the animals fought.\n\nThe Two Men fought.\n\nThey threw light and darkness at one another.\n\nDaylight and night came into balance.\n\nWinter follows summer.\n\nTwo halves.\n\nLight and darkness complete one another.\n\nLife and death complete one another.\n\nYou and I complete one another.\n\nOutside, the _Tuunbaq_ walks in night.\n\nWhere we touch,\n\nthere is light.\n\nEverything is in balance.\n\nCROZIER\n\nThey leave on their long sledge trip shortly after the sun makes its first hesitant, midday, and only-minutes-long appearance on the southern horizon.\n\nBut Crozier understands that it is not the return of the sun that has determined their time for action and his own time of decision; it is the violence in the skies the other twenty-three and a half hours each day that has decided Silence that the time has come. As they sledge away from their snow-house forever, shimmering bands of colored light coil and uncoil above them like fingers opening out from a fist. The aurora grows stronger in the dark sky every day and night.\n\nThe sledge is a more serious device for this longer trip. Almost twice as long as the jury-rigged fish-runnered six-foot sled Silence had used to transport him when he could not walk, this vehicle has runners made up of small and carefully shaped pieces of scavenged wood interlinked with walrus ivory. It uses shoes of whalebone and flattened ivory rather than just a layer of peat paste on its runners, although Silence and Crozier still reapply a layer of ice to the runners several times a day. The cross sections are made up of antlers and the last bits of wood they had, including the sleeping-shelf slat; the rising rear posts are composed of heavily lashed antlers and walrus ivory.\n\nThe leather straps are now rigged for both of them to pull \u2014 neither will ride unless there is an injury or illness \u2014 but Crozier knows that Silence has built this sledge with great care in the hopes that it may be pulled by a dog team before this year is over.\n\nShe is with child. She has not told Crozier this \u2014 by the strings or by a glance or by any other visible means \u2014 but he knows it and she knows he does. If all goes well, he estimates that the baby will be born in the month he used to think of as July.\n\nThe sledge carries all of their robes and skins and cooking gear and tools and skin-sealed Goldner tins to hold water once thawed and a supply of frozen fish, seal, walrus, fox, hare, and ptarmigan. But Crozier knows that some of this food is for a time that may not come \u2014 at least for him. And some of it may be for presents, depending upon what he decides and what then happens out on the ice. He knows that, depending upon what he decides, they will both be fasting soon in preparation \u2014 although, as he understands it, he is the only one who _must_ fast. Silence will join him in the fast simply because she is his wife now and will not eat when he doesn't. But if he dies, she will take the food and the sledge and come back to land to live her life and continue her duties here.\n\nFor days they travel north along a coastline, skirting cliffs and too-steep hills. A few times the severe topography forces them out onto the ice, but they do not want to be out there for long. Not yet.\n\nThe ice is breaking up here and there, but only into small leads. They do not stop to fish at these leads or to pause at _polynyas,_ but press on, pulling ten hours a day or more, moving back to land as soon as they can to continue the hauling there even though it means much more frequent refreshening of the ice on the runners.\n\nOn the evening of the eighth night, they pause on a hill and look down at a cluster of lighted snow-domes.\n\nSilence has been careful to approach this little village from the downwind side, but still one of the dogs staked into the ice or earth below begins to bark madly. But the others do not join him.\n\nCrozier stares at the lighted structures \u2014 one is a multiple dome made up of at least one large and four small snow-houses connected by common passageways. Just the thought, much less the sight, of such community makes Crozier ache inside.\n\nFrom far below, muffled by snow blocks and caribou skin, comes the sound of human laughter.\n\nHe could go down there now, he knows, and ask this group to help him find his way to Rescue Camp and then to find his men; Crozier knows this is the village of the band belonging to the shaman who escaped the massacre of eight Esquimaux on the other side of King William Island and it is also Silence's extended family, as were the eight murdered men and women.\n\nHe could go down and ask them to help, and he knows that Silence will follow and translate with string-signs. She is his wife. He also knows the odds are great that unless he does what he will be asked to do out on the ice \u2014 Silence's husband or no and whatever their reverence and awe and love for her \u2014 these Esquimaux may well greet him with smiles and nods and laughter and then, when he is eating or asleep or unwary, will slip tight thongs over his wrists and a skin bag over his head and then stab him again and again, women stabbing along with the hunters, until he is dead. He has dreamt about his blood flowing red on white snow.\n\nOr perhaps not. Perhaps Silence does not know what will happen. If she has dreamt that particular future, she has not stringed the outcome to him nor shared those dreams.\n\nHe doesn't want to find out now anyway. This village, this night, tomorrow \u2014 before he has decided about the other thing \u2014 is not his immediate future, whatever else his future and his fate may or may not be.\n\nHe nods to her in the darkness and they turn away from the village and drag the sledge north along the coast.\n\nDuring the days and nights of travel \u2014 they rig only a protective caribou skin to hang above them from the sledge-antlers as they huddle together under hides for the few hours they sleep \u2014 Crozier has much time to think.\n\nIn the last few months, perhaps because he has had no one to speak to \u2014 or at least no interlocutor who can respond with actual out-loud speech \u2014 he has learned how to let different parts of his mind and heart speak within him as if they were different souls with their own arguments. One soul, his older, more-tired soul, knows that he has been a failure in every way a man can be tested. His men \u2014 the men who trusted him to lead them to safety \u2014 are all dead or scattered. His mind hopes that some have survived, but in his heart, in his soul of his heart, he knows that any men so scattered in the land of the _Tuunbaq_ are already dead, their bones bleaching some unnamed beach or empty ice floe. He has failed them all.\n\nHe can, at the very least, follow them.\n\nCrozier does not yet know where he is, although he suspects more each day that they have wintered on the western coast of a large island northeast of King William Island, at a point on almost the same latitude as Terror Camp and _Terror_ herself, although those sites would be a hundred miles or more west from here across the frozen sea. If he wanted to return to _Terror_ he would have to travel west across this sea and perhaps across more islands and then across all of the north of King William Island itself and then twenty-five more miles out onto the ice to reach the ship he abandoned more than ten months earlier.\n\nHe does not want to return to _Terror._\n\nCrozier has learned enough about survival in the past months that he thinks he can find his way back to Rescue Camp and even to Back's River given enough time, hunting as he goes, building snow-houses or skin tents when the inevitable storms arise. He can seek out his scattered men this summer, ten months after he abandoned them, and find some trace of them, even it if takes years.\n\nSilence will follow him if he chooses this path \u2014 he knows she will \u2014 even though it means the death of everything she is and everything she lives for here.\n\nBut he wouldn't ask her to. If he were to go south after his crew, he would go alone because he suspects that, despite all his new knowledge and skills, he would die on such a search. If he doesn't die on the ice, there will be an injury on the river he would have to follow south. If the river or injury or illness along the way don't kill him, he might encounter hostile Esquimaux groups or the even more savage Indians farther south. Englishmen \u2014 especially the old arctic hands \u2014 love to believe that Esquimaux are primitive but peaceful people, slow to anger, always resistant to war and strife. But Crozier has seen the truth in his dreams: they are human beings, as unpredictable as any other race of man, and often descend into warfare and murder and, in hard times, even cannibalism.\n\nA much shorter and surer route to rescue than going south, he knows, would be to head due east from here across the ice before the ice pack opens for the summer \u2014 if it opens at all \u2014 hunting and trapping as he goes, then crossing the Boothia Peninsula to its eastern coast, traveling north to Fury Beach or the old expedition sites there. Once at Fury Beach he could just wait for a whaler or rescue ship. The chances for his survival and rescue in that direction are excellent.\n\nBut what if he makes it to civilization... back to England? Alone. He will always be the captain who let all his men die. The court-martial will be inevitable, its outcome predetermined. Whatever the court's punishment might be, the shame will be a lifelong sentence.\n\nBut this is not what dissuades him from heading east or south.\n\nThe woman next to him is carrying his child.\n\nOf all his failures, it is Francis Crozier's failures as a man which hurt and haunt him the most.\n\nHe is almost fifty-three years old and he has loved only once before this \u2014 proposing marriage to a spoiled child, a mean-spirited girl-woman who had teased him and then used him for her pleasure the way his sailors used dockside chippies. _No,_ he thought, _the way_ I _used dockside chippies._\n\nEvery morning now and often in the night he awakens next to Silence after sharing her dreams, knowing that she has shared his, feeling her warmth against him, feeling himself responding to that warmth. Every day they go out into the cold and fight for life together \u2014 using her craft and knowledge to prey on other souls, to eat other souls, so that their two life-spirit souls can live awhile longer.\n\n_She is carrying our child._ My _child._\n\nBut that is irrelevant to the decision he must make in the next few days.\n\nHe is almost fifty-three years old and he is now being asked to believe in something so preposterous that the very thought of it should make him laugh. He is asked \u2014 if he understands the strings and the dreams, and he believes he does at long last \u2014 to _do_ something so terrible and so painful that if the experience does not kill him, it may drive him mad.\n\nHe has to _believe_ that such counterintuitive insanity is the right thing to do. He has to _believe_ that his dreams \u2014 mere dreams \u2014 and that his love for this woman should make him surrender a lifetime of rationality to become...\n\nBecome what?\n\nSomeone and something else.\n\nPulling the sledge next to Silence under a sky filled with violent color, he reminds himself that Francis Rawdon Moira Crozier believes in nothing.\n\nOr rather, if he believes in anything, it is in Hobbes's _Leviathan._\n\n_Life is solitary, poor, nasty, brutish, and short._\n\nThis cannot be denied by any rational man. Francis Crozier, in spite of his dreams and headaches and strange new will to believe, remains a rational man.\n\nIf a man in a smoking jacket in a coal-fire-heated library in his manor house in London can understand that life is solitary, poor, nasty, brutish, and short, then how can it be denied by a man pulling a sledge stacked with frozen meat and furs across an unnamed island, through the arctic night under a sky gone mad, toward a frozen sea a thousand miles and more from any civilized hearth?\n\nAnd toward a fate too frightening to imagine.\n\nOn their fifth day pulling along the coast, they come to the end of the island and Silence leads them northeast out onto the ice. The going is slower here \u2014 there are the inevitable pressure ridges and shifting floes \u2014 and they have to work much harder. They also travel more slowly so as not to break the sledge. They use their blubber stove to melt snow for drinking water but do not pause to catch fresh meat, despite the many breathing-hole domes Silence points out in the ice.\n\nThe sun now rises for thirty minutes or so each day. Crozier cannot be sure of the time. His watch disappeared with his clothing after Hickey shot him and after Silence rescued him... however she did that. She has never told him.\n\n_That was the first time I died,_ he thinks.\n\nNow he is being asked to die again \u2014 to die as what he was in order to become something else.\n\nBut how many men get such a second chance? How many captains who have watched one hundred twenty-five men in their expedition die or disappear would want it?\n\n_I could disappear._\n\nCrozier has seen the mass of scars on his arm, chest, belly, and leg each night when he strips to crawl beneath the sleeping robes, and he can feel and imagine how terrible the bullet and shotgun-pellet scars are on his back. They could be an explanation and excuse for a lifetime of silence about his past.\n\nHe can hike east across Boothia, hunt and fish in the rich, warming waters off the east coast there, hide from Royal Navy and other English rescue ships, and wait for an American whaling ship. If it takes two or three years there before one comes, he can survive that long. He is sure of it now.\n\nAnd then, instead of going home to England \u2014 has England ever been home for him? \u2014 he can tell his American rescuers that he has no memory of what has happened to him or what ship he belonged to \u2014 he can show his terrible wounds as evidence \u2014 and go to America with them at the end of the whaling season. There he can start a new life.\n\nHow many men get a chance to start such a new life at his age? Many men would want to.\n\nWould Silence go with him? Would Silence bear the stares and laughter of sailors and the harsher stares and whispers of \"civilized\" Americans in some New England city or New York? Would she trade her furs in for calico dresses and whalebone corsets, knowing that she would always be the ultimate stranger in the ultimate strange land?\n\nShe would.\n\nCrozier knows this as surely as he knows anything.\n\nShe would follow him there. And she would die there \u2014 and die soon. Of misery and of the strangeness and of all the vicious, petty, alien, and unbridled thoughts that would pour into her like the poison from the Goldner tins poured into Fitzjames \u2014 unseen, vile, deadly.\n\nHe knows this as well.\n\nBut Crozier could raise his son in America and have a new life in that almost-civilized country, perhaps captain a private sailing ship there. He has been a total failure as a Royal Navy and Discovery Service captain and as an officer and as a gentleman \u2014 well, he was never a gentleman \u2014 but no one in America would ever need to know that.\n\nNo, no, a serious sailing ship would take him to places and ports where he might be known. If he is recognized by any English Naval officer, he would be hanged as a deserter. But a small fishing ship... fishing out of some small New England harbour village, perhaps, with an American wife waiting in port to raise his child with him after Silence dies.\n\n_An American wife?_\n\nCrozier glances at Silence straining in the sledge harness to his right, pulling with him. The crimson and red and purple and white light from the aurora overhead paints her furred hood and shoulders. She does not look at him. But he is sure that she knows what he is thinking. Or if she does not know now, she will when they curl up together later in the night and dream.\n\nHe cannot go home to England. He cannot go to America.\n\nBut the alternative...\n\nHe shivers and pulls his hood forward so that the polar-bear fur on either side of his face can better capture the warmth of his breath and body.\n\nFrancis Crozier believes in nothing. _Life is solitary, poor, nasty, brutish, and short._ It has no plan, no point, no hidden mysteries that make up for the oh-so-obvious miseries and banalities. Nothing he has learned in the last six months has persuaded him otherwise.\n\nHas it?\n\nTogether, they pull the sledge farther out onto the pack ice.\n\nOn the eighth day they stop.\n\nThis place looks no different than most of the other pack ice they have crossed in the previous week \u2014 a bit flatter, perhaps, fewer large ice blocks and pressure ridges, perhaps, but essentially just pack ice. Crozier can see a few small _polynyas_ in the distance \u2014 their dark water like blemishes in the white ice \u2014 and the ice has broken up here and there into several small, impermanent going-nowhere leads. If the spring breakup is not actually coming two months earlier this year, it is doing a good impersonation of it. But Crozier has seen such false spring thaws many times before in his arctic experience and knows that the real breakup of pack ice will not begin until late April or later.\n\nIn the meantime, they have patches of open water and seal breathing holes galore, perhaps even the chance to hunt walrus or narwhal should they appear, but Silence is not interested in hunting.\n\nBoth of them get out of their harness and look around. They have stopped hauling in the brief interlude of midday southern twilight that passes for daytime.\n\nSilence steps in front of Crozier, removes his mittens, and then removes her own. The wind is very cold and their hands should not be exposed for more than a minute, but in that minute she holds his hands in hers and looks at him. She moves her gaze to the east, then looks south, and then looks back at him.\n\nThe question is clear.\n\nCrozier feels his heart pounding. He cannot remember any time in his adult life \u2014 certainly not the night that Hickey ambushed him \u2014 when he has felt so frightened.\n\n\"Yes,\" he says.\n\nSilence puts her mittens back on and begins unpacking the sledge.\n\nAs Crozier helps her unpack things onto the ice and then break down parts of the sledge itself, he wonders again how she has found this place. He has learned that while she sometimes uses the stars or moon to navigate by, more often than not she just pays great attention to the landscape. Even on seemingly barren snowy terrain, she is counting the mathematically precise snow ridges and snow mounds created by the wind, even while noting which way these ridges run. Like Silence, Crozier has begun measuring time not so much in days as in sleepings \u2014 how many times they have stopped to sleep, whatever time of day or night that might have been.\n\nOut here on the ice, he has been more aware than ever \u2014 that is, he has shared some of Silence's awareness \u2014 of the subtleties of hummocked ice and old winter ice and new pressure ridges and thick pack ice and dangerous new ice. He now can see a lead many miles away just by the slight darkening of clouds above it. He now avoids dangerous but almost invisible fissures and rotten ice without actively noticing that he is doing so.\n\nBut why this place? How did she know to come here for what they are about to do?\n\n_I_ am _about to do it,_ he realizes and his heart pounds more wildly.\n\nBut not yet.\n\nIn the quickly dimming light, they connect some of the slats on the sledge and the unlashed vertical posts to build a crude framework for a small tent. They will be here only a few days \u2014 unless Crozier remains here forever \u2014 so they do not try to find a drift in which to construct a snow-house, nor do they spend energy on making the tent fancy. It will serve as shelter.\n\nSome of the skins are set in place for the outer wall of the tent, most go inside.\n\nWhile Crozier is arranging their floor furs and sleeping furs, Silence is outside, quickly and efficiently cutting blocks of ice from some nearby jumble block and building a low wall on the windward side of the tent. That will help some.\n\nOnce inside, she helps Crozier rig the blubber-flame cooking lamp and antler frame in the caribou-skin vestibule of the tent and they begin melting snow for drinking. They will also use the frame and flame for drying their outer clothes. The wind blows snow around the abandoned and empty sledge, which is little more than runners now.\n\nFor three days they both fast. They eat nothing, drinking water in an attempt to quell their belly's rumblings; they leave the tent for long hours each day, even when the snow comes, to exercise and relieve tension.\n\nCrozier takes turns throwing both harpoons and both lances at a large snow-and-ice block; Silence had recovered them from her dead family members at the massacre site and prepared one heavy harpoon with its long cord and one lighter throwing lance for each of them months ago.\n\nNow he throws the harpoon with such force that it buries itself ten inches into the block of ice.\n\nSilence walks closer and removes her hood, peering at him in the shifting light from the aurora.\n\nHe shakes his head and tries to smile.\n\nHe has no signs for _Isn't this what you do to your enemy?_ Instead, he reassures her with a clumsy hug that he is not leaving or planning to use the harpoon on anything or anyone anytime soon.\n\nHe has never seen the aurora like this.\n\nAll day and night the cascading curtains of color dance from horizon to horizon with the center of the displays directly overhead. Not in all of his years of expedition near the north or south poles has Crozier seen anything remotely resembling this explosion of light. The hour or so of wan daylight does almost nothing to lessen the intensity of the aerial display.\n\nAnd there is ample acoustical accompaniment to the visual fireworks.\n\nAll around them, the ice groans, cracks, moans, and grinds from pressure, while long series of explosions under the ice begin like scattered artillery fire and quickly move to an unceasing cannonade.\n\nAlready unnerved by anticipation, Crozier is more deeply shaken by the noise and movement of the ice pack under them. He sleeps now in his parka \u2014 perspiration be damned \u2014 and is out of the tent and onto the ice a half dozen times each sleeping period, sure that their broad floe is breaking up.\n\nIt never does, although cracks open here and there within fifty yards of their tent and send fissures racing faster than a man could run through seemingly solid ice. Then the cracks close and disappear. But the explosions continue, as does the violence in the sky.\n\nIn his last night in this life, Crozier sleeps fitfully \u2014 his fasting-hunger makes him cold in a way that even Silence's body heat cannot compensate for \u2014 and he dreams that Silence is singing.\n\nThe ice explosions resolve themselves into steady drumbeats that serve as background for her high, sweet, sad, lost voice:\n\n_Ayaa, yaa, yapape!_\n\n_Ayaa, yaa, yapape!_\n\n_Aj\u00e2-j\u00e2, aj\u00e2-j\u00e2-j\u00e2..._\n\n_Aji, jai, j\u00e2..._\n\n_Tell me, was life so beautiful on earth?_\n\n_Here I am filled with joy_\n\n_Whene'er the dawn comes up above the earth_\n\n_And the great sun_\n\n_Glides up into the sky._\n\n_But there where you are_\n\n_I lie in fear and trembling_\n\n_Of maggots and teeming vermin_\n\n_Or sea creatures with no souls_\n\n_That eat into the hollow of my collar bone_\n\n_And bore out my eyes._\n\n_Aji, jai, j\u00e2..._\n\n_Aj\u00e2-j\u00e2, aj\u00e2-j\u00e2-j\u00e2..._\n\n_Ayaa, yaa, yapape!_\n\n_Ayaa, yaa, yapape!_\n\nCrozier awakes trembling. He sees that Silence is already awake, staring at him with her dark, unblinking eyes, and in a moment of pure terror deeper than terror, he realizes that it was not her voice that he has just heard singing this dead man's song to him \u2014 literally a song from a dead man to his previous living self \u2014 but the voice of his unborn son.\n\nCrozier and his wife rise and dress in mutual ceremonial silence. Outside, though perhaps morning, it is still night, but a night of a thousand thrusting colors laid over the shaking stars.\n\nThe shattering ice still sounds like a drumbeat.\n\nThe only paths left now are surrender or death. Or both.\n\nAll of his life, the boy and man he was and has been for fifty years would rather die than surrender. The man he is _now_ would rather die than surrender.\n\nBut what is death itself other than the ultimate surrender? The blue flame in his chest will accept neither choice.\n\nIn their snow-house the past weeks, under their sleeping robes, he has learned about another type of surrender. A sort of death. A change from being one to being something else that is neither self nor not-self.\n\nIf two such different people who have no words at all in common can dream the same dreams, then perhaps \u2014 even with all dreams set aside and all other beliefs ignored \u2014 other realities can merge as well.\n\nHe is very frightened.\n\nThey leave the tent wearing only their boots, shorts, leggings, and the thin caribou-skin shirts they sometimes wear under their parkas. It is very cold tonight, but the wind has died down since the day's brief glimpse of midday sun.\n\nHe has no idea what time it is. The sun has been set for many hours, and they have not slept yet.\n\nThe ice breaks under pressure with the steady beat of drums. New leads are opening nearby.\n\nThe aurora casts curtains of light from the starry zenith to the white-ice horizon, sending shimmers to the north, to the east, to the south, and to the west. All things, including the white man and brown woman, are tinted alternately in crimson, violet, yellow, and blue.\n\nHe goes to his knees and raises his face.\n\nShe stands over him, bending slightly as if watching a breathing hole for a seal.\n\nAs taught, he keeps his arms at his side, but she grips him firmly by his upper arms. Her hands are bare in the cold.\n\nShe lowers her head and opens her mouth wide. He opens his. Their lips are almost touching.\n\nShe inhales deeply, seals her mouth over his, and begins blowing into his open mouth, down his throat.\n\nThis is where \u2014 in their practice during the long winter darkness \u2014 he had so much trouble. Breathing in another person's breath is like drowning.\n\nHis body tense, he concentrates fiercely on not gagging, on not pulling away. He thinks \u2014 _surrender._\n\n_Kattajjaq. Pirkusirtuk. Nipaquhiit._ All clack-clack names he half remembers from his dreams. All names the Real People around the world's circle of northern ice have for what they are doing now.\n\nShe begins with a short rhythmic series of notes.\n\nShe is playing his vocal cords like a bank of woodwinds' reeds.\n\nThe low notes rise out over the ice and blend with the pressure-cracking and the pulsing aurora light.\n\nShe repeats the rhythmic motif but this time leaves a short gap of silence between the notes.\n\nHe takes her breath from his lungs, adds his own, and blows back into her open mouth.\n\nShe has no tongue, but her vocal cords are intact. The notes they produce with his breath fluttering them are high and pure.\n\nShe blows music from his throat. He brings music from hers. The opening rhythmic motif quickens, overlaps, hurries itself. The range of notes becomes more complex \u2014 as much flute as oboe, as distinctly human as any voice, the throat-song can be heard for miles across the aurora-painted ice.\n\nEvery three minutes or so in the first half hour, they pause and gasp for breath. Many times in practice they have broken up in laughter here \u2014 he understands through her string-signs that this was part of the fun when it was only a woman's game, making the other throat-singer laugh \u2014 but there can be no laughter tonight.\n\nThe notes begin again.\n\nThe song takes on the quality of a single human voice singing, simultaneously bass-deep and flute-high. They can shape words by breathing through each other's vocal cords like this and now she does \u2014 speaking words in song through the night; she plays his throat and vocal cords like a complex instrument and the words take shape.\n\nThey improvise. When one changes rhythm, the other must always follow. In that sense, he knows now, it is very much like making love.\n\nHe finds the secret space to breathe in between sounds so that they can go longer and make deeper, purer notes. The rhythm quickens toward an almost climactic point, then slows, then quickens again. It is follow the leader, back and forth, one changing the tempo and rhythm, the other following like a lover responding, then the other taking the lead. They throat-sing each other this way for an hour, then two hours, sometimes going twenty minutes and more without stopping for a breath.\n\nThe muscles of his diaphragm hurt. His throat is on fire. The notes and rhythm now are as complicated as those created by any dozen instruments, as interlaced, complex, and ascending as the crescendo of a sonata or symphony.\n\nHe lets her lead. The single voice the two of them make, the sounds and words the two of them speak, are hers, through him. He surrenders.\n\nEventually she stops and falls to her knees next to him. They are both too exhausted to hold their heads up. They pant and wheeze like dogs after a six-mile run.\n\nThe ice has stopped its noises. The wind has ceased its hum. The aurora pulses more slowly overhead.\n\nShe touches his face, gets to her feet, and goes away from him, pulling the tent flap shut behind her.\n\nHe finds enough strength to stand and to shed the rest of his clothes. Naked, he does not feel the cold.\n\nA lead has opened to within thirty feet of where they made their music, and now he walks toward this. His heart will not slow its pounding.\n\nSix feet from the edge of the water he goes to both knees again and raises his face to the sky and closes his eyes.\n\nHe hears the thing rising from the water not five feet from him and hears the scraping of its claws on ice and the huff of its breath as it pulls itself out of the sea onto the ice and hears the ice groaning under its weight, but he does not lower his head nor open his eyes to look. Not yet.\n\nWater from its coming out of the sea laps against his bare knees and threatens to freeze him to the ice he kneels on. He does not move.\n\nHe smells the wet fur, the wet flesh, the bottom-of-the-ocean stink of it, and senses its aurora shadow falling over him, but he does not open his eyes to look. Not yet.\n\nOnly when his skin prickles and goose bumps rise at the heavy-mass presence seeming to surround him and only when its meat-eater's breath envelops him does he open his eyes.\n\nFur dripping like a priest's wet and clinging white vestments. Burn scars raw amid the white. Teeth. Black eyes not three feet from his own and looking deep into him, predator's eyes searching for his soul... searching to see if he has a soul. The massive triangular head bobs lower and blots out the throbbing sky.\n\nSurrendering only to the human being he wants to be with and to the human being he wants to become \u2014 never to the _Tuunbaq_ or to the universe that would extinguish the blue flame in his chest \u2014 he closes his eyes again, tilts back his head, opens his mouth, and extends his tongue exactly as Memo Moira taught him to do for Holy Communion.\n\nTALIRIKTUG\n\n_Lat. 68\u00b0 30\u2032 N., Long. 99\u00b0 W._\n\n_28 May, 1851_\n\nIn the spring of the year that their second child was born, a girl, they were visiting Silna's family in the God-Walking People's band headed by the old shaman Asiajuk when word came from a visiting hunter named Inupijuk that a band of the Real People far to the south had received _aituserk,_ gifts, of wood, metal, and other precious objects from dead _kabloona_ \u2014 white men.\n\nTaliriktug signed to Asiajuk, who translated the signs into questions for Inupijuk. It sounded as if the treasure might be knives, forks, and other artifacts from _Erebus_ 's and _Terror_ 's ship's boats.\n\nAsiajuk whispered to Taliriktug and Silna that Inupijuk was a _qavac_ \u2014 literally, \"a man from the south,\" but also a term in Inuktitut that denoted stupidity. Taliriktug nodded his understanding but continued to sign questions that the sour shaman passed on to the stupidly grinning hunter. Part of Inupijuk's social discomfort, Taliriktug knew, was that the hunter from the south had never been in the presence of _sixam ieua_ spirit-governors before and was not quite sure if Taliriktug and Silna were human beings or not.\n\nIt sounded as if the artifacts were real. Taliriktug and his wife went back to their guests' _iglu,_ where she nursed the baby and he thought about it. When he looked up, she was using string to sign.\n\n_We should go south,_ said the strings between her fingers. _If you want to._\n\nHe nodded.\n\nIn the end, Inupijuk agreed to guide them to the southeastern village and Asiajuk decided to come with them \u2014 very unusual, since the old shaman rarely traveled far these days. Asiajuk brought his best wife, Seagull \u2014 young Nauja of the _amooq_ big tits \u2014 who also carried her scars from the band's lethal encounter with the _kabloona_ three years earlier. She and the shaman were the only survivors of that massacre, but the girl showed no resentment toward Taliriktug. She was curious about the fates of the final _kabloona_ whom everyone knew had headed south across the ice three summers ago.\n\nSix hunters of the God-Walking People's band also wanted to come along \u2014 mostly out of curiosity and to hunt along the way, since the ice was breaking up very early in the strait this spring \u2014 so eventually they set out in several boats since leads were opening along the coastline.\n\nTaliriktug, Silna, and their two children chose to travel \u2014 as did four of the hunters \u2014 in their long double _qayaq,_ but Asiajuk was too old and had too much dignity to paddle a _qayaq_ anymore. He sat with Nauja in the center of a spacious, open _umiak_ as two of the young hunters paddled for him. No one minded waiting for the _umiak_ when there was no wind for its sails since the thirty-foot-long craft carried enough fresh food in it that they rarely had to stop to hunt or fish unless they wanted to. This way they could also bring their own _kamatik_ sledge in case they needed to travel across land. Inupijuk, the southern hunter, rode in the _umiak,_ as did six _Qimmiq_ \u2014 dogs.\n\nAlthough Asiajuk generously offered to let Silna and her children ride in his now-crowded _umiak,_ she string-messaged her preference for the _qayaq._ Taliriktug knew that his wife would never want any child of hers \u2014 certainly not Kanneyuk, the two-month-old \u2014 to be so close to the vicious dogs in such a tight space. Their two-year-old son, Tuugaq \u2014 \"Raven\" \u2014 had no fear of dogs, but he also had no choice in the matter. He rode in the niche in the _qayaq_ between Taliriktug and Silna. The baby, Kanneyuk (whose secret _sixam ieua_ name was Arnaaluk), rode in Silna's _amoutiq,_ an oversized babycarrying hood.\n\nThe morning they left was cold but clear and as they shoved off from the gravel beach the fifteen remaining members of the God-Walking band chanted their farewell-come-back song:\n\n_Ai yei yai ya na_\n\n_Ye he ye ye yi yan e ya quana_\n\n_Ai ye yi yai yana._\n\nOn their second night, the last before paddling and sailing south through leads from the _angilak qikiqtaq,_ or \"biggest island,\" that James Ross had named King William Land so long ago, ignoring the fact that the natives who had told him about it had kept calling it _qikiqtaq, qikiqtaq, qikiqtak_ \u2014 they camped less than a mile from the site of Rescue Camp.\n\nTaliriktug walked there alone.\n\nHe'd been back before. Two summers ago, only weeks after Raven was born, he and Silna had come here. That was only a little less than one year after the man Taliriktug used to be had been betrayed and ambushed and shot down like a dog, but already there was little sign that this had been a major campsite for more than sixty Englishmen. Except for a few tatters of canvas frozen into the gravel, the Holland tents had torn and blown away. All that remained were campfire rings and a few stone tent rings.\n\nAnd some bones.\n\nHe had found some long bones, bits of chewed vertebrae, only one skull \u2014 the lower jaw missing. Holding the skull in his hands two summers ago, he had prayed to God that this was not Dr. Goodsir.\n\nThese scattered and _nanuq_ -gnawed bones he had gathered up and interred with the skull in a simple stone tomb, setting a fork he'd found among the stones atop the heap of rocks the way the Real People, even the God-Walking People he'd spent the summer with, liked to do, sending helpful tools and beloved-by-the-dead possessions to the spirit world with the dead.\n\nEven as he did this, he'd realized that the _Inuit_ would have thought this an obscene waste of precious metal.\n\nHe'd then tried to think of a silent prayer he could say.\n\nThe prayers in Inuktitut he'd heard in the past three months were not appropriate. But in his awkward attempt to learn the language \u2014 even though he would never be able to utter a syllable of it aloud \u2014 he'd played a game that summer trying to translate the Lord's Prayer into Inuktitut.\n\nThat evening, standing by the cairn holding his crewmates' bones, he'd tried to think the prayer.\n\n_N\u00e2legauv\u00eet kailaule. Pijornajat pinatuale nuname sorlo kilangme..._\n\n_Our father, which art in heaven, hallowed be thy name._\n\nThat was as far as he'd been able to get two summers ago, but it felt like enough.\n\nNow, almost two years later, walking back to his wife from a Rescue Camp that was even emptier \u2014 the fork was gone and the cairn had been opened and plundered by Real People from the south, even the bones scattered where he could not find them \u2014 Taliriktug had to smile at his dawning realization that even if he were granted his biblical threescore and ten years, he was never going to master this language of the Real People.\n\nEvery word \u2014 even the simple nouns \u2014 seemed to have a score of variants, and the subtleties of the syntax were far beyond a middle-aged man who'd gone to sea as a boy and never learned even his Latin. Thank God he would never have to speak this language aloud. Straining to understand the click-clack flow of it gave him the kind of headaches he used to have when Silna first shared her dreams with him.\n\nThe Great Bear, for instance. The simple white bear. The God-Walking People and the other Real People he'd encountered in the past two years called it _nanuq,_ which was simple enough, but he also heard variants that might be written down \u2014 in English, since the Real People had no written language \u2014 as _nanoq, n\u00e4nuvak, nannuraluk, takoaq, pisugtooq,_ and _ayualunaq._ And now, from Inupijuk, this hunter from the south (who, he now knew, was not as stupid as Asiajuk insisted), he had learned that the Great Bear was also called _T\u00f4rn\u00e2rssuk_ by many of the southern bands of the Real People.\n\nFor a period of a few painful months \u2014 he was still healing then and learning how to eat and swallow all over again \u2014 he had been perfectly satisfied to have no name at all. When Asiajuk's band began calling him Taliriktug \u2014 \"Strong Arm\" \u2014 after an incident during a white-bear hunt that first summer when he had single-handedly hauled the carcass of the dead bear out of the water when a team of dogs and three hunters had failed (it had not been his superhuman strength, he knew, just that he'd been the only one to see where they'd snarled their harpoon line on a projection of ice), he hadn't minded the new name even though he had been happier without one. Asiajuk told him that he now carried the soul-memory of an earlier \"Strong Arm\" who had died by the hand of the _kabloona._\n\nMonths earlier, when he and Silence had come to the _iglu_ village so that she could have the help of the women during the birth of Raven, he had not been surprised to learn that the Real People Inuktitut name of his wife was Silna. He could see how she embodied the spirit of both Sila, the goddess of the air, and Sedna, the goddess of the sea. Of her secret _sixam ieua_ spirit-governor name, she would not or could not share it with him through her string-signing or dreams.\n\nHe knew his own secret name. On that first night of great misery after the _Tuunbaq_ had taken his tongue and older life away, he had dreamt his secret name. But he would never tell anyone, even Silna, whom he still called Silence in his sent-thoughts during their lovemaking and in their dreams.\n\nThe village was named Taloyoak and consisted of about sixty people and a scattering of more tents than snow-houses. There were even some snow-covered sod homes projecting out from the cliffs that would have grassy roofs come summer.\n\nThe people here were called Oleekataliks, which he thought meant \"Men with Capes,\" although the outer skins they wore on their shoulders looked more to him like Englishmen's woolen comforters than real capes. The head man was about Taliriktug's age and was handsome enough, although he had no teeth left, which made him look older than his years. The man was named Ikpakhuak, which Asiajuk told him meant \"the Dirty One,\" although as far as Taliriktug could see and smell, Ikpakhuak was no dirtier than the rest of them and cleaner than some.\n\nIkpakhuak's much-younger wife was named Higilak, which Asiajuk smirkingly explained meant \"the Ice House.\" But Higilak's manner was not in any way cold toward the strangers; she helped her husband welcome Taliriktug's band with warmth and an outpouring of hot food and gifts.\n\nHe realized that he would never understand these people.\n\nIkpakhuak and Higilak and their family served them _umingmak,_ musk ox steak, as their feast meal, which Taliriktug enjoyed well enough but which Silna, Asiajuk, Nauja, and the rest of his band had to choke down, since they were _Netsilik,_ \"People of the Seal.\" After all the meeting ceremonies and meals were over, he managed through his interpreted signs to move the conversation to the _kabloona_ gifts.\n\nIkpakhuak acknowledged that the People of the Cape had such treasures, but before showing his guests, he asked that Silna and Taliriktug show everyone in the village their magic. The Oleekataliks had not met any _sixam ieua_ in the lifetimes of most of the villagers \u2014 although Ikpakhuak had known Silna's father, Aja, decades ago \u2014 and Ikpakhuak politely asked if Silna and Taliriktug would fly around the village a little bit and perhaps turn themselves into seals, not bears, please.\n\nSilna explained \u2014 through her string-signs interpreted by Asiajuk \u2014 that the two spirit-governors-of-the-sky chose not to do that, but that they would both show the hospitable Oleekataliks where the _Tuunbaqs_ had taken their tongues, and that her _kabloona sixam ieua_ husband would give them the rare treat of seeing his scars... scars inflicted in a terrible battle with evil spirits years ago.\n\nThis completely satisfied Ikpakhuak and his people.\n\nAfter this dog-and-pony cum scar show was over, Taliriktug managed to get Asiajuk to bring the subject back to the _kabloona_ gifts.\n\nIkpakhuak instantly nodded, clapped his hands, and sent boys to gather the treasures. They were handed around the circle.\n\nThere were various pieces of wood, one of them split from a well-handled marlin's spike.\n\nThere were gold buttons bearing the Naval anchor motif of the Discovery Service.\n\nThere was a fragment of a lovingly embroidered man's undervest.\n\nThere was a gold watch, the chain it may have hung from, and a handful of coins. The initials on the back of the watch, CFDV \u2014 Charles Des Voeux.\n\nThere was a silver pencil case with the initals EC on the inside.\n\nThere was a gold-medal citation once presented to Sir John Franklin from the Admiralty.\n\nThere were silver forks and spoons bearing the crests of Franklin's various officers.\n\nThere was a small china plate with the name SIR JOHN FRANKLIN written out on it in colored enamel.\n\nThere was a surgeon's knife.\n\nThere was a mahogany portable lap writing desk that the man now holding it recognized because it had been his own.\n\n_Did we really haul all this shit hundreds of miles in our boats?_ thought Crozier. _And before that, thousands of miles from England? What were we thinking?_ He felt that he might throw up and had to close his eyes until the nausea passed.\n\nSilence touched his wrist. She had sensed the tilt and shift in him. He looked into her eyes to assure her that he was still there, although he wasn't. Not really. Not completely.\n\nThey paddled along the coast to the west, toward the mouth of Back's River.\n\nIkpakhuak's Oleekataliks had been vague, even evasive, about where they had found their _kabloona_ treasures \u2014 some said they were from a place called Keenuna, which sounded like one of a series of islets in the strait just south of King William Island \u2014 but the majority of hunters said they had come across the riches west of Taloyoak at a place called Kugluktuk, which Asiajuk translated as \"Place of Falling Waters.\"\n\nTo Crozier, it sounded like the first small waterfall he'd read about that Back had said was just upriver from the mouth of Back's Great Fish River.\n\nThey spent a week searching there. Asiajuk and his wife and three of the hunters stayed with the _umiak_ at the mouth of the river, but Crozier and Silence with their children, the still-curious hunter Inupijuk, and the other hunters paddled their _qayaqs_ upriver the three miles or so to the first low falls.\n\nHe found some barrel staves there. A leather boot sole with holes where screws had been driven through. Buried in the sand and mud of the riverbank, he uncovered an eight-foot length of curved and once-polished oak that might have been from the gunwale of one of the cutters. (It would have been pure treasure to the Oleekataliks.) Nothing else.\n\nThey were leaving in defeat, paddling downstream to the coast, when they came upon an older man, his three wives, and their four runny-nosed children. Their tent and caribou skins were on the wives' backs and they had come to the river, so the man said, to fish. He had never seen a _kabloona_ before, much less two _sixam ieua_ spirit-governors without tongues and was very frightened, but one of the hunters with Crozier calmed his fears. The old man was named Puhtoorak and was a member of the Qikiqtarqjuaq band of the Real People.\n\nAfter food and pleasantries had been exchanged, the old man asked what they were doing so far from the God-Walking People's northern lands, and when one of the hunters explained that they were looking for living or dead _kabloona_ who might have come this way \u2014 or their treasures \u2014 Puhtoorak said that he hadn't heard of _kabloona_ on this river but mentioned between large bites of their gift of seal meat, \"Last winter I saw a big _kabloona_ boat \u2014 as large as an iceberg \u2014 with three sticks coming up out of it, stuck in the ice just off Utjulik. I think there were dead _kabloona_ in its stomach. Some of our younger men went into the thing \u2014 they had to use their star-shit stone axes to chop a hole in its side \u2014 but they left all the wood and metal treasures where they were because they said the three-sticks house was haunted.\"\n\nCrozier looked at Silence. _Did I understand him correctly?_\n\n_Yes._ She nodded. Kanneyuk began to cry, and Silna parted her summer parka and gave the baby her breast.\n\nCrozier stood on a cliff and looked out at the ship in the ice. It was HMS _Terror._\n\nIt had taken them eight days of travel from the mouth of the Back River west to this part of the coast of Utjulik. Through the God-Walking People hunters who understood his signs, Crozier had offered bribes to Puhtoorak if the old man would agree to bring his family with him and come along to show them the way to the _kabloona_ boat with the three sticks rising from its roof, but the old Qikiqtarqjuaq wanted nothing more to do with the haunted _kabloona_ three-stick house. Even though he had not gone in with the young men last winter, he had seen that the thing was tainted with _piifixaaq_ \u2014 the kind of unhealthy ghost-spirits that haunted a bad place.\n\nUtjulik was an Inuit name for what Crozier had known from maps as the west coast of the Adelaide Peninsula. The open-water leads had ended not very far west of the inlet leading south to Back's River \u2014 the narrowing strait there was solid ice pack \u2014 so they'd had to beach and hide the _qayaqs_ and Asiajuk's _umiak_ and continue on with the six dogs pulling the heavily solid thirteen-foot _kamatik._ Using the kind of inland dead reckoning that Crozier knew he would never master, Silence led them the twenty-five miles or so straight across the interior neck of the peninsula to the area of the west coast where Puhtoorak had said he'd seen the ship... even, he confessed, stood on its deck.\n\nAsiajuk had not wanted to leave his comfortable boat when it came time to head cross-country. If Silna, one of the God-Walking People's most revered spirit-governors, had not signed her sincere request that he join them \u2014 a request from a _sixam ieua_ was a command to even the surliest of shamans \u2014 Asiajuk would have ordered his hunters to take him home. As it was, he rode in style under furs in the _kamatik_ and even helped from time to time by throwing pebbles at the straining dogs and shouting, \"Haw! Haw! Haw!\" when he wanted them to go left and \"Gee! Gee! Gee!\" when he wanted them to go right. Crozier wondered if the old shaman was rediscovering the youthful pleasures of sledge travel by dog team.\n\nNow it was late afternoon of their eighth day and they were looking down at HMS _Terror._ Even Asiajuk seemed intimidated and subdued.\n\nPuhtoorak's best description of the precise location had been that the three-stick house \"was frozen in the ice near an island about five miles due west\" of a certain point and that he and his hunting party \"then had to walk about three miles north across smooth ice to reach the ship after crossing several islands on their walk from the point. They could see the ship from a cliff at the north end of the large island.\"\n\nOf course, Puhtoorak had not used the term \"miles\" nor \"ship\" nor even \"point.\" What the old man had said was that the three-stick _kabloona_ house with an _umiak's_ hull was a certain number of hours' walk west of _tikerqat,_ which means \"Two Fingers,\" which the Real People called two narrow points along this stretch of the Utjulik coastline, and then somewhere close to the north end of a large island there.\n\nCrozier and his band of ten people \u2014 the hunter from the south, Inupijuk, was sticking with them to the bitter end \u2014 had walked due west across rough ice from the Two Fingers and crossed two small islands before reaching a much larger one. They found a cliff dropping almost a hundred feet to the ice pack at the north end of that large island.\n\nTwo or three miles out in the ice, the three masts of HMS _Terror_ rose at a raked angle toward the low clouds.\n\nCrozier wished he had his old telescope, but he didn't need it to identify the masts of his old command.\n\nPuhtoorak had been right \u2014 the ice for this last part of the walk was much smoother than the jumbled shore and pack ice between the mainland and the islands. Crozier's captain's eye saw why: there lay a string of smaller islands to the east and north, creating a sort of natural seawall sheltering this fifteen- or twenty-nautical-square-mile patch of sea from the prevailing winds out of the northwest.\n\nHow _Terror_ could have ended up here, almost two hundred miles south of where she had been frozen fast near _Erebus_ for almost three years, was beyond Crozier's powers of speculation.\n\nHe would not have to speculate much longer.\n\nThe Real People, including the God-Walking People, who lived in the shadow of a living monster year in and year out, approached the ship with obvious anxiety. All of Puhtoorak's talk of haunting ghosts and bad spirits had worked its effect on them \u2014 even on Asiajuk, Nauja, and the hunters who'd not been there to hear the old man. Asiajuk himself was muttering incantations, ghost-chasing chants, and keeping-safe prayers all during their walk out onto the ice, which added to no one's sense of security. When a shaman gets nervous, Crozier knew, everyone gets nervous.\n\nThe only one who would walk next to Crozier at the lead of the procession was Silence, carrying both the children.\n\n_Terror_ was listing about twenty degrees to port, her bow aimed toward the northeast and her masts raking to the northwest, with too much of her starboard-side hull showing above the ice. Surprisingly, there was one anchor deployed \u2014 the port-side bow anchor \u2014 its hawser disappearing into the thick ice. Crozier was surprised because he guessed the bottom to be at least twenty fathoms deep here \u2014 perhaps much more \u2014 and because there were little inlets all along the northern curves of the islands behind him. At the very least \u2014 unless there was a storm \u2014 a prudent captain seeking safe harbour would have brought the ship into the strait on the east side of the large island he'd just walked from, dropping anchor between the big island \u2014 whose cliffs would have blocked the wind \u2014 and the three smaller islands, none more than about two miles long, east of there.\n\nBut _Terror_ was here, about two and a half miles out from the north end of that large island, with her anchor dropped into deep water and all of her exposed to the inevitable storms from the northwest.\n\nOne walk around the ship and a look up at her canted deck from the lower northwest side solved the mystery of why Puhtoorak's hunting band had been forced to chop their way through the hull on the raised starboard side, probably a splintered and battered and already near-breached hull, in order to gain entry: all the top-deck hatches were battened and sealed.\n\nCrozier returned to the man-sized hole the band had smashed into the exposed and weathered hull. He thought he could squeeze through. He remembered Puhtoorak saying that his young hunters had used their star-shit axes to force their way in here, and he had to smile to himself despite the surge of painful emotions he was feeling.\n\n\"Star shit\" was what the Real People called falling stars and the metal they used from the falling stars they found lying on the ice. Crozier had heard Asiajuk talk about _uluriak anoktok_ \u2014 \"star shit falling from the sky.\"\n\nCrozier wished he had a star-shit blade or axe with him right now. The only weapon he carried was a basic work knife with a blade made from walrus ivory. There were harpoons on the _kamatik_ but they weren't his \u2014 he and Silence had left theirs with their _qayaq_ a week ago \u2014 and he didn't want to ask to borrow one just to go into the ship with it.\n\nBack at that sledge, forty feet behind them, the _Qimmiq_ \u2014 the large dogs with their uncanny blue and yellow eyes and souls they shared with their masters \u2014 were barking and growling and howling and snapping at one another and at anyone who came close to them. They did not like this place.\n\nCrozier signed to Silence, _Sign Asiajuk to ask them if anyone wants to come in with me._\n\nShe did so quickly, using just her fingers without string. Even so, the old shaman always understood her much more quickly than he could make out Crozier's clumsy signs.\n\nNone of the Real People wanted to go through that hole.\n\n_I will see you in a few minutes,_ Crozier signed to Silence.\n\nShe actually smiled. _Do not be stupid,_ she signed. _Your children and I are coming with you._\n\nHe squeezed in and Silence followed a second later, carrying Raven in her arms and Kanneyuk in the soft-hide baby-holder she sometimes carried on straps against her chest. Both children were sleeping.\n\nIt was very dark.\n\nCrozier realized that Puhtoorak's young hunters had hacked their way in to the orlop deck. This was lucky for them since if they'd tried a bit lower amidships here, they would have run into the iron of the coal bins and the water-storage tanks on the hold deck and never could have chopped their way through, even with star-shit heads on their axes.\n\nTen feet in from the hole in the hull it was too dark to see, so Crozier found his way by memory, holding Silence's hand as they walked ahead down the canted deck and then turned aft.\n\nAs his eyes adapted to the dark, there was just enough light filtering in for Crozier to make out that the heavy padlocked door to the Spirit Room and to the Gunner's Storeroom farther aft had been smashed open. He had no idea whether this had been the work of Puhtoorak's men, but he doubted it. Those doors had been left padlocked for a reason and they were the first place any white men returning to _Terror_ would want to go.\n\nThe rum casks \u2014 they'd actually had so much rum they'd had to leave casks of it behind when they took to the ice \u2014 were empty. But casks of gunpowder remained, as well as boxes and barrels of shot, canvas bags of cartridges, almost two bulkheads' lengths of muskets still set in their grooved places \u2014 they'd had too many to carry \u2014 and two hundred bayonets still hanging from their fittings along the rafters and beams.\n\nThe metal in this room alone would make Asiajuk's band of Real People the richest men in their world.\n\nThe remaining gunpowder and shot would feed a dozen large bands of the Real People for twenty years and make them undisputed lords of the arctic.\n\nSilence touched his bare wrist. It was too dark to sign, so she thought-sent. _Do you feel it?_\n\nCrozier was astonished to hear that \u2014 for the first time \u2014 her shared thoughts were in English. She had either dreamt his dreams even more deeply than he'd imagined, or she had been very attentive during her months aboard this very ship. It was the first time they'd shared thoughts in words while awake.\n\n_Ii,_ he thought back to her. _Yes._\n\nThis place was bad. Memories haunted it like a bad smell.\n\nTo lighten the tension, he led her forward again, pointed toward the bow, and thought-sent her an image of the forward cable locker on the deck below.\n\n_I was always waiting for you,_ she sent. The words were so clear that he thought they might have been spoken aloud in the darkness, except for the fact that neither of the children awoke.\n\nHis body began to shake with emotion at the thought of what she had just told him.\n\nThey went up the main ladder to the lower deck.\n\nIt was much brighter up here. Crozier realized that \u2014 finally \u2014 daylight was actually coming through the Preston Patent Illuminators that punctuated the deck above them. The curved glass was opaque with ice, but \u2014 for once \u2014 not covered with snow or tarps.\n\nThe deck looked empty. All of the men's hammocks had been carefully folded and stowed away, their mess tables cranked up between the beams to the overhead deck, and their sea chests pushed aside and carefully stowed. The huge Frazer's Patent Stove in the center of the forward berthing area was dark and cold.\n\nCrozier tried to recall if Mr. Diggle was still alive when he, the captain, had been lured onto the ice and shot. It was the first time he'd thought of that name \u2014 _Mr. Diggle_ \u2014 in a long time.\n\n_It's the first time I've thought in my own tongue for a long time._\n\nCrozier had to smile at that. \"In my own tongue.\" If there really was a goddess like Sedna who ruled the world, her real name was Bitch Irony.\n\nSilence tugged him aft.\n\nThe first officers' cabins and mess rooms they looked into were empty.\n\nCrozier found himself wondering which men could have possibly reached _Terror_ and sailed her south.\n\nDes Voeux and his men from Rescue Camp?\n\nHe felt almost certain that Mr. Des Voeux and the others would have continued south in the boats toward Big Fish River.\n\nHickey and his men?\n\nFor Dr. Goodsir's sake, he hoped so, but he did not believe it. Except for Lieutenant Hodgson, and Crozier suspected he had not lived for very long in that company of cutthroats, there was hardly a man in that pack who could sail, much less navigate, _Terror._ He doubted if they had been able to sail and navigate the one small boat he'd given them.\n\nThat left the three men who had left Rescue Camp to hike overland \u2014 Reuben Male, Robert Sinclair, and Samuel Honey. Could a captain of the fo'c'sle, a captain of the foretop, and a blacksmith sail HMS _Terror_ almost two hundred miles south through a maze of leads?\n\nCrozier felt dizzy and a little nauseated from thinking about the men's names and faces again. He could almost hear their voices. He _could_ hear their voices.\n\nPuhtoorak had been correct: this place was now home to _piifixaaq_ \u2014 resentful ghosts that stayed behind to haunt the living.\n\nThere was a corpse in Francis Rawdon Moira Crozier's bunk.\n\nAs far as they could tell without lighting lamps and going down into the hold and orlop deck, this was the only dead body on board.\n\n_Why did he decide to die in my bunk?_ wondered Crozier.\n\nHe had been a man about Crozier's height. His clothes \u2014 he'd died under blankets in a peacoat and watch cap and wool trousers, which was odd since they must have been sailing in full summer \u2014 gave no clue to his identity. Crozier had no wish at all to go through his pockets.\n\nThe man's hands, exposed wrists, and neck were brown and mummified, shriveled, but it was his face that made Crozier wish that the Preston Patent Illuminator overhead was not allowing in as much light as it did.\n\nThe dead man's eyes were brown marbles. His hair and beard were so long and wild that it seemed quite possible that they had continued growing for months after the man's death. His lips had shriveled away to nothing and been pulled back far from the teeth and gums by tendons stretching and contracting.\n\nIt was the teeth that were so upsetting. Rather than having fallen out from scurvy, the front teeth were all there and very broad and an ivory yellow and impossibly long \u2014 three inches long, at least \u2014 as if they had grown the way a rabbit's or rat's teeth continue growing until, unless worn down by gnawing something solid, they curve in and cut the creature's own throat.\n\nThese dead man's rodent teeth were impossible, but Crozier was looking at them in the clear, grey evening light coming down through the domed skylight of his old cabin. It was not, he realized, the first impossible thing he had seen or experienced in the last few years. He suspected it might not be his last.\n\n_Let's go,_ he signed to Silence. He did not want to thought-send here where things were listening.\n\nHe had to use a fire axe to hack his way up through the sealed and nailed-shut main hatch. Rather than ask himself who had sealed it and why \u2014 or if the corpse below had been a living man when the hatch had been sealed so tightly above him \u2014 he threw the axe aside, clambered up, and helped Silence up the ladder.\n\nRaven was fussing himself awake, but Silence rocked him and he began to snore softly again.\n\n_Wait here,_ he signed and went below again.\n\nFirst he brought the heavy theodolite and several of his old manuals up, took a quick reading of the sun, and jotted his bearings in the margin of the salt-stained book. Then he carried theodolite and books below and tossed them aside, knowing that fixing this ship's position one last time was perhaps the most useless thing he'd ever done in a long life of doing useless things. But he also knew he'd had to do it.\n\nJust as he had to do what he did next.\n\nIn the dark Gunner's Storeroom on the orlop deck he split open three successive kegs of gunpowder \u2014 pouring the contents of the first on the orlop deck and down the ladder into the hold deck (he would not go down there), the contents of the second keg everywhere on the lower deck (and especially inside the open door of his own cabin), and the contents of the third keg in black trails along the canted upper deck where Silence waited with his children. Asiajuk and the others on the ice had come around to the port side and now watched from thirty yards away. The dogs continued to howl and strain to get away, but Asiajuk or one of the hunters had staked them to the ice.\n\nCrozier wanted to stay in the open air, even with the afternoon light waning, but he made himself go below to the orlop deck again.\n\nCarrying the last keg of lamp oil left on the ship, he spilled a trail of it on all three decks, taking care to douse the door and bulkhead of his own cabin. His only hesitation was at the entrance to the Great Room where hundreds upon hundreds of spines of books stared back at him.\n\n_Dear God, would it hurt if I took just a few of those to help get through the dark winters ahead?_\n\nBut they now carried the dark _inua_ of the death-ship in them. Almost weeping, he dashed lamp oil across them.\n\nWhen he was finished pouring the last of the fuel on the upper deck, he flung the empty cask far out over the ice.\n\n_One last trip below,_ he promised Silence with his fingers. _Go on to the ice now with the children, my beloved._\n\nThe Lucifer matches were where he had left them in the drawer of his desk three years earlier.\n\nFor a second he was sure that he could hear the bunk creak and the nest of frozen blankets stir as the mummified thing behind him reached for him. He could hear the dry tendons in the dead arm stretching and snapping as the brown hand with its long brown fingers and too-long yellow nails slowly rose.\n\nCrozier did not turn to look. He did not run. He did not look back. Carrying the matches, he left his cabin slowly, stepping over the lines of black gunpowder and deck boards stained by the whale oil.\n\nHe had to go down the main ladder to throw the first match. The air was so bad here that the match almost refused to light. Then the gunpowder lit with a _whump,_ ignited a bulkhead he'd soaked with oil, and raced forward and aft in the dark along its own trail of fire.\n\nKnowing that the orlop deck fire alone would have been enough \u2014 these timbers were dried to tinder after six years in this arctic desert \u2014 he still took time to light the lines of powder on the lower deck and open upper deck.\n\nThen he jumped the ten feet to the ramp of ice on the west side of the ship and cursed as his never fully recovered left leg announced its pain. He should have clambered down the rigged rope ladders here as Silence obviously had had the sense to do.\n\nLimping like the old man he was sure he would soon be, Crozier walked out onto the ice to join the others.\n\nThe ship burned for almost an hour and a half before it sank.\n\nIt was an incredible conflagration. Guy Fawkes Day above the Arctic Circle.\n\nHe definitely wouldn't have needed the gunpowder or lamp oil, he realized while watching. The timbers and canvas and boards were so leached of moisture that the entire ship went up like one of the incendiary mortar bombs it had been designed to launch so many decades ago.\n\n_Terror_ would have sunk anyway, as soon as the ice thawed here in a few weeks or months. The axe-hole in its side had been its death wound.\n\nBut that is not why he burned it. If asked \u2014 which he never would be \u2014 he could not have explained why it had to be burned. He knew that he did not want \"rescuers\" from British ships poring over the abandoned ship, carrying tales of it home to frighten the ghoulish citizens of England and to spur Mr. Dickens or Mr. Tennyson on to new heights of maudlin eloquence. He also knew that it wouldn't have been only tales these rescuers would have brought back to England with them. Whatever had taken possession of the ship was as virulent as the plague. He had seen that with the eyes of his soul and smelled it with all his human and _sixam ieua_ senses.\n\nThe Real People cheered when the burning masts collapsed.\n\nThey'd all been forced to move back a hundred yards. _Terror_ burned its own death-hole in the ice, and shortly after the flaming masts and rigging fell, the burning ship began hissing and bubbling its way to the depths.\n\nThe noise from the fire woke the children and the flames so heated the air out here on the ice that all of them \u2014 his wife, scowling Asiajuk, big-titted Nauja, the hunters, happily grinning Inupijuk, even Taliriktug \u2014 took off their outer parkas and piled them onto the _kamatik._\n\nWhen the show was over and the ship was sunk and the sun was also sinking toward the south so that their shadows leapt long across the greying ice, still they stayed to point and enjoy the steam rising and celebrate the bits of burning debris still scattered here and there on the ice.\n\nThen the band finally turned back toward the big island and then the smaller islands, planning to cross the ice to the mainland before they would make camp for the night. The sunlight shining until after midnight helped their march. All of them wanted to be off the ice and away from this place before the few hours of dimness and full darkness came. Even the dogs quit barking and snarling and seemed to pull harder when they passed the smaller island on their way back in to the land. Asiajuk was asleep and snoring under his robes on the sledge, but both the babies were wide awake and ready to play.\n\nTaliriktug took the squirming Kanneyuk in his left arm and put his right arm around Silna-Silence. Raven, still being carried by his mother, was petulantly trying to slap her arms away and force her to put him down so he could try to walk on his own.\n\nTaliriktug wondered, not for the first time, how a father and mother without tongues were going to discipline a headstrong boy. Then he remembered, not for the first time, that he now belonged to one of the few cultures in the world that did not bother to discipline their headstrong boys or girls. Raven already had an _inua_ of some worthy adult in him. His father would just have to wait to see just how worthy it was.\n\nThe Francis Crozier _inua_ still alive and well in Taliriktug had no illusions about life being anything but poor, nasty, brutish, and short.\n\nBut perhaps it did not have to be solitary.\n\nHis arm around Silna, trying to ignore the raucous snores from the shaman and the fact that baby Kanneyuk had just pissed on her father's best summer parka, while also ignoring the petulant swats and mewling noises from his squirming son, Taliriktug and Crozier continued walking east across the ice toward solid ground.\nACKNOWLEDGMENTS\n\nI wish to acknowledge the following sources for providing information in my writing of _The Terror:_\n\nThe idea to write about this era of Arctic exploration came from a short comment, almost a footnote, about the Franklin Expedition that I encountered in Sir Ranulph Fiennes _Race to the Pole: Tragedy, Heroism, and Scott's Antarctic Quest_ (Hyperion, \u00a9 2004), the pole being raced to in this instance being the South Pole.\n\nThree books that were especially important to me in the early stages of research were _Ice Blink: The Tragic Fate of Sir John Franklin's Lost Polar Expedition_ by Scott Cookman (John Wiley & Sons, Inc., \u00a9 2000); _Frozen in Time: The Fate of the Franklin Expedition_ by Owen Beattie and John Geiger (Greystone Books, Douglas & McIntyre, \u00a9 1987); and _The Arctic Grail: The Quest for the Northwest Passage and the North Pole, 1818\u20131909_ by Pierre Berton (Second Lyons Press Edition, \u00a9 2000).\n\nThese books led me to some of their invaluable sources, including _Narrative of a Journey to the Shores of the Polar Sea_ (John Murray, \u00a9 1823) and _Narrative of a Second Expedition to the Shores of the Polar Sea_ (John Murray, \u00a9 1828), both by Sir John Franklin; _Sir John Franklin's Last Arctic Expedition_ by Richard Cyriax (ASM Press, \u00a9 1939); _The Bomb Vessel_ by Chris Ware (Naval Institute Press, \u00a9 1994); _A Narrative of the Discovery of the Fate of Sir John Franklin_ by F. L. M'Clintock (John Murray, \u00a9 1859); _In Quest of the Northwest Passage_ (Longmans, Green & Co, \u00a9 1958); _Journal of a Voyage in Baffin's Bay and Barrow Straits, in the Years 1850\u201351, Performed by H.M. Ships \"Lady Franklin\" and \"Sophia\" Under the Command of Mr. William Penny, in Search of the Missing Crew of H.M. Ships \"Erebus\" and \"Terror\"_ by Peter Sutherland (Longman, Grown, Green, and Longmans, \u00a9 1852); and _Arctic Expeditions in Search of Sir John Franklin_ by Elisha Kent Kane (T. Nelson & Sons, \u00a9 1898).\n\nOther sources frequently consulted include _Prisoners of the North: Portraits of Five Arctic Immortals_ by Pierre Berton (Carroll & Graff, \u00a9 2004); _Ninety Degrees North: The Quest for the North Pole_ by Fergus Fleming (Grove Press, \u00a9 2001); _The Last Voyage of the Karluk: A Survivor's Memoir of Arctic Disaster_ by William Laird McKinlay (St. Martin's Griffin Edition, \u00a9 1976); _A Sea of Words: A Lexicon and Companion for Patrick O'Brian's Seafaring Tales_ by Dean King (Henry Holt & Co., \u00a9 1995); _The Ice Master: The Doomed 1913 Voyage of the_ Karluk by Jennifer Niven (Hyperion, \u00a9 2000); _Rowing to Latitude: Journeys Along the Arctic's Edge_ by Jill Fredston (North Point Press, a Division of Fartar, Straus and Giroux, \u00a9 2001); _Weird and Tragic Shores: The Story of Charles Francis Hall, Explorer_ by Chauncey Loomis (Modern Library Paperback Edition, \u00a9 2000); _The Crystal Desert: Summers in Antarctica_ by David G. Campbell (Mariner Books, Houghton Mifflin, \u00a9 1992); _The Last Place on Earth: Scott and Amundsen's Race to the South Pole_ by Roland Huntford (The Modern Library, \u00a9 1999); _North to the Night: A Spiritual Odyssey in the Arctic_ by Alvah Simon (Broadway Books, \u00a9 1998); _In the Land of White Death: An Epic Story of Survival in the Siberian Arctic_ by Valerian Albanov (Modern Library, \u00a9 2000); _End of the Earth: Voyages to Antarctica_ by Peter Matthiessen (National Geographic, \u00a9 2003); _Fatal Passage: The Story of John Rae, the Arctic Hero Time Forgot_ by Ken McGoogan (Carrol & Graf, \u00a9 2001); _The Worst Journey in the World_ by Apsley Cherry-Garrard (National Geographic, \u00a9 1992 and 2000); and _Shackleton_ by Roland Huntford (Fawcett Columbine, \u00a9 1985).\n\nOther sources consulted include _The Inuit_ by Nancy Bonvillain (Chelsea House Publications, \u00a9 1995); _Eskimos_ by Kaj Birket-Smith (Crown, \u00a9 1971); _The Fourth World_ by Sam Hall (Knopf, \u00a9 1987); _Ancient Land: Sacred Whale \u2014 The Inuit Hunt and Its Rituals_ by Tom Lowenstein (Farrar, Straus and Giroux, \u00a9 1993); _The Igloo_ by Charlotte and David Yue (Houghton Mifflin, \u00a9 1988); _Arctic Crossing_ by Jonathan Waterman (Knopf, \u00a9 2001); _Hunters of the Polar North \u2014 The Eskimos_ by Wally Herbert (Time-Life Books, \u00a9 1981); _The Eskimos_ by Ernest S. Burch Jr. (University of Oklahoma Press, \u00a9 1988); and _Inuit: When Words Take Shape_ by Raymond Brousseau (Editions Gl\u00e9nat, \u00a9 2002).\n\nMy sincere thanks to Karen Simmons for finding... and returning... many of these later sources.\n\nInternet sources were too many to list, but they include The Aujaqsquittuq Project: Documenting Arctic Climate Change; Spiritism On Line; The Franklin Trial; Enchanted Learning: Animals \u2014 Polar Bear (Ursus martimus); Collections Canada; Digital Library Upenn; Radiworks.cbe; Wordgumbo \u2014 Canadian Inuit-English Dictionary; Alaskool English to I\u00f1upia; Inuktitut Language Phrases; Darwin Wars; Cangeo.ca Special Feature \u2014 Sir John Franklin Expedition; and SirJohnFranklin.com.\n\nThe Internet was also my primary access route to primary source materials, including the Francis Crozier Collection, held at Scott Polar Research Institute, University of Cambridge; the Sophia Cracroft Collection (ibid); Sophia Cracroft correspondence; Notes for the Memoir of Jane Franklin. Also included are details of ships' musters, dates, and official documents from the Records of the British Admiralty, Naval Forces, and Royal Marines; records from the Home Office (UK), and legal documents concerning the investigation into the Goldner food canning irregularities from the Supreme Court of Judicature (UK).\n\nUseful illustrations and maps came from _Harper's Weekly_ (April 1851), _The Athenaeum_ (February 1849), _Blackwood's Edinburgh Magazine_ (November 1855), and other sources.\n\nThe letter from Dr. Harry D. S. Goodsir to his uncle, 2 July 1845, is in the collection of the Royal Scottish Geographical Society and was quoted in _Frozen in Time: The Fate of the Franklin Expedition_ by Owen Beattie and John Geiger.\n\nFinally, my sincere thanks to my agent, Richard Curtis; to my first editor at Little, Brown, Michael Mezzo; to my current editor, Reagan Arthur; and \u2014 as always \u2014 to Karen and Jane Simmons for encouraging me to go on and then waiting for me while I _was_ on this particularly long Arctic expedition.\n ABOUT THE AUTHOR\n\nDan Simmons is the Hugo Award\u2013winning author of _Hyperion_ and _The Fall of Hyperion_ and their sequels, _Endymion_ and _The Rise of Endymion._ He has written the critically acclaimed suspense novels _Darwin's Blade_ and _The Crook Factory,_ as well as other highly respected works including _Summer of Night,_ its sequel, _A Winter Haunting,_ and _Song of Kali, Carrion Comfort,_ _Ilium,_ and _Olympos._ Simmons makes his home in Colorado.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nGuide to\n\nAmor Towles's\n\nA Gentleman in Moscow\n\nA Novel\n\nby\n\nInstaread\nPlease Note\n\nThis is a companion to the original book.\n\nCopyright \u00a9 2016 by Instaread. All rights reserved worldwide. No part of this publication may be reproduced or transmitted in any form without the prior written consent of the publisher.\n\nLimit of Liability\/Disclaimer of Warranty: The publisher and author make no representations or warranties with respect to the accuracy or completeness of these contents and disclaim all warranties such as warranties of fitness for a particular purpose. The author or publisher is not liable for any damages whatsoever. The fact that an individual or organization is referred to in this document as a citation or source of information does not imply that the author or publisher endorses the information that the individual or organization provided. This concise companion is unofficial and is not authorized, approved, licensed, or endorsed by the original book's author or publisher.\nTable of Contents\n\n****Summary\n\nMain Characters\n\nAnalysis\n\nCharacter Analysis\n\nRelationships\n\nThemes\n\nAuthor's Style\n\nReferences\n\n# Summary\n\n_A Gentleman in Moscow_ by Amor Towles is the story of a Russian aristocrat-turned-waiter who lives 32 years of his life under house arrest at the Hotel Metropol in Moscow. Set in post-revolutionary Russia, the novel follows its protagonist, Count Alexander Ilyich Rostov, as he develops new friendships, family, and loves, all while confined within the walls of the Metropol.\n\nIn 1922, in the wake of the Russian Revolution, Rostov, originally a gentleman from Russia's Nizhny Novgorod province, is deemed a threat to the Communist Party and sentenced to house arrest at the hotel where he has been living in luxury. The party is suspicious of Count Rostov, who left Russia after the tsar's execution in 1918, but returned four years later. What saves Rostov from being executed is a single poem published in 1913 espousing revolutionary ideals, to which he claims authorship.\n\nUpon sentencing, the Count is forced to move his things from his suite to a tiny cell off a narrow corridor on the hotel's sixth floor. In his new room, there's little space for more than his bed. Among the things the Count brings are a portrait of his deceased sister Helena and his desk, which has a hollow leg filled with gold pieces.\n\nOne day in the Piazza, the glass-ceilinged hotel courtyard, Rostov meets a precocious Ukrainian nine-year-old, Nina Kulikova. Nina sits down at his table and asks whether it's true that he's a count and whether he's ever met any princesses or fought in any duels. Thus begins their long friendship.\n\nNina, whose widower father is posted in Moscow just for a time, is not enrolled in school. Under a kind of house arrest herself, Nina takes the Count on excursions throughout the hotel. They explore the furnace room, electrical room, and silver-service room. Nina, it turns out, possesses a passkey to the hotel.\n\nBack in his cell, the Count discovers that his closet was built over an old doorframe. He removes the planks over the door opening and enters the vacant cell next door, which is identical to his own. The Count transforms the spare room into a secret study.\n\nOne day, the Count's old friend Mikhail \"Mishka\" Fyodorovich Mindich, whom he knows from their university days in St. Petersburg, arrives at the hotel. Mishka is there to attend the Russian Association of Proletariat Writers. He fashions himself a poet of the new age and the \"art of action.\" [1] The two men reconnect and talk about their favorite subject, philosophy of history.\n\nOn Christmas Eve, Nina gives the Count a package. She is about to go away with her father for a while. When the Count opens the package, he finds another one inside. He opens that one to find another. He continues to unwrap packages until finally, in the last, tiny box, he finds his gift\u2014the hotel passkey.\n\nA year after his sentencing, the Count meets a \"willowy\" actress, Anna Urbanova, in the hotel. She invites him to her suite, and soon they become lovers. Around this same time, Nina returns to Moscow and starts school. She becomes increasingly curious. The Count assists his young friend with mathematics and physics experiments in the ballroom.\n\nAt the hotel bar, the Count meets a British aristocrat who has a particular interest in Russia. The Brit asks the Count why he left and returned to Russia before and after the revolution. The Count explains that the circumstances were personal, not political. A Hussar lieutenant had seduced his sister Helena as a form of revenge on the Count for wooing the Hussar's girlfriend, a princess, years before. After seducing Helena, the lieutenant slept with the girl's handmaiden. Infuriated, the Count shot the lieutenant in the shoulder but didn't kill him. The Count fled to Paris, and the lieutenant died in battle during the First World War. While the Count was still in Paris, Helena died of scarlet fever.\n\nBy 1930, the Count has joined the Boyarsky hotel restaurant staff and is part of \"the Triumvirate\" that runs the restaurant\u2014Andrey, the maitre d', Emile, the chef, and the Count himself. In the same year, Nina, who had gone away, comes back to the hotel. Nearly grown, she explains that she and some of her comrades are headed to the Kady District, the agricultural center of the Ivanovo Province, to collectivize it.\n\nThe Count continues to carry on with Anna. Her acting career has stalled after one of her films was decried by the Party. Now poor, Anna makes dates with directors as an excuse to visit the Metropol and then sneaks away to see the Count.\n\nOne day, a former Red Army colonel and current officer of the Communist Party, Osip Ivanovich Glebnikov, comes to visit the Count. Osip admits he's there to keep an eye on the Count. He also tells the Count that the Party is working on diplomatic relations with the French, British, and Americans and that Russia is becoming a center of European manufacturing. Osip wants the Count to help him learn the languages and cultures of the Western world so he can become a better diplomat. The Count agrees to dine with and tutor Osip once a month.\n\nTimes change. Beyond the Metropol's walls, the Party scrambles to build power stations, steel mills, and factories and to cultivate the agricultural regions. To do so, they drive out the uncooperative kulaks, or peasants. In 1932, famine kills millions of peasants, and many who are spared migrate to Russia's urban centers.\n\nIn 1938, Nina comes back again. She confesses hastily that she has gotten married, but her husband has been arrested and sentenced to five years of labor in Sevvostlag. Nina plans to follow him but needs to leave behind her five-year-old daughter Sofia in the Count's care. She promises she'll return for Sofia.\n\nThen Mishka returns, distraught. He explains that he's editing a volume of Chekhov's letters. His senior editor at Goslitizdat has instructed him to cut a line of Chekhov's about the amazing bread in Berlin. Exasperated, Mishka has adhered to his editor's request, but the next day, he changes his mind and returns to his editor's office. He barges in and accuses the editor of being cozy with Stalin. Not long after, Mishka is sent to Leningrad to be questioned and then, in 1939, to Siberia. Meanwhile, Nina, too, has disappeared.\n\nIn 1946, Sofia is 13 and the Count nearly 60. Mishka returns, emaciated and worn down. He admits to the Count that he's in Moscow illegally while doing research for a new writing project.\n\nOne day, Sofia falls from the service stairs and gashes her forehead. The Count carries her outside, entering the light of day for the first time in more than 20 years, and rushes her to the nearest hospital, which has fallen into disrepair. Luckily, a surgeon arrives to treat Sofia. Osip shows up at the hospital too, and the Count realizes it was Osip who recruited the surgeon. Sofia makes a full recovery.\n\nIn 1950, when Sofia is 17, the Count comes upon the young woman in the ballroom with the Piazza orchestra's conductor, Viktor Stepanovich Skadovksy. Sofia informs the Count that Skadovksy is her piano instructor. The Count is blown away by Sofia's ability to play Chopin.\n\nTwo years later, Anna returns to the hotel. Now a stage actress, she can reside in Moscow for months at a time, to the Count's delight. Around this time, an American acquaintance of the Count's, Richard Vanderwhile, confides that Americans are worried about what will happen after Stalin dies. He asks the Count to work for him as a spy, but the Count declines. Nine months later, in 1953, Stalin dies of a stroke. Georgi Malenkov and Nikita Khrushchev take over the Party.\n\nThen, Katerina Litvinova, Mishka's lover, shows up with news that Mishka is dead. She presents the Count with Mishka's manuscript, titled _Bread and Salt_ , as Mishka asked her to do. The Count confesses to her that the revolutionary poem with which he was credited was actually Mishka's. He had claimed authorship in order to protect his friend from persecution in tsarist Russia. Ironically, it was Mishka's poem that saved the Count's life when it came time for his sentencing by the Communist Party.\n\nIn 1954, Sofia gets the opportunity to travel to Paris with the conservatory where she is a student. At this point, the Count starts planning their emigration; he steals a Finnish passport from a hotel guest. Sofia departs for Paris. The night he plans to make his escape, the Count finds his nemesis, a hotel staff member whom he calls the Bishop, in his study. The Bishop has discovered the good-bye letters the Count has written his friends along with a map charting the route from the Palais Garnier, where Sofia is performing, to the American embassy in Paris. So the Count locks the Bishop in the boiler room; he knows the hotel staff will find him soon enough. Then, thanks to the help of Vanderwhile, all 30 telephones in the hotel begin to ring at once. Amid the commotion, the Count flees.\n\nAfter her performance in Paris, Sofia sneaks off to the American embassy, where Vanderwhile welcomes her. She presents Vanderwhile with a book by Montaigne, a gift from the Count. Vanderwhile discovers that the book has been hollowed out and filled with gold pieces.\n\nIn the summer of 1954, an incognito Count stops over in Nizhny Novgorod province. In a little village, he ducks into an inn. There, at the back of the inn's bar, his willowy Anna awaits him.\n\n# Main Characters\n\nCount Alexander Ilyich Rostov is a Russian aristocrat-turned-waiter, sentenced to house arrest at the Metropol Hotel.\n\nNina Kulikova is a Ukrainian girl who befriends the Count and grows up to become a Communist Party activist.\n\nSofia is Nina's daughter. When she is five, her mother leaves her with the Count.\n\nMikhail \"Mishka\" Fyodorovich Mindich is the Count's friend from the university. He is a poet and revolutionary.\n\nAnna Urbanova is a beautiful actress with whom the Count has a decades-long affair.\n\nEmile Zhukovsky is the chef at the Metropol Boyarsky restaurant.\n\nAndrey is the maitre d' at the Boyarsky.\n\nOsip Ivanovich Glebnikov is a Party officer who is tasked with keeping an eye on the Count. They become good friends.\n\nRichard Vanderwhile is an American who eventually helps the Count and Sofia emigrate from Russia.\n\nMarina is the Metropol seamstress. She helps the Count raise Sofia.\n\nThe Bishop is a cranky waiter at the Boyarsky.\n\n# Analysis\n\n# Character Analysis\n\nCount Alexander Ilyich Rostov\n\nThe Count is a good-natured gentleman and intellectual who doesn't care much for politics but finds himself under scrutiny by the Communist Party. Sentenced to house arrest for the rest of his life, the Count must reinvent himself as a permanent resident at the Hotel Metropol.\n\nThe Count, passive by nature, is critical of the Party's idealism. Based on his readings in history, philosophy, and literature, the Count believes that history is largely cyclical and that, as an individual, he stands mostly helpless amid its ebbs and flows. So the Count resigns himself to his circumstances and tries to make the best of his situation by befriending hotel staff and guests and learning about the restaurant business. But when Sofia has the opportunity to leave Russia, the Count's worldview changes. His elaborate scheme to orchestrate his and Sofia's escape to the American embassy in Paris reflects his embrace of his own version of revolutionary ideals. Propelled by his love for Sofia, the Count realizes that he can alter the course of his and Sofia's personal histories. He risks everything to revolutionize his and Sofia's lives by escaping to freedom.\n\nNina Kulikova\n\nNina Kulikova is a curious child who tends to challenge authority and established rules. As a young adult, she is swept up in the Party's rush to collectivize Russia's agricultural provinces. In the process, Nina's ability to think for herself is compromised; her passionate nature leads her to support the Soviet Communist regime uncritically. What might be read as Nina's fatal flaw\u2014her inability to discern between revolutionary thought and action and conformist thought and action\u2014leads to her downfall. In the end, the Party to which she gave everything betrays her.\n\nNina's character suggests the ways in which, throughout history, resistance movements have often been co-opted by the same authoritarian tendencies that they initially set out to abolish. Nina's tragic story offers a warning to be critical of all forms of power\u2014even those that claim to fight on behalf of the powerless.\n\nMikhail \"Mishka\" Fyodorovich Mindich\n\nThe Count's old friend Mishka is an idealist despite claiming to subscribe to the materialist philosophy of the Bolsheviks. He believes that literature can participate in the revolution of material life in Russia and help to usher in the new age. But as Stalin's regime becomes increasingly repressive, Mishka begins to despair. What triggers Mishka's break\u2014his refusal to remain silent in the face of Stalinism\u2014is his editor's attempt to censor his manuscript of Chekhov's collected letters. Mishka suffers a breakdown because he can't bear the thought of Russia's great literature\u2014long a symbol of the country's independent, resistant, and resilient spirit\u2014being reduced to government propaganda. Like Nina, Mishka is a tragic figure whose flaw\u2014his idealization of literature and corresponding belief that it can alter or halt the often unjust, capricious unfolding of history\u2014leads to his own demise. But unlike Nina, Mishka is critically aware of his flaw; in fact, he embraces it. Through his final work, a manuscript of collected quotations from works of literature throughout history that mention bread, Mishka insists adamantly that literature can play a critical role in the movement of history.\n\nThe Hotel Metropol\n\nBased on a real, historical hotel in Moscow, the Hotel Metropol in _A Gentleman in Moscow_ is a character in and of itself. Throughout the novel, readers become intimately familiar with its rooms, from the grand Piazza, elegant Boyarsky restaurant, and American-style bar, the Shalyapin, to the basement furnace room, electrical room, and silver-service room. In some ways, the Metropol could be any twentieth-century luxury hotel. But through the diversity of people, both Russian and foreign, who pass through its rooms and corridors, the Metropol becomes a figure for post-revolutionary Moscow itself\u2014dynamic and rife with contradictions.\n\nDespite the fact that the hotel is a container\u2014and for the Count, a prison\u2014life proliferates and progresses within. Ultimately, the hotel, despite its luxuries, cannot contain a human life forever. Passion, ambition, and other the human desires cultivated within it inevitably overflow its boundaries. In this way, the Metropol represents how humanity, in its irrepressible drive to learn, love, and evolve, cannot be forced to conform to rigid ideological principles and restrictions. Despite an oppressive regime, the people of Moscow and all of Russia, for that matter, continue to live and fight for what they love most\u2014whether it be good food and wine, friendship across difference, or freedom itself.\n\n# Relationships\n\nThe Count and Sofia\n\nWhen Sofia comes to live with the Count at the Metropol, his life changes in an instant. Before he becomes a surrogate parent, the Count enjoys a life of the mind. Discussions of history, philosophy, food, wine, and the meaning of life with the fellow intellectuals who wander in and out of his life occupy his time. When Sofia arrives, the Count must restructure his schedule around the child's wants and needs.\n\nThe Count also learns that a life of inaction\u2014of passively standing by as time marches on\u2014is no longer justifiable or possible. With Sofia suddenly at the center of his life, the Count begins to feel a new desire to not only think about the world, but to become an actor in the world. Upon learning that Sofia is a gifted pianist, the Count becomes motivated to ensure freedom and opportunity for them both. He plans their exodus from his beloved homeland and chooses Sofia, his family, over what has long been most comfortable and familiar to him, Russia. In the process, both the Count and Sofia acquire a sense of their own power and agency.\n\nThe Count and Mishka\n\nIn some ways, the Count and Mishka couldn't be more different. The Count prefers to stay out of politics, instead enjoying the aesthetic pleasures of life. He is content to spend his days inside the Metropol, where there is plenty of good food, wine, and conversation with the hotel's cosmopolitan clientele to keep him occupied. Mishka, on the other hand, lives according to his political principles. He is always on the move and eager to unearth the realities of life for the people of post-Revolutionary Russia.\n\nWhile Mishka's worldview is a tragic one because he believes that Russians have a propensity to undermine or destroy what they work so hard to create, the Count's is comic. He believes that people everywhere are generally the same and, for the most part, want to do good. Despite their differences, the two men spend their lives learning from each other.\n\nWhat binds the two men together is their love of Russia. Both the Count and Mishka believe that what defines the Russian people is their deep intuition of the relationship between the individual and all of history.\n\nThe Count and the Hotel Metropol\n\nThe Count refuses to think of the hotel to which he is confined as a prison. Instead, he imagines the Metropol as a kind of country\u2014his very own Russia. With Nina's help, the Count learns to appreciate all of the hotel's secrets. He begins to respect the hotel for its elaborate rooms and corridors and sets out to learn about them slowly over time.\n\nAs years pass, the Count realizes that there is much the Metropol can teach him. Even though he is no longer a free and wealthy man, he can still enjoy the pleasures of fine dining and culture. After all, Emile is a gifted chef, and the Metropol itself draws writers, actors, politicians, and other interesting types of every sort. The Metropol is his window looking out into the rapidly changing modern world. For example, when jazz takes the world by storm, the Count encounters it live in the halls of his very own hotel. He learns that while he may be confined to a single building, if he remains open to the world around him\u2014if he keeps seeing, hearing, tasting, and touching his immediate environment\u2014he will have experiences and encounters that challenge and enrich him.\n\n# Themes\n\nUpheaval and Change\n\n_A Gentleman in Moscow_ explores how, throughout history, sometimes things change for the better and other times for the worse. The Count believes that despite time's incessant movement, history tends to repeat itself. As the Count and Nina stare at the unused silver in a storeroom in the bowels of the Metropol, he wonders how long it will be before Bolsheviks start having banquets of their own. The Count realizes that regimes change, but human nature stays largely the same, incapable of repressing its own vanity. Traditions evolve; customs and objects become obsolete. Human innovation transforms material life on earth, as the invention of things like steam engines, printing presses, and airplanes lays bare. Even so, the most essential aspects of humanity persist: the hunger for power, the need to feel loved and desired, hindsight but never foresight. The idea of cyclical history is figured in the novel at one point by the Metropol rooftop bees, who return periodically to make honey.\n\nThe novel presents two conflicting views of history through the Count and Mishka. While Mishka believes heritage must be destroyed before there can be progress, the Count believes that since history repeats itself, willful destruction is futile, even wasteful. Mishka's philosophy is that radical rupture is a historical necessity, while the Count's is that gradual, non-transgressive change from within an existing system is possible and more effective in the long run. These two philosophies of history reflect, broadly, two major, differing views in Western political thought: historical materialism, or the revolutionary theory that derives from Karl Marx, and agonism, or democratic theory.\n\nUtopia\n\nWhile the Proletariat Bolshevik Revolution serves as the backdrop to the novel, the Metropol Hotel is a utopian space where class distinctions are fluid and people learn to empathize with each other's varying experiences in a class society. In this way, the Metropol becomes a place for social experiment and nonviolent, non-revolutionary social reform. The Count, for example, is an aristocrat who slips seamlessly into the role of waiter at the Boyarsky restaurant. He works closely and becomes fast friends with the chef and maitre d'. He develops friendships with people who come from very different class backgrounds including the Party officer Osip Glebnikov and the peasant-turned-actress Anna Urbanova. What the Count discovers is that he has much to learn from people and experiences that are different from those he has previously known. Through his new experiences and relationships, the Count is able to further enrich his palate, his intellect, and his romantic life. Transcending class boundaries in the utopia of the Metropol ultimately makes him a better, smarter, stronger human.\n\nThe Metropol is also utopian in that it is a space that permits the Count to transcend gender norms of his culture and time. When Nina leaves Sofia in his care, the Count must immediately adapt to his new role as primary guardian. He must dress, feed, entertain, educate, and even sleep in the same room as the small child. The Count's entire life becomes structured around his domestic duties. Because there is no one who will do the job for him, the Count soon comes to play an almost maternal role in Sofia's life.\n\nThe utopian Metropol figures the way in which utopian experiments within existing societies are possible and exist. _A Gentleman in Moscow_ suggests that these types of experiments are carried out most successfully in private spaces as opposed to public or collectivized ones. The novel's depiction of secrets and private places, like the Count's hidden study, as promising sites for utopian experimentation contrasts starkly with its representation of the Communist Party's brutal decimation of the agricultural provinces of Russia in the name of collectivization.\n\nAesthetics\n\nWhat the Count values almost more than anything else are his aesthetic experience. He revels in the wordless poetry of a good bottle of wine. He condemns the Bolsheviks for removing the wine labels at the restaurant and eradicating the Russian cultural history that the particulars of each bottle contain. For the Count, history is in the details of food and drink, literature, philosophy, and art. Therefore, for the Count, being a gentleman is about much more than good manners and etiquette. It is about the preservation of tradition and culture. From the Count's perspective, the problem with the Communist Party is not so much its rejection of aristocracy. Rather, it is the fact that, by invalidating the cultural traditions kept by the Russian aristocracy, the Party refuses much of Russian history itself.\n\nThe Count learns through his experiences that one can be a gentleman\u2014a person with taste and an appreciation for history and culture\u2014without being an aristocrat. He learns that being a gentleman means, more than anything else, possessing the ability to think carefully and independently about the particulars of one's time, place, and experiences, and to act with dignity and care toward other people.\n\n# Author's Style\n\nAmor Towles's _A Gentleman in Moscow_ is told from the perspective of a comical, almost whimsical omniscient narrator. This narrator is partial to the worldview of the Count. Often the novel seems to unfold through the eyes of the Count despite being narrated in the third person. At times, the novel's comic, light-hearted tone makes the bleak historical context\u2014post-revolutionary Russia under Stalin, and widespread famine across the Russian countryside\u2014more bearable. At other times, the humorous tone contrasts dramatically with the story's tragic plot, which throws into further relief the nightmares of Stalinist Russia. For example, when Mishka finally loses it and goes storming into the office of his editor, whom he calls \"the ferret,\" and derisively accuses him of taking baths with Stalin, the image of a frazzled, indignant Mishka is downright funny. But immediately afterward, readers learn that the consequences for Mishka are grave\u2014in short order, he's on a train to Siberia. Additionally, readers learn that Nina never returns from the Russian East. The sudden disappearance of two of the novel's main characters is intended to be upsetting. The author's use of humorous anecdote to frame these tragic losses makes them even more affecting.\n\nWhile _A Gentleman in Moscow_ progresses chronologically, often the narrator digresses to account for previous moments of a character's life. Sometimes the narrator even pauses to draw attention to these digressions, as in the instance of Anna Urbanova's return to the Metropol after many years away. \"What's this!\" exclaims the narrator after surprising readers with Anna's sudden presence in the hotel again. The narrator then backtracks in order to tell the story of how Anna came to find herself once again in bed with the Count at the Metropol.\n\nBy self-reflexively referencing the fact of its own constructedness as a novel, _A Gentleman in Moscow_ invites readers to draw connections between it and the broader tradition of the Russian novel. Through its emphasis on grand, universal themes such as human nature, love, and loss, which are paramount to works by such greats as Chekhov and Tolstoy, _A Gentleman in Moscow_ nods to this Russian tradition. The novel also shares some similarities with _The Count of Monte Cristo_ , an 1845 novel by Alexandre Dumas.\n\nIn the end, _A Gentleman in Moscow_ is more sympathetic toward the Count's political philosophy than any other. With a healthy dose of humor and a keen sense of intellectual history, the Count's take on life is the one that is conveyed most complexly and compellingly to readers. Perhaps this is no surprise, given that the novel's author was himself someone who presumably saw the best of what capitalism and Western democracy had to offer throughout his career working for an investment firm and traveling internationally. [2]\n\nThank you for purchasing this Instaread.\n\nDownload our app to get unlimited text & audio notes on bestselling books.\n\nVisit instaread.co to learn more.\n~~~~~~ END OF INSTAREAD~~~~~~\n\n# References\n\n[1] Towles, Amor. _A Gentleman in Moscow_. New York: Viking, 2016, p. 85.\n\n2] Neary, Lynn. \"Idea For 'Gentleman In Moscow' Came From Many Nights In Luxury Hotels,\" NPR, Sept. 6, 2016. Accessed October 5, 2016. [http:\/\/www.npr.org\/2016\/09\/06\/492434255\/idea-for-gentleman-in-moscow-came-from-many-nights-in-luxury-hotels.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## CONTENTS\n\n1. A PRESENT FOR WILLA JEAN\n\n2. SLACKS FOR ELLA FUNT\n\n3. NOBODY LIKES RAMONA\n\n4. THE QUARREL\n\n5. THE GREAT HAIR ARGUMENT\n\n6. RAMONA'S NEW PAJAMAS\n\n7. THE TELEPHONE CALL\n\nEXCERPT FROM _RAMONA QUIMBY, AGE 8_\n\n1. THE FIRST DAY OF SCHOOL\n\nABOUT THE AUTHOR\n\nBACK AD\n\nOTHER BOOKS BY BEVERLY CLEARY\n\nCREDITS\n\nCOPYRIGHT\n\nABOUT THE PUBLISHER\n\n#\n\n## A PRESENT FOR WILLA JEAN\n\n\"When will they be here?\" asked Ramona Quimby, who was supposed to be dusting the living room but instead was twirling around trying to make herself dizzy. She was much too excited to dust.\n\n\"In half an hour,\" cried her mother from the kitchen, where she and Ramona's big sister Beatrice were opening and closing the refrigerator and oven doors, bumping into one another, forgetting where they had laid the pot holders, finding them and losing the measuring spoons.\n\nThe Quimbys were about to entertain their neighbors at a New Year's Day brunch to celebrate Mr. Quimby's finding a job at the ShopRite Market after being out of work for several months. Ramona liked the word _brunch_ , half breakfast and half lunch, and secretly felt the family had cheated because they had eaten their real breakfast earlier. They needed their strength to get ready for the party.\n\n\"And Ramona,\" said Mrs. Quimby as she hastily laid out silverware on the dining-room table, \"be nice to Willa Jean, will you? Try to keep her out of everyone's hair.\"\n\n\"Ramona, watch what you're doing!\" said Mr. Quimby, who was laying a fire in the fireplace. \"You almost knocked over the lamp.\"\n\nRamona stopped twirling, staggered from dizziness, and made a face. Willa Jean, the messy little sister of her friend Howie Kemp, was sticky, crumby, into everything, and always had to have her own way.\n\n\"And behave yourself,\" said Mr. Quimby. \"Willa Jean is company.\"\n\nNot my company, thought Ramona, who saw quite enough of Willa Jean when she played at Howie's house. \"If Howie can't come to the brunch because he has a cold, why can't Willa Jean stay home with their grandmother, too?\" Ramona asked.\n\n\"I really don't know,\" said Ramona's mother. \"That isn't the way things worked out. When the Kemps asked if they could bring Willa Jean, I could hardly say no.\"\n\nI could, thought Ramona, deciding that since Willa Jean, welcome or not, was coming to the brunch, she had better prepare to defend her possessions. She went to her room, where she swept her best crayons and drawing paper into a drawer and covered them with her pajamas. Her Christmas roller skates and favorite toys, battered stuffed animals that she rarely played with but still loved, went into the corner of her closet. There she hid them under her bathrobe and shut the door tight.\n\nBut what could she find to amuse Willa Jean? If Willa Jean did not have something to play with, she would run tattling to the grown-ups. \"Ramona hid her toys!\" Ramona laid a stuffed snake on her bed, then doubted if even Willa Jean could love a stuffed snake.\n\nWhat Ramona needed was a present for Willa Jean, a present wrapped and tied with a good hard knot, a present that would take a long time to unwrap. Next to receiving presents, Ramona liked to give presents, and if she gave Willa Jean a present today, she would not only have the fun of giving, but of knowing the grown-ups would think, Isn't Ramona kind, isn't she generous to give Willa Jean a present? And so soon after Christmas, too. They would look at Ramona in her new red-and-green-plaid slacks and red turtleneck sweater and say, Ramona is one of Santa's helpers, a regular little Christmas elf.\n\nRamona smiled at herself in the mirror and was pleased. Two of her most important teeth were only halfway in, which made her look like a jack-o'-lantern, but she did not mind. If she had grown-up teeth, the rest of her face would catch up someday.\n\nOver her shoulder she saw reflected in the mirror a half-empty box of Kleenex on the floor beside her bed. Kleenex! That was the answer to a present for Willa Jean. She ran into the kitchen, where Beezus was beating muffin batter while her father fried sausages and her mother struggled to unmold a large gelatine salad onto a plate covered with lettuce.\n\n\"A present is a good idea,\" agreed Mrs. Quimby when Ramona asked permission, \"but a box of Kleenex doesn't seem like much of a present.\" She shook the mold. The salad refused to slide out. Her face was flushed and she glanced at the clock on the stove.\n\nRamona was insistent. \"Willa Jean would like it. I know she would.\" There was no time for explaining what Willa Jean was to do with the Kleenex.\n\nMrs. Quimby was having her problems with the stubborn salad. \"All right,\" she consented. \"There's an extra box in the bathroom cupboard.\" The salad slid slowly from the mold and rested, green and shimmering, on the lettuce.\n\nBy the time Ramona had wrapped a large box of Kleenex in leftover Christmas paper, the guests had begun to arrive. First came the Hugginses and McCarthys and little Mrs. Swink in a bright-green pants suit. Umbrellas were leaned outside the front door, coats taken into the bedroom, and the usual grown-up remarks exchanged. \"Happy New Year!\" \"Good to see you!\" \"We thought we would have to swim over, it's raining so hard.\" \"Do you think this rain will ever stop?\" \"Who says it's raining?\" \"This is good old Oregon sunshine!\" Ramona felt she had heard that joke one million times, and she was only in the second grade.\n\nThen Mr. Huggins said to Ramona's father, \"Congratulations! I hear you have a new job.\"\n\n\"That's right,\" said Mr. Quimby. \"Starts tomorrow.\"\n\n\"Great,\" said Mr. Huggins, and Ramona silently agreed. Having a father without a job had been hard on the whole family.\n\nThen Mrs. Swink smiled at Ramona and said, \"My, Juanita, you're getting to be a big girl. How old are you? I can't keep track.\"\n\nShould Ramona tell Mrs. Swink her name was not Juanita? No, Mrs. Swink was very old and should be treated with courtesy. Last year Ramona would have spoken up and said, My name is not Juanita, it's Ramona. Not this year. The room fell silent as Ramona answered, \"I'm seven and a half right now.\" She was proud of herself for speaking so politely.\n\nThere was soft laughter from the grown-ups, which embarrassed Ramona. Why did they have to laugh? She _was_ seven and a half right now. She would not be seven and a half forever.\n\nThen the Grumbies arrived, followed by Howie's mother and father, the Kemps, and of course Willa Jean. Although Willa Jean was perfectly capable of walking, her father was carrying her so she would not get her little white shoes and socks wet. Willa Jean in turn was carrying a big stuffed bear. When Mr. Kemp set his daughter down, her mother peeled off her coat, one arm at a time so Willa Jean would not have to let go of her bear.\n\nThere stood usually messy Willa Jean in a pink dress with tiny flowers embroidered on the collar. Her curly blond hair, freshly washed, stood out like a halo. Her blue eyes were the color of the plastic handle on Ramona's toothbrush. When she smiled, she showed her pearly little baby teeth. Willa Jean was not messy at all.\n\nRamona in her corduroy slacks and turtleneck sweater suddenly felt big and awkward beside her little guest and embarrassed to have jack-o'-lantern teeth.\n\nAnd the things those grown-ups said to Willa Jean! \"Why, hello there, sweetheart!\" \"My, don't you look like a little angel!\" \"Bless your little heart. Did Santa bring you the great big bear?\" Willa Jean smiled and hugged her bear. Ramona noticed she had lace ruffles sewn to the seat of her underpants.\n\n\"What is your bear's name, dear?\" asked Mrs. Swink.\n\n\"Woger,\" answered Willa Jean.\n\nMrs. Kemp smiled as if Willa Jean had said something clever and explained, \"She named her bear Roger after the milkman.\"\n\nMrs. Quimby said with amusement, \"I remember when Ramona named one of her dolls Chevrolet after the car.\" Everyone laughed.\n\nShe didn't have to go and tell that, thought Ramona, feeling that her mother had betrayed her by telling, as if it were funny, something she had done a long time ago. She still thought Chevrolet was a beautiful name, even though she was old enough to know that dolls were not usually named after cars.\n\n\"See my bear?\" Willa Jean held Woger up for Ramona to admire. Because everyone was watching, Ramona said politely, \"He's a nice bear.\" And he was a nice bear, the nicest bear Ramona had ever seen. He was big and soft with a kindly look on his furry face and\u2014this was the best part\u2014each of his four big paws had five furry toes. You could count them, five on each paw. Even though Ramona felt she should be outgrowing bears, she longed to hold that bear, to put her arms around him, hug him close and love him. \"Would you like me to hold the bear for you?\" she asked.\n\n\"No,\" said Willa Jean.\n\n\"Ramona,\" whispered Mrs. Quimby, \"take Willa Jean into the kitchen and sit her at the table so she won't spill orange juice on the carpet.\" Ramona gave her mother a balky look, which was returned with her mother's you-do-it-or-you'll-catch-it look. Mrs. Quimby was not at her best when about to serve a meal to a living room full of guests.\n\nIn the kitchen Willa Jean set Woger carefully on the chair before she climbed up beside him, displaying her ruffled underpants, and grasped her orange juice with both hands, dribbling some down the front of her fresh pink dress.\n\nMrs. Quimby, assisted by Beezus, set out a platter of scrambled eggs and another of bacon and sausage beside the gelatine salad. Hastily she snatched two small plates from the cupboard and dished out two servings of brunch, which she set in front of Ramona and Willa Jean. Beezus, acting like a grown-up, filled a basket with muffins and carried it into the dining room. Guests took plates from the stack at the end of the table and began to serve themselves.\n\nRamona scowled. If Beezus got to eat in the living room with the grown-ups, why couldn't she? She was no baby. She would not spill.\n\n\"Be a good girl!\" whispered Mrs. Quimby, who had forgotten the marmalade.\n\nI'm trying, thought Ramona, but her mother was too flurried to notice her efforts. Willa Jean took one bite of scrambled eggs and then went to work, patting the rest flat on her plate with the back of her spoon.\n\nRamona watched her charge give her egg a final pat with the back of her spoon, pick up her bear, and trot off to the living room, leaving Ramona alone to nibble a muffin, think, and look at her artwork, arithmetic papers, and some cartoons her father had drawn, which had been taped to the refrigerator door for the family to admire. Nobody missed Ramona, all alone out there in the kitchen. Conversation from the living room was boring, all about high prices and who would be the next president, with no mention of children or anything interesting until someone said, \"Oops. Careful, Willa Jean.\"\n\nThen Mrs. Kemp said, \"No-no, Willa Jean. Mustn't put your fingers in Mr. Grumbie's marmalade. It's sticky.\"\n\nMrs. Quimby slipped into the kitchen to see if the coffee was ready. \"Ramona, it's time to take Willa Jean to your room and give her your present,\" she whispered.\n\n\"I changed my mind,\" said Ramona.\n\nMr. Quimby, refilling the muffin basket, overheard. \"Do as your mother says,\" he ordered in a whisper, \"so that kid will give us a little peace.\"\n\nRamona considered. Should she make a fuss? What would a fuss accomplish? On the other hand, if she gave Willa Jean her present, maybe she would have a chance to hold that lovable bear for a little while.\n\n\"OK,\" Ramona agreed without enthusiasm.\n\nMrs. Quimby followed Ramona into the living room. \"Willa Jean,\" she said. \"Ramona has a present for you. In her room.\"\n\nWilla Jean's attention was caught.\n\n\"Go with Ramona,\" Mrs. Quimby said firmly.\n\nWilla Jean, still clutching her bear, went.\n\n\"Here.\" Ramona thrust the package at Willa Jean, and when her guest set her bear on the bed, Ramona started to pick him up.\n\nWilla Jean dropped the package. \"Woger's my bear,\" she said, and ran off to the living room with him. In a moment she returned bearless to pull and yank and tear the wrapping from the package. \"That's not a present.\" Willa Jean looked cross. \"That's Kleenex.\"\n\n\"But it's your very own,\" said Ramona. \"Sit down and I'll show you what to do.\" She broke the perforation in the top of the box and pulled out one pink sheet and then another. \"See. You can sit here and pull out all you want because it's your very own. You can pull out the whole box if you want.\" She did not bother telling Willa Jean that she had always wanted to pull out a whole box of Kleenex, one sheet after another.\n\nWilla Jean looked interested. Slowly she pulled out one sheet and then a second. And another and another. She began to pull faster. Soon she was pulling out sheet after sheet and having such a good time that Ramona wanted to join the fun.\n\n\"It's mine,\" said Willa Jean when Ramona reached for a tissue. Willa Jean got to her feet and, pulling and flinging, ran down the hall to the living room. Ramona followed.\n\n\"See me!\" Willa Jean ordered the grown-ups as she ran around pulling and flinging Kleenex all over the room. Guests grabbed their coffee mugs and held them high for safety.\n\n\"No-no, Willa Jean,\" said Mrs. Kemp. \"Mrs. Quimby won't like you wasting her Kleenex.\"\n\n\"It's mine!\" Willa Jean was carried away by the joy of wasting Kleenex and being the center of attention at the same time. \"Ramona gave it to me.\"\n\nRamona looked around for the bear, which was sitting on Mr. Grumbie's lap. \"Would you like me to hold Roger?\" Ramona asked, careful not to say Woger.\n\n\"No.\" The bear's owner saw through Ramona's scheme. \"Woger wants to sit _there_.\" Mr. Grumbie did not look particularly pleased.\n\nWilla Jean's parents made no effort to stop their daughter's spree of pull and fling. Ramona watched, feeling much older than she had earlier in the day. She also felt awkward while Beezus moved around the living room, dodging Willa Jean and pouring coffee as if she were a grown-up herself.\n\nAt first guests were amused by Willa Jean. But amusement faded as coffee mugs had to be rescued every time Willa Jean passed by. Pink Kleenex littered the room. Ramona heard Mr. Huggins whisper, \"How much Kleenex in a box anyway?\"\n\nMrs. McCarthy answered, \"Two hundred and fifty sheets.\"\n\n\"That's a lot of Kleenex,\" said Mr. Huggins.\n\nWhen Willa Jean came to the last piece of Kleenex, she climbed on the couch and carefully laid it on Mr. Grumbie's bald head. \"Now you have a hat,\" she said.\n\nConversation died, and the party died, too. No one called Willa Jean an angel now or blessed her little heart.\n\nThe Grumbies were first to leave. Mr. Grumbie handed the bear to Willa Jean's mother as Willa Jean filled her arms with pink tissues and tossed them into the air. \"Whee!\" she cried, and scooped up another armful. \"Whee!\"\n\nTheir departure seemed to be a signal for everyone to leave. \"Don't you want to take the Kleenex with you?\" Mrs. Quimby asked Willa Jean's mother. \"We can put it in a bag.\"\n\n\"That's all right. Willa Jean has had her fun.\" Mrs. Kemp was helping Willa Jean into her coat.\n\n\"Bye-bye,\" said Willa Jean prettily as her father carried her and Woger out the door.\n\nOther guests were telling Mr. and Mrs. Quimby how much they had enjoyed the brunch. Beezus was standing beside them as if it had been her party, too. Mrs. McCarthy smiled. \"I can see you are your mother's girl,\" she said.\n\n\"I couldn't get along without her,\" Mrs. Quimby replied generously.\n\n\"Good-bye, Juanita,\" said little Mrs. Swink.\n\n\"Good-bye, Mrs. Swink,\" answered Ramona, polite to the end.\n\nShe tossed an armful of Kleenex into the air so that her mother might notice her, too. Somehow tossing someone else's pulled-out Kleenex was not much fun, and Mrs. Quimby was so busy saying good-bye to other guests she did not pay attention.\n\nAt last the door was closed, and from the porch where the neighbors were opening umbrellas, Ramona's sharp ears caught her name. \"Willa Jean certainly reminds me of Ramona when she was Willa Jean's age,\" someone said.\n\nAnd someone else answered, \"She's Ramona all over again, all right.\"\n\nRamona was filled with indignation. Willa Jean is _not_ me all over again, she thought fiercely. I was never such a pest.\n\n\"Whew!\" said Mr. Quimby. \"That's over. What's the matter with those people, letting the kid show off like that?\"\n\n\"Too much grandmother, I suppose,\" answered Mrs. Quimby. \"Or maybe it's easier for them to ignore her behavior.\"\n\n\"Come on, let's all pitch in and clean up this place,\" said Mr. Quimby. \"Ramona, you find a bag and pick up all the Kleenex.\"\n\n\"Kleenex is made of trees,\" said Beezus, already helping her mother collect coffee mugs from the living room. \"We shouldn't waste it.\" Lately Beezus had become a friend of trees.\n\n\"Put the bag of Kleenex in the cupboard in the bathroom,\" said Mrs. Quimby, \"and let's all remember to use it.\"\n\nI never was as awful as Willa Jean, Ramona told herself as she went to work collecting two hundred and fifty pieces of scattered pink Kleenex. I just know I wasn't. She followed the trail of Kleenex back to her bedroom, and when the two hundred and fiftieth piece was stuffed in the bag, she leaned against her dresser to study herself in the mirror.\n\nHow come nobody ever calls me my mother's girl? Ramona thought. How come Mother never says she couldn't get along without me?\n\n#\n\n## SLACKS FOR ELLA FUNT\n\nThe day Ramona's father went to work at the checkout counter of the ShopRite Market, life in the Quimby household changed. Sometimes Mr. Quimby worked all day; sometimes he worked afternoons and evenings. Sometimes he took the car to work while Mrs. Quimby took the bus to her job in Dr. Hobson's office. Sometimes she drove the car while he took the bus, or one would drop the other off at work.\n\nLife was different for Ramona, too. She now went home with Howie Kemp after school. The Quimbys paid Howie's grandmother to look after Ramona until one of her parents could come for her after work. Mrs. Quimby said she could not hold a job unless she knew where Ramona was. Every single minute. Beezus also went to the Kemps' house after school unless she telephoned her mother for permission to go to a friend's house. Ramona had no choice.\n\nOne rainy Saturday morning, when Mr. Quimby had worked at the market for several weeks, Ramona asked her mother, \"Where do you have to go today?\" Mr. Quimby always worked Saturday, the busiest day at the market, which meant Mrs. Quimby had the use of the car to run errands. Ramona was concerned about her mother's errands because she always had to go with her, or as she thought of it, be dragged along. Most of them were of no interest to her.\n\nMrs. Quimby thought a moment before she said in surprise, \"Well, what do you know? No place. We have groceries in the cupboard. No one needs to go shopping for shoes. No one needs a present to take to a birthday party. I can stay home.\"\n\n\"Then what are you going to do?\" asked Ramona. She hoped her mother would not decide to clean house.\n\n\"Sew,\" answered Mrs. Quimby. \"I've been trying to finish a blouse for weeks.\"\n\nSewing seemed like a cozy way to spend a rainy morning. Ramona watched her mother get out the portable sewing machine and set it on the dining-room table along with a pattern and bundle of fabric.\n\n\"I'm going to wash my hair,\" announced Beezus.\n\n\"Again?\" inquired Mrs. Quimby. \"You washed it only the day before yesterday.\"\n\n\"But it's so oily,\" complained Beezus.\n\n\"Don't worry, it's just your age,\" reassured Mrs. Quimby. \"You'll outgrow it.\"\n\n\"Yes,\" said Beezus gloomily. \"In about a million years when I'm too old to care.\"\n\n\"You'll never be that old,\" said Mrs. Quimby. \"I promise.\"\n\nRamona, bored with her sister's daily complaints about oily hair, leaned on the dining-room table to watch her mother.\n\n\"I like it when you stay home,\" she remarked, thinking of the days before her mother had gone to work when the house had smelled of baking cookies or homemade bread on Saturday morning. \"Can I sew, too?\" she asked, picturing a companionable morning close to her mother. She imagined a neighbor dropping in and saying, Ramona is her mother's girl, as the two of them stitched away together. Yes, her mother would answer, I can't get along without Ramona.\n\n\"Of course you may sew. You know where the scrap bag is,\" answered her mother with a smile. \"What would you like to make?\"\n\n\"I'll have to think,\" Ramona said. She went to her room and in a moment returned with a tired-looking stuffed elephant and a large piece of red-and-white checked cloth left over from a dress her mother had made for her when she was in kindergarten. Ramona had liked that dress because it matched the red plastic hat the firemen gave her when the kindergarten class visited the fire department.\n\nMrs. Quimby looked up from the sewing machine. \"I haven't seen Ella Funt for a long time,\" she remarked as Ramona stood the elephant on its four feet on the table.\n\n\"I'm going to make her some slacks,\" said Ramona as she spread out the fabric. \"All my other animals have clothes. Except that snake.\"\n\nMrs. Quimby considered. \"Slacks for an elephant won't be easy. Why not make slacks for Chevrolet?\"\n\n\"She's too beat up,\" said Ramona critically.\n\n\"If I were you, I would\u2014\" began Mrs. Quimby as Ramona studied the checked cloth.\n\n\"I can _do_ it,\" interrupted Ramona, who was impatient with instructions. Mrs. Quimby said no more. The sewing machine began to hum. Ramona picked up her mother's pinking shears and began to cut. There was something satisfying about using pinking shears and watching fabric part in such a neat zigzag line. Working quietly at the table with her mother was even more satisfying. Even the grouchy old cat, Picky-picky, sitting on a corner of the dining-room carpet washing his paws, was behaving like a satisfactory pet.\n\nThis morning was a time for sharing confidences. \"Mother, did I used to be like Willa Jean?\" Ramona asked the question that had worried her since the brunch.\n\nMrs. Quimby answered when she came to the end of her seam. \"You were a lively little girl with a lot of imagination. And you still are.\"\n\nRamona was reassured by her mother's words. \"I would never throw Kleenex all over the living room when someone had a party,\" she said virtuously.\n\nIf only every Saturday could be like this one: no errands to run, and the two of them sewing and talking together. \"Are you going to stop working now that Daddy has a job?\" Ramona asked as Beezus, her hair wet but combed, came into the dining room with a paper bag in her hand.\n\nMrs. Quimby looked up from the collar she was pinning to her blouse. \"Why, no,\" she answered as if she were surprised by the question.\n\n\"Why not?\" demanded Ramona as Beezus pulled some sewing out of the bag.\n\n\"Because we got behind on our bills when Daddy was out of work,\" explained Mrs. Quimby, \"and because we have plenty of ways to use money. Beezus will be ready for college in five years, and in a few more years you will be ready, too.\"\n\nCollege, to Ramona, was a faraway school for young grown-ups. When they went to college, their mothers worried about them and mailed them boxes of cookies that they called Care Packages. She had learned this from listening to some of the neighbors talk to her mother. Ramona was surprised to learn that she and her sister would be expected to go to such a school someday.\n\n\"Besides,\" continued Mrs. Quimby, \"I like my job. The people are interesting, and Dr. Hobson is pleasant to work for.\"\n\n\"I wish Daddy liked his job.\" Beezus's head was bent over the skirt she was basting together.\n\nRamona understood what Beezus meant, because she felt sad, too, and her stomach felt tight when her father came home tired and discouraged after a day in the checkout line. People were in a hurry, many were cross because the line was long, and some customers acted as if he were to blame because prices were so high.\n\n\"So do I wish he liked his job.\" There was a hint of sadness in Mrs. Quimby's voice. \"But maybe when he has worked at ShopRite longer, he will like it better. New jobs take getting used to.\"\n\nRamona held up the two pieces of cloth she had cut out for the front and back of the slacks. Both were the same. Until now she had fastened together whatever she was making with Scotch tape or a stapler. Now Ramona felt the time had come for her to advance. Beezus had been using the sewing machine for several years. \"Can I sew on the machine?\" she asked.\n\n\"If you're careful.\" Mrs. Quimby demonstrated the use of the machine. Even though Ramona had to stand up to stitch, she found the machine easier to use than she had expected. She followed her mother's instructions carefully and watched the needle move up and down, leaving behind a trail of tiny, even stitches. Ramona was filled with pleasure at the sight. The sewing machine was much more satisfactory than the stapler, which often stuck or ran out of staples. All sorts of uses for the sewing machine began to fly through Ramona's imagination. Maybe she could even sew paper and make a book. Quickly, but still carefully, she finished the seams of Ella Funt's slacks.\n\n\"See? I can use the sewing machine, too,\" Ramona bragged to Beezus.\n\nBeezus did not bother to answer. She was busy pulling on her skirt while Mrs. Quimby stood by to see how it fit.\n\nRamona picked up her gray-flannel elephant and shoved its hind legs into the legs of the slacks.\n\nMrs. Quimby pinned the waistband for Beezus and stood back to look at the skirt. \"It fits nicely,\" she said. \"You can go ahead and stitch it.\" Pleased, Beezus went off to the bedroom to admire her work in the mirror.\n\nRamona tugged and tugged at Ella Funt's slacks, but no matter how hard she tugged she could not make them come up to the elephant's waist, or to what she guessed was the elephant's waist. Ella Funt's bottom was too big, or the slacks were too small. At the same time, the front of the slacks seemed way too big. They bunched under Ella Funt's paunch. Ramona scowled.\n\nBeezus twirled into the living room to make her skirt stand out. Anyone could see she was pleased with what she had accomplished.\n\nRamona scowled harder, but no one noticed. No one even cared. The sewing machine hummed. Beezus slipped out of her skirt. Ramona heaved a gusty sigh. The sewing machine stopped humming.\n\n\"Having problems, Ramona?\" inquired Mrs. Quimby.\n\n\"These slacks look terrible.\" Ramona glowered. \"They look awful!\"\n\nMrs. Quimby considered Ella Funt and her slacks. \"Well,\" she said after a moment, \"maybe you could find something easier to sew. Slacks for an elephant are very hard to make. I'm sure I couldn't do it.\"\n\nRamona could not scowl any harder. \"I _like_ to do hard things.\"\n\n\"I know you do, and I admire you for it,\" answered Ramona's mother, \"but sometimes it's better to start with something easy and work up.\"\n\n\"Why don't you make a skirt?\" suggested Beezus. \"Ella Funt is a girl elephant.\"\n\n\"I don't want to make a skirt!\" Ramona's voice was rising. \"I want to make pants!\" She looked at her mother and sister, so calm and happy with their sewing. Why couldn't her sewing turn out right the way theirs did? Ramona felt shut out from something she longed to share. Picky-picky stopped washing, gave Ramona a long stare, and slowly and disdainfully left the room.\n\n\"Now Ramona,\" said Mrs. Quimby gently, \"I know you are disappointed, but life is full of little disappointments. You'll get over it. Why don't you try something else? A skirt the way Beezus suggested.\"\n\n\"I won't either get over it!\" Nobody had to tell Ramona that life was full of disappointments. She already knew. She was disappointed almost every evening because she had to go to bed at eight-thirty and never got to see the end of the eight o'clock movie on television. She had seen many beginnings but no endings. And even though she had outgrown her tricycle, she was still disappointed because she never could find a tricycle license plate with her name printed on it. Didn't the people who made those license plates care about little girls named Ramona? And then there was that time she had gone to the Easter egg hunt in the park with a big paper bag and had found only two little candy eggs, one of which had been stepped on. Nobody had to tell Ramona about disappointment.\n\nThe disappointment of Ella Funt's slacks was not one of life's little disappointments to Ramona. It was a big disappointment because she had failed at something she wanted to do and because she no longer felt she was sharing with her mother. Beezus was doing so instead. \"I don't want to do something easier!\" yelled Ramona, and hurled poor old Ella Funt and her slacks across the room. As the elephant bounced off the wall, a thought flashed through Ramona's mind. Her mother had not actually said she was not like Willa Jean.\n\nMrs. Quimby spoke sharply. \"That's enough, Ramona. Calm down.\"\n\n\"I won't calm down!\" shouted Ramona, bursting into tears. She fled to that haven of anyone in the family who had tears to shed, the bathroom, where she sat on the edge of the tub sniffling miserably. Nothing was fair. Her mother was always saying everyone must be patient with Beezus when she was cross because Beezus had reached a difficult age, but what about Ramona? Her age was difficult, too\u2014not old enough to sit down with her mother and sew something she wanted to sew and too old to go pulling out a whole box of Kleenex and flinging it all over the house like Willa Jean. People should not think being seven and a half years old was easy, because it wasn't.\n\nAs Ramona sat on the hard edge of the tub, feeling sorry for herself and trying to sort out her thoughts, she noticed a brand-new red-white-and-blue tube of toothpaste lying beside the washbasin. How smooth and shiny it looked with only one little dent where someone had squeezed it once. That tube was as good as new, and it was the large economy size.\n\nRamona was suddenly filled with longing. All her life she had wanted to squeeze toothpaste, really squeeze it, not just one little squirt on her toothbrush but a whole tube, a large economy size tube, all at one time just as she had longed to pull out a whole box of Kleenex.\n\nI'll give it one little squeeze, thought Ramona. Just one teeny squeeze to make me feel better. She seized the tube. How fat and smooth it felt in her hand. She unscrewed the cap and laid it on the counter. Then she squeezed that tube the way she had been told she must never squeeze it, right in the middle. White paste shot out faster than she had expected.\n\nThat squirt really did make Ramona feel better. She squeezed again. Another satisfying squirt. She felt even better. This was more fun than finger painting or modeling turtles out of clay. Suddenly Ramona no longer cared what anyone thought. She squeezed and squirted, squeezed and squirted. She forgot about Ella Funt's slacks, she forgot about her mother and Beezus, she forgot that no one ever called her her mother's girl. The paste coiled and swirled and mounded in the washbasin. Ramona decorated the mound with toothpaste roses as if it were a toothpaste birthday cake. When the tube was almost empty, she rolled it properly from the bottom and squeezed some more. Tighter and tighter she rolled it until not another speck of toothpaste could be squeezed out.\n\nThere, thought Ramona with pleasure and satisfaction, which unfortunately lasted only a moment. With the rolled-up tube in her hand Ramona stood looking at the white mound. What would her mother say? What could she do with it? Wash it down the drain? It might foam all over the bathroom. Or it might stop up the sink. Then she really would be in trouble.\n\nOf course just at that moment Beezus came down the hall toward the bathroom. Ramona started to slam the door, but Beezus blocked it with her foot.\n\nRamona tried to hide the toothpaste by standing tall in front of the washbasin. Unfortunately, she was not tall enough.\n\nBeezus looked over her shoulder. \"Is that _toothpaste_?\" she asked in disbelief.\n\nRamona scowled because she did not know what else to do.\n\n\"Mother!\" Beezus had seen the rolled-up squeezed-out tube. \"Ramona has wasted a whole tube of toothpaste!\" Now apparently Beezus was not only a friend of trees, she was a friend of toothpaste as well.\n\n\"You keep quiet!\" ordered Ramona. \"I'll pick it up.\"\n\n\"How?\" asked Beezus. \"How can you pick up toothpaste?\"\n\n\"With a spoon. I can put it in a plastic bag with a spoon,\" said Ramona. \"We can dip our toothbrushes in the bag.\"\n\n\"Yuck,\" said Beezus rudely. \"Everybody's germs will get mixed up.\"\n\n\"Picky,\" said Ramona.\n\n\"Girls!\" Mrs. Quimby came to see what the argument was about. \"Ramona, what on earth got into you?\" she asked in exasperation when she saw the toothpaste birthday cake.\n\nThe whole thing was much too difficult to explain. Really impossible, so Ramona said cockily, like the funny man on television, \"The devil made me do it.\"\n\n\"That's not funny, Ramona.\" Mrs. Quimby meant what she said.\n\nRamona did not understand. Everyone laughed when the man on television said the devil made him do something. Why wasn't the remark funny when she said it? Because she was seven and a half (right now!). That was why. Grown-ups could get away with anything. It wasn't fair.\n\n\"Get a spoon and a jar from the kitchen,\" directed Mrs. Quimby, \"and scoop up the toothpaste.\" Then she said to Beezus, \"She can use it herself, and the rest of us can use a fresh tube.\"\n\nSomehow Ramona felt sad knowing she was to be excluded from the family tube of toothpaste for a long time. And she wished her mother would not speak about her to Beezus as if she were not in the room.\n\n\"Ramona,\" said her mother, \"don't you ever let me catch you squeezing out a whole tube of toothpaste again.\"\n\n\"I won't,\" promised Ramona, and as she went off to the kitchen for a jar and a spoon she felt unexpectedly cheerful. She had done something she had always wanted to do. _Of course_ she would never squeeze out a whole tube of toothpaste again. She had done it once. She did not need to do it again.\n\n#\n\n## NOBODY LIKES RAMONA\n\nIn February there came a day for Ramona when everything went wrong, one thing after another, like a row of dominoes falling over. Ramona's mother set it off. \"By the way, Ramona,\" said Mrs. Quimby after breakfast as she hastily tossed potatoes, carrots, and stew meat into the Crock-Pot to simmer while the family was away all day. \"Please don't run in the hall in your socks. You might slip and fall.\"\n\nRamona's father was next. \"And Ramona,\" he said, pulling strings off celery before slicing it and adding it to the stew, \"when you wash your hands, don't leave the dirt on the cake of soap.\"\n\nThen Beezus. \"Or wipe it on the towel.\"\n\n\"I haven't had time to get dirty,\" said Ramona, who had finished her breakfast. \"And I have my shoes on.\"\n\n\"We are talking about yesterday,\" said her father.\n\nRamona thought yesterday was a long time ago, hardly worth mentioning. \"Everybody picks on me,\" she said.\n\n\"Poor kid.\" Mr. Quimby kissed his wife on the cheek and each daughter on the top of her head. Then he said, singing the first few words, \"Hi-ho, hi-ho, it's off to work I go and at least forty-six changes in produce prices to remember.\"\n\nRamona knew her father dreaded Wednesdays, the day prices were changed on fruits and vegetables.\n\n\"Maybe things will be easier when you are more used to the job,\" said Mrs. Quimby. After her husband left to catch the bus, she kissed the girls as if she were thinking about something else and handed Ramona her lunch box and Beezus the brown paper bag that held her sandwich. Seventh graders thought lunch boxes were babyish. \"Scoot along,\" Mrs. Quimby said. \"I have to leave early to take the car for a brake adjustment, and then I have to take the bus to work.\"\n\n\"I wish Daddy would get used to his job,\" remarked Beezus as the sisters plodded toward Glenwood School. Clouds hung low, and the wind was cold. For days the sidewalks had been too wet for roller skating.\n\n\"Me, too,\" said Ramona, who wanted her parents to be happy so their children could be happy, too. \"Why doesn't he find another job?\"\n\n\"Jobs aren't that easy to get,\" Beezus explained. \"Remember how long he was out of work before he found a job at the market.\"\n\nRamona did remember. She remembered how discouraged her father had been after a day of job-hunting and how he had disliked standing in line to collect unemployment insurance.\n\n\"It might be easier if he had finished college,\" said Beezus.\n\n\"Why didn't he?\" asked Ramona.\n\n\"Because he and Mother got married,\" Beezus explained. \"And then they had me.\" Beezus sounded smug, as if being the first born made her more important to her parents than Ramona.\n\nBut I'm the baby, thought Ramona. She was glad when school started; maybe her day would improve in school. Ramona liked Mrs. Rudge, her new teacher, who had taken over the second grade when the former teacher left after Christmas to have a baby. She _thought_ Mrs. Rudge liked her, as she liked all the children, but she was not sure exactly where she stood with her. The classroom buzzed softly and busily with the sound of children learning about Indians and cursive writing.\n\nWhen the morning was half over, Ramona finished her work sheet and was busy filling all the double _oo_ 's she could find with crossed eyes and frowns:\n\nMrs. Rudge paused beside her desk. Ramona did not have time to hide the frowning _oo_ 's.\n\nMrs. Rudge glanced over Ramona's work sheet. \"Why don't you look again?\" she suggested. \"Is _like_ spelled _l-i-c-k_?\"\n\n\"I can't spell,\" said Ramona. \"I'm terrible at spelling.\" That's what her family said whenever Ramona wrote a note. They always laughed and said, \"Ramona is no speller. See how she spells _much m-u-c-k_.\" They behaved as if she had done something clever.\n\nRamona learned right there that Mrs. Rudge was a teacher who did not accept excuses. \"There is no such a word as _can't_ ,\" she said, and went on to inspect Becky's work sheet.\n\nHow can there be no such word as _can't_? Ramona wondered. Mrs. Rudge had just said _can't_. If there was no such word as _can't_ , Mrs. Rudge could not have said there was no such word as _can't_. Therefore, what Mrs. Rudge had said could not be true. Ramona was left with a vague feeling that Mrs. Rudge did not like her because she did not offer to give Ramona extra help in spelling.\n\nAt lunchtime when Ramona went into the multipurpose room with her lunch box, she found that she had a leftover-pot-roast sandwich in her lunch. She did not like a leftover-pot-roast sandwich because the meat slid out in big pieces when she bit into it. After chewing awhile she thought, I won't eat it, and she stuffed the rest of her sandwich into the hole in her milk carton and threw it into the trash can. She sat there arguing with herself about how there had to be a word _can't_ because she had just thought it. This was not a good day.\n\nAfter school at the Kemps' house, Ramona and Howie drank the same old apple juice and ate the same old graham crackers that Mrs. Kemp always set out for them. Sticky Willa Jean, holding Woger by one paw, stood and watched. She was wearing a T-shirt with _Grandma Loves Me_ printed on the front. The shirt had shrunk so much it showed her navel\u2014tummy button, Mrs. Kemp called it.\n\nThen Howie got out the checkerboard, which he placed on the carpet. Kneeling, he and Ramona began to divide the red and black checkers.\n\n\"I want to play.\" Willa Jean plunked herself down on the carpet, sitting on Woger as if he were a cushion, which was no way to treat a bear, especially a bear like Woger.\n\n\"Aw, Willa Jean\u2014\" protested Howie, who had his problems with his little sister.\n\n\"Now Howie,\" said Mrs. Kemp, busy with her endless knitting, \"play nicely with your sister. She's little, you know.\"\n\nHowie knew all right.\n\nWilla Jean, pleased to have her grandmother on her side, set a red checker on top of a black checker. \"Your turn,\" she said to Ramona as if she were being generous.\n\nRamona and Howie shared one hopeless look. They were familiar with Willa Jean's original rules for checkers. Ramona set a black checker on top of Willa Jean's red checker. Howie added red and so on in turn until the tower of checkers grew high and crooked. At last, when Willa Jean set a checker on top, the tower tumbled.\n\n\"I won!\" crowed Willa Jean as Howie tried to prevent scattered checkers from rolling under the couch. \"Grandma, I beat Howie and Wamona!\"\n\n\"Smart girl!\" Mrs. Kemp paused in her knitting to smile down upon her granddaughter.\n\nThe situation was hopeless. \"Let's go down in the basement and see if we can think of something to build,\" said Howie, and Ramona agreed. They would be undisturbed in the basement. Willa Jean was afraid of the furnace.\n\nSafe from interruption, Howie and Ramona decided to build a boat of the scrap lumber Mr. Kemp collected for Howie to work with. They had already built a dog, a cat, and a duck decoy that Mr. Kemp said would never fool a real live duck. Now they sawed and pounded until they had a boat with two decks. They were so good at nailing by now they did not even pound their fingers. Next they found a dowel and sawed off two pieces for smokestacks, which Howie studied. \"It's going to be hard to nail them to our boat,\" he said.\n\n\"We could use Scotch tape,\" said Ramona, who felt that almost anything could be accomplished with Scotch tape.\n\n\"I don't think it's strong enough for wood,\" was Howie's objection. \"Glue might be better.\"\n\n\"Scotch tape would work if we use lots of it.\" Ramona was an experienced user of Scotch tape.\n\nIn the end, glue won out because Howie thought a boat should look neat. Very, very carefully they spread glue on the ends of the dowels and pressed them in place. They put the top back on the tube of glue, and each held a dowel in place, waiting for the glue to dry. They had not spilled a drop. Fortunately, the glue was quick drying.\n\n\"Let's see if our boat will float,\" said Howie. He pressed the plug of the laundry tub in place and turned on the faucet.\n\n\"Howie, what are you two doing down there?\" Mrs. Kemp called from the top of the stairs.\n\n\"Just seeing if our boat will float,\" he answered.\n\n\"All right,\" said Mrs. Kemp. \"Just don't let the tub overflow.\"\n\n\"We won't,\" promised Howie.\n\nThe boat floated. Howie and Ramona stirred up a storm at sea to make things interesting and watched their boat ride the waves. As it bobbed up and down, Ramona happened to glance up at a shelf above the laundry tub. There she spotted a blue plastic bottle with the picture of a nice old-fashioned lady's face on the label. Bluing!\n\nRamona knew all about bluing because her mother had used it to make white washing look whiter back in the days before she had gone to work. \"If we could get that bottle, we could turn the water blue like a real ocean,\" she suggested. \"It only takes a little bit.\"\n\nHowie was enthusiastic, but how were they to reach a bottle on such a high shelf? For some reason, Mrs. Rudge's words, there's no such word as _can't_ , ran through Ramona's mind. Of course they could get that bottle of bluing.\n\nRamona managed to balance on her stomach on the edge of the tub. Then she got one knee up and with a boost from Howie was able to climb up onto the edge of the tub. She stood teetering on the narrow edge clinging to the front of the shelf with one hand while she managed to grasp the bottle with the other and hand it down to Howie. As she did so, the top flew off. Bluing splashed over Howie, who tried to catch the top only to have the bottle slip from his fingers into the tub of water, where it poured forth swirls of beautiful deep blue. Ramona was so startled she lost her balance and landed standing up to her knees in blue water.\n\n\"Boy, Ramona, see what you've done.\" Howie looked down at his shirt and jeans, now streaked with blue.\n\nRamona felt Howie was being most unfair. She did not spill the bluing on purpose. Besides, why wasn't the top of the bottle screwed on tight? Because some grown-up had not screwed it on, that's why. Children weren't the only people who did things wrong. She fumbled through the blue water, now much bluer than any ocean, and pulled the plug. As the water drained out, she and Howie looked at one another. Now what should they do?\n\nMrs. Kemp called down the stairway. \"It's awfully quiet down there. What are you two up to?\"\n\n\"We had\u2014sort of an accident,\" confessed Howie.\n\nMrs. Kemp came running down the stairs. \"Oh, my land!\" she cried. \"Oh, my goodness!\"\n\nWilla Jean began to howl at the top of the stairs.\n\n\"Grandma won't let the furnace get you, darling,\" said Mrs. Kemp. Willa Jean sat down at the top of the stairs and wept.\n\nMrs. Kemp lifted dripping Ramona out of the tub. Then, right there in front of Howie, she pulled off Ramona's socks, slacks, and blouse and dumped them in the washing machine. Then she pulled off Howie's socks, shirt, and jeans and dumped those in the washing machine, too. Ramona and Howie did not know where to look, they were so embarrassed to be standing there in their underwear. Two years ago they would not have minded, but now that they were in the second grade, they felt that underwear was private.\n\nMrs. Kemp filled the tub with a few inches of clear water and lifted Ramona back in. Without a word she began to scrub Ramona's feet with a bar of yellow soap. When it was plain that Ramona's feet were going to stay blue, she lifted Ramona out again, pulled a towel out of the dryer, and handed it to her. Then Mrs. Kemp went to work on Howie's blue hands.\n\nWhen Ramona's blue feet were dry, she asked politely, \"What will I wear?\" Of course she could not go around in her underwear.\n\n\"We'll find something.\" Mrs. Kemp, rinsing Ramona's shoes, sounded grim. She held up the shoes, now a strange greenish brown, to let the water drain off them before she leaned them against the furnace to dry.\n\nSuddenly Mrs. Kemp missed Willa Jean. \"Oh, my goodness!\" she cried, and dashed up the stairs. Ramona and Howie, careful not to look at one another, followed. What Ramona saw made tears come to her eyes. There sat Willa Jean under the dining-room table holding a pair of scissors, sharp scissors, and Woger, who now had only one leg. Willa Jean had cut off Woger's leg! That lovable bear. How could Willa Jean do such a terrible thing? Ramona felt like crying, she loved Woger so.\n\n\"Give Grandma the scissors,\" coaxed Mrs. Kemp. \"We don't want the scissors to hurt Willa Jean.\"\n\n\"Boy, Willa Jean.\" Howie was disgusted. \"What did you have to go and do a dumb thing like that for?\"\n\nWilla Jean looked as if her brother had said something unkind. \"I wanted to see if Woger had bones,\" she said.\n\n\"He is so soft you should know he doesn't have bones,\" said Howie. \"You didn't have to wreck him.\"\n\nWilla Jean looked at the stuffing coming out of her bear's wounds and began to cry.\n\n\"Never mind, darling,\" said Mrs. Kemp. \"Grandma will sew Woger's leg back on after she finds some clothes for Howie and Ramona.\"\n\nRamona was soon bundled into Howie's old shirt and jeans and a pair of ragged sneakers much too big for her. She sat on one end of the couch while Howie sat on the other.\n\nRamona was cross because she did not like wearing Howie's old clothes. Howie was cross because Ramona had thought of dying the water blue. Both were cross with Willa Jean for spoiling the checker game. Mrs. Kemp, who was sewing Woger's leg back on, was cross with Ramona and Howie, but of course she was not cross with Willa Jean. Only Willa Jean, lying on her back under the coffee table and sucking her thumb, was happy.\n\nThis afternoon was not the first time Ramona had been in trouble at the Kemps' house. There was that day she and Howie found Mrs. Kemp's pinking shears. Ramona had been pinking Howie's hair when Mrs. Kemp discovered what they were up to. Ramona had thought she was unreasonably displeased, because Howie's hair was so curly the pinking did not show.\n\nNow Ramona worried. If she got into any more trouble, maybe Mrs. Kemp would not want to look after her. Then her mother could no longer work in Dr. Hobson's office and would have to stay home. Ramona quickly squashed a deep-down thought that she would like to have her mother stay home again. She waited anxiously for Beezus to come. She waited and waited. No Beezus.\n\nHowie looked at a sporting-goods catalog, turning the pages with blue hands. Boots, quilted jackets with many pockets, and those tents that folded into tiny packages interested Howie. He did not offer to share the catalog with Ramona, even though he knew she liked pictures of duck decoys.\n\nRamona heaved a gusty sigh. She wished she had brought her Betsy book with her. She enjoyed reading about Betsy because everyone in the book was so nice to her.\n\nWhen Woger's wounds were mended, Mrs. Kemp started supper. The fragrance of pork chops floated from the kitchen. The younger Mrs. Kemp, Howie's mother, came home with packages and bags of groceries. \"Why, hello, Ramona,\" she said. \"I didn't know you were still here.\"\n\nMr. Kemp came home from work. \"Hello there,\" he said. \"Are you still here?\"\n\nRamona did not know how to answer such a question. She felt embarrassed, in the way, unwanted. Where was Beezus? What had happened to her parents? Her ears strained for familiar footsteps or the sound of the Quimby car.\n\n\"Your mother and father are late today,\" remarked Howie's mother as she set the table.\n\nOnce more Ramona did not know how to answer. Cars were now driving with their lights on. Why didn't someone come? What if her mother and father had been in an accident? Who would take care of Ramona? It seemed as if she might have to sit here on the couch in Howie's old clothes forever.\n\nRamona began to feel hungry. How good a pork chop would taste! She knew she would not be asked to share the Kemps' supper. With the price of meat these days there would not be an extra chop. Ramona's mouth watered so much she had to swallow. She thought of the pot-roast sandwich she had not finished at lunchtime.\n\n\"Ramona, could I fix you some peanut butter and crackers?\" asked Howie's mother.\n\n\"No, thank you.\" Ramona pictured a brown chop with mashed potatoes and pool of gravy.\n\nThe Kemps sat down at the table with Willa Jean perched on two cushions beside her grandmother, who began to cut her meat for her. And she won't even eat a whole chop, thought Ramona, who felt like a stranger, an intruder in the lives of others. The Kemps said little as they ate. Perhaps they did not want to talk in front of an outsider. Ramona listened to the clink of knives and forks against plates as the Kemps ate their pork chops. She was profoundly embarrassed.\n\n\"Willa Jean, darlin', we don't chew with our mouth open,\" said Willa Jean's grandmother.\n\nAt last, when Ramona was blinking back tears because she was sure her parents would never come, the old familiar car turned into the driveway.\n\n\"Good-bye!\" cried Ramona, pulling on her car coat as she ran out the door.\n\nMr. Quimby was driving, Mrs. Quimby sat next to him, and Beezus was in the back seat. She must have been picked up at a friend's house. \"You were late,\" Ramona informed her family, her voice stern. \"You kept me waiting.\"\n\n\"I'm sorry.\" Mrs. Quimby sounded tired. \"It was one of those days. After work when I went to catch a bus to the garage to pick up the car, the bus was late, and when I finally got to the garage, the mechanics hadn't finished the job and I had to wait some more. And I had to keep your father waiting, too.\"\n\n\"What a day!\" said Mr. Quimby. \"Price changes to remember, and I worked the express line besides.\" Ramona knew her father disliked the express line in which customers were not supposed to have more than nine items in each basket. Many people tried to slip through with ten or eleven items. Everyone in line was in a hurry and counted the items in one another's baskets. There were arguments. All this unpleasantness took a lot out of Mr. Quimby.\n\nPlease, please like your job, prayed Ramona, forgetting her own troubles for a moment.\n\nMrs. Quimby turned in the front seat to look at her daughters. Of course she noticed Ramona was wearing Howie's old clothes.\n\n\"Ramona, why are you wearing . . . ?\" Mrs. Quimby seemed too tired to finish the question.\n\n\"Howie and I sort of spilled some stuff and Mrs. Kemp washed our clothes and they aren't dry yet,\" explained Ramona. Her mother could discover her blue feet later. \"It was Willa Jean's fault. She wrecked our checker game so we had to go down in the basement to get away from her.\"\n\n\"Sounds like you,\" said Beezus. \"I can remember when you used to bump the coffee table with your tricycle when I was playing checkers with a friend.\"\n\n\"I did not!\" Ramona was indignant.\n\n\"You did, too,\" said Beezus. \"You just can't remember.\"\n\n\"Girls!\" said Mrs. Quimby. \"It doesn't really matter who wrecked whose checker game or where or when.\"\n\nRain slanted through the beams of the car lights, the windshield wipers _splip-splopped_ , the family was silent. Ramona, huddled in the corner of the back seat, wondered if she really had been as awful as Willa Jean. Nobody loved Ramona\u2014well, maybe her father a little bit sometimes. If her mother really loved her, she would say to Beezus that Ramona was never anything like Willa Jean.\n\nRamona not only felt unloved, she was so hungry her stomach growled. As Mr. Quimby turned their car into the driveway, she thought of the stew that had been simmering away in the Crock-Pot all day. How good it would smell when they opened the door! The Crock-Pot always gave out a warm and welcoming fragrance as if Ramona's mother had been home all day preparing supper to greet them. One whiff of stew, Ramona was sure, and everything would be all right again. Her mother would forget her troubles with the car, her father would begin to make jokes again, she and Beezus would set the table, and they would all sit down to a nice warm dinner.\n\n#\n\n## THE QUARREL\n\nAs soon as Ramona stepped through the back door, she knew something was wrong. There was a chill about the house, and it had the faint mustiness of a place that had been closed and unoccupied all day. There was no welcoming fragrance of simmering meat and vegetables. The tiny light on the Crock-Pot was dark, the pot cold.\n\n\"Oh, no!\" cried Mrs. Quimby, noticing.\n\n\"What's wrong?\" asked Mr. Quimby, coming in from the hall where he had gone to turn up the thermostat of the furnace.\n\n\"Wrong!\" Mrs. Quimby lifted the lid of the electric casserole on the kitchen counter. \"Someone forgot to plug in the Crock-Pot this morning, that's what's wrong.\"\n\nThe family gathered to peer in at the cold vegetables and raw meat.\n\n\"I'm starving!\" wailed Beezus.\n\n\"Me, too,\" said Ramona.\n\n\"I thought you turned it on,\" said Mrs. Quimby to her husband as she shoved the plug into the socket. The stew could cook overnight and be warmed up for the next evening.\n\n\"Don't look at me,\" said Mr. Quimby to his wife. \"I thought you turned it on.\" There was an edge to his voice.\n\nFor some reason his remark annoyed Mrs. Quimby. \"I suppose you think turning on a Crock-Pot is woman's work.\" The edge in her voice matched the edge in his.\n\n\"Not exactly,\" said Mr. Quimby, \"but now that you mention it\u2014\"\n\n\"Don't forget the time you forgot to fork the potatoes you put in to bake and they exploded,\" his wife reminded him.\n\nRamona stifled a laugh at that memory. Her father had looked so surprised the evening the potatoes exploded\u2014 _poof_!\u2014when he had opened the oven door.\n\nMr. Quimby was not going to be drawn into a discussion of past baked potatoes. \"Why not just throw the stuff into the frying pan and cook it?\" he asked. His idea of cooking was to toss everything into a pan and stir until done. Sometimes he invented interesting dishes with ground meat and eggs, zucchini and cheese. Other times the family tried to be good sports at dinner.\n\n\"Because you can't fry stew meat.\" Mrs. Quimby sounded annoyed as she looked into the cupboard and the refrigerator. \"It's too tough. You know that. Did you bring groceries?\"\n\n\"No. I thought we were having stew for dinner,\" answered Mr. Quimby. Crossly, Ramona thought. \"I didn't see anything on the grocery list.\"\n\nPicky-picky, the cat, rubbed against Mrs. Quimby's legs, telling her how hungry he was. \"Scat,\" said Mrs. Quimby.\n\nPicky-picky went to Beezus, not Ramona. He did not like Ramona, had never liked her because she was too noisy.\n\n\"I'm practically dying of hunger,\" said Beezus as she picked up the old cat and rubbed her cheek against him.\n\n\"Me, too,\" said Ramona.\n\n\"You girls are no help,\" Mrs. Quimby told her daughters. \"We have a couple of eggs, not enough for an omelet, two strips of bacon, three carrots, and some tired old lettuce. That's it.\" She looked at her husband. \"We don't have to let the cupboard get completely bare before we buy groceries.\"\n\nThis remark gave Ramona a cue. \"Old Mother Hubbard went to the cupboard\u2014\" she began, but she did not finish the rhyme because she could see no one was listening.\n\n\"Anytime we are low on groceries, just make a list,\" said Mr. Quimby. \"That's all you have to do.\"\n\n\"I could make carrot salad,\" suggested Beezus, as if carrot salad might smooth things over.\n\n\"We could have pancakes,\" said Mr. Quimby, \"with half a strip of bacon apiece.\"\n\n\"Not a very nutritious meal,\" said Mrs. Quimby, \"but better than starvation.\" She reached for a mixing bowl while Beezus, who had dropped Picky-picky and washed her hands, began to grate carrots onto a sheet of waxed paper. Ramona leaned against the counter to watch. She wanted to make sure her sister did not grate her fingers into the salad.\n\n\"Ramona, don't just stand there,\" said Mr. Quimby as he laid the bacon in a frying pan. \"Get busy and set the table. As my grandmother used to say, 'Every kettle must rest on its own bottom,' so do your part.\"\n\nRamona made a face as she reached for the place mats. \"Daddy, I bet your grandmother didn't really say all the things you say she said.\"\n\n\"If she did, she must have been a dreadful bore,\" said Mrs. Quimby, who was beating batter as if she were angry with it.\n\nMr. Quimby looked hurt. \"You didn't know my grandmother.\"\n\n\"If she went around spouting wisdom all the time, I can't say I'm sorry.\" Mrs. Quimby was on her knees, dragging the griddle from behind the pots and pans in the bottom of the cupboard.\n\nRamona paused in laying the silverware to make sure there was no blood on the carrots. She felt the muscles of her stomach tighten as they always tightened when her mother was cross with her father.\n\n\"My grandmother was a wonderful woman,\" said Mr. Quimby. \"She had a hard life out there in the country, but she was good to us kids and we learned a lot from her.\"\n\n\"Well, my grandmother wasn't so bad herself.\" With an angry sounding crash the griddle knocked over two pans and a double boiler as Mrs. Quimby yanked it from the cupboard. \"And I learned a lot from her.\"\n\nRamona and Beezus exchanged an anxious look.\n\n\"Just what did you learn from your grandmother?\" asked Mr. Quimby. \"As far as I could see, all she ever did was gad around and play bridge.\"\n\nRamona and Beezus exchanged another look. Were their parents quarreling? Really quarreling? Yes, the sisters' eyes agreed. Both girls were worried.\n\nMrs. Quimby set the griddle on the stove with more noise than necessary. She was plainly trying to think what she had learned from her grandmother. Finally she said, \"My grandmother taught me to pick flowers with long stems and to pick a few leaves to put in with them.\"\n\n\"Very useful,\" said Mr. Quimby.\n\nThe hint of sarcasm in his voice must have annoyed Mrs. Quimby because she said, \"My grandmother didn't have much money, but she had a sense of beauty.\" The drop of water she flicked on the griddle refused to dance.\n\n\"No matter how much my grandmother had to scrimp and pinch to make ends meet,\" said Mr. Quimby, \"she always managed to find money to buy paper for me to draw on.\"\n\nScrimp and pinch to make ends meet, thought Ramona, liking the sound of the words. She would remember them. The smell of bacon sizzling made her feel better. It also made her hungrier.\n\n\"My grandmother taught me useful things, too.\" Mrs. Quimby had had time to think. \"She taught me that a dab of spit would stop a run in a stocking.\" She flicked another drop of water on the griddle. This one danced. The griddle was hot.\n\n\"Some grandmother,\" said Mr. Quimby, \"spitting on her stockings.\"\n\n\"You're both being silly,\" Beezus burst out. \"Just plain silly!\"\n\n\"Young lady, you keep out of this,\" ordered Mr. Quimby.\n\nBeezus glared at her father. \"Well, you are,\" she muttered.\n\nMrs. Quimby silently poured four puddles of batter on the griddle. Ramona prayed that the quarrel, whatever it was about, was over.\n\nBeezus stirred mayonnaise into the blood-free carrots, which she then divided on four limp lettuce leaves on four salad plates. Mr. Quimby turned the bacon. Mrs. Quimby flipped the pancakes. Ramona's stomach relaxed. In a moment her mother would slide the pancakes onto a platter and start another four cooking. Ramona could hardly wait, she was so hungry.\n\n\"Are you sure those pancakes are done?\" asked Mr. Quimby as his wife slid the pancake turner under them. \"They don't look done to me.\"\n\n\"They bubbled in the middle before I turned them,\" said Mrs. Quimby, \"and they look done to me.\"\n\nMr. Quimby took the pancake turner from his wife. Using it as a weapon, he slashed each pancake in the center. Ramona and Beezus exchanged a shocked look. Their father had slashed their mother's pancakes! He had gone too far. Frightened, they watched raw batter ooze from four gashes in the pancakes. Their father was right. The cakes were not done. Now what would their mother do?\n\nMrs. Quimby was furious. She snatched back the pancake turner, scooped up the oozing cakes, and tossed them into the garbage.\n\n\"You didn't need to do that.\" Mr. Quimby looked amused. He had won. \"You could have turned them again and let them finish cooking.\"\n\n\"And I suppose your grandmother made absolutely perfect pancakes,\" said Mrs. Quimby in a voice stiff with anger.\n\nMr. Quimby looked calm and even more amused. \"As a matter of fact, she did,\" he said. \"Brown and lacy, cooked all the way through, and with crisp edges.\"\n\n\"The best pancakes you ever ate,\" stated Mrs. Quimby in a voice that made Ramona silently pray. Mother, be nice again. Please, please be nice again.\n\n\"Right,\" said Mr. Quimby. \"Light enough to melt in your mouth.\"\n\nBe quiet, Daddy, prayed Ramona. You'll make things worse.\n\n\"Oh\u2014you!\" Mrs. Quimby gave Mr. Quimby a swat on the seat of his pants with the pancake turner before she threw it on the counter. \"Bake them yourself since you learned so much from that noble grandmother of yours!\"\n\nRamona and Beezus stood frozen with shock. Their mother had hit their father with a pancake turner. Ramona wanted to fly at her mother, to strike her and cry out, You hit my daddy! She dared not.\n\nMr. Quimby tucked a dish towel in his belt for an apron and calmly ladled batter onto the griddle while his wife stalked into the living room and sat down with the newspaper. If only he wouldn't whistle so cheerfully as he deftly turned the cakes and drained the bacon.\n\n\"Dinner is served,\" Mr. Quimby announced as he set a platter of hot cakes and bacon on the table and pulled the dish towel from his belt. Silently Mrs. Quimby joined the family.\n\nEven though her mother was usually a much better cook than her father, Ramona had to admit her father made excellent pancakes. Unfortunately, she was no longer very hungry. She felt all churned up inside, as if she didn't know whether to cry or to burst out of the house shouting, My mother and father had a fight!\n\n\"Please pass the butter.\" Mrs. Quimby might have been speaking to a stranger.\n\n\"May I please have the syrup?\" Mr. Quimby asked politely.\n\n\"The funniest thing happened at school,\" said Beezus, and Ramona understood that her sister was anxious to start a conversation that would smooth things over and make their parents forget their quarrel, perhaps make them laugh.\n\nAfter a moment of silence Mrs. Quimby said, \"Tell me.\"\n\n\"You'll never guess how a boy spelled _relief_ in a spelling test,\" said Beezus.\n\n\"How?\" asked Ramona to help the conversation along. Mr. Quimby silently served himself two more hot cakes.\n\n\"He spelled it _r-o-l-a-i-d-s_ ,\" said Beezus, looking anxiously at her parents, who actually smiled.\n\nRamona did not smile. \"But the man on television spells _relief_ that way. He said _r-o-l-a-i-d-s_ spells _relief_. I've heard him.\"\n\n\"Silly,\" said Beezus, but this time she spoke with affection. \"That's just a slogan. _Relief_ is _r-e-l-i-e-f_.\"\n\n\"Oh.\" Ramona was glad to know. Tabletalk sank back into silence while Ramona thought about spelling. Spelling was full of traps\u2014blends and silent letters and letters that sounded one way in one word and a different way in another\u2014and having a man stand there on television fooling children was no help. She was glad she had a big sister who understood those things.\n\nThe evening was quiet. Mr. Quimby dozed in front of the television set. Mrs. Quimby took a shower and went to bed to read. Beezus did her homework in her room. Ramona tried to draw a monster eating a mouthful of people, but she could not make the picture on paper match the one in her imagination. Her monster looked as if he were eating paper dolls instead of real people. The house was unnaturally quiet. The television droned on. Both girls went to bed without being told.\n\nUnhappy thoughts kept Ramona awake. What if her mother and father did not love one another anymore? What if they decided to get a divorce like her friend Davy's parents? What would happen to her? Who would take care of her? Beezus was closer to being a grown-up, but what about Ramona? She wanted to cry but could not. She felt too tight inside to cry. Tears teetered on her eyelashes but would not give her the relief of falling.\n\nFinally Ramona could stand her fear and loneliness no longer. She slipped out of bed and tiptoed into her sister's room.\n\n\"Ramona?\" Beezus too was awake.\n\n\"I can't go to sleep,\" whispered Ramona.\n\n\"Neither can I,\" said Beezus. \"Come on, get in bed with me.\"\n\nThis invitation was what Ramona had been hoping for. Gratefully she slipped beneath the covers and snuggled against her sister. \"Do you think they'll get a divorce?\" she whispered. \"They won't talk to each other.\"\n\n\"Of course not,\" said Beezus. \"At least I don't think so.\"\n\n\"Who would take care of me if they did?\" Ramona felt she had to have the answer from someone. \"I'm still little.\" Beezus, of course, was her mother's girl, but what about Ramona?\n\nBeezus seemed to be considering the question. \"I'll try,\" she said at last.\n\n\"You aren't grown up enough,\" said Ramona, nevertheless comforted. Beezus cared.\n\n\"I know,\" admitted Beezus. \"I read a book about a girl who took care of her brothers and sisters when their father died, but that was off in the mountains someplace where they all picked herbs and things. It wouldn't work in the city.\"\n\n\"Mother and Daddy won't be dead.\" Ramona was consoled by this knowledge.\n\nBeezus was silent awhile. \"They could have been joking,\" she said. \"Sort of.\"\n\n\"But Mother hit Daddy,\" Ramona pointed out. \"On the seat of his pants with a pancake turner.\"\n\n\"I don't think that's the same as if she had hit him with something hard,\" said Beezus. \"After all, she didn't really hurt him.\"\n\nRamona tried to find a bright side. \"And he didn't hit her back,\" she said. \"But if they loved us, they wouldn't fight.\" She silently said her prayers, ending with, \"Please, please don't let Mother and Daddy fight.\"\n\nFrom the kitchen came a whiff of the stew that would simmer through the night for their supper the next evening. Soothed by the homey fragrance, the sisters fell asleep.\n\nIn the morning, a few seconds after she awoke and found herself in her sister's bed, a dull, unhappy feeling settled over Ramona. Her parents had quarreled. She dreaded facing them at breakfast. She did not know what to say to them. Beezus looked unhappy, too. Getting dressed took longer than usual, and when they finally went into the kitchen, they were surprised to see their parents sharing the morning paper as they ate breakfast together.\n\n\"Good morning, girls,\" said Mr. Quimby with his usual cheerfulness.\n\n\"There is oatmeal on the stove.\" Mrs. Quimby smiled fondly at her daughters. \"Did you sleep well?\"\n\nBeezus was suddenly angry. \"No, we didn't!\"\n\n\"No, we didn't,\" echoed Ramona, encouraged by her sister's anger. How could her mother expect them to sleep well when they were so worried?\n\nStartled, both parents laid down the newspaper.\n\n\"And it's all your fault,\" Beezus informed them.\n\n\"What on earth are you talking about?\" asked Mrs. Quimby.\n\nBeezus was near tears. \"Your big fight, that's what.\"\n\nRamona blinked back tears, too. \"You wouldn't even talk to each other. And you hit Daddy!\"\n\n\"Of course we were speaking,\" said Mrs. Quimby. \"Where did you get the idea we weren't? We were just tired is all. We had one of those days when everything seemed to go wrong.\"\n\nSo did I, thought Ramona.\n\n\"I went to bed and read,\" continued Mrs. Quimby, \"and your father watched television. That was all there was to it.\"\n\nRamona felt almost limp with relief. At the same time she was angry with her parents for causing so much worry. \"Grown-ups aren't supposed to fight,\" she informed them.\n\n\"Oh, for heaven's sake,\" said Mrs. Quimby. \"Why not?\"\n\nRamona was stern. \"Grown-ups are supposed to be perfect.\"\n\nBoth her parents laughed. \"Well, they are,\" Ramona insisted, annoyed by their laughter.\n\n\"Name one perfect grown-up,\" challenged Mr. Quimby. \"You can't do it.\"\n\n\"Haven't you noticed grown-ups aren't perfect?\" asked Mrs. Quimby. \"Especially when they're tired.\"\n\n\"Then how come you expect us kids to be so perfect all the time?\" demanded Ramona.\n\n\"Good question,\" said Mr. Quimby. \"I'll have to think of an answer.\"\n\n\"We want you to be perfect so you won't grow up to bicker about your grandmothers and their pancakes,\" said Mrs. Quimby. Both parents thought her reply was funny.\n\nRamona felt the way Picky-picky looked when someone rumpled his fur. Maybe grown-ups weren't perfect, but they should be, her parents most of all. They should be cheerful, patient, loving, never sick and never tired. And fun, too.\n\n\"You kids fight,\" said Mr. Quimby. \"Why shouldn't we?\"\n\n\"It isn't dignified,\" said Beezus, giving Ramona another word to add to her list. \"Especially when you hit someone with a pancake turner.\"\n\n\"Oh, you silly little girls,\" said Mrs. Quimby with amusement and affection.\n\n\"Why should we let you kids have all the fun?\" asked Mr. Quimby.\n\n\"We don't quarrel for fun,\" Ramona informed her father.\n\n\"You could fool me,\" said Mr. Quimby.\n\nRamona refused to smile. \"Don't you ever do it again,\" she ordered her parents in her sternest voice.\n\n\"Yes, ma'am,\" answered Mrs. Quimby with mock meekness, as if she were poking a little fun at Ramona.\n\n\"Yes, _ma'am_!\" said her father, and saluted as if she were somebody important.\n\nThis time Ramona had to laugh.\n\n#\n\n## THE GREAT HAIR ARGUMENT\n\n\"Ramona, stand on both feet and hold still,\" said Mrs. Quimby one Saturday morning. \"I can't cut your bangs straight when you wiggle.\"\n\n\"I'm trying,\" said Ramona. Bits of falling hair made her nose tickle. She blew upward, fanning out her bangs from her forehead, to rid herself of the tickle.\n\n\"Now see what you've done.\" Mrs. Quimby recombed the bangs.\n\nRamona stood perfectly still in an agony of itching, twitching her nose to get rid of snips of falling hair, until her mother finally said, \"There, little rabbit, we're finished.\" She removed the towel from Ramona's shoulders and shook it over the kitchen wastebasket. Ramona, who liked being called a little rabbit, continued to twitch her nose and think of the warm and cozy picture books about bears and rabbits her mother used to read to her at bedtime before she kissed her good-night. She had loved those books. They made her feel safe. During the daytime she had preferred books about steam shovels, the noisier the better, but at night\u2014bears, nice bears, and bunnies.\n\n\"Next!\" Mrs. Quimby called out to Beezus, who had just washed her hair. These days Beezus spent a lot of time locked in the bathroom with a bottle of shampoo.\n\n\"Beezus, don't keep me waiting,\" said Mrs. Quimby. \"I have a lot to do this morning.\" The washing machine had broken down, and because no one had been able to stay home during the week to admit a repairman, Mrs. Quimby had to drive to a laundromat with three loads of washing. Repairmen did not work on Saturdays.\n\n\"I'm waiting,\" repeated Mrs. Quimby.\n\nBeezus, rubbing her hair with a towel, appeared in the doorway. \"Mother, I don't want you to cut my hair,\" she announced.\n\nRamona, about to leave the kitchen, decided to stay. She sensed an interesting argument.\n\n\"But Beezus, you're so shaggy,\" protested Mrs. Quimby. \"You look untidy.\"\n\n\"I don't want to look tidy,\" said Beezus. \"I want to look nice.\"\n\n\"You look nice when you're neat.\" Mrs. Quimby's voice told Ramona her mother's patience was stretched thin. \"And don't forget, how you look is not as important as how you behave.\"\n\n\"Mother, you're so old-fashioned,\" said Beezus.\n\nMrs. Quimby looked both annoyed and amused. \"That's news to me.\"\n\nBeezus plainly resented her mother's amusement. \"Well you _are_.\"\n\n\"All right. I'm old-fashioned,\" said Mrs. Quimby in a way that told Ramona she did not mean what she was saying. \"But what are we going to do about your shaggy hair?\"\n\n\"I am not a sheep dog,\" said Beezus. \"You make me sound like one.\"\n\nMrs. Quimby chose silence while Ramona, fascinated, waited to see what would happen next. Deep down she was pleased, and guilty because she was pleased, that her mother was annoyed with Beezus. At the same time, their disagreement worried her. She wanted her family to be happy.\n\n\"I want to get my hair cut in a beauty shop,\" said Beezus. \"Like all the other girls.\"\n\n\"Why Beezus, you know we can't afford a luxury like that,\" said Mrs. Quimby. \"Your hair is sensible and easy to care for.\"\n\n\"I'm practically the only girl in my whole class who gets a home haircut,\" persisted Beezus, ignoring her mother's little speech.\n\n\"Now you're exaggerating.\" Mrs. Quimby looked tired.\n\nRamona did not like to see her mother look tired so she tried to help. \"Karen in my room at school says her mother cuts her hair and her sister's too, and her sister is in your class.\"\n\nBeezus turned on her sister. \"You keep out of this!\"\n\n\"Let's not get all worked up,\" said Mrs. Quimby.\n\n\"I'm not worked up,\" said Beezus. \"I just don't want to have a home haircut, and I'm not going to have one.\"\n\n\"Be sensible,\" said Mrs. Quimby.\n\nBeezus scowled. \"I've been good old sensible Beezus all my life, and I'm tired of being sensible.\" She underlined this announcement by adding, \"Ramona can get away with anything, but not me. No. I always have to be good old sensible Beezus.\"\n\n\"That's not so.\" Ramona was indignant. \"I never get away with anything.\"\n\nAfter a thoughtful moment, Mrs. Quimby spoke. \"So am I tired of being sensible all the time.\"\n\nBoth sisters were surprised, Ramona most of all. Mothers were supposed to be sensible. That was what mothers were for.\n\nMrs. Quimby continued. \"Once in a while I would like to do something that isn't sensible.\"\n\n\"Like what?\" asked Beezus.\n\n\"Oh\u2014I don't know.\" Mrs. Quimby looked at the breakfast dishes in the sink and at the rain spattering against the windows. \"Sit on a cushion in the sunshine, I guess, and blow the fluff off dandelions.\"\n\nBeezus looked as if she did not quite believe her mother. \"Weeds don't bloom this time of year,\" she pointed out.\n\nRamona felt suddenly close to her mother and a little shy. \"I would like to sit on a cushion and blow dandelion fluff with you,\" she confided, thinking what fun it would be, just the two of them, sitting in warm sunshine, blowing on the yellow blossoms, sending dandelion down dancing off into the sunlight. She leaned against her mother, who put her arm around her and gave her a little hug. Ramona twitched her nose with pleasure.\n\n\"But Mother,\" said Beezus, \"you always said we shouldn't blow on dandelions because we would scatter seeds and they would get started in the lawn and are hard to dig out.\"\n\n\"I know,\" admitted Mrs. Quimby, her moment of fantasy at an end. \"Very sensible of me.\"\n\nBeezus was silenced for the time being.\n\n\"I like your hair, Mother,\" said Ramona, and she did. Her mother's short hair was straight, parted on one side and usually tucked behind her left ear. It always smelled good and looked, Ramona felt, the way a mother's hair should look, at least the way her mother's hair should look. \"I think your hair looks nice,\" she said, \"and I don't mind when you cut my hair.\" In the interest of truth she added, \"Except when my nose tickles.\"\n\nBeezus flared up once more. \"Well, goody-goody for you, you little twerp,\" she said, and flounced out of the kitchen. In a moment the door of her room slammed.\n\nRamona's feelings were hurt. \"I'm not a little twerp, am I?\" she asked, wondering if her mother agreed.\n\nMrs. Quimby reached for the broom to sweep bits of hair from the kitchen floor. \"Of course not,\" she said. \"I don't bring up my daughters to be twerps.\"\n\nRamona twitched her nose like a rabbit.\n\nAfterward neither Mrs. Quimby nor Beezus mentioned hair. Beezus's hair grew shaggier and Ramona decided that if her sister did not look like a sheepdog yet, she soon would. She also sensed that, as much as her mother wanted to say something about Beezus's hair, she was determined not to.\n\nBeezus, on the other hand, looked defiant. She sat at the dinner table with a you-can't-make-me-if-I-don't-want-to look on her face.\n\nRamona discovered that the tiny part of herself, deep down inside, that had been pleased because her mother was angry with her sister was no longer pleased. Anger over one person's hair was not worth upsetting the family.\n\n\"Women,\" muttered Mr. Quimby every evening at supper. He also remarked, as if he had hair on his mind, that he thought he was getting a little thin on top and maybe he should massage his scalp.\n\nConversation was strained. Beezus avoided speaking to her mother. Mrs. Quimby tried to look as if nothing had happened. She said calmly, \"Beezus, when the shampoo bottle is almost empty, don't forget to add shampoo to the grocery list. We use it, too, you know.\"\n\n\"Yes, Mother,\" said Beezus.\n\nRamona felt like yelling, Stop it, both of you! She tried to think of interesting things to talk about at the dinner table to make her family forget about hair.\n\nOne evening, to distract her family from hair, Ramona was telling how her teacher had explained that the class should not be afraid of big words because big words were often made up of little words: _dishcloth_ meant a cloth for washing dishes and _pancake_ meant a cake cooked in a pan.\n\n\"But I bake cakes in pans\u2014or used to\u2014and this does not make them pancakes,\" Mrs. Quimby pointed out. \"If I bake an angelfood cake in a pan, it is not a pancake.\"\n\n\"I know,\" said Ramona. \"I don't understand it because _carpet_ does not mean a pet that rides in a car. Picky-picky is not a carpet when we take him to the vet.\" At this example her parents laughed, which pleased Ramona until she noticed that Beezus was neither laughing nor listening.\n\nBeezus took a deep breath. \"Mother,\" she said in a determined way that told Ramona her sister was about to say something her mother might not like. The words came out in a rush. \"Some of the girls at school get their hair cut at Robert's School of Hair Design. People who are learning to cut hair do the work, but a teacher watches to see that they do it right. It doesn't cost as much as a regular beauty shop. I've saved my allowance, and there's this lady named Dawna who is really good and can cut hair so it looks like that girl who ice skates on TV. You know, the one with the hair that sort of floats when she twirls around and then falls in place when she stops. Please, Mother, I have enough money saved.\" When Beezus had finished this speech she sat back in her chair with an anxious, pleading expression on her face.\n\nMrs. Quimby, who had looked tense when Beezus first began to speak, relaxed. \"That seems reasonable. Where is Robert's School of Hair Design?\"\n\n\"In that new shopping center on the other side of town,\" Beezus explained. \"Please, Mother, I'll do anything you want if you'll let me go.\"\n\nRamona did not take this promise seriously.\n\nIn the interests of family peace, Mrs. Quimby relented. \"All right,\" she said with a small sigh. \"But I'll have to drive you over. If you can hold out until Saturday, we'll go see what Dawna can do about your hair after I drive your father to work.\"\n\n\"Oh, thank you, Mother!\" Beezus looked happier than she had since the beginning of the great hair argument.\n\nRamona was pleased, too, even though she knew she would have to be dragged along. Peace in the family was worth a boring morning.\n\nSaturday turned out to be cold, raw, and wet. Ramona despaired of ever using her roller skates. The Quimbys hurried through breakfast, stacked the dishes in the sink, piled into the car and drove off, windshield wipers flopping furiously, to deliver Mr. Quimby to the ShopRite Market. Ramona, resigned to a tiresome morning, could feel Beezus's excitement and see how tightly she clutched her allowance in the drawstring bag she had crocheted.\n\nWhen Mr. Quimby had been dropped off at the market, Beezus joined her mother in the front seat. She always gets to sit in the front seat, thought Ramona.\n\nMrs. Quimby started up the on-ramp to the freeway that cut the city in two. \"Beezus, watch for the signs. I have to keep my eyes on my driving,\" she directed.\n\nRamona thought, I can read, too, if the words aren't too long.\n\nMrs. Quimby looked back over her shoulder for a space in which to merge with the heavy morning traffic. A space came down the freeway, and Mrs. Quimby managed to fit the car into it. In no time they were crossing the river, which looked cold and gray between the black girders of the bridge. Green signs spanned the freeway.\n\n\"Do I turn left?\" asked Mrs. Quimby, uncertain of the way to the shopping center.\n\n\"Right,\" said Beezus.\n\nMrs. Quimby turned right onto the off-ramp.\n\n\"Mother,\" cried Beezus. \"You were supposed to turn left.\"\n\n\"Then why did you tell me to turn right?\" Mrs. Quimby sounded angry.\n\n\"You asked if you should turn left,\" said Beezus, \"and I meant, 'Right, you should turn left.'\"\n\n\"After this, use your head,\" said Mrs. Quimby. \"Now how do I get back on the freeway?\" She drove through a maze of unfamiliar one-way streets looking for an on-ramp sign. Finally she asked for directions from a man at a service station. He looked disagreeable because he had to come out in the rain.\n\nRamona sighed. The whole world seemed gray and cross, and it was most unfair that she should have to be dragged along on a dreary ride just because Beezus wanted her hair cut by Dawna. Her mother would never go to all this trouble for Ramona's hair. Huddled in the back seat, she began to feel carsick. The Quimby car, which they had bought from someone who had owned a large dog, began to smell like a dog. \"Oh-h,\" moaned Ramona, feeling sick. She thought about the oatmeal she had eaten for breakfast and quickly tried not to think about it.\n\nMrs. Quimby glanced in the rear-view mirror. \"Are you all right, Ramona?\" Her voice was anxious.\n\nRamona did not answer. She was afraid to open her mouth.\n\n\"I think she's going to upchuck,\" said Beezus, who, since she was in the seventh grade, said _upchuck_ instead of _throw up_. She felt the new word was more sophisticated.\n\n\"Hang on, Ramona!\" said Mrs. Quimby. \"I can't stop on the freeway, and there's no way to get off.\"\n\n\"Mother!\" cried Beezus. \"She's turning green!\"\n\n\"Ramona, open the window and hang on!\" ordered Mrs. Quimby.\n\nRamona was too miserable to move. Beezus understood. She unbuckled her seat belt, which buzzed angrily. \"Oh, shut up,\" she said to her seat belt as she leaned over and lowered a window for Ramona.\n\nCold air swept away the doggy smell, and drops of rain against her face made Ramona feel better, but she kept her mouth shut and did not move. Hanging on was not easy.\n\n\"How did I ever get into this?\" Mrs. Quimby wondered aloud as she turned onto the off-ramp that led from the freeway.\n\nWhen the haircut expedition finally reached the shopping center and parked near Robert's School of Hair Design, the three Quimbys splashed through the rain. Ramona, who had quickly recovered when the car stopped, found a certain grim pleasure in stomping in puddles with her boots.\n\nAfter the cold, the air inside the beauty school seemed too warm and too fragrant. Pee-you, thought Ramona as she listened to running water, snipping scissors, and the hushed roar of hair dryers.\n\nA man, probably Robert himself, asked, \"What can I do to help you ladies?\" as perspiring Ramona began to wiggle out of her car coat.\n\nBeezus was suddenly shy. \"I\u2014I would like Dawna to cut my hair,\" she said in almost a whisper.\n\n\"Dawna graduated last week,\" said Robert, glancing behind the screen that hid the activity of the school, \"but Lester can take you.\"\n\n\"Go ahead,\" said Mrs. Quimby, answering Beezus's questioning eyes. \"You want your hair cut.\"\n\nWhen Robert asked for payment in advance, Beezus pulled open her crocheted bag and unfolded the bills she had saved. As Robert led her behind the screen, Mrs. Quimby sank with a little sigh into one of the plastic chairs and picked up a shabby magazine. Ramona tried to amuse herself by drawing pictures with her toe in the damp and muddy spots their boots had left on the linoleum.\n\n\"Ramona, please don't do that,\" said Mrs. Quimby, glancing up from her magazine.\n\nRamona flopped back in a chair and sighed. Her booted feet were beginning to feel hot. To pass the time, she studied pictures of hair styles mounted on the wall. \"Is Beezus going to look like _that_?\" she whispered.\n\nMrs. Quimby glanced up again. \"I hope not,\" she whispered back.\n\nRamona peeked behind the screen and reported to her mother. \"A man is washing Beezus's hair, and she's lying back with her head in a sink. He's using gobs of shampoo. He's wasting it.\"\n\n\"Mm-mm.\" Mrs. Quimby did not raise her eyes from the magazine. Ramona twisted her head to see what her mother found so interesting. Recipes.\n\nRamona returned for another look. \"He's rubbing her hair with a towel,\" she reported.\n\n\"Mm-mm.\" Ramona disliked her mother's mm-mming. She walked quietly behind the screen to watch. Lester was studying Beezus's hair, one lock at a time, while a woman, probably a teacher, watched.\n\n\"Ramona, come back here,\" Mrs. Quimby whispered from the edge of the screen.\n\nOnce more Ramona flopped down in the plastic chair and swung her legs back and forth. How nice it would be if she could have her hair shampooed, too. She raised her eyebrows as high as she could to make her bangs look longer and thought of her quarter, two nickels, and eight pennies at home in a Q-tip box.\n\n\"Little girl, would you like to have your hair cut?\" asked Robert, as if he had read her mind\u2014or was tired of watching her swing her legs.\n\nRamona stopped swinging her legs and answered politely, \"No, thank you. We are scrimping and pinching to make ends meet.\" Using \"scrimping and pinching\" made her feel grown up.\n\nAn exasperated sigh escaped Mrs. Quimby. She glanced at her watch. Beezus's haircut was taking longer than she had planned.\n\n\"Haircuts for children under ten are half price,\" said Robert, \"and no waiting. We aren't very busy on a wet morning like this.\"\n\nMrs. Quimby studied Ramona's hair while Ramona tried to push her eyebrows still higher. \"All right, Ramona,\" she said. \"Your hair does need cutting again, and it will help to have one more Saturday chore out of the way.\"\n\nIn a moment Ramona found herself draped with a poodle-printed plastic sheet and lying back with her hair buried under mounds of lather while a young woman named Denise rubbed her scalp. Such bliss! Washing hair at home was never like this. No soap in her eyes, no having to complain that the water was too hot or too cold, no bumping her head on the kitchen faucet while her knees ached from kneeling on a chair, no one telling her to stop wiggling, no water dribbling down her neck. The shampoo was over much too soon. Denise rubbed Ramona's hair with a towel and guided her to a chair in front of a mirror. On the other side of the row of mirrors, she could hear Beezus's hair being snipped with long pauses between snips.\n\n\"She's definitely the pixie type,\" said the teacher to Denise.\n\nMe? thought Ramona, surprised and pleased. Ramona the pixie sounded much nicer than Ramona the pest as she had so often been called by Beezus and her friends.\n\n\"A little off the bangs,\" said the teacher, \"and the ends tapered.\"\n\nDenise went to work. Her scissors flashed and snipped. Unlike Lester on the other side of the mirror, Denise was sure of what she was doing. Perhaps she had studied longer.\n\nRamona closed her eyes. _Snip-snip-snip_ went her bangs. When she opened her eyes she was surprised to discover they were a tiny bit longer in the center of her forehead. Like the top of a heart, thought Ramona, like a valentine.\n\nDenise lifted locks of wet hair between her fingers and snipped with flying scissors. Lift and snip, all the way around Ramona's head. Flicks of a comb, and Denise aimed a hand-held hair dryer at Ramona's head with one hand while she guided Ramona's hair into place with a brush held in the other. In no time Ramona's hair was dry. More flicks of the comb, the plastic sheet was whisked away, and there sat Ramona with shining hair neatly shaped to her head.\n\n\"Excellent,\" said the teacher to Denise. \"She looks adorable.\"\n\nStudents who had no customers gathered around. Ramona could not believe the words she was hearing. \"Darling.\" \"Cute as a bug.\" \"A real little pixie.\" The dryer was humming on the other side of the mirror.\n\nRamona felt light and happy when she returned to her mother.\n\n\"Why, Ramona!\" said Mrs. Quimby, laying aside her magazine. \"Your hair looks lovely. So neat and shiny.\"\n\nRamona couldn't stop smiling, she was so happy. She twitched her nose with joy.\n\nBut something made the smile on Mrs. Quimby's face fade. Ramona turned and stared at Beezus standing beside the screen. Her sister's hair had been teased and sprayed until it stood up three inches above her face. Her bangs were plastered in a curve across her forehead. Beezus did not look like an ice skater on television. She looked like an unhappy seventh-grade girl with forty-year-old hair.\n\nRamona did not know what to say. No one knew what to say except Robert. \"You look lovely, dear,\" he said, but no one answered. Beezus's face looked as stiff as her hair.\n\nRamona thought of the allowance Beezus had saved and wanted to shout at Robert, \"She does not look lovely! My sister looks terrible!\" For once she kept still. She felt sorry for her sister and sad about the allowance she had saved for so long, but deep inside, where she was ashamed of her feeling, she felt a tiny triumph. Ramona looked nicer than Beezus.\n\nRamona walked carefully to the car, not wanting to disturb her hair by running and hopping. Beezus walked in stony silence. When all three had buckled their seat belts, Beezus could no longer hold back her feelings. \"Well, go ahead and say it!\" she burst out in anger and in tears. \"Tell me my hair looks terrible. Tell me my hair looks stiff and horrible, like a wig. A _cheap_ wig!\"\n\n\"Now Beezus.\" Mrs. Quimby spoke gently.\n\n\"Well, it _does_! You know it does,\" Beezus went on. \"I tried to tell the man I didn't want my hair to stand up, but he said I would be pleased when he finished, and now I've wasted your whole morning and all my allowance. I look terrible and can't go to school because everyone will laugh at me.\" She began to sob.\n\n\"Dear girl\u2014\" Mrs. Quimby took Beezus in her arms and let her weep against her shoulder.\n\nTears came into Ramona's eyes. She felt she could not bear her sister's unhappiness even if she did look nicer than Beezus. That awful stiff hair, the wasted allowance . . . Ramona no longer triumphed in looking nicer. She did not want to look nicer. She wanted them to look the same so people would say, There goes that nice-looking Beatrice Quimby and her nice-looking little sister.\n\n\"I j-just wanted to look nice.\" Beezus's voice was muffled by her mother's coat. \"I know th-that what I do is more important than how I look, but I just wanted to look nice.\"\n\n\"Of course you do,\" soothed Mrs. Quimby. \"No matter what we say, we all want to look nice.\"\n\nRamona sniffed, she felt so sad.\n\n\"And you will look nice,\" Mrs. Quimby continued, \"once you wash out all that hair spray and comb your hair. Don't forget Lester cut your hair, and that's what counts.\"\n\nBeezus raised her soggy tearstained face. \"Do you really think it will look all right when it's washed?\"\n\n\"Yes, I do,\" said Mrs. Quimby. \"It just needs to be washed and combed.\"\n\nBeezus sat up and let out an exhausted sigh.\n\nMother and daughter had forgotten their adorable pixie buckled down in the corner of the back seat. Ramona hoped she could make it home without upchucking. She did not want to muss her hair.\n\n#\n\n## RAMONA'S NEW PAJAMAS\n\nAs Mrs. Quimby had predicted, once Beezus washed her hair she looked like Beezus again. Because they were so glad to see her looking like a seventh grader, Ramona and her mother did not point out that her new haircut did not look much different from the cuts her mother had given her.\n\nAs for Ramona, for a few days grown-ups said, \"Why, how nice your hair looks,\" as if they were surprised that her hair could look nice.\n\nChildren asked, \"How come your bangs are longer in the middle?\"\n\n\"Because I'm a pixie,\" Ramona answered, or sometimes, \"because I'm a valentine.\" In a few days everyone forgot about her hair, including Ramona.\n\nClearly Ramona's parents had something more important on their minds. At first Ramona did not know what it was. She heard long, serious conversations coming from their bedroom, and when she knelt by the furnace outlet to try to catch what they were saying, she could make out only a few words. \"I don't . . . school . . . why don't . . . we could . . . teacher . . . school.\" They sounded as if they might be arguing.\n\n\"I told you not to fight anymore!\" Ramona yelled through the furnace pipes. There was a startled silence, then laughter from the bedroom. Afterward Ramona could hear only whispers.\n\nRamona decided her parents must be talking about her. What could they say about Beezus and school? Nothing. What could they say about Ramona and school? To begin with, there was her spelling. . . .\n\nFor a while Ramona expected her parents to have one of those little talks with her about really working at her spelling or being a better girl. When they did not, she put their conversations out of her mind and went back to twitching her nose, pretending she was her mother's little rabbit, warm and snug and loved like little bears and bunnies in the books her mother read to her at bedtime when she was little.\n\nOne evening, when Ramona had turned from a pixie into a rabbit, she held her feet close together and, twitching her nose, went hopping down the hall. _Thud. Thud. Thud_.\n\n\"Ramona, do you have to do that?\" asked her mother, who was watching the evening news on television while she let down a hem on a dress for Beezus.\n\nRamona stopped being her mother's little rabbit, but she did not answer. Of course she did not have to hop. She wanted to. Her mother should know that.\n\nMrs. Quimby glanced up from her sewing. \"Why, Ramona,\" she remarked, \"those pajamas are way too small for you.\"\n\nAnd so they were. Ramona, who had been outgrowing clothes all her life, discovered that the sleeves reached only halfway to her wrists, the legs halfway to her ankles, and the seat was too tight. Her pajamas had been washed so often that the fuzz had worn off the flannel.\n\n\"I have another pair put away for you,\" said Mrs. Quimby. \"I'll get them and you can change.\"\n\n\"Did Beezus outgrow them?\" Ramona was all too familiar with her mother's habit of putting away for Ramona the clothes that Beezus had outgrown several years before.\n\nMrs. Quimby went to the linen closet. \"Not this time. I bought them on sale.\" She handed Ramona a pair of white pajamas printed with colored balloons. They were so new they were still folded and pinned together.\n\nRamona quickly pulled out the pins and changed from too-small pajamas into too-big pajamas. The sleeves covered her hands, the legs rumpled around her ankles, and the seat bagged, but oh, how soft and warm and cozy they felt, like the fur of a baby rabbit.\n\n\"Just fold up the bottoms so you won't trip,\" said Mrs. Quimby. \"They'll shrink when they're washed, and you'll grow into them before you know it.\"\n\nRamona did as she was told and discovered that, now that her pajamas were no longer tight, she could stoop lower and jump higher. Twitching her nose, she became a rabbit once more and _thump, thump, thumped_ down the hall to bed, where she snuggled down, warm and cozy as a little rabbit in a nest, in the pajamas that had never been worn by her sister.\n\nThe next morning she awoke still feeling warm and cozy. She lay in bed, not wanting to take off the pajamas, they felt so good.\n\n\"Ramona, come along and eat your oatmeal while it's still hot,\" her mother called to her.\n\nReluctantly Ramona got out of bed, dabbed a damp washcloth in the middle of her face, and, still in her pajamas, went to breakfast.\n\n\"Why, Ramona, you aren't even dressed.\" Mrs. Quimby, having finished her breakfast, was rinsing her dishes. Mr. Quimby and Beezus were carrying theirs to the sink.\n\n\"Don't worry, Mother,\" said Ramona. \"I'm not going to school in my pajamas.\" As soon as she had spoken Ramona thought how pleasant it would be if she could go to school in her pajamas and feel the soft fuzz against her skin all day.\n\n\"Don't dawdle.\" Mr. Quimby kissed the top of Ramona's head and left for work. Ramona twitched her nose.\n\nRamona quickly ate her oatmeal\u2014this was easy because oatmeal did not require much chewing\u2014and as she ate she thought about wearing her pajamas to school. Suddenly she recalled seeing the kindergarten class in their red plastic fire hats trooping back from a visit to the fire station, which made her think of her own visit to the firehouse when she was in kindergarten and how she had loved her fire hat. For days afterward, whenever she found even two newspapers piled together, she had called her parents' attention to a fire hazard. She also recalled how astonished she had been to learn that firemen slept in their underwear so that they could jump out of bed and into their clothes if they were called out in the night. Of course, Ramona did not sleep in her underwear, but if she put her clothes on over her pajamas she could pretend to be a fireman anyway.\n\nAs Ramona rinsed her dishes she stopped being a rabbit and became a fireman. She raced down the hall and pulled her slacks on over her pajama bottoms. Fortunately, she was not really on her way to fight a fire because she had a hard time stuffing the folded-up legs into her slacks. Then she jerked on her turtleneck sweater over the pajama top. The knitted neck and wristbands hid the flannel nicely. Ramona felt stuffed, but cozy and warm. She remembered to brush her teeth and was ready for school. Like a fireman she pulled on her boots, grabbed her raincoat and hat, and raced into the kitchen for her lunch box.\n\n\"Bye, Mother,\" she called out as she ran out the back door.\n\n\"Where's the fire?\" her mother called after her.\n\nHow did she guess? Ramona wondered as she ran toward school. Then she decided her mother had not really guessed because she often asked where the fire was when Ramona was in a hurry.\n\nA warm, misty spring rain was falling. Bits of green tipped the black branches of trees. Ramona slowed down to investigate crocus buds like tiny yellow and blue Easter eggs that were pushing up through a neighbor's lawn. Then she ran on as fast as she could in her stuffed condition, her mouth open, wailing like a fire engine, her boots clomping on the sidewalk. She paid no attention to the people walking to the bus stop who looked at her in surprise. Firemen must get awfully hot, thought Ramona, when she arrived panting and sweating at Glenwood School.\n\nRamona was glad to sit down on the floor of the cloakroom and pull off her boots. At least her feet felt cooler. She flopped down at her desk. Her face was flushed, and her pajamas no longer felt as soft as a baby rabbit. They were damp with sweat. Maybe pretending to be a fireman wasn't such a good idea after all, thought Ramona, and wondered if anyone would think she looked different. As it turned out, only Davy noticed because Davy always kept an eye on Ramona, who had been chasing him ever since kindergarten. \"You look fat,\" he said.\n\n\"I ate a big breakfast,\" answered Ramona. Then she added, \"Davy-in-the-gravy\" to keep Davy quiet. She knew he did not like to be called Davy-in-the-gravy.\n\nThe classroom seemed unbearably hot, and her clothes felt as tight as the skin on a sausage. As Ramona stood for the flag salute, she wished she had something to unbutton. Later, as she bent over her workbook, she could not help trying to squirm inside her damp clothes.\n\nMrs. Rudge walked slowly up and down between the desks, looking over shoulders at workbooks. Ramona, finding it difficult to think about her work when she was so uncomfortable, noticed that Davy crooked his arm around his page and bent his head low to hide his work while Becky sat up straight so Mrs. Rudge would be sure to see how perfect her work was. \"I like the way Davy keeps his eyes on his own work,\" said Mrs. Rudge. Davy's ears turned pink with pleasure.\n\nRamona quickly lowered her eyes to her workbook and remembered that her parents had had more serious talks in their bedroom about school. What was wrong? she wondered again. Mrs. Rudge paused beside her desk to look, not at Ramona's workbook, but at Ramona whose pajamas felt so damp she thought they might be shrinking.\n\n\"Ramona, how do you feel this morning?\" whispered Mrs. Rudge.\n\n\"Fine,\" answered Ramona, trying to sound as if she spoke the truth.\n\n\"Your cheeks are very pink,\" said Mrs. Rudge. \"I think you had better go to the office and ask Mrs. Miller to take your temperature.\"\n\n\"Now?\" asked Ramona.\n\n\"Yes,\" said Mrs. Rudge. \"Run along.\"\n\nRamona laid down her pencil and tried to look thin as she walked out of the room to a rustle of whispers from the class. What was the matter with Ramona? Was she sick? Would she have to be sent home?\n\nOnce in the hall she grasped her sweater and pajama top and pulled them up an instant to feel the relief of cool air against her sweaty skin. Then she took hold of both her elastic waistbands and pulled them out and in several times to fan a little cool air inside her slacks.\n\nIn the office Mrs. Miller, the school secretary, had Ramona sit on a chair and poked a thermometer under her tongue. \"Be sure to keep your lips closed,\" she said. \"We don't want any thermometers falling on the floor and breaking.\"\n\nRamona sat still while Mrs. Miller answered the telephone and carried on a long conversation with a mother who was worried about her child's schoolwork and was anxious to talk to the principal. She sat still while a sixth-grade boy came in to use the telephone to call his mother to tell her he had forgotten his lunch money. She sat still while a mother came in to deliver a lunch to a fourth grader who had gone off without it.\n\nRamona sat and sat. She thought of the long day ahead, of recess and of lunchtime, and began to wish she really were sick. Maybe she was. Maybe she had a fever, a fever so high Mrs. Miller would telephone her mother at work, and her mother would come and take her home and put her to bed between cool white sheets. They would be alone in the house, just the two of them. Her mother would lay her hand on Ramona's hot forehead and give her little treats\u2014ice cream between meals and cold orange juice, not fresh-frozen orange juice but fresh-fresh orange juice squeezed out of real oranges and not dumped out of a can and thinned with water. Her mother would read aloud stories from library books and would find in the bookcase the books Ramona had loved so much when she was little, especially the one about the little bear whose mother looked so soft and kind and loving in her long white apron and the book about the bunny snug in bed who said good-night to everything, mittens, a mouse, the moon, and the stars. Later, when Ramona was feeling better, her mother would tuck her upon the couch in the living room so she could watch television and even get to see the ends of old movies.\n\nPursing her lips tight around the thermometer, Ramona sighed through her nose. Mrs. Miller, her back turned, was busy with the ditto machine.\n\nFinally, when Ramona could not sit still another second, she made a sort of angry humming noise. \"M-m-m! M-m-m!\"\n\n\"Oh, my goodness, Ramona,\" said Mrs. Miller. \"You were so quiet I forgot all about you. Thank you for buzzing like a little bee to remind me.\" She pulled the thermometer from Ramona's mouth, turned it until she found the silver line that told the temperature, and then said, \"Run along back to your room, and tell Mrs. Rudge you're just fine. OK?\"\n\n\"OK.\" Ramona was disappointed. Now there would be no rescuing telephone call to her mother, only a long, sweaty day. Oh, well, she knew she would not really have been rescued by her mother, who could not leave her work. Howie's grandmother, accompanied by Willa Jean and probably Woger, would have come for her.\n\nRamona paused at the drinking fountain for a long, cool drink of water and fanned more air under her clothes before she returned to Room 2.\n\n\"What did Mrs. Miller say?\" asked Mrs. Rudge.\n\n\"She says I'm fine,\" said Ramona.\n\nMinutes dragged. The seconds between each click of the electric clock seemed to stretch longer and longer. Ramona felt so sleepy she wanted to put her head down on her arms and take a nap.\n\nWhen the recess bell finally rang, Mrs. Rudge said, \"Ramona, would you please come here a minute?\"\n\nReluctantly Ramona walked to Mrs. Rudge's desk.\n\n\"Is there something you would like to tell me?\" asked the teacher.\n\nRamona looked up into Mrs. Rudge's brown eyes, then down at the floor, shook her head, and looked up at Mrs. Rudge once more. Her teacher seemed so kind, so soft and plump, that Ramona longed to lean against her and tell her all her troubles, how hot she was and how no one ever said she was her mother's girl and how she wanted her mother to love her like a little rabbit and how somehow all these feelings had led to pretending to be a fireman.\n\n\"I can keep a secret,\" said Mrs. Rudge. \"I promise.\"\n\nThis encouragement was all Ramona needed. \"I\u2014I'm too warm,\" she confessed. \"I've got my pajamas on.\" Please, please, Mrs. Rudge, don't make me tell why, she prayed, because now that she had confessed she felt that wearing pajamas to school was a silly thing to do. A second grader pretending to be a fireman\u2014it was the dumbest thing she had ever imagined.\n\n\"Why, that's no problem,\" said Mrs. Rudge. \"Just go to the girls' bathroom and take off your pajamas.\" She reached into a drawer and pulled out a paper bag. \"Roll up your pajamas and put them in this bag and hide them in your desk.\"\n\nRamona shook her head. \"I can't.\" As soon as she had spoken, she realized she had chosen the wrong words. Now Mrs. Rudge would say, There's no such word as _can't_ , and Ramona would argue with herself all over again. How could there not be such a word as _can't_? Mrs. Rudge had just said _can't_ so _can't_ had to be a word.\n\nTo Ramona's relief, Mrs. Rudge merely said, \"Why not?\"\n\n\"I don't have any underwear on,\" confessed Ramona. Was there amusement in Mrs. Rudge's warm brown eyes? There better not be. No, it was all right. Mrs. Rudge was not laughing at her.\n\n\"I see,\" said the teacher. \"That _is_ a problem, but I don't think you need to worry about it. Your slacks and sweater are warm enough on a day like this.\"\n\n\"You mean go without any underwear?\" Ramona was a little shocked at the suggestion. In summer she did not wear an undershirt, but she had always worn underpants, even in the hottest weather.\n\n\"Why not?\" asked Mrs. Rudge with a wave of her hand, as if she were waving away underwear as unimportant. Underwear\u2014pooh!\n\n\"Well . . .\" said Ramona, halfway agreeing. \"But . . . promise you won't tell my mother what I did?\"\n\n\"I promise,\" said Mrs. Rudge with a big smile. \"Now run along before you melt into a puddle right here on the floor.\"\n\nRamona did as she was told, and, oh, the relief she felt in the girls' bathroom when she shut herself in a cubicle and peeled off those damp pajamas, which, to her surprise, had not shrunk at all. She quickly pulled on her clothes and rolled up the pajamas as tight as she could and hid them in the paper bag. Even though skipping in the halls was forbidden, Ramona skipped. The halls were empty, recess was over, and she was late, but still she skipped because she felt as light and as cool as a spring breeze. And who would know she was not wearing underwear? Nobody, that's who. Maybe wearing underwear wasn't so important after all. Maybe after today Ramona would skip underwear\u2014at least in summer when she was wearing slacks.\n\nBack in Room 2, Ramona lifted the lid of her desk and hid her package way at the back behind her books. She pretended not to notice the curious stares of the boys and girls, who were wondering why Mrs. Rudge said nothing about Ramona's being late. Instead she looked at Mrs. Rudge, who gave her a tiny smile that said quite plainly, We have a secret, just the two of us.\n\nRamona's heart was warm with love for her teacher. She smiled back and twitched her nose like a bunny.\n\n#\n\n## THE TELEPHONE CALL\n\n**B** y the time school was over Ramona had forgotten about the pajamas in her desk, and that evening she was so busy practicing her name in cursive writing that they remained forgotten. No more babyish printing for Ramona. Mrs. Rudge had taught her to write in what Ramona used to call \"that rumply stuff.\" And write she did.\n\nShe wrote in pencil, ballpoint pen, and crayon on any paper she could find\u2014paper bags, old envelopes, the backs of arithmetic papers, around the edge of the newspaper. She wrote her name with her finger in steam on the bathroom mirror when her father had taken a shower after work. Before supper she wrote her name in dust on the top of the television set. After supper she went outside where, beneath the porch light, she wrote Ramona Quimby in chalk on each of the front steps. When she came back into the house, she found her mother and Beezus on the couch studying pictures in a paperback book.\n\n\"Let me see, too,\" said Ramona, wiping her chalky fingers on the seat of her slacks and twitching her nose.\n\n\"It's just a book on how to cut hair that I ran across,\" said Mrs. Quimby. \"I thought I would try to learn to cut Beezus's hair so it would look like the ice skater on television.\"\n\n\"See, Mother,\" said Beezus. \"First you twist the top hair up out of the way and cut the bottom hair first.\"\n\n\"I see,\" said Mrs. Quimby. \"That doesn't look so difficult.\"\n\nRamona felt left out. Somehow that trip to the beauty school had brought her mother and Beezus closer together. They were friends again, close friends.\n\n\"Bedtime, Ramona,\" said Mrs. Quimby, still studying the book.\n\nA terrible thought crossed Ramona's mind. Her new pajamas! She had left them rolled up in her desk at school. Then Ramona had an even worse thought. This was Friday. She could not bring her pajamas home until Monday. How could she explain if her family found out?\n\nRamona made up her mind right then and there that neither her parents nor Beezus would find out because she was going to keep those pajamas a secret. Without waiting for a second reminder Ramona was in and out of the bathtub in no time. She could not locate the old pajamas she had taken off the night before, but she did come across another too-small pair in a drawer. She put them on, turned off the light, hopped into bed, and pulled the covers up tight around her ears. But what if she fell asleep before her parents came in to kiss her good-night?\n\nRamona took no chances. \"Come and kiss me good-night,\" she sang out while hanging on tight to the sheet and blankets.\n\nMr. Quimby was first. \"What's got into you, Ramona?\" he asked after he had kissed her. \"You forgot to beg to stay up just a little longer to watch TV or finish drawing a picture or read another chapter. You forgot to remind us you don't have to go to school tomorrow. Don't you feel good?\"\n\nRamona giggled. \"Daddy, you're so silly. I feel fine.\" She was pleased that her father had noticed she now read books with chapters.\n\nMrs. Quimby was next. Ramona pulled the covers tight around her ears when she heard her mother coming down the hall. Mrs. Quimby kissed Ramona and then looked at her in the dim light from the hall. \"Are you cold?\" she asked.\n\n\"No,\" answered Ramona.\n\n\"If you are, I can get another blanket from the linen closet,\" said Mrs. Quimby. Then she added, \"Nighty-night. Sweet dreams.\"\n\nThat was close, thought Ramona with a twitch of her nose. When she said her prayers, she added a request at the end. Please, God, do not let anyone find out I wore my pajamas to school. She felt that although God was probably too busy to think about her pajamas, asking would not hurt and might even help.\n\nSaturday morning she dressed in the closet and hid the too-small pajamas in her bottom drawer. She was happy to discover that her father was home for the morning, even though he would have to work at the ShopRite Market Saturday afternoon and evening. Today was going to be a good day. The sun was shining, the sidewalk dry, and her father could watch her skate.\n\nThat is, she was happy until Mr. Quimby looked around the living room and said, \"This is a home, not a base camp.\" He had recently watched a television program on mountain climbing. \"Let's all pitch in and clean this place up. Ramona, pick up all the newspapers and magazines and dust the living room. Beezus, you can run the vacuum cleaner. Then both of you tackle your rooms. Change the sheets and straighten up. Every kettle must rest on its own bottom around here.\" He did not mention that this was one of his grandmother's sayings.\n\nExcept for washing the egg beater in sudsy water so she could beat up a lot of suds, Ramona did not care much for housework, and this morning she longed to be outside racing up and down the sidewalk on her roller skates. However, she carried the old newspapers out to the garage without complaining and hastily flicked a dustcloth around the living room while Beezus plugged in the vacuum cleaner and made it growl back and forth across the carpet. Mr. Quimby went off to clean the bathroom while Mrs. Quimby was busy in the kitchen.\n\nIn a playful mood Beezus pushed the vacuum cleaner right up to Ramona's shoes. Ramona squealed as if she expected the vacuum cleaner to nibble her toes. Beezus pursued her with the vacuum cleaner. Around the carpet they went until Ramona said, \"Ha-ha, you can't catch me!\" and crawled behind the couch. Beezus returned to running the vacuum cleaner properly, back and forth in straight lines, the way their father mowed the lawn.\n\nRamona sat hugging her knees behind the couch. She was in no hurry, as her father put it, to tackle her room.\n\nRamona sat there behind the couch, a kettle resting on its bottom, thinking. She thought how embarrassed she would be if her family found out she had worn her pajamas to school. She thought about her mother and Beezus and what good friends they had become, almost as if they were the same age.\n\nAs Ramona sat letting these thoughts slide through her mind, the telephone rang in the kitchen. Above the growl of the vacuum cleaner, she heard her mother say, \"Why hel _lo_!\" as if she were surprised to hear from the person calling.\n\nWho had surprised her mother? Ramona listened hard. Beezus must have been curious, too, for she turned off the vacuum cleaner, which made eavesdropping easier for Ramona.\n\n\"Oh . . . ? Yes . . . Yes . . . Oh, does she?\" Mrs. Quimby went on in a friendly polite voice quite different from the friendly voice she used when she talked to her sister, the girls' Aunt Beatrice.\n\nWho does what? Ramona wondered, alert, since she was usually the one who had done something. Her mother laughed. Ramona felt indignant without knowing why. She could not think of anything she had done that anyone would telephone her mother about.\n\nMrs. Quimby continued for some time, but Ramona could make no sense out of the conversation. Finally her mother said, \"Thank you for telling me, Mrs. Rudge.\" Then she hung up.\n\nThe sound of her teacher's name gave Ramona a strange feeling, as if she were in an elevator that had suddenly gone down when she expected it to go up. When she stopped feeling as if the floor had dropped beneath her, she was furious. So that was why her mother had laughed. Mrs. Rudge had told! She had telephoned her mother and tattled. And her mother thought it was funny! Ramona would never forgive either of them. Never, never, _never_.\n\nBeezus turned on the vacuum cleaner again. Ramona crawled on her hands and knees from behind the couch and was surprised to see that her mother had come into the living room. \"What have you been doing back there?\" asked Mrs. Quimby.\n\n\"Resting on my bottom,\" said Ramona with a scowl.\n\nBeezus switched off the vacuum cleaner again. Her turn had come to foresee an interesting argument.\n\nRamona faced her mother. \"Mrs. Rudge told!\" she shouted. \"And she promised she would never tell. And then you had to go and laugh!\"\n\n\"Now calm down.\" Mrs. Quimby plucked a fluff of dust from Ramona's sleeve.\n\n\"I _won't_ calm down!\" yelled Ramona so loud her father came down the hall to see what was going on. \"I _hate_ Mrs. Rudge! She's a tattletale. She doesn't love me and she tells fibs!\" Ramona saw her mother and father exchange a familiar look that said, Which of us is going to handle this one?\n\n\" _Hate_ is a strong word, Ramona,\" said Mrs. Quimby quietly.\n\n\"Not strong enough,\" said Ramona.\n\n\"This looks like nine on the Richter scale,\" said Mr. Quimby, as if Ramona were an earthquake.\n\n\"And you and Daddy talk about me in your room at night,\" Ramona stormed at her mother.\n\n\"Someday, Ramona,\" said her father, \"you are going to have to learn that the world does not revolve around you.\"\n\n\"I don't care what Mrs. Rudge says,\" shouted Ramona. \"I didn't leave my pajamas at school on purpose. I forgot.\"\n\nMrs. Quimby looked astonished. \"Left your pajamas\u2014What on earth are your pajamas doing at school?\" She was plainly trying to stifle a laugh.\n\nRamona was both surprised and bewildered. If her mother did not know about her pajamas, what could Mrs. Rudge have said?\n\n\"What on earth are your pajamas doing at school?\" Ramona's mother asked again.\n\nThe whole story\u2014her feeling that the flannel was as soft as bunny fur and how she pretended to be a fireman so she wouldn't have to take her pajamas off\u2014flashed through Ramona's mind and embarrassed her. \"I won't tell,\" she said, folding her arms defiantly.\n\n\"She probably took them for Show and Tell,\" volunteered Beezus.\n\nRamona gave her sister a look of contempt. Second graders in Mrs. Rudge's room did not have Show and Tell every day, only when someone had something really important and educational to bring such as a butterfly that had hatched out of a cocoon in a jar. And Beezus should know that no second grader would take pajamas to school for Show and Tell. That would be too babyish even for kindergarten. Beezus knew these things. She had been through them all. She was just trying to make Ramona look babyish.\n\nRamona was about to shout, I did not! but decided this would be unwise. Beezus had supplied a reason, a very weak reason, why she might have taken her pajamas to school.\n\nApparently Mrs. Quimby did not accept Beezus's explanation either, for she said, \"Your pajamas did not get out of bed and run along beside you to school. Oh, well, I don't suppose it matters.\"\n\nRamona scowled. Her mother need not think she could win her over by being funny. She was mad and she was going to stay mad. She was mad at Beezus for always being her mother's girl. She was mad at her teacher for telling her mother something ( _what_?). She was mad at her parents for not being upset because she was mad. She was mad at herself for letting it out that she had left her pajamas at school.\n\n\"Nobody likes me. Nobody in the whole world,\" said Ramona, warming to her subject as the cat walked disdainfully through the room on his way to peace on Beezus's bed. \"Not even my own mother and father. Not even the cat. Beezus gets all the attention around here. Even Picky-picky likes Beezus more than he likes me!\" She was pleased that her father stayed in the living room and she didn't lose any of her audience. \"You'll be sorry someday when I'm rich and famous.\"\n\n\"I didn't know you were planning to be rich and famous,\" said Mr. Quimby.\n\nNeither had Ramona until that moment.\n\n\"What do you mean, I get all the attention around here?\" demanded Beezus. \"Nobody tapes my schoolwork to the refrigerator door. We can hardly find the refrigerator, it is so buried under all your drawings and junk!\"\n\nBoth parents looked at Beezus in surprise. \"Why, Beezus,\" said Mrs. Quimby, \"I had no idea you minded.\"\n\n\"Well, I do,\" said Beezus crossly. \"And Ramona always gets out of things like washing dishes because she is too little. She'll probably still be too little when she's eighty.\"\n\n\"See?\" said Ramona. \"Beezus doesn't like me because my artwork is stuck to the refrigerator.\" Her parents weren't supposed to feel sorry for Beezus. They were supposed to feel sorry for Ramona.\n\n\"I'm always in the way,\" said Ramona. \"You have to park me with Howie's grandmother so you can go to work, and Howie's grandmother doesn't like me. She thinks I'm so terrible she probably won't want me around anymore, and then there won't be anybody to look after me and you can't go to work. So there!\" Ramona flopped down on the couch, waiting for someone to tell her she was wrong.\n\nRamona's mother and father said nothing.\n\n\"Everybody picks on me all the time,\" said Ramona. Maybe she really would be so bad Mrs. Kemp would say, I simply cannot put up with Ramona another day.\n\nSilence.\n\nRamona made up her mind to shock her parents, really shock them. \"I am going to run away,\" she announced.\n\n\"I'm sorry to hear that,\" said Mr. Quimby as if running away were a perfectly natural thing to do.\n\n\"When are you leaving?\" inquired Ramona's mother politely. The question was almost more than Ramona could bear. Her mother was supposed to say, Oh, Ramona, please, please don't leave me!\n\n\"Today,\" Ramona managed to say with quivering lips. \"This morning.\"\n\n\"She just wants you to feel sorry for her,\" said heartless Beezus. \"She wants you to stop her.\"\n\nRamona waited for her mother or father to say something, but neither spoke. Finally there was nothing for Ramona to do but get up from the couch. \"I guess I'll go pack,\" she said, and started slowly toward her room.\n\nNo one tried to prevent her. When she reached her room, tears began to fall. She got out her Q-tip box with all her money, forty-three cents, in it. Still no one came to beg her not to leave. She looked around for something in which to pack, but all she could find was an old doll's nursing kit. Ramona unzipped it and placed her Q-tip box inside. She added her best box of crayons and a pair of clean socks. Outside she heard the cheerful _ching-chong, ching-chong_ of roller skates on cement. Some children were happy.\n\nIf nobody stopped her, where would she run to? Not Howie's house, even though Howie was no longer mad at her. His grandmother was not paid to look after her on Saturday. She could take the bus to Aunt Beatrice's apartment house, but Aunt Beatrice would bring her back home. Maybe she could live in the park and sleep under the bushes in the cold. Poor little Ramona, all alone in the park, shivering in the dark. Well, at least it was not raining. That was something. And there were no big wild animals, just chipmunks.\n\nShe heard her mother coming down the hall. Tears stopped. Ramona was about to be rescued. Now her mother would say, Please don't run away. We love you and want you to stay.\n\nInstead Mrs. Quimby walked into the bedroom with a suitcase in one hand and two bananas in the other. \"You will need something to pack in,\" she told Ramona. \"Let me help.\" She opened the suitcase on the unmade bed and placed the bananas inside. \"In case you get hungry,\" she explained.\n\nRamona was too shocked to say anything. Mothers weren't supposed to help their children run away. \"You'll need your roller skates in case you want to travel fast,\" said Mrs. Quimby. \"Where are they?\"\n\nAs if she were walking in her sleep, Ramona pulled her roller skates from a jumble of toys in the bottom of her closet and handed them to her mother, who placed them at the bottom of the suitcase. How could her mother not love a little girl like Ramona?\n\n\"Always pack heavy things at the bottom,\" advised Mrs. Quimby. \"Now where are your boots in case it rains?\" She looked around the room. \"And don't forget your Betsy book. And your little box of baby teeth. You wouldn't want to leave your teeth behind.\"\n\nRamona felt she could run away without her old baby teeth, and she was hurt that her mother did not want to keep them to remember her by. She stood watching while her mother packed briskly and efficiently.\n\n\"You will want Ella Funt in case you get lonely,\" said Mrs. Quimby.\n\nWhen Ramona said her mother did not love her, she had no idea her mother would do a terrible thing like this. And her father. Didn't he care either? Apparently not. He was too busy scrubbing the bathroom to care that Ramona was in despair. And what about Beezus? She was probably secretly glad Ramona was going to run away because she could have her parents all to herself. Even Picky-picky would be glad to see the last of her.\n\nAs Ramona watched her mother fold underwear for her to take away, she began to understand that deep down inside in the place where her secret thoughts were hidden, she had never really doubted her mother's love for her. Not until now. . . . She thought of all the things her mother had done for her, the way she had sat up most of the night when Ramona had an earache, the birthday cake she had made in the shape of a cowboy boot all frosted with chocolate with lines of white icing that looked like stitching. That was the year she was four and had wanted cowboy boots more than anything, and her parents had given her real ones as well. She thought of the way her mother reminded her to brush her teeth. Her mother would not do that unless she cared about her teeth, would she? She thought of the time her mother let her get her hair cut at the beauty school, even though they had to scrimp and pinch. She thought of the gentle books about bears and bunnies her mother had read at bedtime when she was little.\n\n\"There.\" Mrs. Quimby closed the suitcase, snapped the latches, and set it on the floor. \"Now you are all packed.\" She sat down on the bed.\n\nRamona pulled her car coat out of the closet and slowly put it on, one arm and then the other. She looked at her mother with sad eyes as she grasped the handle of her suitcase and lifted. The suitcase would not budge. Ramona grasped it with both hands. Still she could not lift it.\n\nHope flowed into Ramona's heart. Had her mother made the suitcase too heavy on purpose? She looked closely at her mother, who was watching her. She saw\u2014didn't she?\u2014a tiny smile in her mother's eyes.\n\n\"You tricked me!\" cried Ramona. \"You made the suitcase too heavy on purpose. You don't want me to run away!\"\n\n\"I couldn't get along without my Ramona,\" said Ramona's mother. She held out her arms. Ramona ran into them. Her mother had said the words she had longed to hear. Her mother could not get along without her. She felt warm and safe and comforted and oh, how good her mother smelled, so clean and sweet like flowers. Better than any mother in the whole world. Ramona's tears dampened her mother's blouse. After a moment Mrs. Quimby handed Ramona a Kleenex. When Ramona had wiped her eyes and nose, she was surprised to discover that her mother had tears in her eyes, too.\n\n\"Mama,\" said Ramona, using a word she had given up as babyish, \"why did you do that?\"\n\n\"Because I could see I couldn't get anyplace arguing with you,\" answered her mother. \"You wouldn't listen.\"\n\nThe truth made Ramona uncomfortable. \"Why did Mrs. Rudge phone?\" she asked, to change the subject.\n\nMrs. Quimby looked concerned. \"She called to say that she had noticed you twitching your nose a lot\u2014Daddy and I have noticed it, too\u2014and she wondered if something was making you nervous. She wondered if you perhaps needed a shorter day in school.\"\n\nAnd a longer day with Howie's grandmother? What a terrible idea. \"School is easy,\" said Ramona, not mentioning spelling, which, after all, might be easy if she paid more attention to it.\n\n\"Have you any idea what makes you twitch your nose?\" asked Mrs. Quimby gently. \"I noticed you twitch it three times during breakfast.\"\n\nRamona was surprised. Maybe she had twitched so much she could twitch without knowing it. \"Of course I know why,\" she said. \"I was pretending I was a rabbit, a baby rabbit, because you call me a little rabbit sometimes.\"\n\nThis time Ramona did not mind when her mother laughed. She laughed a bit, too, to show that she now thought pretending to be a baby rabbit seemed silly, as if it were something she had done a long time ago when she was little.\n\n\"Rabbits are nice,\" said Mrs. Quimby, \"but I prefer a little girl. My little girl.\"\n\n\"Really?\" said Ramona, even though she knew her mother spoke the truth.\n\n\"I am glad to know you were a little rabbit,\" said Ramona's mother. \"I was afraid my working full time might be too much for you, and just when we have decided Daddy will quit his job at the market and go back to school.\"\n\nRamona was astonished. \"School! You mean do homework and stuff like that? Daddy?\"\n\n\"I expect so,\" answered Mrs. Quimby.\n\n\"Why does he want to go and do a thing like that?\" Ramona could not understand.\n\n\"To finish college,\" her mother explained. \"So he can get a better job, he hopes. One that he likes.\"\n\nSo this was what her parents had been talking about at night in their room. \"Will he have to go away?\" asked Ramona.\n\n\"No. He can go to Portland State right here in town,\" explained Mrs. Quimby. \"But I will have to go on working full time, which I want to do anyway because I like my job. Do you think you can manage to get along with Mrs. Kemp?\"\n\nRamona thought how much happier her family would be if her father never came home tired from working in the express line again. \"Of course I can,\" she agreed with courage. \"I've gotten along\u2014sort of\u2014so far.\" After this she would stay away from pinking shears and bluing. As for Willa Jean\u2014maybe she would go to nursery school and learn to shape up. Yes, Ramona could manage. \"And I guess we'll have to scrimp and pinch some more,\" she said.\n\n\"That's right. Scrimp and pinch and save as much money as we can while Daddy is studying, even though he hopes to find part-time work after school starts,\" said Mrs. Quimby. \"And by the way, you don't have to tell me if you don't want to, but I am curious. Why are your pajamas at school?\"\n\n\"Oh.\" Ramona made a face; it all seemed so ridiculous now. She gave her mother the shortest possible explanation.\n\nMrs. Quimby did not seem upset. She merely said, \"What next?\" and laughed.\n\n\"Did Mrs. Rudge say anything about my spelling?\" Ramona hesitated to ask the question, but she did want to know the answer.\n\n\"Why, no,\" said Mrs. Quimby. \"She didn't even mention spelling, but she did say you were one of her little sparklers who made teaching interesting.\" And with that Ramona's mother left the room.\n\nA little sparkler! Ramona liked that. She thought of the last Fourth of July when she had twirled through the dusk, a sparkler fizzing and spitting in each hand and leaving circles of light and figure eights as she had spun across the front yard until she had fallen to the grass with dizziness. And now she was one of Mrs. Rudge's little sparklers!\n\nRamona held out her arms and twirled across the room, pretending she was holding sparklers. Then she seized a pencil and paper that were lying on her bureau and wrote her name in good, bold cursive:\n\nThere. A girl who was a sparkler needed a name that looked like a sparkler. And that was the way Ramona Quimby was going to write her name.\n\n_Ching-chong, ching-chong_ went the roller skates out on the sidewalk. Ramona opened the suitcase and pulled out her skates.\n\n## EXCERPT FROM _RAMONA QUIMBY, AGE 8_\n\nVisit\n\nRAMONA QUIMBY\n\nand all of her friends in\n\nThe World of Beverly Cleary\n\nat www.beverlycleary.com\n\nAnd turn the page\n\nfor a SNEAK PEEK at\n\n**_RAMONA QUIMBY, AGE 8_**\n\n#\n\n## THE FIRST DAY OF SCHOOL\n\nRamona Quimby hoped her parents would forget to give her a little talking-to. She did not want anything to spoil this exciting day.\n\n\"Ha-ha, I get to ride the bus to school all by myself,\" Ramona bragged to her big sister, Beatrice, at breakfast. Her stomach felt quivery with excitement at the day ahead, a day that would begin with a bus ride just the right length to make her feel a long way from home but not long enough\u2014she hoped\u2014to make her feel carsick. Ramona was going to ride the bus, because changes had been made in the schools in the Quimbys' part of the city during the summer. Glenwood, the girls' old school, had become an intermediate school, which meant Ramona had to go to Cedarhurst Primary School.\n\n\"Ha-ha yourself.\" Beezus was too excited to be annoyed with her little sister. \"Today I start high school.\"\n\n\" _Junior_ high school,\" corrected Ramona, who was not going to let her sister get away with acting older than she really was. \"Rosemont Junior High School is not the same as high school, and besides you have to walk.\"\n\nRamona had reached the age of demanding accuracy from everyone, even herself. All summer, whenever a grown-up asked what grade she was in, she felt as if she were fibbing when she answered, \"third,\" because she had not actually started the third grade. Still, she could not say she was in the second grade since she had finished that grade last June. Grown-ups did not understand that summers were free from grades.\n\n\"Ha-ha to both of you,\" said Mr. Quimby, as he carried his breakfast dishes into the kitchen. \"You're not the only ones going to school today.\" Yesterday had been his last day working at the checkout counter of the ShopRite Market. Today he was returning to college to become what he called \"a real, live school teacher.\" He was also going to work one day a week in the frozen-food warehouse of the chain of ShopRite Markets to help the family \"squeak by,\" as the grown-ups put it, until he finished his schooling.\n\n\"Ha-ha to all of you if you don't hurry up,\" said Mrs. Quimby, as she swished suds in the dishpan. She stood back from the sink so she would not spatter the white uniform she wore in the doctor's office where she worked as a receptionist.\n\n\"Daddy, will you have to do homework?\" Ramona wiped off her milk moustache and gathered up her dishes.\n\n\"That's right.\" Mr. Quimby flicked a dish towel at Ramona as she passed him. She giggled and dodged, happy because he was happy. Never again would he stand all day at a cash register, ringing up groceries for a long line of people who were always in a hurry.\n\nRamona slid her plate into the dishwater. \"And will Mother have to sign your progress reports?\"\n\nMrs. Quimby laughed. \"I hope so.\"\n\nBeezus was last to bring her dishes into the kitchen. \"Daddy, what do you have to study to learn to be a teacher?\" she asked.\n\nRamona had been wondering the same thing. Her father knew how to read and do arithmetic. He also knew about Oregon pioneers and about two pints making one quart.\n\nMr. Quimby wiped a plate and stacked it in the cupboard. \"I'm taking an art course, because I want to teach art. And I'll study child development\u2014\"\n\nRamona interrupted. \"What's child development?\"\n\n\"How kids grow,\" answered her father.\n\nWhy does anyone have to go to school to study a thing like that? wondered Ramona. All her life she had been told that the way to grow was to eat good food, usually food she did not like, and get plenty of sleep, usually when she had more interesting things to do than go to bed.\n\nMrs. Quimby hung up the dishcloth, scooped up Picky-picky, the Quimbys' old yellow cat, and dropped him at the top of the basement steps. \"Scat, all of you,\" she said, \"or you'll be late for school.\"\n\nAfter the family's rush to brush teeth, Mr. Quimby said to his daughters, \"Hold out your hands,\" and into each waiting pair he dropped a new pink eraser. \"Just for luck,\" he said, \"not because I expect you to make mistakes.\"\n\n\"Thank you,\" said the girls. Even a small present was appreciated, because presents of any kind had been scarce while the family tried to save money so Mr. Quimby could return to school. Ramona, who liked to draw as much as her father, especially treasured the new eraser, smooth, pearly pink, smelling softly of rubber, and just right for erasing pencil lines.\n\nMrs. Quimby handed each member of her family a lunch, two in paper bags and one in a lunch box for Ramona. \"Now, Ramona\u2014\" she began.\n\nRamona sighed. Here it was, that little talking-to she always dreaded.\n\n\"Please remember,\" said her mother, \"you really must be nice to Willa Jean.\"\n\nRamona made a face. \"I try, but it's awfully hard.\"\n\nBeing nice to Willa Jean was the part of Ramona's life that was not changing, the part she wished would change. Every day after school she had to go to her friend Howie Kemp's house, where her parents paid Howie's grandmother to look after her until one of them could come for her. Both of Howie's parents, too, went off to work each day. She liked Howie, but after spending most of the summer, except for swimming lessons in the park, at the Kemps' house, she was tired of having to play with four-year-old Willa Jean. She was also tired of apple juice and graham crackers for a snack every single day.\n\n\"No matter what Willa Jean does,\" complained Ramona, \"her grandmother thinks it's my fault because I'm bigger. Like the time Willa Jean wore her flippers when she ran under the sprinkler, pretending she was the mermaid on the tuna-fish can, and then left big wet footprints on the kitchen floor. Mrs. Kemp said I should have stopped her because Willa Jean didn't know any better!\"\n\nMrs. Quimby gave Ramona a quick hug. \"I know it isn't easy, but keep trying.\"\n\nWhen Ramona sighed, her father hugged her and said, \"Remember, kid, we're counting on you.\" Then he began to sing, \"We've got high hopes, try hopes, buy cherry pie-in-July hopes\u2014\"\n\nRamona enjoyed her father's making up new words for the song about the little old ant moving the rubber tree plant, and she liked being big enough to be counted on, but sometimes when she went to the Kemps' she felt as if everything depended on her. If Howie's grandmother did not look after her, her mother could not work full time. If her mother did not work full time, her father could not go to school. If her father did not go to school, he might have to go back to being a checker, the work that made him tired and cross.\n\nStill, Ramona had too many interesting things to think about to let her responsibility worry her as she walked through the autumn sunshine toward her school bus stop, her new eraser in hand, new sandals on her feet, that quivery feeling of excitement in her stomach, and the song about high hopes running through her head.\n\nShe thought about her father's new part-time job zipping around in a warehouse on a fork-lift truck, filling orders for orange juice, peas, fish sticks, and all the other frozen items the markets carried. He called himself Santa's Little Helper, because the temperature of the warehouse was way below zero, and he would have to wear heavy padded clothing to keep from freezing. The job sounded like fun to Ramona. She wondered how she was going to feel about her father's teaching art to other people's children and decided not to think about that for a while.\n\nInstead, Ramona thought about Beezus going off to another school, where she would get to take a cooking class and where she could not come to the rescue if her little sister got into trouble. As Ramona approached her bus stop, she thought about one of the best parts of her new school: none of her teachers in her new school would know she was Beatrice's little sister. Teachers always like Beezus; she was so prompt and neat. When both girls had gone to Glenwood School, Ramona often felt as if teachers were thinking, I wonder why Ramona Quimby isn't more like her big sister.\n\nWhen Ramona reached the bus stop, she found Howie Kemp already waiting with his grandmother and Willa Jean, who had come to wave good-bye.\n\nHowie looked up from his lunch box, which he had opened to see what he was going to have for lunch, and said to Ramona, \"Those new sandals make your feet look awfully big.\"\n\n\"Why, Howie,\" said his grandmother, \"that's not a nice thing to say.\"\n\nRamona studied her feet. Howie was right, but why shouldn't her new sandals make her feet look big? Her feet had grown since her last pair. She was not offended.\n\n\"Today I'm going to _kidnergarten_ ,\" boasted Willa Jean, who was wearing new coveralls and T-shirt and a pair of her mother's old earrings. Willa Jean was convinced she was beautiful, because her grandmother said so. Ramona's mother said Mrs. Kemp was right. Willa Jean was beautiful when she was clean, because she was a healthy child. Willa Jean did not feel she was beautiful like a healthy child. She felt she was beautiful like a grown-up lady on TV.\n\nRamona tried to act kindly toward little Willa Jean. After all, her family was depending on her. \"Not _kidnergarten_ , Willa Jean,\" she said. \"You mean nursery school.\"\n\nWilla Jean gave Ramona a cross, stubborn look that Ramona knew too well. \"I am too going to _kid_ nergarten,\" she said. \" _Kid_ nergarten is where the kids are.\"\n\n\"Bless her little heart,\" said her grandmother, admiring as always.\n\nThe bus, the little yellow school bus Ramona had waited all summer to ride, pulled up at the curb. Ramona and Howie climbed aboard as if they were used to getting on buses by themselves. I did it just like a grown-up, thought Ramona.\n\n\"Good morning. I am Mrs. Hanna, your bus aide,\" said a woman sitting behind the driver. \"Take the first empty seats toward the back.\" Ramona and Howie took window seats on opposite sides of the bus, which had a reassuring new smell. Ramona always dreaded the people-and-fumes smell of the big city buses.\n\n\"Bye-byee,\" called Mrs. Kemp and Willa Jean, waving as if Ramona and Howie were going on a long, long journey. \"Bye-byee.\" Howie pretended not to know them.\n\nAs soon as the bus pulled away from the curb, Ramona felt someone kick the back of her seat. She turned and faced a sturdy boy wearing a baseball cap with the visor turned up and a white T-shirt with a long word printed across the front. She studied the word to see if she could find short words in it, as she had learned to do in second grade. _Earth. Quakes. Earthquakes_. Some kind of team. Yes, he looked like the sort of boy whose father would take him to ball games. He did not have a lunch box, which meant he was going to buy his lunch in the cafeteria.\n\nA grown-up would not call him a purple cootie. Ramona faced front without speaking. This boy was not going to spoil her first day in the third grade.\n\n_Thump, thump, thump_ against the back of Ramona's seat. The bus stopped for other children, some excited and some anxious. Still the kicking continued. Ramona ignored it as the bus passed her former school. Good old Glenwood, thought Ramona, as if she had gone there a long, long time ago.\n\n\"All right, Danny,\" said the bus aide to the kicking boy. \"As long as I'm riding shotgun on this bus, we won't have anyone kicking the seats. Understand?\"\n\nRamona smiled to herself as she heard Danny mutter an answer. How funny\u2014the bus aide saying she was riding shotgun as if she were guarding a shipment of gold on a stagecoach instead of making children behave on a little yellow school bus.\n\nRamona pretended she was riding a stagecoach pursued by robbers until she discovered her eraser, her beautiful pink eraser, was missing. \"Did you see my eraser?\" she asked a second-grade girl, who had taken the seat beside her. The two searched the seat and the floor. No eraser.\n\nRamona felt a tap on her shoulder and turned. \"Was it a pink eraser?\" asked the boy in the baseball cap.\n\n\"Yes.\" Ramona was ready to forgive him for kicking her seat. \"Have you seen it?\"\n\n\"Nope.\" The boy grinned as he jerked down the visor of his baseball cap.\n\nThat grin was too much for Ramona. \"Liar!\" she said with her most ferocious glare, and faced front once more, angry at the loss of her new eraser, angry with herself for dropping it so the boy could find it. Purple cootie, she thought, and hoped the cafeteria would serve him fish portions and those canned green beans with the strings left on. And apple wedges, the soft mushy kind with tough skins, for dessert.\n\nThe bus stopped at Cedarhurst, Ramona's new school, a two-story red-brick building very much like her old school. As the children hopped out of the bus, Ramona felt a little thrill of triumph. She had not been carsick. She now discovered she felt as if she had grown even more than her feet. Third graders were the biggest people\u2014except teachers, of course\u2014at this school. All the little first and second graders running around the playground, looking so young, made Ramona feel tall, grown up, and sort of . . . well, wise in the ways of the world.\n\n## ABOUT THE AUTHOR\n\nALAN MCEWAN\n\n**BEVERLY CLEARY** is one of America's most beloved authors. As a child, she struggled with reading and writing. But by third grade, after spending much time in her public library in Portland, she found her skills had greatly improved. Before long, her school librarian was saying that she should write children's books when she grew up.\n\nInstead she became a librarian. When a young boy asked her, \"Where are the books about kids like us?\" she remembered her teacher's encouragement and was inspired to write the books she'd longed to read but couldn't find when she was younger. She based her funny stories on her own neighborhood experiences and the sort of children she knew. And so, the Klickitat Street gang was born!\n\nMrs. Cleary's books have earned her many prestigious awards, including the American Library Association's Laura Ingalls Wilder Award, presented to her in recognition of her lasting contribution to children's literature. RAMONA QUIMBY, AGE 8 and RAMONA AND HER FATHER have also been named Newbery Honor Books. Her characters, including Beezus and Ramona Quimby, Henry Huggins, and Ralph, the motorcycle-riding mouse, have delighted children for generations.\n\nGet to know all the kids in Ramona's neighborhood in The World of Beverly Cleary at WWW.BEVERLYCLEARY.COM\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors and artists.\n\n## BACK AD\n\n## OTHER BOOKS BY BEVERLY CLEARY\n\nEnjoy all of BEVERLY CLEARY'S books\n\nFEATURING\n\nRAMONA QUIMBY:\n\nBeezus and Ramona\n\nRamona the Pest\n\nRamona the Brave\n\nRamona and Her Father\n\nRamona and Her Mother\n\nRamona Quimby, Age 8\n\nRamona Forever\n\nRamona's World\n\n\u2022 \u2022 \u2022 \u2022\n\nFEATURING\n\nHENRY HUGGINS:\n\nHenry Huggins\n\nHenry and Beezus\n\nHenry and Ribsy\n\nHenry and the Paper Route\n\nHenry and the Clubhouse\n\nRibsy\n\n\u2022 \u2022 \u2022 \u2022\n\nFEATURING\n\nRALPH MOUSE:\n\nThe Mouse and the Motorcycle\n\nRunaway Ralph\n\nRalph S. Mouse\n\n\u2022 \u2022 \u2022 \u2022\n\nMORE GREAT FICTION BY\n\nBEVERLY CLEARY:\n\nEllen Tebbits\n\nOtis Spofford\n\nFifteen\n\nThe Luckiest Girl\n\nJean and Johnny\n\nEmily's Runaway\n\nImagination\n\nSister of the Bride\n\nMitch and Amy\n\nSocks\n\nDear Mr. Henshaw\n\nMuggie Maggie\n\nStrider\n\nTwo Times the Fun\n\n\u2022 \u2022 \u2022 \u2022\n\nAND DON'T MISS\n\nBEVERLY CLEARY'S\n\nAUTOBIOGRAPHIES:\n\nA Girl from Yamhill\n\nMy Own Two Feet\n\n## CREDITS\n\nCover art by Jacqueline Rogers \u00a9 2013 by HarperCollins Publishers\n\nCover design and hand lettering by Cara E. Petrus\n\n## COPYRIGHT\n\nRAMONA AND HER MOTHER. Copyright \u00a9 1979 by Beverly Cleary. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage 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 HarperCollins e-books.\n\nwww.harpercollinschildrens.com\n\nLibrary of Congress Catalog Card Number: 2012948578\n\nISBN 978-0-380-70952-6 (pbk.)\n\nISBN 978-0-688-22195-9 (hardcover)\n\nEPub Edition February 2013 ISBN 9780061972324\n\nVersion 03012013\n\nReillustrated edition, 2013\n\n13 14 15 16 17 LP\/BR 72 71 70 69 68 67 66\n\n## ABOUT THE PUBLISHER\n\n**Australia**\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\n<http:\/\/www.harpercollins.com.au\/ebooks>\n\n**Canada**\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n<http:\/\/www.harpercollins.ca>\n\n**New Zealand**\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n<http:\/\/www.harpercollins.co.nz>\n\n**United Kingdom**\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n<http:\/\/www.harpercollins.co.uk>\n\n**United States**\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n<http:\/\/www.harpercollins.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Marilynda Fraser-Cunliffe, David Garcia, Julia\nNeufeld and the Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net (This file was produced from images\ngenerously made available by The Internet Archive)\n\n\n\n\n\nTranscriber's note:\n\nText enclosed by underscores is in italics (_italics_).\n\nSmall capital text has been replaced with all capitals.\n\nPage 3: \"+JOHN JOSEPH LYNCH\"--the + denotes a cross symbol.\n\n       *       *       *       *       *\n\n[Illustration: cover]\n\n[Illustration: Ryerson statue]\n\n\n\n\n  RYERSON MEMORIAL VOLUME:\n\n  PREPARED ON THE OCCASION\n\n  OF THE\n\n  UNVEILING OF THE RYERSON STATUE\n\n  IN THE GROUNDS OF THE EDUCATION DEPARTMENT\n\n  ON THE\n\n  QUEEN'S BIRTHDAY, 1889.\n\n\n  BY J. GEORGE HODGINS, M.A., LL.D.,\n  BARRISTER-AT-LAW, AND DEPUTY MINISTER OF EDUCATION.\n\n\n  _TORONTO_:\n  PRINTED BY WARWICK & SONS 68 AND 70 FRONT STREET WEST.\n  1889.\n\n\n\n\nPREFATORY NOTE.\n\n\nI have two reasons to give for the part which I have taken in the\npreparation of the latter part of this Memorial Volume. The first is\nmentioned in the following paragraph from the brief _resum\u00e9_ of the\nhistorical and personal facts given in the volume, and which I read\non the day of the unveiling of the statue, as follows:--\n\n     \"It devolves upon me, as Chairman of the (Ryerson Memorial\n     Statue) Committee, and at the kind request of my colleagues--no\n     less than as the life-long friend and fellow-labourer of him\n     whose deeds and memory we honour to-day--to trace back to their\n     source the origin and underlying principles of our system of\n     education, and to show that these underlying principles and\n     other vital forces were so combined by a master-hand as to form\n     the groundwork, as they have, in their combination, become the\n     charter of our educational system of to-day.\"\n\nThe second reason is contained in the following\nparagraphs--containing a brief record of Dr. Ryerson's thirty-two\nyears in the Public Service, taken from _The Story of My Life_, page\n351.\n\n     \"During my connection with the Education Department--from\n     1844 to 1876--I made five educational tours of inspection\n     and enquiry to educating countries in Europe and the United\n     States. I made an official tour through each county in Upper\n     Canada, once in every five years, to hold a County Convention\n     of municipal councillors, clergy, school trustees, teachers and\n     local superintendents, and thus developed the School system as\n     the result of repeated inquiries in foreign countries, and the\n     freest consultation with my fellow-citizens of all classes,\n     in the several County Conventions, as well as on many other\n     occasions.\n\n     \"During the nearly thirty-two years of my administration of the\n     Education Department, I met with strong opposition at first from\n     individuals--some on personal, others on religious and political\n     grounds; but that opposition was, for most part, partial and\n     evanescent. During these years I had the support of each\n     successive administration of Government, whether of one party\n     or the other, and, at length, the co-operation of all religious\n     persuasions; so that in 1876 I was allowed to retire, with the\n     good-will of all political parties and religious denominations,\n     and without diminution of my public means of subsistence.\n\n     \"I leave to Dr. J. George Hodgins, my devoted friend of over\n     forty years, and my able colleague for over thirty of these\n     years, the duty of filling up the details of our united labours\n     in founding a system of education for my native Province which\n     is spoken of in terms of strong commendation, not only within,\n     but by people outside of the Dominion.\"[1]\n\n  [1] It is the purpose of the writer of this Retrospect (in\n  accordance with Dr. Ryerson's oft expressed wish) to prepare another\n  volume, giving, from private letters, memoranda, and various\n  documents, a personal history of the founding and vicissitudes of\n  our educational system from 1844 to 1876 inclusive.\n\nMy own estimate of Dr. Ryerson's educational life and labours is\ncontained in the following paragraphs, which were written by me in\n1883:--\n\nDr. Ryerson's fame in the future will mainly rest upon the fact\nthat he was a distinguished Canadian Educationist, and the Founder\nof a great system of Public Education for Upper Canada. What makes\nthis distinguishing excellence in his case the more marked, was the\nfact that the soil on which he had to labour was unprepared, and\nthe social condition of the country was unpropitious. English ideas\nof schools for the poor, supported by subscriptions and voluntary\nofferings, prevailed in Upper Canada; free schools were unknown;\nthe very principle on which they rest--that is, that the ratable\nproperty of the country is responsible for the education of the\nyouth of the land--was denounced as communistic and an invasion of\nthe rights of property; while \"compulsory education\"--the proper\nand necessary complement of free schools--was equally denounced\nas of the essence of \"Prussian despotism,\" and an impertinent and\nunjustifiable interference with \"the rights of British subjects.\"\n\nIt was a reasonable boast at the time that only systems of popular\neducation, based upon the principle of free schools, were possible\nin the republican American States, where the wide diffusion of\neducation was regarded as a prime necessity for the stability and\nsuccess of republican institutions, and, therefore, was fostered\nwith unceasing care. It was the theme on which the popular orator\nloved to dilate to a people on whose sympathies with the subject he\ncould always confidently reckon. The practical mind of Dr. Ryerson,\nhowever, at once saw that the American idea of free schools was\nthe true one. He moreover perceived that by giving his countrymen\nfacilities for freely discussing the question among the ratepayers\nonce a year, they would educate themselves into the idea, without\nany interference from the State. These facilities were provided\nin 1850; and for twenty-one years the question of free schools\n_versus_ rate-bill schools (fees, etc.) was discussed every January\nin from 3,000 to 5,000 school sections, until free schools became\nvoluntarily the rule, and rate-bill schools the exception. In 1871,\nby common consent, the free school principle was incorporated into\nour school system by the Legislature, and has ever since been the\nuniversal practice. In the adoption of this principle, and in the\nsuccessful administration of the Education Department, Dr. Ryerson\nat length demonstrated that a popular (or, as it had been held in\nthe United States, the democratic) system of public schools was\nadmirably adapted to our monarchical institutions. In point of\nfact, leading American educationists have often pointed out that\nthe Canadian system of public education was more efficient in all\nof its details, most practically successful in its results, than\nwas the ordinary American school system in any one of the States of\nthe Union. Thus it is that the fame of Dr. Ryerson as a successful\nfounder of our educational system, rests upon a solid basis. What\nhas been done by him will not be undone; and the ground gone over\nby him will not require to be traversed again. But I forbear, as I\nhope to devote a volume to the private and personal history of our\neducational system.\n\n       *       *       *       *       *\n\nThe first part of this volume contains an interesting account from\nthe leading daily papers of Toronto of the ceremony of unveiling\nthe great statue to a distinguished native of Ontario and a truly\nrepresentative man--representative of her enterprize, energy and\nprogress. As such, many of the leading men of the Province assembled\non the Queen's Birthday to do honour to his memory. It also contains\nthe addresses in full of those gentlemen who were appointed by\ntheir respective institutions, etc., to that duty, and who kindly\nconsented to take part in the ceremonies and proceedings of the day.\n\nThe second part of the volume contains a statement of the origin,\nwith illustrations, of the underlying principles of our system of\neducation--primary, secondary and university.\n\nThere has also been added at the end some historical and personal\nsketches--some of them of a humorous cast--but all illustrative of\nthe early days of education in this Province, and its vicissitudes\nof light and shade. They admirably serve to bring out in strong\nrelief the present state of efficiency of our system of public\ninstruction, as well as the substantial progress which has been\nmade by it in its various departments since the early educational\npioneers of Upper Canada first attempted to give it form and\nsubstance, over fifty years ago.\n\nThe photograph from which the frontispiece is printed was by Mr. J.\nBruce, of 118 King street west, Toronto.\n\n  J. G. H.\n\nTORONTO, 24th May, 1889.\n\n\nPERSONAL NOTE.\n\nA few days after the ceremony of unveiling the Statue, I received\nthe following very kind note from Rev. Dr. Ryerson's only son:--\n\n  27 CECIL ST., TORONTO,\n  May 28th, 1889.\n\nMY DEAR DR. HODGINS,--The 24th of May was indeed a red-letter day to\nme and to my family; and one I shall never forget.\n\nThe Statue and Pedestal are beyond anything I expected; and the\nlikeness is excellent.\n\nAllow me to thank you very heartily for your eloquent Historical\nPaper, and the touching references to my dear Father.\n\nI know that all you did was a labour of love. But I cannot allow\nthis event to pass without expressing to you our deep gratitude for\nthe time and pains you have taken in successfully carrying out this\nsplendid memorial to my revered Father. Mrs. Ryerson joins me in\nvery kind regards.\n\n  Believe me,\n  Yours very faithfully,\n  C. EGERTON RYERSON.\n\n  J. GEORGE HODGINS, Esq., LL.D.,\n  Toronto.\n\n\n\n\nCONTENTS.\n\n\n                                                                 PAGE.\n\n  Title and Prefatory Note                                        iii.\n\n\n  CHAPTER I.\n\n  Preliminary Remarks                                                1\n\n  Appeal for Funds for the Erection of the Statue                    3\n\n  The Financial Results of the Appeal Made--Particulars of the\n      Statue                                                         5\n\n  Programme of Arrangements for Unveiling the Statue                 5\n\n  Inscription on the Statue Pedestal                                 6\n\n  Record of Rev. Dr. Ryerson's Services                              7\n\n\n  CHAPTER II.\n\n  Report of Ceremony of Unveiling the Statue (from _The Globe_)      8\n\n  Address of the Hon. G. W. Ross, Minister of Education              9\n\n  The Statue Unveiled by Sir Alexander Campbell,\n      Lieutenant-Governor                                           11\n\n  Report of the unveiling by _The Empire_ and _The Mail_            13\n\n  Comments of the Press on the Unveiling of the Statue              15\n\n\n  CHAPTER III.\n\n  THE ADDRESSES DELIVERED AT THE UNVEILING OF THE STATUE, VIZ.:\n\n      1. By Mr. Robert McQueen, President of the Teachers'\n             Association of Ontario                                 17\n\n      2. By Alderman McMillan, Acting Mayor of Toronto              19\n\n      3. By Hon. Senator Macdonald, representing the University\n             of Toronto                                             20\n\n      4. By Rev. Dr. Burwash, representing the University of\n             Victoria College                                       22\n\n      5. By Sandford Fleming, LL.D., C.M.G., representing the\n             University of Queen's College                          23\n\n      6. By Rev. Professor Clark, M.A., representing the\n             University of Trinity College                          24\n\n      7. By Professor T. H. Rand, D.C.L., representing McMaster\n             University                                             25\n\n\n  CHAPTER IV.\n\n  EDUCATION IN ONTARIO, PAST AND PRESENT--AN HISTORICAL RETROSPECT,\n  BY J. GEORGE HODGINS, M.A., LL.D., VIZ.:\n\n  Significance of the Event of the Day                              27\n\n  The Ontario System of Education--Its Influence Abroad             27\n\n  Comprehensive Character of the Ontario Educational System         28\n\n  Character and System of Education Abroad, and Lessons Therefrom   29\n\n  Educational Lessons to be Learned Outside of Ontario              29\n\n  Three Educational Periods in the History of Ontario               30\n\n  Colonial Chapter in the History of American Education             30\n\n  The Nine British Colonial Universities in the Thirteen Colonies   32\n\n  The United Empire Loyalist Period in Upper Canada                 36\n\n  Governor Simcoe's Educational Views in 1795                       37\n\n  Early Beginnings of Education in Upper Canada, 1785-1805          37\n\n  State of Education in Upper Canada, 1795-1799                     38\n\n  First Official Educational Movements in Upper Canada, 1797,\n      1798                                                          38\n\n  Educational Pioneers in Upper Canada                              39\n\n  Early Efforts to Establish Common Schools, 1816-1820              40\n\n  State of Education in Upper Canada, 1784-1819                     41\n\n  Fitful Educational Progress from 1822 to 1829                     41\n\n  State of Education in Upper Canada, 1827-1829                     42\n\n  Rev. Dr. Strachan's Course of Study in Grammar Schools, 1829      43\n\n  Rev. Dr. Strachan's System of School Management                   44\n\n  Rev. Dr. Strachan's Career as a Teacher                           45\n\n  Mr. Joseph Hume's Essay on Education, edited by Mr. W. L.\n      MacKenzie                                                     46\n\n  Vicissitudes of Education in Upper Canada, 1830-1839              46\n\n  Educational Efforts in the House of Assembly, by Mr. M. Burwell,\n      1831-1836                                                     47\n\n  Efforts at Educational Legislation, by Dr. Charles Duncombe,\n      1831-1836                                                     48\n\n  Continued Educational Efforts of Mr. Burwell in the House\n      of Assembly                                                   50\n\n  Early Opinions on the Necessity for Manual or Industrial\n      Education in our Schools                                      51\n\n  Later Opinions (on the same subject)                              51\n\n  Further Educational Efforts in the House of Assembly, 1835,\n      1836                                                          52\n\n  Analysis of Dr. Charles Duncombe's Report on Education, 1836      53\n\n  Summary of, and Reflection on, these Educational Efforts from\n      1830 to 1839                                                  54\n\n  Extracts from Official Reports on Education in Upper Canada\n      in 1838                                                       55\n\n  Influences by American Teachers and School Books Deprecated       55\n\n  Extracts from Report of an Education Commission in 1839           57\n\n  Educational Opinions of Prominent Public Men in 1839              58\n\n  Separate Educational Forces Shaping Themselves in Upper Canada    59\n\n  Noted Educational Leaders--Dr. Strachan and Dr. Ryerson           59\n\n  The Educational Efforts of the U. E. Loyalists and the\n      Ruling Party                                                  60\n\n  An Educational Glance Backwards                                   60\n\n  Provision for Higher Education in Upper Canada by the Imperial\n      Government                                                    62\n\n  Rev. Dr. Strachan as an Educator                                  62\n\n  Rev. Dr. Strachan's Reasons for Establishing a University\n      in Upper Canada                                               64\n\n  Rev. Dr. Strachan, the Founder of Two Universities in Toronto     65\n\n  The University of Toronto                                         66\n\n  The University of Victoria College                                66\n\n  The Queen's College University                                    69\n\n  The University of Trinity College                                 70\n\n  The R. C. University College at Ottawa                            70\n\n  The Western University, London                                    70\n\n  The McMaster University                                           71\n\n  Upper Canada College--Albert College--Woodstock\n      College--The School of Practical Science, and\n      various colleges and schools, etc.                            71\n\n  Rev. Dr. Ryerson's advocacy of Popular Rights, 1827-1841          72\n\n  Educational Legislation in the United Parliament of 1841\n      and 1843                                                      72\n\n  Origin of the annual grant of $200,000 for Common Schools\n      in 1841                                                       73\n\n  Educational efforts of Rev. Dr. Ryerson up to this time           74\n\n  First appointment of a Superintendent of Education for Upper\n      Canada, 1842                                                  74\n\n  Appointment of Rev. Dr. Ryerson as Superintendent of\n      Education, 1844                                               75\n\n  Rev. Dr. Ryerson's Report on a System of Public Instruction for\n      Upper Canada                                                  75\n\n  Chief features of Dr. Ryerson's first report and School Bill,\n      1846                                                          77\n\n  Objections to Dr. Ryerson's School Bill of 1846, answered.        77\n\n  First and Second Councils of Public Instruction, 1846 and 1850.   78\n\n  Religious Instruction in the Common Schools, 1846.                79\n\n  State of Common School Education in Upper Canada, 1845.           80\n\n  School Houses and School Teachers in 1845-1850.                   81\n\n  Combined opposition to the projected system of Education.         82\n\n  Educational Proceedings of District Councils in 1847, 1848.       83\n\n  Estimate of Lord Elgin's character by Hon. W. H. Draper.          84\n\n  Invaluable assistance given to Dr. Ryerson by Lord Elgin.         85\n\n  Proceedings of the First Council of Public Instruction. The\n      Normal School.                                                86\n\n  Laying the corner stone of the New Normal School Buildings,\n      1851.                                                         87\n\n  The County Model Schools of 1843-1850.                            88\n\n  Fundamental Principles of Dr. Ryerson's Scheme of Education.      90\n\n  Can Upper Canada Emulate the State of New York in Educational\n      Matters?                                                      90\n\n  Establishment of the Educational Depository and its Results.      92\n\n  Abstract of Depository Schedule Presented to the Legislature\n      in 1877.                                                      92\n\n  Dr. Ryerson a Commissioner on King's College, New Brunswick,\n      in 1854.                                                      93\n\n  Chronological Sketch of Dr. Ryerson's Educational Work,\n      1855, etc.                                                    94\n\n  Bishop Fraser's Estimate of the Upper Canada System of\n      Education in 1863.                                            95\n\n  Character of the Important School Legislation of 1871.            97\n\n  Review of the School Legislation of 1871.                         98\n\n  Objections to Improve our School System Answered.                 98\n\n  Necessity for the Change in the School Law of Ontario in 1871.   100\n\n  Hon. Adam Crooks on the School Inspection Legislation of 1871.   101\n\n  Inspector Harcourt's opinion of the effect of the School Act\n      of 1871.                                                     101\n\n  Inspector McKee, of the County of Simcoe, on the School Act\n      of 1871.                                                     101\n\n\n  CHAPTER V.\n\n  A SPECIAL CHAPTER ON THE STATE OF EDUCATION IN THE OLDEN\n      TIME IN UPPER CANADA.                                        103\n\n  Hon. J. Sandfield Macdonald's School Days--His Reminiscences\n      of them.                                                     103\n\n  Hon. Charles Clarke on Education in the County of Wellington\n      under Dr. Ryerson's Administration.                          104\n\n  Early School Legislation in 1841, 1843 and 1846.                 104\n\n  Inferior Qualification of Teachers and Varied Methods of\n      Teaching.                                                    105\n\n  Dr. Ryerson's Test of the Intelligence of a School Section.      105\n\n  The Character of the School-House, also a Test.                  106\n\n  School Condition of the County of Wellington in 1847.            106\n\n  Great Educational Advance made by the Province of Ontario\n      since 1847.                                                  107\n\n  Great Advance also in the Standard of Teaching Ability.          108\n\n  Rev. W. H. Landon on the State of Education in Upper Canada\n      in 1847-1849.                                                109\n\n  The Old Log School-House and its Belongings.                     110\n\n  The Pioneer Teacher and the Trials of \"Boarding-Round\".          111\n\n  The Old School House (Poetry).                                   113\n\n  Mr. Canniff Haight on the Schools Fifty Years Ago.               113\n\n  A School Teacher's Personal Experience in 1865.                  114\n\n  Mr. James Cumming's Reminiscences of Education in Hamilton\n      in 1847-1852.                                                116\n\n  Education in the County of Simcoe, 1852-1872.                    117\n\n\n  CHAPTER VI.\n\n  PERSONAL CHAPTER RELATING TO THE REV. DR. RYERSON                119\n\n  His Early Life, as Sketched by Himself                           119\n\n  Rev. Dr. Ryerson as a Teacher                                    121\n\n  The Rev. Dr. Ryerson and His Native County of Norfolk            122\n\n  Closing Official Acts and Utterances of Dr. Ryerson              124\n\n  Reasons for Dr. Ryerson's Retirement as Chief Superintendent\n      of Education                                                 125\n\n  Dr. Ryerson's Letter of Resignation in 1868 and Reply to it      126\n\n  Dr. Ryerson's Letter of Resignation in 1872 and Reply to it      128\n\n  A FEW WORDS PERSONAL TO THE WRITER OF THIS RETROSPECT            130\n\n\n\n\nTHE RYERSON MEMORIAL VOLUME.\n\nPRELIMINARY.\n\n\nThe Rev. Dr. Ryerson's death occurred on the 19th of February, 1882.\nEarly in the next month the following circular was issued:--\n\nA preliminary meeting of trustees, inspectors and teachers connected\nwith Public, Separate and High Schools--past and present--will be\nheld in the theatre, or public hall, of the Education Department,\non Tuesday afternoon, the 14th instant, at 4.30, to consider the\nproposal to erect a monument or other tribute of love and esteem to\nthe memory of the late revered founder of the educational system of\nOntario.\n\n  J. GEORGE HODGINS,\n  Toronto, March, 1882.      Convener.\n\nThe following account of the meeting appeared in _The Mail_\nnewspaper of the 15th March:--\n\nA meeting of those connected with educational matters was held in\nthe theatre of the Normal School yesterday afternoon, to consider\nthe proposal to erect a monument or other token of esteem to the\nmemory of the late Dr. Ryerson.\n\nThere were present, Drs. Hodgins, Davies, Carlyle, Tassie;\nInspectors Hughes, McKinnon, McBrien, Little, Fotheringham; Messrs.\nJames Bain, G. McMurrich, and Crombie, Public School trustees,\nToronto; Thomas Kirkland, M.A., Normal School; Mrs. Riches, Misses\nE. A. Scarlett, M. L. Williams, Boulton and Tomlinson; Messrs.\nMcAllister, Doan, Lewis, Spence, McCausland, Martin, Clarke, Coyne,\nParker, Cassidy, Campbell, teachers of Toronto city schools. Dr.\nHodgins was appointed chairman, and Mr. James L. Hughes secretary.\n\nLetters expressing regret at inability to attend were read from Mr.\nG. W. Ross, M.P., Inspectors Scarlett, Bigg and Platt, and Mr. W. J.\nGage. Also a letter from the Minister of Education, to the effect\nthat every inspector and teacher was at full liberty to take such\naction as he might think proper in connection with so laudable an\nobject as the erection of a memorial to the late superintendent.\n\nDr. Hodgins related the circumstances which had led to the calling\nof a meeting, and said that the object was to unite all persons in\nany way connected with schools in a tribute of affection to the late\nchief, even the children might contribute a mite. There was some\ndifference of opinion as to whether the tribute should take the form\nof a monument in the cemetery, or a statue in the Normal School\ngrounds. He mentioned Dr. Ormiston as having taken a very strong\ninterest in the matter.\n\nOn motion of Mr. McAllister, seconded by Mr. McMurchy, a resolution\nexpressing the approval, by the meeting, of the proposal to erect a\nmemorial was unanimously adopted.\n\nMr. Hughes suggested that a central committee of organization be\nappointed, and that local associations be formed in every county.\n\nMessrs. Fotheringham, McMurchy, George McMurrich, McAllister and\nBrother Odo, were appointed to nominate the central committee.\nTheir report was presented to the meeting, and adopted with some\namendments.\n\nIn the discussion which took place, the general opinion was that\nthe memorial should take the form of a statue of the late Chief\nSuperintendent, to be erected in the Normal School grounds, Mr. Bain\nbeing one of those who strongly urged this view.\n\nMr. Inspector Fotheringham thought that the amount of the\ncontribution should be limited so as to make it general, and\nsuggested one dollar for each teacher and trustee, and ten cents for\neach child, corporate bodies being left to their own discretion.\n\nThe chairman having called for suggestions as to the formation of\nlocal committees, Mr. Little, Inspector for Halton, thought that the\ninspector for each county should appoint the teachers and trustees\nof each school to open a subscription list.\n\nMr. McBrien, Inspector for Ontario County, suggested that the\ninspectors should send a postcard to every teacher in the county,\nrequesting him to convene a meeting in his section.\n\nMr. McKinnon, Inspector for Peel, was in favor of Mr. Little's plan.\n\nThere was also considerable discussion as to whether others besides\nthose directly connected with the schools should be asked to\ncontribute. It was urged on the one hand that the tribute would come\nmore fittingly from those more peculiarly interested in education,\nand on the other that it should be made national in its character,\nespecially as a large majority of the people had been educated, and\ntheir characters formed under the school system of which Dr. Ryerson\nwas the founder.\n\nThese questions were left for future consideration, and, after a\nvote of thanks to the chairman, the meeting adjourned.\n\n       *       *       *       *       *\n\nAs the result of that meeting the following circular was issued by\nthe secretary on the 15th March:--\n\nAt a preliminary meeting of trustees, inspectors and teachers\nconnected with Public, Separate and High Schools, held in the public\nhall of the Education Department on the 14th instant, the following\ngentlemen were appointed members of a central committee to carry\nout the resolution unanimously agreed to by the meeting, viz.: to\ncollect funds with which to erect a monument or other tribute of\nlove and esteem to the memory of the late revered founder of the\neducational system of Ontario, viz.:--\n\nDr. _Hodgins_, chairman; Rev. Principal _Davies_; Principal\nMcCabe; Rev. Dr. Ormiston, of New York; President Wilson and Prof.\nG. P. Young, Toronto University; Archbishop Lynch; Rev. Provost\nBody, Trinity College; Rev. Principal _Caven_, Knox College; Rev.\nPresident Castle, Toronto Baptist College; Rev. Father Vincent,\nSuperior, St. Michael's College; Rev. President Nelles, Victoria\nUniversity; Very Rev. Principal Grant, Queen's University; _A.\nMcMurchy_, M.A., President of Ontario Teachers' Association; Very\nRev. Dean Grassett, Chairman of Collegiate Institute Board; Edward\nGalley, Esq., Chairman, Public School Board; Vicar-General Rooney,\nChairman, Separate School Board; Dr. McLellan, Inspector, High\nSchools; Mr. White, Inspector, Separate Schools for Ontario; Rev.\nBrother Tobias, city Inspector of Separate Schools; J. S. Carson,\nEsq., Chairman of inspectors' section Ontario Teachers' Association;\nR. Lewis, Esq., chairman of Public School section; D. C. McHenry,\nM.A., chairman of High School section; also the Public School\nInspectors throughout the Province as _ex officio_ members, (Messrs.\n_D. Fotheringham_ and _J. R. Miller_). Mr. _James L. Hughes_ was\nappointed secretary of the committee, and Mr. _Walter S. Lee_,\ntreasurer.[2]\n\n  [2] James Carlyle, Esq., M.D., Master in the Normal School, was\n  subsequently appointed joint secretary with Mr. Hughes. Both\n  rendered most valuable service in promoting the object in view.--J.\n  G. H.\n\n  As some gentlemen here named declined, and others ceased to act,\n  the committee was finally reduced to nineteen members, including\n  the chairman, secretaries, treasurer and the new appointments.\n  Only those whose names are printed in italics were members of the\n  committee at the time the statue was unveiled in 1889--seven years\n  after their appointment. The following were the members of the\n  committee at the time of the unveiling, viz.:--\n\n  Rev. Principal Caven, D.D., Rev. Dr. Potts, Hon. G. W. Ross,\n  Minister, Rev. H. W. Davies, D.D., Hon. Senator Macdonald, Principal\n  Kirkland, M.A., Rev. W. H. Withrow, D.D., Principal Dickson, M.A.,\n  Rev. Hugh Johnston, D.D., Rector McMurchy, M.A., Mr. G. H. Robinson,\n  M.A., ex head master Collegiate Institute; Mr. David Fotheringham,\n  Inspector of North York; Mr. R. Doan, Mr. S. McAllister, public\n  school teacher, Toronto, and Mr. J. R. Miller, ex-inspector.\n  Chairman, J. George Hodgins, LL.D.; joint secretaries, Mr. James\n  L. Hughes and James Carlyle, M.D.; treasurer, Mr. Walter S. Lee.\n  The artist-sculptor was Mr. Hamilton McCarthy, R.C.A., and the\n  contractor for pedestal, Mr. F. B. Gullett, monumental sculptor.\n\n\nAPPEAL FOR FUNDS FOR ERECTION OF THE STATUE.\n\nAt a subsequent meeting of the committee, Rev. Dr. Ormiston and Dr.\nHodgins were requested to draw up an appeal soliciting aid for the\nproposed memorial. Dr. Ormiston did so as follows:--\n\n    _Appeal to Trustees, Inspectors, Teachers and Pupils--past and\n       present--connected with Public, Separate[3] and High Schools,\n       and to the other friends of Education in the Province of\n       Ontario; from the General Committee appointed at Toronto on\n       the 14th March, 1882, for the collection of funds with which\n       to erect a Monument, or other Tribute of Esteem and Admiration\n       to the memory of the late Rev. Dr. Ryerson, founder of the\n       Educational System of Ontario:_\n\n  [3] Having asked Archbishop Lynch to commend this and subsequent\n  appeals to the teachers of the Separate Schools, he replied as\n  follows:--\n\n    \"ST. MICHAEL'S PALACE, Toronto, December 12, 1882.\n\n       \"MY DEAR DR. HODGINS,--I do not like to assume a prominent part\n       in writing to the teachers of the Separate Schools outside of\n       my own diocese, or to set an example which I fear would be\n       criticized. However, I send you my subscription ($10) towards\n       the erection of the statue to the late lamented Dr. Ryerson.\n\n    \"I am, yours very sincerely,\n    \"+JOHN JOSEPH LYNCH,\n    \"Archbishop of Toronto.\n\n    \"J. GEORGE HODGINS, ESQ., LL.D.,\n    Chairman, etc.\"\n\n       (The Very Reverend Vicar-General Rooney also sent $10, as his\n       subscription to the fund).\n\n  I also wrote to the Rev. Father Stafford, of Lindsay, on the\n  subject. In his reply, dated March 9th, 1882, he said:--\n\n       \"You ask my opinion as to whether the erection of a monument\n       to the late Dr. Ryerson will receive support from the Separate\n       Schools?\n\n       \"Not much from Separate Schools as such, nor much from Separate\n       School supporters. The recollection of the old controversy with\n       the bishops of our Church, is still fresh in the memories of\n       many.\n\n       \"I think some of the Separate School teachers will\n       subscribe--perhaps many of them.\n\n       \"Personally, I must give my mite. I always found Dr. Ryerson,\n       as you are aware, very kind with me, and very attentive to any\n       suggestions I had to offer, and very just in all his dealings\n       with me.\n\n       \"I admired his ability and his love and enthusiasm for his\n       work. No one knows better than you my admiration for that\n       man. My idea is that the monument ought to be something very\n       respectable--say, got up something like the one to Grattan, or\n       Moore, or Burke, near Trinity College, Dublin; and it ought to\n       be erected in the Normal School grounds.\n\n       \"I wished to send a word of sympathy to Dr. Ryerson's family,\n       but I did not know where to address them. Will you kindly say a\n       word for me to the proper person?\n\n    \"Yours faithfully,\n    \"M. STAFFORD, Pr.\n\n    \"DR. HODGINS,\n    \"Toronto.\"\n\n\"Although still young our Province has already been called to mourn\nthe removal of not a few of her gifted sons, who have severally\nadorned the different walks of public life. In weight of character,\nwealth of manhood, and width of human sympathy, the late Chief\nSuperintendent of Education, stood amongst the foremost and\nmightiest of them all.\n\nEgerton Ryerson was a man of rare diversity of gifts, of remarkable\nenergy, and of abundant mental resources. It would have been easy\nfor him to have excelled in any one sphere of human greatness,\nbut it was his to stand high in several. He was a many-sided man;\nrichly endowed in various ways. He was a laborious farmer--a zealous\nstudent--a successful teacher--an eminent preacher--a prominent\necclesiastic--an influential editor--a forcible writer--a sagacious\ncounsellor--a most efficient principal and professor--but he was\nchiefly noted as a great public educationist.\n\nFor a third of a century he was the head and inspiring genius of our\nschool system, establishing, moulding, adapting, controlling it;\nand this, the main work of his life, will endure and command in the\nfuture, as it has in the past, the admiration of all, both at home\nand abroad. During all these years he was the teacher's true friend,\nand the ardent well-wisher of the young. His sympathies--tender and\ntrue--as helpful as they were healthy, went out to every earnest\nworker, whether in acquiring or imparting knowledge. The enquiring\nleft his presence directed; the downcast, cheered; the doubtful,\nconfirmed.\n\nUnselfish, generous, disinterested, he devoted himself wholly to\nhis work. How often did his lip quiver and his eye fill when he\naddressed the gatherings of teachers and pupils, upon whom he looked\nnot only with the eye of a patriot, but of a parent,--\"Ye are my\nchildren all.\"\n\nWe can never forget him; we profoundly mourn our loss; we fondly\ncherish his memory. Affection, gratitude, a sense of what is due\nto so eminent a man, impel us to perpetuate that memory in some\nsuitable way, which will render such a noble life an inspiring\nexample to young men now and in the coming days.\n\nIn obedience then, to one of the purest and loftiest instincts of\nour nature, let us unite in paying a common tribute of admiration\nand regard to the memory of him to whom we all sustained a common\nrelationship, and to whom we also, without distinction as to\nnationality, political preferences, or religious belief, can\npay sincere homage, as the founder of our present excellent and\ncomprehensive system of education.\n\nIn honoring him we do honor to our common country, and recognize our\nobligation to pay fitting homage to the great men of our Dominion,\nwhose names, with his, are inscribed high upon the roll of Canada's\nfamous sons.\"\n\nAt intervals, during the years 1882-1886, circulars were issued by\nthe committee to inspectors, trustees and masters of High, Public\nand Separate Schools, urging the collection of the necessary funds\nto erect the proposed memorial. In order to aid in this work, 7,500\ncopies of a biographical sketch of Dr. Ryerson and his educational\nwork, prepared by the chairman, was sent to the inspectors for\ndistribution. The chairman also made the following suggestions to\ninspectors (with a view to facilitate the collections from pupils),\nwhich was generally acted upon, viz.:--\n\n     \"Permit me to suggest a simple way of securing a response from\n     each school: You might request the teacher to give notice that,\n     on the following week, he would devote _five minutes_ at noon\n     of each day to taking down a list of contributions (from a cent\n     upwards) to the fund.\n\n     \"In this way the pupils--and everyone in the locality, through\n     the children--would have an opportunity of contributing his or\n     her mite to the erection of a statue to one of Canada's most\n     honored sons.\"\n\nThe final circular issued by the committee was as follows:--\n\n     The appeal on behalf of the Ryerson Memorial Fund has been\n     responded to by about two-thirds of the public, and less than\n     one-third of the High-Schools in Ontario. The sum thus received\n     amounts to $4,425.00, including accrued interest on the moneys\n     received and invested.\n\n     The 7,520 masters and teachers now employed in the Public and\n     High Schools of Ontario, have not yet been appealed to, as\n     a body, to contribute to this most desirable and patriotic\n     object, although many of them have sent in their subscriptions.\n     The General Committee have, therefore, decided to make this\n     appeal to them through the various teachers' associations. The\n     committee trust, therefore, that the individual masters and\n     teachers concerned (if they have not already done so) will\n     heartily and promptly respond to this appeal.\n\n     The words with which Dr. Ormiston closes his appeal on behalf of\n     this fund we would heartily commend to your sympathy and kind\n     consideration. We do so with the earnest hope that you will\n     give them a substantial and practical application. Dr. Ormiston\n     says:--\n\n     \"In obedience then, to one of the purest and loftiest instincts\n     of our nature, let us unite in paying a common tribute of\n     admiration and regard to the memory of him to whom we all\n     sustained a common relationship, and to whom we also, without\n     distinction as to nationality, political preferences, or\n     religious belief, can pay sincere homage, as the founder of our\n     present excellent and comprehensive system of education.\n\n     \"In honouring him we do honour our common country, and recognize\n     our obligation to pay fitting homage to the great men of our\n     Dominion, whose names, with his, are inscribed high upon the\n     roll of Canada's famous sons.\"\n\n     The Rev. T. Bowman Stephenson, L.L.D., delegate from the British\n     to the General Conference of the Methodist Church in Canada, in\n     his recent address to that Conference, said, referring to the\n     late Rev. Dr. Ryerson:--\n\n     That gentleman \"visited us in England twice. Old man as he then\n     was, he seemed younger than most of us. I take him to have been\n     one of those rare men who are never young and never old--old in\n     wisdom whilst young in years--young in heart and feeling when\n     already the snow is on the head. Eloquent, logical, far sighted,\n     generous, independent, courageous, with an unhesitating faith in\n     duty, and a boundless love of freedom and justice, he 'served\n     his generation.'--O how well the inspired words describe him--by\n     the will of God, 'he fell on sleep.'\"\n\n\nTHE FINANCIAL RESULTS OF THE APPEALS MADE--PARTICULARS OF THE STATUE.\n\nThe following is the financial result of the labours of the\ncommittee up to the date of its final meeting on the 1st of June,\n1889, viz.:--\n\n  Subscriptions received            $4,647 95\n  Legislative grant                  2,000 00\n  City of Toronto grant                500 00\n  Interest on deposits               1,119 14\n                                    --------- $8,267 09\n\n  Cost of bronze statue             $5,100 00\n  Coat of granite pedestal           2,600 00\n  Fees and incidentals                 381 09\n                                    --------- $8,081 09\n                                              ---------\n                                                $186 00\n  To be expended on the Memorial Volume          186 00\n                                                -------\n\n  Height of bronze figure                9 feet 6 inches.\n  Height of granite pedestal            10 feet 6 inches.\n\nThe granite of the pedestal is from a quarry at St. George, in\nNew Brunswick--a Province which was the first early home of Dr.\nRyerson's father and mother, after the close of the American\nRevolutionary War. Dr. Ryerson's mother was a native of New\nBrunswick as were his elder brothers and sisters.\n\n\nPROGRAMME OF ARRANGEMENTS FOR UNVEILING THE STATUE.\n\nThe following was the programme of arrangements agreed to by the\nCommittee to be observed on the Queen's Birthday, 1889, at the\nceremony of unveiling of the statue of the Rev. Egerton Ryerson,\nD.D., LL.D., founder of the school system of Ontario, 1844-1876,\nceremony to commence at two o'clock p.m.:--\n\n_Chairman for the Day._--The Hon. George W. Ross, LL.D., Minister\nof Education for Ontario. _Dedicatory Hymn._--(\"All People that\non Earth do Dwell,\" Old Hundred) to be announced by the Rev. John\nBurton, B.D. _Selection of Scripture._--To be read by the Rev.\nJohn Potts, D.D., Secretary of Education of the Methodist General\nConference. _Dedicatory Prayer._--By the Rev. G. M. Milligan,\nB.A., Minister of Old St. Andrew's Church, Toronto. _Opening\nAddress._--By the Hon. George W. Ross, LL.D., Chairman of the Day.\n_Unveiling of the Statue._--By the Hon. Sir Alexander Campbell,\nK.C.M.G., Lieutenant-Governor of Ontario. _Patriotic School Song\nby the City School Children._--\"Hurrah, Hurrah, for Canada!\" to be\nled by Mr. Perrin, Music Teacher, City Schools. _Historical Paper\non Education in Ontario._--The abstract only was read by J. George\nHodgins, M.A., LL.D., Deputy Minister of Education for Ontario.\n_Address on behalf of the Ontario Teachers' Association._--By Mr.\nMcQueen, President of the Association, 1889. _Address on behalf of\nthe Citizens of Toronto._--By His Worship the Mayor, E. F. Clarke,\nEsq., M.P.P. (Mr. Clarke having gone to England, the address\nwas read by Alderman McMillan, President of the City Council,\nand Acting Mayor _pro tem_). _Patriotic Song by the City School\nChildren._--\"The Maple Leaf for Ever!\" _Address on behalf of the\nUniversity of Toronto._--By the Hon. John Macdonald, Senator.\n_Address on behalf of Victoria University._--By the Rev. N. Burwash,\nS.T.D., Chancellor of Victoria University. _Address on behalf of\nQueen's University._--By Sandford Fleming, Esq., LL.D., C.M.G.,\nChancellor of Queen's University. _Address on behalf of Trinity\nUniversity._--By the Rev. Professor William Clark, M.A. _Address\non behalf of McMaster University._--By T. H. Rand, Esq., D.C.L.\n_The National Anthem. Benediction._--Pronounced by the Right Rev.\nArthur Sweatman, D.D., D.C.L., Bishop of Toronto.\n\n_Representatives present._--The Mayor and Corporation of the City\nof Toronto, Chairman and members of the High School and Collegiate\nInstitute Board of Toronto, Chairman and members of the Public\nSchool Board of Toronto. Upper Canada College, Hon. John Beverley\nRobinson; Knox College, Rev. William McLaren, D.D.; Wycliffe\nCollege, Colonel Gzowski, A.D.C.; McMaster Divinity Hall, Rev.\nChancellor McVicar, D.D. LL. D.; Brantford Ladies' College, T. M.\nMacintyre, Esq., Ph. D.; Alma Ladies' College, Colin Macdougall,\nEsq., Q.C.[4]; Oshawa Ladies' College, Rev. A. B. Demill.\n\n  [4] In a kind note received from Mr. Macdougall, dated St. Thomas,\n  May 29th, 1889, he said:--\"The ceremonies of unveiling were\n  everything that could be looked for, and were well carried out.\n  The speaking was good and quite sufficient of it.... It must be to\n  you and to your family a gratifying reflection that you will be\n  remembered in history as having largely contributed by personal\n  exertion towards the erection of a monument to the memory of one of\n  Canada's greatest sons.\"\n\nThe following replies from other Colleges were received by the\nSecretaries, viz.:--\n\n  \"ASSUMPTION COLLEGE, SANDWICH, March 12, 1889.\n\n     \"DEAR SIR,--I beg to return thanks for your invitation to the\n     unveiling of the Statue to the late Doctor Ryerson.\n\n     \"I do not think it will be possible for any representative of\n     this College to be present on that occasion.\n\n  \"I remain, dear Sir, yours respectfully,\n  \"(Sgd) DENNIS O'CONNOR.\n\n     \"J. CARLYLE, Esq., Secretary.\"\n\n  \"ST. MICHAEL'S COLLEGE, TORONTO, 15th March, 1889.\n\n     \"DEAR SIR,--I received in due time your letter inviting me to\n     the unveiling of the Statue to the late Dr. Ryerson on the 24th\n     of May next. Your invitation I must respectfully decline, and\n     thanking you for it.\n\n  \"I remain yours very sincerely,\n  \"(Sgd) P. VINCENT.\n\n     \"JAMES CARLYLE, Esq., Secretary.\"\n\n  \"COLLEGE OF OTTAWA, March 21, 1889.\n\n     \"DEAR SIR,--I am in receipt at your circular, dated March 12th,\n     with which you kindly favored me. Please accept my best thanks\n     for your cordial invitation to send a representative of our\n     College to the unveiling of the Statue of the late Dr. Ryerson.\n     I am greatly sorry to state that it will be hardly possible to\n     anybody of us to go on the 24th of May. Please excuse us and\n     believe me.\n\n  \"Yours sincerely,\n  \"(Sgd.) J. M. FEAYARD, O.M.J.\n\n     \"Mr. J. CARLYLE, Secretary.\"\n\nNo replies were received from the other Colleges in Ontario to\nwhich invitations had been sent by the Secretaries. The following\nrepresentatives were also present:\n\nOntario Teachers' Association, Public School Section, Mr. Robert\nAlexander; Inspectors' Section, Mr. David Fotheringham; High School\nSection, Mr. Archibald McMurchy, M.A.\n\nInscription on the pedestal of the bronze statue of Rev. Dr.\nRyerson, as approved by the General Committee, November, 1887, to be\nplaced on the front of the pedestal, facing Bond Street:--\n\n  EGERTON RYERSON,\n  FOUNDER\n  of the\n  SCHOOL SYSTEM OF ONTARIO.\n\nTo be placed on the rear of the pedestal:--\n\n  BORN\n  IN CHARLOTTEVILLE, COUNTY OF NORFOLK, ONTARIO,\n  MARCH 24, 1803.\n  DIED\n  AT TORONTO, FEBRUARY 19, 1882.\n\nRecord of Rev. Dr. Ryerson's services, as approved by the General\nCommittee, November, 1887, and intended to have been engraved on the\nPedestal. It was afterwards decided not to do so, but to insert the\nname only, as founder of the Ontario School System.[5]\n\n  [5] Having submitted the draft of this inscription to several of Dr.\n  Ryerson's clerical friends, I received the following in reply:--\n\n  From the Rev. Dr. Douglass, Montreal:--\n\n       \"Thanks for your very kind favor. The name of Dr. Ryerson will\n       be forever sacred in my heart's best affection. I have read the\n       proposed inscription with very much care and interest. I think\n       it is comprehensive in its scope, and accurate and elegant in\n       detail. I really can offer no suggestion. I congratulate you\n       upon the completeness with which you have executed your talk.\n       With best wishes, ever and truly yours, G. D.\"\n\n  From the Rev. J. A. Williams, D.D., Toronto:--\n\n       \"I very much approve of the proposed inscription for the\n       memorial to the late lamented Dr. Ryerson. It recognizes\n       his worth and distinguished ability, as a writer, as an\n       educationist, and as a patriot. It is a fitting tribute from one\n       who knew him well and so long. With much and sincere respect, I\n       am, yours very truly, J. A. W.\"\n\n  From the Rev. John Potts, D.D., Toronto:--\n\n       \"No pen but yours should write the inscription for Dr. Ryerson's\n       monument. What you enclose is an elegant and eloquent tribute\n       to our dear departed father. The committee will accept the\n       inscription with thanks. Ever yours, J. P.\"\n\n  From the late Rev. Dr. Nelles, Cobourg (whom the committee wished to\n  be consulted on the matter). He said:\n\n       \"The inscription, in its present form, pleases me best of all,\n       and is well nigh perfect, doing 'credit to your head and heart,'\n       as the phrase is.\n\n       \"I am sure we shall get the thing perfect before we leave it.\n       And the old Doctor deserves that two such loving sons should\n       bestow their best efforts in such a matter.\n\n       \"The 'final revise' [subsequently sent] may, I think, now\n       be accepted. It will bear criticism.... You have in you an\n       illimitable power of improvement, and this last is still better\n       than any other. Affectionately yours, S. S. N.\"\n\n  From the Rev. Alexander Sutherland, D.D., Toronto:--\n\n       \"I have read the draft of the proposed inscription for the\n       Ryerson Statue with great care and a great deal of interest. It\n       covers the whole ground, and contains nothing that could well\n       be omitted. The extracts from the 'Letter of Acceptance' and\n       'Official Circular' give it a completeness which leaves nothing\n       to be desired. Yours faithfully, A. S.\"\n\n  From the Rev. Ephraim B. Harper, D.D., Brampton:--\n\n       \"I have read over and over again with all possible attention\n       and care your proposed inscription for Dr. Ryerson's Memorial,\n       and candidly allow that I could not propose the change of a\n       word, or even of a letter in it. I prepared something similar,\n       twenty-three years ago, for a gravestone for the late Rev. Dr.\n       Stinson, and know well the difficulty of compressing important\n       facts within the limits permissible for an inscription. It could\n       not have fallen into better hands, when it fell to your lot to\n       write it. As ever, dear doctor, yours affectionately, E. B. H.\"\n\n  From Rev. W. H. Withrow, D.D., Toronto:--\n\n       \"I think the inscription admirable. Yours, W. H. D.\"\n\n\n  THIS STATUE\n\n  IS\n\n  ERECTED\n\n  AS A MEMORIAL\n\n  OF THE GREAT PUBLIC SERVICES OF THE\n\n  REV. EGERTON RYERSON, D.D., LL.D,\n\n  SON OF COLONEL JOSEPH RYERSON,\n\n  A BRITISH OFFICER WHO SERVED DURING THE AMERICAN REVOLUTION,\n  AND WHO WAS ONE OF\n\n  THE UNITED EMPIRE LOYALISTS\n\n  WHO SETTLED IN THIS PROVINCE.\n\n         *       *       *       *       *\n\n  A DISTINGUISHED MINISTER OF THE METHODIST CHURCH,\n\n  1825-1852.\n\n  HE OBTAINED FOR THAT CHURCH A ROYAL CHARTER IN ENGLAND\n  FOR THE ESTABLISHMENT OF\n\n  UPPER CANADA ACADEMY AT COBOURG,\n\n  1828-1841.\n\n  AFTERWARDS\n\n  THE UNIVERSITY OF VICTORIA COLLEGE,\n\n  OF WHICH HE WAS THE FIRST PRESIDENT.\n\n         *       *       *       *       *\n\n  IN FOUNDING\n\n  THE SCHOOL SYSTEM OF HIS NATIVE PROVINCE,\n\n  AND IN PROMOTING THE\n\n  ESTABLISHMENT OF FREE SCHOOLS,\n\n  HE DISPLAYED THE RARE GIFTS OF A\n\n  FAR-SEEING AND ENLIGHTENED STATESMAN,\n\n  AND FOR THIRTY-TWO YEARS WAS\n\n  THE ABLE ADMINISTRATOR OF THAT SYSTEM,\n\n  1844-1876.\n\n         *       *       *       *       *\n\n  ERECTED BY CONTRIBUTIONS FROM SCHOOL TRUSTEES INSPECTORS, TEACHERS,\n  PUPILS AND OTHERS;\n\n  AIDED BY A GRANT FROM THE CITY OF TORONTO AND\n  THE LEGISLATURE OF ONTARIO.\n\n\n\n\nCHAPTER II.\n\nCEREMONY OF UNVEILING THE STATUE, 24TH MAY, 1889.\n\n\nThe ceremony of unveiling of the Statue is thus described by _The\nGlobe_ of May the 25th (abridged):--\n\nThe number of truly great men is not large in any country.\n\nOntario is not old yet in its physical and intellectual development,\nand yet it is with pride her people recall the memory of a few great\nmen who are now with the overwhelming majority. Among the greatest\nof Canadian public men was Rev. Dr. Egerton Ryerson, the founder of\nthe Ontario Public School system of education. Posterity recognizes\nthis, and posterity seeks to perpetuate his memory in that loving\nmanner which bespeaks gratitude, thankfulness and patriotism. The\ngeneration that now is speaks affectionately and reverently of him,\nwho, by sheer force of character, founded a system of education\nwhich places the child of the poor man on an equal equality with\nthat of the rich, and who so admirably developed his system that\nevery office in the State is open through a complete system of\nelementary and secondary education to all classes in the Province.\nBut this generation has done more. It erected a monument to the\ngreat man, so that generations yet unborn may not be unmindful of\nthe heritage which shall be theirs, as the result of the untiring\nzeal and ability displayed by the Chief Superintendent of Education\nin Ontario for the moral and intellectual advancement of his country.\n\nThe unveiling of this monument, fittingly erected in a commanding\nposition of the Normal School grounds, which were the scene of\nthe labors of the grand teacher, took place yesterday afternoon\nbefore a large concourse of people. There were there statesmen\nand politicians, presidents of universities and eminent divines,\nmen learned in the law and merchant princes, manufacturers and\nagriculturists, teachers and pupils--all being assembled to do\nhonor to the name of him whose monument was unveiled and whose\nvirtues were extolled. The gathering was truly historical and\nunique in its character--there being seen representatives of the\nold class of teachers who presided over the school, houses of the\ncountry when there was no system of education in Ontario, and who,\ntherefore, could the more appreciate the revolution wrought by the\nmaster mind of Dr. Ryerson, when he undertook to mould into shape\nthe heterogeneous elements of public instruction over forty years\nago. Then, again, it is seldom in the history of a nation that all\nclasses, creeds and colors could be got together to do honor to the\nmemory of one man, and seldom could there be seen such an array of\nintellectual leaders in all the walks of life as held seats on the\nplatform when Her Majesty's representative unveiled the form of\nhim whose memory is sought by it to be perpetuated. The sky itself\nseemed to favor the auspicious occasion. The weather could not have\nbeen finer if it had been designed to gladden and rejoice the hearts\nof those who were present, and thereby to assist in making the\nproceedings pass off as pleasantly as possible.\n\nA temporary platform was erected nearly in front of the monument\ncommanding an admirable view of it, while seats were placed on both\nsides of the centre of attraction.\n\n(Among those present not mentioned in the programme were:\nEx-Governor Aikens, of Manitoba, Sir Daniel Wilson, Hon. Oliver\nMowat, Rev. Dr. Scadding, Rev. Dr. Rose, Judge McDougall, Rev. Leroy\nHooker, Thomas Hodgins, Q.C., Mr. F. E. Hodgins, Rev. Dr. Parker,\nRev. Dr. D. G. Sutherland, Mr. Wm. Houston, Lt.-Col. Allan, Rev.\nJohn Hunt, Mr. D. Rose, Dr. H. H. Wright, Professor Ashley, Mr.\nJ. J. Withrow, O. A. Howland, Mr. A. Marling, James Beatty, Q.C.,\nRev. Dr. Thomas, Mr. J. E. Bryant, Mr. F. B. Hodgins, and Members\nof the City Corporation, etc. A very large number of ladies were\nalso present, and many unenumerated old friends of the venerable\nex-Chief.)\n\nThe proceedings were opened by Rev. John Burton giving out Psalm\n100, which was sung by the audience. Rev. John Potts, D.D., read a\nportion of Scripture, and the dedicatory prayer was offered by Rev.\nG. M. Milligan.\n\n\nTHE HON. G. W. ROSS' ADDRESS.\n\nHon. G. W. Ross, Minister of Education, spoke as follows:--\n\nWe are assembled to-day to do honor to the founder of the school\nsystem of the Province of Ontario. On the 18th of October, 1844, the\nRev. Egerton Ryerson received a commission from Sir Charles Metcalfe\nas Superintendent of what was then called the Common Schools of\nUpper Canada. At the time he entered upon his duties there were in\nexistence 2,885 Common Schools, with a registered attendance of\n96,756 children of school age. The entire revenue from all sources\nfor school purposes amounted to $340,000. When he retired in 1876\nthere were 5,092 schools, with a registered attendance of 489,664\npupils, and a revenue of $3,373,035. Besides the Common Schools in\nexistence there were 25 Grammar Schools attended by 958 pupils, and\nmaintained at an expense of $16,320 annually. At the close of his\nlong career there were 104 High Schools attended by 8,541 pupils,\nand maintained at an annual expense of $304,948. The accommodation\nfor the pupils attending the Common Schools was supplied by 2,887\nschool houses, of which 213 were brick or stone, 1,008 frame and\n1,666 log. The teachers numbered 3,086, and were possessed of such\nvaried qualifications as might be expected when I tell you that\nthey obtained their certificates in most cases from the Boards of\nTrustees which employed them. When he surrendered his commission\nthere were 4,926 school-houses, of which 1,931 were brick or stone,\n2,253 frame, and only 742 log, all in charge of a staff of highly\neducated and accomplished teachers, numbering 6,185.\n\nThe school law in existence at the time of Dr. Ryerson's appointment\nto office consisted of 71 sections, and was as crude in many\nrespects as the education which was obtained under it. There were\npractically no authorized text-books, no Boards of Examiners, no\nInspectors, no Department of Education. It was an era of primitive\nsimplicity, but an era, nevertheless, the possibilities of which\nno man could estimate, the development of which no man could\nforesee. \"The deep surge of nations yet to be\" had struck our\nshores. Thousands of sturdy pioneers were at work hewing down our\nforests and wrestling with such social and political problems as\nare incident to a primitive order of things. The materials out of\nwhich to organize society on a higher plane were abundant though\nundeveloped. It was a great opportunity for a man possessed of a\ngenius for organization. In the appointment of Dr. Ryerson the\nopportunity and the man met face to face, and the splendid system\nof education which we to-day enjoy is the best proof that the man\nwas as great, if not greater, than the opportunity. But, while the\nopportunity was a great one, it must not be forgotten that the\ndifficulties to be overcome, to a less vigorous and courageous man,\nwould have been overwhelming.\n\nThe executive machinery for administering the affairs of several\nthousand distinct corporations, with all the complex details\nnecessarily connected with electing trustees, collecting rates,\nappointing teachers, framing a curriculum of studies, regulating\nthe discipline of pupils and supplying text-books, had all to be\nre-cast, if not invented, and put into operation. Cabinet Ministers,\nMembers of Parliament and Municipal Councils had to be indoctrinated\nwith the new education. The press had to be directed, and the\nwhole people educated to receive with favor a school system which\nignored well established theories and deeply founded prejudices.\nEven popular indignation had to be set at defiance, and amid\nmisrepresentation and calumny the master builder had often to do his\nwork and to await the verdict of posterity for the vindication of\nhis wisdom and foresight.\n\nIt is well known, when Dr. Ryerson first proposed to make all the\nCommon Schools of Upper Canada free alike to rich and poor, to\ncitizen and alien, that he was charged with encroaching upon the\nrights of the subject, that he was charged with appropriating the\nmoney of the taxpayer who perhaps had no children to be educated\nfor the benefit of the thriftless and pauperised classes of the\ncommunity. What was his answer? It was this:--\"The education of the\npeople irrespective of rank or race or creed is a better investment\neven for the taxpayer than houses or lands, because it guarantees\nthe safe possession of all his goods--it does even more--it\nguarantees his personal liberty and therefore the taxpayer must be\nmade to pay for the common safety of the people.\"\n\nWhen he asked authority for trustees to erect school houses\nwherever, in their opinion, the public interests required them, he\nwas told such a law would be arbitrary and harsh, that it would\nplace too much power in the hands of a few men. His answer was:\n\"School houses are cheaper than gaols, teachers are cheaper than\npolice officers, the taxpayer must be made to pay for the common\nmorality of the people.\" When he said: \"Teachers must be educated\nand trained for their work, the success of thousands of children\ndepend upon skilful handling and discipline in the school room, we\nmust have Grammar Schools and Normal School and Township libraries\nand Boards of Examiners,\" he was told that the country could not\nafford such luxuries, that he must wait till the people were\nricher. His answer was: \"Efficiency is the highest economy. If the\nspringtime of life is wasted, life's greatest opportunities are\nwasted. The taxpayer must be made to pay for the common intelligence\nof the people.\" As a result of all this courage--may I not call it\nheroism--in the defence of sound principles of education he placed\nhis native Province in the van of all the States of America and all\nthe Colonies of the British Empire. Well may we to-day assemble to\ndo honor to his memory. Not only Ontario, but Canada, owes much\nto his breadth of mind, his sagacity and his tremendous force of\ncharacter.\n\nFor thirty-two years his active brain and busy pen were devoted to\nthe work of propagating sounder views on popular education. For\nthirty-two years he labored to establish the democracy of mind--the\ncommon citizenship of every child attending a Public School. With\na patriotism which no man ever questioned, with talents which no\nman could fail to appreciate, with a tenacity of purpose which no\ndifficulty could daunt, he devoted his life to one purpose, the\nestablishment of a school system which would fully meet the wants\nof a free, strong and progressive people. (Applause). It is said\nof Augustus that he found Rome brick and left it marble. It may be\nsaid of Dr. Ryerson that he found our school system without any\ndefinite organisation, he left it highly organized. He found it weak\nin influence and poor in circumstances, he left it endowed with\nhouses and lands and millions of treasure. He found it tolerated as\ntraditionally respectable, he left it enthroned in the affections of\na free people.\n\nWell may we honor his greatness, for we share in all it has\nproduced. Well may we search our quarries for a fit emblem of the\ndurability of his work, on which to carve his name, that generations\nyet unborn may recall the record of his life and be stirred to\nemulate his example. And yet when we have done all this, when we\nhave committed his memory to the keeping of the bronze and granite\nnow before us, I believe the judgment of those who know his work\nwill be that all the monuments which mortal hands can erect, and\nall the eulogies which affection or admiration can prompt his\ncontemporaries to utter, will be ephemeral and perishable compared\nwith the educational edifice which his own hands builded or the\nintellectual life which of his own genius he imparted to his fellow\ncountrymen.\n\n\nTHE STATUE UNVEILED BY SIR ALEXANDER CAMPBELL.\n\nAt the close of his address Mr. Ross invited the Governor to unveil\nthe statue. Before doing so the Governor, turning to the audience,\nsaid in feeling terms:--\n\n\"Dr. Ryerson was known throughout the length and breadth of this\nProvince. No Representative of Her Majesty has had ever as pleading\na duty given to him to discharge as that which falls to my lot in\nunveiling the monument of that great man.\"\n\nSir Alexander, accompanied by the Minister and Deputy Minister of\nEducation, proceeded then to the statue, and the work of unveiling\nwas only the question of a few moments. As soon as the British\nCanadian flag, which covered the massive form of the statue, was\nraised, the audience raised a cheer which is rarely heard within the\nNormal School grounds, It was the reflex of the inner gratitude of\nthe sharers in a great heritage.\n\nThe sculptor, Mr. McCarthy, who did his work well, was then\nintroduced to the Lieutenant-Governor by the Minister of Education.\n\nThe statue having been exposed to full view the song \"Hurrah for\nCanada\" was sung by city school children, led by Mr. Perrin, music\nteacher, city schools. The children acquitted themselves admirably.\n\n\nEDUCATIONAL RETROSPECT BY THE DEPUTY MINISTER OF EDUCATION.\n\nIt was fitting that the next speaker should be a gentleman so\nlong and so closely connected with Dr. Ryerson in moulding the\neducational institutions of the Province. The Chairman therefore\nintroduced Dr. Hodgins, Deputy Minister of Education, who read a\nmasterly and comprehensive historical paper on Education in Ontario.\nDr. Hodgins traced the growth of education in the Province from\n1841, dealing minutely and with a thorough knowledge of details\nwith the difficulties encountered by Dr. Ryerson in gaining public\nendorsation for his scheme of popular education, and the phases\nthrough which the system passed until at the Centennial Exhibition,\nheld in Philadelphia in 1876, the Chief Superintendent of Education\nwas gratified by the Commissioners making the following award:--\n\nFor a quite complete and admirably arranged exhibition, illustrating\nthe Ontario system of education and its excellent results. Also\nfor the efficiency of an administration, which has gained for the\nOntario department a most honorable distinction among Government\neducational agencies.\n\nDr. Hodgins concluded as follows:--Having been intimately concerned\nin all of the events and educational matters to which I have\nreferred, it may not be out of place for me to add a few words of a\npersonal character in conclusion. At the end of this year I shall\nhave completed my more than 45 years' service, as chief of the staff\nof the Education Department of Ontario. For over 40 years I enjoyed\nthe personal friendship of the distinguished man whose memory we\nhonor here to-day--32 years of which were passed in active and\npleasant service under him. How can I, therefore, regard without\nemotion the events of to-day? They bring vividly to my recollection\nmany memorable incidents and interesting events of our educational\npast known only to myself. They also deeply impress me with the\nfleeting and transitory nature of all things human. The chief and\nsixteen councillors, appointed and elected to assist him, have all\npassed away. His great work remains, however, and his invaluable\nservices to the country we all gratefully recall to-day, while\nhis native land lovingly acknowledges those services in erecting\nthis noble monument to his memory. Truly indeed and faithfully\ndid Egerton Ryerson make good his promise to the people of this\nProvince, when he solemnly pledged himself, on accepting office in\n1844--\n\nTo provide for my native country a system of education, and\nfacilities for intellectual improvement, not second to those of any\ncountry in the world.\n\nGod grant that the seed sown and the foundations thus laid, with\nsuch anxious toil and care--and yet in faith--may prove to be one of\nour richest heritages, so that in the future, wisdom and knowledge,\nin the highest and truest sense, may be the stability of our times!\n\n\nTHE TEACHERS' REPRESENTATIVE.\n\nThe audience was then introduced to Mr. R. McQueen, President of the\nOntario Teachers' Association, who eulogised Dr. Ryerson for his\nmarked individuality, tenacity of purpose, and the grand results\nhe achieved through his influence for popular education in his\nnative country. He deserved a monument from a thankful posterity\nbecause his whole aim in life was to ameliorate the condition of his\nfellow-countrymen. He was there in behalf of the teachers of Ontario\nto express their joy at the tribute paid to the memory of a great\nteacher and founder of a system of education.\n\n\nTHE ACTING-MAYOR RESPONDS.\n\nIn the absence of Mayor Clarke, Ald. McMillan was called on to reply\nin behalf of the citizens of Toronto. The Acting-Mayor said that\nCanada owes Dr. Ryerson a deep debt of gratitude. His life was grand\nand versatile; he was a teacher and a Christian; a combination of\nthe scholar and gentleman; a leader of men. He impressed his genius\non the educational institutions of the Province, he was loyal to\nthe country of his birth, and he abhorred falsehood and oppression.\nTherefore it was eminently fitting that his greatness and worth\nshould be commemorated by a public tribute.\n\nThe school children at this stage sang the well-known Canadian song:\n\"The Maple Leaf for Ever.\"\n\n\nTHE UNIVERSITIES OF ONTARIO.\n\nThe Universities of Ontario owe much to the painstaking care of the\nlate Dr. Ryerson over the development of Public and High School\neducation. The efficiency of the University and its possible\ninfluence depend on a sound and thorough system of elementary\neducation. The High School depends on the Public School and the\nUniversity on the High School. Our Universities have made tremendous\nstrides during the last ten years, and without a doubt the cause\nof this has been due to the fact of the elementary schools putting\nforth the fruit of the good seed sowed thirty years ago. The\nUniversities therefore were not unmindful of what they owe to\nthe genius of Dr. Ryerson at yesterday's re-union. These five\ninstitutions were well and ably represented on the platform. The\naddresses of these representatives were one series of eulogies on\nthe life and labors of him, whose memory they met to do honor to.\n\nSenator Macdonald, after praising the Ontario system of education,\nturned to the young people of the audience and said: \"You are\nforcibly reminded to-day that Canada will cherish the memory of all\nthose of her sons who will work patriotically and nobly for the good\nof their country.\"\n\nRev. Dr. Burwash spoke of Dr. Ryerson as a public educator, but he\nwould be great, the speaker said, in any other profession. Ontario,\nhowever, remembers him as an educator, statesman, philanthropist and\nChristian teacher in the highest sense of the word.\n\nChancellor Fleming honored the name of Dr. Ryerson for having laid\nsuch a broad and national system of education as enables Ontario to\nrank among the first of enlightened nations.\n\nRev. Professor Clark thought that the spirit of Dr. Ryerson was to\nprovide such a system of education as would make men of earnestness,\ncharacter and patriotic ardor. The University with which he was\nidentified honored the name of Dr. Ryerson, and he was there to add\nhis tribute to the worth of so great a Canadian.\n\nProfessor Rand, eulogised also the elementary and secondary system\nof education in Ontario, declaring that its founder richly deserved\nto be commemorated by a public monument.[6]\n\n  [6] These addresses are given in full, commencing on page 17.\n\nThe audience then sang the national anthem. Bishop Sweatman\npronounced the benediction, and the statue of the great educationist\nwas left to posterity to admire and to preserve intact and inviolate.\n\nThe report of _The Empire_ necessarily traversed the same ground.\nI can therefore only give the salient points in addition to those\nreferred to by the _Globe_, it said:--\n\nThe great educational lights of the Province were present in front\nof the Education Department building yesterday afternoon, when the\nstatue of the Rev. Dr. Ryerson was unveiled. The day was appropriate\nfor doing honor to the memory of a man who had so ably served his\nQueen and country. A large crowd of citizens witnessed the unveiling\nand listened to the addresses that were delivered.\n\nThe following relatives of Dr. Ryerson were present: Mr. Charles\nEgerton Ryerson, (his only son), Mrs. C. E. Ryerson and their\ntwo sons, (Egerton and Stanley), Mrs. Edward Harris, (his only\ndaughter), Dr. G. S. Ryerson (his nephew) was absent with the\nGrenadiers at Berlin, Mrs. G. S. Ryerson and son were present; also,\nMrs. Hardy (his niece) and her daughter, Miss Ethel Hardy, Dr. John\nBeatty, of Cobourg, and Mr. James R. Armstrong, (brothers-in-law),\nMrs. J. R. Armstrong, Mrs. George Duggan, (sister-in-law), His Honor\nJudge McDougall, (a connection by marriage).\"\n\nThe main points of _The Mail_ report were as follows:--\n\nIt would perhaps be too much to say that, while the gay and\nthoughtless were seeking amusements in other parts of the city,\nit was only the wise who repaired to the grounds of the Education\nDepartment to take part in the unveiling of the Statue of the late\nEgerton Ryerson; yet it cannot but be admitted that those who\nassembled to witness and assist in this ceremony were men and women\nworthy to have the privilege of publicly honoring the memory of\nCanada's greatest educationist. Among those present were men who\nhave attained to eminence in every department of public life, and\nit was but right that they should pay the tribute they did to the\nmemory of him who was the founder and for many years the head of\nthe greatest of all departments. There were men present who for\nyears were associated with Dr. Ryerson in his great work; men whose\ncharacters were to a great extent moulded by his example, and men\nwhose ambitions have been wakened and whose purposes have been\ninspired by the contemplation of his achievements.\n\nEarly in the afternoon the crowd begin to gather around the statue,\nthe front part of which was veiled by a large British flag, the\nfolds of which hung almost to the foot of the pedestal. In front\nof the statue, under the shade of a couple of the maples that help\nto make the grounds of the Education Department and Normal School\nso attractive, a platform had been erected for the use of those\nwho were to take part in the ceremony, and around it were placed a\nnumber of seats.\n\nAt the close of his address (given on pages 9-11), Hon. Mr. Ross\ninvited Sir Alexander Campbell to unveil the statue. Before\nproceeding with the ceremony Sir Alexander advanced to the front of\nthe platform and briefly expressed himself as feeling highly honored\nby being called upon to perform such a task as the one that on\nthis occasion had devolved upon him. He thought no pleasanter duty\ncould fall to the lot of a Lieutenant-Governor of any province than\nthat of assisting in honoring one of the province's noblest men.\nHe then stepped down, and taking hold of the cords that kept the\nflag in place, drew them aside and the drapery fell to the ground.\nAs the sunlight flashed on the exquisitely chiselled features, and\nthe form so well known to many of those present stood out for the\nfirst time as it will stand, it is hoped, for many years to come, a\nprolonged cheer burst from those who had up to this moment watched\nSir Alexander's movements with almost breathless interest. After a\npause of a few moments, during which the naturalness and finish of\nthe statue was freely commented on, Hon. Mr. Ross called forward Mr.\nHamilton McCarthy, the sculptor, and amid great applause introduced\nhim to Sir Alexander, who spoke in flattering terms of the pleasure\nhe felt in meeting a man who had shown himself capable of producing\nso excellent a work of art.\n\nThe bronze statue, nine feet six inches in height, represents the\nlate Dr. Ryerson in the attitude of addressing an audience in the\ncause of education. The head is turned a little to the right, with\nthe lips slightly parted, and with the massive brow and flowing\nlocks, give a correct and forcible expression, in harmony with the\naction of the advanced arm and firm position of the right leg. The\nproportions of the figure are well kept through the ample folds of\nthe Doctor's gown, which in their various lines, lend richness and\ninterest to the work, and take away the stiffness of the modern\ncostume. The left hand is raised nearly to the breast, and in it is\ngrasped a book. A little to the left and rear of the figure stands\na short pedestal bearing three books, carelessly laid one upon\nanother; and on one of the panels of the pedestal is the arms of the\nDepartment of Education under Dr. Ryerson's _regime_. Dignity of\nbearing, repose and action, and distinct force of character, eminent\nqualities in the personality of the late doctor, mark the expression\nof the figure; and it is evident that no pains have been spared by\nthe artist in the modelling of the details.[7] Mr. Hamilton McCarthy\nhas also been very successful in the design of the pedestal, which\nhas excited general admiration. It is 10 ft. 6 in. in height, and\nis of New Brunswick granite. The conception is unique in character.\nPilasters at the four angles terminate in buttresses to the ground,\nand support above beautifully designed capitols with dentils on the\nface. The pyramidal form of the whole work gives it an effect of\nrising out of the ground. The finely polished panels of the die, in\neach of which a classic shield is outlined, contain the inscriptions.\n\n  [7] In a note from Sir John Macdonald, he said:--\n\n       \"Many thanks for your note, and for the photographic model of\n       our dear old friend, Dr. Ryerson. The apparent frown on the brow\n       is perhaps too pronounced, [the expression was modified after\n       receipt of this note]. The pose seems to me very good.\"\n\n  Rev. J. K. Smith, D.D., of Galt, in a note to me also said:--\n\n       \"Please except my sincere thanks for the excellent photo of the\n       lamented Dr. Ryerson which is an admirable likeness. It is I\n       suppose, as nearly perfect in every respect as a statue could\n       be. I am persuaded that the statue will be a splendid one, and\n       I rejoice that his great name is thus to be handed down to\n       successive generations of our Canadian youth as a sacred memory\n       and a powerful stimulus.\"\n\nMr. McCarthy can be congratulated upon the success of his work, and\nthe province can be congratulated upon the possession of so noble an\naddition to the few works of art now in the country. Mr. Gullett,\nthe contractor for the erection of the pedestal also performed\nhis duties carefully and faithfully, as not the slightest hitch\noccurred, and no damage was sustained by the granite.\n\n\nCOMMENTS OF THE PRESS ON THE UNVEILING OF THE STATUE.\n\nThe _Evangelical Churchman_ of May the 9th, anticipating the\nunveiling of Dr. Ryerson's statue, said:--\n\n     \"On the 24th of this month, the Queen's Birthday, Ontario will\n     do honor to one of her most distinguished sons. On that day\n     will be unveiled the statue to the memory of Egerton Ryerson,\n     the founder of the school system of his native province. The\n     ceremony will be unique in many ways, not the least interesting\n     fact in connection therewith being that the statue of Dr.\n     Ryerson will be the first one erected by the Province of Ontario\n     to one of its own sons. Dr Ryerson was a thorough Canadian\n     and was born in Ontario. Thus this signal honor to his memory\n     acquires additional lustre, and does much to redeem Ontario from\n     the reproach an often uttered that a prophet is not without\n     honor save in his own country. It reveals, indeed, another fact\n     which, in a new country, is not without a peculiar significance.\n     It is this, that national life is commencing in earnest, and\n     that national characteristics are developing themselves. A\n     country which can step aside, as it were, in the rush and hurry\n     of existence to do honor to one of its sons, is not without\n     aspirations after a national existence, is not wholly given up\n     to considerations of material interest, and possesses within\n     it something that is full of promise of permanence and true\n     greatness.\"\n\nThe Hamilton _Times_ of the 25th of May, under the beading of \"The\nMemory of a Great Canadian,\" said:--\n\n     \"The unveiling of the Ryerson statue in the Normal School\n     grounds, at Toronto, yesterday, was the occasion of recalling\n     the achievements of the late Dr. Ryerson in connection with\n     Ontario's educational system. From 1844 until 1876 Rev.\n     Egerton Ryerson was Chief Superintendent of Education in\n     this Province.... But Dr. Ryerson's services to Canada did\n     not begin in 1844. He was a great man before he touched the\n     educational System. He was born in the County of Norfolk in\n     1803, and when he was about 20 years of age he was studying\n     in Hamilton in a little house on Jackson Street, not far from\n     the place where the new Y.M.C.A. building is in the course of\n     construction.... In 1826 Archdeacon Strachan preached a sermon\n     on the death of Bishop Mountain. The Methodists at that time\n     were the most numerous religious body in upper Canada but Dr.\n     Strachan set forth the claim of the Church of England to the\n     Clergy Reserves.... Mr. Ryerson was junior preacher under the\n     late Rev. James Richardson, who had his arm shot off while in\n     naval service near Sackett's Harbour during the war.... When\n     Dr. Strachan's sermon was published, it was agreed that Mr.\n     Richardson and Mr. Ryerson should each write a reply to it.\n     They separated, each going to a different part of their large\n     circuit, and when they met a few weeks later young Ryerson had\n     prepared his paper, but Mr. Richardson had nothing ready. It was\n     read before the other preachers and published. The battle had\n     now begun, and it did not end until the Clergy Reserves were\n     secularised in 1854. During nearly all that time Mr. Ryerson\n     was a leading character in Canadian public life. He wrote, he\n     spoke, he worked, appearing before Parliamentary committees,\n     interviewing the statesmen of Great Britain and occasionally\n     taking his stand upon the hustings.... Dr. Ryerson was born in\n     1803 and became Superintendent of Education in 1844. He was\n     only 42 years of age at the time of his appointment, yet he had\n     performed a greater share of work, and had attained a greater\n     degree of prominence in those forty-two years than most public\n     men can boast of as the achievements of a lifetime. How many\n     men in this latter end of the century get into the thick of the\n     fight and make their influence felt while under 40 years of\n     age? The point we wish to impress is this: Had Egerton Ryerson\n     died in 1844, instead of becoming Superintendent of Education\n     and living until 1882, his history would still have been worth\n     writing, and he would have deserved a monument For the services\n     he performed for his native Province. His long connection with\n     educational affairs to a great extent blotted out the memory\n     of his earlier work and struggles in another connection. He\n     had much to do with founding Victoria College and getting that\n     institution fairly established.... The impression remains with\n     us to this day that if Dr. Ryerson had been a lawyer he would\n     have made all other Canadian lawyers look small; if he had gone\n     into politics he would have been perpetual Premier: in short,\n     he was the ablest native Canadian who has so far helped to make\n     history.\n\nThe _Christian Guardian_ of the 29th May, said:--\n\n     The unveiling of the statue of the late D. Egerton Ryerson,\n     last Friday, in the Normal School grounds in this city, recalls\n     the memory of a worthy and honored Canadian, widely known as\n     a successful journalist, a gifted and learned divine, and\n     an eminent educationist. It will hardly be questioned that\n     the principle of perpetuating the memory of benefactors of a\n     country is a laudable one, or that the individual in this case\n     was worthy of this honor. No one who has travelled in Britain\n     or other European countries, has failed to have his attention\n     arrested by statues, or other memorials, of eminent men whom the\n     county delighted to honor. It is well adapted to inspire the\n     young with high purpose to note that however partisan strife may\n     obscure the patriotic services of public men during active life,\n     when the work of life is over, as a general rule, men of all\n     parties cheerfully recognize the value of the service rendered\n     by those who have faithfully labored for the public good. Owing\n     to the intensity of political feeling in Canada, there is a\n     strong tendency to underestimate the work of our statesmen and\n     politicians, until they have gone where human praise or blame\n     cannot affect them.\n\n     Though Dr. Ryerson passed through many fierce controversies, and\n     at times came into conflict with hostile opponents, to-day men\n     of all creeds and parties are ready to give him his due meed of\n     praise as one of the greatest of Canada's sons, who achieved a\n     work in organizing and building up a system of public education\n     that shall tell powerfully for good through all coming time.\n     He founded no cities; he led no armies to victory; he had no\n     special influence on the material prosperity of the land; but in\n     organizing a system of public schools, which placed the elements\n     of a sound education within the reach of every boy and girl in\n     this Province, he has exercised an undying influence over the\n     future intellectual life of the country, that shall largely\n     determine its place in the scale of civilization.\n\n     It is not only since his death that the strife of tongues\n     has ceased, and the value of his work has been generally\n     acknowledged. For several years before his death the echoes of\n     old battles had become silent; old strifes were healed; and he\n     lived in a peaceful Beulah land awaiting the Master's call to\n     cross the dark river. In the beginning of 1879 at the request\n     of the editor of the _Guardian_, he wrote an article for the\n     Jubilee number of this paper, of which he was the first editor.\n     After giving an interesting account of the origin and growth\n     of the paper, he concluded by saying; \"May the success of the\n     past be as a dim dawn to the success of the future! Such is\n     the prayer and hope of the first editor of the _Guardian_--now\n     retired from all office in Church and State, near the\n     fifty-fifth year of his ministry and the seventy-seventh year\n     of his age--_looking_ for a better country and _waiting_ for a\n     heavenly home.\"\n\nThe _Presbyterian Review_ of the 30th May, said:--\n\n     The various speakers dwelt upon the immense service which the\n     late Dr. Ryerson rendered to the country in laying broad and\n     deep the foundations of our educational system, and testified\n     their satisfaction that gratitude and veneration had found\n     expression in the noble work of art before them, which would\n     perpetuate his name to many generations of students and\n     scholars.... Dr. Hodgins and the other gentlemen associated\n     with him on the Ryerson Statue Committee are to be heartily\n     congratulated on the result of their well-directed efforts and\n     well-sustained efforts to assist in perpetuating the memory\n     of a native-born Canadian who, notwithstanding some errors\n     of judgment, proved himself worthy to be held in grateful\n     remembrance by his countrymen.\n\n_The Week_ of the 31st May, said:--\n\n     That was a grand purpose to which Rev. Egerton Ryerson pledged\n     himself on accepting office as the first Superintendent of\n     Education for Ontario in 1844, \"To provide for my native\n     country a system of education, and facilities for intellectual\n     improvement, not second to those of any country in the world.\"\n     The form and loftiness of the promise marked the courage,\n     individuality and conscious strength of the man who made it.\n     The statue in the Toronto Normal School grounds, which was\n     unveiled with appropriate ceremonies on the 24th inst., will\n     henceforth stand as the testimony of the people of Ontario,\n     especially of its teachers and others interested in educational\n     work, to the faithfulness and ability with which the pledge was\n     redeemed through thirty-two years of indefatigable toil and\n     struggle. The artistically wrought monument in bronze will also\n     serve as a fitting reminder to all who visit the Educational\n     Department that the people of Ontario do not mean to let those\n     who faithfully served their country in its earlier days be\n     forgotten. A monument \"more enduring than bronze\" stands out to\n     view wherever a free public school is efficiently doing its work\n     in training the young of both sexes and of all classes to become\n     intelligent and patriotic citizens of this growing commonwealth.\n     Whether it be literally true or not that Dr. Ryerson \"placed\n     his native Province in the van of all the States of America and\n     all the colonies of the British Empire,\" as the Minister of\n     Education avouches, his plan was certainly comprehensive and\n     statesmanlike, and was followed out with a courage, perseverance\n     and success, for which the Province must ever remain his debtor.\n\nThe Toronto correspondent of the _Montreal Witness_, under date of\n31st of May, says:--\n\n     One of the noblest public tributes ever paid to the memory of\n     any man in Canada was paid the other day to the memory of the\n     late Rev. Egerton Ryerson. From the time of his death, early\n     in 1882, till now, the work of collecting subscriptions for\n     the erection of a statue has gone steadily on. The amounts\n     contributed were individually small, but the contributors were\n     numerous, and now in front of the Departmental Buildings, in\n     St. James' Square, stands a memorial of him which will fairly\n     convey to future generations some idea of what the man himself\n     was in personal appearance. The massiveness and rugged strength\n     are there, and there were, after all, the most marked traits of\n     Dr. Ryerson's personality, though he was by no means lacking\n     in sympathy and intellectual ability.... The addresses were\n     admirable alike for brevity and good taste, and nothing occurred\n     to mar the success of the ceremony.\n\nThe _Educational Journal_ of June 1st, said:--\n\n     The statue of the late Dr. Ryerson, which has been so long in\n     course of preparation, has been set up on the Normal School\n     grounds, and was unveiled, with appropriate ceremonies, on the\n     24th ult.\n\n     The ceremony of unveiling was performed by Sir Alexander\n     Campbell, the Lieutenant-Governor, who said that he thought no\n     pleasanter duty could fall to the lot of any Lieutenant-Governor\n     than that of assisting in honoring one of the Province's noblest\n     men.\n\n     The status is of bronze, nine feet six inches in height, and\n     stands upon a pedestal of New Brunswick granite, ten feet six\n     inches high. It represents Dr. Ryerson in the attitude of\n     addressing an audience in the cause of education. The head is\n     turned a little to the right, with the lips slightly parted, and\n     with the massive brow and flowing locks, gives a correct and\n     forcible expression, in harmony with the action of the advanced\n     arm and firm position of the right leg. The proportions of\n     the figure are very well kept through the ample folds of the\n     doctor's gown, which in their various lines, lend richness and\n     interest to the work, and take away the stiffness of the modern\n     costume. The left hand is raised nearly to the breast, and in it\n     is grasped a book. A little to the left and rear of the figure\n     stands a short pedestal bearing three books, carelessly laid one\n     upon another; and on one of the panels of the pedestal is the\n     arms of the Department of Education. Dignity of bearing, repose\n     and action, and distinct force of character, eminent qualities\n     in the personality of the late doctor, mark the expression of\n     the figure; and it is evident that no pains have been spared\n     by the artist, Mr. Hamilton McCarthy, in the modelling of the\n     details of both statue and pedestal. The statue stands in a\n     commanding position in the Normal School grounds. It will add\n     a new object of interest to the many attractions which these\n     grounds present to teachers and others visiting the Department.\n\nThe _Irish Canadian_ of the 6th of June, under the heading of \"A\nGraceful Tribute,\" said:--\n\n     On the 24th of May (the Queen's Birthday), was unveiled the\n     statue erected in the Normal School grounds to the memory of the\n     Rev. Egerton Ryerson, the founder of the common school system of\n     education in Ontario, and its Superintendent from its inception\n     in 1844 till 1876, when he retired in the fullness of years,\n     and after his labors had been crowned with signal success. The\n     Catholics of this Province, in the matter of education, have\n     nothing for which they should be thankful to the distinguished\n     divine.... For all that, Dr. Ryerson was a man of great and good\n     parts; and, from a Common School point of view, he has left a\n     noble heritage in a system of education that will bear favorable\n     comparison with the best of any land.\n\n     It was the occasion of the unveiling of his statue that\n     his co-laborer in the Education Department--Dr. J. George\n     Hodgins--paid the memory of Dr. Ryerson a graceful tribute.\n     Who so capable for so delicate a task as he who had been Dr.\n     Ryerson's right-hand man, his able support, during his long and\n     varied career in the Education office? And happily has the story\n     of the ups and downs of the Common School system been told by\n     the learned Deputy Minister, to whose ripe judgment, in no small\n     degree, was due the system's unmeasured success. The part that\n     Dr. Hodgins played, however, is kept in the background; and we\n     see only what Dr. Ryerson done during his lengthened incumbency,\n     and the difficulties with which he had to contend in maturing\n     his plans and bringing them as nearly as possible to his own\n     ideal of perfection.\n\n     Dr. Hodgins' retrospect goes back to the period of the U. E.\n     Loyalists, and thence downward to 1876. It leads us by degrees\n     from the primitive system in vogue prior to the grammar schools\n     (in one of which the late venerable Bishop Strachan taught\n     as master), through a series of changes aiming at higher\n     education, till we arrive at the year in which the foundations\n     of the present system were laid. The corner-stone having been\n     placed, the superstructure rose in fair proportions; and the\n     edifice having been completed, to furnish it with all the\n     adjuncts necessary to the best educational training was the\n     Superintendent's constant care. How Dr. Ryerson finally overcame\n     every obstacle to his darling object is told with tender\n     affection by Dr. Hodgins, who, in laying a chaplet on the grave\n     of his dead chief, does honor not only to the memory of a good\n     man, but also to his own generous instincts.\n\nThe _Canada Educational Monthly_ for June-July, said:--\n\n     The Rev. Dr. Ryerson has long been widely known as a gifted and\n     learned divine, as well as a successful journalist, who took a\n     prominent part in the religious and moral development of our\n     country in its early days, ... but the fitting memorial which\n     was unveiled on Her Majesty's seventieth birthday, is erected to\n     him chiefly as a worthy Canadian and an eminent educationist....\n\n     The life work of this able man has now passed into other\n     hands; in itself it forms a whole superstructure, and if the\n     enlightened principles which he laid down and acted upon are\n     carried out in their integrity, they must exercise an undying\n     influence for good upon the intellectual life of the country,\n     upon its gradual advance in the scale of civilization and\n     refinement, and upon its moral and religious life.\n\n     The ceremony of unveiling the statue brought together many true,\n     patriotic and representative men. Some of his personal friends\n     and fellow-workers were there, and others who remembered him\n     with affection and gratitude. The Government, the city, the\n     public and secondary schools, the colleges and universities were\n     all represented, and all united in honoring the memory of the\n     founder of the Ontario school system.\n\n\n\n\nCHAPTER III.\n\nTHE ADDRESSES DELIVERED AT THE UNVEILING.[8]\n\n  [8] The Historical Paper--an abstract of which was read by Dr.\n  Hodgins, will be found on page 26.\n\n\nTHE TEACHERS' ASSOCIATION OF ONTARIO.\n\nMr. Robert McQueen, President of the Ontario Teachers' association,\nspoke as follows:--\n\n_Your Honor_, _Mr. Chairman_, _Ladies and Gentlemen_:\n\nWe are gathered here to-day to do honor to the memory of one to whom\nthis country owes a debt of deep and lasting gratitude; and should\nI fail entirely to give expression to any thought worthy of the man\nor the occasion, I feel certain that all will be overlooked, when I\nsimply utter the two words \"Egerton Ryerson.\" He was a man of marked\nindividuality of character, of energetic action, of great power of\nwill, and tenacity of purpose. His life-work bears the marks of his\nindividuality and energy, the results he achieved are evidences of\nhis power of will and his tenacity of purpose. And, in view of the\nresults which he achieved, it is a meet and a becoming thing, that,\nas a people, we should meet and do honor to his memory, as we have\ngathered here this day to do.\n\nMen, even great men, only act permanently, as they act upon the\ninstitutions of their country. The greatness that centres in the\nindividual is dependent upon and contingent on the existence of the\nindividual, and the influence exercised by him as it centres in him,\ndies with him, passes away when he leaves the scene. But the man who\nleaves his impress on the institutions of his country, exercises an\ninfluence that is permanent in its action, pervasive in its power,\nand, within the range of these institutions, is universal in its\nbeneficence. Every amelioration of these institutions is an abiding\ngood, shared in by all who come within their sphere of operation. It\nbecomes the common heritage of all.\n\nSuch was the impress made, such still is the influence exercised by\nthe individual to whose memory and life-work we have met this day to\ndo honor. His connection with, his charge over, and his influence\nupon the educational institutions of this country, reaching\nbackward nearly half a century from the present time, and lasting\nover a period of nearly thirty years, was that of a formative and\nmeliorating kind. From the time of his acceptance of office in 1844,\ndown to the period when he resigned his trust, his energies were\ndirected, his powers brought to bear on organizing, modelling and\nconsolidating the school system of this province.\n\nLet us for a few moments trace briefly the history of that system\nfor the last fifty years. The Education Bill of 1837 may be said\nto have been the first legislative attempt at organization. This\nbill provided for the annual expenditure of fifteen thousand pounds\nfor common school purposes, and that as soon as the permanently\navailable Public School Fund of this Province amounted to ten\nthousand pounds per annum, a superintendent of common schools should\nbe appointed by the Governor under the Seal of the Province, whose\nduty it should be to report annually to the Legislature on all\nmatters pertaining to the administration of the public schools.\nThe union of the Provinces brought the Act of 1841, introduced\nby the Hon. Mr. Day. This Act was simplified and improved by the\nBill of 1843, introduced by the Hon. F. Hincks. By the provisions\nof that Act the Secretary of the Province was _ex-officio_ Chief\nSuperintendent of Schools. In 1844 that office was tendered to the\nlate Dr. Ryerson, and in the autumn of that year it was accepted by\nhim on two conditions: (1st), that the administration of the school\nsystem should constitute a separate and distinct non-political\ndepartment; (2nd), that he should be permitted to act by a deputy\nfor one year and have leave of absence for that period, in order to\nenable him to visit and examine the educational systems of other\ncountries, in Europe and America, before attempting to lay the\nfoundations of a system in Upper Canada. The whole of 1845 was spent\nin these inquiries and investigations. The results were embodied\nin a report to the Legislature in 1846. Along with this report was\na Draft School Bill, which was introduced and carried through by\nthe Hon. W. Draper, and became law in June, 1846. In a few mouths\na draft bill for cities and incorporated towns was submitted and\ncarried through by the Hon. J. H. Cameron, and became law in June,\n1847; these two Acts, with modifications and improvements, suggested\nby time and experience, were incorporated in the School Bill of\n1850, which was introduced by the Hon. F. Hincks, and was the first\nAct to which the Earl of Elgin gave the Royal Assent after the\nremoval of the Seat of Government to Upper Canada.\n\nThe provision of this bill embodied the basis of our present system.\nIt introduced the principle of free schools, leaving the adoption\nof that principle to the option of each school section. Subsequent\nchanges and modifications were embodied in the Act of 1870, which\nabolished the office of township superintendent, introduced the\ncounty inspectorate, and made free schools compulsory. This Act\nmay be said to be the last touch of the \"Master Hand.\" In order to\napprehend the true nature and extent of that impress which he made,\nof that influence which he exercised and exerted, we have only to\nlook around, to look on that noble structure, whose foundations he\nwas permitted and honored to lay, and whose superstructure he was so\nlargely enabled to rear. Our school system is his true memorial.\n\nI have already said that it is a meet and a becoming thing for a\nnation to honor the memory of its public benefactors. The nation\nthat ceases to cherish and manifest a grateful remembrance of those\nwho have devoted their energies to the promotion of its welfare and\nspent their lives in its service, gives evidence of deterioration\nand decay, gives evidence that it is on the down grade of national\nexistence. To-day we have gathered here to give expression to our\ngratitude in tangible form, by the erection in this public place,\nof this costly and enduring memorial of the individual and his\nlife-work, that by its very presence it may speak to all who behold\nof the grateful sense of benefits received, cherished by the people\nof this Province. Yet there is another way in which we may manifest\nour gratitude, viz., by seeking to conserve, to consolidate, to\nameliorate, our school system, which has thus come down to us from\nthe moulding hand of the departed; and not only seeking to preserve\nwhat has thus been handed down to us, so far as the school system\nitself is concerned, but to seek to make it practically effective\nin its operation, so that all who live under its shadow, may share\nto the full of the blessings and privileges which it is fitted to\nconfer and intended to bestow. Our sense of gratitude will thus\nassume a practical and beneficent form. That the management of our\nschool system has fallen into, and now is in good and wise hands, we\nall feel confident. That its success depends largely on the energy,\nthe zeal and the faithfulness of those who are entrusted with the\neducation of the youth of this Province, our high and public school\nteachers, together with the co-operation of the people at large, of\nall classes and of all creeds, all will be prepared to admit. In one\nword: It is only by the united and concerted action of all three,\nthe executive skill, the zeal and energy of the teachers, and the\npopular sympathy and support that we can hope to secure the full\nbenefit of our school system on the one hand; and on the other, show\nourselves worthy of the heritage that has been handed down to us by\nthose who have gone before us.\n\n\nADDRESS OF THE ACTING MAYOR.\n\nAlderman McMillan, President of the City Council and Acting Mayor of\nToronto, in the absence of Mayor E. F. Clarke, M.P.P., in England,\nspoke as follows:--\n\nI regret very much that His Worship the Mayor is not with us to-day\nto represent the citizens of Toronto on this interesting occasion.\n\nIt is his loss, however, and I fear it will be your loss also, that\nit has fallen to my lot to act as his substitute and to speak on his\nbehalf and on behalf of the citizens of Toronto.\n\nKnowing my own unfitness for the task, I have hesitated; feeling my\nown inability, I have shrunk from the duty imposed on me. Deeply\nconscious as I am of the fact that it requires more eloquence than I\npossess to do justice to the occasion, or to speak in fitting terms\nof the great work of the eminent divine and educationist to whose\nmemory this statue has been erected.\n\nNo more appropriate place could have been selected for its erection\nthan here on the grounds of the department which is the creation of\nhis genius, and in front of a building which stands a monument to\nthe energy of its founder.\n\nHe will be but a poor student of the history of his country who\nhas not yet learned from its pages the deep debt of gratitude\nwhich every Canadian owes to the man whose memory is held in most\naffectionate rememberance by all classes of the community.\n\nHe will be a poor judge of mental power or moral worth who has\nnot yet learned to prize the grandeur of his life-work or the\nversatility of the attainments of the man who was not only a\ndeep thinker and a notable teacher but an earnest and a humble\nChristian--a happy combination of the scholar and the gentleman.\n\nDuring a long and eventful career, and at a crucial period in our\ncountry's history the active part he played in public affairs has\nleft on the institutions of our province the impress of his vigorous\nintellect. Prompted by pure motives, and guided by sound judgment,\nhe gave evidences of an uncommon genius, which he devoted to the\nservice of his country and the best interests of the people, and\nthus he became a leader among men.\n\nIn the school system of this province he built for himself a more\nlasting monument than the granite and bronze we now raise to his\nmemory.\n\nWith fluent speech and ready pen he has oft been the defender of\nour most sacred rights and cherished privileges. A lover of truth,\nhe abhorred falsehood. A lover of freedom, he hated oppression; and\nthe cause of truth and of freedom found in him an able and willing\nchampion. A staunch defender of British connection, he yet manfully\nbattled for equal rights and privileges to all classes of the\npeople.\n\nHe loved the land of his birth with no ordinary affection and,\nduring a long and busy life, he helped to mould her destiny and\nshape her course--guiding her feeble and often erratic steps,\nleading her into the paths of truth and righteousness which, we are\nassured, most surely exalteth the nation.\n\nIn the pages of Canadian history the \"Story of his Life\" and labors\nwill ever be instructive reading.\n\nIn the chronicles of the Methodist Church of Canada (the church of\nhis choice) his name will always stand pre-eminently conspicuous as\none of her ablest scholars and one of her most eminent divines--an\nearnest preacher and a devoted missionary. Filled with a fervent\nlove for his Lord and Master, he labored earnestly in His vineyard\nseeking souls for His hire.\n\nI am proud of the privilege I enjoy of being here on behalf of the\ncitizens of this great city--pleased to be able to bear testimony\nto the high appreciation we have of his services and the strong\naffection we bear to the memory of the Rev. Egerton Ryerson. I am\nalso pleased to have this opportunity of being allowed to tender my\nown humble tribute of respect to the memory of him who was both a\nstatesman and a scholar, a patriot and a Christian.\n\n\nREPRESENTATIVE OF THE UNIVERSITY OF TORONTO.\n\nThe Honourable Senator John Macdonald, speaking on behalf of the\nUniversity of Toronto, said:--\n\nI wish first to express my regret--a regret I have no doubt in which\nyou share--that some one better fitted than myself had not been\nselected to represent the University of Toronto upon this important\noccassion. My embarrassment is lessened however, and possibly your\ndisappointment, in view of the handsome tribute paid to the memory\nof the great man by the Minister of Education in his admirable\naddress, by the presentation of the historic paper by the Deputy\nMinister, and last, though not least, by the speech of Alderman\nMcMillan on behalf of the city of Toronto.\n\nPatriotism is that passion which aims to serve one's country,\neither in defending it from invasion or protecting its rights and\nmaintaining its laws and institutions in vigour and purity. If\nthen we accord, as we ought to do, a place among the patriots of\nour country to those who readily respond to its call in the hour\nof danger, to those who bring their wisdom and judgment to bear\nin the making of laws for its good and healthful government, to\nthose, also, who as diplomats in the carrying on of delicate and\nsubtle international negotiations do so in such a way as not only to\nmaintain their country's honor but to make their country respected,\nwhat place shall we assign to him who devoted his life to the best\ninterests of the young of his own province in order that they might\nbe fitted rightly to take their part in life, to do this all the\nbetter by reason of that educational system of which he was the\nmoving spirit and which it was his to found? What place I ask, if it\nbe not the very first place in the front rank of that distinguished\nclass. Egerton Ryerson may fairly be regarded as the founder of\nthe school system of his own province and as a consequence must,\nthroughout all time, occupy a foremost place in the history of his\ncountry.\n\nHis was a life spent not in the promotion of his personal ends.\nIndeed it may be affirmed that his devotion to his life-work\nso absorbed his time, his thoughts and his energies as to have\ndisqualified him for making that suitable provision for the close of\nlife for which his great abilities so eminently fitted him.\n\nIt would scarcely be fair to claim for him all the honour of\nperfecting the school system of Ontario, scarcely fair to say that\nto him exclusively belongs the results seen to-day which give to\nthe school system of Ontario so prominent a place among the school\nsystems of other countries. A measure of the praise is doubtless due\nto the able staff of workers by which he was supported. His was the\ndirecting mind; 'twas theirs to carry out his plans.\n\nIt would not be fair to ascribe to the architect all the credit\nfor the grace, symmetry and safety of the most magnificent public\nbuildings. True, he it was who planned the foundations, made them\ndeep and broad, as that they might be safe and enduring. True, he\nit was who gave grace and beauty to the elevation, as that it might\nnot only answer its purpose, but that it might be at the same time\n\"a thing of beauty;\" but how easily might not only the safety of\nthe building be imperilled but its beauty marred by careless and by\nignorant treatment; but skilful treatment has produced the needed\nstrength, and has secured the grace of outline, and the building is\nperfect and harmonious in all its parts.\n\nWhat the architect is to the building that was Egerton Ryerson to\nour school system. His it was to lay the foundation upon which a\nstructure which might be at once the pride and the glory of our\nProvince could be erected; his it was to lay these deep and broad\nand enduring. How wisely and how well he did his work. How well\nhis efforts have been supplemented by the able band of workers who\nwere associated with him the splendid school system of our Province\nto-day abundantly testifies.\n\nIt is fitting, therefore, that his statue should be placed on these\ngrounds, so that the coming generations may be made familiar with\nthe general appearance of the man who has done so much for the\neducational interests of his country.\n\nBut it is not here, faithful as the bronze may speak of the man,\nthat his most fitting and most enduring monument must be found.\nThe group of happy faced children which throng our sidewalks\nwending their way each morning to our schools, the pupils of\nour Model Schools, our High Schools, the under-graduates of our\nUniversities, and these seen not only in their school period, but\nin their subsequent career taking their place in our country as\nits legislators, its professional men, its merchants, mechanics,\nfarmers, its matrons, taking their places in life, and taking them\nall the better for their own good and that of their country for the\ntraining, sound, thorough and scholarly, which they have received in\nthe schools, colleges and universities of their own country, helping\nthem to make their homes homes of comfort, elegance and refinement.\n\nHere must be found the true and abiding monument of the man; here\nthe enduring fruit of his life-work, more imperishable also than\neither brass or marble.\n\nStatistics have been freely given, and these need not be repeated.\n\nNot often does it fall to the lot of one man to leave behind him\nsuch a record as that left by Dr. Ryerson in his life-work of\nthirty-three years.\n\nTo see the old-log school house, with its imperfect appliances,\nsupplanted by the palatial school buildings with their perfect\nequipment of our own day.\n\nTo see the work done by our High Schools, our Colleges and\nUniversities, which challenges not the attention only, but the\nadmiration of other countries, that is an honour, even with the\ncapabilities which exist in a young country like ours, reserved\nbut for few, and yet among that number he stands out promptly\nwhose statue has this day been unveiled by His Honour the\nLieutenant-Governor.\n\nAny words of mine are poor and weak indeed in dealing with a matter\nso full of interest, not to the people of Ontario only but to the\npeople of Canada, but full of interest to my own mind, as it must\nbe to the mind of others, is the estimate placed upon Dr. Ryerson's\nwork by many eminent men, all well qualified to judge from an\neducationalist standpoint. Dr. Hodgins has spoken of the late Dr.\nFraser, Bishop of Manchester.\n\nFew men were so well able to form an opinion upon any educational\nsystem. Himself a great scholar, an Oriel man, an enthusiast on all\nmatters connected with the educational system of his own country,\nappointed by the Royal Commission as an assistant commissioner\nto visit and report upon the educational system of the United\nStates and Canada, thus speaks of our school system and of Dr.\nRyerson. After referring to the system says: \"It shows what can be\naccomplished by the energy, determination and devotion of a single\nearnest man. Through evil report and good report, he has found\nothers to support him in the resolution that free education shall be\nplaced within the reach of every Canadian parent for every Canadian\nchild.\"\n\nEgerton Ryerson has deserved well of his country. His best days\nand his best energies were given to the upbuilding of its grandest\ninstitution. Well should his country guard and cherish his memory,\nso that the young who are here assembled to-day may learn this\nlesson, that he who devotes his life for his country's good, his\ncountry will hold his memory not in fragrant only, but in perpetual\nremembrance.\n\n\nREPRESENTATIVE OF VICTORIA UNIVERSITY, COBOURG.\n\nThe Rev. N. Burwash, S.T.D., President and Chancellor of Victoria\nUniversity, spoke as follows:\n\n_Ladies and Gentlemen_:\n\nIt is particularly appropriate that on this occasion Victoria\nUniversity should speak. We meet to do honor to one of the greatest\nof Canada's sons. Dr. Ryerson has written his name indelibly upon\nthe pages of his country's history. He has done so not merely\nby superior gifts of intellect, although to few men have more\nnoble talents been entrusted. Nor has he done so merely as the\nsuccessful leader of a great party in Church or State, although he\nwas a leader, one of the first three in one of Canada's greatest\nmovements towards a perfect political constitution. The success of\nthis movement, after a struggle stretching through the lifetime\nof a full generation, and its results of which all good men now\napprove, might well ennoble the name of any man. But that for which\nwe honor Dr. Ryerson's name and memory to-day is his work as an\neducator; and from Victoria University he first received the call\nto consecrate his life to his work. He was the first President of\nVictoria College. He, a Canadian of the Canadians, was the first\nCanadian to occupy that position in our Province; and Victoria has\nmaintained the Canadian succession unbroken from his day to the\npresent. Victoria was the first institution in this Province in\nactive operation as a teaching institution with University powers.\nAnd it fell to Dr. Ryerson to shape its character and curriculum,\nand to give it a form, many features of which it has retained to\nthis present hour.\n\nIt was from this duty of laying the foundations of our University in\nthis young Province that Dr. Ryerson was called to the wider field\nof fashioning the primary and intermediate education of the whole\npeople.\n\nIt would be presumption on my part to attempt to speak to-day of\nthe difficulties which beset him in his task, and of the skill and\njudgment with which those difficulties were met and overcome. That\nhas already been the more appropriate duty of one who has just\nspoken, and who was for long years his most valued and honored\nassociate in his life-work.\n\nBut, as a Canadian of the fourth generation, and as a Canadian\nschool boy who has enjoyed the advantages of the great work which\nto-day we commit to the perpetual memory of our country, I may\nventure to refer to the peculiarly Canadian character of the system\nfounded by Dr. Ryerson. The early schools of this country were very\nvaried in their type. Prepossession and usage rule imperiously in\neducation. Each little colony or settlement, as it was called,\nhad its national prepossessions. Here was an attempt to reproduce\na miniature English Eton or Rugby. There was a genuine Scottish\nparish school with its Bible and Catechism. Here was an Irish school\nwith its predilection for difficult problems in arithmetic and\nalgebra. There was a Yankee school with its spelling-matches and\ndialogues on examination day. These were the heterogeneous elements\nof forty-five and fifty years ago. To-day, we have everywhere the\nCanadian school as unique in its character as any of these, and\nas well-known in its results all over this continent. The skilful\nmind that took possession of these materials, that carefully\nseparated the good from the bad, that patiently and wisely removed\nor overcame prejudices, that calmly waited till the public mind was\nready for each progressive movement, and then with vigor pushed it\nforward to speedy completion--this was the ----le gift which Dr.\nRyerson devoted to his country's service. Gathering his materials\nfor building up a perfect educational system from all lands, and\nfrom the wisdom and experience of all ages, this great man wrought\nout his life-task in the face of political prejudices, of national\nprejudices and of sectarian prejudices. I know of no man of his day\nwho rose more fully than did he above the narrowness of all these.\nFrom the elevation of a broad catholicity, he grasped the great\noutlines of a comprehensive and national system of education for\nthe Upper Canadian people, and patiently did he work toward that as\nhis ideal. It would be too much to say that he completed the ideal.\nSuch is not often given to mortal man. There are problems in this\nwork still unsolved. There is still something for us to do. But in\nthe solution of these problems we may well thank God for the broad,\nstrong foundations, and structures planned, and so nearly completed\nby this master workman.\n\nIn the inaugural address with which Dr. Ryerson opened his work in\nVictoria College, I find this passage \"The education imparted in\nthis college is to be British and Canadian. Youth should be educated\nfor their country as well as for themselves.\" This motto, never\nforgotten, has given character to his great life-work, and has\ngiven us a system of public schools and intermediate schools which\nmore enduring than any monument will perpetuate to our children for\ngenerations to come, the name of Egerton Ryerson.[9]\n\n  [9] W. Kerr, Q.C., Vice-Chancellor, of Victoria University, in\n  explaining the causes of his absence from the ceremonies of\n  unveiling, said:--\"I thank you very much for your ever thoughtful\n  kindness in sending me a programme of the ceremonies of unveiling\n  of the statue of our late great chief founder of the peerless\n  school system of Ontario. I should very much like to have the\n  privilege of being present on the occasion and of listening to the\n  speeches and addresses, but especially your 'Historical Paper on\n  Education in Ontario,' which no man now living so well understands\n  as yourself.... How often I think of the late chief's farsightedness\n  and patriotic efforts in connection with the Upper Canada Academy\n  and subsequently with Victoria University.\"...\n\n\nREPRESENTATIVE OF QUEEN'S UNIVERSITY, KINGSTON.\n\nSandford Fleming, C.E., LL.D, C.M.G., Chancellor of Queen's\nUniversity, spoke as follows:--\n\n_Your Honor_, _Mr. Chairman_, _Ladies and Gentlemen_:\n\nIn the name of Queen's University, and at the special request of\nthe Senate of that institution, I come here to-day to take part in\nthese interesting proceedings. On behalf of higher education in\nEastern Ontario I have the honor to bear tribute to the memory of\nDr. Egerton Ryerson.\n\nHowever unworthy the individual whom Queen's has sent on this\noccasion, I am warranted in stating that no institution in this\ncountry is more thoroughly alive to the importance of sound\neducation for all classes of the community than the University I\ncome here to represent. Moreover, I venture to say that there is no\none here present who more fully appreciates the incalculable value\nof the school system of Ontario and the work accomplished in its\nestablishment, by him to whose memory we are this day assembled to\ndo honor.\n\nIt is not simply an agreeable duty I am called upon to perform, I\nfeel it to be a high privilege to be allowed to take part by my\npresence on this auspicious occasion. I have but to look back over\na period of forty years to recall the living form of the sculptured\nfigure before us, and to remember the time when in the zenith of\nhis strength and intellectual power, he brought to bear on the\ngreat work of his life that wisdom and foresight, that indomitable\nperseverance and patriotism, that zeal and devotion with which\nhe was gifted. I have but to recollect his persistent efforts to\ninitiate and put in successful operation a comprehensive system of\ncommon school education in this province, to express my unalloyed\nsatisfaction that those efforts--those great and sustained efforts\nwere not in vain. I rejoiced then, as I rejoice now, that the noble\nwork in which he took so conspicuous a part has been crowned with\nsignal success. I thought then, and I think now, that the people of\nthis province, I may indeed say the people of the whole of Canada,\nof all ages, of all classes, of all colors and of all creeds, owe a\ndeep debt of gratitude to Dr. Ryerson, and I cannot be wrong in the\nfirm opinion that we all do well to revere and perpetuate his memory.\n\nWhile Dr. Egerton Ryerson attached most importance to the\nestablishment of the common schools of the country on a sound and\nefficient basis, he also warmly sympathized with every effort\nto promote higher education. He took an active part in founding\nVictoria University, of which he was chosen the first president.\nHe was a strenuous supporter of that institution up to the day\nof his death, firmly believing that the resources of the country\ncould support, and that the people of Ontario should possess, well\nendowed, independent seats of learning of different types.\n\nAs a member of the community I have always had the highest esteem\nand veneration for this great pioneer of education in Canada. I feel\nnow and have always felt with unnumbered thousands that his life has\nindeed been that of a foremost public benefactor. I am, therefore,\ngreatly gratified that it has fallen to my lot, on behalf of\nQueen's University and higher education in Eastern Ontario to bear\ntribute to the memory of the founder and first administrator of our\nsystem of public instruction, a far-seeing Canadian, an enlightened\nstatesman, a man who in his distinguished career rendered the most\nimportant services to the country of his birth.\n\nI am glad to have an opportunity of taking part in the formal\ninauguration of the work of art which we see before us. At the same\ntime I cannot forget that Dr. Egerton Ryerson has left behind him\nan inheritance to unborn generations of Canadians in the schools\nwhich we behold everywhere throughout the land and the free public\ninstruction which they represent. These are now and must always be\nrecognized as his best and most enduring monument.[10]\n\n  [10] In his letter enclosing the manuscript of his address,\n  Chancellor Fleming, said:--\"I write to congratulate you on the\n  complete success of the affair of last Friday. Even the weather was\n  every thing we could desire. It was a genuine pleasure for me to be\n  present on the occasion. In the few remarks I offered I meant every\n  word I said, The only omission was the absence of any reference, or\n  sufficient reference, to the right hand man of Dr. Ryerson during\n  all the years he laboured. This often happens; but you have the\n  happy consciousness that your work and your life has so largely\n  entered into the imperishable monument which he has raised in the\n  school system of the century.\"\n\n\nREPRESENTATIVE OF TRINITY UNIVERSITY, TORONTO.\n\nThe Representative of Trinity University, Rev. Professor Clark,\nremarked that he had the honor of representing the smallest of the\nuniversities of Ontario, but one in which they strove to do their\nwork in a spirit of loyalty to their country as well as to their\nown convictions. He had naturally prepared to make some remarks on\nthe distinguished and illustrious man in whose honor they were then\nassembled; but, as he supposed, nearly everything which had occurred\nto him as being suitable to be said had been anticipated by previous\nspeakers.\n\nAs he looked upon those Normal Schools by which they were imprinted,\nhe could not help being reminded of the words inscribed on the\ninterior of St. Paul's in memory of its founder, Sir Christopher\nWren, \"_Si monumentum requiris circumspice_.\" \"If you ask for his\nmemorial look around you.\" With equal propriety we might point to\nthose schools as a monument and memorial of Dr. Ryerson, the founder\nof the educational system of this Province, no less fitting than the\nstatue which had just been uncovered. But more enduring than the\nbuilding or the effigy was the intellectual and moral work which\nhe had accomplished in our educational institutions, for that work\nwas eternal. Its effects and influences would never pass away, but\nwould go on leavening generations yet unborn. Whatever changes or\nrevolutions might occur, his work and its consequences would still\nlive on.\n\nIn one respect, perhaps, it was fortunate that others should have\nborne testimony to Dr. Ryerson's work and ability, and to the\ngreatness of the work which he had done. His own knowledge of the\nman and of the work was only second-hand, and he could not speak\nwith the freshness and vividness of those who had personal knowledge\nof them. \"_Segnius irritant animos demissa per aurem._\" But although\nhe had not direct and immediate knowledge of Dr. Ryerson, he had\nthe opportunity of studying several of the publications which\ngave an account of his history, and especially of his educational\nwork--more particularly some of those written by Dr. Hodgins. From\nthese he had learnt something of the spirit in which the work had\nbeen accomplished, and the principles which were embodied in it;\nand these would account for the eulogiums which had already been\nbestowed upon the work and its principal doer.\n\nThey might learn from such investigations something that would\nhelp to guard them from dangers which attended their educational\nwork. It seemed to him that Dr. Ryerson's conception of the work of\neducation was singularly simple, earnest, deep and comprehensive,\nfree from affectation and one-sidedness. They were in danger of\nforgetting some of these elements, of making education showy instead\nof solid,--of forgetting that it was the education and discipline\nof the man and of the mind that we had to accomplish, and not the\noutward adornment of him, or the mere imparting of knowledge. Some\nof our dangers have been forcibly pointed out by the President of\nUniversity College, Sir Daniel Wilson, in the March number of the\n_Canada Educational Monthly_. It would not be proper to go into\ndetails on such an occasion, but he would recommend those who were\ninterested in these questions to study that article.\n\nThere was great danger of their being one-sided in education, of\ntheir taking up cries on one side or another. We have often heard\nof a foolish Anglomania by which some people were possessed. But\nthere are other manias which are quite as silly. There was an\nAmericano-mania and a Canada mania (_mania Canadensis_, said the\nMinister), and they were all equally foolish. It was egregious\nfolly merely to imitate an Englishman or an American. But it was\nequally foolish to imagine that we had everything and could do\neverything by ourselves. Our business was not to make Englishmen or\nAmericans or Canadians, but to make men, furnished with sufficient\nknowledge, with cultivated, disciplined minds, with vigorous wills.\nThis is our work, and we must do it with our might, remembering the\nlimitations of our position and realizing what we could and could\nnot accomplish. We must not at present expect the results of the\neducation given at the great German universities or at Oxford and\nCambridge, but we might make the best with the materials at our\ndisposal. We might foster in the rising generations a love of truth\nand goodness, and instil into them a deep sense of duty, and thus\nhelp to qualify them for the position they would have to occupy, and\nthe work which they had to perform.\n\n\nREPRESENTATIVE OF MCMASTER UNIVERSITY.[11]\n\n  [11] Dr. Rand, in sending the manuscript of his address, said:--\"I\n  think the exercises were very successful indeed, and that the\n  memorial volume, if brought out with some expedition, will prove\n  very helpful in quickening a true appreciation of the great work\n  done by Dr. Ryerson in building up the educational system of\n  Ontario.\"\n\nProfessor Theodore H. Rand, M.A., D.C.L., representative of McMaster\nUniversity spoke as follows:--\n\nTwice before it has been my privilege to unite in doing honor by a\npublic memorial to the name of distinguished educationalists,--that\nof Horace Mann, of Massachusetts, and Alexander Forrester, of Nova\nScotia; but in view of the breadth of the area over which Egerton\nRyerson wrought, and the really national character of his work, this\nhas been an occasion of surpassing interest to me, and one, I am\nsure, which marks an epoch in the educational history of Ontario.\nIn speaking as a representative of McMaster University, which is\njust now being organized as one of the instruments of the higher\neducation of this country, I may be permitted to say that the\nChristian denomination which controls the destiny of that University\nhas in all parts of the world been in active sympathy with popular\neducation, and in two of our Canadian provinces has been foremost\nin efforts to secure the efficient organization of systems of free\neducation under government control.\n\nWhen a country has risen to the position of making adequate\npublic provision against the blighting and destructive influences\nof ignorance, it has undertaken the discharge of one of those\ncontingent and great obligations which, perhaps, will always await\nany people who are pressing forward to the attainment of the\npossibilities of Christian civilization. With all its imperfections\nour system of public education in an eminent degree commands the\naffectionate regard of our people and the admiration of strangers.\nWhile Ontario was not the first of our Canadian provinces to\norganize a free system of public schools, and while she has not\nmaintained intact the principle which lies at the basis of the\ncommon schools, the grandeur of the outline of our system and the\ngeneral completeness of its details are, I believe, unsurpassed\nby those of any other system on this continent or throughout\nthe empire. This is especially true of the completeness of the\nprovision made for passing from the elementary schools into the work\nof the higher education. Ontario occupies this advanced position\nto-day, with all its immeasurable advantages, largely because of\nEgerton Ryerson. He was possessed of a profound conviction that\nmind is the great creative power by which all resources of nature\nare to be turned to account. He had no idea that material good is a\ngood at all only as it is a means for the development of the moral\nand social possibilities of the individual and of the nation. He\ndid not argue, as many others in the country did, that since the\narea of the Province is vast, its population widely scattered, its\nforests waiting to be felled, its lands to be cleared and drained,\ntherefore the organization of an efficient school system was a thing\nof the far future. On the contrary, having before him such examples\nas Prussia, Scotland, Ireland, the New England, Middle and Western\nStates, and believing that our civic institutions should afford\nsocial conditions inferior to those of no country in the world, he\npoured all the energy of his great heart and mind into the effort\nto make available even to the remotest hamlets of the Province the\nblessings of knowledge. Intelligence, industry and morality were\nfelt to be inseparably bound up with the progress of education.\nA system good enough for the rich and poor alike, and supported\nat the public expense, was his aim and his final achievement. The\nChristian communism underlying our systems of public education on\nthis continent is proving one of the great safeguards of society\nagainst the forms of a false communism; and there is yet room, in my\njudgment, for a still wider application of kindred principles in our\nsocial system.\n\nThe work of Egerton Ryerson furnishes an additional illustration of\nthe truth that systems of popular education are, so to speak, the\ngift to the people of the Colleges and Universities. His relations\nto the higher education enabled him to grasp all the elements\ninvolved in the great problem he undertook to solve for Ontario,\nand to bring all parts of the educational system into helpful and\nsympathetic relations. Our Universities must always be sources of\nstimulus and enrichment to the schools of the Province at large, if\nthey discharge in any original and adequate measure their functions\nto society.\n\nIn attributing so important a share to Egerton Ryerson in the\nestablishment of our school system, I am not, of course, unmindful\nof the public men who seconded his efforts, and above all, of the\nteachers and inspectors by whose self-sacrificing toils educational\nadvance was rendered possible, and has been sustained. Were he whom\nwe honor in our midst to-day he would be the first to speak thus,\nand especially of his friend, Dr. Hodgins, who for so many years\nwas his able assistant and valued confidant. This bronze memorial\nis well--may it long testify to the patriotic virtues of a noble\nman--but it is as nothing, I trust, in comparison with the living\nand imperishable memorials of enriched and ennobled human lives.\nAs time witnesses the increasing development of the material and\nspiritual forces of the present and coming generations of our\npeople, as our social and national institutions are more fully\nperfected and widely recognized, we shall have hereby and herein\nperennial memorials of the founder of the school system of Ontario\nthroughout all coming time.\n\n\n\n\nCHAPTER IV.\n\nEDUCATION IN ONTARIO--PAST AND PRESENT.\n\nAN HISTORICAL RETROSPECT BY J. GEORGE HODGINS, M.A., LL.D.,\nBARRISTER-AT-LAW, AND DEPUTY MINISTER OF EDUCATION FOR ONTARIO.\n\n\nTo-day will long be memorable in the educational history of\nOntario--for to-day has been unveiled the first statue ever erected\nin this province to one of its own sons.\n\nIt will be still more memorable from the fact that that special\nsubject of public interest and national concern which has been\nsignally honoured to-day, is the pre-eminently important one of\npopular education. These two facts combined give to the celebration\nand pleasant incidents of the day a peculiar significance and a\nspecial interest.\n\n\nSIGNIFICANCE OF THE EVENT OF TO-DAY.\n\nOne of the first indications of a growing national life and a\npatriotic national spirit is the erection of statues to noble\nsons who have rendered such valuable services to the state as are\nrecognized and honoured here to-day.\n\nIt is a most hopeful sign, as well as an assuring and happy augury\nfor the future of a country, when its patriotism takes the grateful\nand graceful form of doing honour to those who have aided in laying\nthe foundation of its future greatness and prosperity. This, we all\nrejoice has been done by Ontario to-day in the unveiling of the\nstatue of the distinguished founder of her educational system. She\nhas reared to-day to one of the sons of her soil a noble monument,\nexpressive of grateful acknowledgment for services of the greatest\nimportance and value to her and to the thousands of her sons and\ndaughters yet unborn.\n\nThe erection of this statue emphasizes in a striking manner a\nnotable fact, chronicled by John Milton, which the mature judgment\nof the nineteenth century has everywhere endorsed, that--\n\n  \"Peace hath her victories no less renowned than war.\"\n\nThat is, that it is not heroic deeds of valor alone which call\nforth a nation's gratitude. It further shows us that unswerving\ndevotion to duty in any of the departments of the public service, or\nprofessional or private life, which have to do with matters which\nconcern a nation's progress and welfare, is equally recognized, if\nnot more signally honoured, than were deeds of prowess in the days\ngone by. We have, at all events on this continent, many notable\nexamples of distinguished honour being done to literary men, to men\nof science and to noted educationists. Any one who has visited the\nchief city of Massachusetts cannot fail to have seen, on the broad\nterrace in front of the capitol, a massive bronze statue to Horace\nMann, the well-known Founder of the Public School system, not only\nof Massachusetts, but practically of the New England States.\n\n\nTHE ONTARIO SYSTEM OF EDUCATION--ITS INFLUENCE ABROAD.\n\nSo, in like manner we unveil to-day the statue, not only of the\nFounder of the School System of Ontario, but of one, the impress\nof whose hand, and the practical suggestions of whose mature\nexperience, may be recognized in the systems of education of some\nof the Maritime Provinces, and in those of Manitoba and British\nColumbia. The first Superintendent of British Columbia, and the\nsecond of Manitoba were trained in the schools of Ontario, and were\nthus experienced pioneers in the new Provinces of their educational\nsystems. Also in the West Indies the educational example of Ontario\nwas felt to be of some value by Sir Francis Hincks, when Governor of\nthe Windward Islands.[12]\n\n  [12] In a letter from Barbados, dated 31st May, 1856, he said: As\n  to education, in which you will take the greatest interest, all I\n  can say is that my own hopes are centered in getting a good Normal\n  School in humble imitation of yours. I think with that all will be\n  well. If we could train good teachers we would have an admirable\n  system. There have been some attempts, but not to much effect. I\n  want your advice as to the establishment of this school. Tell me\n  how to go to work to get good men, etc. I must have your plan of\n  boarding the Normal School pupils at the public expense, which I\n  think essential. I also want to introduce the national books (as you\n  did). Any advice or information will be conducive to good results.\n\nEven the grand old Mother Country has not failed to acknowledge her\nindebtedness to him whom we honour to-day, for practical suggestions\nin the solution of the educational problems which confronted her\npublic men, notably the Duke of Newcastle and the Right Hon. W. E.\nForster,[13] during the years reaching from 1860 to 1870.\n\n  [13] Full information in regard to the working of our system of\n  education was communicated from time to time to the Privy Council\n  Committee on Education in England. This was of great practical value\n  (as he assured us) to the Right Hon. W. E. Forster, promoter of\n  the noted English School Bill of 1870. In 1875 Mr. Forster visited\n  the Education Department of Ontario. The _Journal of Education_ of\n  April, 1876, thus refers to Mr. Forster's visit:--\n\n  \"During the recent visits of the Right Hon. W. E. Forster and\n  Hepworth Dixon, Esq., to the Ontario Education Department, they were\n  kind enough to explain and discuss some of the new problems in the\n  English Educational system, and made enquiries as to the success\n  of our attempts at a practical solution of the same question. The\n  two principal subjects referred to by Mr. Forster were compulsory\n  education and denominational schools, and on these two points\n  full explanation of our Ontario system were given.\"--_\"Journal of\n  Education,\" Province of Ontario, Volume xxviii., page 49._\n\nIn 1860, at the request of the Duke of Newcastle, who accompanied\nthe Prince of Wales to Canada, Dr. Ryerson prepared an elaborate\nsketch of the system of education in Upper Canada, and contrasted\nit with the English and other European systems of education. This\nreport was embodied in a letter to the Duke, dated 12th October,\n1860.\n\nAs to the appropriateness of our erecting a Statue to Egerton\nRyerson in Ontario, as was done to Horace Mann, in Massachusetts,\nI may here quote a reference to the equal value of the labors of\nthose two noted men which was made twenty-five years ago by that\nacute observer and experienced educational commissioner, the late\nwell-known and distinguished Bishop Fraser, of Manchester. He said:\n\n     \"What Education in New England owes to Horace Mann, Education in\n     Canada owes to Egerton Ryerson.\"\n\nTo-day we honour ourselves by seeking to discharge that obligation,\nat least in part.\n\nThere is one circumstance connected with the erection of this statue\nwhich, to my mind, gives it a peculiar value and significance. The\nerection of statues by popular vote, or by the Legislature, gives\na _quasi_, if not a real national character to such erection,\nbut, a statue erected from the proceeds of thousands of small\ncontributions, as in this case, shows that deep down in the hearts\nof the people of this country there must have been genuine regard\nfor the man whom they thus seek to honour. When a memorial takes\nsuch a form as that we may well regard it as more enduring and\nprecious than either the bronze or marble which constitute the\nmaterial of its structure.\n\n\nCOMPREHENSIVE CHARACTER OF THE ONTARIO EDUCATIONAL SYSTEM.\n\nIt devolves upon me, as Chairman of the Committee having charge of\nthis work, and at the kind request of my colleagues,--no less than\nas the life-long friend and fellow-laborer of him whose deeds and\nmemory we honor to-day--to trace back to their source the origin and\nunderlying principles of our system of education, and to show that\nthese underlying principles and other vital forces were so combined\nby a master-hand an to form the groundwork, as they have, in their\ncombination, become the charter of our educational system of to-day.\n\nAnd here, in this connection, a thought or two strikes me; and each\nthought contains for us a moral and a lesson.\n\nThe first is that educational systems are essentially progressive in\ntheir character and purposes, and truly they \"never continue in one\nstay.\"\n\nThe second is that the earliest sources of what might be called our\neducational inspiration are now uncertain guides, and, as such, are\nto-day of doubtful authority.\n\nNo one will venture to affirm that even--as it was then\nconsidered--the broad and comprehensive scheme of public education\nsketched by Dr. Ryerson in 1846, should be considered as the acme\nof our educational achievements of to-day. Nor would any one at all\nconversant with the condition and progress of education on this\ncontinent alone be content to draw his inspiration from, or limit\nhis range of observation to, the New England States as formerly. The\nexamples to be seen, and the experience to be consulted, and the\nsystems to be studied, must to-day, so far as the United States is\nconcerned--be sought for in the far-off Western States.\n\nIn this matter I speak of what I know; and I speak, therefore, with\nthe more emphasis on this point, because of the primary importance\nof keeping this Province and the Dominion educationally abreast of\nthe most advanced States of the American Union--our near neighbors,\nand our energetic and actively progressive educational rivals.\n\nAs an illustration of these notable facts, I may state that having\nbeen selected by the United States Bureau of Education to act as one\nof the seven international educational jurors, at the New Orleans\nExhibition in 1885, it was, during six weeks, my duty with others,\nto examine into and report upon the condition and results of the\nvarious state systems of education in the Union, and in other\ncountries.\n\n\nCHARACTER OF SYSTEMS OF EDUCATION ABROAD, AND LESSONS THEREFROM.\n\nI need not more than state, what you likely anticipate, that France,\nby her enlightened educational legislation of 1881--providing for\nmanual, or industrial, training in all of her schools--and Germany,\nby her earlier and more systematized educational legislation, stand\nat the head of European States, as does Japan at the head of the\nwhole Eastern World. But, in this connection, the interest to us\nshould be to note the fact that the educational centre in the United\nStates has within the last few years been gradually shifted from the\neast to the west. As an illustration, I may say that the highest\naward for the extent, variety and completeness of its educational\nsystem in all its details, was unanimously made by the jurors to\nMinnesota, while Massachusetts and other New England States, with\nNew York, Pennsylvania, etc., were entitled to only second and third\nclass honors. France and Japan justly received first-class honors,\nwhile England and other countries (omitting Germany) had to be\nplaced in the second and third class ranks as educating countries.\n\nA revelation of these and other suggestive facts in regard to\nthe progress of education in countries outside of our own, more\nthan ever convinced me of the wisdom of Dr. Ryerson's policy of\nobservation while head of the Education Department. He laid it down,\nnot so much an educational axiom, as a wise dictum--the result of\nhis educational experience, that--\n\n     \"There is no department of civil government in which careful\n     preparation, varied study and observation, and independent and\n     uniform action, are so important to success and efficiency,\n     as in founding, maturing and developing a system of public\n     instruction.\"\n\nHe, therefore, wisely devoted a large portion of his time to this\n\"careful preparation,\" as well as to \"varied study and observation\"\nof systems of education in Europe and America. And this fact\nlargely accounts for the \"success and efficiency\" of his efforts in\n\"founding, maturing and developing\" our system of public instruction.\n\nIn a reply to a resolution from the Council of the County of\nNorfolk, in 1851, Dr. Ryerson thus referred to this subject:--\n\n\"There is no poetry in the establishment and development of a public\nschool system; it is a matter-of-fact-work from beginning to end;\nand its progress, like the growth of body and mind in an individual,\nis gradual, and is the joint result of time and labour. I am happy,\nhowever, to know that our school system has already become so far\ndeveloped in its principles, objects and character as to command the\nattention and almost unanimous approbation of the country. I have\nlaid it down as a first principle to educate the people through the\npeople themselves, by their own voluntary co-operation and exertion,\nthrough the usual elective municipalities and other acknowledged and\nresponsible organs of a free people.\"\n\n\nEDUCATIONAL LESSONS TO BE LEARNED OUTSIDE OF ONTARIO.\n\nWhen we reflect upon the fact of the immense growth, and the\ncomprehensive character of the educational machinery in operation\non this continent alone, and the vast sums expended to keep it\nin motion, we cannot fail to be profoundly impressed with the\nserious and grave responsibility which is constantly imposed upon\nour educational leaders, of being forever on the watch-tower of\nobservation, to note the changes, improvements and advances which\nare continually taking place in the educational world outside. We\nare too apt to be content with our own progress, and to measure\nourselves by ourselves. In this connection the words recently\naddressed to the Kingston Board of School Trustees by the Very Rev.\nPrincipal Grant, are of special value as an apt illustration of my\nmeaning:--\n\n     \"During my absence I have studied the school systems of many\n     countries, and have learned lessons that ought to assist me in\n     coming to right conclusions. The world is wider than Canada,\n     or than America. The British Empire itself is wider than\n     this continent, and within its boundaries there are so many\n     educational systems and methods that a man who travels with eyes\n     and ears open cannot help learning many things that confirm\n     opinions previously held, and suggest improvements on what he\n     may have thought perfect, or the necessity of revising his\n     former judgments. He gets new points of view, and that of itself\n     is a great matter.\"\n\nOur American neighbors became fully alive years ago to the evils of\nthe fluctuating and uncertain character of the prevailing system\nof educational administration in vogue amongst them. They saw that\nnew and officially untrained men, of merely local experience and\nknowledge, were constantly being elected to take charge of the\nadministrative department of the schools of a state. Such men were\noften able educators, but by no means experienced educationists,\nor masters of systems of education. The American people, shrewd\nand practical as they are, felt the absolute necessity, therefore,\nof furnishing such men, and the vast army of their educationists\nand educators, with full and accurate information on systems and\nplans of education all over the world. With this object in view,\nthey established a central observatory, or Bureau of Education at\nWashington. I need hardly say how ably the work of this Bureau was\nsystematized and most efficiently performed under the direction\nof the Hon. John Eaton, Commissioner of Education. His successive\nreports and periodical circulars of information are mines of\neducational wealth. Their fullness and comprehensiveness have been\na marvel. They have aroused and stimulated educational workers\neverywhere. They are largely welcomed, and are highly prized in\nthese Provinces and elsewhere, as suggestive, and as invaluable\nstorehouses of information, and of the practical details of\neducation all over the world. They have, therefore, largely supplied\nthe place of personal inquiry and research, and yet have greatly\nstimulated both.\n\nIt was Dr. Ryerson's ideal that sooner or later a similar Bureau\nwould be established by the Central Government at Ottawa, the object\nof which would be, not only the supplying of abundant and reliable\ninformation to each province on the subject of systems and plans\nof education, but also, by intercommunication, to secure a general\nharmony of aim and purpose. And that further, without attempting any\ninterference in local administration, the Bureau would be the means\nof keeping up an active yet friendly intercolonial rivalry; and\nthus, on Dominion and national lines, to build up the confederacy,\nand to stimulate and encourage the efforts made in each province for\nthe promotion of substantial educational progress, combined with\nefficiency and economy.\n\n\nTHREE EDUCATIONAL PERIODS IN THE HISTORY OF ONTARIO.\n\nThe educational history of Ontario naturally divides itself in three\nperiods, viz.:--\n\n1. The early settlement, or United Empire Loyalist period.\n\n2. The period preceding the union of Upper and Lower Canada in 1840.\n\n3. The period since that union, and including the administration of\nthe Education Department by the Rev. Dr. Ryerson, down to 1876.\n\nThe United Empire Loyalists period takes us back to a period\nantecedent to that of their historical prominence as a factor in the\nevents of the war of the American Revolution. In order, therefore,\nto estimate the value of the educational influence of those times\non the future of the provinces in which the U. E. Loyalists settle,\nwe must take a glance at the Colonial chapter in the History of\nAmerican Education.\n\n\nCOLONIAL CHAPTER IN THE HISTORY OF AMERICAN EDUCATION.\n\nIt is, therefore, interesting in taking note of our educational\nprogress to give a brief glance at what was done by our fellow\ncolonists at a corresponding early period in the history of the \"old\nthirteen colonies,\" which formed the nucleus of the present American\nConfederation.\n\nIt has been the custom, probably unwittingly, but chiefly on the\npart of certain American writers, to exalt every good in their\npolitical and social condition, as of revolutionary origin, and\nreluctantly to admit that anything which was really excellent in\nboth, in the early colonial times, was of British origin. One\nunacquainted with the processes and progress of civilization\nin America would, on consulting such writers, suppose that,\nMinerva-like, the young Republic had sprung from the head of\nRevolutionary Jove, fully equipped, if not fully armed for the\nbattle of life, into the arena of the new world, and that this\nphenomenon happened just at the extinction of British power in the\nold colonies, and as the result of it. The policy of these writers\nhas been either to ignore the facts of history, or to keep entirely\nout of view the forces which had been operating in the British\ncolonial mind, before and at the time of the Revolution. They have\nnever stopped to enquire as to the source whence they derived their\nidea of political freedom, but have attributed it to their own\nsagacity, or regard it as the outgrowth of their own enlightened\nspeculations and thinkings when emancipated from British control.\nThere never was a greater mistake as to fact, or a greater wrong\ndone to the memory and example of such noble English patriots as\nHampden and his compeers, who laid down their lives for political\nprinciples which, considering the times in which they lived, were\neven more exalted and ennobling than those which were professed by\nthe American revolutionists of 1776. In fact, no proper parallel can\nbe instituted between them. John Hampden, in our humble judgment,\nwas as far superior to John Hancock, \"President of the Continental\nCongress,\" in the purity of his political motives and aspirations,\nas Cromwell was above Jack Cade.[14] However, it is not our purpose\nto discuss this question, but rather to vindicate the sagacity\nof the old colonists, who (at a time when loyalty was the rule,\nand not the exception), laid the foundation of those educational\ninstitutions, which to this day are the glory of the American\nRepublic.\n\n  [14] Thus, in regard to the chivalrous destruction of tea in Boston\n  harbour, in 1773, an American historian says:\n\n  \"The object of the mother country in imposing a duty of three\n  pence per pound on tea imported by the East India Company into\n  America, while it was _twelve_ pence per pound in England, was\n  mainly to break up the contraband trade of the colonial merchants\n  with Holland and her possessions.\"... \"Sons of the merchants [of\n  Boston] had become rich in the traffic, and a considerable part\n  of the large fortune which Hancock [President of the Insurgent\n  Congress] inherited from his uncle was thus acquired.\"... \"It was\n  fit, then, that Hancock, was ... was respondent in the Admiralty\n  Courts, in suits of the Crown, to recover nearly half a million\n  of dollars, ... should be the first to affix his name to the\n  [declaration of independence] which, if made good, would save him\n  from ruin.\"...--_Sabine's American Loyalists_, Vol. I. (Boston,\n  1865), pages 8, 9, 13.\n\n  So much for the much-valued patriotic act, which was a vast\n  pecuniary gain to Hancock and other contraband tea merchants of\n  Boston.\n\nNor were the British colonists into those early times peculiar in\ntheir zeal for the promotion of Education. The Dutch, Swedish, and\nIrish colonists who settled in Pennsylvania, New York, and Maryland\ndid their part in his great work, and on the whole did it well,\naccording to the spirit of the times.\n\nIn 1633, the first schoolmaster opened his school in the Dutch\nColony of New Amsterdam; and in 1638, the \"articles for the\ncolonization and trade of New Netherlands,\" provided that, \"each\nhouseholder and inhabitant shall bear such tax and public charge\nas shall hereafter be considered proper for the maintenance of\nschoolmasters.\" General Eaton, the United States Commissioner of\nEducation, in his valuable report for 1875, says:\n\n     \"We find, in numerous instances, the civil authorities of these\n     Dutch colonies acknowledging, (1) The duty of educating the\n     young, (2) The care of the qualification of the teacher, (3)\n     Provision for the payment of his services, and (4) The provision\n     of the school-house. When, in 1653, municipal privileges were\n     granted to the New Amsterdam [New York], the support of schools\n     was included.\"\n\nIn 1642, the instruction sent to the Governor of New Sweden\n[Pennsylvania], was \"to urge instruction and virtuous education of\nyouth and children.\" In 1693-6, large numbers of primers, tracts\nand cathechisms were received from Sweden, for these schools on the\nDelaware. This was the educational state of the Swedish settlement\nin what was afterwards known as Pennsylvania, on the arrival of its\nnoble English founder, William Penn. His views on education were\nwell expressed in the following declaration:\n\n     \"That which makes a good Constitution must keep it, viz.: men\n     of wisdom and virtue; qualities which, because they descend\n     not with worldly inheritance must be carefully propagated by\n     virtuous education of youth, for which spare no cost; for by\n     such parsimony, all that is loved is lost.\"\n\nThe first real systematic efforts to promote popular education began\nin New England, from thence it has spread in all directions. In 1635\nthe first school was opened at Boston, Massachusetts, and brother\nPhilemon Purmount was appointed schoolmaster by the Town Committee.\nThirty acres of land were given for his support. In 1642 the General\nCourt, (or Legislature) passed a resolution enjoining on the local\nauthorities:\n\n     \"To keep a watchful eye on their brothers and neighbors, and\n     above all things to see that there be no family in so barbarous\n     a state, that the head thereof do not himself, or by the help\n     of others, impart instruction to his children and servants,\n     to enable them to read fluently the English language, and to\n     acquire a knowledge of the penal laws, under a penalty of twenty\n     shillings for such neglect.\"\n\nSpeaking on this subject, in his inaugural address in 1853,\nPresident Walker of Harvard University said:\n\nWhat most distinguishes the early settlers of Massachusetts, is the\ninterest and care they took in education, and especially in the\ninstitution of a system of common schools, to be sustained at the\npublic charge.\n\nHere they were first. In other things they thought wisely and acted\nnobly; but in this, and perhaps in this alone, they were original.\nHonor, immortal honor to the men, who, while still struggling for a\nscanty and bare subsistence, could yet find the means and the heart\nto do what had never been done or attempted before: placing the\nadvantages of a competent instruction within the reach of all. By\ntaking this course, what a noble confidence they manifested in the\ntruth of their principles and in the justice of their measures....\nBut the founders and early settlers of Massachusetts did not limit\ntheir views of education to common schools. Many of their leading\nmen had studied at the English Universities and were imbued with,\nor at least, could appreciate the highest scholarship of that day.\nThey also knew, on general grounds and as practical men, that the\npublic good requires the advancement, as well as the diffusion, of\nknowledge; in short, that both must go together; that the streams\nwill soon cease, if the fountains fail.--Pages 33, 34.\n\nTo be brief on this point I may state that in 1847, the first\nlegislative enactment in favor of schools was made in Massachusetts;\nand in 1670, the Governor of Connecticut declared that \"one-fourth\nof her revenue was devoted to schools.\"\n\n       *       *       *       *       *\n\nGeneral Eaton in his comprehensive report of 1875 says:\n\n     \"History, with hardly a dissenting voice, accords to the English\n     Colonists of New England, the credit of having developed those\n     forms of action, in reference to the education of children,\n     which contained more than any other the distinct features of the\n     systems adopted in this cuntry.\"\n\nIn the early colonial times, before the revolution, there were nine\ncolleges established in seven out of the thirteen colonies.\n\nThese colleges, with the date of their foundation, are as follow:--\n\n  1. Harvard--Massachusetts, in                   1638\n  2. William and Mary--Virginia, in               1693\n  3. Yale--Connecticut, in                        1700\n  4. Nassau Hall (now Princeton)--New Jersey, in  1748\n  5. Kings (now Columbia)--New York, in           1754\n  6. Brown--Rhode Island, in                      1765\n  7. Dartmouth--New Hampshire, in                 1770\n  8. Queen's (now Rutgers)--New Jersey, in        1771\n  9. Hampden--Sydney, Virginia, in                1775\n\nThe Legislature of Massachusetts, aided by the Rev. John Harvard,\nfounded Harvard Congregational College, in 1638, and the colonists\nof Connecticut, established the Yale Congregation College in\n1700.[15]\n\n  [15] \"The project of founding a College in Connecticut was early\n  taken up (in 1652), but was checked by well-founded remonstrance\n  from Massachusetts, who (sic), very justly observed that the whole\n  population of New England was scarcely sufficient to support one\n  institution.\"--President Dwight's _Travels in New England_, vol. I.\n  p. 168.\n\n  The Legislature made a grant of \u00a350 a year to Yale College, from\n  1701 to 1750, when \"it was discontinued on account of the heavy\n  taxes occasioned by the late _Canadian_ War.\"--C. K. Adams, in\n  _North American Review_ for October, 1875, p. 381.\n\nThe New Hampshire colonists endowed the Congregational College at\nDartmouth with 44,000 acres of land in 1770. The Episcopalians of\nthe English colony of New York, aided by the Legislature, founded\nKing's now Columbia College, in 1753. Indeed, so true were the\nEnglish colonists to the educational instincts of the mother land,\nthat when the Dutch Province of New Netherlands fell into their\nhands in 1644, the King's Commissioners were instructed \"to\nmake due enquiry as to what progress hath been made towards ye\nfounda\u00e7on and maintenance of any College Schools for the educa\u00e7on of\nyouth.\"--(Colonial History of N. Y., Vol. III. p. 53.)\n\nThe English Province, _par excellence_, of Virginia made various\npraiseworthy efforts to promote education. In 1619, soon after the\nsettlement of Jamestown, Sir Edwin Sandys, President of the Virginia\nCompany, had 10,000 acres of land set apart for the establishment\nof a University at Henrico for the colonists and Indians. The\nchurches in England gave \u00a31,500 sterling in the same year to aid in\nthe education of the Indians. In 1621, 1,000 acres of land as an\nendowment, and \u00a3150 were granted to establish a school at Charles\ncity. Other efforts were made in the same direction in 1660 and\n1688. The colony also nobly determined to establish a University;\nand in 1692-3, the project was practically realized by the founding\nby the King and Queen, under royal charter, of the Church of England\nCollege at Williamsburgh, of William and Mary. To this College the\nKing gave nearly \u00a32,000, besides 20,000 acres of land, and one penny\nper pound on all the tobacco exported from Maryland. The Legislature\nalso gave it in 1693, the duty on skins and furs exported, and on\nliquors imported.[16] The plans of the College were prepared by\nSir Christopher Wren. Among the first donors to the College was\nthe celebrated Robert Boyle.[17] Of all the colonial Colleges few\nexercised a greater educational influence among the leading men than\ndid this royal college. Jefferson, Munroe, Marshall (afterwards\nChief Justice of the United States), the two Randolphs, and Governor\nTyler, of Virginia, received their education here.\n\n  [16] Circular of Information, U. S. Bureau of Education, No. 1,\n  1887, page 15.\n\n  [17] General Eaton, United States Commissioner of Education, in an\n  educational retrospect in his Report for 1875, speaking of this\n  college, says:--\"The first commencement, in 1700, was a noted event.\n  Several planters came in their coaches, others in sloops, from New\n  York, Pennsylvania, and Maryland. Even Indians had the curiosity to\n  visit Williamsburgh,\" the seat of the College.--Page xix.\n\nThe Irish Roman Catholic Province of Maryland was not, at least in\npurpose, much behind her English sister. In 1671 an Act was passed\nby one of the Houses of the Legislature for the establishment of\na School or College, but owing to religious differences the other\nHouse did not concur. In 1692, the Legislature passed an Act for the\nencouragement of learning; and in 1696, King William's Free School,\nAnnapolis (afterwards St. John's College), was established.\n\nNew Jersey was one of the colonies which early promoted higher\neducation by founding the Presbyterian College at Princeton, under\nthe name of Nassau Hall, in 1746, and the Dutch Reformed College\nat New Brunswick (N.J.), under the name of Queen's, now Rutger's\nCollege, in 1770.\n\nThe little colony of Rhode Island did not fail in its duty to higher\neducation, for in 1764 it founded the Rhode Island College, now\nBrown University.\n\nThe Quaker colony of William Penn, following the example of the\nAnglicized Dutch colony of New York, established the University of\nPennsylvania at Philadelphia--the metropolis of the colonies in 1755.\n\nOf these nine ante-revolution Colleges, Harvard, Yale, Columbia\nand Princeton maintain an equally high reputation, while Brown\nUniversity, the University of Pennsylvania, Rutgers, William and\nMary and Dartmouth Colleges are more or less about the average\nstandard of American Colleges.\n\nGovernor Oglethorpe, the founder of Georgia, was a graduate of\nOxford. He, with other English University colonists, conceived\nthe idea of a College for this, the then youngest of the English\ncolonies. The project of his friend, the Irish Bishop Berkeley, of\nCloyne, of founding a College in the Bermudas having failed, he\nsecured \u00a310,000 of the Bishop's funds to aid him in his settlement\nof the colony. The seed sown by Oglethorpe bore fruit; and while\nGeorgia was still a colony, provision was made for a generous system\nof education.\n\nD. C. Gilman, Esq. (President of the John Hopkins' University,\nBaltimore), in his admirable sketch of the growth of education in\nthe United States during the last century, pays a high tribute to\nthe nine Colonial Colleges to which we have referred. He says:--\n\n     \"These nine Colleges were nurseries of virtue, intelligence,\n     liberality and patriotism, as well as learning; so that when\n     the revolution began, scores of the most enlightened leaders,\n     both in council and upon the field (on both sides) were found\n     among their graduates. The influence of academic culture may be\n     distinctly traced in the formation of the Constitution of the\n     United States, and in the political writings of Adams, Hamilton,\n     Jefferson, Madison, Munroe and many other leading statesmen of\n     the period. A careful student of American politics has remarked\n     that nothing more strikingly indicates the education given at\n     Cambridge than the masterly manner in which different problems\n     of law and government were handled by those who had received\n     their instruction only from that source.\"[18]\n\n  [18] In illustrating the fact that college-bred graduates are\n  considerably less numerous and less conspicuous in the professions\n  and in political life than were men of a similar education 50 or\n  100 years ago, Mr. C. K. Adams, in the _North American Review_ for\n  October, 1875, says that, \"of the 56 signers of the Declaration of\n  Independence, 36 were college-bred, and 15 of the 26 Senators in\n  the first Congress; while now there are only 7 of the 26 Senators\n  'college-bred.'\" He thinks that the comparison, if extended to the\n  House of Representatives and the State Legislatures, would be still\n  less favorable as to the number of college-bred men in these bodies.\n\nProf. Charles Sprague Smith, A.M., (of Columbia College) in his\nessay on _The American University_, read June, 1887, thus refers to\nthe character of these colonial colleges:--\n\n     \"In New England the higher system at general education, brought\n     over from Old England, was divided here, as there, into the\n     two studies of the College and the Grammar School; the latter\n     being superceded in quite recent times by the so-called Academy.\n     The curriculum of the American (or Colonial) College was, in\n     the main, modelled upon that of the parent country, special\n     consideration being given to theological science, etc.\"--Page 13.\n\nA recent American publication on revolutionary topics, thus deals\nwith the question of the superior education of the British colonists\nwho formed the first American Congress:--\n\n     \"An examination of the Continental Congress, composed as it was\n     of leading men of all the colonies, affords some light upon\n     the topic of popular education at that period. The Congress,\n     whose sessions extended through some ten years, comprised in\n     all some three hundred and fifty members, of whom one-third\n     were graduates of colleges. A recent writer in one of the most\n     intelligent and accurate of American journals has[19] taken\n     pains to collect and array a paragraph of important statistics\n     upon this subject, which we have taken leave to insert here,\n     though without verification, that, however, being hardly\n     necessary for our present purpose.\n\n  [19] New York _Evening Post_, January, 1876.\n\n     \"There were in the Continental Congress during its existence,\n     350 members, of these 118, or about one-third of the whole,\n     were graduates from Colleges. Of these, 28 were graduates from\n     the College of New Jersey in Princeton, 23 from Harvard, 23\n     from Yale, 11 from William and Mary, 8 from the University of\n     Pennsylvania, 4 from Columbia College, 1 from Brown University\n     and 1 from Rutger's College, and 21 were educated in foreign\n     Universities. These 118 graduates were distributed in the\n     Colonies as follows:--New Hampshire had 4 College graduates\n     among her delegates; Massachusetts had 17; Rhode Island had\n     4 graduates; Connecticut had 18 graduates; New York out\n     of her large delegation had but 8 graduates; New Jersey\n     had 11 graduates; Pennsylvania had 13 graduates; Delaware\n     had 2 graduates; Maryland had 7 graduates; Virginia had 19\n     graduates; North Carolina had 4 graduates; South Carolina had 7\n     graduates; Georgia had 5 graduates. We find that Princeton had\n     representatives from 10 of the colonies; Yale from 6; Harvard\n     from 5; the University of Pennsylvania from 3; William and Mary\n     from 2; and Columbia, Brown and Rutger's from 1 each. Fifty-six\n     delegates signed the Declaration of Independence. Of these, 28,\n     or just one-half, were College graduates.\"\n\nIncidentally, and as illustrative of the influence of college-bred\nmen in the Legislature, Mr. Adams, speaking of the great liberality\nof South Carolina in founding a college in that State, says:--\n\n     \"But no State ever made a better investment. During the first\n     part of this century the general accomplishments and political\n     ability of the statesmen of South Carolina were the just pride\n     of the State, and would have been the pride of any State. In\n     forming this high standard of intellectual and political power\n     the influence of the College was immeasurable.\"--_North American\n     Review_, January, 1876, pages 215, 216.\n\nIt is gratifying to us, British Colonists, and to the descendants\nof the U. E. Loyalists, thus to have from so important a source, an\nacknowledgment so candid and so honorable to men, many of whom were\nthe founders of Ontario and the Maritime Provinces of the Dominion.\nIt is an historical fact of equal significance, and an element of\nsocial and political strength to us in these British provinces,\nto know that it was to the thoroughness and breadth of culture\nwhich the American \"Revolutionary heroes\" received in early days\nin British colonial institutions which fitted them afterwards to\ntake so prominent and effective an intellectual part in the great\nstruggle which took place when they were in the prime of manhood.\nAnother gratifying reflection arises out of the fact that the high\nplace which the United States has taken in later years as a great\neducating nation is due to her following out the traditional policy\nof the Colonists of ante-revolution times.\n\nThis fact is clearly brought out by Mr. Gilman in the _North\nAmerican Review_ for January, 1876. We only quote the following\nremarks on this point, he says:--\n\n     \"When the new constitution of Massachusetts was adopted in 1780\n     public education received full recognition. An article (the\n     spirit of which was fully in accordance with the legislation\n     of 1647 [more than 200 years before]) was adopted, and _still\n     remains the fundamental law of the State_.... The constitution\n     of New Hampshire, as amended in 1784, transcribes very\n     nearly the same words of that section of the constitution of\n     Massachusetts already quoted,\" etc.--Pages 198, 199.\n\nThus Andrew Ten-Brook, Esq., in his _American State Universities_,\nsays:--\n\n     \"The introduction of an educational system into the New England\n     Colonies may be deemed substantially contemporaneous with\n     their settlement. It was of such a character, too, and so\n     energetically prosecuted, that education suffered little if\n     any deterioration in passing from Old to New England. It was\n     even more on this side than the other side of the ocean....\n     Thus Common School instruction at least was provided for all.\n     Higher schools, too, had an early beginning. What afterwards\n     was Harvard College was established but six years after the\n     settlement of Boston.... Every town [township] of fifty families\n     was obliged to support a school, and the same general state of\n     facts existed throughout New England. Classical schools followed\n     in regular succession. These were modelled after the Grammar\n     Schools of England, in which the founders of the Colleges had\n     themselves received their first classical training.... As early\n     as 1701, the law of Connecticut required every parent to see\n     that he had no child or apprentice in his household who could\n     not read the Word of God, and 'the good laws of the colony.'\n     The system embraced a High School in every town [township] of\n     seventy families, a Grammar School in the four chief county\n     towns to fit pupils for college, and a College to which the\n     general court [Legislature] made an annual appropriation of\n     \u00a3120.\"--Pages 1-3.\n\nMr. Ten-Brook, speaking of these New England schools, which were\nafterwards transplanted to each of Western States, says:--\n\n     \"They were the elements of that noble system out of which has\n     grown the present one, by the natural laws of development,\"\n     etc.--Page 18.\n\nMr. C. K. Adams, in his interesting paper on State Universities in\nthe _North American Review_ for October, 1875, in speaking of the\neducational policy of the colonies, \"pursued up to the time of the\nRevolution, says:--\n\n     \"In general terms it may be stated that, through all the dark\n     periods of our Colonial history, the encouragement of higher\n     education was regarded as one of the great interests of the\n     State. It was no doctrine of the Fathers that higher education\n     was less entitled to the fostering care of the commonwealth than\n     was the education offered by the Common Schools.\"--Page 374.\n\nThe \"Free School\" idea, of which we hear so much as the outgrowth\nof \"modern American civilization and enlightenment,\" was due to\nColonial thought and foresight. It was first broached by Jefferson,\nthree or four years before the treaty with Great Britain was\nsigned by which the United States became a nation. His plan was so\ncomprehensive that we reproduce it here. In a letter to the veteran\nphilosopher, Dr. Priestley, he thus unfolds it:--\n\n     \"I drew a bill for our [Virginia] Legislature, which proposed\n     to lay off every county into hundreds, or townships, of five\n     or six miles square. In the centre of each of them was to be a\n     free English School [to be supported, as his bill provided, \"by\n     taxation according to property.\"]\n\nThe whole Commonwealth was further laid off into ten districts,\nin each of which was to be a college for teaching the languages,\ngeography, surveying, and other useful things of that grade, and\nthen a single University for the sciences. It was received with\nenthusiasm (he goes on to say), but as he had proposed to make\nthe Episcopal College of William and Mary the University, \"the\ndissenters after a while began to apprehend some secret design,\"\netc.--_Ten-Brook's American State Universities_, pages 9, 10.\n\nA writer in the _North American Review_ for October, 1875, in\nreferring to Jefferson's scheme, says:--\n\n     \"The view entertained by Jefferson was by no means exceptional.\n     Indeed, a similar spirit had pervaded the whole history of our\n     Colonial life.\"--Page 379.\n\nThus this comprehensive scheme of public instruction for Virginia\nunfortunately failed; and that noble \"Old Dominion\" is in\nconsequence to-day immeasurably behind even the youngest of her then\nNew England contemporaries in the matter of public education.\n\nAs to the abiding influence of the old Colonial ideas in regard to\nhigher education, we quote the following additional remarks from Mr.\nGilman, in the _North American Review_, he says:--\n\n     \"In reviewing the history of the century, it is easy to see\n     how the colonial notions of college organization have affected\n     ... the higher education of the country, even down to our own\n     times. The graduates of the older colleges have migrated to the\n     Western States, and have transplanted with them the college\n     germs ... and every Western State can bear witness to the zeal\n     for learning which has been manifested within its borders by\n     enthusiastic teachers from the East.\"--Page 217.\n\nMr. Ten-Brook, in his _American State Universities_, also says:--\n\n     \"The New England colonists left the mother country in quest\n     of greater religious freedom. Their religious system was put\n     first, and carried with it a school system as perfect in\n     organization, and administered with equal vigor. This formed an\n     active leaven, which, at a later day, was to spread to other\n     parts.... Everywhere there was a considerable infusion of men\n     who had received in the European universities a liberal culture,\n     which they desired to reproduce on those shores. Early action\n     was full of promise. Probably, at a period from just before\n     the Revolution to the end of it, the average position of the\n     colonies in regard to lighter education relatively as to age,\n     and to the population and wealth, was quite as good as it is at\n     the present time.\"--Pages 16, 17.\n\nThis opinion of the writer is a virtual admission that in reality\nhigher education in the United States has not advanced in quality,\nthough it has in quantity. To be in 1876 merely where education\nwas \"relatively\" in 1776, is no advance at all, but rather\nretrogression. The cause of this declension, the writer thus\nincidentally admits:--\n\n     \"Most of the colonies established, or aided, the\n     (ante-revolution colleges named). The principle of the State\n     support to higher learning was not merely accepted, but was the\n     prevalent one.\"--Page 17.\n\nMr. Gilman, President of the Johns Hopkins' University, touching on\nthe same point, says:--\n\n     \"There was a civil as well as an ecclesiastical element in\n     most of these foundations. Harvard and Yale were chartered,\n     and, to some extent, controlled by colonial government of\n     Massachusetts and Connecticut, and were for a long time nurtured\n     by appropriations from the public chest....--Page 215.\n\n     \"These institutions were colleges of an English parentage and\n     model, not Scotch nor continental universities.... They were\n     disciplinary in their aim, and had more regard for the general\n     culture of large numbers, than for the advanced and special\n     instruction of the chosen few. They were also, to a considerable\n     extent, ecclesiastical foundations--finding the churches and\n     ministers their constant, and sometimes their only efficient\n     supporters. Harvard, Yale and Dartmouth were controlled by the\n     Congregationalists; Princeton was founded by the Presbyterians;\n     and New Brunswick, N.J. (Queen's, now Rutgers) by the Dutch\n     Reformed; William and Mary was emphatically a child of the\n     Church of England; and King's College (now Brown University) was\n     under the patronage of the Baptists....\n\n     \"The declaration of the original supporters of these colleges\n     indicate a desire to train up young men for service of the\n     State, not less distinctly and emphatically than to desire to\n     provide an educated ministry. Individual aid was also expected\n     and invited, and the names of Harvard and Yale perpetuate the\n     remembrance of such generous gifts.\"\n\nThen follows a eulogy upon these colonial colleges, and a tribute to\nthe intellectual vitality of their teaching, as shown in the mental\nequipment and breadth of culture exhibited by men who took part in\nthe perilous and stormy times of the American revolution. To this we\nhave already referred. Mr. Gilman, in following up his remarks in\nthe extract which we have just given, says:--\n\n     \"Hence these nine colleges were nurseries of virtue,\n     intelligence, liberality and patriotism, as well as of learning;\n     so that when the revolution began, scores of enlightened\n     leaders, both in council and in the field (and on both\n     sides), were found among their graduates. The influence of\n     academic culture may be distinctly traced in the formation of\n     the Constitution of the United States, and in the political\n     writings of Adams, Jefferson, Madison, Munroe, and many other\n     leading statesman of the period. A careful student of American\n     politics has remarked that nothing more strikingly indicates the\n     influence of the education given at Harvard, 'than the masterly\n     manner in which difficult problems of law and government were\n     handled by those who had received their instruction only from\n     that source.'\"--Pages 215, 216.\n\n\nTHE UNITED EMPIRE LOYALIST PERIOD IN UPPER CANADA.\n\nWe might pursue this branch of our subject further, were it\ndesirable. But that is not necessary. Our object was to show that\nto British Colonial foresight, zeal, and self-sacrifice, was due,\nnot only the foundation of the best colleges and universities on the\ncontinent, but the introduction and diffusion of the principle of\n\"free and universal education for the masses of the people.\" This\nwe have done on the authority of American writers themselves. We\nmight multiply examples on the subject; but the fact is already\nsufficiently established. We should rather seek to draw lessons of\ninstruction from the noble example of the devotion to education on\nthe part of our British colonial progenitors, whose descendants have\nshed such a lustre of heroic self-sacrifice and patriotism on the\nhistory and exploits of the United Empire Loyalists of the thirteen\ncolonies. To the Americans they have left a rich legacy from the\ncolonial times in such universities as Harvard, Yale, Columbia and\nPrinceton--of which the descendants of the expatriated Loyalists,\nno less than those of the victorious revolutionists, are so justly\nproud. Let us, as worthy representatives of these clear-headed and\nfar-seeing Loyalists, bequeath to our children as noble a heritage\nas the fathers of the founders of this Province did to New England,\nand indeed to the whole Republic.\n\nTrained in such an educational school, and animated with the\neducational zeal of these old colonial times, the \"United Empire\nLoyalists\" brought with them into Canada their love for education\nand their devotion to the sovereign.\n\nIn order to keep up the historical sequence of this Retrospect, I\nshall now refer to the early beginning of Educational life in Upper\nCanada, and then take up the thread of the narrative at the point\nwhere the educational forces--afterwards directed by the Rev. Dr.\nStrachan and the Rev. Dr. Ryerson--took practical form and shape.\n(See page 59.)\n\n\nGOVERNOR SIMCOE'S EDUCATIONAL VIEWS IN 1795.\n\nLieutenant-General J. Graves Simcoe, the first Governor of Upper\nCanada, arrived here in 1792. He was a man of comprehensive\nviews and noble impulses in regard to university education.\nHe was educated at Eton College and partly at Merton College,\nOxford, but entered the army before taking his degree. He served\nwith distinction under Wolfe at Quebec, and during the American\nrevolutionary war.\n\nIn April, 1795, Governor Simcoe addressed a letter to the Protestant\nEpiscopal Bishop of Quebec.[20] In that letter Governor Simcoe uses\nthe following striking language in describing the social condition\nof the people in the rural parts of Upper Canada, and the utter\nabsence of schools and churches, as contrasted with their existence\non the United States side of the lines. He said:--\n\n     \"There was nothing, in my late progress, that has given my equal\n     uneasiness with the general application of all ranks of the most\n     loyal inhabitants of the Province, that I would obtain for them\n     churches and ministers. They say that the rising generation\n     (of the U. E. Loyalist settlers) is rapidly returning into\n     barbarism. They state that the Sabbath, so wisely set apart for\n     devotion, is literally unknown to their children, who are busily\n     employed in searching for amusements in which they may consume\n     the day. And it is of serious consideration that on the approach\n     of the settlements of the United States, particularly on the\n     St. Lawrence frontier, these people, who, by experience, have\n     found that schools and churches are essential to their rapid\n     establishment (as a nation), may probably allure many of our\n     most respectable settlers to emigrate to them, while in this\n     respect we suffer a disgraceful deficiency.\"\n\n  [20] For fuller details on this point, see the section on the\n  universities, page 61.\n\nThe remedy which Governor Simcoe suggested for the state of things\nwhich he so graphically described is thus set forth in the same\nletter to the Bishop of Quebec. It was, as will be seen, entirely\ngeneral in its character:--\n\n     \"Nothing has happened since I left England, in the least, to\n     invalidate, to my own conception, the policy of the measure I\n     then proposed. And as far as may be now in the power of His\n     Majesty's ministers, I most earnestly hope that what remains to\n     be effected--that is by giving the means of proper education in\n     this Province, both in its rudiments and in its completion, that\n     from ourselves we may raise up a loyal, and, in due progress, a\n     learned clergy.\"\n\n\nEARLY BEGINNINGS OF EDUCATION IN UPPER CANADA, 1785-1805.\n\nA few particulars as to the kind of schools which existed in Upper\nCanada before and after the date of this letter may be interesting.\nFor instance, the first school opened (so far as I have been able\nto learn) was by the Rev. Dr. John Stuart, a Protestant Episcopal\nclergyman, and a United Empire Loyalist, who had been chaplain to\nthe provincial volunteers, and came into Upper Canada with them as a\nrefugee.[21]\n\n  [21] Rev. John Stuart. D.D., was born in Virginia in 1736. In 1769\n  he went to England to be ordained, and returned to Philadelphia in\n  1770. For seven years he labored as a missionary among the Iroquois\n  Indians at Fort Hunter. He was then aided by the famous Brant in\n  translating the New Testament into Mohawk. In 1781 he came to Upper\n  Canada and labored in this province as a missionary among the\n  refugee loyalists and Iroquois. He subsequently became rector of\n  Cataraqui (Kingston), and chaplain to the Legislative Council. He\n  died in 1811, aged 75 years. One of his sons was the late Archdeacon\n  Stuart, of Kingston; another was the late chief Justice, Sir James\n  Stuart of Quebec.\n\nIn the year 1785 Dr. Stuart opened a select classical school at\nCataraqui, (Kingston); and a Mr. Donovan taught the Garrison school\nthere. In 1786, Mr. J. Clarke taught a school in Frederickburg,\nand Mr. Smith one in Ernestown. In 1789, Mr Lyons kept school in\nAdolphustown. In the same year, Deacon Trayes, a Baptist, opened\none at Port Rowan. In 1792, Rev. Mr. Addison, an Episcopalian,\nopened a school at Newark (Niagara), then the seat of government.\nIn 1794, the Rev. Mr. Burns, a Presbyterian, (father of the late\nJudge Burns) opened a school at the same place; and in 1796, Mr.\nRichard Cockrel opened an evening school in Newark; Mr. Cockrel\nshortly afterwards transferred his school to the Rev. Mr. Arthur\nand removed to Ancaster, where he opened another school. A notice\nin the _York Gazette_ in 1796 stated that \"as schools were now\nopened, ignorance would be no longer tolerated.\" In 1797, Mr. James\nBlayney opened a school at Niagara. In 1798, Mr. Wm. Cooper opened\na school in George St., little York, (Toronto). In 1800, the late\nBishop Strachan opened a private school at Kingston, and in 1804,\none at Cornwall. In 1802, Mr. and Mrs. Tyler opened a school near\nNiagara; and in the same year, Dr. Baldwin, (father of the late Hon.\nRobert Baldwin) opened a classical school at York, and in 1803, the\nfirst school in Prince Edward district was opened at \"High Shore,\"\nSophiasburgh; another at \"Grassy Point,\" was taught by John James.\nRev. Wm. Wright, (Presbyterian) kept the first school at Meyer's\nCreek, (Belleville) in 1805. He was followed by Mr. Leslie. In that\nyear, Rev. Mr. Strachan held the first public examination of his\nschool at Cornwall.\n\n\nSTATE OF EDUCATION IN UPPER CANADA, 1795-1799.\n\nAs to the actual state of education in Upper Canada at this time, we\nget a brief glimpse from the travels of the Duc de la Rochefoucauld,\nwho visited Kingston in July, 1795. He says:--\n\n     \"In this district there are some schools, but they are few in\n     number. The children are instructed in reading and writing, and\n     pay each a dollar a month. One of the masters taught Latin, but\n     he has left, without being succeeded by another instructor in\n     the same language.\"\n\nAs to the character of the private schools thus established, and the\nfacilities of education which they afforded, we learn incidentally\nfrom letters and early books of travel, what they were. In a \"_Tour\nthrough Upper Canada, by a Citizen of the United States_,\" published\nin 1799, we learn that the policy of the government of that day, was\nto exclude \"schoolmasters from the States, lest they should instil\nRepublicanism into the tender minds of the youth of the province.\"\n\n\nFIRST OFFICIAL EDUCATIONAL MOVEMENTS IN UPPER CANADA, 1797, 1798.\n\nAs the result of the correspondence between the Governor and Bishop\nMountain, the question of a University and free grammar schools was\ndiscussed. The Governor referred the matter to the Upper Canada\nLegislature, which, in 1797 memorialized King George III, soliciting\na grant of land for the endowment of a grammar school in each\ndistrict, and a University for the whole Province. To this request\nthe King gave his assent, and, in 1798, the \"chief civil officers\"\nin Upper Canada recommended that \"500,000 acres of land be set apart\nfor the establishment of a grammar school in each district and a\ncentral University for the whole Province.\" They also recommended\na grant for the erection of a \"plain but solid and substantial\nbuilding for a grammar school in each district, containing a school\nroom capable of holding 100 boys without danger to their health\nfrom too many being crowded together, and also a set of apartments\nfor the master, large enough for his family and from ten to twenty\nboarders.\"\n\nThe salaries proposed to be given were: \u00a3100 for the head master,\n\u00a350 for the assistant master; and \u00a330 for repairs, etc., Kingston\nand Newark (Niagara) were recommended as eligible sites for schools;\nafter which, when the funds were sufficient, schools were to be\nestablished at Cornwall and Sandwich. York (Toronto) was recommended\nas entitled to the University, and for the establishment and support\nof which a sum at least equal to that granted to the four schools\nwas named.\n\nIn 1799, an act was passed by the Upper Canada Legislature \"to\nprovide for the education and support of orphan children.\"\nIt authorized the township wardens, with the consent of two\nmagistrates, to bind and apprentice, until they became of age,\nchildren deserted by their parents. In the same year an orphan\nschool was opened near St. Catherines.\n\n\nEDUCATIONAL PIONEERS IN UPPER CANADA.\n\nGovernor Simcoe authorized the Hon. Messrs. Cartwright and Hamilton,\nto select a person to take charge of the proposed college. The\ncelebrated Rev. Dr. Chalmers having declined the appointment, it was\naccepted by Mr. (late the Right Reverend Doctor) Strachan (Bishop of\nToronto) then a schoolmaster at Kettle, Scotland; but his arrival at\nKingston, on the 31st of December, 1799, he found that the project\nof a college had been abandoned, Governor Simcoe, in the meantime,\nhaving left for England.\n\nIt was soon discovered that half a million of acres of land would\nendow but few grammar schools, land being then only worth a shilling\nper acre: the schema had, therefore, to be abandoned. Meanwhile\nthe Hon. Mr. Cartwright made an arrangement with Mr. Strachan to\ninstruct his sons, and a select number of pupils for three years. In\n1804, Mr. Strachan, having been ordained, removed to the mission of\nCornwall, where, at the request of the parents of his former pupils,\nhe opened a private school.\n\nFor several years this school was the only one of any note in\nUpper Canada; and in it and in Mr. Strachan's school at York, were\neducated many of those gentlemen who have filled some of the most\nimportant positions in the province. Subsequently Mr. Strachan's\nschool was constituted the grammar school of the Eastern district.\nIn 1806, a temporary act was passed by the Legislature and made\npermanent in 1808, establishing a classical and mathematical school\nin each of the eight districts into which Upper Canada was then\ndivided. In the same year(1806) at the suggestion of Dr. Strachan\nan Act was passed, granting \u00a3400 for the purchase of apparatus for\nillustrating the principles of Natural Philosophy, which were to be\ndeposited in the hands of a person employed in the instruction of\nyouth. In 1807 an appropriation of \u00a3800 a year for four years was\nmade to provide for the salaries of masters in the Grammar Schools\nto be maintained in each of the districts into which Upper Canada\nwas divided. These masters were to be engaged by trustees appointed\nby the governor, and the governor's sanction was also necessary for\nthe teacher's appointment. There is still in existence the letter,\ndated, April 16th, 1807, signed by Governor Gore, appointing the\nRev. George Okill Stewart, D.D., Archdeacon of Kingston, first Head\nMaster of the Home District Grammar School at York (Toronto).\n\nNorth of what is now Adelaide Street (formerly Newgate Street),\nbounded westward by Church Street, and eastward by Jarvis Street,\nwas a large field, almost square, containing about six acres--for\nmany years the playground of the District Grammar School.\n\nIn the south-west corner of it, some hundred feet or more from the\nstreet boundaries, was elected the plain wooden building, about\nfifty-five feet long by forty wide in which, on the first Monday\nof June, 1807, when the population of the town was only about five\nhundred, the Grammar School was opened. It was attended by the sons\nand daughters of the well-to-do citizens of York, and on the few\nexisting records may be found many a well known name.\n\nIn 1812, the Rev. John Strachan. D.D., was appointed Rector of\nYork, and succeeded the Rev. Mr. Stewart as Head Master of this\nschool.[22] Mr. Barnabas Bidwell (father of the late Hon. M. S.\nBidwell) kept a good Latin School at Bath, on the Bay of Quint\u00e9, in\n1811. In 1813, he removed to Kingston, where he taught for twenty\nyears until he died in 1833.\n\n  [22] _Canadian Educational Monthly_, February 1889, page 58.\n\nIn 1820, the \"Central School at York\" was opened under the\nmastership of Mr. Joseph Spragge, father of the late Chief Justice\nSpragge. Lieutenant-Governor and Lady Sarah Maitland took a special\ninterest in the success of this school.\n\nIn a _View of Upper Canada_, published at Baltimore in 1814, by Mr.\nM. Smith, (who resided in the Province from 1808 until the breaking\nout of the War in 1812) he said:--\n\n     \"The greater part of the inhabitants are not well educated,\n     for, as they were poor when they came to the Province and the\n     country being but thinly settled for a number of years, they\n     had but little chance for the benefit of schools. But since the\n     country has become settled, or the inhabitants rich, or in a\n     good way of living, which is almost universally the case, they\n     pay considerable attention to learning. Ten dollars a year is\n     the common price given for the tuition of each scholar by good\n     teachers.\"--Page 52.\n\nIn 1813, Rev. John Langhorn (a Church of England missionary at\nEarnestown and Bath from 1787 to 1812, and teacher of a school)\nmade a present of his library to the inhabitants of the Bay of\nQuinte district. In 1814 Rev. Robert Baldwin was appointed Grammar\nschoolmaster at Cornwall, vice Rev. John (afterwards Dean) Bethune\nresigned. In 1815 the Midland District School, so-called, was\nincorporated.\n\nDr. Strachan resigned the head mastership of the District School\non July 1st, 1823. He was succeeded by Mr. Samuel Armour, M.A., a\ngraduate of Glasgow University, who afterwards became a clergyman\nof the English Church, and officiated many years in the township of\nCavan.\n\nThe Rev. Thomas Phillips, D.D., an accomplished scholar, came out\nfrom England in 1825 to take charge of the school, and remained in\nthe position of headmaster, much honored and beloved by his pupils,\nuntil, in 1830, chiefly by the exertions of the Governor, Sir John\nColborne, Upper Canada College was established and the work of the\ncollege began in the old District Grammar School building. Classes\nwere opened in the new buildings erected in another part of the\ncity for the college in 1831, and the Grammar School was closed,\nthe building being removed from its original site to the line of\nNelson street (now Jarvis street), and fenced into a plot about 70 \u00d7\n120 feet. The remaining portion of the six acres was handed over to\nUpper Canada College.\n\nOn the active remonstrance of the citizens living in the eastern\npart of Toronto, the school was re-opened and secured to the city,\nMr. Charles N. B. Cosens being appointed headmaster in 1836, and\nsucceeded by Mr. Marcus C. Crombie in 1838.[23]\n\n  [23] _Ibid._, page 59.\n\n\nEARLY EFFORTS TO ESTABLISH COMMON SCHOOLS 1816-1820.\n\nIn 1816, seven years after the establishment of District Grammar\nSchools, a praiseworthy effort was made to provide for the\nestablishment and maintenance of common schools.\"[24] A liberal\ngrant of $24,000 a year, for four years, was made as an experiment.\nWhether the experiment was a success, or not, does not appear, but\nin 1820, the grant was reduced to $10,000 a year.\n\n  [24] In 1816, an Act was passed granting \u00a3800 for the purchase\n  of a library for the use of the Legislative Council and house of\n  Assembly.\n\n\nSTATE OF EDUCATION IN UPPER CANADA, 1784-1819.\n\nIn regard to the state of education of Upper Canada in 1817, and\nthe fluctuating character of its progress since the settlement of\nthe Province, in 1784, up to that time, Mr. Gourlay, a well-known\nCanadian politician and author, writes as follows:--\n\n     \"There is no college in Upper Canada, but there are said to be\n     several townships of land set apart for the purpose of endowing\n     such institution, when the population and circumstances of the\n     Province shall require it.\n\n     \"No provision is made by law for free schools. The inhabitants\n     of the severe! townships are left to a voluntary support of\n     schools, according to their own discretion.\n\n     \"An Act of the Provincial Legislature, in 1807, granted a\n     hundred pounds a year to the teacher of one school, in each\n     of the eight districts under the direction of trustees. In\n     some districts the school thus provided for is made a free\n     school; but in other districts the salary is considered as a\n     public encouragement to a teacher of literary eminence, in\n     addition to the compensation received for the tuition of each\n     scholar.\"--_Statistical Account of Upper Canada, etc., by Robert\n     Gourlay, 2 vols., London, 1822._\n\nThe Rev. Dr. Strachan became a master of one of these schools, and\nRev. George Ryerson and his brother, Egerton, master and usher of\nanother.\n\nAs to the state of feeling in the rural parts of the oldest settled\nportions of Upper Canada, we make the following extracts from a\nletter written to Mr. Gourlay from the township of Grimsby, in 1818,\nby a highly respected resident there, William Crooks, Esq. Mr.\nCrooks remarks:--\n\n     \"The state of education is at a very low ebb, not only in the\n     township, but generally throughout the [Niagara] district;\n     although the liberality of the legislature has been great in\n     support of the district (Grammar) schools, (giving to the\n     teachers of each \u00a3100 per annum), yet they have been productive\n     of little or no good hitherto, for this obvious cause, they\n     are looked upon as seminaries exclusively instituted for the\n     education of the children of the more wealthy classes of\n     society, and to which the poor man's child is considered as\n     unfit to be admitted. From such causes, instead of their being\n     a benefit to the Province, they are sunk into obscurity, and\n     the heads of most of them are at this moment enjoying their\n     situations as comfortable sinecures. Another class of schools\n     has, within a short time, been likewise founded upon the\n     liberality of the legislative purse, denominated common, or\n     parish, schools, but like the preceding, the anxiety of the\n     teacher employed, seems more alive to his stipend than the\n     advancement of the education of those placed under his care:\n     from the pecuniary advantages thus held out, we have been\n     inundated with the worthless scum, under the character of school\n     masters, not only of this, but of every other country where\n     the knowledge has been promulgated, of the easy means our laws\n     afford of getting a living here, by obtaining a parish school,\n     which is done upon the recommendation of some few freeholders,\n     getting his salary from the public, and making his employers\n     contribute handsomely beside.\n\n     \"It is true, rules are laid down for their government, and the\n     proper books prescribed for their use; but scarcely in one case\n     in ten are they adhered to, for in the same class you will\n     frequently see one child with Noah Webster's spelling book in\n     this hand, and the next with Lindley Murray's. However prone\n     the teachers are to variety in their schools, much blame is to\n     be attributed to the trustees, who are in many instances too\n     careless and I might almost add too ignorant to discriminate\n     right from wrong, in the trust they have undertaken for the\n     public benefit. It is, therefore, not to be wondered at why\n     the parish school system should meet with almost universal\n     reprobation from most discerning men.\n\n     \"Of these parish schools, we are burdened with a liberal share,\n     having no less than three of them. If the establishment of this\n     system was meant by the legislature to abbreviate the present\n     enormous price of education, they have been miserably deceived;\n     for I can see no alteration or reduction from the charge made\n     before the passing of the act. The price then was 12_s._ 6_d._\n     [_i.e._ $2.50,] and is now the same, per quarter.\"\n\nIn July, 1819, provisions was made for an additional district\ngrammar school; for holding annual public examinations; for\nreporting the condition of the schools to the governor, and for\neducating ten common school pupils, free of charge, at each of the\nnine public grammar schools already established; but the provincial\nallowance to teachers of grammar schools was reduced to \u00a350 in all\ncases where the numbers of pupils did not exceed ten.\n\n\nFITFUL EDUCATIONAL PROGRESS FROM 1822 TO 1829.\n\nIn 1822, Sir Peregrine Maitland, the Lieutenant-Governor, submitted\nto the Imperial Government a plan for organizing a general system of\neducation, including elementary schools; and, in 1823, he obtained\npermission from England to establish a Board of Education for\nthe general superintendence of this system of education, and for\nthe management of the university and school lands throughout the\nProvince. The members of this Board, with Rev. Dr. Strachan at its\nhead, as chairman, were: Hon. Joseph Wells, Hon. G. H. Markland,\nRev. Robert Addison, Hon. J. B. Robinson, and Thomas Ridout, Esq.\nThis Board prepared some general regulations in regard to the\nschools and proposed a plan by which to exchange 225,944 acres of\nthe less valuable of the school lands for the more productive\nClergy Reserve lands. The plan having been approved by the Home\nGovernment, was carried into effect by the Governor soon after. In\n1824, the first attempts towards providing the public with general\nreading books, in connection with the Common and Sunday schools,\nwere made. The sum of \u00a3150 was annually appropriated for this\nobject, and authorized to be expended by the Provincial Board of\nEducation in the purchase of \"books and tracts designed to afford\nmoral and religions instruction,\" and distributed equally among all\nthe districts of Upper Canada.\n\nThus were presented the dim outlines of a system of public\ninstruction which it was clear the necessities of the country\nrequired, but which for want of a vigorous and systematic\nsupervision was gradually permitted to languish, and the legislative\nenactments themselves were suffered to become almost obsolete on the\nstatute book.\n\nIn January, 1824, the Common School Act was made to apply \"to all\nschools that are now or may hereafter be established and kept among\nthe Indians who shall be resident within the limits of any organized\ncounty or township within this Province, excepting such schools as\nshall or may be otherwise provided for.\"[25] Provision was also made\nfor the examination of Common School teachers by County Boards of\nEducation.\n\n  [25] All the Indian schools of the Province, which are chiefly\n  sustained by various religious bodies are now under the control\n  of the Indian Department at Ottawa. The following is an account\n  of a typical Indian school at the Credit in 1830: The school room\n  in a large and commodious apartment with tiers of raised benches\n  in the rear: on one division of which sit the girls, and the boys\n  on the other. There are also desks, and slates for ciphering and\n  copy-books and copper-plate lines for whose who write. The Bible\n  and Testaments, and some of the other books, are English printed\n  and some American. No sectarian intolerance prevails in that way.\n  Among the school furniture are a handsome map of the world, the\n  arithmecon, attractive alphabets in pasteboard, regular figures\n  illustrative of geometry, some of them cut out of wood and some\n  of them made of pasteboard: the picture of Elijah fed by ravens,\n  figures of birds, fishes and quadrupeds on pasteboard, ,\n  accompanied with the history of each animal: the figure of a clock\n  in pasteboard, by which to explain the principles of the time-piece.\n  The walls are adorned with good, moral maxims; and I perceived that\n  one of the rules was rather novel, though doubtful in place here. It\n  was, \"No blankets to be worn in school.\" The attendance is about 50\n  Indian children. The girls are taught by Miss Rolph, sister of the\n  late Member of Middlesex: the boys by Mr. Edwy Ryerson, a younger\n  brother of the late editor of the _Christian Guardian_, Rev. Egerton\n  Ryerson. The translating office is occupied by Mr. Peter Jones, the\n  Indian minister.\n\n\nSTATE OF EDUCATION IN UPPER CANADA, 1827-1829, FROM THE OFFICIAL\nREPORTS, ETC.\n\nThe number and condition of the Common and other schools in Upper\nCanada in 1827 may be gathered from \"An appeal to the Friends of\nReligion, and Literature, in behalf of the University of Upper\nCanada,\" published in London in 1827 (of which I have an original\nMS. copy). Dr. Strachan says:--\n\n     _School in Upper Canada, 1827._--\"In about 340 Common Schools\n     in Upper Canada from 12,000 to 14,000 children are taught\n     reading and writing, the elements of arithmetic, and the first\n     principles of religion. The people, scattered as they are\n     over a vast wilderness, are thus becoming alive to the great\n     advantage of educating their children ... insomuch so, that\n     schools supported by subscriptions are more in number than\n     those established by law. Provision is made by statute for the\n     translation of some of the more promising scholars from the\n     Common to the District Schools, where the classics and practical\n     mathematics are taught. In these schools (eleven in number)\n     there are at present 300 young men acquiring an education to\n     qualify them for the different professions....\n\n     _Niagara District Board of Education, 1828._--\"It must be\n     admitted generally that after the approval and appointment of\n     teachers by the Trustees the Board have not rejected teachers\n     however incompetent, from a regards to the wishes of their\n     employers, the terms for tuition being so very low as not to\n     induce men of sufficient qualifications generally to engage in\n     the humble and ill-requited duties.\n\n     Reading, writing and arithmetic have been uniformly taught\n     in all schools. Grammar and geography in a few. Religious\n     instruction has not been overlooked. Upon the whole, the system\n     of education in this district may be considered as efficient as\n     can be expected upon the footing on which it is placed.\"--_Rev.\n     Thomas Creen._\n\n     _Provincial Board of Education, 1828._--The number of District\n     Grammar Schools in operation in the Province is 11; pupils in\n     attendance, 372. Number of Common Schools, 401; number of pupils\n     in attendance, 10,712--increase over 1827 of nearly 2,000 pupils.\n\n     The President of the Board (Rev. Dr. Strachan) last summer\n     visited in person all the districts of the Province, and not\n     only inspected the grammar schools, but examined minutely the\n     systems of management adopted by their respective teachers.\n\n     In order to produce a greater uniformity of the system to\n     supply, in some measure, the want of experience to younger\n     teachers, the President has submitted an outline of study for\n     the grammar schools, the adoption of which, the board cannot but\n     think would be highly beneficial, and produce a higher standard\n     of education throughout the Province.[26]\n\n  [26] This course of study is appended herewith on the next page.\n\n     In some places girls are admitted to the district schools,\n     for want of good female schools, but the admission of female\n     children interferes with the government which is required in\n     classical seminaries.\n\n     In a population of nearly 200,000, at least one-fifth, or\n     40,000, is composed of children between the ages of 5 and 15\n     who should be going to school. Taking the number of those who\n     are benefited in the Common and Sunday schools at 15,000, then\n     the expense to the Province is about 3_s._ 9_d._ each. In\n     some districts the salaries allowed to the schoolmasters of\n     the Common Schools are exceedingly small. They range from \u00a312\n     10_s._ downwards. In some little more than \u00a35, and in one less\n     than that trifling sum. Many schools continue only six months;\n     others eight months in the year. The natural consequence of this\n     state of things is that superior teachers desert the Common\n     Schools as soon as they can procure any other employment, and\n     many persons resort to the occupation of teachers merely as a\n     temporary expedient. These latter are without experience, which\n     is all-important to an instructor of youth and can have little\n     desire to establish a reputation in an employment to which they\n     have only recourse for present convenience.\n\n     In the sister Colony of Nova Scotia, the sum of \u00a31,000 is\n     annually appropriated to the Common Schools and divided among 12\n     counties, not equally, but in proportion to population. One law,\n     by giving the same sum to each district, whether populous or\n     not, in that particular requires alteration.\n\n     The Board would submit that in addition to the public allowance,\n     a power should be given to the townships to assess themselves\n     for the schools. In Nova Scotia it is provided in the statute\n     that two-thirds of the freeholders may, under certain\n     conditions, tax themselves according to their ability for the\n     support of education, and that no school of 30 scholars shall\n     be entitled to the stipulated aid of \u00a320, unless the teacher\n     receives _bona fide_ from his employers \u00a310, together with this\n     sum, exclusive of and in addition to board, etc.; and that no\n     school of 15 scholars shall be entitled to the stipulated sum of\n     \u00a315 unless the teacher receives from his employers \u00a325 per annum\n     as aforesaid. The Board had distributed to the Common Schools a\n     large quantity of useful school books including Mavor's Spelling\n     Book. They have also contracted for 2,000 copies of this\n     excellent work to be executed on cards for the township schools.\n     There appears to be a great scarcity of arithmetic books in\n     the Province, and those in use are in general too difficult\n     or deficient in matter, etc. The President has, therefore,\n     undertaken to draw up a short manual on the subject, suitable to\n     the state and business of the country.\n\n     Neither the sick nor the destitute have higher claims upon the\n     public than the ignorant. The want of knowledge brings all other\n     wants in its train; and if education be regarded as a charity,\n     it is a charity of which the blessings are without alloy. It\n     demands no jealous scrutiny of the claims of its applicants,\n     nor does it require to be so stinted as not to multiply their\n     number.--_Rev. John Strachan, President, York, 5th February,\n     1829._\n\n       *       *       *       *       *\n\n     _Midland District Grammar School, 1829._--It is to be lamented\n     that so few of the wealthy farmers of the Midland District\n     avail themselves of the means of giving their sons that degree\n     of education which the public school can readily afford, which\n     would qualify young persons to discharge the duties of the\n     magistracy and other public situations with credit to themselves.\n\n     The trustees regret that no poor children are educated gratis\n     under the patriotic and very benevolent provision of the\n     statute, in consequence of no returns being made to them of the\n     most promising scholars from the Common School. The trustees,\n     therefore, submit, whether in order to encourage native genius\n     in humble circumstances, some means might not be devised of\n     maintaining all the 10 children whom the statute authorizes the\n     trustee to select for gratuitous instruction.--_Rev. George\n     O'Kill Stuart, Thomas Markland and John Macaulay, Kingston,\n     December 30th, 1829._\n\n       *       *       *       *       *\n\n     NOTE.--The local reports for the succeeding year are chiefly\n     statistical and explanatory and contain no general remarks. From\n     a report signed by the Rev. Dr. Strachan and Hon. Wm. Allan in\n     1839, it would appear that things had for years remained in\n     _statu quo_. They say:--\n\n     \"For many years elaborate reports were sent from this (Home\n     District) Board of Education detailing what were believed to be\n     the alterations necessary to render the present Common School\n     Act efficient. In consequence of these, and like reports from\n     other districts, a measure for the establishment of such schools\n     has been for more than six years before the Legislature, which\n     purposes to provide remedies for the defects which are met with\n     in the working of the present system.\"\n\n\nCOURSE OF STUDY SUGGESTED BY REV. DR. STRACHAN AS SUITABLE FOR THE\nDISTRICT GRAMMAR SCHOOLS THROUGHOUT THE PROVINCE, 1829.\n\nIn a \"letter from the Rev. Dr. Strachan to Rev. A. N. Bethune,\nRector of Cobourg,\" dated October 6th, 1829, he thus sketched a\ncourse of study for Grammar Schools:\n\n\n_First Year--Boys from 7 to 9._\n\n1st. _Latin._--Eton Grammar; Vocabulary; Corderius; Select\u00e6 e\nProfanis.\n\n2nd. _English._--Mavor's Spelling Book; Enfield's Lessons; Walker's\nLessons; Murray's Lessons; Blair's Class Book; English Grammar;\nWriting; Arithmetic, chiefly mental.\n\n\n_Second Year--Boys from 9 to 11._\n\n1st. _Latin._--Grammar; Valpy's Delectas; New Testament; Daley's\nExercises; Exampla Minora; Entropius; Ph\u00e6drus; Cornelius Nepos.\n\n2nd. _English._--Grammar and Reading, as before; Writing and\nArithmetic (mental and mixed); Geography; Civil and Natural History\nand Elocution.\n\n3rd.--To commence French.\n\n\n_Third Year--Boys from 11 to 13._\n\n1st. _Latin_--Grammar; Bailey's Exercises; Cornelius Nepos; C\u00e6sar;\nOvid's Metamorphoses; Nonsense Verses; Psalms into Latin Verse;\nExampla Moralis; Versions or rendering English into Latin.\n\n2nd. _Greek_--To commence about the middle of the third year: Eton\nGrammar, or Nelson's edition of Moore's Grammar; Greek Vocabulary;\nNew Testament; Greek Exercises.\n\n3rd. _English._--Grammar; Writing; Elocution; Civil and Natural\nHistory; Geography, Ancient and Modern; English Composition.\n\n4th. _Arithmetic._--And to commence Algebra.\n\n5th. _French._\n\n\n_Fourth Year--Boys from 12 to 14._\n\n1st. _Latin._--Grammar; Terence Virgil; Horace; Sallust; Cicero;\nLivy; Latin composition, verse and prose; Grotius de Veritate\nExampla Moralia.\n\n2nd. _Greek._--Eton Grammar; Gr\u00e6ca Minora; Greek and Latin\nTestament; Xenophon; Homer.\n\n3rd. _English._--Grammar and Composition; Civil and Natural History;\nGeography, Ancient and Modern, use of the globes; construction of\nmaps.\n\n4th. _Mathematics._--Arithmetic; Book-keeping; Algebra; Euclid.\n\n5th. _French._\n\n\n_Fifth Year--Boys from 14 to 16._\n\n1st. _Latin._--Virgil; Horace; Livy, Juvenal; Tacitus; Composition,\nin prose and verse.\n\n2nd. _Greek._--Gr\u00e6ca Majora; Homer; Thucidides; Composition, in\nprose and verse.\n\n3rd. _English._--Grammar and Composition; Elocution; Civil and\nNatural History; Geography, Ancient and Modern; use of the globes;\nconstruction of maps.\n\n4th. _Mathematics._--Algebra; Euclid; Trigonometry, Application to\nheights and distances; Surveying; Navigation; Dialling; Elements of\nAstronomy, etc.\n\n5th. _French._\n\n\nREV. DR. STRACHAN'S SYSTEM OF SCHOOL MANAGEMENT.\n\nRev. Dr. Scadding, in his sketch of Dr. Strachan, \"The first Bishop\nof Toronto--a Review and a Study,\" says:\n\n     \"The system pursued in Dr. Stachan's school at Cornwall and\n     afterwards at York, exhibited features that would have gratified\n     the advanced educationists of the present age. In that system\n     the practical and the useful were by no means sacrificed to the\n     ornamental and theoretical, or the merely conventional. Things\n     were regarded as well as words.... In regard to things--the\n     science of common objects--we doubt if in the most complete of\n     our modern schools there was ever awakened a greater interest\n     or intelligence in relation to such matters. Who, that had once\n     participated in the excitement of its natural history class,\n     ever forgot it? Or in that of the historical or geographical\n     exercises? We venture to think that, in many an instance, the\n     fullest experience of after life, in travel or otherwise, had\n     often their associations with ideas awakened then; and often\n     compared satisfactorily and pleasurably with the pictures of\n     places, animals and persons given, rudely it may be, in text\n     books, ransacked and conned in a fervour of emulation then.\n     The manner of study in these subjects was this: each lad was\n     required to prepare a set of questions, to be put by himself\n     to his fellows in the class. If a reply was not forthcoming,\n     and the information furnished by the questioner was judged\n     correct the latter 'went up' and took the place of the other.\n     This process, besides being instructive and stimulating to the\n     pupils, possessed the advantage of being, as it often proved,\n     highly diverting to the teacher.\"\n\nOn this system Dr. Strachan himself remarks:\n\n     \"The method of instruction by question and answer possesses\n     many advantages over any other, and is not only the very best\n     and shortest, but the most satisfactory. In this system the\n     deficiencies of each scholar becomes manifest, and the teacher\n     knows to what particular points he must direct his explanations.\n     There is no time for inattention or wandering; the question\n     and necessity for reply compel attention and recollection. The\n     children, if the teacher proceed with conciliatory firmness,\n     acquire lively interest in the lesson, for each is particularly\n     addressed and brought forward with action.\"[27]\n\n  [27] The _Christian Recorder_, edited by Rev. Dr. Strachan, York,\n  1830, vol. 1, page 182.\n\nThe late Bishop Fuller, who was also one of Dr. Strachan's pupils,\nalso states that:--\n\n     \"He had a remarkable talent for interesting boys in their work;\n     and, by taking a deep interest in it himself, he led them to\n     do the same. He was very original in many of his plans for\n     promoting the good of his school. Amongst others, which I never\n     met with elsewhere, was one of making the boys question one\n     another on certain of the lessons. This made the boys quick at\n     seizing on the leading points in the lessons, ready at shaping\n     questions, and deeply interested in the questions and answers.\n     The Bishop took as deep an interest in the questioning and\n     answering of the boys as they did themselves; and thus this\n     plan, whilst it was of great service to the boys in various\n     ways, tended strongly to bind master and scholars together.\"[28]\n\n  [28] Sermon on the Death of Bishop Strachan, _Journal of Education_\n  for U. C., vol. xx. (1868), page 182.\n\nAs to his method of teaching arithmetic, he explains it in the\nfollowing words:\n\n     \"In a new country like this, a variety of branches must be\n     taught in every respectable school. Young men ... are anxious\n     to get forward as fast as possible, and even those destined for\n     the learned professions are seldom allowed the time requisite\n     for acquiring the knowledge previously necessary. These\n     considerations induced me to turn my thoughts to the discovery\n     of some sure, and at the same time, expeditious method of\n     teaching arithmetic. This object I have accomplished with a much\n     greater degree of success than I dared to promise myself.\n\n     \"I divide my pupils into separate classes according to their\n     progress. Each class has one or more sums to produce every\n     day, neatly wrought upon their slates. The work is carefully\n     examined, after which I command every figure to be blotted out,\n     and the sums to be wrought under my eye. The one whom I happen\n     to pitch upon first gives, with an audible voice, the rules and\n     reasons for every step, and as he proceeds the rest silently\n     work along with him figure for figure, but ready to correct him\n     if he blunder that they may get his place. As soon as this one\n     is finished, the work is again blotted out and another called\n     upon to work the question aloud as before, while the rest\n     proceed along with him in silence, and so on round the whole\n     class.... This method of teaching arithmetic possesses this\n     important advantage, that it may be pursued without interrupting\n     the pupils' progress in any other useful study. The same\n     method of teaching Algebra has been used with equal success.\n     Such a plan is certainly very laborious, but it will be found\n     successful, _and he that is anxious to spare labor ought not to\n     be a public teacher_.\"[29]\n\n  [29] Preface to \"A Concise Introduction to Practical Arithmetic, for\n  the use of Schools: By the Rev. John Strachan, Montreal. Printed by\n  Nahum Mower, 1809.\"\n\nDesiring to give a local interest to the exercises in his book,\nDr. Strachan gave several examples from Canadian subjects. Thus a\nquestion is addition reads:--\n\n     \"From Quebec to Montreal is 180 miles--from thence to Kingston\n     200--from thence to York 149--from thence to Niagara 78\n     miles--from thence to Detroit, 210. Required the distance from\n     Quebec to Detroit. _Answer_--317 miles.\"\n\nAgain a question in multiplication reads:--\n\n     \"The distance from Quebec to Montreal is 180 miles, supposing\n     the road 17 yards broad, how many square yards does it contain?\n     _Answer_--5,385,600 yards.\"\n\nAs to his diligence as a student, while yet a teacher. Dr. Fuller\nremarks:--\n\n     \"The late Bishop said to me on one occasion: 'I had to study\n     every night quite as hard as the boys; for I was not much in\n     advance of the highest class in school. These and parochial\n     duties demanded sixteen hours every day,--and yet these nine\n     years were the happiest years of my life.\"\n\n\nREV. DR. STRACHAN'S CAREER AS A TEACHER.\n\nHaving been appointed Minister at Cornwall in 1803, Dr. Fuller\nstates that there he was:--\n\n     \"Induced to resume his school, at the solicitation of the\n     parents of those boys who had been in his school at Kingston,\n     and of others, both in Lower and Upper Canada, who were desirous\n     of placing their sons under a master so practical, wise and\n     successful, as he had proved himself to be. Thus he commenced\n     the school at Cornwall, which afterwards became so celebrated,\n     and at which were educated the first men that Canada has\n     produced, and of whom she may well be proud--such men as the\n     late Sir J. B. Robinson, Judge Maclean, Sir J. B. Macaulay, Sir\n     Allan MacNab, Judge Jones, Mr. Stanton, the Bethunes (Alexander,\n     John and Donald), Sir James Stuart, and his brother Andrew\n     Stuart, besides many others who have reflected credit on our\n     country.\n\n     \"The Bishop had a great faculty for not only attaching his\n     scholars to him, but also for inducing them to apply themselves\n     most assiduously to their studies. He told me he made it a rule,\n     during the time he kept school, to watch closely every new\n     boy, and at the end of a fortnight, to note down in a book his\n     estimate of the boys who had passed through his hands.\n\n     \"He was never afraid of having his dignity lowered by liberties\n     taken with him, and he always felt every confidence in his\n     position and entered warmly and personally into many of the\n     boys' amusements, and thus gained an immense influence over\n     them. The influence over his pupils has been shown in the\n     fact, that almost all of them embraced his principles; and the\n     love and affection for him of his celebrated Cornwall school\n     was shown many years ago, when the surviving members thereof\n     presented him with an address[30] and a most beautiful and\n     costly candelabra. Nor did his more recent scholars entertain\n     less affection for him, though they never proved it so\n     substantially as did those of his Cornwall School.... He was\n     an excellent teacher. His scholars were well grounded in their\n     work. The grammar was well mastered, and every rule thereof\n     deeply impressed on the memory. Every lesson was thoroughly\n     dissected, and everything connected with it thoroughly\n     understood, before we passed on to another lesson.\"[31]\n\n  [30] The principal signers of the address were Sir J. B. Robinson,\n  Sir J. B. Macaulay, Very Rev. Dean Bethune, Right Rev. Bishop\n  Bethune, Hon. Chief Justice McLean, Hon. Justice Jones, Hon. W.\n  B. Robinson, Hon. G. S. Boulton, Rev W. Macaulay, Judge (George)\n  Ridout, Surveyor-General Chewett, Col. Gregg, Capt. Macaulay, R.A.,\n  Inspector-General Markland, Sheriff McLean, Messrs. T. G. Ridout,\n  P. Vankoughnet, S. P. Jarvis, J. Radenhurst, R. G. Anderson, R.\n  Stanton, and others.\n\n  [31] _Journal of Education_ for U. C., Vol. xx. (1868), page 183.\n  For further reference to Dr. Strachan's educational efforts see the\n  sections on universities, page 59 _et seq._\n\n\nMR. JOSEPH HUME'S ESSAY ON EDUCATION EDITED AS A CATECHISM BY MR.\nWM. LYON MACKENZIE IN 1830.\n\nIn 1830 Mr. Mackenzie republished at York (Toronto), in pamphlet\nform, the first part of a Catechism of Education, prepared by Mr.\nJoseph Hume, M.P., in England. The pamphlet in my possession is\nworn and weather-stained. It is inscribed to David Thorburn, of\nQueenston, and extends to 46 pages. In his preface Mr. Mackenzie\nsays:\n\n\"To Mr. Joseph Hume.--The compiler is indebted for an Essay on\nEducation, which lays down and explains principles of vital\nimportance to the best interest of the Canadians, the perusal of\nwhich first suggested the design of this catechism.\n\n\"In the first parts, under the heads Domestic, Technical, Social\nand Political Instruction, it has been attempted to shew chiefly\nwhat the means are by which the human mind may be endowed with those\nqualities on which the generation of happiness depends.\"\n\n\nVICISSITUDES OF EDUCATION IN UPPER CANADA, 1830-1839.\n\nFor many years subsequently spasmodic efforts were made from time to\ntime by progressive and earnest men in the Legislature to establish\na system of schools. Enquiries were instituted and reports made,\nchiefly but not wholly, by the House of Assembly. A vigorous contest\nwas maintained between that body and the Legislative Council on\nthe subject. Bills were passed by the Assembly and rejected by the\nCouncil. The contest continued until the Rebellion occurred, and\nthis event turned all men's thoughts into another channel for the\ntime.\n\nOf the able and zealous men who, almost single handed, fought the\nbattle of elementary education in the Legislature, prior to the\nrebellion of 1837, I may refer to the efforts in this direction of\nDr. Charles Duncombe and Mr. Malhon Burwell, who did good service in\nthe cause, as also did Archdeacon Strachan, Hon. William Morris and\nothers for higher education.\n\nIn the light of the growth and educational progress of to-day,\nthe miserable condition of public education in the days of the\neducational pioneers to whom I have referred, can hardly be\ncredited. And were it not on record in the proceedings of the\nLegislature, the statements there made would appear to apply to some\nother country rather than to ours.\n\nThere were in the House of Assembly in those days (as I have\nintimated) men of rare power and ability, who did noble service\nin the popular cause, and in behalf of general education. They\npassed school bills, founded on elaborate reports, year after\nyear, only to see them defeated by a majority in the Legislative\nCouncil. This state of things continued for some years, and with\ndisastrous effects on the intellectual life of the country. This\nfact is illustrated in the proceedings of the House of Assembly. For\nexample: In a petition of the United Presbytery of Upper Canada,\npresented to the House in 1830, the signers say:--\n\n     \"It is with deep regret that your petitioners (in their\n     ministerial capacity, connected with a very large portion of\n     His Majesty's subjects in this Province) are compelled to say\n     that the state of education is, in general, in a deplorable\n     condition.\"\n\nThe reason for this state of things is thus clearly set forth by the\nHouse of Assembly in an address to the Lieutenant-Governor, adopted\nin the same year:--\n\n     \"We the Commons of Upper Canada, in Parliament assembled,\n     most respectfully represent that there is in this Province a\n     very general want of education; that the insufficiency of the\n     school fund to support competent, respectable and well-educated\n     teachers, has degraded common school teaching from a regular\n     business to a mere matter of convenience to transient persons,\n     or common idlers, who often teach school one season and leave\n     it vacant until it accommodates some other like person, whereby\n     the minds of our youth are left without cultivation, or, what\n     is still worse, frequently with vulgar, low-bred, vicious or\n     intemperate examples before them, in the capacity of monitors.\"\n\n\nEDUCATIONAL EFFORTS OF MR. MAHLON BURWELL IN THE HOUSE OF ASSEMBLY,\n1831-1836.\n\nFew men exerted themselves more or to better purpose in the cause of\neducation than did Mr. Burwell during the time he was a member of\nthe old Upper Canada Legislature, in 1831-1838.[32]\n\n  [32] Col. Mahlon Burwell was born in the State of New Jersey, but\n  early in life came to Upper Canada. He settled first at Fort Erie,\n  then at Long Point, and finally removed to the Talbot Settlement. He\n  was near neighbor, and for a long time, right-hand man of the noted\n  Col. Talbot, of Port Talbot. He was a surveyor by profession, and\n  in 1810 surveyed the townships of Malahide, Bayham, and part of the\n  then village of London.\n\nAmongst the many motions relating to education which were moved by\nMr. Burwell in the House of Assembly from time to time, was the\nfollowing important one, which was concurred in by the House in\nFebruary, 1831:--\n\n     \"That a standing committee be appointed on the subject of\n     education generally in this Province....\n\n     \"That it be a principal duty and business of the committee to\n     enquire whether an appropriation of 500,000 acres of land was\n     not made, in virtue of a joint address of both houses of the\n     Provincial Parliament, adopted at their session of 1797, or\n     1798, and whether the same is not subject to the control of the\n     Legislature of this Province; to enquire if anything, and what,\n     has been done with the lands or any part of them, and what is\n     their present situation.\n\n     \"That the said committee do enquire in what way the several\n     district schools of the Province can best be endowed with\n     portions of the said lands, so as to render them more efficient\n     and fitting for the improvement of the rising generation than\n     they are at present.\"...\n\nSuch were the comprehensive terms of a motion which gave to the\nsubject of education a status in the House of Assembly at the time\nby making a committee on the subject a Standing Committee of the\nHouse, and clothing it with important powers. Mr. Burwell also, of\nthe same month, moved for the production of all the despatches,\nreports, and other documents relating to the royal grant of lands\nby George III. for grammar schools and colleges in Upper Canada.\nIn response to this latter motion, the Lieut.-Governor, Sir John\nColborne (Lord Seaton), sent down to the House a mass of papers\nof great value, showing what steps had been taken by the Imperial\nand Provincial Governments during the intervening years for the\npromotion of public education. These papers were printed at the\ntime, but little is now known of their contents.\n\nIn April, 1831, Mr. Burwell, as chairman of the Quarter Sessions\nof the London District, presented to the Lieut-Governor a memorial\nsetting forth the advantages to that locality of endowing a college\nat London. Amongst the reasons given are the following:--\n\n     \"Your memorialists are aware that education of a superior kind\n     cannot be brought to every man's door, and that under any\n     arrangements, the inhabitants of the Province generally must\n     send their children a short distance from home; but such is the\n     extent of the several districts, that the school can seldom be\n     a day's journey from any part of them; and the scholars can\n     return to their homes without expense during the holidays; and,\n     if sick, they can be visited by their parents in a few hours,\n     and removed to their habitations without difficulty. Added to\n     all this the cheapness at which board can be obtained in country\n     places, and the easiness with which, in most cases, it can be\n     paid for by produce from their farms.\"\n\nThese reasons are somewhat primitive in their character; but they\nthrow light on the social condition of the people in these days, and\nillustrate the common practice then of paying even for education\n\"in kind,\" or by \"produce from the farms.\" The object of the\nmemorialists was to obtain such an endowment for the London District\nGrammar School--\n\n     \"As shall render it efficient as a classical seminary, and a\n     nursery (as such schools are intended to be) for the University\n     of King's College....\n\n     \"The endowment should be such a one as would furnish a good\n     school-house, a commodious residence for the head master--to\n     enable him to keep boarders and produce an income of four or\n     five hundred pounds.\"\n\nIn the following June a similar, but a much longer and more strongly\nworded, memorial was presented to the Governor from the trustees\nof the Kingston \"Royal Grammar School,\" protesting against the\nwithdrawal from that school of an extra grant of \u00a3200 a year and\ngiving it to Upper Canada College, thus reducing the rank of the\nKingston \"Royal Grammar School\" to that of a district grammar school.\n\n\nEFFORTS AT EDUCATIONAL LEGISLATION BY DR. CHARLES DUNCOMBE,\n1831-1836.\n\nAs one of those who took a prominent part in the troublesome events\nof 1837-38, in Upper Canada, Dr. Duncombe acquired considerable\nnotoriety. He was, nevertheless, a man of broad views, of\ncomprehensive aims and large sympathies.[33]\n\n  [33] Dr. Charles Duncombe was an American by birth, and win born\n  in the State of New Jersey in or about the year 1796. He came to\n  Upper Canada with his parents during the progress, or immediately\n  after the close, of the war of 1812-15, and settled in the \"London\n  District.\" Charles Duncombe studied medicine and surgery, and in\n  1827 and 1828 began to practice his profession on the town line\n  between the townships of Burford and Brantford, near Bishopsgate. He\n  soon obtained a large practice and with it an extended influence.\n  During the rebellion of 1837-8, Dr. Duncombe took part and went to\n  the United States, and remained there until 1843, when he received\n  a pardon from Sir Charles (afterwards Lord) Metcalfe. He did not,\n  however, remain long in Upper Canada after his return, but soon\n  left for the Western States, whence he subsequently removed to\n  California, and died there.\n\nFrom his first entry into the House of Assembly, Dr. Charles\nDuncombe, M.P.P. for the county of Norfolk, took up warmly the cause\nof popular education. In this he was actively supported by two other\nmedical gentlemen--Dr. Thomas D. Morrison and Dr. Thomas Bruce--who\nwere also members of the House of Assembly at that time.\n\nDr. Charles Duncombe's first motion in the House of Assembly (on\nthe 13th December, 1831) was for an address to the Lieut.-Governor\nurging the setting apart of a sufficient quantity of the public\nlands of the Province to form a permanent fund for the support and\nmaintenance of common schools. His motion was, however, defeated.\n\nAs Dr. Duncombe's motion is of historical interest, so far as the\nfacts which it alleges are concerned, I give some extracts from it.\nThe motion stated:--\n\n     \"That there is in this Province a very general want of\n     education; that the insufficiency of the Common School Fund\n     to support competent, respectable and well-educated teachers,\n     has degraded common school teaching from a regular business to\n     a mere matter of convenience to transient persons, or common\n     idlers, who often stay but for one season, and leave the schools\n     vacant until they accommodate some other like person, whereby\n     the minds of the youth of this Province are left without\n     due cultivation, or, what is worse, frequently with vulgar,\n     low-bred, vicious and intemperate examples before them in the\n     persons of their monitors,\" (_i.e._, teachers).\n\nThe motion goes on to say that:--\n\n     \"If provision were made for the liberal and punctual payment of\n     common school teachers ... the teaching of common schools would\n     soon become a regular and respectable calling, gentlemanly,\n     well-educated persons would not be ashamed to take charge of\n     youth, the schools would be no longer vacant, nor the scholars\n     ignorant. Upper Canada would then form a national character that\n     would command respect abroad and ensure peace, prosperity and\n     happiness at home, perpetuate attachment to British principles\n     and British institutions, and enable posterity to value, as they\n     ought, the inestimable blessings of our glorious constitution.\"\n\nThe motion went on to urge the Lieut.-Governor to represent to the\nColonial Secretary the important necessity--in view of the facts\ncited--of entreating\n\n     \"That His Majesty, William IV., be graciously pleased to place\n     at the disposal of the Provincial Legislature a portion of the\n     waste lands of the Crown as a permanent fund for the support of\n     common schools within the same.\"\n\nDr. Charles Duncombe, with a prescience of the future, and of the\nnecessities of the case, (which were not then recognized, nor for\nmany years afterwards,) strongly urged, as did other members of\nthe Assembly, that at least one million acres of the \"waste lands\"\nof the Province should be set apart for the support of common\nschools.[34]\n\n  [34] It is gratifying to know that, although defeated at the time,\n  Dr. Duncombe's efforts bore fruit nearly twenty years afterwards--in\n  1850--when Hon. Wm. Hamilton Merritt, then President of the Council,\n  introduced and had a Bill passed by the Legislature setting apart\n  1,000,000 acres of the Crown Lands for the permanent endowment of\n  public schools in United Canada.\n\nThe motion was negatived. Dr. Duncombe was, however, determined not\nto be beaten. Mr. David Burn and other friends of his in the county\nof Oxford--no doubt on his suggestion--got up a petition to the\nLegislature on the subject, and on the 21st December--a week after\nhis motion was defeated--Dr. Duncombe read this petition and had it\nreferred to a select committee for report thereon.\n\nOn the 26th December an elaborate report on the petition was\nbrought in by Dr. Duncombe himself, as chairman of the committee.\nIn that report the whole subject was gone into fully, and a scheme\nelaborated by which the 1,000,000 acres of land were proposed to\nbe hypothecated in advance, so that by the issue of debentures for\n$500,000, redeemable in ten, fifteen and twenty years, a sufficient\nsum would be at once realized on the prospective value of these\nlands to form a permanent fund for the support of common schools.\n\nThis report (as did the rejected motion) placed on record a few\nfacts and principles which are interesting in the light of to-day.\nThe report stated that--\n\n     \"The common schools of this Province are generally in so\n     deplorable a state that they scarcely deserve the name of\n     schools.\"\n\nIt recommended that the common school law of the Province be so\namended that hereafter the school grant be paid only to--\n\n     \"Organized schools, taught by a person who had a certificate\n     from the District Board of Education, or school inspector, of\n     his or her ability to teach a common school.\"\n\nIt also urged that the Common School Fund should be large enough,\nwith the local contributions, to provide an ample stipend for good\nteachers, instead of \"transient persons\" and \"common idlers\" then so\noften employed as teachers--\n\n     \"So that common school teaching, instead of being a mere\n     matter of convenience to transient persons, or common idlers,\n     would become a regular, respectable business in the hands of\n     gentlemanly, well-educated persons. For surely the foundation of\n     the minds of our children (on which must depend the happiness or\n     misery we are to enjoy with them) and their own success in life,\n     is a business worthy to be respectable, worthy of the patronage\n     of men in the highest walks of life.\"\n\nThe report then laid down an important principle in regard to\nthe necessity for a certain and permanent endowment for public\neducation. It said:--\n\n     \"Funds and appropriation for the support of education should\n     be permanent. They should not depend upon the annual vote of\n     the Legislature, nor on any other casualty that might, by\n     possibility fail, and thereby check the regular progress of\n     education.\"\n\nDr. Duncombe, in stating this principle, had no doubt in view the\nexample (then well known) of the fickleness of the Legislature in\nthe matter of school grants. In 1816 the vote for the support of\ncommon schools was $24,000. In three subsequent years the same vote\nwas repeated; but, in 1820, it was reduced to $10,000--closing\nschools here and there all over the Province, and inflicting\ngrievous hardship on many worthy (and, in the language of the day\nand of the report, unworthy) teachers. This miserable state of\nthings continued for many years, and, as I stated on this subject in\n1863--\n\n     \"Thus ebbed and flowed, without a master hand to stay the\n     current, that tide which, in other lands, is regarded as the\n     nation's life's blood; and thus was permitted to ensue that\n     state of living death by which Upper Canada, in the significant\n     and popular metaphor of the day was likened to a 'girdled tree,'\n     destitute alike of life, of beauty, or of stately growth.\"[35]\n\n  [35] Historical Sketch of Education in Upper Canada, by J. George\n  Hodgins, M.A., LL.B., F.R.G.S. in \"Eighty Years' Progress of British\n  North America,\" 1863.\n\nNo wonder that in these degenerate days the young men, with\nstirrings within them of noble impulses and patriotic devotion to\ntheir country, should have been compelled to depend upon themselves\nfor intellectual enlightenment and advancement. The flippant sneer\nof many persons of to-day at such \"self-made\" men is unworthy of\nthose who enjoy the advantages which these self-made men laboured to\nsecure. They belonged to that noble band of pioneers, who achieved\nfor us the civil and religious freedom which we now so richly enjoy.\nAll honour to them, therefore!\n\n\nCONTINUED EDUCATIONAL EFFORTS BY MR. BURWELL IN THE HOUSE OF\nASSEMBLY.\n\nIn January, 1832, Mr. Burwell made a motion similar to the defeated\none of Dr. C. Duncombe, which led to considerable discussion. It was\nas follows:--\n\n     \"That this House do address His Majesty, humbly beseeching that\n     His Majesty will be graciously pleased to grant an appropriation\n     of one million of acres of waste lands of the Crown in this\n     Province for the maintenance and support of common schools\n     within the same.\"...[36]\n\nIn the same month Mr. Burwell introduced a bill \"for the\nestablishment and support of common schools throughout the\nProvince.\" It was printed but was not proceeded with that session.\nMr. Burwell's object clearly was to keep the subject before the\nHouse and to promote discussion on it. In this he succeeded. The\nHouse of Assembly was alive to the importance of the question,\nbut the Legislative Council was obstructive in regard to the same\nsubject.\n\nIn November, 1832, Mr. Burwell again had a committee of the House of\nAssembly appointed to enquire into the manner in which the King's\nwishes had been carried out in regard to the royal grant of lands\nfor educational purposes in 1798. To expedite this enquiry the\nimportant despatches and reports formerly asked for by him and sent\ndown to the House by the Governor, with others, were printed and\ndistributed.\n\nMr. Burwell also introduced a bill \"for the establishment,\nmaintenance and regulation of common schools,\" in the Province. He\nmade several motions, too, on the subject of the King's College\ncharter and school lands. On the 21st November he submitted the\nfirst report of his \"Select Committee on the Subject of Education.\"\nThe historical part of this report being somewhat interesting in its\nstatements, I quote it as follows:--\n\n     \"The committee have been forcibly struck with the uniform\n     anxiety which has been manifested at all times by the\n     Legislature and Provincial authorities for the establishment of\n     a university.\n\n     \"It formed part of the prayer of both Houses in their address to\n     the King in 1797.\n\n     \"It was strongly recommended by the Executive Government, the\n     judges, and law officers of the Crown, in 1798.\n\n     \"In 1806 the Legislature, to show that something more was even\n     then required than grammar schools, did all their limited means\n     permitted, in providing a small apparatus for the instruction of\n     youth in physical science, that they might enter the world with\n     something more than a common district school education; such an\n     institution was again noticed in 1820, and an earnest desire\n     expressed by the Legislature, which knew best the wants of the\n     Province, for its speedy establishment.\n\n     \"In 1825 so many young men were found turning their attention to\n     the learned professions that the Executive Government thought\n     that the establishment of a university could be no longer\n     delayed without the greatest detriment to the Province, and,\n     therefore, applied to His Majesty for a Royal Charter, which\n     was granted in 1827, in terms as liberal, it is said, as the\n     then Government would allow; but such has proved by no means\n     satisfactory to your Honorable House.\"[36]\n\n[36] See also the opinion of Archdeacon Strachan on this subject, in\na subsequent part of this Retrospect.\n\nAbout the middle of December, 1832, Mr. Burwell brought in the\nsecond and very elaborate report of the Select Committee on\nEducation. This report was chiefly based upon the opinions of\nseveral witnesses examined by the committee on the subject of\nschool lands, King's College charter, U. C. College, and education\ngenerally. The witnesses examined were Chief Justice Robinson,\nArchdeacon Strachan, Chairman, and the Hon. G. H. Markland,\nSecretary to the Provincial Board of Education; Hon. Joseph Wells, a\nmember of the Board, and Treasurer of U. C. College; Rev. Dr. Joseph\nH. Harris, Principal of U. C. College; Rev. Dr. Thomas Phillips,\nVice-Principal, and Mr. S. P. Hurd, Surveyor-General of the Province.\n\nThe general views of these noted men on the subject of education\nare both interesting and instructive in the light of to-day. The\nreport itself deals with the then pressing question of the extension\nof educational facilities to the entire Province. It points out\nin strong language the undesirability of continuing a system of\ndistrict, or grammar, schools which were quite adequate to the wants\nof the Province when the population was only 50,000, but which was\nnot at all equal to the requirements of Upper Canada when that\npopulation had increased to nearly 300,000. These references show\nhow wonderfully the Province has progressed in population and in its\neducational advantages since that time.\n\n\nEARLY OPINIONS ON THE NECESSITY FOR MANUAL, OR INDUSTRIAL, EDUCATION.\n\nThe following passage from the report of 1832 is prophetical in its\nanticipation of the future. By way of illustration I may mention the\nfact that a somewhat similar utterance was made by Sir Lyon Playfair\nin his address as president of the British Association, at Aberdeen,\nin 1887. The passage in the report of 1832 is as follows:--\n\n     \"That the situation of the Province in wealth and commerce,\n     and in its demand for superior attainments in the various\n     professions is very different from what it formerly was; and\n     that unless opportunities are immediately furnished by the\n     establishment of superior schools for the instruction of our\n     youth in the higher branches of science, we must fall behind the\n     age in which we live.\"\n\nWhat was thus put forth as a local thought, but yet as an\neducational axiom, by these educational pioneers of Upper Canada,\nupwards of fifty years ago, is thus forcibly and beautifully\namplified by the president of the British Association in 1887.\nSpeaking generally, and contrasting the educational policy of the\ncolonies and that of the mother country, he said:--\n\n     \"The colonies, being young countries, value their raw materials\n     as their chief source of wealth. When they become older they\n     will discover it is not in these, but in the culture of\n     scientific intellect, that their future prosperity depends....\n     Jules Simon tersely puts it:--'The nation which most educates\n     her people will become the greatest nation, if not to-day,\n     certainly to-morrow.' Higher education is the condition of\n     higher prosperity, and the nation which neglects to develop the\n     intellectual factor of production must degenerate, for it cannot\n     stand still.... The illustrious consort of our Queen was not the\n     first prince who saw how closely science is bound up with the\n     welfare of states.... How unwise it is for England to lag in the\n     onward march of science, when most other European powers are\n     using the resources of their states to promote higher education\n     and to advance the boundaries of knowledge. [She] alone fails\n     to grasp the fact that the competition of the world has become\n     a competition of intellect.... A nation in its industrial\n     progress, when the competition of the world is keen, cannot\n     stand still.... I contend that in public education there should\n     be a free play to the scientific faculty, so that the youths who\n     possess it should learn the richness of their possession during\n     the educative process.... Science has impressed itself upon the\n     age in which we live; and as science is not stationary, but\n     progressive, men are required to advance its boundaries, acting\n     as pioneers in the onward march of states. Human progress is so\n     identified with scientific thought, both in its conception and\n     realization, that it seems as if they were alternative terms in\n     the history of civilization.\"\n\nIn giving these extracts so fully I have done so for two reasons:\nFirst, I desire to do honour to the zeal and to acknowledge the\nforethought and prescience of those members of the House of Assembly\nwho, in 1832, placed so strong an emphasis upon the value of \"the\ninstruction of our youth in the higher branches of science;\" and\nsecondly, to point out, in the weighty words of Sir Lyon Playfair,\nthe immense importance (in the light of past experience) which he\nand other leaders of thought in regard to England's industrial life\nand practical progress, attach to the teaching of elementary science\nin the schools. He touches upon this point in another part of his\naddress, in pointing out the absurdity of requiring all pupils to\nstudy the same subjects. He says:--\n\n     \"In a school a boy should be aided to discover the class of\n     knowledge that is best suited to his mental capacities, so\n     that in the upper forms of the school, and in the university,\n     knowledge may be specialized in order to cultivate the powers\n     of the man to the fullest extent.... The adaptation of public\n     schools to a scientific age does not involve a contest as\n     to whether science or classics shall prevail, for both are\n     indispensable to true education. The real question is, whether\n     schools will undertake the duty of moulding the minds of boys\n     according to their mental varieties.\"\n\nLATER OPINIONS ON THE NECESSITY FOR MANUAL TRAINING IN OUR SCHOOLS.\n\nSo deeply impressed was I of the immense importance of this subject,\nand of the necessity of providing in our school system for a\npractical solution of the question which was then, and is now, of\npressing importance--viz., manual training in our schools--that in\n1876 I prepared and delivered a lecture on the subject, in various\nparts of the Province. The lecture was founded on the industrial\nlessons taught to us so impressively at the Centennial Exhibition,\nPhiladelphia, in 1876. These lessons, in their educational aspects,\nwere even more forcibly impressed upon me at the great Industrial\nExhibition held in New Orleans, in 1885. Having been there six\nweeks, as an Educational Juror, on behalf of the United States\nBureau of Education, I had abundant and admirable facilities for\nstudying the whole question, and for seeing how it was being worked\nout (more or less effectually) in the various national school\nsystems which came under review during that enquiry--especially in\nFrance. Thus the French school law of 1882 provides that \"primary\neducation includes [among other things] the elements of the natural,\nphysical and mathematical sciences, and their application to\nagriculture, to hygiene, and to the industrial art; manual work, and\nthe use of tools of the principal trades, the elements of drawing,\nmodelling, etc.\" Apprenticeship schools have also been established,\nthe object of which is to form workmen, as distinguished from\nforemen, and in which various trades are taught. An official report,\npublished by the United States Bureau of Education in 1882, states\nthat the apprentices of these schools \"find employment readily after\nthey have left the workshops, at wages, it is said, varying from\nfive to even as much as eight francs per day.\"\n\nIn discussing this question in the lecture to which I have referred,\nthese passages occur:--\n\n\"It is not assumed that every pupil in our schools is qualified, or\nthat he should be compelled, as a matter of course, to engage in\nthe study of elementary science or practical drawing. Far from it.\nBut what I do say is, that those pupils who exhibit a taste for any\nof the various subjects of natural history, elementary science or\npractical mechanics, should have an opportunity in the Public and\nHigh Schools (of cities and large towns) of learning something about\nthem. In an address by Mr. Gladstone on this subject, he stated that\nthe boys of the English schools, and it is so in our schools, had\nnot yet had fair play in the study of elementary science and natural\nhistory....\n\n\"There are few schools in which there are not boys possessing\ntalent scientific, inventive, or industrial talent, or constructive\ngenius, which are never evoked, much less aroused or stimulated. As\nto the question whether for the few the country should be put to\nthe expense of their special training, I answer it in the words of\nProfessor Huxley, who says:--\n\n\"To the lad of genius, even to the one in a million, I would make\naccessible the highest and most complete training the country could\nafford. Whatever it might cost, depend upon it the investment\nwould be invaluable. I weigh my words when I say: that if the\nnation could purchase a potential Watt, or Davy, or Faraday, at the\ncost of a hundred thousand pounds down, he would be dirt cheap at\nthe money.... It is a mere commonplace and an every-day piece of\nknowledge to say that, what these three men did, (in their special\ndepartments of practical science), has produced untold millions\nof wealth for England and the world, speaking in its narrowest\neconomical sense of the word.\"\n\nThe educational mind of the United States, as well as Europe, is\nbeing constantly directed to the consideration of this interesting\npractical subject. Magazines and reviews, as well as educational\njournals, freely discuss it. One of the most useful articles on\n\"Manual Training in the Public schools,\" will be found in the\n_Andover Review_ for October, 1888. The United States Bureau\nof Education has also published various reports and papers on\nthe subject. One of the most valuable is an elaborate report on\n_Industrial Education in the United States_, published in 1883. Some\nof the more important railways in that country have also established\ntraining schools for their employ\u00e9s.\n\n\nFURTHER EDUCATIONAL EFFORTS IN THE HOUSE OF ASSEMBLY, 1835, 1836.\n\nFor the four years during which Dr. Duncombe was a member the\nLegislature of Upper Canada, his efforts to promote the cause of\neducation were unceasing. With the exception of Mr. Burwell, who\ndevoted himself almost entirely to the interests of education in\nthe House, none excelled Dr. Duncombe in his zeal for the cause\nof public education. His efforts were chiefly directed to awaken\nan interest amongst his fellow members in the subject generally,\nand especially on behalf of the education of the deaf and dumb,\nin asylums for the insane, in prison discipline and similar\nmatters. At length his efforts in the session of 1835 culminated\nin the appointment, by resolution of the House of Assembly, of\nDoctors Charles Duncombe, Thomas D. Morrison and William Bruce,\nCommissioners, to enquire, amongst other things, into \"the system\nand management of schools and colleges\" in the United States and\nelsewhere. Two of these commissioners deputed their colleague, Dr.\nDuncombe, to \"go on a journey to the United States, or elsewhere, to\nobtain such information as is desired by a resolution\" of the House\nof Assembly in that behalf. Six hundred dollars were granted by the\nHouse to defray the expenses of this enquiry.\n\nLate in 1835 Dr. Duncombe went on his mission of enquiry to the\nUnited States, and visited literary institutions in the Western,\nMiddle, Eastern and some of the Southern States of the Union. He\nalso obtained detailed information as to education in England,\nFrance and Prussia, and embodied the result in an elaborate\nreport of nearly sixty pages and an appendix of one hundred and\nsixty pages. To this report he annexed the draft of a School\nBill, extending to twenty-two pages, with a variety of forms and\ninstructions appended. The whole document embraced two hundred and\nsixty pages of printed matter. The report is minute and exhaustive\nin its treatment of the subject in hand, although somewhat\ndiscursive and speculative in many parts. It is, nevertheless, in\nthe light of to-day, both interesting and instructive. It presents\na vivid picture, and not a very flattering one, of the condition of\neducation in the United States and in Europe. Its discussions of\nspecial subjects--such as female education, classical studies, the\nmanagement of colleges and universities, etc., etc.--are fair and\nenlightened, and, on the whole, intelligent and practical in their\ncharacter.\n\nIt is clear that the Legislative Council of the day did not\nsympathize with Dr. Duncombe and his colleagues in their zeal for\npopular education.\n\nThe bill which he had so carefully prepared, although adopted by the\nHouse of Assembly by a vote of 35 to 10, early in 1836 failed to\nreceive the sanction of the Council. His proposition to increase the\ncommon school grant from $22,600 to $80,000 per annum was considered\ntoo great a step in advance, and was not therefore pressed to a\nvote in the House of Assembly. He, however, got two influential\ncommittees appointed to deal with the questions of public education\nand school lands. These committees were subsequently united and\nenlarged. They did good service and kept public interest awakened as\nto the value of the important subjects entrusted to them.\n\n\nANALYSIS OF DR. DUNCOMBE'S REPORT ON EDUCATION, 1836.\n\nThe report, be it remembered, speaks of events and educational facts\nof more than fifty years ago. They are of special interest to us\nof to-day, since they form the background, so to speak, of our own\neducational history and progress. I shall make a few extracts from\nthe report:--Dr. Duncombe says:--\n\n     \"The first principles of the system recommended in this report\n     with regard to common schools, schools for the education of\n     the poorer classes, and for the education of teachers (or the\n     normal schools) made their appearance almost simultaneously in\n     Great Britain and on the Continent, as appears by the voluminous\n     reports of Lord Brougham ... and by Mr. Dick's very able and\n     splendid report upon the common schools ... of Scotland, and by\n     M. Cousin's reports of the schools in Prussia and Germany, and\n     Bulver's observations upon education as a prevention of crime\n     in France.... The glimmering of this beacon light was soon seen\n     across the ocean, and lighted up a similar flame in the United\n     Slates. Commissioner after commissioner was sent to Scotland and\n     to England by the authority of their State Legislatures to light\n     their lamps at the fountain of science, that the whole continent\n     of America might be ignited by the flame.\"\n\nDr. Duncombe's observations in regard to the state of education in\nthe United States are interesting, as by contrast they illustrate\nthe remarkable progress made in that country during the last half\ncentury in the matter of public education. He says:--\n\n     \"In the United States, where they devote much time and expense\n     towards the promotion of literature, they are equally destitute\n     of a system of national education with ourselves: and although\n     by their greater exertion to import the improvements made in\n     Great Britain and on the Continent, and their numerous attempts\n     at systematizing these modern modes of education ... they have\n     placed themselves in advance of us in their common school\n     system. Yet, after all, their schools seemed to me to be good\n     schools on bad or imperfect systems; they seem groping in\n     the dark, no instruction in the past to guide the future, no\n     beacon light, no council of wise men to guide them, more than\n     we have, upon the subject of common schools. Our schools want\n     in character, they want respectability, they want permanency\n     in their character and in their support.... It should be so\n     arranged that all the inhabitants should contribute something\n     towards the [maintainance of the school fund], and all those who\n     are benefited directly by it should pay, in proportion to such\n     benefit, a small sum, but quite enough to interest them in the\n     prudent expenditure of their share of the school moneys.\"\n\nThe objection to a liberal education being too freely given for the\nbenefit of the learned professions seems to have been urged even in\nthese days. Dr. Duncombe answers it in the following language:--\n\n     \"It has been supposed that there are too many in the learned\n     professions already, and that, therefore, there are too many who\n     obtain a liberal education. But this opinion is founded upon two\n     errors: One is, that every liberally educated man must be above\n     manual labor, and must, therefore, enter one of the learned\n     professions; and the other is, that all who do enter these\n     professions do it, and have a right to do it, from personal and\n     family interests, and not for the public good--whereas a liberal\n     education ought not to unfit a man, whether in his physical\n     constitution or his feelings, for active business in any honest\n     employment; and neither ought men who enter any of the learned\n     professions to excuse themselves from labor and privation for\n     the good of the world. There is a great and pernicious error on\n     this subject.\"\n\nThe question of free education is thus discussed by Dr. Duncombe:--\n\n     \"Nothing is more important in the formation of an enterprising\n     character than to let the youth early learn his own powers;\n     and in order to this, he must be put upon his own recourses,\n     and must understand, if he is ever [to be] anything, he must\n     make himself, and that he has within himself all the means\n     for his own advancement. It is not desirable, therefore, that\n     institutions should be so richly endowed as to furnish the\n     means of education free of expense to those who are of an age\n     to help themselves; nor is it desirable that any man, or any\n     society of men, should furnish an entirely gratuitous education\n     to the youth of the Province. All the necessary advantages for\n     educating himself ought to be put within the reach of the young\n     man, and if, with these advantages, he cannot do much towards it\n     he is not worthy of an education.\"\n\nAfter discussing several other topics in his report, Dr. Duncombe\nmade a striking forecast of the educational future of Upper Canada.\nHe said:--\n\n     \"Was there ever a more auspicious period than the present for\n     literary reform? If I rightly understand the signs of the\n     times, we stand upon the threshold of a new dispensation in\n     the science of education, and especially in the history of\n     common schools, colleges and universities in this Province. The\n     flattering prospects of our being permitted legally to dispose\n     of the school lands of this Province, so long dormant--the\n     sale and appropriation of the clergy reserves for the purposes\n     of education, and, above all by our having control of the\n     other natural resources of the Province, we shall be enabled\n     to provide respectably and permanently for the support of\n     literary institutions _in every part of the Province_, while\n     by remodelling the charter of King's College, so as to adapt\n     the institution to the present state of science of education,\n     and the wishes and wants of the people of this Province....\n     With such charming prospects before us, with what alacrity\n     and delight can we approach the subject of education to make\n     liberal, permanent and efficient provision for the education of\n     all the youth of Upper Canada, to cause 'the blind to see, the\n     deaf to hear and the dumb to speak,' and, above all, to make\n     certain and extensive provision for the support of schools for\n     teachers and tutoresses.\"...\n\nIt is sad to think that to the writer of these cheering, hopeful\nwords, the future so vividly pictured by him became suddenly\ndarkened, and the pleasant hopes in which he then indulged were\nnever realized by him, or by many of those who, more than half a\ncentury ago, were like him so active in promoting the great cause of\npopular and collegiate education in this Province. Within one year\nDr. Duncombe was a \"proscribed rebel,\" as were many others who with\nhim saw as in a vision the future which, he then pictured for Upper\nCanada.\n\n\nSUMMARY OF, AND REFLECTIONS ON, THESE EDUCATIONAL EFFORTS, FROM 1830\nTO 1839.\n\nDuring the next session of the Legislature, in the winter of\n1836, Mr. Burwell sought to give effect to Dr. Duncombe's liberal\nresolution of the preceding session, viz., to provide, out of the\npublic in revenue, a grant of $80,000 a year for support of the\ncommon schools. He proposed two resolutions: one was to the effect\nthat $40,000 a year be granted out of the public revenue for the\nsupport of these schools; the other was as follows:--\n\n     \"That the sum of ten thousand pounds ($40,000), be raised\n     annually by assessment, by order of the Quarter Sessions in the\n     several districts on the ratable property of the inhabitants, in\n     aid of the Provincial grant for the common school fund, in the\n     same manner as other assessments are now made.\"\n\nWhen the matter came before the House of Assembly in February, 1837,\nthe committee of supply reported a grant of only $22,400 for the\nyear. The assessment proposition was not adopted, as the question of\nlocal taxation for school purposes though often before it had not\nyet been practically entertained by the Legislature.\n\nNext year, however, another effort was made to provide somewhat\nliberally for the common schools. But as the Bill as passed by the\nHouse of Assembly, embodied in it the principle of local taxation\nfor schools for the first time, it was not concurred in by the\nLegislative Council. That body proposed a conference to explain the\nreason, and appointed the Honorable Messrs. Allan and Hamilton as\nits conferees. The House of Assembly nominated Messrs. Boulton,\nCartwright, Thompson and Rykert as its representatives at the\nconference.\n\nThe Legislative Council stated that:--\n\n     \"It could not pass the Bill, because it proposes to levy an\n     assessment at the discretion of the Justice of the Peace, to\n     the extent of 1-1\/2d. [3 cents] in the \u00a3 [$4] to support Common\n     Schools; and as acts have lately passed imposing rates on the\n     inhabitants of several of the districts, for the purpose of\n     defraying the expense of building jails and court-houses and for\n     the construction of macadamized roads, the Council fear that\n     the proposed assessment for common school education might be\n     burthensome,\" etc.\n\nThus, because jails, court-houses and roads were considered more\nnecessary and important than schools, the last Act for the promotion\nof education ever passed by the Upper Canada House of Assembly was\nrejected by the Legislative Council! Such was the untoward state of\naffairs when the Legislative Union of Upper and Lower Canada took\nplace in 1840.\n\n\nEXTRACTS FROM OFFICIAL REPORTS ON EDUCATION IN UPPER CANADA IN 1838.\n\nI shall now give a few extracts from official returns illustrative\nof the state of education in the Province in 1838 and 1839, while\nthese efforts of improvements were being made. They also show how\nentirely practical were the views of those who took an active part\nin educational affairs of the time, how keenly alive they were to\nthe educational deficiencies of those days.\n\nOther extracts from official reports and proceedings of the House\nof Assembly might be given to illustrate this part of the subject,\nbut they would make this retrospect too long. They are all of a most\ninteresting and instructive character, and well deserve publication\nin a connected form. They throw a vivid light upon the educational\nchaos which existed at the time. They also show how enlightened\ncomparatively, as well as how darkened also, were the views of\nthose who took part on both sides in the educational debates and\nproceedings of those years--especially during 1830,--1836, 1838 and\n1839. What were then problematical theories and merely tentative\nschemes, are to-day educational truisms and successful fields of\noperation.\n\nThe growth of the schools under the fitful system which prevailed\nin Upper Canada from 1816 to 1842 was painfully slow. The number\nof what was called \"schools\" was small, and the quality of them,\nwith rare exceptions, was exceedingly inferior. Anything beyond the\nthree R's was generally taught by itinerants. Dr. Ryerson mentions\nin an autobiographical letter, as an example, that his knowledge\nof English grammar was derived entirely from the \"lectures\" of a\nperipatetic teacher of that subject. He also mentions several of\nhis after contemporaries who acquired a knowledge of grammar and\nother special subjects in the same way. No one, as he said to me,\never heard in these days of the possibility of a \"royal road to\nlearning.\" It was hard work, of the hardest kind, and few ever\ndreamt of reaching a higher eminence than that of mastering the\nfirst two R's--Reading and 'Riting. Arithmetic was approached with\ncaution, and its higher \"developments\" with consternation. How could\nsuch a state of things be otherwise when \"transient persons\" and\n\"common idlers\" were with rare exceptions, the kind of \"teachers\"\nemployed. Education had no money value then, except in so far as\nit receded from a rate of payment to that of a day laborer or a\npensioner.\n\nThe following are the extracts from the official reports:--\n\n\nINFLUENCES BY AMERICAN TEACHERS AND SCHOOL BOOKS DEPRECATED.\n\n_Schools in the Home District--No United States books\npermitted._--The schoolmasters, with the exception of two Americans\nwho have been long in the Province, are all British subjects--that\nthey have all taken the oath of allegiance--that during the last\nyear the salary allowed was \u00a310 (ten pounds each), and no books from\nthe United States are permitted to be used in the schools.--_John\nStrachan, Wm. Allan, Toronto, 8th March, 1839._\n\n_Schools in the Eastern District--Transitory Teachers._--Were\nthe allowance to be increased teachers would come forward better\nprepared and be induced to remain. Many at present seem to continue\nfor a few months, as a matter of convenience, and to assist\nthemselves in following other occupations, which greatly <DW44>s\nthe improvement of the children.--_Joseph Aderson, D. McDonell,\nCornwall, 9th May, 1839._\n\n_Schools in the Western District: Their State and Suggestions for\nImprovement._--The situation of the school houses is not always\njudiciously chosen, it being situated often more for the convenience\nof some one influential person than for that of the inhabitants\ngenerally of the settlement.\n\nThe school-house is often a wretched log hut, or a ruinous building\naltogether unfit for the purpose--especially in the winter season.\n\nIn too many cases the teachers are badly qualified for the task\nwhich they undertake; and some of them having taken up the\nprofession more from necessity than choice are seldom permanent, and\nconsequently very ineffectual teachers.\n\nThe remuneration which the teachers of common schools receive for\ntheir services are by no means sufficient to induce respectable and\nwell qualified teachers to undertake the irksome and laborious task.\n\n_Hints for the Improvement of the Schools of the Province\n(condensed)._--1. The school should be erected in a dry and healthy\nsituation if possible, and situated so as to suit the majority of\nthe inhabitants of the settlement in which it is erected. It should\nbe a neat and commodious building, sufficiently large to render it\nairy and healthy in the summer season and well finished inside and\nout to cause it to be comfortable in the winter.\n\n2. A comfortable dwelling should be erected for the accommodation of\nthe teacher and his family.\n\n3. Teachers throughout the Province might be divided into three\nclasses, allowance from Government to be not less than \u00a3100, \u00a375 and\n\u00a350 respectively.\n\n4. Every teacher, previous to receiving any appointment, should be\nexamined as to his literary acquirements, his political opinions and\nhis moral character.\n\n5. A uniform set of elementary books should be compiled and\npublished for the use of the common schools of the Province, and\nthose republican productions that tend to poison the minds of the\nyouth of the country should be driven out of the Province.\n\n6. A discreet and competent person should be appointed by his\nExcellency the Lieutenant-Governor to visit the schools in each\ndistrict eight, or at least four, times in the year to examine the\nscholars and the internal economy of the schools and to report\nthereon--_W. Johnson, Sandwich, 21st February, 1839._\n\n_Schools in the London District--\"Boarding Round\"--American\nBooks._--The Board of Education cannot abstain from remarking upon\na system commonly practiced by teachers and generally encouraged by\nthe employers in the country, of receiving the teachers as members\nor lodgers with each family who are subscribers to the school in\nsuccession for the period of engagement, which in its influence\nand consequence has not hitherto been productive of good; and more\nespecially in cases where the teachers have been Americans, a system\nthan which none can be more mischievous in its effects, added to\nwhich the circumstance, as will be seen by reference to the books\nused in the schools, that a portion of American books, particularly\ngeographies, have been permitted to be used (notwithstanding the\nBoard have the power to order the discontinuance of such) because\nothers could not be procured in the country, nor has any provision\nbeen made by the legislature for the formation of depots where\nproper books could be had.--_John B. Askin, London, 12th February,\n1838._\n\nNames of text-books used in the common schools of Upper Canada in\n1838, viz.:--\n\n     Old and New Testaments.\n\n     _Readers_--English, Murray's, Canadian and Reading Made Easy.\n\n     _Spelling Books_--Mavor's, Cobb's, Webster's, Graham's,\n     Universal.\n\n     _Grammars_--Murray's, Lennie's, Kirkham's, McCulloch's.\n\n     _Arithmetics_--Walkingame's, Gray's, Dillworth's, Daboll's,\n     Watson's, Pike's, Adams', Morrison's, Hutton's, Rogers',\n     Bailey's, Hall's, Joyce's, Keith's, Allison's, Bonnycastle's.\n\n     _Geographies_--Goldsmith's, Hutton's, Olney's, Woodbridge's,\n     Willett's, Evans', Stewart's, Parley's, Elvey's.\n\n     _History_--English, Goldsmith's, Tytler's, Hume's, Simpson's.\n\n     _Geometry and Euclid_--Ingram's, Hutton's.\n\n     _Dictionaries_--Walker's, Cobb's, Walker Johnson's.\n\n     _Miscellaneous_--Dillworth's Teachers' Assistant, Burham's\n     Primer, Mason's Primer, Child's A, B, C, Scott's Lessons,\n     Morrison's Book-keeping, Blake's Natural Philosophy, Blair's\n     Rhetoric.\n\n  Number of schools in Upper Canada in 1838 (estimated), 835.\n\n  Number of pupils in attendance,                \"     23,776.\n\n\nEXTRACTS FROM THE REPORT OF A COMMISSION APPOINTED TO ENQUIRE INTO\nTHE SUBJECT OF EDUCATION IN UPPER CANADA IN 1839.\n\nThe Commissioners appointed to conduct this enquiry were the Rev.\nDr. McCaul, Rev. H. J. Grasett and S. B. Harrison, Esq. James\nHopkirk, Esq., was appointed Secretary. The following are the\nextracts:\n\n_Preliminary Educational History._--In 1797 both Houses of the\nLegislature petitioned the King for an appropriation of waste lands\nof the Crown to form a fund for the support of a Grammar School in\neach district and a College or University for instruction in the\ndifferent branches of a liberal education. In 1807 an Act, limited\nto four years, was passed granting \u00a3800 for the support of eight\ndistrict Grammar Schools. In 1808 the limiting clause (to four\nyears) was repealed. In 1816 an Act establishing common schools\nwas passed and \u00a36,000 were granted for their support. In 1819\nan amending Act was passed requiring annual examinations in the\nschools; that reports to the district Board of Education should be\nmade each year; that \"ten children of the poorer inhabitants,\" to\nbe selected by ballot, should receive free tuition in each Grammar\nSchool, and that trustees should give certificates to teachers. In\n1820 the grant to common schools was reduced from \u00a36,000 to \u00a32,500\nper annum. In 1824 \u00a3150 per annum was granted for the supply of\ncommon schools, with books, tracts, etc., and that teachers must be\nexamined and licensed by the District Board of Education, one member\nof which might certify as to the ability of the teacher before the\npayment to him of the public grant. In 1833, the annual grant to\ncommon schools was increased from \u00a32,500 to \u00a35,650. No grants to a\nteacher is to be made \"unless the trustees shall make it appear that\nthey have made provision for his support so as to secure him for his\nservices in a sum at least equal to double the amount which may be\nallotted by the Board of Education from the public money.\" No school\nlegislation took place during the years from 1833 to 1841.\n\n_District Grammar Schools._--The Commissioners made several\nrecommendations for the improvement of the schools, viz:--1.\nUniformity in the system applicable to all the schools. 2.\nExamination of teacher, so as to test his qualification for the\noffice of teaching. 3. Assistant in each school where there are 30\npupils. 4. School-house built on a uniform plan. 5. Admission of a\ncertain number of free pupils. 6. Quarterly reports from each school\nand systematic inspection of them.\n\n_Common Schools._--The Commissioners also made recommendation for\nthe improvement of these schools, viz.:--1. That there should be\na model school with two rooms in each township, and at least two\nacres of land attached thereto for the use of the master. 2. In\neach of these schools there should be a male and female teacher\n(married desirable), and, in addition, other \"teachers licensed to\nitinerate through the township, beyond the sphere of the permanent\nschool,\" say at places \"more than two miles distant from it.\" \"Thus\nprovision is made for one permanent and four occasional schools in\neach township.\" 3. Fees to be $2 per quarter, while one pupil in\nfive might be admitted free. 4. The subjects of the instruction\nshould be: Spelling, reading, writing, the Holy Scriptures,\ngeography, history, arithmetic, book-keeping, mensuration, and in\ngirl's school sewing and knitting. 5. Books should be provided\nat a cheaper rate from Britain, or a series of compilations, or\nrepublications should be prepared and printed here, and that they\nshould be appointed to be used in all the schools of the Province.\n6. The general control of the schools should be vested in a Board of\nCommissioners, with a secretary at Toronto. One of the Board should\nbe chairman and inspector general of the schools--having control\nover the Grammar and Common Schools, and should be the medium of\ncommunication between the District Boards and the Council of King's\nCollege. 7. There should be elected township director of schools.\nThe Commissioners add:--\n\n_Normal School._--\"No plan of education can be efficiently carried\nout without the establishment of schools for the training of\nteachers.\" They, therefore, recommended that the Central School of\nToronto should be a Normal School--others to be added afterwards.\n\n_Grants._--The Commissioners recommend that \u00a321,410 be granted for\nDistrict Grammar Schools--\u00a312,000 from the sale of Grammar School\nlands, and \u00a324,300 for Common Schools--\u00a315,000 of the latter to be\nraised by taxation at the rate of 3\/4d. in the \u00a3.\n\n\nEDUCATIONAL OPINIONS OF PROMINENT PUBLIC MEN IN 1839.\n\n_Hon. G. S. Boulton._--In his replies to the Commissioners, he\nsaid:--Teachers should be British subjects and should be examined by\nthe Board of Education and approved previous to appointment. Each\nteacher should receive at least $20 per annum, exclusive of fees\nfrom pupils.... I recommend the passage of an Act appropriating\n500,000 acres of land for the support of Common Schools, as proposed\nin the last session of the Legislature by a joint committee of both\nHouses.\"\n\n_Hon. Wm. Morris,_ in his reply to the Commissioners, said:--The\nhundreds of the youth of the country who, for want of convenient\ninstitutions of learning, have been sent to and educated in the\nneighboring Republic, where, if they have not imbibed a predilection\nfor that form of government, have been greatly exposed to the\ndanger of losing that attachment to monarchical government, and the\nprinciples of the British Constitution, which is the essential duty\nof those who administer the affairs of this colony to cherish in the\nminds of the rising generation.\n\n_Hon. James Crooks._--The system of Common Schools, although in\nsome instances abused by the employment of improper persons,\nindeed sometimes aliens, as teachers, yet, on the whole, I think\nhighly beneficial; perhaps were the system of parochial schools,\nas established in Scotland, with such modification as would be\nnecessary under the different circumstances of this Province,\nengrafted upon the Common School system, it might be found to work\nwell.\n\n_Hon. P. B. De Blacquiere._--The present condition of teachers is\ntruly wretched, and reflects great disgrace upon the nation, and\nwhat but the actual results can or could be expected? I think a\ndifficulty will arise as to finding inspectors properly qualified,\nor who, in the present state of the country, can be trusted....\n\n_Rev. Robert McGill._--I know the qualifications of nearly all\nthe Common School teachers in this (Niagara) District, and do not\nhesitate to say, that there is not more than one in ten fully\nqualified to instruct the young in this the humblest department. I\nshould doubt, therefore, whether the money granted to them being an\nequivalent good, or whether the state of education in this Province\nwould be worse were those funds entirely withdrawn.\n\n_Rev. Robert Murray._[37]--The great difficulty attending any change\nin the present wretched system of education in the Province is to\nensure the efficiency of that scheme which may be adopted in its\nroom. To leave the supervision in the hands of the electors in\neach district, or to a few individuals appointed by them, probably\nthemselves without education, would certainly tend to perpetuate the\nsystem of gross oppression to which teachers have been subjected,\nand to disappoint the reasonable expectations of the Government....\nIt appears absolutely necessary to ensure the efficiency of a system\n(as suggested) that men of education, who themselves have had\nlarge experience in the education of youth should be appointed to\nsuperintend the whole system of operation....\n\n  [37] First Superintendent of Education for Upper Canada, and the\n  immediate predecessor of Rev. Dr. Ryerson.\n\n_Malhon Burwell, Esq._--I cannot conceive anything more wanting in\nefficiency than our present system for Common School education. I\nannex for the notice of the Commission of Investigation a copy of\na Common School bill, which I have several times endeavored to get\npassed through the House of Assembly.\n\n(NOTE.--See Bishop Strachan's estimate of this bill in next extract.)\n\n_Right Rev. Bishop Strachan._--The Common School Bill, drawn up\nby Mr. Burwell, appears to be an able performance; it has several\ntimes been entertained by the House of Assembly, and once passed\nthat body, but was unfortunately lost in the Legislative Council.\nIt is based on true principles, and contains within it the power of\nexpansion as new townships, counties and districts are organized. It\nmay, perhaps, admit of a few modifications, but is, on the whole, by\nfar the best measure for the establishment of common schools which I\nhave seen.\n\n\nSEPARATE EDUCATIONAL FORCES SHAPING THEMSELVES IN UPPER CANADA.\n\nI will now take up the thread of the historical narrative of\neducation in Upper Canada from page -- of this Retrospect.\n\nDuring the early settlement period, and that preceding the union\nof Upper and Lower Canada in 1840, two social forces (which took\nan educational form later on), were slowly shaping themselves into\nan antagonism to each other which culminated in the events or\npolitical crisis of 1837-38. This was apparent from the position\nwhich the representatives of these forces assumed on the religious,\npolitical and other questions of the day. As yet the question of an\neducational system for the Province--beyond that of a University and\ndistrict Grammar Schools--had, down to 1836, taken no definite shape\nin the public mind. Indeed, such a thing, as we now regard it, was\nnot deemed practical, except by a few leading men, as I have shown,\nwho were years in advance of their times.\n\n\nNOTED REPRESENTATIVE EDUCATIONAL LEADERS--DR. STRACHAN AND DR.\nRYERSON.\n\nIt will simplify my statement of the case if I revert back to the\ntransition period between the establishment of the district Grammar\nSchools in 1809 and the university charter of 1827; and from thence\ntake a somewhat prospective view of events in the order in which\nthey afterwards transpired. For convenience I would, therefore,\nselect two noted men of their times as representatives of the two\nsocial forces to which I have referred, and of the opposite opinions\non education and other subjects which then prevailed.\n\nThe first was the Rev. Dr. Strachan (afterwards first Church of\nEngland Bishop of Toronto), and the other was the Rev. Dr. Ryerson,\nthe representative and trusted leader of the members of the\nMethodist Church in the Province.\n\nDr. Strachan was the undoubted representative of the English and\nparticularly the Scotch views on educational matters. Dr. Ryerson,\non the other hand, was the equally true and faithful exponent of the\nBritish Colonial, or United Empire Loyalist, views and opinions on\nthe same subject. What these latter views and opinions were may be\ngathered from a reference to the educational chapter in the colonial\nhistory of the thirteen colonies, as given in an earlier portion of\nthis Retrospect.\n\n\nTHE EDUCATIONAL EFFORTS OF THE U. E. LOYALISTS AND THE RULING PARTY.\n\nThe first settlers of Upper Canada were \"exiled tories,\" so called,\nfrom the revolted colonies. In that, and in the other Provinces,\nthey were warmly received and welcomed as the heroic defenders of\nthe royal cause. They sacrificed everything but their principles and\ntheir honor in maintaining \"the unity of the Empire.\" Even after\nthe struggle was ended, they adhered to the \"lost cause\" with the\nsame devotion as they had shown in following the royal standard,\nnot only to victory, but even to disaster and defeat. They were\nmen of wonderful resolution and daring, as well as of superior\nintelligence. Such were the first settlers of Upper Canada.\n\nSoon after the arrival of the \"U. E. Loyalists\" in Upper Canada, a\ntide of emigration set in, chiefly from the three kingdoms. These\nimmigrants brought with them the feelings and habits of home life\nin the old world, with the opinions and prejudices of their class,\nillustrating the truth of the old Latin quotation, \"_C\u00e6lum, non\nanimum, mutant qui trans mare currunt_.\"\n\nBy degrees portions of the U. E. Loyalists and of these immigrants,\nwhose views on \"Church and State\" coincided, united their forces and\nformed a powerful and dominant party. They ruled the Province with\na high hand for many years. From their social position and frequent\nintermarriage they became a compact and exclusive party, and were\ndistinguished by the _sobriquet_ of the \"Family Compact.\" Against\nthis powerful party was arrayed many of the U. E. Loyalists and\ntheir descendants, and the entire liberal and progressive party.\n\nIt is sufficient to say in this connection that under the skillful\nleadership of Dr. Ryerson and other prominent men of moderate views\nwho acted with him, the power of the Family Compact was broken, the\ncompact itself was gradually dissolved. Its opponents became in turn\nthe ruling party in this Province, a position which their legitimate\nsuccessors still occupy.\n\nThe Family Compact party, in the heyday of their power and\ninfluence, were not averse to education. Far from it; for they were\nmen of education themselves. But it took the form of zeal for higher\neducation and for the higher classes. Rev. Dr. Strachan, who was the\nmost energetic and powerful leader of this party, occupied a seat\nin the Legislative Council (Senate) by appointment of the Governor.\nHe devoted all his energies to the establishment of a university,\nwith district classical schools as feeders. He practically ignored\nelementary schools, or rather made no provision for them; and it was\nnot until nine years after these district classical schools were\nestablished that the U. E. Loyalists, (combined with the progressive\nparty of which it formed no inconsiderable portion), were able to\nget a measure passed by the Legislature for the establishment and\nmaintenance of common schools.[38]\n\n  [38] See remarks on this anachronism on page 64.\n\n\nAN EDUCATIONAL GLANCE BACKWARDS.\n\nBut in order to understand more fully the sequence of events which\nled to the development of the educational spirit in this Province,\nit will be necessary to give a condensed summary of the facts. With\nthis historical background in prospective view, the distinguishing\nfeatures of that comprehensive system of education which, in later\nyears, Dr. Ryerson was privileged to found, can be more clearly seen.\n\nThe U. E. Loyalists removed to British America in 1783, the year\nof their exile. Most of them settled in Upper Canada, along the\nnorth shore of the Upper St. Lawrence, and the corresponding margin\nof Lakes Ontario and Erie. They brought with them from the old\ncolonies their educational traditions and their devotion to the\nflag of the Empire. Those of them who had settled along the Bay of\nQuint\u00e9, united in 1789, in framing a memorial to Governor-General\nLord Dorchester (Sir Guy Carleton), in which lamenting the\neducational privations which they had endured since their settlement\nin Canada, they prayed the Governor to establish a \"seminary of\nlearning\" at Frontenac (Kingston). Their prayer was granted, so\nfar as the setting apart of lands for the support of the seminary\nwas concerned, as well as the support of schools wherever the\nexpatriated colonials had settled, or might settle, in the country.\n\nImmediately after the passing of the Constitutional or Quebec Act,\nof 1791, by which, among other things, Upper Canada was separated\nfrom Quebec, the Governor of the new Province (J. Graves Simcoe),\nsought the co-operation of the Church of England Bishop (Mountain),\nof Quebec, who had ecclesiastical jurisdiction over both Provinces,\nin urging upon the Home Government the necessity of providing for\na University and for classical schools in Upper Canada. Provision\nfor elementary schools formed no part of this scheme. The British\nColonial idea of providing for such schools never crossed the\nminds of the leaders of public opinion in these days nor that of\nthe bishop. They were chiefly Englishmen, with the old-fashioned\nEnglish ideas of those times, that the education of the masses was\nunnecessary, for it would tend to revolution and the upsetting of\nthe established order of things.\n\nIn April, 1795, Governor Simcoe addressed a letter to the Protestant\nEpiscopal Bishop of Quebec--then having jurisdiction in Upper\nCanada--urging him to seek to promote the establishment of a\n\"Protestant Episcopal University\" in Upper Canada. The reasons which\nhe gave for this appeal were characteristic of the English Churchmen\nand of the times, and reveal somewhat of the social and religious\nstate of the colony. They showed, too, that he was a statesman as\nwell as a Churchman. He said:\n\n     The people of this Province enjoy the forms as well as the\n     privileges of the British constitution. They have the means\n     of governing themselves; and, having nothing to ask, must\n     ever remain a part of the British Empire, provided they shall\n     become sufficiently capable and enlightened to understand\n     their relative situation and to manage their own power to the\n     public interest. Liberal education seems to me, therefore, to\n     be indispensably necessary; and the completion of it by the\n     establishment of a university in the capital of the country, * *\n     * would be most useful to inculcate just principles, habits, and\n     manners into the rising generation; to coalesce the different\n     customs of the various descriptions of settlers * * * into one\n     form. In short, from distinct parts and ancient prejudices to\n     new-form, as it were, and establish one nation, and thereby\n     strengthen the union with Great Britain and preserve a lasting\n     obedience to His Majesty's authority.\n\n       *       *       *       *       *\n\n     I naturally should wish that the clergy requisite for offices\n     in the university, in the first instance, should be Englishmen,\n     if possible. * * * I most earnestly hope that * * * by giving\n     the means of proper education in this Province, both in its\n     rudiments and in its completion, that from ourselves we may\n     raise up a loyal, and in due progress, a learned clergy, which\n     will speedily tend to unite not only the Puritans within the\n     Province, but the clergy of the Episcopal Church, however\n     dispersed * * * and on all sides, to bring within the pale [of\n     the Episcopal Church] in Upper Canada a very great body of\n     sectaries, who in my judgment, as it were, offer themselves to\n     its protection and re-union.\n\n     These objects would be materially promoted by a university in\n     Upper Canada, which might, in due progress, acquire such a\n     character as to become the place of education to many persons\n     beyond the extent of the King's dominions. * * * The Episcopal\n     clergy in Great Britain, from pious motives as well as policy,\n     are materially interested that the Church should increase in\n     this Province. I will venture to prophesy its preservation\n     depends upon a university being erected therein. * * * I have\n     not the smallest hesitation in saying that I believe if a\n     Protestant Episcopal university should be proposed to be erected\n     (even in the United States) the British nation would liberally\n     subscribe to the undertaking. * * * The universities of England,\n     I make no doubt, would contribute to the planting of a scion\n     from their respectable stock in this distant colony.\n\nThere are two or three things worth noticing in this vigorous letter\nof the Governor:--\n\n(1) Among the objects sought to be attained by the establishment\nof a university was the conservation of \"the privileges of the\nBritish Constitution\"; (2) the fusing of the various nationalities\nrepresented in the colony; (3) the absorption of \"Puritans\" and\n\"sectaries\" into the Episcopal Church; (4) the growth and spread of\nloyalty to the King's authority.\n\nTwo things also are noticeable: First, the Governor did not ignore,\nor underestimate, the necessity of popular education, or \"education\nin the rudiments;\" second, he gives no hint of a desire to\nappropriate the public domain to the building up of an \"Episcopal\nuniversity.\" On the other hand, he assumes that, if done at all, it\nis to be aided by contributions from England. I call attention to\nthese two points, from the fact that they were quite lost sight of\nby those who afterwards took up the cause of university education in\nUpper Canada where he had left it.\n\n\nPROVISION FOR HIGHER EDUCATION IN U. C. BY THE IMPERIAL GOVERNMENT.\n\nGovernor Simcoe, having received a higher appointment in the\ncolonial service, left soon after. The Bishop of Quebec, however,\nacted upon his suggestion and wrote to the Colonial Minister on the\nsubject, in June, 1796. In November, 1797, the Legislature of Upper\nCanada addressed a memorial to King George III, asking:\n\n     \"That His Majesty would be graciously pleased to direct his\n     Government in this Province to appropriate a certain portion of\n     the waste lands of the Crown as a fund for the establishment\n     and support of a respectable grammar school in each district\n     thereof, and also of a college, or university, for the\n     instruction of the youth in the different branches of liberal\n     knowledge.\"\n\nTo this memorial the King directed a gracious answer to be sent. The\nduke of Portland, Colonial Minister, therefore instructed the acting\nGovernor, President Russell, to give practical effect to the prayer\nof the petitioners. In doing so he used the following language:\n\n     [His Majesty] being always ready ... to assist and encourage the\n     exertions of his Province in laying the foundation for promoting\n     sound learning and a religious education, has condescended\n     to express his [desire] to comply with the wishes of the\n     Legislature ... in such a manner as shall be judged to be most\n     effectual--\n\n     _First_, by the establishment of free grammar [classical]\n     schools in those districts in which they are called for, and--\n\n     _Secondly_, in due process of time, by establishing other\n     seminaries of a larger and more comprehensive nature, for the\n     promotion of religious and moral learning, and the study of the\n     arts and sciences.\n\nSuch were the terms in which the King, through his Colonial Minister,\nintimated his desire that classical and university learning should be\npromoted in this Province. The very comprehensiveness and express\nterms of the duke of Portland's dispatch on this subject gave rise\nto a protracted controversy in after years, especially as the\ncontroverted expressions were embodied in substance in the royal\ncharter for a university obtained in 1827 by Rev. Dr. Strachan\n(afterwards first Church of England Bishop of Toronto). Around the\nexpressions--\"religious education,\" \"religious and moral learning,\"\nand \"other seminaries of a larger and more comprehensive nature,\"\netc., a fierce war was waged for many years, which, though virtually\nover now, has yet left traces of the bitter conflict.\n\nThe result of the instructions to President Russell was, that\n549,217 acres of crown lands was set apart for the twofold purpose\nset forth in the Colonial Minister's dispatch. Of these acres,\n225,944 were, in 1827, devoted to the university that was virtually\nestablished, on paper, in that year, and by royal charter in 1828.\n\nAs these lands thus set apart were, in those early days,\nunproductive of revenue, nothing could be done to give practical\neffect to the gracious act of the King. A principal for the proposed\nuniversity was, however, selected in Scotland. The position was\nfirst offered to the afterwards justly celebrated Rev. Dr. Thomas\nChalmers, but declined. It was then offered to a successful parish\nschoolmaster, Mr. (afterwards so distinguished in this Province as\nthe Rev. Dr.) Strachan.\n\n\nTHE REVEREND DOCTOR STRACHAN AS AN EDUCATOR.\n\nRev. Dr. (afterwards Bishop) Strachan, though not a versatile man,\nwas in many respects a many-sided one. In his day he had to do with\nall of one great public questions which came before the country.\nOn many of them (and in their settlement), he has left the impress\nof his active mind and persistent will. This was particularly the\ncase in regard to those questions which more deeply touched the best\ninterests of Canadian life, in its religious and social aspects. And\nit was a singular yet characteristic fact, that the more he was\nopposed by those who differed _in toto_ from the policy of his acts,\nthe more strenuously he persevered in his purpose--even against the\nwiser counsels and calmer judgment of many leading public men of his\ntime. But this opens up a question which it is not my purpose to\ndiscuss.\n\nDr. Strachan, as I have said,--although not versatile,--was a\nmany-sided man. And this was quite true in regard to that department\nof his career which I desire to illustrate. He was both an educator\nand an educationist. In the former capacity he was successively the\nparish schoolmaster, near St. Andrew's, and at Kettle, (Scotland).\nHe had there as a pupil the afterwards celebrated Sir David Wilkie.\nIn Canada, he was first a tutor in the family of the Hon. Richard\nCartwright, at Kingston; then master of the Cornwall Grammar School,\nat which most of the distinguished public men of the Bishop's later\nyears were educated. Subsequently he was Chairman of the Provincial\nBoard of Education at York. He was named by the late Hon. Peter\nMcGill as first Principal of McGill College, Montreal--although he\nnever was in a position to undertake its duties. He was afterwards\nPresident of King's College, Toronto, and subsequently President of\nTrinity College University.\n\nIn his capacity as an educator, Dr. Strachan was considered one of\nthe most successful teachers which this Province has yet produced.\nHis aim was to call into active play the varied mental powers of\nhis pupils, and to stimulate any desire which they had to excel in\nknowledge and virtue. One of his earliest _brochures_ is _a Letter\nto his Pupils_, and is in the nature of an appeal on behalf of the\nChristian religion. This, he inscribed, \"as a mark of esteem to\nMr. Andrew Stuart and Mr. James Cartwright, students-at-law.\" This\nletter was printed at Montreal, in 1807, in the quaint old type of\nthe time. It is evidently a warning appeal against the infidelity\nand excesses of the French revolutionists.\n\nDr. Strachan's early and practical experience as a teacher gave to\nhim an additional and keen sense of the educational wants of the\ncountry. His success as an educator proved to him what could be\ndone in that direction. It also enlisted his feelings and fired his\nambition to be the founder of an institution of superior learning,\nin which the young men of the Province could be thoroughly educated.\nThe education of the masses was not provided for by him, but in\nan Act passed in 1819 and relating to classical schools (which he\npromoted), it was agreed--\n\n     That in order to extend the benefits of a liberal education\n     to promising children of the poorer inhabitants, trustees [of\n     common schools wherever established] shall have the power of\n     sending scholars, not exceeding ten in number, to be chosen by\n     lot every four years, to be taught gratis at the [classical]\n     schools.\n\nThus, in this exceptional manner, provision was made so that, should\na limited number of the children of the poorer inhabitants develop\nability or taste for learning, they should not be wholly excluded\nfrom the privileges so liberally provided for children of the richer\nclasses. These class distinctions have, happily, forever disappeared\nfrom our statute book. They were no doubt conceived in a benevolent\nspirit, and were characteristic of the social ethics of the times,\nbut they were pernicious as a principle to embody in a school law.\n\nIn his \"Appeal\" in behalf of a university in Upper Canada, published\nearly in 1827, Dr. Strachan gave a fuller expression to this idea of\nproviding education only for the wealthier classes. He said:--\n\n     It is indeed quite evident that the consequences of a university\n     ... possessing in itself sufficient recommendations to attract\n     to it the sons of the most opulent families, would soon\n     be visible in the greater intelligence and more confirmed\n     principles of loyalty of those who would be called to various\n     public duties required in the country--_i.e._, the governing\n     classes.\n\nIn justice to Dr. Strachan, it is proper to state that a few years\nafterwards (in reply to a question put to him by a committee of\nthe House of Assembly) he laid down a broader, a nobler and a more\ncomprehensive principle in regard to a system of national education.\nHe said:--\n\n     The whole expense [of education] in a free country like\n     this should be defrayed by the public; that promising boys,\n     giving indication of high talent, though poor, might have an\n     opportunity of cultivating their faculties, and, if able and\n     virtuous, taking a lead in the community.\n\n\nLACK OF COMPREHENSIVENESS IN THE EDUCATIONAL POLICY OF THE TIMES.\n\nThe policy of the country in regard to education in these early\ntimes was further marked by a lack of comprehensiveness in its aims.\n\nThe framework of the educational system, then projected, was\nconstructed on a principle the very reverse of natural. And this\nfact led to the existence, subsequently, and for many years, of\na singular anachronism as the result of its application of that\nprinciple. Thus, in 1797, lands were set apart in Upper Canada by\nthe Crown for the establishment of district grammar schools and a\nuniversity. But no provision was thought of for the establishment of\nelementary schools. These grammar schools were first established in\n1807--eight in all, viz., at Sandwich, Townsend (London District),\nNiagara, York, Cobourg, Kingston, Augusta (District of Johnstown),\nand Cornwall. But no provision was made for elementary schools (and\nthen only for four years) until 1816--nine years after the district\ngrammar schools were established.\n\nDr. Strachan's feelings in this matter were evidently in harmony\nwith this spirit of the times, and he directed his efforts\nexclusively to the establishment of these higher institutions of\nlearning. He never lost sight, however, of the crowning institution\nof all--the university. His speeches and addresses on education all\npointed to \"this consummation, devoutly to be wished.\"\n\nReferring to this educational anomaly, or anachronism, of\nestablishing higher institutions of learning before providing for\nelementary schools, an English resident in a book entitled \"Three\nYears in Canada,\" and published in 1839, thus forcibly points out\nthe singular want of foresight in this matter. He says:--\n\n     \"The Provincial Board of Education either assumed that\n     elementary education in Upper Canada had attained its zenith\n     or deemed it better to begin at the apex and work downward to\n     the base of the structure they were called upon to rear than to\n     follow the old-fashioned custom of first laying the foundation\n     and then working upwards.\"\n\nThe fact was, and the chief reason for the perpetration of this\neducational anachronism was, that the friends of popular education,\nwhile all-powerful in the House of Assembly, were few and\nconsequently uninfluential in the Legislative Council. They were,\ntherefore, not able at all times to influence that body so as to\nsecure its assent to the elementary education bills passed by the\npopular branch. We have seen that, in 1838, the Council refused to\nconcur in the Common School Bill passed in that year by the House of\nAssembly. Before another School Act was introduced, both Houses had\nceased to exist in the Union of the two provinces in 1840.\n\n\nREV. DR. STRACHAN'S REASONS FOR ESTABLISHING A UNIVERSITY IN UPPER\nCANADA.\n\nThe reasons which Dr. Strachan gave for urging the early\nestablishment of a Provincial University were reasonable and\nweighty in themselves, had the other necessary kind of school been\nestablished and provided for. I shall give these reasons in Dr.\nStrachan's own words. They are characteristic of the Bishop's own\nfeelings in regard to American institutions and their influence on\nthe young. He said:--\n\n     \"There is not in either province any English seminary ... at\n     which a liberal education can be obtained. Thus the youth of\n     300,000 Englishmen have no opportunity of receiving instruction\n     within the Canadas in law, medicine or divinity.\n\n     \"The consequence is that many young men ... are obliged to look\n     beyond the province for the last two or three years of their\n     education--undoubtedly the most important and critical period\n     of their whole lives ... The youth are, therefore, in some\n     degree, compelled to look towards the United States, where means\n     of education, though of a description far inferior to those of\n     Great Britain, are yet superior to anything within the province,\n     and a growing necessity is arising of sending them to finish\n     their education in that country.\"\n\nDr. Strachan then proceeds to point out in his own graphic language,\nthe peculiarly adverse influences to which loyal Canadians from\nyouth were then subjected while attending schools and universities\nin the United States. He says:--\n\n     \"Now, in the United States a custom prevails unknown to or\n     unpractised in any other nation; in all other countries morals\n     and religion are made the basis of public instruction, and\n     the first books put into the hands of children teach them\n     the domestic, the social and religious virtues; but in the\n     United States politics pervade the whole system of education;\n     the school books, from the very first elements, are stuffed\n     with praises of their own institutions, and breathe hatred to\n     everything English.\"\n\nDr. Ryerson came to the same conclusions as did Dr. Strachan in\nregard to the character of American school books. Speaking on the\nsame subject, twenty years afterwards, he said:--\n\n     \"With very few exceptions American school books abound in\n     statements and allusions prejudicial to the institutions and\n     character of the British nation.\"\n\nDr. Strachan still further refers to the anti-British influences of\neducation obtained by Canadian youth in the United States. He said:--\n\n     \"To such a country our youth may go, strongly attached to their\n     native land ... but by hearing its institutions continually\n     depreciated, and those of the United States praised ... some may\n     become fascinated with that liberty which has degenerated into\n     licentiousness, and imbibe, perhaps unconsciously, sentiments\n     unfriendly to things of which Englishmen are proud.\"\n\nDr. Strachan then proceeded to point out the advantages of having\nthe youth of the province \"carefully nurtured within the British\nDominions.\" He said:--\n\n     \"The establishment of an university at the seat of Government\n     will complete a system of education in Upper Canada from the\n     letters of the alphabet to the most profound investigations of\n     science.... This establishment, by collecting all the promising\n     youth of the colony into one place, will gradually give a new\n     tone to public sentiment and feelings ... producing the most\n     beneficial effects through the whole province. It is, indeed,\n     quite evident that the consequences of an university ...\n     possessing in itself sufficient recommendations to attract to\n     it the sons of the most opulent families would soon be visible\n     in the greater intelligence and more confirmed principles of\n     loyalty of those who would be called to various public duties\n     required in the country.\"\n\nFrom these wise and practical remarks, it will be seen how truly\nBishop Strachan estimated the great advantages to the youth of the\ncountry of university training obtained within our borders. In\nthis view he was far-seeing enough. But yet his range of vision,\nas to its beneficial effects, did not extend beyond \"the sons of\nthe most opulent families\"--which was another indication of the\nprevailing feeling of the times, that higher education in the form\nof university training was not thought of even for \"the promising\nchildren of the younger inhabitants.\" Happily our public men,\nand the Bishop himself, outgrew this narrow feeling and social\nprejudice. He even lived to see, and with great satisfaction as\nto the results, that, under the fostering care of men of large\nsympathies and more generous impulses, the doors of the educational\ninstitutions of the country, from the highest to the lowest, were\nthrown wide open to every boy, rich and poor, high and low, and\nto all the youth of the province, without distinction of race, or\ncreed, or social rank.\n\nRev. Dr. Strachan succeeded in getting a Royal Charter for the\nuniversity in 1828. This charter virtually placed the proposed\nuniversity under the control of the Episcopal Church. When its terms\nwere known in Upper Canada it was fiercely assailed. The charter\nwas subsequently modified, in deference to public opinion; but it\nwas not until many years afterwards that the university was, by\nstatute, declared to be free from denominational control. Out of the\ncontroversy which the Duke of Portland's despatch and the charter\ncaused, arose other colleges and universities, viz, Victoria and\nQueen's.\n\n\nREV. DR. STRACHAN, THE FOUNDER OF TWO UNIVERSITIES IN TORONTO.\n\nIn the Rev. Dr. Scadding's most interesting sketch of the \"First\nBishop of Toronto,--a Review and a Study\"--occurs the following\nstriking passage in regard to the founding of the \"Twins of\nLearning\" in Toronto:--\n\nThe results of the life of the first Anglican Bishop of Toronto\nare tangible realities.... He built the principal church edifice\nappertaining to his own communion four times.... \"Twins of Learning\"\nwitness for him; he founded two universities in succession\n(1842 and 1852), both invested with the character borne by such\ninstitutions as originally instituted, by Royal charter--procured\nin both instances by his own personal travail; the later of the two\nby an individual and solitary effort, to which it is not easy to\nfind a parallel. He saw both of them in operation, investigating,\nconserving and propagating truth, on somewhat different lines\nindeed, but probably with co-ordinate utility, as things are. The\nvery park, with its widely renowned avenue, the Champs Elys\u00e9es\nof Toronto, in which the bourgeoisie of the place love to take\ntheir pastime, are a provision of his--that property having been\nspecially selected by him, as President of King's College, with the\nsame judiciousness and the same careful prescience of the need of\namplitude for such purposes which guided him also in choosing the\nfine site and grounds of Trinity College.\n\n\nTHE UNIVERSITY OF TORONTO.\n\nThis university was originally established under the charter\nobtained by Rev. Dr. Strachan in 1828. But it only existed on paper\nuntil 1842-43.\n\nIn April, 1842, the corner-stone of the new institution was laid\nby Governor-General Sir Charles Bagot, (M.A. of Christ Church,\nOxford). In June, 1843, it was opened under the style and title of\nthe \"University of King's College,\" Toronto, by the Right Rev. John\nStrachan, D.D., LL. D., President of the University. In October\nof that year, an effort was made by Hon. Attorney-General Baldwin\nto introduce a comprehensive scheme of university reform, but it\nwas defeated in the Legislature. In 1845 and 1847 other abortive\nattempts were made to \"reform\" the university; but in 1849 a\ncomprehensive measure was introduced into the Legislature and passed\ninto a law, by which it was reincorporated under the name of the\n\"University of Toronto,\" and made a purely provincial institution,\nby placing it under the sole control of the Government, and of a\nsenate and officers appointed by the Government.\n\nIn 1853 another Act was passed, under which the University was\nconstituted with two corporations, \"The University of Toronto,\" and\n\"University College,\" the functions of the former being limited to\nthe examination of candidates for degrees in the several faculties,\nor for scholarships and honors, and the granting of such degrees,\netc.; those of the latter being confined to the teaching of subjects\nin the Faculty of Arts.[39] By this Act certain institutions,\nfrom which students might be examined, were affiliated with the\nUniversity.\n\n  [39] By recent legislation University College has been merged in the\n  University of Toronto.\n\nIn 1873 further amendments were made in the constitution of the\nUniversity. The Chancellor was made elective for a period of three\nyears by Convocation, which was then re-established. By this Act the\npowers of the Senate were extended to all branches of knowledge,\nliterature, science and arts, and also to granting certificates\nof proficiency to women; the power of affiliation was likewise\nextended; the Senate was also empowered to provide for local\nexaminations.\n\nLatterly, the faculties of law and medicine have been restored and\nother extensions of the University course have been made.\n\n\nTHE UNIVERSITY OF VICTORIA COLLEGE, COBOURG.\n\nThe Rev. Dr. Ryerson, who was the founder of this University, thus\nspeaks of its early history, in an address to the students when he\nwas appointed its first principal in 1841. He said:--\n\nHis late Most Gracious Majesty William IV., of precious memory,\nfirst invested this institution, in 1836, with a corporate\ncharter as the Upper Canada Academy--the first institution of the\nkind established by Royal Charter unconnected with the Church\nof England, throughout the British colonies. It is a cause of\nrenewed satisfaction and congratulation that, after five year's\noperation as an academy, it has been incorporated as a university\nand financially assisted by the unanimous vote of both branches of\nthe Provincial Legislature--sanctioned by more than an official\ncordiality, in Her Majesty's name, by the late lamented Lord\nSydenham, Governor-General, one of whose last messages to the\nLegislative Assembly was a recommendation to grant \u00a3500 as an aid to\nthe Victoria College.... We have buoyant hopes for our country when\nour rulers and legislators direct their earliest and most liberal\nattention to in literary institutions and educational interests. A\nfoundation for a common school system in this Province his been laid\nby the Legislature, which I believe will, at no distant day, exceed\nin efficiency any yet established on the American continent;[40]\nand I have reason to believe that the attention of the Government\nis earnestly directed to make permanent provision for the support\nof colleges also, that they may be rendered efficient in their\noperation and accessible to as large a number of the enterprising\nyouth of our country as possible.\n\n  [40] This memorable prophecy, made by Dr. Ryerson in 1841, was\n  abundantly verified in after years, chiefly as the result of his own\n  labors in maturing the school system, of which he was the founder.\n\nThis institution originated with the Wesleyan Methodists in 1828-30.\nThe conference in the latter year agreed to establish it as an\nAcademy, and the following year, Dr. Ryerson, in the _Christian\nGuardian_ newspaper, of which he was then editor, issued a strong\nappeal in behalf of the proposed institution on the 21st April,\n1831. On the 7th June, 1832, the foundation stone of the Academy\nwas laid; and on the 18th June, 1836, it was formerly opened under\nthe designation of \"Upper Canada Academy.\" In the previous year\nDr. Ryerson was deputed to go to England to collect subscriptions\non behalf of the institution. He was there enabled to obtain a\nRoyal Charter for the Academy and a grant of $16,400 from the Local\nLegislature.\n\nAmongst the last public acts performed by Lord Sydenham was the\ngiving of the Royal assent to a Bill for the erection of the Upper\nCanada Academy into a College with University powers. This he did on\nthe 27th August, 1841. Dr. Ryerson thus refers to the event, in a\nletter written from Kingston on that day:--\n\nThe establishment of such an institution by the members of the\nWesleyan Methodist Church in Canada attests their estimate of\neducation and science; and the passing of such an Act unanimously by\nboth Houses of the Legislature, and the Royal assent to it by His\nExcellency in Her Majesty's name, is an ample refutation of recent\nstatements and proceedings of the Wesleyan Committee in London\n... while the Act itself will advance the paramount interests of\nliterary education amongst Her Majesty's Canadian subjects.... For\nthe accomplishment of this purpose, a grant must be added to the\ncharter--a measure ... honorable to the enlightened liberality of\nthe Government and Legislature. When they are securely laying a\nbroad foundation for popular government, and devising comprehensive\nschemes for the development of the latent resources of the country,\nand the improvement of its internal communication, and proposing a\nliberal system of common school education, free from the domination\nof every church, and aiding colleges which may have been established\nby any church, we may rationally and confidently anticipate the\narrival of a long-looked for era of civil government and civil\nliberty, social harmony, and public prosperity.\n\nThe Academy was thus incorporated as a University, in August, 1841.\nIn October, 1841, Rev. Dr. Ryerson was appointed the first president\nof the University, a position which he held until he was appointed\nChief Superintendent of Education for Upper Canada in 1844. He was\nsucceeded by the Rev. Dr. Macnab, now rector of Darlington. In 1850\nthe late accomplished president (Rev. S. S. Nelles, D.D., LL.D.) was\nappointed. He had been a pupil under Dr. Ryerson, but finished his\nuniversity education at the Wesleyan University, Middletown, Conn.,\nand graduated there. He received the degree of D.D. from the Queen's\nUniversity, Kingston, and that of LL.D. from his own university.\nHis career was an unusually long and prosperous one; and under his\nadministration the university has taken high rank amongst the sister\nuniversities of Ontario.[41]\n\n  [41] It it a gratifying fact that Victoria College was the first\n  university in Upper Canada whose doors were open to receive\n  students. The first session commenced in October, 1841; that of\n  Queen's College University in March, 1842, and King's College\n  University in June, 1843. The first graduate in arts who received\n  a diploma in Upper Canada was sent out from Victoria College in\n  1845-46.\n\nIn the original appeal made by Dr. Ryerson in England on behalf\nof the Academy (in 1835), he stated the \"specific objects of the\ninstitution\" to be as follows:--\n\n1. To educate, upon terms equally moderate with similar institutions\nin the neighboring republic of the United States, and with strict\nattention to their morals, the youth of Canada generally.\n\n2. To educate for common school masters, free of charge, poor young\nmen of Christian principles and character, and of promising talent,\nwho have an ardent thirst for knowledge.\n\n3. To educate the most promising youth of the recently converted\nIndian tribes of Canada as teachers to their aboriginal\ncountrymen.[42]\n\n  [42] Several promising Indian youth were educated at Victoria\n  College, and some of them became useful teachers and missionaries.\n\nThese extracts are highly interesting, as showing the noble and\ncomprehensive aims, in these early days of educational effort, which\nDr. Ryerson had in view in founding this valuable institution of\nlearning. He goes on then (apart from these objects) to show the\ngrave necessity which existed for the early establishment of such an\ninstitution. He said:--\n\nFor want of such an institution upwards of sixty of the youth of\nCanada are now attending seminaries of learning, under a similar\nmanagement, in the United States, where nearly two hundred Canadian\nyouth have been taught the elementary branches of a professional\neducation during the last eight years. There is good reason to\nbelieve that nearly, if not quite, all the Canadian youth now being\ntaught in the United States seminaries of learning, will return to\nCanada as soon as this institution shall have been brought into\noperation....\n\nIn behalf, therefore, of this institution--most important to\nthe best interests of a healthy, fertile and rapidly improving\nBritish colonial possession, the inhabitants of which have in\nthis, as in other instances, shown the strongest desire to help\nthemselves to the utmost of their very limited means--a respectful\nand earnest appeal is made to British liberality, an appeal which\nit is devotedly hoped will be responded to in a manner that will\ncontribute to draw still closer the bonds by which the loyal\nProvince of Upper and the British population of Lower Canada are\nunited to the Mother Country.\n\nThis appeal was endorsed by the Governor of the Province, Sir John\nColborne (afterwards Lord Seaton), in the following terms:--\n\nThe Rev. Egerton Ryerson proceeds to England ... to solicit\nsubscriptions ... to enable [the conference here] to bring into\noperation a seminary established at Cobourg, in Upper Canada.... As\nI am persuaded this colony will derive the greatest advantage from\nthe institution and from the exertions of the conference to diffuse\nreligions instruction, I cannot but strongly recommend that it may\nreceive encouragement and support from all persons interested in the\nwelfare of Upper Canada.\n\nThe \"appeal\" was also heartily endorsed by the Hon. Peter McGill,\nfounder of McGill College University, Montreal, and by other\ndistinguished gentlemen and merchants in Montreal. In his letter Mr.\nMcGill referred to Dr. Ryerson as \"a gentleman who has distinguished\nhimself in Upper Canada by his writings in defense of religion,\norder, and good government.\"\n\nAfter much delay and great discouragement, Dr. Ryerson succeeded in\nthe objects of his mission--money and a royal charter; but at the\nclose of his mission he writes to the Academy Committee as follows:--\n\nThus terminated this protracted [business], ... though I had to\nencounter successive, discouraging and almost insurmountable\ndifficulties [in obtaining the charter]. Not having been able to\neffect any loan ... on account of the agitated state of the Canadas,\nand being in suspense as to the result of my application to the\nGovernment, I was several months pressed down with anxiety and fear,\nby this suspense and by reason of the failure of my efforts to\nobtain relief. In this anxiety and fear my own unassisted resolution\nand fortitude could not sustain me. I had to rely upon the unfailing\nsupport of the Lord my God.\n\nI have given these particulars somewhat in detail, as they afford\na striking narrative illustration of the almost insurmountable\ndifficulties which the early pioneers of education in this Province\nencountered in endeavoring to found these valuable institutions\nwhich have been so useful to this country, and which have shed\nsuch lustre upon their founders' names. It is also due to Victoria\nUniversity, and (as I shall show) to Queen's University also to\nstate these particulars, from the fact that the first practical,\nyet entirely abortive, attempt to make King's College a provincial\nuniversity, was made in 1843, two years after the Methodists\nand Presbyterians had, in self-defence, been compelled to found\nuniversities of their own. This they did at a great sacrifice.\n\nBy the time that the liberation of King's College took place, in\n1849-'53, the really provincial universities at Cobourg and Kingston\nhad become recognized as most important factors in our educational\nsystem; and from them alone, up to that time, could students of all\ndenominations obtain a university education.\n\nIn connection with the university, Faraday Hall, or School of\nPractical Science, was erected in 1877. It is a handsome and\nspacious building, and is admirably fitted up for the purpose of\nscience teaching.\n\n\nTHE QUEEN'S COLLEGE UNIVERSITY, KINGSTON.\n\nAs early as 1829 it was felt among the members of the United\nPresbytery of Upper Canada that a seminary, or college, for the\ntraining of their ministers was highly desirable. As the management\nof King's College at Toronto was in the hands of the adherents of\nthe Church of England, it was felt that such an institution could\nnot be made available for Presbyterian theological instruction.\nA committee of the British House of Commons, to which had been\nreferred petitions from Canada in 1828 and 1830 against the\nexclusive character of the charter of King's College, Toronto, were\ndisposed to solve the difficulty by suggesting that two theological\nchairs be established in King's College (and did so recommend)--one\nfor students of the Churches of England and Scotland, respectively.\nNothing, however, of the kind was done; nor was there any arts\ncollege then open on equal terms to all the youth of the country.\nThe Presbyterians, like the Methodists, had, therefore, to found\nan institution of their own. Steps were taken by the synod of the\nChurch in 1831 and 1839 to found such an institution. At a meeting\nheld in Hamilton, in November, 1839, the commission appointed for\nthat purpose prepared the draft of a charter for the proposed\ncollege. Kingston was selected by the synod as the site for the new\ninstitution.\n\nAn Act embodying the charter was passed by the Provincial\nLegislature in February, 1840, incorporating the \"University\nof Kingston.\" The Act was, however, disallowed by the imperial\nauthorities, on the ground that it conflicted with the royal\nprerogative of granting charters. A royal charter was, however,\nissued in 1841, incorporating the institution under the name of\nQueen's College, with \"the style and privileges of a university.\"\n\nThe opening of Queen's took place on the 7th of March, 1842. Rev.\nThomas Liddell, D.D., of Edinburgh, was the Principal and Professor\nof Divinity, and Rev. P. C. Campbell, of Brockville, Professor\nof Classics. Rev. James Williamson, D.D., LL.D., in 1842, became\nProfessor of Mathematics and Natural Philosophy. He is, therefore,\nthe oldest college professor in Ontario.\n\nAfter the opening of King's College, Toronto, in 1843, an agitation\ncommenced with the view to unite the three universities then in\noperation into a single provincial institution. Many plans were\nproposed, and several measures tending to that end were introduced\ninto Parliament and fully discussed. In 1843 the Hon. Robert Baldwin\nintroduced a university bill, which, though it presented many\npopular features, was strongly objected to by the churches named and\nothers also, because it was deficient in providing for religious\ninstruction.\n\nA bill was introduced by Hon. W. H. Draper, in 1845, to amend the\nlaw so as to make it more generally acceptable to the religious\nbodies of the country; and in 1847 the late Hon. John Hillyard\nCameron introduced a measure in which it was proposed to devote a\nlarge part of the endowment to increased support of high schools and\nalso to largely subsidize the denominational colleges. The measure\nfailed to carry in Parliament, however, and this practically ended\nthe agitation for the union of colleges for many years.\n\nIn 1846 Dr. Liddell resigned his position as Principal and returned\nto Scotland. Rev. J. Machar, D.D., was next appointed Principal, and\nunder his administration there was slow but real improvement.\n\nRev. Dr. Cook, of Quebec occupied the position of Principal for\na time, but he refused to accept the position permanently. Rev.\nDr. Leitch was next appointed, but his early death deprived the\ninstitution of his services. He was followed by the Rev. Dr.\nSnodgrass, and on his retirement the Rev. George Monro Grant, D.D.,\nof Halifax, was appointed. Dr. Grant entered on his arduous duties\nwith his accustomed energy, and occupies that position with great\nacceptance. He is an able speaker and a wise administrator. Queen's\nCollege has now faculties of arts, theology, and law, and there are\naffiliated with it the Royal College of Physicians and Surgeons,\nalso in a prosperous condition, and the Kingston Women's Medical\nCollege.\n\nIn 1869 it was resolved to make an appeal to the country for aid.\nThe people of Kingston raised about $25,000, and the result of the\nwhole effort was that about $103,000 was raised for the equipment of\nthe college.\n\nIn 1878 Principal Grant made the proposition to raise $150,000,\nin order to provide new buildings, additional professors, and\napparatus. The appeal was successful; additional ground of about\ntwenty acres was at once purchased--a site of rare beauty and\nconvenience--and the present noble building was erected.\n\n\nTHE UNIVERSITY OF TRINITY COLLEGE, TORONTO.\n\nThe immediate cause of the founding of this College and University\nwas the suppression, in 1849, of the Faculty of Divinity in King's\nCollege, now the University of Toronto. In consequence of this the\nRight Rev. J. Strachan D.D., Bishop of Toronto, issued in February,\n1850, a pastoral appeal to members of the Church of England for\nfunds to enable him to establish a Church University and College. In\nresponse to this pastoral, the Bishop succeeded in raising a large\nendowment from voluntary subscriptions from churchmen in Canada,\nEngland, and the United States, so that on April 30, 1851, the\nfoundation stone of the college building was laid, and on January\n15, 1852, the work of instruction was begun, the staff consisting\nof four professors in arts, besides those in the faculties of law\nand medicine. During the last thirty years the endowment has been\nlargely increased by liberal contributions made from time to time,\nso that the original amount is now about trebled. In 1878 a large\nand handsome convocation hall was erected, and in 1884 a long felt\nwant was supplied by the erection of a finely proportioned and\nbeautiful chapel.\n\nThe University of Trinity College at present consists of the\nfaculty of arts and divinity, of an affiliated Medical School with\na commodious building and a large staff of professors, and an\naffiliated Women's Medical College. Provision is also made for the\nhigher education of women in connection with the Bishop Strachan\nSchool in Toronto, and connected with the University is a large\nschool for boys at Port Hope.\n\n\nTHE R. C. UNIVERSITY COLLEGE AT OTTAWA.\n\nThe College (or University) of Ottawa is under the direction of\nthe Roman Catholic Church. It was founded in 1848 by the Right\nReverend Joseph Eugene Guignes, O.M.I., D.D., first R. C. Bishop of\nOttawa. In 1856, the Bishop confided the direction of the college\nto the \"Society of the Oblate Fathers of Mary Immaculate.\" The\ntotal value of the college building and grounds is about $75,000.\nIt has also a good library and cabinet of natural philosophy (or\nphysics), and of chemistry and natural history. The college obtained\nuniversity powers in 1866. It confers degrees in arts, science, and\nliterature--B. A., B. Sc., B. L., as well as M. A.\n\n\nTHE WESTERN UNIVERSITY, LONDON.\n\nThis institution, in connection with the Church of England in\nCanada, was incorporated in 1878, with power to affiliate with\nHuron Theological College and to confer Degrees in Arts, Divinity,\nMedicine and Law. The affiliation between that College and the\nUniversity took place in 1881, and the University was inaugurated in\nthe month of October of that year. The object of its establishment\nwas, as a Church of England Institution in the Diocese of Huron, to\nobtain the same power of conferring Degrees as was possessed by the\nsister University of Trinity College; also, that a liberal Education\nin Arts, Science and Literature might be extended to that extensive\nportion of the Province of which London is the geographical centre.\n\n\nTHE MCMASTER UNIVERSITY.\n\nBy the munificence of the late Hon. Wm. McMaster, McMaster\nUniversity is being established on a sound financial basis. McMaster\nHall, Woodstock College and Moulton College (for ladies) are\naffiliated institutions.\n\n\nUPPER CANADA COLLEGE.\n\nUpper Canada College was founded in 1828 upon the model of the\ngreat public schools of England, and was endowed with a grant of\n66,000 acres of public lands, from which it now derives an annual\nincome of $15,000, in addition to its building and grounds in the\ncity of Toronto. It is governed by a committee of the senate of the\nProvincial University. The curriculum extends over a six years'\ncourse of study in the same number of forms, and embraces the usual\nsubjects.\n\nIn other forms, known as the lower and upper modern, commercial and\nscientific training can be obtained. Scholarships may be established\nby the different county councils, while four exhibitions have\nbeen founded out of the University funds. This college and the\nhigh schools constitute the principal feeders of the Collegiate\nInstitutes and provincial University.\n\n\nALBERT COLLEGE, BELLEVILLE.\n\nThis institution, founded in 1854, was the product of the zeal\nof the Methodism of that early day. Accordingly, the General\nConference of the Methodist Episcopal Church, in 1854, adopted a\nscheme--initiated in the Bay of Quint\u00e9 Conference in the preceding\nyear--for the erection and maintenance of an educational institution\nof high grade in Belleville. Having been chartered by Parliament in\n1857 as \"Belleville Seminary,\" it was opened in July of the same\nyear, and entered upon its work under very favorable auspices.\n\nIn the year 1886, by Act of Parliament, the name was changed to\n\"Albert College,\" and a senate created with ample powers. By the\nterms of the union of the Methodist Churches of Canada, Albert\nCollege was retained in Belleville, and adopted by the General\nConference of the United Church as a church school. The charter was\namended and the college was affiliated to the Victoria University,\nCobourg. The senate has full powers to examine, grant prizes,\nscholarships, medals, honor certificates, and diplomas in music,\nfine arts, commercial science, collegiate courses, etc.\n\n\nWOODSTOCK COLLEGE.\n\nWoodstock College, formerly \"The Canadian Literary Institute,\" was\nfounded in 1867, principally through the exertions of the late R. A.\nFife, D.D. Under his presidency, ably assisted for eighteen years\nby Prof. J. E. Wells, M.A., the school constantly increased in\nefficiency and power, until from a small beginning it has attained\nits present large proportions and wide influence.\n\n\nTHE SCHOOL OF PRACTICAL SCIENCE, TORONTO.\n\nPrior to the year 1871 there was no institution in the Province\nfor practical instruction in the industrial sciences. In 1870 the\nGovernment of the Province issued a commission to Dr. Hodgins,\nDeputy Superintendent of Education, and to Dr. Machatt of London,\ndirecting them to proceed to the United States for the purpose of\ninspecting and reporting upon any Technical or Science Schools or\nColleges there established, as to their buildings, departments of\nstudy and general appliances. On their return a report was submitted\nto the Government, with full details as to the cost of the proposed\ninstitution. The Government acted upon the information contained in\ntheir report, and with a grant of $50,000 established a \"College of\nTechnology\" in Toronto. In 1877 the name was changed to the School\nof Practical Science, and the Hon. Adam Crooks, Q.C., Minister of\nEducation, had a suitable building for it erected close to the\nProvincial University, four of the University Professors are engaged\nin Departments of the School. The new building was opened for\nstudents in September, 1878.\n\n\nVARIOUS OTHER COLLEGES AND SCHOOLS, ETC.\n\nThere are numerous superior colleges and schools for boys and\ncolleges for ladies in Ontario, but the limits of this paper forbids\na further reference to them, or to the other numerous educational\ninstitutions--theological, literary and commercial--in the Province.\n\n\nREV. DR. RYERSON'S ADVOCACY OF POPULAR RIGHTS, 1827-1841.[43]\n\n  [43] In the preface to the _Story of My Life_, I thus referred to\n  this period of Dr. Ryerson's labours:--\"Public men of the present\n  day looked upon Dr. Ryerson practically as one of their own\n  contemporaries--noted for his zeal and energy in the successful\n  management of a great Public Department, and as the founder of a\n  system of Popular Education.... In this estimate of Dr. Ryerson's\n  labours they were quite correct. And in their appreciation of the\n  statesmanlike qualities of mind, which devised and developed such a\n  system in the midst of difficulties which would have appalled less\n  resolute hearts, they were equally correct.\n\n\"But, after all, how immeasurably does this partial historical view\nof his character and labours fall short of a true estimate of that\ncharacter and of those labours!\n\n\"In a point of fact, Dr. Ryerson's great struggle for the civil\nand religious freedom which we now enjoy, was almost over when he\nassumed the position of Chief Director of our Educational System.\nNo one can read the record of his labours from 1825 to 1845, as\ndetailed in the pages of this 'story' without being impressed with\nthe fact that, had he done no more for his native country than that\nwhich is therein recorded, he would have accomplished a great work,\nand have earned the gratitude of his fellow-countrymen.\"\n\nDuring all this time the friends of popular education were not idle.\nFrom 1827 and for many years Dr. Ryerson was engaged in waging war\nwith the opponents of liberal institutions and religious equality.\nHis chief antagonist was Dr. Strachan. The subjects in dispute\nrelated to a dominant church, the application of clergy-reserved\nlands to the purposes of education, and the liberation of the\nprovincial university from exclusive control under the presidency\nof Dr. Strachan, first as archdeacon and afterwards as bishop. Not\nbeing eligible to the popular branch of the Legislature (being a\nminister), Dr. Ryerson had to develop his powers of resistance to\nthe dominant and ruling party in other directions; and this he\ndid with wonderful success. As a writer and debater few equalled\nhim in his presentation of facts, and in his skill in detecting\nthe weak points of his adversary's position or argument. As a\ncontroversialist and pamphleteer he had confessedly no rival. He,\ntherefore, was able to furnish his friends in the House of Assembly\nwith facts and arguments which were irresistible. They passed\nresolutions and school bills time and again, but could not always\ninduce the Legislative Council (Senate) to concur in their adoption.\nThis state of things continued for many years, and with disastrous\neffects on the intellectual growth and well-being of the province.\nThis fact is attested by indubitable witnesses, and is recorded in\nthe proceedings of the House of Assembly of the time, as is shown in\nthe extracts from its proceedings which I have already given.\n\n\nEDUCATIONAL LEGISLATION IN THE UNITED PARLIAMENT OF 1841 AND 1843.\n\nIn 1840 the House of Assembly and Legislative Council of Upper\nCanada ceased to exist, and the two Provinces of Upper and Lower\nwere united under one Legislature.\n\nThe momentous political events which preceded this union, and\nwhich led to the total disruption of all political parties and\ncombinations, were very salutary in their effects. Under the\nliberal policy pursued by the Home Government, after the publication\nof Lord Durham's report, grievances were redressed, and a broad and\ncomprehensive scheme of popular government inaugurated. The result\nwas that the wise and statesmanlike measures, designed to promote\npublic tranquility and local self-government, were proposed to and\nadopted by the Legislature.\n\nAmongst these was a measure providing for the establishment of a\nmunicipal council in each local division of the Province of Upper\nCanada (and partly so in Lower Canada) for the regulation of\ninternal matters.\n\nIn recommending the scheme of Common School Education to the\nfavorable consideration of the first Parliament of United Canada, in\n1841, Lord Sydenham, the first Governor-General, used the following\nlanguage:--\n\n     A due provision for the education of the people is one of the\n     first duties of the State, and, in this province especially,\n     the want of it is grievously felt. The establishment of an\n     efficient system, by which the blessings of instruction may be\n     placed within the reach of all is a work of difficulty, but its\n     overwhelming importance demands that it should be undertaken.\n     I recommend the consideration of that subject to your best\n     attention, and I shall be most anxious to afford you, in your\n     labours, all the co-operation in my power. If it should be found\n     impossible so to reconcile conflicting opinions as to obtain a\n     measure which may meet the approbation of all, I trust that, at\n     least, steps may be taken by which an advance to a more perfect\n     system may be made, and the difficulty under which the people of\n     this province now labor may be greatly diminished, subject to\n     such improvements hereafter as time and experience may point out.\n\nThe enlightened expectations of the Governor-General were, happily,\nrealized. But so diverse were the populations of the two Canadas\nthus united, and so different were their social conditions, that the\nSchool Act then passed was repealed two years afterward (in 1843),\nand a school bill for each Province was passed by the Legislature\nin that year. Provision for Roman Catholic and Protestant Separate\nSchools was made in both Acts.[44]\n\n  [44] This Retrospect would not be complete without reference, in\n  fuller detail, to the history of the Separate School question and\n  to legislation on it in Upper Canada. It was found, however, to be\n  so extensive a subject that no adequate justice could be done to\n  it in this somewhat brief Retrospect. The writer has, therefore,\n  prepared a full and exhaustive paper on the subject, which will be\n  published separately, should it be considered desirable. The details\n  given are largely personal, and, therefore, of special interest.\n  In addition to private letters bearing on the subject, the paper\n  contains official and other authentic information in regard to the\n  whole question.\n\nOn this system was ingrafted, by means of a separate Act applicable\nto the whole Province, a scheme of public education, with a liberal\nprovision ($200,000 per annum) for its maintenance.\n\n\nORIGIN OF THE ANNUAL GRANT OF $200,000 FOR THE COMMON SCHOOLS IN\n1841.\n\nIn a letter to the writer of this Retrospect, from the Hon. Issac\nBuchanan, dated 11th April. 1883, in reply to some enquiries in\nregard to the appointment of Dr. Ryerson, Mr. Buchanan thus related\nthe circumstances under which the munificent sum of $200,000 a year\nwas granted by the Legislature in 1841 for the support of the then\nnewly-established Common Schools in Upper and Lower Canada. He\nsaid:--\"This first attempt of mine to get an endowment for education\n(out of the Clergy Reserve Fund), failed as there was no responsible\ngovernment then. But five years afterwards when my election for\nToronto had carried Responsible Government, and before the first\nparliament met, I was talking to the Governor-General (C. Poulett\nThompson, Lord Sydenham). He felt under considerable obligation to\nme for standing in the breach when Mr. Robert Baldwin found that\nhe could not succeed in carrying Toronto.... He spoke of Canada as\n'a drag upon the mother country.' I replied warmly ... for I felt\nsure (as I told him), that if we were allowed to throw the affairs\nof the Province into regular books ... we would show a surplus over\nexpenditure. His Excellency agreed to my proposal, and I stipulated\nthat, if we showed a yearly surplus, one half would be given as an\nendowment for an educational system. Happily we found that Upper\nCanada had a surplus revenue of about \u00a3100,000 ($400,000), one half\nof which the parliament of 1841 laid aside for education, the law\nstipulating that every District Council getting a share of it would\ntax locally for as much more, and this constituted the fund of your\neducational system.\"\n\n\nEDUCATIONAL EFFORTS OF DR. RYERSON UP TO THIS TIME.\n\nUp to this time Dr. Ryerson's energies, as I have shown, were wholly\nengrossed in contending for the civil and religious rights of the\npeople. He had also, ten years before, projected and collected money\nfor the establishment of an academy or college for higher education\nat Cobourg, on the north shore of Lake Ontario. His efforts in this,\nand in the establishment of the Victoria College at Cobourg, as a\nuniversity, in 1840, aroused a widespread interest in education\ngenerally, which bore good fruit afterward. This university has now\nbeen in operation forty-eight years, and from it the first arts\ngraduate in Upper Canada was sent forth in 1846. Its first president\nwas the Rev. Dr. Ryerson; the second was the Rev. Dr. Macnab, now\nof Bowmanville. Its late distinguished president, the Rev. Dr.\nNelles, was Dr. Ryerson's pupil. He held his position with honor to\nhimself for thirty-six years, and died in October, 1887, deeply and\nuniversally regretted by the entire community.\n\n\nFIRST APPOINTMENT OF A SUPERINTENDENT OF EDUCATION FOR UPPER CANADA.\n\nIn the _Canada Gazette_ of May, 1842, the following announcement was\nmade:--\n\n  \"SECRETARY'S OFFICE, KINGSTON, 11th May, 1842.\n\n     \"His Excellency the Governor-General has been pleased to make\n     the following appointments:--\n\n     \"The Honorable Robert Sympson Jameson, Vice-Chancellor, to be\n     Superintendent of Education, under the Provincial Act, 4th and\n     5th Victoria, chapter 18.\n\n     \"The Reverend Robert Murray and Jean Baptiste Meilleur, Esquire,\n     to be Assistant Superintendents of Education for Western and\n     Eastern Canada, respectively.\n\n  \"By command,\n\n  \"S. B. HARRISON, Provincial Secretary.\"\n\nHad the Governor-General, Lord Sydenham, not met with the fatal\naccident which terminated his life, in September, 1841, the Rev.\nDr. Ryerson would, without doubt, as I have shown in \"The Story of\nmy Life,\" have been appointed in that year. Mr. Murray, who was\nneighbor and friend of the Hon. S. B. Harrison, of Bronte, near\nOakville, then Provincial Secretary, was nominated by him, and\nreceived his appointment from Lord Sydenham's successor, Sir Charles\nBagot.\n\nIn point of fact, the appointment was first spoken of to Dr. Ryerson\nby Lord Sydenham himself, in the autumn of 1841. The particulars of\nthat circumstance are mentioned in detail in a letter written by Dr.\nRyerson to T. W. C. Murdoch, Esq., Private Secretary to Sir Charles\nBagot, on the 14th January, 1842. Dr. Ryerson said:--\n\n\"In the last interview with which I was honoured by [Lord Sydenham],\nhe intimated that he thought I might be more usefully employed\nfor this country than in my present limited sphere; and whether\nthere was not some position in which I could more advantageously\nserve the country at large. I remarked that I could not resign my\npresent official position in the Church, with the advocacy of whose\ninterests I had been entrusted, until their final and satisfactory\nadjustment by the Government, as I might thereby be represented as\nhaving abandoned or sacrificed their interests; but that after such\nadjustment I should feel myself very differently situated, and free\nto do anything which might be beneficial to the country, and which\ninvolved no compromise of my professional character; that I knew\nof no such position likely to be at the disposal of the Government\nexcept the superintendency of Common Schools (provided for in the\nBill then before the Legislature), which office would afford the\nincumbent a most favorable opportunity, by his communications,\npreparation and recommendation of books for libraries, etc., to\nabolish differences and jealousies on minor points; to promote\nagreement on great principles and interests; to introduce the best\nkind of reading for the youth of the country; and the not onerous\nduties of which office would also afford him leisure to prepare\npublications calculated to teach the people at large to appreciate,\nupon high moral and social considerations, the institutions\nestablished amongst them; and to furnish, from time to time, such\nexpositions of great principles and measures of the administration\nas would secure the proper appreciation and support of them on the\npart of the people at large. Lord Sydenham expressed himself as\nhighly gratified at this expression of my views and feelings; but\nthe passing of the Bill was then doubtful, although His Lordship\nexpressed his determination to get it passed if possible, and\ngive effect to what he had proposed to me, and which was then\ncontemplated by him.\"\n\n\"What afterwards grew to be the Department of Education was (under\nthe first general school law, passed in 1841, for the whole of\nthe Province of Canada) originally a subordinate branch of the\nProvincial Secretary's office at Kingston. That for Upper Canada was\nmanaged by Assistant Superintendent Rev. Robert Murray, M.A., and\none clerk (Mr. Robert Richardson). The nominal Chief Superintendent,\nas above noted, was the Hon. Vice-Chancellor Jameson. On the repeal\nof the General School Act of 1841 and the passing of a separate Act\nfor each province in 1843 the Education branch of the Provincial\nSecretary's office was divided and reconstructed. The divisions\nthen made were designated respectively \"Education Office (East) and\nEducation Office (West).\"[45]\n\n  [45] The _Evangelical Churchman_ of the 21st February, 1887,\n  thus refers to the vicissitudes of the Education office:--The\n  Education Department for Ontario, or rather Upper Canada as it\n  was then called, had its first Toronto office in Bay Street, in\n  the building now occupied by the publishers of the _Evangelical\n  Churchman_. From 1841, when the first Provincial School Law was\n  passed, until 1844, the office was a mere adjunct to the Provincial\n  Secretary's Department at Kingston. In that year Rev. Dr. Ryerson\n  was appointed to the office which he so ably filled until 1876, when\n  he retired. In 1844 the Education office was removed to Cobourg,\n  when the present Deputy Minister of Education became its chief and\n  sole officer under Dr. Ryerson. In 1846 it was again removed and\n  transferred to Toronto, and was placed in a room over the front\n  door of the present _Evangelical Churchman_ office. The first\n  Council of Public Instruction for the Province held its meetings\n  in the west portion of the printing office, upstairs. In the room\n  over the door the first reliable statistical report of the number\n  of schools, etc., in this province was compiled. It was printed in\n  the shape of a broad sheet, 12x15, on light blue paper, and bears\n  date \"September, 1846.\" This report, which is full of interesting\n  statistics, has long since been out of print, but we have been\n  fortunate enough to obtain a copy. The Education office had various\n  vicissitudes in those early days of its existence. In 1847 it was\n  removed from this building to the Secretary's office at the old\n  Government House--long since demolished. In 1849, when the seat of\n  Government was removed from Montreal to Toronto, it was transferred\n  to the \"Albany Chambers,\" now the Revere House, on King street.\n  Thence, in 1852, it was finally removed to its present handsome\n  quarters in St. James' square.\n\n\nAPPOINTMENT OF REV. DR. RYERSON AS SUPERINTENDENT OF EDUCATION.\n\nIn 1844 Mr. Murray was made Professor of Mathematics in the\nUniversity of King's College, and the Rev. Dr. Ryerson was appointed\nas Superintendent of Schools in his place. The announcement of this\nappointment appeared in the CANADA GAZETTE of October, 1844, as\nfollows:--\n\n  \"SECRETARY'S OFFICE, MONTREAL, 18th October, 1844.\n\n     \"His Excellency the Governor-General has been pleased to\n     appoint:--\n\n     \"The Reverend Egerton Ryerson, D.D., to be Assistant\n     Superintendent of Education for that part of the province\n     formerly Upper Canada, in place of the Reverend Robert Murray,\n     appointed a Professor in the University of King's College, and\n     all communications connected with the Education office for Upper\n     Canada are to be addressed to him at Cobourg.\n\n  \"By command.\n\n  \"D. DALY,\n  \"Secretary of the Province.\"\n\nDr. Ryerson was notified of the appointment by letter in September,\n1844, but was not gazetted until the 18th of the next mouth. It\nwas my good fortune to be associated with him from the time of his\nappointment in 1844 until he retired from office in 1876.\n\n\nDR. RYERSON'S REPORT ON A SYSTEM OF PUBLIC INSTRUCTION FOR UPPER\nCANADA.\n\nImmediately after his appointment Dr. Ryerson went to Europe,\nand remained away for over a year in familiarizing himself with\nthe systems of education there. On his return he published an\nelaborate report on his projected scheme of \"Public Instruction for\nUpper Canada.\" That report was approved by the Governor-General\nin Council, and he was directed to prepare a bill to give effect\nto his recommendations, which he did in 1846. A brief analysis of\nthat report may be interesting:--It is divided into two parts: 1.\nPrinciples of the system and subjects to be taught; 2. machinery of\nthe system.\n\nAfter defining what was \"meant by education,\" the principles of the\nsystem were laid down as follows:--\n\n1. It should be universal.\n\n2. It should be practical.\n\n3. It should be founded on religion and morality.\n\n4. It should develop all the intellectual and physical powers.\n\n5. It should provide for the efficient teaching of the following\nsubjects: Biblical history and morality, reading and spelling,\nwriting, arithmetic, grammar, geography, linear drawing, vocal\nmusic, history, natural history, natural philosophy, agriculture,\nhuman physiology, civil government, political economy. Each of these\ntopics was fully discussed and illustrated in the first part of the\nreport.\n\n       *       *       *       *       *\n\nThe second part explained the machinery of the system, which was\nsummarized as follows:--\n\n1. Schools--their gradation and system.\n\n2. The teacher and his training.\n\n3. The text-books recommended.\n\n4. Control and inspection on the part of the Government.\n\n5. Individual and local efforts.\n\nThese several topics wore also fully discussed and illustrated, so\nthat the whole comprehensive scheme of education proposed by Dr.\nRyerson was clearly and fully understood. The report occupied nearly\n200 pages.\n\nThe school law founded upon this report provided, amongst other\nthings, for--\n\n1. A general Board of Education for the Province, to take charge\nof a normal school, and to aid the Chief Superintendent in certain\nmatters.\n\n2. A normal school, with practice or model schools attached.\n\n3. The regulation of school libraries.\n\n4. Plans of school houses.\n\n5. Appointment of district, instead of county and township, school\nsuperintendents.\n\n6. Apportionment of school moneys to each school according to the\naverage number of children in each school district, as compared with\nthose in the whole township.\n\n7. Levy of a school rate by each district (county) municipal\ncouncil, of a sum at least equal to the legislative grant to each\nsuch district.\n\n8. The collection, by the local school trustees, of the balance\nrequired to defray the expenses of their school, in any way which\nthe school ratepayers (at the annual meeting) might determine.\n\n9. The recommendation of a uniform series of text books, with the\nproviso that no aid would be given to any school in which books\ndisapproved of by the general Board of Education might be used.\n\n10. The establishment of district model schools (re\u00ebnacted from the\nAct of 1843).\n\n11. Examination and licensing of teachers.\n\n12. Visitation of schools by clergymen, magistrates, municipal\ncouncillors, etc.\n\n13. Protection of children (re\u00ebnacted from the Act of 1843) from\nbeing \"required to read or study in or from any religious book, or\njoin in any religious exercise or devotion, objected to by parents.\"\n\n14. Establishment (re\u00ebnacted from the laws of 1841 and 1843) of\nRoman Catholic Separate Schools, where the teacher of the locality\nwas a Protestant, and _vice versa_. (These schools received grants\nin accordance with their average attendance of pupils.)\n\n15. Levy of rates by district municipal councils, at their\ndiscretion, for the erection of school houses and teachers'\nresidences.\n\nSuch were the principal provisions of the first School Act, proposed\nand adapted from other school laws by Dr. Ryerson in 1846, so far\nas rural schools were concerned. In the following year he prepared\na comprehensive measure in regard to schools in cities, towns and\nincorporated villages.\n\n\nCHIEF FEATURES OF DR. RYERSON'S FIRST REPORT AND SCHOOL BILL OF 1846.\n\nIn sending his draft of School Bill to the Government, early in\nMarch, 1846, Dr. Ryerson, in a private letter to Hon. Attorney\nGeneral Draper, dated 30th of that month, thus explained its general\nfeatures:--\n\n\"I thank you sincerely for your kind favor of the 23rd instant, and\nfeel not a little gratified that you approve of the draft of Bill\nwhich I had prepared for your consideration. I feel the justice\nof the high ground on which you place the moral qualifications of\nteachers and their duties....\n\n\"That to which I attach the highest importance in the measure is the\nauthority of trustees to levy a rate-bill upon all the inhabitants\nof a school section. The rate-bill will thus be a second edition\nof the school tax imposed by the District Council. The principle,\nonce established, it will be seen after a while that the Council may\nas well impose the whole of the school tax at once as to have the\nimposition of it divided between the Council and the trustees....\n\n\"I attach the greatest importance to the Normal School. I have no\ndoubt the Legislature will be disposed to support it when once\nestablished.... I hope, however, that you will have in view the\nproviding for it hereafter, and some appropriation for school\nlibraries ... from which an offer to a district or township, five\npounds for example, upon the condition that it would also contribute\nso as to purchase books from a list recommended by the Provincial\nBoard of Education, and, therefore, the most suitable for the young\nand grown-up people of the country....\n\n\"It has been mentioned to me, and I have thought that the term\ninspector, instead of superintendent, would be the better\ndesignation of the District overseer of Schools....\n\n\"I this day transmit to Mr. Secretary Daly my 'Report on a system\nof Elementary Instruction for Upper Canada.'... I have introduced\nno debatable topics, except that of Christianity, the principle of\nmaintaining which as the basis and cement of a system of public\ninstruction, I have discussed at large. On the defective modes of\nteaching those branches (viz., reading, writing, arithmetic, grammar\nand geography), which are taught in the Common Schools, I have also\ndwelt at some length in order to furnish District Superintendents\nand other persons concerned with a proper standard of teaching\nand examination, and in order to inculcate the true principles of\nteaching which are applicable to all subjects. I hope the Report\nwill be the means of laying a good foundation in the leading minds\nof the country on the great work of public instruction.\n\n\"I should add in respect to my Report that, in pointing out defects\nin systems of instruction and modes of teaching, I have almost\ninvariably quoted American authors, and have thus incidentally\nexposed the defects of almost every part of the American system,\nand have practically shown that every redeeming feature of the\nAmerican school system has been or is being borrowed from European\ngovernments.\"\n\n\nOBJECTIONS TO DR. RYERSON'S SCHOOL BILL OF 1846 ANSWERED.\n\nIn a private letter to Hon. Attorney-General Draper, dated 20th\nApril, 1846, Dr. Ryerson replies to several objections made in the\nHouse of Assembly and by the press to his first School Act of 1846.\nI quote the following:--\n\n\"The Montreal _Pilot_ objects to appointing trustees for three\nyears. This is one of the improvements adopted in the New York law\nof 1843. The superintendents, in their reports, speak largely of\nthe evils of the annual system, and strongly on the advantages of\nthe triennial one. The opposition to the Bill seems to be based on\nnotions derived from what the State of New York system was several\nyears ago. The opponents do not seem to be aware that it was amended\nin 1841, and amended again in 1843. Messrs. Price, Roblin, etc.,\nseem to be where the Americans were ten years ago.\n\n\"I anticipate the objection to the rate-bill clause. I look upon\nthat above all others to be the poor man's clause, and at the very\nfoundation of a system of public education. It is objected to by\nprecisely the class of persons or rather by the individuals that\nI expected. I have heard of one rich man objecting to it, who\neducates his own children at colleges and ladies' seminaries, but\nwho looks not beyond his own family. He says, I am told, that 'he\ndoes not wish to be compelled to educate _all the brats_ in the\nneighborhood.' Now, to educate 'all the brats in every neighborhood'\nis just the very object of this clause; and, in order to do so, it\nis proposed to compel selfish, rich men to do what they ought to do,\nbut what they will not do voluntarily.[46]\n\n  [46] In this matter of trustees' rate bill or school rate on the\n  property of the school section, Dr. Ryerson was quite in advance of\n  his times. The rate-bill clause, as he had prepared it, was rejected\n  by the House of Assembly and a school fee substituted for it. In\n  a brief, private note from Mr. Draper, dated 22nd April, 1846, he\n  said:--\n\n  \"Last night, or rather this morning at one, I got the School Bill\n  through Committee of the Whole. I have been forced to submit to some\n  changes, none very serious.... The rate-bill is to be on people\n  sending children to school--not on the whole section. I fought this,\n  but was well beaten.\"\n\n  In his reply, Dr. Ryerson said:--\n\n  \"I deeply regret the loss of the original rate-bill clause. It\n  involved a new and important principle.... I am persuaded that it\n  will on a future occasion pass by a strong majority.\"\n\n  It did so pass in 1850; but it was not until 1871 that the Municipal\n  Council was authorized by the Legislature to impose the whole of the\n  school tax as desired and predicted by Dr. Ryerson in 1846.\n\n\"Mr. Gowan's statements as to the evils of not extending the period\nof keeping a school open in each district beyond three months of the\nyear are substantially what have been communicated to me in many\nreports and letters. In several of the annual district reports which\nI have received, it is stated that giving public money to districts\nin which a school is not taught more than three months of the year\nis an actual injury rather than a benefit, and an abuse of the\nintentions of the Legislature.\"\n\n\nFIRST AND SECOND COUNCILS OF PUBLIC INSTRUCTION, 1846 AND 1850.\n\nDr. Ryerson was assisted in his important work by an able council of\nrepresentative men, who were appointed in 1846. The members of this\nfirst council were as follows:--\n\nRev. Egerton Ryerson, D.D., Chief Superintendent of Schools; Right\nRev. Michael Power, D.D., Roman Catholic Bishop of Toronto; Rev.\nHenry James Grasett, M.A., Rector of Toronto; Hon. Samuel Beaty\nHarrison, Q.C., Judge, County of York; Joseph Curran Morrison, Q.C.,\nM.P.P.; Hugh Scobie, Esq., Editor of the _British Colonist_; James\nScott Howard, Esq., Treasurer, County of York.\n\nDr. Ryerson proposed to Bishop Strachan that he should represent the\nChurch of England on the new Board. The Bishop was quite pleased at\nhis request, and so expressed himself. He declined, however, on the\nground that he feared his appointment might embarrass, rather than\naid, in the promotion of the new scheme of education. He suggested\nthat Rev. H. J. Grasett be appointed in his place.[47] He also gave\nfriendly advice to Dr. Ryerson to be careful not to recommend a\npersonal enemy for appointment on such a board.\n\n  [47] In a letter to the writer of this Retrospect from the Very Rev.\n  Dean Grasett in 1875, he said:--\"I esteem it an honour that I should\n  have been associated with Dr. Ryerson in his Council for so many\n  years (30), and a privilege if I have been of the least assistance\n  in upholding his hands in performing a work, the credit of which is\n  exclusively his own.\n\n  \"I shall carry with me to the end of life the liveliest feelings of\n  respect for the public character and regard for the private worth\n  of one who has rendered to his country services which entitle him\n  to her lasting gratitude. My venerable friend has had from time to\n  time many cheering recognitions of his valuable public services\n  from the heads of our Government ...; but I think that in his case,\n  as in others that are familiar to us, it must be left to future\n  generations adequately to appreciate their value when they shall be\n  reaping the full benefit of them.\"\n\nTwo more members were added in 1850, viz., Rev. John Jenning, D.D.,\nPresbyterian Minister; Rev. Adam Lillie, D.D., Congregational\nMinister. Not one of the gentlemen named survive; but, in their day,\nthey rendered effective service to the country as members of the\nfirst and second Councils of Public Instruction.\n\nThe Hon. S. B. Harrison (afterwards Judge of the County of York)\nwas nominated by Rev. Dr. Ryerson, as Chairman of the reconstructed\nBoard of Education in 1850 (then named the Council of Public\nInstruction), as successor to the lamented Bishop Power, who died in\n1847.[48] Mr. Harrison held that position until his death, in 1862.\n\n  [48] In his address at the beginning of the Normal School for Upper\n  Canada, in November, 1847, Dr. Ryerson thus referred to the then\n  recent decease of Bishop Power. He said, referring to the harmony\n  which had characterized the meetings of the Provincial Board of\n  Education:--\n\n  \"One event indeed has occurred, over which the members of the Board\n  have reason to mourn--the decease of the Right Reverend Prelate,\n  who, by his colleagues had been unanimously chosen Chairman of\n  the Board, and whose conduct as Chairman and Member of the Board\n  was marked by a punctuality, a courtesy, a fairness, a zeal\n  and intelligence which entitle his memory to the affectionate\n  remembrance of his colleagues and the grateful esteem of every\n  member of the community.... I cannot reflect upon the full and\n  frequent conversations which I have had with him on subjects of\n  public instruction, and with the scrupulous regard which ever\n  manifested for the views, and rights and wishes of Protestants,\n  without feelings of the deepest respect for his character and\n  memory.\"\n\nDr. Ryerson did not enter practically upon the duties of his office\nuntil about the middle of 1846. In the meantime, the correspondence\nand routine duties of the Education Office were (on Dr. Ryerson's\nsuggestion) placed in the hands of his friend, the Rev. Alexander\nMacNab, D.D., now Rector of Darlington (Bowmanville).\n\n\nRELIGIOUS INSTRUCTION IN THE COMMON SCHOOLS, 1846.\n\nAmong the first duties of the new Provincial Board of Education was\nthe establishment of a Normal School, the authorization of a series\nof text-books, and the preparation of regulations for the government\nof the Common Schools. The most important of these regulations was\nthe one relating to religious instruction in the schools. Before\nsubmitting it to the Board for adoption Dr. Ryerson consulted\nvarious representative persons on the subject. In a private letter\nto Hon. Attorney-General Draper, dated December 17th, 1846, Dr.\nRyerson thus explained his proceedings in regard to the preparation\nof the clause in the new regulations relating to religious\ninstruction in the schools. He said:--\n\n\"I submitted the clause first to Rev. Mr. Grasett. He quite approved\nof it, as he felt exceedingly anxious that there should be such an\nexplicit recognition of Christianity in our school system. I then\nwaited on the Roman Catholic Bishop, who examined it and concurred\nin it.... I showed it also to the Bishop of Toronto. After he had\nread the section he said he believed I had done all that could be\ndone on that subject, and that ... he would write a circular to\nhis clergy, recommending them to act as school visitors, and to do\nall in their power to promote the efficiency and usefulness of the\nCommon Schools.\"\n\nIn his report for 1847, Dr. Ryerson stated that he consulted other\nministers on the subject. The Hon. Mr. Draper, in a private note to\nDr. Ryerson, dated 1st January, 1847, said:--\n\n\"I am more gratified than I can express that you have so\nsuccessfully met the difficulty about the religious instruction of\nchildren in Common Schools. You (to whom I expressed myself about\nthree years ago on the subject of the importance of not dividing\nreligion from secular instruction) will readily understand the\npleasure I feel that in Common Schools at least the principle and\npromoted application of it, for mixed schools, has been approved by\nthe Bishop of my own Church and by the Roman Catholic Prelate.\"\n\nIn a letter to the late Hon. Robert Baldwin, written in 1849, Dr.\nRyerson thus refers to the question of religious instruction and the\nBible in school: \"Be assured that no system of popular education\nwill flourish in a country which does violence to the religious\nsentiments and feelings of the churches of that country. Be assured,\nthat every such system will droop and wither which does not take\nroot in the Christian and patriotic sympathies of the people--which\ndoes not command the respect and confidence of the several religious\npersuasions, both ministers and laity--for these in fact make the\naggregate of the Christianity of the country.\"\n\nSpeaking in a subsequent letter of another feature of the question\nof the Bible in schools, Dr. Ryerson says: \"The principal opposition\nwhich, in 1846 and for several years afterwards, I encountered was\nthat I did not make the use of the Bible compulsory in the schools,\nbut simply recognized the right of Protestants to use it in the\nschool (not as an ordinary reading book), as it was not given to\nteach us how to read but to teach us the way to Heaven), as a\nbook of religious instruction, without the right or the power of\ncompelling any others to use it. The recognition of the right has\nbeen maintained inviolate to the present time; facilities for the\nexercise of it have been provided, and recommendations for that\npurpose have been given, but no compulsory authority assumed, or\nright of compulsion acknowledged; and the religious exercises in\neach school have been left to the decision of the authorities of\nsuch school, and the religious instruction of each child has always\nbeen under the absolute authority of the parents or guardian of each\nchild.\"\n\nTo the objection urged against the reading of the Bible in the\nschools because \"a majority of the teachers are utterly unfit to\ngive religious instruction,\" Dr. Ryerson replied: \"The reading of\nthe Bible and giving religious instruction from it are two very\ndifferent things. The question is not the competency of teachers to\ngive religious instruction, but the right of a Protestant to the\nreading of the Bible by his child in the school as a text-book of\nreligious instruction. That right I hold to be sacred and divine.\"\n\n\nSTATE OF COMMON SCHOOL EDUCATION IN UPPER CANADA, 1845.\n\nFrom the reports then made to him by the County Superintendents\nof Schools, I select the following extracts, showing what was the\nactual state of education in the province when Dr. Ryerson commenced\nhis labors as Superintendent of Education.\n\nMr. Hamilton Hunter, Superintendent of Schools in the Home District\n(County of York), in his report for 1845, says:--\n\n     \"There is one fact with which I have been forcibly struck,\n     in my visits to the schools, which shows, in the clearest\n     manner, the great necessity that existed in this Colony for the\n     establishment of a system of Common School education. It is\n     this: That in our schools the amount of attainment, on the part\n     of the pupils, is generally in an inverse ratio to their size\n     and age, after they have reached their twelfth or thirteenth\n     year. The largest scholars that attend our schools are by\n     far the lowest in point of attainment, which shows how sadly\n     the education of that portion of the community, now about to\n     attain the years of manhood and womanhood has been neglected.\n     In many of our country schools, it is a very common thing\n     to find persons advanced to the age of young men and women\n     commencing to learn the first rudiments. The mind feels pained\n     upon contemplating this; but it is gratifying to think that a\n     remedy has been provided against it in the establishment of our\n     Common Schools, by which the elementary branches of education\n     are brought within the grasp of all. It leads us to reflect upon\n     the melancholy state of ignorance that must have existed at no\n     distant period in this Province had no means been provided other\n     than those which formerly existed for placing the elements of\n     knowledge within the reach of the rising generation.\"\n\nHon. Hamnett Pinkey, Superintendent of Schools in the District of\nDalhousie (Carleton, etc.), in his report, says:--\n\n     \"The Common Schools are very indifferently conducted, and the\n     masters in general very inadequately perform the duties required\n     of them; a reform is expected from the establishment of the\n     District Model School.\"\n\nRev. Alexander Mann, M.A., Superintendent of Schools in the Bathurst\nDistrict (Lanark, etc) says:--\n\n     \"In existing circumstances I have declined giving a regular\n     certificate to any teacher.... I made an effort on my own\n     responsibility, and at my own expense to improve teachers, by\n     opening a private school, solely for their benefit, but as I did\n     not meet with proper encouragement I was obliged to relinquish\n     my purpose.\"\n\nRichey Waugh, Esq., Superintendent of Schools in the Johnstown\nDistrict (Leeds and Grenville), says:--\n\n     \"The trustees of many schools employ teachers only for whatever\n     time the school fund will pay their wages, and they receive but\n     little benefit from the public money thus expended.\"\n\nPatrick Thornton, Esq., Superintendent of Schools in the Gore\nDistrict (Wentworth, etc.), says:--\n\n     \"It is a matter of regret that the old parrot system of\n     repeating words without attaching ideas to them, does still in\n     too many instances prevail; and the dregs must remain till some\n     of the old formal teachers are off the field.\"\n\nRev. Newton Bosworth, F.R.S., Superintendent of Schools in the Brock\nDistrict (Oxford, etc.), says:--\n\n     \"The diversity of books and modes of teaching referred to in\n     my last report, still exists, nearly to the same extent; and\n     in the qualifications of teachers also, as great a variety was\n     observable as before.... It appears to me that parents should\n     be impressed, to a much greater extent than at present, with a\n     sense of the necessity and importance of education for their\n     children.\"\n\nGeorge Duck, jr., Esq., Superintendent of Schools in the Western\nDistrict (Kent, etc.), says:--\n\n     \"In many townships little or nothing was raised by rate-bill.\n     In many places the poverty of the settlements prevented it; and\n     the only school that was kept open in these districts was just\n     during the time that the allowance from the aggregate fund was\n     sufficient to pay the teacher. This course is, in fact, of very\n     doubtful benefit, as the school is seldom kept open for more\n     than three months in the year, and the children lose so much\n     benefit continuous education produces.\"\n\n\nSCHOOL HOUSES AND SCHOOL TEACHERS IN 1845-1850.\n\nThe Rev. Dr. Ryerson, in his report for 1845-46, speaking of school\nhouses in the Province, says:--\n\n\"With a few exceptions, the school houses are deficient in\nalmost every essential quality of places adopted for elementary\ninstruction. Very few are furnished with any thing more than desks\nand forms of the most ordinary kind, and have no apparatus for\ninstruction, nor appendages, or conveniences either for exercise or\nsuch as are required for the sake of modesty and decency.\"\n\nIn his annual Report for 1847, Dr. Ryerson incidentally refers to\nthe character of teachers in some districts. He says:--\n\n\"In one district, where intemperance heretofore prevailed to a\nconsiderable extent, even among school teachers, the Superintendent\ngave notice that he would not give a certificate of qualification\nto any but strictly sober candidates, and that, at the end of six\nmonths he would cancel the certificates of all teachers who suffered\nthemselves at any time to become intoxicated.... I know of two\nother districts in which the Superintendents have acted thoroughly\non the same principle, with the same happy results.... In a note\nin reference to it in the printed Form of Regulations, I remarked\nthat 'no intemperate or profane person should be intrusted with the\ninstruction of youth.'... No one will doubt that there are fewer\nunqualified and immoral teachers employed now than there were before\nthe passing of the present School Act (of 1846).\" Pages 8 and 9.\n\nIn his circular issued to the newly formed County Boards of\nExaminers, dated 8th October, 1850, Dr. Ryerson thus referred to\nthis matter:--\n\n\"Many representations have been made to this Department respecting\nintemperate and profane and Sabbath-breaking teachers.... I cannot\nbut regard it as your special mission to rid the profession of\ncommon school teachers of unworthy character, and ... to protect the\nyouth against the poison of a vicious teacher's example.\" _Report\nfor 1850, pages 305, 306._\n\n\nDR. RYERSON'S PRACTICAL AGENCIES TO GIVE INFORMATION AND REMOVE\nPREJUDICE.\n\nIt will thus be seen from the foregoing, that at this time\neducational affairs were at a low ebb. Dr. Ryerson, therefore,\nsought in every practical way to overcome this educational apathy\nand inertia. His pen and personal effort were freely used. The first\ncircular to municipal councils--prepared by him--was issued by the\nnew Provincial Board of Education in August, 1846. This he followed\nup by one from himself addressed to county councils, in which he\nexplained fully and at length the scope and objects of the new\nscheme of popular education. This was done under three heads:--1.\nThat it was \"based upon the principles of our common Christianity.\"\n2. That \"upon the duty of educating the youth of our country there\nexists but one opinion, and, therefore, there should be but one\nparty.\" 3. That \"the system of elementary education is public, not\nprivate.\"\n\nAnother agency Dr. Ryerson sought to employ to aid the Department in\nits great work. And by it he hoped to educate and rightly influence\npublic opinion in favor of the new departure then in progress. The\nplan he proposed to the Government in 1846, to authorize the issue,\nunder his direction, of a departmental _Journal of Education_,\n\"to be devoted,\" among other things, to the exposition of every\npart of our school system,\" then new to the people, ... \"and to\nthe discussion of the various means of promoting the efficiency of\nthe schools.\" This the Government felt unwilling at the time to\ndo. He, therefore, undertook the expense and responsibility of the\npublication himself in January, 1848. And it was not until years had\ndemonstrated the practical value and success of the proposed agency\nthat the expense of the publication was provided for by an annual\nvote of the Legislature.\n\nA third agency which Dr. Ryerson successfully employed to aid the\nDepartment was that of personally holding county school conventions.\nIn explaining this project to the Government in 1846, he said:--\n\n\"I propose ... to visit and employ one or two days in school\ndiscourse and deliberation with the Superintendent, Visitors,\nTrustees and Teachers in each of the several Districts of Upper\nCanada. I know of no means so effectual to remove prejudice, to\ncreate unanimity of views and feelings, and to excite a general\ninterest in the cause of popular education,\" etc.\n\nThis project was concurred in by the Government, on condition that\nthe expense of the proposed nearly three months' visitation \"should\nnot exceed \u00a375.\"\n\nThus was inaugurated, in 1846, a series of county school conventions\nwhich, at intervals of about five years each, were held all over\nthe country. The early ones involved travelling in all kinds of\nweather and in all kinds of conveyances, so as to keep engagements\nmade weeks before. They were, however, of immense service to the\nDepartment in removing prejudice, settling difficulties and solving\ndoubts as to the practicability of plans proposed for improving the\ncondition of the schools and raising the intellectual and social\nstatus of the teacher.\n\n\nCOMBINED OPPOSITION TO THE PROJECTED SYSTEM OF EDUCATION.\n\nIt was not to be expected that so comprehensive a scheme of\neducation as that proposed by Dr. Ryerson in 1846 and 1847 would\nmeet with general acceptance. The very reverse was the fact.\nIt was assailed as revolutionary and oppressive. It certainly\nwas revolutionary in the best sense; but not oppressive, for\nit was largely permissive and wholly tentative. And, for many\nyears the Town of Richmond, in the County of Carleton, refused\nto establish schools under its provisions. The new measures were\nso far revolutionary that they differed almost wholly from the\nformer projected school acts. The system proposed was composite.\nIts machinery was adopted chiefly from the State of New York. The\nprinciple upon which the schools were to be supported was taken from\nNew England--Normal schools, from Germany, and the uniform series of\nschool books, from Ireland. All were, however, so blended together\nand harmonized, to meet the requirements and circumstances of the\ncountry that they became, in Dr. Ryerson's moulding hands, \"racy of\nthe soil.\"\n\nUp to this time no one but Dr. Ryerson had been able to give a\npractical turn to the rather crude theories which had been held\non the subject of popular education. He, however, had to pay the\npenalty of all such reformers; but yet he lived to see the fuller\ndetails of his system of education worked out on his own lines.\n\nIt is needless to say that Dr. Ryerson's scheme was assailed\nas impracticable. This I have explained. It was held to be too\ncomprehensive for the country. Even his reference to the compact\nand systematized plan adopted in Prussia was seized upon as an\nindication of his covert design to introduce the baneful system of\nso-called \"Prussian despotism.\" His commendation of \"free schools,\"\nas a prospective feature of our educational system was denounced as\nan attempt to legalize an \"outrageous robbery,\" and as communistic\n\"war against property.\" As an example of the injustice of these\ncriticisms on Dr. Ryerson's scheme of education, he said in a\nlecture on education in 1847:--\n\n\"I have seen in certain of the public prints, a provision of our\nschool law ascribed to Prussia, which was borrowed from the school\nlaw of the City of Buffalo; and another provision declared to be\nincompatible with the rights of man which forms the basis and glory\nof the common school system of Massachusetts.\"\n\nAlthough opposition to Dr. Ryerson's educational plans, as embodied\nin his school acts, was somewhat general, yet it was singularly\nillogical. The sore point was that it touched men's pockets in the\nform of school rates.\n\n\nEDUCATIONAL PROCEEDINGS OF DISTRICT COUNCILS IN 1847, 1848.\n\nAn influential district council in the West sought to influence\nall of the others against the new system, especially against the\nestablishment of a Normal School. In a memorial to the Legislature\n(which it sent broadcast), dated Hamilton, 10th November, 1847, and\nsigned by James Little, Chairman of the Education Committee, John\nWhite and Francis Cameron, and adopted by the Gore District Council,\nthe following passages occur:\n\n     \"With respect to the necessity of establishing a Normal, with\n     Elementary Model Schools in this Province, memorialists are of\n     opinion that, however well adapted, such an institution might\n     be to the wants of the old and densely populated countries of\n     Europe, where services in almost every vocation will scarcely\n     yield the common necessaries of life, they are, so far as this\n     object expected to be gained is concerned, altogether unsuited\n     to a country like Upper Canada.... Nor do your memorialists hope\n     to provide qualified teachers by any other means, in the present\n     circumstances of the country than securing as heretofore, the\n     services of those whose physical disabilities from age render\n     this mode of obtaining a livelihood the only one suited to their\n     decaying energies, or by employing such of the newly arrived\n     emigrants as are qualified for common school teachers year by\n     year as they come amongst us, and who will adopt this as a means\n     of temporary support, until their character and ability are\n     known and turned to better account for themselves.\"\n\nThis memorial having been sent to each of the district councils in\nUpper Canada for their concurrence, and with a view to procure the\nrepeal of the School Act, 9 Vic., ch. 20, the Colborne District\nCouncil not only refused to concur in it but subjected it to severe\ncriticism. In regard to the foregoing extract from the Gore District\nMemorial, the Colborne Council (now Peterboro' and Victoria) in its\nreport on that memorial said:\n\n     \"That the moneys required to pay for the establishment and\n     support of Normal and Model Schools are little less than a waste\n     of so much of the Legislative grant, is an opinion in which your\n     committee are so far from concurring, that they believe it is\n     from these sources must mainly arise the instrumentality through\n     which the friends of education can alone hope for the first\n     considerable amelioration of the evils they lament.... Nor can\n     your committee reconcile it either with their just expectations,\n     or their sense of duty to rest satisfied with the services of\n     those whose physical disabilities from age and decaying energies\n     render them unfit, or of those 'newly arrived emigrants,' whose\n     'unknown character and abilities' render them unable to procure\n     a livelihood by any other means than by becoming the preceptors\n     of our children; the dictators of their sentiments and manners;\n     the guardians of their virtue; and in a high degree the masters\n     of their future destinies in this world and the next.\"\n\nThis report was prepared by Mr. Thomas Benson, Chairman of the\nEducation Committee, and Warden of the District (father of Judge\nBenson of Port Hope). It was adopted by the Colborne District\nCouncil in February, 1848.[49]\n\n  [49] Correspondence between members of the Government and the Chief\n  Superintendent, 1850, pp. 17-20.\n\nThe Western District Council, in its memorial to the Legislature\nagainst the School Act, represented that \"spite, hatred and malice\nbetween neighbors and friends,\" existed, and was \"occasioned by the\npresent School Act.\" It added:\n\n     \"So numerous are the petitions on the subject that more than\n     half of the time of the Council is taken up in endeavoring to\n     settle the differences, but unfortunately without any beneficial\n     effect.\"\n\nThe Chief Superintendent, in his report, referring to this statement\nsays:\n\n     \"Now, in examining the printed report of the committee, to\n     whom all these petitions were referred, I find that of the\n     twenty-nine petitions presented to the Council, one prayed for\n     the establishment of a female school in one of the sections\n     (which was granted); one prayed for a local school tax in a\n     section; two related to the formation of new school sections,\n     and the remaining twenty-five related to the disputes as to the\n     boundaries of school sections and the non-payment of school\n     moneys by township superintendents. Thus not one of these\n     disputes could have arisen out of the School Act, but must have\n     all been caused by an improper division of the school sections,\n     either by the township superintendents under the late Act, or by\n     the Council under the present statute.\"\n\nIn this (Western) District the Council says:\n\n     \"We well know that a very large number of the trustees can\n     neither read nor write, and, therefore, it must be obvious that\n     the greater part of the requirements of the law remain undone.\"\n\nOn this statement the Chief Superintendent remarks:\n\n     \"In other districts where the trustees can read and write,\n     and where the councillors an correspondingly intelligent\n     and discreet in their school proceedings, no disputes or\n     inconvenience have, as far as I am aware, occurred on this\n     subject.\"\n\nIn the District of Dalhousie, the Chief Superintendent states:\n\n     \"Still greater dissatisfaction and confusion were created by\n     the mode of proceeding adopted by the council. Before the\n     passing of the present School Act the council of this district\n     never imposed a school assessment.... The introduction of a\n     district assessment (under the new Act) would naturally excite\n     some dissatisfaction, and especially in a district bordering on\n     counties in Lower Canada where the school assessment had been\n     resisted.... In addition the Chief Superintendent adds:\n\n     \"The Council in the autumn of 1847 passed a by-law to this\n     effect:\n\n     \"Whereas the school section division mode by this Council at\n     its last session, are in many instances, discordant to the\n     convenience and wishes of the inhabitants, and that to correct\n     them satisfactorily this present session is impracticable. The\n     District Superintendent is empowered and required to make a\n     distribution of the school fund (legislative grant and county\n     assessment) 'share and share alike,' among qualified teachers\n     without reference to the number of scholars under their tuition,\n     but in proportion to the time such teachers may have been\n     teaching, etc.\"\n\n     Thus the Superintendent remarks, \"this by-law contemplated\n     the abolition of the statute requiring the school grant to be\n     distributed according to the school population of each section.\n     It made no distinction between the able teacher who taught\n     sixty scholars and the young one who taught twenty; it had no\n     regard to the engagements which may have been made by trustees\n     according to law; it required of teachers conditions which the\n     law had not enjoined, and proposed to deprive many of them of\n     advantages which the law had conferred.... Of course I pointed\n     out the illegality and injustice of the by-law and it was\n     not acted upon. At the session of the Council lately held, a\n     resolution was adopted praying the Governor-General to dissolve\n     the Council that the sense of the inhabitants of the Dalhousie\n     District might be taken on the school law.... It is doubtless\n     probable that many of the inhabitants have not distinguished\n     between the provisions of the law and the proceedings of their\n     own Council--attributing to the former what has been occasioned\n     by the latter.\"[50]\n\n  [50] Chief Superintendent's Annual Report for 1847-8, page 6.\n\nContrast the enlightened discussion of such questions to-day with\nthe ignorant dogmatism of that day, and you can form some idea of\nthe magnitude of Dr. Ryerson's labors,--not only in laying broad\nand deep the foundations for his superstructure, but in seeking\nto overcome the deep-rooted and unreasoning prejudices of those\ndays--days indeed of anxiety and toil and fierce opposition, which I\nso well remember.\n\n\nESTIMATE OF LORD ELGIN'S CHARACTER BY THE HON. W. H. DRAPER.\n\nOn the arrival in Canada of Lord Elgin, as Governor-General, Dr.\nRyerson wrote confidentially to Hon. Attorney-General Draper (on\nthe 16th February, 1847), asking him for his \"opinion of the new\nGovernor-General.\" Mr. Draper replied on the 22nd as follows:--\n\n\"As far as my opportunities of judging go, I think Canada will\nfind cause of satisfaction in having Lord Elgin for a Governor. He\nis industrious in habit, pleasing in manner, extremely courteous\nand affable in bearing. I find him also diligent and shrewd in\ninquiry; and the observations which fall from him show that he has\nstudiously kept pace with the great questions of the day (I do not\nmean our Canadian politics simply), and besides the cultivation\nof classical education in its broader sense, he possesses a mind\nstored with facts bearing on and illustrative of those questions.\nIn these respects, or more correctly speaking in the latter, and\nas regards trade and finance, he reminds me more of Lord Sydenham\nthan any other governor of my time. I think he possesses also\ncaution and firmness;--that he will not resolve hastily, that he\nmay not have to change his resolves. He has large ideas of the\ncapabilities and resources both of Canada and of the British North\nAmerican Provinces, and, as it strikes me, without any reference to\na political union of these provinces, thinks that a course might\nbe taken to develop the whole, by separate parts taking a common\ncourse in matters in which they have a common interest--internal\ncommunication, favorable to our European commerce and connections\nwill serve as an illustration of the sort of questions to which I\nallude.... All this, of course, is mere opinion, but such are my\nfirst impressions, and as such, and no more, I readily give them to\nyou in reply to your enquiry, etc.\"\n\nSubsequently, Dr. Ryerson met Lord Elgin in Montreal, and, in a\nletter to me, dated 24th July, 1847, he says:--\n\n\"At his own request I have had an interesting interview with Lord\nElgin. He is exceedingly well versed in systems of education, and is\na thoroughly practical man on the subject.\"\n\n\nINVALUABLE ASSISTANCE GIVEN TO DR. RYERSON BY LORD ELGIN.\n\nIt was fortunate that just at this crisis Canada was favored with\nthe presence of one of the most accomplished, in every sense of\nthe term, of the Queen's representatives, the Earl of Elgin and\nKincardine.\n\nThat distinguished statesman, who afterwards filled with great\ndignity the highest post in the civil service of Great Britain,\nthat of Governor-General of India, reached Canada at a critical\ntransitional period in our history. Few can recall the incidents of\nthose days without a feeling of admiration for the fearlessness,\ntact, and ability with which he discharged the delicate and\ndifficult duties of his high office.\n\nWhen Lord Elgin arrived in Canada in 1847, and when he removed\nto Toronto, after the riot and burning of the Parliament House\nin Montreal in 1849, educational affairs were fiercely discussed\nand were yet almost at the low ebb at which Dr. Ryerson had found\nthem. Not that they had previously reached a higher plane and had\ngradually settled down to a lower one. The reverse was the fact, but\nthe question of education had only then (in Dr. Ryerson's hands)\nbegun to attract serious public attention. It was, however, as I\nhave explained, in an adverse direction, for the whole subject, in\nthe advanced form in which it was presented by Dr. Ryerson, was\nunpopular. It involved taxation and other unpalatable \"burdens,\" as\nits opponents averred. Notwithstanding the zeal and ability with\nwhich Dr. Ryerson had collected and arranged his facts, analyzed the\nvarious systems of education in Europe and America, and fortified\nhimself with the opinions of the most experienced educationists in\nthese countries, the system which he projected, and the school law\nwhich embodied it, continued to be fiercely assailed by a portion of\nthe press, and by hostile politicians. This hostility culminated in\nan event which brought things to a crisis in 1849.\n\nAt this time, an administration was in office, one or two members\nof which were personally unfavorable to Dr. Ryerson's continuation\nin office. One of these, a prominent and popular member of the\ncabinet (Hon. Malcolm Cameron, who afterward became a warm friend of\nDr. Ryerson) induced his colleagues to assent to the passage of a\nschool bill which practically legislated Dr. Ryerson out of office,\nbesides being objectionable in other respects. He at once tendered\nhis resignation. The Hon. Robert Baldwin, Attorney-General, declined\nto recommend its acceptance. By advice of the Cabinet, the operation\nof the bill was suspended until a new one, framed by Dr. Ryerson,\ncould be prepared and passed. The result was the passage of the\nSchool Act of 1850--popular in its character and comprehensive in\nits provisions. It now forms the broad basis of the present school\nsystem of Ontario.\n\nIt was at this period of our educational history that Lord Elgin\nfirst came into official contact with our educational system. Being\nfamiliar with the Scottish parochial school system, he soon mastered\nthe whole subject, and perceived the great importance to the whole\ncountry of the question which was then being so fiercely discussed.\n\nBeing in England in 1853, Dr. Ryerson wrote to me there:--\n\n\"I was glad to learn that Lord Elgin was to go in the same\nsteamship with you from Boston. I have no doubt it will have proved\ninteresting to him as well as to you, and perhaps useful to you.\nI miss you very much from the office, but I do not like to employ\nany more aid without sanction of the Government, though I could get\nno one to take your place. I would wish you to write me what Lord\nElgin may have thought or have said as to our doings and plans of\nproceeding. If the library plan succeeds, it will achieve noble\nresults.[51] I feel that our success and happiness in the Department\nare inseparably united.\"\n\n  [51] Lord Elgin always referred to Dr. Ryerson's library scheme\n  in his educational addresses as the \"Crown and Glory of the\n  Institutions of the Province.\"\n\nIt was indeed fortunate for my mission that I was on the same Cunard\nsteamer to England with His Excellency the late lamented Lord Elgin,\nto whom I entered into full detail in regard to the objects of that\nmission. Before leaving the steamer, Lord Elgin most kindly promised\nto aid me in every way he could while in England, and wrote me his\naddress as \"Broom Hall, Dumfermline,\" in case I should have occasion\nto refer to him. He also added the following paragraph to the\nletter of your instructions and authority, which, in more than one\ninstance, I found to be of essential service to me:--\n\n     \"I believe the object of Mr. Hodgins' mission to be most\n     important to Canada, and I trust that he will meet with all\n     support and encouragement.\n\n  (Signed)      \"ELGIN AND KINCARDINE,\n  \"Governor-General.\n\n  \"September, 1853.\"\n\nOne of my letters, reporting to Dr. Ryerson as far as I had gone, my\nproceedings in England, having been enclosed to the Hon. Mr. Hincks,\nhe said in reply:--\"I return you Mr. Hodgins' interesting letter,\nwith thanks for its perusal. It was fortunate he went by the same\nsteamer as Lord Elgin. I am much interested in the success of your\nlibraries, which is beyond my most sanguine expectations.\n\n  \"QUEBEC, 11th Oct. 1853.\"\n\nI recall with pleasure the great services which Lord Elgin then\nrendered to the cause of education at a critical period of its\nhistory in this Province. His speeches and addresses on the subject\nat that time had a wonderful effect in moderating the opposition\nwhich Dr. Ryerson received while laying the foundations of our\nsystem of education. They had also the potent effect of popularizing\nthat system in the estimation of the people which it was designed to\nbenefit. That popularity has, happily, continued to this day, thanks\nin a great degree to the dignity imparted to the subject by the\npersuasive eloquence of Lord Elgin. His eminence as a distinguished\ngraduate of Oxford, and his general knowledge of European systems\nof education, enabled him to speak with a precision and certainty\nwhich few could gainsay. It was a gratifying fact that he identified\nhimself personally, as well as officially, throughout the whole of\nhis seven years' administration, with the general education and\nintellectual improvement of the people of Canada. The first bill to\nwhich His Excellency assented in the Queen's name was the School Act\nof 1850, to which I have referred.\n\n\nPROCEEDINGS OF THE FIRST COUNCIL OF PUBLIC INSTRUCTION--THE NORMAL\nSCHOOL.\n\nOne of the first and necessary acts of the Provincial Board of\nEducation, or first council of public instruction, was the adoption\nof a uniform series of text-books--one only on each subject. Those\nchosen were the Irish National Series, with two additions. The\nnext important step taken by the Board was the establishment, in\nNovember, 1847, of a Normal School, with the necessary adjunct\nof a Model School. The old Government House was fitted up as a\nNormal School, and the stable connected with it was renovated\nand converted into a Model School, or school of practice for\nteachers-in-training.[52] On the removal of the seat of government\nto Toronto, in 1849, the Normal School was held in the Temperance\nHall, and other arrangements were made.\n\n  [52] _Apropros_ of this, Dr. Ryerson, in a private note to Hon. W.\n  H. Draper, in April, 1846, said:--\n\n  \"The stables of the Government House may be fitted up for Model\n  Schools, etc. It is a curious and not interesting fact that the\n  stables of Louis the Fourteenth, at Versailles, are now used for the\n  great National Normal School of France, and is the most splendid\n  establishment of the kind on the Continent of Europe.\"\n\nSo successful were these schools in raising the status of the\nteaching profession that the government of the day--the memorable\nBaldwin-Lafontaine administration--willingly listened to a\nproposition of the Provincial Board of Education to grant funds for\nthe purchase of a site and the erection of suitable buildings for\nthese schools. The Hon. Francis Hincks, who was Inspector General,\nhad (upon Dr. Ryerson's estimate) a proposed grant of \u00a315,000 put\nin the estimates of 1850 for the purposes named. A site of seven\nacres and a half of land was purchased from the estate of the Hon.\nPeter M'Gill. The writer of this retrospect had the pleasure (in\nthe absence of Dr. Ryerson in Europe) of signing the cheque for the\npurchase money, \u00a34,500, and of seeing that the deed was duly made\nout in the name of Her Majesty the Queen and her successors, and\ntransferred for safe keeping to the Crown Lands Department.\n\nAfter the plans for the buildings had been approved, certain\nimportant additions were considered desirable (chiefly a theatre,\nor central lecture hall, etc.). As the grant already made was\nquite insufficient for the proposed additions, Mr. Hincks was once\nmore appealed to. He responded very promptly and heartily, and\nrecommended to his colleagues that a further grant be made, which\nwas done, and an item of \u00a310,000 additional was placed in the\nestimates of 1851 and concurred in by the Legislature. The work then\nproceeded and near the close of the second year it was brought to a\nconclusion.\n\nSo carefully had these two grants been husbanded that when the\nbuildings were completed and furnished, there was a balance\nleft over of \u00a390. With this sum the expense of fitting up the\nDepartmental Library was defrayed. The result was highly gratifying\nto Mr. Hincks, and he so expressed himself at the opening of the\nbuildings in the following year.\n\n\nLAYING THE CORNER STONE OF THE NORMAL SCHOOL BUILDING, 1851.\n\nOn Wednesday, the 2nd of July, 1851, the imposing ceremony of laying\nthe corner stone of the new edifice took place. The guard of honor\nwas the 71st Highlanders, under Sir Hew Dalrymple. Ministers of\nboth Houses of Parliament, the city corporation, etc., attended.\nThe inscription on the brass plate--I quote from the original, as\nwritten by Dr. Ryerson--was as follows:\n\n     \"This Institution, Erected by the Enlightened Liberality of\n     Parliament, is Designed for the Instruction and Training of\n     School Teachers upon Christian Principles.\"\n\nRight Rev. Bishop Charbonnell, to whom was assigned the duty of\npresenting the Governor-General with the silver trowel, spoke with\ngreat cordiality, and with French grace and eloquence. He said:\n\n     \"MONSEIGNEUR,--Je suis tr\u00e8s heureux et tr\u00e8s honor\u00e9 d'avoir \u00e8t\u00e9\n     choisi par le Conseil de l'Instruction Publique, dont votre\n     Excellence a daign\u00e8 me faire membre, pour lui pr\u00e8senter cette\n     truelle d'argent aux industrieuses embl\u00e8mes du blazon des Bruces.\n\n     \"L'etablissement dont votre Excellence va poser la pierre\n     angulaire, Monseigneur, sera un des plus glorieux monuments\n     de tout ce que son lib\u00e9ral gouvernment aura fait pour la\n     prosp\u00e9rit\u00e9, de ce pays: _ad \u00e6dificationem_.\"\n\nThis in substance is as follows:--\n\n     \"MY LORD,--I am very happy and am highly honored to have been\n     chosen by the Council of Public Instruction--of which your\n     Excellency has condescended to make me a member--to present to\n     you, on their behalf, this silver trowel emblazoned with the\n     industrial emblems which form the arms of the Bruces.\n\n     \"The institution, of which your Excellency is about to lay\n     the corner stone, is destined to be, my Lord, one of the most\n     glorious monuments amongst all of those which your liberal\n     administration has devised for the welfare of this country.\"\n\nIn laying the corner-stone, Lord Elgin was particularly happy in his\nreply to these remarks, and to the address of the newly-constituted\nCouncil of Public Instruction. He said, addressing Dr. Ryerson:--\n\n     \"It appears to me, sir, ... that this young country has\n     had the advantage of profiting by the experience of older\n     countries--by their failures and disappointments, as well as\n     by their successes; and that experience, improved by your\n     diligent exertions and excellent judgment ... and fortified by\n     the support of the Council of Education, and the Government and\n     Parliament of the Province, has enabled Upper Canada to place\n     herself in the van among the nations in the great and important\n     work of providing an efficient system of general education for\n     the whole community.... I do not think that I shall be charged\n     with exaggeration when I affirm that this work is _the_ work of\n     our day and generation--that it is the problem in our modern\n     society which is most difficult of solution.... How has Upper\n     Canada addressed herself to the execution of this great work?...\n     Sir, I understand from your statements--and I come to the same\n     conclusions from my own investigation and observation--that it\n     is the principle of our educational system that its foundation\n     be laid deep in the firm rock of our common Christianity....\n     Permit me to say, both as an humble Christian man and as the\n     head of the Civil Government of the Province, that it gives me\n     unfeigned pleasure to perceive that the youth of this country,\n     ... who are destined in their maturer years to meet in the\n     discharge of the duties of civil life upon terms of perfect\n     civil and religious equality--I say it gives me pleasure to hear\n     and to know that they are receiving an education which is fitted\n     so well to qualify them for the discharge of these important\n     duties; and that while their hearts are yet tender ... they are\n     associated under conditions which are likely to provoke amongst\n     them the growth of those truly Christian graces--mutual respect,\n     forbearance and charity.\"\n\nOne of His Excellency's last acts in Toronto, when about to\nleave the country, was to visit those buildings and express his\nsatisfaction with the several departments of the system therein\nconducted.\n\n\nTHE COUNTY MODEL SCHOOLS OF 1843-1850.\n\nThe necessity of these schools was felt more than forty years ago,\nand provision was then made for their establishment. Thus, in the\nfirst School Act passed in 1843 to regulate Common Schools in this\nprovince, section 57 of that Act declares:--\n\n\"That it shall and may be lawful for the court of wardens of any\ncounty in Upper Canada ... to raise and levy by county rate a sum\nnot exceeding \u00a3200 ($800), and to appropriate and expend the same\nfor the maintenance of one or more _County Model Schools_, within\nsuch county and to constitute, by by-law, or by-laws, to that\neffect, any township, town, or city school, or schools within the\ncounty, to be, for any term not less than one year, such County\nModel School or Schools,\" etc.\n\n\"A sum not less than \u00a340\" was appropriated to each such school\ntowards 'the payment of the teachers and the purchase of books and\napparatus.' The 66th section of the same Act also declared:--\n\n\"That in every such township, town or city Model School gratuitous\ninstruction shall be given to teachers of Common Schools within the\ntownship, town or city, wherein such Model School may be established\nduring such periods and under such regulations of the township, town\nor city superintendent may from time to time direct.\"\n\n\"Again, in the first Common School Act prepared by Dr. Ryerson, and\npassed in 1846, after providing for the establishment of District\nModel Schools--it was declared (sec. 40):--\n\n\"That at every such District Model School gratuitous instruction\nshall be afforded to all teachers of Common Schools within the\ndistrict in which such Model School may be established during such\nperiod and under such regulations as the district superintendent may\nfrom time to time direct.\"\n\nThese County Model Schools (as will be seen) had higher functions\nthan have the County Model Schools of the present day. They were\ndesigned to afford instruction to persons who were already teachers,\nand were thus in Dr. Ryerson's views constituted local Normal\nSchools for that purpose. So much importance did Dr. Ryerson attach\nto the value of training institutions for teaching, and so much did\nhe anticipate a demand for them that on page 162 of his \"Report on\na System of Public Elementary Instruction,\" published in 1845, he\nsaid:--\n\n\"As soon as examples of the advantages of trained teachers can be\ngiven, I believe the ratio of demand will increase faster than\nthat of supply, and that an additional Normal School will soon be\nrequired in each of the most populous districts.\"\n\nThen again so jealously was the efficiency of these District or\nCounty Model Schools guarded that in the same Act, 9 Victoria,\nchapter 20, it was provided that no teacher could be appointed\nto such school without the approval in writing of the district\nsuperintendent, and unless he held a certificate from the Normal\nSchool (which was established in 1847). In addition to these\nrequirements power was given to the district superintendent to\nsuspend or dismiss Model School teachers and to appoint others in\ntheir places, in case the local trustees neglected or refused to do\nso. This district superintendent was also authorized to examine (as\nthey often did at the Model School) all \"candidates for teaching\nin Common Schools\" and to give them certificates of qualification,\nspecial or general, at his discretion.\n\nThe question may here be asked, \"Of what practical value were these\nCounty Model Schools in the work of training school teachers, and\ndid they at all discharge the higher functions to which reference is\nmade?\"\n\nIt was clear that these schools were regarded in those early days as\na necessary adjunct to our system of education, for the very purpose\nof aiding teachers in their professional work. Thus, Mr. Hamilton\nHunter, in his report as School Superintendent of the Home District\nfor the year 1844 says:--\n\n\"The deficiency in the qualification of teachers could be remedied\nby establishing in each district a Model School upon a good scale,\nand having it under the management of a superior teacher or\nteachers.... The School Bill makes provision for this, etc.\"\n\nIn his report for 1847 Dr. Ryerson thus speaks of the operation and\nsuccess of these schools wherever they had been established:--\n\n\"The School Superintendent of Dalhousie District says: 'In this\n[County Model School] I have there held public examinations of\nCommon School teachers; and on some occasions, when reluctant\nto give them certificates, I have sent them to the Model School\nMaster for information and examination.... [These teachers] did\nnot make any permanent stay except one, merely learning the mode\nof instruction, the value of the studies and discipline of the\nschool.'... The Superintendent of the Johnstone District says:-- ...\n'Much good has been done by the establishment of the Model School\nin this district. Several teachers, whose education was by no means\ngood, have acquired a sound knowledge of the subjects which are\nrequired to be taught in the Common Schools.' The Superintendent of\nSchools in the Midland District says:--'Almost every teacher who has\nattended the Model School for any length of time is now teaching\nwith good success.'\"\n\nIn the Act which was hurriedly passed in 1849, but which, by\nOrder-in-Council, never went into operation, provision was made to\nestablish, or continue the County Model Schools \"in any township,\ntown, or city,\" and granting to each of them \"\u00a325 over and above the\nsum to which such schools would be entitled as a Common School ...\nwhich sum shall be expended in the payment of a teacher or teachers,\nand for no other purpose.\"\n\nIn the Act of 1850, provision for the establishment and maintenance\nof Township Model Schools was made. Township councils were\nauthorized to raise a special tax for the support and efficiency\nof these schools; and it was \"provided likewise, that tuition to\nstudent-teachers in such Model Schools should be free.\"\n\nThe reason why Township Model Schools were substituted for county\nones, is given by Dr. Ryerson in his circular to town reeves, dated\n12th August, 1850. Other reasons contributed to this change, but the\ncircular gives the chief reason.\n\n\"The attempts of district councils to establish Model Schools have\nthus far proved entire failures...: The late district councils have\nin every instance, except one, abandoned the attempt.... To the\nsuccess and usefulness of a Model School, a model teacher, at any\nexpense, is indispensable, and then a Model School-house, properly\nfurnished, and their judicious and energetic management.\"\n\nIn addition, I may say that the causes of failure of these valuable\ntraining institutions in 1850, may be incidentally learned from\nthe very words here used by Dr. Ryerson by way of suggestions\nto town reeves. These schools had neither model teachers, nor\nwere the buildings \"model school-houses.\" Besides, the district\nsuperintendents of that day, and after them, the township\nsuperintendents, had, as a rule, no experience as trained teachers\nthemselves.\n\nBy the Act of 1871, the status and qualifications of these most\nimportant officers were raised to their present high standard. The\nvery name was changed, and that of inspector was substituted.\n\nIt was felt by Dr. Ryerson that until these new officers had secured\nsome degree of popular favor, and had proved their efficiency as\norganizers of schools, and as practical judges of the necessary\nqualifications of teachers, it would be useless for him to attempt\nthe re-establishment of the County Model Schools. Before that time\nhad fully arrived he retired from office--leaving this important\nand necessary duty to be undertaken (as it was efficiently) by his\nsuccessor, Hon. Adam Crooks, as Minister of Education.\n\n\nFUNDAMENTAL PRINCIPLES OF DR. RYERSON'S SCHEME OF EDUCATION.\n\nIn founding the system of public instruction for Upper Canada,\nDr. Ryerson wisely laid down certain fundamental principles which\nhe believed to be essential to the success and stability of that\nsystem. These general principles may be thus summarized:\n\n1. That the machinery of education should be in the hands of the\npeople themselves, and should be managed through their own agency;\nthey should, therefore, be held, be consulted, by means of public\nmeetings and conferences, in regard to all school legislation. This\nhe himself did every few years.\n\n2. That the aid of the Government should only be given where it\ncould be used most effectually to stimulate and assist local effort\nin this great work.\n\n3. That the property of the country is responsible for, and should\ncontribute toward the education of the entire youth of the country;\nand that, as a complement to this, \"compulsory education\" should\nnecessarily be enforced.\n\n4. That a thorough and systematic inspection of the schools is\nessential to their vitality and efficiency.\n\nThese and other important principles, Dr. Ryerson kept steadily in\nview during his long administration of the school system of his\nnative Province. He was not able to embody them all at once in his\nearlier school bills, but he did so in the final legislation on the\nsubject with which he was connected in 1870-1874. Their judicious\napplication to the school system contributed largely, under the\nDivine blessing, which he ever sought, to the wonderful success of\nhis labors.\n\n\nCAN UPPER CANADA EMULATE THE STATE OF NEW YORK IN EDUCATIONAL\nMATTERS?\n\nIn his \"Address to the People of Upper Canada\" on school affairs in\n1850, Dr. Ryerson thus answers this question:--\n\n\"Another ground of encouragement in our country's educational work\nis the practical proof already acquired of the possibility of not\nonly improving our schools, but of successfully emulating our\nAmerican neighbours in this respect. Often have we heard this, both\npublicly and privately, pronounced utopian; and often have we sought\nin friendly discussion to prove that it was neither impracticable\nnor extravagant to aim in rivalling our New York neighbours in our\nCommon Schools.\"--_Journal of Education for Upper Canada_, vol. 3,\npage 2.\n\nIn his report for 1851, Dr. Ryerson returns to this subject. He\nsays:--\n\n\"The period is very recent when the [subject of educational\ncomparison with the State of New York] would have been an\nabsurdity--when the word 'contrast' must have been employed instead\nof the word 'comparison,' when not a few of our fellow countrymen,\nand some of our public men, considered the project, or the idea\nof emulating the Common School doings of our New York neighbours,\nas presumptuous and chimerical. I have not viewed the noble and\npatriotic exertions of the American people in any spirit of\njealousy.... I hold up their example to the admiration and imitation\nof the people of Canada; but I have not despaired of, much less\ndepreciated my own country; and have had, and have still in a higher\ndegree than ever, a strong conviction that there are qualities in\nthe people of Upper Canada, which, under a proper and possible\norganization, and with judicious counsel, would place schools and\neducation in this country upon more than a level with what we have\nwitnessed and admired in the State of New York. It is true our\nAmerican neighbours have had more than thirty years the start of us;\nbut I am persuaded we shall not require half that time to overtake\nthem--profiting, as we have done, and doubtless will do, by their\nmistakes and failures, as well as by their ingenuity and success.\nTo rebuke an unpatriotic spirit of Canadian degradation in which\nsome Canadians indulge,[53] and to animate the hopes and exertions\nof the true friends of our intellectual and social progress, I will\nshow what has already been accomplished in Upper Canada in respect\nto Common Schools by a comparison, in a few particulars, with what\nhas been done in the State of New York.\" (The particulars which\nDr. Ryerson points out are seven in number).--_Report for 1851_,\n(written in 1852), page 17.\n\n  [53] Dr. Ryerson constantly deprecated, in these early years, this\n  want of the spirit of Canadian Patriotism. In an eloquent paper\n  (in the third volume of the _Journal of Education_) he shows that\n  \"_Canadian Patriotism (is) the Lever of Canadian Greatness_.\" He\n  sums it up in these words: \"It cannot be too strongly impressed\n  upon every mind that it is on Canadian energy, Canadian ambition,\n  Canadian self-reliance, skill and enterprize--in a word, on Canadian\n  patriotism--that depends Canadian prosperity, elevation and\n  happiness.\"--_Page 40._\n\nIn his reply to a complimentary letter from the Municipal Council of\nthe County of Norfolk, in 1851, Dr. Ryerson thus referred to this\nsubject:--\n\n\"No person who has at all studied the subject of comparative school\nlegislation between Canada and other countries, can fail to observe\nthat there is an extent of local discretion and power in each of\nour School and County Municipalities not found in any one of the\nneighbouring States, while there are other elements incorporated\ninto our school system, which secure to the remotest municipality\nof Upper Canada the information and facilities which can alone be\nacquired and provided by a Public Department. But the rational\nconviction and voluntary co-operation of the people themselves have\nbeen relied upon and appealed to as the basis of exertion and the\ninstrument of success. When, therefore, steps were taken to improve\nthe text-books of the schools, a set of the books recommended was\nprocured and furnished to each County Municipality in Upper Canada,\nthat the people might examine and judge of the desirableness of\nthe books proposed, in regard to both excellence and cheapness.\nIn promoting an improvement in the condition and character of\nschool-houses, plans and illustrations of school-houses and premises\nwere procured and placed in the hands of the local councils, and\nseveral of them were published in the _Journal of Education_. The\nsame course has been adopted in respect to School Maps, etc. And in\npressing upon the public mind the necessity and advantage of duly\nqualified School Teachers, an institution has been established to\ntrain them; and the specimens of Teachers thus trained (though but\npartially trained in most instances, from the short period of their\ntraining) have excited a desire and demand for improved teachers in\nevery County in Upper Canada. I trust this year will witness the\nintroduction of Libraries--thus completing the establishment of\nevery branch of our school system.\n\n\"In all this there has been no coercion--but a perfect blending of\nfreedom and unity, of conviction and action; and the entire absence\nof any opposition to the school system during the recent elections\nthroughout Upper Canada, shows how general and cordial is the\nconviction of the people as to its adaptation to their circumstances\nand interests.\n\n\"I have the deepest conviction of the strong common sense and\npatriotism of the Canadian people at large--a conviction founded\non long observation and comparison between the people of Canada\nand those of many other countries; and I have a faith, little short\nof full assurance, as to the advancing and glorious future of our\ncountry. With this conviction and faith, and animated with the\nconsciousness of general approval and co-operation on the part of\nthe people, I shall renew my humble contributions of labour to the\ncommon treasury of Canadian progress and civilization.\"\n\n\nESTABLISHMENT OF THE EDUCATIONAL DEPOSITORY, AND ITS RESULTS.\n\nIn 1850-51, Dr. Ryerson, while in England, made arrangements for\nestablishing a library, a prize book and an apparatus and map\ndepository, in connection with his Department. His reasons for doing\nso may be thus briefly stated:\n\n1. He felt it to be practically useless to train teachers in the\nbest methods of imparting instruction, and in the use of apparatus\nand other school appliances in the normal school and not provide for\nthem, when in charge of schools, a constant and abundant supply of\nthese necessary appliances at the very cheapest rates.\n\n2. He held it to be equally necessary that the pupils, who had\nacquired a taste for reading and knowledge in the schools, should\nhave an equally abundant and perennial supply of the best and purest\nliterature as it is issued from the press; otherwise they would be\nsure to procure reading matter (often pernicious, as he had painful\nproof) for themselves.\n\n3. He could see no distinction, and therefore could not admit of\nany, in the principle of providing such a two-fold supply of school\nmaterial and reading matter, and in that of providing trained\nteachers and skilled inspectors at the expense of the Province, as\nwell as a money bonus to aid in maintaining the schools in a state\nof efficiency.\n\n4. He further felt that it was immaterial whether the money voted by\nParliament was expended in one direction or the other, so long as in\neach department of the system the best interests and necessities of\nthe schools were consulted, and the symmetry and efficiency of the\nschool system, as a whole, were preserved and promoted.\n\n5. He projected this plan of supply on a purely commercial basis,\nand so arranged and successfully carried out his scheme that\nwhile there was distributed nearly a million dollars' worth of\nschool material and books up to the time when the depository was\nclosed, it did not cost the country anything for the expenses of\nits management, as it more than paid its way. An elaborate report\non this subject was prepared by Mr. James Brown, an experienced\naccountant, under the direction of Hon. Adam Crooks, the first\nMinister of Education. It more than sustained the statement here\nmade. The particulars are as follows:--\n\n\nABSTRACT OF DEPOSITORY SCHEDULE PRESENTED TO THE LEGISLATURE IN 1877.\n\n  Total amount of legislative grants to the depository for all\n      purposes, viz.:\n      (1) Purchase of stock, and (2) Salaries and the\n      entire cost of management, etc., 1850 to 1875\n      inclusive                                            $811,523 72\n\n  Total value of books, maps and apparatus despatched\n      from the depository, 1850 to 1875 inclusive           803,067 86\n                                                            __________\n  Difference to be accounted for                              8,455 86\n\n  Net value of the stock on hand at the end of 1875,\n      after paying all expenses of management, etc           79,509 41\n\n  Deduct the difference to be accounted for (as above)        8,455 86\n                                                            ----------\n  Grand total of profits made by the depository after\n      paying all charges, as above, during the years\n      1850-1875                                              71,054 55\n\n\nDR. RYERSON A COMMISSIONER ON KING'S COLLEGE, ETC., NEW BRUNSWICK IN\n1854.\n\nOn the 1st of May, 1854, the Legislature of New Brunswick passed\nan Act empowering the Lieutenant-Governor to appoint a Royal\nCommission:--\n\n\"To enquire into the present state of King's College, its management\nand utility, with a view of improving the same, and rendering that\ninstitution more generally useful, and of suggesting the best mode\nof effecting that desirable object,\" etc.\n\nIn accordance with this Act, Sir Edmund Head, the then\nLieutenant-Governor, in August, 1854, appointed the following\ngentlemen as commissioners, viz.:--Hon. John Hamilton Gray, (late\nJudge of the High Court of British Columbia), Rev. Dr. Egerton\nRyerson, John William (now Sir Wm.) Dawson, Hon. John Simcoe\nSaunders, and Hon. James Brown.\n\nIn accepting the position of commissioner, Dr. Ryerson, at the close\nof his letter to Provincial Secretary Partelow, said:--\n\n\"When I mentioned to the head of the Canadian Administration the\nrequest which had been made to me from New Brunswick, and the\nprobability that a compliance with it would cause my absence for two\nor three weeks from the duties of my department, he thought I ought,\nby all means, to go--that it was part of my appropriate work, and\nthat we should regard each Province of British North America as a\npart of our own country.\n\n\"New Brunswick is so to me, in a peculiar sense, as the birth-place\nof my sainted mother and my elder brothers and sisters.\"\n\nThe commission met first at Fredericton, and afterwards at Toronto.\nTo Dr. Ryerson was entrusted the principal duty of drawing up the\nelaborate report, and in Hon. J. H. Gray's letter as chairman,\naccompanying the report in December, 1854, he says:\n\n\"I beg to express, with the full concurrence of my fellow\ncommissioners, our acknowledgements of the very valuable assistance\nafforded us by Dr. Ryerson. His great experience and unquestioned\nproficiency on all subjects connected with education, justly entitle\nhis opinion to great weight.\"\n\nSir Wm. Dawson, in a letter to Mr. Gray, thus summarizes the\ncontents of the report:--\n\n\"1st. The improvement of the College course of instruction and\nits extension by the introduction of special courses. 2ndly, The\ndefinition of the true place of the Provincial College in its\nrelations to the other educational institutions of the Province,\nand to the religious beliefs of the people; and 3rdly, The union of\nall the educational institutions in a Provincial university system,\nunder official supervision.\"\n\nA change in the Government of New Brunswick in 1854, prevented the\nreport being considered in the Legislature at that time. In a letter\nfrom Mr. Gray to Dr. Ryerson, dated May 15, 1855, he says:--\n\n\"The change of Government prevented our report being adopted and\nacted upon, but it met with universal approbation, and from every\nportion of the Province the voice of praise has gone up. I give\nyou credit for it all; and in my remarks in the House, I made my\nacknowledgements publicly to you and Mr. Dawson.\"\n\nIn a confidential letter to me, on Separate School matters, from Dr.\nRyerson, dated Quebec, January 30, 1858, he said:--\n\n\"Sir Edmund Head (now Governor-General), highly approved of my\nReport, etc., on the New Brunswick College question and has sent it\nto the authorities of McGill College to see if they cannot adopt\nsomething of the same kind.\"\n\nMr. Gray had hoped that the comprehensive bill proposed by the\ncommission in 1854, and to give effect to their recommendations\nrelating to King's College, Normal and Model Schools, and a Chief\nSuperintendent of Education, would be passed in the following year,\n1855. In this he was disappointed, for the bill did not pass until\n1860. In a letter to Dr. Ryerson from the Hon. Charles Fisher, dated\nFredericton, 14th May, 1860, he said:--\n\n\"After years of controversy and difficulty we have passed an Act\nto remodel King's College on the plan proposed by your commission,\nunder the title of the University of New Brunswick. We have not\nconnected the College or the head of it with the other educational\ninterests in the Province, but confined him to the University,\nand he must be a layman. This provision was inserted to prevent\ndifficulty.\"[54]\n\n  [54] In 1858 Mr. Henry Fisher (Brother of Hon. Charles Fisher) was\n  appointed Superintendent of Education for New Brunswick. He visited\n  Dr. Ryerson in that year to confer with him before undertaking\n  the duties of his new office. His death occurred in 1860, and in\n  communicating the sad news to Dr. Ryerson, Hon. Charles Fisher,\n  referring to his brother, said:--\n\n  \"He wished particularly (just before his death) to be remembered\n  to you, and that I should thank you for your kindness to him on\n  all occasions. He was succeeding in his efforts to improve the\n  educational interests of the Province, and had been enabled to\n  secure the support of all parties.\"\n\n\nPARTIAL CHRONOLOGICAL SKETCH OF DR. RYERSON'S EDUCATIONAL WORK,\n1855, ETC.\n\nI will now give a brief summary, in chronological order, of the\nsuccessive steps which Dr. Ryerson took to develop the system of\neducation which he had founded.\n\nIn 1855 Dr. Ryerson established meteorological stations in\nconnection with twelve selected county grammar schools, ten\nfollowing the coast line of the lakes and on the large rivers, and\ntwo entirely inland. In this he was aided by Colonel now General Sir\n(J. H.) Lefroy, R.E., for many years director of the Provincial (now\nDominion) Magnetical Observatory at Toronto. Sets of instruments,\nhaving been purchased in London and tested at the Kew Observatory,\nwere sent out to the twelve stations, duly equipped and provided\nwith all necessary appliances.\n\nIn 1857 Dr. Ryerson made his third educational tour in Europe,\nwhere he procured, at Antwerp, Brussels, Florence, Rome, Paris and\nLondon, an admirable collection of copies of paintings by the Old\nMasters, statues, busts, etc., besides various other articles of a\ntypical character for an educational museum in connection with the\nDepartment. In 1867 I was deputed to largely add to this museum\ncollection, which I did in Paris, London, etc., especially in the\ndirection of Egyptian and Assyrian antiquities, busts, casts,\nfictile ivory, etc.\n\nIn 1858-61 Dr. Ryerson took a leading part in a protracted public\ndiscussion before a committee of the House of Assembly, in favor\nof grants to the various \"outlying\" denominational universities,\nchiefly in terms of Hon. Robert Baldwin's liberalized University\nAct of 1853. He maintained that these colleges \"did the State\nsome service,\" and that it was right that their claims should\nbe recognised in a substantial manner, as colleges of a central\nuniversity. He deprecated the multiplication of universities in the\nProvince, which he held would be the result of a rejection of the\nproposed scheme. His plan was not adopted, and universities were\nincreased from five to eight subsequently. Twenty-five years after\nthe close of that discussion a scheme for the confederation of these\ncolleges was again considered but without much effect.\n\nIn 1862, Dr. Ryerson addressed a circular to boards of trustees\nin cities and towns, deploring the \"numbers of children in these\ncenters of population, growing up with no other education than a\ntraining in idleness, vagrancy and crime.\" He added: \"I have, at\ndifferent times, submitted three propositions or plans for the\naccomplishment of the object of free schools in cities and towns.\nFirst, that as the property of all is taxed for the common school\neducation of all, all should be compelled to allow their children\nthe means of such education at either public or private schools.\nOr, secondly, that each municipality should be empowered to deal\nwith the vagrancy of children of school age, or the neglect of their\neducation, as a crime, subject to such penalties and such measures\nfor its prevention as each municipality, in its own discretion,\nmight from time to time adopt. Or, thirdly, that the aid of\nreligious benevolence should be invoked and encouraged to supplement\nthe agency of our present school system.\"\n\nBefore bringing the matter again before the Government, Dr. Ryerson\nsolicited the opinion and suggestions of the school boards on the\nsubject.\n\n\nBISHOP FRASER'S ESTIMATE OF THE U. C. SYSTEM OF EDUCATION IN 1863.\n\nIn 1863, Rev. James (afterwards Bishop) Fraser (of Manchester),\nwas appointed a Royal Commission to enquire into the American and\nCanadian systems of education. From his report, published after his\nreturn to England, I quote the following passages:--\n\n\"The Canadian system of education, in those main features of it\nwhich are common to both Provinces, makes no pretence of being\noriginal. It confesses to a borrowed and eclectic character. The\nneighboring States of New York and Massachusetts, the Irish, English\nand Prussian systems, have all contributed elements, which have\nbeen combined with considerable skill, and the whole administered\nwith remarkable energy, by those to whom its construction was\nconfided. It appears to me, however, that its fundamental ideas\nwere first developed by Mr. (now, I believe, Sir Arthur) Buller,\nin the masterly report on the state of education in Canada,\nwhich he addressed in the year 1838 to Lord Durham, the then\nGovernor-General, in which he sketched the programme of a system,\n'making,' as he candidly admitted, 'no attempt at originality,\nbut keeping constantly in view, as models, the system in force in\nPrussia and the United States, particularly the latter, as being\nmost adapted to the circumstances of the colony.'\n\n\"As a result of Mr. Buller's recommendations, (not, however, till\nafter the legislative union of the Provinces which Lord Durham\nhad suggested, as the best remedy for the various political ills\nunder which they severally laboured,) a law was passed in 1841,\ncovering both Provinces in its range, for the establishment and\nmaintenance of public schools. It provided for the appointment of\na Superintendent of Education for the whole Province, with two\nAssistant Superintendents under him, one for each of the sections. A\nsum of $200,000 was appropriated for the support of schools, which\nwas to be distributed among the several municipal districts, in\nproportion to the number of children of school age in each of them;\n$80,000 being assigned to Upper and $120,000 to Lower Canada, such\nbeing the then ratio of their respective populations.\n\n\"The circumstances of the two sections, however, particularly in\nthe proportions of Roman Catholics to Protestants in each, and\nthe extent to which the Roman Catholic religion may be said to be\nestablished in Lower Canada, were soon found to be so different\nthat insuperable difficulties were encountered in working a\ncombined system under one central administration, and in 1845 the\nlaw was changed. The nominal office of Chief Superintendent was\nabolished, and the entire executive administration of the system\nwas confined to the sectional superintendents, and the Provinces,\nfor all educational purposes, again became separated. The law\nitself was thoroughly revised and adapted to the peculiar wants of\neach Province, as ascertained by experience; and ever since there\nhave been two systems at work, identical in their leading idea,\ndiffering, sometimes widely, in their details, administered by\nindependent executives, and without any organic relations at all.\n\n\"Before we proceed to observe the manner and record the results\nof its practical working, it is proper to premise that it is a\npurely permissive, not a compulsory system, and its adoption by\nany municipality is entirely voluntary.... Entering a Canadian\nschool, with American impressions fresh upon the mind, the first\nfeeling is one of disappointment. One misses the life, the motion,\nthe vivacity, the precision--in a word, the brilliancy. But as you\nstay, and pass both teacher and pupils in review, the feeling of\ndisappointment gives way to a feeling of surprise. You find that\nthis plain, unpretending teacher has the power, and has successfully\nused the power, of communicating real, solid knowledge and good\nsense to those youthful minds, which, if they do not move rapidly,\nat least grasp, when they do take hold, firmly. If there is an\nappearance of what the Americans call \"loose ends\" in the school,\nit is only an appearance. The knowledge is stowed away compactly\nenough in its proper compartments, and is at hand, not perhaps very\npromptly, but pretty surely, when wanted. To set off against their\nquickness, I heard many random answers in American schools; while,\nper contra to the slowness of the Canadian scholar, I seldom got\na reply very wide of the mark. The whole teaching was homely, but\nit was sound. I chanced to meet a schoolmaster at Toronto, who had\nkept school in Canada, and was then keeping school at Haarlem, New\nYork, and he gave Canadian education the preference for thoroughness\nand solid results. Each system--or rather, I should say, the result\nof each system--seems to harmonize best with the character of the\nrespective peoples. The Canadian chooses his type of school as the\nVicar of Wakefield's wife chose her wedding-gown, and as the Vicar\nof Wakefield chose his wife, \"not for a fine, glossy surface, but\nfor such qualities as will wear well.\" I cannot say, judging from\nthe schools which I have seen--which I take to be types of their\nbest schools--that their choice has been misplaced, or that they\nhave any reason to be disappointed with the results. I speak of the\ngeneral character of education to which they evidently lean. That\nthe actual results should be unequal, often in the widest possible\ndegree, is true of education under all systems, everywhere.\n\n\"One of the most interesting features in the Canadian system is the\nway in which it has endeavored to deal with what we find to be one\nof our most formidable difficulties, the religious difficulty. In\nCanada it has been dealt with by the use of two expedients; one, by\nprescribing certain rules and regulations, which it was hoped would\nallow of religious instruction being given in the schools without\nintroducing sectarianism or hurting consciences; the other, by\npermitting, in certain cases, the establishment of \"separate,\" which\nare practically denominational, and in fact Roman Catholic schools.\n\n\"The permission under certain circumstances to establish separate,\nthat is, denominational schools, is a peculiar feature of the\nsystem both of Upper and Lower Canada. Dr. Ryerson thinks that the\nadmission of the principle is a thing to be regretted, though at the\nsame time he considers that the advantages which it entails entirely\nrest with those who avail themselves of its provisions, and he would\nnot desire to see any coercion used either to repeal or modify them.\n\n\"Such, in all its main features, is the school system of Upper\nCanada. A system, in the eyes of its administrators, who regard\nit with justifiable self-complacency, not perfect but yet far in\nadvance, as a system of national education, of anything that we can\nshow at home. It is indeed very remarkable to me that in a country,\noccupied in the greater part of its area by a sparse and anything\nbut wealthy population, whose predominant characteristic is as far\nas possible removed from the spirit of enterprise, an educational\nsystem so complete in its theory and so capable of adaptation in\npractice should have been originally organized, and have been\nmaintained in what, with all allowances, must still be called\nsuccessful operation for so long a period as twenty-five years. It\nshows what can be accomplished by the energy, determination and\ndevotion of a single earnest man. What national education in Great\nBritain owes to Sir James Kay Shuttleworth, what education in New\nEngland owes to Horace Mann, that debt education in Canada owes to\nEgerton Ryerson. He has been the object of bitter abuse, of not a\nlittle misrepresentation; but he has not swerved from his policy or\nfrom his fixed ideas. Through evil report and good report he has\nresolved, and he has found others to support him in the resolution,\nthat free education shall be placed within the reach of every\nCanadian parent for every Canadian child. I hope I have not been\nungenerous in dwelling sometimes upon the deficiencies in this noble\nwork. To point out a defect is sometimes the first step towards\nrepairing it; and if this report should ever cross the ocean and be\nread by those of whom it speaks, I hope, not with too great freedom,\nthey will perhaps accept the assurance that, while I desired to\nappreciate, I was bound, above all, to be true; and that even where\nI could not wholly praise) I never meant to blame. Honest criticism\nis not hostility.\"\n\n       *       *       *       *       *\n\nIn a letter addressed to Dr. Ryerson in 1875, the Bishop says:--\n\n\"I take it very kindly in you that you remember an old acquaintance,\nand I have read with interest your last report. I am glad to observe\nprogress in the old lines almost everywhere. I was flattered also to\nfind that some words of mine, written in 1865, are thought worthy\nof being quoted.... It is pleasant to find a public servant now in\nthe thirty-second year of his incumbency, still so hopeful and so\nvigorous. Few men have lived a more useful or active life than you,\nand your highest reward must be to look back upon what you have been\npermitted to achieve.\"\n\nSpeaking of the character of Dr. Ryerson's educational work and of\nthe way in which he met difficulties in accomplishing it, Mr. J.\nAntisell Allen, of Kingston, in his paper on \"_Dr. Ryerson, a Review\nand a Study_,\" says:--\n\n\"There is hardly a foot-length of our civilization on which he\nhas not left his mark. For those who believe that, on the grounds\nof expediency, a government is justified in interfering with the\nordinary working of the great human life-struggle, and so, in\ntaking one man's money to benefit another man's children, that is\nto a majority so overwhelming as to come almost under the category\nof universal, as to be every one's belief--what system of general\neducation can recommend itself more fully, or work more smoothly,\nthan does his? In his struggles in this direction, neither seduced\nby friends, nor cowed by enemies, nor damped in his ardour by the\nvastness of the undertaking--turning neither to the right hand nor\nto the left--he has raised to himself a \"_monumentum perennius\n\u00e6re_,\" and has bequeathed to us and posterity a system of public\nand high school education second to none anywhere, and, making some\ndeduction for possible mistakes incident to our weak humanity,\na system almost as perfect as we in this generation are perhaps\ncapable of generally acquiescing in.\"\n\nIn 1867 Dr. Ryerson made his fourth and final educational tour in\nEurope and America. On his return he submitted to the Government a\nhighly valuable \"Special Report on the Systems and State of Popular\nEducation in the several countries in Europe and the United States\nof America, with practical suggestions for the improvement of\npublic instruction in Upper Canada.\" He also made a separate and\ninteresting \"Report on the Institutions for the Deaf and Dumb in\nvarious countries.\" A few years afterwards he had the happiness of\nseeing institutions of a similar kind in successful operation in\nthis Province.\n\n\nCHARACTER OF THE IMPORTANT SCHOOL LEGISLATION OF 1871.\n\nThe fifth and last series of conventions was held in 1869, and\non the results of the consultations and deliberations of these\nconventions, Dr. Ryerson framed that crowning measure of his\nadministration, which received the sanction of the legislature in\n1871--twenty-one years after the first great departure in school\nlegislation--that of 1850.\n\nFor the various objects which he had recommended during the years\nfrom 1850 to 1871, liberal grants were made by the Legislature.\nThe policy of the Government during those years was to sustain\nDr. Ryerson and to second his efforts to build up and consolidate\nthe system of public instruction which he had taken such pains to\nestablish. The result was that our school system expanded and grew\nin every direction, and became firmly rooted in the affections of\nthe people. In this way it came to be regarded as one of the most\nsuccessful and popular systems of education on the continent. And\nyet, as I have shown, he was continually suggesting improvements in\nit, for he always held that there was room, as well as a necessity,\nfor them.\n\nSchool legislation, chiefly in regard to high schools and matters of\ndetail, took place at intervals during the intervening years, but it\nwas is 1871 and 1874 that the final legislation under Dr. Ryerson's\nauspices took place. That of 1871 was strikingly progressive and\ntook a wide range. That of 1874 was largely supplemental and\nremedial.\n\nThe Act of 1871 introduced into our school law for the first\ntime some important principles, which, as yet, had not received\nlegislative sanction. They were chiefly those which related, among\nothers, to the following matters:\n\n1. Governmental, combined with improved local, inspection of schools.\n\n2. A high and fixed standard of qualifications for inspectors of\npublic schools.\n\n3. The abolition of non-certificated township superintendents of\nschools, and the substitution therefor of duly licensed county\ninspectors.\n\n4. The institution of simultaneous and uniform examinations\nin the several counties for teachers desiring certificates\nof qualification. This principle was soon extended to other\nexaminations, including competitive examinations in counties, etc.\n\n\nREVIEW OF THE SCHOOL LEGISLATION OF 1871.\n\nAt Dr. Ryerson's request I prepared for him and wrote the text of\nhis education report for 1870. In that report I reviewed in detail\nthe various provisions and improvements introduced into our school\nsystem by the School Act of 1871. I reproduce here the more salient\npoints of that report, touching upon the reasons for the passing of\nthat progressive measure, and indicating some of its main features.\nI said:--\n\nSo many and important have been the changes recently made in the law\naffecting our System of Public Instruction, that it may be well, as\na preliminary to a discussion of those changes, briefly to refer\nto a few facts relating to the history and progress of our School\nSystem.\n\nIn 1844, our municipal system (on which our then elementary School\nLaw was engrafted), was in its infancy. The principle of local\nself-government was new, and much opposition was experienced in\ngiving effect to the School Law then in operation. The theory\nof local taxation for the support of schools was in some places\nvigorously opposed, and in others regarded as a doubtful experiment.\nEven as late as 1850, some municipalities refused to accept the\nimproved law enacted that year, or act under its provisions, and\nthus deprived their constituents of the great boon of popular\neducation. It is only six years since the last disability, caused by\nsuch refusal, was removed,--thus uniting the entire Province in a\ncordial acceptance of the School Law.\n\nThe following brief statistical references will illustrate the\ngrowth and advance of our School System:--\n\nIn 1844, there were but 2,610 Public Schools, in 1870, there were\n4,566. In that year, (1844), the school population was 183,539--of\nwhich 96,756 children attended the Public Schools, while 86,783\n(or nearly as many more) were reported as not in attendance at any\nschool whatever.\n\nIn 1870, the school population was 483,966--of which 420,488\nchildren were in attendance in our schools, and 63,478 reported\nas not in attendance--not one-seventh, instead of nearly one-half\nof the children of school age, as in 1844. In 1844, the whole\nsum available for the support of the Public Schools was about\n$280,000--of which, approximately $190,000 were raised by local\ntaxation.[55] In 1870, the whole sum available for Public Schools\nwas $1,712,060--of which $1,336,383 were raised by local taxation\nand fees--an increase of more than seven hundred per cent over 1844!\n\n  [55] In 1850, (the first year in which we have positive information\n  on this subject), we find that the total sum expended in this\n  Province for public elementary education, was $410,472; of which\n  $326,472 were raised by local rates and fees.\n\nThere are few Canadians who do not now refer with an mixed pride\nand satisfaction to the vastly improved condition of our Public\nSchools under the operation of the present law, revised in 1850,\nand now revised and extended.[56] On no one point have we greater\ncause for thankfulness and congratulation, than in the fact of the\nunanimity and cordiality with which our School System is supported\nby all classes of the community, by men of all shades of political\nfeeling, and, with a single exception (and that in part only), of\nall religious persuasions in the Province.\n\n  [56] No one is more sensible than I am of the numerous defects of\n  our School system, and for this reason I have labored all the more\n  assiduously to have these detects removed by our recent school\n  legislation. As I have stated further on, I have even had to combat\n  the views of those friends of the system who had thought that it was\n  not susceptible of much improvement.\n\n\nOBJECTIONS TO IMPROVE OUR SCHOOL SYSTEM ANSWERED.\n\nIt is a singular and gratifying (yet in some respect it has proved\nan embarrassing) fact that the chief difficulty experienced in\npromoting the improvement of our School System has arisen from the\nsomewhat over-sensitiveness of the friends of our Schools, lest the\nproposed changes should disturb the foundations of a system which\nthey had learned to regard with so much favor and affection. This\nsolicitude arose partly from a mistaken view of the condition and\nnecessities of our system, and partly from a misapprehension of\nthe scope and objects of the proposed ameliorations in our School\nLaw. It will be my aim, however, in the following remarks to justify\nand illustrate the principles and policy involved in the recent\nimportant changes which have been made in our School Law.\n\nI would, in the first place, remark that were we, in making\nimprovements in our School System, to confine our observation and\nexperience to our own Province alone, we might be disposed to look\nwith complacency upon that system, and to rest satisfied with the\nprogress which we have already made. The effect of such a state\nof feeling would be that we would seek to profit little by the\neducational experience and advancement of other countries. But such\na short-sighted and unpatriotic course, though approved by some on\nthe principle of \"let well-alone,\" yet would not commend itself to\nthe maturer judgment of those who are accustomed to look at the\n\"stern logic of facts,\" and to take a comprehensive and practical\nview of the underlying causes of the social progress in other\ncountries.\n\n5. The fixing and rendering uniform of a higher standard of\nqualification for public and high school teachers.\n\n6. Giving the profession of teaching a fixed legal status, and\nproviding more fully and equitably for the retirement and united\nsupport, by the profession and the legislature, of worn out or\ndisabled teachers.\n\n7. The establishment by law of a national system of free schools.\n\n8. Declaring the right by law, as well as the necessity, of every\nchild to attend some school, thus recognizing the principle of, and\nproviding for, \"compulsory education.\"\n\n9. Requiring, by law, that adequate school accommodation, in regard\nto school house, playground and site, be provided by the trustees,\nfor all of the resident children of school age in their localities.\n\n10. Prescribing a more systematic and practical course of study for\neach of the classes in the public schools.\n\n11. Discriminating, by a clearly defined line, the course of study\nin public and high schools respectively.\n\n12. Providing for the establishment and support of collegiate\ninstitutes, or local colleges.\n\n13. Requiring municipalities to maintain high schools and collegiate\ninstitutes, equally with the public schools, and as part of the\ngeneral school system.\n\n14. Providing, at the option of the ratepayers, for the substitution\nof township boards of education, in place of local trustee boards.\n\n15. Authorizing the establishment of industrial schools.\n\n       *       *       *       *       *\n\nSuch were the main features of the comprehensive and progressive\nSchool Act passed in 1871. In many respects it revolutionized\nthe existing state of things. It gave a wonderful impetus to the\nschools, and to every department of school system--the effects of\nwhich we feel to this day.\n\nWe are a young country, placed in close proximity to a large and\nwonderfully progressive people. In the good providence of God, we\nare permitted to construct on the broad and deep foundations of\nBritish liberty, the corner-stone of a new nationality, leaving\nto those who come after us to raise the stately edifice itself.\nApart from the vital Christianity of our people, what more lasting\nbond and cement of society in that new nationality, than a free\nand comprehensive system of Christian education for the youth of\nthe land, such as we have sought to establish? Our aim should,\ntherefore, be to make that system commensurate with the wants of\nour people, in harmony with the progressive spirit of the times,\nand comprehensive enough to embrace the various branches of human\nknowledge which are now continually being called into requisition in\nthe daily life of the farmer, the artizan, and the man of business.\nIn no department of social and national progress have our neighbors\nmade greater advances, or prided themselves more justly, than, in\nthat of free popular education. On the other hand, in no feature of\nprogress under British institutions up to a late period has there\nbeen less satisfaction, as a whole, or less positive advancement\nthan in that of public education. By many of our neighbors on the\nother side of the lines, such inertness and non-appreciation of\na vital part of national life has been regarded as inherent in\nmonarchical institutions. The fact, however, has been overlooked\nthat the lingering effects of the long prevalence in Britain of\nthe feudal theory, on which her social and political institutions\nwere originally founded, has, in spite of various ameliorations in\nthe condition of her people, exercised a sure but silent influence\nagainst the earlier adoption of the principle of the free and\nuniversal education of the people. But so surely and certainly has\nthis latent feeling of opposition to popular education given way\nbefore the prevalence of more enlightened views, that, even in\nthe most monarchical countries of Europe, the desire felt and the\nefforts put forth for the diffusion of public education in all its\ncomprehensiveness and fulness have been remarkable. Nevertheless,\neven among ourselves, that principle of latent opposition to popular\neducation did exist in the earlier stages of our educational\nhistory. Its gradual removal, therefore, under the beneficent\noperation of our School Laws, and the prevalence of juster and more\npatriotic views in matter of education are subjects of sincere\ncongratulation to our people.\n\n\nNECESSITY FOR THE CHANGES IN THE SCHOOL LAW OF ONTARIO IN 1871.\n\nWe will now proceed, in the light of the educational facts and\nillustrations which we have given from other countries, to discuss\nthe recent improvements which have been effected in our own law.\n\nThe population of this Province, according to the recent census,\nis 1,620,842. The number of children of school age is 483,966, or\na little over one-fourth of the whole. The number of Elementary\nSchools is not much below 5,000, and are maintained at an annual\ncost of nearly $1,800,000, or one dollar per head of the population.\nSuch being the magnitude to which our Educational System has grown,\nevery man will feel how imperative it is upon us to see that\nthat system is as thorough and complete in all of its details as\npossible; and that in no respect should it be allowed to fall below\nthe standard now reached by the other educating countries to which\nwe have referred.\n\nSo long as our system of schools was in its infancy, and might be\nfairly regarded as yet an experiment, so long as we confine our\nefforts to mere elementary organization and be content with very\nmoderate results. Experience has shown, however, that without great\ncare and constant effort the tendency of all systems of education,\nand ours among the rest, is to a state of equilibrium, or to a\nuniform dead level of passable respectability. This is the stage\nin its history, as elsewhere, at which our system has arrived, and\nat which, as we have explained, many of its friends are disposed\nto leave it. But those who have carefully studied the subject in\nall its bearings, and have looked more closely into the educational\nhistory, the progress and failures of other countries, know full\nwell that our school system would fall behind that of other\ncountries and become stationary, unless it embodies within itself\nfrom time to time the true elements of progress, and provides fully\nand on a sufficient scale for the educational wants of the youth of\nthe country.\n\nSince 1850 it was left to the ratepayers in each school division\nto decide annually whether the schools should be free or partly\nsupported by rate-bill on pupils attending the school. The\nprinciple, that a Public School education is the right of every\nchild in the land, and that every man should contribute, according\nto his property, to the education of every child in the community,\nby whose influence and labors such property is protected and\nrendered valuable, had greatly obtained, so that Free Schools had\nincreased from one hundred to five hundred per annum, until upwards\nof four thousand of the four thousand four hundred Public Schools\nwere made free by actual experiments, and by the annual discussions\nand votes in these primary meetings of the people. The demand was\nvery general for several years, that all the Public Schools should\nnow be made free by law, and all local disputes on the subject be\nthus terminated. This has now been happily accomplished by the new\nlaw.\n\nIt is not necessary to go farther into detail in this retrospect, as\nthe foregoing extracts indicate the scope and spirit of the improved\nAct of 1871.\n\n\nHON. ADAM CROOKS ON THE SCHOOL INSPECTION LEGISLATION OF 1871.\n\nIn his speech before the Legislative Assembly on the 18th of\nFebruary, 1877, the first Minister of Education for Ontario, in\nreferring to the improved system of school inspection introduced by\nthe Act of 1871, and the more certain tenure of office secured to\nCounty Inspectors under that Act, said:\n\n     \"I have also been ready to say that most valuable results were\n     secured by the change in the law in 1871, under which the\n     present mode of school inspection took the place of the old\n     plan of local superintendence. Inspectors now must possess high\n     qualifications, both as teachers and in scholarship, while\n     the emoluments of the office make it an object of ambition to\n     every school teacher; and we have many teachers in the Province\n     who posses qualifications of the high standard prescribed for\n     Public School Inspectors. The tenure of the office of County\n     Inspector is such as should secure their impartiality. So long\n     as an Inspector discharges his duties efficiently, he can be\n     removed only by a two-thirds majority of the County Council. It\n     is unlikely that such two-thirds majority would be found unless\n     the Inspector had given reasonable cause for his dismissal. It\n     would not be wise therefore to alter the tenure by which County\n     Inspectors hold office.\"\n\n\nEFFECT OF THE SCHOOL ACT OF 1871 IN THE COUNTY OF HALDIMAND.\n\nThe following valuable testimony as to the great improvement in our\nschools which was wrought through the agency of the School Act of\n1871, is highly suggestive and practical in its character. What is\ntrue of Haldimand, as here expressed, is also true of other parts of\nthe Province.\n\nIn an address to the teachers of Haldimand in 1873, Mr. Inspector\nHarcourt, M.P.P., said:--\n\n\"No one, whose attention has been called to the matter, could\nimagine the miserable condition of the majority of the school-houses\nof 1871. At that time there were not ten properly furnished\nbuildings in Haldimand. Many of them with low ceilings, broken\nfloors and damaged windows, had for seats nothing better than the\nantiquated bench facing the wall. Too cold or too hot by turns in\nwinter, and suffocating in summer. With nothing to attract and\neverything to discourage scholars, we wonder that an intelligent\npublic has so long tolerated their existence.... In the main,\nhowever, I am especially gratified at the improvements effected. In\ntwo years sixteen brick buildings have been erected; all of them\nsubstantial and well furnished--some of them models of neatness\nand finish. In a dozen sections preparations are being made for\nreplacing the old houses, so that we have good reason to hope\nthat in a year or two, at furthest, our country will no longer be\nnoticeable for the miserable style of its school-houses.\"\n\n\"Connected with the question of progress in certain branches of\nstudy, in relation of which I might say of cause and effect, are the\ntwo items of Examination of Teachers and School Accommodation. The\nprovisions now in force for the examination of teachers are such\nthat, if wisely carried out, the standard of the profession must be\nraised, and along with it the status of our schools.... The fact\nthat somehow or another teachers received first and second class\ncertificates, three or four years ago, who could not now obtain a\nthird; that while it was exceptional for an applicant to fail then,\nthose who succeed now are but thirty per cent. of the whole is known\nto all of us....\n\n\"To summarize the foregoing statements we HAVE progressed since\n1871, swiftly in one particular, slowly and steadily in several\nothers.\"--_Address, pages 5-7._\n\n\nEFFECT OF THE SCHOOL LAW OF 1871 IN THE COUNTY OF SIMCOE (SOUTH.)\n\nAt the inauguration of the new school-house in Barrie in 1872, the\nRev. Wm. McKee, B.A., Inspector of Schools in South Simcoe, stated\nwhat had been the salutary effect of the School Law of 1871 in his\ncounty. He said:--\n\nDuring my visits to the schools I found many of the school-houses\nof a very inferior description--being rude log buildings, old and\ndilapidated, with seats and desks of a corresponding character,\noften situated on the edge of the road, and without wells, offices,\nplaygrounds or fencing of any kind; so that it is quite certain and\nplain the requirements of the new School Law have not come into\nforce at all too soon, so far as the interests and advancement\nof education in this part of Ontario are concerned. Indeed truth\nobliges me to state that in the Riding which forms my field of\nlabour--and I believe the remark will hold true with still greater\nforce in regard to North Simcoe--the school-houses which are\nsufficiently large, well ventilated, fully furnished, and provided\nwith an adequate supply of requisites are very few--perhaps less\nthan half-a-dozen all told. It is true, however, that since the New\nSchool Law and Regulations came into operation there are indications\nof a change for the better in regard to the matters to which I\nhave alluded. I could mention not less than twelve or fourteen\nschool sections in which steps have _already_ been, or are being\ntaken for the erection of new school-houses which are designed to\nreplace the old buildings, and which, in regard to adequate school\naccommodation, are also intended to meet the requirements of the New\nSchool Law, and to be in every way suitable for school purposes.\nAnd it is to be distinctly noticed that in all the cases to which\nI have referred, the _initiative_ has been taken by the people or\nthe trustees themselves; and I, for my part, feel that I cannot\nbut regard this as a very significant fact--a very hopeful and\nencouraging symptom. I look upon it as an omen for good, and as an\nimportant and gratifying evidence of the favourable and successful\nworking of the New School Law and Regulations. For being intimately\nacquainted with the southern part of the county for the last fifteen\nyears, I have no hesitation in maintaining that the effects spoken\nof, or the action taken by school trustees or the people, can be\nfairly traced to no other cause than to the working and influence of\nthe New School Law and Regulations. I can testify that latterly--I\nmean particularly since the passing of the New School Act--I have\nmarked among the people of these townships a deepening sense of\nthe importance of a sound education, and likewise an increasing\ndesire to encourage and promote it. I have noticed, also, I think,\nboth among trustees and parents, a growing conviction that not\nonly the efficiency of the teacher, but, also the discipline and\nspirit of a school, the progress of children in their studies,\ntheir proper training, and their successful education, are far more\nintimately connected than it was one time imagined, with the style\nand character of the schoolroom in which the work of instruction is\ncarried on, and with the kind of school accommodations provided for\nand enjoyed by pupils.[57]\n\n  [57] These two extracts are given simply as illustrative examples,\n  and as they were public utterances of the Inspectors named. Similar\n  testimony was received by the Department from other Inspectors, but,\n  from the nature of the case, and their non-publication in the local\n  newspapers, they were not subject to the same criticism as were the\n  statements publicly made and published in the localities concerned.\n\n\n\n\nCHAPTER V.\n\nA SPECIAL CHAPTER ON THE STATE EDUCATION IN THE \"OLDEN TIME\" IN\nUPPER CANADA.\n\n\nIn this special chapter, I insert a series of interesting papers,\nillustrative of what may be called the interior or domestic\ncharacter of school life in its various phases, in Upper Canada,\nfrom the time of the passing of Dr. Ryerson's first School Act,\nin 1846, to the passing of his last School Act, in 1871. These\npapers give a graphic, bird's-eye view of the state of the school,\nschool houses and school teachers in the years gone by. They are by\nrepresentative men--inspectors, public men and school teachers, and\nthey are, therefore, interesting and valuable, as giving the result\nof the personal observation of these gentlemen.\n\nThe personal experience of the Hon. J. Sandfield Macdonald is of an\nearlier date than those which follow, and refers to one of the most\nnoted of the old District Grammar Schools. It is highly interesting\nas a reminiscence of the early training of one of our most noted\npublic men, and one, under whose auspices, the important School Act\nof 1871 was passed.\n\n\nHON. J. SANDFIELD MACDONALD'S SCHOOL DAYS--HIS REMINISCENCES OF THEM.\n\nAt a public dinner given to the Hon. J. Sandfield Macdonald in 1870,\nhe thus referred to his early school days:--\n\n\"My friend, Judge Jarvis, has referred to my early life, and has\nvery properly remarked that this is the country that offers the\nwidest field to the industrious, or to a man of energy if he only\npossesses a modicum of brains.... It is true what the Judge states\nthat I arrived in Cornwall forty years ago next autumn.... I was\nengaged in a dry goods store. But the Judge has told you that I was\nnot satisfied with that state of things. I went to the school here,\nwhich has had a reputation it may be proud of ever since the time\nof the late Bishop Strachan. It was the school that educated the\nBoultons, the McGills, and the Jarvises. In the school I entered,\nand there I had to strive with those who were able to be maintained\nby their parents. I worked against them at a great disadvantage,\nand would have succumbed but that I was cheered on by my venerable\npreceptor. Many others have struggled in that school of whom Canada\nshould be proud. One of them particularly. He was one of the\nbrightest and most talented of the men our eastern district can\nboast of. But providence has thought proper to take him away from\nhis sphere of usefulness. Need I say that I refer to that ornament\nof the Bench, the late Chancellor Vankoughnet.--Were Dr. Urquhart\nable to boast of no other pupil but that honourable gentleman, he\nmight have retired on his laurels. If that old gentleman had not\nsent me a letter of encouragement I would not have been here, as I\nwas about to break down for want of means. This letter was written\nin 1835, and ... I cannot help shewing what was thought of me by one\nwho had the most perceptive idea of the ability of his pupils. This\nletter had the effect of making me bear up in my struggle with my\nsuperiors in position and was as follows:\n\n\"'These certify that the bearer, Mr. John S. McDonald, was a pupil\nin the Eastern District School, from the 19th Nov., 1832, to the\n23rd Dec. last; that during that period his industry and application\nwere close and assiduous, and that his progress in the several\nbranches of study, to which he directed his attention, was highly\nrespectable, and very considerably exceeded what is usually made\nin the same space of time; that the perseverance manifested in\novercoming the difficulties to be encountered at the outset of a\nclassical and mathematical education, called forth the particular\nremark and approval of his teacher, as indicating considerable\nenergy of character, and as an earnest of future success in the\nprosecution of his studies. Moreover, that his general deportment\nduring the same period, was most exemplary, and becoming, evincing\nat all times a kindly disposition towards his fellow students and\na most respectful deference to the discipline of the school; and\nthat, if the good opinion and good wishes of his teacher can on any\noccasion profit him, he is justly entitled to both.'\"\n\n\"I owe all the spirit of independence which I have maintained\nthroughout my career, to my learning in that school. After I left\nschool I went into the study of the law, in the office of the late\nMr. McLean.\"\n\nEDUCATION IN THE COUNTY OF WELLINGTON UNDER DR. RYERSON'S\nADMINISTRATION.\n\nThe following admirable resum\u00e9 of the former state of education and\nof educational progress in the county of Wellington, and indeed\nthroughout Ontario generally, was given in an address on \"Then and\nNow,\" delivered before a public meeting in Guelph, in May, 1880, by\nHon. Charles Clarke, ex-Speaker of the House of Assembly. After some\npreliminary remarks, Mr. Clarke said:--\n\nLittle more than thirty years ago, much of the country, now known\nas the county of Wellington, was a wilderness. The territory north\nof Guelph was almost unsettled, beyond the townships of Nichol, and\nthe settled portion possessed but few residents. In Wellington, the\nupper tier of townships had scarcely been entered upon, and names of\nplaces now \"familiar as household words\" were unknown. Such roads as\nthere were had been simply cut through the bush, and had experienced\nlittle other improvement than that which the axe, the handspike,\nthe logging chain and fire had afforded. Peel was in the early\nstage of settlement; Maryboro was almost unknown; Minto was really\na _terra incognito_; Luther was, in popular estimation, a vast and\nirreclaimable swamp; Arthur had a mere handful of settlers; Mount\nForest was a nameless and unbroken government reserve for a town\nplot, covered with virgin forest; Elora possessed some half-dozen\nhouses; such places as Harriston, Palmerston and Drayton were not\neven a dream of the future; and the gravel roads, thrifty villages,\nand smiling farms which now make pleasant travel from the northern\nbank of the Grand River to the utmost bounds of Wellington, were\ncovered with thick and luxuriant growth of maple, hemlock, elm\nand cedar. Everything was in primitive shape, and yet the mark of\nfuture progress was made, here and there, and coming events cast\ntheir shadow. Oxen were far more numerous than teams of horses,\nand neither could be regarded as plentiful. The axe was more busy\nthan the plough, and regularly prepared more acres for the annual\nsowing. Money was scarce, produce was low in price, barter was the\nrule and not the exception, postal communication was defective,\nwages were poor, and \"hard times\" were as commonly talked about and\nas earnestly believed in as to-day, when, measured by the past,\nthe term is comparatively meaningless. There was a feeling of\ndespondency throughout the community, and people were divided as to\nthe cause of the general depression. Some blamed the Rebellion of a\nfew years before; others said that the effects of Family Compactism\nhad not yet died away; and still others attributed all evils to the\nnewly effected Union between Upper and Lower Canada. There is little\nwonder that, at such a time, schools and schoolmasters were under\nthe weather, and reckoned as but of \"small account\" by many of our\npeople.\n\nEARLY SCHOOL LEGISLATION IN 1841, 1843 AND 1846.\n\nThanks to the energy, however, of a noble few, prominent amongst\nwhom stood Egerton Ryerson, the Government of that day took steps\nto obtain information as to the system of public education in force\nin some of the states of the American Union and in Europe, and,\ntaking Massachussetts and Prussia as a guide, enacted a sweeping\namendment to School Act for Upper Canada, in the ninth year of Her\nMajesty's reign, and put it into operation in 1847. In 1841, the\nfirst Common School Law had been passed, and in 1843 it was amended,\nbut the system was defective and unproductive of expected results.\nUnder it, townships were divided into school sections, by township\nsuperintendents, who were practically uncontrolled, and therefore,\nin many instances, arbitrary, and these divisions were unequal in\nsize, often unnecessarily small, and frequently unfairly made. The\nconsequence of this state of things was unpopularity of the law,\nand a pretty general conviction that common schools were too often\ncommon nuisances. The report of the Superintendent of Education, for\n1847, tells us that the system produced \"miserable school-houses,\npoor and cheap teachers, interrupted and temporary instruction and\nheavy rate-bills.\" In some districts, before the passage of the\namending School Act, 9 Vic. Chap. 20, the District Council had\nnever imposed a school assessment, depending for school maintenance\nentirely upon the small legislative grant apportioned to each\ndistrict, and an equivalent raised solely by rate-bills or voluntary\ncontributions. No uniformity existed in the use of class books, the\ntownship superintendent or the teacher, or even the parent dictating\nwhat should be employed in each particular section. In 1846, no\nfewer than 13 different spelling books, 107 readers, 35 arithmetics,\n20 geographies, 21 histories, and 16 grammars, were used in our\nCommon Schools, besides varying class books on other subjects.\n\nINFERIOR QUALIFICATIONS OF TEACHERS AND VARIED METHODS OF TEACHING.\n\nThe methods of teaching were almost as numerous as the teachers,\nand followed no specified rule. Sometimes it was by classes, often\nby individuals, and in other cases by an extensive use of monitors,\nbeing generally a mixture of the three styles, and nearly always\na higgledy-piggledy, go-as-you-please arrangement, as easy as\npossible to the teacher, and as unproductive of good results to\nthe pupil as such indefinite work might be fully expected to be.\nAnd the character of the teachers, speaking in general terms, and\nnot forgetting many bright exceptions, was not above suspicion.\nCertificates were granted by township superintendents, who too often\nrelieved the charitable, and the district council, by thrusting into\nthe school-house the ne'er-do-wells, the infirm, the crippled, the\nsickly and the unfortunate, who, under ordinary circumstances, would\nhave become dependent upon the good nature and benevolence of their\nfellow-citizens. In one district a superintendent, after the passage\nof the new School Law, was compelled to give notice that he would\nnot grant certificates to any candidates unless they were strictly\nsober, and that he would cancel the certificates of all teachers who\nsuffered themselves at any time to become intoxicated. And, we are\ngravely informed, the result was that a majority, not all, of the\nhitherto intemperate teachers became thoroughly temperate men, and\nthat the incorrigible were dismissed. The quality of the teachers\nmay be guessed at very fairly, it is safe to say, from the salaries\npaid to them. In 1845, the average was \u00a326 2s, or $104.40; in 1846,\n\u00a326 4s, or $104.80; and in 1847, \u00a328 10s, or $114, and this, too,\nfor the most part, exclusive of board. Had the schools been kept\nopen during the whole of the teaching months of these years, the\nsalaries would have averaged $134 in 1845, $147 in 1846, and $148\nin 1847. It must be borne in mind that, in those days, male were\nmuch more numerous than female teachers, so that the smaller amounts\ngenerally paid to those of the gentler sex had comparatively little\ninfluence in lessening the general average. The parsimony and\npoverty of the people had much to do, of course, with the quality\nof the teacher, for men who could obtain higher wages at almost any\nother occupation, through physical or intellectual superiority,\nwould not waste time and opportunity to earn more than the paltry\npittance paid to the pedagogue, simply through philanthropic desire\nto advance the interests of the rising generation.\n\nDR. RYERSON'S TEST OF THE INTELLIGENCE OF A SCHOOL SECTION.\n\nSays Dr. Ryerson, in the report to which I am indebted for these\nfacts: \"This small compensation of teachers is the great source of\ninefficiency in the common schools. Persons of good abilities and\nattainments will not teach for little or nothing so long as they\ncan obtain a more ample remuneration in other pursuits.\" He adds,\nin language as truthful, and as worthy of notice to-day, as when\nit was written: \"People cannot obtain good teachers any more than\ngood lawyers or physicians without paying for their services.\" And,\nas he says in the next sentence, so say we all, and so I am happy\nto observe are many of our school corporations saying all over the\nprovince: \"The intelligence of any school section or corporation of\ntrustees may be tested by the amount of salary they are disposed to\ngive a good teacher.\" If Egerton Ryerson had said and done nothing\nmore than this, he would have deserved the gratitude of every\nteacher in Ontario, simply because he had the courage to put upon\nrecord a sentiment which, at the time when he used the words, was\neminently unpopular, and a direct and stinging rebuke to nearly\nevery school-board then existent. In those days, cheap teachers were\nwanted, and the supply equalled the demand, while the pockets of the\ncharitable were saved, a semblance of education was kept up, and the\ncounty poorhouses were not required so long as every other school\nsection provided for one, at least, of those who would, in these\ndays, be generally regarded as eligible candidates for admission\nthereto. The amount of interest taken in educational matters was not\nevidenced in small salaries alone.\n\n\nTHE CHARACTER OF THE SCHOOL-HOUSE ALSO A TEST.\n\nThe school-house, in its quality, too often matched the teacher. Of\n2,572 school-houses in Upper Canada in 1847, 49 only were of brick,\nand 84 of stone, the others being frame and log. Of the 2,500,\n800, or about one-third, were in good repair; 98 had more than one\nroom; 1,125, or less than half, were properly furnished with desks\nand seats; only 367 were provided with a suitable play-ground;\nand not more than 163, out of 2,572, had necessary outbuildings.\nComing nearer home, we find that the municipalities now comprised\nin the county of Wellington contained, in 1847, 43 school-houses,\nof which one was built of stone, 9 were frame, and 33 were log, and\nthe report states that only 13 were good, 25 were middling, and the\nbalance were inferior. When we remember the standard of \"goodness\"\nin those days, when school authorities at Toronto were thankful for\nsmall favors in rural districts, we can have some faint idea of the\ncharacter of the buildings pronounced inferior. It is probable that\nthey came up to the style of accommodation of the Mapleton school,\nin Manitoba, which I find described in the last report of the\nSuperintendent of Protestant Schools for that Province, as follows:\n\"Found that since my last visit the school-house has been floored;\nit still required plastering and ceiling and weather-boarding.\" What\nsort of a building it was before these improvements were effected,\nit doesn't require a very active brain to imagine, and when you have\nthe picture in your mind's eye you will have some conception of the\npleasures of teaching in the \"good old times,\" of less than half a\ncentury ago, in Upper Canada.\n\n\nSCHOOL CONDITION OF THE COUNTY OF WELLINGTON IN 1847.\n\nReturning to 1847, we are told that in the whole of Wellington\nDistrict, composed of the territory now forming the three counties\nof Wellington, Waterloo and Grey, there were 102 schools, of which\nonly 22 possessed good buildings. Let us glance for a moment at the\nthen state of finances of the school corporations in which we feel\nmost interested. Guelph township, including the village of Guelph,\nraised $507.38 by the municipal assessment, for school purposes,\nrealized $556.75 from rate-bills, and received $416.69 from the\nlegislative grant, or a total of $1,480.82, wherewith to pay seven\nteachers, maintain, more or less efficiently, ten schools, and\nafford instruction, good, bad or indifferent, as the case might\nbe, to 517 scholars. The township of Puslinch was nearly abreast\nof Guelph, and kept up 10 schools, paid 13 teachers, and had 558\nscholars on the roll, at an outlay of $1,381.86, but it must be\nremembered that if two or three teachers were employed, at different\nportions of the year, in one school, they increased the grand total\nof teachers for the year. It may have been that, while thirteen\nappear to have been engaged, there were not more, and probably less\nthan ten employed for the full teaching year. In 1847, Erin had\nthe highest number of scholars of any municipality in the county,\nhaving returned a total of 585, in six schools, and with eleven\nteachers, at an outlay of $1,039.06. Amaranth was at the foot of the\nlist, with one school, one teacher, thirty-eight scholars, and an\noutlay, made up from rate-bill, assessment and legislative grant,\nof $68.04. Peel and Wellesley, combined, had one school, three\nteachers,--employed at some portion or other of the year,--and spent\n$80.52. Nichol (including Fergus and Elora), Eramosa and Garafraxa\nmade returns,--the name Garafraxa being spelt with a double r, as I\nhave found it to be in all old official documents,--but Pilkington,\nArthur, Maryboro, Luther and Minto do not appear to have had school\norganization, not even municipal existence, while, of the whole\ncounty of Grey, Derby and Sydenham were alone mentioned in the\nreturn. It may be interesting to know--although I am aware, from\npainful experience, that listening to strings of figures is not the\nmost enlivening occupation in the world,--that the whole amount paid\nfor school purposes, in the county of Wellington, for that year,\nwas $5,862, of which $5,763 was given to teachers, and that the\naverage cost for each pupil taught was $2.10. One other fact may be\nadduced which will enable you to form a still clearer estimate of\nthe educational status of Upper Canada at the date referred to. The\nChief Superintendent had, in forms and regulations issued by him,\nspecified the lowest general standard of qualification for teachers,\nbut was forced to believe that a much lower standard had been acted\nupon by school visitors. These visitors were clergymen, magistrates\nand district councillors,--equivalent to our reeves,--and any two of\nthem could examine a teacher, test his or her qualification, pretty\nmuch as they deemed best, and grant a certificate, available only\nfor one school and one year, it is true, but nevertheless renewable,\nand answering every purpose of the certificate of to-day. It is not\ndifficult to imagine a much more easy and varying examination, under\nsuch circumstances, than that which an improved system soon rendered\nnecessary, and the quality of teachers so produced need not be\nfurther particularized.\n\nWe have thus obtained some glimpse of the THEN of our educational\nfacilities of a generation ago. The picture might be elaborated.\nIt would be easy to fill in details from memory; to tell how the\nblind oft times led the blind; how the ignorant teacher insured the\nignorant pupil; and how \"schooling\" was frequently a farce, and mere\nwaste of time....\n\n\nGREAT EDUCATIONAL ADVANCE MADE BY THE PROVINCE OF ONTARIO SINCE 1847.\n\nThat the Province has made enormous strides in population, wealth,\nintelligence and importance, during the last thirty years, admits\nof no doubt. Our forests have disappeared, an improved system of\nagriculture has followed, manufactories have sprung up, railways\nhave connected every county, a daily press has become an established\nand indispensable institution, the telegraph has economized time\nby practically annihilating distance, while numerous inventions\nand discoveries have created new wants, and supplied as rapidly\nas they have made them. Without losing our characteristic love\nof hard work--I here speak of everybody in general, and nobody\nin particular, and purposely avoid all personal allusions--and\nthat industrial enterprise which springs from it, we have become\na reading and much more cultured people. The scholar, endowed\nwith physical capacity equal to that possessed by an illiterate\ncompetitor, is worth more than he in the factory, the workshop,\nthe store, the mill, the mine, or on the sea or farm. Cultivated\nbrain has a market value, and book learning is no longer despised,\nor regarded with half contempt, as the mark distinguishing the\nmere dreamer from the worker. To possess the \"Reason Why\" is no\nproof now-a-days of physical and practical inferiority; to know\na little of everything, and everything of something, is not now\nthe peculiar privilege of the English Gentleman. Little wonder is\nthere, therefore, that what the school has helped to bring about,\nshould tend to make the school more valued. That such has been its\neffect we have but to look around to see. Where in 1847, we, in\nUpper Canada, had 2,863 school houses, our last returns show that\nwe possess more than 5,000, and while the number has so largely\nincreased the advance in value has been in much greater proportion.\nIn 1847 we had, in all Upper Canada, but 49 school-houses built of\nbrick; now we boast of 1,569 built of that material, or over thirty\ntimes as many. In 1847 we had eighty-four constructed of stone;\nnow we claim more than 500. In 1847 half of our school buildings\nwere of logs; now not more than a seventh are of that primitive\ncharacter. There are no returns of money cost of buildings or of\namount expended in their erection in 1847, but we find that the\nexpenditure for all school purposes in that year, inclusive of\nteachers' salaries, was $350,000, while for 1877, for erection and\nrepairs of school-houses, fuel, etc., alone we paid $1,035,390, and\na total for school purposes of $3,073,489, or, in round numbers,\nnine times as much as in 1847. The improved financial value of the\nteacher is another strong testimony, willingly borne by the people,\nto their increased interest in education, for, as a rule, a free\npeople will not pay for that which they fail to appreciate. In 1847\nthere was paid for teachers' salaries a total sum of $310,398. In\n1877 the amount was $2,038,099. In 1847 there were 3,028 teachers\nemployed, while in 1877 there were 6,468. In 1847, board was often\ngiven in addition to the nominal salary, and was, in fact, part of\nthe teacher's remuneration. Grant that the teachers here enumerated\nas serving in 1847 were employed eight months in that year--which\nis more than the average--and put board at $2 per week, which was\nhigher than was the average rate in those days--the average payment\nto each teacher would not exceed $170, and this was fully equal to,\nif not greater than was actually allowed. In 1877 the average amount\npaid to each teacher was $315. The larger amount willingly paid in\n1877 for the support of Free Schools, than was unwillingly given in\n1847, for the maintenance of rate-supported schools--for payment\nwas then made under protest, and the school law was exceedingly\nunpopular, while rate-bills and contributions were nearly everywhere\nnecessary, in addition to municipal assessments, to make up the\nteachers' salaries--is yet another proof of the hold which the\neducational movement has taken upon the judgment and sympathy of the\npeople of Ontario. In 1847, too, pupils were grudgingly taught, at\na cost of $2.80 per head, while in 1877 the average was $6.20. And\nwhen we add to all these things the fact that, in 1847, only 124,829\npupils attended our common schools, out of a school population of\n230,975, or scarcely one in two, while in 1877, out of a school\npopulation of 494,804, not less than 490,860 names were entered on\nthe roll, it is needless to say anything further in illustration\nof the marked contrast between the two periods, of the immense\nsuperiority of the present over the past condition of our schools,\nand of the public opinion which is necessary to their effective\nmaintenance.\n\n\nGREAT ADVANCE ALSO IN THE STANDARD OF TEACHING ABILITY.\n\nAnd the standard of teaching ability, in so far as literary\nacquirements go, has kept pace with the progress which has\notherwise characterized the history of a scholastic generation.\nWe have long got past the period when any two magistrates, any\ntwo reeves, or even any two clergymen, could grant permission\nto teach, and annually invest the teacher with legal status. We\nsubject our examiners themselves to examinations, have uniformity\nin the character of our examination papers, and propound questions\nto candidates which fully and fairly test their educational\nattainments. We have gone beyond _that_, and instituted county\nNormal Schools--for such our Model Schools may be fairly termed--at\nwhich we require applicants for a certificate to still further\nestablish their fitness for the work upon which they seek to\nenter. We have not reached perfection, but we have travelled a\nlong distance in the direction in which it lies. We have made\nevery school practically free, built up a High School system which\nopens up to all seekers after higher education ample opportunity\nto prepare for the University course, at a minimum of cost, and\nplaced our University upon such a footing that its advantages are\nnot the exclusive privilege of the well-to-do, but are proffered\nto even the poorest student who cares to submit to a period of\nself-denial, and lose a little extra time in early life, for the\npurpose of securing them. As a people we have done no more than,\nprobably not so much as, we ought to do, with the view of placing\neducational facilities within the reach of every child born or\nbrought into the Province, but we have, nevertheless, ventured\nand effected more than has been attempted in many older and more\nwealthy lands. We have the consciousness of having done our duty,\naccording to our lights. In our long-settled sections of country\nthe school-house bell is within the hearing, or the school-house\nitself is within sight of nearly every family. In newer portions\nof the Province, wherever half-a-dozen or so of clearances are\ncommenced, in the wilds of Muskoka or Algoma, provision is made for\nthe instruction of the little ones who bless those backwoods' homes.\nThe school-master is abroad throughout the land, and is doing much\nto ensure a glowing future for our country, and when his work is\ndone, and he is compelled to retire from his labors, we willingly\nopen the public purse and give to him that which keeps him above\nabsolute penury, and assures him that, while Ontario cares only to\nhelp those who possess the disposition to help themselves, she is\nneither ungrateful nor forgetful. And, seeing all these things, we\ncannot help feeling that our youthful Province may modestly and yet\nproudly lift her head amongst the nations of the earth, assured that\nthere are none who can reproach her with neglect of the first and\nbest interests of those little children whom God has entrusted to\nher keeping.\n\n\nSTATE OF EDUCATION IN UPPER CANADA IN 1847-1849.\n\nFrom an elaborate report prepared by the Rev. W. H. Landon, School\nSuperintendent, to the Municipal Council of the district of Brock\n(county of Oxford) in 1849, I make the following extracts. The first\nrefers to the educational supineness of the people:--\n\n\"Up to a recent period (say the last two years), the people,\ngenerally, seems to have entered upon no enquiries, and to have\nformed no just conclusion on the subject of education, or the proper\nmeans of imparting it. They seemed to think, if they thought at\nall, that all schools were equal, and that all teachers who could\nread, write, etc., in a better manner than their pupils were equally\ngood.... As to books, it was supposed that any one, or any ten of\nthe fifty different varieties of spelling books in use, with the\nEnglish Reader, was all that was requisite for the reading classes,\nwhile a few treatises on arithmetic taken at random from the almost\nendless variety with which the country was flooded, would supply the\nmeans of imparting & knowledge of the science of numbers, and two or\nthree grammars by as many different authors, would supply material\nfor the grammar class and complete the stock of text books for the\nschools.\"...\n\nMr. Landon draws the following graphic picture of the school-house\n\"shanties\" of those days. He also gives a vivid view of the interior:\n\n\"The school-houses in many instances (though not in all) are\nmiserable shanties made of logs loosely and roughly put together;\nthe interstices filled with clay, portions of which are from time\nto time crumbling down filling the place with filth and dust. Under\nyour feet are loose boards, without nails, across which, when one\nwalks, a clatter is produced equal to that heard in a lumber yard.\nOver your head are the naked rafters, stained with smoke and hung\nwith cobwebs and dust. Two or three little windows, generally half\nway up the walls, admit the light; and a rough door which does not\nfit the opening, creaks upon its wooden hinges.... The writing\ndesks are generally long sloping shelves pinned up against the\nwalls as high as the breasts of the pupils who sit before them. The\nseats are without backs and from eighteen inches to two feet high.\nSometimes we have a master's desk, but awkwardly constructed, for\nthe most part too high for the sitting posture and too low for the\nstanding one.... We have no blackboards, no maps and no illustrative\napparatus of any kind.\n\n\"When we enter one of these schools, we behold a picture of\ndiscomfort and misery. The children are perched upon the benches\nbefore described: but as they have no support for their backs, and\nas only the taller of them can reach the floor with their feet,\nmarks of weariness and pain are visible in their features and\npostures. Some to procure rest and ease to their aching frames have\ndrawn up both feet upon the bench and are sitting cross-legged like\na tailor on the shop board. Others stooping forward, rest their\nelbows upon their knees, with one hand supporting their chins and\nwith the other holding up their books before their weary eyes;\nwhile all avail themselves of every possible excuse to change their\nposition and so obtain relief. Some asking permission to go out,\nothers to get a drink, and many constantly flocking to the teacher's\ndesk with words to be pronounced, sums to be examined and corrected,\npens to be mended, or difficulties to be explained, in connection\nwith grammar, or other lesson, etc. So that the place is filled with\nnoise and disorder, rendering study impossible and anything like the\ncultivation of cheerful and benevolent affections entirely out of\nthe question.\"...\n\nThen follows an example of the character of the teaching \"in a\nschool in the centre of one of the largest and wealthiest townships\"\nin the district of Brock:--\n\n\"This school was taught by a person who in his youth had enjoyed\nwhat we term superior advantages, being connected with a family of\nhighest respectability. Notice of my intended visit had several\ndays before been sent to the teacher. The female pupils had ...\ndecorated the place with evergreens and bouquets of flowers. The\nroom, though humble and coarse, was neat and tidy. When I entered,\nthe class in the fourth book of lessons was reading. A book was\nput into my hands and I desired them to proceed.... When they\nwere done reading I proceeded to examine them on the lesson.\nGreat Britain was mentioned in the lesson and allusion made to her\npeople and institutions. My first question, therefore was--Where\nis Great Britain? From the vacant and surprised stare with which\nthis question was received I was satisfied that they had no clear\nconception of what Great Britain was.... I finally asked what is the\nform of Government in Great Britain? As no answer was given, I ...\nasked whether a King, Queen or President governed in Great Britain?\nTo this question a pupil, aided by the teacher, who whispered in her\near, replied a Queen. I than asked her name.... After a good deal\nof hesitation, a young woman of eighteen or twenty years of age,\nreplied \"Queen Elizabeth!\"\n\n\nTHE OLD LOG SCHOOL HOUSE AND ITS BELONGINGS.\n\nIn connection with the realistic picture of education in the County\nof Oxford, in 1847-49, sketched by the Rev. W. H. Landon, District\nSuperintendent, the following dual pictures of \"The Old Log School\"\nand \"The Pioneer Teachers,\" taken from the Toronto _Globe_ of 1887,\nwill be found to be highly interesting. The pictures are graphically\ndrawn by a teacher, and from a teacher's standpoint. Speaking of the\nrepresentative teacher of a former generation recalling the past, he\nsaid:--\n\nThe old days come up vividly before him, when he first engaged\nin the work in some country district, engaging to devote to it\nthe best energies of body and mind, for, it may be, some such\nmunificent salary as eight or ten dollars a month, said salary\nto be supplemented by the saving in expense effected through the\nprocess formerly so much in vogue of boarding around. How well he\nremembers the old log school house, with its low ceiling on which\na tall man could easily lay his hand; the narrow apertures, fitted\nwith a few panes of 7\u00d79 glass which served for windows; the floor\nof unplaned boards, whose crevices were either compactly filled\nwith accumulations of dust and litter characteristic of the school\nroom, or worse still, yawning to swallow up pen-knife and slate\npencil as they would ever and anon drop from the fingers of some\nluckless wight, started from the half slumber into which the drowsy\nmonotony of the ill-ventilated school room had beguiled him, by the\nstentorian tones, or possibly the vigorous cuffs, of the master of\nceremonies. Very distinctly the vision of such a school room of the\nold type, though at a date much less than fifty years ago, rises\nbefore the writer as memory carries him back to the little Canadian\nhamlet in which his boyhood was passed. The desks, so as far any\nwere provided, consisted of a wide shelf fixed at a pretty sharp\nangle against the wall, and extending all around the room, with an\nintermission only at the narrow space occupied by the door. This\nprimitive arrangement was sometimes supplemented with a long, flat\ntable composed of three or four loose planks in the rough, supported\nby wooden benches or horses placed transversely beneath. The seats\nwere of planks or slabs, likewise unsmoothed, constructed by driving\nrudely hewn legs into holes bored with a large augur, at a suitable\nangle, in the lower surface of the plank or slab. These legs often\nprojected an inch or so above the surface of the seat. It could\nnot be said of these rude structures as in Cowper's \"Evolution of\nthe sofa,\" that \"the slippery seat betrayed the sliding part that\npressed it,\" for between the projecting legs and the innumerable\n\"splinters\" the unhappy occupant was in much greater danger of being\nimpaled and pinned fast than of slipping off. Perhaps it was better\nso, for in view of the great height usually given them, the fall,\nfor a small child, while it would most surely have been a \"laughing\nmatter,\" might yet have proved a serious one.... It was certainly a\nstrange and cruel infatuation which constrained our grandfathers to\nthink that the proper position for a boy or girl at school was upon\na narrow perch, without back or arm support of any kind, and with\nthe feet dangling some six or eight inches above the floor.\n\nThe picture of the old school bench would not be accurate without\nreference to the warping of the plank which was pretty sure soon to\ntake place, with the result of raising one or other of the diagonal\nlegs an inch or two from the floor, thus converting the seat,\nwhen filled with its living, aching load, into a tilting board,\nprovocation of many a trick from the omnipresent mischievous boy of\nthe school, and resulting in many a blow from the palm or cane of\nthe irate master, which would, of course, generally descend upon\ninnocent ears or shoulders.\n\nWhat a picture did the wooden desks and walls of those old-time\nschool houses present, worn smooth with the elbows, smeared with\nthe jackets, variegated with the ink, carved with the jack-knives\nand stained with the tears of boisterous and blubbering boys and\nrosy-cheeked, hoydenish or timid girls. What burlesque, too, upon\nevery intelligent idea of education were the processes carried on in\nthem. From nine o'clock to twelve, and from one till four, six long\nhours, as marked by the sun's shadow on the rude dial marked out on\nthe windowsill, did the work go on.\n\nMurray's Reader, and in the most ambitious districts his Grammar,\nWalker's Dictionary, Walkingame's Arithmetic, Goldsmith's Geography\nand somebody's spelling-book, with slate and pencil, a scanty supply\nof paper and ink, and pen shaped with a keen pen-knife from a quill\npicked up from the wayside or plucked ruthlessly from the wing of\nsome reluctant \"squawking\" goose, would complete the scholar's\noutfit. It is the hour for preparation of the reading lesson. The\nschool room resounds with the loud hum of a score or two of boys\nand girls all \"studying aloud\" with a most distracting din, and all\nthe heads and bodies swaying constantly and simultaneously back and\nforth as an accompaniment to the voice. This voice in the case of\nperhaps a majority would be modulated without the slightest relation\nto the contents of the printed page, while the thoughts of the\nostentatiously industrious student would be busy with some projected\ngame or trick for the coming recess. And yet how often would the\nScotch school master's eye gleam with pride and pleasure when he\nhad, by dint of persuasion or threat, succeeded in getting every boy\nand girl engaged in the horrible, monotonous chant.\n\nThen the recitation--what a scene of confusion and stripes, tears\nand bellowings. Perhaps it was the column of spellings. A few,\nfitted by nature with memories adapted for that kind of work, would\nmake their way in triumph to the head of the long semicircular\nclass. But woe, woe to the dullards and the dunces, under a regime\nwhose penalty for missing a word a foot and a half long would\nbe, very likely, two or three strokes on the tingling fingers or\naching palm with the pitiless hardwood ferule, this process being\noccasionally varied as some noisy or idling youngster was called up\nfrom a back seat to be visited with a still sterner chastisement\nfor some trifling misdemeanor.... The writer can recall instances\nand experiences innumerable, the infliction being sometimes\naccompanied with a caution to tell no tales at home under pain of\na worse infliction. In his own case he well remembers the wrath\nof his father, who would have thought it wrong and encouraging\ninsubordination to listen to any complaints against the master,\nwhen, on occasion of the victim of a tendency to juvenile pranks\nbeing dangerously ill with scarlet fever, and that father being\ncalled on by the doctor to annoint his back with some soothing\nlotion, he found said back striped and checked with a network of\n\"black and blue.\" It is needless to add that at this point ended\nboth the writer's experience under that schoolmaster and the\nschoolmaster's term of engagement in that district.\n\nAs a significant comment upon the moral effects of the regime of\nthe schoolmasters of the old school the writer may add that one of\nhis most vivid memories of the mental status produced by the school\ntraining referred to is that of an intense longing for the day when\nhe should be large enough to repay that old schoolmaster in his\nown coin. That day came. The flagellated boy, transformed into a\ntolerably lusty youth, found himself face to face with his quondam\ntormenter. But his long cherished wrath speedily gave place to pity\nfor the decrepit, friendless and lonely old bachelor, whose days\nwere drawing to a close, with no loving hand of wife or daughter to\nminister to their feebleness.\n\n\nTHE PIONEER TEACHERS, AND THE TRIALS OF \"BOARDING ROUND.\"\n\nThe writer of the foregoing paper pictured roughly the rural\nCanadian school of forty or fifty years ago. It may not be without\ninterest to have that picture supplemented with a glimpse of\nrural Canadian life as seen by the schoolmaster of the period.\nThe \"boarding 'round\" system afforded him excellent facilities\nfor observation. The venerable custom of boarding round died, no\ndoubt, a good many years ago, so far as Canada is concerned....\nBut thirty or forty years ago it was, in some parts of Canada at\nleast, almost a matter of course that the teacher should \"board\n'round.\" When a school became vacant or a new and ambitious\nsettlement had reached the pitch of development at which a school\nwas deemed a necessary sequent to the carpenter's shop, the smithy\nand the shoemaker's shanty, one of the first steps was, of course,\nto pitch upon a suitable candidate for the scanty honors and still\nmore scanty emoluments of the village pedagogue. Probably some\ninfluential member of the community had a son or a daughter in the\nteens, who was thought pretty well up in \"the three R's.\" If so, it\nwould usually be deemed quite unnecessary to look farther. In fact\nit would, in such a case, be useless for an outsider to contest\nthe constituency. It never entered the unsophisticated heads of\nthe trustees to invite competition by advertising for candidates\n\"to state salary expected.\" This method of putting up professional\ntalent in a kind of Dutch auction is an evolution of our present\n\"best educational system on earth.\" Our grandfathers went about the\nbusiness in a different way. The coming teacher being fixed upon,\nthe next step was for the candidate himself, or some interested\nrelative or friend on his behalf, to circulate a subscription sheet.\nA form of heading would be prepared somewhat in the following\nstyle:--\"We, the undersigned residents of Smithton District, being\ndesirous of securing the services of Henry Schoolman as teacher of\nthe district school, hereby agree to engage the services of said\nH. S. for the period of six months, and to pay him at the rate of\n\u00a31 2s 6d for each and every pupil we hereunto subscribe or send\nto said school. We further agree to supply the teacher with board\nin proportion to our several subscriptions; also to furnish our\nproportions of wood for the use of said school. Signed, etc.\"\n\nThe average juvenile Canadian made, no doubt, a much more merciful,\nand often much more efficient, teacher than the ex-soldier, or\nbroken-down tradesman from the Old Country. One of the first\nduties of the newly-fledged teacher would be to go carefully over\nhis treasured list of subscribers and ascertain, by a careful\narithmetical calculation, the exact number of weeks and days for\nwhich he was entitled to board and lodging at the house of each\nof his respective patrons. The next step would be to find out,\nby personal or written enquiries, at what time it would be most\nconvenient for each family to open its doors to him. This process,\nand the subsequent installation in each home would, it may well be\nimagined, be trying ordeals to the young and bashful pedagogue....\nThe receptions accorded the poor itinerant would be, of course,\nas various as the feelings, dispositions and circumstances of\nthe householders, or, more strictly speaking, of the presiding\ndivinities of the parlor and the kitchen, especially the kitchen.\nIn many cases he would quickly feel at home. The welcome would be\ncordial, the hospitality ungrudging, the companionship agreeable.\nIn such cases the bashful beginner would soon be able to shake off\nthe intolerably humiliating dread of being regarded as an intruder,\nan interloper, or half-mendicant. But in numerous instances, as\nmay readily be conceived, the situation would be most galling to a\nsensitive nature. The over-worked, perpetually tired and fretful\nmistress of the house would receive him with an ill-concealed frown\nor an involuntary sigh. To her he represented just so much addition,\nfor so long a time, to her hourly toils and cares, already too heavy\nto be borne.\n\nUnwashed specimens of \"the heritage of the poor\" would swarm in\nevery corner. The fear and awe which secured him immunity for a time\nwould soon wear away, and as they were replaced with the familiarity\nthat breeds contempt, he would be exposed to all manner of\nwell-meant advances and indignities. The scorn at the roughly-spread\nsupper table would be a scramble, and the twilight hour, which, if\nhe happened to have a spice of romance in his composition, he would\nfain have consecrated to quiet thought or fancy, would be made\nhideous by a juvenile pandemonium, as amidst stripes and cuffs and\nyells and tears the unruly flock would finally be got to bed. These,\nof course, were the ill-regulated householders, but they exist.\n\nWell does the writer remember some personal experiences in this\ndelightful phase of the professional life of an earlier day--not\nmuch more than half a semi-centennial distant. The old, dingy\nfarmhouse, the bare floors, the hard seats, the utter absence of\neverything in the shape of books or other literature, the teeming\nolive branches at every age and stage of development, the little\n\"spare\" bedroom, whose sole furnishings consisted of the bed and\nbedding, on whose hard floor he reclined for lack of chair and table\nevening after evening for hours after he was supposed to be in bed,\nreading by the feeble rays of a tallow candle the ponderous volumes\nof Dr. Dick's philosophies, which had been kindly loaned him by a\nfriend, and which were devoured with an eagerness begotten of a\ngenuine hunger, though out of all proportion to the literary merits\nof the works.\n\n\nTHE OLD SCHOOL HOUSE.\n\nRude and unfinished and uncomfortable as it was, \"The Old School\nHouse\" would be sure to bring up to many an \"old boy\" tender\nmemories, which would be recalled in after years in words somewhat\nlike those in poetic form as follows:\n\n    It stood on a bleak country corner,\n      The houses were distant and few;\n    A meadow lay back in the distance,\n      Beyond rose the hills to our view.\n    The road, crossing there at right angles,\n      Untraversed by pomp and array;\n    Were cropped by the cows in the summer,\n      I've watched them there many a day.\n\n    In memory's hall hangs the picture,\n      And years of sad care are between;\n    It hangs with a beautiful gilding,\n      And well do I love it, I ween.\n    It stood on a bleak country corner,\n      But boyhood's young heart made it warm;\n    It glowed in the sunshine of summer,\n      'Twas cheerful in winter and storm.\n\n    The teacher, O well I remember,\n      My heart has long kept him in place;\n    Perhaps by the world he's forgotten,\n      His memory no touch can efface.\n    He met us with smiles on the threshold,\n      And in that rude temple of art,\n    He left, with the skill of a workman,\n      His touch on the mind and the heart.\n\n    Oh! gay were the sports of the noontide,\n      When winter winds frolicked with snow;\n    We laughed at the freaks of the storm-king,\n      And shouted him on all aglow.\n    We flashed at his beautiful sculpture,\n      Regardless of all its array;\n    We plunged in the feathery snow-drifts,\n      And sported the winter away.\n\n    We sat on the old-fashioned benches,\n      Beguiled with our pencil and slate;\n    We thought of the opening future,\n      And dreamed of our manhood's estate.\n    I cast a fond glance o'er the meadow,\n      The hills just behind it I see;\n    Away in the charm of the distance,\n      Old school house! a blessing on thee!\n\n_Mr. Canniff Haight_, in Canada of \"Fifty Years Ago,\" gives the\nfollowing account of the common school education of his day:--\"The\nschoolhouse was close at hand, and its aspect is deeply graven in\nmy memory. It was a small, square structure, with low ceiling. In\nthe centre of the room was a box stove, around which the long wooden\nbenches, without backs, were ranged. Next the wall were the desks,\nraised a little from the floor. In the summer time the pupils were\nall of tender years, the elder ones being kept at home to help with\nthe work. I was one of a lot of little urchins ranged daily on hard\nwooden seats, with our feet dangling in the air for seven or eight\nhours a day. In such a plight we were expected to be very good\nchildren, to make no noise, and to learn our lessons. It is a marvel\nthat so many years had to elapse before parents and teachers could\nbe brought to see that keeping children in such a position for so\nmany hours was an act of great cruelty. The terror of the rod was\nthe only thing that could keep us still, and at that often failed.\nSometimes, tired and weary, we fell asleep and tumbled off the\nbench, to be awakened by the fall of the rod. In the winter time,\nthe small school was filled to overflowing with the larger boys and\ngirls. This did not improve our condition, for we were more closely\npacked together, and were either shivering with the cold or being\nroasted with a red-hot stove.... I next sat under the rod of an\nIrish pedagogue--an old man who evidently believed that the only way\nto get anything into a boy's head was to pound it in with a stick\nthrough his back. There was no discipline, and the noise we made\nseemed to rival a bedlam.--_pp. 17, 18._\n\n\"As far as my recollection goes, the teachers were generally of a\nvery inferior order, and rarely possessed more than a smattering\nof the rudiments of grammar and arithmetic. They were poorly paid,\nand \"boarded round\" the neighbourhood. But it is not improbable\nthat they generally received all that their services were worth....\nThe school-houses where the youth were taught were in keeping\nwith the extent of instruction received within them. They were\ninvariably small, with low ceilings, badly lighted, and without\nventilation.\"--_pp. 157, 158._\n\n\nA SCHOOL TEACHER'S PERSONAL EXPERIENCE IN 1865.\n\nFrom the Toronto _Mail_ of November 28th, 1888, I select the\nfollowing graphic account of the personal experience of a school\nteacher in 1865:--\n\n\"'Yes,\" said an old teacher to a representative of the _Mail_\nyesterday, \"education has been wonderfully revolutionized in Ontario\nduring the last twenty-five years. It was in January, 1865, that I\nfirst took up the birch, swaying it until I cautiously made up my\nmind to quit when the new Act requiring higher qualifications came\ninto force in 1871. At that time the school houses in my county\nwere all constructed of logs, and more uninviting buildings than\nthese were not to be found in the country. It did appear that the\nratepayers were more led to educate their children out of a feeling\nof latent and legal compulsion, than out of duty and parental\nregard. The life of a teacher in those days was not the high-toned\none of to-day. Let me give my own experience, and when I have\nrelated it you will see how much there is in the complaints and\ngrumblings of the existing generation of school teachers.\n\n\"'A school became vacant in the neighboring township, and I made up\nmy mind, armed as I was with a first-class certificate awarded me by\nthe County Board of Examaminers, that I would apply for the position.\n\n\"'I went to two influential men in the neighborhood and succeeded in\ncoaxing them to go with me to the trustees of the school. We arrived\nat the section in due time, and after making due enquiries proceeded\nto the house of one of the trustees, who had the reputation of\nknowing everything worth discovering in the school law of the\nperiod. I felt an awful dread and confusion come over me when in\nthe presence of that trustee. I was introduced, and I immediately\ntold my errand. The horny-handed son of toil gave me one of those\ninscrutable looks that nearly sunk me to the earth. He coughed\nslightly, jerked his head back, put his two hands in the pockets of\nhis trousers, and immediately proceeded to business.'\n\n\"'So yous wants the school, does you?'\n\n\"'I do, sir.'\n\n\"'Well, I might as well tell ye at once that the teacher we intend\nhiring must be better than the present one. He is a curse to the\nchildren of this section, with his grammar and his jography, and all\nhis other fal da rals. Why, sir; my son Bill comes home the other\nnight and says he,\n\n\"'Father, what is grammar?'\n\n\"I says, Bill, I never studied grammar, and you see how I am able to\nget along without it. Grammar is no good for ploughing or cutting up\nthat slash fornent the house.\"\n\n\"Well,\" says the boy, \"would you tell me what our teacher meant by\nsaying that Berlin is on the Spree?\" I then got mad, and says I,\n\"Bill, never let me hear ye say anything more about these things.\nSure they were never taught to us from the New Testament when I was\nin school.\"\n\nBut that is not all, the boy began to say to himself, orthography,\netymology, syntax, and prosody, going over them again and again,\nuntil I says to Moriah, \"is the child getting mad, wife!\"\n\nNo sooner had he heard that than he began to say, \"Noun,\nadjective, pronoun, adverb, preposition, noun, adjective, pronoun,\npreposition,\" as before. I then got afeard the child was clean\ncrazy, and when I was in the act of rising to catch hold of him,\nsays he, \"Father, the earth is not flat, it is a round ball and goes\nround.\"\n\n\"Och, och,\" says I, \"that spalpeen of a school master has driven my\nson mad.\"...\n\nSo I calls a meeting, and my two colleages, Thomas Ginty and Edward\nCrawford, and myself, met at the school house and dismissed the\nrascal.\n\nWhen the leader of education of school section No. 5 ---- stopped,\nI turned to my companions, not knowing what qualifications were\nexpected of me.\n\nThe trustee continued:--\"Can you read?\" \"Yes.\" \"Can you cipher?\"\n\"Yes.\" \"Do you know how to count by long division?\" \"I do.\" \"You\ndon't know grammar?\"\n\nHere was the crucial test. I resolved, however, to get out of it the\nbest way possible, and replied:--\"I don't like grammar, and don't\nknow much about it,\" which was true. \"Good.\"\n\nThe trustee smiled sweetly and said:--\"The school is yours; but,\nremember, no grammar or jography, or out you go.\"\n\nI was taken to the other trustees, and in their presence I was put\nthrough the same examination as above recorded.\n\nThe bargain was struck there and then. I was promised fifteen\ndollars a month board from house to house, with the condition that\nI should put on the fires in winter, and keep the school house\nclean--a herculean task in those days.\n\nOn the morning that I was to assume my pedagogic duties, I arrived\nat the school house about 8 a.m., and at once proceeded to build a\nfire. I never can forget the feeling of utter loneliness which came\nover me, as I stood opposite a fireplace wide enough to accommodate\nan ox. On the walls of the building were three maps, of what\ncountries I do not now remember, because geography was proscribed\nduring my _regime_. Daylight was seen through the walls. The seats\nwere of the most primitive character, while my own desk resembled\nmore the top of a toboggan slide than the rostrum of one who was\nteaching the young idea how to shoot. There was no fence round the\nbuilding, the play ground being illimitable in its dimensions and\ncapacities. In short the whole of the surroundings of this rural\nacademy congealed all literary aspirations, and it was little wonder\nthat the boys and girls meeting there, and pretending to be drinking\nat the fountain of knowledge, grew up utterly destitute of the first\nprinciples of a rational and suitable education.\n\nThe children arrived, and I immediately began to assert my authority\nas the head of the institution. I made a short speech, telling the\nchildren that nothing would be taught in my school but reading,\nwriting and 'rithmetic, adding that the ten commandments would be\nincluded in the curriculum. I got along famously that day with the\npupils, and as I was told afterwards, when my charges went home\nand told their parents the new order of things, I was universally\npronounced the greatest teacher of the age. I lodged that night with\nTrustee Fallis, my examiner. He was delighted with the reports given\nby Bill of my mode of teaching, and in a moment of confidence told\nme that the school was mine as long as I liked to keep it. I humbly\nthanked him, and retired to my room, where I found Bill, the heir\nof the house, in innocent slumber. Bill did not by any means prove a\npleasant bed-fellow. Occasionally his feet were found where his head\nshould be, and he repeatedly called out, \"No grammar! Hurrah!\" Next\nmorning I rose, and after breakfast majestically strode through two\nfeet of snow to the log school house, and then put in another day's\ngrinding.\n\nAt four o'clock the trustees met and handed me a badly written and\nworse spelled manuscript, which on perusal I found to be the list\nof houses I was to board at during the coming three months.... Now,\nsir, continued the dominie, I put in a whole year of that kind\nof work; listened to all the gossip of the neighborhood, slept\noccasionally on the floor, nursed all the babies of the section, and\njust as I thought that my position was secure, disaster overcame me.\nOne of the trustees had a daughter of prepossessing appearance, and\nI could not resist the temptation of falling in love with her. When\nthe news got abroad that I was paying attentions to her, the section\nbegan to talk adversely of myself. Complaints were soon heard as to\nmy teaching. Some said that I was teaching grammar on the sly, and\nthat I was partial to some of the children. The result of it all was\nthat a meeting of the board took place, and, despite the protests of\nThomas Ginty, I was, in the language of modern times, \"fired.\"\n\n\"Now,\" continued the old teacher, \"the moral of what I told you is\nthis:--Compare the log school house, its internal arrangements and\nplay grounds, with those of to-day throughout the whole country, and\nyou have the greatest exemplification possible of the progress made\nby Ontario during the last quarter of a century. There is scarcely\na section now but has its brick school house, with surroundings\nattractive and neat. The teacher is given ample opportunity to\ncultivate the minds of his pupils to the highest pitch possible,\nat least as far as requisite for the general affairs of life. The\nold prejudices against teaching anything else but the three R's are\ngone. You have the walls of the buildings ornamented with maps,\ncharts and other appliances. Text-books are provided of the most\nmodern character, while a rigid system of examinations acts as\nan incentive to the teacher and makes him an instructor in fact\nand reality. Lastly, we have an intelligent and cultured class of\nteachers, and taking all things into consideration, they are well\npaid, and still better, their services are year by year becoming\nmore appreciated. A better class of people now act in the capacity\nof trustees, and as years roll by a still more intelligent class\nwill be found in these positions.\n\n\nREMINISCENCES OF EDUCATION IN THE CITY OF HAMILTON IN 1852.\n\nIn conversation with a Reporter of the _Hamilton Times_ in 1888, Mr.\nJames Cummings, who, after thirty-seven years service on the City\nBoard of Education, thus related his early educational experience:--\n\n\"At the time I first became connected with the board in 1852,\"\nsaid Mr. Cummings, \"education was at rather a low ebb, not only in\nHamilton, but throughout the country. Dr. Ryerson had just commenced\nhis excellent common school scheme and in 1847, had established a\ntraining school for teachers in Toronto. Hamilton had decided to\nadopt the common school system and the reorganization of the schools\nwas just being effected. Previous to that time the education of\nthe children of the better class was almost wholly in the hands of\nprivate school teachers. There were a few small school rooms in\nthe different wards, but they were wretched places, where usually\nfifty children were crowded into a small room reeking with fetid\natmosphere. On the site of the present Cannon street school was a\nsmall frame grammar school, at which many of the present leading\ncitizens received the ground-work of their education. A great reform\nwas inaugurated here in the year 1852. Under the new Ryerson Act\nof 1850 trustees were clothed with new powers, and we immediately\nproceeded to reorganize the schools here on the new plan. Tenders\nwere let for the erection of the present Central school and for ward\nschools in different parts of the city. Mr. (now Dr. J. H.) Sangster\nwas appointed head master, and to him was entrusted the appointment\nof teachers, and he was made responsible for the efficiency of\nthe schools. He had been a teacher in the Ryerson model school in\nToronto, and he appointed as his assistants thoroughly trained\ngraduates of that excellent institution. This was the founding of\nHamilton's excellent common school system. It kept improving from\nyear to year and soon closed out the private schools. Larger ward\nschools were erected, and since that day Hamilton has taken a high\nposition educationally in the Province.\"\n\n\nEDUCATION IN THE COUNTY OF SIMCOE, 1852-1872.\n\nDr. Ryerson was invited to open the new Public School in the town of\nBarrie in 1872. Before the ceremony was proceeded with a letter was\nread from Judge (now the Hon. Senator) Gowan, who was unable to be\npresent, referring to his long services in the cause of education in\nthe County of Simcoe. He said:--\n\n\"Ever since I came to this country, nearly thirty years ago, I\nhave been connected with the school system, having held the office\nof Trustee of the Grammar School, and the position of Chairman of\nthe Board of Public Instruction from its first institution till\nsuperseded by recent enactment, and, with the exception of my\nfriend, Mr. Dallas, I am the only member of the original Board now\nliving.\n\n\"I have seen the gradual improvement in the school system, and\nthe improvement in the schools in this country from very small\nbeginnings to the present advanced and most prosperous condition,\nso you will understand my disappointment in not being able to be\npresent on the interesting occasion of laying the corner-stone of\nthe Public School house of Barrie, by the Chief Superintendent of\nEducation.\n\n\"My position as Secretary and Treasurer of the Grammar School,\nand Chairman of the Board of Public Instruction, in this, the\nlargest county in Ontario, brought me in constant communication\nwith the Education Office in Toronto; and I can say that the able,\nzealous, and wise administration of the school law by Dr. Ryerson\nand his assistant, Dr. Hodgins, has, here at least, had a happy\neffect--fostering the increase of schools, securing their better\nmanagement, giving them efficient teachers, and providing the means,\nwithin easy access to all, of securing a good common education to\nthe youth of this country, and a very superior education in the\nGrammar Schools.\"\n\nMr. (now Judge) Boys, gave a sketch of the educational history of\nBarrie as follows:\n\n\"Twenty years ago there was no Public or Common School, not,\nhowever, without school accommodation, as we were then included in\nwhat was known as School Section No. 1 of the adjoining Township of\nVespra. We had no building specially set apart as a school house,\nbut a rented room then sufficed to carry on the daily teaching\nembraced within the section.... Twenty years ago one teacher took\ncharge of all our scholars--both male and female--and if there is\nany doubt as to his labor having been great, there can be none as to\nhis salary having been small, for he subsisted on a sum of \u00a360 per\nannum.\n\n\"In January, 1854, Barrie became possessed of a school of its own,\nand built a school house of frame 24 \u00d7 36, just about large enough\nto fill up one room in the building we are now erecting. It was,\nno doubt, at the time it was built, amply large, yet I find, from\nthe record of the school, that such was the growth of the town\nby September, 1854, non-residents were refused admittance to the\nBarrie school on the ground of its over-crowded state, the average\nattendance of males being seventy--the females were then taught\nin another building by a female teacher. This state of things\ncontinued for nearly a year, when a separate school was established\nfor Barrie, which brought some relief to the over-crowded building.\nBut it was evident that more school accommodation would have to\nbe supplied, and I see by the minute book of the school, that a\nnew school house was talked of so far back as January, 1855. The\nnew school house, however, never came. The difficulty at last was\nsettled by an enlargement of the old building, which then assumed\nthe appearance it now presents. With the enlarged school house,\nsupplemented by some rented rooms, the schools of Barrie have ever\nsince continued to the present time. It took time to convince our\npeople of the imperative necessity there was for a large outlay in\nproviding a new school house. But the ratepayers became convinced\nat last, and gave their hearty approval to an expenditure which\nwill enable us, during the next year, to erect a school building\nsuitable to the place, and one worthy of the trouble you, sir, have\ntaken to be present at its official commencement. During the time\nI refer to, a Grammar School building of brick was erected and\nenlarged, and a Separate School building put up.\n\n\"To-day, with your kind assistance, we have inaugurated a system\nof Public School accommodation which, with our school known as the\nBarrie School, Separate and High Schools, will ultimately provide\nfor the educational wants of the neighborhood. I use the expression\n'inaugurated a system,' because I hope and trust that our efforts\nin this direction will not be slackened on the completion of this\nbuilding.... We believe this building will be worthy of the honor\nyou have done us in coming here to-day, we also believe at some\nfuture day, we shall have a system of Public School accommodation\nworthy of the life-long and successful efforts you have made to give\nto Ontario an almost perfect system of education. It is seldom that\npublic men are asked to assist in building a monument to themselves,\nbut I have asked you to do so on this occasion, for I look upon\nbuildings of this nature as memorials of your well-directed public\nwork during the last thirty years, and when you have gone to your\nlong home, and the envy--aye--and the malice of your enemies are\nforgotten, your name associated with the noble work you have\naccomplished, will be handed down from generation to generation, and\neach school section throughout the country will contain a monument\nto your memory, as enduring as the foundations of this continent.\"\n\n\n\n\nCHAPTER VI.\n\nPERSONAL CHAPTER RELATING TO THE REV. DR. RYERSON.\n\n\nIn Dr. Ryerson's personal sketch of his early history he gives the\nfollowing particulars as to his student and teacher life:\n\nI was born on the 24th of March, 1803, in the Township of\nCharlotteville, near the Village of Vittoria, in the then London\nDistrict, now the County of Norfolk.... The district grammar-school\nwas then kept within half-a-mile of my father's residence, by Mr.\nJames Mitchell (afterwards Judge Mitchell), an excellent classical\nscholar; he came from Scotland with the late Rt. Rev. Dr. Strachan,\nfirst Bishop of Toronto. He treated me with much kindness. When I\nrecited to him my lessons in English grammar he often said that he\nhad never studied the English grammar himself, that he wrote and\nspoke English by the Latin grammar. At the age of fourteen I had\nthe opportunity of attending a course of instruction in the English\nlanguage given by two professors, the one an Englishman, and the\nother an American, who taught nothing but English grammar. They\nprofessed in one course of instruction, by lectures, to enable a\ndiligent pupil to parse any sentence in the English language. I was\nsent to attend these lectures, the only boarding abroad for school\ninstruction I ever enjoyed. My previous knowledge of the _letter_\nof the grammar was of great service to me, and gave me an advantage\nover other pupils, so that before the end of the course I was\ngenerally called up to give visitors an illustration of the success\nof the system, which was certainly the most effective I have ever\nsince witnessed, having charts, etc., to illustrate the agreement\nand government of words.\n\nThis whole course of instruction by two able men, who did nothing\nbut teach grammar from one week's end to another had to me all\nthe attraction of a charm and a new discovery. It gratified both\ncuriosity and ambition, and I pursued it with absorbing interest,\nuntil I had gone through Murray's two volumes of \"Expositions\nand Exercises,\" Lord Kames' \"Elements of Criticism,\" and Blair's\n\"Lectures on Rhetoric,\" of which I still have the notes which I\nthen made. The same professors obtained sufficient encouragement to\ngive a second course of instruction and lectures at Vittoria, and\none of them becoming ill, the other solicited my father to allow me\nto assist him, as it would be useful to me, while it would enable\nhim to fulfil his engagements. Thus, before I was sixteen, I was\ninducted as a teacher, by lecturing on my native language. This\ncourse of instruction, and exercises in English, have proved of the\ngreatest advantage to me, not less in enabling me to study foreign\nlanguages than in using my own.\n\nWhile working on the farm I did more than ordinary day's work,\nthat I might show how industrious, instead of lazy, as some said,\nreligion made a person. I studied between three and six o'clock in\nthe morning, carried a book in my pocket during the day to improve\nodd moments by reading or learning, and then reviewed my studies of\nthe day aloud while walking out in the evening.... A kind friend\noffered to give me any book that I would commit to memory, and\nsubmit to his examination of the same. In this way I obtained my\nfirst Latin grammar, \"Watts on the Mind,\" and \"Watts' Logic.\"\n\nMy eldest brother, George, after the war, went to Union College,\nU. S., where he finished his collegiate studies. He was a\nfellow-student with the late Dr. Wayland, and afterwards succeeded\nmy brother-in-law as Master of the London District Grammar School.\nHis counsels, examinations, and ever kind assistance were a great\nencouragement and of immense service to me.\n\nI felt a strong desire to pursue further my classical studies, and\ndetermined, with the kind counsel and aid of my eldest brother, to\nproceed to Hamilton, and place myself for a year under the tuition\nof a man of high reputation both as a scholar and a teacher,\nthe late John Law, Esq., then head master of the Gore District\nGrammar School. I applied myself with such ardour, and prepared\nsuch an amount of work in both Latin and Greek, that Mr. Law said\nit was impossible for him to give the time and hear me read all\nthat I had prepared, and that he would, therefore, examine me on\nthe translation and construction of the more difficult passages,\nremarking more than once that it was impossible for any human mind\nto sustain long the strain that I was imposing upon mine.[58] In\nthe course of some six months his apprehensions were realized, as\nI was seized with a brain fever, and on partially recovering took\ncold, which resulted in inflammation of the lungs by which I was so\nreduced that my physician, the late Dr. James Graham, of Norfolk,\npronounced my case hopeless, and my death was hourly expected.\n\n  [58] Having written to the late Hon. Samuel Mills for his\n  recollections of these school days, Mr. Mills replied as follows:\n  \"I have a distinct recollection of having had the honor of being\n  at the Hamilton Grammar School with yourself in the years 1823 and\n  1824, and that the late John Law was head master at the time. He\n  was considered a highly educated and accomplished scholar, and was\n  so well qualified for the position he held, that the school had a\n  provincial reputation and was patronized by many parties living at\n  a great distance by sending their sons to it; and the very fact\n  of your attending the school gave \u00e9clat to it, as you were then\n  considered a well educated young man, far in advance of the rest of\n  us. Your studies, if my recollection serves me right, were confined\n  entirely to reading Latin and Greek, and I know Mr. Law and the\n  whole school looked upon you as being a credit to it.\"\n\nAfter a severe illness Dr. Ryerson happily recovered.\n\nHis narrative states that, \"the next day after my recovery, I\nleft home and became usher in the London District Grammar School,\napplying myself to my new work with much diligence and earnestness,\nso that I soon succeeded in gaining the good-will of parents and\npupils, and they were quite satisfied with my services,--leaving the\nhead master to his favorite pursuits of gardening and building!\n\nIn 1872, Dr. Ryerson wrote to Mr. Simpson McCall, of Vittoria and\nasked: \"Will you have the kindness to let me know what is your own\nrecollection as to the attendance at the school, especially in the\nwinter months, and the impression of the neighborhood generally\nas to its efficiency during the two years that I taught it?\" Mr.\nMcCall replied as follows: I can assure you that I have a vivid\nrecollection of the London District School during the winters of\n1821 and 1822, being an attendant myself. I also remember several of\nthe scholars with whom I associated, viz: H. V. A. Rapelje, Esq.,\nlate Sheriff of the County of Norfolk; Capt. Joseph Bostwick, of\nPort Stanley; James and Hannah Moore.\n\nThe number generally attending during the winters of those two\nyears, if I remember correctly, were from forty to fifty.\n\nThe school while under your charge was well and efficiently\nconducted, and was so considered and appreciated throughout\nthe neighborhood at the time; and after you left the charge of\nthe London District School it was generally regretted in the\nneighborhood.\n\nI remember hearing this frequently remarked not only by pupils who\nattended the school under your tuition but also by their parents.\n\n\"During two years I was thus teacher and student, advancing\nconsiderably in classical studies, I took great delight in \"Locke on\nthe Human Understanding,\" Paley's \"Moral and Political Philosophy,\"\nand \"Blackstone's Commentaries,\" especially the sections of the\nlatter on the Prerogatives of the Crown, the Rights of the subject,\nand the Province of Parliament.\"\n\nIn an address before the Ontario Teachers' Association in 1872,\nDr. Ryerson said: \"As it has of late been stated, so confidently\nand largely, that he had yet to learn the elements of his native\ntongue. Such had been the representations on the subject, that he\nhad begun to suspect his own identity, and to ask himself whether\nit was not a delusion that he had in boyhood not only studied, but,\nas he supposed, had mastered Murray's two octavo volumes of English\nGrammar and Kame's Elements of Criticism and Blair's Rhetoric, of\nwhich he still had the notes that he made in early life; and had\nbeen called to assist teaching a special class of young persons in\nEnglish Grammar when he was only fifteen years of age; and whether\nit was not a fancy that he had taught, as he supposed, with some\ndegree of acceptance and success, what was then known as the London\nDistrict Grammar School for two years, and had subsequently placed\nhimself for a year under an accomplished scholar in order to read\nLatin and Greek. Somewhat disturbed by these doubts, he thought he\nwould satisfy himself by writing to the only two gentlemen with whom\nhe was now acquainted, who knew him in these early relations. In\nreference to these statements he would read the correspondence on\nthe subject.\" (See the foot notes appended.)\n\n\nTHE REV. DR. RYERSON AS A TEACHER.\n\nAs to Dr. Ryerson's influence as a teacher, Rev. Dr. Ormiston thus\nreferred to it at the Ontario Teacher's Convention in 1872. He said:\n\"The teacher has a reward peculiar to his work--a living, lasting\nmemorial of his worth. The feelings of loving reverence which we\nentertain for those who have awakened our intellectual life, and\nguided us in our earliest attempts at the acquisition of knowledge,\nare as enduring as they are grateful. I shall never forget, as I can\nnever repay, the obligations under which I lie to the venerable and\nhonorable Chief Superintendent, Dr. Ryerson, not only for the kindly\npaternal greeting with which, as principal, he welcomed me, a raw,\ntimid, untutored lad, on my first entrance into Victoria College,\nwhen words of encouragement fell like dew-drops on my heart, and for\nthe many acts of thoughtful generosity which aided me in my early\ncareer, and for the faithful friendship and Christian sympathy which\nhas extended over nearly thirty years, unbroken and unclouded, a\nfriendship which, strengthened and intensified by prolonged and\nendearing intimacy, I now cherish as one of the highest honors and\ndearest delights of my life; but especially for the quickening,\nenergizing influence of his instructions as professor, when he\ntaught me how to think, to reason and to learn. How I enjoyed the\nhours spent in his lecture-room--hours of mental and moral growth\nnever to be forgotten! I owe him much, and but for his presence here\nto-day, I would say more of what I think and feel of his character\nand worth. He has won for himself a place in the heart of many\na young Canadian, and his name will be ever associated with the\neducational advantages and history of Ontario. May he be spared for\nmany years to see the result of his labors, in the growing prospects\nand success of the common schools and educational institutions of\nthis noble and prosperous province, whose best interests he has\npatriotically done so much to promote.\"\n\nIn 1882, after Dr. Ryerson's death, Dr. Ormiston thus referred\nto his experience at Victoria College, then under Dr. Ryerson's\npresidency. He said:--\n\n\"In the autumn of 1843, I went to Victoria College, doubting much\nwhether I was prepared to matriculate as a freshman. Though my\nattainments in some of the subjects prescribed for examination\nwere far in advance of the requirements, in other subjects I knew\nI was sadly deficient. On the evening of my arrival, while my\nmind was burdened with the importance of the step I had taken,\nand by no means free from anxiety about the issue, Dr. Ryerson,\nat that time Principal of the College, visited me in my room. I\nshall never forget that interview. He took me by the hand; and few\nmen could express as much by a mere hand-shake as he. It was a\nwelcome, an encouragement, an inspiration, and an earnest of future\nfellowship and friendship. It lessened the timid awe I naturally\nfelt towards one in such an elevated position--I had never before\nseen a Principal of a College--it dissipated all boyish awkwardness\nand awakened filial confidence. He spoke of Scotland, my native\nland, and of her noble sons, distinguished in every branch of\nphilosophy and literature; specially of the number, the diligence,\nthe frugality, self-denial and success of her college students. In\nthis way he soon led me to tell him of my parentage, past life and\nefforts, present hopes and aspirations. His manner was so gracious\nand paternal--his sympathy so quick and genuine--his counsel so\nready and cheering--his assurances so grateful and inspiriting, that\nnot only was my heart _his_ from that hour, but my future career\nseemed brighter and more certain than it had ever appeared before.\n\n\"Many times in after years have I been instructed, and guided, and\ndelighted with his conversation, always replete with interest and\ninformation; but that first interview I can never forget, it is as\nfresh and clear to me to-day as it was on the morning after it took\nplace. It has exerted a profound, enduring, moulding influence on\nmy whole life. For what, under God, I am, and have been enabled to\nachieve, I owe more to that noble, unselfish, kind-hearted man than\nto any one else.\n\n\"As a teacher he was earnest and efficient, eloquent and inspiring,\nbut he expected and exacted rather too much work from the average\nstudent. His own ready and affluent mind sympathized keenly with\nthe apt, bright scholar, to whom his praise was warmly given, but\nhe scarcely made sufficient allowance for the dullness or lack\nof previous preparation which failed to keep pace with him in\nhis long and rapid strides; hence his censures were occasionally\nsevere. His methods of examination furnished the very best kind\nof mental discipline, fitted alike to cultivate the memory and to\nstrengthen the judgment. All the students revered him, but the best\nof the class appreciated him most. His counsels were faithful and\njudicious; his admonitions paternal and discriminating; his rebukes\nseldom administered, but scathingly severe. No student ever left his\npresence, without resolving to do better, to aim higher, and to win\nhis approval.\"\n\n\nTHE REV. DR. RYERSON AND HIS NATIVE COUNTY OF NORFOLK.\n\nMr. P. K. Olyne, in the _New Dominion Monthly_ for July, 1869, in an\narticle on \"Norfolk, or the Long-Point County,\" thus referrs to its\nsettlement and to the boyhood there of Dr. Ryerson:--\n\n\"After undergoing many hardships which were only a foretaste of\nwhat they had to endure in the future, a company arrived in the\nLong Point region about the year 1780. This was then a solitary\nwilderness. These pioneer Loyalists went to work with zeal\nunsurpassed in clearing away the forest, in building roads and\nerecting houses as commodious as it was possible to erect out\nof rude materials. Among those who first came to the Long Point\ncountry, worthy of particular notice, were Colonel Ryerson, Colonel\nBackhouse, Walsh and Tisdale. In the pioneer home of Joseph Ryerson\nmight have been seen a remarkably bright lad. Being extremely fond\nof books, he spent his spare moments in studying. So regular was his\nhabits in this respect, that when a neighbour would drop in and ask\nfor Egerton, the answer was sure to be: \"You will find him in such\na place, with a book.\" Notwithstanding he was placed in a position\nwhere opportunities for gaining an education were very meagre\nindeed, yet he overcame all obstacles--obstacles that he could not\nforget in after life, and which, like a true patriot, he set himself\nto remove. How much Dr. Egerton Ryerson, Chief Superintendent of\nEducation, has done for the educational interests of Canada the\nreader is left to judge for himself. Of late the Doctor has made a\npractice of visiting the home of his childhood annually. Not always\nby rail and stage has he accomplished the journey from Toronto,\nbut still clinging to the sport of his youthful days he would set\nforward in an open boat, and paddling it himself along the shores of\nthe lakes would finally reach the place so dear to him, and which,\nno doubt, brought afresh to his memory many recollections both\njoyous and sad.\n\n\"A rude log schoolhouse was constructed by the early settlers as\nsoon as they could do so conveniently. A fire-place extended along\nnearly a whole side of the building. Logs of considerable length\nwere rolled into this in cold weather for fuel, before which rude\nbenches or hewed logs were placed as seats for the instructor and\npupils. The close of the teacher's term was denominated \"the last\nday.\" It was customary on this occasion for the children to turn\nthe pedagogue out of doors by force, and for this purpose some\nwhiskey was generally provided as a stimulant. Such was the state\nof educational institutions in the days of young Ryerson. What\nadvancement has education made since? We trace it step by step as\nonward it has advanced, until to-day Norfolk can proudly boast of\ninstitutions and teachers second to none of the kind in the world.\"\n\nIn 1851, Dr. Ryerson sent to each County Council specimens of maps,\ncharts, natural history, prints, etc., to the value of $30, the\nCouncil of the County of Norfolk, acknowledged the gift in a very\nhearty manner. In reply to the County Council, Dr. Ryerson said:--\n\nFrom the Municipal Council of my native county, I have never\nexperienced unkind opposition, but have been encouraged by its\npatriotic co-operation: and it affords me no small satisfaction,\nthat that same Council is the first in Upper Canada to acknowledge\nthe receipt of the documents and maps referred to--that the\nresolution of the Council was seconded by an old school-fellow,[59]\nand couched in terms to me the most gratifying and encouraging; and\nthat my first official letter of a new year, relates to topics which\ncall up the earliest associations of my youth, and are calculated to\nprompt and impel me to renewed exertions for the intellectual and\nsocial advancement of my native land.\n\n  [59] Mr. I. W. Powell, M.P.P., father of Colonel Powell,\n  Adjutant-General of Canada.\n\nTo the County Board of Public Instruction he said:\n\n\"I hope the poorest boy in my native County may have access to a\nbetter common school than existed there when I was a lad. What\nI witnessed and felt in my boyhood, gave birth to the strongest\nimpulses of my own mind, to do what I could to place the means and\nfacilities of mental development and culture within the reach of\nevery youth in the land.\"\n\n\"I am more than gratified, I am profoundly impressed, that such\nefforts are made for the interests of the young, and of future\ngenerations in the County of Norfolk. That county is dear to me by a\nthousand tender recollections; and I still seem to hear in the midst\nof it, a voice issuing from a mother's grave, as was wont formerly\nfrom the living tongue, telling me that the only life worthy the\nname, is that which makes man one with his fellow-men, and with his\ncountry.\"\n\nIn September, 1864, Dr. Ryerson thus referred to the trip in his\nfrail skiff to his native county of Norfolk in the preceding month:\n\n\"In my lonely voyage from Toronto to Port Ryerse the scene was\noften enchanting and the solitude sweet beyond expression. I have\nwitnessed the setting sun amidst the Swiss and Tyrolese Alps from\nlofty elevations, on the plains of Lombardy, from the highest\neminence of the Appenines, between Bologna and Florence, and from\nthe crater summit of Vesuvius, but I was never more delighted and\nimpressed (owing, perhaps, in part to the susceptible state of\nmy feelings) with the beauty, effulgence, and even sublimity of\natmospheric phenomena, and the softened magnificence of surrounding\nobjects, than in witnessing the setting sun on the 23rd of June,\nfrom the unruffled bosom of Lake Erie, a few miles east of Port\nDover, and about a mile from the thickly wooded shore, with its\ndeepening and variously reflected shadows. And when the silent\ndarkness enveloped all this beauty, and grandeur, and magnificence\nin undistinguishable gloom, my mind experienced that wonderful sense\nof freedom and relief which come from all that suggests the idea of\nboundlessness--the deep sky, the dark night, the endless circle, the\nillimitable waters. The world with its tumult of cares seemed to\nhave retired, and God and His works appeared all in all, suggesting\nthe enquiry which faith and experience promptly answered in the\naffirmative--\n\n    With glorious clouds encompassed round\n      Whom angels dimly see;\n    Will the unsearchable be found;\n      Will God appear to me?\n\n\"My last remark is the vivifying influence and unspeakable pleasure\nof visiting scenes endeared to me by many tender, and comparatively\nfew painful recollections. Amid the fields, woods, out-door\nexercises, and associations of the first twenty years of my life,\nI have seemed to forget the sorrows, labors and burdens of more\nthan two score years, and be transported back to what was youthful,\nsimple, healthy, active, and happy. I can heartily sympathise with\nthe feelings of Sir Walter Scott when, in reply to Washington\nIrving, who had expressed disapprobation in the scenery of the\nTweed, immortalized by the genius of the border minstrel, he said:\n\n\"'It may be partiality, but to my eyes these gray hills and all this\nwild border country have beauties peculiar to themselves. I like\nthe very nakedness of the land. It has something bold, and stern,\nand solitary about it. When I have been for some time in the rich\nscenery of Edinburgh, which is ornamented garden land, I begin to\nwish myself back again among my honest gray hills, and if I did not\nsee the heather at least once a year I think I should die.'\n\n\"Last autumn I lodged two weeks on the farm on which I was born,\nwith the family of Mr. Joseph Duncan, where the meals were taken\ndaily in a room the wood-work of which I, as an amateur carpenter,\nhad finished more than forty years ago, while recovering from a long\nand serious illness.\"[60]\n\n  [60] The island within Long Point, which Mr. Ryerson's father\n  obtained from the Crown, but which then belonged to him, was marked\n  on old maps as Pottshawk Point, but designated on later maps, and\n  more generally known, as \"Ryerson's Island.\"\n\n\nCLOSING OFFICIAL ACTS AND UTTERANCES OF DR. RYERSON.\n\nAn entire revision and consolidation of the laws relating to public\nand high schools took place in 1874, in which Dr. Ryerson took a\nleading part. But the revision related chiefly to details and to the\nsupply of former omissions in the law.\n\nThe last important official act of Dr. Ryerson was to arrange\nfor the educational exhibit of the department at the Centennial\nExhibition in 1876. That was most successfully carried out; and,\nat the close of that exhibition, the following highly gratifying\n\"award\" was communicated to the then venerable ex-chief, after\nhe had retired from office. The award was made by the American\nCentennial Commission, and was to the following effect:\n\n     For a quite complete and admirably arranged exhibition,\n     illustrating the Ontario system of education and its excellent\n     results; also for the efficiency of an administration which has\n     gained for the Ontario Department a most honorable distinction\n     among government educational agencies.\n\nThis award was quite a gratification to the now retired chief of the\nDepartment, then in his seventy-third year, and amply repaid him, as\nhe said, for many years of anxious toil and solicitude, while it was\na gratifying and unlooked-for compensation for all of the undeserved\nopposition which he had encountered while laying the foundations of\nour educational system.\n\nIn a letter to a friend toward the close of his official career,\nDr. Ryerson thus explained the principles upon which he had\nconducted the educational affairs of the Province during his long\nadministration of them. He said:--\n\nDuring these many years I have organized and administered the\nEducation Department upon the broad and impartial principles which I\nhave always advocated. During the long period of my administration\nof the Department I knew neither religious sect nor political party;\nI knew no party other than that of the country at large; I never\nexercised any patronage for personal or party purposes; I never\nmade or recommended one of the numerous appointments of teachers\nin the Normal or Model Schools, or clerk in the education office,\nexcept upon the ground of testimonials as to personal characters and\nqualifications, and on a probationary trial of six months.\n\n       *       *       *       *       *\n\nI believe this is the true method of managing all the public\ndepartments, and every branch of the public service. I believe it\nwould contribute immensely to both the efficiency and economy of\nthe public service. * * * It would greatly elevate the standard of\naction and attainments and stimulate the ambition of the young men\nof the country, when they know that their selection and advancement\nin their country's service depended upon their individual merits,\nirrespective of sect or party, and not as the reward of zeal as\npolitical partisans in elections or otherwise, on their own part, or\non that of their fathers or relatives.\n\nThe power of a government in a country is immense for good or ill.\nIt is designed by the Supreme Being to be a \"minister of God for\ngood\" to a whole people, without partiality, as well as without\nhypocrisy, like the rays of the sun; and the administration of\ninfinite wisdom and justice and truth and purity.\n\n       *       *       *       *       *\n\nI know it has been contended that party patronage * * * is an\nessential element in the existence of a government. * * * The\nEducation Department has existed--and it is the highest public\ndepartment in Upper Canada--for more than thirty years without such\nan element, with increasing efficiency and increasing strength, in\nthe public estimation, during the whole of that period. Justice, and\nvirtue, and patriotism, and intelligence are stronger elements of\npower and usefulness than those of rewarding partisans; and if the\nrivalship and competition of public men should consist in devising\nand promoting measures for the advancement of the country and in\nexercising the executive power most impartially and intelligently\nfor the best interests of all classes, then the moral standard of\ngovernment and of public men would be greatly exalted, and the\nhighest civilization of the whole country be advanced.\n\nIn a series of letters published in defence of his administration of\nthe Education Department in 1872, he thus pointed out the character\nof his difficult and delicate task. He also gave a brief glance at\nwhat had been accomplished by the Department since he took office in\n1844. He said:--\n\nThe Department of which I have had charge since 1844, and during\nseveral administrations of government, is confessedly the most\ndifficult and complicated, if not the most important, of any\ndepartment of the public service. Since 1844, it has devolved on me\nto frame laws, and to devise, develop, and administer a system of\npublic instruction for the people of this Province. That system has\nbeen more eulogized by both English and American educationists, and\nmore largely adopted in other British colonies, on both sides of the\nglobe, than any other system of public instruction in America.\n\nThe system of popular education in Ontario has opened a free\nschool to every child in the land, and proclaimed his right to\nits advantages; it has planted a school house in nearly every\nneighborhood, and in hundreds of instances, made the school house\nthe best building in the neighborhood; it has superseded the topers\nand broken-down characters, so common as teachers of a former age,\nby a class of teachers not excelled in morals by the teachers\nof any other country, and who, as a whole, compare favorably in\nqualifications with those of any State in America; it has achieved\na uniformity of excellent text-books, earnestly prayed for by\neducators in the neighboring States, and has spread throughout\nthe land books of useful and entertaining knowledge to the number\nof nearly a million of volumes; it is the nearest approach to a\nvoluntary system of any public school system in the world; and it\nhas developed larger resources than that of any other State in\nAmerica, in proportion to the wealth and number of inhabitants.\n\nThis unparalleled success is due to the Christian feeling, the\nenergy, patriotism and liberality of the people of this Province;\nbut it has been imposed on me to construct the machinery,\ndevise the facilities and agencies by which so great a work\nhas been accomplished, and to do what I could to encourage my\nfellow-countrymen in its promotion.\n\nThe administration of laws generally is by learned judges, by the\npleadings of learned counsel, and the deliberations of selected\njuries; but the administration of the school law and system is\nthrough the agency of several hundred elected councils, and\nnearly twenty thousand elected trustees,--thus embodying, not\nthe learned professions, but the intelligence, common sense,\nfeelings and interests of the people at large in the work of school\nadministration and local self-government.\n\n\nREASONS FOR DR. RYERSON'S RETIREMENT AS CHIEF SUPERINTENDENT OF\nEDUCATION.\n\nIn \"The Story of My Life,\" I gave the following reasons, amongst\nothers, which induced Dr. Ryerson to propose a change in the\nheadship of the Education Department. I said: \"For many years after\nconfederation Dr. Ryerson felt that the new political condition of\nthe Province, which localized, as well as circumscribed, its civil\nadministration of affairs, required a change in the management of\nthe Education Department. He, therefore, (as early as in 1868,\nand again in 1872) urged upon the Government the desirability\nof relieving him from the anomalous position in which he found\nhimself placed under the new system. The reasons which he urged\nfor his retirement are given in a pamphlet devoted to a 'defence'\nof the system of education, which he published in 1872, and are as\nfollows:--\n\n\"When political men have made attacks upon the school law, or the\nschool system and myself, and I have answered them, then the cry\nhas been raised by my assailants and their abettors, that I was\ninterfering with politics. They would assail me without stint, in\nhopes of crushing me, and then gag me against all defence or reply.\n\n\"So deeply did I feel the disadvantage and growing evil of this\nstate of things to the department and school system itself, that\nI proposed, four years ago last December, to retire from the\ndepartment, and recommended the creation and appointment of a\nMinister of Public Instruction. My resignation was not accepted, nor\nmy recommendation adopted; when, two months later, I proposed that,\nat the commencement of each session of the legislature, a committee\nof seven or nine (including the Provincial Secretary for the time\nbeing) should be elected by ballot, or by mutual agreement of the\nleading men of both parties, on the Education Department; which\ncommittee should examine into all the operations of the department\nfor the year then ending, consider the school estimates, and any\nbill or recommendations which might be submitted for the advancement\nof the school system, and report to the house accordingly. By many\nthoughtful men, this system has been considered more safe, more\nlikely to secure a competent and working head of the department,\nand less liable to make the school system a tool of party politics,\nthan for the head of it to have a seat in Parliament, and thus\nleave the educational interests of the country dependent upon the\nvotes of a majority of electors in one riding. This recommendation,\nsubmitted on the 30th of January, 1869, has not yet been adopted;\nand I am left isolated, responsible in the estimation of legislators\nand everybody else for the department--the target of every attack,\nwhether in the newspapers or in the Legislative Assembly, yet\nwithout any access to it or to its members, except through the\npress, and no other support than the character of my work and the\ngeneral confidence of the public.\"\n\n\nDR. RYERSON'S LETTER OF RESIGNATION IN 1868 AND REPLY TO IT.\n\nThe salient points of Dr. Ryerson's letter of resignation, dated\n7th December, 1868, and addressed to Hon. M. C. Cameron, Provincial\nSecretary, are as follows:--\n\n\"I have the honor to submit to the favourable consideration of the\nLieutenant-Governor in Council what, some three weeks since, I\nsubmitted to the individual members of the Government, namely, that\n\n\"The Department of Public Instruction shall be under the management\nof a member of the Executive Council, to be designated 'Minister\nof Public Instruction,' who shall be an _ex officio_ member of\nthe Toronto University and of the Council of Public Instruction,\nand who, in addition to the powers and functions vested in the\nChief Superintendent of Education, shall have the oversight of\nall educational institutions which are or may be, aided by public\nendowment or legislative grant, to inspect and examine, from time to\ntime, personally or any person appointed by him, into the character\nand working of such institution; and by him shall all public moneys\nbe paid in support or aid of such institutions, and to him they will\nreport at such times and in such manner as he shall direct....\n\n\"Our system of public instruction has acquired such gigantic\ndimensions, and the network of its operations so pervades every\nmunicipality of the land, and is so interwoven with our municipal\nand judicial systems of government, that I think its administration\nshould now be vested in a responsible Minister of the Crown, with a\nseat in Parliament; and that I should not stand in the way of the\napplication to our varied educational interests of that ministerial\nresponsibility, which is sound in principle and wise in policy.\nDuring the past year I have presented a report on school systems\nin other countries, with a view of improving my own; and the\nLegislative Assembly has appointed a Select Committee for the same\npurpose. I have, therefore, thought this was the proper time to\nsuggest the modification and extension of the Department of Public\nInstruction....\n\n\"While, in addition to the duties imposed upon me by law as Chief\nSuperintendent of Education, I have voluntarily established a system\nof providing the municipal and school authorities with libraries,\ntext-books and every description of school furniture and school\napparatus--devising and developing their domestic manufacture. I\nhave thus saved the country very many thousands of dollars in the\nprices as well as the quality of the books, maps, etc., etc. I can\ntruly say that I have not derived one farthing's advantage from\nany of these arrangements, beyond the consciousness of conferring\nmaterial, intellectual and social benefits upon the country.\"...\n\nTo this letter the Government of the Hon. J. Sandfield Macdonald\nreplied, through Provincial Secretary Cameron, on the 30th of\nJanuary, 1869, as follows:--\n\n\"In acknowledging your letter of the 7th December last, placing your\nresignation of the office of Chief Superintendent of Education in\nthe hands of His Excellency the Lieutenant-Governor, and suggesting\nthat the Department of Public Instruction should be placed under the\nmore direct management of the Government, through a Minister, to be\ndesignated 'the Minister of Public Instruction,' holding a place\nin the Executive Council and a seat in the Legislative Assembly,\nthus bringing this department, in common with all other branches\nof the Government, within the control of the people through the\nresponsible advisers of the Crown. I am directed by His Excellency\nthe Lieutenant-Governor, to thank you for the valuable suggestions\ncontained in your letter, and to request that you will continue to\ndischarge those important duties, which you have performed for a\nquarter of a century with so much credit to yourself and benefit to\nthe people of this Province, until His Excellency's advisers shall\nhave more fully considered your suggestions, and matured a measure\nfor placing your department under the direct supervision of a member\nof the Executive Council.\n\n\"The services that you have rendered your country, and your now\nadvanced age, fully warrant your asking to be relieved from the\nfurther discharge of your arduous duties; but knowing your vigor of\nmind and energy of character, His Excellency ventures to hope that\ncompliance with the request now made will not prove too great a tax\nupon your energies, or interfere seriously with any other plans you\nmay have formed for the employment of the remaining years of a life\ndevoted to the moral and intellectual improvement of your fellowmen.\"\n\nTo this letter Dr. Ryerson replied on the 30th January, 1869, as\nfollows:--\n\n\"I have the honour to acknowledge the receipt of your letter of\nthis date conveying the most kind expression of his Excellency the\nLieutenant-Governor in regard to myself and my past humble services,\nand the request that I would continue in my present office until His\nExcellency's advisers should be able to mature a measure to give\neffect to the recommendations of my letter of 7th of December last\nrespecting the direct responsibility of the Education Department to\nParliament, and the creation of the office of Minister of Public\nInstruction to be filled by a responsible Minister of the Crown,\nhaving a seat in Parliament. The more than kind reference to myself\non the part of His Excellency has deeply affected me, and for which\nI desire to express my most heartfelt thanks.\n\n\"I beg to assure you for the satisfaction of His Excellency that I\nwill subordinate every inclination and contemplated engagement, to\nthe great work of the Education Department and the system of Public\nInstruction, as long as I have strength and may be desired by the\nconstituted authorities to do so.\n\n\"I have found that the apprehensions first expressed by the Hon.\nM. C. Cameron, as Chairman of the Education Committee of the\nLegislative Assembly during the late session, that connecting the\nDepartment of Public Instruction with the political Ministry of the\nday might draw the system of Public Instruction into the arena of\nparty politics and thus impede its progress, is largely shared in by\nthoughtful men, and that my recommendation has been coldly received\ngenerally, and strongly objected to in many quarters.\n\n\"Under these circumstances I have been led to review the whole\nquestion; and aided by the experience which the recent session of\nthe Legislature has afforded, I would respectfully suggest that,\nuntil a better system can be devised, a committee of say seven or\nnine members of the Legislative Assembly (to be presided over by the\nProvincial Secretary) be elected by ballot, (or if not by ballot, by\nthe mutual agreement of the leaders of both parties in the House,)\nat the commencement of each session, to examine into the working,\nand report upon all matters relating to the Education Department\nand its administration, as well as upon any measures which might be\nsuggested for the promotion of Public Instruction. The Provincial\nSecretary, being _ex officio_ Chairman of such Committee, would\nbe able to bring before it any thing that had required the\ninterposition of, or had been brought before the Government during\nthe year and meriting the attention of the Committee. The Committee\nbeing chosen by ballot, or by mutual agreement on both sides of\nthe House, would preclude the character of party in its mode of\nappointment, and give weight and influence to its recommendations.\nIn this way the Education Department, necessarily so identified with\nmatters affecting popular progress and enlightenment would, it its\nwhole administration be more directly responsible to Parliament and\nthrough it to the people than any other Public Department is now,\nand that without being identified or connected with any political\nparty, and on the occasion of a vacancy in the Administration of the\nDepartment, a selection and appointment could be made free from the\nexigencies of party or of party elections, upon the simple and sole\nground of qualifications for the office and with a view of promoting\nthe interests of public education irrespective of sect or party.\"\n\n\nDR. RYERSON'S LETTER OF RESIGNATION IN 1872 AND REPLY TO IT.\n\nOn the 10th of February, 1872, Dr. Ryerson addressed a letter to the\nHon. Edward Blake, then Premier of Ontario, in which he said:\n\n\"After much deliberation, I have thought it advisable to address\nyou in respect to my long desired retirement from the Education\nDepartment, of which I have had charge longer than any Judge has\never occupied the Bench in Canada, and to a greater age....\n\n\"The infirmities of age must compel me to retire before long; and\nI have thought my immediate or early retirement would enable the\nGovernment to exercise its discretion more freely in regard to the\nDepartment, and system of Public Instruction....\n\n\"In case you concur in what I have above intimated, I would suggest\nthe creation of the office of Minister of Public Instruction, and\nthe appointment of yourself to it, as is the Premier in Lower\nCanada, bringing the University, U. C. College, Institutions of Deaf\nand Blind, as well as the Normal, High and Public Schools, under\ndirect governmental supervision.\n\n\"In the practical administration of the Education Department an\nabler, more judicious and reliable man cannot be found than Dr.\nHodgins, who has been in the Department twenty-seven years--who was\nfirst educated to business in a retail store in Galt, and afterwards\nin a wholesale establishment in Hamilton with the Stinsons--clerk\nin the same establishment with Mr. Charles McGill, M.P., and was\noffered to be set up in business by the Stinsons or admitted as\na partner within a year or so if he would remain, but he chose\nliterature and went to Victoria College in 1840, where I found him;\nand on account of his punctuality, thoroughness, neatness, method,\nand excellent conduct, I appointed him on trial first clerk in my\noffice in 1844; and having proved his ability, I wrote to him when\nI was in Europe, to come home to his widowed mother in Dublin, and\nspend a year in the great education office there, to learn the whole\nsystem and management--I having arranged with the late Archbishop\nWhately and other members of the National Board, to admit Mr.\nH. into their office to study the principles and details of its\nmanagement and of the Normal and Model Schools connected with it.\nMr. Hodgins did so at his own expense, and losing the salary for\nthe year; at the end of which he returned to my office with the\ntestimonials of the Irish National Board, as to his diligence and\nthe thorough manner in which he had mastered the modes of proceeding\nin the several branches of that great Education Department. He also\nbrought drawings, of his own make, of the Dublin Education Offices,\nNormal and Model Schools. Then since you know that Mr. Hodgins\nhaving taken his degree of M. A., has proceeded regularly to his\ndegree of law in the Toronto University, and has been admitted to\nthe Bar as Barrister at Law. He is, therefore, the most thoroughly\ntrained man in all Canada for the Education Department; and is\nthe ablest, most thorough administrator of a public department of\nany man with whom I have met. I think he has not been appreciated\naccording to his merits; but should you create and fill the office\nof Minister of Public Instruction, you may safely confide the\nordinary administration of the Education Department to Dr. Hodgins,\nwith the title of my office.\n\n\"In the meantime you can make yourself familiar with the principles\nand branches, and modes of its arrangement. Whatever you may find\nto approve of in my policy and course of procedure, I have no doubt\nyou will have the fairness to avow and the patriotism to maintain,\nwhatever may be your views and feelings in regard to myself,\npersonally; and if you find defects in, and can improve upon, my\nplans or proceedings, no one will rejoice at your success more than\nmyself. I enclose a printed paper, which will afford information of\nthe details of the Department....\n\n\"I may add that should I retire from my present office, I would\nhave no objections, if desired, to be appointed Member of the\nCouncil of Public Instruction and give any assistance I could in its\nproceedings as the result of my experience.\"\n\n       *       *       *       *       *\n\nIn his reply, dated 12th of February, 1872, Mr. Blake said:--\n\n\"I have your note of the 10th instant, marked private, proposing\nyour retirement and the reconstruction of the Education Office,\nand enclosing copies of a former official correspondence on the\nsame subject. At this late stage of the session, and under the\npresent pressure of public business, there is no probability of\nour giving this matter the consideration which it deserves, and it\nmust therefore be postponed till the recess, when, if you will have\nthe goodness to put yourself in communication with the Provincial\nSecretary, as on the former occasion, the subject will receive the\nearly and earnest attention of the Government.\"...\n\n       *       *       *       *       *\n\nNothing further was done in this matter until 1876, when Dr. Ryerson\nfinally retired from office on full salary, after having filled his\nresponsible post for nearly thirty-two years.\n\n\nA FEW WORDS, PERSONAL TO THE WRITER OF THIS RETROSPECT.\n\nHaving been intimately concerned in all of the events and\neducational matters to which I have referred in the foregoing\nRetrospect, it may not be out of place for me to add a few words of\na personal character in conclusion.\n\nAt the end of this year I shall have completed my more than 45\nyears' service, as chief of the staff of the Education Department of\nOntario.\n\nFor over 40 years I enjoyed the personal friendship of the\ndistinguished man whose memory we honor here to-day--32 years of\nwhich were passed in active and pleasant service under him.\n\nThe day on which he took official leave of the Department was indeed\na memorable one. As he bade farewell to each of his assistants in\nthe office, he and they were deeply moved. He could not, however,\nbring himself to utter a word to me at our official parting, but\nas soon as he reached home he wrote to me the following tender and\nloving note:--\n\n  171 VICTORIA STREET, TORONTO,\n    MONDAY EVENING, FEBRUARY 21ST, 1876.\n\nMY DEAR HODGINS,--I felt too deeply to-day when parting with you\nin the Office to be able to say a word. I was quite overcome\nwith the thought of severing our official connection, which has\nexisted between us for thirty-two years, during the whole of\nwhich time, without interruption, we have labored as one mind and\nheart in two bodies, and I believe with a single eye to promote\nthe best interests of our country, irrespective of religious sect\nor political party--to devise, develop, and mature a system of\ninstruction which embraces and provides for every child in the\nland a good education; good teachers to teach; good inspectors to\noversee the Schools; good maps, globes, and text-books; good books\nto read; and every provision whereby Municipal Councils and Trustees\ncan provide suitable accommodation, teachers, and facilities for\nimparting education and knowledge to the rising generation of the\nland.\n\nWhile I devoted the year 1845 to visiting educating countries and\ninvestigating their system of instruction, in order to devise\none for our country, you devoted the same time in Dublin in\nmastering, under the special auspices of the Board of education\nthere, the several different branches of their Education Office,\nin administering the system of National Education in Ireland, so\nthat in the details of our Education Office here, as well as in our\ngeneral school system, we have been enabled to build up the most\nextensive establishment in the country, leaving nothing, as far\nas I know, to be devised in the completeness of its arrangements,\nand in the good character and efficiency of its officers. Whatever\ncredit or satisfaction may attach to the accomplishment of this\nwork, I feel that you are entitled to share equally with myself.\nAlthough I know that you have been opposed to the change, yet could\nI have believed that I might have been of any service to you, or to\nothers with whom I have labored so cordially, or that I could have\nadvanced the school system, I would not have voluntarily retired\nfrom office.[61] But all circumstances considered, and entering\nwithin a few days upon my 74th year, I have felt that this was the\ntime for me to commit to other hands the reins of the government of\nthe public school system, and labor during the last hours of my day\nand life, in a more retired sphere.\n\n  [61] This remark evidently refers to the oft expression of my\n  dissent from Dr. Ryerson's views in regard to the important\n  change which he had proposed to the Government for the future\n  administration of the Education Department. It was one of the very\n  few subjects on which I had occasion to differ from the views of my\n  venerated friend.\n\nBut my heart is, and ever will be, with you in its sympathies and\nprayers, and neither you nor yours will more truly rejoice in you\nsuccess and happiness, than\n\n  Your old life-long Friend and Fellow-laborer,\n    E. RYERSON.\n\nJ. GEORGE HODGINS, Esq., LL.D.\n\nWhile in England, in reply to a retrospective letter from me at the\nclose of that eventful year, Dr. Ryerson wrote as follows:--\n\n  \"LONDON, December 12th, 1876.\n\n  \"DEAR HODGINS,--\n\n   .     .     .     .     .     .     .     .     .     .\n\n\"Had we been enabled to work together, as in former years, we would\nhave done great things for our country, and I could have died in the\nharness with you. But it was not to be so.... I have no doubt it\nwill be seen that the hand of God is in this, as it has been in all\nour work together for more than thirty years.\n\n  \"Your ever Affectionate Friend,\n    \"E. RYERSON.\n\n\"J. GEORGE HODGINS, Esq., LL.D.\"\n\nUnder these circumstances how can I, therefore, regard without\nemotion the events of to-day? They bring vividly to my recollection\nmany memorable incidents and interesting events of our educational\npast known only to myself. They also deeply impress me with the\nfleeting and transitory nature of all things human. The Chief and\nsixteen counsellors, appointed and elected to assist him, (besides\nmore than twenty persons connected with various branches of the\nDepartment) have all passed away since my first connection with\nthe Department in 1844.[62] His great work remains, however, and\nhis invaluable services to the country we all gratefully recall\nto-day, while his native land lovingly acknowledges these services\nin erecting this noble monument to his memory. Truly indeed and\nfaithfully did Egerton Ryerson make good his promise to the people\nof this Province, when he solemnly pledged himself, on accepting\noffice in 1844--\n\n\"To provide for my native country a system of education, and\nfacilities for intellectual improvement not second to those of any\ncountry in the world.\"\n\nGod grant that the seed sown and the foundations thus laid with such\nanxious toil and care--and yet in faith--may prove to be one of our\nrichest heritages, so that in the future, wisdom and knowledge, in\nthe highest and truest sense, may be the stability of our times!\n\n  J. G. H.\n\n  Toronto, 24th May, 1889.\n\n  [62] These sixteen were:--\n\n    1. The Right Reverend Michael Power, D.D.\n    2. Hugh Scobie, Esquire.\n    3. Hon. Samuel Bealey Harrison, Q.C.\n    4. The Reverend Adam Lillie, D.D.\n    5. James Scott Howard, Esquire.\n    6. The Reverend John Jennings, D.D.\n    7. The Very Reverend Henry James Grasett, D.D.\n    8. The Hon. Mr. Justice Morrison.\n    9. The Reverend John Ambery, M.A.\n    10. The Right Reverend Thomas Brock Fuller, D.D.\n    11. The Reverend J. Tabarat, D.D.\n    12. The Reverend John McCaul, LL.D.\n    13. The Reverend John Barclay, D.D.\n    14. The Honorable William McMaster.\n    15. The Reverend Samuel S. Nelles, D.D., LL.D.\n    16. The Most Reverend John Joseph Lynch, D.D.\n\n\n\n       *       *       *       *       *\n\n\nTranscriber's note:\n\nMinor typographical and punctuation errors have been corrected\nwithout note. Irregularities and inconsistencies in the text have\nbeen retained as printed.\n\nMismatched quotes are not fixed if it's not sufficiently clear where\nthe missing quote should be placed.\n\nIn the table of contents the page number for the Title and Prefatory\nnote has been changed from i. to iii. to match the book.\n\nPage 17: \"and the difficultiee with which he had to\ncontend\"--changed to \"difficulties\".\n\nPage 22: \"this was the ----le gift which Dr. Ryerson devoted\"--The\ndash (----le) has been inserted where there was a blank area in the\nprinted book.\n\nPage 64: \"the social and religions virtues\"--\"religions\" changed to\n\"religious\".\n\n\n\n\n\nEnd of Project Gutenberg's Ryerson Memorial Volume, by J. George Hodgins\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Bitblocks (CURRENCY:BBK) traded up 5% against the US dollar during the 24-hour period ending at 19:00 PM E.T. on February 11th. Bitblocks has a market cap of $85,299.00 and approximately $787.00 worth of Bitblocks was traded on exchanges in the last day. One Bitblocks coin can currently be purchased for $0.0007 or 0.00000020 BTC on cryptocurrency exchanges including Stocks.Exchange, CryptoBridge, Cryptohub and CoinExchange. In the last week, Bitblocks has traded up 12.3% against the US dollar.\nTokenPay (TPAY) traded 0.6% lower against the dollar and now trades at $0.71 or 0.00019698 BTC.\nSaluS (SLS) traded up 1.6% against the dollar and now trades at $10.55 or 0.00291185 BTC.\nNectar (NEC) traded down 19.3% against the dollar and now trades at $0.12 or 0.00003334 BTC.\nParkinGo (GOT) traded down 26.4% against the dollar and now trades at $0.61 or 0.00016876 BTC.\nHempCoin (THC) traded 7% lower against the dollar and now trades at $0.0112 or 0.00000309 BTC.\nConsensus (SEN) traded down 3.5% against the dollar and now trades at $0.0017 or 0.00000046 BTC.\nVeriCoin (VRC) traded 3.1% lower against the dollar and now trades at $0.0741 or 0.00002045 BTC.\nECC (ECC) traded down 33.8% against the dollar and now trades at $0.0001 or 0.00000002 BTC.\nAuroracoin (AUR) traded up 7.8% against the dollar and now trades at $0.10 or 0.00002867 BTC.\nBBK is a PoW\/PoS coin that uses the Scrypt hashing algorithm. It was first traded on February 5th, 2018. Bitblocks' total supply is 162,575,170 coins and its circulating supply is 117,736,710 coins. Bitblocks' official website is bitblocksproject.com. Bitblocks' official Twitter account is @BitBlocks_.\nBitblocks can be bought or sold on these cryptocurrency exchanges: CryptoBridge, CoinExchange, Cryptohub, Stocks.Exchange and Trade Satoshi. It is usually not presently possible to purchase alternative cryptocurrencies such as Bitblocks directly using US dollars. Investors seeking to trade Bitblocks should first purchase Bitcoin or Ethereum using an exchange that deals in US dollars such as Gemini, Coinbase or GDAX. Investors can then use their newly-acquired Bitcoin or Ethereum to purchase Bitblocks using one of the aforementioned exchanges.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yummy, a Hot Turkey Dinner!\nWhat could be better than a nice hot turkey dinner with the family at Christmas?\nWe all love this traditional event, but after the meal, we suffer from the feeling of being full, an enlarged stomach and are eager to lie down. Well, that is exactly what Fluffy or Fido feels when they indulge in a turkey dinner with some fixings, but much worse.\nTurkey, gravy and potatoes are not digestible to your pet like they are to us. These foods are high in fat and cause stomach upset, allergies and even severe pancreatitis.\nPancreatitis is inflammation to the pancreas that causes painful abdominal cramps, repetitive vomiting and diarrhea. The majority of animals can suffer great pain and may need medication to calm the pain and inflammation. In many situations, some animals require intensive hospitalization with IV fluids, IV medication and treatments.\nInstead of giving harmful food to your precious pet, go for a walk and share the special time. Give Fluffy or Fido healthy treats that are great for them and safe.\nThe staff at Lomsnes Veterinary Hospital would be delighted to inform you of a variety of safe treats that your family pet will love.\nThis Christmas, have a safe and happy holiday season and give thanks for your family, your pets and the health of all!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I sat on the veranda of the Hotel Santa Catalina sipping 'un t\u00e9 brit\u00e1nico' (a British tea) and looking out on the manicured lawns that seemed to stretch down to the sea. Despite the heat in Las Palmas (a tempting 28C), the Brit in me was happy to sip on a good cuppa, even if I did have to ask for it 'con leche fria aparte' (with cold milk on the side).\nMy afternoon companion was Angie Cabrera, a local English teacher and native of the island of Gran Canaria, who has researched the history of the British in the Canary Islands (which have been part of Spain since the 15th Century) and uses it as a cultural, historical and linguistic lesson for her secondary school students.\n\"The hotel was built by the British,\" she told me over the brim of her teacup. \"British architects and everything. The Hotel Metropol in front of us as well was a British build, although it is now the council offices. The Metropol was a favourite of Agatha Christie.\" I found out later the crime writer is thought to have penned more than one of her novels there.\nIt turns out, Las Palmas de Gran Canaria, to afford the city its proper title, was the holiday destination for discerning British tourists long before the package holiday boom of the 1960s and '70s turned the island's hotter, drier south into the sun-worshipping holidaymaker mecca that it is today (some 858,118 Brits visited Gran Canaria in 2017 alone). But how did this tiny island (just more than an hour top to bottom by car) off the coast of West Africa become such a hotspot for British tourists around the turn of the 20th Century?\nMy tea stop was just the latest part of the story Cabrera had been regaling me with all day. We started our tour down near the city's main port, Puerto de La Luz, on Alfredo L Jones Street \ufffdC or as the locals say, 'Al-freh-doh ehleh chon-ess street'. The Mr Jones in question was not Canarian or even Spanish, but was, as his surname might suggest, born Alfred Lewis Jones in Carmarthenshire, South Wales, in 1845. What he was to do for this mid-Atlantic city, however, more than justifies this prominent epitaph. You see, while Jones' story is virtually unknown outside of the islands, some might say that the Welshman put the Canaries on the map.\nBack when steam was king, the Canaries were strategically important for passage from Britain to the Americas, being the last fuelling port before sailing on across the Atlantic. A constant supply of good-quality coal was needed to power ships on the final leg of their journey, and coal from the collieries of the UK was brought over in ships to be stored in the port of Las Palmas. Jones owned several collieries, including one in Maesteg, South Wales, but his main business was shipping. Having been part of many of the most prominent shipping and trading companies of the late 1800s as a shareholder and owner, he founded The Grand Canary Coaling Company in 1886. Due to Gran Canaria's lucrative location and his easy access to coal, supplying the port of Las Palmas was an obvious business opportunity.\nBringing coal from the UK was all well and good, but returning empty ships made no business sense to Jones, so he looked for a way to make the return voyage more profitable. He came up with the idea of taking local produce back home.\nBananas were considered exotic in the UK at that time, but steadily became commonplace in the British diet as the banana boats became more frequent. Tomatoes had a similar destiny. Considered bad for the health in the islands, according to Burgess, they were increasingly cultivated due to their appreciation abroad.\nIt was this constant stream of fruit ships arriving into the South Quay Import Dock in London's docklands that led to the renaming of one of the dock berths. Let to Fruit Lines Limited in 1937, it was named after the place of the fruits' origin, the Canary Islands, and what we now know as Canary Wharf came into being.\nWith ships making such regular journeys between the UK and Las Palmas, plus a solid foundation of British people living in the city (some 437 were registered as residing in the city in 1910), an unofficial British colony was created, bringing with it investment, infrastructure advances and social culture.\nThe very first mass wave of tourists started to reach the archipelago's shores in the late 1800s thanks to reduced fares negotiated by Jones on his ships, and hotels were built to cater to this new influx of visitors. Those with bronchial problems particularly favoured Las Palmas, as the temperate climate was thought to be beneficial to health.\n\"Look at the street names,\" Cabrera told me as we left the port behind and made our way to the Ciudad Jard\u00edn neighbourhood of the city \ufffdC the 'home' of the British back in the early 1900s. I spotted 'Calle Lord Byron' among other street names as we wandered up to the brightly whitewashed Holy Trinity Church. I leaned in to read the plaque on the wall. It was built by British-born, Las Palmas-based architect Norman Wright in 1892 through the generosity of Jones, among other benefactors, and opened for Anglican worship in 1893. Services here are still carried out in English.\nReligion wasn't the only thing the British brought with them; they introduced the telephone and telegraph, the first banks, an animal protection society, British-style sandwich loaves, frozen meat, the first piped water supply, and a dedicated social club, The British Club, which still exists today.\nThe current Metropole Swimming Club, which stands next to what was the Hotel Metropol, has its origins as the hotel's recreational pool, thought to be the first on the island.\nAs Cabrera and I continued to make our way through the city, I found more clues to the unique British history. Even in the branch of clothes shop Mango in Triana high street, I saw enormous wooden doors with thick iron hinges emblazoned with a British ironmonger's details.\n\"We even use our own versions of English words that were overheard by the locals at the time,\" Cabrera told me. \"Queque (cake), and naife (knife) for a start\". We took a seat at a bar and I glanced at the menu. \"What's bistec?\" I asked. \"What does it sound like?\" she replied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brice was born in Brest, Britanny, France. When he was 19, Brice enlisted in the French Army and fought in the First Indochina War. While patrolling in Indochina, one of his team triggered a mine and its explosion sent Brice whirling through the air, but left him virtually unhurt. Member or the Commandos Marine, special forces units of the French Navy, he served later as a paratrooper during the Algerian War.\nBesides theatre productions, he was mainly seen in TV-series, including Ein Schlo\u00df am W\u00f6rthersee (A Castle by the Woerthersee) and Die H\u00fctte am See (The Cottage by the Lake). In 1979 Brice again played Winnetou in a 14-part TV series called Mein Freund Winnetou (My friend Winnetou \u2013 Winnetou le Mescalero), which did not originate from Karl May material. In 1997 he appeared in a two-part TV mini series Winnetous R\u00fcckkehr [de] (The Return of Winnetou), which earned devastating criticism from the fans, since the character had died in the movie Winnetou III and now suddenly returned to life. Again, this did not originate from writings by Karl May.\nBrice tried to escape the Winnetou character in a 1976 TV series, Star Maidens, and in several movies for the big screen, playing Zorro in the Italian Zorro contro Maciste (1963). He also worked with Terence Hill (still called Mario Girotti at the time) in Shots in Threequarter Time (1965), with Lex Barker in a non-Karl May film The Hell of Manitoba (a.k.a. A Place Called Glory) (1965) and in the anthology Killer's Carnival (1966).\nPierre Brice died of pneumonia on 6 June 2015 in a Paris hospital.\nLike Lex Barker (who recorded two tracks as a singer), Brice tried to sing with the help of German composer Martin Boettcher, and even managed to issue several singles and CDs. Most of the songs were in German and, as Brice did not understand the language at the time of recording, he had to sing them phonetically.\ncontains the above songs, as well as Lex Barker songs.\n^ Obituaries in the Performing Arts, 2015 -Harris M. Lentz III - 0786476672 2016 -Page 40 French actor Pierre Brice, who was best known for his role as the Indian chief Winnetou in a series of westerns based on the works of German author Karl May, died of pneumonia in a Paris, France, hospital on June 6, 2015. He was 86.\n^ \"Ein Leben als Winnetou\". Sueddeutsche.de. Retrieved 6 June 2015.\nThis page is based on the Wikipedia article Pierre Brice; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Start an active and healthy 2016 with a FREE public walk on the 9th January as part of Operation Transformation!\nIf you are looking for something to get you on the road to a healthy and active 2016, D\u00fan Laoghaire-Rathdown Sports Partnership has the perfect event for you. As part of our aim to encourage residents of the County to get active, a FREE 4km public walk will be held on the 9th January starting in the People's Park at 10am (registration from 9.30am).\nThis event is being held in conjunction with Sport Ireland and RTE's Operation Transformation programme. On this day every County in Ireland will have a similar walking event to cater for all levels of fitness, but in particular those looking to start exercising again.\nMeeting from 9.30am beside the bandstand in the Peoples Park, the 4km route will bring participants done the beautiful East Pier and back, is based on pathways and suitable for all levels of walker. Supported by D\u00fan Laoghaire-Rathdown County Council, this event will provide the perfect chance to put those New Years resolutions into action.\nProviding the perfect chance to put those New Years resolutions into action, all are very welcome and encouraged to attend. Trained walking leaders will be present on the day for a helping hand and there will be a hot cuppa at the finish line for all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A process paper is a paper that describes how to do something or how something has already occurred. It utilizes a systematic step-by-step style of writing that is easy for the reader to follow. Here is a reliable guide on how to write a process paper.\nRemember that the process paper should be easy for the reader to follow. In this paper you are either expecting your reader to duplicate your procedure, or you are expecting your reader to follow and understand your method well. Therefore, keep your writing style on the simple side. Your main focus is for your reader to follow your process easily and quickly.\nUse transition words at the beginning of each area that introduces a new step. This is done so that the overall writing and direction flows naturally. If you pay attention to the way you naturally speak when explaining something to a friend for instance, you will notice that you naturally insert words like initially, during, afterward, etc.\nAvoid repetition of the same transition words. Process papers often contain the words: first, then, next, then, then, then for example. Yes, these words transition each step, but they are redundant and make for boring reading. Try coming up with better and different transition terms, such as: after step 1, simultaneously, immediately after and lastly.\nDon't write your process paper as if you are trying to win some sort of literary prize in literature. This is not the intent of a process paper. The writing should be focused and to the point. Therefore, avoid using extra adverbs or adjectives, and omit overall excessive wordiness. Each sentence should be direct.\nIt is not always recommended to use subheadings or bullet points or other elements used to break up a paper, such as this handy guideline you are reading right now. So, ask your teacher what he or she prefers. While the use of elements to break up chunks of writing is encouraged, in a process paper this isn't necessarily the case. You want your reader to stay focused and read your procedure all in one focused read.\nIf you follow the points outlined in this guide, you will see that writing a process paper is actually one of the easier types of papers to write. Simply write how you speak when explaining a process out loud. Also, do not forget to proofread your paper for spelling, punctuation, and grammar errors and overall sentence structure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Chinese internet company Xiaomi was founded in 2010. In 2014 and 2015 it was amongst the top five (sometimes even top three) smartphone makers world wide.\nAnd with a valuation of over USD 46billion it is one of the world's most valuable technology start-ups. Xiaomi might be known for its low-budget and stylish smartphones, targeting China's middle class and its youth, but that is only part of what gave it that valuation. Investors believe that Xiaomi can build and ecosystem with its smartphones, smart home products and services consisting of games, apps and others. Its potential lies in bringing revenue through hardware-purchases as well as recurring purchases through services.\nXiaomi's portfolio represents that idea. Its offerings include a set-top box, rice cookers, a water and an air purifier, a drone, a fitness tracker,TVs, notebooks and a virtual reality headset as well as accessories such as power banks and audio devices.\nIn China, Xiaomi's market share in the smartphone sector fell almost 43% from Q3 2015 to Q3 2016 losing to its competitors Huawei, OPPO and vivo.\nIn regards to that Xiaomi recently said that sinking smartphone sales do not affect the company that much because they care more about recurring revenue streams than one-time payments.\nIn the same breath, Xiaomi highlighted its focus on smart home devices. In April, Liu De, Xiaomi's Vice President stated that its smart home business is expected to double to 10 billion yuan ($1.5 billion) this year.\nThat statement leaves it open to interpretation how the company sees the position of its smartphones in their overall strategy. Xiaomi wants to profit from its smartphone users through apps and other services. That is only possible if they ship smartphones is not happening. Moreover, the amount of smartphones introduced makes that statement even more confusing. (There are currently twelve smartphones in Xiaomoi's portfolio and at least seven introduced in 2015 and 2016).\nWith that in mind, lets look at what Xiaomi can do to solve its growth problem.\nTraditionally, Xiaomi has limited its distribution to online sales. However, tradition retail is still very important. When you look at the distribution of online and offline retail sales, this makes perfect sense.\nIn 2013 online accounted for a mere 8% of total retail sales. Nevertheless, e-commerce is not to be disregarded fully, especially as online retail sales are predicted to grow about 20% annually over the next two years. Because OPPO and vivo have invested in offline retail they were able to surpass Xiaomi.\nXiaomi should serve its customers where they like to shop, which in this case is online as well as offline. Also, new product releases if they aim at a mass market, require people to test and see them. Your \"average\" customer is unlikely to spend $600 on a phone she has not seen or tested before. Further, that give them opportunity for after-sales services (which has been a topic of complaint by its customers).\nIn a study conducted this year, only 37% of Xiaomi owners reported that they would buy an Xiaomi phone again. As a reference, 74% of Apple users reported that they would buy an iPhone again. Further, Xiaomi users have stated its dislikes with the brand. In India, for example, customers reported\"poor after-sales support, non-availability of spare parts\" and long repair times. Also, users of Mi phones reported cracked screens, issues with the headphones jack and overheating. Another sentiment that is hurting its brand is that people perceive it as \"cheap iPhone copies\". With a strengthened brand and the right channels Xiaomi has to have the right products to deliver to the right customers.\nThis is especially important in the smart home sector. A cracked screen is something customers can forgive, but a malfunctioning security camera will be far more harmful to their brand reputation.\nChina has a market saturation of about 65% in total. Whereas 85% of 18\u201334-years-olds have a smartphone, only 43% of all 35+ have one. This gives us a potential market growth of about 320 million (280 million 35-years-olds and 50 million 18\u201334-years-olds).\nCalculation: Considering only the 35 to 64 years old to be buyers of smartphones, we get a total of about 570 Millions (of a total of 1332810869). Assuming 43% of that already covered, we get a potential market growth of about 280 million. (Assuming the population stays the same). The same principle applies to the other age group.\nAlso, after initial decline in average selling per smartphone, Chinese have started spending more on their more on their smartphones.\nFrom 2010 to 2013 the average selling price's (ASP) CAGR was about \u201311%, from 2013 to 2015 it rose on average 55% annually and was at $255 in 2015. This is still below the global ASP of about $300. This becomes extremely interesting when you look at the ASPs of oppo, Huawei and Xiaomi. Xiamoi's phones average at about $215, oppo's at about $280 and Huawei's at about $470. However, with its newly released, about $600 expenise Mi Note 2, is embarking into the premium segment. Nevertheless, even if Xiaomi makes premium-priced smartphones, they won't be successful if customers do not trust the firm.\nXiaomi's smartphone prices only cover its costs without margin. Also, these smartphones are high-end devices that compete with the high-end portfolio of its competitors.Considering financial resources needed to develop its retail activities, not raising prices is rather unrealistic. As a consequence, they would have to raise its prices to compensate for the costs of retail.\nFurthermore, first-time buyers are more and more delaying repurchases. However, if they repurchase, they not only tend to go for higher-priced smartphones but also to avoid Xiaomi due to its hurt brand.\nIf Xiaomi wants to reach the 35+ years old flash might not work that well amongst the older generation. They have to switch to alternative ways of marketing.\nXiaomi must not forget that also the Chines smartphone market will get saturated. Therefore, it has to look at several ways on how to \"beat\" the product life cycle. One way is simply put innovation.\nOne way for Xiaomi to boost sales is to introduce new smartphones with substantial innovation. If you look at the chart of the iPhone's market share and new iPhone releases you see that each new release ignited the market share.\nOne way to do so is to focus on currently arising trends. Some of which Xiaomi could focus on are VR-ready smartphones and ubiquitous computing.\nBesides innovating the smartphone, there are several complementary products smartphones users are increasingly using with smartphones. These are devices such as wearables and smart home appliances.\nThe wearables sales revenue is expected to grow about 20% annually.\nSimilarly, the global smart homes market is expected to grow 20% annually.\nAnother interesting category might be megatablets, especially as competitions is not that strong in that sector yet. Whichever product category Xiaomi chooses to focus on, they have to evaluate its potential in detail. They also have to do that with is current portfolio.\nXiaomi wants to build an ecosystem and be present in your whole house. Whereas that might be a nice thought, I think it is unrealistic in the speed Xiaomi is trying to achieve it. Xiaomi has a very wide, but in some lines not very deep portfolio.\nThey have twelve phones, but they also have one tablet, one notebook, four TVs, two sound systems, several smaller devices such as routers and set-top boxes a segway, a rice cooker one air and one water purifier and two types of security cameras.\nIf you look at how the competition in each of these sectors is (take Samsung for TV, Netgear for routers, Apple for tablets, JBL for sound systems etc.) it is easy to see that they are fighting on a lot of fronts. Each of these products require heavy R&D costs which can only be compensate over time through economies of scale, learning and experience curves and other volume advantages, I strongly advice them to focus on a few promising categories.\nSo fare I have only slightly touched internationalization (I quoted globales forecasts under \"Boost sales through new product categories\"). There are several things to consider when going international. I will elaborate on Xiaomi's internationalization strategy in depth in a separate post, here I want to touch only on a few things.\nXiaomi has started going global already, but it has and will further encounter several issues. Products have to be changed hardware- (charing adapters) as well as software-side (Xiaomi's smartphones do not have Google's Android appstore inside China). Further, they have to invest in IP rights to be allowed to license and sell outside China (the have encounter legal issues in that area already).\nAlso, they are very little known outside Asia, therefore, they would have to invest in marketing (advertising as well es distribution).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each student, faculty and staff member has a part in the College becoming a more sustainable campus.\nBy changing just a few actions in your day-to-day routine, you can contribute to a greener, more sustainable campus. Become a Green Grizzly by implementing the conservation tips listed below.\nUse the sleep\/hibernate mode on your computer to save energy when not in use.\nTurn off computers, monitors and desktop printers at the end of the day and when not in use (during lunch or class).\nTurn off projectors at the end of class.\nTurn off the lights if you'll be gone more than five minutes.\nUnplug \"vampire\" electronics like cell phone chargers and laptops when not in use. Unplug televisions and stereos when away for break.\nReduce food waste in the Dining Hall.\nStop smoking and support a tobacco-free campus.\nTake the stairs instead of elevators.\nLower the sash on your fume hood when not in use.\nRemove yourself from junk email and catalog lists.\nSwitch as many bulbs as possible to compact fluorescent bulbs or LEDs.\nSet the thermostat at 74 degrees in the summer and 68 degrees in the winter, and use a programmable thermostat.\nDrive less aggressively and more slowly.\nKeep your engine tuned properly.\nHave your wheels aligned and keep your tires properly inflated.\nCombine trips when running errands.\nAvoid a single-occupancy car at least once a week.\nWalk, carpool or bicycle on campus instead of driving alone.\nWhile washing your hands or brushing your teeth, turn off the water until you're ready to rinse.\nDo full loads of laundry in cold water.\nPrint on both sides of the paper or re-use the back sides of old documents.\nBring cutlery and dishware to the office instead of using disposable products.\nCarry a reusable coffee mug or water bottle instead of using disposable cups and buying bottled water.\nGo paperless for banking, utility bills and credit card payment.\nReduce paper use, and think before you print.\nBring reusable bags when shopping.\nRecycle. Put recyclables in designated containers, not the trash.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The SPL Series are engineered with a segmented diamond rim and a slightly softer bonded segment for fast, aggressive drilling.\nLackmond's SPL Series Wet Concrete Core Bits are our most price conscious offering, combining quality and drilling speed for an excellent cost-performance value. They are designed for use on reinforced concrete with a variety of aggregate hardness. The SPL Series are engineered with a segmented diamond rim and a slightly softer bonded segment for fast, aggressive drilling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Tatari is expanding, and we are looking for you to join! The position is located in Santa Monica, CA (no remote, no exceptions). Tatari pays competitive salaries, and you will be eligible for all Tatari perks (including full healthcare coverage, equity, FSA and Commuter benefits).\nPurchase all types of TV inventory: local\/national, broadcast, cable, OTT, etc.\nCultivate relationships with the media industry and TV networks, at both junior (i.e. account executives) and senior levels. Maintain daily contact and continuously update inventory pricing and availability.\nBe able to negotiate pricing to benefit Tatari clients.\nWork closely together with the client strategy and analytics teams. Review every single media purchase through a data & analytics lens.\nMonitor stations for competitive activities, economic and demographic changes, etc. Seek out unique opportunities e.g. firesales, packages, etc.\nParticipate, actively engage, and be the expert voice on TV in Tatari client meetings.\nHelp automate and optimize media-buying with our technology and product team.\n5+ years of experience with TV media-buying at all levels: local\/national, remnant\/upfront, cable\/broadcast.\nExisting relationships with media companies across junior and senior levels.\nPassionate about TV \u2013 doesn't just know networks, but also very familiar with programs, shows, and personalities.\nStrong and fast with numbers. A streetsmart dealmaker.\nPro-active: doesn't wait for media companies to provide pricing but continuously seeking new opportunities.\nGood instinct about the TV market place.\nHigh attention to detail. Strong communicator, verbally and orally.\nAppreciate engineering, data and analytics.\nUnderstanding of marketing and sales funnel KPIs.\nSkilled at Excel and Google Sheets and a generally fast learner when comes to software.\nEmbraces automation and can engage with engineering team to enhance tools used to increase efficiency and accuracy of media buys.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Materials Science and Technology with special reference to Exploration, development and production of special\/New functional Glasses and transparent nano glass-ceramics for Opto-electronic and photonic applications.\nA. D. Sontakke, K.Biswas, A. K. Mandal, K. Annapurna, J. FLUORESC., 20 (2010) 425.\nK.Biswas, A. D. Sontakke, J. Ghosh, K. Annapurna, J. AM CERAM. SOC., 93(2010) 1010.\nK.Annapurna, Maumita Das, P.Kundu, R.N.Dwivedi, S.Buddhudu, J.MOL.STRUCT.741 (2005)53.\nK Annapurna, RN Dwivedi & S Buddhudu, OPTICAL MATER. 13(4) (2000) 381.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Canadian tour of a global torch relay to raise awareness abouthuman rights violations in China ignited in Halifax yesterday with arally, run and live music.\nThe Canadian tour of a global torch relay to raise awareness about human rights violations in China ignited in Halifax yesterday with a rally, run and live music.\n\"We do not boycott our Olympics, but we want Olympics with human rights,\" said Muhong Wang of Halifax, one of the local organizers.\nThe global grassroots effort has touched down in 30 countries and will span six continents and 40 nations. Its goal is to raise awareness of human rights violations in China leading up to the Beijing Olympics, which begin in August.\nChina has also been criticized for its tactics in quelling anti-government riots and protests in Tibet.\nHalifax NDP MP Alexa McDonough lit the torch yesterday in front of the Spring Garden Road Memorial Library and about 50 onlookers.\n\"Our plea in coming together in gatherings large and small across the planet is to say to the Chinese government that it is an abuse into the very agreement in which you entered to host the Olympics, for you not to have seized the opportunity to clean up your act,\" she said.\nMcDonough then lit two more torches carried by women dressed as Olympic goddesses. The women ran 31\/2 kilometres to Grande Parade Square, where there was live music and more speeches.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The French have an interesting phrase for dusk: entre chien et loup. Literally translated, it means \"between dog and wolf\". More precisely, the phrase might be rendered as \"twilight\" in English: when the level of light is so low that one is unable to distinguish between these two similar, yet very different, four-legged creatures.\nBut the French phrase means more than that, implying that it's also the time between what is comfortable and familiar and what is dangerous. What is unknown and frightening.\nReworking a text that is familiar, but needs major work was, for me, entre chien et loup. Terrifying because you can see\u2014and not see\u2014how to proceed. You can still make out the shape of the old narrative, but the new chapters are dissolving.\nI had been working on the rewrite of No Angel Hotel for months; the more I played with the \"canvas\" of the text, the muddier (and uglier) it became. I was about abandon the project forever when one of my students gave me some wise advice. Gwendolyn Kosten Modder had read the published version and loved it and she told me, It's of its time, Anne. Don't tinker with it too much.\nSo I went back to the drawing board. I took the original and compared it, word by word, with the muddy, reworked version, keeping the best of them both and correcting some major plot inconsistencies.\nNot many writers have the chance to rework their prose once it's been in print, but I was lucky enough to be able to. The result is the same book, but one that is completely different and, hopefully, much better. And if you don't believe me, buy the new one and get the old out-of-print one on Amazon for one pence (plus postage).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here you can find a little collection of all the pictures taken 2015. I hope you enjoy them.\nI also hope that these pictures give you a good and close impression of our foundation.\nVisitors are always very welcome!\nOn the 1.April 2015 we celebrated our 16th Anniversary. A big Thank You to everyone who supported us and therefore has helped to turn Koh Samui into an animal friendly island!\nThank you very much again to all the sponsors and helpers from all over the world. Because of your help, since 16 years every dog and cat can be helped for free on Koh Samui. Stray animals which have not found a place to survive on their own on Koh Samui can be brought into the shelter.\nBecause of your donations, we can still neuter at least six animals every day. In the last 16 years, over 20,000 dogs and cats have been neutered and countless vaccinations and treatments have been carried out. We offer free medical treatment for ALL stray cats and dogs, with a pick-up service even beyond our opening hours (as long as we have a helper available at this time). For dogs and cats with an owner, we must ask for coverage of the medicine costs. Medicine in Thailand is rather inexpensive and those who have no money will still get medicine for free.\nOur animals love when visitors come by!\nAnd more visitors.... Wow we are spoiled with love and cuddles!\nA big thank you to Karin and Doris.\nPetra and Dirk are big PUG-FANS - We all love pugs!\nMartina is here once again!\nTotally vegetarian! But they still taste delicious.\nin the Russian community for us!\nI wonder what's in the bag!\nThank you Nicole & Martin!\nPhilip and Felicity have bought a lot of puppy food for us, A huge THANK YOU to you both!\nFoxy had only hair on his head when he first came to us.\nAnd after 2 months....he looks like this now.\nDogbed? who said it was a dogbed?\nThank you so much for the lovely beds!\nOur cats and dogs love to rest in them.\nThis is Samuis' favourite position. She will soon fly to Claar in the Netherlande.\nBranco is our 'smallest dog', he is 11 years old.\nRelaxing time on the bed.\nMet preparing the fresh cooked dinner for our cats!\nas well as at the shelter.\nWuschel is getting a hair cut!\nCome on, jump in too, we want to play!\nHans-Peter?, When are you going to bring me home?\nI will follow you everywhere!\nHans-Peter, of course, was present at Dr Sith's 10-year jubilee celebration. He has been living for quite a time already in Kannom on the continent, which is rather close to the ferryslot. He is an IT expert and has taken care of our computer- mostly by telemaintenance- for over two years now. Presently we have a second computer linked to the one in our bedroom.\nThe new working place is in a small room which had rarely been used ; up to now the computer has been in our bedroom, so the new room is an ideal solution, as Werner and I tend to have a little afternoon nap : having some privacy at least isn't that bad, is it?\nMaxime used to live at the Imperial Boatshouse for many years and been with us for treatment several times.\nDam is offering Maxime all kind of different dog food and is making sure that she can eat in peace.\nI'm not gonna bite you!\nAnd of course the ears are also important, stay still!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Do you make New Year's resolutions? I do. I make a resolution to NOT make a resolution. I always disappoint myself because I can never keep them. I have a few things in mind I would like to accomplish but if I tell you then I have to be accountable and I'm not sure I'm ready for that responsibility.\nLast night I had a house full of giggly girls ages 8 and younger. I had decided (why?) that I would have an evening to celebrate New Years by hosting a little craft and movie night.\nOur refrigerator broke down on Christmas Eve and finally the parts arrived yesterday. Of course, the tech appeared 10 minutes after all the girls arrived to install them. Naturally! My plan was to have the girls decorate some sugar cookies, make individual pizzas and then watch Ramona and Beezus together, complete with brownies and popcorn.\nSoon, our refrigerator was literally parted out all over the kitchen floor and the tech was completely folded up inside the bottom freezer compartment. In the midst of all the activity though, I was able to wipe and scrub down my freezer when he crawled out. How is that for some multi-tasking? Our cat decided he liked the tech and proceeded to settle in for a nap on his jacket. This was after milling through and sniffing all of his equipment. Nothing like a little cat hair getting stuck in our new motherboard or worse, the ice machine.\nI wish I had some sordid details to share with you over what the girls said or did but surprisingly it was pretty uneventful. Aside from some spilled drinks, sugar highs and tales of what their dads wear to bed at night (ewww!), they seemed to have a good time.\nStill, I'm hesitant to call this a new annual tradition. Happy New Year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CNN's Senior Editor of Middle East Affairs, Octavia Nasr, is coming under increasing pressure for a comment she made on Twitter.\nShe tweeted: \"Sad to hear of the passing of Sayyed Mohammed Hussein Fadlallah\u2026 One of Hezbollah's giants I respect a lot,\" following the spiritual leader's death on Sunday. Israel-apologists are accusing her of supporting terrorism. And if they get their way, they'll soon hound her out of her job.\nIndeed. And may Octavia keep her job.\nThe Syria News Wire is written from Damascus and London. It was the fourth Syrian blog to appear on the internet \u2013 back in 2004. It is a Lonely Planet favourite, and award nominated, Toot-ified blog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"from photos provided by Mary George.\nBAGSHAW, Nancy Sheila. 2 February 1918 - 12 September 2005.\nBAGSHAW, William Garlick. 26 May 1909 - 28 April 1970.\nBLACKWELL, John S. 10 July 1985 aged 69.\nCAKEBREAD, Catherine Ellen. Wife of Leslie, mother of Ronald, Marilyn and Gregory. 6 July 1922 - 8 December 2010.\nCAKEBREAD, Leslie. 2 April 1920 - 7 December 1999. Husband of Catherine, father of Ronald, Marilyn, Gregory.\nDODD, Hilda Winnie. 3 October 1964 aged 71 years. Wife of Thomas.\nDODD, Thomas. 12 January 1981 aged 93. Husband of Hilda Winnie.\nEWEN, Arthur Frank. 21 June 1967 aged 80.\nEWEN, Emily E. 17 November 1985 aged 84.\nFRANK, Roy. 5 November 2006 aged 78 years.\nHEADING, Karen. 8 June 1962 - 13 November 2002. Wife of Peter, mother of Laura and Morgan.\nHENDRIE, Edward Douglas. 7 October 1971 aged 52. Husband of Wanda, father of James, David and Kym.\nHENDRIE, Wanda Jean. 20 January 1986 aged 62. Wife of Edward, mother of James, David and Kym.\nKITTO, Reginald Edward. Husband of Ina, father of Nita, Elma, Verna and John. 28 September 1968 aged 75 years.\nNICHOLLS ,Vivienne Violet. Wife of G. H. 11 September 1986 aged 82.\nNORTON, Sharon Louise (nee Watson). 27 May 1965 - 11 September 2011. Wife of Andrew, mother of Ryan, James and Sarah.\nOKE, Reginald Murvyn. Husband of Evelyn Grace. 3 November 1921 - 21 January 1993. Father of Peter.\nTOLL, Ruby V. 25 June 2001 aged 90. Wife of Fred.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for Hero hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for The Rebirth Of Kirk Franklin hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for The Nu Nation Project hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for God's Property hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for Whatcha Lookin' 4 hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for Kirk Franklin And The Family Christmas hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.\nNOTE: We're looking for a knowledgeable Kirk Franklin nerd! A review for Kirk Franklin And The Family hasn't been published \u2014 yet. We need someone who can write a full track-by-track review of this album (at least a couple paragraphs per song); if you know the music, you can submit a review. You'll be compensated when visitors make purchases through vendor links on their pages \u2014 for as long as your review remains on the site. Get more details in the FAQ.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":".Rod Ingle's Free Annual Christmas Gift!!\nMerry Christmas Dinar Recap Readers!!\nWell here we go folks! A very Merry Christmas to all my friends at home and abroad!\n2017's almost gone but there's always time for some more A cappella Christmas cheer!\nAgain, I'm sending out my free a cappella musical Christmas gift! It's going through all of my email and social media outlets so you may get it twice!\n#1 For best effect, listen with earbuds or larger speakers.\n#2 Turn on the Christmas tree and have a warm holiday beverage while listening.\n#3 Rinse and repeat lol.\n#4 (& most importantly) SHARE & SHARE AGAIN with friends!\nThe Jive Aces present: \"Bring Me Sunshine\"\nThis music video should put a smile on your face.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This week in Kentucky politics, Rand Paul was one of the few politicians to defend Donald Trump's summit with Russian President Vladimir Putin. Gov. Matt Bevin's administration restored vision and dental benefits to almost 400,000 people on Medicaid after taking them away earlier this month. And Kentucky's bourbon industry ramped up its warnings about how a trade war would impact the state's signature industry. Listen to this week's edition of Kentucky Politics Distilled in the player above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Titan Souls has come a long way from humble beginnings at Game Jam to now a multiplatform release title. So the developers over at Acid Nerve have released a trailer showcasing the evolution of the game.\nThe Shadow Of The Colossus-inspired title with some Dark Souls level difficulty in there has fans pretty interested in what Titan Souls will offer upon release. The players goal is to collect shards of a Titan's soul, the little obstacle that is in their way is that each shard is guarded by a massive Titan. Players are expected to die, again, and again.\nThe video compares the 2013 Titan Souls prototype to the finished product that is set to release later this month. You can clearly see that strides have been made over time to make it a very polished looking title that fans will be happy to get their hands on. The video shows off several parts of the game and fans are able to get an early look at the title as well as compare it to the 2013 version. Check out the trailer below.\nIt was also recently announced that Titan Souls now has a demo available on PC that players can download via Steam. Titan Souls is set to release on April 14th for PC, PlayStation 4 and Vita.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Encore Property Group we provide a comprehensive property services around Perth, and we are offering highly professional assistance on various aspects of real estate properties. To be able to grow and provide you with top real estate services, we would like to hear from you and your opionon how we can improve our assistance to you.\nWe welcome your feedback and comments about our services.\nDo you have something nice to say, something not so nice or even something bad to say about us? We love to hear it as that is the only way, we can improve our service. We can learn from the honest feedback.\nWe always collect feedback in order to improve our services via our forms. Now we would like to collect feedback online in order to save environment and use less paper.\nEncore Property Groupo 1 years ago.\nThank you Damian. It is a pleasure to work with you!\nProperty Management. Truly Amazing Service!\nHaving owned a number of different investment properties, I have had dealings with numerous estate agents and property managers over the years.My experience has been that most agents are focused on the selling of properties and while you might get the \"red carpet\" treatment as a buyer or seller, this quickly changes when you are a landlord. The other problem with estate agencies is that rental managers usually have many hundreds of properties so even the most conscientious manager can give your property only a little time. The approach at Encore Property is completely different and extremely refreshing. Michelle and Paula have handled my properties and they are by far the best property managers I have ever worked with. They understand the market as well as the needs of both owner and tenant. They have found several tenants for my properties and on all occasions I have been extremely happy with the tenants and also the level of rent even in this trying economy .\nWhen there have been maintenance issues Michelle has found quality, professional tradesmen and no job has required re-work.\nI would not hesitate to recommend Encore Property Group to anyone seeking a managing agent.\nI have been using Encore Property Group for more than 4 years for property management services, initially through a referral from a friend. Encore Property Group have always gone the extra mile and taken on the management of my property as if it were their own; far exceeding my expectations, year after year. Michelle and the team keep me well informed, works closely with my tenants, and is always proactive and positive. Nothing is too much trouble and I constantly refer Encore Property Group to friends and colleagues. I would have no hesitation in recommending them to anyone looking for a reliable, hardworking property manager. It is rare these days to have a business relationship where the supplier is a true business partner and understands your needs \u2013 Encore Property ticks both of these boxes.\nThank you Colin so much for your involvement during the sale and settlement of our new property. Even though I was very stressed, it was nice to have you around for support. I hope our relationship will grow bigger and better as time goes on. I am looking forward to working with you in the future and will always recommend you and Encore Property.\nChris and I were very impressed with the professionalism and the expertise that was shown to us from the moment we meet Colin Anderson and the team at Encore Property Group. We soon learnt that any issues that arose were quickly resolved and over the weeks we received weekly phone calls to let us know how the Sale was progressing and to see how things were going with us.\nFrom day one, the staff at Encore Property have been fantastic.\nAt first we weren't too sure if we were going to rent or sell our property. I contacted Encore in the morning and I had Colin and Michelle over at my place before lunch go through both options.\nFast forward a few months and we ended up decided to rent our property with Karen as our property manager. Again, when I rang through to the office, Karen was out to us in no time.\nThroughout this whole process, Karen has delivered exceptional service, providing updates and returning my calls within a couple hours. Karen assisted in getting the property prepare for rent and found a tenant in less time than I have expected. This was a super stressful time and Karen made it so much easier.\nI strongly recommend Encore property as the service is absolutely fantastic. i was a tenant with them for 6 months and i dealt with Karen and Michelle, I have to say they have wonderful personalities and they're very understanding. Thank you for having us.\nOur property management services will save you hassles!\nYou don't want this happen to you - watch the video below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Space for lectures, seminars, e-learning, debates, group work, group discussion, research projects, problem-based learning, reports and presentations, videos, computer simulations and workshops.\nThe School and its facilities are based in the landmark building K2 in the heart of our Southwark campus. K2 features specially designed lecturing and teaching rooms.\nWe have a range of skills labs suitable for a wide variety of professional courses, including practice skills learning. Our simulation equipment and facilities are available for use within many courses.\nPrimary and Social Care's practice-based learning approach centres on raising students' awareness, knowledge and conceptual understanding. This is combined with the learning they achieve on placement; time is then set aside for them to reflect on their performance with teaching staff.\nOur rooms have been specifically designed to create the ultimate venue for effective lectures, seminars, e-learning, debates, group work, group discussion, research projects, problem-based learning, reports and presentations, videos, computer simulations and workshops.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Corporate gifts- big and small \u2013 are also something she is happy to make available as well as customized items.\nLocated in the Pinery Plaza. Shan Home & Gift is now open!!\nDrop in \u2013 you won't be disappointed!\nDrop in - you won't be disappointed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvxhs b/data_all_eng_slimpj/shuffled/split2/finalzzvxhs
new file mode 100644
index 0000000000000000000000000000000000000000..edc1fb2703d8c4f07453c5ed1e9c318dadff7b12
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+{"text":"Jehu was anointed to wipe off the house of Ahab for his atrocities. The time Jehu was supposed to carry out the assignment was when Ahaziah, Jehoshaphat's grandson, travelled to Joram. Little did he know that death was prepared for him there in Joram. Ahaziah had no body to succeed him because after he was killed, the mother, Athaliah, killed all other royal heirs of the house of Judah except Joash who was stolen away by Jehoshabeath. Athaliah was Jezebel's daughter; exercising evil influence on men in power was something she inherited from her mother. Athaliah reigned in Judah for six years even though she was not eligible to do that but had her way because of the spirit of Jezebel, her mother, which was in her. Such spirit forcefully acquired what belongs to another.\nLet us seek for the old path where we could find rest. Let us listen to the trumpet's sound, wherewith we could escape the calamity to come. Our reading for today is 2Chronicles25-27. Keep the fire burning and alive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rules for Intrepid Living: Rule #63.\nRule #63. Don't take yourself too seriously.\nThere are a lot of very serious things happening in the world right now. We, as a society and culture, have a lot of problems that we need to solve, and we can't agree on how best to do that. As individuals we have problems as well. If nothing else, we have the day to day pressure of paying bills, going to work, and being responsible citizens. It's serious business, and it can stress you out, wear you down, and burn you out if you're not careful. But there is something that can help you through the tough times.\nYeah man, that's good stuff. See, the world is a serious place, and there are lots of things we should take seriously. But we shouldn't take ourselves seriously.\nHow often are you laughing?\nI try to laugh as much as possible. If something makes me happy, I laugh about it. If something makes me sad, I laugh about it. If something makes me mad, upset, afraid, nervous\u2026 I laugh about it. Basically, I believe in laughing about everything. I understand that there are times, places, and situations where it could be disrespectful to crack a joke, and I believe in being polite and respecting others. But taking yourself too seriously isn't a healthy practice.\nLower blood pressure, which reduces the risk of stroke and heart attack.\nReduced levels of stress hormones such as cortisol.\nImproved immune system through boosted T-cell activation.\nReduction in chronic pain by triggering the release of endorphins.\nAll these health benefits could be yours, all you have to do is laugh! Some studies suggest that laughter can even improve alertness, memory, and creativity. Just as important, laughing is fun. It's hard work to deal with serious business. It feels a lot better to embrace a little bit of funny business and laugh at things.\nSo whenever you're getting stressed, overwhelmed, upset, or sad, find something funny to laugh about. Maybe you pull up a favorite comedy on Netflix or call that goofy friend of yours that can always make you chuckle. If you're lucky, you'll have someone like me in your life who will make fart jokes until you're laughing so hard you can't speak.\nWhatever you do, don't take yourself too seriously. Life is both long and short; it's better to enjoy the ride as much as you can.\n\"Rules for Intrepid Living\" is a recurring post suggesting how we might all live more adventurous lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have a home bar that's missing that certain something and happen to love Maker's Mark bourbon whiskey at the same time, look no further than getting one of these cool new yet aged Authentic Maker's Mark Bourbon Whiskey Barrel. This genuine, one-of-a-kind white oak Maker's Mark bourbon whiskey barrel comes straight from the distillery in Kentucky and it's naturally aged wood is protected with a clear coat seal. It's perfect for use as table, shelf, decorative item, jump over it with ice skates on, or whatever you can think up while sharing (responsibly) a glass or two with friends. The only downside... it's not filled with Maker's Mark bourbon and I wish you could get the top dipped in the iconic red wax.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"alrighty then ......THIS IS BEGINNING TO SOUND LIKE A DISNEY MOVIE!!!!\nI've been listen to The Killers so much lately that I've kind of abandoned everyone else... BUT NOT ANYMORE I TELL YOU........ James Blunt is once again blasting tunes from my headphones!\nI LOVE THIS PART THE MOST!\nWhich artist or artists have been monopolising your play list recently ?????","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can make easy, fun and unique crafts for your baby's nursery!\nWaiting for your baby to arrive while you're pregnant can feel like FOREVER, but one way to take your mind off the wait is to delve into decorating your baby's nursery.\nI have an easy project idea to decorate your baby's room: A hand painted picture.\nIn my example we did our baby's initial, but you can paint whatever design you feel inspired to!\nAll you need is to visit your local craft store.\nYou need a blank canvas (any size!), at least 2 paint colors (I used 3) and some ribbon.\nYou start by painting the canvas (try to incorporate nursery design themes and colors). Once you've achieved your design,let the paint dry.\nThen attach the ribbon to the back so you can hang the painting on the wall. We hung ours above my baby's crib!\nSo cute & unique and I cannot tell you how many compliments I've gotten from it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..683f2290a3f3748978226c2f84636c7c60804ef9
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@@ -0,0 +1,5 @@
+{"text":"Mary Carroll-Hackett took the MFA in Literature & Writing from Bennington College. Her work has appeared in Carolina Quarterly, Clackamas Literary Review, Pedestal Magazine, Praxilla and Drunken Boat, among numerous others. Her collection of prose poems, The Real Politics of Lipstick, was released in August 2010 as the winner of the 2010 Slipstream Press Poetry competition, and she has another forthcoming, Animal Soul, from Kattywompus Press. She teaches Creative Writing at Longwood University where she also edits The Dos Passos Review, administers the Liam Rector First Book Prize for Poetry, and edits SPACES, an online journal of art and literature.\nHere, touch here, just beneath the ribs you kissed when you were nearly twenty-three and I was a girl knee-deep in meadow, fescue, timothy, mustard, the wild ginger I had just learned from my grandmother a few weeks before we met. Dancing in that bar, that night, I wanted to tell you how to spot wild ginger, how Carolina, where we both grew, banned spreading cornflower seeds, and I most wanted to tell that you smelled of sweet clover, and sun. In my silence, I named you shining one. Later, while you slept, I whispered secrets into your hair\u2014how black snake root would keep you strong, rue and thistle protect you, how the plant I named as spikenard\u2014lavender\u2014would lead you in my direction, when our time came. In the morning, I left rosemary behind, that you might, just might, remember. Because as false at times as desire may seem, it isn't. Nor is the humid wish for love, steaming, sweating, our second skins, those made of glass, the ones we fear the most, will shatter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ode to my Auntie Majbritt.\nbright red lipstick, a beret and a scarf.\nProduct care: Dry clean or gently hand wash in warm water with mild soap.\nGently squeeze to remove excess water and hang to dry. Iron while still slightly damp on low temperature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carmel is a highly specialised musculoskeletal physiotherapist working both for Anston Pyhsiotherapy and the NHS. After qualifying in 2005 from Sheffield Hallam University, Carmel worked in a variety of community settings including GP practices, learning disabilities and paediatrics.\nShe has specialised as a senior neuromusculoskeletal therapist for the last 5 years assessing and treating a wide variety of musculoskeletal conditions. Carmel is qualified in the advanced use of manual therapy techniques and adopts this concept in the management of musculoskeletal disorders.\nFollowing completion of post graduate training, Carmel's areas of particular interest are whiplash and spinal conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UBGB Officer Scale-I 2019 Exam Syllabus. An online Instructor's Resource Guide with testing material is available for each volume. To facilitate a one-semester course, McGraw-Hill's Essentials of Federal Taxation folds the key topics from the investments, compensation, retirement savings, and home ownership chapters that are found in Taxation of Individuals into three individual taxation chapters that discuss gross income and exclusions, for AGI deductions, and from AGI deductions, respectively. The exercises encompass many levels of biology, and sufficient introduction is given to each topic so that it may be used as a stand-alone item. For introductory courses in Computer Integrated Manufacturing and Automated Manufacturing. For a one- or two-term introductory course in discrete mathematics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SpaceVR aims to send its \"Overview One\" camera to the International Space Station to provide 3D, 360-degree views from the orbiting lab.\nUpdate for Sept. 15: SpaceVR has launched a new Kickstarter campaign to fund its \"Overview One\" project. You can learn more about it here.\nSAN FRANCISCO -- A new crowdfunding campaign aims to raise enough money to put a virtual-reality camera aboard the International Space Station.\nSan Francisco-based company SpaceVR launched a Kickstarter campaign today (Aug. 10) to help fund its high-definition camera, known as \"Overview One.\" SpaceVR plans to launch Overview One to the International Space Station and thereby provide 360-degree 3D views to users back on Earth from the orbiting lab's large-windowed Cupola module.\n\"Our project gives everyone the opportunity to become an astronaut and to realistically experience seeing the Earth in its entirety \u2014 recognizing its preciousness and fragility in our universe,\" Holmes added. \"It is our goal to offer this experience to anyone through every virtual-reality platform.\"\nThe 30-day Kickstarter campaign, which runs through Sept. 9, aims to raise $500,000. SpaceVR will use the money to cover launch costs and the first year of Overview One's operation, company representatives said.\nFor Overview One, SpaceVR is partnering with Nanoracks, a company that helps commercial customers make use of the International Space Station (ISS), and space manufacturing startup Made in Space. Made in Space built the orbiting lab's 3D printer, and the company will use a follow-on commercial machine to manufacture Overview One's camera casing, SpaceVR representatives said.\nSpaceVR is targeting late 2015 for Overview One's launch. Once the camera is up and running, it will stream live footage back to virtual-reality headsets in museums, classrooms and people's homes, company representatives said.\n\"When I traveled on the ISS and looked at Earth, my conception of the scale of our planet went from unimaginably large to absolutely finite and, in fact, small,\" SpaceVR board member Richard Garriott, a video game developer who visited the orbiting lab on his own dime in 2008, said in the same statement.\n\"I later learned that this sensation is common among those who have been to space, and is known as the 'overview effect,'\" Garriott added. \"We would all benefit if we had a shared experience of this kind. I am excited to see SpaceVR launch its Kickstarter project and take the first steps to making this happen.\"\nSpaceVR's ambitions don't end in Earth orbit. The company also hopes to send one virtual-reality camera to the moon in 2017, another to an asteroid in 2022 and a third to Mars, perhaps as early as 2026, company representatives said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxliu b/data_all_eng_slimpj/shuffled/split2/finalzzxliu
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index 0000000000000000000000000000000000000000..15b537d101e4f593ae9976a0bb168cafe53bc6e1
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@@ -0,0 +1,5 @@
+{"text":"Boxing Hand wraps provide optimal protection for your hands and wrist during training, combat and competition.\nBoxing Hand wraps are here to hold your hand together by providing support for your wrists, fingers, knuckles and the whole hand itself.\nMany boxers mistakenly think that boxing tapes are an extra cushion or protection for your knuckles.\nBut on the contrary, the purpose of the boxing bandage is NOT to cushion the impact; That's what boxing gloves are for.\nBoxing bandages are there to secure all your loose joints and your mobile bones. Boxing Hand wraps keep all your joints together so that the shock is better distributed throughout your hand.\nIn the end the boxing bandages are designed to prevent injury, which is essential in boxing because you have to protect your main asset and work tools: your hands!\nYou will find many Leone 1947 boxing bandage on our site Barbarians Fight Wear based in Strasbourg in Alsace, Bas-Rhin. We have many color boxing bands: pink, green, blue, red, black, gray camouflage, green camouflage, purple, orange, white, yellow, Maori, fluo green, fluo yellow, italian flag, french flag and thai flag. If with that you do not find your happiness !!!\nWe also have three band sizes: 2.5 meters (100 inches) | 3.5 meters (140 inches) and 4.5 meters (180 inches).\nWe recommend, size 2.5 meters for children and juniors or to use the bands with glove MMA or glove punch bag as smaller. For the boxing glove we prune you to take 3.5 meters or 4.5 meters.\nDo not hesitate to ask us to explain how to put boxing tapes. Either by going to our store in Strasbourg, our boxing club \"KFCE Sanda \/ MMA Schilitigheim and Bischheim\" or by sending you tutorials.\nWe are both boxers (Marie-Pierre and Michel ANSTETT) so we are able to show you the good method to put the boxing band.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delving into their imaginations, they come up with weird, wonderful (and, sometimes, a little wonky) doodles which will then publically auctioned on 21 September, raising much needed cash for Epilepsy Action. It's a great way to support an even greater cause. It's also the perfect moment to pause and remember the incredible power of creative daydreaming.\nAt OKIDO, we believe everyone should doodle. Not only with words and pictures, but with science too. The most magical results emerge when you mix them up together, and doodle with them all at once.\nAfter all, the next generation of scientists will need to think creatively to come up with inventions, medicines and solutions to make the world an even more wonderful place. That's where OKIDO comes in.\nAll our materials are designed to help tomorrow's creative scientists explore their curiosity and critical thinking. Fun, imaginative STEAM learning (science, technology, engineering, arts and maths) is at the heart of everything we do.\nMake a pendulum painting \u2013 the perfect combination of science, art and doodling!\nCressida Cowell, author of the How to Train Your Dragon series of children's books, has recently launched a campaign with the National Literacy Trust.\nCalled Free Writing Friday, it suggests giving children a notebook in which they can write or draw for 15 minutes a week. Adults are banned from making corrections or suggestions. Instead, children's imaginations are given free reign, and doodles rule OK.\nWe've teamed up with the Institute of Imagination for an extra special Lab Live S.T.E.A.M. Fair event. We're running the event over two days \u2013 so you've got double the chance to come and join us and the iOi for all the tinkering and experimenting you can handle!\nThink you need a lab to get your own small scientist's brain bubbling away? Think again.\nDid you know that Nobel Prize winning scientists also happen to be 17 times more likely to be painters than the average Joe?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SAN GABRIEL \u2014 Totally dominating.\nThat's how the Burbank High football team looked in the first half against San Gabriel.\nThat's the way the Bulldogs looked in the second half against the Matadors.\nBurbank was able to jump out to an early 37-point lead Friday in its nonleague game on the road. It appeared as if the Bulldogs would be on the giving end of a monumental blowout, and San Gabriel looked powerless to stop the onslaught.\nBut Burbank began to self-destruct. However, despite a valiant comeback attempt by the Matadors in the second half, the Bulldogs were able to hold on to a 44-36 victory.\nBurbank Coach Hector Valencia said he didn't think it was complacency that led to his team's struggles late against San Gabriel (0-3).\n\"No, I think it was just all the little mistakes that we were making,\" said Valencia, whose team committed four turnovers \u2014 all in the second half. \"Those are the things that really killed us. But we were the ones that shot ourselves in the foot.\"\nThe Bulldogs (2-1), ranked No. 4 in the CIF Southern Section Southeast Division, did receive a fine effort from senior running back Ulises Ochoa. Ochoa ran for 184 yards in 22 carries and had touchdowns of five, 22 and 12 yards.\nOchoa is one of the main reasons why Burbank jumped out to a 37-0 lead, as he ran for 145 yards in the first half.\nThe Burbank defense also played well in the first two quarters. It held San Gabriel to minus-11 yards of offense in the first quarter.\nBut the Matadors began to come back in the second half. They scored two touchdowns to cut the lead to 37-14 at the half.\nBulldogs senior quarterback Adam Colman also played well. He completed 14 of 22 passes for 224 yards and two touchdowns. However, he also had two interceptions.\nHis two touchdown throws of 20 and 46 yards were both caught by senior Ryan Thanaratnam. He ended with 133 yards receiving in six catches.\nThe Matadors continued to claw their way back in the game in the second half. When they converted a two-point conversion attempt with 7:31 remaining, San Gabriel was within eight-points of the lead.\n\"We just didn't execute well,\" Valencia said. \"We can't play like that against other teams. Our execution is something that we definitely have to work on.\"\nThe Matadors had one last position with 5:24 left. However, a fourth-and-nine play from their own 42-yard line fell short.\nWhen Burbank got the ball back, Colman ran for a first down with 2:29 remaining to allow the Bulldogs to run out the clock.\nWith a bye next week, the Bulldogs, who won a share of the league crown in 2009, open Pacific League play Oct. 1 at Muir.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For a lot of people, February is just a drag because it means that the shops are stocking up on Valentine's day things to sell to you and you're reminded quite bluntly that you're the single friend while everyone has plans for February 14th and all you have, for yet another year, is your TV and a box of tissues (you could even blame my fellow Scorpios for this because apparently we created this lovely day to scam everyone into conceiving more Scorpios. I love this idea better). I really want to say that I feel your pain and that we can all virtually hold hands around an imaginary fire and sing some anti-love songs but, firstly, I'm here with a lot of self-love posts lined up for you and secondly, I'm manifesting a Valentine's day date so I can't cry with you this year. Sorry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new designed folding mechanism is from up until today an unknown perfection. To unlock the legs simply push on the wide locking profile and the legs slide with a lateral movement when turned down.\nAn outstanding and innovative creation of black and white Vases AMALFI in drop shape by the designer Christian Haas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyxiy b/data_all_eng_slimpj/shuffled/split2/finalzzyxiy
new file mode 100644
index 0000000000000000000000000000000000000000..eb66d94ed9c065ae3312fb134995f82f0c8a24da
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@@ -0,0 +1,5 @@
+{"text":"I don't need to wear pants!\nI can talk to my friends IN PERSON!\nI can have a shower any day, since Mom won't be the one helping!\nSo thanks for keeping me company. You have no idea how isolated being stuck on the farm makes me feel. You have taken my mind off of it all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for visting the Utah State University Luminescence Laboratory homepage! We have been up and running since January 2007. We currently have two RISO TL\/OSL Readers and one with a single-grain attachment.\nWe specialize in the analysis of the optically stimulated luminescence (OSL) signals from quartz sand grains in geomorphological and geoarchaeological applications. We are also equipped to measure Infrared Stimulated Luminescence (IRSL) on potassium feldpsar. Please explore the 'Projects' and 'Publications' pages for examples of internal and external projects that the Lab has been involved with.\nPlease visit our How To Collect page for our guidelines and requirements on sample collection.\nShort course: Dr. Tammy Rittenour runs a two-week OSL training course for students and professionals, starting on Memorial Day. Spots are limited, click on 'Summer short course' in the menu to the left for more information.\nGeological Society of America happenings (2018) - GSA Events Page (external link) for upcoming regional and national meeting sessions related to luminescence dating.\nTammy Rittenour and Kerry Riley organized a successful 2016 Rocky Mountain Friends of the Pleistocene field trip to southern Utah in mid-October. Please visit our 2016 RM FOP Field Trip page for details from the trip.\nThank you to everyone who made the 2013 9th NWLDW in Logan a great success!\nSee the conference proceedings in Quaternary International.\n9th New World Luminescence Dating Workshop attendees outside of the USU Geology Building.\nAbove Horseshoe Canyon, UT -- the location of USU research regarding the relationship of its alluvial stratigraphy and characteristic \"Barrier Canyon\" rock art.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Japanese have a word for the act of acquiring books and letting them pile up unread.\nTsundoku\u2014a play on the words tsunde (\"to stack things\"), oku (\"to leave for a while\") and doku(\"to read\")\u2014is recognizable to book hoarders worldwide. You see a title and you just have to have it, even though you already have more unread books on your bedside table than you could possibly read in a year \u2026 but you're going to get to it eventually, right? Tsundoku is such a relatable but untranslatable concept that it regularly re-enters our Western consciousness through online articles. In fact, Googling the word might lead you straight into another problem. I call it \"Tab-sundoku,\" or the even more recognizable act of opening tabs and letting them pile up unread.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chile is home to a number of unique buildings, but the country's seaside dwellings steal the show when it comes to originality. Casa Cox, designed by the locale's very own Zeta Arquitectos, bridges the gap between a forest-oriented cabin and a stilted surf shack \u2014 two things that you don't often pair in the same sentence. To develop the minimalist space, Zeta took to the drawing board \u2014 looking to create a friendly, cozy common area that could be used as both a sociable retreat and a secluded getaway. An exterior adorned in slatted wood siding, intriguing stilts, and a raised patio area give the home its very own character, allowing denizens to spend their evenings admiring the surf. Within the structure, a single solitary living area plays host to the dwelling's primary living areas, complete with a modernized kitchen, wood-burning stove and chimney, and a large, front-facing window that perfectly frames the home's oceanfront view. An upper level that can only be accessed by ladder features four separate beds, as well as its very own private deck, giving family and friends a splendid view of the surrounding area.\nHere is the final film for the Mars Habitation competition for NASA A wonderful project to work on and our personal endeavours were completely matched by the HASSELL and Eckersley O'Callaghan Teams. All of us at LightField London hope you enjoy it as much as we enjoyed making it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The African Storybook (ASb) visit to Nigeria to facilitate the quality assurance of the Hausa storybooks on the website was a follow-up workshop to one that Tony Lelliott held in Abuja in May, 2017. Its purpose was two-fold: to introduce the initiative, storybooks and website tools for translation and adaptation to college lecturers from Katsina, Jigawa and Zamfara states, and State School Improvement Team (SSIT) from Jigawa state and to carry out editiorial quality assurance of storybooks translated into Nigerian Hausa. Tony Lelliott and Dorcas Wephukulu facilitated the process.\nIt was a very successful visit. We held a two and half day workshop for nine college lecturers from the aforementioned colleges and the State School Improvement Team. The group started off by translating into Hausa (Nigeria) Africa unity race, Kalabushe the talkative, Two little friends, Big blue bus and Byantaka and the dead pot. Each of these storybooks was edited and quality assured by a different group and approved except Two little friends. Although participants worked in pairs, it was a consultative process with often one pair checking out something with another across the table. From time to time, spontaneous discussions involving the whole team were held to reach consensus.\nBefore the workshop, there were 46 Hausa (Nigeria) storybooks on the website none of which was ASb approved. Our analysis showed that seven titles were duplicates, 11 had been translated from unapproved storybooks and three had been created in Hausa. The goal of the workshop was to have between 30 and 35 storybooks quality assured and approved by the end of the workshop and to have a team in place who would take charge of future ongoing quality assurance and approval of Hausa storybooks on the website. The team managed to quality assure the 25 of the 46 that had been translated from approved ones as well as four of the ones they translated on that day. They also quality assured one Hausa created storybook. By the end of the workshop, we had 30 Hausa (Nigeria) ASb storybooks approved. Here is a link to one of the approved storybooks for our Hausa experts to comment on.\nThree people were chosen to take forward the process of quality assurance which includes working to get the Hausa special characters inserted in the approved storybooks. They also have approval rights of Hausa translations and created storybooks on the website. All participants were registered on the website and can edit their own storybooks.\nThe Nigeria Educational Research Development Council (NERDC) is responsible for the Nigerian school curriculum, and other development organisations implementing literacy initiatives in Nigeria. We managed to hold very productive meetings with their Head of English for Education Systems Southern Africa and Nigeria at the British Council and the Literacy Coordinator from Reading and Numeracy Activity (RANA). All the discussions centred on how we can collaborate not only in Nigeria but eleswhere in Africa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It seems different now. We advertise our products on TV. Look at my new car. How many did you get? Our office is on the northern side of the building. Gabriel won't go with you. This interpretation is subjective. You can get it for free. He argued me into going. The ship's hull is damaged.\nWe were astonished to hear what had happened. Who is to blame for the failure? Herb smiled at the audience and took a bow. You've been infected. Albert is living with his parents. I wonder which of you will win. I'm pretty sure that Dori just made up that story. Jacques is really good at it. Don't you walk away from me. Kyung doesn't eat any dark meat.\nDid you watch the Oscars? Strange my foot! It will not make much difference whether you go today or tomorrow. The dictionary is incomplete. It only goes to the letter J. A contract with that company is worth next to nothing. With all the teaching and research, I have no time for myself. It is difficult to give up a long love suddenly. You are very nice. I came here to learn. That was a horrible shock.\nI've heard that you can't alter your destiny. As I told you earlier, I can't help you. I'm not here to put pressure on you. Dates and walnuts are commonly used in the filling for baklava made in the Levant. Venkata built up a good name for himself. Whenever I go by, Mt. Fuji is in the clouds. What did Boyce give you? Please explain the procedure. I have many Vietnamese learning books. I can take the bus.\nJackson is my family name. It could've been better. Suu is very pious. I know you were born in Boston. What is the poorest country in the European Union? I don't know if George is coming. This is bad news. I sure am cold. I think she will succeed. Instead of returning home, come with us.\nPardon me, is there an ATM in this area? I was thinking maybe I should study French. You should dump him. Thomas closed the shop. From a scientific point of view, I'm not entirely sure about that. Can it be that a white horse is not a horse? Was I supposed to ignore Tanya? I made a horrible decision. He wrote a letter. You need to try to relax.\nAre you sure there isn't rat meat in that burger? Pink is not a natural hair color. We all know that Rik is guilty. I hate the sound that cellophane makes when you crinkle it. The plane departs at 5:30 PM. The English are a polite people. Did you have a good time at the party? We'll never forget Micheal. The ship set sail only to be wrecked two days after. I see no reason.\nMan always thinks about the past before he dies, as if he were frantically searching for proof that he truly lived. Smokers foul up the air. I thought you had come on business. Everything is very expensive in this store. Delbert is the best barber in town. Even if the villages of this region have been destroyed, their names will never be erased from history. Please don't tell Van about what happened. I've got to see Sundaresan. I felt my phone vibrate in my pocket. You're in pain, aren't you?\nWe got on the bus at Shinjuku. I just now signed the divorce papers, and with this I'm finally free! I was on a bike. Lukas put the documents through the shredding machine. That writer is well known all over the world. Bye for now. Don't overthink it. Think twice before translating. She is my best friend. Everything is just beginning.\nWe have what we deserve. Even an expert driver can make a mistake. In fact, a group of their ancestors peopled a continent on the planet Earth about 250 000 of your Earth years ago. I'm trying to get your parents to like me. I'd like to talk to you about what happened. The Loyalists fled to Canada. You really are very beautiful, you know. I'm afraid we are out of stock. This was his first job. The stick is sticky.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have been blessed to have 2 little bundle of joys then why not check our the Baby Elegance range of twin travel systems. With the Cupla duo you can easily adapt the system into a huge verity of configurations to suit your lifestyle. The Cupla duo is also compatible with the Baby Elegance car seats and the Maxi cosi range so you have loads of options to choose from.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who wants cushions when you can have flat beanbags?\nThese giant 1x 1 metre beanbags are like cushions in Giant land.\nUse them indoors and out, by the pool, at the sea, by the river, or by the TV. Buy two, or three, or four, and take them everywhere you go.\nSoft but durable, our beanbags can be put together to make a day bed, or comfortable camping mattress. Try it.\nThis beanbag features the very popular \"Garrima\" pattern by Aboriginal Artist Christine Slabb, a bespoked design for the Commonwealth Games 2018 on the Gold Coast highlighting sports culture, and celebrations and.\nWhile the Garrima beanbag was not featured as part of the styling featured in the \"Jarjum\" (meaning, \"All our children\") Learning & Play Space at the Commonwealth Games, other products showcasing the design were. Since then, we've had hundreds of enquiries for Garrima products, and the newest one, the beanbags, have been flying out the door.\nThis beanbag design is eye-catching in Navy Blue hues.\nFill them with your choice of recyclable materials - scrunched up newspaper, old clothing, or clean rags. Or try the newest trick - stuffing the beanbag case with a bunch of your kid's soft toys that are hanging about in the cupboard collecting dust.\nDon't forget to add one of our best-selling banquet picnic tables in Blue Bay design. These handmade etched picnic tables, made on the Gold Coast of Australia, fit a wine bucket, have portholes for wine glasses, and plenty of room for a yummy cheese platter too.\nBeanbags come unfilled. Fill with soft toys or used clothing for a planet friendly alternative.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With our high stressed, faced paced existence, the last thing we want to do is think about the negative of a natural disaster. This is especially true if you live in an area where natural disasters are not that prevalent. I can tell you from first hand experience, that the time you take in preparing for natural disasters is well spent and could save your life, the lives of your family and help the repair process of healing.\nThe first and foremost thing you need to consider for natural disaster preparation is knowing and understanding what your home insurance covers and specifically what it will not cover. Many states that are prone to hurricanes and earthquakes require additional home owner insurance coverage. You also need to know that your standard home insurance usually does not cover flood damage. You therefore might need two additional policies. The cost of earthquake and hurricane or wind damage insurance can be higher, but, if you don't live in a flood zone, the flood insurance can be very reasonable.\nSomething that you may not be aware of is the attitude of insurance companies and the disaster crews they send in when a natural disaster happens. Most of us want to try to save our homes and go to extra lengths to protect them. Since we have done so, the disaster crews will put us at the top of the list. If you have not shown any effort in trying to protect you, your family and your home, you will be at the bottom of the list for assistance.\nWe live in Florida, and take a hurricane emergency kit very seriously. While this topic should include anyone that lives in areas that experience tornados and earthquakes, not all people prepare ahead of time. Those in tornado stricken areas should have an underground emergency safe area that is stocked with all of the necessities for at least three to four days for the people that may be trapped there. Earthquake and hurricane emergency kits should be complete. If you have camping gear, keep that as part of your emergency kit. Make sure you check for everything from dry matches and candles to cooking utensils and emergency medical items. When Hurricane Charley came through, we were out of power for an entire week. Candles were our light and fresh water was the main concern. Thankfully, I had set aside plenty of both.\nKeep all of your important documents together and easily accessible. We have all of our insurance, birth, marriage and home ownership information in one place. It's in a sealed plastic container and we can grab it and take it with us. Without your information, you will be at the mercy of the insurance companies and the authorities.\nVideo or picture proof is your best insurance. Each year I go around the house and property and take pictures of everything. I copy these to disks and make sure that I send a copy of the disk to a family member or friend. I also have a detailed notebook with proof of any expenses of home upgrades over the last five years. Now you may think this is an extreme measure, but, if anything happened, all I have to do is submit the pictures and receipts. Most people lose out because they don't have any proof of the worth and value of their property and its contents. This leaves an open door for an insurance company to create their own value. Thats usually lower than reality.\nHave a travel and communication plan. In preparing for natural disasters, we have a complete travel plan along with a main contact person. We will keep in contact with the main contact person and all other friends and family will get updates from them. This is the smartest thing to do because you don't know what type of communication methods will be down.\nWhatever your natural disaster, the answer is to be prepared and check it annually. Be familiar with your local emergency shelters and if they take animals. Thankfully, many across the country now do so.\nPosted on June 18th under Natural Disasters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WHEN I read a report of the World Economic Forum (WEF) that Saudi Arabia ranked seventh among countries that immigrants wish to move to, it did not surprise me. During the last three years, I have got an opportunity to acquaint with students of various nationalities in Chinese universities, who are pursuing higher studies in different disciplines such as solar engineering, electronics engineering, train technology, automated control, medicine, physics and literature.\nI don't exaggerate when I say that many of these students asked me when they knew my nationality how to get a job in Saudi Arabia and migrate to the country. A final year PhD student, who is conducting research on digital security engineering, requested me to give him my telephone number and invited me to attend some events in an attempt to strengthen friendship.\nHe surprised me by sending his curriculum vitae, explaining his expertise, courses attended, certificates received and specialization and requested me to help him find a job at a Saudi university or a company. In the meantime, I met with a Singaporean businessman by chance at a coffee shop. He told me that he had learned that there are many modern universities in Saudi Arabia.\nChina has recently announced granting of permanent residency to foreigners on the basis of certain terms and conditions. Many foreign students in China have been trying to meet the requirements to stay in China permanently. We, the Saudi students, were not attracted by that announcement as every one of us wanted to return home to serve our nation.\nSaudi Arabia, thanks to its wise leadership, has been able to manage its political and economic crises. It was successful in overcoming many dangers that hit the Middle East and the Arab World. It has succeeded in building a strong economy and achieving economic and political stability. The Kingdom has become a prominent member of G20 while upholding its Islamic values and culture.\nSaudi Arabia's dreams embrace the sky while I can see its bright future through the eyes of my children. Its roadmap is designed by Custodian of the Two Holy Mosques King Salman and his strong and brave son, Crown Prince Muhammad Bin Salman.\nI am proud and grateful for being the son of this great nation, which has never neglected its people. Rather it has been taking care of its citizens with generosity, providing them with free education, giving scholarships to pursue higher studies in reputable international universities and providing free treatment and expensive surgeries at government's cost.\nMy son was admitted to a hospital in China where he received treatment for three weeks. The cultural attach\u00e9 at the Saudi Embassy in Beijing paid the cost of his treatment that reached more than $50,000. At the same time, I saw many Chinese patients standing at the hospital's entrance as they could not find anyone to treat them because they did not have money.\nOn the other hand, being a foreign student living among them in China I find my government standing with me at the time of difficulty. Moreover, officials at the Saudi Cultural Attach\u00e9 provided me and my family necessary moral and material support to overcome the difficult situation. Dr. Fahd Al-Sharief called me personally to get reassured about my son's health. He deals with all Saudi students in China in the same way to ensure their security, safety and welfare.\nThe World Economic Forum's announcement that Saudi Arabia is ranked seventh in the ranking of countries that migrants wish to move to increases our responsibility toward our country. It's our duty to safeguard all the achievements the Kingdom has made over the past decades. We aspire to make it a better place than it is now.\nWe have not yet reached the peak of our ambitions. What we are going to achieve in the future would be much more beautiful than what we already have. The gains that we make through the efforts of our hands would be as greater as the achievements of our forefathers. Our aim should be to build a better and greater Saudi Arabia with tremendous progress in all sectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..5dec169eba06235e1afb72c5f570b6fa40b960b4
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+{"text":"Stand-alone shower cabins. They are three sided ones.\nCorner shower cabins. They are two sided ones.\nThese cabins may have one or two sliding door leaves and be with or without stationary panels.\nThe ordinary hinged shower doors are convenient and easy to use. However, you do not always have an opportunity to install them. If there is not enough free space in front of the glass shower cubicle, you cannot install these doors. In order not to renounce the idea to purchase the frameless shower unit, you can pick up the one with frameless glass sliding doors. Moreover, the choice of such cabins is extremely wide today.\nAs for the most popular options for the sliding glass doors, they include the shower doors for the niche glass shower cubicles. Besides, they include the doors for the shower cubicles where only one or two walls are the glass one(s). So the rest of the shower unit's walls are the load-bearing ones or the partitions made of different materials.\nThe frameless sliding glass shower door with a fixed panel is the best option for the niche shower cubicle. In this case, the main elements of the glass construction are two glass sheets (a fixed panel and a sliding door), door fittings, glass wall fittings for a fixed panel, a handle, and a sealant kit. It is necessary to use the toughened glass to make the frameless glass sliding doors. The sliding door fitting kit includes an upper guide track, track-to-wall fasteners, top rollers, door stop, and glass-to-track fasteners.\nThe frameless sliding glass shower door with two fixed panels is suitable for the corner shower cabin. In this case, it is necessary to install three glass panels. One of which is a sliding door and the other two ones are stationary panels. The components kit includes sliding door hardware, glass-to-wall fasteners, handle, and sealants. The toughened glass is the most popular material for making the shower units with frameless sliding shower doors. However, the fitting kit differs since the guide track for the sliding door has to be secured not only to the walls but also to the glass. Besides, the additional glass-to-wall fasteners are required because there are more glass panels in this glass structure.\nThe most widespread options for the shower units with frameless glass sliding doors are the most typical designs though manufactured to size. That is why it is possible to produce these shower units in a short time and install them quickly. However, there is still one catch. Such shower cubicles are similar to hundreds of other ones. So if you need an exclusive shower cabin with frameless sliding door, you have to order a bespoke one.\nIf you place an individual order, you have an opportunity to get the glass construction that will excellently fit into your bathroom in every sense. Thus, such a glass construction can be large or small, high or low, and with or without decorative elements. However, it is also very important to understand what you need to consider for proper frameless glass shower doors mounting.\nIn some cases, the installation of a glass shower unit with frameless glass sliding doors is much more preferable than the installation of a conventional bath. Or such a glass construction complements the bath. Moreover, when it comes to choosing a suitable place for the installation of a future glass shower cabin, it turns out that it is not so easy to find such a place. That is because there are many nuances you have to take into account. And the aesthetic appeal is one of them.\nThe aesthetics of the chosen installation site is usually the first thing that comes to mind. All of us always think where the frameless glass shower units with custom shower doors will look more beautiful. However, very quickly, everyone realises that the practical side of the issue is more important than the aesthetic appeal. So, first of all, the connection to the sewerage system is crucial because it is not always possible to change the initial location of drain pipes in the bathroom.\nAs you can see, the possibility to connect the shower cabins with frameless glass sliding doors to water pipes and drain pipes comes to the forefront. And if it is impossible to \"move\" the soil stack, then you have two options. Thus, you have to either install your glass shower cubicle next to the sewerage system or choose another installation site connecting the shower cabin to a water pipe and a drain pipe with the help of flexible hoses or additional pipes.\nAnother important issue that you have to consider when choosing the installation site for glass shower units with frameless glass sliding doors is a possibility to mount the structural elements of the shower cubicle to the walls, floor, and ceiling. The matter is that at the fixing points, the supporting surfaces must meet certain requirements in order to ensure the safety and convenience of the frameless glass shower enclosure's use. In particular, the walls must withstand all the necessary weight loads.\nAlso, you should make sure that there is enough free space for the door to move sideways.\nThe minimally possible size of the glass shower enclosure is 80\u00d780 cm. Therefore, you should ensure that this space is available for mounting your frameless glass shower unit. Moreover, according to the rules of ergonomics, the distance to the nearest surface (wall or any sanitary ware) from the exit from the frameless glass sliding doors should be not less than 60 cm, or better even 75 cm. Consequently, it becomes clear that the space required for the installation of the glass shower unit exceeds the dimensions of the shower cubicle itself.\nTo make yourself safe from unpleasant surprises when purchasing and installing the frameless glass shower unit, it is important to ask the experts in advance. Hence, in every case, you will know whether the installation of the particular glass construction is possible or it is better to choose something else. It is possible to get expert advice at the companies which manufacture and install the shower cabins with frameless glass sliding doors according to the customers' wishes and requirements.\nAt the Creative Glass Studio, we provide customers with detailed information. You will know on which glass shower units with frameless sliding doors they should select and install under specified conditions. Bathroom size, the location of a sewerage system, availability of any restrictions, etc. Thus, obtaining this information, you will get a convenient, safe, and of course unique design solution. 10 mm thick toughened glass and 13 mm thick laminated toughened glass are the glass types that we use in our custom glass shower doors and shower cubicles production. Besides, we use not only simple, clear glass. However, also low iron glass, tinted glass (bronze and grey ones), frosted glass, and sandblasted glass.\nFurthermore, we can sandblast the glass fully or partially, creating exquisite patterns. Besides, we ensure perfect protection. We will protect your glass shower doors and shower enclosures against limescale by applying the special coating. This protection will last up to 3 years and is free of charge. Finally, we use the modern sealants. Almost invisible and do not spoil the appearance of the shower doors and the shower unit as a whole. In 7-10 working days after payment, you will become a lucky owner of the bespoke frameless glass sliding doors with the ultra modern design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Compliance PhD: Reporting a HIPAA Violation is Easy. Are you Prepared?\nReporting a HIPAA Violation is Easy. Are you Prepared?\nDid you know that anyone can file a complaint against your practice? It's true. Current and former employees, as well as current and former patients have the right to file a HIPAA Compliant against you as long as they comply with the following 3 easy requirements.\nIt's really that easy. As a reminder, HIPAA Security & HIPAA Privacy Training is required at least annually. This annual training is mandatory for ALL Staff, and not just new hires. If you have not completed training in awhile, your office may be primed for a violation, and a complaint may not be far behind.\nIf you are unsure when your staff last completed HIPAA Training, or don't know how to begin training your staff. Visit http:\/\/www.compliancephd.com\/ to find out how we can help protect your practice by training your staff in current HIPAA regulations.\nHIPAA protects employees and patients from retaliation and retribution by Covered Entities accused of violating HIPAA Law. If you believe a HIPAA Violation has occurred, you can submit a complaint to your OCR Regional Office. Click HERE to view a list of Regional Offices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Weddings are our specialty at City Limo San Francisco, also having more cars than ever before.\nOur wedding cars are immaculately presented and thoroughly dressed with all the blossoms and ribbons of your choice. When you reserve with City Limo SF we plan to create every second of your 'Big Day' memorable, and in case you have any particular requirements we will do are very best to adapt them.\nCity Limo San Francisco specialize in wedding limos. We pride ourselves on the quality of the Wedding Limousines and also our Chauffeurs. Your chauffeur will endeavor to aid at all possible, by spraying on your flowers with water, keeping your own umbrella, taking photos, and assisting you in and from your Wedding Car or Limo.\nTo receive a quote on your own wedding limousines please call or email one of our friendly employees now, or please do not hesitate to fill in your info on the inquiry form and we'll contact you personally. We look forward to speaking with you soon about your wedding requirements.\nCity Limo San Francisco provides exceptional chauffeured transportation with value-based pricing and skillful customer services. Our transport business is dedicated and committed to supplying our clients actual quality at affordable prices with ours limo solutions, charter services and contract services across Bay Area. We deliver you in style and personally make sure your every requirement is met.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are going to love this super sassy, trendy, light weight and tank! The Twisted Tank is a must-have in your wardrobe. It is so awesome, that we have stocked it in two color options; both Dusty Orange and Blue. Shop away, you sassy thing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By now the British Army has pressed the German Army back beyond the villages of Mametz and Fricourt and is now facing them across the shallow valley to the south of Mametz Wood. XV Corps is given the task of capturing Mametz Wood this morning. The 38th (Welsh) Division forms part of XV Corps and by the 5th July they had moved into trenches from Bottom Wood to the edge of Caterpillar Wood and from there they can see the task that faced them. Mametz Wood is the largest Wood on the Somme covering an area of over 200 acres with a maximum length of about a mile. It is situated on a low spur of the Bazentin Ridge and overlooks the valley separating the two forces. As a result any attack made by the British forces would be, of necessity, initially go downhill from their positions and then uphill to make contact with the enemy in the Wood itself. The Germans are also able to bring flanking fire to bear on any approach to the Wood from Flatiron and Sabot Copses to the east. Further, the German second line is 300 yards beyond the northern edge of the Wood and hence it can be reinforced easily since much of the movement would go unobserved through the mass of the Wood. To the west is the area held by the 17th Division. To the east is the 18th Division from Caterpillar Wood eastwards. The capture of Mametz Wood is to be the responsibility of the 38th Division though support from the 17th Division is absolutely necessary.\nThe plan for the attack is that the 17th Division will attack Acid Drop Copse and the left flank of the Wood and act as support for the 38th Division whose task is to attack the Hammerhead. Both divisions are to advance and then turn north to sweep through the Wood. In the 38th Division the responsibility for the attack is to rest on the shoulders of Brigadier-General Evans and the 115th Brigade. His reconnaissance of the ground has made it clear to him that an attack on a narrow front is all that is possible and he decides that this would be no more than one battalion wide. He chose the 16th Welsh Regiment.\nThe C.O. of the 16th Welsh Regiment, Lieutenant Colonel F W Smith, realizes immediately that his right flank will be exposed to fire from the area of Flatiron Copse and he suggests that the plan be altered to allow him to attack at first light. This is not possible because the attack has to be coordinated with the 17th Division. When the attack takes place two battalions are used. The 11th South Wales Borderers attacks with its left flank as close to Caterpillar Wood as possible and the 16th Welsh Regiment are tight against their right flank. Each battalion will have no more than a 250 yard frontage for the attack. It is planned that the problem of flanking fire is to be eased by the use of a smoke screen.\nSeveral things conspire against the attack. It rains prior to the attack and movement becomes very difficult to move in the resulting mud. The attack is planned for 08:30 and the 38th Division attack though the planned smoke screen does not appear. The South Wales Borderers and Welsh Regiment rise to their task as soon as the artillery stops and immediately come under fire from the German machine guns in Flatiron and Sabot Copses. Casualties mount and the attack stops 250 yards short of the Wood.\nBy 10:00 the situation deteriorates as the Germans increase shell and machine gun fire across the approaches to the Wood. The 10th South Wales Borderers are ordered up in support but do not reach the battle area until after noon. Artillery support is offered but actually falls on the men of the 16th Welsh Regiment as they try to push home the attack. The arrival of the 10th South Wales Borderers does help a little in the afternoon but their C.O., Lieutenant Colonel Sidney John Wilkinson DSO, is killed at age 38 in Caterpillar Wood as he brings his men forward. Gradually the attack slows. Evans received another order telling him that an attack is to be carried out at 17:00 when the Wood is to be entered at all costs. Evans reports the situation to Divisional Headquarters and they in turn report to Corps Headquarters who orders a withdrawal to allow reorganization. The attack has failed and cost 400 casualties in the three battalions engaged and nothing has been achieved as these battle weary units are withdrawn.\nBrothers and Lieutenants in the Welsh Regiment Arthur and Leonard Tregaskis are killed together in Mametz Wood. The 33-year old Leonard is one year older than his brother and they will be buried next to one another in Flatiron Copse Cemetery, Mametz Somme.\nCompany Sergeant Major Richard Thomas (Welsh Regiment) is killed in action. He is a Mountain Ash Rugby Football player and Welsh Rugby International.\nPrivate Albert Thomas Oliver (Welsh Regiment) is killed at age 23 three days before his brother is killed.\nPrivates Charles and Henry John Morgan are killed together. Charles dies at 22 while Henry is 26.\nSouth Wales Borderer's losses include Lieutenant Thomas Pryce Hamer (South Wales Borderers) who is killed at age 33. He is the Captain of the Llamidloes Football Club and is a Welsh Amateur International.\nHaig is unimpressed by the performance of the Division and at the end of the day Lieutenant General Horne (XV Corps) informs Haig that he is not happy with the conduct of General Phillips who will be removed from command in two days.\nThe trawler Albatross sinks the German submarine U-77 in the North Sea.\nTanga is occupied almost without opposition by a British force. The bodies of over 200 officers and men who have fallen in an earlier attack are found buried in various places and reburied in Tanga Memorial Cemetery.\nLieutenant Colonel Arthur Mervyn Holdsworth (commanding 2nd Royal Berkshire Regiment) dies of wounds received on the 1st at age 40. He is the eldest son of Arthur Frederick Holdsworth JP and he played cricket for Berkshire and Devon. He was on the Regimental Polo Team which played at Madrid on the occasion of King Alfonso's Coronation.\nCaptain Oswald Eric Wreford-Brown (Northumberland Fusiliers) is dies of wounds at age 39 from an artillery shell in the Quadrangle trench Fricourt received two days prior. He was a member of the Middlesex Cricket Club and a stockbroker and his brother was killed in May 1915.\nCaptain Frederic Grainger Hall (Cheshire Regiment) is killed in action at age 25. He is the son of the Reverend Frederic Hall.\nCaptain Arthur Owen Trefusis (North Lancashire Regiment) is killed at age 32. His brother will be killed four months and they are sons of the Right Reverend Robert Edward Trefusis Bishop of Crediton.\nCaptain George Guy Hermon-Hodge (Royal Horse Artillery attached Royal Field Artillery) dies of shrapnel wounds received in action at age 32. He is one of two sons of the 1st Baron Wyfold to be killed in the Great War. His younger brother was killed in May 1915.\nCaptain Geoffrey Asteley Burney (Scottish Horse attached Royal Flying Corps) is killed at age 32. When the war broke out he was working for the British Museum in Siberia.\nCaptain Robert Leavitt Knubley MC (Wiltshire Regiment) dies of wounds at age 28. He is the son of the Reverend Canon Edward Ponsonby Knubley.\nCaptain John Hamilton Betts MC (Manchester Regiment) is killed at age 18. He is the son of the Reverend John Arthur Betts Rector of Stokesby who will lose another son in September 1917.\nCaptain Herbert Graham Barber MC (York and Lancs Regiment) is killed at age 31. His brother will be killed in November 1917.\nLieutenant John Gerard Harold-Barry (Munster Fusiliers) is killed at age 20. He is the grandson of John Harold Barry JP DL who lost a son last year.\nLieutenant James John Frost (Northumberland Fusiliers) is killed at age 21. His brother was killed last September.\nLieutenant Ernest St Clair Tulloch (Northumberland Fusiliers) a Rhodes Scholar is killed at age 22.\nLieutenant William Nimmo Brown (South African Regiment) is killed becoming the first South African officer killed in France. His brother will die on service in September 1919 and they are sons of the Reverend William Nimmo Brown.\nLieutenant Wilfred Ernest Waud (Northumberland Fusiliers) is killed at age 40. He is the son of the Reverend S W Waud.\nLieutenant Philip Crowley (Lancashire Regiment) is killed at age 31. He is the son of the Reverend Henry Ernest Crowley Rector of Cambria.\nLieutenant Neil Grant Crawhall (Manchester attached East Lancashire Regiment) is killed in action. He is the son of the Reverend Edward Isaac Laroche Crawhall Vicar of Ganton who lost another son in March of this year.\nSecond Lieutenant Ronald Woodhouse Taylor (Northumberland Fusiliers) is killed at age 20. He is the son of the Reverend George Robert Taylor Vicar of St Michael's Byker.\nSecond Lieutenant Arthur Whitmore Isaac (Worcestershire Regiment) is killed at age 42. He played first class cricket for Worcestershire from 1899 to 1911. His brother was killed in May 1915. They are sons of the late John Swinton Isaac DL. For more than a decade Isaac played for Worcestershire although only in 1903 did he have a long run in the side making fourteen appearances. However, he made only 279 runs (at an average of 13.28), and indeed this was the only time he reached even 200 runs in a season. He made three half-centuries in his career, the highest being the 60 he hit against Hampshire in 1904.\nSecond Lieutenant Roy Duncanson (Duke of Wellington's Regiment) is killed in action. His brother will be killed in October 1917 while his sister will drown in the sinking of the Osmanieh in December 1917.\nSecond Lieutenant Hubert Vernon Anchitel Corfield (East Lancashire Regiment) is killed at age 20. His brother will be killed next July and they are both sons of the Reverend Egerton Corfield Rector of Finchampstead Berks and grandsons of the late Reverend T A Anson.\nSecond Lieutenant Charles Millington Sing (Royal Sussex Regiment) is killed at age 27. He is the son of the Reverend E J Sing and the Assistance Native Commissioner for Northern Rhodiesia.\nSecond Lieutenant Francis John Hollowell (Worcestershire Regiment) is killed in action at age 20. He is the son of the Reverend William Hollowell.\nSergeant Edward Walter Mofflin (Australian Infantry) dies of wounds. His brother was killed on Gallipoli in June 1915.\nSergeant John Joseph Bell DCM (Cheshire Regiment) is killed at age 26. He is the second of 4 brothers to lose their lives in the Great War.\nSergeant John William Griffiths (Manchester Regiment) is killed at age 21. His brother will be killed in October 1918.\nCorporal Vivian Drake (Wiltshire Regiment) is killed at age 19. His brother will be killed in October.\nLance Corporal Wilfred Keeling (Sherwood Foresters) is killed at age 21. His brother will be killed in July 1917.\nLance Corporal Harry Brain (Sussex Regiment) is killed at age 17. His father died of wounds in April 1915 at age 35.\nPrivate Arthur Chilton (Sherwood Foresters) is killed. His two brothers will also lose their lives in the Great War the first in only seven days.\nPrivate Archibald Alfred Thorpe (Army Veterinary Corps) dies of meningitis at home at age 20. His brother will be killed in December 1917.\nPrivate Harry Badcock (Suffolk Regiment) dies of wounds received on the 1st at age 18. His brother will die on service in November of this year.\nPrivate James Boulton (Cheshire Regiment) dies of wounds received on the 3rd. His brother will be killed in April 1917 in Mesopotamia.\nPrivate Harry Seabridge (North Lancashire Regiment) is killed at age 26 His brother will be killed in eight days. Private Charles Stuart (York and Lancaster Regiment) is killed. His brother will be killed in September.\nMunitions worker Dorothy Maud Willis dies at age 23. Her sister will die serving in the Women's Royal Air Force in October 1919 and be buried in the same grave.\nCaptain Arthur Murdo Maxwell Robertson-Walker (Royal Fusiliers) is killed at age 35. He is the second son of James Robertson-Walker JP, of Gilgarran, Distington, Cumberland and a Member of the London Stock Exchange. He was an excellent cricketer, and well-known golfer, especially at Sandwich. Captain Robertson- Walker joined the Royal Fusiliers in December, 1914, became Adjutant of his Regiment in July, 1915, and was promoted Captain in April, 1916. He went to the Front in May, 1915. He was mentioned in Sir Douglas Haig's Despatches in 1916, and recommended for promotion. He is killed while leading with his Colonel an attack on the German trenches at Ovillers. The Battalion is caught by enfilading machine-gun fire, and both he and the Colonel fall, being killed instantaneously. The Battalion reaches their objective, and, though there were only 70 of them left, they return with over 80 prisoners.\nSecond Lieutenant Alban Charles Philias Arnold (Royal Fusiliers) is killed in action at age 23. He is the son of the Reverend Charles Lowther Arnold Vicar of Holy Trinity Fareham and he played cricket for Cambridge University and the Hampshire Cricket Club and it was stated by Wisden that \"he would probably have developed into a cricketer of very high class\". His brother will be killed in March 1918.\nSecond Lieutenant Charles Plantagenet Burdett (Royal Fusiliers) is killed at age 26. He is the grandson of the Reverend Henry Burdett.\nCSM William Philo (Royal Fusiliers) is killed at age 34. He is a Middleweight Professional boxer who won a bronze medal in at the 1908 Summer Olympics.\nLance Corporal Thomas Charles Simpson (Royal Fusiliers) is killed at age 30. His brother was killed last January.\nPrivate Thomas Howarth (Royal Fusiliers) is killed at age 24. His younger brother will be killed serving in the Manchester Regiment in two days.\nPrivate Edward John Richard Thomas (Royal Fusiliers) is killed at age 32. He is a Welsh International Rugby union player for seven clubs who earned four caps between 1906 and 1908.\nPrivate John Joseph Paulson (Sherwood Foresters) is killed at age 23. His brother will die on service in India in September 1918.\nCaptain 'the Honorable' Roland Erasmus Philipps MC (Royal Fusiliers) is killed at age 26. He is the son of the 1st Viscount St David's who has lost another son in 1915.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadhpp b/data_all_eng_slimpj/shuffled/split2/finalzzzadhpp
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+{"text":"Abstract: Damage mechanisms in a proton exchange membrane (PEM) fuel cell are accelerated by mechanical stresses arising during fuel cell assembly (bolt assembling), and the stresses arise during fuel cell running, because it consists of the materials with different thermal expansion and swelling coefficients. Therefore, in order to acquire a complete understanding of the damage mechanisms in the membrane and gas diffusion layers, mechanical response under steady-state hygro-thermal stresses should be studied under real cell operating conditions and in real cell geometry (three-dimensional). In this work, full three-dimensional, non-isothermal computational fluid dynamics model of a PEM fuel cell has been developed to simulate the hygro and thermal stresses in PEM fuel cell, which are occurring during the cell operation due to the changes of temperature and relative humidity. A unique feature of the present model is to incorporate the effect of hygro and thermal stresses into actual three-dimensional fuel cell model. The mechanical behaviour of the membrane, catalyst layers, and gas diffusion layers during the operation of a unit cell has been studied and investigated. The model is shown to be able to understand the many interacting, complex electrochemical, transport phenomena, and stresses distribution that have limited experimental data. The results show that the non-uniform distribution of stresses, caused by the temperature gradient in the cell, induces localized bending stresses, which can contribute to delaminating between the membrane and the gas diffusion layers. These results may explain the occurrence of cracks and pinholes in the membrane during regular cell operation. This model is used to study the effect of operating, design, and material parameters on fuel cell hygro-thermal stresses in polymer membrane, catalyst layers, and gas diffusion layers. Detailed analyses of the fuel cell durability under various operating conditions have been conducted and examined. The analysis helped identifying critical parameters and shed insight into the physical mechanisms leading to a fuel cell durability under various operating conditions. Optimization study of a PEM fuel cell durability has been performed. To achieve long cell life, the results show that the cell must be operate at lower cell operating temperature, higher cell operating pressure, higher stoichiometric flow ratio, and must have higher GDL porosity, higher GDL thermal conductivity, higher membrane thermal conductivity, narrower gases channels, thicker gas diffusion layers, and thinner membrane. In these optimum conditions, the maximum deformation (displacement) reduction by about 50% than the base case operating conditions.\n1 Department of Civil, Mining and Environmental Engineering, Lule\u00e5 University of Technology, SE-971 87 Lule\u00e5, Sweden.\n2 Department of Applied Physics and Mechanical Engineering, Lule\u00e5 University of Technology, SE-971 87 Lule\u00e5, Sweden.\nAbstract: The thermal borehole resistance in a groundwater-filled borehole heat exchanger (BHE) is affected of both conductive and convective heat transfer through the borehole water. To calculate this heat transport, different models are required compared to calculation of only conductive heat transfer in a back-filled BHE. In this paper some modelling approximations for groundwater-filled, single U-pipe BHEs were investigated using a 3D CFD model. The purpose is to find approximations that enable to construct a fast, simple model including the convective heat transfer that may be used in thermal response test analyses and BHE design programs. Both total heat transfer calculations (including convective and conductive heat transport) and only conductive heat transfer calculations were performed for comparison purposes. The approximations that are investigated are the choice of boundary condition at the U-pipe wall and using a single pipe in the middle of the borehole instead of the U-pipe. For the total heat transfer case, it is shown that the choice of boundary condition hardly affects the calculated borehole thermal resistance. For the only conductive heat transfer case, the choice of boundary condition at the pipe wall gives large differences in the result. It is also shown that using an annulus model (single pipe in the middle of the borehole) results in similar heat transfer as the U-pipe model provided that the equivalent radius is chosen appropriately. This approximation can radically decrease the number of calculation cells needed.\n1 Product Development, Ashok Leyland Technical Centre, Chennai, Tamilnadu, India.\n2 Department of Mechanical Engineering, Veltech Engineering College, Chennai, Tamilnadu, India.\n3 Internal Combustion Engineering Division, Department of Mechanical Engineering, College of Engineering, Anna University Chennai, Tamilnadu, India.\nAbstract: Many studies have reported that exhaust from biodiesel fuel gives higher oxides of nitrogen or lower, while HC and smoke emissions are significantly lower than that of diesel fuel. Possible explanations are: the physical properties and fatty acid composition of biodiesel affecting the spray and the mixture formation with reduced heat losses. The aim of this present investigation is to study the effect of unsaturated fatty acid composition of biodiesel on combustion, performance and emissions characteristics of a diesel engine. For this experiment thirteen different biodiesel fuels with different fatty acid compositions were selected. The performance and emissions tests on a single cylinder DI diesel engine were conducted using same biodiesel fuels. The results showed that biodiesel having more unsaturated fatty acids emit more oxides of nitrogen and exhibit lower thermal efficiency compared to biodiesel having more saturated acids. No significant differences in HC and smoke emissions among the biodiesel fuels were noticed. Thermal efficiency and NOX emission of saturated biodiesel is comparatively better than other biodiesel. Combustion analysis results show that high unsaturated fatty acid biodiesel has longer premixed combustion and high peak pressure compared to that of high saturated fatty acid biodiesel.\nAbstract: The present work comprises a research about hydraulic machines with the aim of optimization and the selection of adequate turbines of low power for exploitation of an available energy still unexplored in water supply systems based on analyses of 3D hydrodynamic flows and on characteristic curves which lead to the best efficiency point. The analysis is carried out based on non-dimensional parameters (i.e., discharge, head, efficiency, runner speed and mechanical power) in order to be possible comparisons. Mathematical models based on the physical principles, associated to the development of volumetric and rotordynamic machines, are developed. New turbines are suggested, which are based on similar theory among turbo machines based on applications in hydraulic systems with guarantee discharge and available head. The hydrodynamic fluid mechanical analysis requires the use of complex advanced models (CFD) which apply the equations of Navier-Stokes by using mathematical models of conservation laws, for the study of the turbulent flow behaviour. To determine the correlation between the flow velocity and pressure fields, the k-? model, is used in this research. Many turbines are evaluated (i.e., positive displacement (PD), pump as turbine (PAT), propeller with volute at inlet, four and five blades tubular propellers) and sensitivity analyses, to the best configurations, as well as comparisons between performance curves and experimental tests. Results are presented with the appropriate range variation for each turbine type and application..\nMechanical Engineering, Tolani Maritime Institute, Induri, Pune 410 507, India.\nAbstract: Latent heat thermal storage (LHTS) system has been attractive over the years as an effective energy storage and retrieval device especially in solar thermal applications. However, the performance of LHTS systems is limited by the poor thermal conductivity of phase change materials (PCMs) employed. A numerical study is carried out to investigate the performance enhancement of a LHTS unit of shell and tube configuration due to the dispersion of high conductivity particles in the PCM during charging process (melting). Temperature based governing equations have been formulated and solved numerically following an alternate iteration between the temperature and thermal resistance. Exergy based performance evaluation is taken as a main aspect. The numerical results are presented for several mass flow rates and inlet temperatures of heat transfer fluid (HTF). The results indicate a significant improvement in the performance of the LHTS unit when high conductivity particles are dispersed.\nSchool of Energy and Environment Engineering, Hebei University of Technology, Tianjin, China.\nAbstract: The present work was performed to apply thermoelectric technology to a low power dehumidifying device as an alternative to the conventional vapor-compression refrigeration systems. The experimental prototype of a small-scale thermoelectric dehumidifier (TED) with rectangular cooling fins was built and its operation performance was studied experimentally. The results showed that the TED experienced two typical thermodynamic processes including the cooling dehumidification and the isothermal dehumidification, where the latter was dominated. It was found that there existed a peak during the variation of the average coefficient of performance (COP) as a function of the input power of the thermoelectric module. Under the present experimental conditions, the COP of the TED reached the maximum of 0.32 and the corresponding dehumidifying rate was 0.0097 g\/min, when the input power was kept at 6.0 W. The rapid elimination of condensed liquid-drops on the cooling fins amounted on the thermoelectric module is a major approach to improving the operation performance of the TED.\n2 Department of Physics, B S College, Daspalla, Nayagarh-752 078 (Orissa), India.\n3 Department of Physics, Stewart Science College, Mission Road, Cuttack-753 001 (Orissa), India.\nAbstract: This paper theoretically analyzes the unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate through a porous medium in presence of constant suction and heat source. Approximate solutions are obtained for velocity field, temperature field, skin friction and rate of heat transfer using multi-parameter perturbation technique. The effects of the flow parameters on the flow field are analyzed with the aid of figures and tables. The problem has some relevance in the geophysical and astrophysical studies.\n1 Department of Mechanical Education, Technical Education Faculty, S\u00fcleyman Demirel University, 32260, Isparta, Turkey.\n2 Department of Mechanical Education, Technical Education Faculty, Pamukkale University, Denizli, Turkey.\nAbstract: In this study, thermoeconomic optimization of the steam power plant with Levelized-cost method was carried out. Aim of thermoeconomy is to minimize exergy cost. With this aim, the first law and the second law of thermodynamics to each component of system were performed. Irreversibility and exergy values were obtained. Economic analysis by using exergy values was carried out. Unit electric cost for each component of system was calculated. Optimum design and operating conditions for minimum exergy cost were obtained.\nDepartment of Electrical Engineering, Maulana Azad National Institute of Technology, Bhopal, India.\nAbstract: A Matlab-Simulink based simulation study of PV cell\/PV module\/PV array is carried out and presented in this paper. The simulation model makes use of basic circuit equations of PV solar cell based on its behaviour as diode and comprehensive behavioural study is performed under varying conditions of solar insolation, temperature, varying diode model parameters, series and shunt resistance etc. The study is helpful in outlining the principle and intricacies of PV cell\/modules and may be used to verify the impact of different topologies and control techniques on the performance of different types of PV systems. The PV module\/Array performance is immensely marred by shading effect and its P-V characteristics exhibit multiple maxima. The Matlab\/simulink based study therefore also points out significance of locating maximum power point for a given Module\/Array. An experimental verification is also carried out in the lab by developing a PC based data acquisition system, which is also briefly discussed.\n10. Which is a better transportation fuel \u2013 butanol or ethanol ?\nDepartment of Economics, Orbita 3, Suleyman Demirel University, Almaty, 050043 Kazakhstan.\nAbstract: This article examines butanol and ethanol as transportation fuels for gasoline-powered engines. This paper examines two aspects. First, the fuel properties of butanol and ethanol are examined and compared to each other. Consequently, butanol overcomes three deficiencies of ethanol. Butanol has a higher energy content, butanol-gasoline blends do not separate in the presence of water, and butanol can be blended with gasoline in any percentage, all the way up to 100%. Second, a review of the fermentation technology is examined for both butanol and ethanol production. Both butanol and ethanol can be fermented from the same feedstocks, which include the sugar and starch crops and lignocellulosic fermentation from wood and crop residues, and fast-growing energy crops like hybrid poplar, switchgrass, and willow. Furthermore, the capital and facilities used to produce ethanol can be switched to butanol fermentation with minimal costs. Thus, society is able to transition away from ethanol and begin to produce butanol with minimal capital and infrastructure costs. Unfortunately, the main drawback to butanol fermentation is its low chemical yield. Until researchers discover or engineer new microorganisms that handle higher butanol concentrations, butanol may not be adapted as an alternative fuel.\nCenter for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi-16, India.\nAbstract: In the design and study of solar energy, information on solar radiation and its components at a given location is very essential. Solar radiation data are required by solar engineers, architects, agriculturists and hydrologists for many applications such as solar heating, cooking, drying and interior illumination of buildings. For this purpose, in the past, several empirical correlations have been developed in order to estimate the solar radiation around the world. The main objective of this study is to review the global solar radiation models available in the literature. There are several formulae which relate global radiation to other climatic parameters such as sunshine hours, relative humidity and maximum temperature. The most commonly used parameter for estimating global solar radiation is sunshine duration. Sunshine duration can be easily and reliably measured and data are widely available.\n12. Auction design for the allocation of carbon emission allowances: uniform or discriminatory price ?\n1 Institute of Policy and Management, Chinese Academy of Sciences, Beijing, 100080, China.\n2 Center for Energy and Environmental Policy Research, Beijing Institute of Technology, Beijing, 100081, China.\n3 School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China.\n4 Graduate School of the Chinese Academy of Sciences, Beijing 100080, China.\nAbstract: Only four states used auction in Phase (2005-2007) of the European Union Emission Trading System, of which four used a uniform-price sealed auction format. Here we discuss whether the auction should adopt a uniform-price or discriminatory-price format using an agent-based carbon allowances auction model established for the purpose. The main conclusions are as follows: (1) when carbon allowances are relatively scarce, the government should use a discriminatory-price auction; when carbon allowances are relatively abundant, the government should use a uniform-price auction. (2) Uncertainty of the generating cost reduces the ability of an auction to know bidders' private values, which will reduce the government's revenue and reduce auction efficiency. (3) Compared with the discriminatory-price auction, the uniform-price auction can prevent large bidders from obtaining excessive profits. (4) The uniform-price auction is relatively insensitive to market structure. However, a monopoly market is more likely to develop under the discriminatory-price auction format. The results of the model have some policy implications for designing carbon market mechanisms in the future.\nDepartment of Applied Sciences & Humanities, Institute of Engineering & Technology, U.P.T.U., Lucknow, India.\nAbstract: To define the diffuse radiation incident on an inclined plane of any orientation and of any tilt angle, we need to know sky diffuse radiation as well as ground-reflected radiation. Here we made a study of ground-reflected component of radiation on inclined surfaces with varying inclination angles for Lucknow (Latitude 26.750, Longitude 80.850), Mumbai (Latitude 19.120 N, Longitude 72.850 E), Calcutta (Latitude 22.650 N, Longitude 88.350 E), and Pune (Latitude 18.530 N, Longitude 73.910 E) cities of India using isotropic and anisotropic models. We have calculated the data for entire year using these models for all considered stations and obtained the percentage contributions in diffuse radiations. It is found that anisotropic model predict lower values than isotropic and contribution of ground-reflected component in diffuse radiation for isotropic model is more than anisotropic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pristine landscape, dedicated play-space and close proximity to the nearby park and school provide a perfect nurturing environment for your family. Spacious two bedroom floor plans are sure to capture your heart with well planned living areas and cozy bedrooms. Pointe Pacific is situated in a quiet residential neighborhood, just moments from boutique shopping, convenient dining and daily necessities such as grocery stores and banks. Located minutes from the CA-22 and I-405 freeways, Pointe Pacific makes your morning commute a breeze!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Incredible little hole in the wall in downtown!\nThe inside is so adorable, the staff are as sweet as can be & the food is plentiful and very tasty!\nMy wife & I stopped in for a quick lunch. We ordered the Garlic Peppered Turkey Adobo Sandwich & the Portabella Veggie Burger both with fries. We swapped a half of each and this was probably the best decision I've made all week.\nThe sandwiches were generous with their portions, the fries crisp & the side of ranch was great too!\nWe can't wait to go back & try all of the sandwiches now!\nThe food is great here! I grabbed the chicken pesto panini and it was delicious! I'll definitely be back to try the other items on the menu..\nThis is a new business in downtown Stockton. It's always nice when something new shows up because there aren't too many options down here. The owner is a sweet woman and the baked goods were fresh. I had the pulled pork adobo with fries and it was hot and fresh.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you cannot afford one of the cars you may want to get a peek at Chip Foose wheels. Either you have a vehicle or planning to purchase one, you need to have understanding of the different car types besides the makes and models of cars. Purchasing a vehicle is regarded as a task by many people. Small cars aren't only fuel efficient, they also give a selection of amazing characteristics and accessories which makes driving more fun and safe.\nYou may have heard about various kinds of cars based on their body styles. A car definitely wants a cover, whether it's parked indoors or outdoors. If you store your vehicle outside, the benefit of experiencing an auto cover can readily be seen. Pay a visit to an accessory shop and buy some attractive upholstery fabric which will agree with your car well. A new car can offer you all of the style, luxury and comfort you demand! Unfortunately, not everybody can afford a new vehicle.\nYou should learn how to customize or modify just about all the pieces of your vehicle, not just for looks but also for performance and handling. If you prefer to make your car seem different and distinctive, altering the interiors is extremely important. You also need to increase your car's hood higher enough in the event you desire the judge to check at your engine. Initially, you should assemble the things you would need to customize your vehicle. To begin with, choose the model of the auto you would like to customize.\nThrough the 1950s and on in the 60s also, a wide array of cars were tapped for police usage, that range from six-cylinder Studebakers to Buicks with huge V-8s. Seldom are you going to find a car which has been traveling mute. Speaking of the Q70, it's a vehicle that isn't exactly certain if it wants to be a midsize auto or a full-sized vehicle. Building such a vehicle is just one of the very best ways to check your designing skills. There are quite a few who are interested in build their very own miniature cars with the assistance of 3D modeling program.\nWhen it has to do with custom mats, you can design them with the direction you desire. Customized car mats can enable you to decorate your auto's interior. Automotive floor mats are offered from $15 to as much as you're ready to spend. Car mats and covers must perfectly fit to prevent any dirt entering the vehicle. Car floor mats are needed to shield your vehicle's interior from spoiling. Custom made car carpets are also offered.\nAt the same time that you will select a garage plan dependent on the architectural style very similar to or complimentary to your residence, you might need to consider other aesthetic materials, like the shade of paint or kind of siding. A good idea to customize your auto is to receive the upholstery or the interior changed. Ideas for improving the plan of grills, tires, windshields and other components of the vehicle can be gotten from such sessions. There are quite a lot of suggestions for painting your car with two colors on each side of the human body line. People have even known to modify the simple appearance of the vehicle or even the performance of the vehicle (by replacing the engine) if they want a custom made car in accordance with their desire.\nChoose the dormer style you want, discuss your choices with a skilled architect and you'll have the dream design for your house. If you wish to make your vehicle design reflect your personality, you must almost rebuild the vehicle. In reality, virtually all sorts and designs do not need any drilling or complicated fixings. When it has to do with automotive logo designs, you have to look at the target of the plan, the quality required, and the point to which your creativity can be unleashed. In addition, there are modern designs of rock bands or music groups that are often specialized and aren't simple to have in stores.\nThere are several different software employed for building a digital vehicle. More than a few companies specialize in detachable speakers that could stick to the owner on a hike. Many businesses produce wide assortments of car seat covers made from all sorts of materials. The most suitable company stipulates the excellent fittings with safety. To attain the satisfaction of the requirements, every glass businesses supply the glasses that are approved and tested. Before you begin with the actual procedure for designing, you ought to do a comprehensive research on the vehicle you want to design. Before you begin with the actual procedure for building a digital 3D car, prepare hard copies of the plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My name is Jacqueline Laurette, I am the face behind the Art House Studios + Baby Boutique in Downtown Red Deer.\nAt The Art House we start result in mind. Our mission is to give our clients and customers an experience they can't get anywhere else. I have been a baby photographer for a few years now, but before that I spent most of my life studying the arts in it many forms. Which is why I have always has such an appreciation for unique, quality and handmade items that you can't find anywhere else.\nBeing a baby photographer, I began to see a gap in the market for such items, especially those that are crafted specifically for babies and toddlers. My clients were constantly telling me how difficult it was to find unique and quality items and clothing for their littles, and that the options for brick and mortar baby boutiques around our city in particular were non existent, leaving them with big box options, or online shopping.\nThis, as well as my love for babies and children were the inspirations behind The Art House Studios + Baby Boutique.\nUnorthodox. The photo studio and boutique combo is relatively unheard of. But not impossible.\nAnd so we embarked on this unusual adventure in hopes of providing my photography clients as well and new customers an opportunity to shop local, shop handmade, shop quality clothing and products that are natural and safe, and of course \u2013 ethically produced. With over 80% of the products sourced locally and the remaining 20% from Canada and other ethical parts of the world such as USA and New Zealand.\nThis is such a personal journey for me, and if it's not obvious yet, once you come into the Studio + Boutique you will see the story of my world through art, photography and quality products.\nI want to create a beautiful experience for customers every time they visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeoft b/data_all_eng_slimpj/shuffled/split2/finalzzzaeoft
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"\"To tire\" the head is to adorn it (2 Kings 9:30). As a noun the word is derived from \"tiara,\" and is the rendering of the Heb. p'er, a \"turban\" or an ornament for the head (Ezekiel 24:17; RSV, \"headtire;\" 24:23). In Isaiah 3:18 the word Saharonim is rendered \"round tires like the moon,\" and in Jud 8:21,26\"ornaments,\" but in both cases \"crescents\" in the Revised Version.\nEaston, Matthew George. \"Entry for 'Tires'\". \"Easton's Bible Dictionary\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"9 Thng Ba 2011 . Ti coreal 12 full serial, download corel 12, CorelDRAW 12 Full Cra ck Link mediafire . : Corel DRAW 12 Full Crack Link mediafire.\nI have the disk but unfortunately it is cracked. . Can't I just download the program and use my purchased key to activate it? I have searched online and tried to download a few but they are incomplete versions when I had the full version. . I have a 'perfectly' good copy of X3, as well as 'perfectly' good copies of 12, 11, 9, 7,.\n30 Jan 2013 . Bagi anda yang memiliki usaha Percetakan, Design Grafis, Reklame, maka Software Corel Draw 12 ini penting bagi anda. Jika Anda berminat.\n28 Jun 2013 . CorelDRAW 12 INCLUDED CRACK full version, CorelDRAW 12 Free Download, CorelDRAW 12 gratis, CorelDRAW 12 patch only,.\n7 Jul 2017 . Corel Draw 12 Crack Plus Keygen And Serial Key Free Download. CrackNest.com Hello All, You All know my team is always provides you full.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sinn F\u00e9in has confirmed that MEP Liadh N\u00ed Riada will stand as its candidate for the presidency.\nThe party's Ard Chomhairle announced the decision after meeting in Dublin to make its final selection.\n\"Of course, it is a huge honour and very humbling and privileged to be even held in such high regard by people to be put forward as a possible candidate,\" she said.\nSpeaking from the European Parliament in Strasbourg, she said it would be a huge honour to be nominated.\n\"I have to wait for the process to be completed and see, in respect to all candidates that are putting themselves forward, we have to see what the outcome is,\" she said.\nThere are already four names officially in the race after President Michael D Higgins confirmed he would stand for re-election and three others received the required number of nominations from local authorities.\nBusinessman Gavin Duffy secured his fourth nomination on Friday \u2013 joining fellow Dragon's Den panellist Sean Gallagher and Senator Joan Freeman on the ballot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On behalf of Rothschild & Ausbrooks, PLLC posted in Foreclosure on Thursday, January 29, 2015.\nThe real estate bubble, and the resulting mortgage crisis and financial crash that followed, was driven, to a great extent, by lending institutions that needed product to sell. With the fail of interest rates on investments like Treasury bonds, investors worldwide needed somewhere to put their money that would return a higher interest rate.\nThat place was securitized mortgage loans, which because of the historically low default of 30-year fixed rate mortgages, were seen as safe an investment as U.S. Treasury's. By the early 2000s, the market had vacuumed-up all of the traditional 30-year FHA-type mortgages.\nHowever, there was still a large market for these bundled mortgage securities. And Wall Street was making staggering sums of money off these transactions.\nThis drove the market to \"invent\" more and more loans that were highly questionable under traditional mortgage underwriting standards. They needed more borrowers and had to sell them to virtually anyone who would walk in the door. They employed such problematic loan devices as NINAs, \"no income no assets\" to expand the number of loans they could book and therefore resell in the securitized bundles of mortgages.\nBy this point, it had become little more than a trillion-dollar Ponzi scheme. It, of course, was not sustainable. For investors to earn any return on their investments, borrowers need to be able to make their payments. Which they could not.\nEventually, the house of cards collapsed, taking millions with it, and causing millions of bankruptcies and foreclosures.\nAnd now, it appears the process is repeating itself, only this time with auto loans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A fanfiction inspired by The Legend of Zelda: Breath of the Wild.\n\"The story before the story began. A hundred and one years before, to be exact.\"\nRead the full work here.\nHis bones ached as he made his way up the massive stone steps, creaking like they were bowstrings, ill-tended to and apt to snap. He had kept himself in peak physical condition even after he had retired, but no man could outrun the march of time. The grave called to him, whispering its coming arrival with every new throb or cramp that greeted him in the morning. He had years yet, but that didn't stop the realization that it was looming before him from sinking into his stomach like cold iron. He was old, and no amount of strength could keep that fact from hurting the young man still trapped somewhere in his jaded mind.\nThe wind whistled through the gaps in his armor, the late afternoon sun glinting off the mirrored surface. He had stayed awake long into the night polishing it, ensuring it was in immaculate condition for his meeting today while he mulled over what it was he wanted to say. Sparks shot out of the edges of his boots as he clanged against the stonework, lifting himself up the last few steps and onto the battlements of the castle. Rhoam was there waiting for him, a regal sight as he stood with impeccable posture, massive hands clamped behind his back. He looked as much a king as he always had, though Vallus had long ago stopped teasing him for being royalty. The joke had been far less funny after Rhoam's father had died and he had ascended to the throne. The King's silver hair and ample beard stirred in the wind as it buffeted the walls of the castle they stood upon, and he watched as his old friend breathed in and out, deeply and slowly.\nVallus took up position next to the him, resting his hands on the bricks before them, mortared together ages ago to create the centerpiece of the kingdom. He gazed out at the view, with all of Hyrule spread before their silent gazes, a gleaming empire of peace and prosperity. There were a handful of clouds dotting the horizon, but otherwise the sun was uninhibited as it cascaded through the clear blue above their heads. Vallus wondered if Rhoam saw the shadows hiding behind the light as much as he did. Did he fear the oncoming storm? Did he fret about it as often, or as deeply?\n\"You look old.\" Rhoam's voice startled him out of his reverie, and he frowned as he turned to glare at him. Rhoam wasn't looking at him yet, his eyes still on the wide-open world that he ruled, but Vallus thought he detected the faintest hint of sarcasm in the words.\n\"You got fat.\" The quip rolled out of Vallus' lips before he rightly knew what he was saying, but he decided before the sound had faded that he would stand by it. If the King of Hyrule had forgotten how to take a joke, Vallus would be happy to remind him.\nThe king rumbled out a laugh that started like drums and ended like thunder. \"I have no recourse to argue with that fact. How are you, my friend?\" Rhoam finally turned to look at him, his eyes crinkling at the edges as he smiled.\n\"I am crushed anew each day that I must rise and face this kingdom without her by my side.\" He spoke fervently, and Vallus had no doubts that he meant the full gravity of the words he spoke. Rhoam had been tempestuous and foolhardy in his youth, rushing out on grand adventures and dragging Vallus along with him. It had seemed he had little care for his own hide until he had found Amalia. She had given him a reason to want to live, a reason to want to be the good man Vallus had known him to be when he wasn't keen on burning through the days with excitement. Vallus had been aware that his friend had been lost from the minute he laid eyes on Amalia, though when he had seen how happy Rhoam was with her he had never been able to bring himself to be disappointed. Then, a year later, Liluth had walked into his life, and his chapter as a knight had come to an end. He had retired, settled down in a town that had barely started to grow from the mountainside, and given up his suit of armor for a good plow horse. He had married Liluth as quickly as she had allowed, and together they had prepared to raise a family.\nFate had other plans, however, and he didn't think he could bring himself to forgive it for such.\n\"You know as well as I do that the only thing that will save our kingdom is her ability to wield her inherited powers.\" Rhoam scowled at the ground, and though his expression was filled with irritation, Vallus could still see the fear.\n\"Oh? And I should hide it from her instead?\" the question was pointed enough that Vallus recognized it as a jab against his own parenting, but he let it slide away, refusing to rise to the bait.\nThey lapsed into silence, both leaning on the comforting presence of the other. At the height of their days they had been strong, and proud. They had given their hearts to exceptional women, and it had seemed the world had granted them the greatest gifts. They had been fools, thinking that they had deserved such things. Fate had not only taken his wife, but it had damned his son in the same turn. Vallus would fight to spare him, if only he had an inkling as to how. He would happily turn his blade against any foe, strike down any villain that would dare lay hands upon his son, but there was no hope for it. He had not the power, and the enemies were invisible and as yet unannounced. As it stood, all he could do was teach him, and he was failing at even that.\n\"That I can relate to.\" Rhoam inhaled slowly, the wind whipping his beard to and fro. \"You hope that throwing them together will awaken their sense of purpose.\" It was a statement, and not a question, though Vallus knew the king well enough to recognize that in truth it was a bit of both.\nVallus nodded, and he could sense that their meeting was over without the customary goodbyes generally said amongst friends. Rhoam and Vallus never said farewell. It had started as a joke when Vallus had been his appointed knight, both of them knowing they would see the other more often than they saw anyone else, so it was ludicrous to say goodbye each time they parted ways. Eventually it had become a custom that neither was willing to breach. Farewells felt like an end, and neither of them were ever capable of admitting that an end had come. In this way, they would never have to admit how many days in their lives had passed, and that the era of their youth had found its sunset years ago.\nVallus left then, their powerlessness making the tips of his ears burn as the sun dipped below the horizon. It was as story as old as the dirt beneath their towns and cities, a story told over and over so many times that no one had bothered to count. The great calamity known as Ganon would rise, intent on destroying the world, and a wise princess and brave warrior would foil his plans. If Rhoam was right, if Liluth had been right in her last moments, then it had happened so many times that it was all but common, though it didn't feel that way.\nHe remembered reading the account of the war ten thousand years ago and thinking that it had sounded exciting. He had been jealous of a hero with a magic sword, capable of saving the world like no other could. He had remembered that story again when Liluth had told him that their son was that hero reborn, the next Link in the cycle that would churn Hyrule into the darkness and back out once more. He had held his son, small and squalling, big blue eyes full of wonder at the world, and he could no longer see the excitement, or the valor. Now he could only feel fear; all-consuming, breathtaking fear that the boy he had raised from a baby would be swallowed by that story and never seen again.\nYet there was no helping it. Liluth had known from the moment he was born, in that way that she had always known things no normal person had a right to. Their plans, their dreams, their precious family had been tossed into the ever-spinning hands of the clock, a time of need for Hyrule dashing their hopes as the insignificant things they were. They had planned on naming him Henry, but she had held him with shaking hands and announced that their child's name was Link, as had been so many Links before him. Vallus knew the name didn't matter, though he had argued against it, argued against the losing battle of accepting his son's fate. Naming him something different would not have changed the life that was destined to find him. It was who he was: the hero of legend that would save the princess and fight the darkness.\nHis only wish in this world was that Link would remember to save himself, as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The facts show that 46% of the sudanese population still live under the poverty line. This sad but sincere fact is due to over three decades of civil war and destruction. The war is over now but the battle rages on with a population that struggles on an everyday basis with basic human needs like food, water, medical treatment, housing and education. We at Project Humanity declare war on extreme poverty in Africa and particularly in Sudan because we believe from the bottom of our hearts that we can change the facts but only if we act with urgency now.\nProject Humanity exists to provide food, clothing, school supplies and healthcare for the poorest of the poor in the slums of Sudan. We believe that every life matters and that nobody should die of hunger or of medical conditions that could be treated almost effortlessly. We work closely with the poor to understand their situation and work alongside them towards a life in dignity.\nProject Humanity e.V. is a Non-Profit organisation. That means that 100% of your donation goes towards fighting extreme poverty in Sudan. Reaching peoples hearts is not always easy but we are committed to our mission. We try to raise public awareness through many social media outlets which we believe, lead to informed understanding and involvement. So please share our mission and be a helping hand to the needy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Al-Rehab is Middle Easts leading perfume manufacturer which creates fine traditional Arabian and oriental perfumes. High quality exotic long longer lasting and is free from alcohol. Comes in a 6ml (.2 Ounce) roll-on vials makes it easy to anoint. One of the most popular perfume oils in the world. Inexpensive price does not sacrifice high quality. Unique blends make many varieties of oil that can be used both by men and women.\nUsed both by men & women.\nIngredients: Citronellol, Limonene, Geraniol, Linalool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Natural gas is the most environmentally friendly fossil fuel. It consists principally of methane between 90 and 99%, while the remainder consists of other gases such as butane and ethane.\nIt takes more than 600 liters of methane gas to make one liter in liquid form, then the benefits to detain in such physical condition are clear: an easier transport, especially for large distances.\nLNG burns more cleanly than coal and oil, for example, emitting significantly smaller quantities of greenhouse gases into the atmosphere in the process.\nAlso reduced emissions of harmful pollutants such as CO2, NOx (nitrogen oxides), SOx (sulfur oxides) and particulate matter means that the passage to LNG will have a positive impact on human and planetary health.\nToday the most common applications for LNG are autogas station \u2013 TiApm manufactures vertical and horizontal storage tanks with thermosiphon for LCNG stations \u2013 industrial applications \u2013 TiApm offers storage units and rigid \/ semitrailers for refueling \u2013 and naval bunkering \u2013 for which TiApm provides specific semitrailers suitable to act in conjunction with high-speed modules.\nLNG road tankers by TiApm are extremely robust and reliable yet having very light own weights to maximize payload.\nLNG tanks by TiApm are available in sizes from 10m3 to 350m3 (volumetric capacity), in vertical or horizontal configuration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The question regarding conviction history collects flawed information produced by a discriminatory system. By asking about past convictions, Princeton is not asking who has committed a crime, but rather who has been arrested and punished for these crimes by a demonstrably inequitable justice system. Thus, this information is not an unbiased indicator of culpability or character.\nThe very presence of this question on the application deters many students with conviction histories from applying to Princeton in the first place. Thus, the Box excludes prospective students long before Princeton admissions officers read applications, and would continue to do so even if admissions officers vowed to not take this information into consideration. As a result, Princeton's applicant pool is less representative of students with conviction histories and thus less representative of students from low income and minority communities.\nThe Box enables discrimination against students with conviction histories and undermines qualified students' efforts to compete equally with other applicants. For reasons stated in #1, this disproportionately impacts low income students and students of color.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MAHER & GROSH (1863-1926) MILLER BROS. CONTRACT--HORSEMAN-3 1\/2 INCHES CLOSED-7 BLADES TOOLS MARKED AND TIGHT MAIN SNAPS BUT PEN IS SLOW-(RARE GENUINE..\nNEAL HAND MADE (LATE 1900's EARLY 2000)--DAGGER DIRK BOOT KNIFE-7 1\/2 INCHES LENGTH-FLUTED ROUND HANDLE-SHEAT NOT ORIGINAL FOR IT BUT ALSO A CUSTOM MA..\nUNION KNIFE CO. NAUGATUCK (1851-1885)--SLOT KNIFE-4 1\/2 INCHES CLOSED-3 BLADES UTENSELS BLADE MARKED AS STATED, FORK IS MARKED \"ARMY KNIFE UNION\"-COCA..\nRUSSELL straight line (1800's)--LITTLE BOY JACK-3 INCHES CLOSED-LONG PULL SPEAR BLADE FULL MARKED ASO HAS R with arrow on back tang TIGHT AND SNAPS-EB..\nJ. RUSSELL & CO. GREEN RIVER WORKS (1800's) one of if not their earlest stmaps--FIGHTING,CORN ??-ABOUT 18 INCHES OVERALL LENGTH-BLADE STAMPED AS STATE..\nBELKNAP LOUISVILLE, KY. (1970's)--BG 1 STOCKMAN-4 INCHES CLOSED-3 BLADES MAIN IS \"ETCHED\"-WHITE HANDLES HAS BLUE GRASS ENAMEL EMBLEM IN IT.. MINT WIT..\nRUSSELL blade always unmarked (LATE 1800's EARLY 1900's)--LITTLE BOY BARLOW-2 5\/8 INCHES CLOSED-SPEAR BLADE TIGHT AND SNAPS-IRON HANDLES FRONT HAS, \"T..\nTHOMAS TURNER SHEFFIELD (1805-1890)--NAVY KNIFE-4 3\/4 INCHES CLOSED-NAVY BLADE MARKED TIGHT AND SNAPS-FINE GENUINE OLD SHEFFIELD STAG HANDLES NO CRACK..\nSAYNOR COOKE & RIDAL (1877-1890)--SWAY BACK NAVY-4 1\/4 INCHES CLOSED-2 BLADES MARKED TIGHT AND SNAP-SUPER STAG HANDLES WITH MARVELOUS OLD BIRDS EYE RI..\nR J. RUSSELL & CO. GREEN RIVER WORKS (LATE 1800's EARLY 1900)--NAVY-4 1\/4 INCHES CLOSED-NAVY BLADE MARKED TIGHT AND SNAPS-GENUINE STAG HANDLES WITH IR..\nJONATHON CROOKS heart & pistol (1780-1827)--NAVY PRUNER-4 3\/8 INCHES CLOSED-LARGE SHEEP FOOT BLADE MARKED TIGHT AND SNAPS-OLD KNARLY STAG HANDLES. W..\nRUSSELL R U.S.A RUSSELL R U.S.A. (OLD)--BOY JACK-3 1\/4 INCHES CLOSED-SHEEP FOOT BLADE MARKED TIGHT AND SNAPS-JIGGED BONE HANDLES SMALL CHIP AT REAR FR..\nF. NEWTON PREMIER SHEFFIELD (BEFORE 1890-1800's)--4 1\/2 INCHES CLOSED-MARKED TIGHT AND SNAPS ALTHOUGH SOME USE BUT WHAT WOULD YOU EXPECT ON A 150 YEAR..\nRODGERS CUTLERS TO HER MAJESTY (1837-1890)--MINATURE-1 1\/2 INCHES CLOSED-GLOVE BUTTON HOOK-IVORY HANDLES WITH PINCHED IRON BOLSTERS. *IN MY COLLE..\nRODGERS CUTLERS TO HER MAJESTY (1837-1891)--MINATURE-1 1\/2 INCHES CLOSED-CLIP BLADE MARKED TIGHT AND SNAPS-IVORY HANDLES WITH PINCHED IRON BOLSTERS.. ..\nRODGERS CUTLERS TO HER MAJESTY (1800's)--WHITTLER-2 1\/2 INCHES CLOSED-3 BLADES FULL MARKED TIGHT AND SNAP-GENUINE PEARL HANDLES WITH BAR SHIELD. *..\nWOSTENHOLM I*X.L. SHEFFIELD (LATE 1800's EARLY 1900)--SLEEVEBOARD WHITTLER-3 3\/4 INCHES CLOSED-3 BLADES MARKED TIGHT AND SNAP 1 PEN BROKE, WHAT A SHAM..\nJ. RUSSELL & CO. GREEN RIVER WORKS (1800's)--SPECIALTY TOOL-7 1\/2 INCHES OVERALL LENGTH-DEEPEST BEST STAMP I HAVE SEEN ON THIS EARLY A RUSSELL OF ANYT..\nJ. A. HENCKELS GERMANY (OLD)--STOCKMAN-4 1\/8 INCHES CLOSED-3 BLADES MARKED TIGHT AND SNAP MAIN \"ETCHED PREMIUM STOCK KNIFE\"-RED STAG HANDLES. SOM..\nROBERT KLASS N.K.C.A. (1980)--DOG LEG JACK-4 INCHES CLOSED-3 BLADES MAIN IS ETCHED, \"N.K.C.A.\"-KNARLY STAG HANDLES. MINT BACK IS BEST KNARLY STAG ..\nROBERTSON BROS. & CO. LOUISVILLE (1880-1926)--ENGLISH JACK-4 1\/2 INCHES CLOSED-2 BLADES MARKED TIGHT AND SNAP, MAIN TARNISH TRACE OF ETCHING, \"DEPENDA..\nRUSSELL straight line (LATE 1800 EARLY 1900)--BARLOW-3 3\/8 INCHES CLOSED-SPEY BLADE MARKED TIGHT & SNAPS-SAW CUT BROWN BONE HANDLES. SELDOM SEE..\nFRANK BUSTER & SON CELEBRATED CUTLERY GERMANY ROOSTERS--STOCKMAN WHITTLER-3 7\/8 INCHES CL;OSED-3 BLADES MAIN MAIN GOLD ETCHED-GENUINE PEARL HANDLES WI..\nBEAR & BULL SOLINGEN GERMANY (OLD)--CONGRESS-4 INCHES CLOSED-4 BLADES MAIN ETCHED, \"BEAR & BULL HAND MADE\"-GENUINE STAG HANDLES. MINT MINT..\nFRANK BUSTER CO. SOLINGEN GERMANY ROOSTERS (1986)--LOCKBACK WHITTLER-3 7\/8 INCHES CLOSED-SPEAR BLADE ETCHED, \"FORT CITY KNIFE COLLECTORS CLUB\"-GENUINE..\nFROM MY RUSSELL COLLECTION <>R<> J. RUSSELL & CO.\nR J. RUSSELL & CO. G.R.W. (1800's EARLY 1900)--FOLDING SPATULA DR. DRUGEST ARTIST KNIFE-SPATULA BLADE FULL MARKED AND SNAPS-EBONY HANDLES.. SUPER CON..\nLITTLETON CELEBRATED CUTLERY (OLD)--EQUAL END-2 7\/8 INCHES CLOSED-4 BLADES MARKED USED TIGHT & SNAP CLOSED BUT NOT OPEN-TORTISE HANDLES PIECES MISSING..\nW SINGLETON CO. 10 BAKER HILL (OLD)--CONGRESS-2 7\/8 INCHES CLOSED-4 BLADES MARKED TIGHT & SNAP 1 PEN SHORT BROKE-GENUINE PEARL HANDLES SLIVER WORN OFF..\nFRIENDS & CO. NAZIRABAD ?? (OLD)--LARGE JACK-4 1\/2 INCHES CLOSED-CLIP BLADE HAS THE RODGERS MALTESE CROSS & STAR ON BACK TANG MARKED TIGHT AND SNAPS D..\nRUSSELL R U.S.A. (OLD)--NAVY OR ROPE-4 1\/4 INCHES CLOSED-NAVY OR ROPE BLADE \"\"ETCHED WATERBURY BLUE MARKED ROPE\"\" RARE ETCHED RUSSELL, REAR OF BLADE A..\nRUSSELL CATALOG OF PRODUCTS---REPRODUCTION OF IT BY DEWEY FERGUSON A COLLECTOR ITEM OF ITS OWN ANY MORE... * MANY THANKS FOR HIM HAVING DID THI9S YEAR..\nR J. RUSSELL & CO. G.R.W. (1800's)--DR. ARTIST-3 1\/4 INCHES CLOSED-SPATULA BLADE MARKED TIGHT AND SNAPS-YELLOW IVORY HANDLES FEW TIGHT HAIRS N..\nCIVIL WAR TYPE HOME MADE GREAT FOR REENACTOR OR WESTERN SHOW--BOWIE TYPE-12 3\/4 INCHES LENGTH-FABUOLIS ELK HORN STAG HANDLE-SUPER LEATHER SHEATH-FITS ..\nCAMILLUS CUTLERY CO. CAMILLUS N.Y. (30's OR 40's ?)--REPLICA OF GEORGE WASHINTON'S KNIFE-2 1\/2 INCHES CLOSED-OLD STILE CLIP BLADE MARKED TIGHT AND FI..\nROBINSON HOLLOW GROUND (OLD MINT)--6 KNIVES STEAK OR OTHER-I BELIEVE CHERRY HANDLES AND IN A SUPER CHERRY WOOD BLOCK.. HAD IN A DRAWER FOR YEARS CLE..\nTRUST WORTHY MAPPIN & WEBB SHEFFIELD (EARLY 1800's)--KNIFE FORK AND STEEL IVORY HANDLES TIGHT CRACK AS IVORY DOES.. RARE RARE NAME OF CUTLERY PREV..\nPOWDER HORN WITH HORN CAP--(OLD ???)--PURCHASED OUT OF A BUNCH OF THINGS SUPOOSEDLY CIVIL WAR ERA, MAY BE OR A REENACTOR.. HAVE A BUTTON LISTED O..\nOLD CARVED YELLOW BONE HANDLES SAILOR ? FRENCH ?\nH. BOKER & CO. SOLINGEN (OLD)--STOCKMAN-4 INCHES CLOSED-3 BLADES MARKED TIGHT AND SNAP, MAIN IS ETCHED BUT OPPS IS SHORT-SOME OF FINEST BROWN BONE HAN..\nVIXE ?? not sure if that is right ?? then SCOVILL ST (FEDERAL EAGEL, LIKE CIVIL WAR)--SMALL BUTTON POSSIBLE SHIRT OR VEST ??..\nCAST STEEL (1800's OR BEFORE)--NAVY SPECIAL PURPOSE MADE OR WORN AND MADE FOR PERSONAL USE-4 1\/4 INCHES CLOSED-TIGHT AND FINE SNAP-**HAND CARVED** SMO..\nR J. RISSELL & CO. GREEN RIVER WORKS (1800\"s)--SMALL NAVY OR LARGE NEW ENGLAND WHALER KNIFE-3 5\/8 INCHES CLOSED-EARLY MARKED BLADE TIGHT AND SNAPS-EBO..\nCATTARAUGUS CUTLERY CO. LITTLE VALLEY N.Y. (1886-1963)--COKE HUNTER LOCK, \"KING OF WOODS TYPE\"-5 3\/8 INCHES CLOSED-CLIP BLADE STAINS & SCRATCHES SNAPS..\nCATTARAUGUS CUTLERY CO. LITTLE VALLEY N.Y. (1886-1963)--SCOUT UTILITY-3 3\/8 INCHES CLOSED-4 BLADES TOOL D2589 SPEAR,LEATHER PUNCH, COPING AND LOCKING ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahvkm b/data_all_eng_slimpj/shuffled/split2/finalzzzahvkm
new file mode 100644
index 0000000000000000000000000000000000000000..3c582952e72ae8581629f10f4e8afa409b46c7fd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahvkm
@@ -0,0 +1,5 @@
+{"text":"Here's some 1969 Atlanta gold! Audio quality is better than FM here on a tape provided to us by Contributor Karl Philips. This is from the original master REEL, which I encoded using a loaned Wollensack courtesy of Contributor and site friend Jack Parnell.\nScoped as always, WQXI is sporting PAMS jingles, some great tunes and tons of commercials. Check out the one for Burger King!\nFans of 60s Atlanta radio will surely remember Randy Robins as a great southern jock with lots of one-liners.\nBe sure to read through the comments, they are sure to bring back some memories!\nI was surprised to hear Roby Yonge in the last three minutes of the aircheck doing a spot for a movie. Roby was at WABC then.\nI think I also heard a spot with The Real Don Steele about a minute later.\nWow! and people say that WABC played too many commercials back in the day. don't get me wrong, this aircheck was loads of fun to listen to, but they played two commercials after almost every song. Even in their hay day WABC didn't play commercials after every single song.\nflattered to see my aircheck listed but my name was spelled with an extra b.also i don't have any old airchecks and wanted to hear this,but when i click on nothing happens.whats up with that.\nRandy, do you have the Real Player installed on your computer? You need it to hear the airchecks.\nIf that's not the problem, try clearing your browser's cache. All these airchecks are encoded to stream flawlessly on a 28.8k dialup modem, so everyone can hear them.\nI just clicked on the link and it works, not sure what the issue is on your end.\nOk -all you Texans need to come out of the shadows and put something in here about \"Big R\" !!!! This is one of the guys that made me want to be a DJ. I answered his phones for the Top 7 at 9 at Quixie before entering the Navy in 1970 and later got the chance to DJ'd with him at Lake 102 in North Atlanta in 2001. He's a funny guy and I wish someone would wise up and hire him somewhere. Randy\u2026.Very entertaining. I got your number brother \u2026.and I'll talk to you soon.\nLoved him here, but at Lake 102 fm in buford he was the only jock with real talent.\nI had the pleasure of signing for the transmitter while I did production when Randy was on the air at WQXI. I eventally did all nights at quizie. I wish some of you had the pleasure of knowing his children or knew what a dedicated father he is. Its too bad the folks in Atlanta radio don't try harder to get him back on the air.\nHello Dave Weiss !!! I was working at Quixie in 69 answering the phones for the Top 7 at 9 for Randy while you were doing all nights. Bubba was my nickname. You took me in the production studio and we cut me an aircheck. Later worked with Randy at Lake 102 in Buford doing all nights on weekends. I am the contributor of these airchecks which I made just for fun back then \u2013 not knowing how much pleasure they would bring others by being posted here. Later.\nHey, does anybody have any airchecks of Brother Love at WQXI, 1969-1971? Followed Randy Robins, 9-midnight. Anyone\u2026.??\npersonality radio. I guess we didn't realize how good Quixie was until it was gone. Wow, what memories.\nRemember your secretary\/community service director? I still have copies of the newsletter that you mostly wrote\u2026..you were pretty darn clever. I have photo of The Mighty Thau promo too, taken by Charles dowdy. Those were great days. I'm not quite that skinny anymore.\nLynne hall vigodsky preminger\"\u2026\u2026\u2026\u2026\u2026not a good air name right?.\nI am doing a documentary on Randy Robins and his days in radio and an era gone by.\nI need any sound checks MP3's or anything else that you good folks may have. I will be in North Georgia filming Randy March 7th-12th if you have any ideas or contributions or stories that you would like to share my email me at [email protected].\nRandy Robins are you the same Jock that use to be known as Sweet Randy at KLIF 1190 in Dallas, Texas? If so Robin Hood Brians, John Bass and Myself Ron Tyler from Tyler, Texas said hello. PLease go to my web site at \/\/www.afproductionsonline and get the phone number and give me a call sometime if your the sa,e guy. When you were at KLIF those were the godd old days.\nWow, Randy sounds exactly like he did when I listened to him at KIMN radio in Denver!\nGod bless you and thanks for being there in my teen years in the 70's!\nI was at WVLD Valdosta at the time and about to go in the Army. I had the private control room number for WQXI and Randy answered the phone. I told him that I wanted to do weekends (fill in) there and how should I go about it. Well, the first thing he told me was I sounded like \"Rick E. Radio\" and I needed to lose that. Well, I did and got a pretty good job at KFJZ Ft Worth on weekends. Then, got shipped back to Ft Benning later on and did a couple of weekends on Quixie before the Army sent me away again. Big R, that little piece of advice really helped, and I really listened. Hope you are doing well.\nI am a close friend to Randy and to my knowledge he never worked at any station in Indianapolis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb lyrics \u00bb music \u00bb [Music + Lyrics] Dunsin Oyekan \u2013 \"Open Up\"\n[Music + Lyrics] Dunsin Oyekan \u2013 \"Open Up\"\nRenowned anointed Gospel music minister, Dunsin Oyekan releases yet another spirit-lifting and atmosphere shifting song titled \"OPEN UP\".\nHe Shared \"There are dimensions and realms in Christ that my generation must see!\nLet every blockage be opened now!\nFor out of my belly shall flow rivers of living waters! I pray this song will birth an encounter in your life as it has for me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi I'm Jenna. I just got off the Salomon Lotus. I'd give this board 3 out of 5 stars. It's really just not the right board for me. This is a solid entry level board it's very soft flexing. It's kind of your pricepoint board. It does have a lot board for the value. Again super soft, great to learn on. It's not a board that you're going to get out there on your more intermediate to advanced level girls and just really charge on. You'd get a lot of chatter and a lot of vibration. This if what I felt when I brought it up to higher speeds but it's very frogiving and an awesome board to learn on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Berlin - a thousand words can't describe this unique city. Art and architecture, bars and business, clubs and cool, design, emotions, fashion, history - you name it. Berlin is more than a city, Berlin is a feeling. Berlin has what you were always looking for.\nBerlin is pure life. Our international team lives and loves the city. We explore new restaurants and venues, we enjoy museums and exhibitions, we haunt the bars and clubs at night, we find the best brunch places. We do it for us - and you can profit from our knowledge.\nDon't feel afraid to contact us - we want to get to know you!\nThe Bavarian capital lives by its tradition - beer gardens and folklore. On the other side of the coin, you find the modern city with its affluent inhabitants. Driving the latest sports car edition with a dirndl and lederhosen to the Oktoberfest is not a contradiction in Munich.\nBavarians taking pride in their independence, it is hard for outsiders to get accepted in this city. We know what counts, and we are well aware of the traditions. We also know the cool and the hip places, and we cheer for Bayern Munich's soccer team. We know where to find the tastiest sausage on Viktualienmarkt, and lead you to the hidden places in the pre-Alps. Join us for a refreshing beer in a mountain hut after a day of team-building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eco-friendly Coffee And also is a formula for individuals who can use some assistance, this because of the visibility of the included eco-friendly tea remove. Our Green Coffee essence is standard on 50% chlorogenic acid. Green tea essence look here as well as Artichoke remove are included in Green Coffee Plus. Suitable for find more vegetarians! ATTENTION: Wonder products do NOT exist, this is a supporting product, you ought to always exercise firstly and consume healthy and differed.\nGreen coffee essence or Eco-friendly coffee is a food supplement made from \"unroasted beans\" from the coffee plant. It includes materials such as cholorogenic acids. You can typically purchase environment-friendly coffee essence through pills, tablets or powder. The capsules have 300mg of remove and are extremely standardized on 50% Cholorogenic acid. Eco-friendly coffee extract includes fairly little high levels of caffeine, but individuals who are sensitive to this have to take care with this. Green tea plus artichoke remove is likewise refined in eco-friendly coffee plus.\nEco-friendly coffee plus is highly dosed and has 300mg (50% chlorogenic acid) per capsule. To support the formula there is likewise 150mg Environment-friendly tea extract and also 150mg Artichoke extract added, the pills are vegetable. Nowadays the term 'eco-friendly coffee' pops up a growing number of. But just what is this really? Green coffee is actually nothing greater than the kind where coffee beans originally occur in nature prior to being baked. Green coffee beans are go for that reason unroasted coffee beans. Environment-friendly coffee is chock full of chlorogenic acid and high levels of caffeine. Green coffee remove is a food supplement that is made from 'unroasted beans' of the coffee plant.\nEco-friendly Coffee And also remove pills from vitaminesperpost.nl appropriate for every person to be able to utilize healthy cells and also tissues. Eco-friendly Coffee Plus with environment-friendly tea essence as well as artichoke remove is packed with anti-oxidants that protect against cost-free radicals. The Environment-friendly Coffee And also pills that you could buy at vitaminesperpost.nl are extremely dosed. Each pill contains 300 mg (50% chlorogenic acid). To sustain the formula, one more 150 mg of environment-friendly tea essence as well as 150 mg of artichoke essence are included. The Eco-friendly Coffee Plus capsules are vegetable.\nStandard eco-friendly coffee is really nothing more or less than coffee that is made from unroasted coffee beans. Coffee beans are naturally brown, dark brown, reddish-brown, environment-friendly or greyish. They turn brown-black right into black through the burning procedure. Due to the fact that environment-friendly coffee beans are not roasted, particular nutrients are maintained. As an example, eco-friendly coffee has a lot more phenols as well as terpenes (including cafestol and kahweol) than baked coffees.\nGreen coffee is really absolutely nothing new; it is simply coffee that is made with unroasted coffee beans. If you choose coffee beans, they are normally gray-green to brown-green in shade. Only after roasting do coffee beans get their normal brown-black to pitch-black shade and also highly fragrant aroma. The original suggestion behind green coffee is that unroasted coffee beans maintain a lot more of their all-natural nutrients.\nEco-friendly coffee could have more nutrients than black coffee; That does not discuss why environment-friendly coffee would help with weight-loss and also weight management. It is not left out that eco-friendly coffee beans accelerate your metabolism, but it is also not clinically established. Phenols and also terpenes are not necessarily valuable in weight-loss, slimming or weight management. The preferred environment-friendly coffee that is noted as slendering coffee (including Leptin Eco-friendly Coffee 800 and LipoLysin) is as a result not made from eco-friendly coffee beans.\nThe green coffee that is mentioned as 'slimming coffee' is not just coffee from environment-friendly coffee beans ... Many prominent types of environment-friendly coffee have nothing to do with eco-friendly coffee beans. Eco-friendly slendering coffee normally includes environment-friendly tea delegates which all kinds of additional components are added. It is these enhancements that give eco-friendly \"coffee\" its slimming impact. Instances of added excipients in slendering coffee are natural herbs, high levels of caffeine, lingzhi, ginseng, cassia seed, guarana, green tea essence, ECGC, Svetol \u00ae and also chromium.\nMuch eco-friendly coffee is as a result no coffee whatsoever. Environment-friendly slendering coffee is typically made from environment-friendly tea with ingredients as well as ingredients contributed to it. These included compounds variety from natural herbs and caffeine to ephedrine as well as sibutramine. Lots of people rely on the functioning of slendering coffee since the component caffeine is called an accelerator of the metabolic rate. Caffeine is refined in all type of fat burners, Stackers, diet regimen tablets as well as other slimming items.\nEnvironment-friendly slendering coffee is in many instances not coffee, but tea. This green \"coffee\" does not help you to lose weight because of the compounds existing in coffee, such as cafestol and also kahweol, as lots of producers do case. These are the included natural as well as\/ or artificial additives that create weight reduction. Environment-friendly coffee is really nothing basically than a powdery slendering tablet where you make a fluid beverage.\n\"Slendering coffee\" and also \"environment-friendly coffee\" are now principles that are made use of for an exceptionally wide range of slendering items that typically have nothing to do with coffee or coffee beans. Every slimming coffee has its own distinct composition of added fabrics. Whether you could in fact lose weight with environment-friendly coffee remains a matter of trying. Although the thought percentages in the initial paragraph are skeptical to claim the least, they are not necessarily excluded.\nThe terms \"slendering coffee\" and \"eco-friendly coffee\" are exceptionally vague ideas made use of for numerous types of slendering items. Traditional kinds of slimming coffee are made from green coffee; green coffee is in concept absolutely nothing essentially compared to coffee made from unroasted coffee beans. When a coffee bean is not roasted, it preserves its original eco-friendly shade. However, the various other active ingredients of slimming coffee differ commonly. In addition, there are also \"weight management coffees\" that do not consist of coffee beans whatsoever which primarily have debatable compounds that you likewise discover in particular medications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Latshang, Tsogyal Daniela; Kaufmann, Barbara; Nussbaumer-Ochsner, Yvonne; Ulrich, Silvia; Furian, Michael; Kohler, Malcolm; Thurnheer, Robert; Saguner, Ardan Muammer; Duru, Firat; Bloch, Konrad Ernst (2016). Patients with obstructive sleep apnea have cardiac repolarization disturbances when travelling to altitude: randomized, placebo-controlled trial of acetazolamide. Sleep, 39(09):1631-1637.\nLatshang, Tsogyal Daniela; Mueller, Daniela Juliana; Lo Cascio, Christian Maurizio; St\u00f6whas, Anne-Christin; Stadelmann, Katrin; Tesler, Noemi; Achermann, Peter; Huber, Reto; Kohler, Malcolm; Bloch, Konrad Ernst (2016). Actigraphy of wrist and ankle for measuring sleep duration in altitude travelers. High Altitude Medicine & Biology, 17(3):194-202.\nThis list was generated on Fri Apr 19 16:37:55 2019 CEST.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Guideline-recommended, blood-based mutation results within 72 hours.\nWhat is the GeneStrat\u00ae test?\nTHE GENES TESTED IN THE GENESTRAT\u00ae TEST ARE COVERED BY MEDICARE AND MANY PRIVATE PAYERS.\nNearly 80% of patients currently do not have mutation results available at their initial oncology consult.\nHow do we change this paradigm with blood-based testing?\nLITTLE AS 8 DAYS2 FROM DIAGNOSTIC BIOPSY.\nReference: Boyle T, et al. Poster presentation: AMP annual meeting. Salt Lake City, UT; 2017.\nLearn how some of the industry's leading experts are using Biodesix Lung Reflex\u00ae testing in their practice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find local bargains in and near Draycott (Derbyshire).\nThe listings below are an automated feed from Ebay for the village \"Draycott\".\nView more listed items with \"Draycott\" in the title. (Problem with any of these listings? Report it to us here).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Born in Leningrad. In 2011, she graduated from the Rimsky-Korsakov St Petersburg State College, majoring in academic vocal (class of the Honoured Artist of Russia T.B. Lebed). In 2016, graduated from the Rimsky-Korsakov St Petersburg State Conservatory (class of the Honoured Artist of Russia G.I. Kiseleva).\nIn August 2016, completed courses at l'Accademia Musicale Chigiana in Siena (Italy) where she studied with La Scala and Metropolitan Opera soloist, the People's Artist of Bulgaria, Raina Kabaivanska.\nIn 2016\u20132017, was a soloist at the St Petersburg Zazerkalye Theatre.\nElizaveta Senatorova is an active participant of the Camerata by Irina Sharapova Project and concerts organised by the composers' organisation MolOt. She is in constant collaboration with young contemporary composers. In 2015-2016 Senatorova took part in the Festival From Avant-Gard to Our Days.\nSince 2017 she has been a soloist of the Primorsky Stage of the MariinskyTheatre.\nIolanta, Laura (Iolanta, P. Tchaikovsky), Cherubino (Le nozze di Figaro, W.A. Mozart), Serena (Porgy and Bess, G. Gershwin), Giannetta (L'elisir d'amore, G. Donizetti), Sergent Smirnova (Old Khottabych, S. Pleshak), La Ciesca (Gianni Schicchi, G. Puccinni), The Witch (Dido and Aeneas, H. Purcell).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was fairly erratic with my blogging in 2012, but was still able to average about 40K of views per month. Not bad, for not blogging much. The real drop off came in 2013, when I stepped away from this blog for much of the entire year. Views quickly dropped to under 10K a month.\nSo why did I stop blogging? Simple answer: It started to feel like work.\nWhen I first started blogging in 2008, there was an initial curiosity that led me to start my own blog. I used it as a way to distract me from all of the work that I was doing in graduate school. I was working on my PhD at the time. This blog was a way to \"get away from it all,\" so to speak. It worked. What started as my lifestream, eventually turned into a niche blog. I blame Gary Vaynerchuk for that. He told me (and the rest of the Internet) to blog about something that I was passionate about, and so I did. My initial love for Threadless T-Shirts spread to encompass all indie tees and designers. When this happened, it was a whole new world, with a whole lot of T-Shirts.\nI pushed, and pushed, and pushed for three years, until it was basically myself and Hide Your Arms at the top of the T-Shirt blogging food chain. Then, just as I was about to break the 100K mark in monthly unique views, I took a step back, and stopped. I'm not even sure exactly why I stopped, other than\u2026it started to feel like work. My passion for T-Shirts had dried up. It would have been disingenuous for me to continue blogging about T-Shirts. I simply didn't care about tees as much as I had. I over consumed. I was over it.\nAnd so I took a step back. My burning passions now include traveling the world, and exploring Hawaii. My new website (now almost three years old), recently peaked a 30K month. Not bad, but far from my 100K days. But, as I've come to realize, it's not entirely about the numbers. It's about the passion.\nTrying to make HYA into work was probably a bad idea for me in the end, chasing pageviews (I was up at 200k at peak) and posting up to a dozen times a day. I'm down to about 60k a month now with about an hour of effort a day. I think it's clear from my posts that the passion is gone but I can't bring myself to quit the site.\nTrying to bring other bloggers in could have made a difference but with them not being paid writers they didn't really have much of an incentive to post and some of them never posted after signing up. That lack of input from them was disheartening but I still made some good friends from that idea.\nI'm pretty sure that I'm the last of the old guard to be regularly posting about tees, I won't give up, but I won't be spending 20+ hours writing list posts anymore.\nI wanted to thank you Coty for all that you have done for the t-shirt community. Without coming across your site last year and reading all about the t-shirt world from 2008-2013, I would not have started my own brand. Watching the videos and reading your excitement when opening up packages from certain brands, seeing what you liked about each shirt, and all the other little things was very insightful. It's great to see that you're blogging again on this site!\nAnd Andy, your site was also instrumental in me launching my brand. Seeing what's out there in the indie brand world helped me to cement the vision I had for Sotare. I'm thankful that you keep posting shirts, and have even been gracious enough to include my own!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Have you heard about sunset in Kuching? Where the sun lowered itself on the horizon with reflections on the river. I was told by my friends that one of the indications of a clean air is a combination of orange-purplish skies when the sun sets. It feels romantic. It looks romantic.\nKuching downtown is large but the old town area is small. There is a lot of old pre-war shop houses, some still maintained its original signage and architectural design. I walked around the old town area to kill time before going to the riverside.\nOnce I reached the riverside, the view was awesome. The skies are filled with orange and purple hues, indication of a very fresh and clean air. There were sampans bringing people from one side of the river to the other. Behind them was the reflection of the trailing edge of the sun's disk disappearing below the horizon. Again, it was romantic.\nWhilst looking at the sunset together with Anis of Five Feet Traveller, I sat quietly, enjoying the moment and thanking God for giving me the opportunity to come to Kuching, to experience the magical sunset of Borneo.\nMagical. Yes, the sunset was truly magical.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ready for some hotel promo codes in Niiza? Welcome on DirectHotels.com! Be prepared for your journey to Japan. There are amazing places nearby Niiza such as Historic House of Old Shimada Family, Nerima Art Museum and Toshimaen. You can't miss them! You can use Haneda Airport which is based 21 kilometers from the center of Niiza. We hope you will enjoy your time in Japan!\nNiiza provide a wide range of hotels as there are about 19 hotels closeby. If you want travel coupons be sure to use DirectHotels.com! This hotel coupons finder is a perfect tool which shows you all the best hotel vouchers. There's nothing easier than using DirectHotels.com!\nwe study all the popular booking sites as well as direct booking possibilities. If a hotel allows any kind of deals for booking directly they will be visible here! So, thanks to DirectHotels.com you can easily look through all the possibilities and choose the cheapest hotel.\nAs stated there are a lot of possibilities in Niiza.\nCheck out our top hotels in Niiza. Do not hesitate and visit Niiza and remember that the best hotel promo codes you can find only on DirectHotels.com!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The beauty of fishing is that every angler has a shot to score a world-record fish. His bait (and patience) could attract an all-timer, no matter his age or level of expertise. After all, records are made to be broken!\nExcept for these. The following four records are gonna be tough to beat.\nSome records\u2014Joe DiMaggio's hit streak (56 games). Cal Ripken Jr.'s consecutive games played streak (2,632), or the number of cake balls I ate during my birthday week (incalculable using modern methods)\u2014may never be broken. George Perry's 22-pound, 4-ounce largemouth bass caught in Georgia's Montgomery Lake belongs on that list.\nIn the 86 years since Perry's massive catch, anglers around the world have been trying to break his record (a Japanese man came close in 2009). There's even a plaque commemorating the event. Perhaps, and we make no guarantees here, but perhaps a homemade custom lure could help the fishing fanatic in your life reel in a plaque of his own?\n\"The Buffett of Striped Bass Fishermen\" is a lofty a nickname. But for a man who owns four world records and struck a deal on Shark Tank, it's probably warranted. His most jaw-dropping achievement (so far) came in 2011, when he broke a nearly 30-year-old record and snagged this 81.88 pound striper. Folks that's a post player on a middle school basketball team.\nIf you know a bass beginner, he must first become a whisperer before he can grow into a Myerson-level maven. And all the fishing fame and fortune that comes with it.\nIt's quite amazing how far we've come since the early days of photographs. In olden times, a man's fishing story was only as good as his word. Now we have dramatic YouTube videos with professional production quality to chronicle the evidence.\nThis intrepid group of anglers ventured south of Cabo San Lucas, Mexico, in 2011 in hopes of snagging a world record yellowfin tuna\u2014and a million-dollar prize. Sure enough, they captured the saga of their 427-pounder in crystal clear HD. A big novelty check was not far behind.\nIn 2016, a group of fishermen from the Deline First Nation brought in a would-be official world record lake trout from Great Bear Lake in Canada's Northwest Territories if not for one small detail: they used a net.\nNo less a fishing feat, in my view. If your back hurts after posing for photos, at the very least you deserve a commemorative medal for your efforts.\nThe heads on these giant lake trout I'm blessed to chase as a guide never cease to blow my mind. Dinosaurs of the arctic.\nWe're sure these guys would love our Troutdoorsman Crate, though we can't promise the grill holder has the capacity for monsters like this one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check Out Our Video Recapping the Process of Developing the Folly Theater Marquee!\nAs you may know, Star Signs recently completed a design\/build project for the Folly Theater. We are proud to have our work displayed on such a beautiful historic building in downtown Kansas City and created a brief video documenting the process through which the sign was created. We hope you enjoy watching as much as we enjoyed making it!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Criminal Appeal No. 1688-SB of 2004. D\/d. 26.5.2017.\nFor the Appellants :- Ms. Mannat Anand, Advocate\/Amicus Curiae.\nFor the Respondent :- Mehardeep Singh, Addl.A.G., Punjab.\nJitendra Chauhan, J. - The present appeal has been preferred by the appellants challenging the judgment and order dated 20.07.2004, passed by the learned Judge, Special Court, Patiala (hereinafter referred to as the trial Court), whereby, they have been convicted for the commission of offence punishable under Section 15 of the Narcotic Drugs and Psychotropic Substances Act, (for short, 'the Act') and sentenced to undergo rigorous imprisonment for a period of ten years and to pay a fine of L1,00,000\/- each along with default clause.\n\"Both the accused face trial for offence under Section 15 of the Narcotic Drugs and Psychotropic Substances Act 1985 (hereinafter called the Act) on the allegations that on 14.10.2002 in the area of G.T. Road, Rajpura, they both were found in possession of three bags containing 106.500 gms of poppy husk, by the police party headed by SI Balkar Singh of CIA Staff, Rajpura.\n2. The case of the prosecution, in brief, is that on 14.10.2002 SI Balkar Singh of CIA Staff Rajpura alongwith his police party was present in front of Focal Point where Bag Singh of village Darian met him and was joined in his police party. When he was having a talk with said Bag Singh, a vehicle TAT 713 bearing No.2D-147858K of Army colour came from Ambala Side. On seeing the police party, the driver of that vehicle i.e. the accused Gannu stopped the vehicle at some distance, while Uttam Singh was sitting with him on the adjoining seat. SI Balkar Singh told both of them of his intention to conduct the search of the vehicle and when he checked from behind, three bags were lying on seat. SI Balkar Singh suspected that these were carrying some contraband and gave option to the accused to get their search conducted before some Gazetted Officer of Magistrate. The accused vide memo Ex.PC and PC\/1 agreed for their search to be conducted by SI Balkar Singh. On the search, all the three bags were found containing churra poppy heads, out of which two samples of 250 gm each were separated and put into parcel. On weighment, the remaining churra poppy heads came to be 35 kgs. Samples and the bags with remaining poppy husk were sealed with the seal bearing impression BS. Sample seal Ex.P.1 was also prepared. The seal after use was handed over to PW Bag Singh. Case property was taken into possession vide memo Ex.PD. Vehicle was also taken into possession. On personal search of the accused Gannu L 150\/- and from accused Udham L 50\/- were recovered and taken into possession vide memos Ex.PE and Ex.PE\/1, respectively. Both the accused were supplied grounds of their arrest vide memos Ex.PF and Ex.PF\/1. Ruqa Ex.PA was sent to the Police Station whereupon formal FIR Ex.PA\/1 was recorded. Site plan of the place of recovery Ex.PG was prepared. On return to the police station, case property and the accused were produced before SHO Jai Kishan of Police Station City Rajpura, who after verifying the factum of recovery put his own seal bearing impression 'JK' on the case property and the sample seal Ex.P.1. On the next day, case property and the accused were produced before Illaqa Magistrate. Vide report of the Chemical Examiner Ex.PJ the sample was found to be that of churra poppy heads and after completion of investigation, the challan against the accused was presented in court.\"\n3. The learned trial Court, after finding prima facie case against the accused, charged them for commission of offence punishable under Sections 15, of the NDPS Act, to which they pleaded not guilty and claimed trial.\n4. In order to substantiate its case against the accused, the prosecution examined SI Jai Kishan as PW-1; ASI Bhinder Singh as PW-2; ASI Baldev Singh as PW-3; SI Balkar Singh as PW-4; Constable Sunil Kumar as PW-5 and ASI Raunak Singh as PW-6.\n5. During their examination under Section 313 Cr.P.C., the accused-appellants denied the prosecution allegations and pleaded false implication. However, they did not lead any evidence in defence.\n6. After hearing learned counsel for both the parties and considering evidence led on record, learned trial Court, convicted the appellants as detailed at the outset of this judgment.\n7. Hence, the present appeal, which was admitted by this Court on 30.08.2004.\n8. It is contended that the prosecution has failed to prove conscious possession of the appellants over the contraband allegedly recovered from them. It is further contended that the alleged recovery had been effected from three bags. The prosecution was required to take out the entire contents of all the three bags over a piece of cloth to ascertain that the entire material recovered was a contraband. It is further argued that in the instant case, the complainant as well as the investigating officer is one and the same person which makes the case of the prosecution highly doubtful. There is an unexplained delay of one day in sending the case property to the Magistrate. The alleged independent witness was previously known to the investigating officer and no other independent witness was joined.\n9. On the other hand, the learned State counsel, has vehemently argued that on the strength of the evidence led on record, the prosecution has been able to bring home guilt to the accused.\n10. I have heard the learned counsel for the parties and perused the record.\n\"20. Under the Narcotic Drugs and Psychotropic Substances Act, not only the very possession of the narcotics, drugs and psychotropic substances have been made an offence but severe punishment without exception has also been provided. The Act also provides for presumption of guilt emerging from possession of Narcotics Drugs and Psychotropic Substances. In case of commercial quantity of the nartocis, drugs and psychotropic substances, the minimum sentence of 10 years rigorous imprisonment besides minimum fine of Rupees one lakh has been provided. Hon'ble the Supreme Court in Noor Aga v. State of Punjab, 2008(4) R.A.J. 381 has held that under the Narcotic Druiigs and Psychotropic Substances Act the extent of burden to prove the foundational facts on the prosecution, i.e., proof beyond all reasonable conduct is more onerous. A heightened scrutiny test would be necessary to be invoked. It is so because whereas, on one hand, the court must strive towards giving effect to the parliamentary object and intent in the light of the international conventions, but on the other hand, it is also necessary to uphold the individual human rights and dignity as provided for under the UN Declaration of Human Rights but insisting upon scrupulous compliance of the provisions of the Act for the purpose of upholding the democratic values, it is necessary for giving effect to the concept of wider civilization. It is further observed that while deciding such cases, the Courts must always remind itself that it is a well settled principle of criminal jurisprudence that more serious the offence, the stricter is the degree of proof. Therefore, a higher degree of assurance would be necessary to convict an accused under the Narcotic Drugs and Psychotropic Substances Act.\n21. Thus, under the Narcotic Drugs and Psychotropic Substances Act, it is the fundamental duty of the prosecution to prove beyond a shadow of reasonable doubt that the investigation conducted in the case is absolutely flawless specifically with regard to the link evidence which is of most significant aspect. It is incumbent upon the prosecution to prove that from the stage of effecting the recovery till the same reach the Chemical Examiner, there was no chance of tampering with it. Once the presumption is stumbling, on this vital aspect, the benefit is to be extended to the accused.\"\n12. In the instant case, Form No.29 was not prepared on the spot and was admittedly filled in after delay of two days, which has not been explained by the prosecution. Therefore, prosecution has failed to prove beyond reasonable doubt that the sample of contraband which was examined by the Forensic Science Laboratory was the same which was alleged to have been recovered from the appellants at the time of the alleged recovery.\n13. The second flaw in this case is that the contraband has not been proved to be recovered from the conscious possession of the appellants as it has been alleged to be recovered from three bags lying on the backside of the truck as mentioned in memo of consent, Ex.PC\/1. On the contrary, SI Balkar Singh (PW4) is the investigating officer who has minced no words in testifying that the three bags were lying on the seat; whereas, ASI Baldev Singh (PW3), an alleged recovery witness has deposed that the bags were found lying under the seat of the vehicle. It is manifestly clear that there is major contradiction as to from where the contraband was recovered; backside of the truck, on the seat or beneath the seat, which makes the recovery highly doubtful.\n14. Another aspect of the matter is that in sheer violation of the principles of fair and impartial investigation, the complainant and the investigating officer is the same person, which makes the prosecution case doubtful. In Laltu Prasad v. State of West Bengal, 2017(2) R.C.R. (Criminal) 237 (Calcutta) (DB), it was held that the complainant himself acting as the investigating officer violating the principles of fair and impartial investigation is a practice, to say the least, should not be resorted to and it is a disturbing feature. To the same effect, is a Division Bench judgment of Hon'ble Himachal Pradesh High Court reported as State of Himachal Pradesh v. Atul Sharma and others, 2015 (6) R.C.R. (Criminal) 949, wherein, it has been held that where the complainant himself conducts investigation, it causes miscarriage of justice to accused qua fair investigation.\n\"15. Though, all the witnesses have stated that one sample from each gunny bag was taken out and then all the gunny bags were sealed with the seal bearing impression 'BS'. It has also come in the evidence of the witnesses that the contents of the gunny bags were not turned in order to know the whole of the contents therein. It would be significant to mention here that the Investigating Officer was supposed to know about about whole of the contents of the bag before bringing the accused into net. As such, this circumstance also seriously affects the prosecution case.\n16. As regards tampering of the case property, he was to turn down all the contents of the bags on a piece of paper or cloth, but in the instant case, the Investigating Officer appears to have not examined the contents of the bags; he took it casually and in an anxiety to involve the accused, sealed the bags without knowing the whole contents thereof. Thus, in the absence of any evidence to the aforesaid extent, it cannot be believed that whole of the contents of the bags were poppy husk. P.K. Rai (PW1) while appearing in the witness box has stated that one of the gunny bags does not bear any seal. Inspector Mohinder Kumar (PW3) states that there is no seal on one bag brought in the court, nor it contained particulars of the case. The second bag from the bottom was tampered with or the bag bearing the seal of 'BS' was not produced in the Court. The argument advanced by the Assistant Advocate General, Punjab that the bag was changed, as it had torn with the passage of time, cannot be accepted. The bag was neither changed in the presence of the accused or any competent authority, much less the Investigating Officer.\"\n16. Besides this, there are certain documents which were allegedly prepared on the spot like search memo, Ex.PE and Ex.PE\/1; recovery memo, Ex.PD; site plan, Ex.PG; arrest memo, Ex.PF and PF\/1. All these documents bear the FIR number, meaning thereby, either the FIR was registered prior to the alleged recovery of the contraband or the number has been inserted in the documents after its registration. In both these situations, it casts serious doubt on the veracity of the prosecution version as has been held in Didar Singh @ Dara's case (supra).\n(1) The Central Government may, having regard to the hazardous nature of any narcotic drugs or psychotropic substances, their vulnerability to theft, substitution, constraints of proper storage space or any other relevant considerations, by notification published in the Official Gazette, specify such narcotic drugs or psychotropic substances or class of narcotic drugs or class of psychotropic substances which shall, as soon as may be after their seizure, be disposed of by such officer and in such manner as the Government may from time to time, determine after following the procedure hereinafter specified.\n(3) Where an application is made under subsection (2), the Magistrate shall as soon as may be allow the application.\n(4) Notwithstanding anything anything contained in the Indian Evidence Act, 1872 (1 of 1872) or the Code of Criminal Procedure, 1973 (2 of 1974), every Court trying an offence under this Act, shall treat the inventory, the photographs of narcotic drugs or psychotropic substances and any list of samples drawn under sub-section (2) and certified by the Magistrate, as primary evidence in respect of such offence.\"\n17. The last nail in the coffin of the prosecution case is that in the present case, the procedure for drawing the representative samples of the contraband in front of Magistrate as laid down in Section 52-A(2)(c) of the Act has not been followed, which makes it obligatory upon the officer concerned that as soon as the contraband is seized, it has to be forwarded to officer-incharge of the police station, who is duty bound to approach the Magistrate for grant of permission to draw the representative samples in his presence. Such samples were then required to be enlisted and the correctness of the list of the samples drawn were to be certified by the Magistrate. In other words, the drawing of the samples has to be in the presence and supervision of the Magistrate, which has not been followed in the present case. Reference in this regard can be made to Union of India v. Mohanlal and another, 2016(1) R.C.R. (Criminal) 858 (SC).\nIn the upshot, the cumulative effect of the discussion made above is that the present appeal succeeds; the impugned judgment of conviction and order of sentence dated 20.07.2004, passed by the learned trial Court, is set aside and the appellants are acquitted of the charge framed against them. The appellants are stated to be on bail. Their bail bonds shall stand discharged.\nNarcotic Drugs and Psychotropic Substances Act, 1985 Sections 15 and 52A Investigation - Violation of principles of fair and impartial investigation of serious offence is must - Where the complainant and investigating officer is the same, it makes the prosecution case doubtful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajakg b/data_all_eng_slimpj/shuffled/split2/finalzzzajakg
new file mode 100644
index 0000000000000000000000000000000000000000..37b77932f438610e1e151a3393424365fb6a9532
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajakg
@@ -0,0 +1,5 @@
+{"text":"This un-insulated door-type that tilts in one single piece is for those who prefer good looks and simple and reliable operation and also reasonable price. Our tilting garage doors and their accessories are manufactured of zinc coated steel. The door is powder coated as standard and on request the frame can be powder coated as well. Tilting garage doors can be ordered in any RAL colour, or with golden oak or Italian walnut wood grain foil finish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fee varies based on activity, either a 1%-10% one-time percentage or $27 per hour if more involved bookkeeping activities are required.\nOther: May include best practices or professional development opportunities pertaining to activity, project or group.\nArts Services Initiative of WNY provides artists and arts groups tools and resources to help with the business side of their artistic career. Our mission is to promote the cultural sector's vital role in economic development and the community through capacity-building, collaboration and advocacy efforts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For those who do not visit North Campus often, and even those who do, Pierpont Commons contains a hub of resources that are worth knowing about! Built in 1965, the goal of Pierpont Commons was to help build the community on North Campus under the guidance of Wilber K. Pierpont. Not only are there great study spots and dining options available, but there are some important offices located within this building.\nThe basement floor of Pierpont Commons offers many services including the MCard center, the Financial Aid Office, the Office of the Registrar, the I-9 Regional Center, and the University of Michigan Police Department. There are also some food options on this floor as well, including Ahmo's, Panda Express, and Hibachi-San. Upstairs on the second floor level is Fireside Cafe, and on the first floor there is a U-Go's, which also serves pizza. The first floor also has a Barnes & Noble bookstore, the Computer Showcase, U-M Credit Union, and ATMs. All of these shops and services, and their hours, can be found on Pierpont Common's website.\nOutside of the food and services within Pierpont Commons, it also has many great study spots, which are conveniently close to the Duderstadt Library (and is connected to Pierpont Commons). There are also many computers and printers available within the building available for student use. If you need a study break, you can check out pool equipment from the Information Desk, or play on the piano that is located within the building. The building is very beautiful with an abundance of natural lighting and flags from all over the world on display.\nRead up on more Pierpont history and be sure to check on the building hours, as they vary each semester.\nOur Michigan Union location is closed while the Union undergoes renovation.\nFind out more at the Re:Union project site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Built for your life and whatever life throws at you, the Outlander is powerful and designed your everyday needs. Equipped with Mitsubishi's All-Wheel Control, this vehicle allows the driver to have an efficiently-running vehicle on various types of terrain by choosing 2-Wheel drive, 4-Wheel drive auto, or 4-Wheel drive lock. The SE Touring Package trim features seven passenger seating, SiriusXM, Apple CarPlay, Android Auto, CD, USB, Bluetooth cellphone and audio connectivity, dual zone automatic climate control, heated front seats and side view mirrors, power sunroof, blind spot warning with rear cross traffic alert, power folding mirrors, rearview camera, 18 inch alloy wheels, and more. This vehicles also has 3M on hood, rear hitch and wiring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"November 28, 2016 \u2013 They're Cuter When You Neuter \u2013 We held a \"Boys Only\" low-cost neuter clinic and provided services to 36 tom cats. Thanks to all responsible pet owners who are doing their part to control our cat population.\nNovember 18, 2016 \u2013 Wine 4 Whiskers \u2013 THANK YOU to the Winery at Seven Springs Farm for hosting our Wine 4 Whiskers Fundraising Event and to our sponsors; Teresa's Bakery, Halls Ingles, Sam's Knoxville Center, Maynardville Fred's, Maynardville Food City, and Party City! Without the support of the community and our wonderful volunteers, we could not help the number of animals that we do. It was a chili-kind of night, and we had a good turn out with great food, music, friends and wine.\nOctober 1, 2016 \u2013 Ride Like an Animal. We held our 10th Annual Ride Like an Animal motorcycle ride on October 1. What a beautiful day! Nearly 60 motorcycles, escorted by Union County Sheriff with a support vehicle bringing up the back. The ride went through Union, Claiborne, and Grainger counties, finishing with a great barbeque at Li'l Jo's in Maynardville with music provided by the Jeanette Weeks Band. Lots of great door prizes and auction items.\nIt is so heartwarming to see everyone gather for the benefit of all the four-legged furry animals that do not have a voice and need our help! Events like this help us care for and adopt out many dogs and cats, as well as provide low-cost spay\/neuter and vaccinations for the pets in our community. THANK YOU!\nWe cannot mention every rider or volunteer by name but we do want to recognize our generous sponsors: Annie, Stoney, Techie, Muppy & BH Ashley, Burning Art by Zophia-Farrier\/Blacksmith\/Welder, Durbin Drywall, Dennis Hall Automotive, d'Vine Inspirations, Fountain City Funding, Jeanette Weeks Band, Lil Jo's BBQ, Knoxville Harley Davidson Clinton Hwy, Ohio Valley Veneer, Ralph & Sue Shick, Rowland Veterinary Services, Advantage Electronics, Aimee Alden & Jemma Daisy, Clinton Highway Wrecker Service, Crossroads Firearms, James, Janet & Jacob, Kathy Pack, Melissa Friend & Savannah Ann, Lisa Foutz & Lacie, Mayor Mike Williams, Roberts Garage, Union County Chiropractic Clinic-Dr. Darrell Johnson, D.C., The Winery at Seven Springs Farm, and Wallace Auto.\nSeptember 12, 2016 \u2013 Low-Cost Cat Spay\/Neuter Day. We operated on 32 cats, many of whom were this year's kittens. These 32 cats could have produced over 300 kittens in the first year alone. Thanks go out to all responsible pet owners who did their part in controlling the cat population!\nSeptember 9-10, 2016 \u2013 Union County Humane Society Yard Sale. We had a great yard sale, bringing in over $2,500 for the shelter. A big thank you to Janet Holloway for allowing us to use the parking lot at Janet's Hair Salon, everyone who donated items, all of the volunteers who worked the yard sale, and every patron who stopped by to support us. A special thank you to Alexis Browning and Judy Najar who developed this fundraiser, now in its second year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakbzw b/data_all_eng_slimpj/shuffled/split2/finalzzzakbzw
new file mode 100644
index 0000000000000000000000000000000000000000..8bcbd8ed17c0cb0ea4f6b290fbf5ad55a326dffa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakbzw
@@ -0,0 +1,5 @@
+{"text":"IRANET was founded in 1993 in order to develop and provide the requirements of IPM's scientists.\nTo provide Internet for Iranian academic, IPM \/ IRANET joined EARN (European Academic & Research Network) via Vienna University in 1993. In 1995, EARN merged with RARE (R\u00e9seaux Associ\u00e9s pour la Recherche Europ\u00e9enne) to form TERENA (Trans-European Research and Education Networking Association).\nToday, after years of hard work, IRANET has become one of the safest internet provider for universities, institutes, research centers and large organizations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watch Camp X-Ray (2014). A young soldier escapes her suffocating small town by joining the military, only to find that she isn't going for a tour of duty in Iraq as she hoped. Instead, she's sent to Guantanamo.\nWatch Heaven Can Wait (1943). Henry Van Cleve presents himself at the gates of Hell only to find he is closely vetted on his qualifications for entry.\nWatch Toxin (2015). A pharmaceutical company recruits a well-known scientist to help develop a vaccine against a deadly virus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have skype installed you can select the link below to call Sally Harding Reg.MBACP. Advanced Dip CBT; Dip Therapeutic Counselling.\nYour future starts now!. Which path will you choose?\nSally Harding Counselling and Hypnotherapy provides holistic, person centred approaches, to dealing with life situations.\nKnowledge is power! How well do you know yourself? When you know what makes you tick, you can start to become the person you want to be.\nYou will be accepted for the unique individual you are, in a non-judgemental and encouraging environment. You will have the opportunity to talk about anything that is concerning you.\nYou will then decide how to start working towards your goals.\nAnxiety, depression, relationship issues, loss, berearvement, stress, identity issues, grief, self-harm, low confidence or self esteem, family difficulties, mental health problems.\nHypnotherapy can be very helpful in reducing the impact of the symptoms of many long term and chronic health conditions.\nAdvanced Diploma in Cognitive Behavioural Therapy.\nAccredited Diploma in Hypnotherapy and Neurolinguistics.\nFlexible appointment times to suit. Please contact for futher details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Something is going on in the chambers of Chief Justice Roberts.\nLast October, in the oral argument for Gill v. Whitford, the Wisconsin political gerrymandering case, the Chief Justice provocatively explained that \"if you're the intelligent man on the street,\" you will know that the case will come down to the fact that the Court simply prefers one political party over the other. Worse yet, this will be true across the country, \"[a]nd that is going to cause very serious harm to the status and integrity of the decisions of this Court in the eyes of the country.\" Best to keep the federal courts out of this inherently political fight.\nA careful reader will recognize this position. This was Justice Frankfurter's view in Baker v. Carr, decided in 1961. Frankfurter argued that the Court must carefully nourish public confidence by \"complete detachment . . . from political entanglements and by abstention from injecting itself into the clash of political forces in political settlements.\" But he lost this argument almost immediately. The Court took on malapportionment and entrenched interests in the name of democracy, and the public sided with the Court. The Chief Justice has no reason to expect a different result this time.\nIn the same Term when Chief Justice Roberts worries about the legitimacy of the Court, the justices appear poised to take on public sector unions for the third time, in Janus v. AFSCME. This is remarkable. On one side are arguments for individual rights and the First Amendment, backed by the right. On the other side are long-standing precedents and the apparent death of labor unions. What would our intelligent man on the street think about this case? How could our intelligent man not see this case but as a Republican attack on Democrats? Yet the Chief Justice seems unfazed.\nFrom recent history, we know the Chief Justice is playing a sophisticated chess game. On January 22, the Court decided Artis v. District of Columbia, a supplemental jurisdiction case. All you need to know from this case comes from Justice Gorsuch's dissent, which argued that the decision \"clears away a fence that once marked a basic boundary between federal and state power.\" This is heady stuff. After Artis, Gorsuch concluded, \"[w]hat boundaries remain then?\" Is our federalism is dead?\nCuriously, Chief Justice Roberts provided the fifth vote in Artis. This is curious because Chief Justice Roberts authored the Shelby County opinion, where the Court went as far as to create a new doctrine \u2013 equality of states \u2013 in order to strike down important parts of the Voting Rights Act. Shelby County is a decision about our federalism on steroids. And if Justice Gorsuch is right, Shelby County stands in direct tension with Artis. Chief Justice Roberts cares more than most people about our federalism; and surely, our intelligent man on the street doesn't care quite as much. His vote in Artis makes no sense, yet he chose not to explain it.\nIt is clear that the Chief Justice has a selective gaze. His upcoming votes in Gill and Janus will tell us a lot about his philosophy and the legacy of his Court. And about his consistency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Code of Business Conduct serves to guide the actions of our employees consistent with our Company values. The Code helps our people do the right thing and play by the rules wherever we operate around the world.\nOn February 12, 2018, The Coca-Cola Company amended its Code of Business Conduct. Revisions to the Code were made primarily to simplify the document and make it more user-friendly, restructure the layout of the Code around commitments to our values, and to provide more guidance and examples on topics of special interest.\nWant to report any issues regarding accounting, internal accounting controls, auditing or other business conduct at The Coca-Cola Company?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzampbp b/data_all_eng_slimpj/shuffled/split2/finalzzzampbp
new file mode 100644
index 0000000000000000000000000000000000000000..88ecbd6ee0ff453222f29666970e8be65b684f77
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzampbp
@@ -0,0 +1,5 @@
+{"text":"Unique, in every sense of the word. Hands down, the most personalized relationship-based building experience offered in our native Austin, TX. We attend every design & selection meeting, large or small, methodically orchestrating detailed project plans via our patented pre-construction processes, with zero disconnect between office & field. In terms of health conciseness & overall sustainability practices, we are without question the Whole Foods of our industry. When it comes to service, VIP treatment is all we know. While remaining client based, our ever evolving design style is termed \"Common-sense Contemporary\" timeless, agile, intuitive, fresh, and free flowing. We are artists with child like imaginations, unrestricted, confident, fun, & creative; full of ideas & opinions, but never overbearing or wasteful.\nLike Yoda, emotional intelligence comes easy to us; so does communication in all forms including verbal, visual, & written. We hold black belts in building science, & stay connected to other industry leaders throughout the world. Technologically, we're aggressive & cutting edge, including being fully BIM capable. Our positive energy & competitive culture is contagious, drawing the absolute best talent around from top to bottom, especially our in-house finish team.\nDuring our first decade of existence, we've have had the pleasure of building some of the most stunning healthy, happy, & harmonious spaces this city has ever seen; subtly & consistently providing their inhabitants with a beautiful thing: the ideal state of mind. And let me tell you, that's what you want.\nCurrently taking clients for pre-construction services in '17, but backlogged for project starts until '18.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"xxManilaSEXPRIDE TessaGoldie livejasmin. DivineHoiny. katherinsenxual. BrianaEbony.\nANGIEHOTSTARAngelicBarbiTimeaStarSaraBlaze .JenniferHopZOESANTODOMINGOOAshleyBartonANGIEHOTSTAR .AnaAndMarkoJackMillerXMelanieReedDebbyeDesire .sexgracekatherinsenxualJenniferHopJackMillerX .JessyAmberhotox84kylielovedAssaye .WetAndHairyyWolverine380KarinaWestMelanieSteff .SirenaFoxAudreyNoirWetAndHairyycanelitax .BeautySansakenjiexx4uxxandreasworldSweetIvyy .danihotsex21BrianaEbonykatherinsenxualKaroba .SoBeautyHelgaUnusualRAMONAStellaMoonKimWonders .SaraBlazeDivineKara4UMilenaSilverDebbyeDesire .\nErminesexgracekatherinsenxualPlayfulChristine .BrieBellaAshleyBartonkatherinsenxualANGIEHOTSTAR .LarissaBellaEddyEverhardKimberleyStarr5jack5 .EmbeerWoodkatherinsenxualCosmosParadiseZOESANTODOMINGOO .CurvyDiamondDHarleyBlack5jack5AngelicBarbi .Wolverine380LarissaBellaBeautyLadyVeronhotshyprincess .MistressTallidaSweetIvyyWolverine380MiaBellaX .hotshyprincessandreasworldLarissaBellaKimberleyStarr .HugeJackyXXLDivineKara4USlutyPrincess1SlutyPrincess1 .cococoWetAndHairyyMelisaGoldnessMiaBellaX .sexgraceAnaAndMarkoMatt3oSLAVEPASSION4YOU .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"23.5 inches , 675 grams .May have the German Giant gene that runs in her lines.She was produced here at Rainbow Bearded Dragons.\nThis is Rague, Father of the babies. A STUNNING Red Black eyed Trans. male. Father of the clutch. He's also 100% het for Hypo, produced by Mike Moran, Galaxy Dragons. He's 20.5 inches long now and 575 grams.\nI produced some nice Breeder and Collector Quality Red Translucents from this pairing in the past.Looks to be the same this year! These are either hypo or 100% het for hypo. All are Translucent and may carry the German Giant genetics that run in this line .They will get very LARGE with good care.All are eating great on 3\/4 inch size crickets and Leafy greens with grated yellow summer squash all mixed together.\nThis Breeder Q. Trans Male is #L3 He's just turning 7 months of age and is 14.5 inches long. He has get color still coming in and he's very tame, easy to handle. He's also 100% het for Hypo. Sale 255.00 plus 35.00 for shipping!\nTo Inquire about one of my dragons. Just click on link to send an e-mail .I will need the number on the dragon you would like to get in the e-mail.Must be at least 18 years of age to inquire about the dragons. All forms of payment are accepted, including credit cards. I'm a Night Owl so late inquiries are welcomed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2026and today, country singer Kip Moore joins them with his efforts!\nLately, I've been interested in charity and how many different ways there are to give back. So many different initiatives, companies, people, causes, and ways to give to just about anything you can imagine!\nWell, our country music superhero from Tifton, Georgia is on the move and is putting his nose to the grindstone on a project that is near and dear to his heart.\nThat project involves opening skate parks to help fund the Comeback Kid Skatepark Project, which is an intiative from his Kip's Kids Fund.\nI know what you're thinking\u2026What do skateparks do?! Well, actually, they help a great deal in urban areas! Off the top of my head, it will reduce inner-city crimes by a landslide as Kip is providing a safe place for high school age kids to go.\nI love this intiative SO MUCH!\nSkating is such a wonderful sport and it encourages kids to hang out and to have a common bond. I love skate life and I can see how this project will help out urban communities! I'm happy that Kip knows how important it is to help give kids something they can be proud of and something that will keep them from negative peer pressure on the streets.\nKip opened the first Comeback Kid Skatepark in Annapolis, Maryland and the second in San Marcos, Texas\u2026AND he's got two more parks that are set to be opened in Nashville and Boston!\nThank you Kip for caring for the kids out there! You're gnarly, dude!\nLet's talk about Justin Bieber, shall we?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Choosing the Best Hiking Shoes \u2013 STABILITY IS KEY!!!\nHeeluxe tells you how to find the perfect hiking boot for stability! Using these tips, your going to be able to buy the best boot possible for your foot! Remember to like us on facebook and follow us on twitter! And don't forget to subscribe!\nTop 3 Survival Tips: Number 2 is NEVER On TV Survival Shows\tThe NEW Flathead Catfish Fishing Rod!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamvpr b/data_all_eng_slimpj/shuffled/split2/finalzzzamvpr
new file mode 100644
index 0000000000000000000000000000000000000000..0a83e2b6c91ce42234bbef2ba91de525ce9c330b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Actor Michael Sheen has hit out at newspaper headlines claiming he has quit acting to become a political activist following an interview with The Times Magazine.\nSheen's apparent comments were made during promotion for his latest film Passengers and published on Saturday.\nThey were picked up by a number of media outlets, including The Times who reported it with the headline: \"Michael Sheen quits acting for activism\".\nSimilar headlines also appeared in the Guardian, Telegraph and Mail Online.\nBut, in a blog post on the same day, Sheen said he \"did not declare that I'm 'quitting acting and leaving Hollywood' to go into politics\".\n\"I did one interview with The Times of London a few weeks ago, parts of which (including a headline that is not a quote) have been picked up by a lot of other outlets,\" he said.\n\"In the actual original interview I said I have become more involved with community issues back at home over the last few years and because of the political situation it's something I would like to focus on more.\n\"The interviewer asked me what that meant for my career and I said it might mean I work less as an actor and maybe even stop for a while at some point. But I don't really know yet.\nThe star's post has prompted fresh headlines clarifying his position.\nThe Times has said it won't comment on the matter. Sheen is understood to have written a letter to the newspaper to be published tomorrow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Santacon San Francisco will take place on Saturday, December 8. While there is no set time for the event, the toy drop-off to benefit Toys for Tots will take place at 12 noon in Union Square. Santa Tom will be there to collect donations of new, unwrapped toys that can be donated to the Toys for Tots program. See the Original Facebook group for details.\nBecause, unfortunately, there is no central coordination this year, SantaCon(s) will start at various nearby locations. We recommend starting at Mayes (1233 Polk Street) if you want join the crowd. Lower Polk Street has an awesome selection of bars including Mayes, Mcteagues, Blur, Jackalope, Vertigo, Route 101, Rusted Mule, R Bar, Lush Lounge, Pour House, Playland and more who will be welcoming all Santas!\nNote 1: All genuine SantaCon gatherings in the city have identified themselves as no cover\/free to attend. Please don't get scammed by profiteers trying to sell you tickets. There is NO registration fee or any registration requirement to attend SantaCon. If anyone is charging you a fee, it's fraudulent.\nNote 2: Santa has become aware of journalists and publications with a vendetta against SantaCon. They will be looking to produce bad stories about you. Please be nice, make Santa proud, and make the city of San Francisco love you.\n-Make new friends and if you see lonely Santas who want to hang out with other Santas, invite them to join your group.\n-Watch out for fellow Santas. If you see a Santa getting wasted or being taken advantage of by someone, look out out for them and help them stay safe.\n-Be nice to bartenders and bar staff. It's usually a crazy time and the bar staff is trying to serve everyone. Be patient. Don't be \"that\" jerk. Tip well.\n-If you or your friends have had too much to drink, pace yourself. Maybe it's time to call it a day and go home. No one likes messy Santa around.\n-Carry enough cash for drinks. Most bars will be \"CASH ONLY\" for SantaCon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download \"Visioning Statement and Guiding Principles\"\n2 Planning for the Future The General Plan The General Plan represents Woodland's shared vision of the future and defines a path to lead the community toward its desired future in 2035 and beyond. The General Plan is the City's primary tool for guiding future development. The vision statement is an aspirational description of what the community would like to be in the future, looking forward to 2035, and represents a compilation of input from the community through the General Plan public process. The Guiding Principles provide a foundation for the goals, policies and implementation programs in the Plan. Each day the City must make decisions about growth, services, and where and how to focus resources. The General Plan provides the guidance for these decisions by establishing long-term goals for the City's future, policies to inform day-to day-decisions, and implementation programs to realize that vision. The General Plan is the foundation for the City's regulatory and community and economic based documents. The General Plan provides for orderly growth and conveys the community's values and expectations into the future. It sets the tone for evaluation of future development projects, planning of future public facilities and services, defining and maintaining economic sustainability, meeting mobility needs and protecting historic and natural resources.\n3 Vision Statement I n 2035 Woodland is a highly desirable community to live, learn, work and recreate. It has maintained its small-town feel while maturing into an attractive, vibrant, and sustainable city that celebrates its architectural heritage and cultural diversity. Woodland is a healthy community with livable neighborhoods, a thriving downtown, well maintained infrastructure, excellent schools and recreational amenities connected by a seamless network of trails and paths. The city is the region s center of agricultural technology and food production and is recognized globally as a leader in sustainable agriculture. The community is prosperous and fiscally sound, offering abundant employment opportunities to its diverse and creative workforce. Woodland has become a destination for visitors seeking to experience its unique agricultural, historical, recreational, cultural and entertainment amenities.\n4 Quality and Character: Retain and enhance Woodland s quality of life, its distinctive identity and small town characteristics. Guiding Principles Preserve Woodland s unique small town charm and quality of life by maintaining the city s distinct urban edge and surrounding agricultural open space, promoting the Downtown and historic resources, and developing a variety of recreational, community and cultural facilities. Preserve and enhance the best qualities of Woodland s existing neighborhoods and promote the development of new neighborhoods with similar qualities while fostering healthy and attractive commercial and employment centers. Promote development that strengthens the physical form of the City, enhances livability, incorporates sustainable design practices and continues to enhance Woodland s unique sense of place. High quality design and pleasing physical form promotes Woodland as offering a high quality of life and a desirable place to live, learn, work and recreate. Orderly Development: Promote new growth while achieving an orderly pattern of community development, consistent with economic, social, fiscal and environmental needs. Provide for urban development and expansion of associated services on a logical and incremental basis to accommodate projected population and employment growth. Growth will be managed to ensure adequate infrastructure, public services, and amenities that the City can sustain, provide and maintain. New growth areas will be carefully planned to enhance and not detract from existing neighborhoods and commercial centers. Infill and adaptive reuse of underutilized and vacant buildings is promoted. Historic Downtown: Strengthen the historic downtown district as the City s center of shopping, dining, entertainment and employment. Promote Downtown as the civic, cultural, and entertainment center of Woodland. The General Plan promotes a broad mix of uses, including increased dining, retail, and entertainment destinations with an array of urban housing and professional office\/technology companies. Provide support to new business ventures and private reinvestment with policies and actions to assist revitalization and re-use of historically significant structures.\n5 Economic Development: Foster economic growth and diversification with a range of employment opportunities for all residents. The General Plan promotes a diversified economic base and seeks to capitalize on Woodland s location and assets access to 1-5, Sacramento International Airport, rail service, prime farmland, and UC Davis-by supporting and assisting business development and mitigating constraints to economic investment. The Plan provides sites in a variety of infill and new growth locations to attract hotel, office, industrial, and research and development uses, which in turn will provide jobs and help the City achieve fiscal sustainability. The General Plan seeks partnerships in higher education, seed research, agricultural technology, food production, and other locally appropriate sectors. It supports linkages with Woodland s strong historical and cultural resources and promotes tourism. Mobility Options: Coordinate land use and transportation planning to provide a range of attractive and viable transportation options such as bicycle, pedestrian and transit. Support development choices and transportation improvements that allow and encourage more residents, workers and visitors to walk, bike or use transit. Promote the development of complete streets to safely and effectively serve the needs of all modes of travel. Promote and seek opportunities to develop a seamless network of trails and paths to support a healthy and active lifestyle, livable neighborhoods and access to schools and amenities. Housing Choice: Provide a variety of housing types to meet the needs for all generations and income levels. Encourage and enable a mix of housing types and densities that will provide Woodland residents with access to a full range of housing opportunities, and enable the City to meet its fair share of the region s housing need. Housing types facilitated by the General Plan range from larger lot to small lot single-family homes, townhomes, apartment buildings and lofts in a variety of Iocations and settings. Infill development in the Downtown and along mixed-use corridors is encouraged. Promote design practices that support high quality design and neighborhood development.\n6 Agricultural Heritage: Preserve Safety: Ensure that Woodland Woodland s surrounding agriculture is an important part of the community s heritage, plays a major role in the city s economy, and endows Woodland with a unique sense of place. To help maintain this important resource, the General Plan maintains the voterapproved Urban Limit Line within which urban development will be contained. Development will occur in an orderly, contiguous manner to preserve agricultural use of land as long as possible. Outside the current city limits, the City will encourage and support continued agricultural use of land. Protect community livelihood and investment through requirements for structures to withstand seismic activity, provide fire protection and minimize flood risk. Ensure a sense of personal and public safety as essential in developing a high quality of life. The General Plan supports appropriate levels of prevention and response. prime agricultural land and uses within and surrounding the community. remains a safe place to live, protected from natural and manmade hazards. Environmental Stewardship: Foster a sustainable community for the next generation and protect and improve the quality of the natural environment. Encourage the preservation of the area s natural resources and continue to support regional efforts. Promote land use patterns that improve air quality and reduce greenhouse gas emissions, minimize vehicle mile s traveled, and encourage conservation. Support proactive solutions to protect areas at risk of flooding.\n7 Public Services: Provide realistic supportable and appropriate levels of public services that are sustainable and fiscally sound. Balance the fiscal realities of providing sustainable public services with community desires for high quality amenities and facilities to ensure that meeting today s needs does not compromise the community s fiscal future. Require new development to pay for itself, including new facilities and on-going operations. Support the continuing maintenance and expansion of existing public facilities as the most efficient and effective means of living within the community s means. The City will continue to strive to improve the efficiency and quality of its public facilities and services. Health and Recreation: Provide all residents with opportunities to live an active, healthy, and green life style. Promote healthy lifestyles by enhancing opportunities for physical activity, healthy eating and sustainable living. The General Plan ensures that adequate parks and recreational amenities are well integrated in new neighborhoods. The Plan promotes creation of a recreational greenbelt and expansion of walking and biking paths to enable residents to use active transportation options to connect to work, schools, grocery stores, and variety of open spaces. Quality Education: Foster quality educational and enrichment opportunities. The General Plan underscores the importance of high quality educational opportunities-including K-12 education, higher education, workforce training, and general community enrichment to Woodland s quality of life. The City recognizes the link between education, safety, and opportunities for economic advancement in the community for youth, their families, and all Woodland residents. To this end, the Plan supports continued partnership with the Woodland Joint Unified School District, the County Office of Education, and Woodland Community College in planning, facility sharing, extracurricular activities and recreation, and promoting academic achievement, as well as linkages between Woodland s growing cluster of agricultural technology and research establishments and higher education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are using your Cisco Catalyst switches to insert DHCP Option 82 information and you are also using your Cisco routers as DHCP relay-agents (via 'ip helper-address'), you'll notice right away that your Option 82 enabled DHCP requests are not being forwarded by your routers.\nAs a security measure, Cisco IOS will not forward DHCP requests that contain Option 82 information and a gateway request set to all zeroes. This is the condition of an initial DHCP request that has been rewritten by a Cisco Catalyst switch.\nTo globally enable these packets to transit all router interfaces, issue the 'ip dhcp relay information trust-all' in configuration mode. If you'd like to maintain this security feature and only trust these requests on certain interfaces, you can issue an interface specific command as seen in the configuration sequence below.\nOnce you've enabled trust for some or all interfaces, your Option 82 enabled DHCP requests should once again be relayed by your Cisco routers.\nAs always, if you have problems or questions, let me know.\nWhat is DHCP Option 82?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This main-dish salad works well with many variations - change the fresh herbs, beans, and cheese to use what you have on hand.\nCombine first 5 ingredients in a large bowl; stir with a whisk. Add lettuce and the next 6 ingredients (lettuce through chickpeas); toss well. Serve with pita wedges.\nI'm not one for thick, creamy dressings. That said, I found the very simple dressing on this salad to be a solid keeper. Exactly as it says: change the herbs, etc & it's outstanding. The 1st time I made it without the dill, onion, capers & cucumbers. I didn't have these on hand & we still loved it! 2nd time around, I included dill...& again, we loved it! Eventually I'll make it with all the suggested ingredients. 'Till then, I'm happy adjusting it to suit my needs. I adore the lightness of the dressing, with the fresh parsley. One note: I used Italian organic lemon juice: delightful!\nWonderful one-dish meal ~ so tasty!!\nThe salad ingredients are fine. I found the dressing to be missing something....just not sure what! I will try again with some variations to the dressing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanwqi b/data_all_eng_slimpj/shuffled/split2/finalzzzanwqi
new file mode 100644
index 0000000000000000000000000000000000000000..5ad3499130cd8097af86498c2e3d4ea60b5ab618
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@@ -0,0 +1,5 @@
+{"text":"The Royal Institution of Chartered Surveyors (RICS) has launched an insight paper which explores the impact of using artificial intelligence (AI) in the built environment, and the urgent need for industry professionals to understand how it will influence their areas of operation, as technology has a far greater role to play in the future.\nToday the National Home Builders Registration Council (NHBRC) announced that it will be spending more than R30 million in the new financial year starting on 1st of April 2017 in its efforts to bring about transformation in the home building industry.\nThe Department of Trade and Industry (dti) plans to spend R216 million on renovations at five industrial parks in the country.\nBuilding activity in the South African market for new housing remained largely subdued in the first three quarters of 2016.\nPPC reports that its half-year net profit fell by more than two-thirds as a result of higher finance costs and weaker currencies in African countries where it has operations.\nDawn announces a 10% drop in revenue to R2.4bn for the six months to end-September.\nThe Consulting Engineers South Africa (CESA) Bi-annual Economic and Capacity Survey for the period January to June 2016, just released indicates that times remain tough.\nIn line with past practice and disclosure, Cashbuild herewith provides its quarterly trading update.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great Britain will face Japan in the battle for fifth place at the 2018 Champions Trophy after a narrow loss to Argentina.\nVictoria Sauze's early strike was enough to secure victory for the South American side, meaning they qualify for the bronze medal game.\nSarah Jones came closest to scoring for David Ralph's side, seeing a fierce shot brilliantly saved, but it was another day of frustration as they couldn't break through a stubborn Argentinean side.\nDespite the final score, Ralph was left feeling positive after an improved showing.\n\"It was an intense game against a high quality Argentina team,\" he reflected.\n\"While we are disappointed with the result we feel we performed well tonight but unfortunately couldn't turn it into a positive result.\n\"Now we prepare for Japan, which will be another tough game for us but that's why this is a great learning experience.\"\nNeeding to win by two goals or more to keep their chances of a medal alive, Great Britain succumbed to a nightmare start as Sauze pounced in the second minute after Sophie Bray had done well to initially block an Argentina corner.\nIn response, Amy Costello drilled a corner just wide after Giselle Ansley had forced Belen Succi into a block, before Jones drew a world class save from the Argentinean 'keeper in the 24th minute.\nAt the other end Lucina von der Heyde, Maria Granatto and Delfina Merino all saw efforts narrowly miss the target while Sabbie Heesh reacted well to keep out an Agustina Habif corner in the final minute of the first half.\nThe third quarter was a tight and nervy affair, with neither side creating any clear-cut chances, while in the final quarter - despite employing a kicking back for the final seven minutes - GB couldn't force an equaliser.\nGreat Britain will play Japan in the 5th\/6th play-off on Sunday 25 November at 0600GMT, a game you can watch live on BT Sport.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twenty three years ago tonight, a masked gunman took Jacob Wetterling as he, his brother and a friend biked along a country road in St. Joseph, MN. The Jacob Wetterling Resource Center is asking that people leave their porch lights on tonight to remember Jacob Wetterling.\nJacob was 11 years old when he was abducted. He hasn't been seen since, and the case remains unsolved. This year, the center is also asking people to honor all missing children by talking to others about child safety.\nThe BCA and the Stearns County Sheriff's Office urge anyone with information about what happened that night to contact the Stearns County Sheriff's Office at 320-251-4240.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Established in 1983, the land known as Las Casas de la Selva is the home of Tropic Ventures Sustainable Forestry & Rainforest Enrichment Project, in Patillas, Puerto Rico. Silvicultural techniques developed and applied at Las Casas de la Selva over the last three decades, demonstrate, that on a small scale in Puerto Rico, secondary forests can be ecologically and economically suitable for sustainable timber production. This eco-technology was implemented to encourage similar practice in the Caribbean and globally, as a contribution to economic development that encourages local protection and sustainable management of secondary tropical forests. For thirty years Las Casas de la Selva has seen people from all over the planet live and work in her forest in all areas of science, art, and management. Learning by doing; gathering valuable experiences about life and human interaction in this forest biome, striving to continue and keep the process of creating ecological and sustainable communities, going, going, and never gone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To get started, dice about half of onion and mince 2 cloves of garlic. Heat 2-3 tbsp of olive oil over medium heat in a large soup\/stock pot. (Did I mention that this is a one pot deal?? Score!) Once the oil is heated, add & saute the onion & garlic. As the onion & garlic is sauteing, chop about a third of a package of baby carrots into bite-sized bits. (Keep those carrots in check and cut gently. They will try to get away from you!) Add the carrot bits to the onions in the pot, and saute another 3 minutes. Stir the onions & carrots often so they do not burn. During the late minute of sauteing, add a little flavor- salt & pepper, some thyme, tarragon, and rosemary. I used all dried herbs, but feel free to use fresh if you have them on hand.\nPour an entire 32 oz carton of chicken broth into the pot. I also add a few cups of water. Turn up the heat to medium-high. This is when I throw in a bay leave. It isn't necessary, but it gives the broth a nice flavor. Heat the broth to a boil. As the broth is heating, cut up 1-2 cooked chicken breasts into bite-sized pieces. (Ah-hem- this is where that rotisserie chicken from earlier this week comes in nicely!) Add the chicken to the pot. As this point, you can throw in any frozen veggies you may have on hand- peas, green beans, whatever suits your fancy. Once the broth is boiling, add your noodles. I used vermicelli & broke them into fourths before throwing them into the pot. But you can use any pasta you have on hand- small shells, linguine, egg noodles, etc. Just make sure they are in easily spoonable sizes. :) Once the noodles are almost done, scoop out the bay leaf. Then throw in any greens you have on hand. I used baby spinach leaves that I tore into smaller pieces. But you could also use kale or swiss chard. Let the soup cook for another 3-5 minutes and then taste the broth. Does it need more salt or pepper? Go ahead and add it.\nThis is an amazingly easy and surprisingly quick dinner, and who isn't impressed with HOMEMADE chicken noodle soup?? I ladled our soup into bowls and topped it with a few tbsp of grated Parmesan cheese. Serve with a crusty bread or your favorite crackers. Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanyhc b/data_all_eng_slimpj/shuffled/split2/finalzzzanyhc
new file mode 100644
index 0000000000000000000000000000000000000000..cdb511d8630e2251c7f87e6e9637aed84b324f2c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanyhc
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+{"text":"We offer a full range of custom graphic design services and will assist you with printing services if required. Our designers are well versed in all forms of graphic design and printing. Whether you need a simple business card designed or a highly complex, color catalog, we have the software skills and creativity to get any size project completed fast and at a very high-quality.\nLogo design is the process of creating a logo, which is the most important visual identifier of your brand. By arranging symbols, text and color, a logo designer creates a unique mark that communicates the essence of your brand. Your logo design is the cornerstone piece of your brand identity. You'll use it for all parts of your business: on your website, on social media, on physical products, packaging, in stores and in marketing materials.\nUnlike other types of design, product packaging is super touchy-feely. That's why your customers deserve easy-to-understand, beautiful packaging that invites them to take your product home.\nA great package will make your product pop. And fall off the shelf (into customers' carts). Get a custom packaging design from us and we will create something you'll love.\nProfessional photography services for your website and digital marketing use. Our team of professional photographers has years of experience shooting people and products that capture our clients' unique personality, show customer engagement and highlight product details. Since you can't meet each potential new online lead in person, professional, high-resolution photos are absolutely essential to making a positive first impression.\nBranding aims to establish a significant and differentiated presence in the market that attracts and retains loyal customers. Here we plan for all of your needs and vision strategies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At New Horizons Toronto we are all about flexibility.\nIn order to provide great flexibility for your company's training needs, we offer Private Classes which can be held at our facility or Onsite at the client's location. This option gives you the ability to customize your training - to tailor the class to fit your requirements.\nAll prices are in $CAD and subject to HST.\nAn Account Representative at New Horizons Toronto is ready to answer your questions about onsite or private training. Contact us today at 855-472-8226 or by completing the form below.\nPrivate onsite training can help your business, whether large or small, by allowing you to train as many or as few employees as you need. Taking advantage of New Horizons Toronto's private onsite training can save you both time and travel expenses. Since everyone is trained at once, New Horizons team training reduces your overall costs.\nNew Horizons Toronto can fit onsite training into your schedule and conduct it at a location convenient for your staff. Depending on your company's specific needs, New Horizons can use standard training options, or customize them for individual topics you require.\nYour employees will be ready to put their new knowledge to use right away with New Horizons Toronto's customized training and real-world examples.\nProper training through New Horizons Toronto can increase employee productivity, while minimizing downtime. Onsite training also allows employees to operate their actual equipment, allowing them to quickly use their new skills.\nPrivate team training with New Horizons Toronto helps build teams, as well as the potential for success.\nStacey has more than 30 years' experience securing, designing and implementing such diverse systems as Microsoft Windows, Novell Netware, DEC VMS, IBM AIX, Solaris, SCO Unix, Linux, and MAC O\/S and Cisco switches, routers and firewalls. He also was systems analyst for several software development projects using assembly language, C, dBase, Access, Visual Basic and Oracle. By providing consulting services he has assisted numerous organizations to improve, automate and secure business processes and their supporting systems.\nBringing his real world experience to the classroom since 1999, his emphasis has been focused on security related training. Whether in the classroom or through virtual online sessions, he enjoys sharing his depth and breadth of experience and his common sense approach to security, technology and business processes. Stacey is an authorized instructor for ISC2, EC-Council, Cisco, Microsoft, Logical Operations, CompTIA and ITIL.\nHaving held industry certifications since 1997, Stacey was one of the first 2000 persons in the world to achieve MCSE certification for Windows 2000. His current certifications include CISSP, CASP, CFR, CEH, ECSA, CHFI, CEI, CCNA, CCSS, CCSI, CTT+, LINUX+, SECURITY+, A+, SCNP, ITIL Foundations, MCT, MCSA, MCITP, MCTS, MCP+I.\nStacey has authored two books in the Laudon MCSE\/MCSA Certification series published by Pearson Prentice Hall: Installing, Configuring, and Administering Windows XP Professional and Planning and Maintaining a Microsoft Windows Server 2003 Network Infrastructure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After working on canvas paintings for months, the last thing you want is for your artwork to get damaged, especially if selling your work is a major source of income. Unframed canvas paintings are susceptible to color fading and scratches when stored with other items in a closet, basement, or garage\u2014and, worse, they can be badly warped by high temperature and humidity. That's why it's important to store them somewhere safe.\nWith a climate-controlled storage unit at a nearby self storage facility, you can store your unframed canvas paintings to ensure that they're in good shape when you go to sell them.\nBelow are some tips for properly prepping your paintings for storage and moderating the temperature and humidity levels in a climate-controlled storage unit.\nEven the smallest things, like dust and dirt, can damage your paintings while they're in storage, which is why you need to cover your paintings, keep your hands off of the canvas, and store paintings in the proper containers.\nAccording to John Rogers, owner of Gallery 72 in Omaha, Neb., artwork needs protection from dust and dirt, which can damage the paint (especially oil paints) on canvas. That's why Rogers suggests using loose plastic to cover canvas paintings, as it allows enough air circulation without letting dust or dirt in.\nWhen covering canvas paintings, make sure not to touch the surface of the canvas since the oils on your hands can damage the paint as well. If you must touch a painting's surface, wear gloves made of 100% cotton\u2014never use latex or any other kind of non-slip material, as these can easily rub off layers of paint.\nAs for storing unframed canvas paintings in specific preservation containers, Rogers suggests rolling large paintings and storing them in diameter tubes, which prevents crushing or distortion. Smaller paintings, however, can be kept in flat file cabinets with a layer of archival paper or foam board between them.\nDespite keeping your paintings in diameter tubes or flat file cabinets, they can still be vulnerable to extreme temperature and humidity levels, which can deteriorate paint and make canvas buckle. For this very reason, you should never keep your artwork anywhere besides a self storage unit equipped with climate control.\nClimate-controlled storage is a great feature to have when storing paintings because it allows you to monitor and maintain the temperature and humidity levels of your unit. Storage renters around the nation use climate control on a daily basis to protect sensitive items, including electronics, antiques, wooden crafts, and\u2014you guessed it\u2014artwork.\n\"Any [unframed canvas painting] should be stored between 70-75\u00b0 with about 50% relative humidity,\" says Vanessa Amor, Business Manager of Museo Vault Fine Art Storage. She adds that these conditions match standards set by the art insurance industry and museum associations.\nAlthough climate-controlled units are typically more expensive to rent than standard units, the additional cost is worth the peace of mind you'll have when storing your life's work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nowadays, most girls are troubled by hair loss problems. Girls use different things to protect their hair from falling, but there is no benefit. Rather, the use of more things in the hair can greatly harm the health of the hair. Today, we are going to tell you about something that will ease your hair loss problem.\nGreen Tea is very beneficial for hair. There are plenty of antioxidants present in Green Tea which help in increasing hair growth. Applying green tea in hair roots improves blood circulation. By which the hair grows fast. Its use also eliminates the problem of hair loss.\nFirst, leave the green tea bag in water and leave it for 10-15 minutes. Now wash your hair with shampoo. Now put green tea in your hair. Afterwards wash your hair with plain water. Wash your hair with green tea water twice or three times a month to get the best results. Doing this will remove the problems of dandruff as well as your hair loss problem will be wiped out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Obscure double glazed window to rear, heated towel rail, low level w.c., p-shaped bath with shower above with two shower heads, pedestal wash hand basin with mixer tap, tiled walls, wood effect flooring.\nAccess to loft, full length storage cupboard, returning stairs down to Sitting Room.\nDouble glazed window to rear, obscure glazed door to rear, range of floor and base units, stainless steel sink unit with mixer tap, 4-ring electric induction hob, electric oven, door to larder cupboard..\nObscure double glazed window to front, entrance door to side, radiator, wood-effect flooring, storage cupboard, 5'4 x 3'3.\nDecked pathway leading to raised decked area to rear, mainly laid to lawn with a range of planting areas, fenced to boundaries, outside tap, pathway to side leading to frontage.\nLawned area to one side, Driveway parking leading to Garage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoegr b/data_all_eng_slimpj/shuffled/split2/finalzzzaoegr
new file mode 100644
index 0000000000000000000000000000000000000000..e36bfc3f518936eeb4c3fd3cd9bd2e8b6ec3cfad
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoegr
@@ -0,0 +1,5 @@
+{"text":"In today's Gospel, Jesus's latest miracles seem to have inspired quite a lot of searching and confusion among his followeRabbi, we have been looking for you! When did you get to Capernaum? Come to think of it, how did you get across the sea without a boat? What is this food of eternal life?\nI am never in this alone, and everything I am called to do is with and through Jesus.\n\u2014Amanda Collins is a Religious Studies teacher at St. Ignatius College Prep in Chicago.\nLoving God, let your words be my beacon when I am lost at sea. Help me to believe in you always, that I may feast on the fooof eternal life. Amen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a few weeks our team will be heading out to the Pacific Rim. Letters to the students at the boarding house are due to the Church office by July 22. I have been asked, 'What do I say?' There is no right or wrong answer to this question. But let me offer a few suggestions.\nAvoid politics or anything related to religious controversy where Christians are in the minority.\nShare your favorite Bible verse and explain why it is so special to you.\nWrite out the words of your favorite hymn and explain what resonates the most with you.\nWrite encouraging words that will motivate the student to complete their studies and live for the glory of God.\nSend a picture of your family. Provide the names of each family member and ask them to pray as you pray for them.\nBut most of all, just tell them that someone in the kingdom loves them and is praying for them. That will be a huge encouragement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our mission is to aggressively and tenaciously defend our clients, even before the case reaches the courtroom. When the stakes are high and you're facing criminal charges, you need lawyers with experience, who are available to take your case immediately, and work for the best possible result in or out of court. We understand that being arrested or charged with a crime is not only a frightening experience, but all-encompassing for you and your family. We represent clients that are witnesses, subjects or targets of criminal investigations. We have represented clients charged with felonies and misdemeanors in both State Court and Federal Court throughout the United States.\nAttorney Robert Eckard has handeled thousands of criminal cases including over 40 trials and 1000 bench trials. He is a former prosecutor for State Attorney's Office, 6th Judicial Circuit, where he prosecuted both felony and misdemeanor criminal cases and knows first-hand the strategies and tactics used by the prosecution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is vital that employers, commissioners and workers have a clear understanding of the risks associated with practice. Vision Rehabilitation is focused on enabling people who are losing (or have lost) their sight to do things they used to do with sight and some of these are intrinsically risky. Poor practice, or practice undertaken by an unqualified worker (where they are untrained for a specific task) creates avoidable risk. As a requirement of any future application to join the list of approved professional registers endorsed by the Professional Standards Authority, RWPN has drafted an assessment of the risks associated with the tasks that Vision Rehabilitation Workers do. Assessment of Professional Risk The final page of the document provides a matrix for understanding the documented risks.\nIf you have any views or comments to make please contact RWPN by 31st March 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Howlin' Hobbit at the Pike Place Market Buskers Festival, 2010. All photos by Cathleen Shattuck (http:\/\/www.flickr.com\/photos\/canopic\/). Fez by Fez-O-Rama (http:\/\/fez-o-rama.com\/). Ukulele by Mainland Ukuleles (http:\/\/www.mainlandukuleles.com\/).\nI seem to be astounding the lady in red!\nbut just *what* is west of here?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaorxc b/data_all_eng_slimpj/shuffled/split2/finalzzzaorxc
new file mode 100644
index 0000000000000000000000000000000000000000..6d75bdc3b001dc9839649ead2fc01365c9e296ed
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaorxc
@@ -0,0 +1,5 @@
+{"text":"Datawiresport is a sports data provider with a strong focus on both proffesional and amateur sports, operating from the Netherlands. We work closely together to sports associations and government bodies maximizing value with sports data and specific digital products.\n2009 - present : Sports-Ads B.V.\nPhilosophy In it for the game: because we love this business.\nProduct Contribution to the awareness regarding management, protection and valuation of sports data.\nDistinction We are one of a kind.\nProduct Processing and publishing quality data via several digital products.\nDistinction Quality, Price and Support.\nValues Personal Contact, Customer Confidentiality, Quality vs Quantity.\nMilestones 2009\/01\/04: Sale of Datawiresport.\n2014\/01\/01: Free of any obligations to previous shareholders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LoveYourBrain invites you to celebrate National Brain Injury Awareness Month with us. In partnership with the Kevin Pearce Fund, we are all about connecting, educating and empowering people to promote a brain healthy lifestyle. LoveYourBrain is the message that embodies our positive approach to brain injury prevention and recovery.\nGet involved, Lend Your Brain: Join our community!\nLoveYourBrain envisions a community that has a greater understanding of the importance of the human brain, a safer playground, and ultimately, a smarter and healthier generation of active people.\nWe have a goal of raising$15,000 during the month of March to support the amazing work the Kevin Pearce Fund is doing and continue to raise awareness about brain injury. We are already halfway to our goal, but we need your help to reach it. What can you do?\nLater this month, Burton will be hosting a LoveYourBrain Day, where a percentage of all sales that day will be donated to the Kevin Pearce Fund. Stay tuned on social media to learn more about this event.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2. jul 2012 Where one of my objects was to determine the upper limit of intelligence reached by this and that class, order, or species of animals, I usually found that In order to do this the octopus had to climb up a vertical partition above the surface of the water and descend the other side.1 According to Schneider, Go to the productFind similar products \u00b7 sublime climbing brush boars hair no klatreutstyr klatrevegger friluft nettbutikk. KLATRESIDEN . climb on! brush bar 3 x0 1oz vertical playground. VPG. 99 kr. Click here to find similar products . uniq one coconut all in hair treatment. E\u2011BEAUTYSHOP. 109 kr. Click here to find similar jenter s\u00f8ker kj\u00e6reste tv 2017 Single side vertical climb Do not insert a conductor (like a metal chopstick) into one end of the power cable while the other end is connected . that secure the LG Logo Bracket to the bottom of the rear side of the monitor using a screwdriver. LG Logo Educating children about the dangers of climbing on furniture to reach the monitor or its controls. 22 Dec 2014 Switzerland is not a large country and I live within a few miles of it, but when I checked it was over the other side. .. I made sure my bottles were topped up with cold water and ate what I could before slowly jogging off to the start of the climb with one handful of cheese and the other full of sausage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A hallmark of Dan and Ashlyn Perry's community involvement is their sense of joyfulness, as they take unabashed delight in contributing to the betterment of the lives of their fellow New Mexicans.\nA hallmark of Dan and Ashlyn Perry's community involvement is their sense of joyfulness, as they take unabashed delight in contributing to the betterment of the lives of their fellow New Mexicans. The Perrys underscore their commitments with both time and money. Veronica Gonzales, Cabinet Secretary for the Department of Cultural Affairs, Charles MacKay, General Director of The Santa Fe Opera, and Jamie Clements, President\/CEO of the Museum of New Mexico Foundation, have all observed that the sheer pleasure of working with the Perrys is inspiring and that their enthusiasm inspires others to get involved as well. Dan Perry's insatiable love of opera and the couple's genuine concern and deep admiration for the performers led them to become cornerstones of The Santa Fe Opera. Perry has been a key member of the Opera board for several years, serving in various positions and working on whatever is needed most. And both Ashlyn and Dan Perry have been committed members of several Opera Gala Committees. As an arts commissioner for the City of Santa Fe, Ashlyn Perry has found a public service outlet for her love of art and design. She is a tireless supporter of Santa Fe's arts and cultural affairs, and her work on the commission includes recommending policies and programs that develop and promote artistic excellence in the community. Their big hearts and willingness to embrace a challenge led the Perrys to accept the position of co-chairs for the New Mexico Museum of Art Centennial Campaign, a $12.5 million endeavor. Their involvement in the campaign has been a major boost for the Museum of New Mexico Foundation, on which Dan Perry currently serves as Vice President of the Board of Trustees. The Perrys were instrumental in securing the major gift for the new Vladem Contemporary. \"Without the Perrys, the new contemporary museum would not be possible,\" said Mary J. Kershaw, director of the New Mexico Museum of Art. 'Not only did they accept the challenge, they have risen to the challenge, and they have been personally active in making sure the campaign moves ahead and succeeds. Sometimes these things are more honorary positions, but Dan and Ashlyn have been truly active, and without their involvement we would not be where we are in the campaign.\" Kershaw also lauded Ashlyn Perry's active commitment to the Centennial Campaign's gala fundraiser in early May. \"Ashlyn was instrumental in the success of the evening, and her talent for design was evident to all who attended,\" Kershaw said. The Perrys have been extraordinarily generous to all four of the state museums in Santa Fe: the Museum of Art, the Museum of Indian Arts and Culture, the History Museum\/Palace of the Governors, and the Museum of International Folk Art. For example, Ashlyn and Dan Perry were responsible for the purchase of the sculpture Valentino Tzigiwhaeno Rivera (2008-2016) for the Museum of Indian Arts and Culture. They are also a force behind the Museum of New Mexico Foundation's development initiatives in Dallas, San Antonio, and Austin, Texas. The Dan and Ashlyn Perry Charitable Foundation supports children's charities, the arts, and conservation efforts. Their financial commitment to the economic development of the Rio Chama Valley is substantial, and through their Trout Stalker Ranch in Chama, the Perrys have aided the New Mexico Land Conservancy with hundreds of acres of conservation easement. Dan and Ashlyn Perry have fortified many educational, artistic, and child welfare organizations in New Mexico, including Big Brothers Big Sisters of Northern New Mexico, the Institute of American Indian Arts, SITE Santa Fe, and Silver Bullet Productions. The philanthropic and artistic landscape of New Mexico changed for the better with the arrival of Dan and Ashlyn Perry, who moved here permanently in 2011 after being longtime visitors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I finished up the project for Shooters and turned it in. I believe they were happy with the results.\nPhil Nolan has been an artist of one sort or another all of his life, after discovering comic book art and animation in high school he decided to continue in college towards becoming an animator. After achieving this goal he now freelances either from home or local studios, always with plans for ever bigger and better projects.\nNew tutorial for Adobe Fuse available!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaowcq b/data_all_eng_slimpj/shuffled/split2/finalzzzaowcq
new file mode 100644
index 0000000000000000000000000000000000000000..c5da82c10828208cedb3c33a9d1a9843d2cca916
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaowcq
@@ -0,0 +1,5 @@
+{"text":"Cities ranging from Boston to New York and Washington are set to swelter through Thursday as high heat and humidity combine to make 95-degree temperatures feel as if they're 100 degrees Fahrenheit or more, boosting energy demand to cool homes and businesses.\nA large area of high pressure has parked across the eastern U.S., bringing in a typical summer pattern that has spread heat advisories and excessive heat warnings from Virginia to Maine, the National Weather Service said. New York, Washington and Philadelphia reached 95 Tuesday and is forecast to stay as hot for Wednesday. Boston, meanwhile, is set to hit 98, one degree shy of a 1948 record.\nPower prices in New York jumped, climbing as high as $175 a megawatt-hour on Long Island, the highest since January, according to Genscape. Demand in the state was forecast to hit about 30,900 megawatts at 4 p.m. Tuesday, nearing the summer record of almost 34,000 megawatts set in 2013.\nOvernight temperatures will stay in the mid-70s, keeping air conditioners running. While temperatures will ease off a little on Thursday, Oravec said the mercury is set to climb again the following week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In order to compete in a global economy we need to make sure our children receive the best education possible. That means starting early. Gary believes we must invest in early education, and he supports efforts to ensure all Virginia Beach children have access to full-day kindergarten. Gary is prepared to be a strong advocate for Hampton Roads' public schools and families. He'll also work to cap tuition and fees at Virginia's public universities and colleges, making higher education more accessible to Virginia students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are happy, at Kingwood Garage Door, to bring every customer 24\/7 availability for service so that you can be sure to have a garage door that is well taken care of and the protective qualities it has by being installed. And we want our customers to know that help is there when it's needed.\nWe offer same day services, at any time, for each and every garage door, and all make and model of them. We are skilled when it comes to remedying the issues you're having and fixing them fast and professionally and we provide our Customer Satisfaction job we will do for you.\nWe are available 24\/7 for Kingwood garage door repairs. If your door is damaged or you garage door opener has stopped working given us a call for same day service. We are also available for evening and weekend appointments for Kingwood garage door repair.\nView Larger Map Our technician will arrive with a fully stocked truck so your garage door will be repaired on our first visit. At our company, our skilled technicians have the tools, knowledge and experience to fix all makes and models of garage door openers. For quality garage door services from a trusted garage door company, call us today. No matter the time, day or night our 24 hour a day emergency service will be available to assist you in all of Kingwood, TX.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kayaking by moonlight is a spectacular way to experience Melbourne from a unique water-level perspective. Melbourne's Yarra River is an excellent location for new paddlers to experience kayaking for the first time, or for experienced kayakers to enjoy a stunning paddle through Melbourne's scenic waterways. No kayaking experience is needed to enjoy the tour, our patient guides will be with you every step of the way. We use a fleet of fibreglass Sea Bear sea kayaks, which are extremely stable and easy to operate.\nThe tour meets just before sunset in Victoria Harbour, part of the modern Docklands precinct in the west end of the city. After a short safety briefing and paddling technique session on land we set off for an exploration of the sights in Victoria Harbour before tying up to a dock in a beautiful marina. The guests stay seated in the boats and are served fish n chips for a very special on-the-water dining experience.\nAfter our meal, we continue our tour upstream, under the Bolte Bridge and through to Melbourne's lively downtown. Kayaking past the Crown Casino and alongside stylish South bank. We continue past Flinders Street Railway Station and beneath historic Princes Bridge before arriving at the approach to Melbourne's sporting precinct. Enjoy the sights and sounds from your unique vantage point as the sun goes down and the city comes to life.\nOur tour ends at the boat landing area at the Rowing Sheds, right in the heart of Melbourne, and directly across the river from Federation Square. What an incredible way to experience Melbourne!\nMeet on the wharf, at Shed 2 Victoria Harbour Promenade, Docklands. The tour ends at the Parks Victoria landing at the rowing sheds directly across the Yarra from Federation Square.\nBookings are available all year round. We need a minimum of 4 people for the tour to run.\nA sun hat, sunscreen, sunglasses and a camera. You can wear your normal street clothes but you may want to bring a change of clothes if you are going out afterwards.\nI have never been kayaking before. Is that OK?\nAbsolutely. Your guide will give you lots of help and the tour begins with a training component.\n10 yrs and older is recommended for all the tours.\nI don't know if I am fit enough?\nIf you occasionally play sport, walk, work out or are somewhat active you will be fine. The paddling is not too strenuous.\nHow many people on each tour?\nRegular tours have a maximum of 12 but larger groups can be accommodated.\nHow much storage area is there in each boat?\nLOTS! We use big stable sea kayaks which are designed to take big loads. The boats are big enough to take a medium size pack if need be. Day packs, hand bags etc are not a problem.\nYou may wish to bring along extra water, hat, snacks and a camera. No special clothing is required - just dress appropriately for the weather. A change of clothes is a good idea if you are going out after the tour.\nAbsolutely. There will be plenty of time for snaps along the way. Disposable waterproof cameras are ideal but we have had very few problems with digital cameras- so far! You can always ask your guide to hang on to your camera until you need it.\nPaddling in the rain is more fun than it sounds! The tours run rain, hail or shine; however, we reserve the right to cancel a tour if it is unsafe. If you are worried about the weather you can check the 4 day forecast for Melbourne before you book or contact us for more information.\nMeet on the wharf at the Community Hub at The Dock, 912 Collins St, Docklands. The tour ends at the Parks Victoria landing at the rowing sheds directly across the Yarra River from Federation Square.\nChildren must be 6 years of age to 11 years inclusively. Children below the age of 6 are not permitted on this kayaking tour in Melbourne.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you believe that your privacy has been breached, please contact our Privacy Officer using the contact link on our website and provide details of the incident so that we can investigate it. We will respond to an inquiry or complaint promptly (usually within 14 to 30 days).\nIf you contact our Privacy Officer but you are not satisfied with the response that you receive you can contact the Commonwealth Privacy Commissioner's hotline on 1300 363 992 or enquires@oaic.gov.au if you are not satisfied with our response.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapgff b/data_all_eng_slimpj/shuffled/split2/finalzzzapgff
new file mode 100644
index 0000000000000000000000000000000000000000..8b7a6bbf5f01892047c06b1db1137ee4d3851222
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapgff
@@ -0,0 +1,5 @@
+{"text":"And so it is that almost exactly 12 months ago I gave up drinking alcohol. People always want to know why. I can cite 17 reasons, according to my mood. I had a premonition my job was going to escalate (it did) and we were going to move house and renovate a wreck (we didn't). But on a more metaphysical level, I was preternaturally tired, exhausted from exhaustive partying that had gone on for at least two decades, and my children were becoming teenagers; they saw more, they knew more, and I wanted them to like what they saw and knew of me. I felt as if I'd fused \u2013 like I'd blown my entire brain and body electrics. I was in continual fight-or-flight mode, my anxiety ran like a fever. This time last year, on my birthday, I had this very tangible flash of who I was as a teenager. I was not the person that teenager thought they would be.\nEnergy. Clarity. This was what mattered.\nIt was a simple irony that continues to amaze me now \u2013 I thought I drank alcohol to quell the anxiety. But it turned out it was the alcohol that gave me the anxiety in the first place. This should have been obvious. It wasn't. Also, when alcohol was out of the picture, it took with it the largest slice of self-loathing, so the pie-chart of my mind turned out to be thrillingly roomy.\nEnergy. Clarity. This was what mattered. For a month I was ecstatic, as if I'd emerged from a rough chrysalis and was fresh to the touch. The month after that was a disaster. My husband had also cut back on his drinking, so the two of us were suddenly nodding off on the sofa at 9pm while watching Inspector Morse. It was too much. I felt like I'd fast-tracked to old age, like I'd been cheated by the system, like I'd made this big exciting decision but I'd also somehow simultaneously walked into the wrong room of my life. Someone at this point said to me: 'The party's over for you. That's what's happened. You've just got to accept it!' I felt as if the floor of the lift I was in had been removed and my entire centre was plummeting down the empty shaft.\nWe partied so hard \u2013 the energy was so extraordinary, the dynamic so intense \u2013 that it completely ripped me open.\nAnd yet, a year on, they weren't exactly right. It took a holiday for me to see this. In fact, it was a tale of two holidays; both took place, crucially, on the same island. The first time was a couple of years ago; we were a big group, united in a desire to have the best time. We partied so hard \u2013 the energy was so extraordinary, the dynamic so intense \u2013 that it completely ripped me open. For a week afterwards I couldn't sleep, and the sleeplessness brought on the panic attacks. I could feel that my breathing was as loose as a severed fan belt. I found it hard to look anyone in the eye. I dreaded night, but also day, with all its corners and confrontations; words, my touchstone, my balm, my connection, were escaping me.\nBut it is a price I will pay again and again.\nThe second holiday, same island. A big group united in a desire to have the best time. The energy was so extraordinary, the dynamic so intense, that it completely filled up my heart. One night I actually laughed so hard and for so long that my cheeks ached. I had actual cheek-ache. One of these holidays was with alcohol, one without. I'm not saying that sometimes I don't feel like I'm on the periphery. I do. And that I don't get paranoid. I do. It turns out conversation is crazy-hard for me. Who knew? Quick chit-chat, amazing. Long, deep and meaningful, amazing. In between those two things, what the hell? Who can do this?\nBut it is a price I will pay again and again. For now, on those mornings when I used to gaze into the void only to have the void stare back at me, I look at the sky. I literally look at the sky. And the sky literally looks back at me. And that is more than enough.\nThis is the new issue of Cond\u00e9 Nast Traveller. For those who know the party takes on a million different shapes and forms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 1982 she was launched at Nikolayev, Black Sea.\nIn December 1987 she was commissioned to the Northern Fleet.\nIn July-December 1988 she operated in the Mediterranean Sea then sailed to Severomorsk, Barents Sea.\nIn 1989-1990 she actively served with Northern Fleet.\nIn October 1990 she was renamed \"Admiral Gorshkov\".\nIn 1990-1991 she made training tours and took part in the test trials of VTOL fighter Yak-41\/141.\nIn February 1992 she anchored for repairing at Rosta, Murmansk and never operated again (except short participation in Navy parade in June 1995).\nIn February 1994 huge conflagration happened aboard.\nIn 1994 selling negotiations with Indian Navy began.\nYakovlev Yak-41\/Yak-141 Freestyle [ 1 ].\nIn 1999 she was towed to Severodvinks for modernization for Indian Navy.\nIn 2000 after years of negotiations, Russia and India signed a deal for the sale.\nIn December 2008 after complete modernization and refurbishing she was officially commissioned to the Indian Navy, however she still under construction yet. Permanent problems with treaty obligations are impeding process.\nIt is planing she will began active duties in 2011-2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ravello has been around for a few years. in its initial guise, it was all about helping organizations \"cloudburst\" \u2013 that is to move peak demand application load from on-premises in infrastructure into the cloud. This was a space where many companies were trying to play, CliQr, AppZero, CloudSwitch, and CloudVelocity where all in the general space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adjustable double head tilts and rotates infinitely. Integral louver lens holders can hold a single glass lens (sold separately) or an eggcrate louver (two included). Two low-voltage, MR16 lamps of up to 35 watts each (not included).Available Finishes: Antique Bronze, Chrome, Satin Nickel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OMNeT++ 3.2 will implement reference counting of encapsulated messages, which has the potential of bringing dramatic performance improvement especially for wireless simulations. Reference counting means the following: when you duplicate a message that contains an encapsulated message, then instead of duplicating the encapsulated message as well, the two messages will hold a pointer to the same encapsulated message. Similarly, when you create 100 copies of the original message, then all of them will hold a pointer to the same encapsulated message instance -- saving the creation and copying of 100 encapsulated messages (or 200, if the encapsulated message also contains another encapsulated message).\nDecapsulation will however cause the message to be copied (because decapsulation will allow the code to can change data in the message object, delete it etc, so it can no longer be shared). That is, if all 100 messages get decapsulated, then those 100 messages we spared at duplication will eventually be created during decapsulation -- so we haven't gained anything. The potential performance gain occurs when most of those messages get deleted without ever being decapsulated. That's exactly the case with wireless simulations -- you'll see that there, reference counting may reduce the total number of messages created\/deleted by 50% or even much more.\nLet us consider wireless simulations in the INET Framework, say, hosts exhanging UDP packets over 802.11 (ad-hoc mode). The INET Framework uses the lower layers of the Mobility Framework, so you'll see hosts sending AirFrames when you run the simulation. AirFrames represent the physical characteristics of the frame (frequency, power, etc). AirFrames encapsulate 802.11 frames (Mac80211Pkt), 802.11 frames encapsulate IPDatagrams, IPDatagrams encapsulate UDPPackets, and finally UDPPackets encapsulate the payload which might be a cMessage. This is altogether 5 levels of encapsulation.\nWhen a host transmits an AirFrame, it will send a copy to every other host within interference range. Each host individually evaluates the physical layer model, and decides (based on the signal-noise ratio) whether the frame is receivable. If not, the AirFrame is discarded. Then the 802.11 frame gets decapsulated from the AirFrame, and the 6-byte MAC address is checked. If the MAC address doesn't match (and it's not broadcast etc), the frame is discarded. Otherwise the IPDatagram gets decapsulated from it and sent up to the IP layer.\nWhat does this mean in practice? Without reference counting, all copies of AirFrames have their own copies of all the higher-layer protocol headers as well. That is, if there are 100 hosts in range, altogether 5x100=500 message objects are created, most of which gets deleted without ever being looked at, because the corresponding AirFrame gets discarded in the physical layer (bit errors or collision) or in the MAC layer (wrong MAC address). If we use reference counting, then all 100 AirFrames will refer to the same 802.11 frame object as encapsulated message, that is, we'll have 104 objects altogether (100 AirFrames, plus 1 of 802.11, IPDatagram, UDPPacket, payload message). Let us assume that out of the 100 recipients, only 83 are able to receive the frame correctly, the rest discards it due to physical layer errors. Those 83 recipients will decapsulate the 802.11 frame, resulting in 82 additional copies of the 802.11 frame message object being created. However, only one host will have a matching MAC address, the rest discard the frame -- so no extra copies of IPDatagram and higher layer messages are created. Thus, we end up with 104+82 = 186 messages used for the transmission, instead of the original 500 -- which is a 62% gain!\nTo get the full picture, we have to take into account the ACKs as well (where isn't that much saving because, being just an AirFrame plus a 802.11 ACK frame it doesn't have several levels of encapsulation), but the overall number of messages is still about half the original. A real simulation experiment (with INET's MobileAdhoc\/80211 example network and 100 hosts) showed 718,275 messages being created, as opposed to 1,393,125 without reference counting -- a 48.44% saving. Profiling of simulations (e.g. using the graphical KCachegrind) show that message creation is responsible for a massive part of the runtime overhead, so reference counting can be a very significant performance improvement factor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaprwp b/data_all_eng_slimpj/shuffled/split2/finalzzzaprwp
new file mode 100644
index 0000000000000000000000000000000000000000..bb4fa25cf4d096933e2224b1c0eb1a8aeafa4712
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaprwp
@@ -0,0 +1,5 @@
+{"text":"Helps level floors prior to the installation of ceramic tile, natural stone, resilient flooring, carpet, wood and other floor coverings. This underlayment can be applied to 1.5\" (3.8 cm) thick in one pour and seeks its own level in minutes. With proper installation, the use of LevelQuik\u00ae ES can achieve an extra heavy rating for high impact use in food plants, dairies, breweries and kitchens. LevelQuik ES may be applied in residential structures with floor joists up to 24\" o.c. Formulated using Controlled Cure\u00ae Technology, Level Quik ES helps eliminate installation problems such as bond failure, crumbling and staining of resilient flooring caused by the free moisture found in traditional underlayments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Competency is one of my personal values. Knowing this, understanding how it's a part of me, and believing that I must make choices that go with this value helps make me a better working mom. Let me tell you why.\nThis means you need to take time to think about what is important to you. Write down your thoughts and then shorten them down to one to two word descriptions. Make a clean list of these values and memorize them. Print them out and put them up around your home. Make them the background of your phone so you'll see them constantly.\nHow did they come to be? What's the story behind them? For example, one of my values is feeling competent. Meaning if someone or something makes me feel like I don't know what I'm doing I get very angry.\nThe earliest story I can remember is working as a cashier. Everyone that I worked with had a way of making me feel incompetent. Gosh, I detested that job. Since my teenage years, I've learned ways to support my value of competency like having self-confidence and understanding I can choose to not give people control over my choices.\nWhen I became a working mom life became extremely frustrating. This is when I began my work on knowing my VP's (values and priorities). It became clear that my competency was being questioned on many fronts. I struggled with making the right choices concerning motherhood and career while being sleep deprived.\nWhen you believe in your values and priorities whole-heartedly your life becomes less confusing and challenging. When you believe things start to click because you have a good picture of who you are. Self-confidence grows when you learn believing in your values helps you make good life choices.\nMy working mom struggles lessened more once I got past this step. I started to look for projects that boosted my competency. If I felt incompetent about something in my personal life I made it a priority to become competent. I gave up on doing it all by myself because that just wasn't working out for me. There was too much incompetence.\nMy coach helped me integrate my work life better with my personal life. I read books, found blogs, signed up for newsletters, watched YouTube videos and discovered content-rich podcasts. I didn't realize the plethora of information and stories available for me to learn from until I made competency a priority. I began to live according to my values. I reached out for help. I found the courage to be vulnerable because I learned that admitting I didn't know was the right path to competency.\nOnce you believe in your values you begin living according to them. You find courage to make audacious choices based on your values and priorities. When you have to make tough decision you can use your values and priorities for strength. When you know, understand, and believe in your values it's easier to confidently say no or be hesitant before committing to something that doesn't feel right. When you know, understand your values and priorities you spend less time trying to figure things out.\nThis process is exactly what working moms need in a lifestyle that is constantly in flux. When you are asked to make many decisions in the snap of a finger, you are prepared and confident.\nHaving your values and priorities sorted out and part of your rock solid personal foundation gives you more time in your day for things that matter to you. You spend less time living in chaos and more time enjoying doing things the way you want to do them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you have finished setting up your on-line diary and services take a few minutes to look over your set-up, if you have entered something incorrectly it is best you spot it now rather than a customer when you are 'live'.\nThe most common problem we get in reported to customer service is that a customer cannot book a service with a certain therapist, the solution is simple - that therapist was not linked to that service in the diary set-up.\nEvery time you enter a new service you must assign staff to it, this is done under the diary tab in 'My Account'.\nThis feature is important as not all staff can necessarily provide all services.\nIn the above example Ann Marie is not trained in Reflexology or Aromatherapy so those services are not ticked against her name. Click on the above screen shot to make it larger.\nYouBookIn is very easy to use and set-up, but the best system in the world -Like Carlsburg we like to think it is probably us - is only as good as the data you input into it.\nSo in summary - a little care and attention to detail at the set-up stage will pay dividends later. Having said that the majority of people set-up YouBookIn without any problems.\nWhat dissapoints me about your system - is it doesn't have a week to view - so I cannot tell how busy my week is going to be. Will you be providing a solution to this issue?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What key concepts and interpretations from the literatures you will use to analyse your data. A deductive analysis \u2013 working from a general theory to the particular \u2013 and an abductive analysis \u2013 putting the general and the specific together to form an explanation \u2013 require extensive use of literatures. The concepts and\/or approaches that are used in analysis must appear first in the literatures review, may then be taken up in the methods chapter, when the analytic approach is outlined, and are referred to again in the chapters which report the analysis. Inductive analysis \u2013 working from the data \u2013 also uses literatures, and these also appear first in the literatures chapter and then again during the analysis report and discussion. Even if you read \"stuff\" after you finished your preliminary analysis, you generally still introduce them earlier. The exception to this rule is if you are writing a chronological thesis, as you might if you are doing action research.\nWhat general approach to the topic that you have taken and where your work sits within the field. Maybe your work sits neatly within an established area. You just need to explain this. You might want to situate yourself in opposition to work which addresses the same topic, but takes a different approach. Both of these options \u2013 work that is like yours, and work that is not \u2013 situate your study in the field and show where your contribution will be made. Your research will add something to work that is like yours, and say something to the work that is not. But some research brings multiple strands of research together, so you need to explain what strands you are using and how they fit together. And then you need to locate that in your field alongside the research that yours seeks to inform.\nSo when you locate, you must move through the review reporting as you go how your work connects with the literatures \u2013 or summarise at the end of each chunk how the literatures inform your research.\n(4) taking an approach \u2013 using the same approach as another scholar (or group) as a stepping stone to somewhere new.\nWhen you take your own perspective on the literatures, and use it to make your case, you can be said to have transformed knowledge. In sum then. Literatures work is a process of locating your work \u2013 situating it in the field, and showing what you build on and talk with \u2013 and this means that you have been, and are transforming knowledge to position and explain your own study.\nBut wait there's even more to locational work than this.\nLocating your work also requires you to imagine yourself and your work on an equal footing with that of others. You have to have the chutzpah to use the work of others, who are more experienced and expert, to make your own. You can't be in awe of the literatures. Equally, you don't need to cut other people's work into ribbons. You simply have to use it critically and appreciatively. After all, that's why the literatures exist \u2013 to inform and to support further knowledge building.\nAnd thats sometimes easier said than done. There is scholarly-identity work going on when you locate your own work in relation to that of other scholars. Locational literatures work require you to adopt the rhetorical position of a fully-fledged scholar able to hold their own in a scholarly conversation. In transformational and locational literatures work, you are asserting and explaining your selection and interpretation of other people's work, and laying claim to using their ideas in particular ways for your own ends. It's all about you, not them. Barbara and I call this text work\/identity work.\nIt is this complex combination of sophisticated argument and authoritative writer that is so hard to do. You are simultaneously transforming knowledge \u2013 using the literatures as a resource from which you make the argument for your study \u2013 while also establishing yourself as a bona fide researcher able to make a worthwhile and worthy contribution to the literatures.\nThis entry was posted in academic writing, Joseph Harris, literature review, text work\/identity work, transforming knowledge and tagged Barbara Kamler, Joseph Harris, literature review, literatures work, Pat Thomson, text work identity work, transforming knowledge. Bookmark the permalink.\nbobbydazzler of decoding that dense talk\u2026.\nThank you for explaining this so clearly \u2013 I've been wrestling with how to explain the \"more\" that a dissertation or thesis needs, and your post covers it perfectly!\nThis explains very good indeed. There is always a continuous struggling process to present the literature rewiew between a chronological order and the purpose of the research study design within a logical yet original manner. This becomes painstakingly engaging thoughtfull process specifically for social sciences where different disciplines earlier works are required to intermingle to use for new purposes.\nDear Pat, thank you so much for this website. It truly is a goldmine of information. So many times when I have struggled with my PhD, I have found helpful hints and encouragement here which have helped me on my way. Your blog is like my second supervisor!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marine Corps - USMC Community > Marine Corps Forum > Open Squad Bay > Too old to be an officer?\nView Full Version : Too old to be an officer?\nI am 27 now, and in 3 years, when I leave the Army National Guard, I will be back in the Marine Reserves, where I never should have left.\nMy question is: how does a commission work? I'd like to be an officer, but not sure how it works. Can anyone give me some info?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqizw b/data_all_eng_slimpj/shuffled/split2/finalzzzaqizw
new file mode 100644
index 0000000000000000000000000000000000000000..ed3f993d1f86240ef9d826a8921c524c67fb9130
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@@ -0,0 +1,5 @@
+{"text":" Newsletters from Seiretto Ltd.\nWeb bots - how to reduce spam and welcome the search engine creepy crawlies.\nTracking your customers through your web site. Which pages did that new customer view prior to buying?\nMeta tags revisited - optimise your pages for search engine submissions.\nThere are vast numbers of companies offering to optimise your web site to get you to the top of google but what works and what doesn't?\nAccepting online payments via credit card. How safe are you as the owner of the online shop?\nOptimisation of osCommerce for Google and other search engines. We show you how to help ensure your dynamic web pages get indexed by Google and other search engines, and we also show you how to add meta tag keywords.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lime Wood Hotel, Hampshire, is a wonderful boutique country house hotel in the heart of the New Forest National Park, close the to the town of Lyndhurst and just a twenty minute drive from the coast.\n\"We like to think of ourselves as a luxury country house hotel with a difference. We've used Lime Wood's classical structure as the basis of our design but we've lovingly renovated it, keeping the spirit of its past, taking inspiration from its extraordinarily beautiful surroundings and adding the odd contemporary twist to bring it to life.\nIf you are considering getting married at Lime Wood Hotel, Hampshire, and would like to discuss your wedding photography requirements then please don't hesitate to get in touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our first and only child was born at ten minutes to midnight on December 23rd.\nI have always loved the season of Advent; that year, when I was expecting a Christmas baby, I found the Advent liturgies almost unbearably moving. In my yearning expectancy, the great 'O' antiphons, with their longing and mounting impatience, resonated with me as never before.\nIt seemed to me that all the mystery of creation was contained within me. I waited for the birth with heart stopping excitement. And, through those final days of pregnancy, the beautiful, sombre rhythms of the O Antiphons seemed almost the rhythms of life itself. They are sung as though with breath held, suffused with reverence and wonder.\n\"O come Lord Jesus\" \u2013 the fervent request is as old as St Paul  and forms the penultimate verse of the book of Revelation and thus of the New Testament.\nAll of these associations were with me in those final days. I felt as old as Eve, participating in the act of Creation.\nI woke before dawn on the eve of Christmas Eve, to the certain knowledge that our baby was on the way. Eighteen hours later, it seemed as though this baby would never be born. \"Labour\" is accurately named. For the previous many hours contractions had been sweeping in like tidal waves, each one picking me up like a tiny piece of flotsam and bearing me down into a new tunnel of pain and exhaustion. I no longer believed that this cruel effort was going to produce anything at all. In the brief respites between contractions I withdrew from time and the world. Inexorably drawn back by the next wave, I was surprised to find myself still there.\nAnd so it went on, hour after hour \u2013 wholly relentless, unavoidable. And then, stunning pain, pain to take the breath away, and I began to know fear. The urge to deliver the baby was inexorable \u2013 but it seemed like an urge to self-destruct. I could no more avoid it than I will be able to avoid death.\nThen the moment came when, at last, with a final crescendo of effort, our baby was there. I held her warm living body against me, looked into her astonishingly calm eyes and instantly, irrevocably, fell in love. She glided around my heartstrings like running water, to crystallise there - so that my heart was no longer mine but was from then on a part of her.\nThis, then, or something like it, Mary went through. In blood and sweat and pain she went through this most elemental of experiences, lying on the mud floor of an outhouse. In the words of the opening prayer of the Dawn Mass on Christmas Day, \"Almighty God\u2026 your Eternal Word leaped down from heaven in the silent watches of the night, and now your Church is filled with wonder at the nearness of her God.\" The maker of the universe passed through a female womb and lay helpless and beloved on his mother's breast as my baby did on mine.\nOn that night in Bethlehem, God became our travelling companion on life's journey. He took on our human condition, and became subject to illness, to aging and to death.\nAs the weeks passed, and as I held our infant daughter in my arms, feeling the warmth of her downy little head and her tiny fingers with the nails I didn't dare to cut moving like small, lovely spiders over my skin, I thought the universe itself could not contain my wondering love. And when she smiled for the first time, the radiance lit my world from end to end. For the first time, I began to have some infinitesimal sense of God's love for his creation.\n Allen Cabaniss, \"A Jewish Provenience of the Advent Antiphons?\" The Jewish Quarterly Review, New Ser., Vol. 66, No. 1. (Jul., 1975), pp. 39-56.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Candidates must have first applied online at grady-ems.org\/academy. Then candidates must send their preferred date\/time for exam to registrar@grady-ems.org and include testing date , name, and phone number to see if their desired date is available as only 10 people per day are allowed to test.\nCandidates must create an online account at www.atitesting.com prior to their exam date. Candidates can find more information and practice exams online at the ATI website \u2013 the fee is $58 and must be paid on the day of the exam with a credit or debit card \u2013 cash and checks are NOT accepted.\nCandidates must arrive at testing location (2284 Marietta Blvd NW Atlanta, GA 30318) at least 15 minutes ahead of their testing time.The exam lasts approximately 4 hours. Latecomers may be allowed to retest on another day depending on availability. Applicants must receive a cumulative score of 60% or higher to be eligible for enrollment in our program. Prior college experience, completed degrees, or prior certification does not allow you to exempt the entrance exam, all new students must take it.\nEmail registrar@grady-ems.org for additional exam dates.\nNote: Scores will remain valid for students who successfully passed the HOBET exam and will not be required to sit for the ATI TEAS for Allied Health. HOBET scores will be valid for 5 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Northern State admissions is somewhat selective with an acceptance rate of 88%. Students that get into Northern State have an average SAT score between 950-1140 or an average ACT score of 19-24. The regular admissions application deadline for Northern State is rolling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqris b/data_all_eng_slimpj/shuffled/split2/finalzzzaqris
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index 0000000000000000000000000000000000000000..a8e040ca0dafb1812002c1a741d4c4d435bec4b9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":" Cuba Web Directory - Solways Cuba, Viajes a Cuba.\nName: Solways Cuba, Viajes a Cuba.\nDescription: Tour operator expert in Cuba Trips with information, rates and online reservations facility for more than 100 Cuba hotels, Rent a cars, Flights, Vacations Packages, Tours and Events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can place calls using the Intermedia Unite desktop application directly.\nYou can locate the contact you would like to place a call to in either the Contacts or the Dial tab. Press the Call button to start the call.\nYou can use the dial pad in the Dial tab to enter the preferred number and start the call directly.\nIn the Call Controller mode, a new call would rind your current device first, and once you pick up the call on your device, the call will be sent to the recipient.\nWhile in the Call controller mode, you can pick up calls on the phone device only. The desktop application allows you to either reject the call or to send it to the voicemail.\nIn the Softphone mode, you can place or receive calls directly from your desktop, using the audio devices of your computer.\nYou can read the Intermedia Unite Desktop App: Call Controller and Softphone modes article for more information on differences between these modes and available options.\nYou can have multiple calls running at the same time. You will see active calls in different tabs.\nYou can switch between these tabs by simply clicking on them.\nNote: 911 emergency calls are not allowed and will be blocked by the application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever since receiving her first box of beads and findings from her grandpa in her younger years, Kimberley Selwood has turned her craft into her career!\nWinning a variety of awards and successfully running her own jewellery brand, TJC are excited to have Kimberley on board to bring you a new and exclusive designer collection filled with simple, elegant and timeless pieces!\nSo, what's new and exclusive?\nWild at heart collection: A collection inspired by wild flowers and growing love incorporating soft pastel shades making the range feminine and pretty, with the extra symbolism of love, romance, tenderness and friendship.\nCrimson Spice collection: Kimberley has drawn inspiration from the Pink City \u2013 Jaipur, its royal history, deep-rooted art and craft vibrancy. Jaipur is perhaps one of the most decroative cities in India and this collection reflets that.\nDistant Shores Collection: Dreaming of being swept away in the moment, these pieces are inspired by the ripples nad waves of the sea and how the sunlight dances on water. One key quote reflecting theese specific pieces are \u2013 'dream higher than the sky and deeper than the ocean'.\nWhat pieces should you have in your collection?\nMulti Butterfly Necklace: This is a beautiful piece, it's a smaller version of my handmade necklace, but it combines a flock of butterflies, known as a kaleidoscope, which then catches the light at different angles. A dramatic piece but due to the delicate butterflies it isn't difficult to wear and sits beautifully around the neck.\nButterfly Stud Earrings: A beautiful simple pair of earrings that shows the light delicateness of the butterfly yet adds extra sparkle and a little luxury to any outfit.\nSimple Bracelet: An easy everyday piece with extra glam, ideal for any age, yet it is still sophisticated.\nBangle: Beautiful hinged bangle that has a flash of zircon to bring extra luxury, also the cross over gives a very elegant look. Easy to wear and great with the other items in the range.\nLattice ring: The open shape of this ring is inspired by the patterns and arch ways in Jaipur, all the entrances and doorways have beautiful shapes which I have tried to emulate in this collection. The lattice middle is seen in many window frame, a lovely charming piece, which is Indian inspired but unique in style.\nOpen Ring: Inspired by the shapes in Jaipur, but this is open to allow the shape to really stand out, a big ring, but wearable and light.\nShop the new Kimberley Selwood range now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sennheiser is launching the MKH 8000 Series of microphones, which combine the company's exclusive RF condenser technology with a compact, modular form factor and a range of application-specific accessories for broadcast, recording and performance use. The MKH 8000 Series is available in three models for speech, vocal and instrumental reproduction with an extremely wide frequency response.\nThe omni-directional MKH 8020, cardioid MKH 8040 and super-cardioid MKH 8050 each comprise two modules: the microphone head and a separate XLR module. For applications such as television, the XLR module may be removed and the mic head directly attached to one of Sennheiser's special remote capsule accessories to create a compact microphone assembly. The black Nextel coating of the MKH 8000 Series promises to eliminate troublesome reflections and further optimize the mics for TV use.\nThe frequency response of the new MKH 8000 microphones ranges from 10 to 60k Hz (MKH 8020) and 30 to 50k Hz (MKH 8040 and MKH 8050). The pickup patterns have been designed for accuracy across the entire frequency response. The maximum SPL is 138 dB for the MKH 8020 and 142 dB for the MKH 8040 and MKH 8050.\nThe MKH 8000 Series is available in three sets ($1,299 each), and each set ships with microphone, wind screen, clip and aluminum transport case.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"loose tea strainer slow brew infuser cute sloth hanging leaf leaves mug steeper set.\nloose tea strainer leaf target.\nloose tea strainer premium infuser stainless steel filter perfect for leaf amazon.\nloose tea strainer novelty infuser swan herb spice filter bk leaf target.\nloose tea strainer long handle stainless steel infuser basket for steeping leaf spoon.\nloose tea strainer use this charming robot stainless steel infuser to prepare your favorite his arms and hands move create a custom fit best leaf strain.\nloose tea strainer apace leaf infuser set of 2 with scoop and drip tray ultra fine stainless steel steeper for a superior brewing experience gift.\nloose tea strainer stainless steel infuser leaf filter herbal spice net ball.\nloose tea strainer deep diver infuser sainsburys.\nloose tea strainer new type stainless steel portable mesh infuser filter metal teapot spice strain leaf without.\nloose tea strainer best infuser for leaf pack combo kits including double handles large steeper single long handle filter no.\nloose tea strainer china infuser stainless steel with handle for leaf amazon.\nloose tea strainer leaf infuser basket stainless steel without.\nloose tea strainer infuser spoons a for gift set.\nloose tea strainer large infuser ball cm set.\nloose tea strainer stainless steel mesh infuser ball reusable leaf herbal drinking without.\nloose tea strainer diver infuser leaf target.\nloose tea strainer loop black ts teavana leaf.\nloose tea strainer infuser set by pack combo kit of 2 leaf.\nloose tea strainer infuser ball 5 cm gift set.\nloose tea strainer cute stainless steel infuser leaf filter herbal spice shell spoon.\nloose tea strainer stainless steel mesh infuser reusable leaf spice filter for strain without.\nloose tea strainer stainless steel cm teavana leaf.\nloose tea strainer stainless steel leaf infuser with lid no.\nloose tea strainer loop white ts leaf.\nloose tea strainer silicone infuser for leaf unicorn herbal filter drinkers ornament amazon.\nloose tea strainer stainless steel locking spice mesh infuser ball herbal filter reusable no.\nloose tea strainer leaf infuser in stainless steel spoon.\nloose tea strainer stainless steel infuser metal cup leaf filter with lid best.\nloose tea strainer bamboo leaf walmart.\nloose tea strainer stainless steel infuser mesh ball spice 3 kettle pot cup leaf target.\nloose tea strainer modern and stylish infuser leaf target.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..58f8f672aa6b89412580c1cfb24a506fba3cd81c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"On this episode of Fault Lines, hosts Garland Nixon and Lee Stranahan discuss how the mainstream media pivots from one story to the next and reframes events to push specific narratives. How does this method of operation impact how the public views the news & world events, and what impact does social media have on the modern flow of information?\nUS President Donald Trump is continuing to step up pressure on the Islamic Republic of Iran and its oil industry. Speaking to Sputnik, Ali Golmoradi, a member of the Iranian Parliament Committee on Energy, noted that Tehran has enough tools of leverage to resist the US economic offensive, including the closure of the Hormuz Strait.\nUS judges scheduled oral arguments on the White House's effort to block a Congressional subpoena for financial records for 14 May, according to Reuters.\nWASHINGTON (Sputnik) \u2014 The United States has pledged that its military will fly and sail wherever the international law allows ensuring the safety of commercial and non-commercial traffic in response to Tehran's warning to close the Strait of Hormuz, US Special Representative for Iran Brian Hook said in a press briefing on Tuesday.\nPolitical leaders from all around the globe are sending their condolences to Sri Lanka over deadly blasts that claimed the lives of 215 people and injured 450 more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Teaviews \u00bb East Pacific Tea Co.\nBelow you'll find extracts of all posts on Teaviews.com related to East Pacific Tea Co.. Click on any post to pull up photographs and full-text.\n\" To the taste, the tightrope act on my tongue was magnificent. Melon and flowers traded places, making mincemeat of my liquid-full mount. Too glib?\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A guide to help users create citations using APA (American Psychological Association) style, 6th edition. Skip to main content. The manual refers to the GPO (U. S. Gov. Printing Office). Canadian equivalents may be: Queens Printer, Ministry of Supply and Services, Canadian Government Publishing, etc.\nNote: References in APA publications are cited in text with an authordate citation system (p. 174). Groups or agencies as the author: (National Institute of Mental Health, 2007) (NIMH, 2007) Note: Spell out the entire name of the agency in the first citation and use the abbreviation or acronym in subsequent citations.\nGovernment documents can be confusing to cite. They can take the form of anything from an informational pamphlet to a Congressional debate and everything in between. Unlike standard publications, these documents do not necessarily follow the pattern of author, title, publisher, date. This guide will try to help you get started building your citation The 2016 edition of the GPO Style Manual is the first revision to be issued under GPO's new name, U.\nS. Government Publishing Office. Since 1894, the GPO Style Manual has served as a guide to the style and form of Federal Government printing and publishing.\nThe Manual has come to be widely recognized by writers and editors both within and By act of Congress the Public Printer of the U. S. Government Printing Office is authorized to determine the form and style of Government printing. The Style Manual is the product of many years of public printing experience, and its rules are based on principles of good usage and custom in the printing trade.\nAPA (American Psychological Association) style is most frequently used within the social sciences, in order to cite various sources. This APA Citation Guide, revised according to the 6th edition of the APA manual, provides the general format for intext citations and the reference page.\nAPA Publication Manual, Sixth Edition. APA Style Guide to Electronic References, Sixth Edition (for parentheticalstyle citations) or commas around the year (for narrative citations). theres an exact copy of the code on the Government Printing Offices (GPOs) website The major difference between the citation styles in the Chicago Manual of Style and the U. S. Government Printing Office (GPO) Style Manual are caused by the needs of the users of the two style guides.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I got a DWI-how do I get my license back?\nIn order to have your driving privileges reinstated, you must complete a SATOP. SATOP consists of an assessment screening and completion of a program. After successful completion of the recommended program, a completion form will be sent to the Department of Revenue. In addition to the SATOP completion certificate, the Department of Revenue has additional requirements. Please contact them at 573-526-2407.\nSATOP programs can be completed at any state certified SATOP agency. (consult OMU listing).\nThe SATOP assessment screening costs $375 ($126 assessment screening fee and $249 supplemental fee). If you receive a recommendation for the Offender Education Program, the cost will be $130. If you receive a recommendation for the Weekend Intervention Program, the cost will be $458.28. If you receive a recommendation for the Clinical Intervention Program, the cost will be $1,046.91. If you receive a recommendation for the Youth Clinical Intervention Program, the cost will be $510.20. The Department of Mental Health's Standard Means Test can be applied to determine a sliding fee scale for individuals participating in the Weekend Intervention Program, Clinical Intervention Program, or Youth Clinical Intervention Program.\nWhat is the supplemental fee?\nThe supplemental fee is assessed to everyone (Missouri statute) entering a SATOP. The supplemental fee is deposited into the Mental Health Earnings Fund and is used to assist individuals who may be unable to fully pay for the cost of the Weekend Intervention Program, Clinical Intervention Program, or Youth Clinical Intervention Program.\nI completed a program a couple of years ago and now the business is no longer there. What can I do?\nThere may be some instances where an agency is no longer in business or has changed locations. You are encouraged to search your local telephone book and contact a certified SATOP agency to determine if they may have record of your completion. The Division of Behavioral Health has been maintaining a database since 1995 and should have records of completions done after that time. It is your responsibility to keep copies of all receipts and completion paperwork you receive from an agency and keep them in a safe place.\nI'm a resident of another state, but got a DWI while in Missouri. Do I have to come back to complete the program in Missouri?\nNo\u2014provided you receive an assessment from a state-certified agency and complete the recommended program. The agency where you receive services must complete a SATOP Comparable Program Completion Form , return it to you, and you must mail the completed form and $249 (supplemental fee) to the Missouri Department of Mental Health.\nI've moved to a different state. How can I meet the SATOP requirements?\nYou must receive an assessment from a state-certified agency and complete the recommended program. The agency where you receive services must complete a SATOP Comparable Program Completion Form, return it to you, and you must mail the completed form and $249 (supplemental fee) to the Department of Mental Health.\nI live in Missouri and completed an inpatient program. Will this meet SATOP requirements?\nIf the completed program was a certified or recognized accredited alcohol and drug treatment and rehabilitation program, and you successfully completed a minimum of 120 hours of treatment (a minimum of 40 hours of individual and\/or group counseling, and the remaining hours must include any combination of the following: driver-related education, individual counseling, group counseling, group education, and family therapy) during a period of no less than 30 calendar days, this may meet SATOP requirements. You must take a completed SATOP Comparable Program Completion to an Offender Management Unit , who will verify your information. If the services you received are accepted as comparable, the Offender Management Unit will electronically notify the Department of Revenue. Some Department of Corrections, Veteran's Affairs Medical Centers, Military, and Federal Bureau of Prisons program will also be accepted as comparable, as long as they meet the above requirements.\nYou can contact the Division of Behavioral Health district offices or utilize the treatment services locator . The completion of a treatment program may be comparable to SATOP if it meets requirements as a state certified or nationally accredited program.\nI got a letter from the Department of Mental Health informing me I have to do a SATOP program. I have never had a DWI (or have already completed a program). Do I still need to do SATOP?\nThe Division of Behavioral Health and Department of Revenue cooperate in providing information to individuals who have their driving privileges suspended or revoked. If you think you received a letter by mistake, we suggest you contact the Department of Revenue at 573-526-2407 and check the status of your driver license and verify that all reinstatement requirements have been fulfilled.\nMy DWI was dismissed (or I was found not guilty) in court. Why do I still need to do SATOP?\nEven though a case may be dismissed in court, there may still be administrative action taken by the Department of Revenue. If you tested with a .08 blood alcohol content or higher or if you refused a breathalyzer or any other chemical test, then your driving privileges will be suspended. In order to get your driving privileges reinstated, you must successfully complete a SATOP. To verify your driver license status, please contact the Department of Revenue at 573-526-2407.\nWhat if I need an interpreter in order to understand the program?\nSATOP services will be provided to every individual in accordance with the Americans with Disabilities Act. If you need interpretive services either signed or spoken, contact a local Offender Management Unit to arrange such services. Sign language interpreters must be certified and licensed. Offender Management Units should be able to provide an interpreter when needed. If someone is unable to find an interpreter, a list can be obtained by contacting the Department of Mental Health at 573-751-4942.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LOTE at St. Carlos is all about fostering an appreciation of the diverse multicultural society we live in.\nAt St. Carlos, students from Prep to Year Six participate in a weekly 45 minute lesson in LOTE (Italian).\nAt the junior levels, the emphasis is on the development and acquisition of listening and speaking skills. Through stories, songs, rhymes and games, students begin to explore the sound and the rhythm of Italian.\nFrom Year Three to Year Six, students begin developing skills in reading and writing, through a range of sequential and developmental activities. Students at the senior levels are encouraged to explore language through an integrated and personalised approach, by selecting tasks that are relevant and purposeful to their particular needs or interests.\nThroughout the year, students at St. Carlos also have the opportunity to take part in external competitions such as the Penola Catholic College Italian Transition Program for students in Years Five and Six and the Dante Alighieri Junior Poster Competition, which is open to all year levels.\nAs a culmination to the year, in term four, the school community celebrates all things Italian, such as art, music, singing and dancing, a parade of costumes and an Italian lunch, with it's annual Italian day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzargyh b/data_all_eng_slimpj/shuffled/split2/finalzzzargyh
new file mode 100644
index 0000000000000000000000000000000000000000..2bf1af10c196892ab0ebaf3e601c3ab5509ebdda
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzargyh
@@ -0,0 +1,5 @@
+{"text":"One of the most exciting and controversial forms of artwork is making caricatures. Caricature by definition is an image that distorts the attributes of a person or thing to create a humorous imitation. Most funny caricatures is about celebrity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Completing a commercial flooring project can be a challenge. It pays to have an experienced and knowledgeable partner in your corner. Someone who sees the big picture. Someone who knows how to anticipate and avoid problems that could potentially delay your projects. Customer satisfaction is our first objective. We can think outside the box to perform consistently during difficult circumstances. We are directly accountable to you during all phases of your project.\nFrom helping you choose the best product to suit your needs to providing timely installation and maintenance. Our staff has a thorough knowledge of the quality, appearance and performance of flooring products and accessories, along with the installation techniques that best apply to each product, and the product lines, guarantees and delivery history of the major mills and manufacturers.\nMember of Starnet since 1997. This gives us access to a wide array of innovative products. Our consultants can help you lay out your ideas and concepts into a cohesive design.\nOur experienced project managers have the talent to adapt and overcome obstacles while moving your project to a successful completion. We bring over 80 years of combined experience to the floor covering industry and draw from all the diversity our team offers.\nWe have specialized equipment that aids in the installation process.\nFloorcraft Floor Covering, Inc. has a group of seasoned installers that are trained and adept in a vast range of flooring products.\nOur goal is to serve our customers well and exceed your expectations. Our team's ability and dedication is the foundation for our long standing client relationships.\nWe've got the lifts and the options that will work best for you. Visit www.hlift.com today!\nFloorcraft Floor Covering, Inc. was founded in 1957 and incorporated in 1966, allowing us to become one of the most experienced firms in the commercial floor covering industry. We have the experience and the capacity to handle the largest jobs, yet our service is personalized so that even the smallest jobs receive our full attention.\nStarnet\u00ae is comprised of over 170 independently owned full-service flooring contractors representing more than 300 locations worldwide. With $2 billion in annual sales, Starnet\u00ae is the single most influential force in the contract flooring industry. How did we get there?\nPower and Data Cable Management Made Easy!\nFreeAxez is an award winning, all steel, quick-connect, low profile access floor that breaks the mold of the traditional post and panel raised access floor. FreeAxez provides unparalleled capacity and flexibility for the modern office and high technology environment, saving life cycle costs and improving workplace efficiency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u2192 Tour & Travel Tips \u2192 What are the Priorities in Indian Trains?\nThe way every organization gives priorities to different employees, every country gives priorities to different cities; in the same way railways in India do give priorities to some trains. Of course, it is always good to know about the trains that enjoy priorities of the authorities.\nThese are the trains that enjoy the utmost priority on the Indian railway network. The trains are counted among the fastest trains in the country. The trains are completely air-conditioned trains that link up New Delhi with other capital cities of different states of India. Some of these trains are like Mumbai Rajdhani Express, Howrah Rajdhani Express and Dibrugarh Rajdhani Express.\nThis is an express train that links up metro Indian cities with other nearby important cities and gets second utmost priority on the network. TheseShatabdi Express trains cross over a short distance and they return to the station of origin the same day itself. A few of the Shatabdi expresses in India are such as New Delhi Habibganj Shatabdi Express, Bangalore Chennai Shatabdi Express and Lucknow New Delhi Shatabdi Express. These trains can be found crowded all the times and the beauty of these trains is that they are mostly on time.\nTalking about Duronto Express, these are the trains that are few stops long-distance trains. The trains are run by the Indian Railways and they link up main cities to other chief cities. These trains are absolutely fast as Rajdhani and Shatabdi express. Duronto Expressistaken among one of the quickest trains in India. There are Duronto trains such as Pune Howrah Duronto Express, Mumbai New Delhi Duronto Express and Sealdah New Delhi Duronto Express.\nThese Express trains are one of the fastest trains in India and get the highest priority on Indian railway network coupled with Rajdhani Express, Gatimaan Express, Shatabdi Express and Duronto Express. The train runs at a general speed of one hundred thirty km\/h and the train is India's first semi-high speed completely AC train that has all the aircraft like features. Some of these trains are namely Mumbai CST \u2013New Delhi \u2013 Chandigarh Tejas Express, Anand Vihar Terminal Tejas Express and Karmali Tejas Express and Lucknow Junction.\nGarib Rath Express full air conditioned train was originated by Indian Railways for long distance travel for people within reasonable budgets. Garib Rath trains are the trains that run on the highest speed of one hundred thirty Kmph, provocatively slower than India's high-speed trains. Some of these trains are like Pune Nagpur Garib Rath Express, Jabalpur Mumbai Garib Rath Express and Sealdah Ranchi Garib Rath Express.\nThus, now you see, it is not just about Food on Train or working place or in organizations alone; priorities are given among trains too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hut34 Entropy(ENTRP) is now trading at $0.461480 and has moved in last 24 hours!\nHut34 Entropy (ENTRP) \u0647\u0648 \u0631\u0645\u0632 \u0645\u0645\u064a\u0632 \u0644\u0639\u0645\u0644\u0629 \u0645\u0634\u0641\u0631\u0629 \u0635\u062f\u0631 \u0639\u0644\u0649 \u0645\u0646\u0635\u0629 \u064a\u062b\u0631\u064a\u0648\u0645. \u0633\u0639\u0631 Hut34 Entropy (ENTRP)\u0627\u0644\u064a\u0648\u0645 \u0647\u0648 $0.461480 \u0628\u062d\u062c\u0645 \u062a\u062f\u0627\u0648\u0644 \u0639\u0644\u0649 \u0645\u062f\u0649 24 \u0633\u0627\u0639\u0629 \u0642\u062f\u0631\u0647 $0.00000000.\u0644\u0647\u0627 \u0645\u0639\u0631\u0648\u0636 \u0645\u062a\u062f\u0627\u0648\u0644 \u0642\u062f\u0631\u0647 11.4 \u0645\u0644\u064a\u0648\u0646 \u0639\u0645\u0644\u0629 \u0648\u0623\u0642\u0635\u0649 \u0645\u0639\u0631\u0648\u0636 \u0642\u062f\u0631\u0647 100 \u0645\u0644\u064a\u0648\u0646 \u0639\u0645\u0644\u0629.\u0623\u0643\u062b\u0631 \u0628\u0648\u0631\u0635\u0629 \u0646\u0634\u0627\u0637\u064b\u0627 \u062a\u062a\u062f\u0627\u0648\u0644 \u0641\u064a Hut34 Entropy \u0647\u064a Token Store.\u0627\u0633\u062a\u0643\u0634\u0641 \u0639\u0646\u0648\u0627\u0646 \u0645\u0639\u0627\u0645\u0644\u0627\u062a Hut34 Entropy \u0639\u0644\u0649 \u0645\u0633\u062a\u0643\u0634\u0641\u0627\u062a \u0627\u0644\u0643\u062a\u0644 \u0645\u062b\u0644etherscan.io \u060c ethplorer.io \u0648 enjinx.io. \u064a\u0645\u0643\u0646 \u0627\u0644\u0639\u062b\u0648\u0631 \u0639\u0644\u0649 \u0645\u0639\u0644\u0648\u0645\u0627\u062a \u0625\u0636\u0627\u0641\u064a\u0629 \u0639\u0646 \u0639\u0645\u0644\u0629 Hut34 Entropy \u0641\u064a https:\/\/hut34.io\/.\nFantastic news to be included in the CoinGecko Beam program! At the Hut34 Project (https:\/\/hut34.io) we're building a crypto future. Our protocol - HutX - allows chatbots, IoT and digital services to connect, interact, and monetise their data and communication. We've also built the worlds first google powered ethereum wallet (https:\/\/wallet.hut34.io) with 0x project powered easy ERC20 token conversions. Lots more enhancements to come!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When \"the Internet\" first started to become a tool used more and more by the public, I would often find myself using websites or directories that just didn't seem to make much sense!\nThe term \"User Experience\" didn't even exist back then and when I decided I wanted to make websites easier to use, I started by learning how to design and develop them. This didn't fully encapsulate what I wanted to achieve though, and so I set out to study Information Architecture and Usability.\nSince then I've gone on to develop, design and eventually architect websites, software applications and mobile apps for some of the biggest and best brands in the world, along with countless others that no-one will have heard of!\nI'm passionate about creating great interfaces for people to use whether its on a mobile device, smart TV, desktop computer or even in-car or home controller systems.\nMy user experience portfolio and CV is available upon request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarxuj b/data_all_eng_slimpj/shuffled/split2/finalzzzarxuj
new file mode 100644
index 0000000000000000000000000000000000000000..3dea2d1ccf66e24818fcf788e30ff0bd36240d1d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarxuj
@@ -0,0 +1,5 @@
+{"text":"Rezvan Habibimarand and Mozaffar Saberi. Picture: PA.\nThe great-grandparents' case was previously raised in the House of Commons and more than 127,000 people had signed an online petition calling for the couple to be allowed to stay in the UK.\nThe couple's family and legal team previously spoke of the pair's distress at the prospect of being separated from their family who all live in Edinburgh.\nA Home Office spokeswoman confirmed: \"Following a review of the case, during which supplementary evidence was provided and considered, Mr Saberi and Mrs Habibimarand have been granted 30 months' leave to remain.\"\nWriting on Facebook, their son Navid Saberi said: \"You did it folks, you did it, thank you does not sound\/feel enough but thank you. My parents will not be deported, just received the confirmation.\n\"Thank you for all your efforts, likes, shares and signatures.\"\nHe continued: \"I should thank John Vassiliou, our solicitor, and his team who stuck with us in the last six years and never gave up hope and kept us motivated.\n\"In addition, I would like to thank our MP Ian Murray who worked incredibly hard, raising the issue in the House of Commons and in the Council of Europe. He has been amazing.\n\"Many thanks also go to the Scottish Minister for Europe, Migration and International Development Ben Macpherson MSP and regional MSPs Alison Johnstone and Andy Wightman who all wrote letters on our behalf, and also to David Lammy MP and Diane Abbott MP for helping to raise the case's profile on social media.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A draft of sorts? The beginning?\nIn a contentious hour-long \"town hall meeting\" called to explain the step, these workers peppered the official who signed the order with often hostile complaints about the largest diplomatic call-up since Vietnam. Announced last week, it will require some diplomats \u2013 under threat of dismissal \u2013 to serve at the embassy in Baghdad and in so-called Provincial Reconstruction Teams in outlying provinces.\nNewspapers have a duty to inform citizens about such democratic events.\nCoordinated antiwar protests in at least 11 American cities this weekend raised anew an interesting question about the nature of news coverage: Are the media ignoring rallies against the Iraq war because of their low turnout or is the turnout dampened by the lack of news coverage?\nI find it unsettling that I even have to consider the question.\nOut of the mouths of young men..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This month is one of the peak times of year for divorce but while starting the divorce process can be done online, there is still a long way to go before full digital divorce becomes a reality.\nNonetheless The Ministry of Justice has committed itself to reducing the stress individuals encounter during divorce proceedings as well of the length of time divorces take. The aim is to put as much as possible online yet there is still no date set for the launch of a full digital divorce service online.\nThe preferred approach is to introduce processes stage by stage starting with the divorce petition. The prospect of the full digital divorce could revolutionise this aspect of family law and bring this area of law into line with other legal processes that can now be done online.\nOne of the difficulties likely to be encountered with a full digital divorce is being able to introduce a process that is fair to both parties in a divorce. The nature of human relationships means that there will be difficulties in developing processes that will provide the same level of protection for those involved as would be the case offline.\nCould January 2017 be a Quieter 'Divorce Month'?\nJanuary is traditionally the month when people make promises to themselves to make a fresh start and this can mean seeking divorce. But will this be a quieter year?\nThe Office for National Statistics revealed a record drop in divorce rates in 2014 when divorces fell to a new 40-year low. 2014 saw 111,169 divorces which was a 3.1% fall on the previous year. The difference becomes even more stark when comparing that figure to 2003 when the number of divorces was 27% higher.\nCuts to legal aid and a rise in cohabitation since the early part of the last decade have both combined to reduce divorce rates.\nSurprisingly the highest number of divorces were among men and women in their 40s rather than younger age groups.\nWhy divorces are more likely to happen in this age group is unclear however economic factors have been blamed in the past for putting pressure on couples. Financial disagreements are often a leading cause of resentments building between married couples and these can become even more pronounced during recessions.\nDespite divorce rates declining, most people involved in family law agree that the workload has changed little with disputes over custody and finances a major part of work undertaken.\nDigital divorce to be introduced in 2017?\nA recent article in the Times newspaper claims that couples will be able to divorce online this year under plans that could open the way for the abolition of fault-based grounds for ending marriage. The article also states that ministers are preparing a pilot project to allow divorce proceedings to be issued digitally for the first time, in order to save time, paperwork and stress for thousands of separating couples. It goes on to say that Sir James Munby, England's most senior family judge, is backing this plan and it will be tested before being introduced across England and Wales in June 2017.\nIt seems, therefore, that the modernisation of divorce is a long way off, however, excitement and expectation is growing amongst family law practitioners as to when the first digital petition or pilot scheme will be released.\nIt is estimated that millions of pounds go unpaid every year as a result of non-compliance with financial remedy orders. The Law Commission has called for a cultural change to ensure that family financial orders are enforced more effectively as family financial orders are an important yet often overlooked area of law. The Law Commission's report has been welcomed by family practitioners.\nFamily financial orders are usually made upon the ending of a marriage or civil partnership that require a payment of money, or the transfer of property, between former spouses or civil partners. Orders can also be made between parents for the benefit of the children of the family. The Law Commission found that people tend to think the process is over once a financial order has been made. The Commission has said that 'Enforcement of that order is unlikely to be at the forefront of people's minds. There is an expectation that people will comply with court orders'.\n3.\tMoney that will become payable to the debtor in the future.\nThe Commission also suggests more effective coercive methods of enforcement such as disqualifying debtors from driving or ban them from travelling out of the UK. A Ministry of Justice spokesperson has said that they 'welcome the report and will consider its recommendations carefully'.\nWe have considerable experience in dealing with the enforcement of family financial orders so if you have failed to comply with the terms of a financial order or your former partner has breached the terms of the order then we recommend that you seek legal advice promptly. Our specialist team will be able to advise you of your options and the potential consequences. Please contact us today on 0161 927 3118.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From chocolatier at world-famous Chef Alain Ducasse in France to his nomination for the coveted James Beard \"Outstanding Pastry Award\" while at Four Seasons New York, Chef Bruno's travels through Shanghai, Washington, DC, San Francisco and Vancouver culminate in his nostalgic powers of great dessert. Here he shares his latest tasty creation of Imperial Keemun Hao Ya Cr\u00e9me Br\u00fble\u00e9.\nEnjoy! We know we did!\nBring to the whipping cream to a boil.\nAdd: Tealeaves Imperial Keemun Hao Ya tea and turn off stove. Infuse for 1 hour.\nBring the mixture back to a boil.\nIn a separate bowl, whisk: egg yolks and sugar together.\nPass the infused cream through a fine mesh strainer. Mix in the egg yolk mixture. Stir well.\nPour the cr\u00e8me br\u00fbl\u00e9e mixture into individual ramequins. Place the ramequins into a water-filled deep tray.\nPlace the tray in the oven and bake for about 40 minutes, or until the cr\u00e8me br\u00fbl\u00e9es set and achieve the consistency of a pudding.\nRemove from the oven and allow to cool down. Place in a refrigerator for 2 hours.\nOnce ready to serve, sprinkle the top of each ramekin with about 2 tablespoons of granulated sugar.\nRemove excess sugar. Using a blow torch, caramelize the top of the cr\u00e8me br\u00fbl\u00e9es. Bon App\u00e9tit!\nSweet and slightly spicy, a highly prized Yunnan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click here to dowload a 4.0.9 version (released September 27 2018).\nVersion 4.0, released April 30 2015, is a \"long-term support\" version, and the 4.0 branch has undergone a full validation process.\nThe latest \"long-term support\" version si version 5.0, and version 4.0 will be maintained until the spring of 2019, when version 6.0 is released. We recommend users using version 4.0 or older to switch to version 5.0.\nA Windows build (based on Cygwin) is now also available, in 64-bit version.\nAs with any version, in case you detect bugs, we appreciate your feedback on the forum and bug-tracker, which will help us to provide you with patch releases for this stable series, as well as improvements for future developement versions.\nReminder: current Code_Saturne developpment cycles are based on a release approximately every 6 months, including a fully validated, \"long-term support\" version every 2 years.\nFor more details, the reader may refer either to the release notes of the version or to the ChangeLog and\/or to the NEWS file.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasnfo b/data_all_eng_slimpj/shuffled/split2/finalzzzasnfo
new file mode 100644
index 0000000000000000000000000000000000000000..dba4d6a3e359fa8231f94e8e18c1c89be09ffdc7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasnfo
@@ -0,0 +1,5 @@
+{"text":"Note: This course is aimed at students needing scores in the range of 5.5 to 7.0. Additional self-study material is available for those who need higher scores.\nThe cost for the four-week course is $840 with all materials included. If you need training in only one skill you can enrol for four weeks in a single skills for $280.\nNote: you have to apply at least one week prior to the commencement of the course you want to participate in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Silicus Azure DataOps help enterprises extract value from their investments in modern cloud data estate infrastructure \u2013 from localized database management to full scale analytics programs. Our Azure DataOps services help industrialize all aspects of insights and intelligence delivery \u2013 data management, analytics, business intelligence and integration with intelligent systems.\nWe offer end to end Azure Data Ops services to automate, streamline and manage the flow of data from ingestion and integration to analytics and reporting. Our Data Ops platform services are delivered through a highly experienced team of data architects, data analysts, data engineers and service desk specialists across the platform, infrastructure and the data estate implementation.\nData estate infrastructure, toolchain and platform development and implementation services for continuous data ingestion, storage, transformation, analysis and reporting with ongoing Managed Services for Data Platform & Infrastructure Ops.\nManaged services for Data Platform Ops covering end to end flow of data along the pipeline \u2013 from data ingestion to data analytics and reporting.\nAzure Managed services for Data Infrastructure Ops provide Azure data related infrastructure configuration management, change, & incident management, and 24x7 monitoring of data infrastructure telemetry.\nOperate your cloud data estate securely, efficiently and optimally with trained cloud help desk professionals.\nOur Azure Data Platform Ops Managed services ensure large enterprise wide Data estate deployments are effectively and pro-actively monitored and managed across the data pipeline. Our experts keep track of key metrics and are equipped to collect performance, usage and security related insights from various parts of the data ecosystem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please apply a Flat Rate to all ADD ON PRODUCTS without Lobsters!\nYou pay a flat rate no matter where the package is going. Consider that a long haul overnight delivery - in most cases, by 1:00 pm the next day. A bit of a specialist in next day delivery!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AquaChek\u00ae TruTest Digital Pool Chemical Test Strip Reader tests for free chlorine\/bromine, pH, and total alkalinity. Now there's no guesswork required to interpret color results. Insert a test strip in your pool or spa, then insert the strip reader and get fast and accurate digital results instantly! Comes with 25 TruTest digital strips.\nDownload the free AquaChek\u00ae Smart app and simplify pool and spa water maintenance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"$49.90 Down and $2.50 per Day!\nThe S&W model 360 Single\/Double Action revolver is an iconic J-Frame design which is lightweight and powerful. Chambered in .357 Magnum this revolver is sure to get the job done. The fixed front and rear sight along with exposed hammer round out the list of features. The frame is constructed from a lightweight scandium alloy and it features a PVD Black Metal Finish.\nRevolver arrived in perfect shape without any cosmetic flaws. Barrel is centered perfect. Action is smooth and crisp. Factory grips are probably good for recoil with the 357. But I installed lower profile wood grips for EDC. This revolver looks great in my opinion with the solid cylinder. Over all a great little gun and was as described by buds. ( Note .357 is alot of power for this size frame and recoil should be expected and in my opinion is not a down side to the gun. We all should know getting into a scandium j frame that its going to have substantial recoil) Buds great as usual as well!!!\nExcellent handgun for EDC. It is easy and confortable to conceal IWB, even for a smaller framed guy. Recoil is stout with .357 loads but still controlable. It is no less accurate than I am at the range firing 5-15 yards with average groups about the size of a paper plate. I am not a fan of the larger coyote tan grips for EDC or the un fluted cylinder but thats purly asthetics and just my opinion. The fit and finish are standard S&W high quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzausuv b/data_all_eng_slimpj/shuffled/split2/finalzzzausuv
new file mode 100644
index 0000000000000000000000000000000000000000..e526f22cfed400ab16c46b78b8b3957ac6f557e6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzausuv
@@ -0,0 +1,5 @@
+{"text":"African American producing theater company whose mission is to entertain, educate and inspire artists and audiences while addressing critical issues facing our community.\nPresent quality affordable theatrical productions, readings and educational workshops for African American\/multi-cultural adults, youth and others in Portland.\nProvide a forum for artists and community members to connect, and heighten cultural and social awareness among residents from an African American perspective.\nParticipate, attend, donate or share.\nPassinArt: A Theatre Company is an African American producing theater company whose mission is to entertain, educate and inspire artists and audiences while addressing critical issues facing our community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(1) Water supply. The absolute discharge is measured in cubic meter per hour (m\u00b3\/hr) and varies from k\u0101r\u0113z to k\u0101r\u0113z, as it depends on the local conditions of supply to the water table and a channel's length. In a study of 2,000 channels in Iran, Beaumont (1989, p. 23) has calculated that the discharge is always lower than 300 m\u00b3\/hr. While two-thirds of the channels discharge less than 60 m\u00b3\/hr, for 45 percent of the channels the discharge is less than 30 m\u00b3\/hr (Beaumont, 1971, p. 47). The latter water flow allows for irrigating 10-20 hectares of land. For Iran an average discharge of 54 m\u00b3\/hr has been calculated, and the highest known discharge is given by a k\u0101r\u0113z at \u0160\u0101hrud, which yields around 3,200 m\u00b3\/hr (Goblot, 1979, p. 41). In addition to its absolute discharge, the productivity of a channel must be assessed in its relation to the region's water resources. A channel's specific discharge measures the liters of water per second per kilometer (l\/sec\/km; cf. Balland, 1992b, p. 106). In Iran the specific discharge is about 3 l\/sec\/km, but about 8 l\/sec\/km is its average in Afghanistan. The particular efficacy of the Afghan k\u0101r\u0113z seems to reflect that the Afghan channels are generally shorter (see ii (2), above), while the area supplying the water is larger. In all other regions, from Turfan to the Sahara, the specific discharge remains in the lower range of 2-3 l\/sec\/km (Balland, 1992b, p. 107, fig. 4).\nThe major significance of the k\u0101r\u0113z lies in its continuous discharge throughout the year. In contrast, irrigation systems that rely on surface water runoff can completely cease to discharge water during the dry season. The continuous discharge, however, needs be distinguished from a constant discharge. Significant seasonal variations can be observed in the discharge of the subterranean aquifers, although these variations are much smaller than those of the rivers. In a study of 24 k\u0101r\u0113z in the region of Mashad in 1964-65, the maximum discharge occurred from February to May, and the lowest discharge was measured in September and October. The same phenomenon has been noted in Afghanistan. A study of 7 k\u0101r\u0113z in the region of Kabul calculated an average of a 29-percent decrease for the discharge between July and September 1963 (Wagner, p. 108). These variations basically mirror the regular pattern of temperatures, although with a slight time lag, as the heat increases evaporation and thus decreases the supply of water to the water tables. But variations in precipitation and local conditions of available runoff water are also important, since they can cause significant differences from one year to another (Beaumont, 1971; Balland, 1992a, p. 103). In general, the seasonal variability reflects the permeability of the surrounding rock and the thickness of the aquifer. The k\u0101r\u0113z that drains a shallow, local water table of small volume covered by a very permeable detrital material shows the greater seasonal variability, and it may even temporarily ceases its discharge. For Afghanistan S. Radojivic described both the occasional drying up, as in the case of the k\u0101r\u0113z of \u1e20ol\u0101m \u1e34\u0101n at Sehg\u0101na in Kataw\u0101z (1978a, p. 51), and the recurrent drying up for 4-6 months every year, which has been observed for the k\u0101r\u0113z at Pol-e Sangi, about 11 km northeast of Kabul (1978b, p. 2). Farmers are of course aware of these variations, since a constant discharge increases the value of the arable land (Taraki, p. 622). Consequently, they distinguish between a k\u0101r\u0113z with constant discharge (Pers. \u1e21arq-\u0101b, \u0101b-\u1e21ari, cf. McLachlan, p.77; Pashto k\u0101r\u0113z p\u0101\u1e35\u0259 in Afghanistan, cf. Balland, 1992a, p. 103, n. 5) and one whose discharge is fluctuating according to rainfall (Pers. h\u0101v\u0101-bin, po\u0161t-\u0101b in Iran; Pashto k\u0101r\u0113z wahri in Afghanistan, cf. Balland, 1992a, p. 103, n. 6). In southern Afghanistan N. E. McClymonds studied the impact of the extreme drought of 1970 and 1971 (Balland, 1992a, p. 105). In the region of Kajakay the average discharge of 73 k\u0101r\u0113z fell 60 percent in comparison with their discharge in 1950, the year for which measurements were available. Because of improvement or repair works, 9 k\u0101r\u0113z had nevertheless experienced a modest increase of their discharge, 6 percent on average, while in 3 k\u0101r\u0113z the discharge remained unchanged. But in the other 61 k\u0101r\u0113z the discharge dropped on average 72 percent, and 4 channels had even dried up completely, so that the dependent villages had been deserted. The number of k\u0101r\u0113z supplying more than 50 l\/sec had fallen from 15 to 1. Since the closest meteorological stations of La\u0161karg\u0101h and Kandahar measured in 1971 a rainfall deficit of 47 and 70 percent, respectively, overexploitation of the water tables could be ruled out, and this fall had to be attributed to the disastrous rainfall situation in 1970 and 1971.\nLastly, many k\u0101r\u0113z are destined to dry up eventually because their water tables are overexploited. Such long-term variations are irreversible without radical modification in human attitudes. For example, an ill-considered multiplication of channels will deplete the water table, which in turn causes the water level to drop permanently.\n(2) Geographical characteristics. Preferred sites for the construction of k\u0101r\u0113z are those that have sufficiently and regularly supplied water tables and are located below abundantly watered mountainous terrain, where the channels can emerge into plains with good cultivable soil. Piedmont areas thus seem to be the best, especially since the nature of the surface materials (regolith) is generally quite favorable for water infiltration. Within these areas the alluvial cones are especially favorable, watered as they are not only by runoff on the main slopes but also by drainage of fluvial basins located at various distances within the mountains (Goblot, 1979, pp. 28-29). The k\u0101r\u0113z tend to be built in networks where fluvial basins are quite numerous. On the Iranian plateau there are numerous areas that fulfill these conditions particularly well, and k\u0101r\u0113z were important fort Iran's development. In general, in the Iranian regions (Figure 3) where the most k\u0101r\u0113z are grouped in networks (Beaumont, 1971, pp. 41-42), the annual precipitation does not exceed 100-200 mm, so that these regions are unsuitable for rainwater cultivation.\nTehran at the foot of the Alborz mountains is an exceptionally well-suited site: a row of converging piedmonts, covered with Pliocene detrital soil, at the foot of a mountain with a height of more than 4,000 m (Figure 4).\nAnother very dense k\u0101r\u0113z network can be found on the plain of Var\u0101min (Beaumont, 1971, p. 42).\nBut in the subarid piedmonts for which k\u0101r\u0113z are particularly well suited, the underground water channels do not occupy the whole area, inasmuch as diversion canals, starting from the mountain slopes, are an effective means for harnessing more or less permanent water runoffs for irrigation (Figure 5). These may be found serving narrow strips of land along rivers and streams, in alternation with the karez that dominate the broad areas between rivers and streams (Figure 5B), as in southern Afghanistan, at the eastern end of the Zamind\u0101war district along the Helmand river and its tributaries (Balland, 1992b, p. 110). The region of Kajakay receives about 200 mm of annual precipitation, and its fluvial irrigation is strictly limited to the very narrow low alluvial plain of the Helmand River. The extensive alluvial slope which rises above it some tens of meters away on the right bank has a dense network of around 80 k\u0101r\u0113z in two tiered groups, so that the more abundant k\u0101r\u0113z are located uphill, while the lower, less productive ones recover the water lost through seepage. On the regional scale, over the whole Zamind\u0101war, the surface area irrigated by k\u0101r\u0113z (30,000 hectares) markedly exceeds that of the fluvial oases (17,500 hectares).\nIn large oases the arrangement, which differs from the preceding one only in details, is based on the opposition between center and periphery. The k\u0101r\u0113z is located on the margins, which extend in both directions, uphill and downhill, from the oasis, so that it is possible to cultivate high plains beyond the reach of fluvial diversion canals and to capture water lost through seepage, taking over from the canals which have been exhausted in these areas due to the withdrawals of successive users. This type is very widespread on the whole Iranian plateau, from the piedmont of Tehran (Beaumont, 1968, p. 173; 1974, p. 422) to K\u0101bolest\u0101n (Gentelle, 1977, p. 251). It is also found in the southeast of Afghanistan, as in the \u1e34\u014dst basin, at the foot of an eastern branch of the Solaym\u0101n mountains (Wald, 1969, p. 29) and on the eastern piedmont of the Haz\u0101raj\u0101t (see HAZ\u0100RA i. HISTORICAL GEOGRAPHY). Between Qal\u0101t and \u1e20azni, along the great longitudinal valleys of the \u1e20azni R\u014dd and the Tarnak, about 1,000 k\u0101r\u0113z irrigate around 16,000 hectares (that is a sixth of the total in Afghan territory) versus 60,000 hectares supplied by the diversion canals in the same region (Balland, 1992b, p. 112).\nA rarer pattern is characterized by a genuine interweaving, or overlapping, of the diversion canals and the k\u0101r\u0113z, and is generally reserved for large oases. This pattern represents an ancient development (Neely, 1974, p. 28), as it is attested since the Sasanian period (224-650 CE; see online SASANIAN DYNASTY). The k\u0101r\u0113z is located within the network of diversion canals in the center of the site, and these internal k\u0101r\u0113z are always parallel to the nearest thalweg and capture the underflow recharged by seepage of the fluvial waters upstream. Examples can be found south of Kandahar (Balland, 1992b, figs. 4B and 5), along the valley of the Tarnak (Balland, 1992b, pp, 110-12).\nThe multiplication of k\u0101r\u0113z in the piedmont sites has had profound impact on the rural landscape of the Iranian plateau, because they have shaped both agrarian patterns and settlement patterns. Their presence has led to a general division of the lands below the channel outlets into long strips parallel to the direction of water flow (Figure 6), which contrasts distinctly with the much more irregular polygonal division of fields irrigated by diversion canals (Bonine, 1989).\nThis general trend in land division can be frequently observed in the orientation of rural roads and even in the layout of roads within villages (Roaf, 1989).\n(3) Social implications. Because the construction cost of these complex facilities is always very high, k\u0101r\u0113z have important repercussions for rural society. Since substantial capital is required to excavate an underground channel, only great landowners, rich merchants, governors, and other powerful persons are normally capable of making the considerable investments required for its construction. There are no examples of k\u0101r\u0113z being built by a community of small farmers. Conversely, this irrigation technique appears, at least at the time of its origin, to be linked to large-scale land ownership, which traditionally dominated the society of the Iranian plateau (Lambton, 1953). In the course of time, however, the great majority of k\u0101r\u0113z have been progressively parceled out through the process of inheritance, which extends to the division of irrigation water among the heirs. Moreover, shares of water can be leased and are frequently sold independently of land ownership. Many k\u0101r\u0113z have become endowment properties (sing. waqf). In the 20th century, before and after the agrarian reform of the 1960s, there was a growing tendency toward such parceling out and to an increase in the number of small owners of shares of water (Bonine, 1989).\nThere are two established methods for the sharing out of the water (Safi-Ne\u017e\u0101d, 1980; 1992, pp. 65-75)\u2014measuring time or volume (Planhol and Rognon, pp. 119-20). In Iran distribution by volume is quite rare, practically limited to the piedmont of Tehran, and division by time is by far the most prevalent method. According to a timetable peculiar to each k\u0101r\u0113z, the total outflow is successively placed at the disposal of each user. Time is measured by a water clock (clepsydra) or more rarely by a sundial, which was, for example, still in use in the Bi\u0101b\u0101nak region in the 1980s. These traditional methods are increasingly abandoned in favor of battery-powered timekeeping. A frequent complication for the same user is the diurnal and nocturnal alternation of successive withdrawals of water, because nocturnal irrigation, although tiring for the farmer, is much more effective due to lower rates of evaporation. The alternation systems are sometimes extremely complex, as their organization aims at insuring a certain advantage for those with the smallest shares in order to avoid their being disadvantaged. The oases with k\u0101r\u0113z are thus veritable hydrologic micro-societies, characterized by the emergence of a whole hierarchy of control authorities, which in Iran are generally villagers. Each k\u0101r\u0113z is placed under the direction of a manager (ra\u02beis), who is almost always the most important shareholder of water and who in turn appoints the water distributor (mir-\u0101b). The latter office is very demanding but highly regarded in village society, and the office holder is always paid, usually in goods, produce, or water, although nowadays more and more often in money.\nXavier de Planhol, \"K\u0100RIZ iii. ECONOMIC AND SOCIAL CONTEXTS,\" Encyclop\u00e6dia Iranica, XV\/6, pp. 569-572, available online at http:\/\/www.iranicaonline.org\/articles\/kariz_3 (accessed on 30 December 2012).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So how can a couple of guys from Columbus, Ohio, equipped with just a drum set and a keyboard, quickly become such an international sensation? While their unique combination of various styles certainly plays a role in their appeal, I think their magnetic sound isn't the only driving factor. Twenty One Pilots is so popular, at least in part, because they have found a way to musically channel the experiences of anxious people in an age dominated by distraction, doubt, and despair.\n\"Car Radio\" echoes Pascal's insight, \"I have often said that man's unhappiness arises from one thing alone: that he cannot remain quietly in his room.\" Silence causes us to face ourselves: our thoughts, our fears, our doubts, and the uncomfortable yet undeniable realities of our existence.\nResorting to distraction to escape silence is nothing new, but distraction has never been so readily accessible to the average person than it is today. Our lives are dominated by social media apps, games, email, streaming services, and news notifications. This constant stimulation creates a fractured consciousness as we experience a wide range of emotions in an instant. We feel frail and spread thin, but we're also too addicted to stop. The silence is too much for our anxious, over-stimulated brains, so we pick up our phones (or turn on the radio) for another dopamine hit.\nTwenty One Pilots articulates the existential despair and doubt that is at the back of our minds; the nagging thoughts and feelings that remain in our periphery no matter how hard we try to drown them out. And while the content of their music voices our concerns and fears, their musical style mirrors the disarray of our distracted lives. The chaotic electric sounds from their tracks resemble the cacophony of stimulation coming from our screens, and their melding of different genres imitates the convergence of disparate narratives thrust upon our psyches.\nBut Twenty One PIlots is probably so enchanting because their music doesn't simply remain discordant, but is coalesced into a coherent stream of sound. They provide just enough order to navigate the chaotic waters we all wade in. As a result, their music doesn't as much offer a respite from our distracted lives, but forces us to confront the pandemonium head-on, in addition to the discomforting thoughts that always haunt the periphery. And that's no small feat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following documents are the Federal Register notices announcing the current and previous versions of the LSC regulations. The supplemental information found in these notices may be useful in understanding the regulations. They are provided for research purposes. Please make sure that you are using the current version of the regulation if you want to know the rules that apply today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are so many neighborhoods in Lancashire that are worth exploring, you'll wish for a car to check out all there is to do in this city. The city buses, while cheap, may not get you close to where you wanted to go. Ride hailing can be convenient, but rural areas may not have timely service. Plus, the cost of going from here to there can start to add up.\nIf you want the simplest way to get around town at a fantastic rate, then you have found the best place. We work with many of the most dependable dealers. Opening its doors in 1974, Alamo introduced the concept of Unlimited Free Mileage. Alamo strives to provide a fun, low-cost, high-value rental experience to both business and family travelers.\nHow do I get the best prices on a Burnley Alamo car rental?\nWhich forms of identification do I need when I pick up my car rental in Burnley?\nWhat is the minimum age to rent a car in Burnley?\nDo I need insurance when I rent a car in the U.K.?\nAlmost all countries require driver's insurance. You might be able to purchase additional insurance through Alamo for your trip, if you think your insurance won't cover you while traveling through the United Kingdom. Although many of the major card issuers, like Visa and MasterCard, offer some insurance, there are usually restrictions. For example, they may not cover your rental car if you get a luxury or larger vehicle. Make sure to check Alamo's policy before you decide.\nDo I need to put down a deposit on my Alamo rental car?\nAlamo, like several other rental car agencies, requires a deposit when you make your reservation. The amount can vary greatly depending on which vehicle you reserve. Additionally, be sure there is enough available on your credit card to make a deposit.\nAt CarRentals.com, we suggest a full tank policy. You pick up your Alamo rental with a full tank, and you should return it filled up. If you want to save some money, it is best to find a filling station before you return the car to Alamo.\nDo I need unlimited miles when I drive in Burnley?\nThe majority of suppliers will offer unlimited miles, but you need to ensure it's part of the package that you are purchasing. It's helpful to know how far you want to drive so you can plan your trip ahead of time. Although most car rental companies offer unlimited mileage, if it isn't part of your rental agreement, you could face additional cost. You may want unlimited miles if you are planning to explore everything that is in Lancashire.\nWhat is there to do around Burnley?\nOnce you've checked out Burnley, you'll be happy that you have your own set of wheels. Not sure where to start your trip? The Keighley Bus Museum Trust Ltd can make for a fascinating day.\nWith everything there is to do in Lancashire, having access to your own auto lets you explore at your own pace. When you are ready to check out more of the U.K., you can go when you like, without trying to wave down a cab or sign up for a ride-share.\nHeading out of town soon? Book your Alamo Burnley car rental and zip off to take in all the items on your itinerary. Book through CarRentals.com and hit the streets in Lancashire, United Kingdom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauwgh b/data_all_eng_slimpj/shuffled/split2/finalzzzauwgh
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+{"text":"It is important to offer testing to the partner to make sure there is no risk to have a baby with a haemoglobin disorder. Ideally, results from both parents' results should be available before week 12 of pregnancy to enable decisions regarding prenatal diagnosis. Information for fathers on haemologlobin E carrier screening, can be downloaded or ordered via PHRD.\nIf patient is found to have reduced MCV and MCH indices, iron supplementation is not required unless the patient's ferritin levels are reduced.\nTest partner, if not done previously (that includes: haemoglobin screen, full blood count and ferritin levels). This can be done in primary care.\nIf partner is not a carrier of any haemoglobin variants, the couple have a 1 in 2 chance (or 50%) to have children who are healthy carriers.\nIf partner is a carrier of Sickle Cell Disease (haemoglobin S) or Beta Thalassaemia, refer URGENTLY TO Clinical Genetics for appropriate counselling and to discuss further testing.\nIf partner is a carrier of any other haemoglobin variant, reassure the couple as there is no other relevant interaction with any other haemoglobin variant.\nIf the couple has other children, only test them if the partner is a carrier of beta thalassaemia or sickle cell disease. Otherwise, there is no need to test them unless they present with health problems. It is normally recommended to postpone testing until the age of 16, to enable them to make their own informed decision.\nMake sure the patient had received his\/her haemoglobinopathy card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amazing well kept 3 storey, 3bedroom, 3bathroom townhome with DOUBLE GARAGE. Open concept, bright and airy, this is a home you will be proud of. Crown moulding throughout, excellent appliances and a cooks dream with a gas stove. A nice island allows for lots of space to entertain and 2 decks allow and a patio allow lots of options to bbq, set up some outdoor tbles and chairs and even grow your own vegetables. Situated near the small playground in the complex, with trees that bloom into an amazing forest like feeling. Close to shopping and transit and extra room on the driveway that can accomidate another 2 vehicles. Lots of visitor parking available as well. Call today to get in and see this property.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Domiciliary care is right for you if you value independence, stability, affordability, and personalised one-to-one care, but it's not right for everyone.\nDomiciliary care is commonly known as home care and both are umbrella terms that refer to the different types of care and support that can be provided in your own home.\nDomiciliary carers can offer varying levels of care depending on your needs. Different carers will specialise in providing different types and intensities of care on a sliding scale from companionship care, to personal assistant care, to specialist care such as looking after dementia sufferers.\nAs such, domiciliary care is appropriate in most situations, as the level of care is so flexible.\nThe biggest advantage to domiciliary care is that it allows your loved one to retain their independence. Even with care as intensive as live in care, it's the ability to choose when to eat, to drink, and when to bathe that is so important.\nThe most luxurious care home in the world still has to cater to all of its residents, which means a set schedule, and a loss of independence.\nIf you choose domiciliary care your loved one can stay in their own home, which allows them to retain a sense of normalcy, and close to friends and family. Keeping your loved one familiar surroundings is especially relevant if they're suffering from memory loss or dementia, as a new environment can cause further confusion and distress. It might be that your loved one is in a fragile state of mind. Old age, accident or injury, or ongoing disease can all cause stress, and being in one's own home can make things a little easier.\nThe nature of domiciliary care is that it's fundamentally completely flexible to their needs. This means it can be the perfect solution if their needs will change over time. A good example would be post-hospital care or dementia care, when the health of the patient is changing on an almost day-by-day basis.\nBecause domiciliary care is so flexible, you only pay for the care you need, when you need it. Care homes are very black and white: either you're in, or you're out. As a general rule, 8 hours or less per day of domiciliary care is less expensive than a care home. And if you choose to opt for live-in care, carers charge by the day instead of by the hour and their rates can be very reasonable.\nThe ratio of staff to residents in care homes is notoriously bad. There are no set guidelines in the UK and care home workers are often overworked and underpaid and can't give individuals the attention they deserve.\nDomiciliary care is the complete opposite \u2013 rather than being 'just a number', patients are cared for on a completely bespoke, individual basis.\nYou're limited to the facilities that already exist in the home. That will often mean having to make appropriate home modifications. These are generally minor (handrails, ramps) but can become more major depending on the home (doorway widening, stair-lifts).\nAlthough the major advantage of domiciliary care is that you can stay in your home, it does mean allowing a carer into your home. Some people find it difficult to have someone in their home and struggle with the idea of sharing their space.\nRelated topic What Elderly Care is Right For Me?\nSaying that, although many people are resistant initially, it's normal for care workers and patients to strike-up a very close friendship within a short amount of time.\nA care home can be easy \u2013 if space becomes available you simply make an appointment and all the care staff are already on hand. If you're looking for domiciliary care, you have to find a carer yourself, and this can take a little longer than it would to organise a care home place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We just finished hearing about the challenges and ministry of Jonah this past week. Jonah, called to preach repentance to the Ninevites, finds that he would rather not, and so attempts to get away from God. That, of course, doesn't work because there is no where that God is not, so he ends up in the belly of a big fish for three days and nights, and is eventually disgorged in Nineveh to do the work he was called to do. This he does, begrudgingly, and the people of Nineveh repent, to the praise and glory of God.\nAnd today we hear that no sign will be given to the people of Jesus' time except this sign of Jonah. And that is true. Jesus is called to preach repentance just like Jonah was, although, praise God, he does it willingly. Jesus too will be covered over for three days and three nights, but this time in the tomb and not a fish. He then is disgorged in the glory of the Resurrection to give the way to repentance, which some have done, to the praise and glory of God. This is the only sign we need.\nBut Jesus berates the people because while the evil people of Nineveh repented, the Jews of Jesus' day not so much. The people of Nineveh didn't have anything near as great a prophet as Jesus is, and they repented, but the people of Jesus' time did not. And so history and eternity will be kinder to the Ninevites than to these people.\nThe Psalmist today sings that the Lord has made known his salvation. This he has done, to the Ninevites, to the people of Jesus' time, and to us. Today we pray for the softening of our hearts so that we might repent of our wickedness in the way that the Ninevites did, and so have eternal life.\nSo, what's the big deal about seeking a sign? Don't we all need a sign from God every now and then so that we can be sure we are on the right path? Otherwise, how are we supposed to know the will of God?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Q: How much should I take? How do I take?\nA: While I cannot give specific recommendations for dosages or anything \u2013 I can share with you what I and most others are doing in the C60 community.\nPortions\/Dose: There is no set Dose, only studies with various amazing results \u2013 Most tend to use about .5 to 2 teaspoons per day according to preference and need.\nPlease see my C60 page for more information.\nA: I do not recommend this, never heard of anyone doing it. I give my pets oral c60 of 1 drop per day.\nOk so, I've only met a few people in this world who were more socially anxious than I am, I am probably Asperger's and definitely the most introverted person I've ever known \u2013 and the few people I did meet who were worse off were socially crippled and unable to perform basic functions. To be quite honest if I hadn't had the life I had had, I wouldn't have ever hit this point of 'not giving a crap' anymore and that, combined with some small measure of natural humor\/looks allowed me to build confidence in the dating world which transitioned into business. I got very lucky, I think, frankly. I've had some PTSD causing Traumas too, Jordan Peterson says those are just signs of a survivor of a trauma. The mind without legs has very strong arms \u2013 we cope.\nSo I say this to let you know I can definitely relate to you on that level. As for drugs \u2013 I don't really ever take them unless I have to, such as in a life threatening situation or after surgery. My go to choices for social anxiety are this, and some may surprise you.\n1. Read Jordan Peterson's 12 rules for life \u2013 He has so many tricks from a master of his profession (psychology and how your brain works with anxiety and stuff) such as following the circadian rhythm (bed by 10pm etc) and how to eat a certain way (including a high protein\/healthy fat breakfast each day) to balance your blood sugars and all these other things that if you don't get right, you can't solve anxiety at all period! I never knew! Take notes!\n2. A bit of overlap \u2013 but the most important thing you can do is eat a healthy organic diet primarily consisting of raw leafy greens (some need steaming like broccoli or cooking, etc \u2013 but lots of raw enzymes are good, I do smoothies and it pre-digests a whole bunch of vegetables into a fast super drink daily. I even grow my own microgreens \u2013 which is a lot easier than it sounds.\n3. Address all common known deficiencies. If you do everything else you will still be deficient in at least this (that I know of, and there's others): Selenium, Magnesium, Iodine \u2013 I've recommended how to make Magnesium it's so simple I don't bother selling it as finished liquid (but I do have amazon links for much of these on my site and I get 4% or something!) and Iodine I do sell and teach how to make on my site. Selenium You can eat mustard seed (yuck) or go with the trace mineral drops from hallelujah diet company \u2013 I want to carry this product and am talking to them about it now.\n7. Figure out how to Love Yourself \u2013 We all carry many scars and weaknesses \u2013 but you made it here. That's a lot.\n8. No person is an island. We all need a social network.\n9. Just start taking steps, and building willpower. Everyday you delay builds a tiny thing into a bigger thing in your minds eye. And it becomes scary and then we delay more and it creates a negative feedback loop. This can get ugly. Been there.\n10. Read Think and Grow Rich, by Napolean Hill. It's old, but timeless. And will inspire you. Changed my life in a dark place once. The day my ambition sparked.\nof course none of this is medical advice, just what I've researched with Aspergers obsession into alt health.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavdzx b/data_all_eng_slimpj/shuffled/split2/finalzzzavdzx
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index 0000000000000000000000000000000000000000..42dd6b5154eda64c171a88fc7366ed64dca93f00
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+{"text":"Radiant Sculpt is a total bodywork with weights, fun music and heat!\nThis class compliments your regular Radiant yoga practice. This class incorporates strength training and specific exercise to create lean muscles, boost your metabolism and to give you the results you want.\nBe ready to move, sweat and feel amazing.\nThe class is heated at 105 degrees, lasting 60 minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Queen visited Lord's Cricket Ground, met the England and Australia teams, and watched the second Ashes Test.\nClick above for video from the BBC.\nThe Flame Lily again! Well, if you're going to do a back-to-back repeat, might as well be this lovely chunk of diamonds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a million versions of this story. You're on a trip. Somewhere far away\u2014the polar opposite of where you live\u2014and you fall in love. It could be with the place, it could be with the food, the person you're with, or even the car you're driving. You adopt this new thing once you get back home, only to discover, disappointed, that it was only a vacation romance.\nI'd be a really crappy journalist if I didn't attempt to find out\u2014so please join me in welcoming this 2018 Ram 2500 Power Wagon to the Motor Trend Garage.\nThe Power Wagon was the winner of our Trans-Labrador Highway comparison because it was downright unstoppable in mud, sand, gravel, and dirt. Where we doubted the abilities of the other trucks, the trusty Ram never gave us pause. Looking at the Power Wagon's standard feature list, it's pretty obvious why. Each Power Wagon starts life as a 2500 Heavy Duty Crew Cab, equipped with a 76-inch bed and four-wheel drive. Under the hood, the base 5.7-liter V-8 is swapped out for a 6.4-liter V-8 that makes 410 hp and 429 lb-ft of torque paired with a six-speed automatic.\nThe most expensive is the Leather and Luxury package, which for $4,995 adds power leather upholstered bench seats (three up front and three in back), heated and cooled front outboard seats, a heated steering wheel, Uconnect infotainment with navigation and Apple CarPlay and Android Auto compatibility, front and rear parking sensors, and power folding mirrors\u2014the latter two will be incredibly useful when parking in L.A. Some other select options on our Power Wagon include the ever-useful RamBox ($1,295), a spray-in bedliner ($495), and power-adjustable pedals with memory ($195) for our shorter staffers. Total price out the door for our 2500 Power Wagon is $63,280.\nIn the month since the Power Wagon arrived in our garage, it's already seen a fair amount of action. It tackled an off-road expedition in the San Francisco volcanic field near Flagstaff, Arizona, hauled some motorcycles and cars, and unfortunately even had a short stint at our local Ram dealer. Clearly, love back home is going to have its ups and downs, but I'm excited to see where this roller-coaster takes us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Three Components of Every Good Cybersecurity Plan The threat landscape facing modern businesses has never been more vast, and the criminals who would exploit it never more sophisticated. In the face of this intimidating reality, it's imperative that you have a plan: because the more you know about what's facing you - the more prepared you are for an attack - the better. But where do you even start?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To find manuals for Bicycles & Mopeds you can select the category of your choice below. Use our search box to directly look up a manual from our database. Fill in the product number or brand name to find the manual you want. In case you didn't find the manual you were looking for, request one through our contact page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"Kioku\" (\u30e8\u30b9\u30ac\u30ce\u30bd\u30e9 \u30e1\u30a4\u30f3\u30c6\u30fc\u30de \"Memory\") is one of the BGMs of \"Yosuga no Sora\"which is a Japanese romance\/drama adult visual novel. Download free Kioku sheet music now!\nSabishii Yoru (\u5bc2\u3057\u3044\u591c) is the OST 12 of Yosuga no Sora which is a Japanese romance\/drama adult visual novel. Download free Sabishii Yoru sheet music now!\nYosuga no Sora is a Japanese romance\/drama adult visual novel developed by CUFFS (Sphere). Download free Yosuga no Sora sheet music.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here in the Pasadena area, you really don't want to be without your car's heating and air conditioning system. Especially in the hot and humid summers - you need to ensure your A\/C is working properly. The ASE-certified experts at Pasadena Auto Service have the skills and expertise to handle any A\/C and heating system repair issue.\nAll our repairs come with a 12-month warranty. Rest easy knowing our ASE-certified experts stand behind their work.\nLooking to help the environment and save a little money besides? Let us convert your old R-12 system to the new R-134A system. It's more eco-friendly, and economical. Come in today for more information. And if we make a repair on your vehicle, your A\/C check is free! We also offer complete transmission service, suspension and alignment, engine repairs, tune-ups, and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There won't be anyone saying they don't like these Brussels Sprouts!\nBring a large pot of salted water to the boil. Add the Brussels sprouts and simmer for 5 minutes. Using a slotted spoon, transfer the sprouts to a bowl of iced water to stop the cooking process.\nPut the teriyaki sauce, garlic, olive oil, honey and chilli flakes in a small bowl and combine thoroughly.\nDrain the sprouts and put in a large bowl, then pour half the teriyaki glaze over. Toss to combine, then transfer the sprouts to a non-stick baking tray.\nRoast in the oven for about 10 minutes, then transfer to a serving bowl and drizzle the remaining teriyaki glaze over. Shake the bowl to coat all the sprouts with the glaze and serve immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Of the three types of forest in West Africa swamp, savanna and closed only in the latter does the stand of timber repay commercial extraction on a large scale. Here, the West African timber trees acquire their valuable characteristic long, clean boles, the first branches being often over 100 feet (30 meters) from the ground.\nKnowing you look good in an outfit can boost your confidence, but putting it together can seem daunting. eHow is here to help you develop your personal style.\nWeight9 20t ApplicationStone,hard rock,ore,etc,slag impact crusher for 2016 shijiazhuang sludge tailing slag slurry pump for tin mining 160x110x56mm or customized Factory price slag tin box for phone mobile. Chat Online. Xin Chuang, Xin Chuang Suppliers and Manufacturers mobile granite extraction of irons for sale china.\nJun 06, 20170183;32;stone amp amp minerals extraction of irons offers 5527 buy ddt products. About 12% of these are storage boxes bins, 4% are integrated circuits, and 1% are aquariums accessories. A wide variety of buy ddt options are available to you, such as field effect transistor, pharmaceutical intermediates, and drive ic.\nCommercial Steam Cleaners Daimer Industries Inc.\nThe KleenJet 174; Mega 1000CVP is a certified (yes we have lab results to prove it) anti bacterial commercial steam cleaner excellent for countless applications. With an onboard vacuum for simultaneous steam and extraction it can tackle even the most difficult applications. Click the tabs below to learn more.\nThis pH RESTORE Alkaline Water pitcher Ionizer is equipped with a multi stage PH001 and PH002 filter that consists of zeolite, stone and ceramic ball blend, micro nets, activated coconut carbon charcoal and ion exchange resin. All of this comes together to increase pH levels in water and also raises the ORP of the drinking water.\nCrusher. A crusher is a machine designed to reduce large rocks into smaller rocks, gravel, or rock dust. The earliest crushers were hand held stones, where the weight of the stone In industry, crushers are machines which use a metal surface to break or ..\nMar 28, 20180183;32;Stone Cutting Chemicals. Cheap P400 Grit, find P400 Grit deals on line at rock extraction of irons engineered. Disc with precise hole design for improved life and dust extraction. Use for . For use with Silicon Carbide grit in a rock tumbler or flat lap . . Hydro engineered.\nThis jig is designed to clamp wood chisels and plane irons in the correct grinding angle. It also ensures that the tool is positioned at 90176;. Two safety stops can be set so that the complete surface of the grindstone so preventing the tool being sharpened slipping of the stone. The jig accommodates chisels and plane irons up to 3 (76mm).\nstone commercial extraction of irons offers 7515 iron wire drawing machine products. High production iron steel wire drawing machine\/luckey draw machine Xingtai City Judu Commercial .. Square Yard\/Square Yards; Stone\/Stones; Strand\/Strands; Tonne\/Tonnes; Tons; Tray\/Trays; Unit\/Units; Volt\/Volts; Watt\/Watts; Wp; Yard\/Yards.\nShop for the most advanced vapor steam cleaners for home, commercial and industrial uses, featuring dry superheated steam with anti bacterial properties.\nTools. Grand Materials and Supply, Inc. is your number one source for all your tool needs. Whether youre a home owner or a contractor, we know you need the right tool to perform the job.\nMatch the following a) Haematite a) Extraction of Iron b) Calcination b) Ore of Iron c) Smelting c) CaSiO3 d) Slag d) Acts as Flux e) Lime Stone e) Type of Iron f) Heating in absence of air. 18. Testing Zone..\nABOUT US. As one of the largest independent distributors of hotel guest amenities, housekeeping supplies, cleaning products, maintenance equipment, laundry vending products (and much more) in the United States, we are dedicated to providing our customers with value, innovation and outstanding service(continued on About Us page).\nFacial Steamers with Aromatherapy, Magnifying Lamp, and Portable Mini Facial Steamers with Ozone. We are the manufacturer for high quality facial spa steamers, and by dealing with us you will save hundreds if not thousands of dollars on your spa steamers, digital facial steamers, and aromatherapy ozone steamers with magnifying lamps.\nThe Extraction of Iron Chemistry LibreTexts Nov 29, 2015 Extracting iron from iron ore using a Blast Furnace is a basic oxide and reacts with acidic oxides such as silicon dioxide present in the rock.\nmini rock extraction of irons 530 tph. China Glass Crusher Machine. Electric Stone Crusher Wholesale, Stone Crusher Suppliers good performance commercial mini rock crusher; mini mobile concrete crushers for sale; Xinhai service personals are experienced professionals in mineral processing. You only need to call us, and the rest is our work.\nBismuth is a chemical element with symbol Bi and atomic number 83. It is a pentavalent post transition metal and one of the pnictogens with chemical properties resembling its lighter homologs arsenic and antimony. Elemental bismuth may occur naturally, although its sulfide and oxide form important commercial ores.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ACL Music Festival has eight stages where musical groups from genres including rock, indie, country, folk, electronic and hip hop perform for fans. The concerts continue from 10 AM to 10 PM on Friday, Saturday, and Sunday during the festival with various stages spread out in the park. Approximately 450,000 people attend the festival each year. In addition to the music performances, there are food and drinks, an art market, a kids area for families, and other activities for attendees. The festival also has various water fill stations for patrons, as well as plenty of bathrooms to support the large crowd.\nFounded in 2002, the festival began as a one-weekend event and remained as such through the 2012 date. On August 16, 2012, Austin City Council members voted unanimously to allow the Austin City Limits Music Festival to expand to two consecutive weekends beginning in 2013.\nThe Austin Eats food court in 2004.\nThe Festival's food court, called Austin Eats, was modeled after the Jazz Fest in New Orleans. Austin Eats features all of Austin's finest restaurants and caters to each individual festivalgoer. With gluten-free and vegetarian options, this food is far from funnel cakes and turkey legs. Restaurants featured include Torchy's Tacos, Stubbs B-B-Q, Chi'Lantro BBQ, and Amy's Ice Creams, all local Austin favorites. All vendors accept cash and there are ATMs located around the facility. Recycling bins are provided and festivalgoers are encouraged to be conscious when disposing of their food wastes.\nThe ACL Art Market features numerous art vendors and is located in the center of Zilker Park; all booths are open throughout the duration of the festival.\nIn 2015 it was announced that Auckland City Limits Music Festival would debut at Western Springs stadium in Auckland, New Zealand in early October 2016. The festival will showcase over 40 artists from a broad spectrum of musical genres, and highlight local culinary, artisans, festival fashion, an area for children, and a new festival forum for speaking on and exchanging cultural and innovative ideas.\nIn spring of 2018 Sydney City Limits made its debut at Brazilian Fields and Parade Grounds at Centennial Park in Sydney, Australia. This festival was one day and featured 28 bands on four stages. It had many of the same activities as Austin City Limits such as Sydney eats, Sydney market, and Sydney kiddie limits.\nAustin City Limits Music Festival, Weekend 2 in 2018.\nAustin City Limits Music Festival was held over 2 weekends: October 5\u20137 and October 12\u201314, 2018. The lineup featured headliners Paul McCartney, Metallica, Arctic Monkeys, Travis Scott, Odesza, and The National. Childish Gambino was forced to cancel his performance after a foot injury, and Justice was promoted from an earlier time slot to replace him as a headliner. Phoenix and Lil Wayne were also added to fill the empty time slots, with the former playing on the first weekend of the festival and the latter on the second weekend.\nAustin City Limits Music Festival was held over 2 weekends: October 6\u20138 and October 13\u201315, 2017. The lineup featured headliners Jay Z, Red Hot Chili Peppers, Chance The Rapper, The Killers, and Gorillaz.\nAustin City Limits Music Festival was held over 2 weekends: September 30 \u2013 October 2 and October 7\u20139, 2016. The lineup featured headliners Radiohead, Kygo, Kendrick Lamar, Mumford & sons, and LCD Soundsystem.\nAustin City Limits Music Festival was held over 2 weekends: October 2\u20134 and October 9\u201311, 2015. The lineup featured headliners Foo Fighters, Drake, the Strokes, Florence and the Machine, and the Weeknd.\nAustin City Limits Music Festival was held over 2 weekends: October 3\u20135 and October 10\u201312, 2014. It was listed in Forbes as one of 5 American Music Festivals to look forward to.\nFor the first time, the Austin City Limits Music Festival was split across two weekends with matching lineups: October 4\u20136 and October 11\u201313, 2013.\nDue to heavy rains and flash floods, the festival was cancelled on Day 3 of Weekend 2.\nAustin City Limits took place from October 12\u201314, 2012. The official lineup was announced on May 22, 2012.\nACL Festival celebrated its 10th Anniversary on September 16\u201318, 2011. The official lineup was announced on May 17, 2011.\nThe Strokes performing at Austin City Limits in 2010.\nThe 2010 festival took place on October 8\u201310, 2010. The performers included The Eagles, Phish, Muse, The Strokes, and M.I.A.\nThe 2009 festival took place on October 2\u20134, 2009. This year's festival is most commonly remembered as the one when torrential rains which started falling on Saturday afternoon turned the new grass turf into slick fields of Dillo Dirt mud.\nThe 2008 edition took place September 26\u201328, 2008.\nThe 2007 Austin City Limits Music Festival occurred September 14, 15, and 16 in Zilker Park. Several acts, including Amy Winehouse, The White Stripes and Rodrigo y Gabriela, cancelled their appearances at the festival due to health reasons, the latter two on very short notice. The scheduled performance by Saturday headliner, The White Stripes was replaced by moving already scheduled Muse into the headlining slot.\nOther notable moments include Friday when a propane tank was ignited and a fire broke out in the service area, burning down two trailers and several port-o-potties. Four people who were working at the festival were injured, two of them seriously. A second fire broke out on the speaker stack at the AT&T stage during Bj\u00f6rk's set, but it was quickly extinguished and no injuries were reported.\nThe 2006 festival took place on September 15, 16, and 17.\nAfter sweltering heat in 2005, festival organizers attempted to relieve festivalgoers from the Texas sun by adding more misting and water stations and more tents for shade. Organizers also added a mobile phone texting feature to the festival. AT&T's Blue Room offered options for fans to watch live streaming bands playing from the comfort of their own homes.\nBen Kweller suffered a nosebleed during his set. He attempted to stem the flow by inserting a tampon, thrown to him by an audience member, into his nostril. The tampon expanded painfully and then he removed it. Kweller performed two more songs until he had to leave the stage. The next day when The Flaming Lips performed, lead singer Wayne Coyne asked the audience to throw tampons at him to help mop up his signature fake blood. It continued to rain tampons on the band for well over two songs.\nMain entrance to the festival in 2005.\nThe 2005 festival took place on September 23, 24, and 25.\nThe 2005 Austin City Limits Festival won Pollstar's Festival of the Year Award. This was also the infamous \"Dust Bowl\" year where dust kicked up by the festival crowd made it difficult for audiences to breathe. The following year, sprinklers were installed in Zilker Park to remedy this problem.Organizers reduced the daily capacity of the event by 10,000 fans because of neighborhood disputes in the previous years. Three-day passes were sold for $105.\nNaturally 7 and numerous others.\nThe 2004 festival took place on September 17, 18, and 19.\nThe 2004 festival had eight stages, and, on the second day of the festival, a top attendance of 75,000 people. This spike in attendance forced the festival promoters to lower capacity per the request of surrounding neighborhood associations.\nSeptember 19\u201321 marked the second year for the ACL Music festival. With 100 bands performing, tickets increase to $65 for a three-day admission pass. Children under 10 were admitted for free when accompanied by an adult with a ticket.\n2002 was the inaugural year of the festival. Unlike subsequent years, it was a 2-day event only. The festival, arranged by Charlie Jones and Charles Attal founders of C3 Presents, was thrown together in a matter of three or four months. The 2-day festival hosted 5 stages and 67 bands. One-day passes were $25. 42,000 people attended the event when only 25,000 were expected. The festival has grown every year since.\n^ \"C3 Presents\". C3 Presents. Retrieved 2013-04-15.\n^ \"Austin City Limits Music Festival\". Topo Chico USA. Retrieved 3 August 2016.\n^ \"Council Approves Two Weekends of ACL Fest\". Austin 360. Retrieved 2014-08-14.\n^ \"ACL Music Fest \u2013 Austin Eats: ACL's food court, 10 years later \u2013 Food \u2013 The Austin Chronicle\". AustinChronicle.com. 2011-09-16. Retrieved 2014-07-21.\n^ \"2014 Austin Eats | ACL Music Festival | Oct. 3-5 & 10-12, 2014 | Zilker Park, Austin, Texas\". Aclfestival.com. Retrieved 2014-06-25.\n^ \"2014 ACL Art Market | ACL Music Festival | Oct. 3-5 & 10-12, 2014 | Zilker Park, Austin, Texas\". Aclfestival.com. Retrieved 2014-06-25.\n^ \"Artists | Austin City Limits\". Acltv.com. Retrieved 2014-06-25.\n^ \"5 American Music Festivals To Look Forward To\". Forbes. 2014-10-05. Retrieved 2014-10-07.\n^ \"Austin City Limits Festival Announces Schedule | News\". Pitchfork. 2012-07-26. Retrieved 2014-06-25.\n^ \"Austin City Limits Festival Celebrates 10th Anniversary\". dailytexanonline.com. 2014-09-14. Retrieved 2014-08-14.\n^ \"Austin Music Source\". www.austin360.com. Retrieved 2014-06-25.\n^ \"Rolling Stone, ''The 10 Best Shows at Austin City Limits''\". Rollingstone.com. Retrieved 2013-04-15.\nWikimedia Commons has media related to Austin City Limits Music Festival.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Louisville Orthopaedic Clinic has installed the Cuattro DR high definition X-ray imaging system. It's Digital Overhead Tube Crane (OTC) and elevating table allow for more flexibility and efficiency creating a more comfortable patient experience. This new system provides superior image quality and is safer for our patients due to reduced radiation exposure.\nThese images are stored on site, as well as, a copy stored off site for disaster recovery. Patients are entitled to one complimentary CD of their images, but there is a fee for an additional CD or filmed images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heavy duty sealant gun Smooth Rod Cartridge Gun with 12:1 mechanical advantage drive. Full sized handle for comfortable, efficient dispensing. Half-cradle carriage to securely hold standard 300ml sealant cartridges.\nB-Line Manual Foil\/Sausage Dispensing Gun with Smooth & Efficient 12:1 Ratio Drive.\nALL METAL BODY, TRIGGER AND SPOUT. ACCURATE APPLICATION OF SMALL JOINT SEALANT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last week, I wrote about Emilie Wapnick's recent TEDxBend talk about multipotentialites. From the number of emails I've received thanking me for that article and sharing the video, it seems that Emilie has given a name to what many people are experiencing, perhaps more now than ever before. And to my surprise, the more I listened to Emilie talk about the qualities and characteristics of multipotentialites, I realized that, in many ways, she was also describing me.\nI started my professional life as a voice teacher on the faculty of a well-known conservatory of music. At the same time, I was also a church organist and choir director. After a few years, I moved on from both of those positions to be a professional singer in New York and a vocal technique teacher for opera and musical theater singers. Within a few years, I was teaching singers who were performing leading roles in opera houses all over the world and on Broadway. Singing and teaching was my primary source of income for more than 20 years.\nYet alongside that career, I was also very interested in how performance in any medium (acting, singing, playing an instrument, or dance) gave voice to the soul. I studied the human energy system and how it relates to life and performance in any chosen field. Studying the human energy system led to deeper study of intuition and ancient wisdom teachings. Soon, in addition to teaching vocal technique, I was also teaching classes in personal and spiritual development, particularly as they related to performance. My dual careers in performing \/ voice teaching and in spiritual development fed each other in beautiful ways.\nIn the late 1990s, just as I was sensing that something new in my professional path might be on the horizon, I heard about personal coaching. In 2000 and 2001, I took a deep dive into coach training. By the summer of 2003, I had shifted my primary income source from teaching singers to personal coaching.\nAs my coaching practice developed, I became curious about quantum physics. I was fascinated by the \"science\" behind the ancient wisdom teachings I had been studying. Learning about the fundamental principles of quantum physics took me to a new level of understanding about life as energy in motion. That led me to create the Transformational Presence work that I do now. As I look forward, I don't know where my path will take me next, yet I trust that more unexpected and amazing opportunities are still to come.\nAll of this makes me wonder if the word \"career\" is actually losing its relevance in today's world, and if \"soul mission\" or \"life purpose\" is perhaps becoming more relevant now than ever before.\nYour soul mission is the energy and passion that fuels your life. It's your reason for being, your purpose in life. Soul mission is also your life-long lesson \u2013 what you are here to learn and become a master of \u2013 as well as the greatest gift you bring to your world.\nWhen you build your life on your soul mission, the roles you play or the career paths you follow become vehicles for living and expressing your soul mission. No one's soul mission is to be a doctor, a lawyer, a teacher, or a parent. Your soul mission is not limited to a specific career \u2013 it's the foundation upon which your whole life can be built.\nAs an example, my soul mission is \"I liberate and empower.\" Throughout my whole life, whether as a singer, teacher, coach, or friend, my greatest passion has been helping people set themselves free and giving them tools and skills to live the life they feel called to live.\nThe soul mission of a multipotentialite might be \"I learn and create,\" or \"I am a steward for greater possibilities,\" or even simply \"I invent.\" Their soul mission is the thread that ties together all of the multipotentialite paths. Within Emilie Wapnick's context, each of the career paths or creative pursuits we undertake becomes a vehicle for developing the many aspects of our greatest potential. Each path feeds our soul in a different way.\nWhile it's our curiosity about a subject or topic that initially draws us into a particular path or pursuit, what actually keeps us engaged is the feeling of aliveness and passion that we experience as we go deeper. It's the passion and aliveness that hooks us. This is an important distinction in creating a fulfilling life. It's not the specific career path or creative pursuit itself that sustains us emotionally, mentally, and spiritually; it's the process of developing our inner potential, our skills, and our talents to the fullest that excites and sustains us. It's who we are becoming as we pursue that interest that brings a sense of reward and fulfillment.\nThe soul thrives on learning, growth, and adventure. It wants to experience itself at its absolute best. To that end, it may seek multiple paths of expression in order to make the most of this life. When you know your soul mission, it then becomes the thread that weaves together your many experiences and pursuits. Therefore, discovering your soul mission is one of the most important steps in building a fulfilling and rewarding life.\nMaking a difference in today's rapidly changing world begins with getting clear about why you are here \u2013 your soul mission \u2013 and then letting your soul mission lead you down the career paths and creative pursuits that most bring you alive.\nIf you enjoyed this blog post and found it helpful or inspiring, please share it with your friends on social media by clicking on the icons below. You are also welcome to make a comment below. You may subscribe to our free weekly newsletter by clicking here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eros Airport is the city airport of Windhoek.\nIt is found just south of the city surrounded by the \"Southern Industrial\" in the north, residential areas in the west and east and another industrial area in the south.\nThe airport is at 1700 m above sea level. It has a runway 01\/19 with a length of 2230 meters and a runway 09\/27 with a length of 1005 meters.\nI do not know when flying started here but the airport was here when the second World War ended. It is today Namibias busiest airport in terms of aircraft movements. It is Namibia's hub for general and leisure aviation but also a hub for domestic flights to destinations such as Ondangwa, Walvis Bay, Luderitz, Katima Mulilo and Oranjemund.\nAir traffic control is provided from a modern tower northeast of where the two runways cross. Next to the tower is the air traffic control center for approach and area control.\n\u2026and like this from the \"landside\".\nIn the 1948 elections in South Africa the main Afrikaner nationalist party campaigned on a policy of apartheid. The National Party won the election and started introducing apartheid. They saw South Africa as divided into four distinct racial groups: white, black, coloured, and Indian. These groups were then further divided into thirteen nations or racial federations. The apartheid politics centred on the separation of these groups.\nThe new politics influenced everything, including the architecture. In these first years South African architects traveled to Brazil for inspiration from the Brazilian modernism. An example of architecture in the Brazilian mode and with apartheid ideas is the terminal building at Eros airport that was built in 1957. In the plans for the building it is clear how the building was divided into areas for \"European\" and \"non-European\".\nEros airport was called J.G. Strijdom. He was the South African Prime Minister between 1954 and 1958 and one of the strongest proponents for apartheid. Look at this old photograph of the terminal building and compare it with the photo below.\nThe second floor has a big balcony, facing the apron and the airfield. Behind it was a large restaurant for \"Europeans\". From the balcony is a long staircase to a large \"promenade deck\".\nOn the other side of the same building is another balcony that also had a restaurant. This side, facing from the airfield was for \"non-Europeans\". The balcony is small with no view over the airfield and the restaurant was about one sixth in size compared to the other. Between the two restaurants was a common kitchen.\nThe bottom floor has been rebuilt since those days. Today it looks like any modern terminal building on any small airport.\n\u2026and a chance to peek into the room that was once a modern restaurant with a bar and with a balcony overlooking the airfield and its departing and arriving aircraft.\nToday this all seems absurd and tragic. And when you consider the fact that the number of \"non-European\" passengers were close to zero you can't but be amazed that the architects still incorporated duplicated and dedicated areas for all functions at the airport terminal building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If there's one thing you should implement to increase mobility, it's a lacrosse ball. It's great for breaking up muscle knots and hard to reach trigger points. Use the Lacrosse ball on your traps and delts after a heavy upper body workout or on glutes and calves after a long run.\nRogue Lacrosse Balls is rated 4.8 out of 5 by 25.\nRated 5 out of 5 by Rice bag from Great product Great price - great product - fast shipping! Excellent experience!\nRated 4 out of 5 by Soblessed from Works well! My second one. Purchased it because it works well and has great grip. Thanks Rogue!\nRated 5 out of 5 by NinjaGigi from I use it for mobility in deep muscle massage. Works great for reaching deep muscle aches. I use it to improve my mobility upper and lower body. Post and pre run.\nRated 5 out of 5 by Wizzle from Great rogue ball I use it for my shoulders and hamstrings. loosens those muscles up really good.\nRated 5 out of 5 by Garbano from Recover Gets the job done! What else more is there to say?\nRated 5 out of 5 by viken from awesome tool great size, I use it for post-wod mobility and it's helping me a lot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"St. John Funeral Homes in Badoc, Ilocos Norte is established on September 2013. It is owned by Mr. Alfredo Quitoriano II. They aim to provide the quality memorial service and comfort to the families at the time of grief.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Because you are a hustler, a lot of people are going to come to you for help with their projects. They will offer you all sorts of compensation for your involvement, from paying you directly to giving you an ownership stake in their project.\nYou will be flattered by these offers. You will be excited by some of them too. Your excitement over these offers stems from the fact that you know how to help these people with their projects, and because you know that you have the ability to create value. In many cases, you will know exactly how to help the person asking you succeed in their venture.\nBut in something very close to 100% of the cases, you shouldn't get involved in other people's projects.\nAnything that isn't part of your mission is a distraction. You aren't going to have enough time to do all the things that you want to do when it comes to your own projects. Helping other people with their projects will distract you from your own projects, and it will push the results you need further into the future\u2014or kill them outright. Avoid distractions by learning to say no.\nThe time you spend helping other people with their thing is the time you are taking away from your thing. You are supposed to be doing your own meaningful work. That takes a massive investment of time and energy. By saying \"yes\" to other people's projects, you are stealing time from your highest priority. Invest your time in your hustle.\nThere will always be more offers and more projects available to you than you have bandwidth. The more successful you are, the more people will ask for your help. The number of people who need your help with interesting projects will always exceed the time you have available. You are not missing out when you turn down the opportunity to help with their project; you are missing out on your project's success.\nYour laser focus on the meaningful work you are here to do should dominate your time, your energy, and your investments. Nothing should take precedence over your mission. Anything that takes your focus from that work\u2014and your projects\u2014is a distraction. Hustle requires focus.\nYou are kinder by saying \"no\" to the people who ask for your help with their projects than you are by saying \"yes\" and not giving their projects the time and attention that someone else could invest.\nStay focused on your hustle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"[erlang-questions] Building 17.0-rc2 (and R16B03-1) with wxWidgets on OS X 10.9.2 64bit Re: [ANN] Erlang\/OTP 17.0-rc2 has been released.\nPrevious message: [erlang-questions] Building 17.0-rc2 (and R16B03-1) with\twxWidgets on OS X 10.9.2 64bit Re: [ANN] Erlang\/OTP 17.0-rc2\thas been released.\nNext message: [erlang-questions] Building 17.0-rc2 (and R16B03-1) with wxWidgets on OS X 10.9.2 64bit Re: [ANN] Erlang\/OTP 17.0-rc2 has been released.\narticle in my Gist. Thanks.\n> and I have erlang 17 64bit running.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Francis Family found Hollora Tribute For Peats Sake. 500th Find.\nHollora found 8 towns her 1600th find.\nWhen and Why post to a Milestone?\nWell just look what Hiram did now!!\nkkcjrlma Reaches 1000 fantastic finds.\nHear Ye! Hear Ye! Hear Ye!!\nAmmo can in the woods!!!\nJobi Wan Kenobi Finds 250!\nMapachi back in the saddle, hits 1500!\nSeriousTool hits 200. Hides, that is.\nDSKG Blasts through 2000 during an out of this world weekend!!\nIampaw! Newest member to the 5000 Club!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week at the XIX International AIDS Conference \u2014 as panelists and pundits debate whether an AIDS-free generation is actually possible \u2014 we must not neglect the other chronic diseases that remain an emerging and alarming threat to both aging HIV-positive and sero-negative populations in these settings. Today, chronic non-communicable diseases (C-NCDs), including cancer, lung and heart disease, and diabetes, kill over 28 million people annually in low and middle-income countries, many of whom are HIV-positive.\nOnce deemed a death sentence, HIV is now considered a manageable chronic condition through the use of lifelong antiretroviral therapy (ART). HIV-positive individuals are now living longer, particularly in resource-limited settings where HIV care and treatment were not previously available. In fact, through the global scale-up of HIV and AIDS services, health systems in low- and middle-income countries are now better prepared to tackle other C-NCDs \u2014 like cancers, diabetes, chronic lung diseases, and cardiovascular diseases \u2014 by leveraging the existing investments, infrastructure, and systems put in place in recent decades.\nOver a decade ago, many critics said that bringing life-saving HIV treatment to the most hard-to-reach areas would be impossible. Yet today, more than 8 million people have access to antiretroviral treatment in low-and middle-income countries. The same models used for lifelong ART can be adapted and used for managing and monitoring patients with other C-NCDs. MSH firmly believes we can apply the lessons learned from our experiences with HIV to the C-NCD epidemic. For example, service delivery models \u2014 e.g. scaling up of ART and prevention of mother-to-child transmission (PMTCT) \u2014 innovations in funding, health care financing models, pricing for drugs and laboratory supplies and equipment, and new technologies for care diagnosis, among other innovations, provide a model for chronic diseases in low- and middle-income countries.\nBy integrating current health systems and leveraging the existing groundwork laid by HIV and AIDS intervention scale-up, we can leverage existing public infrastructure, pharmaceutical supply chains, and human resources management, among other developments, to benefit patients with chronic diseases.\nHIV and other C-NCDs have serious socioeconomic consequences, often creating a financial barrier for individuals in need of proper care and treatment, and forcing them to pay high out-of-pocket fees. Despite advancements in service delivery, only twenty countries worldwide currently have Universal Health Coverage (UHC) plans in which everyone can receive basic health services.\nThe long-term nature of chronic diseases, including HIV, poses many challenges for the health system, but it is crucial that the prevention, care, and treatment of chronic disease be integrated in order to save many more lives.\nMSH believes that in order to effectively combat the rise of C-NCDs \u2014 and turn the tide against HIV and AIDS \u2014 we must strengthen current health systems while leveraging existing platforms and ensuring access at an affordable cost in the context of UHC.\nFor a more in depth discussion on this topic, watch the webcast of MSH's recent panel at the International AIDS Conference (via Kaiser Family Foundation).\nGloria Sangiwa, MD, is Management Sciences for Health's global technical lead for chronic non-communicable diseases and the director of technical quality and innovation in MSH's Center for Health Services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"September is always an interesting month for me for many reasons. I have big time commitments I have to meet every year as well as help work on things like planning my daughter's birthday party the first week in October. This year is no different except that it may be even busier this year due to the added responsibility of coaching my daughter's soccer team. Even with everything I have to do both as a father and as as professional, there is one other thing that has to get on my plate. That other thing is this blog\/website\/thing.\nThe blog came into existence in 2003 but I had been toying with different websites and domains on a personal level for some time before that point. Without going into the history of the thing, it became a blog shortly before my daughter was born. Over the years since the amount of content has ebbed and flowed along with its overall value to myself and the world in general. That is IF any one personal website can be said to have any value at all. It is a concept that is at least extremely debatable. Anyway, over that time I have trashed the site more times than I can count and flirted with taking it down entirely. The one or two of you out there that have been around that long probably remember all the other times I have written similar posts about bringing the site down for one reason or another and those issues I wrote about then are still relevant today. The overriding concern, however, is that I just don't have anything to write about\u2026or can write about. Oh I have plenty of things to say but I have decided that this is not the forum for much of what I REALLY want to talk about. It is just too public. So, where does that leave me? I don't know but it is time to find out.\nI am giving myself the month of September to either get things around here busy living or get busy dying. The renewal is coming up soon and money is tight so if I can't find any personal value (what the rest of the world thinks is a non-issue) in the site then I think it is time to kill it and move on to something else. There are some things I want to explore in the mean time. The first is moving the site to WordPress. As much as I feel connected to MovableType, I just don't feel like there is a community around the software like there is for WordPress. Maybe joining that community will give this blog a reason to live. I also want to get back to the things I used to enjoy about having the site. Things like sharing photos, opinions on film, and thoughts on philosophical issues. Again, none of that may have any value to the rest of the world but I enjoyed it and really what is a personal blog than an exercise in vanity?\nThe long and short comes down to this\u2026I have 30 days to decide whether or not to continue here. So if there was ever anything you wanted me to write about, now is the time to ask. I can't guarantee I can address everything but I will leave that open. Personally I hope that I find my passion for blogging and the internet in general again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The kitchen is usually the backbone of a home since this is where food is prepared. Food helps in body building, to provide energy and also to protect the body from infections, therefore this is why food normally made in the kitchen is very vital to every individual in the world. The kitchen is normally the place where food is prepared, and it should, therefore, be clean and neat away from dust. A good kitchen will ensure that good food will be made.\nRemodeling of a kitchen would need a professional in this area which is normally referred to as a kitchen renovation contractor. The chances of chaos happening is usually high especially when a person keeps changing their budget in the middle of the contract. The purpose for remodeling the kitchen should reason should be clear because the kitchen could be meant to be resold by investors when it is the main area of concern or if the person who owns the place wants a better place to use as a kitchen.\nDue to investment issues in kitchen remodeling it is very essential that you ensure that the budget and how remodeling will be done will create ways for cash returns to be brought back to the investing investors. If the whole house is not fancy then the kitchen which is just one part of the house should not be remodeled in a way that it will be fancy as well. It is essential that you look at the social class of a person because it will help you decide whether he or she can afford an expensive remodeled kitchen or not. The contractor should also put into consideration the method of payment that will be used that is cash or credit.\nKnowing how long the project will take will help in making important decisions such as whether food will be bought from outside the premises and whether you will find another room to use as kitchen meanwhile. It will also help one know whether they can still leave in the same house as the project goes on or whether to relocate to another place for a short while as the project is being completed. It is also important that if the kitchen is close to where customers are, they then should avoid that place for a while since it is likely to cause delays. One should be aware of costs such as transport of equipment, costs of obtaining equipment and materials as well as the cost to be paid to the laborers because they are likely to be incurred.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Estimations of apoplastic concentrations of K+ and Ca2+ in the vicinity of stomatal guard cells.\nA method which determines the null point for stomatal aperture has been used to estimate the apoplastic concentrations of potassium and calcium adjacent to the stomatal guard cells of Commelina communis. These two ions are contributors to important aspects of stomatal physiology: the determination of guard cell turgor (K+) and intracellular signalling (Ca2+) when the guard cells respond to the various stimuli that effect changes in stomatal aperture. We obtained estimates of apoplastic K+ concentrations in the range of 50\u201375 mol m-3, which are in general agreement with those of Bowling (1987). Ca2+ concentrations appear to be in the region of 0\u00b705 mol m-3 adjacent to the guard cells even though much higher concentrations (up to 4 mol m-3) may be delivered to the leaf in the xylem sap. Thus it is suggested that the gradient in apoplastic Ca2+ may be very large over short distances, and may be strictly controlled by cells within the epidermis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enterprise-level organic search engine marketing is all about optimizing different bits and pieces of web content that make up a large, complex website. Because each piece of an average enterprise-level \"Frankensite\" has dissimilar templates, disparate branding standards, and different stakeholders, it can be quite challenging to facilitate change and produce improved search engine visibility.\nGiven the nature of enterprise-level search engine marketing, what qualities should you look for when assembling an organic search engine marketing team?\nShould you seek out extraordinary writers and creative marketers?\nOr should you make certain every member of your team can write optimal code?\nPerhaps someone more politically oriented would be best suited to your team?\nWhile creative, technical, and political skills might be in good order, you should consider some offline skills when assembling an efficient enterprise-level SEO team.\nThere is a reason why Simplicity remains the definitive resource for sewing patterns, tools, and supplies for the past 85 years. Simplicity lives up to its brand name \u2013 it knows how to keep things simple.\nJust like simple paper patterns can be used as a foundation to construct intricate articles of clothing, so too can concise, consistent communication go far toward weaving fundamental principles of organic search engine optimization into large organizations that manage complex websites.\nWhether walking through a complex, technically oriented deliverable or just talking about how infographics work, it's important that enterprise-level search marketers are capable of communicating complex search engine marketing issues plain-English. Some people that we will work with might not know what a title tag is, let alone ever seen a robots.txt file or have never heard the word canonical used in a sentence.\nIf each team member strives to keep written and verbal communication simple, we can start to stitch together small patches of SEO success within a large, multifaceted organization because we're all speaking the same language. It also helps if the actual presentations are written in a way that they can be readily shared with almost anyone in the enterprise.\nIt isn't enough to be able to tell a good story. We need to be able to write a compelling narrative, too.\nJust as every good book has engaging plot that has a definitive beginning, middle, and end, enterprise-level SEO team members should be able to prepare documents and deliverables that read like a good book.\nSearch engine marketers usually have an engaging message to land with our audience, yet we don't always know where the document or email will end up. Readers could be SEO novices, digital marketing executives, or contracted professional service providers.\nWe need to speak to all levels of SEO knowledge in our potential audience in order to ensure that everyone understands the moral of the story. That's why each and every deliverable we create needs to have a beginning, middle, and end.\nThere should be an introduction that includes a statement about the purpose of the deliverable. Even a simple email message should have an introduction that clearly states the primary mission of why the message is important to its readers. Only then can we convey a message that will educate, illuminate, and inspire our readers to make the right decisions and take appropriate actions.\nFinally, SEO team members should be comfortable outlining a brief summary or next steps that encapsulate the primary purpose of the message. There is nothing worse than producing a spot-on, well thought-through deliverable that fails to remind our audience to take action \u2013 and take action now!\nWhen search marketers commit to telling the entire story each and every time we communicate with our extended SEO team, we're dedicating ourselves to telling the entire story that explains not only what we desire to happen, but why it needs to happen in the way we advise.\nIf it sounds like I'm getting a bit repetitive, indeed I am. With good reason, of course.\nWhile written and verbal communication skills are essential qualities we all desire in an enterprise-level search engine marketer, we must also get used to saying the same things over and over again.\nThe fact of the matter is that patience and persistence are essential qualities for successful enterprise-level search engine marketers to possess.\nThe CEO's third cousin twice removed who happens to know \"a little something\" about search engines.\nIf we are to optimize an enterprise-level \"Frankensite\" that has been stitched together from outliers and acquisitions, then it doesn't hurt to have SEO team members that know how to weave, knit, and sew.\nIf we want to create enterprise-level SEO strategies, then we should be prepared to communicate our thoughts in a concise manner that always explains why the message is important.\nAnd if we do these things often enough and frequently enough, then we just might facilitate positive change in search results, because these offline attributes are three essential qualities of enterprise-level search marketers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Exude your femininity with the new Chance Super One by Mauboussin, a collection that evokes gentleness and of delicate beauty. Designed with a diamond set on a lucky symbol \u2013 the clover, and the band with paved diamonds, Chance Super One collection bestows luck and beauty upon you with options of white, yellow or rose gold to choose from.\nFor the launch of Chance Super One, a limited time exclusive offer awaits you from now till 21 Apr 2019 at our store at Wisma Atria! Purchase a Chance Super One ring at S$990* only (usual price S$1,390).\nVisit us now at Wisma Atria #02-14\/15 Tel: +65 6733 0571.\nThe new 'Right Time Man' chronograph for men is a striking piece of watch meant to create an impression. The silver\/black layered dial against the sleekness of polished steel and the texture of brown aged leather strap easily catches the radar of the human eye. Equipped with an automatic movement, 'Right Time Man' possesses everything you need for all dress code.\nLife Time, Mon Amour timepiece for men by Mauboussin A matte layered dial in dark grey with clous de Paris pattern against the sleekness of polished steel is a result of a great contrast \u2013 distinguishing and classy. The new Life Time, Mon Amour timepiece with automatic movement for men by Mauboussin encapsulates this distinctiveness.\nA matte layered dial in dark grey with clous de Paris pattern against the sleekness of polished steel is a result of a great contrast \u2013 distinguishing and classy. The new Life Time, Mon Amour timepiece with automatic movement for men by Mauboussin encapsulates this distinctiveness.\nLe Temps ne s'arr\u00eate jamais \u2013 the new automatic timepiece of superiority and precision for the modern and discerning gentlemen. A classic white or blue dial in automatic movement with power reserve indicator, this elegant timepiece is a revelation of creativity of Mauboussin, demonstrating its savoir-faire in today's watch-making industry.\nOne design, seven stunning colours, three choices of gold, your capsules in life begin with the new \"Capsule d'Emotions\" collection by Mauboussin.\nGet your perfect dose of emotion now; or fulfill your whims by stacking up the colours on your finger.\nThe new \"78 Tours\" ultra-slim watch is inspired by phonograph record. The watch case embodies the vinyl in the form of a flat disc with an inscribed, modulated spiral groove as seen on the bezel of the timepiece. Spin \"78 Tours\" on your wrist now!\nEtoilement Divine The divine star\u2026 A symbol of beauty\u2026 The new 'Etoilement Divine' by Mauboussin embraces the stunning and heavenly emblem.\nThe divine star\u2026 A symbol of beauty\u2026 The new 'Etoilement Divine' by Mauboussin embraces the stunning and heavenly emblem.\nWhen love is felt every moment, temperature rises, passion soars! This overwhelming state of emotion is the flow of lava from a volcano that floods a peaceful valley. The new 'La Passion est un Volcan' collection by Mauboussin epitomizes this passion.\nThe new Etoile Sublime series is an elevation of a level of purity and excellence.\nA beautiful design in combination of black ceramic and white gold with an iconic star paved with diamonds, embedded on a cushion-like black enamel. The band is designed with a space to remind us the necessity to pause and breathe the exuberance of life! A perfect epitome of the women of today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The recuperating focus Management system gives the upsides of streamlined undertakings, updated association and control, strict cost control, awesome patient thought and upgraded benefit.\nOur Hospital Management Software is to a great degree fruitful, earth shattering, versatile, easy to use and made to pass on veritable conceivable focal points to focuses and recuperating focuses. The Hospital Management Software is planned for multispecialty specialist's offices, to cover a wide range mending office association and the officials shapes. It is a ground-breaking Hospital Management system that gives appropriate information over the specialist's office to help practical essential administration for patient thought, recuperating focus association and fundamental cash related accounting, in a reliable stream.\nWe are the principle web and versatile application enhancement association which is having capacity in working up the recuperating office the board programming which joins diverse fundamental segments in it. This will be of mind blowing help in managing the fundamental features of an extensive variety of specialist's office errands, for instance, the remedial, real, cash related, advantage getting ready, administrative and various distinctive functionalities, for instance, OPD and IPD the administrators, docrtor game plan, etc.\nAt Abstractsoftweb, our fashioners have unimaginable inclusion in working up the restorative administrations helpful programming which can perform distinctive exercises of remedial offices and moreover engages the customers to take the patient economics, plan the courses of action, keeping purposeful records of electronic prosperity information for the individual patient. All of these records are done in the propelled way and these can moreover be shared from different human administrations settings too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The California Community Colleges Online Education Initiative created a series of Online Readiness Tutorials that help prospective online students decide whether they should take classes in this mode. The tutorials have been available for a few years and I and other faculty at my college have used them within our online classes. Some of us tie assessments to those tutorials so that students earn some points by taking quizzes after viewing them.\nI have always wanted one place for my college's students to access those tutorials for a couple of reasons. One is to know which students have accessed them, making future research more authentic. As we ask about so many of the things we do for students, does this make a difference? The second reason is to prevent students from being asked to complete the same assignment for multiple classes. While repeating the same task over and over might reinforce learning, it will also be perceived as busywork.\nI solved the first of these problems by creating a self-enroll class in Canvas. It contains the readiness tutorials as well as a quiz so students can demonstrate they understand the content. I put the link to that class on my public-facing \"Distance Education\" web page, and I contact our counselors, librarians, and faculty teaching online classes and encourage them to refer prospective online students to that resource. I hope in a few years to have a large enough n (or is it N?) to answer that question about difference.\nThe second problem has been tougher. I am the instructor for the self-enroll class so *could* let the other online instructors know which of their students completed the quiz. While I like to be helpful, I also fear those requests would take up all my time. I would rather spend my time fighting to make readiness activities a requirement before taking an online class.\nMy solution for now is to link a Google Form in the self-enroll class. Thanks to the requirements options for Modules, the link to the form is not visible until students have passed the quiz. The form asks for a preferred name, collects the form-filler's email address, is limited to people who have user accounts in my college's Canvas instance, and emails the form results back to the form-filler. The form-filler can then forward the email as needed. Fortunately we are a Google Suite for Education client, and Google Forms created within that domain can meet all of these interests. There is also an option to limit to one response, which mitigates against the benefit of unauthorized collaboration among students. The form I designed has a congratulatory message and an image that indicates success, which means it looks nice if printed. Just don't call it a \"badge.\"\nHowever, students have other Google accounts and might already be logged in to those accounts before they access our Canvas instance. Google does not yet perfectly manage that situation, especially when a Google Suite file has its sharing settings limited to users in its domain. This is not so bad for Google Documents, Slides, or Sheets. If a user tries to access a domain-only file while logged in to another Google account, there is a handy button to \"Switch accounts.\" Unfortunately this option does not appear for Google Forms. These display \"Need permission\" and prompt the user to ask the owner of the form to grant access. This less-than-helpful message is a reason why I have not been an active user of Google Forms when it matters to record who fills out the form. I have used regular expressions to require a properly formatted username that matches our system's format, but that does not prevent one user from typing in another's username.\nSingle sign-on came to the rescue! We enabled single sign-on for our Canvas and Google systems among others, and now our students cannot log on to Canvas without concurrently being logged on to our Google Suite domain. So now I can set up my self-enroll class with a module requirement to manage access to a Google Form. As originally intended, that form only asks students to type a preferred name and sends them a congratulatory email that they can use to document their readiness to learn online.\nStudents should need no permission to learn.\nIn the end I feel confident about the Google Forms solution to my problems with documenting student readiness. The benefits of centralizing data collection on student access to readiness tutorials and the ability for students to share their completion of those tutorials with all instructors hopefully outweighs the risk of having to ask for permission. And this way I'll get a fair idea of how many instructors share direct Canvas links *and* use domain-only Google Forms in the same class.\nNice job. We use a lot of Google forms, sheets, and drive in our courses. It is great to see other institutions are doing with Google products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why Clean your Cooling Tower?\nIs Superheating a Potable Water System After Positive Legionella Results Effective?\nConEd Multifamily Gas Conversion Incentive Program \u2013 Must Apply Before May 18th!\nWhat are Auditors asking for when looking at Water Management Plans to reduce the risk of Legionella?\nWait\u2026 Water has a 2nd Liquid State?\nWhat Exactly is a British Thermal Unit (BTU)?\nWhat is a Load Letter and Why Is It Important?\nWhy Do You Need to Test For Legionella?\nDoes my Healthcare Facilities Accreditation require Legionella Testing?\nWhat is a Boiler (ECB) Violation and Why Did I Get One?\nIs Your Cooling Tower Compliant for 2017?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you very much Karen for an excellent tutorial. I didn't have any trouble following your instructions and your photos were very clear. I had a lot of fun making this little pillow and will be using this technique again. :) The design is a freebie from Stitchy Kitty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your vote counts here. Unions embody the true spirit of democracy, where people with common interests can come together and effect real, meaningful change where it counts. In a time when many Americans feel their voice is drowned out by the money and influence of special interests, every American still has the chance to exercise real democracy where it matters most: Where we earn our livelihoods.\nMany of the most immediate issues that impact our daily work and our quality of life are within the sphere of union activity. Important concerns like job security, wages, health insurance, and retirement are at the heart of achieving financial stability and improving one's quality of life, and union members get to have a say in these and much more.\nThe process is known as Collective Bargaining. Collective bargaining is when a group of workers negotiate their terms of employment with an employer through a union. These terms often include things like wages, benefits, hours, overtime, leave, health and safety, grievance procedures, and any other concerns that either side may wish to address. This negotiation results in a collective bargaining agreement, which serves as an enforceable labor contract between the union and the employer. Collective bargaining is a very effective way to solve \u2014 or prevent \u2014 workplace issues, and promote peace and productivity.\nA collective bargaining agreement is like an insurance policy for your job \u2014 not very different than the insurance policy you buy for your house, car, life, etc.. And it's your job, after all, that provides for those things; why not have a contract \u2014 an insurance policy \u2014 that protects your employment, as well?\nWe all have a stake in the economy and in the future. And most working people share the same basic concerns: they want job security; a voice in their workplace \u2014 deciding their wages, hours, and benefits; they want safe working conditions; and they want to be treated with dignity. Working people want straight talk \u2014 about terms and conditions, and about the future \u2014 so they can make decisions about their own employment. Workers deserve a way to speak up for their common interests, negotiate the terms of their employment, and, when necessary, to have a fair system for resolving problems.\nIn other words, American workers expect the principles and values of democracy and due process to carry through into their workplaces, not be placed on hold between 9 and 5.\nWorkers must always strike a balance between earning a fair wage and ensuring their employer remains profitable; between keeping their jobs and protecting their fringe benefits. But very often today's corporate executives earn many times more than their workers, top management receives bonuses for laying off workers, and companies announce record profits after cutting jobs or reducing employee wages. And what workers want most of all is mutual trust and respect, clear terms and conditions of employment, and a fair share of the prosperity they help create. In short, they want the chance to make the American Dream a reality.\nWorking people don't always have the voice they need in their workplace or the tools necessary to resolve these issues alone. That's why unions matter. That's where we come in.\nUnion members enjoy wages and benefits that surpass industry standards. Salary surveys demonstrate that union workers earn substantially more than their non-union peers (up to 30% more). And, with the experience of USWU's representatives, you are sure to make impressive gains at the negotiating table.\nUnion members consistently have better benefits than non-union workers. Union membership can provide you with additional benefits that will protect you and your family and enhance your quality of life.\nA union contract will provide you with a forum in which you can dispute company practices that adversely affect your job. Should a problem arise, your union will help you to resolve it fairly.\nUSWU members are guaranteed to have a say in their working conditions. The terms and conditions of your job will be clearly defined in your collective bargaining agreement.\nAre you always treated with the respect you deserve? Is the distribution of work always equal? Can you afford to be at the whim of your manager? If you answered \"no\" to any of these questions, a contract with USWU can improve your position.\nA union contract will defend you against unfair layoffs or discharges. USWU will be there to represent your interests in any situation in which your contract is being violated. This gives you the security of knowing that your job is protected. As a non-union worker, under the law you were an \"at will\" employee, which means you may be fired any time the employer chooses, without justified cause. With a union contract, your job is safe and you are protected from unfair termination.\nThe workers and the company share one very important thing in common: The success of the company.\nWithout a reliable, competent, and productive workforce, the company cannot thrive. Without a thriving company, there are no jobs or wages for the workers.\nWorkers and their families and friends are the consumers of the very goods and services the company provides, spending their hard-earned dollars on the products that profit the company.\nWhen workers have job security, a fair wage, healthcare, a secure future, etc. they have an investment in the success of their employer because it's their future as well.\nWhen employers and workers realize that what they have in common \u2014 the success of the company \u2014 is most important, they can come together and negotiate an agreement that best benefits everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It doesn't actually sound like fresh evidence at all. It sounds more like that which you were saying when I first joined, but said slightly differently. If this really was false evidence, it certainly wouldn't be privy only to members of a forum. I don't believe that ANY evidence of worth was withheld from the jury.\nYou can't even work that out, what happened to the missing 358 crime scene photographs? When did they get disclosed? Did the jury get to see them all secretly?\nAnd what about the missing 26th bullet case?\nOh, it's irrelevant according to you!\nWhy didn't cops do any ballistic tests when the investigation was one of four murders and a suicide?\nThey got rid of all the bodies before they did any ballistic tests!\nIt was a dishonest prosecution, they made out a false case in order to prosecute Jeremy Bamber for killing somebody that the cops killed themselves!\nThat has been the case ever since their sudden change of minds in chorus with Paul Harrison years back. From what I have heard the only people that ever seem to have taken them seriously since then were Maggie and Susan. Who have now come to their senses.\nWhy give these two the time of day Mike?\nWow! How sad is that? Two faced? This from the same guy who wrote the post below ..... Of course this was before he learned who to brown nose!\nI can't believe that so called ordinary people can't see what's been happening in this case!\nIt's a really serious situation that there was a 26th bullet fired, but only 25 bullet cases recovered!\nWhat happened to the 26th bullet case?\nSurely, the cops in Essex as thick and corrupt as they are, should have dealt with this anomaly!\nThere must have been a second gun used in these shootings, and the missing bullet case (the 26th one) was obviously whisked away from the scene in the other gun from which the 26th bullet had been fired!\nStep one: Frame him for murder.\n\"We discussed the implication of how this silencer could be in the gun cupboard with blood and paint on it. Obviously if it was being alleged that somebody had had a brainstorm and shot dead four people they would surely not have stopped to remove the silencer, put it back in the gun cupboard, go back upstairs and shoot herself dead. Contact was made with the police about the discovery of the blood and paint stained silencer.\"\n\"Ann concedes that the conversations regarding Mabel's will were conducted partly 'to stop Jeremy benefitting through aunt June' but the family were unanimous that Jeremy showed no interest in calling on his grandmother.\nTwo days after Mabel was told about the murders, Robert asked her solicitor to call at Vaulty. Six persons were present: Mabel, Robert, Pamela, Dr Ellis and Mr Peek and his secretary. Robert recalled that the meeting 'resulted in a solicitor from this firm and myself being made joint executors to her estate. The new form of the will was that Pam was made the sole beneficiary.' Why Jeremy's name didn't come up at the meeting, and particularly why Mabel apparently failed to ask about him at this point (given that Ann recollected she made no mention of Jeremy until December), is unclear.\"\nStep three: Get Ainsley on the case because Taff Jones had a brain.\n\"The 1986 internal review affirmed that 'it was known to the Boutflours that had Jeremy inherited the estate he intended to sell what he could, thereby disposing of what had been part of the Speakman family estate. In addition to this, he would also have sold an area of land which [Nevill] Bamber had purchased intending to sell it at a later date to Peter and Ann Eaton when they had sufficient funds.' The review noted that while it was not suggested that this interest had in any way influenced the Boutflours in their statements to the police, it was known to DCI Jones during the initial stages 'and may have been a factor which affected the level of credence he placed upon the information given by the relatives'\".\nStep Four? It would not suprise me if once they heard Julie and Jeremy broke up they took the first opportunity to con her into believing what they wanted people to believe also.\nOperation Poppycock more like! You give these people far too much power than they could possibly ave held!\nIf the silencer was used on the guns barrel when Sheila was shot dead on the main bedroom floor, and if the gun which fired the fatal shot was the anshuzt rifle, then how could the anshuzt rifle which was spotted leaning against the inside of a first floor box room window just before the raid team entered the premises, Jeremy Bamber have shot dead his sister, and staged her death scene on the bedroom floor shot twice in the neck any time sooner than the rifle in question being seen by Julia Jeapes resting at the box room window?\nHow did the only rifle found upstairs get moved from its position at the first floor box room window, and did it end up on the body of Sheila Caffell, and Jeremy Bamber be her killer?\nCops would have known if there was a silencer fitted to the end of the guns barrel at the time Sheila was shot dead on the bedroom floor!\nI believe DS Jones seized the Pargeter silencer (SBJ\/1) on the first morning of the police investigation (7th August 1985), and that there was no silencer at all fitted to the anshuzt rifle..\nWho contaminated the silencer with blood and who created those scratch marks?\nMy prime suspect = Ann Eaton.\nI am not the first to arrive at this conclusion and I won't be the last either.\nNot much point trying to distract the forum from the obvious Caroline. You need credibility in order to achieve that.\nIt's absolutely stupid to suggest that AE would have the thought and confidence to not only attempt to falsify evidence but succeed in fooling both EP and the lab. It would take someone with knowledge of such things.\nWhat credibility do you IMAGINE you have here or anywhere else for that matter? By the way, do tell us where AE might have managed to acquire enough of Sheila's blood to contaminate the silencer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modern wall light range infusing stone & metal features creating smooth clean lines.\nVersatile exterior die-cast aluminium wall light range.\nDecorative Range of Exterior Spot Wall Mounts in IP54.\nThe Mirabella Pandora collection of Crystal wall light mixes timeless designs with unique crystal and metal finished, to give our clients a collection that can adapt to any interior.\nThe Mirabella Pandora collection of Crystal wall lights mixes timeless designs with unique crystal and metal finished, to give our clients a collection that can adapt to any interior.\nDecorative modern exterior up\/down wall light.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy 1st Preview to the Crew and Cast of HAMLET! \u2022 #costumeshop #costumemaker #draper #csthamlet #costumerendering #costumedesigner #costumedesign \u2022 Costume Designer: @mssmickey \u2022 \u2022 #repost @chicagoshakes \u30fb\u30fb\u30fb A glimpse into the costume shop as our artisans select fabrics for Susan E. Mickey's design for Gertrude in #cstHAMLET. Performances begin April 17!\nExpanding on my previous rendering of the trans soldier: Category is: Cocktail Realness. They look quite dapper if I do say so myself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High-performance, diametrically magnetised Neodymium (N42) ring magnets. These diametrically magnetised magnets have their north and south pole on opposite curved sides, cutting the flat surface in half.\n6mm outer diameter x 3mm inner diameter x 1mm thick.\nEach magnet's north and south pole are on opposite curved sides.\nUnusually, diametrically magnetised magnets are not designed to hold the maximum possible weight for the size of the magnet but instead are used to provide rotational movement. A diametrically magnetised magnet is magnetised across its diameter so that the north pole in on one curved side and the south pole is on the opposite curved side. Within motors, diametrically magnetised magnets are commonly used on the end of shafts to provide drive. They can also be used to produce a magnetic rack and pinion system by using a line of north and south in alternating polarities. It is also possible to use diametrically magnetised magnets for reed switch applications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While photographing the volleyball game of my daughter (yes you can photograph sports with a manual RF!), I caught this little girl at the game, peeking through the glass . Her curiosity and facial expression moved me, so I quickly shot this picture. At an aperture of F 1.4, shooting these kind of pictures makes a good exercise in manual focusing as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfkka b/data_all_eng_slimpj/shuffled/split2/finalzzzbfkka
new file mode 100644
index 0000000000000000000000000000000000000000..e4ae3408e6fe513cebf370f1681002a670d8e43b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Talk of The Villages Florida \/ Talk of the Villages Forums \/ Computer questions \/ Local agency expert on text scams?\nLocal agency expert on text scams?\nThe sheriff was no help. Anyone know of a local agency with knowledge on the latest scams? In this case, using a text message exchange to assist in gaining access to someone's online account of some kind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carving in the wood is an art, but carving human and animal figures out of wood is a whole new ball game. Can you imagine a horse carved out of wood? We have a collection here of the 10 best carvings you will ever see. Get ready to blow your mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are overdue for a tribute to poet Philip Levine, who died February 14. He grew up in industrial Detroit during the Depression and worked factory jobs for Cadillac and Chevrolet. Levine didn't publish his first volume of verse until his mid-30s, but over time he became one of the country's most honored poets.\nAfter teaching for many years at California State University at Fresno, he became a visiting professor at Princeton, Brown, Columbia, New York University, and other prestigious schools. His worst students, he said, were Ivy Leaguers who were shocked to learn that their poems were no good. He preferred the working-class students of Fresno State, who seemed more receptive to the notion that a poem, like a car's transmission, sometimes needed to be rebuilt.\nAbout the following \"imaginary art lesson\" Levine explained that he placed it in 1942 because that was when, as a student at Detroit's Durfee High School, he discovered poetry. To hear him read the poem aloud, click here.\nin particular, \"what have I done?\"\nof chalk.\" Messieur Degas did not smile.\n\"what have I done\" he repeated.\nof an isosceles triangle.\" Degas mused.\nbe incorrect. \"It is possible,\"\nto separate the dark from the dark.\"\nknew this could go on forever.\nW. H. Auden Forges \"The Shield of Achilles\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul Edmondson\/Stocksy\/Design by Erik Mace for Yahoo Travel.\nSometimes your vacation getaway home is best with no frills.\nSure, a stay at an oceanfront 5-star resort sounds like the perfect place to get away from it all. But, what if you swapped all the fancy amenities for something a little simpler? \"In search of more authentic connection and a need to slow down, more and more people are choosing to spend their vacations down on the farm,\" says Anne Banas, Executive Editor of Smarter Travel. \"Experiential travel is a huge trend right now, and farm stays offer a chance to live like a local (and more simply) while being able to participate in the action. These ranch vacations break up your typical escape plans creating a unique blend of relaxation and first-time experiences for many of the guests. \"A lot of people from the city have never been in the quiet, pet a potbelly pig, or seen stars,\" says ranch owner Galiena Jacobs. \"It's truly therapy to be surrounded by nature and away from the city noise.\" Trade in stress at home for a crowing rooster at one of these unique ranches Airbnb has to offer.\nStay in a quaint covered wagon on a sheep farm in Montana.\nBack in the day on the Oregon Trail, amenities included dying oxen and dysentery. Luckily, this antique covered wagon only comes with a sky full of stars and grazing animals. Cozy up for only $99 a night (it sleeps two) and enjoy the peaceful surroundings of rolling pastures and mountains. Just minutes from Bozeman, you're conveniently located near humanity yet surrounded farm animals.\nThe best of both worlds: an adorable cottage on an island farm, just minutes from downtown Portland.\nWhat do you get when you combine organic farm-to-table meals, stunning views, a private casita, and a Redbone Coonhoud pooch? A modern farmhouse to rent for $79 a night, located just 15 minutes from downtown Portland on Sauvie Island. According to the listing, \"It is the perfect place to write, relax, read, take photos, celebrate, and\/or get away and recharge for a bit.\" Situated on five acres of a working farm, two cats, 10 chickens, six ducks, and two bunnies are also included.\nWe love everything about this rental. Especially the mini horse.\nWe've heard of the love shack, but this is the Love Nest, a.k.a., a private room for $125 a night that sits above a barn and has a deck where you can take in the desert surroundings. Part of the Genuine Draft Horse Ranch, you can get a taste of Western life with a nearby old-fashion saloon and horses roaming the property. \"Visitors really get a taste of the ranch life,\" says owner Jacobs. But, it's not just farm chores they have to offer; massages and facials are available upon request and you can rent the whole property out for a wedding.\nA cozy cabin on a farm in the Adirondacks sounds like the perfect escape.\nYou don't have to trek out to the wild West to get a sense of the farm life. Situated on a hundred acres overlooking the Hudson Valley in New York's Adirondack mountains, a cozy off-the-grid cabin is available for $90 a night. Along with access to a private dock for kayaking\/canoeing and country roads for cycling, you'll be able to watch the sunrise from bed. But, like the good ol' days, you'll have to rely on an unheated outhouse for your business.\nThis luxury farmhouse sleeps 12, making it the perfect spot for a weekend retreat with friends.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your lawn has seen better days, it may be time to replace it. While there is no hard and fast rule, generally if less than half of your grass is green and healthy, you may want to consider starting over by laying new sod.\nThe best time to lay sod is when the weather is cooler, but the grass is still growing. For us Floridians, this means the ideal time for sod replacement is during spring time.\nYou'll need to choose the right type of sod based on the type of soil you have and the climate where you live. Once you have picked your new grass, you'll need to prepare its new home to be as welcoming as possible.\nBegin by removing the old sod along with any weeds, rocks, and other debris. Add new soil and grade it. This is the ideal time to fill in any low spots with soil from any high spots. You can also take care of any drainage issues you may have had and make sure water will be flowing away from your home, driveway, or other structures. Remember to keep the level of the soil lower than sidewalks, walkways, or other landscaping features so once the new sod is laid down, it will be flush with them. Now is also the time to add amendments to the soil. Adding lime or sulfur can adjust the pH of your soil, if needed. Mixing in organic compost or manure will provide your new sod with nutrients and help it retain water.\nWith the soil prepared, lay down the new sod in straight rows, and stagger the pieces to make a brick pattern in the grass. Make sure the pieces of sod are pushed closely against each other and press them down so they are seated firmly on the soil. You can use a knife to cut the sod around landscaping features or sprinkler heads if you have them. Once the new sod is down, it needs to be watered to encourage the roots to grow down into the new soil. Two to three shorter watering over the course of the day will work better than one long soaking.\nWhile laying new sod can be a Do-It-Yourself project, it can also be labor intensive and time consuming. Consider contacting the professionals at Ludlow's to see how their knowledge and experience can take care of your sod replacement during spring time and keep your lawn looking full and beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Chicago, Illinois -- The Eastland Disaster Historical Society hosted the mock retrial fundraising event \"The Chicago Trial That Never Was\" last Thursday, June 18 at the Harold Washington Library Center. Proceeds from the fundraiser benefitted EDHS as the organization strives to support the historical research and preservation of the history of the Eastland Disaster \u2013 Chicago's greatest loss-of-life tragedy.\nAfter the Eastland Disaster killed 844 of its 2,500+ passengers on July 24, 1915, the criminal case was tried by a bench trial (no jury, only a judge). In August 1915, a grand jury indicted six men for their roles in the Disaster, including the president, vice-president and two administrators of the St. Joseph-Chicago Steamship Co., as well as the ship's captain and engineer. In February 1916, a federal judge in Grand Rapids, Michigan declared the defendants \"not guilty.\" Following the criminal trial, the court denied extradition of the defendants from Michigan to Illinois.\nThe mock retrial chose to charge three of the six men with involuntary manslaughter \u2014 the president, vice-president and secretary-treasurer of the St. Joseph-Chicago Steamship Co.\nDan Webb of Winston & Strawn LLP represented the defendants, Bob Clifford of Clifford Law Offices was the prosecutor, and Illinois Supreme Court Justice Anne M. Burke served as the judge. Larry Yellen from Fox 32 Chicago News was the evening's host. The event committee included Donna J. Pugh, partner at Foley & Lardner LLP, Frank J. Saibert, partner at Nixon Peabody LLP, John Lupton, Executive Director at the Illinois Supreme Court Historic Preservation Commission, and Tom Panoplos, Managing Director at Mesirow Financial. Draft trial scripts were written by professors from The John Marshall Law School, who met weekly since April to go over relevant documents and case facts. The facts from 1915 were then reopened using modern law.\nDan Webb objected to the letter stating that it was hearsay. He claimed that because his clients had purchased the ship after the letter was written, they never read it. He also added that the letter's author was not a witness and could not be cross-examined. The defense instead focused on seven \"undisputed facts inconsistent with recklessness,\" which included the U.S. government certifying the Eastland as \"seaworthy\" and safe to carry 2,500 passengers.\nEDHS is grateful for the event's sponsors, including lead sponsor Sidley Austin LLP. Other event sponsors included 88 Brand Partners, Clifford Law Offices, Winston & Strawn LLP, McCorkle Litigation Services, Drost Kivlahan McMahon & O'Connor LLC, Foley & Lardner LLP, Bohemian Lawyers Association of Chicago, River Roast, Teddy Waffles Video Production, Grant Staff, and Uline. EDHS also wishes to thank the event committee, case\/trial prep committee, actors, set and staging committee, event supporters, and volunteers.\nPhoto credit Rich Kolar Photography.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We value children's online privacy.\nIn fact, at this time, our educational apps do not use ANY third parties in any way.\nWe do not have access to your typing in the keyboard app!\nLinks to social media sharing services are per the users' discretion.\nAll pre-installed artwork is copyrighted by appikiko, LLC and cannot be sold without our explicit permission.\nUsers are free to use the pre-installed artwork for their own personal use, or for promotional purposes.\nAppikiko, LLC does not have permission to reproduce or sell users' artwork, unless permission is explicitly granted by the artist.\nAs parents and app makers, we are proud to be members of the KNOW What's Inside community (formerly known as, MOMs With Apps). We fully support the KNOW What's Inside campaign to inform parents and educators about mobile app privacy policies.\nPlease join us in supporting educational apps that follow these guidelines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you haven't yet heard about my Project Smile, let me give you the condensed version.\nLife is too short to let life's annoyances get the best of us or be depressed.\nThis project is about finding our smiles in spite of the hiccups and the heartaches.\nBut I don't want this to be a chore. I REPEAT, I don't want this to be a chore.\nI don't want you to feel like it's yet another thing on your list of things to do.\nJust mentally be present in your life. Be aware. Pay attention. Enjoy the moments. Don't take life for granted. And jot notes in your head. Or your smartphone, whichever works for you.\nThere are really no rules.\nIf you want to grab this button I'd love it, but no pressure.\nI have no set guidelines for how to do it. You can pick your level of commitment to the project. Daily. Weekly. Monthly. Every other month. Or not at all. You can share as many or as few smiles as you want. You choose.\nI will be doing it ONCE per month. I'll provide a linky the last day of every month, starting Sept 30th when I will share my smiles.\nI'd love for you to stop by and join me, but no pressure. If you can't get around to it, don't sweat it. Maybe I'll catch ya the next month.\nSmiles come in all forms. Some are outward while others just give you that warm fuzzy feeling. Even for just a moment.\nAs I've perused through my wordless wednesday linkups the last couple weeks, I can't tell you how happy it makes me to see so many of my Project Smile buttons scattered around the blogosphere, knowing that you have accepted the challenge and are actively seeking out your smiles.\nTy from Ty's Adventures shared her take on project smile when it first launched the end of August. She wrote a short, colorful blurb and her excitement is contagious. She even sent out a tweet.\nHelen from Lady Helen has an entire blog dedicated to finding joy in life. Her smile crusade post made my day.\nThanks ladies! Smiles are contagious.\nSo, please join Ty, Helen and I (and all the others) on this crusade. Let's make the world brighter, one smile at a time.\nI like this a lot....I was just thinking tonight on my run - Stacey you have so much to be happy about - focus focus focus on the good...and then I read your blog post...perfect to document all the smiles I get in my weeks....Thanks Alicia!\nso it doesn't have to be smiles, but things that bring us joy??\nVery alluring button...I am thinking about it.\nGuess what? I'm smiling RIGHT NOW!\nI think this is wonderful. With the craziness of back to school, I missed the original post. Ironic, as I needed to be reminded to take it easy! I would love to participate. And if I can ever figure out how to put things on my new blog, I would even take your button.\nI am looking forward to reading these other posts! Smiles are a good thing and remembering to think of all the good in our lives is really important especially on the hard days! Hope your weekend has been a great one!\nI'm still keeping track of the little things that make me happy everyday, and Lish, I've gotta tell ya...this has been a fantastic thing for me to do to brighten my mood :) Thanking for coming up with it and I can't wait to post every month!!\nOh my gosh! I loved this when I read about it in August and decided to join in and then completely forgot in amongst getting caught up in my current frustrations - so thank you so much for this post to remind me to look for the things that make me smile, I so need it currently!\nI'm looking forward to seeing what and how others share their \"smiles\".\nThank you for this! I saw this on a friend's site at Honesty (who is a follower of yours) and I did it today on my blog! It made me smile!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Song lyrics by Private Line. Check-out these awesome song lyrics by the artist, learn every word of your favourite song and sing it like Private Line. Get one of the 36 lyrics and watch the video by artist Private Line.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Berlin Open Access Conference Series will be held in North America for the first time this year! The conference will take place in Washington DC November 9-10, 2011.\n\"In 2003, a landmark meeting organized by the Max Planck Society and the European Cultural Heritage Online project brought together international experts with the aim of developing a new web-based research environment using the Open Access paradigm as a mechanism for having scientific knowledge and cultural heritage accessible worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhdbd b/data_all_eng_slimpj/shuffled/split2/finalzzzbhdbd
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index 0000000000000000000000000000000000000000..98aa175623e6f65877f509f375f7210e3384dd9d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"\"Plan to be without power for days\", Cooper said.\nWhere we're headed: It's too early to tell exactly how Florence will swing through the mid-Atlantic coast, but Jefferies breaks down which companies are in its potential path.\nThe NHC is warning it will bring the risk of \"life-threatening storm surge and rainfall\" and \"catastrophic flash flooding and significant river flooding\" to parts of the mid-Atlantic states.\nPeople fleeing North and SC clogged coastal highways early Wednesday as Hurricane Florence, a monster Category 4 storm, bore down on the United States east coast for a direct hit in a low-lying region dense with beachfront vacation homes.\nThe storm is still on track to make landfall in the coastal carolina region, but is starting to drift south. Officials say it is possible SC will bear the brunt of Hurricane Florence.\nA unsafe storm surge will come before the massive storm makes landfall late Friday or early Saturday.\nAs the hurricane became larger and better organized Tuesday afternoon, mandatory evacuations were ordered for parts of South Carolina, North Carolina and Virginia. Cars and trucks full of people and belongings streamed inland. And there will be flooding in the inland areas as well. It is poised to slow to a crawl there and then drift to the southwest, unloading disastrous amounts of rain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apical - Apex, the uppermost part, tip.\nAreole- A small specialized area where spines are produced.\nAxil - Angle at which a leaf joins its stem.\nClavate - Club-like or club shaped.\nDiurnal - During the daytime.\nIndefinite - Slight, barely noticeable.\nGlochida - A cluster of many small bristles, usually mound shaped.\nHydroponic - Growing in a water\/nutrient solution without soil.\nRamified - To divide into branches.\nStomates- An epidermal pore for gas exchange.\nUmbricated - Having a cup-like depression.\nVegetative reproduction - Propagation by cuttings, grafts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hepelwa, Aloyce. 2014. \"Dynamics of Watershed Ecosystem Values and Sustainability: An Integrated Assessment Approach .\" International Journal of Ecosystem 4 (2): 43-52.\nThis is a new approach proposed to be used to gauge the sustainability of ecosystems goods and services by taking into account the economic values to the local community in place. A conservation idea as implied in the concept of payment for environmental services (PES) through the willingness to pay (WTP) is used to define the sustainability indicators. We employed the DPSIR - Driving forces - Pressure \u2013 State \u2013 Impact - Response and the Total Economic Value (TEV) frameworks in the process of establishing the VBSI. A multi stage regression analysis was employed to establish environmental, economic and demographic attributes influencing the willingness to pay. The study estimated the marginal effect of each variable on the ecosystem value and these were aggregated to establish the VBSI. We defined and classified watershed resource sustainability indicators based on predictors of ecosystem values represented by the WTP premium in the study area. We argue that extending the horizon to include the economic values when instituting the sustainable watershed resource management is paramount. We estimated the VBSI (=0.25), which was below the average VBSI (=0.5) threshold for the sustainable state. This result indicated that, the watershed ecosystem in the study area not sustainable. Deliberate efforts were needed to address the water resource management in the area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Construction of a new tyne tunnel crossing was completed in 2011 to help alleviate congestion at the original Tyne Tunnel which was originally opened in 1967.\nIn total over 3000 cu-m of Expanded polystyrene was in the construction of the required void formers.\nFact \u2013 Mr Henry Smith (C.E.O) Father worked on the first tunnel back in 1951 with Henry Smith and Son Darren Smith following suit in 2011 on Tyne tunnel 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About Action Heating & Cooling, Inc.\nAction Heating and Cooling was established in 1991 with the goal of providing \"good old fashion\" service for people with heating and\/or cooling needs. To us this means treating everyone equally with the best service and products possible. Since then, our customer list has grown remarkably, primarily by word of mouth, and taking care of our customers comfort needs. By utilizing over 41 years of experience, we are able to provide honest information, fair pricing, and top notch equipment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhhmu b/data_all_eng_slimpj/shuffled/split2/finalzzzbhhmu
new file mode 100644
index 0000000000000000000000000000000000000000..52e52e7080d9b9da5d616ab71ab855646d957492
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+{"text":"How funny is that, really? Or an exchange such as \"LOL theres only one slice left\" \/ \"don't deprive me LOL\" \u2014 text exchanges often drip with these LOL's the way normal writing drips with commas. Let's face it \u2014 no mentally composed human being spend his or her entire life immersed in ceaseless hilarity. The LOLs must mean something else.\n[LOLs] signal basic empathy between texters. What began as signifying laughter morphed into easing tension and creating a sense of equality\u2026 That is, \"LOL\" no longer \"means\" anything. Rather, it \"does something\" \u2014 conveying an attitude \u2014 just as the ending \"-ed\" doesn't \"mean\" anything but conveys past tense. LOL is, of all things, grammar.\nWell put. He concludes: All indications are that America's youth are doing it quite well. Texting is not the mangling of language \u2014 it's the birth of a new one.\nIf anything, then, texting will keep Lexicide going for years to come.\nThis entry was posted in commentary, sightings on April 30, 2013 by mark.\nLexicide is accepting comments once again. We had been receiving an enormity of spam through commenting, so we turned off that feature to encourage a minimalist amount of junk we would have to delete. Alas, we realized Lexicide is not about being politically correct, but about serving you, our stakeholders. So comment away. We look forward to answering your questions and telling you you're wrong.\nThis entry was posted in Uncategorized on April 22, 2013 by mark.\nCorporate Americans love philosophy. We try to make our software agnostic, our designs minimalist and our business plans conform to a schema. Bookshelves and Kindles must be bursting with Russell, Van der Rohe and Kant! Cogito ergo vendo!\nAdd Jeremy Bentham to that library, because managers love utilitarianism! You know, the philosophy that promotes policies proffering \"the greatest good for the greatest number of people.\"* What? That's not what you meant when you demanded the website have a \"utilitarian look?\" You meant you wanted it to be functional but not florid, easy to use but not necessarily pretty?\nOh, pardonez-moi! And here I thought you were asking for a site that offered navigation and interface that appealed to the greatest number of potential visitors, which is actually a noble goal. (Except that we fired our user experience architect, figuring the sales manager could do her job.) No no. What you meant to say was \"I want the site to be minimalistic.\" Ha. Just kidding. What you want is to fire your creative director and get your UX expert back. Then you'll have what you want. Won't be pretty, though, especially with what you'll have to pay her.\nThis entry was posted in words on April 9, 2013 by mark.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cut from 100% Habotai silk, our featherweight Cecilia cover-up dress walks, drapes, and feels like a million, making it perfect for those who must have a stylish quick-change in the beach bag for impromptu invites or essential errands. Coral embroidery in a floral motif frames the neck, dropping the waist to the hip with a dip dye of ocean color around the gently gathered skirt. Feminine and pretty, your beach bag desperately wants this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cotswold Airport (formerly Kemble Airfield), is a private airport located near the village of Kemble. It is used for a variety of aviation purposes including aircraft storage and dismantling, flying training, flying clubs, private aviation, aircraft maintenance, air shows and as a base for historic aircraft.\nIt is also used for a range of non-aviation activities including film and TV shoots, corporate events, Formula 1 car testing and industrial use.\nClick here for the Kemble Out of Hours Indemnity Form.\nOaksey Park is an unlicensed, grass airfield operated by Austen Associates, and is PPR by phone to the airfield manager.\nThe airfield is situated approximately 3NM SE of Kemble, just outside the Kemble ATZ.\nNo circuit flying is permitted because of local noise sensitive areas, except as dictated by the requirements of good airmanship.\nAn AGCS is established on frequency 132.225, although this is not always manned.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every day tens of millions, perhaps even hundreds of millions, of containers are filled with viscous products from a variety of industries, ranging from automotive, to food, cosmetics, pharmaceuticals, and even aerospace. You may have never thought about how your shampoo was placed into its container in such an accurate manner that the weight of each bottle is almost always exact, or how drug dosages are divided in such a precise way, at rapid speeds, when even the smallest error could have major consequences.\nThis is all done through the use of sophisticated pump and dosing technology. One of the leaders within this field is ViscoTec. A company whose roots stem back to 1980, Germany-based ViscoTec considers themselves a premium brand for dosing systems and components. They cater to numerous industries and pride themselves on the reliability, efficiency, and precision of their products.\nIf you think about it, dosing technology has a lot in common with the components used within FFF\/FDM 3D printers. Just like an extruder on a printer slowly releases a specific amount of molten plastic, dosing technology does the same basic thing with viscous materials, only that the process has to be even more precise. Realizing the overlap in the technologies, ViscoTec has announced that they are entering the 3D printing space.\nAs you can see from the video and images posted throughout this article, ViscoTec seemingly has managed incredibly precise deposition of viscous materials such as silicone, UV-curing adhesives, and two component adhesives, among others. We have seen several different paste extruding machines come to market recently; however, none of them I have seen had the accuracy, precision, and fine nozzle diameters which are obviously present within ViscoTec's technology, judging from the video below.\nThe company states that they feel they possess several advantages within the 3D printing industry because of their vast experience working with dosing technologies. They are currently able to print at a layer thickness of 0.2mm, at a speed as as fast as 1 ml\/minute. This of course depends on the thickness of the material being extruded, but the company claims that they can even handle some of the thicker materials out there with ease.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published: Nov. 18, 2014 at 04:48 p.m.\nUpdated: Nov. 18, 2014 at 06:22 p.m.\nThis was a banner week for quitting and rattling cages in the NFL. LeGarrette Blount pouted his way off the field Monday night after not getting any carries in Tennessee, guaranteeing that he would never touch the ball for Pittsburgh again.\nNiners linebacker Ahmad Brooks essentially quit on his team, taking off his cleats and cutting his ankle tape because he didn't want to be part of a rotation in San Francisco. And Marshawn Lynch called out the Seattle front office in comments to NFL Media columnist Michael Silver, a rare case in which Lynch was about more than just action.\nBrooks and Lynch aren't gong anywhere this season because NFL teams have a sliding scale of transaction justice. Brooks and Lynch are too valuable to release, at least until February.\nAndy Reid's Coach of the Year candidacy: OK, so every coach is battling for second place behind Bruce Arians. But the job Reid is doing should not be ignored. The 7-3 Chiefs beat the defending champs in a game in which they lost the turnover battle (2-0) and threw just 16 passes, with just 18 yards gained from their wide receivers.\nIt's strange to see a Reid-led team win with the running game, including clever uses of De'Anthony Thomas. Reid even made a smart, game-changing challenge Sunday against Seattle. The end of days are near.\nNew England's chances to win the No. 1 seed: They have beaten the other three AFC division leaders by 70 points combined. That's hard to believe.\nChris Borland: He's gone from a total unknown to a legitimate Defensive Rookie of the Year candidate in the span of a month. He is making as big an impact right now as any linebacker in the league. The 49ers have found foundation pieces in Borland and outside linebacker Aaron Lynch.\nSan Diego's clock operator: Oakland wasn't going to get their first victory in nearly a year if they were given two more seconds on Sunday, but give the Chargers' clock operator credit for making sure.\nIndianapolis' Confetti Boss: Premature confetti on Sunday night spawned a lot of great jokes on Twitter. If Peyton Manning was still with the Colts, this guy (or girl) would have been chastised publicly after the game.\nSeattle Seahawks' urgency: I never expected the Seahawks' season to be on the line in Week 12, but that's how Sunday's game against Arizona feels. A loss would erase any chance to win the division. And if the Seahawks can't win tough games at home anymore, how are the going to win in Arizona, San Francisco and Philadelphia in games still on the schedule?\nCleveland Browns' playoff chances: Week 12 was supposed to be the turning point for this team. Get Josh Gordon back, and things were going to look up for this team. Instead, the Browns are coming off an ugly home loss against Houston with devastating injuries to underrated linebackers Karlos Dansby and Jabaal Sheard.\nAnyone that bought an RGIII Redskins jersey: That thing might be out of date as early as next season. It would not be a huge surprise if Griffin is sent elsewhere, unless owner Dan Snyder wants to find a different coach to work with him.\nSan Diego Chargers: It's rare for a team to make this list after a win, but that's what the Chargers get for producing one of the most unwatchable games of the season coming off their bye.\nSaints' home invincibility: All of those winning streaks that dip back into previous years sound impressive -- until you realize this Saints team is a lot more like the 7-9 squad of 2012 that Sean Payton missed out on. (Yes, that season still counts.) The reality is that this Saints squad needed a miracle to beat Tampa at home and now could be facing a three-game home losing streak with Baltimore coming to town.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CBS Radio has launched CBSPhilly.com, a website consolidating News KYW-A, Talk WPHT-A (1210 AM, The Big Talker), and Sports WIP-A with O&O KYW-TV (CBS 3). The site itself features local news, sports and video.\nHowever, accessing the station's online simulcasts through a large \"Listen Live\" link brings users away from the Philadelphia site and to Radio.com \u2014 CBS Interactive's platform for CBS Radio stations, Last.fm, AOL Radio and other streams. Users can there listen to KYW 1060, WIP 610 or 1210AM, or easily browse other simulcasts and pureplay streams. This should help introduce Philadelphia listeners to CBS Radio's wide range of Internet radio offerings \u2014 a smart move on the company's part.\nFormer Ando Media CEO Bob Maccini, Aritaur Communications CEO Joseph Gallager and Media Services Group Director Stephan Sloan have formed Angel Street Capital LLC. The new company will provide consulting and invest capital for promising digital media start-ups and radio groups. The company will reportedly look to make investments of $50-500k, though is open to raising funds from outside sources as well. RBR has more coverage here.\nThe upcoming RAIN Summit East (which kicks off in Washington D.C. in less than a week!) will feature the exclusive debut of Coleman Insights' \"Successful Streaming Audio Strategies\" presentation \u2014 a major study of the streaming audio marketplace. The study \"will focus on areas such as branding, consumer loyalty and product attributes and how those findings vary by consumer segments and between Internet-only and AM\/FM radio streams,\" said Coleman Insights VP Sam Milkman.\nThe comparison of Internet-only and AM\/FM streams should be particularly fascinating in the wake of recent data from Bridge Ratings which showed consumers' dissatisfaction with simulcasts (RAIN coverage here). More information about RAIN Summit East can be found here.\niAds, Apple's advertising platform for iPhones, iPads and other iOS devices, is earning mixed reactions from app developers advertising their products with the service. Apple says 75% of developers who advertise through iAds have either renewed their trial or increased their spending, indicating some level of success. And nearly a dozen developers The Los Angeles Times spoke with were enthusiastic about iAds.\nHowever, some developers like David Smith who used iAds to advertise his app, found the system to be \"disappointing\" and \"ineffective.\" You can read more on the story from The L.A. Times here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disney will sell three different types of streaming services, according to CEO Bob Iger: A family-friendly movie and TV subscription, a sports streaming platform and a more adult-focused Hulu.\nWith the controlling share of Hulu, Disney hopes to \"direct\" the company and invest in more content, Iger said.\nDisney wants to win the streaming wars by having three distinct products: A family-friendly movie and TV service, a sports streaming platform and an adult-focused Hulu.\n\"We think it's quite a compelling opportunity for the company, and we think it's really compelling from a consumer perspective,\" Disney CEO Bob Iger told CNBC's David Faber on \"Squawk on the Street.\"\nDisney will purchase many parts of Twenty-First Century Fox for $52.4 billion in stock in a deal announced Thursday. Part of the deal includes Fox's movie studios, networks Nat Geo and FX, Asian pay-TV operator Star TV, and stakes in Sky, Endemol Shine Group and Hulu, as well as several regional sports networks. It will also give it more content to launch its direct-to-consumer services, as it prepares to remove its content from Netflix by 2019.\nDisney's streaming plans involve launching an over-the-top subscription plan for Disney, Pixar, Marvel and Lucasfilm content in 2019. It will start a sports-focused ESPN streaming service next year, and hopes to make Hulu it's more \"adult-oriented product.\"\nThe Disney-Fox deal will give Disney a 60 percent stake of Hulu. The company things its an \"interesting\" opportunity to grow the platform alongside the other minority owners which include Comcast and Time Warner, Iger said.\n\"With that we'll have the ability to direct Hulu in ways that we haven't been able to, as essentially equal partners,\" Iger said. \"But we'll also be able to infuse Hulu with even more content. We will actually invest in content from both entities for Hulu.\"\nDisclosure: CNBC parent company NBCUniversal is owned by Comcast.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As we look at our present nihilistic culture, malnourished in the absence of fides et ratio and living only on a meager diet of panem et circenses, it is hard to perceive any sign of true progress, unless we see progress as synonymous with suicide. Whether the homicide, genocide, and infanticide of secular fundamentalism can be seen as its own ultimate suicide, there is no need for the rest of us to follow such self-destructive notions of \"progress\". On the contrary, as Chesterton reminds us, \"true progress consists in looking for a place where we can stop\", which, in the dark ages in which we now live, means a place where we can stop the rot.\n\"The foundation of the true doctrine of progress is that all things tend to get worse,\" wrote Chesterton. \"Man must perpetually interfere to resist a natural degeneration; if man does not reform a thing Nature will deform it. He must always be altering the thing even in order to keep it the same.\" Chesterton used the example of a gatepost to illustrate this point, stating that we cannot preserve a gatepost by leaving it alone. If we leave it alone we will be leaving it to rot. If we wish to preserve the gatepost we have to be continually painting it. If this principle of conservation is the \"foundation of the true doctrine of progress\" it means that conservatism (properly understood) is progressive! It seems that the laws of \"natural degeneration\" imply that there are only two possible paths upon which humanity can progress, the path of conservatism (conservationism), or the path of decay.\nIt is noteworthy that Chesterton builds the \"true doctrine of progress\" on the principle of entropy, thereby uniting a fundamental law of physics with a fundamental law of metaphysics. Just as the laws of thermodynamics dictate that matter and energy ultimately \"decay\" towards an ultimate state of inertia, so the laws of Christian metaphysics dictate that matter and energy ultimately \"decay\" towards ultimate inertia through the effects of evil (sin). Just as entropy can only be overcome physically through the input of material energy from beyond the closed system, so it can only be overcome metaphysically through the input of spiritual energy (grace) from beyond the closed system. Chesterton, as a metaphysician of the first order, connects the physical and metaphysical with congruous precision.\nThis spiritual entropy, which theologians prefer to call Original Sin, is a via media between the fallacy of progressive optimism, otherwise known as the ascent of man, and the opposing fallacy of nihilistic pessimism, which could be called man's descent. \"One class of popular writers are perpetually telling us that the world has always been growing better and better,\" wrote Chesterton; \"others, rather less popular, that it has for some time been growing steadily worse. Personally, I cannot understand anybody thinking it has ever grown steadily anything.\" Reality is not in the process of becoming, whether we wish to perceive it as becoming better or worse, but, on the contrary, it simply is. The laws that govern reality are unchanging and unchangeable. Original Sin is as much an integral and integrated part of the cosmos as are the laws of thermodynamics and, in consequence, it cannot be extricated or repaired by any man-made progress. We cannot progress beyond what we are, i.e. fallen beings made in the image of God, and no technological progress or ideological innovation is going to alter the fact. Plus \u00e7a change, plus c'est la m\u00eame chose!\nOne of the most astonishing paradoxes at the heart of reality is that change itself is subject to unchangeable laws. Mutability is an immutable law of the cosmos! Even time does not change, in the sense that it is settled. A clock can only measure the passage of time because time itself does not change. If time altered its speed, if it became faster or slower, a clock would be useless as a means of measuring it. A clock only works because it measures something unchanging.\nIt seems, therefore, that progress can be perceived in three distinct ways. It can be seen by the optimistic materialist as an inexorable ascent towards a golden age in the future; it can be seen by the pessimistic nihilist as an inevitable descent into the primeval soup of deconstructed meaning; and it can be seen by the Christian realist as the recurrence in every age of the unchanging struggle between spiritual entropy (sin) and spiritual energy (virtue energized by grace). We should note, by the way, that the juxtaposition of \"Christian\" and \"realist\" is not an act of supercilious and patronizing self-justification on the part of the author but is the technical phrase applied by philosophy to the orthodox position of Christian philosophers such as St. Augustine and St. Thomas Aquinas as distinct from the nominalism and de facto relativism of William of Ockham and his followers. In the final analysis, the choice is between the delirious daydreams of the optimist, the nihilistic nightmares of the nihilist, or the perennial realism of the Christian.\n\"Faith and Reason\". Christian philosophy and orthodoxy are rooted in the inextricable connection between faith and reason. Irrational faith is ultimately heterodox, and faithless reason is ultimately irrational.\n\"Bread and Circuses\". Juvenal, the Roman poet and satirist, lamented that the Romans, once rulers of the world, had become so decadent that they only cared for hand-outs (bread) and spectacles (circuses). In such a period of decadence, the Roman leaders could keep the allegiance of the masses merely by offering them an endless supply of bread and circuses. The situation is much the same today. The welfare state supplies the hand-outs of \"bread\", and the latest gadgets of technology supply the mindless distractions (circuses) of \"virtual reality\".\n\"The middle way\". The moderate course between two extremes is recommended in Latin proverbs as the path of wisdom.\nThis is true regardless of the theory of relativity. Time may be said to move more slowly in higher gravitational fields but it does so in accordance with unchanging laws connecting it consistently and inextricably to gravitation. Men may be said to fly in the absence of gravity but to argue from this \"relative\" truth that men do not walk and jump in accordance with unchanging gravitational laws would be absurd. Gravitational force may change but gravitational laws do not.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With the start of March's round of #30Lists just a few days away, I decided it was time to get my base together.\nThis year I decided to go with a book that I made and bound myself with my Cinch Binding Tool.\nTo create the base for my cover I used some gelli prints that I had previously pulled onto deli paper. I really like the translucent quality you get with deli paper.\nFor the inside of my book I decided to use tags and I pulled easy, fast prints off the gelli plate with them. My goal was to keep the base really basic so that I have the room that I need to list. I can always add more details and layers as I go along each day, this was just so that I'm not staring at a blank tag each morning.\nNext I bound my book with the Cinch, flipping the tags the opposite way halfway through the book, so that the book would lay flat when closed.\nTo give you an idea how I might add additional artistic elements to my tags on any given day, I added a few circles and some stamping on the tags for days 1 & 2. My goal is to keep it all really easy and simple.\nI hope you're as excited to begin listing as I am! If you still aren't sure or if you just haven't registered, you can do so HERE. This really is a fun, fast and easy way to do something creative every day, in just a few short minutes. Hope you'll join me!\nAs always, thanks for reading and I wish you a beautiful week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I know you guys posted on Gamedev.net, but you guys should also try asking for some help on LinkedIn in some of the GameDev groups. People are usually looking for projects there, and there are quite a few members as well.\nI'm glad to hear this game is still alive! I've been gone since like, 07 or 08.\nA warrior should not say something fainthearted even casually. He should set his mind to this beforehand. Even in trifling matters the depths of one's heart can be seen.\nLocation: In a place so much less simple than Feudal Japan.\nReally happy to see that things are getting worked on more actively! I still want to play a Samurai MMO, and this is the only option for me, and it seems the best option, at that!\nReally appreciate the update even if it's not the best of news.\nThank you for the update Dex. I just came across a new Idea to help get you guys some money and get some interest in the site again. I am now an editor\/talent scout for Resiliant Publishing, A company who's purpose it to give people a chance and a helping hand to break into the literary world. I'll post a new thread with contat info for me for anyone interested. And i will offer kickbacks in the form of donations to T:law for any author we pick up or publish anything that the are refered from this site or another T:law member.\nThats great news and im glad you want to help out the EoE team, but did you need to post it 4 time?\nBeing enthusiastic is awalys a good thing, and im sure every one will see the posts who need to see them. Plus congratz on the job hope you will work hard and have fun at it.\nSo as an update I have some better news this week, I promise.\nI am a little late to actually make it a week, but o well so sue me! I have finally finished tweaking the basics of the engine for screenshots, all that is left is to get some gui images made up (I may make them so they may be incredibly ugly, but will still show the functionality we currently have), wait until our temporary character model is finished animating and implement that, and put some content in to show off.\nScreenshots are coming as soon as I can get them, I promise this is a priority.\nScreenshots are coming, I am excited! It seems like since you all switched programs EoE is running 300x more efficient lol!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As it turns out, cycling is beginning to take over North America as an extra-curricular athletic hobby.\nAccording to the National Sporting Goods Association the number of Americans who ride bicycles is greater than all those who ski, golf, and play tennis combined. According to Outdoor Foundation there were 1.2 billion cycling outings in America in 2011 second only to jogging and running.\nThis got me thinking about one of my favourite personal passions (cycling) and our corporate organizations.\nMaybe if we were to act like a peloton in our organizations, we might see higher levels of employee engagement.\nIn cycling speak, it's what a pack of cyclists are called when they ride together. Check out the photo to the right for an example.\nA peloton is a massive group of riders who ultimately work together \u2014 as a team \u2014 to move from one distance to another. Take away competitive cycling competitions for a moment (eg. Giro d'Italia or Tour de France) and think about amateur cyclists going out for weekend rides or events like the GranFondo between Vancouver and Whistler.\nThese women and men ride together as a team but what happens along the journey?\nIt would be nice if our organizations thought like and acted similar to a cycling peloton.\nIt sure might assist efforts to drive up employee engagement, creating a culture of sharing and connected leadership.\nEvery peloton is different. General constraints hold the thing together (if it's not held together, and it disintegrates, it's arguably not a peloton anymore) \u2013 like the total number of riders, the conditions, the ability of the riders \u2013 and of course, the broader context of the purpose of the peloton to start with.\nA pack of cyclists out for a Sunday club ride, may exhibit some of the more \"friendly\" attributes that you've got listed above, Dan. But the peloton that I spent time towards the peak of my abilities, was about as Darwinian as it gets. We were in competition with each other. We were trying to inflict pain and suffering on each other, through tactics and strategies, each of us playing a role as an individual, sometimes as a member of a team, within the context of a race. That race may last one day, multiple days, it may part of a race series over a number of events that season.\nSo for me, that's where the metaphor breaks down a bit. The temporary truce of the breakaway companions, a small group that has escaped off the front in search of glory, TV coverage, or just as part of their team duties and responsibilities, is a far cry often from the camaraderie of a group of peers or club members out for a weekend jaunt.\nAnd drafting may be seen as freeloading, not doing one's work \u2013 at which point the rules of the peloton are broken. It's the prisoner's dilemma, the tragedy of the commons and Machiavelli, all rolled into one giant chess game at 50kmh, with 100 guys playing at the same time.\nComplex system? Absolutely. Passes the test of agents, interactions, non-linearity, \"system level behaviours\" that emerge, and some kind of resilience \/ robustness \/ ability to adapt to external forces (changing conditions).\nIs it the right metaphor to model your org after? It's a beautiful thing, the peloton, with its flow and motion, speed and seemingly effortless coordination \u2013 the closest thing I've come to understand the flocking behaviours of fish and birds \u2013 and it's exhilarating to be inside.\nBut it's also a nasty place, filled with rivalries, ego's, doping, and fierce competition. One that requires the ability at the highest levels of the sport, to commit oneself fully and completely to the goal of being able to suffer more than your competitors.\nOne where sometimes, nice guys don't finish first.\nMy experience is based on 'friendly pelotons' with friends, add-ons, Italy\/Spain\/France trips, etc.\nAnd your comments certainly help others realize that not all pelotons are as I depicted. Thank you.\nAnd this is why I shouldn't leave blog post comments at 11pm at night after a long day\u2026 Apparently I missed that bit about ignoring the competitive peloton part.\nLooking forward to getting tucked into Flat Army.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Joyce Laurie Simmons, age 77, of Middletown, Delaware, formerly of Dover, passed away peacefully on Saturday, April 21, 2018, surrounded by her family. Born in Morgantown, West Virginia on October 7, 1940, she was a daughter of the late Frank and Mary (Lovio) Laurie. Joyce dedicated 35 years of service as an insurance agent with State Farm and was the first female agent to be employed at the Eastern Seaboard location. Active in her community and a friend to many, Joyce was a valued member of many organizations, including the MOT Rotary Club, where she was a two time past president; served as a board of director for the MOT Jean Birch Senior Center; volunteered for the Women's Club of Odessa; and a member of the St. Paul's United Methodist Church in Odessa. She was a major Rotary Foundation contributor, a proud Paul Harris Award recipient and in 2017 received the Outstanding Rotarian of the Year award. Joyce was the driving force and main inspiration for the MOT Rotary Can-Do playground that will open in a few weeks at the Charles Price Memorial Park in Middletown. More than anything, she was a grandmother extraordinaire to her four grandchildren. In addition to her parents, Joyce was preceded in death by her husband, Clyde Simmons. She is survived by her children, Lisa Shepherd-Wilson (Mike) of Ashburn, Va., Charles Collins Jr. of Horsham, Pa. and Curtis Collins (Kate) of Middletown; brother, David Laurie of Morgantown, W.Va.; sister, Sylvia Dial of Morgantown, W.Va.; grandchildren, Collin, Victoria and Morgan Shepherd and Chase Collins; and many nieces, nephews and dear friends. Family and friends may gather from 11 a.m. until 1 p.m. on Saturday, May 5, 2018 at Odessa Fire Company Memorial Hall, 304 Main St., Odessa, where a Celebration of Life will begin at 1 p.m. Interment will be held privately in Arlington National Cemetery in Arlington, Va. In lieu of flowers, contributions may be made in Joyce's memory to Blanchette Rockefeller Neurosciences Institute, One Waterfront Place, 7th Floor, P.O. Box 1650, Morgantown, WV 26507. To sign the guest book, visit spicermullikin.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wishing You A Very Happy Diwali Full Of Love & Mithai From Team So Delhi!\nA festival that's celebrated with the most amount of zest and zeal in India, no matter which caste, creed or city you belong to, Diwali is something everyone celebrates! From the smallest thela wala to the fanciest houses, all have a little candle or light of hope.\nThus, our happy little bunch here at So Delhi would like to wish all our readers a very happy and safe Diwali! This year, let's give back to those who need it and make it a 'Khushiyo Ki Diwali' for as many people as possible. Put up some fairy lights, distribute some mithai, make a rangoli and light up a diya, because hey, it's celebration time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SOUTHEAST ASIA - is a sub region of Asia, consisting of the countries that are geographically south of China, east of India, west of New Guinea and north of Australia. The region lies on the intersection of geological plates, with heavy seismic and volcanic activity. Southeast Asia consists of two geographic regions: Mainland Southeast Asia, also known as Indochina, comprises Cambodia, Laos, Burma (Myanmar), Thailand, Vietnam and Peninsular Malaysia, and Maritime Southeast Asia, comprises Brunei, East Malaysia, East Timor, Indonesia, Philippines, Christmas Island, and Singapore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parkinson's Victoria has a resource library at its office in Canterbury Road, Surrey Hills.\nThis library contains books, DVDs, and CDs with content specifically about Parkinson's and many aspects relevant to living with, or caring for someone with Parkinson's.\nAll current members of Parkinson's Victoria can borrow resources from this library for a period of one month at a time, for free. Resources can be picked up from the office, or posted out to you \u2013 the only cost you will incur is returning the resource to us (via post or in person) at the end of the lending period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Charlie Johnson is the director of the Community Integration for Training and Employment (CITE) program at Toledo GROW's Botanical Garden in Toledo, OH. He developed the program to employ adjudicated youth in urban agriculture, helping them to gain the skills they need to get jobs when they finish the program. For the past 10 years, Charlie has been involved in implementing an urban agriculture training center, introduced small chickens and small livestock to Toledo, and helped start dozens of community gardens. This program has been sponsored by Heritage Foods USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the past five years and seven months, I worked at Crowd Favorite. However, I left the organization effectively last Friday, September 2nd, 2016, as a full-time employee. I move forward into new opportunities and possibilities, leaving behind a job I loved at a company that, for many years, felt like my home, and I do so with no specific destination in mind, ready to begin a journey into the unknown. To understand why I made this decision, I'd like to share my history at the company and my thoughts on my departure.\nAfter the interview, as I waited for the bus, I wrote an email to Alex again. I thanked him for the interview, and told him that his team impressed me enough that I wanted to lower my requested salary. I did not believe, after my interview, that the skills I brought to the table were worth the amount I initially believed, and I wanted to impress upon him that the team mattered more to me than a meager difference in my asking rate. I heard back from him quickly, the same month, with an offer of employment that I accepted.\nOn my first day, Alex told me he wanted unit tests written in PHP Unit for a project the team wrote from the ground up. He could not get PHP Unit to work, nor could the other developers, and he needed to leave for a conference that week, so I had to do it all on my own. That set the tone for how Crowd Favorite positioned itself as an employer. They threw developers into the deep end, and the developers either sank or swam, but did so on their own skills and merits. We had support from the rest of the team when we needed it, but trying and failing until we succeeded taught us the most.\nAround 2014, Alex informed the team of his diagnosis of stage four colon cancer. In order to ensure the company could survive as his health deteriorated, he decided to sell the company to Karim Marucchi, then-CEO of VeloMedia. Much of our team had difficulty accepting the decision, and felt conflicted regarding the new owners and the management team they brought in, and many team members left immediately or shortly after Alex announced the change. We appreciated the effort to ensure the survival of the company, and our positions within it, but did not trust the changes we foresaw. Under Alex, the company was very developer-focused, working on interesting projects more than worrying about the income, while the new management team brought a lot more focus on traditional business models and practices.\nAs we transitioned to the new corporate identity, merging VeloMedia and Crowd Favorite, the culture of the company shifted. Much of the quirkier, unique nature of the organization disappeared with the Star Wars decorations and references as they left the office. The company became, over time, a more distributed and remote company. While we positioned ourselves as a technical solutions company more than a WordPress development company, most hiring came from WordPress conferences and individuals known within that ecosystem. Under Alex, we received very few hires in that manner, as he preferred people who did not identify as WordPress developers, but rather developers with a background in a variety of technologies. I found myself feeling a bit adrift and isolated as my friends and coworkers moved on to other opportunities where they felt they fit in better than they would now.\nShortly after Alex passed away in September 2015, Crowd Favorite ran into some difficult times, and it added a lot of stress to the entire team. More benefits had to be shifted, the office in Denver closed, and we moved to a fully-remote team. I found working remote full-time very difficult for me, I would get lonely and demotivated more easily. I did not expect the company to go back to the old structure, but believed that I could at least stick around enough to allow other developers a chance for a graceful exit, and knew that, if there were any chance that things could turn around, they would need me there to help.\nLuckily, Crowd Favorite made it through that tough time. However, the stress and isolation had taken their toll on me. I realized that I had not really grown much in the past year, and that the situation still left me stagnant in my salary and my career path. My motivation decreased, and I seldom had the energy to focus on my work to be nearly as productive as they needed. Making matters worse, the lack of energy carried over into other efforts, and even the act of seeking other work fatigued me enough that it would not happen if I didn't make an active change. I needed to leave.\nI met with Jason Rosenbaum, the COO of the company, and told him about my motivation issues. I knew that the company hadn't seen it, but I did, and I knew they needed their team to be operating at 100% to continue their success and desired growth. We agreed that it may be best for both parties if I seek out new challenges elsewhere. I agreed to stay on a bit longer than two weeks in order to assist with the transition, since I had a long legacy at Crowd Favorite, and a large amount of institutional knowledge to pass on to their team. We decided that my final day should be Friday, September 2nd, 2016.\nI sit here now, in my living room, on Tuesday, September 6th, 2016, newly-unemployed. I agreed to a contract with Crowd Favorite to work part-time through the end of the month, and I sent my updated resume out to several recruiters. I've heard a couple responses, and interviews and discussions are coming together rapidly. I don't know where I'm going yet, but I do know my path. It's time to start this journey again.\nIn any case, and by any terminology, failure states will occur. The key is to design both for success, and for failure.\nIn software development, an \"exception\" is an unanticipated error state in which the program hit a condition that cannot be fulfilled within its constraints. There are many causes for exceptions. In an ideal case, these will truly come from unanticipated conditions. Perhaps a file we were relying on has gone missing. Perhaps a remote server is no longer responding, or the database has shut off unexpectedly. It's even possible that, despite our best efforts, a user has found a way to get the program into a state we didn't plan for which it cannot handle on its own.\nWhen we're designing a building, a car, a rocket, or a website, it's very easy to focus on how things should look when everything is working properly. This is the state we're most concerned about, because we want this to be the state that the most people will see and interact with. As an artist, it is very easy to get focused on all the ways you want something to work without delving further into all the ways it should behave when it doesn't work.\nHowever, it is na\u00efve to operate on the assumption that exceptions will not occur, or even that they will not be reasonably common. Treat exceptions as the norm. Assume that something can go wrong in any case and condition where it is possible to do so. Write code that allows the software to degrade gracefully. Give the building redundant supports so that a structural failure of one does not bring it down.\nIf we operate on the assumption that our project will reach a failure state, we quickly identify that there must be some way to allow the project to continue as best it can while informing the relevant parties that a failure occurred. Total shutdown should be a final resort, not the first line of defense.\nConsider Amazon as an example. This is a very complicated eCommerce site, and one of the strongest digital marketplaces in the world. They operate with millions of users, some as strict consumers and others as businesses, pushing transactions of many kinds in great volume throughout the day. The extreme majority of transactions will have no issues. However, in a complex system there are multiple points of failure.\nIt is possible that they will get enough traffic or orders to shut down their orders database. Even though that database is down, they do their best to never prevent the user from interacting with the site as much as possible. So, even when orders are offline a user may be able to go through the store, read reviews, explore products, search, and add items to their cart or wishlist. It is only when the user reaches the failure point that they are stopped, and when that occurs the user is notified that their desired interaction cannot be completed, but that they can continue to interact with other portions of the site while the system administrators address the issue.\nIn architecture, this can include stress fracture planning. When you look at a sidewalk or driveway, there are lines through the cement called \"contraction joints\". They aren't structurally necessary and they aren't particularly appealing visually. In an ideal world and ideal design, they probably wouldn't be there, so why do those lines exist? They exist for failure, for that undesired state. Due to natural processes, the land under and around a driveway or sidewalk will shift over time, adding stress to the cement. The contraction joints are intentional, structured weaknesses designed to be a controlled point of failure. By having them, the majority of fractures will occur along and down those lines, in relatively straight lines that can be easily patched and filled.\nIn many cases, success can be defined as \"working exactly as intended.\" An application is successful if the user can do everything they're supposed to do, and if it looks and acts the way it is intended to.\nSuch a definition of success is insufficient, though, because it only tells half the story. Success does not exist in a vacuum, and is not an absolute state. If everything is working properly, we're certainly doing well, but we haven't reached success just yet.\nSuccess needs to be a broader term. Rather than considering success to be everything working as intended, we need to redefine it. Success is not the absolute inability to fail, but the ability to continue when failure occurs. Only the simplest projects can even be presumed to exist in a state where no error is possible, and even then it is still na\u00efve to assume none could occur. When there is no possibility of existence without failure, we must consider success to mean the ability to fail in a controlled fashion and get back into a functional state to continue operating.\nFor over 4 years, I've worked at Crowd Favorite. When I began working here, the company was solely-owned by Alex King. Though he is no longer with us, he serves as a role-model and an inspiration for much of my approach to my career, and I would feel remiss not expressing my thoughts on how he has influenced me.\nAlex was one of the best developers I've had the privilege to work with. He worked through the .com bubble in Silicon Valley, invented the share icon, helped develop the core blogging framework that would become WordPress, and built a reputation for excellence and pride both in his own work and in the work that was done within his company.\nAs a developer, and as an employer, Alex had high expectations on himself and his employees. He pushed everyone he worked with to do their best, almost always beyond their comfort and sometimes even beyond their limits. Very little that was less than perfect was good enough. More than once, I witnessed products we branded get delayed due to perfection paralysis. If it didn't meet his vision, it wasn't yet ready for it to be seen by others. Alex built his reputation on his excellence, on his demands, and that he would go to the same lengths for a client project that he would go to for his own. Nothing less than the best was good enough.\nDespite that demand for excellence, Alex was more understanding than it may sound. He pushed for perfection, and he demanded excellence, but recognized that you cannot have such success without failures. He knew that failure can be a success, because we can learn from it, and accepted that those he hired may do things in ways that didn't work out. This was especially true as he pushed us to expand our horizons, to go beyond the way things are done into the way things could be done better. Sometimes, we would iterate for significant time trying to figure something out, and still settle on doing it the way it's been done before. This was valuable time, because we explored other options, we tested our assumptions, and we grew in our knowledge of our applications and our limitations.\nUnfortunately, in 2013 Alex was diagnosed with cancer, and had to step away from the business, and from development as a whole. He fought a long battle, and ultimately passed away in September of this year. He left behind a wonderful family, and the remembrances and his ongoing influence among those who were fortunate enough to have worked with him.\nOne of Alex's most memorable sayings was \"Let's make better mistakes tomorrow,\" and I've tried to keep that in mind as a I grow and develop professionally, and as a person. As demanding as Alex ever was on me, I have always had the tendency to be equally or more demanding on myself. I've had projects that intimidated me, where I felt myself failing before I began, and in those cases I then had to fight both my own fears and the demanded growth simultaneously. That fear only added a greater chance to fail without motivation to grow. Accepting the possibility of that failure, moving past that fear, was where I was able to grow.\nI'm still not perfect, of course, nor do I want to be. Perfection has no room for growth, and represents stagnation. I don't want to be perfect because I don't want to stop growing, learning, adapting, and experiencing new things. Sometimes it can be difficult to push past that fear, but even that is another opportunity for growth and there is success just in getting past that point.\nIt can be a very big challenge to accept that things won't always go well, or that we don't start out fully competent at everything we do to the same level of excellence as others who have done it for years. It can be intimidating to put ourselves out there to be judged by those we consider superior within a field. It can be difficult to forgive ourselves the easy mistakes, and the missed opportunities, that come up as we develop. We can't expect to be perfect. We can't expect to ever know everything.\nAll we can demand of ourselves, all we can really push for, is to do our best and never stop trying. Let's not stay in our comfort zone. Let's get out there, learn, try new things, and make better mistakes tomorrow.\nFor over two years now I've owned the devbyday.com domain. For over two years it has sat dormant. There is so much potential, and so much available that I want to achieve, that I wanted to hold off until I got everything just right before revealing it to the world at large. And therein lies the problem. In seeking perfection, I allowed myself to slip into the comfortable view that having nothing is better than having something imperfect. That is the paralysis of perfection.\nIt's easy, as a developer, designer, or client, to get into such a mindset. We want the world to see something in a very specific way, and have such expectations, that we would rather reveal only when it's ready. I find that, too often, this leads to endless iteration, particularly when perfection is not something with clearly defined expectation and goals. Over the past two years I've gone through several cycles of evaluating designs, and implementing my own, but never finding the one that truly stood out to me as the perfect model of my design, and my unclear vision.\nAt a Mexican restaurant near my office, they have a saying painted on the wall: \"Perfeccion tiene su precio.\" (\"Perfection has a price\" in English). This is very true. Apart from the obvious financial components of this price, there are more intangible costs that we pay. On top of the simple costs of hours spent working on a design for this site and researching options, on top of the costs incurred by hosting and owning a domain but not providing any content, there was a price in knowledge.\nUntil now, I've not written anything on this site. However, that does not mean nothing has happened in the last two years. That does not mean there is nothing I've learned in the past two years. On the contrary, I've grown significantly, in both professional and personal capacities. I've had great ideas for posts and articles to write on my experiences. Those ideas are all lost now, casualties to my search for an ideal, and a more substantial cost to that quest. Perfection has cost me money, and cost me knowledge. The worst part is that paying the price for perfection does not mean perfection can be purchased. Perfection may never be achieved, but striving for it without practical concerns will ensure significant ongoing costs.\nStriving for perfection has a cost, and perfection may not ever be achieved, but that does not mean we should not try anyway. By reaching for the stars, we learn what our limits are. If we give ourselves measures that are impossible to match, we have a lasting gauge for our continued growth and personal improvement.\nIn that sense, desiring perfection is a laudable behavior. It comes down to the difference between recognizing when that perfection is being used as a measure for growth, and when it is being used instead to push us down and hold us back. For too long, perfection has become my expectation instead of my standard of measurement when it comes to this site, its contents, and its look and feel.\nThe price of perfection, as an expectation, has gotten too high, and it's time to start the journey, and share that journey as it progresses, rather than only being willing to provide the results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lavender water is scented water. The recipe produced at the museum is an 1840's English preparation. It can be used in much the same manner as perfume. It was also used as a flavoring for drinks as well. Recently lavender has become a popular flavoring added to ice cream, cookies, wine and even mead. As a fragrance it is also wonderful when sprayed on linens.\nAll of the products produced for resale are reproduced using recipes found in an 1845 encyclopedia and used by a doctor from the 1840's The products are typical of the sort of non medicinal items sold by pharmacies during the 1840's.\nMade in Dr.Thralls Pharmacy at The Farmers' Museum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I did pay a premium, but still did not account. Withdrew money from my credit card. When the upgrade is done. Waiting for you urgent reply.\nAs of this writing upon checking your account I can see that it already is upgraded to the Premium status. Can you try to relogin to your account there should be increase in the limits in the quota applicable to that of a Premium subscription this is displayed in the Account Status section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many beginning writers are particularly desperate to get their work out there for the public. They hear published authors going on and on about sales and book signings and reviews they've received, flaunting their \"published\" status as if they were royalty. Speaking of which, their \"royalties\" are often a mere pittance. Beginning writers can't always see the truth beyond the veneer of big talk, and they become infected with the desire to publish at all costs \u2014 all costs except one; that of hiring a good copy-editor.\nBesides being a writer myself, I do a lot of freelance copy-editing and so, as I read, I often see work that is prematurely published. I believe that if you publish your writing (that is, put it out there for the world to see and read), it should be as good as you can make it with as few errors as possible.\nOne writer told me, \"I don't care if it has a few mistakes. I just want to get it published.\" I cringed. She wanted the free copy-editing I offered her just to help her out, but she didn't feel that she needed to make any changes or corrections. She was convinced that her writing was excellent. In fact, it was quite poor and needed a fair bit of work. This woman was an extreme case, displaying slovenly writing habits and a poor attitude. Most writers care a lot more about the quality of their work.\nI understand that the cost of having work copy-edited can be onerous for some, especially when they have not yet made their millions on that bestselling novel, but an investment in a good copy-editing job will be worthwhile in the long run. The copy-editor spends many, many hours reading, correcting, and making suggestions for improvements to the author's work. Unlike reading for pleasure, copy-editing involves careful scrutiny to find grammar, punctuation, and word usage problems. The job comes with a lot of responsibility.\nIn order to be good copy-editors, we have to be a bit pedantic. I try not to overlook even the smallest of errors. For me, it is precisely because I care about writing so much, that I can do a good job of copy-editing.\nWhat Does the Reader Look for?\nWhen I am choosing a novel to read for pleasure, like most readers, I go to the first few pages of the paperback or the e-book sample to look for certain indicators of the writing quality. I want to be \"hooked\" on the first page. I do not want to read about scenery as the character drives by in a car. Nor do I want him to wake up to an alarm clock, or look out a window at the view with the description following. I don't want to read about the character's dream either.\nI look for the first instances of dialogue to give me an idea of the author's skill in writing it. If a large variety of dialogue tags are used (responded, replied, answered, retorted, inquired) rather than \"said\" and \"asked,\" I lose interest, as this indicates either a very dated writing style or an inexperienced writer.\nIf I see a pattern developing where, after each bit of dialogue, the speaker is doing something (for example, \"Wait for me,\" John said, turning around to grab his suitcase), especially if it uses an \"ing\" word, for me that is often the book's death knell.\nIncorrect usage of words makes me shudder. I cringe when I see \"lay\" and \"lie\" misused. I'm sure many readers feel the same when they see the wrong word used.\nIf you are a writer, a good copy-editor can save you from yourself. DON'T publish that book yet! Get it copy-edited properly and then you don't have to worry about mistakes in your book, and tarnish your reputation forever.\nIf you are in the market for a good copy-editor, please contact me. I will do three pages of copy-editing for you for free and you can decide whether this is what you need for your novel, or article, or whatever form your writing takes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Elegant new property 33 Parkside provides a sophisticated environment in which to live. Relax in our spacious social areas, including a stunning private cinema room and fitness suite, unwind on the exclusive roof terrace and in the landscaped courtyards, or work until your heart's content in our modern study spaces. This brand new propety sets a new standard in exclusive student living that you shouldn't miss out on! This prestigious accommodation will appeal to students looking for an unrivalled luxury living experience in a prime location. the neighbourhood There has arguably never been a more exciting time to live and study in Coventry, which saw off stiff competition to secure the title of UK City of Culture 2021. Revered for its resilience and ability to reinvent itself after being decimated in the Second World War, the city is now home to iconic sights and a thriving student population. 33 Parkside is conveniently situated just seven minutes' walk away from Coventry University, and a 10-minute stroll from the shops, restaurants and bars that line the city centre streets. The area is steeped in history, with Coventry Cathedral and War Memorial Park frequented by visitors throughout the year. Coventry's prosperous nightlife highlights include Aardvark, a sports bar that is particularly popular among students, and nightclubs Kasbah and Catch Twenty-Two Club Lounge. Local transport links provide a straightforward journey to Lower Precinct Shopping Centre, which hosts more than a quarter of the city's retail stores. Key Features \u2022\tUp to 100Mbps WiFi \u2022\tWalk to Uni \u2022\tState of the Art Security \u2022\tLarge Beds \u2022\tCinema and Games Room \u2022\tFitness Suite \u2022\tRoof Top Terrace \u2022\tAll Utility Bills Included **Special Offer** Refer a friend and receive \u00a350 in vouchers!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Connexus Group, we manage claims as a core offering for Insurers, Lloyd's Syndicates, MGAs, Brokers and Portfolio run off specialists, as well as serving the claims requirements of fleet managers and accident aftercare programs for manufacturers and dealerships.\nFor many businesses in this sector, the term 'outsourcing' brings with it a connotation of fundamentally changing traditional processes. We recognise that in some cases this is a step too far.\nFor some businesses, end-to-end outsourcing on a white-label basis is the right solution whereas for others, integrating elements of an external process that are compatible with an existing internal process is what is needed. Whatever solution is chosen, it must integrate seamlessly with existing back office functions.\nWe work with you to build a tailored solution, enabling you to use either some or all of our services. We also understand that, if you do choose to outsource, then the solution must be transparent and cost-effective.\nOur FileView facility enables real-time access to our systems to give clients the ability to review claims performance, answer enquiries and conduct external audits.\nWe provide access to a national supply chain for household claims, recovery and salvage services, engineering, fund management, fraud investigation, legal service provision and subrogation.\nOur complete claims solution is manned by our own staff 24\/7, 365 days a year.\nWith over 230 staff in our insurance, digital, legal, accounting and engineering teams, we are able to service our clients with first notice of loss (FNOL), engineering, replacement vehicles, medico legal reporting, diagnostics and rehabilitation.\nWe provide extensive internal and external training with an emphasis on continuing education. Many of our insurance staff are pursuing Chartered Insurance Institute and legal qualifications.\nWe assume responsibility under delegated authority arrangements from both insurers and MGAs on a wide variety of motor, household, property and casualty risks.\nOur data analysts are able to build solutions to import voluminous data and present it in a database format that enables the production of management information, and to enable claims to be managed swiftly and efficiently.\nWe are regularly audited by insurers and Lloyds, receiving exemplary scores. Our existing delegated authorities range from \u00a310,000 to \u00a3250,000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 4Gr8 Kidz Racing Series Kids 12 Bin toy organizer is just $27.99 with free shipping at Amazon (retail $79.99). Looks like good reviews too. You can click here to go directly to this product.\nOh, this is good for all the last minute shoppers! =) Go here to be directed to Amazon's Free 1 day shipping deals!\nPrint your coupon so your kids 12 & younger can eat free at Chili's on Monday (12\/20) at participating Chili's restaurants. This is with the purchase of one adult entree. Note: two children can eat free with the purchase of just one adult entree.\nThis Rudolph DVD\/Board Game looks like fun and is only $4.99 right now (retail $19.99). Looks like good reviews too. Click here to be directed to this Amazon page.\nClick here to be directed to the Amazon sale of outwear (75 % off). This coat pictured is $26.99 (retail $110).\nCheck out all the sales on Melissa & Doug Toys at Amazon here.\nOld Navy has their fleece scarves on sale for only $1.00 today (December 18, in-store).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few years back, I used to come across folks who would go to Thailand two, sometimes three or more times a year, and I'll be honest when I say I used to judge them. It's not that I didn't enjoy my initial visit to Bangkok- in fact I loved it- but my travel mantra had always been that with so much to see in the world, why would anyone bother going to the same place repeatedly?\nFast forward to last year and I found myself (willingly) going to Thailand twice in the span of a year, with no shame or regrets.\nIn my defense, my last vacation plan in the late fall\/early winter didn't start off with Thailand as the destination at all but when it all just fell together naturally to settle on the land of smiles, I went with it and had no qualms. I walked around, stuff my face, rested up, stuffed my face some more, and did everything in my power to not think about the bitter winter that awaited me back in Seoul.\nWhat is it about Thailand that draws me back so repeatedly? I'm not so certain myself. But when I put all the duration I've spent in Thailand over my lifetime, I realize it's fourth in the ranking of countries I've spent the most time in (after the US, Korea, and the UK) so, at the very least, there's a special connection I've established there, and in particular with Bangkok.\nSo without further ado, here's part III of my Bangkok eats series from November, 2014 (a continuation of the series which includes part I as well as part II from my earlier visit in March of the same year and a post about Don Mueang Airport and its food court). I don't think I can make this one as detailed as the previous BKK installments as it's just so much and, in fact, I'm going to have to wrap Bangkok up with a fourth installment- not to mention separate posts for Chiang Mai and Phuket. But for a broad overview of some of the places I ate and things I did, here's part III!\nAs always, my mantra is try to be as local as possible and, with the help of some of my stellar Thai friends, here's some of the places we hit up, eats and drinks we had, and unforgettable memories.\nJust in case I decided to change plans, I didn't really tell my Bangkok friends I'd be coming until last minute. Given how everyone are busy bodies, I was able to get some time in on my first full day in Bangkok with my buddy Shaun before he jetted off for some travels during my time there. He's a foodie and cook so I can always count on him for some excellent eats and this was no exception.\nAs one of Thai cuisine's iconic dishes, somtum itself has long been more of an everyday dish until recently. Many eateries, just like Somtum Der, are taking somtum beyond the generally mainstream, sweeter Bangkok flavors and here, the flavors lean Issan (the northeastern part of Thailand) and closer to the original flavors of sumtum. I'm told Issan restaurants and eateries, even in Bangkok, are often more homely and rustic in appearance and style but Somtum Der is definitely modern and spacious and feels more akin to a bistro. It's been steadily growing in popularity and recognition to the point it's even opened a branch in New York!\nThe magic happens in the open kitchen where many of the ingredients are sourced from the owner's home regions in the northwest. As the restaurant name suggests, the somtums are some of the spot's best dishes with a long range of somtum made Issan-style and pounded fresh to order. Unlike the sweeter palates of Bangkok, the flavors are fiercer and stronger which Somtum Der doesn't compromise with the exception of their fish sauce which is pasteurized here for a general appeal (which is not the case in the northeast and more pungent).\nI always leave the ordering when dining in the more-than-trustworthy hands of Mr. Shaun and he went ahead ordering a delicious storm of food that smacked me straight in the taste buds to let me know I was truly back in Thailand.\nGrilled pork neck. A very Issan dish that was grilled perfectly and so juicy.\nSomtums... the first is the more generic one while the second one has fish cakes and sausages(?). Definitely less sweet but not lacking in any flavors! And you know I love somtum to death so I was reaching for both these somtums quite often.\nDeep fried spicy mince pork. Laab itself is another iconic dish in Issan cuisine and here the take is a sinfully deep fried version that's perfectly crisp on the outside and tender inside.\nThe fried chicken also had the perfect crust to lock in the moist thigh meat inside. What really took it to the next level though, was the sauce accompanying it.\nSpicy tub tim fish salad. Those herbs pack quite a punch!\nSpicy grilled beef. So tender and well grilled! I didn't mind eating it even without the delicious side that came with it.\nAs usual, it was another feast for me with Shaun. Thanks again, buddy!\nPS- Excuse the odd perm hair I tried out last fall which will continue to pop up in my Thailand pictures. Sigh.\nI then returned to meet up with my buddy Pop (whose house I was crashing at yet again) where we had a lazy Saturday chilling at a cafe, getting massages after, and catching up. Perfect weekend!\nAs the sun set, Pop and I headed over to Ari to be reunited with another dear friend, the flawless Ms. Noo, as well as to meet her friend Ppong. Why all the way out in Ari? Well the plan was to have dinner at the locally trending Summer Street where plates of seafood are prepped and brought to the outdoor tables for grilling.\n2. Grilling outdoors in Bangkok heat (even if it's winter) didn't sound that appealing.\nBut it definitely was a fun and pretty atmosphere and definitely one of THE places to be apparently for the locals on a Saturday night.\nLuckily, Noo knew a spot nearby which we relocated to after a 3-5 minute walk.\nBaan Pheung Chom certainly doesn't lose out in the atmosphere department from Summer Street and is in many ways more charming. The old Thai house belonging to the owner's grandmother was renovated into this restaurant and the property has a large yard filled with trees, a pond in the back, and lots of interesting antiques and decoration from all over the world. It was jam packed both inside and out though we were lucky to score a table inside within just a few minutes.\nBaan Pheung Chom exudes a homely yet rustic grace to it such as in the old fashioned beer mugs that I was told was just like the ones used to serve beer at home for Thais. The prices are extremely low, servers are friendly and attentive, and you have the feeling like you're almost having a meal at someone's home (which is technically true).\nSome standouts included the stir-fried kajong flowers with shrimp (dok kajong pad kung) which was certainly unusual and addictive. The sauce on the side is cloyingly sweet which pairs well with the flowers but frankly, I preferred the crunchy bits on their own.\nMy regular favorite stir fried greens. I think this is pad pak boong?\nAnother representative dish here is the \"three smells\" stir fry noodles (woon sen pad saam men). This is a dish that really packs a wallop (thrice, in fact) in flavor from the garlic, stinky bean, and sharp cha-om (an herby vegetable with quite the strong flavor). It's an intriguing mesh of flavors that come in waves from the bitter to sharp and pungent but I still found enjoyable. If stronger flavors aren't your forte and\/or if you're still adjusting to the bolder flavors of Thai cuisine though, you may want to save this dish until you build some more tolerance. Otherwise, delicious! Though, definitely not a dish to be eaten on a date hehe.\nI've tried an extensive amount of Thai curries now with all sorts of ingredients but the red salmon curry with lotus and bamboo bits was certainly different take for me and a first. I was initially concerned thinking the red curry would drown out the oilier flavors of the salmon but they surprisingly paired well while the crunchy bots of lotus roots and bamboo serving as a nice contrast in texture. With just a bit of spice to it, it was lovely spooned atop the flaky rice.\nThe wing bean salad tasted like most other wing bean salads I've had. I've mentioned before this is one of the rare Thai dishes that I don't quite have an affinity for but very fresh and obviously made to order!\nIt wouldn't be a full Thai meal without dessert and I was soon whisked away to get some street side sweets at Pang Ya Ari. The roadside stall has a few tables around and patrons can either make orders to go or sit down and enjoy. Judging by the pace and energy level, it was definitely a popular joint.\nAn assortment of drinks, sweets, and desserts are offered including their famed milk and Thai iced teas.\nTheir thick slices of bread can be topped with riches such as butter, sugar, condensed milk, pandan leaves concentrate and more. And just off those ingredients alone, you can tell that it certainly can't taste bad, right? :P Here's one with butter and sugar and one with condensed milk and pandan leaves concentrate.\nDon't we just look happy?\nOur last stop for the night was at the quaint Fatbird. The vintage meets hipster decor and vibes is evident upon entry but, like the glowing lights, the atmosphere warms as time passes. We were too stuffed to enjoy any of the food (which I'm told is pretty good) but taking in a few Singha bottles in the cozy spot and catching up was just the thing to cap a pretty memorable first night back in Bangkok.\nOne of Bangkok's most famed restaurants for home-style Thai, I'm surprised it took me this long to come here but I'm ever so grateful to Pop for taking us here as it has become one of my favorite spots in Bangkok.\nThe flagship location is in the owner's original childhood home but is not so easy to find as it takes a bit of walking from the nearest Thonglor BTS. But they do a pickup service in their own tuktuk from the entrance of Soi 34 it seems if you call ahead. They also have five other locations scattered about if the one off Sukhumvit isn't close to you.\nThe first thing that strikes you about the flagship restaurant is its whimsical and modern decor. The check board floors, colorful chairs, and light fixtures gives it a modern, swanky hotel feel which could come off as artificial if it weren't for the enormous glass walls that lets in natural sunlight. The modern redesign comes from the owner's daughter and I have to say she did a swell job without making the place come off as tacky.\nDrinks are fresh and made to order with real fruit in the slushies.\nCrispy fish, crunchy cashew and lots and lots of lemongrass. So good. What is it about lemongrass that pairs so well with fish?\nFried fish cake rolls with a sweet, cucumber sauce. Would be boring on its own but the bright and sweet flavors of the sauce is what elevates the dish for sure.\nLook at how perfectly that's grilled!\nCrab fried rice. So flaky and delish.\nCrab roe and crab meat dip with fresh vegetables. I've fallen in love with these nam phrik style crudites and dipping platters in Thai cuisine. This version has a lovely creamy dimension to it from the crab roe and meat.\nStuffed to the max but couldn't resist getting dessert. Taro balls and young coconut in coconut milk.\nPart chewy, crunchy, and tamely sweet. A perfect way to cap off a rich meal.\nAnd then we took the restaurant's tuktuk back. :) Thanks for the rec, Pop!\nOn a side note, we decided late minute to take part in one of those trending mystery rooms and specifically, the Escape Hunt in Bangkok. For those who don't know, you and your friends choose one of several themed mysteries to solve and escape the room using the clues around you. The catch is that you're given only an hour to solve the mystery and also find a way to escape the room. We chose a murder mystery one which proved a lot of fun. There's a lot of red herrings scattered about to throw you off and forces you to work together.\nAll in all, it was a lot of fun and enjoyable experience. Though certainly not unique to Bangkok, for a repeated visitor to the city, it was a nice little side thing to do. Prices aren't exactly cheap and I don't think I'd be doing it again but the prices do drop the more people you have in your group. If your group is large enough, you can even split into two rooms and compete against each other to see who has the fastest time. Afterwards, they serve you tea, take pictures of you in Sherlock Holmes attire, and give you certificates. Gimmicky but fun.\nThonburi is generally more associated with Wat Arun and not much else but the area has been burgeoned in recent years with a growing number of hip spots including the Jam Factory compound. In addition to a cool gallery, shop, and whatnot, the compound houses the definitely hi so, highbrow restaurant, Never Ending Summer.\nBrought forth by famed Thai architect Duangrit Bunnag and his partner, Naree Boonyakiat, The Never Ending Summer is a culinary homage of sorts to the upper class daily Bangkokian eats they used to enjoy as children.\nOutside, the building is housed in an old warehouse which doesn't look like much but the inside is a spacious, airy, open kitchen style restaurant that's modern but still manages to bring about some homely vibes.\nThe main chefs of the restaurant were apparently made to study old cookbooks from Bangkok's high society ladies and combines old favorites and rare treats.\nWatermelon with 'dry' dressing of sugar, dried fish, deep-fried shallots. I'm told it's an old classic and was certainly unique. I found it enjoyable and the savory fish and shallot bits paired well with the fruit but it could have cut down on the sugar as the watermelon was plenty sweet on its own.\nCloseup of the dry dressing.\nNam phrik ka pi included plenty of fresh vegetables and greens, grilled fish, egg omelette, etc. The nam phrik was very fresh and not overpowering but I thought could be a little bolder and less sweet.\nKang liang goong sod with squash, baby corn, shrimp, greens, and such. I enjoyed this soup a lot more than I thought it would. I haven't had a lot of non-spicy soups in Thai cuisine and this one had plenty the flavorful broth that still let the gentle flavors of the ingredients come through.\nPad woon sen. I liked it wasn't oily like many places but it was rather unexciting in taste.\nAll in all, I had a good meal and enjoyed the vibes and chill atmosphere of the place. I'm not certain the food quite matches the level of the restaurant's feel and look and I certainly wasn't blown away by any particular dish. Still, it's worth a visit if you're already out in the Thonburi area and I certainly would love to check out more of the Jam Factory the next time I'm in town.\nWe capped off the night by heading over to the historic neighborhood of Rattanakosin district in the old town of Bangkok to enjoy cocktails at Eagle's Nest, a rooftop bar right off the Chao Phraya river. You're spoiled for choice in rooftop bars in Bangkok but the Eagle's nest is well worth a visit for its spectacular views of not only Wat Arun but Wat Pho and even Wat Phra Kaew.\nInterestingly, the bar itself is located on the rooftop of a hotel called Sala Arun. There's no direct access to the rooftop so you have to enter the hotel and then climb the stairs to the top (no elevator).\nI should mention that the rooftop is also on something like the 4th or 5th floor and the hotel's floor halls don't seem to turn on their AC so the climb can be rather tedious for those who aren't in the best of shape. It doesn't help that the bar's restroom is only located on the first floor so anytime you need to take a wizz, you'll have to down to the lobby and then climb the stairs all over again. The last set of stairs is particularly memorable as it's one of those small, spiral kind which I haven't seen since my days in Europe.\nAnd here's a bit of the Grand Palace and Wat Phra Kaew on the other side of the bar.\nThe bar itself has a warm, friendly vibe and gets especially popular as the sun begins to set. The tables closest to the river are especially popular.\nDrinks are pretty extensive from classic cocktails to variations and all moderately priced. Being an outdoor venue, it's definitely warm but the gentle breezes along the river banks definitely helps.\nAnd off I went the next day to Phuket and then Chiang Mai later on with some back and forth to Bangkok in between.\nStay tuned for part four of the Bangkok eats installment as well as posts for Phuket and Chiang Mai!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I got a new M300 w\/ elevating seat in June 2015. In March 2017 it stopped dead on the parking lot of a store. lights on the joy stick panel continued to flash, but no response. It went to the vendor\/repair my insurance uses. Received a new logic board. Was returned, but did it again within a couple of weeks. Back to repair, got a new joystick. Returned to me. I had to go see my 98yr. mom 1200 miles away - drove in my lift van, the chair got very little actual use during the trip, but on the way back, the electronics gave out again. Back to the repair shop. I picked it up yesterday, 5\/26\/17. Already, the chair has stopped dead 5 times with joystick lights flashing. If I turn power off, wait 5 seconds and turn back on, the chair runs fine until next time it shorts out. If it follows the pattern since March's first repair, it will run with increasing number of cut outs and eventually, just stop responding. So March, April and May, so far, this chair has been in the shop, then given unreliable service, and stranded me so ofter. I'm 67, I don't use it inappropriately, treat it gently - but it seems to be a lemon - maybe this individual chair, maybe the M300 model. Can't seen to get any help, as responsibility for this expensive chair just rolls around from vendor to insurance company. I'm told Permobil was called by vendor and asked for help. The chair itself, compared to my previous rehab chairs, is rattly , flavors on on front castor start up a week after being adjusted - it isn't made nearly as tight and solid as chairs I had in the past. Not even 2 yrs. old and already, an unreliable nuisance.\nThe wheelchair itself is good but we cannot get the parts. We need to replace the guide wheels and the foot rest but the manufacturer said to me that they do not provide these to customers directly. You have to ask your local supplier. We have called them and waited for at least three or four weeks. We had a promise for today but it is getting on for six in the evening and no sign of anyone. They said they would call first but they haven't got that far. So the problem is we cannot safely use the wheelchair because parts need replacing. They also seem very expensive but we are willing to pay the price that was quoted for the four little stabilizing wheels and the foot rest. We have waited for about a month to get a response, had to call up to remind the local guy, then waited in all day. At this time we are supposing they will not come, and there has been no apology or explanation. We seem to be dealing with a local monopoly so, if that is the case where you live, I would get parts up front or try something else.","meta":{"redpajama_set_name":"RedPajamaC4"}}